LMFDB - Embedded newform 378.4.k.b.269.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(215,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 86 x^{14} + 5225 x^{12} - 158916 x^{10} + 3517046 x^{8} - 29955345 x^{6} + 190411550 x^{4} + \cdots + 1500625 $ x^16 - 86x^14 + 5225x^12 - 158916x^10 + 3517046x^8 - 29955345x^6 + 190411550x^4 - 17021375*x^2 + 1500625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.2
Root			$0.258948 - 0.149504i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.4.k.b.215.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.04238 - 7.00160i) q^{5} +(17.8823 + 4.81922i) q^{7} +8.00000i q^{8} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.04238 - 7.00160i) q^{5} +(17.8823 + 4.81922i) q^{7} +8.00000i q^{8} +(14.0032 + 8.08475i) q^{10} +(-8.85614 - 5.11310i) q^{11} +59.8112i q^{13} +(-35.7922 + 9.53512i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(17.8813 - 30.9714i) q^{17} +(-30.4587 + 17.5854i) q^{19} -32.3390 q^{20} +20.4524 q^{22} +(-11.4706 + 6.62254i) q^{23} +(29.8184 - 51.6470i) q^{25} +(-59.8112 - 103.596i) q^{26} +(52.4588 - 52.3075i) q^{28} -56.7331i q^{29} +(199.360 + 115.100i) q^{31} +(27.7128 + 16.0000i) q^{32} +71.5254i q^{34} +(-38.5445 - 144.685i) q^{35} +(6.66566 + 11.5453i) q^{37} +(35.1707 - 60.9175i) q^{38} +(56.0128 - 32.3390i) q^{40} +176.870 q^{41} +354.814 q^{43} +(-35.4246 + 20.4524i) q^{44} +(13.2451 - 22.9412i) q^{46} +(-76.6146 - 132.700i) q^{47} +(296.550 + 172.357i) q^{49} +119.274i q^{50} +(207.192 + 119.622i) q^{52} +(28.8496 + 16.6563i) q^{53} +82.6762i q^{55} +(-38.5538 + 143.058i) q^{56} +(56.7331 + 98.2647i) q^{58} +(-106.848 + 185.067i) q^{59} +(362.200 - 209.116i) q^{61} -460.401 q^{62} -64.0000 q^{64} +(418.774 - 241.779i) q^{65} +(137.515 - 238.182i) q^{67} +(-71.5254 - 123.886i) q^{68} +(211.447 + 212.058i) q^{70} +1145.65i q^{71} +(169.760 + 98.0113i) q^{73} +(-23.0905 - 13.3313i) q^{74} +140.683i q^{76} +(-133.727 - 134.113i) q^{77} +(357.735 + 619.615i) q^{79} +(-64.6780 + 112.026i) q^{80} +(-306.348 + 176.870i) q^{82} +1406.35 q^{83} -289.132 q^{85} +(-614.556 + 354.814i) q^{86} +(40.9048 - 70.8491i) q^{88} +(296.781 + 514.040i) q^{89} +(-288.243 + 1069.56i) q^{91} +52.9803i q^{92} +(265.401 + 153.229i) q^{94} +(246.251 + 142.173i) q^{95} -325.405i q^{97} +(-685.997 - 1.98097i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) $ 16 * q + 32 * q^4 - 52 * q^7	$ 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$2$	−1.73205	+	1.00000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	2.00000	−	3.46410i	0.250000	−	0.433013i
$5$	−4.04238	−	7.00160i	−0.361561	−	0.626242i	0.626657	−	0.779295i	$-0.284423\pi$
$5$	−4.04238	−	7.00160i	−0.361561	−	0.626242i	−0.988218	+	0.153053i	$0.951089\pi$
$6$		0			0
$7$	17.8823	+	4.81922i	0.965551	+	0.260213i
$7$	17.8823	+	4.81922i	0.965551	+	0.260213i
$8$			8.00000i			0.353553i
$9$		0			0
$10$	14.0032	+	8.08475i	0.442820	+	0.255662i
$11$	−8.85614	−	5.11310i	−0.242748	−	0.140151i	0.373691	−	0.927553i	$-0.378092\pi$
$11$	−8.85614	−	5.11310i	−0.242748	−	0.140151i	−0.616439	+	0.787403i	$0.711425\pi$
$12$		0			0
$13$			59.8112i			1.27605i	0.770016	+	0.638025i	$0.220248\pi$
$13$			59.8112i			1.27605i	−0.770016	+	0.638025i	$0.779752\pi$
$14$	−35.7922	+	9.53512i	−0.683276	+	0.182026i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	17.8813	−	30.9714i	0.255110	−	0.441863i	−0.709816	−	0.704387i	$-0.751222\pi$
$17$	17.8813	−	30.9714i	0.255110	−	0.441863i	0.964925	+	0.262525i	$0.0845551\pi$
$18$		0			0
$19$	−30.4587	+	17.5854i	−0.367775	+	0.212335i	−0.672486	−	0.740110i	$-0.734773\pi$
$19$	−30.4587	+	17.5854i	−0.367775	+	0.212335i	0.304711	+	0.952445i	$0.401440\pi$
$20$	−32.3390			−0.361561
$21$		0			0
$22$	20.4524			0.198203
$23$	−11.4706	+	6.62254i	−0.103990	+	0.0600389i	−0.551093	−	0.834444i	$-0.685789\pi$
$23$	−11.4706	+	6.62254i	−0.103990	+	0.0600389i	0.447103	+	0.894483i	$0.352456\pi$
$24$		0			0
$25$	29.8184	−	51.6470i	0.238547	−	0.413176i
$26$	−59.8112	−	103.596i	−0.451152	−	0.781418i
$27$		0			0
$28$	52.4588	−	52.3075i	0.354064	−	0.353043i
$29$		−	56.7331i		−	0.363279i	−0.983365	−	0.181639i	$-0.941860\pi$
$29$		−	56.7331i		−	0.363279i	0.983365	−	0.181639i	$-0.0581403\pi$
$30$		0			0
$31$	199.360	+	115.100i	1.15503	+	0.666859i	0.950109	−	0.311919i	$-0.100972\pi$
$31$	199.360	+	115.100i	1.15503	+	0.666859i	0.204925	+	0.978778i	$0.434305\pi$
$32$	27.7128	+	16.0000i	0.153093	+	0.0883883i
$33$		0			0
$34$			71.5254i			0.360779i
$35$	−38.5445	−	144.685i	−0.186149	−	0.698752i
$36$		0			0
$37$	6.66566	+	11.5453i	0.0296170	+	0.0512981i	0.880454	−	0.474132i	$-0.157238\pi$
$37$	6.66566	+	11.5453i	0.0296170	+	0.0512981i	−0.850837	+	0.525430i	$0.823905\pi$
$38$	35.1707	−	60.9175i	0.150143	−	0.260056i
$39$		0			0
$40$	56.0128	−	32.3390i	0.221410	−	0.127831i
$41$	176.870			0.673719			0.336859	−	0.941555i	$-0.390635\pi$
$41$	176.870			0.673719			0.336859	+	0.941555i	$0.390635\pi$
$42$		0			0
$43$	354.814			1.25834			0.629170	−	0.777268i	$-0.283395\pi$
$43$	354.814			1.25834			0.629170	+	0.777268i	$0.283395\pi$
$44$	−35.4246	+	20.4524i	−0.121374	+	0.0700753i
$45$		0			0
$46$	13.2451	−	22.9412i	0.0424539	−	0.0735324i
$47$	−76.6146	−	132.700i	−0.237774	−	0.411837i	0.722301	−	0.691579i	$-0.243084\pi$
$47$	−76.6146	−	132.700i	−0.237774	−	0.411837i	−0.960075	+	0.279742i	$0.909751\pi$
$48$		0			0
$49$	296.550	+	172.357i	0.864578	+	0.502499i
$50$			119.274i			0.337357i
$51$		0			0
$52$	207.192	+	119.622i	0.552546	+	0.319012i
$53$	28.8496	+	16.6563i	0.0747699	+	0.0431684i	0.536919	−	0.843634i	$-0.319588\pi$
$53$	28.8496	+	16.6563i	0.0747699	+	0.0431684i	−0.462149	+	0.886802i	$0.652921\pi$
$54$		0			0
$55$			82.6762i			0.202692i
$56$	−38.5538	+	143.058i	−0.0919993	+	0.341374i
$57$		0			0
$58$	56.7331	+	98.2647i	0.128438	+	0.222462i
$59$	−106.848	+	185.067i	−0.235771	+	0.408367i	−0.959496	−	0.281721i	$-0.909095\pi$
$59$	−106.848	+	185.067i	−0.235771	+	0.408367i	0.723726	+	0.690088i	$0.242428\pi$
$60$		0			0
$61$	362.200	−	209.116i	0.760246	−	0.438928i	−0.0691383	−	0.997607i	$-0.522025\pi$
$61$	362.200	−	209.116i	0.760246	−	0.438928i	0.829384	+	0.558679i	$0.188692\pi$
$62$	−460.401			−0.943081
$63$		0			0
$64$	−64.0000			−0.125000
$65$	418.774	−	241.779i	0.799116	−	0.461370i
$66$		0			0
$67$	137.515	−	238.182i	0.250747	−	0.434307i	−0.712984	−	0.701180i	$-0.752657\pi$
$67$	137.515	−	238.182i	0.250747	−	0.434307i	0.963732	+	0.266873i	$0.0859903\pi$
$68$	−71.5254	−	123.886i	−0.127555	−	0.220931i
$69$		0			0
$70$	211.447	+	212.058i	0.361039	+	0.362083i
$71$			1145.65i			1.91498i	0.288474	+	0.957488i	$0.406852\pi$
$71$			1145.65i			1.91498i	−0.288474	+	0.957488i	$0.593148\pi$
$72$		0			0
$73$	169.760	+	98.0113i	0.272178	+	0.157142i	0.629877	−	0.776695i	$-0.283105\pi$
$73$	169.760	+	98.0113i	0.272178	+	0.157142i	−0.357699	+	0.933837i	$0.616439\pi$
$74$	−23.0905	−	13.3313i	−0.0362732	−	0.0209424i
$75$		0			0
$76$			140.683i			0.212335i
$77$	−133.727	−	134.113i	−0.197917	−	0.198489i
$78$		0			0
$79$	357.735	+	619.615i	0.509473	+	0.882433i	0.999940	+	0.0109729i	$0.00349284\pi$
$79$	357.735	+	619.615i	0.509473	+	0.882433i	−0.490467	+	0.871460i	$0.663174\pi$
$80$	−64.6780	+	112.026i	−0.0903903	+	0.156561i
$81$		0			0
$82$	−306.348	+	176.870i	−0.412567	+	0.238195i
$83$	1406.35			1.85985			0.929923	−	0.367755i	$-0.119874\pi$
$83$	1406.35			1.85985			0.929923	+	0.367755i	$0.119874\pi$
$84$		0			0
$85$	−289.132			−0.368951
$86$	−614.556	+	354.814i	−0.770573	+	0.444890i
$87$		0			0
$88$	40.9048	−	70.8491i	0.0495507	−	0.0858244i
$89$	296.781	+	514.040i	0.353469	+	0.612226i	0.986855	−	0.161610i	$-0.0516687\pi$
$89$	296.781	+	514.040i	0.353469	+	0.612226i	−0.633386	+	0.773836i	$0.718335\pi$
$90$		0			0
$91$	−288.243	+	1069.56i	−0.332045	+	1.23209i
$92$			52.9803i			0.0600389i
$93$		0			0
$94$	265.401	+	153.229i	0.291213	+	0.168132i
$95$	246.251	+	142.173i	0.265946	+	0.153544i
$96$		0			0
$97$		−	325.405i		−	0.340617i	−0.985391	−	0.170308i	$-0.945524\pi$
$97$		−	325.405i		−	0.340617i	0.985391	−	0.170308i	$-0.0544764\pi$
$98$	−685.997	−	1.98097i	−0.707104	−	0.00204192i
$99$		0			0
$100$	−119.274	−	206.588i	−0.119274	−	0.206588i
$101$	836.023	−	1448.03i	0.823637	−	1.42658i	−0.0793189	−	0.996849i	$-0.525275\pi$
$101$	836.023	−	1448.03i	0.823637	−	1.42658i	0.902956	−	0.429732i	$-0.141392\pi$
$102$		0			0
$103$	1350.21	−	779.547i	1.29166	−	0.745738i	0.312709	−	0.949849i	$-0.398764\pi$
$103$	1350.21	−	779.547i	1.29166	−	0.745738i	0.978948	+	0.204111i	$0.0654303\pi$
$104$	−478.490			−0.451152
$105$		0			0
$106$	−66.6254			−0.0610493
$107$	1649.43	−	952.300i	1.49025	−	0.860395i	0.490311	−	0.871548i	$-0.336883\pi$
$107$	1649.43	−	952.300i	1.49025	−	0.860395i	0.999938	+	0.0111523i	$0.00354996\pi$
$108$		0			0
$109$	−31.2268	+	54.0865i	−0.0274403	+	0.0475279i	−0.879419	−	0.476048i	$-0.842069\pi$
$109$	−31.2268	+	54.0865i	−0.0274403	+	0.0475279i	0.851979	+	0.523576i	$0.175402\pi$
$110$	−82.6762	−	143.199i	−0.0716624	−	0.124123i
$111$		0			0
$112$	−76.2810	−	286.338i	−0.0643560	−	0.241575i
$113$			1229.81i			1.02381i	0.859042	+	0.511905i	$0.171060\pi$
$113$			1229.81i			1.02381i	−0.859042	+	0.511905i	$0.828940\pi$
$114$		0			0
$115$	92.7367	+	53.5416i	0.0751978	+	0.0434155i
$116$	−196.529	−	113.466i	−0.157304	−	0.0908196i
$117$		0			0
$118$		−	427.393i		−	0.333430i
$119$	469.017	−	467.664i	0.361300	−	0.360258i
$120$		0			0
$121$	−613.212	−	1062.12i	−0.460716	−	0.797983i
$122$	−418.233	+	724.400i	−0.310369	+	0.537575i
$123$		0			0
$124$	797.438	−	460.401i	0.577517	−	0.333429i
$125$	−1492.74			−1.06812
$126$		0			0
$127$	−2002.74			−1.39933			−0.699664	−	0.714472i	$-0.746667\pi$
$127$	−2002.74			−1.39933			−0.699664	+	0.714472i	$0.746667\pi$
$128$	110.851	−	64.0000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−483.559	+	837.548i	−0.326238	+	0.565060i
$131$	−164.370	−	284.697i	−0.109627	−	0.189879i	0.805992	−	0.591926i	$-0.201632\pi$
$131$	−164.370	−	284.697i	−0.109627	−	0.189879i	−0.915619	+	0.402047i	$0.868299\pi$
$132$		0			0
$133$	−629.419	+	167.679i	−0.410357	+	0.109320i
$134$			550.058i			0.354610i
$135$		0			0
$136$	247.771	+	143.051i	0.156222	+	0.0901948i
$137$	1143.69	+	660.310i	0.713227	+	0.411782i	0.812255	−	0.583303i	$-0.198240\pi$
$137$	1143.69	+	660.310i	0.713227	+	0.411782i	−0.0990277	+	0.995085i	$0.531573\pi$
$138$		0			0
$139$			53.3563i			0.0325584i	0.999867	+	0.0162792i	$0.00518206\pi$
$139$			53.3563i			0.0325584i	−0.999867	+	0.0162792i	$0.994818\pi$
$140$	−578.294	−	155.849i	−0.349106	−	0.0940830i
$141$		0			0
$142$	−1145.65	−	1984.32i	−0.677046	−	1.17268i
$143$	305.821	−	529.697i	0.178839	−	0.309759i
$144$		0			0
$145$	−397.223	+	229.337i	−0.227500	+	0.131347i
$146$	−392.045			−0.222232
$147$		0			0
$148$	53.3253			0.0296170
$149$	−1102.14	+	636.318i	−0.605976	+	0.349860i	−0.771389	−	0.636364i	$-0.780438\pi$
$149$	−1102.14	+	636.318i	−0.605976	+	0.349860i	0.165413	+	0.986224i	$0.447104\pi$
$150$		0			0
$151$	−867.110	+	1501.88i	−0.467314	+	0.809412i	−0.999303	−	0.0373400i	$-0.988112\pi$
$151$	−867.110	+	1501.88i	−0.467314	+	0.809412i	0.531989	+	0.846751i	$0.321445\pi$
$152$	−140.683	−	243.670i	−0.0750717	−	0.130028i
$153$		0			0
$154$	365.735	+	98.5646i	0.191375	+	0.0515751i
$155$		−	1861.11i		−	0.964441i
$156$		0			0
$157$	−2112.06	−	1219.40i	−1.07363	−	0.619863i	−0.144461	−	0.989510i	$-0.546145\pi$
$157$	−2112.06	−	1219.40i	−1.07363	−	0.619863i	−0.929172	+	0.369648i	$0.879478\pi$
$158$	−1239.23	−	715.470i	−0.623974	−	0.360252i
$159$		0			0
$160$		−	258.712i		−	0.127831i
$161$	−237.035	+	63.1467i	−0.116031	+	0.0309109i
$162$		0			0
$163$	−1685.62	−	2919.57i	−0.809986	−	1.40294i	−0.912873	−	0.408244i	$-0.866141\pi$
$163$	−1685.62	−	2919.57i	−0.809986	−	1.40294i	0.102887	−	0.994693i	$-0.467192\pi$
$164$	353.740	−	612.696i	0.168430	−	0.291729i
$165$		0			0
$166$	−2435.87	+	1406.35i	−1.13892	+	0.657555i
$167$	−3671.83			−1.70140			−0.850702	−	0.525649i	$-0.823823\pi$
$167$	−3671.83			−1.70140			−0.850702	+	0.525649i	$0.823823\pi$
$168$		0			0
$169$	−1380.38			−0.628303
$170$	500.792	−	289.132i	0.225935	−	0.130444i
$171$		0			0
$172$	709.628	−	1229.11i	0.314585	−	0.544877i
$173$	870.888	+	1508.42i	0.382730	+	0.662908i	0.991451	−	0.130476i	$-0.0416505\pi$
$173$	870.888	+	1508.42i	0.382730	+	0.662908i	−0.608721	+	0.793384i	$0.708317\pi$
$174$		0			0
$175$	782.119	−	779.863i	0.337844	−	0.336869i
$176$			163.619i			0.0700753i
$177$		0			0
$178$	−1028.08	−	593.562i	−0.432909	−	0.249940i
$179$	−536.713	−	309.872i	−0.224111	−	0.129390i	0.383741	−	0.923441i	$-0.374635\pi$
$179$	−536.713	−	309.872i	−0.224111	−	0.129390i	−0.607852	+	0.794050i	$0.707969\pi$
$180$		0			0
$181$			547.087i			0.224667i	0.993671	+	0.112333i	$0.0358324\pi$
$181$			547.087i			0.224667i	−0.993671	+	0.112333i	$0.964168\pi$
$182$	−570.307	−	2140.78i	−0.232275	−	0.871895i
$183$		0			0
$184$	−52.9803	−	91.7646i	−0.0212270	−	0.0367662i
$185$	53.8902	−	93.3405i	0.0214167	−	0.0370948i
$186$		0			0
$187$	−316.719	+	182.858i	−0.123855	+	0.0715075i
$188$	−612.917			−0.237774
$189$		0			0
$190$	−568.693			−0.217144
$191$	4105.01	−	2370.03i	1.55512	−	0.897850i	0.557410	−	0.830237i	$-0.311795\pi$
$191$	4105.01	−	2370.03i	1.55512	−	0.897850i	0.997712	−	0.0676124i	$-0.0215381\pi$
$192$		0			0
$193$	−307.380	+	532.397i	−0.114641	+	0.198564i	−0.917636	−	0.397422i	$-0.869905\pi$
$193$	−307.380	+	532.397i	−0.114641	+	0.198564i	0.802995	+	0.595985i	$0.203238\pi$
$194$	325.405	+	563.618i	0.120426	+	0.208584i
$195$		0			0
$196$	1190.16	−	682.566i	0.433733	−	0.248749i
$197$			1863.52i			0.673960i	0.941512	+	0.336980i	$0.109405\pi$
$197$			1863.52i			0.673960i	−0.941512	+	0.336980i	$0.890595\pi$
$198$		0			0
$199$	−152.370	−	87.9710i	−0.0542776	−	0.0313372i	0.472616	−	0.881269i	$-0.343310\pi$
$199$	−152.370	−	87.9710i	−0.0542776	−	0.0313372i	−0.526893	+	0.849931i	$0.676643\pi$
$200$	413.176	+	238.547i	0.146080	+	0.0843392i
$201$		0			0
$202$			3344.09i			1.16480i
$203$	273.409	−	1014.52i	0.0945300	−	0.350764i
$204$		0			0
$205$	−714.975	−	1238.37i	−0.243590	−	0.421911i
$206$	−1559.09	+	2700.43i	−0.527317	+	0.913339i
$207$		0			0
$208$	828.769	−	478.490i	0.276273	−	0.159506i
$209$	359.663			0.119035
$210$		0			0
$211$	−2171.71			−0.708563			−0.354281	−	0.935139i	$-0.615274\pi$
$211$	−2171.71			−0.708563			−0.354281	+	0.935139i	$0.615274\pi$
$212$	115.399	−	66.6254i	0.0373849	−	0.0215842i
$213$		0			0
$214$	−1904.60	+	3298.86i	−0.608391	+	1.05376i
$215$	−1434.29	−	2484.26i	−0.454967	−	0.788025i
$216$		0			0
$217$	3010.30	+	3019.01i	0.941718	+	0.944441i
$218$		−	124.907i		−	0.0388064i
$219$		0			0
$220$	286.399	+	165.352i	0.0877682	+	0.0506730i
$221$	1852.44	+	1069.50i	0.563839	+	0.325532i
$222$		0			0
$223$		−	890.870i		−	0.267521i	−0.991014	−	0.133760i	$-0.957295\pi$
$223$		−	890.870i		−	0.267521i	0.991014	−	0.133760i	$-0.0427052\pi$
$224$	418.460	+	419.670i	0.124819	+	0.125180i
$225$		0			0
$226$	−1229.81	−	2130.09i	−0.361971	−	0.626953i
$227$	1956.43	−	3388.64i	0.572040	−	0.990802i	−0.424317	−	0.905514i	$-0.639486\pi$
$227$	1956.43	−	3388.64i	0.572040	−	0.990802i	0.996356	−	0.0852878i	$-0.0271810\pi$
$228$		0			0
$229$	4004.17	−	2311.81i	1.15547	−	0.667112i	0.205257	−	0.978708i	$-0.434197\pi$
$229$	4004.17	−	2311.81i	1.15547	−	0.667112i	0.950215	+	0.311597i	$0.100864\pi$
$230$	−214.166			−0.0613987
$231$		0			0
$232$	453.865			0.128438
$233$	−2733.67	+	1578.29i	−0.768621	+	0.443764i	−0.832383	−	0.554201i	$-0.813024\pi$
$233$	−2733.67	+	1578.29i	−0.768621	+	0.443764i	0.0637613	+	0.997965i	$0.479690\pi$
$234$		0			0
$235$	−619.410	+	1072.85i	−0.171940	+	0.297808i
$236$	427.393	+	740.267i	0.117885	+	0.204183i
$237$		0			0
$238$	−344.696	+	1279.03i	−0.0938796	+	0.348351i
$239$			1674.99i			0.453331i	0.973973	+	0.226666i	$0.0727824\pi$
$239$			1674.99i			0.453331i	−0.973973	+	0.226666i	$0.927218\pi$
$240$		0			0
$241$	−2914.98	−	1682.96i	−0.779129	−	0.449830i	0.0569925	−	0.998375i	$-0.481849\pi$
$241$	−2914.98	−	1682.96i	−0.779129	−	0.449830i	−0.836122	+	0.548544i	$0.815182\pi$
$242$	2124.23	+	1226.42i	0.564259	+	0.325775i
$243$		0			0
$244$		−	1672.93i		−	0.438928i
$245$	8.00783	−	2773.06i	0.00208817	−	0.723119i
$246$		0			0
$247$	−1051.80	−	1821.77i	−0.270950	−	0.469299i
$248$	−920.802	+	1594.88i	−0.235770	+	0.408366i
$249$		0			0
$250$	2585.51	−	1492.74i	0.654087	−	0.377637i
$251$	−2051.87			−0.515988			−0.257994	−	0.966147i	$-0.583061\pi$
$251$	−2051.87			−0.515988			−0.257994	+	0.966147i	$0.583061\pi$
$252$		0			0
$253$	135.447			0.0336580
$254$	3468.85	−	2002.74i	0.856909	−	0.494737i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	3784.64	+	6555.19i	0.918597	+	1.59106i	0.801548	+	0.597930i	$0.204010\pi$
$257$	3784.64	+	6555.19i	0.918597	+	1.59106i	0.117049	+	0.993126i	$0.462657\pi$
$258$		0			0
$259$	63.5579	+	238.579i	0.0152482	+	0.0572376i
$260$		−	1934.24i		−	0.461370i
$261$		0			0
$262$	569.394	+	328.740i	0.134265	+	0.0775177i
$263$	246.408	+	142.264i	0.0577726	+	0.0333550i	0.528608	−	0.848866i	$-0.322714\pi$
$263$	246.408	+	142.264i	0.0577726	+	0.0333550i	−0.470836	+	0.882221i	$0.656047\pi$
$264$		0			0
$265$		−	269.325i		−	0.0624320i
$266$	922.507	−	919.847i	0.212641	−	0.212028i
$267$		0			0
$268$	−550.058	−	952.729i	−0.125374	−	0.217154i
$269$	−2154.92	+	3732.43i	−0.488430	+	0.845986i	−0.999911	−	0.0133086i	$-0.995764\pi$
$269$	−2154.92	+	3732.43i	−0.488430	+	0.845986i	0.511481	+	0.859294i	$0.329097\pi$
$270$		0			0
$271$	5937.36	−	3427.94i	1.33088	−	0.768385i	0.345447	−	0.938438i	$-0.387727\pi$
$271$	5937.36	−	3427.94i	1.33088	−	0.768385i	0.985435	+	0.170053i	$0.0543940\pi$
$272$	−572.203			−0.127555
$273$		0			0
$274$	−2641.24			−0.582347
$275$	−528.152	+	304.929i	−0.115814	+	0.0668651i
$276$		0			0
$277$	−3965.62	+	6868.65i	−0.860183	+	1.48988i	0.0115687	+	0.999933i	$0.496317\pi$
$277$	−3965.62	+	6868.65i	−0.860183	+	1.48988i	−0.871752	+	0.489948i	$0.837016\pi$
$278$	−53.3563	−	92.4158i	−0.0115111	−	0.0199379i
$279$		0			0
$280$	1157.48	−	308.356i	0.247046	−	0.0658136i
$281$			4758.17i			1.01014i	0.863079	+	0.505069i	$0.168533\pi$
$281$			4758.17i			1.01014i	−0.863079	+	0.505069i	$0.831467\pi$
$282$		0			0
$283$	−3608.93	−	2083.62i	−0.758053	−	0.437662i	0.0705435	−	0.997509i	$-0.477527\pi$
$283$	−3608.93	−	2083.62i	−0.758053	−	0.437662i	−0.828596	+	0.559847i	$0.810860\pi$
$284$	3968.64	+	2291.29i	0.829209	+	0.478744i
$285$		0			0
$286$			1223.28i			0.252917i
$287$	3162.84	+	852.376i	0.650510	+	0.175311i
$288$		0			0
$289$	1817.02	+	3147.16i	0.369838	+	0.640579i
$290$	458.673	−	794.445i	0.0928766	−	0.160867i
$291$		0			0
$292$	679.042	−	392.045i	0.136089	−	0.0785709i
$293$	−7684.65			−1.53223			−0.766113	−	0.642706i	$-0.777812\pi$
$293$	−7684.65			−1.53223			−0.766113	+	0.642706i	$0.777812\pi$
$294$		0			0
$295$	1727.68			0.340982
$296$	−92.3621	+	53.3253i	−0.0181366	+	0.0104712i
$297$		0			0
$298$	1272.64	−	2204.27i	0.247389	−	0.428490i
$299$	−396.102	−	686.069i	−0.0766126	−	0.132697i
$300$		0			0
$301$	6344.87	+	1709.93i	1.21499	+	0.327437i
$302$		−	3468.44i		−	0.660882i
$303$		0			0
$304$	487.340	+	281.366i	0.0919436	+	0.0530837i
$305$	−2928.30	−	1690.65i	−0.549750	−	0.317399i
$306$		0			0
$307$			7109.98i			1.32178i	0.750481	+	0.660892i	$0.229822\pi$
$307$			7109.98i			1.32178i	−0.750481	+	0.660892i	$0.770178\pi$
$308$	−732.036	+	195.016i	−0.135427	+	0.0360781i
$309$		0			0
$310$	1861.11	+	3223.54i	0.340981	+	0.590597i
$311$	963.289	−	1668.47i	0.175637	−	0.304212i	−0.764745	−	0.644334i	$-0.777135\pi$
$311$	963.289	−	1668.47i	0.175637	−	0.304212i	0.940382	+	0.340121i	$0.110468\pi$
$312$		0			0
$313$	6614.43	−	3818.84i	1.19447	−	0.689629i	0.235155	−	0.971958i	$-0.424440\pi$
$313$	6614.43	−	3818.84i	1.19447	−	0.689629i	0.959318	+	0.282329i	$0.0911070\pi$
$314$	4877.59			0.876618
$315$		0			0
$316$	2861.88			0.509473
$317$	−8159.53	+	4710.91i	−1.44569	+	0.834672i	−0.998221	−	0.0596251i	$-0.981009\pi$
$317$	−8159.53	+	4710.91i	−1.44569	+	0.834672i	−0.447474	+	0.894297i	$0.647676\pi$
$318$		0			0
$319$	−290.082	+	502.437i	−0.0509137	+	0.0881851i
$320$	258.712	+	448.102i	0.0451951	+	0.0782803i
$321$		0			0
$322$	347.410	−	346.409i	0.0601255	−	0.0599522i
$323$			1257.80i			0.216674i
$324$		0			0
$325$	3089.07	+	1783.48i	0.527233	+	0.304398i
$326$	5839.15	+	3371.23i	0.992026	+	0.572747i
$327$		0			0
$328$			1414.96i			0.238195i
$329$	−730.529	−	2742.20i	−0.122418	−	0.459522i
$330$		0			0
$331$	−806.117	−	1396.24i	−0.133862	−	0.231855i	0.791300	−	0.611428i	$-0.209404\pi$
$331$	−806.117	−	1396.24i	−0.133862	−	0.231855i	−0.925162	+	0.379572i	$0.876071\pi$
$332$	2812.70	−	4871.75i	0.464961	−	0.805337i
$333$		0			0
$334$	6359.79	−	3671.83i	1.04189	−	0.601537i
$335$	−2223.54			−0.362642
$336$		0			0
$337$	6357.99			1.02772			0.513860	−	0.857874i	$-0.328215\pi$
$337$	6357.99			1.02772			0.513860	+	0.857874i	$0.328215\pi$
$338$	2390.89	−	1380.38i	0.384756	−	0.222139i
$339$		0			0
$340$	−578.265	+	1001.58i	−0.0922377	+	0.159760i
$341$	−1177.04	−	2038.69i	−0.186921	−	0.323757i
$342$		0			0
$343$	4472.36	+	4511.27i	0.704037	+	0.710163i
$344$			2838.51i			0.444890i
$345$		0			0
$346$	−3016.84	−	1741.78i	−0.468747	−	0.270631i
$347$	−5058.61	−	2920.59i	−0.782595	−	0.451831i	0.0547545	−	0.998500i	$-0.482562\pi$
$347$	−5058.61	−	2920.59i	−0.782595	−	0.451831i	−0.837349	+	0.546669i	$0.815896\pi$
$348$		0			0
$349$			3896.74i			0.597673i	0.954304	+	0.298836i	$0.0965985\pi$
$349$			3896.74i			0.597673i	−0.954304	+	0.298836i	$0.903402\pi$
$350$	−574.806	+	2132.88i	−0.0877848	+	0.325735i
$351$		0			0
$352$	−163.619	−	283.397i	−0.0247754	−	0.0429122i
$353$	−1718.14	+	2975.91i	−0.259058	+	0.448702i	−0.965990	−	0.258580i	$-0.916746\pi$
$353$	−1718.14	+	2975.91i	−0.259058	+	0.448702i	0.706932	+	0.707282i	$0.250079\pi$
$354$		0			0
$355$	8021.36	−	4631.13i	1.19924	−	0.692380i
$356$	2374.25			0.353469
$357$		0			0
$358$	1239.49			0.182986
$359$	6159.46	−	3556.17i	0.905526	−	0.522806i	0.0265373	−	0.999648i	$-0.491552\pi$
$359$	6159.46	−	3556.17i	0.905526	−	0.522806i	0.878989	+	0.476842i	$0.158219\pi$
$360$		0			0
$361$	−2811.01	+	4868.81i	−0.409828	+	0.709843i
$362$	−547.087	−	947.583i	−0.0794317	−	0.137580i
$363$		0			0
$364$	3128.58	+	3137.62i	0.450500	+	0.451803i
$365$		−	1584.79i		−	0.227265i
$366$		0			0
$367$	8697.86	+	5021.71i	1.23712	+	0.714254i	0.968505	−	0.248993i	$-0.0800995\pi$
$367$	8697.86	+	5021.71i	1.23712	+	0.714254i	0.268619	+	0.963247i	$0.413433\pi$
$368$	183.529	+	105.961i	0.0259976	+	0.0150097i
$369$		0			0
$370$			215.561i			0.0302877i
$371$	435.626	+	436.886i	0.0609611	+	0.0611374i
$372$		0			0
$373$	−4850.34	−	8401.03i	−0.673300	−	1.16619i	−0.976963	−	0.213410i	$-0.931543\pi$
$373$	−4850.34	−	8401.03i	−0.673300	−	1.16619i	0.303663	−	0.952780i	$-0.401790\pi$
$374$	365.716	−	633.439i	0.0505634	−	0.0875785i
$375$		0			0
$376$	1061.60	−	612.917i	0.145606	−	0.0840659i
$377$	3393.28			0.463562
$378$		0			0
$379$	−6250.00			−0.847073			−0.423537	−	0.905879i	$-0.639212\pi$
$379$	−6250.00			−0.847073			−0.423537	+	0.905879i	$0.639212\pi$
$380$	985.005	−	568.693i	0.132973	−	0.0767720i
$381$		0			0
$382$	−4740.06	+	8210.02i	−0.634876	+	1.09964i
$383$	−2028.42	−	3513.32i	−0.270620	−	0.468727i	0.698401	−	0.715707i	$-0.253895\pi$
$383$	−2028.42	−	3513.32i	−0.270620	−	0.468727i	−0.969021	+	0.246980i	$0.920562\pi$
$384$		0			0
$385$	−398.435	+	1478.44i	−0.0527432	+	0.195709i
$386$		−	1229.52i		−	0.162127i
$387$		0			0
$388$	−1127.24	−	650.809i	−0.147491	−	0.0851542i
$389$	−9836.97	−	5679.38i	−1.28214	−	0.740247i	−0.304905	−	0.952383i	$-0.598625\pi$
$389$	−9836.97	−	5679.38i	−1.28214	−	0.740247i	−0.977240	+	0.212136i	$0.931958\pi$
$390$		0			0
$391$			473.680i			0.0612660i
$392$	−1378.86	+	2372.40i	−0.177660	+	0.305674i
$393$		0			0
$394$	−1863.52	−	3227.71i	−0.238281	−	0.412714i
$395$	2892.20	−	5009.44i	0.368411	−	0.638106i
$396$		0			0
$397$	−4768.46	+	2753.07i	−0.602826	+	0.348042i	−0.770153	−	0.637860i	$-0.779820\pi$
$397$	−4768.46	+	2753.07i	−0.602826	+	0.348042i	0.167327	+	0.985902i	$0.446487\pi$
$398$	351.884			0.0443174
$399$		0			0
$400$	−954.189			−0.119274
$401$	1061.58	−	612.902i	0.132201	−	0.0763263i	−0.432441	−	0.901662i	$-0.642348\pi$
$401$	1061.58	−	612.902i	0.132201	−	0.0763263i	0.564642	+	0.825336i	$0.309014\pi$
$402$		0			0
$403$	−6884.29	+	11923.9i	−0.850945	+	1.47388i
$404$	−3344.09	−	5792.14i	−0.411819	−	0.713291i
$405$		0			0
$406$	540.957	+	2030.60i	0.0661263	+	0.248220i
$407$		−	136.329i		−	0.0166033i
$408$		0			0
$409$	−243.031	−	140.314i	−0.0293817	−	0.0169635i	0.485237	−	0.874383i	$-0.338733\pi$
$409$	−243.031	−	140.314i	−0.0293817	−	0.0169635i	−0.514619	+	0.857419i	$0.672067\pi$
$410$	2476.75	+	1429.95i	0.298336	+	0.172244i
$411$		0			0
$412$		−	6236.37i		−	0.745738i
$413$	−2802.57	+	2794.48i	−0.333911	+	0.332948i
$414$		0			0
$415$	−5685.00	−	9846.71i	−0.672448	−	1.16471i
$416$	−956.980	+	1657.54i	−0.112788	+	0.195354i
$417$		0			0
$418$	−622.954	+	359.663i	−0.0728940	+	0.0420854i
$419$	1270.45			0.148128			0.0740640	−	0.997253i	$-0.476403\pi$
$419$	1270.45			0.148128			0.0740640	+	0.997253i	$0.476403\pi$
$420$		0			0
$421$	3850.89			0.445798			0.222899	−	0.974842i	$-0.428448\pi$
$421$	3850.89			0.445798			0.222899	+	0.974842i	$0.428448\pi$
$422$	3761.51	−	2171.71i	0.433904	−	0.250515i
$423$		0			0
$424$	−133.251	+	230.797i	−0.0152623	+	0.0264351i
$425$	−1066.39	−	1847.03i	−0.121711	−	0.210810i
$426$		0			0
$427$	7484.74	−	1993.95i	0.848271	−	0.225981i
$428$		−	7618.40i		−	0.860395i
$429$		0			0
$430$	4968.53	+	2868.58i	0.557218	+	0.321710i
$431$	−5941.28	−	3430.20i	−0.663994	−	0.383357i	0.129803	−	0.991540i	$-0.458565\pi$
$431$	−5941.28	−	3430.20i	−0.663994	−	0.383357i	−0.793797	+	0.608183i	$0.791899\pi$
$432$		0			0
$433$		−	571.708i		−	0.0634516i	−0.999497	−	0.0317258i	$-0.989900\pi$
$433$		−	571.708i		−	0.0634516i	0.999497	−	0.0317258i	$-0.0101003\pi$
$434$	−8233.01	−	2218.77i	−0.910593	−	0.245402i
$435$		0			0
$436$	124.907	+	216.346i	0.0137201	+	0.0237640i
$437$	232.920	−	403.429i	0.0254967	−	0.0441616i
$438$		0			0
$439$	8677.87	−	5010.17i	0.943445	−	0.544698i	0.0524063	−	0.998626i	$-0.483311\pi$
$439$	8677.87	−	5010.17i	0.943445	−	0.544698i	0.891039	+	0.453928i	$0.149978\pi$
$440$	−661.410			−0.0716624
$441$		0			0
$442$	−4278.02			−0.460372
$443$	−12589.7	+	7268.66i	−1.35024	+	0.779559i	−0.988282	−	0.152637i	$-0.951223\pi$
$443$	−12589.7	+	7268.66i	−1.35024	+	0.779559i	−0.361953	+	0.932196i	$0.617890\pi$
$444$		0			0
$445$	2399.40	−	4155.88i	0.255601	−	0.442714i
$446$	890.870	+	1543.03i	0.0945828	+	0.163822i
$447$		0			0
$448$	−1144.46	−	308.430i	−0.120694	−	0.0325267i
$449$			16345.4i			1.71801i	0.511966	+	0.859006i	$0.328917\pi$
$449$			16345.4i			1.71801i	−0.511966	+	0.859006i	$0.671083\pi$
$450$		0			0
$451$	−1566.39	−	904.354i	−0.163544	−	0.0944221i
$452$	4260.18	+	2459.61i	0.443323	+	0.255952i
$453$		0			0
$454$			7825.73i			0.808986i
$455$	8653.82	−	2305.40i	0.891642	−	0.237535i
$456$		0			0
$457$	−9116.19	−	15789.7i	−0.933124	−	1.61622i	−0.777946	−	0.628331i	$-0.783738\pi$
$457$	−9116.19	−	15789.7i	−0.933124	−	1.61622i	−0.155178	−	0.987887i	$-0.549595\pi$
$458$	−4623.62	+	8008.34i	−0.471719	+	0.817042i
$459$		0			0
$460$	370.947	−	214.166i	0.0375989	−	0.0217077i
$461$	6201.75			0.626560			0.313280	−	0.949661i	$-0.398572\pi$
$461$	6201.75			0.626560			0.313280	+	0.949661i	$0.398572\pi$
$462$		0			0
$463$	−5557.97			−0.557885			−0.278943	−	0.960308i	$-0.589984\pi$
$463$	−5557.97			−0.557885			−0.278943	+	0.960308i	$0.589984\pi$
$464$	−786.117	+	453.865i	−0.0786521	+	0.0454098i
$465$		0			0
$466$	3156.57	−	5467.34i	0.313788	−	0.543497i
$467$	−1952.17	−	3381.25i	−0.193438	−	0.335045i	0.752949	−	0.658079i	$-0.228631\pi$
$467$	−1952.17	−	3381.25i	−0.193438	−	0.335045i	−0.946387	+	0.323034i	$0.895297\pi$
$468$		0			0
$469$	3606.92	−	3596.52i	0.355122	−	0.354098i
$470$		−	2477.64i		−	0.243159i
$471$		0			0
$472$	−1480.53	−	854.786i	−0.144379	−	0.0833575i
$473$	−3142.28	−	1814.20i	−0.305459	−	0.176357i
$474$		0			0
$475$			2097.47i			0.202607i
$476$	−682.003	−	2560.05i	−0.0656713	−	0.246512i
$477$		0			0
$478$	−1674.99	−	2901.17i	−0.160277	−	0.277607i
$479$	1810.25	−	3135.44i	0.172677	−	0.299085i	−0.766678	−	0.642032i	$-0.778092\pi$
$479$	1810.25	−	3135.44i	0.172677	−	0.299085i	0.939355	+	0.342947i	$0.111425\pi$
$480$		0			0
$481$	−690.536	+	398.681i	−0.0654589	+	0.0377927i
$482$	6731.85			0.636156
$483$		0			0
$484$	−4905.70			−0.460716
$485$	−2278.35	+	1315.41i	−0.213309	+	0.123154i
$486$		0			0
$487$	−75.3936	+	130.585i	−0.00701521	+	0.0121507i	−0.869512	−	0.493912i	$-0.835566\pi$
$487$	−75.3936	+	130.585i	−0.00701521	+	0.0121507i	0.862496	+	0.506063i	$0.168900\pi$
$488$	1672.93	+	2897.60i	0.155184	+	0.268787i
$489$		0			0
$490$	2759.19	+	4811.08i	0.254382	+	0.443556i
$491$		−	14822.1i		−	1.36235i	−0.732122	−	0.681173i	$-0.761470\pi$
$491$		−	14822.1i		−	1.36235i	0.732122	−	0.681173i	$-0.238530\pi$
$492$		0			0
$493$	−1757.10	−	1014.46i	−0.160519	−	0.0926758i
$494$	3643.55	+	2103.60i	0.331844	+	0.191590i
$495$		0			0
$496$		−	3683.21i		−	0.333429i
$497$	−5521.12	+	20486.7i	−0.498302	+	1.84901i
$498$		0			0
$499$	−2609.83	−	4520.35i	−0.234132	−	0.405528i	0.724888	−	0.688867i	$-0.241891\pi$
$499$	−2609.83	−	4520.35i	−0.234132	−	0.405528i	−0.959020	+	0.283338i	$0.908558\pi$
$500$	−2985.49	+	5171.01i	−0.267030	+	0.462509i
$501$		0			0
$502$	3553.94	−	2051.87i	0.315977	−	0.182429i
$503$	11822.9			1.04803			0.524013	−	0.851710i	$-0.324434\pi$
$503$	11822.9			1.04803			0.524013	+	0.851710i	$0.324434\pi$
$504$		0			0
$505$	−13518.1			−1.19118
$506$	−234.601	+	135.447i	−0.0206112	+	0.0118999i
$507$		0			0
$508$	−4005.48	+	6937.70i	−0.349832	+	0.605926i
$509$	6499.36	+	11257.2i	0.565970	+	0.980290i	0.996959	+	0.0779320i	$0.0248317\pi$
$509$	6499.36	+	11257.2i	0.565970	+	0.980290i	−0.430988	+	0.902358i	$0.641835\pi$
$510$		0			0
$511$	2563.36	+	2570.78i	0.221911	+	0.222553i
$512$		−	512.000i		−	0.0441942i
$513$		0			0
$514$	−13110.4	−	7569.28i	−1.12505	−	0.649546i
$515$	−10916.1	−	6302.44i	−0.934025	−	0.539260i
$516$		0			0
$517$			1566.95i			0.133297i
$518$	−348.664	−	349.672i	−0.0295742	−	0.0296597i
$519$		0			0
$520$	1934.24	+	3350.19i	0.163119	+	0.282530i
$521$	4522.32	−	7832.89i	0.380281	−	0.658667i	−0.610821	−	0.791769i	$-0.709160\pi$
$521$	4522.32	−	7832.89i	0.380281	−	0.658667i	0.991102	+	0.133102i	$0.0424938\pi$
$522$		0			0
$523$	−6013.69	+	3472.00i	−0.502792	+	0.290287i	−0.729866	−	0.683591i	$-0.760417\pi$
$523$	−6013.69	+	3472.00i	−0.502792	+	0.290287i	0.227074	+	0.973878i	$0.427084\pi$
$524$	−1314.96			−0.109627
$525$		0			0
$526$	−569.055			−0.0471711
$527$	7129.63	−	4116.29i	0.589320	−	0.340244i
$528$		0			0
$529$	−5995.78	+	10385.0i	−0.492791	+	0.853538i
$530$	269.325	+	466.484i	0.0220731	+	0.0382317i
$531$		0			0
$532$	−677.982	+	2515.73i	−0.0552523	+	0.205020i
$533$			10578.8i			0.859698i
$534$		0			0
$535$	−13335.2	−	7699.11i	−1.07763	−	0.622171i
$536$	1905.46	+	1100.12i	0.153551	+	0.0886526i
$537$		0			0
$538$		−	8619.67i		−	0.690745i
$539$	−1745.01	−	3042.71i	−0.139449	−	0.243152i
$540$		0			0
$541$	−5623.22	−	9739.70i	−0.446878	−	0.774016i	0.551303	−	0.834305i	$-0.314131\pi$
$541$	−5623.22	−	9739.70i	−0.446878	−	0.774016i	−0.998181	+	0.0602895i	$0.980798\pi$
$542$	−6855.87	+	11874.7i	−0.543330	+	0.941076i
$543$		0			0
$544$	991.084	−	572.203i	0.0781110	−	0.0450974i
$545$	504.922			0.0396853
$546$		0			0
$547$	−22163.7			−1.73245			−0.866225	−	0.499653i	$-0.833461\pi$
$547$	−22163.7			−1.73245			−0.866225	+	0.499653i	$0.833461\pi$
$548$	4574.76	−	2641.24i	0.356614	−	0.205891i
$549$		0			0
$550$	609.858	−	1056.30i	0.0472808	−	0.0818927i
$551$	997.673	+	1728.02i	0.0771367	+	0.133605i
$552$		0			0
$553$	3411.05	+	12804.1i	0.262301	+	0.984605i
$554$		−	15862.5i		−	1.21648i
$555$		0			0
$556$	184.832	+	106.713i	0.0140982	+	0.00813960i
$557$	12594.9	+	7271.69i	0.958106	+	0.553163i	0.895590	−	0.444881i	$-0.146754\pi$
$557$	12594.9	+	7271.69i	0.958106	+	0.553163i	0.0625162	+	0.998044i	$0.480087\pi$
$558$		0			0
$559$			21221.9i			1.60570i
$560$	−1696.46	+	1691.57i	−0.128016	+	0.127646i
$561$		0			0
$562$	−4758.17	−	8241.40i	−0.357138	−	0.618581i
$563$	7903.60	−	13689.4i	0.591646	−	1.02476i	−0.402364	−	0.915480i	$-0.631812\pi$
$563$	7903.60	−	13689.4i	0.591646	−	1.02476i	0.994011	−	0.109282i	$-0.0348552\pi$
$564$		0			0
$565$	8610.62	−	4971.34i	0.641153	−	0.370170i
$566$	8334.48			0.618947
$567$		0			0
$568$	−9165.17			−0.677046
$569$	−12273.5	+	7086.11i	−0.904274	+	0.522083i	−0.878584	−	0.477587i	$-0.841511\pi$
$569$	−12273.5	+	7086.11i	−0.904274	+	0.522083i	−0.0256894	+	0.999670i	$0.508178\pi$
$570$		0			0
$571$	8389.84	−	14531.6i	0.614893	−	1.06503i	−0.375510	−	0.926818i	$-0.622533\pi$
$571$	8389.84	−	14531.6i	0.614893	−	1.06503i	0.990403	−	0.138208i	$-0.0441341\pi$
$572$	−1223.28	−	2118.79i	−0.0894196	−	0.154879i
$573$		0			0
$574$	−6330.57	+	1686.48i	−0.460336	+	0.122635i
$575$			789.894i			0.0572885i
$576$		0			0
$577$	−13630.4	−	7869.54i	−0.983437	−	0.567787i	−0.0801307	−	0.996784i	$-0.525534\pi$
$577$	−13630.4	−	7869.54i	−0.983437	−	0.567787i	−0.903306	+	0.428997i	$0.858867\pi$
$578$	−6294.33	−	3634.03i	−0.452958	−	0.261515i
$579$		0			0
$580$			1834.69i			0.131347i
$581$	25148.7	+	6777.52i	1.79578	+	0.483957i
$582$		0			0
$583$	−170.331	−	295.022i	−0.0121002	−	0.0209581i
$584$	−784.090	+	1358.08i	−0.0555580	+	0.0962293i
$585$		0			0
$586$	13310.2	−	7684.65i	0.938293	−	0.541724i
$587$	13571.7			0.954281			0.477140	−	0.878827i	$-0.341673\pi$
$587$	13571.7			0.954281			0.477140	+	0.878827i	$0.341673\pi$
$588$		0			0
$589$	−8096.32			−0.566389
$590$	−2992.44	+	1727.68i	−0.208808	+	0.120555i
$591$		0			0
$592$	106.651	−	184.724i	0.00740424	−	0.0128245i
$593$	−6581.83	−	11400.1i	−0.455790	−	0.789451i	0.542943	−	0.839769i	$-0.317310\pi$
$593$	−6581.83	−	11400.1i	−0.455790	−	0.789451i	−0.998733	+	0.0503182i	$0.983976\pi$
$594$		0			0
$595$	−5170.34	−	1393.39i	−0.356241	−	0.0960059i
$596$			5090.55i			0.349860i
$597$		0			0
$598$	1372.14	+	792.204i	0.0938309	+	0.0541733i
$599$	11588.7	+	6690.73i	0.790486	+	0.456387i	0.840134	−	0.542380i	$-0.182477\pi$
$599$	11588.7	+	6690.73i	0.790486	+	0.456387i	−0.0496478	+	0.998767i	$0.515810\pi$
$600$		0			0
$601$		−	24999.7i		−	1.69677i	−0.529380	−	0.848385i	$-0.677575\pi$
$601$		−	24999.7i		−	1.69677i	0.529380	−	0.848385i	$-0.322425\pi$
$602$	−12699.6	+	3383.19i	−0.859794	+	0.229051i
$603$		0			0
$604$	3468.44	+	6007.51i	0.233657	+	0.404706i
$605$	−4957.67	+	8586.94i	−0.333154	+	0.577039i
$606$		0			0
$607$	−5366.56	+	3098.38i	−0.358850	+	0.207182i	−0.668576	−	0.743644i	$-0.733096\pi$
$607$	−5366.56	+	3098.38i	−0.358850	+	0.207182i	0.309726	+	0.950826i	$0.399763\pi$
$608$	−1125.46			−0.0750717
$609$		0			0
$610$	6762.61			0.448869
$611$	7936.97	−	4582.41i	0.525524	−	0.303412i
$612$		0			0
$613$	−7454.48	+	12911.5i	−0.491164	+	0.850721i	−0.999948	−	0.0101733i	$-0.996762\pi$
$613$	−7454.48	+	12911.5i	−0.491164	+	0.850721i	0.508784	+	0.860894i	$0.330095\pi$
$614$	−7109.98	−	12314.8i	−0.467321	−	0.809424i
$615$		0			0
$616$	1072.91	−	1069.81i	0.0701764	−	0.0699741i
$617$		−	4203.83i		−	0.274295i	−0.990551	−	0.137147i	$-0.956207\pi$
$617$		−	4203.83i		−	0.274295i	0.990551	−	0.137147i	$-0.0437934\pi$
$618$		0			0
$619$	15644.0	+	9032.05i	1.01581	+	0.586476i	0.912887	−	0.408213i	$-0.133848\pi$
$619$	15644.0	+	9032.05i	1.01581	+	0.586476i	0.102920	+	0.994690i	$0.467181\pi$
$620$	−6447.09	−	3722.23i	−0.417615	−	0.241110i
$621$		0			0
$622$			3853.15i			0.248388i
$623$	2829.84	+	10622.4i	0.181983	+	0.683113i
$624$		0			0
$625$	2306.92	+	3995.71i	0.147643	+	0.255725i
$626$	−7637.69	+	13228.9i	−0.487641	+	0.844619i
$627$		0			0
$628$	−8448.23	+	4877.59i	−0.536817	+	0.309931i
$629$	476.764			0.0302223
$630$		0			0
$631$	13551.8			0.854976			0.427488	−	0.904021i	$-0.359399\pi$
$631$	13551.8			0.854976			0.427488	+	0.904021i	$0.359399\pi$
$632$	−4956.92	+	2861.88i	−0.311987	+	0.180126i
$633$		0			0
$634$	9421.82	−	16319.1i	0.590202	−	1.02226i
$635$	8095.83	+	14022.4i	0.505942	+	0.876317i
$636$		0			0
$637$	−10308.9	+	17737.0i	−0.641213	+	1.10324i
$638$		−	1160.33i		−	0.0720029i
$639$		0			0
$640$	−896.205	−	517.424i	−0.0553525	−	0.0319578i
$641$	−1093.17	−	631.139i	−0.0673595	−	0.0388900i	0.465942	−	0.884815i	$-0.345716\pi$
$641$	−1093.17	−	631.139i	−0.0673595	−	0.0388900i	−0.533301	+	0.845925i	$0.679049\pi$
$642$		0			0
$643$		−	3682.91i		−	0.225878i	−0.993602	−	0.112939i	$-0.963973\pi$
$643$		−	3682.91i		−	0.225878i	0.993602	−	0.112939i	$-0.0360265\pi$
$644$	−255.324	+	947.408i	−0.0156229	+	0.0579706i
$645$		0			0
$646$	−1257.80	−	2178.57i	−0.0766060	−	0.132685i
$647$	15163.2	−	26263.5i	0.921372	−	1.59586i	0.124079	−	0.992272i	$-0.460402\pi$
$647$	15163.2	−	26263.5i	0.921372	−	1.59586i	0.797294	−	0.603592i	$-0.206264\pi$
$648$		0			0
$649$	1892.53	−	1092.65i	0.114466	−	0.0660868i
$650$	−7133.90			−0.430484
$651$		0			0
$652$	−13484.9			−0.809986
$653$	−19035.7	+	10990.3i	−1.14077	+	0.658625i	−0.946621	−	0.322347i	$-0.895528\pi$
$653$	−19035.7	+	10990.3i	−1.14077	+	0.658625i	−0.194150	+	0.980972i	$0.562195\pi$
$654$		0			0
$655$	−1328.89	+	2301.71i	−0.0792734	+	0.137305i
$656$	−1414.96	−	2450.78i	−0.0842148	−	0.145864i
$657$		0			0
$658$	4007.52	+	4019.11i	0.237431	+	0.238117i
$659$		−	12850.7i		−	0.759625i	−0.925063	−	0.379813i	$-0.875988\pi$
$659$		−	12850.7i		−	0.759625i	0.925063	−	0.379813i	$-0.124012\pi$
$660$		0			0
$661$	7607.66	+	4392.29i	0.447661	+	0.258457i	0.706842	−	0.707372i	$-0.250119\pi$
$661$	7607.66	+	4392.29i	0.447661	+	0.258457i	−0.259181	+	0.965829i	$0.583453\pi$
$662$	2792.47	+	1612.23i	0.163946	+	0.0946544i
$663$		0			0
$664$			11250.8i			0.657555i
$665$	3718.37	+	3729.12i	0.216830	+	0.217457i
$666$		0			0
$667$	375.717	+	650.762i	0.0218109	+	0.0377775i
$668$	−7343.65	+	12719.6i	−0.425351	+	0.736729i
$669$		0			0
$670$	3851.29	−	2223.54i	0.222072	−	0.128213i
$671$	−4276.93			−0.246064
$672$		0			0
$673$	6585.03			0.377168			0.188584	−	0.982057i	$-0.439610\pi$
$673$	6585.03			0.377168			0.188584	+	0.982057i	$0.439610\pi$
$674$	−11012.4	+	6357.99i	−0.629348	+	0.363354i
$675$		0			0
$676$	−2760.76	+	4781.78i	−0.157076	+	0.272063i
$677$	−2469.92	−	4278.03i	−0.140217	−	0.242863i	0.787361	−	0.616492i	$-0.211447\pi$
$677$	−2469.92	−	4278.03i	−0.140217	−	0.242863i	−0.927578	+	0.373629i	$0.878113\pi$
$678$		0			0
$679$	1568.20	−	5818.97i	0.0886331	−	0.328883i
$680$		−	2313.06i		−	0.130444i
$681$		0			0
$682$	4077.38	+	2354.08i	0.228931	+	0.132173i
$683$	−24397.6	−	14086.0i	−1.36684	−	0.789144i	−0.376315	−	0.926492i	$-0.622809\pi$
$683$	−24397.6	−	14086.0i	−1.36684	−	0.789144i	−0.990523	+	0.137348i	$0.956142\pi$
$684$		0			0
$685$		−	10676.9i		−	0.595537i
$686$	−12257.6	−	3341.40i	−0.682214	−	0.185969i
$687$		0			0
$688$	−2838.51	−	4916.45i	−0.157292	−	0.272439i
$689$	−996.236	+	1725.53i	−0.0550850	+	0.0954101i
$690$		0			0
$691$	15105.0	−	8720.89i	0.831580	−	0.480113i	−0.0228131	−	0.999740i	$-0.507262\pi$
$691$	15105.0	−	8720.89i	0.831580	−	0.480113i	0.854393	+	0.519627i	$0.173929\pi$
$692$	6967.10			0.382730
$693$		0			0
$694$	11682.4			0.638986
$695$	373.579	−	215.686i	0.0203895	−	0.0117719i
$696$		0			0
$697$	3162.67	−	5477.91i	0.171872	−	0.297691i
$698$	−3896.74	−	6749.35i	−0.211309	−	0.365998i
$699$		0			0
$700$	−1137.29	−	4269.07i	−0.0614078	−	0.230508i
$701$			3076.03i			0.165735i	0.996561	+	0.0828675i	$0.0264078\pi$
$701$			3076.03i			0.165735i	−0.996561	+	0.0828675i	$0.973592\pi$
$702$		0			0
$703$	−406.055	−	234.436i	−0.0217847	−	0.0125774i
$704$	566.793	+	327.238i	0.0303435	+	0.0175188i
$705$		0			0
$706$		−	6872.58i		−	0.366364i
$707$	21928.4	−	21865.1i	1.16648	−	1.16312i
$708$		0			0
$709$	−8416.52	−	14577.8i	−0.445824	−	0.772190i	0.552285	−	0.833655i	$-0.313756\pi$
$709$	−8416.52	−	14577.8i	−0.445824	−	0.772190i	−0.998109	+	0.0614656i	$0.980423\pi$
$710$	−9262.27	+	16042.7i	−0.489587	+	0.847989i
$711$		0			0
$712$	−4112.32	+	2374.25i	−0.216455	+	0.124970i
$713$	−3049.02			−0.160150
$714$		0			0
$715$	−4944.97			−0.258645
$716$	−2146.85	+	1239.49i	−0.112055	+	0.0646952i
$717$		0			0
$718$	−7112.33	+	12318.9i	−0.369680	+	0.640304i
$719$	8802.66	+	15246.7i	0.456584	+	0.790826i	0.998778	−	0.0494269i	$-0.0157395\pi$
$719$	8802.66	+	15246.7i	0.456584	+	0.790826i	−0.542194	+	0.840253i	$0.682406\pi$
$720$		0			0
$721$	27901.7	−	7433.07i	1.44121	−	0.383942i
$722$		−	11244.0i		−	0.579584i
$723$		0			0
$724$	1895.17	+	1094.17i	0.0972835	+	0.0561667i
$725$	−2930.10	−	1691.69i	−0.150098	−	0.0866591i
$726$		0			0
$727$			2775.92i			0.141614i	0.997490	+	0.0708068i	$0.0225574\pi$
$727$			2775.92i			0.141614i	−0.997490	+	0.0708068i	$0.977443\pi$
$728$	−8556.48	−	2305.95i	−0.435610	−	0.117396i
$729$		0			0
$730$	1584.79	+	2744.94i	0.0803504	+	0.139171i
$731$	6344.55	−	10989.1i	0.321014	−	0.556013i
$732$		0			0
$733$	−11338.9	+	6546.50i	−0.571365	+	0.329878i	−0.757695	−	0.652609i	$-0.773674\pi$
$733$	−11338.9	+	6546.50i	−0.571365	+	0.329878i	0.186329	+	0.982487i	$0.440341\pi$
$734$	−20086.8			−1.01011
$735$		0			0
$736$	−423.843			−0.0212270
$737$	−2435.70	+	1406.25i	−0.121737	+	0.0702848i
$738$		0			0
$739$	2662.96	−	4612.38i	0.132556	−	0.229593i	−0.792105	−	0.610384i	$-0.791015\pi$
$739$	2662.96	−	4612.38i	0.132556	−	0.229593i	0.924661	+	0.380791i	$0.124348\pi$
$740$	−215.561	−	373.362i	−0.0107083	−	0.0185474i
$741$		0			0
$742$	−1191.41	−	321.082i	−0.0589462	−	0.0158859i
$743$		−	23458.8i		−	1.15831i	−0.815219	−	0.579153i	$-0.803383\pi$
$743$		−	23458.8i		−	1.15831i	0.815219	−	0.579153i	$-0.196617\pi$
$744$		0			0
$745$	8910.49	+	5144.47i	0.438195	+	0.252992i
$746$	16802.1	+	9700.67i	0.824621	+	0.476095i
$747$		0			0
$748$			1462.86i			0.0715075i
$749$	34084.9	−	9080.30i	1.66280	−	0.442973i
$750$		0			0
$751$	2312.47	+	4005.31i	0.112361	+	0.194615i	0.916722	−	0.399526i	$-0.130825\pi$
$751$	2312.47	+	4005.31i	0.112361	+	0.194615i	−0.804361	+	0.594141i	$0.797492\pi$
$752$	−1225.83	+	2123.21i	−0.0594435	+	0.102959i
$753$		0			0
$754$	−5877.33	+	3393.28i	−0.283872	+	0.163894i
$755$	14020.7			0.675850
$756$		0			0
$757$	24438.8			1.17337			0.586686	−	0.809815i	$-0.300432\pi$
$757$	24438.8			1.17337			0.586686	+	0.809815i	$0.300432\pi$
$758$	10825.3	−	6250.00i	0.518724	−	0.299486i
$759$		0			0
$760$	−1137.39	+	1970.01i	−0.0542860	+	0.0940261i
$761$	−4006.59	−	6939.62i	−0.190853	−	0.330566i	0.754680	−	0.656093i	$-0.227792\pi$
$761$	−4006.59	−	6939.62i	−0.190853	−	0.330566i	−0.945533	+	0.325526i	$0.894459\pi$
$762$		0			0
$763$	−819.061	+	816.699i	−0.0388624	+	0.0387503i
$764$		−	18960.2i		−	0.897850i
$765$		0			0
$766$	7026.64	+	4056.83i	0.331440	+	0.191357i
$767$	−11069.1	−	6390.73i	−0.521096	−	0.300855i
$768$		0			0
$769$			31846.2i			1.49337i	0.665178	+	0.746685i	$0.268356\pi$
$769$			31846.2i			1.49337i	−0.665178	+	0.746685i	$0.731644\pi$
$770$	−788.328	−	2959.16i	−0.0368953	−	0.138495i
$771$		0			0
$772$	1229.52	+	2129.59i	0.0573204	+	0.0992819i
$773$	601.416	−	1041.68i	0.0279837	−	0.0484693i	−0.851694	−	0.524039i	$-0.824425\pi$
$773$	601.416	−	1041.68i	0.0279837	−	0.0484693i	0.879678	+	0.475570i	$0.157758\pi$
$774$		0			0
$775$	11889.2	−	6864.21i	0.551060	−	0.318155i
$776$	2603.24			0.120426
$777$		0			0
$778$	22717.5			1.04687
$779$	−5387.24	+	3110.32i	−0.247777	+	0.143054i
$780$		0			0
$781$	5857.80	−	10146.0i	0.268385	−	0.464856i
$782$	−473.680	−	820.437i	−0.0216608	−	0.0375176i
$783$		0			0
$784$	15.8478	−	5487.98i	0.000721929	−	0.249999i
$785$			19717.0i			0.896472i
$786$		0			0
$787$	5953.65	+	3437.34i	0.269663	+	0.155690i	0.628734	−	0.777620i	$-0.283573\pi$
$787$	5953.65	+	3437.34i	0.269663	+	0.155690i	−0.359072	+	0.933310i	$0.616907\pi$
$788$	6455.41	+	3727.03i	0.291833	+	0.168490i
$789$		0			0
$790$			11568.8i			0.521012i
$791$	−5926.71	+	21991.7i	−0.266409	+	0.988541i
$792$		0			0
$793$	12507.5	+	21663.6i	0.560094	+	0.970111i
$794$	5506.14	−	9536.91i	0.246103	−	0.426262i
$795$		0			0
$796$	−609.481	+	351.884i	−0.0271388	+	0.0156686i
$797$	−30846.8			−1.37095			−0.685477	−	0.728094i	$-0.740406\pi$
$797$	−30846.8			−1.37095			−0.685477	+	0.728094i	$0.740406\pi$
$798$		0			0
$799$	−5479.89			−0.242634
$800$	1652.70	−	954.189i	0.0730399	−	0.0421696i
$801$		0			0
$802$	−1225.80	+	2123.15i	−0.0539709	+	0.0934803i
$803$	−1002.28	−	1736.00i	−0.0440470	−	0.0762917i
$804$		0			0
$805$	1400.31	+	1404.36i	0.0613100	+	0.0614873i
$806$		−	27537.2i		−	1.20342i
$807$		0			0
$808$	11584.3	+	6688.18i	0.504373	+	0.291200i
$809$	−21591.0	−	12465.6i	−0.938319	−	0.541739i	−0.0488861	−	0.998804i	$-0.515567\pi$
$809$	−21591.0	−	12465.6i	−0.938319	−	0.541739i	−0.889433	+	0.457066i	$0.848900\pi$
$810$		0			0
$811$		−	3480.14i		−	0.150684i	−0.997158	−	0.0753418i	$-0.975995\pi$
$811$		−	3480.14i		−	0.150684i	0.997158	−	0.0753418i	$-0.0240048\pi$
$812$	−2967.57	−	2976.15i	−0.128253	−	0.128624i
$813$		0			0
$814$	136.329	+	236.128i	0.00587017	+	0.0101674i
$815$	−13627.8	+	23604.0i	−0.585719	+	1.01449i
$816$		0			0
$817$	−10807.2	+	6239.53i	−0.462785	+	0.267189i
$818$	561.256			0.0239900
$819$		0			0
$820$	−5719.80			−0.243590
$821$	23231.5	−	13412.7i	0.987559	−	0.570168i	0.0830154	−	0.996548i	$-0.473545\pi$
$821$	23231.5	−	13412.7i	0.987559	−	0.570168i	0.904544	+	0.426381i	$0.140212\pi$
$822$		0			0
$823$	−9350.70	+	16195.9i	−0.396045	+	0.685970i	−0.993234	−	0.116131i	$-0.962951\pi$
$823$	−9350.70	+	16195.9i	−0.396045	+	0.685970i	0.597189	+	0.802101i	$0.296284\pi$
$824$	6236.37	+	10801.7i	0.263658	+	0.456670i
$825$		0			0
$826$	2059.70	−	7642.75i	0.0867629	−	0.321944i
$827$			14508.6i			0.610052i	0.952344	+	0.305026i	$0.0986651\pi$
$827$			14508.6i			0.610052i	−0.952344	+	0.305026i	$0.901335\pi$
$828$		0			0
$829$	21771.4	+	12569.7i	0.912124	+	0.526615i	0.881114	−	0.472904i	$-0.156794\pi$
$829$	21771.4	+	12569.7i	0.912124	+	0.526615i	0.0310102	+	0.999519i	$0.490128\pi$
$830$	19693.4	+	11370.0i	0.823577	+	0.475492i
$831$		0			0
$832$		−	3827.92i		−	0.159506i
$833$	10640.9	−	6102.60i	0.442597	−	0.253832i
$834$		0			0
$835$	14842.9	+	25708.6i	0.615161	+	1.06549i
$836$	719.325	−	1245.91i	0.0297588	−	0.0515438i
$837$		0			0
$838$	−2200.49	+	1270.45i	−0.0907095	+	0.0523711i
$839$	21527.0			0.885809			0.442905	−	0.896569i	$-0.353948\pi$
$839$	21527.0			0.885809			0.442905	+	0.896569i	$0.353948\pi$
$840$		0			0
$841$	21170.4			0.868029
$842$	−6669.94	+	3850.89i	−0.272995	+	0.157614i
$843$		0			0
$844$	−4343.42	+	7523.03i	−0.177141	+	0.306817i
$845$	5580.02	+	9664.88i	0.227170	+	0.393470i
$846$		0			0
$847$	−5847.06	−	21948.2i	−0.237199	−	0.890378i
$848$		−	533.003i		−	0.0215842i
$849$		0			0
$850$	3694.07	+	2132.77i	0.149065	+	0.0860629i
$851$	−152.918	−	88.2872i	−0.00615976	−	0.00355634i
$852$		0			0
$853$		−	21878.2i		−	0.878188i	−0.898441	−	0.439094i	$-0.855300\pi$
$853$		−	21878.2i		−	0.878188i	0.898441	−	0.439094i	$-0.144700\pi$
$854$	−10970.0	+	10938.4i	−0.439561	+	0.438294i
$855$		0			0
$856$	7618.40	+	13195.5i	0.304196	+	0.526882i
$857$	12233.5	−	21189.0i	0.487618	−	0.844578i	−0.512281	−	0.858818i	$-0.671199\pi$
$857$	12233.5	−	21189.0i	0.487618	−	0.844578i	0.999899	+	0.0142395i	$0.00453274\pi$
$858$		0			0
$859$	−2162.83	+	1248.71i	−0.0859079	+	0.0495989i	−0.542339	−	0.840160i	$-0.682461\pi$
$859$	−2162.83	+	1248.71i	−0.0859079	+	0.0495989i	0.456431	+	0.889759i	$0.349128\pi$
$860$	−11474.3			−0.454967
$861$		0			0
$862$	13720.8			0.542149
$863$	−40031.0	+	23111.9i	−1.57899	+	0.911632i	−0.583993	+	0.811759i	$0.698510\pi$
$863$	−40031.0	+	23111.9i	−1.57899	+	0.911632i	−0.995000	+	0.0998731i	$0.968156\pi$
$864$		0			0
$865$	7040.91	−	12195.2i	0.276761	−	0.479364i
$866$	571.708	+	990.228i	0.0224335	+	0.0388560i
$867$		0			0
$868$	16478.8	−	4389.98i	0.644385	−	0.171666i
$869$		−	7316.54i		−	0.285612i
$870$		0			0
$871$	14246.0	+	8224.92i	0.554198	+	0.319966i
$872$	−432.692	−	249.815i	−0.0168037	−	0.00970159i
$873$		0			0
$874$			931.678i			0.0360578i
$875$	−26693.6	−	7193.86i	−1.03132	−	0.277939i
$876$		0			0
$877$	24859.9	+	43058.7i	0.957195	+	1.65791i	0.729262	+	0.684234i	$0.239863\pi$
$877$	24859.9	+	43058.7i	0.957195	+	1.65791i	0.227933	+	0.973677i	$0.426803\pi$
$878$	−10020.3	+	17355.7i	−0.385160	+	0.667116i
$879$		0			0
$880$	1145.60	−	661.410i	0.0438841	−	0.0253365i
$881$	45275.9			1.73142			0.865711	−	0.500543i	$-0.166866\pi$
$881$	45275.9			1.73142			0.865711	+	0.500543i	$0.166866\pi$
$882$		0			0
$883$	39588.7			1.50880			0.754398	−	0.656417i	$-0.227929\pi$
$883$	39588.7			1.50880			0.754398	+	0.656417i	$0.227929\pi$
$884$	7409.75	−	4278.02i	0.281919	−	0.162766i
$885$		0			0
$886$	14537.3	−	25179.4i	0.551231	−	0.954761i
$887$	23199.8	+	40183.2i	0.878210	+	1.52110i	0.853303	+	0.521415i	$0.174596\pi$
$887$	23199.8	+	40183.2i	0.878210	+	1.52110i	0.0249069	+	0.999690i	$0.492071\pi$
$888$		0			0
$889$	−35813.5	−	9651.65i	−1.35112	−	0.364124i
$890$			9597.60i			0.361475i
$891$		0			0
$892$	−3086.07	−	1781.74i	−0.115840	−	0.0668802i
$893$	4667.17	+	2694.59i	0.174895	+	0.100975i
$894$		0			0
$895$			5010.47i			0.187130i
$896$	2290.70	−	610.248i	0.0854095	−	0.0227533i
$897$		0			0
$898$	−16345.4	−	28311.1i	−0.607409	−	1.05206i
$899$	6530.00	−	11310.3i	0.242256	−	0.419599i
$900$		0			0
$901$	1031.74	−	595.676i	0.0381490	−	0.0220253i
$902$	3617.41			0.133533
$903$		0			0
$904$	−9838.46			−0.361971
$905$	3830.48	−	2211.53i	0.140696	−	0.0812307i
$906$		0			0
$907$	−4077.57	+	7062.56i	−0.149276	+	0.258554i	−0.930960	−	0.365121i	$-0.881028\pi$
$907$	−4077.57	+	7062.56i	−0.149276	+	0.258554i	0.781684	+	0.623675i	$0.214361\pi$
$908$	−7825.73	−	13554.6i	−0.286020	−	0.495401i
$909$		0			0
$910$	−12683.5	+	12646.9i	−0.462036	+	0.460703i
$911$			36014.7i			1.30979i	0.755719	+	0.654896i	$0.227288\pi$
$911$			36014.7i			1.30979i	−0.755719	+	0.654896i	$0.772712\pi$
$912$		0			0
$913$	−12454.9	−	7190.81i	−0.451474	−	0.260658i
$914$	31579.4	+	18232.4i	1.14284	+	0.659818i
$915$		0			0
$916$		−	18494.5i		−	0.667112i
$917$	−1567.29	−	5883.16i	−0.0564410	−	0.211864i
$918$		0			0
$919$	−6642.63	−	11505.4i	−0.238433	−	0.412979i	0.721832	−	0.692069i	$-0.243301\pi$
$919$	−6642.63	−	11505.4i	−0.238433	−	0.412979i	−0.960265	+	0.279090i	$0.909967\pi$
$920$	−428.333	+	741.894i	−0.0153497	+	0.0265864i
$921$		0			0
$922$	−10741.7	+	6201.75i	−0.383688	+	0.221522i
$923$	−68522.5			−2.44360
$924$		0			0
$925$	795.037			0.0282602
$926$	9626.69	−	5557.97i	0.341634	−	0.197242i
$927$		0			0
$928$	907.730	−	1572.23i	0.0321096	−	0.0556154i
$929$	−14335.1	−	24829.1i	−0.506263	−	0.876873i	−0.999974	−	0.00724698i	$-0.997693\pi$
$929$	−14335.1	−	24829.1i	−0.506263	−	0.876873i	0.493711	−	0.869626i	$-0.335640\pi$
$930$		0			0
$931$	−12063.5	−	34.8361i	−0.424668	−	0.00122632i
$932$			12626.3i			0.443764i
$933$		0			0
$934$	6762.51	+	3904.34i	0.236912	+	0.136781i
$935$	2560.60	+	1478.36i	0.0895620	+	0.0517087i
$936$		0			0
$937$			5963.38i			0.207914i	0.994582	+	0.103957i	$0.0331504\pi$
$937$			5963.38i			0.207914i	−0.994582	+	0.103957i	$0.966850\pi$
$938$	−2650.85	+	9836.29i	−0.0922744	+	0.342395i
$939$		0			0
$940$	2477.64	+	4291.40i	0.0859699	+	0.148904i
$941$	−12512.7	+	21672.6i	−0.433477	+	0.750804i	−0.997170	−	0.0751803i	$-0.976047\pi$
$941$	−12512.7	+	21672.6i	−0.433477	+	0.750804i	0.563693	+	0.825984i	$0.309380\pi$
$942$		0			0
$943$	−2028.80	+	1171.33i	−0.0700603	+	0.0404493i
$944$	3419.14			0.117885
$945$		0			0
$946$	7256.79			0.249407
$947$	38955.0	−	22490.7i	1.33671	−	0.771752i	0.350395	−	0.936602i	$-0.386047\pi$
$947$	38955.0	−	22490.7i	1.33671	−	0.771752i	0.986319	+	0.164850i	$0.0527141\pi$
$948$		0			0
$949$	−5862.17	+	10153.6i	−0.200521	+	0.347312i
$950$	−2097.47	−	3632.93i	−0.0716326	−	0.124071i
$951$		0			0
$952$	3741.31	+	3752.13i	0.127370	+	0.127739i
$953$			27121.8i			0.921891i	0.887428	+	0.460945i	$0.152490\pi$
$953$			27121.8i			0.921891i	−0.887428	+	0.460945i	$0.847510\pi$
$954$		0			0
$955$	−33188.0	−	19161.1i	−1.12454	−	0.649255i
$956$	5802.34	+	3349.98i	0.196298	+	0.113333i
$957$		0			0
$958$			7240.98i			0.244202i
$959$	17269.6	+	17319.5i	0.581506	+	0.583188i
$960$		0			0
$961$	11600.7	+	20092.9i	0.389401	+	0.674463i
$962$	797.362	−	1381.07i	0.0267235	−	0.0462864i
$963$		0			0
$964$	−11659.9	+	6731.85i	−0.389565	+	0.224915i
$965$	4970.18			0.165799
$966$		0			0
$967$	−53685.8			−1.78533			−0.892667	−	0.450716i	$-0.851169\pi$
$967$	−53685.8			−1.78533			−0.892667	+	0.450716i	$0.851169\pi$
$968$	8496.92	−	4905.70i	0.282130	−	0.162888i
$969$		0			0
$970$	2630.82	−	4556.71i	0.0870829	−	0.150832i
$971$	−12297.2	−	21299.4i	−0.406422	−	0.703943i	0.588064	−	0.808814i	$-0.299890\pi$
$971$	−12297.2	−	21299.4i	−0.406422	−	0.703943i	−0.994486	+	0.104871i	$0.966557\pi$
$972$		0			0
$973$	−257.136	+	954.131i	−0.00847214	+	0.0314368i
$974$		−	301.574i		−	0.00992101i
$975$		0			0
$976$	−5795.20	−	3345.86i	−0.190061	−	0.109732i
$977$	−22836.8	−	13184.8i	−0.747813	−	0.431750i	0.0770904	−	0.997024i	$-0.475437\pi$
$977$	−22836.8	−	13184.8i	−0.747813	−	0.431750i	−0.824903	+	0.565274i	$0.808770\pi$
$978$		0			0
$979$		−	6069.88i		−	0.198156i
$980$	−9590.14	−	5573.86i	−0.312598	−	0.181684i
$981$		0			0
$982$	14822.1	+	25672.6i	0.481662	+	0.834263i
$983$	12796.0	−	22163.3i	0.415188	−	0.719126i	−0.580261	−	0.814431i	$-0.697049\pi$
$983$	12796.0	−	22163.3i	0.415188	−	0.719126i	0.995448	+	0.0953049i	$0.0303826\pi$
$984$		0			0
$985$	13047.6	−	7533.03i	0.422062	−	0.243678i
$986$	4057.86			0.131063
$987$		0			0
$988$	−8414.42			−0.270950
$989$	−4069.92	+	2349.77i	−0.130855	+	0.0755494i
$990$		0			0
$991$	23380.2	−	40495.8i	0.749443	−	1.29807i	−0.198647	−	0.980071i	$-0.563655\pi$
$991$	23380.2	−	40495.8i	0.749443	−	1.29807i	0.948090	−	0.318002i	$-0.103012\pi$
$992$	3683.21	+	6379.51i	0.117885	+	0.204183i
$993$		0			0
$994$	−10923.9	−	41005.2i	−0.348576	−	1.30846i
$995$			1422.45i			0.0453212i
$996$		0			0
$997$	−46071.2	−	26599.2i	−1.46348	−	0.844941i	−0.464310	−	0.885673i	$-0.653698\pi$
$997$	−46071.2	−	26599.2i	−1.46348	−	0.844941i	−0.999170	+	0.0407322i	$0.987031\pi$
$998$	9040.70	+	5219.65i	0.286752	+	0.165556i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.k.b.269.2	yes	16
3.2	odd	2	inner	378.4.k.b.269.7	yes	16
7.5	odd	6	inner	378.4.k.b.215.7	yes	16
21.5	even	6	inner	378.4.k.b.215.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.k.b.215.2	✓	16	21.5	even	6	inner
378.4.k.b.215.7	yes	16	7.5	odd	6	inner
378.4.k.b.269.2	yes	16	1.1	even	1	trivial
378.4.k.b.269.7	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.04238 - 7.00160i) q^{5} +(17.8823 + 4.81922i) q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} +(2.00000 - 3.46410i) q^{4} +(-4.04238 - 7.00160i) q^{5} +(17.8823 + 4.81922i) q^{7} +8.00000i q^{8} +(14.0032 + 8.08475i) q^{10} +(-8.85614 - 5.11310i) q^{11} +59.8112i q^{13} +(-35.7922 + 9.53512i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(17.8813 - 30.9714i) q^{17} +(-30.4587 + 17.5854i) q^{19} -32.3390 q^{20} +20.4524 q^{22} +(-11.4706 + 6.62254i) q^{23} +(29.8184 - 51.6470i) q^{25} +(-59.8112 - 103.596i) q^{26} +(52.4588 - 52.3075i) q^{28} -56.7331i q^{29} +(199.360 + 115.100i) q^{31} +(27.7128 + 16.0000i) q^{32} +71.5254i q^{34} +(-38.5445 - 144.685i) q^{35} +(6.66566 + 11.5453i) q^{37} +(35.1707 - 60.9175i) q^{38} +(56.0128 - 32.3390i) q^{40} +176.870 q^{41} +354.814 q^{43} +(-35.4246 + 20.4524i) q^{44} +(13.2451 - 22.9412i) q^{46} +(-76.6146 - 132.700i) q^{47} +(296.550 + 172.357i) q^{49} +119.274i q^{50} +(207.192 + 119.622i) q^{52} +(28.8496 + 16.6563i) q^{53} +82.6762i q^{55} +(-38.5538 + 143.058i) q^{56} +(56.7331 + 98.2647i) q^{58} +(-106.848 + 185.067i) q^{59} +(362.200 - 209.116i) q^{61} -460.401 q^{62} -64.0000 q^{64} +(418.774 - 241.779i) q^{65} +(137.515 - 238.182i) q^{67} +(-71.5254 - 123.886i) q^{68} +(211.447 + 212.058i) q^{70} +1145.65i q^{71} +(169.760 + 98.0113i) q^{73} +(-23.0905 - 13.3313i) q^{74} +140.683i q^{76} +(-133.727 - 134.113i) q^{77} +(357.735 + 619.615i) q^{79} +(-64.6780 + 112.026i) q^{80} +(-306.348 + 176.870i) q^{82} +1406.35 q^{83} -289.132 q^{85} +(-614.556 + 354.814i) q^{86} +(40.9048 - 70.8491i) q^{88} +(296.781 + 514.040i) q^{89} +(-288.243 + 1069.56i) q^{91} +52.9803i q^{92} +(265.401 + 153.229i) q^{94} +(246.251 + 142.173i) q^{95} -325.405i q^{97} +(-685.997 - 1.98097i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 52 q^{7}+O(q^{10}) \) 16 * q + 32 * q^4 - 52 * q^7	\( 16 q + 32 q^{4} - 52 q^{7} - 24 q^{10} - 128 q^{16} - 186 q^{19} - 216 q^{22} - 44 q^{25} - 200 q^{28} + 408 q^{31} - 704 q^{37} - 96 q^{40} - 2036 q^{43} + 144 q^{46} - 20 q^{49} + 480 q^{52} - 672 q^{58} - 1242 q^{61} - 1024 q^{64} + 596 q^{67} - 48 q^{70} + 852 q^{73} - 2914 q^{79} + 1344 q^{82} - 10980 q^{85} - 432 q^{88} + 4134 q^{91} - 492 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 - 52 * q^7 - 24 * q^10 - 128 * q^16 - 186 * q^19 - 216 * q^22 - 44 * q^25 - 200 * q^28 + 408 * q^31 - 704 * q^37 - 96 * q^40 - 2036 * q^43 + 144 * q^46 - 20 * q^49 + 480 * q^52 - 672 * q^58 - 1242 * q^61 - 1024 * q^64 + 596 * q^67 - 48 * q^70 + 852 * q^73 - 2914 * q^79 + 1344 * q^82 - 10980 * q^85 - 432 * q^88 + 4134 * q^91 - 492 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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