LMFDB - Embedded newform 585.4.b.c.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,4,Mod(181,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("585.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	585.b (of order $2$, degree $1$, 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:	$34.5161173534$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	585.181
Dual form			585.4.b.c.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +7.00000 q^{4} -5.00000i q^{5} -16.0000i q^{7} +15.0000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} +7.00000 q^{4} -5.00000i q^{5} -16.0000i q^{7} +15.0000i q^{8} +5.00000 q^{10} +60.0000i q^{11} +(-39.0000 + 26.0000i) q^{13} +16.0000 q^{14} +41.0000 q^{16} -116.000 q^{17} +4.00000i q^{19} -35.0000i q^{20} -60.0000 q^{22} -142.000 q^{23} -25.0000 q^{25} +(-26.0000 - 39.0000i) q^{26} -112.000i q^{28} +220.000 q^{29} +290.000i q^{31} +161.000i q^{32} -116.000i q^{34} -80.0000 q^{35} +44.0000i q^{37} -4.00000 q^{38} +75.0000 q^{40} +370.000i q^{41} +212.000 q^{43} +420.000i q^{44} -142.000i q^{46} +176.000i q^{47} +87.0000 q^{49} -25.0000i q^{50} +(-273.000 + 182.000i) q^{52} +378.000 q^{53} +300.000 q^{55} +240.000 q^{56} +220.000i q^{58} -804.000i q^{59} -578.000 q^{61} -290.000 q^{62} +167.000 q^{64} +(130.000 + 195.000i) q^{65} +294.000i q^{67} -812.000 q^{68} -80.0000i q^{70} +140.000i q^{71} +662.000i q^{73} -44.0000 q^{74} +28.0000i q^{76} +960.000 q^{77} -1400.00 q^{79} -205.000i q^{80} -370.000 q^{82} -792.000i q^{83} +580.000i q^{85} +212.000i q^{86} -900.000 q^{88} -294.000i q^{89} +(416.000 + 624.000i) q^{91} -994.000 q^{92} -176.000 q^{94} +20.0000 q^{95} -286.000i q^{97} +87.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4}+O(q^{10}) $ 2 * q + 14 * q^4	$ 2 q + 14 q^{4} + 10 q^{10} - 78 q^{13} + 32 q^{14} + 82 q^{16} - 232 q^{17} - 120 q^{22} - 284 q^{23} - 50 q^{25} - 52 q^{26} + 440 q^{29} - 160 q^{35} - 8 q^{38} + 150 q^{40} + 424 q^{43} + 174 q^{49} - 546 q^{52} + 756 q^{53} + 600 q^{55} + 480 q^{56} - 1156 q^{61} - 580 q^{62} + 334 q^{64} + 260 q^{65} - 1624 q^{68} - 88 q^{74} + 1920 q^{77} - 2800 q^{79} - 740 q^{82} - 1800 q^{88} + 832 q^{91} - 1988 q^{92} - 352 q^{94} + 40 q^{95}+O(q^{100}) $ 2 * q + 14 * q^4 + 10 * q^10 - 78 * q^13 + 32 * q^14 + 82 * q^16 - 232 * q^17 - 120 * q^22 - 284 * q^23 - 50 * q^25 - 52 * q^26 + 440 * q^29 - 160 * q^35 - 8 * q^38 + 150 * q^40 + 424 * q^43 + 174 * q^49 - 546 * q^52 + 756 * q^53 + 600 * q^55 + 480 * q^56 - 1156 * q^61 - 580 * q^62 + 334 * q^64 + 260 * q^65 - 1624 * q^68 - 88 * q^74 + 1920 * q^77 - 2800 * q^79 - 740 * q^82 - 1800 * q^88 + 832 * q^91 - 1988 * q^92 - 352 * q^94 + 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			1.00000i			0.353553i	0.984251	+	0.176777i	$0.0565670\pi$
$2$			1.00000i			0.353553i	−0.984251	+	0.176777i	$0.943433\pi$
$3$		0			0
$3$		0			0
$4$	7.00000			0.875000
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$		−	16.0000i		−	0.863919i	−0.901893	−	0.431959i	$-0.857822\pi$
$7$		−	16.0000i		−	0.863919i	0.901893	−	0.431959i	$-0.142178\pi$
$8$			15.0000i			0.662913i
$9$		0			0
$10$	5.00000			0.158114
$11$			60.0000i			1.64461i	0.569049	+	0.822304i	$0.307311\pi$
$11$			60.0000i			1.64461i	−0.569049	+	0.822304i	$0.692689\pi$
$12$		0			0
$13$	−39.0000	+	26.0000i	−0.832050	+	0.554700i
$13$	−39.0000	+	26.0000i	−0.832050	+	0.554700i
$14$	16.0000			0.305441
$15$		0			0
$16$	41.0000			0.640625
$17$	−116.000			−1.65495			−0.827474	−	0.561503i	$-0.810223\pi$
$17$	−116.000			−1.65495			−0.827474	+	0.561503i	$0.810223\pi$
$18$		0			0
$19$			4.00000i			0.0482980i	0.999708	+	0.0241490i	$0.00768762\pi$
$19$			4.00000i			0.0482980i	−0.999708	+	0.0241490i	$0.992312\pi$
$20$		−	35.0000i		−	0.391312i
$21$		0			0
$22$	−60.0000			−0.581456
$23$	−142.000			−1.28735			−0.643675	−	0.765299i	$-0.722591\pi$
$23$	−142.000			−1.28735			−0.643675	+	0.765299i	$0.722591\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$	−26.0000	−	39.0000i	−0.196116	−	0.294174i
$27$		0			0
$28$		−	112.000i		−	0.755929i
$29$	220.000			1.40872			0.704362	−	0.709841i	$-0.251233\pi$
$29$	220.000			1.40872			0.704362	+	0.709841i	$0.251233\pi$
$30$		0			0
$31$			290.000i			1.68018i	0.542448	+	0.840089i	$0.317498\pi$
$31$			290.000i			1.68018i	−0.542448	+	0.840089i	$0.682502\pi$
$32$			161.000i			0.889408i
$33$		0			0
$34$		−	116.000i		−	0.585113i
$35$	−80.0000			−0.386356
$36$		0			0
$37$			44.0000i			0.195501i	0.995211	+	0.0977507i	$0.0311648\pi$
$37$			44.0000i			0.195501i	−0.995211	+	0.0977507i	$0.968835\pi$
$38$	−4.00000			−0.0170759
$39$		0			0
$40$	75.0000			0.296464
$41$			370.000i			1.40937i	0.709519	+	0.704686i	$0.248912\pi$
$41$			370.000i			1.40937i	−0.709519	+	0.704686i	$0.751088\pi$
$42$		0			0
$43$	212.000			0.751853			0.375927	−	0.926649i	$-0.377324\pi$
$43$	212.000			0.751853			0.375927	+	0.926649i	$0.377324\pi$
$44$			420.000i			1.43903i
$45$		0			0
$46$		−	142.000i		−	0.455147i
$47$			176.000i			0.546218i	0.961983	+	0.273109i	$0.0880519\pi$
$47$			176.000i			0.546218i	−0.961983	+	0.273109i	$0.911948\pi$
$48$		0			0
$49$	87.0000			0.253644
$50$		−	25.0000i		−	0.0707107i
$51$		0			0
$52$	−273.000	+	182.000i	−0.728044	+	0.485363i
$53$	378.000			0.979666			0.489833	−	0.871816i	$-0.337058\pi$
$53$	378.000			0.979666			0.489833	+	0.871816i	$0.337058\pi$
$54$		0			0
$55$	300.000			0.735491
$56$	240.000			0.572703
$57$		0			0
$58$			220.000i			0.498059i
$59$		−	804.000i		−	1.77410i	−0.461674	−	0.887050i	$-0.652751\pi$
$59$		−	804.000i		−	1.77410i	0.461674	−	0.887050i	$-0.347249\pi$
$60$		0			0
$61$	−578.000			−1.21320			−0.606601	−	0.795006i	$-0.707467\pi$
$61$	−578.000			−1.21320			−0.606601	+	0.795006i	$0.707467\pi$
$62$	−290.000			−0.594033
$63$		0			0
$64$	167.000			0.326172
$65$	130.000	+	195.000i	0.248069	+	0.372104i
$66$		0			0
$67$			294.000i			0.536087i	0.963407	+	0.268043i	$0.0863771\pi$
$67$			294.000i			0.536087i	−0.963407	+	0.268043i	$0.913623\pi$
$68$	−812.000			−1.44808
$69$		0			0
$70$		−	80.0000i		−	0.136598i
$71$			140.000i			0.234013i	0.993131	+	0.117007i	$0.0373299\pi$
$71$			140.000i			0.234013i	−0.993131	+	0.117007i	$0.962670\pi$
$72$		0			0
$73$			662.000i			1.06139i	0.847564	+	0.530693i	$0.178068\pi$
$73$			662.000i			1.06139i	−0.847564	+	0.530693i	$0.821932\pi$
$74$	−44.0000			−0.0691202
$75$		0			0
$76$			28.0000i			0.0422608i
$77$	960.000			1.42081
$78$		0			0
$79$	−1400.00			−1.99383			−0.996913	−	0.0785100i	$-0.974984\pi$
$79$	−1400.00			−1.99383			−0.996913	+	0.0785100i	$0.974984\pi$
$80$		−	205.000i		−	0.286496i
$81$		0			0
$82$	−370.000			−0.498289
$83$		−	792.000i		−	1.04739i	−0.851906	−	0.523695i	$-0.824553\pi$
$83$		−	792.000i		−	1.04739i	0.851906	−	0.523695i	$-0.175447\pi$
$84$		0			0
$85$			580.000i			0.740116i
$86$			212.000i			0.265820i
$87$		0			0
$88$	−900.000			−1.09023
$89$		−	294.000i		−	0.350157i	−0.984555	−	0.175078i	$-0.943982\pi$
$89$		−	294.000i		−	0.350157i	0.984555	−	0.175078i	$-0.0560179\pi$
$90$		0			0
$91$	416.000	+	624.000i	0.479216	+	0.718824i
$92$	−994.000			−1.12643
$93$		0			0
$94$	−176.000			−0.193117
$95$	20.0000			0.0215995
$96$		0			0
$97$		−	286.000i		−	0.299370i	−0.988734	−	0.149685i	$-0.952174\pi$
$97$		−	286.000i		−	0.299370i	0.988734	−	0.149685i	$-0.0478260\pi$
$98$			87.0000i			0.0896768i
$99$		0			0
$100$	−175.000			−0.175000
$101$	488.000			0.480770			0.240385	−	0.970678i	$-0.422726\pi$
$101$	488.000			0.480770			0.240385	+	0.970678i	$0.422726\pi$
$102$		0			0
$103$	1612.00			1.54209			0.771044	−	0.636782i	$-0.219735\pi$
$103$	1612.00			1.54209			0.771044	+	0.636782i	$0.219735\pi$
$104$	−390.000	−	585.000i	−0.367718	−	0.551577i
$105$		0			0
$106$			378.000i			0.346364i
$107$	−176.000			−0.159015			−0.0795073	−	0.996834i	$-0.525335\pi$
$107$	−176.000			−0.159015			−0.0795073	+	0.996834i	$0.525335\pi$
$108$		0			0
$109$			1934.00i			1.69948i	0.527200	+	0.849741i	$0.323242\pi$
$109$			1934.00i			1.69948i	−0.527200	+	0.849741i	$0.676758\pi$
$110$			300.000i			0.260035i
$111$		0			0
$112$		−	656.000i		−	0.553448i
$113$	−1852.00			−1.54178			−0.770892	−	0.636966i	$-0.780189\pi$
$113$	−1852.00			−1.54178			−0.770892	+	0.636966i	$0.780189\pi$
$114$		0			0
$115$			710.000i			0.575720i
$116$	1540.00			1.23263
$117$		0			0
$118$	804.000			0.627239
$119$			1856.00i			1.42974i
$120$		0			0
$121$	−2269.00			−1.70473
$122$		−	578.000i		−	0.428932i
$123$		0			0
$124$			2030.00i			1.47016i
$125$			125.000i			0.0894427i
$126$		0			0
$127$	−2064.00			−1.44213			−0.721065	−	0.692868i	$-0.756347\pi$
$127$	−2064.00			−1.44213			−0.721065	+	0.692868i	$0.756347\pi$
$128$			1455.00i			1.00473i
$129$		0			0
$130$	−195.000	+	130.000i	−0.131559	+	0.0877058i
$131$	978.000			0.652277			0.326138	−	0.945322i	$-0.394252\pi$
$131$	978.000			0.652277			0.326138	+	0.945322i	$0.394252\pi$
$132$		0			0
$133$	64.0000			0.0417256
$134$	−294.000			−0.189535
$135$		0			0
$136$		−	1740.00i		−	1.09709i
$137$			786.000i			0.490164i	0.969502	+	0.245082i	$0.0788149\pi$
$137$			786.000i			0.490164i	−0.969502	+	0.245082i	$0.921185\pi$
$138$		0			0
$139$	280.000			0.170858			0.0854291	−	0.996344i	$-0.472774\pi$
$139$	280.000			0.170858			0.0854291	+	0.996344i	$0.472774\pi$
$140$	−560.000			−0.338062
$141$		0			0
$142$	−140.000			−0.0827362
$143$	−1560.00	−	2340.00i	−0.912264	−	1.36840i
$144$		0			0
$145$		−	1100.00i		−	0.630000i
$146$	−662.000			−0.375257
$147$		0			0
$148$			308.000i			0.171064i
$149$			586.000i			0.322195i	0.986939	+	0.161097i	$0.0515033\pi$
$149$			586.000i			0.322195i	−0.986939	+	0.161097i	$0.948497\pi$
$150$		0			0
$151$		−	2270.00i		−	1.22338i	−0.791099	−	0.611689i	$-0.790491\pi$
$151$		−	2270.00i		−	1.22338i	0.791099	−	0.611689i	$-0.209509\pi$
$152$	−60.0000			−0.0320174
$153$		0			0
$154$			960.000i			0.502331i
$155$	1450.00			0.751399
$156$		0			0
$157$	266.000			0.135217			0.0676086	−	0.997712i	$-0.478463\pi$
$157$	266.000			0.135217			0.0676086	+	0.997712i	$0.478463\pi$
$158$		−	1400.00i		−	0.704924i
$159$		0			0
$160$	805.000			0.397755
$161$			2272.00i			1.11217i
$162$		0			0
$163$			2142.00i			1.02929i	0.857403	+	0.514645i	$0.172076\pi$
$163$			2142.00i			1.02929i	−0.857403	+	0.514645i	$0.827924\pi$
$164$			2590.00i			1.23320i
$165$		0			0
$166$	792.000			0.370308
$167$		−	1904.00i		−	0.882251i	−0.897445	−	0.441126i	$-0.854579\pi$
$167$		−	1904.00i		−	0.882251i	0.897445	−	0.441126i	$-0.145421\pi$
$168$		0			0
$169$	845.000	−	2028.00i	0.384615	−	0.923077i
$170$	−580.000			−0.261670
$171$		0			0
$172$	1484.00			0.657872
$173$	−2782.00			−1.22261			−0.611305	−	0.791395i	$-0.709355\pi$
$173$	−2782.00			−1.22261			−0.611305	+	0.791395i	$0.709355\pi$
$174$		0			0
$175$			400.000i			0.172784i
$176$			2460.00i			1.05358i
$177$		0			0
$178$	294.000			0.123799
$179$	1530.00			0.638869			0.319435	−	0.947608i	$-0.396507\pi$
$179$	1530.00			0.638869			0.319435	+	0.947608i	$0.396507\pi$
$180$		0			0
$181$	−1458.00			−0.598742			−0.299371	−	0.954137i	$-0.596777\pi$
$181$	−1458.00			−0.598742			−0.299371	+	0.954137i	$0.596777\pi$
$182$	−624.000	+	416.000i	−0.254143	+	0.169428i
$183$		0			0
$184$		−	2130.00i		−	0.853400i
$185$	220.000			0.0874309
$186$		0			0
$187$		−	6960.00i		−	2.72174i
$188$			1232.00i			0.477941i
$189$		0			0
$190$			20.0000i			0.00763659i
$191$	−292.000			−0.110620			−0.0553099	−	0.998469i	$-0.517615\pi$
$191$	−292.000			−0.110620			−0.0553099	+	0.998469i	$0.517615\pi$
$192$		0			0
$193$		−	418.000i		−	0.155898i	−0.996957	−	0.0779490i	$-0.975163\pi$
$193$		−	418.000i		−	0.155898i	0.996957	−	0.0779490i	$-0.0248371\pi$
$194$	286.000			0.105843
$195$		0			0
$196$	609.000			0.221939
$197$		−	3614.00i		−	1.30704i	−0.756909	−	0.653520i	$-0.773292\pi$
$197$		−	3614.00i		−	1.30704i	0.756909	−	0.653520i	$-0.226708\pi$
$198$		0			0
$199$	2880.00			1.02592			0.512959	−	0.858413i	$-0.328549\pi$
$199$	2880.00			1.02592			0.512959	+	0.858413i	$0.328549\pi$
$200$		−	375.000i		−	0.132583i
$201$		0			0
$202$			488.000i			0.169978i
$203$		−	3520.00i		−	1.21702i
$204$		0			0
$205$	1850.00			0.630291
$206$			1612.00i			0.545210i
$207$		0			0
$208$	−1599.00	+	1066.00i	−0.533032	+	0.355355i
$209$	−240.000			−0.0794313
$210$		0			0
$211$	−768.000			−0.250575			−0.125287	−	0.992120i	$-0.539985\pi$
$211$	−768.000			−0.250575			−0.125287	+	0.992120i	$0.539985\pi$
$212$	2646.00			0.857208
$213$		0			0
$214$		−	176.000i		−	0.0562202i
$215$		−	1060.00i		−	0.336239i
$216$		0			0
$217$	4640.00			1.45154
$218$	−1934.00			−0.600858
$219$		0			0
$220$	2100.00			0.643554
$221$	4524.00	−	3016.00i	1.37700	−	0.918001i
$222$		0			0
$223$			2452.00i			0.736314i	0.929764	+	0.368157i	$0.120011\pi$
$223$			2452.00i			0.736314i	−0.929764	+	0.368157i	$0.879989\pi$
$224$	2576.00			0.768376
$225$		0			0
$226$		−	1852.00i		−	0.545103i
$227$			3336.00i			0.975410i	0.873008	+	0.487705i	$0.162166\pi$
$227$			3336.00i			0.975410i	−0.873008	+	0.487705i	$0.837834\pi$
$228$		0			0
$229$		−	3926.00i		−	1.13291i	−0.824091	−	0.566457i	$-0.808314\pi$
$229$		−	3926.00i		−	1.13291i	0.824091	−	0.566457i	$-0.191686\pi$
$230$	−710.000			−0.203548
$231$		0			0
$232$			3300.00i			0.933860i
$233$	1528.00			0.429625			0.214812	−	0.976655i	$-0.431086\pi$
$233$	1528.00			0.429625			0.214812	+	0.976655i	$0.431086\pi$
$234$		0			0
$235$	880.000			0.244276
$236$		−	5628.00i		−	1.55234i
$237$		0			0
$238$	−1856.00			−0.505490
$239$			5336.00i			1.44417i	0.691803	+	0.722086i	$0.256817\pi$
$239$			5336.00i			1.44417i	−0.691803	+	0.722086i	$0.743183\pi$
$240$		0			0
$241$			1860.00i			0.497150i	0.968613	+	0.248575i	$0.0799622\pi$
$241$			1860.00i			0.497150i	−0.968613	+	0.248575i	$0.920038\pi$
$242$		−	2269.00i		−	0.602714i
$243$		0			0
$244$	−4046.00			−1.06155
$245$		−	435.000i		−	0.113433i
$246$		0			0
$247$	−104.000	−	156.000i	−0.0267909	−	0.0401864i
$248$	−4350.00			−1.11381
$249$		0			0
$250$	−125.000			−0.0316228
$251$	−4722.00			−1.18745			−0.593725	−	0.804668i	$-0.702343\pi$
$251$	−4722.00			−1.18745			−0.593725	+	0.804668i	$0.702343\pi$
$252$		0			0
$253$		−	8520.00i		−	2.11719i
$254$		−	2064.00i		−	0.509870i
$255$		0			0
$256$	−119.000			−0.0290527
$257$	4284.00			1.03980			0.519900	−	0.854227i	$-0.325969\pi$
$257$	4284.00			1.03980			0.519900	+	0.854227i	$0.325969\pi$
$258$		0			0
$259$	704.000			0.168897
$260$	910.000	+	1365.00i	0.217061	+	0.325591i
$261$		0			0
$262$			978.000i			0.230615i
$263$	198.000			0.0464228			0.0232114	−	0.999731i	$-0.492611\pi$
$263$	198.000			0.0464228			0.0232114	+	0.999731i	$0.492611\pi$
$264$		0			0
$265$		−	1890.00i		−	0.438120i
$266$			64.0000i			0.0147522i
$267$		0			0
$268$			2058.00i			0.469076i
$269$	5840.00			1.32368			0.661842	−	0.749643i	$-0.269775\pi$
$269$	5840.00			1.32368			0.661842	+	0.749643i	$0.269775\pi$
$270$		0			0
$271$			4790.00i			1.07370i	0.843679	+	0.536848i	$0.180385\pi$
$271$			4790.00i			1.07370i	−0.843679	+	0.536848i	$0.819615\pi$
$272$	−4756.00			−1.06020
$273$		0			0
$274$	−786.000			−0.173299
$275$		−	1500.00i		−	0.328921i
$276$		0			0
$277$	946.000			0.205197			0.102599	−	0.994723i	$-0.467284\pi$
$277$	946.000			0.205197			0.102599	+	0.994723i	$0.467284\pi$
$278$			280.000i			0.0604075i
$279$		0			0
$280$		−	1200.00i		−	0.256120i
$281$		−	4530.00i		−	0.961698i	−0.876803	−	0.480849i	$-0.840329\pi$
$281$		−	4530.00i		−	0.961698i	0.876803	−	0.480849i	$-0.159671\pi$
$282$		0			0
$283$	2612.00			0.548648			0.274324	−	0.961637i	$-0.411546\pi$
$283$	2612.00			0.548648			0.274324	+	0.961637i	$0.411546\pi$
$284$			980.000i			0.204762i
$285$		0			0
$286$	2340.00	−	1560.00i	0.483801	−	0.322534i
$287$	5920.00			1.21758
$288$		0			0
$289$	8543.00			1.73886
$290$	1100.00			0.222739
$291$		0			0
$292$			4634.00i			0.928713i
$293$		−	4422.00i		−	0.881693i	−0.897583	−	0.440846i	$-0.854678\pi$
$293$		−	4422.00i		−	0.881693i	0.897583	−	0.440846i	$-0.145322\pi$
$294$		0			0
$295$	−4020.00			−0.793402
$296$	−660.000			−0.129600
$297$		0			0
$298$	−586.000			−0.113913
$299$	5538.00	−	3692.00i	1.07114	−	0.714093i
$300$		0			0
$301$		−	3392.00i		−	0.649540i
$302$	2270.00			0.432529
$303$		0			0
$304$			164.000i			0.0309409i
$305$			2890.00i			0.542560i
$306$		0			0
$307$		−	8306.00i		−	1.54413i	−0.635542	−	0.772066i	$-0.719224\pi$
$307$		−	8306.00i		−	1.54413i	0.635542	−	0.772066i	$-0.280776\pi$
$308$	6720.00			1.24321
$309$		0			0
$310$			1450.00i			0.265660i
$311$	−2812.00			−0.512713			−0.256357	−	0.966582i	$-0.582522\pi$
$311$	−2812.00			−0.512713			−0.256357	+	0.966582i	$0.582522\pi$
$312$		0			0
$313$	−1978.00			−0.357199			−0.178599	−	0.983922i	$-0.557157\pi$
$313$	−1978.00			−0.357199			−0.178599	+	0.983922i	$0.557157\pi$
$314$			266.000i			0.0478065i
$315$		0			0
$316$	−9800.00			−1.74460
$317$			7666.00i			1.35825i	0.734022	+	0.679125i	$0.237641\pi$
$317$			7666.00i			1.35825i	−0.734022	+	0.679125i	$0.762359\pi$
$318$		0			0
$319$			13200.0i			2.31680i
$320$		−	835.000i		−	0.145868i
$321$		0			0
$322$	−2272.00			−0.393210
$323$		−	464.000i		−	0.0799308i
$324$		0			0
$325$	975.000	−	650.000i	0.166410	−	0.110940i
$326$	−2142.00			−0.363909
$327$		0			0
$328$	−5550.00			−0.934291
$329$	2816.00			0.471888
$330$		0			0
$331$		−	160.000i		−	0.0265692i	−0.999912	−	0.0132846i	$-0.995771\pi$
$331$		−	160.000i		−	0.0265692i	0.999912	−	0.0132846i	$-0.00422874\pi$
$332$		−	5544.00i		−	0.916465i
$333$		0			0
$334$	1904.00			0.311923
$335$	1470.00			0.239745
$336$		0			0
$337$	6786.00			1.09690			0.548452	−	0.836182i	$-0.315217\pi$
$337$	6786.00			1.09690			0.548452	+	0.836182i	$0.315217\pi$
$338$	2028.00	+	845.000i	0.326357	+	0.135982i
$339$		0			0
$340$			4060.00i			0.647601i
$341$	−17400.0			−2.76323
$342$		0			0
$343$		−	6880.00i		−	1.08305i
$344$			3180.00i			0.498413i
$345$		0			0
$346$		−	2782.00i		−	0.432258i
$347$	−16.0000			−0.00247529			−0.00123764	−	0.999999i	$-0.500394\pi$
$347$	−16.0000			−0.00247529			−0.00123764	+	0.999999i	$0.500394\pi$
$348$		0			0
$349$		−	2986.00i		−	0.457985i	−0.973428	−	0.228993i	$-0.926457\pi$
$349$		−	2986.00i		−	0.457985i	0.973428	−	0.228993i	$-0.0735432\pi$
$350$	−400.000			−0.0610883
$351$		0			0
$352$	−9660.00			−1.46273
$353$			678.000i			0.102227i	0.998693	+	0.0511137i	$0.0162771\pi$
$353$			678.000i			0.102227i	−0.998693	+	0.0511137i	$0.983723\pi$
$354$		0			0
$355$	700.000			0.104654
$356$		−	2058.00i		−	0.306387i
$357$		0			0
$358$			1530.00i			0.225874i
$359$			5416.00i			0.796227i	0.917336	+	0.398114i	$0.130335\pi$
$359$			5416.00i			0.796227i	−0.917336	+	0.398114i	$0.869665\pi$
$360$		0			0
$361$	6843.00			0.997667
$362$		−	1458.00i		−	0.211687i
$363$		0			0
$364$	2912.00	+	4368.00i	0.419314	+	0.628971i
$365$	3310.00			0.474667
$366$		0			0
$367$	−7464.00			−1.06163			−0.530814	−	0.847488i	$-0.678114\pi$
$367$	−7464.00			−1.06163			−0.530814	+	0.847488i	$0.678114\pi$
$368$	−5822.00			−0.824709
$369$		0			0
$370$			220.000i			0.0309115i
$371$		−	6048.00i		−	0.846352i
$372$		0			0
$373$	8542.00			1.18576			0.592879	−	0.805291i	$-0.297991\pi$
$373$	8542.00			1.18576			0.592879	+	0.805291i	$0.297991\pi$
$374$	6960.00			0.962281
$375$		0			0
$376$	−2640.00			−0.362095
$377$	−8580.00	+	5720.00i	−1.17213	+	0.781419i
$378$		0			0
$379$			10224.0i			1.38568i	0.721093	+	0.692839i	$0.243640\pi$
$379$			10224.0i			1.38568i	−0.721093	+	0.692839i	$0.756360\pi$
$380$	140.000			0.0188996
$381$		0			0
$382$		−	292.000i		−	0.0391100i
$383$		−	6032.00i		−	0.804754i	−0.915474	−	0.402377i	$-0.868184\pi$
$383$		−	6032.00i		−	0.804754i	0.915474	−	0.402377i	$-0.131816\pi$
$384$		0			0
$385$		−	4800.00i		−	0.635404i
$386$	418.000			0.0551182
$387$		0			0
$388$		−	2002.00i		−	0.261949i
$389$	−3820.00			−0.497896			−0.248948	−	0.968517i	$-0.580085\pi$
$389$	−3820.00			−0.497896			−0.248948	+	0.968517i	$0.580085\pi$
$390$		0			0
$391$	16472.0			2.13050
$392$			1305.00i			0.168144i
$393$		0			0
$394$	3614.00			0.462108
$395$			7000.00i			0.891666i
$396$		0			0
$397$		−	11156.0i		−	1.41034i	−0.709040	−	0.705168i	$-0.750871\pi$
$397$		−	11156.0i		−	1.41034i	0.709040	−	0.705168i	$-0.249129\pi$
$398$			2880.00i			0.362717i
$399$		0			0
$400$	−1025.00			−0.128125
$401$		−	1030.00i		−	0.128269i	−0.997941	−	0.0641343i	$-0.979571\pi$
$401$		−	1030.00i		−	0.128269i	0.997941	−	0.0641343i	$-0.0204286\pi$
$402$		0			0
$403$	−7540.00	−	11310.0i	−0.931995	−	1.39799i
$404$	3416.00			0.420674
$405$		0			0
$406$	3520.00			0.430282
$407$	−2640.00			−0.321523
$408$		0			0
$409$		−	16296.0i		−	1.97014i	−0.172168	−	0.985068i	$-0.555077\pi$
$409$		−	16296.0i		−	1.97014i	0.172168	−	0.985068i	$-0.444923\pi$
$410$			1850.00i			0.222841i
$411$		0			0
$412$	11284.0			1.34933
$413$	−12864.0			−1.53268
$414$		0			0
$415$	−3960.00			−0.468407
$416$	−4186.00	−	6279.00i	−0.493355	−	0.740032i
$417$		0			0
$418$		−	240.000i		−	0.0280832i
$419$	−30.0000			−0.00349784			−0.00174892	−	0.999998i	$-0.500557\pi$
$419$	−30.0000			−0.00349784			−0.00174892	+	0.999998i	$0.500557\pi$
$420$		0			0
$421$		−	11990.0i		−	1.38802i	−0.719965	−	0.694011i	$-0.755842\pi$
$421$		−	11990.0i		−	1.38802i	0.719965	−	0.694011i	$-0.244158\pi$
$422$		−	768.000i		−	0.0885916i
$423$		0			0
$424$			5670.00i			0.649433i
$425$	2900.00			0.330990
$426$		0			0
$427$			9248.00i			1.04811i
$428$	−1232.00			−0.139138
$429$		0			0
$430$	1060.00			0.118878
$431$		−	1260.00i		−	0.140817i	−0.997518	−	0.0704084i	$-0.977570\pi$
$431$		−	1260.00i		−	0.140817i	0.997518	−	0.0704084i	$-0.0224303\pi$
$432$		0			0
$433$	−1678.00			−0.186234			−0.0931172	−	0.995655i	$-0.529683\pi$
$433$	−1678.00			−0.186234			−0.0931172	+	0.995655i	$0.529683\pi$
$434$			4640.00i			0.513196i
$435$		0			0
$436$			13538.0i			1.48705i
$437$		−	568.000i		−	0.0621765i
$438$		0			0
$439$	7360.00			0.800168			0.400084	−	0.916479i	$-0.368981\pi$
$439$	7360.00			0.800168			0.400084	+	0.916479i	$0.368981\pi$
$440$			4500.00i			0.487566i
$441$		0			0
$442$	3016.00	+	4524.00i	0.324562	+	0.486843i
$443$	8628.00			0.925347			0.462673	−	0.886529i	$-0.346890\pi$
$443$	8628.00			0.925347			0.462673	+	0.886529i	$0.346890\pi$
$444$		0			0
$445$	−1470.00			−0.156595
$446$	−2452.00			−0.260326
$447$		0			0
$448$		−	2672.00i		−	0.281786i
$449$			11586.0i			1.21777i	0.793260	+	0.608883i	$0.208382\pi$
$449$			11586.0i			1.21777i	−0.793260	+	0.608883i	$0.791618\pi$
$450$		0			0
$451$	−22200.0			−2.31787
$452$	−12964.0			−1.34906
$453$		0			0
$454$	−3336.00			−0.344860
$455$	3120.00	−	2080.00i	0.321468	−	0.214312i
$456$		0			0
$457$			12994.0i			1.33005i	0.746820	+	0.665026i	$0.231580\pi$
$457$			12994.0i			1.33005i	−0.746820	+	0.665026i	$0.768420\pi$
$458$	3926.00			0.400546
$459$		0			0
$460$			4970.00i			0.503755i
$461$		−	1330.00i		−	0.134369i	−0.997741	−	0.0671847i	$-0.978598\pi$
$461$		−	1330.00i		−	0.134369i	0.997741	−	0.0671847i	$-0.0214017\pi$
$462$		0			0
$463$		−	14348.0i		−	1.44019i	−0.693875	−	0.720095i	$-0.744098\pi$
$463$		−	14348.0i		−	1.44019i	0.693875	−	0.720095i	$-0.255902\pi$
$464$	9020.00			0.902463
$465$		0			0
$466$			1528.00i			0.151895i
$467$	2484.00			0.246137			0.123068	−	0.992398i	$-0.460727\pi$
$467$	2484.00			0.246137			0.123068	+	0.992398i	$0.460727\pi$
$468$		0			0
$469$	4704.00			0.463135
$470$			880.000i			0.0863646i
$471$		0			0
$472$	12060.0			1.17607
$473$			12720.0i			1.23650i
$474$		0			0
$475$		−	100.000i		−	0.00965961i
$476$			12992.0i			1.25102i
$477$		0			0
$478$	−5336.00			−0.510592
$479$		−	2424.00i		−	0.231222i	−0.993295	−	0.115611i	$-0.963117\pi$
$479$		−	2424.00i		−	0.231222i	0.993295	−	0.115611i	$-0.0368826\pi$
$480$		0			0
$481$	−1144.00	−	1716.00i	−0.108445	−	0.162667i
$482$	−1860.00			−0.175769
$483$		0			0
$484$	−15883.0			−1.49164
$485$	−1430.00			−0.133882
$486$		0			0
$487$			10584.0i			0.984819i	0.870364	+	0.492409i	$0.163884\pi$
$487$			10584.0i			0.984819i	−0.870364	+	0.492409i	$0.836116\pi$
$488$		−	8670.00i		−	0.804247i
$489$		0			0
$490$	435.000			0.0401047
$491$	−1622.00			−0.149083			−0.0745416	−	0.997218i	$-0.523749\pi$
$491$	−1622.00			−0.149083			−0.0745416	+	0.997218i	$0.523749\pi$
$492$		0			0
$493$	−25520.0			−2.33137
$494$	156.000	−	104.000i	0.0142080	−	0.00947203i
$495$		0			0
$496$			11890.0i			1.07636i
$497$	2240.00			0.202168
$498$		0			0
$499$			17884.0i			1.60440i	0.597052	+	0.802202i	$0.296338\pi$
$499$			17884.0i			1.60440i	−0.597052	+	0.802202i	$0.703662\pi$
$500$			875.000i			0.0782624i
$501$		0			0
$502$		−	4722.00i		−	0.419827i
$503$	−11282.0			−1.00008			−0.500039	−	0.866003i	$-0.666681\pi$
$503$	−11282.0			−1.00008			−0.500039	+	0.866003i	$0.666681\pi$
$504$		0			0
$505$		−	2440.00i		−	0.215007i
$506$	8520.00			0.748538
$507$		0			0
$508$	−14448.0			−1.26186
$509$		−	8754.00i		−	0.762307i	−0.924512	−	0.381153i	$-0.875527\pi$
$509$		−	8754.00i		−	0.762307i	0.924512	−	0.381153i	$-0.124473\pi$
$510$		0			0
$511$	10592.0			0.916952
$512$			11521.0i			0.994455i
$513$		0			0
$514$			4284.00i			0.367625i
$515$		−	8060.00i		−	0.689643i
$516$		0			0
$517$	−10560.0			−0.898314
$518$			704.000i			0.0597143i
$519$		0			0
$520$	−2925.00	+	1950.00i	−0.246673	+	0.164448i
$521$	−4642.00			−0.390345			−0.195172	−	0.980769i	$-0.562527\pi$
$521$	−4642.00			−0.390345			−0.195172	+	0.980769i	$0.562527\pi$
$522$		0			0
$523$	−6028.00			−0.503989			−0.251994	−	0.967729i	$-0.581086\pi$
$523$	−6028.00			−0.503989			−0.251994	+	0.967729i	$0.581086\pi$
$524$	6846.00			0.570742
$525$		0			0
$526$			198.000i			0.0164129i
$527$		−	33640.0i		−	2.78061i
$528$		0			0
$529$	7997.00			0.657270
$530$	1890.00			0.154899
$531$		0			0
$532$	448.000			0.0365099
$533$	−9620.00	−	14430.0i	−0.781779	−	1.17267i
$534$		0			0
$535$			880.000i			0.0711135i
$536$	−4410.00			−0.355379
$537$		0			0
$538$			5840.00i			0.467993i
$539$			5220.00i			0.417145i
$540$		0			0
$541$			2630.00i			0.209007i	0.994525	+	0.104503i	$0.0333253\pi$
$541$			2630.00i			0.209007i	−0.994525	+	0.104503i	$0.966675\pi$
$542$	−4790.00			−0.379609
$543$		0			0
$544$		−	18676.0i		−	1.47192i
$545$	9670.00			0.760032
$546$		0			0
$547$	6236.00			0.487444			0.243722	−	0.969845i	$-0.421631\pi$
$547$	6236.00			0.487444			0.243722	+	0.969845i	$0.421631\pi$
$548$			5502.00i			0.428894i
$549$		0			0
$550$	1500.00			0.116291
$551$			880.000i			0.0680386i
$552$		0			0
$553$			22400.0i			1.72250i
$554$			946.000i			0.0725482i
$555$		0			0
$556$	1960.00			0.149501
$557$			16026.0i			1.21911i	0.792744	+	0.609554i	$0.208652\pi$
$557$			16026.0i			1.21911i	−0.792744	+	0.609554i	$0.791348\pi$
$558$		0			0
$559$	−8268.00	+	5512.00i	−0.625580	+	0.417053i
$560$	−3280.00			−0.247509
$561$		0			0
$562$	4530.00			0.340012
$563$	7068.00			0.529095			0.264548	−	0.964373i	$-0.414777\pi$
$563$	7068.00			0.529095			0.264548	+	0.964373i	$0.414777\pi$
$564$		0			0
$565$			9260.00i			0.689506i
$566$			2612.00i			0.193976i
$567$		0			0
$568$	−2100.00			−0.155130
$569$	5970.00			0.439851			0.219926	−	0.975517i	$-0.429419\pi$
$569$	5970.00			0.439851			0.219926	+	0.975517i	$0.429419\pi$
$570$		0			0
$571$	−3568.00			−0.261499			−0.130750	−	0.991415i	$-0.541738\pi$
$571$	−3568.00			−0.261499			−0.130750	+	0.991415i	$0.541738\pi$
$572$	−10920.0	−	16380.0i	−0.798231	−	1.19735i
$573$		0			0
$574$			5920.00i			0.430481i
$575$	3550.00			0.257470
$576$		0			0
$577$			8954.00i			0.646031i	0.946394	+	0.323016i	$0.104697\pi$
$577$			8954.00i			0.646031i	−0.946394	+	0.323016i	$0.895303\pi$
$578$			8543.00i			0.614778i
$579$		0			0
$580$		−	7700.00i		−	0.551250i
$581$	−12672.0			−0.904859
$582$		0			0
$583$			22680.0i			1.61117i
$584$	−9930.00			−0.703607
$585$		0			0
$586$	4422.00			0.311725
$587$			16876.0i			1.18662i	0.804973	+	0.593311i	$0.202180\pi$
$587$			16876.0i			1.18662i	−0.804973	+	0.593311i	$0.797820\pi$
$588$		0			0
$589$	−1160.00			−0.0811494
$590$		−	4020.00i		−	0.280510i
$591$		0			0
$592$			1804.00i			0.125243i
$593$			19778.0i			1.36962i	0.728721	+	0.684810i	$0.240115\pi$
$593$			19778.0i			1.36962i	−0.728721	+	0.684810i	$0.759885\pi$
$594$		0			0
$595$	9280.00			0.639400
$596$			4102.00i			0.281920i
$597$		0			0
$598$	3692.00	+	5538.00i	0.252470	+	0.378705i
$599$	16880.0			1.15142			0.575708	−	0.817655i	$-0.304727\pi$
$599$	16880.0			1.15142			0.575708	+	0.817655i	$0.304727\pi$
$600$		0			0
$601$	−4918.00			−0.333793			−0.166896	−	0.985974i	$-0.553375\pi$
$601$	−4918.00			−0.333793			−0.166896	+	0.985974i	$0.553375\pi$
$602$	3392.00			0.229647
$603$		0			0
$604$		−	15890.0i		−	1.07046i
$605$			11345.0i			0.762380i
$606$		0			0
$607$	−27944.0			−1.86855			−0.934277	−	0.356549i	$-0.883953\pi$
$607$	−27944.0			−1.86855			−0.934277	+	0.356549i	$0.883953\pi$
$608$	−644.000			−0.0429567
$609$		0			0
$610$	−2890.00			−0.191824
$611$	−4576.00	−	6864.00i	−0.302987	−	0.454481i
$612$		0			0
$613$		−	2668.00i		−	0.175790i	−0.996130	−	0.0878952i	$-0.971986\pi$
$613$		−	2668.00i		−	0.175790i	0.996130	−	0.0878952i	$-0.0280141\pi$
$614$	8306.00			0.545933
$615$		0			0
$616$			14400.0i			0.941871i
$617$		−	2134.00i		−	0.139241i	−0.997574	−	0.0696204i	$-0.977821\pi$
$617$		−	2134.00i		−	0.139241i	0.997574	−	0.0696204i	$-0.0221788\pi$
$618$		0			0
$619$			10744.0i			0.697638i	0.937190	+	0.348819i	$0.113417\pi$
$619$			10744.0i			0.697638i	−0.937190	+	0.348819i	$0.886583\pi$
$620$	10150.0			0.657474
$621$		0			0
$622$		−	2812.00i		−	0.181272i
$623$	−4704.00			−0.302507
$624$		0			0
$625$	625.000			0.0400000
$626$		−	1978.00i		−	0.126289i
$627$		0			0
$628$	1862.00			0.118315
$629$		−	5104.00i		−	0.323545i
$630$		0			0
$631$			3310.00i			0.208826i	0.994534	+	0.104413i	$0.0332963\pi$
$631$			3310.00i			0.208826i	−0.994534	+	0.104413i	$0.966704\pi$
$632$		−	21000.0i		−	1.32173i
$633$		0			0
$634$	−7666.00			−0.480214
$635$			10320.0i			0.644940i
$636$		0			0
$637$	−3393.00	+	2262.00i	−0.211045	+	0.140697i
$638$	−13200.0			−0.819111
$639$		0			0
$640$	7275.00			0.449328
$641$	30618.0			1.88664			0.943322	−	0.331880i	$-0.107683\pi$
$641$	30618.0			1.88664			0.943322	+	0.331880i	$0.107683\pi$
$642$		0			0
$643$		−	13138.0i		−	0.805773i	−0.915250	−	0.402887i	$-0.868007\pi$
$643$		−	13138.0i		−	0.805773i	0.915250	−	0.402887i	$-0.131993\pi$
$644$			15904.0i			0.973145i
$645$		0			0
$646$	464.000			0.0282598
$647$	9794.00			0.595119			0.297559	−	0.954703i	$-0.403827\pi$
$647$	9794.00			0.595119			0.297559	+	0.954703i	$0.403827\pi$
$648$		0			0
$649$	48240.0			2.91770
$650$	650.000	+	975.000i	0.0392232	+	0.0588348i
$651$		0			0
$652$			14994.0i			0.900629i
$653$	12098.0			0.725010			0.362505	−	0.931982i	$-0.381922\pi$
$653$	12098.0			0.725010			0.362505	+	0.931982i	$0.381922\pi$
$654$		0			0
$655$		−	4890.00i		−	0.291707i
$656$			15170.0i			0.902880i
$657$		0			0
$658$			2816.00i			0.166838i
$659$	8310.00			0.491216			0.245608	−	0.969369i	$-0.421012\pi$
$659$	8310.00			0.491216			0.245608	+	0.969369i	$0.421012\pi$
$660$		0			0
$661$			4930.00i			0.290098i	0.989424	+	0.145049i	$0.0463340\pi$
$661$			4930.00i			0.290098i	−0.989424	+	0.145049i	$0.953666\pi$
$662$	160.000			0.00939362
$663$		0			0
$664$	11880.0			0.694327
$665$		−	320.000i		−	0.0186603i
$666$		0			0
$667$	−31240.0			−1.81352
$668$		−	13328.0i		−	0.771970i
$669$		0			0
$670$			1470.00i			0.0847628i
$671$		−	34680.0i		−	1.99524i
$672$		0			0
$673$	18982.0			1.08722			0.543612	−	0.839336i	$-0.317056\pi$
$673$	18982.0			1.08722			0.543612	+	0.839336i	$0.317056\pi$
$674$			6786.00i			0.387814i
$675$		0			0
$676$	5915.00	−	14196.0i	0.336538	−	0.807692i
$677$	21894.0			1.24292			0.621458	−	0.783447i	$-0.286540\pi$
$677$	21894.0			1.24292			0.621458	+	0.783447i	$0.286540\pi$
$678$		0			0
$679$	−4576.00			−0.258631
$680$	−8700.00			−0.490632
$681$		0			0
$682$		−	17400.0i		−	0.976951i
$683$		−	52.0000i		−	0.00291321i	−0.999999	−	0.00145661i	$-0.999536\pi$
$683$		−	52.0000i		−	0.00291321i	0.999999	−	0.00145661i	$-0.000463652\pi$
$684$		0			0
$685$	3930.00			0.219208
$686$	6880.00			0.382915
$687$		0			0
$688$	8692.00			0.481656
$689$	−14742.0	+	9828.00i	−0.815131	+	0.543421i
$690$		0			0
$691$			8440.00i			0.464649i	0.972638	+	0.232325i	$0.0746332\pi$
$691$			8440.00i			0.464649i	−0.972638	+	0.232325i	$0.925367\pi$
$692$	−19474.0			−1.06978
$693$		0			0
$694$		−	16.0000i		−	0.000875147i
$695$		−	1400.00i		−	0.0764101i
$696$		0			0
$697$		−	42920.0i		−	2.33244i
$698$	2986.00			0.161922
$699$		0			0
$700$			2800.00i			0.151186i
$701$	3108.00			0.167457			0.0837286	−	0.996489i	$-0.473317\pi$
$701$	3108.00			0.167457			0.0837286	+	0.996489i	$0.473317\pi$
$702$		0			0
$703$	−176.000			−0.00944234
$704$			10020.0i			0.536425i
$705$		0			0
$706$	−678.000			−0.0361429
$707$		−	7808.00i		−	0.415347i
$708$		0			0
$709$		−	10246.0i		−	0.542731i	−0.962476	−	0.271366i	$-0.912525\pi$
$709$		−	10246.0i		−	0.542731i	0.962476	−	0.271366i	$-0.0874753\pi$
$710$			700.000i			0.0370007i
$711$		0			0
$712$	4410.00			0.232123
$713$		−	41180.0i		−	2.16298i
$714$		0			0
$715$	−11700.0	+	7800.00i	−0.611965	+	0.407977i
$716$	10710.0			0.559010
$717$		0			0
$718$	−5416.00			−0.281509
$719$	−30840.0			−1.59964			−0.799818	−	0.600243i	$-0.795071\pi$
$719$	−30840.0			−1.59964			−0.799818	+	0.600243i	$0.795071\pi$
$720$		0			0
$721$		−	25792.0i		−	1.33224i
$722$			6843.00i			0.352729i
$723$		0			0
$724$	−10206.0			−0.523899
$725$	−5500.00			−0.281745
$726$		0			0
$727$	3836.00			0.195694			0.0978469	−	0.995201i	$-0.468804\pi$
$727$	3836.00			0.195694			0.0978469	+	0.995201i	$0.468804\pi$
$728$	−9360.00	+	6240.00i	−0.476517	+	0.317678i
$729$		0			0
$730$			3310.00i			0.167820i
$731$	−24592.0			−1.24428
$732$		0			0
$733$		−	10728.0i		−	0.540584i	−0.962778	−	0.270292i	$-0.912880\pi$
$733$		−	10728.0i		−	0.540584i	0.962778	−	0.270292i	$-0.0871202\pi$
$734$		−	7464.00i		−	0.375342i
$735$		0			0
$736$		−	22862.0i		−	1.14498i
$737$	−17640.0			−0.881652
$738$		0			0
$739$			11544.0i			0.574632i	0.957836	+	0.287316i	$0.0927630\pi$
$739$			11544.0i			0.574632i	−0.957836	+	0.287316i	$0.907237\pi$
$740$	1540.00			0.0765021
$741$		0			0
$742$	6048.00			0.299231
$743$		−	7232.00i		−	0.357088i	−0.983932	−	0.178544i	$-0.942861\pi$
$743$		−	7232.00i		−	0.357088i	0.983932	−	0.178544i	$-0.0571387\pi$
$744$		0			0
$745$	2930.00			0.144090
$746$			8542.00i			0.419229i
$747$		0			0
$748$		−	48720.0i		−	2.38152i
$749$			2816.00i			0.137376i
$750$		0			0
$751$	−17648.0			−0.857503			−0.428751	−	0.903422i	$-0.641046\pi$
$751$	−17648.0			−0.857503			−0.428751	+	0.903422i	$0.641046\pi$
$752$			7216.00i			0.349921i
$753$		0			0
$754$	−5720.00	−	8580.00i	−0.276273	−	0.414410i
$755$	−11350.0			−0.547111
$756$		0			0
$757$	5666.00			0.272040			0.136020	−	0.990706i	$-0.456569\pi$
$757$	5666.00			0.272040			0.136020	+	0.990706i	$0.456569\pi$
$758$	−10224.0			−0.489911
$759$		0			0
$760$			300.000i			0.0143186i
$761$			7390.00i			0.352020i	0.984388	+	0.176010i	$0.0563191\pi$
$761$			7390.00i			0.352020i	−0.984388	+	0.176010i	$0.943681\pi$
$762$		0			0
$763$	30944.0			1.46821
$764$	−2044.00			−0.0967923
$765$		0			0
$766$	6032.00			0.284524
$767$	20904.0	+	31356.0i	0.984093	+	1.47614i
$768$		0			0
$769$		−	7576.00i		−	0.355263i	−0.984097	−	0.177632i	$-0.943156\pi$
$769$		−	7576.00i		−	0.355263i	0.984097	−	0.177632i	$-0.0568436\pi$
$770$	4800.00			0.224649
$771$		0			0
$772$		−	2926.00i		−	0.136411i
$773$		−	15042.0i		−	0.699901i	−0.936768	−	0.349950i	$-0.886198\pi$
$773$		−	15042.0i		−	0.699901i	0.936768	−	0.349950i	$-0.113802\pi$
$774$		0			0
$775$		−	7250.00i		−	0.336036i
$776$	4290.00			0.198456
$777$		0			0
$778$		−	3820.00i		−	0.176033i
$779$	−1480.00			−0.0680700
$780$		0			0
$781$	−8400.00			−0.384860
$782$			16472.0i			0.753245i
$783$		0			0
$784$	3567.00			0.162491
$785$		−	1330.00i		−	0.0604710i
$786$		0			0
$787$		−	23686.0i		−	1.07283i	−0.843955	−	0.536413i	$-0.819779\pi$
$787$		−	23686.0i		−	1.07283i	0.843955	−	0.536413i	$-0.180221\pi$
$788$		−	25298.0i		−	1.14366i
$789$		0			0
$790$	−7000.00			−0.315252
$791$			29632.0i			1.33198i
$792$		0			0
$793$	22542.0	−	15028.0i	1.00945	−	0.672963i
$794$	11156.0			0.498629
$795$		0			0
$796$	20160.0			0.897679
$797$	−1666.00			−0.0740436			−0.0370218	−	0.999314i	$-0.511787\pi$
$797$	−1666.00			−0.0740436			−0.0370218	+	0.999314i	$0.511787\pi$
$798$		0			0
$799$		−	20416.0i		−	0.903963i
$800$		−	4025.00i		−	0.177882i
$801$		0			0
$802$	1030.00			0.0453498
$803$	−39720.0			−1.74556
$804$		0			0
$805$	11360.0			0.497376
$806$	11310.0	−	7540.00i	0.494265	−	0.329510i
$807$		0			0
$808$			7320.00i			0.318709i
$809$	23010.0			0.999986			0.499993	−	0.866030i	$-0.333336\pi$
$809$	23010.0			0.999986			0.499993	+	0.866030i	$0.333336\pi$
$810$		0			0
$811$			19220.0i			0.832189i	0.909321	+	0.416095i	$0.136601\pi$
$811$			19220.0i			0.832189i	−0.909321	+	0.416095i	$0.863399\pi$
$812$		−	24640.0i		−	1.06489i
$813$		0			0
$814$		−	2640.00i		−	0.113676i
$815$	10710.0			0.460313
$816$		0			0
$817$			848.000i			0.0363131i
$818$	16296.0			0.696548
$819$		0			0
$820$	12950.0			0.551504
$821$			15210.0i			0.646568i	0.946302	+	0.323284i	$0.104787\pi$
$821$			15210.0i			0.646568i	−0.946302	+	0.323284i	$0.895213\pi$
$822$		0			0
$823$	252.000			0.0106734			0.00533668	−	0.999986i	$-0.498301\pi$
$823$	252.000			0.0106734			0.00533668	+	0.999986i	$0.498301\pi$
$824$			24180.0i			1.02227i
$825$		0			0
$826$		−	12864.0i		−	0.541884i
$827$			27596.0i			1.16035i	0.814493	+	0.580173i	$0.197015\pi$
$827$			27596.0i			1.16035i	−0.814493	+	0.580173i	$0.802985\pi$
$828$		0			0
$829$	−12350.0			−0.517410			−0.258705	−	0.965956i	$-0.583296\pi$
$829$	−12350.0			−0.517410			−0.258705	+	0.965956i	$0.583296\pi$
$830$		−	3960.00i		−	0.165607i
$831$		0			0
$832$	−6513.00	+	4342.00i	−0.271391	+	0.180928i
$833$	−10092.0			−0.419768
$834$		0			0
$835$	−9520.00			−0.394555
$836$	−1680.00			−0.0695024
$837$		0			0
$838$		−	30.0000i		−	0.00123667i
$839$		−	29544.0i		−	1.21570i	−0.794052	−	0.607850i	$-0.792032\pi$
$839$		−	29544.0i		−	1.21570i	0.794052	−	0.607850i	$-0.207968\pi$
$840$		0			0
$841$	24011.0			0.984501
$842$	11990.0			0.490740
$843$		0			0
$844$	−5376.00			−0.219253
$845$	−10140.0	−	4225.00i	−0.412813	−	0.172005i
$846$		0			0
$847$			36304.0i			1.47275i
$848$	15498.0			0.627598
$849$		0			0
$850$			2900.00i			0.117023i
$851$		−	6248.00i		−	0.251679i
$852$		0			0
$853$			33372.0i			1.33955i	0.742564	+	0.669775i	$0.233609\pi$
$853$			33372.0i			1.33955i	−0.742564	+	0.669775i	$0.766391\pi$
$854$	−9248.00			−0.370562
$855$		0			0
$856$		−	2640.00i		−	0.105413i
$857$	−5276.00			−0.210297			−0.105149	−	0.994457i	$-0.533532\pi$
$857$	−5276.00			−0.210297			−0.105149	+	0.994457i	$0.533532\pi$
$858$		0			0
$859$	14840.0			0.589446			0.294723	−	0.955583i	$-0.404773\pi$
$859$	14840.0			0.589446			0.294723	+	0.955583i	$0.404773\pi$
$860$		−	7420.00i		−	0.294209i
$861$		0			0
$862$	1260.00			0.0497863
$863$		−	792.000i		−	0.0312399i	−0.999878	−	0.0156199i	$-0.995028\pi$
$863$		−	792.000i		−	0.0312399i	0.999878	−	0.0156199i	$-0.00497218\pi$
$864$		0			0
$865$			13910.0i			0.546768i
$866$		−	1678.00i		−	0.0658438i
$867$		0			0
$868$	32480.0			1.27010
$869$		−	84000.0i		−	3.27906i
$870$		0			0
$871$	−7644.00	−	11466.0i	−0.297367	−	0.446051i
$872$	−29010.0			−1.12661
$873$		0			0
$874$	568.000			0.0219827
$875$	2000.00			0.0772712
$876$		0			0
$877$			13264.0i			0.510711i	0.966847	+	0.255355i	$0.0821925\pi$
$877$			13264.0i			0.510711i	−0.966847	+	0.255355i	$0.917808\pi$
$878$			7360.00i			0.282902i
$879$		0			0
$880$	12300.0			0.471174
$881$	27018.0			1.03321			0.516606	−	0.856223i	$-0.327195\pi$
$881$	27018.0			1.03321			0.516606	+	0.856223i	$0.327195\pi$
$882$		0			0
$883$	−26348.0			−1.00417			−0.502084	−	0.864819i	$-0.667433\pi$
$883$	−26348.0			−1.00417			−0.502084	+	0.864819i	$0.667433\pi$
$884$	31668.0	−	21112.0i	1.20488	−	0.803250i
$885$		0			0
$886$			8628.00i			0.327160i
$887$	26854.0			1.01654			0.508269	−	0.861198i	$-0.330286\pi$
$887$	26854.0			1.01654			0.508269	+	0.861198i	$0.330286\pi$
$888$		0			0
$889$			33024.0i			1.24588i
$890$		−	1470.00i		−	0.0553646i
$891$		0			0
$892$			17164.0i			0.644275i
$893$	−704.000			−0.0263813
$894$		0			0
$895$		−	7650.00i		−	0.285711i
$896$	23280.0			0.868002
$897$		0			0
$898$	−11586.0			−0.430545
$899$			63800.0i			2.36691i
$900$		0			0
$901$	−43848.0			−1.62130
$902$		−	22200.0i		−	0.819489i
$903$		0			0
$904$		−	27780.0i		−	1.02207i
$905$			7290.00i			0.267766i
$906$		0			0
$907$	44836.0			1.64141			0.820703	−	0.571355i	$-0.193582\pi$
$907$	44836.0			1.64141			0.820703	+	0.571355i	$0.193582\pi$
$908$			23352.0i			0.853484i
$909$		0			0
$910$	2080.00	+	3120.00i	0.0757707	+	0.113656i
$911$	20368.0			0.740749			0.370374	−	0.928883i	$-0.379229\pi$
$911$	20368.0			0.740749			0.370374	+	0.928883i	$0.379229\pi$
$912$		0			0
$913$	47520.0			1.72254
$914$	−12994.0			−0.470244
$915$		0			0
$916$		−	27482.0i		−	0.991300i
$917$		−	15648.0i		−	0.563514i
$918$		0			0
$919$	−36920.0			−1.32522			−0.662611	−	0.748964i	$-0.730552\pi$
$919$	−36920.0			−1.32522			−0.662611	+	0.748964i	$0.730552\pi$
$920$	−10650.0			−0.381652
$921$		0			0
$922$	1330.00			0.0475067
$923$	−3640.00	−	5460.00i	−0.129807	−	0.194711i
$924$		0			0
$925$		−	1100.00i		−	0.0391003i
$926$	14348.0			0.509184
$927$		0			0
$928$			35420.0i			1.25293i
$929$		−	16214.0i		−	0.572620i	−0.958137	−	0.286310i	$-0.907571\pi$
$929$		−	16214.0i		−	0.572620i	0.958137	−	0.286310i	$-0.0924287\pi$
$930$		0			0
$931$			348.000i			0.0122505i
$932$	10696.0			0.375922
$933$		0			0
$934$			2484.00i			0.0870224i
$935$	−34800.0			−1.21720
$936$		0			0
$937$	−17974.0			−0.626665			−0.313332	−	0.949644i	$-0.601445\pi$
$937$	−17974.0			−0.626665			−0.313332	+	0.949644i	$0.601445\pi$
$938$			4704.00i			0.163743i
$939$		0			0
$940$	6160.00			0.213742
$941$		−	13450.0i		−	0.465948i	−0.972483	−	0.232974i	$-0.925154\pi$
$941$		−	13450.0i		−	0.465948i	0.972483	−	0.232974i	$-0.0748458\pi$
$942$		0			0
$943$		−	52540.0i		−	1.81436i
$944$		−	32964.0i		−	1.13653i
$945$		0			0
$946$	−12720.0			−0.437170
$947$		−	10584.0i		−	0.363182i	−0.983374	−	0.181591i	$-0.941875\pi$
$947$		−	10584.0i		−	0.363182i	0.983374	−	0.181591i	$-0.0581248\pi$
$948$		0			0
$949$	−17212.0	−	25818.0i	−0.588751	−	0.883127i
$950$	100.000			0.00341519
$951$		0			0
$952$	−27840.0			−0.947794
$953$	5208.00			0.177024			0.0885119	−	0.996075i	$-0.471789\pi$
$953$	5208.00			0.177024			0.0885119	+	0.996075i	$0.471789\pi$
$954$		0			0
$955$			1460.00i			0.0494707i
$956$			37352.0i			1.26365i
$957$		0			0
$958$	2424.00			0.0817494
$959$	12576.0			0.423462
$960$		0			0
$961$	−54309.0			−1.82300
$962$	1716.00	−	1144.00i	0.0575115	−	0.0383410i
$963$		0			0
$964$			13020.0i			0.435006i
$965$	−2090.00			−0.0697197
$966$		0			0
$967$			42464.0i			1.41215i	0.708136	+	0.706076i	$0.249536\pi$
$967$			42464.0i			1.41215i	−0.708136	+	0.706076i	$0.750464\pi$
$968$		−	34035.0i		−	1.13009i
$969$		0			0
$970$		−	1430.00i		−	0.0473346i
$971$	−2622.00			−0.0866570			−0.0433285	−	0.999061i	$-0.513796\pi$
$971$	−2622.00			−0.0866570			−0.0433285	+	0.999061i	$0.513796\pi$
$972$		0			0
$973$		−	4480.00i		−	0.147608i
$974$	−10584.0			−0.348186
$975$		0			0
$976$	−23698.0			−0.777207
$977$			21406.0i			0.700961i	0.936570	+	0.350480i	$0.113982\pi$
$977$			21406.0i			0.700961i	−0.936570	+	0.350480i	$0.886018\pi$
$978$		0			0
$979$	17640.0			0.575870
$980$		−	3045.00i		−	0.0992540i
$981$		0			0
$982$		−	1622.00i		−	0.0527089i
$983$			368.000i			0.0119404i	0.999982	+	0.00597018i	$0.00190038\pi$
$983$			368.000i			0.0119404i	−0.999982	+	0.00597018i	$0.998100\pi$
$984$		0			0
$985$	−18070.0			−0.584526
$986$		−	25520.0i		−	0.824262i
$987$		0			0
$988$	−728.000	−	1092.00i	−0.0234421	−	0.0351631i
$989$	−30104.0			−0.967898
$990$		0			0
$991$	33352.0			1.06908			0.534541	−	0.845142i	$-0.320484\pi$
$991$	33352.0			1.06908			0.534541	+	0.845142i	$0.320484\pi$
$992$	−46690.0			−1.49436
$993$		0			0
$994$			2240.00i			0.0714773i
$995$		−	14400.0i		−	0.458805i
$996$		0			0
$997$	3666.00			0.116453			0.0582264	−	0.998303i	$-0.481455\pi$
$997$	3666.00			0.116453			0.0582264	+	0.998303i	$0.481455\pi$
$998$	−17884.0			−0.567243
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.4.b.c.181.2		2
3.2	odd	2		195.4.b.c.181.1	✓	2
13.12	even	2	inner	585.4.b.c.181.1		2
39.38	odd	2		195.4.b.c.181.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.4.b.c.181.1	✓	2	3.2	odd	2
195.4.b.c.181.2	yes	2	39.38	odd	2
585.4.b.c.181.1		2	13.12	even	2	inner
585.4.b.c.181.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	585.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +7.00000 q^{4} -5.00000i q^{5} -16.0000i q^{7} +15.0000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +7.00000 q^{4} -5.00000i q^{5} -16.0000i q^{7} +15.0000i q^{8} +5.00000 q^{10} +60.0000i q^{11} +(-39.0000 + 26.0000i) q^{13} +16.0000 q^{14} +41.0000 q^{16} -116.000 q^{17} +4.00000i q^{19} -35.0000i q^{20} -60.0000 q^{22} -142.000 q^{23} -25.0000 q^{25} +(-26.0000 - 39.0000i) q^{26} -112.000i q^{28} +220.000 q^{29} +290.000i q^{31} +161.000i q^{32} -116.000i q^{34} -80.0000 q^{35} +44.0000i q^{37} -4.00000 q^{38} +75.0000 q^{40} +370.000i q^{41} +212.000 q^{43} +420.000i q^{44} -142.000i q^{46} +176.000i q^{47} +87.0000 q^{49} -25.0000i q^{50} +(-273.000 + 182.000i) q^{52} +378.000 q^{53} +300.000 q^{55} +240.000 q^{56} +220.000i q^{58} -804.000i q^{59} -578.000 q^{61} -290.000 q^{62} +167.000 q^{64} +(130.000 + 195.000i) q^{65} +294.000i q^{67} -812.000 q^{68} -80.0000i q^{70} +140.000i q^{71} +662.000i q^{73} -44.0000 q^{74} +28.0000i q^{76} +960.000 q^{77} -1400.00 q^{79} -205.000i q^{80} -370.000 q^{82} -792.000i q^{83} +580.000i q^{85} +212.000i q^{86} -900.000 q^{88} -294.000i q^{89} +(416.000 + 624.000i) q^{91} -994.000 q^{92} -176.000 q^{94} +20.0000 q^{95} -286.000i q^{97} +87.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4}+O(q^{10}) \) 2 * q + 14 * q^4	\( 2 q + 14 q^{4} + 10 q^{10} - 78 q^{13} + 32 q^{14} + 82 q^{16} - 232 q^{17} - 120 q^{22} - 284 q^{23} - 50 q^{25} - 52 q^{26} + 440 q^{29} - 160 q^{35} - 8 q^{38} + 150 q^{40} + 424 q^{43} + 174 q^{49} - 546 q^{52} + 756 q^{53} + 600 q^{55} + 480 q^{56} - 1156 q^{61} - 580 q^{62} + 334 q^{64} + 260 q^{65} - 1624 q^{68} - 88 q^{74} + 1920 q^{77} - 2800 q^{79} - 740 q^{82} - 1800 q^{88} + 832 q^{91} - 1988 q^{92} - 352 q^{94} + 40 q^{95}+O(q^{100}) \) 2 * q + 14 * q^4 + 10 * q^10 - 78 * q^13 + 32 * q^14 + 82 * q^16 - 232 * q^17 - 120 * q^22 - 284 * q^23 - 50 * q^25 - 52 * q^26 + 440 * q^29 - 160 * q^35 - 8 * q^38 + 150 * q^40 + 424 * q^43 + 174 * q^49 - 546 * q^52 + 756 * q^53 + 600 * q^55 + 480 * q^56 - 1156 * q^61 - 580 * q^62 + 334 * q^64 + 260 * q^65 - 1624 * q^68 - 88 * q^74 + 1920 * q^77 - 2800 * q^79 - 740 * q^82 - 1800 * q^88 + 832 * q^91 - 1988 * q^92 - 352 * q^94 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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