LMFDB - Embedded newform 810.4.e.p.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not 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:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.4.e.p.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} -8.00000 q^{8} -10.0000 q^{10} +(-24.0000 - 41.5692i) q^{11} +(-1.00000 + 1.73205i) q^{13} +(-4.00000 + 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} +114.000 q^{17} +140.000 q^{19} +(-10.0000 - 17.3205i) q^{20} +(48.0000 - 83.1384i) q^{22} +(36.0000 - 62.3538i) q^{23} +(-12.5000 - 21.6506i) q^{25} -4.00000 q^{26} -16.0000 q^{28} +(105.000 + 181.865i) q^{29} +(-136.000 + 235.559i) q^{31} +(16.0000 - 27.7128i) q^{32} +(114.000 + 197.454i) q^{34} -20.0000 q^{35} -334.000 q^{37} +(140.000 + 242.487i) q^{38} +(20.0000 - 34.6410i) q^{40} +(-99.0000 + 171.473i) q^{41} +(134.000 + 232.095i) q^{43} +192.000 q^{44} +144.000 q^{46} +(108.000 + 187.061i) q^{47} +(163.500 - 283.190i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-4.00000 - 6.92820i) q^{52} +78.0000 q^{53} +240.000 q^{55} +(-16.0000 - 27.7128i) q^{56} +(-210.000 + 363.731i) q^{58} +(120.000 - 207.846i) q^{59} +(-151.000 - 261.540i) q^{61} -544.000 q^{62} +64.0000 q^{64} +(-5.00000 - 8.66025i) q^{65} +(-298.000 + 516.151i) q^{67} +(-228.000 + 394.908i) q^{68} +(-20.0000 - 34.6410i) q^{70} +768.000 q^{71} -478.000 q^{73} +(-334.000 - 578.505i) q^{74} +(-280.000 + 484.974i) q^{76} +(96.0000 - 166.277i) q^{77} +(320.000 + 554.256i) q^{79} +80.0000 q^{80} -396.000 q^{82} +(-174.000 - 301.377i) q^{83} +(-285.000 + 493.634i) q^{85} +(-268.000 + 464.190i) q^{86} +(192.000 + 332.554i) q^{88} -210.000 q^{89} -8.00000 q^{91} +(144.000 + 249.415i) q^{92} +(-216.000 + 374.123i) q^{94} +(-350.000 + 606.218i) q^{95} +(767.000 + 1328.48i) q^{97} +654.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} - 16 q^{8} - 20 q^{10} - 48 q^{11} - 2 q^{13} - 8 q^{14} - 16 q^{16} + 228 q^{17} + 280 q^{19} - 20 q^{20} + 96 q^{22} + 72 q^{23} - 25 q^{25} - 8 q^{26} - 32 q^{28}+ \cdots + 1308 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 - 16 * q^8 - 20 * q^10 - 48 * q^11 - 2 * q^13 - 8 * q^14 - 16 * q^16 + 228 * q^17 + 280 * q^19 - 20 * q^20 + 96 * q^22 + 72 * q^23 - 25 * q^25 - 8 * q^26 - 32 * q^28 + 210 * q^29 - 272 * q^31 + 32 * q^32 + 228 * q^34 - 40 * q^35 - 668 * q^37 + 280 * q^38 + 40 * q^40 - 198 * q^41 + 268 * q^43 + 384 * q^44 + 288 * q^46 + 216 * q^47 + 327 * q^49 + 50 * q^50 - 8 * q^52 + 156 * q^53 + 480 * q^55 - 32 * q^56 - 420 * q^58 + 240 * q^59 - 302 * q^61 - 1088 * q^62 + 128 * q^64 - 10 * q^65 - 596 * q^67 - 456 * q^68 - 40 * q^70 + 1536 * q^71 - 956 * q^73 - 668 * q^74 - 560 * q^76 + 192 * q^77 + 640 * q^79 + 160 * q^80 - 792 * q^82 - 348 * q^83 - 570 * q^85 - 536 * q^86 + 384 * q^88 - 420 * q^89 - 16 * q^91 + 288 * q^92 - 432 * q^94 - 700 * q^95 + 1534 * q^97 + 1308 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	0.914956	−	0.403554i	$-0.132225\pi$
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	−0.806966	+	0.590598i	$0.798892\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−10.0000			−0.316228
$11$	−24.0000	−	41.5692i	−0.657843	−	1.13942i	−0.981173	−	0.193131i	$-0.938136\pi$
$11$	−24.0000	−	41.5692i	−0.657843	−	1.13942i	0.323330	−	0.946286i	$-0.395198\pi$
$12$		0			0
$13$	−1.00000	+	1.73205i	−0.0213346	+	0.0369527i	−0.876496	−	0.481410i	$-0.840125\pi$
$13$	−1.00000	+	1.73205i	−0.0213346	+	0.0369527i	0.855161	+	0.518363i	$0.173458\pi$
$14$	−4.00000	+	6.92820i	−0.0763604	+	0.132260i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	114.000			1.62642			0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000			1.62642			0.813208	+	0.581974i	$0.197719\pi$
$18$		0			0
$19$	140.000			1.69043			0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000			1.69043			0.845216	+	0.534425i	$0.179472\pi$
$20$	−10.0000	−	17.3205i	−0.111803	−	0.193649i
$21$		0			0
$22$	48.0000	−	83.1384i	0.465165	−	0.805690i
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	−0.655418	−	0.755266i	$-0.727508\pi$
$23$	36.0000	−	62.3538i	0.326370	−	0.565290i	0.981789	+	0.189976i	$0.0608410\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−4.00000			−0.0301717
$27$		0			0
$28$	−16.0000			−0.107990
$29$	105.000	+	181.865i	0.672345	+	1.16454i	0.977237	+	0.212149i	$0.0680463\pi$
$29$	105.000	+	181.865i	0.672345	+	1.16454i	−0.304892	+	0.952387i	$0.598620\pi$
$30$		0			0
$31$	−136.000	+	235.559i	−0.787946	+	1.36476i	0.139278	+	0.990253i	$0.455522\pi$
$31$	−136.000	+	235.559i	−0.787946	+	1.36476i	−0.927223	+	0.374509i	$0.877811\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	114.000	+	197.454i	0.575025	+	0.995972i
$35$	−20.0000			−0.0965891
$36$		0			0
$37$	−334.000			−1.48403			−0.742017	−	0.670381i	$-0.766131\pi$
$37$	−334.000			−1.48403			−0.742017	+	0.670381i	$0.766131\pi$
$38$	140.000	+	242.487i	0.597658	+	1.03517i
$39$		0			0
$40$	20.0000	−	34.6410i	0.0790569	−	0.136931i
$41$	−99.0000	+	171.473i	−0.377102	+	0.653161i	−0.990639	−	0.136505i	$-0.956413\pi$
$41$	−99.0000	+	171.473i	−0.377102	+	0.653161i	0.613537	+	0.789666i	$0.289746\pi$
$42$		0			0
$43$	134.000	+	232.095i	0.475228	+	0.823119i	0.999597	−	0.0283717i	$-0.00903221\pi$
$43$	134.000	+	232.095i	0.475228	+	0.823119i	−0.524369	+	0.851491i	$0.675699\pi$
$44$	192.000			0.657843
$45$		0			0
$46$	144.000			0.461557
$47$	108.000	+	187.061i	0.335179	+	0.580547i	0.983519	−	0.180804i	$-0.0578698\pi$
$47$	108.000	+	187.061i	0.335179	+	0.580547i	−0.648340	+	0.761351i	$0.724536\pi$
$48$		0			0
$49$	163.500	−	283.190i	0.476676	−	0.825628i
$50$	25.0000	−	43.3013i	0.0707107	−	0.122474i
$51$		0			0
$52$	−4.00000	−	6.92820i	−0.0106673	−	0.0184763i
$53$	78.0000			0.202153			0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000			0.202153			0.101077	+	0.994879i	$0.467771\pi$
$54$		0			0
$55$	240.000			0.588393
$56$	−16.0000	−	27.7128i	−0.0381802	−	0.0661300i
$57$		0			0
$58$	−210.000	+	363.731i	−0.475420	+	0.823451i
$59$	120.000	−	207.846i	0.264791	−	0.458631i	−0.702718	−	0.711469i	$-0.748030\pi$
$59$	120.000	−	207.846i	0.264791	−	0.458631i	0.967509	+	0.252837i	$0.0813637\pi$
$60$		0			0
$61$	−151.000	−	261.540i	−0.316944	−	0.548963i	0.662905	−	0.748704i	$-0.269323\pi$
$61$	−151.000	−	261.540i	−0.316944	−	0.548963i	−0.979849	+	0.199741i	$0.935990\pi$
$62$	−544.000			−1.11432
$63$		0			0
$64$	64.0000			0.125000
$65$	−5.00000	−	8.66025i	−0.00954113	−	0.0165257i
$66$		0			0
$67$	−298.000	+	516.151i	−0.543381	+	0.941163i	0.455326	+	0.890325i	$0.349523\pi$
$67$	−298.000	+	516.151i	−0.543381	+	0.941163i	−0.998707	+	0.0508381i	$0.983811\pi$
$68$	−228.000	+	394.908i	−0.406604	+	0.704259i
$69$		0			0
$70$	−20.0000	−	34.6410i	−0.0341494	−	0.0591485i
$71$	768.000			1.28373			0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000			1.28373			0.641865	+	0.766818i	$0.278161\pi$
$72$		0			0
$73$	−478.000			−0.766379			−0.383190	−	0.923670i	$-0.625174\pi$
$73$	−478.000			−0.766379			−0.383190	+	0.923670i	$0.625174\pi$
$74$	−334.000	−	578.505i	−0.524685	−	0.908782i
$75$		0			0
$76$	−280.000	+	484.974i	−0.422608	+	0.731978i
$77$	96.0000	−	166.277i	0.142081	−	0.246091i
$78$		0			0
$79$	320.000	+	554.256i	0.455732	+	0.789351i	0.998730	−	0.0503832i	$-0.0160443\pi$
$79$	320.000	+	554.256i	0.455732	+	0.789351i	−0.542998	+	0.839734i	$0.682711\pi$
$80$	80.0000			0.111803
$81$		0			0
$82$	−396.000			−0.533303
$83$	−174.000	−	301.377i	−0.230108	−	0.398559i	0.727732	−	0.685862i	$-0.240575\pi$
$83$	−174.000	−	301.377i	−0.230108	−	0.398559i	−0.957840	+	0.287303i	$0.907241\pi$
$84$		0			0
$85$	−285.000	+	493.634i	−0.363678	+	0.629908i
$86$	−268.000	+	464.190i	−0.336037	+	0.582033i
$87$		0			0
$88$	192.000	+	332.554i	0.232583	+	0.402845i
$89$	−210.000			−0.250112			−0.125056	−	0.992150i	$-0.539911\pi$
$89$	−210.000			−0.250112			−0.125056	+	0.992150i	$0.539911\pi$
$90$		0			0
$91$	−8.00000			−0.00921569
$92$	144.000	+	249.415i	0.163185	+	0.282645i
$93$		0			0
$94$	−216.000	+	374.123i	−0.237007	+	0.410509i
$95$	−350.000	+	606.218i	−0.377992	+	0.654701i
$96$		0			0
$97$	767.000	+	1328.48i	0.802856	+	1.39059i	0.917729	+	0.397207i	$0.130021\pi$
$97$	767.000	+	1328.48i	0.802856	+	1.39059i	−0.114873	+	0.993380i	$0.536646\pi$
$98$	654.000			0.674122
$99$		0			0
$100$	100.000			0.100000
$101$	861.000	+	1491.30i	0.848245	+	1.46920i	0.882773	+	0.469799i	$0.155674\pi$
$101$	861.000	+	1491.30i	0.848245	+	1.46920i	−0.0345288	+	0.999404i	$0.510993\pi$
$102$		0			0
$103$	−526.000	+	911.059i	−0.503188	+	0.871546i	0.496806	+	0.867862i	$0.334506\pi$
$103$	−526.000	+	911.059i	−0.503188	+	0.871546i	−0.999993	+	0.00368461i	$0.998827\pi$
$104$	8.00000	−	13.8564i	0.00754293	−	0.0130647i
$105$		0			0
$106$	78.0000	+	135.100i	0.0714720	+	0.123793i
$107$	564.000			0.509570			0.254785	−	0.966998i	$-0.417995\pi$
$107$	564.000			0.509570			0.254785	+	0.966998i	$0.417995\pi$
$108$		0			0
$109$	−610.000			−0.536031			−0.268016	−	0.963415i	$-0.586368\pi$
$109$	−610.000			−0.536031			−0.268016	+	0.963415i	$0.586368\pi$
$110$	240.000	+	415.692i	0.208028	+	0.360315i
$111$		0			0
$112$	32.0000	−	55.4256i	0.0269975	−	0.0467610i
$113$	651.000	−	1127.57i	0.541955	−	0.938694i	−0.456837	−	0.889551i	$-0.651018\pi$
$113$	651.000	−	1127.57i	0.541955	−	0.938694i	0.998792	−	0.0491432i	$-0.0156491\pi$
$114$		0			0
$115$	180.000	+	311.769i	0.145957	+	0.252805i
$116$	−840.000			−0.672345
$117$		0			0
$118$	480.000			0.374471
$119$	228.000	+	394.908i	0.175636	+	0.304211i
$120$		0			0
$121$	−486.500	+	842.643i	−0.365515	+	0.633090i
$122$	302.000	−	523.079i	0.224113	−	0.388175i
$123$		0			0
$124$	−544.000	−	942.236i	−0.393973	−	0.682381i
$125$	125.000			0.0894427
$126$		0			0
$127$	−124.000			−0.0866395			−0.0433198	−	0.999061i	$-0.513793\pi$
$127$	−124.000			−0.0866395			−0.0433198	+	0.999061i	$0.513793\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	10.0000	−	17.3205i	0.00674660	−	0.0116855i
$131$	96.0000	−	166.277i	0.0640272	−	0.110898i	−0.832235	−	0.554423i	$-0.812939\pi$
$131$	96.0000	−	166.277i	0.0640272	−	0.110898i	0.896262	+	0.443525i	$0.146272\pi$
$132$		0			0
$133$	280.000	+	484.974i	0.182549	+	0.316185i
$134$	−1192.00			−0.768456
$135$		0			0
$136$	−912.000			−0.575025
$137$	−1257.00	−	2177.19i	−0.783889	−	1.35774i	−0.929661	−	0.368417i	$-0.879900\pi$
$137$	−1257.00	−	2177.19i	−0.783889	−	1.35774i	0.145772	−	0.989318i	$-0.453433\pi$
$138$		0			0
$139$	−670.000	+	1160.47i	−0.408839	+	0.708130i	−0.994760	−	0.102238i	$-0.967400\pi$
$139$	−670.000	+	1160.47i	−0.408839	+	0.708130i	0.585921	+	0.810368i	$0.300733\pi$
$140$	40.0000	−	69.2820i	0.0241473	−	0.0418243i
$141$		0			0
$142$	768.000	+	1330.22i	0.453867	+	0.786121i
$143$	96.0000			0.0561393
$144$		0			0
$145$	−1050.00			−0.601364
$146$	−478.000	−	827.920i	−0.270956	−	0.469309i
$147$		0			0
$148$	668.000	−	1157.01i	0.371009	−	0.642606i
$149$	705.000	−	1221.10i	0.387623	−	0.671383i	−0.604506	−	0.796600i	$-0.706630\pi$
$149$	705.000	−	1221.10i	0.387623	−	0.671383i	0.992129	+	0.125217i	$0.0399629\pi$
$150$		0			0
$151$	1064.00	+	1842.90i	0.573424	+	0.993200i	0.996211	+	0.0869709i	$0.0277187\pi$
$151$	1064.00	+	1842.90i	0.573424	+	0.993200i	−0.422786	+	0.906229i	$0.638948\pi$
$152$	−1120.00			−0.597658
$153$		0			0
$154$	384.000			0.200932
$155$	−680.000	−	1177.79i	−0.352380	−	0.610340i
$156$		0			0
$157$	−1513.00	+	2620.59i	−0.769112	+	1.33214i	0.168933	+	0.985627i	$0.445968\pi$
$157$	−1513.00	+	2620.59i	−0.769112	+	1.33214i	−0.938045	+	0.346513i	$0.887366\pi$
$158$	−640.000	+	1108.51i	−0.322251	+	0.558155i
$159$		0			0
$160$	80.0000	+	138.564i	0.0395285	+	0.0684653i
$161$	288.000			0.140979
$162$		0			0
$163$	2612.00			1.25514			0.627569	−	0.778561i	$-0.284050\pi$
$163$	2612.00			1.25514			0.627569	+	0.778561i	$0.284050\pi$
$164$	−396.000	−	685.892i	−0.188551	−	0.326580i
$165$		0			0
$166$	348.000	−	602.754i	0.162711	−	0.281824i
$167$	−12.0000	+	20.7846i	−0.00556041	+	0.00963091i	−0.868792	−	0.495177i	$-0.835103\pi$
$167$	−12.0000	+	20.7846i	−0.00556041	+	0.00963091i	0.863232	+	0.504808i	$0.168437\pi$
$168$		0			0
$169$	1096.50	+	1899.19i	0.499090	+	0.864449i
$170$	−1140.00			−0.514318
$171$		0			0
$172$	−1072.00			−0.475228
$173$	981.000	+	1699.14i	0.431122	+	0.746725i	0.996970	−	0.0777846i	$-0.0247846\pi$
$173$	981.000	+	1699.14i	0.431122	+	0.746725i	−0.565849	+	0.824509i	$0.691451\pi$
$174$		0			0
$175$	50.0000	−	86.6025i	0.0215980	−	0.0374088i
$176$	−384.000	+	665.108i	−0.164461	+	0.284854i
$177$		0			0
$178$	−210.000	−	363.731i	−0.0884279	−	0.153162i
$179$	120.000			0.0501074			0.0250537	−	0.999686i	$-0.492024\pi$
$179$	120.000			0.0501074			0.0250537	+	0.999686i	$0.492024\pi$
$180$		0			0
$181$	902.000			0.370415			0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000			0.370415			0.185208	+	0.982699i	$0.440704\pi$
$182$	−8.00000	−	13.8564i	−0.00325824	−	0.00564344i
$183$		0			0
$184$	−288.000	+	498.831i	−0.115389	+	0.199860i
$185$	835.000	−	1446.26i	0.331840	−	0.574764i
$186$		0			0
$187$	−2736.00	−	4738.89i	−1.06993	−	1.85317i
$188$	−864.000			−0.335179
$189$		0			0
$190$	−1400.00			−0.534561
$191$	−84.0000	−	145.492i	−0.0318221	−	0.0551175i	0.849676	−	0.527306i	$-0.176798\pi$
$191$	−84.0000	−	145.492i	−0.0318221	−	0.0551175i	−0.881498	+	0.472188i	$0.843464\pi$
$192$		0			0
$193$	659.000	−	1141.42i	0.245782	−	0.425706i	−0.716569	−	0.697516i	$-0.754289\pi$
$193$	659.000	−	1141.42i	0.245782	−	0.425706i	0.962351	+	0.271809i	$0.0876220\pi$
$194$	−1534.00	+	2656.97i	−0.567705	+	0.983294i
$195$		0			0
$196$	654.000	+	1132.76i	0.238338	+	0.412814i
$197$	4014.00			1.45170			0.725852	−	0.687851i	$-0.241446\pi$
$197$	4014.00			1.45170			0.725852	+	0.687851i	$0.241446\pi$
$198$		0			0
$199$	2000.00			0.712443			0.356222	−	0.934401i	$-0.384065\pi$
$199$	2000.00			0.712443			0.356222	+	0.934401i	$0.384065\pi$
$200$	100.000	+	173.205i	0.0353553	+	0.0612372i
$201$		0			0
$202$	−1722.00	+	2982.59i	−0.599799	+	1.03888i
$203$	−420.000	+	727.461i	−0.145213	+	0.251516i
$204$		0			0
$205$	−495.000	−	857.365i	−0.168645	−	0.292102i
$206$	−2104.00			−0.711615
$207$		0			0
$208$	32.0000			0.0106673
$209$	−3360.00	−	5819.69i	−1.11204	−	1.92611i
$210$		0			0
$211$	1934.00	−	3349.79i	0.631005	−	1.09293i	−0.356342	−	0.934356i	$-0.615976\pi$
$211$	1934.00	−	3349.79i	0.631005	−	1.09293i	0.987347	−	0.158577i	$-0.0506906\pi$
$212$	−156.000	+	270.200i	−0.0505383	+	0.0875349i
$213$		0			0
$214$	564.000	+	976.877i	0.180160	+	0.312046i
$215$	−1340.00			−0.425057
$216$		0			0
$217$	−1088.00			−0.340361
$218$	−610.000	−	1056.55i	−0.189516	−	0.328251i
$219$		0			0
$220$	−480.000	+	831.384i	−0.147098	+	0.254781i
$221$	−114.000	+	197.454i	−0.0346990	+	0.0601004i
$222$		0			0
$223$	1574.00	+	2726.25i	0.472658	+	0.818668i	0.999510	−	0.0312886i	$-0.00996110\pi$
$223$	1574.00	+	2726.25i	0.472658	+	0.818668i	−0.526852	+	0.849957i	$0.676628\pi$
$224$	128.000			0.0381802
$225$		0			0
$226$	2604.00			0.766440
$227$	1278.00	+	2213.56i	0.373673	+	0.647221i	0.990128	−	0.140170i	$-0.0447648\pi$
$227$	1278.00	+	2213.56i	0.373673	+	0.647221i	−0.616454	+	0.787391i	$0.711431\pi$
$228$		0			0
$229$	305.000	−	528.275i	0.0880130	−	0.152443i	−0.818658	−	0.574281i	$-0.805282\pi$
$229$	305.000	−	528.275i	0.0880130	−	0.152443i	0.906671	+	0.421838i	$0.138615\pi$
$230$	−360.000	+	623.538i	−0.103207	+	0.178760i
$231$		0			0
$232$	−840.000	−	1454.92i	−0.237710	−	0.411726i
$233$	2058.00			0.578644			0.289322	−	0.957232i	$-0.406570\pi$
$233$	2058.00			0.578644			0.289322	+	0.957232i	$0.406570\pi$
$234$		0			0
$235$	−1080.00			−0.299793
$236$	480.000	+	831.384i	0.132396	+	0.229316i
$237$		0			0
$238$	−456.000	+	789.815i	−0.124194	+	0.215110i
$239$	2460.00	−	4260.84i	0.665792	−	1.15318i	−0.313279	−	0.949661i	$-0.601427\pi$
$239$	2460.00	−	4260.84i	0.665792	−	1.15318i	0.979070	−	0.203523i	$-0.0652393\pi$
$240$		0			0
$241$	719.000	+	1245.34i	0.192178	+	0.332862i	0.945972	−	0.324249i	$-0.105112\pi$
$241$	719.000	+	1245.34i	0.192178	+	0.332862i	−0.753794	+	0.657111i	$0.771778\pi$
$242$	−1946.00			−0.516916
$243$		0			0
$244$	1208.00			0.316944
$245$	817.500	+	1415.95i	0.213176	+	0.369232i
$246$		0			0
$247$	−140.000	+	242.487i	−0.0360647	+	0.0624659i
$248$	1088.00	−	1884.47i	0.278581	−	0.482516i
$249$		0			0
$250$	125.000	+	216.506i	0.0316228	+	0.0547723i
$251$	−792.000			−0.199166			−0.0995829	−	0.995029i	$-0.531751\pi$
$251$	−792.000			−0.199166			−0.0995829	+	0.995029i	$0.531751\pi$
$252$		0			0
$253$	−3456.00			−0.858802
$254$	−124.000	−	214.774i	−0.0306317	−	0.0530557i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1083.00	−	1875.81i	0.262863	−	0.455291i	−0.704139	−	0.710062i	$-0.748667\pi$
$257$	1083.00	−	1875.81i	0.262863	−	0.455291i	0.967001	+	0.254771i	$0.0820001\pi$
$258$		0			0
$259$	−668.000	−	1157.01i	−0.160261	−	0.277580i
$260$	40.0000			0.00954113
$261$		0			0
$262$	384.000			0.0905481
$263$	1596.00	+	2764.35i	0.374196	+	0.648127i	0.990206	−	0.139611i	$-0.0445854\pi$
$263$	1596.00	+	2764.35i	0.374196	+	0.648127i	−0.616010	+	0.787738i	$0.711252\pi$
$264$		0			0
$265$	−195.000	+	337.750i	−0.0452028	+	0.0782936i
$266$	−560.000	+	969.948i	−0.129082	+	0.223577i
$267$		0			0
$268$	−1192.00	−	2064.60i	−0.271690	−	0.470581i
$269$	−5490.00			−1.24435			−0.622177	−	0.782877i	$-0.713752\pi$
$269$	−5490.00			−1.24435			−0.622177	+	0.782877i	$0.713752\pi$
$270$		0			0
$271$	−6328.00			−1.41845			−0.709223	−	0.704985i	$-0.750954\pi$
$271$	−6328.00			−1.41845			−0.709223	+	0.704985i	$0.750954\pi$
$272$	−912.000	−	1579.63i	−0.203302	−	0.352129i
$273$		0			0
$274$	2514.00	−	4354.38i	0.554293	−	0.960064i
$275$	−600.000	+	1039.23i	−0.131569	+	0.227883i
$276$		0			0
$277$	287.000	+	497.099i	0.0622533	+	0.107826i	0.895472	−	0.445117i	$-0.146838\pi$
$277$	287.000	+	497.099i	0.0622533	+	0.107826i	−0.833219	+	0.552943i	$0.813505\pi$
$278$	−2680.00			−0.578186
$279$		0			0
$280$	160.000			0.0341494
$281$	2121.00	+	3673.68i	0.450278	+	0.779905i	0.998403	−	0.0564915i	$-0.0179914\pi$
$281$	2121.00	+	3673.68i	0.450278	+	0.779905i	−0.548125	+	0.836397i	$0.684658\pi$
$282$		0			0
$283$	314.000	−	543.864i	0.0659553	−	0.114238i	−0.831162	−	0.556030i	$-0.812324\pi$
$283$	314.000	−	543.864i	0.0659553	−	0.114238i	0.897117	+	0.441792i	$0.145657\pi$
$284$	−1536.00	+	2660.43i	−0.320933	+	0.555871i
$285$		0			0
$286$	96.0000	+	166.277i	0.0198482	+	0.0343782i
$287$	−792.000			−0.162893
$288$		0			0
$289$	8083.00			1.64523
$290$	−1050.00	−	1818.65i	−0.212614	−	0.368259i
$291$		0			0
$292$	956.000	−	1655.84i	0.191595	−	0.331852i
$293$	−279.000	+	483.242i	−0.0556292	+	0.0963526i	−0.892499	−	0.451049i	$-0.851050\pi$
$293$	−279.000	+	483.242i	−0.0556292	+	0.0963526i	0.836870	+	0.547402i	$0.184383\pi$
$294$		0			0
$295$	600.000	+	1039.23i	0.118418	+	0.205106i
$296$	2672.00			0.524685
$297$		0			0
$298$	2820.00			0.548182
$299$	72.0000	+	124.708i	0.0139260	+	0.0241205i
$300$		0			0
$301$	−536.000	+	928.379i	−0.102640	+	0.177777i
$302$	−2128.00	+	3685.80i	−0.405472	+	0.702299i
$303$		0			0
$304$	−1120.00	−	1939.90i	−0.211304	−	0.365989i
$305$	1510.00			0.283483
$306$		0			0
$307$	−6964.00			−1.29465			−0.647323	−	0.762216i	$-0.724112\pi$
$307$	−6964.00			−1.29465			−0.647323	+	0.762216i	$0.724112\pi$
$308$	384.000	+	665.108i	0.0710404	+	0.123046i
$309$		0			0
$310$	1360.00	−	2355.59i	0.249170	−	0.431576i
$311$	1416.00	−	2452.58i	0.258180	−	0.447181i	−0.707574	−	0.706639i	$-0.750211\pi$
$311$	1416.00	−	2452.58i	0.258180	−	0.447181i	0.965754	+	0.259458i	$0.0835439\pi$
$312$		0			0
$313$	−4321.00	−	7484.19i	−0.780311	−	1.35154i	−0.931761	−	0.363073i	$-0.881727\pi$
$313$	−4321.00	−	7484.19i	−0.780311	−	1.35154i	0.151449	−	0.988465i	$-0.451606\pi$
$314$	−6052.00			−1.08769
$315$		0			0
$316$	−2560.00			−0.455732
$317$	−1107.00	−	1917.38i	−0.196137	−	0.339719i	0.751136	−	0.660148i	$-0.229506\pi$
$317$	−1107.00	−	1917.38i	−0.196137	−	0.339719i	−0.947273	+	0.320429i	$0.896173\pi$
$318$		0			0
$319$	5040.00	−	8729.54i	0.884595	−	1.53216i
$320$	−160.000	+	277.128i	−0.0279508	+	0.0484123i
$321$		0			0
$322$	288.000	+	498.831i	0.0498435	+	0.0863315i
$323$	15960.0			2.74934
$324$		0			0
$325$	50.0000			0.00853385
$326$	2612.00	+	4524.12i	0.443759	+	0.768612i
$327$		0			0
$328$	792.000	−	1371.78i	0.133326	−	0.230927i
$329$	−432.000	+	748.246i	−0.0723919	+	0.125386i
$330$		0			0
$331$	−5386.00	−	9328.83i	−0.894385	−	1.54912i	−0.834564	−	0.550911i	$-0.814281\pi$
$331$	−5386.00	−	9328.83i	−0.894385	−	1.54912i	−0.0598204	−	0.998209i	$-0.519053\pi$
$332$	1392.00			0.230108
$333$		0			0
$334$	−48.0000			−0.00786360
$335$	−1490.00	−	2580.76i	−0.243007	−	0.420901i
$336$		0			0
$337$	827.000	−	1432.41i	0.133678	−	0.231537i	−0.791414	−	0.611281i	$-0.790654\pi$
$337$	827.000	−	1432.41i	0.133678	−	0.231537i	0.925092	+	0.379744i	$0.123988\pi$
$338$	−2193.00	+	3798.39i	−0.352910	+	0.611258i
$339$		0			0
$340$	−1140.00	−	1974.54i	−0.181839	−	0.314954i
$341$	13056.0			2.07338
$342$		0			0
$343$	2680.00			0.421885
$344$	−1072.00	−	1856.76i	−0.168019	−	0.291017i
$345$		0			0
$346$	−1962.00	+	3398.28i	−0.304849	+	0.528014i
$347$	1098.00	−	1901.79i	0.169867	−	0.294218i	−0.768506	−	0.639842i	$-0.779000\pi$
$347$	1098.00	−	1901.79i	0.169867	−	0.294218i	0.938373	+	0.345625i	$0.112333\pi$
$348$		0			0
$349$	−4135.00	−	7162.03i	−0.634216	−	1.09849i	−0.986681	−	0.162670i	$-0.947990\pi$
$349$	−4135.00	−	7162.03i	−0.634216	−	1.09849i	0.352464	−	0.935825i	$-0.385344\pi$
$350$	200.000			0.0305441
$351$		0			0
$352$	−1536.00			−0.232583
$353$	5151.00	+	8921.79i	0.776657	+	1.34521i	0.933858	+	0.357643i	$0.116420\pi$
$353$	5151.00	+	8921.79i	0.776657	+	1.34521i	−0.157201	+	0.987567i	$0.550247\pi$
$354$		0			0
$355$	−1920.00	+	3325.54i	−0.287051	+	0.497186i
$356$	420.000	−	727.461i	0.0625280	−	0.108302i
$357$		0			0
$358$	120.000	+	207.846i	0.0177156	+	0.0306844i
$359$	2280.00			0.335192			0.167596	−	0.985856i	$-0.446400\pi$
$359$	2280.00			0.335192			0.167596	+	0.985856i	$0.446400\pi$
$360$		0			0
$361$	12741.0			1.85756
$362$	902.000	+	1562.31i	0.130962	+	0.226832i
$363$		0			0
$364$	16.0000	−	27.7128i	0.00230392	−	0.00399051i
$365$	1195.00	−	2069.80i	0.171368	−	0.296817i
$366$		0			0
$367$	4382.00	+	7589.85i	0.623266	+	1.07953i	0.988873	+	0.148759i	$0.0475278\pi$
$367$	4382.00	+	7589.85i	0.623266	+	1.07953i	−0.365608	+	0.930769i	$0.619139\pi$
$368$	−1152.00			−0.163185
$369$		0			0
$370$	3340.00			0.469293
$371$	156.000	+	270.200i	0.0218305	+	0.0378115i
$372$		0			0
$373$	659.000	−	1141.42i	0.0914792	−	0.158447i	−0.816655	−	0.577127i	$-0.804174\pi$
$373$	659.000	−	1141.42i	0.0914792	−	0.158447i	0.908134	+	0.418680i	$0.137507\pi$
$374$	5472.00	−	9477.78i	0.756552	−	1.31039i
$375$		0			0
$376$	−864.000	−	1496.49i	−0.118504	−	0.205254i
$377$	−420.000			−0.0573769
$378$		0			0
$379$	1100.00			0.149085			0.0745425	−	0.997218i	$-0.476250\pi$
$379$	1100.00			0.149085			0.0745425	+	0.997218i	$0.476250\pi$
$380$	−1400.00	−	2424.87i	−0.188996	−	0.327351i
$381$		0			0
$382$	168.000	−	290.985i	0.0225016	−	0.0389740i
$383$	−1764.00	+	3055.34i	−0.235343	+	0.407625i	−0.959372	−	0.282144i	$-0.908955\pi$
$383$	−1764.00	+	3055.34i	−0.235343	+	0.407625i	0.724030	+	0.689769i	$0.242288\pi$
$384$		0			0
$385$	480.000	+	831.384i	0.0635404	+	0.110055i
$386$	2636.00			0.347588
$387$		0			0
$388$	−6136.00			−0.802856
$389$	−4815.00	−	8339.82i	−0.627584	−	1.08701i	−0.988035	−	0.154229i	$-0.950711\pi$
$389$	−4815.00	−	8339.82i	−0.627584	−	1.08701i	0.360451	−	0.932778i	$-0.382623\pi$
$390$		0			0
$391$	4104.00	−	7108.34i	0.530814	−	0.919396i
$392$	−1308.00	+	2265.52i	−0.168531	+	0.291903i
$393$		0			0
$394$	4014.00	+	6952.45i	0.513255	+	0.888983i
$395$	−3200.00			−0.407619
$396$		0			0
$397$	−3094.00			−0.391142			−0.195571	−	0.980690i	$-0.562656\pi$
$397$	−3094.00			−0.391142			−0.195571	+	0.980690i	$0.562656\pi$
$398$	2000.00	+	3464.10i	0.251887	+	0.436281i
$399$		0			0
$400$	−200.000	+	346.410i	−0.0250000	+	0.0433013i
$401$	−819.000	+	1418.55i	−0.101992	+	0.176656i	−0.912505	−	0.409065i	$-0.865855\pi$
$401$	−819.000	+	1418.55i	−0.101992	+	0.176656i	0.810513	+	0.585721i	$0.199188\pi$
$402$		0			0
$403$	−272.000	−	471.118i	−0.0336211	−	0.0582334i
$404$	−6888.00			−0.848245
$405$		0			0
$406$	−1680.00			−0.205362
$407$	8016.00	+	13884.1i	0.976261	+	1.69093i
$408$		0			0
$409$	6875.00	−	11907.8i	0.831166	−	1.43962i	−0.0659483	−	0.997823i	$-0.521007\pi$
$409$	6875.00	−	11907.8i	0.831166	−	1.43962i	0.897114	−	0.441799i	$-0.145659\pi$
$410$	990.000	−	1714.73i	0.119250	−	0.206548i
$411$		0			0
$412$	−2104.00	−	3644.23i	−0.251594	−	0.435773i
$413$	960.000			0.114379
$414$		0			0
$415$	1740.00			0.205815
$416$	32.0000	+	55.4256i	0.00377146	+	0.00653237i
$417$		0			0
$418$	6720.00	−	11639.4i	0.786330	−	1.36196i
$419$	−6240.00	+	10808.0i	−0.727551	+	1.26016i	0.230364	+	0.973105i	$0.426008\pi$
$419$	−6240.00	+	10808.0i	−0.727551	+	1.26016i	−0.957915	+	0.287051i	$0.907325\pi$
$420$		0			0
$421$	−3631.00	−	6289.08i	−0.420342	−	0.728054i	0.575631	−	0.817710i	$-0.304757\pi$
$421$	−3631.00	−	6289.08i	−0.420342	−	0.728054i	−0.995973	+	0.0896557i	$0.971423\pi$
$422$	7736.00			0.892376
$423$		0			0
$424$	−624.000			−0.0714720
$425$	−1425.00	−	2468.17i	−0.162642	−	0.281703i
$426$		0			0
$427$	604.000	−	1046.16i	0.0684534	−	0.118565i
$428$	−1128.00	+	1953.75i	−0.127392	+	0.220650i
$429$		0			0
$430$	−1340.00	−	2320.95i	−0.150280	−	0.260293i
$431$	−9792.00			−1.09435			−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00			−1.09435			−0.547174	+	0.837019i	$0.684296\pi$
$432$		0			0
$433$	1802.00			0.199997			0.0999984	−	0.994988i	$-0.468116\pi$
$433$	1802.00			0.199997			0.0999984	+	0.994988i	$0.468116\pi$
$434$	−1088.00	−	1884.47i	−0.120336	−	0.208427i
$435$		0			0
$436$	1220.00	−	2113.10i	0.134008	−	0.232108i
$437$	5040.00	−	8729.54i	0.551707	−	0.955584i
$438$		0			0
$439$	1160.00	+	2009.18i	0.126113	+	0.218435i	0.922168	−	0.386791i	$-0.126416\pi$
$439$	1160.00	+	2009.18i	0.126113	+	0.218435i	−0.796054	+	0.605225i	$0.793083\pi$
$440$	−1920.00			−0.208028
$441$		0			0
$442$	−456.000			−0.0490717
$443$	5586.00	+	9675.24i	0.599095	+	1.03766i	0.992955	+	0.118492i	$0.0378060\pi$
$443$	5586.00	+	9675.24i	0.599095	+	1.03766i	−0.393861	+	0.919170i	$0.628861\pi$
$444$		0			0
$445$	525.000	−	909.327i	0.0559267	−	0.0968679i
$446$	−3148.00	+	5452.50i	−0.334220	+	0.578886i
$447$		0			0
$448$	128.000	+	221.703i	0.0134987	+	0.0233805i
$449$	−6810.00			−0.715777			−0.357888	−	0.933764i	$-0.616503\pi$
$449$	−6810.00			−0.715777			−0.357888	+	0.933764i	$0.616503\pi$
$450$		0			0
$451$	9504.00			0.992297
$452$	2604.00	+	4510.26i	0.270978	+	0.469347i
$453$		0			0
$454$	−2556.00	+	4427.12i	−0.264227	+	0.457654i
$455$	20.0000	−	34.6410i	0.00206069	−	0.00356922i
$456$		0			0
$457$	−8533.00	−	14779.6i	−0.873429	−	1.51282i	−0.858427	−	0.512935i	$-0.828558\pi$
$457$	−8533.00	−	14779.6i	−0.873429	−	1.51282i	−0.0150014	−	0.999887i	$-0.504775\pi$
$458$	1220.00			0.124469
$459$		0			0
$460$	−1440.00			−0.145957
$461$	−9459.00	−	16383.5i	−0.955639	−	1.65522i	−0.732899	−	0.680337i	$-0.761833\pi$
$461$	−9459.00	−	16383.5i	−0.955639	−	1.65522i	−0.222740	−	0.974878i	$-0.571500\pi$
$462$		0			0
$463$	−526.000	+	911.059i	−0.0527976	+	0.0914481i	−0.891216	−	0.453579i	$-0.850147\pi$
$463$	−526.000	+	911.059i	−0.0527976	+	0.0914481i	0.838419	+	0.545027i	$0.183480\pi$
$464$	1680.00	−	2909.85i	0.168086	−	0.291134i
$465$		0			0
$466$	2058.00	+	3564.56i	0.204582	+	0.354346i
$467$	−11076.0			−1.09751			−0.548754	−	0.835984i	$-0.684898\pi$
$467$	−11076.0			−1.09751			−0.548754	+	0.835984i	$0.684898\pi$
$468$		0			0
$469$	−2384.00			−0.234718
$470$	−1080.00	−	1870.61i	−0.105993	−	0.183585i
$471$		0			0
$472$	−960.000	+	1662.77i	−0.0936178	+	0.162151i
$473$	6432.00	−	11140.6i	0.625251	−	1.08297i
$474$		0			0
$475$	−1750.00	−	3031.09i	−0.169043	−	0.292791i
$476$	−1824.00			−0.175636
$477$		0			0
$478$	9840.00			0.941571
$479$	−4500.00	−	7794.23i	−0.429249	−	0.743481i	0.567558	−	0.823334i	$-0.307888\pi$
$479$	−4500.00	−	7794.23i	−0.429249	−	0.743481i	−0.996807	+	0.0798526i	$0.974555\pi$
$480$		0			0
$481$	334.000	−	578.505i	0.0316613	−	0.0548390i
$482$	−1438.00	+	2490.69i	−0.135890	+	0.235369i
$483$		0			0
$484$	−1946.00	−	3370.57i	−0.182757	−	0.316545i
$485$	−7670.00			−0.718096
$486$		0			0
$487$	−8764.00			−0.815472			−0.407736	−	0.913100i	$-0.633682\pi$
$487$	−8764.00			−0.815472			−0.407736	+	0.913100i	$0.633682\pi$
$488$	1208.00	+	2092.32i	0.112057	+	0.194088i
$489$		0			0
$490$	−1635.00	+	2831.90i	−0.150738	+	0.261086i
$491$	2796.00	−	4842.81i	0.256989	−	0.445118i	−0.708445	−	0.705766i	$-0.750603\pi$
$491$	2796.00	−	4842.81i	0.256989	−	0.445118i	0.965434	+	0.260648i	$0.0839362\pi$
$492$		0			0
$493$	11970.0	+	20732.6i	1.09351	+	1.89402i
$494$	−560.000			−0.0510032
$495$		0			0
$496$	4352.00			0.393973
$497$	1536.00	+	2660.43i	0.138630	+	0.240114i
$498$		0			0
$499$	−2350.00	+	4070.32i	−0.210823	+	0.365155i	−0.951972	−	0.306184i	$-0.900948\pi$
$499$	−2350.00	+	4070.32i	−0.210823	+	0.365155i	0.741150	+	0.671340i	$0.234281\pi$
$500$	−250.000	+	433.013i	−0.0223607	+	0.0387298i
$501$		0			0
$502$	−792.000	−	1371.78i	−0.0704157	−	0.121964i
$503$	11808.0			1.04671			0.523353	−	0.852116i	$-0.324681\pi$
$503$	11808.0			1.04671			0.523353	+	0.852116i	$0.324681\pi$
$504$		0			0
$505$	−8610.00			−0.758693
$506$	−3456.00	−	5985.97i	−0.303632	−	0.525907i
$507$		0			0
$508$	248.000	−	429.549i	0.0216599	−	0.0375160i
$509$	585.000	−	1013.25i	0.0509424	−	0.0882348i	−0.839430	−	0.543468i	$-0.817111\pi$
$509$	585.000	−	1013.25i	0.0509424	−	0.0882348i	0.890372	+	0.455233i	$0.150444\pi$
$510$		0			0
$511$	−956.000	−	1655.84i	−0.0827612	−	0.143347i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	4332.00			0.371744
$515$	−2630.00	−	4555.29i	−0.225032	−	0.389767i
$516$		0			0
$517$	5184.00	−	8978.95i	0.440990	−	0.763818i
$518$	1336.00	−	2314.02i	0.113321	−	0.196278i
$519$		0			0
$520$	40.0000	+	69.2820i	0.00337330	+	0.00584273i
$521$	16638.0			1.39909			0.699543	−	0.714590i	$-0.253387\pi$
$521$	16638.0			1.39909			0.699543	+	0.714590i	$0.253387\pi$
$522$		0			0
$523$	15692.0			1.31198			0.655988	−	0.754771i	$-0.272252\pi$
$523$	15692.0			1.31198			0.655988	+	0.754771i	$0.272252\pi$
$524$	384.000	+	665.108i	0.0320136	+	0.0554492i
$525$		0			0
$526$	−3192.00	+	5528.71i	−0.264597	+	0.458295i
$527$	−15504.0	+	26853.7i	−1.28153	+	2.21967i
$528$		0			0
$529$	3491.50	+	6047.46i	0.286965	+	0.497038i
$530$	−780.000			−0.0639265
$531$		0			0
$532$	−2240.00			−0.182549
$533$	−198.000	−	342.946i	−0.0160907	−	0.0278699i
$534$		0			0
$535$	−1410.00	+	2442.19i	−0.113943	+	0.197355i
$536$	2384.00	−	4129.21i	0.192114	−	0.332751i
$537$		0			0
$538$	−5490.00	−	9508.96i	−0.439946	−	0.762008i
$539$	−15696.0			−1.25431
$540$		0			0
$541$	−22018.0			−1.74977			−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0			−1.74977			−0.874887	+	0.484327i	$0.839064\pi$
$542$	−6328.00	−	10960.4i	−0.501496	−	0.868617i
$543$		0			0
$544$	1824.00	−	3159.26i	0.143756	−	0.248993i
$545$	1525.00	−	2641.38i	0.119860	−	0.207604i
$546$		0			0
$547$	2282.00	+	3952.54i	0.178375	+	0.308955i	0.941324	−	0.337504i	$-0.109583\pi$
$547$	2282.00	+	3952.54i	0.178375	+	0.308955i	−0.762949	+	0.646459i	$0.776249\pi$
$548$	10056.0			0.783889
$549$		0			0
$550$	−2400.00			−0.186066
$551$	14700.0	+	25461.1i	1.13655	+	1.96857i
$552$		0			0
$553$	−1280.00	+	2217.03i	−0.0984288	+	0.170484i
$554$	−574.000	+	994.197i	−0.0440197	+	0.0762444i
$555$		0			0
$556$	−2680.00	−	4641.90i	−0.204420	−	0.354065i
$557$	7734.00			0.588331			0.294165	−	0.955755i	$-0.404958\pi$
$557$	7734.00			0.588331			0.294165	+	0.955755i	$0.404958\pi$
$558$		0			0
$559$	−536.000			−0.0405552
$560$	160.000	+	277.128i	0.0120736	+	0.0209121i
$561$		0			0
$562$	−4242.00	+	7347.36i	−0.318395	+	0.551476i
$563$	−10074.0	+	17448.7i	−0.754118	+	1.30617i	0.191693	+	0.981455i	$0.438602\pi$
$563$	−10074.0	+	17448.7i	−0.754118	+	1.30617i	−0.945812	+	0.324716i	$0.894731\pi$
$564$		0			0
$565$	3255.00	+	5637.83i	0.242370	+	0.419797i
$566$	1256.00			0.0932749
$567$		0			0
$568$	−6144.00			−0.453867
$569$	−12015.0	−	20810.6i	−0.885228	−	1.53326i	−0.845452	−	0.534052i	$-0.820669\pi$
$569$	−12015.0	−	20810.6i	−0.885228	−	1.53326i	−0.0397769	−	0.999209i	$-0.512665\pi$
$570$		0			0
$571$	−1186.00	+	2054.21i	−0.0869222	+	0.150554i	−0.906209	−	0.422831i	$-0.861036\pi$
$571$	−1186.00	+	2054.21i	−0.0869222	+	0.150554i	0.819287	+	0.573384i	$0.194370\pi$
$572$	−192.000	+	332.554i	−0.0140348	+	0.0243090i
$573$		0			0
$574$	−792.000	−	1371.78i	−0.0575914	−	0.0997512i
$575$	−1800.00			−0.130548
$576$		0			0
$577$	8546.00			0.616594			0.308297	−	0.951290i	$-0.400241\pi$
$577$	8546.00			0.616594			0.308297	+	0.951290i	$0.400241\pi$
$578$	8083.00	+	14000.2i	0.581676	+	1.00749i
$579$		0			0
$580$	2100.00	−	3637.31i	0.150341	−	0.260398i
$581$	696.000	−	1205.51i	0.0496987	−	0.0860807i
$582$		0			0
$583$	−1872.00	−	3242.40i	−0.132985	−	0.230337i
$584$	3824.00			0.270956
$585$		0			0
$586$	−1116.00			−0.0786716
$587$	−7722.00	−	13374.9i	−0.542966	−	0.940445i	−0.998732	−	0.0503450i	$-0.983968\pi$
$587$	−7722.00	−	13374.9i	−0.542966	−	0.940445i	0.455766	−	0.890100i	$-0.349365\pi$
$588$		0			0
$589$	−19040.0	+	32978.2i	−1.33197	+	2.30704i
$590$	−1200.00	+	2078.46i	−0.0837343	+	0.145032i
$591$		0			0
$592$	2672.00	+	4628.04i	0.185504	+	0.321303i
$593$	−18342.0			−1.27018			−0.635089	−	0.772439i	$-0.719037\pi$
$593$	−18342.0			−1.27018			−0.635089	+	0.772439i	$0.719037\pi$
$594$		0			0
$595$	−2280.00			−0.157094
$596$	2820.00	+	4884.38i	0.193812	+	0.335692i
$597$		0			0
$598$	−144.000	+	249.415i	−0.00984715	+	0.0170558i
$599$	12300.0	−	21304.2i	0.839006	−	1.45320i	−0.0517213	−	0.998662i	$-0.516471\pi$
$599$	12300.0	−	21304.2i	0.839006	−	1.45320i	0.890727	−	0.454539i	$-0.150196\pi$
$600$		0			0
$601$	4499.00	+	7792.50i	0.305354	+	0.528889i	0.977340	−	0.211675i	$-0.0678917\pi$
$601$	4499.00	+	7792.50i	0.305354	+	0.528889i	−0.671986	+	0.740564i	$0.734558\pi$
$602$	−2144.00			−0.145154
$603$		0			0
$604$	−8512.00			−0.573424
$605$	−2432.50	−	4213.21i	−0.163463	−	0.283126i
$606$		0			0
$607$	−2038.00	+	3529.92i	−0.136277	+	0.236038i	−0.926084	−	0.377316i	$-0.876847\pi$
$607$	−2038.00	+	3529.92i	−0.136277	+	0.236038i	0.789808	+	0.613354i	$0.210180\pi$
$608$	2240.00	−	3879.79i	0.149414	−	0.258793i
$609$		0			0
$610$	1510.00	+	2615.40i	0.100226	+	0.173597i
$611$	−432.000			−0.0286037
$612$		0			0
$613$	−4078.00			−0.268693			−0.134347	−	0.990934i	$-0.542894\pi$
$613$	−4078.00			−0.268693			−0.134347	+	0.990934i	$0.542894\pi$
$614$	−6964.00	−	12062.0i	−0.457727	−	0.792806i
$615$		0			0
$616$	−768.000	+	1330.22i	−0.0502331	+	0.0870063i
$617$	5043.00	−	8734.73i	0.329049	−	0.569930i	−0.653274	−	0.757122i	$-0.726605\pi$
$617$	5043.00	−	8734.73i	0.329049	−	0.569930i	0.982324	+	0.187191i	$0.0599384\pi$
$618$		0			0
$619$	−4390.00	−	7603.70i	−0.285055	−	0.493730i	0.687568	−	0.726120i	$-0.258679\pi$
$619$	−4390.00	−	7603.70i	−0.285055	−	0.493730i	−0.972622	+	0.232391i	$0.925345\pi$
$620$	5440.00			0.352380
$621$		0			0
$622$	5664.00			0.365122
$623$	−420.000	−	727.461i	−0.0270095	−	0.0467819i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	8642.00	−	14968.4i	0.551763	−	0.955682i
$627$		0			0
$628$	−6052.00	−	10482.4i	−0.384556	−	0.666070i
$629$	−38076.0			−2.41366
$630$		0			0
$631$	2792.00			0.176145			0.0880727	−	0.996114i	$-0.471929\pi$
$631$	2792.00			0.176145			0.0880727	+	0.996114i	$0.471929\pi$
$632$	−2560.00	−	4434.05i	−0.161126	−	0.279078i
$633$		0			0
$634$	2214.00	−	3834.76i	0.138690	−	0.240217i
$635$	310.000	−	536.936i	0.0193732	−	0.0335553i
$636$		0			0
$637$	327.000	+	566.381i	0.0203394	+	0.0352289i
$638$	20160.0			1.25101
$639$		0			0
$640$	−640.000			−0.0395285
$641$	3801.00	+	6583.53i	0.234213	+	0.405669i	0.959044	−	0.283258i	$-0.0914154\pi$
$641$	3801.00	+	6583.53i	0.234213	+	0.405669i	−0.724831	+	0.688927i	$0.758082\pi$
$642$		0			0
$643$	−12106.0	+	20968.2i	−0.742479	+	1.28601i	0.208884	+	0.977940i	$0.433017\pi$
$643$	−12106.0	+	20968.2i	−0.742479	+	1.28601i	−0.951363	+	0.308071i	$0.900317\pi$
$644$	−576.000	+	997.661i	−0.0352447	+	0.0610456i
$645$		0			0
$646$	15960.0	+	27643.5i	0.972040	+	1.68362i
$647$	−9456.00			−0.574581			−0.287290	−	0.957844i	$-0.592754\pi$
$647$	−9456.00			−0.574581			−0.287290	+	0.957844i	$0.592754\pi$
$648$		0			0
$649$	−11520.0			−0.696764
$650$	50.0000	+	86.6025i	0.00301717	+	0.00522589i
$651$		0			0
$652$	−5224.00	+	9048.23i	−0.313785	+	0.543491i
$653$	−4779.00	+	8277.47i	−0.286396	+	0.496053i	−0.972947	−	0.231029i	$-0.925791\pi$
$653$	−4779.00	+	8277.47i	−0.286396	+	0.496053i	0.686551	+	0.727082i	$0.259124\pi$
$654$		0			0
$655$	480.000	+	831.384i	0.0286338	+	0.0495952i
$656$	3168.00			0.188551
$657$		0			0
$658$	−1728.00			−0.102378
$659$	−14640.0	−	25357.2i	−0.865392	−	1.49890i	−0.866657	−	0.498904i	$-0.833736\pi$
$659$	−14640.0	−	25357.2i	−0.865392	−	1.49890i	0.00126511	−	0.999999i	$-0.499597\pi$
$660$		0			0
$661$	14549.0	−	25199.6i	0.856113	−	1.48283i	−0.0194961	−	0.999810i	$-0.506206\pi$
$661$	14549.0	−	25199.6i	0.856113	−	1.48283i	0.875609	−	0.483021i	$-0.160460\pi$
$662$	10772.0	−	18657.7i	0.632425	−	1.09539i
$663$		0			0
$664$	1392.00	+	2411.01i	0.0813555	+	0.140912i
$665$	−2800.00			−0.163277
$666$		0			0
$667$	15120.0			0.877734
$668$	−48.0000	−	83.1384i	−0.00278020	−	0.00481545i
$669$		0			0
$670$	2980.00	−	5161.51i	0.171832	−	0.297622i
$671$	−7248.00	+	12553.9i	−0.416998	+	0.722262i
$672$		0			0
$673$	5819.00	+	10078.8i	0.333293	+	0.577280i	0.983155	−	0.182772i	$-0.0585069\pi$
$673$	5819.00	+	10078.8i	0.333293	+	0.577280i	−0.649863	+	0.760052i	$0.725174\pi$
$674$	3308.00			0.189050
$675$		0			0
$676$	−8772.00			−0.499090
$677$	1713.00	+	2967.00i	0.0972466	+	0.168436i	0.910544	−	0.413412i	$-0.135663\pi$
$677$	1713.00	+	2967.00i	0.0972466	+	0.168436i	−0.813297	+	0.581848i	$0.802330\pi$
$678$		0			0
$679$	−3068.00	+	5313.93i	−0.173401	+	0.300339i
$680$	2280.00	−	3949.08i	0.128579	−	0.222706i
$681$		0			0
$682$	13056.0	+	22613.7i	0.733050	+	1.26968i
$683$	20148.0			1.12876			0.564379	−	0.825516i	$-0.309116\pi$
$683$	20148.0			1.12876			0.564379	+	0.825516i	$0.309116\pi$
$684$		0			0
$685$	12570.0			0.701131
$686$	2680.00	+	4641.90i	0.149159	+	0.258350i
$687$		0			0
$688$	2144.00	−	3713.52i	0.118807	−	0.205780i
$689$	−78.0000	+	135.100i	−0.00431286	+	0.00747010i
$690$		0			0
$691$	14714.0	+	25485.4i	0.810053	+	1.40305i	0.912826	+	0.408349i	$0.133895\pi$
$691$	14714.0	+	25485.4i	0.810053	+	1.40305i	−0.102772	+	0.994705i	$0.532771\pi$
$692$	−7848.00			−0.431122
$693$		0			0
$694$	4392.00			0.240228
$695$	−3350.00	−	5802.37i	−0.182838	−	0.316686i
$696$		0			0
$697$	−11286.0	+	19547.9i	−0.613325	+	1.06231i
$698$	8270.00	−	14324.1i	0.448459	−	0.776753i
$699$		0			0
$700$	200.000	+	346.410i	0.0107990	+	0.0187044i
$701$	−16242.0			−0.875110			−0.437555	−	0.899192i	$-0.644155\pi$
$701$	−16242.0			−0.875110			−0.437555	+	0.899192i	$0.644155\pi$
$702$		0			0
$703$	−46760.0			−2.50866
$704$	−1536.00	−	2660.43i	−0.0822304	−	0.142427i
$705$		0			0
$706$	−10302.0	+	17843.6i	−0.549180	+	0.951207i
$707$	−3444.00	+	5965.18i	−0.183204	+	0.317318i
$708$		0			0
$709$	−1015.00	−	1758.03i	−0.0537646	−	0.0931231i	0.837890	−	0.545838i	$-0.183789\pi$
$709$	−1015.00	−	1758.03i	−0.0537646	−	0.0931231i	−0.891655	+	0.452715i	$0.850455\pi$
$710$	−7680.00			−0.405951
$711$		0			0
$712$	1680.00			0.0884279
$713$	9792.00	+	16960.2i	0.514324	+	0.890836i
$714$		0			0
$715$	−240.000	+	415.692i	−0.0125531	+	0.0217427i
$716$	−240.000	+	415.692i	−0.0125268	+	0.0216971i
$717$		0			0
$718$	2280.00	+	3949.08i	0.118508	+	0.205262i
$719$	−6960.00			−0.361007			−0.180504	−	0.983574i	$-0.557773\pi$
$719$	−6960.00			−0.361007			−0.180504	+	0.983574i	$0.557773\pi$
$720$		0			0
$721$	−4208.00			−0.217357
$722$	12741.0	+	22068.1i	0.656746	+	1.13752i
$723$		0			0
$724$	−1804.00	+	3124.62i	−0.0926038	+	0.160394i
$725$	2625.00	−	4546.63i	0.134469	−	0.232907i
$726$		0			0
$727$	−9298.00	−	16104.6i	−0.474338	−	0.821578i	0.525230	−	0.850960i	$-0.323979\pi$
$727$	−9298.00	−	16104.6i	−0.474338	−	0.821578i	−0.999568	+	0.0293826i	$0.990646\pi$
$728$	64.0000			0.00325824
$729$		0			0
$730$	4780.00			0.242350
$731$	15276.0	+	26458.8i	0.772918	+	1.33873i
$732$		0			0
$733$	−10621.0	+	18396.1i	−0.535192	+	0.926979i	0.463962	+	0.885855i	$0.346427\pi$
$733$	−10621.0	+	18396.1i	−0.535192	+	0.926979i	−0.999154	+	0.0411244i	$0.986906\pi$
$734$	−8764.00	+	15179.7i	−0.440715	+	0.763342i
$735$		0			0
$736$	−1152.00	−	1995.32i	−0.0576947	−	0.0999301i
$737$	28608.0			1.42984
$738$		0			0
$739$	−340.000			−0.0169244			−0.00846218	−	0.999964i	$-0.502694\pi$
$739$	−340.000			−0.0169244			−0.00846218	+	0.999964i	$0.502694\pi$
$740$	3340.00	+	5785.05i	0.165920	+	0.287382i
$741$		0			0
$742$	−312.000	+	540.400i	−0.0154365	+	0.0267368i
$743$	−10944.0	+	18955.6i	−0.540372	+	0.935952i	0.458510	+	0.888689i	$0.348383\pi$
$743$	−10944.0	+	18955.6i	−0.540372	+	0.935952i	−0.998882	+	0.0472628i	$0.984950\pi$
$744$		0			0
$745$	3525.00	+	6105.48i	0.173350	+	0.300252i
$746$	2636.00			0.129371
$747$		0			0
$748$	21888.0			1.06993
$749$	1128.00	+	1953.75i	0.0550283	+	0.0953119i
$750$		0			0
$751$	−8896.00	+	15408.3i	−0.432250	+	0.748679i	−0.997067	−	0.0765376i	$-0.975613\pi$
$751$	−8896.00	+	15408.3i	−0.432250	+	0.748679i	0.564817	+	0.825216i	$0.308947\pi$
$752$	1728.00	−	2992.98i	0.0837948	−	0.145137i
$753$		0			0
$754$	−420.000	−	727.461i	−0.0202858	−	0.0351360i
$755$	−10640.0			−0.512886
$756$		0			0
$757$	37346.0			1.79308			0.896541	−	0.442960i	$-0.146072\pi$
$757$	37346.0			1.79308			0.896541	+	0.442960i	$0.146072\pi$
$758$	1100.00	+	1905.26i	0.0527095	+	0.0912955i
$759$		0			0
$760$	2800.00	−	4849.74i	0.133640	−	0.231472i
$761$	−5679.00	+	9836.32i	−0.270517	+	0.468550i	−0.968994	−	0.247083i	$-0.920528\pi$
$761$	−5679.00	+	9836.32i	−0.270517	+	0.468550i	0.698477	+	0.715632i	$0.253861\pi$
$762$		0			0
$763$	−1220.00	−	2113.10i	−0.0578859	−	0.100261i
$764$	672.000			0.0318221
$765$		0			0
$766$	−7056.00			−0.332825
$767$	240.000	+	415.692i	0.0112984	+	0.0195695i
$768$		0			0
$769$	17135.0	−	29678.7i	0.803516	−	1.39173i	−0.113772	−	0.993507i	$-0.536293\pi$
$769$	17135.0	−	29678.7i	0.803516	−	1.39173i	0.917288	−	0.398224i	$-0.130373\pi$
$770$	−960.000	+	1662.77i	−0.0449299	+	0.0778208i
$771$		0			0
$772$	2636.00	+	4565.69i	0.122891	+	0.212853i
$773$	13278.0			0.617822			0.308911	−	0.951091i	$-0.400035\pi$
$773$	13278.0			0.617822			0.308911	+	0.951091i	$0.400035\pi$
$774$		0			0
$775$	6800.00			0.315178
$776$	−6136.00	−	10627.9i	−0.283853	−	0.491647i
$777$		0			0
$778$	9630.00	−	16679.6i	0.443769	−	0.768630i
$779$	−13860.0	+	24006.2i	−0.637466	+	1.10412i
$780$		0			0
$781$	−18432.0	−	31925.2i	−0.844493	−	1.46270i
$782$

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} -8.00000 q^{8} -10.0000 q^{10} +(-24.0000 - 41.5692i) q^{11} +(-1.00000 + 1.73205i) q^{13} +(-4.00000 + 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} +114.000 q^{17} +140.000 q^{19} +(-10.0000 - 17.3205i) q^{20} +(48.0000 - 83.1384i) q^{22} +(36.0000 - 62.3538i) q^{23} +(-12.5000 - 21.6506i) q^{25} -4.00000 q^{26} -16.0000 q^{28} +(105.000 + 181.865i) q^{29} +(-136.000 + 235.559i) q^{31} +(16.0000 - 27.7128i) q^{32} +(114.000 + 197.454i) q^{34} -20.0000 q^{35} -334.000 q^{37} +(140.000 + 242.487i) q^{38} +(20.0000 - 34.6410i) q^{40} +(-99.0000 + 171.473i) q^{41} +(134.000 + 232.095i) q^{43} +192.000 q^{44} +144.000 q^{46} +(108.000 + 187.061i) q^{47} +(163.500 - 283.190i) q^{49} +(25.0000 - 43.3013i) q^{50} +(-4.00000 - 6.92820i) q^{52} +78.0000 q^{53} +240.000 q^{55} +(-16.0000 - 27.7128i) q^{56} +(-210.000 + 363.731i) q^{58} +(120.000 - 207.846i) q^{59} +(-151.000 - 261.540i) q^{61} -544.000 q^{62} +64.0000 q^{64} +(-5.00000 - 8.66025i) q^{65} +(-298.000 + 516.151i) q^{67} +(-228.000 + 394.908i) q^{68} +(-20.0000 - 34.6410i) q^{70} +768.000 q^{71} -478.000 q^{73} +(-334.000 - 578.505i) q^{74} +(-280.000 + 484.974i) q^{76} +(96.0000 - 166.277i) q^{77} +(320.000 + 554.256i) q^{79} +80.0000 q^{80} -396.000 q^{82} +(-174.000 - 301.377i) q^{83} +(-285.000 + 493.634i) q^{85} +(-268.000 + 464.190i) q^{86} +(192.000 + 332.554i) q^{88} -210.000 q^{89} -8.00000 q^{91} +(144.000 + 249.415i) q^{92} +(-216.000 + 374.123i) q^{94} +(-350.000 + 606.218i) q^{95} +(767.000 + 1328.48i) q^{97} +654.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} - 16 q^{8} - 20 q^{10} - 48 q^{11} - 2 q^{13} - 8 q^{14} - 16 q^{16} + 228 q^{17} + 280 q^{19} - 20 q^{20} + 96 q^{22} + 72 q^{23} - 25 q^{25} - 8 q^{26} - 32 q^{28}+ \cdots + 1308 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 - 16 * q^8 - 20 * q^10 - 48 * q^11 - 2 * q^13 - 8 * q^14 - 16 * q^16 + 228 * q^17 + 280 * q^19 - 20 * q^20 + 96 * q^22 + 72 * q^23 - 25 * q^25 - 8 * q^26 - 32 * q^28 + 210 * q^29 - 272 * q^31 + 32 * q^32 + 228 * q^34 - 40 * q^35 - 668 * q^37 + 280 * q^38 + 40 * q^40 - 198 * q^41 + 268 * q^43 + 384 * q^44 + 288 * q^46 + 216 * q^47 + 327 * q^49 + 50 * q^50 - 8 * q^52 + 156 * q^53 + 480 * q^55 - 32 * q^56 - 420 * q^58 + 240 * q^59 - 302 * q^61 - 1088 * q^62 + 128 * q^64 - 10 * q^65 - 596 * q^67 - 456 * q^68 - 40 * q^70 + 1536 * q^71 - 956 * q^73 - 668 * q^74 - 560 * q^76 + 192 * q^77 + 640 * q^79 + 160 * q^80 - 792 * q^82 - 348 * q^83 - 570 * q^85 - 536 * q^86 + 384 * q^88 - 420 * q^89 - 16 * q^91 + 288 * q^92 - 432 * q^94 - 700 * q^95 + 1534 * q^97 + 1308 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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