LMFDB - Embedded newform 500.3.b.a.251.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,3,Mod(251,500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("500.251");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 500 = 2^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	500.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:	$13.6240132180$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.6
Root			$0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	500.251
Dual form			500.3.b.a.251.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -3.01719i q^{3} -4.00000 q^{4} +6.03437 q^{6} +4.64990i q^{7} -8.00000i q^{8} -0.103412 q^{9} +O(q^{10})$	$q+2.00000i q^{2} -3.01719i q^{3} -4.00000 q^{4} +6.03437 q^{6} +4.64990i q^{7} -8.00000i q^{8} -0.103412 q^{9} +12.0687i q^{12} -9.29980 q^{14} +16.0000 q^{16} -0.206823i q^{18} +14.0296 q^{21} -27.7108i q^{23} -24.1375 q^{24} -26.8427i q^{27} -18.5996i q^{28} +57.8374 q^{29} +32.0000i q^{32} +0.413647 q^{36} +70.1981 q^{41} +28.0592i q^{42} +14.7940i q^{43} +55.4216 q^{46} -88.0823i q^{47} -48.2750i q^{48} +27.3784 q^{49} +53.6853 q^{54} +37.1992 q^{56} +115.675i q^{58} +110.011 q^{61} -0.480854i q^{63} -64.0000 q^{64} -116.000i q^{67} -83.6086 q^{69} +0.827294i q^{72} -81.9200 q^{81} +140.396i q^{82} +148.231i q^{83} -56.1184 q^{84} -29.5880 q^{86} -174.506i q^{87} -51.7927 q^{89} +110.843i q^{92} +176.165 q^{94} +96.5500 q^{96} +54.7569i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9}+O(q^{10}) $ 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9	$ 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9} + 16 q^{14} + 128 q^{16} + 32 q^{21} - 64 q^{24} - 44 q^{29} + 304 q^{36} - 124 q^{41} + 176 q^{46} - 556 q^{49} + 32 q^{54} - 64 q^{56} + 116 q^{61} - 512 q^{64} + 352 q^{69} + 1000 q^{81} - 128 q^{84} - 304 q^{86} - 284 q^{89} + 16 q^{94} + 256 q^{96}+O(q^{100}) $ 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9 + 16 * q^14 + 128 * q^16 + 32 * q^21 - 64 * q^24 - 44 * q^29 + 304 * q^36 - 124 * q^41 + 176 * q^46 - 556 * q^49 + 32 * q^54 - 64 * q^56 + 116 * q^61 - 512 * q^64 + 352 * q^69 + 1000 * q^81 - 128 * q^84 - 304 * q^86 - 284 * q^89 + 16 * q^94 + 256 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$.

$n$	$251$	$377$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000i	1.00000i
$2$	2.00000i	1.00000i
$3$	− 3.01719i	− 1.00573i	−0.864365	−	0.502864i	$-0.832280\pi$
$3$	− 3.01719i	− 1.00573i	0.864365	−	0.502864i	$-0.167720\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	6.03437	1.00573
$7$	4.64990i	0.664271i	0.943232	+	0.332136i	$0.107769\pi$
$7$	4.64990i	0.664271i	−0.943232	+	0.332136i	$0.892231\pi$
$8$	− 8.00000i	− 1.00000i
$9$	−0.103412	−0.0114902
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	12.0687i	1.00573i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−9.29980	−0.664271
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 0.206823i	− 0.0114902i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	14.0296	0.668077
$22$	0	0
$23$	− 27.7108i	− 1.20482i	−0.798188	−	0.602408i	$-0.794208\pi$
$23$	− 27.7108i	− 1.20482i	0.798188	−	0.602408i	$-0.205792\pi$
$24$	−24.1375	−1.00573
$25$	0	0
$26$	0	0
$27$	− 26.8427i	− 0.994173i
$28$	− 18.5996i	− 0.664271i
$29$	57.8374	1.99439	0.997197	−	0.0748216i	$-0.0238387\pi$
$29$	57.8374	1.99439	0.997197	+	0.0748216i	$0.0238387\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.0000i	1.00000i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0.413647	0.0114902
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	70.1981	1.71215	0.856074	−	0.516853i	$-0.172896\pi$
$41$	70.1981	1.71215	0.856074	+	0.516853i	$0.172896\pi$
$42$	28.0592i	0.668077i
$43$	14.7940i	0.344046i	0.985093	+	0.172023i	$0.0550304\pi$
$43$	14.7940i	0.344046i	−0.985093	+	0.172023i	$0.944970\pi$
$44$	0	0
$45$	0	0
$46$	55.4216	1.20482
$47$	− 88.0823i	− 1.87409i	−0.349208	−	0.937045i	$-0.613549\pi$
$47$	− 88.0823i	− 1.87409i	0.349208	−	0.937045i	$-0.386451\pi$
$48$	− 48.2750i	− 1.00573i
$49$	27.3784	0.558744
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	53.6853	0.994173
$55$	0	0
$56$	37.1992	0.664271
$57$	0	0
$58$	115.675i	1.99439i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	110.011	1.80345	0.901727	−	0.432306i	$-0.142300\pi$
$61$	110.011	1.80345	0.901727	+	0.432306i	$0.142300\pi$
$62$	0	0
$63$	− 0.480854i	− 0.00763261i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	− 116.000i	− 1.73134i	−0.500612	−	0.865672i	$-0.666892\pi$
$67$	− 116.000i	− 1.73134i	0.500612	−	0.865672i	$-0.333108\pi$
$68$	0	0
$69$	−83.6086	−1.21172
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0.827294i	0.0114902i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−81.9200	−1.01136
$82$	140.396i	1.71215i
$83$	148.231i	1.78591i	0.450141	+	0.892957i	$0.351374\pi$
$83$	148.231i	1.78591i	−0.450141	+	0.892957i	$0.648626\pi$
$84$	−56.1184	−0.668077
$85$	0	0
$86$	−29.5880	−0.344046
$87$	− 174.506i	− 2.00582i
$88$	0	0
$89$	−51.7927	−0.581940	−0.290970	−	0.956732i	$-0.593978\pi$
$89$	−51.7927	−0.581940	−0.290970	+	0.956732i	$0.593978\pi$
$90$	0	0
$91$	0	0
$92$	110.843i	1.20482i
$93$	0	0
$94$	176.165	1.87409
$95$	0	0
$96$	96.5500	1.00573
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	54.7569i	0.558744i
$99$	0	0
$100$	0	0
$101$	−193.332	−1.91417	−0.957087	−	0.289799i	$-0.906411\pi$
$101$	−193.332	−1.91417	−0.957087	+	0.289799i	$0.906411\pi$
$102$	0	0
$103$	44.0000i	0.427184i	0.976923	+	0.213592i	$0.0685164\pi$
$103$	44.0000i	0.427184i	−0.976923	+	0.213592i	$0.931484\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	204.195i	1.90836i	0.299226	+	0.954182i	$0.403272\pi$
$107$	204.195i	1.90836i	−0.299226	+	0.954182i	$0.596728\pi$
$108$	107.371i	0.994173i
$109$	−215.899	−1.98072	−0.990362	−	0.138506i	$-0.955770\pi$
$109$	−215.899	−1.98072	−0.990362	+	0.138506i	$0.955770\pi$
$110$	0	0
$111$	0	0
$112$	74.3984i	0.664271i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	−231.350	−1.99439
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	220.021i	1.80345i
$123$	− 211.801i	− 1.72196i
$124$	0	0
$125$	0	0
$126$	0.961708	0.00763261
$127$	− 162.246i	− 1.27753i	−0.769402	−	0.638765i	$-0.779446\pi$
$127$	− 162.246i	− 1.27753i	0.769402	−	0.638765i	$-0.220554\pi$
$128$	− 128.000i	− 1.00000i
$129$	44.6362	0.346017
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	232.000	1.73134
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	− 167.217i	− 1.21172i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−265.761	−1.88483
$142$	0	0
$143$	0	0
$144$	−1.65459	−0.0114902
$145$	0	0
$146$	0	0
$147$	− 82.6059i	− 0.561945i
$148$	0	0
$149$	287.994	1.93285	0.966424	−	0.256951i	$-0.0827180\pi$
$149$	287.994	1.93285	0.966424	+	0.256951i	$0.0827180\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	128.852	0.800325
$162$	− 163.840i	− 1.01136i
$163$	32.9265i	0.202003i	0.994886	+	0.101002i	$0.0322047\pi$
$163$	32.9265i	0.202003i	−0.994886	+	0.101002i	$0.967795\pi$
$164$	−280.792	−1.71215
$165$	0	0
$166$	−296.462	−1.78591
$167$	− 141.516i	− 0.847400i	−0.905803	−	0.423700i	$-0.860731\pi$
$167$	− 141.516i	− 0.847400i	0.905803	−	0.423700i	$-0.139269\pi$
$168$	− 112.237i	− 0.668077i
$169$	−169.000	−1.00000
$170$	0	0
$171$	0	0
$172$	− 59.1760i	− 0.344046i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	349.013	2.00582
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 103.585i	− 0.581940i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	−59.5890	−0.329221	−0.164611	−	0.986359i	$-0.552637\pi$
$181$	−59.5890	−0.329221	−0.164611	+	0.986359i	$0.552637\pi$
$182$	0	0
$183$	− 331.923i	− 1.81379i
$184$	−221.686	−1.20482
$185$	0	0
$186$	0	0
$187$	0	0
$188$	352.329i	1.87409i
$189$	124.816	0.660400
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	193.100i	1.00573i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	−109.514	−0.558744
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	−349.994	−1.74126
$202$	− 386.663i	− 1.91417i
$203$	268.938i	1.32482i
$204$	0	0
$205$	0	0
$206$	−88.0000	−0.427184
$207$	2.86562i	0.0138436i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	−408.390	−1.90836
$215$	0	0
$216$	−214.741	−0.994173
$217$	0	0
$218$	− 431.798i	− 1.98072i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 224.050i	− 1.00471i	−0.864662	−	0.502354i	$-0.832468\pi$
$223$	− 224.050i	− 1.00471i	0.864662	−	0.502354i	$-0.167532\pi$
$224$	−148.797	−0.664271
$225$	0	0
$226$	0	0
$227$	453.615i	1.99831i	0.0411555	+	0.999153i	$0.486896\pi$
$227$	453.615i	1.99831i	−0.0411555	+	0.999153i	$0.513104\pi$
$228$	0	0
$229$	8.84461	0.0386228	0.0193114	−	0.999814i	$-0.493853\pi$
$229$	8.84461	0.0386228	0.0193114	+	0.999814i	$0.493853\pi$
$230$	0	0
$231$	0	0
$232$	− 462.699i	− 1.99439i
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−23.5161	−0.0975770	−0.0487885	−	0.998809i	$-0.515536\pi$
$241$	−23.5161	−0.0975770	−0.0487885	+	0.998809i	$0.515536\pi$
$242$	242.000i	1.00000i
$243$	5.58396i	0.0229793i
$244$	−440.043	−1.80345
$245$	0	0
$246$	423.601	1.72196
$247$	0	0
$248$	0	0
$249$	447.240	1.79615
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.92342i	0.00763261i
$253$	0	0
$254$	324.493	1.27753
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	89.2725i	0.346017i
$259$	0	0
$260$	0	0
$261$	−5.98107	−0.0229160
$262$	0	0
$263$	508.833i	1.93473i	0.253394	+	0.967363i	$0.418453\pi$
$263$	508.833i	1.93473i	−0.253394	+	0.967363i	$0.581547\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	156.268i	0.585274i
$268$	464.000i	1.73134i
$269$	−38.0000	−0.141264	−0.0706320	−	0.997502i	$-0.522502\pi$
$269$	−38.0000	−0.141264	−0.0706320	+	0.997502i	$0.522502\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	334.434	1.21172
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	558.976	1.98924	0.994620	−	0.103595i	$-0.0330347\pi$
$281$	558.976	1.98924	0.994620	+	0.103595i	$0.0330347\pi$
$282$	− 531.521i	− 1.88483i
$283$	− 316.000i	− 1.11661i	−0.829637	−	0.558304i	$-0.811452\pi$
$283$	− 316.000i	− 1.11661i	0.829637	−	0.558304i	$-0.188548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	326.414i	1.13733i
$288$	− 3.30918i	− 0.0114902i
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	165.212	0.561945
$295$	0	0
$296$	0	0
$297$	0	0
$298$	575.989i	1.93285i
$299$	0	0
$300$	0	0
$301$	−68.7906	−0.228540
$302$	0	0
$303$	583.318i	1.92514i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	568.920i	1.85316i	0.376099	+	0.926579i	$0.377265\pi$
$307$	568.920i	1.85316i	−0.376099	+	0.926579i	$0.622735\pi$
$308$	0	0
$309$	132.756	0.429632
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	616.094	1.91930
$322$	257.705i	0.800325i
$323$	0	0
$324$	327.680	1.01136
$325$	0	0
$326$	−65.8530	−0.202003
$327$	651.407i	1.99207i
$328$	− 561.585i	− 1.71215i
$329$	409.574	1.24490
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 592.924i	− 1.78591i
$333$	0	0
$334$	283.032	0.847400
$335$	0	0
$336$	224.474	0.668077
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	− 338.000i	− 1.00000i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	355.152i	1.03543i
$344$	118.352	0.344046
$345$	0	0
$346$	0	0
$347$	496.030i	1.42948i	0.699389	+	0.714741i	$0.253455\pi$
$347$	496.030i	1.42948i	−0.699389	+	0.714741i	$0.746545\pi$
$348$	698.025i	2.00582i
$349$	−427.869	−1.22598	−0.612992	−	0.790089i	$-0.710034\pi$
$349$	−427.869	−1.22598	−0.612992	+	0.790089i	$0.710034\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	207.171	0.581940
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	− 119.178i	− 0.329221i
$363$	− 365.080i	− 1.00573i
$364$	0	0
$365$	0	0
$366$	663.846	1.81379
$367$	− 656.702i	− 1.78938i	−0.446689	−	0.894689i	$-0.647397\pi$
$367$	− 656.702i	− 1.78938i	0.446689	−	0.894689i	$-0.352603\pi$
$368$	− 443.372i	− 1.20482i
$369$	−7.25931	−0.0196729
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	−704.658	−1.87409
$377$	0	0
$378$	249.631i	0.660400i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−489.527	−1.28485
$382$	0	0
$383$	− 713.710i	− 1.86347i	−0.363137	−	0.931736i	$-0.618294\pi$
$383$	− 713.710i	− 1.86347i	0.363137	−	0.931736i	$-0.381706\pi$
$384$	−386.200	−1.00573
$385$	0	0
$386$	0	0
$387$	− 1.52987i	− 0.00395316i
$388$	0	0
$389$	−652.125	−1.67641	−0.838207	−	0.545352i	$-0.816396\pi$
$389$	−652.125	−1.67641	−0.838207	+	0.545352i	$0.816396\pi$
$390$	0	0
$391$	0	0
$392$	− 219.028i	− 0.558744i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	765.237	1.90832	0.954160	−	0.299296i	$-0.0967519\pi$
$401$	765.237	1.90832	0.954160	+	0.299296i	$0.0967519\pi$
$402$	− 699.987i	− 1.74126i
$403$	0	0
$404$	773.327	1.91417
$405$	0	0
$406$	−537.876	−1.32482
$407$	0	0
$408$	0	0
$409$	−743.463	−1.81776	−0.908879	−	0.417059i	$-0.863061\pi$
$409$	−743.463	−1.81776	−0.908879	+	0.417059i	$0.863061\pi$
$410$	0	0
$411$	0	0
$412$	− 176.000i	− 0.427184i
$413$	0	0
$414$	−5.73124	−0.0138436
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	65.8191	0.156340	0.0781699	−	0.996940i	$-0.475092\pi$
$421$	65.8191	0.156340	0.0781699	+	0.996940i	$0.475092\pi$
$422$	0	0
$423$	9.10874i	0.0215337i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	511.539i	1.19798i
$428$	− 816.780i	− 1.90836i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 429.483i	− 0.994173i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	863.595	1.98072
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−2.83125	−0.00642007
$442$	0	0
$443$	415.284i	0.937437i	0.883348	+	0.468718i	$0.155284\pi$
$443$	415.284i	0.937437i	−0.883348	+	0.468718i	$0.844716\pi$
$444$	0	0
$445$	0	0
$446$	448.099	1.00471
$447$	− 868.933i	− 1.94392i
$448$	− 297.594i	− 0.664271i
$449$	−398.000	−0.886414	−0.443207	−	0.896419i	$-0.646159\pi$
$449$	−398.000	−0.886414	−0.443207	+	0.896419i	$0.646159\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	−907.231	−1.99831
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	17.6892i	0.0386228i
$459$	0	0
$460$	0	0
$461$	−460.385	−0.998666	−0.499333	−	0.866410i	$-0.666422\pi$
$461$	−460.385	−0.998666	−0.499333	+	0.866410i	$0.666422\pi$
$462$	0	0
$463$	− 925.642i	− 1.99923i	−0.0278164	−	0.999613i	$-0.508855\pi$
$463$	− 925.642i	− 1.99923i	0.0278164	−	0.999613i	$-0.491145\pi$
$464$	925.399	1.99439
$465$	0	0
$466$	0	0
$467$	− 644.450i	− 1.37998i	−0.723820	−	0.689989i	$-0.757615\pi$
$467$	− 644.450i	− 1.37998i	0.723820	−	0.689989i	$-0.242385\pi$
$468$	0	0
$469$	539.388	1.15008
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	− 47.0321i	− 0.0975770i
$483$	− 388.771i	− 0.804910i
$484$	−484.000	−1.00000
$485$	0	0
$486$	−11.1679	−0.0229793
$487$	953.429i	1.95776i	0.204435	+	0.978880i	$0.434464\pi$
$487$	953.429i	1.95776i	−0.204435	+	0.978880i	$0.565536\pi$
$488$	− 880.086i	− 1.80345i
$489$	99.3454	0.203160
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	847.203i	1.72196i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	894.481i	1.79615i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−426.980	−0.852255
$502$	0	0
$503$	985.525i	1.95929i	0.200730	+	0.979647i	$0.435669\pi$
$503$	985.525i	1.95929i	−0.200730	+	0.979647i	$0.564331\pi$
$504$	−3.84683	−0.00763261
$505$	0	0
$506$	0	0
$507$	509.904i	1.00573i
$508$	648.985i	1.27753i
$509$	982.000	1.92927	0.964637	−	0.263584i	$-0.0849045\pi$
$509$	982.000	1.92927	0.964637	+	0.263584i	$0.0849045\pi$
$510$	0	0
$511$	0	0
$512$	512.000i	1.00000i
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−178.545	−0.346017
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−491.436	−0.943256	−0.471628	−	0.881798i	$-0.656333\pi$
$521$	−491.436	−0.943256	−0.471628	+	0.881798i	$0.656333\pi$
$522$	− 11.9621i	− 0.0229160i
$523$	− 931.823i	− 1.78169i	−0.454309	−	0.890844i	$-0.650114\pi$
$523$	− 931.823i	− 1.78169i	0.454309	−	0.890844i	$-0.349886\pi$
$524$	0	0
$525$	0	0
$526$	−1017.67	−1.93473
$527$	0	0
$528$	0	0
$529$	−238.887	−0.451583
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−312.536	−0.585274
$535$	0	0
$536$	−928.000	−1.73134
$537$	0	0
$538$	− 76.0000i	− 0.141264i
$539$	0	0
$540$	0	0
$541$	−857.878	−1.58573	−0.792863	−	0.609400i	$-0.791410\pi$
$541$	−857.878	−1.58573	−0.792863	+	0.609400i	$0.791410\pi$
$542$	0	0
$543$	179.791i	0.331107i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 790.229i	− 1.44466i	−0.691549	−	0.722330i	$-0.743071\pi$
$547$	− 790.229i	− 1.44466i	0.691549	−	0.722330i	$-0.256929\pi$
$548$	0	0
$549$	−11.3764	−0.0207220
$550$	0	0
$551$	0	0
$552$	668.869i	1.21172i
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1117.95i	1.98924i
$563$	1124.00i	1.99645i	0.0595755	+	0.998224i	$0.481025\pi$
$563$	1124.00i	1.99645i	−0.0595755	+	0.998224i	$0.518975\pi$
$564$	1063.04	1.88483
$565$	0	0
$566$	632.000	1.11661
$567$	− 380.920i	− 0.671816i
$568$	0	0
$569$	1023.00	1.79788	0.898941	−	0.438069i	$-0.144337\pi$
$569$	1023.00	1.79788	0.898941	+	0.438069i	$0.144337\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−652.828	−1.13733
$575$	0	0
$576$	6.61835	0.0114902
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 578.000i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	−689.259	−1.18633
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1076.00i	− 1.83305i	−0.399978	−	0.916525i	$-0.630982\pi$
$587$	− 1076.00i	− 1.83305i	0.399978	−	0.916525i	$-0.369018\pi$
$588$	330.423i	0.561945i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−1151.98	−1.93285
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	942.651	1.56847	0.784235	−	0.620463i	$-0.213055\pi$
$601$	942.651	1.56847	0.784235	+	0.620463i	$0.213055\pi$
$602$	− 137.581i	− 0.228540i
$603$	11.9958i	0.0198935i
$604$	0	0
$605$	0	0
$606$	−1166.64	−1.92514
$607$	964.000i	1.58814i	0.607827	+	0.794069i	$0.292041\pi$
$607$	964.000i	1.58814i	−0.607827	+	0.794069i	$0.707959\pi$
$608$	0	0
$609$	811.436	1.33241
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−1137.84	−1.85316
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	265.512i	0.429632i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−743.831	−1.19780
$622$	0	0
$623$	− 240.831i	− 0.386566i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	209.768	0.327251	0.163626	−	0.986522i	$-0.447681\pi$
$641$	209.768	0.327251	0.163626	+	0.986522i	$0.447681\pi$
$642$	1232.19i	1.91930i
$643$	390.780i	0.607745i	0.952713	+	0.303873i	$0.0982797\pi$
$643$	390.780i	0.607745i	−0.952713	+	0.303873i	$0.901720\pi$
$644$	−515.409	−0.800325
$645$	0	0
$646$	0	0
$647$	− 956.000i	− 1.47759i	−0.673931	−	0.738794i	$-0.735395\pi$
$647$	− 956.000i	− 1.47759i	0.673931	−	0.738794i	$-0.264605\pi$
$648$	655.360i	1.01136i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 131.706i	− 0.202003i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−1302.81	−1.99207
$655$	0	0
$656$	1123.17	1.71215
$657$	0	0
$658$	819.147i	1.24490i
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1317.02	−1.99247	−0.996236	−	0.0866803i	$-0.972374\pi$
$661$	−1317.02	−1.99247	−0.996236	+	0.0866803i	$0.972374\pi$
$662$	0	0
$663$	0	0
$664$	1185.85	1.78591
$665$	0	0
$666$	0	0
$667$	− 1602.72i	− 2.40288i
$668$	566.063i	0.847400i
$669$	−676.000	−1.01046
$670$	0	0
$671$	0	0
$672$	448.948i	0.668077i
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	676.000	1.00000
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1368.64	2.00975
$682$	0	0
$683$	1014.84i	1.48586i	0.669367	+	0.742931i	$0.266565\pi$
$683$	1014.84i	1.48586i	−0.669367	+	0.742931i	$0.733435\pi$
$684$	0	0
$685$	0	0
$686$	−710.304	−1.03543
$687$	− 26.6858i	− 0.0388440i
$688$	236.704i	0.344046i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	−992.061	−1.42948
$695$	0	0
$696$	−1396.05	−2.00582
$697$	0	0
$698$	− 855.737i	− 1.22598i
$699$	0	0
$700$	0	0
$701$	902.000	1.28673	0.643367	−	0.765558i	$-0.277537\pi$
$701$	902.000	1.28673	0.643367	+	0.765558i	$0.277537\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 898.973i	− 1.27153i
$708$	0	0
$709$	−160.815	−0.226820	−0.113410	−	0.993548i	$-0.536177\pi$
$709$	−160.815	−0.226820	−0.113410	+	0.993548i	$0.536177\pi$
$710$	0	0
$711$	0	0
$712$	414.341i	0.581940i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−204.596	−0.283766
$722$	722.000i	1.00000i
$723$	70.9523i	0.0981360i
$724$	238.356	0.329221
$725$	0	0
$726$	730.159	1.00573
$727$	− 660.664i	− 0.908754i	−0.890809	−	0.454377i	$-0.849862\pi$
$727$	− 660.664i	− 0.908754i	0.890809	−	0.454377i	$-0.150138\pi$
$728$	0	0
$729$	−720.432	−0.988247
$730$	0	0
$731$	0	0
$732$	1327.69i	1.81379i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	1313.40	1.78938
$735$	0	0
$736$	886.745	1.20482
$737$	0	0
$738$	− 14.5186i	− 0.0196729i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	764.000i	1.02826i	0.857711	+	0.514132i	$0.171886\pi$
$743$	764.000i	1.02826i	−0.857711	+	0.514132i	$0.828114\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 15.3288i	− 0.0205205i
$748$	0	0
$749$	−949.486	−1.26767
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 1409.32i	− 1.87409i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−499.263	−0.660400
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1079.01	−1.41788	−0.708942	−	0.705266i	$-0.750827\pi$
$761$	−1079.01	−1.41788	−0.708942	+	0.705266i	$0.750827\pi$
$762$	− 979.055i	− 1.28485i
$763$	− 1003.91i	− 1.31574i
$764$	0	0
$765$	0	0
$766$	1427.42	1.86347
$767$	0	0
$768$	− 772.400i	− 1.00573i
$769$	−299.846	−0.389917	−0.194958	−	0.980812i	$-0.562457\pi$
$769$	−299.846	−0.389917	−0.194958	+	0.980812i	$0.562457\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	3.05975	0.00395316
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 1304.25i	− 1.67641i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 1552.51i	− 1.98277i
$784$	438.055	0.558744
$785$	0	0
$786$	0	0
$787$	− 1528.74i	− 1.94249i	−0.238086	−	0.971244i	$-0.576520\pi$
$787$	− 1528.74i	− 1.94249i	0.238086	−	0.971244i	$-0.423480\pi$
$788$	0	0
$789$	1535.24	1.94581
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5.35597	0.00668661
$802$	1530.47i	1.90832i
$803$	0	0
$804$	1399.97	1.74126
$805$	0	0
$806$	0	0
$807$	114.653i	0.142073i
$808$	1546.65i	1.91417i
$809$	482.314	0.596186	0.298093	−	0.954537i	$-0.403649\pi$
$809$	482.314	0.596186	0.298093	+	0.954537i	$0.403649\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 1075.75i	− 1.32482i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 1486.93i	− 1.81776i
$819$	0	0
$820$	0	0
$821$	−1418.80	−1.72813	−0.864067	−	0.503377i	$-0.832091\pi$
$821$	−1418.80	−1.72813	−0.864067	+	0.503377i	$0.832091\pi$
$822$	0	0
$823$	− 1396.00i	− 1.69623i	−0.529810	−	0.848117i	$-0.677737\pi$
$823$	− 1396.00i	− 1.69623i	0.529810	−	0.848117i	$-0.322263\pi$
$824$	352.000	0.427184
$825$	0	0
$826$	0	0
$827$	− 596.000i	− 0.720677i	−0.932822	−	0.360339i	$-0.882661\pi$
$827$	− 596.000i	− 0.720677i	0.932822	−	0.360339i	$-0.117339\pi$
$828$	− 11.4625i	− 0.0138436i
$829$	−1171.27	−1.41288	−0.706438	−	0.707775i	$-0.749699\pi$
$829$	−1171.27	−1.41288	−0.706438	+	0.707775i	$0.749699\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2504.17	2.97761
$842$	131.638i	0.156340i
$843$	− 1686.54i	− 2.00063i
$844$	0	0
$845$	0	0
$846$	−18.2175	−0.0215337
$847$	562.638i	0.664271i
$848$	0	0
$849$	−953.431	−1.12300
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	−1023.08	−1.19798
$855$	0	0
$856$	1633.56	1.90836
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	984.852	1.14385
$862$	0	0
$863$	1646.88i	1.90832i	0.299304	+	0.954158i	$0.403246\pi$
$863$	1646.88i	1.90832i	−0.299304	+	0.954158i	$0.596754\pi$
$864$	858.965	0.994173
$865$	0	0
$866$	0	0
$867$	871.967i	1.00573i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	1727.19i	1.98072i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	898.919	1.02034	0.510170	−	0.860074i	$-0.329583\pi$
$881$	898.919	1.02034	0.510170	+	0.860074i	$0.329583\pi$
$882$	− 5.66251i	− 0.00642007i
$883$	1749.93i	1.98180i	0.134603	+	0.990900i	$0.457024\pi$
$883$	1749.93i	1.98180i	−0.134603	+	0.990900i	$0.542976\pi$
$884$	0	0
$885$	0	0
$886$	−830.569	−0.937437
$887$	95.4499i	0.107610i	0.998551	+	0.0538049i	$0.0171349\pi$
$887$	95.4499i	0.107610i	−0.998551	+	0.0538049i	$0.982865\pi$
$888$	0	0
$889$	754.429	0.848627
$890$	0	0
$891$	0	0
$892$	896.199i	1.00471i
$893$	0	0
$894$	1737.87	1.94392
$895$	0	0
$896$	595.187	0.664271
$897$	0	0
$898$	− 796.000i	− 0.886414i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	207.554i	0.229849i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1303.16i	1.43678i	0.695640	+	0.718391i	$0.255121\pi$
$907$	1303.16i	1.43678i	−0.695640	+	0.718391i	$0.744879\pi$
$908$	− 1814.46i	− 1.99831i
$909$	19.9928	0.0219942
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−35.3785	−0.0386228
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1716.54	1.86378
$922$	− 920.770i	− 0.998666i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1851.28	1.99923
$927$	− 4.55012i	− 0.00490843i
$928$	1850.80i	1.99439i
$929$	−1495.62	−1.60992	−0.804960	−	0.593329i	$-0.797813\pi$
$929$	−1495.62	−1.60992	−0.804960	+	0.593329i	$0.797813\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	1288.90	1.37998
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	1078.78i	1.15008i
$939$	0	0
$940$	0	0
$941$	−118.000	−0.125399	−0.0626993	−	0.998032i	$-0.519971\pi$
$941$	−118.000	−0.125399	−0.0626993	+	0.998032i	$0.519971\pi$
$942$	0	0
$943$	− 1945.24i	− 2.06283i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1106.14i	1.16804i	0.811738	+	0.584021i	$0.198522\pi$
$947$	1106.14i	1.16804i	−0.811738	+	0.584021i	$0.801478\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	− 21.1162i	− 0.0219275i
$964$	94.0643	0.0975770
$965$	0	0
$966$	777.543	0.804910
$967$	1900.05i	1.96489i	0.186559	+	0.982444i	$0.440266\pi$
$967$	1900.05i	1.96489i	−0.186559	+	0.982444i	$0.559734\pi$
$968$	− 968.000i	− 1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 22.3358i	− 0.0229793i
$973$	0	0
$974$	−1906.86	−1.95776
$975$	0	0
$976$	1760.17	1.80345
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	198.691i	0.203160i
$979$	0	0
$980$	0	0
$981$	22.3265	0.0227589
$982$	0	0
$983$	284.000i	0.288911i	0.989511	+	0.144456i	$0.0461431\pi$
$983$	284.000i	0.288911i	−0.989511	+	0.144456i	$0.953857\pi$
$984$	−1694.41	−1.72196
$985$	0	0
$986$	0	0
$987$	− 1235.76i	− 1.25204i
$988$	0	0
$989$	409.953	0.414513
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−1788.96	−1.79615
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	500.3.b.a.251.6	yes	8
4.3	odd	2	inner	500.3.b.a.251.3	✓	8
5.2	odd	4		500.3.d.a.499.2		4
5.3	odd	4		500.3.d.b.499.3		4
5.4	even	2	inner	500.3.b.a.251.3	✓	8
20.3	even	4		500.3.d.a.499.2		4
20.7	even	4		500.3.d.b.499.3		4
20.19	odd	2	CM	500.3.b.a.251.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.3.b.a.251.3	✓	8	4.3	odd	2	inner
500.3.b.a.251.3	✓	8	5.4	even	2	inner
500.3.b.a.251.6	yes	8	1.1	even	1	trivial
500.3.b.a.251.6	yes	8	20.19	odd	2	CM
500.3.d.a.499.2		4	5.2	odd	4
500.3.d.a.499.2		4	20.3	even	4
500.3.d.b.499.3		4	5.3	odd	4
500.3.d.b.499.3		4	20.7	even	4

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:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	500.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -3.01719i q^{3} -4.00000 q^{4} +6.03437 q^{6} +4.64990i q^{7} -8.00000i q^{8} -0.103412 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -3.01719i q^{3} -4.00000 q^{4} +6.03437 q^{6} +4.64990i q^{7} -8.00000i q^{8} -0.103412 q^{9} +12.0687i q^{12} -9.29980 q^{14} +16.0000 q^{16} -0.206823i q^{18} +14.0296 q^{21} -27.7108i q^{23} -24.1375 q^{24} -26.8427i q^{27} -18.5996i q^{28} +57.8374 q^{29} +32.0000i q^{32} +0.413647 q^{36} +70.1981 q^{41} +28.0592i q^{42} +14.7940i q^{43} +55.4216 q^{46} -88.0823i q^{47} -48.2750i q^{48} +27.3784 q^{49} +53.6853 q^{54} +37.1992 q^{56} +115.675i q^{58} +110.011 q^{61} -0.480854i q^{63} -64.0000 q^{64} -116.000i q^{67} -83.6086 q^{69} +0.827294i q^{72} -81.9200 q^{81} +140.396i q^{82} +148.231i q^{83} -56.1184 q^{84} -29.5880 q^{86} -174.506i q^{87} -51.7927 q^{89} +110.843i q^{92} +176.165 q^{94} +96.5500 q^{96} +54.7569i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9}+O(q^{10}) \) 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9	\( 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9} + 16 q^{14} + 128 q^{16} + 32 q^{21} - 64 q^{24} - 44 q^{29} + 304 q^{36} - 124 q^{41} + 176 q^{46} - 556 q^{49} + 32 q^{54} - 64 q^{56} + 116 q^{61} - 512 q^{64} + 352 q^{69} + 1000 q^{81} - 128 q^{84} - 304 q^{86} - 284 q^{89} + 16 q^{94} + 256 q^{96}+O(q^{100}) \) 8 * q - 32 * q^4 + 16 * q^6 - 76 * q^9 + 16 * q^14 + 128 * q^16 + 32 * q^21 - 64 * q^24 - 44 * q^29 + 304 * q^36 - 124 * q^41 + 176 * q^46 - 556 * q^49 + 32 * q^54 - 64 * q^56 + 116 * q^61 - 512 * q^64 + 352 * q^69 + 1000 * q^81 - 128 * q^84 - 304 * q^86 - 284 * q^89 + 16 * q^94 + 256 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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