LMFDB - Embedded newform 500.3.b.a.251.4

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			251.4
Root			$-0.587785 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	500.251
Dual form			500.3.b.a.251.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +5.86472i q^{3} -4.00000 q^{4} +11.7294 q^{6} +11.5237i q^{7} +8.00000i q^{8} -25.3950 q^{9} +O(q^{10})$	$q-2.00000i q^{2} +5.86472i q^{3} -4.00000 q^{4} +11.7294 q^{6} +11.5237i q^{7} +8.00000i q^{8} -25.3950 q^{9} -23.4589i q^{12} +23.0474 q^{14} +16.0000 q^{16} +50.7900i q^{18} -67.5833 q^{21} -0.836986i q^{23} -46.9178 q^{24} -96.1520i q^{27} -46.0948i q^{28} +13.7455 q^{29} -32.0000i q^{32} +101.580 q^{36} -18.6152 q^{41} +135.167i q^{42} -85.1432i q^{43} -1.67397 q^{46} +58.4378i q^{47} +93.8356i q^{48} -83.7955 q^{49} -192.304 q^{54} -92.1896 q^{56} -27.4910i q^{58} -120.001 q^{61} -292.644i q^{63} -64.0000 q^{64} +116.000i q^{67} +4.90869 q^{69} -203.160i q^{72} +335.350 q^{81} +37.2304i q^{82} +163.843i q^{83} +270.333 q^{84} -170.286 q^{86} +80.6135i q^{87} -58.1977 q^{89} +3.34794i q^{92} +116.876 q^{94} +187.671 q^{96} +167.591i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.86472i	1.95491i	0.211149	+	0.977454i	$0.432280\pi$
$3$	5.86472i	1.95491i	−0.211149	+	0.977454i	$0.567720\pi$
$4$	−4.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	11.7294	1.95491
$7$	11.5237i	1.64624i	0.567866	+	0.823121i	$0.307769\pi$
$7$	11.5237i	1.64624i	−0.567866	+	0.823121i	$0.692231\pi$
$8$	8.00000i	1.00000i
$9$	−25.3950	−2.82166
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 23.4589i	− 1.95491i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	23.0474	1.64624
$15$	0	0
$16$	16.0000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	50.7900i	2.82166i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	−67.5833	−3.21825
$22$	0	0
$23$	− 0.836986i	− 0.0363907i	−0.999834	−	0.0181953i	$-0.994208\pi$
$23$	− 0.836986i	− 0.0363907i	0.999834	−	0.0181953i	$-0.00579208\pi$
$24$	−46.9178	−1.95491
$25$	0	0
$26$	0	0
$27$	− 96.1520i	− 3.56119i
$28$	− 46.0948i	− 1.64624i
$29$	13.7455	0.473983	0.236991	−	0.971512i	$-0.423839\pi$
$29$	13.7455	0.473983	0.236991	+	0.971512i	$0.423839\pi$
$30$	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
$36$	101.580	2.82166
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−18.6152	−0.454029	−0.227014	−	0.973891i	$-0.572896\pi$
$41$	−18.6152	−0.454029	−0.227014	+	0.973891i	$0.572896\pi$
$42$	135.167i	3.21825i
$43$	− 85.1432i	− 1.98007i	−0.140807	−	0.990037i	$-0.544970\pi$
$43$	− 85.1432i	− 1.98007i	0.140807	−	0.990037i	$-0.455030\pi$
$44$	0	0
$45$	0	0
$46$	−1.67397	−0.0363907
$47$	58.4378i	1.24336i	0.783272	+	0.621679i	$0.213549\pi$
$47$	58.4378i	1.24336i	−0.783272	+	0.621679i	$0.786451\pi$
$48$	93.8356i	1.95491i
$49$	−83.7955	−1.71011
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−192.304	−3.56119
$55$	0	0
$56$	−92.1896	−1.64624
$57$	0	0
$58$	− 27.4910i	− 0.473983i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−120.001	−1.96723	−0.983615	−	0.180279i	$-0.942300\pi$
$61$	−120.001	−1.96723	−0.983615	+	0.180279i	$0.942300\pi$
$62$	0	0
$63$	− 292.644i	− 4.64514i
$64$	−64.0000	−1.00000
$65$	0	0
$66$	0	0
$67$	116.000i	1.73134i	0.500612	+	0.865672i	$0.333108\pi$
$67$	116.000i	1.73134i	−0.500612	+	0.865672i	$0.666892\pi$
$68$	0	0
$69$	4.90869	0.0711404
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 203.160i	− 2.82166i
$73$	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
$81$	335.350	4.14013
$82$	37.2304i	0.454029i
$83$	163.843i	1.97401i	0.160695	+	0.987004i	$0.448626\pi$
$83$	163.843i	1.97401i	−0.160695	+	0.987004i	$0.551374\pi$
$84$	270.333	3.21825
$85$	0	0
$86$	−170.286	−1.98007
$87$	80.6135i	0.926592i
$88$	0	0
$89$	−58.1977	−0.653907	−0.326953	−	0.945041i	$-0.606022\pi$
$89$	−58.1977	−0.653907	−0.326953	+	0.945041i	$0.606022\pi$
$90$	0	0
$91$	0	0
$92$	3.34794i	0.0363907i
$93$	0	0
$94$	116.876	1.24336
$95$	0	0
$96$	187.671	1.95491
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	167.591i	1.71011i
$99$	0	0
$100$	0	0
$101$	190.817	1.88928	0.944640	−	0.328109i	$-0.106411\pi$
$101$	190.817	1.88928	0.944640	+	0.328109i	$0.106411\pi$
$102$	0	0
$103$	− 44.0000i	− 0.427184i	−0.976923	−	0.213592i	$-0.931484\pi$
$103$	− 44.0000i	− 0.427184i	0.976923	−	0.213592i	$-0.0685164\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.19946i	− 0.0205557i	−0.999947	−	0.0102778i	$-0.996728\pi$
$107$	− 2.19946i	− 0.0205557i	0.999947	−	0.0102778i	$-0.00327160\pi$
$108$	384.608i	3.56119i
$109$	−95.4328	−0.875530	−0.437765	−	0.899089i	$-0.644230\pi$
$109$	−95.4328	−0.875530	−0.437765	+	0.899089i	$0.644230\pi$
$110$	0	0
$111$	0	0
$112$	184.379i	1.64624i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	−54.9820	−0.473983
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	240.002i	1.96723i
$123$	− 109.173i	− 0.887585i
$124$	0	0
$125$	0	0
$126$	−585.288	−4.64514
$127$	− 135.726i	− 1.06871i	−0.845260	−	0.534355i	$-0.820554\pi$
$127$	− 135.726i	− 1.06871i	0.845260	−	0.534355i	$-0.179446\pi$
$128$	128.000i	1.00000i
$129$	499.341	3.87086
$130$	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	$0.5\pi$
$138$	− 9.81738i	− 0.0711404i
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−342.722	−2.43065
$142$	0	0
$143$	0	0
$144$	−406.320	−2.82166
$145$	0	0
$146$	0	0
$147$	− 491.438i	− 3.34311i
$148$	0	0
$149$	−187.985	−1.26164	−0.630822	−	0.775928i	$-0.717282\pi$
$149$	−187.985	−1.26164	−0.630822	+	0.775928i	$0.717282\pi$
$150$	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
$161$	9.64517	0.0599079
$162$	− 670.701i	− 4.14013i
$163$	217.276i	1.33298i	0.745513	+	0.666491i	$0.232205\pi$
$163$	217.276i	1.33298i	−0.745513	+	0.666491i	$0.767795\pi$
$164$	74.4607	0.454029
$165$	0	0
$166$	327.685	1.97401
$167$	331.462i	1.98480i	0.123054	+	0.992400i	$0.460731\pi$
$167$	331.462i	1.98480i	−0.123054	+	0.992400i	$0.539269\pi$
$168$	− 540.666i	− 3.21825i
$169$	−169.000	−1.00000
$170$	0	0
$171$	0	0
$172$	340.573i	1.98007i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	161.227	0.926592
$175$	0	0
$176$	0	0
$177$	0	0
$178$	116.395i	0.653907i
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	321.172	1.77443	0.887215	−	0.461356i	$-0.152637\pi$
$181$	321.172	1.77443	0.887215	+	0.461356i	$0.152637\pi$
$182$	0	0
$183$	− 703.773i	− 3.84576i
$184$	6.69589	0.0363907
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 233.751i	− 1.24336i
$189$	1108.03	5.86258
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 375.342i	− 1.95491i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	335.182	1.71011
$197$	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
$201$	−680.308	−3.38462
$202$	− 381.634i	− 1.88928i
$203$	158.399i	0.780290i
$204$	0	0
$205$	0	0
$206$	−88.0000	−0.427184
$207$	21.2552i	0.102682i
$208$	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
$214$	−4.39892	−0.0205557
$215$	0	0
$216$	769.216	3.56119
$217$	0	0
$218$	190.866i	0.875530i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 297.530i	− 1.33421i	−0.744962	−	0.667107i	$-0.767532\pi$
$223$	− 297.530i	− 1.33421i	0.744962	−	0.667107i	$-0.232468\pi$
$224$	368.758	1.64624
$225$	0	0
$226$	0	0
$227$	377.965i	1.66504i	0.553992	+	0.832522i	$0.313104\pi$
$227$	377.965i	1.66504i	−0.553992	+	0.832522i	$0.686896\pi$
$228$	0	0
$229$	−276.311	−1.20660	−0.603299	−	0.797515i	$-0.706147\pi$
$229$	−276.311	−1.20660	−0.603299	+	0.797515i	$0.706147\pi$
$230$	0	0
$231$	0	0
$232$	109.964i	0.473983i
$233$	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
$241$	−263.950	−1.09523	−0.547615	−	0.836731i	$-0.684464\pi$
$241$	−263.950	−1.09523	−0.547615	+	0.836731i	$0.684464\pi$
$242$	− 242.000i	− 1.00000i
$243$	1101.37i	4.53238i
$244$	480.004	1.96723
$245$	0	0
$246$	−218.346	−0.887585
$247$	0	0
$248$	0	0
$249$	−960.892	−3.85900
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1170.58i	4.64514i
$253$	0	0
$254$	−271.452	−1.06871
$255$	0	0
$256$	256.000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	− 998.683i	− 3.87086i
$259$	0	0
$260$	0	0
$261$	−349.067	−1.33742
$262$	0	0
$263$	− 30.4761i	− 0.115879i	−0.998320	−	0.0579393i	$-0.981547\pi$
$263$	− 30.4761i	− 0.115879i	0.998320	−	0.0579393i	$-0.0184530\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 341.313i	− 1.27833i
$268$	− 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
$276$	−19.6348	−0.0711404
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−486.442	−1.73111	−0.865556	−	0.500812i	$-0.833035\pi$
$281$	−486.442	−1.73111	−0.865556	+	0.500812i	$0.833035\pi$
$282$	685.443i	2.43065i
$283$	316.000i	1.11661i	0.829637	+	0.558304i	$0.188548\pi$
$283$	316.000i	1.11661i	−0.829637	+	0.558304i	$0.811452\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 214.516i	− 0.747442i
$288$	812.639i	2.82166i
$289$	−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	$0.5\pi$
$294$	−982.875	−3.34311
$295$	0	0
$296$	0	0
$297$	0	0
$298$	375.970i	1.26164i
$299$	0	0
$300$	0	0
$301$	981.164	3.25968
$302$	0	0
$303$	1119.09i	3.69337i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	324.532i	1.05711i	0.848900	+	0.528553i	$0.177265\pi$
$307$	324.532i	1.05711i	−0.848900	+	0.528553i	$0.822735\pi$
$308$	0	0
$309$	258.048	0.835106
$310$	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
$321$	12.8992	0.0401845
$322$	− 19.2903i	− 0.0599079i
$323$	0	0
$324$	−1341.40	−4.14013
$325$	0	0
$326$	434.552	1.33298
$327$	− 559.687i	− 1.71158i
$328$	− 148.921i	− 0.454029i
$329$	−673.420	−2.04687
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	− 655.371i	− 1.97401i
$333$	0	0
$334$	662.923	1.98480
$335$	0	0
$336$	−1081.33	−3.21825
$337$	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
$343$	− 400.973i	− 1.16902i
$344$	681.146	1.98007
$345$	0	0
$346$	0	0
$347$	686.594i	1.97866i	0.145704	+	0.989328i	$0.453455\pi$
$347$	686.594i	1.97866i	−0.145704	+	0.989328i	$0.546545\pi$
$348$	− 322.454i	− 0.926592i
$349$	670.306	1.92065	0.960324	−	0.278887i	$-0.0899656\pi$
$349$	670.306	1.92065	0.960324	+	0.278887i	$0.0899656\pi$
$350$	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
$356$	232.791	0.653907
$357$	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
$362$	− 642.344i	− 1.77443i
$363$	709.632i	1.95491i
$364$	0	0
$365$	0	0
$366$	−1407.55	−3.84576
$367$	− 338.566i	− 0.922524i	−0.887264	−	0.461262i	$-0.847397\pi$
$367$	− 338.566i	− 0.922524i	0.887264	−	0.461262i	$-0.152603\pi$
$368$	− 13.3918i	− 0.0363907i
$369$	472.732	1.28112
$370$	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
$376$	−467.503	−1.24336
$377$	0	0
$378$	− 2216.05i	− 5.86258i
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	795.997	2.08923
$382$	0	0
$383$	485.097i	1.26657i	0.773918	+	0.633286i	$0.218294\pi$
$383$	485.097i	1.26657i	−0.773918	+	0.633286i	$0.781706\pi$
$384$	−750.685	−1.95491
$385$	0	0
$386$	0	0
$387$	2162.21i	5.58711i
$388$	0	0
$389$	−605.036	−1.55536	−0.777681	−	0.628659i	$-0.783604\pi$
$389$	−605.036	−1.55536	−0.777681	+	0.628659i	$0.783604\pi$
$390$	0	0
$391$	0	0
$392$	− 670.364i	− 1.71011i
$393$	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
$401$	−760.179	−1.89571	−0.947854	−	0.318706i	$-0.896752\pi$
$401$	−760.179	−1.89571	−0.947854	+	0.318706i	$0.896752\pi$
$402$	1360.62i	3.38462i
$403$	0	0
$404$	−763.269	−1.88928
$405$	0	0
$406$	316.798	0.780290
$407$	0	0
$408$	0	0
$409$	400.949	0.980315	0.490157	−	0.871634i	$-0.336939\pi$
$409$	400.949	0.980315	0.490157	+	0.871634i	$0.336939\pi$
$410$	0	0
$411$	0	0
$412$	176.000i	0.427184i
$413$	0	0
$414$	42.5105	0.102682
$415$	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
$421$	818.678	1.94460	0.972302	−	0.233727i	$-0.0750923\pi$
$421$	818.678	1.94460	0.972302	+	0.233727i	$0.0750923\pi$
$422$	0	0
$423$	− 1484.03i	− 3.50834i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1382.86i	− 3.23854i
$428$	8.79784i	0.0205557i
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 1538.43i	− 3.56119i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	381.731	0.875530
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	2127.99	4.82536
$442$	0	0
$443$	− 124.056i	− 0.280035i	−0.990149	−	0.140018i	$-0.955284\pi$
$443$	− 124.056i	− 0.280035i	0.990149	−	0.140018i	$-0.0447159\pi$
$444$	0	0
$445$	0	0
$446$	−595.059	−1.33421
$447$	− 1102.48i	− 2.46640i
$448$	− 737.516i	− 1.64624i
$449$	−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
$454$	755.930	1.66504
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	552.622i	1.20660i
$459$	0	0
$460$	0	0
$461$	−97.0810	−0.210588	−0.105294	−	0.994441i	$-0.533578\pi$
$461$	−97.0810	−0.210588	−0.105294	+	0.994441i	$0.533578\pi$
$462$	0	0
$463$	− 733.720i	− 1.58471i	−0.610062	−	0.792354i	$-0.708855\pi$
$463$	− 733.720i	− 1.58471i	0.610062	−	0.792354i	$-0.291145\pi$
$464$	219.928	0.473983
$465$	0	0
$466$	0	0
$467$	− 918.742i	− 1.96733i	−0.180017	−	0.983663i	$-0.557615\pi$
$467$	− 918.742i	− 1.96733i	0.180017	−	0.983663i	$-0.442385\pi$
$468$	0	0
$469$	−1336.75	−2.85021
$470$	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
$482$	527.900i	1.09523i
$483$	56.5662i	0.117114i
$484$	−484.000	−1.00000
$485$	0	0
$486$	2202.74	4.53238
$487$	− 105.252i	− 0.216123i	−0.994144	−	0.108061i	$-0.965536\pi$
$487$	− 105.252i	− 0.216123i	0.994144	−	0.108061i	$-0.0344643\pi$
$488$	− 960.009i	− 1.96723i
$489$	−1274.26	−2.60586
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	436.692i	0.887585i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	1921.78i	3.85900i
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−1943.93	−3.88010
$502$	0	0
$503$	678.612i	1.34913i	0.738216	+	0.674565i	$0.235669\pi$
$503$	678.612i	1.34913i	−0.738216	+	0.674565i	$0.764331\pi$
$504$	2341.15	4.64514
$505$	0	0
$506$	0	0
$507$	− 991.138i	− 1.95491i
$508$	542.905i	1.06871i
$509$	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
$516$	−1997.37	−3.87086
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1025.72	−1.96876	−0.984380	−	0.176054i	$-0.943667\pi$
$521$	−1025.72	−1.96876	−0.984380	+	0.176054i	$0.943667\pi$
$522$	698.133i	1.33742i
$523$	739.898i	1.41472i	0.706854	+	0.707360i	$0.250114\pi$
$523$	739.898i	1.41472i	−0.706854	+	0.707360i	$0.749886\pi$
$524$	0	0
$525$	0	0
$526$	−60.9521	−0.115879
$527$	0	0
$528$	0	0
$529$	528.299	0.998676
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−682.627	−1.27833
$535$	0	0
$536$	−928.000	−1.73134
$537$	0	0
$538$	76.0000i	0.141264i
$539$	0	0
$540$	0	0
$541$	−892.198	−1.64916	−0.824582	−	0.565743i	$-0.808590\pi$
$541$	−892.198	−1.64916	−0.824582	+	0.565743i	$0.808590\pi$
$542$	0	0
$543$	1883.58i	3.46885i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 194.617i	− 0.355790i	−0.984049	−	0.177895i	$-0.943071\pi$
$547$	− 194.617i	− 0.355790i	0.984049	−	0.177895i	$-0.0569287\pi$
$548$	0	0
$549$	3047.43	5.55087
$550$	0	0
$551$	0	0
$552$	39.2695i	0.0711404i
$553$	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
$562$	972.885i	1.73111i
$563$	− 1124.00i	− 1.99645i	−0.0595755	−	0.998224i	$-0.518975\pi$
$563$	− 1124.00i	− 1.99645i	0.0595755	−	0.998224i	$-0.481025\pi$
$564$	1370.89	2.43065
$565$	0	0
$566$	632.000	1.11661
$567$	3864.47i	6.81565i
$568$	0	0
$569$	790.246	1.38883	0.694416	−	0.719573i	$-0.255663\pi$
$569$	790.246	1.38883	0.694416	+	0.719573i	$0.255663\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−429.031	−0.747442
$575$	0	0
$576$	1625.28	2.82166
$577$	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
$581$	−1888.07	−3.24970
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1076.00i	1.83305i	0.399978	+	0.916525i	$0.369018\pi$
$587$	1076.00i	1.83305i	−0.399978	+	0.916525i	$0.630982\pi$
$588$	1965.75i	3.34311i
$589$	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
$596$	751.939	1.26164
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	1000.59	1.66488	0.832438	−	0.554118i	$-0.186945\pi$
$601$	1000.59	1.66488	0.832438	+	0.554118i	$0.186945\pi$
$602$	− 1962.33i	− 3.25968i
$603$	− 2945.82i	− 4.88527i
$604$	0	0
$605$	0	0
$606$	2238.18	3.69337
$607$	− 964.000i	− 1.58814i	−0.607827	−	0.794069i	$-0.707959\pi$
$607$	− 964.000i	− 1.58814i	0.607827	−	0.794069i	$-0.292041\pi$
$608$	0	0
$609$	−928.965	−1.52539
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	649.063	1.05711
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	− 516.096i	− 0.835106i
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−80.4779	−0.129594
$622$	0	0
$623$	− 670.652i	− 1.07649i
$624$	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
$641$	1267.64	1.97760	0.988802	−	0.149235i	$-0.0476811\pi$
$641$	1267.64	1.97760	0.988802	+	0.149235i	$0.0476811\pi$
$642$	− 25.7984i	− 0.0401845i
$643$	1036.30i	1.61166i	0.592149	+	0.805829i	$0.298280\pi$
$643$	1036.30i	1.61166i	−0.592149	+	0.805829i	$0.701720\pi$
$644$	−38.5807	−0.0599079
$645$	0	0
$646$	0	0
$647$	956.000i	1.47759i	0.673931	+	0.738794i	$0.264605\pi$
$647$	956.000i	1.47759i	−0.673931	+	0.738794i	$0.735395\pi$
$648$	2682.80i	4.14013i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 869.105i	− 1.33298i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−1119.37	−1.71158
$655$	0	0
$656$	−297.843	−0.454029
$657$	0	0
$658$	1346.84i	2.04687i
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−515.966	−0.780584	−0.390292	−	0.920691i	$-0.627626\pi$
$661$	−515.966	−0.780584	−0.390292	+	0.920691i	$0.627626\pi$
$662$	0	0
$663$	0	0
$664$	−1310.74	−1.97401
$665$	0	0
$666$	0	0
$667$	− 11.5048i	− 0.0172485i
$668$	− 1325.85i	− 1.98480i
$669$	1744.93	2.60826
$670$	0	0
$671$	0	0
$672$	2162.66i	3.21825i
$673$	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
$681$	−2216.66	−3.25501
$682$	0	0
$683$	− 1183.21i	− 1.73237i	−0.499724	−	0.866185i	$-0.666565\pi$
$683$	− 1183.21i	− 1.73237i	0.499724	−	0.866185i	$-0.333435\pi$
$684$	0	0
$685$	0	0
$686$	−801.946	−1.16902
$687$	− 1620.49i	− 2.35879i
$688$	− 1362.29i	− 1.98007i
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	1373.19	1.97866
$695$	0	0
$696$	−644.908	−0.926592
$697$	0	0
$698$	− 1340.61i	− 1.92065i
$699$	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
$707$	2198.92i	3.11021i
$708$	0	0
$709$	958.204	1.35149	0.675743	−	0.737137i	$-0.263823\pi$
$709$	958.204	1.35149	0.675743	+	0.737137i	$0.263823\pi$
$710$	0	0
$711$	0	0
$712$	− 465.581i	− 0.653907i
$713$	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
$721$	507.043	0.703249
$722$	− 722.000i	− 1.00000i
$723$	− 1547.99i	− 2.14107i
$724$	−1284.69	−1.77443
$725$	0	0
$726$	1419.26	1.95491
$727$	− 1027.69i	− 1.41360i	−0.707414	−	0.706800i	$-0.750138\pi$
$727$	− 1027.69i	− 1.41360i	0.707414	−	0.706800i	$-0.249862\pi$
$728$	0	0
$729$	−3441.07	−4.72026
$730$	0	0
$731$	0	0
$732$	2815.09i	3.84576i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	−677.132	−0.922524
$735$	0	0
$736$	−26.7835	−0.0363907
$737$	0	0
$738$	− 945.465i	− 1.28112i
$739$	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	$-0.828114\pi$
$743$	− 764.000i	− 1.02826i	0.857711	−	0.514132i	$-0.171886\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4160.78i	− 5.56999i
$748$	0	0
$749$	25.3459	0.0338397
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	935.005i	1.24336i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	−4432.11	−5.86258
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1503.88	1.97618	0.988092	−	0.153867i	$-0.0491726\pi$
$761$	1503.88	1.97618	0.988092	+	0.153867i	$0.0491726\pi$
$762$	− 1591.99i	− 2.08923i
$763$	− 1099.74i	− 1.44134i
$764$	0	0
$765$	0	0
$766$	970.194	1.26657
$767$	0	0
$768$	1501.37i	1.95491i
$769$	−1527.31	−1.98611	−0.993053	−	0.117671i	$-0.962457\pi$
$769$	−1527.31	−1.98611	−0.993053	+	0.117671i	$0.962457\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	4324.42	5.58711
$775$	0	0
$776$	0	0
$777$	0	0
$778$	1210.07i	1.55536i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 1321.66i	− 1.68794i
$784$	−1340.73	−1.71011
$785$	0	0
$786$	0	0
$787$	828.812i	1.05313i	0.850135	+	0.526564i	$0.176520\pi$
$787$	828.812i	1.05313i	−0.850135	+	0.526564i	$0.823480\pi$
$788$	0	0
$789$	178.734	0.226532
$790$	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
$801$	1477.93	1.84511
$802$	1520.36i	1.89571i
$803$	0	0
$804$	2721.23	3.38462
$805$	0	0
$806$	0	0
$807$	− 222.859i	− 0.276158i
$808$	1526.54i	1.88928i
$809$	517.599	0.639801	0.319900	−	0.947451i	$-0.396351\pi$
$809$	517.599	0.639801	0.319900	+	0.947451i	$0.396351\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 633.595i	− 0.780290i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 801.898i	− 0.980315i
$819$	0	0
$820$	0	0
$821$	1633.66	1.98984	0.994922	−	0.100645i	$-0.0320906\pi$
$821$	1633.66	1.98984	0.994922	+	0.100645i	$0.0320906\pi$
$822$	0	0
$823$	1396.00i	1.69623i	0.529810	+	0.848117i	$0.322263\pi$
$823$	1396.00i	1.69623i	−0.529810	+	0.848117i	$0.677737\pi$
$824$	352.000	0.427184
$825$	0	0
$826$	0	0
$827$	596.000i	0.720677i	0.932822	+	0.360339i	$0.117339\pi$
$827$	596.000i	0.720677i	−0.932822	+	0.360339i	$0.882661\pi$
$828$	− 85.0209i	− 0.102682i
$829$	754.113	0.909666	0.454833	−	0.890577i	$-0.349699\pi$
$829$	754.113	0.909666	0.454833	+	0.890577i	$0.349699\pi$
$830$	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
$841$	−652.061	−0.775341
$842$	− 1637.36i	− 1.94460i
$843$	− 2852.85i	− 3.38416i
$844$	0	0
$845$	0	0
$846$	−2968.06	−3.50834
$847$	1394.37i	1.64624i
$848$	0	0
$849$	−1853.25	−2.18287
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	−2765.71	−3.23854
$855$	0	0
$856$	17.5957	0.0205557
$857$	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
$861$	1258.08	1.46118
$862$	0	0
$863$	1028.70i	1.19201i	0.802982	+	0.596004i	$0.203246\pi$
$863$	1028.70i	1.19201i	−0.802982	+	0.596004i	$0.796754\pi$
$864$	−3076.87	−3.56119
$865$	0	0
$866$	0	0
$867$	− 1694.91i	− 1.95491i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 763.463i	− 0.875530i
$873$	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
$881$	163.518	0.185605	0.0928026	−	0.995685i	$-0.470417\pi$
$881$	163.518	0.185605	0.0928026	+	0.995685i	$0.470417\pi$
$882$	− 4255.97i	− 4.82536i
$883$	1555.44i	1.76154i	0.473540	+	0.880772i	$0.342976\pi$
$883$	1555.44i	1.76154i	−0.473540	+	0.880772i	$0.657024\pi$
$884$	0	0
$885$	0	0
$886$	−248.111	−0.280035
$887$	1118.44i	1.26093i	0.776219	+	0.630463i	$0.217135\pi$
$887$	1118.44i	1.26093i	−0.776219	+	0.630463i	$0.782865\pi$
$888$	0	0
$889$	1564.07	1.75936
$890$	0	0
$891$	0	0
$892$	1190.12i	1.33421i
$893$	0	0
$894$	−2204.96	−2.46640
$895$	0	0
$896$	−1475.03	−1.64624
$897$	0	0
$898$	796.000i	0.886414i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	5754.26i	6.37238i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	312.559i	0.344608i	0.985044	+	0.172304i	$0.0551211\pi$
$907$	312.559i	0.344608i	−0.985044	+	0.172304i	$0.944879\pi$
$908$	− 1511.86i	− 1.66504i
$909$	−4845.80	−5.33091
$910$	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
$916$	1105.24	1.20660
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1903.29	−2.06654
$922$	194.162i	0.210588i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1467.44	−1.58471
$927$	1117.38i	1.20537i
$928$	− 439.856i	− 0.473983i
$929$	1857.96	1.99995	0.999976	−	0.00687017i	$-0.00218686\pi$
$929$	1857.96	1.99995	0.999976	+	0.00687017i	$0.00218686\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−1837.48	−1.96733
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	2673.50i	2.85021i
$939$	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
$943$	15.5806i	0.0165224i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1120.37i	1.18307i	0.806278	+	0.591536i	$0.201478\pi$
$947$	1120.37i	1.18307i	−0.806278	+	0.591536i	$0.798522\pi$
$948$	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
$963$	55.8552i	0.0580013i
$964$	1055.80	1.09523
$965$	0	0
$966$	113.132	0.117114
$967$	− 930.293i	− 0.962040i	−0.876710	−	0.481020i	$-0.840266\pi$
$967$	− 930.293i	− 0.962040i	0.876710	−	0.481020i	$-0.159734\pi$
$968$	968.000i	1.00000i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 4405.47i	− 4.53238i
$973$	0	0
$974$	−210.503	−0.216123
$975$	0	0
$976$	−1920.02	−1.96723
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	2548.53i	2.60586i
$979$	0	0
$980$	0	0
$981$	2423.51	2.47045
$982$	0	0
$983$	− 284.000i	− 0.288911i	−0.989511	−	0.144456i	$-0.953857\pi$
$983$	− 284.000i	− 0.288911i	0.989511	−	0.144456i	$-0.0461431\pi$
$984$	873.383	0.887585
$985$	0	0
$986$	0	0
$987$	− 3949.42i	− 4.00144i
$988$	0	0
$989$	−71.2636	−0.0720562
$990$	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
$996$	3843.57	3.85900
$997$	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.251.4	✓	8
4.3	odd	2	inner	500.3.b.a.251.5	yes	8
5.2	odd	4		500.3.d.b.499.4		4
5.3	odd	4		500.3.d.a.499.1		4
5.4	even	2	inner	500.3.b.a.251.5	yes	8
20.3	even	4		500.3.d.b.499.4		4
20.7	even	4		500.3.d.a.499.1		4
20.19	odd	2	CM	500.3.b.a.251.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.3.b.a.251.4	✓	8	1.1	even	1	trivial
500.3.b.a.251.4	✓	8	20.19	odd	2	CM
500.3.b.a.251.5	yes	8	4.3	odd	2	inner
500.3.b.a.251.5	yes	8	5.4	even	2	inner
500.3.d.a.499.1		4	5.3	odd	4
500.3.d.a.499.1		4	20.7	even	4
500.3.d.b.499.4		4	5.2	odd	4
500.3.d.b.499.4		4	20.3	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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +5.86472i q^{3} -4.00000 q^{4} +11.7294 q^{6} +11.5237i q^{7} +8.00000i q^{8} -25.3950 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +5.86472i q^{3} -4.00000 q^{4} +11.7294 q^{6} +11.5237i q^{7} +8.00000i q^{8} -25.3950 q^{9} -23.4589i q^{12} +23.0474 q^{14} +16.0000 q^{16} +50.7900i q^{18} -67.5833 q^{21} -0.836986i q^{23} -46.9178 q^{24} -96.1520i q^{27} -46.0948i q^{28} +13.7455 q^{29} -32.0000i q^{32} +101.580 q^{36} -18.6152 q^{41} +135.167i q^{42} -85.1432i q^{43} -1.67397 q^{46} +58.4378i q^{47} +93.8356i q^{48} -83.7955 q^{49} -192.304 q^{54} -92.1896 q^{56} -27.4910i q^{58} -120.001 q^{61} -292.644i q^{63} -64.0000 q^{64} +116.000i q^{67} +4.90869 q^{69} -203.160i q^{72} +335.350 q^{81} +37.2304i q^{82} +163.843i q^{83} +270.333 q^{84} -170.286 q^{86} +80.6135i q^{87} -58.1977 q^{89} +3.34794i q^{92} +116.876 q^{94} +187.671 q^{96} +167.591i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 16 q^{6} - 76 q^{9}+O(q^{10}) \) 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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Database

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