LMFDB - Embedded newform 1600.3.b.v.1151.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1151,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1600.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.b (of order $2$, degree $1$, 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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.6
Root			$2.65109i$ of defining polynomial
Character	$\chi$	$=$	1600.1151
Dual form			1600.3.b.v.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.30219i q^{3} -0.206625i q^{7} -19.1132 q^{9} +O(q^{10})$	$q+5.30219i q^{3} -0.206625i q^{7} -19.1132 q^{9} +15.0176i q^{11} -11.6999 q^{13} -18.1911 q^{17} +19.3999i q^{19} +1.09556 q^{21} +27.2242i q^{23} -53.6220i q^{27} +44.4175 q^{29} -20.3822i q^{31} -79.6262 q^{33} +18.1089 q^{37} -62.0352i q^{39} -32.3043 q^{41} -4.06244i q^{43} +5.37588i q^{47} +48.9573 q^{49} -96.4527i q^{51} -79.1703 q^{53} -102.862 q^{57} -83.3999i q^{59} +36.7486 q^{61} +3.94925i q^{63} +4.51518i q^{67} -144.348 q^{69} -41.6530i q^{71} -41.5910 q^{73} +3.10301 q^{77} -15.5473i q^{79} +112.295 q^{81} -50.9862i q^{83} +235.510i q^{87} -10.8885 q^{89} +2.41749i q^{91} +108.070 q^{93} -12.1559 q^{97} -287.035i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{9}+O(q^{10}) $ 6 * q - 18 * q^9	$ 6 q - 18 q^{9} - 80 q^{17} - 8 q^{21} + 44 q^{29} - 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} - 400 q^{57} + 100 q^{61} - 184 q^{69} - 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} - 208 q^{97}+O(q^{100}) $ 6 * q - 18 * q^9 - 80 * q^17 - 8 * q^21 + 44 * q^29 - 144 * q^33 + 208 * q^37 - 68 * q^41 + 62 * q^49 + 64 * q^53 - 400 * q^57 + 100 * q^61 - 184 * q^69 - 80 * q^73 + 400 * q^77 + 238 * q^81 - 76 * q^89 + 320 * q^93 - 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	5.30219i	1.76740i	0.468058	+	0.883698i	$0.344954\pi$
$3$	5.30219i	1.76740i	−0.468058	+	0.883698i	$0.655046\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.206625i	− 0.0295178i	−0.999891	−	0.0147589i	$-0.995302\pi$
$7$	− 0.206625i	− 0.0295178i	0.999891	−	0.0147589i	$-0.00469807\pi$
$8$	0	0
$9$	−19.1132	−2.12369
$10$	0	0
$11$	15.0176i	1.36524i	0.730774	+	0.682619i	$0.239159\pi$
$11$	15.0176i	1.36524i	−0.730774	+	0.682619i	$0.760841\pi$
$12$	0	0
$13$	−11.6999	−0.899995	−0.449998	−	0.893030i	$-0.648575\pi$
$13$	−11.6999	−0.899995	−0.449998	+	0.893030i	$0.648575\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−18.1911	−1.07007	−0.535033	−	0.844831i	$-0.679701\pi$
$17$	−18.1911	−1.07007	−0.535033	+	0.844831i	$0.679701\pi$
$18$	0	0
$19$	19.3999i	1.02105i	0.859864	+	0.510523i	$0.170548\pi$
$19$	19.3999i	1.02105i	−0.859864	+	0.510523i	$0.829452\pi$
$20$	0	0
$21$	1.09556	0.0521696
$22$	0	0
$23$	27.2242i	1.18366i	0.806062	+	0.591831i	$0.201595\pi$
$23$	27.2242i	1.18366i	−0.806062	+	0.591831i	$0.798405\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 53.6220i	− 1.98600i
$28$	0	0
$29$	44.4175	1.53164	0.765819	−	0.643056i	$-0.222334\pi$
$29$	44.4175	1.53164	0.765819	+	0.643056i	$0.222334\pi$
$30$	0	0
$31$	− 20.3822i	− 0.657492i	−0.944418	−	0.328746i	$-0.893374\pi$
$31$	− 20.3822i	− 0.657492i	0.944418	−	0.328746i	$-0.106626\pi$
$32$	0	0
$33$	−79.6262	−2.41292
$34$	0	0
$35$	0	0
$36$	0	0
$37$	18.1089	0.489431	0.244715	−	0.969595i	$-0.421305\pi$
$37$	18.1089	0.489431	0.244715	+	0.969595i	$0.421305\pi$
$38$	0	0
$39$	− 62.0352i	− 1.59065i
$40$	0	0
$41$	−32.3043	−0.787910	−0.393955	−	0.919130i	$-0.628893\pi$
$41$	−32.3043	−0.787910	−0.393955	+	0.919130i	$0.628893\pi$
$42$	0	0
$43$	− 4.06244i	− 0.0944753i	−0.998884	−	0.0472377i	$-0.984958\pi$
$43$	− 4.06244i	− 0.0944753i	0.998884	−	0.0472377i	$-0.0150418\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.37588i	0.114380i	0.998363	+	0.0571902i	$0.0182142\pi$
$47$	5.37588i	0.114380i	−0.998363	+	0.0571902i	$0.981786\pi$
$48$	0	0
$49$	48.9573	0.999129
$50$	0	0
$51$	− 96.4527i	− 1.89123i
$52$	0	0
$53$	−79.1703	−1.49378	−0.746890	−	0.664948i	$-0.768454\pi$
$53$	−79.1703	−1.49378	−0.746890	+	0.664948i	$0.768454\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−102.862	−1.80459
$58$	0	0
$59$	− 83.3999i	− 1.41356i	−0.707435	−	0.706779i	$-0.750148\pi$
$59$	− 83.3999i	− 1.41356i	0.707435	−	0.706779i	$-0.249852\pi$
$60$	0	0
$61$	36.7486	0.602435	0.301218	−	0.953555i	$-0.402607\pi$
$61$	36.7486	0.602435	0.301218	+	0.953555i	$0.402607\pi$
$62$	0	0
$63$	3.94925i	0.0626866i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.51518i	0.0673907i	0.999432	+	0.0336954i	$0.0107276\pi$
$67$	4.51518i	0.0673907i	−0.999432	+	0.0336954i	$0.989272\pi$
$68$	0	0
$69$	−144.348	−2.09200
$70$	0	0
$71$	− 41.6530i	− 0.586662i	−0.956011	−	0.293331i	$-0.905236\pi$
$71$	− 41.6530i	− 0.586662i	0.956011	−	0.293331i	$-0.0947638\pi$
$72$	0	0
$73$	−41.5910	−0.569740	−0.284870	−	0.958566i	$-0.591950\pi$
$73$	−41.5910	−0.569740	−0.284870	+	0.958566i	$0.591950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.10301	0.0402988
$78$	0	0
$79$	− 15.5473i	− 0.196801i	−0.995147	−	0.0984004i	$-0.968627\pi$
$79$	− 15.5473i	− 0.196801i	0.995147	−	0.0984004i	$-0.0313726\pi$
$80$	0	0
$81$	112.295	1.38636
$82$	0	0
$83$	− 50.9862i	− 0.614291i	−0.951663	−	0.307146i	$-0.900626\pi$
$83$	− 50.9862i	− 0.614291i	0.951663	−	0.307146i	$-0.0993739\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	235.510i	2.70701i
$88$	0	0
$89$	−10.8885	−0.122343	−0.0611713	−	0.998127i	$-0.519484\pi$
$89$	−10.8885	−0.122343	−0.0611713	+	0.998127i	$0.519484\pi$
$90$	0	0
$91$	2.41749i	0.0265659i
$92$	0	0
$93$	108.070	1.16205
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.1559	−0.125318	−0.0626592	−	0.998035i	$-0.519958\pi$
$97$	−12.1559	−0.125318	−0.0626592	+	0.998035i	$0.519958\pi$
$98$	0	0
$99$	− 287.035i	− 2.89934i
$100$	0	0
$101$	−127.723	−1.26459	−0.632294	−	0.774728i	$-0.717887\pi$
$101$	−127.723	−1.26459	−0.632294	+	0.774728i	$0.717887\pi$
$102$	0	0
$103$	− 4.77575i	− 0.0463665i	−0.999731	−	0.0231833i	$-0.992620\pi$
$103$	− 4.77575i	− 0.0463665i	0.999731	−	0.0231833i	$-0.00738012\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 107.213i	− 1.00199i	−0.865449	−	0.500997i	$-0.832967\pi$
$107$	− 107.213i	− 1.00199i	0.865449	−	0.500997i	$-0.167033\pi$
$108$	0	0
$109$	53.6689	0.492376	0.246188	−	0.969222i	$-0.420822\pi$
$109$	53.6689	0.492376	0.246188	+	0.969222i	$0.420822\pi$
$110$	0	0
$111$	96.0170i	0.865018i
$112$	0	0
$113$	−20.5063	−0.181471	−0.0907356	−	0.995875i	$-0.528922\pi$
$113$	−20.5063	−0.181471	−0.0907356	+	0.995875i	$0.528922\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	223.623	1.91131
$118$	0	0
$119$	3.75873i	0.0315860i
$120$	0	0
$121$	−104.529	−0.863876
$122$	0	0
$123$	− 171.283i	− 1.39255i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 138.477i	− 1.09037i	−0.838316	−	0.545184i	$-0.816460\pi$
$127$	− 138.477i	− 1.09037i	0.838316	−	0.545184i	$-0.183540\pi$
$128$	0	0
$129$	21.5398	0.166975
$130$	0	0
$131$	219.105i	1.67256i	0.548304	+	0.836279i	$0.315274\pi$
$131$	219.105i	1.67256i	−0.548304	+	0.836279i	$0.684726\pi$
$132$	0	0
$133$	4.00849	0.0301390
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−59.7821	−0.436366	−0.218183	−	0.975908i	$-0.570013\pi$
$137$	−59.7821	−0.436366	−0.218183	+	0.975908i	$0.570013\pi$
$138$	0	0
$139$	26.6524i	0.191744i	0.995394	+	0.0958718i	$0.0305639\pi$
$139$	26.6524i	0.191744i	−0.995394	+	0.0958718i	$0.969436\pi$
$140$	0	0
$141$	−28.5039	−0.202155
$142$	0	0
$143$	− 175.705i	− 1.22871i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	259.581i	1.76586i
$148$	0	0
$149$	143.463	0.962838	0.481419	−	0.876490i	$-0.340121\pi$
$149$	143.463	0.962838	0.481419	+	0.876490i	$0.340121\pi$
$150$	0	0
$151$	− 83.4937i	− 0.552939i	−0.961023	−	0.276469i	$-0.910836\pi$
$151$	− 83.4937i	− 0.552939i	0.961023	−	0.276469i	$-0.0891644\pi$
$152$	0	0
$153$	347.690	2.27249
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−169.673	−1.08072	−0.540360	−	0.841434i	$-0.681712\pi$
$157$	−169.673	−1.08072	−0.540360	+	0.841434i	$0.681712\pi$
$158$	0	0
$159$	− 419.776i	− 2.64010i
$160$	0	0
$161$	5.62520	0.0349391
$162$	0	0
$163$	275.478i	1.69005i	0.534726	+	0.845026i	$0.320415\pi$
$163$	275.478i	1.69005i	−0.534726	+	0.845026i	$0.679585\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	132.481i	0.793299i	0.917970	+	0.396650i	$0.129827\pi$
$167$	132.481i	0.793299i	−0.917970	+	0.396650i	$0.870173\pi$
$168$	0	0
$169$	−32.1115	−0.190009
$170$	0	0
$171$	− 370.793i	− 2.16838i
$172$	0	0
$173$	272.614	1.57580	0.787901	−	0.615801i	$-0.211168\pi$
$173$	272.614	1.57580	0.787901	+	0.615801i	$0.211168\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	442.202	2.49831
$178$	0	0
$179$	157.523i	0.880014i	0.897994	+	0.440007i	$0.145024\pi$
$179$	157.523i	0.880014i	−0.897994	+	0.440007i	$0.854976\pi$
$180$	0	0
$181$	335.063	1.85118	0.925590	−	0.378529i	$-0.123570\pi$
$181$	335.063	1.85118	0.925590	+	0.378529i	$0.123570\pi$
$182$	0	0
$183$	194.848i	1.06474i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 273.187i	− 1.46090i
$188$	0	0
$189$	−11.0796	−0.0586223
$190$	0	0
$191$	− 298.575i	− 1.56322i	−0.623766	−	0.781611i	$-0.714398\pi$
$191$	− 298.575i	− 1.56322i	0.623766	−	0.781611i	$-0.285602\pi$
$192$	0	0
$193$	191.915	0.994376	0.497188	−	0.867643i	$-0.334366\pi$
$193$	191.915	0.994376	0.497188	+	0.867643i	$0.334366\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−59.2472	−0.300747	−0.150374	−	0.988629i	$-0.548048\pi$
$197$	−59.2472	−0.300747	−0.150374	+	0.988629i	$0.548048\pi$
$198$	0	0
$199$	− 309.100i	− 1.55327i	−0.629953	−	0.776633i	$-0.716926\pi$
$199$	− 309.100i	− 1.55327i	0.629953	−	0.776633i	$-0.283074\pi$
$200$	0	0
$201$	−23.9403	−0.119106
$202$	0	0
$203$	− 9.17775i	− 0.0452106i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 520.342i	− 2.51373i
$208$	0	0
$209$	−291.340	−1.39397
$210$	0	0
$211$	− 205.693i	− 0.974850i	−0.873165	−	0.487425i	$-0.837936\pi$
$211$	− 205.693i	− 0.974850i	0.873165	−	0.487425i	$-0.162064\pi$
$212$	0	0
$213$	220.852	1.03686
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.21147	−0.0194077
$218$	0	0
$219$	− 220.523i	− 1.00696i
$220$	0	0
$221$	212.835	0.963054
$222$	0	0
$223$	228.723i	1.02567i	0.858488	+	0.512833i	$0.171404\pi$
$223$	228.723i	1.02567i	−0.858488	+	0.512833i	$0.828596\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	282.403i	1.24407i	0.782991	+	0.622033i	$0.213693\pi$
$227$	282.403i	1.24407i	−0.782991	+	0.622033i	$0.786307\pi$
$228$	0	0
$229$	49.2525	0.215076	0.107538	−	0.994201i	$-0.465703\pi$
$229$	49.2525	0.215076	0.107538	+	0.994201i	$0.465703\pi$
$230$	0	0
$231$	16.4527i	0.0712240i
$232$	0	0
$233$	−124.273	−0.533363	−0.266681	−	0.963785i	$-0.585927\pi$
$233$	−124.273	−0.533363	−0.266681	+	0.963785i	$0.585927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	82.4345	0.347825
$238$	0	0
$239$	80.4527i	0.336622i	0.985734	+	0.168311i	$0.0538313\pi$
$239$	80.4527i	0.336622i	−0.985734	+	0.168311i	$0.946169\pi$
$240$	0	0
$241$	−1.20979	−0.00501988	−0.00250994	−	0.999997i	$-0.500799\pi$
$241$	−1.20979	−0.00501988	−0.00250994	+	0.999997i	$0.500799\pi$
$242$	0	0
$243$	112.812i	0.464247i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 226.977i	− 0.918936i
$248$	0	0
$249$	270.338	1.08570
$250$	0	0
$251$	211.853i	0.844034i	0.906588	+	0.422017i	$0.138678\pi$
$251$	211.853i	0.844034i	−0.906588	+	0.422017i	$0.861322\pi$
$252$	0	0
$253$	−408.843	−1.61598
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−182.646	−0.710685	−0.355342	−	0.934736i	$-0.615636\pi$
$257$	−182.646	−0.710685	−0.355342	+	0.934736i	$0.615636\pi$
$258$	0	0
$259$	− 3.74175i	− 0.0144469i
$260$	0	0
$261$	−848.960	−3.25272
$262$	0	0
$263$	− 74.6636i	− 0.283892i	−0.989874	−	0.141946i	$-0.954664\pi$
$263$	− 74.6636i	− 0.283892i	0.989874	−	0.141946i	$-0.0453359\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 57.7329i	− 0.216228i
$268$	0	0
$269$	−184.089	−0.684344	−0.342172	−	0.939637i	$-0.611163\pi$
$269$	−184.089	−0.684344	−0.342172	+	0.939637i	$0.611163\pi$
$270$	0	0
$271$	− 234.746i	− 0.866222i	−0.901341	−	0.433111i	$-0.857416\pi$
$271$	− 234.746i	− 0.866222i	0.901341	−	0.433111i	$-0.142584\pi$
$272$	0	0
$273$	−12.8180	−0.0469524
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−452.208	−1.63252	−0.816260	−	0.577684i	$-0.803957\pi$
$277$	−452.208	−1.63252	−0.816260	+	0.577684i	$0.803957\pi$
$278$	0	0
$279$	389.570i	1.39631i
$280$	0	0
$281$	−196.110	−0.697900	−0.348950	−	0.937141i	$-0.613462\pi$
$281$	−196.110	−0.697900	−0.348950	+	0.937141i	$0.613462\pi$
$282$	0	0
$283$	418.449i	1.47862i	0.673366	+	0.739309i	$0.264848\pi$
$283$	418.449i	1.47862i	−0.673366	+	0.739309i	$0.735152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.67486i	0.0232574i
$288$	0	0
$289$	41.9170	0.145042
$290$	0	0
$291$	− 64.4527i	− 0.221487i
$292$	0	0
$293$	−286.666	−0.978383	−0.489191	−	0.872176i	$-0.662708\pi$
$293$	−286.666	−0.978383	−0.489191	+	0.872176i	$0.662708\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	805.275	2.71136
$298$	0	0
$299$	− 318.522i	− 1.06529i
$300$	0	0
$301$	−0.839400	−0.00278870
$302$	0	0
$303$	− 677.214i	− 2.23503i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	261.715i	0.852493i	0.904607	+	0.426247i	$0.140164\pi$
$307$	261.715i	0.852493i	−0.904607	+	0.426247i	$0.859836\pi$
$308$	0	0
$309$	25.3219	0.0819480
$310$	0	0
$311$	578.904i	1.86143i	0.365747	+	0.930714i	$0.380813\pi$
$311$	578.904i	1.86143i	−0.365747	+	0.930714i	$0.619187\pi$
$312$	0	0
$313$	99.3124	0.317292	0.158646	−	0.987336i	$-0.449287\pi$
$313$	99.3124	0.317292	0.158646	+	0.987336i	$0.449287\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−191.623	−0.604489	−0.302245	−	0.953230i	$-0.597736\pi$
$317$	−191.623	−0.604489	−0.302245	+	0.953230i	$0.597736\pi$
$318$	0	0
$319$	667.045i	2.09105i
$320$	0	0
$321$	568.466	1.77092
$322$	0	0
$323$	− 352.905i	− 1.09259i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	284.563i	0.870222i
$328$	0	0
$329$	1.11079	0.00337626
$330$	0	0
$331$	530.187i	1.60177i	0.598816	+	0.800886i	$0.295638\pi$
$331$	530.187i	1.60177i	−0.598816	+	0.800886i	$0.704362\pi$
$332$	0	0
$333$	−346.120	−1.03940
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−487.427	−1.44637	−0.723186	−	0.690653i	$-0.757323\pi$
$337$	−487.427	−1.44637	−0.723186	+	0.690653i	$0.757323\pi$
$338$	0	0
$339$	− 108.728i	− 0.320731i
$340$	0	0
$341$	306.093	0.897633
$342$	0	0
$343$	− 20.2404i	− 0.0590099i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	310.497i	0.894804i	0.894333	+	0.447402i	$0.147651\pi$
$347$	310.497i	0.894804i	−0.894333	+	0.447402i	$0.852349\pi$
$348$	0	0
$349$	−253.004	−0.724941	−0.362471	−	0.931995i	$-0.618067\pi$
$349$	−253.004	−0.724941	−0.362471	+	0.931995i	$0.618067\pi$
$350$	0	0
$351$	627.374i	1.78739i
$352$	0	0
$353$	−322.639	−0.913992	−0.456996	−	0.889469i	$-0.651075\pi$
$353$	−322.639	−0.913992	−0.456996	+	0.889469i	$0.651075\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−19.9295	−0.0558250
$358$	0	0
$359$	− 254.975i	− 0.710236i	−0.934822	−	0.355118i	$-0.884441\pi$
$359$	− 254.975i	− 0.710236i	0.934822	−	0.355118i	$-0.115559\pi$
$360$	0	0
$361$	−15.3550	−0.0425347
$362$	0	0
$363$	− 554.232i	− 1.52681i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 207.935i	− 0.566581i	−0.959034	−	0.283291i	$-0.908574\pi$
$367$	− 207.935i	− 0.566581i	0.959034	−	0.283291i	$-0.0914261\pi$
$368$	0	0
$369$	617.438	1.67327
$370$	0	0
$371$	16.3585i	0.0440931i
$372$	0	0
$373$	−203.826	−0.546450	−0.273225	−	0.961950i	$-0.588090\pi$
$373$	−203.826	−0.546450	−0.273225	+	0.961950i	$0.588090\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−519.682	−1.37847
$378$	0	0
$379$	454.099i	1.19815i	0.800692	+	0.599076i	$0.204465\pi$
$379$	454.099i	1.19815i	−0.800692	+	0.599076i	$0.795535\pi$
$380$	0	0
$381$	734.229	1.92711
$382$	0	0
$383$	541.569i	1.41402i	0.707205	+	0.707009i	$0.249956\pi$
$383$	541.569i	1.41402i	−0.707205	+	0.707009i	$0.750044\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	77.6462i	0.200636i
$388$	0	0
$389$	−423.431	−1.08851	−0.544256	−	0.838919i	$-0.683188\pi$
$389$	−423.431	−1.08851	−0.544256	+	0.838919i	$0.683188\pi$
$390$	0	0
$391$	− 495.240i	− 1.26660i
$392$	0	0
$393$	−1161.74	−2.95607
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.7772	0.0296654	0.0148327	−	0.999890i	$-0.495278\pi$
$397$	11.7772	0.0296654	0.0148327	+	0.999890i	$0.495278\pi$
$398$	0	0
$399$	21.2538i	0.0532676i
$400$	0	0
$401$	127.442	0.317809	0.158905	−	0.987294i	$-0.449204\pi$
$401$	127.442	0.317809	0.158905	+	0.987294i	$0.449204\pi$
$402$	0	0
$403$	238.471i	0.591739i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	271.953i	0.668190i
$408$	0	0
$409$	−608.012	−1.48658	−0.743290	−	0.668969i	$-0.766736\pi$
$409$	−608.012	−1.48658	−0.743290	+	0.668969i	$0.766736\pi$
$410$	0	0
$411$	− 316.976i	− 0.771231i
$412$	0	0
$413$	−17.2325	−0.0417251
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−141.316	−0.338887
$418$	0	0
$419$	565.630i	1.34995i	0.737840	+	0.674976i	$0.235846\pi$
$419$	565.630i	1.34995i	−0.737840	+	0.674976i	$0.764154\pi$
$420$	0	0
$421$	−711.356	−1.68968	−0.844841	−	0.535018i	$-0.820305\pi$
$421$	−711.356	−1.68968	−0.844841	+	0.535018i	$0.820305\pi$
$422$	0	0
$423$	− 102.750i	− 0.242908i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.59316i	− 0.0177826i
$428$	0	0
$429$	931.622	2.17161
$430$	0	0
$431$	309.254i	0.717526i	0.933429	+	0.358763i	$0.116801\pi$
$431$	309.254i	0.717526i	−0.933429	+	0.358763i	$0.883199\pi$
$432$	0	0
$433$	187.374	0.432735	0.216368	−	0.976312i	$-0.430579\pi$
$433$	187.374	0.432735	0.216368	+	0.976312i	$0.430579\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−528.147	−1.20857
$438$	0	0
$439$	− 289.657i	− 0.659811i	−0.944014	−	0.329906i	$-0.892983\pi$
$439$	− 289.657i	− 0.659811i	0.944014	−	0.329906i	$-0.107017\pi$
$440$	0	0
$441$	−935.730	−2.12184
$442$	0	0
$443$	− 295.516i	− 0.667079i	−0.942736	−	0.333539i	$-0.891757\pi$
$443$	− 295.516i	− 0.667079i	0.942736	−	0.333539i	$-0.108243\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	760.667i	1.70172i
$448$	0	0
$449$	604.409	1.34612	0.673061	−	0.739587i	$-0.264979\pi$
$449$	604.409	1.34612	0.673061	+	0.739587i	$0.264979\pi$
$450$	0	0
$451$	− 485.134i	− 1.07569i
$452$	0	0
$453$	442.699	0.977262
$454$	0	0
$455$	0	0
$456$	0	0
$457$	392.507	0.858877	0.429438	−	0.903096i	$-0.358712\pi$
$457$	392.507	0.858877	0.429438	+	0.903096i	$0.358712\pi$
$458$	0	0
$459$	975.444i	2.12515i
$460$	0	0
$461$	400.277	0.868279	0.434139	−	0.900846i	$-0.357053\pi$
$461$	400.277	0.868279	0.434139	+	0.900846i	$0.357053\pi$
$462$	0	0
$463$	− 732.679i	− 1.58246i	−0.611518	−	0.791230i	$-0.709441\pi$
$463$	− 732.679i	− 1.58246i	0.611518	−	0.791230i	$-0.290559\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 592.126i	− 1.26794i	−0.773360	−	0.633968i	$-0.781425\pi$
$467$	− 592.126i	− 1.26794i	0.773360	−	0.633968i	$-0.218575\pi$
$468$	0	0
$469$	0.932947	0.00198923
$470$	0	0
$471$	− 899.639i	− 1.91006i
$472$	0	0
$473$	61.0082	0.128981
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1513.20	3.17232
$478$	0	0
$479$	− 309.151i	− 0.645409i	−0.946500	−	0.322705i	$-0.895408\pi$
$479$	− 309.151i	− 0.645409i	0.946500	−	0.322705i	$-0.104592\pi$
$480$	0	0
$481$	−211.873	−0.440485
$482$	0	0
$483$	29.8259i	0.0617513i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 570.769i	− 1.17201i	−0.810307	−	0.586005i	$-0.800700\pi$
$487$	− 570.769i	− 1.17201i	0.810307	−	0.586005i	$-0.199300\pi$
$488$	0	0
$489$	−1460.64	−2.98699
$490$	0	0
$491$	301.659i	0.614378i	0.951649	+	0.307189i	$0.0993883\pi$
$491$	301.659i	0.614378i	−0.951649	+	0.307189i	$0.900612\pi$
$492$	0	0
$493$	−808.004	−1.63895
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.60653	−0.0173170
$498$	0	0
$499$	− 517.758i	− 1.03759i	−0.854898	−	0.518795i	$-0.826381\pi$
$499$	− 517.758i	− 1.03759i	0.854898	−	0.518795i	$-0.173619\pi$
$500$	0	0
$501$	−702.439	−1.40207
$502$	0	0
$503$	406.671i	0.808491i	0.914650	+	0.404246i	$0.132466\pi$
$503$	406.671i	0.808491i	−0.914650	+	0.404246i	$0.867534\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 170.261i	− 0.335821i
$508$	0	0
$509$	627.097	1.23202	0.616009	−	0.787739i	$-0.288748\pi$
$509$	627.097	1.23202	0.616009	+	0.787739i	$0.288748\pi$
$510$	0	0
$511$	8.59372i	0.0168175i
$512$	0	0
$513$	1040.26	2.02780
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−80.7330	−0.156157
$518$	0	0
$519$	1445.45i	2.78507i
$520$	0	0
$521$	−111.743	−0.214478	−0.107239	−	0.994233i	$-0.534201\pi$
$521$	−111.743	−0.214478	−0.107239	+	0.994233i	$0.534201\pi$
$522$	0	0
$523$	769.813i	1.47192i	0.677027	+	0.735959i	$0.263268\pi$
$523$	769.813i	1.47192i	−0.677027	+	0.735959i	$0.736732\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	370.776i	0.703560i
$528$	0	0
$529$	−212.160	−0.401058
$530$	0	0
$531$	1594.04i	3.00195i
$532$	0	0
$533$	377.958	0.709115
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−835.214	−1.55533
$538$	0	0
$539$	735.222i	1.36405i
$540$	0	0
$541$	225.558	0.416927	0.208463	−	0.978030i	$-0.433154\pi$
$541$	225.558	0.416927	0.208463	+	0.978030i	$0.433154\pi$
$542$	0	0
$543$	1776.57i	3.27177i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 882.346i	− 1.61306i	−0.591190	−	0.806532i	$-0.701342\pi$
$547$	− 882.346i	− 1.61306i	0.591190	−	0.806532i	$-0.298658\pi$
$548$	0	0
$549$	−702.382	−1.27938
$550$	0	0
$551$	861.694i	1.56387i
$552$	0	0
$553$	−3.21245	−0.00580913
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−303.119	−0.544199	−0.272100	−	0.962269i	$-0.587718\pi$
$557$	−303.119	−0.544199	−0.272100	+	0.962269i	$0.587718\pi$
$558$	0	0
$559$	47.5303i	0.0850273i
$560$	0	0
$561$	1448.49	2.58198
$562$	0	0
$563$	− 344.003i	− 0.611017i	−0.952189	−	0.305508i	$-0.901174\pi$
$563$	− 344.003i	− 0.611017i	0.952189	−	0.305508i	$-0.0988264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 23.2029i	− 0.0409223i
$568$	0	0
$569$	228.925	0.402329	0.201165	−	0.979557i	$-0.435527\pi$
$569$	228.925	0.402329	0.201165	+	0.979557i	$0.435527\pi$
$570$	0	0
$571$	− 371.169i	− 0.650033i	−0.945708	−	0.325017i	$-0.894630\pi$
$571$	− 371.169i	− 0.650033i	0.945708	−	0.325017i	$-0.105370\pi$
$572$	0	0
$573$	1583.10	2.76283
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−580.289	−1.00570	−0.502850	−	0.864374i	$-0.667715\pi$
$577$	−580.289	−1.00570	−0.502850	+	0.864374i	$0.667715\pi$
$578$	0	0
$579$	1017.57i	1.75746i
$580$	0	0
$581$	−10.5350	−0.0181325
$582$	0	0
$583$	− 1188.95i	− 2.03936i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 65.0801i	− 0.110869i	−0.998462	−	0.0554345i	$-0.982346\pi$
$587$	− 65.0801i	− 0.110869i	0.998462	−	0.0554345i	$-0.0176544\pi$
$588$	0	0
$589$	395.413	0.671329
$590$	0	0
$591$	− 314.140i	− 0.531539i
$592$	0	0
$593$	−1002.90	−1.69124	−0.845619	−	0.533787i	$-0.820768\pi$
$593$	−1002.90	−1.69124	−0.845619	+	0.533787i	$0.820768\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1638.91	2.74524
$598$	0	0
$599$	888.567i	1.48342i	0.670722	+	0.741709i	$0.265984\pi$
$599$	888.567i	1.48342i	−0.670722	+	0.741709i	$0.734016\pi$
$600$	0	0
$601$	132.065	0.219742	0.109871	−	0.993946i	$-0.464956\pi$
$601$	132.065	0.219742	0.109871	+	0.993946i	$0.464956\pi$
$602$	0	0
$603$	− 86.2995i	− 0.143117i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	700.090i	1.15336i	0.816970	+	0.576680i	$0.195652\pi$
$607$	700.090i	1.15336i	−0.816970	+	0.576680i	$0.804348\pi$
$608$	0	0
$609$	48.6621	0.0799050
$610$	0	0
$611$	− 62.8975i	− 0.102942i
$612$	0	0
$613$	727.420	1.18666	0.593328	−	0.804961i	$-0.297814\pi$
$613$	727.420	1.18666	0.593328	+	0.804961i	$0.297814\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	99.7137	0.161611	0.0808053	−	0.996730i	$-0.474251\pi$
$617$	99.7137	0.161611	0.0808053	+	0.996730i	$0.474251\pi$
$618$	0	0
$619$	− 721.349i	− 1.16535i	−0.812707	−	0.582673i	$-0.802007\pi$
$619$	− 721.349i	− 1.16535i	0.812707	−	0.582673i	$-0.197993\pi$
$620$	0	0
$621$	1459.82	2.35075
$622$	0	0
$623$	2.24983i	0.00361129i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1544.74i	− 2.46370i
$628$	0	0
$629$	−329.422	−0.523723
$630$	0	0
$631$	796.856i	1.26285i	0.775438	+	0.631423i	$0.217529\pi$
$631$	796.856i	1.26285i	−0.775438	+	0.631423i	$0.782471\pi$
$632$	0	0
$633$	1090.62	1.72295
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−572.797	−0.899211
$638$	0	0
$639$	796.121i	1.24589i
$640$	0	0
$641$	205.013	0.319833	0.159917	−	0.987131i	$-0.448877\pi$
$641$	205.013	0.319833	0.159917	+	0.987131i	$0.448877\pi$
$642$	0	0
$643$	− 495.044i	− 0.769897i	−0.922938	−	0.384949i	$-0.874219\pi$
$643$	− 495.044i	− 0.769897i	0.922938	−	0.384949i	$-0.125781\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1121.84i	− 1.73391i	−0.498389	−	0.866953i	$-0.666075\pi$
$647$	− 1121.84i	− 1.73391i	0.498389	−	0.866953i	$-0.333925\pi$
$648$	0	0
$649$	1252.47	1.92984
$650$	0	0
$651$	− 22.3300i	− 0.0343011i
$652$	0	0
$653$	622.987	0.954038	0.477019	−	0.878893i	$-0.341717\pi$
$653$	622.987	0.954038	0.477019	+	0.878893i	$0.341717\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	794.936	1.20995
$658$	0	0
$659$	264.030i	0.400653i	0.979729	+	0.200326i	$0.0642002\pi$
$659$	264.030i	0.400653i	−0.979729	+	0.200326i	$0.935800\pi$
$660$	0	0
$661$	−1285.15	−1.94425	−0.972124	−	0.234469i	$-0.924665\pi$
$661$	−1285.15	−1.94425	−0.972124	+	0.234469i	$0.924665\pi$
$662$	0	0
$663$	1128.49i	1.70210i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1209.23i	1.81294i
$668$	0	0
$669$	−1212.73	−1.81276
$670$	0	0
$671$	551.876i	0.822468i
$672$	0	0
$673$	1244.16	1.84868	0.924342	−	0.381565i	$-0.124615\pi$
$673$	1244.16	1.84868	0.924342	+	0.381565i	$0.124615\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	963.335	1.42295	0.711473	−	0.702713i	$-0.248028\pi$
$677$	963.335	1.42295	0.711473	+	0.702713i	$0.248028\pi$
$678$	0	0
$679$	2.51170i	0.00369912i
$680$	0	0
$681$	−1497.35	−2.19876
$682$	0	0
$683$	770.819i	1.12858i	0.825577	+	0.564289i	$0.190850\pi$
$683$	770.819i	1.12858i	−0.825577	+	0.564289i	$0.809150\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	261.146i	0.380125i
$688$	0	0
$689$	926.287	1.34439
$690$	0	0
$691$	− 408.765i	− 0.591555i	−0.955257	−	0.295778i	$-0.904421\pi$
$691$	− 408.765i	− 0.591555i	0.955257	−	0.295778i	$-0.0955788\pi$
$692$	0	0
$693$	−59.3084	−0.0855821
$694$	0	0
$695$	0	0
$696$	0	0
$697$	587.652	0.843116
$698$	0	0
$699$	− 658.921i	− 0.942663i
$700$	0	0
$701$	−1335.62	−1.90530	−0.952650	−	0.304068i	$-0.901655\pi$
$701$	−1335.62	−1.90530	−0.952650	+	0.304068i	$0.901655\pi$
$702$	0	0
$703$	351.311i	0.499731i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.3908i	0.0373279i
$708$	0	0
$709$	362.956	0.511927	0.255964	−	0.966686i	$-0.417607\pi$
$709$	362.956	0.511927	0.255964	+	0.966686i	$0.417607\pi$
$710$	0	0
$711$	297.158i	0.417943i
$712$	0	0
$713$	554.891	0.778249
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−426.575	−0.594945
$718$	0	0
$719$	648.098i	0.901388i	0.892679	+	0.450694i	$0.148823\pi$
$719$	648.098i	0.901388i	−0.892679	+	0.450694i	$0.851177\pi$
$720$	0	0
$721$	−0.986788	−0.00136864
$722$	0	0
$723$	− 6.41453i	− 0.00887211i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	431.123i	0.593017i	0.955030	+	0.296508i	$0.0958222\pi$
$727$	431.123i	0.593017i	−0.955030	+	0.296508i	$0.904178\pi$
$728$	0	0
$729$	412.506	0.565852
$730$	0	0
$731$	73.9003i	0.101095i
$732$	0	0
$733$	−1464.87	−1.99846	−0.999231	−	0.0392126i	$-0.987515\pi$
$733$	−1464.87	−1.99846	−0.999231	+	0.0392126i	$0.987515\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−67.8073	−0.0920044
$738$	0	0
$739$	99.1951i	0.134229i	0.997745	+	0.0671144i	$0.0213793\pi$
$739$	99.1951i	0.134229i	−0.997745	+	0.0671144i	$0.978621\pi$
$740$	0	0
$741$	1203.48	1.62412
$742$	0	0
$743$	− 602.719i	− 0.811196i	−0.914052	−	0.405598i	$-0.867063\pi$
$743$	− 602.719i	− 0.811196i	0.914052	−	0.405598i	$-0.132937\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	974.508i	1.30456i
$748$	0	0
$749$	−22.1529	−0.0295767
$750$	0	0
$751$	− 541.472i	− 0.721001i	−0.932759	−	0.360501i	$-0.882606\pi$
$751$	− 541.472i	− 0.721001i	0.932759	−	0.360501i	$-0.117394\pi$
$752$	0	0
$753$	−1123.28	−1.49174
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1092.79	−1.44358	−0.721789	−	0.692113i	$-0.756680\pi$
$757$	−1092.79	−1.44358	−0.721789	+	0.692113i	$0.756680\pi$
$758$	0	0
$759$	− 2167.76i	− 2.85608i
$760$	0	0
$761$	−18.7706	−0.0246657	−0.0123328	−	0.999924i	$-0.503926\pi$
$761$	−18.7706	−0.0246657	−0.0123328	+	0.999924i	$0.503926\pi$
$762$	0	0
$763$	− 11.0893i	− 0.0145338i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	975.773i	1.27219i
$768$	0	0
$769$	−39.0830	−0.0508231	−0.0254116	−	0.999677i	$-0.508090\pi$
$769$	−39.0830	−0.0508231	−0.0254116	+	0.999677i	$0.508090\pi$
$770$	0	0
$771$	− 968.423i	− 1.25606i
$772$	0	0
$773$	−31.2171	−0.0403843	−0.0201922	−	0.999796i	$-0.506428\pi$
$773$	−31.2171	−0.0403843	−0.0201922	+	0.999796i	$0.506428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	19.8395	0.0255334
$778$	0	0
$779$	− 626.699i	− 0.804492i
$780$	0	0
$781$	625.529	0.800933
$782$	0	0
$783$	− 2381.75i	− 3.04183i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	988.563i	1.25612i	0.778167	+	0.628058i	$0.216150\pi$
$787$	988.563i	1.25612i	−0.778167	+	0.628058i	$0.783850\pi$
$788$	0	0
$789$	395.880	0.501750
$790$	0	0
$791$	4.23710i	0.00535663i
$792$	0	0
$793$	−429.956	−0.542189
$794$	0	0
$795$	0	0
$796$	0	0
$797$	118.920	0.149210	0.0746048	−	0.997213i	$-0.476230\pi$
$797$	118.920	0.149210	0.0746048	+	0.997213i	$0.476230\pi$
$798$	0	0
$799$	− 97.7933i	− 0.122395i
$800$	0	0
$801$	208.114	0.259818
$802$	0	0
$803$	− 624.598i	− 0.777831i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 976.072i	− 1.20951i
$808$	0	0
$809$	−1214.04	−1.50067	−0.750337	−	0.661056i	$-0.770109\pi$
$809$	−1214.04	−1.50067	−0.750337	+	0.661056i	$0.770109\pi$
$810$	0	0
$811$	− 706.666i	− 0.871351i	−0.900104	−	0.435676i	$-0.856509\pi$
$811$	− 706.666i	− 0.871351i	0.900104	−	0.435676i	$-0.143491\pi$
$812$	0	0
$813$	1244.67	1.53096
$814$	0	0
$815$	0	0
$816$	0	0
$817$	78.8108	0.0964636
$818$	0	0
$819$	− 46.2060i	− 0.0564176i
$820$	0	0
$821$	−991.775	−1.20801	−0.604004	−	0.796981i	$-0.706429\pi$
$821$	−991.775	−1.20801	−0.604004	+	0.796981i	$0.706429\pi$
$822$	0	0
$823$	− 523.998i	− 0.636693i	−0.947974	−	0.318347i	$-0.896872\pi$
$823$	− 523.998i	− 0.636693i	0.947974	−	0.318347i	$-0.103128\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 318.794i	− 0.385483i	−0.981250	−	0.192741i	$-0.938262\pi$
$827$	− 318.794i	− 0.385483i	0.981250	−	0.192741i	$-0.0617379\pi$
$828$	0	0
$829$	−1034.11	−1.24742	−0.623711	−	0.781655i	$-0.714376\pi$
$829$	−1034.11	−1.24742	−0.623711	+	0.781655i	$0.714376\pi$
$830$	0	0
$831$	− 2397.69i	− 2.88531i
$832$	0	0
$833$	−890.588	−1.06913
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1092.94	−1.30578
$838$	0	0
$839$	445.284i	0.530732i	0.964148	+	0.265366i	$0.0854928\pi$
$839$	445.284i	0.530732i	−0.964148	+	0.265366i	$0.914507\pi$
$840$	0	0
$841$	1131.91	1.34591
$842$	0	0
$843$	− 1039.81i	− 1.23346i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.5983i	0.0254997i
$848$	0	0
$849$	−2218.69	−2.61330
$850$	0	0
$851$	493.002i	0.579321i
$852$	0	0
$853$	−1200.49	−1.40737	−0.703684	−	0.710513i	$-0.748463\pi$
$853$	−1200.49	−1.40737	−0.703684	+	0.710513i	$0.748463\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1223.15	−1.42724	−0.713622	−	0.700531i	$-0.752946\pi$
$857$	−1223.15	−1.42724	−0.713622	+	0.700531i	$0.752946\pi$
$858$	0	0
$859$	210.033i	0.244509i	0.992499	+	0.122254i	$0.0390124\pi$
$859$	210.033i	0.244509i	−0.992499	+	0.122254i	$0.960988\pi$
$860$	0	0
$861$	−35.3914	−0.0411050
$862$	0	0
$863$	1560.47i	1.80819i	0.427333	+	0.904095i	$0.359453\pi$
$863$	1560.47i	1.80819i	−0.427333	+	0.904095i	$0.640547\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	222.252i	0.256346i
$868$	0	0
$869$	233.483	0.268680
$870$	0	0
$871$	− 52.8273i	− 0.0606513i
$872$	0	0
$873$	232.338	0.266137
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1011.66	1.15354	0.576771	−	0.816906i	$-0.304313\pi$
$877$	1011.66	1.15354	0.576771	+	0.816906i	$0.304313\pi$
$878$	0	0
$879$	− 1519.96i	− 1.72919i
$880$	0	0
$881$	266.455	0.302446	0.151223	−	0.988500i	$-0.451679\pi$
$881$	266.455	0.302446	0.151223	+	0.988500i	$0.451679\pi$
$882$	0	0
$883$	− 469.871i	− 0.532130i	−0.963955	−	0.266065i	$-0.914276\pi$
$883$	− 469.871i	− 0.532130i	0.963955	−	0.266065i	$-0.0857237\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 764.559i	− 0.861961i	−0.902361	−	0.430980i	$-0.858168\pi$
$887$	− 764.559i	− 0.861961i	0.902361	−	0.430980i	$-0.141832\pi$
$888$	0	0
$889$	−28.6127	−0.0321853
$890$	0	0
$891$	1686.41i	1.89271i
$892$	0	0
$893$	−104.291	−0.116788
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1688.86	1.88279
$898$	0	0
$899$	− 905.328i	− 1.00704i
$900$	0	0
$901$	1440.20	1.59844
$902$	0	0
$903$	− 4.45065i	− 0.00492874i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1333.20i	1.46990i	0.678123	+	0.734948i	$0.262794\pi$
$907$	1333.20i	1.46990i	−0.678123	+	0.734948i	$0.737206\pi$
$908$	0	0
$909$	2441.20	2.68559
$910$	0	0
$911$	1496.11i	1.64227i	0.570736	+	0.821134i	$0.306658\pi$
$911$	1496.11i	1.64227i	−0.570736	+	0.821134i	$0.693342\pi$
$912$	0	0
$913$	765.691	0.838654
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45.2725	0.0493702
$918$	0	0
$919$	− 564.228i	− 0.613959i	−0.951716	−	0.306980i	$-0.900682\pi$
$919$	− 564.228i	− 0.613959i	0.951716	−	0.306980i	$-0.0993183\pi$
$920$	0	0
$921$	−1387.66	−1.50669
$922$	0	0
$923$	487.337i	0.527993i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	91.2798i	0.0984680i
$928$	0	0
$929$	−1449.16	−1.55991	−0.779957	−	0.625833i	$-0.784759\pi$
$929$	−1449.16	−1.55991	−0.779957	+	0.625833i	$0.784759\pi$
$930$	0	0
$931$	949.765i	1.02016i
$932$	0	0
$933$	−3069.46	−3.28988
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−258.795	−0.276196	−0.138098	−	0.990419i	$-0.544099\pi$
$937$	−258.795	−0.276196	−0.138098	+	0.990419i	$0.544099\pi$
$938$	0	0
$939$	526.573i	0.560781i
$940$	0	0
$941$	−20.3444	−0.0216200	−0.0108100	−	0.999942i	$-0.503441\pi$
$941$	−20.3444	−0.0216200	−0.0108100	+	0.999942i	$0.503441\pi$
$942$	0	0
$943$	− 879.461i	− 0.932620i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 613.539i	− 0.647876i	−0.946078	−	0.323938i	$-0.894993\pi$
$947$	− 613.539i	− 0.647876i	0.946078	−	0.323938i	$-0.105007\pi$
$948$	0	0
$949$	486.612	0.512763
$950$	0	0
$951$	− 1016.02i	− 1.06837i
$952$	0	0
$953$	81.6126	0.0856376	0.0428188	−	0.999083i	$-0.486366\pi$
$953$	81.6126	0.0856376	0.0428188	+	0.999083i	$0.486366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3536.80	−3.69571
$958$	0	0
$959$	12.3525i	0.0128806i
$960$	0	0
$961$	545.564	0.567704
$962$	0	0
$963$	2049.19i	2.12792i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 127.482i	− 0.131833i	−0.997825	−	0.0659165i	$-0.979003\pi$
$967$	− 127.482i	− 0.131833i	0.997825	−	0.0659165i	$-0.0209971\pi$
$968$	0	0
$969$	1871.17	1.93103
$970$	0	0
$971$	1122.62i	1.15615i	0.815983	+	0.578076i	$0.196196\pi$
$971$	1122.62i	1.15615i	−0.815983	+	0.578076i	$0.803804\pi$
$972$	0	0
$973$	5.50703	0.00565985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	424.837	0.434838	0.217419	−	0.976078i	$-0.430236\pi$
$977$	424.837	0.434838	0.217419	+	0.976078i	$0.430236\pi$
$978$	0	0
$979$	− 163.519i	− 0.167027i
$980$	0	0
$981$	−1025.78	−1.04565
$982$	0	0
$983$	− 663.324i	− 0.674795i	−0.941362	−	0.337398i	$-0.890453\pi$
$983$	− 663.324i	− 0.674795i	0.941362	−	0.337398i	$-0.109547\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.88961i	0.00596719i
$988$	0	0
$989$	110.597	0.111827
$990$	0	0
$991$	1771.36i	1.78745i	0.448616	+	0.893724i	$0.351917\pi$
$991$	1771.36i	1.78745i	−0.448616	+	0.893724i	$0.648083\pi$
$992$	0	0
$993$	−2811.15	−2.83097
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−447.066	−0.448411	−0.224205	−	0.974542i	$-0.571979\pi$
$997$	−447.066	−0.448411	−0.224205	+	0.974542i	$0.571979\pi$
$998$	0	0
$999$	− 971.038i	− 0.972010i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.b.v.1151.6		6
4.3	odd	2	inner	1600.3.b.v.1151.1		6
5.2	odd	4		320.3.h.g.319.5		6
5.3	odd	4		320.3.h.f.319.2		6
5.4	even	2		1600.3.b.w.1151.1		6
8.3	odd	2		800.3.b.h.351.6		6
8.5	even	2		800.3.b.h.351.1		6
20.3	even	4		320.3.h.g.319.6		6
20.7	even	4		320.3.h.f.319.1		6
20.19	odd	2		1600.3.b.w.1151.6		6
40.3	even	4		160.3.h.a.159.1	✓	6
40.13	odd	4		160.3.h.b.159.5	yes	6
40.19	odd	2		800.3.b.i.351.1		6
40.27	even	4		160.3.h.b.159.6	yes	6
40.29	even	2		800.3.b.i.351.6		6
40.37	odd	4		160.3.h.a.159.2	yes	6
80.3	even	4		1280.3.e.i.639.6		6
80.13	odd	4		1280.3.e.h.639.1		6
80.27	even	4		1280.3.e.h.639.6		6
80.37	odd	4		1280.3.e.i.639.1		6
80.43	even	4		1280.3.e.g.639.1		6
80.53	odd	4		1280.3.e.f.639.6		6
80.67	even	4		1280.3.e.f.639.1		6
80.77	odd	4		1280.3.e.g.639.6		6
120.53	even	4		1440.3.j.b.1279.4		6
120.77	even	4		1440.3.j.a.1279.3		6
120.83	odd	4		1440.3.j.a.1279.4		6
120.107	odd	4		1440.3.j.b.1279.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.h.a.159.1	✓	6	40.3	even	4
160.3.h.a.159.2	yes	6	40.37	odd	4
160.3.h.b.159.5	yes	6	40.13	odd	4
160.3.h.b.159.6	yes	6	40.27	even	4
320.3.h.f.319.1		6	20.7	even	4
320.3.h.f.319.2		6	5.3	odd	4
320.3.h.g.319.5		6	5.2	odd	4
320.3.h.g.319.6		6	20.3	even	4
800.3.b.h.351.1		6	8.5	even	2
800.3.b.h.351.6		6	8.3	odd	2
800.3.b.i.351.1		6	40.19	odd	2
800.3.b.i.351.6		6	40.29	even	2
1280.3.e.f.639.1		6	80.67	even	4
1280.3.e.f.639.6		6	80.53	odd	4
1280.3.e.g.639.1		6	80.43	even	4
1280.3.e.g.639.6		6	80.77	odd	4
1280.3.e.h.639.1		6	80.13	odd	4
1280.3.e.h.639.6		6	80.27	even	4
1280.3.e.i.639.1		6	80.37	odd	4
1280.3.e.i.639.6		6	80.3	even	4
1440.3.j.a.1279.3		6	120.77	even	4
1440.3.j.a.1279.4		6	120.83	odd	4
1440.3.j.b.1279.3		6	120.107	odd	4
1440.3.j.b.1279.4		6	120.53	even	4
1600.3.b.v.1151.1		6	4.3	odd	2	inner
1600.3.b.v.1151.6		6	1.1	even	1	trivial
1600.3.b.w.1151.1		6	5.4	even	2
1600.3.b.w.1151.6		6	20.19	odd	2

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.30219i q^{3} -0.206625i q^{7} -19.1132 q^{9} +O(q^{10})\)	\(q+5.30219i q^{3} -0.206625i q^{7} -19.1132 q^{9} +15.0176i q^{11} -11.6999 q^{13} -18.1911 q^{17} +19.3999i q^{19} +1.09556 q^{21} +27.2242i q^{23} -53.6220i q^{27} +44.4175 q^{29} -20.3822i q^{31} -79.6262 q^{33} +18.1089 q^{37} -62.0352i q^{39} -32.3043 q^{41} -4.06244i q^{43} +5.37588i q^{47} +48.9573 q^{49} -96.4527i q^{51} -79.1703 q^{53} -102.862 q^{57} -83.3999i q^{59} +36.7486 q^{61} +3.94925i q^{63} +4.51518i q^{67} -144.348 q^{69} -41.6530i q^{71} -41.5910 q^{73} +3.10301 q^{77} -15.5473i q^{79} +112.295 q^{81} -50.9862i q^{83} +235.510i q^{87} -10.8885 q^{89} +2.41749i q^{91} +108.070 q^{93} -12.1559 q^{97} -287.035i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{9}+O(q^{10}) \) 6 * q - 18 * q^9	\( 6 q - 18 q^{9} - 80 q^{17} - 8 q^{21} + 44 q^{29} - 144 q^{33} + 208 q^{37} - 68 q^{41} + 62 q^{49} + 64 q^{53} - 400 q^{57} + 100 q^{61} - 184 q^{69} - 80 q^{73} + 400 q^{77} + 238 q^{81} - 76 q^{89} + 320 q^{93} - 208 q^{97}+O(q^{100}) \) 6 * q - 18 * q^9 - 80 * q^17 - 8 * q^21 + 44 * q^29 - 144 * q^33 + 208 * q^37 - 68 * q^41 + 62 * q^49 + 64 * q^53 - 400 * q^57 + 100 * q^61 - 184 * q^69 - 80 * q^73 + 400 * q^77 + 238 * q^81 - 76 * q^89 + 320 * q^93 - 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more