LMFDB - Embedded newform 5700.2.f.o.3649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5700,2,Mod(3649,5700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5700.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5700.f (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:	$45.5147291521$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.4
Root			$1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	5700.3649
Dual form			5700.2.f.o.3649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +4.16228 q^{11} -6.32456i q^{13} -4.00000i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.16228i q^{23} -1.00000i q^{27} -2.16228 q^{29} +5.32456 q^{31} +4.16228i q^{33} -2.00000i q^{37} +6.32456 q^{39} -6.32456 q^{41} +10.3246i q^{43} -6.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -6.48683i q^{53} +1.00000i q^{57} -10.3246 q^{59} +0.675445 q^{61} +4.00000i q^{63} +5.32456i q^{67} +4.16228 q^{69} -3.67544 q^{71} +3.32456i q^{73} -16.6491i q^{77} -3.00000 q^{79} +1.00000 q^{81} +14.8114i q^{83} -2.16228i q^{87} -8.16228 q^{89} -25.2982 q^{91} +5.32456i q^{93} +14.6491i q^{97} -4.16228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 4 q^{11} + 4 q^{19} + 16 q^{21} + 4 q^{29} - 4 q^{31} - 36 q^{49} + 16 q^{51} - 16 q^{59} + 28 q^{61} + 4 q^{69} - 40 q^{71} - 12 q^{79} + 4 q^{81} - 20 q^{89} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^11 + 4 * q^19 + 16 * q^21 + 4 * q^29 - 4 * q^31 - 36 * q^49 + 16 * q^51 - 16 * q^59 + 28 * q^61 + 4 * q^69 - 40 * q^71 - 12 * q^79 + 4 * q^81 - 20 * q^89 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$.

$n$	$1901$	$2851$	$3877$	$4201$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.16228	1.25497	0.627487	−	0.778627i	$-0.284084\pi$
$11$	4.16228	1.25497	0.627487	+	0.778627i	$0.284084\pi$
$12$	0	0
$13$	− 6.32456i	− 1.75412i	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	− 6.32456i	− 1.75412i	0.480384	−	0.877058i	$-0.340497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	− 4.16228i	− 0.867895i	−0.900938	−	0.433947i	$-0.857120\pi$
$23$	− 4.16228i	− 0.867895i	0.900938	−	0.433947i	$-0.142880\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−2.16228	−0.401525	−0.200762	−	0.979640i	$-0.564342\pi$
$29$	−2.16228	−0.401525	−0.200762	+	0.979640i	$0.564342\pi$
$30$	0	0
$31$	5.32456	0.956318	0.478159	−	0.878273i	$-0.341304\pi$
$31$	5.32456	0.956318	0.478159	+	0.878273i	$0.341304\pi$
$32$	0	0
$33$	4.16228i	0.724560i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	6.32456	1.01274
$40$	0	0
$41$	−6.32456	−0.987730	−0.493865	−	0.869539i	$-0.664416\pi$
$41$	−6.32456	−0.987730	−0.493865	+	0.869539i	$0.664416\pi$
$42$	0	0
$43$	10.3246i	1.57448i	0.616647	+	0.787240i	$0.288491\pi$
$43$	10.3246i	1.57448i	−0.616647	+	0.787240i	$0.711509\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	− 6.48683i	− 0.891035i	−0.895273	−	0.445518i	$-0.853020\pi$
$53$	− 6.48683i	− 0.891035i	0.895273	−	0.445518i	$-0.146980\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−10.3246	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$59$	−10.3246	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$60$	0	0
$61$	0.675445	0.0864818	0.0432409	−	0.999065i	$-0.486232\pi$
$61$	0.675445	0.0864818	0.0432409	+	0.999065i	$0.486232\pi$
$62$	0	0
$63$	4.00000i	0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.32456i	0.650498i	0.945628	+	0.325249i	$0.105448\pi$
$67$	5.32456i	0.650498i	−0.945628	+	0.325249i	$0.894552\pi$
$68$	0	0
$69$	4.16228	0.501079
$70$	0	0
$71$	−3.67544	−0.436195	−0.218098	−	0.975927i	$-0.569985\pi$
$71$	−3.67544	−0.436195	−0.218098	+	0.975927i	$0.569985\pi$
$72$	0	0
$73$	3.32456i	0.389110i	0.980892	+	0.194555i	$0.0623262\pi$
$73$	3.32456i	0.389110i	−0.980892	+	0.194555i	$0.937674\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.6491i	− 1.89734i
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.8114i	1.62576i	0.582430	+	0.812881i	$0.302102\pi$
$83$	14.8114i	1.62576i	−0.582430	+	0.812881i	$0.697898\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.16228i	− 0.231820i
$88$	0	0
$89$	−8.16228	−0.865200	−0.432600	−	0.901586i	$-0.642404\pi$
$89$	−8.16228	−0.865200	−0.432600	+	0.901586i	$0.642404\pi$
$90$	0	0
$91$	−25.2982	−2.65197
$92$	0	0
$93$	5.32456i	0.552131i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.6491i	1.48739i	0.668518	+	0.743696i	$0.266929\pi$
$97$	14.6491i	1.48739i	−0.668518	+	0.743696i	$0.733071\pi$
$98$	0	0
$99$	−4.16228	−0.418325
$100$	0	0
$101$	−5.67544	−0.564728	−0.282364	−	0.959307i	$-0.591119\pi$
$101$	−5.67544	−0.564728	−0.282364	+	0.959307i	$0.591119\pi$
$102$	0	0
$103$	1.00000i	0.0985329i	0.998786	+	0.0492665i	$0.0156884\pi$
$103$	1.00000i	0.0985329i	−0.998786	+	0.0492665i	$0.984312\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.6491i	− 1.41618i	−0.706121	−	0.708091i	$-0.749556\pi$
$107$	− 14.6491i	− 1.41618i	0.706121	−	0.708091i	$-0.250444\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 9.83772i	− 0.925455i	−0.886501	−	0.462728i	$-0.846871\pi$
$113$	− 9.83772i	− 0.925455i	0.886501	−	0.462728i	$-0.153129\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.32456i	0.584705i
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	6.32456	0.574960
$122$	0	0
$123$	− 6.32456i	− 0.570266i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 15.3246i	− 1.35983i	−0.733289	−	0.679917i	$-0.762016\pi$
$127$	− 15.3246i	− 1.35983i	0.733289	−	0.679917i	$-0.237984\pi$
$128$	0	0
$129$	−10.3246	−0.909026
$130$	0	0
$131$	6.48683	0.566757	0.283379	−	0.959008i	$-0.408545\pi$
$131$	6.48683	0.566757	0.283379	+	0.959008i	$0.408545\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.3246i	1.22383i	0.790924	+	0.611915i	$0.209600\pi$
$137$	14.3246i	1.22383i	−0.790924	+	0.611915i	$0.790400\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	− 26.3246i	− 2.20137i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	4.64911	0.380870	0.190435	−	0.981700i	$-0.439010\pi$
$149$	4.64911	0.380870	0.190435	+	0.981700i	$0.439010\pi$
$150$	0	0
$151$	12.6491	1.02937	0.514685	−	0.857379i	$-0.327909\pi$
$151$	12.6491	1.02937	0.514685	+	0.857379i	$0.327909\pi$
$152$	0	0
$153$	4.00000i	0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.6491i	1.16913i	0.811348	+	0.584563i	$0.198734\pi$
$157$	14.6491i	1.16913i	−0.811348	+	0.584563i	$0.801266\pi$
$158$	0	0
$159$	6.48683	0.514439
$160$	0	0
$161$	−16.6491	−1.31213
$162$	0	0
$163$	3.67544i	0.287883i	0.989586	+	0.143942i	$0.0459777\pi$
$163$	3.67544i	0.287883i	−0.989586	+	0.143942i	$0.954022\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.9737i	1.31346i	0.754125	+	0.656731i	$0.228061\pi$
$167$	16.9737i	1.31346i	−0.754125	+	0.656731i	$0.771939\pi$
$168$	0	0
$169$	−27.0000	−2.07692
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	− 12.1623i	− 0.924681i	−0.886703	−	0.462340i	$-0.847010\pi$
$173$	− 12.1623i	− 0.924681i	0.886703	−	0.462340i	$-0.152990\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.3246i	− 0.776041i
$178$	0	0
$179$	8.64911	0.646465	0.323232	−	0.946320i	$-0.395230\pi$
$179$	8.64911	0.646465	0.323232	+	0.946320i	$0.395230\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	0.675445i	0.0499303i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.6491i	− 1.21750i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−14.1623	−1.02475	−0.512373	−	0.858763i	$-0.671234\pi$
$191$	−14.1623	−1.02475	−0.512373	+	0.858763i	$0.671234\pi$
$192$	0	0
$193$	− 7.67544i	− 0.552491i	−0.961087	−	0.276245i	$-0.910910\pi$
$193$	− 7.67544i	− 0.552491i	0.961087	−	0.276245i	$-0.0890902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3246i	0.735594i	0.929906	+	0.367797i	$0.119888\pi$
$197$	10.3246i	0.735594i	−0.929906	+	0.367797i	$0.880112\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−5.32456	−0.375565
$202$	0	0
$203$	8.64911i	0.607049i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.16228i	0.289298i
$208$	0	0
$209$	4.16228	0.287911
$210$	0	0
$211$	1.32456	0.0911861	0.0455931	−	0.998960i	$-0.485482\pi$
$211$	1.32456	0.0911861	0.0455931	+	0.998960i	$0.485482\pi$
$212$	0	0
$213$	− 3.67544i	− 0.251837i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 21.2982i	− 1.44582i
$218$	0	0
$219$	−3.32456	−0.224653
$220$	0	0
$221$	−25.2982	−1.70174
$222$	0	0
$223$	− 7.64911i	− 0.512222i	−0.966647	−	0.256111i	$-0.917559\pi$
$223$	− 7.64911i	− 0.512222i	0.966647	−	0.256111i	$-0.0824413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.3246i	− 1.08350i	−0.840540	−	0.541749i	$-0.817762\pi$
$227$	− 16.3246i	− 1.08350i	0.840540	−	0.541749i	$-0.182238\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	16.6491	1.09543
$232$	0	0
$233$	− 18.9737i	− 1.24301i	−0.783412	−	0.621503i	$-0.786522\pi$
$233$	− 18.9737i	− 1.24301i	0.783412	−	0.621503i	$-0.213478\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.00000i	− 0.194871i
$238$	0	0
$239$	−30.6491	−1.98253	−0.991263	−	0.131900i	$-0.957892\pi$
$239$	−30.6491	−1.98253	−0.991263	+	0.131900i	$0.957892\pi$
$240$	0	0
$241$	26.9737	1.73753	0.868763	−	0.495228i	$-0.164915\pi$
$241$	26.9737	1.73753	0.868763	+	0.495228i	$0.164915\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.32456i	− 0.402422i
$248$	0	0
$249$	−14.8114	−0.938634
$250$	0	0
$251$	10.6491	0.672166	0.336083	−	0.941832i	$-0.390898\pi$
$251$	10.6491	0.672166	0.336083	+	0.941832i	$0.390898\pi$
$252$	0	0
$253$	− 17.3246i	− 1.08919i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.1623i	1.63196i	0.578082	+	0.815979i	$0.303802\pi$
$257$	26.1623i	1.63196i	−0.578082	+	0.815979i	$0.696198\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.16228	0.133842
$262$	0	0
$263$	− 14.1623i	− 0.873283i	−0.899635	−	0.436642i	$-0.856168\pi$
$263$	− 14.1623i	− 0.873283i	0.899635	−	0.436642i	$-0.143832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8.16228i	− 0.499523i
$268$	0	0
$269$	10.3246	0.629499	0.314750	−	0.949175i	$-0.398079\pi$
$269$	10.3246	0.629499	0.314750	+	0.949175i	$0.398079\pi$
$270$	0	0
$271$	−24.9737	−1.51704	−0.758521	−	0.651648i	$-0.774078\pi$
$271$	−24.9737	−1.51704	−0.758521	+	0.651648i	$0.774078\pi$
$272$	0	0
$273$	− 25.2982i	− 1.53112i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 0.350889i	− 0.0210829i	−0.999944	−	0.0105414i	$-0.996644\pi$
$277$	− 0.350889i	− 0.0210829i	0.999944	−	0.0105414i	$-0.00335551\pi$
$278$	0	0
$279$	−5.32456	−0.318773
$280$	0	0
$281$	0.486833	0.0290420	0.0145210	−	0.999895i	$-0.495378\pi$
$281$	0.486833	0.0290420	0.0145210	+	0.999895i	$0.495378\pi$
$282$	0	0
$283$	18.3246i	1.08928i	0.838669	+	0.544641i	$0.183334\pi$
$283$	18.3246i	1.08928i	−0.838669	+	0.544641i	$0.816666\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.2982i	1.49331i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−14.6491	−0.858746
$292$	0	0
$293$	− 3.51317i	− 0.205241i	−0.994721	−	0.102621i	$-0.967277\pi$
$293$	− 3.51317i	− 0.205241i	0.994721	−	0.102621i	$-0.0327228\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.16228i	− 0.241520i
$298$	0	0
$299$	−26.3246	−1.52239
$300$	0	0
$301$	41.2982	2.38039
$302$	0	0
$303$	− 5.67544i	− 0.326046i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	26.6491	1.51113	0.755566	−	0.655072i	$-0.227362\pi$
$311$	26.6491	1.51113	0.755566	+	0.655072i	$0.227362\pi$
$312$	0	0
$313$	− 25.3246i	− 1.43143i	−0.698393	−	0.715714i	$-0.746101\pi$
$313$	− 25.3246i	− 1.43143i	0.698393	−	0.715714i	$-0.253899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.5132i	0.646644i	0.946289	+	0.323322i	$0.104800\pi$
$317$	11.5132i	0.646644i	−0.946289	+	0.323322i	$0.895200\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	14.6491	0.817634
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	27.0000	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$331$	27.0000	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 24.6491i	− 1.34272i	−0.741130	−	0.671361i	$-0.765710\pi$
$337$	− 24.6491i	− 1.34272i	0.741130	−	0.671361i	$-0.234290\pi$
$338$	0	0
$339$	9.83772	0.534312
$340$	0	0
$341$	22.1623	1.20015
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.35089i	− 0.501982i	−0.967989	−	0.250991i	$-0.919244\pi$
$347$	− 9.35089i	− 0.501982i	0.967989	−	0.250991i	$-0.0807565\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−6.32456	−0.337580
$352$	0	0
$353$	30.9737i	1.64856i	0.566181	+	0.824281i	$0.308420\pi$
$353$	30.9737i	1.64856i	−0.566181	+	0.824281i	$0.691580\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 16.0000i	− 0.846810i
$358$	0	0
$359$	18.6491	0.984262	0.492131	−	0.870521i	$-0.336218\pi$
$359$	18.6491	0.984262	0.492131	+	0.870521i	$0.336218\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.32456i	0.331953i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 26.6491i	− 1.39107i	−0.718491	−	0.695536i	$-0.755167\pi$
$367$	− 26.6491i	− 1.39107i	0.718491	−	0.695536i	$-0.244833\pi$
$368$	0	0
$369$	6.32456	0.329243
$370$	0	0
$371$	−25.9473	−1.34712
$372$	0	0
$373$	10.3246i	0.534585i	0.963615	+	0.267293i	$0.0861290\pi$
$373$	10.3246i	0.534585i	−0.963615	+	0.267293i	$0.913871\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.6754i	0.704321i
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	15.3246	0.785101
$382$	0	0
$383$	− 36.9737i	− 1.88927i	−0.328129	−	0.944633i	$-0.606418\pi$
$383$	− 36.9737i	− 1.88927i	0.328129	−	0.944633i	$-0.393582\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.3246i	− 0.524827i
$388$	0	0
$389$	−24.9737	−1.26622	−0.633108	−	0.774064i	$-0.718221\pi$
$389$	−24.9737	−1.26622	−0.633108	+	0.774064i	$0.718221\pi$
$390$	0	0
$391$	−16.6491	−0.841982
$392$	0	0
$393$	6.48683i	0.327217i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 19.6491i	− 0.986161i	−0.869984	−	0.493080i	$-0.835871\pi$
$397$	− 19.6491i	− 0.986161i	0.869984	−	0.493080i	$-0.164129\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.4868	−0.923188	−0.461594	−	0.887091i	$-0.652722\pi$
$401$	−18.4868	−0.923188	−0.461594	+	0.887091i	$0.652722\pi$
$402$	0	0
$403$	− 33.6754i	− 1.67749i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.32456i	− 0.412633i
$408$	0	0
$409$	7.67544	0.379526	0.189763	−	0.981830i	$-0.439228\pi$
$409$	7.67544	0.379526	0.189763	+	0.981830i	$0.439228\pi$
$410$	0	0
$411$	−14.3246	−0.706578
$412$	0	0
$413$	41.2982i	2.03215i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.00000i	− 0.293821i
$418$	0	0
$419$	−9.35089	−0.456821	−0.228410	−	0.973565i	$-0.573353\pi$
$419$	−9.35089	−0.456821	−0.228410	+	0.973565i	$0.573353\pi$
$420$	0	0
$421$	23.6754	1.15387	0.576935	−	0.816790i	$-0.304248\pi$
$421$	23.6754	1.15387	0.576935	+	0.816790i	$0.304248\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.70178i	− 0.130748i
$428$	0	0
$429$	26.3246	1.27096
$430$	0	0
$431$	−14.6491	−0.705623	−0.352811	−	0.935694i	$-0.614774\pi$
$431$	−14.6491	−0.705623	−0.352811	+	0.935694i	$0.614774\pi$
$432$	0	0
$433$	− 6.64911i	− 0.319536i	−0.987155	−	0.159768i	$-0.948925\pi$
$433$	− 6.64911i	− 0.319536i	0.987155	−	0.159768i	$-0.0510746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.16228i	− 0.199109i
$438$	0	0
$439$	4.35089	0.207657	0.103828	−	0.994595i	$-0.466891\pi$
$439$	4.35089	0.207657	0.103828	+	0.994595i	$0.466891\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 15.8377i	− 0.752473i	−0.926524	−	0.376236i	$-0.877218\pi$
$443$	− 15.8377i	− 0.752473i	0.926524	−	0.376236i	$-0.122782\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.64911i	0.219895i
$448$	0	0
$449$	−7.18861	−0.339252	−0.169626	−	0.985509i	$-0.554256\pi$
$449$	−7.18861	−0.339252	−0.169626	+	0.985509i	$0.554256\pi$
$450$	0	0
$451$	−26.3246	−1.23957
$452$	0	0
$453$	12.6491i	0.594307i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 30.0000i	− 1.40334i	−0.712502	−	0.701670i	$-0.752438\pi$
$457$	− 30.0000i	− 1.40334i	0.712502	−	0.701670i	$-0.247562\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	9.67544	0.450630	0.225315	−	0.974286i	$-0.427659\pi$
$461$	9.67544	0.450630	0.225315	+	0.974286i	$0.427659\pi$
$462$	0	0
$463$	30.9737i	1.43947i	0.694250	+	0.719734i	$0.255736\pi$
$463$	30.9737i	1.43947i	−0.694250	+	0.719734i	$0.744264\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.4605i	1.91856i	0.282452	+	0.959282i	$0.408852\pi$
$467$	41.4605i	1.91856i	−0.282452	+	0.959282i	$0.591148\pi$
$468$	0	0
$469$	21.2982	0.983460
$470$	0	0
$471$	−14.6491	−0.674995
$472$	0	0
$473$	42.9737i	1.97593i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.48683i	0.297012i
$478$	0	0
$479$	26.8114	1.22504	0.612522	−	0.790454i	$-0.290155\pi$
$479$	26.8114	1.22504	0.612522	+	0.790454i	$0.290155\pi$
$480$	0	0
$481$	−12.6491	−0.576750
$482$	0	0
$483$	− 16.6491i	− 0.757561i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.64911i	0.210671i	0.994437	+	0.105336i	$0.0335917\pi$
$487$	4.64911i	0.210671i	−0.994437	+	0.105336i	$0.966408\pi$
$488$	0	0
$489$	−3.67544	−0.166209
$490$	0	0
$491$	39.2982	1.77350	0.886752	−	0.462246i	$-0.152956\pi$
$491$	39.2982	1.77350	0.886752	+	0.462246i	$0.152956\pi$
$492$	0	0
$493$	8.64911i	0.389536i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.7018i	0.659465i
$498$	0	0
$499$	−35.6228	−1.59469	−0.797347	−	0.603521i	$-0.793764\pi$
$499$	−35.6228	−1.59469	−0.797347	+	0.603521i	$0.793764\pi$
$500$	0	0
$501$	−16.9737	−0.758327
$502$	0	0
$503$	− 34.0000i	− 1.51599i	−0.652263	−	0.757993i	$-0.726180\pi$
$503$	− 34.0000i	− 1.51599i	0.652263	−	0.757993i	$-0.273820\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 27.0000i	− 1.19911i
$508$	0	0
$509$	3.51317	0.155718	0.0778592	−	0.996964i	$-0.475192\pi$
$509$	3.51317	0.155718	0.0778592	+	0.996964i	$0.475192\pi$
$510$	0	0
$511$	13.2982	0.588279
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 24.9737i	− 1.09834i
$518$	0	0
$519$	12.1623	0.533865
$520$	0	0
$521$	42.8114	1.87560	0.937800	−	0.347175i	$-0.112859\pi$
$521$	42.8114	1.87560	0.937800	+	0.347175i	$0.112859\pi$
$522$	0	0
$523$	− 40.6491i	− 1.77746i	−0.458430	−	0.888731i	$-0.651588\pi$
$523$	− 40.6491i	− 1.77746i	0.458430	−	0.888731i	$-0.348412\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 21.2982i	− 0.927765i
$528$	0	0
$529$	5.67544	0.246758
$530$	0	0
$531$	10.3246	0.448048
$532$	0	0
$533$	40.0000i	1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.64911i	0.373237i
$538$	0	0
$539$	−37.4605	−1.61354
$540$	0	0
$541$	−13.3246	−0.572867	−0.286434	−	0.958100i	$-0.592470\pi$
$541$	−13.3246	−0.572867	−0.286434	+	0.958100i	$0.592470\pi$
$542$	0	0
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 45.6491i	− 1.95182i	−0.218185	−	0.975908i	$-0.570013\pi$
$547$	− 45.6491i	− 1.95182i	0.218185	−	0.975908i	$-0.429987\pi$
$548$	0	0
$549$	−0.675445	−0.0288273
$550$	0	0
$551$	−2.16228	−0.0921161
$552$	0	0
$553$	12.0000i	0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	65.2982	2.76182
$560$	0	0
$561$	16.6491	0.702926
$562$	0	0
$563$	− 41.6228i	− 1.75419i	−0.480316	−	0.877095i	$-0.659478\pi$
$563$	− 41.6228i	− 1.75419i	0.480316	−	0.877095i	$-0.340522\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	10.3246	0.432828	0.216414	−	0.976302i	$-0.430564\pi$
$569$	10.3246	0.432828	0.216414	+	0.976302i	$0.430564\pi$
$570$	0	0
$571$	8.97367	0.375536	0.187768	−	0.982213i	$-0.439875\pi$
$571$	8.97367	0.375536	0.187768	+	0.982213i	$0.439875\pi$
$572$	0	0
$573$	− 14.1623i	− 0.591638i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20.6754i	0.860730i	0.902655	+	0.430365i	$0.141615\pi$
$577$	20.6754i	0.860730i	−0.902655	+	0.430365i	$0.858385\pi$
$578$	0	0
$579$	7.67544	0.318981
$580$	0	0
$581$	59.2456	2.45792
$582$	0	0
$583$	− 27.0000i	− 1.11823i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.18861i	0.214157i	0.994251	+	0.107078i	$0.0341496\pi$
$587$	5.18861i	0.214157i	−0.994251	+	0.107078i	$0.965850\pi$
$588$	0	0
$589$	5.32456	0.219394
$590$	0	0
$591$	−10.3246	−0.424695
$592$	0	0
$593$	− 21.2982i	− 0.874613i	−0.899312	−	0.437307i	$-0.855932\pi$
$593$	− 21.2982i	− 0.874613i	0.899312	−	0.437307i	$-0.144068\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.00000i	0.0818546i
$598$	0	0
$599$	−14.9737	−0.611807	−0.305904	−	0.952062i	$-0.598959\pi$
$599$	−14.9737	−0.611807	−0.305904	+	0.952062i	$0.598959\pi$
$600$	0	0
$601$	−28.3246	−1.15538	−0.577691	−	0.816255i	$-0.696046\pi$
$601$	−28.3246	−1.15538	−0.577691	+	0.816255i	$0.696046\pi$
$602$	0	0
$603$	− 5.32456i	− 0.216833i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.9737i	0.891884i	0.895062	+	0.445942i	$0.147131\pi$
$607$	21.9737i	0.891884i	−0.895062	+	0.445942i	$0.852869\pi$
$608$	0	0
$609$	−8.64911	−0.350480
$610$	0	0
$611$	−37.9473	−1.53518
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.6228i	1.83670i	0.395765	+	0.918352i	$0.370480\pi$
$617$	45.6228i	1.83670i	−0.395765	+	0.918352i	$0.629520\pi$
$618$	0	0
$619$	2.32456	0.0934318	0.0467159	−	0.998908i	$-0.485124\pi$
$619$	2.32456	0.0934318	0.0467159	+	0.998908i	$0.485124\pi$
$620$	0	0
$621$	−4.16228	−0.167026
$622$	0	0
$623$	32.6491i	1.30806i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.16228i	0.166225i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−33.2982	−1.32558	−0.662791	−	0.748805i	$-0.730628\pi$
$631$	−33.2982	−1.32558	−0.662791	+	0.748805i	$0.730628\pi$
$632$	0	0
$633$	1.32456i	0.0526463i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	56.9210i	2.25529i
$638$	0	0
$639$	3.67544	0.145398
$640$	0	0
$641$	−34.9737	−1.38138	−0.690688	−	0.723153i	$-0.742692\pi$
$641$	−34.9737	−1.38138	−0.690688	+	0.723153i	$0.742692\pi$
$642$	0	0
$643$	31.2982i	1.23428i	0.786853	+	0.617141i	$0.211709\pi$
$643$	31.2982i	1.23428i	−0.786853	+	0.617141i	$0.788291\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.81139i	− 0.110527i	−0.998472	−	0.0552635i	$-0.982400\pi$
$647$	− 2.81139i	− 0.110527i	0.998472	−	0.0552635i	$-0.0175999\pi$
$648$	0	0
$649$	−42.9737	−1.68686
$650$	0	0
$651$	21.2982	0.834743
$652$	0	0
$653$	− 21.2982i	− 0.833464i	−0.909029	−	0.416732i	$-0.863175\pi$
$653$	− 21.2982i	− 0.833464i	0.909029	−	0.416732i	$-0.136825\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.32456i	− 0.129703i
$658$	0	0
$659$	24.6491	0.960193	0.480097	−	0.877216i	$-0.340602\pi$
$659$	24.6491	0.960193	0.480097	+	0.877216i	$0.340602\pi$
$660$	0	0
$661$	20.6491	0.803157	0.401579	−	0.915825i	$-0.368462\pi$
$661$	20.6491	0.803157	0.401579	+	0.915825i	$0.368462\pi$
$662$	0	0
$663$	− 25.2982i	− 0.982502i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000i	0.348481i
$668$	0	0
$669$	7.64911	0.295732
$670$	0	0
$671$	2.81139	0.108532
$672$	0	0
$673$	− 20.0000i	− 0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$	− 20.0000i	− 0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.486833i	0.0187105i	0.999956	+	0.00935526i	$0.00297791\pi$
$677$	0.486833i	0.0187105i	−0.999956	+	0.00935526i	$0.997022\pi$
$678$	0	0
$679$	58.5964	2.24873
$680$	0	0
$681$	16.3246	0.625558
$682$	0	0
$683$	− 40.3246i	− 1.54298i	−0.636244	−	0.771488i	$-0.719513\pi$
$683$	− 40.3246i	− 1.54298i	0.636244	−	0.771488i	$-0.280487\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.00000i	− 0.0381524i
$688$	0	0
$689$	−41.0263	−1.56298
$690$	0	0
$691$	−39.9473	−1.51967	−0.759834	−	0.650117i	$-0.774720\pi$
$691$	−39.9473	−1.51967	−0.759834	+	0.650117i	$0.774720\pi$
$692$	0	0
$693$	16.6491i	0.632447i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25.2982i	0.958238i
$698$	0	0
$699$	18.9737	0.717650
$700$	0	0
$701$	−50.3246	−1.90073	−0.950366	−	0.311134i	$-0.899291\pi$
$701$	−50.3246	−1.90073	−0.950366	+	0.311134i	$0.899291\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22.7018i	0.853788i
$708$	0	0
$709$	32.6228	1.22517	0.612587	−	0.790403i	$-0.290129\pi$
$709$	32.6228	1.22517	0.612587	+	0.790403i	$0.290129\pi$
$710$	0	0
$711$	3.00000	0.112509
$712$	0	0
$713$	− 22.1623i	− 0.829984i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 30.6491i	− 1.14461i
$718$	0	0
$719$	−32.4868	−1.21155	−0.605777	−	0.795634i	$-0.707138\pi$
$719$	−32.4868	−1.21155	−0.605777	+	0.795634i	$0.707138\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	26.9737i	1.00316i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 42.2719i	− 1.56778i	−0.620901	−	0.783889i	$-0.713233\pi$
$727$	− 42.2719i	− 1.56778i	0.620901	−	0.783889i	$-0.286767\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	41.2982	1.52747
$732$	0	0
$733$	− 29.9737i	− 1.10710i	−0.832815	−	0.553551i	$-0.813272\pi$
$733$	− 29.9737i	− 1.10710i	0.832815	−	0.553551i	$-0.186728\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.1623i	0.816358i
$738$	0	0
$739$	−6.32456	−0.232653	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	−6.32456	−0.232653	−0.116326	+	0.993211i	$0.537112\pi$
$740$	0	0
$741$	6.32456	0.232338
$742$	0	0
$743$	− 42.9737i	− 1.57655i	−0.615323	−	0.788275i	$-0.710974\pi$
$743$	− 42.9737i	− 1.57655i	0.615323	−	0.788275i	$-0.289026\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 14.8114i	− 0.541920i
$748$	0	0
$749$	−58.5964	−2.14107
$750$	0	0
$751$	29.9473	1.09279	0.546397	−	0.837526i	$-0.315999\pi$
$751$	29.9473	1.09279	0.546397	+	0.837526i	$0.315999\pi$
$752$	0	0
$753$	10.6491i	0.388075i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.0000i	1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$	29.0000i	1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$	0	0
$759$	17.3246	0.628842
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	65.2982i	2.35778i
$768$	0	0
$769$	−3.32456	−0.119887	−0.0599433	−	0.998202i	$-0.519092\pi$
$769$	−3.32456	−0.119887	−0.0599433	+	0.998202i	$0.519092\pi$
$770$	0	0
$771$	−26.1623	−0.942211
$772$	0	0
$773$	9.02633i	0.324655i	0.986737	+	0.162327i	$0.0519000\pi$
$773$	9.02633i	0.324655i	−0.986737	+	0.162327i	$0.948100\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	−6.32456	−0.226601
$780$	0	0
$781$	−15.2982	−0.547413
$782$	0	0
$783$	2.16228i	0.0772735i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.32456i	0.118508i	0.998243	+	0.0592538i	$0.0188721\pi$
$787$	3.32456i	0.118508i	−0.998243	+	0.0592538i	$0.981128\pi$
$788$	0	0
$789$	14.1623	0.504190
$790$	0	0
$791$	−39.3509	−1.39916
$792$	0	0
$793$	− 4.27189i	− 0.151699i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.3246i	− 0.507402i	−0.967283	−	0.253701i	$-0.918352\pi$
$797$	− 14.3246i	− 0.507402i	0.967283	−	0.253701i	$-0.0816479\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	8.16228	0.288400
$802$	0	0
$803$	13.8377i	0.488323i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.3246i	0.363442i
$808$	0	0
$809$	−41.2982	−1.45197	−0.725984	−	0.687712i	$-0.758615\pi$
$809$	−41.2982	−1.45197	−0.725984	+	0.687712i	$0.758615\pi$
$810$	0	0
$811$	20.2982	0.712767	0.356383	−	0.934340i	$-0.384010\pi$
$811$	20.2982	0.712767	0.356383	+	0.934340i	$0.384010\pi$
$812$	0	0
$813$	− 24.9737i	− 0.875865i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.3246i	0.361210i
$818$	0	0
$819$	25.2982	0.883991
$820$	0	0
$821$	−0.324555	−0.0113271	−0.00566353	−	0.999984i	$-0.501803\pi$
$821$	−0.324555	−0.0113271	−0.00566353	+	0.999984i	$0.501803\pi$
$822$	0	0
$823$	− 29.6228i	− 1.03259i	−0.856412	−	0.516293i	$-0.827312\pi$
$823$	− 29.6228i	− 1.03259i	0.856412	−	0.516293i	$-0.172688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 19.6228i	− 0.682351i	−0.940000	−	0.341175i	$-0.889175\pi$
$827$	− 19.6228i	− 0.682351i	0.940000	−	0.341175i	$-0.110825\pi$
$828$	0	0
$829$	5.35089	0.185844	0.0929220	−	0.995673i	$-0.470379\pi$
$829$	5.35089	0.185844	0.0929220	+	0.995673i	$0.470379\pi$
$830$	0	0
$831$	0.350889	0.0121722
$832$	0	0
$833$	36.0000i	1.24733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.32456i	− 0.184044i
$838$	0	0
$839$	29.6754	1.02451	0.512255	−	0.858833i	$-0.328810\pi$
$839$	29.6754	1.02451	0.512255	+	0.858833i	$0.328810\pi$
$840$	0	0
$841$	−24.3246	−0.838778
$842$	0	0
$843$	0.486833i	0.0167674i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25.2982i	− 0.869257i
$848$	0	0
$849$	−18.3246	−0.628897
$850$	0	0
$851$	−8.32456	−0.285362
$852$	0	0
$853$	48.5964i	1.66391i	0.554843	+	0.831955i	$0.312778\pi$
$853$	48.5964i	1.66391i	−0.554843	+	0.831955i	$0.687222\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.6754i	0.877056i	0.898717	+	0.438528i	$0.144500\pi$
$857$	25.6754i	0.877056i	−0.898717	+	0.438528i	$0.855500\pi$
$858$	0	0
$859$	−8.97367	−0.306178	−0.153089	−	0.988212i	$-0.548922\pi$
$859$	−8.97367	−0.306178	−0.153089	+	0.988212i	$0.548922\pi$
$860$	0	0
$861$	−25.2982	−0.862161
$862$	0	0
$863$	− 19.3509i	− 0.658712i	−0.944206	−	0.329356i	$-0.893168\pi$
$863$	− 19.3509i	− 0.658712i	0.944206	−	0.329356i	$-0.106832\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	−12.4868	−0.423587
$870$	0	0
$871$	33.6754	1.14105
$872$	0	0
$873$	− 14.6491i	− 0.495797i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.6228i	1.54057i	0.637699	+	0.770286i	$0.279886\pi$
$877$	45.6228i	1.54057i	−0.637699	+	0.770286i	$0.720114\pi$
$878$	0	0
$879$	3.51317	0.118496
$880$	0	0
$881$	37.9473	1.27848	0.639239	−	0.769008i	$-0.279249\pi$
$881$	37.9473	1.27848	0.639239	+	0.769008i	$0.279249\pi$
$882$	0	0
$883$	11.6754i	0.392910i	0.980513	+	0.196455i	$0.0629430\pi$
$883$	11.6754i	0.392910i	−0.980513	+	0.196455i	$0.937057\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	−61.2982	−2.05588
$890$	0	0
$891$	4.16228	0.139442
$892$	0	0
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 26.3246i	− 0.878951i
$898$	0	0
$899$	−11.5132	−0.383986
$900$	0	0
$901$	−25.9473	−0.864431
$902$	0	0
$903$	41.2982i	1.37432i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	45.9473i	1.52566i	0.646601	+	0.762828i	$0.276190\pi$
$907$	45.9473i	1.52566i	−0.646601	+	0.762828i	$0.723810\pi$
$908$	0	0
$909$	5.67544	0.188243
$910$	0	0
$911$	−54.6491	−1.81061	−0.905303	−	0.424767i	$-0.860356\pi$
$911$	−54.6491	−1.81061	−0.905303	+	0.424767i	$0.860356\pi$
$912$	0	0
$913$	61.6491i	2.04029i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25.9473i	− 0.856856i
$918$	0	0
$919$	−9.62278	−0.317426	−0.158713	−	0.987325i	$-0.550734\pi$
$919$	−9.62278	−0.317426	−0.158713	+	0.987325i	$0.550734\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	23.2456i	0.765137i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.00000i	− 0.0328443i
$928$	0	0
$929$	−53.2982	−1.74866	−0.874329	−	0.485334i	$-0.838698\pi$
$929$	−53.2982	−1.74866	−0.874329	+	0.485334i	$0.838698\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	0	0
$933$	26.6491i	0.872453i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	25.3246	0.826436
$940$	0	0
$941$	−36.1096	−1.17714	−0.588570	−	0.808446i	$-0.700309\pi$
$941$	−36.1096	−1.17714	−0.588570	+	0.808446i	$0.700309\pi$
$942$	0	0
$943$	26.3246i	0.857245i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.6491i	− 1.12594i	−0.826476	−	0.562972i	$-0.809658\pi$
$947$	− 34.6491i	− 1.12594i	0.826476	−	0.562972i	$-0.190342\pi$
$948$	0	0
$949$	21.0263	0.682544
$950$	0	0
$951$	−11.5132	−0.373340
$952$	0	0
$953$	40.4868i	1.31150i	0.754979	+	0.655749i	$0.227647\pi$
$953$	40.4868i	1.31150i	−0.754979	+	0.655749i	$0.772353\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.00000i	− 0.290929i
$958$	0	0
$959$	57.2982	1.85026
$960$	0	0
$961$	−2.64911	−0.0854552
$962$	0	0
$963$	14.6491i	0.472061i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.70178i	− 0.151199i	−0.997138	−	0.0755995i	$-0.975913\pi$
$967$	− 4.70178i	− 0.151199i	0.997138	−	0.0755995i	$-0.0240871\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	39.6228	1.27156	0.635778	−	0.771872i	$-0.280680\pi$
$971$	39.6228	1.27156	0.635778	+	0.771872i	$0.280680\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.67544i	0.181574i	0.995870	+	0.0907868i	$0.0289382\pi$
$977$	5.67544i	0.181574i	−0.995870	+	0.0907868i	$0.971062\pi$
$978$	0	0
$979$	−33.9737	−1.08580
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	33.6228i	1.07240i	0.844091	+	0.536200i	$0.180141\pi$
$983$	33.6228i	1.07240i	−0.844091	+	0.536200i	$0.819859\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	42.9737	1.36648
$990$	0	0
$991$	37.2719	1.18398	0.591990	−	0.805945i	$-0.298342\pi$
$991$	37.2719	1.18398	0.591990	+	0.805945i	$0.298342\pi$
$992$	0	0
$993$	27.0000i	0.856819i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 57.9210i	− 1.83438i	−0.398454	−	0.917188i	$-0.630453\pi$
$997$	− 57.9210i	− 1.83438i	0.398454	−	0.917188i	$-0.369547\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5700.2.f.o.3649.4		4
5.2	odd	4		5700.2.a.v.1.2	yes	2
5.3	odd	4		5700.2.a.s.1.2	✓	2
5.4	even	2	inner	5700.2.f.o.3649.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5700.2.a.s.1.2	✓	2	5.3	odd	4
5700.2.a.v.1.2	yes	2	5.2	odd	4
5700.2.f.o.3649.2		4	5.4	even	2	inner
5700.2.f.o.3649.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5700.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +4.16228 q^{11} -6.32456i q^{13} -4.00000i q^{17} +1.00000 q^{19} +4.00000 q^{21} -4.16228i q^{23} -1.00000i q^{27} -2.16228 q^{29} +5.32456 q^{31} +4.16228i q^{33} -2.00000i q^{37} +6.32456 q^{39} -6.32456 q^{41} +10.3246i q^{43} -6.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -6.48683i q^{53} +1.00000i q^{57} -10.3246 q^{59} +0.675445 q^{61} +4.00000i q^{63} +5.32456i q^{67} +4.16228 q^{69} -3.67544 q^{71} +3.32456i q^{73} -16.6491i q^{77} -3.00000 q^{79} +1.00000 q^{81} +14.8114i q^{83} -2.16228i q^{87} -8.16228 q^{89} -25.2982 q^{91} +5.32456i q^{93} +14.6491i q^{97} -4.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 4 q^{11} + 4 q^{19} + 16 q^{21} + 4 q^{29} - 4 q^{31} - 36 q^{49} + 16 q^{51} - 16 q^{59} + 28 q^{61} + 4 q^{69} - 40 q^{71} - 12 q^{79} + 4 q^{81} - 20 q^{89} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^11 + 4 * q^19 + 16 * q^21 + 4 * q^29 - 4 * q^31 - 36 * q^49 + 16 * q^51 - 16 * q^59 + 28 * q^61 + 4 * q^69 - 40 * q^71 - 12 * q^79 + 4 * q^81 - 20 * q^89 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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