LMFDB - Embedded newform 3744.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.a (trivial)

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:	yes
Analytic conductor:	$29.8959905168$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{7} +O(q^{10})$	$q+2.82843 q^{7} -2.82843 q^{11} -1.00000 q^{13} +2.82843 q^{19} -5.65685 q^{23} -5.00000 q^{25} -4.00000 q^{29} -8.48528 q^{31} +6.00000 q^{37} -12.0000 q^{41} -2.82843 q^{47} +1.00000 q^{49} -4.00000 q^{53} +14.1421 q^{59} +6.00000 q^{61} +8.48528 q^{67} +8.48528 q^{71} -10.0000 q^{73} -8.00000 q^{77} -11.3137 q^{79} -14.1421 q^{83} +4.00000 q^{89} -2.82843 q^{91} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 2 q^{13} - 10 q^{25} - 8 q^{29} + 12 q^{37} - 24 q^{41} + 2 q^{49} - 8 q^{53} + 12 q^{61} - 20 q^{73} - 16 q^{77} + 8 q^{89} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^13 - 10 * q^25 - 8 * q^29 + 12 * q^37 - 24 * q^41 + 2 * q^49 - 8 * q^53 + 12 * q^61 - 20 * q^73 - 16 * q^77 + 8 * q^89 - 4 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.1421	1.84115	0.920575	−	0.390567i	$-0.127721\pi$
$59$	14.1421	1.84115	0.920575	+	0.390567i	$0.127721\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.1421	−1.55230	−0.776151	−	0.630548i	$-0.782830\pi$
$83$	−14.1421	−1.55230	−0.776151	+	0.630548i	$0.782830\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−5.65685	−0.479808	−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685	−0.479808	−0.239904	+	0.970797i	$0.577116\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	8.48528	0.690522	0.345261	−	0.938507i	$-0.387790\pi$
$151$	8.48528	0.690522	0.345261	+	0.938507i	$0.387790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−14.1421	−1.10770	−0.553849	−	0.832617i	$-0.686841\pi$
$163$	−14.1421	−1.10770	−0.553849	+	0.832617i	$0.686841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.7990	1.53209	0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990	1.53209	0.766046	+	0.642786i	$0.222221\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	−14.1421	−1.06904
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.3137	−0.845626	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	−11.3137	−0.845626	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.3137	−0.794067
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	11.3137	0.778868	0.389434	−	0.921054i	$-0.372671\pi$
$211$	11.3137	0.778868	0.389434	+	0.921054i	$0.372671\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.1421	0.947027	0.473514	−	0.880786i	$-0.342985\pi$
$223$	14.1421	0.947027	0.473514	+	0.880786i	$0.342985\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.1421	−0.914779	−0.457389	−	0.889267i	$-0.651215\pi$
$239$	−14.1421	−0.914779	−0.457389	+	0.889267i	$0.651215\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.82843	−0.179969
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.2843	−1.78529	−0.892644	−	0.450763i	$-0.851152\pi$
$251$	−28.2843	−1.78529	−0.892644	+	0.450763i	$0.851152\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	0	0
$259$	16.9706	1.05450
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.2843	−1.74408	−0.872041	−	0.489432i	$-0.837204\pi$
$263$	−28.2843	−1.74408	−0.872041	+	0.489432i	$0.837204\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	−8.48528	−0.515444	−0.257722	−	0.966219i	$-0.582972\pi$
$271$	−8.48528	−0.515444	−0.257722	+	0.966219i	$0.582972\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.1421	0.852803
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−28.2843	−1.68133	−0.840663	−	0.541559i	$-0.817834\pi$
$283$	−28.2843	−1.68133	−0.840663	+	0.541559i	$0.817834\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−33.9411	−2.00348
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−31.1127	−1.77570	−0.887848	−	0.460137i	$-0.847800\pi$
$307$	−31.1127	−1.77570	−0.887848	+	0.460137i	$0.847800\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.65685	−0.320771	−0.160385	−	0.987054i	$-0.551274\pi$
$311$	−5.65685	−0.320771	−0.160385	+	0.987054i	$0.551274\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	11.3137	0.633446
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.00000	0.277350
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	8.48528	0.466393	0.233197	−	0.972430i	$-0.425081\pi$
$331$	8.48528	0.466393	0.233197	+	0.972430i	$0.425081\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6274	1.21470	0.607352	−	0.794433i	$-0.292232\pi$
$347$	22.6274	1.21470	0.607352	+	0.794433i	$0.292232\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.1127	1.64207	0.821033	−	0.570881i	$-0.193398\pi$
$359$	31.1127	1.64207	0.821033	+	0.570881i	$0.193398\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.65685	−0.295285	−0.147643	−	0.989041i	$-0.547169\pi$
$367$	−5.65685	−0.295285	−0.147643	+	0.989041i	$0.547169\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.3137	−0.587378
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−8.48528	−0.435860	−0.217930	−	0.975964i	$-0.569930\pi$
$379$	−8.48528	−0.435860	−0.217930	+	0.975964i	$0.569930\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.1421	0.722629	0.361315	−	0.932444i	$-0.382328\pi$
$383$	14.1421	0.722629	0.361315	+	0.932444i	$0.382328\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	8.48528	0.422682
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	40.0000	1.96827
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.65685	0.276355	0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685	0.276355	0.138178	+	0.990407i	$0.455875\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.9706	0.821263
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.82843	0.136241	0.0681203	−	0.997677i	$-0.478300\pi$
$431$	2.82843	0.136241	0.0681203	+	0.997677i	$0.478300\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−11.3137	−0.539974	−0.269987	−	0.962864i	$-0.587019\pi$
$439$	−11.3137	−0.539974	−0.269987	+	0.962864i	$0.587019\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.65685	−0.268765	−0.134383	−	0.990930i	$-0.542905\pi$
$443$	−5.65685	−0.268765	−0.134383	+	0.990930i	$0.542905\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	33.9411	1.59823
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	2.82843	0.131448	0.0657241	−	0.997838i	$-0.479064\pi$
$463$	2.82843	0.131448	0.0657241	+	0.997838i	$0.479064\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.9706	0.785304	0.392652	−	0.919687i	$-0.371558\pi$
$467$	16.9706	0.785304	0.392652	+	0.919687i	$0.371558\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−14.1421	−0.648886
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.4558	1.16311	0.581554	−	0.813508i	$-0.302445\pi$
$479$	25.4558	1.16311	0.581554	+	0.813508i	$0.302445\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	25.4558	1.15351	0.576757	−	0.816916i	$-0.304318\pi$
$487$	25.4558	1.15351	0.576757	+	0.816916i	$0.304318\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.65685	0.255290	0.127645	−	0.991820i	$-0.459258\pi$
$491$	5.65685	0.255290	0.127645	+	0.991820i	$0.459258\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	31.1127	1.39280	0.696398	−	0.717656i	$-0.254785\pi$
$499$	31.1127	1.39280	0.696398	+	0.717656i	$0.254785\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	−28.2843	−1.25122
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	−5.65685	−0.247357	−0.123678	−	0.992322i	$-0.539469\pi$
$523$	−5.65685	−0.247357	−0.123678	+	0.992322i	$0.539469\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.65685	0.241870	0.120935	−	0.992660i	$-0.461411\pi$
$547$	5.65685	0.241870	0.120935	+	0.992660i	$0.461411\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.0000	1.69485	0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000	1.69485	0.847427	+	0.530912i	$0.178150\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.3137	−0.476816	−0.238408	−	0.971165i	$-0.576626\pi$
$563$	−11.3137	−0.476816	−0.238408	+	0.971165i	$0.576626\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	28.2843	1.18366	0.591830	−	0.806063i	$-0.298406\pi$
$571$	28.2843	1.18366	0.591830	+	0.806063i	$0.298406\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.2843	1.17954
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−40.0000	−1.65948
$582$	0	0
$583$	11.3137	0.468566
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.1421	−0.583708	−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421	−0.583708	−0.291854	+	0.956463i	$0.594272\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	28.2843	1.15566	0.577832	−	0.816156i	$-0.303899\pi$
$599$	28.2843	1.15566	0.577832	+	0.816156i	$0.303899\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.82843	0.114426
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	−8.48528	−0.341052	−0.170526	−	0.985353i	$-0.554547\pi$
$619$	−8.48528	−0.341052	−0.170526	+	0.985353i	$0.554547\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.3137	0.453274
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−25.4558	−1.01338	−0.506691	−	0.862128i	$-0.669131\pi$
$631$	−25.4558	−1.01338	−0.506691	+	0.862128i	$0.669131\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−36.7696	−1.45005	−0.725025	−	0.688723i	$-0.758172\pi$
$643$	−36.7696	−1.45005	−0.725025	+	0.688723i	$0.758172\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.9411	−1.33436	−0.667182	−	0.744895i	$-0.732500\pi$
$647$	−33.9411	−1.33436	−0.667182	+	0.744895i	$0.732500\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.3137	0.440720	0.220360	−	0.975419i	$-0.429277\pi$
$659$	11.3137	0.440720	0.220360	+	0.975419i	$0.429277\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	22.6274	0.876137
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.9706	−0.655141
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	−5.65685	−0.217090
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.48528	0.324680	0.162340	−	0.986735i	$-0.448096\pi$
$683$	8.48528	0.324680	0.162340	+	0.986735i	$0.448096\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	−36.7696	−1.39878	−0.699390	−	0.714740i	$-0.746545\pi$
$691$	−36.7696	−1.39878	−0.699390	+	0.714740i	$0.746545\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.9411	−1.27649
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.9706	0.632895	0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706	0.632895	0.316448	+	0.948610i	$0.397510\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.0000	0.742781
$726$	0	0
$727$	−39.5980	−1.46861	−0.734304	−	0.678821i	$-0.762491\pi$
$727$	−39.5980	−1.46861	−0.734304	+	0.678821i	$0.762491\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−42.4264	−1.56068	−0.780340	−	0.625355i	$-0.784954\pi$
$739$	−42.4264	−1.56068	−0.780340	+	0.625355i	$0.784954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.7696	−1.34894	−0.674472	−	0.738300i	$-0.735629\pi$
$743$	−36.7696	−1.34894	−0.674472	+	0.738300i	$0.735629\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	33.9411	1.23853	0.619265	−	0.785182i	$-0.287431\pi$
$751$	33.9411	1.23853	0.619265	+	0.785182i	$0.287431\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	5.65685	0.204792
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.1421	−0.510643
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	42.4264	1.52400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−33.9411	−1.21607
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.4558	0.907403	0.453701	−	0.891154i	$-0.350103\pi$
$787$	25.4558	0.907403	0.453701	+	0.891154i	$0.350103\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−45.2548	−1.60908
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.2843	0.998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.0000	1.40633	0.703163	−	0.711029i	$-0.251771\pi$
$809$	40.0000	1.40633	0.703163	+	0.711029i	$0.251771\pi$
$810$	0	0
$811$	36.7696	1.29115	0.645577	−	0.763695i	$-0.276617\pi$
$811$	36.7696	1.29115	0.645577	+	0.763695i	$0.276617\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.0000	−1.67521	−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000	−1.67521	−0.837606	+	0.546275i	$0.816045\pi$
$822$	0	0
$823$	−11.3137	−0.394371	−0.197186	−	0.980366i	$-0.563180\pi$
$823$	−11.3137	−0.394371	−0.197186	+	0.980366i	$0.563180\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.7990	0.688478	0.344239	−	0.938882i	$-0.388137\pi$
$827$	19.7990	0.688478	0.344239	+	0.938882i	$0.388137\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	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
$839$	25.4558	0.878833	0.439417	−	0.898283i	$-0.355185\pi$
$839$	25.4558	0.878833	0.439417	+	0.898283i	$0.355185\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.48528	−0.291558
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−33.9411	−1.16349
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32.0000	1.09310	0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000	1.09310	0.546550	+	0.837427i	$0.315941\pi$
$858$	0	0
$859$	22.6274	0.772038	0.386019	−	0.922491i	$-0.373850\pi$
$859$	22.6274	0.772038	0.386019	+	0.922491i	$0.373850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.1421	0.481404	0.240702	−	0.970599i	$-0.422622\pi$
$863$	14.1421	0.481404	0.240702	+	0.970599i	$0.422622\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−8.48528	−0.287513
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	0	0
$883$	−39.5980	−1.33258	−0.666289	−	0.745694i	$-0.732118\pi$
$883$	−39.5980	−1.33258	−0.666289	+	0.745694i	$0.732118\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.6274	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$887$	22.6274	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$888$	0	0
$889$	48.0000	1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.9411	1.13200
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.9706	−0.563498	−0.281749	−	0.959488i	$-0.590915\pi$
$907$	−16.9706	−0.563498	−0.281749	+	0.959488i	$0.590915\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.3137	0.374840	0.187420	−	0.982280i	$-0.439987\pi$
$911$	11.3137	0.374840	0.187420	+	0.982280i	$0.439987\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	−56.5685	−1.86602	−0.933012	−	0.359845i	$-0.882829\pi$
$919$	−56.5685	−1.86602	−0.933012	+	0.359845i	$0.882829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.48528	−0.279296
$924$	0	0
$925$	−30.0000	−0.986394
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	67.8823	2.21055
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.7990	−0.643381	−0.321690	−	0.946845i	$-0.604251\pi$
$947$	−19.7990	−0.643381	−0.321690	+	0.946845i	$0.604251\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.9411	−1.09602
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.82843	0.0909561	0.0454780	−	0.998965i	$-0.485519\pi$
$967$	2.82843	0.0909561	0.0454780	+	0.998965i	$0.485519\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−45.2548	−1.45230	−0.726148	−	0.687538i	$-0.758691\pi$
$971$	−45.2548	−1.45230	−0.726148	+	0.687538i	$0.758691\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.0000	−1.66363	−0.831814	−	0.555055i	$-0.812697\pi$
$977$	−52.0000	−1.66363	−0.831814	+	0.555055i	$0.812697\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.7990	−0.631490	−0.315745	−	0.948844i	$-0.602254\pi$
$983$	−19.7990	−0.631490	−0.315745	+	0.948844i	$0.602254\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	28.2843	0.898479	0.449240	−	0.893411i	$-0.351695\pi$
$991$	28.2843	0.898479	0.449240	+	0.893411i	$0.351695\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.a.t.1.2	yes	2
3.2	odd	2		3744.2.a.u.1.2	yes	2
4.3	odd	2	inner	3744.2.a.t.1.1	✓	2
8.3	odd	2		7488.2.a.cn.1.1		2
8.5	even	2		7488.2.a.cn.1.2		2
12.11	even	2		3744.2.a.u.1.1	yes	2
24.5	odd	2		7488.2.a.cm.1.2		2
24.11	even	2		7488.2.a.cm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3744.2.a.t.1.1	✓	2	4.3	odd	2	inner
3744.2.a.t.1.2	yes	2	1.1	even	1	trivial
3744.2.a.u.1.1	yes	2	12.11	even	2
3744.2.a.u.1.2	yes	2	3.2	odd	2
7488.2.a.cm.1.1		2	24.11	even	2
7488.2.a.cm.1.2		2	24.5	odd	2
7488.2.a.cn.1.1		2	8.3	odd	2
7488.2.a.cn.1.2		2	8.5	even	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{7} +O(q^{10})\)	\(q+2.82843 q^{7} -2.82843 q^{11} -1.00000 q^{13} +2.82843 q^{19} -5.65685 q^{23} -5.00000 q^{25} -4.00000 q^{29} -8.48528 q^{31} +6.00000 q^{37} -12.0000 q^{41} -2.82843 q^{47} +1.00000 q^{49} -4.00000 q^{53} +14.1421 q^{59} +6.00000 q^{61} +8.48528 q^{67} +8.48528 q^{71} -10.0000 q^{73} -8.00000 q^{77} -11.3137 q^{79} -14.1421 q^{83} +4.00000 q^{89} -2.82843 q^{91} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 2 q^{13} - 10 q^{25} - 8 q^{29} + 12 q^{37} - 24 q^{41} + 2 q^{49} - 8 q^{53} + 12 q^{61} - 20 q^{73} - 16 q^{77} + 8 q^{89} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^13 - 10 * q^25 - 8 * q^29 + 12 * q^37 - 24 * q^41 + 2 * q^49 - 8 * q^53 + 12 * q^61 - 20 * q^73 - 16 * q^77 + 8 * q^89 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more