LMFDB - Embedded newform 8424.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8424,2,Mod(1,8424)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8424, 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("8424.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8424 = 2^{3} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8424.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:	$67.2659786627$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.9225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 7x + 19 $ x^4 - x^3 - 10x^2 + 7x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.69932$ of defining polynomial
Character	$\chi$	$=$	8424.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.69932 q^{5} -2.69932 q^{7} +3.28630 q^{11} +1.00000 q^{13} +1.51348 q^{17} +3.57261 q^{19} +6.75844 q^{23} +8.68493 q^{25} -7.75844 q^{29} -10.1715 q^{31} +9.98562 q^{35} -9.68493 q^{37} -9.67055 q^{41} +6.08609 q^{43} +0.486518 q^{47} +0.286303 q^{49} +5.95865 q^{53} -12.1571 q^{55} +3.57261 q^{59} +2.12671 q^{61} -3.69932 q^{65} -4.25414 q^{67} +4.89953 q^{71} -2.33951 q^{73} -8.87077 q^{77} -4.43998 q^{79} -16.9425 q^{83} -5.59885 q^{85} +7.78540 q^{89} -2.69932 q^{91} -13.2162 q^{95} +15.2982 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} + q^{7} + 5 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} + q^{23} + 3 q^{25} - 5 q^{29} - 11 q^{31} + 20 q^{35} - 7 q^{37} + 13 q^{41} + 6 q^{43} + 4 q^{47} - 7 q^{49} + 8 q^{53} + q^{55} - 2 q^{59}+ \cdots + 36 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 + q^7 + 5 * q^11 + 4 * q^13 + 4 * q^17 - 2 * q^19 + q^23 + 3 * q^25 - 5 * q^29 - 11 * q^31 + 20 * q^35 - 7 * q^37 + 13 * q^41 + 6 * q^43 + 4 * q^47 - 7 * q^49 + 8 * q^53 + q^55 - 2 * q^59 + 13 * q^61 - 3 * q^65 - 19 * q^67 + 18 * q^71 + 6 * q^77 - 10 * q^79 + 12 * q^83 - 9 * q^85 + q^89 + q^91 + 11 * q^95 + 36 * q^97

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
$5$	−3.69932	−1.65438	−0.827192	−	0.561919i	$-0.810063\pi$
$5$	−3.69932	−1.65438	−0.827192	+	0.561919i	$0.810063\pi$
$6$	0	0
$7$	−2.69932	−1.02025	−0.510123	−	0.860102i	$-0.670400\pi$
$7$	−2.69932	−1.02025	−0.510123	+	0.860102i	$0.670400\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.28630	0.990857	0.495429	−	0.868649i	$-0.335011\pi$
$11$	3.28630	0.990857	0.495429	+	0.868649i	$0.335011\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.51348	0.367073	0.183537	−	0.983013i	$-0.441245\pi$
$17$	1.51348	0.367073	0.183537	+	0.983013i	$0.441245\pi$
$18$	0	0
$19$	3.57261	0.819612	0.409806	−	0.912173i	$-0.365596\pi$
$19$	3.57261	0.819612	0.409806	+	0.912173i	$0.365596\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.75844	1.40923	0.704616	−	0.709589i	$-0.251119\pi$
$23$	6.75844	1.40923	0.704616	+	0.709589i	$0.251119\pi$
$24$	0	0
$25$	8.68493	1.73699
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.75844	−1.44071	−0.720353	−	0.693608i	$-0.756020\pi$
$29$	−7.75844	−1.44071	−0.720353	+	0.693608i	$0.756020\pi$
$30$	0	0
$31$	−10.1715	−1.82685	−0.913423	−	0.407011i	$-0.866571\pi$
$31$	−10.1715	−1.82685	−0.913423	+	0.407011i	$0.866571\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.98562	1.68788
$36$	0	0
$37$	−9.68493	−1.59219	−0.796097	−	0.605170i	$-0.793105\pi$
$37$	−9.68493	−1.59219	−0.796097	+	0.605170i	$0.793105\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.67055	−1.51029	−0.755143	−	0.655560i	$-0.772433\pi$
$41$	−9.67055	−1.51029	−0.755143	+	0.655560i	$0.772433\pi$
$42$	0	0
$43$	6.08609	0.928120	0.464060	−	0.885804i	$-0.346392\pi$
$43$	6.08609	0.928120	0.464060	+	0.885804i	$0.346392\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.486518	0.0709660	0.0354830	−	0.999370i	$-0.488703\pi$
$47$	0.486518	0.0709660	0.0354830	+	0.999370i	$0.488703\pi$
$48$	0	0
$49$	0.286303	0.0409004
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.95865	0.818484	0.409242	−	0.912426i	$-0.365793\pi$
$53$	5.95865	0.818484	0.409242	+	0.912426i	$0.365793\pi$
$54$	0	0
$55$	−12.1571	−1.63926
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.57261	0.465114	0.232557	−	0.972583i	$-0.425291\pi$
$59$	3.57261	0.465114	0.232557	+	0.972583i	$0.425291\pi$
$60$	0	0
$61$	2.12671	0.272297	0.136149	−	0.990688i	$-0.456528\pi$
$61$	2.12671	0.272297	0.136149	+	0.990688i	$0.456528\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.69932	−0.458844
$66$	0	0
$67$	−4.25414	−0.519726	−0.259863	−	0.965645i	$-0.583677\pi$
$67$	−4.25414	−0.519726	−0.259863	+	0.965645i	$0.583677\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.89953	0.581467	0.290734	−	0.956804i	$-0.406101\pi$
$71$	4.89953	0.581467	0.290734	+	0.956804i	$0.406101\pi$
$72$	0	0
$73$	−2.33951	−0.273819	−0.136909	−	0.990584i	$-0.543717\pi$
$73$	−2.33951	−0.273819	−0.136909	+	0.990584i	$0.543717\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.87077	−1.01092
$78$	0	0
$79$	−4.43998	−0.499536	−0.249768	−	0.968306i	$-0.580354\pi$
$79$	−4.43998	−0.499536	−0.249768	+	0.968306i	$0.580354\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.9425	−1.85968	−0.929839	−	0.367967i	$-0.880054\pi$
$83$	−16.9425	−1.85968	−0.929839	+	0.367967i	$0.880054\pi$
$84$	0	0
$85$	−5.59885	−0.607280
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.78540	0.825251	0.412625	−	0.910901i	$-0.364612\pi$
$89$	7.78540	0.825251	0.412625	+	0.910901i	$0.364612\pi$
$90$	0	0
$91$	−2.69932	−0.282965
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.2162	−1.35595
$96$	0	0
$97$	15.2982	1.55329	0.776646	−	0.629937i	$-0.216919\pi$
$97$	15.2982	1.55329	0.776646	+	0.629937i	$0.216919\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.8708	1.47970	0.739848	−	0.672774i	$-0.234897\pi$
$101$	14.8708	1.47970	0.739848	+	0.672774i	$0.234897\pi$
$102$	0	0
$103$	−7.71370	−0.760053	−0.380027	−	0.924976i	$-0.624085\pi$
$103$	−7.71370	−0.760053	−0.380027	+	0.924976i	$0.624085\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0979	1.45957	0.729787	−	0.683675i	$-0.239619\pi$
$107$	15.0979	1.45957	0.729787	+	0.683675i	$0.239619\pi$
$108$	0	0
$109$	−17.0160	−1.62983	−0.814917	−	0.579577i	$-0.803218\pi$
$109$	−17.0160	−1.62983	−0.814917	+	0.579577i	$0.803218\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.88515	−0.271412	−0.135706	−	0.990749i	$-0.543330\pi$
$113$	−2.88515	−0.271412	−0.135706	+	0.990749i	$0.543330\pi$
$114$	0	0
$115$	−25.0016	−2.33141
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.08536	−0.374505
$120$	0	0
$121$	−0.200216	−0.0182014
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−13.6317	−1.21926
$126$	0	0
$127$	10.3717	0.920337	0.460168	−	0.887832i	$-0.347789\pi$
$127$	10.3717	0.920337	0.460168	+	0.887832i	$0.347789\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.9180	−1.21602	−0.608012	−	0.793928i	$-0.708033\pi$
$131$	−13.9180	−1.21602	−0.608012	+	0.793928i	$0.708033\pi$
$132$	0	0
$133$	−9.64359	−0.836205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.94680	0.508069	0.254035	−	0.967195i	$-0.418242\pi$
$137$	5.94680	0.508069	0.254035	+	0.967195i	$0.418242\pi$
$138$	0	0
$139$	−11.2450	−0.953785	−0.476892	−	0.878962i	$-0.658237\pi$
$139$	−11.2450	−0.953785	−0.476892	+	0.878962i	$0.658237\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.28630	0.274814
$144$	0	0
$145$	28.7009	2.38348
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.19175	0.753018	0.376509	−	0.926413i	$-0.377124\pi$
$149$	9.19175	0.753018	0.376509	+	0.926413i	$0.377124\pi$
$150$	0	0
$151$	8.35981	0.680312	0.340156	−	0.940369i	$-0.389520\pi$
$151$	8.35981	0.680312	0.340156	+	0.940369i	$0.389520\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.6274	3.02231
$156$	0	0
$157$	−18.8715	−1.50611	−0.753054	−	0.657958i	$-0.771420\pi$
$157$	−18.8715	−1.50611	−0.753054	+	0.657958i	$0.771420\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−18.2432	−1.43776
$162$	0	0
$163$	10.7973	0.845707	0.422853	−	0.906198i	$-0.361029\pi$
$163$	10.7973	0.845707	0.422853	+	0.906198i	$0.361029\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.87077	0.686441	0.343220	−	0.939255i	$-0.388482\pi$
$167$	8.87077	0.686441	0.343220	+	0.939255i	$0.388482\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.445172	0.0338458	0.0169229	−	0.999857i	$-0.494613\pi$
$173$	0.445172	0.0338458	0.0169229	+	0.999857i	$0.494613\pi$
$174$	0	0
$175$	−23.4434	−1.77215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.5379	1.75931	0.879653	−	0.475615i	$-0.157775\pi$
$179$	23.5379	1.75931	0.879653	+	0.475615i	$0.157775\pi$
$180$	0	0
$181$	−22.8420	−1.69783	−0.848916	−	0.528527i	$-0.822744\pi$
$181$	−22.8420	−1.69783	−0.848916	+	0.528527i	$0.822744\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	35.8276	2.63410
$186$	0	0
$187$	4.97376	0.363717
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.65797	0.337039	0.168519	−	0.985698i	$-0.446101\pi$
$191$	4.65797	0.337039	0.168519	+	0.985698i	$0.446101\pi$
$192$	0	0
$193$	−12.5870	−0.906031	−0.453016	−	0.891503i	$-0.649652\pi$
$193$	−12.5870	−0.906031	−0.453016	+	0.891503i	$0.649652\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.8833	−1.20289	−0.601444	−	0.798915i	$-0.705408\pi$
$197$	−16.8833	−1.20289	−0.601444	+	0.798915i	$0.705408\pi$
$198$	0	0
$199$	1.73814	0.123213	0.0616067	−	0.998101i	$-0.480378\pi$
$199$	1.73814	0.123213	0.0616067	+	0.998101i	$0.480378\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.9425	1.46987
$204$	0	0
$205$	35.7744	2.49859
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.7407	0.812119
$210$	0	0
$211$	18.4678	1.27138	0.635688	−	0.771946i	$-0.280716\pi$
$211$	18.4678	1.27138	0.635688	+	0.771946i	$0.280716\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−22.5144	−1.53547
$216$	0	0
$217$	27.4560	1.86383
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.51348	0.101808
$222$	0	0
$223$	−21.7829	−1.45869	−0.729345	−	0.684146i	$-0.760175\pi$
$223$	−21.7829	−1.45869	−0.729345	+	0.684146i	$0.760175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3665	0.688047	0.344023	−	0.938961i	$-0.388210\pi$
$227$	10.3665	0.688047	0.344023	+	0.938961i	$0.388210\pi$
$228$	0	0
$229$	21.5735	1.42562	0.712808	−	0.701359i	$-0.247423\pi$
$229$	21.5735	1.42562	0.712808	+	0.701359i	$0.247423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.9417	0.978866	0.489433	−	0.872041i	$-0.337204\pi$
$233$	14.9417	0.978866	0.489433	+	0.872041i	$0.337204\pi$
$234$	0	0
$235$	−1.79978	−0.117405
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.4696	1.19470	0.597350	−	0.801981i	$-0.296220\pi$
$239$	18.4696	1.19470	0.597350	+	0.801981i	$0.296220\pi$
$240$	0	0
$241$	−1.81756	−0.117079	−0.0585397	−	0.998285i	$-0.518644\pi$
$241$	−1.81756	−0.117079	−0.0585397	+	0.998285i	$0.518644\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.05912	−0.0676649
$246$	0	0
$247$	3.57261	0.227319
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.1858	0.895402	0.447701	−	0.894183i	$-0.352243\pi$
$251$	14.1858	0.895402	0.447701	+	0.894183i	$0.352243\pi$
$252$	0	0
$253$	22.2103	1.39635
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.83014	−0.114161	−0.0570806	−	0.998370i	$-0.518179\pi$
$257$	−1.83014	−0.114161	−0.0570806	+	0.998370i	$0.518179\pi$
$258$	0	0
$259$	26.1427	1.62443
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−0.185833	−0.0114590	−0.00572949	−	0.999984i	$-0.501824\pi$
$263$	−0.185833	−0.0114590	−0.00572949	+	0.999984i	$0.501824\pi$
$264$	0	0
$265$	−22.0429	−1.35409
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.0023	1.76830	0.884151	−	0.467201i	$-0.154737\pi$
$269$	29.0023	1.76830	0.884151	+	0.467201i	$0.154737\pi$
$270$	0	0
$271$	3.89773	0.236770	0.118385	−	0.992968i	$-0.462228\pi$
$271$	3.89773	0.236770	0.118385	+	0.992968i	$0.462228\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	28.5413	1.72111
$276$	0	0
$277$	29.1039	1.74868	0.874341	−	0.485312i	$-0.161294\pi$
$277$	29.1039	1.74868	0.874341	+	0.485312i	$0.161294\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.05985	0.0632251	0.0316125	−	0.999500i	$-0.489936\pi$
$281$	1.05985	0.0632251	0.0316125	+	0.999500i	$0.489936\pi$
$282$	0	0
$283$	10.6502	0.633092	0.316546	−	0.948577i	$-0.397477\pi$
$283$	10.6502	0.633092	0.316546	+	0.948577i	$0.397477\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.1039	1.54086
$288$	0	0
$289$	−14.7094	−0.865257
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.5726	0.734499	0.367250	−	0.930122i	$-0.380299\pi$
$293$	12.5726	0.734499	0.367250	+	0.930122i	$0.380299\pi$
$294$	0	0
$295$	−13.2162	−0.769476
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.75844	0.390851
$300$	0	0
$301$	−16.4283	−0.946910
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.86737	−0.450484
$306$	0	0
$307$	−13.6233	−0.777522	−0.388761	−	0.921339i	$-0.627097\pi$
$307$	−13.6233	−0.777522	−0.388761	+	0.921339i	$0.627097\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.1427	−0.688549	−0.344274	−	0.938869i	$-0.611875\pi$
$311$	−12.1427	−0.688549	−0.344274	+	0.938869i	$0.611875\pi$
$312$	0	0
$313$	−1.11825	−0.0632070	−0.0316035	−	0.999500i	$-0.510061\pi$
$313$	−1.11825	−0.0632070	−0.0316035	+	0.999500i	$0.510061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0067	0.562030	0.281015	−	0.959703i	$-0.409329\pi$
$317$	10.0067	0.562030	0.281015	+	0.959703i	$0.409329\pi$
$318$	0	0
$319$	−25.4966	−1.42753
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.40707	0.300858
$324$	0	0
$325$	8.68493	0.481753
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.31327	−0.0724027
$330$	0	0
$331$	10.7399	0.590320	0.295160	−	0.955448i	$-0.404627\pi$
$331$	10.7399	0.590320	0.295160	+	0.955448i	$0.404627\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.7374	0.859827
$336$	0	0
$337$	30.1350	1.64156	0.820778	−	0.571247i	$-0.193540\pi$
$337$	30.1350	1.64156	0.820778	+	0.571247i	$0.193540\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−33.4265	−1.81014
$342$	0	0
$343$	18.1224	0.978517
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.7913	−1.22350	−0.611751	−	0.791050i	$-0.709535\pi$
$347$	−22.7913	−1.22350	−0.611751	+	0.791050i	$0.709535\pi$
$348$	0	0
$349$	20.7728	1.11194	0.555972	−	0.831201i	$-0.312346\pi$
$349$	20.7728	1.11194	0.555972	+	0.831201i	$0.312346\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.1527	−1.17907	−0.589536	−	0.807742i	$-0.700689\pi$
$353$	−22.1527	−1.17907	−0.589536	+	0.807742i	$0.700689\pi$
$354$	0	0
$355$	−18.1249	−0.961970
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.28450	−0.490017	−0.245009	−	0.969521i	$-0.578791\pi$
$359$	−9.28450	−0.490017	−0.245009	+	0.969521i	$0.578791\pi$
$360$	0	0
$361$	−6.23649	−0.328236
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.65457	0.453001
$366$	0	0
$367$	7.38425	0.385455	0.192727	−	0.981252i	$-0.438267\pi$
$367$	7.38425	0.385455	0.192727	+	0.981252i	$0.438267\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16.0843	−0.835055
$372$	0	0
$373$	3.49658	0.181046	0.0905229	−	0.995894i	$-0.471146\pi$
$373$	3.49658	0.181046	0.0905229	+	0.995894i	$0.471146\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.75844	−0.399580
$378$	0	0
$379$	−24.6891	−1.26819	−0.634096	−	0.773255i	$-0.718627\pi$
$379$	−24.6891	−1.26819	−0.634096	+	0.773255i	$0.718627\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.3488	−0.988680	−0.494340	−	0.869269i	$-0.664590\pi$
$383$	−19.3488	−0.988680	−0.494340	+	0.869269i	$0.664590\pi$
$384$	0	0
$385$	32.8158	1.67245
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.2831	0.774882	0.387441	−	0.921895i	$-0.373359\pi$
$389$	15.2831	0.774882	0.387441	+	0.921895i	$0.373359\pi$
$390$	0	0
$391$	10.2288	0.517291
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.4249	0.826425
$396$	0	0
$397$	33.7751	1.69513	0.847563	−	0.530695i	$-0.178069\pi$
$397$	33.7751	1.69513	0.847563	+	0.530695i	$0.178069\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0633	0.902036	0.451018	−	0.892515i	$-0.351061\pi$
$401$	18.0633	0.902036	0.451018	+	0.892515i	$0.351061\pi$
$402$	0	0
$403$	−10.1715	−0.506676
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−31.8276	−1.57764
$408$	0	0
$409$	21.1486	1.04573	0.522866	−	0.852415i	$-0.324863\pi$
$409$	21.1486	1.04573	0.522866	+	0.852415i	$0.324863\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.64359	−0.474530
$414$	0	0
$415$	62.6755	3.07662
$416$	0	0
$417$	0	0
$418$	0	0
$419$	40.0379	1.95598	0.977990	−	0.208654i	$-0.0669082\pi$
$419$	40.0379	1.95598	0.977990	+	0.208654i	$0.0669082\pi$
$420$	0	0
$421$	−2.24243	−0.109290	−0.0546448	−	0.998506i	$-0.517403\pi$
$421$	−2.24243	−0.109290	−0.0546448	+	0.998506i	$0.517403\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.1445	0.637601
$426$	0	0
$427$	−5.74066	−0.277810
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.7541	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$431$	−17.7541	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$432$	0	0
$433$	26.4223	1.26978	0.634889	−	0.772604i	$-0.281046\pi$
$433$	26.4223	1.26978	0.634889	+	0.772604i	$0.281046\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.1452	1.15502
$438$	0	0
$439$	−0.408874	−0.0195145	−0.00975724	−	0.999952i	$-0.503106\pi$
$439$	−0.408874	−0.0195145	−0.00975724	+	0.999952i	$0.503106\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.2820	1.05865	0.529324	−	0.848420i	$-0.322445\pi$
$443$	22.2820	1.05865	0.529324	+	0.848420i	$0.322445\pi$
$444$	0	0
$445$	−28.8007	−1.36528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.0573	1.13534	0.567668	−	0.823258i	$-0.307846\pi$
$449$	24.0573	1.13534	0.567668	+	0.823258i	$0.307846\pi$
$450$	0	0
$451$	−31.7804	−1.49648
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.98562	0.468133
$456$	0	0
$457$	−4.98042	−0.232974	−0.116487	−	0.993192i	$-0.537163\pi$
$457$	−4.98042	−0.232974	−0.116487	+	0.993192i	$0.537163\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	38.9551	1.81432	0.907159	−	0.420788i	$-0.138247\pi$
$461$	38.9551	1.81432	0.907159	+	0.420788i	$0.138247\pi$
$462$	0	0
$463$	−10.4418	−0.485271	−0.242635	−	0.970118i	$-0.578012\pi$
$463$	−10.4418	−0.485271	−0.242635	+	0.970118i	$0.578012\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.3876	−0.526957	−0.263479	−	0.964665i	$-0.584870\pi$
$467$	−11.3876	−0.526957	−0.263479	+	0.964665i	$0.584870\pi$
$468$	0	0
$469$	11.4833	0.530248
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.0007	0.919634
$474$	0	0
$475$	31.0278	1.42365
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−18.6377	−0.851577	−0.425788	−	0.904823i	$-0.640003\pi$
$479$	−18.6377	−0.851577	−0.425788	+	0.904823i	$0.640003\pi$
$480$	0	0
$481$	−9.68493	−0.441595
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−56.5927	−2.56974
$486$	0	0
$487$	21.1849	0.959980	0.479990	−	0.877274i	$-0.340640\pi$
$487$	21.1849	0.959980	0.479990	+	0.877274i	$0.340640\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.1664	−1.81268	−0.906342	−	0.422544i	$-0.861137\pi$
$491$	−40.1664	−1.81268	−0.906342	+	0.422544i	$0.861137\pi$
$492$	0	0
$493$	−11.7423	−0.528845
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.2254	−0.593239
$498$	0	0
$499$	14.1409	0.633033	0.316516	−	0.948587i	$-0.397487\pi$
$499$	14.1409	0.633033	0.316516	+	0.948587i	$0.397487\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.72808	0.0770513	0.0385256	−	0.999258i	$-0.487734\pi$
$503$	1.72808	0.0770513	0.0385256	+	0.999258i	$0.487734\pi$
$504$	0	0
$505$	−55.0117	−2.44799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.8808	1.76769	0.883843	−	0.467783i	$-0.154947\pi$
$509$	39.8808	1.76769	0.883843	+	0.467783i	$0.154947\pi$
$510$	0	0
$511$	6.31507	0.279362
$512$	0	0
$513$	0	0
$514$	0	0
$515$	28.5354	1.25742
$516$	0	0
$517$	1.59885	0.0703172
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.714420	0.0312993	0.0156497	−	0.999878i	$-0.495018\pi$
$521$	0.714420	0.0312993	0.0156497	+	0.999878i	$0.495018\pi$
$522$	0	0
$523$	−33.0109	−1.44347	−0.721734	−	0.692171i	$-0.756654\pi$
$523$	−33.0109	−1.44347	−0.721734	+	0.692171i	$0.756654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.3943	−0.670586
$528$	0	0
$529$	22.6765	0.985934
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.67055	−0.418878
$534$	0	0
$535$	−55.8521	−2.41470
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.940877	0.0405264
$540$	0	0
$541$	−4.57855	−0.196847	−0.0984235	−	0.995145i	$-0.531380\pi$
$541$	−4.57855	−0.196847	−0.0984235	+	0.995145i	$0.531380\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	62.9475	2.69637
$546$	0	0
$547$	14.3141	0.612028	0.306014	−	0.952027i	$-0.401005\pi$
$547$	14.3141	0.612028	0.306014	+	0.952027i	$0.401005\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.7178	−1.18082
$552$	0	0
$553$	11.9849	0.509650
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.6022	0.830574	0.415287	−	0.909691i	$-0.363681\pi$
$557$	19.6022	0.830574	0.415287	+	0.909691i	$0.363681\pi$
$558$	0	0
$559$	6.08609	0.257414
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.443371	0.0186859	0.00934294	−	0.999956i	$-0.497026\pi$
$563$	0.443371	0.0186859	0.00934294	+	0.999956i	$0.497026\pi$
$564$	0	0
$565$	10.6731	0.449020
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.2288	1.35110	0.675550	−	0.737314i	$-0.263906\pi$
$569$	32.2288	1.35110	0.675550	+	0.737314i	$0.263906\pi$
$570$	0	0
$571$	7.93169	0.331931	0.165965	−	0.986132i	$-0.446926\pi$
$571$	7.93169	0.331931	0.165965	+	0.986132i	$0.446926\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	58.6966	2.44782
$576$	0	0
$577$	26.7769	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$577$	26.7769	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	45.7331	1.89733
$582$	0	0
$583$	19.5819	0.811001
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.50916	−0.309936	−0.154968	−	0.987919i	$-0.549527\pi$
$587$	−7.50916	−0.309936	−0.154968	+	0.987919i	$0.549527\pi$
$588$	0	0
$589$	−36.3386	−1.49731
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.3530	0.507275	0.253638	−	0.967299i	$-0.418373\pi$
$593$	12.3530	0.507275	0.253638	+	0.967299i	$0.418373\pi$
$594$	0	0
$595$	15.1131	0.619575
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−33.1797	−1.35569	−0.677843	−	0.735207i	$-0.737085\pi$
$599$	−33.1797	−1.35569	−0.677843	+	0.735207i	$0.737085\pi$
$600$	0	0
$601$	−21.9898	−0.896981	−0.448490	−	0.893788i	$-0.648038\pi$
$601$	−21.9898	−0.896981	−0.448490	+	0.893788i	$0.648038\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.740661	0.0301122
$606$	0	0
$607$	19.6799	0.798782	0.399391	−	0.916781i	$-0.369222\pi$
$607$	19.6799	0.798782	0.399391	+	0.916781i	$0.369222\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.486518	0.0196824
$612$	0	0
$613$	−7.39431	−0.298653	−0.149327	−	0.988788i	$-0.547711\pi$
$613$	−7.39431	−0.298653	−0.149327	+	0.988788i	$0.547711\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.5507	−1.67277	−0.836383	−	0.548145i	$-0.815334\pi$
$617$	−41.5507	−1.67277	−0.836383	+	0.548145i	$0.815334\pi$
$618$	0	0
$619$	5.88695	0.236616	0.118308	−	0.992977i	$-0.462253\pi$
$619$	5.88695	0.236616	0.118308	+	0.992977i	$0.462253\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21.0153	−0.841958
$624$	0	0
$625$	7.00340	0.280136
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.6580	−0.584452
$630$	0	0
$631$	−2.73992	−0.109074	−0.0545372	−	0.998512i	$-0.517368\pi$
$631$	−2.73992	−0.109074	−0.0545372	+	0.998512i	$0.517368\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−38.3681	−1.52259
$636$	0	0
$637$	0.286303	0.0113437
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.3116	−0.723265	−0.361633	−	0.932321i	$-0.617781\pi$
$641$	−18.3116	−0.723265	−0.361633	+	0.932321i	$0.617781\pi$
$642$	0	0
$643$	36.1324	1.42492	0.712462	−	0.701710i	$-0.247580\pi$
$643$	36.1324	1.42492	0.712462	+	0.701710i	$0.247580\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.594707	0.0233803	0.0116902	−	0.999932i	$-0.496279\pi$
$647$	0.594707	0.0233803	0.0116902	+	0.999932i	$0.496279\pi$
$648$	0	0
$649$	11.7407	0.460861
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.30843	0.0512027	0.0256013	−	0.999672i	$-0.491850\pi$
$653$	1.30843	0.0512027	0.0256013	+	0.999672i	$0.491850\pi$
$654$	0	0
$655$	51.4872	2.01177
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.6459	1.73916	0.869579	−	0.493794i	$-0.164390\pi$
$659$	44.6459	1.73916	0.869579	+	0.493794i	$0.164390\pi$
$660$	0	0
$661$	−38.4036	−1.49373	−0.746864	−	0.664977i	$-0.768441\pi$
$661$	−38.4036	−1.49373	−0.746864	+	0.664977i	$0.768441\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	35.6747	1.38340
$666$	0	0
$667$	−52.4349	−2.03029
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.98901	0.269808
$672$	0	0
$673$	−22.9173	−0.883397	−0.441699	−	0.897163i	$-0.645624\pi$
$673$	−22.9173	−0.883397	−0.441699	+	0.897163i	$0.645624\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.1064	0.388420	0.194210	−	0.980960i	$-0.437786\pi$
$677$	10.1064	0.388420	0.194210	+	0.980960i	$0.437786\pi$
$678$	0	0
$679$	−41.2946	−1.58474
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.0818	−0.462296	−0.231148	−	0.972919i	$-0.574248\pi$
$683$	−12.0818	−0.462296	−0.231148	+	0.972919i	$0.574248\pi$
$684$	0	0
$685$	−21.9991	−0.840541
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.95865	0.227007
$690$	0	0
$691$	6.36719	0.242219	0.121110	−	0.992639i	$-0.461355\pi$
$691$	6.36719	0.242219	0.121110	+	0.992639i	$0.461355\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	41.5986	1.57793
$696$	0	0
$697$	−14.6362	−0.554386
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.6030	0.513778	0.256889	−	0.966441i	$-0.417303\pi$
$701$	13.6030	0.513778	0.256889	+	0.966441i	$0.417303\pi$
$702$	0	0
$703$	−34.6004	−1.30498
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.1409	−1.50965
$708$	0	0
$709$	−5.97051	−0.224227	−0.112114	−	0.993695i	$-0.535762\pi$
$709$	−5.97051	−0.224227	−0.112114	+	0.993695i	$0.535762\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−68.7431	−2.57445
$714$	0	0
$715$	−12.1571	−0.454649
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.09527	0.0408468	0.0204234	−	0.999791i	$-0.493499\pi$
$719$	1.09527	0.0408468	0.0204234	+	0.999791i	$0.493499\pi$
$720$	0	0
$721$	20.8217	0.775441
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−67.3815	−2.50249
$726$	0	0
$727$	−5.09056	−0.188798	−0.0943992	−	0.995534i	$-0.530093\pi$
$727$	−5.09056	−0.188798	−0.0943992	+	0.995534i	$0.530093\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.21118	0.340688
$732$	0	0
$733$	8.90617	0.328957	0.164478	−	0.986381i	$-0.447406\pi$
$733$	8.90617	0.328957	0.164478	+	0.986381i	$0.447406\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.9804	−0.514975
$738$	0	0
$739$	−7.99480	−0.294094	−0.147047	−	0.989130i	$-0.546977\pi$
$739$	−7.99480	−0.294094	−0.147047	+	0.989130i	$0.546977\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.5530	0.900763	0.450382	−	0.892836i	$-0.351288\pi$
$743$	24.5530	0.900763	0.450382	+	0.892836i	$0.351288\pi$
$744$	0	0
$745$	−34.0032	−1.24578
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−40.7541	−1.48912
$750$	0	0
$751$	−17.4315	−0.636085	−0.318042	−	0.948076i	$-0.603025\pi$
$751$	−17.4315	−0.636085	−0.318042	+	0.948076i	$0.603025\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.9256	−1.12550
$756$	0	0
$757$	21.3041	0.774310	0.387155	−	0.922015i	$-0.373458\pi$
$757$	21.3041	0.774310	0.387155	+	0.922015i	$0.373458\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.3470	−1.02758	−0.513789	−	0.857916i	$-0.671759\pi$
$761$	−28.3470	−1.02758	−0.513789	+	0.857916i	$0.671759\pi$
$762$	0	0
$763$	45.9315	1.66283
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.57261	0.128999
$768$	0	0
$769$	−8.65025	−0.311936	−0.155968	−	0.987762i	$-0.549850\pi$
$769$	−8.65025	−0.311936	−0.155968	+	0.987762i	$0.549850\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−44.3715	−1.59593	−0.797965	−	0.602703i	$-0.794090\pi$
$773$	−44.3715	−1.59593	−0.797965	+	0.602703i	$0.794090\pi$
$774$	0	0
$775$	−88.3384	−3.17321
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34.5491	−1.23785
$780$	0	0
$781$	16.1013	0.576151
$782$	0	0
$783$	0	0
$784$	0	0
$785$	69.8116	2.49168
$786$	0	0
$787$	−44.3438	−1.58068	−0.790342	−	0.612665i	$-0.790097\pi$
$787$	−44.3438	−1.58068	−0.790342	+	0.612665i	$0.790097\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.78793	0.276907
$792$	0	0
$793$	2.12671	0.0755217
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.3244	−0.542818	−0.271409	−	0.962464i	$-0.587490\pi$
$797$	−15.3244	−0.542818	−0.271409	+	0.962464i	$0.587490\pi$
$798$	0	0
$799$	0.736336	0.0260497
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.68833	−0.271315
$804$	0	0
$805$	67.4872	2.37861
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.4610	−0.438105	−0.219053	−	0.975713i	$-0.570297\pi$
$809$	−12.4610	−0.438105	−0.219053	+	0.975713i	$0.570297\pi$
$810$	0	0
$811$	−4.53453	−0.159229	−0.0796144	−	0.996826i	$-0.525369\pi$
$811$	−4.53453	−0.159229	−0.0796144	+	0.996826i	$0.525369\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−39.9425	−1.39912
$816$	0	0
$817$	21.7432	0.760698
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.1519	0.703304	0.351652	−	0.936131i	$-0.385620\pi$
$821$	20.1519	0.703304	0.351652	+	0.936131i	$0.385620\pi$
$822$	0	0
$823$	−35.6317	−1.24204	−0.621022	−	0.783793i	$-0.713282\pi$
$823$	−35.6317	−1.24204	−0.621022	+	0.783793i	$0.713282\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0235	1.87858	0.939291	−	0.343122i	$-0.111485\pi$
$827$	54.0235	1.87858	0.939291	+	0.343122i	$0.111485\pi$
$828$	0	0
$829$	42.1874	1.46523	0.732615	−	0.680643i	$-0.238300\pi$
$829$	42.1874	1.46523	0.732615	+	0.680643i	$0.238300\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.433314	0.0150134
$834$	0	0
$835$	−32.8158	−1.13564
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−56.8783	−1.96366	−0.981829	−	0.189770i	$-0.939226\pi$
$839$	−56.8783	−1.96366	−0.981829	+	0.189770i	$0.939226\pi$
$840$	0	0
$841$	31.1934	1.07563
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.69932	−0.127260
$846$	0	0
$847$	0.540445	0.0185699
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−65.4550	−2.24377
$852$	0	0
$853$	−29.6452	−1.01503	−0.507516	−	0.861642i	$-0.669436\pi$
$853$	−29.6452	−1.01503	−0.507516	+	0.861642i	$0.669436\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.50756	0.119816	0.0599080	−	0.998204i	$-0.480919\pi$
$857$	3.50756	0.119816	0.0599080	+	0.998204i	$0.480919\pi$
$858$	0	0
$859$	−45.5337	−1.55359	−0.776795	−	0.629753i	$-0.783156\pi$
$859$	−45.5337	−1.55359	−0.776795	+	0.629753i	$0.783156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.7009	0.466384	0.233192	−	0.972431i	$-0.425083\pi$
$863$	13.7009	0.466384	0.233192	+	0.972431i	$0.425083\pi$
$864$	0	0
$865$	−1.64683	−0.0559940
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.5911	−0.494969
$870$	0	0
$871$	−4.25414	−0.144146
$872$	0	0
$873$	0	0
$874$	0	0
$875$	36.7963	1.24394
$876$	0	0
$877$	22.2180	0.750248	0.375124	−	0.926975i	$-0.377600\pi$
$877$	22.2180	0.750248	0.375124	+	0.926975i	$0.377600\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.8007	1.03770	0.518850	−	0.854865i	$-0.326360\pi$
$881$	30.8007	1.03770	0.518850	+	0.854865i	$0.326360\pi$
$882$	0	0
$883$	7.46354	0.251168	0.125584	−	0.992083i	$-0.459919\pi$
$883$	7.46354	0.251168	0.125584	+	0.992083i	$0.459919\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.0327	1.31059	0.655295	−	0.755373i	$-0.272544\pi$
$887$	39.0327	1.31059	0.655295	+	0.755373i	$0.272544\pi$
$888$	0	0
$889$	−27.9964	−0.938969
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.73814	0.0581646
$894$	0	0
$895$	−87.0742	−2.91057
$896$	0	0
$897$	0	0
$898$	0	0
$899$	78.9146	2.63195
$900$	0	0
$901$	9.01831	0.300444
$902$	0	0
$903$	0	0
$904$	0	0
$905$	84.4998	2.80887
$906$	0	0
$907$	−30.1763	−1.00199	−0.500994	−	0.865451i	$-0.667032\pi$
$907$	−30.1763	−1.00199	−0.500994	+	0.865451i	$0.667032\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.0877	−0.831192	−0.415596	−	0.909549i	$-0.636427\pi$
$911$	−25.0877	−0.831192	−0.415596	+	0.909549i	$0.636427\pi$
$912$	0	0
$913$	−55.6781	−1.84268
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.5692	1.24064
$918$	0	0
$919$	−55.7058	−1.83756	−0.918782	−	0.394765i	$-0.870826\pi$
$919$	−55.7058	−1.83756	−0.918782	+	0.394765i	$0.870826\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.89953	0.161270
$924$	0	0
$925$	−84.1130	−2.76562
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.634974	−0.0208328	−0.0104164	−	0.999946i	$-0.503316\pi$
$929$	−0.634974	−0.0208328	−0.0104164	+	0.999946i	$0.503316\pi$
$930$	0	0
$931$	1.02285	0.0335224
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.3995	−0.601728
$936$	0	0
$937$	22.1079	0.722232	0.361116	−	0.932521i	$-0.382396\pi$
$937$	22.1079	0.722232	0.361116	+	0.932521i	$0.382396\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.9171	1.69245	0.846225	−	0.532826i	$-0.178870\pi$
$941$	51.9171	1.69245	0.846225	+	0.532826i	$0.178870\pi$
$942$	0	0
$943$	−65.3578	−2.12834
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.7608	−1.19456	−0.597282	−	0.802031i	$-0.703753\pi$
$947$	−36.7608	−1.19456	−0.597282	+	0.802031i	$0.703753\pi$
$948$	0	0
$949$	−2.33951	−0.0759436
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7.77534	−0.251868	−0.125934	−	0.992039i	$-0.540193\pi$
$953$	−7.77534	−0.251868	−0.125934	+	0.992039i	$0.540193\pi$
$954$	0	0
$955$	−17.2313	−0.557592
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.0523	−0.518355
$960$	0	0
$961$	72.4584	2.33737
$962$	0	0
$963$	0	0
$964$	0	0
$965$	46.5632	1.49892
$966$	0	0
$967$	50.4450	1.62220	0.811100	−	0.584907i	$-0.198869\pi$
$967$	50.4450	1.62220	0.811100	+	0.584907i	$0.198869\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.6628	0.759376	0.379688	−	0.925115i	$-0.376031\pi$
$971$	23.6628	0.759376	0.379688	+	0.925115i	$0.376031\pi$
$972$	0	0
$973$	30.3537	0.973094
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.8370	1.59443	0.797213	−	0.603698i	$-0.206307\pi$
$977$	49.8370	1.59443	0.797213	+	0.603698i	$0.206307\pi$
$978$	0	0
$979$	25.5852	0.817706
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.6427	−0.530820	−0.265410	−	0.964136i	$-0.585507\pi$
$983$	−16.6427	−0.530820	−0.265410	+	0.964136i	$0.585507\pi$
$984$	0	0
$985$	62.4568	1.99004
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.1324	1.30794
$990$	0	0
$991$	16.9096	0.537150	0.268575	−	0.963259i	$-0.413447\pi$
$991$	16.9096	0.537150	0.268575	+	0.963259i	$0.413447\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.42992	−0.203842
$996$	0	0
$997$	−50.7245	−1.60646	−0.803230	−	0.595669i	$-0.796887\pi$
$997$	−50.7245	−1.60646	−0.803230	+	0.595669i	$0.796887\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8424.2.a.o.1.1	✓	4
3.2	odd	2		8424.2.a.r.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8424.2.a.o.1.1	✓	4	1.1	even	1	trivial
8424.2.a.r.1.4	yes	4	3.2	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:	\( N \)	\(=\)	\( 8424 = 2^{3} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8424.a (trivial)

\(f(q)\)	\(=\)	\(q-3.69932 q^{5} -2.69932 q^{7} +3.28630 q^{11} +1.00000 q^{13} +1.51348 q^{17} +3.57261 q^{19} +6.75844 q^{23} +8.68493 q^{25} -7.75844 q^{29} -10.1715 q^{31} +9.98562 q^{35} -9.68493 q^{37} -9.67055 q^{41} +6.08609 q^{43} +0.486518 q^{47} +0.286303 q^{49} +5.95865 q^{53} -12.1571 q^{55} +3.57261 q^{59} +2.12671 q^{61} -3.69932 q^{65} -4.25414 q^{67} +4.89953 q^{71} -2.33951 q^{73} -8.87077 q^{77} -4.43998 q^{79} -16.9425 q^{83} -5.59885 q^{85} +7.78540 q^{89} -2.69932 q^{91} -13.2162 q^{95} +15.2982 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} + q^{7} + 5 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} + q^{23} + 3 q^{25} - 5 q^{29} - 11 q^{31} + 20 q^{35} - 7 q^{37} + 13 q^{41} + 6 q^{43} + 4 q^{47} - 7 q^{49} + 8 q^{53} + q^{55} - 2 q^{59}+ \cdots + 36 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 + q^7 + 5 * q^11 + 4 * q^13 + 4 * q^17 - 2 * q^19 + q^23 + 3 * q^25 - 5 * q^29 - 11 * q^31 + 20 * q^35 - 7 * q^37 + 13 * q^41 + 6 * q^43 + 4 * q^47 - 7 * q^49 + 8 * q^53 + q^55 - 2 * q^59 + 13 * q^61 - 3 * q^65 - 19 * q^67 + 18 * q^71 + 6 * q^77 - 10 * q^79 + 12 * q^83 - 9 * q^85 + q^89 + q^91 + 11 * q^95 + 36 * q^97

Introduction

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