LMFDB - Embedded newform 8352.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8352,2,Mod(1,8352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8352, 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("8352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8352 = 2^{5} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8352.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:	$66.6910557682$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	8352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.23607 q^{5} -2.35114 q^{7} +O(q^{10})$	$q+4.23607 q^{5} -2.35114 q^{7} +4.53077 q^{11} -1.76393 q^{13} +5.23607 q^{17} -6.15537 q^{19} -3.80423 q^{23} +12.9443 q^{25} -1.00000 q^{29} -0.726543 q^{31} -9.95959 q^{35} +2.47214 q^{37} +7.23607 q^{41} +5.98385 q^{43} -5.42882 q^{47} -1.47214 q^{49} +3.76393 q^{53} +19.1926 q^{55} +6.71040 q^{59} -2.76393 q^{61} -7.47214 q^{65} +12.3107 q^{67} -3.24920 q^{71} -4.94427 q^{73} -10.6525 q^{77} +2.17963 q^{79} +16.1150 q^{83} +22.1803 q^{85} -1.70820 q^{89} +4.14725 q^{91} -26.0746 q^{95} +9.70820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^5	$ 4 q + 8 q^{5} - 16 q^{13} + 12 q^{17} + 16 q^{25} - 4 q^{29} - 8 q^{37} + 20 q^{41} + 12 q^{49} + 24 q^{53} - 20 q^{61} - 12 q^{65} + 16 q^{73} + 20 q^{77} + 44 q^{85} + 20 q^{89} + 12 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 16 * q^13 + 12 * q^17 + 16 * q^25 - 4 * q^29 - 8 * q^37 + 20 * q^41 + 12 * q^49 + 24 * q^53 - 20 * q^61 - 12 * q^65 + 16 * q^73 + 20 * q^77 + 44 * q^85 + 20 * q^89 + 12 * 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
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	−2.35114	−0.888648	−0.444324	−	0.895866i	$-0.646556\pi$
$7$	−2.35114	−0.888648	−0.444324	+	0.895866i	$0.646556\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.53077	1.36608	0.683039	−	0.730382i	$-0.260658\pi$
$11$	4.53077	1.36608	0.683039	+	0.730382i	$0.260658\pi$
$12$	0	0
$13$	−1.76393	−0.489227	−0.244613	−	0.969621i	$-0.578661\pi$
$13$	−1.76393	−0.489227	−0.244613	+	0.969621i	$0.578661\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	−6.15537	−1.41214	−0.706069	−	0.708143i	$-0.749533\pi$
$19$	−6.15537	−1.41214	−0.706069	+	0.708143i	$0.749533\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.80423	−0.793236	−0.396618	−	0.917984i	$-0.629816\pi$
$23$	−3.80423	−0.793236	−0.396618	+	0.917984i	$0.629816\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.726543	−0.130491	−0.0652454	−	0.997869i	$-0.520783\pi$
$31$	−0.726543	−0.130491	−0.0652454	+	0.997869i	$0.520783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.95959	−1.68348
$36$	0	0
$37$	2.47214	0.406417	0.203208	−	0.979136i	$-0.434863\pi$
$37$	2.47214	0.406417	0.203208	+	0.979136i	$0.434863\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.23607	1.13008	0.565042	−	0.825062i	$-0.308860\pi$
$41$	7.23607	1.13008	0.565042	+	0.825062i	$0.308860\pi$
$42$	0	0
$43$	5.98385	0.912529	0.456265	−	0.889844i	$-0.349187\pi$
$43$	5.98385	0.912529	0.456265	+	0.889844i	$0.349187\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.42882	−0.791875	−0.395938	−	0.918277i	$-0.629580\pi$
$47$	−5.42882	−0.791875	−0.395938	+	0.918277i	$0.629580\pi$
$48$	0	0
$49$	−1.47214	−0.210305
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.76393	0.517016	0.258508	−	0.966009i	$-0.416769\pi$
$53$	3.76393	0.517016	0.258508	+	0.966009i	$0.416769\pi$
$54$	0	0
$55$	19.1926	2.58794
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.71040	0.873619	0.436810	−	0.899554i	$-0.356108\pi$
$59$	6.71040	0.873619	0.436810	+	0.899554i	$0.356108\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.47214	−0.926804
$66$	0	0
$67$	12.3107	1.50400	0.751998	−	0.659166i	$-0.229090\pi$
$67$	12.3107	1.50400	0.751998	+	0.659166i	$0.229090\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.24920	−0.385609	−0.192804	−	0.981237i	$-0.561758\pi$
$71$	−3.24920	−0.385609	−0.192804	+	0.981237i	$0.561758\pi$
$72$	0	0
$73$	−4.94427	−0.578683	−0.289342	−	0.957226i	$-0.593436\pi$
$73$	−4.94427	−0.578683	−0.289342	+	0.957226i	$0.593436\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.6525	−1.21396
$78$	0	0
$79$	2.17963	0.245227	0.122614	−	0.992454i	$-0.460872\pi$
$79$	2.17963	0.245227	0.122614	+	0.992454i	$0.460872\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.1150	1.76885	0.884423	−	0.466685i	$-0.154552\pi$
$83$	16.1150	1.76885	0.884423	+	0.466685i	$0.154552\pi$
$84$	0	0
$85$	22.1803	2.40580
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.70820	−0.181069	−0.0905346	−	0.995893i	$-0.528858\pi$
$89$	−1.70820	−0.181069	−0.0905346	+	0.995893i	$0.528858\pi$
$90$	0	0
$91$	4.14725	0.434750
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−26.0746	−2.67519
$96$	0	0
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.4164	1.53399	0.766995	−	0.641653i	$-0.221751\pi$
$101$	15.4164	1.53399	0.766995	+	0.641653i	$0.221751\pi$
$102$	0	0
$103$	−3.24920	−0.320153	−0.160076	−	0.987105i	$-0.551174\pi$
$103$	−3.24920	−0.320153	−0.160076	+	0.987105i	$0.551174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.35114	0.227293	0.113647	−	0.993521i	$-0.463747\pi$
$107$	2.35114	0.227293	0.113647	+	0.993521i	$0.463747\pi$
$108$	0	0
$109$	−1.76393	−0.168954	−0.0844770	−	0.996425i	$-0.526922\pi$
$109$	−1.76393	−0.168954	−0.0844770	+	0.996425i	$0.526922\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.94427	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$113$	2.94427	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$114$	0	0
$115$	−16.1150	−1.50273
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.3107	−1.12852
$120$	0	0
$121$	9.52786	0.866169
$122$	0	0
$123$	0	0
$124$	0	0
$125$	33.6525	3.00997
$126$	0	0
$127$	−6.15537	−0.546201	−0.273100	−	0.961986i	$-0.588049\pi$
$127$	−6.15537	−0.546201	−0.273100	+	0.961986i	$0.588049\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.35926	−0.380870	−0.190435	−	0.981700i	$-0.560990\pi$
$131$	−4.35926	−0.380870	−0.190435	+	0.981700i	$0.560990\pi$
$132$	0	0
$133$	14.4721	1.25489
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.4164	−0.975370	−0.487685	−	0.873020i	$-0.662158\pi$
$137$	−11.4164	−0.975370	−0.487685	+	0.873020i	$0.662158\pi$
$138$	0	0
$139$	−0.898056	−0.0761721	−0.0380861	−	0.999274i	$-0.512126\pi$
$139$	−0.898056	−0.0761721	−0.0380861	+	0.999274i	$0.512126\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.99197	−0.668322
$144$	0	0
$145$	−4.23607	−0.351786
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.23607	0.674725	0.337362	−	0.941375i	$-0.390465\pi$
$149$	8.23607	0.674725	0.337362	+	0.941375i	$0.390465\pi$
$150$	0	0
$151$	24.0664	1.95850	0.979250	−	0.202658i	$-0.0649581\pi$
$151$	24.0664	1.95850	0.979250	+	0.202658i	$0.0649581\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.07768	−0.247205
$156$	0	0
$157$	−9.70820	−0.774799	−0.387400	−	0.921912i	$-0.626627\pi$
$157$	−9.70820	−0.774799	−0.387400	+	0.921912i	$0.626627\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.94427	0.704907
$162$	0	0
$163$	−1.28157	−0.100380	−0.0501902	−	0.998740i	$-0.515983\pi$
$163$	−1.28157	−0.100380	−0.0501902	+	0.998740i	$0.515983\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.8576	−0.840190	−0.420095	−	0.907480i	$-0.638003\pi$
$167$	−10.8576	−0.840190	−0.420095	+	0.907480i	$0.638003\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.4721	1.55647	0.778234	−	0.627975i	$-0.216116\pi$
$173$	20.4721	1.55647	0.778234	+	0.627975i	$0.216116\pi$
$174$	0	0
$175$	−30.4338	−2.30058
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.3723	−1.59744	−0.798719	−	0.601704i	$-0.794489\pi$
$179$	−21.3723	−1.59744	−0.798719	+	0.601704i	$0.794489\pi$
$180$	0	0
$181$	−19.1803	−1.42566	−0.712832	−	0.701335i	$-0.752588\pi$
$181$	−19.1803	−1.42566	−0.712832	+	0.701335i	$0.752588\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4721	0.769927
$186$	0	0
$187$	23.7234	1.73483
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.2784	1.75673	0.878363	−	0.477994i	$-0.158636\pi$
$191$	24.2784	1.75673	0.878363	+	0.477994i	$0.158636\pi$
$192$	0	0
$193$	8.76393	0.630842	0.315421	−	0.948952i	$-0.397854\pi$
$193$	8.76393	0.630842	0.315421	+	0.948952i	$0.397854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9443	0.779747	0.389874	−	0.920868i	$-0.372519\pi$
$197$	10.9443	0.779747	0.389874	+	0.920868i	$0.372519\pi$
$198$	0	0
$199$	19.3642	1.37269	0.686344	−	0.727277i	$-0.259214\pi$
$199$	19.3642	1.37269	0.686344	+	0.727277i	$0.259214\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.35114	0.165018
$204$	0	0
$205$	30.6525	2.14086
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−27.8885	−1.92909
$210$	0	0
$211$	4.53077	0.311911	0.155955	−	0.987764i	$-0.450154\pi$
$211$	4.53077	0.311911	0.155955	+	0.987764i	$0.450154\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	25.3480	1.72872
$216$	0	0
$217$	1.70820	0.115960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.23607	−0.621285
$222$	0	0
$223$	−27.8707	−1.86636	−0.933179	−	0.359412i	$-0.882977\pi$
$223$	−27.8707	−1.86636	−0.933179	+	0.359412i	$0.882977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.8576	−0.720647	−0.360324	−	0.932827i	$-0.617334\pi$
$227$	−10.8576	−0.720647	−0.360324	+	0.932827i	$0.617334\pi$
$228$	0	0
$229$	21.7082	1.43452	0.717259	−	0.696806i	$-0.245396\pi$
$229$	21.7082	1.43452	0.717259	+	0.696806i	$0.245396\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.9443	1.17557	0.587784	−	0.809018i	$-0.300000\pi$
$233$	17.9443	1.17557	0.587784	+	0.809018i	$0.300000\pi$
$234$	0	0
$235$	−22.9969	−1.50015
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.898056	0.0580904	0.0290452	−	0.999578i	$-0.490753\pi$
$239$	0.898056	0.0580904	0.0290452	+	0.999578i	$0.490753\pi$
$240$	0	0
$241$	−3.94427	−0.254073	−0.127036	−	0.991898i	$-0.540547\pi$
$241$	−3.94427	−0.254073	−0.127036	+	0.991898i	$0.540547\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.23607	−0.398408
$246$	0	0
$247$	10.8576	0.690856
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.89002	−0.561133	−0.280567	−	0.959835i	$-0.590522\pi$
$251$	−8.89002	−0.561133	−0.280567	+	0.959835i	$0.590522\pi$
$252$	0	0
$253$	−17.2361	−1.08362
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.47214	−0.0918293	−0.0459147	−	0.998945i	$-0.514620\pi$
$257$	−1.47214	−0.0918293	−0.0459147	+	0.998945i	$0.514620\pi$
$258$	0	0
$259$	−5.81234	−0.361161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.726543	0.0448005	0.0224003	−	0.999749i	$-0.492869\pi$
$263$	0.726543	0.0448005	0.0224003	+	0.999749i	$0.492869\pi$
$264$	0	0
$265$	15.9443	0.979449
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.70820	0.469977	0.234989	−	0.971998i	$-0.424495\pi$
$269$	7.70820	0.469977	0.234989	+	0.971998i	$0.424495\pi$
$270$	0	0
$271$	−13.3803	−0.812796	−0.406398	−	0.913696i	$-0.633215\pi$
$271$	−13.3803	−0.812796	−0.406398	+	0.913696i	$0.633215\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	58.6475	3.53658
$276$	0	0
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.5279	0.628040	0.314020	−	0.949416i	$-0.398324\pi$
$281$	10.5279	0.628040	0.314020	+	0.949416i	$0.398324\pi$
$282$	0	0
$283$	−0.898056	−0.0533839	−0.0266919	−	0.999644i	$-0.508497\pi$
$283$	−0.898056	−0.0533839	−0.0266919	+	0.999644i	$0.508497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.0130	−1.00425
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.41641	−0.550112	−0.275056	−	0.961428i	$-0.588696\pi$
$293$	−9.41641	−0.550112	−0.275056	+	0.961428i	$0.588696\pi$
$294$	0	0
$295$	28.4257	1.65501
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.71040	0.388072
$300$	0	0
$301$	−14.0689	−0.810917
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.7082	−0.670410
$306$	0	0
$307$	27.6992	1.58087	0.790437	−	0.612543i	$-0.209853\pi$
$307$	27.6992	1.58087	0.790437	+	0.612543i	$0.209853\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.5599	−0.882323	−0.441161	−	0.897428i	$-0.645433\pi$
$311$	−15.5599	−0.882323	−0.441161	+	0.897428i	$0.645433\pi$
$312$	0	0
$313$	14.4164	0.814864	0.407432	−	0.913236i	$-0.366424\pi$
$313$	14.4164	0.814864	0.407432	+	0.913236i	$0.366424\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.291796	−0.0163889	−0.00819445	−	0.999966i	$-0.502608\pi$
$317$	−0.291796	−0.0163889	−0.00819445	+	0.999966i	$0.502608\pi$
$318$	0	0
$319$	−4.53077	−0.253674
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−32.2299	−1.79332
$324$	0	0
$325$	−22.8328	−1.26654
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.7639	0.703698
$330$	0	0
$331$	−0.171513	−0.00942723	−0.00471362	−	0.999989i	$-0.501500\pi$
$331$	−0.171513	−0.00942723	−0.00471362	+	0.999989i	$0.501500\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	52.1491	2.84921
$336$	0	0
$337$	−27.1246	−1.47757	−0.738786	−	0.673940i	$-0.764601\pi$
$337$	−27.1246	−1.47757	−0.738786	+	0.673940i	$0.764601\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.29180	−0.178261
$342$	0	0
$343$	19.9192	1.07553
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.4661	−0.991312	−0.495656	−	0.868519i	$-0.665072\pi$
$347$	−18.4661	−0.991312	−0.495656	+	0.868519i	$0.665072\pi$
$348$	0	0
$349$	−4.34752	−0.232718	−0.116359	−	0.993207i	$-0.537122\pi$
$349$	−4.34752	−0.232718	−0.116359	+	0.993207i	$0.537122\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.94427	−0.156708	−0.0783539	−	0.996926i	$-0.524966\pi$
$353$	−2.94427	−0.156708	−0.0783539	+	0.996926i	$0.524966\pi$
$354$	0	0
$355$	−13.7638	−0.730508
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.383516	−0.0202412	−0.0101206	−	0.999949i	$-0.503222\pi$
$359$	−0.383516	−0.0202412	−0.0101206	+	0.999949i	$0.503222\pi$
$360$	0	0
$361$	18.8885	0.994134
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−20.9443	−1.09627
$366$	0	0
$367$	−10.8576	−0.566765	−0.283382	−	0.959007i	$-0.591457\pi$
$367$	−10.8576	−0.566765	−0.283382	+	0.959007i	$0.591457\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.84953	−0.459445
$372$	0	0
$373$	1.76393	0.0913329	0.0456665	−	0.998957i	$-0.485459\pi$
$373$	1.76393	0.0913329	0.0456665	+	0.998957i	$0.485459\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.76393	0.0908471
$378$	0	0
$379$	−27.8707	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$379$	−27.8707	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.3804	1.19468	0.597341	−	0.801987i	$-0.296224\pi$
$383$	23.3804	1.19468	0.597341	+	0.801987i	$0.296224\pi$
$384$	0	0
$385$	−45.1246	−2.29976
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	−19.9192	−1.00736
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.23305	0.464565
$396$	0	0
$397$	−19.7639	−0.991923	−0.495962	−	0.868344i	$-0.665184\pi$
$397$	−19.7639	−0.991923	−0.495962	+	0.868344i	$0.665184\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	1.28157	0.0638396
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.2007	0.555197
$408$	0	0
$409$	11.8885	0.587851	0.293925	−	0.955828i	$-0.405038\pi$
$409$	11.8885	0.587851	0.293925	+	0.955828i	$0.405038\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.7771	−0.776340
$414$	0	0
$415$	68.2641	3.35095
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.4791	1.73327	0.866634	−	0.498944i	$-0.166279\pi$
$419$	35.4791	1.73327	0.866634	+	0.498944i	$0.166279\pi$
$420$	0	0
$421$	−8.29180	−0.404117	−0.202059	−	0.979373i	$-0.564763\pi$
$421$	−8.29180	−0.404117	−0.202059	+	0.979373i	$0.564763\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	67.7771	3.28767
$426$	0	0
$427$	6.49839	0.314479
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.60034	0.269759	0.134879	−	0.990862i	$-0.456935\pi$
$431$	5.60034	0.269759	0.134879	+	0.990862i	$0.456935\pi$
$432$	0	0
$433$	21.8885	1.05190	0.525948	−	0.850517i	$-0.323711\pi$
$433$	21.8885	1.05190	0.525948	+	0.850517i	$0.323711\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.4164	1.12016
$438$	0	0
$439$	−13.2088	−0.630421	−0.315211	−	0.949022i	$-0.602075\pi$
$439$	−13.2088	−0.630421	−0.315211	+	0.949022i	$0.602075\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.1684	1.10076	0.550382	−	0.834913i	$-0.314482\pi$
$443$	23.1684	1.10076	0.550382	+	0.834913i	$0.314482\pi$
$444$	0	0
$445$	−7.23607	−0.343023
$446$	0	0
$447$	0	0
$448$	0	0
$449$	37.4164	1.76579	0.882895	−	0.469571i	$-0.155591\pi$
$449$	37.4164	1.76579	0.882895	+	0.469571i	$0.155591\pi$
$450$	0	0
$451$	32.7849	1.54378
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17.5680	0.823603
$456$	0	0
$457$	−24.8328	−1.16163	−0.580815	−	0.814036i	$-0.697266\pi$
$457$	−24.8328	−1.16163	−0.580815	+	0.814036i	$0.697266\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−35.8885	−1.67150	−0.835748	−	0.549113i	$-0.814966\pi$
$461$	−35.8885	−1.67150	−0.835748	+	0.549113i	$0.814966\pi$
$462$	0	0
$463$	20.8172	0.967459	0.483730	−	0.875217i	$-0.339282\pi$
$463$	20.8172	0.967459	0.483730	+	0.875217i	$0.339282\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.23305	0.427255	0.213627	−	0.976915i	$-0.431472\pi$
$467$	9.23305	0.427255	0.213627	+	0.976915i	$0.431472\pi$
$468$	0	0
$469$	−28.9443	−1.33652
$470$	0	0
$471$	0	0
$472$	0	0
$473$	27.1115	1.24659
$474$	0	0
$475$	−79.6767	−3.65582
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.5649	−1.85346	−0.926729	−	0.375730i	$-0.877392\pi$
$479$	−40.5649	−1.85346	−0.926729	+	0.375730i	$0.877392\pi$
$480$	0	0
$481$	−4.36068	−0.198830
$482$	0	0
$483$	0	0
$484$	0	0
$485$	41.1246	1.86737
$486$	0	0
$487$	−15.7719	−0.714695	−0.357347	−	0.933972i	$-0.616319\pi$
$487$	−15.7719	−0.714695	−0.357347	+	0.933972i	$0.616319\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.32688	−0.285528	−0.142764	−	0.989757i	$-0.545599\pi$
$491$	−6.32688	−0.285528	−0.142764	+	0.989757i	$0.545599\pi$
$492$	0	0
$493$	−5.23607	−0.235821
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.63932	0.342670
$498$	0	0
$499$	−22.6134	−1.01231	−0.506156	−	0.862442i	$-0.668934\pi$
$499$	−22.6134	−1.01231	−0.506156	+	0.862442i	$0.668934\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.08580	0.226765	0.113382	−	0.993551i	$-0.463832\pi$
$503$	5.08580	0.226765	0.113382	+	0.993551i	$0.463832\pi$
$504$	0	0
$505$	65.3050	2.90603
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.18034	−0.0523176	−0.0261588	−	0.999658i	$-0.508328\pi$
$509$	−1.18034	−0.0523176	−0.0261588	+	0.999658i	$0.508328\pi$
$510$	0	0
$511$	11.6247	0.514246
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.7638	−0.606506
$516$	0	0
$517$	−24.5967	−1.08176
$518$	0	0
$519$	0	0
$520$	0	0
$521$	44.8885	1.96660	0.983301	−	0.181984i	$-0.0582521\pi$
$521$	44.8885	1.96660	0.983301	+	0.181984i	$0.0582521\pi$
$522$	0	0
$523$	25.7315	1.12516	0.562581	−	0.826742i	$-0.309809\pi$
$523$	25.7315	1.12516	0.562581	+	0.826742i	$0.309809\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.80423	−0.165715
$528$	0	0
$529$	−8.52786	−0.370777
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.7639	−0.552867
$534$	0	0
$535$	9.95959	0.430591
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.66991	−0.287293
$540$	0	0
$541$	40.3607	1.73524	0.867621	−	0.497227i	$-0.165648\pi$
$541$	40.3607	1.73524	0.867621	+	0.497227i	$0.165648\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.47214	−0.320071
$546$	0	0
$547$	−22.2703	−0.952210	−0.476105	−	0.879388i	$-0.657952\pi$
$547$	−22.2703	−0.952210	−0.476105	+	0.879388i	$0.657952\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.15537	0.262227
$552$	0	0
$553$	−5.12461	−0.217921
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.3607	−1.28642	−0.643212	−	0.765688i	$-0.722398\pi$
$557$	−30.3607	−1.28642	−0.643212	+	0.765688i	$0.722398\pi$
$558$	0	0
$559$	−10.5551	−0.446434
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.07768	0.129709	0.0648544	−	0.997895i	$-0.479342\pi$
$563$	3.07768	0.129709	0.0648544	+	0.997895i	$0.479342\pi$
$564$	0	0
$565$	12.4721	0.524707
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	28.6377	1.19845	0.599225	−	0.800581i	$-0.295475\pi$
$571$	28.6377	1.19845	0.599225	+	0.800581i	$0.295475\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−49.2429	−2.05357
$576$	0	0
$577$	7.12461	0.296601	0.148301	−	0.988942i	$-0.452620\pi$
$577$	7.12461	0.296601	0.148301	+	0.988942i	$0.452620\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−37.8885	−1.57188
$582$	0	0
$583$	17.0535	0.706284
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.6296	1.09912	0.549560	−	0.835454i	$-0.314795\pi$
$587$	26.6296	1.09912	0.549560	+	0.835454i	$0.314795\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.8328	−0.650176	−0.325088	−	0.945684i	$-0.605394\pi$
$593$	−15.8328	−0.650176	−0.325088	+	0.945684i	$0.605394\pi$
$594$	0	0
$595$	−52.1491	−2.13790
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.8141	−1.79020	−0.895098	−	0.445869i	$-0.852895\pi$
$599$	−43.8141	−1.79020	−0.895098	+	0.445869i	$0.852895\pi$
$600$	0	0
$601$	−32.7639	−1.33647	−0.668234	−	0.743951i	$-0.732950\pi$
$601$	−32.7639	−1.33647	−0.668234	+	0.743951i	$0.732950\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.3607	1.64089
$606$	0	0
$607$	28.9402	1.17465	0.587324	−	0.809352i	$-0.300181\pi$
$607$	28.9402	1.17465	0.587324	+	0.809352i	$0.300181\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.57608	0.387407
$612$	0	0
$613$	1.65248	0.0667429	0.0333714	−	0.999443i	$-0.489376\pi$
$613$	1.65248	0.0667429	0.0333714	+	0.999443i	$0.489376\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.3607	−0.980724	−0.490362	−	0.871519i	$-0.663135\pi$
$617$	−24.3607	−0.980724	−0.490362	+	0.871519i	$0.663135\pi$
$618$	0	0
$619$	−37.1037	−1.49132	−0.745662	−	0.666324i	$-0.767867\pi$
$619$	−37.1037	−1.49132	−0.745662	+	0.666324i	$0.767867\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.01623	0.160907
$624$	0	0
$625$	77.8328	3.11331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.9443	0.516122
$630$	0	0
$631$	−36.5892	−1.45659	−0.728296	−	0.685263i	$-0.759687\pi$
$631$	−36.5892	−1.45659	−0.728296	+	0.685263i	$0.759687\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−26.0746	−1.03474
$636$	0	0
$637$	2.59675	0.102887
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.2361	0.680784	0.340392	−	0.940284i	$-0.389440\pi$
$641$	17.2361	0.680784	0.340392	+	0.940284i	$0.389440\pi$
$642$	0	0
$643$	4.14725	0.163552	0.0817758	−	0.996651i	$-0.473941\pi$
$643$	4.14725	0.163552	0.0817758	+	0.996651i	$0.473941\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.8303	−1.48726	−0.743630	−	0.668591i	$-0.766898\pi$
$647$	−37.8303	−1.48726	−0.743630	+	0.668591i	$0.766898\pi$
$648$	0	0
$649$	30.4033	1.19343
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.2361	−0.439701	−0.219851	−	0.975534i	$-0.570557\pi$
$653$	−11.2361	−0.439701	−0.219851	+	0.975534i	$0.570557\pi$
$654$	0	0
$655$	−18.4661	−0.721530
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−25.9030	−1.00904	−0.504520	−	0.863400i	$-0.668330\pi$
$659$	−25.9030	−1.00904	−0.504520	+	0.863400i	$0.668330\pi$
$660$	0	0
$661$	31.3050	1.21762	0.608811	−	0.793315i	$-0.291647\pi$
$661$	31.3050	1.21762	0.608811	+	0.793315i	$0.291647\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	61.3050	2.37730
$666$	0	0
$667$	3.80423	0.147300
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.5227	−0.483435
$672$	0	0
$673$	−17.8328	−0.687405	−0.343702	−	0.939079i	$-0.611681\pi$
$673$	−17.8328	−0.687405	−0.343702	+	0.939079i	$0.611681\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.3607	−0.398193	−0.199097	−	0.979980i	$-0.563801\pi$
$677$	−10.3607	−0.398193	−0.199097	+	0.979980i	$0.563801\pi$
$678$	0	0
$679$	−22.8254	−0.875957
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.59222	−0.137453	−0.0687263	−	0.997636i	$-0.521894\pi$
$683$	−3.59222	−0.137453	−0.0687263	+	0.997636i	$0.521894\pi$
$684$	0	0
$685$	−48.3607	−1.84777
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.63932	−0.252938
$690$	0	0
$691$	18.8091	0.715533	0.357766	−	0.933811i	$-0.383538\pi$
$691$	18.8091	0.715533	0.357766	+	0.933811i	$0.383538\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.80423	−0.144303
$696$	0	0
$697$	37.8885	1.43513
$698$	0	0
$699$	0	0
$700$	0	0
$701$	48.0132	1.81343	0.906716	−	0.421742i	$-0.138581\pi$
$701$	48.0132	1.81343	0.906716	+	0.421742i	$0.138581\pi$
$702$	0	0
$703$	−15.2169	−0.573916
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.2461	−1.36318
$708$	0	0
$709$	−35.6525	−1.33896	−0.669478	−	0.742832i	$-0.733482\pi$
$709$	−35.6525	−1.33896	−0.669478	+	0.742832i	$0.733482\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.76393	0.103510
$714$	0	0
$715$	−33.8545	−1.26609
$716$	0	0
$717$	0	0
$718$	0	0
$719$	51.8061	1.93204	0.966020	−	0.258466i	$-0.0832170\pi$
$719$	51.8061	1.93204	0.966020	+	0.258466i	$0.0832170\pi$
$720$	0	0
$721$	7.63932	0.284503
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.9443	−0.480738
$726$	0	0
$727$	−24.2784	−0.900438	−0.450219	−	0.892918i	$-0.648654\pi$
$727$	−24.2784	−0.900438	−0.450219	+	0.892918i	$0.648654\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31.3319	1.15885
$732$	0	0
$733$	−4.94427	−0.182621	−0.0913104	−	0.995822i	$-0.529106\pi$
$733$	−4.94427	−0.182621	−0.0913104	+	0.995822i	$0.529106\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	55.7771	2.05458
$738$	0	0
$739$	−39.6669	−1.45917	−0.729585	−	0.683891i	$-0.760287\pi$
$739$	−39.6669	−1.45917	−0.729585	+	0.683891i	$0.760287\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.1038	1.72807	0.864035	−	0.503431i	$-0.167929\pi$
$743$	47.1038	1.72807	0.864035	+	0.503431i	$0.167929\pi$
$744$	0	0
$745$	34.8885	1.27822
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.52786	−0.201984
$750$	0	0
$751$	26.0746	0.951474	0.475737	−	0.879588i	$-0.342181\pi$
$751$	26.0746	0.951474	0.475737	+	0.879588i	$0.342181\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	101.947	3.71023
$756$	0	0
$757$	−21.8197	−0.793049	−0.396525	−	0.918024i	$-0.629784\pi$
$757$	−21.8197	−0.793049	−0.396525	+	0.918024i	$0.629784\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.8885	0.575959	0.287980	−	0.957637i	$-0.407016\pi$
$761$	15.8885	0.575959	0.287980	+	0.957637i	$0.407016\pi$
$762$	0	0
$763$	4.14725	0.150141
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.8367	−0.427398
$768$	0	0
$769$	17.2361	0.621549	0.310774	−	0.950484i	$-0.399412\pi$
$769$	17.2361	0.621549	0.310774	+	0.950484i	$0.399412\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.2361	1.12348	0.561742	−	0.827313i	$-0.310132\pi$
$773$	31.2361	1.12348	0.561742	+	0.827313i	$0.310132\pi$
$774$	0	0
$775$	−9.40456	−0.337822
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−44.5407	−1.59583
$780$	0	0
$781$	−14.7214	−0.526772
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−41.1246	−1.46780
$786$	0	0
$787$	39.4953	1.40786	0.703929	−	0.710271i	$-0.251428\pi$
$787$	39.4953	1.40786	0.703929	+	0.710271i	$0.251428\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.92240	−0.246132
$792$	0	0
$793$	4.87539	0.173130
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.2361	−1.03559	−0.517797	−	0.855503i	$-0.673248\pi$
$797$	−29.2361	−1.03559	−0.517797	+	0.855503i	$0.673248\pi$
$798$	0	0
$799$	−28.4257	−1.00563
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.4014	−0.790527
$804$	0	0
$805$	37.8885	1.33540
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11.0557	−0.388699	−0.194349	−	0.980932i	$-0.562260\pi$
$809$	−11.0557	−0.388699	−0.194349	+	0.980932i	$0.562260\pi$
$810$	0	0
$811$	31.6749	1.11226	0.556128	−	0.831097i	$-0.312286\pi$
$811$	31.6749	1.11226	0.556128	+	0.831097i	$0.312286\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.42882	−0.190163
$816$	0	0
$817$	−36.8328	−1.28862
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.2361	−1.47405	−0.737024	−	0.675866i	$-0.763770\pi$
$821$	−42.2361	−1.47405	−0.737024	+	0.675866i	$0.763770\pi$
$822$	0	0
$823$	9.06154	0.315865	0.157933	−	0.987450i	$-0.449517\pi$
$823$	9.06154	0.315865	0.157933	+	0.987450i	$0.449517\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.5245	1.75691	0.878455	−	0.477826i	$-0.158575\pi$
$827$	50.5245	1.75691	0.878455	+	0.477826i	$0.158575\pi$
$828$	0	0
$829$	15.7082	0.545568	0.272784	−	0.962075i	$-0.412055\pi$
$829$	15.7082	0.545568	0.272784	+	0.962075i	$0.412055\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.70820	−0.267073
$834$	0	0
$835$	−45.9937	−1.59168
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.5519	−0.813102	−0.406551	−	0.913628i	$-0.633269\pi$
$839$	−23.5519	−0.813102	−0.406551	+	0.913628i	$0.633269\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−41.8885	−1.44101
$846$	0	0
$847$	−22.4014	−0.769720
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.40456	−0.322384
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.0557	0.958365	0.479183	−	0.877715i	$-0.340933\pi$
$857$	28.0557	0.958365	0.479183	+	0.877715i	$0.340933\pi$
$858$	0	0
$859$	27.3561	0.933379	0.466689	−	0.884421i	$-0.345447\pi$
$859$	27.3561	0.933379	0.466689	+	0.884421i	$0.345447\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.5729	1.10880	0.554398	−	0.832252i	$-0.312949\pi$
$863$	32.5729	1.10880	0.554398	+	0.832252i	$0.312949\pi$
$864$	0	0
$865$	86.7214	2.94861
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.87539	0.335000
$870$	0	0
$871$	−21.7153	−0.735795
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−79.1217	−2.67480
$876$	0	0
$877$	−6.81966	−0.230284	−0.115142	−	0.993349i	$-0.536732\pi$
$877$	−6.81966	−0.230284	−0.115142	+	0.993349i	$0.536732\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.3607	0.753350	0.376675	−	0.926345i	$-0.377067\pi$
$881$	22.3607	0.753350	0.376675	+	0.926345i	$0.377067\pi$
$882$	0	0
$883$	−13.9758	−0.470324	−0.235162	−	0.971956i	$-0.575562\pi$
$883$	−13.9758	−0.470324	−0.235162	+	0.971956i	$0.575562\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.3318	0.716251	0.358126	−	0.933673i	$-0.383416\pi$
$887$	21.3318	0.716251	0.358126	+	0.933673i	$0.383416\pi$
$888$	0	0
$889$	14.4721	0.485380
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.4164	1.11824
$894$	0	0
$895$	−90.5344	−3.02623
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.726543	0.0242315
$900$	0	0
$901$	19.7082	0.656575
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−81.2492	−2.70082
$906$	0	0
$907$	35.4791	1.17806	0.589032	−	0.808109i	$-0.299509\pi$
$907$	35.4791	1.17806	0.589032	+	0.808109i	$0.299509\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.8141	1.45163	0.725813	−	0.687892i	$-0.241464\pi$
$911$	43.8141	1.45163	0.725813	+	0.687892i	$0.241464\pi$
$912$	0	0
$913$	73.0132	2.41638
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.2492	0.338459
$918$	0	0
$919$	7.95148	0.262295	0.131148	−	0.991363i	$-0.458134\pi$
$919$	7.95148	0.262295	0.131148	+	0.991363i	$0.458134\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.73136	0.188650
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	9.06154	0.296980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	100.494	3.28650
$936$	0	0
$937$	34.3607	1.12251	0.561257	−	0.827641i	$-0.310318\pi$
$937$	34.3607	1.12251	0.561257	+	0.827641i	$0.310318\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50.0132	−1.63038	−0.815191	−	0.579192i	$-0.803368\pi$
$941$	−50.0132	−1.63038	−0.815191	+	0.579192i	$0.803368\pi$
$942$	0	0
$943$	−27.5276	−0.896423
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.5600	0.830589	0.415294	−	0.909687i	$-0.363679\pi$
$947$	25.5600	0.830589	0.415294	+	0.909687i	$0.363679\pi$
$948$	0	0
$949$	8.72136	0.283107
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.4164	0.402207	0.201103	−	0.979570i	$-0.435547\pi$
$953$	12.4164	0.402207	0.201103	+	0.979570i	$0.435547\pi$
$954$	0	0
$955$	102.845	3.32799
$956$	0	0
$957$	0	0
$958$	0	0
$959$	26.8416	0.866760
$960$	0	0
$961$	−30.4721	−0.982972
$962$	0	0
$963$	0	0
$964$	0	0
$965$	37.1246	1.19508
$966$	0	0
$967$	−40.2219	−1.29345	−0.646724	−	0.762724i	$-0.723862\pi$
$967$	−40.2219	−1.29345	−0.646724	+	0.762724i	$0.723862\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.45309	0.0466317	0.0233159	−	0.999728i	$-0.492578\pi$
$971$	1.45309	0.0466317	0.0233159	+	0.999728i	$0.492578\pi$
$972$	0	0
$973$	2.11146	0.0676902
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.8328	1.40234	0.701168	−	0.712996i	$-0.252662\pi$
$977$	43.8328	1.40234	0.701168	+	0.712996i	$0.252662\pi$
$978$	0	0
$979$	−7.73948	−0.247355
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.52265	−0.0804602	−0.0402301	−	0.999190i	$-0.512809\pi$
$983$	−2.52265	−0.0804602	−0.0402301	+	0.999190i	$0.512809\pi$
$984$	0	0
$985$	46.3607	1.47717
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−22.7639	−0.723851
$990$	0	0
$991$	−10.8576	−0.344905	−0.172452	−	0.985018i	$-0.555169\pi$
$991$	−10.8576	−0.344905	−0.172452	+	0.985018i	$0.555169\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	82.0279	2.60046
$996$	0	0
$997$	0.944272	0.0299054	0.0149527	−	0.999888i	$-0.495240\pi$
$997$	0.944272	0.0299054	0.0149527	+	0.999888i	$0.495240\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8352.2.a.ba.1.3		4
3.2	odd	2		928.2.a.f.1.4	yes	4
4.3	odd	2	inner	8352.2.a.ba.1.4		4
12.11	even	2		928.2.a.f.1.1	✓	4
24.5	odd	2		1856.2.a.z.1.1		4
24.11	even	2		1856.2.a.z.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.f.1.1	✓	4	12.11	even	2
928.2.a.f.1.4	yes	4	3.2	odd	2
1856.2.a.z.1.1		4	24.5	odd	2
1856.2.a.z.1.4		4	24.11	even	2
8352.2.a.ba.1.3		4	1.1	even	1	trivial
8352.2.a.ba.1.4		4	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 8352 = 2^{5} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8352.a (trivial)

\(f(q)\)	\(=\)	\(q+4.23607 q^{5} -2.35114 q^{7} +O(q^{10})\)	\(q+4.23607 q^{5} -2.35114 q^{7} +4.53077 q^{11} -1.76393 q^{13} +5.23607 q^{17} -6.15537 q^{19} -3.80423 q^{23} +12.9443 q^{25} -1.00000 q^{29} -0.726543 q^{31} -9.95959 q^{35} +2.47214 q^{37} +7.23607 q^{41} +5.98385 q^{43} -5.42882 q^{47} -1.47214 q^{49} +3.76393 q^{53} +19.1926 q^{55} +6.71040 q^{59} -2.76393 q^{61} -7.47214 q^{65} +12.3107 q^{67} -3.24920 q^{71} -4.94427 q^{73} -10.6525 q^{77} +2.17963 q^{79} +16.1150 q^{83} +22.1803 q^{85} -1.70820 q^{89} +4.14725 q^{91} -26.0746 q^{95} +9.70820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^5	\( 4 q + 8 q^{5} - 16 q^{13} + 12 q^{17} + 16 q^{25} - 4 q^{29} - 8 q^{37} + 20 q^{41} + 12 q^{49} + 24 q^{53} - 20 q^{61} - 12 q^{65} + 16 q^{73} + 20 q^{77} + 44 q^{85} + 20 q^{89} + 12 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 16 * q^13 + 12 * q^17 + 16 * q^25 - 4 * q^29 - 8 * q^37 + 20 * q^41 + 12 * q^49 + 24 * q^53 - 20 * q^61 - 12 * q^65 + 16 * q^73 + 20 * q^77 + 44 * q^85 + 20 * q^89 + 12 * q^97

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