LMFDB - Embedded newform 8464.2.a.bx.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08816 q^{3} -1.22871 q^{5} -0.627214 q^{7} -1.81592 q^{9} -5.89037 q^{11} -2.35194 q^{13} -1.33703 q^{15} +5.33992 q^{17} +0.270925 q^{19} -0.682507 q^{21} -3.49028 q^{25} -5.24047 q^{27} -1.98287 q^{29} -6.60353 q^{31} -6.40964 q^{33} +0.770663 q^{35} +10.0704 q^{37} -2.55928 q^{39} +10.8126 q^{41} -4.59725 q^{43} +2.23123 q^{45} -6.97259 q^{47} -6.60660 q^{49} +5.81067 q^{51} -2.63694 q^{53} +7.23754 q^{55} +0.294808 q^{57} -1.64927 q^{59} +9.13485 q^{61} +1.13897 q^{63} +2.88984 q^{65} +6.19401 q^{67} +9.06233 q^{71} -5.93852 q^{73} -3.79797 q^{75} +3.69452 q^{77} +6.00714 q^{79} -0.254696 q^{81} +1.95305 q^{83} -6.56120 q^{85} -2.15768 q^{87} +4.59045 q^{89} +1.47517 q^{91} -7.18567 q^{93} -0.332887 q^{95} +10.1844 q^{97} +10.6964 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} + 8 q^{5} - 7 q^{7} - q^{9} - 5 q^{11} - 7 q^{13} + q^{15} + 13 q^{17} - 12 q^{19} + 6 q^{21} + q^{25} + 20 q^{27} - 4 q^{29} - 6 q^{31} + 9 q^{33} - 9 q^{35} + 14 q^{37} + 6 q^{39} + q^{41}+ \cdots + 34 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 + 8 * q^5 - 7 * q^7 - q^9 - 5 * q^11 - 7 * q^13 + q^15 + 13 * q^17 - 12 * q^19 + 6 * q^21 + q^25 + 20 * q^27 - 4 * q^29 - 6 * q^31 + 9 * q^33 - 9 * q^35 + 14 * q^37 + 6 * q^39 + q^41 - 11 * q^43 + 5 * q^45 - 9 * q^47 - 12 * q^49 + 14 * q^51 + 6 * q^53 + 14 * q^55 - 7 * q^57 - 3 * q^59 + 13 * q^61 + 19 * q^63 - 9 * q^65 - 10 * q^67 - 8 * q^71 - 4 * q^73 - 4 * q^75 + 18 * q^77 + 18 * q^79 + 33 * q^81 - 2 * q^83 + 23 * q^85 - 6 * q^87 + 10 * q^89 + 12 * q^91 - 20 * q^93 - 28 * q^95 + 17 * q^97 + 34 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.08816	0.628247	0.314124	−	0.949382i	$-0.398289\pi$
$3$	1.08816	0.628247	0.314124	+	0.949382i	$0.398289\pi$
$4$	0	0
$5$	−1.22871	−0.549495	−0.274747	−	0.961516i	$-0.588594\pi$
$5$	−1.22871	−0.549495	−0.274747	+	0.961516i	$0.588594\pi$
$6$	0	0
$7$	−0.627214	−0.237065	−0.118532	−	0.992950i	$-0.537819\pi$
$7$	−0.627214	−0.237065	−0.118532	+	0.992950i	$0.537819\pi$
$8$	0	0
$9$	−1.81592	−0.605306
$10$	0	0
$11$	−5.89037	−1.77601	−0.888006	−	0.459831i	$-0.847910\pi$
$11$	−5.89037	−1.77601	−0.888006	+	0.459831i	$0.847910\pi$
$12$	0	0
$13$	−2.35194	−0.652310	−0.326155	−	0.945316i	$-0.605753\pi$
$13$	−2.35194	−0.652310	−0.326155	+	0.945316i	$0.605753\pi$
$14$	0	0
$15$	−1.33703	−0.345218
$16$	0	0
$17$	5.33992	1.29512	0.647561	−	0.762014i	$-0.275789\pi$
$17$	5.33992	1.29512	0.647561	+	0.762014i	$0.275789\pi$
$18$	0	0
$19$	0.270925	0.0621544	0.0310772	−	0.999517i	$-0.490106\pi$
$19$	0.270925	0.0621544	0.0310772	+	0.999517i	$0.490106\pi$
$20$	0	0
$21$	−0.682507	−0.148935
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.49028	−0.698056
$26$	0	0
$27$	−5.24047	−1.00853
$28$	0	0
$29$	−1.98287	−0.368211	−0.184105	−	0.982907i	$-0.558939\pi$
$29$	−1.98287	−0.368211	−0.184105	+	0.982907i	$0.558939\pi$
$30$	0	0
$31$	−6.60353	−1.18603	−0.593014	−	0.805192i	$-0.702062\pi$
$31$	−6.60353	−1.18603	−0.593014	+	0.805192i	$0.702062\pi$
$32$	0	0
$33$	−6.40964	−1.11577
$34$	0	0
$35$	0.770663	0.130266
$36$	0	0
$37$	10.0704	1.65556	0.827782	−	0.561050i	$-0.189602\pi$
$37$	10.0704	1.65556	0.827782	+	0.561050i	$0.189602\pi$
$38$	0	0
$39$	−2.55928	−0.409812
$40$	0	0
$41$	10.8126	1.68865	0.844324	−	0.535833i	$-0.180002\pi$
$41$	10.8126	1.68865	0.844324	+	0.535833i	$0.180002\pi$
$42$	0	0
$43$	−4.59725	−0.701074	−0.350537	−	0.936549i	$-0.614001\pi$
$43$	−4.59725	−0.701074	−0.350537	+	0.936549i	$0.614001\pi$
$44$	0	0
$45$	2.23123	0.332612
$46$	0	0
$47$	−6.97259	−1.01706	−0.508528	−	0.861045i	$-0.669810\pi$
$47$	−6.97259	−1.01706	−0.508528	+	0.861045i	$0.669810\pi$
$48$	0	0
$49$	−6.60660	−0.943800
$50$	0	0
$51$	5.81067	0.813656
$52$	0	0
$53$	−2.63694	−0.362212	−0.181106	−	0.983464i	$-0.557968\pi$
$53$	−2.63694	−0.362212	−0.181106	+	0.983464i	$0.557968\pi$
$54$	0	0
$55$	7.23754	0.975909
$56$	0	0
$57$	0.294808	0.0390483
$58$	0	0
$59$	−1.64927	−0.214717	−0.107358	−	0.994220i	$-0.534239\pi$
$59$	−1.64927	−0.214717	−0.107358	+	0.994220i	$0.534239\pi$
$60$	0	0
$61$	9.13485	1.16960	0.584799	−	0.811178i	$-0.301173\pi$
$61$	9.13485	1.16960	0.584799	+	0.811178i	$0.301173\pi$
$62$	0	0
$63$	1.13897	0.143497
$64$	0	0
$65$	2.88984	0.358441
$66$	0	0
$67$	6.19401	0.756718	0.378359	−	0.925659i	$-0.376488\pi$
$67$	6.19401	0.756718	0.378359	+	0.925659i	$0.376488\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.06233	1.07550	0.537750	−	0.843104i	$-0.319274\pi$
$71$	9.06233	1.07550	0.537750	+	0.843104i	$0.319274\pi$
$72$	0	0
$73$	−5.93852	−0.695051	−0.347525	−	0.937671i	$-0.612978\pi$
$73$	−5.93852	−0.695051	−0.347525	+	0.937671i	$0.612978\pi$
$74$	0	0
$75$	−3.79797	−0.438551
$76$	0	0
$77$	3.69452	0.421030
$78$	0	0
$79$	6.00714	0.675856	0.337928	−	0.941172i	$-0.390274\pi$
$79$	6.00714	0.675856	0.337928	+	0.941172i	$0.390274\pi$
$80$	0	0
$81$	−0.254696	−0.0282996
$82$	0	0
$83$	1.95305	0.214375	0.107187	−	0.994239i	$-0.465816\pi$
$83$	1.95305	0.214375	0.107187	+	0.994239i	$0.465816\pi$
$84$	0	0
$85$	−6.56120	−0.711662
$86$	0	0
$87$	−2.15768	−0.231327
$88$	0	0
$89$	4.59045	0.486587	0.243293	−	0.969953i	$-0.421772\pi$
$89$	4.59045	0.486587	0.243293	+	0.969953i	$0.421772\pi$
$90$	0	0
$91$	1.47517	0.154640
$92$	0	0
$93$	−7.18567	−0.745119
$94$	0	0
$95$	−0.332887	−0.0341535
$96$	0	0
$97$	10.1844	1.03406	0.517032	−	0.855966i	$-0.327037\pi$
$97$	10.1844	1.03406	0.517032	+	0.855966i	$0.327037\pi$
$98$	0	0
$99$	10.6964	1.07503
$100$	0	0
$101$	−14.2352	−1.41646	−0.708229	−	0.705983i	$-0.750505\pi$
$101$	−14.2352	−1.41646	−0.708229	+	0.705983i	$0.750505\pi$
$102$	0	0
$103$	0.753717	0.0742659	0.0371330	−	0.999310i	$-0.488177\pi$
$103$	0.753717	0.0742659	0.0371330	+	0.999310i	$0.488177\pi$
$104$	0	0
$105$	0.838601	0.0818391
$106$	0	0
$107$	−0.593036	−0.0573310	−0.0286655	−	0.999589i	$-0.509126\pi$
$107$	−0.593036	−0.0573310	−0.0286655	+	0.999589i	$0.509126\pi$
$108$	0	0
$109$	−6.86224	−0.657283	−0.328642	−	0.944455i	$-0.606591\pi$
$109$	−6.86224	−0.657283	−0.328642	+	0.944455i	$0.606591\pi$
$110$	0	0
$111$	10.9582	1.04010
$112$	0	0
$113$	−1.24628	−0.117240	−0.0586202	−	0.998280i	$-0.518670\pi$
$113$	−1.24628	−0.117240	−0.0586202	+	0.998280i	$0.518670\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.27092	0.394847
$118$	0	0
$119$	−3.34928	−0.307028
$120$	0	0
$121$	23.6964	2.15422
$122$	0	0
$123$	11.7658	1.06089
$124$	0	0
$125$	10.4321	0.933072
$126$	0	0
$127$	7.18567	0.637625	0.318812	−	0.947818i	$-0.396716\pi$
$127$	7.18567	0.637625	0.318812	+	0.947818i	$0.396716\pi$
$128$	0	0
$129$	−5.00252	−0.440448
$130$	0	0
$131$	0.658240	0.0575107	0.0287553	−	0.999586i	$-0.490846\pi$
$131$	0.658240	0.0575107	0.0287553	+	0.999586i	$0.490846\pi$
$132$	0	0
$133$	−0.169928	−0.0147346
$134$	0	0
$135$	6.43900	0.554181
$136$	0	0
$137$	−12.5241	−1.07001	−0.535003	−	0.844850i	$-0.679689\pi$
$137$	−12.5241	−1.07001	−0.535003	+	0.844850i	$0.679689\pi$
$138$	0	0
$139$	−17.0602	−1.44703	−0.723515	−	0.690309i	$-0.757475\pi$
$139$	−17.0602	−1.44703	−0.723515	+	0.690309i	$0.757475\pi$
$140$	0	0
$141$	−7.58726	−0.638963
$142$	0	0
$143$	13.8538	1.15851
$144$	0	0
$145$	2.43637	0.202330
$146$	0	0
$147$	−7.18901	−0.592940
$148$	0	0
$149$	−18.4139	−1.50853	−0.754264	−	0.656571i	$-0.772006\pi$
$149$	−18.4139	−1.50853	−0.754264	+	0.656571i	$0.772006\pi$
$150$	0	0
$151$	11.5003	0.935883	0.467941	−	0.883760i	$-0.344996\pi$
$151$	11.5003	0.935883	0.467941	+	0.883760i	$0.344996\pi$
$152$	0	0
$153$	−9.69685	−0.783944
$154$	0	0
$155$	8.11380	0.651716
$156$	0	0
$157$	3.22814	0.257633	0.128817	−	0.991668i	$-0.458882\pi$
$157$	3.22814	0.257633	0.128817	+	0.991668i	$0.458882\pi$
$158$	0	0
$159$	−2.86940	−0.227559
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.0637	−0.866576	−0.433288	−	0.901256i	$-0.642647\pi$
$163$	−11.0637	−0.866576	−0.433288	+	0.901256i	$0.642647\pi$
$164$	0	0
$165$	7.87557	0.613112
$166$	0	0
$167$	21.9317	1.69713	0.848563	−	0.529094i	$-0.177468\pi$
$167$	21.9317	1.69713	0.848563	+	0.529094i	$0.177468\pi$
$168$	0	0
$169$	−7.46838	−0.574491
$170$	0	0
$171$	−0.491976	−0.0376224
$172$	0	0
$173$	11.0953	0.843561	0.421780	−	0.906698i	$-0.361405\pi$
$173$	11.0953	0.843561	0.421780	+	0.906698i	$0.361405\pi$
$174$	0	0
$175$	2.18915	0.165484
$176$	0	0
$177$	−1.79466	−0.134895
$178$	0	0
$179$	17.5377	1.31083	0.655414	−	0.755270i	$-0.272494\pi$
$179$	17.5377	1.31083	0.655414	+	0.755270i	$0.272494\pi$
$180$	0	0
$181$	20.4060	1.51676	0.758382	−	0.651810i	$-0.225990\pi$
$181$	20.4060	1.51676	0.758382	+	0.651810i	$0.225990\pi$
$182$	0	0
$183$	9.94014	0.734797
$184$	0	0
$185$	−12.3736	−0.909723
$186$	0	0
$187$	−31.4541	−2.30015
$188$	0	0
$189$	3.28690	0.239087
$190$	0	0
$191$	9.46247	0.684680	0.342340	−	0.939576i	$-0.388781\pi$
$191$	9.46247	0.684680	0.342340	+	0.939576i	$0.388781\pi$
$192$	0	0
$193$	25.5609	1.83991	0.919956	−	0.392022i	$-0.128224\pi$
$193$	25.5609	1.83991	0.919956	+	0.392022i	$0.128224\pi$
$194$	0	0
$195$	3.14460	0.225190
$196$	0	0
$197$	5.26954	0.375439	0.187719	−	0.982223i	$-0.439890\pi$
$197$	5.26954	0.375439	0.187719	+	0.982223i	$0.439890\pi$
$198$	0	0
$199$	25.0160	1.77333	0.886667	−	0.462409i	$-0.153015\pi$
$199$	25.0160	1.77333	0.886667	+	0.462409i	$0.153015\pi$
$200$	0	0
$201$	6.74004	0.475406
$202$	0	0
$203$	1.24369	0.0872898
$204$	0	0
$205$	−13.2856	−0.927903
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.59585	−0.110387
$210$	0	0
$211$	8.86632	0.610383	0.305191	−	0.952291i	$-0.401280\pi$
$211$	8.86632	0.610383	0.305191	+	0.952291i	$0.401280\pi$
$212$	0	0
$213$	9.86123	0.675680
$214$	0	0
$215$	5.64867	0.385236
$216$	0	0
$217$	4.14183	0.281166
$218$	0	0
$219$	−6.46203	−0.436664
$220$	0	0
$221$	−12.5592	−0.844821
$222$	0	0
$223$	16.9830	1.13727	0.568633	−	0.822592i	$-0.307473\pi$
$223$	16.9830	1.13727	0.568633	+	0.822592i	$0.307473\pi$
$224$	0	0
$225$	6.33805	0.422537
$226$	0	0
$227$	17.9952	1.19438	0.597192	−	0.802098i	$-0.296283\pi$
$227$	17.9952	1.19438	0.597192	+	0.802098i	$0.296283\pi$
$228$	0	0
$229$	7.49653	0.495385	0.247692	−	0.968839i	$-0.420328\pi$
$229$	7.49653	0.495385	0.247692	+	0.968839i	$0.420328\pi$
$230$	0	0
$231$	4.02022	0.264511
$232$	0	0
$233$	5.52307	0.361829	0.180914	−	0.983499i	$-0.442094\pi$
$233$	5.52307	0.361829	0.180914	+	0.983499i	$0.442094\pi$
$234$	0	0
$235$	8.56727	0.558867
$236$	0	0
$237$	6.53671	0.424605
$238$	0	0
$239$	−5.21901	−0.337590	−0.168795	−	0.985651i	$-0.553988\pi$
$239$	−5.21901	−0.337590	−0.168795	+	0.985651i	$0.553988\pi$
$240$	0	0
$241$	−1.55390	−0.100095	−0.0500476	−	0.998747i	$-0.515937\pi$
$241$	−1.55390	−0.100095	−0.0500476	+	0.998747i	$0.515937\pi$
$242$	0	0
$243$	15.4443	0.990749
$244$	0	0
$245$	8.11758	0.518613
$246$	0	0
$247$	−0.637198	−0.0405439
$248$	0	0
$249$	2.12522	0.134680
$250$	0	0
$251$	8.92657	0.563440	0.281720	−	0.959497i	$-0.409095\pi$
$251$	8.92657	0.563440	0.281720	+	0.959497i	$0.409095\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−7.13961	−0.447100
$256$	0	0
$257$	−3.64256	−0.227216	−0.113608	−	0.993526i	$-0.536241\pi$
$257$	−3.64256	−0.227216	−0.113608	+	0.993526i	$0.536241\pi$
$258$	0	0
$259$	−6.31630	−0.392476
$260$	0	0
$261$	3.60074	0.222880
$262$	0	0
$263$	−16.1369	−0.995044	−0.497522	−	0.867451i	$-0.665757\pi$
$263$	−16.1369	−0.995044	−0.497522	+	0.867451i	$0.665757\pi$
$264$	0	0
$265$	3.24003	0.199034
$266$	0	0
$267$	4.99513	0.305697
$268$	0	0
$269$	−4.49392	−0.273999	−0.137000	−	0.990571i	$-0.543746\pi$
$269$	−4.49392	−0.273999	−0.137000	+	0.990571i	$0.543746\pi$
$270$	0	0
$271$	3.50249	0.212761	0.106381	−	0.994325i	$-0.466074\pi$
$271$	3.50249	0.212761	0.106381	+	0.994325i	$0.466074\pi$
$272$	0	0
$273$	1.60521	0.0971520
$274$	0	0
$275$	20.5590	1.23976
$276$	0	0
$277$	−12.0692	−0.725166	−0.362583	−	0.931951i	$-0.618105\pi$
$277$	−12.0692	−0.725166	−0.362583	+	0.931951i	$0.618105\pi$
$278$	0	0
$279$	11.9915	0.717909
$280$	0	0
$281$	−6.26602	−0.373799	−0.186900	−	0.982379i	$-0.559844\pi$
$281$	−6.26602	−0.373799	−0.186900	+	0.982379i	$0.559844\pi$
$282$	0	0
$283$	−31.6306	−1.88024	−0.940121	−	0.340842i	$-0.889288\pi$
$283$	−31.6306	−1.88024	−0.940121	+	0.340842i	$0.889288\pi$
$284$	0	0
$285$	−0.362233	−0.0214568
$286$	0	0
$287$	−6.78183	−0.400319
$288$	0	0
$289$	11.5148	0.677339
$290$	0	0
$291$	11.0822	0.649648
$292$	0	0
$293$	6.77777	0.395961	0.197981	−	0.980206i	$-0.436562\pi$
$293$	6.77777	0.395961	0.197981	+	0.980206i	$0.436562\pi$
$294$	0	0
$295$	2.02647	0.117986
$296$	0	0
$297$	30.8683	1.79116
$298$	0	0
$299$	0	0
$300$	0	0
$301$	2.88346	0.166200
$302$	0	0
$303$	−15.4901	−0.889885
$304$	0	0
$305$	−11.2241	−0.642688
$306$	0	0
$307$	6.07347	0.346631	0.173316	−	0.984866i	$-0.444552\pi$
$307$	6.07347	0.346631	0.173316	+	0.984866i	$0.444552\pi$
$308$	0	0
$309$	0.820161	0.0466574
$310$	0	0
$311$	−10.2293	−0.580052	−0.290026	−	0.957019i	$-0.593664\pi$
$311$	−10.2293	−0.580052	−0.290026	+	0.957019i	$0.593664\pi$
$312$	0	0
$313$	4.02016	0.227233	0.113616	−	0.993525i	$-0.463756\pi$
$313$	4.02016	0.227233	0.113616	+	0.993525i	$0.463756\pi$
$314$	0	0
$315$	−1.39946	−0.0788506
$316$	0	0
$317$	4.99304	0.280437	0.140218	−	0.990121i	$-0.455220\pi$
$317$	4.99304	0.280437	0.140218	+	0.990121i	$0.455220\pi$
$318$	0	0
$319$	11.6799	0.653947
$320$	0	0
$321$	−0.645316	−0.0360180
$322$	0	0
$323$	1.44672	0.0804974
$324$	0	0
$325$	8.20892	0.455349
$326$	0	0
$327$	−7.46719	−0.412936
$328$	0	0
$329$	4.37331	0.241108
$330$	0	0
$331$	−4.90967	−0.269860	−0.134930	−	0.990855i	$-0.543081\pi$
$331$	−4.90967	−0.269860	−0.134930	+	0.990855i	$0.543081\pi$
$332$	0	0
$333$	−18.2870	−1.00212
$334$	0	0
$335$	−7.61062	−0.415813
$336$	0	0
$337$	29.6329	1.61421	0.807105	−	0.590408i	$-0.201033\pi$
$337$	29.6329	1.61421	0.807105	+	0.590408i	$0.201033\pi$
$338$	0	0
$339$	−1.35615	−0.0736560
$340$	0	0
$341$	38.8972	2.10640
$342$	0	0
$343$	8.53426	0.460807
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.8579	0.690248	0.345124	−	0.938557i	$-0.387837\pi$
$347$	12.8579	0.690248	0.345124	+	0.938557i	$0.387837\pi$
$348$	0	0
$349$	−23.5370	−1.25990	−0.629952	−	0.776634i	$-0.716926\pi$
$349$	−23.5370	−1.25990	−0.629952	+	0.776634i	$0.716926\pi$
$350$	0	0
$351$	12.3253	0.657874
$352$	0	0
$353$	8.31289	0.442450	0.221225	−	0.975223i	$-0.428994\pi$
$353$	8.31289	0.442450	0.221225	+	0.975223i	$0.428994\pi$
$354$	0	0
$355$	−11.1349	−0.590982
$356$	0	0
$357$	−3.64453	−0.192889
$358$	0	0
$359$	19.3351	1.02047	0.510234	−	0.860036i	$-0.329559\pi$
$359$	19.3351	1.02047	0.510234	+	0.860036i	$0.329559\pi$
$360$	0	0
$361$	−18.9266	−0.996137
$362$	0	0
$363$	25.7854	1.35338
$364$	0	0
$365$	7.29670	0.381927
$366$	0	0
$367$	−30.4419	−1.58906	−0.794528	−	0.607228i	$-0.792282\pi$
$367$	−30.4419	−1.58906	−0.794528	+	0.607228i	$0.792282\pi$
$368$	0	0
$369$	−19.6348	−1.02215
$370$	0	0
$371$	1.65393	0.0858677
$372$	0	0
$373$	19.2962	0.999120	0.499560	−	0.866279i	$-0.333495\pi$
$373$	19.2962	0.999120	0.499560	+	0.866279i	$0.333495\pi$
$374$	0	0
$375$	11.3517	0.586200
$376$	0	0
$377$	4.66360	0.240188
$378$	0	0
$379$	−32.3335	−1.66086	−0.830430	−	0.557123i	$-0.811905\pi$
$379$	−32.3335	−1.66086	−0.830430	+	0.557123i	$0.811905\pi$
$380$	0	0
$381$	7.81912	0.400586
$382$	0	0
$383$	7.18995	0.367389	0.183695	−	0.982983i	$-0.441194\pi$
$383$	7.18995	0.367389	0.183695	+	0.982983i	$0.441194\pi$
$384$	0	0
$385$	−4.53949	−0.231354
$386$	0	0
$387$	8.34822	0.424364
$388$	0	0
$389$	26.2852	1.33271	0.666357	−	0.745633i	$-0.267853\pi$
$389$	26.2852	1.33271	0.666357	+	0.745633i	$0.267853\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0.716268	0.0361309
$394$	0	0
$395$	−7.38102	−0.371379
$396$	0	0
$397$	−9.27172	−0.465334	−0.232667	−	0.972556i	$-0.574745\pi$
$397$	−9.27172	−0.465334	−0.232667	+	0.972556i	$0.574745\pi$
$398$	0	0
$399$	−0.184908	−0.00925697
$400$	0	0
$401$	−28.0321	−1.39986	−0.699929	−	0.714213i	$-0.746785\pi$
$401$	−28.0321	−1.39986	−0.699929	+	0.714213i	$0.746785\pi$
$402$	0	0
$403$	15.5311	0.773659
$404$	0	0
$405$	0.312947	0.0155505
$406$	0	0
$407$	−59.3184	−2.94030
$408$	0	0
$409$	0.0443103	0.00219100	0.00109550	−	0.999999i	$-0.499651\pi$
$409$	0.0443103	0.00219100	0.00109550	+	0.999999i	$0.499651\pi$
$410$	0	0
$411$	−13.6282	−0.672228
$412$	0	0
$413$	1.03445	0.0509017
$414$	0	0
$415$	−2.39972	−0.117798
$416$	0	0
$417$	−18.5642	−0.909093
$418$	0	0
$419$	−12.3373	−0.602714	−0.301357	−	0.953511i	$-0.597440\pi$
$419$	−12.3373	−0.602714	−0.301357	+	0.953511i	$0.597440\pi$
$420$	0	0
$421$	2.79149	0.136049	0.0680244	−	0.997684i	$-0.478330\pi$
$421$	2.79149	0.136049	0.0680244	+	0.997684i	$0.478330\pi$
$422$	0	0
$423$	12.6616	0.615630
$424$	0	0
$425$	−18.6378	−0.904067
$426$	0	0
$427$	−5.72951	−0.277270
$428$	0	0
$429$	15.0751	0.727832
$430$	0	0
$431$	−3.49668	−0.168429	−0.0842146	−	0.996448i	$-0.526838\pi$
$431$	−3.49668	−0.168429	−0.0842146	+	0.996448i	$0.526838\pi$
$432$	0	0
$433$	−9.42923	−0.453140	−0.226570	−	0.973995i	$-0.572751\pi$
$433$	−9.42923	−0.453140	−0.226570	+	0.973995i	$0.572751\pi$
$434$	0	0
$435$	2.65115	0.127113
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−22.5828	−1.07782	−0.538910	−	0.842363i	$-0.681164\pi$
$439$	−22.5828	−1.07782	−0.538910	+	0.842363i	$0.681164\pi$
$440$	0	0
$441$	11.9970	0.571288
$442$	0	0
$443$	6.08056	0.288896	0.144448	−	0.989512i	$-0.453859\pi$
$443$	6.08056	0.288896	0.144448	+	0.989512i	$0.453859\pi$
$444$	0	0
$445$	−5.64032	−0.267377
$446$	0	0
$447$	−20.0372	−0.947729
$448$	0	0
$449$	−12.0641	−0.569338	−0.284669	−	0.958626i	$-0.591884\pi$
$449$	−12.0641	−0.569338	−0.284669	+	0.958626i	$0.591884\pi$
$450$	0	0
$451$	−63.6903	−2.99906
$452$	0	0
$453$	12.5141	0.587965
$454$	0	0
$455$	−1.81255	−0.0849737
$456$	0	0
$457$	−20.1601	−0.943050	−0.471525	−	0.881853i	$-0.656296\pi$
$457$	−20.1601	−0.943050	−0.471525	+	0.881853i	$0.656296\pi$
$458$	0	0
$459$	−27.9837	−1.30617
$460$	0	0
$461$	34.1368	1.58991	0.794954	−	0.606670i	$-0.207495\pi$
$461$	34.1368	1.58991	0.794954	+	0.606670i	$0.207495\pi$
$462$	0	0
$463$	−11.6716	−0.542424	−0.271212	−	0.962520i	$-0.587425\pi$
$463$	−11.6716	−0.542424	−0.271212	+	0.962520i	$0.587425\pi$
$464$	0	0
$465$	8.82908	0.409439
$466$	0	0
$467$	−4.74513	−0.219579	−0.109789	−	0.993955i	$-0.535018\pi$
$467$	−4.74513	−0.219579	−0.109789	+	0.993955i	$0.535018\pi$
$468$	0	0
$469$	−3.88497	−0.179391
$470$	0	0
$471$	3.51272	0.161857
$472$	0	0
$473$	27.0795	1.24512
$474$	0	0
$475$	−0.945602	−0.0433872
$476$	0	0
$477$	4.78847	0.219249
$478$	0	0
$479$	29.7540	1.35950	0.679748	−	0.733445i	$-0.262089\pi$
$479$	29.7540	1.35950	0.679748	+	0.733445i	$0.262089\pi$
$480$	0	0
$481$	−23.6850	−1.07994
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.5136	−0.568213
$486$	0	0
$487$	8.71408	0.394873	0.197436	−	0.980316i	$-0.436738\pi$
$487$	8.71408	0.394873	0.197436	+	0.980316i	$0.436738\pi$
$488$	0	0
$489$	−12.0390	−0.544424
$490$	0	0
$491$	−15.0620	−0.679738	−0.339869	−	0.940473i	$-0.610383\pi$
$491$	−15.0620	−0.679738	−0.339869	+	0.940473i	$0.610383\pi$
$492$	0	0
$493$	−10.5884	−0.476877
$494$	0	0
$495$	−13.1428	−0.590723
$496$	0	0
$497$	−5.68402	−0.254963
$498$	0	0
$499$	−31.5993	−1.41458	−0.707290	−	0.706923i	$-0.750083\pi$
$499$	−31.5993	−1.41458	−0.707290	+	0.706923i	$0.750083\pi$
$500$	0	0
$501$	23.8651	1.06621
$502$	0	0
$503$	33.4242	1.49031	0.745157	−	0.666890i	$-0.232375\pi$
$503$	33.4242	1.49031	0.745157	+	0.666890i	$0.232375\pi$
$504$	0	0
$505$	17.4909	0.778336
$506$	0	0
$507$	−8.12677	−0.360922
$508$	0	0
$509$	−17.9178	−0.794191	−0.397096	−	0.917777i	$-0.629982\pi$
$509$	−17.9178	−0.794191	−0.397096	+	0.917777i	$0.629982\pi$
$510$	0	0
$511$	3.72472	0.164772
$512$	0	0
$513$	−1.41977	−0.0626844
$514$	0	0
$515$	−0.926098	−0.0408087
$516$	0	0
$517$	41.0711	1.80631
$518$	0	0
$519$	12.0734	0.529965
$520$	0	0
$521$	8.20662	0.359538	0.179769	−	0.983709i	$-0.442465\pi$
$521$	8.20662	0.359538	0.179769	+	0.983709i	$0.442465\pi$
$522$	0	0
$523$	6.49965	0.284210	0.142105	−	0.989852i	$-0.454613\pi$
$523$	6.49965	0.284210	0.142105	+	0.989852i	$0.454613\pi$
$524$	0	0
$525$	2.38214	0.103965
$526$	0	0
$527$	−35.2623	−1.53605
$528$	0	0
$529$	0	0
$530$	0	0
$531$	2.99494	0.129969
$532$	0	0
$533$	−25.4306	−1.10152
$534$	0	0
$535$	0.728668	0.0315031
$536$	0	0
$537$	19.0837	0.823524
$538$	0	0
$539$	38.9153	1.67620
$540$	0	0
$541$	−26.3569	−1.13317	−0.566585	−	0.824003i	$-0.691736\pi$
$541$	−26.3569	−1.13317	−0.566585	+	0.824003i	$0.691736\pi$
$542$	0	0
$543$	22.2049	0.952903
$544$	0	0
$545$	8.43168	0.361174
$546$	0	0
$547$	20.2475	0.865721	0.432861	−	0.901461i	$-0.357504\pi$
$547$	20.2475	0.865721	0.432861	+	0.901461i	$0.357504\pi$
$548$	0	0
$549$	−16.5881	−0.707964
$550$	0	0
$551$	−0.537209	−0.0228859
$552$	0	0
$553$	−3.76777	−0.160222
$554$	0	0
$555$	−13.4644	−0.571531
$556$	0	0
$557$	8.15671	0.345611	0.172806	−	0.984956i	$-0.444717\pi$
$557$	8.15671	0.345611	0.172806	+	0.984956i	$0.444717\pi$
$558$	0	0
$559$	10.8124	0.457318
$560$	0	0
$561$	−34.2270	−1.44506
$562$	0	0
$563$	−31.4513	−1.32551	−0.662757	−	0.748835i	$-0.730614\pi$
$563$	−31.4513	−1.32551	−0.662757	+	0.748835i	$0.730614\pi$
$564$	0	0
$565$	1.53132	0.0644230
$566$	0	0
$567$	0.159749	0.00670884
$568$	0	0
$569$	−26.6983	−1.11925	−0.559625	−	0.828746i	$-0.689055\pi$
$569$	−26.6983	−1.11925	−0.559625	+	0.828746i	$0.689055\pi$
$570$	0	0
$571$	28.8560	1.20759	0.603793	−	0.797141i	$-0.293655\pi$
$571$	28.8560	1.20759	0.603793	+	0.797141i	$0.293655\pi$
$572$	0	0
$573$	10.2966	0.430148
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.0824	−1.16909	−0.584544	−	0.811362i	$-0.698726\pi$
$577$	−28.0824	−1.16909	−0.584544	+	0.811362i	$0.698726\pi$
$578$	0	0
$579$	27.8142	1.15592
$580$	0	0
$581$	−1.22498	−0.0508207
$582$	0	0
$583$	15.5326	0.643293
$584$	0	0
$585$	−5.24772	−0.216966
$586$	0	0
$587$	28.0401	1.15734	0.578669	−	0.815563i	$-0.303572\pi$
$587$	28.0401	1.15734	0.578669	+	0.815563i	$0.303572\pi$
$588$	0	0
$589$	−1.78906	−0.0737168
$590$	0	0
$591$	5.73408	0.235868
$592$	0	0
$593$	−7.57300	−0.310986	−0.155493	−	0.987837i	$-0.549697\pi$
$593$	−7.57300	−0.310986	−0.155493	+	0.987837i	$0.549697\pi$
$594$	0	0
$595$	4.11528	0.168710
$596$	0	0
$597$	27.2213	1.11409
$598$	0	0
$599$	12.5238	0.511708	0.255854	−	0.966715i	$-0.417643\pi$
$599$	12.5238	0.511708	0.255854	+	0.966715i	$0.417643\pi$
$600$	0	0
$601$	−18.2262	−0.743463	−0.371732	−	0.928340i	$-0.621236\pi$
$601$	−18.2262	−0.743463	−0.371732	+	0.928340i	$0.621236\pi$
$602$	0	0
$603$	−11.2478	−0.458046
$604$	0	0
$605$	−29.1160	−1.18373
$606$	0	0
$607$	8.86155	0.359679	0.179840	−	0.983696i	$-0.442442\pi$
$607$	8.86155	0.359679	0.179840	+	0.983696i	$0.442442\pi$
$608$	0	0
$609$	1.35333	0.0548395
$610$	0	0
$611$	16.3991	0.663437
$612$	0	0
$613$	−21.3136	−0.860849	−0.430425	−	0.902627i	$-0.641636\pi$
$613$	−21.3136	−0.860849	−0.430425	+	0.902627i	$0.641636\pi$
$614$	0	0
$615$	−14.4568	−0.582953
$616$	0	0
$617$	35.7387	1.43879	0.719393	−	0.694603i	$-0.244420\pi$
$617$	35.7387	1.43879	0.719393	+	0.694603i	$0.244420\pi$
$618$	0	0
$619$	7.37359	0.296370	0.148185	−	0.988960i	$-0.452657\pi$
$619$	7.37359	0.296370	0.148185	+	0.988960i	$0.452657\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.87920	−0.115353
$624$	0	0
$625$	4.63343	0.185337
$626$	0	0
$627$	−1.73653	−0.0693503
$628$	0	0
$629$	53.7752	2.14416
$630$	0	0
$631$	19.2251	0.765338	0.382669	−	0.923886i	$-0.375005\pi$
$631$	19.2251	0.765338	0.382669	+	0.923886i	$0.375005\pi$
$632$	0	0
$633$	9.64794	0.383471
$634$	0	0
$635$	−8.82908	−0.350371
$636$	0	0
$637$	15.5383	0.615651
$638$	0	0
$639$	−16.4564	−0.651006
$640$	0	0
$641$	15.5293	0.613371	0.306686	−	0.951811i	$-0.400780\pi$
$641$	15.5293	0.613371	0.306686	+	0.951811i	$0.400780\pi$
$642$	0	0
$643$	−12.8573	−0.507043	−0.253522	−	0.967330i	$-0.581589\pi$
$643$	−12.8573	−0.507043	−0.253522	+	0.967330i	$0.581589\pi$
$644$	0	0
$645$	6.14664	0.242024
$646$	0	0
$647$	35.2209	1.38468	0.692339	−	0.721572i	$-0.256580\pi$
$647$	35.2209	1.38468	0.692339	+	0.721572i	$0.256580\pi$
$648$	0	0
$649$	9.71480	0.381339
$650$	0	0
$651$	4.50695	0.176641
$652$	0	0
$653$	28.7710	1.12590	0.562948	−	0.826492i	$-0.309667\pi$
$653$	28.7710	1.12590	0.562948	+	0.826492i	$0.309667\pi$
$654$	0	0
$655$	−0.808784	−0.0316018
$656$	0	0
$657$	10.7839	0.420718
$658$	0	0
$659$	−3.01164	−0.117317	−0.0586584	−	0.998278i	$-0.518682\pi$
$659$	−3.01164	−0.117317	−0.0586584	+	0.998278i	$0.518682\pi$
$660$	0	0
$661$	9.58560	0.372837	0.186418	−	0.982470i	$-0.440312\pi$
$661$	9.58560	0.372837	0.186418	+	0.982470i	$0.440312\pi$
$662$	0	0
$663$	−13.6663	−0.530756
$664$	0	0
$665$	0.208792	0.00809659
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.4801	0.714484
$670$	0	0
$671$	−53.8076	−2.07722
$672$	0	0
$673$	15.9623	0.615301	0.307650	−	0.951499i	$-0.400457\pi$
$673$	15.9623	0.615301	0.307650	+	0.951499i	$0.400457\pi$
$674$	0	0
$675$	18.2907	0.704009
$676$	0	0
$677$	26.1072	1.00338	0.501690	−	0.865048i	$-0.332712\pi$
$677$	26.1072	1.00338	0.501690	+	0.865048i	$0.332712\pi$
$678$	0	0
$679$	−6.38777	−0.245140
$680$	0	0
$681$	19.5816	0.750368
$682$	0	0
$683$	41.3823	1.58345	0.791726	−	0.610877i	$-0.209183\pi$
$683$	41.3823	1.58345	0.791726	+	0.610877i	$0.209183\pi$
$684$	0	0
$685$	15.3884	0.587962
$686$	0	0
$687$	8.15739	0.311224
$688$	0	0
$689$	6.20193	0.236275
$690$	0	0
$691$	−40.6203	−1.54527	−0.772635	−	0.634850i	$-0.781062\pi$
$691$	−40.6203	−1.54527	−0.772635	+	0.634850i	$0.781062\pi$
$692$	0	0
$693$	−6.70895	−0.254852
$694$	0	0
$695$	20.9620	0.795135
$696$	0	0
$697$	57.7386	2.18700
$698$	0	0
$699$	6.00997	0.227318
$700$	0	0
$701$	1.58900	0.0600157	0.0300078	−	0.999550i	$-0.490447\pi$
$701$	1.58900	0.0600157	0.0300078	+	0.999550i	$0.490447\pi$
$702$	0	0
$703$	2.72832	0.102900
$704$	0	0
$705$	9.32253	0.351107
$706$	0	0
$707$	8.92853	0.335792
$708$	0	0
$709$	−4.32980	−0.162609	−0.0813046	−	0.996689i	$-0.525909\pi$
$709$	−4.32980	−0.162609	−0.0813046	+	0.996689i	$0.525909\pi$
$710$	0	0
$711$	−10.9085	−0.409100
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−17.0222	−0.636596
$716$	0	0
$717$	−5.67910	−0.212090
$718$	0	0
$719$	25.9486	0.967718	0.483859	−	0.875146i	$-0.339235\pi$
$719$	25.9486	0.967718	0.483859	+	0.875146i	$0.339235\pi$
$720$	0	0
$721$	−0.472742	−0.0176058
$722$	0	0
$723$	−1.69088	−0.0628845
$724$	0	0
$725$	6.92078	0.257031
$726$	0	0
$727$	−17.3672	−0.644114	−0.322057	−	0.946720i	$-0.604374\pi$
$727$	−17.3672	−0.644114	−0.322057	+	0.946720i	$0.604374\pi$
$728$	0	0
$729$	17.5698	0.650735
$730$	0	0
$731$	−24.5489	−0.907976
$732$	0	0
$733$	47.1889	1.74296	0.871480	−	0.490431i	$-0.163161\pi$
$733$	47.1889	1.74296	0.871480	+	0.490431i	$0.163161\pi$
$734$	0	0
$735$	8.83319	0.325817
$736$	0	0
$737$	−36.4850	−1.34394
$738$	0	0
$739$	−15.0987	−0.555416	−0.277708	−	0.960666i	$-0.589575\pi$
$739$	−15.0987	−0.555416	−0.277708	+	0.960666i	$0.589575\pi$
$740$	0	0
$741$	−0.693371	−0.0254716
$742$	0	0
$743$	11.8979	0.436490	0.218245	−	0.975894i	$-0.429967\pi$
$743$	11.8979	0.436490	0.218245	+	0.975894i	$0.429967\pi$
$744$	0	0
$745$	22.6253	0.828928
$746$	0	0
$747$	−3.54657	−0.129762
$748$	0	0
$749$	0.371961	0.0135912
$750$	0	0
$751$	−7.94974	−0.290090	−0.145045	−	0.989425i	$-0.546333\pi$
$751$	−7.94974	−0.290090	−0.145045	+	0.989425i	$0.546333\pi$
$752$	0	0
$753$	9.71351	0.353980
$754$	0	0
$755$	−14.1305	−0.514262
$756$	0	0
$757$	22.2062	0.807100	0.403550	−	0.914958i	$-0.367776\pi$
$757$	22.2062	0.807100	0.403550	+	0.914958i	$0.367776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.39284	0.122991	0.0614953	−	0.998107i	$-0.480413\pi$
$761$	3.39284	0.122991	0.0614953	+	0.998107i	$0.480413\pi$
$762$	0	0
$763$	4.30410	0.155819
$764$	0	0
$765$	11.9146	0.430773
$766$	0	0
$767$	3.87898	0.140062
$768$	0	0
$769$	4.15320	0.149768	0.0748842	−	0.997192i	$-0.476141\pi$
$769$	4.15320	0.149768	0.0748842	+	0.997192i	$0.476141\pi$
$770$	0	0
$771$	−3.96367	−0.142748
$772$	0	0
$773$	−28.2415	−1.01578	−0.507888	−	0.861423i	$-0.669574\pi$
$773$	−28.2415	−1.01578	−0.507888	+	0.861423i	$0.669574\pi$
$774$	0	0
$775$	23.0481	0.827914
$776$	0	0
$777$	−6.87312	−0.246572
$778$	0	0
$779$	2.92941	0.104957
$780$	0	0
$781$	−53.3804	−1.91010
$782$	0	0
$783$	10.3912	0.371351
$784$	0	0
$785$	−3.96644	−0.141568
$786$	0	0
$787$	−21.8912	−0.780336	−0.390168	−	0.920744i	$-0.627583\pi$
$787$	−21.8912	−0.780336	−0.390168	+	0.920744i	$0.627583\pi$
$788$	0	0
$789$	−17.5595	−0.625133
$790$	0	0
$791$	0.781687	0.0277936
$792$	0	0
$793$	−21.4846	−0.762941
$794$	0	0
$795$	3.52566	0.125042
$796$	0	0
$797$	−10.7266	−0.379956	−0.189978	−	0.981788i	$-0.560842\pi$
$797$	−10.7266	−0.379956	−0.189978	+	0.981788i	$0.560842\pi$
$798$	0	0
$799$	−37.2331	−1.31721
$800$	0	0
$801$	−8.33588	−0.294534
$802$	0	0
$803$	34.9801	1.23442
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.89009	−0.172139
$808$	0	0
$809$	33.6941	1.18462	0.592312	−	0.805709i	$-0.298215\pi$
$809$	33.6941	1.18462	0.592312	+	0.805709i	$0.298215\pi$
$810$	0	0
$811$	−34.7974	−1.22190	−0.610952	−	0.791668i	$-0.709213\pi$
$811$	−34.7974	−1.22190	−0.610952	+	0.791668i	$0.709213\pi$
$812$	0	0
$813$	3.81125	0.133667
$814$	0	0
$815$	13.5940	0.476179
$816$	0	0
$817$	−1.24551	−0.0435748
$818$	0	0
$819$	−2.67879	−0.0936043
$820$	0	0
$821$	−23.8353	−0.831857	−0.415929	−	0.909397i	$-0.636543\pi$
$821$	−23.8353	−0.831857	−0.415929	+	0.909397i	$0.636543\pi$
$822$	0	0
$823$	23.7843	0.829069	0.414534	−	0.910034i	$-0.363944\pi$
$823$	23.7843	0.829069	0.414534	+	0.910034i	$0.363944\pi$
$824$	0	0
$825$	22.3714	0.778873
$826$	0	0
$827$	−30.8236	−1.07184	−0.535921	−	0.844268i	$-0.680035\pi$
$827$	−30.8236	−1.07184	−0.535921	+	0.844268i	$0.680035\pi$
$828$	0	0
$829$	−30.9416	−1.07465	−0.537324	−	0.843376i	$-0.680565\pi$
$829$	−30.9416	−1.07465	−0.537324	+	0.843376i	$0.680565\pi$
$830$	0	0
$831$	−13.1331	−0.455584
$832$	0	0
$833$	−35.2787	−1.22234
$834$	0	0
$835$	−26.9476	−0.932562
$836$	0	0
$837$	34.6056	1.19614
$838$	0	0
$839$	39.3250	1.35765	0.678825	−	0.734300i	$-0.262489\pi$
$839$	39.3250	1.35765	0.678825	+	0.734300i	$0.262489\pi$
$840$	0	0
$841$	−25.0682	−0.864421
$842$	0	0
$843$	−6.81841	−0.234838
$844$	0	0
$845$	9.17646	0.315680
$846$	0	0
$847$	−14.8627	−0.510690
$848$	0	0
$849$	−34.4190	−1.18126
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−5.80760	−0.198848	−0.0994242	−	0.995045i	$-0.531700\pi$
$853$	−5.80760	−0.198848	−0.0994242	+	0.995045i	$0.531700\pi$
$854$	0	0
$855$	0.604495	0.0206733
$856$	0	0
$857$	−21.1686	−0.723107	−0.361553	−	0.932351i	$-0.617753\pi$
$857$	−21.1686	−0.723107	−0.361553	+	0.932351i	$0.617753\pi$
$858$	0	0
$859$	−2.69259	−0.0918698	−0.0459349	−	0.998944i	$-0.514627\pi$
$859$	−2.69259	−0.0918698	−0.0459349	+	0.998944i	$0.514627\pi$
$860$	0	0
$861$	−7.37969	−0.251499
$862$	0	0
$863$	−20.1950	−0.687447	−0.343724	−	0.939071i	$-0.611688\pi$
$863$	−20.1950	−0.687447	−0.343724	+	0.939071i	$0.611688\pi$
$864$	0	0
$865$	−13.6329	−0.463532
$866$	0	0
$867$	12.5299	0.425537
$868$	0	0
$869$	−35.3843	−1.20033
$870$	0	0
$871$	−14.5679	−0.493615
$872$	0	0
$873$	−18.4939	−0.625925
$874$	0	0
$875$	−6.54314	−0.221199
$876$	0	0
$877$	21.0342	0.710273	0.355137	−	0.934814i	$-0.384434\pi$
$877$	21.0342	0.710273	0.355137	+	0.934814i	$0.384434\pi$
$878$	0	0
$879$	7.37527	0.248762
$880$	0	0
$881$	−24.6353	−0.829983	−0.414992	−	0.909825i	$-0.636215\pi$
$881$	−24.6353	−0.829983	−0.414992	+	0.909825i	$0.636215\pi$
$882$	0	0
$883$	1.16589	0.0392352	0.0196176	−	0.999808i	$-0.493755\pi$
$883$	1.16589	0.0392352	0.0196176	+	0.999808i	$0.493755\pi$
$884$	0	0
$885$	2.20512	0.0741241
$886$	0	0
$887$	10.1795	0.341793	0.170897	−	0.985289i	$-0.445334\pi$
$887$	10.1795	0.341793	0.170897	+	0.985289i	$0.445334\pi$
$888$	0	0
$889$	−4.50695	−0.151158
$890$	0	0
$891$	1.50026	0.0502605
$892$	0	0
$893$	−1.88905	−0.0632145
$894$	0	0
$895$	−21.5487	−0.720293
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.0940	0.436708
$900$	0	0
$901$	−14.0811	−0.469108
$902$	0	0
$903$	3.13765	0.104415
$904$	0	0
$905$	−25.0730	−0.833454
$906$	0	0
$907$	−19.6527	−0.652556	−0.326278	−	0.945274i	$-0.605795\pi$
$907$	−19.6527	−0.652556	−0.326278	+	0.945274i	$0.605795\pi$
$908$	0	0
$909$	25.8500	0.857389
$910$	0	0
$911$	47.8643	1.58581	0.792907	−	0.609343i	$-0.208567\pi$
$911$	47.8643	1.58581	0.792907	+	0.609343i	$0.208567\pi$
$912$	0	0
$913$	−11.5042	−0.380732
$914$	0	0
$915$	−12.2135	−0.403767
$916$	0	0
$917$	−0.412858	−0.0136338
$918$	0	0
$919$	35.7413	1.17900	0.589499	−	0.807769i	$-0.299325\pi$
$919$	35.7413	1.17900	0.589499	+	0.807769i	$0.299325\pi$
$920$	0	0
$921$	6.60888	0.217770
$922$	0	0
$923$	−21.3140	−0.701560
$924$	0	0
$925$	−35.1485	−1.15568
$926$	0	0
$927$	−1.36869	−0.0449536
$928$	0	0
$929$	−22.9339	−0.752435	−0.376218	−	0.926531i	$-0.622776\pi$
$929$	−22.9339	−0.752435	−0.376218	+	0.926531i	$0.622776\pi$
$930$	0	0
$931$	−1.78989	−0.0586613
$932$	0	0
$933$	−11.1311	−0.364416
$934$	0	0
$935$	38.6479	1.26392
$936$	0	0
$937$	−19.6756	−0.642773	−0.321386	−	0.946948i	$-0.604149\pi$
$937$	−19.6756	−0.642773	−0.321386	+	0.946948i	$0.604149\pi$
$938$	0	0
$939$	4.37456	0.142758
$940$	0	0
$941$	−52.2897	−1.70460	−0.852299	−	0.523056i	$-0.824792\pi$
$941$	−52.2897	−1.70460	−0.852299	+	0.523056i	$0.824792\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.03863	−0.131377
$946$	0	0
$947$	26.3714	0.856956	0.428478	−	0.903552i	$-0.359050\pi$
$947$	26.3714	0.856956	0.428478	+	0.903552i	$0.359050\pi$
$948$	0	0
$949$	13.9670	0.453389
$950$	0	0
$951$	5.43320	0.176184
$952$	0	0
$953$	41.4107	1.34142	0.670711	−	0.741718i	$-0.265989\pi$
$953$	41.4107	1.34142	0.670711	+	0.741718i	$0.265989\pi$
$954$	0	0
$955$	−11.6266	−0.376228
$956$	0	0
$957$	12.7095	0.410840
$958$	0	0
$959$	7.85529	0.253661
$960$	0	0
$961$	12.6066	0.406663
$962$	0	0
$963$	1.07690	0.0347028
$964$	0	0
$965$	−31.4068	−1.01102
$966$	0	0
$967$	24.8632	0.799546	0.399773	−	0.916614i	$-0.369089\pi$
$967$	24.8632	0.799546	0.399773	+	0.916614i	$0.369089\pi$
$968$	0	0
$969$	1.57425	0.0505723
$970$	0	0
$971$	52.7969	1.69433	0.847166	−	0.531328i	$-0.178307\pi$
$971$	52.7969	1.69433	0.847166	+	0.531328i	$0.178307\pi$
$972$	0	0
$973$	10.7004	0.343040
$974$	0	0
$975$	8.93258	0.286072
$976$	0	0
$977$	−13.1353	−0.420236	−0.210118	−	0.977676i	$-0.567385\pi$
$977$	−13.1353	−0.420236	−0.210118	+	0.977676i	$0.567385\pi$
$978$	0	0
$979$	−27.0394	−0.864184
$980$	0	0
$981$	12.4613	0.397857
$982$	0	0
$983$	53.7978	1.71588	0.857942	−	0.513746i	$-0.171743\pi$
$983$	53.7978	1.71588	0.857942	+	0.513746i	$0.171743\pi$
$984$	0	0
$985$	−6.47472	−0.206302
$986$	0	0
$987$	4.75884	0.151476
$988$	0	0
$989$	0	0
$990$	0	0
$991$	43.6587	1.38686	0.693432	−	0.720522i	$-0.256098\pi$
$991$	43.6587	1.38686	0.693432	+	0.720522i	$0.256098\pi$
$992$	0	0
$993$	−5.34249	−0.169539
$994$	0	0
$995$	−30.7373	−0.974438
$996$	0	0
$997$	−53.3789	−1.69053	−0.845263	−	0.534350i	$-0.820556\pi$
$997$	−53.3789	−1.69053	−0.845263	+	0.534350i	$0.820556\pi$
$998$	0	0
$999$	−52.7736	−1.66968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bx.1.4		5
4.3	odd	2		1058.2.a.m.1.2		5
12.11	even	2		9522.2.a.bp.1.5		5
23.3	even	11		368.2.m.b.193.1		10
23.8	even	11		368.2.m.b.225.1		10
23.22	odd	2		8464.2.a.bw.1.4		5
92.3	odd	22		46.2.c.a.9.1	✓	10
92.31	odd	22		46.2.c.a.41.1	yes	10
92.91	even	2		1058.2.a.l.1.2		5
276.95	even	22		414.2.i.f.55.1		10
276.215	even	22		414.2.i.f.271.1		10
276.275	odd	2		9522.2.a.bu.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.a.9.1	✓	10	92.3	odd	22
46.2.c.a.41.1	yes	10	92.31	odd	22
368.2.m.b.193.1		10	23.3	even	11
368.2.m.b.225.1		10	23.8	even	11
414.2.i.f.55.1		10	276.95	even	22
414.2.i.f.271.1		10	276.215	even	22
1058.2.a.l.1.2		5	92.91	even	2
1058.2.a.m.1.2		5	4.3	odd	2
8464.2.a.bw.1.4		5	23.22	odd	2
8464.2.a.bx.1.4		5	1.1	even	1	trivial
9522.2.a.bp.1.5		5	12.11	even	2
9522.2.a.bu.1.1		5	276.275	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.08816 q^{3} -1.22871 q^{5} -0.627214 q^{7} -1.81592 q^{9} -5.89037 q^{11} -2.35194 q^{13} -1.33703 q^{15} +5.33992 q^{17} +0.270925 q^{19} -0.682507 q^{21} -3.49028 q^{25} -5.24047 q^{27} -1.98287 q^{29} -6.60353 q^{31} -6.40964 q^{33} +0.770663 q^{35} +10.0704 q^{37} -2.55928 q^{39} +10.8126 q^{41} -4.59725 q^{43} +2.23123 q^{45} -6.97259 q^{47} -6.60660 q^{49} +5.81067 q^{51} -2.63694 q^{53} +7.23754 q^{55} +0.294808 q^{57} -1.64927 q^{59} +9.13485 q^{61} +1.13897 q^{63} +2.88984 q^{65} +6.19401 q^{67} +9.06233 q^{71} -5.93852 q^{73} -3.79797 q^{75} +3.69452 q^{77} +6.00714 q^{79} -0.254696 q^{81} +1.95305 q^{83} -6.56120 q^{85} -2.15768 q^{87} +4.59045 q^{89} +1.47517 q^{91} -7.18567 q^{93} -0.332887 q^{95} +10.1844 q^{97} +10.6964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} + 8 q^{5} - 7 q^{7} - q^{9} - 5 q^{11} - 7 q^{13} + q^{15} + 13 q^{17} - 12 q^{19} + 6 q^{21} + q^{25} + 20 q^{27} - 4 q^{29} - 6 q^{31} + 9 q^{33} - 9 q^{35} + 14 q^{37} + 6 q^{39} + q^{41}+ \cdots + 34 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 + 8 * q^5 - 7 * q^7 - q^9 - 5 * q^11 - 7 * q^13 + q^15 + 13 * q^17 - 12 * q^19 + 6 * q^21 + q^25 + 20 * q^27 - 4 * q^29 - 6 * q^31 + 9 * q^33 - 9 * q^35 + 14 * q^37 + 6 * q^39 + q^41 - 11 * q^43 + 5 * q^45 - 9 * q^47 - 12 * q^49 + 14 * q^51 + 6 * q^53 + 14 * q^55 - 7 * q^57 - 3 * q^59 + 13 * q^61 + 19 * q^63 - 9 * q^65 - 10 * q^67 - 8 * q^71 - 4 * q^73 - 4 * q^75 + 18 * q^77 + 18 * q^79 + 33 * q^81 - 2 * q^83 + 23 * q^85 - 6 * q^87 + 10 * q^89 + 12 * q^91 - 20 * q^93 - 28 * q^95 + 17 * q^97 + 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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