LMFDB - Embedded newform 8464.2.a.bv.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,2,0,-9,0,7,0,-1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
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			$-1.68251$ 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)

$f(q)$	$=$	$q+1.11546 q^{3} +1.08816 q^{5} -0.863693 q^{7} -1.75575 q^{9} +0.657415 q^{11} +6.20009 q^{13} +1.21379 q^{15} -5.69389 q^{17} -6.45317 q^{19} -0.963415 q^{21} -3.81592 q^{25} -5.30485 q^{27} +0.101861 q^{29} +1.54399 q^{31} +0.733320 q^{33} -0.939833 q^{35} +7.33771 q^{37} +6.91595 q^{39} +6.09398 q^{41} -5.37582 q^{43} -1.91053 q^{45} +7.84594 q^{47} -6.25403 q^{49} -6.35131 q^{51} +4.75696 q^{53} +0.715370 q^{55} -7.19825 q^{57} -5.68391 q^{59} -0.224463 q^{61} +1.51643 q^{63} +6.74666 q^{65} +2.37419 q^{67} -6.42031 q^{71} -4.17196 q^{73} -4.25650 q^{75} -0.567805 q^{77} -16.9996 q^{79} -0.650094 q^{81} -16.9937 q^{83} -6.19584 q^{85} +0.113622 q^{87} -5.06206 q^{89} -5.35498 q^{91} +1.72226 q^{93} -7.02206 q^{95} +9.83304 q^{97} -1.15426 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9} - q^{11} + q^{13} - 11 q^{15} - q^{17} - 10 q^{19} - 11 q^{25} + 10 q^{29} + 11 q^{33} + 3 q^{35} + 20 q^{37} + 22 q^{39} - 3 q^{41} - 27 q^{43} + 5 q^{45} + 11 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9 - q^11 + q^13 - 11 * q^15 - q^17 - 10 * q^19 - 11 * q^25 + 10 * q^29 + 11 * q^33 + 3 * q^35 + 20 * q^37 + 22 * q^39 - 3 * q^41 - 27 * q^43 + 5 * q^45 + 11 * q^47 + 12 * q^49 - 2 * q^53 + 4 * q^55 + 11 * q^57 + 17 * q^59 + 7 * q^61 - 39 * q^63 - 15 * q^65 - 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 - 16 * q^79 + q^81 - 18 * q^83 - 7 * q^85 - 22 * q^87 - 14 * q^89 - 26 * q^91 - 44 * q^93 - 26 * q^95 + 47 * q^97 - 8 * 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
$3$	1.11546	0.644011	0.322005	−	0.946738i	$-0.395643\pi$
$3$	1.11546	0.644011	0.322005	+	0.946738i	$0.395643\pi$
$4$	0	0
$5$	1.08816	0.486638	0.243319	−	0.969946i	$-0.421764\pi$
$5$	1.08816	0.486638	0.243319	+	0.969946i	$0.421764\pi$
$6$	0	0
$7$	−0.863693	−0.326445	−0.163223	−	0.986589i	$-0.552189\pi$
$7$	−0.863693	−0.326445	−0.163223	+	0.986589i	$0.552189\pi$
$8$	0	0
$9$	−1.75575	−0.585250
$10$	0	0
$11$	0.657415	0.198218	0.0991091	−	0.995077i	$-0.468401\pi$
$11$	0.657415	0.198218	0.0991091	+	0.995077i	$0.468401\pi$
$12$	0	0
$13$	6.20009	1.71960	0.859798	−	0.510635i	$-0.170590\pi$
$13$	6.20009	1.71960	0.859798	+	0.510635i	$0.170590\pi$
$14$	0	0
$15$	1.21379	0.313400
$16$	0	0
$17$	−5.69389	−1.38097	−0.690486	−	0.723346i	$-0.742603\pi$
$17$	−5.69389	−1.38097	−0.690486	+	0.723346i	$0.742603\pi$
$18$	0	0
$19$	−6.45317	−1.48046	−0.740229	−	0.672354i	$-0.765283\pi$
$19$	−6.45317	−1.48046	−0.740229	+	0.672354i	$0.765283\pi$
$20$	0	0
$21$	−0.963415	−0.210234
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.81592	−0.763183
$26$	0	0
$27$	−5.30485	−1.02092
$28$	0	0
$29$	0.101861	0.0189151	0.00945756	−	0.999955i	$-0.496990\pi$
$29$	0.101861	0.0189151	0.00945756	+	0.999955i	$0.496990\pi$
$30$	0	0
$31$	1.54399	0.277308	0.138654	−	0.990341i	$-0.455722\pi$
$31$	1.54399	0.277308	0.138654	+	0.990341i	$0.455722\pi$
$32$	0	0
$33$	0.733320	0.127655
$34$	0	0
$35$	−0.939833	−0.158861
$36$	0	0
$37$	7.33771	1.20631	0.603156	−	0.797623i	$-0.293910\pi$
$37$	7.33771	1.20631	0.603156	+	0.797623i	$0.293910\pi$
$38$	0	0
$39$	6.91595	1.10744
$40$	0	0
$41$	6.09398	0.951720	0.475860	−	0.879521i	$-0.342137\pi$
$41$	6.09398	0.951720	0.475860	+	0.879521i	$0.342137\pi$
$42$	0	0
$43$	−5.37582	−0.819805	−0.409903	−	0.912129i	$-0.634437\pi$
$43$	−5.37582	−0.819805	−0.409903	+	0.912129i	$0.634437\pi$
$44$	0	0
$45$	−1.91053	−0.284805
$46$	0	0
$47$	7.84594	1.14445	0.572224	−	0.820097i	$-0.306081\pi$
$47$	7.84594	1.14445	0.572224	+	0.820097i	$0.306081\pi$
$48$	0	0
$49$	−6.25403	−0.893433
$50$	0	0
$51$	−6.35131	−0.889361
$52$	0	0
$53$	4.75696	0.653418	0.326709	−	0.945125i	$-0.394060\pi$
$53$	4.75696	0.653418	0.326709	+	0.945125i	$0.394060\pi$
$54$	0	0
$55$	0.715370	0.0964605
$56$	0	0
$57$	−7.19825	−0.953432
$58$	0	0
$59$	−5.68391	−0.739982	−0.369991	−	0.929035i	$-0.620639\pi$
$59$	−5.68391	−0.739982	−0.369991	+	0.929035i	$0.620639\pi$
$60$	0	0
$61$	−0.224463	−0.0287395	−0.0143697	−	0.999897i	$-0.504574\pi$
$61$	−0.224463	−0.0287395	−0.0143697	+	0.999897i	$0.504574\pi$
$62$	0	0
$63$	1.51643	0.191052
$64$	0	0
$65$	6.74666	0.836821
$66$	0	0
$67$	2.37419	0.290053	0.145027	−	0.989428i	$-0.453673\pi$
$67$	2.37419	0.290053	0.145027	+	0.989428i	$0.453673\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.42031	−0.761950	−0.380975	−	0.924585i	$-0.624412\pi$
$71$	−6.42031	−0.761950	−0.380975	+	0.924585i	$0.624412\pi$
$72$	0	0
$73$	−4.17196	−0.488291	−0.244146	−	0.969739i	$-0.578507\pi$
$73$	−4.17196	−0.488291	−0.244146	+	0.969739i	$0.578507\pi$
$74$	0	0
$75$	−4.25650	−0.491498
$76$	0	0
$77$	−0.567805	−0.0647074
$78$	0	0
$79$	−16.9996	−1.91261	−0.956304	−	0.292373i	$-0.905555\pi$
$79$	−16.9996	−1.91261	−0.956304	+	0.292373i	$0.905555\pi$
$80$	0	0
$81$	−0.650094	−0.0722327
$82$	0	0
$83$	−16.9937	−1.86530	−0.932651	−	0.360781i	$-0.882510\pi$
$83$	−16.9937	−1.86530	−0.932651	+	0.360781i	$0.882510\pi$
$84$	0	0
$85$	−6.19584	−0.672034
$86$	0	0
$87$	0.113622	0.0121815
$88$	0	0
$89$	−5.06206	−0.536577	−0.268289	−	0.963339i	$-0.586458\pi$
$89$	−5.06206	−0.536577	−0.268289	+	0.963339i	$0.586458\pi$
$90$	0	0
$91$	−5.35498	−0.561354
$92$	0	0
$93$	1.72226	0.178590
$94$	0	0
$95$	−7.02206	−0.720448
$96$	0	0
$97$	9.83304	0.998394	0.499197	−	0.866488i	$-0.333628\pi$
$97$	9.83304	0.998394	0.499197	+	0.866488i	$0.333628\pi$
$98$	0	0
$99$	−1.15426	−0.116007
$100$	0	0
$101$	0.613509	0.0610465	0.0305232	−	0.999534i	$-0.490283\pi$
$101$	0.613509	0.0610465	0.0305232	+	0.999534i	$0.490283\pi$
$102$	0	0
$103$	−8.57993	−0.845405	−0.422703	−	0.906268i	$-0.638919\pi$
$103$	−8.57993	−0.845405	−0.422703	+	0.906268i	$0.638919\pi$
$104$	0	0
$105$	−1.04835	−0.102308
$106$	0	0
$107$	−18.0072	−1.74082	−0.870409	−	0.492330i	$-0.836145\pi$
$107$	−18.0072	−1.74082	−0.870409	+	0.492330i	$0.836145\pi$
$108$	0	0
$109$	9.63966	0.923312	0.461656	−	0.887059i	$-0.347255\pi$
$109$	9.63966	0.923312	0.461656	+	0.887059i	$0.347255\pi$
$110$	0	0
$111$	8.18492	0.776878
$112$	0	0
$113$	9.67107	0.909778	0.454889	−	0.890548i	$-0.349679\pi$
$113$	9.67107	0.909778	0.454889	+	0.890548i	$0.349679\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.8858	−1.00639
$118$	0	0
$119$	4.91778	0.450812
$120$	0	0
$121$	−10.5678	−0.960710
$122$	0	0
$123$	6.79759	0.612918
$124$	0	0
$125$	−9.59309	−0.858032
$126$	0	0
$127$	−13.8867	−1.23225	−0.616123	−	0.787650i	$-0.711297\pi$
$127$	−13.8867	−1.23225	−0.616123	+	0.787650i	$0.711297\pi$
$128$	0	0
$129$	−5.99651	−0.527964
$130$	0	0
$131$	−9.52232	−0.831969	−0.415984	−	0.909372i	$-0.636563\pi$
$131$	−9.52232	−0.831969	−0.415984	+	0.909372i	$0.636563\pi$
$132$	0	0
$133$	5.57356	0.483289
$134$	0	0
$135$	−5.77250	−0.496818
$136$	0	0
$137$	−0.366843	−0.0313415	−0.0156707	−	0.999877i	$-0.504988\pi$
$137$	−0.366843	−0.0313415	−0.0156707	+	0.999877i	$0.504988\pi$
$138$	0	0
$139$	−1.12658	−0.0955552	−0.0477776	−	0.998858i	$-0.515214\pi$
$139$	−1.12658	−0.0955552	−0.0477776	+	0.998858i	$0.515214\pi$
$140$	0	0
$141$	8.75183	0.737037
$142$	0	0
$143$	4.07603	0.340855
$144$	0	0
$145$	0.110841	0.00920482
$146$	0	0
$147$	−6.97612	−0.575381
$148$	0	0
$149$	−13.7182	−1.12384	−0.561918	−	0.827193i	$-0.689936\pi$
$149$	−13.7182	−1.12384	−0.561918	+	0.827193i	$0.689936\pi$
$150$	0	0
$151$	6.32863	0.515017	0.257508	−	0.966276i	$-0.417098\pi$
$151$	6.32863	0.515017	0.257508	+	0.966276i	$0.417098\pi$
$152$	0	0
$153$	9.99705	0.808214
$154$	0	0
$155$	1.68010	0.134949
$156$	0	0
$157$	−11.8281	−0.943986	−0.471993	−	0.881602i	$-0.656465\pi$
$157$	−11.8281	−0.943986	−0.471993	+	0.881602i	$0.656465\pi$
$158$	0	0
$159$	5.30619	0.420809
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.47807	−0.429075	−0.214538	−	0.976716i	$-0.568824\pi$
$163$	−5.47807	−0.429075	−0.214538	+	0.976716i	$0.568824\pi$
$164$	0	0
$165$	0.797967	0.0621216
$166$	0	0
$167$	14.4648	1.11932	0.559660	−	0.828723i	$-0.310932\pi$
$167$	14.4648	1.11932	0.559660	+	0.828723i	$0.310932\pi$
$168$	0	0
$169$	25.4411	1.95701
$170$	0	0
$171$	11.3302	0.866438
$172$	0	0
$173$	−8.19826	−0.623302	−0.311651	−	0.950197i	$-0.600882\pi$
$173$	−8.19826	−0.623302	−0.311651	+	0.950197i	$0.600882\pi$
$174$	0	0
$175$	3.29578	0.249138
$176$	0	0
$177$	−6.34017	−0.476557
$178$	0	0
$179$	16.8855	1.26209	0.631043	−	0.775748i	$-0.282627\pi$
$179$	16.8855	1.26209	0.631043	+	0.775748i	$0.282627\pi$
$180$	0	0
$181$	−16.6704	−1.23910	−0.619550	−	0.784958i	$-0.712685\pi$
$181$	−16.6704	−1.23910	−0.619550	+	0.784958i	$0.712685\pi$
$182$	0	0
$183$	−0.250379	−0.0185086
$184$	0	0
$185$	7.98457	0.587037
$186$	0	0
$187$	−3.74325	−0.273734
$188$	0	0
$189$	4.58176	0.333274
$190$	0	0
$191$	16.5264	1.19581	0.597903	−	0.801569i	$-0.296001\pi$
$191$	16.5264	1.19581	0.597903	+	0.801569i	$0.296001\pi$
$192$	0	0
$193$	−12.3726	−0.890600	−0.445300	−	0.895382i	$-0.646903\pi$
$193$	−12.3726	−0.890600	−0.445300	+	0.895382i	$0.646903\pi$
$194$	0	0
$195$	7.52563	0.538922
$196$	0	0
$197$	−1.04034	−0.0741211	−0.0370606	−	0.999313i	$-0.511799\pi$
$197$	−1.04034	−0.0741211	−0.0370606	+	0.999313i	$0.511799\pi$
$198$	0	0
$199$	11.1980	0.793803	0.396901	−	0.917861i	$-0.370086\pi$
$199$	11.1980	0.793803	0.396901	+	0.917861i	$0.370086\pi$
$200$	0	0
$201$	2.64831	0.186798
$202$	0	0
$203$	−0.0879767	−0.00617476
$204$	0	0
$205$	6.63120	0.463143
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.24241	−0.293454
$210$	0	0
$211$	8.55695	0.589085	0.294542	−	0.955638i	$-0.404833\pi$
$211$	8.55695	0.589085	0.294542	+	0.955638i	$0.404833\pi$
$212$	0	0
$213$	−7.16159	−0.490704
$214$	0	0
$215$	−5.84973	−0.398948
$216$	0	0
$217$	−1.33353	−0.0905261
$218$	0	0
$219$	−4.65365	−0.314465
$220$	0	0
$221$	−35.3026	−2.37471
$222$	0	0
$223$	0.498819	0.0334034	0.0167017	−	0.999861i	$-0.494683\pi$
$223$	0.498819	0.0334034	0.0167017	+	0.999861i	$0.494683\pi$
$224$	0	0
$225$	6.69979	0.446653
$226$	0	0
$227$	−21.0494	−1.39710	−0.698550	−	0.715561i	$-0.746171\pi$
$227$	−21.0494	−1.39710	−0.698550	+	0.715561i	$0.746171\pi$
$228$	0	0
$229$	−1.00627	−0.0664965	−0.0332483	−	0.999447i	$-0.510585\pi$
$229$	−1.00627	−0.0664965	−0.0332483	+	0.999447i	$0.510585\pi$
$230$	0	0
$231$	−0.633364	−0.0416723
$232$	0	0
$233$	15.4176	1.01004	0.505020	−	0.863107i	$-0.331485\pi$
$233$	15.4176	1.01004	0.505020	+	0.863107i	$0.331485\pi$
$234$	0	0
$235$	8.53760	0.556932
$236$	0	0
$237$	−18.9624	−1.23174
$238$	0	0
$239$	4.20623	0.272078	0.136039	−	0.990703i	$-0.456563\pi$
$239$	4.20623	0.272078	0.136039	+	0.990703i	$0.456563\pi$
$240$	0	0
$241$	15.7101	1.01198	0.505989	−	0.862540i	$-0.331128\pi$
$241$	15.7101	1.01198	0.505989	+	0.862540i	$0.331128\pi$
$242$	0	0
$243$	15.1894	0.974400
$244$	0	0
$245$	−6.80536	−0.434779
$246$	0	0
$247$	−40.0102	−2.54579
$248$	0	0
$249$	−18.9558	−1.20127
$250$	0	0
$251$	9.52734	0.601361	0.300680	−	0.953725i	$-0.402786\pi$
$251$	9.52734	0.601361	0.300680	+	0.953725i	$0.402786\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.91121	−0.432797
$256$	0	0
$257$	6.65848	0.415344	0.207672	−	0.978198i	$-0.433411\pi$
$257$	6.65848	0.415344	0.207672	+	0.978198i	$0.433411\pi$
$258$	0	0
$259$	−6.33753	−0.393795
$260$	0	0
$261$	−0.178843	−0.0110701
$262$	0	0
$263$	−11.7940	−0.727250	−0.363625	−	0.931546i	$-0.618461\pi$
$263$	−11.7940	−0.727250	−0.363625	+	0.931546i	$0.618461\pi$
$264$	0	0
$265$	5.17631	0.317978
$266$	0	0
$267$	−5.64652	−0.345562
$268$	0	0
$269$	11.5650	0.705128	0.352564	−	0.935788i	$-0.385310\pi$
$269$	11.5650	0.705128	0.352564	+	0.935788i	$0.385310\pi$
$270$	0	0
$271$	25.4681	1.54707	0.773537	−	0.633751i	$-0.218485\pi$
$271$	25.4681	1.54707	0.773537	+	0.633751i	$0.218485\pi$
$272$	0	0
$273$	−5.97326	−0.361518
$274$	0	0
$275$	−2.50864	−0.151277
$276$	0	0
$277$	25.8732	1.55457	0.777285	−	0.629149i	$-0.216596\pi$
$277$	25.8732	1.55457	0.777285	+	0.629149i	$0.216596\pi$
$278$	0	0
$279$	−2.71086	−0.162295
$280$	0	0
$281$	−28.5451	−1.70286	−0.851430	−	0.524468i	$-0.824264\pi$
$281$	−28.5451	−1.70286	−0.851430	+	0.524468i	$0.824264\pi$
$282$	0	0
$283$	−0.192540	−0.0114453	−0.00572265	−	0.999984i	$-0.501822\pi$
$283$	−0.192540	−0.0114453	−0.00572265	+	0.999984i	$0.501822\pi$
$284$	0	0
$285$	−7.83282	−0.463976
$286$	0	0
$287$	−5.26333	−0.310685
$288$	0	0
$289$	15.4204	0.907084
$290$	0	0
$291$	10.9684	0.642977
$292$	0	0
$293$	−26.7110	−1.56047	−0.780237	−	0.625484i	$-0.784902\pi$
$293$	−26.7110	−1.56047	−0.780237	+	0.625484i	$0.784902\pi$
$294$	0	0
$295$	−6.18498	−0.360103
$296$	0	0
$297$	−3.48749	−0.202365
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.64306	0.267622
$302$	0	0
$303$	0.684345	0.0393146
$304$	0	0
$305$	−0.244250	−0.0139857
$306$	0	0
$307$	−9.10262	−0.519514	−0.259757	−	0.965674i	$-0.583642\pi$
$307$	−9.10262	−0.519514	−0.259757	+	0.965674i	$0.583642\pi$
$308$	0	0
$309$	−9.57056	−0.544450
$310$	0	0
$311$	10.5193	0.596495	0.298248	−	0.954489i	$-0.403598\pi$
$311$	10.5193	0.596495	0.298248	+	0.954489i	$0.403598\pi$
$312$	0	0
$313$	−11.0628	−0.625306	−0.312653	−	0.949867i	$-0.601218\pi$
$313$	−11.0628	−0.625306	−0.312653	+	0.949867i	$0.601218\pi$
$314$	0	0
$315$	1.65011	0.0929732
$316$	0	0
$317$	−13.4320	−0.754418	−0.377209	−	0.926128i	$-0.623116\pi$
$317$	−13.4320	−0.754418	−0.377209	+	0.926128i	$0.623116\pi$
$318$	0	0
$319$	0.0669650	0.00374932
$320$	0	0
$321$	−20.0863	−1.12111
$322$	0	0
$323$	36.7437	2.04447
$324$	0	0
$325$	−23.6590	−1.31237
$326$	0	0
$327$	10.7527	0.594623
$328$	0	0
$329$	−6.77648	−0.373600
$330$	0	0
$331$	−20.1397	−1.10698	−0.553489	−	0.832856i	$-0.686704\pi$
$331$	−20.1397	−1.10698	−0.553489	+	0.832856i	$0.686704\pi$
$332$	0	0
$333$	−12.8832	−0.705994
$334$	0	0
$335$	2.58349	0.141151
$336$	0	0
$337$	−3.89740	−0.212305	−0.106153	−	0.994350i	$-0.533853\pi$
$337$	−3.89740	−0.212305	−0.106153	+	0.994350i	$0.533853\pi$
$338$	0	0
$339$	10.7877	0.585907
$340$	0	0
$341$	1.01504	0.0549676
$342$	0	0
$343$	11.4474	0.618103
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.96397	−0.266480	−0.133240	−	0.991084i	$-0.542538\pi$
$347$	−4.96397	−0.266480	−0.133240	+	0.991084i	$0.542538\pi$
$348$	0	0
$349$	−15.8510	−0.848483	−0.424242	−	0.905549i	$-0.639459\pi$
$349$	−15.8510	−0.848483	−0.424242	+	0.905549i	$0.639459\pi$
$350$	0	0
$351$	−32.8905	−1.75557
$352$	0	0
$353$	−23.3603	−1.24335	−0.621673	−	0.783277i	$-0.713547\pi$
$353$	−23.3603	−1.24335	−0.621673	+	0.783277i	$0.713547\pi$
$354$	0	0
$355$	−6.98629	−0.370794
$356$	0	0
$357$	5.48558	0.290328
$358$	0	0
$359$	−11.0561	−0.583517	−0.291759	−	0.956492i	$-0.594240\pi$
$359$	−11.0561	−0.583517	−0.291759	+	0.956492i	$0.594240\pi$
$360$	0	0
$361$	22.6434	1.19176
$362$	0	0
$363$	−11.7880	−0.618707
$364$	0	0
$365$	−4.53974	−0.237621
$366$	0	0
$367$	−26.5936	−1.38818	−0.694088	−	0.719890i	$-0.744192\pi$
$367$	−26.5936	−1.38818	−0.694088	+	0.719890i	$0.744192\pi$
$368$	0	0
$369$	−10.6995	−0.556994
$370$	0	0
$371$	−4.10855	−0.213305
$372$	0	0
$373$	13.7906	0.714049	0.357025	−	0.934095i	$-0.383791\pi$
$373$	13.7906	0.714049	0.357025	+	0.934095i	$0.383791\pi$
$374$	0	0
$375$	−10.7007	−0.552582
$376$	0	0
$377$	0.631548	0.0325264
$378$	0	0
$379$	−4.10337	−0.210776	−0.105388	−	0.994431i	$-0.533608\pi$
$379$	−4.10337	−0.210776	−0.105388	+	0.994431i	$0.533608\pi$
$380$	0	0
$381$	−15.4901	−0.793580
$382$	0	0
$383$	−4.02696	−0.205768	−0.102884	−	0.994693i	$-0.532807\pi$
$383$	−4.02696	−0.205768	−0.102884	+	0.994693i	$0.532807\pi$
$384$	0	0
$385$	−0.617861	−0.0314891
$386$	0	0
$387$	9.43860	0.479791
$388$	0	0
$389$	−9.14337	−0.463587	−0.231794	−	0.972765i	$-0.574459\pi$
$389$	−9.14337	−0.463587	−0.231794	+	0.972765i	$0.574459\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.6218	−0.535797
$394$	0	0
$395$	−18.4983	−0.930748
$396$	0	0
$397$	−16.8808	−0.847225	−0.423613	−	0.905843i	$-0.639238\pi$
$397$	−16.8808	−0.847225	−0.423613	+	0.905843i	$0.639238\pi$
$398$	0	0
$399$	6.21708	0.311243
$400$	0	0
$401$	−27.8215	−1.38934	−0.694669	−	0.719330i	$-0.744449\pi$
$401$	−27.8215	−1.38934	−0.694669	+	0.719330i	$0.744449\pi$
$402$	0	0
$403$	9.57287	0.476858
$404$	0	0
$405$	−0.707404	−0.0351512
$406$	0	0
$407$	4.82392	0.239113
$408$	0	0
$409$	7.88614	0.389944	0.194972	−	0.980809i	$-0.437538\pi$
$409$	7.88614	0.389944	0.194972	+	0.980809i	$0.437538\pi$
$410$	0	0
$411$	−0.409198	−0.0201843
$412$	0	0
$413$	4.90916	0.241564
$414$	0	0
$415$	−18.4918	−0.907727
$416$	0	0
$417$	−1.25665	−0.0615386
$418$	0	0
$419$	8.56517	0.418436	0.209218	−	0.977869i	$-0.432908\pi$
$419$	8.56517	0.418436	0.209218	+	0.977869i	$0.432908\pi$
$420$	0	0
$421$	24.5681	1.19738	0.598688	−	0.800983i	$-0.295689\pi$
$421$	24.5681	1.19738	0.598688	+	0.800983i	$0.295689\pi$
$422$	0	0
$423$	−13.7755	−0.669788
$424$	0	0
$425$	21.7274	1.05393
$426$	0	0
$427$	0.193867	0.00938188
$428$	0	0
$429$	4.54665	0.219514
$430$	0	0
$431$	1.81375	0.0873654	0.0436827	−	0.999045i	$-0.486091\pi$
$431$	1.81375	0.0873654	0.0436827	+	0.999045i	$0.486091\pi$
$432$	0	0
$433$	32.1937	1.54713	0.773565	−	0.633717i	$-0.218472\pi$
$433$	32.1937	1.54713	0.773565	+	0.633717i	$0.218472\pi$
$434$	0	0
$435$	0.123638	0.00592801
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−13.0953	−0.625003	−0.312502	−	0.949917i	$-0.601167\pi$
$439$	−13.0953	−0.625003	−0.312502	+	0.949917i	$0.601167\pi$
$440$	0	0
$441$	10.9805	0.522882
$442$	0	0
$443$	−27.5077	−1.30693	−0.653464	−	0.756958i	$-0.726685\pi$
$443$	−27.5077	−1.30693	−0.653464	+	0.756958i	$0.726685\pi$
$444$	0	0
$445$	−5.50831	−0.261119
$446$	0	0
$447$	−15.3021	−0.723762
$448$	0	0
$449$	−30.1186	−1.42139	−0.710693	−	0.703502i	$-0.751619\pi$
$449$	−30.1186	−1.42139	−0.710693	+	0.703502i	$0.751619\pi$
$450$	0	0
$451$	4.00628	0.188648
$452$	0	0
$453$	7.05934	0.331677
$454$	0	0
$455$	−5.82705	−0.273176
$456$	0	0
$457$	−18.8163	−0.880189	−0.440095	−	0.897951i	$-0.645055\pi$
$457$	−18.8163	−0.880189	−0.440095	+	0.897951i	$0.645055\pi$
$458$	0	0
$459$	30.2052	1.40986
$460$	0	0
$461$	8.21514	0.382617	0.191309	−	0.981530i	$-0.438727\pi$
$461$	8.21514	0.382617	0.191309	+	0.981530i	$0.438727\pi$
$462$	0	0
$463$	37.6613	1.75027	0.875135	−	0.483880i	$-0.160773\pi$
$463$	37.6613	1.75027	0.875135	+	0.483880i	$0.160773\pi$
$464$	0	0
$465$	1.87408	0.0869086
$466$	0	0
$467$	−4.44458	−0.205670	−0.102835	−	0.994698i	$-0.532791\pi$
$467$	−4.44458	−0.205670	−0.102835	+	0.994698i	$0.532791\pi$
$468$	0	0
$469$	−2.05057	−0.0946866
$470$	0	0
$471$	−13.1938	−0.607938
$472$	0	0
$473$	−3.53415	−0.162500
$474$	0	0
$475$	24.6248	1.12986
$476$	0	0
$477$	−8.35203	−0.382413
$478$	0	0
$479$	−28.6970	−1.31120	−0.655599	−	0.755109i	$-0.727584\pi$
$479$	−28.6970	−1.31120	−0.655599	+	0.755109i	$0.727584\pi$
$480$	0	0
$481$	45.4945	2.07437
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.6999	0.485857
$486$	0	0
$487$	32.5590	1.47539	0.737695	−	0.675134i	$-0.235914\pi$
$487$	32.5590	1.47539	0.737695	+	0.675134i	$0.235914\pi$
$488$	0	0
$489$	−6.11056	−0.276329
$490$	0	0
$491$	36.1185	1.63001	0.815003	−	0.579456i	$-0.196735\pi$
$491$	36.1185	1.63001	0.815003	+	0.579456i	$0.196735\pi$
$492$	0	0
$493$	−0.579986	−0.0261213
$494$	0	0
$495$	−1.25601	−0.0564535
$496$	0	0
$497$	5.54518	0.248735
$498$	0	0
$499$	12.7414	0.570384	0.285192	−	0.958470i	$-0.407943\pi$
$499$	12.7414	0.570384	0.285192	+	0.958470i	$0.407943\pi$
$500$	0	0
$501$	16.1349	0.720854
$502$	0	0
$503$	4.09979	0.182800	0.0914002	−	0.995814i	$-0.470866\pi$
$503$	4.09979	0.182800	0.0914002	+	0.995814i	$0.470866\pi$
$504$	0	0
$505$	0.667594	0.0297075
$506$	0	0
$507$	28.3785	1.26033
$508$	0	0
$509$	10.1111	0.448168	0.224084	−	0.974570i	$-0.428061\pi$
$509$	10.1111	0.448168	0.224084	+	0.974570i	$0.428061\pi$
$510$	0	0
$511$	3.60329	0.159400
$512$	0	0
$513$	34.2331	1.51143
$514$	0	0
$515$	−9.33630	−0.411407
$516$	0	0
$517$	5.15804	0.226850
$518$	0	0
$519$	−9.14483	−0.401413
$520$	0	0
$521$	−3.18180	−0.139397	−0.0696985	−	0.997568i	$-0.522204\pi$
$521$	−3.18180	−0.139397	−0.0696985	+	0.997568i	$0.522204\pi$
$522$	0	0
$523$	13.3261	0.582711	0.291355	−	0.956615i	$-0.405894\pi$
$523$	13.3261	0.582711	0.291355	+	0.956615i	$0.405894\pi$
$524$	0	0
$525$	3.67631	0.160447
$526$	0	0
$527$	−8.79131	−0.382955
$528$	0	0
$529$	0	0
$530$	0	0
$531$	9.97952	0.433074
$532$	0	0
$533$	37.7832	1.63657
$534$	0	0
$535$	−19.5946	−0.847148
$536$	0	0
$537$	18.8351	0.812797
$538$	0	0
$539$	−4.11150	−0.177095
$540$	0	0
$541$	−39.0800	−1.68018	−0.840089	−	0.542448i	$-0.817498\pi$
$541$	−39.0800	−1.68018	−0.840089	+	0.542448i	$0.817498\pi$
$542$	0	0
$543$	−18.5951	−0.797993
$544$	0	0
$545$	10.4895	0.449319
$546$	0	0
$547$	−33.5400	−1.43407	−0.717034	−	0.697038i	$-0.754501\pi$
$547$	−33.5400	−1.43407	−0.717034	+	0.697038i	$0.754501\pi$
$548$	0	0
$549$	0.394100	0.0168198
$550$	0	0
$551$	−0.657327	−0.0280031
$552$	0	0
$553$	14.6825	0.624362
$554$	0	0
$555$	8.90647	0.378059
$556$	0	0
$557$	35.0746	1.48616	0.743080	−	0.669202i	$-0.233364\pi$
$557$	35.0746	1.48616	0.743080	+	0.669202i	$0.233364\pi$
$558$	0	0
$559$	−33.3306	−1.40973
$560$	0	0
$561$	−4.17545	−0.176288
$562$	0	0
$563$	8.52310	0.359206	0.179603	−	0.983739i	$-0.442519\pi$
$563$	8.52310	0.359206	0.179603	+	0.983739i	$0.442519\pi$
$564$	0	0
$565$	10.5236	0.442733
$566$	0	0
$567$	0.561482	0.0235800
$568$	0	0
$569$	−21.1986	−0.888691	−0.444345	−	0.895856i	$-0.646564\pi$
$569$	−21.1986	−0.888691	−0.444345	+	0.895856i	$0.646564\pi$
$570$	0	0
$571$	34.9280	1.46169	0.730847	−	0.682542i	$-0.239125\pi$
$571$	34.9280	1.46169	0.730847	+	0.682542i	$0.239125\pi$
$572$	0	0
$573$	18.4345	0.770112
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.9882	−0.499077	−0.249538	−	0.968365i	$-0.580279\pi$
$577$	−11.9882	−0.499077	−0.249538	+	0.968365i	$0.580279\pi$
$578$	0	0
$579$	−13.8011	−0.573556
$580$	0	0
$581$	14.6773	0.608919
$582$	0	0
$583$	3.12730	0.129519
$584$	0	0
$585$	−11.8455	−0.489749
$586$	0	0
$587$	−39.0013	−1.60976	−0.804878	−	0.593441i	$-0.797769\pi$
$587$	−39.0013	−1.60976	−0.804878	+	0.593441i	$0.797769\pi$
$588$	0	0
$589$	−9.96362	−0.410544
$590$	0	0
$591$	−1.16046	−0.0477348
$592$	0	0
$593$	32.3079	1.32673	0.663364	−	0.748297i	$-0.269128\pi$
$593$	32.3079	1.32673	0.663364	+	0.748297i	$0.269128\pi$
$594$	0	0
$595$	5.35131	0.219382
$596$	0	0
$597$	12.4909	0.511218
$598$	0	0
$599$	−15.1294	−0.618171	−0.309085	−	0.951034i	$-0.600023\pi$
$599$	−15.1294	−0.618171	−0.309085	+	0.951034i	$0.600023\pi$
$600$	0	0
$601$	−28.1895	−1.14987	−0.574937	−	0.818198i	$-0.694973\pi$
$601$	−28.1895	−1.14987	−0.574937	+	0.818198i	$0.694973\pi$
$602$	0	0
$603$	−4.16848	−0.169754
$604$	0	0
$605$	−11.4994	−0.467518
$606$	0	0
$607$	14.9696	0.607597	0.303798	−	0.952736i	$-0.401745\pi$
$607$	14.9696	0.607597	0.303798	+	0.952736i	$0.401745\pi$
$608$	0	0
$609$	−0.0981345	−0.00397661
$610$	0	0
$611$	48.6455	1.96799
$612$	0	0
$613$	−23.2654	−0.939680	−0.469840	−	0.882752i	$-0.655688\pi$
$613$	−23.2654	−0.939680	−0.469840	+	0.882752i	$0.655688\pi$
$614$	0	0
$615$	7.39684	0.298269
$616$	0	0
$617$	−3.50364	−0.141051	−0.0705257	−	0.997510i	$-0.522468\pi$
$617$	−3.50364	−0.141051	−0.0705257	+	0.997510i	$0.522468\pi$
$618$	0	0
$619$	22.5589	0.906721	0.453360	−	0.891327i	$-0.350225\pi$
$619$	22.5589	0.906721	0.453360	+	0.891327i	$0.350225\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.37207	0.175163
$624$	0	0
$625$	8.64080	0.345632
$626$	0	0
$627$	−4.73224	−0.188987
$628$	0	0
$629$	−41.7801	−1.66588
$630$	0	0
$631$	−13.2803	−0.528681	−0.264341	−	0.964429i	$-0.585154\pi$
$631$	−13.2803	−0.528681	−0.264341	+	0.964429i	$0.585154\pi$
$632$	0	0
$633$	9.54493	0.379377
$634$	0	0
$635$	−15.1109	−0.599658
$636$	0	0
$637$	−38.7756	−1.53634
$638$	0	0
$639$	11.2725	0.445931
$640$	0	0
$641$	21.9810	0.868196	0.434098	−	0.900866i	$-0.357067\pi$
$641$	21.9810	0.868196	0.434098	+	0.900866i	$0.357067\pi$
$642$	0	0
$643$	28.4536	1.12210	0.561049	−	0.827782i	$-0.310398\pi$
$643$	28.4536	1.12210	0.561049	+	0.827782i	$0.310398\pi$
$644$	0	0
$645$	−6.52514	−0.256927
$646$	0	0
$647$	4.85510	0.190874	0.0954368	−	0.995435i	$-0.469575\pi$
$647$	4.85510	0.190874	0.0954368	+	0.995435i	$0.469575\pi$
$648$	0	0
$649$	−3.73669	−0.146678
$650$	0	0
$651$	−1.48750	−0.0582998
$652$	0	0
$653$	26.9031	1.05280	0.526399	−	0.850237i	$-0.323542\pi$
$653$	26.9031	1.05280	0.526399	+	0.850237i	$0.323542\pi$
$654$	0	0
$655$	−10.3618	−0.404868
$656$	0	0
$657$	7.32492	0.285772
$658$	0	0
$659$	−14.0740	−0.548244	−0.274122	−	0.961695i	$-0.588387\pi$
$659$	−14.0740	−0.548244	−0.274122	+	0.961695i	$0.588387\pi$
$660$	0	0
$661$	−8.41468	−0.327293	−0.163647	−	0.986519i	$-0.552326\pi$
$661$	−8.41468	−0.327293	−0.163647	+	0.986519i	$0.552326\pi$
$662$	0	0
$663$	−39.3787	−1.52934
$664$	0	0
$665$	6.06490	0.235187
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0.556412	0.0215121
$670$	0	0
$671$	−0.147565	−0.00569669
$672$	0	0
$673$	15.4924	0.597189	0.298594	−	0.954380i	$-0.403482\pi$
$673$	15.4924	0.597189	0.298594	+	0.954380i	$0.403482\pi$
$674$	0	0
$675$	20.2429	0.779148
$676$	0	0
$677$	−12.9692	−0.498447	−0.249224	−	0.968446i	$-0.580175\pi$
$677$	−12.9692	−0.498447	−0.249224	+	0.968446i	$0.580175\pi$
$678$	0	0
$679$	−8.49273	−0.325921
$680$	0	0
$681$	−23.4798	−0.899747
$682$	0	0
$683$	−0.822814	−0.0314841	−0.0157421	−	0.999876i	$-0.505011\pi$
$683$	−0.822814	−0.0314841	−0.0157421	+	0.999876i	$0.505011\pi$
$684$	0	0
$685$	−0.399182	−0.0152520
$686$	0	0
$687$	−1.12246	−0.0428245
$688$	0	0
$689$	29.4936	1.12362
$690$	0	0
$691$	10.6220	0.404079	0.202039	−	0.979377i	$-0.435243\pi$
$691$	10.6220	0.404079	0.202039	+	0.979377i	$0.435243\pi$
$692$	0	0
$693$	0.996924	0.0378700
$694$	0	0
$695$	−1.22589	−0.0465008
$696$	0	0
$697$	−34.6985	−1.31430
$698$	0	0
$699$	17.1977	0.650477
$700$	0	0
$701$	14.1981	0.536256	0.268128	−	0.963383i	$-0.413595\pi$
$701$	14.1981	0.536256	0.268128	+	0.963383i	$0.413595\pi$
$702$	0	0
$703$	−47.3515	−1.78590
$704$	0	0
$705$	9.52335	0.358670
$706$	0	0
$707$	−0.529884	−0.0199283
$708$	0	0
$709$	0.881160	0.0330927	0.0165463	−	0.999863i	$-0.494733\pi$
$709$	0.881160	0.0330927	0.0165463	+	0.999863i	$0.494733\pi$
$710$	0	0
$711$	29.8471	1.11935
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.43536	0.165873
$716$	0	0
$717$	4.69188	0.175221
$718$	0	0
$719$	30.3690	1.13257	0.566287	−	0.824208i	$-0.308379\pi$
$719$	30.3690	1.13257	0.566287	+	0.824208i	$0.308379\pi$
$720$	0	0
$721$	7.41043	0.275979
$722$	0	0
$723$	17.5240	0.651725
$724$	0	0
$725$	−0.388693	−0.0144357
$726$	0	0
$727$	−21.6876	−0.804349	−0.402174	−	0.915563i	$-0.631745\pi$
$727$	−21.6876	−0.804349	−0.402174	+	0.915563i	$0.631745\pi$
$728$	0	0
$729$	18.8934	0.699757
$730$	0	0
$731$	30.6094	1.13213
$732$	0	0
$733$	14.9194	0.551061	0.275530	−	0.961292i	$-0.411147\pi$
$733$	14.9194	0.551061	0.275530	+	0.961292i	$0.411147\pi$
$734$	0	0
$735$	−7.59111	−0.280002
$736$	0	0
$737$	1.56083	0.0574938
$738$	0	0
$739$	−27.6514	−1.01717	−0.508586	−	0.861011i	$-0.669832\pi$
$739$	−27.6514	−1.01717	−0.508586	+	0.861011i	$0.669832\pi$
$740$	0	0
$741$	−44.6298	−1.63952
$742$	0	0
$743$	−11.3882	−0.417792	−0.208896	−	0.977938i	$-0.566987\pi$
$743$	−11.3882	−0.417792	−0.208896	+	0.977938i	$0.566987\pi$
$744$	0	0
$745$	−14.9275	−0.546901
$746$	0	0
$747$	29.8367	1.09167
$748$	0	0
$749$	15.5527	0.568282
$750$	0	0
$751$	44.8862	1.63792	0.818961	−	0.573849i	$-0.194550\pi$
$751$	44.8862	1.63792	0.818961	+	0.573849i	$0.194550\pi$
$752$	0	0
$753$	10.6274	0.387283
$754$	0	0
$755$	6.88654	0.250627
$756$	0	0
$757$	−17.1715	−0.624108	−0.312054	−	0.950064i	$-0.601017\pi$
$757$	−17.1715	−0.624108	−0.312054	+	0.950064i	$0.601017\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.27625	0.263764	0.131882	−	0.991265i	$-0.457898\pi$
$761$	7.27625	0.263764	0.131882	+	0.991265i	$0.457898\pi$
$762$	0	0
$763$	−8.32571	−0.301411
$764$	0	0
$765$	10.8784	0.393308
$766$	0	0
$767$	−35.2407	−1.27247
$768$	0	0
$769$	35.5696	1.28267	0.641337	−	0.767260i	$-0.278380\pi$
$769$	35.5696	1.28267	0.641337	+	0.767260i	$0.278380\pi$
$770$	0	0
$771$	7.42726	0.267486
$772$	0	0
$773$	27.3785	0.984736	0.492368	−	0.870387i	$-0.336132\pi$
$773$	27.3785	0.984736	0.492368	+	0.870387i	$0.336132\pi$
$774$	0	0
$775$	−5.89173	−0.211637
$776$	0	0
$777$	−7.06926	−0.253608
$778$	0	0
$779$	−39.3255	−1.40898
$780$	0	0
$781$	−4.22081	−0.151032
$782$	0	0
$783$	−0.540357	−0.0193108
$784$	0	0
$785$	−12.8708	−0.459380
$786$	0	0
$787$	−7.78788	−0.277608	−0.138804	−	0.990320i	$-0.544326\pi$
$787$	−7.78788	−0.277608	−0.138804	+	0.990320i	$0.544326\pi$
$788$	0	0
$789$	−13.1557	−0.468357
$790$	0	0
$791$	−8.35284	−0.296993
$792$	0	0
$793$	−1.39169	−0.0494203
$794$	0	0
$795$	5.77397	0.204782
$796$	0	0
$797$	12.6641	0.448586	0.224293	−	0.974522i	$-0.427993\pi$
$797$	12.6641	0.448586	0.224293	+	0.974522i	$0.427993\pi$
$798$	0	0
$799$	−44.6739	−1.58045
$800$	0	0
$801$	8.88771	0.314032
$802$	0	0
$803$	−2.74271	−0.0967882
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.9002	0.454110
$808$	0	0
$809$	−21.4316	−0.753496	−0.376748	−	0.926316i	$-0.622958\pi$
$809$	−21.4316	−0.753496	−0.376748	+	0.926316i	$0.622958\pi$
$810$	0	0
$811$	−15.8738	−0.557404	−0.278702	−	0.960378i	$-0.589904\pi$
$811$	−15.8738	−0.557404	−0.278702	+	0.960378i	$0.589904\pi$
$812$	0	0
$813$	28.4086	0.996333
$814$	0	0
$815$	−5.96099	−0.208804
$816$	0	0
$817$	34.6911	1.21369
$818$	0	0
$819$	9.40200	0.328532
$820$	0	0
$821$	16.6377	0.580660	0.290330	−	0.956927i	$-0.406235\pi$
$821$	16.6377	0.580660	0.290330	+	0.956927i	$0.406235\pi$
$822$	0	0
$823$	37.3137	1.30067	0.650336	−	0.759647i	$-0.274628\pi$
$823$	37.3137	1.30067	0.650336	+	0.759647i	$0.274628\pi$
$824$	0	0
$825$	−2.79829	−0.0974239
$826$	0	0
$827$	0.531764	0.0184913	0.00924563	−	0.999957i	$-0.497057\pi$
$827$	0.531764	0.0184913	0.00924563	+	0.999957i	$0.497057\pi$
$828$	0	0
$829$	23.5271	0.817131	0.408565	−	0.912729i	$-0.366029\pi$
$829$	23.5271	0.817131	0.408565	+	0.912729i	$0.366029\pi$
$830$	0	0
$831$	28.8605	1.00116
$832$	0	0
$833$	35.6098	1.23381
$834$	0	0
$835$	15.7399	0.544703
$836$	0	0
$837$	−8.19062	−0.283109
$838$	0	0
$839$	−40.0857	−1.38391	−0.691955	−	0.721940i	$-0.743250\pi$
$839$	−40.0857	−1.38391	−0.691955	+	0.721940i	$0.743250\pi$
$840$	0	0
$841$	−28.9896	−0.999642
$842$	0	0
$843$	−31.8409	−1.09666
$844$	0	0
$845$	27.6839	0.952355
$846$	0	0
$847$	9.12734	0.313619
$848$	0	0
$849$	−0.214770	−0.00737090
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−10.3877	−0.355670	−0.177835	−	0.984060i	$-0.556909\pi$
$853$	−10.3877	−0.355670	−0.177835	+	0.984060i	$0.556909\pi$
$854$	0	0
$855$	12.3290	0.421642
$856$	0	0
$857$	−53.9545	−1.84305	−0.921525	−	0.388320i	$-0.873056\pi$
$857$	−53.9545	−1.84305	−0.921525	+	0.388320i	$0.873056\pi$
$858$	0	0
$859$	−50.5432	−1.72451	−0.862256	−	0.506472i	$-0.830949\pi$
$859$	−50.5432	−1.72451	−0.862256	+	0.506472i	$0.830949\pi$
$860$	0	0
$861$	−5.87104	−0.200084
$862$	0	0
$863$	6.19287	0.210808	0.105404	−	0.994429i	$-0.466386\pi$
$863$	6.19287	0.210808	0.105404	+	0.994429i	$0.466386\pi$
$864$	0	0
$865$	−8.92099	−0.303323
$866$	0	0
$867$	17.2009	0.584172
$868$	0	0
$869$	−11.1758	−0.379114
$870$	0	0
$871$	14.7202	0.498774
$872$	0	0
$873$	−17.2644	−0.584310
$874$	0	0
$875$	8.28549	0.280101
$876$	0	0
$877$	30.0802	1.01574	0.507868	−	0.861435i	$-0.330434\pi$
$877$	30.0802	1.01574	0.507868	+	0.861435i	$0.330434\pi$
$878$	0	0
$879$	−29.7951	−1.00496
$880$	0	0
$881$	−17.0221	−0.573490	−0.286745	−	0.958007i	$-0.592573\pi$
$881$	−17.0221	−0.573490	−0.286745	+	0.958007i	$0.592573\pi$
$882$	0	0
$883$	−6.15095	−0.206996	−0.103498	−	0.994630i	$-0.533004\pi$
$883$	−6.15095	−0.206996	−0.103498	+	0.994630i	$0.533004\pi$
$884$	0	0
$885$	−6.89910	−0.231911
$886$	0	0
$887$	−4.83249	−0.162259	−0.0811295	−	0.996704i	$-0.525853\pi$
$887$	−4.83249	−0.162259	−0.0811295	+	0.996704i	$0.525853\pi$
$888$	0	0
$889$	11.9939	0.402261
$890$	0	0
$891$	−0.427382	−0.0143178
$892$	0	0
$893$	−50.6312	−1.69431
$894$	0	0
$895$	18.3741	0.614179
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.157272	0.00524532
$900$	0	0
$901$	−27.0856	−0.902353
$902$	0	0
$903$	5.17915	0.172351
$904$	0	0
$905$	−18.1400	−0.602993
$906$	0	0
$907$	26.6831	0.885998	0.442999	−	0.896522i	$-0.353915\pi$
$907$	26.6831	0.885998	0.442999	+	0.896522i	$0.353915\pi$
$908$	0	0
$909$	−1.07717	−0.0357274
$910$	0	0
$911$	59.5419	1.97271	0.986356	−	0.164627i	$-0.0526420\pi$
$911$	59.5419	1.97271	0.986356	+	0.164627i	$0.0526420\pi$
$912$	0	0
$913$	−11.1719	−0.369737
$914$	0	0
$915$	−0.272451	−0.00900697
$916$	0	0
$917$	8.22436	0.271592
$918$	0	0
$919$	−25.3496	−0.836206	−0.418103	−	0.908400i	$-0.637305\pi$
$919$	−25.3496	−0.836206	−0.418103	+	0.908400i	$0.637305\pi$
$920$	0	0
$921$	−10.1536	−0.334573
$922$	0	0
$923$	−39.8065	−1.31025
$924$	0	0
$925$	−28.0001	−0.920637
$926$	0	0
$927$	15.0642	0.494773
$928$	0	0
$929$	−40.6950	−1.33516	−0.667579	−	0.744539i	$-0.732669\pi$
$929$	−40.6950	−1.33516	−0.667579	+	0.744539i	$0.732669\pi$
$930$	0	0
$931$	40.3583	1.32269
$932$	0	0
$933$	11.7339	0.384150
$934$	0	0
$935$	−4.07324	−0.133209
$936$	0	0
$937$	−15.4609	−0.505086	−0.252543	−	0.967586i	$-0.581267\pi$
$937$	−15.4609	−0.505086	−0.252543	+	0.967586i	$0.581267\pi$
$938$	0	0
$939$	−12.3401	−0.402704
$940$	0	0
$941$	−11.1007	−0.361871	−0.180936	−	0.983495i	$-0.557913\pi$
$941$	−11.1007	−0.361871	−0.180936	+	0.983495i	$0.557913\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.98567	0.162184
$946$	0	0
$947$	−32.9785	−1.07166	−0.535828	−	0.844327i	$-0.680001\pi$
$947$	−32.9785	−1.07166	−0.535828	+	0.844327i	$0.680001\pi$
$948$	0	0
$949$	−25.8665	−0.839663
$950$	0	0
$951$	−14.9829	−0.485853
$952$	0	0
$953$	8.55887	0.277249	0.138624	−	0.990345i	$-0.455732\pi$
$953$	8.55887	0.277249	0.138624	+	0.990345i	$0.455732\pi$
$954$	0	0
$955$	17.9833	0.581925
$956$	0	0
$957$	0.0746968	0.00241460
$958$	0	0
$959$	0.316839	0.0102313
$960$	0	0
$961$	−28.6161	−0.923100
$962$	0	0
$963$	31.6161	1.01881
$964$	0	0
$965$	−13.4633	−0.433400
$966$	0	0
$967$	11.0128	0.354148	0.177074	−	0.984197i	$-0.443337\pi$
$967$	11.0128	0.354148	0.177074	+	0.984197i	$0.443337\pi$
$968$	0	0
$969$	40.9861	1.31666
$970$	0	0
$971$	39.8078	1.27749	0.638746	−	0.769418i	$-0.279454\pi$
$971$	39.8078	1.27749	0.638746	+	0.769418i	$0.279454\pi$
$972$	0	0
$973$	0.973018	0.0311935
$974$	0	0
$975$	−26.3907	−0.845178
$976$	0	0
$977$	−55.7219	−1.78270	−0.891351	−	0.453313i	$-0.850242\pi$
$977$	−55.7219	−1.78270	−0.891351	+	0.453313i	$0.850242\pi$
$978$	0	0
$979$	−3.32788	−0.106359
$980$	0	0
$981$	−16.9248	−0.540368
$982$	0	0
$983$	−7.29642	−0.232720	−0.116360	−	0.993207i	$-0.537123\pi$
$983$	−7.29642	−0.232720	−0.116360	+	0.993207i	$0.537123\pi$
$984$	0	0
$985$	−1.13205	−0.0360702
$986$	0	0
$987$	−7.55890	−0.240602
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−38.9939	−1.23868	−0.619341	−	0.785122i	$-0.712600\pi$
$991$	−38.9939	−1.23868	−0.619341	+	0.785122i	$0.712600\pi$
$992$	0	0
$993$	−22.4650	−0.712906
$994$	0	0
$995$	12.1851	0.386295
$996$	0	0
$997$	3.39347	0.107472	0.0537361	−	0.998555i	$-0.482887\pi$
$997$	3.39347	0.107472	0.0537361	+	0.998555i	$0.482887\pi$
$998$	0	0
$999$	−38.9254	−1.23155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bv.1.4		5
4.3	odd	2		1058.2.a.k.1.2		5
12.11	even	2		9522.2.a.bw.1.2		5
23.11	odd	22		368.2.m.a.305.1		10
23.21	odd	22		368.2.m.a.257.1		10
23.22	odd	2		8464.2.a.bu.1.4		5
92.11	even	22		46.2.c.b.29.1	yes	10
92.67	even	22		46.2.c.b.27.1	✓	10
92.91	even	2		1058.2.a.j.1.2		5
276.11	odd	22		414.2.i.c.397.1		10
276.251	odd	22		414.2.i.c.73.1		10
276.275	odd	2		9522.2.a.bz.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.b.27.1	✓	10	92.67	even	22
46.2.c.b.29.1	yes	10	92.11	even	22
368.2.m.a.257.1		10	23.21	odd	22
368.2.m.a.305.1		10	23.11	odd	22
414.2.i.c.73.1		10	276.251	odd	22
414.2.i.c.397.1		10	276.11	odd	22
1058.2.a.j.1.2		5	92.91	even	2
1058.2.a.k.1.2		5	4.3	odd	2
8464.2.a.bu.1.4		5	23.22	odd	2
8464.2.a.bv.1.4		5	1.1	even	1	trivial
9522.2.a.bw.1.2		5	12.11	even	2
9522.2.a.bz.1.4		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+1.11546 q^{3} +1.08816 q^{5} -0.863693 q^{7} -1.75575 q^{9} +0.657415 q^{11} +6.20009 q^{13} +1.21379 q^{15} -5.69389 q^{17} -6.45317 q^{19} -0.963415 q^{21} -3.81592 q^{25} -5.30485 q^{27} +0.101861 q^{29} +1.54399 q^{31} +0.733320 q^{33} -0.939833 q^{35} +7.33771 q^{37} +6.91595 q^{39} +6.09398 q^{41} -5.37582 q^{43} -1.91053 q^{45} +7.84594 q^{47} -6.25403 q^{49} -6.35131 q^{51} +4.75696 q^{53} +0.715370 q^{55} -7.19825 q^{57} -5.68391 q^{59} -0.224463 q^{61} +1.51643 q^{63} +6.74666 q^{65} +2.37419 q^{67} -6.42031 q^{71} -4.17196 q^{73} -4.25650 q^{75} -0.567805 q^{77} -16.9996 q^{79} -0.650094 q^{81} -16.9937 q^{83} -6.19584 q^{85} +0.113622 q^{87} -5.06206 q^{89} -5.35498 q^{91} +1.72226 q^{93} -7.02206 q^{95} +9.83304 q^{97} -1.15426 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{5} - 9 q^{7} + 7 q^{9} - q^{11} + q^{13} - 11 q^{15} - q^{17} - 10 q^{19} - 11 q^{25} + 10 q^{29} + 11 q^{33} + 3 q^{35} + 20 q^{37} + 22 q^{39} - 3 q^{41} - 27 q^{43} + 5 q^{45} + 11 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q + 2 * q^5 - 9 * q^7 + 7 * q^9 - q^11 + q^13 - 11 * q^15 - q^17 - 10 * q^19 - 11 * q^25 + 10 * q^29 + 11 * q^33 + 3 * q^35 + 20 * q^37 + 22 * q^39 - 3 * q^41 - 27 * q^43 + 5 * q^45 + 11 * q^47 + 12 * q^49 - 2 * q^53 + 4 * q^55 + 11 * q^57 + 17 * q^59 + 7 * q^61 - 39 * q^63 - 15 * q^65 - 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 - 16 * q^79 + q^81 - 18 * q^83 - 7 * q^85 - 22 * q^87 - 14 * q^89 - 26 * q^91 - 44 * q^93 - 26 * q^95 + 47 * q^97 - 8 * q^99

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