LMFDB - Embedded newform 5824.2.a.cl.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5824,2,Mod(1,5824)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5824.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5824 = 2^{6} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5824.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,3,0,5,0,6,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.5048741372$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1025428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 2 $ x^5 - 8x^3 + 13x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2912)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.37452$ of defining polynomial
Character	$\chi$	$=$	5824.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37452 q^{3} +3.96568 q^{5} +1.00000 q^{7} +2.63834 q^{9} -1.90120 q^{11} +1.00000 q^{13} +9.41658 q^{15} -0.790495 q^{17} +6.58575 q^{19} +2.37452 q^{21} -2.28588 q^{23} +10.7266 q^{25} -0.858778 q^{27} -4.83083 q^{29} +5.20535 q^{31} -4.51442 q^{33} +3.96568 q^{35} -4.52126 q^{37} +2.37452 q^{39} +7.73678 q^{41} +4.64755 q^{43} +10.4628 q^{45} +2.38166 q^{47} +1.00000 q^{49} -1.87705 q^{51} -10.0328 q^{53} -7.53953 q^{55} +15.6380 q^{57} +2.77764 q^{59} -7.73678 q^{61} +2.63834 q^{63} +3.96568 q^{65} +9.94867 q^{67} -5.42787 q^{69} +9.41658 q^{71} -9.55571 q^{73} +25.4705 q^{75} -1.90120 q^{77} -7.41389 q^{79} -9.95419 q^{81} -2.41866 q^{83} -3.13485 q^{85} -11.4709 q^{87} +12.1585 q^{89} +1.00000 q^{91} +12.3602 q^{93} +26.1170 q^{95} -9.60401 q^{97} -5.01599 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 6 q^{9} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 7 q^{19} + 5 q^{21} + 4 q^{23} + 2 q^{25} + 23 q^{27} - 3 q^{29} - 2 q^{31} + 17 q^{33} + 3 q^{35} + q^{37} + 5 q^{39} - 7 q^{41}+ \cdots + 40 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 6 * q^9 + 5 * q^11 + 5 * q^13 + 4 * q^17 + 7 * q^19 + 5 * q^21 + 4 * q^23 + 2 * q^25 + 23 * q^27 - 3 * q^29 - 2 * q^31 + 17 * q^33 + 3 * q^35 + q^37 + 5 * q^39 - 7 * q^41 + 15 * q^43 + q^45 - 6 * q^47 + 5 * q^49 + 16 * q^51 - 11 * q^53 - 16 * q^55 + 18 * q^57 + 20 * q^59 + 7 * q^61 + 6 * q^63 + 3 * q^65 + 11 * q^67 - 5 * q^69 - 2 * q^73 + 19 * q^75 + 5 * q^77 - 10 * q^79 + 33 * q^81 + 9 * q^83 - 20 * q^85 - 2 * q^87 + 11 * q^89 + 5 * q^91 + 13 * q^93 + 11 * q^95 - 24 * q^97 + 40 * 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
$3$	2.37452	1.37093	0.685464	−	0.728106i	$-0.259599\pi$
$3$	2.37452	1.37093	0.685464	+	0.728106i	$0.259599\pi$
$4$	0	0
$5$	3.96568	1.77351	0.886753	−	0.462244i	$-0.152956\pi$
$5$	3.96568	1.77351	0.886753	+	0.462244i	$0.152956\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.63834	0.879445
$10$	0	0
$11$	−1.90120	−0.573232	−0.286616	−	0.958046i	$-0.592530\pi$
$11$	−1.90120	−0.573232	−0.286616	+	0.958046i	$0.592530\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	9.41658	2.43135
$16$	0	0
$17$	−0.790495	−0.191723	−0.0958616	−	0.995395i	$-0.530561\pi$
$17$	−0.790495	−0.191723	−0.0958616	+	0.995395i	$0.530561\pi$
$18$	0	0
$19$	6.58575	1.51087	0.755437	−	0.655221i	$-0.227425\pi$
$19$	6.58575	1.51087	0.755437	+	0.655221i	$0.227425\pi$
$20$	0	0
$21$	2.37452	0.518162
$22$	0	0
$23$	−2.28588	−0.476640	−0.238320	−	0.971187i	$-0.576597\pi$
$23$	−2.28588	−0.476640	−0.238320	+	0.971187i	$0.576597\pi$
$24$	0	0
$25$	10.7266	2.14532
$26$	0	0
$27$	−0.858778	−0.165272
$28$	0	0
$29$	−4.83083	−0.897062	−0.448531	−	0.893767i	$-0.648053\pi$
$29$	−4.83083	−0.897062	−0.448531	+	0.893767i	$0.648053\pi$
$30$	0	0
$31$	5.20535	0.934908	0.467454	−	0.884017i	$-0.345171\pi$
$31$	5.20535	0.934908	0.467454	+	0.884017i	$0.345171\pi$
$32$	0	0
$33$	−4.51442	−0.785860
$34$	0	0
$35$	3.96568	0.670322
$36$	0	0
$37$	−4.52126	−0.743291	−0.371646	−	0.928375i	$-0.621206\pi$
$37$	−4.52126	−0.743291	−0.371646	+	0.928375i	$0.621206\pi$
$38$	0	0
$39$	2.37452	0.380227
$40$	0	0
$41$	7.73678	1.20828	0.604141	−	0.796877i	$-0.293516\pi$
$41$	7.73678	1.20828	0.604141	+	0.796877i	$0.293516\pi$
$42$	0	0
$43$	4.64755	0.708744	0.354372	−	0.935104i	$-0.384695\pi$
$43$	4.64755	0.708744	0.354372	+	0.935104i	$0.384695\pi$
$44$	0	0
$45$	10.4628	1.55970
$46$	0	0
$47$	2.38166	0.347400	0.173700	−	0.984799i	$-0.444428\pi$
$47$	2.38166	0.347400	0.173700	+	0.984799i	$0.444428\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.87705	−0.262839
$52$	0	0
$53$	−10.0328	−1.37812	−0.689059	−	0.724706i	$-0.741976\pi$
$53$	−10.0328	−1.37812	−0.689059	+	0.724706i	$0.741976\pi$
$54$	0	0
$55$	−7.53953	−1.01663
$56$	0	0
$57$	15.6380	2.07130
$58$	0	0
$59$	2.77764	0.361618	0.180809	−	0.983518i	$-0.442128\pi$
$59$	2.77764	0.361618	0.180809	+	0.983518i	$0.442128\pi$
$60$	0	0
$61$	−7.73678	−0.990593	−0.495297	−	0.868724i	$-0.664941\pi$
$61$	−7.73678	−0.990593	−0.495297	+	0.868724i	$0.664941\pi$
$62$	0	0
$63$	2.63834	0.332399
$64$	0	0
$65$	3.96568	0.491882
$66$	0	0
$67$	9.94867	1.21542	0.607712	−	0.794158i	$-0.292088\pi$
$67$	9.94867	1.21542	0.607712	+	0.794158i	$0.292088\pi$
$68$	0	0
$69$	−5.42787	−0.653439
$70$	0	0
$71$	9.41658	1.11754	0.558771	−	0.829322i	$-0.311273\pi$
$71$	9.41658	1.11754	0.558771	+	0.829322i	$0.311273\pi$
$72$	0	0
$73$	−9.55571	−1.11841	−0.559206	−	0.829029i	$-0.688894\pi$
$73$	−9.55571	−1.11841	−0.559206	+	0.829029i	$0.688894\pi$
$74$	0	0
$75$	25.4705	2.94108
$76$	0	0
$77$	−1.90120	−0.216661
$78$	0	0
$79$	−7.41389	−0.834128	−0.417064	−	0.908877i	$-0.636941\pi$
$79$	−7.41389	−0.834128	−0.417064	+	0.908877i	$0.636941\pi$
$80$	0	0
$81$	−9.95419	−1.10602
$82$	0	0
$83$	−2.41866	−0.265482	−0.132741	−	0.991151i	$-0.542378\pi$
$83$	−2.41866	−0.265482	−0.132741	+	0.991151i	$0.542378\pi$
$84$	0	0
$85$	−3.13485	−0.340022
$86$	0	0
$87$	−11.4709	−1.22981
$88$	0	0
$89$	12.1585	1.28880	0.644398	−	0.764690i	$-0.277108\pi$
$89$	12.1585	1.28880	0.644398	+	0.764690i	$0.277108\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	12.3602	1.28169
$94$	0	0
$95$	26.1170	2.67954
$96$	0	0
$97$	−9.60401	−0.975140	−0.487570	−	0.873084i	$-0.662117\pi$
$97$	−9.60401	−0.975140	−0.487570	+	0.873084i	$0.662117\pi$
$98$	0	0
$99$	−5.01599	−0.504126
$100$	0	0
$101$	12.4145	1.23529	0.617645	−	0.786457i	$-0.288087\pi$
$101$	12.4145	1.23529	0.617645	+	0.786457i	$0.288087\pi$
$102$	0	0
$103$	−15.2080	−1.49849	−0.749246	−	0.662292i	$-0.769584\pi$
$103$	−15.2080	−1.49849	−0.749246	+	0.662292i	$0.769584\pi$
$104$	0	0
$105$	9.41658	0.918964
$106$	0	0
$107$	−3.37512	−0.326285	−0.163142	−	0.986603i	$-0.552163\pi$
$107$	−3.37512	−0.326285	−0.163142	+	0.986603i	$0.552163\pi$
$108$	0	0
$109$	−8.37750	−0.802419	−0.401209	−	0.915986i	$-0.631410\pi$
$109$	−8.37750	−0.802419	−0.401209	+	0.915986i	$0.631410\pi$
$110$	0	0
$111$	−10.7358	−1.01900
$112$	0	0
$113$	0.0930891	0.00875708	0.00437854	−	0.999990i	$-0.498606\pi$
$113$	0.0930891	0.00875708	0.00437854	+	0.999990i	$0.498606\pi$
$114$	0	0
$115$	−9.06508	−0.845323
$116$	0	0
$117$	2.63834	0.243914
$118$	0	0
$119$	−0.790495	−0.0724646
$120$	0	0
$121$	−7.38546	−0.671405
$122$	0	0
$123$	18.3711	1.65647
$124$	0	0
$125$	22.7099	2.03124
$126$	0	0
$127$	−11.4773	−1.01844	−0.509221	−	0.860636i	$-0.670066\pi$
$127$	−11.4773	−1.01844	−0.509221	+	0.860636i	$0.670066\pi$
$128$	0	0
$129$	11.0357	0.971638
$130$	0	0
$131$	18.9018	1.65146	0.825729	−	0.564067i	$-0.190764\pi$
$131$	18.9018	1.65146	0.825729	+	0.564067i	$0.190764\pi$
$132$	0	0
$133$	6.58575	0.571057
$134$	0	0
$135$	−3.40564	−0.293111
$136$	0	0
$137$	−1.83559	−0.156825	−0.0784123	−	0.996921i	$-0.524985\pi$
$137$	−1.83559	−0.156825	−0.0784123	+	0.996921i	$0.524985\pi$
$138$	0	0
$139$	17.8899	1.51740	0.758701	−	0.651439i	$-0.225835\pi$
$139$	17.8899	1.51740	0.758701	+	0.651439i	$0.225835\pi$
$140$	0	0
$141$	5.65529	0.476261
$142$	0	0
$143$	−1.90120	−0.158986
$144$	0	0
$145$	−19.1575	−1.59095
$146$	0	0
$147$	2.37452	0.195847
$148$	0	0
$149$	20.8751	1.71015	0.855077	−	0.518502i	$-0.173510\pi$
$149$	20.8751	1.71015	0.855077	+	0.518502i	$0.173510\pi$
$150$	0	0
$151$	−0.0957723	−0.00779384	−0.00389692	−	0.999992i	$-0.501240\pi$
$151$	−0.0957723	−0.00779384	−0.00389692	+	0.999992i	$0.501240\pi$
$152$	0	0
$153$	−2.08559	−0.168610
$154$	0	0
$155$	20.6427	1.65806
$156$	0	0
$157$	−8.74868	−0.698220	−0.349110	−	0.937082i	$-0.613516\pi$
$157$	−8.74868	−0.698220	−0.349110	+	0.937082i	$0.613516\pi$
$158$	0	0
$159$	−23.8232	−1.88930
$160$	0	0
$161$	−2.28588	−0.180153
$162$	0	0
$163$	1.88186	0.147399	0.0736993	−	0.997281i	$-0.476519\pi$
$163$	1.88186	0.147399	0.0736993	+	0.997281i	$0.476519\pi$
$164$	0	0
$165$	−17.9028	−1.39373
$166$	0	0
$167$	−4.39005	−0.339713	−0.169856	−	0.985469i	$-0.554330\pi$
$167$	−4.39005	−0.339713	−0.169856	+	0.985469i	$0.554330\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.3754	1.32873
$172$	0	0
$173$	16.5100	1.25523	0.627615	−	0.778524i	$-0.284031\pi$
$173$	16.5100	1.25523	0.627615	+	0.778524i	$0.284031\pi$
$174$	0	0
$175$	10.7266	0.810856
$176$	0	0
$177$	6.59556	0.495753
$178$	0	0
$179$	−14.1116	−1.05475	−0.527375	−	0.849633i	$-0.676824\pi$
$179$	−14.1116	−1.05475	−0.527375	+	0.849633i	$0.676824\pi$
$180$	0	0
$181$	−18.6222	−1.38418	−0.692089	−	0.721812i	$-0.743310\pi$
$181$	−18.6222	−1.38418	−0.692089	+	0.721812i	$0.743310\pi$
$182$	0	0
$183$	−18.3711	−1.35803
$184$	0	0
$185$	−17.9299	−1.31823
$186$	0	0
$187$	1.50289	0.109902
$188$	0	0
$189$	−0.858778	−0.0624669
$190$	0	0
$191$	8.73886	0.632321	0.316161	−	0.948706i	$-0.397606\pi$
$191$	8.73886	0.632321	0.316161	+	0.948706i	$0.397606\pi$
$192$	0	0
$193$	22.2572	1.60211	0.801055	−	0.598591i	$-0.204273\pi$
$193$	22.2572	1.60211	0.801055	+	0.598591i	$0.204273\pi$
$194$	0	0
$195$	9.41658	0.674335
$196$	0	0
$197$	−17.4950	−1.24647	−0.623235	−	0.782035i	$-0.714182\pi$
$197$	−17.4950	−1.24647	−0.623235	+	0.782035i	$0.714182\pi$
$198$	0	0
$199$	17.0519	1.20878	0.604388	−	0.796690i	$-0.293418\pi$
$199$	17.0519	1.20878	0.604388	+	0.796690i	$0.293418\pi$
$200$	0	0
$201$	23.6233	1.66626
$202$	0	0
$203$	−4.83083	−0.339058
$204$	0	0
$205$	30.6816	2.14290
$206$	0	0
$207$	−6.03093	−0.419179
$208$	0	0
$209$	−12.5208	−0.866081
$210$	0	0
$211$	19.2736	1.32685	0.663426	−	0.748242i	$-0.269102\pi$
$211$	19.2736	1.32685	0.663426	+	0.748242i	$0.269102\pi$
$212$	0	0
$213$	22.3598	1.53207
$214$	0	0
$215$	18.4307	1.25696
$216$	0	0
$217$	5.20535	0.353362
$218$	0	0
$219$	−22.6902	−1.53326
$220$	0	0
$221$	−0.790495	−0.0531745
$222$	0	0
$223$	−14.9712	−1.00255	−0.501273	−	0.865289i	$-0.667135\pi$
$223$	−14.9712	−1.00255	−0.501273	+	0.865289i	$0.667135\pi$
$224$	0	0
$225$	28.3004	1.88669
$226$	0	0
$227$	−25.2115	−1.67334	−0.836672	−	0.547704i	$-0.815502\pi$
$227$	−25.2115	−1.67334	−0.836672	+	0.547704i	$0.815502\pi$
$228$	0	0
$229$	20.4488	1.35130	0.675648	−	0.737225i	$-0.263864\pi$
$229$	20.4488	1.35130	0.675648	+	0.737225i	$0.263864\pi$
$230$	0	0
$231$	−4.51442	−0.297027
$232$	0	0
$233$	−10.8339	−0.709750	−0.354875	−	0.934914i	$-0.615477\pi$
$233$	−10.8339	−0.709750	−0.354875	+	0.934914i	$0.615477\pi$
$234$	0	0
$235$	9.44488	0.616116
$236$	0	0
$237$	−17.6044	−1.14353
$238$	0	0
$239$	−23.9563	−1.54961	−0.774804	−	0.632202i	$-0.782151\pi$
$239$	−23.9563	−1.54961	−0.774804	+	0.632202i	$0.782151\pi$
$240$	0	0
$241$	−18.9415	−1.22013	−0.610064	−	0.792352i	$-0.708856\pi$
$241$	−18.9415	−1.22013	−0.610064	+	0.792352i	$0.708856\pi$
$242$	0	0
$243$	−21.0601	−1.35100
$244$	0	0
$245$	3.96568	0.253358
$246$	0	0
$247$	6.58575	0.419041
$248$	0	0
$249$	−5.74315	−0.363958
$250$	0	0
$251$	−23.6184	−1.49078	−0.745389	−	0.666630i	$-0.767736\pi$
$251$	−23.6184	−1.49078	−0.745389	+	0.666630i	$0.767736\pi$
$252$	0	0
$253$	4.34591	0.273225
$254$	0	0
$255$	−7.44376	−0.466146
$256$	0	0
$257$	11.8696	0.740403	0.370201	−	0.928952i	$-0.379289\pi$
$257$	11.8696	0.740403	0.370201	+	0.928952i	$0.379289\pi$
$258$	0	0
$259$	−4.52126	−0.280938
$260$	0	0
$261$	−12.7453	−0.788917
$262$	0	0
$263$	−4.39404	−0.270948	−0.135474	−	0.990781i	$-0.543256\pi$
$263$	−4.39404	−0.270948	−0.135474	+	0.990781i	$0.543256\pi$
$264$	0	0
$265$	−39.7870	−2.44410
$266$	0	0
$267$	28.8705	1.76685
$268$	0	0
$269$	−16.3848	−0.998999	−0.499500	−	0.866314i	$-0.666483\pi$
$269$	−16.3848	−0.998999	−0.499500	+	0.866314i	$0.666483\pi$
$270$	0	0
$271$	22.7377	1.38122	0.690608	−	0.723229i	$-0.257343\pi$
$271$	22.7377	1.38122	0.690608	+	0.723229i	$0.257343\pi$
$272$	0	0
$273$	2.37452	0.143712
$274$	0	0
$275$	−20.3934	−1.22977
$276$	0	0
$277$	−16.9124	−1.01617	−0.508085	−	0.861307i	$-0.669646\pi$
$277$	−16.9124	−1.01617	−0.508085	+	0.861307i	$0.669646\pi$
$278$	0	0
$279$	13.7335	0.822200
$280$	0	0
$281$	−10.9725	−0.654562	−0.327281	−	0.944927i	$-0.606132\pi$
$281$	−10.9725	−0.654562	−0.327281	+	0.944927i	$0.606132\pi$
$282$	0	0
$283$	−8.23740	−0.489662	−0.244831	−	0.969566i	$-0.578733\pi$
$283$	−8.23740	−0.489662	−0.244831	+	0.969566i	$0.578733\pi$
$284$	0	0
$285$	62.0152	3.67346
$286$	0	0
$287$	7.73678	0.456688
$288$	0	0
$289$	−16.3751	−0.963242
$290$	0	0
$291$	−22.8049	−1.33685
$292$	0	0
$293$	−18.3459	−1.07178	−0.535888	−	0.844289i	$-0.680023\pi$
$293$	−18.3459	−1.07178	−0.535888	+	0.844289i	$0.680023\pi$
$294$	0	0
$295$	11.0152	0.641332
$296$	0	0
$297$	1.63271	0.0947392
$298$	0	0
$299$	−2.28588	−0.132196
$300$	0	0
$301$	4.64755	0.267880
$302$	0	0
$303$	29.4785	1.69349
$304$	0	0
$305$	−30.6816	−1.75682
$306$	0	0
$307$	9.81762	0.560321	0.280161	−	0.959953i	$-0.409612\pi$
$307$	9.81762	0.560321	0.280161	+	0.959953i	$0.409612\pi$
$308$	0	0
$309$	−36.1117	−2.05433
$310$	0	0
$311$	6.62691	0.375778	0.187889	−	0.982190i	$-0.439836\pi$
$311$	6.62691	0.375778	0.187889	+	0.982190i	$0.439836\pi$
$312$	0	0
$313$	−19.4947	−1.10190	−0.550952	−	0.834537i	$-0.685735\pi$
$313$	−19.4947	−1.10190	−0.550952	+	0.834537i	$0.685735\pi$
$314$	0	0
$315$	10.4628	0.589512
$316$	0	0
$317$	5.36607	0.301389	0.150694	−	0.988580i	$-0.451849\pi$
$317$	5.36607	0.301389	0.150694	+	0.988580i	$0.451849\pi$
$318$	0	0
$319$	9.18435	0.514225
$320$	0	0
$321$	−8.01428	−0.447313
$322$	0	0
$323$	−5.20600	−0.289670
$324$	0	0
$325$	10.7266	0.595005
$326$	0	0
$327$	−19.8925	−1.10006
$328$	0	0
$329$	2.38166	0.131305
$330$	0	0
$331$	−13.9398	−0.766198	−0.383099	−	0.923707i	$-0.625143\pi$
$331$	−13.9398	−0.766198	−0.383099	+	0.923707i	$0.625143\pi$
$332$	0	0
$333$	−11.9286	−0.653684
$334$	0	0
$335$	39.4532	2.15556
$336$	0	0
$337$	−31.7343	−1.72868	−0.864339	−	0.502910i	$-0.832263\pi$
$337$	−31.7343	−1.72868	−0.864339	+	0.502910i	$0.832263\pi$
$338$	0	0
$339$	0.221042	0.0120053
$340$	0	0
$341$	−9.89638	−0.535919
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−21.5252	−1.15888
$346$	0	0
$347$	34.2464	1.83844	0.919221	−	0.393741i	$-0.128819\pi$
$347$	34.2464	1.83844	0.919221	+	0.393741i	$0.128819\pi$
$348$	0	0
$349$	−23.5784	−1.26212	−0.631061	−	0.775733i	$-0.717380\pi$
$349$	−23.5784	−1.26212	−0.631061	+	0.775733i	$0.717380\pi$
$350$	0	0
$351$	−0.858778	−0.0458382
$352$	0	0
$353$	−21.2783	−1.13253	−0.566266	−	0.824223i	$-0.691612\pi$
$353$	−21.2783	−1.13253	−0.566266	+	0.824223i	$0.691612\pi$
$354$	0	0
$355$	37.3431	1.98197
$356$	0	0
$357$	−1.87705	−0.0993438
$358$	0	0
$359$	−5.17067	−0.272897	−0.136449	−	0.990647i	$-0.543569\pi$
$359$	−5.17067	−0.272897	−0.136449	+	0.990647i	$0.543569\pi$
$360$	0	0
$361$	24.3721	1.28274
$362$	0	0
$363$	−17.5369	−0.920448
$364$	0	0
$365$	−37.8949	−1.98351
$366$	0	0
$367$	−31.3473	−1.63632	−0.818158	−	0.574994i	$-0.805004\pi$
$367$	−31.3473	−1.63632	−0.818158	+	0.574994i	$0.805004\pi$
$368$	0	0
$369$	20.4122	1.06262
$370$	0	0
$371$	−10.0328	−0.520879
$372$	0	0
$373$	−11.6509	−0.603260	−0.301630	−	0.953425i	$-0.597531\pi$
$373$	−11.6509	−0.603260	−0.301630	+	0.953425i	$0.597531\pi$
$374$	0	0
$375$	53.9251	2.78468
$376$	0	0
$377$	−4.83083	−0.248800
$378$	0	0
$379$	35.6833	1.83293	0.916465	−	0.400116i	$-0.131030\pi$
$379$	35.6833	1.83293	0.916465	+	0.400116i	$0.131030\pi$
$380$	0	0
$381$	−27.2530	−1.39621
$382$	0	0
$383$	−10.3909	−0.530949	−0.265474	−	0.964118i	$-0.585529\pi$
$383$	−10.3909	−0.530949	−0.265474	+	0.964118i	$0.585529\pi$
$384$	0	0
$385$	−7.53953	−0.384250
$386$	0	0
$387$	12.2618	0.623302
$388$	0	0
$389$	24.9453	1.26478	0.632388	−	0.774652i	$-0.282075\pi$
$389$	24.9453	1.26478	0.632388	+	0.774652i	$0.282075\pi$
$390$	0	0
$391$	1.80698	0.0913829
$392$	0	0
$393$	44.8827	2.26403
$394$	0	0
$395$	−29.4011	−1.47933
$396$	0	0
$397$	−7.27215	−0.364979	−0.182489	−	0.983208i	$-0.558416\pi$
$397$	−7.27215	−0.364979	−0.182489	+	0.983208i	$0.558416\pi$
$398$	0	0
$399$	15.6380	0.782878
$400$	0	0
$401$	10.8834	0.543490	0.271745	−	0.962369i	$-0.412399\pi$
$401$	10.8834	0.543490	0.271745	+	0.962369i	$0.412399\pi$
$402$	0	0
$403$	5.20535	0.259297
$404$	0	0
$405$	−39.4751	−1.96153
$406$	0	0
$407$	8.59581	0.426078
$408$	0	0
$409$	8.93010	0.441565	0.220783	−	0.975323i	$-0.429139\pi$
$409$	8.93010	0.441565	0.220783	+	0.975323i	$0.429139\pi$
$410$	0	0
$411$	−4.35863	−0.214995
$412$	0	0
$413$	2.77764	0.136679
$414$	0	0
$415$	−9.59163	−0.470835
$416$	0	0
$417$	42.4799	2.08025
$418$	0	0
$419$	16.6842	0.815077	0.407538	−	0.913188i	$-0.366387\pi$
$419$	16.6842	0.815077	0.407538	+	0.913188i	$0.366387\pi$
$420$	0	0
$421$	−4.61025	−0.224690	−0.112345	−	0.993669i	$-0.535836\pi$
$421$	−4.61025	−0.224690	−0.112345	+	0.993669i	$0.535836\pi$
$422$	0	0
$423$	6.28361	0.305520
$424$	0	0
$425$	−8.47933	−0.411308
$426$	0	0
$427$	−7.73678	−0.374409
$428$	0	0
$429$	−4.51442	−0.217958
$430$	0	0
$431$	27.8637	1.34215	0.671073	−	0.741392i	$-0.265834\pi$
$431$	27.8637	1.34215	0.671073	+	0.741392i	$0.265834\pi$
$432$	0	0
$433$	−24.7976	−1.19170	−0.595848	−	0.803097i	$-0.703184\pi$
$433$	−24.7976	−1.19170	−0.595848	+	0.803097i	$0.703184\pi$
$434$	0	0
$435$	−45.4899	−2.18107
$436$	0	0
$437$	−15.0543	−0.720143
$438$	0	0
$439$	8.53846	0.407519	0.203759	−	0.979021i	$-0.434684\pi$
$439$	8.53846	0.407519	0.203759	+	0.979021i	$0.434684\pi$
$440$	0	0
$441$	2.63834	0.125635
$442$	0	0
$443$	31.8091	1.51130	0.755649	−	0.654977i	$-0.227322\pi$
$443$	31.8091	1.51130	0.755649	+	0.654977i	$0.227322\pi$
$444$	0	0
$445$	48.2166	2.28569
$446$	0	0
$447$	49.5683	2.34450
$448$	0	0
$449$	20.5752	0.971005	0.485502	−	0.874235i	$-0.338637\pi$
$449$	20.5752	0.971005	0.485502	+	0.874235i	$0.338637\pi$
$450$	0	0
$451$	−14.7091	−0.692626
$452$	0	0
$453$	−0.227413	−0.0106848
$454$	0	0
$455$	3.96568	0.185914
$456$	0	0
$457$	−11.8398	−0.553840	−0.276920	−	0.960893i	$-0.589314\pi$
$457$	−11.8398	−0.553840	−0.276920	+	0.960893i	$0.589314\pi$
$458$	0	0
$459$	0.678860	0.0316865
$460$	0	0
$461$	6.18393	0.288014	0.144007	−	0.989577i	$-0.454001\pi$
$461$	6.18393	0.288014	0.144007	+	0.989577i	$0.454001\pi$
$462$	0	0
$463$	4.12908	0.191895	0.0959473	−	0.995386i	$-0.469412\pi$
$463$	4.12908	0.191895	0.0959473	+	0.995386i	$0.469412\pi$
$464$	0	0
$465$	49.0165	2.27309
$466$	0	0
$467$	−30.6662	−1.41906	−0.709531	−	0.704675i	$-0.751093\pi$
$467$	−30.6662	−1.41906	−0.709531	+	0.704675i	$0.751093\pi$
$468$	0	0
$469$	9.94867	0.459387
$470$	0	0
$471$	−20.7739	−0.957210
$472$	0	0
$473$	−8.83590	−0.406275
$474$	0	0
$475$	70.6428	3.24131
$476$	0	0
$477$	−26.4700	−1.21198
$478$	0	0
$479$	1.38445	0.0632571	0.0316285	−	0.999500i	$-0.489931\pi$
$479$	1.38445	0.0632571	0.0316285	+	0.999500i	$0.489931\pi$
$480$	0	0
$481$	−4.52126	−0.206152
$482$	0	0
$483$	−5.42787	−0.246977
$484$	0	0
$485$	−38.0864	−1.72942
$486$	0	0
$487$	−12.2846	−0.556668	−0.278334	−	0.960484i	$-0.589782\pi$
$487$	−12.2846	−0.556668	−0.278334	+	0.960484i	$0.589782\pi$
$488$	0	0
$489$	4.46851	0.202073
$490$	0	0
$491$	19.8910	0.897668	0.448834	−	0.893615i	$-0.351840\pi$
$491$	19.8910	0.897668	0.448834	+	0.893615i	$0.351840\pi$
$492$	0	0
$493$	3.81875	0.171988
$494$	0	0
$495$	−19.8918	−0.894071
$496$	0	0
$497$	9.41658	0.422391
$498$	0	0
$499$	25.4751	1.14042	0.570210	−	0.821499i	$-0.306862\pi$
$499$	25.4751	1.14042	0.570210	+	0.821499i	$0.306862\pi$
$500$	0	0
$501$	−10.4243	−0.465722
$502$	0	0
$503$	−38.5306	−1.71800	−0.858998	−	0.511979i	$-0.828912\pi$
$503$	−38.5306	−1.71800	−0.858998	+	0.511979i	$0.828912\pi$
$504$	0	0
$505$	49.2319	2.19079
$506$	0	0
$507$	2.37452	0.105456
$508$	0	0
$509$	15.1099	0.669734	0.334867	−	0.942265i	$-0.391309\pi$
$509$	15.1099	0.669734	0.334867	+	0.942265i	$0.391309\pi$
$510$	0	0
$511$	−9.55571	−0.422720
$512$	0	0
$513$	−5.65570	−0.249705
$514$	0	0
$515$	−60.3102	−2.65758
$516$	0	0
$517$	−4.52799	−0.199141
$518$	0	0
$519$	39.2032	1.72083
$520$	0	0
$521$	0.999094	0.0437711	0.0218856	−	0.999760i	$-0.493033\pi$
$521$	0.999094	0.0437711	0.0218856	+	0.999760i	$0.493033\pi$
$522$	0	0
$523$	−10.8117	−0.472762	−0.236381	−	0.971660i	$-0.575961\pi$
$523$	−10.8117	−0.472762	−0.236381	+	0.971660i	$0.575961\pi$
$524$	0	0
$525$	25.4705	1.11163
$526$	0	0
$527$	−4.11480	−0.179244
$528$	0	0
$529$	−17.7747	−0.772815
$530$	0	0
$531$	7.32835	0.318023
$532$	0	0
$533$	7.73678	0.335117
$534$	0	0
$535$	−13.3846	−0.578668
$536$	0	0
$537$	−33.5082	−1.44599
$538$	0	0
$539$	−1.90120	−0.0818903
$540$	0	0
$541$	−30.2777	−1.30174	−0.650870	−	0.759189i	$-0.725596\pi$
$541$	−30.2777	−1.30174	−0.650870	+	0.759189i	$0.725596\pi$
$542$	0	0
$543$	−44.2188	−1.89761
$544$	0	0
$545$	−33.2225	−1.42309
$546$	0	0
$547$	1.73334	0.0741123	0.0370562	−	0.999313i	$-0.488202\pi$
$547$	1.73334	0.0741123	0.0370562	+	0.999313i	$0.488202\pi$
$548$	0	0
$549$	−20.4122	−0.871173
$550$	0	0
$551$	−31.8146	−1.35535
$552$	0	0
$553$	−7.41389	−0.315271
$554$	0	0
$555$	−42.5748	−1.80720
$556$	0	0
$557$	22.4334	0.950532	0.475266	−	0.879842i	$-0.342352\pi$
$557$	22.4334	0.950532	0.475266	+	0.879842i	$0.342352\pi$
$558$	0	0
$559$	4.64755	0.196570
$560$	0	0
$561$	3.56863	0.150668
$562$	0	0
$563$	−8.72756	−0.367823	−0.183911	−	0.982943i	$-0.558876\pi$
$563$	−8.72756	−0.367823	−0.183911	+	0.982943i	$0.558876\pi$
$564$	0	0
$565$	0.369161	0.0155307
$566$	0	0
$567$	−9.95419	−0.418037
$568$	0	0
$569$	34.4553	1.44444	0.722221	−	0.691662i	$-0.243121\pi$
$569$	34.4553	1.44444	0.722221	+	0.691662i	$0.243121\pi$
$570$	0	0
$571$	36.3358	1.52061	0.760303	−	0.649569i	$-0.225050\pi$
$571$	36.3358	1.52061	0.760303	+	0.649569i	$0.225050\pi$
$572$	0	0
$573$	20.7506	0.866868
$574$	0	0
$575$	−24.5198	−1.02255
$576$	0	0
$577$	−6.35988	−0.264765	−0.132383	−	0.991199i	$-0.542263\pi$
$577$	−6.35988	−0.264765	−0.132383	+	0.991199i	$0.542263\pi$
$578$	0	0
$579$	52.8502	2.19638
$580$	0	0
$581$	−2.41866	−0.100343
$582$	0	0
$583$	19.0744	0.789981
$584$	0	0
$585$	10.4628	0.432583
$586$	0	0
$587$	26.1171	1.07797	0.538984	−	0.842316i	$-0.318809\pi$
$587$	26.1171	1.07797	0.538984	+	0.842316i	$0.318809\pi$
$588$	0	0
$589$	34.2811	1.41253
$590$	0	0
$591$	−41.5423	−1.70882
$592$	0	0
$593$	−16.9390	−0.695600	−0.347800	−	0.937569i	$-0.613071\pi$
$593$	−16.9390	−0.695600	−0.347800	+	0.937569i	$0.613071\pi$
$594$	0	0
$595$	−3.13485	−0.128516
$596$	0	0
$597$	40.4900	1.65715
$598$	0	0
$599$	30.0609	1.22825	0.614127	−	0.789207i	$-0.289508\pi$
$599$	30.0609	1.22825	0.614127	+	0.789207i	$0.289508\pi$
$600$	0	0
$601$	17.9141	0.730733	0.365366	−	0.930864i	$-0.380944\pi$
$601$	17.9141	0.730733	0.365366	+	0.930864i	$0.380944\pi$
$602$	0	0
$603$	26.2479	1.06890
$604$	0	0
$605$	−29.2883	−1.19074
$606$	0	0
$607$	−43.5432	−1.76737	−0.883683	−	0.468086i	$-0.844944\pi$
$607$	−43.5432	−1.76737	−0.883683	+	0.468086i	$0.844944\pi$
$608$	0	0
$609$	−11.4709	−0.464824
$610$	0	0
$611$	2.38166	0.0963515
$612$	0	0
$613$	7.90262	0.319184	0.159592	−	0.987183i	$-0.448982\pi$
$613$	7.90262	0.319184	0.159592	+	0.987183i	$0.448982\pi$
$614$	0	0
$615$	72.8540	2.93776
$616$	0	0
$617$	24.0162	0.966856	0.483428	−	0.875384i	$-0.339392\pi$
$617$	24.0162	0.966856	0.483428	+	0.875384i	$0.339392\pi$
$618$	0	0
$619$	−39.0428	−1.56926	−0.784631	−	0.619963i	$-0.787148\pi$
$619$	−39.0428	−1.56926	−0.784631	+	0.619963i	$0.787148\pi$
$620$	0	0
$621$	1.96307	0.0787752
$622$	0	0
$623$	12.1585	0.487119
$624$	0	0
$625$	36.4271	1.45709
$626$	0	0
$627$	−29.7309	−1.18734
$628$	0	0
$629$	3.57404	0.142506
$630$	0	0
$631$	41.9395	1.66958	0.834792	−	0.550566i	$-0.185588\pi$
$631$	41.9395	1.66958	0.834792	+	0.550566i	$0.185588\pi$
$632$	0	0
$633$	45.7656	1.81902
$634$	0	0
$635$	−45.5151	−1.80621
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	24.8441	0.982817
$640$	0	0
$641$	−47.6545	−1.88224	−0.941121	−	0.338071i	$-0.890226\pi$
$641$	−47.6545	−1.88224	−0.941121	+	0.338071i	$0.890226\pi$
$642$	0	0
$643$	39.2730	1.54878	0.774388	−	0.632711i	$-0.218058\pi$
$643$	39.2730	1.54878	0.774388	+	0.632711i	$0.218058\pi$
$644$	0	0
$645$	43.7640	1.72321
$646$	0	0
$647$	33.6183	1.32167	0.660836	−	0.750531i	$-0.270202\pi$
$647$	33.6183	1.32167	0.660836	+	0.750531i	$0.270202\pi$
$648$	0	0
$649$	−5.28084	−0.207291
$650$	0	0
$651$	12.3602	0.484434
$652$	0	0
$653$	9.52793	0.372857	0.186428	−	0.982469i	$-0.440309\pi$
$653$	9.52793	0.372857	0.186428	+	0.982469i	$0.440309\pi$
$654$	0	0
$655$	74.9585	2.92887
$656$	0	0
$657$	−25.2112	−0.983582
$658$	0	0
$659$	−49.0365	−1.91019	−0.955096	−	0.296296i	$-0.904248\pi$
$659$	−49.0365	−1.91019	−0.955096	+	0.296296i	$0.904248\pi$
$660$	0	0
$661$	−29.7000	−1.15520	−0.577599	−	0.816321i	$-0.696010\pi$
$661$	−29.7000	−1.15520	−0.577599	+	0.816321i	$0.696010\pi$
$662$	0	0
$663$	−1.87705	−0.0728984
$664$	0	0
$665$	26.1170	1.01277
$666$	0	0
$667$	11.0427	0.427576
$668$	0	0
$669$	−35.5494	−1.37442
$670$	0	0
$671$	14.7091	0.567840
$672$	0	0
$673$	2.94599	0.113560	0.0567798	−	0.998387i	$-0.481917\pi$
$673$	2.94599	0.113560	0.0567798	+	0.998387i	$0.481917\pi$
$674$	0	0
$675$	−9.21178	−0.354562
$676$	0	0
$677$	32.7963	1.26046	0.630231	−	0.776407i	$-0.282960\pi$
$677$	32.7963	1.26046	0.630231	+	0.776407i	$0.282960\pi$
$678$	0	0
$679$	−9.60401	−0.368568
$680$	0	0
$681$	−59.8651	−2.29404
$682$	0	0
$683$	30.3772	1.16235	0.581175	−	0.813778i	$-0.302593\pi$
$683$	30.3772	1.16235	0.581175	+	0.813778i	$0.302593\pi$
$684$	0	0
$685$	−7.27934	−0.278129
$686$	0	0
$687$	48.5561	1.85253
$688$	0	0
$689$	−10.0328	−0.382221
$690$	0	0
$691$	6.83537	0.260030	0.130015	−	0.991512i	$-0.458498\pi$
$691$	6.83537	0.260030	0.130015	+	0.991512i	$0.458498\pi$
$692$	0	0
$693$	−5.01599	−0.190542
$694$	0	0
$695$	70.9456	2.69112
$696$	0	0
$697$	−6.11589	−0.231656
$698$	0	0
$699$	−25.7252	−0.973016
$700$	0	0
$701$	−34.2701	−1.29436	−0.647182	−	0.762335i	$-0.724053\pi$
$701$	−34.2701	−1.29436	−0.647182	+	0.762335i	$0.724053\pi$
$702$	0	0
$703$	−29.7759	−1.12302
$704$	0	0
$705$	22.4270	0.844652
$706$	0	0
$707$	12.4145	0.466895
$708$	0	0
$709$	13.7877	0.517808	0.258904	−	0.965903i	$-0.416639\pi$
$709$	13.7877	0.517808	0.258904	+	0.965903i	$0.416639\pi$
$710$	0	0
$711$	−19.5603	−0.733570
$712$	0	0
$713$	−11.8988	−0.445614
$714$	0	0
$715$	−7.53953	−0.281962
$716$	0	0
$717$	−56.8848	−2.12440
$718$	0	0
$719$	−14.5153	−0.541331	−0.270665	−	0.962674i	$-0.587244\pi$
$719$	−14.5153	−0.541331	−0.270665	+	0.962674i	$0.587244\pi$
$720$	0	0
$721$	−15.2080	−0.566377
$722$	0	0
$723$	−44.9769	−1.67271
$724$	0	0
$725$	−51.8184	−1.92449
$726$	0	0
$727$	−13.8926	−0.515247	−0.257623	−	0.966245i	$-0.582939\pi$
$727$	−13.8926	−0.515247	−0.257623	+	0.966245i	$0.582939\pi$
$728$	0	0
$729$	−20.1449	−0.746109
$730$	0	0
$731$	−3.67386	−0.135883
$732$	0	0
$733$	−25.6742	−0.948299	−0.474149	−	0.880444i	$-0.657244\pi$
$733$	−25.6742	−0.948299	−0.474149	+	0.880444i	$0.657244\pi$
$734$	0	0
$735$	9.41658	0.347336
$736$	0	0
$737$	−18.9144	−0.696719
$738$	0	0
$739$	27.1160	0.997477	0.498738	−	0.866753i	$-0.333797\pi$
$739$	27.1160	0.997477	0.498738	+	0.866753i	$0.333797\pi$
$740$	0	0
$741$	15.6380	0.574475
$742$	0	0
$743$	21.8443	0.801389	0.400694	−	0.916212i	$-0.368769\pi$
$743$	21.8443	0.801389	0.400694	+	0.916212i	$0.368769\pi$
$744$	0	0
$745$	82.7839	3.03297
$746$	0	0
$747$	−6.38124	−0.233477
$748$	0	0
$749$	−3.37512	−0.123324
$750$	0	0
$751$	−45.6037	−1.66410	−0.832051	−	0.554699i	$-0.812833\pi$
$751$	−45.6037	−1.66410	−0.832051	+	0.554699i	$0.812833\pi$
$752$	0	0
$753$	−56.0822	−2.04375
$754$	0	0
$755$	−0.379802	−0.0138224
$756$	0	0
$757$	−17.0427	−0.619428	−0.309714	−	0.950830i	$-0.600233\pi$
$757$	−17.0427	−0.619428	−0.309714	+	0.950830i	$0.600233\pi$
$758$	0	0
$759$	10.3194	0.374572
$760$	0	0
$761$	18.3994	0.666977	0.333489	−	0.942754i	$-0.391774\pi$
$761$	18.3994	0.666977	0.333489	+	0.942754i	$0.391774\pi$
$762$	0	0
$763$	−8.37750	−0.303286
$764$	0	0
$765$	−8.27079	−0.299031
$766$	0	0
$767$	2.77764	0.100295
$768$	0	0
$769$	−20.0522	−0.723102	−0.361551	−	0.932352i	$-0.617753\pi$
$769$	−20.0522	−0.723102	−0.361551	+	0.932352i	$0.617753\pi$
$770$	0	0
$771$	28.1845	1.01504
$772$	0	0
$773$	47.0932	1.69383	0.846913	−	0.531732i	$-0.178459\pi$
$773$	47.0932	1.69383	0.846913	+	0.531732i	$0.178459\pi$
$774$	0	0
$775$	55.8357	2.00568
$776$	0	0
$777$	−10.7358	−0.385146
$778$	0	0
$779$	50.9525	1.82556
$780$	0	0
$781$	−17.9028	−0.640611
$782$	0	0
$783$	4.14861	0.148259
$784$	0	0
$785$	−34.6944	−1.23830
$786$	0	0
$787$	−19.4363	−0.692831	−0.346415	−	0.938081i	$-0.612601\pi$
$787$	−19.4363	−0.692831	−0.346415	+	0.938081i	$0.612601\pi$
$788$	0	0
$789$	−10.4337	−0.371451
$790$	0	0
$791$	0.0930891	0.00330987
$792$	0	0
$793$	−7.73678	−0.274741
$794$	0	0
$795$	−94.4751	−3.35069
$796$	0	0
$797$	15.0716	0.533864	0.266932	−	0.963715i	$-0.413990\pi$
$797$	15.0716	0.533864	0.266932	+	0.963715i	$0.413990\pi$
$798$	0	0
$799$	−1.88269	−0.0666047
$800$	0	0
$801$	32.0781	1.13343
$802$	0	0
$803$	18.1673	0.641109
$804$	0	0
$805$	−9.06508	−0.319502
$806$	0	0
$807$	−38.9060	−1.36956
$808$	0	0
$809$	4.09322	0.143910	0.0719550	−	0.997408i	$-0.477076\pi$
$809$	4.09322	0.143910	0.0719550	+	0.997408i	$0.477076\pi$
$810$	0	0
$811$	−40.9519	−1.43802	−0.719008	−	0.695002i	$-0.755403\pi$
$811$	−40.9519	−1.43802	−0.719008	+	0.695002i	$0.755403\pi$
$812$	0	0
$813$	53.9910	1.89355
$814$	0	0
$815$	7.46285	0.261412
$816$	0	0
$817$	30.6076	1.07082
$818$	0	0
$819$	2.63834	0.0921909
$820$	0	0
$821$	−22.3599	−0.780365	−0.390183	−	0.920737i	$-0.627588\pi$
$821$	−22.3599	−0.780365	−0.390183	+	0.920737i	$0.627588\pi$
$822$	0	0
$823$	1.95660	0.0682026	0.0341013	−	0.999418i	$-0.489143\pi$
$823$	1.95660	0.0682026	0.0341013	+	0.999418i	$0.489143\pi$
$824$	0	0
$825$	−48.4245	−1.68592
$826$	0	0
$827$	−19.2020	−0.667718	−0.333859	−	0.942623i	$-0.608351\pi$
$827$	−19.2020	−0.667718	−0.333859	+	0.942623i	$0.608351\pi$
$828$	0	0
$829$	35.8253	1.24426	0.622132	−	0.782912i	$-0.286267\pi$
$829$	35.8253	1.24426	0.622132	+	0.782912i	$0.286267\pi$
$830$	0	0
$831$	−40.1589	−1.39310
$832$	0	0
$833$	−0.790495	−0.0273890
$834$	0	0
$835$	−17.4095	−0.602482
$836$	0	0
$837$	−4.47024	−0.154514
$838$	0	0
$839$	−15.6024	−0.538653	−0.269327	−	0.963049i	$-0.586801\pi$
$839$	−15.6024	−0.538653	−0.269327	+	0.963049i	$0.586801\pi$
$840$	0	0
$841$	−5.66310	−0.195279
$842$	0	0
$843$	−26.0543	−0.897358
$844$	0	0
$845$	3.96568	0.136424
$846$	0	0
$847$	−7.38546	−0.253767
$848$	0	0
$849$	−19.5598	−0.671292
$850$	0	0
$851$	10.3351	0.354282
$852$	0	0
$853$	26.1752	0.896222	0.448111	−	0.893978i	$-0.352097\pi$
$853$	26.1752	0.896222	0.448111	+	0.893978i	$0.352097\pi$
$854$	0	0
$855$	68.9053	2.35651
$856$	0	0
$857$	41.9691	1.43364	0.716818	−	0.697260i	$-0.245598\pi$
$857$	41.9691	1.43364	0.716818	+	0.697260i	$0.245598\pi$
$858$	0	0
$859$	−36.9085	−1.25930	−0.629651	−	0.776878i	$-0.716802\pi$
$859$	−36.9085	−1.25930	−0.629651	+	0.776878i	$0.716802\pi$
$860$	0	0
$861$	18.3711	0.626086
$862$	0	0
$863$	−45.7213	−1.55637	−0.778185	−	0.628035i	$-0.783859\pi$
$863$	−45.7213	−1.55637	−0.778185	+	0.628035i	$0.783859\pi$
$864$	0	0
$865$	65.4732	2.22616
$866$	0	0
$867$	−38.8830	−1.32054
$868$	0	0
$869$	14.0953	0.478149
$870$	0	0
$871$	9.94867	0.337098
$872$	0	0
$873$	−25.3386	−0.857582
$874$	0	0
$875$	22.7099	0.767735
$876$	0	0
$877$	−7.23508	−0.244311	−0.122156	−	0.992511i	$-0.538981\pi$
$877$	−7.23508	−0.244311	−0.122156	+	0.992511i	$0.538981\pi$
$878$	0	0
$879$	−43.5626	−1.46933
$880$	0	0
$881$	15.7086	0.529238	0.264619	−	0.964353i	$-0.414754\pi$
$881$	15.7086	0.529238	0.264619	+	0.964353i	$0.414754\pi$
$882$	0	0
$883$	1.79507	0.0604088	0.0302044	−	0.999544i	$-0.490384\pi$
$883$	1.79507	0.0604088	0.0302044	+	0.999544i	$0.490384\pi$
$884$	0	0
$885$	26.1559	0.879220
$886$	0	0
$887$	32.1949	1.08100	0.540499	−	0.841345i	$-0.318235\pi$
$887$	32.1949	1.08100	0.540499	+	0.841345i	$0.318235\pi$
$888$	0	0
$889$	−11.4773	−0.384935
$890$	0	0
$891$	18.9249	0.634007
$892$	0	0
$893$	15.6850	0.524878
$894$	0	0
$895$	−55.9621	−1.87061
$896$	0	0
$897$	−5.42787	−0.181231
$898$	0	0
$899$	−25.1461	−0.838671
$900$	0	0
$901$	7.93092	0.264217
$902$	0	0
$903$	11.0357	0.367245
$904$	0	0
$905$	−73.8497	−2.45485
$906$	0	0
$907$	7.19237	0.238819	0.119409	−	0.992845i	$-0.461900\pi$
$907$	7.19237	0.238819	0.119409	+	0.992845i	$0.461900\pi$
$908$	0	0
$909$	32.7536	1.08637
$910$	0	0
$911$	−32.2856	−1.06967	−0.534835	−	0.844956i	$-0.679626\pi$
$911$	−32.2856	−1.06967	−0.534835	+	0.844956i	$0.679626\pi$
$912$	0	0
$913$	4.59835	0.152183
$914$	0	0
$915$	−72.8540	−2.40848
$916$	0	0
$917$	18.9018	0.624192
$918$	0	0
$919$	53.3704	1.76053	0.880263	−	0.474486i	$-0.157366\pi$
$919$	53.3704	1.76053	0.880263	+	0.474486i	$0.157366\pi$
$920$	0	0
$921$	23.3121	0.768160
$922$	0	0
$923$	9.41658	0.309950
$924$	0	0
$925$	−48.4978	−1.59460
$926$	0	0
$927$	−40.1239	−1.31784
$928$	0	0
$929$	−14.4388	−0.473720	−0.236860	−	0.971544i	$-0.576118\pi$
$929$	−14.4388	−0.473720	−0.236860	+	0.971544i	$0.576118\pi$
$930$	0	0
$931$	6.58575	0.215839
$932$	0	0
$933$	15.7357	0.515164
$934$	0	0
$935$	5.95996	0.194912
$936$	0	0
$937$	25.4009	0.829812	0.414906	−	0.909864i	$-0.363814\pi$
$937$	25.4009	0.829812	0.414906	+	0.909864i	$0.363814\pi$
$938$	0	0
$939$	−46.2905	−1.51063
$940$	0	0
$941$	−24.2366	−0.790090	−0.395045	−	0.918662i	$-0.629271\pi$
$941$	−24.2366	−0.790090	−0.395045	+	0.918662i	$0.629271\pi$
$942$	0	0
$943$	−17.6854	−0.575915
$944$	0	0
$945$	−3.40564	−0.110785
$946$	0	0
$947$	−7.62584	−0.247807	−0.123903	−	0.992294i	$-0.539541\pi$
$947$	−7.62584	−0.247807	−0.123903	+	0.992294i	$0.539541\pi$
$948$	0	0
$949$	−9.55571	−0.310192
$950$	0	0
$951$	12.7418	0.413182
$952$	0	0
$953$	41.6062	1.34776	0.673878	−	0.738843i	$-0.264627\pi$
$953$	41.6062	1.34776	0.673878	+	0.738843i	$0.264627\pi$
$954$	0	0
$955$	34.6555	1.12143
$956$	0	0
$957$	21.8084	0.704966
$958$	0	0
$959$	−1.83559	−0.0592742
$960$	0	0
$961$	−3.90437	−0.125947
$962$	0	0
$963$	−8.90469	−0.286950
$964$	0	0
$965$	88.2650	2.84135
$966$	0	0
$967$	38.7669	1.24666	0.623329	−	0.781959i	$-0.285780\pi$
$967$	38.7669	1.24666	0.623329	+	0.781959i	$0.285780\pi$
$968$	0	0
$969$	−12.3617	−0.397117
$970$	0	0
$971$	−19.4635	−0.624615	−0.312307	−	0.949981i	$-0.601102\pi$
$971$	−19.4635	−0.624615	−0.312307	+	0.949981i	$0.601102\pi$
$972$	0	0
$973$	17.8899	0.573524
$974$	0	0
$975$	25.4705	0.815710
$976$	0	0
$977$	−41.5896	−1.33057	−0.665284	−	0.746590i	$-0.731690\pi$
$977$	−41.5896	−1.33057	−0.665284	+	0.746590i	$0.731690\pi$
$978$	0	0
$979$	−23.1156	−0.738779
$980$	0	0
$981$	−22.1027	−0.705683
$982$	0	0
$983$	−50.2222	−1.60184	−0.800920	−	0.598772i	$-0.795656\pi$
$983$	−50.2222	−1.60184	−0.800920	+	0.598772i	$0.795656\pi$
$984$	0	0
$985$	−69.3797	−2.21062
$986$	0	0
$987$	5.65529	0.180010
$988$	0	0
$989$	−10.6238	−0.337816
$990$	0	0
$991$	−32.8893	−1.04476	−0.522382	−	0.852712i	$-0.674956\pi$
$991$	−32.8893	−1.04476	−0.522382	+	0.852712i	$0.674956\pi$
$992$	0	0
$993$	−33.1002	−1.05040
$994$	0	0
$995$	67.6223	2.14377
$996$	0	0
$997$	59.9060	1.89724	0.948620	−	0.316417i	$-0.102480\pi$
$997$	59.9060	1.89724	0.948620	+	0.316417i	$0.102480\pi$
$998$	0	0
$999$	3.88276	0.122845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5824.2.a.cl.1.4		5
4.3	odd	2		5824.2.a.ci.1.2		5
8.3	odd	2		2912.2.a.t.1.4	yes	5
8.5	even	2		2912.2.a.q.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.q.1.2	✓	5	8.5	even	2
2912.2.a.t.1.4	yes	5	8.3	odd	2
5824.2.a.ci.1.2		5	4.3	odd	2
5824.2.a.cl.1.4		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5824.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37452 q^{3} +3.96568 q^{5} +1.00000 q^{7} +2.63834 q^{9} -1.90120 q^{11} +1.00000 q^{13} +9.41658 q^{15} -0.790495 q^{17} +6.58575 q^{19} +2.37452 q^{21} -2.28588 q^{23} +10.7266 q^{25} -0.858778 q^{27} -4.83083 q^{29} +5.20535 q^{31} -4.51442 q^{33} +3.96568 q^{35} -4.52126 q^{37} +2.37452 q^{39} +7.73678 q^{41} +4.64755 q^{43} +10.4628 q^{45} +2.38166 q^{47} +1.00000 q^{49} -1.87705 q^{51} -10.0328 q^{53} -7.53953 q^{55} +15.6380 q^{57} +2.77764 q^{59} -7.73678 q^{61} +2.63834 q^{63} +3.96568 q^{65} +9.94867 q^{67} -5.42787 q^{69} +9.41658 q^{71} -9.55571 q^{73} +25.4705 q^{75} -1.90120 q^{77} -7.41389 q^{79} -9.95419 q^{81} -2.41866 q^{83} -3.13485 q^{85} -11.4709 q^{87} +12.1585 q^{89} +1.00000 q^{91} +12.3602 q^{93} +26.1170 q^{95} -9.60401 q^{97} -5.01599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 6 q^{9} + 5 q^{11} + 5 q^{13} + 4 q^{17} + 7 q^{19} + 5 q^{21} + 4 q^{23} + 2 q^{25} + 23 q^{27} - 3 q^{29} - 2 q^{31} + 17 q^{33} + 3 q^{35} + q^{37} + 5 q^{39} - 7 q^{41}+ \cdots + 40 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 6 * q^9 + 5 * q^11 + 5 * q^13 + 4 * q^17 + 7 * q^19 + 5 * q^21 + 4 * q^23 + 2 * q^25 + 23 * q^27 - 3 * q^29 - 2 * q^31 + 17 * q^33 + 3 * q^35 + q^37 + 5 * q^39 - 7 * q^41 + 15 * q^43 + q^45 - 6 * q^47 + 5 * q^49 + 16 * q^51 - 11 * q^53 - 16 * q^55 + 18 * q^57 + 20 * q^59 + 7 * q^61 + 6 * q^63 + 3 * q^65 + 11 * q^67 - 5 * q^69 - 2 * q^73 + 19 * q^75 + 5 * q^77 - 10 * q^79 + 33 * q^81 + 9 * q^83 - 20 * q^85 - 2 * q^87 + 11 * q^89 + 5 * q^91 + 13 * q^93 + 11 * q^95 - 24 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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