LMFDB - Embedded newform 3328.2.a.bg.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,-3,0,1,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24914 q^{3} -1.00000 q^{5} +3.24914 q^{7} +7.55691 q^{9} +1.05863 q^{11} -1.00000 q^{13} -3.24914 q^{15} +1.00000 q^{17} +4.00000 q^{19} +10.5569 q^{21} -2.94137 q^{23} -4.00000 q^{25} +14.8061 q^{27} +7.43965 q^{29} -5.05863 q^{31} +3.43965 q^{33} -3.24914 q^{35} -6.55691 q^{37} -3.24914 q^{39} +9.43965 q^{41} -0.307774 q^{43} -7.55691 q^{45} -6.80605 q^{47} +3.55691 q^{49} +3.24914 q^{51} +1.55691 q^{53} -1.05863 q^{55} +12.9966 q^{57} -5.67418 q^{59} +9.67418 q^{61} +24.5535 q^{63} +1.00000 q^{65} +1.50172 q^{67} -9.55691 q^{69} -5.36641 q^{71} -3.55691 q^{73} -12.9966 q^{75} +3.43965 q^{77} +6.73281 q^{79} +25.4362 q^{81} -2.49828 q^{83} -1.00000 q^{85} +24.1725 q^{87} -2.11727 q^{89} -3.24914 q^{91} -16.4362 q^{93} -4.00000 q^{95} +7.67418 q^{97} +8.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} + q^{7} + 6 q^{9} + 4 q^{11} - 3 q^{13} - q^{15} + 3 q^{17} + 12 q^{19} + 15 q^{21} - 8 q^{23} - 12 q^{25} + 19 q^{27} + 4 q^{29} - 16 q^{31} - 8 q^{33} - q^{35} - 3 q^{37} - q^{39}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 + q^7 + 6 * q^9 + 4 * q^11 - 3 * q^13 - q^15 + 3 * q^17 + 12 * q^19 + 15 * q^21 - 8 * q^23 - 12 * q^25 + 19 * q^27 + 4 * q^29 - 16 * q^31 - 8 * q^33 - q^35 - 3 * q^37 - q^39 + 10 * q^41 + 7 * q^43 - 6 * q^45 + 5 * q^47 - 6 * q^49 + q^51 - 12 * q^53 - 4 * q^55 + 4 * q^57 - 2 * q^59 + 14 * q^61 + 22 * q^63 + 3 * q^65 + 22 * q^67 - 12 * q^69 - 9 * q^71 + 6 * q^73 - 4 * q^75 - 8 * q^77 + 6 * q^79 + 23 * q^81 + 10 * q^83 - 3 * q^85 + 40 * q^87 - 8 * q^89 - q^91 + 4 * q^93 - 12 * q^95 + 8 * q^97 + 24 * 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
$3$	3.24914	1.87589	0.937946	−	0.346781i	$-0.112725\pi$
$3$	3.24914	1.87589	0.937946	+	0.346781i	$0.112725\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	3.24914	1.22806	0.614030	−	0.789283i	$-0.289547\pi$
$7$	3.24914	1.22806	0.614030	+	0.789283i	$0.289547\pi$
$8$	0	0
$9$	7.55691	2.51897
$10$	0	0
$11$	1.05863	0.319190	0.159595	−	0.987183i	$-0.448981\pi$
$11$	1.05863	0.319190	0.159595	+	0.987183i	$0.448981\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.24914	−0.838924
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	10.5569	2.30371
$22$	0	0
$23$	−2.94137	−0.613317	−0.306659	−	0.951820i	$-0.599211\pi$
$23$	−2.94137	−0.613317	−0.306659	+	0.951820i	$0.599211\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	14.8061	2.84943
$28$	0	0
$29$	7.43965	1.38151	0.690754	−	0.723090i	$-0.257279\pi$
$29$	7.43965	1.38151	0.690754	+	0.723090i	$0.257279\pi$
$30$	0	0
$31$	−5.05863	−0.908557	−0.454279	−	0.890860i	$-0.650103\pi$
$31$	−5.05863	−0.908557	−0.454279	+	0.890860i	$0.650103\pi$
$32$	0	0
$33$	3.43965	0.598766
$34$	0	0
$35$	−3.24914	−0.549205
$36$	0	0
$37$	−6.55691	−1.07795	−0.538975	−	0.842322i	$-0.681188\pi$
$37$	−6.55691	−1.07795	−0.538975	+	0.842322i	$0.681188\pi$
$38$	0	0
$39$	−3.24914	−0.520279
$40$	0	0
$41$	9.43965	1.47423	0.737113	−	0.675770i	$-0.236189\pi$
$41$	9.43965	1.47423	0.737113	+	0.675770i	$0.236189\pi$
$42$	0	0
$43$	−0.307774	−0.0469350	−0.0234675	−	0.999725i	$-0.507471\pi$
$43$	−0.307774	−0.0469350	−0.0234675	+	0.999725i	$0.507471\pi$
$44$	0	0
$45$	−7.55691	−1.12652
$46$	0	0
$47$	−6.80605	−0.992765	−0.496383	−	0.868104i	$-0.665339\pi$
$47$	−6.80605	−0.992765	−0.496383	+	0.868104i	$0.665339\pi$
$48$	0	0
$49$	3.55691	0.508131
$50$	0	0
$51$	3.24914	0.454971
$52$	0	0
$53$	1.55691	0.213859	0.106929	−	0.994267i	$-0.465898\pi$
$53$	1.55691	0.213859	0.106929	+	0.994267i	$0.465898\pi$
$54$	0	0
$55$	−1.05863	−0.142746
$56$	0	0
$57$	12.9966	1.72144
$58$	0	0
$59$	−5.67418	−0.738715	−0.369358	−	0.929287i	$-0.620422\pi$
$59$	−5.67418	−0.738715	−0.369358	+	0.929287i	$0.620422\pi$
$60$	0	0
$61$	9.67418	1.23865	0.619326	−	0.785134i	$-0.287406\pi$
$61$	9.67418	1.23865	0.619326	+	0.785134i	$0.287406\pi$
$62$	0	0
$63$	24.5535	3.09345
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	1.50172	0.183464	0.0917321	−	0.995784i	$-0.470760\pi$
$67$	1.50172	0.183464	0.0917321	+	0.995784i	$0.470760\pi$
$68$	0	0
$69$	−9.55691	−1.15052
$70$	0	0
$71$	−5.36641	−0.636875	−0.318438	−	0.947944i	$-0.603158\pi$
$71$	−5.36641	−0.636875	−0.318438	+	0.947944i	$0.603158\pi$
$72$	0	0
$73$	−3.55691	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$73$	−3.55691	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$74$	0	0
$75$	−12.9966	−1.50071
$76$	0	0
$77$	3.43965	0.391984
$78$	0	0
$79$	6.73281	0.757501	0.378750	−	0.925499i	$-0.376354\pi$
$79$	6.73281	0.757501	0.378750	+	0.925499i	$0.376354\pi$
$80$	0	0
$81$	25.4362	2.82625
$82$	0	0
$83$	−2.49828	−0.274222	−0.137111	−	0.990556i	$-0.543782\pi$
$83$	−2.49828	−0.274222	−0.137111	+	0.990556i	$0.543782\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	24.1725	2.59156
$88$	0	0
$89$	−2.11727	−0.224430	−0.112215	−	0.993684i	$-0.535795\pi$
$89$	−2.11727	−0.224430	−0.112215	+	0.993684i	$0.535795\pi$
$90$	0	0
$91$	−3.24914	−0.340602
$92$	0	0
$93$	−16.4362	−1.70436
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	7.67418	0.779195	0.389597	−	0.920985i	$-0.372614\pi$
$97$	7.67418	0.779195	0.389597	+	0.920985i	$0.372614\pi$
$98$	0	0
$99$	8.00000	0.804030
$100$	0	0
$101$	−3.11383	−0.309838	−0.154919	−	0.987927i	$-0.549512\pi$
$101$	−3.11383	−0.309838	−0.154919	+	0.987927i	$0.549512\pi$
$102$	0	0
$103$	−17.6121	−1.73537	−0.867686	−	0.497112i	$-0.834394\pi$
$103$	−17.6121	−1.73537	−0.867686	+	0.497112i	$0.834394\pi$
$104$	0	0
$105$	−10.5569	−1.03025
$106$	0	0
$107$	10.9414	1.05774	0.528871	−	0.848702i	$-0.322616\pi$
$107$	10.9414	1.05774	0.528871	+	0.848702i	$0.322616\pi$
$108$	0	0
$109$	−10.3224	−0.988705	−0.494352	−	0.869262i	$-0.664595\pi$
$109$	−10.3224	−0.988705	−0.494352	+	0.869262i	$0.664595\pi$
$110$	0	0
$111$	−21.3043	−2.02212
$112$	0	0
$113$	−17.1138	−1.60993	−0.804967	−	0.593320i	$-0.797817\pi$
$113$	−17.1138	−1.60993	−0.804967	+	0.593320i	$0.797817\pi$
$114$	0	0
$115$	2.94137	0.274284
$116$	0	0
$117$	−7.55691	−0.698637
$118$	0	0
$119$	3.24914	0.297848
$120$	0	0
$121$	−9.87930	−0.898118
$122$	0	0
$123$	30.6707	2.76549
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	15.9379	1.41426	0.707131	−	0.707082i	$-0.249989\pi$
$127$	15.9379	1.41426	0.707131	+	0.707082i	$0.249989\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	7.69223	0.672073	0.336036	−	0.941849i	$-0.390913\pi$
$131$	7.69223	0.672073	0.336036	+	0.941849i	$0.390913\pi$
$132$	0	0
$133$	12.9966	1.12694
$134$	0	0
$135$	−14.8061	−1.27430
$136$	0	0
$137$	−5.32238	−0.454722	−0.227361	−	0.973811i	$-0.573010\pi$
$137$	−5.32238	−0.454722	−0.227361	+	0.973811i	$0.573010\pi$
$138$	0	0
$139$	−12.4802	−1.05856	−0.529280	−	0.848447i	$-0.677538\pi$
$139$	−12.4802	−1.05856	−0.529280	+	0.848447i	$0.677538\pi$
$140$	0	0
$141$	−22.1138	−1.86232
$142$	0	0
$143$	−1.05863	−0.0885274
$144$	0	0
$145$	−7.43965	−0.617829
$146$	0	0
$147$	11.5569	0.953198
$148$	0	0
$149$	9.11383	0.746634	0.373317	−	0.927704i	$-0.378220\pi$
$149$	9.11383	0.746634	0.373317	+	0.927704i	$0.378220\pi$
$150$	0	0
$151$	−15.4216	−1.25499	−0.627496	−	0.778620i	$-0.715920\pi$
$151$	−15.4216	−1.25499	−0.627496	+	0.778620i	$0.715920\pi$
$152$	0	0
$153$	7.55691	0.610940
$154$	0	0
$155$	5.05863	0.406319
$156$	0	0
$157$	−3.79145	−0.302590	−0.151295	−	0.988489i	$-0.548344\pi$
$157$	−3.79145	−0.302590	−0.151295	+	0.988489i	$0.548344\pi$
$158$	0	0
$159$	5.05863	0.401176
$160$	0	0
$161$	−9.55691	−0.753190
$162$	0	0
$163$	4.17246	0.326812	0.163406	−	0.986559i	$-0.447752\pi$
$163$	4.17246	0.326812	0.163406	+	0.986559i	$0.447752\pi$
$164$	0	0
$165$	−3.43965	−0.267776
$166$	0	0
$167$	14.2897	1.10577	0.552886	−	0.833257i	$-0.313526\pi$
$167$	14.2897	1.10577	0.552886	+	0.833257i	$0.313526\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	30.2277	2.31157
$172$	0	0
$173$	6.99656	0.531939	0.265969	−	0.963981i	$-0.414308\pi$
$173$	6.99656	0.531939	0.265969	+	0.963981i	$0.414308\pi$
$174$	0	0
$175$	−12.9966	−0.982448
$176$	0	0
$177$	−18.4362	−1.38575
$178$	0	0
$179$	17.7474	1.32650	0.663252	−	0.748396i	$-0.269176\pi$
$179$	17.7474	1.32650	0.663252	+	0.748396i	$0.269176\pi$
$180$	0	0
$181$	−8.11727	−0.603352	−0.301676	−	0.953411i	$-0.597546\pi$
$181$	−8.11727	−0.603352	−0.301676	+	0.953411i	$0.597546\pi$
$182$	0	0
$183$	31.4328	2.32358
$184$	0	0
$185$	6.55691	0.482074
$186$	0	0
$187$	1.05863	0.0774149
$188$	0	0
$189$	48.1070	3.49927
$190$	0	0
$191$	19.9379	1.44266	0.721329	−	0.692593i	$-0.243532\pi$
$191$	19.9379	1.44266	0.721329	+	0.692593i	$0.243532\pi$
$192$	0	0
$193$	4.11727	0.296367	0.148184	−	0.988960i	$-0.452657\pi$
$193$	4.11727	0.296367	0.148184	+	0.988960i	$0.452657\pi$
$194$	0	0
$195$	3.24914	0.232676
$196$	0	0
$197$	10.7914	0.768859	0.384429	−	0.923154i	$-0.374398\pi$
$197$	10.7914	0.768859	0.384429	+	0.923154i	$0.374398\pi$
$198$	0	0
$199$	0.615547	0.0436350	0.0218175	−	0.999762i	$-0.493055\pi$
$199$	0.615547	0.0436350	0.0218175	+	0.999762i	$0.493055\pi$
$200$	0	0
$201$	4.87930	0.344159
$202$	0	0
$203$	24.1725	1.69657
$204$	0	0
$205$	−9.43965	−0.659294
$206$	0	0
$207$	−22.2277	−1.54493
$208$	0	0
$209$	4.23453	0.292909
$210$	0	0
$211$	1.80949	0.124571	0.0622853	−	0.998058i	$-0.480161\pi$
$211$	1.80949	0.124571	0.0622853	+	0.998058i	$0.480161\pi$
$212$	0	0
$213$	−17.4362	−1.19471
$214$	0	0
$215$	0.307774	0.0209900
$216$	0	0
$217$	−16.4362	−1.11576
$218$	0	0
$219$	−11.5569	−0.780944
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	6.80605	0.455767	0.227884	−	0.973688i	$-0.426819\pi$
$223$	6.80605	0.455767	0.227884	+	0.973688i	$0.426819\pi$
$224$	0	0
$225$	−30.2277	−2.01518
$226$	0	0
$227$	−6.67074	−0.442753	−0.221376	−	0.975188i	$-0.571055\pi$
$227$	−6.67074	−0.442753	−0.221376	+	0.975188i	$0.571055\pi$
$228$	0	0
$229$	−18.1138	−1.19700	−0.598498	−	0.801124i	$-0.704235\pi$
$229$	−18.1138	−1.19700	−0.598498	+	0.801124i	$0.704235\pi$
$230$	0	0
$231$	11.1759	0.735320
$232$	0	0
$233$	−11.4362	−0.749211	−0.374606	−	0.927184i	$-0.622222\pi$
$233$	−11.4362	−0.749211	−0.374606	+	0.927184i	$0.622222\pi$
$234$	0	0
$235$	6.80605	0.443978
$236$	0	0
$237$	21.8759	1.42099
$238$	0	0
$239$	−16.8613	−1.09066	−0.545332	−	0.838220i	$-0.683596\pi$
$239$	−16.8613	−1.09066	−0.545332	+	0.838220i	$0.683596\pi$
$240$	0	0
$241$	−5.23109	−0.336964	−0.168482	−	0.985705i	$-0.553887\pi$
$241$	−5.23109	−0.336964	−0.168482	+	0.985705i	$0.553887\pi$
$242$	0	0
$243$	38.2277	2.45231
$244$	0	0
$245$	−3.55691	−0.227243
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−8.11727	−0.514411
$250$	0	0
$251$	−2.05520	−0.129723	−0.0648614	−	0.997894i	$-0.520661\pi$
$251$	−2.05520	−0.129723	−0.0648614	+	0.997894i	$0.520661\pi$
$252$	0	0
$253$	−3.11383	−0.195765
$254$	0	0
$255$	−3.24914	−0.203469
$256$	0	0
$257$	−21.2277	−1.32414	−0.662072	−	0.749440i	$-0.730323\pi$
$257$	−21.2277	−1.32414	−0.662072	+	0.749440i	$0.730323\pi$
$258$	0	0
$259$	−21.3043	−1.32379
$260$	0	0
$261$	56.2208	3.47998
$262$	0	0
$263$	23.5569	1.45258	0.726291	−	0.687388i	$-0.241243\pi$
$263$	23.5569	1.45258	0.726291	+	0.687388i	$0.241243\pi$
$264$	0	0
$265$	−1.55691	−0.0956405
$266$	0	0
$267$	−6.87930	−0.421006
$268$	0	0
$269$	−6.32582	−0.385692	−0.192846	−	0.981229i	$-0.561772\pi$
$269$	−6.32582	−0.385692	−0.192846	+	0.981229i	$0.561772\pi$
$270$	0	0
$271$	−3.45769	−0.210040	−0.105020	−	0.994470i	$-0.533491\pi$
$271$	−3.45769	−0.210040	−0.105020	+	0.994470i	$0.533491\pi$
$272$	0	0
$273$	−10.5569	−0.638934
$274$	0	0
$275$	−4.23453	−0.255352
$276$	0	0
$277$	28.5535	1.71561	0.857806	−	0.513974i	$-0.171827\pi$
$277$	28.5535	1.71561	0.857806	+	0.513974i	$0.171827\pi$
$278$	0	0
$279$	−38.2277	−2.28863
$280$	0	0
$281$	−24.9966	−1.49117	−0.745585	−	0.666411i	$-0.767830\pi$
$281$	−24.9966	−1.49117	−0.745585	+	0.666411i	$0.767830\pi$
$282$	0	0
$283$	21.8207	1.29710	0.648552	−	0.761170i	$-0.275375\pi$
$283$	21.8207	1.29710	0.648552	+	0.761170i	$0.275375\pi$
$284$	0	0
$285$	−12.9966	−0.769850
$286$	0	0
$287$	30.6707	1.81044
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	24.9345	1.46169
$292$	0	0
$293$	27.4362	1.60284	0.801420	−	0.598102i	$-0.204078\pi$
$293$	27.4362	1.60284	0.801420	+	0.598102i	$0.204078\pi$
$294$	0	0
$295$	5.67418	0.330364
$296$	0	0
$297$	15.6742	0.909508
$298$	0	0
$299$	2.94137	0.170104
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−10.1173	−0.581222
$304$	0	0
$305$	−9.67418	−0.553942
$306$	0	0
$307$	−26.6707	−1.52218	−0.761090	−	0.648646i	$-0.775335\pi$
$307$	−26.6707	−1.52218	−0.761090	+	0.648646i	$0.775335\pi$
$308$	0	0
$309$	−57.2242	−3.25537
$310$	0	0
$311$	6.87930	0.390089	0.195045	−	0.980794i	$-0.437515\pi$
$311$	6.87930	0.390089	0.195045	+	0.980794i	$0.437515\pi$
$312$	0	0
$313$	−17.6707	−0.998809	−0.499405	−	0.866369i	$-0.666448\pi$
$313$	−17.6707	−0.998809	−0.499405	+	0.866369i	$0.666448\pi$
$314$	0	0
$315$	−24.5535	−1.38343
$316$	0	0
$317$	−1.11383	−0.0625588	−0.0312794	−	0.999511i	$-0.509958\pi$
$317$	−1.11383	−0.0625588	−0.0312794	+	0.999511i	$0.509958\pi$
$318$	0	0
$319$	7.87586	0.440963
$320$	0	0
$321$	35.5500	1.98421
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−33.5389	−1.85470
$328$	0	0
$329$	−22.1138	−1.21917
$330$	0	0
$331$	−18.4983	−1.01676	−0.508379	−	0.861134i	$-0.669755\pi$
$331$	−18.4983	−1.01676	−0.508379	+	0.861134i	$0.669755\pi$
$332$	0	0
$333$	−49.5500	−2.71533
$334$	0	0
$335$	−1.50172	−0.0820477
$336$	0	0
$337$	−17.4362	−0.949811	−0.474905	−	0.880037i	$-0.657518\pi$
$337$	−17.4362	−0.949811	−0.474905	+	0.880037i	$0.657518\pi$
$338$	0	0
$339$	−55.6052	−3.02006
$340$	0	0
$341$	−5.35524	−0.290002
$342$	0	0
$343$	−11.1871	−0.604045
$344$	0	0
$345$	9.55691	0.514527
$346$	0	0
$347$	8.98539	0.482361	0.241181	−	0.970480i	$-0.422465\pi$
$347$	8.98539	0.482361	0.241181	+	0.970480i	$0.422465\pi$
$348$	0	0
$349$	−10.9931	−0.588448	−0.294224	−	0.955736i	$-0.595061\pi$
$349$	−10.9931	−0.588448	−0.294224	+	0.955736i	$0.595061\pi$
$350$	0	0
$351$	−14.8061	−0.790289
$352$	0	0
$353$	23.2311	1.23647	0.618233	−	0.785995i	$-0.287849\pi$
$353$	23.2311	1.23647	0.618233	+	0.785995i	$0.287849\pi$
$354$	0	0
$355$	5.36641	0.284819
$356$	0	0
$357$	10.5569	0.558731
$358$	0	0
$359$	−11.2863	−0.595668	−0.297834	−	0.954618i	$-0.596264\pi$
$359$	−11.2863	−0.595668	−0.297834	+	0.954618i	$0.596264\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−32.0992	−1.68477
$364$	0	0
$365$	3.55691	0.186177
$366$	0	0
$367$	−13.6121	−0.710546	−0.355273	−	0.934763i	$-0.615612\pi$
$367$	−13.6121	−0.710546	−0.355273	+	0.934763i	$0.615612\pi$
$368$	0	0
$369$	71.3346	3.71353
$370$	0	0
$371$	5.05863	0.262631
$372$	0	0
$373$	−33.1070	−1.71421	−0.857107	−	0.515139i	$-0.827740\pi$
$373$	−33.1070	−1.71421	−0.857107	+	0.515139i	$0.827740\pi$
$374$	0	0
$375$	29.2423	1.51006
$376$	0	0
$377$	−7.43965	−0.383161
$378$	0	0
$379$	−33.1690	−1.70378	−0.851889	−	0.523722i	$-0.824543\pi$
$379$	−33.1690	−1.70378	−0.851889	+	0.523722i	$0.824543\pi$
$380$	0	0
$381$	51.7846	2.65300
$382$	0	0
$383$	−33.0698	−1.68979	−0.844894	−	0.534934i	$-0.820337\pi$
$383$	−33.0698	−1.68979	−0.844894	+	0.534934i	$0.820337\pi$
$384$	0	0
$385$	−3.43965	−0.175301
$386$	0	0
$387$	−2.32582	−0.118228
$388$	0	0
$389$	19.5569	0.991575	0.495787	−	0.868444i	$-0.334880\pi$
$389$	19.5569	0.991575	0.495787	+	0.868444i	$0.334880\pi$
$390$	0	0
$391$	−2.94137	−0.148751
$392$	0	0
$393$	24.9931	1.26074
$394$	0	0
$395$	−6.73281	−0.338765
$396$	0	0
$397$	34.2208	1.71749	0.858746	−	0.512402i	$-0.171244\pi$
$397$	34.2208	1.71749	0.858746	+	0.512402i	$0.171244\pi$
$398$	0	0
$399$	42.2277	2.11403
$400$	0	0
$401$	−32.6707	−1.63150	−0.815750	−	0.578405i	$-0.803675\pi$
$401$	−32.6707	−1.63150	−0.815750	+	0.578405i	$0.803675\pi$
$402$	0	0
$403$	5.05863	0.251988
$404$	0	0
$405$	−25.4362	−1.26394
$406$	0	0
$407$	−6.94137	−0.344071
$408$	0	0
$409$	6.20855	0.306993	0.153497	−	0.988149i	$-0.450947\pi$
$409$	6.20855	0.306993	0.153497	+	0.988149i	$0.450947\pi$
$410$	0	0
$411$	−17.2932	−0.853009
$412$	0	0
$413$	−18.4362	−0.907187
$414$	0	0
$415$	2.49828	0.122636
$416$	0	0
$417$	−40.5500	−1.98574
$418$	0	0
$419$	0.369845	0.0180681	0.00903405	−	0.999959i	$-0.497124\pi$
$419$	0.369845	0.0180681	0.00903405	+	0.999959i	$0.497124\pi$
$420$	0	0
$421$	−3.87930	−0.189065	−0.0945327	−	0.995522i	$-0.530136\pi$
$421$	−3.87930	−0.189065	−0.0945327	+	0.995522i	$0.530136\pi$
$422$	0	0
$423$	−51.4328	−2.50075
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	31.4328	1.52114
$428$	0	0
$429$	−3.43965	−0.166068
$430$	0	0
$431$	−0.516327	−0.0248706	−0.0124353	−	0.999923i	$-0.503958\pi$
$431$	−0.516327	−0.0248706	−0.0124353	+	0.999923i	$0.503958\pi$
$432$	0	0
$433$	30.7846	1.47941	0.739706	−	0.672930i	$-0.234965\pi$
$433$	30.7846	1.47941	0.739706	+	0.672930i	$0.234965\pi$
$434$	0	0
$435$	−24.1725	−1.15898
$436$	0	0
$437$	−11.7655	−0.562819
$438$	0	0
$439$	30.4362	1.45264	0.726321	−	0.687356i	$-0.241229\pi$
$439$	30.4362	1.45264	0.726321	+	0.687356i	$0.241229\pi$
$440$	0	0
$441$	26.8793	1.27997
$442$	0	0
$443$	−3.45769	−0.164280	−0.0821400	−	0.996621i	$-0.526175\pi$
$443$	−3.45769	−0.164280	−0.0821400	+	0.996621i	$0.526175\pi$
$444$	0	0
$445$	2.11727	0.100368
$446$	0	0
$447$	29.6121	1.40060
$448$	0	0
$449$	14.7880	0.697889	0.348945	−	0.937143i	$-0.386540\pi$
$449$	14.7880	0.697889	0.348945	+	0.937143i	$0.386540\pi$
$450$	0	0
$451$	9.99312	0.470558
$452$	0	0
$453$	−50.1070	−2.35423
$454$	0	0
$455$	3.24914	0.152322
$456$	0	0
$457$	−31.8759	−1.49109	−0.745545	−	0.666455i	$-0.767811\pi$
$457$	−31.8759	−1.49109	−0.745545	+	0.666455i	$0.767811\pi$
$458$	0	0
$459$	14.8061	0.691087
$460$	0	0
$461$	−18.5569	−0.864282	−0.432141	−	0.901806i	$-0.642242\pi$
$461$	−18.5569	−0.864282	−0.432141	+	0.901806i	$0.642242\pi$
$462$	0	0
$463$	24.8241	1.15367	0.576837	−	0.816859i	$-0.304287\pi$
$463$	24.8241	1.15367	0.576837	+	0.816859i	$0.304287\pi$
$464$	0	0
$465$	16.4362	0.762211
$466$	0	0
$467$	40.2829	1.86407	0.932034	−	0.362371i	$-0.118033\pi$
$467$	40.2829	1.86407	0.932034	+	0.362371i	$0.118033\pi$
$468$	0	0
$469$	4.87930	0.225305
$470$	0	0
$471$	−12.3189	−0.567627
$472$	0	0
$473$	−0.325819	−0.0149812
$474$	0	0
$475$	−16.0000	−0.734130
$476$	0	0
$477$	11.7655	0.538704
$478$	0	0
$479$	4.07324	0.186111	0.0930556	−	0.995661i	$-0.470337\pi$
$479$	4.07324	0.186111	0.0930556	+	0.995661i	$0.470337\pi$
$480$	0	0
$481$	6.55691	0.298970
$482$	0	0
$483$	−31.0518	−1.41290
$484$	0	0
$485$	−7.67418	−0.348467
$486$	0	0
$487$	15.0518	0.682060	0.341030	−	0.940052i	$-0.389224\pi$
$487$	15.0518	0.682060	0.341030	+	0.940052i	$0.389224\pi$
$488$	0	0
$489$	13.5569	0.613065
$490$	0	0
$491$	3.92676	0.177212	0.0886061	−	0.996067i	$-0.471759\pi$
$491$	3.92676	0.177212	0.0886061	+	0.996067i	$0.471759\pi$
$492$	0	0
$493$	7.43965	0.335065
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−17.4362	−0.782121
$498$	0	0
$499$	−32.3189	−1.44679	−0.723397	−	0.690432i	$-0.757420\pi$
$499$	−32.3189	−1.44679	−0.723397	+	0.690432i	$0.757420\pi$
$500$	0	0
$501$	46.4293	2.07431
$502$	0	0
$503$	−16.1104	−0.718327	−0.359163	−	0.933275i	$-0.616938\pi$
$503$	−16.1104	−0.718327	−0.359163	+	0.933275i	$0.616938\pi$
$504$	0	0
$505$	3.11383	0.138564
$506$	0	0
$507$	3.24914	0.144299
$508$	0	0
$509$	−16.2277	−0.719278	−0.359639	−	0.933091i	$-0.617100\pi$
$509$	−16.2277	−0.719278	−0.359639	+	0.933091i	$0.617100\pi$
$510$	0	0
$511$	−11.5569	−0.511248
$512$	0	0
$513$	59.2242	2.61481
$514$	0	0
$515$	17.6121	0.776082
$516$	0	0
$517$	−7.20512	−0.316881
$518$	0	0
$519$	22.7328	0.997860
$520$	0	0
$521$	8.32238	0.364610	0.182305	−	0.983242i	$-0.441644\pi$
$521$	8.32238	0.364610	0.182305	+	0.983242i	$0.441644\pi$
$522$	0	0
$523$	32.2829	1.41163	0.705815	−	0.708396i	$-0.250581\pi$
$523$	32.2829	1.41163	0.705815	+	0.708396i	$0.250581\pi$
$524$	0	0
$525$	−42.2277	−1.84297
$526$	0	0
$527$	−5.05863	−0.220358
$528$	0	0
$529$	−14.3484	−0.623842
$530$	0	0
$531$	−42.8793	−1.86080
$532$	0	0
$533$	−9.43965	−0.408877
$534$	0	0
$535$	−10.9414	−0.473037
$536$	0	0
$537$	57.6639	2.48838
$538$	0	0
$539$	3.76547	0.162190
$540$	0	0
$541$	2.32238	0.0998470	0.0499235	−	0.998753i	$-0.484102\pi$
$541$	2.32238	0.0998470	0.0499235	+	0.998753i	$0.484102\pi$
$542$	0	0
$543$	−26.3741	−1.13182
$544$	0	0
$545$	10.3224	0.442162
$546$	0	0
$547$	9.89390	0.423033	0.211516	−	0.977374i	$-0.432160\pi$
$547$	9.89390	0.423033	0.211516	+	0.977374i	$0.432160\pi$
$548$	0	0
$549$	73.1070	3.12013
$550$	0	0
$551$	29.7586	1.26776
$552$	0	0
$553$	21.8759	0.930256
$554$	0	0
$555$	21.3043	0.904319
$556$	0	0
$557$	−30.3155	−1.28451	−0.642255	−	0.766491i	$-0.722001\pi$
$557$	−30.3155	−1.28451	−0.642255	+	0.766491i	$0.722001\pi$
$558$	0	0
$559$	0.307774	0.0130174
$560$	0	0
$561$	3.43965	0.145222
$562$	0	0
$563$	38.7440	1.63286	0.816432	−	0.577441i	$-0.195949\pi$
$563$	38.7440	1.63286	0.816432	+	0.577441i	$0.195949\pi$
$564$	0	0
$565$	17.1138	0.719984
$566$	0	0
$567$	82.6458	3.47080
$568$	0	0
$569$	19.2345	0.806354	0.403177	−	0.915122i	$-0.367906\pi$
$569$	19.2345	0.806354	0.403177	+	0.915122i	$0.367906\pi$
$570$	0	0
$571$	10.8320	0.453307	0.226653	−	0.973976i	$-0.427222\pi$
$571$	10.8320	0.453307	0.226653	+	0.973976i	$0.427222\pi$
$572$	0	0
$573$	64.7811	2.70627
$574$	0	0
$575$	11.7655	0.490654
$576$	0	0
$577$	−45.1329	−1.87891	−0.939454	−	0.342674i	$-0.888667\pi$
$577$	−45.1329	−1.87891	−0.939454	+	0.342674i	$0.888667\pi$
$578$	0	0
$579$	13.3776	0.555953
$580$	0	0
$581$	−8.11727	−0.336761
$582$	0	0
$583$	1.64820	0.0682615
$584$	0	0
$585$	7.55691	0.312440
$586$	0	0
$587$	−5.05863	−0.208792	−0.104396	−	0.994536i	$-0.533291\pi$
$587$	−5.05863	−0.208792	−0.104396	+	0.994536i	$0.533291\pi$
$588$	0	0
$589$	−20.2345	−0.833749
$590$	0	0
$591$	35.0629	1.44230
$592$	0	0
$593$	29.4328	1.20866	0.604330	−	0.796734i	$-0.293441\pi$
$593$	29.4328	1.20866	0.604330	+	0.796734i	$0.293441\pi$
$594$	0	0
$595$	−3.24914	−0.133202
$596$	0	0
$597$	2.00000	0.0818546
$598$	0	0
$599$	−40.4293	−1.65190	−0.825949	−	0.563745i	$-0.809360\pi$
$599$	−40.4293	−1.65190	−0.825949	+	0.563745i	$0.809360\pi$
$600$	0	0
$601$	22.9931	0.937909	0.468955	−	0.883222i	$-0.344631\pi$
$601$	22.9931	0.937909	0.468955	+	0.883222i	$0.344631\pi$
$602$	0	0
$603$	11.3484	0.462141
$604$	0	0
$605$	9.87930	0.401650
$606$	0	0
$607$	−17.7846	−0.721853	−0.360927	−	0.932594i	$-0.617540\pi$
$607$	−17.7846	−0.721853	−0.360927	+	0.932594i	$0.617540\pi$
$608$	0	0
$609$	78.5397	3.18259
$610$	0	0
$611$	6.80605	0.275344
$612$	0	0
$613$	−24.2277	−0.978546	−0.489273	−	0.872131i	$-0.662738\pi$
$613$	−24.2277	−0.978546	−0.489273	+	0.872131i	$0.662738\pi$
$614$	0	0
$615$	−30.6707	−1.23676
$616$	0	0
$617$	18.2277	0.733818	0.366909	−	0.930257i	$-0.380416\pi$
$617$	18.2277	0.733818	0.366909	+	0.930257i	$0.380416\pi$
$618$	0	0
$619$	−43.1982	−1.73628	−0.868142	−	0.496316i	$-0.834686\pi$
$619$	−43.1982	−1.73628	−0.868142	+	0.496316i	$0.834686\pi$
$620$	0	0
$621$	−43.5500	−1.74760
$622$	0	0
$623$	−6.87930	−0.275613
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	13.7586	0.549465
$628$	0	0
$629$	−6.55691	−0.261441
$630$	0	0
$631$	−7.07668	−0.281718	−0.140859	−	0.990030i	$-0.544986\pi$
$631$	−7.07668	−0.281718	−0.140859	+	0.990030i	$0.544986\pi$
$632$	0	0
$633$	5.87930	0.233681
$634$	0	0
$635$	−15.9379	−0.632477
$636$	0	0
$637$	−3.55691	−0.140930
$638$	0	0
$639$	−40.5535	−1.60427
$640$	0	0
$641$	−28.2277	−1.11493	−0.557463	−	0.830202i	$-0.688225\pi$
$641$	−28.2277	−1.11493	−0.557463	+	0.830202i	$0.688225\pi$
$642$	0	0
$643$	2.73281	0.107772	0.0538858	−	0.998547i	$-0.482839\pi$
$643$	2.73281	0.107772	0.0538858	+	0.998547i	$0.482839\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	0	0
$647$	−14.5604	−0.572427	−0.286213	−	0.958166i	$-0.592397\pi$
$647$	−14.5604	−0.572427	−0.286213	+	0.958166i	$0.592397\pi$
$648$	0	0
$649$	−6.00688	−0.235790
$650$	0	0
$651$	−53.4036	−2.09305
$652$	0	0
$653$	29.3415	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$653$	29.3415	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$654$	0	0
$655$	−7.69223	−0.300560
$656$	0	0
$657$	−26.8793	−1.04866
$658$	0	0
$659$	−32.2829	−1.25756	−0.628781	−	0.777583i	$-0.716446\pi$
$659$	−32.2829	−1.25756	−0.628781	+	0.777583i	$0.716446\pi$
$660$	0	0
$661$	−24.4102	−0.949448	−0.474724	−	0.880135i	$-0.657452\pi$
$661$	−24.4102	−0.949448	−0.474724	+	0.880135i	$0.657452\pi$
$662$	0	0
$663$	−3.24914	−0.126186
$664$	0	0
$665$	−12.9966	−0.503985
$666$	0	0
$667$	−21.8827	−0.847303
$668$	0	0
$669$	22.1138	0.854970
$670$	0	0
$671$	10.2414	0.395365
$672$	0	0
$673$	11.8793	0.457913	0.228957	−	0.973437i	$-0.426469\pi$
$673$	11.8793	0.457913	0.228957	+	0.973437i	$0.426469\pi$
$674$	0	0
$675$	−59.2242	−2.27954
$676$	0	0
$677$	8.76891	0.337016	0.168508	−	0.985700i	$-0.446105\pi$
$677$	8.76891	0.337016	0.168508	+	0.985700i	$0.446105\pi$
$678$	0	0
$679$	24.9345	0.956898
$680$	0	0
$681$	−21.6742	−0.830556
$682$	0	0
$683$	−3.38445	−0.129502	−0.0647512	−	0.997901i	$-0.520625\pi$
$683$	−3.38445	−0.129502	−0.0647512	+	0.997901i	$0.520625\pi$
$684$	0	0
$685$	5.32238	0.203358
$686$	0	0
$687$	−58.8544	−2.24543
$688$	0	0
$689$	−1.55691	−0.0593137
$690$	0	0
$691$	11.2242	0.426989	0.213495	−	0.976944i	$-0.431515\pi$
$691$	11.2242	0.426989	0.213495	+	0.976944i	$0.431515\pi$
$692$	0	0
$693$	25.9931	0.987397
$694$	0	0
$695$	12.4802	0.473402
$696$	0	0
$697$	9.43965	0.357552
$698$	0	0
$699$	−37.1579	−1.40544
$700$	0	0
$701$	48.6707	1.83827	0.919134	−	0.393944i	$-0.128890\pi$
$701$	48.6707	1.83827	0.919134	+	0.393944i	$0.128890\pi$
$702$	0	0
$703$	−26.2277	−0.989195
$704$	0	0
$705$	22.1138	0.832855
$706$	0	0
$707$	−10.1173	−0.380499
$708$	0	0
$709$	11.7586	0.441603	0.220802	−	0.975319i	$-0.429133\pi$
$709$	11.7586	0.441603	0.220802	+	0.975319i	$0.429133\pi$
$710$	0	0
$711$	50.8793	1.90812
$712$	0	0
$713$	14.8793	0.557234
$714$	0	0
$715$	1.05863	0.0395906
$716$	0	0
$717$	−54.7846	−2.04597
$718$	0	0
$719$	38.7191	1.44398	0.721989	−	0.691905i	$-0.243228\pi$
$719$	38.7191	1.44398	0.721989	+	0.691905i	$0.243228\pi$
$720$	0	0
$721$	−57.2242	−2.13114
$722$	0	0
$723$	−16.9966	−0.632109
$724$	0	0
$725$	−29.7586	−1.10521
$726$	0	0
$727$	−8.93449	−0.331362	−0.165681	−	0.986179i	$-0.552982\pi$
$727$	−8.93449	−0.331362	−0.165681	+	0.986179i	$0.552982\pi$
$728$	0	0
$729$	47.8984	1.77401
$730$	0	0
$731$	−0.307774	−0.0113834
$732$	0	0
$733$	36.3484	1.34256	0.671279	−	0.741205i	$-0.265745\pi$
$733$	36.3484	1.34256	0.671279	+	0.741205i	$0.265745\pi$
$734$	0	0
$735$	−11.5569	−0.426283
$736$	0	0
$737$	1.58977	0.0585599
$738$	0	0
$739$	−8.44309	−0.310584	−0.155292	−	0.987869i	$-0.549632\pi$
$739$	−8.44309	−0.310584	−0.155292	+	0.987869i	$0.549632\pi$
$740$	0	0
$741$	−12.9966	−0.477441
$742$	0	0
$743$	22.5354	0.826745	0.413372	−	0.910562i	$-0.364351\pi$
$743$	22.5354	0.826745	0.413372	+	0.910562i	$0.364351\pi$
$744$	0	0
$745$	−9.11383	−0.333905
$746$	0	0
$747$	−18.8793	−0.690757
$748$	0	0
$749$	35.5500	1.29897
$750$	0	0
$751$	−2.43621	−0.0888986	−0.0444493	−	0.999012i	$-0.514153\pi$
$751$	−2.43621	−0.0888986	−0.0444493	+	0.999012i	$0.514153\pi$
$752$	0	0
$753$	−6.67762	−0.243346
$754$	0	0
$755$	15.4216	0.561250
$756$	0	0
$757$	43.1329	1.56769	0.783847	−	0.620955i	$-0.213255\pi$
$757$	43.1329	1.56769	0.783847	+	0.620955i	$0.213255\pi$
$758$	0	0
$759$	−10.1173	−0.367234
$760$	0	0
$761$	−30.0191	−1.08819	−0.544096	−	0.839023i	$-0.683127\pi$
$761$	−30.0191	−1.08819	−0.544096	+	0.839023i	$0.683127\pi$
$762$	0	0
$763$	−33.5389	−1.21419
$764$	0	0
$765$	−7.55691	−0.273221
$766$	0	0
$767$	5.67418	0.204883
$768$	0	0
$769$	−35.3484	−1.27469	−0.637347	−	0.770577i	$-0.719968\pi$
$769$	−35.3484	−1.27469	−0.637347	+	0.770577i	$0.719968\pi$
$770$	0	0
$771$	−68.9716	−2.48395
$772$	0	0
$773$	−47.2277	−1.69866	−0.849330	−	0.527862i	$-0.822994\pi$
$773$	−47.2277	−1.69866	−0.849330	+	0.527862i	$0.822994\pi$
$774$	0	0
$775$	20.2345	0.726846
$776$	0	0
$777$	−69.2208	−2.48328
$778$	0	0
$779$	37.7586	1.35284
$780$	0	0
$781$	−5.68106	−0.203284
$782$	0	0
$783$	110.152	3.93651
$784$	0	0
$785$	3.79145	0.135323
$786$	0	0
$787$	−24.7880	−0.883597	−0.441799	−	0.897114i	$-0.645659\pi$
$787$	−24.7880	−0.883597	−0.441799	+	0.897114i	$0.645659\pi$
$788$	0	0
$789$	76.5397	2.72489
$790$	0	0
$791$	−55.6052	−1.97709
$792$	0	0
$793$	−9.67418	−0.343540
$794$	0	0
$795$	−5.05863	−0.179411
$796$	0	0
$797$	−32.1104	−1.13741	−0.568704	−	0.822542i	$-0.692555\pi$
$797$	−32.1104	−1.13741	−0.568704	+	0.822542i	$0.692555\pi$
$798$	0	0
$799$	−6.80605	−0.240781
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	−3.76547	−0.132880
$804$	0	0
$805$	9.55691	0.336837
$806$	0	0
$807$	−20.5535	−0.723517
$808$	0	0
$809$	−4.55004	−0.159971	−0.0799854	−	0.996796i	$-0.525487\pi$
$809$	−4.55004	−0.159971	−0.0799854	+	0.996796i	$0.525487\pi$
$810$	0	0
$811$	−48.3449	−1.69762	−0.848810	−	0.528698i	$-0.822680\pi$
$811$	−48.3449	−1.69762	−0.848810	+	0.528698i	$0.822680\pi$
$812$	0	0
$813$	−11.2345	−0.394012
$814$	0	0
$815$	−4.17246	−0.146155
$816$	0	0
$817$	−1.23109	−0.0430706
$818$	0	0
$819$	−24.5535	−0.857968
$820$	0	0
$821$	−24.3484	−0.849764	−0.424882	−	0.905249i	$-0.639684\pi$
$821$	−24.3484	−0.849764	−0.424882	+	0.905249i	$0.639684\pi$
$822$	0	0
$823$	−28.4431	−0.991464	−0.495732	−	0.868476i	$-0.665100\pi$
$823$	−28.4431	−0.991464	−0.495732	+	0.868476i	$0.665100\pi$
$824$	0	0
$825$	−13.7586	−0.479013
$826$	0	0
$827$	34.6448	1.20472	0.602358	−	0.798226i	$-0.294228\pi$
$827$	34.6448	1.20472	0.602358	+	0.798226i	$0.294228\pi$
$828$	0	0
$829$	−34.2208	−1.18854	−0.594268	−	0.804267i	$-0.702558\pi$
$829$	−34.2208	−1.18854	−0.594268	+	0.804267i	$0.702558\pi$
$830$	0	0
$831$	92.7743	3.21830
$832$	0	0
$833$	3.55691	0.123240
$834$	0	0
$835$	−14.2897	−0.494516
$836$	0	0
$837$	−74.8984	−2.58887
$838$	0	0
$839$	12.9345	0.446548	0.223274	−	0.974756i	$-0.428325\pi$
$839$	12.9345	0.446548	0.223274	+	0.974756i	$0.428325\pi$
$840$	0	0
$841$	26.3484	0.908564
$842$	0	0
$843$	−81.2173	−2.79727
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−32.0992	−1.10294
$848$	0	0
$849$	70.8984	2.43323
$850$	0	0
$851$	19.2863	0.661126
$852$	0	0
$853$	31.6448	1.08350	0.541748	−	0.840541i	$-0.317763\pi$
$853$	31.6448	1.08350	0.541748	+	0.840541i	$0.317763\pi$
$854$	0	0
$855$	−30.2277	−1.03376
$856$	0	0
$857$	−38.8724	−1.32786	−0.663928	−	0.747796i	$-0.731112\pi$
$857$	−38.8724	−1.32786	−0.663928	+	0.747796i	$0.731112\pi$
$858$	0	0
$859$	29.8207	1.01747	0.508734	−	0.860924i	$-0.330114\pi$
$859$	29.8207	1.01747	0.508734	+	0.860924i	$0.330114\pi$
$860$	0	0
$861$	99.6536	3.39618
$862$	0	0
$863$	−24.5163	−0.834545	−0.417273	−	0.908781i	$-0.637014\pi$
$863$	−24.5163	−0.834545	−0.417273	+	0.908781i	$0.637014\pi$
$864$	0	0
$865$	−6.99656	−0.237890
$866$	0	0
$867$	−51.9862	−1.76555
$868$	0	0
$869$	7.12758	0.241787
$870$	0	0
$871$	−1.50172	−0.0508838
$872$	0	0
$873$	57.9931	1.96277
$874$	0	0
$875$	29.2423	0.988569
$876$	0	0
$877$	4.34836	0.146834	0.0734169	−	0.997301i	$-0.476610\pi$
$877$	4.34836	0.146834	0.0734169	+	0.997301i	$0.476610\pi$
$878$	0	0
$879$	89.1441	3.00676
$880$	0	0
$881$	27.6967	0.933126	0.466563	−	0.884488i	$-0.345492\pi$
$881$	27.6967	0.933126	0.466563	+	0.884488i	$0.345492\pi$
$882$	0	0
$883$	−14.0440	−0.472619	−0.236310	−	0.971678i	$-0.575938\pi$
$883$	−14.0440	−0.472619	−0.236310	+	0.971678i	$0.575938\pi$
$884$	0	0
$885$	18.4362	0.619726
$886$	0	0
$887$	−3.66730	−0.123136	−0.0615680	−	0.998103i	$-0.519610\pi$
$887$	−3.66730	−0.123136	−0.0615680	+	0.998103i	$0.519610\pi$
$888$	0	0
$889$	51.7846	1.73680
$890$	0	0
$891$	26.9276	0.902109
$892$	0	0
$893$	−27.2242	−0.911024
$894$	0	0
$895$	−17.7474	−0.593231
$896$	0	0
$897$	9.55691	0.319096
$898$	0	0
$899$	−37.6344	−1.25518
$900$	0	0
$901$	1.55691	0.0518683
$902$	0	0
$903$	−3.24914	−0.108125
$904$	0	0
$905$	8.11727	0.269827
$906$	0	0
$907$	9.39239	0.311869	0.155935	−	0.987767i	$-0.450161\pi$
$907$	9.39239	0.311869	0.155935	+	0.987767i	$0.450161\pi$
$908$	0	0
$909$	−23.5309	−0.780472
$910$	0	0
$911$	−3.64496	−0.120763	−0.0603815	−	0.998175i	$-0.519232\pi$
$911$	−3.64496	−0.120763	−0.0603815	+	0.998175i	$0.519232\pi$
$912$	0	0
$913$	−2.64476	−0.0875289
$914$	0	0
$915$	−31.4328	−1.03914
$916$	0	0
$917$	24.9931	0.825346
$918$	0	0
$919$	12.6516	0.417339	0.208670	−	0.977986i	$-0.433087\pi$
$919$	12.6516	0.417339	0.208670	+	0.977986i	$0.433087\pi$
$920$	0	0
$921$	−86.6570	−2.85544
$922$	0	0
$923$	5.36641	0.176637
$924$	0	0
$925$	26.2277	0.862360
$926$	0	0
$927$	−133.093	−4.37135
$928$	0	0
$929$	37.1070	1.21744	0.608720	−	0.793385i	$-0.291683\pi$
$929$	37.1070	1.21744	0.608720	+	0.793385i	$0.291683\pi$
$930$	0	0
$931$	14.2277	0.466293
$932$	0	0
$933$	22.3518	0.731765
$934$	0	0
$935$	−1.05863	−0.0346210
$936$	0	0
$937$	−50.8724	−1.66193	−0.830965	−	0.556325i	$-0.812211\pi$
$937$	−50.8724	−1.66193	−0.830965	+	0.556325i	$0.812211\pi$
$938$	0	0
$939$	−57.4147	−1.87366
$940$	0	0
$941$	−21.4622	−0.699647	−0.349824	−	0.936816i	$-0.613759\pi$
$941$	−21.4622	−0.699647	−0.349824	+	0.936816i	$0.613759\pi$
$942$	0	0
$943$	−27.7655	−0.904168
$944$	0	0
$945$	−48.1070	−1.56492
$946$	0	0
$947$	61.1690	1.98773	0.993863	−	0.110617i	$-0.0352827\pi$
$947$	61.1690	1.98773	0.993863	+	0.110617i	$0.0352827\pi$
$948$	0	0
$949$	3.55691	0.115462
$950$	0	0
$951$	−3.61899	−0.117354
$952$	0	0
$953$	−13.2345	−0.428709	−0.214354	−	0.976756i	$-0.568765\pi$
$953$	−13.2345	−0.428709	−0.214354	+	0.976756i	$0.568765\pi$
$954$	0	0
$955$	−19.9379	−0.645176
$956$	0	0
$957$	25.5898	0.827200
$958$	0	0
$959$	−17.2932	−0.558425
$960$	0	0
$961$	−5.41023	−0.174524
$962$	0	0
$963$	82.6830	2.66442
$964$	0	0
$965$	−4.11727	−0.132539
$966$	0	0
$967$	48.4441	1.55786	0.778929	−	0.627112i	$-0.215763\pi$
$967$	48.4441	1.55786	0.778929	+	0.627112i	$0.215763\pi$
$968$	0	0
$969$	12.9966	0.417510
$970$	0	0
$971$	−5.69910	−0.182893	−0.0914464	−	0.995810i	$-0.529149\pi$
$971$	−5.69910	−0.182893	−0.0914464	+	0.995810i	$0.529149\pi$
$972$	0	0
$973$	−40.5500	−1.29997
$974$	0	0
$975$	12.9966	0.416223
$976$	0	0
$977$	2.65164	0.0848334	0.0424167	−	0.999100i	$-0.486494\pi$
$977$	2.65164	0.0848334	0.0424167	+	0.999100i	$0.486494\pi$
$978$	0	0
$979$	−2.24141	−0.0716357
$980$	0	0
$981$	−78.0054	−2.49052
$982$	0	0
$983$	27.2491	0.869113	0.434556	−	0.900645i	$-0.356905\pi$
$983$	27.2491	0.869113	0.434556	+	0.900645i	$0.356905\pi$
$984$	0	0
$985$	−10.7914	−0.343844
$986$	0	0
$987$	−71.8509	−2.28704
$988$	0	0
$989$	0.905275	0.0287861
$990$	0	0
$991$	−22.7552	−0.722841	−0.361421	−	0.932403i	$-0.617708\pi$
$991$	−22.7552	−0.722841	−0.361421	+	0.932403i	$0.617708\pi$
$992$	0	0
$993$	−60.1035	−1.90733
$994$	0	0
$995$	−0.615547	−0.0195142
$996$	0	0
$997$	−4.79488	−0.151856	−0.0759278	−	0.997113i	$-0.524192\pi$
$997$	−4.79488	−0.151856	−0.0759278	+	0.997113i	$0.524192\pi$
$998$	0	0
$999$	−97.0820	−3.07154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bg.1.3		3
4.3	odd	2		3328.2.a.be.1.1		3
8.3	odd	2		3328.2.a.bh.1.3		3
8.5	even	2		3328.2.a.bf.1.1		3
16.3	odd	4		104.2.b.c.53.3	✓	6
16.5	even	4		416.2.b.c.209.1		6
16.11	odd	4		104.2.b.c.53.4	yes	6
16.13	even	4		416.2.b.c.209.6		6
48.5	odd	4		3744.2.g.c.1873.4		6
48.11	even	4		936.2.g.c.469.3		6
48.29	odd	4		3744.2.g.c.1873.1		6
48.35	even	4		936.2.g.c.469.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.c.53.3	✓	6	16.3	odd	4
104.2.b.c.53.4	yes	6	16.11	odd	4
416.2.b.c.209.1		6	16.5	even	4
416.2.b.c.209.6		6	16.13	even	4
936.2.g.c.469.3		6	48.11	even	4
936.2.g.c.469.4		6	48.35	even	4
3328.2.a.be.1.1		3	4.3	odd	2
3328.2.a.bf.1.1		3	8.5	even	2
3328.2.a.bg.1.3		3	1.1	even	1	trivial
3328.2.a.bh.1.3		3	8.3	odd	2
3744.2.g.c.1873.1		6	48.29	odd	4
3744.2.g.c.1873.4		6	48.5	odd	4

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24914 q^{3} -1.00000 q^{5} +3.24914 q^{7} +7.55691 q^{9} +1.05863 q^{11} -1.00000 q^{13} -3.24914 q^{15} +1.00000 q^{17} +4.00000 q^{19} +10.5569 q^{21} -2.94137 q^{23} -4.00000 q^{25} +14.8061 q^{27} +7.43965 q^{29} -5.05863 q^{31} +3.43965 q^{33} -3.24914 q^{35} -6.55691 q^{37} -3.24914 q^{39} +9.43965 q^{41} -0.307774 q^{43} -7.55691 q^{45} -6.80605 q^{47} +3.55691 q^{49} +3.24914 q^{51} +1.55691 q^{53} -1.05863 q^{55} +12.9966 q^{57} -5.67418 q^{59} +9.67418 q^{61} +24.5535 q^{63} +1.00000 q^{65} +1.50172 q^{67} -9.55691 q^{69} -5.36641 q^{71} -3.55691 q^{73} -12.9966 q^{75} +3.43965 q^{77} +6.73281 q^{79} +25.4362 q^{81} -2.49828 q^{83} -1.00000 q^{85} +24.1725 q^{87} -2.11727 q^{89} -3.24914 q^{91} -16.4362 q^{93} -4.00000 q^{95} +7.67418 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} + q^{7} + 6 q^{9} + 4 q^{11} - 3 q^{13} - q^{15} + 3 q^{17} + 12 q^{19} + 15 q^{21} - 8 q^{23} - 12 q^{25} + 19 q^{27} + 4 q^{29} - 16 q^{31} - 8 q^{33} - q^{35} - 3 q^{37} - q^{39}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 + q^7 + 6 * q^9 + 4 * q^11 - 3 * q^13 - q^15 + 3 * q^17 + 12 * q^19 + 15 * q^21 - 8 * q^23 - 12 * q^25 + 19 * q^27 + 4 * q^29 - 16 * q^31 - 8 * q^33 - q^35 - 3 * q^37 - q^39 + 10 * q^41 + 7 * q^43 - 6 * q^45 + 5 * q^47 - 6 * q^49 + q^51 - 12 * q^53 - 4 * q^55 + 4 * q^57 - 2 * q^59 + 14 * q^61 + 22 * q^63 + 3 * q^65 + 22 * q^67 - 12 * q^69 - 9 * q^71 + 6 * q^73 - 4 * q^75 - 8 * q^77 + 6 * q^79 + 23 * q^81 + 10 * q^83 - 3 * q^85 + 40 * q^87 - 8 * q^89 - q^91 + 4 * q^93 - 12 * q^95 + 8 * q^97 + 24 * q^99

Introduction

L-functions

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Database

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