LMFDB - Embedded newform 3328.2.a.bi.1.2

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 = [4,0,-2,0,-2,0,0,0,6] 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:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.38651$ 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)

$f(q)$	$=$	$q-1.38651 q^{3} -1.38651 q^{5} +2.44247 q^{7} -1.07759 q^{9} -1.05596 q^{11} +1.00000 q^{13} +1.92241 q^{15} +6.96254 q^{17} -0.408139 q^{19} -3.38651 q^{21} -3.57603 q^{23} -3.07759 q^{25} +5.65362 q^{27} -4.34904 q^{29} +5.74705 q^{31} +1.46410 q^{33} -3.38651 q^{35} -7.49843 q^{37} -1.38651 q^{39} +1.57603 q^{41} -2.27145 q^{43} +1.49409 q^{45} +4.33055 q^{47} -1.03433 q^{49} -9.65362 q^{51} +0.647823 q^{53} +1.46410 q^{55} +0.565889 q^{57} +3.82898 q^{59} +4.80301 q^{61} -2.63199 q^{63} -1.38651 q^{65} -11.4050 q^{67} +4.95819 q^{69} +7.66945 q^{71} -7.68795 q^{73} +4.26711 q^{75} -2.57916 q^{77} +15.8131 q^{79} -4.60602 q^{81} +15.4050 q^{83} -9.65362 q^{85} +6.02999 q^{87} -2.88494 q^{89} +2.44247 q^{91} -7.96833 q^{93} +0.565889 q^{95} -3.30892 q^{97} +1.13790 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * 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
$3$	−1.38651	−0.800501	−0.400251	−	0.916406i	$-0.631077\pi$
$3$	−1.38651	−0.800501	−0.400251	+	0.916406i	$0.631077\pi$
$4$	0	0
$5$	−1.38651	−0.620066	−0.310033	−	0.950726i	$-0.600340\pi$
$5$	−1.38651	−0.620066	−0.310033	+	0.950726i	$0.600340\pi$
$6$	0	0
$7$	2.44247	0.923167	0.461584	−	0.887097i	$-0.347281\pi$
$7$	2.44247	0.923167	0.461584	+	0.887097i	$0.347281\pi$
$8$	0	0
$9$	−1.07759	−0.359198
$10$	0	0
$11$	−1.05596	−0.318385	−0.159192	−	0.987248i	$-0.550889\pi$
$11$	−1.05596	−0.318385	−0.159192	+	0.987248i	$0.550889\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.92241	0.496363
$16$	0	0
$17$	6.96254	1.68866	0.844331	−	0.535821i	$-0.179998\pi$
$17$	6.96254	1.68866	0.844331	+	0.535821i	$0.179998\pi$
$18$	0	0
$19$	−0.408139	−0.0936335	−0.0468168	−	0.998903i	$-0.514908\pi$
$19$	−0.408139	−0.0936335	−0.0468168	+	0.998903i	$0.514908\pi$
$20$	0	0
$21$	−3.38651	−0.738997
$22$	0	0
$23$	−3.57603	−0.745653	−0.372827	−	0.927901i	$-0.621611\pi$
$23$	−3.57603	−0.745653	−0.372827	+	0.927901i	$0.621611\pi$
$24$	0	0
$25$	−3.07759	−0.615519
$26$	0	0
$27$	5.65362	1.08804
$28$	0	0
$29$	−4.34904	−0.807597	−0.403799	−	0.914848i	$-0.632310\pi$
$29$	−4.34904	−0.807597	−0.403799	+	0.914848i	$0.632310\pi$
$30$	0	0
$31$	5.74705	1.03220	0.516100	−	0.856528i	$-0.327383\pi$
$31$	5.74705	1.03220	0.516100	+	0.856528i	$0.327383\pi$
$32$	0	0
$33$	1.46410	0.254867
$34$	0	0
$35$	−3.38651	−0.572425
$36$	0	0
$37$	−7.49843	−1.23273	−0.616367	−	0.787459i	$-0.711396\pi$
$37$	−7.49843	−1.23273	−0.616367	+	0.787459i	$0.711396\pi$
$38$	0	0
$39$	−1.38651	−0.222019
$40$	0	0
$41$	1.57603	0.246134	0.123067	−	0.992398i	$-0.460727\pi$
$41$	1.57603	0.246134	0.123067	+	0.992398i	$0.460727\pi$
$42$	0	0
$43$	−2.27145	−0.346393	−0.173197	−	0.984887i	$-0.555410\pi$
$43$	−2.27145	−0.346393	−0.173197	+	0.984887i	$0.555410\pi$
$44$	0	0
$45$	1.49409	0.222726
$46$	0	0
$47$	4.33055	0.631675	0.315838	−	0.948813i	$-0.397715\pi$
$47$	4.33055	0.631675	0.315838	+	0.948813i	$0.397715\pi$
$48$	0	0
$49$	−1.03433	−0.147762
$50$	0	0
$51$	−9.65362	−1.35178
$52$	0	0
$53$	0.647823	0.0889854	0.0444927	−	0.999010i	$-0.485833\pi$
$53$	0.647823	0.0889854	0.0444927	+	0.999010i	$0.485833\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	0	0
$57$	0.565889	0.0749538
$58$	0	0
$59$	3.82898	0.498491	0.249245	−	0.968440i	$-0.419817\pi$
$59$	3.82898	0.498491	0.249245	+	0.968440i	$0.419817\pi$
$60$	0	0
$61$	4.80301	0.614962	0.307481	−	0.951554i	$-0.400514\pi$
$61$	4.80301	0.614962	0.307481	+	0.951554i	$0.400514\pi$
$62$	0	0
$63$	−2.63199	−0.331599
$64$	0	0
$65$	−1.38651	−0.171975
$66$	0	0
$67$	−11.4050	−1.39334	−0.696672	−	0.717390i	$-0.745337\pi$
$67$	−11.4050	−1.39334	−0.696672	+	0.717390i	$0.745337\pi$
$68$	0	0
$69$	4.95819	0.596896
$70$	0	0
$71$	7.66945	0.910197	0.455098	−	0.890441i	$-0.349604\pi$
$71$	7.66945	0.910197	0.455098	+	0.890441i	$0.349604\pi$
$72$	0	0
$73$	−7.68795	−0.899807	−0.449903	−	0.893077i	$-0.648542\pi$
$73$	−7.68795	−0.899807	−0.449903	+	0.893077i	$0.648542\pi$
$74$	0	0
$75$	4.26711	0.492723
$76$	0	0
$77$	−2.57916	−0.293922
$78$	0	0
$79$	15.8131	1.77912	0.889559	−	0.456820i	$-0.151012\pi$
$79$	15.8131	1.77912	0.889559	+	0.456820i	$0.151012\pi$
$80$	0	0
$81$	−4.60602	−0.511780
$82$	0	0
$83$	15.4050	1.69092	0.845460	−	0.534039i	$-0.179327\pi$
$83$	15.4050	1.69092	0.845460	+	0.534039i	$0.179327\pi$
$84$	0	0
$85$	−9.65362	−1.04708
$86$	0	0
$87$	6.02999	0.646483
$88$	0	0
$89$	−2.88494	−0.305803	−0.152902	−	0.988241i	$-0.548862\pi$
$89$	−2.88494	−0.305803	−0.152902	+	0.988241i	$0.548862\pi$
$90$	0	0
$91$	2.44247	0.256041
$92$	0	0
$93$	−7.96833	−0.826277
$94$	0	0
$95$	0.565889	0.0580589
$96$	0	0
$97$	−3.30892	−0.335970	−0.167985	−	0.985790i	$-0.553726\pi$
$97$	−3.30892	−0.335970	−0.167985	+	0.985790i	$0.553726\pi$
$98$	0	0
$99$	1.13790	0.114363
$100$	0	0
$101$	12.2238	1.21632	0.608159	−	0.793815i	$-0.291908\pi$
$101$	12.2238	1.21632	0.608159	+	0.793815i	$0.291908\pi$
$102$	0	0
$103$	13.7012	1.35002	0.675011	−	0.737808i	$-0.264139\pi$
$103$	13.7012	1.35002	0.675011	+	0.737808i	$0.264139\pi$
$104$	0	0
$105$	4.69543	0.458227
$106$	0	0
$107$	−8.26711	−0.799212	−0.399606	−	0.916687i	$-0.630853\pi$
$107$	−8.26711	−0.799212	−0.399606	+	0.916687i	$0.630853\pi$
$108$	0	0
$109$	−11.7223	−1.12279	−0.561396	−	0.827548i	$-0.689735\pi$
$109$	−11.7223	−1.12279	−0.561396	+	0.827548i	$0.689735\pi$
$110$	0	0
$111$	10.3966	0.986806
$112$	0	0
$113$	−3.58930	−0.337653	−0.168826	−	0.985646i	$-0.553998\pi$
$113$	−3.58930	−0.337653	−0.168826	+	0.985646i	$0.553998\pi$
$114$	0	0
$115$	4.95819	0.462354
$116$	0	0
$117$	−1.07759	−0.0996235
$118$	0	0
$119$	17.0058	1.55892
$120$	0	0
$121$	−9.88494	−0.898631
$122$	0	0
$123$	−2.18518	−0.197031
$124$	0	0
$125$	11.1997	1.00173
$126$	0	0
$127$	6.02999	0.535075	0.267538	−	0.963547i	$-0.413790\pi$
$127$	6.02999	0.535075	0.267538	+	0.963547i	$0.413790\pi$
$128$	0	0
$129$	3.14939	0.277288
$130$	0	0
$131$	15.2429	1.33178	0.665890	−	0.746050i	$-0.268052\pi$
$131$	15.2429	1.33178	0.665890	+	0.746050i	$0.268052\pi$
$132$	0	0
$133$	−0.996868	−0.0864394
$134$	0	0
$135$	−7.83879	−0.674656
$136$	0	0
$137$	20.8233	1.77905	0.889527	−	0.456883i	$-0.151034\pi$
$137$	20.8233	1.77905	0.889527	+	0.456883i	$0.151034\pi$
$138$	0	0
$139$	0.314712	0.0266936	0.0133468	−	0.999911i	$-0.495751\pi$
$139$	0.314712	0.0266936	0.0133468	+	0.999911i	$0.495751\pi$
$140$	0	0
$141$	−6.00434	−0.505657
$142$	0	0
$143$	−1.05596	−0.0883040
$144$	0	0
$145$	6.02999	0.500763
$146$	0	0
$147$	1.43411	0.118284
$148$	0	0
$149$	−10.2238	−0.837570	−0.418785	−	0.908085i	$-0.637544\pi$
$149$	−10.2238	−0.837570	−0.418785	+	0.908085i	$0.637544\pi$
$150$	0	0
$151$	0.741250	0.0603221	0.0301610	−	0.999545i	$-0.490398\pi$
$151$	0.741250	0.0603221	0.0301610	+	0.999545i	$0.490398\pi$
$152$	0	0
$153$	−7.50278	−0.606564
$154$	0	0
$155$	−7.96833	−0.640032
$156$	0	0
$157$	20.6681	1.64949	0.824747	−	0.565502i	$-0.191318\pi$
$157$	20.6681	1.64949	0.824747	+	0.565502i	$0.191318\pi$
$158$	0	0
$159$	−0.898213	−0.0712329
$160$	0	0
$161$	−8.73434	−0.688363
$162$	0	0
$163$	15.1679	1.18804	0.594020	−	0.804450i	$-0.297540\pi$
$163$	15.1679	1.18804	0.594020	+	0.804450i	$0.297540\pi$
$164$	0	0
$165$	−2.02999	−0.158035
$166$	0	0
$167$	21.4050	1.65637	0.828185	−	0.560455i	$-0.189374\pi$
$167$	21.4050	1.65637	0.828185	+	0.560455i	$0.189374\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.439808	0.0336329
$172$	0	0
$173$	−14.6091	−1.11071	−0.555357	−	0.831612i	$-0.687418\pi$
$173$	−14.6091	−1.11071	−0.555357	+	0.831612i	$0.687418\pi$
$174$	0	0
$175$	−7.51693	−0.568227
$176$	0	0
$177$	−5.30892	−0.399043
$178$	0	0
$179$	6.20279	0.463618	0.231809	−	0.972761i	$-0.425536\pi$
$179$	6.20279	0.463618	0.231809	+	0.972761i	$0.425536\pi$
$180$	0	0
$181$	21.6263	1.60747	0.803735	−	0.594988i	$-0.202843\pi$
$181$	21.6263	1.60747	0.803735	+	0.594988i	$0.202843\pi$
$182$	0	0
$183$	−6.65941	−0.492278
$184$	0	0
$185$	10.3966	0.764377
$186$	0	0
$187$	−7.35218	−0.537644
$188$	0	0
$189$	13.8088	1.00444
$190$	0	0
$191$	−12.6162	−0.912873	−0.456436	−	0.889756i	$-0.650874\pi$
$191$	−12.6162	−0.912873	−0.456436	+	0.889756i	$0.650874\pi$
$192$	0	0
$193$	−15.2040	−1.09441	−0.547204	−	0.836999i	$-0.684308\pi$
$193$	−15.2040	−1.09441	−0.547204	+	0.836999i	$0.684308\pi$
$194$	0	0
$195$	1.92241	0.137666
$196$	0	0
$197$	23.7488	1.69203	0.846017	−	0.533156i	$-0.178994\pi$
$197$	23.7488	1.69203	0.846017	+	0.533156i	$0.178994\pi$
$198$	0	0
$199$	−0.0686650	−0.00486753	−0.00243377	−	0.999997i	$-0.500775\pi$
$199$	−0.0686650	−0.00486753	−0.00243377	+	0.999997i	$0.500775\pi$
$200$	0	0
$201$	15.8131	1.11537
$202$	0	0
$203$	−10.6224	−0.745548
$204$	0	0
$205$	−2.18518	−0.152619
$206$	0	0
$207$	3.85350	0.267837
$208$	0	0
$209$	0.430980	0.0298115
$210$	0	0
$211$	−6.61349	−0.455291	−0.227646	−	0.973744i	$-0.573103\pi$
$211$	−6.61349	−0.455291	−0.227646	+	0.973744i	$0.573103\pi$
$212$	0	0
$213$	−10.6338	−0.728614
$214$	0	0
$215$	3.14939	0.214787
$216$	0	0
$217$	14.0370	0.952893
$218$	0	0
$219$	10.6594	0.720297
$220$	0	0
$221$	6.96254	0.468351
$222$	0	0
$223$	−20.2989	−1.35931	−0.679657	−	0.733530i	$-0.737871\pi$
$223$	−20.2989	−1.35931	−0.679657	+	0.733530i	$0.737871\pi$
$224$	0	0
$225$	3.31639	0.221093
$226$	0	0
$227$	7.09922	0.471192	0.235596	−	0.971851i	$-0.424296\pi$
$227$	7.09922	0.471192	0.235596	+	0.971851i	$0.424296\pi$
$228$	0	0
$229$	4.22819	0.279407	0.139703	−	0.990193i	$-0.455385\pi$
$229$	4.22819	0.279407	0.139703	+	0.990193i	$0.455385\pi$
$230$	0	0
$231$	3.57603	0.235285
$232$	0	0
$233$	−8.14626	−0.533679	−0.266840	−	0.963741i	$-0.585979\pi$
$233$	−8.14626	−0.533679	−0.266840	+	0.963741i	$0.585979\pi$
$234$	0	0
$235$	−6.00434	−0.391680
$236$	0	0
$237$	−21.9251	−1.42419
$238$	0	0
$239$	29.3390	1.89778	0.948891	−	0.315603i	$-0.102207\pi$
$239$	29.3390	1.89778	0.948891	+	0.315603i	$0.102207\pi$
$240$	0	0
$241$	3.04881	0.196391	0.0981956	−	0.995167i	$-0.468693\pi$
$241$	3.04881	0.196391	0.0981956	+	0.995167i	$0.468693\pi$
$242$	0	0
$243$	−10.5746	−0.678359
$244$	0	0
$245$	1.43411	0.0916220
$246$	0	0
$247$	−0.408139	−0.0259693
$248$	0	0
$249$	−21.3592	−1.35358
$250$	0	0
$251$	11.4508	0.722770	0.361385	−	0.932417i	$-0.382304\pi$
$251$	11.4508	0.722770	0.361385	+	0.932417i	$0.382304\pi$
$252$	0	0
$253$	3.77615	0.237405
$254$	0	0
$255$	13.3848	0.838191
$256$	0	0
$257$	−13.7788	−0.859499	−0.429749	−	0.902948i	$-0.641398\pi$
$257$	−13.7788	−0.859499	−0.429749	+	0.902948i	$0.641398\pi$
$258$	0	0
$259$	−18.3147	−1.13802
$260$	0	0
$261$	4.68650	0.290087
$262$	0	0
$263$	15.7629	0.971981	0.485990	−	0.873964i	$-0.338459\pi$
$263$	15.7629	0.971981	0.485990	+	0.873964i	$0.338459\pi$
$264$	0	0
$265$	−0.898213	−0.0551768
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	0.193860	0.0118198	0.00590992	−	0.999983i	$-0.498119\pi$
$269$	0.193860	0.0118198	0.00590992	+	0.999983i	$0.498119\pi$
$270$	0	0
$271$	24.3446	1.47883	0.739413	−	0.673252i	$-0.235103\pi$
$271$	24.3446	1.47883	0.739413	+	0.673252i	$0.235103\pi$
$272$	0	0
$273$	−3.38651	−0.204961
$274$	0	0
$275$	3.24982	0.195972
$276$	0	0
$277$	29.9753	1.80104	0.900522	−	0.434811i	$-0.143185\pi$
$277$	29.9753	1.80104	0.900522	+	0.434811i	$0.143185\pi$
$278$	0	0
$279$	−6.19297	−0.370764
$280$	0	0
$281$	33.1890	1.97989	0.989946	−	0.141443i	$-0.0451743\pi$
$281$	33.1890	1.97989	0.989946	+	0.141443i	$0.0451743\pi$
$282$	0	0
$283$	−23.0771	−1.37179	−0.685896	−	0.727699i	$-0.740590\pi$
$283$	−23.0771	−1.37179	−0.685896	+	0.727699i	$0.740590\pi$
$284$	0	0
$285$	−0.784610	−0.0464763
$286$	0	0
$287$	3.84940	0.227223
$288$	0	0
$289$	31.4769	1.85158
$290$	0	0
$291$	4.58784	0.268944
$292$	0	0
$293$	3.34325	0.195315	0.0976573	−	0.995220i	$-0.468865\pi$
$293$	3.34325	0.195315	0.0976573	+	0.995220i	$0.468865\pi$
$294$	0	0
$295$	−5.30892	−0.309097
$296$	0	0
$297$	−5.97001	−0.346415
$298$	0	0
$299$	−3.57603	−0.206807
$300$	0	0
$301$	−5.54796	−0.319779
$302$	0	0
$303$	−16.9485	−0.973665
$304$	0	0
$305$	−6.65941	−0.381317
$306$	0	0
$307$	−17.6854	−1.00936	−0.504679	−	0.863307i	$-0.668389\pi$
$307$	−17.6854	−1.00936	−0.504679	+	0.863307i	$0.668389\pi$
$308$	0	0
$309$	−18.9969	−1.08069
$310$	0	0
$311$	28.9485	1.64152	0.820759	−	0.571275i	$-0.193551\pi$
$311$	28.9485	1.64152	0.820759	+	0.571275i	$0.193551\pi$
$312$	0	0
$313$	9.51170	0.537633	0.268817	−	0.963191i	$-0.413367\pi$
$313$	9.51170	0.537633	0.268817	+	0.963191i	$0.413367\pi$
$314$	0	0
$315$	3.64928	0.205613
$316$	0	0
$317$	20.6727	1.16109	0.580547	−	0.814227i	$-0.302839\pi$
$317$	20.6727	1.16109	0.580547	+	0.814227i	$0.302839\pi$
$318$	0	0
$319$	4.59243	0.257127
$320$	0	0
$321$	11.4624	0.639770
$322$	0	0
$323$	−2.84168	−0.158115
$324$	0	0
$325$	−3.07759	−0.170714
$326$	0	0
$327$	16.2531	0.898796
$328$	0	0
$329$	10.5772	0.583142
$330$	0	0
$331$	21.0110	1.15487	0.577435	−	0.816437i	$-0.304054\pi$
$331$	21.0110	1.15487	0.577435	+	0.816437i	$0.304054\pi$
$332$	0	0
$333$	8.08026	0.442795
$334$	0	0
$335$	15.8131	0.863964
$336$	0	0
$337$	13.5117	0.736029	0.368015	−	0.929820i	$-0.380038\pi$
$337$	13.5117	0.736029	0.368015	+	0.929820i	$0.380038\pi$
$338$	0	0
$339$	4.97659	0.270291
$340$	0	0
$341$	−6.06866	−0.328637
$342$	0	0
$343$	−19.6236	−1.05958
$344$	0	0
$345$	−6.87458	−0.370115
$346$	0	0
$347$	−8.15953	−0.438026	−0.219013	−	0.975722i	$-0.570284\pi$
$347$	−8.15953	−0.438026	−0.219013	+	0.975722i	$0.570284\pi$
$348$	0	0
$349$	17.5417	0.938985	0.469493	−	0.882936i	$-0.344437\pi$
$349$	17.5417	0.938985	0.469493	+	0.882936i	$0.344437\pi$
$350$	0	0
$351$	5.65362	0.301768
$352$	0	0
$353$	1.52576	0.0812080	0.0406040	−	0.999175i	$-0.487072\pi$
$353$	1.52576	0.0812080	0.0406040	+	0.999175i	$0.487072\pi$
$354$	0	0
$355$	−10.6338	−0.564382
$356$	0	0
$357$	−23.5787	−1.24792
$358$	0	0
$359$	−17.4050	−0.918601	−0.459301	−	0.888281i	$-0.651900\pi$
$359$	−17.4050	−0.918601	−0.459301	+	0.888281i	$0.651900\pi$
$360$	0	0
$361$	−18.8334	−0.991233
$362$	0	0
$363$	13.7056	0.719356
$364$	0	0
$365$	10.6594	0.557939
$366$	0	0
$367$	−25.5830	−1.33542	−0.667712	−	0.744420i	$-0.732726\pi$
$367$	−25.5830	−1.33542	−0.667712	+	0.744420i	$0.732726\pi$
$368$	0	0
$369$	−1.69831	−0.0884107
$370$	0	0
$371$	1.58229	0.0821484
$372$	0	0
$373$	−8.77302	−0.454250	−0.227125	−	0.973866i	$-0.572933\pi$
$373$	−8.77302	−0.454250	−0.227125	+	0.973866i	$0.572933\pi$
$374$	0	0
$375$	−15.5284	−0.801884
$376$	0	0
$377$	−4.34904	−0.223987
$378$	0	0
$379$	32.2945	1.65886	0.829429	−	0.558611i	$-0.188666\pi$
$379$	32.2945	1.65886	0.829429	+	0.558611i	$0.188666\pi$
$380$	0	0
$381$	−8.36064	−0.428328
$382$	0	0
$383$	10.0052	0.511243	0.255622	−	0.966777i	$-0.417720\pi$
$383$	10.0052	0.511243	0.255622	+	0.966777i	$0.417720\pi$
$384$	0	0
$385$	3.57603	0.182251
$386$	0	0
$387$	2.44770	0.124424
$388$	0	0
$389$	17.6625	0.895527	0.447763	−	0.894152i	$-0.352221\pi$
$389$	17.6625	0.895527	0.447763	+	0.894152i	$0.352221\pi$
$390$	0	0
$391$	−24.8982	−1.25916
$392$	0	0
$393$	−21.1344	−1.06609
$394$	0	0
$395$	−21.9251	−1.10317
$396$	0	0
$397$	6.81628	0.342099	0.171050	−	0.985262i	$-0.445284\pi$
$397$	6.81628	0.342099	0.171050	+	0.985262i	$0.445284\pi$
$398$	0	0
$399$	1.38217	0.0691949
$400$	0	0
$401$	−17.5530	−0.876557	−0.438279	−	0.898839i	$-0.644412\pi$
$401$	−17.5530	−0.876557	−0.438279	+	0.898839i	$0.644412\pi$
$402$	0	0
$403$	5.74705	0.286281
$404$	0	0
$405$	6.38628	0.317337
$406$	0	0
$407$	7.91807	0.392484
$408$	0	0
$409$	−28.8833	−1.42819	−0.714093	−	0.700051i	$-0.753161\pi$
$409$	−28.8833	−1.42819	−0.714093	+	0.700051i	$0.753161\pi$
$410$	0	0
$411$	−28.8717	−1.42413
$412$	0	0
$413$	9.35218	0.460191
$414$	0	0
$415$	−21.3592	−1.04848
$416$	0	0
$417$	−0.436352	−0.0213682
$418$	0	0
$419$	−22.7687	−1.11232	−0.556161	−	0.831074i	$-0.687726\pi$
$419$	−22.7687	−1.11232	−0.556161	+	0.831074i	$0.687726\pi$
$420$	0	0
$421$	−23.4668	−1.14370	−0.571850	−	0.820358i	$-0.693774\pi$
$421$	−23.4668	−1.14370	−0.571850	+	0.820358i	$0.693774\pi$
$422$	0	0
$423$	−4.66656	−0.226896
$424$	0	0
$425$	−21.4278	−1.03940
$426$	0	0
$427$	11.7312	0.567713
$428$	0	0
$429$	1.46410	0.0706875
$430$	0	0
$431$	−22.3421	−1.07618	−0.538091	−	0.842886i	$-0.680855\pi$
$431$	−22.3421	−1.07618	−0.538091	+	0.842886i	$0.680855\pi$
$432$	0	0
$433$	9.55496	0.459182	0.229591	−	0.973287i	$-0.426261\pi$
$433$	9.55496	0.459182	0.229591	+	0.973287i	$0.426261\pi$
$434$	0	0
$435$	−8.36064	−0.400862
$436$	0	0
$437$	1.45952	0.0698181
$438$	0	0
$439$	0.966878	0.0461466	0.0230733	−	0.999734i	$-0.492655\pi$
$439$	0.966878	0.0461466	0.0230733	+	0.999734i	$0.492655\pi$
$440$	0	0
$441$	1.11459	0.0530757
$442$	0	0
$443$	−27.2164	−1.29309	−0.646545	−	0.762876i	$-0.723787\pi$
$443$	−27.2164	−1.29309	−0.646545	+	0.762876i	$0.723787\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	14.1755	0.670476
$448$	0	0
$449$	7.55931	0.356746	0.178373	−	0.983963i	$-0.442917\pi$
$449$	7.55931	0.356746	0.178373	+	0.983963i	$0.442917\pi$
$450$	0	0
$451$	−1.66423	−0.0783653
$452$	0	0
$453$	−1.02775	−0.0482879
$454$	0	0
$455$	−3.38651	−0.158762
$456$	0	0
$457$	−27.7012	−1.29581	−0.647904	−	0.761722i	$-0.724354\pi$
$457$	−27.7012	−1.29581	−0.647904	+	0.761722i	$0.724354\pi$
$458$	0	0
$459$	39.3635	1.83733
$460$	0	0
$461$	−33.8025	−1.57434	−0.787171	−	0.616735i	$-0.788455\pi$
$461$	−33.8025	−1.57434	−0.787171	+	0.616735i	$0.788455\pi$
$462$	0	0
$463$	−1.97090	−0.0915953	−0.0457977	−	0.998951i	$-0.514583\pi$
$463$	−1.97090	−0.0915953	−0.0457977	+	0.998951i	$0.514583\pi$
$464$	0	0
$465$	11.0482	0.512346
$466$	0	0
$467$	15.2410	0.705269	0.352635	−	0.935761i	$-0.385286\pi$
$467$	15.2410	0.705269	0.352635	+	0.935761i	$0.385286\pi$
$468$	0	0
$469$	−27.8564	−1.28629
$470$	0	0
$471$	−28.6565	−1.32042
$472$	0	0
$473$	2.39857	0.110286
$474$	0	0
$475$	1.25609	0.0576332
$476$	0	0
$477$	−0.698090	−0.0319633
$478$	0	0
$479$	−10.4795	−0.478819	−0.239410	−	0.970919i	$-0.576954\pi$
$479$	−10.4795	−0.478819	−0.239410	+	0.970919i	$0.576954\pi$
$480$	0	0
$481$	−7.49843	−0.341899
$482$	0	0
$483$	12.1102	0.551035
$484$	0	0
$485$	4.58784	0.208323
$486$	0	0
$487$	−42.0714	−1.90644	−0.953219	−	0.302280i	$-0.902252\pi$
$487$	−42.0714	−1.90644	−0.953219	+	0.302280i	$0.902252\pi$
$488$	0	0
$489$	−21.0304	−0.951028
$490$	0	0
$491$	17.8025	0.803417	0.401709	−	0.915768i	$-0.368416\pi$
$491$	17.8025	0.803417	0.401709	+	0.915768i	$0.368416\pi$
$492$	0	0
$493$	−30.2804	−1.36376
$494$	0	0
$495$	−1.57770	−0.0709126
$496$	0	0
$497$	18.7324	0.840264
$498$	0	0
$499$	−40.1534	−1.79751	−0.898756	−	0.438450i	$-0.855528\pi$
$499$	−40.1534	−1.79751	−0.898756	+	0.438450i	$0.855528\pi$
$500$	0	0
$501$	−29.6782	−1.32593
$502$	0	0
$503$	5.88808	0.262536	0.131268	−	0.991347i	$-0.458095\pi$
$503$	5.88808	0.262536	0.131268	+	0.991347i	$0.458095\pi$
$504$	0	0
$505$	−16.9485	−0.754197
$506$	0	0
$507$	−1.38651	−0.0615770
$508$	0	0
$509$	−26.6064	−1.17931	−0.589655	−	0.807655i	$-0.700736\pi$
$509$	−26.6064	−1.17931	−0.589655	+	0.807655i	$0.700736\pi$
$510$	0	0
$511$	−18.7776	−0.830672
$512$	0	0
$513$	−2.30746	−0.101877
$514$	0	0
$515$	−18.9969	−0.837102
$516$	0	0
$517$	−4.57289	−0.201116
$518$	0	0
$519$	20.2557	0.889127
$520$	0	0
$521$	32.6638	1.43103	0.715513	−	0.698600i	$-0.246193\pi$
$521$	32.6638	1.43103	0.715513	+	0.698600i	$0.246193\pi$
$522$	0	0
$523$	2.88494	0.126150	0.0630749	−	0.998009i	$-0.479909\pi$
$523$	2.88494	0.126150	0.0630749	+	0.998009i	$0.479909\pi$
$524$	0	0
$525$	10.4223	0.454866
$526$	0	0
$527$	40.0140	1.74304
$528$	0	0
$529$	−10.2120	−0.444001
$530$	0	0
$531$	−4.12608	−0.179057
$532$	0	0
$533$	1.57603	0.0682653
$534$	0	0
$535$	11.4624	0.495564
$536$	0	0
$537$	−8.60022	−0.371127
$538$	0	0
$539$	1.09222	0.0470451
$540$	0	0
$541$	31.4984	1.35422	0.677112	−	0.735880i	$-0.263231\pi$
$541$	31.4984	1.35422	0.677112	+	0.735880i	$0.263231\pi$
$542$	0	0
$543$	−29.9851	−1.28678
$544$	0	0
$545$	16.2531	0.696204
$546$	0	0
$547$	−4.28817	−0.183349	−0.0916745	−	0.995789i	$-0.529222\pi$
$547$	−4.28817	−0.183349	−0.0916745	+	0.995789i	$0.529222\pi$
$548$	0	0
$549$	−5.17569	−0.220893
$550$	0	0
$551$	1.77502	0.0756182
$552$	0	0
$553$	38.6232	1.64242
$554$	0	0
$555$	−14.4150	−0.611885
$556$	0	0
$557$	−23.9640	−1.01539	−0.507693	−	0.861538i	$-0.669502\pi$
$557$	−23.9640	−1.01539	−0.507693	+	0.861538i	$0.669502\pi$
$558$	0	0
$559$	−2.27145	−0.0960722
$560$	0	0
$561$	10.1939	0.430385
$562$	0	0
$563$	39.1247	1.64891	0.824455	−	0.565927i	$-0.191482\pi$
$563$	39.1247	1.64891	0.824455	+	0.565927i	$0.191482\pi$
$564$	0	0
$565$	4.97659	0.209367
$566$	0	0
$567$	−11.2501	−0.472458
$568$	0	0
$569$	−1.17593	−0.0492975	−0.0246488	−	0.999696i	$-0.507847\pi$
$569$	−1.17593	−0.0492975	−0.0246488	+	0.999696i	$0.507847\pi$
$570$	0	0
$571$	19.0295	0.796361	0.398180	−	0.917307i	$-0.369642\pi$
$571$	19.0295	0.796361	0.398180	+	0.917307i	$0.369642\pi$
$572$	0	0
$573$	17.4924	0.730756
$574$	0	0
$575$	11.0056	0.458963
$576$	0	0
$577$	−26.5245	−1.10423	−0.552115	−	0.833768i	$-0.686179\pi$
$577$	−26.5245	−1.10423	−0.552115	+	0.833768i	$0.686179\pi$
$578$	0	0
$579$	21.0805	0.876075
$580$	0	0
$581$	37.6263	1.56100
$582$	0	0
$583$	−0.684077	−0.0283316
$584$	0	0
$585$	1.49409	0.0617731
$586$	0	0
$587$	46.1418	1.90447	0.952237	−	0.305359i	$-0.0987764\pi$
$587$	46.1418	1.90447	0.952237	+	0.305359i	$0.0987764\pi$
$588$	0	0
$589$	−2.34559	−0.0966485
$590$	0	0
$591$	−32.9280	−1.35448
$592$	0	0
$593$	−37.3575	−1.53409	−0.767044	−	0.641594i	$-0.778273\pi$
$593$	−37.3575	−1.53409	−0.767044	+	0.641594i	$0.778273\pi$
$594$	0	0
$595$	−23.5787	−0.966632
$596$	0	0
$597$	0.0952046	0.00389647
$598$	0	0
$599$	−15.4757	−0.632320	−0.316160	−	0.948706i	$-0.602394\pi$
$599$	−15.4757	−0.632320	−0.316160	+	0.948706i	$0.602394\pi$
$600$	0	0
$601$	44.5226	1.81611	0.908057	−	0.418846i	$-0.137565\pi$
$601$	44.5226	1.81611	0.908057	+	0.418846i	$0.137565\pi$
$602$	0	0
$603$	12.2900	0.500485
$604$	0	0
$605$	13.7056	0.557210
$606$	0	0
$607$	14.2908	0.580047	0.290024	−	0.957019i	$-0.406337\pi$
$607$	14.2908	0.580047	0.290024	+	0.957019i	$0.406337\pi$
$608$	0	0
$609$	14.7281	0.596812
$610$	0	0
$611$	4.33055	0.175195
$612$	0	0
$613$	27.4827	1.11002	0.555008	−	0.831845i	$-0.312715\pi$
$613$	27.4827	1.11002	0.555008	+	0.831845i	$0.312715\pi$
$614$	0	0
$615$	3.02977	0.122172
$616$	0	0
$617$	7.79287	0.313729	0.156865	−	0.987620i	$-0.449861\pi$
$617$	7.79287	0.313729	0.156865	+	0.987620i	$0.449861\pi$
$618$	0	0
$619$	−41.9092	−1.68447	−0.842237	−	0.539107i	$-0.818762\pi$
$619$	−41.9092	−1.68447	−0.842237	+	0.539107i	$0.818762\pi$
$620$	0	0
$621$	−20.2175	−0.811300
$622$	0	0
$623$	−7.04639	−0.282308
$624$	0	0
$625$	−0.140462	−0.00561847
$626$	0	0
$627$	−0.597557	−0.0238641
$628$	0	0
$629$	−52.2081	−2.08167
$630$	0	0
$631$	−24.3905	−0.970972	−0.485486	−	0.874245i	$-0.661357\pi$
$631$	−24.3905	−0.970972	−0.485486	+	0.874245i	$0.661357\pi$
$632$	0	0
$633$	9.16967	0.364461
$634$	0	0
$635$	−8.36064	−0.331782
$636$	0	0
$637$	−1.03433	−0.0409817
$638$	0	0
$639$	−8.26455	−0.326940
$640$	0	0
$641$	−2.12978	−0.0841213	−0.0420606	−	0.999115i	$-0.513392\pi$
$641$	−2.12978	−0.0841213	−0.0420606	+	0.999115i	$0.513392\pi$
$642$	0	0
$643$	30.7475	1.21256	0.606281	−	0.795251i	$-0.292661\pi$
$643$	30.7475	1.21256	0.606281	+	0.795251i	$0.292661\pi$
$644$	0	0
$645$	−4.36666	−0.171937
$646$	0	0
$647$	−11.0471	−0.434308	−0.217154	−	0.976137i	$-0.569677\pi$
$647$	−11.0471	−0.434308	−0.217154	+	0.976137i	$0.569677\pi$
$648$	0	0
$649$	−4.04326	−0.158712
$650$	0	0
$651$	−19.4624	−0.762793
$652$	0	0
$653$	−3.16700	−0.123934	−0.0619672	−	0.998078i	$-0.519737\pi$
$653$	−3.16700	−0.123934	−0.0619672	+	0.998078i	$0.519737\pi$
$654$	0	0
$655$	−21.1344	−0.825791
$656$	0	0
$657$	8.28448	0.323208
$658$	0	0
$659$	−18.5572	−0.722886	−0.361443	−	0.932394i	$-0.617716\pi$
$659$	−18.5572	−0.722886	−0.361443	+	0.932394i	$0.617716\pi$
$660$	0	0
$661$	12.2758	0.477473	0.238737	−	0.971084i	$-0.423267\pi$
$661$	12.2758	0.477473	0.238737	+	0.971084i	$0.423267\pi$
$662$	0	0
$663$	−9.65362	−0.374916
$664$	0	0
$665$	1.38217	0.0535981
$666$	0	0
$667$	15.5523	0.602187
$668$	0	0
$669$	28.1446	1.08813
$670$	0	0
$671$	−5.07180	−0.195795
$672$	0	0
$673$	−33.4484	−1.28934	−0.644670	−	0.764461i	$-0.723005\pi$
$673$	−33.4484	−1.28934	−0.644670	+	0.764461i	$0.723005\pi$
$674$	0	0
$675$	−17.3995	−0.669708
$676$	0	0
$677$	−20.5429	−0.789528	−0.394764	−	0.918783i	$-0.629174\pi$
$677$	−20.5429	−0.789528	−0.394764	+	0.918783i	$0.629174\pi$
$678$	0	0
$679$	−8.08193	−0.310156
$680$	0	0
$681$	−9.84314	−0.377190
$682$	0	0
$683$	0.878823	0.0336272	0.0168136	−	0.999859i	$-0.494648\pi$
$683$	0.878823	0.0336272	0.0168136	+	0.999859i	$0.494648\pi$
$684$	0	0
$685$	−28.8717	−1.10313
$686$	0	0
$687$	−5.86243	−0.223666
$688$	0	0
$689$	0.647823	0.0246801
$690$	0	0
$691$	30.0864	1.14454	0.572270	−	0.820065i	$-0.306063\pi$
$691$	30.0864	1.14454	0.572270	+	0.820065i	$0.306063\pi$
$692$	0	0
$693$	2.77928	0.105576
$694$	0	0
$695$	−0.436352	−0.0165518
$696$	0	0
$697$	10.9731	0.415637
$698$	0	0
$699$	11.2949	0.427211
$700$	0	0
$701$	−21.7312	−0.820777	−0.410388	−	0.911911i	$-0.634607\pi$
$701$	−21.7312	−0.820777	−0.410388	+	0.911911i	$0.634607\pi$
$702$	0	0
$703$	3.06040	0.115425
$704$	0	0
$705$	8.32508	0.313540
$706$	0	0
$707$	29.8564	1.12287
$708$	0	0
$709$	−4.42230	−0.166083	−0.0830414	−	0.996546i	$-0.526463\pi$
$709$	−4.42230	−0.166083	−0.0830414	+	0.996546i	$0.526463\pi$
$710$	0	0
$711$	−17.0401	−0.639055
$712$	0	0
$713$	−20.5516	−0.769663
$714$	0	0
$715$	1.46410	0.0547543
$716$	0	0
$717$	−40.6788	−1.51918
$718$	0	0
$719$	−44.1427	−1.64624	−0.823122	−	0.567865i	$-0.807770\pi$
$719$	−44.1427	−1.64624	−0.823122	+	0.567865i	$0.807770\pi$
$720$	0	0
$721$	33.4648	1.24630
$722$	0	0
$723$	−4.22721	−0.157212
$724$	0	0
$725$	13.3846	0.497091
$726$	0	0
$727$	−14.9519	−0.554536	−0.277268	−	0.960793i	$-0.589429\pi$
$727$	−14.9519	−0.554536	−0.277268	+	0.960793i	$0.589429\pi$
$728$	0	0
$729$	28.4798	1.05481
$730$	0	0
$731$	−15.8151	−0.584941
$732$	0	0
$733$	17.4465	0.644401	0.322200	−	0.946671i	$-0.395578\pi$
$733$	17.4465	0.644401	0.322200	+	0.946671i	$0.395578\pi$
$734$	0	0
$735$	−1.98841	−0.0733436
$736$	0	0
$737$	12.0433	0.443619
$738$	0	0
$739$	24.4406	0.899060	0.449530	−	0.893265i	$-0.351591\pi$
$739$	24.4406	0.899060	0.449530	+	0.893265i	$0.351591\pi$
$740$	0	0
$741$	0.565889	0.0207884
$742$	0	0
$743$	10.5290	0.386271	0.193136	−	0.981172i	$-0.438134\pi$
$743$	10.5290	0.386271	0.193136	+	0.981172i	$0.438134\pi$
$744$	0	0
$745$	14.1755	0.519349
$746$	0	0
$747$	−16.6003	−0.607374
$748$	0	0
$749$	−20.1922	−0.737806
$750$	0	0
$751$	−37.4411	−1.36625	−0.683123	−	0.730303i	$-0.739379\pi$
$751$	−37.4411	−1.36625	−0.683123	+	0.730303i	$0.739379\pi$
$752$	0	0
$753$	−15.8767	−0.578578
$754$	0	0
$755$	−1.02775	−0.0374037
$756$	0	0
$757$	15.1423	0.550358	0.275179	−	0.961393i	$-0.411263\pi$
$757$	15.1423	0.550358	0.275179	+	0.961393i	$0.411263\pi$
$758$	0	0
$759$	−5.23567	−0.190043
$760$	0	0
$761$	15.6828	0.568502	0.284251	−	0.958750i	$-0.408255\pi$
$761$	15.6828	0.568502	0.284251	+	0.958750i	$0.408255\pi$
$762$	0	0
$763$	−28.6313	−1.03652
$764$	0	0
$765$	10.4027	0.376109
$766$	0	0
$767$	3.82898	0.138256
$768$	0	0
$769$	−8.81870	−0.318010	−0.159005	−	0.987278i	$-0.550829\pi$
$769$	−8.81870	−0.318010	−0.159005	+	0.987278i	$0.550829\pi$
$770$	0	0
$771$	19.1045	0.688030
$772$	0	0
$773$	47.6819	1.71500	0.857500	−	0.514484i	$-0.172017\pi$
$773$	47.6819	1.71500	0.857500	+	0.514484i	$0.172017\pi$
$774$	0	0
$775$	−17.6871	−0.635338
$776$	0	0
$777$	25.3935	0.910987
$778$	0	0
$779$	−0.643238	−0.0230464
$780$	0	0
$781$	−8.09866	−0.289793
$782$	0	0
$783$	−24.5878	−0.878698
$784$	0	0
$785$	−28.6565	−1.02279
$786$	0	0
$787$	38.0874	1.35767	0.678835	−	0.734291i	$-0.262485\pi$
$787$	38.0874	1.35767	0.678835	+	0.734291i	$0.262485\pi$
$788$	0	0
$789$	−21.8554	−0.778072
$790$	0	0
$791$	−8.76675	−0.311710
$792$	0	0
$793$	4.80301	0.170560
$794$	0	0
$795$	1.24538	0.0441691
$796$	0	0
$797$	41.6950	1.47691	0.738456	−	0.674302i	$-0.235555\pi$
$797$	41.6950	1.47691	0.738456	+	0.674302i	$0.235555\pi$
$798$	0	0
$799$	30.1516	1.06669
$800$	0	0
$801$	3.10879	0.109844
$802$	0	0
$803$	8.11819	0.286485
$804$	0	0
$805$	12.1102	0.426830
$806$	0	0
$807$	−0.268788	−0.00946180
$808$	0	0
$809$	0.353387	0.0124244	0.00621221	−	0.999981i	$-0.498023\pi$
$809$	0.353387	0.0124244	0.00621221	+	0.999981i	$0.498023\pi$
$810$	0	0
$811$	−22.6257	−0.794497	−0.397248	−	0.917711i	$-0.630035\pi$
$811$	−22.6257	−0.794497	−0.397248	+	0.917711i	$0.630035\pi$
$812$	0	0
$813$	−33.7540	−1.18380
$814$	0	0
$815$	−21.0304	−0.736663
$816$	0	0
$817$	0.927069	0.0324340
$818$	0	0
$819$	−2.63199	−0.0919691
$820$	0	0
$821$	−12.9263	−0.451130	−0.225565	−	0.974228i	$-0.572423\pi$
$821$	−12.9263	−0.451130	−0.225565	+	0.974228i	$0.572423\pi$
$822$	0	0
$823$	46.3631	1.61611	0.808057	−	0.589104i	$-0.200519\pi$
$823$	46.3631	1.61611	0.808057	+	0.589104i	$0.200519\pi$
$824$	0	0
$825$	−4.50591	−0.156876
$826$	0	0
$827$	−26.0458	−0.905702	−0.452851	−	0.891586i	$-0.649593\pi$
$827$	−26.0458	−0.905702	−0.452851	+	0.891586i	$0.649593\pi$
$828$	0	0
$829$	10.9969	0.381937	0.190969	−	0.981596i	$-0.438837\pi$
$829$	10.9969	0.381937	0.190969	+	0.981596i	$0.438837\pi$
$830$	0	0
$831$	−41.5611	−1.44174
$832$	0	0
$833$	−7.20158	−0.249520
$834$	0	0
$835$	−29.6782	−1.02706
$836$	0	0
$837$	32.4916	1.12307
$838$	0	0
$839$	7.82730	0.270228	0.135114	−	0.990830i	$-0.456860\pi$
$839$	7.82730	0.270228	0.135114	+	0.990830i	$0.456860\pi$
$840$	0	0
$841$	−10.0858	−0.347787
$842$	0	0
$843$	−46.0169	−1.58491
$844$	0	0
$845$	−1.38651	−0.0476974
$846$	0	0
$847$	−24.1437	−0.829587
$848$	0	0
$849$	31.9966	1.09812
$850$	0	0
$851$	26.8146	0.919193
$852$	0	0
$853$	−25.6157	−0.877064	−0.438532	−	0.898715i	$-0.644501\pi$
$853$	−25.6157	−0.877064	−0.438532	+	0.898715i	$0.644501\pi$
$854$	0	0
$855$	−0.609797	−0.0208546
$856$	0	0
$857$	−22.4477	−0.766799	−0.383399	−	0.923583i	$-0.625247\pi$
$857$	−22.4477	−0.766799	−0.383399	+	0.923583i	$0.625247\pi$
$858$	0	0
$859$	33.1025	1.12944	0.564722	−	0.825281i	$-0.308983\pi$
$859$	33.1025	1.12944	0.564722	+	0.825281i	$0.308983\pi$
$860$	0	0
$861$	−5.33723	−0.181892
$862$	0	0
$863$	−2.22908	−0.0758787	−0.0379394	−	0.999280i	$-0.512079\pi$
$863$	−2.22908	−0.0758787	−0.0379394	+	0.999280i	$0.512079\pi$
$864$	0	0
$865$	20.2557	0.688715
$866$	0	0
$867$	−43.6430	−1.48219
$868$	0	0
$869$	−16.6981	−0.566444
$870$	0	0
$871$	−11.4050	−0.386444
$872$	0	0
$873$	3.56566	0.120679
$874$	0	0
$875$	27.3548	0.924762
$876$	0	0
$877$	17.3405	0.585548	0.292774	−	0.956182i	$-0.405422\pi$
$877$	17.3405	0.585548	0.292774	+	0.956182i	$0.405422\pi$
$878$	0	0
$879$	−4.63545	−0.156350
$880$	0	0
$881$	−41.7063	−1.40512	−0.702561	−	0.711624i	$-0.747960\pi$
$881$	−41.7063	−1.40512	−0.702561	+	0.711624i	$0.747960\pi$
$882$	0	0
$883$	4.06788	0.136895	0.0684475	−	0.997655i	$-0.478195\pi$
$883$	4.06788	0.136895	0.0684475	+	0.997655i	$0.478195\pi$
$884$	0	0
$885$	7.36086	0.247433
$886$	0	0
$887$	45.8571	1.53973	0.769866	−	0.638205i	$-0.220323\pi$
$887$	45.8571	1.53973	0.769866	+	0.638205i	$0.220323\pi$
$888$	0	0
$889$	14.7281	0.493964
$890$	0	0
$891$	4.86378	0.162943
$892$	0	0
$893$	−1.76747	−0.0591460
$894$	0	0
$895$	−8.60022	−0.287474
$896$	0	0
$897$	4.95819	0.165549
$898$	0	0
$899$	−24.9942	−0.833602
$900$	0	0
$901$	4.51049	0.150266
$902$	0	0
$903$	7.69229	0.255984
$904$	0	0
$905$	−29.9851	−0.996737
$906$	0	0
$907$	−28.6621	−0.951709	−0.475854	−	0.879524i	$-0.657861\pi$
$907$	−28.6621	−0.951709	−0.475854	+	0.879524i	$0.657861\pi$
$908$	0	0
$909$	−13.1723	−0.436899
$910$	0	0
$911$	13.4041	0.444098	0.222049	−	0.975035i	$-0.428725\pi$
$911$	13.4041	0.444098	0.222049	+	0.975035i	$0.428725\pi$
$912$	0	0
$913$	−16.2671	−0.538363
$914$	0	0
$915$	9.23334	0.305245
$916$	0	0
$917$	37.2304	1.22946
$918$	0	0
$919$	−32.7124	−1.07908	−0.539541	−	0.841959i	$-0.681402\pi$
$919$	−32.7124	−1.07908	−0.539541	+	0.841959i	$0.681402\pi$
$920$	0	0
$921$	24.5210	0.807993
$922$	0	0
$923$	7.66945	0.252443
$924$	0	0
$925$	23.0771	0.758771
$926$	0	0
$927$	−14.7643	−0.484924
$928$	0	0
$929$	−0.0167204	−0.000548578 0	−0.000274289	−	1.00000i	$-0.500087\pi$
$929$	−0.0167204	−0.000548578 0	−0.000274289		1.00000i	$0.500087\pi$
$930$	0	0
$931$	0.422152	0.0138355
$932$	0	0
$933$	−40.1373	−1.31404
$934$	0	0
$935$	10.1939	0.333375
$936$	0	0
$937$	−12.5139	−0.408813	−0.204406	−	0.978886i	$-0.565526\pi$
$937$	−12.5139	−0.408813	−0.204406	+	0.978886i	$0.565526\pi$
$938$	0	0
$939$	−13.1881	−0.430376
$940$	0	0
$941$	48.0150	1.56524	0.782622	−	0.622497i	$-0.213882\pi$
$941$	48.0150	1.56524	0.782622	+	0.622497i	$0.213882\pi$
$942$	0	0
$943$	−5.63591	−0.183531
$944$	0	0
$945$	−19.1460	−0.622820
$946$	0	0
$947$	3.81113	0.123845	0.0619225	−	0.998081i	$-0.480277\pi$
$947$	3.81113	0.123845	0.0619225	+	0.998081i	$0.480277\pi$
$948$	0	0
$949$	−7.68795	−0.249561
$950$	0	0
$951$	−28.6629	−0.929457
$952$	0	0
$953$	0.770354	0.0249542	0.0124771	−	0.999922i	$-0.496028\pi$
$953$	0.770354	0.0249542	0.0124771	+	0.999922i	$0.496028\pi$
$954$	0	0
$955$	17.4924	0.566041
$956$	0	0
$957$	−6.36744	−0.205830
$958$	0	0
$959$	50.8603	1.64236
$960$	0	0
$961$	2.02854	0.0654367
$962$	0	0
$963$	8.90858	0.287075
$964$	0	0
$965$	21.0805	0.678605
$966$	0	0
$967$	−2.10043	−0.0675454	−0.0337727	−	0.999430i	$-0.510752\pi$
$967$	−2.10043	−0.0675454	−0.0337727	+	0.999430i	$0.510752\pi$
$968$	0	0
$969$	3.94002	0.126572
$970$	0	0
$971$	−57.1733	−1.83478	−0.917389	−	0.397991i	$-0.869708\pi$
$971$	−57.1733	−1.83478	−0.917389	+	0.397991i	$0.869708\pi$
$972$	0	0
$973$	0.768676	0.0246426
$974$	0	0
$975$	4.26711	0.136657
$976$	0	0
$977$	−33.4216	−1.06925	−0.534626	−	0.845089i	$-0.679547\pi$
$977$	−33.4216	−1.06925	−0.534626	+	0.845089i	$0.679547\pi$
$978$	0	0
$979$	3.04639	0.0973631
$980$	0	0
$981$	12.6318	0.403304
$982$	0	0
$983$	20.3905	0.650357	0.325178	−	0.945653i	$-0.394576\pi$
$983$	20.3905	0.650357	0.325178	+	0.945653i	$0.394576\pi$
$984$	0	0
$985$	−32.9280	−1.04917
$986$	0	0
$987$	−14.6654	−0.466806
$988$	0	0
$989$	8.12277	0.258289
$990$	0	0
$991$	−11.2702	−0.358011	−0.179006	−	0.983848i	$-0.557288\pi$
$991$	−11.2702	−0.358011	−0.179006	+	0.983848i	$0.557288\pi$
$992$	0	0
$993$	−29.1320	−0.924475
$994$	0	0
$995$	0.0952046	0.00301819
$996$	0	0
$997$	27.3662	0.866696	0.433348	−	0.901227i	$-0.357332\pi$
$997$	27.3662	0.866696	0.433348	+	0.901227i	$0.357332\pi$
$998$	0	0
$999$	−42.3933	−1.34126

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bi.1.2		4
4.3	odd	2		3328.2.a.bm.1.3		4
8.3	odd	2		3328.2.a.bj.1.2		4
8.5	even	2		3328.2.a.bn.1.3		4
16.3	odd	4		832.2.b.c.417.6	yes	8
16.5	even	4		832.2.b.d.417.6	yes	8
16.11	odd	4		832.2.b.c.417.3	✓	8
16.13	even	4		832.2.b.d.417.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.3	✓	8	16.11	odd	4
832.2.b.c.417.6	yes	8	16.3	odd	4
832.2.b.d.417.3	yes	8	16.13	even	4
832.2.b.d.417.6	yes	8	16.5	even	4
3328.2.a.bi.1.2		4	1.1	even	1	trivial
3328.2.a.bj.1.2		4	8.3	odd	2
3328.2.a.bm.1.3		4	4.3	odd	2
3328.2.a.bn.1.3		4	8.5	even	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-1.38651 q^{3} -1.38651 q^{5} +2.44247 q^{7} -1.07759 q^{9} -1.05596 q^{11} +1.00000 q^{13} +1.92241 q^{15} +6.96254 q^{17} -0.408139 q^{19} -3.38651 q^{21} -3.57603 q^{23} -3.07759 q^{25} +5.65362 q^{27} -4.34904 q^{29} +5.74705 q^{31} +1.46410 q^{33} -3.38651 q^{35} -7.49843 q^{37} -1.38651 q^{39} +1.57603 q^{41} -2.27145 q^{43} +1.49409 q^{45} +4.33055 q^{47} -1.03433 q^{49} -9.65362 q^{51} +0.647823 q^{53} +1.46410 q^{55} +0.565889 q^{57} +3.82898 q^{59} +4.80301 q^{61} -2.63199 q^{63} -1.38651 q^{65} -11.4050 q^{67} +4.95819 q^{69} +7.66945 q^{71} -7.68795 q^{73} +4.26711 q^{75} -2.57916 q^{77} +15.8131 q^{79} -4.60602 q^{81} +15.4050 q^{83} -9.65362 q^{85} +6.02999 q^{87} -2.88494 q^{89} +2.44247 q^{91} -7.96833 q^{93} +0.565889 q^{95} -3.30892 q^{97} +1.13790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} + 18 q^{15} - 2 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{25} - 14 q^{27} + 16 q^{29} + 10 q^{31} - 8 q^{33} - 10 q^{35} - 14 q^{37} - 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 + 18 * q^15 - 2 * q^17 + 6 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^25 - 14 * q^27 + 16 * q^29 + 10 * q^31 - 8 * q^33 - 10 * q^35 - 14 * q^37 - 2 * q^39 - 20 * q^41 + 14 * q^43 - 20 * q^45 + 20 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^55 + 4 * q^57 + 2 * q^59 + 22 * q^63 - 2 * q^65 - 6 * q^67 - 20 * q^69 + 28 * q^71 + 8 * q^73 - 16 * q^75 - 16 * q^77 + 16 * q^79 + 20 * q^81 + 22 * q^83 - 2 * q^85 + 12 * q^87 + 8 * q^89 + 36 * q^93 + 4 * q^95 - 20 * q^97 - 2 * q^99

Introduction

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