LMFDB - Embedded newform 3328.2.a.bc.1.1

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 = [2,0,4,0,0,0,-6,0,2] 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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+2.00000 q^{3} -3.46410 q^{5} -4.73205 q^{7} +1.00000 q^{9} +1.26795 q^{11} -1.00000 q^{13} -6.92820 q^{15} -1.46410 q^{17} -2.73205 q^{19} -9.46410 q^{21} +4.00000 q^{23} +7.00000 q^{25} -4.00000 q^{27} -2.00000 q^{29} +3.26795 q^{31} +2.53590 q^{33} +16.3923 q^{35} +4.92820 q^{37} -2.00000 q^{39} +4.92820 q^{41} +7.46410 q^{43} -3.46410 q^{45} -3.26795 q^{47} +15.3923 q^{49} -2.92820 q^{51} +10.9282 q^{53} -4.39230 q^{55} -5.46410 q^{57} -0.196152 q^{59} +10.9282 q^{61} -4.73205 q^{63} +3.46410 q^{65} -2.73205 q^{67} +8.00000 q^{69} +2.19615 q^{71} +0.535898 q^{73} +14.0000 q^{75} -6.00000 q^{77} +1.46410 q^{79} -11.0000 q^{81} +6.73205 q^{83} +5.07180 q^{85} -4.00000 q^{87} -17.3205 q^{89} +4.73205 q^{91} +6.53590 q^{93} +9.46410 q^{95} -14.3923 q^{97} +1.26795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 6 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + 4 q^{17} - 2 q^{19} - 12 q^{21} + 8 q^{23} + 14 q^{25} - 8 q^{27} - 4 q^{29} + 10 q^{31} + 12 q^{33} + 12 q^{35} - 4 q^{37} - 4 q^{39} - 4 q^{41}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 6 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + 4 * q^17 - 2 * q^19 - 12 * q^21 + 8 * q^23 + 14 * q^25 - 8 * q^27 - 4 * q^29 + 10 * q^31 + 12 * q^33 + 12 * q^35 - 4 * q^37 - 4 * q^39 - 4 * q^41 + 8 * q^43 - 10 * q^47 + 10 * q^49 + 8 * q^51 + 8 * q^53 + 12 * q^55 - 4 * q^57 + 10 * q^59 + 8 * q^61 - 6 * q^63 - 2 * q^67 + 16 * q^69 - 6 * q^71 + 8 * q^73 + 28 * q^75 - 12 * q^77 - 4 * q^79 - 22 * q^81 + 10 * q^83 + 24 * q^85 - 8 * q^87 + 6 * q^91 + 20 * q^93 + 12 * q^95 - 8 * q^97 + 6 * 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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−6.92820	−1.78885
$16$	0	0
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	0	0
$19$	−2.73205	−0.626775	−0.313388	−	0.949625i	$-0.601464\pi$
$19$	−2.73205	−0.626775	−0.313388	+	0.949625i	$0.601464\pi$
$20$	0	0
$21$	−9.46410	−2.06524
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	3.26795	0.586941	0.293471	−	0.955968i	$-0.405190\pi$
$31$	3.26795	0.586941	0.293471	+	0.955968i	$0.405190\pi$
$32$	0	0
$33$	2.53590	0.441443
$34$	0	0
$35$	16.3923	2.77081
$36$	0	0
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	4.92820	0.769656	0.384828	−	0.922988i	$-0.374261\pi$
$41$	4.92820	0.769656	0.384828	+	0.922988i	$0.374261\pi$
$42$	0	0
$43$	7.46410	1.13826	0.569132	−	0.822246i	$-0.307279\pi$
$43$	7.46410	1.13826	0.569132	+	0.822246i	$0.307279\pi$
$44$	0	0
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−3.26795	−0.476679	−0.238340	−	0.971182i	$-0.576603\pi$
$47$	−3.26795	−0.476679	−0.238340	+	0.971182i	$0.576603\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	−2.92820	−0.410030
$52$	0	0
$53$	10.9282	1.50110	0.750552	−	0.660811i	$-0.229788\pi$
$53$	10.9282	1.50110	0.750552	+	0.660811i	$0.229788\pi$
$54$	0	0
$55$	−4.39230	−0.592258
$56$	0	0
$57$	−5.46410	−0.723738
$58$	0	0
$59$	−0.196152	−0.0255369	−0.0127684	−	0.999918i	$-0.504064\pi$
$59$	−0.196152	−0.0255369	−0.0127684	+	0.999918i	$0.504064\pi$
$60$	0	0
$61$	10.9282	1.39921	0.699607	−	0.714528i	$-0.253359\pi$
$61$	10.9282	1.39921	0.699607	+	0.714528i	$0.253359\pi$
$62$	0	0
$63$	−4.73205	−0.596182
$64$	0	0
$65$	3.46410	0.429669
$66$	0	0
$67$	−2.73205	−0.333773	−0.166887	−	0.985976i	$-0.553371\pi$
$67$	−2.73205	−0.333773	−0.166887	+	0.985976i	$0.553371\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	2.19615	0.260635	0.130318	−	0.991472i	$-0.458400\pi$
$71$	2.19615	0.260635	0.130318	+	0.991472i	$0.458400\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	14.0000	1.61658
$76$	0	0
$77$	−6.00000	−0.683763
$78$	0	0
$79$	1.46410	0.164724	0.0823622	−	0.996602i	$-0.473754\pi$
$79$	1.46410	0.164724	0.0823622	+	0.996602i	$0.473754\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.73205	0.738939	0.369469	−	0.929243i	$-0.379539\pi$
$83$	6.73205	0.738939	0.369469	+	0.929243i	$0.379539\pi$
$84$	0	0
$85$	5.07180	0.550114
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−17.3205	−1.83597	−0.917985	−	0.396615i	$-0.870185\pi$
$89$	−17.3205	−1.83597	−0.917985	+	0.396615i	$0.870185\pi$
$90$	0	0
$91$	4.73205	0.496054
$92$	0	0
$93$	6.53590	0.677741
$94$	0	0
$95$	9.46410	0.970996
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	1.26795	0.127434
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	0	0
$105$	32.7846	3.19945
$106$	0	0
$107$	8.92820	0.863122	0.431561	−	0.902084i	$-0.357963\pi$
$107$	8.92820	0.863122	0.431561	+	0.902084i	$0.357963\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	9.85641	0.935529
$112$	0	0
$113$	9.46410	0.890308	0.445154	−	0.895454i	$-0.353149\pi$
$113$	9.46410	0.890308	0.445154	+	0.895454i	$0.353149\pi$
$114$	0	0
$115$	−13.8564	−1.29212
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	9.85641	0.888722
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	14.9282	1.31436
$130$	0	0
$131$	−7.85641	−0.686417	−0.343209	−	0.939259i	$-0.611514\pi$
$131$	−7.85641	−0.686417	−0.343209	+	0.939259i	$0.611514\pi$
$132$	0	0
$133$	12.9282	1.12102
$134$	0	0
$135$	13.8564	1.19257
$136$	0	0
$137$	0.928203	0.0793018	0.0396509	−	0.999214i	$-0.487375\pi$
$137$	0.928203	0.0793018	0.0396509	+	0.999214i	$0.487375\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−6.53590	−0.550422
$142$	0	0
$143$	−1.26795	−0.106031
$144$	0	0
$145$	6.92820	0.575356
$146$	0	0
$147$	30.7846	2.53907
$148$	0	0
$149$	0.928203	0.0760414	0.0380207	−	0.999277i	$-0.487895\pi$
$149$	0.928203	0.0760414	0.0380207	+	0.999277i	$0.487895\pi$
$150$	0	0
$151$	17.1244	1.39356	0.696780	−	0.717285i	$-0.254615\pi$
$151$	17.1244	1.39356	0.696780	+	0.717285i	$0.254615\pi$
$152$	0	0
$153$	−1.46410	−0.118366
$154$	0	0
$155$	−11.3205	−0.909285
$156$	0	0
$157$	3.07180	0.245156	0.122578	−	0.992459i	$-0.460884\pi$
$157$	3.07180	0.245156	0.122578	+	0.992459i	$0.460884\pi$
$158$	0	0
$159$	21.8564	1.73333
$160$	0	0
$161$	−18.9282	−1.49175
$162$	0	0
$163$	13.2679	1.03923	0.519613	−	0.854402i	$-0.326076\pi$
$163$	13.2679	1.03923	0.519613	+	0.854402i	$0.326076\pi$
$164$	0	0
$165$	−8.78461	−0.683881
$166$	0	0
$167$	11.6603	0.902298	0.451149	−	0.892449i	$-0.351014\pi$
$167$	11.6603	0.902298	0.451149	+	0.892449i	$0.351014\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.73205	−0.208925
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	−33.1244	−2.50397
$176$	0	0
$177$	−0.392305	−0.0294874
$178$	0	0
$179$	10.3923	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$180$	0	0
$181$	4.92820	0.366310	0.183155	−	0.983084i	$-0.441369\pi$
$181$	4.92820	0.366310	0.183155	+	0.983084i	$0.441369\pi$
$182$	0	0
$183$	21.8564	1.61567
$184$	0	0
$185$	−17.0718	−1.25514
$186$	0	0
$187$	−1.85641	−0.135754
$188$	0	0
$189$	18.9282	1.37682
$190$	0	0
$191$	21.4641	1.55309	0.776544	−	0.630063i	$-0.216971\pi$
$191$	21.4641	1.55309	0.776544	+	0.630063i	$0.216971\pi$
$192$	0	0
$193$	−22.3923	−1.61183	−0.805917	−	0.592029i	$-0.798327\pi$
$193$	−22.3923	−1.61183	−0.805917	+	0.592029i	$0.798327\pi$
$194$	0	0
$195$	6.92820	0.496139
$196$	0	0
$197$	−16.9282	−1.20608	−0.603042	−	0.797709i	$-0.706045\pi$
$197$	−16.9282	−1.20608	−0.603042	+	0.797709i	$0.706045\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	0	0
$201$	−5.46410	−0.385408
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	−17.0718	−1.19235
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	19.8564	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$211$	19.8564	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$212$	0	0
$213$	4.39230	0.300956
$214$	0	0
$215$	−25.8564	−1.76339
$216$	0	0
$217$	−15.4641	−1.04977
$218$	0	0
$219$	1.07180	0.0724253
$220$	0	0
$221$	1.46410	0.0984861
$222$	0	0
$223$	−10.1962	−0.682785	−0.341392	−	0.939921i	$-0.610899\pi$
$223$	−10.1962	−0.682785	−0.341392	+	0.939921i	$0.610899\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	27.1244	1.80031	0.900153	−	0.435573i	$-0.143454\pi$
$227$	27.1244	1.80031	0.900153	+	0.435573i	$0.143454\pi$
$228$	0	0
$229$	29.3205	1.93755	0.968777	−	0.247934i	$-0.0797517\pi$
$229$	29.3205	1.93755	0.968777	+	0.247934i	$0.0797517\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−3.07180	−0.201240	−0.100620	−	0.994925i	$-0.532083\pi$
$233$	−3.07180	−0.201240	−0.100620	+	0.994925i	$0.532083\pi$
$234$	0	0
$235$	11.3205	0.738469
$236$	0	0
$237$	2.92820	0.190207
$238$	0	0
$239$	−15.2679	−0.987602	−0.493801	−	0.869575i	$-0.664393\pi$
$239$	−15.2679	−0.987602	−0.493801	+	0.869575i	$0.664393\pi$
$240$	0	0
$241$	−9.60770	−0.618886	−0.309443	−	0.950918i	$-0.600143\pi$
$241$	−9.60770	−0.618886	−0.309443	+	0.950918i	$0.600143\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−53.3205	−3.40652
$246$	0	0
$247$	2.73205	0.173836
$248$	0	0
$249$	13.4641	0.853253
$250$	0	0
$251$	14.3923	0.908434	0.454217	−	0.890891i	$-0.349919\pi$
$251$	14.3923	0.908434	0.454217	+	0.890891i	$0.349919\pi$
$252$	0	0
$253$	5.07180	0.318861
$254$	0	0
$255$	10.1436	0.635216
$256$	0	0
$257$	3.85641	0.240556	0.120278	−	0.992740i	$-0.461621\pi$
$257$	3.85641	0.240556	0.120278	+	0.992740i	$0.461621\pi$
$258$	0	0
$259$	−23.3205	−1.44907
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	7.32051	0.451402	0.225701	−	0.974197i	$-0.427533\pi$
$263$	7.32051	0.451402	0.225701	+	0.974197i	$0.427533\pi$
$264$	0	0
$265$	−37.8564	−2.32550
$266$	0	0
$267$	−34.6410	−2.12000
$268$	0	0
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	0	0
$271$	9.80385	0.595541	0.297771	−	0.954637i	$-0.403757\pi$
$271$	9.80385	0.595541	0.297771	+	0.954637i	$0.403757\pi$
$272$	0	0
$273$	9.46410	0.572793
$274$	0	0
$275$	8.87564	0.535221
$276$	0	0
$277$	−25.8564	−1.55356	−0.776780	−	0.629771i	$-0.783149\pi$
$277$	−25.8564	−1.55356	−0.776780	+	0.629771i	$0.783149\pi$
$278$	0	0
$279$	3.26795	0.195647
$280$	0	0
$281$	25.3205	1.51049	0.755247	−	0.655440i	$-0.227517\pi$
$281$	25.3205	1.51049	0.755247	+	0.655440i	$0.227517\pi$
$282$	0	0
$283$	−12.5359	−0.745182	−0.372591	−	0.927996i	$-0.621531\pi$
$283$	−12.5359	−0.745182	−0.372591	+	0.927996i	$0.621531\pi$
$284$	0	0
$285$	18.9282	1.12121
$286$	0	0
$287$	−23.3205	−1.37657
$288$	0	0
$289$	−14.8564	−0.873906
$290$	0	0
$291$	−28.7846	−1.68738
$292$	0	0
$293$	19.0718	1.11419	0.557093	−	0.830450i	$-0.311917\pi$
$293$	19.0718	1.11419	0.557093	+	0.830450i	$0.311917\pi$
$294$	0	0
$295$	0.679492	0.0395615
$296$	0	0
$297$	−5.07180	−0.294295
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−35.3205	−2.03584
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	−37.8564	−2.16765
$306$	0	0
$307$	−2.73205	−0.155926	−0.0779632	−	0.996956i	$-0.524842\pi$
$307$	−2.73205	−0.155926	−0.0779632	+	0.996956i	$0.524842\pi$
$308$	0	0
$309$	−13.8564	−0.788263
$310$	0	0
$311$	14.9282	0.846501	0.423250	−	0.906013i	$-0.360889\pi$
$311$	14.9282	0.846501	0.423250	+	0.906013i	$0.360889\pi$
$312$	0	0
$313$	20.3923	1.15264	0.576321	−	0.817224i	$-0.304488\pi$
$313$	20.3923	1.15264	0.576321	+	0.817224i	$0.304488\pi$
$314$	0	0
$315$	16.3923	0.923602
$316$	0	0
$317$	3.46410	0.194563	0.0972817	−	0.995257i	$-0.468985\pi$
$317$	3.46410	0.194563	0.0972817	+	0.995257i	$0.468985\pi$
$318$	0	0
$319$	−2.53590	−0.141983
$320$	0	0
$321$	17.8564	0.996647
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−7.00000	−0.388290
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	15.4641	0.852564
$330$	0	0
$331$	27.5167	1.51245	0.756226	−	0.654310i	$-0.227041\pi$
$331$	27.5167	1.51245	0.756226	+	0.654310i	$0.227041\pi$
$332$	0	0
$333$	4.92820	0.270064
$334$	0	0
$335$	9.46410	0.517079
$336$	0	0
$337$	−5.46410	−0.297649	−0.148824	−	0.988864i	$-0.547549\pi$
$337$	−5.46410	−0.297649	−0.148824	+	0.988864i	$0.547549\pi$
$338$	0	0
$339$	18.9282	1.02804
$340$	0	0
$341$	4.14359	0.224388
$342$	0	0
$343$	−39.7128	−2.14429
$344$	0	0
$345$	−27.7128	−1.49201
$346$	0	0
$347$	20.9282	1.12348	0.561742	−	0.827312i	$-0.310131\pi$
$347$	20.9282	1.12348	0.561742	+	0.827312i	$0.310131\pi$
$348$	0	0
$349$	−30.3923	−1.62686	−0.813431	−	0.581661i	$-0.802403\pi$
$349$	−30.3923	−1.62686	−0.813431	+	0.581661i	$0.802403\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−24.9282	−1.32679	−0.663397	−	0.748267i	$-0.730886\pi$
$353$	−24.9282	−1.32679	−0.663397	+	0.748267i	$0.730886\pi$
$354$	0	0
$355$	−7.60770	−0.403775
$356$	0	0
$357$	13.8564	0.733359
$358$	0	0
$359$	−13.5167	−0.713382	−0.356691	−	0.934222i	$-0.616095\pi$
$359$	−13.5167	−0.713382	−0.356691	+	0.934222i	$0.616095\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	−18.7846	−0.985936
$364$	0	0
$365$	−1.85641	−0.0971688
$366$	0	0
$367$	−26.2487	−1.37017	−0.685086	−	0.728462i	$-0.740235\pi$
$367$	−26.2487	−1.37017	−0.685086	+	0.728462i	$0.740235\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	−51.7128	−2.68480
$372$	0	0
$373$	26.7846	1.38685	0.693427	−	0.720527i	$-0.256100\pi$
$373$	26.7846	1.38685	0.693427	+	0.720527i	$0.256100\pi$
$374$	0	0
$375$	−13.8564	−0.715542
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	16.5885	0.852092	0.426046	−	0.904702i	$-0.359906\pi$
$379$	16.5885	0.852092	0.426046	+	0.904702i	$0.359906\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	27.6603	1.41337	0.706686	−	0.707527i	$-0.250189\pi$
$383$	27.6603	1.41337	0.706686	+	0.707527i	$0.250189\pi$
$384$	0	0
$385$	20.7846	1.05928
$386$	0	0
$387$	7.46410	0.379422
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−5.85641	−0.296171
$392$	0	0
$393$	−15.7128	−0.792607
$394$	0	0
$395$	−5.07180	−0.255190
$396$	0	0
$397$	11.4641	0.575367	0.287683	−	0.957726i	$-0.407115\pi$
$397$	11.4641	0.575367	0.287683	+	0.957726i	$0.407115\pi$
$398$	0	0
$399$	25.8564	1.29444
$400$	0	0
$401$	11.4641	0.572490	0.286245	−	0.958156i	$-0.407593\pi$
$401$	11.4641	0.572490	0.286245	+	0.958156i	$0.407593\pi$
$402$	0	0
$403$	−3.26795	−0.162788
$404$	0	0
$405$	38.1051	1.89346
$406$	0	0
$407$	6.24871	0.309737
$408$	0	0
$409$	0.928203	0.0458967	0.0229483	−	0.999737i	$-0.492695\pi$
$409$	0.928203	0.0458967	0.0229483	+	0.999737i	$0.492695\pi$
$410$	0	0
$411$	1.85641	0.0915698
$412$	0	0
$413$	0.928203	0.0456739
$414$	0	0
$415$	−23.3205	−1.14476
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	10.7846	0.526863	0.263431	−	0.964678i	$-0.415146\pi$
$419$	10.7846	0.526863	0.263431	+	0.964678i	$0.415146\pi$
$420$	0	0
$421$	−24.2487	−1.18181	−0.590905	−	0.806741i	$-0.701229\pi$
$421$	−24.2487	−1.18181	−0.590905	+	0.806741i	$0.701229\pi$
$422$	0	0
$423$	−3.26795	−0.158893
$424$	0	0
$425$	−10.2487	−0.497136
$426$	0	0
$427$	−51.7128	−2.50256
$428$	0	0
$429$	−2.53590	−0.122434
$430$	0	0
$431$	−14.8756	−0.716535	−0.358267	−	0.933619i	$-0.616632\pi$
$431$	−14.8756	−0.716535	−0.358267	+	0.933619i	$0.616632\pi$
$432$	0	0
$433$	34.7846	1.67164	0.835821	−	0.549002i	$-0.184992\pi$
$433$	34.7846	1.67164	0.835821	+	0.549002i	$0.184992\pi$
$434$	0	0
$435$	13.8564	0.664364
$436$	0	0
$437$	−10.9282	−0.522767
$438$	0	0
$439$	−10.9282	−0.521575	−0.260787	−	0.965396i	$-0.583982\pi$
$439$	−10.9282	−0.521575	−0.260787	+	0.965396i	$0.583982\pi$
$440$	0	0
$441$	15.3923	0.732967
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	60.0000	2.84427
$446$	0	0
$447$	1.85641	0.0878050
$448$	0	0
$449$	38.3923	1.81184	0.905922	−	0.423444i	$-0.139179\pi$
$449$	38.3923	1.81184	0.905922	+	0.423444i	$0.139179\pi$
$450$	0	0
$451$	6.24871	0.294240
$452$	0	0
$453$	34.2487	1.60914
$454$	0	0
$455$	−16.3923	−0.768483
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	5.85641	0.273354
$460$	0	0
$461$	−7.07180	−0.329366	−0.164683	−	0.986347i	$-0.552660\pi$
$461$	−7.07180	−0.329366	−0.164683	+	0.986347i	$0.552660\pi$
$462$	0	0
$463$	15.6603	0.727794	0.363897	−	0.931439i	$-0.381446\pi$
$463$	15.6603	0.727794	0.363897	+	0.931439i	$0.381446\pi$
$464$	0	0
$465$	−22.6410	−1.04995
$466$	0	0
$467$	−12.2487	−0.566803	−0.283401	−	0.959001i	$-0.591463\pi$
$467$	−12.2487	−0.566803	−0.283401	+	0.959001i	$0.591463\pi$
$468$	0	0
$469$	12.9282	0.596969
$470$	0	0
$471$	6.14359	0.283082
$472$	0	0
$473$	9.46410	0.435160
$474$	0	0
$475$	−19.1244	−0.877486
$476$	0	0
$477$	10.9282	0.500368
$478$	0	0
$479$	−24.7321	−1.13004	−0.565018	−	0.825078i	$-0.691131\pi$
$479$	−24.7321	−1.13004	−0.565018	+	0.825078i	$0.691131\pi$
$480$	0	0
$481$	−4.92820	−0.224707
$482$	0	0
$483$	−37.8564	−1.72253
$484$	0	0
$485$	49.8564	2.26386
$486$	0	0
$487$	16.4449	0.745188	0.372594	−	0.927994i	$-0.378468\pi$
$487$	16.4449	0.745188	0.372594	+	0.927994i	$0.378468\pi$
$488$	0	0
$489$	26.5359	1.19999
$490$	0	0
$491$	−24.2487	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$491$	−24.2487	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$492$	0	0
$493$	2.92820	0.131880
$494$	0	0
$495$	−4.39230	−0.197419
$496$	0	0
$497$	−10.3923	−0.466159
$498$	0	0
$499$	−10.7321	−0.480433	−0.240216	−	0.970719i	$-0.577218\pi$
$499$	−10.7321	−0.480433	−0.240216	+	0.970719i	$0.577218\pi$
$500$	0	0
$501$	23.3205	1.04188
$502$	0	0
$503$	37.4641	1.67044	0.835221	−	0.549915i	$-0.185340\pi$
$503$	37.4641	1.67044	0.835221	+	0.549915i	$0.185340\pi$
$504$	0	0
$505$	41.5692	1.84981
$506$	0	0
$507$	2.00000	0.0888231
$508$	0	0
$509$	6.39230	0.283334	0.141667	−	0.989914i	$-0.454754\pi$
$509$	6.39230	0.283334	0.141667	+	0.989914i	$0.454754\pi$
$510$	0	0
$511$	−2.53590	−0.112182
$512$	0	0
$513$	10.9282	0.482492
$514$	0	0
$515$	24.0000	1.05757
$516$	0	0
$517$	−4.14359	−0.182235
$518$	0	0
$519$	13.8564	0.608229
$520$	0	0
$521$	−37.1769	−1.62875	−0.814375	−	0.580339i	$-0.802920\pi$
$521$	−37.1769	−1.62875	−0.814375	+	0.580339i	$0.802920\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	−66.2487	−2.89133
$526$	0	0
$527$	−4.78461	−0.208421
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−0.196152	−0.00851229
$532$	0	0
$533$	−4.92820	−0.213464
$534$	0	0
$535$	−30.9282	−1.33714
$536$	0	0
$537$	20.7846	0.896922
$538$	0	0
$539$	19.5167	0.840642
$540$	0	0
$541$	16.9282	0.727800	0.363900	−	0.931438i	$-0.381445\pi$
$541$	16.9282	0.727800	0.363900	+	0.931438i	$0.381445\pi$
$542$	0	0
$543$	9.85641	0.422979
$544$	0	0
$545$	−6.92820	−0.296772
$546$	0	0
$547$	15.8564	0.677971	0.338985	−	0.940792i	$-0.389916\pi$
$547$	15.8564	0.677971	0.338985	+	0.940792i	$0.389916\pi$
$548$	0	0
$549$	10.9282	0.466404
$550$	0	0
$551$	5.46410	0.232779
$552$	0	0
$553$	−6.92820	−0.294617
$554$	0	0
$555$	−34.1436	−1.44931
$556$	0	0
$557$	6.78461	0.287473	0.143737	−	0.989616i	$-0.454088\pi$
$557$	6.78461	0.287473	0.143737	+	0.989616i	$0.454088\pi$
$558$	0	0
$559$	−7.46410	−0.315698
$560$	0	0
$561$	−3.71281	−0.156755
$562$	0	0
$563$	−0.535898	−0.0225854	−0.0112927	−	0.999936i	$-0.503595\pi$
$563$	−0.535898	−0.0225854	−0.0112927	+	0.999936i	$0.503595\pi$
$564$	0	0
$565$	−32.7846	−1.37926
$566$	0	0
$567$	52.0526	2.18600
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	−37.3205	−1.56181	−0.780907	−	0.624647i	$-0.785243\pi$
$571$	−37.3205	−1.56181	−0.780907	+	0.624647i	$0.785243\pi$
$572$	0	0
$573$	42.9282	1.79335
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−20.9282	−0.871253	−0.435626	−	0.900128i	$-0.643473\pi$
$577$	−20.9282	−0.871253	−0.435626	+	0.900128i	$0.643473\pi$
$578$	0	0
$579$	−44.7846	−1.86118
$580$	0	0
$581$	−31.8564	−1.32163
$582$	0	0
$583$	13.8564	0.573874
$584$	0	0
$585$	3.46410	0.143223
$586$	0	0
$587$	21.6603	0.894014	0.447007	−	0.894530i	$-0.352490\pi$
$587$	21.6603	0.894014	0.447007	+	0.894530i	$0.352490\pi$
$588$	0	0
$589$	−8.92820	−0.367880
$590$	0	0
$591$	−33.8564	−1.39267
$592$	0	0
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	−49.5692	−2.02873
$598$	0	0
$599$	17.1769	0.701830	0.350915	−	0.936407i	$-0.385871\pi$
$599$	17.1769	0.701830	0.350915	+	0.936407i	$0.385871\pi$
$600$	0	0
$601$	−16.3923	−0.668656	−0.334328	−	0.942457i	$-0.608509\pi$
$601$	−16.3923	−0.668656	−0.334328	+	0.942457i	$0.608509\pi$
$602$	0	0
$603$	−2.73205	−0.111258
$604$	0	0
$605$	32.5359	1.32277
$606$	0	0
$607$	27.3205	1.10891	0.554453	−	0.832215i	$-0.312928\pi$
$607$	27.3205	1.10891	0.554453	+	0.832215i	$0.312928\pi$
$608$	0	0
$609$	18.9282	0.767010
$610$	0	0
$611$	3.26795	0.132207
$612$	0	0
$613$	−44.6410	−1.80303	−0.901517	−	0.432744i	$-0.857545\pi$
$613$	−44.6410	−1.80303	−0.901517	+	0.432744i	$0.857545\pi$
$614$	0	0
$615$	−34.1436	−1.37680
$616$	0	0
$617$	−6.67949	−0.268906	−0.134453	−	0.990920i	$-0.542928\pi$
$617$	−6.67949	−0.268906	−0.134453	+	0.990920i	$0.542928\pi$
$618$	0	0
$619$	−12.5885	−0.505973	−0.252986	−	0.967470i	$-0.581413\pi$
$619$	−12.5885	−0.505973	−0.252986	+	0.967470i	$0.581413\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	81.9615	3.28372
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−6.92820	−0.276686
$628$	0	0
$629$	−7.21539	−0.287696
$630$	0	0
$631$	−3.94744	−0.157145	−0.0785726	−	0.996908i	$-0.525036\pi$
$631$	−3.94744	−0.157145	−0.0785726	+	0.996908i	$0.525036\pi$
$632$	0	0
$633$	39.7128	1.57844
$634$	0	0
$635$	−13.8564	−0.549875
$636$	0	0
$637$	−15.3923	−0.609865
$638$	0	0
$639$	2.19615	0.0868784
$640$	0	0
$641$	26.2487	1.03676	0.518381	−	0.855150i	$-0.326535\pi$
$641$	26.2487	1.03676	0.518381	+	0.855150i	$0.326535\pi$
$642$	0	0
$643$	−9.26795	−0.365492	−0.182746	−	0.983160i	$-0.558499\pi$
$643$	−9.26795	−0.365492	−0.182746	+	0.983160i	$0.558499\pi$
$644$	0	0
$645$	−51.7128	−2.03619
$646$	0	0
$647$	10.1436	0.398786	0.199393	−	0.979920i	$-0.436103\pi$
$647$	10.1436	0.398786	0.199393	+	0.979920i	$0.436103\pi$
$648$	0	0
$649$	−0.248711	−0.00976277
$650$	0	0
$651$	−30.9282	−1.21217
$652$	0	0
$653$	16.9282	0.662452	0.331226	−	0.943551i	$-0.392538\pi$
$653$	16.9282	0.662452	0.331226	+	0.943551i	$0.392538\pi$
$654$	0	0
$655$	27.2154	1.06339
$656$	0	0
$657$	0.535898	0.0209074
$658$	0	0
$659$	14.0000	0.545363	0.272681	−	0.962104i	$-0.412090\pi$
$659$	14.0000	0.545363	0.272681	+	0.962104i	$0.412090\pi$
$660$	0	0
$661$	−17.3205	−0.673690	−0.336845	−	0.941560i	$-0.609360\pi$
$661$	−17.3205	−0.673690	−0.336845	+	0.941560i	$0.609360\pi$
$662$	0	0
$663$	2.92820	0.113722
$664$	0	0
$665$	−44.7846	−1.73667
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	−20.3923	−0.788412
$670$	0	0
$671$	13.8564	0.534921
$672$	0	0
$673$	−29.1769	−1.12469	−0.562344	−	0.826904i	$-0.690100\pi$
$673$	−29.1769	−1.12469	−0.562344	+	0.826904i	$0.690100\pi$
$674$	0	0
$675$	−28.0000	−1.07772
$676$	0	0
$677$	34.9282	1.34240	0.671200	−	0.741276i	$-0.265779\pi$
$677$	34.9282	1.34240	0.671200	+	0.741276i	$0.265779\pi$
$678$	0	0
$679$	68.1051	2.61363
$680$	0	0
$681$	54.2487	2.07882
$682$	0	0
$683$	28.1962	1.07890	0.539448	−	0.842019i	$-0.318633\pi$
$683$	28.1962	1.07890	0.539448	+	0.842019i	$0.318633\pi$
$684$	0	0
$685$	−3.21539	−0.122854
$686$	0	0
$687$	58.6410	2.23729
$688$	0	0
$689$	−10.9282	−0.416331
$690$	0	0
$691$	46.8372	1.78177	0.890885	−	0.454229i	$-0.150085\pi$
$691$	46.8372	1.78177	0.890885	+	0.454229i	$0.150085\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−34.6410	−1.31401
$696$	0	0
$697$	−7.21539	−0.273302
$698$	0	0
$699$	−6.14359	−0.232372
$700$	0	0
$701$	−28.6410	−1.08176	−0.540878	−	0.841101i	$-0.681908\pi$
$701$	−28.6410	−1.08176	−0.540878	+	0.841101i	$0.681908\pi$
$702$	0	0
$703$	−13.4641	−0.507808
$704$	0	0
$705$	22.6410	0.852710
$706$	0	0
$707$	56.7846	2.13561
$708$	0	0
$709$	−5.32051	−0.199816	−0.0999079	−	0.994997i	$-0.531855\pi$
$709$	−5.32051	−0.199816	−0.0999079	+	0.994997i	$0.531855\pi$
$710$	0	0
$711$	1.46410	0.0549081
$712$	0	0
$713$	13.0718	0.489543
$714$	0	0
$715$	4.39230	0.164263
$716$	0	0
$717$	−30.5359	−1.14038
$718$	0	0
$719$	−17.0718	−0.636671	−0.318335	−	0.947978i	$-0.603124\pi$
$719$	−17.0718	−0.636671	−0.318335	+	0.947978i	$0.603124\pi$
$720$	0	0
$721$	32.7846	1.22096
$722$	0	0
$723$	−19.2154	−0.714628
$724$	0	0
$725$	−14.0000	−0.519947
$726$	0	0
$727$	−9.07180	−0.336454	−0.168227	−	0.985748i	$-0.553804\pi$
$727$	−9.07180	−0.336454	−0.168227	+	0.985748i	$0.553804\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−10.9282	−0.404194
$732$	0	0
$733$	−2.67949	−0.0989693	−0.0494846	−	0.998775i	$-0.515758\pi$
$733$	−2.67949	−0.0989693	−0.0494846	+	0.998775i	$0.515758\pi$
$734$	0	0
$735$	−106.641	−3.93351
$736$	0	0
$737$	−3.46410	−0.127602
$738$	0	0
$739$	−53.7654	−1.97779	−0.988896	−	0.148612i	$-0.952519\pi$
$739$	−53.7654	−1.97779	−0.988896	+	0.148612i	$0.952519\pi$
$740$	0	0
$741$	5.46410	0.200729
$742$	0	0
$743$	−21.8038	−0.799906	−0.399953	−	0.916536i	$-0.630973\pi$
$743$	−21.8038	−0.799906	−0.399953	+	0.916536i	$0.630973\pi$
$744$	0	0
$745$	−3.21539	−0.117803
$746$	0	0
$747$	6.73205	0.246313
$748$	0	0
$749$	−42.2487	−1.54373
$750$	0	0
$751$	−14.9282	−0.544738	−0.272369	−	0.962193i	$-0.587807\pi$
$751$	−14.9282	−0.544738	−0.272369	+	0.962193i	$0.587807\pi$
$752$	0	0
$753$	28.7846	1.04897
$754$	0	0
$755$	−59.3205	−2.15889
$756$	0	0
$757$	−0.784610	−0.0285171	−0.0142586	−	0.999898i	$-0.504539\pi$
$757$	−0.784610	−0.0285171	−0.0142586	+	0.999898i	$0.504539\pi$
$758$	0	0
$759$	10.1436	0.368189
$760$	0	0
$761$	22.7846	0.825941	0.412971	−	0.910744i	$-0.364491\pi$
$761$	22.7846	0.825941	0.412971	+	0.910744i	$0.364491\pi$
$762$	0	0
$763$	−9.46410	−0.342623
$764$	0	0
$765$	5.07180	0.183371
$766$	0	0
$767$	0.196152	0.00708265
$768$	0	0
$769$	24.5359	0.884787	0.442394	−	0.896821i	$-0.354129\pi$
$769$	24.5359	0.884787	0.442394	+	0.896821i	$0.354129\pi$
$770$	0	0
$771$	7.71281	0.277770
$772$	0	0
$773$	20.5359	0.738625	0.369312	−	0.929305i	$-0.379593\pi$
$773$	20.5359	0.738625	0.369312	+	0.929305i	$0.379593\pi$
$774$	0	0
$775$	22.8756	0.821717
$776$	0	0
$777$	−46.6410	−1.67324
$778$	0	0
$779$	−13.4641	−0.482402
$780$	0	0
$781$	2.78461	0.0996412
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−10.6410	−0.379794
$786$	0	0
$787$	−4.87564	−0.173798	−0.0868990	−	0.996217i	$-0.527696\pi$
$787$	−4.87564	−0.173798	−0.0868990	+	0.996217i	$0.527696\pi$
$788$	0	0
$789$	14.6410	0.521234
$790$	0	0
$791$	−44.7846	−1.59236
$792$	0	0
$793$	−10.9282	−0.388072
$794$	0	0
$795$	−75.7128	−2.68526
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	4.78461	0.169267
$800$	0	0
$801$	−17.3205	−0.611990
$802$	0	0
$803$	0.679492	0.0239787
$804$	0	0
$805$	65.5692	2.31101
$806$	0	0
$807$	−39.7128	−1.39796
$808$	0	0
$809$	−1.46410	−0.0514751	−0.0257375	−	0.999669i	$-0.508193\pi$
$809$	−1.46410	−0.0514751	−0.0257375	+	0.999669i	$0.508193\pi$
$810$	0	0
$811$	−52.5885	−1.84663	−0.923315	−	0.384043i	$-0.874531\pi$
$811$	−52.5885	−1.84663	−0.923315	+	0.384043i	$0.874531\pi$
$812$	0	0
$813$	19.6077	0.687672
$814$	0	0
$815$	−45.9615	−1.60996
$816$	0	0
$817$	−20.3923	−0.713436
$818$	0	0
$819$	4.73205	0.165351
$820$	0	0
$821$	0.248711	0.00868008	0.00434004	−	0.999991i	$-0.498619\pi$
$821$	0.248711	0.00868008	0.00434004	+	0.999991i	$0.498619\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	17.7513	0.618021
$826$	0	0
$827$	−5.26795	−0.183185	−0.0915923	−	0.995797i	$-0.529196\pi$
$827$	−5.26795	−0.183185	−0.0915923	+	0.995797i	$0.529196\pi$
$828$	0	0
$829$	−12.7846	−0.444028	−0.222014	−	0.975043i	$-0.571263\pi$
$829$	−12.7846	−0.444028	−0.222014	+	0.975043i	$0.571263\pi$
$830$	0	0
$831$	−51.7128	−1.79390
$832$	0	0
$833$	−22.5359	−0.780823
$834$	0	0
$835$	−40.3923	−1.39783
$836$	0	0
$837$	−13.0718	−0.451827
$838$	0	0
$839$	30.9808	1.06957	0.534787	−	0.844987i	$-0.320392\pi$
$839$	30.9808	1.06957	0.534787	+	0.844987i	$0.320392\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	50.6410	1.74417
$844$	0	0
$845$	−3.46410	−0.119169
$846$	0	0
$847$	44.4449	1.52714
$848$	0	0
$849$	−25.0718	−0.860462
$850$	0	0
$851$	19.7128	0.675747
$852$	0	0
$853$	7.17691	0.245733	0.122866	−	0.992423i	$-0.460791\pi$
$853$	7.17691	0.245733	0.122866	+	0.992423i	$0.460791\pi$
$854$	0	0
$855$	9.46410	0.323665
$856$	0	0
$857$	−19.8564	−0.678282	−0.339141	−	0.940736i	$-0.610136\pi$
$857$	−19.8564	−0.678282	−0.339141	+	0.940736i	$0.610136\pi$
$858$	0	0
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	−46.6410	−1.58952
$862$	0	0
$863$	−4.73205	−0.161081	−0.0805404	−	0.996751i	$-0.525665\pi$
$863$	−4.73205	−0.161081	−0.0805404	+	0.996751i	$0.525665\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	−29.7128	−1.00910
$868$	0	0
$869$	1.85641	0.0629743
$870$	0	0
$871$	2.73205	0.0925720
$872$	0	0
$873$	−14.3923	−0.487106
$874$	0	0
$875$	32.7846	1.10832
$876$	0	0
$877$	16.5359	0.558378	0.279189	−	0.960236i	$-0.409934\pi$
$877$	16.5359	0.558378	0.279189	+	0.960236i	$0.409934\pi$
$878$	0	0
$879$	38.1436	1.28655
$880$	0	0
$881$	21.7128	0.731523	0.365762	−	0.930709i	$-0.380809\pi$
$881$	21.7128	0.731523	0.365762	+	0.930709i	$0.380809\pi$
$882$	0	0
$883$	24.5359	0.825699	0.412849	−	0.910799i	$-0.364534\pi$
$883$	24.5359	0.825699	0.412849	+	0.910799i	$0.364534\pi$
$884$	0	0
$885$	1.35898	0.0456817
$886$	0	0
$887$	−18.2487	−0.612732	−0.306366	−	0.951914i	$-0.599113\pi$
$887$	−18.2487	−0.612732	−0.306366	+	0.951914i	$0.599113\pi$
$888$	0	0
$889$	−18.9282	−0.634832
$890$	0	0
$891$	−13.9474	−0.467257
$892$	0	0
$893$	8.92820	0.298771
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−6.53590	−0.217984
$900$	0	0
$901$	−16.0000	−0.533037
$902$	0	0
$903$	−70.6410	−2.35079
$904$	0	0
$905$	−17.0718	−0.567486
$906$	0	0
$907$	54.1051	1.79653	0.898265	−	0.439453i	$-0.144828\pi$
$907$	54.1051	1.79653	0.898265	+	0.439453i	$0.144828\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	31.3205	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$911$	31.3205	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$912$	0	0
$913$	8.53590	0.282497
$914$	0	0
$915$	−75.7128	−2.50299
$916$	0	0
$917$	37.1769	1.22769
$918$	0	0
$919$	−0.679492	−0.0224144	−0.0112072	−	0.999937i	$-0.503567\pi$
$919$	−0.679492	−0.0224144	−0.0112072	+	0.999937i	$0.503567\pi$
$920$	0	0
$921$	−5.46410	−0.180048
$922$	0	0
$923$	−2.19615	−0.0722872
$924$	0	0
$925$	34.4974	1.13427
$926$	0	0
$927$	−6.92820	−0.227552
$928$	0	0
$929$	27.4641	0.901068	0.450534	−	0.892759i	$-0.351234\pi$
$929$	27.4641	0.901068	0.450534	+	0.892759i	$0.351234\pi$
$930$	0	0
$931$	−42.0526	−1.37822
$932$	0	0
$933$	29.8564	0.977455
$934$	0	0
$935$	6.43078	0.210309
$936$	0	0
$937$	−41.7128	−1.36270	−0.681349	−	0.731959i	$-0.738606\pi$
$937$	−41.7128	−1.36270	−0.681349	+	0.731959i	$0.738606\pi$
$938$	0	0
$939$	40.7846	1.33096
$940$	0	0
$941$	16.2487	0.529693	0.264846	−	0.964291i	$-0.414679\pi$
$941$	16.2487	0.529693	0.264846	+	0.964291i	$0.414679\pi$
$942$	0	0
$943$	19.7128	0.641938
$944$	0	0
$945$	−65.5692	−2.13297
$946$	0	0
$947$	10.4449	0.339412	0.169706	−	0.985495i	$-0.445718\pi$
$947$	10.4449	0.339412	0.169706	+	0.985495i	$0.445718\pi$
$948$	0	0
$949$	−0.535898	−0.0173960
$950$	0	0
$951$	6.92820	0.224662
$952$	0	0
$953$	4.14359	0.134224	0.0671121	−	0.997745i	$-0.478621\pi$
$953$	4.14359	0.134224	0.0671121	+	0.997745i	$0.478621\pi$
$954$	0	0
$955$	−74.3538	−2.40603
$956$	0	0
$957$	−5.07180	−0.163948
$958$	0	0
$959$	−4.39230	−0.141835
$960$	0	0
$961$	−20.3205	−0.655500
$962$	0	0
$963$	8.92820	0.287707
$964$	0	0
$965$	77.5692	2.49704
$966$	0	0
$967$	42.9808	1.38217	0.691084	−	0.722774i	$-0.257133\pi$
$967$	42.9808	1.38217	0.691084	+	0.722774i	$0.257133\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	43.8564	1.40742	0.703710	−	0.710488i	$-0.251526\pi$
$971$	43.8564	1.40742	0.703710	+	0.710488i	$0.251526\pi$
$972$	0	0
$973$	−47.3205	−1.51703
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−37.6077	−1.20318	−0.601588	−	0.798806i	$-0.705465\pi$
$977$	−37.6077	−1.20318	−0.601588	+	0.798806i	$0.705465\pi$
$978$	0	0
$979$	−21.9615	−0.701893
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	5.80385	0.185114	0.0925570	−	0.995707i	$-0.470496\pi$
$983$	5.80385	0.185114	0.0925570	+	0.995707i	$0.470496\pi$
$984$	0	0
$985$	58.6410	1.86846
$986$	0	0
$987$	30.9282	0.984456
$988$	0	0
$989$	29.8564	0.949378
$990$	0	0
$991$	43.3205	1.37612	0.688061	−	0.725653i	$-0.258462\pi$
$991$	43.3205	1.37612	0.688061	+	0.725653i	$0.258462\pi$
$992$	0	0
$993$	55.0333	1.74643
$994$	0	0
$995$	85.8564	2.72183
$996$	0	0
$997$	−49.5692	−1.56987	−0.784936	−	0.619576i	$-0.787304\pi$
$997$	−49.5692	−1.56987	−0.784936	+	0.619576i	$0.787304\pi$
$998$	0	0
$999$	−19.7128	−0.623686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bc.1.1		2
4.3	odd	2		3328.2.a.n.1.1		2
8.3	odd	2		3328.2.a.bd.1.2		2
8.5	even	2		3328.2.a.m.1.2		2
16.3	odd	4		104.2.b.b.53.3	✓	4
16.5	even	4		416.2.b.b.209.1		4
16.11	odd	4		104.2.b.b.53.4	yes	4
16.13	even	4		416.2.b.b.209.4		4
48.5	odd	4		3744.2.g.b.1873.4		4
48.11	even	4		936.2.g.b.469.1		4
48.29	odd	4		3744.2.g.b.1873.2		4
48.35	even	4		936.2.g.b.469.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.b.53.3	✓	4	16.3	odd	4
104.2.b.b.53.4	yes	4	16.11	odd	4
416.2.b.b.209.1		4	16.5	even	4
416.2.b.b.209.4		4	16.13	even	4
936.2.g.b.469.1		4	48.11	even	4
936.2.g.b.469.2		4	48.35	even	4
3328.2.a.m.1.2		2	8.5	even	2
3328.2.a.n.1.1		2	4.3	odd	2
3328.2.a.bc.1.1		2	1.1	even	1	trivial
3328.2.a.bd.1.2		2	8.3	odd	2
3744.2.g.b.1873.2		4	48.29	odd	4
3744.2.g.b.1873.4		4	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} -3.46410 q^{5} -4.73205 q^{7} +1.00000 q^{9} +1.26795 q^{11} -1.00000 q^{13} -6.92820 q^{15} -1.46410 q^{17} -2.73205 q^{19} -9.46410 q^{21} +4.00000 q^{23} +7.00000 q^{25} -4.00000 q^{27} -2.00000 q^{29} +3.26795 q^{31} +2.53590 q^{33} +16.3923 q^{35} +4.92820 q^{37} -2.00000 q^{39} +4.92820 q^{41} +7.46410 q^{43} -3.46410 q^{45} -3.26795 q^{47} +15.3923 q^{49} -2.92820 q^{51} +10.9282 q^{53} -4.39230 q^{55} -5.46410 q^{57} -0.196152 q^{59} +10.9282 q^{61} -4.73205 q^{63} +3.46410 q^{65} -2.73205 q^{67} +8.00000 q^{69} +2.19615 q^{71} +0.535898 q^{73} +14.0000 q^{75} -6.00000 q^{77} +1.46410 q^{79} -11.0000 q^{81} +6.73205 q^{83} +5.07180 q^{85} -4.00000 q^{87} -17.3205 q^{89} +4.73205 q^{91} +6.53590 q^{93} +9.46410 q^{95} -14.3923 q^{97} +1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 6 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + 4 q^{17} - 2 q^{19} - 12 q^{21} + 8 q^{23} + 14 q^{25} - 8 q^{27} - 4 q^{29} + 10 q^{31} + 12 q^{33} + 12 q^{35} - 4 q^{37} - 4 q^{39} - 4 q^{41}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 6 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + 4 * q^17 - 2 * q^19 - 12 * q^21 + 8 * q^23 + 14 * q^25 - 8 * q^27 - 4 * q^29 + 10 * q^31 + 12 * q^33 + 12 * q^35 - 4 * q^37 - 4 * q^39 - 4 * q^41 + 8 * q^43 - 10 * q^47 + 10 * q^49 + 8 * q^51 + 8 * q^53 + 12 * q^55 - 4 * q^57 + 10 * q^59 + 8 * q^61 - 6 * q^63 - 2 * q^67 + 16 * q^69 - 6 * q^71 + 8 * q^73 + 28 * q^75 - 12 * q^77 - 4 * q^79 - 22 * q^81 + 10 * q^83 + 24 * q^85 - 8 * q^87 + 6 * q^91 + 20 * q^93 + 12 * q^95 - 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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