LMFDB - Embedded newform 6160.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$49.1878476451$
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 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} +O(q^{10})$	$q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} -1.00000 q^{11} +0.941367 q^{13} +2.24914 q^{15} +6.49828 q^{17} +4.36641 q^{19} +2.24914 q^{21} -6.24914 q^{23} +1.00000 q^{25} -2.11727 q^{27} +8.74742 q^{29} +9.55691 q^{31} -2.24914 q^{33} +1.00000 q^{35} +4.24914 q^{37} +2.11727 q^{39} -2.13187 q^{41} -7.67418 q^{43} +2.05863 q^{45} -11.1138 q^{47} +1.00000 q^{49} +14.6155 q^{51} -4.74742 q^{53} -1.00000 q^{55} +9.82066 q^{57} +1.88273 q^{59} +9.11383 q^{61} +2.05863 q^{63} +0.941367 q^{65} -12.9966 q^{67} -14.0552 q^{69} +14.6155 q^{71} -10.4983 q^{73} +2.24914 q^{75} -1.00000 q^{77} -8.36641 q^{79} -10.9379 q^{81} +8.49828 q^{83} +6.49828 q^{85} +19.6742 q^{87} +12.3810 q^{89} +0.941367 q^{91} +21.4948 q^{93} +4.36641 q^{95} -15.3630 q^{97} -2.05863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9 - 3 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 + 6 * q^19 - 2 * q^21 - 10 * q^23 + 3 * q^25 - 8 * q^27 + 12 * q^31 + 2 * q^33 + 3 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^49 + 28 * q^51 + 12 * q^53 - 3 * q^55 - 8 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 - 4 * q^67 - 8 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 - 3 * q^77 - 18 * q^79 + 3 * q^81 + 8 * q^83 + 2 * q^85 + 44 * q^87 + 18 * q^89 + 2 * q^91 + 12 * q^93 + 6 * q^95 - 4 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.24914	1.29854	0.649271	−	0.760557i	$-0.275074\pi$
$3$	2.24914	1.29854	0.649271	+	0.760557i	$0.275074\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.05863	0.686211
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.941367	0.261088	0.130544	−	0.991443i	$-0.458328\pi$
$13$	0.941367	0.261088	0.130544	+	0.991443i	$0.458328\pi$
$14$	0	0
$15$	2.24914	0.580726
$16$	0	0
$17$	6.49828	1.57606	0.788032	−	0.615634i	$-0.211100\pi$
$17$	6.49828	1.57606	0.788032	+	0.615634i	$0.211100\pi$
$18$	0	0
$19$	4.36641	1.00172	0.500861	−	0.865528i	$-0.333017\pi$
$19$	4.36641	1.00172	0.500861	+	0.865528i	$0.333017\pi$
$20$	0	0
$21$	2.24914	0.490803
$22$	0	0
$23$	−6.24914	−1.30304	−0.651518	−	0.758633i	$-0.725867\pi$
$23$	−6.24914	−1.30304	−0.651518	+	0.758633i	$0.725867\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.11727	−0.407468
$28$	0	0
$29$	8.74742	1.62436	0.812178	−	0.583410i	$-0.198282\pi$
$29$	8.74742	1.62436	0.812178	+	0.583410i	$0.198282\pi$
$30$	0	0
$31$	9.55691	1.71647	0.858236	−	0.513255i	$-0.171560\pi$
$31$	9.55691	1.71647	0.858236	+	0.513255i	$0.171560\pi$
$32$	0	0
$33$	−2.24914	−0.391525
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	4.24914	0.698554	0.349277	−	0.937019i	$-0.386427\pi$
$37$	4.24914	0.698554	0.349277	+	0.937019i	$0.386427\pi$
$38$	0	0
$39$	2.11727	0.339034
$40$	0	0
$41$	−2.13187	−0.332943	−0.166471	−	0.986046i	$-0.553237\pi$
$41$	−2.13187	−0.332943	−0.166471	+	0.986046i	$0.553237\pi$
$42$	0	0
$43$	−7.67418	−1.17030	−0.585151	−	0.810925i	$-0.698965\pi$
$43$	−7.67418	−1.17030	−0.585151	+	0.810925i	$0.698965\pi$
$44$	0	0
$45$	2.05863	0.306883
$46$	0	0
$47$	−11.1138	−1.62112	−0.810559	−	0.585657i	$-0.800837\pi$
$47$	−11.1138	−1.62112	−0.810559	+	0.585657i	$0.800837\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.6155	2.04659
$52$	0	0
$53$	−4.74742	−0.652109	−0.326054	−	0.945351i	$-0.605719\pi$
$53$	−4.74742	−0.652109	−0.326054	+	0.945351i	$0.605719\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	9.82066	1.30078
$58$	0	0
$59$	1.88273	0.245111	0.122556	−	0.992462i	$-0.460891\pi$
$59$	1.88273	0.245111	0.122556	+	0.992462i	$0.460891\pi$
$60$	0	0
$61$	9.11383	1.16691	0.583453	−	0.812147i	$-0.301701\pi$
$61$	9.11383	1.16691	0.583453	+	0.812147i	$0.301701\pi$
$62$	0	0
$63$	2.05863	0.259363
$64$	0	0
$65$	0.941367	0.116762
$66$	0	0
$67$	−12.9966	−1.58778	−0.793891	−	0.608060i	$-0.791948\pi$
$67$	−12.9966	−1.58778	−0.793891	+	0.608060i	$0.791948\pi$
$68$	0	0
$69$	−14.0552	−1.69205
$70$	0	0
$71$	14.6155	1.73455	0.867273	−	0.497833i	$-0.165871\pi$
$71$	14.6155	1.73455	0.867273	+	0.497833i	$0.165871\pi$
$72$	0	0
$73$	−10.4983	−1.22873	−0.614365	−	0.789022i	$-0.710588\pi$
$73$	−10.4983	−1.22873	−0.614365	+	0.789022i	$0.710588\pi$
$74$	0	0
$75$	2.24914	0.259708
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.36641	−0.941294	−0.470647	−	0.882322i	$-0.655980\pi$
$79$	−8.36641	−0.941294	−0.470647	+	0.882322i	$0.655980\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	8.49828	0.932808	0.466404	−	0.884572i	$-0.345549\pi$
$83$	8.49828	0.932808	0.466404	+	0.884572i	$0.345549\pi$
$84$	0	0
$85$	6.49828	0.704838
$86$	0	0
$87$	19.6742	2.10929
$88$	0	0
$89$	12.3810	1.31238	0.656192	−	0.754594i	$-0.272166\pi$
$89$	12.3810	1.31238	0.656192	+	0.754594i	$0.272166\pi$
$90$	0	0
$91$	0.941367	0.0986821
$92$	0	0
$93$	21.4948	2.22891
$94$	0	0
$95$	4.36641	0.447984
$96$	0	0
$97$	−15.3630	−1.55987	−0.779937	−	0.625859i	$-0.784749\pi$
$97$	−15.3630	−1.55987	−0.779937	+	0.625859i	$0.784749\pi$
$98$	0	0
$99$	−2.05863	−0.206900
$100$	0	0
$101$	−16.8793	−1.67955	−0.839776	−	0.542932i	$-0.817314\pi$
$101$	−16.8793	−1.67955	−0.839776	+	0.542932i	$0.817314\pi$
$102$	0	0
$103$	6.61555	0.651849	0.325925	−	0.945396i	$-0.394324\pi$
$103$	6.61555	0.651849	0.325925	+	0.945396i	$0.394324\pi$
$104$	0	0
$105$	2.24914	0.219494
$106$	0	0
$107$	14.5535	1.40694	0.703469	−	0.710726i	$-0.251633\pi$
$107$	14.5535	1.40694	0.703469	+	0.710726i	$0.251633\pi$
$108$	0	0
$109$	−0.249141	−0.0238633	−0.0119317	−	0.999929i	$-0.503798\pi$
$109$	−0.249141	−0.0238633	−0.0119317	+	0.999929i	$0.503798\pi$
$110$	0	0
$111$	9.55691	0.907102
$112$	0	0
$113$	10.9966	1.03447	0.517235	−	0.855844i	$-0.326961\pi$
$113$	10.9966	1.03447	0.517235	+	0.855844i	$0.326961\pi$
$114$	0	0
$115$	−6.24914	−0.582735
$116$	0	0
$117$	1.93793	0.179162
$118$	0	0
$119$	6.49828	0.595696
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.79488	−0.432340
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.88273	0.522008	0.261004	−	0.965338i	$-0.415946\pi$
$127$	5.88273	0.522008	0.261004	+	0.965338i	$0.415946\pi$
$128$	0	0
$129$	−17.2603	−1.51969
$130$	0	0
$131$	1.25258	0.109438	0.0547191	−	0.998502i	$-0.482574\pi$
$131$	1.25258	0.109438	0.0547191	+	0.998502i	$0.482574\pi$
$132$	0	0
$133$	4.36641	0.378615
$134$	0	0
$135$	−2.11727	−0.182225
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−10.9820	−0.931477	−0.465739	−	0.884922i	$-0.654211\pi$
$139$	−10.9820	−0.931477	−0.465739	+	0.884922i	$0.654211\pi$
$140$	0	0
$141$	−24.9966	−2.10509
$142$	0	0
$143$	−0.941367	−0.0787210
$144$	0	0
$145$	8.74742	0.726434
$146$	0	0
$147$	2.24914	0.185506
$148$	0	0
$149$	0.0146079	0.00119673	0.000598363	−	1.00000i	$-0.499810\pi$
$149$	0.0146079	0.00119673	0.000598363		1.00000i	$0.499810\pi$
$150$	0	0
$151$	15.2457	1.24068	0.620339	−	0.784334i	$-0.286995\pi$
$151$	15.2457	1.24068	0.620339	+	0.784334i	$0.286995\pi$
$152$	0	0
$153$	13.3776	1.08151
$154$	0	0
$155$	9.55691	0.767630
$156$	0	0
$157$	−5.50172	−0.439085	−0.219542	−	0.975603i	$-0.570456\pi$
$157$	−5.50172	−0.439085	−0.219542	+	0.975603i	$0.570456\pi$
$158$	0	0
$159$	−10.6776	−0.846790
$160$	0	0
$161$	−6.24914	−0.492501
$162$	0	0
$163$	18.2277	1.42770	0.713850	−	0.700298i	$-0.246950\pi$
$163$	18.2277	1.42770	0.713850	+	0.700298i	$0.246950\pi$
$164$	0	0
$165$	−2.24914	−0.175095
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−12.1138	−0.931833
$170$	0	0
$171$	8.98883	0.687393
$172$	0	0
$173$	0.117266	0.00891559	0.00445780	−	0.999990i	$-0.498581\pi$
$173$	0.117266	0.00891559	0.00445780	+	0.999990i	$0.498581\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	4.23453	0.318287
$178$	0	0
$179$	22.5535	1.68573	0.842863	−	0.538128i	$-0.180868\pi$
$179$	22.5535	1.68573	0.842863	+	0.538128i	$0.180868\pi$
$180$	0	0
$181$	20.8793	1.55195	0.775973	−	0.630766i	$-0.217259\pi$
$181$	20.8793	1.55195	0.775973	+	0.630766i	$0.217259\pi$
$182$	0	0
$183$	20.4983	1.51528
$184$	0	0
$185$	4.24914	0.312403
$186$	0	0
$187$	−6.49828	−0.475201
$188$	0	0
$189$	−2.11727	−0.154008
$190$	0	0
$191$	−3.11383	−0.225309	−0.112654	−	0.993634i	$-0.535935\pi$
$191$	−3.11383	−0.225309	−0.112654	+	0.993634i	$0.535935\pi$
$192$	0	0
$193$	−6.17246	−0.444304	−0.222152	−	0.975012i	$-0.571308\pi$
$193$	−6.17246	−0.444304	−0.222152	+	0.975012i	$0.571308\pi$
$194$	0	0
$195$	2.11727	0.151621
$196$	0	0
$197$	6.73281	0.479693	0.239847	−	0.970811i	$-0.422903\pi$
$197$	6.73281	0.479693	0.239847	+	0.970811i	$0.422903\pi$
$198$	0	0
$199$	13.2311	0.937927	0.468964	−	0.883217i	$-0.344627\pi$
$199$	13.2311	0.937927	0.468964	+	0.883217i	$0.344627\pi$
$200$	0	0
$201$	−29.2311	−2.06180
$202$	0	0
$203$	8.74742	0.613949
$204$	0	0
$205$	−2.13187	−0.148897
$206$	0	0
$207$	−12.8647	−0.894158
$208$	0	0
$209$	−4.36641	−0.302031
$210$	0	0
$211$	−23.1138	−1.59122	−0.795611	−	0.605808i	$-0.792850\pi$
$211$	−23.1138	−1.59122	−0.795611	+	0.605808i	$0.792850\pi$
$212$	0	0
$213$	32.8724	2.25238
$214$	0	0
$215$	−7.67418	−0.523375
$216$	0	0
$217$	9.55691	0.648766
$218$	0	0
$219$	−23.6121	−1.59556
$220$	0	0
$221$	6.11727	0.411492
$222$	0	0
$223$	−10.3810	−0.695164	−0.347582	−	0.937650i	$-0.612997\pi$
$223$	−10.3810	−0.695164	−0.347582	+	0.937650i	$0.612997\pi$
$224$	0	0
$225$	2.05863	0.137242
$226$	0	0
$227$	−16.1104	−1.06928	−0.534642	−	0.845079i	$-0.679554\pi$
$227$	−16.1104	−1.06928	−0.534642	+	0.845079i	$0.679554\pi$
$228$	0	0
$229$	24.2897	1.60511	0.802555	−	0.596578i	$-0.203473\pi$
$229$	24.2897	1.60511	0.802555	+	0.596578i	$0.203473\pi$
$230$	0	0
$231$	−2.24914	−0.147983
$232$	0	0
$233$	6.88617	0.451128	0.225564	−	0.974228i	$-0.427578\pi$
$233$	6.88617	0.451128	0.225564	+	0.974228i	$0.427578\pi$
$234$	0	0
$235$	−11.1138	−0.724986
$236$	0	0
$237$	−18.8172	−1.22231
$238$	0	0
$239$	11.1353	0.720283	0.360142	−	0.932898i	$-0.382728\pi$
$239$	11.1353	0.720283	0.360142	+	0.932898i	$0.382728\pi$
$240$	0	0
$241$	2.36641	0.152434	0.0762168	−	0.997091i	$-0.475716\pi$
$241$	2.36641	0.152434	0.0762168	+	0.997091i	$0.475716\pi$
$242$	0	0
$243$	−18.2491	−1.17068
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	4.11039	0.261538
$248$	0	0
$249$	19.1138	1.21129
$250$	0	0
$251$	−25.1070	−1.58474	−0.792368	−	0.610043i	$-0.791152\pi$
$251$	−25.1070	−1.58474	−0.792368	+	0.610043i	$0.791152\pi$
$252$	0	0
$253$	6.24914	0.392880
$254$	0	0
$255$	14.6155	0.915261
$256$	0	0
$257$	−2.86469	−0.178694	−0.0893472	−	0.996001i	$-0.528478\pi$
$257$	−2.86469	−0.178694	−0.0893472	+	0.996001i	$0.528478\pi$
$258$	0	0
$259$	4.24914	0.264029
$260$	0	0
$261$	18.0077	1.11465
$262$	0	0
$263$	−3.76547	−0.232189	−0.116094	−	0.993238i	$-0.537037\pi$
$263$	−3.76547	−0.232189	−0.116094	+	0.993238i	$0.537037\pi$
$264$	0	0
$265$	−4.74742	−0.291632
$266$	0	0
$267$	27.8466	1.70419
$268$	0	0
$269$	8.94137	0.545165	0.272582	−	0.962132i	$-0.412122\pi$
$269$	8.94137	0.545165	0.272582	+	0.962132i	$0.412122\pi$
$270$	0	0
$271$	−21.4948	−1.30572	−0.652859	−	0.757479i	$-0.726431\pi$
$271$	−21.4948	−1.30572	−0.652859	+	0.757479i	$0.726431\pi$
$272$	0	0
$273$	2.11727	0.128143
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	7.64820	0.459536	0.229768	−	0.973245i	$-0.426203\pi$
$277$	7.64820	0.459536	0.229768	+	0.973245i	$0.426203\pi$
$278$	0	0
$279$	19.6742	1.17786
$280$	0	0
$281$	−28.6155	−1.70706	−0.853530	−	0.521043i	$-0.825543\pi$
$281$	−28.6155	−1.70706	−0.853530	+	0.521043i	$0.825543\pi$
$282$	0	0
$283$	2.87930	0.171156	0.0855782	−	0.996331i	$-0.472726\pi$
$283$	2.87930	0.171156	0.0855782	+	0.996331i	$0.472726\pi$
$284$	0	0
$285$	9.82066	0.581726
$286$	0	0
$287$	−2.13187	−0.125841
$288$	0	0
$289$	25.2277	1.48398
$290$	0	0
$291$	−34.5535	−2.02556
$292$	0	0
$293$	−12.9414	−0.756043	−0.378021	−	0.925797i	$-0.623395\pi$
$293$	−12.9414	−0.756043	−0.378021	+	0.925797i	$0.623395\pi$
$294$	0	0
$295$	1.88273	0.109617
$296$	0	0
$297$	2.11727	0.122856
$298$	0	0
$299$	−5.88273	−0.340207
$300$	0	0
$301$	−7.67418	−0.442332
$302$	0	0
$303$	−37.9639	−2.18097
$304$	0	0
$305$	9.11383	0.521856
$306$	0	0
$307$	−0.498281	−0.0284384	−0.0142192	−	0.999899i	$-0.504526\pi$
$307$	−0.498281	−0.0284384	−0.0142192	+	0.999899i	$0.504526\pi$
$308$	0	0
$309$	14.8793	0.846454
$310$	0	0
$311$	23.4396	1.32914	0.664570	−	0.747226i	$-0.268615\pi$
$311$	23.4396	1.32914	0.664570	+	0.747226i	$0.268615\pi$
$312$	0	0
$313$	9.36984	0.529615	0.264807	−	0.964301i	$-0.414692\pi$
$313$	9.36984	0.529615	0.264807	+	0.964301i	$0.414692\pi$
$314$	0	0
$315$	2.05863	0.115991
$316$	0	0
$317$	−9.97852	−0.560449	−0.280225	−	0.959934i	$-0.590409\pi$
$317$	−9.97852	−0.560449	−0.280225	+	0.959934i	$0.590409\pi$
$318$	0	0
$319$	−8.74742	−0.489762
$320$	0	0
$321$	32.7328	1.82697
$322$	0	0
$323$	28.3741	1.57878
$324$	0	0
$325$	0.941367	0.0522176
$326$	0	0
$327$	−0.560352	−0.0309875
$328$	0	0
$329$	−11.1138	−0.612725
$330$	0	0
$331$	20.4362	1.12328	0.561638	−	0.827383i	$-0.310171\pi$
$331$	20.4362	1.12328	0.561638	+	0.827383i	$0.310171\pi$
$332$	0	0
$333$	8.74742	0.479356
$334$	0	0
$335$	−12.9966	−0.710078
$336$	0	0
$337$	7.88273	0.429400	0.214700	−	0.976680i	$-0.431123\pi$
$337$	7.88273	0.429400	0.214700	+	0.976680i	$0.431123\pi$
$338$	0	0
$339$	24.7328	1.34330
$340$	0	0
$341$	−9.55691	−0.517536
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−14.0552	−0.756706
$346$	0	0
$347$	−13.5569	−0.727773	−0.363887	−	0.931443i	$-0.618550\pi$
$347$	−13.5569	−0.727773	−0.363887	+	0.931443i	$0.618550\pi$
$348$	0	0
$349$	−10.7328	−0.574514	−0.287257	−	0.957853i	$-0.592743\pi$
$349$	−10.7328	−0.574514	−0.287257	+	0.957853i	$0.592743\pi$
$350$	0	0
$351$	−1.99312	−0.106385
$352$	0	0
$353$	−20.8578	−1.11015	−0.555075	−	0.831801i	$-0.687310\pi$
$353$	−20.8578	−1.11015	−0.555075	+	0.831801i	$0.687310\pi$
$354$	0	0
$355$	14.6155	0.775713
$356$	0	0
$357$	14.6155	0.773537
$358$	0	0
$359$	−0.366407	−0.0193382	−0.00966911	−	0.999953i	$-0.503078\pi$
$359$	−0.366407	−0.0193382	−0.00966911	+	0.999953i	$0.503078\pi$
$360$	0	0
$361$	0.0655089	0.00344783
$362$	0	0
$363$	2.24914	0.118049
$364$	0	0
$365$	−10.4983	−0.549505
$366$	0	0
$367$	−20.8432	−1.08801	−0.544003	−	0.839083i	$-0.683092\pi$
$367$	−20.8432	−1.08801	−0.544003	+	0.839083i	$0.683092\pi$
$368$	0	0
$369$	−4.38875	−0.228469
$370$	0	0
$371$	−4.74742	−0.246474
$372$	0	0
$373$	−5.37758	−0.278440	−0.139220	−	0.990261i	$-0.544460\pi$
$373$	−5.37758	−0.278440	−0.139220	+	0.990261i	$0.544460\pi$
$374$	0	0
$375$	2.24914	0.116145
$376$	0	0
$377$	8.23453	0.424100
$378$	0	0
$379$	−3.53093	−0.181372	−0.0906860	−	0.995880i	$-0.528906\pi$
$379$	−3.53093	−0.181372	−0.0906860	+	0.995880i	$0.528906\pi$
$380$	0	0
$381$	13.2311	0.677850
$382$	0	0
$383$	−1.38445	−0.0707422	−0.0353711	−	0.999374i	$-0.511261\pi$
$383$	−1.38445	−0.0707422	−0.0353711	+	0.999374i	$0.511261\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−15.7983	−0.803074
$388$	0	0
$389$	15.7294	0.797511	0.398756	−	0.917057i	$-0.369442\pi$
$389$	15.7294	0.797511	0.398756	+	0.917057i	$0.369442\pi$
$390$	0	0
$391$	−40.6087	−2.05367
$392$	0	0
$393$	2.81722	0.142110
$394$	0	0
$395$	−8.36641	−0.420960
$396$	0	0
$397$	−17.6121	−0.883926	−0.441963	−	0.897033i	$-0.645718\pi$
$397$	−17.6121	−0.883926	−0.441963	+	0.897033i	$0.645718\pi$
$398$	0	0
$399$	9.82066	0.491648
$400$	0	0
$401$	−32.8172	−1.63881	−0.819407	−	0.573212i	$-0.805697\pi$
$401$	−32.8172	−1.63881	−0.819407	+	0.573212i	$0.805697\pi$
$402$	0	0
$403$	8.99656	0.448151
$404$	0	0
$405$	−10.9379	−0.543510
$406$	0	0
$407$	−4.24914	−0.210622
$408$	0	0
$409$	−33.3561	−1.64935	−0.824676	−	0.565605i	$-0.808643\pi$
$409$	−33.3561	−1.64935	−0.824676	+	0.565605i	$0.808643\pi$
$410$	0	0
$411$	22.4914	1.10942
$412$	0	0
$413$	1.88273	0.0926433
$414$	0	0
$415$	8.49828	0.417164
$416$	0	0
$417$	−24.7000	−1.20956
$418$	0	0
$419$	−12.3449	−0.603089	−0.301544	−	0.953452i	$-0.597502\pi$
$419$	−12.3449	−0.603089	−0.301544	+	0.953452i	$0.597502\pi$
$420$	0	0
$421$	−8.87930	−0.432750	−0.216375	−	0.976310i	$-0.569423\pi$
$421$	−8.87930	−0.432750	−0.216375	+	0.976310i	$0.569423\pi$
$422$	0	0
$423$	−22.8793	−1.11243
$424$	0	0
$425$	6.49828	0.315213
$426$	0	0
$427$	9.11383	0.441049
$428$	0	0
$429$	−2.11727	−0.102223
$430$	0	0
$431$	−18.2784	−0.880437	−0.440219	−	0.897891i	$-0.645099\pi$
$431$	−18.2784	−0.880437	−0.440219	+	0.897891i	$0.645099\pi$
$432$	0	0
$433$	6.86469	0.329896	0.164948	−	0.986302i	$-0.447254\pi$
$433$	6.86469	0.329896	0.164948	+	0.986302i	$0.447254\pi$
$434$	0	0
$435$	19.6742	0.943305
$436$	0	0
$437$	−27.2863	−1.30528
$438$	0	0
$439$	11.8466	0.565409	0.282705	−	0.959207i	$-0.408768\pi$
$439$	11.8466	0.565409	0.282705	+	0.959207i	$0.408768\pi$
$440$	0	0
$441$	2.05863	0.0980302
$442$	0	0
$443$	−17.8827	−0.849634	−0.424817	−	0.905279i	$-0.639662\pi$
$443$	−17.8827	−0.849634	−0.424817	+	0.905279i	$0.639662\pi$
$444$	0	0
$445$	12.3810	0.586916
$446$	0	0
$447$	0.0328552	0.00155400
$448$	0	0
$449$	−16.8172	−0.793654	−0.396827	−	0.917893i	$-0.629889\pi$
$449$	−16.8172	−0.793654	−0.396827	+	0.917893i	$0.629889\pi$
$450$	0	0
$451$	2.13187	0.100386
$452$	0	0
$453$	34.2897	1.61107
$454$	0	0
$455$	0.941367	0.0441320
$456$	0	0
$457$	−12.2277	−0.571986	−0.285993	−	0.958232i	$-0.592323\pi$
$457$	−12.2277	−0.571986	−0.285993	+	0.958232i	$0.592323\pi$
$458$	0	0
$459$	−13.7586	−0.642196
$460$	0	0
$461$	−16.6155	−0.773863	−0.386932	−	0.922108i	$-0.626465\pi$
$461$	−16.6155	−0.773863	−0.386932	+	0.922108i	$0.626465\pi$
$462$	0	0
$463$	−18.4837	−0.859009	−0.429505	−	0.903065i	$-0.641312\pi$
$463$	−18.4837	−0.859009	−0.429505	+	0.903065i	$0.641312\pi$
$464$	0	0
$465$	21.4948	0.996799
$466$	0	0
$467$	−24.3956	−1.12889	−0.564447	−	0.825469i	$-0.690911\pi$
$467$	−24.3956	−1.12889	−0.564447	+	0.825469i	$0.690911\pi$
$468$	0	0
$469$	−12.9966	−0.600125
$470$	0	0
$471$	−12.3741	−0.570170
$472$	0	0
$473$	7.67418	0.352859
$474$	0	0
$475$	4.36641	0.200344
$476$	0	0
$477$	−9.77320	−0.447484
$478$	0	0
$479$	27.7655	1.26864	0.634318	−	0.773072i	$-0.281281\pi$
$479$	27.7655	1.26864	0.634318	+	0.773072i	$0.281281\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−14.0552	−0.639534
$484$	0	0
$485$	−15.3630	−0.697596
$486$	0	0
$487$	−10.0958	−0.457484	−0.228742	−	0.973487i	$-0.573461\pi$
$487$	−10.0958	−0.457484	−0.228742	+	0.973487i	$0.573461\pi$
$488$	0	0
$489$	40.9966	1.85393
$490$	0	0
$491$	32.1104	1.44912	0.724561	−	0.689211i	$-0.242043\pi$
$491$	32.1104	1.44912	0.724561	+	0.689211i	$0.242043\pi$
$492$	0	0
$493$	56.8432	2.56009
$494$	0	0
$495$	−2.05863	−0.0925287
$496$	0	0
$497$	14.6155	0.655597
$498$	0	0
$499$	8.79488	0.393713	0.196857	−	0.980432i	$-0.436927\pi$
$499$	8.79488	0.393713	0.196857	+	0.980432i	$0.436927\pi$
$500$	0	0
$501$	17.9931	0.803874
$502$	0	0
$503$	15.0034	0.668970	0.334485	−	0.942401i	$-0.391438\pi$
$503$	15.0034	0.668970	0.334485	+	0.942401i	$0.391438\pi$
$504$	0	0
$505$	−16.8793	−0.751119
$506$	0	0
$507$	−27.2457	−1.21002
$508$	0	0
$509$	21.8759	0.969630	0.484815	−	0.874617i	$-0.338887\pi$
$509$	21.8759	0.969630	0.484815	+	0.874617i	$0.338887\pi$
$510$	0	0
$511$	−10.4983	−0.464417
$512$	0	0
$513$	−9.24485	−0.408170
$514$	0	0
$515$	6.61555	0.291516
$516$	0	0
$517$	11.1138	0.488786
$518$	0	0
$519$	0.263748	0.0115773
$520$	0	0
$521$	14.0292	0.614631	0.307316	−	0.951608i	$-0.400569\pi$
$521$	14.0292	0.614631	0.307316	+	0.951608i	$0.400569\pi$
$522$	0	0
$523$	−1.14992	−0.0502825	−0.0251412	−	0.999684i	$-0.508004\pi$
$523$	−1.14992	−0.0502825	−0.0251412	+	0.999684i	$0.508004\pi$
$524$	0	0
$525$	2.24914	0.0981605
$526$	0	0
$527$	62.1035	2.70527
$528$	0	0
$529$	16.0518	0.697902
$530$	0	0
$531$	3.87586	0.168198
$532$	0	0
$533$	−2.00688	−0.0869274
$534$	0	0
$535$	14.5535	0.629202
$536$	0	0
$537$	50.7259	2.18899
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−6.47680	−0.278459	−0.139230	−	0.990260i	$-0.544463\pi$
$541$	−6.47680	−0.278459	−0.139230	+	0.990260i	$0.544463\pi$
$542$	0	0
$543$	46.9605	2.01527
$544$	0	0
$545$	−0.249141	−0.0106720
$546$	0	0
$547$	−12.9966	−0.555693	−0.277846	−	0.960626i	$-0.589621\pi$
$547$	−12.9966	−0.555693	−0.277846	+	0.960626i	$0.589621\pi$
$548$	0	0
$549$	18.7620	0.800744
$550$	0	0
$551$	38.1948	1.62715
$552$	0	0
$553$	−8.36641	−0.355776
$554$	0	0
$555$	9.55691	0.405668
$556$	0	0
$557$	−16.4914	−0.698763	−0.349382	−	0.936981i	$-0.613608\pi$
$557$	−16.4914	−0.698763	−0.349382	+	0.936981i	$0.613608\pi$
$558$	0	0
$559$	−7.22422	−0.305552
$560$	0	0
$561$	−14.6155	−0.617069
$562$	0	0
$563$	16.7620	0.706435	0.353218	−	0.935541i	$-0.385088\pi$
$563$	16.7620	0.706435	0.353218	+	0.935541i	$0.385088\pi$
$564$	0	0
$565$	10.9966	0.462629
$566$	0	0
$567$	−10.9379	−0.459350
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−12.7328	−0.532852	−0.266426	−	0.963855i	$-0.585843\pi$
$571$	−12.7328	−0.532852	−0.266426	+	0.963855i	$0.585843\pi$
$572$	0	0
$573$	−7.00344	−0.292573
$574$	0	0
$575$	−6.24914	−0.260607
$576$	0	0
$577$	−11.8613	−0.493790	−0.246895	−	0.969042i	$-0.579410\pi$
$577$	−11.8613	−0.493790	−0.246895	+	0.969042i	$0.579410\pi$
$578$	0	0
$579$	−13.8827	−0.576947
$580$	0	0
$581$	8.49828	0.352568
$582$	0	0
$583$	4.74742	0.196618
$584$	0	0
$585$	1.93793	0.0801235
$586$	0	0
$587$	8.60094	0.354999	0.177499	−	0.984121i	$-0.443199\pi$
$587$	8.60094	0.354999	0.177499	+	0.984121i	$0.443199\pi$
$588$	0	0
$589$	41.7294	1.71943
$590$	0	0
$591$	15.1430	0.622902
$592$	0	0
$593$	10.7328	0.440744	0.220372	−	0.975416i	$-0.429273\pi$
$593$	10.7328	0.440744	0.220372	+	0.975416i	$0.429273\pi$
$594$	0	0
$595$	6.49828	0.266404
$596$	0	0
$597$	29.7586	1.21794
$598$	0	0
$599$	−16.0812	−0.657059	−0.328529	−	0.944494i	$-0.606553\pi$
$599$	−16.0812	−0.657059	−0.328529	+	0.944494i	$0.606553\pi$
$600$	0	0
$601$	−34.0889	−1.39052	−0.695258	−	0.718760i	$-0.744710\pi$
$601$	−34.0889	−1.39052	−0.695258	+	0.718760i	$0.744710\pi$
$602$	0	0
$603$	−26.7552	−1.08955
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−11.6742	−0.473840	−0.236920	−	0.971529i	$-0.576138\pi$
$607$	−11.6742	−0.473840	−0.236920	+	0.971529i	$0.576138\pi$
$608$	0	0
$609$	19.6742	0.797238
$610$	0	0
$611$	−10.4622	−0.423255
$612$	0	0
$613$	−34.2637	−1.38390	−0.691950	−	0.721946i	$-0.743248\pi$
$613$	−34.2637	−1.38390	−0.691950	+	0.721946i	$0.743248\pi$
$614$	0	0
$615$	−4.79488	−0.193348
$616$	0	0
$617$	39.3707	1.58500	0.792502	−	0.609869i	$-0.208778\pi$
$617$	39.3707	1.58500	0.792502	+	0.609869i	$0.208778\pi$
$618$	0	0
$619$	−16.1104	−0.647531	−0.323766	−	0.946137i	$-0.604949\pi$
$619$	−16.1104	−0.647531	−0.323766	+	0.946137i	$0.604949\pi$
$620$	0	0
$621$	13.2311	0.530946
$622$	0	0
$623$	12.3810	0.496035
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.82066	−0.392199
$628$	0	0
$629$	27.6121	1.10097
$630$	0	0
$631$	−13.8827	−0.552663	−0.276331	−	0.961062i	$-0.589119\pi$
$631$	−13.8827	−0.552663	−0.276331	+	0.961062i	$0.589119\pi$
$632$	0	0
$633$	−51.9862	−2.06627
$634$	0	0
$635$	5.88273	0.233449
$636$	0	0
$637$	0.941367	0.0372983
$638$	0	0
$639$	30.0881	1.19026
$640$	0	0
$641$	−39.3415	−1.55390	−0.776948	−	0.629565i	$-0.783233\pi$
$641$	−39.3415	−1.55390	−0.776948	+	0.629565i	$0.783233\pi$
$642$	0	0
$643$	−38.3595	−1.51275	−0.756376	−	0.654137i	$-0.773032\pi$
$643$	−38.3595	−1.51275	−0.756376	+	0.654137i	$0.773032\pi$
$644$	0	0
$645$	−17.2603	−0.679624
$646$	0	0
$647$	−25.7294	−1.01153	−0.505763	−	0.862672i	$-0.668789\pi$
$647$	−25.7294	−1.01153	−0.505763	+	0.862672i	$0.668789\pi$
$648$	0	0
$649$	−1.88273	−0.0739038
$650$	0	0
$651$	21.4948	0.842449
$652$	0	0
$653$	18.4768	0.723053	0.361526	−	0.932362i	$-0.382256\pi$
$653$	18.4768	0.723053	0.361526	+	0.932362i	$0.382256\pi$
$654$	0	0
$655$	1.25258	0.0489423
$656$	0	0
$657$	−21.6121	−0.843169
$658$	0	0
$659$	39.3776	1.53393	0.766966	−	0.641687i	$-0.221765\pi$
$659$	39.3776	1.53393	0.766966	+	0.641687i	$0.221765\pi$
$660$	0	0
$661$	−34.2208	−1.33103	−0.665517	−	0.746383i	$-0.731789\pi$
$661$	−34.2208	−1.33103	−0.665517	+	0.746383i	$0.731789\pi$
$662$	0	0
$663$	13.7586	0.534339
$664$	0	0
$665$	4.36641	0.169322
$666$	0	0
$667$	−54.6639	−2.11659
$668$	0	0
$669$	−23.3484	−0.902700
$670$	0	0
$671$	−9.11383	−0.351835
$672$	0	0
$673$	−24.4622	−0.942948	−0.471474	−	0.881880i	$-0.656278\pi$
$673$	−24.4622	−0.942948	−0.471474	+	0.881880i	$0.656278\pi$
$674$	0	0
$675$	−2.11727	−0.0814936
$676$	0	0
$677$	−39.1070	−1.50300	−0.751501	−	0.659732i	$-0.770670\pi$
$677$	−39.1070	−1.50300	−0.751501	+	0.659732i	$0.770670\pi$
$678$	0	0
$679$	−15.3630	−0.589577
$680$	0	0
$681$	−36.2345	−1.38851
$682$	0	0
$683$	−23.3776	−0.894518	−0.447259	−	0.894404i	$-0.647600\pi$
$683$	−23.3776	−0.894518	−0.447259	+	0.894404i	$0.647600\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	0	0
$687$	54.6310	2.08430
$688$	0	0
$689$	−4.46907	−0.170258
$690$	0	0
$691$	−8.49828	−0.323290	−0.161645	−	0.986849i	$-0.551680\pi$
$691$	−8.49828	−0.323290	−0.161645	+	0.986849i	$0.551680\pi$
$692$	0	0
$693$	−2.05863	−0.0782010
$694$	0	0
$695$	−10.9820	−0.416569
$696$	0	0
$697$	−13.8535	−0.524739
$698$	0	0
$699$	15.4880	0.585809
$700$	0	0
$701$	14.9751	0.565601	0.282800	−	0.959179i	$-0.408737\pi$
$701$	14.9751	0.565601	0.282800	+	0.959179i	$0.408737\pi$
$702$	0	0
$703$	18.5535	0.699758
$704$	0	0
$705$	−24.9966	−0.941425
$706$	0	0
$707$	−16.8793	−0.634811
$708$	0	0
$709$	40.7259	1.52949	0.764747	−	0.644330i	$-0.222864\pi$
$709$	40.7259	1.52949	0.764747	+	0.644330i	$0.222864\pi$
$710$	0	0
$711$	−17.2234	−0.645927
$712$	0	0
$713$	−59.7225	−2.23663
$714$	0	0
$715$	−0.941367	−0.0352051
$716$	0	0
$717$	25.0449	0.935318
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	6.61555	0.246376
$722$	0	0
$723$	5.32238	0.197942
$724$	0	0
$725$	8.74742	0.324871
$726$	0	0
$727$	51.6413	1.91527	0.957635	−	0.287984i	$-0.0929848\pi$
$727$	51.6413	1.91527	0.957635	+	0.287984i	$0.0929848\pi$
$728$	0	0
$729$	−8.23109	−0.304855
$730$	0	0
$731$	−49.8690	−1.84447
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	0	0
$735$	2.24914	0.0829608
$736$	0	0
$737$	12.9966	0.478735
$738$	0	0
$739$	49.1070	1.80643	0.903214	−	0.429190i	$-0.141201\pi$
$739$	49.1070	1.80643	0.903214	+	0.429190i	$0.141201\pi$
$740$	0	0
$741$	9.24485	0.339618
$742$	0	0
$743$	53.2311	1.95286	0.976430	−	0.215836i	$-0.0692475\pi$
$743$	53.2311	1.95286	0.976430	+	0.215836i	$0.0692475\pi$
$744$	0	0
$745$	0.0146079	0.000535192 0
$746$	0	0
$747$	17.4948	0.640103
$748$	0	0
$749$	14.5535	0.531772
$750$	0	0
$751$	31.6121	1.15354	0.576771	−	0.816906i	$-0.304312\pi$
$751$	31.6121	1.15354	0.576771	+	0.816906i	$0.304312\pi$
$752$	0	0
$753$	−56.4691	−2.05785
$754$	0	0
$755$	15.2457	0.554848
$756$	0	0
$757$	−1.24570	−0.0452758	−0.0226379	−	0.999744i	$-0.507206\pi$
$757$	−1.24570	−0.0452758	−0.0226379	+	0.999744i	$0.507206\pi$
$758$	0	0
$759$	14.0552	0.510171
$760$	0	0
$761$	−10.6009	−0.384284	−0.192142	−	0.981367i	$-0.561543\pi$
$761$	−10.6009	−0.384284	−0.192142	+	0.981367i	$0.561543\pi$
$762$	0	0
$763$	−0.249141	−0.00901949
$764$	0	0
$765$	13.3776	0.483667
$766$	0	0
$767$	1.77234	0.0639956
$768$	0	0
$769$	−19.1284	−0.689789	−0.344895	−	0.938641i	$-0.612085\pi$
$769$	−19.1284	−0.689789	−0.344895	+	0.938641i	$0.612085\pi$
$770$	0	0
$771$	−6.44309	−0.232042
$772$	0	0
$773$	39.9931	1.43845	0.719226	−	0.694776i	$-0.244496\pi$
$773$	39.9931	1.43845	0.719226	+	0.694776i	$0.244496\pi$
$774$	0	0
$775$	9.55691	0.343294
$776$	0	0
$777$	9.55691	0.342852
$778$	0	0
$779$	−9.30863	−0.333516
$780$	0	0
$781$	−14.6155	−0.522985
$782$	0	0
$783$	−18.5206	−0.661873
$784$	0	0
$785$	−5.50172	−0.196365
$786$	0	0
$787$	−37.9931	−1.35431	−0.677154	−	0.735841i	$-0.736787\pi$
$787$	−37.9931	−1.35431	−0.677154	+	0.735841i	$0.736787\pi$
$788$	0	0
$789$	−8.46907	−0.301507
$790$	0	0
$791$	10.9966	0.390993
$792$	0	0
$793$	8.57946	0.304665
$794$	0	0
$795$	−10.6776	−0.378696
$796$	0	0
$797$	−20.3810	−0.721933	−0.360966	−	0.932579i	$-0.617553\pi$
$797$	−20.3810	−0.721933	−0.360966	+	0.932579i	$0.617553\pi$
$798$	0	0
$799$	−72.2208	−2.55499
$800$	0	0
$801$	25.4880	0.900573
$802$	0	0
$803$	10.4983	0.370476
$804$	0	0
$805$	−6.24914	−0.220253
$806$	0	0
$807$	20.1104	0.707919
$808$	0	0
$809$	−31.7294	−1.11555	−0.557773	−	0.829994i	$-0.688344\pi$
$809$	−31.7294	−1.11555	−0.557773	+	0.829994i	$0.688344\pi$
$810$	0	0
$811$	−43.5095	−1.52782	−0.763912	−	0.645321i	$-0.776724\pi$
$811$	−43.5095	−1.52782	−0.763912	+	0.645321i	$0.776724\pi$
$812$	0	0
$813$	−48.3449	−1.69553
$814$	0	0
$815$	18.2277	0.638487
$816$	0	0
$817$	−33.5086	−1.17232
$818$	0	0
$819$	1.93793	0.0677167
$820$	0	0
$821$	−24.7766	−0.864711	−0.432355	−	0.901703i	$-0.642317\pi$
$821$	−24.7766	−0.864711	−0.432355	+	0.901703i	$0.642317\pi$
$822$	0	0
$823$	−28.8647	−1.00616	−0.503080	−	0.864240i	$-0.667800\pi$
$823$	−28.8647	−1.00616	−0.503080	+	0.864240i	$0.667800\pi$
$824$	0	0
$825$	−2.24914	−0.0783050
$826$	0	0
$827$	−34.2277	−1.19021	−0.595106	−	0.803647i	$-0.702890\pi$
$827$	−34.2277	−1.19021	−0.595106	+	0.803647i	$0.702890\pi$
$828$	0	0
$829$	23.6381	0.820985	0.410492	−	0.911864i	$-0.365357\pi$
$829$	23.6381	0.820985	0.410492	+	0.911864i	$0.365357\pi$
$830$	0	0
$831$	17.2019	0.596727
$832$	0	0
$833$	6.49828	0.225152
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−20.2345	−0.699408
$838$	0	0
$839$	24.9053	0.859826	0.429913	−	0.902870i	$-0.358544\pi$
$839$	24.9053	0.859826	0.429913	+	0.902870i	$0.358544\pi$
$840$	0	0
$841$	47.5174	1.63853
$842$	0	0
$843$	−64.3604	−2.21669
$844$	0	0
$845$	−12.1138	−0.416728
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	6.47594	0.222254
$850$	0	0
$851$	−26.5535	−0.910241
$852$	0	0
$853$	22.9966	0.787387	0.393694	−	0.919242i	$-0.371197\pi$
$853$	22.9966	0.787387	0.393694	+	0.919242i	$0.371197\pi$
$854$	0	0
$855$	8.98883	0.307411
$856$	0	0
$857$	−31.2603	−1.06783	−0.533916	−	0.845538i	$-0.679280\pi$
$857$	−31.2603	−1.06783	−0.533916	+	0.845538i	$0.679280\pi$
$858$	0	0
$859$	−20.3449	−0.694160	−0.347080	−	0.937836i	$-0.612827\pi$
$859$	−20.3449	−0.694160	−0.347080	+	0.937836i	$0.612827\pi$
$860$	0	0
$861$	−4.79488	−0.163409
$862$	0	0
$863$	58.3595	1.98658	0.993291	−	0.115644i	$-0.0368930\pi$
$863$	58.3595	1.98658	0.993291	+	0.115644i	$0.0368930\pi$
$864$	0	0
$865$	0.117266	0.00398717
$866$	0	0
$867$	56.7405	1.92701
$868$	0	0
$869$	8.36641	0.283811
$870$	0	0
$871$	−12.2345	−0.414551
$872$	0	0
$873$	−31.6267	−1.07040
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	11.9639	0.403992	0.201996	−	0.979386i	$-0.435257\pi$
$877$	11.9639	0.403992	0.201996	+	0.979386i	$0.435257\pi$
$878$	0	0
$879$	−29.1070	−0.981753
$880$	0	0
$881$	5.88961	0.198426	0.0992130	−	0.995066i	$-0.468367\pi$
$881$	5.88961	0.198426	0.0992130	+	0.995066i	$0.468367\pi$
$882$	0	0
$883$	19.4880	0.655822	0.327911	−	0.944709i	$-0.393655\pi$
$883$	19.4880	0.655822	0.327911	+	0.944709i	$0.393655\pi$
$884$	0	0
$885$	4.23453	0.142342
$886$	0	0
$887$	−50.4622	−1.69435	−0.847177	−	0.531310i	$-0.821700\pi$
$887$	−50.4622	−1.69435	−0.847177	+	0.531310i	$0.821700\pi$
$888$	0	0
$889$	5.88273	0.197301
$890$	0	0
$891$	10.9379	0.366434
$892$	0	0
$893$	−48.5275	−1.62391
$894$	0	0
$895$	22.5535	0.753880
$896$	0	0
$897$	−13.2311	−0.441773
$898$	0	0
$899$	83.5984	2.78816
$900$	0	0
$901$	−30.8501	−1.02777
$902$	0	0
$903$	−17.2603	−0.574387
$904$	0	0
$905$	20.8793	0.694051
$906$	0	0
$907$	16.8432	0.559269	0.279635	−	0.960106i	$-0.409787\pi$
$907$	16.8432	0.559269	0.279635	+	0.960106i	$0.409787\pi$
$908$	0	0
$909$	−34.7483	−1.15253
$910$	0	0
$911$	38.6155	1.27939	0.639695	−	0.768629i	$-0.279061\pi$
$911$	38.6155	1.27939	0.639695	+	0.768629i	$0.279061\pi$
$912$	0	0
$913$	−8.49828	−0.281252
$914$	0	0
$915$	20.4983	0.677652
$916$	0	0
$917$	1.25258	0.0413638
$918$	0	0
$919$	−17.2818	−0.570074	−0.285037	−	0.958517i	$-0.592006\pi$
$919$	−17.2818	−0.570074	−0.285037	+	0.958517i	$0.592006\pi$
$920$	0	0
$921$	−1.12070	−0.0369285
$922$	0	0
$923$	13.7586	0.452870
$924$	0	0
$925$	4.24914	0.139711
$926$	0	0
$927$	13.6190	0.447306
$928$	0	0
$929$	2.91539	0.0956508	0.0478254	−	0.998856i	$-0.484771\pi$
$929$	2.91539	0.0956508	0.0478254	+	0.998856i	$0.484771\pi$
$930$	0	0
$931$	4.36641	0.143103
$932$	0	0
$933$	52.7191	1.72594
$934$	0	0
$935$	−6.49828	−0.212517
$936$	0	0
$937$	22.5795	0.737639	0.368819	−	0.929501i	$-0.379762\pi$
$937$	22.5795	0.737639	0.368819	+	0.929501i	$0.379762\pi$
$938$	0	0
$939$	21.0741	0.687727
$940$	0	0
$941$	−21.7655	−0.709534	−0.354767	−	0.934955i	$-0.615440\pi$
$941$	−21.7655	−0.709534	−0.354767	+	0.934955i	$0.615440\pi$
$942$	0	0
$943$	13.3224	0.433836
$944$	0	0
$945$	−2.11727	−0.0688747
$946$	0	0
$947$	1.49484	0.0485759	0.0242879	−	0.999705i	$-0.492268\pi$
$947$	1.49484	0.0485759	0.0242879	+	0.999705i	$0.492268\pi$
$948$	0	0
$949$	−9.88273	−0.320807
$950$	0	0
$951$	−22.4431	−0.727767
$952$	0	0
$953$	−4.83098	−0.156491	−0.0782453	−	0.996934i	$-0.524932\pi$
$953$	−4.83098	−0.156491	−0.0782453	+	0.996934i	$0.524932\pi$
$954$	0	0
$955$	−3.11383	−0.100761
$956$	0	0
$957$	−19.6742	−0.635976
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	60.3346	1.94628
$962$	0	0
$963$	29.9603	0.965456
$964$	0	0
$965$	−6.17246	−0.198699
$966$	0	0
$967$	51.9278	1.66989	0.834943	−	0.550336i	$-0.185501\pi$
$967$	51.9278	1.66989	0.834943	+	0.550336i	$0.185501\pi$
$968$	0	0
$969$	63.8174	2.05011
$970$	0	0
$971$	−59.6933	−1.91565	−0.957824	−	0.287354i	$-0.907224\pi$
$971$	−59.6933	−1.91565	−0.957824	+	0.287354i	$0.907224\pi$
$972$	0	0
$973$	−10.9820	−0.352065
$974$	0	0
$975$	2.11727	0.0678068
$976$	0	0
$977$	31.2311	0.999171	0.499586	−	0.866265i	$-0.333486\pi$
$977$	31.2311	0.999171	0.499586	+	0.866265i	$0.333486\pi$
$978$	0	0
$979$	−12.3810	−0.395699
$980$	0	0
$981$	−0.512889	−0.0163753
$982$	0	0
$983$	22.6155	0.721324	0.360662	−	0.932697i	$-0.382551\pi$
$983$	22.6155	0.721324	0.360662	+	0.932697i	$0.382551\pi$
$984$	0	0
$985$	6.73281	0.214525
$986$	0	0
$987$	−24.9966	−0.795649
$988$	0	0
$989$	47.9570	1.52494
$990$	0	0
$991$	−22.4102	−0.711884	−0.355942	−	0.934508i	$-0.615840\pi$
$991$	−22.4102	−0.711884	−0.355942	+	0.934508i	$0.615840\pi$
$992$	0	0
$993$	45.9639	1.45862
$994$	0	0
$995$	13.2311	0.419454
$996$	0	0
$997$	−27.3415	−0.865914	−0.432957	−	0.901415i	$-0.642530\pi$
$997$	−27.3415	−0.865914	−0.432957	+	0.901415i	$0.642530\pi$
$998$	0	0
$999$	−8.99656	−0.284639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bf.1.3		3
4.3	odd	2		770.2.a.m.1.1	✓	3
12.11	even	2		6930.2.a.ce.1.2		3
20.3	even	4		3850.2.c.ba.1849.1		6
20.7	even	4		3850.2.c.ba.1849.6		6
20.19	odd	2		3850.2.a.bt.1.3		3
28.27	even	2		5390.2.a.ca.1.3		3
44.43	even	2		8470.2.a.ci.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.m.1.1	✓	3	4.3	odd	2
3850.2.a.bt.1.3		3	20.19	odd	2
3850.2.c.ba.1849.1		6	20.3	even	4
3850.2.c.ba.1849.6		6	20.7	even	4
5390.2.a.ca.1.3		3	28.27	even	2
6160.2.a.bf.1.3		3	1.1	even	1	trivial
6930.2.a.ce.1.2		3	12.11	even	2
8470.2.a.ci.1.1		3	44.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} +O(q^{10})\)	\(q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} -1.00000 q^{11} +0.941367 q^{13} +2.24914 q^{15} +6.49828 q^{17} +4.36641 q^{19} +2.24914 q^{21} -6.24914 q^{23} +1.00000 q^{25} -2.11727 q^{27} +8.74742 q^{29} +9.55691 q^{31} -2.24914 q^{33} +1.00000 q^{35} +4.24914 q^{37} +2.11727 q^{39} -2.13187 q^{41} -7.67418 q^{43} +2.05863 q^{45} -11.1138 q^{47} +1.00000 q^{49} +14.6155 q^{51} -4.74742 q^{53} -1.00000 q^{55} +9.82066 q^{57} +1.88273 q^{59} +9.11383 q^{61} +2.05863 q^{63} +0.941367 q^{65} -12.9966 q^{67} -14.0552 q^{69} +14.6155 q^{71} -10.4983 q^{73} +2.24914 q^{75} -1.00000 q^{77} -8.36641 q^{79} -10.9379 q^{81} +8.49828 q^{83} +6.49828 q^{85} +19.6742 q^{87} +12.3810 q^{89} +0.941367 q^{91} +21.4948 q^{93} +4.36641 q^{95} -15.3630 q^{97} -2.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9 - 3 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 + 6 * q^19 - 2 * q^21 - 10 * q^23 + 3 * q^25 - 8 * q^27 + 12 * q^31 + 2 * q^33 + 3 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^49 + 28 * q^51 + 12 * q^53 - 3 * q^55 - 8 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 - 4 * q^67 - 8 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 - 3 * q^77 - 18 * q^79 + 3 * q^81 + 8 * q^83 + 2 * q^85 + 44 * q^87 + 18 * q^89 + 2 * q^91 + 12 * q^93 + 6 * q^95 - 4 * q^97 - 7 * q^99

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