LMFDB - Embedded newform 4160.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25088.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 2 $ x^4 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.595188$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.36028 q^{3} +1.00000 q^{5} -1.19038 q^{7} +8.29150 q^{9} +O(q^{10})$	$q-3.36028 q^{3} +1.00000 q^{5} -1.19038 q^{7} +8.29150 q^{9} +2.16991 q^{11} +1.00000 q^{13} -3.36028 q^{15} +6.00000 q^{17} +2.16991 q^{19} +4.00000 q^{21} +7.70010 q^{23} +1.00000 q^{25} -17.7809 q^{27} +5.29150 q^{29} +4.55066 q^{31} -7.29150 q^{33} -1.19038 q^{35} -9.29150 q^{37} -3.36028 q^{39} +12.5830 q^{41} -10.0808 q^{43} +8.29150 q^{45} -1.19038 q^{47} -5.58301 q^{49} -20.1617 q^{51} +8.58301 q^{53} +2.16991 q^{55} -7.29150 q^{57} +11.2712 q^{59} -2.70850 q^{61} -9.87000 q^{63} +1.00000 q^{65} +5.53019 q^{67} -25.8745 q^{69} -6.50972 q^{71} -1.29150 q^{73} -3.36028 q^{75} -2.58301 q^{77} -4.33981 q^{79} +34.8745 q^{81} -1.19038 q^{83} +6.00000 q^{85} -17.7809 q^{87} -4.58301 q^{89} -1.19038 q^{91} -15.2915 q^{93} +2.16991 q^{95} -2.00000 q^{97} +17.9918 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 12 * q^9	$ 4 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{17} + 16 q^{21} + 4 q^{25} - 8 q^{33} - 16 q^{37} + 8 q^{41} + 12 q^{45} + 20 q^{49} - 8 q^{53} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 40 q^{69} + 16 q^{73} + 32 q^{77} + 76 q^{81} + 24 q^{85} + 24 q^{89} - 40 q^{93} - 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^17 + 16 * q^21 + 4 * q^25 - 8 * q^33 - 16 * q^37 + 8 * q^41 + 12 * q^45 + 20 * q^49 - 8 * q^53 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 40 * q^69 + 16 * q^73 + 32 * q^77 + 76 * q^81 + 24 * q^85 + 24 * q^89 - 40 * q^93 - 8 * q^97

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
$3$	−3.36028	−1.94006	−0.970030	−	0.242984i	$-0.921874\pi$
$3$	−3.36028	−1.94006	−0.970030	+	0.242984i	$0.921874\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.19038	−0.449920	−0.224960	−	0.974368i	$-0.572225\pi$
$7$	−1.19038	−0.449920	−0.224960	+	0.974368i	$0.572225\pi$
$8$	0	0
$9$	8.29150	2.76383
$10$	0	0
$11$	2.16991	0.654252	0.327126	−	0.944981i	$-0.393920\pi$
$11$	2.16991	0.654252	0.327126	+	0.944981i	$0.393920\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.36028	−0.867621
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.16991	0.497811	0.248905	−	0.968528i	$-0.419929\pi$
$19$	2.16991	0.497811	0.248905	+	0.968528i	$0.419929\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	7.70010	1.60558	0.802791	−	0.596261i	$-0.203348\pi$
$23$	7.70010	1.60558	0.802791	+	0.596261i	$0.203348\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−17.7809	−3.42194
$28$	0	0
$29$	5.29150	0.982607	0.491304	−	0.870988i	$-0.336521\pi$
$29$	5.29150	0.982607	0.491304	+	0.870988i	$0.336521\pi$
$30$	0	0
$31$	4.55066	0.817322	0.408661	−	0.912686i	$-0.365996\pi$
$31$	4.55066	0.817322	0.408661	+	0.912686i	$0.365996\pi$
$32$	0	0
$33$	−7.29150	−1.26929
$34$	0	0
$35$	−1.19038	−0.201210
$36$	0	0
$37$	−9.29150	−1.52751	−0.763757	−	0.645504i	$-0.776647\pi$
$37$	−9.29150	−1.52751	−0.763757	+	0.645504i	$0.776647\pi$
$38$	0	0
$39$	−3.36028	−0.538076
$40$	0	0
$41$	12.5830	1.96514	0.982568	−	0.185905i	$-0.0595218\pi$
$41$	12.5830	1.96514	0.982568	+	0.185905i	$0.0595218\pi$
$42$	0	0
$43$	−10.0808	−1.53732	−0.768658	−	0.639661i	$-0.779075\pi$
$43$	−10.0808	−1.53732	−0.768658	+	0.639661i	$0.779075\pi$
$44$	0	0
$45$	8.29150	1.23602
$46$	0	0
$47$	−1.19038	−0.173634	−0.0868171	−	0.996224i	$-0.527670\pi$
$47$	−1.19038	−0.173634	−0.0868171	+	0.996224i	$0.527670\pi$
$48$	0	0
$49$	−5.58301	−0.797572
$50$	0	0
$51$	−20.1617	−2.82320
$52$	0	0
$53$	8.58301	1.17897	0.589483	−	0.807781i	$-0.299331\pi$
$53$	8.58301	1.17897	0.589483	+	0.807781i	$0.299331\pi$
$54$	0	0
$55$	2.16991	0.292590
$56$	0	0
$57$	−7.29150	−0.965783
$58$	0	0
$59$	11.2712	1.46739	0.733694	−	0.679480i	$-0.237794\pi$
$59$	11.2712	1.46739	0.733694	+	0.679480i	$0.237794\pi$
$60$	0	0
$61$	−2.70850	−0.346788	−0.173394	−	0.984853i	$-0.555473\pi$
$61$	−2.70850	−0.346788	−0.173394	+	0.984853i	$0.555473\pi$
$62$	0	0
$63$	−9.87000	−1.24350
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	5.53019	0.675620	0.337810	−	0.941214i	$-0.390314\pi$
$67$	5.53019	0.675620	0.337810	+	0.941214i	$0.390314\pi$
$68$	0	0
$69$	−25.8745	−3.11492
$70$	0	0
$71$	−6.50972	−0.772562	−0.386281	−	0.922381i	$-0.626240\pi$
$71$	−6.50972	−0.772562	−0.386281	+	0.922381i	$0.626240\pi$
$72$	0	0
$73$	−1.29150	−0.151159	−0.0755795	−	0.997140i	$-0.524081\pi$
$73$	−1.29150	−0.151159	−0.0755795	+	0.997140i	$0.524081\pi$
$74$	0	0
$75$	−3.36028	−0.388012
$76$	0	0
$77$	−2.58301	−0.294361
$78$	0	0
$79$	−4.33981	−0.488267	−0.244134	−	0.969742i	$-0.578504\pi$
$79$	−4.33981	−0.488267	−0.244134	+	0.969742i	$0.578504\pi$
$80$	0	0
$81$	34.8745	3.87495
$82$	0	0
$83$	−1.19038	−0.130661	−0.0653304	−	0.997864i	$-0.520810\pi$
$83$	−1.19038	−0.130661	−0.0653304	+	0.997864i	$0.520810\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	−17.7809	−1.90632
$88$	0	0
$89$	−4.58301	−0.485798	−0.242899	−	0.970052i	$-0.578098\pi$
$89$	−4.58301	−0.485798	−0.242899	+	0.970052i	$0.578098\pi$
$90$	0	0
$91$	−1.19038	−0.124785
$92$	0	0
$93$	−15.2915	−1.58565
$94$	0	0
$95$	2.16991	0.222628
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	17.9918	1.80824
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−12.0399	−1.18633	−0.593164	−	0.805082i	$-0.702121\pi$
$103$	−12.0399	−1.18633	−0.593164	+	0.805082i	$0.702121\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	5.74103	0.555007	0.277503	−	0.960725i	$-0.410493\pi$
$107$	5.74103	0.555007	0.277503	+	0.960725i	$0.410493\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	31.2221	2.96347
$112$	0	0
$113$	−8.58301	−0.807421	−0.403711	−	0.914887i	$-0.632280\pi$
$113$	−8.58301	−0.807421	−0.403711	+	0.914887i	$0.632280\pi$
$114$	0	0
$115$	7.70010	0.718038
$116$	0	0
$117$	8.29150	0.766550
$118$	0	0
$119$	−7.14226	−0.654729
$120$	0	0
$121$	−6.29150	−0.571955
$122$	0	0
$123$	−42.2825	−3.81248
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0399	1.06837	0.534185	−	0.845368i	$-0.320619\pi$
$127$	12.0399	1.06837	0.534185	+	0.845368i	$0.320619\pi$
$128$	0	0
$129$	33.8745	2.98248
$130$	0	0
$131$	15.8219	1.38236	0.691182	−	0.722681i	$-0.257090\pi$
$131$	15.8219	1.38236	0.691182	+	0.722681i	$0.257090\pi$
$132$	0	0
$133$	−2.58301	−0.223975
$134$	0	0
$135$	−17.7809	−1.53034
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.33981	−0.368098	−0.184049	−	0.982917i	$-0.558921\pi$
$139$	−4.33981	−0.368098	−0.184049	+	0.982917i	$0.558921\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	2.16991	0.181457
$144$	0	0
$145$	5.29150	0.439435
$146$	0	0
$147$	18.7605	1.54734
$148$	0	0
$149$	−8.58301	−0.703147	−0.351574	−	0.936160i	$-0.614353\pi$
$149$	−8.58301	−0.703147	−0.351574	+	0.936160i	$0.614353\pi$
$150$	0	0
$151$	−15.6110	−1.27041	−0.635204	−	0.772344i	$-0.719084\pi$
$151$	−15.6110	−1.27041	−0.635204	+	0.772344i	$0.719084\pi$
$152$	0	0
$153$	49.7490	4.02197
$154$	0	0
$155$	4.55066	0.365518
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−28.8413	−2.28727
$160$	0	0
$161$	−9.16601	−0.722383
$162$	0	0
$163$	21.3521	1.67242	0.836212	−	0.548407i	$-0.184765\pi$
$163$	21.3521	1.67242	0.836212	+	0.548407i	$0.184765\pi$
$164$	0	0
$165$	−7.29150	−0.567643
$166$	0	0
$167$	−5.95188	−0.460570	−0.230285	−	0.973123i	$-0.573966\pi$
$167$	−5.95188	−0.460570	−0.230285	+	0.973123i	$0.573966\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.9918	1.37587
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−1.19038	−0.0899840
$176$	0	0
$177$	−37.8745	−2.84682
$178$	0	0
$179$	4.76150	0.355891	0.177946	−	0.984040i	$-0.443055\pi$
$179$	4.76150	0.355891	0.177946	+	0.984040i	$0.443055\pi$
$180$	0	0
$181$	5.29150	0.393314	0.196657	−	0.980472i	$-0.436991\pi$
$181$	5.29150	0.393314	0.196657	+	0.980472i	$0.436991\pi$
$182$	0	0
$183$	9.10132	0.672789
$184$	0	0
$185$	−9.29150	−0.683125
$186$	0	0
$187$	13.0194	0.952076
$188$	0	0
$189$	21.1660	1.53960
$190$	0	0
$191$	15.8219	1.14483	0.572416	−	0.819964i	$-0.306006\pi$
$191$	15.8219	1.14483	0.572416	+	0.819964i	$0.306006\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	−3.36028	−0.240635
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−22.1208	−1.56810	−0.784050	−	0.620698i	$-0.786849\pi$
$199$	−22.1208	−1.56810	−0.784050	+	0.620698i	$0.786849\pi$
$200$	0	0
$201$	−18.5830	−1.31074
$202$	0	0
$203$	−6.29888	−0.442095
$204$	0	0
$205$	12.5830	0.878835
$206$	0	0
$207$	63.8454	4.43756
$208$	0	0
$209$	4.70850	0.325694
$210$	0	0
$211$	15.8219	1.08922	0.544612	−	0.838688i	$-0.316677\pi$
$211$	15.8219	1.08922	0.544612	+	0.838688i	$0.316677\pi$
$212$	0	0
$213$	21.8745	1.49882
$214$	0	0
$215$	−10.0808	−0.687508
$216$	0	0
$217$	−5.41699	−0.367730
$218$	0	0
$219$	4.33981	0.293257
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	−17.0123	−1.13923	−0.569613	−	0.821913i	$-0.692907\pi$
$223$	−17.0123	−1.13923	−0.569613	+	0.821913i	$0.692907\pi$
$224$	0	0
$225$	8.29150	0.552767
$226$	0	0
$227$	−7.91094	−0.525068	−0.262534	−	0.964923i	$-0.584558\pi$
$227$	−7.91094	−0.525068	−0.262534	+	0.964923i	$0.584558\pi$
$228$	0	0
$229$	16.5830	1.09584	0.547918	−	0.836532i	$-0.315421\pi$
$229$	16.5830	1.09584	0.547918	+	0.836532i	$0.315421\pi$
$230$	0	0
$231$	8.67963	0.571078
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−1.19038	−0.0776516
$236$	0	0
$237$	14.5830	0.947268
$238$	0	0
$239$	11.2712	0.729075	0.364537	−	0.931189i	$-0.381227\pi$
$239$	11.2712	0.729075	0.364537	+	0.931189i	$0.381227\pi$
$240$	0	0
$241$	4.58301	0.295217	0.147609	−	0.989046i	$-0.452842\pi$
$241$	4.58301	0.295217	0.147609	+	0.989046i	$0.452842\pi$
$242$	0	0
$243$	−63.8454	−4.09568
$244$	0	0
$245$	−5.58301	−0.356685
$246$	0	0
$247$	2.16991	0.138068
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	28.8413	1.82045	0.910224	−	0.414116i	$-0.135909\pi$
$251$	28.8413	1.82045	0.910224	+	0.414116i	$0.135909\pi$
$252$	0	0
$253$	16.7085	1.05045
$254$	0	0
$255$	−20.1617	−1.26257
$256$	0	0
$257$	16.5830	1.03442	0.517210	−	0.855859i	$-0.326971\pi$
$257$	16.5830	1.03442	0.517210	+	0.855859i	$0.326971\pi$
$258$	0	0
$259$	11.0604	0.687259
$260$	0	0
$261$	43.8745	2.71576
$262$	0	0
$263$	−7.70010	−0.474808	−0.237404	−	0.971411i	$-0.576297\pi$
$263$	−7.70010	−0.474808	−0.237404	+	0.971411i	$0.576297\pi$
$264$	0	0
$265$	8.58301	0.527250
$266$	0	0
$267$	15.4002	0.942477
$268$	0	0
$269$	3.41699	0.208338	0.104169	−	0.994560i	$-0.466782\pi$
$269$	3.41699	0.208338	0.104169	+	0.994560i	$0.466782\pi$
$270$	0	0
$271$	28.6305	1.73918	0.869589	−	0.493776i	$-0.164384\pi$
$271$	28.6305	1.73918	0.869589	+	0.493776i	$0.164384\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	2.16991	0.130850
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	37.7318	2.25894
$280$	0	0
$281$	20.5830	1.22788	0.613940	−	0.789353i	$-0.289584\pi$
$281$	20.5830	1.22788	0.613940	+	0.789353i	$0.289584\pi$
$282$	0	0
$283$	−14.8424	−0.882286	−0.441143	−	0.897437i	$-0.645427\pi$
$283$	−14.8424	−0.882286	−0.441143	+	0.897437i	$0.645427\pi$
$284$	0	0
$285$	−7.29150	−0.431911
$286$	0	0
$287$	−14.9785	−0.884153
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	6.72057	0.393967
$292$	0	0
$293$	−31.8745	−1.86213	−0.931064	−	0.364855i	$-0.881119\pi$
$293$	−31.8745	−1.86213	−0.931064	+	0.364855i	$0.881119\pi$
$294$	0	0
$295$	11.2712	0.656236
$296$	0	0
$297$	−38.5830	−2.23881
$298$	0	0
$299$	7.70010	0.445308
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	20.1617	1.15826
$304$	0	0
$305$	−2.70850	−0.155088
$306$	0	0
$307$	23.3111	1.33044	0.665218	−	0.746649i	$-0.268338\pi$
$307$	23.3111	1.33044	0.665218	+	0.746649i	$0.268338\pi$
$308$	0	0
$309$	40.4575	2.30155
$310$	0	0
$311$	24.9232	1.41327	0.706633	−	0.707581i	$-0.250213\pi$
$311$	24.9232	1.41327	0.706633	+	0.707581i	$0.250213\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	−9.87000	−0.556112
$316$	0	0
$317$	−2.70850	−0.152124	−0.0760622	−	0.997103i	$-0.524235\pi$
$317$	−2.70850	−0.152124	−0.0760622	+	0.997103i	$0.524235\pi$
$318$	0	0
$319$	11.4821	0.642872
$320$	0	0
$321$	−19.2915	−1.07675
$322$	0	0
$323$	13.0194	0.724421
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	6.72057	0.371648
$328$	0	0
$329$	1.41699	0.0781214
$330$	0	0
$331$	−33.8137	−1.85857	−0.929284	−	0.369366i	$-0.879575\pi$
$331$	−33.8137	−1.85857	−0.929284	+	0.369366i	$0.879575\pi$
$332$	0	0
$333$	−77.0405	−4.22179
$334$	0	0
$335$	5.53019	0.302147
$336$	0	0
$337$	−0.583005	−0.0317583	−0.0158792	−	0.999874i	$-0.505055\pi$
$337$	−0.583005	−0.0317583	−0.0158792	+	0.999874i	$0.505055\pi$
$338$	0	0
$339$	28.8413	1.56645
$340$	0	0
$341$	9.87451	0.534735
$342$	0	0
$343$	14.9785	0.808763
$344$	0	0
$345$	−25.8745	−1.39304
$346$	0	0
$347$	−1.40122	−0.0752215	−0.0376107	−	0.999292i	$-0.511975\pi$
$347$	−1.40122	−0.0752215	−0.0376107	+	0.999292i	$0.511975\pi$
$348$	0	0
$349$	−11.1660	−0.597703	−0.298851	−	0.954300i	$-0.596603\pi$
$349$	−11.1660	−0.597703	−0.298851	+	0.954300i	$0.596603\pi$
$350$	0	0
$351$	−17.7809	−0.949077
$352$	0	0
$353$	26.4575	1.40819	0.704096	−	0.710105i	$-0.251353\pi$
$353$	26.4575	1.40819	0.704096	+	0.710105i	$0.251353\pi$
$354$	0	0
$355$	−6.50972	−0.345500
$356$	0	0
$357$	24.0000	1.27021
$358$	0	0
$359$	17.9918	0.949570	0.474785	−	0.880102i	$-0.342526\pi$
$359$	17.9918	0.949570	0.474785	+	0.880102i	$0.342526\pi$
$360$	0	0
$361$	−14.2915	−0.752184
$362$	0	0
$363$	21.1412	1.10963
$364$	0	0
$365$	−1.29150	−0.0676003
$366$	0	0
$367$	−9.65916	−0.504204	−0.252102	−	0.967701i	$-0.581122\pi$
$367$	−9.65916	−0.504204	−0.252102	+	0.967701i	$0.581122\pi$
$368$	0	0
$369$	104.332	5.43131
$370$	0	0
$371$	−10.2170	−0.530440
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	−3.36028	−0.173524
$376$	0	0
$377$	5.29150	0.272526
$378$	0	0
$379$	4.55066	0.233752	0.116876	−	0.993147i	$-0.462712\pi$
$379$	4.55066	0.233752	0.116876	+	0.993147i	$0.462712\pi$
$380$	0	0
$381$	−40.4575	−2.07270
$382$	0	0
$383$	9.87000	0.504334	0.252167	−	0.967684i	$-0.418857\pi$
$383$	9.87000	0.504334	0.252167	+	0.967684i	$0.418857\pi$
$384$	0	0
$385$	−2.58301	−0.131642
$386$	0	0
$387$	−83.5854	−4.24888
$388$	0	0
$389$	−20.5830	−1.04360	−0.521800	−	0.853068i	$-0.674739\pi$
$389$	−20.5830	−1.04360	−0.521800	+	0.853068i	$0.674739\pi$
$390$	0	0
$391$	46.2006	2.33646
$392$	0	0
$393$	−53.1660	−2.68187
$394$	0	0
$395$	−4.33981	−0.218360
$396$	0	0
$397$	6.70850	0.336690	0.168345	−	0.985728i	$-0.446158\pi$
$397$	6.70850	0.336690	0.168345	+	0.985728i	$0.446158\pi$
$398$	0	0
$399$	8.67963	0.434525
$400$	0	0
$401$	−20.5830	−1.02787	−0.513933	−	0.857830i	$-0.671812\pi$
$401$	−20.5830	−1.02787	−0.513933	+	0.857830i	$0.671812\pi$
$402$	0	0
$403$	4.55066	0.226684
$404$	0	0
$405$	34.8745	1.73293
$406$	0	0
$407$	−20.1617	−0.999378
$408$	0	0
$409$	−8.58301	−0.424402	−0.212201	−	0.977226i	$-0.568063\pi$
$409$	−8.58301	−0.424402	−0.212201	+	0.977226i	$0.568063\pi$
$410$	0	0
$411$	−20.1617	−0.994503
$412$	0	0
$413$	−13.4170	−0.660207
$414$	0	0
$415$	−1.19038	−0.0584333
$416$	0	0
$417$	14.5830	0.714133
$418$	0	0
$419$	35.9836	1.75791	0.878957	−	0.476902i	$-0.158240\pi$
$419$	35.9836	1.75791	0.878957	+	0.476902i	$0.158240\pi$
$420$	0	0
$421$	−31.1660	−1.51894	−0.759469	−	0.650543i	$-0.774541\pi$
$421$	−31.1660	−1.51894	−0.759469	+	0.650543i	$0.774541\pi$
$422$	0	0
$423$	−9.87000	−0.479896
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	3.22413	0.156027
$428$	0	0
$429$	−7.29150	−0.352037
$430$	0	0
$431$	−12.8086	−0.616968	−0.308484	−	0.951229i	$-0.599822\pi$
$431$	−12.8086	−0.616968	−0.308484	+	0.951229i	$0.599822\pi$
$432$	0	0
$433$	−19.1660	−0.921060	−0.460530	−	0.887644i	$-0.652341\pi$
$433$	−19.1660	−0.921060	−0.460530	+	0.887644i	$0.652341\pi$
$434$	0	0
$435$	−17.7809	−0.852531
$436$	0	0
$437$	16.7085	0.799276
$438$	0	0
$439$	−17.3593	−0.828512	−0.414256	−	0.910160i	$-0.635958\pi$
$439$	−17.3593	−0.828512	−0.414256	+	0.910160i	$0.635958\pi$
$440$	0	0
$441$	−46.2915	−2.20436
$442$	0	0
$443$	−12.8833	−0.612104	−0.306052	−	0.952015i	$-0.599008\pi$
$443$	−12.8833	−0.612104	−0.306052	+	0.952015i	$0.599008\pi$
$444$	0	0
$445$	−4.58301	−0.217255
$446$	0	0
$447$	28.8413	1.36415
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	27.3040	1.28569
$452$	0	0
$453$	52.4575	2.46467
$454$	0	0
$455$	−1.19038	−0.0558057
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	−106.686	−4.97966
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−39.5547	−1.83826	−0.919132	−	0.393951i	$-0.871108\pi$
$463$	−39.5547	−1.83826	−0.919132	+	0.393951i	$0.871108\pi$
$464$	0	0
$465$	−15.2915	−0.709126
$466$	0	0
$467$	39.3439	1.82062	0.910308	−	0.413930i	$-0.135844\pi$
$467$	39.3439	1.82062	0.910308	+	0.413930i	$0.135844\pi$
$468$	0	0
$469$	−6.58301	−0.303975
$470$	0	0
$471$	−47.0440	−2.16767
$472$	0	0
$473$	−21.8745	−1.00579
$474$	0	0
$475$	2.16991	0.0995622
$476$	0	0
$477$	71.1660	3.25847
$478$	0	0
$479$	−13.6520	−0.623775	−0.311887	−	0.950119i	$-0.600961\pi$
$479$	−13.6520	−0.623775	−0.311887	+	0.950119i	$0.600961\pi$
$480$	0	0
$481$	−9.29150	−0.423656
$482$	0	0
$483$	30.8004	1.40147
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−5.53019	−0.250597	−0.125298	−	0.992119i	$-0.539989\pi$
$487$	−5.53019	−0.250597	−0.125298	+	0.992119i	$0.539989\pi$
$488$	0	0
$489$	−71.7490	−3.24460
$490$	0	0
$491$	10.6387	0.480117	0.240059	−	0.970758i	$-0.422833\pi$
$491$	10.6387	0.480117	0.240059	+	0.970758i	$0.422833\pi$
$492$	0	0
$493$	31.7490	1.42990
$494$	0	0
$495$	17.9918	0.808671
$496$	0	0
$497$	7.74902	0.347591
$498$	0	0
$499$	−11.2712	−0.504569	−0.252285	−	0.967653i	$-0.581182\pi$
$499$	−11.2712	−0.504569	−0.252285	+	0.967653i	$0.581182\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	0	0
$503$	−40.8812	−1.82280	−0.911402	−	0.411517i	$-0.864999\pi$
$503$	−40.8812	−1.82280	−0.911402	+	0.411517i	$0.864999\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	−3.36028	−0.149235
$508$	0	0
$509$	44.3320	1.96498	0.982491	−	0.186309i	$-0.0596527\pi$
$509$	44.3320	1.96498	0.982491	+	0.186309i	$0.0596527\pi$
$510$	0	0
$511$	1.53737	0.0680094
$512$	0	0
$513$	−38.5830	−1.70348
$514$	0	0
$515$	−12.0399	−0.530542
$516$	0	0
$517$	−2.58301	−0.113600
$518$	0	0
$519$	73.9262	3.24500
$520$	0	0
$521$	34.4575	1.50961	0.754806	−	0.655949i	$-0.227731\pi$
$521$	34.4575	1.50961	0.754806	+	0.655949i	$0.227731\pi$
$522$	0	0
$523$	14.8424	0.649011	0.324505	−	0.945884i	$-0.394802\pi$
$523$	14.8424	0.649011	0.324505	+	0.945884i	$0.394802\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	27.3040	1.18938
$528$	0	0
$529$	36.2915	1.57789
$530$	0	0
$531$	93.4554	4.05562
$532$	0	0
$533$	12.5830	0.545030
$534$	0	0
$535$	5.74103	0.248207
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	−12.1146	−0.521813
$540$	0	0
$541$	−36.5830	−1.57283	−0.786413	−	0.617701i	$-0.788064\pi$
$541$	−36.5830	−1.57283	−0.786413	+	0.617701i	$0.788064\pi$
$542$	0	0
$543$	−17.7809	−0.763053
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−11.6182	−0.496759	−0.248380	−	0.968663i	$-0.579898\pi$
$547$	−11.6182	−0.496759	−0.248380	+	0.968663i	$0.579898\pi$
$548$	0	0
$549$	−22.4575	−0.958463
$550$	0	0
$551$	11.4821	0.489153
$552$	0	0
$553$	5.16601	0.219681
$554$	0	0
$555$	31.2221	1.32530
$556$	0	0
$557$	−17.2915	−0.732664	−0.366332	−	0.930484i	$-0.619387\pi$
$557$	−17.2915	−0.732664	−0.366332	+	0.930484i	$0.619387\pi$
$558$	0	0
$559$	−10.0808	−0.426374
$560$	0	0
$561$	−43.7490	−1.84708
$562$	0	0
$563$	−23.5220	−0.991333	−0.495667	−	0.868513i	$-0.665076\pi$
$563$	−23.5220	−0.991333	−0.495667	+	0.868513i	$0.665076\pi$
$564$	0	0
$565$	−8.58301	−0.361090
$566$	0	0
$567$	−41.5138	−1.74341
$568$	0	0
$569$	−22.4575	−0.941468	−0.470734	−	0.882275i	$-0.656011\pi$
$569$	−22.4575	−0.941468	−0.470734	+	0.882275i	$0.656011\pi$
$570$	0	0
$571$	−11.4821	−0.480510	−0.240255	−	0.970710i	$-0.577231\pi$
$571$	−11.4821	−0.480510	−0.240255	+	0.970710i	$0.577231\pi$
$572$	0	0
$573$	−53.1660	−2.22104
$574$	0	0
$575$	7.70010	0.321116
$576$	0	0
$577$	22.7085	0.945367	0.472684	−	0.881232i	$-0.343285\pi$
$577$	22.7085	0.945367	0.472684	+	0.881232i	$0.343285\pi$
$578$	0	0
$579$	6.72057	0.279297
$580$	0	0
$581$	1.41699	0.0587868
$582$	0	0
$583$	18.6243	0.771341
$584$	0	0
$585$	8.29150	0.342811
$586$	0	0
$587$	18.9713	0.783030	0.391515	−	0.920172i	$-0.371951\pi$
$587$	18.9713	0.783030	0.391515	+	0.920172i	$0.371951\pi$
$588$	0	0
$589$	9.87451	0.406872
$590$	0	0
$591$	6.72057	0.276447
$592$	0	0
$593$	19.1660	0.787054	0.393527	−	0.919313i	$-0.371255\pi$
$593$	19.1660	0.787054	0.393527	+	0.919313i	$0.371255\pi$
$594$	0	0
$595$	−7.14226	−0.292804
$596$	0	0
$597$	74.3320	3.04221
$598$	0	0
$599$	−22.5425	−0.921060	−0.460530	−	0.887644i	$-0.652341\pi$
$599$	−22.5425	−0.921060	−0.460530	+	0.887644i	$0.652341\pi$
$600$	0	0
$601$	19.4170	0.792036	0.396018	−	0.918243i	$-0.370392\pi$
$601$	19.4170	0.792036	0.396018	+	0.918243i	$0.370392\pi$
$602$	0	0
$603$	45.8536	1.86730
$604$	0	0
$605$	−6.29150	−0.255786
$606$	0	0
$607$	13.9990	0.568201	0.284100	−	0.958795i	$-0.408305\pi$
$607$	13.9990	0.568201	0.284100	+	0.958795i	$0.408305\pi$
$608$	0	0
$609$	21.1660	0.857690
$610$	0	0
$611$	−1.19038	−0.0481575
$612$	0	0
$613$	35.1660	1.42034	0.710171	−	0.704029i	$-0.248618\pi$
$613$	35.1660	1.42034	0.710171	+	0.704029i	$0.248618\pi$
$614$	0	0
$615$	−42.2825	−1.70499
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−11.6929	−0.469978	−0.234989	−	0.971998i	$-0.575505\pi$
$619$	−11.6929	−0.469978	−0.234989	+	0.971998i	$0.575505\pi$
$620$	0	0
$621$	−136.915	−5.49421
$622$	0	0
$623$	5.45550	0.218570
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−15.8219	−0.631865
$628$	0	0
$629$	−55.7490	−2.22286
$630$	0	0
$631$	−8.46878	−0.337137	−0.168568	−	0.985690i	$-0.553914\pi$
$631$	−8.46878	−0.337137	−0.168568	+	0.985690i	$0.553914\pi$
$632$	0	0
$633$	−53.1660	−2.11316
$634$	0	0
$635$	12.0399	0.477789
$636$	0	0
$637$	−5.58301	−0.221207
$638$	0	0
$639$	−53.9754	−2.13523
$640$	0	0
$641$	−25.7490	−1.01702	−0.508512	−	0.861055i	$-0.669804\pi$
$641$	−25.7490	−1.01702	−0.508512	+	0.861055i	$0.669804\pi$
$642$	0	0
$643$	17.0123	0.670898	0.335449	−	0.942058i	$-0.391112\pi$
$643$	17.0123	0.670898	0.335449	+	0.942058i	$0.391112\pi$
$644$	0	0
$645$	33.8745	1.33381
$646$	0	0
$647$	30.2425	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$647$	30.2425	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$648$	0	0
$649$	24.4575	0.960041
$650$	0	0
$651$	18.2026	0.713418
$652$	0	0
$653$	−11.4170	−0.446782	−0.223391	−	0.974729i	$-0.571713\pi$
$653$	−11.4170	−0.446782	−0.223391	+	0.974729i	$0.571713\pi$
$654$	0	0
$655$	15.8219	0.618212
$656$	0	0
$657$	−10.7085	−0.417778
$658$	0	0
$659$	−35.9836	−1.40172	−0.700861	−	0.713298i	$-0.747201\pi$
$659$	−35.9836	−1.40172	−0.700861	+	0.713298i	$0.747201\pi$
$660$	0	0
$661$	−15.1660	−0.589889	−0.294945	−	0.955514i	$-0.595301\pi$
$661$	−15.1660	−0.589889	−0.294945	+	0.955514i	$0.595301\pi$
$662$	0	0
$663$	−20.1617	−0.783015
$664$	0	0
$665$	−2.58301	−0.100165
$666$	0	0
$667$	40.7451	1.57766
$668$	0	0
$669$	57.1660	2.21017
$670$	0	0
$671$	−5.87719	−0.226886
$672$	0	0
$673$	27.1660	1.04717	0.523586	−	0.851973i	$-0.324594\pi$
$673$	27.1660	1.04717	0.523586	+	0.851973i	$0.324594\pi$
$674$	0	0
$675$	−17.7809	−0.684389
$676$	0	0
$677$	24.5830	0.944802	0.472401	−	0.881384i	$-0.343387\pi$
$677$	24.5830	0.944802	0.472401	+	0.881384i	$0.343387\pi$
$678$	0	0
$679$	2.38075	0.0913649
$680$	0	0
$681$	26.5830	1.01866
$682$	0	0
$683$	−9.44832	−0.361530	−0.180765	−	0.983526i	$-0.557857\pi$
$683$	−9.44832	−0.361530	−0.180765	+	0.983526i	$0.557857\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	−55.7236	−2.12599
$688$	0	0
$689$	8.58301	0.326986
$690$	0	0
$691$	19.9509	0.758966	0.379483	−	0.925199i	$-0.376102\pi$
$691$	19.9509	0.758966	0.379483	+	0.925199i	$0.376102\pi$
$692$	0	0
$693$	−21.4170	−0.813564
$694$	0	0
$695$	−4.33981	−0.164619
$696$	0	0
$697$	75.4980	2.85969
$698$	0	0
$699$	−33.6028	−1.27098
$700$	0	0
$701$	32.5830	1.23064	0.615322	−	0.788276i	$-0.289026\pi$
$701$	32.5830	1.23064	0.615322	+	0.788276i	$0.289026\pi$
$702$	0	0
$703$	−20.1617	−0.760413
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	7.14226	0.268612
$708$	0	0
$709$	40.5830	1.52413	0.762063	−	0.647502i	$-0.224186\pi$
$709$	40.5830	1.52413	0.762063	+	0.647502i	$0.224186\pi$
$710$	0	0
$711$	−35.9836	−1.34949
$712$	0	0
$713$	35.0405	1.31228
$714$	0	0
$715$	2.16991	0.0811499
$716$	0	0
$717$	−37.8745	−1.41445
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	14.3320	0.533752
$722$	0	0
$723$	−15.4002	−0.572739
$724$	0	0
$725$	5.29150	0.196521
$726$	0	0
$727$	10.5025	0.389518	0.194759	−	0.980851i	$-0.437608\pi$
$727$	10.5025	0.389518	0.194759	+	0.980851i	$0.437608\pi$
$728$	0	0
$729$	109.915	4.07093
$730$	0	0
$731$	−60.4851	−2.23712
$732$	0	0
$733$	43.1660	1.59437	0.797186	−	0.603733i	$-0.206321\pi$
$733$	43.1660	1.59437	0.797186	+	0.603733i	$0.206321\pi$
$734$	0	0
$735$	18.7605	0.691991
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−26.6714	−0.981124	−0.490562	−	0.871406i	$-0.663208\pi$
$739$	−26.6714	−0.981124	−0.490562	+	0.871406i	$0.663208\pi$
$740$	0	0
$741$	−7.29150	−0.267860
$742$	0	0
$743$	−12.2508	−0.449437	−0.224718	−	0.974424i	$-0.572146\pi$
$743$	−12.2508	−0.449437	−0.224718	+	0.974424i	$0.572146\pi$
$744$	0	0
$745$	−8.58301	−0.314457
$746$	0	0
$747$	−9.87000	−0.361125
$748$	0	0
$749$	−6.83399	−0.249709
$750$	0	0
$751$	−24.0798	−0.878685	−0.439343	−	0.898320i	$-0.644789\pi$
$751$	−24.0798	−0.878685	−0.439343	+	0.898320i	$0.644789\pi$
$752$	0	0
$753$	−96.9150	−3.53178
$754$	0	0
$755$	−15.6110	−0.568144
$756$	0	0
$757$	−0.833990	−0.0303119	−0.0151559	−	0.999885i	$-0.504824\pi$
$757$	−0.833990	−0.0303119	−0.0151559	+	0.999885i	$0.504824\pi$
$758$	0	0
$759$	−56.1453	−2.03794
$760$	0	0
$761$	7.16601	0.259768	0.129884	−	0.991529i	$-0.458540\pi$
$761$	7.16601	0.259768	0.129884	+	0.991529i	$0.458540\pi$
$762$	0	0
$763$	2.38075	0.0861890
$764$	0	0
$765$	49.7490	1.79868
$766$	0	0
$767$	11.2712	0.406980
$768$	0	0
$769$	−43.1660	−1.55661	−0.778303	−	0.627889i	$-0.783919\pi$
$769$	−43.1660	−1.55661	−0.778303	+	0.627889i	$0.783919\pi$
$770$	0	0
$771$	−55.7236	−2.00684
$772$	0	0
$773$	−33.2915	−1.19741	−0.598706	−	0.800969i	$-0.704318\pi$
$773$	−33.2915	−1.19741	−0.598706	+	0.800969i	$0.704318\pi$
$774$	0	0
$775$	4.55066	0.163464
$776$	0	0
$777$	−37.1660	−1.33332
$778$	0	0
$779$	27.3040	0.978266
$780$	0	0
$781$	−14.1255	−0.505450
$782$	0	0
$783$	−94.0879	−3.36243
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	40.6704	1.44974	0.724872	−	0.688884i	$-0.241899\pi$
$787$	40.6704	1.44974	0.724872	+	0.688884i	$0.241899\pi$
$788$	0	0
$789$	25.8745	0.921157
$790$	0	0
$791$	10.2170	0.363275
$792$	0	0
$793$	−2.70850	−0.0961816
$794$	0	0
$795$	−28.8413	−1.02290
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−7.14226	−0.252675
$800$	0	0
$801$	−38.0000	−1.34266
$802$	0	0
$803$	−2.80244	−0.0988960
$804$	0	0
$805$	−9.16601	−0.323059
$806$	0	0
$807$	−11.4821	−0.404188
$808$	0	0
$809$	−25.2915	−0.889202	−0.444601	−	0.895729i	$-0.646654\pi$
$809$	−25.2915	−0.889202	−0.444601	+	0.895729i	$0.646654\pi$
$810$	0	0
$811$	15.1894	0.533370	0.266685	−	0.963784i	$-0.414072\pi$
$811$	15.1894	0.533370	0.266685	+	0.963784i	$0.414072\pi$
$812$	0	0
$813$	−96.2065	−3.37411
$814$	0	0
$815$	21.3521	0.747931
$816$	0	0
$817$	−21.8745	−0.765292
$818$	0	0
$819$	−9.87000	−0.344886
$820$	0	0
$821$	−37.7490	−1.31745	−0.658725	−	0.752384i	$-0.728904\pi$
$821$	−37.7490	−1.31745	−0.658725	+	0.752384i	$0.728904\pi$
$822$	0	0
$823$	−25.4810	−0.888213	−0.444107	−	0.895974i	$-0.646479\pi$
$823$	−25.4810	−0.888213	−0.444107	+	0.895974i	$0.646479\pi$
$824$	0	0
$825$	−7.29150	−0.253858
$826$	0	0
$827$	50.1934	1.74540	0.872698	−	0.488261i	$-0.162368\pi$
$827$	50.1934	1.74540	0.872698	+	0.488261i	$0.162368\pi$
$828$	0	0
$829$	−10.7085	−0.371921	−0.185961	−	0.982557i	$-0.559540\pi$
$829$	−10.7085	−0.371921	−0.185961	+	0.982557i	$0.559540\pi$
$830$	0	0
$831$	−33.6028	−1.16567
$832$	0	0
$833$	−33.4980	−1.16064
$834$	0	0
$835$	−5.95188	−0.205973
$836$	0	0
$837$	−80.9150	−2.79683
$838$	0	0
$839$	15.6110	0.538953	0.269476	−	0.963007i	$-0.413149\pi$
$839$	15.6110	0.538953	0.269476	+	0.963007i	$0.413149\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	−69.1647	−2.38216
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	7.48925	0.257334
$848$	0	0
$849$	49.8745	1.71169
$850$	0	0
$851$	−71.5455	−2.45255
$852$	0	0
$853$	−10.7085	−0.366652	−0.183326	−	0.983052i	$-0.558686\pi$
$853$	−10.7085	−0.366652	−0.183326	+	0.983052i	$0.558686\pi$
$854$	0	0
$855$	17.9918	0.615306
$856$	0	0
$857$	3.41699	0.116722	0.0583612	−	0.998296i	$-0.481413\pi$
$857$	3.41699	0.116722	0.0583612	+	0.998296i	$0.481413\pi$
$858$	0	0
$859$	−39.9017	−1.36143	−0.680714	−	0.732549i	$-0.738331\pi$
$859$	−39.9017	−1.36143	−0.680714	+	0.732549i	$0.738331\pi$
$860$	0	0
$861$	50.3320	1.71531
$862$	0	0
$863$	−30.4534	−1.03665	−0.518323	−	0.855185i	$-0.673443\pi$
$863$	−30.4534	−1.03665	−0.518323	+	0.855185i	$0.673443\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	0	0
$867$	−63.8454	−2.16830
$868$	0	0
$869$	−9.41699	−0.319450
$870$	0	0
$871$	5.53019	0.187383
$872$	0	0
$873$	−16.5830	−0.561250
$874$	0	0
$875$	−1.19038	−0.0402420
$876$	0	0
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	0	0
$879$	107.107	3.61264
$880$	0	0
$881$	18.4575	0.621849	0.310925	−	0.950435i	$-0.399361\pi$
$881$	18.4575	0.621849	0.310925	+	0.950435i	$0.399361\pi$
$882$	0	0
$883$	−9.23747	−0.310866	−0.155433	−	0.987846i	$-0.549677\pi$
$883$	−9.23747	−0.310866	−0.155433	+	0.987846i	$0.549677\pi$
$884$	0	0
$885$	−37.8745	−1.27314
$886$	0	0
$887$	30.6642	1.02960	0.514802	−	0.857309i	$-0.327865\pi$
$887$	30.6642	1.02960	0.514802	+	0.857309i	$0.327865\pi$
$888$	0	0
$889$	−14.3320	−0.480681
$890$	0	0
$891$	75.6744	2.53519
$892$	0	0
$893$	−2.58301	−0.0864370
$894$	0	0
$895$	4.76150	0.159160
$896$	0	0
$897$	−25.8745	−0.863925
$898$	0	0
$899$	24.0798	0.803107
$900$	0	0
$901$	51.4980	1.71565
$902$	0	0
$903$	−40.3234	−1.34188
$904$	0	0
$905$	5.29150	0.175895
$906$	0	0
$907$	25.0594	0.832082	0.416041	−	0.909346i	$-0.363417\pi$
$907$	25.0594	0.832082	0.416041	+	0.909346i	$0.363417\pi$
$908$	0	0
$909$	−49.7490	−1.65007
$910$	0	0
$911$	−2.38075	−0.0788778	−0.0394389	−	0.999222i	$-0.512557\pi$
$911$	−2.38075	−0.0788778	−0.0394389	+	0.999222i	$0.512557\pi$
$912$	0	0
$913$	−2.58301	−0.0854850
$914$	0	0
$915$	9.10132	0.300880
$916$	0	0
$917$	−18.8340	−0.621953
$918$	0	0
$919$	26.4606	0.872854	0.436427	−	0.899740i	$-0.356244\pi$
$919$	26.4606	0.872854	0.436427	+	0.899740i	$0.356244\pi$
$920$	0	0
$921$	−78.3320	−2.58113
$922$	0	0
$923$	−6.50972	−0.214270
$924$	0	0
$925$	−9.29150	−0.305503
$926$	0	0
$927$	−99.8290	−3.27881
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	−12.1146	−0.397040
$932$	0	0
$933$	−83.7490	−2.74182
$934$	0	0
$935$	13.0194	0.425781
$936$	0	0
$937$	−11.1660	−0.364778	−0.182389	−	0.983226i	$-0.558383\pi$
$937$	−11.1660	−0.364778	−0.182389	+	0.983226i	$0.558383\pi$
$938$	0	0
$939$	6.72057	0.219317
$940$	0	0
$941$	13.7490	0.448205	0.224103	−	0.974566i	$-0.428055\pi$
$941$	13.7490	0.448205	0.224103	+	0.974566i	$0.428055\pi$
$942$	0	0
$943$	96.8904	3.15518
$944$	0	0
$945$	21.1660	0.688530
$946$	0	0
$947$	59.7164	1.94052	0.970261	−	0.242060i	$-0.0778231\pi$
$947$	59.7164	1.94052	0.970261	+	0.242060i	$0.0778231\pi$
$948$	0	0
$949$	−1.29150	−0.0419239
$950$	0	0
$951$	9.10132	0.295130
$952$	0	0
$953$	37.7490	1.22281	0.611405	−	0.791318i	$-0.290605\pi$
$953$	37.7490	1.22281	0.611405	+	0.791318i	$0.290605\pi$
$954$	0	0
$955$	15.8219	0.511984
$956$	0	0
$957$	−38.5830	−1.24721
$958$	0	0
$959$	−7.14226	−0.230635
$960$	0	0
$961$	−10.2915	−0.331984
$962$	0	0
$963$	47.6018	1.53395
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−28.4943	−0.916316	−0.458158	−	0.888871i	$-0.651491\pi$
$967$	−28.4943	−0.916316	−0.458158	+	0.888871i	$0.651491\pi$
$968$	0	0
$969$	−43.7490	−1.40542
$970$	0	0
$971$	−50.5404	−1.62192	−0.810959	−	0.585103i	$-0.801054\pi$
$971$	−50.5404	−1.62192	−0.810959	+	0.585103i	$0.801054\pi$
$972$	0	0
$973$	5.16601	0.165615
$974$	0	0
$975$	−3.36028	−0.107615
$976$	0	0
$977$	−54.4575	−1.74225	−0.871125	−	0.491061i	$-0.836609\pi$
$977$	−54.4575	−1.74225	−0.871125	+	0.491061i	$0.836609\pi$
$978$	0	0
$979$	−9.94470	−0.317834
$980$	0	0
$981$	−16.5830	−0.529455
$982$	0	0
$983$	22.8894	0.730060	0.365030	−	0.930996i	$-0.381059\pi$
$983$	22.8894	0.730060	0.365030	+	0.930996i	$0.381059\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−4.76150	−0.151560
$988$	0	0
$989$	−77.6235	−2.46828
$990$	0	0
$991$	10.2170	0.324554	0.162277	−	0.986745i	$-0.448116\pi$
$991$	10.2170	0.324554	0.162277	+	0.986745i	$0.448116\pi$
$992$	0	0
$993$	113.624	3.60573
$994$	0	0
$995$	−22.1208	−0.701275
$996$	0	0
$997$	19.4170	0.614942	0.307471	−	0.951557i	$-0.400517\pi$
$997$	19.4170	0.614942	0.307471	+	0.951557i	$0.400517\pi$
$998$	0	0
$999$	165.212	5.22707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bv.1.1		4
4.3	odd	2	inner	4160.2.a.bv.1.4		4
8.3	odd	2		2080.2.a.r.1.1	✓	4
8.5	even	2		2080.2.a.r.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.r.1.1	✓	4	8.3	odd	2
2080.2.a.r.1.4	yes	4	8.5	even	2
4160.2.a.bv.1.1		4	1.1	even	1	trivial
4160.2.a.bv.1.4		4	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q-3.36028 q^{3} +1.00000 q^{5} -1.19038 q^{7} +8.29150 q^{9} +O(q^{10})\)	\(q-3.36028 q^{3} +1.00000 q^{5} -1.19038 q^{7} +8.29150 q^{9} +2.16991 q^{11} +1.00000 q^{13} -3.36028 q^{15} +6.00000 q^{17} +2.16991 q^{19} +4.00000 q^{21} +7.70010 q^{23} +1.00000 q^{25} -17.7809 q^{27} +5.29150 q^{29} +4.55066 q^{31} -7.29150 q^{33} -1.19038 q^{35} -9.29150 q^{37} -3.36028 q^{39} +12.5830 q^{41} -10.0808 q^{43} +8.29150 q^{45} -1.19038 q^{47} -5.58301 q^{49} -20.1617 q^{51} +8.58301 q^{53} +2.16991 q^{55} -7.29150 q^{57} +11.2712 q^{59} -2.70850 q^{61} -9.87000 q^{63} +1.00000 q^{65} +5.53019 q^{67} -25.8745 q^{69} -6.50972 q^{71} -1.29150 q^{73} -3.36028 q^{75} -2.58301 q^{77} -4.33981 q^{79} +34.8745 q^{81} -1.19038 q^{83} +6.00000 q^{85} -17.7809 q^{87} -4.58301 q^{89} -1.19038 q^{91} -15.2915 q^{93} +2.16991 q^{95} -2.00000 q^{97} +17.9918 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 12 * q^9	\( 4 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{17} + 16 q^{21} + 4 q^{25} - 8 q^{33} - 16 q^{37} + 8 q^{41} + 12 q^{45} + 20 q^{49} - 8 q^{53} - 8 q^{57} - 32 q^{61} + 4 q^{65} - 40 q^{69} + 16 q^{73} + 32 q^{77} + 76 q^{81} + 24 q^{85} + 24 q^{89} - 40 q^{93} - 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 12 * q^9 + 4 * q^13 + 24 * q^17 + 16 * q^21 + 4 * q^25 - 8 * q^33 - 16 * q^37 + 8 * q^41 + 12 * q^45 + 20 * q^49 - 8 * q^53 - 8 * q^57 - 32 * q^61 + 4 * q^65 - 40 * q^69 + 16 * q^73 + 32 * q^77 + 76 * q^81 + 24 * q^85 + 24 * q^89 - 40 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

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Database

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