LMFDB - Embedded newform 2960.2.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.935504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 $ x^5 - x^4 - 8x^3 + 4x^2 + 8*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.92141$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.92141 q^{3} -1.00000 q^{5} -0.236809 q^{7} +5.53464 q^{9} +O(q^{10})$	$q+2.92141 q^{3} -1.00000 q^{5} -0.236809 q^{7} +5.53464 q^{9} +4.24650 q^{11} +3.08106 q^{13} -2.92141 q^{15} +2.28814 q^{17} +1.26563 q^{19} -0.691817 q^{21} -3.84282 q^{23} +1.00000 q^{25} +7.40471 q^{27} -2.47362 q^{29} -2.28110 q^{31} +12.4058 q^{33} +0.236809 q^{35} +1.00000 q^{37} +9.00104 q^{39} +6.56293 q^{41} -7.54289 q^{43} -5.53464 q^{45} +7.93688 q^{47} -6.94392 q^{49} +6.68460 q^{51} -5.18216 q^{53} -4.24650 q^{55} +3.69743 q^{57} +7.56172 q^{59} -5.49787 q^{61} -1.31065 q^{63} -3.08106 q^{65} +9.10845 q^{67} -11.2265 q^{69} +6.18704 q^{71} +2.34422 q^{73} +2.92141 q^{75} -1.00561 q^{77} +3.35952 q^{79} +5.02830 q^{81} -1.55715 q^{83} -2.28814 q^{85} -7.22645 q^{87} -16.6072 q^{89} -0.729623 q^{91} -6.66404 q^{93} -1.26563 q^{95} +16.9522 q^{97} +23.5028 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	$ 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) $ 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * 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
$3$	2.92141	1.68668	0.843338	−	0.537383i	$-0.180587\pi$
$3$	2.92141	1.68668	0.843338	+	0.537383i	$0.180587\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.236809	−0.0895055	−0.0447527	−	0.998998i	$-0.514250\pi$
$7$	−0.236809	−0.0895055	−0.0447527	+	0.998998i	$0.514250\pi$
$8$	0	0
$9$	5.53464	1.84488
$10$	0	0
$11$	4.24650	1.28037	0.640183	−	0.768222i	$-0.278858\pi$
$11$	4.24650	1.28037	0.640183	+	0.768222i	$0.278858\pi$
$12$	0	0
$13$	3.08106	0.854532	0.427266	−	0.904126i	$-0.359477\pi$
$13$	3.08106	0.854532	0.427266	+	0.904126i	$0.359477\pi$
$14$	0	0
$15$	−2.92141	−0.754305
$16$	0	0
$17$	2.28814	0.554956	0.277478	−	0.960732i	$-0.410501\pi$
$17$	2.28814	0.554956	0.277478	+	0.960732i	$0.410501\pi$
$18$	0	0
$19$	1.26563	0.290356	0.145178	−	0.989406i	$-0.453625\pi$
$19$	1.26563	0.290356	0.145178	+	0.989406i	$0.453625\pi$
$20$	0	0
$21$	−0.691817	−0.150967
$22$	0	0
$23$	−3.84282	−0.801283	−0.400642	−	0.916235i	$-0.631213\pi$
$23$	−3.84282	−0.801283	−0.400642	+	0.916235i	$0.631213\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.40471	1.42504
$28$	0	0
$29$	−2.47362	−0.459339	−0.229670	−	0.973269i	$-0.573765\pi$
$29$	−2.47362	−0.459339	−0.229670	+	0.973269i	$0.573765\pi$
$30$	0	0
$31$	−2.28110	−0.409698	−0.204849	−	0.978794i	$-0.565670\pi$
$31$	−2.28110	−0.409698	−0.204849	+	0.978794i	$0.565670\pi$
$32$	0	0
$33$	12.4058	2.15956
$34$	0	0
$35$	0.236809	0.0400281
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	9.00104	1.44132
$40$	0	0
$41$	6.56293	1.02496	0.512479	−	0.858700i	$-0.328727\pi$
$41$	6.56293	1.02496	0.512479	+	0.858700i	$0.328727\pi$
$42$	0	0
$43$	−7.54289	−1.15028	−0.575140	−	0.818055i	$-0.695052\pi$
$43$	−7.54289	−1.15028	−0.575140	+	0.818055i	$0.695052\pi$
$44$	0	0
$45$	−5.53464	−0.825055
$46$	0	0
$47$	7.93688	1.15771	0.578857	−	0.815429i	$-0.303499\pi$
$47$	7.93688	1.15771	0.578857	+	0.815429i	$0.303499\pi$
$48$	0	0
$49$	−6.94392	−0.991989
$50$	0	0
$51$	6.68460	0.936031
$52$	0	0
$53$	−5.18216	−0.711825	−0.355912	−	0.934519i	$-0.615830\pi$
$53$	−5.18216	−0.711825	−0.355912	+	0.934519i	$0.615830\pi$
$54$	0	0
$55$	−4.24650	−0.572597
$56$	0	0
$57$	3.69743	0.489736
$58$	0	0
$59$	7.56172	0.984452	0.492226	−	0.870467i	$-0.336183\pi$
$59$	7.56172	0.984452	0.492226	+	0.870467i	$0.336183\pi$
$60$	0	0
$61$	−5.49787	−0.703930	−0.351965	−	0.936013i	$-0.614486\pi$
$61$	−5.49787	−0.703930	−0.351965	+	0.936013i	$0.614486\pi$
$62$	0	0
$63$	−1.31065	−0.165127
$64$	0	0
$65$	−3.08106	−0.382158
$66$	0	0
$67$	9.10845	1.11277	0.556387	−	0.830923i	$-0.312187\pi$
$67$	9.10845	1.11277	0.556387	+	0.830923i	$0.312187\pi$
$68$	0	0
$69$	−11.2265	−1.35151
$70$	0	0
$71$	6.18704	0.734267	0.367133	−	0.930168i	$-0.380339\pi$
$71$	6.18704	0.734267	0.367133	+	0.930168i	$0.380339\pi$
$72$	0	0
$73$	2.34422	0.274370	0.137185	−	0.990545i	$-0.456194\pi$
$73$	2.34422	0.274370	0.137185	+	0.990545i	$0.456194\pi$
$74$	0	0
$75$	2.92141	0.337335
$76$	0	0
$77$	−1.00561	−0.114600
$78$	0	0
$79$	3.35952	0.377975	0.188988	−	0.981979i	$-0.439479\pi$
$79$	3.35952	0.377975	0.188988	+	0.981979i	$0.439479\pi$
$80$	0	0
$81$	5.02830	0.558700
$82$	0	0
$83$	−1.55715	−0.170919	−0.0854596	−	0.996342i	$-0.527236\pi$
$83$	−1.55715	−0.170919	−0.0854596	+	0.996342i	$0.527236\pi$
$84$	0	0
$85$	−2.28814	−0.248184
$86$	0	0
$87$	−7.22645	−0.774757
$88$	0	0
$89$	−16.6072	−1.76036	−0.880181	−	0.474638i	$-0.842579\pi$
$89$	−16.6072	−1.76036	−0.880181	+	0.474638i	$0.842579\pi$
$90$	0	0
$91$	−0.729623	−0.0764853
$92$	0	0
$93$	−6.66404	−0.691028
$94$	0	0
$95$	−1.26563	−0.129851
$96$	0	0
$97$	16.9522	1.72123	0.860616	−	0.509254i	$-0.170079\pi$
$97$	16.9522	1.72123	0.860616	+	0.509254i	$0.170079\pi$
$98$	0	0
$99$	23.5028	2.36212
$100$	0	0
$101$	−9.84620	−0.979733	−0.489867	−	0.871797i	$-0.662955\pi$
$101$	−9.84620	−0.979733	−0.489867	+	0.871797i	$0.662955\pi$
$102$	0	0
$103$	6.84951	0.674902	0.337451	−	0.941343i	$-0.390435\pi$
$103$	6.84951	0.674902	0.337451	+	0.941343i	$0.390435\pi$
$104$	0	0
$105$	0.691817	0.0675144
$106$	0	0
$107$	0.666135	0.0643977	0.0321988	−	0.999481i	$-0.489749\pi$
$107$	0.666135	0.0643977	0.0321988	+	0.999481i	$0.489749\pi$
$108$	0	0
$109$	2.29079	0.219418	0.109709	−	0.993964i	$-0.465008\pi$
$109$	2.29079	0.219418	0.109709	+	0.993964i	$0.465008\pi$
$110$	0	0
$111$	2.92141	0.277288
$112$	0	0
$113$	−5.29302	−0.497926	−0.248963	−	0.968513i	$-0.580090\pi$
$113$	−5.29302	−0.497926	−0.248963	+	0.968513i	$0.580090\pi$
$114$	0	0
$115$	3.84282	0.358345
$116$	0	0
$117$	17.0525	1.57651
$118$	0	0
$119$	−0.541853	−0.0496716
$120$	0	0
$121$	7.03272	0.639338
$122$	0	0
$123$	19.1730	1.72877
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	20.4837	1.81763	0.908816	−	0.417197i	$-0.136987\pi$
$127$	20.4837	1.81763	0.908816	+	0.417197i	$0.136987\pi$
$128$	0	0
$129$	−22.0359	−1.94015
$130$	0	0
$131$	10.7199	0.936605	0.468303	−	0.883568i	$-0.344866\pi$
$131$	10.7199	0.936605	0.468303	+	0.883568i	$0.344866\pi$
$132$	0	0
$133$	−0.299713	−0.0259884
$134$	0	0
$135$	−7.40471	−0.637296
$136$	0	0
$137$	2.67582	0.228611	0.114305	−	0.993446i	$-0.463536\pi$
$137$	2.67582	0.228611	0.114305	+	0.993446i	$0.463536\pi$
$138$	0	0
$139$	8.69519	0.737517	0.368758	−	0.929525i	$-0.379783\pi$
$139$	8.69519	0.737517	0.368758	+	0.929525i	$0.379783\pi$
$140$	0	0
$141$	23.1869	1.95269
$142$	0	0
$143$	13.0837	1.09411
$144$	0	0
$145$	2.47362	0.205423
$146$	0	0
$147$	−20.2860	−1.67316
$148$	0	0
$149$	−2.43016	−0.199086	−0.0995432	−	0.995033i	$-0.531738\pi$
$149$	−2.43016	−0.199086	−0.0995432	+	0.995033i	$0.531738\pi$
$150$	0	0
$151$	5.64062	0.459027	0.229513	−	0.973305i	$-0.426286\pi$
$151$	5.64062	0.459027	0.229513	+	0.973305i	$0.426286\pi$
$152$	0	0
$153$	12.6640	1.02383
$154$	0	0
$155$	2.28110	0.183223
$156$	0	0
$157$	18.1488	1.44843	0.724215	−	0.689574i	$-0.242202\pi$
$157$	18.1488	1.44843	0.724215	+	0.689574i	$0.242202\pi$
$158$	0	0
$159$	−15.1392	−1.20062
$160$	0	0
$161$	0.910015	0.0717193
$162$	0	0
$163$	−17.8308	−1.39662	−0.698309	−	0.715797i	$-0.746064\pi$
$163$	−17.8308	−1.39662	−0.698309	+	0.715797i	$0.746064\pi$
$164$	0	0
$165$	−12.4058	−0.965787
$166$	0	0
$167$	−19.1403	−1.48112	−0.740560	−	0.671990i	$-0.765440\pi$
$167$	−19.1403	−1.48112	−0.740560	+	0.671990i	$0.765440\pi$
$168$	0	0
$169$	−3.50707	−0.269775
$170$	0	0
$171$	7.00481	0.535671
$172$	0	0
$173$	−5.64135	−0.428904	−0.214452	−	0.976735i	$-0.568797\pi$
$173$	−5.64135	−0.428904	−0.214452	+	0.976735i	$0.568797\pi$
$174$	0	0
$175$	−0.236809	−0.0179011
$176$	0	0
$177$	22.0909	1.66045
$178$	0	0
$179$	19.6014	1.46508	0.732540	−	0.680724i	$-0.238335\pi$
$179$	19.6014	1.46508	0.732540	+	0.680724i	$0.238335\pi$
$180$	0	0
$181$	1.24873	0.0928173	0.0464086	−	0.998923i	$-0.485222\pi$
$181$	1.24873	0.0928173	0.0464086	+	0.998923i	$0.485222\pi$
$182$	0	0
$183$	−16.0615	−1.18730
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	9.71658	0.710547
$188$	0	0
$189$	−1.75350	−0.127549
$190$	0	0
$191$	−13.7238	−0.993022	−0.496511	−	0.868030i	$-0.665386\pi$
$191$	−13.7238	−0.993022	−0.496511	+	0.868030i	$0.665386\pi$
$192$	0	0
$193$	−13.5169	−0.972967	−0.486484	−	0.873690i	$-0.661721\pi$
$193$	−13.5169	−0.972967	−0.486484	+	0.873690i	$0.661721\pi$
$194$	0	0
$195$	−9.00104	−0.644578
$196$	0	0
$197$	−15.0991	−1.07577	−0.537884	−	0.843019i	$-0.680776\pi$
$197$	−15.0991	−1.07577	−0.537884	+	0.843019i	$0.680776\pi$
$198$	0	0
$199$	−1.46783	−0.104052	−0.0520260	−	0.998646i	$-0.516568\pi$
$199$	−1.46783	−0.104052	−0.0520260	+	0.998646i	$0.516568\pi$
$200$	0	0
$201$	26.6095	1.87689
$202$	0	0
$203$	0.585776	0.0411134
$204$	0	0
$205$	−6.56293	−0.458375
$206$	0	0
$207$	−21.2686	−1.47827
$208$	0	0
$209$	5.37449	0.371762
$210$	0	0
$211$	−17.1185	−1.17849	−0.589243	−	0.807956i	$-0.700574\pi$
$211$	−17.1185	−1.17849	−0.589243	+	0.807956i	$0.700574\pi$
$212$	0	0
$213$	18.0749	1.23847
$214$	0	0
$215$	7.54289	0.514421
$216$	0	0
$217$	0.540186	0.0366702
$218$	0	0
$219$	6.84843	0.462774
$220$	0	0
$221$	7.04990	0.474228
$222$	0	0
$223$	−14.9780	−1.00300	−0.501501	−	0.865157i	$-0.667219\pi$
$223$	−14.9780	−1.00300	−0.501501	+	0.865157i	$0.667219\pi$
$224$	0	0
$225$	5.53464	0.368976
$226$	0	0
$227$	11.0070	0.730558	0.365279	−	0.930898i	$-0.380974\pi$
$227$	11.0070	0.730558	0.365279	+	0.930898i	$0.380974\pi$
$228$	0	0
$229$	−8.52285	−0.563206	−0.281603	−	0.959531i	$-0.590866\pi$
$229$	−8.52285	−0.563206	−0.281603	+	0.959531i	$0.590866\pi$
$230$	0	0
$231$	−2.93780	−0.193293
$232$	0	0
$233$	−24.3122	−1.59275	−0.796374	−	0.604805i	$-0.793251\pi$
$233$	−24.3122	−1.59275	−0.796374	+	0.604805i	$0.793251\pi$
$234$	0	0
$235$	−7.93688	−0.517745
$236$	0	0
$237$	9.81452	0.637522
$238$	0	0
$239$	−15.6847	−1.01456	−0.507281	−	0.861781i	$-0.669349\pi$
$239$	−15.6847	−1.01456	−0.507281	+	0.861781i	$0.669349\pi$
$240$	0	0
$241$	3.33540	0.214852	0.107426	−	0.994213i	$-0.465739\pi$
$241$	3.33540	0.214852	0.107426	+	0.994213i	$0.465739\pi$
$242$	0	0
$243$	−7.52443	−0.482692
$244$	0	0
$245$	6.94392	0.443631
$246$	0	0
$247$	3.89948	0.248118
$248$	0	0
$249$	−4.54907	−0.288286
$250$	0	0
$251$	−25.2344	−1.59278	−0.796390	−	0.604783i	$-0.793260\pi$
$251$	−25.2344	−1.59278	−0.796390	+	0.604783i	$0.793260\pi$
$252$	0	0
$253$	−16.3185	−1.02594
$254$	0	0
$255$	−6.68460	−0.418606
$256$	0	0
$257$	13.9594	0.870760	0.435380	−	0.900247i	$-0.356614\pi$
$257$	13.9594	0.870760	0.435380	+	0.900247i	$0.356614\pi$
$258$	0	0
$259$	−0.236809	−0.0147146
$260$	0	0
$261$	−13.6906	−0.847426
$262$	0	0
$263$	24.3521	1.50162	0.750809	−	0.660520i	$-0.229664\pi$
$263$	24.3521	1.50162	0.750809	+	0.660520i	$0.229664\pi$
$264$	0	0
$265$	5.18216	0.318338
$266$	0	0
$267$	−48.5165	−2.96916
$268$	0	0
$269$	−6.28320	−0.383094	−0.191547	−	0.981483i	$-0.561350\pi$
$269$	−6.28320	−0.383094	−0.191547	+	0.981483i	$0.561350\pi$
$270$	0	0
$271$	15.3608	0.933102	0.466551	−	0.884494i	$-0.345496\pi$
$271$	15.3608	0.933102	0.466551	+	0.884494i	$0.345496\pi$
$272$	0	0
$273$	−2.13153	−0.129006
$274$	0	0
$275$	4.24650	0.256073
$276$	0	0
$277$	−22.0958	−1.32761	−0.663805	−	0.747906i	$-0.731059\pi$
$277$	−22.0958	−1.32761	−0.663805	+	0.747906i	$0.731059\pi$
$278$	0	0
$279$	−12.6251	−0.755843
$280$	0	0
$281$	30.6287	1.82715	0.913577	−	0.406666i	$-0.133309\pi$
$281$	30.6287	1.82715	0.913577	+	0.406666i	$0.133309\pi$
$282$	0	0
$283$	−15.9311	−0.947003	−0.473501	−	0.880793i	$-0.657010\pi$
$283$	−15.9311	−0.947003	−0.473501	+	0.880793i	$0.657010\pi$
$284$	0	0
$285$	−3.69743	−0.219017
$286$	0	0
$287$	−1.55416	−0.0917394
$288$	0	0
$289$	−11.7644	−0.692024
$290$	0	0
$291$	49.5243	2.90316
$292$	0	0
$293$	−16.9550	−0.990524	−0.495262	−	0.868744i	$-0.664928\pi$
$293$	−16.9550	−0.990524	−0.495262	+	0.868744i	$0.664928\pi$
$294$	0	0
$295$	−7.56172	−0.440260
$296$	0	0
$297$	31.4441	1.82457
$298$	0	0
$299$	−11.8400	−0.684723
$300$	0	0
$301$	1.78623	0.102956
$302$	0	0
$303$	−28.7648	−1.65249
$304$	0	0
$305$	5.49787	0.314807
$306$	0	0
$307$	−12.1939	−0.695941	−0.347970	−	0.937506i	$-0.613129\pi$
$307$	−12.1939	−0.695941	−0.347970	+	0.937506i	$0.613129\pi$
$308$	0	0
$309$	20.0102	1.13834
$310$	0	0
$311$	−6.36621	−0.360994	−0.180497	−	0.983575i	$-0.557771\pi$
$311$	−6.36621	−0.360994	−0.180497	+	0.983575i	$0.557771\pi$
$312$	0	0
$313$	−10.7665	−0.608558	−0.304279	−	0.952583i	$-0.598415\pi$
$313$	−10.7665	−0.608558	−0.304279	+	0.952583i	$0.598415\pi$
$314$	0	0
$315$	1.31065	0.0738469
$316$	0	0
$317$	−31.0506	−1.74398	−0.871989	−	0.489526i	$-0.837170\pi$
$317$	−31.0506	−1.74398	−0.871989	+	0.489526i	$0.837170\pi$
$318$	0	0
$319$	−10.5042	−0.588123
$320$	0	0
$321$	1.94605	0.108618
$322$	0	0
$323$	2.89594	0.161135
$324$	0	0
$325$	3.08106	0.170906
$326$	0	0
$327$	6.69233	0.370087
$328$	0	0
$329$	−1.87953	−0.103622
$330$	0	0
$331$	1.41995	0.0780475	0.0390237	−	0.999238i	$-0.487575\pi$
$331$	1.41995	0.0780475	0.0390237	+	0.999238i	$0.487575\pi$
$332$	0	0
$333$	5.53464	0.303296
$334$	0	0
$335$	−9.10845	−0.497648
$336$	0	0
$337$	−22.7338	−1.23839	−0.619194	−	0.785238i	$-0.712540\pi$
$337$	−22.7338	−1.23839	−0.619194	+	0.785238i	$0.712540\pi$
$338$	0	0
$339$	−15.4631	−0.839840
$340$	0	0
$341$	−9.68669	−0.524564
$342$	0	0
$343$	3.30205	0.178294
$344$	0	0
$345$	11.2265	0.604412
$346$	0	0
$347$	−15.6262	−0.838858	−0.419429	−	0.907788i	$-0.637770\pi$
$347$	−15.6262	−0.838858	−0.419429	+	0.907788i	$0.637770\pi$
$348$	0	0
$349$	−3.12713	−0.167392	−0.0836958	−	0.996491i	$-0.526672\pi$
$349$	−3.12713	−0.167392	−0.0836958	+	0.996491i	$0.526672\pi$
$350$	0	0
$351$	22.8144	1.21774
$352$	0	0
$353$	−6.11891	−0.325677	−0.162838	−	0.986653i	$-0.552065\pi$
$353$	−6.11891	−0.325677	−0.162838	+	0.986653i	$0.552065\pi$
$354$	0	0
$355$	−6.18704	−0.328374
$356$	0	0
$357$	−1.58298	−0.0837799
$358$	0	0
$359$	−15.5046	−0.818301	−0.409150	−	0.912467i	$-0.634175\pi$
$359$	−15.5046	−0.818301	−0.409150	+	0.912467i	$0.634175\pi$
$360$	0	0
$361$	−17.3982	−0.915694
$362$	0	0
$363$	20.5455	1.07836
$364$	0	0
$365$	−2.34422	−0.122702
$366$	0	0
$367$	14.8391	0.774595	0.387298	−	0.921955i	$-0.373409\pi$
$367$	14.8391	0.774595	0.387298	+	0.921955i	$0.373409\pi$
$368$	0	0
$369$	36.3235	1.89092
$370$	0	0
$371$	1.22718	0.0637122
$372$	0	0
$373$	30.5440	1.58151	0.790755	−	0.612133i	$-0.209688\pi$
$373$	30.5440	1.58151	0.790755	+	0.612133i	$0.209688\pi$
$374$	0	0
$375$	−2.92141	−0.150861
$376$	0	0
$377$	−7.62137	−0.392520
$378$	0	0
$379$	5.03941	0.258857	0.129429	−	0.991589i	$-0.458686\pi$
$379$	5.03941	0.258857	0.129429	+	0.991589i	$0.458686\pi$
$380$	0	0
$381$	59.8412	3.06576
$382$	0	0
$383$	−18.8066	−0.960971	−0.480485	−	0.877003i	$-0.659539\pi$
$383$	−18.8066	−0.960971	−0.480485	+	0.877003i	$0.659539\pi$
$384$	0	0
$385$	1.00561	0.0512506
$386$	0	0
$387$	−41.7472	−2.12213
$388$	0	0
$389$	−8.73528	−0.442896	−0.221448	−	0.975172i	$-0.571078\pi$
$389$	−8.73528	−0.442896	−0.221448	+	0.975172i	$0.571078\pi$
$390$	0	0
$391$	−8.79292	−0.444677
$392$	0	0
$393$	31.3173	1.57975
$394$	0	0
$395$	−3.35952	−0.169036
$396$	0	0
$397$	28.4846	1.42960	0.714800	−	0.699329i	$-0.246518\pi$
$397$	28.4846	1.42960	0.714800	+	0.699329i	$0.246518\pi$
$398$	0	0
$399$	−0.875585	−0.0438341
$400$	0	0
$401$	−27.3810	−1.36734	−0.683672	−	0.729789i	$-0.739618\pi$
$401$	−27.3810	−1.36734	−0.683672	+	0.729789i	$0.739618\pi$
$402$	0	0
$403$	−7.02821	−0.350100
$404$	0	0
$405$	−5.02830	−0.249858
$406$	0	0
$407$	4.24650	0.210491
$408$	0	0
$409$	2.18812	0.108196	0.0540979	−	0.998536i	$-0.482772\pi$
$409$	2.18812	0.108196	0.0540979	+	0.998536i	$0.482772\pi$
$410$	0	0
$411$	7.81717	0.385593
$412$	0	0
$413$	−1.79068	−0.0881138
$414$	0	0
$415$	1.55715	0.0764374
$416$	0	0
$417$	25.4022	1.24395
$418$	0	0
$419$	20.7703	1.01470	0.507349	−	0.861741i	$-0.330626\pi$
$419$	20.7703	1.01470	0.507349	+	0.861741i	$0.330626\pi$
$420$	0	0
$421$	−6.84080	−0.333400	−0.166700	−	0.986008i	$-0.553311\pi$
$421$	−6.84080	−0.333400	−0.166700	+	0.986008i	$0.553311\pi$
$422$	0	0
$423$	43.9278	2.13584
$424$	0	0
$425$	2.28814	0.110991
$426$	0	0
$427$	1.30195	0.0630056
$428$	0	0
$429$	38.2229	1.84542
$430$	0	0
$431$	−0.401350	−0.0193323	−0.00966616	−	0.999953i	$-0.503077\pi$
$431$	−0.401350	−0.0193323	−0.00966616	+	0.999953i	$0.503077\pi$
$432$	0	0
$433$	15.5731	0.748396	0.374198	−	0.927349i	$-0.377918\pi$
$433$	15.5731	0.748396	0.374198	+	0.927349i	$0.377918\pi$
$434$	0	0
$435$	7.22645	0.346482
$436$	0	0
$437$	−4.86359	−0.232657
$438$	0	0
$439$	−20.1366	−0.961067	−0.480534	−	0.876976i	$-0.659557\pi$
$439$	−20.1366	−0.961067	−0.480534	+	0.876976i	$0.659557\pi$
$440$	0	0
$441$	−38.4321	−1.83010
$442$	0	0
$443$	−1.07761	−0.0511988	−0.0255994	−	0.999672i	$-0.508149\pi$
$443$	−1.07761	−0.0511988	−0.0255994	+	0.999672i	$0.508149\pi$
$444$	0	0
$445$	16.6072	0.787258
$446$	0	0
$447$	−7.09949	−0.335795
$448$	0	0
$449$	−30.7500	−1.45118	−0.725591	−	0.688127i	$-0.758433\pi$
$449$	−30.7500	−1.45118	−0.725591	+	0.688127i	$0.758433\pi$
$450$	0	0
$451$	27.8695	1.31232
$452$	0	0
$453$	16.4786	0.774230
$454$	0	0
$455$	0.729623	0.0342053
$456$	0	0
$457$	−19.0907	−0.893026	−0.446513	−	0.894777i	$-0.647334\pi$
$457$	−19.0907	−0.893026	−0.446513	+	0.894777i	$0.647334\pi$
$458$	0	0
$459$	16.9430	0.790833
$460$	0	0
$461$	−7.75953	−0.361397	−0.180699	−	0.983539i	$-0.557836\pi$
$461$	−7.75953	−0.361397	−0.180699	+	0.983539i	$0.557836\pi$
$462$	0	0
$463$	10.4022	0.483430	0.241715	−	0.970347i	$-0.422290\pi$
$463$	10.4022	0.483430	0.241715	+	0.970347i	$0.422290\pi$
$464$	0	0
$465$	6.66404	0.309037
$466$	0	0
$467$	37.4276	1.73194	0.865971	−	0.500094i	$-0.166701\pi$
$467$	37.4276	1.73194	0.865971	+	0.500094i	$0.166701\pi$
$468$	0	0
$469$	−2.15697	−0.0995994
$470$	0	0
$471$	53.0200	2.44303
$472$	0	0
$473$	−32.0309	−1.47278
$474$	0	0
$475$	1.26563	0.0580711
$476$	0	0
$477$	−28.6814	−1.31323
$478$	0	0
$479$	−21.7867	−0.995460	−0.497730	−	0.867332i	$-0.665833\pi$
$479$	−21.7867	−0.995460	−0.497730	+	0.867332i	$0.665833\pi$
$480$	0	0
$481$	3.08106	0.140484
$482$	0	0
$483$	2.65853	0.120967
$484$	0	0
$485$	−16.9522	−0.769759
$486$	0	0
$487$	19.0281	0.862247	0.431124	−	0.902293i	$-0.358117\pi$
$487$	19.0281	0.862247	0.431124	+	0.902293i	$0.358117\pi$
$488$	0	0
$489$	−52.0911	−2.35564
$490$	0	0
$491$	1.10728	0.0499707	0.0249854	−	0.999688i	$-0.492046\pi$
$491$	1.10728	0.0499707	0.0249854	+	0.999688i	$0.492046\pi$
$492$	0	0
$493$	−5.65999	−0.254913
$494$	0	0
$495$	−23.5028	−1.05637
$496$	0	0
$497$	−1.46515	−0.0657209
$498$	0	0
$499$	24.0136	1.07500	0.537498	−	0.843265i	$-0.319370\pi$
$499$	24.0136	1.07500	0.537498	+	0.843265i	$0.319370\pi$
$500$	0	0
$501$	−55.9167	−2.49817
$502$	0	0
$503$	31.5338	1.40602	0.703010	−	0.711180i	$-0.251839\pi$
$503$	31.5338	1.40602	0.703010	+	0.711180i	$0.251839\pi$
$504$	0	0
$505$	9.84620	0.438150
$506$	0	0
$507$	−10.2456	−0.455023
$508$	0	0
$509$	−3.54190	−0.156992	−0.0784959	−	0.996914i	$-0.525012\pi$
$509$	−3.54190	−0.156992	−0.0784959	+	0.996914i	$0.525012\pi$
$510$	0	0
$511$	−0.555133	−0.0245576
$512$	0	0
$513$	9.37163	0.413768
$514$	0	0
$515$	−6.84951	−0.301826
$516$	0	0
$517$	33.7039	1.48230
$518$	0	0
$519$	−16.4807	−0.723422
$520$	0	0
$521$	32.5131	1.42443	0.712213	−	0.701963i	$-0.247693\pi$
$521$	32.5131	1.42443	0.712213	+	0.701963i	$0.247693\pi$
$522$	0	0
$523$	39.2673	1.71704	0.858519	−	0.512782i	$-0.171385\pi$
$523$	39.2673	1.71704	0.858519	+	0.512782i	$0.171385\pi$
$524$	0	0
$525$	−0.691817	−0.0301934
$526$	0	0
$527$	−5.21949	−0.227364
$528$	0	0
$529$	−8.23273	−0.357945
$530$	0	0
$531$	41.8514	1.81619
$532$	0	0
$533$	20.2208	0.875860
$534$	0	0
$535$	−0.666135	−0.0287995
$536$	0	0
$537$	57.2638	2.47112
$538$	0	0
$539$	−29.4873	−1.27011
$540$	0	0
$541$	37.6480	1.61861	0.809307	−	0.587386i	$-0.199843\pi$
$541$	37.6480	1.61861	0.809307	+	0.587386i	$0.199843\pi$
$542$	0	0
$543$	3.64805	0.156553
$544$	0	0
$545$	−2.29079	−0.0981266
$546$	0	0
$547$	38.6713	1.65346	0.826732	−	0.562596i	$-0.190197\pi$
$547$	38.6713	1.65346	0.826732	+	0.562596i	$0.190197\pi$
$548$	0	0
$549$	−30.4287	−1.29867
$550$	0	0
$551$	−3.13069	−0.133372
$552$	0	0
$553$	−0.795564	−0.0338308
$554$	0	0
$555$	−2.92141	−0.124007
$556$	0	0
$557$	−11.2977	−0.478699	−0.239349	−	0.970934i	$-0.576934\pi$
$557$	−11.2977	−0.478699	−0.239349	+	0.970934i	$0.576934\pi$
$558$	0	0
$559$	−23.2401	−0.982952
$560$	0	0
$561$	28.3861	1.19846
$562$	0	0
$563$	12.3115	0.518868	0.259434	−	0.965761i	$-0.416464\pi$
$563$	12.3115	0.518868	0.259434	+	0.965761i	$0.416464\pi$
$564$	0	0
$565$	5.29302	0.222679
$566$	0	0
$567$	−1.19075	−0.0500067
$568$	0	0
$569$	−16.3305	−0.684611	−0.342305	−	0.939589i	$-0.611208\pi$
$569$	−16.3305	−0.684611	−0.342305	+	0.939589i	$0.611208\pi$
$570$	0	0
$571$	−4.51336	−0.188878	−0.0944391	−	0.995531i	$-0.530106\pi$
$571$	−4.51336	−0.188878	−0.0944391	+	0.995531i	$0.530106\pi$
$572$	0	0
$573$	−40.0930	−1.67491
$574$	0	0
$575$	−3.84282	−0.160257
$576$	0	0
$577$	3.94813	0.164363	0.0821814	−	0.996617i	$-0.473811\pi$
$577$	3.94813	0.164363	0.0821814	+	0.996617i	$0.473811\pi$
$578$	0	0
$579$	−39.4884	−1.64108
$580$	0	0
$581$	0.368747	0.0152982
$582$	0	0
$583$	−22.0060	−0.911396
$584$	0	0
$585$	−17.0525	−0.705036
$586$	0	0
$587$	−16.2556	−0.670942	−0.335471	−	0.942050i	$-0.608896\pi$
$587$	−16.2556	−0.670942	−0.335471	+	0.942050i	$0.608896\pi$
$588$	0	0
$589$	−2.88703	−0.118958
$590$	0	0
$591$	−44.1108	−1.81447
$592$	0	0
$593$	−35.8650	−1.47280	−0.736400	−	0.676547i	$-0.763476\pi$
$593$	−35.8650	−1.47280	−0.736400	+	0.676547i	$0.763476\pi$
$594$	0	0
$595$	0.541853	0.0222138
$596$	0	0
$597$	−4.28814	−0.175502
$598$	0	0
$599$	26.4561	1.08097	0.540483	−	0.841355i	$-0.318242\pi$
$599$	26.4561	1.08097	0.540483	+	0.841355i	$0.318242\pi$
$600$	0	0
$601$	28.8380	1.17633	0.588163	−	0.808742i	$-0.299851\pi$
$601$	28.8380	1.17633	0.588163	+	0.808742i	$0.299851\pi$
$602$	0	0
$603$	50.4120	2.05293
$604$	0	0
$605$	−7.03272	−0.285921
$606$	0	0
$607$	−26.3467	−1.06938	−0.534691	−	0.845048i	$-0.679572\pi$
$607$	−26.3467	−1.06938	−0.534691	+	0.845048i	$0.679572\pi$
$608$	0	0
$609$	1.71129	0.0693450
$610$	0	0
$611$	24.4540	0.989303
$612$	0	0
$613$	3.87527	0.156521	0.0782604	−	0.996933i	$-0.475063\pi$
$613$	3.87527	0.156521	0.0782604	+	0.996933i	$0.475063\pi$
$614$	0	0
$615$	−19.1730	−0.773131
$616$	0	0
$617$	−29.7703	−1.19851	−0.599253	−	0.800560i	$-0.704536\pi$
$617$	−29.7703	−1.19851	−0.599253	+	0.800560i	$0.704536\pi$
$618$	0	0
$619$	−1.08911	−0.0437750	−0.0218875	−	0.999760i	$-0.506968\pi$
$619$	−1.08911	−0.0437750	−0.0218875	+	0.999760i	$0.506968\pi$
$620$	0	0
$621$	−28.4550	−1.14186
$622$	0	0
$623$	3.93274	0.157562
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.7011	0.627042
$628$	0	0
$629$	2.28814	0.0912342
$630$	0	0
$631$	−38.2649	−1.52330	−0.761651	−	0.647987i	$-0.775611\pi$
$631$	−38.2649	−1.52330	−0.761651	+	0.647987i	$0.775611\pi$
$632$	0	0
$633$	−50.0102	−1.98773
$634$	0	0
$635$	−20.4837	−0.812870
$636$	0	0
$637$	−21.3946	−0.847686
$638$	0	0
$639$	34.2430	1.35463
$640$	0	0
$641$	−14.3890	−0.568332	−0.284166	−	0.958775i	$-0.591717\pi$
$641$	−14.3890	−0.568332	−0.284166	+	0.958775i	$0.591717\pi$
$642$	0	0
$643$	−28.3119	−1.11651	−0.558256	−	0.829669i	$-0.688529\pi$
$643$	−28.3119	−1.11651	−0.558256	+	0.829669i	$0.688529\pi$
$644$	0	0
$645$	22.0359	0.867662
$646$	0	0
$647$	−5.68745	−0.223597	−0.111798	−	0.993731i	$-0.535661\pi$
$647$	−5.68745	−0.223597	−0.111798	+	0.993731i	$0.535661\pi$
$648$	0	0
$649$	32.1108	1.26046
$650$	0	0
$651$	1.57811	0.0618508
$652$	0	0
$653$	27.7953	1.08772	0.543858	−	0.839178i	$-0.316963\pi$
$653$	27.7953	1.08772	0.543858	+	0.839178i	$0.316963\pi$
$654$	0	0
$655$	−10.7199	−0.418863
$656$	0	0
$657$	12.9744	0.506180
$658$	0	0
$659$	13.9817	0.544650	0.272325	−	0.962205i	$-0.412207\pi$
$659$	13.9817	0.544650	0.272325	+	0.962205i	$0.412207\pi$
$660$	0	0
$661$	−1.90995	−0.0742886	−0.0371443	−	0.999310i	$-0.511826\pi$
$661$	−1.90995	−0.0742886	−0.0371443	+	0.999310i	$0.511826\pi$
$662$	0	0
$663$	20.5957	0.799869
$664$	0	0
$665$	0.299713	0.0116224
$666$	0	0
$667$	9.50567	0.368061
$668$	0	0
$669$	−43.7569	−1.69174
$670$	0	0
$671$	−23.3467	−0.901289
$672$	0	0
$673$	28.2915	1.09056	0.545279	−	0.838255i	$-0.316424\pi$
$673$	28.2915	1.09056	0.545279	+	0.838255i	$0.316424\pi$
$674$	0	0
$675$	7.40471	0.285008
$676$	0	0
$677$	32.1699	1.23639	0.618194	−	0.786025i	$-0.287865\pi$
$677$	32.1699	1.23639	0.618194	+	0.786025i	$0.287865\pi$
$678$	0	0
$679$	−4.01443	−0.154060
$680$	0	0
$681$	32.1558	1.23221
$682$	0	0
$683$	−28.5551	−1.09263	−0.546316	−	0.837579i	$-0.683970\pi$
$683$	−28.5551	−1.09263	−0.546316	+	0.837579i	$0.683970\pi$
$684$	0	0
$685$	−2.67582	−0.102238
$686$	0	0
$687$	−24.8987	−0.949946
$688$	0	0
$689$	−15.9665	−0.608277
$690$	0	0
$691$	−18.9684	−0.721592	−0.360796	−	0.932645i	$-0.617495\pi$
$691$	−18.9684	−0.721592	−0.360796	+	0.932645i	$0.617495\pi$
$692$	0	0
$693$	−5.56568	−0.211423
$694$	0	0
$695$	−8.69519	−0.329827
$696$	0	0
$697$	15.0169	0.568807
$698$	0	0
$699$	−71.0260	−2.68645
$700$	0	0
$701$	−50.3717	−1.90251	−0.951256	−	0.308401i	$-0.900206\pi$
$701$	−50.3717	−1.90251	−0.951256	+	0.308401i	$0.900206\pi$
$702$	0	0
$703$	1.26563	0.0477342
$704$	0	0
$705$	−23.1869	−0.873269
$706$	0	0
$707$	2.33167	0.0876915
$708$	0	0
$709$	49.7236	1.86741	0.933705	−	0.358044i	$-0.116556\pi$
$709$	49.7236	1.86741	0.933705	+	0.358044i	$0.116556\pi$
$710$	0	0
$711$	18.5937	0.697318
$712$	0	0
$713$	8.76587	0.328284
$714$	0	0
$715$	−13.0837	−0.489303
$716$	0	0
$717$	−45.8215	−1.71124
$718$	0	0
$719$	−32.3221	−1.20541	−0.602706	−	0.797964i	$-0.705911\pi$
$719$	−32.3221	−1.20541	−0.602706	+	0.797964i	$0.705911\pi$
$720$	0	0
$721$	−1.62203	−0.0604075
$722$	0	0
$723$	9.74406	0.362386
$724$	0	0
$725$	−2.47362	−0.0918679
$726$	0	0
$727$	−14.1578	−0.525084	−0.262542	−	0.964921i	$-0.584561\pi$
$727$	−14.1578	−0.525084	−0.262542	+	0.964921i	$0.584561\pi$
$728$	0	0
$729$	−37.0668	−1.37285
$730$	0	0
$731$	−17.2592	−0.638355
$732$	0	0
$733$	−9.32734	−0.344513	−0.172257	−	0.985052i	$-0.555106\pi$
$733$	−9.32734	−0.344513	−0.172257	+	0.985052i	$0.555106\pi$
$734$	0	0
$735$	20.2860	0.748262
$736$	0	0
$737$	38.6790	1.42476
$738$	0	0
$739$	17.8281	0.655818	0.327909	−	0.944709i	$-0.393656\pi$
$739$	17.8281	0.655818	0.327909	+	0.944709i	$0.393656\pi$
$740$	0	0
$741$	11.3920	0.418495
$742$	0	0
$743$	32.1725	1.18030	0.590148	−	0.807295i	$-0.299069\pi$
$743$	32.1725	1.18030	0.590148	+	0.807295i	$0.299069\pi$
$744$	0	0
$745$	2.43016	0.0890342
$746$	0	0
$747$	−8.61825	−0.315325
$748$	0	0
$749$	−0.157747	−0.00576394
$750$	0	0
$751$	−42.6063	−1.55472	−0.777362	−	0.629053i	$-0.783443\pi$
$751$	−42.6063	−1.55472	−0.777362	+	0.629053i	$0.783443\pi$
$752$	0	0
$753$	−73.7200	−2.68651
$754$	0	0
$755$	−5.64062	−0.205283
$756$	0	0
$757$	42.3269	1.53840	0.769199	−	0.639009i	$-0.220655\pi$
$757$	42.3269	1.53840	0.769199	+	0.639009i	$0.220655\pi$
$758$	0	0
$759$	−47.6731	−1.73042
$760$	0	0
$761$	−16.8746	−0.611702	−0.305851	−	0.952079i	$-0.598941\pi$
$761$	−16.8746	−0.611702	−0.305851	+	0.952079i	$0.598941\pi$
$762$	0	0
$763$	−0.542480	−0.0196391
$764$	0	0
$765$	−12.6640	−0.457869
$766$	0	0
$767$	23.2981	0.841246
$768$	0	0
$769$	−47.9816	−1.73026	−0.865131	−	0.501546i	$-0.832765\pi$
$769$	−47.9816	−1.73026	−0.865131	+	0.501546i	$0.832765\pi$
$770$	0	0
$771$	40.7810	1.46869
$772$	0	0
$773$	3.55338	0.127806	0.0639031	−	0.997956i	$-0.479645\pi$
$773$	3.55338	0.127806	0.0639031	+	0.997956i	$0.479645\pi$
$774$	0	0
$775$	−2.28110	−0.0819396
$776$	0	0
$777$	−0.691817	−0.0248188
$778$	0	0
$779$	8.30625	0.297602
$780$	0	0
$781$	26.2732	0.940130
$782$	0	0
$783$	−18.3164	−0.654576
$784$	0	0
$785$	−18.1488	−0.647757
$786$	0	0
$787$	39.8879	1.42185	0.710926	−	0.703267i	$-0.248276\pi$
$787$	39.8879	1.42185	0.710926	+	0.703267i	$0.248276\pi$
$788$	0	0
$789$	71.1426	2.53274
$790$	0	0
$791$	1.25344	0.0445671
$792$	0	0
$793$	−16.9393	−0.601531
$794$	0	0
$795$	15.1392	0.536933
$796$	0	0
$797$	−21.5677	−0.763967	−0.381984	−	0.924169i	$-0.624759\pi$
$797$	−21.5677	−0.763967	−0.381984	+	0.924169i	$0.624759\pi$
$798$	0	0
$799$	18.1607	0.642480
$800$	0	0
$801$	−91.9150	−3.24766
$802$	0	0
$803$	9.95472	0.351294
$804$	0	0
$805$	−0.910015	−0.0320738
$806$	0	0
$807$	−18.3558	−0.646155
$808$	0	0
$809$	−8.85085	−0.311179	−0.155590	−	0.987822i	$-0.549728\pi$
$809$	−8.85085	−0.311179	−0.155590	+	0.987822i	$0.549728\pi$
$810$	0	0
$811$	28.2324	0.991374	0.495687	−	0.868501i	$-0.334916\pi$
$811$	28.2324	0.991374	0.495687	+	0.868501i	$0.334916\pi$
$812$	0	0
$813$	44.8752	1.57384
$814$	0	0
$815$	17.8308	0.624586
$816$	0	0
$817$	−9.54652	−0.333990
$818$	0	0
$819$	−4.03820	−0.141106
$820$	0	0
$821$	22.5904	0.788410	0.394205	−	0.919022i	$-0.371020\pi$
$821$	22.5904	0.788410	0.394205	+	0.919022i	$0.371020\pi$
$822$	0	0
$823$	26.6233	0.928030	0.464015	−	0.885827i	$-0.346408\pi$
$823$	26.6233	0.928030	0.464015	+	0.885827i	$0.346408\pi$
$824$	0	0
$825$	12.4058	0.431913
$826$	0	0
$827$	−28.2784	−0.983338	−0.491669	−	0.870782i	$-0.663613\pi$
$827$	−28.2784	−0.983338	−0.491669	+	0.870782i	$0.663613\pi$
$828$	0	0
$829$	7.20241	0.250150	0.125075	−	0.992147i	$-0.460083\pi$
$829$	7.20241	0.250150	0.125075	+	0.992147i	$0.460083\pi$
$830$	0	0
$831$	−64.5510	−2.23925
$832$	0	0
$833$	−15.8887	−0.550510
$834$	0	0
$835$	19.1403	0.662377
$836$	0	0
$837$	−16.8909	−0.583835
$838$	0	0
$839$	45.8629	1.58336	0.791682	−	0.610933i	$-0.209206\pi$
$839$	45.8629	1.58336	0.791682	+	0.610933i	$0.209206\pi$
$840$	0	0
$841$	−22.8812	−0.789007
$842$	0	0
$843$	89.4789	3.08182
$844$	0	0
$845$	3.50707	0.120647
$846$	0	0
$847$	−1.66541	−0.0572243
$848$	0	0
$849$	−46.5411	−1.59729
$850$	0	0
$851$	−3.84282	−0.131730
$852$	0	0
$853$	24.3403	0.833395	0.416698	−	0.909045i	$-0.363187\pi$
$853$	24.3403	0.833395	0.416698	+	0.909045i	$0.363187\pi$
$854$	0	0
$855$	−7.00481	−0.239559
$856$	0	0
$857$	−6.62514	−0.226310	−0.113155	−	0.993577i	$-0.536096\pi$
$857$	−6.62514	−0.226310	−0.113155	+	0.993577i	$0.536096\pi$
$858$	0	0
$859$	−47.4734	−1.61977	−0.809887	−	0.586587i	$-0.800471\pi$
$859$	−47.4734	−1.61977	−0.809887	+	0.586587i	$0.800471\pi$
$860$	0	0
$861$	−4.54035	−0.154735
$862$	0	0
$863$	31.7080	1.07935	0.539676	−	0.841873i	$-0.318547\pi$
$863$	31.7080	1.07935	0.539676	+	0.841873i	$0.318547\pi$
$864$	0	0
$865$	5.64135	0.191812
$866$	0	0
$867$	−34.3687	−1.16722
$868$	0	0
$869$	14.2662	0.483947
$870$	0	0
$871$	28.0637	0.950902
$872$	0	0
$873$	93.8241	3.17547
$874$	0	0
$875$	0.236809	0.00800561
$876$	0	0
$877$	−41.4417	−1.39939	−0.699693	−	0.714444i	$-0.746680\pi$
$877$	−41.4417	−1.39939	−0.699693	+	0.714444i	$0.746680\pi$
$878$	0	0
$879$	−49.5326	−1.67069
$880$	0	0
$881$	−18.6518	−0.628394	−0.314197	−	0.949358i	$-0.601735\pi$
$881$	−18.6518	−0.628394	−0.314197	+	0.949358i	$0.601735\pi$
$882$	0	0
$883$	14.7785	0.497336	0.248668	−	0.968589i	$-0.420007\pi$
$883$	14.7785	0.497336	0.248668	+	0.968589i	$0.420007\pi$
$884$	0	0
$885$	−22.0909	−0.742577
$886$	0	0
$887$	−45.2730	−1.52012	−0.760060	−	0.649853i	$-0.774830\pi$
$887$	−45.2730	−1.52012	−0.760060	+	0.649853i	$0.774830\pi$
$888$	0	0
$889$	−4.85072	−0.162688
$890$	0	0
$891$	21.3526	0.715340
$892$	0	0
$893$	10.0452	0.336148
$894$	0	0
$895$	−19.6014	−0.655204
$896$	0	0
$897$	−34.5894	−1.15491
$898$	0	0
$899$	5.64258	0.188190
$900$	0	0
$901$	−11.8575	−0.395031
$902$	0	0
$903$	5.21830	0.173654
$904$	0	0
$905$	−1.24873	−0.0415091
$906$	0	0
$907$	−34.7017	−1.15225	−0.576126	−	0.817361i	$-0.695436\pi$
$907$	−34.7017	−1.15225	−0.576126	+	0.817361i	$0.695436\pi$
$908$	0	0
$909$	−54.4951	−1.80749
$910$	0	0
$911$	−52.6957	−1.74588	−0.872942	−	0.487823i	$-0.837791\pi$
$911$	−52.6957	−1.74588	−0.872942	+	0.487823i	$0.837791\pi$
$912$	0	0
$913$	−6.61242	−0.218839
$914$	0	0
$915$	16.0615	0.530978
$916$	0	0
$917$	−2.53858	−0.0838313
$918$	0	0
$919$	23.6925	0.781542	0.390771	−	0.920488i	$-0.372208\pi$
$919$	23.6925	0.781542	0.390771	+	0.920488i	$0.372208\pi$
$920$	0	0
$921$	−35.6233	−1.17383
$922$	0	0
$923$	19.0626	0.627455
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	37.9096	1.24511
$928$	0	0
$929$	−43.8495	−1.43866	−0.719328	−	0.694671i	$-0.755550\pi$
$929$	−43.8495	−1.43866	−0.719328	+	0.694671i	$0.755550\pi$
$930$	0	0
$931$	−8.78844	−0.288029
$932$	0	0
$933$	−18.5983	−0.608881
$934$	0	0
$935$	−9.71658	−0.317766
$936$	0	0
$937$	−18.9535	−0.619185	−0.309592	−	0.950869i	$-0.600193\pi$
$937$	−18.9535	−0.619185	−0.309592	+	0.950869i	$0.600193\pi$
$938$	0	0
$939$	−31.4533	−1.02644
$940$	0	0
$941$	54.0151	1.76084	0.880421	−	0.474194i	$-0.157260\pi$
$941$	54.0151	1.76084	0.880421	+	0.474194i	$0.157260\pi$
$942$	0	0
$943$	−25.2202	−0.821282
$944$	0	0
$945$	1.75350	0.0570415
$946$	0	0
$947$	49.1455	1.59701	0.798507	−	0.601985i	$-0.205623\pi$
$947$	49.1455	1.59701	0.798507	+	0.601985i	$0.205623\pi$
$948$	0	0
$949$	7.22268	0.234458
$950$	0	0
$951$	−90.7117	−2.94153
$952$	0	0
$953$	49.6923	1.60969	0.804845	−	0.593484i	$-0.202248\pi$
$953$	49.6923	1.60969	0.804845	+	0.593484i	$0.202248\pi$
$954$	0	0
$955$	13.7238	0.444093
$956$	0	0
$957$	−30.6871	−0.991973
$958$	0	0
$959$	−0.633659	−0.0204619
$960$	0	0
$961$	−25.7966	−0.832148
$962$	0	0
$963$	3.68681	0.118806
$964$	0	0
$965$	13.5169	0.435124
$966$	0	0
$967$	30.4451	0.979049	0.489524	−	0.871990i	$-0.337170\pi$
$967$	30.4451	0.979049	0.489524	+	0.871990i	$0.337170\pi$
$968$	0	0
$969$	8.46024	0.271782
$970$	0	0
$971$	10.0309	0.321908	0.160954	−	0.986962i	$-0.448543\pi$
$971$	10.0309	0.321908	0.160954	+	0.986962i	$0.448543\pi$
$972$	0	0
$973$	−2.05910	−0.0660118
$974$	0	0
$975$	9.00104	0.288264
$976$	0	0
$977$	−1.41215	−0.0451785	−0.0225893	−	0.999745i	$-0.507191\pi$
$977$	−1.41215	−0.0451785	−0.0225893	+	0.999745i	$0.507191\pi$
$978$	0	0
$979$	−70.5225	−2.25391
$980$	0	0
$981$	12.6787	0.404799
$982$	0	0
$983$	−28.9889	−0.924603	−0.462302	−	0.886723i	$-0.652976\pi$
$983$	−28.9889	−0.924603	−0.462302	+	0.886723i	$0.652976\pi$
$984$	0	0
$985$	15.0991	0.481098
$986$	0	0
$987$	−5.49087	−0.174776
$988$	0	0
$989$	28.9860	0.921701
$990$	0	0
$991$	40.5149	1.28700	0.643499	−	0.765447i	$-0.277482\pi$
$991$	40.5149	1.28700	0.643499	+	0.765447i	$0.277482\pi$
$992$	0	0
$993$	4.14825	0.131641
$994$	0	0
$995$	1.46783	0.0465334
$996$	0	0
$997$	52.0416	1.64817	0.824086	−	0.566464i	$-0.191689\pi$
$997$	52.0416	1.64817	0.824086	+	0.566464i	$0.191689\pi$
$998$	0	0
$999$	7.40471	0.234275

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.z.1.5		5
4.3	odd	2		1480.2.a.h.1.1	✓	5
20.19	odd	2		7400.2.a.q.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.h.1.1	✓	5	4.3	odd	2
2960.2.a.z.1.5		5	1.1	even	1	trivial
7400.2.a.q.1.5		5	20.19	odd	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+2.92141 q^{3} -1.00000 q^{5} -0.236809 q^{7} +5.53464 q^{9} +O(q^{10})\)	\(q+2.92141 q^{3} -1.00000 q^{5} -0.236809 q^{7} +5.53464 q^{9} +4.24650 q^{11} +3.08106 q^{13} -2.92141 q^{15} +2.28814 q^{17} +1.26563 q^{19} -0.691817 q^{21} -3.84282 q^{23} +1.00000 q^{25} +7.40471 q^{27} -2.47362 q^{29} -2.28110 q^{31} +12.4058 q^{33} +0.236809 q^{35} +1.00000 q^{37} +9.00104 q^{39} +6.56293 q^{41} -7.54289 q^{43} -5.53464 q^{45} +7.93688 q^{47} -6.94392 q^{49} +6.68460 q^{51} -5.18216 q^{53} -4.24650 q^{55} +3.69743 q^{57} +7.56172 q^{59} -5.49787 q^{61} -1.31065 q^{63} -3.08106 q^{65} +9.10845 q^{67} -11.2265 q^{69} +6.18704 q^{71} +2.34422 q^{73} +2.92141 q^{75} -1.00561 q^{77} +3.35952 q^{79} +5.02830 q^{81} -1.55715 q^{83} -2.28814 q^{85} -7.22645 q^{87} -16.6072 q^{89} -0.729623 q^{91} -6.66404 q^{93} -1.26563 q^{95} +16.9522 q^{97} +23.5028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9	\( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) \) 5 * q + q^3 - 5 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 4 * q^13 - q^15 + 10 * q^19 - 5 * q^21 + 8 * q^23 + 5 * q^25 + 7 * q^27 - 8 * q^29 + 2 * q^31 - 11 * q^33 - q^35 + 5 * q^37 + 2 * q^39 - 13 * q^41 + 18 * q^43 - 2 * q^45 + 9 * q^47 - 6 * q^49 + 22 * q^51 - 13 * q^53 - 7 * q^55 + 4 * q^57 + 24 * q^59 - 2 * q^61 + 20 * q^63 - 4 * q^65 + 22 * q^67 - 32 * q^69 + 21 * q^71 + 29 * q^73 + q^75 + 11 * q^77 + 6 * q^79 + 5 * q^81 + 33 * q^83 - 12 * q^87 - 26 * q^89 + 38 * q^91 + 14 * q^93 - 10 * q^95 + 26 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

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