LMFDB - Embedded newform 2960.2.a.y.1.1

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:	5.5.6397264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 14x - 4 $ x^5 - 10x^3 - 2x^2 + 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$-2.65435$ 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)

$f(q)$	$=$	$q-2.65435 q^{3} -1.00000 q^{5} -0.0991320 q^{7} +4.04557 q^{9} +O(q^{10})$	$q-2.65435 q^{3} -1.00000 q^{5} -0.0991320 q^{7} +4.04557 q^{9} +0.220081 q^{11} +4.84730 q^{13} +2.65435 q^{15} +5.26565 q^{17} +5.23852 q^{19} +0.263131 q^{21} -7.30870 q^{23} +1.00000 q^{25} -2.77530 q^{27} -9.42165 q^{29} -7.40783 q^{31} -0.584172 q^{33} +0.0991320 q^{35} +1.00000 q^{37} -12.8664 q^{39} -1.28688 q^{41} +7.72704 q^{43} -4.04557 q^{45} -10.9366 q^{47} -6.99017 q^{49} -13.9769 q^{51} +11.5288 q^{53} -0.220081 q^{55} -13.9049 q^{57} +9.52078 q^{59} +11.6728 q^{61} -0.401045 q^{63} -4.84730 q^{65} -3.23852 q^{67} +19.3998 q^{69} -8.58417 q^{71} -3.89287 q^{73} -2.65435 q^{75} -0.0218171 q^{77} +3.42965 q^{79} -4.77009 q^{81} +1.92982 q^{83} -5.26565 q^{85} +25.0083 q^{87} +11.0243 q^{89} -0.480523 q^{91} +19.6630 q^{93} -5.23852 q^{95} +16.0058 q^{97} +0.890352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * 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
$3$	−2.65435	−1.53249	−0.766244	−	0.642549i	$-0.777877\pi$
$3$	−2.65435	−1.53249	−0.766244	+	0.642549i	$0.777877\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.0991320	−0.0374684	−0.0187342	−	0.999824i	$-0.505964\pi$
$7$	−0.0991320	−0.0374684	−0.0187342	+	0.999824i	$0.505964\pi$
$8$	0	0
$9$	4.04557	1.34852
$10$	0	0
$11$	0.220081	0.0663569	0.0331785	−	0.999449i	$-0.489437\pi$
$11$	0.220081	0.0663569	0.0331785	+	0.999449i	$0.489437\pi$
$12$	0	0
$13$	4.84730	1.34440	0.672200	−	0.740370i	$-0.265350\pi$
$13$	4.84730	1.34440	0.672200	+	0.740370i	$0.265350\pi$
$14$	0	0
$15$	2.65435	0.685350
$16$	0	0
$17$	5.26565	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$17$	5.26565	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$18$	0	0
$19$	5.23852	1.20180	0.600899	−	0.799325i	$-0.294809\pi$
$19$	5.23852	1.20180	0.600899	+	0.799325i	$0.294809\pi$
$20$	0	0
$21$	0.263131	0.0574198
$22$	0	0
$23$	−7.30870	−1.52397	−0.761984	−	0.647595i	$-0.775775\pi$
$23$	−7.30870	−1.52397	−0.761984	+	0.647595i	$0.775775\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.77530	−0.534106
$28$	0	0
$29$	−9.42165	−1.74956	−0.874778	−	0.484524i	$-0.838993\pi$
$29$	−9.42165	−1.74956	−0.874778	+	0.484524i	$0.838993\pi$
$30$	0	0
$31$	−7.40783	−1.33049	−0.665243	−	0.746627i	$-0.731672\pi$
$31$	−7.40783	−1.33049	−0.665243	+	0.746627i	$0.731672\pi$
$32$	0	0
$33$	−0.584172	−0.101691
$34$	0	0
$35$	0.0991320	0.0167564
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−12.8664	−2.06028
$40$	0	0
$41$	−1.28688	−0.200977	−0.100488	−	0.994938i	$-0.532041\pi$
$41$	−1.28688	−0.200977	−0.100488	+	0.994938i	$0.532041\pi$
$42$	0	0
$43$	7.72704	1.17836	0.589181	−	0.808001i	$-0.299450\pi$
$43$	7.72704	1.17836	0.589181	+	0.808001i	$0.299450\pi$
$44$	0	0
$45$	−4.04557	−0.603077
$46$	0	0
$47$	−10.9366	−1.59527	−0.797634	−	0.603142i	$-0.793915\pi$
$47$	−10.9366	−1.59527	−0.797634	+	0.603142i	$0.793915\pi$
$48$	0	0
$49$	−6.99017	−0.998596
$50$	0	0
$51$	−13.9769	−1.95715
$52$	0	0
$53$	11.5288	1.58360	0.791800	−	0.610781i	$-0.209144\pi$
$53$	11.5288	1.58360	0.791800	+	0.610781i	$0.209144\pi$
$54$	0	0
$55$	−0.220081	−0.0296757
$56$	0	0
$57$	−13.9049	−1.84174
$58$	0	0
$59$	9.52078	1.23950	0.619750	−	0.784799i	$-0.287234\pi$
$59$	9.52078	1.23950	0.619750	+	0.784799i	$0.287234\pi$
$60$	0	0
$61$	11.6728	1.49455	0.747274	−	0.664516i	$-0.231362\pi$
$61$	11.6728	1.49455	0.747274	+	0.664516i	$0.231362\pi$
$62$	0	0
$63$	−0.401045	−0.0505269
$64$	0	0
$65$	−4.84730	−0.601234
$66$	0	0
$67$	−3.23852	−0.395648	−0.197824	−	0.980238i	$-0.563387\pi$
$67$	−3.23852	−0.395648	−0.197824	+	0.980238i	$0.563387\pi$
$68$	0	0
$69$	19.3998	2.33547
$70$	0	0
$71$	−8.58417	−1.01875	−0.509377	−	0.860544i	$-0.670124\pi$
$71$	−8.58417	−1.01875	−0.509377	+	0.860544i	$0.670124\pi$
$72$	0	0
$73$	−3.89287	−0.455626	−0.227813	−	0.973705i	$-0.573157\pi$
$73$	−3.89287	−0.455626	−0.227813	+	0.973705i	$0.573157\pi$
$74$	0	0
$75$	−2.65435	−0.306498
$76$	0	0
$77$	−0.0218171	−0.00248628
$78$	0	0
$79$	3.42965	0.385865	0.192933	−	0.981212i	$-0.438200\pi$
$79$	3.42965	0.385865	0.192933	+	0.981212i	$0.438200\pi$
$80$	0	0
$81$	−4.77009	−0.530010
$82$	0	0
$83$	1.92982	0.211826	0.105913	−	0.994375i	$-0.466224\pi$
$83$	1.92982	0.211826	0.105913	+	0.994375i	$0.466224\pi$
$84$	0	0
$85$	−5.26565	−0.571140
$86$	0	0
$87$	25.0083	2.68117
$88$	0	0
$89$	11.0243	1.16858	0.584289	−	0.811546i	$-0.301374\pi$
$89$	11.0243	1.16858	0.584289	+	0.811546i	$0.301374\pi$
$90$	0	0
$91$	−0.480523	−0.0503725
$92$	0	0
$93$	19.6630	2.03895
$94$	0	0
$95$	−5.23852	−0.537461
$96$	0	0
$97$	16.0058	1.62514	0.812572	−	0.582860i	$-0.198067\pi$
$97$	16.0058	1.62514	0.812572	+	0.582860i	$0.198067\pi$
$98$	0	0
$99$	0.890352	0.0894838
$100$	0	0
$101$	−11.6630	−1.16051	−0.580254	−	0.814436i	$-0.697047\pi$
$101$	−11.6630	−1.16051	−0.580254	+	0.814436i	$0.697047\pi$
$102$	0	0
$103$	16.2788	1.60400	0.801998	−	0.597327i	$-0.203771\pi$
$103$	16.2788	1.60400	0.801998	+	0.597327i	$0.203771\pi$
$104$	0	0
$105$	−0.263131	−0.0256789
$106$	0	0
$107$	5.04374	0.487597	0.243798	−	0.969826i	$-0.421607\pi$
$107$	5.04374	0.487597	0.243798	+	0.969826i	$0.421607\pi$
$108$	0	0
$109$	−3.48514	−0.333816	−0.166908	−	0.985972i	$-0.553378\pi$
$109$	−3.48514	−0.333816	−0.166908	+	0.985972i	$0.553378\pi$
$110$	0	0
$111$	−2.65435	−0.251940
$112$	0	0
$113$	5.07800	0.477698	0.238849	−	0.971057i	$-0.423230\pi$
$113$	5.07800	0.477698	0.238849	+	0.971057i	$0.423230\pi$
$114$	0	0
$115$	7.30870	0.681540
$116$	0	0
$117$	19.6101	1.81295
$118$	0	0
$119$	−0.521994	−0.0478511
$120$	0	0
$121$	−10.9516	−0.995597
$122$	0	0
$123$	3.41583	0.307995
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	22.4943	1.99605	0.998025	−	0.0628185i	$-0.0200089\pi$
$127$	22.4943	1.99605	0.998025	+	0.0628185i	$0.0200089\pi$
$128$	0	0
$129$	−20.5103	−1.80583
$130$	0	0
$131$	−1.05088	−0.0918155	−0.0459078	−	0.998946i	$-0.514618\pi$
$131$	−1.05088	−0.0918155	−0.0459078	+	0.998946i	$0.514618\pi$
$132$	0	0
$133$	−0.519305	−0.0450294
$134$	0	0
$135$	2.77530	0.238860
$136$	0	0
$137$	−7.89287	−0.674333	−0.337167	−	0.941445i	$-0.609469\pi$
$137$	−7.89287	−0.674333	−0.337167	+	0.941445i	$0.609469\pi$
$138$	0	0
$139$	4.64845	0.394277	0.197138	−	0.980376i	$-0.436835\pi$
$139$	4.64845	0.394277	0.197138	+	0.980376i	$0.436835\pi$
$140$	0	0
$141$	29.0296	2.44473
$142$	0	0
$143$	1.06680	0.0892102
$144$	0	0
$145$	9.42165	0.782425
$146$	0	0
$147$	18.5544	1.53034
$148$	0	0
$149$	−16.6873	−1.36708	−0.683538	−	0.729915i	$-0.739560\pi$
$149$	−16.6873	−1.36708	−0.683538	+	0.729915i	$0.739560\pi$
$150$	0	0
$151$	15.3998	1.25322	0.626610	−	0.779333i	$-0.284442\pi$
$151$	15.3998	1.25322	0.626610	+	0.779333i	$0.284442\pi$
$152$	0	0
$153$	21.3025	1.72221
$154$	0	0
$155$	7.40783	0.595011
$156$	0	0
$157$	−18.3444	−1.46405	−0.732023	−	0.681280i	$-0.761424\pi$
$157$	−18.3444	−1.46405	−0.732023	+	0.681280i	$0.761424\pi$
$158$	0	0
$159$	−30.6014	−2.42685
$160$	0	0
$161$	0.724525	0.0571006
$162$	0	0
$163$	−11.7110	−0.917280	−0.458640	−	0.888622i	$-0.651663\pi$
$163$	−11.7110	−0.917280	−0.458640	+	0.888622i	$0.651663\pi$
$164$	0	0
$165$	0.584172	0.0454777
$166$	0	0
$167$	−2.18764	−0.169285	−0.0846425	−	0.996411i	$-0.526975\pi$
$167$	−2.18764	−0.169285	−0.0846425	+	0.996411i	$0.526975\pi$
$168$	0	0
$169$	10.4963	0.807411
$170$	0	0
$171$	21.1928	1.62065
$172$	0	0
$173$	21.0830	1.60291	0.801457	−	0.598053i	$-0.204059\pi$
$173$	21.0830	1.60291	0.801457	+	0.598053i	$0.204059\pi$
$174$	0	0
$175$	−0.0991320	−0.00749367
$176$	0	0
$177$	−25.2715	−1.89952
$178$	0	0
$179$	17.6048	1.31584	0.657921	−	0.753087i	$-0.271436\pi$
$179$	17.6048	1.31584	0.657921	+	0.753087i	$0.271436\pi$
$180$	0	0
$181$	−18.7784	−1.39579	−0.697895	−	0.716200i	$-0.745880\pi$
$181$	−18.7784	−1.39579	−0.697895	+	0.716200i	$0.745880\pi$
$182$	0	0
$183$	−30.9836	−2.29038
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	1.15887	0.0847449
$188$	0	0
$189$	0.275121	0.0200121
$190$	0	0
$191$	9.07059	0.656325	0.328162	−	0.944621i	$-0.393571\pi$
$191$	9.07059	0.656325	0.328162	+	0.944621i	$0.393571\pi$
$192$	0	0
$193$	1.11968	0.0805960	0.0402980	−	0.999188i	$-0.487169\pi$
$193$	1.11968	0.0805960	0.0402980	+	0.999188i	$0.487169\pi$
$194$	0	0
$195$	12.8664	0.921384
$196$	0	0
$197$	10.4242	0.742691	0.371345	−	0.928495i	$-0.378897\pi$
$197$	10.4242	0.742691	0.371345	+	0.928495i	$0.378897\pi$
$198$	0	0
$199$	−3.19102	−0.226206	−0.113103	−	0.993583i	$-0.536079\pi$
$199$	−3.19102	−0.226206	−0.113103	+	0.993583i	$0.536079\pi$
$200$	0	0
$201$	8.59616	0.606326
$202$	0	0
$203$	0.933986	0.0655530
$204$	0	0
$205$	1.28688	0.0898796
$206$	0	0
$207$	−29.5678	−2.05511
$208$	0	0
$209$	1.15290	0.0797477
$210$	0	0
$211$	8.16583	0.562159	0.281079	−	0.959685i	$-0.409308\pi$
$211$	8.16583	0.562159	0.281079	+	0.959685i	$0.409308\pi$
$212$	0	0
$213$	22.7854	1.56123
$214$	0	0
$215$	−7.72704	−0.526980
$216$	0	0
$217$	0.734353	0.0498511
$218$	0	0
$219$	10.3330	0.698242
$220$	0	0
$221$	25.5242	1.71694
$222$	0	0
$223$	11.0148	0.737604	0.368802	−	0.929508i	$-0.379768\pi$
$223$	11.0148	0.737604	0.368802	+	0.929508i	$0.379768\pi$
$224$	0	0
$225$	4.04557	0.269704
$226$	0	0
$227$	2.20946	0.146647	0.0733235	−	0.997308i	$-0.476639\pi$
$227$	2.20946	0.146647	0.0733235	+	0.997308i	$0.476639\pi$
$228$	0	0
$229$	1.60485	0.106051	0.0530257	−	0.998593i	$-0.483113\pi$
$229$	1.60485	0.106051	0.0530257	+	0.998593i	$0.483113\pi$
$230$	0	0
$231$	0.0579101	0.00381020
$232$	0	0
$233$	28.0609	1.83833	0.919164	−	0.393875i	$-0.128866\pi$
$233$	28.0609	1.83833	0.919164	+	0.393875i	$0.128866\pi$
$234$	0	0
$235$	10.9366	0.713426
$236$	0	0
$237$	−9.10348	−0.591334
$238$	0	0
$239$	1.93061	0.124881	0.0624403	−	0.998049i	$-0.480112\pi$
$239$	1.93061	0.124881	0.0624403	+	0.998049i	$0.480112\pi$
$240$	0	0
$241$	5.06121	0.326021	0.163011	−	0.986624i	$-0.447880\pi$
$241$	5.06121	0.326021	0.163011	+	0.986624i	$0.447880\pi$
$242$	0	0
$243$	20.9874	1.34634
$244$	0	0
$245$	6.99017	0.446586
$246$	0	0
$247$	25.3927	1.61570
$248$	0	0
$249$	−5.12242	−0.324620
$250$	0	0
$251$	−2.26285	−0.142830	−0.0714150	−	0.997447i	$-0.522751\pi$
$251$	−2.26285	−0.142830	−0.0714150	+	0.997447i	$0.522751\pi$
$252$	0	0
$253$	−1.60851	−0.101126
$254$	0	0
$255$	13.9769	0.875265
$256$	0	0
$257$	−29.9960	−1.87110	−0.935549	−	0.353196i	$-0.885095\pi$
$257$	−29.9960	−1.87110	−0.935549	+	0.353196i	$0.885095\pi$
$258$	0	0
$259$	−0.0991320	−0.00615976
$260$	0	0
$261$	−38.1159	−2.35931
$262$	0	0
$263$	21.9585	1.35402	0.677008	−	0.735976i	$-0.263276\pi$
$263$	21.9585	1.35402	0.677008	+	0.735976i	$0.263276\pi$
$264$	0	0
$265$	−11.5288	−0.708207
$266$	0	0
$267$	−29.2624	−1.79083
$268$	0	0
$269$	27.0309	1.64811	0.824053	−	0.566513i	$-0.191708\pi$
$269$	27.0309	1.64811	0.824053	+	0.566513i	$0.191708\pi$
$270$	0	0
$271$	1.25444	0.0762020	0.0381010	−	0.999274i	$-0.487869\pi$
$271$	1.25444	0.0762020	0.0381010	+	0.999274i	$0.487869\pi$
$272$	0	0
$273$	1.27547	0.0771952
$274$	0	0
$275$	0.220081	0.0132714
$276$	0	0
$277$	28.3833	1.70539	0.852693	−	0.522412i	$-0.174968\pi$
$277$	28.3833	1.70539	0.852693	+	0.522412i	$0.174968\pi$
$278$	0	0
$279$	−29.9689	−1.79419
$280$	0	0
$281$	15.4963	0.924434	0.462217	−	0.886767i	$-0.347054\pi$
$281$	15.4963	0.924434	0.462217	+	0.886767i	$0.347054\pi$
$282$	0	0
$283$	11.7807	0.700290	0.350145	−	0.936695i	$-0.386132\pi$
$283$	11.7807	0.700290	0.350145	+	0.936695i	$0.386132\pi$
$284$	0	0
$285$	13.9049	0.823653
$286$	0	0
$287$	0.127571	0.00753028
$288$	0	0
$289$	10.7270	0.631002
$290$	0	0
$291$	−42.4850	−2.49052
$292$	0	0
$293$	4.21870	0.246459	0.123230	−	0.992378i	$-0.460675\pi$
$293$	4.21870	0.246459	0.123230	+	0.992378i	$0.460675\pi$
$294$	0	0
$295$	−9.52078	−0.554321
$296$	0	0
$297$	−0.610790	−0.0354416
$298$	0	0
$299$	−35.4275	−2.04882
$300$	0	0
$301$	−0.765997	−0.0441513
$302$	0	0
$303$	30.9576	1.77847
$304$	0	0
$305$	−11.6728	−0.668382
$306$	0	0
$307$	−19.8328	−1.13192	−0.565958	−	0.824434i	$-0.691494\pi$
$307$	−19.8328	−1.13192	−0.565958	+	0.824434i	$0.691494\pi$
$308$	0	0
$309$	−43.2095	−2.45811
$310$	0	0
$311$	−15.7501	−0.893106	−0.446553	−	0.894757i	$-0.647349\pi$
$311$	−15.7501	−0.893106	−0.446553	+	0.894757i	$0.647349\pi$
$312$	0	0
$313$	5.79132	0.327345	0.163673	−	0.986515i	$-0.447666\pi$
$313$	5.79132	0.327345	0.163673	+	0.986515i	$0.447666\pi$
$314$	0	0
$315$	0.401045	0.0225963
$316$	0	0
$317$	−22.8425	−1.28296	−0.641482	−	0.767138i	$-0.721680\pi$
$317$	−22.8425	−1.28296	−0.641482	+	0.767138i	$0.721680\pi$
$318$	0	0
$319$	−2.07353	−0.116095
$320$	0	0
$321$	−13.3878	−0.747236
$322$	0	0
$323$	27.5842	1.53483
$324$	0	0
$325$	4.84730	0.268880
$326$	0	0
$327$	9.25079	0.511570
$328$	0	0
$329$	1.08417	0.0597721
$330$	0	0
$331$	19.7119	1.08346	0.541732	−	0.840551i	$-0.317769\pi$
$331$	19.7119	1.08346	0.541732	+	0.840551i	$0.317769\pi$
$332$	0	0
$333$	4.04557	0.221696
$334$	0	0
$335$	3.23852	0.176939
$336$	0	0
$337$	−11.3948	−0.620714	−0.310357	−	0.950620i	$-0.600449\pi$
$337$	−11.3948	−0.620714	−0.310357	+	0.950620i	$0.600449\pi$
$338$	0	0
$339$	−13.4788	−0.732067
$340$	0	0
$341$	−1.63032	−0.0882869
$342$	0	0
$343$	1.38687	0.0748841
$344$	0	0
$345$	−19.3998	−1.04445
$346$	0	0
$347$	21.4817	1.15320	0.576600	−	0.817027i	$-0.304379\pi$
$347$	21.4817	1.15320	0.576600	+	0.817027i	$0.304379\pi$
$348$	0	0
$349$	11.1367	0.596134	0.298067	−	0.954545i	$-0.403658\pi$
$349$	11.1367	0.596134	0.298067	+	0.954545i	$0.403658\pi$
$350$	0	0
$351$	−13.4527	−0.718052
$352$	0	0
$353$	1.83417	0.0976232	0.0488116	−	0.998808i	$-0.484457\pi$
$353$	1.83417	0.0976232	0.0488116	+	0.998808i	$0.484457\pi$
$354$	0	0
$355$	8.58417	0.455601
$356$	0	0
$357$	1.38555	0.0733313
$358$	0	0
$359$	−17.8350	−0.941293	−0.470647	−	0.882322i	$-0.655979\pi$
$359$	−17.8350	−0.941293	−0.470647	+	0.882322i	$0.655979\pi$
$360$	0	0
$361$	8.44209	0.444321
$362$	0	0
$363$	29.0693	1.52574
$364$	0	0
$365$	3.89287	0.203762
$366$	0	0
$367$	24.0692	1.25640	0.628202	−	0.778050i	$-0.283791\pi$
$367$	24.0692	1.25640	0.628202	+	0.778050i	$0.283791\pi$
$368$	0	0
$369$	−5.20616	−0.271022
$370$	0	0
$371$	−1.14287	−0.0593349
$372$	0	0
$373$	−5.39983	−0.279593	−0.139796	−	0.990180i	$-0.544645\pi$
$373$	−5.39983	−0.279593	−0.139796	+	0.990180i	$0.544645\pi$
$374$	0	0
$375$	2.65435	0.137070
$376$	0	0
$377$	−45.6696	−2.35210
$378$	0	0
$379$	12.8207	0.658555	0.329277	−	0.944233i	$-0.393195\pi$
$379$	12.8207	0.658555	0.329277	+	0.944233i	$0.393195\pi$
$380$	0	0
$381$	−59.7078	−3.05892
$382$	0	0
$383$	3.79132	0.193728	0.0968638	−	0.995298i	$-0.469119\pi$
$383$	3.79132	0.193728	0.0968638	+	0.995298i	$0.469119\pi$
$384$	0	0
$385$	0.0218171	0.00111190
$386$	0	0
$387$	31.2603	1.58905
$388$	0	0
$389$	−10.3604	−0.525295	−0.262647	−	0.964892i	$-0.584596\pi$
$389$	−10.3604	−0.525295	−0.262647	+	0.964892i	$0.584596\pi$
$390$	0	0
$391$	−38.4850	−1.94627
$392$	0	0
$393$	2.78939	0.140706
$394$	0	0
$395$	−3.42965	−0.172564
$396$	0	0
$397$	−0.975666	−0.0489673	−0.0244836	−	0.999700i	$-0.507794\pi$
$397$	−0.975666	−0.0489673	−0.0244836	+	0.999700i	$0.507794\pi$
$398$	0	0
$399$	1.37842	0.0690071
$400$	0	0
$401$	1.32276	0.0660557	0.0330279	−	0.999454i	$-0.489485\pi$
$401$	1.32276	0.0660557	0.0330279	+	0.999454i	$0.489485\pi$
$402$	0	0
$403$	−35.9080	−1.78870
$404$	0	0
$405$	4.77009	0.237028
$406$	0	0
$407$	0.220081	0.0109090
$408$	0	0
$409$	−16.9214	−0.836710	−0.418355	−	0.908284i	$-0.637393\pi$
$409$	−16.9214	−0.836710	−0.418355	+	0.908284i	$0.637393\pi$
$410$	0	0
$411$	20.9504	1.03341
$412$	0	0
$413$	−0.943814	−0.0464420
$414$	0	0
$415$	−1.92982	−0.0947313
$416$	0	0
$417$	−12.3386	−0.604225
$418$	0	0
$419$	−6.51951	−0.318499	−0.159249	−	0.987238i	$-0.550907\pi$
$419$	−6.51951	−0.318499	−0.159249	+	0.987238i	$0.550907\pi$
$420$	0	0
$421$	31.9854	1.55887	0.779436	−	0.626482i	$-0.215506\pi$
$421$	31.9854	1.55887	0.779436	+	0.626482i	$0.215506\pi$
$422$	0	0
$423$	−44.2448	−2.15125
$424$	0	0
$425$	5.26565	0.255421
$426$	0	0
$427$	−1.15715	−0.0559982
$428$	0	0
$429$	−2.83166	−0.136714
$430$	0	0
$431$	4.61498	0.222296	0.111148	−	0.993804i	$-0.464547\pi$
$431$	4.61498	0.222296	0.111148	+	0.993804i	$0.464547\pi$
$432$	0	0
$433$	−1.85965	−0.0893689	−0.0446844	−	0.999001i	$-0.514228\pi$
$433$	−1.85965	−0.0893689	−0.0446844	+	0.999001i	$0.514228\pi$
$434$	0	0
$435$	−25.0083	−1.19906
$436$	0	0
$437$	−38.2868	−1.83150
$438$	0	0
$439$	4.91634	0.234644	0.117322	−	0.993094i	$-0.462569\pi$
$439$	4.91634	0.234644	0.117322	+	0.993094i	$0.462569\pi$
$440$	0	0
$441$	−28.2792	−1.34663
$442$	0	0
$443$	32.8401	1.56028	0.780140	−	0.625605i	$-0.215148\pi$
$443$	32.8401	1.56028	0.780140	+	0.625605i	$0.215148\pi$
$444$	0	0
$445$	−11.0243	−0.522604
$446$	0	0
$447$	44.2939	2.09503
$448$	0	0
$449$	2.14035	0.101010	0.0505048	−	0.998724i	$-0.483917\pi$
$449$	2.14035	0.101010	0.0505048	+	0.998724i	$0.483917\pi$
$450$	0	0
$451$	−0.283218	−0.0133362
$452$	0	0
$453$	−40.8765	−1.92055
$454$	0	0
$455$	0.480523	0.0225272
$456$	0	0
$457$	5.38842	0.252060	0.126030	−	0.992026i	$-0.459776\pi$
$457$	5.38842	0.252060	0.126030	+	0.992026i	$0.459776\pi$
$458$	0	0
$459$	−14.6137	−0.682111
$460$	0	0
$461$	35.9688	1.67523	0.837617	−	0.546258i	$-0.183948\pi$
$461$	35.9688	1.67523	0.837617	+	0.546258i	$0.183948\pi$
$462$	0	0
$463$	27.9523	1.29906	0.649528	−	0.760338i	$-0.274967\pi$
$463$	27.9523	1.29906	0.649528	+	0.760338i	$0.274967\pi$
$464$	0	0
$465$	−19.6630	−0.911848
$466$	0	0
$467$	−26.9562	−1.24739	−0.623693	−	0.781669i	$-0.714368\pi$
$467$	−26.9562	−1.24739	−0.623693	+	0.781669i	$0.714368\pi$
$468$	0	0
$469$	0.321041	0.0148243
$470$	0	0
$471$	48.6925	2.24363
$472$	0	0
$473$	1.70058	0.0781925
$474$	0	0
$475$	5.23852	0.240360
$476$	0	0
$477$	46.6404	2.13552
$478$	0	0
$479$	−19.7279	−0.901391	−0.450695	−	0.892678i	$-0.648824\pi$
$479$	−19.7279	−0.901391	−0.450695	+	0.892678i	$0.648824\pi$
$480$	0	0
$481$	4.84730	0.221018
$482$	0	0
$483$	−1.92314	−0.0875060
$484$	0	0
$485$	−16.0058	−0.726787
$486$	0	0
$487$	−26.2630	−1.19009	−0.595045	−	0.803693i	$-0.702866\pi$
$487$	−26.2630	−1.19009	−0.595045	+	0.803693i	$0.702866\pi$
$488$	0	0
$489$	31.0852	1.40572
$490$	0	0
$491$	24.9923	1.12789	0.563944	−	0.825813i	$-0.309283\pi$
$491$	24.9923	1.12789	0.563944	+	0.825813i	$0.309283\pi$
$492$	0	0
$493$	−49.6111	−2.23437
$494$	0	0
$495$	−0.890352	−0.0400184
$496$	0	0
$497$	0.850966	0.0381710
$498$	0	0
$499$	−34.2383	−1.53272	−0.766359	−	0.642412i	$-0.777934\pi$
$499$	−34.2383	−1.53272	−0.766359	+	0.642412i	$0.777934\pi$
$500$	0	0
$501$	5.80677	0.259427
$502$	0	0
$503$	−3.60906	−0.160920	−0.0804600	−	0.996758i	$-0.525639\pi$
$503$	−3.60906	−0.160920	−0.0804600	+	0.996758i	$0.525639\pi$
$504$	0	0
$505$	11.6630	0.518995
$506$	0	0
$507$	−27.8609	−1.23735
$508$	0	0
$509$	−39.0854	−1.73243	−0.866215	−	0.499672i	$-0.833454\pi$
$509$	−39.0854	−1.73243	−0.866215	+	0.499672i	$0.833454\pi$
$510$	0	0
$511$	0.385908	0.0170716
$512$	0	0
$513$	−14.5385	−0.641888
$514$	0	0
$515$	−16.2788	−0.717329
$516$	0	0
$517$	−2.40694	−0.105857
$518$	0	0
$519$	−55.9617	−2.45645
$520$	0	0
$521$	7.79053	0.341309	0.170655	−	0.985331i	$-0.445412\pi$
$521$	7.79053	0.341309	0.170655	+	0.985331i	$0.445412\pi$
$522$	0	0
$523$	19.5176	0.853444	0.426722	−	0.904383i	$-0.359668\pi$
$523$	19.5176	0.853444	0.426722	+	0.904383i	$0.359668\pi$
$524$	0	0
$525$	0.263131	0.0114840
$526$	0	0
$527$	−39.0070	−1.69917
$528$	0	0
$529$	30.4171	1.32248
$530$	0	0
$531$	38.5169	1.67149
$532$	0	0
$533$	−6.23790	−0.270193
$534$	0	0
$535$	−5.04374	−0.218060
$536$	0	0
$537$	−46.7292	−2.01651
$538$	0	0
$539$	−1.53840	−0.0662638
$540$	0	0
$541$	−1.54384	−0.0663748	−0.0331874	−	0.999449i	$-0.510566\pi$
$541$	−1.54384	−0.0663748	−0.0331874	+	0.999449i	$0.510566\pi$
$542$	0	0
$543$	49.8445	2.13903
$544$	0	0
$545$	3.48514	0.149287
$546$	0	0
$547$	27.6568	1.18252	0.591259	−	0.806481i	$-0.298631\pi$
$547$	27.6568	1.18252	0.591259	+	0.806481i	$0.298631\pi$
$548$	0	0
$549$	47.2230	2.01543
$550$	0	0
$551$	−49.3555	−2.10261
$552$	0	0
$553$	−0.339988	−0.0144577
$554$	0	0
$555$	2.65435	0.112671
$556$	0	0
$557$	−3.38398	−0.143384	−0.0716918	−	0.997427i	$-0.522840\pi$
$557$	−3.38398	−0.143384	−0.0716918	+	0.997427i	$0.522840\pi$
$558$	0	0
$559$	37.4553	1.58419
$560$	0	0
$561$	−3.07604	−0.129871
$562$	0	0
$563$	−13.4849	−0.568322	−0.284161	−	0.958777i	$-0.591715\pi$
$563$	−13.4849	−0.568322	−0.284161	+	0.958777i	$0.591715\pi$
$564$	0	0
$565$	−5.07800	−0.213633
$566$	0	0
$567$	0.472869	0.0198586
$568$	0	0
$569$	21.8628	0.916536	0.458268	−	0.888814i	$-0.348470\pi$
$569$	21.8628	0.916536	0.458268	+	0.888814i	$0.348470\pi$
$570$	0	0
$571$	30.7199	1.28559	0.642795	−	0.766039i	$-0.277775\pi$
$571$	30.7199	1.28559	0.642795	+	0.766039i	$0.277775\pi$
$572$	0	0
$573$	−24.0765	−1.00581
$574$	0	0
$575$	−7.30870	−0.304794
$576$	0	0
$577$	−12.8276	−0.534022	−0.267011	−	0.963693i	$-0.586036\pi$
$577$	−12.8276	−0.534022	−0.267011	+	0.963693i	$0.586036\pi$
$578$	0	0
$579$	−2.97201	−0.123513
$580$	0	0
$581$	−0.191307	−0.00793676
$582$	0	0
$583$	2.53726	0.105083
$584$	0	0
$585$	−19.6101	−0.810777
$586$	0	0
$587$	12.0533	0.497494	0.248747	−	0.968569i	$-0.419981\pi$
$587$	12.0533	0.497494	0.248747	+	0.968569i	$0.419981\pi$
$588$	0	0
$589$	−38.8061	−1.59898
$590$	0	0
$591$	−27.6694	−1.13817
$592$	0	0
$593$	19.6325	0.806210	0.403105	−	0.915154i	$-0.367931\pi$
$593$	19.6325	0.806210	0.403105	+	0.915154i	$0.367931\pi$
$594$	0	0
$595$	0.521994	0.0213997
$596$	0	0
$597$	8.47008	0.346657
$598$	0	0
$599$	−18.7296	−0.765269	−0.382635	−	0.923900i	$-0.624983\pi$
$599$	−18.7296	−0.765269	−0.382635	+	0.923900i	$0.624983\pi$
$600$	0	0
$601$	−22.4616	−0.916229	−0.458114	−	0.888893i	$-0.651475\pi$
$601$	−22.4616	−0.916229	−0.458114	+	0.888893i	$0.651475\pi$
$602$	0	0
$603$	−13.1016	−0.533540
$604$	0	0
$605$	10.9516	0.445244
$606$	0	0
$607$	−17.0840	−0.693419	−0.346710	−	0.937973i	$-0.612701\pi$
$607$	−17.0840	−0.693419	−0.346710	+	0.937973i	$0.612701\pi$
$608$	0	0
$609$	−2.47913	−0.100459
$610$	0	0
$611$	−53.0130	−2.14468
$612$	0	0
$613$	4.99327	0.201676	0.100838	−	0.994903i	$-0.467848\pi$
$613$	4.99327	0.201676	0.100838	+	0.994903i	$0.467848\pi$
$614$	0	0
$615$	−3.41583	−0.137739
$616$	0	0
$617$	42.4996	1.71097	0.855486	−	0.517827i	$-0.173259\pi$
$617$	42.4996	1.71097	0.855486	+	0.517827i	$0.173259\pi$
$618$	0	0
$619$	−27.8458	−1.11922	−0.559609	−	0.828757i	$-0.689049\pi$
$619$	−27.8458	−1.11922	−0.559609	+	0.828757i	$0.689049\pi$
$620$	0	0
$621$	20.2838	0.813961
$622$	0	0
$623$	−1.09286	−0.0437847
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.06020	−0.122212
$628$	0	0
$629$	5.26565	0.209955
$630$	0	0
$631$	21.6791	0.863032	0.431516	−	0.902105i	$-0.357979\pi$
$631$	21.6791	0.863032	0.431516	+	0.902105i	$0.357979\pi$
$632$	0	0
$633$	−21.6750	−0.861502
$634$	0	0
$635$	−22.4943	−0.892661
$636$	0	0
$637$	−33.8835	−1.34251
$638$	0	0
$639$	−34.7278	−1.37381
$640$	0	0
$641$	−17.6079	−0.695471	−0.347736	−	0.937593i	$-0.613049\pi$
$641$	−17.6079	−0.695471	−0.347736	+	0.937593i	$0.613049\pi$
$642$	0	0
$643$	−11.1535	−0.439851	−0.219925	−	0.975517i	$-0.570581\pi$
$643$	−11.1535	−0.439851	−0.219925	+	0.975517i	$0.570581\pi$
$644$	0	0
$645$	20.5103	0.807591
$646$	0	0
$647$	−34.6543	−1.36240	−0.681200	−	0.732097i	$-0.738542\pi$
$647$	−34.6543	−1.36240	−0.681200	+	0.732097i	$0.738542\pi$
$648$	0	0
$649$	2.09534	0.0822494
$650$	0	0
$651$	−1.94923	−0.0763963
$652$	0	0
$653$	20.5116	0.802683	0.401341	−	0.915929i	$-0.368544\pi$
$653$	20.5116	0.802683	0.401341	+	0.915929i	$0.368544\pi$
$654$	0	0
$655$	1.05088	0.0410611
$656$	0	0
$657$	−15.7489	−0.614422
$658$	0	0
$659$	−3.67372	−0.143108	−0.0715540	−	0.997437i	$-0.522796\pi$
$659$	−3.67372	−0.143108	−0.0715540	+	0.997437i	$0.522796\pi$
$660$	0	0
$661$	43.6886	1.69929	0.849645	−	0.527354i	$-0.176816\pi$
$661$	43.6886	1.69929	0.849645	+	0.527354i	$0.176816\pi$
$662$	0	0
$663$	−67.7501	−2.63120
$664$	0	0
$665$	0.519305	0.0201378
$666$	0	0
$667$	68.8600	2.66627
$668$	0	0
$669$	−29.2371	−1.13037
$670$	0	0
$671$	2.56896	0.0991735
$672$	0	0
$673$	7.14752	0.275517	0.137758	−	0.990466i	$-0.456010\pi$
$673$	7.14752	0.275517	0.137758	+	0.990466i	$0.456010\pi$
$674$	0	0
$675$	−2.77530	−0.106821
$676$	0	0
$677$	25.9643	0.997891	0.498945	−	0.866633i	$-0.333721\pi$
$677$	25.9643	0.997891	0.498945	+	0.866633i	$0.333721\pi$
$678$	0	0
$679$	−1.58669	−0.0608915
$680$	0	0
$681$	−5.86468	−0.224735
$682$	0	0
$683$	−43.0530	−1.64738	−0.823688	−	0.567043i	$-0.808087\pi$
$683$	−43.0530	−1.64738	−0.823688	+	0.567043i	$0.808087\pi$
$684$	0	0
$685$	7.89287	0.301571
$686$	0	0
$687$	−4.25983	−0.162523
$688$	0	0
$689$	55.8835	2.12899
$690$	0	0
$691$	43.6279	1.65968	0.829842	−	0.557999i	$-0.188431\pi$
$691$	43.6279	1.65968	0.829842	+	0.557999i	$0.188431\pi$
$692$	0	0
$693$	−0.0882624	−0.00335281
$694$	0	0
$695$	−4.64845	−0.176326
$696$	0	0
$697$	−6.77626	−0.256669
$698$	0	0
$699$	−74.4833	−2.81722
$700$	0	0
$701$	10.1255	0.382436	0.191218	−	0.981548i	$-0.438756\pi$
$701$	10.1255	0.382436	0.191218	+	0.981548i	$0.438756\pi$
$702$	0	0
$703$	5.23852	0.197575
$704$	0	0
$705$	−29.0296	−1.09332
$706$	0	0
$707$	1.15617	0.0434823
$708$	0	0
$709$	−3.93709	−0.147861	−0.0739303	−	0.997263i	$-0.523554\pi$
$709$	−3.93709	−0.147861	−0.0739303	+	0.997263i	$0.523554\pi$
$710$	0	0
$711$	13.8749	0.520348
$712$	0	0
$713$	54.1416	2.02762
$714$	0	0
$715$	−1.06680	−0.0398960
$716$	0	0
$717$	−5.12451	−0.191378
$718$	0	0
$719$	−3.31794	−0.123738	−0.0618691	−	0.998084i	$-0.519706\pi$
$719$	−3.31794	−0.123738	−0.0618691	+	0.998084i	$0.519706\pi$
$720$	0	0
$721$	−1.61375	−0.0600991
$722$	0	0
$723$	−13.4342	−0.499624
$724$	0	0
$725$	−9.42165	−0.349911
$726$	0	0
$727$	9.10368	0.337637	0.168818	−	0.985647i	$-0.446005\pi$
$727$	9.10368	0.337637	0.168818	+	0.985647i	$0.446005\pi$
$728$	0	0
$729$	−41.3975	−1.53324
$730$	0	0
$731$	40.6879	1.50490
$732$	0	0
$733$	−24.8060	−0.916232	−0.458116	−	0.888892i	$-0.651476\pi$
$733$	−24.8060	−0.916232	−0.458116	+	0.888892i	$0.651476\pi$
$734$	0	0
$735$	−18.5544	−0.684388
$736$	0	0
$737$	−0.712737	−0.0262540
$738$	0	0
$739$	−11.1655	−0.410729	−0.205364	−	0.978686i	$-0.565838\pi$
$739$	−11.1655	−0.410729	−0.205364	+	0.978686i	$0.565838\pi$
$740$	0	0
$741$	−67.4011	−2.47604
$742$	0	0
$743$	−46.6202	−1.71033	−0.855165	−	0.518356i	$-0.826544\pi$
$743$	−46.6202	−1.71033	−0.855165	+	0.518356i	$0.826544\pi$
$744$	0	0
$745$	16.6873	0.611375
$746$	0	0
$747$	7.80723	0.285652
$748$	0	0
$749$	−0.499996	−0.0182694
$750$	0	0
$751$	−36.1590	−1.31946	−0.659730	−	0.751503i	$-0.729329\pi$
$751$	−36.1590	−1.31946	−0.659730	+	0.751503i	$0.729329\pi$
$752$	0	0
$753$	6.00640	0.218885
$754$	0	0
$755$	−15.3998	−0.560457
$756$	0	0
$757$	44.5753	1.62012	0.810059	−	0.586348i	$-0.199435\pi$
$757$	44.5753	1.62012	0.810059	+	0.586348i	$0.199435\pi$
$758$	0	0
$759$	4.26953	0.154974
$760$	0	0
$761$	5.79211	0.209964	0.104982	−	0.994474i	$-0.466522\pi$
$761$	5.79211	0.209964	0.104982	+	0.994474i	$0.466522\pi$
$762$	0	0
$763$	0.345489	0.0125075
$764$	0	0
$765$	−21.3025	−0.770194
$766$	0	0
$767$	46.1501	1.66638
$768$	0	0
$769$	32.1730	1.16019	0.580094	−	0.814549i	$-0.303016\pi$
$769$	32.1730	1.16019	0.580094	+	0.814549i	$0.303016\pi$
$770$	0	0
$771$	79.6198	2.86744
$772$	0	0
$773$	−15.3846	−0.553346	−0.276673	−	0.960964i	$-0.589232\pi$
$773$	−15.3846	−0.553346	−0.276673	+	0.960964i	$0.589232\pi$
$774$	0	0
$775$	−7.40783	−0.266097
$776$	0	0
$777$	0.263131	0.00943976
$778$	0	0
$779$	−6.74135	−0.241534
$780$	0	0
$781$	−1.88921	−0.0676014
$782$	0	0
$783$	26.1479	0.934449
$784$	0	0
$785$	18.3444	0.654741
$786$	0	0
$787$	−35.3001	−1.25831	−0.629156	−	0.777279i	$-0.716600\pi$
$787$	−35.3001	−1.25831	−0.629156	+	0.777279i	$0.716600\pi$
$788$	0	0
$789$	−58.2854	−2.07501
$790$	0	0
$791$	−0.503392	−0.0178986
$792$	0	0
$793$	56.5815	2.00927
$794$	0	0
$795$	30.6014	1.08532
$796$	0	0
$797$	32.1361	1.13832	0.569160	−	0.822227i	$-0.307268\pi$
$797$	32.1361	1.13832	0.569160	+	0.822227i	$0.307268\pi$
$798$	0	0
$799$	−57.5883	−2.03733
$800$	0	0
$801$	44.5997	1.57585
$802$	0	0
$803$	−0.856746	−0.0302339
$804$	0	0
$805$	−0.724525	−0.0255362
$806$	0	0
$807$	−71.7496	−2.52570
$808$	0	0
$809$	34.9610	1.22916	0.614582	−	0.788853i	$-0.289325\pi$
$809$	34.9610	1.22916	0.614582	+	0.788853i	$0.289325\pi$
$810$	0	0
$811$	−22.1294	−0.777068	−0.388534	−	0.921434i	$-0.627018\pi$
$811$	−22.1294	−0.777068	−0.388534	+	0.921434i	$0.627018\pi$
$812$	0	0
$813$	−3.32973	−0.116779
$814$	0	0
$815$	11.7110	0.410220
$816$	0	0
$817$	40.4783	1.41616
$818$	0	0
$819$	−1.94399	−0.0679284
$820$	0	0
$821$	−39.7487	−1.38724	−0.693620	−	0.720341i	$-0.743985\pi$
$821$	−39.7487	−1.38724	−0.693620	+	0.720341i	$0.743985\pi$
$822$	0	0
$823$	−9.13715	−0.318501	−0.159251	−	0.987238i	$-0.550908\pi$
$823$	−9.13715	−0.318501	−0.159251	+	0.987238i	$0.550908\pi$
$824$	0	0
$825$	−0.584172	−0.0203382
$826$	0	0
$827$	16.8060	0.584403	0.292202	−	0.956357i	$-0.405612\pi$
$827$	16.8060	0.584403	0.292202	+	0.956357i	$0.405612\pi$
$828$	0	0
$829$	30.7489	1.06796	0.533978	−	0.845499i	$-0.320697\pi$
$829$	30.7489	1.06796	0.533978	+	0.845499i	$0.320697\pi$
$830$	0	0
$831$	−75.3391	−2.61348
$832$	0	0
$833$	−36.8078	−1.27531
$834$	0	0
$835$	2.18764	0.0757065
$836$	0	0
$837$	20.5589	0.710621
$838$	0	0
$839$	20.7298	0.715671	0.357835	−	0.933785i	$-0.383515\pi$
$839$	20.7298	0.715671	0.357835	+	0.933785i	$0.383515\pi$
$840$	0	0
$841$	59.7674	2.06095
$842$	0	0
$843$	−41.1327	−1.41669
$844$	0	0
$845$	−10.4963	−0.361085
$846$	0	0
$847$	1.08565	0.0373034
$848$	0	0
$849$	−31.2701	−1.07319
$850$	0	0
$851$	−7.30870	−0.250539
$852$	0	0
$853$	−8.26852	−0.283109	−0.141554	−	0.989930i	$-0.545210\pi$
$853$	−8.26852	−0.283109	−0.141554	+	0.989930i	$0.545210\pi$
$854$	0	0
$855$	−21.1928	−0.724778
$856$	0	0
$857$	−23.0606	−0.787736	−0.393868	−	0.919167i	$-0.628863\pi$
$857$	−23.0606	−0.787736	−0.393868	+	0.919167i	$0.628863\pi$
$858$	0	0
$859$	52.0956	1.77748	0.888739	−	0.458414i	$-0.151582\pi$
$859$	52.0956	1.77748	0.888739	+	0.458414i	$0.151582\pi$
$860$	0	0
$861$	−0.338618	−0.0115401
$862$	0	0
$863$	−0.369200	−0.0125677	−0.00628385	−	0.999980i	$-0.502000\pi$
$863$	−0.369200	−0.0125677	−0.00628385	+	0.999980i	$0.502000\pi$
$864$	0	0
$865$	−21.0830	−0.716844
$866$	0	0
$867$	−28.4733	−0.967004
$868$	0	0
$869$	0.754800	0.0256048
$870$	0	0
$871$	−15.6981	−0.531909
$872$	0	0
$873$	64.7526	2.19154
$874$	0	0
$875$	0.0991320	0.00335127
$876$	0	0
$877$	−0.590335	−0.0199342	−0.00996711	−	0.999950i	$-0.503173\pi$
$877$	−0.590335	−0.0199342	−0.00996711	+	0.999950i	$0.503173\pi$
$878$	0	0
$879$	−11.1979	−0.377696
$880$	0	0
$881$	−1.15191	−0.0388089	−0.0194045	−	0.999812i	$-0.506177\pi$
$881$	−1.15191	−0.0388089	−0.0194045	+	0.999812i	$0.506177\pi$
$882$	0	0
$883$	35.0790	1.18050	0.590252	−	0.807219i	$-0.299029\pi$
$883$	35.0790	1.18050	0.590252	+	0.807219i	$0.299029\pi$
$884$	0	0
$885$	25.2715	0.849491
$886$	0	0
$887$	11.1339	0.373839	0.186920	−	0.982375i	$-0.440150\pi$
$887$	11.1339	0.373839	0.186920	+	0.982375i	$0.440150\pi$
$888$	0	0
$889$	−2.22991	−0.0747887
$890$	0	0
$891$	−1.04981	−0.0351698
$892$	0	0
$893$	−57.2916	−1.91719
$894$	0	0
$895$	−17.6048	−0.588463
$896$	0	0
$897$	94.0368	3.13980
$898$	0	0
$899$	69.7940	2.32776
$900$	0	0
$901$	60.7065	2.02243
$902$	0	0
$903$	2.03322	0.0676614
$904$	0	0
$905$	18.7784	0.624216
$906$	0	0
$907$	23.3139	0.774127	0.387063	−	0.922053i	$-0.373489\pi$
$907$	23.3139	0.774127	0.387063	+	0.922053i	$0.373489\pi$
$908$	0	0
$909$	−47.1833	−1.56497
$910$	0	0
$911$	4.25089	0.140838	0.0704192	−	0.997517i	$-0.477566\pi$
$911$	4.25089	0.140838	0.0704192	+	0.997517i	$0.477566\pi$
$912$	0	0
$913$	0.424717	0.0140561
$914$	0	0
$915$	30.9836	1.02429
$916$	0	0
$917$	0.104175	0.00344018
$918$	0	0
$919$	41.7765	1.37808	0.689039	−	0.724724i	$-0.258033\pi$
$919$	41.7765	1.37808	0.689039	+	0.724724i	$0.258033\pi$
$920$	0	0
$921$	52.6431	1.73465
$922$	0	0
$923$	−41.6101	−1.36961
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	65.8569	2.16302
$928$	0	0
$929$	27.6688	0.907783	0.453891	−	0.891057i	$-0.350035\pi$
$929$	27.6688	0.907783	0.453891	+	0.891057i	$0.350035\pi$
$930$	0	0
$931$	−36.6182	−1.20011
$932$	0	0
$933$	41.8063	1.36868
$934$	0	0
$935$	−1.15887	−0.0378991
$936$	0	0
$937$	−46.3821	−1.51524	−0.757619	−	0.652698i	$-0.773637\pi$
$937$	−46.3821	−1.51524	−0.757619	+	0.652698i	$0.773637\pi$
$938$	0	0
$939$	−15.3722	−0.501653
$940$	0	0
$941$	−43.4417	−1.41616	−0.708080	−	0.706132i	$-0.750438\pi$
$941$	−43.4417	−1.41616	−0.708080	+	0.706132i	$0.750438\pi$
$942$	0	0
$943$	9.40542	0.306283
$944$	0	0
$945$	−0.275121	−0.00894968
$946$	0	0
$947$	−34.4145	−1.11832	−0.559161	−	0.829059i	$-0.688877\pi$
$947$	−34.4145	−1.11832	−0.559161	+	0.829059i	$0.688877\pi$
$948$	0	0
$949$	−18.8699	−0.612543
$950$	0	0
$951$	60.6320	1.96613
$952$	0	0
$953$	4.95326	0.160452	0.0802259	−	0.996777i	$-0.474436\pi$
$953$	4.95326	0.160452	0.0802259	+	0.996777i	$0.474436\pi$
$954$	0	0
$955$	−9.07059	−0.293517
$956$	0	0
$957$	5.50386	0.177914
$958$	0	0
$959$	0.782436	0.0252662
$960$	0	0
$961$	23.8759	0.770191
$962$	0	0
$963$	20.4048	0.657535
$964$	0	0
$965$	−1.11968	−0.0360436
$966$	0	0
$967$	14.4788	0.465607	0.232804	−	0.972524i	$-0.425210\pi$
$967$	14.4788	0.465607	0.232804	+	0.972524i	$0.425210\pi$
$968$	0	0
$969$	−73.2181	−2.35210
$970$	0	0
$971$	−35.5736	−1.14161	−0.570806	−	0.821085i	$-0.693369\pi$
$971$	−35.5736	−1.14161	−0.570806	+	0.821085i	$0.693369\pi$
$972$	0	0
$973$	−0.460810	−0.0147729
$974$	0	0
$975$	−12.8664	−0.412056
$976$	0	0
$977$	−9.38842	−0.300362	−0.150181	−	0.988658i	$-0.547986\pi$
$977$	−9.38842	−0.300362	−0.150181	+	0.988658i	$0.547986\pi$
$978$	0	0
$979$	2.42625	0.0775432
$980$	0	0
$981$	−14.0994	−0.450159
$982$	0	0
$983$	51.5878	1.64539	0.822697	−	0.568480i	$-0.192468\pi$
$983$	51.5878	1.64539	0.822697	+	0.568480i	$0.192468\pi$
$984$	0	0
$985$	−10.4242	−0.332141
$986$	0	0
$987$	−2.87776	−0.0916000
$988$	0	0
$989$	−56.4746	−1.79579
$990$	0	0
$991$	−41.7281	−1.32554	−0.662769	−	0.748824i	$-0.730619\pi$
$991$	−41.7281	−1.32554	−0.662769	+	0.748824i	$0.730619\pi$
$992$	0	0
$993$	−52.3223	−1.66040
$994$	0	0
$995$	3.19102	0.101162
$996$	0	0
$997$	−46.6676	−1.47798	−0.738989	−	0.673717i	$-0.764697\pi$
$997$	−46.6676	−1.47798	−0.738989	+	0.673717i	$0.764697\pi$
$998$	0	0
$999$	−2.77530	−0.0878065

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.i.1.5	✓	5	4.3	odd	2
2960.2.a.y.1.1		5	1.1	even	1	trivial
7400.2.a.p.1.1		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)

\(f(q)\)	\(=\)	\(q-2.65435 q^{3} -1.00000 q^{5} -0.0991320 q^{7} +4.04557 q^{9} +O(q^{10})\)	\(q-2.65435 q^{3} -1.00000 q^{5} -0.0991320 q^{7} +4.04557 q^{9} +0.220081 q^{11} +4.84730 q^{13} +2.65435 q^{15} +5.26565 q^{17} +5.23852 q^{19} +0.263131 q^{21} -7.30870 q^{23} +1.00000 q^{25} -2.77530 q^{27} -9.42165 q^{29} -7.40783 q^{31} -0.584172 q^{33} +0.0991320 q^{35} +1.00000 q^{37} -12.8664 q^{39} -1.28688 q^{41} +7.72704 q^{43} -4.04557 q^{45} -10.9366 q^{47} -6.99017 q^{49} -13.9769 q^{51} +11.5288 q^{53} -0.220081 q^{55} -13.9049 q^{57} +9.52078 q^{59} +11.6728 q^{61} -0.401045 q^{63} -4.84730 q^{65} -3.23852 q^{67} +19.3998 q^{69} -8.58417 q^{71} -3.89287 q^{73} -2.65435 q^{75} -0.0218171 q^{77} +3.42965 q^{79} -4.77009 q^{81} +1.92982 q^{83} -5.26565 q^{85} +25.0083 q^{87} +11.0243 q^{89} -0.480523 q^{91} +19.6630 q^{93} -5.23852 q^{95} +16.0058 q^{97} +0.890352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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