LMFDB - Embedded newform 2960.2.a.x.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:	$1$
Dimension:	$5$
Coefficient field:	5.5.998068.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 3x - 2 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 3x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.350931$ 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)

$f(q)$	$=$	$q-3.08134 q^{3} -1.00000 q^{5} -1.91593 q^{7} +6.49464 q^{9} +O(q^{10})$	$q-3.08134 q^{3} -1.00000 q^{5} -1.91593 q^{7} +6.49464 q^{9} -1.79551 q^{11} -0.701862 q^{13} +3.08134 q^{15} +0.463548 q^{17} +0.120416 q^{19} +5.90361 q^{21} -0.408979 q^{23} +1.00000 q^{25} -10.7682 q^{27} +1.24357 q^{29} +2.54603 q^{31} +5.53257 q^{33} +1.91593 q^{35} -1.00000 q^{37} +2.16267 q^{39} +9.54921 q^{41} +4.56366 q^{43} -6.49464 q^{45} +7.05671 q^{47} -3.32923 q^{49} -1.42835 q^{51} -3.77089 q^{53} +1.79551 q^{55} -0.371042 q^{57} -1.54876 q^{59} +8.89448 q^{61} -12.4432 q^{63} +0.701862 q^{65} -3.19332 q^{67} +1.26020 q^{69} +4.19650 q^{71} +6.19650 q^{73} -3.08134 q^{75} +3.44007 q^{77} +11.5187 q^{79} +13.6964 q^{81} -13.6530 q^{83} -0.463548 q^{85} -3.83185 q^{87} +1.48188 q^{89} +1.34472 q^{91} -7.84517 q^{93} -0.120416 q^{95} -14.5637 q^{97} -11.6612 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9	$ 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 4 q^{13} + q^{15} - q^{17} - 10 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 11 q^{29} + 3 q^{31} - 3 q^{33} - 2 q^{35} - 5 q^{37} - 18 q^{39} + 16 q^{41} - 31 q^{43} - 6 q^{45} - 5 q^{47} + 7 q^{49} - 34 q^{51} + 8 q^{53} + 8 q^{55} - 8 q^{57} - 24 q^{59} - 15 q^{61} - 19 q^{63} + 4 q^{65} - 12 q^{67} + 10 q^{69} - 5 q^{71} + 5 q^{73} - q^{75} - 4 q^{77} - 4 q^{79} + 17 q^{81} - 27 q^{83} + q^{85} + 4 q^{87} + 16 q^{89} - 26 q^{91} + 14 q^{93} + 10 q^{95} - 19 q^{97} - 37 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 4 * q^13 + q^15 - q^17 - 10 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 11 * q^29 + 3 * q^31 - 3 * q^33 - 2 * q^35 - 5 * q^37 - 18 * q^39 + 16 * q^41 - 31 * q^43 - 6 * q^45 - 5 * q^47 + 7 * q^49 - 34 * q^51 + 8 * q^53 + 8 * q^55 - 8 * q^57 - 24 * q^59 - 15 * q^61 - 19 * q^63 + 4 * q^65 - 12 * q^67 + 10 * q^69 - 5 * q^71 + 5 * q^73 - q^75 - 4 * q^77 - 4 * q^79 + 17 * q^81 - 27 * q^83 + q^85 + 4 * q^87 + 16 * q^89 - 26 * q^91 + 14 * q^93 + 10 * q^95 - 19 * q^97 - 37 * 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
$3$	−3.08134	−1.77901	−0.889505	−	0.456925i	$-0.848951\pi$
$3$	−3.08134	−1.77901	−0.889505	+	0.456925i	$0.848951\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.91593	−0.724152	−0.362076	−	0.932149i	$-0.617932\pi$
$7$	−1.91593	−0.724152	−0.362076	+	0.932149i	$0.617932\pi$
$8$	0	0
$9$	6.49464	2.16488
$10$	0	0
$11$	−1.79551	−0.541367	−0.270683	−	0.962668i	$-0.587250\pi$
$11$	−1.79551	−0.541367	−0.270683	+	0.962668i	$0.587250\pi$
$12$	0	0
$13$	−0.701862	−0.194662	−0.0973308	−	0.995252i	$-0.531030\pi$
$13$	−0.701862	−0.194662	−0.0973308	+	0.995252i	$0.531030\pi$
$14$	0	0
$15$	3.08134	0.795598
$16$	0	0
$17$	0.463548	0.112427	0.0562135	−	0.998419i	$-0.482097\pi$
$17$	0.463548	0.112427	0.0562135	+	0.998419i	$0.482097\pi$
$18$	0	0
$19$	0.120416	0.0276253	0.0138126	−	0.999905i	$-0.495603\pi$
$19$	0.120416	0.0276253	0.0138126	+	0.999905i	$0.495603\pi$
$20$	0	0
$21$	5.90361	1.28827
$22$	0	0
$23$	−0.408979	−0.0852779	−0.0426390	−	0.999091i	$-0.513577\pi$
$23$	−0.408979	−0.0852779	−0.0426390	+	0.999091i	$0.513577\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−10.7682	−2.07233
$28$	0	0
$29$	1.24357	0.230925	0.115462	−	0.993312i	$-0.463165\pi$
$29$	1.24357	0.230925	0.115462	+	0.993312i	$0.463165\pi$
$30$	0	0
$31$	2.54603	0.457280	0.228640	−	0.973511i	$-0.426572\pi$
$31$	2.54603	0.457280	0.228640	+	0.973511i	$0.426572\pi$
$32$	0	0
$33$	5.53257	0.963097
$34$	0	0
$35$	1.91593	0.323851
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	2.16267	0.346305
$40$	0	0
$41$	9.54921	1.49134	0.745668	−	0.666318i	$-0.232131\pi$
$41$	9.54921	1.49134	0.745668	+	0.666318i	$0.232131\pi$
$42$	0	0
$43$	4.56366	0.695952	0.347976	−	0.937503i	$-0.386869\pi$
$43$	4.56366	0.695952	0.347976	+	0.937503i	$0.386869\pi$
$44$	0	0
$45$	−6.49464	−0.968163
$46$	0	0
$47$	7.05671	1.02933	0.514664	−	0.857392i	$-0.327917\pi$
$47$	7.05671	1.02933	0.514664	+	0.857392i	$0.327917\pi$
$48$	0	0
$49$	−3.32923	−0.475604
$50$	0	0
$51$	−1.42835	−0.200009
$52$	0	0
$53$	−3.77089	−0.517971	−0.258986	−	0.965881i	$-0.583388\pi$
$53$	−3.77089	−0.517971	−0.258986	+	0.965881i	$0.583388\pi$
$54$	0	0
$55$	1.79551	0.242107
$56$	0	0
$57$	−0.371042	−0.0491457
$58$	0	0
$59$	−1.54876	−0.201632	−0.100816	−	0.994905i	$-0.532145\pi$
$59$	−1.54876	−0.201632	−0.100816	+	0.994905i	$0.532145\pi$
$60$	0	0
$61$	8.89448	1.13882	0.569411	−	0.822053i	$-0.307171\pi$
$61$	8.89448	1.13882	0.569411	+	0.822053i	$0.307171\pi$
$62$	0	0
$63$	−12.4432	−1.56770
$64$	0	0
$65$	0.701862	0.0870553
$66$	0	0
$67$	−3.19332	−0.390126	−0.195063	−	0.980791i	$-0.562491\pi$
$67$	−3.19332	−0.390126	−0.195063	+	0.980791i	$0.562491\pi$
$68$	0	0
$69$	1.26020	0.151710
$70$	0	0
$71$	4.19650	0.498033	0.249016	−	0.968499i	$-0.419893\pi$
$71$	4.19650	0.498033	0.249016	+	0.968499i	$0.419893\pi$
$72$	0	0
$73$	6.19650	0.725245	0.362623	−	0.931936i	$-0.381881\pi$
$73$	6.19650	0.725245	0.362623	+	0.931936i	$0.381881\pi$
$74$	0	0
$75$	−3.08134	−0.355802
$76$	0	0
$77$	3.44007	0.392032
$78$	0	0
$79$	11.5187	1.29595	0.647976	−	0.761661i	$-0.275616\pi$
$79$	11.5187	1.29595	0.647976	+	0.761661i	$0.275616\pi$
$80$	0	0
$81$	13.6964	1.52182
$82$	0	0
$83$	−13.6530	−1.49861	−0.749305	−	0.662225i	$-0.769612\pi$
$83$	−13.6530	−1.49861	−0.749305	+	0.662225i	$0.769612\pi$
$84$	0	0
$85$	−0.463548	−0.0502789
$86$	0	0
$87$	−3.83185	−0.410818
$88$	0	0
$89$	1.48188	0.157079	0.0785396	−	0.996911i	$-0.474974\pi$
$89$	1.48188	0.157079	0.0785396	+	0.996911i	$0.474974\pi$
$90$	0	0
$91$	1.34472	0.140965
$92$	0	0
$93$	−7.84517	−0.813506
$94$	0	0
$95$	−0.120416	−0.0123544
$96$	0	0
$97$	−14.5637	−1.47872	−0.739358	−	0.673313i	$-0.764871\pi$
$97$	−14.5637	−1.47872	−0.739358	+	0.673313i	$0.764871\pi$
$98$	0	0
$99$	−11.6612	−1.17199
$100$	0	0
$101$	7.49464	0.745744	0.372872	−	0.927883i	$-0.378373\pi$
$101$	7.49464	0.745744	0.372872	+	0.927883i	$0.378373\pi$
$102$	0	0
$103$	−9.34472	−0.920762	−0.460381	−	0.887721i	$-0.652287\pi$
$103$	−9.34472	−0.920762	−0.460381	+	0.887721i	$0.652287\pi$
$104$	0	0
$105$	−5.90361	−0.576134
$106$	0	0
$107$	−8.94701	−0.864941	−0.432470	−	0.901648i	$-0.642358\pi$
$107$	−8.94701	−0.864941	−0.432470	+	0.901648i	$0.642358\pi$
$108$	0	0
$109$	−13.1599	−1.26049	−0.630247	−	0.776395i	$-0.717046\pi$
$109$	−13.1599	−1.26049	−0.630247	+	0.776395i	$0.717046\pi$
$110$	0	0
$111$	3.08134	0.292468
$112$	0	0
$113$	−7.45282	−0.701102	−0.350551	−	0.936544i	$-0.614006\pi$
$113$	−7.45282	−0.701102	−0.350551	+	0.936544i	$0.614006\pi$
$114$	0	0
$115$	0.408979	0.0381375
$116$	0	0
$117$	−4.55834	−0.421419
$118$	0	0
$119$	−0.888124	−0.0814142
$120$	0	0
$121$	−7.77614	−0.706922
$122$	0	0
$123$	−29.4243	−2.65310
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.3169	1.09295	0.546475	−	0.837476i	$-0.315969\pi$
$127$	12.3169	1.09295	0.546475	+	0.837476i	$0.315969\pi$
$128$	0	0
$129$	−14.0622	−1.23811
$130$	0	0
$131$	12.1044	1.05757	0.528785	−	0.848756i	$-0.322648\pi$
$131$	12.1044	1.05757	0.528785	+	0.848756i	$0.322648\pi$
$132$	0	0
$133$	−0.230708	−0.0200049
$134$	0	0
$135$	10.7682	0.926775
$136$	0	0
$137$	6.80197	0.581132	0.290566	−	0.956855i	$-0.406156\pi$
$137$	6.80197	0.581132	0.290566	+	0.956855i	$0.406156\pi$
$138$	0	0
$139$	4.97264	0.421774	0.210887	−	0.977510i	$-0.432365\pi$
$139$	4.97264	0.421774	0.210887	+	0.977510i	$0.432365\pi$
$140$	0	0
$141$	−21.7441	−1.83118
$142$	0	0
$143$	1.26020	0.105383
$144$	0	0
$145$	−1.24357	−0.103273
$146$	0	0
$147$	10.2585	0.846104
$148$	0	0
$149$	−11.9635	−0.980089	−0.490044	−	0.871698i	$-0.663019\pi$
$149$	−11.9635	−0.980089	−0.490044	+	0.871698i	$0.663019\pi$
$150$	0	0
$151$	−0.240832	−0.0195986	−0.00979930	−	0.999952i	$-0.503119\pi$
$151$	−0.240832	−0.0195986	−0.00979930	+	0.999952i	$0.503119\pi$
$152$	0	0
$153$	3.01058	0.243391
$154$	0	0
$155$	−2.54603	−0.204502
$156$	0	0
$157$	−13.1070	−1.04605	−0.523025	−	0.852318i	$-0.675196\pi$
$157$	−13.1070	−1.04605	−0.523025	+	0.852318i	$0.675196\pi$
$158$	0	0
$159$	11.6194	0.921476
$160$	0	0
$161$	0.783573	0.0617542
$162$	0	0
$163$	−10.8100	−0.846702	−0.423351	−	0.905966i	$-0.639146\pi$
$163$	−10.8100	−0.846702	−0.423351	+	0.905966i	$0.639146\pi$
$164$	0	0
$165$	−5.53257	−0.430710
$166$	0	0
$167$	4.24083	0.328165	0.164083	−	0.986447i	$-0.447534\pi$
$167$	4.24083	0.328165	0.164083	+	0.986447i	$0.447534\pi$
$168$	0	0
$169$	−12.5074	−0.962107
$170$	0	0
$171$	0.782057	0.0598054
$172$	0	0
$173$	4.17987	0.317789	0.158895	−	0.987296i	$-0.449207\pi$
$173$	4.17987	0.317789	0.158895	+	0.987296i	$0.449207\pi$
$174$	0	0
$175$	−1.91593	−0.144830
$176$	0	0
$177$	4.77226	0.358705
$178$	0	0
$179$	−11.6622	−0.871673	−0.435836	−	0.900026i	$-0.643547\pi$
$179$	−11.6622	−0.871673	−0.435836	+	0.900026i	$0.643547\pi$
$180$	0	0
$181$	−12.3126	−0.915188	−0.457594	−	0.889161i	$-0.651289\pi$
$181$	−12.3126	−0.915188	−0.457594	+	0.889161i	$0.651289\pi$
$182$	0	0
$183$	−27.4069	−2.02598
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−0.832306	−0.0608642
$188$	0	0
$189$	20.6310	1.50068
$190$	0	0
$191$	−5.35348	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$191$	−5.35348	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$192$	0	0
$193$	−6.66164	−0.479515	−0.239758	−	0.970833i	$-0.577068\pi$
$193$	−6.66164	−0.479515	−0.239758	+	0.970833i	$0.577068\pi$
$194$	0	0
$195$	−2.16267	−0.154872
$196$	0	0
$197$	−23.9983	−1.70980	−0.854902	−	0.518789i	$-0.826383\pi$
$197$	−23.9983	−1.70980	−0.854902	+	0.518789i	$0.826383\pi$
$198$	0	0
$199$	6.54854	0.464214	0.232107	−	0.972690i	$-0.425438\pi$
$199$	6.54854	0.464214	0.232107	+	0.972690i	$0.425438\pi$
$200$	0	0
$201$	9.83969	0.694038
$202$	0	0
$203$	−2.38259	−0.167225
$204$	0	0
$205$	−9.54921	−0.666945
$206$	0	0
$207$	−2.65617	−0.184616
$208$	0	0
$209$	−0.216208	−0.0149554
$210$	0	0
$211$	−14.3567	−0.988353	−0.494176	−	0.869362i	$-0.664530\pi$
$211$	−14.3567	−0.988353	−0.494176	+	0.869362i	$0.664530\pi$
$212$	0	0
$213$	−12.9308	−0.886005
$214$	0	0
$215$	−4.56366	−0.311239
$216$	0	0
$217$	−4.87800	−0.331140
$218$	0	0
$219$	−19.0935	−1.29022
$220$	0	0
$221$	−0.325347	−0.0218852
$222$	0	0
$223$	−19.9950	−1.33896	−0.669481	−	0.742829i	$-0.733483\pi$
$223$	−19.9950	−1.33896	−0.669481	+	0.742829i	$0.733483\pi$
$224$	0	0
$225$	6.49464	0.432976
$226$	0	0
$227$	24.0592	1.59687	0.798433	−	0.602083i	$-0.205663\pi$
$227$	24.0592	1.59687	0.798433	+	0.602083i	$0.205663\pi$
$228$	0	0
$229$	21.0090	1.38831	0.694156	−	0.719824i	$-0.255778\pi$
$229$	21.0090	1.38831	0.694156	+	0.719824i	$0.255778\pi$
$230$	0	0
$231$	−10.6000	−0.697429
$232$	0	0
$233$	−0.861731	−0.0564539	−0.0282269	−	0.999602i	$-0.508986\pi$
$233$	−0.861731	−0.0564539	−0.0282269	+	0.999602i	$0.508986\pi$
$234$	0	0
$235$	−7.05671	−0.460329
$236$	0	0
$237$	−35.4929	−2.30551
$238$	0	0
$239$	7.28035	0.470927	0.235463	−	0.971883i	$-0.424339\pi$
$239$	7.28035	0.470927	0.235463	+	0.971883i	$0.424339\pi$
$240$	0	0
$241$	−6.69491	−0.431257	−0.215628	−	0.976475i	$-0.569180\pi$
$241$	−6.69491	−0.431257	−0.215628	+	0.976475i	$0.569180\pi$
$242$	0	0
$243$	−9.89873	−0.635004
$244$	0	0
$245$	3.32923	0.212696
$246$	0	0
$247$	−0.0845153	−0.00537758
$248$	0	0
$249$	42.0695	2.66604
$250$	0	0
$251$	−22.4521	−1.41717	−0.708583	−	0.705628i	$-0.750665\pi$
$251$	−22.4521	−1.41717	−0.708583	+	0.705628i	$0.750665\pi$
$252$	0	0
$253$	0.734326	0.0461666
$254$	0	0
$255$	1.42835	0.0894466
$256$	0	0
$257$	7.80027	0.486568	0.243284	−	0.969955i	$-0.421775\pi$
$257$	7.80027	0.486568	0.243284	+	0.969955i	$0.421775\pi$
$258$	0	0
$259$	1.91593	0.119050
$260$	0	0
$261$	8.07652	0.499924
$262$	0	0
$263$	−19.2488	−1.18693	−0.593466	−	0.804859i	$-0.702241\pi$
$263$	−19.2488	−1.18693	−0.593466	+	0.804859i	$0.702241\pi$
$264$	0	0
$265$	3.77089	0.231644
$266$	0	0
$267$	−4.56618	−0.279446
$268$	0	0
$269$	8.43641	0.514377	0.257188	−	0.966361i	$-0.417204\pi$
$269$	8.43641	0.514377	0.257188	+	0.966361i	$0.417204\pi$
$270$	0	0
$271$	−15.3109	−0.930070	−0.465035	−	0.885292i	$-0.653958\pi$
$271$	−15.3109	−0.930070	−0.465035	+	0.885292i	$0.653958\pi$
$272$	0	0
$273$	−4.14352	−0.250777
$274$	0	0
$275$	−1.79551	−0.108273
$276$	0	0
$277$	−26.9491	−1.61921	−0.809606	−	0.586974i	$-0.800319\pi$
$277$	−26.9491	−1.61921	−0.809606	+	0.586974i	$0.800319\pi$
$278$	0	0
$279$	16.5355	0.989956
$280$	0	0
$281$	5.25473	0.313471	0.156735	−	0.987641i	$-0.449903\pi$
$281$	5.25473	0.313471	0.156735	+	0.987641i	$0.449903\pi$
$282$	0	0
$283$	26.0631	1.54929	0.774644	−	0.632398i	$-0.217929\pi$
$283$	26.0631	1.54929	0.774644	+	0.632398i	$0.217929\pi$
$284$	0	0
$285$	0.371042	0.0219786
$286$	0	0
$287$	−18.2956	−1.07995
$288$	0	0
$289$	−16.7851	−0.987360
$290$	0	0
$291$	44.8755	2.63065
$292$	0	0
$293$	12.9621	0.757256	0.378628	−	0.925549i	$-0.376396\pi$
$293$	12.9621	0.757256	0.378628	+	0.925549i	$0.376396\pi$
$294$	0	0
$295$	1.54876	0.0901725
$296$	0	0
$297$	19.3343	1.12189
$298$	0	0
$299$	0.287047	0.0166003
$300$	0	0
$301$	−8.74364	−0.503975
$302$	0	0
$303$	−23.0935	−1.32669
$304$	0	0
$305$	−8.89448	−0.509297
$306$	0	0
$307$	32.9036	1.87791	0.938953	−	0.344046i	$-0.111798\pi$
$307$	32.9036	1.87791	0.938953	+	0.344046i	$0.111798\pi$
$308$	0	0
$309$	28.7942	1.63805
$310$	0	0
$311$	−8.28374	−0.469728	−0.234864	−	0.972028i	$-0.575464\pi$
$311$	−8.28374	−0.469728	−0.234864	+	0.972028i	$0.575464\pi$
$312$	0	0
$313$	−21.2891	−1.20333	−0.601666	−	0.798748i	$-0.705496\pi$
$313$	−21.2891	−1.20333	−0.601666	+	0.798748i	$0.705496\pi$
$314$	0	0
$315$	12.4432	0.701097
$316$	0	0
$317$	0.160156	0.00899525	0.00449762	−	0.999990i	$-0.498568\pi$
$317$	0.160156	0.00899525	0.00449762	+	0.999990i	$0.498568\pi$
$318$	0	0
$319$	−2.23284	−0.125015
$320$	0	0
$321$	27.5688	1.53874
$322$	0	0
$323$	0.0558185	0.00310583
$324$	0	0
$325$	−0.701862	−0.0389323
$326$	0	0
$327$	40.5502	2.24243
$328$	0	0
$329$	−13.5201	−0.745390
$330$	0	0
$331$	−10.1986	−0.560564	−0.280282	−	0.959918i	$-0.590428\pi$
$331$	−10.1986	−0.560564	−0.280282	+	0.959918i	$0.590428\pi$
$332$	0	0
$333$	−6.49464	−0.355904
$334$	0	0
$335$	3.19332	0.174470
$336$	0	0
$337$	34.1363	1.85952	0.929761	−	0.368162i	$-0.120013\pi$
$337$	34.1363	1.85952	0.929761	+	0.368162i	$0.120013\pi$
$338$	0	0
$339$	22.9646	1.24727
$340$	0	0
$341$	−4.57142	−0.247556
$342$	0	0
$343$	19.7900	1.06856
$344$	0	0
$345$	−1.26020	−0.0678469
$346$	0	0
$347$	−8.39300	−0.450560	−0.225280	−	0.974294i	$-0.572330\pi$
$347$	−8.39300	−0.450560	−0.225280	+	0.974294i	$0.572330\pi$
$348$	0	0
$349$	1.78527	0.0955636	0.0477818	−	0.998858i	$-0.484785\pi$
$349$	1.78527	0.0955636	0.0477818	+	0.998858i	$0.484785\pi$
$350$	0	0
$351$	7.55776	0.403403
$352$	0	0
$353$	−36.5472	−1.94521	−0.972607	−	0.232457i	$-0.925323\pi$
$353$	−36.5472	−1.94521	−0.972607	+	0.232457i	$0.925323\pi$
$354$	0	0
$355$	−4.19650	−0.222727
$356$	0	0
$357$	2.73661	0.144837
$358$	0	0
$359$	30.3826	1.60353	0.801766	−	0.597638i	$-0.203894\pi$
$359$	30.3826	1.60353	0.801766	+	0.597638i	$0.203894\pi$
$360$	0	0
$361$	−18.9855	−0.999237
$362$	0	0
$363$	23.9609	1.25762
$364$	0	0
$365$	−6.19650	−0.324340
$366$	0	0
$367$	8.94769	0.467065	0.233533	−	0.972349i	$-0.424971\pi$
$367$	8.94769	0.467065	0.233533	+	0.972349i	$0.424971\pi$
$368$	0	0
$369$	62.0186	3.22856
$370$	0	0
$371$	7.22474	0.375090
$372$	0	0
$373$	−20.5273	−1.06286	−0.531432	−	0.847101i	$-0.678346\pi$
$373$	−20.5273	−1.06286	−0.531432	+	0.847101i	$0.678346\pi$
$374$	0	0
$375$	3.08134	0.159120
$376$	0	0
$377$	−0.872814	−0.0449522
$378$	0	0
$379$	−27.7551	−1.42569	−0.712843	−	0.701324i	$-0.752593\pi$
$379$	−27.7551	−1.42569	−0.712843	+	0.701324i	$0.752593\pi$
$380$	0	0
$381$	−37.9526	−1.94437
$382$	0	0
$383$	−24.2761	−1.24045	−0.620225	−	0.784424i	$-0.712959\pi$
$383$	−24.2761	−1.24045	−0.620225	+	0.784424i	$0.712959\pi$
$384$	0	0
$385$	−3.44007	−0.175322
$386$	0	0
$387$	29.6393	1.50665
$388$	0	0
$389$	21.2907	1.07948	0.539740	−	0.841832i	$-0.318523\pi$
$389$	21.2907	1.07948	0.539740	+	0.841832i	$0.318523\pi$
$390$	0	0
$391$	−0.189581	−0.00958754
$392$	0	0
$393$	−37.2978	−1.88143
$394$	0	0
$395$	−11.5187	−0.579567
$396$	0	0
$397$	24.6205	1.23567	0.617833	−	0.786309i	$-0.288011\pi$
$397$	24.6205	1.23567	0.617833	+	0.786309i	$0.288011\pi$
$398$	0	0
$399$	0.710888	0.0355889
$400$	0	0
$401$	24.3148	1.21423	0.607113	−	0.794616i	$-0.292328\pi$
$401$	24.3148	1.21423	0.607113	+	0.794616i	$0.292328\pi$
$402$	0	0
$403$	−1.78696	−0.0890148
$404$	0	0
$405$	−13.6964	−0.680579
$406$	0	0
$407$	1.79551	0.0890002
$408$	0	0
$409$	−14.3222	−0.708185	−0.354093	−	0.935210i	$-0.615210\pi$
$409$	−14.3222	−0.708185	−0.354093	+	0.935210i	$0.615210\pi$
$410$	0	0
$411$	−20.9592	−1.03384
$412$	0	0
$413$	2.96732	0.146012
$414$	0	0
$415$	13.6530	0.670199
$416$	0	0
$417$	−15.3224	−0.750340
$418$	0	0
$419$	−11.6354	−0.568424	−0.284212	−	0.958761i	$-0.591732\pi$
$419$	−11.6354	−0.568424	−0.284212	+	0.958761i	$0.591732\pi$
$420$	0	0
$421$	12.9400	0.630658	0.315329	−	0.948982i	$-0.397885\pi$
$421$	12.9400	0.630658	0.315329	+	0.948982i	$0.397885\pi$
$422$	0	0
$423$	45.8308	2.22837
$424$	0	0
$425$	0.463548	0.0224854
$426$	0	0
$427$	−17.0412	−0.824680
$428$	0	0
$429$	−3.88310	−0.187478
$430$	0	0
$431$	26.9441	1.29785	0.648925	−	0.760853i	$-0.275219\pi$
$431$	26.9441	1.29785	0.648925	+	0.760853i	$0.275219\pi$
$432$	0	0
$433$	−30.4510	−1.46338	−0.731691	−	0.681636i	$-0.761269\pi$
$433$	−30.4510	−1.46338	−0.731691	+	0.681636i	$0.761269\pi$
$434$	0	0
$435$	3.83185	0.183723
$436$	0	0
$437$	−0.0492475	−0.00235583
$438$	0	0
$439$	27.9251	1.33279	0.666397	−	0.745597i	$-0.267836\pi$
$439$	27.9251	1.33279	0.666397	+	0.745597i	$0.267836\pi$
$440$	0	0
$441$	−21.6221	−1.02962
$442$	0	0
$443$	15.9334	0.757021	0.378510	−	0.925597i	$-0.376436\pi$
$443$	15.9334	0.757021	0.378510	+	0.925597i	$0.376436\pi$
$444$	0	0
$445$	−1.48188	−0.0702480
$446$	0	0
$447$	36.8636	1.74359
$448$	0	0
$449$	18.1818	0.858053	0.429027	−	0.903292i	$-0.358857\pi$
$449$	18.1818	0.858053	0.429027	+	0.903292i	$0.358857\pi$
$450$	0	0
$451$	−17.1457	−0.807360
$452$	0	0
$453$	0.742083	0.0348661
$454$	0	0
$455$	−1.34472	−0.0630413
$456$	0	0
$457$	30.4691	1.42528	0.712642	−	0.701528i	$-0.247499\pi$
$457$	30.4691	1.42528	0.712642	+	0.701528i	$0.247499\pi$
$458$	0	0
$459$	−4.99156	−0.232986
$460$	0	0
$461$	−40.9402	−1.90678	−0.953388	−	0.301746i	$-0.902431\pi$
$461$	−40.9402	−1.90678	−0.953388	+	0.301746i	$0.902431\pi$
$462$	0	0
$463$	−29.6926	−1.37993	−0.689966	−	0.723841i	$-0.742375\pi$
$463$	−29.6926	−1.37993	−0.689966	+	0.723841i	$0.742375\pi$
$464$	0	0
$465$	7.84517	0.363811
$466$	0	0
$467$	21.8407	1.01067	0.505334	−	0.862924i	$-0.331369\pi$
$467$	21.8407	1.01067	0.505334	+	0.862924i	$0.331369\pi$
$468$	0	0
$469$	6.11817	0.282511
$470$	0	0
$471$	40.3870	1.86093
$472$	0	0
$473$	−8.19410	−0.376765
$474$	0	0
$475$	0.120416	0.00552506
$476$	0	0
$477$	−24.4905	−1.12134
$478$	0	0
$479$	−39.9852	−1.82697	−0.913486	−	0.406870i	$-0.866620\pi$
$479$	−39.9852	−1.82697	−0.913486	+	0.406870i	$0.866620\pi$
$480$	0	0
$481$	0.701862	0.0320022
$482$	0	0
$483$	−2.41445	−0.109861
$484$	0	0
$485$	14.5637	0.661302
$486$	0	0
$487$	15.2716	0.692022	0.346011	−	0.938230i	$-0.387536\pi$
$487$	15.2716	0.692022	0.346011	+	0.938230i	$0.387536\pi$
$488$	0	0
$489$	33.3091	1.50629
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0.576454	0.0259622
$494$	0	0
$495$	11.6612	0.524131
$496$	0	0
$497$	−8.04018	−0.360651
$498$	0	0
$499$	−26.8677	−1.20276	−0.601381	−	0.798962i	$-0.705383\pi$
$499$	−26.8677	−1.20276	−0.601381	+	0.798962i	$0.705383\pi$
$500$	0	0
$501$	−13.0674	−0.583810
$502$	0	0
$503$	28.1198	1.25380	0.626900	−	0.779100i	$-0.284324\pi$
$503$	28.1198	1.25380	0.626900	+	0.779100i	$0.284324\pi$
$504$	0	0
$505$	−7.49464	−0.333507
$506$	0	0
$507$	38.5395	1.71160
$508$	0	0
$509$	17.8035	0.789126	0.394563	−	0.918869i	$-0.370896\pi$
$509$	17.8035	0.789126	0.394563	+	0.918869i	$0.370896\pi$
$510$	0	0
$511$	−11.8720	−0.525188
$512$	0	0
$513$	−1.29666	−0.0572487
$514$	0	0
$515$	9.34472	0.411777
$516$	0	0
$517$	−12.6704	−0.557244
$518$	0	0
$519$	−12.8796	−0.565351
$520$	0	0
$521$	−1.85260	−0.0811638	−0.0405819	−	0.999176i	$-0.512921\pi$
$521$	−1.85260	−0.0811638	−0.0405819	+	0.999176i	$0.512921\pi$
$522$	0	0
$523$	−18.0289	−0.788349	−0.394175	−	0.919036i	$-0.628969\pi$
$523$	−18.0289	−0.788349	−0.394175	+	0.919036i	$0.628969\pi$
$524$	0	0
$525$	5.90361	0.257655
$526$	0	0
$527$	1.18021	0.0514106
$528$	0	0
$529$	−22.8327	−0.992728
$530$	0	0
$531$	−10.0587	−0.436509
$532$	0	0
$533$	−6.70223	−0.290306
$534$	0	0
$535$	8.94701	0.386813
$536$	0	0
$537$	35.9351	1.55072
$538$	0	0
$539$	5.97766	0.257476
$540$	0	0
$541$	29.9270	1.28666	0.643331	−	0.765588i	$-0.277552\pi$
$541$	29.9270	1.28666	0.643331	+	0.765588i	$0.277552\pi$
$542$	0	0
$543$	37.9392	1.62813
$544$	0	0
$545$	13.1599	0.563710
$546$	0	0
$547$	−4.04761	−0.173063	−0.0865317	−	0.996249i	$-0.527578\pi$
$547$	−4.04761	−0.173063	−0.0865317	+	0.996249i	$0.527578\pi$
$548$	0	0
$549$	57.7664	2.46541
$550$	0	0
$551$	0.149745	0.00637936
$552$	0	0
$553$	−22.0689	−0.938466
$554$	0	0
$555$	−3.08134	−0.130795
$556$	0	0
$557$	20.9231	0.886540	0.443270	−	0.896388i	$-0.353818\pi$
$557$	20.9231	0.886540	0.443270	+	0.896388i	$0.353818\pi$
$558$	0	0
$559$	−3.20306	−0.135475
$560$	0	0
$561$	2.56461	0.108278
$562$	0	0
$563$	0.579643	0.0244291	0.0122145	−	0.999925i	$-0.496112\pi$
$563$	0.579643	0.0244291	0.0122145	+	0.999925i	$0.496112\pi$
$564$	0	0
$565$	7.45282	0.313542
$566$	0	0
$567$	−26.2413	−1.10203
$568$	0	0
$569$	−12.2089	−0.511823	−0.255912	−	0.966700i	$-0.582376\pi$
$569$	−12.2089	−0.511823	−0.255912	+	0.966700i	$0.582376\pi$
$570$	0	0
$571$	−18.7312	−0.783878	−0.391939	−	0.919991i	$-0.628196\pi$
$571$	−18.7312	−0.783878	−0.391939	+	0.919991i	$0.628196\pi$
$572$	0	0
$573$	16.4959	0.689124
$574$	0	0
$575$	−0.408979	−0.0170556
$576$	0	0
$577$	−31.0472	−1.29251	−0.646257	−	0.763120i	$-0.723667\pi$
$577$	−31.0472	−1.29251	−0.646257	+	0.763120i	$0.723667\pi$
$578$	0	0
$579$	20.5268	0.853063
$580$	0	0
$581$	26.1581	1.08522
$582$	0	0
$583$	6.77067	0.280412
$584$	0	0
$585$	4.55834	0.188464
$586$	0	0
$587$	14.9781	0.618213	0.309106	−	0.951027i	$-0.399970\pi$
$587$	14.9781	0.618213	0.309106	+	0.951027i	$0.399970\pi$
$588$	0	0
$589$	0.306582	0.0126325
$590$	0	0
$591$	73.9467	3.04176
$592$	0	0
$593$	17.6107	0.723186	0.361593	−	0.932336i	$-0.382233\pi$
$593$	17.6107	0.723186	0.361593	+	0.932336i	$0.382233\pi$
$594$	0	0
$595$	0.888124	0.0364095
$596$	0	0
$597$	−20.1783	−0.825842
$598$	0	0
$599$	−25.0646	−1.02411	−0.512056	−	0.858952i	$-0.671116\pi$
$599$	−25.0646	−1.02411	−0.512056	+	0.858952i	$0.671116\pi$
$600$	0	0
$601$	1.01286	0.0413154	0.0206577	−	0.999787i	$-0.493424\pi$
$601$	1.01286	0.0413154	0.0206577	+	0.999787i	$0.493424\pi$
$602$	0	0
$603$	−20.7394	−0.844576
$604$	0	0
$605$	7.77614	0.316145
$606$	0	0
$607$	1.95623	0.0794008	0.0397004	−	0.999212i	$-0.487360\pi$
$607$	1.95623	0.0794008	0.0397004	+	0.999212i	$0.487360\pi$
$608$	0	0
$609$	7.34155	0.297495
$610$	0	0
$611$	−4.95284	−0.200370
$612$	0	0
$613$	−0.0547440	−0.00221109	−0.00110555	−	0.999999i	$-0.500352\pi$
$613$	−0.0547440	−0.00221109	−0.00110555	+	0.999999i	$0.500352\pi$
$614$	0	0
$615$	29.4243	1.18650
$616$	0	0
$617$	11.9830	0.482419	0.241210	−	0.970473i	$-0.422456\pi$
$617$	11.9830	0.482419	0.241210	+	0.970473i	$0.422456\pi$
$618$	0	0
$619$	−28.6765	−1.15261	−0.576303	−	0.817236i	$-0.695505\pi$
$619$	−28.6765	−1.15261	−0.576303	+	0.817236i	$0.695505\pi$
$620$	0	0
$621$	4.40394	0.176724
$622$	0	0
$623$	−2.83918	−0.113749
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.666209	0.0266058
$628$	0	0
$629$	−0.463548	−0.0184829
$630$	0	0
$631$	−35.7406	−1.42281	−0.711405	−	0.702782i	$-0.751941\pi$
$631$	−35.7406	−1.42281	−0.711405	+	0.702782i	$0.751941\pi$
$632$	0	0
$633$	44.2377	1.75829
$634$	0	0
$635$	−12.3169	−0.488782
$636$	0	0
$637$	2.33666	0.0925817
$638$	0	0
$639$	27.2547	1.07818
$640$	0	0
$641$	33.0530	1.30552	0.652758	−	0.757566i	$-0.273612\pi$
$641$	33.0530	1.30552	0.652758	+	0.757566i	$0.273612\pi$
$642$	0	0
$643$	10.5516	0.416115	0.208058	−	0.978117i	$-0.433286\pi$
$643$	10.5516	0.416115	0.208058	+	0.978117i	$0.433286\pi$
$644$	0	0
$645$	14.0622	0.553698
$646$	0	0
$647$	19.3274	0.759839	0.379919	−	0.925020i	$-0.375952\pi$
$647$	19.3274	0.759839	0.379919	+	0.925020i	$0.375952\pi$
$648$	0	0
$649$	2.78082	0.109157
$650$	0	0
$651$	15.0308	0.589102
$652$	0	0
$653$	32.0180	1.25296	0.626480	−	0.779438i	$-0.284495\pi$
$653$	32.0180	1.25296	0.626480	+	0.779438i	$0.284495\pi$
$654$	0	0
$655$	−12.1044	−0.472959
$656$	0	0
$657$	40.2440	1.57007
$658$	0	0
$659$	−3.65222	−0.142270	−0.0711352	−	0.997467i	$-0.522662\pi$
$659$	−3.65222	−0.142270	−0.0711352	+	0.997467i	$0.522662\pi$
$660$	0	0
$661$	−6.53040	−0.254003	−0.127001	−	0.991903i	$-0.540535\pi$
$661$	−6.53040	−0.254003	−0.127001	+	0.991903i	$0.540535\pi$
$662$	0	0
$663$	1.00250	0.0389340
$664$	0	0
$665$	0.230708	0.00894647
$666$	0	0
$667$	−0.508593	−0.0196928
$668$	0	0
$669$	61.6112	2.38203
$670$	0	0
$671$	−15.9701	−0.616520
$672$	0	0
$673$	−28.5840	−1.10183	−0.550917	−	0.834560i	$-0.685722\pi$
$673$	−28.5840	−1.10183	−0.550917	+	0.834560i	$0.685722\pi$
$674$	0	0
$675$	−10.7682	−0.414466
$676$	0	0
$677$	−20.4781	−0.787036	−0.393518	−	0.919317i	$-0.628742\pi$
$677$	−20.4781	−0.787036	−0.393518	+	0.919317i	$0.628742\pi$
$678$	0	0
$679$	27.9029	1.07082
$680$	0	0
$681$	−74.1346	−2.84084
$682$	0	0
$683$	−33.0243	−1.26364	−0.631820	−	0.775115i	$-0.717692\pi$
$683$	−33.0243	−1.26364	−0.631820	+	0.775115i	$0.717692\pi$
$684$	0	0
$685$	−6.80197	−0.259890
$686$	0	0
$687$	−64.7357	−2.46982
$688$	0	0
$689$	2.64664	0.100829
$690$	0	0
$691$	−10.4553	−0.397737	−0.198869	−	0.980026i	$-0.563727\pi$
$691$	−10.4553	−0.397737	−0.198869	+	0.980026i	$0.563727\pi$
$692$	0	0
$693$	22.3420	0.848702
$694$	0	0
$695$	−4.97264	−0.188623
$696$	0	0
$697$	4.42652	0.167666
$698$	0	0
$699$	2.65528	0.100432
$700$	0	0
$701$	14.2889	0.539684	0.269842	−	0.962905i	$-0.413029\pi$
$701$	14.2889	0.539684	0.269842	+	0.962905i	$0.413029\pi$
$702$	0	0
$703$	−0.120416	−0.00454157
$704$	0	0
$705$	21.7441	0.818931
$706$	0	0
$707$	−14.3592	−0.540032
$708$	0	0
$709$	−5.58939	−0.209914	−0.104957	−	0.994477i	$-0.533470\pi$
$709$	−5.58939	−0.209914	−0.104957	+	0.994477i	$0.533470\pi$
$710$	0	0
$711$	74.8095	2.80558
$712$	0	0
$713$	−1.04127	−0.0389959
$714$	0	0
$715$	−1.26020	−0.0471288
$716$	0	0
$717$	−22.4332	−0.837784
$718$	0	0
$719$	−2.73489	−0.101994	−0.0509970	−	0.998699i	$-0.516240\pi$
$719$	−2.73489	−0.101994	−0.0509970	+	0.998699i	$0.516240\pi$
$720$	0	0
$721$	17.9038	0.666772
$722$	0	0
$723$	20.6293	0.767210
$724$	0	0
$725$	1.24357	0.0461850
$726$	0	0
$727$	−13.4349	−0.498274	−0.249137	−	0.968468i	$-0.580147\pi$
$727$	−13.4349	−0.498274	−0.249137	+	0.968468i	$0.580147\pi$
$728$	0	0
$729$	−10.5878	−0.392142
$730$	0	0
$731$	2.11548	0.0782437
$732$	0	0
$733$	41.0456	1.51606	0.758028	−	0.652222i	$-0.226163\pi$
$733$	41.0456	1.51606	0.758028	+	0.652222i	$0.226163\pi$
$734$	0	0
$735$	−10.2585	−0.378389
$736$	0	0
$737$	5.73364	0.211201
$738$	0	0
$739$	−3.85680	−0.141875	−0.0709374	−	0.997481i	$-0.522599\pi$
$739$	−3.85680	−0.141875	−0.0709374	+	0.997481i	$0.522599\pi$
$740$	0	0
$741$	0.260420	0.00956677
$742$	0	0
$743$	14.0004	0.513626	0.256813	−	0.966461i	$-0.417327\pi$
$743$	14.0004	0.513626	0.256813	+	0.966461i	$0.417327\pi$
$744$	0	0
$745$	11.9635	0.438309
$746$	0	0
$747$	−88.6712	−3.24431
$748$	0	0
$749$	17.1418	0.626349
$750$	0	0
$751$	−22.3482	−0.815498	−0.407749	−	0.913094i	$-0.633686\pi$
$751$	−22.3482	−0.815498	−0.407749	+	0.913094i	$0.633686\pi$
$752$	0	0
$753$	69.1825	2.52115
$754$	0	0
$755$	0.240832	0.00876476
$756$	0	0
$757$	23.7603	0.863582	0.431791	−	0.901974i	$-0.357882\pi$
$757$	23.7603	0.863582	0.431791	+	0.901974i	$0.357882\pi$
$758$	0	0
$759$	−2.26270	−0.0821310
$760$	0	0
$761$	17.1070	0.620127	0.310063	−	0.950716i	$-0.399650\pi$
$761$	17.1070	0.620127	0.310063	+	0.950716i	$0.399650\pi$
$762$	0	0
$763$	25.2135	0.912789
$764$	0	0
$765$	−3.01058	−0.108848
$766$	0	0
$767$	1.08702	0.0392500
$768$	0	0
$769$	8.12526	0.293004	0.146502	−	0.989210i	$-0.453198\pi$
$769$	8.12526	0.293004	0.146502	+	0.989210i	$0.453198\pi$
$770$	0	0
$771$	−24.0353	−0.865609
$772$	0	0
$773$	−21.4585	−0.771807	−0.385904	−	0.922539i	$-0.626110\pi$
$773$	−21.4585	−0.771807	−0.385904	+	0.922539i	$0.626110\pi$
$774$	0	0
$775$	2.54603	0.0914560
$776$	0	0
$777$	−5.90361	−0.211791
$778$	0	0
$779$	1.14988	0.0411986
$780$	0	0
$781$	−7.53486	−0.269618
$782$	0	0
$783$	−13.3909	−0.478553
$784$	0	0
$785$	13.1070	0.467808
$786$	0	0
$787$	25.4807	0.908289	0.454144	−	0.890928i	$-0.349945\pi$
$787$	25.4807	0.908289	0.454144	+	0.890928i	$0.349945\pi$
$788$	0	0
$789$	59.3121	2.11157
$790$	0	0
$791$	14.2791	0.507705
$792$	0	0
$793$	−6.24270	−0.221685
$794$	0	0
$795$	−11.6194	−0.412097
$796$	0	0
$797$	33.4910	1.18631	0.593157	−	0.805087i	$-0.297881\pi$
$797$	33.4910	1.18631	0.593157	+	0.805087i	$0.297881\pi$
$798$	0	0
$799$	3.27113	0.115724
$800$	0	0
$801$	9.62429	0.340057
$802$	0	0
$803$	−11.1259	−0.392624
$804$	0	0
$805$	−0.783573	−0.0276173
$806$	0	0
$807$	−25.9954	−0.915082
$808$	0	0
$809$	6.12435	0.215321	0.107660	−	0.994188i	$-0.465664\pi$
$809$	6.12435	0.215321	0.107660	+	0.994188i	$0.465664\pi$
$810$	0	0
$811$	−25.2378	−0.886218	−0.443109	−	0.896468i	$-0.646125\pi$
$811$	−25.2378	−0.886218	−0.443109	+	0.896468i	$0.646125\pi$
$812$	0	0
$813$	47.1780	1.65461
$814$	0	0
$815$	10.8100	0.378657
$816$	0	0
$817$	0.549537	0.0192259
$818$	0	0
$819$	8.73344	0.305171
$820$	0	0
$821$	33.9233	1.18393	0.591965	−	0.805963i	$-0.298352\pi$
$821$	33.9233	1.18393	0.591965	+	0.805963i	$0.298352\pi$
$822$	0	0
$823$	7.35828	0.256493	0.128247	−	0.991742i	$-0.459065\pi$
$823$	7.35828	0.256493	0.128247	+	0.991742i	$0.459065\pi$
$824$	0	0
$825$	5.53257	0.192619
$826$	0	0
$827$	−36.5174	−1.26984	−0.634918	−	0.772580i	$-0.718966\pi$
$827$	−36.5174	−1.26984	−0.634918	+	0.772580i	$0.718966\pi$
$828$	0	0
$829$	12.2821	0.426575	0.213287	−	0.976990i	$-0.431583\pi$
$829$	12.2821	0.426575	0.213287	+	0.976990i	$0.431583\pi$
$830$	0	0
$831$	83.0391	2.88059
$832$	0	0
$833$	−1.54326	−0.0534707
$834$	0	0
$835$	−4.24083	−0.146760
$836$	0	0
$837$	−27.4160	−0.947636
$838$	0	0
$839$	0.867205	0.0299392	0.0149696	−	0.999888i	$-0.495235\pi$
$839$	0.867205	0.0299392	0.0149696	+	0.999888i	$0.495235\pi$
$840$	0	0
$841$	−27.4535	−0.946674
$842$	0	0
$843$	−16.1916	−0.557668
$844$	0	0
$845$	12.5074	0.430267
$846$	0	0
$847$	14.8985	0.511919
$848$	0	0
$849$	−80.3091	−2.75620
$850$	0	0
$851$	0.408979	0.0140196
$852$	0	0
$853$	21.6528	0.741376	0.370688	−	0.928757i	$-0.379122\pi$
$853$	21.6528	0.741376	0.370688	+	0.928757i	$0.379122\pi$
$854$	0	0
$855$	−0.782057	−0.0267458
$856$	0	0
$857$	−29.5784	−1.01038	−0.505190	−	0.863008i	$-0.668578\pi$
$857$	−29.5784	−1.01038	−0.505190	+	0.863008i	$0.668578\pi$
$858$	0	0
$859$	−3.20196	−0.109249	−0.0546247	−	0.998507i	$-0.517396\pi$
$859$	−3.20196	−0.109249	−0.0546247	+	0.998507i	$0.517396\pi$
$860$	0	0
$861$	56.3748	1.92125
$862$	0	0
$863$	−32.1348	−1.09388	−0.546940	−	0.837172i	$-0.684207\pi$
$863$	−32.1348	−1.09388	−0.546940	+	0.837172i	$0.684207\pi$
$864$	0	0
$865$	−4.17987	−0.142120
$866$	0	0
$867$	51.7206	1.75652
$868$	0	0
$869$	−20.6819	−0.701585
$870$	0	0
$871$	2.24127	0.0759425
$872$	0	0
$873$	−94.5857	−3.20124
$874$	0	0
$875$	1.91593	0.0647701
$876$	0	0
$877$	−24.1442	−0.815291	−0.407645	−	0.913140i	$-0.633650\pi$
$877$	−24.1442	−0.815291	−0.407645	+	0.913140i	$0.633650\pi$
$878$	0	0
$879$	−39.9407	−1.34717
$880$	0	0
$881$	−17.8928	−0.602823	−0.301412	−	0.953494i	$-0.597458\pi$
$881$	−17.8928	−0.602823	−0.301412	+	0.953494i	$0.597458\pi$
$882$	0	0
$883$	−21.3985	−0.720116	−0.360058	−	0.932930i	$-0.617243\pi$
$883$	−21.3985	−0.720116	−0.360058	+	0.932930i	$0.617243\pi$
$884$	0	0
$885$	−4.77226	−0.160418
$886$	0	0
$887$	5.95423	0.199923	0.0999617	−	0.994991i	$-0.468128\pi$
$887$	5.95423	0.199923	0.0999617	+	0.994991i	$0.468128\pi$
$888$	0	0
$889$	−23.5983	−0.791462
$890$	0	0
$891$	−24.5920	−0.823863
$892$	0	0
$893$	0.849740	0.0284355
$894$	0	0
$895$	11.6622	0.389824
$896$	0	0
$897$	−0.884487	−0.0295322
$898$	0	0
$899$	3.16616	0.105597
$900$	0	0
$901$	−1.74799	−0.0582339
$902$	0	0
$903$	26.9421	0.896577
$904$	0	0
$905$	12.3126	0.409284
$906$	0	0
$907$	−12.6647	−0.420524	−0.210262	−	0.977645i	$-0.567432\pi$
$907$	−12.6647	−0.420524	−0.210262	+	0.977645i	$0.567432\pi$
$908$	0	0
$909$	48.6749	1.61445
$910$	0	0
$911$	−49.9395	−1.65457	−0.827284	−	0.561784i	$-0.810115\pi$
$911$	−49.9395	−1.65457	−0.827284	+	0.561784i	$0.810115\pi$
$912$	0	0
$913$	24.5141	0.811298
$914$	0	0
$915$	27.4069	0.906044
$916$	0	0
$917$	−23.1912	−0.765841
$918$	0	0
$919$	−30.2648	−0.998344	−0.499172	−	0.866503i	$-0.666362\pi$
$919$	−30.2648	−0.998344	−0.499172	+	0.866503i	$0.666362\pi$
$920$	0	0
$921$	−101.387	−3.34081
$922$	0	0
$923$	−2.94536	−0.0969478
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−60.6905	−1.99334
$928$	0	0
$929$	8.30552	0.272495	0.136248	−	0.990675i	$-0.456496\pi$
$929$	8.30552	0.272495	0.136248	+	0.990675i	$0.456496\pi$
$930$	0	0
$931$	−0.400891	−0.0131387
$932$	0	0
$933$	25.5250	0.835651
$934$	0	0
$935$	0.832306	0.0272193
$936$	0	0
$937$	−47.7679	−1.56051	−0.780255	−	0.625462i	$-0.784911\pi$
$937$	−47.7679	−1.56051	−0.780255	+	0.625462i	$0.784911\pi$
$938$	0	0
$939$	65.5989	2.14074
$940$	0	0
$941$	34.0730	1.11075	0.555375	−	0.831600i	$-0.312575\pi$
$941$	34.0730	1.11075	0.555375	+	0.831600i	$0.312575\pi$
$942$	0	0
$943$	−3.90542	−0.127178
$944$	0	0
$945$	−20.6310	−0.671126
$946$	0	0
$947$	−31.8869	−1.03619	−0.518093	−	0.855324i	$-0.673358\pi$
$947$	−31.8869	−1.03619	−0.518093	+	0.855324i	$0.673358\pi$
$948$	0	0
$949$	−4.34909	−0.141177
$950$	0	0
$951$	−0.493494	−0.0160026
$952$	0	0
$953$	−0.560584	−0.0181591	−0.00907954	−	0.999959i	$-0.502890\pi$
$953$	−0.560584	−0.0181591	−0.00907954	+	0.999959i	$0.502890\pi$
$954$	0	0
$955$	5.35348	0.173234
$956$	0	0
$957$	6.88013	0.222403
$958$	0	0
$959$	−13.0321	−0.420828
$960$	0	0
$961$	−24.5177	−0.790895
$962$	0	0
$963$	−58.1076	−1.87249
$964$	0	0
$965$	6.66164	0.214446
$966$	0	0
$967$	−43.2850	−1.39195	−0.695975	−	0.718066i	$-0.745028\pi$
$967$	−43.2850	−1.39195	−0.695975	+	0.718066i	$0.745028\pi$
$968$	0	0
$969$	−0.171996	−0.00552530
$970$	0	0
$971$	−4.88659	−0.156818	−0.0784090	−	0.996921i	$-0.524984\pi$
$971$	−4.88659	−0.156818	−0.0784090	+	0.996921i	$0.524984\pi$
$972$	0	0
$973$	−9.52721	−0.305428
$974$	0	0
$975$	2.16267	0.0692610
$976$	0	0
$977$	−43.3930	−1.38826	−0.694132	−	0.719847i	$-0.744212\pi$
$977$	−43.3930	−1.38826	−0.694132	+	0.719847i	$0.744212\pi$
$978$	0	0
$979$	−2.66074	−0.0850375
$980$	0	0
$981$	−85.4690	−2.72882
$982$	0	0
$983$	−19.2832	−0.615038	−0.307519	−	0.951542i	$-0.599499\pi$
$983$	−19.2832	−0.615038	−0.307519	+	0.951542i	$0.599499\pi$
$984$	0	0
$985$	23.9983	0.764648
$986$	0	0
$987$	41.6601	1.32606
$988$	0	0
$989$	−1.86644	−0.0593493
$990$	0	0
$991$	−15.1799	−0.482204	−0.241102	−	0.970500i	$-0.577509\pi$
$991$	−15.1799	−0.482204	−0.241102	+	0.970500i	$0.577509\pi$
$992$	0	0
$993$	31.4252	0.997250
$994$	0	0
$995$	−6.54854	−0.207603
$996$	0	0
$997$	−30.5532	−0.967630	−0.483815	−	0.875170i	$-0.660749\pi$
$997$	−30.5532	−0.967630	−0.483815	+	0.875170i	$0.660749\pi$
$998$	0	0
$999$	10.7682	0.340689

Twists

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

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

\(f(q)\)	\(=\)	\(q-3.08134 q^{3} -1.00000 q^{5} -1.91593 q^{7} +6.49464 q^{9} +O(q^{10})\)	\(q-3.08134 q^{3} -1.00000 q^{5} -1.91593 q^{7} +6.49464 q^{9} -1.79551 q^{11} -0.701862 q^{13} +3.08134 q^{15} +0.463548 q^{17} +0.120416 q^{19} +5.90361 q^{21} -0.408979 q^{23} +1.00000 q^{25} -10.7682 q^{27} +1.24357 q^{29} +2.54603 q^{31} +5.53257 q^{33} +1.91593 q^{35} -1.00000 q^{37} +2.16267 q^{39} +9.54921 q^{41} +4.56366 q^{43} -6.49464 q^{45} +7.05671 q^{47} -3.32923 q^{49} -1.42835 q^{51} -3.77089 q^{53} +1.79551 q^{55} -0.371042 q^{57} -1.54876 q^{59} +8.89448 q^{61} -12.4432 q^{63} +0.701862 q^{65} -3.19332 q^{67} +1.26020 q^{69} +4.19650 q^{71} +6.19650 q^{73} -3.08134 q^{75} +3.44007 q^{77} +11.5187 q^{79} +13.6964 q^{81} -13.6530 q^{83} -0.463548 q^{85} -3.83185 q^{87} +1.48188 q^{89} +1.34472 q^{91} -7.84517 q^{93} -0.120416 q^{95} -14.5637 q^{97} -11.6612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9	\( 5 q - q^{3} - 5 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 4 q^{13} + q^{15} - q^{17} - 10 q^{19} + 5 q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 11 q^{29} + 3 q^{31} - 3 q^{33} - 2 q^{35} - 5 q^{37} - 18 q^{39} + 16 q^{41} - 31 q^{43} - 6 q^{45} - 5 q^{47} + 7 q^{49} - 34 q^{51} + 8 q^{53} + 8 q^{55} - 8 q^{57} - 24 q^{59} - 15 q^{61} - 19 q^{63} + 4 q^{65} - 12 q^{67} + 10 q^{69} - 5 q^{71} + 5 q^{73} - q^{75} - 4 q^{77} - 4 q^{79} + 17 q^{81} - 27 q^{83} + q^{85} + 4 q^{87} + 16 q^{89} - 26 q^{91} + 14 q^{93} + 10 q^{95} - 19 q^{97} - 37 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 + 2 * q^7 + 6 * q^9 - 8 * q^11 - 4 * q^13 + q^15 - q^17 - 10 * q^19 + 5 * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 11 * q^29 + 3 * q^31 - 3 * q^33 - 2 * q^35 - 5 * q^37 - 18 * q^39 + 16 * q^41 - 31 * q^43 - 6 * q^45 - 5 * q^47 + 7 * q^49 - 34 * q^51 + 8 * q^53 + 8 * q^55 - 8 * q^57 - 24 * q^59 - 15 * q^61 - 19 * q^63 + 4 * q^65 - 12 * q^67 + 10 * q^69 - 5 * q^71 + 5 * q^73 - q^75 - 4 * q^77 - 4 * q^79 + 17 * q^81 - 27 * q^83 + q^85 + 4 * q^87 + 16 * q^89 - 26 * q^91 + 14 * q^93 + 10 * q^95 - 19 * q^97 - 37 * q^99

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