LMFDB - Embedded newform 1160.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,2,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, 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("1160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1160.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:	$9.26264663447$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.90321 q^{3} +1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} -5.52543 q^{11} -0.622216 q^{13} -2.90321 q^{15} +3.52543 q^{17} -1.09679 q^{19} -2.62222 q^{21} +5.33185 q^{23} +1.00000 q^{25} -7.05086 q^{27} -1.00000 q^{29} -1.65878 q^{31} +16.0415 q^{33} +0.903212 q^{35} -2.28100 q^{37} +1.80642 q^{39} -7.67307 q^{41} -1.09679 q^{43} +5.42864 q^{45} -1.65878 q^{47} -6.18421 q^{49} -10.2351 q^{51} -2.42864 q^{53} -5.52543 q^{55} +3.18421 q^{57} +9.28592 q^{59} -10.9906 q^{61} +4.90321 q^{63} -0.622216 q^{65} -11.1383 q^{67} -15.4795 q^{69} -7.18421 q^{71} +11.9541 q^{73} -2.90321 q^{75} -4.99063 q^{77} -15.3319 q^{79} +4.18421 q^{81} -7.95407 q^{83} +3.52543 q^{85} +2.90321 q^{87} -16.6637 q^{89} -0.561993 q^{91} +4.81579 q^{93} -1.09679 q^{95} -11.9541 q^{97} -29.9956 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 10 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{21} - 4 q^{23} + 3 q^{25} - 8 q^{27} - 3 q^{29} + 2 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 10 q^{41}+ \cdots - 30 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 10 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^21 - 4 * q^23 + 3 * q^25 - 8 * q^27 - 3 * q^29 + 2 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 10 * q^41 - 10 * q^43 + 3 * q^45 + 2 * q^47 - 5 * q^49 - 4 * q^51 + 6 * q^53 - 10 * q^55 - 4 * q^57 - 12 * q^59 - 6 * q^61 + 8 * q^63 - 2 * q^65 - 20 * q^69 - 8 * q^71 + 16 * q^73 - 2 * q^75 + 12 * q^77 - 26 * q^79 - q^81 - 4 * q^83 + 4 * q^85 + 2 * q^87 - 10 * q^89 + 12 * q^91 + 28 * q^93 - 10 * q^95 - 16 * q^97 - 30 * q^99

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
$3$	−2.90321	−1.67617	−0.838085	−	0.545540i	$-0.816325\pi$
$3$	−2.90321	−1.67617	−0.838085	+	0.545540i	$0.816325\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.903212	0.341382	0.170691	−	0.985325i	$-0.445400\pi$
$7$	0.903212	0.341382	0.170691	+	0.985325i	$0.445400\pi$
$8$	0	0
$9$	5.42864	1.80955
$10$	0	0
$11$	−5.52543	−1.66598	−0.832990	−	0.553289i	$-0.813373\pi$
$11$	−5.52543	−1.66598	−0.832990	+	0.553289i	$0.813373\pi$
$12$	0	0
$13$	−0.622216	−0.172572	−0.0862858	−	0.996270i	$-0.527500\pi$
$13$	−0.622216	−0.172572	−0.0862858	+	0.996270i	$0.527500\pi$
$14$	0	0
$15$	−2.90321	−0.749606
$16$	0	0
$17$	3.52543	0.855042	0.427521	−	0.904005i	$-0.359387\pi$
$17$	3.52543	0.855042	0.427521	+	0.904005i	$0.359387\pi$
$18$	0	0
$19$	−1.09679	−0.251620	−0.125810	−	0.992054i	$-0.540153\pi$
$19$	−1.09679	−0.251620	−0.125810	+	0.992054i	$0.540153\pi$
$20$	0	0
$21$	−2.62222	−0.572214
$22$	0	0
$23$	5.33185	1.11177	0.555884	−	0.831260i	$-0.312380\pi$
$23$	5.33185	1.11177	0.555884	+	0.831260i	$0.312380\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−7.05086	−1.35694
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.65878	−0.297926	−0.148963	−	0.988843i	$-0.547593\pi$
$31$	−1.65878	−0.297926	−0.148963	+	0.988843i	$0.547593\pi$
$32$	0	0
$33$	16.0415	2.79246
$34$	0	0
$35$	0.903212	0.152671
$36$	0	0
$37$	−2.28100	−0.374993	−0.187497	−	0.982265i	$-0.560037\pi$
$37$	−2.28100	−0.374993	−0.187497	+	0.982265i	$0.560037\pi$
$38$	0	0
$39$	1.80642	0.289259
$40$	0	0
$41$	−7.67307	−1.19833	−0.599166	−	0.800625i	$-0.704501\pi$
$41$	−7.67307	−1.19833	−0.599166	+	0.800625i	$0.704501\pi$
$42$	0	0
$43$	−1.09679	−0.167259	−0.0836293	−	0.996497i	$-0.526651\pi$
$43$	−1.09679	−0.167259	−0.0836293	+	0.996497i	$0.526651\pi$
$44$	0	0
$45$	5.42864	0.809254
$46$	0	0
$47$	−1.65878	−0.241958	−0.120979	−	0.992655i	$-0.538603\pi$
$47$	−1.65878	−0.241958	−0.120979	+	0.992655i	$0.538603\pi$
$48$	0	0
$49$	−6.18421	−0.883458
$50$	0	0
$51$	−10.2351	−1.43320
$52$	0	0
$53$	−2.42864	−0.333599	−0.166800	−	0.985991i	$-0.553343\pi$
$53$	−2.42864	−0.333599	−0.166800	+	0.985991i	$0.553343\pi$
$54$	0	0
$55$	−5.52543	−0.745049
$56$	0	0
$57$	3.18421	0.421759
$58$	0	0
$59$	9.28592	1.20892	0.604462	−	0.796634i	$-0.293388\pi$
$59$	9.28592	1.20892	0.604462	+	0.796634i	$0.293388\pi$
$60$	0	0
$61$	−10.9906	−1.40721	−0.703603	−	0.710593i	$-0.748427\pi$
$61$	−10.9906	−1.40721	−0.703603	+	0.710593i	$0.748427\pi$
$62$	0	0
$63$	4.90321	0.617747
$64$	0	0
$65$	−0.622216	−0.0771764
$66$	0	0
$67$	−11.1383	−1.36076	−0.680378	−	0.732861i	$-0.738185\pi$
$67$	−11.1383	−1.36076	−0.680378	+	0.732861i	$0.738185\pi$
$68$	0	0
$69$	−15.4795	−1.86351
$70$	0	0
$71$	−7.18421	−0.852609	−0.426304	−	0.904580i	$-0.640185\pi$
$71$	−7.18421	−0.852609	−0.426304	+	0.904580i	$0.640185\pi$
$72$	0	0
$73$	11.9541	1.39912	0.699559	−	0.714575i	$-0.253380\pi$
$73$	11.9541	1.39912	0.699559	+	0.714575i	$0.253380\pi$
$74$	0	0
$75$	−2.90321	−0.335234
$76$	0	0
$77$	−4.99063	−0.568735
$78$	0	0
$79$	−15.3319	−1.72497	−0.862484	−	0.506084i	$-0.831092\pi$
$79$	−15.3319	−1.72497	−0.862484	+	0.506084i	$0.831092\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	0	0
$83$	−7.95407	−0.873072	−0.436536	−	0.899687i	$-0.643795\pi$
$83$	−7.95407	−0.873072	−0.436536	+	0.899687i	$0.643795\pi$
$84$	0	0
$85$	3.52543	0.382386
$86$	0	0
$87$	2.90321	0.311257
$88$	0	0
$89$	−16.6637	−1.76635	−0.883174	−	0.469045i	$-0.844598\pi$
$89$	−16.6637	−1.76635	−0.883174	+	0.469045i	$0.844598\pi$
$90$	0	0
$91$	−0.561993	−0.0589128
$92$	0	0
$93$	4.81579	0.499374
$94$	0	0
$95$	−1.09679	−0.112528
$96$	0	0
$97$	−11.9541	−1.21375	−0.606876	−	0.794797i	$-0.707577\pi$
$97$	−11.9541	−1.21375	−0.606876	+	0.794797i	$0.707577\pi$
$98$	0	0
$99$	−29.9956	−3.01467
$100$	0	0
$101$	8.66370	0.862071	0.431035	−	0.902335i	$-0.358148\pi$
$101$	8.66370	0.862071	0.431035	+	0.902335i	$0.358148\pi$
$102$	0	0
$103$	−17.6271	−1.73685	−0.868427	−	0.495817i	$-0.834868\pi$
$103$	−17.6271	−1.73685	−0.868427	+	0.495817i	$0.834868\pi$
$104$	0	0
$105$	−2.62222	−0.255902
$106$	0	0
$107$	3.95407	0.382254	0.191127	−	0.981565i	$-0.438786\pi$
$107$	3.95407	0.382254	0.191127	+	0.981565i	$0.438786\pi$
$108$	0	0
$109$	−2.81579	−0.269704	−0.134852	−	0.990866i	$-0.543056\pi$
$109$	−2.81579	−0.269704	−0.134852	+	0.990866i	$0.543056\pi$
$110$	0	0
$111$	6.62222	0.628553
$112$	0	0
$113$	14.7096	1.38377	0.691883	−	0.722010i	$-0.256781\pi$
$113$	14.7096	1.38377	0.691883	+	0.722010i	$0.256781\pi$
$114$	0	0
$115$	5.33185	0.497198
$116$	0	0
$117$	−3.37778	−0.312276
$118$	0	0
$119$	3.18421	0.291896
$120$	0	0
$121$	19.5303	1.77549
$122$	0	0
$123$	22.2766	2.00861
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.34122	−0.207750	−0.103875	−	0.994590i	$-0.533124\pi$
$127$	−2.34122	−0.207750	−0.103875	+	0.994590i	$0.533124\pi$
$128$	0	0
$129$	3.18421	0.280354
$130$	0	0
$131$	−5.65878	−0.494410	−0.247205	−	0.968963i	$-0.579512\pi$
$131$	−5.65878	−0.494410	−0.247205	+	0.968963i	$0.579512\pi$
$132$	0	0
$133$	−0.990632	−0.0858987
$134$	0	0
$135$	−7.05086	−0.606841
$136$	0	0
$137$	9.62714	0.822502	0.411251	−	0.911522i	$-0.365092\pi$
$137$	9.62714	0.822502	0.411251	+	0.911522i	$0.365092\pi$
$138$	0	0
$139$	6.66370	0.565208	0.282604	−	0.959237i	$-0.408802\pi$
$139$	6.66370	0.565208	0.282604	+	0.959237i	$0.408802\pi$
$140$	0	0
$141$	4.81579	0.405563
$142$	0	0
$143$	3.43801	0.287501
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	17.9541	1.48083
$148$	0	0
$149$	−17.2257	−1.41118	−0.705592	−	0.708618i	$-0.749319\pi$
$149$	−17.2257	−1.41118	−0.705592	+	0.708618i	$0.749319\pi$
$150$	0	0
$151$	5.93978	0.483372	0.241686	−	0.970354i	$-0.422300\pi$
$151$	5.93978	0.483372	0.241686	+	0.970354i	$0.422300\pi$
$152$	0	0
$153$	19.1383	1.54724
$154$	0	0
$155$	−1.65878	−0.133236
$156$	0	0
$157$	−15.5669	−1.24238	−0.621188	−	0.783662i	$-0.713349\pi$
$157$	−15.5669	−1.24238	−0.621188	+	0.783662i	$0.713349\pi$
$158$	0	0
$159$	7.05086	0.559169
$160$	0	0
$161$	4.81579	0.379538
$162$	0	0
$163$	10.5161	0.823681	0.411841	−	0.911256i	$-0.364886\pi$
$163$	10.5161	0.823681	0.411841	+	0.911256i	$0.364886\pi$
$164$	0	0
$165$	16.0415	1.24883
$166$	0	0
$167$	12.6494	0.978841	0.489420	−	0.872048i	$-0.337208\pi$
$167$	12.6494	0.978841	0.489420	+	0.872048i	$0.337208\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	−5.95407	−0.455319
$172$	0	0
$173$	6.13335	0.466310	0.233155	−	0.972440i	$-0.425095\pi$
$173$	6.13335	0.466310	0.233155	+	0.972440i	$0.425095\pi$
$174$	0	0
$175$	0.903212	0.0682764
$176$	0	0
$177$	−26.9590	−2.02636
$178$	0	0
$179$	15.2257	1.13802	0.569011	−	0.822330i	$-0.307326\pi$
$179$	15.2257	1.13802	0.569011	+	0.822330i	$0.307326\pi$
$180$	0	0
$181$	3.93978	0.292841	0.146421	−	0.989222i	$-0.453225\pi$
$181$	3.93978	0.292841	0.146421	+	0.989222i	$0.453225\pi$
$182$	0	0
$183$	31.9081	2.35872
$184$	0	0
$185$	−2.28100	−0.167702
$186$	0	0
$187$	−19.4795	−1.42448
$188$	0	0
$189$	−6.36842	−0.463234
$190$	0	0
$191$	−7.06515	−0.511216	−0.255608	−	0.966781i	$-0.582276\pi$
$191$	−7.06515	−0.511216	−0.255608	+	0.966781i	$0.582276\pi$
$192$	0	0
$193$	1.98571	0.142935	0.0714673	−	0.997443i	$-0.477232\pi$
$193$	1.98571	0.142935	0.0714673	+	0.997443i	$0.477232\pi$
$194$	0	0
$195$	1.80642	0.129361
$196$	0	0
$197$	−16.2766	−1.15966	−0.579828	−	0.814739i	$-0.696880\pi$
$197$	−16.2766	−1.15966	−0.579828	+	0.814739i	$0.696880\pi$
$198$	0	0
$199$	22.5303	1.59713	0.798567	−	0.601906i	$-0.205592\pi$
$199$	22.5303	1.59713	0.798567	+	0.601906i	$0.205592\pi$
$200$	0	0
$201$	32.3368	2.28086
$202$	0	0
$203$	−0.903212	−0.0633930
$204$	0	0
$205$	−7.67307	−0.535911
$206$	0	0
$207$	28.9447	2.01180
$208$	0	0
$209$	6.06022	0.419194
$210$	0	0
$211$	−3.33185	−0.229374	−0.114687	−	0.993402i	$-0.536587\pi$
$211$	−3.33185	−0.229374	−0.114687	+	0.993402i	$0.536587\pi$
$212$	0	0
$213$	20.8573	1.42912
$214$	0	0
$215$	−1.09679	−0.0748003
$216$	0	0
$217$	−1.49823	−0.101707
$218$	0	0
$219$	−34.7052	−2.34516
$220$	0	0
$221$	−2.19358	−0.147556
$222$	0	0
$223$	−13.8938	−0.930401	−0.465200	−	0.885205i	$-0.654018\pi$
$223$	−13.8938	−0.930401	−0.465200	+	0.885205i	$0.654018\pi$
$224$	0	0
$225$	5.42864	0.361909
$226$	0	0
$227$	14.4558	0.959468	0.479734	−	0.877414i	$-0.340733\pi$
$227$	14.4558	0.959468	0.479734	+	0.877414i	$0.340733\pi$
$228$	0	0
$229$	15.3274	1.01286	0.506432	−	0.862280i	$-0.330964\pi$
$229$	15.3274	1.01286	0.506432	+	0.862280i	$0.330964\pi$
$230$	0	0
$231$	14.4889	0.953297
$232$	0	0
$233$	−11.7146	−0.767446	−0.383723	−	0.923448i	$-0.625358\pi$
$233$	−11.7146	−0.767446	−0.383723	+	0.923448i	$0.625358\pi$
$234$	0	0
$235$	−1.65878	−0.108207
$236$	0	0
$237$	44.5116	2.89134
$238$	0	0
$239$	−2.53035	−0.163675	−0.0818374	−	0.996646i	$-0.526079\pi$
$239$	−2.53035	−0.163675	−0.0818374	+	0.996646i	$0.526079\pi$
$240$	0	0
$241$	4.75557	0.306333	0.153167	−	0.988200i	$-0.451053\pi$
$241$	4.75557	0.306333	0.153167	+	0.988200i	$0.451053\pi$
$242$	0	0
$243$	9.00492	0.577666
$244$	0	0
$245$	−6.18421	−0.395095
$246$	0	0
$247$	0.682439	0.0434225
$248$	0	0
$249$	23.0923	1.46342
$250$	0	0
$251$	−6.76986	−0.427310	−0.213655	−	0.976909i	$-0.568537\pi$
$251$	−6.76986	−0.427310	−0.213655	+	0.976909i	$0.568537\pi$
$252$	0	0
$253$	−29.4608	−1.85218
$254$	0	0
$255$	−10.2351	−0.640945
$256$	0	0
$257$	−13.0923	−0.816678	−0.408339	−	0.912830i	$-0.633892\pi$
$257$	−13.0923	−0.816678	−0.408339	+	0.912830i	$0.633892\pi$
$258$	0	0
$259$	−2.06022	−0.128016
$260$	0	0
$261$	−5.42864	−0.336024
$262$	0	0
$263$	−10.1289	−0.624575	−0.312288	−	0.949988i	$-0.601095\pi$
$263$	−10.1289	−0.624575	−0.312288	+	0.949988i	$0.601095\pi$
$264$	0	0
$265$	−2.42864	−0.149190
$266$	0	0
$267$	48.3783	2.96070
$268$	0	0
$269$	−4.91750	−0.299825	−0.149913	−	0.988699i	$-0.547899\pi$
$269$	−4.91750	−0.299825	−0.149913	+	0.988699i	$0.547899\pi$
$270$	0	0
$271$	24.5763	1.49290	0.746451	−	0.665440i	$-0.231756\pi$
$271$	24.5763	1.49290	0.746451	+	0.665440i	$0.231756\pi$
$272$	0	0
$273$	1.63158	0.0987479
$274$	0	0
$275$	−5.52543	−0.333196
$276$	0	0
$277$	−24.7971	−1.48991	−0.744955	−	0.667114i	$-0.767529\pi$
$277$	−24.7971	−1.48991	−0.744955	+	0.667114i	$0.767529\pi$
$278$	0	0
$279$	−9.00492	−0.539111
$280$	0	0
$281$	−16.5303	−0.986118	−0.493059	−	0.869996i	$-0.664121\pi$
$281$	−16.5303	−0.986118	−0.493059	+	0.869996i	$0.664121\pi$
$282$	0	0
$283$	25.2400	1.50036	0.750181	−	0.661233i	$-0.229966\pi$
$283$	25.2400	1.50036	0.750181	+	0.661233i	$0.229966\pi$
$284$	0	0
$285$	3.18421	0.188616
$286$	0	0
$287$	−6.93041	−0.409089
$288$	0	0
$289$	−4.57136	−0.268904
$290$	0	0
$291$	34.7052	2.03445
$292$	0	0
$293$	8.63651	0.504550	0.252275	−	0.967656i	$-0.418821\pi$
$293$	8.63651	0.504550	0.252275	+	0.967656i	$0.418821\pi$
$294$	0	0
$295$	9.28592	0.540647
$296$	0	0
$297$	38.9590	2.26063
$298$	0	0
$299$	−3.31756	−0.191860
$300$	0	0
$301$	−0.990632	−0.0570991
$302$	0	0
$303$	−25.1526	−1.44498
$304$	0	0
$305$	−10.9906	−0.629322
$306$	0	0
$307$	14.6365	0.835350	0.417675	−	0.908597i	$-0.362845\pi$
$307$	14.6365	0.835350	0.417675	+	0.908597i	$0.362845\pi$
$308$	0	0
$309$	51.1753	2.91126
$310$	0	0
$311$	−1.65878	−0.0940608	−0.0470304	−	0.998893i	$-0.514976\pi$
$311$	−1.65878	−0.0940608	−0.0470304	+	0.998893i	$0.514976\pi$
$312$	0	0
$313$	18.3368	1.03646	0.518228	−	0.855243i	$-0.326592\pi$
$313$	18.3368	1.03646	0.518228	+	0.855243i	$0.326592\pi$
$314$	0	0
$315$	4.90321	0.276265
$316$	0	0
$317$	22.5763	1.26801	0.634005	−	0.773329i	$-0.281410\pi$
$317$	22.5763	1.26801	0.634005	+	0.773329i	$0.281410\pi$
$318$	0	0
$319$	5.52543	0.309365
$320$	0	0
$321$	−11.4795	−0.640723
$322$	0	0
$323$	−3.86665	−0.215146
$324$	0	0
$325$	−0.622216	−0.0345143
$326$	0	0
$327$	8.17484	0.452070
$328$	0	0
$329$	−1.49823	−0.0826001
$330$	0	0
$331$	−35.1066	−1.92964	−0.964818	−	0.262920i	$-0.915314\pi$
$331$	−35.1066	−1.92964	−0.964818	+	0.262920i	$0.915314\pi$
$332$	0	0
$333$	−12.3827	−0.678568
$334$	0	0
$335$	−11.1383	−0.608549
$336$	0	0
$337$	−8.60793	−0.468904	−0.234452	−	0.972128i	$-0.575329\pi$
$337$	−8.60793	−0.468904	−0.234452	+	0.972128i	$0.575329\pi$
$338$	0	0
$339$	−42.7052	−2.31943
$340$	0	0
$341$	9.16547	0.496338
$342$	0	0
$343$	−11.9081	−0.642979
$344$	0	0
$345$	−15.4795	−0.833388
$346$	0	0
$347$	−18.6178	−0.999454	−0.499727	−	0.866183i	$-0.666566\pi$
$347$	−18.6178	−0.999454	−0.499727	+	0.866183i	$0.666566\pi$
$348$	0	0
$349$	−28.1017	−1.50425	−0.752125	−	0.659020i	$-0.770971\pi$
$349$	−28.1017	−1.50425	−0.752125	+	0.659020i	$0.770971\pi$
$350$	0	0
$351$	4.38715	0.234169
$352$	0	0
$353$	15.9398	0.848389	0.424194	−	0.905571i	$-0.360557\pi$
$353$	15.9398	0.848389	0.424194	+	0.905571i	$0.360557\pi$
$354$	0	0
$355$	−7.18421	−0.381298
$356$	0	0
$357$	−9.24443	−0.489267
$358$	0	0
$359$	21.5669	1.13826	0.569129	−	0.822248i	$-0.307280\pi$
$359$	21.5669	1.13826	0.569129	+	0.822248i	$0.307280\pi$
$360$	0	0
$361$	−17.7971	−0.936687
$362$	0	0
$363$	−56.7007	−2.97602
$364$	0	0
$365$	11.9541	0.625704
$366$	0	0
$367$	−15.3733	−0.802482	−0.401241	−	0.915973i	$-0.631421\pi$
$367$	−15.3733	−0.802482	−0.401241	+	0.915973i	$0.631421\pi$
$368$	0	0
$369$	−41.6543	−2.16844
$370$	0	0
$371$	−2.19358	−0.113885
$372$	0	0
$373$	−29.9081	−1.54858	−0.774292	−	0.632828i	$-0.781894\pi$
$373$	−29.9081	−1.54858	−0.774292	+	0.632828i	$0.781894\pi$
$374$	0	0
$375$	−2.90321	−0.149921
$376$	0	0
$377$	0.622216	0.0320457
$378$	0	0
$379$	−31.1195	−1.59850	−0.799252	−	0.600996i	$-0.794771\pi$
$379$	−31.1195	−1.59850	−0.799252	+	0.600996i	$0.794771\pi$
$380$	0	0
$381$	6.79706	0.348224
$382$	0	0
$383$	7.09679	0.362629	0.181314	−	0.983425i	$-0.441965\pi$
$383$	7.09679	0.362629	0.181314	+	0.983425i	$0.441965\pi$
$384$	0	0
$385$	−4.99063	−0.254346
$386$	0	0
$387$	−5.95407	−0.302662
$388$	0	0
$389$	33.0005	1.67319	0.836595	−	0.547821i	$-0.184543\pi$
$389$	33.0005	1.67319	0.836595	+	0.547821i	$0.184543\pi$
$390$	0	0
$391$	18.7971	0.950608
$392$	0	0
$393$	16.4286	0.828715
$394$	0	0
$395$	−15.3319	−0.771429
$396$	0	0
$397$	12.2766	0.616142	0.308071	−	0.951363i	$-0.400316\pi$
$397$	12.2766	0.616142	0.308071	+	0.951363i	$0.400316\pi$
$398$	0	0
$399$	2.87601	0.143981
$400$	0	0
$401$	30.6321	1.52969	0.764846	−	0.644213i	$-0.222815\pi$
$401$	30.6321	1.52969	0.764846	+	0.644213i	$0.222815\pi$
$402$	0	0
$403$	1.03212	0.0514135
$404$	0	0
$405$	4.18421	0.207915
$406$	0	0
$407$	12.6035	0.624731
$408$	0	0
$409$	−29.7462	−1.47085	−0.735427	−	0.677603i	$-0.763019\pi$
$409$	−29.7462	−1.47085	−0.735427	+	0.677603i	$0.763019\pi$
$410$	0	0
$411$	−27.9496	−1.37865
$412$	0	0
$413$	8.38715	0.412705
$414$	0	0
$415$	−7.95407	−0.390450
$416$	0	0
$417$	−19.3461	−0.947385
$418$	0	0
$419$	12.3368	0.602691	0.301345	−	0.953515i	$-0.402564\pi$
$419$	12.3368	0.602691	0.301345	+	0.953515i	$0.402564\pi$
$420$	0	0
$421$	−13.0094	−0.634038	−0.317019	−	0.948419i	$-0.602682\pi$
$421$	−13.0094	−0.634038	−0.317019	+	0.948419i	$0.602682\pi$
$422$	0	0
$423$	−9.00492	−0.437834
$424$	0	0
$425$	3.52543	0.171008
$426$	0	0
$427$	−9.92687	−0.480395
$428$	0	0
$429$	−9.98126	−0.481900
$430$	0	0
$431$	13.6316	0.656610	0.328305	−	0.944572i	$-0.393523\pi$
$431$	13.6316	0.656610	0.328305	+	0.944572i	$0.393523\pi$
$432$	0	0
$433$	21.8938	1.05215	0.526075	−	0.850438i	$-0.323663\pi$
$433$	21.8938	1.05215	0.526075	+	0.850438i	$0.323663\pi$
$434$	0	0
$435$	2.90321	0.139198
$436$	0	0
$437$	−5.84791	−0.279744
$438$	0	0
$439$	13.2573	0.632738	0.316369	−	0.948636i	$-0.397536\pi$
$439$	13.2573	0.632738	0.316369	+	0.948636i	$0.397536\pi$
$440$	0	0
$441$	−33.5718	−1.59866
$442$	0	0
$443$	−25.2716	−1.20069	−0.600346	−	0.799741i	$-0.704970\pi$
$443$	−25.2716	−1.20069	−0.600346	+	0.799741i	$0.704970\pi$
$444$	0	0
$445$	−16.6637	−0.789935
$446$	0	0
$447$	50.0098	2.36538
$448$	0	0
$449$	21.9496	1.03587	0.517933	−	0.855421i	$-0.326702\pi$
$449$	21.9496	1.03587	0.517933	+	0.855421i	$0.326702\pi$
$450$	0	0
$451$	42.3970	1.99640
$452$	0	0
$453$	−17.2444	−0.810214
$454$	0	0
$455$	−0.561993	−0.0263466
$456$	0	0
$457$	−9.45091	−0.442095	−0.221048	−	0.975263i	$-0.570948\pi$
$457$	−9.45091	−0.442095	−0.221048	+	0.975263i	$0.570948\pi$
$458$	0	0
$459$	−24.8573	−1.16024
$460$	0	0
$461$	24.3970	1.13628	0.568141	−	0.822931i	$-0.307663\pi$
$461$	24.3970	1.13628	0.568141	+	0.822931i	$0.307663\pi$
$462$	0	0
$463$	−8.17929	−0.380124	−0.190062	−	0.981772i	$-0.560869\pi$
$463$	−8.17929	−0.380124	−0.190062	+	0.981772i	$0.560869\pi$
$464$	0	0
$465$	4.81579	0.223327
$466$	0	0
$467$	−30.0370	−1.38995	−0.694974	−	0.719035i	$-0.744584\pi$
$467$	−30.0370	−1.38995	−0.694974	+	0.719035i	$0.744584\pi$
$468$	0	0
$469$	−10.0602	−0.464538
$470$	0	0
$471$	45.1941	2.08243
$472$	0	0
$473$	6.06022	0.278649
$474$	0	0
$475$	−1.09679	−0.0503241
$476$	0	0
$477$	−13.1842	−0.603663
$478$	0	0
$479$	21.3131	0.973821	0.486911	−	0.873452i	$-0.338124\pi$
$479$	21.3131	0.973821	0.486911	+	0.873452i	$0.338124\pi$
$480$	0	0
$481$	1.41927	0.0647132
$482$	0	0
$483$	−13.9813	−0.636170
$484$	0	0
$485$	−11.9541	−0.542806
$486$	0	0
$487$	2.14764	0.0973190	0.0486595	−	0.998815i	$-0.484505\pi$
$487$	2.14764	0.0973190	0.0486595	+	0.998815i	$0.484505\pi$
$488$	0	0
$489$	−30.5303	−1.38063
$490$	0	0
$491$	−10.1289	−0.457111	−0.228556	−	0.973531i	$-0.573400\pi$
$491$	−10.1289	−0.457111	−0.228556	+	0.973531i	$0.573400\pi$
$492$	0	0
$493$	−3.52543	−0.158777
$494$	0	0
$495$	−29.9956	−1.34820
$496$	0	0
$497$	−6.48886	−0.291065
$498$	0	0
$499$	24.4701	1.09543	0.547717	−	0.836664i	$-0.315497\pi$
$499$	24.4701	1.09543	0.547717	+	0.836664i	$0.315497\pi$
$500$	0	0
$501$	−36.7239	−1.64070
$502$	0	0
$503$	37.5669	1.67503	0.837513	−	0.546417i	$-0.184009\pi$
$503$	37.5669	1.67503	0.837513	+	0.546417i	$0.184009\pi$
$504$	0	0
$505$	8.66370	0.385530
$506$	0	0
$507$	36.6178	1.62625
$508$	0	0
$509$	19.9398	0.883815	0.441908	−	0.897061i	$-0.354302\pi$
$509$	19.9398	0.883815	0.441908	+	0.897061i	$0.354302\pi$
$510$	0	0
$511$	10.7971	0.477634
$512$	0	0
$513$	7.73329	0.341433
$514$	0	0
$515$	−17.6271	−0.776744
$516$	0	0
$517$	9.16547	0.403097
$518$	0	0
$519$	−17.8064	−0.781615
$520$	0	0
$521$	19.5526	0.856616	0.428308	−	0.903633i	$-0.359110\pi$
$521$	19.5526	0.856616	0.428308	+	0.903633i	$0.359110\pi$
$522$	0	0
$523$	9.23999	0.404036	0.202018	−	0.979382i	$-0.435250\pi$
$523$	9.23999	0.404036	0.202018	+	0.979382i	$0.435250\pi$
$524$	0	0
$525$	−2.62222	−0.114443
$526$	0	0
$527$	−5.84791	−0.254739
$528$	0	0
$529$	5.42864	0.236028
$530$	0	0
$531$	50.4099	2.18760
$532$	0	0
$533$	4.77430	0.206798
$534$	0	0
$535$	3.95407	0.170949
$536$	0	0
$537$	−44.2034	−1.90752
$538$	0	0
$539$	34.1704	1.47182
$540$	0	0
$541$	6.56199	0.282122	0.141061	−	0.990001i	$-0.454949\pi$
$541$	6.56199	0.282122	0.141061	+	0.990001i	$0.454949\pi$
$542$	0	0
$543$	−11.4380	−0.490852
$544$	0	0
$545$	−2.81579	−0.120615
$546$	0	0
$547$	−11.3921	−0.487090	−0.243545	−	0.969890i	$-0.578310\pi$
$547$	−11.3921	−0.487090	−0.243545	+	0.969890i	$0.578310\pi$
$548$	0	0
$549$	−59.6642	−2.54641
$550$	0	0
$551$	1.09679	0.0467247
$552$	0	0
$553$	−13.8479	−0.588873
$554$	0	0
$555$	6.62222	0.281097
$556$	0	0
$557$	43.9813	1.86355	0.931773	−	0.363041i	$-0.118262\pi$
$557$	43.9813	1.86355	0.931773	+	0.363041i	$0.118262\pi$
$558$	0	0
$559$	0.682439	0.0288641
$560$	0	0
$561$	56.5531	2.38767
$562$	0	0
$563$	30.3323	1.27836	0.639178	−	0.769059i	$-0.279275\pi$
$563$	30.3323	1.27836	0.639178	+	0.769059i	$0.279275\pi$
$564$	0	0
$565$	14.7096	0.618839
$566$	0	0
$567$	3.77923	0.158713
$568$	0	0
$569$	−19.3778	−0.812359	−0.406179	−	0.913793i	$-0.633139\pi$
$569$	−19.3778	−0.812359	−0.406179	+	0.913793i	$0.633139\pi$
$570$	0	0
$571$	3.34614	0.140032	0.0700158	−	0.997546i	$-0.477695\pi$
$571$	3.34614	0.140032	0.0700158	+	0.997546i	$0.477695\pi$
$572$	0	0
$573$	20.5116	0.856885
$574$	0	0
$575$	5.33185	0.222354
$576$	0	0
$577$	−31.9956	−1.33199	−0.665996	−	0.745955i	$-0.731993\pi$
$577$	−31.9956	−1.33199	−0.665996	+	0.745955i	$0.731993\pi$
$578$	0	0
$579$	−5.76494	−0.239583
$580$	0	0
$581$	−7.18421	−0.298051
$582$	0	0
$583$	13.4193	0.555769
$584$	0	0
$585$	−3.37778	−0.139654
$586$	0	0
$587$	24.5575	1.01360	0.506799	−	0.862064i	$-0.330829\pi$
$587$	24.5575	1.01360	0.506799	+	0.862064i	$0.330829\pi$
$588$	0	0
$589$	1.81933	0.0749642
$590$	0	0
$591$	47.2543	1.94378
$592$	0	0
$593$	−16.7971	−0.689772	−0.344886	−	0.938645i	$-0.612082\pi$
$593$	−16.7971	−0.689772	−0.344886	+	0.938645i	$0.612082\pi$
$594$	0	0
$595$	3.18421	0.130540
$596$	0	0
$597$	−65.4104	−2.67707
$598$	0	0
$599$	25.9126	1.05876	0.529380	−	0.848385i	$-0.322425\pi$
$599$	25.9126	1.05876	0.529380	+	0.848385i	$0.322425\pi$
$600$	0	0
$601$	−30.8158	−1.25700	−0.628501	−	0.777809i	$-0.716331\pi$
$601$	−30.8158	−1.25700	−0.628501	+	0.777809i	$0.716331\pi$
$602$	0	0
$603$	−60.4657	−2.46235
$604$	0	0
$605$	19.5303	0.794022
$606$	0	0
$607$	−28.1476	−1.14248	−0.571239	−	0.820784i	$-0.693537\pi$
$607$	−28.1476	−1.14248	−0.571239	+	0.820784i	$0.693537\pi$
$608$	0	0
$609$	2.62222	0.106258
$610$	0	0
$611$	1.03212	0.0417551
$612$	0	0
$613$	37.5496	1.51661	0.758306	−	0.651898i	$-0.226027\pi$
$613$	37.5496	1.51661	0.758306	+	0.651898i	$0.226027\pi$
$614$	0	0
$615$	22.2766	0.898277
$616$	0	0
$617$	42.8212	1.72392	0.861958	−	0.506980i	$-0.169238\pi$
$617$	42.8212	1.72392	0.861958	+	0.506980i	$0.169238\pi$
$618$	0	0
$619$	−17.5540	−0.705555	−0.352778	−	0.935707i	$-0.614763\pi$
$619$	−17.5540	−0.705555	−0.352778	+	0.935707i	$0.614763\pi$
$620$	0	0
$621$	−37.5941	−1.50860
$622$	0	0
$623$	−15.0509	−0.603000
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.5941	−0.702641
$628$	0	0
$629$	−8.04149	−0.320635
$630$	0	0
$631$	30.9304	1.23132	0.615660	−	0.788012i	$-0.288889\pi$
$631$	30.9304	1.23132	0.615660	+	0.788012i	$0.288889\pi$
$632$	0	0
$633$	9.67307	0.384470
$634$	0	0
$635$	−2.34122	−0.0929085
$636$	0	0
$637$	3.84791	0.152460
$638$	0	0
$639$	−39.0005	−1.54284
$640$	0	0
$641$	−37.1842	−1.46869	−0.734344	−	0.678778i	$-0.762510\pi$
$641$	−37.1842	−1.46869	−0.734344	+	0.678778i	$0.762510\pi$
$642$	0	0
$643$	−42.0054	−1.65653	−0.828266	−	0.560336i	$-0.810672\pi$
$643$	−42.0054	−1.65653	−0.828266	+	0.560336i	$0.810672\pi$
$644$	0	0
$645$	3.18421	0.125378
$646$	0	0
$647$	−30.6464	−1.20483	−0.602416	−	0.798182i	$-0.705795\pi$
$647$	−30.6464	−1.20483	−0.602416	+	0.798182i	$0.705795\pi$
$648$	0	0
$649$	−51.3087	−2.01404
$650$	0	0
$651$	4.34968	0.170477
$652$	0	0
$653$	1.85236	0.0724883	0.0362442	−	0.999343i	$-0.488461\pi$
$653$	1.85236	0.0724883	0.0362442	+	0.999343i	$0.488461\pi$
$654$	0	0
$655$	−5.65878	−0.221107
$656$	0	0
$657$	64.8943	2.53177
$658$	0	0
$659$	−39.1985	−1.52696	−0.763478	−	0.645833i	$-0.776510\pi$
$659$	−39.1985	−1.52696	−0.763478	+	0.645833i	$0.776510\pi$
$660$	0	0
$661$	−43.6356	−1.69723	−0.848614	−	0.529012i	$-0.822562\pi$
$661$	−43.6356	−1.69723	−0.848614	+	0.529012i	$0.822562\pi$
$662$	0	0
$663$	6.36842	0.247329
$664$	0	0
$665$	−0.990632	−0.0384151
$666$	0	0
$667$	−5.33185	−0.206450
$668$	0	0
$669$	40.3368	1.55951
$670$	0	0
$671$	60.7279	2.34438
$672$	0	0
$673$	−36.8671	−1.42112	−0.710562	−	0.703635i	$-0.751559\pi$
$673$	−36.8671	−1.42112	−0.710562	+	0.703635i	$0.751559\pi$
$674$	0	0
$675$	−7.05086	−0.271388
$676$	0	0
$677$	−0.137799	−0.00529604	−0.00264802	−	0.999996i	$-0.500843\pi$
$677$	−0.137799	−0.00529604	−0.00264802	+	0.999996i	$0.500843\pi$
$678$	0	0
$679$	−10.7971	−0.414353
$680$	0	0
$681$	−41.9684	−1.60823
$682$	0	0
$683$	−7.61729	−0.291468	−0.145734	−	0.989324i	$-0.546554\pi$
$683$	−7.61729	−0.291468	−0.145734	+	0.989324i	$0.546554\pi$
$684$	0	0
$685$	9.62714	0.367834
$686$	0	0
$687$	−44.4987	−1.69773
$688$	0	0
$689$	1.51114	0.0575698
$690$	0	0
$691$	19.4795	0.741035	0.370517	−	0.928826i	$-0.379180\pi$
$691$	19.4795	0.741035	0.370517	+	0.928826i	$0.379180\pi$
$692$	0	0
$693$	−27.0923	−1.02915
$694$	0	0
$695$	6.66370	0.252769
$696$	0	0
$697$	−27.0509	−1.02462
$698$	0	0
$699$	34.0098	1.28637
$700$	0	0
$701$	−37.0005	−1.39749	−0.698744	−	0.715371i	$-0.746257\pi$
$701$	−37.0005	−1.39749	−0.698744	+	0.715371i	$0.746257\pi$
$702$	0	0
$703$	2.50177	0.0943560
$704$	0	0
$705$	4.81579	0.181373
$706$	0	0
$707$	7.82516	0.294295
$708$	0	0
$709$	44.6133	1.67549	0.837744	−	0.546063i	$-0.183874\pi$
$709$	44.6133	1.67549	0.837744	+	0.546063i	$0.183874\pi$
$710$	0	0
$711$	−83.2311	−3.12141
$712$	0	0
$713$	−8.84437	−0.331224
$714$	0	0
$715$	3.43801	0.128574
$716$	0	0
$717$	7.34614	0.274347
$718$	0	0
$719$	−31.6543	−1.18051	−0.590254	−	0.807218i	$-0.700972\pi$
$719$	−31.6543	−1.18051	−0.590254	+	0.807218i	$0.700972\pi$
$720$	0	0
$721$	−15.9210	−0.592931
$722$	0	0
$723$	−13.8064	−0.513466
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−1.65878	−0.0615208	−0.0307604	−	0.999527i	$-0.509793\pi$
$727$	−1.65878	−0.0615208	−0.0307604	+	0.999527i	$0.509793\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	0	0
$731$	−3.86665	−0.143013
$732$	0	0
$733$	44.9358	1.65974	0.829871	−	0.557955i	$-0.188414\pi$
$733$	44.9358	1.65974	0.829871	+	0.557955i	$0.188414\pi$
$734$	0	0
$735$	17.9541	0.662246
$736$	0	0
$737$	61.5437	2.26699
$738$	0	0
$739$	43.9639	1.61724	0.808619	−	0.588332i	$-0.200215\pi$
$739$	43.9639	1.61724	0.808619	+	0.588332i	$0.200215\pi$
$740$	0	0
$741$	−1.98126	−0.0727836
$742$	0	0
$743$	7.58565	0.278291	0.139145	−	0.990272i	$-0.455565\pi$
$743$	7.58565	0.278291	0.139145	+	0.990272i	$0.455565\pi$
$744$	0	0
$745$	−17.2257	−0.631101
$746$	0	0
$747$	−43.1798	−1.57986
$748$	0	0
$749$	3.57136	0.130495
$750$	0	0
$751$	−48.5259	−1.77074	−0.885368	−	0.464891i	$-0.846093\pi$
$751$	−48.5259	−1.77074	−0.885368	+	0.464891i	$0.846093\pi$
$752$	0	0
$753$	19.6543	0.716244
$754$	0	0
$755$	5.93978	0.216171
$756$	0	0
$757$	−20.9032	−0.759740	−0.379870	−	0.925040i	$-0.624031\pi$
$757$	−20.9032	−0.759740	−0.379870	+	0.925040i	$0.624031\pi$
$758$	0	0
$759$	85.5308	3.10457
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	−2.54326	−0.0920721
$764$	0	0
$765$	19.1383	0.691946
$766$	0	0
$767$	−5.77784	−0.208626
$768$	0	0
$769$	15.5397	0.560377	0.280188	−	0.959945i	$-0.409603\pi$
$769$	15.5397	0.560377	0.280188	+	0.959945i	$0.409603\pi$
$770$	0	0
$771$	38.0098	1.36889
$772$	0	0
$773$	38.1891	1.37357	0.686784	−	0.726862i	$-0.259022\pi$
$773$	38.1891	1.37357	0.686784	+	0.726862i	$0.259022\pi$
$774$	0	0
$775$	−1.65878	−0.0595852
$776$	0	0
$777$	5.98126	0.214577
$778$	0	0
$779$	8.41573	0.301525
$780$	0	0
$781$	39.6958	1.42043
$782$	0	0
$783$	7.05086	0.251977
$784$	0	0
$785$	−15.5669	−0.555607
$786$	0	0
$787$	27.9541	0.996455	0.498227	−	0.867046i	$-0.333984\pi$
$787$	27.9541	0.996455	0.498227	+	0.867046i	$0.333984\pi$
$788$	0	0
$789$	29.4064	1.04689
$790$	0	0
$791$	13.2859	0.472393
$792$	0	0
$793$	6.83854	0.242844
$794$	0	0
$795$	7.05086	0.250068
$796$	0	0
$797$	2.50622	0.0887747	0.0443874	−	0.999014i	$-0.485866\pi$
$797$	2.50622	0.0887747	0.0443874	+	0.999014i	$0.485866\pi$
$798$	0	0
$799$	−5.84791	−0.206884
$800$	0	0
$801$	−90.4612	−3.19629
$802$	0	0
$803$	−66.0513	−2.33090
$804$	0	0
$805$	4.81579	0.169734
$806$	0	0
$807$	14.2766	0.502558
$808$	0	0
$809$	−44.0928	−1.55022	−0.775110	−	0.631826i	$-0.782306\pi$
$809$	−44.0928	−1.55022	−0.775110	+	0.631826i	$0.782306\pi$
$810$	0	0
$811$	−29.8894	−1.04956	−0.524779	−	0.851238i	$-0.675852\pi$
$811$	−29.8894	−1.04956	−0.524779	+	0.851238i	$0.675852\pi$
$812$	0	0
$813$	−71.3502	−2.50236
$814$	0	0
$815$	10.5161	0.368362
$816$	0	0
$817$	1.20294	0.0420857
$818$	0	0
$819$	−3.05086	−0.106606
$820$	0	0
$821$	−6.65386	−0.232221	−0.116111	−	0.993236i	$-0.537043\pi$
$821$	−6.65386	−0.232221	−0.116111	+	0.993236i	$0.537043\pi$
$822$	0	0
$823$	36.7926	1.28251	0.641255	−	0.767328i	$-0.278414\pi$
$823$	36.7926	1.28251	0.641255	+	0.767328i	$0.278414\pi$
$824$	0	0
$825$	16.0415	0.558493
$826$	0	0
$827$	−42.8657	−1.49059	−0.745294	−	0.666736i	$-0.767691\pi$
$827$	−42.8657	−1.49059	−0.745294	+	0.666736i	$0.767691\pi$
$828$	0	0
$829$	41.5339	1.44253	0.721266	−	0.692658i	$-0.243561\pi$
$829$	41.5339	1.44253	0.721266	+	0.692658i	$0.243561\pi$
$830$	0	0
$831$	71.9911	2.49734
$832$	0	0
$833$	−21.8020	−0.755394
$834$	0	0
$835$	12.6494	0.437751
$836$	0	0
$837$	11.6958	0.404267
$838$	0	0
$839$	20.5763	0.710372	0.355186	−	0.934796i	$-0.384418\pi$
$839$	20.5763	0.710372	0.355186	+	0.934796i	$0.384418\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	47.9911	1.65290
$844$	0	0
$845$	−12.6128	−0.433895
$846$	0	0
$847$	17.6400	0.606119
$848$	0	0
$849$	−73.2770	−2.51486
$850$	0	0
$851$	−12.1619	−0.416906
$852$	0	0
$853$	−57.1481	−1.95671	−0.978357	−	0.206923i	$-0.933655\pi$
$853$	−57.1481	−1.95671	−0.978357	+	0.206923i	$0.933655\pi$
$854$	0	0
$855$	−5.95407	−0.203625
$856$	0	0
$857$	−10.6035	−0.362208	−0.181104	−	0.983464i	$-0.557967\pi$
$857$	−10.6035	−0.362208	−0.181104	+	0.983464i	$0.557967\pi$
$858$	0	0
$859$	0.934855	0.0318968	0.0159484	−	0.999873i	$-0.494923\pi$
$859$	0.934855	0.0318968	0.0159484	+	0.999873i	$0.494923\pi$
$860$	0	0
$861$	20.1204	0.685703
$862$	0	0
$863$	18.7096	0.636883	0.318442	−	0.947942i	$-0.396840\pi$
$863$	18.7096	0.636883	0.318442	+	0.947942i	$0.396840\pi$
$864$	0	0
$865$	6.13335	0.208540
$866$	0	0
$867$	13.2716	0.450728
$868$	0	0
$869$	84.7150	2.87376
$870$	0	0
$871$	6.93041	0.234828
$872$	0	0
$873$	−64.8943	−2.19634
$874$	0	0
$875$	0.903212	0.0305341
$876$	0	0
$877$	21.5714	0.728413	0.364207	−	0.931318i	$-0.381340\pi$
$877$	21.5714	0.728413	0.364207	+	0.931318i	$0.381340\pi$
$878$	0	0
$879$	−25.0736	−0.845712
$880$	0	0
$881$	24.2636	0.817463	0.408731	−	0.912655i	$-0.365971\pi$
$881$	24.2636	0.817463	0.408731	+	0.912655i	$0.365971\pi$
$882$	0	0
$883$	−35.1798	−1.18389	−0.591947	−	0.805977i	$-0.701640\pi$
$883$	−35.1798	−1.18389	−0.591947	+	0.805977i	$0.701640\pi$
$884$	0	0
$885$	−26.9590	−0.906217
$886$	0	0
$887$	−46.4242	−1.55877	−0.779386	−	0.626544i	$-0.784469\pi$
$887$	−46.4242	−1.55877	−0.779386	+	0.626544i	$0.784469\pi$
$888$	0	0
$889$	−2.11462	−0.0709220
$890$	0	0
$891$	−23.1195	−0.774534
$892$	0	0
$893$	1.81933	0.0608816
$894$	0	0
$895$	15.2257	0.508939
$896$	0	0
$897$	9.63158	0.321589
$898$	0	0
$899$	1.65878	0.0553234
$900$	0	0
$901$	−8.56199	−0.285241
$902$	0	0
$903$	2.87601	0.0957078
$904$	0	0
$905$	3.93978	0.130963
$906$	0	0
$907$	6.34122	0.210557	0.105278	−	0.994443i	$-0.466427\pi$
$907$	6.34122	0.210557	0.105278	+	0.994443i	$0.466427\pi$
$908$	0	0
$909$	47.0321	1.55996
$910$	0	0
$911$	−39.9353	−1.32312	−0.661558	−	0.749894i	$-0.730105\pi$
$911$	−39.9353	−1.32312	−0.661558	+	0.749894i	$0.730105\pi$
$912$	0	0
$913$	43.9496	1.45452
$914$	0	0
$915$	31.9081	1.05485
$916$	0	0
$917$	−5.11108	−0.168783
$918$	0	0
$919$	0.815792	0.0269105	0.0134552	−	0.999909i	$-0.495717\pi$
$919$	0.815792	0.0269105	0.0134552	+	0.999909i	$0.495717\pi$
$920$	0	0
$921$	−42.4929	−1.40019
$922$	0	0
$923$	4.47013	0.147136
$924$	0	0
$925$	−2.28100	−0.0749987
$926$	0	0
$927$	−95.6914	−3.14292
$928$	0	0
$929$	11.4479	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$929$	11.4479	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$930$	0	0
$931$	6.78277	0.222296
$932$	0	0
$933$	4.81579	0.157662
$934$	0	0
$935$	−19.4795	−0.637048
$936$	0	0
$937$	14.4286	0.471363	0.235682	−	0.971830i	$-0.424268\pi$
$937$	14.4286	0.471363	0.235682	+	0.971830i	$0.424268\pi$
$938$	0	0
$939$	−53.2355	−1.73728
$940$	0	0
$941$	−33.8163	−1.10238	−0.551189	−	0.834380i	$-0.685826\pi$
$941$	−33.8163	−1.10238	−0.551189	+	0.834380i	$0.685826\pi$
$942$	0	0
$943$	−40.9117	−1.33227
$944$	0	0
$945$	−6.36842	−0.207165
$946$	0	0
$947$	49.0879	1.59514	0.797571	−	0.603225i	$-0.206118\pi$
$947$	49.0879	1.59514	0.797571	+	0.603225i	$0.206118\pi$
$948$	0	0
$949$	−7.43801	−0.241448
$950$	0	0
$951$	−65.5437	−2.12540
$952$	0	0
$953$	32.1847	1.04256	0.521282	−	0.853384i	$-0.325454\pi$
$953$	32.1847	1.04256	0.521282	+	0.853384i	$0.325454\pi$
$954$	0	0
$955$	−7.06515	−0.228623
$956$	0	0
$957$	−16.0415	−0.518548
$958$	0	0
$959$	8.69535	0.280787
$960$	0	0
$961$	−28.2484	−0.911240
$962$	0	0
$963$	21.4652	0.691707
$964$	0	0
$965$	1.98571	0.0639223
$966$	0	0
$967$	−30.1003	−0.967961	−0.483981	−	0.875079i	$-0.660809\pi$
$967$	−30.1003	−0.967961	−0.483981	+	0.875079i	$0.660809\pi$
$968$	0	0
$969$	11.2257	0.360621
$970$	0	0
$971$	−61.0593	−1.95949	−0.979743	−	0.200257i	$-0.935822\pi$
$971$	−61.0593	−1.95949	−0.979743	+	0.200257i	$0.935822\pi$
$972$	0	0
$973$	6.01874	0.192952
$974$	0	0
$975$	1.80642	0.0578519
$976$	0	0
$977$	−37.6829	−1.20558	−0.602792	−	0.797899i	$-0.705945\pi$
$977$	−37.6829	−1.20558	−0.602792	+	0.797899i	$0.705945\pi$
$978$	0	0
$979$	92.0741	2.94270
$980$	0	0
$981$	−15.2859	−0.488042
$982$	0	0
$983$	59.0977	1.88493	0.942463	−	0.334312i	$-0.108504\pi$
$983$	59.0977	1.88493	0.942463	+	0.334312i	$0.108504\pi$
$984$	0	0
$985$	−16.2766	−0.518614
$986$	0	0
$987$	4.34968	0.138452
$988$	0	0
$989$	−5.84791	−0.185953
$990$	0	0
$991$	−36.3654	−1.15518	−0.577592	−	0.816326i	$-0.696007\pi$
$991$	−36.3654	−1.15518	−0.577592	+	0.816326i	$0.696007\pi$
$992$	0	0
$993$	101.922	3.23440
$994$	0	0
$995$	22.5303	0.714260
$996$	0	0
$997$	28.3412	0.897575	0.448788	−	0.893638i	$-0.351856\pi$
$997$	28.3412	0.897575	0.448788	+	0.893638i	$0.351856\pi$
$998$	0	0
$999$	16.0830	0.508843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.2.a.f.1.1	✓	3
4.3	odd	2		2320.2.a.t.1.3		3
5.4	even	2		5800.2.a.s.1.3		3
8.3	odd	2		9280.2.a.bi.1.1		3
8.5	even	2		9280.2.a.bs.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.f.1.1	✓	3	1.1	even	1	trivial
2320.2.a.t.1.3		3	4.3	odd	2
5800.2.a.s.1.3		3	5.4	even	2
9280.2.a.bi.1.1		3	8.3	odd	2
9280.2.a.bs.1.3		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.90321 q^{3} +1.00000 q^{5} +0.903212 q^{7} +5.42864 q^{9} -5.52543 q^{11} -0.622216 q^{13} -2.90321 q^{15} +3.52543 q^{17} -1.09679 q^{19} -2.62222 q^{21} +5.33185 q^{23} +1.00000 q^{25} -7.05086 q^{27} -1.00000 q^{29} -1.65878 q^{31} +16.0415 q^{33} +0.903212 q^{35} -2.28100 q^{37} +1.80642 q^{39} -7.67307 q^{41} -1.09679 q^{43} +5.42864 q^{45} -1.65878 q^{47} -6.18421 q^{49} -10.2351 q^{51} -2.42864 q^{53} -5.52543 q^{55} +3.18421 q^{57} +9.28592 q^{59} -10.9906 q^{61} +4.90321 q^{63} -0.622216 q^{65} -11.1383 q^{67} -15.4795 q^{69} -7.18421 q^{71} +11.9541 q^{73} -2.90321 q^{75} -4.99063 q^{77} -15.3319 q^{79} +4.18421 q^{81} -7.95407 q^{83} +3.52543 q^{85} +2.90321 q^{87} -16.6637 q^{89} -0.561993 q^{91} +4.81579 q^{93} -1.09679 q^{95} -11.9541 q^{97} -29.9956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 10 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{21} - 4 q^{23} + 3 q^{25} - 8 q^{27} - 3 q^{29} + 2 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 10 q^{41}+ \cdots - 30 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 10 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 - 10 * q^19 - 8 * q^21 - 4 * q^23 + 3 * q^25 - 8 * q^27 - 3 * q^29 + 2 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 10 * q^41 - 10 * q^43 + 3 * q^45 + 2 * q^47 - 5 * q^49 - 4 * q^51 + 6 * q^53 - 10 * q^55 - 4 * q^57 - 12 * q^59 - 6 * q^61 + 8 * q^63 - 2 * q^65 - 20 * q^69 - 8 * q^71 + 16 * q^73 - 2 * q^75 + 12 * q^77 - 26 * q^79 - q^81 - 4 * q^83 + 4 * q^85 + 2 * q^87 - 10 * q^89 + 12 * q^91 + 28 * q^93 - 10 * q^95 - 16 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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