LMFDB - Embedded newform 1160.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1160,2,Mod(1,1160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,-3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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			1.3
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.70928 q^{3} -1.00000 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -3.70928 q^{11} -3.07838 q^{13} -1.70928 q^{15} -6.04945 q^{17} -1.36910 q^{19} -1.07838 q^{21} +1.70928 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -4.78765 q^{31} -6.34017 q^{33} +0.630898 q^{35} -2.63090 q^{37} -5.26180 q^{39} +8.34017 q^{41} +1.12783 q^{43} +0.0783777 q^{45} +10.3896 q^{47} -6.60197 q^{49} -10.3402 q^{51} +9.02052 q^{53} +3.70928 q^{55} -2.34017 q^{57} +5.75872 q^{59} -9.60197 q^{61} +0.0494483 q^{63} +3.07838 q^{65} -8.44748 q^{67} +2.92162 q^{69} -9.75872 q^{71} -4.29072 q^{73} +1.70928 q^{75} +2.34017 q^{77} +6.23287 q^{79} -8.75872 q^{81} -11.1545 q^{83} +6.04945 q^{85} +1.70928 q^{87} +2.58145 q^{89} +1.94214 q^{91} -8.18342 q^{93} +1.36910 q^{95} +13.6514 q^{97} +0.290725 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{19} - 2 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} - 4 q^{37} - 8 q^{39} + 14 q^{41} - 18 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^19 - 2 * q^23 + 3 * q^25 - 8 * q^27 + 3 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 - 4 * q^37 - 8 * q^39 + 14 * q^41 - 18 * q^43 - 3 * q^45 + 2 * q^47 - q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 4 * q^57 - 8 * q^59 - 10 * q^61 - 18 * q^63 + 6 * q^65 - 26 * q^67 + 12 * q^69 - 4 * q^71 - 20 * q^73 - 2 * q^75 - 4 * q^77 - 4 * q^79 - q^81 - 14 * q^83 - 2 * q^87 + 22 * q^89 - 24 * q^91 - 20 * q^93 + 8 * q^95 + 4 * q^97 + 8 * 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
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.630898	−0.238457	−0.119228	−	0.992867i	$-0.538042\pi$
$7$	−0.630898	−0.238457	−0.119228	+	0.992867i	$0.538042\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	−3.70928	−1.11839	−0.559194	−	0.829037i	$-0.688889\pi$
$11$	−3.70928	−1.11839	−0.559194	+	0.829037i	$0.688889\pi$
$12$	0	0
$13$	−3.07838	−0.853788	−0.426894	−	0.904302i	$-0.640392\pi$
$13$	−3.07838	−0.853788	−0.426894	+	0.904302i	$0.640392\pi$
$14$	0	0
$15$	−1.70928	−0.441333
$16$	0	0
$17$	−6.04945	−1.46721	−0.733603	−	0.679578i	$-0.762163\pi$
$17$	−6.04945	−1.46721	−0.733603	+	0.679578i	$0.762163\pi$
$18$	0	0
$19$	−1.36910	−0.314094	−0.157047	−	0.987591i	$-0.550197\pi$
$19$	−1.36910	−0.314094	−0.157047	+	0.987591i	$0.550197\pi$
$20$	0	0
$21$	−1.07838	−0.235321
$22$	0	0
$23$	1.70928	0.356409	0.178204	−	0.983994i	$-0.442971\pi$
$23$	1.70928	0.356409	0.178204	+	0.983994i	$0.442971\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.26180	−1.01263
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.78765	−0.859888	−0.429944	−	0.902856i	$-0.641467\pi$
$31$	−4.78765	−0.859888	−0.429944	+	0.902856i	$0.641467\pi$
$32$	0	0
$33$	−6.34017	−1.10368
$34$	0	0
$35$	0.630898	0.106641
$36$	0	0
$37$	−2.63090	−0.432517	−0.216258	−	0.976336i	$-0.569385\pi$
$37$	−2.63090	−0.432517	−0.216258	+	0.976336i	$0.569385\pi$
$38$	0	0
$39$	−5.26180	−0.842562
$40$	0	0
$41$	8.34017	1.30252	0.651258	−	0.758856i	$-0.274242\pi$
$41$	8.34017	1.30252	0.651258	+	0.758856i	$0.274242\pi$
$42$	0	0
$43$	1.12783	0.171992	0.0859959	−	0.996295i	$-0.472593\pi$
$43$	1.12783	0.171992	0.0859959	+	0.996295i	$0.472593\pi$
$44$	0	0
$45$	0.0783777	0.0116839
$46$	0	0
$47$	10.3896	1.51548	0.757741	−	0.652555i	$-0.226303\pi$
$47$	10.3896	1.51548	0.757741	+	0.652555i	$0.226303\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	−10.3402	−1.44791
$52$	0	0
$53$	9.02052	1.23906	0.619532	−	0.784972i	$-0.287322\pi$
$53$	9.02052	1.23906	0.619532	+	0.784972i	$0.287322\pi$
$54$	0	0
$55$	3.70928	0.500159
$56$	0	0
$57$	−2.34017	−0.309963
$58$	0	0
$59$	5.75872	0.749722	0.374861	−	0.927081i	$-0.377690\pi$
$59$	5.75872	0.749722	0.374861	+	0.927081i	$0.377690\pi$
$60$	0	0
$61$	−9.60197	−1.22941	−0.614703	−	0.788759i	$-0.710724\pi$
$61$	−9.60197	−1.22941	−0.614703	+	0.788759i	$0.710724\pi$
$62$	0	0
$63$	0.0494483	0.00622990
$64$	0	0
$65$	3.07838	0.381826
$66$	0	0
$67$	−8.44748	−1.03202	−0.516012	−	0.856581i	$-0.672584\pi$
$67$	−8.44748	−1.03202	−0.516012	+	0.856581i	$0.672584\pi$
$68$	0	0
$69$	2.92162	0.351722
$70$	0	0
$71$	−9.75872	−1.15815	−0.579074	−	0.815275i	$-0.696586\pi$
$71$	−9.75872	−1.15815	−0.579074	+	0.815275i	$0.696586\pi$
$72$	0	0
$73$	−4.29072	−0.502191	−0.251096	−	0.967962i	$-0.580791\pi$
$73$	−4.29072	−0.502191	−0.251096	+	0.967962i	$0.580791\pi$
$74$	0	0
$75$	1.70928	0.197370
$76$	0	0
$77$	2.34017	0.266687
$78$	0	0
$79$	6.23287	0.701252	0.350626	−	0.936516i	$-0.385969\pi$
$79$	6.23287	0.701252	0.350626	+	0.936516i	$0.385969\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	−11.1545	−1.22436	−0.612182	−	0.790717i	$-0.709708\pi$
$83$	−11.1545	−1.22436	−0.612182	+	0.790717i	$0.709708\pi$
$84$	0	0
$85$	6.04945	0.656155
$86$	0	0
$87$	1.70928	0.183254
$88$	0	0
$89$	2.58145	0.273633	0.136817	−	0.990596i	$-0.456313\pi$
$89$	2.58145	0.273633	0.136817	+	0.990596i	$0.456313\pi$
$90$	0	0
$91$	1.94214	0.203592
$92$	0	0
$93$	−8.18342	−0.848581
$94$	0	0
$95$	1.36910	0.140467
$96$	0	0
$97$	13.6514	1.38609	0.693046	−	0.720894i	$-0.256268\pi$
$97$	13.6514	1.38609	0.693046	+	0.720894i	$0.256268\pi$
$98$	0	0
$99$	0.290725	0.0292189
$100$	0	0
$101$	14.0989	1.40289	0.701446	−	0.712722i	$-0.252538\pi$
$101$	14.0989	1.40289	0.701446	+	0.712722i	$0.252538\pi$
$102$	0	0
$103$	−0.447480	−0.0440915	−0.0220458	−	0.999757i	$-0.507018\pi$
$103$	−0.447480	−0.0440915	−0.0220458	+	0.999757i	$0.507018\pi$
$104$	0	0
$105$	1.07838	0.105239
$106$	0	0
$107$	−16.0494	−1.55156	−0.775779	−	0.631004i	$-0.782643\pi$
$107$	−16.0494	−1.55156	−0.775779	+	0.631004i	$0.782643\pi$
$108$	0	0
$109$	20.4391	1.95771	0.978854	−	0.204561i	$-0.0655765\pi$
$109$	20.4391	1.95771	0.978854	+	0.204561i	$0.0655765\pi$
$110$	0	0
$111$	−4.49693	−0.426830
$112$	0	0
$113$	5.86603	0.551830	0.275915	−	0.961182i	$-0.411019\pi$
$113$	5.86603	0.551830	0.275915	+	0.961182i	$0.411019\pi$
$114$	0	0
$115$	−1.70928	−0.159391
$116$	0	0
$117$	0.241276	0.0223060
$118$	0	0
$119$	3.81658	0.349866
$120$	0	0
$121$	2.75872	0.250793
$122$	0	0
$123$	14.2557	1.28539
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.70928	−0.506616	−0.253308	−	0.967386i	$-0.581519\pi$
$127$	−5.70928	−0.506616	−0.253308	+	0.967386i	$0.581519\pi$
$128$	0	0
$129$	1.92777	0.169730
$130$	0	0
$131$	−18.1483	−1.58563	−0.792814	−	0.609463i	$-0.791385\pi$
$131$	−18.1483	−1.58563	−0.792814	+	0.609463i	$0.791385\pi$
$132$	0	0
$133$	0.863763	0.0748978
$134$	0	0
$135$	5.26180	0.452863
$136$	0	0
$137$	−12.8865	−1.10097	−0.550486	−	0.834844i	$-0.685558\pi$
$137$	−12.8865	−1.10097	−0.550486	+	0.834844i	$0.685558\pi$
$138$	0	0
$139$	−9.94214	−0.843281	−0.421641	−	0.906763i	$-0.638546\pi$
$139$	−9.94214	−0.843281	−0.421641	+	0.906763i	$0.638546\pi$
$140$	0	0
$141$	17.7587	1.49555
$142$	0	0
$143$	11.4186	0.954867
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−11.2846	−0.930737
$148$	0	0
$149$	10.3135	0.844916	0.422458	−	0.906383i	$-0.361167\pi$
$149$	10.3135	0.844916	0.422458	+	0.906383i	$0.361167\pi$
$150$	0	0
$151$	−24.2823	−1.97607	−0.988033	−	0.154240i	$-0.950707\pi$
$151$	−24.2823	−1.97607	−0.988033	+	0.154240i	$0.950707\pi$
$152$	0	0
$153$	0.474142	0.0383321
$154$	0	0
$155$	4.78765	0.384554
$156$	0	0
$157$	1.76713	0.141033	0.0705163	−	0.997511i	$-0.477535\pi$
$157$	1.76713	0.141033	0.0705163	+	0.997511i	$0.477535\pi$
$158$	0	0
$159$	15.4186	1.22277
$160$	0	0
$161$	−1.07838	−0.0849881
$162$	0	0
$163$	−16.9132	−1.32474	−0.662372	−	0.749175i	$-0.730450\pi$
$163$	−16.9132	−1.32474	−0.662372	+	0.749175i	$0.730450\pi$
$164$	0	0
$165$	6.34017	0.493582
$166$	0	0
$167$	6.97107	0.539438	0.269719	−	0.962939i	$-0.413069\pi$
$167$	6.97107	0.539438	0.269719	+	0.962939i	$0.413069\pi$
$168$	0	0
$169$	−3.52359	−0.271045
$170$	0	0
$171$	0.107307	0.00820598
$172$	0	0
$173$	5.23513	0.398020	0.199010	−	0.979997i	$-0.436227\pi$
$173$	5.23513	0.398020	0.199010	+	0.979997i	$0.436227\pi$
$174$	0	0
$175$	−0.630898	−0.0476914
$176$	0	0
$177$	9.84324	0.739864
$178$	0	0
$179$	2.52359	0.188622	0.0943110	−	0.995543i	$-0.469935\pi$
$179$	2.52359	0.188622	0.0943110	+	0.995543i	$0.469935\pi$
$180$	0	0
$181$	−2.76487	−0.205511	−0.102755	−	0.994707i	$-0.532766\pi$
$181$	−2.76487	−0.205511	−0.102755	+	0.994707i	$0.532766\pi$
$182$	0	0
$183$	−16.4124	−1.21324
$184$	0	0
$185$	2.63090	0.193427
$186$	0	0
$187$	22.4391	1.64091
$188$	0	0
$189$	3.31965	0.241469
$190$	0	0
$191$	−2.81432	−0.203637	−0.101818	−	0.994803i	$-0.532466\pi$
$191$	−2.81432	−0.203637	−0.101818	+	0.994803i	$0.532466\pi$
$192$	0	0
$193$	−9.95055	−0.716256	−0.358128	−	0.933672i	$-0.616585\pi$
$193$	−9.95055	−0.716256	−0.358128	+	0.933672i	$0.616585\pi$
$194$	0	0
$195$	5.26180	0.376805
$196$	0	0
$197$	−9.73206	−0.693381	−0.346690	−	0.937980i	$-0.612695\pi$
$197$	−9.73206	−0.693381	−0.346690	+	0.937980i	$0.612695\pi$
$198$	0	0
$199$	19.6020	1.38955	0.694773	−	0.719229i	$-0.255505\pi$
$199$	19.6020	1.38955	0.694773	+	0.719229i	$0.255505\pi$
$200$	0	0
$201$	−14.4391	−1.01845
$202$	0	0
$203$	−0.630898	−0.0442803
$204$	0	0
$205$	−8.34017	−0.582503
$206$	0	0
$207$	−0.133969	−0.00931150
$208$	0	0
$209$	5.07838	0.351279
$210$	0	0
$211$	−2.76099	−0.190074	−0.0950372	−	0.995474i	$-0.530297\pi$
$211$	−2.76099	−0.190074	−0.0950372	+	0.995474i	$0.530297\pi$
$212$	0	0
$213$	−16.6803	−1.14292
$214$	0	0
$215$	−1.12783	−0.0769171
$216$	0	0
$217$	3.02052	0.205046
$218$	0	0
$219$	−7.33403	−0.495588
$220$	0	0
$221$	18.6225	1.25268
$222$	0	0
$223$	−11.2846	−0.755671	−0.377836	−	0.925873i	$-0.623332\pi$
$223$	−11.2846	−0.755671	−0.377836	+	0.925873i	$0.623332\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	0	0
$227$	−4.76099	−0.315998	−0.157999	−	0.987439i	$-0.550504\pi$
$227$	−4.76099	−0.315998	−0.157999	+	0.987439i	$0.550504\pi$
$228$	0	0
$229$	2.31351	0.152881	0.0764406	−	0.997074i	$-0.475644\pi$
$229$	2.31351	0.152881	0.0764406	+	0.997074i	$0.475644\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	2.68035	0.175595	0.0877977	−	0.996138i	$-0.472017\pi$
$233$	2.68035	0.175595	0.0877977	+	0.996138i	$0.472017\pi$
$234$	0	0
$235$	−10.3896	−0.677744
$236$	0	0
$237$	10.6537	0.692031
$238$	0	0
$239$	22.0722	1.42773	0.713867	−	0.700281i	$-0.246942\pi$
$239$	22.0722	1.42773	0.713867	+	0.700281i	$0.246942\pi$
$240$	0	0
$241$	14.9939	0.965839	0.482920	−	0.875665i	$-0.339576\pi$
$241$	14.9939	0.965839	0.482920	+	0.875665i	$0.339576\pi$
$242$	0	0
$243$	0.814315	0.0522383
$244$	0	0
$245$	6.60197	0.421784
$246$	0	0
$247$	4.21461	0.268169
$248$	0	0
$249$	−19.0661	−1.20826
$250$	0	0
$251$	2.81432	0.177638	0.0888190	−	0.996048i	$-0.471691\pi$
$251$	2.81432	0.177638	0.0888190	+	0.996048i	$0.471691\pi$
$252$	0	0
$253$	−6.34017	−0.398603
$254$	0	0
$255$	10.3402	0.647527
$256$	0	0
$257$	−5.91548	−0.368997	−0.184499	−	0.982833i	$-0.559066\pi$
$257$	−5.91548	−0.368997	−0.184499	+	0.982833i	$0.559066\pi$
$258$	0	0
$259$	1.65983	0.103137
$260$	0	0
$261$	−0.0783777	−0.00485146
$262$	0	0
$263$	−11.9649	−0.737789	−0.368895	−	0.929471i	$-0.620264\pi$
$263$	−11.9649	−0.737789	−0.368895	+	0.929471i	$0.620264\pi$
$264$	0	0
$265$	−9.02052	−0.554126
$266$	0	0
$267$	4.41241	0.270035
$268$	0	0
$269$	14.4969	0.883893	0.441947	−	0.897041i	$-0.354288\pi$
$269$	14.4969	0.883893	0.441947	+	0.897041i	$0.354288\pi$
$270$	0	0
$271$	5.28458	0.321015	0.160508	−	0.987035i	$-0.448687\pi$
$271$	5.28458	0.321015	0.160508	+	0.987035i	$0.448687\pi$
$272$	0	0
$273$	3.31965	0.200915
$274$	0	0
$275$	−3.70928	−0.223678
$276$	0	0
$277$	−25.3340	−1.52217	−0.761087	−	0.648650i	$-0.775334\pi$
$277$	−25.3340	−1.52217	−0.761087	+	0.648650i	$0.775334\pi$
$278$	0	0
$279$	0.375245	0.0224654
$280$	0	0
$281$	26.2290	1.56469	0.782345	−	0.622845i	$-0.214023\pi$
$281$	26.2290	1.56469	0.782345	+	0.622845i	$0.214023\pi$
$282$	0	0
$283$	2.97107	0.176612	0.0883059	−	0.996093i	$-0.471855\pi$
$283$	2.97107	0.176612	0.0883059	+	0.996093i	$0.471855\pi$
$284$	0	0
$285$	2.34017	0.138620
$286$	0	0
$287$	−5.26180	−0.310594
$288$	0	0
$289$	19.5958	1.15270
$290$	0	0
$291$	23.3340	1.36787
$292$	0	0
$293$	−15.9071	−0.929301	−0.464650	−	0.885494i	$-0.653820\pi$
$293$	−15.9071	−0.929301	−0.464650	+	0.885494i	$0.653820\pi$
$294$	0	0
$295$	−5.75872	−0.335286
$296$	0	0
$297$	19.5174	1.13252
$298$	0	0
$299$	−5.26180	−0.304297
$300$	0	0
$301$	−0.711543	−0.0410126
$302$	0	0
$303$	24.0989	1.38445
$304$	0	0
$305$	9.60197	0.549807
$306$	0	0
$307$	9.44134	0.538846	0.269423	−	0.963022i	$-0.413167\pi$
$307$	9.44134	0.538846	0.269423	+	0.963022i	$0.413167\pi$
$308$	0	0
$309$	−0.764867	−0.0435117
$310$	0	0
$311$	11.2123	0.635794	0.317897	−	0.948125i	$-0.397023\pi$
$311$	11.2123	0.635794	0.317897	+	0.948125i	$0.397023\pi$
$312$	0	0
$313$	4.12556	0.233190	0.116595	−	0.993180i	$-0.462802\pi$
$313$	4.12556	0.233190	0.116595	+	0.993180i	$0.462802\pi$
$314$	0	0
$315$	−0.0494483	−0.00278610
$316$	0	0
$317$	14.3630	0.806704	0.403352	−	0.915045i	$-0.367845\pi$
$317$	14.3630	0.806704	0.403352	+	0.915045i	$0.367845\pi$
$318$	0	0
$319$	−3.70928	−0.207680
$320$	0	0
$321$	−27.4329	−1.53116
$322$	0	0
$323$	8.28231	0.460840
$324$	0	0
$325$	−3.07838	−0.170758
$326$	0	0
$327$	34.9360	1.93197
$328$	0	0
$329$	−6.55479	−0.361377
$330$	0	0
$331$	−5.95055	−0.327072	−0.163536	−	0.986537i	$-0.552290\pi$
$331$	−5.95055	−0.327072	−0.163536	+	0.986537i	$0.552290\pi$
$332$	0	0
$333$	0.206204	0.0112999
$334$	0	0
$335$	8.44748	0.461535
$336$	0	0
$337$	21.2846	1.15945	0.579723	−	0.814814i	$-0.303161\pi$
$337$	21.2846	1.15945	0.579723	+	0.814814i	$0.303161\pi$
$338$	0	0
$339$	10.0267	0.544574
$340$	0	0
$341$	17.7587	0.961689
$342$	0	0
$343$	8.58145	0.463355
$344$	0	0
$345$	−2.92162	−0.157295
$346$	0	0
$347$	−17.2123	−0.924007	−0.462004	−	0.886878i	$-0.652869\pi$
$347$	−17.2123	−0.924007	−0.462004	+	0.886878i	$0.652869\pi$
$348$	0	0
$349$	3.47641	0.186088	0.0930440	−	0.995662i	$-0.470340\pi$
$349$	3.47641	0.186088	0.0930440	+	0.995662i	$0.470340\pi$
$350$	0	0
$351$	16.1978	0.864574
$352$	0	0
$353$	−19.2306	−1.02354	−0.511771	−	0.859122i	$-0.671010\pi$
$353$	−19.2306	−1.02354	−0.511771	+	0.859122i	$0.671010\pi$
$354$	0	0
$355$	9.75872	0.517939
$356$	0	0
$357$	6.52359	0.345265
$358$	0	0
$359$	−7.79380	−0.411341	−0.205670	−	0.978621i	$-0.565937\pi$
$359$	−7.79380	−0.411341	−0.205670	+	0.978621i	$0.565937\pi$
$360$	0	0
$361$	−17.1256	−0.901345
$362$	0	0
$363$	4.71542	0.247495
$364$	0	0
$365$	4.29072	0.224587
$366$	0	0
$367$	−4.81432	−0.251305	−0.125653	−	0.992074i	$-0.540102\pi$
$367$	−4.81432	−0.251305	−0.125653	+	0.992074i	$0.540102\pi$
$368$	0	0
$369$	−0.653684	−0.0340294
$370$	0	0
$371$	−5.69102	−0.295463
$372$	0	0
$373$	−32.9893	−1.70812	−0.854061	−	0.520173i	$-0.825868\pi$
$373$	−32.9893	−1.70812	−0.854061	+	0.520173i	$0.825868\pi$
$374$	0	0
$375$	−1.70928	−0.0882666
$376$	0	0
$377$	−3.07838	−0.158545
$378$	0	0
$379$	2.76099	0.141823	0.0709113	−	0.997483i	$-0.477409\pi$
$379$	2.76099	0.141823	0.0709113	+	0.997483i	$0.477409\pi$
$380$	0	0
$381$	−9.75872	−0.499955
$382$	0	0
$383$	27.6658	1.41366	0.706828	−	0.707385i	$-0.250125\pi$
$383$	27.6658	1.41366	0.706828	+	0.707385i	$0.250125\pi$
$384$	0	0
$385$	−2.34017	−0.119266
$386$	0	0
$387$	−0.0883965	−0.00449344
$388$	0	0
$389$	31.8576	1.61525	0.807623	−	0.589700i	$-0.200754\pi$
$389$	31.8576	1.61525	0.807623	+	0.589700i	$0.200754\pi$
$390$	0	0
$391$	−10.3402	−0.522925
$392$	0	0
$393$	−31.0205	−1.56478
$394$	0	0
$395$	−6.23287	−0.313610
$396$	0	0
$397$	−22.7792	−1.14326	−0.571629	−	0.820512i	$-0.693688\pi$
$397$	−22.7792	−1.14326	−0.571629	+	0.820512i	$0.693688\pi$
$398$	0	0
$399$	1.47641	0.0739129
$400$	0	0
$401$	30.9093	1.54354	0.771769	−	0.635903i	$-0.219372\pi$
$401$	30.9093	1.54354	0.771769	+	0.635903i	$0.219372\pi$
$402$	0	0
$403$	14.7382	0.734162
$404$	0	0
$405$	8.75872	0.435224
$406$	0	0
$407$	9.75872	0.483722
$408$	0	0
$409$	−4.07223	−0.201359	−0.100680	−	0.994919i	$-0.532102\pi$
$409$	−4.07223	−0.201359	−0.100680	+	0.994919i	$0.532102\pi$
$410$	0	0
$411$	−22.0267	−1.08650
$412$	0	0
$413$	−3.63317	−0.178776
$414$	0	0
$415$	11.1545	0.547552
$416$	0	0
$417$	−16.9939	−0.832193
$418$	0	0
$419$	7.50307	0.366549	0.183275	−	0.983062i	$-0.441330\pi$
$419$	7.50307	0.366549	0.183275	+	0.983062i	$0.441330\pi$
$420$	0	0
$421$	−9.91548	−0.483251	−0.241625	−	0.970370i	$-0.577681\pi$
$421$	−9.91548	−0.483251	−0.241625	+	0.970370i	$0.577681\pi$
$422$	0	0
$423$	−0.814315	−0.0395934
$424$	0	0
$425$	−6.04945	−0.293441
$426$	0	0
$427$	6.05786	0.293160
$428$	0	0
$429$	19.5174	0.942311
$430$	0	0
$431$	0.680346	0.0327711	0.0163856	−	0.999866i	$-0.494784\pi$
$431$	0.680346	0.0327711	0.0163856	+	0.999866i	$0.494784\pi$
$432$	0	0
$433$	−25.6826	−1.23423	−0.617114	−	0.786874i	$-0.711698\pi$
$433$	−25.6826	−1.23423	−0.617114	+	0.786874i	$0.711698\pi$
$434$	0	0
$435$	−1.70928	−0.0819535
$436$	0	0
$437$	−2.34017	−0.111946
$438$	0	0
$439$	5.49079	0.262061	0.131030	−	0.991378i	$-0.458171\pi$
$439$	5.49079	0.262061	0.131030	+	0.991378i	$0.458171\pi$
$440$	0	0
$441$	0.517447	0.0246404
$442$	0	0
$443$	−23.9649	−1.13861	−0.569304	−	0.822127i	$-0.692787\pi$
$443$	−23.9649	−1.13861	−0.569304	+	0.822127i	$0.692787\pi$
$444$	0	0
$445$	−2.58145	−0.122372
$446$	0	0
$447$	17.6286	0.833806
$448$	0	0
$449$	41.4329	1.95534	0.977670	−	0.210144i	$-0.0673934\pi$
$449$	41.4329	1.95534	0.977670	+	0.210144i	$0.0673934\pi$
$450$	0	0
$451$	−30.9360	−1.45672
$452$	0	0
$453$	−41.5052	−1.95008
$454$	0	0
$455$	−1.94214	−0.0910490
$456$	0	0
$457$	−0.142380	−0.00666024	−0.00333012	−	0.999994i	$-0.501060\pi$
$457$	−0.142380	−0.00666024	−0.00333012	+	0.999994i	$0.501060\pi$
$458$	0	0
$459$	31.8310	1.48574
$460$	0	0
$461$	−16.5692	−0.771703	−0.385851	−	0.922561i	$-0.626092\pi$
$461$	−16.5692	−0.771703	−0.385851	+	0.922561i	$0.626092\pi$
$462$	0	0
$463$	18.0228	0.837590	0.418795	−	0.908081i	$-0.362453\pi$
$463$	18.0228	0.837590	0.418795	+	0.908081i	$0.362453\pi$
$464$	0	0
$465$	8.18342	0.379497
$466$	0	0
$467$	−33.2267	−1.53755	−0.768775	−	0.639520i	$-0.779133\pi$
$467$	−33.2267	−1.53755	−0.768775	+	0.639520i	$0.779133\pi$
$468$	0	0
$469$	5.32950	0.246093
$470$	0	0
$471$	3.02052	0.139178
$472$	0	0
$473$	−4.18342	−0.192354
$474$	0	0
$475$	−1.36910	−0.0628187
$476$	0	0
$477$	−0.707008	−0.0323717
$478$	0	0
$479$	−22.7442	−1.03921	−0.519604	−	0.854407i	$-0.673920\pi$
$479$	−22.7442	−1.03921	−0.519604	+	0.854407i	$0.673920\pi$
$480$	0	0
$481$	8.09890	0.369278
$482$	0	0
$483$	−1.84324	−0.0838705
$484$	0	0
$485$	−13.6514	−0.619879
$486$	0	0
$487$	−2.30510	−0.104454	−0.0522270	−	0.998635i	$-0.516632\pi$
$487$	−2.30510	−0.104454	−0.0522270	+	0.998635i	$0.516632\pi$
$488$	0	0
$489$	−28.9093	−1.30733
$490$	0	0
$491$	18.7298	0.845264	0.422632	−	0.906301i	$-0.361106\pi$
$491$	18.7298	0.845264	0.422632	+	0.906301i	$0.361106\pi$
$492$	0	0
$493$	−6.04945	−0.272453
$494$	0	0
$495$	−0.290725	−0.0130671
$496$	0	0
$497$	6.15676	0.276168
$498$	0	0
$499$	16.8781	0.755569	0.377785	−	0.925893i	$-0.376686\pi$
$499$	16.8781	0.755569	0.377785	+	0.925893i	$0.376686\pi$
$500$	0	0
$501$	11.9155	0.532344
$502$	0	0
$503$	−16.2784	−0.725820	−0.362910	−	0.931824i	$-0.618217\pi$
$503$	−16.2784	−0.725820	−0.362910	+	0.931824i	$0.618217\pi$
$504$	0	0
$505$	−14.0989	−0.627393
$506$	0	0
$507$	−6.02279	−0.267481
$508$	0	0
$509$	−38.2290	−1.69447	−0.847235	−	0.531218i	$-0.821734\pi$
$509$	−38.2290	−1.69447	−0.847235	+	0.531218i	$0.821734\pi$
$510$	0	0
$511$	2.70701	0.119751
$512$	0	0
$513$	7.20394	0.318062
$514$	0	0
$515$	0.447480	0.0197183
$516$	0	0
$517$	−38.5380	−1.69490
$518$	0	0
$519$	8.94828	0.392786
$520$	0	0
$521$	26.4801	1.16011	0.580057	−	0.814576i	$-0.303030\pi$
$521$	26.4801	1.16011	0.580057	+	0.814576i	$0.303030\pi$
$522$	0	0
$523$	35.4824	1.55154	0.775768	−	0.631018i	$-0.217363\pi$
$523$	35.4824	1.55154	0.775768	+	0.631018i	$0.217363\pi$
$524$	0	0
$525$	−1.07838	−0.0470643
$526$	0	0
$527$	28.9627	1.26163
$528$	0	0
$529$	−20.0784	−0.872973
$530$	0	0
$531$	−0.451356	−0.0195872
$532$	0	0
$533$	−25.6742	−1.11207
$534$	0	0
$535$	16.0494	0.693878
$536$	0	0
$537$	4.31351	0.186142
$538$	0	0
$539$	24.4885	1.05480
$540$	0	0
$541$	−29.1050	−1.25132	−0.625662	−	0.780095i	$-0.715171\pi$
$541$	−29.1050	−1.25132	−0.625662	+	0.780095i	$0.715171\pi$
$542$	0	0
$543$	−4.72592	−0.202809
$544$	0	0
$545$	−20.4391	−0.875514
$546$	0	0
$547$	−43.5136	−1.86051	−0.930253	−	0.366919i	$-0.880413\pi$
$547$	−43.5136	−1.86051	−0.930253	+	0.366919i	$0.880413\pi$
$548$	0	0
$549$	0.752581	0.0321194
$550$	0	0
$551$	−1.36910	−0.0583257
$552$	0	0
$553$	−3.93230	−0.167218
$554$	0	0
$555$	4.49693	0.190884
$556$	0	0
$557$	−41.0349	−1.73870	−0.869352	−	0.494193i	$-0.835464\pi$
$557$	−41.0349	−1.73870	−0.869352	+	0.494193i	$0.835464\pi$
$558$	0	0
$559$	−3.47187	−0.146845
$560$	0	0
$561$	38.3545	1.61933
$562$	0	0
$563$	−26.5874	−1.12053	−0.560263	−	0.828315i	$-0.689300\pi$
$563$	−26.5874	−1.12053	−0.560263	+	0.828315i	$0.689300\pi$
$564$	0	0
$565$	−5.86603	−0.246786
$566$	0	0
$567$	5.52586	0.232064
$568$	0	0
$569$	−23.6430	−0.991166	−0.495583	−	0.868560i	$-0.665046\pi$
$569$	−23.6430	−0.991166	−0.495583	+	0.868560i	$0.665046\pi$
$570$	0	0
$571$	−42.3545	−1.77248	−0.886241	−	0.463224i	$-0.846693\pi$
$571$	−42.3545	−1.77248	−0.886241	+	0.463224i	$0.846693\pi$
$572$	0	0
$573$	−4.81044	−0.200959
$574$	0	0
$575$	1.70928	0.0712817
$576$	0	0
$577$	13.2534	0.551746	0.275873	−	0.961194i	$-0.411033\pi$
$577$	13.2534	0.551746	0.275873	+	0.961194i	$0.411033\pi$
$578$	0	0
$579$	−17.0082	−0.706838
$580$	0	0
$581$	7.03734	0.291958
$582$	0	0
$583$	−33.4596	−1.38575
$584$	0	0
$585$	−0.241276	−0.00997555
$586$	0	0
$587$	−10.2907	−0.424744	−0.212372	−	0.977189i	$-0.568119\pi$
$587$	−10.2907	−0.424744	−0.212372	+	0.977189i	$0.568119\pi$
$588$	0	0
$589$	6.55479	0.270085
$590$	0	0
$591$	−16.6348	−0.684263
$592$	0	0
$593$	26.6947	1.09622	0.548110	−	0.836406i	$-0.315347\pi$
$593$	26.6947	1.09622	0.548110	+	0.836406i	$0.315347\pi$
$594$	0	0
$595$	−3.81658	−0.156465
$596$	0	0
$597$	33.5052	1.37127
$598$	0	0
$599$	−27.4413	−1.12122	−0.560611	−	0.828079i	$-0.689434\pi$
$599$	−27.4413	−1.12122	−0.560611	+	0.828079i	$0.689434\pi$
$600$	0	0
$601$	−8.22446	−0.335483	−0.167741	−	0.985831i	$-0.553647\pi$
$601$	−8.22446	−0.335483	−0.167741	+	0.985831i	$0.553647\pi$
$602$	0	0
$603$	0.662094	0.0269626
$604$	0	0
$605$	−2.75872	−0.112158
$606$	0	0
$607$	−27.4824	−1.11548	−0.557738	−	0.830017i	$-0.688330\pi$
$607$	−27.4824	−1.11548	−0.557738	+	0.830017i	$0.688330\pi$
$608$	0	0
$609$	−1.07838	−0.0436981
$610$	0	0
$611$	−31.9832	−1.29390
$612$	0	0
$613$	−42.1445	−1.70220	−0.851100	−	0.525004i	$-0.824064\pi$
$613$	−42.1445	−1.70220	−0.851100	+	0.525004i	$0.824064\pi$
$614$	0	0
$615$	−14.2557	−0.574843
$616$	0	0
$617$	30.6453	1.23373	0.616866	−	0.787068i	$-0.288402\pi$
$617$	30.6453	1.23373	0.616866	+	0.787068i	$0.288402\pi$
$618$	0	0
$619$	−4.29072	−0.172459	−0.0862294	−	0.996275i	$-0.527482\pi$
$619$	−4.29072	−0.172459	−0.0862294	+	0.996275i	$0.527482\pi$
$620$	0	0
$621$	−8.99386	−0.360911
$622$	0	0
$623$	−1.62863	−0.0652497
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.68035	0.346660
$628$	0	0
$629$	15.9155	0.634592
$630$	0	0
$631$	29.4596	1.17277	0.586384	−	0.810033i	$-0.300551\pi$
$631$	29.4596	1.17277	0.586384	+	0.810033i	$0.300551\pi$
$632$	0	0
$633$	−4.71929	−0.187575
$634$	0	0
$635$	5.70928	0.226566
$636$	0	0
$637$	20.3234	0.805241
$638$	0	0
$639$	0.764867	0.0302577
$640$	0	0
$641$	8.14238	0.321605	0.160802	−	0.986987i	$-0.448592\pi$
$641$	8.14238	0.321605	0.160802	+	0.986987i	$0.448592\pi$
$642$	0	0
$643$	14.6576	0.578038	0.289019	−	0.957323i	$-0.406671\pi$
$643$	14.6576	0.578038	0.289019	+	0.957323i	$0.406671\pi$
$644$	0	0
$645$	−1.92777	−0.0759057
$646$	0	0
$647$	−43.2990	−1.70226	−0.851129	−	0.524957i	$-0.824081\pi$
$647$	−43.2990	−1.70226	−0.851129	+	0.524957i	$0.824081\pi$
$648$	0	0
$649$	−21.3607	−0.838481
$650$	0	0
$651$	5.16290	0.202350
$652$	0	0
$653$	−7.75485	−0.303471	−0.151735	−	0.988421i	$-0.548486\pi$
$653$	−7.75485	−0.303471	−0.151735	+	0.988421i	$0.548486\pi$
$654$	0	0
$655$	18.1483	0.709115
$656$	0	0
$657$	0.336297	0.0131202
$658$	0	0
$659$	7.25792	0.282728	0.141364	−	0.989958i	$-0.454851\pi$
$659$	7.25792	0.282728	0.141364	+	0.989958i	$0.454851\pi$
$660$	0	0
$661$	−28.3234	−1.10165	−0.550825	−	0.834621i	$-0.685687\pi$
$661$	−28.3234	−1.10165	−0.550825	+	0.834621i	$0.685687\pi$
$662$	0	0
$663$	31.8310	1.23621
$664$	0	0
$665$	−0.863763	−0.0334953
$666$	0	0
$667$	1.70928	0.0661834
$668$	0	0
$669$	−19.2885	−0.745735
$670$	0	0
$671$	35.6163	1.37495
$672$	0	0
$673$	−10.2679	−0.395800	−0.197900	−	0.980222i	$-0.563412\pi$
$673$	−10.2679	−0.395800	−0.197900	+	0.980222i	$0.563412\pi$
$674$	0	0
$675$	−5.26180	−0.202527
$676$	0	0
$677$	7.12783	0.273945	0.136972	−	0.990575i	$-0.456263\pi$
$677$	7.12783	0.273945	0.136972	+	0.990575i	$0.456263\pi$
$678$	0	0
$679$	−8.61265	−0.330523
$680$	0	0
$681$	−8.13784	−0.311843
$682$	0	0
$683$	−43.3835	−1.66002	−0.830011	−	0.557747i	$-0.811666\pi$
$683$	−43.3835	−1.66002	−0.830011	+	0.557747i	$0.811666\pi$
$684$	0	0
$685$	12.8865	0.492370
$686$	0	0
$687$	3.95443	0.150871
$688$	0	0
$689$	−27.7686	−1.05790
$690$	0	0
$691$	4.81044	0.182998	0.0914989	−	0.995805i	$-0.470834\pi$
$691$	4.81044	0.182998	0.0914989	+	0.995805i	$0.470834\pi$
$692$	0	0
$693$	−0.183417	−0.00696745
$694$	0	0
$695$	9.94214	0.377127
$696$	0	0
$697$	−50.4534	−1.91106
$698$	0	0
$699$	4.58145	0.173286
$700$	0	0
$701$	−18.2290	−0.688499	−0.344250	−	0.938878i	$-0.611867\pi$
$701$	−18.2290	−0.688499	−0.344250	+	0.938878i	$0.611867\pi$
$702$	0	0
$703$	3.60197	0.135851
$704$	0	0
$705$	−17.7587	−0.668832
$706$	0	0
$707$	−8.89496	−0.334529
$708$	0	0
$709$	−9.54864	−0.358607	−0.179303	−	0.983794i	$-0.557384\pi$
$709$	−9.54864	−0.358607	−0.179303	+	0.983794i	$0.557384\pi$
$710$	0	0
$711$	−0.488518	−0.0183209
$712$	0	0
$713$	−8.18342	−0.306471
$714$	0	0
$715$	−11.4186	−0.427030
$716$	0	0
$717$	37.7275	1.40896
$718$	0	0
$719$	18.8059	0.701342	0.350671	−	0.936499i	$-0.385954\pi$
$719$	18.8059	0.701342	0.350671	+	0.936499i	$0.385954\pi$
$720$	0	0
$721$	0.282314	0.0105139
$722$	0	0
$723$	25.6286	0.953139
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−5.32562	−0.197516	−0.0987581	−	0.995111i	$-0.531487\pi$
$727$	−5.32562	−0.197516	−0.0987581	+	0.995111i	$0.531487\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	−6.82273	−0.252348
$732$	0	0
$733$	40.3051	1.48870	0.744351	−	0.667788i	$-0.232759\pi$
$733$	40.3051	1.48870	0.744351	+	0.667788i	$0.232759\pi$
$734$	0	0
$735$	11.2846	0.416238
$736$	0	0
$737$	31.3340	1.15420
$738$	0	0
$739$	−14.2472	−0.524093	−0.262047	−	0.965055i	$-0.584397\pi$
$739$	−14.2472	−0.524093	−0.262047	+	0.965055i	$0.584397\pi$
$740$	0	0
$741$	7.20394	0.264643
$742$	0	0
$743$	38.7031	1.41988	0.709940	−	0.704262i	$-0.248722\pi$
$743$	38.7031	1.41988	0.709940	+	0.704262i	$0.248722\pi$
$744$	0	0
$745$	−10.3135	−0.377858
$746$	0	0
$747$	0.874264	0.0319876
$748$	0	0
$749$	10.1256	0.369980
$750$	0	0
$751$	31.1955	1.13834	0.569170	−	0.822220i	$-0.307264\pi$
$751$	31.1955	1.13834	0.569170	+	0.822220i	$0.307264\pi$
$752$	0	0
$753$	4.81044	0.175302
$754$	0	0
$755$	24.2823	0.883724
$756$	0	0
$757$	17.1233	0.622357	0.311178	−	0.950352i	$-0.399276\pi$
$757$	17.1233	0.622357	0.311178	+	0.950352i	$0.399276\pi$
$758$	0	0
$759$	−10.8371	−0.393362
$760$	0	0
$761$	−25.1506	−0.911709	−0.455854	−	0.890054i	$-0.650666\pi$
$761$	−25.1506	−0.911709	−0.455854	+	0.890054i	$0.650666\pi$
$762$	0	0
$763$	−12.8950	−0.466829
$764$	0	0
$765$	−0.474142	−0.0171426
$766$	0	0
$767$	−17.7275	−0.640104
$768$	0	0
$769$	27.2618	0.983085	0.491543	−	0.870853i	$-0.336433\pi$
$769$	27.2618	0.983085	0.491543	+	0.870853i	$0.336433\pi$
$770$	0	0
$771$	−10.1112	−0.364145
$772$	0	0
$773$	−1.75276	−0.0630423	−0.0315212	−	0.999503i	$-0.510035\pi$
$773$	−1.75276	−0.0630423	−0.0315212	+	0.999503i	$0.510035\pi$
$774$	0	0
$775$	−4.78765	−0.171978
$776$	0	0
$777$	2.83710	0.101780
$778$	0	0
$779$	−11.4186	−0.409112
$780$	0	0
$781$	36.1978	1.29526
$782$	0	0
$783$	−5.26180	−0.188041
$784$	0	0
$785$	−1.76713	−0.0630717
$786$	0	0
$787$	−31.2001	−1.11216	−0.556081	−	0.831128i	$-0.687696\pi$
$787$	−31.2001	−1.11216	−0.556081	+	0.831128i	$0.687696\pi$
$788$	0	0
$789$	−20.4514	−0.728088
$790$	0	0
$791$	−3.70086	−0.131588
$792$	0	0
$793$	29.5585	1.04965
$794$	0	0
$795$	−15.4186	−0.546840
$796$	0	0
$797$	21.8660	0.774535	0.387267	−	0.921967i	$-0.373419\pi$
$797$	21.8660	0.774535	0.387267	+	0.921967i	$0.373419\pi$
$798$	0	0
$799$	−62.8515	−2.22353
$800$	0	0
$801$	−0.202328	−0.00714891
$802$	0	0
$803$	15.9155	0.561645
$804$	0	0
$805$	1.07838	0.0380078
$806$	0	0
$807$	24.7792	0.872271
$808$	0	0
$809$	−5.22076	−0.183552	−0.0917760	−	0.995780i	$-0.529254\pi$
$809$	−5.22076	−0.183552	−0.0917760	+	0.995780i	$0.529254\pi$
$810$	0	0
$811$	6.05786	0.212720	0.106360	−	0.994328i	$-0.466080\pi$
$811$	6.05786	0.212720	0.106360	+	0.994328i	$0.466080\pi$
$812$	0	0
$813$	9.03281	0.316794
$814$	0	0
$815$	16.9132	0.592444
$816$	0	0
$817$	−1.54411	−0.0540215
$818$	0	0
$819$	−0.152221	−0.00531902
$820$	0	0
$821$	22.3668	0.780608	0.390304	−	0.920686i	$-0.372370\pi$
$821$	22.3668	0.780608	0.390304	+	0.920686i	$0.372370\pi$
$822$	0	0
$823$	14.7031	0.512519	0.256259	−	0.966608i	$-0.417510\pi$
$823$	14.7031	0.512519	0.256259	+	0.966608i	$0.417510\pi$
$824$	0	0
$825$	−6.34017	−0.220736
$826$	0	0
$827$	53.4778	1.85961	0.929803	−	0.368057i	$-0.119977\pi$
$827$	53.4778	1.85961	0.929803	+	0.368057i	$0.119977\pi$
$828$	0	0
$829$	−36.8515	−1.27990	−0.639952	−	0.768415i	$-0.721046\pi$
$829$	−36.8515	−1.27990	−0.639952	+	0.768415i	$0.721046\pi$
$830$	0	0
$831$	−43.3028	−1.50216
$832$	0	0
$833$	39.9383	1.38378
$834$	0	0
$835$	−6.97107	−0.241244
$836$	0	0
$837$	25.1917	0.870751
$838$	0	0
$839$	−40.3896	−1.39440	−0.697202	−	0.716874i	$-0.745572\pi$
$839$	−40.3896	−1.39440	−0.697202	+	0.716874i	$0.745572\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	44.8326	1.54412
$844$	0	0
$845$	3.52359	0.121215
$846$	0	0
$847$	−1.74047	−0.0598033
$848$	0	0
$849$	5.07838	0.174289
$850$	0	0
$851$	−4.49693	−0.154153
$852$	0	0
$853$	15.5714	0.533156	0.266578	−	0.963813i	$-0.414107\pi$
$853$	15.5714	0.533156	0.266578	+	0.963813i	$0.414107\pi$
$854$	0	0
$855$	−0.107307	−0.00366983
$856$	0	0
$857$	−47.0784	−1.60817	−0.804083	−	0.594517i	$-0.797343\pi$
$857$	−47.0784	−1.60817	−0.804083	+	0.594517i	$0.797343\pi$
$858$	0	0
$859$	29.2678	0.998603	0.499302	−	0.866428i	$-0.333590\pi$
$859$	29.2678	0.998603	0.499302	+	0.866428i	$0.333590\pi$
$860$	0	0
$861$	−8.99386	−0.306510
$862$	0	0
$863$	−13.4635	−0.458302	−0.229151	−	0.973391i	$-0.573595\pi$
$863$	−13.4635	−0.458302	−0.229151	+	0.973391i	$0.573595\pi$
$864$	0	0
$865$	−5.23513	−0.178000
$866$	0	0
$867$	33.4947	1.13754
$868$	0	0
$869$	−23.1194	−0.784272
$870$	0	0
$871$	26.0045	0.881130
$872$	0	0
$873$	−1.06997	−0.0362129
$874$	0	0
$875$	0.630898	0.0213282
$876$	0	0
$877$	39.8576	1.34590	0.672948	−	0.739690i	$-0.265028\pi$
$877$	39.8576	1.34590	0.672948	+	0.739690i	$0.265028\pi$
$878$	0	0
$879$	−27.1896	−0.917081
$880$	0	0
$881$	−1.33403	−0.0449446	−0.0224723	−	0.999747i	$-0.507154\pi$
$881$	−1.33403	−0.0449446	−0.0224723	+	0.999747i	$0.507154\pi$
$882$	0	0
$883$	4.47868	0.150719	0.0753597	−	0.997156i	$-0.475989\pi$
$883$	4.47868	0.150719	0.0753597	+	0.997156i	$0.475989\pi$
$884$	0	0
$885$	−9.84324	−0.330877
$886$	0	0
$887$	25.0745	0.841919	0.420960	−	0.907079i	$-0.361693\pi$
$887$	25.0745	0.841919	0.420960	+	0.907079i	$0.361693\pi$
$888$	0	0
$889$	3.60197	0.120806
$890$	0	0
$891$	32.4885	1.08841
$892$	0	0
$893$	−14.2245	−0.476003
$894$	0	0
$895$	−2.52359	−0.0843543
$896$	0	0
$897$	−8.99386	−0.300296
$898$	0	0
$899$	−4.78765	−0.159677
$900$	0	0
$901$	−54.5692	−1.81796
$902$	0	0
$903$	−1.21622	−0.0404734
$904$	0	0
$905$	2.76487	0.0919073
$906$	0	0
$907$	−40.9588	−1.36001	−0.680007	−	0.733206i	$-0.738023\pi$
$907$	−40.9588	−1.36001	−0.680007	+	0.733206i	$0.738023\pi$
$908$	0	0
$909$	−1.10504	−0.0366519
$910$	0	0
$911$	−36.8865	−1.22211	−0.611053	−	0.791590i	$-0.709254\pi$
$911$	−36.8865	−1.22211	−0.611053	+	0.791590i	$0.709254\pi$
$912$	0	0
$913$	41.3751	1.36931
$914$	0	0
$915$	16.4124	0.542577
$916$	0	0
$917$	11.4497	0.378104
$918$	0	0
$919$	−43.7464	−1.44306	−0.721531	−	0.692382i	$-0.756561\pi$
$919$	−43.7464	−1.44306	−0.721531	+	0.692382i	$0.756561\pi$
$920$	0	0
$921$	16.1378	0.531760
$922$	0	0
$923$	30.0410	0.988813
$924$	0	0
$925$	−2.63090	−0.0865034
$926$	0	0
$927$	0.0350725	0.00115193
$928$	0	0
$929$	26.5646	0.871557	0.435779	−	0.900054i	$-0.356473\pi$
$929$	26.5646	0.871557	0.435779	+	0.900054i	$0.356473\pi$
$930$	0	0
$931$	9.03877	0.296234
$932$	0	0
$933$	19.1650	0.627434
$934$	0	0
$935$	−22.4391	−0.733836
$936$	0	0
$937$	−16.0267	−0.523568	−0.261784	−	0.965126i	$-0.584311\pi$
$937$	−16.0267	−0.523568	−0.261784	+	0.965126i	$0.584311\pi$
$938$	0	0
$939$	7.05172	0.230124
$940$	0	0
$941$	23.1629	0.755089	0.377544	−	0.925991i	$-0.376769\pi$
$941$	23.1629	0.755089	0.377544	+	0.925991i	$0.376769\pi$
$942$	0	0
$943$	14.2557	0.464228
$944$	0	0
$945$	−3.31965	−0.107988
$946$	0	0
$947$	−44.5464	−1.44756	−0.723781	−	0.690029i	$-0.757598\pi$
$947$	−44.5464	−1.44756	−0.723781	+	0.690029i	$0.757598\pi$
$948$	0	0
$949$	13.2085	0.428765
$950$	0	0
$951$	24.5503	0.796096
$952$	0	0
$953$	46.8781	1.51853	0.759266	−	0.650780i	$-0.225558\pi$
$953$	46.8781	1.51853	0.759266	+	0.650780i	$0.225558\pi$
$954$	0	0
$955$	2.81432	0.0910691
$956$	0	0
$957$	−6.34017	−0.204949
$958$	0	0
$959$	8.13009	0.262534
$960$	0	0
$961$	−8.07838	−0.260593
$962$	0	0
$963$	1.25792	0.0405359
$964$	0	0
$965$	9.95055	0.320320
$966$	0	0
$967$	19.4824	0.626511	0.313255	−	0.949669i	$-0.398580\pi$
$967$	19.4824	0.626511	0.313255	+	0.949669i	$0.398580\pi$
$968$	0	0
$969$	14.1568	0.454781
$970$	0	0
$971$	55.5090	1.78137	0.890685	−	0.454621i	$-0.150225\pi$
$971$	55.5090	1.78137	0.890685	+	0.454621i	$0.150225\pi$
$972$	0	0
$973$	6.27247	0.201086
$974$	0	0
$975$	−5.26180	−0.168512
$976$	0	0
$977$	11.2474	0.359837	0.179918	−	0.983682i	$-0.442417\pi$
$977$	11.2474	0.359837	0.179918	+	0.983682i	$0.442417\pi$
$978$	0	0
$979$	−9.57531	−0.306028
$980$	0	0
$981$	−1.60197	−0.0511469
$982$	0	0
$983$	11.9194	0.380168	0.190084	−	0.981768i	$-0.439124\pi$
$983$	11.9194	0.380168	0.190084	+	0.981768i	$0.439124\pi$
$984$	0	0
$985$	9.73206	0.310089
$986$	0	0
$987$	−11.2039	−0.356625
$988$	0	0
$989$	1.92777	0.0612994
$990$	0	0
$991$	−18.6081	−0.591106	−0.295553	−	0.955326i	$-0.595504\pi$
$991$	−18.6081	−0.591106	−0.295553	+	0.955326i	$0.595504\pi$
$992$	0	0
$993$	−10.1711	−0.322771
$994$	0	0
$995$	−19.6020	−0.621424
$996$	0	0
$997$	−43.2099	−1.36847	−0.684236	−	0.729261i	$-0.739864\pi$
$997$	−43.2099	−1.36847	−0.684236	+	0.729261i	$0.739864\pi$
$998$	0	0
$999$	13.8432	0.437981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.2.a.e.1.3	✓	3
4.3	odd	2		2320.2.a.r.1.1		3
5.4	even	2		5800.2.a.r.1.1		3
8.3	odd	2		9280.2.a.bl.1.3		3
8.5	even	2		9280.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.e.1.3	✓	3	1.1	even	1	trivial
2320.2.a.r.1.1		3	4.3	odd	2
5800.2.a.r.1.1		3	5.4	even	2
9280.2.a.bl.1.3		3	8.3	odd	2
9280.2.a.bv.1.1		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.70928 q^{3} -1.00000 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -3.70928 q^{11} -3.07838 q^{13} -1.70928 q^{15} -6.04945 q^{17} -1.36910 q^{19} -1.07838 q^{21} +1.70928 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -4.78765 q^{31} -6.34017 q^{33} +0.630898 q^{35} -2.63090 q^{37} -5.26180 q^{39} +8.34017 q^{41} +1.12783 q^{43} +0.0783777 q^{45} +10.3896 q^{47} -6.60197 q^{49} -10.3402 q^{51} +9.02052 q^{53} +3.70928 q^{55} -2.34017 q^{57} +5.75872 q^{59} -9.60197 q^{61} +0.0494483 q^{63} +3.07838 q^{65} -8.44748 q^{67} +2.92162 q^{69} -9.75872 q^{71} -4.29072 q^{73} +1.70928 q^{75} +2.34017 q^{77} +6.23287 q^{79} -8.75872 q^{81} -11.1545 q^{83} +6.04945 q^{85} +1.70928 q^{87} +2.58145 q^{89} +1.94214 q^{91} -8.18342 q^{93} +1.36910 q^{95} +13.6514 q^{97} +0.290725 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{19} - 2 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} - 4 q^{37} - 8 q^{39} + 14 q^{41} - 18 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 2 * q^15 - 8 * q^19 - 2 * q^23 + 3 * q^25 - 8 * q^27 + 3 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 - 4 * q^37 - 8 * q^39 + 14 * q^41 - 18 * q^43 - 3 * q^45 + 2 * q^47 - q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 4 * q^57 - 8 * q^59 - 10 * q^61 - 18 * q^63 + 6 * q^65 - 26 * q^67 + 12 * q^69 - 4 * q^71 - 20 * q^73 - 2 * q^75 - 4 * q^77 - 4 * q^79 - q^81 - 14 * q^83 - 2 * q^87 + 22 * q^89 - 24 * q^91 - 20 * q^93 + 8 * q^95 + 4 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more