LMFDB - Embedded newform 1503.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.38569.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 5x^{3} + 4x - 1 $ x^5 - 5x^3 + 4x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.15098$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15098 q^{2} -0.675235 q^{4} -0.0593478 q^{5} -1.53510 q^{7} +3.07915 q^{8} +O(q^{10})$	\(q-1.15098 q^{2} -0.675235 q^{4} -0.0593478 q^{5} -1.53510 q^{7} +3.07915 q^{8} +0.0683084 q^{10} -0.726798 q^{11} -1.12339 q^{13} +1.76687 q^{14} -2.19359 q^{16} +6.50878 q^{17} -4.67524 q^{19} +0.0400737 q^{20} +0.836533 q^{22} +3.23604 q^{23} -4.99648 q^{25} +1.29301 q^{26} +1.03655 q^{28} +3.01149 q^{29} -0.738105 q^{31} -3.63352 q^{32} -7.49150 q^{34} +0.0911046 q^{35} +8.63949 q^{37} +5.38112 q^{38} -0.182741 q^{40} -10.3776 q^{41} +7.89407 q^{43} +0.490760 q^{44} -3.72463 q^{46} +1.39496 q^{47} -4.64348 q^{49} +5.75087 q^{50} +0.758555 q^{52} -13.9475 q^{53} +0.0431339 q^{55} -4.72680 q^{56} -3.46618 q^{58} +6.92291 q^{59} -13.9518 q^{61} +0.849548 q^{62} +8.56930 q^{64} +0.0666709 q^{65} -8.78089 q^{67} -4.39496 q^{68} -0.104860 q^{70} +9.28259 q^{71} +9.04640 q^{73} -9.94392 q^{74} +3.15688 q^{76} +1.11571 q^{77} -9.62559 q^{79} +0.130185 q^{80} +11.9445 q^{82} -6.72942 q^{83} -0.386282 q^{85} -9.08594 q^{86} -2.23792 q^{88} -7.03744 q^{89} +1.72452 q^{91} -2.18509 q^{92} -1.60557 q^{94} +0.277465 q^{95} -0.754292 q^{97} +5.34457 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{5} - 4 q^{7}+O(q^{10}) $ 5 * q + q^5 - 4 * q^7	$ 5 q + q^{5} - 4 q^{7} - 9 q^{10} + 7 q^{11} - 8 q^{13} + q^{14} - 6 q^{16} - 5 q^{17} - 20 q^{19} + 3 q^{20} - q^{22} - q^{23} - 6 q^{25} + 3 q^{26} - 11 q^{28} + 5 q^{29} - 18 q^{31} + 5 q^{32} - 12 q^{34} + 6 q^{35} - 17 q^{37} - 2 q^{40} - 6 q^{41} - 10 q^{43} + 9 q^{44} - q^{46} + 3 q^{47} - 13 q^{49} - 9 q^{50} - 15 q^{53} - 9 q^{55} - 13 q^{56} + 21 q^{58} + 17 q^{59} - 2 q^{61} - 10 q^{64} - 26 q^{65} - 24 q^{67} - 18 q^{68} + 25 q^{70} - q^{71} + 2 q^{73} - 17 q^{74} + 14 q^{76} - 26 q^{77} - 6 q^{79} - 12 q^{80} + 17 q^{82} + 7 q^{83} - 11 q^{85} - 15 q^{86} + 4 q^{88} - 15 q^{91} - 45 q^{92} - 2 q^{94} - q^{95} - 13 q^{97} + 16 q^{98}+O(q^{100}) $ 5 * q + q^5 - 4 * q^7 - 9 * q^10 + 7 * q^11 - 8 * q^13 + q^14 - 6 * q^16 - 5 * q^17 - 20 * q^19 + 3 * q^20 - q^22 - q^23 - 6 * q^25 + 3 * q^26 - 11 * q^28 + 5 * q^29 - 18 * q^31 + 5 * q^32 - 12 * q^34 + 6 * q^35 - 17 * q^37 - 2 * q^40 - 6 * q^41 - 10 * q^43 + 9 * q^44 - q^46 + 3 * q^47 - 13 * q^49 - 9 * q^50 - 15 * q^53 - 9 * q^55 - 13 * q^56 + 21 * q^58 + 17 * q^59 - 2 * q^61 - 10 * q^64 - 26 * q^65 - 24 * q^67 - 18 * q^68 + 25 * q^70 - q^71 + 2 * q^73 - 17 * q^74 + 14 * q^76 - 26 * q^77 - 6 * q^79 - 12 * q^80 + 17 * q^82 + 7 * q^83 - 11 * q^85 - 15 * q^86 + 4 * q^88 - 15 * q^91 - 45 * q^92 - 2 * q^94 - q^95 - 13 * q^97 + 16 * q^98

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 $

$2$	−1.15098	−0.813869	−0.406934	−	0.913457i	$-0.633402\pi$
$2$	−1.15098	−0.813869	−0.406934	+	0.913457i	$0.633402\pi$
$3$	0	0
$3$	0	0
$4$	−0.675235	−0.337618
$5$	−0.0593478	−0.0265411	−0.0132706	−	0.999912i	$-0.504224\pi$
$5$	−0.0593478	−0.0265411	−0.0132706	+	0.999912i	$0.504224\pi$
$6$	0	0
$7$	−1.53510	−0.580212	−0.290106	−	0.956995i	$-0.593691\pi$
$7$	−1.53510	−0.580212	−0.290106	+	0.956995i	$0.593691\pi$
$8$	3.07915	1.08865
$9$	0	0
$10$	0.0683084	0.0216010
$11$	−0.726798	−0.219138	−0.109569	−	0.993979i	$-0.534947\pi$
$11$	−0.726798	−0.219138	−0.109569	+	0.993979i	$0.534947\pi$
$12$	0	0
$13$	−1.12339	−0.311573	−0.155787	−	0.987791i	$-0.549791\pi$
$13$	−1.12339	−0.311573	−0.155787	+	0.987791i	$0.549791\pi$
$14$	1.76687	0.472216
$15$	0	0
$16$	−2.19359	−0.548397
$17$	6.50878	1.57861	0.789305	−	0.614001i	$-0.210441\pi$
$17$	6.50878	1.57861	0.789305	+	0.614001i	$0.210441\pi$
$18$	0	0
$19$	−4.67524	−1.07257	−0.536286	−	0.844036i	$-0.680173\pi$
$19$	−4.67524	−1.07257	−0.536286	+	0.844036i	$0.680173\pi$
$20$	0.0400737	0.00896076
$21$	0	0
$22$	0.836533	0.178349
$23$	3.23604	0.674761	0.337380	−	0.941368i	$-0.390459\pi$
$23$	3.23604	0.674761	0.337380	+	0.941368i	$0.390459\pi$
$24$	0	0
$25$	−4.99648	−0.999296
$26$	1.29301	0.253580
$27$	0	0
$28$	1.03655	0.195890
$29$	3.01149	0.559219	0.279610	−	0.960114i	$-0.409795\pi$
$29$	3.01149	0.559219	0.279610	+	0.960114i	$0.409795\pi$
$30$	0	0
$31$	−0.738105	−0.132568	−0.0662838	−	0.997801i	$-0.521114\pi$
$31$	−0.738105	−0.132568	−0.0662838	+	0.997801i	$0.521114\pi$
$32$	−3.63352	−0.642322
$33$	0	0
$34$	−7.49150	−1.28478
$35$	0.0911046	0.0153995
$36$	0	0
$37$	8.63949	1.42032	0.710162	−	0.704038i	$-0.248622\pi$
$37$	8.63949	1.42032	0.710162	+	0.704038i	$0.248622\pi$
$38$	5.38112	0.872933
$39$	0	0
$40$	−0.182741	−0.0288939
$41$	−10.3776	−1.62071	−0.810354	−	0.585940i	$-0.800725\pi$
$41$	−10.3776	−1.62071	−0.810354	+	0.585940i	$0.800725\pi$
$42$	0	0
$43$	7.89407	1.20383	0.601917	−	0.798559i	$-0.294404\pi$
$43$	7.89407	1.20383	0.601917	+	0.798559i	$0.294404\pi$
$44$	0.490760	0.0739848
$45$	0	0
$46$	−3.72463	−0.549167
$47$	1.39496	0.203475	0.101738	−	0.994811i	$-0.467560\pi$
$47$	1.39496	0.203475	0.101738	+	0.994811i	$0.467560\pi$
$48$	0	0
$49$	−4.64348	−0.663354
$50$	5.75087	0.813295
$51$	0	0
$52$	0.758555	0.105193
$53$	−13.9475	−1.91584	−0.957919	−	0.287038i	$-0.907329\pi$
$53$	−13.9475	−1.91584	−0.957919	+	0.287038i	$0.907329\pi$
$54$	0	0
$55$	0.0431339	0.00581617
$56$	−4.72680	−0.631645
$57$	0	0
$58$	−3.46618	−0.455131
$59$	6.92291	0.901286	0.450643	−	0.892704i	$-0.351195\pi$
$59$	6.92291	0.901286	0.450643	+	0.892704i	$0.351195\pi$
$60$	0	0
$61$	−13.9518	−1.78634	−0.893171	−	0.449717i	$-0.851525\pi$
$61$	−13.9518	−1.78634	−0.893171	+	0.449717i	$0.851525\pi$
$62$	0.849548	0.107893
$63$	0	0
$64$	8.56930	1.07116
$65$	0.0666709	0.00826951
$66$	0	0
$67$	−8.78089	−1.07276	−0.536378	−	0.843978i	$-0.680208\pi$
$67$	−8.78089	−1.07276	−0.536378	+	0.843978i	$0.680208\pi$
$68$	−4.39496	−0.532967
$69$	0	0
$70$	−0.104860	−0.0125332
$71$	9.28259	1.10164	0.550820	−	0.834624i	$-0.314315\pi$
$71$	9.28259	1.10164	0.550820	+	0.834624i	$0.314315\pi$
$72$	0	0
$73$	9.04640	1.05880	0.529401	−	0.848372i	$-0.322417\pi$
$73$	9.04640	1.05880	0.529401	+	0.848372i	$0.322417\pi$
$74$	−9.94392	−1.15596
$75$	0	0
$76$	3.15688	0.362120
$77$	1.11571	0.127146
$78$	0	0
$79$	−9.62559	−1.08296	−0.541482	−	0.840713i	$-0.682136\pi$
$79$	−9.62559	−1.08296	−0.541482	+	0.840713i	$0.682136\pi$
$80$	0.130185	0.0145551
$81$	0	0
$82$	11.9445	1.31904
$83$	−6.72942	−0.738650	−0.369325	−	0.929300i	$-0.620411\pi$
$83$	−6.72942	−0.738650	−0.369325	+	0.929300i	$0.620411\pi$
$84$	0	0
$85$	−0.386282	−0.0418981
$86$	−9.08594	−0.979763
$87$	0	0
$88$	−2.23792	−0.238563
$89$	−7.03744	−0.745967	−0.372984	−	0.927838i	$-0.621665\pi$
$89$	−7.03744	−0.745967	−0.372984	+	0.927838i	$0.621665\pi$
$90$	0	0
$91$	1.72452	0.180778
$92$	−2.18509	−0.227811
$93$	0	0
$94$	−1.60557	−0.165602
$95$	0.277465	0.0284673
$96$	0	0
$97$	−0.754292	−0.0765867	−0.0382934	−	0.999267i	$-0.512192\pi$
$97$	−0.754292	−0.0765867	−0.0382934	+	0.999267i	$0.512192\pi$
$98$	5.34457	0.539883
$99$	0	0
$100$	3.37380	0.337380
$101$	−10.6059	−1.05532	−0.527661	−	0.849455i	$-0.676931\pi$
$101$	−10.6059	−1.05532	−0.527661	+	0.849455i	$0.676931\pi$
$102$	0	0
$103$	−5.52621	−0.544513	−0.272257	−	0.962225i	$-0.587770\pi$
$103$	−5.52621	−0.544513	−0.272257	+	0.962225i	$0.587770\pi$
$104$	−3.45910	−0.339193
$105$	0	0
$106$	16.0534	1.55924
$107$	−0.273314	−0.0264222	−0.0132111	−	0.999913i	$-0.504205\pi$
$107$	−0.273314	−0.0264222	−0.0132111	+	0.999913i	$0.504205\pi$
$108$	0	0
$109$	−15.6916	−1.50298	−0.751492	−	0.659742i	$-0.770666\pi$
$109$	−15.6916	−1.50298	−0.751492	+	0.659742i	$0.770666\pi$
$110$	−0.0496464	−0.00473360
$111$	0	0
$112$	3.36737	0.318186
$113$	−20.1726	−1.89768	−0.948839	−	0.315760i	$-0.897740\pi$
$113$	−20.1726	−1.89768	−0.948839	+	0.315760i	$0.897740\pi$
$114$	0	0
$115$	−0.192052	−0.0179089
$116$	−2.03346	−0.188802
$117$	0	0
$118$	−7.96816	−0.733529
$119$	−9.99160	−0.915929
$120$	0	0
$121$	−10.4718	−0.951979
$122$	16.0583	1.45385
$123$	0	0
$124$	0.498395	0.0447572
$125$	0.593269	0.0530636
$126$	0	0
$127$	−5.57961	−0.495111	−0.247555	−	0.968874i	$-0.579627\pi$
$127$	−5.57961	−0.495111	−0.247555	+	0.968874i	$0.579627\pi$
$128$	−2.59608	−0.229463
$129$	0	0
$130$	−0.0767372	−0.00673029
$131$	19.9390	1.74208	0.871041	−	0.491210i	$-0.163445\pi$
$131$	19.9390	1.74208	0.871041	+	0.491210i	$0.163445\pi$
$132$	0	0
$133$	7.17694	0.622319
$134$	10.1067	0.873083
$135$	0	0
$136$	20.0415	1.71855
$137$	−6.58107	−0.562259	−0.281129	−	0.959670i	$-0.590709\pi$
$137$	−6.58107	−0.562259	−0.281129	+	0.959670i	$0.590709\pi$
$138$	0	0
$139$	−1.54196	−0.130788	−0.0653938	−	0.997860i	$-0.520830\pi$
$139$	−1.54196	−0.130788	−0.0653938	+	0.997860i	$0.520830\pi$
$140$	−0.0615170	−0.00519914
$141$	0	0
$142$	−10.6841	−0.896591
$143$	0.816480	0.0682775
$144$	0	0
$145$	−0.178725	−0.0148423
$146$	−10.4123	−0.861725
$147$	0	0
$148$	−5.83369	−0.479527
$149$	−1.57899	−0.129356	−0.0646778	−	0.997906i	$-0.520602\pi$
$149$	−1.57899	−0.129356	−0.0646778	+	0.997906i	$0.520602\pi$
$150$	0	0
$151$	9.20724	0.749274	0.374637	−	0.927171i	$-0.377767\pi$
$151$	9.20724	0.749274	0.374637	+	0.927171i	$0.377767\pi$
$152$	−14.3958	−1.16765
$153$	0	0
$154$	−1.28416	−0.103480
$155$	0.0438049	0.00351850
$156$	0	0
$157$	−8.37016	−0.668012	−0.334006	−	0.942571i	$-0.608401\pi$
$157$	−8.37016	−0.668012	−0.334006	+	0.942571i	$0.608401\pi$
$158$	11.0789	0.881390
$159$	0	0
$160$	0.215642	0.0170480
$161$	−4.96763	−0.391504
$162$	0	0
$163$	−17.8913	−1.40135	−0.700676	−	0.713480i	$-0.747118\pi$
$163$	−17.8913	−1.40135	−0.700676	+	0.713480i	$0.747118\pi$
$164$	7.00732	0.547180
$165$	0	0
$166$	7.74546	0.601164
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.7380	−0.902922
$170$	0.444604	0.0340996
$171$	0	0
$172$	−5.33035	−0.406436
$173$	8.56831	0.651436	0.325718	−	0.945467i	$-0.394394\pi$
$173$	8.56831	0.651436	0.325718	+	0.945467i	$0.394394\pi$
$174$	0	0
$175$	7.67008	0.579803
$176$	1.59429	0.120174
$177$	0	0
$178$	8.09998	0.607120
$179$	13.3026	0.994284	0.497142	−	0.867669i	$-0.334383\pi$
$179$	13.3026	0.994284	0.497142	+	0.867669i	$0.334383\pi$
$180$	0	0
$181$	8.95629	0.665716	0.332858	−	0.942977i	$-0.391987\pi$
$181$	8.95629	0.665716	0.332858	+	0.942977i	$0.391987\pi$
$182$	−1.98489	−0.147130
$183$	0	0
$184$	9.96426	0.734575
$185$	−0.512735	−0.0376970
$186$	0	0
$187$	−4.73057	−0.345933
$188$	−0.941925	−0.0686969
$189$	0	0
$190$	−0.319358	−0.0231686
$191$	−6.73177	−0.487094	−0.243547	−	0.969889i	$-0.578311\pi$
$191$	−6.73177	−0.487094	−0.243547	+	0.969889i	$0.578311\pi$
$192$	0	0
$193$	3.59253	0.258596	0.129298	−	0.991606i	$-0.458728\pi$
$193$	3.59253	0.258596	0.129298	+	0.991606i	$0.458728\pi$
$194$	0.868178	0.0623316
$195$	0	0
$196$	3.13544	0.223960
$197$	−5.03819	−0.358956	−0.179478	−	0.983762i	$-0.557441\pi$
$197$	−5.03819	−0.358956	−0.179478	+	0.983762i	$0.557441\pi$
$198$	0	0
$199$	−18.8821	−1.33852	−0.669259	−	0.743029i	$-0.733388\pi$
$199$	−18.8821	−1.33852	−0.669259	+	0.743029i	$0.733388\pi$
$200$	−15.3849	−1.08788
$201$	0	0
$202$	12.2072	0.858893
$203$	−4.62293	−0.324466
$204$	0	0
$205$	0.615888	0.0430155
$206$	6.36058	0.443162
$207$	0	0
$208$	2.46426	0.170866
$209$	3.39795	0.235041
$210$	0	0
$211$	10.7029	0.736815	0.368408	−	0.929664i	$-0.379903\pi$
$211$	10.7029	0.736815	0.368408	+	0.929664i	$0.379903\pi$
$212$	9.41786	0.646821
$213$	0	0
$214$	0.314580	0.0215042
$215$	−0.468495	−0.0319511
$216$	0	0
$217$	1.13306	0.0769173
$218$	18.0608	1.22323
$219$	0	0
$220$	−0.0291255	−0.00196364
$221$	−7.31192	−0.491853
$222$	0	0
$223$	−7.13210	−0.477601	−0.238800	−	0.971069i	$-0.576754\pi$
$223$	−7.13210	−0.477601	−0.238800	+	0.971069i	$0.576754\pi$
$224$	5.57781	0.372683
$225$	0	0
$226$	23.2183	1.54446
$227$	−11.6284	−0.771803	−0.385902	−	0.922540i	$-0.626110\pi$
$227$	−11.6284	−0.771803	−0.385902	+	0.922540i	$0.626110\pi$
$228$	0	0
$229$	−12.9008	−0.852507	−0.426254	−	0.904604i	$-0.640167\pi$
$229$	−12.9008	−0.852507	−0.426254	+	0.904604i	$0.640167\pi$
$230$	0.221048	0.0145755
$231$	0	0
$232$	9.27284	0.608792
$233$	8.14215	0.533410	0.266705	−	0.963778i	$-0.414065\pi$
$233$	8.14215	0.533410	0.266705	+	0.963778i	$0.414065\pi$
$234$	0	0
$235$	−0.0827877	−0.00540047
$236$	−4.67460	−0.304290
$237$	0	0
$238$	11.5002	0.745446
$239$	−16.6237	−1.07530	−0.537648	−	0.843169i	$-0.680687\pi$
$239$	−16.6237	−1.07530	−0.537648	+	0.843169i	$0.680687\pi$
$240$	0	0
$241$	−7.91630	−0.509934	−0.254967	−	0.966950i	$-0.582065\pi$
$241$	−7.91630	−0.509934	−0.254967	+	0.966950i	$0.582065\pi$
$242$	12.0528	0.774786
$243$	0	0
$244$	9.42073	0.603101
$245$	0.275580	0.0176062
$246$	0	0
$247$	5.25213	0.334185
$248$	−2.27274	−0.144319
$249$	0	0
$250$	−0.682843	−0.0431868
$251$	21.6125	1.36417	0.682084	−	0.731274i	$-0.261074\pi$
$251$	21.6125	1.36417	0.682084	+	0.731274i	$0.261074\pi$
$252$	0	0
$253$	−2.35195	−0.147866
$254$	6.42205	0.402955
$255$	0	0
$256$	−14.1506	−0.884410
$257$	15.0536	0.939019	0.469509	−	0.882928i	$-0.344431\pi$
$257$	15.0536	0.939019	0.469509	+	0.882928i	$0.344431\pi$
$258$	0	0
$259$	−13.2625	−0.824089
$260$	−0.0450186	−0.00279193
$261$	0	0
$262$	−22.9495	−1.41783
$263$	−27.4806	−1.69453	−0.847265	−	0.531171i	$-0.821752\pi$
$263$	−27.4806	−1.69453	−0.847265	+	0.531171i	$0.821752\pi$
$264$	0	0
$265$	0.827754	0.0508485
$266$	−8.26054	−0.506486
$267$	0	0
$268$	5.92917	0.362182
$269$	9.53477	0.581345	0.290673	−	0.956823i	$-0.406121\pi$
$269$	9.53477	0.581345	0.290673	+	0.956823i	$0.406121\pi$
$270$	0	0
$271$	14.1985	0.862494	0.431247	−	0.902234i	$-0.358074\pi$
$271$	14.1985	0.862494	0.431247	+	0.902234i	$0.358074\pi$
$272$	−14.2776	−0.865705
$273$	0	0
$274$	7.57471	0.457605
$275$	3.63143	0.218983
$276$	0	0
$277$	18.0059	1.08187	0.540936	−	0.841064i	$-0.318070\pi$
$277$	18.0059	1.08187	0.540936	+	0.841064i	$0.318070\pi$
$278$	1.77478	0.106444
$279$	0	0
$280$	0.280525	0.0167646
$281$	−8.99125	−0.536373	−0.268186	−	0.963367i	$-0.586424\pi$
$281$	−8.99125	−0.536373	−0.268186	+	0.963367i	$0.586424\pi$
$282$	0	0
$283$	−27.2768	−1.62144	−0.810718	−	0.585436i	$-0.800923\pi$
$283$	−27.2768	−1.62144	−0.810718	+	0.585436i	$0.800923\pi$
$284$	−6.26793	−0.371933
$285$	0	0
$286$	−0.939756	−0.0555689
$287$	15.9306	0.940355
$288$	0	0
$289$	25.3642	1.49201
$290$	0.205710	0.0120797
$291$	0	0
$292$	−6.10845	−0.357470
$293$	7.59228	0.443545	0.221773	−	0.975098i	$-0.428816\pi$
$293$	7.59228	0.443545	0.221773	+	0.975098i	$0.428816\pi$
$294$	0	0
$295$	−0.410860	−0.0239212
$296$	26.6023	1.54623
$297$	0	0
$298$	1.81739	0.105278
$299$	−3.63534	−0.210237
$300$	0	0
$301$	−12.1182	−0.698479
$302$	−10.5974	−0.609811
$303$	0	0
$304$	10.2555	0.588195
$305$	0.828007	0.0474115
$306$	0	0
$307$	−2.34933	−0.134083	−0.0670416	−	0.997750i	$-0.521356\pi$
$307$	−2.34933	−0.134083	−0.0670416	+	0.997750i	$0.521356\pi$
$308$	−0.753364	−0.0429269
$309$	0	0
$310$	−0.0504188	−0.00286359
$311$	9.03371	0.512255	0.256127	−	0.966643i	$-0.417553\pi$
$311$	9.03371	0.512255	0.256127	+	0.966643i	$0.417553\pi$
$312$	0	0
$313$	−4.58324	−0.259060	−0.129530	−	0.991576i	$-0.541347\pi$
$313$	−4.58324	−0.259060	−0.129530	+	0.991576i	$0.541347\pi$
$314$	9.63393	0.543674
$315$	0	0
$316$	6.49954	0.365628
$317$	0.511404	0.0287233	0.0143616	−	0.999897i	$-0.495428\pi$
$317$	0.511404	0.0287233	0.0143616	+	0.999897i	$0.495428\pi$
$318$	0	0
$319$	−2.18874	−0.122546
$320$	−0.508569	−0.0284299
$321$	0	0
$322$	5.71766	0.318633
$323$	−30.4301	−1.69317
$324$	0	0
$325$	5.61301	0.311354
$326$	20.5925	1.14052
$327$	0	0
$328$	−31.9542	−1.76438
$329$	−2.14139	−0.118059
$330$	0	0
$331$	18.9587	1.04206	0.521031	−	0.853538i	$-0.325548\pi$
$331$	18.9587	1.04206	0.521031	+	0.853538i	$0.325548\pi$
$332$	4.54395	0.249381
$333$	0	0
$334$	−1.15098	−0.0629791
$335$	0.521126	0.0284722
$336$	0	0
$337$	31.5383	1.71800	0.859001	−	0.511973i	$-0.171085\pi$
$337$	31.5383	1.71800	0.859001	+	0.511973i	$0.171085\pi$
$338$	13.5102	0.734860
$339$	0	0
$340$	0.260831	0.0141455
$341$	0.536454	0.0290506
$342$	0	0
$343$	17.8739	0.965098
$344$	24.3070	1.31055
$345$	0	0
$346$	−9.86199	−0.530184
$347$	26.9408	1.44626	0.723129	−	0.690713i	$-0.242703\pi$
$347$	26.9408	1.44626	0.723129	+	0.690713i	$0.242703\pi$
$348$	0	0
$349$	18.3800	0.983857	0.491928	−	0.870636i	$-0.336292\pi$
$349$	18.3800	0.983857	0.491928	+	0.870636i	$0.336292\pi$
$350$	−8.82814	−0.471884
$351$	0	0
$352$	2.64084	0.140757
$353$	14.2109	0.756370	0.378185	−	0.925730i	$-0.376548\pi$
$353$	14.2109	0.756370	0.378185	+	0.925730i	$0.376548\pi$
$354$	0	0
$355$	−0.550901	−0.0292388
$356$	4.75193	0.251852
$357$	0	0
$358$	−15.3111	−0.809217
$359$	11.5677	0.610520	0.305260	−	0.952269i	$-0.401257\pi$
$359$	11.5677	0.610520	0.305260	+	0.952269i	$0.401257\pi$
$360$	0	0
$361$	2.85783	0.150412
$362$	−10.3086	−0.541805
$363$	0	0
$364$	−1.16445	−0.0610340
$365$	−0.536884	−0.0281018
$366$	0	0
$367$	6.06166	0.316416	0.158208	−	0.987406i	$-0.449428\pi$
$367$	6.06166	0.316416	0.158208	+	0.987406i	$0.449428\pi$
$368$	−7.09853	−0.370036
$369$	0	0
$370$	0.590150	0.0306804
$371$	21.4108	1.11159
$372$	0	0
$373$	−34.2760	−1.77474	−0.887371	−	0.461056i	$-0.847471\pi$
$373$	−34.2760	−1.77474	−0.887371	+	0.461056i	$0.847471\pi$
$374$	5.44481	0.281544
$375$	0	0
$376$	4.29529	0.221513
$377$	−3.38309	−0.174238
$378$	0	0
$379$	9.45850	0.485851	0.242925	−	0.970045i	$-0.421893\pi$
$379$	9.45850	0.485851	0.242925	+	0.970045i	$0.421893\pi$
$380$	−0.187354	−0.00961106
$381$	0	0
$382$	7.74816	0.396430
$383$	20.8981	1.06784	0.533921	−	0.845534i	$-0.320718\pi$
$383$	20.8981	1.06784	0.533921	+	0.845534i	$0.320718\pi$
$384$	0	0
$385$	−0.0662146	−0.00337461
$386$	−4.13495	−0.210463
$387$	0	0
$388$	0.509325	0.0258570
$389$	16.7753	0.850542	0.425271	−	0.905066i	$-0.360179\pi$
$389$	16.7753	0.850542	0.425271	+	0.905066i	$0.360179\pi$
$390$	0	0
$391$	21.0627	1.06518
$392$	−14.2980	−0.722157
$393$	0	0
$394$	5.79888	0.292143
$395$	0.571257	0.0287431
$396$	0	0
$397$	5.00650	0.251269	0.125635	−	0.992077i	$-0.459903\pi$
$397$	5.00650	0.251269	0.125635	+	0.992077i	$0.459903\pi$
$398$	21.7330	1.08938
$399$	0	0
$400$	10.9602	0.548010
$401$	−1.41427	−0.0706252	−0.0353126	−	0.999376i	$-0.511243\pi$
$401$	−1.41427	−0.0706252	−0.0353126	+	0.999376i	$0.511243\pi$
$402$	0	0
$403$	0.829183	0.0413045
$404$	7.16145	0.356295
$405$	0	0
$406$	5.32091	0.264073
$407$	−6.27917	−0.311247
$408$	0	0
$409$	−24.7588	−1.22425	−0.612123	−	0.790763i	$-0.709684\pi$
$409$	−24.7588	−1.22425	−0.612123	+	0.790763i	$0.709684\pi$
$410$	−0.708877	−0.0350089
$411$	0	0
$412$	3.73149	0.183837
$413$	−10.6273	−0.522937
$414$	0	0
$415$	0.399376	0.0196046
$416$	4.08188	0.200130
$417$	0	0
$418$	−3.91099	−0.191293
$419$	30.6592	1.49780	0.748901	−	0.662682i	$-0.230582\pi$
$419$	30.6592	1.49780	0.748901	+	0.662682i	$0.230582\pi$
$420$	0	0
$421$	34.5471	1.68372	0.841862	−	0.539693i	$-0.181460\pi$
$421$	34.5471	1.68372	0.841862	+	0.539693i	$0.181460\pi$
$422$	−12.3188	−0.599671
$423$	0	0
$424$	−42.9465	−2.08567
$425$	−32.5210	−1.57750
$426$	0	0
$427$	21.4173	1.03646
$428$	0.184551	0.00892062
$429$	0	0
$430$	0.539231	0.0260040
$431$	−22.4664	−1.08217	−0.541084	−	0.840969i	$-0.681986\pi$
$431$	−22.4664	−1.08217	−0.541084	+	0.840969i	$0.681986\pi$
$432$	0	0
$433$	3.35851	0.161400	0.0806999	−	0.996738i	$-0.474284\pi$
$433$	3.35851	0.161400	0.0806999	+	0.996738i	$0.474284\pi$
$434$	−1.30414	−0.0626006
$435$	0	0
$436$	10.5955	0.507434
$437$	−15.1292	−0.723730
$438$	0	0
$439$	17.9538	0.856886	0.428443	−	0.903569i	$-0.359062\pi$
$439$	17.9538	0.856886	0.428443	+	0.903569i	$0.359062\pi$
$440$	0.132816	0.00633174
$441$	0	0
$442$	8.41590	0.400304
$443$	−18.7542	−0.891041	−0.445520	−	0.895272i	$-0.646981\pi$
$443$	−18.7542	−0.891041	−0.445520	+	0.895272i	$0.646981\pi$
$444$	0	0
$445$	0.417657	0.0197988
$446$	8.20894	0.388704
$447$	0	0
$448$	−13.1547	−0.621501
$449$	4.79294	0.226193	0.113096	−	0.993584i	$-0.463923\pi$
$449$	4.79294	0.226193	0.113096	+	0.993584i	$0.463923\pi$
$450$	0	0
$451$	7.54242	0.355159
$452$	13.6213	0.640690
$453$	0	0
$454$	13.3841	0.628147
$455$	−0.102346	−0.00479807
$456$	0	0
$457$	−29.5638	−1.38294	−0.691468	−	0.722407i	$-0.743036\pi$
$457$	−29.5638	−1.38294	−0.691468	+	0.722407i	$0.743036\pi$
$458$	14.8486	0.693829
$459$	0	0
$460$	0.129680	0.00604637
$461$	−7.12099	−0.331657	−0.165829	−	0.986155i	$-0.553030\pi$
$461$	−7.12099	−0.331657	−0.165829	+	0.986155i	$0.553030\pi$
$462$	0	0
$463$	−8.15201	−0.378856	−0.189428	−	0.981895i	$-0.560663\pi$
$463$	−8.15201	−0.378856	−0.189428	+	0.981895i	$0.560663\pi$
$464$	−6.60596	−0.306674
$465$	0	0
$466$	−9.37149	−0.434126
$467$	3.56520	0.164978	0.0824889	−	0.996592i	$-0.473713\pi$
$467$	3.56520	0.164978	0.0824889	+	0.996592i	$0.473713\pi$
$468$	0	0
$469$	13.4795	0.622426
$470$	0.0952873	0.00439527
$471$	0	0
$472$	21.3167	0.981181
$473$	−5.73739	−0.263806
$474$	0	0
$475$	23.3597	1.07182
$476$	6.74668	0.309234
$477$	0	0
$478$	19.1336	0.875150
$479$	−24.8281	−1.13442	−0.567212	−	0.823572i	$-0.691978\pi$
$479$	−24.8281	−1.13442	−0.567212	+	0.823572i	$0.691978\pi$
$480$	0	0
$481$	−9.70555	−0.442535
$482$	9.11154	0.415019
$483$	0	0
$484$	7.07091	0.321405
$485$	0.0447656	0.00203270
$486$	0	0
$487$	29.4523	1.33461	0.667306	−	0.744783i	$-0.267447\pi$
$487$	29.4523	1.33461	0.667306	+	0.744783i	$0.267447\pi$
$488$	−42.9597	−1.94469
$489$	0	0
$490$	−0.317188	−0.0143291
$491$	26.1084	1.17825	0.589127	−	0.808040i	$-0.299472\pi$
$491$	26.1084	1.17825	0.589127	+	0.808040i	$0.299472\pi$
$492$	0	0
$493$	19.6011	0.882790
$494$	−6.04512	−0.271983
$495$	0	0
$496$	1.61910	0.0726996
$497$	−14.2497	−0.639185
$498$	0	0
$499$	−22.7386	−1.01792	−0.508960	−	0.860790i	$-0.669970\pi$
$499$	−22.7386	−1.01792	−0.508960	+	0.860790i	$0.669970\pi$
$500$	−0.400596	−0.0179152
$501$	0	0
$502$	−24.8756	−1.11025
$503$	11.4082	0.508668	0.254334	−	0.967116i	$-0.418144\pi$
$503$	11.4082	0.508668	0.254334	+	0.967116i	$0.418144\pi$
$504$	0	0
$505$	0.629434	0.0280094
$506$	2.70705	0.120343
$507$	0	0
$508$	3.76755	0.167158
$509$	−33.2584	−1.47415	−0.737077	−	0.675809i	$-0.763795\pi$
$509$	−33.2584	−1.47415	−0.737077	+	0.675809i	$0.763795\pi$
$510$	0	0
$511$	−13.8871	−0.614329
$512$	21.4792	0.949257
$513$	0	0
$514$	−17.3265	−0.764238
$515$	0.327968	0.0144520
$516$	0	0
$517$	−1.01385	−0.0445892
$518$	15.2649	0.670700
$519$	0	0
$520$	0.205290	0.00900256
$521$	−21.6845	−0.950017	−0.475008	−	0.879981i	$-0.657555\pi$
$521$	−21.6845	−0.950017	−0.475008	+	0.879981i	$0.657555\pi$
$522$	0	0
$523$	14.4527	0.631973	0.315986	−	0.948764i	$-0.397665\pi$
$523$	14.4527	0.631973	0.315986	+	0.948764i	$0.397665\pi$
$524$	−13.4636	−0.588158
$525$	0	0
$526$	31.6298	1.37912
$527$	−4.80416	−0.209273
$528$	0	0
$529$	−12.5281	−0.544698
$530$	−0.952732	−0.0413840
$531$	0	0
$532$	−4.84612	−0.210106
$533$	11.6581	0.504969
$534$	0	0
$535$	0.0162206	0.000701276 0
$536$	−27.0377	−1.16785
$537$	0	0
$538$	−10.9744	−0.473139
$539$	3.37487	0.145366
$540$	0	0
$541$	28.1002	1.20812	0.604061	−	0.796938i	$-0.293548\pi$
$541$	28.1002	1.20812	0.604061	+	0.796938i	$0.293548\pi$
$542$	−16.3422	−0.701957
$543$	0	0
$544$	−23.6498	−1.01398
$545$	0.931263	0.0398909
$546$	0	0
$547$	−13.7841	−0.589367	−0.294684	−	0.955595i	$-0.595214\pi$
$547$	−13.7841	−0.589367	−0.294684	+	0.955595i	$0.595214\pi$
$548$	4.44377	0.189829
$549$	0	0
$550$	−4.17972	−0.178224
$551$	−14.0794	−0.599803
$552$	0	0
$553$	14.7762	0.628348
$554$	−20.7246	−0.880502
$555$	0	0
$556$	1.04119	0.0441562
$557$	−8.80962	−0.373276	−0.186638	−	0.982429i	$-0.559759\pi$
$557$	−8.80962	−0.373276	−0.186638	+	0.982429i	$0.559759\pi$
$558$	0	0
$559$	−8.86814	−0.375082
$560$	−0.199846	−0.00844503
$561$	0	0
$562$	10.3488	0.436537
$563$	−23.8194	−1.00387	−0.501934	−	0.864906i	$-0.667378\pi$
$563$	−23.8194	−1.00387	−0.501934	+	0.864906i	$0.667378\pi$
$564$	0	0
$565$	1.19720	0.0503665
$566$	31.3952	1.31964
$567$	0	0
$568$	28.5825	1.19930
$569$	−23.1724	−0.971436	−0.485718	−	0.874116i	$-0.661442\pi$
$569$	−23.1724	−0.971436	−0.485718	+	0.874116i	$0.661442\pi$
$570$	0	0
$571$	−7.50566	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$571$	−7.50566	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$572$	−0.551316	−0.0230517
$573$	0	0
$574$	−18.3359	−0.765325
$575$	−16.1688	−0.674285
$576$	0	0
$577$	−33.7172	−1.40366	−0.701832	−	0.712342i	$-0.747634\pi$
$577$	−33.7172	−1.40366	−0.701832	+	0.712342i	$0.747634\pi$
$578$	−29.1938	−1.21430
$579$	0	0
$580$	0.120682	0.00501103
$581$	10.3303	0.428574
$582$	0	0
$583$	10.1370	0.419833
$584$	27.8553	1.15266
$585$	0	0
$586$	−8.73859	−0.360988
$587$	16.9458	0.699429	0.349715	−	0.936856i	$-0.386279\pi$
$587$	16.9458	0.699429	0.349715	+	0.936856i	$0.386279\pi$
$588$	0	0
$589$	3.45082	0.142188
$590$	0.472893	0.0194687
$591$	0	0
$592$	−18.9515	−0.778901
$593$	−42.1996	−1.73293	−0.866465	−	0.499238i	$-0.833613\pi$
$593$	−42.1996	−1.73293	−0.866465	+	0.499238i	$0.833613\pi$
$594$	0	0
$595$	0.592980	0.0243098
$596$	1.06619	0.0436727
$597$	0	0
$598$	4.18422	0.171106
$599$	26.0476	1.06428	0.532139	−	0.846657i	$-0.321388\pi$
$599$	26.0476	1.06428	0.532139	+	0.846657i	$0.321388\pi$
$600$	0	0
$601$	17.0571	0.695773	0.347887	−	0.937537i	$-0.386899\pi$
$601$	17.0571	0.695773	0.347887	+	0.937537i	$0.386899\pi$
$602$	13.9478	0.568470
$603$	0	0
$604$	−6.21705	−0.252968
$605$	0.621476	0.0252666
$606$	0	0
$607$	35.7563	1.45130	0.725651	−	0.688063i	$-0.241539\pi$
$607$	35.7563	1.45130	0.725651	+	0.688063i	$0.241539\pi$
$608$	16.9876	0.688937
$609$	0	0
$610$	−0.953023	−0.0385868
$611$	−1.56709	−0.0633975
$612$	0	0
$613$	−5.83164	−0.235538	−0.117769	−	0.993041i	$-0.537574\pi$
$613$	−5.83164	−0.235538	−0.117769	+	0.993041i	$0.537574\pi$
$614$	2.70404	0.109126
$615$	0	0
$616$	3.43543	0.138417
$617$	−12.1589	−0.489499	−0.244750	−	0.969586i	$-0.578706\pi$
$617$	−12.1589	−0.489499	−0.244750	+	0.969586i	$0.578706\pi$
$618$	0	0
$619$	−43.5173	−1.74911	−0.874555	−	0.484927i	$-0.838846\pi$
$619$	−43.5173	−1.74911	−0.874555	+	0.484927i	$0.838846\pi$
$620$	−0.0295786	−0.00118791
$621$	0	0
$622$	−10.3977	−0.416908
$623$	10.8032	0.432819
$624$	0	0
$625$	24.9472	0.997887
$626$	5.27524	0.210841
$627$	0	0
$628$	5.65183	0.225533
$629$	56.2326	2.24214
$630$	0	0
$631$	−8.33034	−0.331626	−0.165813	−	0.986157i	$-0.553025\pi$
$631$	−8.33034	−0.331626	−0.165813	+	0.986157i	$0.553025\pi$
$632$	−29.6387	−1.17896
$633$	0	0
$634$	−0.588618	−0.0233770
$635$	0.331138	0.0131408
$636$	0	0
$637$	5.21645	0.206683
$638$	2.51921	0.0997365
$639$	0	0
$640$	0.154072	0.00609022
$641$	−18.6439	−0.736392	−0.368196	−	0.929748i	$-0.620024\pi$
$641$	−18.6439	−0.736392	−0.368196	+	0.929748i	$0.620024\pi$
$642$	0	0
$643$	−50.0060	−1.97204	−0.986022	−	0.166616i	$-0.946716\pi$
$643$	−50.0060	−1.97204	−0.986022	+	0.166616i	$0.946716\pi$
$644$	3.35432	0.132179
$645$	0	0
$646$	35.0245	1.37802
$647$	28.7022	1.12840	0.564201	−	0.825638i	$-0.309184\pi$
$647$	28.7022	1.12840	0.564201	+	0.825638i	$0.309184\pi$
$648$	0	0
$649$	−5.03156	−0.197506
$650$	−6.46048	−0.253401
$651$	0	0
$652$	12.0808	0.473121
$653$	−3.90284	−0.152730	−0.0763650	−	0.997080i	$-0.524331\pi$
$653$	−3.90284	−0.152730	−0.0763650	+	0.997080i	$0.524331\pi$
$654$	0	0
$655$	−1.18334	−0.0462369
$656$	22.7642	0.888791
$657$	0	0
$658$	2.46471	0.0960845
$659$	−37.9732	−1.47923	−0.739614	−	0.673032i	$-0.764992\pi$
$659$	−37.9732	−1.47923	−0.739614	+	0.673032i	$0.764992\pi$
$660$	0	0
$661$	−14.0911	−0.548080	−0.274040	−	0.961718i	$-0.588360\pi$
$661$	−14.0911	−0.548080	−0.274040	+	0.961718i	$0.588360\pi$
$662$	−21.8211	−0.848101
$663$	0	0
$664$	−20.7209	−0.804128
$665$	−0.425935	−0.0165171
$666$	0	0
$667$	9.74529	0.377339
$668$	−0.675235	−0.0261256
$669$	0	0
$670$	−0.599808	−0.0231726
$671$	10.1401	0.391455
$672$	0	0
$673$	−2.00760	−0.0773874	−0.0386937	−	0.999251i	$-0.512320\pi$
$673$	−2.00760	−0.0773874	−0.0386937	+	0.999251i	$0.512320\pi$
$674$	−36.3001	−1.39823
$675$	0	0
$676$	7.92591	0.304843
$677$	14.0908	0.541554	0.270777	−	0.962642i	$-0.412719\pi$
$677$	14.0908	0.541554	0.270777	+	0.962642i	$0.412719\pi$
$678$	0	0
$679$	1.15791	0.0444365
$680$	−1.18942	−0.0456122
$681$	0	0
$682$	−0.617450	−0.0236434
$683$	−15.3972	−0.589158	−0.294579	−	0.955627i	$-0.595179\pi$
$683$	−15.3972	−0.589158	−0.294579	+	0.955627i	$0.595179\pi$
$684$	0	0
$685$	0.390572	0.0149230
$686$	−20.5725	−0.785463
$687$	0	0
$688$	−17.3163	−0.660178
$689$	15.6685	0.596924
$690$	0	0
$691$	8.96256	0.340952	0.170476	−	0.985362i	$-0.445470\pi$
$691$	8.96256	0.340952	0.170476	+	0.985362i	$0.445470\pi$
$692$	−5.78562	−0.219936
$693$	0	0
$694$	−31.0084	−1.17706
$695$	0.0915121	0.00347125
$696$	0	0
$697$	−67.5455	−2.55847
$698$	−21.1550	−0.800730
$699$	0	0
$700$	−5.17911	−0.195752
$701$	51.4198	1.94210	0.971049	−	0.238881i	$-0.0767807\pi$
$701$	51.4198	1.94210	0.971049	+	0.238881i	$0.0767807\pi$
$702$	0	0
$703$	−40.3917	−1.52340
$704$	−6.22815	−0.234732
$705$	0	0
$706$	−16.3565	−0.615586
$707$	16.2810	0.612310
$708$	0	0
$709$	14.1777	0.532456	0.266228	−	0.963910i	$-0.414223\pi$
$709$	14.1777	0.532456	0.266228	+	0.963910i	$0.414223\pi$
$710$	0.634079	0.0237965
$711$	0	0
$712$	−21.6694	−0.812094
$713$	−2.38854	−0.0894514
$714$	0	0
$715$	−0.0484563	−0.00181216
$716$	−8.98239	−0.335688
$717$	0	0
$718$	−13.3142	−0.496883
$719$	−18.9176	−0.705507	−0.352754	−	0.935716i	$-0.614755\pi$
$719$	−18.9176	−0.705507	−0.352754	+	0.935716i	$0.614755\pi$
$720$	0	0
$721$	8.48326	0.315933
$722$	−3.28931	−0.122416
$723$	0	0
$724$	−6.04761	−0.224758
$725$	−15.0468	−0.558826
$726$	0	0
$727$	−4.52329	−0.167760	−0.0838798	−	0.996476i	$-0.526731\pi$
$727$	−4.52329	−0.167760	−0.0838798	+	0.996476i	$0.526731\pi$
$728$	5.31005	0.196804
$729$	0	0
$730$	0.617945	0.0228712
$731$	51.3807	1.90038
$732$	0	0
$733$	−29.3400	−1.08370	−0.541849	−	0.840476i	$-0.682276\pi$
$733$	−29.3400	−1.08370	−0.541849	+	0.840476i	$0.682276\pi$
$734$	−6.97687	−0.257521
$735$	0	0
$736$	−11.7582	−0.433414
$737$	6.38193	0.235082
$738$	0	0
$739$	−44.8214	−1.64878	−0.824391	−	0.566021i	$-0.808482\pi$
$739$	−44.8214	−1.64878	−0.824391	+	0.566021i	$0.808482\pi$
$740$	0.346217	0.0127272
$741$	0	0
$742$	−24.6435	−0.904690
$743$	−20.6928	−0.759145	−0.379573	−	0.925162i	$-0.623929\pi$
$743$	−20.6928	−0.759145	−0.379573	+	0.925162i	$0.623929\pi$
$744$	0	0
$745$	0.0937093	0.00343324
$746$	39.4511	1.44441
$747$	0	0
$748$	3.19425	0.116793
$749$	0.419563	0.0153305
$750$	0	0
$751$	−21.3810	−0.780205	−0.390103	−	0.920771i	$-0.627560\pi$
$751$	−21.3810	−0.780205	−0.390103	+	0.920771i	$0.627560\pi$
$752$	−3.05996	−0.111585
$753$	0	0
$754$	3.89388	0.141807
$755$	−0.546429	−0.0198866
$756$	0	0
$757$	8.80621	0.320067	0.160034	−	0.987112i	$-0.448840\pi$
$757$	8.80621	0.320067	0.160034	+	0.987112i	$0.448840\pi$
$758$	−10.8866	−0.395419
$759$	0	0
$760$	0.854357	0.0309908
$761$	11.7609	0.426333	0.213166	−	0.977016i	$-0.431622\pi$
$761$	11.7609	0.426333	0.213166	+	0.977016i	$0.431622\pi$
$762$	0	0
$763$	24.0881	0.872049
$764$	4.54553	0.164451
$765$	0	0
$766$	−24.0534	−0.869083
$767$	−7.77715	−0.280817
$768$	0	0
$769$	−51.4539	−1.85548	−0.927738	−	0.373232i	$-0.878250\pi$
$769$	−51.4539	−1.85548	−0.927738	+	0.373232i	$0.878250\pi$
$770$	0.0762120	0.00274649
$771$	0	0
$772$	−2.42581	−0.0873066
$773$	32.0957	1.15440	0.577201	−	0.816602i	$-0.304145\pi$
$773$	32.0957	1.15440	0.577201	+	0.816602i	$0.304145\pi$
$774$	0	0
$775$	3.68793	0.132474
$776$	−2.32258	−0.0833758
$777$	0	0
$778$	−19.3081	−0.692230
$779$	48.5177	1.73833
$780$	0	0
$781$	−6.74657	−0.241411
$782$	−24.2428	−0.866920
$783$	0	0
$784$	10.1859	0.363781
$785$	0.496751	0.0177298
$786$	0	0
$787$	45.4030	1.61844	0.809220	−	0.587505i	$-0.199890\pi$
$787$	45.4030	1.61844	0.809220	+	0.587505i	$0.199890\pi$
$788$	3.40196	0.121190
$789$	0	0
$790$	−0.657508	−0.0233931
$791$	30.9669	1.10106
$792$	0	0
$793$	15.6733	0.556576
$794$	−5.76240	−0.204500
$795$	0	0
$796$	12.7499	0.451907
$797$	0.658673	0.0233314	0.0116657	−	0.999932i	$-0.496287\pi$
$797$	0.658673	0.0233314	0.0116657	+	0.999932i	$0.496287\pi$
$798$	0	0
$799$	9.07947	0.321209
$800$	18.1548	0.641870
$801$	0	0
$802$	1.62780	0.0574797
$803$	−6.57491	−0.232023
$804$	0	0
$805$	0.294818	0.0103910
$806$	−0.954376	−0.0336165
$807$	0	0
$808$	−32.6571	−1.14887
$809$	5.32511	0.187221	0.0936105	−	0.995609i	$-0.470159\pi$
$809$	5.32511	0.187221	0.0936105	+	0.995609i	$0.470159\pi$
$810$	0	0
$811$	−12.2598	−0.430501	−0.215251	−	0.976559i	$-0.569057\pi$
$811$	−12.2598	−0.430501	−0.215251	+	0.976559i	$0.569057\pi$
$812$	3.12156	0.109545
$813$	0	0
$814$	7.22722	0.253314
$815$	1.06181	0.0371935
$816$	0	0
$817$	−36.9066	−1.29120
$818$	28.4970	0.996375
$819$	0	0
$820$	−0.415869	−0.0145228
$821$	7.24036	0.252690	0.126345	−	0.991986i	$-0.459675\pi$
$821$	7.24036	0.252690	0.126345	+	0.991986i	$0.459675\pi$
$822$	0	0
$823$	14.5934	0.508694	0.254347	−	0.967113i	$-0.418139\pi$
$823$	14.5934	0.508694	0.254347	+	0.967113i	$0.418139\pi$
$824$	−17.0160	−0.592782
$825$	0	0
$826$	12.2319	0.425602
$827$	−41.5316	−1.44419	−0.722097	−	0.691792i	$-0.756822\pi$
$827$	−41.5316	−1.44419	−0.722097	+	0.691792i	$0.756822\pi$
$828$	0	0
$829$	−4.01369	−0.139401	−0.0697006	−	0.997568i	$-0.522204\pi$
$829$	−4.01369	−0.139401	−0.0697006	+	0.997568i	$0.522204\pi$
$830$	−0.459676	−0.0159556
$831$	0	0
$832$	−9.62669	−0.333746
$833$	−30.2234	−1.04718
$834$	0	0
$835$	−0.0593478	−0.00205382
$836$	−2.29442	−0.0793541
$837$	0	0
$838$	−35.2883	−1.21901
$839$	−7.96730	−0.275062	−0.137531	−	0.990497i	$-0.543917\pi$
$839$	−7.96730	−0.275062	−0.137531	+	0.990497i	$0.543917\pi$
$840$	0	0
$841$	−19.9309	−0.687274
$842$	−39.7632	−1.37033
$843$	0	0
$844$	−7.22695	−0.248762
$845$	0.696624	0.0239646
$846$	0	0
$847$	16.0752	0.552349
$848$	30.5951	1.05064
$849$	0	0
$850$	37.4311	1.28388
$851$	27.9577	0.958379
$852$	0	0
$853$	31.0202	1.06211	0.531056	−	0.847337i	$-0.321795\pi$
$853$	31.0202	1.06211	0.531056	+	0.847337i	$0.321795\pi$
$854$	−24.6510	−0.843540
$855$	0	0
$856$	−0.841575	−0.0287644
$857$	−14.6173	−0.499318	−0.249659	−	0.968334i	$-0.580319\pi$
$857$	−14.6173	−0.499318	−0.249659	+	0.968334i	$0.580319\pi$
$858$	0	0
$859$	32.1234	1.09604	0.548018	−	0.836467i	$-0.315383\pi$
$859$	32.1234	1.09604	0.548018	+	0.836467i	$0.315383\pi$
$860$	0.316345	0.0107873
$861$	0	0
$862$	25.8585	0.880743
$863$	−40.7770	−1.38806	−0.694032	−	0.719944i	$-0.744167\pi$
$863$	−40.7770	−1.38806	−0.694032	+	0.719944i	$0.744167\pi$
$864$	0	0
$865$	−0.508510	−0.0172899
$866$	−3.86560	−0.131358
$867$	0	0
$868$	−0.765084	−0.0259687
$869$	6.99586	0.237318
$870$	0	0
$871$	9.86439	0.334242
$872$	−48.3169	−1.63622
$873$	0	0
$874$	17.4135	0.589021
$875$	−0.910725	−0.0307881
$876$	0	0
$877$	−37.5229	−1.26706	−0.633528	−	0.773719i	$-0.718394\pi$
$877$	−37.5229	−1.26706	−0.633528	+	0.773719i	$0.718394\pi$
$878$	−20.6645	−0.697393
$879$	0	0
$880$	−0.0946178	−0.00318957
$881$	−38.0924	−1.28336	−0.641682	−	0.766971i	$-0.721763\pi$
$881$	−38.0924	−1.28336	−0.641682	+	0.766971i	$0.721763\pi$
$882$	0	0
$883$	19.0442	0.640890	0.320445	−	0.947267i	$-0.396168\pi$
$883$	19.0442	0.640890	0.320445	+	0.947267i	$0.396168\pi$
$884$	4.93727	0.166058
$885$	0	0
$886$	21.5858	0.725190
$887$	−27.3252	−0.917491	−0.458745	−	0.888568i	$-0.651701\pi$
$887$	−27.3252	−0.917491	−0.458745	+	0.888568i	$0.651701\pi$
$888$	0	0
$889$	8.56525	0.287269
$890$	−0.480716	−0.0161136
$891$	0	0
$892$	4.81585	0.161246
$893$	−6.52176	−0.218242
$894$	0	0
$895$	−0.789481	−0.0263894
$896$	3.98524	0.133137
$897$	0	0
$898$	−5.51659	−0.184091
$899$	−2.22280	−0.0741344
$900$	0	0
$901$	−90.7813	−3.02436
$902$	−8.68121	−0.289053
$903$	0	0
$904$	−62.1145	−2.06590
$905$	−0.531536	−0.0176689
$906$	0	0
$907$	−54.3781	−1.80559	−0.902797	−	0.430067i	$-0.858490\pi$
$907$	−54.3781	−1.80559	−0.902797	+	0.430067i	$0.858490\pi$
$908$	7.85190	0.260575
$909$	0	0
$910$	0.117799	0.00390500
$911$	39.5068	1.30892	0.654460	−	0.756097i	$-0.272896\pi$
$911$	39.5068	1.30892	0.654460	+	0.756097i	$0.272896\pi$
$912$	0	0
$913$	4.89093	0.161866
$914$	34.0275	1.12553
$915$	0	0
$916$	8.71106	0.287821
$917$	−30.6084	−1.01078
$918$	0	0
$919$	39.2491	1.29471	0.647354	−	0.762190i	$-0.275876\pi$
$919$	39.2491	1.29471	0.647354	+	0.762190i	$0.275876\pi$
$920$	−0.591357	−0.0194965
$921$	0	0
$922$	8.19614	0.269926
$923$	−10.4280	−0.343242
$924$	0	0
$925$	−43.1670	−1.41932
$926$	9.38283	0.308339
$927$	0	0
$928$	−10.9423	−0.359199
$929$	3.32230	0.109001	0.0545006	−	0.998514i	$-0.482643\pi$
$929$	3.32230	0.109001	0.0545006	+	0.998514i	$0.482643\pi$
$930$	0	0
$931$	21.7094	0.711495
$932$	−5.49787	−0.180089
$933$	0	0
$934$	−4.10349	−0.134270
$935$	0.280749	0.00918147
$936$	0	0
$937$	2.16258	0.0706484	0.0353242	−	0.999376i	$-0.488754\pi$
$937$	2.16258	0.0706484	0.0353242	+	0.999376i	$0.488754\pi$
$938$	−15.5147	−0.506573
$939$	0	0
$940$	0.0559012	0.00182329
$941$	22.0637	0.719255	0.359628	−	0.933096i	$-0.382904\pi$
$941$	22.0637	0.719255	0.359628	+	0.933096i	$0.382904\pi$
$942$	0	0
$943$	−33.5823	−1.09359
$944$	−15.1860	−0.494262
$945$	0	0
$946$	6.60365	0.214703
$947$	−54.8587	−1.78267	−0.891334	−	0.453347i	$-0.850230\pi$
$947$	−54.8587	−1.78267	−0.891334	+	0.453347i	$0.850230\pi$
$948$	0	0
$949$	−10.1627	−0.329894
$950$	−26.8867	−0.872318
$951$	0	0
$952$	−30.7657	−0.997121
$953$	−35.6512	−1.15486	−0.577428	−	0.816441i	$-0.695944\pi$
$953$	−35.6512	−1.15486	−0.577428	+	0.816441i	$0.695944\pi$
$954$	0	0
$955$	0.399516	0.0129280
$956$	11.2249	0.363039
$957$	0	0
$958$	28.5767	0.923271
$959$	10.1026	0.326229
$960$	0	0
$961$	−30.4552	−0.982426
$962$	11.1709	0.360165
$963$	0	0
$964$	5.34537	0.172163
$965$	−0.213209	−0.00686344
$966$	0	0
$967$	54.3232	1.74692	0.873458	−	0.486899i	$-0.161872\pi$
$967$	54.3232	1.74692	0.873458	+	0.486899i	$0.161872\pi$
$968$	−32.2442	−1.03637
$969$	0	0
$970$	−0.0515244	−0.00165435
$971$	−7.33160	−0.235282	−0.117641	−	0.993056i	$-0.537533\pi$
$971$	−7.33160	−0.235282	−0.117641	+	0.993056i	$0.537533\pi$
$972$	0	0
$973$	2.36706	0.0758845
$974$	−33.8992	−1.08620
$975$	0	0
$976$	30.6044	0.979624
$977$	−4.90524	−0.156933	−0.0784663	−	0.996917i	$-0.525002\pi$
$977$	−4.90524	−0.156933	−0.0784663	+	0.996917i	$0.525002\pi$
$978$	0	0
$979$	5.11480	0.163470
$980$	−0.186082	−0.00594416
$981$	0	0
$982$	−30.0503	−0.958944
$983$	45.3449	1.44628	0.723138	−	0.690703i	$-0.242699\pi$
$983$	45.3449	1.44628	0.723138	+	0.690703i	$0.242699\pi$
$984$	0	0
$985$	0.299005	0.00952710
$986$	−22.5606	−0.718475
$987$	0	0
$988$	−3.54642	−0.112827
$989$	25.5455	0.812300
$990$	0	0
$991$	62.1348	1.97378	0.986889	−	0.161402i	$-0.0516016\pi$
$991$	62.1348	1.97378	0.986889	+	0.161402i	$0.0516016\pi$
$992$	2.68192	0.0851512
$993$	0	0
$994$	16.4011	0.520213
$995$	1.12061	0.0355258
$996$	0	0
$997$	28.6058	0.905955	0.452978	−	0.891522i	$-0.350362\pi$
$997$	28.6058	0.905955	0.452978	+	0.891522i	$0.350362\pi$
$998$	26.1718	0.828453
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.c.1.2		5
3.2	odd	2		501.2.a.c.1.4	✓	5
12.11	even	2		8016.2.a.u.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.c.1.4	✓	5	3.2	odd	2
1503.2.a.c.1.2		5	1.1	even	1	trivial
8016.2.a.u.1.3		5	12.11	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q-1.15098 q^{2} -0.675235 q^{4} -0.0593478 q^{5} -1.53510 q^{7} +3.07915 q^{8} +O(q^{10})\)	\(q-1.15098 q^{2} -0.675235 q^{4} -0.0593478 q^{5} -1.53510 q^{7} +3.07915 q^{8} +0.0683084 q^{10} -0.726798 q^{11} -1.12339 q^{13} +1.76687 q^{14} -2.19359 q^{16} +6.50878 q^{17} -4.67524 q^{19} +0.0400737 q^{20} +0.836533 q^{22} +3.23604 q^{23} -4.99648 q^{25} +1.29301 q^{26} +1.03655 q^{28} +3.01149 q^{29} -0.738105 q^{31} -3.63352 q^{32} -7.49150 q^{34} +0.0911046 q^{35} +8.63949 q^{37} +5.38112 q^{38} -0.182741 q^{40} -10.3776 q^{41} +7.89407 q^{43} +0.490760 q^{44} -3.72463 q^{46} +1.39496 q^{47} -4.64348 q^{49} +5.75087 q^{50} +0.758555 q^{52} -13.9475 q^{53} +0.0431339 q^{55} -4.72680 q^{56} -3.46618 q^{58} +6.92291 q^{59} -13.9518 q^{61} +0.849548 q^{62} +8.56930 q^{64} +0.0666709 q^{65} -8.78089 q^{67} -4.39496 q^{68} -0.104860 q^{70} +9.28259 q^{71} +9.04640 q^{73} -9.94392 q^{74} +3.15688 q^{76} +1.11571 q^{77} -9.62559 q^{79} +0.130185 q^{80} +11.9445 q^{82} -6.72942 q^{83} -0.386282 q^{85} -9.08594 q^{86} -2.23792 q^{88} -7.03744 q^{89} +1.72452 q^{91} -2.18509 q^{92} -1.60557 q^{94} +0.277465 q^{95} -0.754292 q^{97} +5.34457 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{5} - 4 q^{7}+O(q^{10}) \) 5 * q + q^5 - 4 * q^7	\( 5 q + q^{5} - 4 q^{7} - 9 q^{10} + 7 q^{11} - 8 q^{13} + q^{14} - 6 q^{16} - 5 q^{17} - 20 q^{19} + 3 q^{20} - q^{22} - q^{23} - 6 q^{25} + 3 q^{26} - 11 q^{28} + 5 q^{29} - 18 q^{31} + 5 q^{32} - 12 q^{34} + 6 q^{35} - 17 q^{37} - 2 q^{40} - 6 q^{41} - 10 q^{43} + 9 q^{44} - q^{46} + 3 q^{47} - 13 q^{49} - 9 q^{50} - 15 q^{53} - 9 q^{55} - 13 q^{56} + 21 q^{58} + 17 q^{59} - 2 q^{61} - 10 q^{64} - 26 q^{65} - 24 q^{67} - 18 q^{68} + 25 q^{70} - q^{71} + 2 q^{73} - 17 q^{74} + 14 q^{76} - 26 q^{77} - 6 q^{79} - 12 q^{80} + 17 q^{82} + 7 q^{83} - 11 q^{85} - 15 q^{86} + 4 q^{88} - 15 q^{91} - 45 q^{92} - 2 q^{94} - q^{95} - 13 q^{97} + 16 q^{98}+O(q^{100}) \) 5 * q + q^5 - 4 * q^7 - 9 * q^10 + 7 * q^11 - 8 * q^13 + q^14 - 6 * q^16 - 5 * q^17 - 20 * q^19 + 3 * q^20 - q^22 - q^23 - 6 * q^25 + 3 * q^26 - 11 * q^28 + 5 * q^29 - 18 * q^31 + 5 * q^32 - 12 * q^34 + 6 * q^35 - 17 * q^37 - 2 * q^40 - 6 * q^41 - 10 * q^43 + 9 * q^44 - q^46 + 3 * q^47 - 13 * q^49 - 9 * q^50 - 15 * q^53 - 9 * q^55 - 13 * q^56 + 21 * q^58 + 17 * q^59 - 2 * q^61 - 10 * q^64 - 26 * q^65 - 24 * q^67 - 18 * q^68 + 25 * q^70 - q^71 + 2 * q^73 - 17 * q^74 + 14 * q^76 - 26 * q^77 - 6 * q^79 - 12 * q^80 + 17 * q^82 + 7 * q^83 - 11 * q^85 - 15 * q^86 + 4 * q^88 - 15 * q^91 - 45 * q^92 - 2 * q^94 - q^95 - 13 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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