LMFDB - Embedded newform 501.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [501,2,Mod(1,501)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(501, 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("501.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 501 = 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	501.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:	$4.00050514127$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.15098$ of defining polynomial
Character	$\chi$	$=$	501.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15098 q^{2} -1.00000 q^{3} -0.675235 q^{4} +0.0593478 q^{5} -1.15098 q^{6} -1.53510 q^{7} -3.07915 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.15098 q^{2} -1.00000 q^{3} -0.675235 q^{4} +0.0593478 q^{5} -1.15098 q^{6} -1.53510 q^{7} -3.07915 q^{8} +1.00000 q^{9} +0.0683084 q^{10} +0.726798 q^{11} +0.675235 q^{12} -1.12339 q^{13} -1.76687 q^{14} -0.0593478 q^{15} -2.19359 q^{16} -6.50878 q^{17} +1.15098 q^{18} -4.67524 q^{19} -0.0400737 q^{20} +1.53510 q^{21} +0.836533 q^{22} -3.23604 q^{23} +3.07915 q^{24} -4.99648 q^{25} -1.29301 q^{26} -1.00000 q^{27} +1.03655 q^{28} -3.01149 q^{29} -0.0683084 q^{30} -0.738105 q^{31} +3.63352 q^{32} -0.726798 q^{33} -7.49150 q^{34} -0.0911046 q^{35} -0.675235 q^{36} +8.63949 q^{37} -5.38112 q^{38} +1.12339 q^{39} -0.182741 q^{40} +10.3776 q^{41} +1.76687 q^{42} +7.89407 q^{43} -0.490760 q^{44} +0.0593478 q^{45} -3.72463 q^{46} -1.39496 q^{47} +2.19359 q^{48} -4.64348 q^{49} -5.75087 q^{50} +6.50878 q^{51} +0.758555 q^{52} +13.9475 q^{53} -1.15098 q^{54} +0.0431339 q^{55} +4.72680 q^{56} +4.67524 q^{57} -3.46618 q^{58} -6.92291 q^{59} +0.0400737 q^{60} -13.9518 q^{61} -0.849548 q^{62} -1.53510 q^{63} +8.56930 q^{64} -0.0666709 q^{65} -0.836533 q^{66} -8.78089 q^{67} +4.39496 q^{68} +3.23604 q^{69} -0.104860 q^{70} -9.28259 q^{71} -3.07915 q^{72} +9.04640 q^{73} +9.94392 q^{74} +4.99648 q^{75} +3.15688 q^{76} -1.11571 q^{77} +1.29301 q^{78} -9.62559 q^{79} -0.130185 q^{80} +1.00000 q^{81} +11.9445 q^{82} +6.72942 q^{83} -1.03655 q^{84} -0.386282 q^{85} +9.08594 q^{86} +3.01149 q^{87} -2.23792 q^{88} +7.03744 q^{89} +0.0683084 q^{90} +1.72452 q^{91} +2.18509 q^{92} +0.738105 q^{93} -1.60557 q^{94} -0.277465 q^{95} -3.63352 q^{96} -0.754292 q^{97} -5.34457 q^{98} +0.726798 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - q^5 - 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - q^{5} - 4 q^{7} + 5 q^{9} - 9 q^{10} - 7 q^{11} - 8 q^{13} - q^{14} + q^{15} - 6 q^{16} + 5 q^{17} - 20 q^{19} - 3 q^{20} + 4 q^{21} - q^{22} + q^{23} - 6 q^{25} - 3 q^{26} - 5 q^{27} - 11 q^{28} - 5 q^{29} + 9 q^{30} - 18 q^{31} - 5 q^{32} + 7 q^{33} - 12 q^{34} - 6 q^{35} - 17 q^{37} + 8 q^{39} - 2 q^{40} + 6 q^{41} + q^{42} - 10 q^{43} - 9 q^{44} - q^{45} - q^{46} - 3 q^{47} + 6 q^{48} - 13 q^{49} + 9 q^{50} - 5 q^{51} + 15 q^{53} - 9 q^{55} + 13 q^{56} + 20 q^{57} + 21 q^{58} - 17 q^{59} + 3 q^{60} - 2 q^{61} - 4 q^{63} - 10 q^{64} + 26 q^{65} + q^{66} - 24 q^{67} + 18 q^{68} - q^{69} + 25 q^{70} + q^{71} + 2 q^{73} + 17 q^{74} + 6 q^{75} + 14 q^{76} + 26 q^{77} + 3 q^{78} - 6 q^{79} + 12 q^{80} + 5 q^{81} + 17 q^{82} - 7 q^{83} + 11 q^{84} - 11 q^{85} + 15 q^{86} + 5 q^{87} + 4 q^{88} - 9 q^{90} - 15 q^{91} + 45 q^{92} + 18 q^{93} - 2 q^{94} + q^{95} + 5 q^{96} - 13 q^{97} - 16 q^{98} - 7 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - q^5 - 4 * q^7 + 5 * q^9 - 9 * q^10 - 7 * q^11 - 8 * q^13 - q^14 + q^15 - 6 * q^16 + 5 * q^17 - 20 * q^19 - 3 * q^20 + 4 * q^21 - q^22 + q^23 - 6 * q^25 - 3 * q^26 - 5 * q^27 - 11 * q^28 - 5 * q^29 + 9 * q^30 - 18 * q^31 - 5 * q^32 + 7 * q^33 - 12 * q^34 - 6 * q^35 - 17 * q^37 + 8 * q^39 - 2 * q^40 + 6 * q^41 + q^42 - 10 * q^43 - 9 * q^44 - q^45 - q^46 - 3 * q^47 + 6 * q^48 - 13 * q^49 + 9 * q^50 - 5 * q^51 + 15 * q^53 - 9 * q^55 + 13 * q^56 + 20 * q^57 + 21 * q^58 - 17 * q^59 + 3 * q^60 - 2 * q^61 - 4 * q^63 - 10 * q^64 + 26 * q^65 + q^66 - 24 * q^67 + 18 * q^68 - q^69 + 25 * q^70 + q^71 + 2 * q^73 + 17 * q^74 + 6 * q^75 + 14 * q^76 + 26 * q^77 + 3 * q^78 - 6 * q^79 + 12 * q^80 + 5 * q^81 + 17 * q^82 - 7 * q^83 + 11 * q^84 - 11 * q^85 + 15 * q^86 + 5 * q^87 + 4 * q^88 - 9 * q^90 - 15 * q^91 + 45 * q^92 + 18 * q^93 - 2 * q^94 + q^95 + 5 * q^96 - 13 * q^97 - 16 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.15098	0.813869	0.406934	−	0.913457i	$-0.366598\pi$
$2$	1.15098	0.813869	0.406934	+	0.913457i	$0.366598\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.675235	−0.337618
$5$	0.0593478	0.0265411	0.0132706	−	0.999912i	$-0.495776\pi$
$5$	0.0593478	0.0265411	0.0132706	+	0.999912i	$0.495776\pi$
$6$	−1.15098	−0.469887
$7$	−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
$9$	1.00000	0.333333
$10$	0.0683084	0.0216010
$11$	0.726798	0.219138	0.109569	−	0.993979i	$-0.465053\pi$
$11$	0.726798	0.219138	0.109569	+	0.993979i	$0.465053\pi$
$12$	0.675235	0.194924
$13$	−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
$15$	−0.0593478	−0.0153235
$16$	−2.19359	−0.548397
$17$	−6.50878	−1.57861	−0.789305	−	0.614001i	$-0.789559\pi$
$17$	−6.50878	−1.57861	−0.789305	+	0.614001i	$0.789559\pi$
$18$	1.15098	0.271290
$19$	−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
$21$	1.53510	0.334986
$22$	0.836533	0.178349
$23$	−3.23604	−0.674761	−0.337380	−	0.941368i	$-0.609541\pi$
$23$	−3.23604	−0.674761	−0.337380	+	0.941368i	$0.609541\pi$
$24$	3.07915	0.628530
$25$	−4.99648	−0.999296
$26$	−1.29301	−0.253580
$27$	−1.00000	−0.192450
$28$	1.03655	0.195890
$29$	−3.01149	−0.559219	−0.279610	−	0.960114i	$-0.590205\pi$
$29$	−3.01149	−0.559219	−0.279610	+	0.960114i	$0.590205\pi$
$30$	−0.0683084	−0.0124713
$31$	−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
$33$	−0.726798	−0.126519
$34$	−7.49150	−1.28478
$35$	−0.0911046	−0.0153995
$36$	−0.675235	−0.112539
$37$	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
$39$	1.12339	0.179887
$40$	−0.182741	−0.0288939
$41$	10.3776	1.62071	0.810354	−	0.585940i	$-0.199275\pi$
$41$	10.3776	1.62071	0.810354	+	0.585940i	$0.199275\pi$
$42$	1.76687	0.272634
$43$	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
$45$	0.0593478	0.00884705
$46$	−3.72463	−0.549167
$47$	−1.39496	−0.203475	−0.101738	−	0.994811i	$-0.532440\pi$
$47$	−1.39496	−0.203475	−0.101738	+	0.994811i	$0.532440\pi$
$48$	2.19359	0.316617
$49$	−4.64348	−0.663354
$50$	−5.75087	−0.813295
$51$	6.50878	0.911411
$52$	0.758555	0.105193
$53$	13.9475	1.91584	0.957919	−	0.287038i	$-0.0926707\pi$
$53$	13.9475	1.91584	0.957919	+	0.287038i	$0.0926707\pi$
$54$	−1.15098	−0.156629
$55$	0.0431339	0.00581617
$56$	4.72680	0.631645
$57$	4.67524	0.619250
$58$	−3.46618	−0.455131
$59$	−6.92291	−0.901286	−0.450643	−	0.892704i	$-0.648805\pi$
$59$	−6.92291	−0.901286	−0.450643	+	0.892704i	$0.648805\pi$
$60$	0.0400737	0.00517350
$61$	−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
$63$	−1.53510	−0.193404
$64$	8.56930	1.07116
$65$	−0.0666709	−0.00826951
$66$	−0.836533	−0.102970
$67$	−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
$69$	3.23604	0.389573
$70$	−0.104860	−0.0125332
$71$	−9.28259	−1.10164	−0.550820	−	0.834624i	$-0.685685\pi$
$71$	−9.28259	−1.10164	−0.550820	+	0.834624i	$0.685685\pi$
$72$	−3.07915	−0.362882
$73$	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
$75$	4.99648	0.576944
$76$	3.15688	0.362120
$77$	−1.11571	−0.127146
$78$	1.29301	0.146404
$79$	−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
$81$	1.00000	0.111111
$82$	11.9445	1.31904
$83$	6.72942	0.738650	0.369325	−	0.929300i	$-0.379589\pi$
$83$	6.72942	0.738650	0.369325	+	0.929300i	$0.379589\pi$
$84$	−1.03655	−0.113097
$85$	−0.386282	−0.0418981
$86$	9.08594	0.979763
$87$	3.01149	0.322866
$88$	−2.23792	−0.238563
$89$	7.03744	0.745967	0.372984	−	0.927838i	$-0.378335\pi$
$89$	7.03744	0.745967	0.372984	+	0.927838i	$0.378335\pi$
$90$	0.0683084	0.00720033
$91$	1.72452	0.180778
$92$	2.18509	0.227811
$93$	0.738105	0.0765380
$94$	−1.60557	−0.165602
$95$	−0.277465	−0.0284673
$96$	−3.63352	−0.370845
$97$	−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
$99$	0.726798	0.0730460
$100$	3.37380	0.337380
$101$	10.6059	1.05532	0.527661	−	0.849455i	$-0.323069\pi$
$101$	10.6059	1.05532	0.527661	+	0.849455i	$0.323069\pi$
$102$	7.49150	0.741769
$103$	−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
$105$	0.0911046	0.00889090
$106$	16.0534	1.55924
$107$	0.273314	0.0264222	0.0132111	−	0.999913i	$-0.495795\pi$
$107$	0.273314	0.0264222	0.0132111	+	0.999913i	$0.495795\pi$
$108$	0.675235	0.0649746
$109$	−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
$111$	−8.63949	−0.820025
$112$	3.36737	0.318186
$113$	20.1726	1.89768	0.948839	−	0.315760i	$-0.102260\pi$
$113$	20.1726	1.89768	0.948839	+	0.315760i	$0.102260\pi$
$114$	5.38112	0.503988
$115$	−0.192052	−0.0179089
$116$	2.03346	0.188802
$117$	−1.12339	−0.103858
$118$	−7.96816	−0.733529
$119$	9.99160	0.915929
$120$	0.182741	0.0166819
$121$	−10.4718	−0.951979
$122$	−16.0583	−1.45385
$123$	−10.3776	−0.935717
$124$	0.498395	0.0447572
$125$	−0.593269	−0.0530636
$126$	−1.76687	−0.157405
$127$	−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
$129$	−7.89407	−0.695034
$130$	−0.0767372	−0.00673029
$131$	−19.9390	−1.74208	−0.871041	−	0.491210i	$-0.836555\pi$
$131$	−19.9390	−1.74208	−0.871041	+	0.491210i	$0.836555\pi$
$132$	0.490760	0.0427152
$133$	7.17694	0.622319
$134$	−10.1067	−0.873083
$135$	−0.0593478	−0.00510784
$136$	20.0415	1.71855
$137$	6.58107	0.562259	0.281129	−	0.959670i	$-0.409291\pi$
$137$	6.58107	0.562259	0.281129	+	0.959670i	$0.409291\pi$
$138$	3.72463	0.317061
$139$	−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
$141$	1.39496	0.117477
$142$	−10.6841	−0.896591
$143$	−0.816480	−0.0682775
$144$	−2.19359	−0.182799
$145$	−0.178725	−0.0148423
$146$	10.4123	0.861725
$147$	4.64348	0.382988
$148$	−5.83369	−0.479527
$149$	1.57899	0.129356	0.0646778	−	0.997906i	$-0.479398\pi$
$149$	1.57899	0.129356	0.0646778	+	0.997906i	$0.479398\pi$
$150$	5.75087	0.469556
$151$	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
$153$	−6.50878	−0.526204
$154$	−1.28416	−0.103480
$155$	−0.0438049	−0.00351850
$156$	−0.758555	−0.0607330
$157$	−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
$159$	−13.9475	−1.10611
$160$	0.215642	0.0170480
$161$	4.96763	0.391504
$162$	1.15098	0.0904299
$163$	−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
$165$	−0.0431339	−0.00335797
$166$	7.74546	0.601164
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	−4.72680	−0.364680
$169$	−11.7380	−0.902922
$170$	−0.444604	−0.0340996
$171$	−4.67524	−0.357524
$172$	−5.33035	−0.406436
$173$	−8.56831	−0.651436	−0.325718	−	0.945467i	$-0.605606\pi$
$173$	−8.56831	−0.651436	−0.325718	+	0.945467i	$0.605606\pi$
$174$	3.46618	0.262770
$175$	7.67008	0.579803
$176$	−1.59429	−0.120174
$177$	6.92291	0.520358
$178$	8.09998	0.607120
$179$	−13.3026	−0.994284	−0.497142	−	0.867669i	$-0.665617\pi$
$179$	−13.3026	−0.994284	−0.497142	+	0.867669i	$0.665617\pi$
$180$	−0.0400737	−0.00298692
$181$	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
$183$	13.9518	1.03134
$184$	9.96426	0.734575
$185$	0.512735	0.0376970
$186$	0.849548	0.0622919
$187$	−4.73057	−0.345933
$188$	0.941925	0.0686969
$189$	1.53510	0.111662
$190$	−0.319358	−0.0231686
$191$	6.73177	0.487094	0.243547	−	0.969889i	$-0.421689\pi$
$191$	6.73177	0.487094	0.243547	+	0.969889i	$0.421689\pi$
$192$	−8.56930	−0.618436
$193$	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
$195$	0.0666709	0.00477440
$196$	3.13544	0.223960
$197$	5.03819	0.358956	0.179478	−	0.983762i	$-0.442559\pi$
$197$	5.03819	0.358956	0.179478	+	0.983762i	$0.442559\pi$
$198$	0.836533	0.0594498
$199$	−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
$201$	8.78089	0.619356
$202$	12.2072	0.858893
$203$	4.62293	0.324466
$204$	−4.39496	−0.307709
$205$	0.615888	0.0430155
$206$	−6.36058	−0.443162
$207$	−3.23604	−0.224920
$208$	2.46426	0.170866
$209$	−3.39795	−0.235041
$210$	0.104860	0.00723602
$211$	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
$213$	9.28259	0.636033
$214$	0.314580	0.0215042
$215$	0.468495	0.0319511
$216$	3.07915	0.209510
$217$	1.13306	0.0769173
$218$	−18.0608	−1.22323
$219$	−9.04640	−0.611299
$220$	−0.0291255	−0.00196364
$221$	7.31192	0.491853
$222$	−9.94392	−0.667392
$223$	−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
$225$	−4.99648	−0.333099
$226$	23.2183	1.54446
$227$	11.6284	0.771803	0.385902	−	0.922540i	$-0.373890\pi$
$227$	11.6284	0.771803	0.385902	+	0.922540i	$0.373890\pi$
$228$	−3.15688	−0.209070
$229$	−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
$231$	1.11571	0.0734080
$232$	9.27284	0.608792
$233$	−8.14215	−0.533410	−0.266705	−	0.963778i	$-0.585935\pi$
$233$	−8.14215	−0.533410	−0.266705	+	0.963778i	$0.585935\pi$
$234$	−1.29301	−0.0845266
$235$	−0.0827877	−0.00540047
$236$	4.67460	0.304290
$237$	9.62559	0.625249
$238$	11.5002	0.745446
$239$	16.6237	1.07530	0.537648	−	0.843169i	$-0.319313\pi$
$239$	16.6237	1.07530	0.537648	+	0.843169i	$0.319313\pi$
$240$	0.130185	0.00840337
$241$	−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
$243$	−1.00000	−0.0641500
$244$	9.42073	0.603101
$245$	−0.275580	−0.0176062
$246$	−11.9445	−0.761551
$247$	5.25213	0.334185
$248$	2.27274	0.144319
$249$	−6.72942	−0.426460
$250$	−0.682843	−0.0431868
$251$	−21.6125	−1.36417	−0.682084	−	0.731274i	$-0.738926\pi$
$251$	−21.6125	−1.36417	−0.682084	+	0.731274i	$0.738926\pi$
$252$	1.03655	0.0652966
$253$	−2.35195	−0.147866
$254$	−6.42205	−0.402955
$255$	0.386282	0.0241899
$256$	−14.1506	−0.884410
$257$	−15.0536	−0.939019	−0.469509	−	0.882928i	$-0.655569\pi$
$257$	−15.0536	−0.939019	−0.469509	+	0.882928i	$0.655569\pi$
$258$	−9.08594	−0.565666
$259$	−13.2625	−0.824089
$260$	0.0450186	0.00279193
$261$	−3.01149	−0.186406
$262$	−22.9495	−1.41783
$263$	27.4806	1.69453	0.847265	−	0.531171i	$-0.178248\pi$
$263$	27.4806	1.69453	0.847265	+	0.531171i	$0.178248\pi$
$264$	2.23792	0.137735
$265$	0.827754	0.0508485
$266$	8.26054	0.506486
$267$	−7.03744	−0.430684
$268$	5.92917	0.362182
$269$	−9.53477	−0.581345	−0.290673	−	0.956823i	$-0.593879\pi$
$269$	−9.53477	−0.581345	−0.290673	+	0.956823i	$0.593879\pi$
$270$	−0.0683084	−0.00415712
$271$	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
$273$	−1.72452	−0.104373
$274$	7.57471	0.457605
$275$	−3.63143	−0.218983
$276$	−2.18509	−0.131527
$277$	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
$279$	−0.738105	−0.0441892
$280$	0.280525	0.0167646
$281$	8.99125	0.536373	0.268186	−	0.963367i	$-0.413576\pi$
$281$	8.99125	0.536373	0.268186	+	0.963367i	$0.413576\pi$
$282$	1.60557	0.0956105
$283$	−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
$285$	0.277465	0.0164356
$286$	−0.939756	−0.0555689
$287$	−15.9306	−0.940355
$288$	3.63352	0.214107
$289$	25.3642	1.49201
$290$	−0.205710	−0.0120797
$291$	0.754292	0.0442174
$292$	−6.10845	−0.357470
$293$	−7.59228	−0.443545	−0.221773	−	0.975098i	$-0.571184\pi$
$293$	−7.59228	−0.443545	−0.221773	+	0.975098i	$0.571184\pi$
$294$	5.34457	0.311702
$295$	−0.410860	−0.0239212
$296$	−26.6023	−1.54623
$297$	−0.726798	−0.0421731
$298$	1.81739	0.105278
$299$	3.63534	0.210237
$300$	−3.37380	−0.194786
$301$	−12.1182	−0.698479
$302$	10.5974	0.609811
$303$	−10.6059	−0.609290
$304$	10.2555	0.588195
$305$	−0.828007	−0.0474115
$306$	−7.49150	−0.428261
$307$	−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
$309$	5.52621	0.314375
$310$	−0.0504188	−0.00286359
$311$	−9.03371	−0.512255	−0.256127	−	0.966643i	$-0.582447\pi$
$311$	−9.03371	−0.512255	−0.256127	+	0.966643i	$0.582447\pi$
$312$	−3.45910	−0.195833
$313$	−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
$315$	−0.0911046	−0.00513316
$316$	6.49954	0.365628
$317$	−0.511404	−0.0287233	−0.0143616	−	0.999897i	$-0.504572\pi$
$317$	−0.511404	−0.0287233	−0.0143616	+	0.999897i	$0.504572\pi$
$318$	−16.0534	−0.900228
$319$	−2.18874	−0.122546
$320$	0.508569	0.0284299
$321$	−0.273314	−0.0152549
$322$	5.71766	0.318633
$323$	30.4301	1.69317
$324$	−0.675235	−0.0375131
$325$	5.61301	0.311354
$326$	−20.5925	−1.14052
$327$	15.6916	0.867748
$328$	−31.9542	−1.76438
$329$	2.14139	0.118059
$330$	−0.0496464	−0.00273294
$331$	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
$333$	8.63949	0.473441
$334$	−1.15098	−0.0629791
$335$	−0.521126	−0.0284722
$336$	−3.36737	−0.183705
$337$	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
$339$	−20.1726	−1.09562
$340$	0.260831	0.0141455
$341$	−0.536454	−0.0290506
$342$	−5.38112	−0.290978
$343$	17.8739	0.965098
$344$	−24.3070	−1.31055
$345$	0.192052	0.0103397
$346$	−9.86199	−0.530184
$347$	−26.9408	−1.44626	−0.723129	−	0.690713i	$-0.757297\pi$
$347$	−26.9408	−1.44626	−0.723129	+	0.690713i	$0.757297\pi$
$348$	−2.03346	−0.109005
$349$	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
$351$	1.12339	0.0599623
$352$	2.64084	0.140757
$353$	−14.2109	−0.756370	−0.378185	−	0.925730i	$-0.623452\pi$
$353$	−14.2109	−0.756370	−0.378185	+	0.925730i	$0.623452\pi$
$354$	7.96816	0.423503
$355$	−0.550901	−0.0292388
$356$	−4.75193	−0.251852
$357$	−9.99160	−0.528812
$358$	−15.3111	−0.809217
$359$	−11.5677	−0.610520	−0.305260	−	0.952269i	$-0.598743\pi$
$359$	−11.5677	−0.610520	−0.305260	+	0.952269i	$0.598743\pi$
$360$	−0.182741	−0.00963129
$361$	2.85783	0.150412
$362$	10.3086	0.541805
$363$	10.4718	0.549625
$364$	−1.16445	−0.0610340
$365$	0.536884	0.0281018
$366$	16.0583	0.839379
$367$	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
$369$	10.3776	0.540236
$370$	0.590150	0.0306804
$371$	−21.4108	−1.11159
$372$	−0.498395	−0.0258406
$373$	−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
$375$	0.593269	0.0306363
$376$	4.29529	0.221513
$377$	3.38309	0.174238
$378$	1.76687	0.0908781
$379$	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
$381$	5.57961	0.285852
$382$	7.74816	0.396430
$383$	−20.8981	−1.06784	−0.533921	−	0.845534i	$-0.679282\pi$
$383$	−20.8981	−1.06784	−0.533921	+	0.845534i	$0.679282\pi$
$384$	−2.59608	−0.132481
$385$	−0.0662146	−0.00337461
$386$	4.13495	0.210463
$387$	7.89407	0.401278
$388$	0.509325	0.0258570
$389$	−16.7753	−0.850542	−0.425271	−	0.905066i	$-0.639821\pi$
$389$	−16.7753	−0.850542	−0.425271	+	0.905066i	$0.639821\pi$
$390$	0.0767372	0.00388574
$391$	21.0627	1.06518
$392$	14.2980	0.722157
$393$	19.9390	1.00579
$394$	5.79888	0.292143
$395$	−0.571257	−0.0287431
$396$	−0.490760	−0.0246616
$397$	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
$399$	−7.17694	−0.359296
$400$	10.9602	0.548010
$401$	1.41427	0.0706252	0.0353126	−	0.999376i	$-0.488757\pi$
$401$	1.41427	0.0706252	0.0353126	+	0.999376i	$0.488757\pi$
$402$	10.1067	0.504075
$403$	0.829183	0.0413045
$404$	−7.16145	−0.356295
$405$	0.0593478	0.00294902
$406$	5.32091	0.264073
$407$	6.27917	0.311247
$408$	−20.0415	−0.992203
$409$	−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
$411$	−6.58107	−0.324620
$412$	3.73149	0.183837
$413$	10.6273	0.522937
$414$	−3.72463	−0.183056
$415$	0.399376	0.0196046
$416$	−4.08188	−0.200130
$417$	1.54196	0.0755103
$418$	−3.91099	−0.191293
$419$	−30.6592	−1.49780	−0.748901	−	0.662682i	$-0.769418\pi$
$419$	−30.6592	−1.49780	−0.748901	+	0.662682i	$0.769418\pi$
$420$	−0.0615170	−0.00300172
$421$	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
$423$	−1.39496	−0.0678252
$424$	−42.9465	−2.08567
$425$	32.5210	1.57750
$426$	10.6841	0.517647
$427$	21.4173	1.03646
$428$	−0.184551	−0.00892062
$429$	0.816480	0.0394200
$430$	0.539231	0.0260040
$431$	22.4664	1.08217	0.541084	−	0.840969i	$-0.318014\pi$
$431$	22.4664	1.08217	0.541084	+	0.840969i	$0.318014\pi$
$432$	2.19359	0.105539
$433$	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
$435$	0.178725	0.00856922
$436$	10.5955	0.507434
$437$	15.1292	0.723730
$438$	−10.4123	−0.497517
$439$	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
$441$	−4.64348	−0.221118
$442$	8.41590	0.400304
$443$	18.7542	0.891041	0.445520	−	0.895272i	$-0.353019\pi$
$443$	18.7542	0.891041	0.445520	+	0.895272i	$0.353019\pi$
$444$	5.83369	0.276855
$445$	0.417657	0.0197988
$446$	−8.20894	−0.388704
$447$	−1.57899	−0.0746835
$448$	−13.1547	−0.621501
$449$	−4.79294	−0.226193	−0.113096	−	0.993584i	$-0.536077\pi$
$449$	−4.79294	−0.226193	−0.113096	+	0.993584i	$0.536077\pi$
$450$	−5.75087	−0.271098
$451$	7.54242	0.355159
$452$	−13.6213	−0.640690
$453$	−9.20724	−0.432594
$454$	13.3841	0.628147
$455$	0.102346	0.00479807
$456$	−14.3958	−0.674144
$457$	−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
$459$	6.50878	0.303804
$460$	0.129680	0.00604637
$461$	7.12099	0.331657	0.165829	−	0.986155i	$-0.446970\pi$
$461$	7.12099	0.331657	0.165829	+	0.986155i	$0.446970\pi$
$462$	1.28416	0.0597445
$463$	−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
$465$	0.0438049	0.00203140
$466$	−9.37149	−0.434126
$467$	−3.56520	−0.164978	−0.0824889	−	0.996592i	$-0.526287\pi$
$467$	−3.56520	−0.164978	−0.0824889	+	0.996592i	$0.526287\pi$
$468$	0.758555	0.0350642
$469$	13.4795	0.622426
$470$	−0.0952873	−0.00439527
$471$	8.37016	0.385677
$472$	21.3167	0.981181
$473$	5.73739	0.263806
$474$	11.0789	0.508871
$475$	23.3597	1.07182
$476$	−6.74668	−0.309234
$477$	13.9475	0.638613
$478$	19.1336	0.875150
$479$	24.8281	1.13442	0.567212	−	0.823572i	$-0.308022\pi$
$479$	24.8281	1.13442	0.567212	+	0.823572i	$0.308022\pi$
$480$	−0.215642	−0.00984265
$481$	−9.70555	−0.442535
$482$	−9.11154	−0.415019
$483$	−4.96763	−0.226035
$484$	7.07091	0.321405
$485$	−0.0447656	−0.00203270
$486$	−1.15098	−0.0522097
$487$	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
$489$	17.8913	0.809070
$490$	−0.317188	−0.0143291
$491$	−26.1084	−1.17825	−0.589127	−	0.808040i	$-0.700528\pi$
$491$	−26.1084	−1.17825	−0.589127	+	0.808040i	$0.700528\pi$
$492$	7.00732	0.315915
$493$	19.6011	0.882790
$494$	6.04512	0.271983
$495$	0.0431339	0.00193872
$496$	1.61910	0.0726996
$497$	14.2497	0.639185
$498$	−7.74546	−0.347082
$499$	−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
$501$	1.00000	0.0446767
$502$	−24.8756	−1.11025
$503$	−11.4082	−0.508668	−0.254334	−	0.967116i	$-0.581856\pi$
$503$	−11.4082	−0.508668	−0.254334	+	0.967116i	$0.581856\pi$
$504$	4.72680	0.210548
$505$	0.629434	0.0280094
$506$	−2.70705	−0.120343
$507$	11.7380	0.521302
$508$	3.76755	0.167158
$509$	33.2584	1.47415	0.737077	−	0.675809i	$-0.236205\pi$
$509$	33.2584	1.47415	0.737077	+	0.675809i	$0.236205\pi$
$510$	0.444604	0.0196874
$511$	−13.8871	−0.614329
$512$	−21.4792	−0.949257
$513$	4.67524	0.206417
$514$	−17.3265	−0.764238
$515$	−0.327968	−0.0144520
$516$	5.33035	0.234656
$517$	−1.01385	−0.0445892
$518$	−15.2649	−0.670700
$519$	8.56831	0.376107
$520$	0.205290	0.00900256
$521$	21.6845	0.950017	0.475008	−	0.879981i	$-0.342445\pi$
$521$	21.6845	0.950017	0.475008	+	0.879981i	$0.342445\pi$
$522$	−3.46618	−0.151710
$523$	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
$525$	−7.67008	−0.334750
$526$	31.6298	1.37912
$527$	4.80416	0.209273
$528$	1.59429	0.0693828
$529$	−12.5281	−0.544698
$530$	0.952732	0.0413840
$531$	−6.92291	−0.300429
$532$	−4.84612	−0.210106
$533$	−11.6581	−0.504969
$534$	−8.09998	−0.350521
$535$	0.0162206	0.000701276 0
$536$	27.0377	1.16785
$537$	13.3026	0.574050
$538$	−10.9744	−0.473139
$539$	−3.37487	−0.145366
$540$	0.0400737	0.00172450
$541$	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
$543$	−8.95629	−0.384351
$544$	−23.6498	−1.01398
$545$	−0.931263	−0.0398909
$546$	−1.98489	−0.0849455
$547$	−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
$549$	−13.9518	−0.595447
$550$	−4.17972	−0.178224
$551$	14.0794	0.599803
$552$	−9.96426	−0.424107
$553$	14.7762	0.628348
$554$	20.7246	0.880502
$555$	−0.512735	−0.0217644
$556$	1.04119	0.0441562
$557$	8.80962	0.373276	0.186638	−	0.982429i	$-0.440241\pi$
$557$	8.80962	0.373276	0.186638	+	0.982429i	$0.440241\pi$
$558$	−0.849548	−0.0359642
$559$	−8.86814	−0.375082
$560$	0.199846	0.00844503
$561$	4.73057	0.199725
$562$	10.3488	0.436537
$563$	23.8194	1.00387	0.501934	−	0.864906i	$-0.332622\pi$
$563$	23.8194	1.00387	0.501934	+	0.864906i	$0.332622\pi$
$564$	−0.941925	−0.0396622
$565$	1.19720	0.0503665
$566$	−31.3952	−1.31964
$567$	−1.53510	−0.0644680
$568$	28.5825	1.19930
$569$	23.1724	0.971436	0.485718	−	0.874116i	$-0.338558\pi$
$569$	23.1724	0.971436	0.485718	+	0.874116i	$0.338558\pi$
$570$	0.319358	0.0133764
$571$	−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
$573$	−6.73177	−0.281224
$574$	−18.3359	−0.765325
$575$	16.1688	0.674285
$576$	8.56930	0.357054
$577$	−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
$579$	−3.59253	−0.149301
$580$	0.120682	0.00501103
$581$	−10.3303	−0.428574
$582$	0.868178	0.0359871
$583$	10.1370	0.419833
$584$	−27.8553	−1.15266
$585$	−0.0666709	−0.00275650
$586$	−8.73859	−0.360988
$587$	−16.9458	−0.699429	−0.349715	−	0.936856i	$-0.613721\pi$
$587$	−16.9458	−0.699429	−0.349715	+	0.936856i	$0.613721\pi$
$588$	−3.13544	−0.129303
$589$	3.45082	0.142188
$590$	−0.472893	−0.0194687
$591$	−5.03819	−0.207243
$592$	−18.9515	−0.778901
$593$	42.1996	1.73293	0.866465	−	0.499238i	$-0.166387\pi$
$593$	42.1996	1.73293	0.866465	+	0.499238i	$0.166387\pi$
$594$	−0.836533	−0.0343234
$595$	0.592980	0.0243098
$596$	−1.06619	−0.0436727
$597$	18.8821	0.772794
$598$	4.18422	0.171106
$599$	−26.0476	−1.06428	−0.532139	−	0.846657i	$-0.678612\pi$
$599$	−26.0476	−1.06428	−0.532139	+	0.846657i	$0.678612\pi$
$600$	−15.3849	−0.628087
$601$	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
$603$	−8.78089	−0.357585
$604$	−6.21705	−0.252968
$605$	−0.621476	−0.0252666
$606$	−12.2072	−0.495882
$607$	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
$609$	−4.62293	−0.187330
$610$	−0.953023	−0.0385868
$611$	1.56709	0.0633975
$612$	4.39496	0.177656
$613$	−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
$615$	−0.615888	−0.0248350
$616$	3.43543	0.138417
$617$	12.1589	0.489499	0.244750	−	0.969586i	$-0.421294\pi$
$617$	12.1589	0.489499	0.244750	+	0.969586i	$0.421294\pi$
$618$	6.36058	0.255860
$619$	−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
$621$	3.23604	0.129858
$622$	−10.3977	−0.416908
$623$	−10.8032	−0.432819
$624$	−2.46426	−0.0986493
$625$	24.9472	0.997887
$626$	−5.27524	−0.210841
$627$	3.39795	0.135701
$628$	5.65183	0.225533
$629$	−56.2326	−2.24214
$630$	−0.104860	−0.00417772
$631$	−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
$633$	−10.7029	−0.425400
$634$	−0.588618	−0.0233770
$635$	−0.331138	−0.0131408
$636$	9.41786	0.373442
$637$	5.21645	0.206683
$638$	−2.51921	−0.0997365
$639$	−9.28259	−0.367214
$640$	0.154072	0.00609022
$641$	18.6439	0.736392	0.368196	−	0.929748i	$-0.379976\pi$
$641$	18.6439	0.736392	0.368196	+	0.929748i	$0.379976\pi$
$642$	−0.314580	−0.0124155
$643$	−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
$645$	−0.468495	−0.0184470
$646$	35.0245	1.37802
$647$	−28.7022	−1.12840	−0.564201	−	0.825638i	$-0.690816\pi$
$647$	−28.7022	−1.12840	−0.564201	+	0.825638i	$0.690816\pi$
$648$	−3.07915	−0.120961
$649$	−5.03156	−0.197506
$650$	6.46048	0.253401
$651$	−1.13306	−0.0444082
$652$	12.0808	0.473121
$653$	3.90284	0.152730	0.0763650	−	0.997080i	$-0.475669\pi$
$653$	3.90284	0.152730	0.0763650	+	0.997080i	$0.475669\pi$
$654$	18.0608	0.706233
$655$	−1.18334	−0.0462369
$656$	−22.7642	−0.888791
$657$	9.04640	0.352934
$658$	2.46471	0.0960845
$659$	37.9732	1.47923	0.739614	−	0.673032i	$-0.235008\pi$
$659$	37.9732	1.47923	0.739614	+	0.673032i	$0.235008\pi$
$660$	0.0291255	0.00113371
$661$	−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
$663$	−7.31192	−0.283971
$664$	−20.7209	−0.804128
$665$	0.425935	0.0165171
$666$	9.94392	0.385319
$667$	9.74529	0.377339
$668$	0.675235	0.0261256
$669$	7.13210	0.275743
$670$	−0.599808	−0.0231726
$671$	−10.1401	−0.391455
$672$	5.57781	0.215169
$673$	−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
$675$	4.99648	0.192315
$676$	7.92591	0.304843
$677$	−14.0908	−0.541554	−0.270777	−	0.962642i	$-0.587281\pi$
$677$	−14.0908	−0.541554	−0.270777	+	0.962642i	$0.587281\pi$
$678$	−23.2183	−0.891695
$679$	1.15791	0.0444365
$680$	1.18942	0.0456122
$681$	−11.6284	−0.445601
$682$	−0.617450	−0.0236434
$683$	15.3972	0.589158	0.294579	−	0.955627i	$-0.404821\pi$
$683$	15.3972	0.589158	0.294579	+	0.955627i	$0.404821\pi$
$684$	3.15688	0.120707
$685$	0.390572	0.0149230
$686$	20.5725	0.785463
$687$	12.9008	0.492195
$688$	−17.3163	−0.660178
$689$	−15.6685	−0.596924
$690$	0.221048	0.00841517
$691$	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
$693$	−1.11571	−0.0423821
$694$	−31.0084	−1.17706
$695$	−0.0915121	−0.00347125
$696$	−9.27284	−0.351486
$697$	−67.5455	−2.55847
$698$	21.1550	0.800730
$699$	8.14215	0.307964
$700$	−5.17911	−0.195752
$701$	−51.4198	−1.94210	−0.971049	−	0.238881i	$-0.923219\pi$
$701$	−51.4198	−1.94210	−0.971049	+	0.238881i	$0.923219\pi$
$702$	1.29301	0.0488014
$703$	−40.3917	−1.52340
$704$	6.22815	0.234732
$705$	0.0827877	0.00311796
$706$	−16.3565	−0.615586
$707$	−16.2810	−0.612310
$708$	−4.67460	−0.175682
$709$	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
$711$	−9.62559	−0.360988
$712$	−21.6694	−0.812094
$713$	2.38854	0.0894514
$714$	−11.5002	−0.430383
$715$	−0.0484563	−0.00181216
$716$	8.98239	0.335688
$717$	−16.6237	−0.620822
$718$	−13.3142	−0.496883
$719$	18.9176	0.705507	0.352754	−	0.935716i	$-0.385245\pi$
$719$	18.9176	0.705507	0.352754	+	0.935716i	$0.385245\pi$
$720$	−0.130185	−0.00485169
$721$	8.48326	0.315933
$722$	3.28931	0.122416
$723$	7.91630	0.294410
$724$	−6.04761	−0.224758
$725$	15.0468	0.558826
$726$	12.0528	0.447323
$727$	−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
$729$	1.00000	0.0370370
$730$	0.617945	0.0228712
$731$	−51.3807	−1.90038
$732$	−9.42073	−0.348200
$733$	−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
$735$	0.275580	0.0101649
$736$	−11.7582	−0.433414
$737$	−6.38193	−0.235082
$738$	11.9445	0.439681
$739$	−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
$741$	−5.25213	−0.192942
$742$	−24.6435	−0.904690
$743$	20.6928	0.759145	0.379573	−	0.925162i	$-0.376071\pi$
$743$	20.6928	0.759145	0.379573	+	0.925162i	$0.376071\pi$
$744$	−2.27274	−0.0833227
$745$	0.0937093	0.00343324
$746$	−39.4511	−1.44441
$747$	6.72942	0.246217
$748$	3.19425	0.116793
$749$	−0.419563	−0.0153305
$750$	0.682843	0.0249339
$751$	−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
$753$	21.6125	0.787603
$754$	3.89388	0.141807
$755$	0.546429	0.0198866
$756$	−1.03655	−0.0376990
$757$	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
$759$	2.35195	0.0853702
$760$	0.854357	0.0309908
$761$	−11.7609	−0.426333	−0.213166	−	0.977016i	$-0.568378\pi$
$761$	−11.7609	−0.426333	−0.213166	+	0.977016i	$0.568378\pi$
$762$	6.42205	0.232646
$763$	24.0881	0.872049
$764$	−4.54553	−0.164451
$765$	−0.386282	−0.0139660
$766$	−24.0534	−0.869083
$767$	7.77715	0.280817
$768$	14.1506	0.510614
$769$	−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
$771$	15.0536	0.542143
$772$	−2.42581	−0.0873066
$773$	−32.0957	−1.15440	−0.577201	−	0.816602i	$-0.695855\pi$
$773$	−32.0957	−1.15440	−0.577201	+	0.816602i	$0.695855\pi$
$774$	9.08594	0.326588
$775$	3.68793	0.132474
$776$	2.32258	0.0833758
$777$	13.2625	0.475788
$778$	−19.3081	−0.692230
$779$	−48.5177	−1.73833
$780$	−0.0450186	−0.00161192
$781$	−6.74657	−0.241411
$782$	24.2428	0.866920
$783$	3.01149	0.107622
$784$	10.1859	0.363781
$785$	−0.496751	−0.0177298
$786$	22.9495	0.818583
$787$	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
$789$	−27.4806	−0.978337
$790$	−0.657508	−0.0233931
$791$	−30.9669	−1.10106
$792$	−2.23792	−0.0795211
$793$	15.6733	0.556576
$794$	5.76240	0.204500
$795$	−0.827754	−0.0293574
$796$	12.7499	0.451907
$797$	−0.658673	−0.0233314	−0.0116657	−	0.999932i	$-0.503713\pi$
$797$	−0.658673	−0.0233314	−0.0116657	+	0.999932i	$0.503713\pi$
$798$	−8.26054	−0.292420
$799$	9.07947	0.321209
$800$	−18.1548	−0.641870
$801$	7.03744	0.248656
$802$	1.62780	0.0574797
$803$	6.57491	0.232023
$804$	−5.92917	−0.209106
$805$	0.294818	0.0103910
$806$	0.954376	0.0336165
$807$	9.53477	0.335640
$808$	−32.6571	−1.14887
$809$	−5.32511	−0.187221	−0.0936105	−	0.995609i	$-0.529841\pi$
$809$	−5.32511	−0.187221	−0.0936105	+	0.995609i	$0.529841\pi$
$810$	0.0683084	0.00240011
$811$	−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
$813$	−14.1985	−0.497961
$814$	7.22722	0.253314
$815$	−1.06181	−0.0371935
$816$	−14.2776	−0.499815
$817$	−36.9066	−1.29120
$818$	−28.4970	−0.996375
$819$	1.72452	0.0602595
$820$	−0.415869	−0.0145228
$821$	−7.24036	−0.252690	−0.126345	−	0.991986i	$-0.540325\pi$
$821$	−7.24036	−0.252690	−0.126345	+	0.991986i	$0.540325\pi$
$822$	−7.57471	−0.264198
$823$	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
$825$	3.63143	0.126430
$826$	12.2319	0.425602
$827$	41.5316	1.44419	0.722097	−	0.691792i	$-0.243178\pi$
$827$	41.5316	1.44419	0.722097	+	0.691792i	$0.243178\pi$
$828$	2.18509	0.0759370
$829$	−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
$831$	−18.0059	−0.624619
$832$	−9.62669	−0.333746
$833$	30.2234	1.04718
$834$	1.77478	0.0614554
$835$	−0.0593478	−0.00205382
$836$	2.29442	0.0793541
$837$	0.738105	0.0255127
$838$	−35.2883	−1.21901
$839$	7.96730	0.275062	0.137531	−	0.990497i	$-0.456083\pi$
$839$	7.96730	0.275062	0.137531	+	0.990497i	$0.456083\pi$
$840$	−0.280525	−0.00967903
$841$	−19.9309	−0.687274
$842$	39.7632	1.37033
$843$	−8.99125	−0.309675
$844$	−7.22695	−0.248762
$845$	−0.696624	−0.0239646
$846$	−1.60557	−0.0552008
$847$	16.0752	0.552349
$848$	−30.5951	−1.05064
$849$	27.2768	0.936137
$850$	37.4311	1.28388
$851$	−27.9577	−0.958379
$852$	−6.26793	−0.214736
$853$	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
$855$	−0.277465	−0.00948910
$856$	−0.841575	−0.0287644
$857$	14.6173	0.499318	0.249659	−	0.968334i	$-0.419681\pi$
$857$	14.6173	0.499318	0.249659	+	0.968334i	$0.419681\pi$
$858$	0.939756	0.0320827
$859$	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
$861$	15.9306	0.542914
$862$	25.8585	0.880743
$863$	40.7770	1.38806	0.694032	−	0.719944i	$-0.255833\pi$
$863$	40.7770	1.38806	0.694032	+	0.719944i	$0.255833\pi$
$864$	−3.63352	−0.123615
$865$	−0.508510	−0.0172899
$866$	3.86560	0.131358
$867$	−25.3642	−0.861413
$868$	−0.765084	−0.0259687
$869$	−6.99586	−0.237318
$870$	0.205710	0.00697422
$871$	9.86439	0.334242
$872$	48.3169	1.63622
$873$	−0.754292	−0.0255289
$874$	17.4135	0.589021
$875$	0.910725	0.0307881
$876$	6.10845	0.206385
$877$	−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
$879$	7.59228	0.256081
$880$	−0.0946178	−0.00318957
$881$	38.0924	1.28336	0.641682	−	0.766971i	$-0.278237\pi$
$881$	38.0924	1.28336	0.641682	+	0.766971i	$0.278237\pi$
$882$	−5.34457	−0.179961
$883$	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
$885$	0.410860	0.0138109
$886$	21.5858	0.725190
$887$	27.3252	0.917491	0.458745	−	0.888568i	$-0.348299\pi$
$887$	27.3252	0.917491	0.458745	+	0.888568i	$0.348299\pi$
$888$	26.6023	0.892716
$889$	8.56525	0.287269
$890$	0.480716	0.0161136
$891$	0.726798	0.0243487
$892$	4.81585	0.161246
$893$	6.52176	0.218242
$894$	−1.81739	−0.0607825
$895$	−0.789481	−0.0263894
$896$	−3.98524	−0.133137
$897$	−3.63534	−0.121381
$898$	−5.51659	−0.184091
$899$	2.22280	0.0741344
$900$	3.37380	0.112460
$901$	−90.7813	−3.02436
$902$	8.68121	0.289053
$903$	12.1182	0.403267
$904$	−62.1145	−2.06590
$905$	0.531536	0.0176689
$906$	−10.5974	−0.352075
$907$	−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
$909$	10.6059	0.351774
$910$	0.117799	0.00390500
$911$	−39.5068	−1.30892	−0.654460	−	0.756097i	$-0.727104\pi$
$911$	−39.5068	−1.30892	−0.654460	+	0.756097i	$0.727104\pi$
$912$	−10.2555	−0.339595
$913$	4.89093	0.161866
$914$	−34.0275	−1.12553
$915$	0.828007	0.0273731
$916$	8.71106	0.287821
$917$	30.6084	1.01078
$918$	7.49150	0.247256
$919$	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
$921$	2.34933	0.0774130
$922$	8.19614	0.269926
$923$	10.4280	0.343242
$924$	−0.753364	−0.0247838
$925$	−43.1670	−1.41932
$926$	−9.38283	−0.308339
$927$	−5.52621	−0.181504
$928$	−10.9423	−0.359199
$929$	−3.32230	−0.109001	−0.0545006	−	0.998514i	$-0.517357\pi$
$929$	−3.32230	−0.109001	−0.0545006	+	0.998514i	$0.517357\pi$
$930$	0.0504188	0.00165330
$931$	21.7094	0.711495
$932$	5.49787	0.180089
$933$	9.03371	0.295750
$934$	−4.10349	−0.134270
$935$	−0.280749	−0.00918147
$936$	3.45910	0.113064
$937$	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
$939$	4.58324	0.149568
$940$	0.0559012	0.00182329
$941$	−22.0637	−0.719255	−0.359628	−	0.933096i	$-0.617096\pi$
$941$	−22.0637	−0.719255	−0.359628	+	0.933096i	$0.617096\pi$
$942$	9.63393	0.313890
$943$	−33.5823	−1.09359
$944$	15.1860	0.494262
$945$	0.0911046	0.00296363
$946$	6.60365	0.214703
$947$	54.8587	1.78267	0.891334	−	0.453347i	$-0.149770\pi$
$947$	54.8587	1.78267	0.891334	+	0.453347i	$0.149770\pi$
$948$	−6.49954	−0.211095
$949$	−10.1627	−0.329894
$950$	26.8867	0.872318
$951$	0.511404	0.0165834
$952$	−30.7657	−0.997121
$953$	35.6512	1.15486	0.577428	−	0.816441i	$-0.304056\pi$
$953$	35.6512	1.15486	0.577428	+	0.816441i	$0.304056\pi$
$954$	16.0534	0.519747
$955$	0.399516	0.0129280
$956$	−11.2249	−0.363039
$957$	2.18874	0.0707521
$958$	28.5767	0.923271
$959$	−10.1026	−0.326229
$960$	−0.508569	−0.0164140
$961$	−30.4552	−0.982426
$962$	−11.1709	−0.360165
$963$	0.273314	0.00880741
$964$	5.34537	0.172163
$965$	0.213209	0.00686344
$966$	−5.71766	−0.183963
$967$	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
$969$	−30.4301	−0.977555
$970$	−0.0515244	−0.00165435
$971$	7.33160	0.235282	0.117641	−	0.993056i	$-0.462467\pi$
$971$	7.33160	0.235282	0.117641	+	0.993056i	$0.462467\pi$
$972$	0.675235	0.0216582
$973$	2.36706	0.0758845
$974$	33.8992	1.08620
$975$	−5.61301	−0.179760
$976$	30.6044	0.979624
$977$	4.90524	0.156933	0.0784663	−	0.996917i	$-0.474998\pi$
$977$	4.90524	0.156933	0.0784663	+	0.996917i	$0.474998\pi$
$978$	20.5925	0.658477
$979$	5.11480	0.163470
$980$	0.186082	0.00594416
$981$	−15.6916	−0.500995
$982$	−30.0503	−0.958944
$983$	−45.3449	−1.44628	−0.723138	−	0.690703i	$-0.757301\pi$
$983$	−45.3449	−1.44628	−0.723138	+	0.690703i	$0.757301\pi$
$984$	31.9542	1.01866
$985$	0.299005	0.00952710
$986$	22.5606	0.718475
$987$	−2.14139	−0.0681613
$988$	−3.54642	−0.112827
$989$	−25.5455	−0.812300
$990$	0.0496464	0.00157787
$991$	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
$993$	−18.9587	−0.601635
$994$	16.4011	0.520213
$995$	−1.12061	−0.0355258
$996$	4.54395	0.143980
$997$	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
$999$	−8.63949	−0.273342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	501.2.a.c.1.4	✓	5
3.2	odd	2		1503.2.a.c.1.2		5
4.3	odd	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	1.1	even	1	trivial
1503.2.a.c.1.2		5	3.2	odd	2
8016.2.a.u.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 501 = 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	501.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15098 q^{2} -1.00000 q^{3} -0.675235 q^{4} +0.0593478 q^{5} -1.15098 q^{6} -1.53510 q^{7} -3.07915 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.15098 q^{2} -1.00000 q^{3} -0.675235 q^{4} +0.0593478 q^{5} -1.15098 q^{6} -1.53510 q^{7} -3.07915 q^{8} +1.00000 q^{9} +0.0683084 q^{10} +0.726798 q^{11} +0.675235 q^{12} -1.12339 q^{13} -1.76687 q^{14} -0.0593478 q^{15} -2.19359 q^{16} -6.50878 q^{17} +1.15098 q^{18} -4.67524 q^{19} -0.0400737 q^{20} +1.53510 q^{21} +0.836533 q^{22} -3.23604 q^{23} +3.07915 q^{24} -4.99648 q^{25} -1.29301 q^{26} -1.00000 q^{27} +1.03655 q^{28} -3.01149 q^{29} -0.0683084 q^{30} -0.738105 q^{31} +3.63352 q^{32} -0.726798 q^{33} -7.49150 q^{34} -0.0911046 q^{35} -0.675235 q^{36} +8.63949 q^{37} -5.38112 q^{38} +1.12339 q^{39} -0.182741 q^{40} +10.3776 q^{41} +1.76687 q^{42} +7.89407 q^{43} -0.490760 q^{44} +0.0593478 q^{45} -3.72463 q^{46} -1.39496 q^{47} +2.19359 q^{48} -4.64348 q^{49} -5.75087 q^{50} +6.50878 q^{51} +0.758555 q^{52} +13.9475 q^{53} -1.15098 q^{54} +0.0431339 q^{55} +4.72680 q^{56} +4.67524 q^{57} -3.46618 q^{58} -6.92291 q^{59} +0.0400737 q^{60} -13.9518 q^{61} -0.849548 q^{62} -1.53510 q^{63} +8.56930 q^{64} -0.0666709 q^{65} -0.836533 q^{66} -8.78089 q^{67} +4.39496 q^{68} +3.23604 q^{69} -0.104860 q^{70} -9.28259 q^{71} -3.07915 q^{72} +9.04640 q^{73} +9.94392 q^{74} +4.99648 q^{75} +3.15688 q^{76} -1.11571 q^{77} +1.29301 q^{78} -9.62559 q^{79} -0.130185 q^{80} +1.00000 q^{81} +11.9445 q^{82} +6.72942 q^{83} -1.03655 q^{84} -0.386282 q^{85} +9.08594 q^{86} +3.01149 q^{87} -2.23792 q^{88} +7.03744 q^{89} +0.0683084 q^{90} +1.72452 q^{91} +2.18509 q^{92} +0.738105 q^{93} -1.60557 q^{94} -0.277465 q^{95} -3.63352 q^{96} -0.754292 q^{97} -5.34457 q^{98} +0.726798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - q^5 - 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - q^{5} - 4 q^{7} + 5 q^{9} - 9 q^{10} - 7 q^{11} - 8 q^{13} - q^{14} + q^{15} - 6 q^{16} + 5 q^{17} - 20 q^{19} - 3 q^{20} + 4 q^{21} - q^{22} + q^{23} - 6 q^{25} - 3 q^{26} - 5 q^{27} - 11 q^{28} - 5 q^{29} + 9 q^{30} - 18 q^{31} - 5 q^{32} + 7 q^{33} - 12 q^{34} - 6 q^{35} - 17 q^{37} + 8 q^{39} - 2 q^{40} + 6 q^{41} + q^{42} - 10 q^{43} - 9 q^{44} - q^{45} - q^{46} - 3 q^{47} + 6 q^{48} - 13 q^{49} + 9 q^{50} - 5 q^{51} + 15 q^{53} - 9 q^{55} + 13 q^{56} + 20 q^{57} + 21 q^{58} - 17 q^{59} + 3 q^{60} - 2 q^{61} - 4 q^{63} - 10 q^{64} + 26 q^{65} + q^{66} - 24 q^{67} + 18 q^{68} - q^{69} + 25 q^{70} + q^{71} + 2 q^{73} + 17 q^{74} + 6 q^{75} + 14 q^{76} + 26 q^{77} + 3 q^{78} - 6 q^{79} + 12 q^{80} + 5 q^{81} + 17 q^{82} - 7 q^{83} + 11 q^{84} - 11 q^{85} + 15 q^{86} + 5 q^{87} + 4 q^{88} - 9 q^{90} - 15 q^{91} + 45 q^{92} + 18 q^{93} - 2 q^{94} + q^{95} + 5 q^{96} - 13 q^{97} - 16 q^{98} - 7 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - q^5 - 4 * q^7 + 5 * q^9 - 9 * q^10 - 7 * q^11 - 8 * q^13 - q^14 + q^15 - 6 * q^16 + 5 * q^17 - 20 * q^19 - 3 * q^20 + 4 * q^21 - q^22 + q^23 - 6 * q^25 - 3 * q^26 - 5 * q^27 - 11 * q^28 - 5 * q^29 + 9 * q^30 - 18 * q^31 - 5 * q^32 + 7 * q^33 - 12 * q^34 - 6 * q^35 - 17 * q^37 + 8 * q^39 - 2 * q^40 + 6 * q^41 + q^42 - 10 * q^43 - 9 * q^44 - q^45 - q^46 - 3 * q^47 + 6 * q^48 - 13 * q^49 + 9 * q^50 - 5 * q^51 + 15 * q^53 - 9 * q^55 + 13 * q^56 + 20 * q^57 + 21 * q^58 - 17 * q^59 + 3 * q^60 - 2 * q^61 - 4 * q^63 - 10 * q^64 + 26 * q^65 + q^66 - 24 * q^67 + 18 * q^68 - q^69 + 25 * q^70 + q^71 + 2 * q^73 + 17 * q^74 + 6 * q^75 + 14 * q^76 + 26 * q^77 + 3 * q^78 - 6 * q^79 + 12 * q^80 + 5 * q^81 + 17 * q^82 - 7 * q^83 + 11 * q^84 - 11 * q^85 + 15 * q^86 + 5 * q^87 + 4 * q^88 - 9 * q^90 - 15 * q^91 + 45 * q^92 + 18 * q^93 - 2 * q^94 + q^95 + 5 * q^96 - 13 * q^97 - 16 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more