LMFDB - Embedded newform 8002.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$95$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.96030 q^{3} +1.00000 q^{4} +3.24617 q^{5} -2.96030 q^{6} -2.38509 q^{7} +1.00000 q^{8} +5.76337 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.96030 q^{3} +1.00000 q^{4} +3.24617 q^{5} -2.96030 q^{6} -2.38509 q^{7} +1.00000 q^{8} +5.76337 q^{9} +3.24617 q^{10} +0.133388 q^{11} -2.96030 q^{12} +4.02237 q^{13} -2.38509 q^{14} -9.60963 q^{15} +1.00000 q^{16} +6.06913 q^{17} +5.76337 q^{18} +3.12723 q^{19} +3.24617 q^{20} +7.06057 q^{21} +0.133388 q^{22} -2.06205 q^{23} -2.96030 q^{24} +5.53760 q^{25} +4.02237 q^{26} -8.18040 q^{27} -2.38509 q^{28} +9.25547 q^{29} -9.60963 q^{30} +6.75906 q^{31} +1.00000 q^{32} -0.394870 q^{33} +6.06913 q^{34} -7.74239 q^{35} +5.76337 q^{36} +6.41078 q^{37} +3.12723 q^{38} -11.9074 q^{39} +3.24617 q^{40} -5.92489 q^{41} +7.06057 q^{42} -0.249635 q^{43} +0.133388 q^{44} +18.7089 q^{45} -2.06205 q^{46} -4.97120 q^{47} -2.96030 q^{48} -1.31136 q^{49} +5.53760 q^{50} -17.9665 q^{51} +4.02237 q^{52} -7.36444 q^{53} -8.18040 q^{54} +0.433001 q^{55} -2.38509 q^{56} -9.25754 q^{57} +9.25547 q^{58} +3.83824 q^{59} -9.60963 q^{60} -7.22904 q^{61} +6.75906 q^{62} -13.7461 q^{63} +1.00000 q^{64} +13.0573 q^{65} -0.394870 q^{66} +2.37458 q^{67} +6.06913 q^{68} +6.10430 q^{69} -7.74239 q^{70} +6.81080 q^{71} +5.76337 q^{72} -8.31338 q^{73} +6.41078 q^{74} -16.3930 q^{75} +3.12723 q^{76} -0.318143 q^{77} -11.9074 q^{78} -4.84361 q^{79} +3.24617 q^{80} +6.92632 q^{81} -5.92489 q^{82} +0.842251 q^{83} +7.06057 q^{84} +19.7014 q^{85} -0.249635 q^{86} -27.3990 q^{87} +0.133388 q^{88} +1.61830 q^{89} +18.7089 q^{90} -9.59369 q^{91} -2.06205 q^{92} -20.0088 q^{93} -4.97120 q^{94} +10.1515 q^{95} -2.96030 q^{96} +19.2378 q^{97} -1.31136 q^{98} +0.768767 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9}+O(q^{10}) $ 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9} + 36 q^{10} + 40 q^{11} + 24 q^{12} + 52 q^{13} + 21 q^{14} + 15 q^{15} + 95 q^{16} + 84 q^{17} + 121 q^{18} + 37 q^{19} + 36 q^{20} + 36 q^{21} + 40 q^{22} + 37 q^{23} + 24 q^{24} + 133 q^{25} + 52 q^{26} + 93 q^{27} + 21 q^{28} + 66 q^{29} + 15 q^{30} + 10 q^{31} + 95 q^{32} + 63 q^{33} + 84 q^{34} + 55 q^{35} + 121 q^{36} + 49 q^{37} + 37 q^{38} + 14 q^{39} + 36 q^{40} + 98 q^{41} + 36 q^{42} + 37 q^{43} + 40 q^{44} + 97 q^{45} + 37 q^{46} + 91 q^{47} + 24 q^{48} + 170 q^{49} + 133 q^{50} + 22 q^{51} + 52 q^{52} + 70 q^{53} + 93 q^{54} - q^{55} + 21 q^{56} + 50 q^{57} + 66 q^{58} + 72 q^{59} + 15 q^{60} + 97 q^{61} + 10 q^{62} + 75 q^{63} + 95 q^{64} + 75 q^{65} + 63 q^{66} + 39 q^{67} + 84 q^{68} + 65 q^{69} + 55 q^{70} + 28 q^{71} + 121 q^{72} + 117 q^{73} + 49 q^{74} + 62 q^{75} + 37 q^{76} + 92 q^{77} + 14 q^{78} + q^{79} + 36 q^{80} + 155 q^{81} + 98 q^{82} + 117 q^{83} + 36 q^{84} + 81 q^{85} + 37 q^{86} + 46 q^{87} + 40 q^{88} + 90 q^{89} + 97 q^{90} + 65 q^{91} + 37 q^{92} + 36 q^{93} + 91 q^{94} + 38 q^{95} + 24 q^{96} + 111 q^{97} + 170 q^{98} + 97 q^{99}+O(q^{100}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9 + 36 * q^10 + 40 * q^11 + 24 * q^12 + 52 * q^13 + 21 * q^14 + 15 * q^15 + 95 * q^16 + 84 * q^17 + 121 * q^18 + 37 * q^19 + 36 * q^20 + 36 * q^21 + 40 * q^22 + 37 * q^23 + 24 * q^24 + 133 * q^25 + 52 * q^26 + 93 * q^27 + 21 * q^28 + 66 * q^29 + 15 * q^30 + 10 * q^31 + 95 * q^32 + 63 * q^33 + 84 * q^34 + 55 * q^35 + 121 * q^36 + 49 * q^37 + 37 * q^38 + 14 * q^39 + 36 * q^40 + 98 * q^41 + 36 * q^42 + 37 * q^43 + 40 * q^44 + 97 * q^45 + 37 * q^46 + 91 * q^47 + 24 * q^48 + 170 * q^49 + 133 * q^50 + 22 * q^51 + 52 * q^52 + 70 * q^53 + 93 * q^54 - q^55 + 21 * q^56 + 50 * q^57 + 66 * q^58 + 72 * q^59 + 15 * q^60 + 97 * q^61 + 10 * q^62 + 75 * q^63 + 95 * q^64 + 75 * q^65 + 63 * q^66 + 39 * q^67 + 84 * q^68 + 65 * q^69 + 55 * q^70 + 28 * q^71 + 121 * q^72 + 117 * q^73 + 49 * q^74 + 62 * q^75 + 37 * q^76 + 92 * q^77 + 14 * q^78 + q^79 + 36 * q^80 + 155 * q^81 + 98 * q^82 + 117 * q^83 + 36 * q^84 + 81 * q^85 + 37 * q^86 + 46 * q^87 + 40 * q^88 + 90 * q^89 + 97 * q^90 + 65 * q^91 + 37 * q^92 + 36 * q^93 + 91 * q^94 + 38 * q^95 + 24 * q^96 + 111 * q^97 + 170 * q^98 + 97 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.96030	−1.70913	−0.854565	−	0.519345i	$-0.826176\pi$
$3$	−2.96030	−1.70913	−0.854565	+	0.519345i	$0.826176\pi$
$4$	1.00000	0.500000
$5$	3.24617	1.45173	0.725865	−	0.687837i	$-0.241440\pi$
$5$	3.24617	1.45173	0.725865	+	0.687837i	$0.241440\pi$
$6$	−2.96030	−1.20854
$7$	−2.38509	−0.901478	−0.450739	−	0.892656i	$-0.648839\pi$
$7$	−2.38509	−0.901478	−0.450739	+	0.892656i	$0.648839\pi$
$8$	1.00000	0.353553
$9$	5.76337	1.92112
$10$	3.24617	1.02653
$11$	0.133388	0.0402181	0.0201091	−	0.999798i	$-0.493599\pi$
$11$	0.133388	0.0402181	0.0201091	+	0.999798i	$0.493599\pi$
$12$	−2.96030	−0.854565
$13$	4.02237	1.11560	0.557802	−	0.829974i	$-0.311645\pi$
$13$	4.02237	1.11560	0.557802	+	0.829974i	$0.311645\pi$
$14$	−2.38509	−0.637441
$15$	−9.60963	−2.48119
$16$	1.00000	0.250000
$17$	6.06913	1.47198	0.735991	−	0.676992i	$-0.236717\pi$
$17$	6.06913	1.47198	0.735991	+	0.676992i	$0.236717\pi$
$18$	5.76337	1.35844
$19$	3.12723	0.717436	0.358718	−	0.933446i	$-0.383214\pi$
$19$	3.12723	0.717436	0.358718	+	0.933446i	$0.383214\pi$
$20$	3.24617	0.725865
$21$	7.06057	1.54074
$22$	0.133388	0.0284385
$23$	−2.06205	−0.429968	−0.214984	−	0.976618i	$-0.568970\pi$
$23$	−2.06205	−0.429968	−0.214984	+	0.976618i	$0.568970\pi$
$24$	−2.96030	−0.604268
$25$	5.53760	1.10752
$26$	4.02237	0.788851
$27$	−8.18040	−1.57432
$28$	−2.38509	−0.450739
$29$	9.25547	1.71870	0.859349	−	0.511390i	$-0.170869\pi$
$29$	9.25547	1.71870	0.859349	+	0.511390i	$0.170869\pi$
$30$	−9.60963	−1.75447
$31$	6.75906	1.21396	0.606981	−	0.794716i	$-0.292380\pi$
$31$	6.75906	1.21396	0.606981	+	0.794716i	$0.292380\pi$
$32$	1.00000	0.176777
$33$	−0.394870	−0.0687380
$34$	6.06913	1.04085
$35$	−7.74239	−1.30870
$36$	5.76337	0.960562
$37$	6.41078	1.05393	0.526963	−	0.849889i	$-0.323331\pi$
$37$	6.41078	1.05393	0.526963	+	0.849889i	$0.323331\pi$
$38$	3.12723	0.507304
$39$	−11.9074	−1.90671
$40$	3.24617	0.513264
$41$	−5.92489	−0.925313	−0.462656	−	0.886538i	$-0.653104\pi$
$41$	−5.92489	−0.925313	−0.462656	+	0.886538i	$0.653104\pi$
$42$	7.06057	1.08947
$43$	−0.249635	−0.0380690	−0.0190345	−	0.999819i	$-0.506059\pi$
$43$	−0.249635	−0.0380690	−0.0190345	+	0.999819i	$0.506059\pi$
$44$	0.133388	0.0201091
$45$	18.7089	2.78895
$46$	−2.06205	−0.304033
$47$	−4.97120	−0.725124	−0.362562	−	0.931960i	$-0.618098\pi$
$47$	−4.97120	−0.725124	−0.362562	+	0.931960i	$0.618098\pi$
$48$	−2.96030	−0.427282
$49$	−1.31136	−0.187337
$50$	5.53760	0.783136
$51$	−17.9665	−2.51581
$52$	4.02237	0.557802
$53$	−7.36444	−1.01158	−0.505792	−	0.862656i	$-0.668800\pi$
$53$	−7.36444	−1.01158	−0.505792	+	0.862656i	$0.668800\pi$
$54$	−8.18040	−1.11321
$55$	0.433001	0.0583859
$56$	−2.38509	−0.318721
$57$	−9.25754	−1.22619
$58$	9.25547	1.21530
$59$	3.83824	0.499697	0.249848	−	0.968285i	$-0.419619\pi$
$59$	3.83824	0.499697	0.249848	+	0.968285i	$0.419619\pi$
$60$	−9.60963	−1.24060
$61$	−7.22904	−0.925583	−0.462792	−	0.886467i	$-0.653152\pi$
$61$	−7.22904	−0.925583	−0.462792	+	0.886467i	$0.653152\pi$
$62$	6.75906	0.858401
$63$	−13.7461	−1.73185
$64$	1.00000	0.125000
$65$	13.0573	1.61956
$66$	−0.394870	−0.0486051
$67$	2.37458	0.290101	0.145051	−	0.989424i	$-0.453666\pi$
$67$	2.37458	0.290101	0.145051	+	0.989424i	$0.453666\pi$
$68$	6.06913	0.735991
$69$	6.10430	0.734871
$70$	−7.74239	−0.925393
$71$	6.81080	0.808293	0.404147	−	0.914694i	$-0.367569\pi$
$71$	6.81080	0.808293	0.404147	+	0.914694i	$0.367569\pi$
$72$	5.76337	0.679220
$73$	−8.31338	−0.973008	−0.486504	−	0.873678i	$-0.661728\pi$
$73$	−8.31338	−0.973008	−0.486504	+	0.873678i	$0.661728\pi$
$74$	6.41078	0.745238
$75$	−16.3930	−1.89290
$76$	3.12723	0.358718
$77$	−0.318143	−0.0362558
$78$	−11.9074	−1.34825
$79$	−4.84361	−0.544949	−0.272474	−	0.962163i	$-0.587842\pi$
$79$	−4.84361	−0.544949	−0.272474	+	0.962163i	$0.587842\pi$
$80$	3.24617	0.362933
$81$	6.92632	0.769592
$82$	−5.92489	−0.654295
$83$	0.842251	0.0924491	0.0462246	−	0.998931i	$-0.485281\pi$
$83$	0.842251	0.0924491	0.0462246	+	0.998931i	$0.485281\pi$
$84$	7.06057	0.770371
$85$	19.7014	2.13692
$86$	−0.249635	−0.0269188
$87$	−27.3990	−2.93748
$88$	0.133388	0.0142193
$89$	1.61830	0.171540	0.0857698	−	0.996315i	$-0.472665\pi$
$89$	1.61830	0.171540	0.0857698	+	0.996315i	$0.472665\pi$
$90$	18.7089	1.97209
$91$	−9.59369	−1.00569
$92$	−2.06205	−0.214984
$93$	−20.0088	−2.07482
$94$	−4.97120	−0.512740
$95$	10.1515	1.04152
$96$	−2.96030	−0.302134
$97$	19.2378	1.95330	0.976652	−	0.214827i	$-0.0689187\pi$
$97$	19.2378	1.95330	0.976652	+	0.214827i	$0.0689187\pi$
$98$	−1.31136	−0.132468
$99$	0.768767	0.0772640
$100$	5.53760	0.553760
$101$	−7.67798	−0.763987	−0.381994	−	0.924165i	$-0.624762\pi$
$101$	−7.67798	−0.763987	−0.381994	+	0.924165i	$0.624762\pi$
$102$	−17.9665	−1.77894
$103$	8.60412	0.847789	0.423895	−	0.905712i	$-0.360663\pi$
$103$	8.60412	0.847789	0.423895	+	0.905712i	$0.360663\pi$
$104$	4.02237	0.394426
$105$	22.9198	2.23674
$106$	−7.36444	−0.715297
$107$	15.7677	1.52432	0.762160	−	0.647389i	$-0.224139\pi$
$107$	15.7677	1.52432	0.762160	+	0.647389i	$0.224139\pi$
$108$	−8.18040	−0.787160
$109$	−10.8946	−1.04352	−0.521758	−	0.853094i	$-0.674724\pi$
$109$	−10.8946	−1.04352	−0.521758	+	0.853094i	$0.674724\pi$
$110$	0.433001	0.0412851
$111$	−18.9778	−1.80129
$112$	−2.38509	−0.225369
$113$	11.1717	1.05094	0.525472	−	0.850811i	$-0.323889\pi$
$113$	11.1717	1.05094	0.525472	+	0.850811i	$0.323889\pi$
$114$	−9.25754	−0.867048
$115$	−6.69377	−0.624198
$116$	9.25547	0.859349
$117$	23.1824	2.14321
$118$	3.83824	0.353339
$119$	−14.4754	−1.32696
$120$	−9.60963	−0.877235
$121$	−10.9822	−0.998383
$122$	−7.22904	−0.654486
$123$	17.5395	1.58148
$124$	6.75906	0.606981
$125$	1.74515	0.156091
$126$	−13.7461	−1.22460
$127$	4.66698	0.414128	0.207064	−	0.978327i	$-0.433609\pi$
$127$	4.66698	0.414128	0.207064	+	0.978327i	$0.433609\pi$
$128$	1.00000	0.0883883
$129$	0.738994	0.0650648
$130$	13.0573	1.14520
$131$	−0.770744	−0.0673402	−0.0336701	−	0.999433i	$-0.510720\pi$
$131$	−0.770744	−0.0673402	−0.0336701	+	0.999433i	$0.510720\pi$
$132$	−0.394870	−0.0343690
$133$	−7.45872	−0.646753
$134$	2.37458	0.205133
$135$	−26.5550	−2.28549
$136$	6.06913	0.520424
$137$	8.17630	0.698549	0.349274	−	0.937020i	$-0.386428\pi$
$137$	8.17630	0.698549	0.349274	+	0.937020i	$0.386428\pi$
$138$	6.10430	0.519632
$139$	−13.8085	−1.17122	−0.585609	−	0.810594i	$-0.699145\pi$
$139$	−13.8085	−1.17122	−0.585609	+	0.810594i	$0.699145\pi$
$140$	−7.74239	−0.654351
$141$	14.7162	1.23933
$142$	6.81080	0.571550
$143$	0.536537	0.0448675
$144$	5.76337	0.480281
$145$	30.0448	2.49508
$146$	−8.31338	−0.688020
$147$	3.88202	0.320184
$148$	6.41078	0.526963
$149$	−1.16046	−0.0950684	−0.0475342	−	0.998870i	$-0.515136\pi$
$149$	−1.16046	−0.0950684	−0.0475342	+	0.998870i	$0.515136\pi$
$150$	−16.3930	−1.33848
$151$	−7.83474	−0.637582	−0.318791	−	0.947825i	$-0.603277\pi$
$151$	−7.83474	−0.637582	−0.318791	+	0.947825i	$0.603277\pi$
$152$	3.12723	0.253652
$153$	34.9787	2.82786
$154$	−0.318143	−0.0256367
$155$	21.9410	1.76235
$156$	−11.9074	−0.953356
$157$	−4.55148	−0.363248	−0.181624	−	0.983368i	$-0.558135\pi$
$157$	−4.55148	−0.363248	−0.181624	+	0.983368i	$0.558135\pi$
$158$	−4.84361	−0.385337
$159$	21.8009	1.72893
$160$	3.24617	0.256632
$161$	4.91818	0.387607
$162$	6.92632	0.544183
$163$	−18.4182	−1.44262	−0.721311	−	0.692611i	$-0.756460\pi$
$163$	−18.4182	−1.44262	−0.721311	+	0.692611i	$0.756460\pi$
$164$	−5.92489	−0.462656
$165$	−1.28181	−0.0997890
$166$	0.842251	0.0653714
$167$	−9.98211	−0.772439	−0.386219	−	0.922407i	$-0.626219\pi$
$167$	−9.98211	−0.772439	−0.386219	+	0.922407i	$0.626219\pi$
$168$	7.06057	0.544735
$169$	3.17944	0.244572
$170$	19.7014	1.51103
$171$	18.0234	1.37828
$172$	−0.249635	−0.0190345
$173$	−3.86663	−0.293974	−0.146987	−	0.989138i	$-0.546958\pi$
$173$	−3.86663	−0.293974	−0.146987	+	0.989138i	$0.546958\pi$
$174$	−27.3990	−2.07711
$175$	−13.2077	−0.998406
$176$	0.133388	0.0100545
$177$	−11.3623	−0.854046
$178$	1.61830	0.121297
$179$	−14.8646	−1.11103	−0.555516	−	0.831506i	$-0.687479\pi$
$179$	−14.8646	−1.11103	−0.555516	+	0.831506i	$0.687479\pi$
$180$	18.7089	1.39448
$181$	0.240071	0.0178444	0.00892218	−	0.999960i	$-0.497160\pi$
$181$	0.240071	0.0178444	0.00892218	+	0.999960i	$0.497160\pi$
$182$	−9.59369	−0.711132
$183$	21.4001	1.58194
$184$	−2.06205	−0.152017
$185$	20.8105	1.53001
$186$	−20.0088	−1.46712
$187$	0.809553	0.0592003
$188$	−4.97120	−0.362562
$189$	19.5110	1.41921
$190$	10.1515	0.736468
$191$	7.91782	0.572913	0.286457	−	0.958093i	$-0.407523\pi$
$191$	7.91782	0.572913	0.286457	+	0.958093i	$0.407523\pi$
$192$	−2.96030	−0.213641
$193$	12.6242	0.908707	0.454353	−	0.890822i	$-0.349870\pi$
$193$	12.6242	0.908707	0.454353	+	0.890822i	$0.349870\pi$
$194$	19.2378	1.38119
$195$	−38.6534	−2.76803
$196$	−1.31136	−0.0936687
$197$	−5.49226	−0.391307	−0.195654	−	0.980673i	$-0.562683\pi$
$197$	−5.49226	−0.391307	−0.195654	+	0.980673i	$0.562683\pi$
$198$	0.768767	0.0546339
$199$	−16.4501	−1.16612	−0.583060	−	0.812429i	$-0.698145\pi$
$199$	−16.4501	−1.16612	−0.583060	+	0.812429i	$0.698145\pi$
$200$	5.53760	0.391568
$201$	−7.02947	−0.495820
$202$	−7.67798	−0.540221
$203$	−22.0751	−1.54937
$204$	−17.9665	−1.25790
$205$	−19.2332	−1.34330
$206$	8.60412	0.599477
$207$	−11.8844	−0.826022
$208$	4.02237	0.278901
$209$	0.417137	0.0288539
$210$	22.9198	1.58162
$211$	−4.71354	−0.324493	−0.162247	−	0.986750i	$-0.551874\pi$
$211$	−4.71354	−0.324493	−0.162247	+	0.986750i	$0.551874\pi$
$212$	−7.36444	−0.505792
$213$	−20.1620	−1.38148
$214$	15.7677	1.07786
$215$	−0.810357	−0.0552659
$216$	−8.18040	−0.556606
$217$	−16.1209	−1.09436
$218$	−10.8946	−0.737877
$219$	24.6101	1.66300
$220$	0.433001	0.0291929
$221$	24.4123	1.64215
$222$	−18.9778	−1.27371
$223$	14.6436	0.980607	0.490303	−	0.871552i	$-0.336886\pi$
$223$	14.6436	0.980607	0.490303	+	0.871552i	$0.336886\pi$
$224$	−2.38509	−0.159360
$225$	31.9153	2.12768
$226$	11.1717	0.743129
$227$	4.00729	0.265973	0.132987	−	0.991118i	$-0.457543\pi$
$227$	4.00729	0.265973	0.132987	+	0.991118i	$0.457543\pi$
$228$	−9.25754	−0.613096
$229$	19.5526	1.29207	0.646036	−	0.763307i	$-0.276426\pi$
$229$	19.5526	1.29207	0.646036	+	0.763307i	$0.276426\pi$
$230$	−6.69377	−0.441374
$231$	0.941799	0.0619658
$232$	9.25547	0.607651
$233$	8.73069	0.571966	0.285983	−	0.958235i	$-0.407680\pi$
$233$	8.73069	0.571966	0.285983	+	0.958235i	$0.407680\pi$
$234$	23.1824	1.51548
$235$	−16.1374	−1.05268
$236$	3.83824	0.249848
$237$	14.3385	0.931388
$238$	−14.4754	−0.938301
$239$	8.56196	0.553827	0.276914	−	0.960895i	$-0.410688\pi$
$239$	8.56196	0.553827	0.276914	+	0.960895i	$0.410688\pi$
$240$	−9.60963	−0.620299
$241$	12.1315	0.781457	0.390729	−	0.920506i	$-0.372223\pi$
$241$	12.1315	0.781457	0.390729	+	0.920506i	$0.372223\pi$
$242$	−10.9822	−0.705963
$243$	4.03721	0.258987
$244$	−7.22904	−0.462792
$245$	−4.25690	−0.271964
$246$	17.5395	1.11827
$247$	12.5789	0.800375
$248$	6.75906	0.429201
$249$	−2.49332	−0.158007
$250$	1.74515	0.110373
$251$	−20.7537	−1.30996	−0.654980	−	0.755646i	$-0.727323\pi$
$251$	−20.7537	−1.30996	−0.654980	+	0.755646i	$0.727323\pi$
$252$	−13.7461	−0.865925
$253$	−0.275054	−0.0172925
$254$	4.66698	0.292833
$255$	−58.3221	−3.65227
$256$	1.00000	0.0625000
$257$	−7.69330	−0.479895	−0.239947	−	0.970786i	$-0.577130\pi$
$257$	−7.69330	−0.479895	−0.239947	+	0.970786i	$0.577130\pi$
$258$	0.738994	0.0460077
$259$	−15.2903	−0.950090
$260$	13.0573	0.809778
$261$	53.3427	3.30183
$262$	−0.770744	−0.0476167
$263$	2.38744	0.147216	0.0736078	−	0.997287i	$-0.476549\pi$
$263$	2.38744	0.147216	0.0736078	+	0.997287i	$0.476549\pi$
$264$	−0.394870	−0.0243026
$265$	−23.9062	−1.46855
$266$	−7.45872	−0.457323
$267$	−4.79066	−0.293183
$268$	2.37458	0.145051
$269$	15.2643	0.930678	0.465339	−	0.885133i	$-0.345932\pi$
$269$	15.2643	0.930678	0.465339	+	0.885133i	$0.345932\pi$
$270$	−26.5550	−1.61608
$271$	9.18499	0.557949	0.278974	−	0.960299i	$-0.410006\pi$
$271$	9.18499	0.557949	0.278974	+	0.960299i	$0.410006\pi$
$272$	6.06913	0.367995
$273$	28.4002	1.71886
$274$	8.17630	0.493949
$275$	0.738653	0.0445424
$276$	6.10430	0.367435
$277$	17.8676	1.07356	0.536779	−	0.843723i	$-0.319641\pi$
$277$	17.8676	1.07356	0.536779	+	0.843723i	$0.319641\pi$
$278$	−13.8085	−0.828176
$279$	38.9549	2.33217
$280$	−7.74239	−0.462696
$281$	25.7365	1.53531	0.767656	−	0.640862i	$-0.221423\pi$
$281$	25.7365	1.53531	0.767656	+	0.640862i	$0.221423\pi$
$282$	14.7162	0.876339
$283$	−4.31288	−0.256374	−0.128187	−	0.991750i	$-0.540916\pi$
$283$	−4.31288	−0.256374	−0.128187	+	0.991750i	$0.540916\pi$
$284$	6.81080	0.404147
$285$	−30.0515	−1.78010
$286$	0.536537	0.0317261
$287$	14.1314	0.834149
$288$	5.76337	0.339610
$289$	19.8344	1.16673
$290$	30.0448	1.76429
$291$	−56.9497	−3.33845
$292$	−8.31338	−0.486504
$293$	−17.4459	−1.01920	−0.509600	−	0.860412i	$-0.670206\pi$
$293$	−17.4459	−1.01920	−0.509600	+	0.860412i	$0.670206\pi$
$294$	3.88202	0.226404
$295$	12.4596	0.725425
$296$	6.41078	0.372619
$297$	−1.09117	−0.0633162
$298$	−1.16046	−0.0672235
$299$	−8.29434	−0.479674
$300$	−16.3930	−0.946448
$301$	0.595401	0.0343183
$302$	−7.83474	−0.450838
$303$	22.7291	1.30575
$304$	3.12723	0.179359
$305$	−23.4667	−1.34370
$306$	34.9787	1.99960
$307$	25.9751	1.48248	0.741239	−	0.671241i	$-0.234238\pi$
$307$	25.9751	1.48248	0.741239	+	0.671241i	$0.234238\pi$
$308$	−0.318143	−0.0181279
$309$	−25.4708	−1.44898
$310$	21.9410	1.24617
$311$	14.4182	0.817582	0.408791	−	0.912628i	$-0.365950\pi$
$311$	14.4182	0.817582	0.408791	+	0.912628i	$0.365950\pi$
$312$	−11.9074	−0.674124
$313$	−29.5183	−1.66847	−0.834236	−	0.551408i	$-0.814091\pi$
$313$	−29.5183	−1.66847	−0.834236	+	0.551408i	$0.814091\pi$
$314$	−4.55148	−0.256855
$315$	−44.6223	−2.51418
$316$	−4.84361	−0.272474
$317$	−25.8186	−1.45012	−0.725058	−	0.688688i	$-0.758187\pi$
$317$	−25.8186	−1.45012	−0.725058	+	0.688688i	$0.758187\pi$
$318$	21.8009	1.22254
$319$	1.23457	0.0691228
$320$	3.24617	0.181466
$321$	−46.6771	−2.60526
$322$	4.91818	0.274079
$323$	18.9796	1.05605
$324$	6.92632	0.384796
$325$	22.2743	1.23555
$326$	−18.4182	−1.02009
$327$	32.2514	1.78350
$328$	−5.92489	−0.327147
$329$	11.8567	0.653683
$330$	−1.28181	−0.0705615
$331$	−19.0588	−1.04757	−0.523784	−	0.851851i	$-0.675480\pi$
$331$	−19.0588	−1.04757	−0.523784	+	0.851851i	$0.675480\pi$
$332$	0.842251	0.0462246
$333$	36.9477	2.02472
$334$	−9.98211	−0.546197
$335$	7.70829	0.421149
$336$	7.06057	0.385186
$337$	−4.37750	−0.238458	−0.119229	−	0.992867i	$-0.538042\pi$
$337$	−4.37750	−0.238458	−0.119229	+	0.992867i	$0.538042\pi$
$338$	3.17944	0.172939
$339$	−33.0715	−1.79620
$340$	19.7014	1.06846
$341$	0.901580	0.0488233
$342$	18.0234	0.974593
$343$	19.8233	1.07036
$344$	−0.249635	−0.0134594
$345$	19.8156	1.06683
$346$	−3.86663	−0.207871
$347$	10.2187	0.548569	0.274284	−	0.961649i	$-0.411559\pi$
$347$	10.2187	0.548569	0.274284	+	0.961649i	$0.411559\pi$
$348$	−27.3990	−1.46874
$349$	23.2504	1.24457	0.622283	−	0.782793i	$-0.286205\pi$
$349$	23.2504	1.24457	0.622283	+	0.782793i	$0.286205\pi$
$350$	−13.2077	−0.705979
$351$	−32.9046	−1.75632
$352$	0.133388	0.00710963
$353$	3.44084	0.183138	0.0915688	−	0.995799i	$-0.470812\pi$
$353$	3.44084	0.183138	0.0915688	+	0.995799i	$0.470812\pi$
$354$	−11.3623	−0.603902
$355$	22.1090	1.17342
$356$	1.61830	0.0857698
$357$	42.8515	2.26794
$358$	−14.8646	−0.785618
$359$	−35.4497	−1.87096	−0.935482	−	0.353373i	$-0.885035\pi$
$359$	−35.4497	−1.87096	−0.935482	+	0.353373i	$0.885035\pi$
$360$	18.7089	0.986044
$361$	−9.22042	−0.485285
$362$	0.240071	0.0126179
$363$	32.5106	1.70636
$364$	−9.59369	−0.502846
$365$	−26.9866	−1.41254
$366$	21.4001	1.11860
$367$	25.1366	1.31212	0.656060	−	0.754709i	$-0.272222\pi$
$367$	25.1366	1.31212	0.656060	+	0.754709i	$0.272222\pi$
$368$	−2.06205	−0.107492
$369$	−34.1473	−1.77764
$370$	20.8105	1.08188
$371$	17.5648	0.911920
$372$	−20.0088	−1.03741
$373$	13.6623	0.707409	0.353704	−	0.935357i	$-0.384922\pi$
$373$	13.6623	0.707409	0.353704	+	0.935357i	$0.384922\pi$
$374$	0.809553	0.0418610
$375$	−5.16618	−0.266780
$376$	−4.97120	−0.256370
$377$	37.2289	1.91739
$378$	19.5110	1.00354
$379$	−23.0480	−1.18390	−0.591949	−	0.805976i	$-0.701641\pi$
$379$	−23.0480	−1.18390	−0.591949	+	0.805976i	$0.701641\pi$
$380$	10.1515	0.520762
$381$	−13.8157	−0.707798
$382$	7.91782	0.405111
$383$	−22.2153	−1.13515	−0.567574	−	0.823322i	$-0.692118\pi$
$383$	−22.2153	−1.13515	−0.567574	+	0.823322i	$0.692118\pi$
$384$	−2.96030	−0.151067
$385$	−1.03275	−0.0526336
$386$	12.6242	0.642553
$387$	−1.43874	−0.0731352
$388$	19.2378	0.976652
$389$	28.1461	1.42706	0.713531	−	0.700624i	$-0.247095\pi$
$389$	28.1461	1.42706	0.713531	+	0.700624i	$0.247095\pi$
$390$	−38.6534	−1.95729
$391$	−12.5149	−0.632905
$392$	−1.31136	−0.0662338
$393$	2.28163	0.115093
$394$	−5.49226	−0.276696
$395$	−15.7232	−0.791118
$396$	0.768767	0.0386320
$397$	6.67404	0.334961	0.167480	−	0.985875i	$-0.446437\pi$
$397$	6.67404	0.334961	0.167480	+	0.985875i	$0.446437\pi$
$398$	−16.4501	−0.824571
$399$	22.0800	1.10538
$400$	5.53760	0.276880
$401$	0.236896	0.0118300	0.00591500	−	0.999983i	$-0.498117\pi$
$401$	0.236896	0.0118300	0.00591500	+	0.999983i	$0.498117\pi$
$402$	−7.02947	−0.350598
$403$	27.1874	1.35430
$404$	−7.67798	−0.381994
$405$	22.4840	1.11724
$406$	−22.0751	−1.09557
$407$	0.855124	0.0423869
$408$	−17.9665	−0.889472
$409$	30.2021	1.49340	0.746699	−	0.665162i	$-0.231638\pi$
$409$	30.2021	1.49340	0.746699	+	0.665162i	$0.231638\pi$
$410$	−19.2332	−0.949860
$411$	−24.2043	−1.19391
$412$	8.60412	0.423895
$413$	−9.15454	−0.450466
$414$	−11.8844	−0.584085
$415$	2.73409	0.134211
$416$	4.02237	0.197213
$417$	40.8772	2.00176
$418$	0.417137	0.0204028
$419$	0.716253	0.0349912	0.0174956	−	0.999847i	$-0.494431\pi$
$419$	0.716253	0.0349912	0.0174956	+	0.999847i	$0.494431\pi$
$420$	22.9198	1.11837
$421$	−12.3108	−0.599990	−0.299995	−	0.953941i	$-0.596985\pi$
$421$	−12.3108	−0.599990	−0.299995	+	0.953941i	$0.596985\pi$
$422$	−4.71354	−0.229451
$423$	−28.6509	−1.39305
$424$	−7.36444	−0.357649
$425$	33.6085	1.63025
$426$	−20.1620	−0.976852
$427$	17.2419	0.834393
$428$	15.7677	0.762160
$429$	−1.58831	−0.0766844
$430$	−0.810357	−0.0390789
$431$	34.1930	1.64702	0.823509	−	0.567303i	$-0.192013\pi$
$431$	34.1930	1.64702	0.823509	+	0.567303i	$0.192013\pi$
$432$	−8.18040	−0.393580
$433$	−26.4538	−1.27129	−0.635645	−	0.771982i	$-0.719266\pi$
$433$	−26.4538	−1.27129	−0.635645	+	0.771982i	$0.719266\pi$
$434$	−16.1209	−0.773830
$435$	−88.9416	−4.26442
$436$	−10.8946	−0.521758
$437$	−6.44852	−0.308475
$438$	24.6101	1.17592
$439$	−11.8580	−0.565954	−0.282977	−	0.959127i	$-0.591322\pi$
$439$	−11.8580	−0.565954	−0.282977	+	0.959127i	$0.591322\pi$
$440$	0.433001	0.0206425
$441$	−7.55787	−0.359898
$442$	24.4123	1.16117
$443$	15.7569	0.748635	0.374317	−	0.927301i	$-0.377877\pi$
$443$	15.7569	0.748635	0.374317	+	0.927301i	$0.377877\pi$
$444$	−18.9778	−0.900647
$445$	5.25328	0.249029
$446$	14.6436	0.693394
$447$	3.43530	0.162484
$448$	−2.38509	−0.112685
$449$	−27.1783	−1.28262	−0.641312	−	0.767280i	$-0.721610\pi$
$449$	−27.1783	−1.28262	−0.641312	+	0.767280i	$0.721610\pi$
$450$	31.9153	1.50450
$451$	−0.790312	−0.0372144
$452$	11.1717	0.525472
$453$	23.1932	1.08971
$454$	4.00729	0.188072
$455$	−31.1427	−1.45999
$456$	−9.25754	−0.433524
$457$	34.8604	1.63070	0.815350	−	0.578969i	$-0.196545\pi$
$457$	34.8604	1.63070	0.815350	+	0.578969i	$0.196545\pi$
$458$	19.5526	0.913632
$459$	−49.6480	−2.31737
$460$	−6.69377	−0.312099
$461$	−30.2221	−1.40759	−0.703793	−	0.710405i	$-0.748512\pi$
$461$	−30.2221	−1.40759	−0.703793	+	0.710405i	$0.748512\pi$
$462$	0.941799	0.0438164
$463$	19.7329	0.917064	0.458532	−	0.888678i	$-0.348375\pi$
$463$	19.7329	0.917064	0.458532	+	0.888678i	$0.348375\pi$
$464$	9.25547	0.429674
$465$	−64.9520	−3.01208
$466$	8.73069	0.404441
$467$	−6.77294	−0.313414	−0.156707	−	0.987645i	$-0.550088\pi$
$467$	−6.77294	−0.313414	−0.156707	+	0.987645i	$0.550088\pi$
$468$	23.1824	1.07161
$469$	−5.66358	−0.261520
$470$	−16.1374	−0.744361
$471$	13.4737	0.620838
$472$	3.83824	0.176669
$473$	−0.0332984	−0.00153106
$474$	14.3385	0.658590
$475$	17.3174	0.794575
$476$	−14.4754	−0.663479
$477$	−42.4440	−1.94338
$478$	8.56196	0.391615
$479$	−35.2879	−1.61235	−0.806174	−	0.591679i	$-0.798465\pi$
$479$	−35.2879	−1.61235	−0.806174	+	0.591679i	$0.798465\pi$
$480$	−9.60963	−0.438617
$481$	25.7865	1.17576
$482$	12.1315	0.552574
$483$	−14.5593	−0.662470
$484$	−10.9822	−0.499191
$485$	62.4492	2.83567
$486$	4.03721	0.183132
$487$	12.1904	0.552400	0.276200	−	0.961100i	$-0.410925\pi$
$487$	12.1904	0.552400	0.276200	+	0.961100i	$0.410925\pi$
$488$	−7.22904	−0.327243
$489$	54.5233	2.46563
$490$	−4.25690	−0.192307
$491$	16.6402	0.750961	0.375480	−	0.926830i	$-0.377478\pi$
$491$	16.6402	0.750961	0.375480	+	0.926830i	$0.377478\pi$
$492$	17.5395	0.790739
$493$	56.1727	2.52989
$494$	12.5789	0.565950
$495$	2.49555	0.112167
$496$	6.75906	0.303491
$497$	−16.2444	−0.728659
$498$	−2.49332	−0.111728
$499$	28.9208	1.29467	0.647337	−	0.762204i	$-0.275882\pi$
$499$	28.9208	1.29467	0.647337	+	0.762204i	$0.275882\pi$
$500$	1.74515	0.0780457
$501$	29.5500	1.32020
$502$	−20.7537	−0.926282
$503$	9.16313	0.408564	0.204282	−	0.978912i	$-0.434514\pi$
$503$	9.16313	0.408564	0.204282	+	0.978912i	$0.434514\pi$
$504$	−13.7461	−0.612302
$505$	−24.9240	−1.10910
$506$	−0.275054	−0.0122277
$507$	−9.41209	−0.418006
$508$	4.66698	0.207064
$509$	−1.13176	−0.0501643	−0.0250821	−	0.999685i	$-0.507985\pi$
$509$	−1.13176	−0.0501643	−0.0250821	+	0.999685i	$0.507985\pi$
$510$	−58.3221	−2.58255
$511$	19.8281	0.877145
$512$	1.00000	0.0441942
$513$	−25.5820	−1.12947
$514$	−7.69330	−0.339337
$515$	27.9304	1.23076
$516$	0.738994	0.0325324
$517$	−0.663101	−0.0291631
$518$	−15.2903	−0.671815
$519$	11.4464	0.502440
$520$	13.0573	0.572600
$521$	−24.9786	−1.09433	−0.547166	−	0.837024i	$-0.684293\pi$
$521$	−24.9786	−1.09433	−0.547166	+	0.837024i	$0.684293\pi$
$522$	53.3427	2.33475
$523$	14.3858	0.629047	0.314524	−	0.949250i	$-0.398155\pi$
$523$	14.3858	0.629047	0.314524	+	0.949250i	$0.398155\pi$
$524$	−0.770744	−0.0336701
$525$	39.0986	1.70640
$526$	2.38744	0.104097
$527$	41.0216	1.78693
$528$	−0.394870	−0.0171845
$529$	−18.7479	−0.815128
$530$	−23.9062	−1.03842
$531$	22.1212	0.959979
$532$	−7.45872	−0.323376
$533$	−23.8321	−1.03228
$534$	−4.79066	−0.207312
$535$	51.1846	2.21290
$536$	2.37458	0.102566
$537$	44.0036	1.89890
$538$	15.2643	0.658089
$539$	−0.174921	−0.00753437
$540$	−26.5550	−1.14274
$541$	−29.5458	−1.27027	−0.635137	−	0.772399i	$-0.719056\pi$
$541$	−29.5458	−1.27027	−0.635137	+	0.772399i	$0.719056\pi$
$542$	9.18499	0.394529
$543$	−0.710683	−0.0304983
$544$	6.06913	0.260212
$545$	−35.3658	−1.51490
$546$	28.4002	1.21542
$547$	15.8406	0.677294	0.338647	−	0.940914i	$-0.390031\pi$
$547$	15.8406	0.677294	0.338647	+	0.940914i	$0.390031\pi$
$548$	8.17630	0.349274
$549$	−41.6636	−1.77816
$550$	0.738653	0.0314963
$551$	28.9440	1.23306
$552$	6.10430	0.259816
$553$	11.5524	0.491259
$554$	17.8676	0.759119
$555$	−61.6052	−2.61499
$556$	−13.8085	−0.585609
$557$	31.5168	1.33541	0.667705	−	0.744426i	$-0.267277\pi$
$557$	31.5168	1.33541	0.667705	+	0.744426i	$0.267277\pi$
$558$	38.9549	1.64909
$559$	−1.00412	−0.0424699
$560$	−7.74239	−0.327176
$561$	−2.39652	−0.101181
$562$	25.7365	1.08563
$563$	−14.5820	−0.614559	−0.307279	−	0.951619i	$-0.599419\pi$
$563$	−14.5820	−0.614559	−0.307279	+	0.951619i	$0.599419\pi$
$564$	14.7162	0.619666
$565$	36.2652	1.52569
$566$	−4.31288	−0.181284
$567$	−16.5199	−0.693770
$568$	6.81080	0.285775
$569$	−11.1962	−0.469368	−0.234684	−	0.972072i	$-0.575406\pi$
$569$	−11.1962	−0.469368	−0.234684	+	0.972072i	$0.575406\pi$
$570$	−30.0515	−1.25872
$571$	38.4926	1.61086	0.805432	−	0.592689i	$-0.201934\pi$
$571$	38.4926	1.61086	0.805432	+	0.592689i	$0.201934\pi$
$572$	0.536537	0.0224338
$573$	−23.4391	−0.979183
$574$	14.1314	0.589832
$575$	−11.4188	−0.476199
$576$	5.76337	0.240140
$577$	14.3634	0.597954	0.298977	−	0.954260i	$-0.403355\pi$
$577$	14.3634	0.597954	0.298977	+	0.954260i	$0.403355\pi$
$578$	19.8344	0.825002
$579$	−37.3713	−1.55310
$580$	30.0448	1.24754
$581$	−2.00884	−0.0833408
$582$	−56.9497	−2.36064
$583$	−0.982331	−0.0406840
$584$	−8.31338	−0.344010
$585$	75.2539	3.11137
$586$	−17.4459	−0.720683
$587$	16.2385	0.670233	0.335117	−	0.942177i	$-0.391224\pi$
$587$	16.2385	0.670233	0.335117	+	0.942177i	$0.391224\pi$
$588$	3.88202	0.160092
$589$	21.1371	0.870941
$590$	12.4596	0.512953
$591$	16.2587	0.668795
$592$	6.41078	0.263481
$593$	4.84490	0.198956	0.0994781	−	0.995040i	$-0.468283\pi$
$593$	4.84490	0.198956	0.0994781	+	0.995040i	$0.468283\pi$
$594$	−1.09117	−0.0447713
$595$	−46.9896	−1.92639
$596$	−1.16046	−0.0475342
$597$	48.6973	1.99305
$598$	−8.29434	−0.339181
$599$	−3.23296	−0.132095	−0.0660476	−	0.997816i	$-0.521039\pi$
$599$	−3.23296	−0.132095	−0.0660476	+	0.997816i	$0.521039\pi$
$600$	−16.3930	−0.669240
$601$	−1.44631	−0.0589964	−0.0294982	−	0.999565i	$-0.509391\pi$
$601$	−1.44631	−0.0589964	−0.0294982	+	0.999565i	$0.509391\pi$
$602$	0.595401	0.0242667
$603$	13.6856	0.557320
$604$	−7.83474	−0.318791
$605$	−35.6501	−1.44938
$606$	22.7291	0.923307
$607$	37.7696	1.53302	0.766510	−	0.642232i	$-0.221991\pi$
$607$	37.7696	1.53302	0.766510	+	0.642232i	$0.221991\pi$
$608$	3.12723	0.126826
$609$	65.3489	2.64807
$610$	−23.4667	−0.950137
$611$	−19.9960	−0.808951
$612$	34.9787	1.41393
$613$	−29.3816	−1.18671	−0.593356	−	0.804940i	$-0.702197\pi$
$613$	−29.3816	−1.18671	−0.593356	+	0.804940i	$0.702197\pi$
$614$	25.9751	1.04827
$615$	56.9360	2.29588
$616$	−0.318143	−0.0128183
$617$	13.6948	0.551333	0.275666	−	0.961253i	$-0.411101\pi$
$617$	13.6948	0.551333	0.275666	+	0.961253i	$0.411101\pi$
$618$	−25.4708	−1.02458
$619$	20.4323	0.821244	0.410622	−	0.911806i	$-0.365312\pi$
$619$	20.4323	0.821244	0.410622	+	0.911806i	$0.365312\pi$
$620$	21.9410	0.881173
$621$	16.8684	0.676907
$622$	14.4182	0.578118
$623$	−3.85979	−0.154639
$624$	−11.9074	−0.476678
$625$	−22.0230	−0.880918
$626$	−29.5183	−1.17979
$627$	−1.23485	−0.0493151
$628$	−4.55148	−0.181624
$629$	38.9079	1.55136
$630$	−44.6223	−1.77779
$631$	28.3564	1.12885	0.564424	−	0.825485i	$-0.309098\pi$
$631$	28.3564	1.12885	0.564424	+	0.825485i	$0.309098\pi$
$632$	−4.84361	−0.192668
$633$	13.9535	0.554601
$634$	−25.8186	−1.02539
$635$	15.1498	0.601202
$636$	21.8009	0.864463
$637$	−5.27478	−0.208994
$638$	1.23457	0.0488772
$639$	39.2532	1.55283
$640$	3.24617	0.128316
$641$	17.1051	0.675609	0.337805	−	0.941216i	$-0.390316\pi$
$641$	17.1051	0.675609	0.337805	+	0.941216i	$0.390316\pi$
$642$	−46.6771	−1.84220
$643$	20.7022	0.816416	0.408208	−	0.912889i	$-0.366154\pi$
$643$	20.7022	0.816416	0.408208	+	0.912889i	$0.366154\pi$
$644$	4.91818	0.193803
$645$	2.39890	0.0944565
$646$	18.9796	0.746742
$647$	18.0462	0.709469	0.354734	−	0.934967i	$-0.384571\pi$
$647$	18.0462	0.709469	0.354734	+	0.934967i	$0.384571\pi$
$648$	6.92632	0.272092
$649$	0.511977	0.0200969
$650$	22.2743	0.873669
$651$	47.7228	1.87040
$652$	−18.4182	−0.721311
$653$	−5.94174	−0.232518	−0.116259	−	0.993219i	$-0.537090\pi$
$653$	−5.94174	−0.232518	−0.116259	+	0.993219i	$0.537090\pi$
$654$	32.2514	1.26113
$655$	−2.50196	−0.0977598
$656$	−5.92489	−0.231328
$657$	−47.9131	−1.86927
$658$	11.8567	0.462224
$659$	13.6732	0.532632	0.266316	−	0.963886i	$-0.414194\pi$
$659$	13.6732	0.532632	0.266316	+	0.963886i	$0.414194\pi$
$660$	−1.28181	−0.0498945
$661$	26.2477	1.02092	0.510459	−	0.859902i	$-0.329475\pi$
$661$	26.2477	1.02092	0.510459	+	0.859902i	$0.329475\pi$
$662$	−19.0588	−0.740742
$663$	−72.2677	−2.80664
$664$	0.842251	0.0326857
$665$	−24.2122	−0.938911
$666$	36.9477	1.43169
$667$	−19.0853	−0.738985
$668$	−9.98211	−0.386219
$669$	−43.3494	−1.67598
$670$	7.70829	0.297797
$671$	−0.964270	−0.0372252
$672$	7.06057	0.272367
$673$	−37.0262	−1.42725	−0.713627	−	0.700526i	$-0.752949\pi$
$673$	−37.0262	−1.42725	−0.713627	+	0.700526i	$0.752949\pi$
$674$	−4.37750	−0.168615
$675$	−45.2998	−1.74359
$676$	3.17944	0.122286
$677$	−13.7744	−0.529392	−0.264696	−	0.964332i	$-0.585272\pi$
$677$	−13.7744	−0.529392	−0.264696	+	0.964332i	$0.585272\pi$
$678$	−33.0715	−1.27010
$679$	−45.8839	−1.76086
$680$	19.7014	0.755515
$681$	−11.8628	−0.454583
$682$	0.901580	0.0345233
$683$	42.7464	1.63565	0.817823	−	0.575471i	$-0.195181\pi$
$683$	42.7464	1.63565	0.817823	+	0.575471i	$0.195181\pi$
$684$	18.0234	0.689142
$685$	26.5417	1.01410
$686$	19.8233	0.756858
$687$	−57.8815	−2.20832
$688$	−0.249635	−0.00951724
$689$	−29.6225	−1.12853
$690$	19.8156	0.754366
$691$	−13.4017	−0.509826	−0.254913	−	0.966964i	$-0.582047\pi$
$691$	−13.4017	−0.509826	−0.254913	+	0.966964i	$0.582047\pi$
$692$	−3.86663	−0.146987
$693$	−1.83358	−0.0696518
$694$	10.2187	0.387897
$695$	−44.8246	−1.70029
$696$	−27.3990	−1.03855
$697$	−35.9590	−1.36204
$698$	23.2504	0.880041
$699$	−25.8454	−0.977564
$700$	−13.2077	−0.499203
$701$	−4.94428	−0.186743	−0.0933714	−	0.995631i	$-0.529764\pi$
$701$	−4.94428	−0.186743	−0.0933714	+	0.995631i	$0.529764\pi$
$702$	−32.9046	−1.24190
$703$	20.0480	0.756124
$704$	0.133388	0.00502727
$705$	47.7714	1.79917
$706$	3.44084	0.129498
$707$	18.3126	0.688718
$708$	−11.3623	−0.427023
$709$	3.53906	0.132912	0.0664561	−	0.997789i	$-0.478831\pi$
$709$	3.53906	0.132912	0.0664561	+	0.997789i	$0.478831\pi$
$710$	22.1090	0.829736
$711$	−27.9155	−1.04691
$712$	1.61830	0.0606484
$713$	−13.9375	−0.521965
$714$	42.8515	1.60368
$715$	1.74169	0.0651355
$716$	−14.8646	−0.555516
$717$	−25.3460	−0.946563
$718$	−35.4497	−1.32297
$719$	−9.08328	−0.338749	−0.169375	−	0.985552i	$-0.554175\pi$
$719$	−9.08328	−0.338749	−0.169375	+	0.985552i	$0.554175\pi$
$720$	18.7089	0.697238
$721$	−20.5216	−0.764263
$722$	−9.22042	−0.343149
$723$	−35.9128	−1.33561
$724$	0.240071	0.00892218
$725$	51.2531	1.90349
$726$	32.5106	1.20658
$727$	−26.9825	−1.00073	−0.500363	−	0.865816i	$-0.666800\pi$
$727$	−26.9825	−1.00073	−0.500363	+	0.865816i	$0.666800\pi$
$728$	−9.59369	−0.355566
$729$	−32.7303	−1.21223
$730$	−26.9866	−0.998820
$731$	−1.51507	−0.0560368
$732$	21.4001	0.790971
$733$	−18.0492	−0.666663	−0.333331	−	0.942810i	$-0.608173\pi$
$733$	−18.0492	−0.666663	−0.333331	+	0.942810i	$0.608173\pi$
$734$	25.1366	0.927809
$735$	12.6017	0.464821
$736$	−2.06205	−0.0760083
$737$	0.316742	0.0116673
$738$	−34.1473	−1.25698
$739$	−13.3199	−0.489980	−0.244990	−	0.969526i	$-0.578785\pi$
$739$	−13.3199	−0.489980	−0.244990	+	0.969526i	$0.578785\pi$
$740$	20.8105	0.765007
$741$	−37.2372	−1.36794
$742$	17.5648	0.644825
$743$	−38.9271	−1.42810	−0.714049	−	0.700096i	$-0.753141\pi$
$743$	−38.9271	−1.42810	−0.714049	+	0.700096i	$0.753141\pi$
$744$	−20.0088	−0.733559
$745$	−3.76704	−0.138014
$746$	13.6623	0.500214
$747$	4.85421	0.177606
$748$	0.809553	0.0296002
$749$	−37.6073	−1.37414
$750$	−5.16618	−0.188642
$751$	−26.4510	−0.965211	−0.482606	−	0.875838i	$-0.660310\pi$
$751$	−26.4510	−0.965211	−0.482606	+	0.875838i	$0.660310\pi$
$752$	−4.97120	−0.181281
$753$	61.4371	2.23889
$754$	37.2289	1.35580
$755$	−25.4329	−0.925597
$756$	19.5110	0.709607
$757$	−38.2441	−1.39001	−0.695003	−	0.719007i	$-0.744597\pi$
$757$	−38.2441	−1.39001	−0.695003	+	0.719007i	$0.744597\pi$
$758$	−23.0480	−0.837142
$759$	0.814243	0.0295551
$760$	10.1515	0.368234
$761$	−28.4369	−1.03084	−0.515418	−	0.856939i	$-0.672363\pi$
$761$	−28.4369	−1.03084	−0.515418	+	0.856939i	$0.672363\pi$
$762$	−13.8157	−0.500489
$763$	25.9846	0.940707
$764$	7.91782	0.286457
$765$	113.547	4.10529
$766$	−22.2153	−0.802672
$767$	15.4388	0.557464
$768$	−2.96030	−0.106821
$769$	10.9691	0.395554	0.197777	−	0.980247i	$-0.436628\pi$
$769$	10.9691	0.395554	0.197777	+	0.980247i	$0.436628\pi$
$770$	−1.03275	−0.0372176
$771$	22.7745	0.820202
$772$	12.6242	0.454353
$773$	6.06194	0.218033	0.109016	−	0.994040i	$-0.465230\pi$
$773$	6.06194	0.218033	0.109016	+	0.994040i	$0.465230\pi$
$774$	−1.43874	−0.0517144
$775$	37.4290	1.34449
$776$	19.2378	0.690597
$777$	45.2637	1.62383
$778$	28.1461	1.00908
$779$	−18.5285	−0.663853
$780$	−38.6534	−1.38402
$781$	0.908482	0.0325081
$782$	−12.5149	−0.447531
$783$	−75.7134	−2.70578
$784$	−1.31136	−0.0468344
$785$	−14.7749	−0.527338
$786$	2.28163	0.0813831
$787$	35.7478	1.27427	0.637136	−	0.770752i	$-0.280119\pi$
$787$	35.7478	1.27427	0.637136	+	0.770752i	$0.280119\pi$
$788$	−5.49226	−0.195654
$789$	−7.06752	−0.251611
$790$	−15.7232	−0.559405
$791$	−26.6454	−0.947402
$792$	0.768767	0.0273170
$793$	−29.0778	−1.03258
$794$	6.67404	0.236853
$795$	70.7695	2.50993
$796$	−16.4501	−0.583060
$797$	32.0344	1.13472	0.567358	−	0.823471i	$-0.307966\pi$
$797$	32.0344	1.13472	0.567358	+	0.823471i	$0.307966\pi$
$798$	22.0800	0.781625
$799$	−30.1709	−1.06737
$800$	5.53760	0.195784
$801$	9.32687	0.329549
$802$	0.236896	0.00836507
$803$	−1.10891	−0.0391326
$804$	−7.02947	−0.247910
$805$	15.9652	0.562700
$806$	27.1874	0.957636
$807$	−45.1867	−1.59065
$808$	−7.67798	−0.270110
$809$	11.2074	0.394032	0.197016	−	0.980400i	$-0.436875\pi$
$809$	11.2074	0.394032	0.197016	+	0.980400i	$0.436875\pi$
$810$	22.4840	0.790008
$811$	1.21401	0.0426297	0.0213148	−	0.999773i	$-0.493215\pi$
$811$	1.21401	0.0426297	0.0213148	+	0.999773i	$0.493215\pi$
$812$	−22.0751	−0.774684
$813$	−27.1903	−0.953606
$814$	0.855124	0.0299721
$815$	−59.7884	−2.09430
$816$	−17.9665	−0.628952
$817$	−0.780666	−0.0273120
$818$	30.2021	1.05599
$819$	−55.2920	−1.93206
$820$	−19.2332	−0.671652
$821$	−21.3150	−0.743898	−0.371949	−	0.928253i	$-0.621310\pi$
$821$	−21.3150	−0.743898	−0.371949	+	0.928253i	$0.621310\pi$
$822$	−24.2043	−0.844222
$823$	−35.8937	−1.25118	−0.625588	−	0.780154i	$-0.715141\pi$
$823$	−35.8937	−1.25118	−0.625588	+	0.780154i	$0.715141\pi$
$824$	8.60412	0.299739
$825$	−2.18663	−0.0761288
$826$	−9.15454	−0.318527
$827$	−18.4852	−0.642792	−0.321396	−	0.946945i	$-0.604152\pi$
$827$	−18.4852	−0.642792	−0.321396	+	0.946945i	$0.604152\pi$
$828$	−11.8844	−0.413011
$829$	16.4226	0.570380	0.285190	−	0.958471i	$-0.407943\pi$
$829$	16.4226	0.570380	0.285190	+	0.958471i	$0.407943\pi$
$830$	2.73409	0.0949016
$831$	−52.8933	−1.83485
$832$	4.02237	0.139450
$833$	−7.95883	−0.275757
$834$	40.8772	1.41546
$835$	−32.4036	−1.12137
$836$	0.417137	0.0144270
$837$	−55.2918	−1.91116
$838$	0.716253	0.0247425
$839$	−18.8057	−0.649245	−0.324623	−	0.945844i	$-0.605237\pi$
$839$	−18.8057	−0.649245	−0.324623	+	0.945844i	$0.605237\pi$
$840$	22.9198	0.790808
$841$	56.6637	1.95392
$842$	−12.3108	−0.424257
$843$	−76.1878	−2.62405
$844$	−4.71354	−0.162247
$845$	10.3210	0.355053
$846$	−28.6509	−0.985037
$847$	26.1935	0.900020
$848$	−7.36444	−0.252896
$849$	12.7674	0.438177
$850$	33.6085	1.15276
$851$	−13.2194	−0.453154
$852$	−20.1620	−0.690739
$853$	−20.4600	−0.700537	−0.350269	−	0.936649i	$-0.613910\pi$
$853$	−20.4600	−0.700537	−0.350269	+	0.936649i	$0.613910\pi$
$854$	17.2419	0.590005
$855$	58.5070	2.00090
$856$	15.7677	0.538929
$857$	43.2663	1.47795	0.738974	−	0.673733i	$-0.235310\pi$
$857$	43.2663	1.47795	0.738974	+	0.673733i	$0.235310\pi$
$858$	−1.58831	−0.0542241
$859$	18.0922	0.617297	0.308649	−	0.951176i	$-0.400123\pi$
$859$	18.0922	0.617297	0.308649	+	0.951176i	$0.400123\pi$
$860$	−0.810357	−0.0276329
$861$	−41.8331	−1.42567
$862$	34.1930	1.16462
$863$	−1.22787	−0.0417973	−0.0208987	−	0.999782i	$-0.506653\pi$
$863$	−1.22787	−0.0417973	−0.0208987	+	0.999782i	$0.506653\pi$
$864$	−8.18040	−0.278303
$865$	−12.5517	−0.426771
$866$	−26.4538	−0.898938
$867$	−58.7157	−1.99409
$868$	−16.1209	−0.547180
$869$	−0.646082	−0.0219168
$870$	−88.9416	−3.01540
$871$	9.55143	0.323638
$872$	−10.8946	−0.368939
$873$	110.875	3.75254
$874$	−6.44852	−0.218124
$875$	−4.16234	−0.140713
$876$	24.6101	0.831498
$877$	−52.2110	−1.76304	−0.881520	−	0.472147i	$-0.843479\pi$
$877$	−52.2110	−1.76304	−0.881520	+	0.472147i	$0.843479\pi$
$878$	−11.8580	−0.400190
$879$	51.6450	1.74194
$880$	0.433001	0.0145965
$881$	−35.3288	−1.19026	−0.595129	−	0.803630i	$-0.702899\pi$
$881$	−35.3288	−1.19026	−0.595129	+	0.803630i	$0.702899\pi$
$882$	−7.55787	−0.254487
$883$	−42.7257	−1.43784	−0.718918	−	0.695095i	$-0.755362\pi$
$883$	−42.7257	−1.43784	−0.718918	+	0.695095i	$0.755362\pi$
$884$	24.4123	0.821074
$885$	−36.8841	−1.23984
$886$	15.7569	0.529365
$887$	−5.26493	−0.176779	−0.0883895	−	0.996086i	$-0.528172\pi$
$887$	−5.26493	−0.176779	−0.0883895	+	0.996086i	$0.528172\pi$
$888$	−18.9778	−0.636854
$889$	−11.1312	−0.373327
$890$	5.25328	0.176090
$891$	0.923892	0.0309515
$892$	14.6436	0.490303
$893$	−15.5461	−0.520230
$894$	3.43530	0.114894
$895$	−48.2530	−1.61292
$896$	−2.38509	−0.0796801
$897$	24.5537	0.819825
$898$	−27.1783	−0.906953
$899$	62.5582	2.08643
$900$	31.9153	1.06384
$901$	−44.6958	−1.48903
$902$	−0.790312	−0.0263145
$903$	−1.76256	−0.0586545
$904$	11.1717	0.371565
$905$	0.779311	0.0259052
$906$	23.1932	0.770541
$907$	21.2761	0.706463	0.353231	−	0.935536i	$-0.385083\pi$
$907$	21.2761	0.706463	0.353231	+	0.935536i	$0.385083\pi$
$908$	4.00729	0.132987
$909$	−44.2510	−1.46771
$910$	−31.1427	−1.03237
$911$	−10.6114	−0.351570	−0.175785	−	0.984429i	$-0.556246\pi$
$911$	−10.6114	−0.351570	−0.175785	+	0.984429i	$0.556246\pi$
$912$	−9.25754	−0.306548
$913$	0.112347	0.00371813
$914$	34.8604	1.15308
$915$	69.4683	2.29655
$916$	19.5526	0.646036
$917$	1.83829	0.0607057
$918$	−49.6480	−1.63863
$919$	−9.60221	−0.316748	−0.158374	−	0.987379i	$-0.550625\pi$
$919$	−9.60221	−0.316748	−0.158374	+	0.987379i	$0.550625\pi$
$920$	−6.69377	−0.220687
$921$	−76.8941	−2.53375
$922$	−30.2221	−0.995313
$923$	27.3955	0.901735
$924$	0.941799	0.0309829
$925$	35.5003	1.16724
$926$	19.7329	0.648463
$927$	49.5887	1.62871
$928$	9.25547	0.303826
$929$	38.4489	1.26147	0.630734	−	0.775999i	$-0.282754\pi$
$929$	38.4489	1.26147	0.630734	+	0.775999i	$0.282754\pi$
$930$	−64.9520	−2.12986
$931$	−4.10093	−0.134403
$932$	8.73069	0.285983
$933$	−42.6822	−1.39735
$934$	−6.77294	−0.221617
$935$	2.62794	0.0859429
$936$	23.1824	0.757740
$937$	−42.0186	−1.37269	−0.686343	−	0.727278i	$-0.740785\pi$
$937$	−42.0186	−1.37269	−0.686343	+	0.727278i	$0.740785\pi$
$938$	−5.66358	−0.184922
$939$	87.3829	2.85163
$940$	−16.1374	−0.526342
$941$	17.3028	0.564054	0.282027	−	0.959406i	$-0.408993\pi$
$941$	17.3028	0.564054	0.282027	+	0.959406i	$0.408993\pi$
$942$	13.4737	0.438998
$943$	12.2174	0.397855
$944$	3.83824	0.124924
$945$	63.3359	2.06032
$946$	−0.0332984	−0.00108262
$947$	−9.86840	−0.320680	−0.160340	−	0.987062i	$-0.551259\pi$
$947$	−9.86840	−0.320680	−0.160340	+	0.987062i	$0.551259\pi$
$948$	14.3385	0.465694
$949$	−33.4395	−1.08549
$950$	17.3174	0.561850
$951$	76.4307	2.47844
$952$	−14.4754	−0.469151
$953$	−23.4093	−0.758300	−0.379150	−	0.925335i	$-0.623784\pi$
$953$	−23.4093	−0.758300	−0.379150	+	0.925335i	$0.623784\pi$
$954$	−42.4440	−1.37417
$955$	25.7026	0.831716
$956$	8.56196	0.276914
$957$	−3.65470	−0.118140
$958$	−35.2879	−1.14010
$959$	−19.5012	−0.629726
$960$	−9.60963	−0.310149
$961$	14.6849	0.473705
$962$	25.7865	0.831390
$963$	90.8750	2.92841
$964$	12.1315	0.390729
$965$	40.9801	1.31920
$966$	−14.5593	−0.468437
$967$	−50.0548	−1.60965	−0.804827	−	0.593510i	$-0.797742\pi$
$967$	−50.0548	−1.60965	−0.804827	+	0.593510i	$0.797742\pi$
$968$	−10.9822	−0.352982
$969$	−56.1853	−1.80493
$970$	62.4492	2.00512
$971$	−42.5873	−1.36669	−0.683346	−	0.730095i	$-0.739476\pi$
$971$	−42.5873	−1.36669	−0.683346	+	0.730095i	$0.739476\pi$
$972$	4.03721	0.129494
$973$	32.9344	1.05583
$974$	12.1904	0.390606
$975$	−65.9385	−2.11172
$976$	−7.22904	−0.231396
$977$	55.0473	1.76112	0.880560	−	0.473935i	$-0.157167\pi$
$977$	55.0473	1.76112	0.880560	+	0.473935i	$0.157167\pi$
$978$	54.5233	1.74346
$979$	0.215863	0.00689901
$980$	−4.25690	−0.135982
$981$	−62.7898	−2.00472
$982$	16.6402	0.531009
$983$	−6.59470	−0.210338	−0.105169	−	0.994454i	$-0.533538\pi$
$983$	−6.59470	−0.210338	−0.105169	+	0.994454i	$0.533538\pi$
$984$	17.5395	0.559137
$985$	−17.8288	−0.568073
$986$	56.1727	1.78890
$987$	−35.0995	−1.11723
$988$	12.5789	0.400187
$989$	0.514761	0.0163684
$990$	2.49555	0.0793137
$991$	48.1794	1.53047	0.765235	−	0.643751i	$-0.222623\pi$
$991$	48.1794	1.53047	0.765235	+	0.643751i	$0.222623\pi$
$992$	6.75906	0.214600
$993$	56.4198	1.79043
$994$	−16.2444	−0.515240
$995$	−53.3999	−1.69289
$996$	−2.49332	−0.0790037
$997$	−21.1130	−0.668656	−0.334328	−	0.942457i	$-0.608509\pi$
$997$	−21.1130	−0.668656	−0.334328	+	0.942457i	$0.608509\pi$
$998$	28.9208	0.915473
$999$	−52.4427	−1.65921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.g.1.4	✓	95

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.g.1.4	✓	95	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.96030 q^{3} +1.00000 q^{4} +3.24617 q^{5} -2.96030 q^{6} -2.38509 q^{7} +1.00000 q^{8} +5.76337 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.96030 q^{3} +1.00000 q^{4} +3.24617 q^{5} -2.96030 q^{6} -2.38509 q^{7} +1.00000 q^{8} +5.76337 q^{9} +3.24617 q^{10} +0.133388 q^{11} -2.96030 q^{12} +4.02237 q^{13} -2.38509 q^{14} -9.60963 q^{15} +1.00000 q^{16} +6.06913 q^{17} +5.76337 q^{18} +3.12723 q^{19} +3.24617 q^{20} +7.06057 q^{21} +0.133388 q^{22} -2.06205 q^{23} -2.96030 q^{24} +5.53760 q^{25} +4.02237 q^{26} -8.18040 q^{27} -2.38509 q^{28} +9.25547 q^{29} -9.60963 q^{30} +6.75906 q^{31} +1.00000 q^{32} -0.394870 q^{33} +6.06913 q^{34} -7.74239 q^{35} +5.76337 q^{36} +6.41078 q^{37} +3.12723 q^{38} -11.9074 q^{39} +3.24617 q^{40} -5.92489 q^{41} +7.06057 q^{42} -0.249635 q^{43} +0.133388 q^{44} +18.7089 q^{45} -2.06205 q^{46} -4.97120 q^{47} -2.96030 q^{48} -1.31136 q^{49} +5.53760 q^{50} -17.9665 q^{51} +4.02237 q^{52} -7.36444 q^{53} -8.18040 q^{54} +0.433001 q^{55} -2.38509 q^{56} -9.25754 q^{57} +9.25547 q^{58} +3.83824 q^{59} -9.60963 q^{60} -7.22904 q^{61} +6.75906 q^{62} -13.7461 q^{63} +1.00000 q^{64} +13.0573 q^{65} -0.394870 q^{66} +2.37458 q^{67} +6.06913 q^{68} +6.10430 q^{69} -7.74239 q^{70} +6.81080 q^{71} +5.76337 q^{72} -8.31338 q^{73} +6.41078 q^{74} -16.3930 q^{75} +3.12723 q^{76} -0.318143 q^{77} -11.9074 q^{78} -4.84361 q^{79} +3.24617 q^{80} +6.92632 q^{81} -5.92489 q^{82} +0.842251 q^{83} +7.06057 q^{84} +19.7014 q^{85} -0.249635 q^{86} -27.3990 q^{87} +0.133388 q^{88} +1.61830 q^{89} +18.7089 q^{90} -9.59369 q^{91} -2.06205 q^{92} -20.0088 q^{93} -4.97120 q^{94} +10.1515 q^{95} -2.96030 q^{96} +19.2378 q^{97} -1.31136 q^{98} +0.768767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9}+O(q^{10}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9	\( 95 q + 95 q^{2} + 24 q^{3} + 95 q^{4} + 36 q^{5} + 24 q^{6} + 21 q^{7} + 95 q^{8} + 121 q^{9} + 36 q^{10} + 40 q^{11} + 24 q^{12} + 52 q^{13} + 21 q^{14} + 15 q^{15} + 95 q^{16} + 84 q^{17} + 121 q^{18} + 37 q^{19} + 36 q^{20} + 36 q^{21} + 40 q^{22} + 37 q^{23} + 24 q^{24} + 133 q^{25} + 52 q^{26} + 93 q^{27} + 21 q^{28} + 66 q^{29} + 15 q^{30} + 10 q^{31} + 95 q^{32} + 63 q^{33} + 84 q^{34} + 55 q^{35} + 121 q^{36} + 49 q^{37} + 37 q^{38} + 14 q^{39} + 36 q^{40} + 98 q^{41} + 36 q^{42} + 37 q^{43} + 40 q^{44} + 97 q^{45} + 37 q^{46} + 91 q^{47} + 24 q^{48} + 170 q^{49} + 133 q^{50} + 22 q^{51} + 52 q^{52} + 70 q^{53} + 93 q^{54} - q^{55} + 21 q^{56} + 50 q^{57} + 66 q^{58} + 72 q^{59} + 15 q^{60} + 97 q^{61} + 10 q^{62} + 75 q^{63} + 95 q^{64} + 75 q^{65} + 63 q^{66} + 39 q^{67} + 84 q^{68} + 65 q^{69} + 55 q^{70} + 28 q^{71} + 121 q^{72} + 117 q^{73} + 49 q^{74} + 62 q^{75} + 37 q^{76} + 92 q^{77} + 14 q^{78} + q^{79} + 36 q^{80} + 155 q^{81} + 98 q^{82} + 117 q^{83} + 36 q^{84} + 81 q^{85} + 37 q^{86} + 46 q^{87} + 40 q^{88} + 90 q^{89} + 97 q^{90} + 65 q^{91} + 37 q^{92} + 36 q^{93} + 91 q^{94} + 38 q^{95} + 24 q^{96} + 111 q^{97} + 170 q^{98} + 97 q^{99}+O(q^{100}) \) 95 * q + 95 * q^2 + 24 * q^3 + 95 * q^4 + 36 * q^5 + 24 * q^6 + 21 * q^7 + 95 * q^8 + 121 * q^9 + 36 * q^10 + 40 * q^11 + 24 * q^12 + 52 * q^13 + 21 * q^14 + 15 * q^15 + 95 * q^16 + 84 * q^17 + 121 * q^18 + 37 * q^19 + 36 * q^20 + 36 * q^21 + 40 * q^22 + 37 * q^23 + 24 * q^24 + 133 * q^25 + 52 * q^26 + 93 * q^27 + 21 * q^28 + 66 * q^29 + 15 * q^30 + 10 * q^31 + 95 * q^32 + 63 * q^33 + 84 * q^34 + 55 * q^35 + 121 * q^36 + 49 * q^37 + 37 * q^38 + 14 * q^39 + 36 * q^40 + 98 * q^41 + 36 * q^42 + 37 * q^43 + 40 * q^44 + 97 * q^45 + 37 * q^46 + 91 * q^47 + 24 * q^48 + 170 * q^49 + 133 * q^50 + 22 * q^51 + 52 * q^52 + 70 * q^53 + 93 * q^54 - q^55 + 21 * q^56 + 50 * q^57 + 66 * q^58 + 72 * q^59 + 15 * q^60 + 97 * q^61 + 10 * q^62 + 75 * q^63 + 95 * q^64 + 75 * q^65 + 63 * q^66 + 39 * q^67 + 84 * q^68 + 65 * q^69 + 55 * q^70 + 28 * q^71 + 121 * q^72 + 117 * q^73 + 49 * q^74 + 62 * q^75 + 37 * q^76 + 92 * q^77 + 14 * q^78 + q^79 + 36 * q^80 + 155 * q^81 + 98 * q^82 + 117 * q^83 + 36 * q^84 + 81 * q^85 + 37 * q^86 + 46 * q^87 + 40 * q^88 + 90 * q^89 + 97 * q^90 + 65 * q^91 + 37 * q^92 + 36 * q^93 + 91 * q^94 + 38 * q^95 + 24 * q^96 + 111 * q^97 + 170 * q^98 + 97 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more