LMFDB - Embedded newform 8002.2.a.d.1.40

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:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.208633 q^{3} +1.00000 q^{4} -3.73927 q^{5} +0.208633 q^{6} +3.61474 q^{7} +1.00000 q^{8} -2.95647 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.208633 q^{3} +1.00000 q^{4} -3.73927 q^{5} +0.208633 q^{6} +3.61474 q^{7} +1.00000 q^{8} -2.95647 q^{9} -3.73927 q^{10} -0.0497564 q^{11} +0.208633 q^{12} +1.11781 q^{13} +3.61474 q^{14} -0.780137 q^{15} +1.00000 q^{16} -4.61800 q^{17} -2.95647 q^{18} -4.65720 q^{19} -3.73927 q^{20} +0.754155 q^{21} -0.0497564 q^{22} +8.02396 q^{23} +0.208633 q^{24} +8.98215 q^{25} +1.11781 q^{26} -1.24272 q^{27} +3.61474 q^{28} -2.69102 q^{29} -0.780137 q^{30} +8.50982 q^{31} +1.00000 q^{32} -0.0103808 q^{33} -4.61800 q^{34} -13.5165 q^{35} -2.95647 q^{36} -2.50971 q^{37} -4.65720 q^{38} +0.233212 q^{39} -3.73927 q^{40} +1.60248 q^{41} +0.754155 q^{42} -2.14191 q^{43} -0.0497564 q^{44} +11.0550 q^{45} +8.02396 q^{46} +6.74907 q^{47} +0.208633 q^{48} +6.06634 q^{49} +8.98215 q^{50} -0.963470 q^{51} +1.11781 q^{52} -3.05825 q^{53} -1.24272 q^{54} +0.186052 q^{55} +3.61474 q^{56} -0.971647 q^{57} -2.69102 q^{58} -5.47311 q^{59} -0.780137 q^{60} -12.6854 q^{61} +8.50982 q^{62} -10.6869 q^{63} +1.00000 q^{64} -4.17979 q^{65} -0.0103808 q^{66} +4.10959 q^{67} -4.61800 q^{68} +1.67407 q^{69} -13.5165 q^{70} -13.5090 q^{71} -2.95647 q^{72} -10.6208 q^{73} -2.50971 q^{74} +1.87398 q^{75} -4.65720 q^{76} -0.179856 q^{77} +0.233212 q^{78} +9.92817 q^{79} -3.73927 q^{80} +8.61014 q^{81} +1.60248 q^{82} -7.46157 q^{83} +0.754155 q^{84} +17.2680 q^{85} -2.14191 q^{86} -0.561436 q^{87} -0.0497564 q^{88} -2.34606 q^{89} +11.0550 q^{90} +4.04059 q^{91} +8.02396 q^{92} +1.77543 q^{93} +6.74907 q^{94} +17.4145 q^{95} +0.208633 q^{96} +11.8188 q^{97} +6.06634 q^{98} +0.147103 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * 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
$3$	0.208633	0.120455	0.0602273	−	0.998185i	$-0.480817\pi$
$3$	0.208633	0.120455	0.0602273	+	0.998185i	$0.480817\pi$
$4$	1.00000	0.500000
$5$	−3.73927	−1.67225	−0.836126	−	0.548537i	$-0.815185\pi$
$5$	−3.73927	−1.67225	−0.836126	+	0.548537i	$0.815185\pi$
$6$	0.208633	0.0851742
$7$	3.61474	1.36624	0.683122	−	0.730305i	$-0.260622\pi$
$7$	3.61474	1.36624	0.683122	+	0.730305i	$0.260622\pi$
$8$	1.00000	0.353553
$9$	−2.95647	−0.985491
$10$	−3.73927	−1.18246
$11$	−0.0497564	−0.0150021	−0.00750105	−	0.999972i	$-0.502388\pi$
$11$	−0.0497564	−0.0150021	−0.00750105	+	0.999972i	$0.502388\pi$
$12$	0.208633	0.0602273
$13$	1.11781	0.310024	0.155012	−	0.987913i	$-0.450458\pi$
$13$	1.11781	0.310024	0.155012	+	0.987913i	$0.450458\pi$
$14$	3.61474	0.966080
$15$	−0.780137	−0.201430
$16$	1.00000	0.250000
$17$	−4.61800	−1.12003	−0.560015	−	0.828482i	$-0.689205\pi$
$17$	−4.61800	−1.12003	−0.560015	+	0.828482i	$0.689205\pi$
$18$	−2.95647	−0.696847
$19$	−4.65720	−1.06843	−0.534217	−	0.845347i	$-0.679393\pi$
$19$	−4.65720	−1.06843	−0.534217	+	0.845347i	$0.679393\pi$
$20$	−3.73927	−0.836126
$21$	0.754155	0.164570
$22$	−0.0497564	−0.0106081
$23$	8.02396	1.67311	0.836556	−	0.547881i	$-0.184565\pi$
$23$	8.02396	1.67311	0.836556	+	0.547881i	$0.184565\pi$
$24$	0.208633	0.0425871
$25$	8.98215	1.79643
$26$	1.11781	0.219220
$27$	−1.24272	−0.239161
$28$	3.61474	0.683122
$29$	−2.69102	−0.499709	−0.249855	−	0.968283i	$-0.580383\pi$
$29$	−2.69102	−0.499709	−0.249855	+	0.968283i	$0.580383\pi$
$30$	−0.780137	−0.142433
$31$	8.50982	1.52841	0.764205	−	0.644974i	$-0.223132\pi$
$31$	8.50982	1.52841	0.764205	+	0.644974i	$0.223132\pi$
$32$	1.00000	0.176777
$33$	−0.0103808	−0.00180707
$34$	−4.61800	−0.791981
$35$	−13.5165	−2.28470
$36$	−2.95647	−0.492745
$37$	−2.50971	−0.412594	−0.206297	−	0.978489i	$-0.566141\pi$
$37$	−2.50971	−0.412594	−0.206297	+	0.978489i	$0.566141\pi$
$38$	−4.65720	−0.755497
$39$	0.233212	0.0373438
$40$	−3.73927	−0.591231
$41$	1.60248	0.250265	0.125132	−	0.992140i	$-0.460064\pi$
$41$	1.60248	0.250265	0.125132	+	0.992140i	$0.460064\pi$
$42$	0.754155	0.116369
$43$	−2.14191	−0.326639	−0.163319	−	0.986573i	$-0.552220\pi$
$43$	−2.14191	−0.326639	−0.163319	+	0.986573i	$0.552220\pi$
$44$	−0.0497564	−0.00750105
$45$	11.0550	1.64799
$46$	8.02396	1.18307
$47$	6.74907	0.984454	0.492227	−	0.870467i	$-0.336183\pi$
$47$	6.74907	0.984454	0.492227	+	0.870467i	$0.336183\pi$
$48$	0.208633	0.0301136
$49$	6.06634	0.866620
$50$	8.98215	1.27027
$51$	−0.963470	−0.134913
$52$	1.11781	0.155012
$53$	−3.05825	−0.420083	−0.210041	−	0.977692i	$-0.567360\pi$
$53$	−3.05825	−0.420083	−0.210041	+	0.977692i	$0.567360\pi$
$54$	−1.24272	−0.169113
$55$	0.186052	0.0250873
$56$	3.61474	0.483040
$57$	−0.971647	−0.128698
$58$	−2.69102	−0.353348
$59$	−5.47311	−0.712538	−0.356269	−	0.934383i	$-0.615951\pi$
$59$	−5.47311	−0.712538	−0.356269	+	0.934383i	$0.615951\pi$
$60$	−0.780137	−0.100715
$61$	−12.6854	−1.62420	−0.812101	−	0.583517i	$-0.801676\pi$
$61$	−12.6854	−1.62420	−0.812101	+	0.583517i	$0.801676\pi$
$62$	8.50982	1.08075
$63$	−10.6869	−1.34642
$64$	1.00000	0.125000
$65$	−4.17979	−0.518439
$66$	−0.0103808	−0.00127779
$67$	4.10959	0.502066	0.251033	−	0.967979i	$-0.419230\pi$
$67$	4.10959	0.502066	0.251033	+	0.967979i	$0.419230\pi$
$68$	−4.61800	−0.560015
$69$	1.67407	0.201534
$70$	−13.5165	−1.61553
$71$	−13.5090	−1.60322	−0.801612	−	0.597845i	$-0.796024\pi$
$71$	−13.5090	−1.60322	−0.801612	+	0.597845i	$0.796024\pi$
$72$	−2.95647	−0.348424
$73$	−10.6208	−1.24307	−0.621533	−	0.783388i	$-0.713490\pi$
$73$	−10.6208	−1.24307	−0.621533	+	0.783388i	$0.713490\pi$
$74$	−2.50971	−0.291748
$75$	1.87398	0.216388
$76$	−4.65720	−0.534217
$77$	−0.179856	−0.0204965
$78$	0.233212	0.0264061
$79$	9.92817	1.11701	0.558503	−	0.829502i	$-0.311376\pi$
$79$	9.92817	1.11701	0.558503	+	0.829502i	$0.311376\pi$
$80$	−3.73927	−0.418063
$81$	8.61014	0.956683
$82$	1.60248	0.176964
$83$	−7.46157	−0.819013	−0.409507	−	0.912307i	$-0.634299\pi$
$83$	−7.46157	−0.819013	−0.409507	+	0.912307i	$0.634299\pi$
$84$	0.754155	0.0822851
$85$	17.2680	1.87297
$86$	−2.14191	−0.230968
$87$	−0.561436	−0.0601923
$88$	−0.0497564	−0.00530404
$89$	−2.34606	−0.248681	−0.124341	−	0.992240i	$-0.539682\pi$
$89$	−2.34606	−0.248681	−0.124341	+	0.992240i	$0.539682\pi$
$90$	11.0550	1.16530
$91$	4.04059	0.423569
$92$	8.02396	0.836556
$93$	1.77543	0.184104
$94$	6.74907	0.696114
$95$	17.4145	1.78669
$96$	0.208633	0.0212936
$97$	11.8188	1.20002	0.600009	−	0.799993i	$-0.295164\pi$
$97$	11.8188	1.20002	0.600009	+	0.799993i	$0.295164\pi$
$98$	6.06634	0.612793
$99$	0.147103	0.0147844
$100$	8.98215	0.898215
$101$	−18.9696	−1.88754	−0.943770	−	0.330601i	$-0.892748\pi$
$101$	−18.9696	−1.88754	−0.943770	+	0.330601i	$0.892748\pi$
$102$	−0.963470	−0.0953977
$103$	−19.8584	−1.95670	−0.978352	−	0.206948i	$-0.933647\pi$
$103$	−19.8584	−1.95670	−0.978352	+	0.206948i	$0.933647\pi$
$104$	1.11781	0.109610
$105$	−2.81999	−0.275203
$106$	−3.05825	−0.297043
$107$	−3.36011	−0.324834	−0.162417	−	0.986722i	$-0.551929\pi$
$107$	−3.36011	−0.324834	−0.162417	+	0.986722i	$0.551929\pi$
$108$	−1.24272	−0.119581
$109$	−10.2528	−0.982043	−0.491021	−	0.871148i	$-0.663376\pi$
$109$	−10.2528	−0.982043	−0.491021	+	0.871148i	$0.663376\pi$
$110$	0.186052	0.0177394
$111$	−0.523609	−0.0496988
$112$	3.61474	0.341561
$113$	3.03324	0.285343	0.142671	−	0.989770i	$-0.454431\pi$
$113$	3.03324	0.285343	0.142671	+	0.989770i	$0.454431\pi$
$114$	−0.971647	−0.0910030
$115$	−30.0038	−2.79787
$116$	−2.69102	−0.249855
$117$	−3.30477	−0.305526
$118$	−5.47311	−0.503841
$119$	−16.6929	−1.53023
$120$	−0.780137	−0.0712164
$121$	−10.9975	−0.999775
$122$	−12.6854	−1.14848
$123$	0.334330	0.0301455
$124$	8.50982	0.764205
$125$	−14.8903	−1.33183
$126$	−10.6869	−0.952063
$127$	16.8635	1.49640	0.748198	−	0.663475i	$-0.230919\pi$
$127$	16.8635	1.49640	0.748198	+	0.663475i	$0.230919\pi$
$128$	1.00000	0.0883883
$129$	−0.446874	−0.0393451
$130$	−4.17979	−0.366592
$131$	4.43417	0.387415	0.193708	−	0.981059i	$-0.437949\pi$
$131$	4.43417	0.387415	0.193708	+	0.981059i	$0.437949\pi$
$132$	−0.0103808	−0.000903536 0
$133$	−16.8346	−1.45974
$134$	4.10959	0.355014
$135$	4.64686	0.399938
$136$	−4.61800	−0.395991
$137$	13.3858	1.14363	0.571813	−	0.820384i	$-0.306240\pi$
$137$	13.3858	1.14363	0.571813	+	0.820384i	$0.306240\pi$
$138$	1.67407	0.142506
$139$	−7.20161	−0.610833	−0.305416	−	0.952219i	$-0.598796\pi$
$139$	−7.20161	−0.610833	−0.305416	+	0.952219i	$0.598796\pi$
$140$	−13.5165	−1.14235
$141$	1.40808	0.118582
$142$	−13.5090	−1.13365
$143$	−0.0556181	−0.00465102
$144$	−2.95647	−0.246373
$145$	10.0624	0.835640
$146$	−10.6208	−0.878981
$147$	1.26564	0.104388
$148$	−2.50971	−0.206297
$149$	−22.8941	−1.87556	−0.937780	−	0.347231i	$-0.887122\pi$
$149$	−22.8941	−1.87556	−0.937780	+	0.347231i	$0.887122\pi$
$150$	1.87398	0.153009
$151$	7.77361	0.632607	0.316304	−	0.948658i	$-0.397558\pi$
$151$	7.77361	0.632607	0.316304	+	0.948658i	$0.397558\pi$
$152$	−4.65720	−0.377748
$153$	13.6530	1.10378
$154$	−0.179856	−0.0144932
$155$	−31.8205	−2.55589
$156$	0.233212	0.0186719
$157$	−9.35849	−0.746889	−0.373445	−	0.927652i	$-0.621823\pi$
$157$	−9.35849	−0.746889	−0.373445	+	0.927652i	$0.621823\pi$
$158$	9.92817	0.789843
$159$	−0.638053	−0.0506009
$160$	−3.73927	−0.295615
$161$	29.0045	2.28588
$162$	8.61014	0.676477
$163$	8.92449	0.699020	0.349510	−	0.936933i	$-0.386348\pi$
$163$	8.92449	0.699020	0.349510	+	0.936933i	$0.386348\pi$
$164$	1.60248	0.125132
$165$	0.0388168	0.00302188
$166$	−7.46157	−0.579130
$167$	−11.3924	−0.881567	−0.440783	−	0.897614i	$-0.645299\pi$
$167$	−11.3924	−0.881567	−0.440783	+	0.897614i	$0.645299\pi$
$168$	0.754155	0.0581843
$169$	−11.7505	−0.903885
$170$	17.2680	1.32439
$171$	13.7689	1.05293
$172$	−2.14191	−0.163319
$173$	−12.1083	−0.920574	−0.460287	−	0.887770i	$-0.652253\pi$
$173$	−12.1083	−0.920574	−0.460287	+	0.887770i	$0.652253\pi$
$174$	−0.561436	−0.0425624
$175$	32.4681	2.45436
$176$	−0.0497564	−0.00375053
$177$	−1.14187	−0.0858285
$178$	−2.34606	−0.175844
$179$	−5.61142	−0.419418	−0.209709	−	0.977764i	$-0.567252\pi$
$179$	−5.61142	−0.419418	−0.209709	+	0.977764i	$0.567252\pi$
$180$	11.0550	0.823995
$181$	6.06421	0.450749	0.225375	−	0.974272i	$-0.427639\pi$
$181$	6.06421	0.450749	0.225375	+	0.974272i	$0.427639\pi$
$182$	4.04059	0.299508
$183$	−2.64660	−0.195643
$184$	8.02396	0.591534
$185$	9.38449	0.689961
$186$	1.77543	0.130181
$187$	0.229775	0.0168028
$188$	6.74907	0.492227
$189$	−4.49210	−0.326753
$190$	17.4145	1.26338
$191$	−19.4900	−1.41025	−0.705125	−	0.709083i	$-0.749109\pi$
$191$	−19.4900	−1.41025	−0.705125	+	0.709083i	$0.749109\pi$
$192$	0.208633	0.0150568
$193$	18.7928	1.35273	0.676367	−	0.736564i	$-0.263553\pi$
$193$	18.7928	1.35273	0.676367	+	0.736564i	$0.263553\pi$
$194$	11.8188	0.848541
$195$	−0.872043	−0.0624483
$196$	6.06634	0.433310
$197$	0.489647	0.0348859	0.0174429	−	0.999848i	$-0.494447\pi$
$197$	0.489647	0.0348859	0.0174429	+	0.999848i	$0.494447\pi$
$198$	0.147103	0.0104542
$199$	−5.64417	−0.400104	−0.200052	−	0.979785i	$-0.564111\pi$
$199$	−5.64417	−0.400104	−0.200052	+	0.979785i	$0.564111\pi$
$200$	8.98215	0.635134
$201$	0.857397	0.0604761
$202$	−18.9696	−1.33469
$203$	−9.72733	−0.682724
$204$	−0.963470	−0.0674564
$205$	−5.99209	−0.418506
$206$	−19.8584	−1.38360
$207$	−23.7226	−1.64884
$208$	1.11781	0.0775061
$209$	0.231725	0.0160288
$210$	−2.81999	−0.194598
$211$	−4.49003	−0.309106	−0.154553	−	0.987984i	$-0.549394\pi$
$211$	−4.49003	−0.309106	−0.154553	+	0.987984i	$0.549394\pi$
$212$	−3.05825	−0.210041
$213$	−2.81843	−0.193116
$214$	−3.36011	−0.229692
$215$	8.00919	0.546222
$216$	−1.24272	−0.0845563
$217$	30.7608	2.08818
$218$	−10.2528	−0.694409
$219$	−2.21585	−0.149733
$220$	0.186052	0.0125437
$221$	−5.16204	−0.347237
$222$	−0.523609	−0.0351424
$223$	−11.7477	−0.786687	−0.393344	−	0.919392i	$-0.628682\pi$
$223$	−11.7477	−0.786687	−0.393344	+	0.919392i	$0.628682\pi$
$224$	3.61474	0.241520
$225$	−26.5555	−1.77036
$226$	3.03324	0.201768
$227$	−10.3619	−0.687742	−0.343871	−	0.939017i	$-0.611738\pi$
$227$	−10.3619	−0.687742	−0.343871	+	0.939017i	$0.611738\pi$
$228$	−0.971647	−0.0643489
$229$	14.2813	0.943735	0.471868	−	0.881669i	$-0.343580\pi$
$229$	14.2813	0.943735	0.471868	+	0.881669i	$0.343580\pi$
$230$	−30.0038	−1.97839
$231$	−0.0375240	−0.00246890
$232$	−2.69102	−0.176674
$233$	0.376380	0.0246574	0.0123287	−	0.999924i	$-0.496076\pi$
$233$	0.376380	0.0246574	0.0123287	+	0.999924i	$0.496076\pi$
$234$	−3.30477	−0.216040
$235$	−25.2366	−1.64626
$236$	−5.47311	−0.356269
$237$	2.07135	0.134548
$238$	−16.6929	−1.08204
$239$	−12.0374	−0.778633	−0.389317	−	0.921104i	$-0.627289\pi$
$239$	−12.0374	−0.778633	−0.389317	+	0.921104i	$0.627289\pi$
$240$	−0.780137	−0.0503576
$241$	15.2264	0.980819	0.490410	−	0.871492i	$-0.336847\pi$
$241$	15.2264	0.980819	0.490410	+	0.871492i	$0.336847\pi$
$242$	−10.9975	−0.706948
$243$	5.52452	0.354398
$244$	−12.6854	−0.812101
$245$	−22.6837	−1.44921
$246$	0.334330	0.0213161
$247$	−5.20585	−0.331241
$248$	8.50982	0.540374
$249$	−1.55673	−0.0986539
$250$	−14.8903	−0.941746
$251$	11.8044	0.745088	0.372544	−	0.928014i	$-0.378486\pi$
$251$	11.8044	0.745088	0.372544	+	0.928014i	$0.378486\pi$
$252$	−10.6869	−0.673210
$253$	−0.399243	−0.0251002
$254$	16.8635	1.05811
$255$	3.60267	0.225608
$256$	1.00000	0.0625000
$257$	−18.4248	−1.14930	−0.574652	−	0.818398i	$-0.694863\pi$
$257$	−18.4248	−1.14930	−0.574652	+	0.818398i	$0.694863\pi$
$258$	−0.446874	−0.0278212
$259$	−9.07195	−0.563703
$260$	−4.17979	−0.259219
$261$	7.95592	0.492459
$262$	4.43417	0.273944
$263$	−8.59051	−0.529713	−0.264857	−	0.964288i	$-0.585325\pi$
$263$	−8.59051	−0.529713	−0.264857	+	0.964288i	$0.585325\pi$
$264$	−0.0103808	−0.000638896 0
$265$	11.4356	0.702485
$266$	−16.8346	−1.03219
$267$	−0.489465	−0.0299548
$268$	4.10959	0.251033
$269$	20.8811	1.27314	0.636571	−	0.771218i	$-0.280352\pi$
$269$	20.8811	1.27314	0.636571	+	0.771218i	$0.280352\pi$
$270$	4.64686	0.282799
$271$	−19.2755	−1.17091	−0.585453	−	0.810707i	$-0.699083\pi$
$271$	−19.2755	−1.17091	−0.585453	+	0.810707i	$0.699083\pi$
$272$	−4.61800	−0.280008
$273$	0.843001	0.0510208
$274$	13.3858	0.808666
$275$	−0.446919	−0.0269502
$276$	1.67407	0.100767
$277$	15.2226	0.914638	0.457319	−	0.889303i	$-0.348810\pi$
$277$	15.2226	0.914638	0.457319	+	0.889303i	$0.348810\pi$
$278$	−7.20161	−0.431924
$279$	−25.1591	−1.50623
$280$	−13.5165	−0.807765
$281$	4.74026	0.282780	0.141390	−	0.989954i	$-0.454843\pi$
$281$	4.74026	0.282780	0.141390	+	0.989954i	$0.454843\pi$
$282$	1.40808	0.0838501
$283$	−12.2352	−0.727305	−0.363653	−	0.931535i	$-0.618471\pi$
$283$	−12.2352	−0.727305	−0.363653	+	0.931535i	$0.618471\pi$
$284$	−13.5090	−0.801612
$285$	3.63325	0.215215
$286$	−0.0556181	−0.00328877
$287$	5.79254	0.341923
$288$	−2.95647	−0.174212
$289$	4.32597	0.254469
$290$	10.0624	0.590887
$291$	2.46580	0.144548
$292$	−10.6208	−0.621533
$293$	−6.09003	−0.355783	−0.177892	−	0.984050i	$-0.556928\pi$
$293$	−6.09003	−0.355783	−0.177892	+	0.984050i	$0.556928\pi$
$294$	1.26564	0.0738137
$295$	20.4654	1.19154
$296$	−2.50971	−0.145874
$297$	0.0618332	0.00358792
$298$	−22.8941	−1.32622
$299$	8.96926	0.518705
$300$	1.87398	0.108194
$301$	−7.74246	−0.446268
$302$	7.77361	0.447321
$303$	−3.95768	−0.227363
$304$	−4.65720	−0.267109
$305$	47.4342	2.71608
$306$	13.6530	0.780490
$307$	19.7516	1.12729	0.563643	−	0.826019i	$-0.309400\pi$
$307$	19.7516	1.12729	0.563643	+	0.826019i	$0.309400\pi$
$308$	−0.179856	−0.0102483
$309$	−4.14312	−0.235694
$310$	−31.8205	−1.80728
$311$	−24.9394	−1.41418	−0.707091	−	0.707123i	$-0.749993\pi$
$311$	−24.9394	−1.41418	−0.707091	+	0.707123i	$0.749993\pi$
$312$	0.233212	0.0132030
$313$	−8.90621	−0.503409	−0.251704	−	0.967804i	$-0.580991\pi$
$313$	−8.90621	−0.503409	−0.251704	+	0.967804i	$0.580991\pi$
$314$	−9.35849	−0.528130
$315$	39.9611	2.25155
$316$	9.92817	0.558503
$317$	−0.924541	−0.0519274	−0.0259637	−	0.999663i	$-0.508265\pi$
$317$	−0.924541	−0.0519274	−0.0259637	+	0.999663i	$0.508265\pi$
$318$	−0.638053	−0.0357802
$319$	0.133895	0.00749669
$320$	−3.73927	−0.209032
$321$	−0.701030	−0.0391277
$322$	29.0045	1.61636
$323$	21.5070	1.19668
$324$	8.61014	0.478341
$325$	10.0403	0.556937
$326$	8.92449	0.494282
$327$	−2.13908	−0.118291
$328$	1.60248	0.0884820
$329$	24.3961	1.34500
$330$	0.0388168	0.00213679
$331$	−24.3479	−1.33828	−0.669140	−	0.743136i	$-0.733337\pi$
$331$	−24.3479	−1.33828	−0.669140	+	0.743136i	$0.733337\pi$
$332$	−7.46157	−0.409507
$333$	7.41989	0.406607
$334$	−11.3924	−0.623362
$335$	−15.3669	−0.839581
$336$	0.754155	0.0411425
$337$	34.7037	1.89043	0.945216	−	0.326445i	$-0.105851\pi$
$337$	34.7037	1.89043	0.945216	+	0.326445i	$0.105851\pi$
$338$	−11.7505	−0.639143
$339$	0.632834	0.0343708
$340$	17.2680	0.936487
$341$	−0.423418	−0.0229294
$342$	13.7689	0.744535
$343$	−3.37493	−0.182229
$344$	−2.14191	−0.115484
$345$	−6.25979	−0.337016
$346$	−12.1083	−0.650944
$347$	−1.73091	−0.0929199	−0.0464600	−	0.998920i	$-0.514794\pi$
$347$	−1.73091	−0.0929199	−0.0464600	+	0.998920i	$0.514794\pi$
$348$	−0.561436	−0.0300961
$349$	−5.21466	−0.279134	−0.139567	−	0.990213i	$-0.544571\pi$
$349$	−5.21466	−0.279134	−0.139567	+	0.990213i	$0.544571\pi$
$350$	32.4681	1.73549
$351$	−1.38912	−0.0741458
$352$	−0.0497564	−0.00265202
$353$	−2.36433	−0.125841	−0.0629203	−	0.998019i	$-0.520041\pi$
$353$	−2.36433	−0.125841	−0.0629203	+	0.998019i	$0.520041\pi$
$354$	−1.14187	−0.0606899
$355$	50.5138	2.68099
$356$	−2.34606	−0.124341
$357$	−3.48269	−0.184324
$358$	−5.61142	−0.296573
$359$	21.8431	1.15283	0.576416	−	0.817156i	$-0.304451\pi$
$359$	21.8431	1.15283	0.576416	+	0.817156i	$0.304451\pi$
$360$	11.0550	0.582652
$361$	2.68948	0.141551
$362$	6.06421	0.318728
$363$	−2.29445	−0.120427
$364$	4.04059	0.211784
$365$	39.7139	2.07872
$366$	−2.64660	−0.138340
$367$	10.7999	0.563748	0.281874	−	0.959451i	$-0.409044\pi$
$367$	10.7999	0.563748	0.281874	+	0.959451i	$0.409044\pi$
$368$	8.02396	0.418278
$369$	−4.73768	−0.246634
$370$	9.38449	0.487876
$371$	−11.0548	−0.573935
$372$	1.77543	0.0920519
$373$	−26.2474	−1.35904	−0.679520	−	0.733657i	$-0.737812\pi$
$373$	−26.2474	−1.35904	−0.679520	+	0.733657i	$0.737812\pi$
$374$	0.229775	0.0118814
$375$	−3.10662	−0.160425
$376$	6.74907	0.348057
$377$	−3.00804	−0.154922
$378$	−4.49210	−0.231049
$379$	7.48676	0.384569	0.192284	−	0.981339i	$-0.438410\pi$
$379$	7.48676	0.384569	0.192284	+	0.981339i	$0.438410\pi$
$380$	17.4145	0.893346
$381$	3.51830	0.180248
$382$	−19.4900	−0.997197
$383$	−7.41227	−0.378749	−0.189375	−	0.981905i	$-0.560646\pi$
$383$	−7.41227	−0.378749	−0.189375	+	0.981905i	$0.560646\pi$
$384$	0.208633	0.0106468
$385$	0.672531	0.0342754
$386$	18.7928	0.956528
$387$	6.33251	0.321899
$388$	11.8188	0.600009
$389$	30.8514	1.56423	0.782113	−	0.623136i	$-0.214142\pi$
$389$	30.8514	1.56423	0.782113	+	0.623136i	$0.214142\pi$
$390$	−0.872043	−0.0441576
$391$	−37.0547	−1.87394
$392$	6.06634	0.306397
$393$	0.925116	0.0466659
$394$	0.489647	0.0246680
$395$	−37.1241	−1.86792
$396$	0.147103	0.00739222
$397$	−9.31182	−0.467347	−0.233673	−	0.972315i	$-0.575075\pi$
$397$	−9.31182	−0.467347	−0.233673	+	0.972315i	$0.575075\pi$
$398$	−5.64417	−0.282917
$399$	−3.51225	−0.175832
$400$	8.98215	0.449107
$401$	−31.5848	−1.57727	−0.788635	−	0.614861i	$-0.789212\pi$
$401$	−31.5848	−1.57727	−0.788635	+	0.614861i	$0.789212\pi$
$402$	0.857397	0.0427631
$403$	9.51235	0.473844
$404$	−18.9696	−0.943770
$405$	−32.1957	−1.59982
$406$	−9.72733	−0.482759
$407$	0.124874	0.00618977
$408$	−0.963470	−0.0476989
$409$	−1.10711	−0.0547432	−0.0273716	−	0.999625i	$-0.508714\pi$
$409$	−1.10711	−0.0547432	−0.0273716	+	0.999625i	$0.508714\pi$
$410$	−5.99209	−0.295928
$411$	2.79273	0.137755
$412$	−19.8584	−0.978352
$413$	−19.7839	−0.973501
$414$	−23.7226	−1.16590
$415$	27.9008	1.36960
$416$	1.11781	0.0548051
$417$	−1.50250	−0.0735775
$418$	0.231725	0.0113340
$419$	7.65441	0.373943	0.186971	−	0.982365i	$-0.440133\pi$
$419$	7.65441	0.373943	0.186971	+	0.982365i	$0.440133\pi$
$420$	−2.81999	−0.137601
$421$	−4.18250	−0.203842	−0.101921	−	0.994792i	$-0.532499\pi$
$421$	−4.18250	−0.203842	−0.101921	+	0.994792i	$0.532499\pi$
$422$	−4.49003	−0.218571
$423$	−19.9534	−0.970170
$424$	−3.05825	−0.148522
$425$	−41.4796	−2.01206
$426$	−2.81843	−0.136553
$427$	−45.8545	−2.21906
$428$	−3.36011	−0.162417
$429$	−0.0116038	−0.000560236 0
$430$	8.00919	0.386237
$431$	−28.3787	−1.36696	−0.683478	−	0.729971i	$-0.739533\pi$
$431$	−28.3787	−1.36696	−0.683478	+	0.729971i	$0.739533\pi$
$432$	−1.24272	−0.0597903
$433$	33.5268	1.61119	0.805597	−	0.592463i	$-0.201845\pi$
$433$	33.5268	1.61119	0.805597	+	0.592463i	$0.201845\pi$
$434$	30.7608	1.47657
$435$	2.09936	0.100657
$436$	−10.2528	−0.491021
$437$	−37.3692	−1.78761
$438$	−2.21585	−0.105877
$439$	−8.71949	−0.416159	−0.208079	−	0.978112i	$-0.566721\pi$
$439$	−8.71949	−0.416159	−0.208079	+	0.978112i	$0.566721\pi$
$440$	0.186052	0.00886970
$441$	−17.9350	−0.854046
$442$	−5.16204	−0.245533
$443$	−24.0633	−1.14328	−0.571640	−	0.820505i	$-0.693693\pi$
$443$	−24.0633	−1.14328	−0.571640	+	0.820505i	$0.693693\pi$
$444$	−0.523609	−0.0248494
$445$	8.77254	0.415858
$446$	−11.7477	−0.556272
$447$	−4.77648	−0.225920
$448$	3.61474	0.170780
$449$	−20.8950	−0.986094	−0.493047	−	0.870003i	$-0.664117\pi$
$449$	−20.8950	−0.986094	−0.493047	+	0.870003i	$0.664117\pi$
$450$	−26.5555	−1.25184
$451$	−0.0797334	−0.00375450
$452$	3.03324	0.142671
$453$	1.62183	0.0762004
$454$	−10.3619	−0.486307
$455$	−15.1088	−0.708314
$456$	−0.971647	−0.0455015
$457$	−19.2602	−0.900956	−0.450478	−	0.892788i	$-0.648746\pi$
$457$	−19.2602	−0.900956	−0.450478	+	0.892788i	$0.648746\pi$
$458$	14.2813	0.667322
$459$	5.73888	0.267868
$460$	−30.0038	−1.39893
$461$	27.5419	1.28275	0.641377	−	0.767226i	$-0.278364\pi$
$461$	27.5419	1.28275	0.641377	+	0.767226i	$0.278364\pi$
$462$	−0.0375240	−0.00174578
$463$	−14.8656	−0.690861	−0.345430	−	0.938444i	$-0.612267\pi$
$463$	−14.8656	−0.690861	−0.345430	+	0.938444i	$0.612267\pi$
$464$	−2.69102	−0.124927
$465$	−6.63882	−0.307868
$466$	0.376380	0.0174354
$467$	−28.9560	−1.33992	−0.669962	−	0.742395i	$-0.733690\pi$
$467$	−28.9560	−1.33992	−0.669962	+	0.742395i	$0.733690\pi$
$468$	−3.30477	−0.152763
$469$	14.8551	0.685944
$470$	−25.2366	−1.16408
$471$	−1.95249	−0.0899662
$472$	−5.47311	−0.251920
$473$	0.106574	0.00490027
$474$	2.07135	0.0951401
$475$	−41.8316	−1.91937
$476$	−16.6929	−0.765117
$477$	9.04163	0.413988
$478$	−12.0374	−0.550577
$479$	−8.04817	−0.367730	−0.183865	−	0.982951i	$-0.558861\pi$
$479$	−8.04817	−0.367730	−0.183865	+	0.982951i	$0.558861\pi$
$480$	−0.780137	−0.0356082
$481$	−2.80538	−0.127914
$482$	15.2264	0.693544
$483$	6.05131	0.275344
$484$	−10.9975	−0.499887
$485$	−44.1937	−2.00673
$486$	5.52452	0.250597
$487$	−1.37705	−0.0624000	−0.0312000	−	0.999513i	$-0.509933\pi$
$487$	−1.37705	−0.0624000	−0.0312000	+	0.999513i	$0.509933\pi$
$488$	−12.6854	−0.574242
$489$	1.86195	0.0842001
$490$	−22.6837	−1.02475
$491$	−14.5222	−0.655376	−0.327688	−	0.944786i	$-0.606270\pi$
$491$	−14.5222	−0.655376	−0.327688	+	0.944786i	$0.606270\pi$
$492$	0.334330	0.0150728
$493$	12.4271	0.559690
$494$	−5.20585	−0.234222
$495$	−0.550059	−0.0247233
$496$	8.50982	0.382102
$497$	−48.8315	−2.19039
$498$	−1.55673	−0.0697588
$499$	23.1215	1.03506	0.517530	−	0.855665i	$-0.326852\pi$
$499$	23.1215	1.03506	0.517530	+	0.855665i	$0.326852\pi$
$500$	−14.8903	−0.665915
$501$	−2.37682	−0.106189
$502$	11.8044	0.526857
$503$	37.2327	1.66012	0.830062	−	0.557671i	$-0.188305\pi$
$503$	37.2327	1.66012	0.830062	+	0.557671i	$0.188305\pi$
$504$	−10.6869	−0.476031
$505$	70.9323	3.15645
$506$	−0.399243	−0.0177485
$507$	−2.45155	−0.108877
$508$	16.8635	0.748198
$509$	3.70392	0.164173	0.0820866	−	0.996625i	$-0.473842\pi$
$509$	3.70392	0.164173	0.0820866	+	0.996625i	$0.473842\pi$
$510$	3.60267	0.159529
$511$	−38.3913	−1.69833
$512$	1.00000	0.0441942
$513$	5.78759	0.255528
$514$	−18.4248	−0.812681
$515$	74.2558	3.27210
$516$	−0.446874	−0.0196726
$517$	−0.335809	−0.0147689
$518$	−9.07195	−0.398599
$519$	−2.52619	−0.110887
$520$	−4.17979	−0.183296
$521$	−20.0181	−0.877009	−0.438505	−	0.898729i	$-0.644492\pi$
$521$	−20.0181	−0.877009	−0.438505	+	0.898729i	$0.644492\pi$
$522$	7.95592	0.348221
$523$	−2.61144	−0.114190	−0.0570952	−	0.998369i	$-0.518184\pi$
$523$	−2.61144	−0.114190	−0.0570952	+	0.998369i	$0.518184\pi$
$524$	4.43417	0.193708
$525$	6.77393	0.295639
$526$	−8.59051	−0.374564
$527$	−39.2984	−1.71187
$528$	−0.0103808	−0.000451768 0
$529$	41.3840	1.79930
$530$	11.4356	0.496732
$531$	16.1811	0.702200
$532$	−16.8346	−0.729870
$533$	1.79126	0.0775882
$534$	−0.489465	−0.0211812
$535$	12.5643	0.543204
$536$	4.10959	0.177507
$537$	−1.17073	−0.0505207
$538$	20.8811	0.900247
$539$	−0.301839	−0.0130011
$540$	4.64686	0.199969
$541$	−14.4200	−0.619966	−0.309983	−	0.950742i	$-0.600323\pi$
$541$	−14.4200	−0.619966	−0.309983	+	0.950742i	$0.600323\pi$
$542$	−19.2755	−0.827955
$543$	1.26520	0.0542948
$544$	−4.61800	−0.197995
$545$	38.3381	1.64222
$546$	0.843001	0.0360771
$547$	−9.14412	−0.390974	−0.195487	−	0.980706i	$-0.562629\pi$
$547$	−9.14412	−0.390974	−0.195487	+	0.980706i	$0.562629\pi$
$548$	13.3858	0.571813
$549$	37.5041	1.60064
$550$	−0.446919	−0.0190567
$551$	12.5326	0.533907
$552$	1.67407	0.0712530
$553$	35.8877	1.52610
$554$	15.2226	0.646747
$555$	1.95792	0.0831089
$556$	−7.20161	−0.305416
$557$	−0.841554	−0.0356578	−0.0178289	−	0.999841i	$-0.505675\pi$
$557$	−0.841554	−0.0356578	−0.0178289	+	0.999841i	$0.505675\pi$
$558$	−25.1591	−1.06507
$559$	−2.39425	−0.101266
$560$	−13.5165	−0.571176
$561$	0.0479387	0.00202398
$562$	4.74026	0.199956
$563$	−9.48608	−0.399791	−0.199895	−	0.979817i	$-0.564060\pi$
$563$	−9.48608	−0.399791	−0.199895	+	0.979817i	$0.564060\pi$
$564$	1.40808	0.0592909
$565$	−11.3421	−0.477165
$566$	−12.2352	−0.514283
$567$	31.1234	1.30706
$568$	−13.5090	−0.566825
$569$	38.7224	1.62333	0.811664	−	0.584125i	$-0.198562\pi$
$569$	38.7224	1.62333	0.811664	+	0.584125i	$0.198562\pi$
$570$	3.63325	0.152180
$571$	−9.42255	−0.394321	−0.197161	−	0.980371i	$-0.563172\pi$
$571$	−9.42255	−0.394321	−0.197161	+	0.980371i	$0.563172\pi$
$572$	−0.0556181	−0.00232551
$573$	−4.06627	−0.169871
$574$	5.79254	0.241776
$575$	72.0724	3.00563
$576$	−2.95647	−0.123186
$577$	19.6067	0.816237	0.408119	−	0.912929i	$-0.366185\pi$
$577$	19.6067	0.816237	0.408119	+	0.912929i	$0.366185\pi$
$578$	4.32597	0.179936
$579$	3.92080	0.162943
$580$	10.0624	0.417820
$581$	−26.9716	−1.11897
$582$	2.46580	0.102211
$583$	0.152167	0.00630213
$584$	−10.6208	−0.439490
$585$	12.3574	0.510917
$586$	−6.09003	−0.251577
$587$	−34.2791	−1.41485	−0.707424	−	0.706789i	$-0.750143\pi$
$587$	−34.2791	−1.41485	−0.707424	+	0.706789i	$0.750143\pi$
$588$	1.26564	0.0521942
$589$	−39.6319	−1.63300
$590$	20.4654	0.842549
$591$	0.102157	0.00420216
$592$	−2.50971	−0.103148
$593$	43.7201	1.79537	0.897684	−	0.440640i	$-0.145248\pi$
$593$	43.7201	1.79537	0.897684	+	0.440640i	$0.145248\pi$
$594$	0.0618332	0.00253704
$595$	62.4192	2.55894
$596$	−22.8941	−0.937780
$597$	−1.17756	−0.0481944
$598$	8.96926	0.366780
$599$	−42.0440	−1.71787	−0.858936	−	0.512082i	$-0.828874\pi$
$599$	−42.0440	−1.71787	−0.858936	+	0.512082i	$0.828874\pi$
$600$	1.87398	0.0765047
$601$	−43.2955	−1.76606	−0.883030	−	0.469317i	$-0.844500\pi$
$601$	−43.2955	−1.76606	−0.883030	+	0.469317i	$0.844500\pi$
$602$	−7.74246	−0.315559
$603$	−12.1499	−0.494781
$604$	7.77361	0.316304
$605$	41.1227	1.67188
$606$	−3.95768	−0.160770
$607$	1.69666	0.0688652	0.0344326	−	0.999407i	$-0.489038\pi$
$607$	1.69666	0.0688652	0.0344326	+	0.999407i	$0.489038\pi$
$608$	−4.65720	−0.188874
$609$	−2.02944	−0.0822373
$610$	47.4342	1.92056
$611$	7.54417	0.305205
$612$	13.6530	0.551890
$613$	−5.55820	−0.224494	−0.112247	−	0.993680i	$-0.535805\pi$
$613$	−5.55820	−0.224494	−0.112247	+	0.993680i	$0.535805\pi$
$614$	19.7516	0.797111
$615$	−1.25015	−0.0504109
$616$	−0.179856	−0.00724662
$617$	9.55321	0.384598	0.192299	−	0.981336i	$-0.438406\pi$
$617$	9.55321	0.384598	0.192299	+	0.981336i	$0.438406\pi$
$618$	−4.14312	−0.166661
$619$	23.5588	0.946907	0.473454	−	0.880819i	$-0.343007\pi$
$619$	23.5588	0.946907	0.473454	+	0.880819i	$0.343007\pi$
$620$	−31.8205	−1.27794
$621$	−9.97153	−0.400144
$622$	−24.9394	−0.999978
$623$	−8.48038	−0.339759
$624$	0.233212	0.00933596
$625$	10.7682	0.430728
$626$	−8.90621	−0.355964
$627$	0.0483456	0.00193074
$628$	−9.35849	−0.373445
$629$	11.5899	0.462118
$630$	39.9611	1.59209
$631$	17.1551	0.682935	0.341467	−	0.939894i	$-0.389076\pi$
$631$	17.1551	0.682935	0.341467	+	0.939894i	$0.389076\pi$
$632$	9.92817	0.394921
$633$	−0.936769	−0.0372332
$634$	−0.924541	−0.0367182
$635$	−63.0573	−2.50235
$636$	−0.638053	−0.0253004
$637$	6.78101	0.268673
$638$	0.133895	0.00530096
$639$	39.9390	1.57996
$640$	−3.73927	−0.147808
$641$	16.3635	0.646319	0.323159	−	0.946345i	$-0.395255\pi$
$641$	16.3635	0.646319	0.323159	+	0.946345i	$0.395255\pi$
$642$	−0.701030	−0.0276675
$643$	26.2248	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$643$	26.2248	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$644$	29.0045	1.14294
$645$	1.67098	0.0657950
$646$	21.5070	0.846180
$647$	4.79256	0.188415	0.0942074	−	0.995553i	$-0.469968\pi$
$647$	4.79256	0.188415	0.0942074	+	0.995553i	$0.469968\pi$
$648$	8.61014	0.338238
$649$	0.272322	0.0106896
$650$	10.0403	0.393814
$651$	6.41773	0.251531
$652$	8.92449	0.349510
$653$	−26.4047	−1.03329	−0.516647	−	0.856198i	$-0.672820\pi$
$653$	−26.4047	−1.03329	−0.516647	+	0.856198i	$0.672820\pi$
$654$	−2.13908	−0.0836447
$655$	−16.5806	−0.647857
$656$	1.60248	0.0625662
$657$	31.4000	1.22503
$658$	24.3961	0.951061
$659$	−17.5207	−0.682508	−0.341254	−	0.939971i	$-0.610852\pi$
$659$	−17.5207	−0.682508	−0.341254	+	0.939971i	$0.610852\pi$
$660$	0.0388168	0.00151094
$661$	−25.9880	−1.01081	−0.505407	−	0.862881i	$-0.668658\pi$
$661$	−25.9880	−1.01081	−0.505407	+	0.862881i	$0.668658\pi$
$662$	−24.3479	−0.946307
$663$	−1.07697	−0.0418262
$664$	−7.46157	−0.289565
$665$	62.9489	2.44106
$666$	7.41989	0.287515
$667$	−21.5926	−0.836070
$668$	−11.3924	−0.440783
$669$	−2.45097	−0.0947600
$670$	−15.3669	−0.593673
$671$	0.631180	0.0243665
$672$	0.754155	0.0290922
$673$	−11.4440	−0.441133	−0.220566	−	0.975372i	$-0.570791\pi$
$673$	−11.4440	−0.441133	−0.220566	+	0.975372i	$0.570791\pi$
$674$	34.7037	1.33674
$675$	−11.1623	−0.429636
$676$	−11.7505	−0.451942
$677$	42.9803	1.65187	0.825934	−	0.563766i	$-0.190648\pi$
$677$	42.9803	1.65187	0.825934	+	0.563766i	$0.190648\pi$
$678$	0.632834	0.0243039
$679$	42.7219	1.63952
$680$	17.2680	0.662196
$681$	−2.16183	−0.0828416
$682$	−0.423418	−0.0162135
$683$	3.76336	0.144001	0.0720004	−	0.997405i	$-0.477062\pi$
$683$	3.76336	0.144001	0.0720004	+	0.997405i	$0.477062\pi$
$684$	13.7689	0.526466
$685$	−50.0532	−1.91243
$686$	−3.37493	−0.128855
$687$	2.97956	0.113677
$688$	−2.14191	−0.0816597
$689$	−3.41854	−0.130236
$690$	−6.25979	−0.238306
$691$	−4.83233	−0.183830	−0.0919152	−	0.995767i	$-0.529299\pi$
$691$	−4.83233	−0.183830	−0.0919152	+	0.995767i	$0.529299\pi$
$692$	−12.1083	−0.460287
$693$	0.531740	0.0201991
$694$	−1.73091	−0.0657043
$695$	26.9288	1.02147
$696$	−0.561436	−0.0212812
$697$	−7.40024	−0.280304
$698$	−5.21466	−0.197378
$699$	0.0785253	0.00297010
$700$	32.4681	1.22718
$701$	16.5005	0.623215	0.311608	−	0.950211i	$-0.399133\pi$
$701$	16.5005	0.623215	0.311608	+	0.950211i	$0.399133\pi$
$702$	−1.38912	−0.0524290
$703$	11.6882	0.440829
$704$	−0.0497564	−0.00187526
$705$	−5.26520	−0.198299
$706$	−2.36433	−0.0889828
$707$	−68.5700	−2.57884
$708$	−1.14187	−0.0429142
$709$	43.6344	1.63872	0.819362	−	0.573276i	$-0.194328\pi$
$709$	43.6344	1.63872	0.819362	+	0.573276i	$0.194328\pi$
$710$	50.5138	1.89575
$711$	−29.3524	−1.10080
$712$	−2.34606	−0.0879221
$713$	68.2825	2.55720
$714$	−3.48269	−0.130336
$715$	0.207971	0.00777768
$716$	−5.61142	−0.209709
$717$	−2.51140	−0.0937899
$718$	21.8431	0.815175
$719$	11.0175	0.410883	0.205441	−	0.978669i	$-0.434137\pi$
$719$	11.0175	0.410883	0.205441	+	0.978669i	$0.434137\pi$
$720$	11.0550	0.411997
$721$	−71.7829	−2.67333
$722$	2.68948	0.100092
$723$	3.17674	0.118144
$724$	6.06421	0.225375
$725$	−24.1711	−0.897692
$726$	−2.29445	−0.0851550
$727$	33.7433	1.25147	0.625735	−	0.780035i	$-0.284799\pi$
$727$	33.7433	1.25147	0.625735	+	0.780035i	$0.284799\pi$
$728$	4.04059	0.149754
$729$	−24.6778	−0.913994
$730$	39.7139	1.46988
$731$	9.89136	0.365845
$732$	−2.64660	−0.0978213
$733$	−14.5589	−0.537744	−0.268872	−	0.963176i	$-0.586651\pi$
$733$	−14.5589	−0.537744	−0.268872	+	0.963176i	$0.586651\pi$
$734$	10.7999	0.398630
$735$	−4.73258	−0.174564
$736$	8.02396	0.295767
$737$	−0.204478	−0.00753204
$738$	−4.73768	−0.174396
$739$	−25.5250	−0.938954	−0.469477	−	0.882945i	$-0.655557\pi$
$739$	−25.5250	−0.938954	−0.469477	+	0.882945i	$0.655557\pi$
$740$	9.38449	0.344981
$741$	−1.08611	−0.0398994
$742$	−11.0548	−0.405834
$743$	40.7339	1.49438	0.747190	−	0.664610i	$-0.231402\pi$
$743$	40.7339	1.49438	0.747190	+	0.664610i	$0.231402\pi$
$744$	1.77543	0.0650905
$745$	85.6073	3.13641
$746$	−26.2474	−0.960986
$747$	22.0599	0.807130
$748$	0.229775	0.00840141
$749$	−12.1459	−0.443802
$750$	−3.10662	−0.113438
$751$	14.7677	0.538883	0.269441	−	0.963017i	$-0.413161\pi$
$751$	14.7677	0.538883	0.269441	+	0.963017i	$0.413161\pi$
$752$	6.74907	0.246113
$753$	2.46280	0.0897493
$754$	−3.00804	−0.109546
$755$	−29.0676	−1.05788
$756$	−4.49210	−0.163376
$757$	44.0679	1.60168	0.800838	−	0.598881i	$-0.204388\pi$
$757$	44.0679	1.60168	0.800838	+	0.598881i	$0.204388\pi$
$758$	7.48676	0.271931
$759$	−0.0832954	−0.00302343
$760$	17.4145	0.631691
$761$	−13.3426	−0.483670	−0.241835	−	0.970317i	$-0.577749\pi$
$761$	−13.3426	−0.483670	−0.241835	+	0.970317i	$0.577749\pi$
$762$	3.51830	0.127454
$763$	−37.0613	−1.34171
$764$	−19.4900	−0.705125
$765$	−51.0523	−1.84580
$766$	−7.41227	−0.267816
$767$	−6.11789	−0.220904
$768$	0.208633	0.00752841
$769$	−33.2064	−1.19745	−0.598727	−	0.800953i	$-0.704327\pi$
$769$	−33.2064	−1.19745	−0.598727	+	0.800953i	$0.704327\pi$
$770$	0.672531	0.0242363
$771$	−3.84402	−0.138439
$772$	18.7928	0.676367
$773$	11.9949	0.431427	0.215713	−	0.976457i	$-0.430792\pi$
$773$	11.9949	0.431427	0.215713	+	0.976457i	$0.430792\pi$
$774$	6.33251	0.227617
$775$	76.4365	2.74568
$776$	11.8188	0.424270
$777$	−1.89271	−0.0679006
$778$	30.8514	1.10608
$779$	−7.46305	−0.267391
$780$	−0.872043	−0.0312242
$781$	0.672159	0.0240517
$782$	−37.0547	−1.32507
$783$	3.34418	0.119511
$784$	6.06634	0.216655
$785$	34.9939	1.24899
$786$	0.925116	0.0329978
$787$	48.4019	1.72534	0.862671	−	0.505765i	$-0.168790\pi$
$787$	48.4019	1.72534	0.862671	+	0.505765i	$0.168790\pi$
$788$	0.489647	0.0174429
$789$	−1.79227	−0.0638064
$790$	−37.1241	−1.32082
$791$	10.9644	0.389848
$792$	0.147103	0.00522709
$793$	−14.1799	−0.503542
$794$	−9.31182	−0.330464
$795$	2.38585	0.0846175
$796$	−5.64417	−0.200052
$797$	26.8314	0.950416	0.475208	−	0.879873i	$-0.342373\pi$
$797$	26.8314	0.950416	0.475208	+	0.879873i	$0.342373\pi$
$798$	−3.51225	−0.124332
$799$	−31.1673	−1.10262
$800$	8.98215	0.317567
$801$	6.93605	0.245073
$802$	−31.5848	−1.11530
$803$	0.528451	0.0186486
$804$	0.857397	0.0302380
$805$	−108.456	−3.82257
$806$	9.51235	0.335058
$807$	4.35649	0.153356
$808$	−18.9696	−0.667346
$809$	−43.8231	−1.54074	−0.770370	−	0.637597i	$-0.779928\pi$
$809$	−43.8231	−1.54074	−0.770370	+	0.637597i	$0.779928\pi$
$810$	−32.1957	−1.13124
$811$	19.3048	0.677883	0.338941	−	0.940807i	$-0.389931\pi$
$811$	19.3048	0.677883	0.338941	+	0.940807i	$0.389931\pi$
$812$	−9.72733	−0.341362
$813$	−4.02152	−0.141041
$814$	0.124874	0.00437683
$815$	−33.3711	−1.16894
$816$	−0.963470	−0.0337282
$817$	9.97531	0.348992
$818$	−1.10711	−0.0387093
$819$	−11.9459	−0.417423
$820$	−5.99209	−0.209253
$821$	2.86270	0.0999091	0.0499545	−	0.998751i	$-0.484092\pi$
$821$	2.86270	0.0999091	0.0499545	+	0.998751i	$0.484092\pi$
$822$	2.79273	0.0974075
$823$	−4.79161	−0.167025	−0.0835126	−	0.996507i	$-0.526614\pi$
$823$	−4.79161	−0.167025	−0.0835126	+	0.996507i	$0.526614\pi$
$824$	−19.8584	−0.691799
$825$	−0.0932422	−0.00324628
$826$	−19.7839	−0.688369
$827$	25.0356	0.870573	0.435287	−	0.900292i	$-0.356647\pi$
$827$	25.0356	0.870573	0.435287	+	0.900292i	$0.356647\pi$
$828$	−23.7226	−0.824418
$829$	−9.41176	−0.326884	−0.163442	−	0.986553i	$-0.552260\pi$
$829$	−9.41176	−0.326884	−0.163442	+	0.986553i	$0.552260\pi$
$830$	27.9008	0.968451
$831$	3.17595	0.110172
$832$	1.11781	0.0387530
$833$	−28.0144	−0.970641
$834$	−1.50250	−0.0520272
$835$	42.5991	1.47420
$836$	0.231725	0.00801438
$837$	−10.5753	−0.365536
$838$	7.65441	0.264417
$839$	−42.7963	−1.47749	−0.738746	−	0.673984i	$-0.764582\pi$
$839$	−42.7963	−1.47749	−0.738746	+	0.673984i	$0.764582\pi$
$840$	−2.81999	−0.0972989
$841$	−21.7584	−0.750291
$842$	−4.18250	−0.144138
$843$	0.988976	0.0340621
$844$	−4.49003	−0.154553
$845$	43.9383	1.51152
$846$	−19.9534	−0.686014
$847$	−39.7532	−1.36594
$848$	−3.05825	−0.105021
$849$	−2.55266	−0.0876072
$850$	−41.4796	−1.42274
$851$	−20.1378	−0.690316
$852$	−2.81843	−0.0965578
$853$	24.2998	0.832010	0.416005	−	0.909362i	$-0.363430\pi$
$853$	24.2998	0.832010	0.416005	+	0.909362i	$0.363430\pi$
$854$	−45.8545	−1.56911
$855$	−51.4855	−1.76077
$856$	−3.36011	−0.114846
$857$	−5.12061	−0.174917	−0.0874583	−	0.996168i	$-0.527874\pi$
$857$	−5.12061	−0.174917	−0.0874583	+	0.996168i	$0.527874\pi$
$858$	−0.0116038	−0.000396147 0
$859$	−12.0365	−0.410679	−0.205339	−	0.978691i	$-0.565830\pi$
$859$	−12.0365	−0.410679	−0.205339	+	0.978691i	$0.565830\pi$
$860$	8.00919	0.273111
$861$	1.20852	0.0411861
$862$	−28.3787	−0.966584
$863$	14.5442	0.495090	0.247545	−	0.968876i	$-0.420376\pi$
$863$	14.5442	0.495090	0.247545	+	0.968876i	$0.420376\pi$
$864$	−1.24272	−0.0422782
$865$	45.2760	1.53943
$866$	33.5268	1.13929
$867$	0.902541	0.0306519
$868$	30.7608	1.04409
$869$	−0.493989	−0.0167574
$870$	2.09936	0.0711750
$871$	4.59373	0.155653
$872$	−10.2528	−0.347205
$873$	−34.9420	−1.18261
$874$	−37.3692	−1.26403
$875$	−53.8246	−1.81960
$876$	−2.21585	−0.0748665
$877$	46.2330	1.56118	0.780589	−	0.625045i	$-0.214919\pi$
$877$	46.2330	1.56118	0.780589	+	0.625045i	$0.214919\pi$
$878$	−8.71949	−0.294269
$879$	−1.27058	−0.0428557
$880$	0.186052	0.00627183
$881$	13.4145	0.451946	0.225973	−	0.974134i	$-0.427444\pi$
$881$	13.4145	0.451946	0.225973	+	0.974134i	$0.427444\pi$
$882$	−17.9350	−0.603902
$883$	−1.61804	−0.0544514	−0.0272257	−	0.999629i	$-0.508667\pi$
$883$	−1.61804	−0.0544514	−0.0272257	+	0.999629i	$0.508667\pi$
$884$	−5.16204	−0.173618
$885$	4.26977	0.143527
$886$	−24.0633	−0.808421
$887$	8.14283	0.273410	0.136705	−	0.990612i	$-0.456349\pi$
$887$	8.14283	0.273410	0.136705	+	0.990612i	$0.456349\pi$
$888$	−0.523609	−0.0175712
$889$	60.9573	2.04444
$890$	8.77254	0.294056
$891$	−0.428409	−0.0143523
$892$	−11.7477	−0.393344
$893$	−31.4318	−1.05182
$894$	−4.77648	−0.159749
$895$	20.9826	0.701372
$896$	3.61474	0.120760
$897$	1.87129	0.0624804
$898$	−20.8950	−0.697274
$899$	−22.9001	−0.763760
$900$	−26.5555	−0.885182
$901$	14.1230	0.470506
$902$	−0.0797334	−0.00265483
$903$	−1.61533	−0.0537550
$904$	3.03324	0.100884
$905$	−22.6757	−0.753767
$906$	1.62183	0.0538818
$907$	36.2084	1.20228	0.601141	−	0.799143i	$-0.294713\pi$
$907$	36.2084	1.20228	0.601141	+	0.799143i	$0.294713\pi$
$908$	−10.3619	−0.343871
$909$	56.0830	1.86015
$910$	−15.1088	−0.500853
$911$	31.5132	1.04408	0.522040	−	0.852921i	$-0.325171\pi$
$911$	31.5132	1.04408	0.522040	+	0.852921i	$0.325171\pi$
$912$	−0.971647	−0.0321744
$913$	0.371260	0.0122869
$914$	−19.2602	−0.637072
$915$	9.89636	0.327164
$916$	14.2813	0.471868
$917$	16.0284	0.529304
$918$	5.73888	0.189411
$919$	−11.1275	−0.367062	−0.183531	−	0.983014i	$-0.558753\pi$
$919$	−11.1275	−0.367062	−0.183531	+	0.983014i	$0.558753\pi$
$920$	−30.0038	−0.989195
$921$	4.12085	0.135787
$922$	27.5419	0.907044
$923$	−15.1005	−0.497038
$924$	−0.0375240	−0.00123445
$925$	−22.5426	−0.741195
$926$	−14.8656	−0.488512
$927$	58.7107	1.92831
$928$	−2.69102	−0.0883370
$929$	−14.2377	−0.467124	−0.233562	−	0.972342i	$-0.575038\pi$
$929$	−14.2377	−0.467124	−0.233562	+	0.972342i	$0.575038\pi$
$930$	−6.63882	−0.217696
$931$	−28.2522	−0.925927
$932$	0.376380	0.0123287
$933$	−5.20319	−0.170345
$934$	−28.9560	−0.947470
$935$	−0.859191	−0.0280986
$936$	−3.30477	−0.108020
$937$	−7.62482	−0.249092	−0.124546	−	0.992214i	$-0.539747\pi$
$937$	−7.62482	−0.249092	−0.124546	+	0.992214i	$0.539747\pi$
$938$	14.8551	0.485036
$939$	−1.85813	−0.0606379
$940$	−25.2366	−0.823128
$941$	−41.2592	−1.34501	−0.672506	−	0.740091i	$-0.734782\pi$
$941$	−41.2592	−1.34501	−0.672506	+	0.740091i	$0.734782\pi$
$942$	−1.95249	−0.0636157
$943$	12.8582	0.418721
$944$	−5.47311	−0.178135
$945$	16.7972	0.546413
$946$	0.106574	0.00346501
$947$	36.2306	1.17734	0.588668	−	0.808375i	$-0.299653\pi$
$947$	36.2306	1.17734	0.588668	+	0.808375i	$0.299653\pi$
$948$	2.07135	0.0672742
$949$	−11.8720	−0.385381
$950$	−41.8316	−1.35720
$951$	−0.192890	−0.00625489
$952$	−16.6929	−0.541019
$953$	−3.31187	−0.107282	−0.0536409	−	0.998560i	$-0.517083\pi$
$953$	−3.31187	−0.107282	−0.0536409	+	0.998560i	$0.517083\pi$
$954$	9.04163	0.292734
$955$	72.8785	2.35829
$956$	−12.0374	−0.389317
$957$	0.0279350	0.000903010 0
$958$	−8.04817	−0.260025
$959$	48.3862	1.56247
$960$	−0.780137	−0.0251788
$961$	41.4171	1.33603
$962$	−2.80538	−0.0904489
$963$	9.93406	0.320121
$964$	15.2264	0.490410
$965$	−70.2713	−2.26211
$966$	6.05131	0.194698
$967$	34.1549	1.09835	0.549174	−	0.835708i	$-0.314942\pi$
$967$	34.1549	1.09835	0.549174	+	0.835708i	$0.314942\pi$
$968$	−10.9975	−0.353474
$969$	4.48707	0.144145
$970$	−44.1937	−1.41897
$971$	43.9222	1.40953	0.704765	−	0.709441i	$-0.251052\pi$
$971$	43.9222	1.40953	0.704765	+	0.709441i	$0.251052\pi$
$972$	5.52452	0.177199
$973$	−26.0319	−0.834546
$974$	−1.37705	−0.0441235
$975$	2.09475	0.0670855
$976$	−12.6854	−0.406051
$977$	12.0474	0.385430	0.192715	−	0.981255i	$-0.438271\pi$
$977$	12.0474	0.385430	0.192715	+	0.981255i	$0.438271\pi$
$978$	1.86195	0.0595385
$979$	0.116731	0.00373074
$980$	−22.6837	−0.724604
$981$	30.3122	0.967794
$982$	−14.5222	−0.463421
$983$	−26.8833	−0.857443	−0.428721	−	0.903437i	$-0.641036\pi$
$983$	−26.8833	−0.857443	−0.428721	+	0.903437i	$0.641036\pi$
$984$	0.334330	0.0106581
$985$	−1.83092	−0.0583380
$986$	12.4271	0.395760
$987$	5.08985	0.162012
$988$	−5.20585	−0.165620
$989$	−17.1866	−0.546503
$990$	−0.550059	−0.0174820
$991$	−3.29891	−0.104793	−0.0523967	−	0.998626i	$-0.516686\pi$
$991$	−3.29891	−0.104793	−0.0523967	+	0.998626i	$0.516686\pi$
$992$	8.50982	0.270187
$993$	−5.07978	−0.161202
$994$	−48.8315	−1.54884
$995$	21.1051	0.669076
$996$	−1.55673	−0.0493269
$997$	−21.5175	−0.681465	−0.340732	−	0.940160i	$-0.610675\pi$
$997$	−21.5175	−0.681465	−0.340732	+	0.940160i	$0.610675\pi$
$998$	23.1215	0.731898
$999$	3.11886	0.0986765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.40	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.40	✓	69	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.208633 q^{3} +1.00000 q^{4} -3.73927 q^{5} +0.208633 q^{6} +3.61474 q^{7} +1.00000 q^{8} -2.95647 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.208633 q^{3} +1.00000 q^{4} -3.73927 q^{5} +0.208633 q^{6} +3.61474 q^{7} +1.00000 q^{8} -2.95647 q^{9} -3.73927 q^{10} -0.0497564 q^{11} +0.208633 q^{12} +1.11781 q^{13} +3.61474 q^{14} -0.780137 q^{15} +1.00000 q^{16} -4.61800 q^{17} -2.95647 q^{18} -4.65720 q^{19} -3.73927 q^{20} +0.754155 q^{21} -0.0497564 q^{22} +8.02396 q^{23} +0.208633 q^{24} +8.98215 q^{25} +1.11781 q^{26} -1.24272 q^{27} +3.61474 q^{28} -2.69102 q^{29} -0.780137 q^{30} +8.50982 q^{31} +1.00000 q^{32} -0.0103808 q^{33} -4.61800 q^{34} -13.5165 q^{35} -2.95647 q^{36} -2.50971 q^{37} -4.65720 q^{38} +0.233212 q^{39} -3.73927 q^{40} +1.60248 q^{41} +0.754155 q^{42} -2.14191 q^{43} -0.0497564 q^{44} +11.0550 q^{45} +8.02396 q^{46} +6.74907 q^{47} +0.208633 q^{48} +6.06634 q^{49} +8.98215 q^{50} -0.963470 q^{51} +1.11781 q^{52} -3.05825 q^{53} -1.24272 q^{54} +0.186052 q^{55} +3.61474 q^{56} -0.971647 q^{57} -2.69102 q^{58} -5.47311 q^{59} -0.780137 q^{60} -12.6854 q^{61} +8.50982 q^{62} -10.6869 q^{63} +1.00000 q^{64} -4.17979 q^{65} -0.0103808 q^{66} +4.10959 q^{67} -4.61800 q^{68} +1.67407 q^{69} -13.5165 q^{70} -13.5090 q^{71} -2.95647 q^{72} -10.6208 q^{73} -2.50971 q^{74} +1.87398 q^{75} -4.65720 q^{76} -0.179856 q^{77} +0.233212 q^{78} +9.92817 q^{79} -3.73927 q^{80} +8.61014 q^{81} +1.60248 q^{82} -7.46157 q^{83} +0.754155 q^{84} +17.2680 q^{85} -2.14191 q^{86} -0.561436 q^{87} -0.0497564 q^{88} -2.34606 q^{89} +11.0550 q^{90} +4.04059 q^{91} +8.02396 q^{92} +1.77543 q^{93} +6.74907 q^{94} +17.4145 q^{95} +0.208633 q^{96} +11.8188 q^{97} +6.06634 q^{98} +0.147103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more