LMFDB - Embedded newform 8031.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02856 q^{2} -1.00000 q^{3} +2.11504 q^{4} -0.0759576 q^{5} +2.02856 q^{6} -4.07042 q^{7} -0.233369 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.02856 q^{2} -1.00000 q^{3} +2.11504 q^{4} -0.0759576 q^{5} +2.02856 q^{6} -4.07042 q^{7} -0.233369 q^{8} +1.00000 q^{9} +0.154084 q^{10} +2.38993 q^{11} -2.11504 q^{12} -3.95048 q^{13} +8.25708 q^{14} +0.0759576 q^{15} -3.75668 q^{16} +5.05321 q^{17} -2.02856 q^{18} -7.37225 q^{19} -0.160654 q^{20} +4.07042 q^{21} -4.84810 q^{22} +1.61126 q^{23} +0.233369 q^{24} -4.99423 q^{25} +8.01378 q^{26} -1.00000 q^{27} -8.60911 q^{28} -0.397165 q^{29} -0.154084 q^{30} +4.85493 q^{31} +8.08738 q^{32} -2.38993 q^{33} -10.2507 q^{34} +0.309179 q^{35} +2.11504 q^{36} +8.28210 q^{37} +14.9550 q^{38} +3.95048 q^{39} +0.0177262 q^{40} -6.38667 q^{41} -8.25708 q^{42} -2.54126 q^{43} +5.05479 q^{44} -0.0759576 q^{45} -3.26853 q^{46} +3.06956 q^{47} +3.75668 q^{48} +9.56833 q^{49} +10.1311 q^{50} -5.05321 q^{51} -8.35544 q^{52} +1.61301 q^{53} +2.02856 q^{54} -0.181533 q^{55} +0.949912 q^{56} +7.37225 q^{57} +0.805671 q^{58} +10.1699 q^{59} +0.160654 q^{60} -2.32954 q^{61} -9.84849 q^{62} -4.07042 q^{63} -8.89234 q^{64} +0.300069 q^{65} +4.84810 q^{66} +5.52407 q^{67} +10.6878 q^{68} -1.61126 q^{69} -0.627188 q^{70} -4.90189 q^{71} -0.233369 q^{72} -11.4881 q^{73} -16.8007 q^{74} +4.99423 q^{75} -15.5926 q^{76} -9.72800 q^{77} -8.01378 q^{78} -1.83938 q^{79} +0.285348 q^{80} +1.00000 q^{81} +12.9557 q^{82} +0.754830 q^{83} +8.60911 q^{84} -0.383830 q^{85} +5.15508 q^{86} +0.397165 q^{87} -0.557735 q^{88} +10.0876 q^{89} +0.154084 q^{90} +16.0801 q^{91} +3.40788 q^{92} -4.85493 q^{93} -6.22677 q^{94} +0.559978 q^{95} -8.08738 q^{96} -5.51199 q^{97} -19.4099 q^{98} +2.38993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * 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 $

$2$	−2.02856	−1.43441	−0.717203	−	0.696864i	$-0.754578\pi$
$2$	−2.02856	−1.43441	−0.717203	+	0.696864i	$0.754578\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.11504	1.05752
$5$	−0.0759576	−0.0339693	−0.0169846	−	0.999856i	$-0.505407\pi$
$5$	−0.0759576	−0.0339693	−0.0169846	+	0.999856i	$0.505407\pi$
$6$	2.02856	0.828155
$7$	−4.07042	−1.53847	−0.769237	−	0.638963i	$-0.779364\pi$
$7$	−4.07042	−1.53847	−0.769237	+	0.638963i	$0.779364\pi$
$8$	−0.233369	−0.0825085
$9$	1.00000	0.333333
$10$	0.154084	0.0487257
$11$	2.38993	0.720590	0.360295	−	0.932839i	$-0.382676\pi$
$11$	2.38993	0.720590	0.360295	+	0.932839i	$0.382676\pi$
$12$	−2.11504	−0.610560
$13$	−3.95048	−1.09567	−0.547833	−	0.836588i	$-0.684547\pi$
$13$	−3.95048	−1.09567	−0.547833	+	0.836588i	$0.684547\pi$
$14$	8.25708	2.20680
$15$	0.0759576	0.0196122
$16$	−3.75668	−0.939170
$17$	5.05321	1.22558	0.612792	−	0.790244i	$-0.290046\pi$
$17$	5.05321	1.22558	0.612792	+	0.790244i	$0.290046\pi$
$18$	−2.02856	−0.478135
$19$	−7.37225	−1.69131	−0.845655	−	0.533731i	$-0.820790\pi$
$19$	−7.37225	−1.69131	−0.845655	+	0.533731i	$0.820790\pi$
$20$	−0.160654	−0.0359232
$21$	4.07042	0.888239
$22$	−4.84810	−1.03362
$23$	1.61126	0.335971	0.167986	−	0.985789i	$-0.446274\pi$
$23$	1.61126	0.335971	0.167986	+	0.985789i	$0.446274\pi$
$24$	0.233369	0.0476363
$25$	−4.99423	−0.998846
$26$	8.01378	1.57163
$27$	−1.00000	−0.192450
$28$	−8.60911	−1.62697
$29$	−0.397165	−0.0737517	−0.0368758	−	0.999320i	$-0.511741\pi$
$29$	−0.397165	−0.0737517	−0.0368758	+	0.999320i	$0.511741\pi$
$30$	−0.154084	−0.0281318
$31$	4.85493	0.871971	0.435985	−	0.899954i	$-0.356400\pi$
$31$	4.85493	0.871971	0.435985	+	0.899954i	$0.356400\pi$
$32$	8.08738	1.42966
$33$	−2.38993	−0.416033
$34$	−10.2507	−1.75799
$35$	0.309179	0.0522609
$36$	2.11504	0.352507
$37$	8.28210	1.36157	0.680785	−	0.732484i	$-0.261639\pi$
$37$	8.28210	1.36157	0.680785	+	0.732484i	$0.261639\pi$
$38$	14.9550	2.42602
$39$	3.95048	0.632583
$40$	0.0177262	0.00280276
$41$	−6.38667	−0.997430	−0.498715	−	0.866766i	$-0.666195\pi$
$41$	−6.38667	−0.997430	−0.498715	+	0.866766i	$0.666195\pi$
$42$	−8.25708	−1.27410
$43$	−2.54126	−0.387538	−0.193769	−	0.981047i	$-0.562071\pi$
$43$	−2.54126	−0.387538	−0.193769	+	0.981047i	$0.562071\pi$
$44$	5.05479	0.762039
$45$	−0.0759576	−0.0113231
$46$	−3.26853	−0.481919
$47$	3.06956	0.447741	0.223870	−	0.974619i	$-0.428131\pi$
$47$	3.06956	0.447741	0.223870	+	0.974619i	$0.428131\pi$
$48$	3.75668	0.542230
$49$	9.56833	1.36690
$50$	10.1311	1.43275
$51$	−5.05321	−0.707591
$52$	−8.35544	−1.15869
$53$	1.61301	0.221564	0.110782	−	0.993845i	$-0.464664\pi$
$53$	1.61301	0.221564	0.110782	+	0.993845i	$0.464664\pi$
$54$	2.02856	0.276052
$55$	−0.181533	−0.0244779
$56$	0.949912	0.126937
$57$	7.37225	0.976478
$58$	0.805671	0.105790
$59$	10.1699	1.32401	0.662007	−	0.749498i	$-0.269705\pi$
$59$	10.1699	1.32401	0.662007	+	0.749498i	$0.269705\pi$
$60$	0.160654	0.0207403
$61$	−2.32954	−0.298266	−0.149133	−	0.988817i	$-0.547648\pi$
$61$	−2.32954	−0.298266	−0.149133	+	0.988817i	$0.547648\pi$
$62$	−9.84849	−1.25076
$63$	−4.07042	−0.512825
$64$	−8.89234	−1.11154
$65$	0.300069	0.0372190
$66$	4.84810	0.596760
$67$	5.52407	0.674872	0.337436	−	0.941348i	$-0.390440\pi$
$67$	5.52407	0.674872	0.337436	+	0.941348i	$0.390440\pi$
$68$	10.6878	1.29608
$69$	−1.61126	−0.193973
$70$	−0.627188	−0.0749633
$71$	−4.90189	−0.581747	−0.290873	−	0.956762i	$-0.593946\pi$
$71$	−4.90189	−0.581747	−0.290873	+	0.956762i	$0.593946\pi$
$72$	−0.233369	−0.0275028
$73$	−11.4881	−1.34458	−0.672290	−	0.740287i	$-0.734689\pi$
$73$	−11.4881	−1.34458	−0.672290	+	0.740287i	$0.734689\pi$
$74$	−16.8007	−1.95304
$75$	4.99423	0.576684
$76$	−15.5926	−1.78860
$77$	−9.72800	−1.10861
$78$	−8.01378	−0.907382
$79$	−1.83938	−0.206947	−0.103473	−	0.994632i	$-0.532996\pi$
$79$	−1.83938	−0.206947	−0.103473	+	0.994632i	$0.532996\pi$
$80$	0.285348	0.0319029
$81$	1.00000	0.111111
$82$	12.9557	1.43072
$83$	0.754830	0.0828534	0.0414267	−	0.999142i	$-0.486810\pi$
$83$	0.754830	0.0828534	0.0414267	+	0.999142i	$0.486810\pi$
$84$	8.60911	0.939331
$85$	−0.383830	−0.0416322
$86$	5.15508	0.555887
$87$	0.397165	0.0425805
$88$	−0.557735	−0.0594548
$89$	10.0876	1.06928	0.534640	−	0.845080i	$-0.320447\pi$
$89$	10.0876	1.06928	0.534640	+	0.845080i	$0.320447\pi$
$90$	0.154084	0.0162419
$91$	16.0801	1.68566
$92$	3.40788	0.355297
$93$	−4.85493	−0.503432
$94$	−6.22677	−0.642242
$95$	0.559978	0.0574525
$96$	−8.08738	−0.825415
$97$	−5.51199	−0.559658	−0.279829	−	0.960050i	$-0.590278\pi$
$97$	−5.51199	−0.559658	−0.279829	+	0.960050i	$0.590278\pi$
$98$	−19.4099	−1.96070
$99$	2.38993	0.240197
$100$	−10.5630	−1.05630
$101$	−8.92906	−0.888475	−0.444237	−	0.895909i	$-0.646525\pi$
$101$	−8.92906	−0.888475	−0.444237	+	0.895909i	$0.646525\pi$
$102$	10.2507	1.01497
$103$	4.48487	0.441907	0.220954	−	0.975284i	$-0.429083\pi$
$103$	4.48487	0.441907	0.220954	+	0.975284i	$0.429083\pi$
$104$	0.921922	0.0904019
$105$	−0.309179	−0.0301728
$106$	−3.27209	−0.317813
$107$	−4.05297	−0.391816	−0.195908	−	0.980622i	$-0.562765\pi$
$107$	−4.05297	−0.391816	−0.195908	+	0.980622i	$0.562765\pi$
$108$	−2.11504	−0.203520
$109$	1.39157	0.133288	0.0666440	−	0.997777i	$-0.478771\pi$
$109$	1.39157	0.133288	0.0666440	+	0.997777i	$0.478771\pi$
$110$	0.368250	0.0351113
$111$	−8.28210	−0.786102
$112$	15.2913	1.44489
$113$	12.3675	1.16344	0.581719	−	0.813390i	$-0.302380\pi$
$113$	12.3675	1.16344	0.581719	+	0.813390i	$0.302380\pi$
$114$	−14.9550	−1.40067
$115$	−0.122388	−0.0114127
$116$	−0.840020	−0.0779939
$117$	−3.95048	−0.365222
$118$	−20.6303	−1.89917
$119$	−20.5687	−1.88553
$120$	−0.0177262	−0.00161817
$121$	−5.28826	−0.480751
$122$	4.72560	0.427835
$123$	6.38667	0.575866
$124$	10.2684	0.922127
$125$	0.759138	0.0678993
$126$	8.25708	0.735599
$127$	13.7465	1.21981	0.609903	−	0.792476i	$-0.291209\pi$
$127$	13.7465	1.21981	0.609903	+	0.792476i	$0.291209\pi$
$128$	1.86387	0.164744
$129$	2.54126	0.223745
$130$	−0.608707	−0.0533872
$131$	19.1850	1.67620	0.838102	−	0.545513i	$-0.183665\pi$
$131$	19.1850	1.67620	0.838102	+	0.545513i	$0.183665\pi$
$132$	−5.05479	−0.439963
$133$	30.0081	2.60204
$134$	−11.2059	−0.968041
$135$	0.0759576	0.00653739
$136$	−1.17927	−0.101121
$137$	11.7884	1.00715	0.503577	−	0.863950i	$-0.332017\pi$
$137$	11.7884	1.00715	0.503577	+	0.863950i	$0.332017\pi$
$138$	3.26853	0.278236
$139$	−11.9364	−1.01243	−0.506217	−	0.862406i	$-0.668956\pi$
$139$	−11.9364	−1.01243	−0.506217	+	0.862406i	$0.668956\pi$
$140$	0.653928	0.0552670
$141$	−3.06956	−0.258503
$142$	9.94375	0.834461
$143$	−9.44136	−0.789526
$144$	−3.75668	−0.313057
$145$	0.0301677	0.00250529
$146$	23.3043	1.92868
$147$	−9.56833	−0.789182
$148$	17.5170	1.43989
$149$	−3.64742	−0.298809	−0.149404	−	0.988776i	$-0.547736\pi$
$149$	−3.64742	−0.298809	−0.149404	+	0.988776i	$0.547736\pi$
$150$	−10.1311	−0.827199
$151$	4.54587	0.369938	0.184969	−	0.982744i	$-0.440782\pi$
$151$	4.54587	0.369938	0.184969	+	0.982744i	$0.440782\pi$
$152$	1.72046	0.139547
$153$	5.05321	0.408528
$154$	19.7338	1.59020
$155$	−0.368769	−0.0296202
$156$	8.35544	0.668970
$157$	0.788070	0.0628948	0.0314474	−	0.999505i	$-0.489988\pi$
$157$	0.788070	0.0628948	0.0314474	+	0.999505i	$0.489988\pi$
$158$	3.73130	0.296846
$159$	−1.61301	−0.127920
$160$	−0.614298	−0.0485645
$161$	−6.55851	−0.516883
$162$	−2.02856	−0.159378
$163$	8.90077	0.697162	0.348581	−	0.937279i	$-0.386664\pi$
$163$	8.90077	0.697162	0.348581	+	0.937279i	$0.386664\pi$
$164$	−13.5081	−1.05480
$165$	0.181533	0.0141323
$166$	−1.53122	−0.118845
$167$	−7.57645	−0.586284	−0.293142	−	0.956069i	$-0.594701\pi$
$167$	−7.57645	−0.586284	−0.293142	+	0.956069i	$0.594701\pi$
$168$	−0.949912	−0.0732873
$169$	2.60631	0.200485
$170$	0.778621	0.0597175
$171$	−7.37225	−0.563770
$172$	−5.37486	−0.409829
$173$	−14.2343	−1.08222	−0.541108	−	0.840953i	$-0.681995\pi$
$173$	−14.2343	−1.08222	−0.541108	+	0.840953i	$0.681995\pi$
$174$	−0.805671	−0.0610778
$175$	20.3286	1.53670
$176$	−8.97819	−0.676756
$177$	−10.1699	−0.764420
$178$	−20.4632	−1.53378
$179$	10.5199	0.786295	0.393147	−	0.919476i	$-0.371386\pi$
$179$	10.5199	0.786295	0.393147	+	0.919476i	$0.371386\pi$
$180$	−0.160654	−0.0119744
$181$	−10.8466	−0.806224	−0.403112	−	0.915151i	$-0.632071\pi$
$181$	−10.8466	−0.806224	−0.403112	+	0.915151i	$0.632071\pi$
$182$	−32.6194	−2.41791
$183$	2.32954	0.172204
$184$	−0.376019	−0.0277205
$185$	−0.629089	−0.0462515
$186$	9.84849	0.722127
$187$	12.0768	0.883143
$188$	6.49224	0.473495
$189$	4.07042	0.296080
$190$	−1.13595	−0.0824103
$191$	−2.91584	−0.210983	−0.105492	−	0.994420i	$-0.533642\pi$
$191$	−2.91584	−0.210983	−0.105492	+	0.994420i	$0.533642\pi$
$192$	8.89234	0.641750
$193$	19.9902	1.43892	0.719461	−	0.694533i	$-0.244389\pi$
$193$	19.9902	1.43892	0.719461	+	0.694533i	$0.244389\pi$
$194$	11.1814	0.802776
$195$	−0.300069	−0.0214884
$196$	20.2374	1.44553
$197$	12.2440	0.872347	0.436174	−	0.899863i	$-0.356333\pi$
$197$	12.2440	0.872347	0.436174	+	0.899863i	$0.356333\pi$
$198$	−4.84810	−0.344539
$199$	−4.21460	−0.298765	−0.149383	−	0.988779i	$-0.547729\pi$
$199$	−4.21460	−0.298765	−0.149383	+	0.988779i	$0.547729\pi$
$200$	1.16550	0.0824133
$201$	−5.52407	−0.389638
$202$	18.1131	1.27443
$203$	1.61663	0.113465
$204$	−10.6878	−0.748293
$205$	0.485116	0.0338820
$206$	−9.09781	−0.633874
$207$	1.61126	0.111990
$208$	14.8407	1.02902
$209$	−17.6191	−1.21874
$210$	0.627188	0.0432801
$211$	22.8371	1.57217	0.786087	−	0.618116i	$-0.212104\pi$
$211$	22.8371	1.57217	0.786087	+	0.618116i	$0.212104\pi$
$212$	3.41159	0.234309
$213$	4.90189	0.335872
$214$	8.22168	0.562023
$215$	0.193028	0.0131644
$216$	0.233369	0.0158788
$217$	−19.7616	−1.34150
$218$	−2.82288	−0.191189
$219$	11.4881	0.776294
$220$	−0.383950	−0.0258859
$221$	−19.9626	−1.34283
$222$	16.8007	1.12759
$223$	2.70283	0.180995	0.0904973	−	0.995897i	$-0.471154\pi$
$223$	2.70283	0.180995	0.0904973	+	0.995897i	$0.471154\pi$
$224$	−32.9190	−2.19950
$225$	−4.99423	−0.332949
$226$	−25.0882	−1.66884
$227$	−23.9529	−1.58981	−0.794905	−	0.606735i	$-0.792479\pi$
$227$	−23.9529	−1.58981	−0.794905	+	0.606735i	$0.792479\pi$
$228$	15.5926	1.03265
$229$	−5.49765	−0.363295	−0.181648	−	0.983364i	$-0.558143\pi$
$229$	−5.49765	−0.363295	−0.181648	+	0.983364i	$0.558143\pi$
$230$	0.248270	0.0163704
$231$	9.72800	0.640056
$232$	0.0926861	0.00608514
$233$	17.9173	1.17380	0.586901	−	0.809658i	$-0.300348\pi$
$233$	17.9173	1.17380	0.586901	+	0.809658i	$0.300348\pi$
$234$	8.01378	0.523877
$235$	−0.233156	−0.0152094
$236$	21.5099	1.40017
$237$	1.83938	0.119481
$238$	41.7248	2.70462
$239$	17.4684	1.12994	0.564969	−	0.825112i	$-0.308888\pi$
$239$	17.4684	1.12994	0.564969	+	0.825112i	$0.308888\pi$
$240$	−0.285348	−0.0184192
$241$	6.11917	0.394171	0.197085	−	0.980386i	$-0.436852\pi$
$241$	6.11917	0.394171	0.197085	+	0.980386i	$0.436852\pi$
$242$	10.7275	0.689592
$243$	−1.00000	−0.0641500
$244$	−4.92707	−0.315423
$245$	−0.726787	−0.0464327
$246$	−12.9557	−0.826026
$247$	29.1239	1.85311
$248$	−1.13299	−0.0719450
$249$	−0.754830	−0.0478354
$250$	−1.53995	−0.0973952
$251$	−1.70829	−0.107826	−0.0539132	−	0.998546i	$-0.517169\pi$
$251$	−1.70829	−0.107826	−0.0539132	+	0.998546i	$0.517169\pi$
$252$	−8.60911	−0.542323
$253$	3.85079	0.242097
$254$	−27.8856	−1.74970
$255$	0.383830	0.0240364
$256$	14.0037	0.875233
$257$	−28.5204	−1.77905	−0.889525	−	0.456886i	$-0.848965\pi$
$257$	−28.5204	−1.77905	−0.889525	+	0.456886i	$0.848965\pi$
$258$	−5.15508	−0.320941
$259$	−33.7116	−2.09474
$260$	0.634659	0.0393599
$261$	−0.397165	−0.0245839
$262$	−38.9179	−2.40436
$263$	12.5347	0.772923	0.386461	−	0.922306i	$-0.373697\pi$
$263$	12.5347	0.772923	0.386461	+	0.922306i	$0.373697\pi$
$264$	0.557735	0.0343262
$265$	−0.122521	−0.00752638
$266$	−60.8732	−3.73238
$267$	−10.0876	−0.617349
$268$	11.6836	0.713692
$269$	8.97180	0.547020	0.273510	−	0.961869i	$-0.411815\pi$
$269$	8.97180	0.547020	0.273510	+	0.961869i	$0.411815\pi$
$270$	−0.154084	−0.00937727
$271$	1.05945	0.0643568	0.0321784	−	0.999482i	$-0.489756\pi$
$271$	1.05945	0.0643568	0.0321784	+	0.999482i	$0.489756\pi$
$272$	−18.9833	−1.15103
$273$	−16.0801	−0.973214
$274$	−23.9135	−1.44467
$275$	−11.9358	−0.719758
$276$	−3.40788	−0.205131
$277$	20.4878	1.23099	0.615497	−	0.788140i	$-0.288955\pi$
$277$	20.4878	1.23099	0.615497	+	0.788140i	$0.288955\pi$
$278$	24.2137	1.45224
$279$	4.85493	0.290657
$280$	−0.0721530	−0.00431197
$281$	−29.5789	−1.76453	−0.882266	−	0.470751i	$-0.843983\pi$
$281$	−29.5789	−1.76453	−0.882266	+	0.470751i	$0.843983\pi$
$282$	6.22677	0.370799
$283$	−7.83366	−0.465663	−0.232831	−	0.972517i	$-0.574799\pi$
$283$	−7.83366	−0.465663	−0.232831	+	0.972517i	$0.574799\pi$
$284$	−10.3677	−0.615210
$285$	−0.559978	−0.0331702
$286$	19.1523	1.13250
$287$	25.9964	1.53452
$288$	8.08738	0.476553
$289$	8.53497	0.502057
$290$	−0.0611969	−0.00359360
$291$	5.51199	0.323118
$292$	−24.2978	−1.42192
$293$	−8.42747	−0.492338	−0.246169	−	0.969227i	$-0.579172\pi$
$293$	−8.42747	−0.492338	−0.246169	+	0.969227i	$0.579172\pi$
$294$	19.4099	1.13201
$295$	−0.772485	−0.0449758
$296$	−1.93279	−0.112341
$297$	−2.38993	−0.138678
$298$	7.39901	0.428613
$299$	−6.36526	−0.368112
$300$	10.5630	0.609856
$301$	10.3440	0.596217
$302$	−9.22156	−0.530641
$303$	8.92906	0.512961
$304$	27.6952	1.58843
$305$	0.176946	0.0101319
$306$	−10.2507	−0.585995
$307$	−15.7827	−0.900766	−0.450383	−	0.892835i	$-0.648713\pi$
$307$	−15.7827	−0.900766	−0.450383	+	0.892835i	$0.648713\pi$
$308$	−20.5751	−1.17238
$309$	−4.48487	−0.255135
$310$	0.748068	0.0424874
$311$	4.53488	0.257150	0.128575	−	0.991700i	$-0.458960\pi$
$311$	4.53488	0.257150	0.128575	+	0.991700i	$0.458960\pi$
$312$	−0.921922	−0.0521935
$313$	−11.7726	−0.665430	−0.332715	−	0.943028i	$-0.607965\pi$
$313$	−11.7726	−0.665430	−0.332715	+	0.943028i	$0.607965\pi$
$314$	−1.59864	−0.0902167
$315$	0.309179	0.0174203
$316$	−3.89038	−0.218851
$317$	6.26394	0.351818	0.175909	−	0.984406i	$-0.443714\pi$
$317$	6.26394	0.351818	0.175909	+	0.984406i	$0.443714\pi$
$318$	3.27209	0.183490
$319$	−0.949194	−0.0531447
$320$	0.675441	0.0377583
$321$	4.05297	0.226215
$322$	13.3043	0.741420
$323$	−37.2535	−2.07284
$324$	2.11504	0.117502
$325$	19.7296	1.09440
$326$	−18.0557	−1.00001
$327$	−1.39157	−0.0769539
$328$	1.49045	0.0822965
$329$	−12.4944	−0.688838
$330$	−0.368250	−0.0202715
$331$	3.76715	0.207061	0.103531	−	0.994626i	$-0.466986\pi$
$331$	3.76715	0.207061	0.103531	+	0.994626i	$0.466986\pi$
$332$	1.59650	0.0876192
$333$	8.28210	0.453856
$334$	15.3693	0.840969
$335$	−0.419595	−0.0229249
$336$	−15.2913	−0.834207
$337$	−13.8155	−0.752579	−0.376289	−	0.926502i	$-0.622800\pi$
$337$	−13.8155	−0.752579	−0.376289	+	0.926502i	$0.622800\pi$
$338$	−5.28705	−0.287577
$339$	−12.3675	−0.671711
$340$	−0.811817	−0.0440269
$341$	11.6029	0.628333
$342$	14.9550	0.808675
$343$	−10.4542	−0.564473
$344$	0.593051	0.0319752
$345$	0.122388	0.00658912
$346$	28.8751	1.55234
$347$	17.7101	0.950728	0.475364	−	0.879789i	$-0.342316\pi$
$347$	17.7101	0.950728	0.475364	+	0.879789i	$0.342316\pi$
$348$	0.840020	0.0450298
$349$	−6.52416	−0.349230	−0.174615	−	0.984637i	$-0.555868\pi$
$349$	−6.52416	−0.349230	−0.174615	+	0.984637i	$0.555868\pi$
$350$	−41.2378	−2.20425
$351$	3.95048	0.210861
$352$	19.3282	1.03020
$353$	26.3992	1.40509	0.702545	−	0.711639i	$-0.252047\pi$
$353$	26.3992	1.40509	0.702545	+	0.711639i	$0.252047\pi$
$354$	20.6303	1.09649
$355$	0.372336	0.0197615
$356$	21.3356	1.13079
$357$	20.5687	1.08861
$358$	−21.3402	−1.12787
$359$	−17.3989	−0.918277	−0.459138	−	0.888365i	$-0.651842\pi$
$359$	−17.3989	−0.918277	−0.459138	+	0.888365i	$0.651842\pi$
$360$	0.0177262	0.000934252 0
$361$	35.3500	1.86053
$362$	22.0030	1.15645
$363$	5.28826	0.277562
$364$	34.0101	1.78262
$365$	0.872609	0.0456744
$366$	−4.72560	−0.247011
$367$	28.5682	1.49125	0.745623	−	0.666368i	$-0.232152\pi$
$367$	28.5682	1.49125	0.745623	+	0.666368i	$0.232152\pi$
$368$	−6.05299	−0.315534
$369$	−6.38667	−0.332477
$370$	1.27614	0.0663435
$371$	−6.56564	−0.340871
$372$	−10.2684	−0.532390
$373$	−8.25837	−0.427602	−0.213801	−	0.976877i	$-0.568584\pi$
$373$	−8.25837	−0.427602	−0.213801	+	0.976877i	$0.568584\pi$
$374$	−24.4985	−1.26679
$375$	−0.759138	−0.0392017
$376$	−0.716340	−0.0369424
$377$	1.56899	0.0808072
$378$	−8.25708	−0.424698
$379$	−0.345966	−0.0177711	−0.00888555	−	0.999961i	$-0.502828\pi$
$379$	−0.345966	−0.0177711	−0.00888555	+	0.999961i	$0.502828\pi$
$380$	1.18438	0.0607573
$381$	−13.7465	−0.704255
$382$	5.91496	0.302635
$383$	22.7307	1.16148	0.580742	−	0.814088i	$-0.302763\pi$
$383$	22.7307	1.16148	0.580742	+	0.814088i	$0.302763\pi$
$384$	−1.86387	−0.0951150
$385$	0.738916	0.0376586
$386$	−40.5512	−2.06400
$387$	−2.54126	−0.129179
$388$	−11.6581	−0.591850
$389$	10.1173	0.512969	0.256485	−	0.966548i	$-0.417436\pi$
$389$	10.1173	0.512969	0.256485	+	0.966548i	$0.417436\pi$
$390$	0.608707	0.0308231
$391$	8.14205	0.411761
$392$	−2.23296	−0.112781
$393$	−19.1850	−0.967757
$394$	−24.8376	−1.25130
$395$	0.139715	0.00702984
$396$	5.05479	0.254013
$397$	17.8481	0.895770	0.447885	−	0.894091i	$-0.352177\pi$
$397$	17.8481	0.895770	0.447885	+	0.894091i	$0.352177\pi$
$398$	8.54956	0.428551
$399$	−30.0081	−1.50229
$400$	18.7617	0.938087
$401$	−16.6625	−0.832084	−0.416042	−	0.909345i	$-0.636583\pi$
$401$	−16.6625	−0.832084	−0.416042	+	0.909345i	$0.636583\pi$
$402$	11.2059	0.558899
$403$	−19.1793	−0.955389
$404$	−18.8853	−0.939581
$405$	−0.0759576	−0.00377436
$406$	−3.27942	−0.162755
$407$	19.7936	0.981133
$408$	1.17927	0.0583823
$409$	−30.1383	−1.49024	−0.745122	−	0.666928i	$-0.767609\pi$
$409$	−30.1383	−1.49024	−0.745122	+	0.666928i	$0.767609\pi$
$410$	−0.984085	−0.0486005
$411$	−11.7884	−0.581481
$412$	9.48568	0.467326
$413$	−41.3960	−2.03696
$414$	−3.26853	−0.160640
$415$	−0.0573351	−0.00281447
$416$	−31.9490	−1.56643
$417$	11.9364	0.584529
$418$	35.7414	1.74817
$419$	3.88798	0.189940	0.0949702	−	0.995480i	$-0.469724\pi$
$419$	3.88798	0.189940	0.0949702	+	0.995480i	$0.469724\pi$
$420$	−0.653928	−0.0319084
$421$	−20.7196	−1.00981	−0.504905	−	0.863175i	$-0.668472\pi$
$421$	−20.7196	−1.00981	−0.504905	+	0.863175i	$0.668472\pi$
$422$	−46.3264	−2.25514
$423$	3.06956	0.149247
$424$	−0.376428	−0.0182809
$425$	−25.2369	−1.22417
$426$	−9.94375	−0.481776
$427$	9.48219	0.458875
$428$	−8.57221	−0.414353
$429$	9.44136	0.455833
$430$	−0.391568	−0.0188831
$431$	−12.0909	−0.582398	−0.291199	−	0.956662i	$-0.594054\pi$
$431$	−12.0909	−0.582398	−0.291199	+	0.956662i	$0.594054\pi$
$432$	3.75668	0.180743
$433$	−25.4139	−1.22131	−0.610657	−	0.791896i	$-0.709094\pi$
$433$	−25.4139	−1.22131	−0.610657	+	0.791896i	$0.709094\pi$
$434$	40.0875	1.92426
$435$	−0.0301677	−0.00144643
$436$	2.94323	0.140955
$437$	−11.8786	−0.568231
$438$	−23.3043	−1.11352
$439$	1.19853	0.0572026	0.0286013	−	0.999591i	$-0.490895\pi$
$439$	1.19853	0.0572026	0.0286013	+	0.999591i	$0.490895\pi$
$440$	0.0423642	0.00201964
$441$	9.56833	0.455635
$442$	40.4953	1.92617
$443$	31.5495	1.49896	0.749481	−	0.662025i	$-0.230303\pi$
$443$	31.5495	1.49896	0.749481	+	0.662025i	$0.230303\pi$
$444$	−17.5170	−0.831320
$445$	−0.766227	−0.0363227
$446$	−5.48283	−0.259620
$447$	3.64742	0.172517
$448$	36.1956	1.71008
$449$	12.6160	0.595385	0.297692	−	0.954662i	$-0.403783\pi$
$449$	12.6160	0.595385	0.297692	+	0.954662i	$0.403783\pi$
$450$	10.1311	0.477584
$451$	−15.2637	−0.718737
$452$	26.1578	1.23036
$453$	−4.54587	−0.213584
$454$	48.5898	2.28043
$455$	−1.22141	−0.0572605
$456$	−1.72046	−0.0805678
$457$	3.90046	0.182456	0.0912279	−	0.995830i	$-0.470921\pi$
$457$	3.90046	0.182456	0.0912279	+	0.995830i	$0.470921\pi$
$458$	11.1523	0.521113
$459$	−5.05321	−0.235864
$460$	−0.258855	−0.0120692
$461$	12.0785	0.562551	0.281276	−	0.959627i	$-0.409242\pi$
$461$	12.0785	0.562551	0.281276	+	0.959627i	$0.409242\pi$
$462$	−19.7338	−0.918100
$463$	−12.1865	−0.566352	−0.283176	−	0.959068i	$-0.591388\pi$
$463$	−12.1865	−0.566352	−0.283176	+	0.959068i	$0.591388\pi$
$464$	1.49202	0.0692654
$465$	0.368769	0.0171012
$466$	−36.3463	−1.68371
$467$	24.8736	1.15101	0.575507	−	0.817797i	$-0.304805\pi$
$467$	24.8736	1.15101	0.575507	+	0.817797i	$0.304805\pi$
$468$	−8.35544	−0.386230
$469$	−22.4853	−1.03827
$470$	0.472970	0.0218165
$471$	−0.788070	−0.0363123
$472$	−2.37335	−0.109242
$473$	−6.07341	−0.279256
$474$	−3.73130	−0.171384
$475$	36.8187	1.68936
$476$	−43.5037	−1.99399
$477$	1.61301	0.0738548
$478$	−35.4357	−1.62079
$479$	−12.0702	−0.551500	−0.275750	−	0.961229i	$-0.588926\pi$
$479$	−12.0702	−0.551500	−0.275750	+	0.961229i	$0.588926\pi$
$480$	0.614298	0.0280387
$481$	−32.7183	−1.49183
$482$	−12.4131	−0.565401
$483$	6.55851	0.298423
$484$	−11.1849	−0.508404
$485$	0.418677	0.0190112
$486$	2.02856	0.0920172
$487$	−6.33917	−0.287255	−0.143628	−	0.989632i	$-0.545877\pi$
$487$	−6.33917	−0.287255	−0.143628	+	0.989632i	$0.545877\pi$
$488$	0.543642	0.0246095
$489$	−8.90077	−0.402507
$490$	1.47433	0.0666034
$491$	−31.1631	−1.40637	−0.703186	−	0.711005i	$-0.748240\pi$
$491$	−31.1631	−1.40637	−0.703186	+	0.711005i	$0.748240\pi$
$492$	13.5081	0.608991
$493$	−2.00696	−0.0903889
$494$	−59.0795	−2.65811
$495$	−0.181533	−0.00815930
$496$	−18.2384	−0.818929
$497$	19.9527	0.895003
$498$	1.53122	0.0686154
$499$	11.5343	0.516347	0.258173	−	0.966099i	$-0.416879\pi$
$499$	11.5343	0.516347	0.258173	+	0.966099i	$0.416879\pi$
$500$	1.60561	0.0718050
$501$	7.57645	0.338491
$502$	3.46536	0.154667
$503$	−16.8029	−0.749203	−0.374601	−	0.927186i	$-0.622220\pi$
$503$	−16.8029	−0.749203	−0.374601	+	0.927186i	$0.622220\pi$
$504$	0.949912	0.0423124
$505$	0.678230	0.0301808
$506$	−7.81155	−0.347266
$507$	−2.60631	−0.115750
$508$	29.0744	1.28997
$509$	16.5058	0.731608	0.365804	−	0.930692i	$-0.380794\pi$
$509$	16.5058	0.731608	0.365804	+	0.930692i	$0.380794\pi$
$510$	−0.778621	−0.0344779
$511$	46.7614	2.06860
$512$	−32.1351	−1.42018
$513$	7.37225	0.325493
$514$	57.8551	2.55188
$515$	−0.340660	−0.0150113
$516$	5.37486	0.236615
$517$	7.33601	0.322637
$518$	68.3860	3.00471
$519$	14.2343	0.624817
$520$	−0.0700270	−0.00307089
$521$	−13.7508	−0.602432	−0.301216	−	0.953556i	$-0.597393\pi$
$521$	−13.7508	−0.602432	−0.301216	+	0.953556i	$0.597393\pi$
$522$	0.805671	0.0352633
$523$	−6.89468	−0.301483	−0.150742	−	0.988573i	$-0.548166\pi$
$523$	−6.89468	−0.301483	−0.150742	+	0.988573i	$0.548166\pi$
$524$	40.5772	1.77262
$525$	−20.3286	−0.887214
$526$	−25.4274	−1.10869
$527$	24.5330	1.06867
$528$	8.97819	0.390725
$529$	−20.4038	−0.887123
$530$	0.248540	0.0107959
$531$	10.1699	0.441338
$532$	63.4685	2.75171
$533$	25.2304	1.09285
$534$	20.4632	0.885529
$535$	0.307854	0.0133097
$536$	−1.28915	−0.0556827
$537$	−10.5199	−0.453967
$538$	−18.1998	−0.784650
$539$	22.8676	0.984977
$540$	0.160654	0.00691343
$541$	−25.5396	−1.09803	−0.549016	−	0.835812i	$-0.684998\pi$
$541$	−25.5396	−1.09803	−0.549016	+	0.835812i	$0.684998\pi$
$542$	−2.14915	−0.0923138
$543$	10.8466	0.465473
$544$	40.8673	1.75217
$545$	−0.105700	−0.00452770
$546$	32.6194	1.39598
$547$	15.2264	0.651033	0.325517	−	0.945536i	$-0.394462\pi$
$547$	15.2264	0.651033	0.325517	+	0.945536i	$0.394462\pi$
$548$	24.9330	1.06509
$549$	−2.32954	−0.0994222
$550$	24.2125	1.03243
$551$	2.92800	0.124737
$552$	0.376019	0.0160044
$553$	7.48707	0.318383
$554$	−41.5607	−1.76574
$555$	0.629089	0.0267033
$556$	−25.2460	−1.07067
$557$	−28.2500	−1.19699	−0.598495	−	0.801127i	$-0.704234\pi$
$557$	−28.2500	−1.19699	−0.598495	+	0.801127i	$0.704234\pi$
$558$	−9.84849	−0.416920
$559$	10.0392	0.424612
$560$	−1.16149	−0.0490819
$561$	−12.0768	−0.509883
$562$	60.0026	2.53106
$563$	20.7910	0.876235	0.438118	−	0.898918i	$-0.355645\pi$
$563$	20.7910	0.876235	0.438118	+	0.898918i	$0.355645\pi$
$564$	−6.49224	−0.273373
$565$	−0.939406	−0.0395211
$566$	15.8910	0.667950
$567$	−4.07042	−0.170942
$568$	1.14395	0.0479991
$569$	15.9318	0.667895	0.333947	−	0.942592i	$-0.391619\pi$
$569$	15.9318	0.667895	0.333947	+	0.942592i	$0.391619\pi$
$570$	1.13595	0.0475796
$571$	5.01421	0.209838	0.104919	−	0.994481i	$-0.466542\pi$
$571$	5.01421	0.209838	0.104919	+	0.994481i	$0.466542\pi$
$572$	−19.9689	−0.834940
$573$	2.91584	0.121811
$574$	−52.7352	−2.20113
$575$	−8.04701	−0.335583
$576$	−8.89234	−0.370514
$577$	−47.8354	−1.99142	−0.995708	−	0.0925515i	$-0.970498\pi$
$577$	−47.8354	−1.99142	−0.995708	+	0.0925515i	$0.970498\pi$
$578$	−17.3137	−0.720154
$579$	−19.9902	−0.830762
$580$	0.0638059	0.00264940
$581$	−3.07248	−0.127468
$582$	−11.1814	−0.463483
$583$	3.85498	0.159657
$584$	2.68097	0.110939
$585$	0.300069	0.0124063
$586$	17.0956	0.706213
$587$	21.3047	0.879341	0.439670	−	0.898159i	$-0.355095\pi$
$587$	21.3047	0.879341	0.439670	+	0.898159i	$0.355095\pi$
$588$	−20.2374	−0.834577
$589$	−35.7917	−1.47477
$590$	1.56703	0.0645136
$591$	−12.2440	−0.503650
$592$	−31.1132	−1.27875
$593$	22.0101	0.903848	0.451924	−	0.892056i	$-0.350738\pi$
$593$	22.0101	0.903848	0.451924	+	0.892056i	$0.350738\pi$
$594$	4.84810	0.198920
$595$	1.56235	0.0640501
$596$	−7.71446	−0.315996
$597$	4.21460	0.172492
$598$	12.9123	0.528023
$599$	−46.2119	−1.88817	−0.944084	−	0.329705i	$-0.893051\pi$
$599$	−46.2119	−1.88817	−0.944084	+	0.329705i	$0.893051\pi$
$600$	−1.16550	−0.0475814
$601$	−35.4940	−1.44783	−0.723916	−	0.689888i	$-0.757660\pi$
$601$	−35.4940	−1.44783	−0.723916	+	0.689888i	$0.757660\pi$
$602$	−20.9834	−0.855218
$603$	5.52407	0.224957
$604$	9.61471	0.391217
$605$	0.401683	0.0163308
$606$	−18.1131	−0.735794
$607$	23.5493	0.955837	0.477918	−	0.878404i	$-0.341391\pi$
$607$	23.5493	0.955837	0.477918	+	0.878404i	$0.341391\pi$
$608$	−59.6221	−2.41800
$609$	−1.61663	−0.0655091
$610$	−0.358945	−0.0145333
$611$	−12.1262	−0.490574
$612$	10.6878	0.432027
$613$	−40.5362	−1.63724	−0.818621	−	0.574334i	$-0.805261\pi$
$613$	−40.5362	−1.63724	−0.818621	+	0.574334i	$0.805261\pi$
$614$	32.0161	1.29206
$615$	−0.485116	−0.0195618
$616$	2.27022	0.0914697
$617$	35.0861	1.41251	0.706257	−	0.707956i	$-0.250382\pi$
$617$	35.0861	1.41251	0.706257	+	0.707956i	$0.250382\pi$
$618$	9.09781	0.365967
$619$	−2.78113	−0.111783	−0.0558916	−	0.998437i	$-0.517800\pi$
$619$	−2.78113	−0.111783	−0.0558916	+	0.998437i	$0.517800\pi$
$620$	−0.779961	−0.0313240
$621$	−1.61126	−0.0646577
$622$	−9.19926	−0.368857
$623$	−41.0606	−1.64506
$624$	−14.8407	−0.594104
$625$	24.9135	0.996540
$626$	23.8815	0.954496
$627$	17.6191	0.703640
$628$	1.66680	0.0665126
$629$	41.8512	1.66872
$630$	−0.627188	−0.0249878
$631$	6.40432	0.254952	0.127476	−	0.991842i	$-0.459312\pi$
$631$	6.40432	0.254952	0.127476	+	0.991842i	$0.459312\pi$
$632$	0.429256	0.0170749
$633$	−22.8371	−0.907695
$634$	−12.7068	−0.504650
$635$	−1.04415	−0.0414359
$636$	−3.41159	−0.135278
$637$	−37.7995	−1.49767
$638$	1.92549	0.0762311
$639$	−4.90189	−0.193916
$640$	−0.141575	−0.00559624
$641$	−17.5902	−0.694773	−0.347386	−	0.937722i	$-0.612931\pi$
$641$	−17.5902	−0.694773	−0.347386	+	0.937722i	$0.612931\pi$
$642$	−8.22168	−0.324484
$643$	−1.22127	−0.0481621	−0.0240811	−	0.999710i	$-0.507666\pi$
$643$	−1.22127	−0.0481621	−0.0240811	+	0.999710i	$0.507666\pi$
$644$	−13.8715	−0.546615
$645$	−0.193028	−0.00760046
$646$	75.5709	2.97330
$647$	13.4523	0.528864	0.264432	−	0.964404i	$-0.414816\pi$
$647$	13.4523	0.528864	0.264432	+	0.964404i	$0.414816\pi$
$648$	−0.233369	−0.00916762
$649$	24.3054	0.954071
$650$	−40.0226	−1.56982
$651$	19.7616	0.774518
$652$	18.8255	0.737264
$653$	−14.3571	−0.561835	−0.280917	−	0.959732i	$-0.590639\pi$
$653$	−14.3571	−0.561835	−0.280917	+	0.959732i	$0.590639\pi$
$654$	2.82288	0.110383
$655$	−1.45725	−0.0569395
$656$	23.9927	0.936756
$657$	−11.4881	−0.448194
$658$	25.3456	0.988073
$659$	−19.9170	−0.775855	−0.387927	−	0.921690i	$-0.626809\pi$
$659$	−19.9170	−0.775855	−0.387927	+	0.921690i	$0.626809\pi$
$660$	0.383950	0.0149452
$661$	−14.2781	−0.555352	−0.277676	−	0.960675i	$-0.589564\pi$
$661$	−14.2781	−0.555352	−0.277676	+	0.960675i	$0.589564\pi$
$662$	−7.64188	−0.297010
$663$	19.9626	0.775284
$664$	−0.176154	−0.00683611
$665$	−2.27935	−0.0883893
$666$	−16.8007	−0.651014
$667$	−0.639936	−0.0247784
$668$	−16.0245	−0.620007
$669$	−2.70283	−0.104497
$670$	0.851172	0.0328837
$671$	−5.56742	−0.214928
$672$	32.9190	1.26988
$673$	−10.6630	−0.411030	−0.205515	−	0.978654i	$-0.565887\pi$
$673$	−10.6630	−0.411030	−0.205515	+	0.978654i	$0.565887\pi$
$674$	28.0255	1.07950
$675$	4.99423	0.192228
$676$	5.51245	0.212017
$677$	−28.6460	−1.10096	−0.550478	−	0.834850i	$-0.685555\pi$
$677$	−28.6460	−1.10096	−0.550478	+	0.834850i	$0.685555\pi$
$678$	25.0882	0.963506
$679$	22.4361	0.861019
$680$	0.0895742	0.00343501
$681$	23.9529	0.917877
$682$	−23.5372	−0.901284
$683$	−35.7677	−1.36861	−0.684307	−	0.729194i	$-0.739895\pi$
$683$	−35.7677	−1.36861	−0.684307	+	0.729194i	$0.739895\pi$
$684$	−15.5926	−0.596198
$685$	−0.895422	−0.0342123
$686$	21.2069	0.809683
$687$	5.49765	0.209748
$688$	9.54669	0.363964
$689$	−6.37218	−0.242761
$690$	−0.248270	−0.00945148
$691$	−33.6568	−1.28037	−0.640183	−	0.768223i	$-0.721141\pi$
$691$	−33.6568	−1.28037	−0.640183	+	0.768223i	$0.721141\pi$
$692$	−30.1062	−1.14447
$693$	−9.72800	−0.369536
$694$	−35.9259	−1.36373
$695$	0.906661	0.0343916
$696$	−0.0926861	−0.00351326
$697$	−32.2732	−1.22243
$698$	13.2346	0.500938
$699$	−17.9173	−0.677695
$700$	42.9959	1.62509
$701$	24.6978	0.932824	0.466412	−	0.884568i	$-0.345546\pi$
$701$	24.6978	0.932824	0.466412	+	0.884568i	$0.345546\pi$
$702$	−8.01378	−0.302461
$703$	−61.0577	−2.30283
$704$	−21.2520	−0.800966
$705$	0.233156	0.00878116
$706$	−53.5524	−2.01547
$707$	36.3450	1.36690
$708$	−21.5099	−0.808390
$709$	−27.3335	−1.02653	−0.513266	−	0.858230i	$-0.671565\pi$
$709$	−27.3335	−1.02653	−0.513266	+	0.858230i	$0.671565\pi$
$710$	−0.755304	−0.0283460
$711$	−1.83938	−0.0689823
$712$	−2.35413	−0.0882247
$713$	7.82255	0.292957
$714$	−41.7248	−1.56151
$715$	0.717143	0.0268196
$716$	22.2500	0.831523
$717$	−17.4684	−0.652370
$718$	35.2946	1.31718
$719$	−1.48282	−0.0552998	−0.0276499	−	0.999618i	$-0.508802\pi$
$719$	−1.48282	−0.0552998	−0.0276499	+	0.999618i	$0.508802\pi$
$720$	0.285348	0.0106343
$721$	−18.2553	−0.679863
$722$	−71.7095	−2.66875
$723$	−6.11917	−0.227574
$724$	−22.9411	−0.852598
$725$	1.98353	0.0736666
$726$	−10.7275	−0.398136
$727$	16.5416	0.613492	0.306746	−	0.951791i	$-0.400760\pi$
$727$	16.5416	0.613492	0.306746	+	0.951791i	$0.400760\pi$
$728$	−3.75261	−0.139081
$729$	1.00000	0.0370370
$730$	−1.77014	−0.0655157
$731$	−12.8415	−0.474960
$732$	4.92707	0.182110
$733$	12.1880	0.450175	0.225087	−	0.974339i	$-0.427733\pi$
$733$	12.1880	0.450175	0.225087	+	0.974339i	$0.427733\pi$
$734$	−57.9521	−2.13905
$735$	0.726787	0.0268080
$736$	13.0309	0.480325
$737$	13.2021	0.486306
$738$	12.9557	0.476906
$739$	−38.5510	−1.41812	−0.709061	−	0.705147i	$-0.750881\pi$
$739$	−38.5510	−1.41812	−0.709061	+	0.705147i	$0.750881\pi$
$740$	−1.33055	−0.0489120
$741$	−29.1239	−1.06989
$742$	13.3188	0.488948
$743$	−43.1955	−1.58469	−0.792344	−	0.610074i	$-0.791140\pi$
$743$	−43.1955	−1.58469	−0.792344	+	0.610074i	$0.791140\pi$
$744$	1.13299	0.0415375
$745$	0.277050	0.0101503
$746$	16.7526	0.613355
$747$	0.754830	0.0276178
$748$	25.5429	0.933943
$749$	16.4973	0.602798
$750$	1.53995	0.0562312
$751$	−51.0709	−1.86360	−0.931801	−	0.362969i	$-0.881763\pi$
$751$	−51.0709	−1.86360	−0.931801	+	0.362969i	$0.881763\pi$
$752$	−11.5313	−0.420505
$753$	1.70829	0.0622536
$754$	−3.18279	−0.115910
$755$	−0.345294	−0.0125665
$756$	8.60911	0.313110
$757$	40.9306	1.48765	0.743824	−	0.668376i	$-0.233010\pi$
$757$	40.9306	1.48765	0.743824	+	0.668376i	$0.233010\pi$
$758$	0.701812	0.0254910
$759$	−3.85079	−0.139775
$760$	−0.130682	−0.00474033
$761$	7.98649	0.289510	0.144755	−	0.989468i	$-0.453761\pi$
$761$	7.98649	0.289510	0.144755	+	0.989468i	$0.453761\pi$
$762$	27.8856	1.01019
$763$	−5.66427	−0.205060
$764$	−6.16713	−0.223119
$765$	−0.383830	−0.0138774
$766$	−46.1105	−1.66604
$767$	−40.1762	−1.45068
$768$	−14.0037	−0.505316
$769$	−4.73725	−0.170830	−0.0854148	−	0.996345i	$-0.527222\pi$
$769$	−4.73725	−0.170830	−0.0854148	+	0.996345i	$0.527222\pi$
$770$	−1.49893	−0.0540178
$771$	28.5204	1.02714
$772$	42.2800	1.52169
$773$	29.3662	1.05623	0.528115	−	0.849173i	$-0.322899\pi$
$773$	29.3662	1.05623	0.528115	+	0.849173i	$0.322899\pi$
$774$	5.15508	0.185296
$775$	−24.2466	−0.870964
$776$	1.28633	0.0461765
$777$	33.7116	1.20940
$778$	−20.5236	−0.735806
$779$	47.0841	1.68696
$780$	−0.634659	−0.0227244
$781$	−11.7151	−0.419201
$782$	−16.5166	−0.590632
$783$	0.397165	0.0141935
$784$	−35.9452	−1.28376
$785$	−0.0598599	−0.00213649
$786$	38.9179	1.38816
$787$	7.18325	0.256055	0.128028	−	0.991771i	$-0.459135\pi$
$787$	7.18325	0.256055	0.128028	+	0.991771i	$0.459135\pi$
$788$	25.8965	0.922526
$789$	−12.5347	−0.446247
$790$	−0.283420	−0.0100836
$791$	−50.3410	−1.78992
$792$	−0.557735	−0.0198183
$793$	9.20279	0.326801
$794$	−36.2059	−1.28490
$795$	0.122521	0.00434536
$796$	−8.91406	−0.315950
$797$	−17.0979	−0.605637	−0.302819	−	0.953048i	$-0.597928\pi$
$797$	−17.0979	−0.605637	−0.302819	+	0.953048i	$0.597928\pi$
$798$	60.8732	2.15489
$799$	15.5111	0.548744
$800$	−40.3902	−1.42801
$801$	10.0876	0.356427
$802$	33.8008	1.19355
$803$	−27.4557	−0.968891
$804$	−11.6836	−0.412050
$805$	0.498169	0.0175581
$806$	38.9063	1.37042
$807$	−8.97180	−0.315822
$808$	2.08377	0.0733067
$809$	−14.8720	−0.522872	−0.261436	−	0.965221i	$-0.584196\pi$
$809$	−14.8720	−0.522872	−0.261436	+	0.965221i	$0.584196\pi$
$810$	0.154084	0.00541397
$811$	19.6129	0.688702	0.344351	−	0.938841i	$-0.388099\pi$
$811$	19.6129	0.688702	0.344351	+	0.938841i	$0.388099\pi$
$812$	3.41924	0.119992
$813$	−1.05945	−0.0371564
$814$	−40.1524	−1.40734
$815$	−0.676081	−0.0236821
$816$	18.9833	0.664549
$817$	18.7348	0.655446
$818$	61.1373	2.13761
$819$	16.0801	0.561885
$820$	1.02604	0.0358309
$821$	−21.8382	−0.762157	−0.381079	−	0.924543i	$-0.624447\pi$
$821$	−21.8382	−0.762157	−0.381079	+	0.924543i	$0.624447\pi$
$822$	23.9135	0.834080
$823$	5.01012	0.174642	0.0873210	−	0.996180i	$-0.472169\pi$
$823$	5.01012	0.174642	0.0873210	+	0.996180i	$0.472169\pi$
$824$	−1.04663	−0.0364611
$825$	11.9358	0.415552
$826$	83.9740	2.92183
$827$	13.3847	0.465433	0.232716	−	0.972545i	$-0.425239\pi$
$827$	13.3847	0.465433	0.232716	+	0.972545i	$0.425239\pi$
$828$	3.40788	0.118432
$829$	7.04326	0.244622	0.122311	−	0.992492i	$-0.460969\pi$
$829$	7.04326	0.244622	0.122311	+	0.992492i	$0.460969\pi$
$830$	0.116308	0.00403709
$831$	−20.4878	−0.710714
$832$	35.1291	1.21788
$833$	48.3508	1.67526
$834$	−24.2137	−0.838452
$835$	0.575489	0.0199156
$836$	−37.2652	−1.28884
$837$	−4.85493	−0.167811
$838$	−7.88699	−0.272452
$839$	−54.5876	−1.88457	−0.942287	−	0.334805i	$-0.891329\pi$
$839$	−54.5876	−1.88457	−0.942287	+	0.334805i	$0.891329\pi$
$840$	0.0721530	0.00248952
$841$	−28.8423	−0.994561
$842$	42.0308	1.44848
$843$	29.5789	1.01875
$844$	48.3015	1.66261
$845$	−0.197969	−0.00681034
$846$	−6.22677	−0.214081
$847$	21.5254	0.739623
$848$	−6.05957	−0.208087
$849$	7.83366	0.268851
$850$	51.1945	1.75596
$851$	13.3446	0.457448
$852$	10.3677	0.355191
$853$	−17.8516	−0.611227	−0.305614	−	0.952156i	$-0.598862\pi$
$853$	−17.8516	−0.611227	−0.305614	+	0.952156i	$0.598862\pi$
$854$	−19.2352	−0.658214
$855$	0.559978	0.0191508
$856$	0.945840	0.0323281
$857$	−57.0510	−1.94883	−0.974413	−	0.224767i	$-0.927838\pi$
$857$	−57.0510	−1.94883	−0.974413	+	0.224767i	$0.927838\pi$
$858$	−19.1523	−0.653850
$859$	−26.0129	−0.887550	−0.443775	−	0.896138i	$-0.646361\pi$
$859$	−26.0129	−0.887550	−0.443775	+	0.896138i	$0.646361\pi$
$860$	0.408262	0.0139216
$861$	−25.9964	−0.885956
$862$	24.5271	0.835395
$863$	6.90726	0.235126	0.117563	−	0.993065i	$-0.462492\pi$
$863$	6.90726	0.235126	0.117563	+	0.993065i	$0.462492\pi$
$864$	−8.08738	−0.275138
$865$	1.08120	0.0367621
$866$	51.5535	1.75186
$867$	−8.53497	−0.289863
$868$	−41.7966	−1.41867
$869$	−4.39599	−0.149124
$870$	0.0611969	0.00207477
$871$	−21.8227	−0.739435
$872$	−0.324749	−0.0109974
$873$	−5.51199	−0.186553
$874$	24.0964	0.815074
$875$	−3.09001	−0.104461
$876$	24.2978	0.820947
$877$	2.46415	0.0832085	0.0416043	−	0.999134i	$-0.486753\pi$
$877$	2.46415	0.0832085	0.0416043	+	0.999134i	$0.486753\pi$
$878$	−2.43128	−0.0820517
$879$	8.42747	0.284252
$880$	0.681962	0.0229889
$881$	20.3161	0.684466	0.342233	−	0.939615i	$-0.388817\pi$
$881$	20.3161	0.684466	0.342233	+	0.939615i	$0.388817\pi$
$882$	−19.4099	−0.653565
$883$	−24.4700	−0.823481	−0.411741	−	0.911301i	$-0.635079\pi$
$883$	−24.4700	−0.823481	−0.411741	+	0.911301i	$0.635079\pi$
$884$	−42.2218	−1.42007
$885$	0.772485	0.0259668
$886$	−64.0000	−2.15012
$887$	−1.95078	−0.0655008	−0.0327504	−	0.999464i	$-0.510427\pi$
$887$	−1.95078	−0.0655008	−0.0327504	+	0.999464i	$0.510427\pi$
$888$	1.93279	0.0648602
$889$	−55.9541	−1.87664
$890$	1.55434	0.0521014
$891$	2.38993	0.0800655
$892$	5.71659	0.191406
$893$	−22.6295	−0.757268
$894$	−7.39901	−0.247460
$895$	−0.799067	−0.0267099
$896$	−7.58672	−0.253455
$897$	6.36526	0.212530
$898$	−25.5922	−0.854023
$899$	−1.92821	−0.0643093
$900$	−10.5630	−0.352100
$901$	8.15090	0.271546
$902$	30.9632	1.03096
$903$	−10.3440	−0.344226
$904$	−2.88620	−0.0959935
$905$	0.823884	0.0273868
$906$	9.22156	0.306366
$907$	−18.4646	−0.613107	−0.306554	−	0.951853i	$-0.599176\pi$
$907$	−18.4646	−0.613107	−0.306554	+	0.951853i	$0.599176\pi$
$908$	−50.6614	−1.68126
$909$	−8.92906	−0.296158
$910$	2.47770	0.0821348
$911$	−43.1389	−1.42926	−0.714628	−	0.699505i	$-0.753404\pi$
$911$	−43.1389	−1.42926	−0.714628	+	0.699505i	$0.753404\pi$
$912$	−27.6952	−0.917079
$913$	1.80399	0.0597033
$914$	−7.91230	−0.261716
$915$	−0.176946	−0.00584965
$916$	−11.6278	−0.384192
$917$	−78.0912	−2.57880
$918$	10.2507	0.338325
$919$	−33.0958	−1.09173	−0.545865	−	0.837873i	$-0.683799\pi$
$919$	−33.0958	−1.09173	−0.545865	+	0.837873i	$0.683799\pi$
$920$	0.0285615	0.000941645 0
$921$	15.7827	0.520058
$922$	−24.5019	−0.806927
$923$	19.3648	0.637401
$924$	20.5751	0.676872
$925$	−41.3627	−1.36000
$926$	24.7209	0.812379
$927$	4.48487	0.147302
$928$	−3.21202	−0.105440
$929$	−48.7909	−1.60078	−0.800388	−	0.599482i	$-0.795373\pi$
$929$	−48.7909	−1.60078	−0.800388	+	0.599482i	$0.795373\pi$
$930$	−0.748068	−0.0245301
$931$	−70.5401	−2.31186
$932$	37.8959	1.24132
$933$	−4.53488	−0.148465
$934$	−50.4575	−1.65102
$935$	−0.917325	−0.0299997
$936$	0.921922	0.0301340
$937$	−36.8444	−1.20365	−0.601826	−	0.798627i	$-0.705560\pi$
$937$	−36.8444	−1.20365	−0.601826	+	0.798627i	$0.705560\pi$
$938$	45.6127	1.48931
$939$	11.7726	0.384186
$940$	−0.493135	−0.0160843
$941$	−35.1886	−1.14711	−0.573557	−	0.819165i	$-0.694437\pi$
$941$	−35.1886	−1.14711	−0.573557	+	0.819165i	$0.694437\pi$
$942$	1.59864	0.0520867
$943$	−10.2906	−0.335108
$944$	−38.2052	−1.24347
$945$	−0.309179	−0.0100576
$946$	12.3203	0.400566
$947$	18.6569	0.606269	0.303135	−	0.952948i	$-0.401967\pi$
$947$	18.6569	0.606269	0.303135	+	0.952948i	$0.401967\pi$
$948$	3.89038	0.126354
$949$	45.3836	1.47321
$950$	−74.6888	−2.42322
$951$	−6.26394	−0.203122
$952$	4.80011	0.155572
$953$	−29.0437	−0.940818	−0.470409	−	0.882448i	$-0.655894\pi$
$953$	−29.0437	−0.940818	−0.470409	+	0.882448i	$0.655894\pi$
$954$	−3.27209	−0.105938
$955$	0.221481	0.00716694
$956$	36.9464	1.19493
$957$	0.949194	0.0306831
$958$	24.4850	0.791075
$959$	−47.9839	−1.54948
$960$	−0.675441	−0.0217998
$961$	−7.42969	−0.239667
$962$	66.3709	2.13988
$963$	−4.05297	−0.130605
$964$	12.9423	0.416844
$965$	−1.51840	−0.0488792
$966$	−13.3043	−0.428059
$967$	−26.8786	−0.864359	−0.432179	−	0.901788i	$-0.642255\pi$
$967$	−26.8786	−0.864359	−0.432179	+	0.901788i	$0.642255\pi$
$968$	1.23412	0.0396660
$969$	37.2535	1.19676
$970$	−0.849311	−0.0272697
$971$	−43.0223	−1.38065	−0.690326	−	0.723498i	$-0.742533\pi$
$971$	−43.0223	−1.38065	−0.690326	+	0.723498i	$0.742533\pi$
$972$	−2.11504	−0.0678400
$973$	48.5862	1.55760
$974$	12.8594	0.412041
$975$	−19.7296	−0.631853
$976$	8.75132	0.280123
$977$	24.5956	0.786883	0.393442	−	0.919350i	$-0.371284\pi$
$977$	24.5956	0.786883	0.393442	+	0.919350i	$0.371284\pi$
$978$	18.0557	0.577358
$979$	24.1085	0.770512
$980$	−1.53719	−0.0491036
$981$	1.39157	0.0444294
$982$	63.2162	2.01731
$983$	−0.0286563	−0.000913995 0	−0.000456998	−	1.00000i	$-0.500145\pi$
$983$	−0.0286563	−0.000913995 0	−0.000456998		1.00000i	$0.500145\pi$
$984$	−1.49045	−0.0475139
$985$	−0.930023	−0.0296330
$986$	4.07123	0.129654
$987$	12.4944	0.397701
$988$	61.5983	1.95970
$989$	−4.09463	−0.130202
$990$	0.368250	0.0117038
$991$	24.2350	0.769850	0.384925	−	0.922948i	$-0.374227\pi$
$991$	24.2350	0.769850	0.384925	+	0.922948i	$0.374227\pi$
$992$	39.2636	1.24662
$993$	−3.76715	−0.119547
$994$	−40.4753	−1.28380
$995$	0.320131	0.0101488
$996$	−1.59650	−0.0505870
$997$	−2.89028	−0.0915361	−0.0457681	−	0.998952i	$-0.514574\pi$
$997$	−2.89028	−0.0915361	−0.0457681	+	0.998952i	$0.514574\pi$
$998$	−23.3980	−0.740651
$999$	−8.28210	−0.262034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.18	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.18	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02856 q^{2} -1.00000 q^{3} +2.11504 q^{4} -0.0759576 q^{5} +2.02856 q^{6} -4.07042 q^{7} -0.233369 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.02856 q^{2} -1.00000 q^{3} +2.11504 q^{4} -0.0759576 q^{5} +2.02856 q^{6} -4.07042 q^{7} -0.233369 q^{8} +1.00000 q^{9} +0.154084 q^{10} +2.38993 q^{11} -2.11504 q^{12} -3.95048 q^{13} +8.25708 q^{14} +0.0759576 q^{15} -3.75668 q^{16} +5.05321 q^{17} -2.02856 q^{18} -7.37225 q^{19} -0.160654 q^{20} +4.07042 q^{21} -4.84810 q^{22} +1.61126 q^{23} +0.233369 q^{24} -4.99423 q^{25} +8.01378 q^{26} -1.00000 q^{27} -8.60911 q^{28} -0.397165 q^{29} -0.154084 q^{30} +4.85493 q^{31} +8.08738 q^{32} -2.38993 q^{33} -10.2507 q^{34} +0.309179 q^{35} +2.11504 q^{36} +8.28210 q^{37} +14.9550 q^{38} +3.95048 q^{39} +0.0177262 q^{40} -6.38667 q^{41} -8.25708 q^{42} -2.54126 q^{43} +5.05479 q^{44} -0.0759576 q^{45} -3.26853 q^{46} +3.06956 q^{47} +3.75668 q^{48} +9.56833 q^{49} +10.1311 q^{50} -5.05321 q^{51} -8.35544 q^{52} +1.61301 q^{53} +2.02856 q^{54} -0.181533 q^{55} +0.949912 q^{56} +7.37225 q^{57} +0.805671 q^{58} +10.1699 q^{59} +0.160654 q^{60} -2.32954 q^{61} -9.84849 q^{62} -4.07042 q^{63} -8.89234 q^{64} +0.300069 q^{65} +4.84810 q^{66} +5.52407 q^{67} +10.6878 q^{68} -1.61126 q^{69} -0.627188 q^{70} -4.90189 q^{71} -0.233369 q^{72} -11.4881 q^{73} -16.8007 q^{74} +4.99423 q^{75} -15.5926 q^{76} -9.72800 q^{77} -8.01378 q^{78} -1.83938 q^{79} +0.285348 q^{80} +1.00000 q^{81} +12.9557 q^{82} +0.754830 q^{83} +8.60911 q^{84} -0.383830 q^{85} +5.15508 q^{86} +0.397165 q^{87} -0.557735 q^{88} +10.0876 q^{89} +0.154084 q^{90} +16.0801 q^{91} +3.40788 q^{92} -4.85493 q^{93} -6.22677 q^{94} +0.559978 q^{95} -8.08738 q^{96} -5.51199 q^{97} -19.4099 q^{98} +2.38993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more