LMFDB - Embedded newform 4031.2.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$0$
Dimension:	$103$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09508 q^{2} -0.524599 q^{3} +2.38936 q^{4} -2.74536 q^{5} +1.09908 q^{6} -3.42586 q^{7} -0.815732 q^{8} -2.72480 q^{9} +O(q^{10})$	\(q-2.09508 q^{2} -0.524599 q^{3} +2.38936 q^{4} -2.74536 q^{5} +1.09908 q^{6} -3.42586 q^{7} -0.815732 q^{8} -2.72480 q^{9} +5.75174 q^{10} +3.66785 q^{11} -1.25345 q^{12} +6.57963 q^{13} +7.17744 q^{14} +1.44021 q^{15} -3.06969 q^{16} +5.80637 q^{17} +5.70866 q^{18} -4.85000 q^{19} -6.55964 q^{20} +1.79720 q^{21} -7.68444 q^{22} +6.09179 q^{23} +0.427932 q^{24} +2.53700 q^{25} -13.7848 q^{26} +3.00322 q^{27} -8.18559 q^{28} -1.00000 q^{29} -3.01736 q^{30} +3.70620 q^{31} +8.06271 q^{32} -1.92415 q^{33} -12.1648 q^{34} +9.40521 q^{35} -6.51051 q^{36} +8.85165 q^{37} +10.1611 q^{38} -3.45167 q^{39} +2.23948 q^{40} -7.23610 q^{41} -3.76528 q^{42} -9.53395 q^{43} +8.76381 q^{44} +7.48054 q^{45} -12.7628 q^{46} -3.47526 q^{47} +1.61036 q^{48} +4.73649 q^{49} -5.31521 q^{50} -3.04601 q^{51} +15.7211 q^{52} -7.04225 q^{53} -6.29199 q^{54} -10.0696 q^{55} +2.79458 q^{56} +2.54431 q^{57} +2.09508 q^{58} -2.99536 q^{59} +3.44118 q^{60} +3.59675 q^{61} -7.76479 q^{62} +9.33476 q^{63} -10.7526 q^{64} -18.0634 q^{65} +4.03125 q^{66} +1.24113 q^{67} +13.8735 q^{68} -3.19575 q^{69} -19.7046 q^{70} -7.71252 q^{71} +2.22270 q^{72} -5.23194 q^{73} -18.5449 q^{74} -1.33091 q^{75} -11.5884 q^{76} -12.5655 q^{77} +7.23151 q^{78} +13.0490 q^{79} +8.42740 q^{80} +6.59890 q^{81} +15.1602 q^{82} -13.1192 q^{83} +4.29415 q^{84} -15.9406 q^{85} +19.9744 q^{86} +0.524599 q^{87} -2.99199 q^{88} -12.8597 q^{89} -15.6723 q^{90} -22.5409 q^{91} +14.5554 q^{92} -1.94427 q^{93} +7.28093 q^{94} +13.3150 q^{95} -4.22969 q^{96} +4.86548 q^{97} -9.92332 q^{98} -9.99415 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) $ 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * 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 $

$2$	−2.09508	−1.48144	−0.740722	−	0.671811i	$-0.765517\pi$
$2$	−2.09508	−1.48144	−0.740722	+	0.671811i	$0.765517\pi$
$3$	−0.524599	−0.302877	−0.151439	−	0.988467i	$-0.548391\pi$
$3$	−0.524599	−0.302877	−0.151439	+	0.988467i	$0.548391\pi$
$4$	2.38936	1.19468
$5$	−2.74536	−1.22776	−0.613881	−	0.789399i	$-0.710393\pi$
$5$	−2.74536	−1.22776	−0.613881	+	0.789399i	$0.710393\pi$
$6$	1.09908	0.448696
$7$	−3.42586	−1.29485	−0.647426	−	0.762128i	$-0.724154\pi$
$7$	−3.42586	−1.29485	−0.647426	+	0.762128i	$0.724154\pi$
$8$	−0.815732	−0.288405
$9$	−2.72480	−0.908265
$10$	5.75174	1.81886
$11$	3.66785	1.10590	0.552950	−	0.833215i	$-0.313502\pi$
$11$	3.66785	1.10590	0.552950	+	0.833215i	$0.313502\pi$
$12$	−1.25345	−0.361841
$13$	6.57963	1.82486	0.912430	−	0.409232i	$-0.134203\pi$
$13$	6.57963	1.82486	0.912430	+	0.409232i	$0.134203\pi$
$14$	7.17744	1.91825
$15$	1.44021	0.371861
$16$	−3.06969	−0.767422
$17$	5.80637	1.40825	0.704125	−	0.710076i	$-0.251339\pi$
$17$	5.80637	1.40825	0.704125	+	0.710076i	$0.251339\pi$
$18$	5.70866	1.34554
$19$	−4.85000	−1.11267	−0.556333	−	0.830959i	$-0.687792\pi$
$19$	−4.85000	−1.11267	−0.556333	+	0.830959i	$0.687792\pi$
$20$	−6.55964	−1.46678
$21$	1.79720	0.392181
$22$	−7.68444	−1.63833
$23$	6.09179	1.27023	0.635113	−	0.772420i	$-0.280954\pi$
$23$	6.09179	1.27023	0.635113	+	0.772420i	$0.280954\pi$
$24$	0.427932	0.0873513
$25$	2.53700	0.507400
$26$	−13.7848	−2.70343
$27$	3.00322	0.577970
$28$	−8.18559	−1.54693
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.01736	−0.550892
$31$	3.70620	0.665654	0.332827	−	0.942988i	$-0.391998\pi$
$31$	3.70620	0.665654	0.332827	+	0.942988i	$0.391998\pi$
$32$	8.06271	1.42530
$33$	−1.92415	−0.334952
$34$	−12.1648	−2.08625
$35$	9.40521	1.58977
$36$	−6.51051	−1.08508
$37$	8.85165	1.45520	0.727601	−	0.686001i	$-0.240635\pi$
$37$	8.85165	1.45520	0.727601	+	0.686001i	$0.240635\pi$
$38$	10.1611	1.64835
$39$	−3.45167	−0.552709
$40$	2.23948	0.354093
$41$	−7.23610	−1.13009	−0.565045	−	0.825060i	$-0.691141\pi$
$41$	−7.23610	−1.13009	−0.565045	+	0.825060i	$0.691141\pi$
$42$	−3.76528	−0.580995
$43$	−9.53395	−1.45391	−0.726957	−	0.686683i	$-0.759066\pi$
$43$	−9.53395	−1.45391	−0.726957	+	0.686683i	$0.759066\pi$
$44$	8.76381	1.32119
$45$	7.48054	1.11513
$46$	−12.7628	−1.88177
$47$	−3.47526	−0.506918	−0.253459	−	0.967346i	$-0.581568\pi$
$47$	−3.47526	−0.506918	−0.253459	+	0.967346i	$0.581568\pi$
$48$	1.61036	0.232435
$49$	4.73649	0.676641
$50$	−5.31521	−0.751684
$51$	−3.04601	−0.426527
$52$	15.7211	2.18012
$53$	−7.04225	−0.967327	−0.483664	−	0.875254i	$-0.660694\pi$
$53$	−7.04225	−0.967327	−0.483664	+	0.875254i	$0.660694\pi$
$54$	−6.29199	−0.856231
$55$	−10.0696	−1.35778
$56$	2.79458	0.373442
$57$	2.54431	0.337001
$58$	2.09508	0.275097
$59$	−2.99536	−0.389962	−0.194981	−	0.980807i	$-0.562465\pi$
$59$	−2.99536	−0.389962	−0.194981	+	0.980807i	$0.562465\pi$
$60$	3.44118	0.444255
$61$	3.59675	0.460517	0.230259	−	0.973129i	$-0.426043\pi$
$61$	3.59675	0.460517	0.230259	+	0.973129i	$0.426043\pi$
$62$	−7.76479	−0.986129
$63$	9.33476	1.17607
$64$	−10.7526	−1.34408
$65$	−18.0634	−2.24049
$66$	4.03125	0.496213
$67$	1.24113	0.151628	0.0758139	−	0.997122i	$-0.475845\pi$
$67$	1.24113	0.151628	0.0758139	+	0.997122i	$0.475845\pi$
$68$	13.8735	1.68241
$69$	−3.19575	−0.384722
$70$	−19.7046	−2.35516
$71$	−7.71252	−0.915308	−0.457654	−	0.889130i	$-0.651310\pi$
$71$	−7.71252	−0.915308	−0.457654	+	0.889130i	$0.651310\pi$
$72$	2.22270	0.261948
$73$	−5.23194	−0.612352	−0.306176	−	0.951975i	$-0.599050\pi$
$73$	−5.23194	−0.612352	−0.306176	+	0.951975i	$0.599050\pi$
$74$	−18.5449	−2.15580
$75$	−1.33091	−0.153680
$76$	−11.5884	−1.32928
$77$	−12.5655	−1.43198
$78$	7.23151	0.818808
$79$	13.0490	1.46813	0.734063	−	0.679081i	$-0.237622\pi$
$79$	13.0490	1.46813	0.734063	+	0.679081i	$0.237622\pi$
$80$	8.42740	0.942212
$81$	6.59890	0.733211
$82$	15.1602	1.67416
$83$	−13.1192	−1.44002	−0.720011	−	0.693963i	$-0.755863\pi$
$83$	−13.1192	−1.44002	−0.720011	+	0.693963i	$0.755863\pi$
$84$	4.29415	0.468531
$85$	−15.9406	−1.72900
$86$	19.9744	2.15389
$87$	0.524599	0.0562429
$88$	−2.99199	−0.318947
$89$	−12.8597	−1.36312	−0.681560	−	0.731762i	$-0.738698\pi$
$89$	−12.8597	−1.36312	−0.681560	+	0.731762i	$0.738698\pi$
$90$	−15.6723	−1.65201
$91$	−22.5409	−2.36292
$92$	14.5554	1.51751
$93$	−1.94427	−0.201612
$94$	7.28093	0.750971
$95$	13.3150	1.36609
$96$	−4.22969	−0.431691
$97$	4.86548	0.494015	0.247007	−	0.969014i	$-0.420553\pi$
$97$	4.86548	0.494015	0.247007	+	0.969014i	$0.420553\pi$
$98$	−9.92332	−1.00241
$99$	−9.99415	−1.00445
$100$	6.06179	0.606179
$101$	18.8933	1.87995	0.939976	−	0.341239i	$-0.110847\pi$
$101$	18.8933	1.87995	0.939976	+	0.341239i	$0.110847\pi$
$102$	6.38164	0.631877
$103$	−17.0282	−1.67783	−0.838917	−	0.544260i	$-0.816811\pi$
$103$	−17.0282	−1.67783	−0.838917	+	0.544260i	$0.816811\pi$
$104$	−5.36721	−0.526299
$105$	−4.93396	−0.481505
$106$	14.7541	1.43304
$107$	−4.93496	−0.477080	−0.238540	−	0.971133i	$-0.576669\pi$
$107$	−4.93496	−0.477080	−0.238540	+	0.971133i	$0.576669\pi$
$108$	7.17577	0.690489
$109$	12.4119	1.18885	0.594424	−	0.804152i	$-0.297380\pi$
$109$	12.4119	1.18885	0.594424	+	0.804152i	$0.297380\pi$
$110$	21.0966	2.01148
$111$	−4.64357	−0.440748
$112$	10.5163	0.993698
$113$	1.64903	0.155128	0.0775639	−	0.996987i	$-0.475286\pi$
$113$	1.64903	0.155128	0.0775639	+	0.996987i	$0.475286\pi$
$114$	−5.33052	−0.499249
$115$	−16.7241	−1.55953
$116$	−2.38936	−0.221846
$117$	−17.9281	−1.65746
$118$	6.27551	0.577707
$119$	−19.8918	−1.82348
$120$	−1.17483	−0.107247
$121$	2.45315	0.223013
$122$	−7.53548	−0.682231
$123$	3.79605	0.342279
$124$	8.85544	0.795242
$125$	6.76183	0.604796
$126$	−19.5571	−1.74228
$127$	13.5665	1.20383	0.601916	−	0.798559i	$-0.294404\pi$
$127$	13.5665	1.20383	0.601916	+	0.798559i	$0.294404\pi$
$128$	6.40219	0.565879
$129$	5.00150	0.440357
$130$	37.8443	3.31917
$131$	21.1845	1.85090	0.925451	−	0.378868i	$-0.123687\pi$
$131$	21.1845	1.85090	0.925451	+	0.378868i	$0.123687\pi$
$132$	−4.59749	−0.400160
$133$	16.6154	1.44074
$134$	−2.60026	−0.224628
$135$	−8.24493	−0.709610
$136$	−4.73644	−0.406146
$137$	4.98542	0.425933	0.212966	−	0.977060i	$-0.431687\pi$
$137$	4.98542	0.425933	0.212966	+	0.977060i	$0.431687\pi$
$138$	6.69534	0.569945
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	22.4724	1.89926
$141$	1.82312	0.153534
$142$	16.1583	1.35598
$143$	24.1331	2.01811
$144$	8.36428	0.697023
$145$	2.74536	0.227990
$146$	10.9613	0.907166
$147$	−2.48476	−0.204939
$148$	21.1497	1.73850
$149$	−6.13438	−0.502548	−0.251274	−	0.967916i	$-0.580850\pi$
$149$	−6.13438	−0.502548	−0.251274	+	0.967916i	$0.580850\pi$
$150$	2.78836	0.227668
$151$	−14.2895	−1.16286	−0.581431	−	0.813596i	$-0.697507\pi$
$151$	−14.2895	−1.16286	−0.581431	+	0.813596i	$0.697507\pi$
$152$	3.95630	0.320898
$153$	−15.8212	−1.27907
$154$	26.3258	2.12139
$155$	−10.1749	−0.817264
$156$	−8.24726	−0.660309
$157$	17.0003	1.35677	0.678387	−	0.734705i	$-0.262679\pi$
$157$	17.0003	1.35677	0.678387	+	0.734705i	$0.262679\pi$
$158$	−27.3387	−2.17495
$159$	3.69436	0.292982
$160$	−22.1350	−1.74993
$161$	−20.8696	−1.64475
$162$	−13.8252	−1.08621
$163$	7.34366	0.575200	0.287600	−	0.957751i	$-0.407143\pi$
$163$	7.34366	0.575200	0.287600	+	0.957751i	$0.407143\pi$
$164$	−17.2896	−1.35009
$165$	5.28249	0.411241
$166$	27.4858	2.13331
$167$	22.2659	1.72298	0.861492	−	0.507770i	$-0.169530\pi$
$167$	22.2659	1.72298	0.861492	+	0.507770i	$0.169530\pi$
$168$	−1.46603	−0.113107
$169$	30.2915	2.33012
$170$	33.3967	2.56141
$171$	13.2153	1.01060
$172$	−22.7800	−1.73696
$173$	−18.3860	−1.39786	−0.698930	−	0.715190i	$-0.746340\pi$
$173$	−18.3860	−1.39786	−0.698930	+	0.715190i	$0.746340\pi$
$174$	−1.09908	−0.0833208
$175$	−8.69139	−0.657007
$176$	−11.2592	−0.848692
$177$	1.57136	0.118111
$178$	26.9420	2.01939
$179$	−16.5204	−1.23479	−0.617397	−	0.786652i	$-0.711813\pi$
$179$	−16.5204	−1.23479	−0.617397	+	0.786652i	$0.711813\pi$
$180$	17.8737	1.33223
$181$	−10.6733	−0.793344	−0.396672	−	0.917960i	$-0.629835\pi$
$181$	−10.6733	−0.793344	−0.396672	+	0.917960i	$0.629835\pi$
$182$	47.2249	3.50054
$183$	−1.88685	−0.139480
$184$	−4.96927	−0.366339
$185$	−24.3010	−1.78664
$186$	4.07340	0.298676
$187$	21.2969	1.55738
$188$	−8.30362	−0.605604
$189$	−10.2886	−0.748386
$190$	−27.8960	−2.02379
$191$	10.9092	0.789363	0.394682	−	0.918818i	$-0.370855\pi$
$191$	10.9092	0.789363	0.394682	+	0.918818i	$0.370855\pi$
$192$	5.64082	0.407091
$193$	7.47944	0.538382	0.269191	−	0.963087i	$-0.413244\pi$
$193$	7.47944	0.538382	0.269191	+	0.963087i	$0.413244\pi$
$194$	−10.1936	−0.731856
$195$	9.47607	0.678595
$196$	11.3172	0.808368
$197$	−9.55699	−0.680907	−0.340454	−	0.940261i	$-0.610581\pi$
$197$	−9.55699	−0.680907	−0.340454	+	0.940261i	$0.610581\pi$
$198$	20.9385	1.48804
$199$	11.5447	0.818382	0.409191	−	0.912449i	$-0.365811\pi$
$199$	11.5447	0.818382	0.409191	+	0.912449i	$0.365811\pi$
$200$	−2.06951	−0.146337
$201$	−0.651094	−0.0459246
$202$	−39.5829	−2.78505
$203$	3.42586	0.240448
$204$	−7.27801	−0.509563
$205$	19.8657	1.38748
$206$	35.6753	2.48562
$207$	−16.5989	−1.15370
$208$	−20.1974	−1.40044
$209$	−17.7891	−1.23050
$210$	10.3370	0.713324
$211$	10.3426	0.712016	0.356008	−	0.934483i	$-0.384138\pi$
$211$	10.3426	0.712016	0.356008	+	0.934483i	$0.384138\pi$
$212$	−16.8264	−1.15565
$213$	4.04598	0.277226
$214$	10.3391	0.706768
$215$	26.1741	1.78506
$216$	−2.44983	−0.166689
$217$	−12.6969	−0.861923
$218$	−26.0040	−1.76121
$219$	2.74467	0.185468
$220$	−24.0598	−1.62211
$221$	38.2037	2.56986
$222$	9.72864	0.652944
$223$	−6.98023	−0.467431	−0.233715	−	0.972305i	$-0.575088\pi$
$223$	−6.98023	−0.467431	−0.233715	+	0.972305i	$0.575088\pi$
$224$	−27.6217	−1.84555
$225$	−6.91280	−0.460853
$226$	−3.45485	−0.229813
$227$	−18.1636	−1.20556	−0.602782	−	0.797906i	$-0.705941\pi$
$227$	−18.1636	−1.20556	−0.602782	+	0.797906i	$0.705941\pi$
$228$	6.07925	0.402608
$229$	−4.11369	−0.271840	−0.135920	−	0.990720i	$-0.543399\pi$
$229$	−4.11369	−0.271840	−0.135920	+	0.990720i	$0.543399\pi$
$230$	35.0384	2.31036
$231$	6.59187	0.433713
$232$	0.815732	0.0535554
$233$	−9.63681	−0.631329	−0.315664	−	0.948871i	$-0.602227\pi$
$233$	−9.63681	−0.631329	−0.315664	+	0.948871i	$0.602227\pi$
$234$	37.5609	2.45543
$235$	9.54082	0.622375
$236$	−7.15697	−0.465879
$237$	−6.84549	−0.444662
$238$	41.6748	2.70138
$239$	−17.2543	−1.11609	−0.558045	−	0.829811i	$-0.688448\pi$
$239$	−17.2543	−1.11609	−0.558045	+	0.829811i	$0.688448\pi$
$240$	−4.42101	−0.285375
$241$	12.7950	0.824197	0.412099	−	0.911139i	$-0.364796\pi$
$241$	12.7950	0.824197	0.412099	+	0.911139i	$0.364796\pi$
$242$	−5.13954	−0.330382
$243$	−12.4714	−0.800044
$244$	8.59393	0.550170
$245$	−13.0034	−0.830754
$246$	−7.95303	−0.507067
$247$	−31.9112	−2.03046
$248$	−3.02327	−0.191978
$249$	6.88233	0.436150
$250$	−14.1666	−0.895972
$251$	−13.4567	−0.849380	−0.424690	−	0.905339i	$-0.639617\pi$
$251$	−13.4567	−0.849380	−0.424690	+	0.905339i	$0.639617\pi$
$252$	22.3041	1.40502
$253$	22.3438	1.40474
$254$	−28.4229	−1.78341
$255$	8.36240	0.523674
$256$	8.09215	0.505760
$257$	−21.0901	−1.31556	−0.657782	−	0.753209i	$-0.728505\pi$
$257$	−21.0901	−1.31556	−0.657782	+	0.753209i	$0.728505\pi$
$258$	−10.4785	−0.652365
$259$	−30.3245	−1.88427
$260$	−43.1600	−2.67667
$261$	2.72480	0.168661
$262$	−44.3833	−2.74201
$263$	−10.0591	−0.620269	−0.310134	−	0.950693i	$-0.600374\pi$
$263$	−10.0591	−0.620269	−0.310134	+	0.950693i	$0.600374\pi$
$264$	1.56959	0.0966018
$265$	19.3335	1.18765
$266$	−34.8106	−2.13437
$267$	6.74616	0.412858
$268$	2.96550	0.181146
$269$	−5.64339	−0.344084	−0.172042	−	0.985090i	$-0.555036\pi$
$269$	−5.64339	−0.344084	−0.172042	+	0.985090i	$0.555036\pi$
$270$	17.2738	1.05125
$271$	−12.4424	−0.755821	−0.377911	−	0.925842i	$-0.623357\pi$
$271$	−12.4424	−0.755821	−0.377911	+	0.925842i	$0.623357\pi$
$272$	−17.8237	−1.08072
$273$	11.8249	0.715676
$274$	−10.4448	−0.630996
$275$	9.30534	0.561133
$276$	−7.63577	−0.459620
$277$	4.24273	0.254921	0.127460	−	0.991844i	$-0.459317\pi$
$277$	4.24273	0.254921	0.127460	+	0.991844i	$0.459317\pi$
$278$	−2.09508	−0.125654
$279$	−10.0986	−0.604590
$280$	−7.67213	−0.458497
$281$	21.5980	1.28843	0.644214	−	0.764845i	$-0.277184\pi$
$281$	21.5980	1.28843	0.644214	+	0.764845i	$0.277184\pi$
$282$	−3.81957	−0.227452
$283$	19.3516	1.15033	0.575167	−	0.818036i	$-0.304937\pi$
$283$	19.3516	1.15033	0.575167	+	0.818036i	$0.304937\pi$
$284$	−18.4280	−1.09350
$285$	−6.98503	−0.413758
$286$	−50.5608	−2.98972
$287$	24.7898	1.46330
$288$	−21.9692	−1.29455
$289$	16.7139	0.983170
$290$	−5.75174	−0.337754
$291$	−2.55243	−0.149626
$292$	−12.5010	−0.731564
$293$	−6.68789	−0.390711	−0.195355	−	0.980733i	$-0.562586\pi$
$293$	−6.68789	−0.390711	−0.195355	+	0.980733i	$0.562586\pi$
$294$	5.20576	0.303606
$295$	8.22333	0.478781
$296$	−7.22057	−0.419687
$297$	11.0154	0.639177
$298$	12.8520	0.744497
$299$	40.0817	2.31798
$300$	−3.18001	−0.183598
$301$	32.6619	1.88260
$302$	29.9376	1.72272
$303$	−9.91140	−0.569395
$304$	14.8880	0.853885
$305$	−9.87438	−0.565405
$306$	33.1466	1.89486
$307$	−3.33975	−0.190609	−0.0953047	−	0.995448i	$-0.530383\pi$
$307$	−3.33975	−0.190609	−0.0953047	+	0.995448i	$0.530383\pi$
$308$	−30.0235	−1.71075
$309$	8.93295	0.508178
$310$	21.3171	1.21073
$311$	18.8033	1.06624	0.533118	−	0.846041i	$-0.321020\pi$
$311$	18.8033	1.06624	0.533118	+	0.846041i	$0.321020\pi$
$312$	2.81564	0.159404
$313$	2.93970	0.166162	0.0830810	−	0.996543i	$-0.473524\pi$
$313$	2.93970	0.166162	0.0830810	+	0.996543i	$0.473524\pi$
$314$	−35.6170	−2.00998
$315$	−25.6273	−1.44393
$316$	31.1787	1.75394
$317$	5.88949	0.330787	0.165393	−	0.986228i	$-0.447111\pi$
$317$	5.88949	0.330787	0.165393	+	0.986228i	$0.447111\pi$
$318$	−7.73997	−0.434036
$319$	−3.66785	−0.205360
$320$	29.5198	1.65021
$321$	2.58887	0.144497
$322$	43.7234	2.43661
$323$	−28.1609	−1.56691
$324$	15.7671	0.875951
$325$	16.6925	0.925934
$326$	−15.3855	−0.852127
$327$	−6.51129	−0.360075
$328$	5.90272	0.325923
$329$	11.9057	0.656384
$330$	−11.0672	−0.609231
$331$	17.9389	0.986013	0.493007	−	0.870026i	$-0.335898\pi$
$331$	17.9389	0.986013	0.493007	+	0.870026i	$0.335898\pi$
$332$	−31.3465	−1.72036
$333$	−24.1189	−1.32171
$334$	−46.6488	−2.55251
$335$	−3.40734	−0.186163
$336$	−5.51685	−0.300969
$337$	5.13027	0.279464	0.139732	−	0.990189i	$-0.455376\pi$
$337$	5.13027	0.279464	0.139732	+	0.990189i	$0.455376\pi$
$338$	−63.4631	−3.45194
$339$	−0.865080	−0.0469847
$340$	−38.0877	−2.06559
$341$	13.5938	0.736146
$342$	−27.6870	−1.49714
$343$	7.75446	0.418702
$344$	7.77715	0.419316
$345$	8.77347	0.472348
$346$	38.5201	2.07085
$347$	−20.1732	−1.08295	−0.541476	−	0.840716i	$-0.682134\pi$
$347$	−20.1732	−1.08295	−0.541476	+	0.840716i	$0.682134\pi$
$348$	1.25345	0.0671922
$349$	4.46243	0.238868	0.119434	−	0.992842i	$-0.461892\pi$
$349$	4.46243	0.238868	0.119434	+	0.992842i	$0.461892\pi$
$350$	18.2091	0.973320
$351$	19.7601	1.05472
$352$	29.5728	1.57624
$353$	18.1175	0.964295	0.482148	−	0.876090i	$-0.339857\pi$
$353$	18.1175	0.964295	0.482148	+	0.876090i	$0.339857\pi$
$354$	−3.29213	−0.174975
$355$	21.1736	1.12378
$356$	−30.7263	−1.62849
$357$	10.4352	0.552290
$358$	34.6116	1.82928
$359$	−14.9192	−0.787407	−0.393704	−	0.919237i	$-0.628806\pi$
$359$	−14.9192	−0.787407	−0.393704	+	0.919237i	$0.628806\pi$
$360$	−6.10212	−0.321610
$361$	4.52249	0.238026
$362$	22.3615	1.17529
$363$	−1.28692	−0.0675457
$364$	−53.8581	−2.82293
$365$	14.3636	0.751823
$366$	3.95311	0.206632
$367$	3.68677	0.192448	0.0962239	−	0.995360i	$-0.469324\pi$
$367$	3.68677	0.192448	0.0962239	+	0.995360i	$0.469324\pi$
$368$	−18.6999	−0.974799
$369$	19.7169	1.02642
$370$	50.9124	2.64681
$371$	24.1257	1.25255
$372$	−4.64556	−0.240861
$373$	6.02644	0.312037	0.156019	−	0.987754i	$-0.450134\pi$
$373$	6.02644	0.312037	0.156019	+	0.987754i	$0.450134\pi$
$374$	−44.6187	−2.30718
$375$	−3.54725	−0.183179
$376$	2.83488	0.146198
$377$	−6.57963	−0.338868
$378$	21.5554	1.10869
$379$	14.2331	0.731106	0.365553	−	0.930790i	$-0.380880\pi$
$379$	14.2331	0.731106	0.365553	+	0.930790i	$0.380880\pi$
$380$	31.8143	1.63204
$381$	−7.11698	−0.364614
$382$	−22.8557	−1.16940
$383$	14.4677	0.739265	0.369633	−	0.929178i	$-0.379484\pi$
$383$	14.4677	0.739265	0.369633	+	0.929178i	$0.379484\pi$
$384$	−3.35858	−0.171392
$385$	34.4969	1.75813
$386$	−15.6700	−0.797583
$387$	25.9781	1.32054
$388$	11.6254	0.590189
$389$	−14.9399	−0.757483	−0.378742	−	0.925502i	$-0.623643\pi$
$389$	−14.9399	−0.757483	−0.378742	+	0.925502i	$0.623643\pi$
$390$	−19.8531	−1.00530
$391$	35.3711	1.78880
$392$	−3.86371	−0.195147
$393$	−11.1134	−0.560596
$394$	20.0226	1.00873
$395$	−35.8242	−1.80251
$396$	−23.8796	−1.19999
$397$	16.6561	0.835947	0.417974	−	0.908459i	$-0.362740\pi$
$397$	16.6561	0.835947	0.417974	+	0.908459i	$0.362740\pi$
$398$	−24.1871	−1.21239
$399$	−8.71642	−0.436367
$400$	−7.78780	−0.389390
$401$	34.5729	1.72649	0.863243	−	0.504788i	$-0.168429\pi$
$401$	34.5729	1.72649	0.863243	+	0.504788i	$0.168429\pi$
$402$	1.36409	0.0680348
$403$	24.3854	1.21473
$404$	45.1428	2.24594
$405$	−18.1164	−0.900209
$406$	−7.17744	−0.356210
$407$	32.4665	1.60931
$408$	2.48473	0.123013
$409$	0.458666	0.0226796	0.0113398	−	0.999936i	$-0.496390\pi$
$409$	0.458666	0.0226796	0.0113398	+	0.999936i	$0.496390\pi$
$410$	−41.6202	−2.05548
$411$	−2.61534	−0.129005
$412$	−40.6863	−2.00447
$413$	10.2617	0.504943
$414$	34.7759	1.70914
$415$	36.0170	1.76800
$416$	53.0496	2.60097
$417$	−0.524599	−0.0256897
$418$	37.2695	1.82291
$419$	13.4675	0.657932	0.328966	−	0.944342i	$-0.393300\pi$
$419$	13.4675	0.657932	0.328966	+	0.944342i	$0.393300\pi$
$420$	−11.7890	−0.575244
$421$	−2.60368	−0.126895	−0.0634477	−	0.997985i	$-0.520210\pi$
$421$	−2.60368	−0.126895	−0.0634477	+	0.997985i	$0.520210\pi$
$422$	−21.6686	−1.05481
$423$	9.46936	0.460416
$424$	5.74459	0.278982
$425$	14.7307	0.714546
$426$	−8.47665	−0.410695
$427$	−12.3220	−0.596301
$428$	−11.7914	−0.569958
$429$	−12.6602	−0.611241
$430$	−54.8368	−2.64447
$431$	−19.4058	−0.934743	−0.467372	−	0.884061i	$-0.654799\pi$
$431$	−19.4058	−0.934743	−0.467372	+	0.884061i	$0.654799\pi$
$432$	−9.21896	−0.443547
$433$	38.4395	1.84728	0.923642	−	0.383256i	$-0.125197\pi$
$433$	38.4395	1.84728	0.923642	+	0.383256i	$0.125197\pi$
$434$	26.6010	1.27689
$435$	−1.44021	−0.0690529
$436$	29.6566	1.42029
$437$	−29.5452	−1.41334
$438$	−5.75030	−0.274760
$439$	−19.3479	−0.923426	−0.461713	−	0.887029i	$-0.652765\pi$
$439$	−19.3479	−0.923426	−0.461713	+	0.887029i	$0.652765\pi$
$440$	8.21408	0.391591
$441$	−12.9060	−0.614570
$442$	−80.0398	−3.80711
$443$	12.4193	0.590059	0.295030	−	0.955488i	$-0.404671\pi$
$443$	12.4193	0.590059	0.295030	+	0.955488i	$0.404671\pi$
$444$	−11.0951	−0.526552
$445$	35.3044	1.67359
$446$	14.6241	0.692473
$447$	3.21809	0.152210
$448$	36.8370	1.74038
$449$	−10.8485	−0.511972	−0.255986	−	0.966680i	$-0.582400\pi$
$449$	−10.8485	−0.511972	−0.255986	+	0.966680i	$0.582400\pi$
$450$	14.4829	0.682729
$451$	−26.5410	−1.24977
$452$	3.94012	0.185328
$453$	7.49625	0.352205
$454$	38.0543	1.78598
$455$	61.8828	2.90111
$456$	−2.07547	−0.0971929
$457$	−33.1767	−1.55194	−0.775969	−	0.630770i	$-0.782739\pi$
$457$	−33.1767	−1.55194	−0.775969	+	0.630770i	$0.782739\pi$
$458$	8.61850	0.402716
$459$	17.4378	0.813927
$460$	−39.9599	−1.86314
$461$	−20.5850	−0.958737	−0.479368	−	0.877614i	$-0.659134\pi$
$461$	−20.5850	−0.958737	−0.479368	+	0.877614i	$0.659134\pi$
$462$	−13.8105	−0.642522
$463$	−30.6323	−1.42361	−0.711803	−	0.702380i	$-0.752121\pi$
$463$	−30.6323	−1.42361	−0.711803	+	0.702380i	$0.752121\pi$
$464$	3.06969	0.142507
$465$	5.33772	0.247531
$466$	20.1899	0.935278
$467$	−3.56347	−0.164898	−0.0824488	−	0.996595i	$-0.526274\pi$
$467$	−3.56347	−0.164898	−0.0824488	+	0.996595i	$0.526274\pi$
$468$	−42.8367	−1.98013
$469$	−4.25192	−0.196336
$470$	−19.9888	−0.922014
$471$	−8.91835	−0.410936
$472$	2.44341	0.112467
$473$	−34.9691	−1.60788
$474$	14.3418	0.658742
$475$	−12.3044	−0.564566
$476$	−47.5285	−2.17847
$477$	19.1887	0.878590
$478$	36.1492	1.65343
$479$	−25.0081	−1.14265	−0.571324	−	0.820724i	$-0.693570\pi$
$479$	−25.0081	−1.14265	−0.571324	+	0.820724i	$0.693570\pi$
$480$	11.6120	0.530013
$481$	58.2406	2.65554
$482$	−26.8065	−1.22100
$483$	10.9482	0.498159
$484$	5.86144	0.266429
$485$	−13.3575	−0.606533
$486$	26.1287	1.18522
$487$	23.5567	1.06745	0.533727	−	0.845657i	$-0.320791\pi$
$487$	23.5567	1.06745	0.533727	+	0.845657i	$0.320791\pi$
$488$	−2.93399	−0.132815
$489$	−3.85248	−0.174215
$490$	27.2431	1.23072
$491$	34.7856	1.56985	0.784927	−	0.619588i	$-0.212700\pi$
$491$	34.7856	1.56985	0.784927	+	0.619588i	$0.212700\pi$
$492$	9.07012	0.408913
$493$	−5.80637	−0.261506
$494$	66.8565	3.00801
$495$	27.4375	1.23323
$496$	−11.3769	−0.510838
$497$	26.4220	1.18519
$498$	−14.4190	−0.646132
$499$	−8.69300	−0.389152	−0.194576	−	0.980887i	$-0.562333\pi$
$499$	−8.69300	−0.389152	−0.194576	+	0.980887i	$0.562333\pi$
$500$	16.1564	0.722537
$501$	−11.6807	−0.521853
$502$	28.1929	1.25831
$503$	−15.4916	−0.690737	−0.345368	−	0.938467i	$-0.612246\pi$
$503$	−15.4916	−0.690737	−0.345368	+	0.938467i	$0.612246\pi$
$504$	−7.61466	−0.339184
$505$	−51.8689	−2.30813
$506$	−46.8120	−2.08105
$507$	−15.8909	−0.705739
$508$	32.4152	1.43819
$509$	37.5875	1.66604	0.833019	−	0.553244i	$-0.186610\pi$
$509$	37.5875	1.66604	0.833019	+	0.553244i	$0.186610\pi$
$510$	−17.5199	−0.775794
$511$	17.9239	0.792906
$512$	−29.7581	−1.31513
$513$	−14.5656	−0.643088
$514$	44.1854	1.94893
$515$	46.7484	2.05998
$516$	11.9504	0.526085
$517$	−12.7467	−0.560600
$518$	63.5322	2.79144
$519$	9.64527	0.423380
$520$	14.7349	0.646170
$521$	11.2405	0.492457	0.246228	−	0.969212i	$-0.420809\pi$
$521$	11.2405	0.492457	0.246228	+	0.969212i	$0.420809\pi$
$522$	−5.70866	−0.249861
$523$	12.0190	0.525553	0.262777	−	0.964857i	$-0.415362\pi$
$523$	12.0190	0.525553	0.262777	+	0.964857i	$0.415362\pi$
$524$	50.6174	2.21123
$525$	4.55950	0.198993
$526$	21.0745	0.918894
$527$	21.5196	0.937407
$528$	5.90655	0.257050
$529$	14.1099	0.613472
$530$	−40.5052	−1.75943
$531$	8.16174	0.354189
$532$	39.7001	1.72122
$533$	−47.6109	−2.06226
$534$	−14.1337	−0.611627
$535$	13.5482	0.585741
$536$	−1.01243	−0.0437302
$537$	8.66659	0.373991
$538$	11.8233	0.509741
$539$	17.3727	0.748297
$540$	−19.7001	−0.847756
$541$	25.4863	1.09574	0.547871	−	0.836563i	$-0.315439\pi$
$541$	25.4863	1.09574	0.547871	+	0.836563i	$0.315439\pi$
$542$	26.0678	1.11971
$543$	5.59923	0.240286
$544$	46.8150	2.00718
$545$	−34.0752	−1.45962
$546$	−24.7741	−1.06023
$547$	−13.3020	−0.568753	−0.284377	−	0.958713i	$-0.591787\pi$
$547$	−13.3020	−0.568753	−0.284377	+	0.958713i	$0.591787\pi$
$548$	11.9119	0.508853
$549$	−9.80042	−0.418272
$550$	−19.4954	−0.831287
$551$	4.85000	0.206617
$552$	2.60687	0.110956
$553$	−44.7040	−1.90101
$554$	−8.88885	−0.377651
$555$	12.7483	0.541133
$556$	2.38936	0.101331
$557$	35.8345	1.51836	0.759178	−	0.650883i	$-0.225601\pi$
$557$	35.8345	1.51836	0.759178	+	0.650883i	$0.225601\pi$
$558$	21.1575	0.895667
$559$	−62.7298	−2.65319
$560$	−28.8711	−1.22002
$561$	−11.1723	−0.471696
$562$	−45.2495	−1.90874
$563$	22.3268	0.940963	0.470481	−	0.882410i	$-0.344080\pi$
$563$	22.3268	0.940963	0.470481	+	0.882410i	$0.344080\pi$
$564$	4.35607	0.183424
$565$	−4.52718	−0.190460
$566$	−40.5431	−1.70416
$567$	−22.6069	−0.949400
$568$	6.29135	0.263979
$569$	4.64125	0.194571	0.0972856	−	0.995257i	$-0.468984\pi$
$569$	4.64125	0.194571	0.0972856	+	0.995257i	$0.468984\pi$
$570$	14.6342	0.612959
$571$	44.4137	1.85866	0.929329	−	0.369254i	$-0.120387\pi$
$571$	44.4137	1.85866	0.929329	+	0.369254i	$0.120387\pi$
$572$	57.6626	2.41099
$573$	−5.72297	−0.239080
$574$	−51.9367	−2.16780
$575$	15.4548	0.644512
$576$	29.2987	1.22078
$577$	−16.3415	−0.680305	−0.340152	−	0.940370i	$-0.610479\pi$
$577$	−16.3415	−0.680305	−0.340152	+	0.940370i	$0.610479\pi$
$578$	−35.0169	−1.45651
$579$	−3.92371	−0.163064
$580$	6.55964	0.272374
$581$	44.9446	1.86461
$582$	5.34754	0.221663
$583$	−25.8299	−1.06977
$584$	4.26786	0.176605
$585$	49.2192	2.03496
$586$	14.0117	0.578816
$587$	33.2750	1.37341	0.686703	−	0.726938i	$-0.259057\pi$
$587$	33.2750	1.37341	0.686703	+	0.726938i	$0.259057\pi$
$588$	−5.93697	−0.244837
$589$	−17.9751	−0.740650
$590$	−17.2285	−0.709287
$591$	5.01359	0.206231
$592$	−27.1718	−1.11675
$593$	4.41411	0.181266	0.0906329	−	0.995884i	$-0.471111\pi$
$593$	4.41411	0.181266	0.0906329	+	0.995884i	$0.471111\pi$
$594$	−23.0781	−0.946906
$595$	54.6101	2.23879
$596$	−14.6572	−0.600383
$597$	−6.05634	−0.247869
$598$	−83.9743	−3.43396
$599$	−20.5239	−0.838584	−0.419292	−	0.907852i	$-0.637722\pi$
$599$	−20.5239	−0.838584	−0.419292	+	0.907852i	$0.637722\pi$
$600$	1.08566	0.0443220
$601$	15.2283	0.621175	0.310588	−	0.950545i	$-0.399474\pi$
$601$	15.2283	0.621175	0.310588	+	0.950545i	$0.399474\pi$
$602$	−68.4293	−2.78897
$603$	−3.38182	−0.137718
$604$	−34.1427	−1.38925
$605$	−6.73477	−0.273807
$606$	20.7652	0.843528
$607$	−21.6830	−0.880084	−0.440042	−	0.897977i	$-0.645037\pi$
$607$	−21.6830	−0.880084	−0.440042	+	0.897977i	$0.645037\pi$
$608$	−39.1041	−1.58588
$609$	−1.79720	−0.0728263
$610$	20.6876	0.837617
$611$	−22.8659	−0.925055
$612$	−37.8024	−1.52807
$613$	−17.5104	−0.707237	−0.353619	−	0.935390i	$-0.615049\pi$
$613$	−17.5104	−0.707237	−0.353619	+	0.935390i	$0.615049\pi$
$614$	6.99703	0.282377
$615$	−10.4215	−0.420237
$616$	10.2501	0.412989
$617$	−34.4839	−1.38827	−0.694136	−	0.719844i	$-0.744213\pi$
$617$	−34.4839	−1.38827	−0.694136	+	0.719844i	$0.744213\pi$
$618$	−18.7152	−0.752837
$619$	41.8666	1.68276	0.841380	−	0.540443i	$-0.181743\pi$
$619$	41.8666	1.68276	0.841380	+	0.540443i	$0.181743\pi$
$620$	−24.3114	−0.976368
$621$	18.2950	0.734153
$622$	−39.3944	−1.57957
$623$	44.0553	1.76504
$624$	10.5955	0.424161
$625$	−31.2486	−1.24995
$626$	−6.15891	−0.246160
$627$	9.33214	0.372690
$628$	40.6198	1.62091
$629$	51.3959	2.04929
$630$	53.6911	2.13911
$631$	−22.5350	−0.897103	−0.448552	−	0.893757i	$-0.648060\pi$
$631$	−22.5350	−0.897103	−0.448552	+	0.893757i	$0.648060\pi$
$632$	−10.6445	−0.423415
$633$	−5.42573	−0.215654
$634$	−12.3389	−0.490042
$635$	−37.2449	−1.47802
$636$	8.82714	0.350019
$637$	31.1643	1.23478
$638$	7.68444	0.304230
$639$	21.0150	0.831342
$640$	−17.5763	−0.694765
$641$	10.6875	0.422131	0.211066	−	0.977472i	$-0.432307\pi$
$641$	10.6875	0.422131	0.211066	+	0.977472i	$0.432307\pi$
$642$	−5.42390	−0.214064
$643$	9.09895	0.358827	0.179414	−	0.983774i	$-0.442580\pi$
$643$	9.09895	0.358827	0.179414	+	0.983774i	$0.442580\pi$
$644$	−49.8649	−1.96495
$645$	−13.7309	−0.540654
$646$	58.9992	2.32129
$647$	−41.9542	−1.64939	−0.824694	−	0.565579i	$-0.808653\pi$
$647$	−41.9542	−1.64939	−0.824694	+	0.565579i	$0.808653\pi$
$648$	−5.38293	−0.211462
$649$	−10.9865	−0.431259
$650$	−34.9721	−1.37172
$651$	6.66079	0.261057
$652$	17.5466	0.687179
$653$	43.1336	1.68795	0.843974	−	0.536384i	$-0.180210\pi$
$653$	43.1336	1.68795	0.843974	+	0.536384i	$0.180210\pi$
$654$	13.6417	0.533432
$655$	−58.1592	−2.27247
$656$	22.2126	0.867256
$657$	14.2560	0.556178
$658$	−24.9434	−0.972396
$659$	33.2616	1.29569	0.647845	−	0.761772i	$-0.275671\pi$
$659$	33.2616	1.29569	0.647845	+	0.761772i	$0.275671\pi$
$660$	12.6218	0.491301
$661$	7.04465	0.274005	0.137003	−	0.990571i	$-0.456253\pi$
$661$	7.04465	0.274005	0.137003	+	0.990571i	$0.456253\pi$
$662$	−37.5835	−1.46072
$663$	−20.0416	−0.778353
$664$	10.7018	0.415309
$665$	−45.6152	−1.76888
$666$	50.5311	1.95804
$667$	−6.09179	−0.235875
$668$	53.2011	2.05841
$669$	3.66182	0.141574
$670$	7.13865	0.275790
$671$	13.1924	0.509286
$672$	14.4903	0.558976
$673$	7.56586	0.291642	0.145821	−	0.989311i	$-0.453418\pi$
$673$	7.56586	0.291642	0.145821	+	0.989311i	$0.453418\pi$
$674$	−10.7483	−0.414010
$675$	7.61917	0.293262
$676$	72.3772	2.78374
$677$	3.45986	0.132973	0.0664867	−	0.997787i	$-0.478821\pi$
$677$	3.45986	0.132973	0.0664867	+	0.997787i	$0.478821\pi$
$678$	1.81241	0.0696052
$679$	−16.6684	−0.639676
$680$	13.0032	0.498651
$681$	9.52863	0.365138
$682$	−28.4801	−1.09056
$683$	42.9224	1.64238	0.821191	−	0.570654i	$-0.193310\pi$
$683$	42.9224	1.64238	0.821191	+	0.570654i	$0.193310\pi$
$684$	31.5760	1.20734
$685$	−13.6868	−0.522944
$686$	−16.2462	−0.620283
$687$	2.15804	0.0823342
$688$	29.2663	1.11577
$689$	−46.3354	−1.76524
$690$	−18.3811	−0.699757
$691$	9.02273	0.343241	0.171620	−	0.985163i	$-0.445100\pi$
$691$	9.02273	0.343241	0.171620	+	0.985163i	$0.445100\pi$
$692$	−43.9306	−1.66999
$693$	34.2385	1.30061
$694$	42.2644	1.60433
$695$	−2.74536	−0.104137
$696$	−0.427932	−0.0162207
$697$	−42.0155	−1.59145
$698$	−9.34915	−0.353870
$699$	5.05546	0.191215
$700$	−20.7668	−0.784912
$701$	38.1325	1.44024	0.720122	−	0.693848i	$-0.244086\pi$
$701$	38.1325	1.44024	0.720122	+	0.693848i	$0.244086\pi$
$702$	−41.3989	−1.56250
$703$	−42.9305	−1.61915
$704$	−39.4391	−1.48642
$705$	−5.00511	−0.188503
$706$	−37.9575	−1.42855
$707$	−64.7257	−2.43426
$708$	3.75454	0.141104
$709$	−1.23807	−0.0464965	−0.0232483	−	0.999730i	$-0.507401\pi$
$709$	−1.23807	−0.0464965	−0.0232483	+	0.999730i	$0.507401\pi$
$710$	−44.3605	−1.66482
$711$	−35.5558	−1.33345
$712$	10.4900	0.393131
$713$	22.5774	0.845530
$714$	−21.8626	−0.818187
$715$	−66.2541	−2.47776
$716$	−39.4731	−1.47518
$717$	9.05161	0.338039
$718$	31.2570	1.16650
$719$	49.3646	1.84099	0.920494	−	0.390756i	$-0.127786\pi$
$719$	49.3646	1.84099	0.920494	+	0.390756i	$0.127786\pi$
$720$	−22.9629	−0.855778
$721$	58.3360	2.17255
$722$	−9.47498	−0.352622
$723$	−6.71224	−0.249631
$724$	−25.5024	−0.947791
$725$	−2.53700	−0.0942217
$726$	2.69620	0.100065
$727$	9.79210	0.363169	0.181585	−	0.983375i	$-0.441877\pi$
$727$	9.79210	0.363169	0.181585	+	0.983375i	$0.441877\pi$
$728$	18.3873	0.681479
$729$	−13.2542	−0.490896
$730$	−30.0928	−1.11378
$731$	−55.3576	−2.04747
$732$	−4.50837	−0.166634
$733$	−26.1499	−0.965869	−0.482935	−	0.875656i	$-0.660429\pi$
$733$	−26.1499	−0.965869	−0.482935	+	0.875656i	$0.660429\pi$
$734$	−7.72408	−0.285101
$735$	6.82155	0.251617
$736$	49.1163	1.81045
$737$	4.55227	0.167685
$738$	−41.3085	−1.52059
$739$	−9.31072	−0.342500	−0.171250	−	0.985228i	$-0.554781\pi$
$739$	−9.31072	−0.342500	−0.171250	+	0.985228i	$0.554781\pi$
$740$	−58.0636	−2.13446
$741$	16.7406	0.614981
$742$	−50.5453	−1.85558
$743$	12.3741	0.453961	0.226980	−	0.973899i	$-0.427115\pi$
$743$	12.3741	0.453961	0.226980	+	0.973899i	$0.427115\pi$
$744$	1.58600	0.0581457
$745$	16.8411	0.617009
$746$	−12.6259	−0.462266
$747$	35.7472	1.30792
$748$	50.8859	1.86057
$749$	16.9065	0.617748
$750$	7.43176	0.271370
$751$	43.7995	1.59827	0.799133	−	0.601154i	$-0.205292\pi$
$751$	43.7995	1.59827	0.799133	+	0.601154i	$0.205292\pi$
$752$	10.6680	0.389020
$753$	7.05938	0.257258
$754$	13.7848	0.502014
$755$	39.2298	1.42772
$756$	−24.5831	−0.894080
$757$	−13.7437	−0.499523	−0.249761	−	0.968307i	$-0.580352\pi$
$757$	−13.7437	−0.499523	−0.249761	+	0.968307i	$0.580352\pi$
$758$	−29.8195	−1.08309
$759$	−11.7215	−0.425464
$760$	−10.8615	−0.393987
$761$	−33.5357	−1.21567	−0.607833	−	0.794065i	$-0.707961\pi$
$761$	−33.5357	−1.21567	−0.607833	+	0.794065i	$0.707961\pi$
$762$	14.9106	0.540155
$763$	−42.5215	−1.53938
$764$	26.0660	0.943035
$765$	43.4348	1.57039
$766$	−30.3110	−1.09518
$767$	−19.7083	−0.711627
$768$	−4.24514	−0.153183
$769$	47.9327	1.72850	0.864249	−	0.503064i	$-0.167794\pi$
$769$	47.9327	1.72850	0.864249	+	0.503064i	$0.167794\pi$
$770$	−72.2738	−2.60457
$771$	11.0638	0.398454
$772$	17.8710	0.643193
$773$	−44.3541	−1.59531	−0.797653	−	0.603116i	$-0.793926\pi$
$773$	−44.3541	−1.59531	−0.797653	+	0.603116i	$0.793926\pi$
$774$	−54.4261	−1.95631
$775$	9.40263	0.337753
$776$	−3.96893	−0.142476
$777$	15.9082	0.570703
$778$	31.3003	1.12217
$779$	35.0951	1.25741
$780$	22.6417	0.810703
$781$	−28.2884	−1.01224
$782$	−74.1053	−2.65000
$783$	−3.00322	−0.107326
$784$	−14.5395	−0.519270
$785$	−46.6720	−1.66580
$786$	23.2834	0.830492
$787$	−38.2673	−1.36408	−0.682041	−	0.731314i	$-0.738908\pi$
$787$	−38.2673	−1.36408	−0.682041	+	0.731314i	$0.738908\pi$
$788$	−22.8350	−0.813465
$789$	5.27698	0.187865
$790$	75.0545	2.67032
$791$	−5.64934	−0.200868
$792$	8.15255	0.289688
$793$	23.6653	0.840379
$794$	−34.8959	−1.23841
$795$	−10.1423	−0.359712
$796$	27.5844	0.977703
$797$	−15.9562	−0.565198	−0.282599	−	0.959238i	$-0.591197\pi$
$797$	−15.9562	−0.565198	−0.282599	+	0.959238i	$0.591197\pi$
$798$	18.2616	0.646453
$799$	−20.1786	−0.713868
$800$	20.4551	0.723196
$801$	35.0399	1.23807
$802$	−72.4329	−2.55769
$803$	−19.1900	−0.677200
$804$	−1.55570	−0.0548652
$805$	57.2945	2.01937
$806$	−51.0894	−1.79955
$807$	2.96052	0.104215
$808$	−15.4119	−0.542188
$809$	7.26456	0.255408	0.127704	−	0.991812i	$-0.459239\pi$
$809$	7.26456	0.255408	0.127704	+	0.991812i	$0.459239\pi$
$810$	37.9552	1.33361
$811$	−6.89645	−0.242167	−0.121083	−	0.992642i	$-0.538637\pi$
$811$	−6.89645	−0.242167	−0.121083	+	0.992642i	$0.538637\pi$
$812$	8.18559	0.287258
$813$	6.52727	0.228921
$814$	−68.0200	−2.38410
$815$	−20.1610	−0.706209
$816$	9.35032	0.327327
$817$	46.2396	1.61772
$818$	−0.960942	−0.0335986
$819$	61.4192	2.14616
$820$	47.4662	1.65759
$821$	27.8394	0.971603	0.485801	−	0.874069i	$-0.338528\pi$
$821$	27.8394	0.971603	0.485801	+	0.874069i	$0.338528\pi$
$822$	5.47935	0.191114
$823$	−7.24494	−0.252543	−0.126271	−	0.991996i	$-0.540301\pi$
$823$	−7.24494	−0.252543	−0.126271	+	0.991996i	$0.540301\pi$
$824$	13.8904	0.483895
$825$	−4.88157	−0.169954
$826$	−21.4990	−0.748046
$827$	−20.0496	−0.697192	−0.348596	−	0.937273i	$-0.613342\pi$
$827$	−20.0496	−0.697192	−0.348596	+	0.937273i	$0.613342\pi$
$828$	−39.6606	−1.37830
$829$	−22.9004	−0.795364	−0.397682	−	0.917523i	$-0.630185\pi$
$829$	−22.9004	−0.795364	−0.397682	+	0.917523i	$0.630185\pi$
$830$	−75.4584	−2.61920
$831$	−2.22573	−0.0772098
$832$	−70.7483	−2.45276
$833$	27.5018	0.952880
$834$	1.09908	0.0380579
$835$	−61.1278	−2.11542
$836$	−42.5045	−1.47005
$837$	11.1306	0.384728
$838$	−28.2156	−0.974690
$839$	−22.0066	−0.759753	−0.379877	−	0.925037i	$-0.624034\pi$
$839$	−22.0066	−0.759753	−0.379877	+	0.925037i	$0.624034\pi$
$840$	4.02479	0.138869
$841$	1.00000	0.0344828
$842$	5.45491	0.187989
$843$	−11.3303	−0.390236
$844$	24.7122	0.850630
$845$	−83.1611	−2.86083
$846$	−19.8391	−0.682081
$847$	−8.40413	−0.288769
$848$	21.6175	0.742349
$849$	−10.1518	−0.348410
$850$	−30.8621	−1.05856
$851$	53.9223	1.84843
$852$	9.66729	0.331196
$853$	−14.8916	−0.509879	−0.254940	−	0.966957i	$-0.582056\pi$
$853$	−14.8916	−0.509879	−0.254940	+	0.966957i	$0.582056\pi$
$854$	25.8155	0.883388
$855$	−36.2806	−1.24077
$856$	4.02560	0.137592
$857$	43.1188	1.47291	0.736456	−	0.676486i	$-0.236498\pi$
$857$	43.1188	1.47291	0.736456	+	0.676486i	$0.236498\pi$
$858$	26.5241	0.905519
$859$	49.1718	1.67772	0.838861	−	0.544346i	$-0.183222\pi$
$859$	49.1718	1.67772	0.838861	+	0.544346i	$0.183222\pi$
$860$	62.5393	2.13257
$861$	−13.0047	−0.443200
$862$	40.6566	1.38477
$863$	28.5346	0.971327	0.485664	−	0.874146i	$-0.338578\pi$
$863$	28.5346	0.971327	0.485664	+	0.874146i	$0.338578\pi$
$864$	24.2141	0.823780
$865$	50.4761	1.71624
$866$	−80.5338	−2.73665
$867$	−8.76809	−0.297780
$868$	−30.3375	−1.02972
$869$	47.8618	1.62360
$870$	3.01736	0.102298
$871$	8.16616	0.276700
$872$	−10.1248	−0.342870
$873$	−13.2574	−0.448697
$874$	61.8994	2.09378
$875$	−23.1650	−0.783121
$876$	6.55800	0.221574
$877$	28.6818	0.968515	0.484258	−	0.874925i	$-0.339090\pi$
$877$	28.6818	0.968515	0.484258	+	0.874925i	$0.339090\pi$
$878$	40.5354	1.36800
$879$	3.50846	0.118337
$880$	30.9105	1.04199
$881$	47.7631	1.60918	0.804590	−	0.593831i	$-0.202385\pi$
$881$	47.7631	1.60918	0.804590	+	0.593831i	$0.202385\pi$
$882$	27.0390	0.910451
$883$	6.77497	0.227996	0.113998	−	0.993481i	$-0.463634\pi$
$883$	6.77497	0.227996	0.113998	+	0.993481i	$0.463634\pi$
$884$	91.2823	3.07016
$885$	−4.31395	−0.145012
$886$	−26.0194	−0.874140
$887$	−7.08779	−0.237985	−0.118992	−	0.992895i	$-0.537966\pi$
$887$	−7.08779	−0.237985	−0.118992	+	0.992895i	$0.537966\pi$
$888$	3.78791	0.127114
$889$	−46.4769	−1.55878
$890$	−73.9654	−2.47933
$891$	24.2038	0.810858
$892$	−16.6783	−0.558430
$893$	16.8550	0.564031
$894$	−6.74215	−0.225491
$895$	45.3545	1.51603
$896$	−21.9330	−0.732730
$897$	−21.0268	−0.702065
$898$	22.7284	0.758458
$899$	−3.70620	−0.123609
$900$	−16.5171	−0.550572
$901$	−40.8899	−1.36224
$902$	55.6054	1.85146
$903$	−17.1344	−0.570198
$904$	−1.34517	−0.0447396
$905$	29.3022	0.974037
$906$	−15.7052	−0.521772
$907$	−46.0999	−1.53072	−0.765360	−	0.643602i	$-0.777439\pi$
$907$	−46.0999	−1.53072	−0.765360	+	0.643602i	$0.777439\pi$
$908$	−43.3994	−1.44026
$909$	−51.4804	−1.70750
$910$	−129.649	−4.29783
$911$	−9.20652	−0.305026	−0.152513	−	0.988301i	$-0.548737\pi$
$911$	−9.20652	−0.305026	−0.152513	+	0.988301i	$0.548737\pi$
$912$	−7.81023	−0.258622
$913$	−48.1194	−1.59252
$914$	69.5077	2.29911
$915$	5.18009	0.171249
$916$	−9.82906	−0.324761
$917$	−72.5751	−2.39664
$918$	−36.5336	−1.20579
$919$	20.7739	0.685268	0.342634	−	0.939469i	$-0.388681\pi$
$919$	20.7739	0.685268	0.342634	+	0.939469i	$0.388681\pi$
$920$	13.6424	0.449777
$921$	1.75203	0.0577313
$922$	43.1271	1.42032
$923$	−50.7455	−1.67031
$924$	15.7503	0.518148
$925$	22.4566	0.738369
$926$	64.1772	2.10899
$927$	46.3982	1.52392
$928$	−8.06271	−0.264671
$929$	40.1740	1.31807	0.659034	−	0.752113i	$-0.270965\pi$
$929$	40.1740	1.31807	0.659034	+	0.752113i	$0.270965\pi$
$930$	−11.1830	−0.366703
$931$	−22.9720	−0.752876
$932$	−23.0258	−0.754234
$933$	−9.86418	−0.322939
$934$	7.46575	0.244287
$935$	−58.4676	−1.91210
$936$	14.6246	0.478019
$937$	−42.9608	−1.40347	−0.701734	−	0.712439i	$-0.747591\pi$
$937$	−42.9608	−1.40347	−0.701734	+	0.712439i	$0.747591\pi$
$938$	8.90811	0.290860
$939$	−1.54217	−0.0503267
$940$	22.7964	0.743538
$941$	−56.0157	−1.82606	−0.913030	−	0.407891i	$-0.866264\pi$
$941$	−56.0157	−1.82606	−0.913030	+	0.407891i	$0.866264\pi$
$942$	18.6847	0.608779
$943$	−44.0808	−1.43547
$944$	9.19481	0.299266
$945$	28.2459	0.918840
$946$	73.2631	2.38199
$947$	13.2004	0.428956	0.214478	−	0.976729i	$-0.431195\pi$
$947$	13.2004	0.428956	0.214478	+	0.976729i	$0.431195\pi$
$948$	−16.3563	−0.531228
$949$	−34.4242	−1.11746
$950$	25.7788	0.836374
$951$	−3.08962	−0.100188
$952$	16.2264	0.525899
$953$	19.3639	0.627259	0.313630	−	0.949545i	$-0.398455\pi$
$953$	19.3639	0.627259	0.313630	+	0.949545i	$0.398455\pi$
$954$	−40.2018	−1.30158
$955$	−29.9497	−0.969150
$956$	−41.2268	−1.33337
$957$	1.92415	0.0621990
$958$	52.3939	1.69277
$959$	−17.0793	−0.551520
$960$	−15.4861	−0.499811
$961$	−17.2641	−0.556905
$962$	−122.019	−3.93404
$963$	13.4468	0.433316
$964$	30.5718	0.984651
$965$	−20.5338	−0.661005
$966$	−22.9373	−0.737994
$967$	58.1224	1.86909	0.934545	−	0.355844i	$-0.115806\pi$
$967$	58.1224	1.86909	0.934545	+	0.355844i	$0.115806\pi$
$968$	−2.00111	−0.0643182
$969$	14.7732	0.474582
$970$	27.9850	0.898545
$971$	−23.5618	−0.756135	−0.378067	−	0.925778i	$-0.623411\pi$
$971$	−23.5618	−0.756135	−0.378067	+	0.925778i	$0.623411\pi$
$972$	−29.7987	−0.955795
$973$	−3.42586	−0.109828
$974$	−49.3531	−1.58138
$975$	−8.75687	−0.280444
$976$	−11.0409	−0.353411
$977$	45.3629	1.45129	0.725644	−	0.688070i	$-0.241542\pi$
$977$	45.3629	1.45129	0.725644	+	0.688070i	$0.241542\pi$
$978$	8.07125	0.258090
$979$	−47.1673	−1.50747
$980$	−31.0697	−0.992484
$981$	−33.8200	−1.07979
$982$	−72.8787	−2.32565
$983$	−16.2565	−0.518503	−0.259251	−	0.965810i	$-0.583476\pi$
$983$	−16.2565	−0.518503	−0.259251	+	0.965810i	$0.583476\pi$
$984$	−3.09656	−0.0987148
$985$	26.2374	0.835992
$986$	12.1648	0.387406
$987$	−6.24573	−0.198804
$988$	−76.2472	−2.42575
$989$	−58.0788	−1.84680
$990$	−57.4838	−1.82696
$991$	11.3619	0.360921	0.180461	−	0.983582i	$-0.442241\pi$
$991$	11.3619	0.360921	0.180461	+	0.983582i	$0.442241\pi$
$992$	29.8820	0.948755
$993$	−9.41075	−0.298641
$994$	−55.3562	−1.75579
$995$	−31.6943	−1.00478
$996$	16.4443	0.521059
$997$	49.5123	1.56807	0.784035	−	0.620717i	$-0.213158\pi$
$997$	49.5123	1.56807	0.784035	+	0.620717i	$0.213158\pi$
$998$	18.2125	0.576508
$999$	26.5835	0.841064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.e.1.18	✓	103

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.e.1.18	✓	103	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09508 q^{2} -0.524599 q^{3} +2.38936 q^{4} -2.74536 q^{5} +1.09908 q^{6} -3.42586 q^{7} -0.815732 q^{8} -2.72480 q^{9} +O(q^{10})\)	\(q-2.09508 q^{2} -0.524599 q^{3} +2.38936 q^{4} -2.74536 q^{5} +1.09908 q^{6} -3.42586 q^{7} -0.815732 q^{8} -2.72480 q^{9} +5.75174 q^{10} +3.66785 q^{11} -1.25345 q^{12} +6.57963 q^{13} +7.17744 q^{14} +1.44021 q^{15} -3.06969 q^{16} +5.80637 q^{17} +5.70866 q^{18} -4.85000 q^{19} -6.55964 q^{20} +1.79720 q^{21} -7.68444 q^{22} +6.09179 q^{23} +0.427932 q^{24} +2.53700 q^{25} -13.7848 q^{26} +3.00322 q^{27} -8.18559 q^{28} -1.00000 q^{29} -3.01736 q^{30} +3.70620 q^{31} +8.06271 q^{32} -1.92415 q^{33} -12.1648 q^{34} +9.40521 q^{35} -6.51051 q^{36} +8.85165 q^{37} +10.1611 q^{38} -3.45167 q^{39} +2.23948 q^{40} -7.23610 q^{41} -3.76528 q^{42} -9.53395 q^{43} +8.76381 q^{44} +7.48054 q^{45} -12.7628 q^{46} -3.47526 q^{47} +1.61036 q^{48} +4.73649 q^{49} -5.31521 q^{50} -3.04601 q^{51} +15.7211 q^{52} -7.04225 q^{53} -6.29199 q^{54} -10.0696 q^{55} +2.79458 q^{56} +2.54431 q^{57} +2.09508 q^{58} -2.99536 q^{59} +3.44118 q^{60} +3.59675 q^{61} -7.76479 q^{62} +9.33476 q^{63} -10.7526 q^{64} -18.0634 q^{65} +4.03125 q^{66} +1.24113 q^{67} +13.8735 q^{68} -3.19575 q^{69} -19.7046 q^{70} -7.71252 q^{71} +2.22270 q^{72} -5.23194 q^{73} -18.5449 q^{74} -1.33091 q^{75} -11.5884 q^{76} -12.5655 q^{77} +7.23151 q^{78} +13.0490 q^{79} +8.42740 q^{80} +6.59890 q^{81} +15.1602 q^{82} -13.1192 q^{83} +4.29415 q^{84} -15.9406 q^{85} +19.9744 q^{86} +0.524599 q^{87} -2.99199 q^{88} -12.8597 q^{89} -15.6723 q^{90} -22.5409 q^{91} +14.5554 q^{92} -1.94427 q^{93} +7.28093 q^{94} +13.3150 q^{95} -4.22969 q^{96} +4.86548 q^{97} -9.92332 q^{98} -9.99415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more