LMFDB - Embedded newform 6035.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6035,2,Mod(1,6035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6035 = 5 \cdot 17 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6035.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:	$48.1897176198$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0473223 q^{2} +3.03750 q^{3} -1.99776 q^{4} +1.00000 q^{5} -0.143741 q^{6} -2.32686 q^{7} +0.189183 q^{8} +6.22638 q^{9} +O(q^{10})$	\(q-0.0473223 q^{2} +3.03750 q^{3} -1.99776 q^{4} +1.00000 q^{5} -0.143741 q^{6} -2.32686 q^{7} +0.189183 q^{8} +6.22638 q^{9} -0.0473223 q^{10} -4.55821 q^{11} -6.06819 q^{12} -3.49636 q^{13} +0.110112 q^{14} +3.03750 q^{15} +3.98657 q^{16} -1.00000 q^{17} -0.294646 q^{18} +6.86241 q^{19} -1.99776 q^{20} -7.06783 q^{21} +0.215705 q^{22} -3.26302 q^{23} +0.574643 q^{24} +1.00000 q^{25} +0.165456 q^{26} +9.80011 q^{27} +4.64851 q^{28} +9.04888 q^{29} -0.143741 q^{30} +0.758596 q^{31} -0.567019 q^{32} -13.8456 q^{33} +0.0473223 q^{34} -2.32686 q^{35} -12.4388 q^{36} -7.60535 q^{37} -0.324745 q^{38} -10.6202 q^{39} +0.189183 q^{40} -8.61253 q^{41} +0.334466 q^{42} -0.884670 q^{43} +9.10622 q^{44} +6.22638 q^{45} +0.154414 q^{46} -4.75307 q^{47} +12.1092 q^{48} -1.58571 q^{49} -0.0473223 q^{50} -3.03750 q^{51} +6.98490 q^{52} -3.93242 q^{53} -0.463763 q^{54} -4.55821 q^{55} -0.440203 q^{56} +20.8445 q^{57} -0.428214 q^{58} -3.82135 q^{59} -6.06819 q^{60} -3.77396 q^{61} -0.0358985 q^{62} -14.4879 q^{63} -7.94630 q^{64} -3.49636 q^{65} +0.655203 q^{66} +0.353322 q^{67} +1.99776 q^{68} -9.91141 q^{69} +0.110112 q^{70} +1.00000 q^{71} +1.17793 q^{72} -0.916236 q^{73} +0.359902 q^{74} +3.03750 q^{75} -13.7095 q^{76} +10.6063 q^{77} +0.502571 q^{78} -2.50384 q^{79} +3.98657 q^{80} +11.0886 q^{81} +0.407564 q^{82} +10.6699 q^{83} +14.1198 q^{84} -1.00000 q^{85} +0.0418646 q^{86} +27.4859 q^{87} -0.862337 q^{88} +5.20487 q^{89} -0.294646 q^{90} +8.13556 q^{91} +6.51874 q^{92} +2.30423 q^{93} +0.224926 q^{94} +6.86241 q^{95} -1.72232 q^{96} +0.787190 q^{97} +0.0750394 q^{98} -28.3812 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10}) $ 36 * q - q^2 - 4 * q^3 + 23 * q^4 + 36 * q^5 - 2 * q^6 - 7 * q^7 - 3 * q^8 + 10 * q^9	\( 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9} - q^{10} - 22 q^{11} - 14 q^{12} - 15 q^{13} - 28 q^{14} - 4 q^{15} + q^{16} - 36 q^{17} - 12 q^{18} - 23 q^{19} + 23 q^{20} - 21 q^{21} + 2 q^{23} - 13 q^{24} + 36 q^{25} - 18 q^{26} - 13 q^{27} - 20 q^{28} - 4 q^{29} - 2 q^{30} - 43 q^{31} - 2 q^{32} - 19 q^{33} + q^{34} - 7 q^{35} - 35 q^{36} - 30 q^{37} - 11 q^{38} - 20 q^{39} - 3 q^{40} - 39 q^{41} + 2 q^{42} - 7 q^{43} - 45 q^{44} + 10 q^{45} - 52 q^{46} - 12 q^{47} - 12 q^{48} - 15 q^{49} - q^{50} + 4 q^{51} - 19 q^{52} - 31 q^{53} + 48 q^{54} - 22 q^{55} - 30 q^{56} + 18 q^{57} - 12 q^{58} - 66 q^{59} - 14 q^{60} - 93 q^{61} - 7 q^{62} - 22 q^{63} - 41 q^{64} - 15 q^{65} - 21 q^{66} - 19 q^{67} - 23 q^{68} - 73 q^{69} - 28 q^{70} + 36 q^{71} - q^{72} - 47 q^{73} - 27 q^{74} - 4 q^{75} - 56 q^{76} - 9 q^{77} - 78 q^{78} - 21 q^{79} + q^{80} - 40 q^{81} - 15 q^{82} - 8 q^{83} - 54 q^{84} - 36 q^{85} - 17 q^{86} - 32 q^{87} - 13 q^{88} - 62 q^{89} - 12 q^{90} - 33 q^{91} + 42 q^{92} - 24 q^{93} - 40 q^{94} - 23 q^{95} + 21 q^{96} - 60 q^{97} + 11 q^{98} - 65 q^{99}+O(q^{100}) \) 36 * q - q^2 - 4 * q^3 + 23 * q^4 + 36 * q^5 - 2 * q^6 - 7 * q^7 - 3 * q^8 + 10 * q^9 - q^10 - 22 * q^11 - 14 * q^12 - 15 * q^13 - 28 * q^14 - 4 * q^15 + q^16 - 36 * q^17 - 12 * q^18 - 23 * q^19 + 23 * q^20 - 21 * q^21 + 2 * q^23 - 13 * q^24 + 36 * q^25 - 18 * q^26 - 13 * q^27 - 20 * q^28 - 4 * q^29 - 2 * q^30 - 43 * q^31 - 2 * q^32 - 19 * q^33 + q^34 - 7 * q^35 - 35 * q^36 - 30 * q^37 - 11 * q^38 - 20 * q^39 - 3 * q^40 - 39 * q^41 + 2 * q^42 - 7 * q^43 - 45 * q^44 + 10 * q^45 - 52 * q^46 - 12 * q^47 - 12 * q^48 - 15 * q^49 - q^50 + 4 * q^51 - 19 * q^52 - 31 * q^53 + 48 * q^54 - 22 * q^55 - 30 * q^56 + 18 * q^57 - 12 * q^58 - 66 * q^59 - 14 * q^60 - 93 * q^61 - 7 * q^62 - 22 * q^63 - 41 * q^64 - 15 * q^65 - 21 * q^66 - 19 * q^67 - 23 * q^68 - 73 * q^69 - 28 * q^70 + 36 * q^71 - q^72 - 47 * q^73 - 27 * q^74 - 4 * q^75 - 56 * q^76 - 9 * q^77 - 78 * q^78 - 21 * q^79 + q^80 - 40 * q^81 - 15 * q^82 - 8 * q^83 - 54 * q^84 - 36 * q^85 - 17 * q^86 - 32 * q^87 - 13 * q^88 - 62 * q^89 - 12 * q^90 - 33 * q^91 + 42 * q^92 - 24 * q^93 - 40 * q^94 - 23 * q^95 + 21 * q^96 - 60 * q^97 + 11 * q^98 - 65 * 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 $

$2$	−0.0473223	−0.0334619	−0.0167309	−	0.999860i	$-0.505326\pi$
$2$	−0.0473223	−0.0334619	−0.0167309	+	0.999860i	$0.505326\pi$
$3$	3.03750	1.75370	0.876849	−	0.480765i	$-0.159641\pi$
$3$	3.03750	1.75370	0.876849	+	0.480765i	$0.159641\pi$
$4$	−1.99776	−0.998880
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.143741	−0.0586821
$7$	−2.32686	−0.879471	−0.439736	−	0.898127i	$-0.644928\pi$
$7$	−2.32686	−0.879471	−0.439736	+	0.898127i	$0.644928\pi$
$8$	0.189183	0.0668863
$9$	6.22638	2.07546
$10$	−0.0473223	−0.0149646
$11$	−4.55821	−1.37435	−0.687176	−	0.726491i	$-0.741150\pi$
$11$	−4.55821	−1.37435	−0.687176	+	0.726491i	$0.741150\pi$
$12$	−6.06819	−1.75174
$13$	−3.49636	−0.969717	−0.484859	−	0.874593i	$-0.661129\pi$
$13$	−3.49636	−0.969717	−0.484859	+	0.874593i	$0.661129\pi$
$14$	0.110112	0.0294288
$15$	3.03750	0.784278
$16$	3.98657	0.996642
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.294646	−0.0694488
$19$	6.86241	1.57435	0.787173	−	0.616733i	$-0.211544\pi$
$19$	6.86241	1.57435	0.787173	+	0.616733i	$0.211544\pi$
$20$	−1.99776	−0.446713
$21$	−7.06783	−1.54233
$22$	0.215705	0.0459884
$23$	−3.26302	−0.680387	−0.340194	−	0.940355i	$-0.610493\pi$
$23$	−3.26302	−0.680387	−0.340194	+	0.940355i	$0.610493\pi$
$24$	0.574643	0.117298
$25$	1.00000	0.200000
$26$	0.165456	0.0324486
$27$	9.80011	1.88603
$28$	4.64851	0.878487
$29$	9.04888	1.68034	0.840168	−	0.542327i	$-0.182457\pi$
$29$	9.04888	1.68034	0.840168	+	0.542327i	$0.182457\pi$
$30$	−0.143741	−0.0262434
$31$	0.758596	0.136248	0.0681239	−	0.997677i	$-0.478299\pi$
$31$	0.758596	0.136248	0.0681239	+	0.997677i	$0.478299\pi$
$32$	−0.567019	−0.100236
$33$	−13.8456	−2.41020
$34$	0.0473223	0.00811570
$35$	−2.32686	−0.393312
$36$	−12.4388	−2.07314
$37$	−7.60535	−1.25031	−0.625156	−	0.780500i	$-0.714965\pi$
$37$	−7.60535	−1.25031	−0.625156	+	0.780500i	$0.714965\pi$
$38$	−0.324745	−0.0526806
$39$	−10.6202	−1.70059
$40$	0.189183	0.0299125
$41$	−8.61253	−1.34505	−0.672525	−	0.740074i	$-0.734790\pi$
$41$	−8.61253	−1.34505	−0.672525	+	0.740074i	$0.734790\pi$
$42$	0.334466	0.0516092
$43$	−0.884670	−0.134911	−0.0674554	−	0.997722i	$-0.521488\pi$
$43$	−0.884670	−0.134911	−0.0674554	+	0.997722i	$0.521488\pi$
$44$	9.10622	1.37281
$45$	6.22638	0.928174
$46$	0.154414	0.0227670
$47$	−4.75307	−0.693307	−0.346654	−	0.937993i	$-0.612682\pi$
$47$	−4.75307	−0.693307	−0.346654	+	0.937993i	$0.612682\pi$
$48$	12.1092	1.74781
$49$	−1.58571	−0.226530
$50$	−0.0473223	−0.00669238
$51$	−3.03750	−0.425334
$52$	6.98490	0.968631
$53$	−3.93242	−0.540159	−0.270080	−	0.962838i	$-0.587050\pi$
$53$	−3.93242	−0.540159	−0.270080	+	0.962838i	$0.587050\pi$
$54$	−0.463763	−0.0631102
$55$	−4.55821	−0.614629
$56$	−0.440203	−0.0588246
$57$	20.8445	2.76093
$58$	−0.428214	−0.0562272
$59$	−3.82135	−0.497497	−0.248749	−	0.968568i	$-0.580019\pi$
$59$	−3.82135	−0.497497	−0.248749	+	0.968568i	$0.580019\pi$
$60$	−6.06819	−0.783400
$61$	−3.77396	−0.483206	−0.241603	−	0.970375i	$-0.577673\pi$
$61$	−3.77396	−0.483206	−0.241603	+	0.970375i	$0.577673\pi$
$62$	−0.0358985	−0.00455911
$63$	−14.4879	−1.82531
$64$	−7.94630	−0.993288
$65$	−3.49636	−0.433671
$66$	0.655203	0.0806499
$67$	0.353322	0.0431651	0.0215826	−	0.999767i	$-0.493130\pi$
$67$	0.353322	0.0431651	0.0215826	+	0.999767i	$0.493130\pi$
$68$	1.99776	0.242264
$69$	−9.91141	−1.19319
$70$	0.110112	0.0131609
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	1.17793	0.138820
$73$	−0.916236	−0.107237	−0.0536186	−	0.998561i	$-0.517076\pi$
$73$	−0.916236	−0.107237	−0.0536186	+	0.998561i	$0.517076\pi$
$74$	0.359902	0.0418378
$75$	3.03750	0.350740
$76$	−13.7095	−1.57258
$77$	10.6063	1.20870
$78$	0.502571	0.0569050
$79$	−2.50384	−0.281703	−0.140852	−	0.990031i	$-0.544984\pi$
$79$	−2.50384	−0.281703	−0.140852	+	0.990031i	$0.544984\pi$
$80$	3.98657	0.445712
$81$	11.0886	1.23207
$82$	0.407564	0.0450079
$83$	10.6699	1.17117	0.585584	−	0.810611i	$-0.300865\pi$
$83$	10.6699	1.17117	0.585584	+	0.810611i	$0.300865\pi$
$84$	14.1198	1.54060
$85$	−1.00000	−0.108465
$86$	0.0418646	0.00451437
$87$	27.4859	2.94680
$88$	−0.862337	−0.0919254
$89$	5.20487	0.551715	0.275858	−	0.961199i	$-0.411038\pi$
$89$	5.20487	0.551715	0.275858	+	0.961199i	$0.411038\pi$
$90$	−0.294646	−0.0310584
$91$	8.13556	0.852838
$92$	6.51874	0.679625
$93$	2.30423	0.238938
$94$	0.224926	0.0231994
$95$	6.86241	0.704069
$96$	−1.72232	−0.175783
$97$	0.787190	0.0799270	0.0399635	−	0.999201i	$-0.487276\pi$
$97$	0.787190	0.0799270	0.0399635	+	0.999201i	$0.487276\pi$
$98$	0.0750394	0.00758012
$99$	−28.3812	−2.85241
$100$	−1.99776	−0.199776
$101$	−15.7581	−1.56799	−0.783994	−	0.620768i	$-0.786821\pi$
$101$	−15.7581	−1.56799	−0.783994	+	0.620768i	$0.786821\pi$
$102$	0.143741	0.0142325
$103$	−6.50390	−0.640848	−0.320424	−	0.947274i	$-0.603825\pi$
$103$	−6.50390	−0.640848	−0.320424	+	0.947274i	$0.603825\pi$
$104$	−0.661453	−0.0648608
$105$	−7.06783	−0.689750
$106$	0.186091	0.0180748
$107$	−18.8391	−1.82124	−0.910621	−	0.413244i	$-0.864396\pi$
$107$	−18.8391	−1.82124	−0.910621	+	0.413244i	$0.864396\pi$
$108$	−19.5783	−1.88392
$109$	−18.7006	−1.79119	−0.895595	−	0.444871i	$-0.853250\pi$
$109$	−18.7006	−1.79119	−0.895595	+	0.444871i	$0.853250\pi$
$110$	0.215705	0.0205667
$111$	−23.1012	−2.19267
$112$	−9.27620	−0.876518
$113$	−6.93392	−0.652288	−0.326144	−	0.945320i	$-0.605750\pi$
$113$	−6.93392	−0.652288	−0.326144	+	0.945320i	$0.605750\pi$
$114$	−0.986411	−0.0923858
$115$	−3.26302	−0.304278
$116$	−18.0775	−1.67845
$117$	−21.7697	−2.01261
$118$	0.180835	0.0166472
$119$	2.32686	0.213303
$120$	0.574643	0.0524575
$121$	9.77731	0.888846
$122$	0.178592	0.0161690
$123$	−26.1605	−2.35881
$124$	−1.51549	−0.136095
$125$	1.00000	0.0894427
$126$	0.685601	0.0610782
$127$	0.548983	0.0487143	0.0243572	−	0.999703i	$-0.492246\pi$
$127$	0.548983	0.0487143	0.0243572	+	0.999703i	$0.492246\pi$
$128$	1.51008	0.133473
$129$	−2.68718	−0.236593
$130$	0.165456	0.0145114
$131$	−3.29685	−0.288047	−0.144024	−	0.989574i	$-0.546004\pi$
$131$	−3.29685	−0.288047	−0.144024	+	0.989574i	$0.546004\pi$
$132$	27.6601	2.40750
$133$	−15.9679	−1.38459
$134$	−0.0167200	−0.00144439
$135$	9.80011	0.843459
$136$	−0.189183	−0.0162223
$137$	17.3979	1.48641	0.743203	−	0.669066i	$-0.233306\pi$
$137$	17.3979	1.48641	0.743203	+	0.669066i	$0.233306\pi$
$138$	0.469030	0.0399265
$139$	−11.4712	−0.972973	−0.486486	−	0.873688i	$-0.661722\pi$
$139$	−11.4712	−0.972973	−0.486486	+	0.873688i	$0.661722\pi$
$140$	4.64851	0.392871
$141$	−14.4374	−1.21585
$142$	−0.0473223	−0.00397120
$143$	15.9372	1.33273
$144$	24.8219	2.06849
$145$	9.04888	0.751469
$146$	0.0433583	0.00358836
$147$	−4.81659	−0.397265
$148$	15.1937	1.24891
$149$	−13.3601	−1.09450	−0.547250	−	0.836969i	$-0.684325\pi$
$149$	−13.3601	−1.09450	−0.547250	+	0.836969i	$0.684325\pi$
$150$	−0.143741	−0.0117364
$151$	−16.3477	−1.33036	−0.665179	−	0.746684i	$-0.731645\pi$
$151$	−16.3477	−1.33036	−0.665179	+	0.746684i	$0.731645\pi$
$152$	1.29825	0.105302
$153$	−6.22638	−0.503373
$154$	−0.501916	−0.0404455
$155$	0.758596	0.0609319
$156$	21.2166	1.69869
$157$	−2.47677	−0.197668	−0.0988338	−	0.995104i	$-0.531511\pi$
$157$	−2.47677	−0.197668	−0.0988338	+	0.995104i	$0.531511\pi$
$158$	0.118487	0.00942633
$159$	−11.9447	−0.947277
$160$	−0.567019	−0.0448268
$161$	7.59260	0.598381
$162$	−0.524740	−0.0412275
$163$	−7.95145	−0.622806	−0.311403	−	0.950278i	$-0.600799\pi$
$163$	−7.95145	−0.622806	−0.311403	+	0.950278i	$0.600799\pi$
$164$	17.2058	1.34354
$165$	−13.8456	−1.07787
$166$	−0.504922	−0.0391895
$167$	6.29875	0.487412	0.243706	−	0.969849i	$-0.421637\pi$
$167$	6.29875	0.487412	0.243706	+	0.969849i	$0.421637\pi$
$168$	−1.33711	−0.103161
$169$	−0.775433	−0.0596487
$170$	0.0473223	0.00362945
$171$	42.7280	3.26749
$172$	1.76736	0.134760
$173$	−3.26393	−0.248152	−0.124076	−	0.992273i	$-0.539597\pi$
$173$	−3.26393	−0.248152	−0.124076	+	0.992273i	$0.539597\pi$
$174$	−1.30070	−0.0986055
$175$	−2.32686	−0.175894
$176$	−18.1716	−1.36974
$177$	−11.6073	−0.872460
$178$	−0.246306	−0.0184614
$179$	−19.3538	−1.44657	−0.723285	−	0.690549i	$-0.757369\pi$
$179$	−19.3538	−1.44657	−0.723285	+	0.690549i	$0.757369\pi$
$180$	−12.4388	−0.927134
$181$	−5.34719	−0.397454	−0.198727	−	0.980055i	$-0.563681\pi$
$181$	−5.34719	−0.397454	−0.198727	+	0.980055i	$0.563681\pi$
$182$	−0.384993	−0.0285376
$183$	−11.4634	−0.847398
$184$	−0.617308	−0.0455086
$185$	−7.60535	−0.559156
$186$	−0.109041	−0.00799531
$187$	4.55821	0.333330
$188$	9.49550	0.692531
$189$	−22.8035	−1.65871
$190$	−0.324745	−0.0235595
$191$	5.00555	0.362189	0.181094	−	0.983466i	$-0.442036\pi$
$191$	5.00555	0.362189	0.181094	+	0.983466i	$0.442036\pi$
$192$	−24.1369	−1.74193
$193$	22.0906	1.59012	0.795059	−	0.606532i	$-0.207440\pi$
$193$	22.0906	1.59012	0.795059	+	0.606532i	$0.207440\pi$
$194$	−0.0372516	−0.00267451
$195$	−10.6202	−0.760528
$196$	3.16787	0.226276
$197$	23.2993	1.66000	0.830002	−	0.557761i	$-0.188339\pi$
$197$	23.2993	1.66000	0.830002	+	0.557761i	$0.188339\pi$
$198$	1.34306	0.0954471
$199$	−6.44481	−0.456861	−0.228430	−	0.973560i	$-0.573359\pi$
$199$	−6.44481	−0.456861	−0.228430	+	0.973560i	$0.573359\pi$
$200$	0.189183	0.0133773
$201$	1.07321	0.0756986
$202$	0.745708	0.0524679
$203$	−21.0555	−1.47781
$204$	6.06819	0.424858
$205$	−8.61253	−0.601525
$206$	0.307779	0.0214440
$207$	−20.3168	−1.41212
$208$	−13.9385	−0.966461
$209$	−31.2803	−2.16371
$210$	0.334466	0.0230803
$211$	−0.494838	−0.0340660	−0.0170330	−	0.999855i	$-0.505422\pi$
$211$	−0.494838	−0.0340660	−0.0170330	+	0.999855i	$0.505422\pi$
$212$	7.85603	0.539555
$213$	3.03750	0.208126
$214$	0.891507	0.0609422
$215$	−0.884670	−0.0603340
$216$	1.85401	0.126150
$217$	−1.76515	−0.119826
$218$	0.884953	0.0599366
$219$	−2.78306	−0.188062
$220$	9.10622	0.613941
$221$	3.49636	0.235191
$222$	1.09320	0.0733709
$223$	15.4521	1.03475	0.517373	−	0.855760i	$-0.326910\pi$
$223$	15.4521	1.03475	0.517373	+	0.855760i	$0.326910\pi$
$224$	1.31938	0.0881545
$225$	6.22638	0.415092
$226$	0.328129	0.0218268
$227$	−9.15037	−0.607331	−0.303666	−	0.952779i	$-0.598211\pi$
$227$	−9.15037	−0.607331	−0.303666	+	0.952779i	$0.598211\pi$
$228$	−41.6424	−2.75784
$229$	−26.3473	−1.74108	−0.870539	−	0.492099i	$-0.836230\pi$
$229$	−26.3473	−1.74108	−0.870539	+	0.492099i	$0.836230\pi$
$230$	0.154414	0.0101817
$231$	32.2167	2.11970
$232$	1.71190	0.112391
$233$	0.0952488	0.00623995	0.00311998	−	0.999995i	$-0.499007\pi$
$233$	0.0952488	0.00623995	0.00311998	+	0.999995i	$0.499007\pi$
$234$	1.03019	0.0673457
$235$	−4.75307	−0.310056
$236$	7.63414	0.496940
$237$	−7.60539	−0.494023
$238$	−0.110112	−0.00713753
$239$	2.09308	0.135390	0.0676949	−	0.997706i	$-0.478436\pi$
$239$	2.09308	0.135390	0.0676949	+	0.997706i	$0.478436\pi$
$240$	12.1092	0.781644
$241$	13.6912	0.881929	0.440964	−	0.897525i	$-0.354636\pi$
$241$	13.6912	0.881929	0.440964	+	0.897525i	$0.354636\pi$
$242$	−0.462684	−0.0297425
$243$	4.28140	0.274652
$244$	7.53947	0.482665
$245$	−1.58571	−0.101307
$246$	1.23797	0.0789303
$247$	−23.9935	−1.52667
$248$	0.143514	0.00911312
$249$	32.4096	2.05388
$250$	−0.0473223	−0.00299292
$251$	−0.111474	−0.00703615	−0.00351808	−	0.999994i	$-0.501120\pi$
$251$	−0.111474	−0.00703615	−0.00351808	+	0.999994i	$0.501120\pi$
$252$	28.9434	1.82326
$253$	14.8735	0.935092
$254$	−0.0259791	−0.00163007
$255$	−3.03750	−0.190215
$256$	15.8211	0.988822
$257$	30.7740	1.91963	0.959814	−	0.280638i	$-0.0905461\pi$
$257$	30.7740	1.91963	0.959814	+	0.280638i	$0.0905461\pi$
$258$	0.127163	0.00791685
$259$	17.6966	1.09961
$260$	6.98490	0.433185
$261$	56.3418	3.48747
$262$	0.156014	0.00963861
$263$	2.44815	0.150959	0.0754797	−	0.997147i	$-0.475951\pi$
$263$	2.44815	0.150959	0.0754797	+	0.997147i	$0.475951\pi$
$264$	−2.61934	−0.161209
$265$	−3.93242	−0.241567
$266$	0.755637	0.0463310
$267$	15.8098	0.967542
$268$	−0.705853	−0.0431168
$269$	9.15027	0.557902	0.278951	−	0.960305i	$-0.410013\pi$
$269$	9.15027	0.557902	0.278951	+	0.960305i	$0.410013\pi$
$270$	−0.463763	−0.0282237
$271$	22.3700	1.35888	0.679440	−	0.733731i	$-0.262223\pi$
$271$	22.3700	1.35888	0.679440	+	0.733731i	$0.262223\pi$
$272$	−3.98657	−0.241721
$273$	24.7117	1.49562
$274$	−0.823309	−0.0497379
$275$	−4.55821	−0.274871
$276$	19.8006	1.19186
$277$	−2.92062	−0.175483	−0.0877415	−	0.996143i	$-0.527965\pi$
$277$	−2.92062	−0.175483	−0.0877415	+	0.996143i	$0.527965\pi$
$278$	0.542842	0.0325575
$279$	4.72331	0.282777
$280$	−0.440203	−0.0263072
$281$	−6.99308	−0.417172	−0.208586	−	0.978004i	$-0.566886\pi$
$281$	−6.99308	−0.417172	−0.208586	+	0.978004i	$0.566886\pi$
$282$	0.683212	0.0406847
$283$	9.15045	0.543938	0.271969	−	0.962306i	$-0.412325\pi$
$283$	9.15045	0.543938	0.271969	+	0.962306i	$0.412325\pi$
$284$	−1.99776	−0.118545
$285$	20.8445	1.23472
$286$	−0.754183	−0.0445958
$287$	20.0402	1.18293
$288$	−3.53048	−0.208035
$289$	1.00000	0.0588235
$290$	−0.428214	−0.0251456
$291$	2.39108	0.140168
$292$	1.83042	0.107117
$293$	−14.7064	−0.859159	−0.429580	−	0.903029i	$-0.641338\pi$
$293$	−14.7064	−0.859159	−0.429580	+	0.903029i	$0.641338\pi$
$294$	0.227932	0.0132933
$295$	−3.82135	−0.222487
$296$	−1.43880	−0.0836287
$297$	−44.6710	−2.59207
$298$	0.632229	0.0366240
$299$	11.4087	0.659783
$300$	−6.06819	−0.350347
$301$	2.05850	0.118650
$302$	0.773611	0.0445163
$303$	−47.8651	−2.74978
$304$	27.3575	1.56906
$305$	−3.77396	−0.216096
$306$	0.294646	0.0168438
$307$	14.7731	0.843144	0.421572	−	0.906795i	$-0.361478\pi$
$307$	14.7731	0.843144	0.421572	+	0.906795i	$0.361478\pi$
$308$	−21.1889	−1.20735
$309$	−19.7556	−1.12385
$310$	−0.0358985	−0.00203890
$311$	32.9924	1.87083	0.935415	−	0.353552i	$-0.115026\pi$
$311$	32.9924	1.87083	0.935415	+	0.353552i	$0.115026\pi$
$312$	−2.00916	−0.113746
$313$	25.3256	1.43149	0.715744	−	0.698363i	$-0.246088\pi$
$313$	25.3256	1.43149	0.715744	+	0.698363i	$0.246088\pi$
$314$	0.117206	0.00661433
$315$	−14.4879	−0.816302
$316$	5.00206	0.281388
$317$	13.8228	0.776368	0.388184	−	0.921582i	$-0.373103\pi$
$317$	13.8228	0.776368	0.388184	+	0.921582i	$0.373103\pi$
$318$	0.565250	0.0316977
$319$	−41.2467	−2.30937
$320$	−7.94630	−0.444212
$321$	−57.2236	−3.19391
$322$	−0.359299	−0.0200230
$323$	−6.86241	−0.381835
$324$	−22.1525	−1.23069
$325$	−3.49636	−0.193943
$326$	0.376281	0.0208402
$327$	−56.8029	−3.14121
$328$	−1.62934	−0.0899655
$329$	11.0598	0.609744
$330$	0.655203	0.0360677
$331$	−25.9235	−1.42488	−0.712442	−	0.701731i	$-0.752411\pi$
$331$	−25.9235	−1.42488	−0.712442	+	0.701731i	$0.752411\pi$
$332$	−21.3158	−1.16986
$333$	−47.3538	−2.59497
$334$	−0.298071	−0.0163097
$335$	0.353322	0.0193040
$336$	−28.1764	−1.53715
$337$	16.8321	0.916905	0.458452	−	0.888719i	$-0.348404\pi$
$337$	16.8321	0.916905	0.458452	+	0.888719i	$0.348404\pi$
$338$	0.0366952	0.00199596
$339$	−21.0618	−1.14392
$340$	1.99776	0.108344
$341$	−3.45784	−0.187253
$342$	−2.02198	−0.109336
$343$	19.9778	1.07870
$344$	−0.167365	−0.00902369
$345$	−9.91141	−0.533613
$346$	0.154456	0.00830363
$347$	9.64357	0.517694	0.258847	−	0.965918i	$-0.416657\pi$
$347$	9.64357	0.517694	0.258847	+	0.965918i	$0.416657\pi$
$348$	−54.9103	−2.94350
$349$	5.35049	0.286405	0.143203	−	0.989693i	$-0.454260\pi$
$349$	5.35049	0.286405	0.143203	+	0.989693i	$0.454260\pi$
$350$	0.110112	0.00588575
$351$	−34.2648	−1.82892
$352$	2.58460	0.137759
$353$	6.48964	0.345409	0.172704	−	0.984974i	$-0.444749\pi$
$353$	6.48964	0.345409	0.172704	+	0.984974i	$0.444749\pi$
$354$	0.549285	0.0291942
$355$	1.00000	0.0530745
$356$	−10.3981	−0.551097
$357$	7.06783	0.374069
$358$	0.915865	0.0484050
$359$	−7.06683	−0.372973	−0.186486	−	0.982458i	$-0.559710\pi$
$359$	−7.06683	−0.372973	−0.186486	+	0.982458i	$0.559710\pi$
$360$	1.17793	0.0620821
$361$	28.0927	1.47856
$362$	0.253041	0.0132996
$363$	29.6985	1.55877
$364$	−16.2529	−0.851884
$365$	−0.916236	−0.0479580
$366$	0.542473	0.0283555
$367$	3.34012	0.174353	0.0871765	−	0.996193i	$-0.472216\pi$
$367$	3.34012	0.174353	0.0871765	+	0.996193i	$0.472216\pi$
$368$	−13.0083	−0.678103
$369$	−53.6248	−2.79160
$370$	0.359902	0.0187104
$371$	9.15020	0.475055
$372$	−4.60330	−0.238670
$373$	21.6585	1.12144	0.560718	−	0.828007i	$-0.310525\pi$
$373$	21.6585	1.12144	0.560718	+	0.828007i	$0.310525\pi$
$374$	−0.215705	−0.0111538
$375$	3.03750	0.156856
$376$	−0.899201	−0.0463728
$377$	−31.6382	−1.62945
$378$	1.07911	0.0555036
$379$	−4.93548	−0.253519	−0.126759	−	0.991933i	$-0.540458\pi$
$379$	−4.93548	−0.253519	−0.126759	+	0.991933i	$0.540458\pi$
$380$	−13.7095	−0.703280
$381$	1.66753	0.0854303
$382$	−0.236874	−0.0121195
$383$	−20.1778	−1.03104	−0.515518	−	0.856879i	$-0.672401\pi$
$383$	−20.1778	−1.03104	−0.515518	+	0.856879i	$0.672401\pi$
$384$	4.58685	0.234072
$385$	10.6063	0.540549
$386$	−1.04538	−0.0532083
$387$	−5.50829	−0.280002
$388$	−1.57262	−0.0798375
$389$	5.24966	0.266168	0.133084	−	0.991105i	$-0.457512\pi$
$389$	5.24966	0.266168	0.133084	+	0.991105i	$0.457512\pi$
$390$	0.502571	0.0254487
$391$	3.26302	0.165018
$392$	−0.299989	−0.0151518
$393$	−10.0142	−0.505148
$394$	−1.10257	−0.0555468
$395$	−2.50384	−0.125982
$396$	56.6988	2.84922
$397$	−0.239576	−0.0120240	−0.00601198	−	0.999982i	$-0.501914\pi$
$397$	−0.239576	−0.0120240	−0.00601198	+	0.999982i	$0.501914\pi$
$398$	0.304983	0.0152874
$399$	−48.5024	−2.42816
$400$	3.98657	0.199328
$401$	−15.3838	−0.768230	−0.384115	−	0.923285i	$-0.625493\pi$
$401$	−15.3838	−0.768230	−0.384115	+	0.923285i	$0.625493\pi$
$402$	−0.0507869	−0.00253302
$403$	−2.65233	−0.132122
$404$	31.4809	1.56623
$405$	11.0886	0.550999
$406$	0.996394	0.0494502
$407$	34.6668	1.71837
$408$	−0.574643	−0.0284490
$409$	12.9402	0.639851	0.319926	−	0.947443i	$-0.396342\pi$
$409$	12.9402	0.639851	0.319926	+	0.947443i	$0.396342\pi$
$410$	0.407564	0.0201282
$411$	52.8461	2.60671
$412$	12.9932	0.640130
$413$	8.89175	0.437534
$414$	0.961437	0.0472521
$415$	10.6699	0.523763
$416$	1.98251	0.0972004
$417$	−34.8437	−1.70630
$418$	1.48026	0.0724017
$419$	−4.36221	−0.213108	−0.106554	−	0.994307i	$-0.533982\pi$
$419$	−4.36221	−0.213108	−0.106554	+	0.994307i	$0.533982\pi$
$420$	14.1198	0.688978
$421$	−0.826941	−0.0403026	−0.0201513	−	0.999797i	$-0.506415\pi$
$421$	−0.826941	−0.0403026	−0.0201513	+	0.999797i	$0.506415\pi$
$422$	0.0234168	0.00113991
$423$	−29.5944	−1.43893
$424$	−0.743947	−0.0361293
$425$	−1.00000	−0.0485071
$426$	−0.143741	−0.00696428
$427$	8.78148	0.424966
$428$	37.6359	1.81920
$429$	48.4091	2.33721
$430$	0.0418646	0.00201889
$431$	13.9053	0.669794	0.334897	−	0.942255i	$-0.391299\pi$
$431$	13.9053	0.669794	0.334897	+	0.942255i	$0.391299\pi$
$432$	39.0688	1.87970
$433$	−39.2186	−1.88473	−0.942364	−	0.334591i	$-0.891402\pi$
$433$	−39.2186	−1.88473	−0.942364	+	0.334591i	$0.891402\pi$
$434$	0.0835308	0.00400961
$435$	27.4859	1.31785
$436$	37.3593	1.78918
$437$	−22.3922	−1.07116
$438$	0.131701	0.00629290
$439$	−13.5511	−0.646760	−0.323380	−	0.946269i	$-0.604819\pi$
$439$	−13.5511	−0.646760	−0.323380	+	0.946269i	$0.604819\pi$
$440$	−0.862337	−0.0411103
$441$	−9.87323	−0.470154
$442$	−0.165456	−0.00786993
$443$	−5.06572	−0.240679	−0.120340	−	0.992733i	$-0.538398\pi$
$443$	−5.06572	−0.240679	−0.120340	+	0.992733i	$0.538398\pi$
$444$	46.1507	2.19022
$445$	5.20487	0.246734
$446$	−0.731226	−0.0346246
$447$	−40.5812	−1.91942
$448$	18.4900	0.873568
$449$	22.7033	1.07143	0.535716	−	0.844398i	$-0.320042\pi$
$449$	22.7033	1.07143	0.535716	+	0.844398i	$0.320042\pi$
$450$	−0.294646	−0.0138898
$451$	39.2577	1.84857
$452$	13.8523	0.651558
$453$	−49.6561	−2.33305
$454$	0.433016	0.0203225
$455$	8.13556	0.381401
$456$	3.94343	0.184668
$457$	31.3511	1.46654	0.733272	−	0.679936i	$-0.237992\pi$
$457$	31.3511	1.46654	0.733272	+	0.679936i	$0.237992\pi$
$458$	1.24681	0.0582598
$459$	−9.80011	−0.457430
$460$	6.51874	0.303938
$461$	−12.4477	−0.579748	−0.289874	−	0.957065i	$-0.593613\pi$
$461$	−12.4477	−0.579748	−0.289874	+	0.957065i	$0.593613\pi$
$462$	−1.52457	−0.0709293
$463$	−34.0792	−1.58379	−0.791897	−	0.610654i	$-0.790907\pi$
$463$	−34.0792	−1.58379	−0.791897	+	0.610654i	$0.790907\pi$
$464$	36.0740	1.67469
$465$	2.30423	0.106856
$466$	−0.00450739	−0.000208801 0
$467$	−30.4128	−1.40734	−0.703669	−	0.710528i	$-0.748456\pi$
$467$	−30.4128	−1.40734	−0.703669	+	0.710528i	$0.748456\pi$
$468$	43.4906	2.01035
$469$	−0.822131	−0.0379625
$470$	0.224926	0.0103751
$471$	−7.52317	−0.346649
$472$	−0.722934	−0.0332757
$473$	4.03251	0.185415
$474$	0.359904	0.0165309
$475$	6.86241	0.314869
$476$	−4.64851	−0.213064
$477$	−24.4847	−1.12108
$478$	−0.0990491	−0.00453040
$479$	−17.9903	−0.821999	−0.410999	−	0.911636i	$-0.634820\pi$
$479$	−17.9903	−0.821999	−0.410999	+	0.911636i	$0.634820\pi$
$480$	−1.72232	−0.0786127
$481$	26.5911	1.21245
$482$	−0.647899	−0.0295110
$483$	23.0625	1.04938
$484$	−19.5327	−0.887851
$485$	0.787190	0.0357444
$486$	−0.202605	−0.00919036
$487$	2.58813	0.117280	0.0586398	−	0.998279i	$-0.481324\pi$
$487$	2.58813	0.117280	0.0586398	+	0.998279i	$0.481324\pi$
$488$	−0.713969	−0.0323199
$489$	−24.1525	−1.09221
$490$	0.0750394	0.00338993
$491$	−25.8211	−1.16529	−0.582645	−	0.812727i	$-0.697982\pi$
$491$	−25.8211	−1.16529	−0.582645	+	0.812727i	$0.697982\pi$
$492$	52.2624	2.35617
$493$	−9.04888	−0.407541
$494$	1.13543	0.0510852
$495$	−28.3812	−1.27564
$496$	3.02420	0.135790
$497$	−2.32686	−0.104374
$498$	−1.53370	−0.0687266
$499$	19.5158	0.873648	0.436824	−	0.899547i	$-0.356103\pi$
$499$	19.5158	0.873648	0.436824	+	0.899547i	$0.356103\pi$
$500$	−1.99776	−0.0893426
$501$	19.1324	0.854774
$502$	0.00527518	0.000235443 0
$503$	23.3578	1.04147	0.520737	−	0.853717i	$-0.325657\pi$
$503$	23.3578	1.04147	0.520737	+	0.853717i	$0.325657\pi$
$504$	−2.74087	−0.122088
$505$	−15.7581	−0.701226
$506$	−0.703850	−0.0312899
$507$	−2.35537	−0.104606
$508$	−1.09674	−0.0486598
$509$	−35.6979	−1.58228	−0.791140	−	0.611635i	$-0.790512\pi$
$509$	−35.6979	−1.58228	−0.791140	+	0.611635i	$0.790512\pi$
$510$	0.143741	0.00636496
$511$	2.13195	0.0943121
$512$	−3.76884	−0.166561
$513$	67.2524	2.96927
$514$	−1.45629	−0.0642343
$515$	−6.50390	−0.286596
$516$	5.36834	0.236328
$517$	21.6655	0.952849
$518$	−0.837443	−0.0367951
$519$	−9.91417	−0.435184
$520$	−0.661453	−0.0290066
$521$	10.5389	0.461719	0.230860	−	0.972987i	$-0.425846\pi$
$521$	10.5389	0.461719	0.230860	+	0.972987i	$0.425846\pi$
$522$	−2.66622	−0.116697
$523$	−0.222100	−0.00971176	−0.00485588	−	0.999988i	$-0.501546\pi$
$523$	−0.222100	−0.00971176	−0.00485588	+	0.999988i	$0.501546\pi$
$524$	6.58632	0.287725
$525$	−7.06783	−0.308466
$526$	−0.115852	−0.00505139
$527$	−0.758596	−0.0330450
$528$	−55.1962	−2.40211
$529$	−12.3527	−0.537073
$530$	0.186091	0.00808328
$531$	−23.7932	−1.03253
$532$	31.9000	1.38304
$533$	30.1125	1.30432
$534$	−0.748154	−0.0323758
$535$	−18.8391	−0.814484
$536$	0.0668425	0.00288716
$537$	−58.7871	−2.53685
$538$	−0.433011	−0.0186684
$539$	7.22801	0.311332
$540$	−19.5783	−0.842515
$541$	−19.5386	−0.840032	−0.420016	−	0.907517i	$-0.637976\pi$
$541$	−19.5386	−0.840032	−0.420016	+	0.907517i	$0.637976\pi$
$542$	−1.05860	−0.0454707
$543$	−16.2421	−0.697014
$544$	0.567019	0.0243108
$545$	−18.7006	−0.801044
$546$	−1.16941	−0.0500463
$547$	10.2449	0.438042	0.219021	−	0.975720i	$-0.429714\pi$
$547$	10.2449	0.438042	0.219021	+	0.975720i	$0.429714\pi$
$548$	−34.7569	−1.48474
$549$	−23.4981	−1.00287
$550$	0.215705	0.00919769
$551$	62.0972	2.64543
$552$	−1.87507	−0.0798083
$553$	5.82608	0.247750
$554$	0.138210	0.00587199
$555$	−23.1012	−0.980592
$556$	22.9167	0.971884
$557$	0.159804	0.00677110	0.00338555	−	0.999994i	$-0.498922\pi$
$557$	0.159804	0.00677110	0.00338555	+	0.999994i	$0.498922\pi$
$558$	−0.223518	−0.00946225
$559$	3.09313	0.130825
$560$	−9.27620	−0.391991
$561$	13.8456	0.584560
$562$	0.330928	0.0139594
$563$	37.2646	1.57052	0.785258	−	0.619169i	$-0.212530\pi$
$563$	37.2646	1.57052	0.785258	+	0.619169i	$0.212530\pi$
$564$	28.8426	1.21449
$565$	−6.93392	−0.291712
$566$	−0.433020	−0.0182012
$567$	−25.8018	−1.08357
$568$	0.189183	0.00793794
$569$	−45.7073	−1.91615	−0.958074	−	0.286522i	$-0.907501\pi$
$569$	−45.7073	−1.91615	−0.958074	+	0.286522i	$0.907501\pi$
$570$	−0.986411	−0.0413162
$571$	43.2806	1.81123	0.905617	−	0.424096i	$-0.139408\pi$
$571$	43.2806	1.81123	0.905617	+	0.424096i	$0.139408\pi$
$572$	−31.8387	−1.33124
$573$	15.2043	0.635170
$574$	−0.948346	−0.0395832
$575$	−3.26302	−0.136077
$576$	−49.4767	−2.06153
$577$	45.8174	1.90740	0.953701	−	0.300756i	$-0.0972391\pi$
$577$	45.8174	1.90740	0.953701	+	0.300756i	$0.0972391\pi$
$578$	−0.0473223	−0.00196835
$579$	67.1002	2.78859
$580$	−18.0775	−0.750627
$581$	−24.8273	−1.03001
$582$	−0.113152	−0.00469028
$583$	17.9248	0.742370
$584$	−0.173336	−0.00717270
$585$	−21.7697	−0.900066
$586$	0.695942	0.0287491
$587$	19.7992	0.817201	0.408601	−	0.912713i	$-0.366017\pi$
$587$	19.7992	0.817201	0.408601	+	0.912713i	$0.366017\pi$
$588$	9.62239	0.396821
$589$	5.20580	0.214501
$590$	0.180835	0.00744485
$591$	70.7714	2.91115
$592$	−30.3193	−1.24611
$593$	14.6995	0.603635	0.301817	−	0.953366i	$-0.402407\pi$
$593$	14.6995	0.603635	0.301817	+	0.953366i	$0.402407\pi$
$594$	2.11393	0.0867357
$595$	2.32686	0.0953921
$596$	26.6902	1.09327
$597$	−19.5761	−0.801196
$598$	−0.539886	−0.0220776
$599$	−7.37509	−0.301338	−0.150669	−	0.988584i	$-0.548143\pi$
$599$	−7.37509	−0.301338	−0.150669	+	0.988584i	$0.548143\pi$
$600$	0.574643	0.0234597
$601$	2.76358	0.112729	0.0563643	−	0.998410i	$-0.482049\pi$
$601$	2.76358	0.112729	0.0563643	+	0.998410i	$0.482049\pi$
$602$	−0.0974131	−0.00397026
$603$	2.19992	0.0895875
$604$	32.6588	1.32887
$605$	9.77731	0.397504
$606$	2.26509	0.0920128
$607$	−27.5810	−1.11948	−0.559738	−	0.828669i	$-0.689098\pi$
$607$	−27.5810	−1.11948	−0.559738	+	0.828669i	$0.689098\pi$
$608$	−3.89112	−0.157806
$609$	−63.9560	−2.59163
$610$	0.178592	0.00723099
$611$	16.6185	0.672312
$612$	12.4388	0.502809
$613$	−10.6824	−0.431458	−0.215729	−	0.976453i	$-0.569213\pi$
$613$	−10.6824	−0.431458	−0.215729	+	0.976453i	$0.569213\pi$
$614$	−0.699096	−0.0282132
$615$	−26.1605	−1.05489
$616$	2.00654	0.0808457
$617$	−34.4090	−1.38525	−0.692627	−	0.721296i	$-0.743547\pi$
$617$	−34.4090	−1.38525	−0.692627	+	0.721296i	$0.743547\pi$
$618$	0.934877	0.0376063
$619$	45.4245	1.82576	0.912881	−	0.408225i	$-0.133852\pi$
$619$	45.4245	1.82576	0.912881	+	0.408225i	$0.133852\pi$
$620$	−1.51549	−0.0608637
$621$	−31.9780	−1.28323
$622$	−1.56128	−0.0626015
$623$	−12.1110	−0.485218
$624$	−42.3381	−1.69488
$625$	1.00000	0.0400000
$626$	−1.19846	−0.0479003
$627$	−95.0139	−3.79449
$628$	4.94799	0.197446
$629$	7.60535	0.303245
$630$	0.685601	0.0273150
$631$	−20.6392	−0.821634	−0.410817	−	0.911718i	$-0.634756\pi$
$631$	−20.6392	−0.821634	−0.410817	+	0.911718i	$0.634756\pi$
$632$	−0.473683	−0.0188421
$633$	−1.50307	−0.0597415
$634$	−0.654128	−0.0259787
$635$	0.548983	0.0217857
$636$	23.8627	0.946216
$637$	5.54422	0.219670
$638$	1.95189	0.0772760
$639$	6.22638	0.246312
$640$	1.51008	0.0596910
$641$	−18.5803	−0.733878	−0.366939	−	0.930245i	$-0.619594\pi$
$641$	−18.5803	−0.733878	−0.366939	+	0.930245i	$0.619594\pi$
$642$	2.70795	0.106874
$643$	−13.0325	−0.513952	−0.256976	−	0.966418i	$-0.582726\pi$
$643$	−13.0325	−0.513952	−0.256976	+	0.966418i	$0.582726\pi$
$644$	−15.1682	−0.597711
$645$	−2.68718	−0.105808
$646$	0.324745	0.0127769
$647$	18.9520	0.745079	0.372540	−	0.928016i	$-0.378487\pi$
$647$	18.9520	0.745079	0.372540	+	0.928016i	$0.378487\pi$
$648$	2.09778	0.0824088
$649$	17.4185	0.683737
$650$	0.165456	0.00648971
$651$	−5.36163	−0.210139
$652$	15.8851	0.622108
$653$	8.12934	0.318126	0.159063	−	0.987268i	$-0.449153\pi$
$653$	8.12934	0.318126	0.159063	+	0.987268i	$0.449153\pi$
$654$	2.68804	0.105111
$655$	−3.29685	−0.128819
$656$	−34.3344	−1.34053
$657$	−5.70483	−0.222567
$658$	−0.523372	−0.0204032
$659$	−8.64288	−0.336679	−0.168339	−	0.985729i	$-0.553840\pi$
$659$	−8.64288	−0.336679	−0.168339	+	0.985729i	$0.553840\pi$
$660$	27.6601	1.07667
$661$	−5.40652	−0.210289	−0.105145	−	0.994457i	$-0.533531\pi$
$661$	−5.40652	−0.210289	−0.105145	+	0.994457i	$0.533531\pi$
$662$	1.22676	0.0476793
$663$	10.6202	0.412454
$664$	2.01856	0.0783352
$665$	−15.9679	−0.619208
$666$	2.24089	0.0868326
$667$	−29.5267	−1.14328
$668$	−12.5834	−0.486866
$669$	46.9356	1.81463
$670$	−0.0167200	−0.000645949 0
$671$	17.2025	0.664095
$672$	4.00760	0.154597
$673$	−31.8435	−1.22748	−0.613739	−	0.789509i	$-0.710335\pi$
$673$	−31.8435	−1.22748	−0.613739	+	0.789509i	$0.710335\pi$
$674$	−0.796534	−0.0306814
$675$	9.80011	0.377206
$676$	1.54913	0.0595819
$677$	14.8334	0.570092	0.285046	−	0.958514i	$-0.407991\pi$
$677$	14.8334	0.570092	0.285046	+	0.958514i	$0.407991\pi$
$678$	0.996690	0.0382776
$679$	−1.83168	−0.0702935
$680$	−0.189183	−0.00725484
$681$	−27.7942	−1.06508
$682$	0.163633	0.00626583
$683$	20.9179	0.800401	0.400201	−	0.916428i	$-0.368940\pi$
$683$	20.9179	0.800401	0.400201	+	0.916428i	$0.368940\pi$
$684$	−85.3603	−3.26383
$685$	17.3979	0.664741
$686$	−0.945393	−0.0360953
$687$	−80.0298	−3.05333
$688$	−3.52680	−0.134458
$689$	13.7492	0.523802
$690$	0.469030	0.0178557
$691$	41.8173	1.59080	0.795402	−	0.606082i	$-0.207260\pi$
$691$	41.8173	1.59080	0.795402	+	0.606082i	$0.207260\pi$
$692$	6.52055	0.247874
$693$	66.0391	2.50862
$694$	−0.456356	−0.0173230
$695$	−11.4712	−0.435127
$696$	5.19987	0.197101
$697$	8.61253	0.326223
$698$	−0.253197	−0.00958367
$699$	0.289318	0.0109430
$700$	4.64851	0.175697
$701$	15.8892	0.600128	0.300064	−	0.953919i	$-0.402992\pi$
$701$	15.8892	0.600128	0.300064	+	0.953919i	$0.402992\pi$
$702$	1.62149	0.0611990
$703$	−52.1910	−1.96842
$704$	36.2209	1.36513
$705$	−14.4374	−0.543745
$706$	−0.307104	−0.0115580
$707$	36.6669	1.37900
$708$	23.1887	0.871483
$709$	19.7750	0.742666	0.371333	−	0.928500i	$-0.378901\pi$
$709$	19.7750	0.742666	0.371333	+	0.928500i	$0.378901\pi$
$710$	−0.0473223	−0.00177597
$711$	−15.5898	−0.584664
$712$	0.984673	0.0369022
$713$	−2.47532	−0.0927013
$714$	−0.334466	−0.0125171
$715$	15.9372	0.596017
$716$	38.6642	1.44495
$717$	6.35771	0.237433
$718$	0.334418	0.0124804
$719$	26.9893	1.00653	0.503265	−	0.864132i	$-0.332132\pi$
$719$	26.9893	1.00653	0.503265	+	0.864132i	$0.332132\pi$
$720$	24.8219	0.925057
$721$	15.1337	0.563607
$722$	−1.32941	−0.0494755
$723$	41.5870	1.54664
$724$	10.6824	0.397009
$725$	9.04888	0.336067
$726$	−1.40540	−0.0521593
$727$	−28.9693	−1.07441	−0.537206	−	0.843451i	$-0.680520\pi$
$727$	−28.9693	−1.07441	−0.537206	+	0.843451i	$0.680520\pi$
$728$	1.53911	0.0570432
$729$	−20.2612	−0.750416
$730$	0.0433583	0.00160476
$731$	0.884670	0.0327207
$732$	22.9011	0.846449
$733$	3.63894	0.134407	0.0672037	−	0.997739i	$-0.478592\pi$
$733$	3.63894	0.134407	0.0672037	+	0.997739i	$0.478592\pi$
$734$	−0.158062	−0.00583418
$735$	−4.81659	−0.177663
$736$	1.85020	0.0681992
$737$	−1.61052	−0.0593241
$738$	2.53765	0.0934121
$739$	−28.6321	−1.05325	−0.526625	−	0.850098i	$-0.676543\pi$
$739$	−28.6321	−1.05325	−0.526625	+	0.850098i	$0.676543\pi$
$740$	15.1937	0.558530
$741$	−72.8801	−2.67732
$742$	−0.433008	−0.0158962
$743$	27.1820	0.997212	0.498606	−	0.866829i	$-0.333845\pi$
$743$	27.1820	0.997212	0.498606	+	0.866829i	$0.333845\pi$
$744$	0.435922	0.0159817
$745$	−13.3601	−0.489475
$746$	−1.02493	−0.0375254
$747$	66.4346	2.43071
$748$	−9.10622	−0.332956
$749$	43.8359	1.60173
$750$	−0.143741	−0.00524868
$751$	30.7342	1.12151	0.560754	−	0.827982i	$-0.310511\pi$
$751$	30.7342	1.12151	0.560754	+	0.827982i	$0.310511\pi$
$752$	−18.9485	−0.690979
$753$	−0.338601	−0.0123393
$754$	1.49719	0.0545245
$755$	−16.3477	−0.594954
$756$	45.5559	1.65685
$757$	−18.8096	−0.683645	−0.341822	−	0.939765i	$-0.611044\pi$
$757$	−18.8096	−0.683645	−0.341822	+	0.939765i	$0.611044\pi$
$758$	0.233558	0.00848322
$759$	45.1783	1.63987
$760$	1.29825	0.0470925
$761$	−47.1903	−1.71065	−0.855324	−	0.518094i	$-0.826642\pi$
$761$	−47.1903	−1.71065	−0.855324	+	0.518094i	$0.826642\pi$
$762$	−0.0789114	−0.00285866
$763$	43.5137	1.57530
$764$	−9.99989	−0.361783
$765$	−6.22638	−0.225115
$766$	0.954858	0.0345004
$767$	13.3608	0.482431
$768$	48.0567	1.73410
$769$	−28.6669	−1.03375	−0.516877	−	0.856060i	$-0.672906\pi$
$769$	−28.6669	−1.03375	−0.516877	+	0.856060i	$0.672906\pi$
$770$	−0.501916	−0.0180878
$771$	93.4758	3.36645
$772$	−44.1318	−1.58834
$773$	−14.0640	−0.505849	−0.252924	−	0.967486i	$-0.581392\pi$
$773$	−14.0640	−0.505849	−0.252924	+	0.967486i	$0.581392\pi$
$774$	0.260665	0.00936940
$775$	0.758596	0.0272496
$776$	0.148923	0.00534602
$777$	53.7534	1.92839
$778$	−0.248426	−0.00890650
$779$	−59.1027	−2.11757
$780$	21.2166	0.759676
$781$	−4.55821	−0.163106
$782$	−0.154414	−0.00552182
$783$	88.6800	3.16917
$784$	−6.32154	−0.225769
$785$	−2.47677	−0.0883996
$786$	0.473893	0.0169032
$787$	8.21551	0.292851	0.146426	−	0.989222i	$-0.453223\pi$
$787$	8.21551	0.292851	0.146426	+	0.989222i	$0.453223\pi$
$788$	−46.5463	−1.65814
$789$	7.43625	0.264737
$790$	0.118487	0.00421558
$791$	16.1343	0.573669
$792$	−5.36923	−0.190787
$793$	13.1951	0.468573
$794$	0.0113373	0.000402344 0
$795$	−11.9447	−0.423635
$796$	12.8752	0.456349
$797$	−39.1324	−1.38614	−0.693070	−	0.720870i	$-0.743742\pi$
$797$	−39.1324	−1.38614	−0.693070	+	0.720870i	$0.743742\pi$
$798$	2.29524	0.0812507
$799$	4.75307	0.168152
$800$	−0.567019	−0.0200472
$801$	32.4075	1.14506
$802$	0.727996	0.0257064
$803$	4.17640	0.147382
$804$	−2.14402	−0.0756139
$805$	7.59260	0.267604
$806$	0.125514	0.00442105
$807$	27.7939	0.978392
$808$	−2.98116	−0.104877
$809$	−37.5892	−1.32157	−0.660783	−	0.750577i	$-0.729776\pi$
$809$	−37.5892	−1.32157	−0.660783	+	0.750577i	$0.729776\pi$
$810$	−0.524740	−0.0184375
$811$	14.6761	0.515347	0.257674	−	0.966232i	$-0.417044\pi$
$811$	14.6761	0.515347	0.257674	+	0.966232i	$0.417044\pi$
$812$	42.0639	1.47615
$813$	67.9487	2.38307
$814$	−1.64051	−0.0574999
$815$	−7.95145	−0.278527
$816$	−12.1092	−0.423906
$817$	−6.07097	−0.212396
$818$	−0.612359	−0.0214106
$819$	50.6551	1.77003
$820$	17.2058	0.600851
$821$	40.7253	1.42132	0.710661	−	0.703534i	$-0.248396\pi$
$821$	40.7253	1.42132	0.710661	+	0.703534i	$0.248396\pi$
$822$	−2.50080	−0.0872254
$823$	−16.4015	−0.571722	−0.285861	−	0.958271i	$-0.592280\pi$
$823$	−16.4015	−0.571722	−0.285861	+	0.958271i	$0.592280\pi$
$824$	−1.23043	−0.0428639
$825$	−13.8456	−0.482040
$826$	−0.420778	−0.0146407
$827$	−18.2721	−0.635383	−0.317691	−	0.948194i	$-0.602908\pi$
$827$	−18.2721	−0.635383	−0.317691	+	0.948194i	$0.602908\pi$
$828$	40.5881	1.41053
$829$	8.18469	0.284266	0.142133	−	0.989848i	$-0.454604\pi$
$829$	8.18469	0.284266	0.142133	+	0.989848i	$0.454604\pi$
$830$	−0.504922	−0.0175261
$831$	−8.87136	−0.307744
$832$	27.7832	0.963208
$833$	1.58571	0.0549416
$834$	1.64888	0.0570961
$835$	6.29875	0.217977
$836$	62.4906	2.16128
$837$	7.43432	0.256968
$838$	0.206429	0.00713098
$839$	−2.78184	−0.0960399	−0.0480199	−	0.998846i	$-0.515291\pi$
$839$	−2.78184	−0.0960399	−0.0480199	+	0.998846i	$0.515291\pi$
$840$	−1.33711	−0.0461348
$841$	52.8823	1.82353
$842$	0.0391327	0.00134860
$843$	−21.2415	−0.731595
$844$	0.988567	0.0340279
$845$	−0.775433	−0.0266757
$846$	1.40048	0.0481493
$847$	−22.7504	−0.781715
$848$	−15.6769	−0.538346
$849$	27.7944	0.953903
$850$	0.0473223	0.00162314
$851$	24.8164	0.850696
$852$	−6.06819	−0.207893
$853$	−28.8821	−0.988905	−0.494453	−	0.869205i	$-0.664631\pi$
$853$	−28.8821	−0.988905	−0.494453	+	0.869205i	$0.664631\pi$
$854$	−0.415560	−0.0142202
$855$	42.7280	1.46127
$856$	−3.56403	−0.121816
$857$	47.3761	1.61834	0.809169	−	0.587577i	$-0.199918\pi$
$857$	47.3761	1.61834	0.809169	+	0.587577i	$0.199918\pi$
$858$	−2.29083	−0.0782076
$859$	9.46443	0.322922	0.161461	−	0.986879i	$-0.448379\pi$
$859$	9.46443	0.322922	0.161461	+	0.986879i	$0.448379\pi$
$860$	1.76736	0.0602664
$861$	60.8719	2.07451
$862$	−0.658029	−0.0224126
$863$	−28.1934	−0.959715	−0.479857	−	0.877346i	$-0.659312\pi$
$863$	−28.1934	−0.959715	−0.479857	+	0.877346i	$0.659312\pi$
$864$	−5.55685	−0.189048
$865$	−3.26393	−0.110977
$866$	1.85591	0.0630665
$867$	3.03750	0.103159
$868$	3.52635	0.119692
$869$	11.4130	0.387160
$870$	−1.30070	−0.0440977
$871$	−1.23534	−0.0418580
$872$	−3.53783	−0.119806
$873$	4.90134	0.165885
$874$	1.05965	0.0358432
$875$	−2.32686	−0.0786623
$876$	5.55989	0.187851
$877$	−25.1333	−0.848690	−0.424345	−	0.905501i	$-0.639496\pi$
$877$	−25.1333	−0.848690	−0.424345	+	0.905501i	$0.639496\pi$
$878$	0.641270	0.0216418
$879$	−44.6707	−1.50671
$880$	−18.1716	−0.612565
$881$	−52.2484	−1.76029	−0.880146	−	0.474702i	$-0.842556\pi$
$881$	−52.2484	−1.76029	−0.880146	+	0.474702i	$0.842556\pi$
$882$	0.467224	0.0157322
$883$	−30.7816	−1.03588	−0.517942	−	0.855415i	$-0.673302\pi$
$883$	−30.7816	−1.03588	−0.517942	+	0.855415i	$0.673302\pi$
$884$	−6.98490	−0.234928
$885$	−11.6073	−0.390176
$886$	0.239721	0.00805359
$887$	31.5151	1.05817	0.529087	−	0.848567i	$-0.322534\pi$
$887$	31.5151	1.05817	0.529087	+	0.848567i	$0.322534\pi$
$888$	−4.37036	−0.146660
$889$	−1.27741	−0.0428429
$890$	−0.246306	−0.00825620
$891$	−50.5444	−1.69330
$892$	−30.8695	−1.03359
$893$	−32.6176	−1.09150
$894$	1.92039	0.0642275
$895$	−19.3538	−0.646926
$896$	−3.51374	−0.117386
$897$	34.6539	1.15706
$898$	−1.07437	−0.0358522
$899$	6.86445	0.228942
$900$	−12.4388	−0.414627
$901$	3.93242	0.131008
$902$	−1.85776	−0.0618568
$903$	6.25270	0.208077
$904$	−1.31178	−0.0436292
$905$	−5.34719	−0.177747
$906$	2.34984	0.0780682
$907$	51.8336	1.72111	0.860553	−	0.509361i	$-0.170118\pi$
$907$	51.8336	1.72111	0.860553	+	0.509361i	$0.170118\pi$
$908$	18.2802	0.606651
$909$	−98.1158	−3.25430
$910$	−0.384993	−0.0127624
$911$	−59.0203	−1.95543	−0.977715	−	0.209935i	$-0.932675\pi$
$911$	−59.0203	−1.95543	−0.977715	+	0.209935i	$0.932675\pi$
$912$	83.0982	2.75166
$913$	−48.6355	−1.60960
$914$	−1.48361	−0.0490733
$915$	−11.4634	−0.378968
$916$	52.6356	1.73913
$917$	7.67132	0.253329
$918$	0.463763	0.0153065
$919$	29.5178	0.973703	0.486851	−	0.873485i	$-0.338145\pi$
$919$	29.5178	0.973703	0.486851	+	0.873485i	$0.338145\pi$
$920$	−0.617308	−0.0203521
$921$	44.8732	1.47862
$922$	0.589054	0.0193995
$923$	−3.49636	−0.115084
$924$	−64.3612	−2.11733
$925$	−7.60535	−0.250062
$926$	1.61270	0.0529968
$927$	−40.4957	−1.33005
$928$	−5.13089	−0.168430
$929$	17.3058	0.567786	0.283893	−	0.958856i	$-0.408374\pi$
$929$	17.3058	0.567786	0.283893	+	0.958856i	$0.408374\pi$
$930$	−0.109041	−0.00357561
$931$	−10.8818	−0.356637
$932$	−0.190284	−0.00623297
$933$	100.214	3.28087
$934$	1.43920	0.0470922
$935$	4.55821	0.149070
$936$	−4.11846	−0.134616
$937$	−48.0383	−1.56934	−0.784672	−	0.619911i	$-0.787169\pi$
$937$	−48.0383	−1.56934	−0.784672	+	0.619911i	$0.787169\pi$
$938$	0.0389051	0.00127030
$939$	76.9264	2.51040
$940$	9.49550	0.309709
$941$	−36.1516	−1.17851	−0.589254	−	0.807948i	$-0.700578\pi$
$941$	−36.1516	−1.17851	−0.589254	+	0.807948i	$0.700578\pi$
$942$	0.356013	0.0115995
$943$	28.1029	0.915155
$944$	−15.2341	−0.495827
$945$	−22.8035	−0.741798
$946$	−0.190828	−0.00620434
$947$	14.7264	0.478545	0.239272	−	0.970952i	$-0.423091\pi$
$947$	14.7264	0.478545	0.239272	+	0.970952i	$0.423091\pi$
$948$	15.1937	0.493470
$949$	3.20349	0.103990
$950$	−0.324745	−0.0105361
$951$	41.9868	1.36151
$952$	0.440203	0.0142671
$953$	25.5064	0.826232	0.413116	−	0.910678i	$-0.364440\pi$
$953$	25.5064	0.826232	0.413116	+	0.910678i	$0.364440\pi$
$954$	1.15867	0.0375134
$955$	5.00555	0.161976
$956$	−4.18146	−0.135238
$957$	−125.287	−4.04995
$958$	0.851343	0.0275056
$959$	−40.4826	−1.30725
$960$	−24.1369	−0.779014
$961$	−30.4245	−0.981437
$962$	−1.25835	−0.0405708
$963$	−117.299	−3.77991
$964$	−27.3518	−0.880941
$965$	22.0906	0.711122
$966$	−1.09137	−0.0351142
$967$	46.5997	1.49854	0.749272	−	0.662262i	$-0.230403\pi$
$967$	46.5997	1.49854	0.749272	+	0.662262i	$0.230403\pi$
$968$	1.84970	0.0594516
$969$	−20.8445	−0.669623
$970$	−0.0372516	−0.00119608
$971$	−42.9506	−1.37835	−0.689176	−	0.724594i	$-0.742027\pi$
$971$	−42.9506	−1.37835	−0.689176	+	0.724594i	$0.742027\pi$
$972$	−8.55321	−0.274344
$973$	26.6919	0.855702
$974$	−0.122476	−0.00392439
$975$	−10.6202	−0.340118
$976$	−15.0451	−0.481583
$977$	33.1655	1.06106	0.530529	−	0.847667i	$-0.321993\pi$
$977$	33.1655	1.06106	0.530529	+	0.847667i	$0.321993\pi$
$978$	1.14295	0.0365475
$979$	−23.7249	−0.758251
$980$	3.16787	0.101194
$981$	−116.437	−3.71754
$982$	1.22191	0.0389928
$983$	32.6117	1.04015	0.520076	−	0.854120i	$-0.325904\pi$
$983$	32.6117	1.04015	0.520076	+	0.854120i	$0.325904\pi$
$984$	−4.94912	−0.157772
$985$	23.2993	0.742376
$986$	0.428214	0.0136371
$987$	33.5939	1.06931
$988$	47.9333	1.52496
$989$	2.88670	0.0917916
$990$	1.34306	0.0426853
$991$	22.7045	0.721233	0.360617	−	0.932714i	$-0.382566\pi$
$991$	22.7045	0.721233	0.360617	+	0.932714i	$0.382566\pi$
$992$	−0.430139	−0.0136569
$993$	−78.7425	−2.49882
$994$	0.110112	0.00349255
$995$	−6.44481	−0.204314
$996$	−64.7467	−2.05158
$997$	−46.4244	−1.47027	−0.735137	−	0.677918i	$-0.762882\pi$
$997$	−46.4244	−1.47027	−0.735137	+	0.677918i	$0.762882\pi$
$998$	−0.923532	−0.0292339
$999$	−74.5333	−2.35813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6035.2.a.b.1.19	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6035.2.a.b.1.19	✓	36	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:	\( N \)	\(=\)	\( 6035 = 5 \cdot 17 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6035.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0473223 q^{2} +3.03750 q^{3} -1.99776 q^{4} +1.00000 q^{5} -0.143741 q^{6} -2.32686 q^{7} +0.189183 q^{8} +6.22638 q^{9} +O(q^{10})\)	\(q-0.0473223 q^{2} +3.03750 q^{3} -1.99776 q^{4} +1.00000 q^{5} -0.143741 q^{6} -2.32686 q^{7} +0.189183 q^{8} +6.22638 q^{9} -0.0473223 q^{10} -4.55821 q^{11} -6.06819 q^{12} -3.49636 q^{13} +0.110112 q^{14} +3.03750 q^{15} +3.98657 q^{16} -1.00000 q^{17} -0.294646 q^{18} +6.86241 q^{19} -1.99776 q^{20} -7.06783 q^{21} +0.215705 q^{22} -3.26302 q^{23} +0.574643 q^{24} +1.00000 q^{25} +0.165456 q^{26} +9.80011 q^{27} +4.64851 q^{28} +9.04888 q^{29} -0.143741 q^{30} +0.758596 q^{31} -0.567019 q^{32} -13.8456 q^{33} +0.0473223 q^{34} -2.32686 q^{35} -12.4388 q^{36} -7.60535 q^{37} -0.324745 q^{38} -10.6202 q^{39} +0.189183 q^{40} -8.61253 q^{41} +0.334466 q^{42} -0.884670 q^{43} +9.10622 q^{44} +6.22638 q^{45} +0.154414 q^{46} -4.75307 q^{47} +12.1092 q^{48} -1.58571 q^{49} -0.0473223 q^{50} -3.03750 q^{51} +6.98490 q^{52} -3.93242 q^{53} -0.463763 q^{54} -4.55821 q^{55} -0.440203 q^{56} +20.8445 q^{57} -0.428214 q^{58} -3.82135 q^{59} -6.06819 q^{60} -3.77396 q^{61} -0.0358985 q^{62} -14.4879 q^{63} -7.94630 q^{64} -3.49636 q^{65} +0.655203 q^{66} +0.353322 q^{67} +1.99776 q^{68} -9.91141 q^{69} +0.110112 q^{70} +1.00000 q^{71} +1.17793 q^{72} -0.916236 q^{73} +0.359902 q^{74} +3.03750 q^{75} -13.7095 q^{76} +10.6063 q^{77} +0.502571 q^{78} -2.50384 q^{79} +3.98657 q^{80} +11.0886 q^{81} +0.407564 q^{82} +10.6699 q^{83} +14.1198 q^{84} -1.00000 q^{85} +0.0418646 q^{86} +27.4859 q^{87} -0.862337 q^{88} +5.20487 q^{89} -0.294646 q^{90} +8.13556 q^{91} +6.51874 q^{92} +2.30423 q^{93} +0.224926 q^{94} +6.86241 q^{95} -1.72232 q^{96} +0.787190 q^{97} +0.0750394 q^{98} -28.3812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10}) \) 36 * q - q^2 - 4 * q^3 + 23 * q^4 + 36 * q^5 - 2 * q^6 - 7 * q^7 - 3 * q^8 + 10 * q^9	\( 36 q - q^{2} - 4 q^{3} + 23 q^{4} + 36 q^{5} - 2 q^{6} - 7 q^{7} - 3 q^{8} + 10 q^{9} - q^{10} - 22 q^{11} - 14 q^{12} - 15 q^{13} - 28 q^{14} - 4 q^{15} + q^{16} - 36 q^{17} - 12 q^{18} - 23 q^{19} + 23 q^{20} - 21 q^{21} + 2 q^{23} - 13 q^{24} + 36 q^{25} - 18 q^{26} - 13 q^{27} - 20 q^{28} - 4 q^{29} - 2 q^{30} - 43 q^{31} - 2 q^{32} - 19 q^{33} + q^{34} - 7 q^{35} - 35 q^{36} - 30 q^{37} - 11 q^{38} - 20 q^{39} - 3 q^{40} - 39 q^{41} + 2 q^{42} - 7 q^{43} - 45 q^{44} + 10 q^{45} - 52 q^{46} - 12 q^{47} - 12 q^{48} - 15 q^{49} - q^{50} + 4 q^{51} - 19 q^{52} - 31 q^{53} + 48 q^{54} - 22 q^{55} - 30 q^{56} + 18 q^{57} - 12 q^{58} - 66 q^{59} - 14 q^{60} - 93 q^{61} - 7 q^{62} - 22 q^{63} - 41 q^{64} - 15 q^{65} - 21 q^{66} - 19 q^{67} - 23 q^{68} - 73 q^{69} - 28 q^{70} + 36 q^{71} - q^{72} - 47 q^{73} - 27 q^{74} - 4 q^{75} - 56 q^{76} - 9 q^{77} - 78 q^{78} - 21 q^{79} + q^{80} - 40 q^{81} - 15 q^{82} - 8 q^{83} - 54 q^{84} - 36 q^{85} - 17 q^{86} - 32 q^{87} - 13 q^{88} - 62 q^{89} - 12 q^{90} - 33 q^{91} + 42 q^{92} - 24 q^{93} - 40 q^{94} - 23 q^{95} + 21 q^{96} - 60 q^{97} + 11 q^{98} - 65 q^{99}+O(q^{100}) \) 36 * q - q^2 - 4 * q^3 + 23 * q^4 + 36 * q^5 - 2 * q^6 - 7 * q^7 - 3 * q^8 + 10 * q^9 - q^10 - 22 * q^11 - 14 * q^12 - 15 * q^13 - 28 * q^14 - 4 * q^15 + q^16 - 36 * q^17 - 12 * q^18 - 23 * q^19 + 23 * q^20 - 21 * q^21 + 2 * q^23 - 13 * q^24 + 36 * q^25 - 18 * q^26 - 13 * q^27 - 20 * q^28 - 4 * q^29 - 2 * q^30 - 43 * q^31 - 2 * q^32 - 19 * q^33 + q^34 - 7 * q^35 - 35 * q^36 - 30 * q^37 - 11 * q^38 - 20 * q^39 - 3 * q^40 - 39 * q^41 + 2 * q^42 - 7 * q^43 - 45 * q^44 + 10 * q^45 - 52 * q^46 - 12 * q^47 - 12 * q^48 - 15 * q^49 - q^50 + 4 * q^51 - 19 * q^52 - 31 * q^53 + 48 * q^54 - 22 * q^55 - 30 * q^56 + 18 * q^57 - 12 * q^58 - 66 * q^59 - 14 * q^60 - 93 * q^61 - 7 * q^62 - 22 * q^63 - 41 * q^64 - 15 * q^65 - 21 * q^66 - 19 * q^67 - 23 * q^68 - 73 * q^69 - 28 * q^70 + 36 * q^71 - q^72 - 47 * q^73 - 27 * q^74 - 4 * q^75 - 56 * q^76 - 9 * q^77 - 78 * q^78 - 21 * q^79 + q^80 - 40 * q^81 - 15 * q^82 - 8 * q^83 - 54 * q^84 - 36 * q^85 - 17 * q^86 - 32 * q^87 - 13 * q^88 - 62 * q^89 - 12 * q^90 - 33 * q^91 + 42 * q^92 - 24 * q^93 - 40 * q^94 - 23 * q^95 + 21 * q^96 - 60 * q^97 + 11 * q^98 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more