LMFDB - Embedded newform 4031.2.a.c.1.48

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:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.48
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)

$f(q)$	$=$	$q+1.71872 q^{2} +1.75528 q^{3} +0.954015 q^{4} -1.95186 q^{5} +3.01685 q^{6} -0.688844 q^{7} -1.79776 q^{8} +0.0810151 q^{9} +O(q^{10})$	\(q+1.71872 q^{2} +1.75528 q^{3} +0.954015 q^{4} -1.95186 q^{5} +3.01685 q^{6} -0.688844 q^{7} -1.79776 q^{8} +0.0810151 q^{9} -3.35471 q^{10} +4.04528 q^{11} +1.67457 q^{12} -1.50039 q^{13} -1.18393 q^{14} -3.42606 q^{15} -4.99789 q^{16} +2.94271 q^{17} +0.139243 q^{18} -7.75532 q^{19} -1.86210 q^{20} -1.20912 q^{21} +6.95273 q^{22} -0.794200 q^{23} -3.15558 q^{24} -1.19025 q^{25} -2.57876 q^{26} -5.12364 q^{27} -0.657168 q^{28} -1.00000 q^{29} -5.88846 q^{30} +2.08878 q^{31} -4.99447 q^{32} +7.10061 q^{33} +5.05770 q^{34} +1.34452 q^{35} +0.0772896 q^{36} +8.60972 q^{37} -13.3293 q^{38} -2.63361 q^{39} +3.50897 q^{40} -5.94982 q^{41} -2.07814 q^{42} +9.77228 q^{43} +3.85926 q^{44} -0.158130 q^{45} -1.36501 q^{46} -5.23972 q^{47} -8.77270 q^{48} -6.52549 q^{49} -2.04572 q^{50} +5.16528 q^{51} -1.43140 q^{52} -10.7066 q^{53} -8.80613 q^{54} -7.89582 q^{55} +1.23838 q^{56} -13.6128 q^{57} -1.71872 q^{58} -5.95819 q^{59} -3.26851 q^{60} -13.6888 q^{61} +3.59003 q^{62} -0.0558067 q^{63} +1.41165 q^{64} +2.92855 q^{65} +12.2040 q^{66} -12.4195 q^{67} +2.80739 q^{68} -1.39404 q^{69} +2.31087 q^{70} -8.41409 q^{71} -0.145646 q^{72} -9.00205 q^{73} +14.7977 q^{74} -2.08923 q^{75} -7.39870 q^{76} -2.78657 q^{77} -4.52646 q^{78} -5.70166 q^{79} +9.75516 q^{80} -9.23648 q^{81} -10.2261 q^{82} +1.22209 q^{83} -1.15351 q^{84} -5.74374 q^{85} +16.7959 q^{86} -1.75528 q^{87} -7.27245 q^{88} +9.54658 q^{89} -0.271782 q^{90} +1.03354 q^{91} -0.757679 q^{92} +3.66639 q^{93} -9.00564 q^{94} +15.1373 q^{95} -8.76670 q^{96} +11.6981 q^{97} -11.2155 q^{98} +0.327729 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.71872	1.21532	0.607661	−	0.794197i	$-0.292108\pi$
$2$	1.71872	1.21532	0.607661	+	0.794197i	$0.292108\pi$
$3$	1.75528	1.01341	0.506706	−	0.862119i	$-0.330863\pi$
$3$	1.75528	1.01341	0.506706	+	0.862119i	$0.330863\pi$
$4$	0.954015	0.477008
$5$	−1.95186	−0.872897	−0.436449	−	0.899729i	$-0.643764\pi$
$5$	−1.95186	−0.872897	−0.436449	+	0.899729i	$0.643764\pi$
$6$	3.01685	1.23162
$7$	−0.688844	−0.260358	−0.130179	−	0.991490i	$-0.541555\pi$
$7$	−0.688844	−0.260358	−0.130179	+	0.991490i	$0.541555\pi$
$8$	−1.79776	−0.635604
$9$	0.0810151	0.0270050
$10$	−3.35471	−1.06085
$11$	4.04528	1.21970	0.609850	−	0.792517i	$-0.291230\pi$
$11$	4.04528	1.21970	0.609850	+	0.792517i	$0.291230\pi$
$12$	1.67457	0.483406
$13$	−1.50039	−0.416134	−0.208067	−	0.978115i	$-0.566717\pi$
$13$	−1.50039	−0.416134	−0.208067	+	0.978115i	$0.566717\pi$
$14$	−1.18393	−0.316419
$15$	−3.42606	−0.884605
$16$	−4.99789	−1.24947
$17$	2.94271	0.713711	0.356855	−	0.934160i	$-0.383849\pi$
$17$	2.94271	0.713711	0.356855	+	0.934160i	$0.383849\pi$
$18$	0.139243	0.0328198
$19$	−7.75532	−1.77919	−0.889596	−	0.456748i	$-0.849014\pi$
$19$	−7.75532	−1.77919	−0.889596	+	0.456748i	$0.849014\pi$
$20$	−1.86210	−0.416379
$21$	−1.20912	−0.263851
$22$	6.95273	1.48233
$23$	−0.794200	−0.165602	−0.0828011	−	0.996566i	$-0.526387\pi$
$23$	−0.794200	−0.165602	−0.0828011	+	0.996566i	$0.526387\pi$
$24$	−3.15558	−0.644129
$25$	−1.19025	−0.238050
$26$	−2.57876	−0.505737
$27$	−5.12364	−0.986045
$28$	−0.657168	−0.124193
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−5.88846	−1.07508
$31$	2.08878	0.375155	0.187578	−	0.982250i	$-0.439936\pi$
$31$	2.08878	0.375155	0.187578	+	0.982250i	$0.439936\pi$
$32$	−4.99447	−0.882906
$33$	7.10061	1.23606
$34$	5.05770	0.867389
$35$	1.34452	0.227266
$36$	0.0772896	0.0128816
$37$	8.60972	1.41543	0.707715	−	0.706498i	$-0.249726\pi$
$37$	8.60972	1.41543	0.707715	+	0.706498i	$0.249726\pi$
$38$	−13.3293	−2.16229
$39$	−2.63361	−0.421715
$40$	3.50897	0.554817
$41$	−5.94982	−0.929205	−0.464603	−	0.885519i	$-0.653803\pi$
$41$	−5.94982	−0.929205	−0.464603	+	0.885519i	$0.653803\pi$
$42$	−2.07814	−0.320663
$43$	9.77228	1.49026	0.745129	−	0.666920i	$-0.232388\pi$
$43$	9.77228	1.49026	0.745129	+	0.666920i	$0.232388\pi$
$44$	3.85926	0.581806
$45$	−0.158130	−0.0235726
$46$	−1.36501	−0.201260
$47$	−5.23972	−0.764292	−0.382146	−	0.924102i	$-0.624815\pi$
$47$	−5.23972	−0.764292	−0.382146	+	0.924102i	$0.624815\pi$
$48$	−8.77270	−1.26623
$49$	−6.52549	−0.932213
$50$	−2.04572	−0.289308
$51$	5.16528	0.723284
$52$	−1.43140	−0.198499
$53$	−10.7066	−1.47067	−0.735334	−	0.677705i	$-0.762975\pi$
$53$	−10.7066	−1.47067	−0.735334	+	0.677705i	$0.762975\pi$
$54$	−8.80613	−1.19836
$55$	−7.89582	−1.06467
$56$	1.23838	0.165485
$57$	−13.6128	−1.80306
$58$	−1.71872	−0.225680
$59$	−5.95819	−0.775691	−0.387845	−	0.921724i	$-0.626781\pi$
$59$	−5.95819	−0.775691	−0.387845	+	0.921724i	$0.626781\pi$
$60$	−3.26851	−0.421963
$61$	−13.6888	−1.75267	−0.876334	−	0.481704i	$-0.840018\pi$
$61$	−13.6888	−1.75267	−0.876334	+	0.481704i	$0.840018\pi$
$62$	3.59003	0.455934
$63$	−0.0558067	−0.00703099
$64$	1.41165	0.176456
$65$	2.92855	0.363242
$66$	12.2040	1.50221
$67$	−12.4195	−1.51728	−0.758641	−	0.651508i	$-0.774137\pi$
$67$	−12.4195	−1.51728	−0.758641	+	0.651508i	$0.774137\pi$
$68$	2.80739	0.340446
$69$	−1.39404	−0.167823
$70$	2.31087	0.276202
$71$	−8.41409	−0.998569	−0.499285	−	0.866438i	$-0.666404\pi$
$71$	−8.41409	−0.998569	−0.499285	+	0.866438i	$0.666404\pi$
$72$	−0.145646	−0.0171645
$73$	−9.00205	−1.05361	−0.526805	−	0.849986i	$-0.676610\pi$
$73$	−9.00205	−1.05361	−0.526805	+	0.849986i	$0.676610\pi$
$74$	14.7977	1.72020
$75$	−2.08923	−0.241243
$76$	−7.39870	−0.848689
$77$	−2.78657	−0.317559
$78$	−4.52646	−0.512520
$79$	−5.70166	−0.641487	−0.320744	−	0.947166i	$-0.603933\pi$
$79$	−5.70166	−0.641487	−0.320744	+	0.947166i	$0.603933\pi$
$80$	9.75516	1.09066
$81$	−9.23648	−1.02628
$82$	−10.2261	−1.12928
$83$	1.22209	0.134142	0.0670710	−	0.997748i	$-0.478635\pi$
$83$	1.22209	0.134142	0.0670710	+	0.997748i	$0.478635\pi$
$84$	−1.15351	−0.125859
$85$	−5.74374	−0.622996
$86$	16.7959	1.81114
$87$	−1.75528	−0.188186
$88$	−7.27245	−0.775246
$89$	9.54658	1.01194	0.505968	−	0.862552i	$-0.331135\pi$
$89$	9.54658	1.01194	0.505968	+	0.862552i	$0.331135\pi$
$90$	−0.271782	−0.0286483
$91$	1.03354	0.108344
$92$	−0.757679	−0.0789935
$93$	3.66639	0.380187
$94$	−9.00564	−0.928861
$95$	15.1373	1.55305
$96$	−8.76670	−0.894748
$97$	11.6981	1.18776	0.593880	−	0.804554i	$-0.297595\pi$
$97$	11.6981	1.18776	0.593880	+	0.804554i	$0.297595\pi$
$98$	−11.2155	−1.13294
$99$	0.327729	0.0329380
$100$	−1.13552	−0.113552
$101$	2.51368	0.250120	0.125060	−	0.992149i	$-0.460088\pi$
$101$	2.51368	0.250120	0.125060	+	0.992149i	$0.460088\pi$
$102$	8.87769	0.879023
$103$	14.0850	1.38783	0.693917	−	0.720055i	$-0.255884\pi$
$103$	14.0850	1.38783	0.693917	+	0.720055i	$0.255884\pi$
$104$	2.69735	0.264497
$105$	2.36002	0.230314
$106$	−18.4017	−1.78734
$107$	−3.73347	−0.360928	−0.180464	−	0.983582i	$-0.557760\pi$
$107$	−3.73347	−0.360928	−0.180464	+	0.983582i	$0.557760\pi$
$108$	−4.88803	−0.470351
$109$	8.37259	0.801949	0.400974	−	0.916089i	$-0.368672\pi$
$109$	8.37259	0.801949	0.400974	+	0.916089i	$0.368672\pi$
$110$	−13.5707	−1.29392
$111$	15.1125	1.43441
$112$	3.44276	0.325310
$113$	8.49458	0.799103	0.399552	−	0.916711i	$-0.369166\pi$
$113$	8.49458	0.799103	0.399552	+	0.916711i	$0.369166\pi$
$114$	−23.3966	−2.19129
$115$	1.55017	0.144554
$116$	−0.954015	−0.0885781
$117$	−0.121554	−0.0112377
$118$	−10.2405	−0.942714
$119$	−2.02706	−0.185821
$120$	6.15923	0.562259
$121$	5.36432	0.487666
$122$	−23.5272	−2.13006
$123$	−10.4436	−0.941668
$124$	1.99272	0.178952
$125$	12.0825	1.08069
$126$	−0.0959164	−0.00854491
$127$	2.12728	0.188765	0.0943827	−	0.995536i	$-0.469912\pi$
$127$	2.12728	0.188765	0.0943827	+	0.995536i	$0.469912\pi$
$128$	12.4152	1.09736
$129$	17.1531	1.51025
$130$	5.03338	0.441456
$131$	9.86967	0.862317	0.431159	−	0.902276i	$-0.358105\pi$
$131$	9.86967	0.862317	0.431159	+	0.902276i	$0.358105\pi$
$132$	6.77410	0.589609
$133$	5.34220	0.463228
$134$	−21.3457	−1.84399
$135$	10.0006	0.860716
$136$	−5.29028	−0.453638
$137$	−6.59722	−0.563639	−0.281819	−	0.959467i	$-0.590938\pi$
$137$	−6.59722	−0.563639	−0.281819	+	0.959467i	$0.590938\pi$
$138$	−2.39598	−0.203959
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	1.28270	0.108408
$141$	−9.19719	−0.774543
$142$	−14.4615	−1.21358
$143$	−6.06951	−0.507558
$144$	−0.404904	−0.0337420
$145$	1.95186	0.162093
$146$	−15.4721	−1.28048
$147$	−11.4541	−0.944717
$148$	8.21381	0.675171
$149$	−20.5444	−1.68306	−0.841530	−	0.540210i	$-0.818345\pi$
$149$	−20.5444	−1.68306	−0.841530	+	0.540210i	$0.818345\pi$
$150$	−3.59081	−0.293188
$151$	9.59495	0.780826	0.390413	−	0.920640i	$-0.372332\pi$
$151$	9.59495	0.780826	0.390413	+	0.920640i	$0.372332\pi$
$152$	13.9422	1.13086
$153$	0.238404	0.0192738
$154$	−4.78935	−0.385936
$155$	−4.07699	−0.327472
$156$	−2.51251	−0.201162
$157$	4.53231	0.361718	0.180859	−	0.983509i	$-0.442112\pi$
$157$	4.53231	0.361718	0.180859	+	0.983509i	$0.442112\pi$
$158$	−9.79959	−0.779613
$159$	−18.7932	−1.49039
$160$	9.74849	0.770686
$161$	0.547080	0.0431159
$162$	−15.8750	−1.24726
$163$	−2.89413	−0.226686	−0.113343	−	0.993556i	$-0.536156\pi$
$163$	−2.89413	−0.226686	−0.113343	+	0.993556i	$0.536156\pi$
$164$	−5.67622	−0.443238
$165$	−13.8594	−1.07895
$166$	2.10044	0.163026
$167$	15.7156	1.21611	0.608054	−	0.793896i	$-0.291950\pi$
$167$	15.7156	1.21611	0.608054	+	0.793896i	$0.291950\pi$
$168$	2.17370	0.167705
$169$	−10.7488	−0.826832
$170$	−9.87191	−0.757141
$171$	−0.628298	−0.0480471
$172$	9.32290	0.710865
$173$	17.8651	1.35826	0.679128	−	0.734020i	$-0.262358\pi$
$173$	17.8651	1.35826	0.679128	+	0.734020i	$0.262358\pi$
$174$	−3.01685	−0.228707
$175$	0.819898	0.0619785
$176$	−20.2179	−1.52398
$177$	−10.4583	−0.786095
$178$	16.4079	1.22983
$179$	−5.19422	−0.388234	−0.194117	−	0.980978i	$-0.562184\pi$
$179$	−5.19422	−0.388234	−0.194117	+	0.980978i	$0.562184\pi$
$180$	−0.150858	−0.0112443
$181$	16.0151	1.19039	0.595197	−	0.803580i	$-0.297074\pi$
$181$	16.0151	1.19039	0.595197	+	0.803580i	$0.297074\pi$
$182$	1.77636	0.131673
$183$	−24.0277	−1.77618
$184$	1.42778	0.105257
$185$	−16.8049	−1.23552
$186$	6.30152	0.462050
$187$	11.9041	0.870513
$188$	−4.99877	−0.364573
$189$	3.52939	0.256725
$190$	26.0168	1.88746
$191$	−7.58766	−0.549024	−0.274512	−	0.961584i	$-0.588516\pi$
$191$	−7.58766	−0.549024	−0.274512	+	0.961584i	$0.588516\pi$
$192$	2.47784	0.178823
$193$	−19.3123	−1.39013	−0.695064	−	0.718948i	$-0.744624\pi$
$193$	−19.3123	−1.39013	−0.695064	+	0.718948i	$0.744624\pi$
$194$	20.1058	1.44351
$195$	5.14044	0.368114
$196$	−6.22542	−0.444673
$197$	−0.492545	−0.0350924	−0.0175462	−	0.999846i	$-0.505585\pi$
$197$	−0.492545	−0.0350924	−0.0175462	+	0.999846i	$0.505585\pi$
$198$	0.563276	0.0400303
$199$	27.1134	1.92202	0.961010	−	0.276515i	$-0.0891796\pi$
$199$	27.1134	1.92202	0.961010	+	0.276515i	$0.0891796\pi$
$200$	2.13979	0.151306
$201$	−21.7997	−1.53763
$202$	4.32032	0.303976
$203$	0.688844	0.0483474
$204$	4.92775	0.345012
$205$	11.6132	0.811101
$206$	24.2082	1.68666
$207$	−0.0643422	−0.00447209
$208$	7.49879	0.519948
$209$	−31.3725	−2.17008
$210$	4.05623	0.279906
$211$	11.6242	0.800240	0.400120	−	0.916463i	$-0.368968\pi$
$211$	11.6242	0.800240	0.400120	+	0.916463i	$0.368968\pi$
$212$	−10.2143	−0.701520
$213$	−14.7691	−1.01196
$214$	−6.41681	−0.438644
$215$	−19.0741	−1.30084
$216$	9.21108	0.626734
$217$	−1.43884	−0.0976749
$218$	14.3902	0.974626
$219$	−15.8011	−1.06774
$220$	−7.53273	−0.507857
$221$	−4.41521	−0.296999
$222$	25.9742	1.74327
$223$	−3.70708	−0.248244	−0.124122	−	0.992267i	$-0.539611\pi$
$223$	−3.70708	−0.248244	−0.124122	+	0.992267i	$0.539611\pi$
$224$	3.44041	0.229872
$225$	−0.0964284	−0.00642856
$226$	14.5999	0.971168
$227$	−13.3848	−0.888381	−0.444190	−	0.895932i	$-0.646509\pi$
$227$	−13.3848	−0.888381	−0.444190	+	0.895932i	$0.646509\pi$
$228$	−12.9868	−0.860072
$229$	20.6123	1.36210	0.681048	−	0.732238i	$-0.261524\pi$
$229$	20.6123	1.36210	0.681048	+	0.732238i	$0.261524\pi$
$230$	2.66431	0.175679
$231$	−4.89121	−0.321818
$232$	1.79776	0.118029
$233$	−10.0789	−0.660293	−0.330147	−	0.943930i	$-0.607098\pi$
$233$	−10.0789	−0.660293	−0.330147	+	0.943930i	$0.607098\pi$
$234$	−0.208919	−0.0136574
$235$	10.2272	0.667148
$236$	−5.68421	−0.370010
$237$	−10.0080	−0.650091
$238$	−3.48397	−0.225832
$239$	−19.3401	−1.25101	−0.625503	−	0.780222i	$-0.715106\pi$
$239$	−19.3401	−1.25101	−0.625503	+	0.780222i	$0.715106\pi$
$240$	17.1231	1.10529
$241$	30.8332	1.98614	0.993071	−	0.117516i	$-0.0374931\pi$
$241$	30.8332	1.98614	0.993071	+	0.117516i	$0.0374931\pi$
$242$	9.21980	0.592671
$243$	−0.841705	−0.0539954
$244$	−13.0593	−0.836036
$245$	12.7368	0.813727
$246$	−17.9497	−1.14443
$247$	11.6360	0.740383
$248$	−3.75512	−0.238450
$249$	2.14512	0.135941
$250$	20.7665	1.31339
$251$	28.9303	1.82607	0.913033	−	0.407885i	$-0.133734\pi$
$251$	28.9303	1.82607	0.913033	+	0.407885i	$0.133734\pi$
$252$	−0.0532405	−0.00335383
$253$	−3.21276	−0.201985
$254$	3.65621	0.229411
$255$	−10.0819	−0.631352
$256$	18.5150	1.15719
$257$	−27.9298	−1.74222	−0.871108	−	0.491092i	$-0.836598\pi$
$257$	−27.9298	−1.74222	−0.871108	+	0.491092i	$0.836598\pi$
$258$	29.4815	1.83544
$259$	−5.93075	−0.368519
$260$	2.79388	0.173269
$261$	−0.0810151	−0.00501471
$262$	16.9633	1.04799
$263$	−17.2556	−1.06402	−0.532012	−	0.846737i	$-0.678564\pi$
$263$	−17.2556	−1.06402	−0.532012	+	0.846737i	$0.678564\pi$
$264$	−12.7652	−0.785644
$265$	20.8978	1.28374
$266$	9.18178	0.562971
$267$	16.7569	1.02551
$268$	−11.8484	−0.723755
$269$	−8.33904	−0.508440	−0.254220	−	0.967146i	$-0.581819\pi$
$269$	−8.33904	−0.508440	−0.254220	+	0.967146i	$0.581819\pi$
$270$	17.1883	1.04605
$271$	0.844323	0.0512890	0.0256445	−	0.999671i	$-0.491836\pi$
$271$	0.844323	0.0512890	0.0256445	+	0.999671i	$0.491836\pi$
$272$	−14.7073	−0.891761
$273$	1.81415	0.109797
$274$	−11.3388	−0.685003
$275$	−4.81491	−0.290350
$276$	−1.32994	−0.0800530
$277$	−18.7343	−1.12564	−0.562818	−	0.826581i	$-0.690283\pi$
$277$	−18.7343	−1.12564	−0.562818	+	0.826581i	$0.690283\pi$
$278$	−1.71872	−0.103082
$279$	0.169222	0.0101311
$280$	−2.41713	−0.144451
$281$	17.7993	1.06182	0.530908	−	0.847429i	$-0.321851\pi$
$281$	17.7993	1.06182	0.530908	+	0.847429i	$0.321851\pi$
$282$	−15.8074	−0.941319
$283$	−12.3650	−0.735025	−0.367512	−	0.930019i	$-0.619790\pi$
$283$	−12.3650	−0.735025	−0.367512	+	0.930019i	$0.619790\pi$
$284$	−8.02717	−0.476325
$285$	26.5702	1.57388
$286$	−10.4318	−0.616847
$287$	4.09850	0.241927
$288$	−0.404627	−0.0238429
$289$	−8.34048	−0.490617
$290$	3.35471	0.196995
$291$	20.5334	1.20369
$292$	−8.58810	−0.502580
$293$	−8.93522	−0.522001	−0.261001	−	0.965339i	$-0.584052\pi$
$293$	−8.93522	−0.522001	−0.261001	+	0.965339i	$0.584052\pi$
$294$	−19.6864	−1.14814
$295$	11.6295	0.677098
$296$	−15.4782	−0.899653
$297$	−20.7266	−1.20268
$298$	−35.3101	−2.04546
$299$	1.19161	0.0689127
$300$	−1.99316	−0.115075
$301$	−6.73157	−0.388001
$302$	16.4911	0.948955
$303$	4.41221	0.253475
$304$	38.7602	2.22305
$305$	26.7185	1.52990
$306$	0.409750	0.0234239
$307$	−14.1899	−0.809861	−0.404931	−	0.914347i	$-0.632704\pi$
$307$	−14.1899	−0.809861	−0.404931	+	0.914347i	$0.632704\pi$
$308$	−2.65843	−0.151478
$309$	24.7231	1.40645
$310$	−7.00723	−0.397984
$311$	1.28307	0.0727565	0.0363782	−	0.999338i	$-0.488418\pi$
$311$	1.28307	0.0727565	0.0363782	+	0.999338i	$0.488418\pi$
$312$	4.73460	0.268044
$313$	−16.8629	−0.953145	−0.476573	−	0.879135i	$-0.658121\pi$
$313$	−16.8629	−0.953145	−0.476573	+	0.879135i	$0.658121\pi$
$314$	7.78980	0.439604
$315$	0.108927	0.00613733
$316$	−5.43947	−0.305994
$317$	11.9231	0.669666	0.334833	−	0.942277i	$-0.391320\pi$
$317$	11.9231	0.669666	0.334833	+	0.942277i	$0.391320\pi$
$318$	−32.3003	−1.81131
$319$	−4.04528	−0.226492
$320$	−2.75534	−0.154028
$321$	−6.55329	−0.365769
$322$	0.940280	0.0523997
$323$	−22.8216	−1.26983
$324$	−8.81174	−0.489541
$325$	1.78585	0.0990609
$326$	−4.97421	−0.275496
$327$	14.6963	0.812705
$328$	10.6963	0.590607
$329$	3.60935	0.198990
$330$	−23.8205	−1.31127
$331$	15.0499	0.827217	0.413608	−	0.910455i	$-0.364268\pi$
$331$	15.0499	0.827217	0.413608	+	0.910455i	$0.364268\pi$
$332$	1.16589	0.0639868
$333$	0.697517	0.0382237
$334$	27.0108	1.47796
$335$	24.2411	1.32443
$336$	6.04302	0.329674
$337$	−7.26395	−0.395693	−0.197846	−	0.980233i	$-0.563395\pi$
$337$	−7.26395	−0.395693	−0.197846	+	0.980233i	$0.563395\pi$
$338$	−18.4743	−1.00487
$339$	14.9104	0.809821
$340$	−5.47962	−0.297174
$341$	8.44969	0.457577
$342$	−1.07987	−0.0583927
$343$	9.31695	0.503068
$344$	−17.5682	−0.947214
$345$	2.72098	0.146492
$346$	30.7051	1.65072
$347$	5.25630	0.282173	0.141086	−	0.989997i	$-0.454940\pi$
$347$	5.25630	0.282173	0.141086	+	0.989997i	$0.454940\pi$
$348$	−1.67457	−0.0897662
$349$	−7.46762	−0.399732	−0.199866	−	0.979823i	$-0.564051\pi$
$349$	−7.46762	−0.399732	−0.199866	+	0.979823i	$0.564051\pi$
$350$	1.40918	0.0753238
$351$	7.68747	0.410327
$352$	−20.2041	−1.07688
$353$	15.8410	0.843129	0.421565	−	0.906798i	$-0.361481\pi$
$353$	15.8410	0.843129	0.421565	+	0.906798i	$0.361481\pi$
$354$	−17.9750	−0.955358
$355$	16.4231	0.871648
$356$	9.10759	0.482701
$357$	−3.55807	−0.188313
$358$	−8.92743	−0.471829
$359$	21.7485	1.14784	0.573921	−	0.818911i	$-0.305422\pi$
$359$	21.7485	1.14784	0.573921	+	0.818911i	$0.305422\pi$
$360$	0.284280	0.0149828
$361$	41.1450	2.16553
$362$	27.5256	1.44671
$363$	9.41590	0.494207
$364$	0.986009	0.0516809
$365$	17.5707	0.919694
$366$	−41.2969	−2.15863
$367$	−21.0567	−1.09915	−0.549576	−	0.835444i	$-0.685211\pi$
$367$	−21.0567	−1.09915	−0.549576	+	0.835444i	$0.685211\pi$
$368$	3.96932	0.206915
$369$	−0.482025	−0.0250932
$370$	−28.8831	−1.50156
$371$	7.37519	0.382901
$372$	3.49779	0.181352
$373$	29.0607	1.50470	0.752352	−	0.658762i	$-0.228919\pi$
$373$	29.0607	1.50470	0.752352	+	0.658762i	$0.228919\pi$
$374$	20.4598	1.05795
$375$	21.2082	1.09519
$376$	9.41976	0.485787
$377$	1.50039	0.0772742
$378$	6.06605	0.312004
$379$	−19.8589	−1.02008	−0.510042	−	0.860149i	$-0.670370\pi$
$379$	−19.8589	−1.02008	−0.510042	+	0.860149i	$0.670370\pi$
$380$	14.4412	0.740818
$381$	3.73397	0.191297
$382$	−13.0411	−0.667240
$383$	7.35515	0.375831	0.187915	−	0.982185i	$-0.439827\pi$
$383$	7.35515	0.375831	0.187915	+	0.982185i	$0.439827\pi$
$384$	21.7921	1.11208
$385$	5.43899	0.277196
$386$	−33.1925	−1.68945
$387$	0.791702	0.0402445
$388$	11.1601	0.566571
$389$	11.7613	0.596321	0.298160	−	0.954516i	$-0.403627\pi$
$389$	11.7613	0.596321	0.298160	+	0.954516i	$0.403627\pi$
$390$	8.83499	0.447377
$391$	−2.33710	−0.118192
$392$	11.7313	0.592519
$393$	17.3241	0.873883
$394$	−0.846550	−0.0426485
$395$	11.1288	0.559952
$396$	0.312658	0.0157117
$397$	−33.2554	−1.66904	−0.834519	−	0.550979i	$-0.814255\pi$
$397$	−33.2554	−1.66904	−0.834519	+	0.550979i	$0.814255\pi$
$398$	46.6005	2.33587
$399$	9.37708	0.469441
$400$	5.94875	0.297437
$401$	−11.2410	−0.561351	−0.280675	−	0.959803i	$-0.590558\pi$
$401$	−11.2410	−0.561351	−0.280675	+	0.959803i	$0.590558\pi$
$402$	−37.4677	−1.86872
$403$	−3.13398	−0.156115
$404$	2.39809	0.119309
$405$	18.0283	0.895833
$406$	1.18393	0.0587576
$407$	34.8288	1.72640
$408$	−9.28593	−0.459722
$409$	−0.662823	−0.0327745	−0.0163872	−	0.999866i	$-0.505216\pi$
$409$	−0.662823	−0.0327745	−0.0163872	+	0.999866i	$0.505216\pi$
$410$	19.9599	0.985749
$411$	−11.5800	−0.571199
$412$	13.4373	0.662007
$413$	4.10426	0.201958
$414$	−0.110586	−0.00543503
$415$	−2.38535	−0.117092
$416$	7.49367	0.367407
$417$	−1.75528	−0.0859565
$418$	−53.9207	−2.63735
$419$	−34.5934	−1.69000	−0.844999	−	0.534767i	$-0.820399\pi$
$419$	−34.5934	−1.69000	−0.844999	+	0.534767i	$0.820399\pi$
$420$	2.25150	0.109862
$421$	24.7211	1.20483	0.602417	−	0.798182i	$-0.294204\pi$
$421$	24.7211	1.20483	0.602417	+	0.798182i	$0.294204\pi$
$422$	19.9787	0.972550
$423$	−0.424496	−0.0206397
$424$	19.2479	0.934763
$425$	−3.50256	−0.169899
$426$	−25.3840	−1.22986
$427$	9.42943	0.456322
$428$	−3.56179	−0.172166
$429$	−10.6537	−0.514366
$430$	−32.7831	−1.58094
$431$	−8.40593	−0.404899	−0.202450	−	0.979293i	$-0.564890\pi$
$431$	−8.40593	−0.404899	−0.202450	+	0.979293i	$0.564890\pi$
$432$	25.6074	1.23204
$433$	−25.8676	−1.24312	−0.621560	−	0.783367i	$-0.713501\pi$
$433$	−25.8676	−1.24312	−0.621560	+	0.783367i	$0.713501\pi$
$434$	−2.47297	−0.118706
$435$	3.42606	0.164267
$436$	7.98758	0.382536
$437$	6.15928	0.294638
$438$	−27.1578	−1.29765
$439$	−9.45027	−0.451037	−0.225518	−	0.974239i	$-0.572408\pi$
$439$	−9.45027	−0.451037	−0.225518	+	0.974239i	$0.572408\pi$
$440$	14.1948	0.676710
$441$	−0.528663	−0.0251744
$442$	−7.58854	−0.360950
$443$	−12.4331	−0.590715	−0.295357	−	0.955387i	$-0.595439\pi$
$443$	−12.4331	−0.590715	−0.295357	+	0.955387i	$0.595439\pi$
$444$	14.4175	0.684226
$445$	−18.6336	−0.883316
$446$	−6.37145	−0.301697
$447$	−36.0612	−1.70563
$448$	−0.972407	−0.0459419
$449$	15.6685	0.739443	0.369722	−	0.929143i	$-0.379453\pi$
$449$	15.6685	0.739443	0.369722	+	0.929143i	$0.379453\pi$
$450$	−0.165734	−0.00781277
$451$	−24.0687	−1.13335
$452$	8.10396	0.381178
$453$	16.8418	0.791299
$454$	−23.0048	−1.07967
$455$	−2.01732	−0.0945732
$456$	24.4725	1.14603
$457$	−6.17990	−0.289083	−0.144542	−	0.989499i	$-0.546171\pi$
$457$	−6.17990	−0.289083	−0.144542	+	0.989499i	$0.546171\pi$
$458$	35.4268	1.65539
$459$	−15.0774	−0.703751
$460$	1.47888	0.0689532
$461$	6.93911	0.323187	0.161593	−	0.986857i	$-0.448337\pi$
$461$	6.93911	0.323187	0.161593	+	0.986857i	$0.448337\pi$
$462$	−8.40665	−0.391113
$463$	−25.1188	−1.16737	−0.583684	−	0.811981i	$-0.698389\pi$
$463$	−25.1188	−1.16737	−0.583684	+	0.811981i	$0.698389\pi$
$464$	4.99789	0.232021
$465$	−7.15627	−0.331864
$466$	−17.3229	−0.802469
$467$	−11.0470	−0.511194	−0.255597	−	0.966783i	$-0.582272\pi$
$467$	−11.0470	−0.511194	−0.255597	+	0.966783i	$0.582272\pi$
$468$	−0.115965	−0.00536047
$469$	8.55509	0.395037
$470$	17.5777	0.810800
$471$	7.95549	0.366570
$472$	10.7114	0.493032
$473$	39.5316	1.81767
$474$	−17.2010	−0.790070
$475$	9.23079	0.423538
$476$	−1.93385	−0.0886379
$477$	−0.867398	−0.0397154
$478$	−33.2403	−1.52038
$479$	16.9012	0.772235	0.386118	−	0.922450i	$-0.373816\pi$
$479$	16.9012	0.772235	0.386118	+	0.922450i	$0.373816\pi$
$480$	17.1114	0.781023
$481$	−12.9180	−0.589008
$482$	52.9938	2.41380
$483$	0.960279	0.0436942
$484$	5.11765	0.232620
$485$	−22.8330	−1.03679
$486$	−1.44666	−0.0656218
$487$	−6.13744	−0.278114	−0.139057	−	0.990284i	$-0.544407\pi$
$487$	−6.13744	−0.278114	−0.139057	+	0.990284i	$0.544407\pi$
$488$	24.6091	1.11400
$489$	−5.08002	−0.229726
$490$	21.8911	0.988940
$491$	−21.6518	−0.977131	−0.488566	−	0.872527i	$-0.662480\pi$
$491$	−21.6518	−0.977131	−0.488566	+	0.872527i	$0.662480\pi$
$492$	−9.96336	−0.449183
$493$	−2.94271	−0.132533
$494$	19.9991	0.899803
$495$	−0.639680	−0.0287515
$496$	−10.4395	−0.468746
$497$	5.79600	0.259986
$498$	3.68686	0.165212
$499$	−24.8528	−1.11256	−0.556282	−	0.830994i	$-0.687773\pi$
$499$	−24.8528	−1.11256	−0.556282	+	0.830994i	$0.687773\pi$
$500$	11.5269	0.515498
$501$	27.5853	1.23242
$502$	49.7233	2.21926
$503$	5.59705	0.249560	0.124780	−	0.992184i	$-0.460177\pi$
$503$	5.59705	0.249560	0.124780	+	0.992184i	$0.460177\pi$
$504$	0.100327	0.00446892
$505$	−4.90634	−0.218329
$506$	−5.52186	−0.245477
$507$	−18.8672	−0.837922
$508$	2.02946	0.0900426
$509$	−7.23324	−0.320608	−0.160304	−	0.987068i	$-0.551247\pi$
$509$	−7.23324	−0.320608	−0.160304	+	0.987068i	$0.551247\pi$
$510$	−17.3280	−0.767296
$511$	6.20101	0.274316
$512$	6.99179	0.308997
$513$	39.7355	1.75436
$514$	−48.0037	−2.11735
$515$	−27.4918	−1.21144
$516$	16.3643	0.720399
$517$	−21.1962	−0.932206
$518$	−10.1933	−0.447869
$519$	31.3582	1.37647
$520$	−5.26483	−0.230878
$521$	34.8298	1.52592	0.762962	−	0.646444i	$-0.223745\pi$
$521$	34.8298	1.52592	0.762962	+	0.646444i	$0.223745\pi$
$522$	−0.139243	−0.00609448
$523$	−29.8191	−1.30390	−0.651950	−	0.758262i	$-0.726049\pi$
$523$	−29.8191	−1.30390	−0.651950	+	0.758262i	$0.726049\pi$
$524$	9.41582	0.411332
$525$	1.43915	0.0628098
$526$	−29.6576	−1.29313
$527$	6.14665	0.267752
$528$	−35.4881	−1.54442
$529$	−22.3692	−0.972576
$530$	35.9176	1.56016
$531$	−0.482703	−0.0209475
$532$	5.09655	0.220963
$533$	8.92706	0.386674
$534$	28.8006	1.24632
$535$	7.28720	0.315053
$536$	22.3273	0.964391
$537$	−9.11732	−0.393441
$538$	−14.3325	−0.617918
$539$	−26.3975	−1.13702
$540$	9.54074	0.410568
$541$	6.66538	0.286567	0.143284	−	0.989682i	$-0.454234\pi$
$541$	6.66538	0.286567	0.143284	+	0.989682i	$0.454234\pi$
$542$	1.45116	0.0623326
$543$	28.1110	1.20636
$544$	−14.6973	−0.630140
$545$	−16.3421	−0.700019
$546$	3.11802	0.133439
$547$	−9.21898	−0.394175	−0.197088	−	0.980386i	$-0.563148\pi$
$547$	−9.21898	−0.394175	−0.197088	+	0.980386i	$0.563148\pi$
$548$	−6.29385	−0.268860
$549$	−1.10900	−0.0473308
$550$	−8.27551	−0.352869
$551$	7.75532	0.330388
$552$	2.50616	0.106669
$553$	3.92755	0.167017
$554$	−32.1991	−1.36801
$555$	−29.4974	−1.25210
$556$	−0.954015	−0.0404593
$557$	17.6281	0.746928	0.373464	−	0.927645i	$-0.378170\pi$
$557$	17.6281	0.746928	0.373464	+	0.927645i	$0.378170\pi$
$558$	0.290847	0.0123125
$559$	−14.6623	−0.620147
$560$	−6.71978	−0.283963
$561$	20.8950	0.882188
$562$	30.5921	1.29045
$563$	−26.9966	−1.13777	−0.568885	−	0.822417i	$-0.692625\pi$
$563$	−26.9966	−1.13777	−0.568885	+	0.822417i	$0.692625\pi$
$564$	−8.77426	−0.369463
$565$	−16.5802	−0.697535
$566$	−21.2521	−0.893292
$567$	6.36249	0.267200
$568$	15.1265	0.634695
$569$	−11.1301	−0.466598	−0.233299	−	0.972405i	$-0.574952\pi$
$569$	−11.1301	−0.466598	−0.233299	+	0.972405i	$0.574952\pi$
$570$	45.6669	1.91277
$571$	8.31786	0.348092	0.174046	−	0.984738i	$-0.444316\pi$
$571$	8.31786	0.348092	0.174046	+	0.984738i	$0.444316\pi$
$572$	−5.79041	−0.242109
$573$	−13.3185	−0.556387
$574$	7.04419	0.294019
$575$	0.945298	0.0394217
$576$	0.114365	0.00476521
$577$	35.1719	1.46423	0.732113	−	0.681183i	$-0.238534\pi$
$577$	35.1719	1.46423	0.732113	+	0.681183i	$0.238534\pi$
$578$	−14.3350	−0.596257
$579$	−33.8985	−1.40877
$580$	1.86210	0.0773196
$581$	−0.841830	−0.0349250
$582$	35.2913	1.46287
$583$	−43.3114	−1.79377
$584$	16.1835	0.669679
$585$	0.237257	0.00980937
$586$	−15.3572	−0.634399
$587$	−22.6398	−0.934443	−0.467221	−	0.884140i	$-0.654745\pi$
$587$	−22.6398	−0.934443	−0.467221	+	0.884140i	$0.654745\pi$
$588$	−10.9274	−0.450637
$589$	−16.1991	−0.667474
$590$	19.9880	0.822892
$591$	−0.864556	−0.0355631
$592$	−43.0304	−1.76854
$593$	12.1079	0.497211	0.248606	−	0.968605i	$-0.420028\pi$
$593$	12.1079	0.497211	0.248606	+	0.968605i	$0.420028\pi$
$594$	−35.6233	−1.46164
$595$	3.95654	0.162202
$596$	−19.5996	−0.802832
$597$	47.5917	1.94780
$598$	2.04805	0.0837511
$599$	−20.7509	−0.847859	−0.423929	−	0.905695i	$-0.639350\pi$
$599$	−20.7509	−0.847859	−0.423929	+	0.905695i	$0.639350\pi$
$600$	3.75593	0.153335
$601$	22.4267	0.914806	0.457403	−	0.889260i	$-0.348780\pi$
$601$	22.4267	0.914806	0.457403	+	0.889260i	$0.348780\pi$
$602$	−11.5697	−0.471547
$603$	−1.00617	−0.0409743
$604$	9.15373	0.372460
$605$	−10.4704	−0.425682
$606$	7.58338	0.308054
$607$	−4.90714	−0.199175	−0.0995873	−	0.995029i	$-0.531752\pi$
$607$	−4.90714	−0.199175	−0.0995873	+	0.995029i	$0.531752\pi$
$608$	38.7337	1.57086
$609$	1.20912	0.0489958
$610$	45.9218	1.85932
$611$	7.86164	0.318048
$612$	0.227441	0.00919374
$613$	−21.6333	−0.873759	−0.436880	−	0.899520i	$-0.643916\pi$
$613$	−21.6333	−0.873759	−0.436880	+	0.899520i	$0.643916\pi$
$614$	−24.3886	−0.984242
$615$	20.3844	0.821980
$616$	5.00958	0.201842
$617$	−33.6290	−1.35385	−0.676926	−	0.736051i	$-0.736688\pi$
$617$	−33.6290	−1.35385	−0.676926	+	0.736051i	$0.736688\pi$
$618$	42.4922	1.70929
$619$	5.29588	0.212859	0.106430	−	0.994320i	$-0.466058\pi$
$619$	5.29588	0.212859	0.106430	+	0.994320i	$0.466058\pi$
$620$	−3.88951	−0.156207
$621$	4.06920	0.163291
$622$	2.20525	0.0884225
$623$	−6.57610	−0.263466
$624$	13.1625	0.526921
$625$	−17.6320	−0.705281
$626$	−28.9826	−1.15838
$627$	−55.0675	−2.19919
$628$	4.32390	0.172542
$629$	25.3359	1.01021
$630$	0.187215	0.00745883
$631$	19.3605	0.770729	0.385364	−	0.922765i	$-0.374076\pi$
$631$	19.3605	0.770729	0.385364	+	0.922765i	$0.374076\pi$
$632$	10.2502	0.407732
$633$	20.4037	0.810974
$634$	20.4925	0.813860
$635$	−4.15214	−0.164773
$636$	−17.9290	−0.710929
$637$	9.79080	0.387926
$638$	−6.95273	−0.275261
$639$	−0.681668	−0.0269664
$640$	−24.2327	−0.957880
$641$	−10.7249	−0.423608	−0.211804	−	0.977312i	$-0.567934\pi$
$641$	−10.7249	−0.423608	−0.211804	+	0.977312i	$0.567934\pi$
$642$	−11.2633	−0.444527
$643$	45.2871	1.78595	0.892973	−	0.450109i	$-0.148615\pi$
$643$	45.2871	1.78595	0.892973	+	0.450109i	$0.148615\pi$
$644$	0.521922	0.0205666
$645$	−33.4804	−1.31829
$646$	−39.2241	−1.54325
$647$	32.4019	1.27385	0.636924	−	0.770927i	$-0.280206\pi$
$647$	32.4019	1.27385	0.636924	+	0.770927i	$0.280206\pi$
$648$	16.6050	0.652305
$649$	−24.1026	−0.946109
$650$	3.06938	0.120391
$651$	−2.52557	−0.0989849
$652$	−2.76105	−0.108131
$653$	21.4058	0.837673	0.418836	−	0.908062i	$-0.362438\pi$
$653$	21.4058	0.837673	0.418836	+	0.908062i	$0.362438\pi$
$654$	25.2588	0.987698
$655$	−19.2642	−0.752715
$656$	29.7365	1.16102
$657$	−0.729302	−0.0284528
$658$	6.20348	0.241837
$659$	22.0479	0.858864	0.429432	−	0.903099i	$-0.358714\pi$
$659$	22.0479	0.858864	0.429432	+	0.903099i	$0.358714\pi$
$660$	−13.2221	−0.514668
$661$	19.3532	0.752753	0.376377	−	0.926467i	$-0.377170\pi$
$661$	19.3532	0.752753	0.376377	+	0.926467i	$0.377170\pi$
$662$	25.8666	1.00533
$663$	−7.74995	−0.300983
$664$	−2.19703	−0.0852612
$665$	−10.4272	−0.404350
$666$	1.19884	0.0464541
$667$	0.794200	0.0307515
$668$	14.9929	0.580093
$669$	−6.50697	−0.251574
$670$	41.6638	1.60961
$671$	−55.3750	−2.13773
$672$	6.03889	0.232955
$673$	−8.23811	−0.317556	−0.158778	−	0.987314i	$-0.550755\pi$
$673$	−8.23811	−0.317556	−0.158778	+	0.987314i	$0.550755\pi$
$674$	−12.4847	−0.480894
$675$	6.09843	0.234729
$676$	−10.2545	−0.394405
$677$	−41.3578	−1.58951	−0.794755	−	0.606931i	$-0.792400\pi$
$677$	−41.3578	−1.58951	−0.794755	+	0.606931i	$0.792400\pi$
$678$	25.6269	0.984194
$679$	−8.05815	−0.309243
$680$	10.3259	0.395979
$681$	−23.4941	−0.900296
$682$	14.5227	0.556103
$683$	14.5462	0.556596	0.278298	−	0.960495i	$-0.410230\pi$
$683$	14.5462	0.556596	0.278298	+	0.960495i	$0.410230\pi$
$684$	−0.599406	−0.0229189
$685$	12.8768	0.491999
$686$	16.0133	0.611390
$687$	36.1803	1.38037
$688$	−48.8407	−1.86203
$689$	16.0641	0.611995
$690$	4.67661	0.178036
$691$	−24.9636	−0.949660	−0.474830	−	0.880077i	$-0.657491\pi$
$691$	−24.9636	−0.949660	−0.474830	+	0.880077i	$0.657491\pi$
$692$	17.0435	0.647898
$693$	−0.225754	−0.00857569
$694$	9.03413	0.342931
$695$	1.95186	0.0740382
$696$	3.15558	0.119612
$697$	−17.5086	−0.663184
$698$	−12.8348	−0.485804
$699$	−17.6914	−0.669149
$700$	0.782195	0.0295642
$701$	3.93939	0.148789	0.0743944	−	0.997229i	$-0.476298\pi$
$701$	3.93939	0.148789	0.0743944	+	0.997229i	$0.476298\pi$
$702$	13.2127	0.498679
$703$	−66.7712	−2.51832
$704$	5.71053	0.215224
$705$	17.9516	0.676097
$706$	27.2263	1.02467
$707$	−1.73153	−0.0651209
$708$	−9.97739	−0.374973
$709$	22.2672	0.836263	0.418131	−	0.908387i	$-0.362685\pi$
$709$	22.2672	0.836263	0.418131	+	0.908387i	$0.362685\pi$
$710$	28.2268	1.05933
$711$	−0.461921	−0.0173234
$712$	−17.1625	−0.643190
$713$	−1.65891	−0.0621265
$714$	−6.11534	−0.228861
$715$	11.8468	0.443046
$716$	−4.95536	−0.185191
$717$	−33.9473	−1.26779
$718$	37.3797	1.39500
$719$	−47.6348	−1.77648	−0.888239	−	0.459381i	$-0.848071\pi$
$719$	−47.6348	−1.77648	−0.888239	+	0.459381i	$0.848071\pi$
$720$	0.790315	0.0294533
$721$	−9.70234	−0.361334
$722$	70.7169	2.63181
$723$	54.1210	2.01278
$724$	15.2787	0.567827
$725$	1.19025	0.0442049
$726$	16.1833	0.600620
$727$	47.0596	1.74534	0.872671	−	0.488308i	$-0.162386\pi$
$727$	47.0596	1.74534	0.872671	+	0.488308i	$0.162386\pi$
$728$	−1.85805	−0.0688639
$729$	26.2320	0.971556
$730$	30.1992	1.11772
$731$	28.7569	1.06361
$732$	−22.9228	−0.847249
$733$	16.5433	0.611041	0.305520	−	0.952186i	$-0.401170\pi$
$733$	16.5433	0.611041	0.305520	+	0.952186i	$0.401170\pi$
$734$	−36.1907	−1.33582
$735$	22.3567	0.824641
$736$	3.96661	0.146211
$737$	−50.2404	−1.85063
$738$	−0.828468	−0.0304963
$739$	−2.88096	−0.105978	−0.0529889	−	0.998595i	$-0.516875\pi$
$739$	−2.88096	−0.105978	−0.0529889	+	0.998595i	$0.516875\pi$
$740$	−16.0322	−0.589355
$741$	20.4245	0.750313
$742$	12.6759	0.465348
$743$	−39.9814	−1.46678	−0.733388	−	0.679810i	$-0.762062\pi$
$743$	−39.9814	−1.46678	−0.733388	+	0.679810i	$0.762062\pi$
$744$	−6.59129	−0.241648
$745$	40.0997	1.46914
$746$	49.9473	1.82870
$747$	0.0990079	0.00362251
$748$	11.3567	0.415241
$749$	2.57178	0.0939707
$750$	36.4510	1.33100
$751$	−1.79666	−0.0655611	−0.0327805	−	0.999463i	$-0.510436\pi$
$751$	−1.79666	−0.0655611	−0.0327805	+	0.999463i	$0.510436\pi$
$752$	26.1875	0.954961
$753$	50.7809	1.85056
$754$	2.57876	0.0939130
$755$	−18.7280	−0.681581
$756$	3.36709	0.122460
$757$	3.49527	0.127038	0.0635190	−	0.997981i	$-0.479768\pi$
$757$	3.49527	0.127038	0.0635190	+	0.997981i	$0.479768\pi$
$758$	−34.1320	−1.23973
$759$	−5.63931	−0.204694
$760$	−27.2132	−0.987126
$761$	−11.3180	−0.410279	−0.205139	−	0.978733i	$-0.565765\pi$
$761$	−11.3180	−0.410279	−0.205139	+	0.978733i	$0.565765\pi$
$762$	6.41767	0.232488
$763$	−5.76741	−0.208794
$764$	−7.23874	−0.261888
$765$	−0.465330	−0.0168240
$766$	12.6415	0.456755
$767$	8.93963	0.322791
$768$	32.4990	1.17271
$769$	51.6575	1.86282	0.931409	−	0.363974i	$-0.118580\pi$
$769$	51.6575	1.86282	0.931409	+	0.363974i	$0.118580\pi$
$770$	9.34812	0.336883
$771$	−49.0248	−1.76558
$772$	−18.4242	−0.663101
$773$	−7.41443	−0.266678	−0.133339	−	0.991070i	$-0.542570\pi$
$773$	−7.41443	−0.266678	−0.133339	+	0.991070i	$0.542570\pi$
$774$	1.36072	0.0489100
$775$	−2.48617	−0.0893059
$776$	−21.0303	−0.754945
$777$	−10.4101	−0.373462
$778$	20.2144	0.724722
$779$	46.1427	1.65324
$780$	4.90405	0.175593
$781$	−34.0374	−1.21795
$782$	−4.01683	−0.143641
$783$	5.12364	0.183104
$784$	32.6137	1.16477
$785$	−8.84643	−0.315743
$786$	29.7753	1.06205
$787$	18.5908	0.662689	0.331344	−	0.943510i	$-0.392498\pi$
$787$	18.5908	0.662689	0.331344	+	0.943510i	$0.392498\pi$
$788$	−0.469896	−0.0167393
$789$	−30.2884	−1.07829
$790$	19.1274	0.680522
$791$	−5.85144	−0.208053
$792$	−0.589178	−0.0209355
$793$	20.5385	0.729345
$794$	−57.1568	−2.02842
$795$	36.6816	1.30096
$796$	25.8666	0.916818
$797$	18.5295	0.656350	0.328175	−	0.944617i	$-0.393566\pi$
$797$	18.5295	0.656350	0.328175	+	0.944617i	$0.393566\pi$
$798$	16.1166	0.570522
$799$	−15.4190	−0.545484
$800$	5.94468	0.210176
$801$	0.773417	0.0273273
$802$	−19.3203	−0.682222
$803$	−36.4159	−1.28509
$804$	−20.7973	−0.733463
$805$	−1.06782	−0.0376358
$806$	−5.38646	−0.189730
$807$	−14.6374	−0.515260
$808$	−4.51899	−0.158977
$809$	10.4069	0.365888	0.182944	−	0.983123i	$-0.441437\pi$
$809$	10.4069	0.365888	0.182944	+	0.983123i	$0.441437\pi$
$810$	30.9857	1.08873
$811$	−34.9136	−1.22598	−0.612992	−	0.790089i	$-0.710034\pi$
$811$	−34.9136	−1.22598	−0.612992	+	0.790089i	$0.710034\pi$
$812$	0.657168	0.0230621
$813$	1.48202	0.0519769
$814$	59.8611	2.09813
$815$	5.64893	0.197873
$816$	−25.8155	−0.903722
$817$	−75.7871	−2.65146
$818$	−1.13921	−0.0398316
$819$	0.0837320	0.00292583
$820$	11.0792	0.386901
$821$	6.21566	0.216928	0.108464	−	0.994100i	$-0.465407\pi$
$821$	6.21566	0.216928	0.108464	+	0.994100i	$0.465407\pi$
$822$	−19.9028	−0.694190
$823$	−8.76219	−0.305431	−0.152715	−	0.988270i	$-0.548802\pi$
$823$	−8.76219	−0.305431	−0.152715	+	0.988270i	$0.548802\pi$
$824$	−25.3214	−0.882112
$825$	−8.45152	−0.294244
$826$	7.05410	0.245444
$827$	26.3299	0.915579	0.457790	−	0.889061i	$-0.348641\pi$
$827$	26.3299	0.915579	0.457790	+	0.889061i	$0.348641\pi$
$828$	−0.0613834	−0.00213322
$829$	−9.05486	−0.314488	−0.157244	−	0.987560i	$-0.550261\pi$
$829$	−9.05486	−0.314488	−0.157244	+	0.987560i	$0.550261\pi$
$830$	−4.09976	−0.142305
$831$	−32.8840	−1.14073
$832$	−2.11803	−0.0734295
$833$	−19.2026	−0.665331
$834$	−3.01685	−0.104465
$835$	−30.6746	−1.06154
$836$	−29.9298	−1.03514
$837$	−10.7021	−0.369920
$838$	−59.4566	−2.05389
$839$	40.0524	1.38276	0.691381	−	0.722490i	$-0.257002\pi$
$839$	40.0524	1.38276	0.691381	+	0.722490i	$0.257002\pi$
$840$	−4.24275	−0.146389
$841$	1.00000	0.0344828
$842$	42.4888	1.46426
$843$	31.2428	1.07606
$844$	11.0896	0.381721
$845$	20.9802	0.721740
$846$	−0.729593	−0.0250839
$847$	−3.69518	−0.126968
$848$	53.5105	1.83756
$849$	−21.7041	−0.744883
$850$	−6.01994	−0.206482
$851$	−6.83784	−0.234398
$852$	−14.0900	−0.482714
$853$	−49.3059	−1.68820	−0.844101	−	0.536183i	$-0.819866\pi$
$853$	−49.3059	−1.68820	−0.844101	+	0.536183i	$0.819866\pi$
$854$	16.2066	0.554578
$855$	1.22635	0.0419402
$856$	6.71188	0.229407
$857$	41.3686	1.41312	0.706562	−	0.707651i	$-0.250245\pi$
$857$	41.3686	1.41312	0.706562	+	0.707651i	$0.250245\pi$
$858$	−18.3108	−0.625120
$859$	−42.8132	−1.46077	−0.730384	−	0.683036i	$-0.760659\pi$
$859$	−42.8132	−1.46077	−0.730384	+	0.683036i	$0.760659\pi$
$860$	−18.1970	−0.620512
$861$	7.19402	0.245171
$862$	−14.4475	−0.492083
$863$	−21.3194	−0.725721	−0.362860	−	0.931844i	$-0.618200\pi$
$863$	−21.3194	−0.725721	−0.362860	+	0.931844i	$0.618200\pi$
$864$	25.5899	0.870585
$865$	−34.8701	−1.18562
$866$	−44.4593	−1.51079
$867$	−14.6399	−0.497197
$868$	−1.37268	−0.0465917
$869$	−23.0648	−0.782421
$870$	5.88846	0.199637
$871$	18.6341	0.631393
$872$	−15.0519	−0.509722
$873$	0.947721	0.0320755
$874$	10.5861	0.358080
$875$	−8.32295	−0.281367
$876$	−15.0745	−0.509321
$877$	34.1767	1.15406	0.577032	−	0.816722i	$-0.304211\pi$
$877$	34.1767	1.15406	0.577032	+	0.816722i	$0.304211\pi$
$878$	−16.2424	−0.548155
$879$	−15.6838	−0.529002
$880$	39.4624	1.33028
$881$	−6.68124	−0.225097	−0.112548	−	0.993646i	$-0.535901\pi$
$881$	−6.68124	−0.225097	−0.112548	+	0.993646i	$0.535901\pi$
$882$	−0.908627	−0.0305951
$883$	−3.04360	−0.102425	−0.0512126	−	0.998688i	$-0.516309\pi$
$883$	−3.04360	−0.102425	−0.0512126	+	0.998688i	$0.516309\pi$
$884$	−4.21218	−0.141671
$885$	20.4131	0.686180
$886$	−21.3691	−0.717909
$887$	35.6468	1.19690	0.598452	−	0.801159i	$-0.295783\pi$
$887$	35.6468	1.19690	0.598452	+	0.801159i	$0.295783\pi$
$888$	−27.1686	−0.911719
$889$	−1.46536	−0.0491467
$890$	−32.0260	−1.07351
$891$	−37.3642	−1.25175
$892$	−3.53661	−0.118414
$893$	40.6357	1.35982
$894$	−61.9792	−2.07289
$895$	10.1384	0.338888
$896$	−8.55212	−0.285706
$897$	2.09161	0.0698370
$898$	26.9299	0.898662
$899$	−2.08878	−0.0696646
$900$	−0.0919942	−0.00306647
$901$	−31.5065	−1.04963
$902$	−41.3675	−1.37739
$903$	−11.8158	−0.393205
$904$	−15.2712	−0.507913
$905$	−31.2592	−1.03909
$906$	28.9465	0.961683
$907$	19.9287	0.661720	0.330860	−	0.943680i	$-0.392661\pi$
$907$	19.9287	0.661720	0.330860	+	0.943680i	$0.392661\pi$
$908$	−12.7693	−0.423764
$909$	0.203646	0.00675450
$910$	−3.46721	−0.114937
$911$	−12.4177	−0.411417	−0.205709	−	0.978613i	$-0.565950\pi$
$911$	−12.4177	−0.411417	−0.205709	+	0.978613i	$0.565950\pi$
$912$	68.0351	2.25287
$913$	4.94371	0.163613
$914$	−10.6215	−0.351329
$915$	46.8986	1.55042
$916$	19.6644	0.649731
$917$	−6.79866	−0.224512
$918$	−25.9139	−0.855285
$919$	5.16525	0.170386	0.0851930	−	0.996364i	$-0.472849\pi$
$919$	5.16525	0.170386	0.0851930	+	0.996364i	$0.472849\pi$
$920$	−2.78682	−0.0918789
$921$	−24.9073	−0.820724
$922$	11.9264	0.392776
$923$	12.6244	0.415539
$924$	−4.66629	−0.153510
$925$	−10.2477	−0.336944
$926$	−43.1723	−1.41873
$927$	1.14109	0.0374785
$928$	4.99447	0.163952
$929$	−54.6203	−1.79203	−0.896017	−	0.444019i	$-0.853552\pi$
$929$	−54.6203	−1.79203	−0.896017	+	0.444019i	$0.853552\pi$
$930$	−12.2997	−0.403322
$931$	50.6073	1.65859
$932$	−9.61546	−0.314965
$933$	2.25216	0.0737323
$934$	−18.9868	−0.621266
$935$	−23.2351	−0.759868
$936$	0.218526	0.00714273
$937$	−2.40151	−0.0784538	−0.0392269	−	0.999230i	$-0.512490\pi$
$937$	−2.40151	−0.0784538	−0.0392269	+	0.999230i	$0.512490\pi$
$938$	14.7038	0.480098
$939$	−29.5991	−0.965929
$940$	9.75690	0.318235
$941$	20.8706	0.680363	0.340181	−	0.940360i	$-0.389512\pi$
$941$	20.8706	0.680363	0.340181	+	0.940360i	$0.389512\pi$
$942$	13.6733	0.445500
$943$	4.72535	0.153878
$944$	29.7784	0.969203
$945$	−6.88886	−0.224095
$946$	67.9440	2.20905
$947$	−21.6776	−0.704426	−0.352213	−	0.935920i	$-0.614571\pi$
$947$	−21.6776	−0.704426	−0.352213	+	0.935920i	$0.614571\pi$
$948$	−9.54781	−0.310098
$949$	13.5066	0.438443
$950$	15.8652	0.514735
$951$	20.9284	0.678648
$952$	3.64418	0.118108
$953$	37.9381	1.22893	0.614467	−	0.788942i	$-0.289371\pi$
$953$	37.9381	1.22893	0.614467	+	0.788942i	$0.289371\pi$
$954$	−1.49082	−0.0482670
$955$	14.8100	0.479241
$956$	−18.4507	−0.596739
$957$	−7.10061	−0.229530
$958$	29.0485	0.938515
$959$	4.54446	0.146748
$960$	−4.83640	−0.156094
$961$	−26.6370	−0.859259
$962$	−22.2024	−0.715835
$963$	−0.302467	−0.00974688
$964$	29.4154	0.947405
$965$	37.6948	1.21344
$966$	1.65046	0.0531025
$967$	−51.7790	−1.66510	−0.832550	−	0.553950i	$-0.813120\pi$
$967$	−51.7790	−1.66510	−0.832550	+	0.553950i	$0.813120\pi$
$968$	−9.64377	−0.309962
$969$	−40.0584	−1.28686
$970$	−39.2436	−1.26004
$971$	−39.0146	−1.25204	−0.626019	−	0.779808i	$-0.715317\pi$
$971$	−39.0146	−1.25204	−0.626019	+	0.779808i	$0.715317\pi$
$972$	−0.803000	−0.0257562
$973$	0.688844	0.0220833
$974$	−10.5486	−0.337998
$975$	3.13466	0.100390
$976$	68.4149	2.18991
$977$	9.80861	0.313805	0.156903	−	0.987614i	$-0.449849\pi$
$977$	9.80861	0.313805	0.156903	+	0.987614i	$0.449849\pi$
$978$	−8.73115	−0.279191
$979$	38.6186	1.23426
$980$	12.1511	0.388154
$981$	0.678306	0.0216566
$982$	−37.2135	−1.18753
$983$	−27.8563	−0.888478	−0.444239	−	0.895908i	$-0.646526\pi$
$983$	−27.8563	−0.888478	−0.444239	+	0.895908i	$0.646526\pi$
$984$	18.7751	0.598528
$985$	0.961378	0.0306320
$986$	−5.05770	−0.161070
$987$	6.33543	0.201659
$988$	11.1009	0.353168
$989$	−7.76114	−0.246790
$990$	−1.09943	−0.0349423
$991$	−39.4984	−1.25471	−0.627354	−	0.778734i	$-0.715862\pi$
$991$	−39.4984	−1.25471	−0.627354	+	0.778734i	$0.715862\pi$
$992$	−10.4323	−0.331227
$993$	26.4168	0.838312
$994$	9.96172	0.315967
$995$	−52.9215	−1.67773
$996$	2.04647	0.0648450
$997$	−18.5795	−0.588420	−0.294210	−	0.955741i	$-0.595056\pi$
$997$	−18.5795	−0.588420	−0.294210	+	0.955741i	$0.595056\pi$
$998$	−42.7151	−1.35212
$999$	−44.1131	−1.39568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.48	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.48	✓	61	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)

\(f(q)\)	\(=\)	\(q+1.71872 q^{2} +1.75528 q^{3} +0.954015 q^{4} -1.95186 q^{5} +3.01685 q^{6} -0.688844 q^{7} -1.79776 q^{8} +0.0810151 q^{9} +O(q^{10})\)	\(q+1.71872 q^{2} +1.75528 q^{3} +0.954015 q^{4} -1.95186 q^{5} +3.01685 q^{6} -0.688844 q^{7} -1.79776 q^{8} +0.0810151 q^{9} -3.35471 q^{10} +4.04528 q^{11} +1.67457 q^{12} -1.50039 q^{13} -1.18393 q^{14} -3.42606 q^{15} -4.99789 q^{16} +2.94271 q^{17} +0.139243 q^{18} -7.75532 q^{19} -1.86210 q^{20} -1.20912 q^{21} +6.95273 q^{22} -0.794200 q^{23} -3.15558 q^{24} -1.19025 q^{25} -2.57876 q^{26} -5.12364 q^{27} -0.657168 q^{28} -1.00000 q^{29} -5.88846 q^{30} +2.08878 q^{31} -4.99447 q^{32} +7.10061 q^{33} +5.05770 q^{34} +1.34452 q^{35} +0.0772896 q^{36} +8.60972 q^{37} -13.3293 q^{38} -2.63361 q^{39} +3.50897 q^{40} -5.94982 q^{41} -2.07814 q^{42} +9.77228 q^{43} +3.85926 q^{44} -0.158130 q^{45} -1.36501 q^{46} -5.23972 q^{47} -8.77270 q^{48} -6.52549 q^{49} -2.04572 q^{50} +5.16528 q^{51} -1.43140 q^{52} -10.7066 q^{53} -8.80613 q^{54} -7.89582 q^{55} +1.23838 q^{56} -13.6128 q^{57} -1.71872 q^{58} -5.95819 q^{59} -3.26851 q^{60} -13.6888 q^{61} +3.59003 q^{62} -0.0558067 q^{63} +1.41165 q^{64} +2.92855 q^{65} +12.2040 q^{66} -12.4195 q^{67} +2.80739 q^{68} -1.39404 q^{69} +2.31087 q^{70} -8.41409 q^{71} -0.145646 q^{72} -9.00205 q^{73} +14.7977 q^{74} -2.08923 q^{75} -7.39870 q^{76} -2.78657 q^{77} -4.52646 q^{78} -5.70166 q^{79} +9.75516 q^{80} -9.23648 q^{81} -10.2261 q^{82} +1.22209 q^{83} -1.15351 q^{84} -5.74374 q^{85} +16.7959 q^{86} -1.75528 q^{87} -7.27245 q^{88} +9.54658 q^{89} -0.271782 q^{90} +1.03354 q^{91} -0.757679 q^{92} +3.66639 q^{93} -9.00564 q^{94} +15.1373 q^{95} -8.76670 q^{96} +11.6981 q^{97} -11.2155 q^{98} +0.327729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more