LMFDB - Embedded newform 4031.2.a.b.1.2

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:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69273 q^{2} -0.478176 q^{3} +5.25077 q^{4} +0.135724 q^{5} +1.28760 q^{6} -4.02730 q^{7} -8.75344 q^{8} -2.77135 q^{9} +O(q^{10})$	\(q-2.69273 q^{2} -0.478176 q^{3} +5.25077 q^{4} +0.135724 q^{5} +1.28760 q^{6} -4.02730 q^{7} -8.75344 q^{8} -2.77135 q^{9} -0.365467 q^{10} +0.734447 q^{11} -2.51079 q^{12} +1.55818 q^{13} +10.8444 q^{14} -0.0648998 q^{15} +13.0691 q^{16} +2.18177 q^{17} +7.46248 q^{18} -2.41920 q^{19} +0.712654 q^{20} +1.92576 q^{21} -1.97766 q^{22} -1.02386 q^{23} +4.18569 q^{24} -4.98158 q^{25} -4.19576 q^{26} +2.75972 q^{27} -21.1464 q^{28} +1.00000 q^{29} +0.174757 q^{30} -1.34287 q^{31} -17.6846 q^{32} -0.351195 q^{33} -5.87490 q^{34} -0.546599 q^{35} -14.5517 q^{36} +9.50487 q^{37} +6.51424 q^{38} -0.745085 q^{39} -1.18805 q^{40} -3.97407 q^{41} -5.18554 q^{42} +11.3999 q^{43} +3.85641 q^{44} -0.376137 q^{45} +2.75697 q^{46} +8.21831 q^{47} -6.24932 q^{48} +9.21912 q^{49} +13.4140 q^{50} -1.04327 q^{51} +8.18166 q^{52} +3.84956 q^{53} -7.43117 q^{54} +0.0996818 q^{55} +35.2527 q^{56} +1.15680 q^{57} -2.69273 q^{58} +1.59107 q^{59} -0.340774 q^{60} -5.94613 q^{61} +3.61597 q^{62} +11.1610 q^{63} +21.4815 q^{64} +0.211482 q^{65} +0.945672 q^{66} +1.93747 q^{67} +11.4560 q^{68} +0.489585 q^{69} +1.47184 q^{70} -0.683946 q^{71} +24.2588 q^{72} -1.48846 q^{73} -25.5940 q^{74} +2.38207 q^{75} -12.7027 q^{76} -2.95784 q^{77} +2.00631 q^{78} -9.37214 q^{79} +1.77378 q^{80} +6.99441 q^{81} +10.7011 q^{82} +4.90153 q^{83} +10.1117 q^{84} +0.296117 q^{85} -30.6967 q^{86} -0.478176 q^{87} -6.42894 q^{88} +2.68667 q^{89} +1.01283 q^{90} -6.27526 q^{91} -5.37605 q^{92} +0.642126 q^{93} -22.1297 q^{94} -0.328342 q^{95} +8.45633 q^{96} -3.69827 q^{97} -24.8246 q^{98} -2.03541 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.69273	−1.90404	−0.952022	−	0.306028i	$-0.901000\pi$
$2$	−2.69273	−1.90404	−0.952022	+	0.306028i	$0.901000\pi$
$3$	−0.478176	−0.276075	−0.138038	−	0.990427i	$-0.544079\pi$
$3$	−0.478176	−0.276075	−0.138038	+	0.990427i	$0.544079\pi$
$4$	5.25077	2.62539
$5$	0.135724	0.0606975	0.0303487	−	0.999539i	$-0.490338\pi$
$5$	0.135724	0.0606975	0.0303487	+	0.999539i	$0.490338\pi$
$6$	1.28760	0.525659
$7$	−4.02730	−1.52218	−0.761088	−	0.648649i	$-0.775334\pi$
$7$	−4.02730	−1.52218	−0.761088	+	0.648649i	$0.775334\pi$
$8$	−8.75344	−3.09481
$9$	−2.77135	−0.923783
$10$	−0.365467	−0.115571
$11$	0.734447	0.221444	0.110722	−	0.993851i	$-0.464684\pi$
$11$	0.734447	0.221444	0.110722	+	0.993851i	$0.464684\pi$
$12$	−2.51079	−0.724804
$13$	1.55818	0.432162	0.216081	−	0.976375i	$-0.430672\pi$
$13$	1.55818	0.432162	0.216081	+	0.976375i	$0.430672\pi$
$14$	10.8444	2.89829
$15$	−0.0648998	−0.0167571
$16$	13.0691	3.26727
$17$	2.18177	0.529156	0.264578	−	0.964364i	$-0.414767\pi$
$17$	2.18177	0.529156	0.264578	+	0.964364i	$0.414767\pi$
$18$	7.46248	1.75892
$19$	−2.41920	−0.555002	−0.277501	−	0.960725i	$-0.589506\pi$
$19$	−2.41920	−0.555002	−0.277501	+	0.960725i	$0.589506\pi$
$20$	0.712654	0.159354
$21$	1.92576	0.420235
$22$	−1.97766	−0.421639
$23$	−1.02386	−0.213489	−0.106745	−	0.994286i	$-0.534043\pi$
$23$	−1.02386	−0.213489	−0.106745	+	0.994286i	$0.534043\pi$
$24$	4.18569	0.854400
$25$	−4.98158	−0.996316
$26$	−4.19576	−0.822856
$27$	2.75972	0.531108
$28$	−21.1464	−3.99630
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.174757	0.0319062
$31$	−1.34287	−0.241186	−0.120593	−	0.992702i	$-0.538480\pi$
$31$	−1.34287	−0.241186	−0.120593	+	0.992702i	$0.538480\pi$
$32$	−17.6846	−3.12622
$33$	−0.351195	−0.0611352
$34$	−5.87490	−1.00754
$35$	−0.546599	−0.0923922
$36$	−14.5517	−2.42529
$37$	9.50487	1.56259	0.781295	−	0.624161i	$-0.214559\pi$
$37$	9.50487	1.56259	0.781295	+	0.624161i	$0.214559\pi$
$38$	6.51424	1.05675
$39$	−0.745085	−0.119309
$40$	−1.18805	−0.187847
$41$	−3.97407	−0.620645	−0.310323	−	0.950631i	$-0.600437\pi$
$41$	−3.97407	−0.620645	−0.310323	+	0.950631i	$0.600437\pi$
$42$	−5.18554	−0.800146
$43$	11.3999	1.73846	0.869232	−	0.494404i	$-0.164614\pi$
$43$	11.3999	1.73846	0.869232	+	0.494404i	$0.164614\pi$
$44$	3.85641	0.581376
$45$	−0.376137	−0.0560713
$46$	2.75697	0.406494
$47$	8.21831	1.19876	0.599382	−	0.800463i	$-0.295413\pi$
$47$	8.21831	1.19876	0.599382	+	0.800463i	$0.295413\pi$
$48$	−6.24932	−0.902011
$49$	9.21912	1.31702
$50$	13.4140	1.89703
$51$	−1.04327	−0.146087
$52$	8.18166	1.13459
$53$	3.84956	0.528778	0.264389	−	0.964416i	$-0.414830\pi$
$53$	3.84956	0.528778	0.264389	+	0.964416i	$0.414830\pi$
$54$	−7.43117	−1.01125
$55$	0.0996818	0.0134411
$56$	35.2527	4.71084
$57$	1.15680	0.153222
$58$	−2.69273	−0.353572
$59$	1.59107	0.207140	0.103570	−	0.994622i	$-0.466973\pi$
$59$	1.59107	0.207140	0.103570	+	0.994622i	$0.466973\pi$
$60$	−0.340774	−0.0439938
$61$	−5.94613	−0.761323	−0.380662	−	0.924714i	$-0.624304\pi$
$61$	−5.94613	−0.761323	−0.380662	+	0.924714i	$0.624304\pi$
$62$	3.61597	0.459229
$63$	11.1610	1.40616
$64$	21.4815	2.68519
$65$	0.211482	0.0262311
$66$	0.945672	0.116404
$67$	1.93747	0.236700	0.118350	−	0.992972i	$-0.462240\pi$
$67$	1.93747	0.236700	0.118350	+	0.992972i	$0.462240\pi$
$68$	11.4560	1.38924
$69$	0.489585	0.0589391
$70$	1.47184	0.175919
$71$	−0.683946	−0.0811695	−0.0405847	−	0.999176i	$-0.512922\pi$
$71$	−0.683946	−0.0811695	−0.0405847	+	0.999176i	$0.512922\pi$
$72$	24.2588	2.85893
$73$	−1.48846	−0.174211	−0.0871053	−	0.996199i	$-0.527762\pi$
$73$	−1.48846	−0.174211	−0.0871053	+	0.996199i	$0.527762\pi$
$74$	−25.5940	−2.97524
$75$	2.38207	0.275058
$76$	−12.7027	−1.45710
$77$	−2.95784	−0.337077
$78$	2.00631	0.227170
$79$	−9.37214	−1.05445	−0.527224	−	0.849726i	$-0.676767\pi$
$79$	−9.37214	−1.05445	−0.527224	+	0.849726i	$0.676767\pi$
$80$	1.77378	0.198315
$81$	6.99441	0.777157
$82$	10.7011	1.18174
$83$	4.90153	0.538012	0.269006	−	0.963138i	$-0.413305\pi$
$83$	4.90153	0.538012	0.269006	+	0.963138i	$0.413305\pi$
$84$	10.1117	1.10328
$85$	0.296117	0.0321184
$86$	−30.6967	−3.31011
$87$	−0.478176	−0.0512659
$88$	−6.42894	−0.685327
$89$	2.68667	0.284787	0.142393	−	0.989810i	$-0.454520\pi$
$89$	2.68667	0.284787	0.142393	+	0.989810i	$0.454520\pi$
$90$	1.01283	0.106762
$91$	−6.27526	−0.657826
$92$	−5.37605	−0.560492
$93$	0.642126	0.0665854
$94$	−22.1297	−2.28250
$95$	−0.328342	−0.0336872
$96$	8.45633	0.863071
$97$	−3.69827	−0.375503	−0.187751	−	0.982217i	$-0.560120\pi$
$97$	−3.69827	−0.375503	−0.187751	+	0.982217i	$0.560120\pi$
$98$	−24.8246	−2.50766
$99$	−2.03541	−0.204566
$100$	−26.1571	−2.61571
$101$	−8.89113	−0.884701	−0.442350	−	0.896842i	$-0.645855\pi$
$101$	−8.89113	−0.884701	−0.442350	+	0.896842i	$0.645855\pi$
$102$	2.80924	0.278156
$103$	11.2080	1.10435	0.552177	−	0.833727i	$-0.313797\pi$
$103$	11.2080	1.10435	0.552177	+	0.833727i	$0.313797\pi$
$104$	−13.6395	−1.33746
$105$	0.261371	0.0255072
$106$	−10.3658	−1.00682
$107$	4.26811	0.412614	0.206307	−	0.978487i	$-0.433855\pi$
$107$	4.26811	0.412614	0.206307	+	0.978487i	$0.433855\pi$
$108$	14.4907	1.39437
$109$	−9.48948	−0.908927	−0.454464	−	0.890765i	$-0.650169\pi$
$109$	−9.48948	−0.908927	−0.454464	+	0.890765i	$0.650169\pi$
$110$	−0.268416	−0.0255924
$111$	−4.54500	−0.431392
$112$	−52.6330	−4.97336
$113$	2.56918	0.241688	0.120844	−	0.992672i	$-0.461440\pi$
$113$	2.56918	0.241688	0.120844	+	0.992672i	$0.461440\pi$
$114$	−3.11495	−0.291742
$115$	−0.138962	−0.0129583
$116$	5.25077	0.487522
$117$	−4.31826	−0.399224
$118$	−4.28432	−0.394403
$119$	−8.78663	−0.805469
$120$	0.568097	0.0518599
$121$	−10.4606	−0.950963
$122$	16.0113	1.44959
$123$	1.90030	0.171345
$124$	−7.05108	−0.633206
$125$	−1.35474	−0.121171
$126$	−30.0536	−2.67739
$127$	−18.4211	−1.63461	−0.817303	−	0.576208i	$-0.804532\pi$
$127$	−18.4211	−1.63461	−0.817303	+	0.576208i	$0.804532\pi$
$128$	−22.4747	−1.98650
$129$	−5.45115	−0.479947
$130$	−0.569463	−0.0499452
$131$	7.91400	0.691450	0.345725	−	0.938336i	$-0.387633\pi$
$131$	7.91400	0.691450	0.345725	+	0.938336i	$0.387633\pi$
$132$	−1.84404	−0.160503
$133$	9.74283	0.844811
$134$	−5.21707	−0.450687
$135$	0.374559	0.0322369
$136$	−19.0980	−1.63764
$137$	5.86293	0.500904	0.250452	−	0.968129i	$-0.419421\pi$
$137$	5.86293	0.500904	0.250452	+	0.968129i	$0.419421\pi$
$138$	−1.31832	−0.112223
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−2.87007	−0.242565
$141$	−3.92980	−0.330949
$142$	1.84168	0.154550
$143$	1.14440	0.0956996
$144$	−36.2189	−3.01825
$145$	0.135724	0.0112712
$146$	4.00801	0.331705
$147$	−4.40836	−0.363596
$148$	49.9079	4.10240
$149$	13.2092	1.08214	0.541072	−	0.840976i	$-0.318019\pi$
$149$	13.2092	1.08214	0.541072	+	0.840976i	$0.318019\pi$
$150$	−6.41427	−0.523723
$151$	1.83730	0.149517	0.0747587	−	0.997202i	$-0.476181\pi$
$151$	1.83730	0.149517	0.0747587	+	0.997202i	$0.476181\pi$
$152$	21.1763	1.71763
$153$	−6.04644	−0.488825
$154$	7.96464	0.641809
$155$	−0.182259	−0.0146394
$156$	−3.91227	−0.313233
$157$	−19.2408	−1.53558	−0.767792	−	0.640699i	$-0.778645\pi$
$157$	−19.2408	−1.53558	−0.767792	+	0.640699i	$0.778645\pi$
$158$	25.2366	2.00772
$159$	−1.84077	−0.145983
$160$	−2.40021	−0.189753
$161$	4.12339	0.324968
$162$	−18.8340	−1.47974
$163$	−2.25118	−0.176326	−0.0881628	−	0.996106i	$-0.528100\pi$
$163$	−2.25118	−0.176326	−0.0881628	+	0.996106i	$0.528100\pi$
$164$	−20.8669	−1.62943
$165$	−0.0476654	−0.00371075
$166$	−13.1985	−1.02440
$167$	16.8373	1.30291	0.651454	−	0.758688i	$-0.274159\pi$
$167$	16.8373	1.30291	0.651454	+	0.758688i	$0.274159\pi$
$168$	−16.8570	−1.30055
$169$	−10.5721	−0.813236
$170$	−0.797363	−0.0611550
$171$	6.70444	0.512701
$172$	59.8582	4.56414
$173$	−0.412141	−0.0313345	−0.0156672	−	0.999877i	$-0.504987\pi$
$173$	−0.412141	−0.0313345	−0.0156672	+	0.999877i	$0.504987\pi$
$174$	1.28760	0.0976125
$175$	20.0623	1.51657
$176$	9.59854	0.723517
$177$	−0.760812	−0.0571861
$178$	−7.23448	−0.542247
$179$	−2.81915	−0.210714	−0.105357	−	0.994434i	$-0.533598\pi$
$179$	−2.81915	−0.210714	−0.105357	+	0.994434i	$0.533598\pi$
$180$	−1.97501	−0.147209
$181$	−1.75653	−0.130562	−0.0652808	−	0.997867i	$-0.520794\pi$
$181$	−1.75653	−0.130562	−0.0652808	+	0.997867i	$0.520794\pi$
$182$	16.8976	1.25253
$183$	2.84330	0.210182
$184$	8.96230	0.660709
$185$	1.29004	0.0948453
$186$	−1.72907	−0.126782
$187$	1.60239	0.117179
$188$	43.1525	3.14722
$189$	−11.1142	−0.808440
$190$	0.884136	0.0641420
$191$	11.6819	0.845276	0.422638	−	0.906299i	$-0.361104\pi$
$191$	11.6819	0.845276	0.422638	+	0.906299i	$0.361104\pi$
$192$	−10.2719	−0.741314
$193$	−7.49104	−0.539217	−0.269608	−	0.962970i	$-0.586894\pi$
$193$	−7.49104	−0.539217	−0.269608	+	0.962970i	$0.586894\pi$
$194$	9.95843	0.714974
$195$	−0.101126	−0.00724176
$196$	48.4075	3.45768
$197$	−2.44275	−0.174039	−0.0870193	−	0.996207i	$-0.527734\pi$
$197$	−2.44275	−0.174039	−0.0870193	+	0.996207i	$0.527734\pi$
$198$	5.48079	0.389503
$199$	7.06206	0.500616	0.250308	−	0.968166i	$-0.419468\pi$
$199$	7.06206	0.500616	0.250308	+	0.968166i	$0.419468\pi$
$200$	43.6060	3.08341
$201$	−0.926452	−0.0653469
$202$	23.9414	1.68451
$203$	−4.02730	−0.282661
$204$	−5.47797	−0.383535
$205$	−0.539375	−0.0376716
$206$	−30.1800	−2.10274
$207$	2.83747	0.197218
$208$	20.3640	1.41199
$209$	−1.77677	−0.122902
$210$	−0.703800	−0.0485668
$211$	−27.4127	−1.88717	−0.943584	−	0.331133i	$-0.892569\pi$
$211$	−27.4127	−1.88717	−0.943584	+	0.331133i	$0.892569\pi$
$212$	20.2132	1.38825
$213$	0.327047	0.0224089
$214$	−11.4929	−0.785635
$215$	1.54723	0.105520
$216$	−24.1571	−1.64368
$217$	5.40812	0.367127
$218$	25.5526	1.73064
$219$	0.711744	0.0480952
$220$	0.523406	0.0352881
$221$	3.39959	0.228681
$222$	12.2384	0.821391
$223$	17.7456	1.18833	0.594167	−	0.804342i	$-0.297482\pi$
$223$	17.7456	1.18833	0.594167	+	0.804342i	$0.297482\pi$
$224$	71.2209	4.75865
$225$	13.8057	0.920379
$226$	−6.91810	−0.460185
$227$	−17.0916	−1.13441	−0.567205	−	0.823577i	$-0.691975\pi$
$227$	−17.0916	−1.13441	−0.567205	+	0.823577i	$0.691975\pi$
$228$	6.07411	0.402268
$229$	−7.22682	−0.477562	−0.238781	−	0.971073i	$-0.576748\pi$
$229$	−7.22682	−0.477562	−0.238781	+	0.971073i	$0.576748\pi$
$230$	0.374186	0.0246731
$231$	1.41437	0.0930585
$232$	−8.75344	−0.574692
$233$	−9.70016	−0.635479	−0.317739	−	0.948178i	$-0.602924\pi$
$233$	−9.70016	−0.635479	−0.317739	+	0.948178i	$0.602924\pi$
$234$	11.6279	0.760140
$235$	1.11542	0.0727619
$236$	8.35435	0.543822
$237$	4.48153	0.291107
$238$	23.6600	1.53365
$239$	8.11827	0.525127	0.262563	−	0.964915i	$-0.415432\pi$
$239$	8.11827	0.525127	0.262563	+	0.964915i	$0.415432\pi$
$240$	−0.848180	−0.0547498
$241$	17.4019	1.12095	0.560477	−	0.828170i	$-0.310618\pi$
$241$	17.4019	1.12095	0.560477	+	0.828170i	$0.310618\pi$
$242$	28.1675	1.81068
$243$	−11.6237	−0.745662
$244$	−31.2218	−1.99877
$245$	1.25125	0.0799396
$246$	−5.11700	−0.326248
$247$	−3.76955	−0.239851
$248$	11.7547	0.746424
$249$	−2.34379	−0.148532
$250$	3.64793	0.230716
$251$	−8.82758	−0.557192	−0.278596	−	0.960408i	$-0.589869\pi$
$251$	−8.82758	−0.557192	−0.278596	+	0.960408i	$0.589869\pi$
$252$	58.6041	3.69171
$253$	−0.751970	−0.0472760
$254$	49.6029	3.11236
$255$	−0.141596	−0.00886710
$256$	17.5552	1.09720
$257$	−17.6231	−1.09930	−0.549651	−	0.835395i	$-0.685239\pi$
$257$	−17.6231	−1.09930	−0.549651	+	0.835395i	$0.685239\pi$
$258$	14.6784	0.913840
$259$	−38.2789	−2.37854
$260$	1.11044	0.0688669
$261$	−2.77135	−0.171542
$262$	−21.3102	−1.31655
$263$	6.75310	0.416414	0.208207	−	0.978085i	$-0.433237\pi$
$263$	6.75310	0.416414	0.208207	+	0.978085i	$0.433237\pi$
$264$	3.07416	0.189202
$265$	0.522477	0.0320955
$266$	−26.2348	−1.60856
$267$	−1.28470	−0.0786226
$268$	10.1732	0.621428
$269$	−26.8587	−1.63761	−0.818803	−	0.574075i	$-0.805362\pi$
$269$	−26.8587	−1.63761	−0.818803	+	0.574075i	$0.805362\pi$
$270$	−1.00859	−0.0613806
$271$	−6.08500	−0.369638	−0.184819	−	0.982773i	$-0.559170\pi$
$271$	−6.08500	−0.369638	−0.184819	+	0.982773i	$0.559170\pi$
$272$	28.5137	1.72890
$273$	3.00068	0.181609
$274$	−15.7873	−0.953744
$275$	−3.65870	−0.220628
$276$	2.57070	0.154738
$277$	−10.6863	−0.642077	−0.321039	−	0.947066i	$-0.604032\pi$
$277$	−10.6863	−0.642077	−0.321039	+	0.947066i	$0.604032\pi$
$278$	−2.69273	−0.161499
$279$	3.72155	0.222803
$280$	4.78463	0.285936
$281$	29.3034	1.74810	0.874048	−	0.485840i	$-0.161486\pi$
$281$	29.3034	1.74810	0.874048	+	0.485840i	$0.161486\pi$
$282$	10.5819	0.630141
$283$	−16.0550	−0.954370	−0.477185	−	0.878803i	$-0.658343\pi$
$283$	−16.0550	−0.954370	−0.477185	+	0.878803i	$0.658343\pi$
$284$	−3.59125	−0.213101
$285$	0.157006	0.00930020
$286$	−3.08156	−0.182216
$287$	16.0048	0.944731
$288$	49.0100	2.88794
$289$	−12.2399	−0.719994
$290$	−0.365467	−0.0214609
$291$	1.76842	0.103667
$292$	−7.81555	−0.457370
$293$	9.26260	0.541127	0.270563	−	0.962702i	$-0.412790\pi$
$293$	9.26260	0.541127	0.270563	+	0.962702i	$0.412790\pi$
$294$	11.8705	0.692303
$295$	0.215946	0.0125729
$296$	−83.2003	−4.83592
$297$	2.02687	0.117611
$298$	−35.5689	−2.06045
$299$	−1.59536	−0.0922620
$300$	12.5077	0.722134
$301$	−45.9107	−2.64625
$302$	−4.94735	−0.284688
$303$	4.25153	0.244244
$304$	−31.6167	−1.81334
$305$	−0.807030	−0.0462104
$306$	16.2814	0.930745
$307$	0.788857	0.0450224	0.0225112	−	0.999747i	$-0.492834\pi$
$307$	0.788857	0.0450224	0.0225112	+	0.999747i	$0.492834\pi$
$308$	−15.5309	−0.884956
$309$	−5.35938	−0.304884
$310$	0.490773	0.0278740
$311$	−2.17784	−0.123494	−0.0617471	−	0.998092i	$-0.519667\pi$
$311$	−2.17784	−0.123494	−0.0617471	+	0.998092i	$0.519667\pi$
$312$	6.52206	0.369239
$313$	−1.71524	−0.0969510	−0.0484755	−	0.998824i	$-0.515436\pi$
$313$	−1.71524	−0.0969510	−0.0484755	+	0.998824i	$0.515436\pi$
$314$	51.8102	2.92382
$315$	1.51482	0.0853503
$316$	−49.2110	−2.76833
$317$	16.9097	0.949745	0.474873	−	0.880055i	$-0.342494\pi$
$317$	16.9097	0.949745	0.474873	+	0.880055i	$0.342494\pi$
$318$	4.95669	0.277957
$319$	0.734447	0.0411211
$320$	2.91555	0.162984
$321$	−2.04091	−0.113912
$322$	−11.1032	−0.618754
$323$	−5.27813	−0.293683
$324$	36.7261	2.04034
$325$	−7.76221	−0.430570
$326$	6.06180	0.335732
$327$	4.53764	0.250932
$328$	34.7868	1.92078
$329$	−33.0976	−1.82473
$330$	0.128350	0.00706543
$331$	21.8567	1.20135	0.600677	−	0.799492i	$-0.294898\pi$
$331$	21.8567	1.20135	0.600677	+	0.799492i	$0.294898\pi$
$332$	25.7368	1.41249
$333$	−26.3413	−1.44349
$334$	−45.3382	−2.48080
$335$	0.262960	0.0143671
$336$	25.1679	1.37302
$337$	−20.0962	−1.09471	−0.547354	−	0.836901i	$-0.684365\pi$
$337$	−20.0962	−1.09471	−0.547354	+	0.836901i	$0.684365\pi$
$338$	28.4677	1.54844
$339$	−1.22852	−0.0667241
$340$	1.55485	0.0843233
$341$	−0.986263	−0.0534092
$342$	−18.0532	−0.976206
$343$	−8.93707	−0.482556
$344$	−99.7881	−5.38022
$345$	0.0664483	0.00357746
$346$	1.10978	0.0596623
$347$	22.5516	1.21063	0.605317	−	0.795984i	$-0.293046\pi$
$347$	22.5516	1.21063	0.605317	+	0.795984i	$0.293046\pi$
$348$	−2.51079	−0.134593
$349$	−5.71643	−0.305993	−0.152997	−	0.988227i	$-0.548892\pi$
$349$	−5.71643	−0.305993	−0.152997	+	0.988227i	$0.548892\pi$
$350$	−54.0223	−2.88761
$351$	4.30015	0.229525
$352$	−12.9884	−0.692282
$353$	14.6337	0.778873	0.389436	−	0.921053i	$-0.372670\pi$
$353$	14.6337	0.778873	0.389436	+	0.921053i	$0.372670\pi$
$354$	2.04866	0.108885
$355$	−0.0928276	−0.00492678
$356$	14.1071	0.747676
$357$	4.20155	0.222370
$358$	7.59121	0.401208
$359$	−2.33153	−0.123053	−0.0615266	−	0.998105i	$-0.519597\pi$
$359$	−2.33153	−0.123053	−0.0615266	+	0.998105i	$0.519597\pi$
$360$	3.29250	0.173530
$361$	−13.1475	−0.691973
$362$	4.72984	0.248595
$363$	5.00200	0.262537
$364$	−32.9500	−1.72705
$365$	−0.202019	−0.0105741
$366$	−7.65622	−0.400197
$367$	8.77530	0.458067	0.229034	−	0.973419i	$-0.426443\pi$
$367$	8.77530	0.458067	0.229034	+	0.973419i	$0.426443\pi$
$368$	−13.3809	−0.697527
$369$	11.0135	0.573341
$370$	−3.47371	−0.180590
$371$	−15.5033	−0.804893
$372$	3.37166	0.174812
$373$	−26.7879	−1.38703	−0.693513	−	0.720444i	$-0.743938\pi$
$373$	−26.7879	−1.38703	−0.693513	+	0.720444i	$0.743938\pi$
$374$	−4.31480	−0.223113
$375$	0.647803	0.0334524
$376$	−71.9385	−3.70994
$377$	1.55818	0.0802504
$378$	29.9275	1.53931
$379$	14.6643	0.753257	0.376628	−	0.926364i	$-0.377083\pi$
$379$	14.6643	0.753257	0.376628	+	0.926364i	$0.377083\pi$
$380$	−1.72405	−0.0884420
$381$	8.80852	0.451274
$382$	−31.4563	−1.60944
$383$	−19.4799	−0.995377	−0.497688	−	0.867356i	$-0.665818\pi$
$383$	−19.4799	−0.995377	−0.497688	+	0.867356i	$0.665818\pi$
$384$	10.7469	0.548424
$385$	−0.401448	−0.0204597
$386$	20.1713	1.02669
$387$	−31.5930	−1.60596
$388$	−19.4188	−0.985839
$389$	−1.13526	−0.0575601	−0.0287801	−	0.999586i	$-0.509162\pi$
$389$	−1.13526	−0.0575601	−0.0287801	+	0.999586i	$0.509162\pi$
$390$	0.272304	0.0137886
$391$	−2.23382	−0.112969
$392$	−80.6991	−4.07592
$393$	−3.78429	−0.190892
$394$	6.57765	0.331377
$395$	−1.27202	−0.0640023
$396$	−10.6875	−0.537065
$397$	25.9440	1.30209	0.651046	−	0.759038i	$-0.274330\pi$
$397$	25.9440	1.30209	0.651046	+	0.759038i	$0.274330\pi$
$398$	−19.0162	−0.953195
$399$	−4.65879	−0.233231
$400$	−65.1046	−3.25523
$401$	−5.45587	−0.272453	−0.136227	−	0.990678i	$-0.543498\pi$
$401$	−5.45587	−0.272453	−0.136227	+	0.990678i	$0.543498\pi$
$402$	2.49468	0.124423
$403$	−2.09243	−0.104231
$404$	−46.6853	−2.32268
$405$	0.949307	0.0471714
$406$	10.8444	0.538199
$407$	6.98082	0.346026
$408$	9.13220	0.452111
$409$	−5.51659	−0.272778	−0.136389	−	0.990655i	$-0.543550\pi$
$409$	−5.51659	−0.272778	−0.136389	+	0.990655i	$0.543550\pi$
$410$	1.45239	0.0717284
$411$	−2.80351	−0.138287
$412$	58.8505	2.89935
$413$	−6.40771	−0.315303
$414$	−7.64053	−0.375512
$415$	0.665253	0.0326560
$416$	−27.5557	−1.35103
$417$	−0.478176	−0.0234164
$418$	4.78436	0.234011
$419$	30.4503	1.48759	0.743797	−	0.668406i	$-0.233023\pi$
$419$	30.4503	1.48759	0.743797	+	0.668406i	$0.233023\pi$
$420$	1.37240	0.0669662
$421$	−23.5449	−1.14751	−0.573753	−	0.819028i	$-0.694513\pi$
$421$	−23.5449	−1.14751	−0.573753	+	0.819028i	$0.694513\pi$
$422$	73.8149	3.59325
$423$	−22.7758	−1.10740
$424$	−33.6969	−1.63647
$425$	−10.8686	−0.527207
$426$	−0.880647	−0.0426675
$427$	23.9468	1.15887
$428$	22.4109	1.08327
$429$	−0.547225	−0.0264203
$430$	−4.16627	−0.200916
$431$	−17.9722	−0.865690	−0.432845	−	0.901468i	$-0.642490\pi$
$431$	−17.9722	−0.865690	−0.432845	+	0.901468i	$0.642490\pi$
$432$	36.0670	1.73527
$433$	−17.5209	−0.842002	−0.421001	−	0.907060i	$-0.638321\pi$
$433$	−17.5209	−0.842002	−0.421001	+	0.907060i	$0.638321\pi$
$434$	−14.5626	−0.699026
$435$	−0.0648998	−0.00311171
$436$	−49.8271	−2.38629
$437$	2.47692	0.118487
$438$	−1.91653	−0.0915755
$439$	26.9134	1.28451	0.642253	−	0.766493i	$-0.278000\pi$
$439$	26.9134	1.28451	0.642253	+	0.766493i	$0.278000\pi$
$440$	−0.872559	−0.0415976
$441$	−25.5494	−1.21664
$442$	−9.15416	−0.435419
$443$	17.0089	0.808118	0.404059	−	0.914733i	$-0.367599\pi$
$443$	17.0089	0.808118	0.404059	+	0.914733i	$0.367599\pi$
$444$	−23.8648	−1.13257
$445$	0.364645	0.0172858
$446$	−47.7841	−2.26264
$447$	−6.31635	−0.298753
$448$	−86.5124	−4.08733
$449$	14.6987	0.693676	0.346838	−	0.937925i	$-0.387255\pi$
$449$	14.6987	0.693676	0.346838	+	0.937925i	$0.387255\pi$
$450$	−37.1749	−1.75244
$451$	−2.91874	−0.137438
$452$	13.4902	0.634525
$453$	−0.878554	−0.0412781
$454$	46.0230	2.15997
$455$	−0.851701	−0.0399284
$456$	−10.1260	−0.474194
$457$	−12.0913	−0.565607	−0.282803	−	0.959178i	$-0.591264\pi$
$457$	−12.0913	−0.565607	−0.282803	+	0.959178i	$0.591264\pi$
$458$	19.4598	0.909299
$459$	6.02107	0.281039
$460$	−0.729658	−0.0340205
$461$	−15.4805	−0.720998	−0.360499	−	0.932760i	$-0.617394\pi$
$461$	−15.4805	−0.720998	−0.360499	+	0.932760i	$0.617394\pi$
$462$	−3.80850	−0.177187
$463$	−10.5504	−0.490320	−0.245160	−	0.969483i	$-0.578840\pi$
$463$	−10.5504	−0.490320	−0.245160	+	0.969483i	$0.578840\pi$
$464$	13.0691	0.606717
$465$	0.0871517	0.00404156
$466$	26.1199	1.20998
$467$	−32.1149	−1.48610	−0.743051	−	0.669235i	$-0.766622\pi$
$467$	−32.1149	−1.48610	−0.743051	+	0.669235i	$0.766622\pi$
$468$	−22.6742	−1.04812
$469$	−7.80277	−0.360298
$470$	−3.00352	−0.138542
$471$	9.20050	0.423936
$472$	−13.9273	−0.641058
$473$	8.37260	0.384973
$474$	−12.0675	−0.554281
$475$	12.0514	0.552957
$476$	−46.1366	−2.11467
$477$	−10.6685	−0.488476
$478$	−21.8603	−0.999865
$479$	−22.7812	−1.04090	−0.520449	−	0.853893i	$-0.674235\pi$
$479$	−22.7812	−1.04090	−0.520449	+	0.853893i	$0.674235\pi$
$480$	1.14772	0.0523862
$481$	14.8103	0.675292
$482$	−46.8585	−2.13435
$483$	−1.97171	−0.0897157
$484$	−54.9262	−2.49664
$485$	−0.501943	−0.0227921
$486$	31.2995	1.41977
$487$	−40.6357	−1.84138	−0.920689	−	0.390297i	$-0.872372\pi$
$487$	−40.6357	−1.84138	−0.920689	+	0.390297i	$0.872372\pi$
$488$	52.0491	2.35615
$489$	1.07646	0.0486791
$490$	−3.36928	−0.152209
$491$	−32.7068	−1.47604	−0.738019	−	0.674780i	$-0.764239\pi$
$491$	−32.7068	−1.47604	−0.738019	+	0.674780i	$0.764239\pi$
$492$	9.97807	0.449846
$493$	2.18177	0.0982619
$494$	10.1504	0.456687
$495$	−0.276253	−0.0124166
$496$	−17.5500	−0.788019
$497$	2.75445	0.123554
$498$	6.31119	0.282811
$499$	18.6630	0.835469	0.417735	−	0.908569i	$-0.362824\pi$
$499$	18.6630	0.835469	0.417735	+	0.908569i	$0.362824\pi$
$500$	−7.11341	−0.318122
$501$	−8.05119	−0.359701
$502$	23.7703	1.06092
$503$	−10.2651	−0.457698	−0.228849	−	0.973462i	$-0.573496\pi$
$503$	−10.2651	−0.457698	−0.228849	+	0.973462i	$0.573496\pi$
$504$	−97.6975	−4.35179
$505$	−1.20674	−0.0536991
$506$	2.02485	0.0900156
$507$	5.05531	0.224514
$508$	−96.7249	−4.29147
$509$	−33.5501	−1.48708	−0.743541	−	0.668690i	$-0.766855\pi$
$509$	−33.5501	−1.48708	−0.743541	+	0.668690i	$0.766855\pi$
$510$	0.381280	0.0168834
$511$	5.99446	0.265179
$512$	−2.32195	−0.102617
$513$	−6.67631	−0.294766
$514$	47.4543	2.09312
$515$	1.52119	0.0670314
$516$	−28.6227	−1.26005
$517$	6.03591	0.265459
$518$	103.075	4.52884
$519$	0.197076	0.00865067
$520$	−1.85120	−0.0811803
$521$	29.0951	1.27468	0.637339	−	0.770584i	$-0.280035\pi$
$521$	29.0951	1.27468	0.637339	+	0.770584i	$0.280035\pi$
$522$	7.46248	0.326624
$523$	−29.6758	−1.29763	−0.648817	−	0.760944i	$-0.724736\pi$
$523$	−29.6758	−1.29763	−0.648817	+	0.760944i	$0.724736\pi$
$524$	41.5546	1.81532
$525$	−9.59331	−0.418686
$526$	−18.1842	−0.792870
$527$	−2.92982	−0.127625
$528$	−4.58979	−0.199745
$529$	−21.9517	−0.954422
$530$	−1.40689	−0.0611113
$531$	−4.40941	−0.191352
$532$	51.1574	2.21795
$533$	−6.19232	−0.268219
$534$	3.45935	0.149701
$535$	0.579284	0.0250446
$536$	−16.9595	−0.732540
$537$	1.34805	0.0581728
$538$	72.3232	3.11807
$539$	6.77095	0.291646
$540$	1.96673	0.0846344
$541$	21.3132	0.916326	0.458163	−	0.888868i	$-0.348508\pi$
$541$	21.3132	0.916326	0.458163	+	0.888868i	$0.348508\pi$
$542$	16.3852	0.703807
$543$	0.839929	0.0360448
$544$	−38.5836	−1.65426
$545$	−1.28795	−0.0551696
$546$	−8.08001	−0.345792
$547$	22.3311	0.954811	0.477405	−	0.878683i	$-0.341577\pi$
$547$	22.3311	0.954811	0.477405	+	0.878683i	$0.341577\pi$
$548$	30.7849	1.31507
$549$	16.4788	0.703297
$550$	9.85189	0.420086
$551$	−2.41920	−0.103061
$552$	−4.28556	−0.182405
$553$	37.7444	1.60506
$554$	28.7753	1.22254
$555$	−0.616864	−0.0261844
$556$	5.25077	0.222682
$557$	−24.6650	−1.04509	−0.522545	−	0.852612i	$-0.675017\pi$
$557$	−24.6650	−1.04509	−0.522545	+	0.852612i	$0.675017\pi$
$558$	−10.0211	−0.424227
$559$	17.7631	0.751298
$560$	−7.14355	−0.301870
$561$	−0.766226	−0.0323501
$562$	−78.9061	−3.32845
$563$	−39.1854	−1.65147	−0.825734	−	0.564060i	$-0.809239\pi$
$563$	−39.1854	−1.65147	−0.825734	+	0.564060i	$0.809239\pi$
$564$	−20.6345	−0.868869
$565$	0.348699	0.0146699
$566$	43.2317	1.81716
$567$	−28.1686	−1.18297
$568$	5.98688	0.251204
$569$	9.93268	0.416399	0.208200	−	0.978086i	$-0.433240\pi$
$569$	9.93268	0.416399	0.208200	+	0.978086i	$0.433240\pi$
$570$	−0.422773	−0.0177080
$571$	−5.63344	−0.235752	−0.117876	−	0.993028i	$-0.537609\pi$
$571$	−5.63344	−0.235752	−0.117876	+	0.993028i	$0.537609\pi$
$572$	6.00899	0.251249
$573$	−5.58603	−0.233360
$574$	−43.0964	−1.79881
$575$	5.10044	0.212703
$576$	−59.5327	−2.48053
$577$	−24.0917	−1.00295	−0.501476	−	0.865172i	$-0.667209\pi$
$577$	−24.0917	−1.00295	−0.501476	+	0.865172i	$0.667209\pi$
$578$	32.9587	1.37090
$579$	3.58204	0.148864
$580$	0.712654	0.0295914
$581$	−19.7399	−0.818949
$582$	−4.76188	−0.197386
$583$	2.82730	0.117095
$584$	13.0291	0.539149
$585$	−0.586090	−0.0242319
$586$	−24.9416	−1.03033
$587$	16.0644	0.663051	0.331525	−	0.943446i	$-0.392437\pi$
$587$	16.0644	0.663051	0.331525	+	0.943446i	$0.392437\pi$
$588$	−23.1473	−0.954580
$589$	3.24866	0.133859
$590$	−0.581483	−0.0239393
$591$	1.16806	0.0480477
$592$	124.220	5.10540
$593$	27.9405	1.14738	0.573689	−	0.819073i	$-0.305512\pi$
$593$	27.9405	1.14738	0.573689	+	0.819073i	$0.305512\pi$
$594$	−5.45780	−0.223936
$595$	−1.19255	−0.0488899
$596$	69.3588	2.84105
$597$	−3.37691	−0.138208
$598$	4.29587	0.175671
$599$	−18.0796	−0.738714	−0.369357	−	0.929288i	$-0.620422\pi$
$599$	−18.0796	−0.738714	−0.369357	+	0.929288i	$0.620422\pi$
$600$	−20.8513	−0.851252
$601$	3.64427	0.148653	0.0743265	−	0.997234i	$-0.476319\pi$
$601$	3.64427	0.148653	0.0743265	+	0.997234i	$0.476319\pi$
$602$	123.625	5.03857
$603$	−5.36940	−0.218659
$604$	9.64725	0.392541
$605$	−1.41975	−0.0577210
$606$	−11.4482	−0.465051
$607$	−23.8803	−0.969272	−0.484636	−	0.874716i	$-0.661048\pi$
$607$	−23.8803	−0.969272	−0.484636	+	0.874716i	$0.661048\pi$
$608$	42.7824	1.73506
$609$	1.92576	0.0780356
$610$	2.17311	0.0879867
$611$	12.8056	0.518060
$612$	−31.7485	−1.28336
$613$	−34.7817	−1.40482	−0.702410	−	0.711772i	$-0.747893\pi$
$613$	−34.7817	−1.40482	−0.702410	+	0.711772i	$0.747893\pi$
$614$	−2.12418	−0.0857247
$615$	0.257916	0.0104002
$616$	25.8912	1.04319
$617$	−30.7967	−1.23983	−0.619914	−	0.784669i	$-0.712833\pi$
$617$	−30.7967	−1.23983	−0.619914	+	0.784669i	$0.712833\pi$
$618$	14.4313	0.580514
$619$	11.3744	0.457174	0.228587	−	0.973523i	$-0.426589\pi$
$619$	11.3744	0.457174	0.228587	+	0.973523i	$0.426589\pi$
$620$	−0.956999	−0.0384340
$621$	−2.82557	−0.113386
$622$	5.86433	0.235138
$623$	−10.8200	−0.433495
$624$	−9.73757	−0.389815
$625$	24.7240	0.988961
$626$	4.61867	0.184599
$627$	0.849610	0.0339302
$628$	−101.029	−4.03150
$629$	20.7374	0.826855
$630$	−4.07899	−0.162511
$631$	8.07851	0.321600	0.160800	−	0.986987i	$-0.448593\pi$
$631$	8.07851	0.321600	0.160800	+	0.986987i	$0.448593\pi$
$632$	82.0385	3.26332
$633$	13.1081	0.521000
$634$	−45.5333	−1.80836
$635$	−2.50017	−0.0992164
$636$	−9.66546	−0.383261
$637$	14.3651	0.569165
$638$	−1.97766	−0.0782965
$639$	1.89545	0.0749829
$640$	−3.05035	−0.120576
$641$	−42.9940	−1.69816	−0.849080	−	0.528265i	$-0.822843\pi$
$641$	−42.9940	−1.69816	−0.849080	+	0.528265i	$0.822843\pi$
$642$	5.49561	0.216894
$643$	3.33470	0.131508	0.0657539	−	0.997836i	$-0.479055\pi$
$643$	3.33470	0.131508	0.0657539	+	0.997836i	$0.479055\pi$
$644$	21.6510	0.853168
$645$	−0.739850	−0.0291316
$646$	14.2126	0.559185
$647$	−22.3258	−0.877720	−0.438860	−	0.898556i	$-0.644618\pi$
$647$	−22.3258	−0.877720	−0.438860	+	0.898556i	$0.644618\pi$
$648$	−61.2252	−2.40515
$649$	1.16856	0.0458698
$650$	20.9015	0.819824
$651$	−2.58603	−0.101355
$652$	−11.8204	−0.462923
$653$	14.6170	0.572006	0.286003	−	0.958229i	$-0.407673\pi$
$653$	14.6170	0.572006	0.286003	+	0.958229i	$0.407673\pi$
$654$	−12.2186	−0.477786
$655$	1.07412	0.0419692
$656$	−51.9374	−2.02781
$657$	4.12503	0.160933
$658$	89.1227	3.47436
$659$	4.42324	0.172305	0.0861524	−	0.996282i	$-0.472543\pi$
$659$	4.42324	0.172305	0.0861524	+	0.996282i	$0.472543\pi$
$660$	−0.250280	−0.00974215
$661$	12.5776	0.489211	0.244606	−	0.969623i	$-0.421342\pi$
$661$	12.5776	0.489211	0.244606	+	0.969623i	$0.421342\pi$
$662$	−58.8542	−2.28743
$663$	−1.62560	−0.0631332
$664$	−42.9052	−1.66505
$665$	1.32233	0.0512779
$666$	70.9299	2.74848
$667$	−1.02386	−0.0396440
$668$	88.4088	3.42064
$669$	−8.48553	−0.328070
$670$	−0.708080	−0.0273555
$671$	−4.36711	−0.168590
$672$	−34.0562	−1.31374
$673$	45.1434	1.74015	0.870075	−	0.492918i	$-0.164070\pi$
$673$	45.1434	1.74015	0.870075	+	0.492918i	$0.164070\pi$
$674$	54.1135	2.08437
$675$	−13.7478	−0.529152
$676$	−55.5115	−2.13506
$677$	−12.0961	−0.464891	−0.232445	−	0.972609i	$-0.574673\pi$
$677$	−12.0961	−0.464891	−0.232445	+	0.972609i	$0.574673\pi$
$678$	3.30807	0.127046
$679$	14.8940	0.571581
$680$	−2.59205	−0.0994005
$681$	8.17279	0.313182
$682$	2.65574	0.101693
$683$	−1.43228	−0.0548048	−0.0274024	−	0.999624i	$-0.508724\pi$
$683$	−1.43228	−0.0548048	−0.0274024	+	0.999624i	$0.508724\pi$
$684$	35.2035	1.34604
$685$	0.795739	0.0304036
$686$	24.0651	0.918809
$687$	3.45569	0.131843
$688$	148.986	5.68003
$689$	5.99832	0.228518
$690$	−0.178927	−0.00681164
$691$	1.31567	0.0500506	0.0250253	−	0.999687i	$-0.492033\pi$
$691$	1.31567	0.0500506	0.0250253	+	0.999687i	$0.492033\pi$
$692$	−2.16406	−0.0822651
$693$	8.19719	0.311385
$694$	−60.7254	−2.30510
$695$	0.135724	0.00514829
$696$	4.18569	0.158658
$697$	−8.67049	−0.328418
$698$	15.3928	0.582625
$699$	4.63838	0.175440
$700$	105.343	3.98158
$701$	12.6527	0.477888	0.238944	−	0.971033i	$-0.423199\pi$
$701$	12.6527	0.477888	0.238944	+	0.971033i	$0.423199\pi$
$702$	−11.5791	−0.437026
$703$	−22.9942	−0.867241
$704$	15.7770	0.594619
$705$	−0.533367	−0.0200878
$706$	−39.4045	−1.48301
$707$	35.8072	1.34667
$708$	−3.99485	−0.150136
$709$	0.707300	0.0265632	0.0132816	−	0.999912i	$-0.495772\pi$
$709$	0.707300	0.0265632	0.0132816	+	0.999912i	$0.495772\pi$
$710$	0.249959	0.00938081
$711$	25.9735	0.974081
$712$	−23.5176	−0.881361
$713$	1.37491	0.0514906
$714$	−11.3136	−0.423402
$715$	0.155322	0.00580873
$716$	−14.8027	−0.553204
$717$	−3.88196	−0.144974
$718$	6.27816	0.234299
$719$	33.6525	1.25503	0.627513	−	0.778606i	$-0.284073\pi$
$719$	33.6525	1.25503	0.627513	+	0.778606i	$0.284073\pi$
$720$	−4.91577	−0.183200
$721$	−45.1378	−1.68102
$722$	35.4026	1.31755
$723$	−8.32117	−0.309468
$724$	−9.22312	−0.342774
$725$	−4.98158	−0.185011
$726$	−13.4690	−0.499882
$727$	7.69807	0.285506	0.142753	−	0.989758i	$-0.454405\pi$
$727$	7.69807	0.285506	0.142753	+	0.989758i	$0.454405\pi$
$728$	54.9301	2.03585
$729$	−15.4250	−0.571298
$730$	0.543981	0.0201336
$731$	24.8719	0.919920
$732$	14.9295	0.551810
$733$	−43.4025	−1.60311	−0.801554	−	0.597923i	$-0.795993\pi$
$733$	−43.4025	−1.60311	−0.801554	+	0.597923i	$0.795993\pi$
$734$	−23.6295	−0.872180
$735$	−0.598319	−0.0220693
$736$	18.1065	0.667414
$737$	1.42297	0.0524157
$738$	−29.6564	−1.09167
$739$	−19.9723	−0.734695	−0.367347	−	0.930084i	$-0.619734\pi$
$739$	−19.9723	−0.734695	−0.367347	+	0.930084i	$0.619734\pi$
$740$	6.77368	0.249006
$741$	1.80251	0.0662168
$742$	41.7462	1.53255
$743$	0.633909	0.0232559	0.0116279	−	0.999932i	$-0.496299\pi$
$743$	0.633909	0.0232559	0.0116279	+	0.999932i	$0.496299\pi$
$744$	−5.62082	−0.206069
$745$	1.79281	0.0656834
$746$	72.1326	2.64096
$747$	−13.5838	−0.497006
$748$	8.41380	0.307639
$749$	−17.1890	−0.628071
$750$	−1.74435	−0.0636948
$751$	−43.7496	−1.59645	−0.798224	−	0.602361i	$-0.794227\pi$
$751$	−43.7496	−1.59645	−0.798224	+	0.602361i	$0.794227\pi$
$752$	107.406	3.91668
$753$	4.22114	0.153827
$754$	−4.19576	−0.152800
$755$	0.249365	0.00907533
$756$	−58.3582	−2.12247
$757$	−16.9913	−0.617561	−0.308780	−	0.951133i	$-0.599921\pi$
$757$	−16.9913	−0.617561	−0.308780	+	0.951133i	$0.599921\pi$
$758$	−39.4871	−1.43423
$759$	0.359574	0.0130517
$760$	2.87413	0.104256
$761$	−37.0734	−1.34391	−0.671955	−	0.740592i	$-0.734545\pi$
$761$	−37.0734	−1.34391	−0.671955	+	0.740592i	$0.734545\pi$
$762$	−23.7189	−0.859246
$763$	38.2170	1.38355
$764$	61.3392	2.21918
$765$	−0.820644	−0.0296705
$766$	52.4541	1.89524
$767$	2.47918	0.0895179
$768$	−8.39448	−0.302910
$769$	−13.5334	−0.488027	−0.244013	−	0.969772i	$-0.578464\pi$
$769$	−13.5334	−0.488027	−0.244013	+	0.969772i	$0.578464\pi$
$770$	1.08099	0.0389562
$771$	8.42697	0.303490
$772$	−39.3338	−1.41565
$773$	−18.8915	−0.679479	−0.339739	−	0.940520i	$-0.610339\pi$
$773$	−18.8915	−0.679479	−0.339739	+	0.940520i	$0.610339\pi$
$774$	85.0713	3.05783
$775$	6.68959	0.240297
$776$	32.3726	1.16211
$777$	18.3041	0.656655
$778$	3.05695	0.109597
$779$	9.61406	0.344459
$780$	−0.530988	−0.0190124
$781$	−0.502322	−0.0179745
$782$	6.01508	0.215099
$783$	2.75972	0.0986244
$784$	120.485	4.30305
$785$	−2.61143	−0.0932060
$786$	10.1900	0.363467
$787$	−49.5485	−1.76621	−0.883107	−	0.469171i	$-0.844553\pi$
$787$	−49.5485	−1.76621	−0.883107	+	0.469171i	$0.844553\pi$
$788$	−12.8263	−0.456919
$789$	−3.22917	−0.114961
$790$	3.42521	0.121863
$791$	−10.3469	−0.367892
$792$	17.8168	0.633093
$793$	−9.26514	−0.329015
$794$	−69.8601	−2.47924
$795$	−0.249836	−0.00886077
$796$	37.0813	1.31431
$797$	−23.5944	−0.835756	−0.417878	−	0.908503i	$-0.637226\pi$
$797$	−23.5944	−0.835756	−0.417878	+	0.908503i	$0.637226\pi$
$798$	12.5448	0.444083
$799$	17.9304	0.634333
$800$	88.0970	3.11470
$801$	−7.44571	−0.263081
$802$	14.6912	0.518763
$803$	−1.09319	−0.0385779
$804$	−4.86459	−0.171561
$805$	0.559641	0.0197248
$806$	5.63434	0.198461
$807$	12.8432	0.452102
$808$	77.8280	2.73798
$809$	−28.4922	−1.00173	−0.500866	−	0.865525i	$-0.666985\pi$
$809$	−28.4922	−1.00173	−0.500866	+	0.865525i	$0.666985\pi$
$810$	−2.55622	−0.0898165
$811$	31.9784	1.12292	0.561458	−	0.827506i	$-0.310241\pi$
$811$	31.9784	1.12292	0.561458	+	0.827506i	$0.310241\pi$
$812$	−21.1464	−0.742094
$813$	2.90970	0.102048
$814$	−18.7974	−0.658850
$815$	−0.305538	−0.0107025
$816$	−13.6346	−0.477305
$817$	−27.5786	−0.964852
$818$	14.8547	0.519381
$819$	17.3909	0.607688
$820$	−2.83214	−0.0989025
$821$	−24.2478	−0.846256	−0.423128	−	0.906070i	$-0.639068\pi$
$821$	−24.2478	−0.846256	−0.423128	+	0.906070i	$0.639068\pi$
$822$	7.54910	0.263305
$823$	9.38687	0.327206	0.163603	−	0.986526i	$-0.447688\pi$
$823$	9.38687	0.327206	0.163603	+	0.986526i	$0.447688\pi$
$824$	−98.1082	−3.41776
$825$	1.74950	0.0609099
$826$	17.2542	0.600351
$827$	−24.6208	−0.856148	−0.428074	−	0.903744i	$-0.640808\pi$
$827$	−24.6208	−0.856148	−0.428074	+	0.903744i	$0.640808\pi$
$828$	14.8989	0.517773
$829$	39.5707	1.37435	0.687173	−	0.726493i	$-0.258851\pi$
$829$	39.5707	1.37435	0.687173	+	0.726493i	$0.258851\pi$
$830$	−1.79134	−0.0621785
$831$	5.10993	0.177262
$832$	33.4721	1.16044
$833$	20.1140	0.696908
$834$	1.28760	0.0445858
$835$	2.28522	0.0790832
$836$	−9.32943	−0.322665
$837$	−3.70593	−0.128096
$838$	−81.9943	−2.83245
$839$	−11.1479	−0.384868	−0.192434	−	0.981310i	$-0.561638\pi$
$839$	−11.1479	−0.384868	−0.192434	+	0.981310i	$0.561638\pi$
$840$	−2.28789	−0.0789399
$841$	1.00000	0.0344828
$842$	63.3998	2.18490
$843$	−14.0122	−0.482606
$844$	−143.938	−4.95455
$845$	−1.43488	−0.0493614
$846$	61.3290	2.10853
$847$	42.1279	1.44753
$848$	50.3102	1.72766
$849$	7.67712	0.263478
$850$	29.2663	1.00383
$851$	−9.73165	−0.333597
$852$	1.71725	0.0588319
$853$	37.9036	1.29780	0.648898	−	0.760876i	$-0.275230\pi$
$853$	37.9036	1.29780	0.648898	+	0.760876i	$0.275230\pi$
$854$	−64.4822	−2.20654
$855$	0.909951	0.0311197
$856$	−37.3607	−1.27696
$857$	−41.1992	−1.40734	−0.703670	−	0.710527i	$-0.748456\pi$
$857$	−41.1992	−1.40734	−0.703670	+	0.710527i	$0.748456\pi$
$858$	1.47353	0.0503054
$859$	−10.3221	−0.352186	−0.176093	−	0.984374i	$-0.556346\pi$
$859$	−10.3221	−0.352186	−0.176093	+	0.984374i	$0.556346\pi$
$860$	8.12417	0.277032
$861$	−7.65309	−0.260817
$862$	48.3942	1.64831
$863$	−38.1589	−1.29895	−0.649473	−	0.760385i	$-0.725010\pi$
$863$	−38.1589	−1.29895	−0.649473	+	0.760385i	$0.725010\pi$
$864$	−48.8044	−1.66036
$865$	−0.0559373	−0.00190192
$866$	47.1791	1.60321
$867$	5.85282	0.198772
$868$	28.3968	0.963851
$869$	−6.88334	−0.233501
$870$	0.174757	0.00592483
$871$	3.01893	0.102293
$872$	83.0656	2.81296
$873$	10.2492	0.346883
$874$	−6.66967	−0.225605
$875$	5.45593	0.184444
$876$	3.73721	0.126269
$877$	−15.8388	−0.534839	−0.267419	−	0.963580i	$-0.586171\pi$
$877$	−15.8388	−0.534839	−0.267419	+	0.963580i	$0.586171\pi$
$878$	−72.4704	−2.44576
$879$	−4.42915	−0.149392
$880$	1.30275	0.0439156
$881$	16.7227	0.563402	0.281701	−	0.959502i	$-0.409101\pi$
$881$	16.7227	0.563402	0.281701	+	0.959502i	$0.409101\pi$
$882$	68.7975	2.31653
$883$	−8.88463	−0.298992	−0.149496	−	0.988762i	$-0.547765\pi$
$883$	−8.88463	−0.298992	−0.149496	+	0.988762i	$0.547765\pi$
$884$	17.8505	0.600377
$885$	−0.103260	−0.00347105
$886$	−45.8003	−1.53869
$887$	1.55200	0.0521111	0.0260555	−	0.999660i	$-0.491705\pi$
$887$	1.55200	0.0521111	0.0260555	+	0.999660i	$0.491705\pi$
$888$	39.7844	1.33508
$889$	74.1871	2.48816
$890$	−0.981890	−0.0329130
$891$	5.13702	0.172097
$892$	93.1782	3.11984
$893$	−19.8817	−0.665316
$894$	17.0082	0.568839
$895$	−0.382626	−0.0127898
$896$	90.5123	3.02380
$897$	0.762863	0.0254712
$898$	−39.5796	−1.32079
$899$	−1.34287	−0.0447871
$900$	72.4905	2.41635
$901$	8.39885	0.279806
$902$	7.85937	0.261688
$903$	21.9534	0.730563
$904$	−22.4892	−0.747979
$905$	−0.238402	−0.00792475
$906$	2.36570	0.0785953
$907$	11.6866	0.388048	0.194024	−	0.980997i	$-0.437846\pi$
$907$	11.6866	0.388048	0.194024	+	0.980997i	$0.437846\pi$
$908$	−89.7441	−2.97826
$909$	24.6404	0.817271
$910$	2.29340	0.0760254
$911$	38.6443	1.28034	0.640172	−	0.768232i	$-0.278863\pi$
$911$	38.6443	1.28034	0.640172	+	0.768232i	$0.278863\pi$
$912$	15.1183	0.500618
$913$	3.59991	0.119140
$914$	32.5585	1.07694
$915$	0.385902	0.0127575
$916$	−37.9464	−1.25378
$917$	−31.8720	−1.05251
$918$	−16.2131	−0.535112
$919$	31.4566	1.03766	0.518829	−	0.854878i	$-0.326368\pi$
$919$	31.4566	1.03766	0.518829	+	0.854878i	$0.326368\pi$
$920$	1.21640	0.0401034
$921$	−0.377212	−0.0124296
$922$	41.6847	1.37281
$923$	−1.06571	−0.0350783
$924$	7.42652	0.244314
$925$	−47.3493	−1.55683
$926$	28.4094	0.933591
$927$	−31.0612	−1.02018
$928$	−17.6846	−0.580524
$929$	−30.1222	−0.988277	−0.494138	−	0.869383i	$-0.664516\pi$
$929$	−30.1222	−0.988277	−0.494138	+	0.869383i	$0.664516\pi$
$930$	−0.234676	−0.00769532
$931$	−22.3029	−0.730948
$932$	−50.9333	−1.66838
$933$	1.04139	0.0340937
$934$	86.4767	2.82960
$935$	0.217482	0.00711244
$936$	37.7997	1.23552
$937$	53.1197	1.73534	0.867672	−	0.497138i	$-0.165616\pi$
$937$	53.1197	1.73534	0.867672	+	0.497138i	$0.165616\pi$
$938$	21.0107	0.686024
$939$	0.820186	0.0267658
$940$	5.85681	0.191028
$941$	6.62931	0.216109	0.108055	−	0.994145i	$-0.465538\pi$
$941$	6.62931	0.216109	0.108055	+	0.994145i	$0.465538\pi$
$942$	−24.7744	−0.807194
$943$	4.06889	0.132501
$944$	20.7938	0.676781
$945$	−1.50846	−0.0490703
$946$	−22.5451	−0.733005
$947$	−35.7338	−1.16119	−0.580596	−	0.814192i	$-0.697180\pi$
$947$	−35.7338	−1.16119	−0.580596	+	0.814192i	$0.697180\pi$
$948$	23.5315	0.764268
$949$	−2.31929	−0.0752872
$950$	−32.4512	−1.05286
$951$	−8.08583	−0.262201
$952$	76.9132	2.49277
$953$	47.0173	1.52304	0.761519	−	0.648142i	$-0.224454\pi$
$953$	47.0173	1.52304	0.761519	+	0.648142i	$0.224454\pi$
$954$	28.7273	0.930080
$955$	1.58552	0.0513061
$956$	42.6272	1.37866
$957$	−0.351195	−0.0113525
$958$	61.3434	1.98192
$959$	−23.6118	−0.762464
$960$	−1.39415	−0.0449959
$961$	−29.1967	−0.941829
$962$	−39.8801	−1.28579
$963$	−11.8284	−0.381166
$964$	91.3734	2.94294
$965$	−1.01671	−0.0327291
$966$	5.30926	0.170823
$967$	47.8946	1.54019	0.770094	−	0.637931i	$-0.220209\pi$
$967$	47.8946	1.54019	0.770094	+	0.637931i	$0.220209\pi$
$968$	91.5661	2.94305
$969$	2.52388	0.0810785
$970$	1.35159	0.0433971
$971$	−46.9138	−1.50554	−0.752768	−	0.658286i	$-0.771282\pi$
$971$	−46.9138	−1.50554	−0.752768	+	0.658286i	$0.771282\pi$
$972$	−61.0335	−1.95765
$973$	−4.02730	−0.129109
$974$	109.421	3.50607
$975$	3.71170	0.118870
$976$	−77.7104	−2.48745
$977$	27.2525	0.871884	0.435942	−	0.899975i	$-0.356415\pi$
$977$	27.2525	0.871884	0.435942	+	0.899975i	$0.356415\pi$
$978$	−2.89861	−0.0926872
$979$	1.97322	0.0630643
$980$	6.57005	0.209872
$981$	26.2987	0.839651
$982$	88.0705	2.81044
$983$	41.8348	1.33432	0.667161	−	0.744913i	$-0.267509\pi$
$983$	41.8348	1.33432	0.667161	+	0.744913i	$0.267509\pi$
$984$	−16.6342	−0.530279
$985$	−0.331539	−0.0105637
$986$	−5.87490	−0.187095
$987$	15.8265	0.503762
$988$	−19.7931	−0.629701
$989$	−11.6719	−0.371144
$990$	0.743873	0.0236418
$991$	22.0360	0.699998	0.349999	−	0.936750i	$-0.386182\pi$
$991$	22.0360	0.699998	0.349999	+	0.936750i	$0.386182\pi$
$992$	23.7480	0.753999
$993$	−10.4514	−0.331664
$994$	−7.41699	−0.235253
$995$	0.958488	0.0303861
$996$	−12.3067	−0.389953
$997$	−42.6431	−1.35052	−0.675260	−	0.737580i	$-0.735969\pi$
$997$	−42.6431	−1.35052	−0.675260	+	0.737580i	$0.735969\pi$
$998$	−50.2542	−1.59077
$999$	26.2308	0.829905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.2	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.2	✓	59	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69273 q^{2} -0.478176 q^{3} +5.25077 q^{4} +0.135724 q^{5} +1.28760 q^{6} -4.02730 q^{7} -8.75344 q^{8} -2.77135 q^{9} +O(q^{10})\)	\(q-2.69273 q^{2} -0.478176 q^{3} +5.25077 q^{4} +0.135724 q^{5} +1.28760 q^{6} -4.02730 q^{7} -8.75344 q^{8} -2.77135 q^{9} -0.365467 q^{10} +0.734447 q^{11} -2.51079 q^{12} +1.55818 q^{13} +10.8444 q^{14} -0.0648998 q^{15} +13.0691 q^{16} +2.18177 q^{17} +7.46248 q^{18} -2.41920 q^{19} +0.712654 q^{20} +1.92576 q^{21} -1.97766 q^{22} -1.02386 q^{23} +4.18569 q^{24} -4.98158 q^{25} -4.19576 q^{26} +2.75972 q^{27} -21.1464 q^{28} +1.00000 q^{29} +0.174757 q^{30} -1.34287 q^{31} -17.6846 q^{32} -0.351195 q^{33} -5.87490 q^{34} -0.546599 q^{35} -14.5517 q^{36} +9.50487 q^{37} +6.51424 q^{38} -0.745085 q^{39} -1.18805 q^{40} -3.97407 q^{41} -5.18554 q^{42} +11.3999 q^{43} +3.85641 q^{44} -0.376137 q^{45} +2.75697 q^{46} +8.21831 q^{47} -6.24932 q^{48} +9.21912 q^{49} +13.4140 q^{50} -1.04327 q^{51} +8.18166 q^{52} +3.84956 q^{53} -7.43117 q^{54} +0.0996818 q^{55} +35.2527 q^{56} +1.15680 q^{57} -2.69273 q^{58} +1.59107 q^{59} -0.340774 q^{60} -5.94613 q^{61} +3.61597 q^{62} +11.1610 q^{63} +21.4815 q^{64} +0.211482 q^{65} +0.945672 q^{66} +1.93747 q^{67} +11.4560 q^{68} +0.489585 q^{69} +1.47184 q^{70} -0.683946 q^{71} +24.2588 q^{72} -1.48846 q^{73} -25.5940 q^{74} +2.38207 q^{75} -12.7027 q^{76} -2.95784 q^{77} +2.00631 q^{78} -9.37214 q^{79} +1.77378 q^{80} +6.99441 q^{81} +10.7011 q^{82} +4.90153 q^{83} +10.1117 q^{84} +0.296117 q^{85} -30.6967 q^{86} -0.478176 q^{87} -6.42894 q^{88} +2.68667 q^{89} +1.01283 q^{90} -6.27526 q^{91} -5.37605 q^{92} +0.642126 q^{93} -22.1297 q^{94} -0.328342 q^{95} +8.45633 q^{96} -3.69827 q^{97} -24.8246 q^{98} -2.03541 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more