LMFDB - Embedded newform 4334.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, 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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.85453 q^{3} +1.00000 q^{4} +3.38544 q^{5} +1.85453 q^{6} +1.90407 q^{7} +1.00000 q^{8} +0.439283 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.85453 q^{3} +1.00000 q^{4} +3.38544 q^{5} +1.85453 q^{6} +1.90407 q^{7} +1.00000 q^{8} +0.439283 q^{9} +3.38544 q^{10} -1.00000 q^{11} +1.85453 q^{12} +5.80723 q^{13} +1.90407 q^{14} +6.27841 q^{15} +1.00000 q^{16} -2.03010 q^{17} +0.439283 q^{18} -0.936970 q^{19} +3.38544 q^{20} +3.53116 q^{21} -1.00000 q^{22} +5.36677 q^{23} +1.85453 q^{24} +6.46122 q^{25} +5.80723 q^{26} -4.74893 q^{27} +1.90407 q^{28} -1.64151 q^{29} +6.27841 q^{30} -4.73759 q^{31} +1.00000 q^{32} -1.85453 q^{33} -2.03010 q^{34} +6.44613 q^{35} +0.439283 q^{36} +4.49263 q^{37} -0.936970 q^{38} +10.7697 q^{39} +3.38544 q^{40} -6.56031 q^{41} +3.53116 q^{42} -6.57003 q^{43} -1.00000 q^{44} +1.48717 q^{45} +5.36677 q^{46} -11.3501 q^{47} +1.85453 q^{48} -3.37450 q^{49} +6.46122 q^{50} -3.76489 q^{51} +5.80723 q^{52} -10.9284 q^{53} -4.74893 q^{54} -3.38544 q^{55} +1.90407 q^{56} -1.73764 q^{57} -1.64151 q^{58} +7.86481 q^{59} +6.27841 q^{60} -5.64178 q^{61} -4.73759 q^{62} +0.836427 q^{63} +1.00000 q^{64} +19.6600 q^{65} -1.85453 q^{66} -7.90173 q^{67} -2.03010 q^{68} +9.95284 q^{69} +6.44613 q^{70} +3.17744 q^{71} +0.439283 q^{72} -8.89795 q^{73} +4.49263 q^{74} +11.9825 q^{75} -0.936970 q^{76} -1.90407 q^{77} +10.7697 q^{78} +15.7206 q^{79} +3.38544 q^{80} -10.1249 q^{81} -6.56031 q^{82} +10.7270 q^{83} +3.53116 q^{84} -6.87280 q^{85} -6.57003 q^{86} -3.04423 q^{87} -1.00000 q^{88} +1.98957 q^{89} +1.48717 q^{90} +11.0574 q^{91} +5.36677 q^{92} -8.78601 q^{93} -11.3501 q^{94} -3.17206 q^{95} +1.85453 q^{96} +10.4072 q^{97} -3.37450 q^{98} -0.439283 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9}+O(q^{10}) $ 24 * q + 24 * q^2 + 8 * q^3 + 24 * q^4 + 8 * q^6 + 11 * q^7 + 24 * q^8 + 28 * q^9	\( 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9} - 24 q^{11} + 8 q^{12} + 9 q^{13} + 11 q^{14} + 6 q^{15} + 24 q^{16} - q^{17} + 28 q^{18} + 35 q^{19} + 21 q^{21} - 24 q^{22} + 13 q^{23} + 8 q^{24} + 38 q^{25} + 9 q^{26} + 23 q^{27} + 11 q^{28} + 15 q^{29} + 6 q^{30} + 31 q^{31} + 24 q^{32} - 8 q^{33} - q^{34} + 28 q^{35} + 28 q^{36} + 13 q^{37} + 35 q^{38} + 31 q^{39} + 16 q^{41} + 21 q^{42} + 35 q^{43} - 24 q^{44} + 13 q^{45} + 13 q^{46} + 46 q^{47} + 8 q^{48} + 55 q^{49} + 38 q^{50} + 18 q^{51} + 9 q^{52} + 6 q^{53} + 23 q^{54} + 11 q^{56} + 18 q^{57} + 15 q^{58} + 13 q^{59} + 6 q^{60} + 60 q^{61} + 31 q^{62} + 52 q^{63} + 24 q^{64} - 9 q^{65} - 8 q^{66} + 28 q^{67} - q^{68} + 21 q^{69} + 28 q^{70} + 10 q^{71} + 28 q^{72} + 3 q^{73} + 13 q^{74} + 48 q^{75} + 35 q^{76} - 11 q^{77} + 31 q^{78} + 47 q^{79} + 16 q^{81} + 16 q^{82} + 66 q^{83} + 21 q^{84} + 37 q^{85} + 35 q^{86} + 34 q^{87} - 24 q^{88} - 5 q^{89} + 13 q^{90} + q^{91} + 13 q^{92} - 16 q^{93} + 46 q^{94} + 9 q^{95} + 8 q^{96} + 24 q^{97} + 55 q^{98} - 28 q^{99}+O(q^{100}) \) 24 * q + 24 * q^2 + 8 * q^3 + 24 * q^4 + 8 * q^6 + 11 * q^7 + 24 * q^8 + 28 * q^9 - 24 * q^11 + 8 * q^12 + 9 * q^13 + 11 * q^14 + 6 * q^15 + 24 * q^16 - q^17 + 28 * q^18 + 35 * q^19 + 21 * q^21 - 24 * q^22 + 13 * q^23 + 8 * q^24 + 38 * q^25 + 9 * q^26 + 23 * q^27 + 11 * q^28 + 15 * q^29 + 6 * q^30 + 31 * q^31 + 24 * q^32 - 8 * q^33 - q^34 + 28 * q^35 + 28 * q^36 + 13 * q^37 + 35 * q^38 + 31 * q^39 + 16 * q^41 + 21 * q^42 + 35 * q^43 - 24 * q^44 + 13 * q^45 + 13 * q^46 + 46 * q^47 + 8 * q^48 + 55 * q^49 + 38 * q^50 + 18 * q^51 + 9 * q^52 + 6 * q^53 + 23 * q^54 + 11 * q^56 + 18 * q^57 + 15 * q^58 + 13 * q^59 + 6 * q^60 + 60 * q^61 + 31 * q^62 + 52 * q^63 + 24 * q^64 - 9 * q^65 - 8 * q^66 + 28 * q^67 - q^68 + 21 * q^69 + 28 * q^70 + 10 * q^71 + 28 * q^72 + 3 * q^73 + 13 * q^74 + 48 * q^75 + 35 * q^76 - 11 * q^77 + 31 * q^78 + 47 * q^79 + 16 * q^81 + 16 * q^82 + 66 * q^83 + 21 * q^84 + 37 * q^85 + 35 * q^86 + 34 * q^87 - 24 * q^88 - 5 * q^89 + 13 * q^90 + q^91 + 13 * q^92 - 16 * q^93 + 46 * q^94 + 9 * q^95 + 8 * q^96 + 24 * q^97 + 55 * q^98 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.85453	1.07071	0.535357	−	0.844626i	$-0.320177\pi$
$3$	1.85453	1.07071	0.535357	+	0.844626i	$0.320177\pi$
$4$	1.00000	0.500000
$5$	3.38544	1.51402	0.757008	−	0.653406i	$-0.226660\pi$
$5$	3.38544	1.51402	0.757008	+	0.653406i	$0.226660\pi$
$6$	1.85453	0.757109
$7$	1.90407	0.719672	0.359836	−	0.933016i	$-0.382833\pi$
$7$	1.90407	0.719672	0.359836	+	0.933016i	$0.382833\pi$
$8$	1.00000	0.353553
$9$	0.439283	0.146428
$10$	3.38544	1.07057
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.85453	0.535357
$13$	5.80723	1.61063	0.805317	−	0.592844i	$-0.201995\pi$
$13$	5.80723	1.61063	0.805317	+	0.592844i	$0.201995\pi$
$14$	1.90407	0.508885
$15$	6.27841	1.62108
$16$	1.00000	0.250000
$17$	−2.03010	−0.492372	−0.246186	−	0.969223i	$-0.579177\pi$
$17$	−2.03010	−0.492372	−0.246186	+	0.969223i	$0.579177\pi$
$18$	0.439283	0.103540
$19$	−0.936970	−0.214956	−0.107478	−	0.994207i	$-0.534277\pi$
$19$	−0.936970	−0.214956	−0.107478	+	0.994207i	$0.534277\pi$
$20$	3.38544	0.757008
$21$	3.53116	0.770563
$22$	−1.00000	−0.213201
$23$	5.36677	1.11905	0.559525	−	0.828814i	$-0.310984\pi$
$23$	5.36677	1.11905	0.559525	+	0.828814i	$0.310984\pi$
$24$	1.85453	0.378554
$25$	6.46122	1.29224
$26$	5.80723	1.13889
$27$	−4.74893	−0.913931
$28$	1.90407	0.359836
$29$	−1.64151	−0.304821	−0.152410	−	0.988317i	$-0.548704\pi$
$29$	−1.64151	−0.304821	−0.152410	+	0.988317i	$0.548704\pi$
$30$	6.27841	1.14628
$31$	−4.73759	−0.850897	−0.425448	−	0.904983i	$-0.639884\pi$
$31$	−4.73759	−0.850897	−0.425448	+	0.904983i	$0.639884\pi$
$32$	1.00000	0.176777
$33$	−1.85453	−0.322832
$34$	−2.03010	−0.348160
$35$	6.44613	1.08960
$36$	0.439283	0.0732139
$37$	4.49263	0.738583	0.369292	−	0.929314i	$-0.379600\pi$
$37$	4.49263	0.738583	0.369292	+	0.929314i	$0.379600\pi$
$38$	−0.936970	−0.151997
$39$	10.7697	1.72453
$40$	3.38544	0.535286
$41$	−6.56031	−1.02455	−0.512274	−	0.858822i	$-0.671197\pi$
$41$	−6.56031	−1.02455	−0.512274	+	0.858822i	$0.671197\pi$
$42$	3.53116	0.544870
$43$	−6.57003	−1.00192	−0.500960	−	0.865470i	$-0.667020\pi$
$43$	−6.57003	−1.00192	−0.500960	+	0.865470i	$0.667020\pi$
$44$	−1.00000	−0.150756
$45$	1.48717	0.221694
$46$	5.36677	0.791287
$47$	−11.3501	−1.65558	−0.827792	−	0.561034i	$-0.810404\pi$
$47$	−11.3501	−1.65558	−0.827792	+	0.561034i	$0.810404\pi$
$48$	1.85453	0.267678
$49$	−3.37450	−0.482072
$50$	6.46122	0.913755
$51$	−3.76489	−0.527190
$52$	5.80723	0.805317
$53$	−10.9284	−1.50113	−0.750567	−	0.660795i	$-0.770219\pi$
$53$	−10.9284	−1.50113	−0.750567	+	0.660795i	$0.770219\pi$
$54$	−4.74893	−0.646247
$55$	−3.38544	−0.456493
$56$	1.90407	0.254443
$57$	−1.73764	−0.230156
$58$	−1.64151	−0.215541
$59$	7.86481	1.02391	0.511955	−	0.859012i	$-0.328921\pi$
$59$	7.86481	1.02391	0.511955	+	0.859012i	$0.328921\pi$
$60$	6.27841	0.810539
$61$	−5.64178	−0.722356	−0.361178	−	0.932497i	$-0.617625\pi$
$61$	−5.64178	−0.722356	−0.361178	+	0.932497i	$0.617625\pi$
$62$	−4.73759	−0.601675
$63$	0.836427	0.105380
$64$	1.00000	0.125000
$65$	19.6600	2.43853
$66$	−1.85453	−0.228277
$67$	−7.90173	−0.965350	−0.482675	−	0.875800i	$-0.660335\pi$
$67$	−7.90173	−0.965350	−0.482675	+	0.875800i	$0.660335\pi$
$68$	−2.03010	−0.246186
$69$	9.95284	1.19818
$70$	6.44613	0.770460
$71$	3.17744	0.377093	0.188547	−	0.982064i	$-0.439622\pi$
$71$	3.17744	0.377093	0.188547	+	0.982064i	$0.439622\pi$
$72$	0.439283	0.0517700
$73$	−8.89795	−1.04143	−0.520713	−	0.853732i	$-0.674334\pi$
$73$	−8.89795	−1.04143	−0.520713	+	0.853732i	$0.674334\pi$
$74$	4.49263	0.522257
$75$	11.9825	1.38362
$76$	−0.936970	−0.107478
$77$	−1.90407	−0.216989
$78$	10.7697	1.21943
$79$	15.7206	1.76871	0.884356	−	0.466814i	$-0.154598\pi$
$79$	15.7206	1.76871	0.884356	+	0.466814i	$0.154598\pi$
$80$	3.38544	0.378504
$81$	−10.1249	−1.12499
$82$	−6.56031	−0.724465
$83$	10.7270	1.17744	0.588719	−	0.808338i	$-0.299632\pi$
$83$	10.7270	1.17744	0.588719	+	0.808338i	$0.299632\pi$
$84$	3.53116	0.385281
$85$	−6.87280	−0.745460
$86$	−6.57003	−0.708465
$87$	−3.04423	−0.326376
$88$	−1.00000	−0.106600
$89$	1.98957	0.210894	0.105447	−	0.994425i	$-0.466373\pi$
$89$	1.98957	0.210894	0.105447	+	0.994425i	$0.466373\pi$
$90$	1.48717	0.156761
$91$	11.0574	1.15913
$92$	5.36677	0.559525
$93$	−8.78601	−0.911067
$94$	−11.3501	−1.17068
$95$	−3.17206	−0.325446
$96$	1.85453	0.189277
$97$	10.4072	1.05669	0.528345	−	0.849030i	$-0.322813\pi$
$97$	10.4072	1.05669	0.528345	+	0.849030i	$0.322813\pi$
$98$	−3.37450	−0.340876
$99$	−0.439283	−0.0441496
$100$	6.46122	0.646122
$101$	0.867122	0.0862818	0.0431409	−	0.999069i	$-0.486264\pi$
$101$	0.867122	0.0862818	0.0431409	+	0.999069i	$0.486264\pi$
$102$	−3.76489	−0.372779
$103$	−1.60832	−0.158472	−0.0792362	−	0.996856i	$-0.525248\pi$
$103$	−1.60832	−0.158472	−0.0792362	+	0.996856i	$0.525248\pi$
$104$	5.80723	0.569445
$105$	11.9545	1.16664
$106$	−10.9284	−1.06146
$107$	−5.94621	−0.574842	−0.287421	−	0.957804i	$-0.592798\pi$
$107$	−5.94621	−0.574842	−0.287421	+	0.957804i	$0.592798\pi$
$108$	−4.74893	−0.456966
$109$	3.87425	0.371086	0.185543	−	0.982636i	$-0.440596\pi$
$109$	3.87425	0.371086	0.185543	+	0.982636i	$0.440596\pi$
$110$	−3.38544	−0.322789
$111$	8.33171	0.790811
$112$	1.90407	0.179918
$113$	−8.36351	−0.786773	−0.393387	−	0.919373i	$-0.628697\pi$
$113$	−8.36351	−0.786773	−0.393387	+	0.919373i	$0.628697\pi$
$114$	−1.73764	−0.162745
$115$	18.1689	1.69426
$116$	−1.64151	−0.152410
$117$	2.55102	0.235842
$118$	7.86481	0.724014
$119$	−3.86547	−0.354347
$120$	6.27841	0.573138
$121$	1.00000	0.0909091
$122$	−5.64178	−0.510783
$123$	−12.1663	−1.09700
$124$	−4.73759	−0.425448
$125$	4.94689	0.442464
$126$	0.836427	0.0745149
$127$	−2.97332	−0.263839	−0.131920	−	0.991260i	$-0.542114\pi$
$127$	−2.97332	−0.263839	−0.131920	+	0.991260i	$0.542114\pi$
$128$	1.00000	0.0883883
$129$	−12.1843	−1.07277
$130$	19.6600	1.72430
$131$	5.33634	0.466238	0.233119	−	0.972448i	$-0.425107\pi$
$131$	5.33634	0.466238	0.233119	+	0.972448i	$0.425107\pi$
$132$	−1.85453	−0.161416
$133$	−1.78406	−0.154698
$134$	−7.90173	−0.682605
$135$	−16.0772	−1.38371
$136$	−2.03010	−0.174080
$137$	−11.8996	−1.01666	−0.508328	−	0.861164i	$-0.669736\pi$
$137$	−11.8996	−1.01666	−0.508328	+	0.861164i	$0.669736\pi$
$138$	9.95284	0.847242
$139$	14.6394	1.24169	0.620847	−	0.783931i	$-0.286789\pi$
$139$	14.6394	1.24169	0.620847	+	0.783931i	$0.286789\pi$
$140$	6.44613	0.544798
$141$	−21.0491	−1.77266
$142$	3.17744	0.266645
$143$	−5.80723	−0.485625
$144$	0.439283	0.0366069
$145$	−5.55724	−0.461504
$146$	−8.89795	−0.736399
$147$	−6.25812	−0.516161
$148$	4.49263	0.369292
$149$	−2.53885	−0.207991	−0.103995	−	0.994578i	$-0.533163\pi$
$149$	−2.53885	−0.207991	−0.103995	+	0.994578i	$0.533163\pi$
$150$	11.9825	0.978370
$151$	0.587966	0.0478480	0.0239240	−	0.999714i	$-0.492384\pi$
$151$	0.587966	0.0478480	0.0239240	+	0.999714i	$0.492384\pi$
$152$	−0.936970	−0.0759983
$153$	−0.891790	−0.0720970
$154$	−1.90407	−0.153435
$155$	−16.0388	−1.28827
$156$	10.7697	0.862264
$157$	8.12331	0.648311	0.324155	−	0.946004i	$-0.394920\pi$
$157$	8.12331	0.648311	0.324155	+	0.946004i	$0.394920\pi$
$158$	15.7206	1.25067
$159$	−20.2671	−1.60728
$160$	3.38544	0.267643
$161$	10.2187	0.805349
$162$	−10.1249	−0.795486
$163$	2.46624	0.193171	0.0965854	−	0.995325i	$-0.469208\pi$
$163$	2.46624	0.193171	0.0965854	+	0.995325i	$0.469208\pi$
$164$	−6.56031	−0.512274
$165$	−6.27841	−0.488773
$166$	10.7270	0.832574
$167$	5.42987	0.420176	0.210088	−	0.977682i	$-0.432625\pi$
$167$	5.42987	0.420176	0.210088	+	0.977682i	$0.432625\pi$
$168$	3.53116	0.272435
$169$	20.7239	1.59414
$170$	−6.87280	−0.527120
$171$	−0.411595	−0.0314755
$172$	−6.57003	−0.500960
$173$	−11.7967	−0.896886	−0.448443	−	0.893811i	$-0.648021\pi$
$173$	−11.7967	−0.896886	−0.448443	+	0.893811i	$0.648021\pi$
$174$	−3.04423	−0.230783
$175$	12.3026	0.929993
$176$	−1.00000	−0.0753778
$177$	14.5855	1.09632
$178$	1.98957	0.149125
$179$	14.9410	1.11674	0.558370	−	0.829592i	$-0.311427\pi$
$179$	14.9410	1.11674	0.558370	+	0.829592i	$0.311427\pi$
$180$	1.48717	0.110847
$181$	5.52843	0.410925	0.205462	−	0.978665i	$-0.434130\pi$
$181$	5.52843	0.410925	0.205462	+	0.978665i	$0.434130\pi$
$182$	11.0574	0.819628
$183$	−10.4629	−0.773437
$184$	5.36677	0.395644
$185$	15.2095	1.11823
$186$	−8.78601	−0.644221
$187$	2.03010	0.148456
$188$	−11.3501	−0.827792
$189$	−9.04231	−0.657731
$190$	−3.17206	−0.230125
$191$	−2.72065	−0.196859	−0.0984296	−	0.995144i	$-0.531382\pi$
$191$	−2.72065	−0.196859	−0.0984296	+	0.995144i	$0.531382\pi$
$192$	1.85453	0.133839
$193$	16.5648	1.19236	0.596178	−	0.802852i	$-0.296685\pi$
$193$	16.5648	1.19236	0.596178	+	0.802852i	$0.296685\pi$
$194$	10.4072	0.747193
$195$	36.4601	2.61096
$196$	−3.37450	−0.241036
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−0.439283	−0.0312185
$199$	−2.42358	−0.171803	−0.0859015	−	0.996304i	$-0.527377\pi$
$199$	−2.42358	−0.171803	−0.0859015	+	0.996304i	$0.527377\pi$
$200$	6.46122	0.456878
$201$	−14.6540	−1.03361
$202$	0.867122	0.0610105
$203$	−3.12556	−0.219371
$204$	−3.76489	−0.263595
$205$	−22.2096	−1.55118
$206$	−1.60832	−0.112057
$207$	2.35753	0.163860
$208$	5.80723	0.402659
$209$	0.936970	0.0648116
$210$	11.9545	0.824942
$211$	−16.9595	−1.16754	−0.583770	−	0.811919i	$-0.698423\pi$
$211$	−16.9595	−1.16754	−0.583770	+	0.811919i	$0.698423\pi$
$212$	−10.9284	−0.750567
$213$	5.89267	0.403759
$214$	−5.94621	−0.406474
$215$	−22.2425	−1.51692
$216$	−4.74893	−0.323124
$217$	−9.02072	−0.612366
$218$	3.87425	0.262397
$219$	−16.5015	−1.11507
$220$	−3.38544	−0.228247
$221$	−11.7893	−0.793032
$222$	8.33171	0.559188
$223$	−4.32158	−0.289394	−0.144697	−	0.989476i	$-0.546221\pi$
$223$	−4.32158	−0.289394	−0.144697	+	0.989476i	$0.546221\pi$
$224$	1.90407	0.127221
$225$	2.83831	0.189220
$226$	−8.36351	−0.556333
$227$	15.1607	1.00625	0.503126	−	0.864213i	$-0.332183\pi$
$227$	15.1607	1.00625	0.503126	+	0.864213i	$0.332183\pi$
$228$	−1.73764	−0.115078
$229$	−15.8060	−1.04449	−0.522244	−	0.852796i	$-0.674905\pi$
$229$	−15.8060	−1.04449	−0.522244	+	0.852796i	$0.674905\pi$
$230$	18.1689	1.19802
$231$	−3.53116	−0.232333
$232$	−1.64151	−0.107770
$233$	−13.4444	−0.880771	−0.440386	−	0.897809i	$-0.645158\pi$
$233$	−13.4444	−0.880771	−0.440386	+	0.897809i	$0.645158\pi$
$234$	2.55102	0.166765
$235$	−38.4252	−2.50658
$236$	7.86481	0.511955
$237$	29.1544	1.89378
$238$	−3.86547	−0.250561
$239$	12.5023	0.808706	0.404353	−	0.914603i	$-0.367497\pi$
$239$	12.5023	0.808706	0.404353	+	0.914603i	$0.367497\pi$
$240$	6.27841	0.405269
$241$	18.3359	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$241$	18.3359	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$242$	1.00000	0.0642824
$243$	−4.53012	−0.290607
$244$	−5.64178	−0.361178
$245$	−11.4242	−0.729865
$246$	−12.1663	−0.775695
$247$	−5.44120	−0.346215
$248$	−4.73759	−0.300837
$249$	19.8935	1.26070
$250$	4.94689	0.312869
$251$	5.72822	0.361562	0.180781	−	0.983523i	$-0.442137\pi$
$251$	5.72822	0.361562	0.180781	+	0.983523i	$0.442137\pi$
$252$	0.836427	0.0526900
$253$	−5.36677	−0.337406
$254$	−2.97332	−0.186563
$255$	−12.7458	−0.798174
$256$	1.00000	0.0625000
$257$	15.1011	0.941979	0.470989	−	0.882139i	$-0.343897\pi$
$257$	15.1011	0.941979	0.470989	+	0.882139i	$0.343897\pi$
$258$	−12.1843	−0.758563
$259$	8.55429	0.531538
$260$	19.6600	1.21926
$261$	−0.721088	−0.0446342
$262$	5.33634	0.329680
$263$	−1.61389	−0.0995165	−0.0497582	−	0.998761i	$-0.515845\pi$
$263$	−1.61389	−0.0995165	−0.0497582	+	0.998761i	$0.515845\pi$
$264$	−1.85453	−0.114138
$265$	−36.9975	−2.27274
$266$	−1.78406	−0.109388
$267$	3.68972	0.225807
$268$	−7.90173	−0.482675
$269$	−23.8635	−1.45498	−0.727492	−	0.686116i	$-0.759314\pi$
$269$	−23.8635	−1.45498	−0.727492	+	0.686116i	$0.759314\pi$
$270$	−16.0772	−0.978429
$271$	2.15340	0.130810	0.0654048	−	0.997859i	$-0.479166\pi$
$271$	2.15340	0.130810	0.0654048	+	0.997859i	$0.479166\pi$
$272$	−2.03010	−0.123093
$273$	20.5063	1.24110
$274$	−11.8996	−0.718884
$275$	−6.46122	−0.389627
$276$	9.95284	0.599091
$277$	16.7556	1.00675	0.503373	−	0.864069i	$-0.332092\pi$
$277$	16.7556	1.00675	0.503373	+	0.864069i	$0.332092\pi$
$278$	14.6394	0.878011
$279$	−2.08114	−0.124595
$280$	6.44613	0.385230
$281$	8.11214	0.483929	0.241965	−	0.970285i	$-0.422208\pi$
$281$	8.11214	0.483929	0.241965	+	0.970285i	$0.422208\pi$
$282$	−21.0491	−1.25346
$283$	27.7080	1.64707	0.823535	−	0.567265i	$-0.191999\pi$
$283$	27.7080	1.64707	0.823535	+	0.567265i	$0.191999\pi$
$284$	3.17744	0.188547
$285$	−5.88268	−0.348460
$286$	−5.80723	−0.343389
$287$	−12.4913	−0.737339
$288$	0.439283	0.0258850
$289$	−12.8787	−0.757569
$290$	−5.55724	−0.326332
$291$	19.3005	1.13141
$292$	−8.89795	−0.520713
$293$	−11.3229	−0.661490	−0.330745	−	0.943720i	$-0.607300\pi$
$293$	−11.3229	−0.661490	−0.330745	+	0.943720i	$0.607300\pi$
$294$	−6.25812	−0.364981
$295$	26.6259	1.55022
$296$	4.49263	0.261129
$297$	4.74893	0.275561
$298$	−2.53885	−0.147072
$299$	31.1661	1.80238
$300$	11.9825	0.691812
$301$	−12.5098	−0.721054
$302$	0.587966	0.0338337
$303$	1.60810	0.0923831
$304$	−0.936970	−0.0537389
$305$	−19.0999	−1.09366
$306$	−0.891790	−0.0509803
$307$	0.353242	0.0201606	0.0100803	−	0.999949i	$-0.496791\pi$
$307$	0.353242	0.0201606	0.0100803	+	0.999949i	$0.496791\pi$
$308$	−1.90407	−0.108495
$309$	−2.98268	−0.169679
$310$	−16.0388	−0.910945
$311$	7.02728	0.398480	0.199240	−	0.979951i	$-0.436153\pi$
$311$	7.02728	0.398480	0.199240	+	0.979951i	$0.436153\pi$
$312$	10.7697	0.609713
$313$	−6.63242	−0.374887	−0.187443	−	0.982275i	$-0.560020\pi$
$313$	−6.63242	−0.374887	−0.187443	+	0.982275i	$0.560020\pi$
$314$	8.12331	0.458425
$315$	2.83168	0.159547
$316$	15.7206	0.884356
$317$	14.5932	0.819635	0.409817	−	0.912168i	$-0.365592\pi$
$317$	14.5932	0.819635	0.409817	+	0.912168i	$0.365592\pi$
$318$	−20.2671	−1.13652
$319$	1.64151	0.0919069
$320$	3.38544	0.189252
$321$	−11.0274	−0.615491
$322$	10.2187	0.569467
$323$	1.90215	0.105838
$324$	−10.1249	−0.562493
$325$	37.5218	2.08133
$326$	2.46624	0.136592
$327$	7.18492	0.397327
$328$	−6.56031	−0.362233
$329$	−21.6115	−1.19148
$330$	−6.27841	−0.345615
$331$	−24.5155	−1.34749	−0.673747	−	0.738962i	$-0.735316\pi$
$331$	−24.5155	−1.34749	−0.673747	+	0.738962i	$0.735316\pi$
$332$	10.7270	0.588719
$333$	1.97354	0.108149
$334$	5.42987	0.297109
$335$	−26.7509	−1.46156
$336$	3.53116	0.192641
$337$	33.4205	1.82053	0.910265	−	0.414025i	$-0.135877\pi$
$337$	33.4205	1.82053	0.910265	+	0.414025i	$0.135877\pi$
$338$	20.7239	1.12723
$339$	−15.5104	−0.842409
$340$	−6.87280	−0.372730
$341$	4.73759	0.256555
$342$	−0.411595	−0.0222565
$343$	−19.7538	−1.06661
$344$	−6.57003	−0.354232
$345$	33.6948	1.81407
$346$	−11.7967	−0.634194
$347$	15.9559	0.856556	0.428278	−	0.903647i	$-0.359120\pi$
$347$	15.9559	0.856556	0.428278	+	0.903647i	$0.359120\pi$
$348$	−3.04423	−0.163188
$349$	−5.57140	−0.298230	−0.149115	−	0.988820i	$-0.547642\pi$
$349$	−5.57140	−0.298230	−0.149115	+	0.988820i	$0.547642\pi$
$350$	12.3026	0.657604
$351$	−27.5781	−1.47201
$352$	−1.00000	−0.0533002
$353$	−23.2510	−1.23753	−0.618764	−	0.785577i	$-0.712366\pi$
$353$	−23.2510	−1.23753	−0.618764	+	0.785577i	$0.712366\pi$
$354$	14.5855	0.775212
$355$	10.7571	0.570925
$356$	1.98957	0.105447
$357$	−7.16862	−0.379404
$358$	14.9410	0.789655
$359$	−15.5936	−0.822998	−0.411499	−	0.911410i	$-0.634995\pi$
$359$	−15.5936	−0.822998	−0.411499	+	0.911410i	$0.634995\pi$
$360$	1.48717	0.0783806
$361$	−18.1221	−0.953794
$362$	5.52843	0.290568
$363$	1.85453	0.0973376
$364$	11.0574	0.579564
$365$	−30.1235	−1.57674
$366$	−10.4629	−0.546902
$367$	−0.145807	−0.00761104	−0.00380552	−	0.999993i	$-0.501211\pi$
$367$	−0.145807	−0.00761104	−0.00380552	+	0.999993i	$0.501211\pi$
$368$	5.36677	0.279762
$369$	−2.88183	−0.150022
$370$	15.2095	0.790706
$371$	−20.8085	−1.08032
$372$	−8.78601	−0.455533
$373$	−12.1227	−0.627690	−0.313845	−	0.949474i	$-0.601617\pi$
$373$	−12.1227	−0.627690	−0.313845	+	0.949474i	$0.601617\pi$
$374$	2.03010	0.104974
$375$	9.17416	0.473752
$376$	−11.3501	−0.585338
$377$	−9.53262	−0.490955
$378$	−9.04231	−0.465086
$379$	9.69559	0.498029	0.249015	−	0.968500i	$-0.419893\pi$
$379$	9.69559	0.498029	0.249015	+	0.968500i	$0.419893\pi$
$380$	−3.17206	−0.162723
$381$	−5.51411	−0.282496
$382$	−2.72065	−0.139201
$383$	−13.8537	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$383$	−13.8537	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$384$	1.85453	0.0946386
$385$	−6.44613	−0.328525
$386$	16.5648	0.843124
$387$	−2.88610	−0.146709
$388$	10.4072	0.528345
$389$	−2.21890	−0.112503	−0.0562513	−	0.998417i	$-0.517915\pi$
$389$	−2.21890	−0.112503	−0.0562513	+	0.998417i	$0.517915\pi$
$390$	36.4601	1.84623
$391$	−10.8951	−0.550989
$392$	−3.37450	−0.170438
$393$	9.89640	0.499207
$394$	−1.00000	−0.0503793
$395$	53.2214	2.67786
$396$	−0.439283	−0.0220748
$397$	39.0355	1.95914	0.979568	−	0.201111i	$-0.0644553\pi$
$397$	39.0355	1.95914	0.979568	+	0.201111i	$0.0644553\pi$
$398$	−2.42358	−0.121483
$399$	−3.30859	−0.165637
$400$	6.46122	0.323061
$401$	−19.2541	−0.961504	−0.480752	−	0.876856i	$-0.659636\pi$
$401$	−19.2541	−0.961504	−0.480752	+	0.876856i	$0.659636\pi$
$402$	−14.6540	−0.730875
$403$	−27.5123	−1.37048
$404$	0.867122	0.0431409
$405$	−34.2772	−1.70325
$406$	−3.12556	−0.155119
$407$	−4.49263	−0.222691
$408$	−3.76489	−0.186390
$409$	8.86995	0.438591	0.219295	−	0.975659i	$-0.429624\pi$
$409$	8.86995	0.438591	0.219295	+	0.975659i	$0.429624\pi$
$410$	−22.2096	−1.09685
$411$	−22.0682	−1.08855
$412$	−1.60832	−0.0792362
$413$	14.9752	0.736880
$414$	2.35753	0.115866
$415$	36.3155	1.78266
$416$	5.80723	0.284723
$417$	27.1492	1.32950
$418$	0.936970	0.0458287
$419$	13.6254	0.665642	0.332821	−	0.942990i	$-0.392000\pi$
$419$	13.6254	0.665642	0.332821	+	0.942990i	$0.392000\pi$
$420$	11.9545	0.583322
$421$	19.7060	0.960411	0.480205	−	0.877156i	$-0.340562\pi$
$421$	19.7060	0.960411	0.480205	+	0.877156i	$0.340562\pi$
$422$	−16.9595	−0.825575
$423$	−4.98592	−0.242424
$424$	−10.9284	−0.530731
$425$	−13.1170	−0.636266
$426$	5.89267	0.285501
$427$	−10.7424	−0.519860
$428$	−5.94621	−0.287421
$429$	−10.7697	−0.519965
$430$	−22.2425	−1.07263
$431$	18.2318	0.878197	0.439098	−	0.898439i	$-0.355298\pi$
$431$	18.2318	0.878197	0.439098	+	0.898439i	$0.355298\pi$
$432$	−4.74893	−0.228483
$433$	6.15244	0.295667	0.147834	−	0.989012i	$-0.452770\pi$
$433$	6.15244	0.295667	0.147834	+	0.989012i	$0.452770\pi$
$434$	−9.02072	−0.433008
$435$	−10.3061	−0.494138
$436$	3.87425	0.185543
$437$	−5.02851	−0.240546
$438$	−16.5015	−0.788473
$439$	23.6701	1.12971	0.564856	−	0.825190i	$-0.308932\pi$
$439$	23.6701	1.12971	0.564856	+	0.825190i	$0.308932\pi$
$440$	−3.38544	−0.161395
$441$	−1.48236	−0.0705887
$442$	−11.7893	−0.560758
$443$	15.7739	0.749439	0.374720	−	0.927138i	$-0.377739\pi$
$443$	15.7739	0.749439	0.374720	+	0.927138i	$0.377739\pi$
$444$	8.33171	0.395406
$445$	6.73558	0.319297
$446$	−4.32158	−0.204633
$447$	−4.70837	−0.222698
$448$	1.90407	0.0899590
$449$	−9.42497	−0.444792	−0.222396	−	0.974956i	$-0.571388\pi$
$449$	−9.42497	−0.444792	−0.222396	+	0.974956i	$0.571388\pi$
$450$	2.83831	0.133799
$451$	6.56031	0.308913
$452$	−8.36351	−0.393387
$453$	1.09040	0.0512315
$454$	15.1607	0.711527
$455$	37.4341	1.75494
$456$	−1.73764	−0.0813724
$457$	−0.725906	−0.0339564	−0.0169782	−	0.999856i	$-0.505405\pi$
$457$	−0.725906	−0.0339564	−0.0169782	+	0.999856i	$0.505405\pi$
$458$	−15.8060	−0.738565
$459$	9.64081	0.449995
$460$	18.1689	0.847129
$461$	−30.8598	−1.43728	−0.718641	−	0.695381i	$-0.755236\pi$
$461$	−30.8598	−1.43728	−0.718641	+	0.695381i	$0.755236\pi$
$462$	−3.53116	−0.164285
$463$	−14.4062	−0.669514	−0.334757	−	0.942304i	$-0.608654\pi$
$463$	−14.4062	−0.669514	−0.334757	+	0.942304i	$0.608654\pi$
$464$	−1.64151	−0.0762052
$465$	−29.7445	−1.37937
$466$	−13.4444	−0.622799
$467$	7.15395	0.331045	0.165523	−	0.986206i	$-0.447069\pi$
$467$	7.15395	0.331045	0.165523	+	0.986206i	$0.447069\pi$
$468$	2.55102	0.117921
$469$	−15.0455	−0.694735
$470$	−38.4252	−1.77242
$471$	15.0649	0.694155
$472$	7.86481	0.362007
$473$	6.57003	0.302090
$474$	29.1544	1.33911
$475$	−6.05398	−0.277775
$476$	−3.86547	−0.177173
$477$	−4.80067	−0.219808
$478$	12.5023	0.571841
$479$	−12.0380	−0.550029	−0.275014	−	0.961440i	$-0.588683\pi$
$479$	−12.0380	−0.550029	−0.275014	+	0.961440i	$0.588683\pi$
$480$	6.27841	0.286569
$481$	26.0897	1.18959
$482$	18.3359	0.835175
$483$	18.9509	0.862298
$484$	1.00000	0.0454545
$485$	35.2330	1.59985
$486$	−4.53012	−0.205490
$487$	19.5761	0.887079	0.443540	−	0.896255i	$-0.353723\pi$
$487$	19.5761	0.887079	0.443540	+	0.896255i	$0.353723\pi$
$488$	−5.64178	−0.255391
$489$	4.57372	0.206831
$490$	−11.4242	−0.516092
$491$	18.3516	0.828195	0.414097	−	0.910233i	$-0.364097\pi$
$491$	18.3516	0.828195	0.414097	+	0.910233i	$0.364097\pi$
$492$	−12.1663	−0.548499
$493$	3.33244	0.150085
$494$	−5.44120	−0.244811
$495$	−1.48717	−0.0668432
$496$	−4.73759	−0.212724
$497$	6.05009	0.271384
$498$	19.8935	0.891449
$499$	−22.9714	−1.02834	−0.514171	−	0.857688i	$-0.671900\pi$
$499$	−22.9714	−1.02834	−0.514171	+	0.857688i	$0.671900\pi$
$500$	4.94689	0.221232
$501$	10.0699	0.449888
$502$	5.72822	0.255663
$503$	18.6328	0.830796	0.415398	−	0.909640i	$-0.363642\pi$
$503$	18.6328	0.830796	0.415398	+	0.909640i	$0.363642\pi$
$504$	0.836427	0.0372574
$505$	2.93559	0.130632
$506$	−5.36677	−0.238582
$507$	38.4331	1.70687
$508$	−2.97332	−0.131920
$509$	−7.50308	−0.332568	−0.166284	−	0.986078i	$-0.553177\pi$
$509$	−7.50308	−0.332568	−0.166284	+	0.986078i	$0.553177\pi$
$510$	−12.7458	−0.564394
$511$	−16.9423	−0.749485
$512$	1.00000	0.0441942
$513$	4.44960	0.196455
$514$	15.1011	0.666080
$515$	−5.44488	−0.239930
$516$	−12.1843	−0.536385
$517$	11.3501	0.499178
$518$	8.55429	0.375854
$519$	−21.8773	−0.960308
$520$	19.6600	0.862150
$521$	41.8665	1.83420	0.917101	−	0.398654i	$-0.130523\pi$
$521$	41.8665	1.83420	0.917101	+	0.398654i	$0.130523\pi$
$522$	−0.721088	−0.0315612
$523$	−7.94164	−0.347264	−0.173632	−	0.984811i	$-0.555550\pi$
$523$	−7.94164	−0.347264	−0.173632	+	0.984811i	$0.555550\pi$
$524$	5.33634	0.233119
$525$	22.8156	0.995756
$526$	−1.61389	−0.0703688
$527$	9.61780	0.418958
$528$	−1.85453	−0.0807081
$529$	5.80225	0.252272
$530$	−36.9975	−1.60707
$531$	3.45488	0.149929
$532$	−1.78406	−0.0773488
$533$	−38.0972	−1.65017
$534$	3.68972	0.159670
$535$	−20.1305	−0.870319
$536$	−7.90173	−0.341303
$537$	27.7085	1.19571
$538$	−23.8635	−1.02883
$539$	3.37450	0.145350
$540$	−16.0772	−0.691854
$541$	−18.5871	−0.799123	−0.399562	−	0.916706i	$-0.630838\pi$
$541$	−18.5871	−0.799123	−0.399562	+	0.916706i	$0.630838\pi$
$542$	2.15340	0.0924964
$543$	10.2526	0.439983
$544$	−2.03010	−0.0870400
$545$	13.1161	0.561830
$546$	20.5063	0.877587
$547$	17.3296	0.740961	0.370480	−	0.928840i	$-0.379193\pi$
$547$	17.3296	0.740961	0.370480	+	0.928840i	$0.379193\pi$
$548$	−11.8996	−0.508328
$549$	−2.47834	−0.105773
$550$	−6.46122	−0.275508
$551$	1.53805	0.0655230
$552$	9.95284	0.423621
$553$	29.9333	1.27289
$554$	16.7556	0.711876
$555$	28.2065	1.19730
$556$	14.6394	0.620847
$557$	−17.2655	−0.731564	−0.365782	−	0.930701i	$-0.619198\pi$
$557$	−17.2655	−0.731564	−0.365782	+	0.930701i	$0.619198\pi$
$558$	−2.08114	−0.0881019
$559$	−38.1537	−1.61373
$560$	6.44613	0.272399
$561$	3.76489	0.158954
$562$	8.11214	0.342190
$563$	−15.4244	−0.650060	−0.325030	−	0.945704i	$-0.605374\pi$
$563$	−15.4244	−0.650060	−0.325030	+	0.945704i	$0.605374\pi$
$564$	−21.0491	−0.886329
$565$	−28.3142	−1.19119
$566$	27.7080	1.16465
$567$	−19.2785	−0.809621
$568$	3.17744	0.133323
$569$	2.82679	0.118505	0.0592527	−	0.998243i	$-0.481128\pi$
$569$	2.82679	0.118505	0.0592527	+	0.998243i	$0.481128\pi$
$570$	−5.88268	−0.246398
$571$	−7.91287	−0.331143	−0.165572	−	0.986198i	$-0.552947\pi$
$571$	−7.91287	−0.331143	−0.165572	+	0.986198i	$0.552947\pi$
$572$	−5.80723	−0.242812
$573$	−5.04553	−0.210780
$574$	−12.4913	−0.521377
$575$	34.6759	1.44609
$576$	0.439283	0.0183035
$577$	−11.8591	−0.493702	−0.246851	−	0.969053i	$-0.579396\pi$
$577$	−11.8591	−0.493702	−0.246851	+	0.969053i	$0.579396\pi$
$578$	−12.8787	−0.535683
$579$	30.7198	1.27667
$580$	−5.55724	−0.230752
$581$	20.4249	0.847369
$582$	19.3005	0.800030
$583$	10.9284	0.452609
$584$	−8.89795	−0.368200
$585$	8.63632	0.357068
$586$	−11.3229	−0.467744
$587$	5.00103	0.206415	0.103207	−	0.994660i	$-0.467089\pi$
$587$	5.00103	0.206415	0.103207	+	0.994660i	$0.467089\pi$
$588$	−6.25812	−0.258081
$589$	4.43898	0.182905
$590$	26.6259	1.09617
$591$	−1.85453	−0.0762852
$592$	4.49263	0.184646
$593$	−2.49790	−0.102576	−0.0512882	−	0.998684i	$-0.516333\pi$
$593$	−2.49790	−0.102576	−0.0512882	+	0.998684i	$0.516333\pi$
$594$	4.74893	0.194851
$595$	−13.0863	−0.536487
$596$	−2.53885	−0.103995
$597$	−4.49460	−0.183952
$598$	31.1661	1.27448
$599$	−1.87420	−0.0765779	−0.0382889	−	0.999267i	$-0.512191\pi$
$599$	−1.87420	−0.0765779	−0.0382889	+	0.999267i	$0.512191\pi$
$600$	11.9825	0.489185
$601$	34.0371	1.38840	0.694201	−	0.719781i	$-0.255758\pi$
$601$	34.0371	1.38840	0.694201	+	0.719781i	$0.255758\pi$
$602$	−12.5098	−0.509862
$603$	−3.47110	−0.141354
$604$	0.587966	0.0239240
$605$	3.38544	0.137638
$606$	1.60810	0.0653247
$607$	−15.7998	−0.641295	−0.320648	−	0.947199i	$-0.603901\pi$
$607$	−15.7998	−0.641295	−0.320648	+	0.947199i	$0.603901\pi$
$608$	−0.936970	−0.0379992
$609$	−5.79644	−0.234884
$610$	−19.0999	−0.773334
$611$	−65.9127	−2.66654
$612$	−0.891790	−0.0360485
$613$	32.0408	1.29412	0.647058	−	0.762441i	$-0.275999\pi$
$613$	32.0408	1.29412	0.647058	+	0.762441i	$0.275999\pi$
$614$	0.353242	0.0142557
$615$	−41.1883	−1.66087
$616$	−1.90407	−0.0767173
$617$	42.0077	1.69117	0.845583	−	0.533844i	$-0.179253\pi$
$617$	42.0077	1.69117	0.845583	+	0.533844i	$0.179253\pi$
$618$	−2.98268	−0.119981
$619$	−37.9891	−1.52691	−0.763456	−	0.645860i	$-0.776499\pi$
$619$	−37.9891	−1.52691	−0.763456	+	0.645860i	$0.776499\pi$
$620$	−16.0388	−0.644136
$621$	−25.4864	−1.02273
$622$	7.02728	0.281768
$623$	3.78829	0.151775
$624$	10.7697	0.431132
$625$	−15.5587	−0.622348
$626$	−6.63242	−0.265085
$627$	1.73764	0.0693947
$628$	8.12331	0.324155
$629$	−9.12049	−0.363658
$630$	2.83168	0.112817
$631$	9.18295	0.365567	0.182784	−	0.983153i	$-0.441489\pi$
$631$	9.18295	0.365567	0.182784	+	0.983153i	$0.441489\pi$
$632$	15.7206	0.625334
$633$	−31.4519	−1.25010
$634$	14.5932	0.579569
$635$	−10.0660	−0.399457
$636$	−20.2671	−0.803642
$637$	−19.5965	−0.776442
$638$	1.64151	0.0649880
$639$	1.39580	0.0552169
$640$	3.38544	0.133821
$641$	−4.78568	−0.189023	−0.0945115	−	0.995524i	$-0.530129\pi$
$641$	−4.78568	−0.189023	−0.0945115	+	0.995524i	$0.530129\pi$
$642$	−11.0274	−0.435218
$643$	25.7751	1.01647	0.508236	−	0.861218i	$-0.330298\pi$
$643$	25.7751	1.01647	0.508236	+	0.861218i	$0.330298\pi$
$644$	10.2187	0.402674
$645$	−41.2493	−1.62419
$646$	1.90215	0.0748389
$647$	3.86471	0.151937	0.0759687	−	0.997110i	$-0.475795\pi$
$647$	3.86471	0.151937	0.0759687	+	0.997110i	$0.475795\pi$
$648$	−10.1249	−0.397743
$649$	−7.86481	−0.308721
$650$	37.5218	1.47173
$651$	−16.7292	−0.655669
$652$	2.46624	0.0965854
$653$	8.65039	0.338516	0.169258	−	0.985572i	$-0.445863\pi$
$653$	8.65039	0.338516	0.169258	+	0.985572i	$0.445863\pi$
$654$	7.18492	0.280952
$655$	18.0659	0.705892
$656$	−6.56031	−0.256137
$657$	−3.90872	−0.152494
$658$	−21.6115	−0.842502
$659$	−5.21062	−0.202977	−0.101488	−	0.994837i	$-0.532360\pi$
$659$	−5.21062	−0.202977	−0.101488	+	0.994837i	$0.532360\pi$
$660$	−6.27841	−0.244387
$661$	−26.7442	−1.04023	−0.520114	−	0.854097i	$-0.674111\pi$
$661$	−26.7442	−1.04023	−0.520114	+	0.854097i	$0.674111\pi$
$662$	−24.5155	−0.952822
$663$	−21.8636	−0.849110
$664$	10.7270	0.416287
$665$	−6.03983	−0.234215
$666$	1.97354	0.0764729
$667$	−8.80961	−0.341110
$668$	5.42987	0.210088
$669$	−8.01450	−0.309858
$670$	−26.7509	−1.03348
$671$	5.64178	0.217799
$672$	3.53116	0.136218
$673$	−26.5261	−1.02251	−0.511254	−	0.859430i	$-0.670819\pi$
$673$	−26.5261	−1.02251	−0.511254	+	0.859430i	$0.670819\pi$
$674$	33.4205	1.28731
$675$	−30.6839	−1.18102
$676$	20.7239	0.797072
$677$	−6.55978	−0.252113	−0.126056	−	0.992023i	$-0.540232\pi$
$677$	−6.55978	−0.252113	−0.126056	+	0.992023i	$0.540232\pi$
$678$	−15.5104	−0.595673
$679$	19.8161	0.760471
$680$	−6.87280	−0.263560
$681$	28.1160	1.07741
$682$	4.73759	0.181412
$683$	−9.98597	−0.382102	−0.191051	−	0.981580i	$-0.561190\pi$
$683$	−9.98597	−0.382102	−0.191051	+	0.981580i	$0.561190\pi$
$684$	−0.411595	−0.0157377
$685$	−40.2856	−1.53923
$686$	−19.7538	−0.754204
$687$	−29.3127	−1.11835
$688$	−6.57003	−0.250480
$689$	−63.4638	−2.41778
$690$	33.6948	1.28274
$691$	−16.6413	−0.633065	−0.316532	−	0.948582i	$-0.602519\pi$
$691$	−16.6413	−0.633065	−0.316532	+	0.948582i	$0.602519\pi$
$692$	−11.7967	−0.448443
$693$	−0.836427	−0.0317733
$694$	15.9559	0.605676
$695$	49.5607	1.87995
$696$	−3.04423	−0.115391
$697$	13.3181	0.504459
$698$	−5.57140	−0.210881
$699$	−24.9330	−0.943054
$700$	12.3026	0.464996
$701$	16.5705	0.625858	0.312929	−	0.949777i	$-0.398690\pi$
$701$	16.5705	0.625858	0.312929	+	0.949777i	$0.398690\pi$
$702$	−27.5781	−1.04087
$703$	−4.20946	−0.158763
$704$	−1.00000	−0.0376889
$705$	−71.2607	−2.68383
$706$	−23.2510	−0.875064
$707$	1.65106	0.0620946
$708$	14.5855	0.548158
$709$	38.1950	1.43444	0.717222	−	0.696845i	$-0.245414\pi$
$709$	38.1950	1.43444	0.717222	+	0.696845i	$0.245414\pi$
$710$	10.7571	0.403705
$711$	6.90582	0.258988
$712$	1.98957	0.0745624
$713$	−25.4256	−0.952195
$714$	−7.16862	−0.268279
$715$	−19.6600	−0.735244
$716$	14.9410	0.558370
$717$	23.1859	0.865892
$718$	−15.5936	−0.581947
$719$	−22.4625	−0.837710	−0.418855	−	0.908053i	$-0.637568\pi$
$719$	−22.4625	−0.837710	−0.418855	+	0.908053i	$0.637568\pi$
$720$	1.48717	0.0554235
$721$	−3.06236	−0.114048
$722$	−18.1221	−0.674434
$723$	34.0044	1.26464
$724$	5.52843	0.205462
$725$	−10.6062	−0.393903
$726$	1.85453	0.0688281
$727$	−37.9601	−1.40786	−0.703931	−	0.710268i	$-0.748574\pi$
$727$	−37.9601	−1.40786	−0.703931	+	0.710268i	$0.748574\pi$
$728$	11.0574	0.409814
$729$	21.9734	0.813830
$730$	−30.1235	−1.11492
$731$	13.3378	0.493318
$732$	−10.4629	−0.386718
$733$	−40.9513	−1.51257	−0.756285	−	0.654242i	$-0.772988\pi$
$733$	−40.9513	−1.51257	−0.756285	+	0.654242i	$0.772988\pi$
$734$	−0.145807	−0.00538182
$735$	−21.1865	−0.781476
$736$	5.36677	0.197822
$737$	7.90173	0.291064
$738$	−2.88183	−0.106082
$739$	20.0560	0.737772	0.368886	−	0.929475i	$-0.379739\pi$
$739$	20.0560	0.737772	0.368886	+	0.929475i	$0.379739\pi$
$740$	15.2095	0.559113
$741$	−10.0909	−0.370697
$742$	−20.8085	−0.763904
$743$	41.7885	1.53307	0.766536	−	0.642201i	$-0.221979\pi$
$743$	41.7885	1.53307	0.766536	+	0.642201i	$0.221979\pi$
$744$	−8.78601	−0.322111
$745$	−8.59513	−0.314901
$746$	−12.1227	−0.443844
$747$	4.71218	0.172410
$748$	2.03010	0.0742279
$749$	−11.3220	−0.413697
$750$	9.17416	0.334993
$751$	−35.7516	−1.30459	−0.652297	−	0.757963i	$-0.726195\pi$
$751$	−35.7516	−1.30459	−0.652297	+	0.757963i	$0.726195\pi$
$752$	−11.3501	−0.413896
$753$	10.6232	0.387129
$754$	−9.53262	−0.347158
$755$	1.99053	0.0724427
$756$	−9.04231	−0.328865
$757$	−45.2332	−1.64403	−0.822015	−	0.569466i	$-0.807150\pi$
$757$	−45.2332	−1.64403	−0.822015	+	0.569466i	$0.807150\pi$
$758$	9.69559	0.352160
$759$	−9.95284	−0.361265
$760$	−3.17206	−0.115063
$761$	42.8182	1.55216	0.776079	−	0.630636i	$-0.217206\pi$
$761$	42.8182	1.55216	0.776079	+	0.630636i	$0.217206\pi$
$762$	−5.51411	−0.199755
$763$	7.37686	0.267060
$764$	−2.72065	−0.0984296
$765$	−3.01911	−0.109156
$766$	−13.8537	−0.500556
$767$	45.6727	1.64915
$768$	1.85453	0.0669196
$769$	−9.41834	−0.339634	−0.169817	−	0.985476i	$-0.554318\pi$
$769$	−9.41834	−0.339634	−0.169817	+	0.985476i	$0.554318\pi$
$770$	−6.44613	−0.232302
$771$	28.0054	1.00859
$772$	16.5648	0.596178
$773$	6.85270	0.246475	0.123237	−	0.992377i	$-0.460672\pi$
$773$	6.85270	0.246475	0.123237	+	0.992377i	$0.460672\pi$
$774$	−2.88610	−0.103739
$775$	−30.6106	−1.09957
$776$	10.4072	0.373597
$777$	15.8642	0.569125
$778$	−2.21890	−0.0795513
$779$	6.14682	0.220233
$780$	36.4601	1.30548
$781$	−3.17744	−0.113698
$782$	−10.8951	−0.389608
$783$	7.79541	0.278585
$784$	−3.37450	−0.120518
$785$	27.5010	0.981553
$786$	9.89640	0.352993
$787$	−34.4799	−1.22908	−0.614538	−	0.788887i	$-0.710658\pi$
$787$	−34.4799	−1.22908	−0.614538	+	0.788887i	$0.710658\pi$
$788$	−1.00000	−0.0356235
$789$	−2.99300	−0.106554
$790$	53.2214	1.89353
$791$	−15.9247	−0.566219
$792$	−0.439283	−0.0156092
$793$	−32.7631	−1.16345
$794$	39.0355	1.38532
$795$	−68.6131	−2.43345
$796$	−2.42358	−0.0859015
$797$	−37.4101	−1.32513	−0.662567	−	0.749003i	$-0.730533\pi$
$797$	−37.4101	−1.32513	−0.662567	+	0.749003i	$0.730533\pi$
$798$	−3.30859	−0.117123
$799$	23.0419	0.815164
$800$	6.46122	0.228439
$801$	0.873986	0.0308808
$802$	−19.2541	−0.679886
$803$	8.89795	0.314002
$804$	−14.6540	−0.516807
$805$	34.5949	1.21931
$806$	−27.5123	−0.969078
$807$	−44.2556	−1.55787
$808$	0.867122	0.0305052
$809$	−17.8706	−0.628298	−0.314149	−	0.949374i	$-0.601719\pi$
$809$	−17.8706	−0.628298	−0.314149	+	0.949374i	$0.601719\pi$
$810$	−34.2772	−1.20438
$811$	42.8017	1.50297	0.751485	−	0.659750i	$-0.229338\pi$
$811$	42.8017	1.50297	0.751485	+	0.659750i	$0.229338\pi$
$812$	−3.12556	−0.109686
$813$	3.99354	0.140060
$814$	−4.49263	−0.157466
$815$	8.34931	0.292464
$816$	−3.76489	−0.131797
$817$	6.15592	0.215369
$818$	8.86995	0.310130
$819$	4.85732	0.169729
$820$	−22.2096	−0.775592
$821$	−52.4414	−1.83022	−0.915108	−	0.403208i	$-0.867895\pi$
$821$	−52.4414	−1.83022	−0.915108	+	0.403208i	$0.867895\pi$
$822$	−22.0682	−0.769719
$823$	30.9154	1.07764	0.538822	−	0.842420i	$-0.318870\pi$
$823$	30.9154	1.07764	0.538822	+	0.842420i	$0.318870\pi$
$824$	−1.60832	−0.0560285
$825$	−11.9825	−0.417178
$826$	14.9752	0.521053
$827$	4.56398	0.158705	0.0793526	−	0.996847i	$-0.474715\pi$
$827$	4.56398	0.158705	0.0793526	+	0.996847i	$0.474715\pi$
$828$	2.35753	0.0819299
$829$	22.6168	0.785512	0.392756	−	0.919643i	$-0.371522\pi$
$829$	22.6168	0.785512	0.392756	+	0.919643i	$0.371522\pi$
$830$	36.3155	1.26053
$831$	31.0737	1.07794
$832$	5.80723	0.201329
$833$	6.85059	0.237359
$834$	27.1492	0.940098
$835$	18.3825	0.636154
$836$	0.936970	0.0324058
$837$	22.4985	0.777661
$838$	13.6254	0.470680
$839$	5.30868	0.183276	0.0916380	−	0.995792i	$-0.470790\pi$
$839$	5.30868	0.183276	0.0916380	+	0.995792i	$0.470790\pi$
$840$	11.9545	0.412471
$841$	−26.3054	−0.907084
$842$	19.7060	0.679113
$843$	15.0442	0.518150
$844$	−16.9595	−0.583770
$845$	70.1595	2.41356
$846$	−4.98592	−0.171419
$847$	1.90407	0.0654247
$848$	−10.9284	−0.375283
$849$	51.3853	1.76354
$850$	−13.1170	−0.449908
$851$	24.1109	0.826511
$852$	5.89267	0.201879
$853$	44.5735	1.52617	0.763085	−	0.646299i	$-0.223684\pi$
$853$	44.5735	1.52617	0.763085	+	0.646299i	$0.223684\pi$
$854$	−10.7424	−0.367596
$855$	−1.39343	−0.0476544
$856$	−5.94621	−0.203237
$857$	12.9352	0.441858	0.220929	−	0.975290i	$-0.429091\pi$
$857$	12.9352	0.441858	0.220929	+	0.975290i	$0.429091\pi$
$858$	−10.7697	−0.367671
$859$	−38.8781	−1.32650	−0.663251	−	0.748397i	$-0.730824\pi$
$859$	−38.8781	−1.32650	−0.663251	+	0.748397i	$0.730824\pi$
$860$	−22.2425	−0.758462
$861$	−23.1655	−0.789479
$862$	18.2318	0.620979
$863$	28.2601	0.961987	0.480993	−	0.876724i	$-0.340276\pi$
$863$	28.2601	0.961987	0.480993	+	0.876724i	$0.340276\pi$
$864$	−4.74893	−0.161562
$865$	−39.9370	−1.35790
$866$	6.15244	0.209068
$867$	−23.8839	−0.811140
$868$	−9.02072	−0.306183
$869$	−15.7206	−0.533286
$870$	−10.3061	−0.349409
$871$	−45.8871	−1.55483
$872$	3.87425	0.131199
$873$	4.57171	0.154729
$874$	−5.02851	−0.170092
$875$	9.41925	0.318429
$876$	−16.5015	−0.557534
$877$	−50.1619	−1.69385	−0.846924	−	0.531714i	$-0.821548\pi$
$877$	−50.1619	−1.69385	−0.846924	+	0.531714i	$0.821548\pi$
$878$	23.6701	0.798827
$879$	−20.9986	−0.708266
$880$	−3.38544	−0.114123
$881$	−48.0165	−1.61772	−0.808858	−	0.588004i	$-0.799914\pi$
$881$	−48.0165	−1.61772	−0.808858	+	0.588004i	$0.799914\pi$
$882$	−1.48236	−0.0499138
$883$	−5.68074	−0.191172	−0.0955860	−	0.995421i	$-0.530472\pi$
$883$	−5.68074	−0.191172	−0.0955860	+	0.995421i	$0.530472\pi$
$884$	−11.7893	−0.396516
$885$	49.3785	1.65984
$886$	15.7739	0.529934
$887$	−30.2342	−1.01517	−0.507583	−	0.861603i	$-0.669461\pi$
$887$	−30.2342	−1.01517	−0.507583	+	0.861603i	$0.669461\pi$
$888$	8.33171	0.279594
$889$	−5.66142	−0.189878
$890$	6.73558	0.225777
$891$	10.1249	0.339196
$892$	−4.32158	−0.144697
$893$	10.6347	0.355877
$894$	−4.70837	−0.157472
$895$	50.5818	1.69076
$896$	1.90407	0.0636106
$897$	57.7984	1.92983
$898$	−9.42497	−0.314515
$899$	7.77681	0.259371
$900$	2.83831	0.0946102
$901$	22.1858	0.739117
$902$	6.56031	0.218434
$903$	−23.1998	−0.772042
$904$	−8.36351	−0.278166
$905$	18.7162	0.622147
$906$	1.09040	0.0362262
$907$	31.7273	1.05349	0.526743	−	0.850025i	$-0.323413\pi$
$907$	31.7273	1.05349	0.526743	+	0.850025i	$0.323413\pi$
$908$	15.1607	0.503126
$909$	0.380912	0.0126341
$910$	37.4341	1.24093
$911$	31.3807	1.03969	0.519844	−	0.854261i	$-0.325990\pi$
$911$	31.3807	1.03969	0.519844	+	0.854261i	$0.325990\pi$
$912$	−1.73764	−0.0575390
$913$	−10.7270	−0.355011
$914$	−0.725906	−0.0240108
$915$	−35.4214	−1.17100
$916$	−15.8060	−0.522244
$917$	10.1608	0.335538
$918$	9.64081	0.318194
$919$	34.8664	1.15014	0.575068	−	0.818105i	$-0.304975\pi$
$919$	34.8664	1.15014	0.575068	+	0.818105i	$0.304975\pi$
$920$	18.1689	0.599011
$921$	0.655098	0.0215862
$922$	−30.8598	−1.01631
$923$	18.4521	0.607360
$924$	−3.53116	−0.116167
$925$	29.0279	0.954430
$926$	−14.4062	−0.473418
$927$	−0.706508	−0.0232048
$928$	−1.64151	−0.0538852
$929$	3.67064	0.120430	0.0602150	−	0.998185i	$-0.480821\pi$
$929$	3.67064	0.120430	0.0602150	+	0.998185i	$0.480821\pi$
$930$	−29.7445	−0.975361
$931$	3.16181	0.103624
$932$	−13.4444	−0.440386
$933$	13.0323	0.426658
$934$	7.15395	0.234084
$935$	6.87280	0.224765
$936$	2.55102	0.0833826
$937$	1.61357	0.0527130	0.0263565	−	0.999653i	$-0.491609\pi$
$937$	1.61357	0.0527130	0.0263565	+	0.999653i	$0.491609\pi$
$938$	−15.0455	−0.491252
$939$	−12.3000	−0.401396
$940$	−38.4252	−1.25329
$941$	−41.9727	−1.36827	−0.684136	−	0.729355i	$-0.739820\pi$
$941$	−41.9727	−1.36827	−0.684136	+	0.729355i	$0.739820\pi$
$942$	15.0649	0.490842
$943$	−35.2077	−1.14652
$944$	7.86481	0.255978
$945$	−30.6122	−0.995815
$946$	6.57003	0.213610
$947$	0.852282	0.0276954	0.0138477	−	0.999904i	$-0.495592\pi$
$947$	0.852282	0.0276954	0.0138477	+	0.999904i	$0.495592\pi$
$948$	29.1544	0.946892
$949$	−51.6724	−1.67736
$950$	−6.05398	−0.196417
$951$	27.0635	0.877594
$952$	−3.86547	−0.125280
$953$	16.8435	0.545616	0.272808	−	0.962069i	$-0.412048\pi$
$953$	16.8435	0.545616	0.272808	+	0.962069i	$0.412048\pi$
$954$	−4.80067	−0.155427
$955$	−9.21060	−0.298048
$956$	12.5023	0.404353
$957$	3.04423	0.0984060
$958$	−12.0380	−0.388929
$959$	−22.6578	−0.731658
$960$	6.27841	0.202635
$961$	−8.55523	−0.275975
$962$	26.0897	0.841166
$963$	−2.61207	−0.0841727
$964$	18.3359	0.590558
$965$	56.0790	1.80525
$966$	18.9509	0.609737
$967$	28.0573	0.902261	0.451130	−	0.892458i	$-0.351021\pi$
$967$	28.0573	0.902261	0.451130	+	0.892458i	$0.351021\pi$
$968$	1.00000	0.0321412
$969$	3.52759	0.113322
$970$	35.2330	1.13126
$971$	30.7110	0.985564	0.492782	−	0.870153i	$-0.335980\pi$
$971$	30.7110	0.985564	0.492782	+	0.870153i	$0.335980\pi$
$972$	−4.53012	−0.145304
$973$	27.8744	0.893613
$974$	19.5761	0.627260
$975$	69.5853	2.22851
$976$	−5.64178	−0.180589
$977$	−51.9599	−1.66235	−0.831173	−	0.556015i	$-0.812330\pi$
$977$	−51.9599	−1.66235	−0.831173	+	0.556015i	$0.812330\pi$
$978$	4.57372	0.146251
$979$	−1.98957	−0.0635870
$980$	−11.4242	−0.364932
$981$	1.70189	0.0543373
$982$	18.3516	0.585622
$983$	31.3527	0.999997	0.499999	−	0.866026i	$-0.333334\pi$
$983$	31.3527	0.999997	0.499999	+	0.866026i	$0.333334\pi$
$984$	−12.1663	−0.387847
$985$	−3.38544	−0.107869
$986$	3.33244	0.106126
$987$	−40.0791	−1.27573
$988$	−5.44120	−0.173108
$989$	−35.2599	−1.12120
$990$	−1.48717	−0.0472653
$991$	−57.5727	−1.82886	−0.914429	−	0.404746i	$-0.867360\pi$
$991$	−57.5727	−1.82886	−0.914429	+	0.404746i	$0.867360\pi$
$992$	−4.73759	−0.150419
$993$	−45.4647	−1.44278
$994$	6.05009	0.191897
$995$	−8.20489	−0.260113
$996$	19.8935	0.630349
$997$	−37.4085	−1.18474	−0.592370	−	0.805666i	$-0.701808\pi$
$997$	−37.4085	−1.18474	−0.592370	+	0.805666i	$0.701808\pi$
$998$	−22.9714	−0.727147
$999$	−21.3352	−0.675014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.f.1.18	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.f.1.18	✓	24	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.85453 q^{3} +1.00000 q^{4} +3.38544 q^{5} +1.85453 q^{6} +1.90407 q^{7} +1.00000 q^{8} +0.439283 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.85453 q^{3} +1.00000 q^{4} +3.38544 q^{5} +1.85453 q^{6} +1.90407 q^{7} +1.00000 q^{8} +0.439283 q^{9} +3.38544 q^{10} -1.00000 q^{11} +1.85453 q^{12} +5.80723 q^{13} +1.90407 q^{14} +6.27841 q^{15} +1.00000 q^{16} -2.03010 q^{17} +0.439283 q^{18} -0.936970 q^{19} +3.38544 q^{20} +3.53116 q^{21} -1.00000 q^{22} +5.36677 q^{23} +1.85453 q^{24} +6.46122 q^{25} +5.80723 q^{26} -4.74893 q^{27} +1.90407 q^{28} -1.64151 q^{29} +6.27841 q^{30} -4.73759 q^{31} +1.00000 q^{32} -1.85453 q^{33} -2.03010 q^{34} +6.44613 q^{35} +0.439283 q^{36} +4.49263 q^{37} -0.936970 q^{38} +10.7697 q^{39} +3.38544 q^{40} -6.56031 q^{41} +3.53116 q^{42} -6.57003 q^{43} -1.00000 q^{44} +1.48717 q^{45} +5.36677 q^{46} -11.3501 q^{47} +1.85453 q^{48} -3.37450 q^{49} +6.46122 q^{50} -3.76489 q^{51} +5.80723 q^{52} -10.9284 q^{53} -4.74893 q^{54} -3.38544 q^{55} +1.90407 q^{56} -1.73764 q^{57} -1.64151 q^{58} +7.86481 q^{59} +6.27841 q^{60} -5.64178 q^{61} -4.73759 q^{62} +0.836427 q^{63} +1.00000 q^{64} +19.6600 q^{65} -1.85453 q^{66} -7.90173 q^{67} -2.03010 q^{68} +9.95284 q^{69} +6.44613 q^{70} +3.17744 q^{71} +0.439283 q^{72} -8.89795 q^{73} +4.49263 q^{74} +11.9825 q^{75} -0.936970 q^{76} -1.90407 q^{77} +10.7697 q^{78} +15.7206 q^{79} +3.38544 q^{80} -10.1249 q^{81} -6.56031 q^{82} +10.7270 q^{83} +3.53116 q^{84} -6.87280 q^{85} -6.57003 q^{86} -3.04423 q^{87} -1.00000 q^{88} +1.98957 q^{89} +1.48717 q^{90} +11.0574 q^{91} +5.36677 q^{92} -8.78601 q^{93} -11.3501 q^{94} -3.17206 q^{95} +1.85453 q^{96} +10.4072 q^{97} -3.37450 q^{98} -0.439283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9}+O(q^{10}) \) 24 * q + 24 * q^2 + 8 * q^3 + 24 * q^4 + 8 * q^6 + 11 * q^7 + 24 * q^8 + 28 * q^9	\( 24 q + 24 q^{2} + 8 q^{3} + 24 q^{4} + 8 q^{6} + 11 q^{7} + 24 q^{8} + 28 q^{9} - 24 q^{11} + 8 q^{12} + 9 q^{13} + 11 q^{14} + 6 q^{15} + 24 q^{16} - q^{17} + 28 q^{18} + 35 q^{19} + 21 q^{21} - 24 q^{22} + 13 q^{23} + 8 q^{24} + 38 q^{25} + 9 q^{26} + 23 q^{27} + 11 q^{28} + 15 q^{29} + 6 q^{30} + 31 q^{31} + 24 q^{32} - 8 q^{33} - q^{34} + 28 q^{35} + 28 q^{36} + 13 q^{37} + 35 q^{38} + 31 q^{39} + 16 q^{41} + 21 q^{42} + 35 q^{43} - 24 q^{44} + 13 q^{45} + 13 q^{46} + 46 q^{47} + 8 q^{48} + 55 q^{49} + 38 q^{50} + 18 q^{51} + 9 q^{52} + 6 q^{53} + 23 q^{54} + 11 q^{56} + 18 q^{57} + 15 q^{58} + 13 q^{59} + 6 q^{60} + 60 q^{61} + 31 q^{62} + 52 q^{63} + 24 q^{64} - 9 q^{65} - 8 q^{66} + 28 q^{67} - q^{68} + 21 q^{69} + 28 q^{70} + 10 q^{71} + 28 q^{72} + 3 q^{73} + 13 q^{74} + 48 q^{75} + 35 q^{76} - 11 q^{77} + 31 q^{78} + 47 q^{79} + 16 q^{81} + 16 q^{82} + 66 q^{83} + 21 q^{84} + 37 q^{85} + 35 q^{86} + 34 q^{87} - 24 q^{88} - 5 q^{89} + 13 q^{90} + q^{91} + 13 q^{92} - 16 q^{93} + 46 q^{94} + 9 q^{95} + 8 q^{96} + 24 q^{97} + 55 q^{98} - 28 q^{99}+O(q^{100}) \) 24 * q + 24 * q^2 + 8 * q^3 + 24 * q^4 + 8 * q^6 + 11 * q^7 + 24 * q^8 + 28 * q^9 - 24 * q^11 + 8 * q^12 + 9 * q^13 + 11 * q^14 + 6 * q^15 + 24 * q^16 - q^17 + 28 * q^18 + 35 * q^19 + 21 * q^21 - 24 * q^22 + 13 * q^23 + 8 * q^24 + 38 * q^25 + 9 * q^26 + 23 * q^27 + 11 * q^28 + 15 * q^29 + 6 * q^30 + 31 * q^31 + 24 * q^32 - 8 * q^33 - q^34 + 28 * q^35 + 28 * q^36 + 13 * q^37 + 35 * q^38 + 31 * q^39 + 16 * q^41 + 21 * q^42 + 35 * q^43 - 24 * q^44 + 13 * q^45 + 13 * q^46 + 46 * q^47 + 8 * q^48 + 55 * q^49 + 38 * q^50 + 18 * q^51 + 9 * q^52 + 6 * q^53 + 23 * q^54 + 11 * q^56 + 18 * q^57 + 15 * q^58 + 13 * q^59 + 6 * q^60 + 60 * q^61 + 31 * q^62 + 52 * q^63 + 24 * q^64 - 9 * q^65 - 8 * q^66 + 28 * q^67 - q^68 + 21 * q^69 + 28 * q^70 + 10 * q^71 + 28 * q^72 + 3 * q^73 + 13 * q^74 + 48 * q^75 + 35 * q^76 - 11 * q^77 + 31 * q^78 + 47 * q^79 + 16 * q^81 + 16 * q^82 + 66 * q^83 + 21 * q^84 + 37 * q^85 + 35 * q^86 + 34 * q^87 - 24 * q^88 - 5 * q^89 + 13 * q^90 + q^91 + 13 * q^92 - 16 * q^93 + 46 * q^94 + 9 * q^95 + 8 * q^96 + 24 * q^97 + 55 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more