LMFDB - Embedded newform 6015.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.0300168158$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12824 q^{2} +1.00000 q^{3} -0.727075 q^{4} +1.00000 q^{5} -1.12824 q^{6} -4.89351 q^{7} +3.07679 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.12824 q^{2} +1.00000 q^{3} -0.727075 q^{4} +1.00000 q^{5} -1.12824 q^{6} -4.89351 q^{7} +3.07679 q^{8} +1.00000 q^{9} -1.12824 q^{10} +4.95894 q^{11} -0.727075 q^{12} +2.22815 q^{13} +5.52105 q^{14} +1.00000 q^{15} -2.01721 q^{16} -2.48086 q^{17} -1.12824 q^{18} -7.75202 q^{19} -0.727075 q^{20} -4.89351 q^{21} -5.59488 q^{22} -0.154661 q^{23} +3.07679 q^{24} +1.00000 q^{25} -2.51389 q^{26} +1.00000 q^{27} +3.55795 q^{28} -6.27073 q^{29} -1.12824 q^{30} +8.31073 q^{31} -3.87769 q^{32} +4.95894 q^{33} +2.79901 q^{34} -4.89351 q^{35} -0.727075 q^{36} +10.5317 q^{37} +8.74614 q^{38} +2.22815 q^{39} +3.07679 q^{40} -2.72465 q^{41} +5.52105 q^{42} -1.74241 q^{43} -3.60553 q^{44} +1.00000 q^{45} +0.174494 q^{46} -5.43472 q^{47} -2.01721 q^{48} +16.9464 q^{49} -1.12824 q^{50} -2.48086 q^{51} -1.62003 q^{52} +2.70179 q^{53} -1.12824 q^{54} +4.95894 q^{55} -15.0563 q^{56} -7.75202 q^{57} +7.07489 q^{58} -3.59171 q^{59} -0.727075 q^{60} -9.11481 q^{61} -9.37650 q^{62} -4.89351 q^{63} +8.40939 q^{64} +2.22815 q^{65} -5.59488 q^{66} -10.3277 q^{67} +1.80377 q^{68} -0.154661 q^{69} +5.52105 q^{70} +3.87685 q^{71} +3.07679 q^{72} -3.06165 q^{73} -11.8823 q^{74} +1.00000 q^{75} +5.63630 q^{76} -24.2666 q^{77} -2.51389 q^{78} -1.84135 q^{79} -2.01721 q^{80} +1.00000 q^{81} +3.07406 q^{82} +8.73045 q^{83} +3.55795 q^{84} -2.48086 q^{85} +1.96585 q^{86} -6.27073 q^{87} +15.2577 q^{88} -7.86411 q^{89} -1.12824 q^{90} -10.9035 q^{91} +0.112450 q^{92} +8.31073 q^{93} +6.13167 q^{94} -7.75202 q^{95} -3.87769 q^{96} +8.73070 q^{97} -19.1196 q^{98} +4.95894 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9} - 5 q^{10} - 13 q^{11} + 9 q^{12} - 18 q^{13} - 6 q^{14} + 23 q^{15} - 11 q^{16} - 34 q^{17} - 5 q^{18} - 35 q^{19} + 9 q^{20} - 16 q^{21} - 11 q^{22} - 14 q^{23} - 12 q^{24} + 23 q^{25} - 6 q^{26} + 23 q^{27} - 26 q^{28} - 43 q^{29} - 5 q^{30} - 21 q^{31} - 14 q^{32} - 13 q^{33} - 12 q^{34} - 16 q^{35} + 9 q^{36} - 18 q^{37} + 6 q^{38} - 18 q^{39} - 12 q^{40} - 45 q^{41} - 6 q^{42} - 43 q^{43} - 11 q^{44} + 23 q^{45} - 29 q^{46} - 14 q^{47} - 11 q^{48} - 25 q^{49} - 5 q^{50} - 34 q^{51} - 20 q^{52} - 3 q^{53} - 5 q^{54} - 13 q^{55} + 3 q^{56} - 35 q^{57} + 10 q^{58} - 9 q^{59} + 9 q^{60} - 67 q^{61} - 7 q^{62} - 16 q^{63} - 8 q^{64} - 18 q^{65} - 11 q^{66} - 32 q^{67} - 24 q^{68} - 14 q^{69} - 6 q^{70} - 8 q^{71} - 12 q^{72} - 39 q^{73} - 16 q^{74} + 23 q^{75} - 48 q^{76} - 26 q^{77} - 6 q^{78} - 59 q^{79} - 11 q^{80} + 23 q^{81} - q^{82} - 23 q^{83} - 26 q^{84} - 34 q^{85} - 7 q^{86} - 43 q^{87} + 17 q^{88} - 51 q^{89} - 5 q^{90} - 37 q^{91} + 11 q^{92} - 21 q^{93} + 8 q^{94} - 35 q^{95} - 14 q^{96} - 29 q^{97} + 32 q^{98} - 13 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9 - 5 * q^10 - 13 * q^11 + 9 * q^12 - 18 * q^13 - 6 * q^14 + 23 * q^15 - 11 * q^16 - 34 * q^17 - 5 * q^18 - 35 * q^19 + 9 * q^20 - 16 * q^21 - 11 * q^22 - 14 * q^23 - 12 * q^24 + 23 * q^25 - 6 * q^26 + 23 * q^27 - 26 * q^28 - 43 * q^29 - 5 * q^30 - 21 * q^31 - 14 * q^32 - 13 * q^33 - 12 * q^34 - 16 * q^35 + 9 * q^36 - 18 * q^37 + 6 * q^38 - 18 * q^39 - 12 * q^40 - 45 * q^41 - 6 * q^42 - 43 * q^43 - 11 * q^44 + 23 * q^45 - 29 * q^46 - 14 * q^47 - 11 * q^48 - 25 * q^49 - 5 * q^50 - 34 * q^51 - 20 * q^52 - 3 * q^53 - 5 * q^54 - 13 * q^55 + 3 * q^56 - 35 * q^57 + 10 * q^58 - 9 * q^59 + 9 * q^60 - 67 * q^61 - 7 * q^62 - 16 * q^63 - 8 * q^64 - 18 * q^65 - 11 * q^66 - 32 * q^67 - 24 * q^68 - 14 * q^69 - 6 * q^70 - 8 * q^71 - 12 * q^72 - 39 * q^73 - 16 * q^74 + 23 * q^75 - 48 * q^76 - 26 * q^77 - 6 * q^78 - 59 * q^79 - 11 * q^80 + 23 * q^81 - q^82 - 23 * q^83 - 26 * q^84 - 34 * q^85 - 7 * q^86 - 43 * q^87 + 17 * q^88 - 51 * q^89 - 5 * q^90 - 37 * q^91 + 11 * q^92 - 21 * q^93 + 8 * q^94 - 35 * q^95 - 14 * q^96 - 29 * q^97 + 32 * q^98 - 13 * 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 $

$2$	−1.12824	−0.797786	−0.398893	−	0.916998i	$-0.630605\pi$
$2$	−1.12824	−0.797786	−0.398893	+	0.916998i	$0.630605\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.727075	−0.363538
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.12824	−0.460602
$7$	−4.89351	−1.84957	−0.924786	−	0.380488i	$-0.875756\pi$
$7$	−4.89351	−1.84957	−0.924786	+	0.380488i	$0.875756\pi$
$8$	3.07679	1.08781
$9$	1.00000	0.333333
$10$	−1.12824	−0.356781
$11$	4.95894	1.49518	0.747589	−	0.664162i	$-0.231211\pi$
$11$	4.95894	1.49518	0.747589	+	0.664162i	$0.231211\pi$
$12$	−0.727075	−0.209889
$13$	2.22815	0.617978	0.308989	−	0.951066i	$-0.400009\pi$
$13$	2.22815	0.617978	0.308989	+	0.951066i	$0.400009\pi$
$14$	5.52105	1.47556
$15$	1.00000	0.258199
$16$	−2.01721	−0.504303
$17$	−2.48086	−0.601697	−0.300849	−	0.953672i	$-0.597270\pi$
$17$	−2.48086	−0.601697	−0.300849	+	0.953672i	$0.597270\pi$
$18$	−1.12824	−0.265929
$19$	−7.75202	−1.77844	−0.889218	−	0.457484i	$-0.848751\pi$
$19$	−7.75202	−1.77844	−0.889218	+	0.457484i	$0.848751\pi$
$20$	−0.727075	−0.162579
$21$	−4.89351	−1.06785
$22$	−5.59488	−1.19283
$23$	−0.154661	−0.0322490	−0.0161245	−	0.999870i	$-0.505133\pi$
$23$	−0.154661	−0.0322490	−0.0161245	+	0.999870i	$0.505133\pi$
$24$	3.07679	0.628048
$25$	1.00000	0.200000
$26$	−2.51389	−0.493014
$27$	1.00000	0.192450
$28$	3.55795	0.672389
$29$	−6.27073	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$29$	−6.27073	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$30$	−1.12824	−0.205987
$31$	8.31073	1.49265	0.746326	−	0.665581i	$-0.231816\pi$
$31$	8.31073	1.49265	0.746326	+	0.665581i	$0.231816\pi$
$32$	−3.87769	−0.685486
$33$	4.95894	0.863241
$34$	2.79901	0.480026
$35$	−4.89351	−0.827154
$36$	−0.727075	−0.121179
$37$	10.5317	1.73140	0.865700	−	0.500563i	$-0.166874\pi$
$37$	10.5317	1.73140	0.865700	+	0.500563i	$0.166874\pi$
$38$	8.74614	1.41881
$39$	2.22815	0.356790
$40$	3.07679	0.486484
$41$	−2.72465	−0.425519	−0.212760	−	0.977105i	$-0.568245\pi$
$41$	−2.72465	−0.425519	−0.212760	+	0.977105i	$0.568245\pi$
$42$	5.52105	0.851916
$43$	−1.74241	−0.265715	−0.132857	−	0.991135i	$-0.542415\pi$
$43$	−1.74241	−0.265715	−0.132857	+	0.991135i	$0.542415\pi$
$44$	−3.60553	−0.543554
$45$	1.00000	0.149071
$46$	0.174494	0.0257278
$47$	−5.43472	−0.792736	−0.396368	−	0.918092i	$-0.629730\pi$
$47$	−5.43472	−0.792736	−0.396368	+	0.918092i	$0.629730\pi$
$48$	−2.01721	−0.291159
$49$	16.9464	2.42092
$50$	−1.12824	−0.159557
$51$	−2.48086	−0.347390
$52$	−1.62003	−0.224658
$53$	2.70179	0.371119	0.185559	−	0.982633i	$-0.440590\pi$
$53$	2.70179	0.371119	0.185559	+	0.982633i	$0.440590\pi$
$54$	−1.12824	−0.153534
$55$	4.95894	0.668664
$56$	−15.0563	−2.01198
$57$	−7.75202	−1.02678
$58$	7.07489	0.928979
$59$	−3.59171	−0.467601	−0.233800	−	0.972285i	$-0.575116\pi$
$59$	−3.59171	−0.467601	−0.233800	+	0.972285i	$0.575116\pi$
$60$	−0.727075	−0.0938650
$61$	−9.11481	−1.16703	−0.583516	−	0.812102i	$-0.698323\pi$
$61$	−9.11481	−1.16703	−0.583516	+	0.812102i	$0.698323\pi$
$62$	−9.37650	−1.19082
$63$	−4.89351	−0.616524
$64$	8.40939	1.05117
$65$	2.22815	0.276368
$66$	−5.59488	−0.688682
$67$	−10.3277	−1.26173	−0.630867	−	0.775891i	$-0.717301\pi$
$67$	−10.3277	−1.26173	−0.630867	+	0.775891i	$0.717301\pi$
$68$	1.80377	0.218740
$69$	−0.154661	−0.0186190
$70$	5.52105	0.659892
$71$	3.87685	0.460097	0.230048	−	0.973179i	$-0.426112\pi$
$71$	3.87685	0.460097	0.230048	+	0.973179i	$0.426112\pi$
$72$	3.07679	0.362604
$73$	−3.06165	−0.358339	−0.179169	−	0.983818i	$-0.557341\pi$
$73$	−3.06165	−0.358339	−0.179169	+	0.983818i	$0.557341\pi$
$74$	−11.8823	−1.38129
$75$	1.00000	0.115470
$76$	5.63630	0.646528
$77$	−24.2666	−2.76544
$78$	−2.51389	−0.284642
$79$	−1.84135	−0.207168	−0.103584	−	0.994621i	$-0.533031\pi$
$79$	−1.84135	−0.207168	−0.103584	+	0.994621i	$0.533031\pi$
$80$	−2.01721	−0.225531
$81$	1.00000	0.111111
$82$	3.07406	0.339473
$83$	8.73045	0.958291	0.479146	−	0.877735i	$-0.340947\pi$
$83$	8.73045	0.958291	0.479146	+	0.877735i	$0.340947\pi$
$84$	3.55795	0.388204
$85$	−2.48086	−0.269087
$86$	1.96585	0.211983
$87$	−6.27073	−0.672293
$88$	15.2577	1.62647
$89$	−7.86411	−0.833594	−0.416797	−	0.908999i	$-0.636847\pi$
$89$	−7.86411	−0.833594	−0.416797	+	0.908999i	$0.636847\pi$
$90$	−1.12824	−0.118927
$91$	−10.9035	−1.14299
$92$	0.112450	0.0117237
$93$	8.31073	0.861783
$94$	6.13167	0.632434
$95$	−7.75202	−0.795341
$96$	−3.87769	−0.395765
$97$	8.73070	0.886468	0.443234	−	0.896406i	$-0.353831\pi$
$97$	8.73070	0.886468	0.443234	+	0.896406i	$0.353831\pi$
$98$	−19.1196	−1.93137
$99$	4.95894	0.498393
$100$	−0.727075	−0.0727075
$101$	−10.2158	−1.01651	−0.508254	−	0.861207i	$-0.669709\pi$
$101$	−10.2158	−1.01651	−0.508254	+	0.861207i	$0.669709\pi$
$102$	2.79901	0.277143
$103$	−6.54817	−0.645210	−0.322605	−	0.946534i	$-0.604559\pi$
$103$	−6.54817	−0.645210	−0.322605	+	0.946534i	$0.604559\pi$
$104$	6.85556	0.672243
$105$	−4.89351	−0.477557
$106$	−3.04826	−0.296073
$107$	17.2433	1.66697	0.833486	−	0.552541i	$-0.186342\pi$
$107$	17.2433	1.66697	0.833486	+	0.552541i	$0.186342\pi$
$108$	−0.727075	−0.0699629
$109$	−8.42962	−0.807411	−0.403705	−	0.914889i	$-0.632278\pi$
$109$	−8.42962	−0.807411	−0.403705	+	0.914889i	$0.632278\pi$
$110$	−5.59488	−0.533451
$111$	10.5317	0.999624
$112$	9.87123	0.932744
$113$	−6.97985	−0.656609	−0.328304	−	0.944572i	$-0.606477\pi$
$113$	−6.97985	−0.656609	−0.328304	+	0.944572i	$0.606477\pi$
$114$	8.74614	0.819151
$115$	−0.154661	−0.0144222
$116$	4.55930	0.423320
$117$	2.22815	0.205993
$118$	4.05231	0.373045
$119$	12.1401	1.11288
$120$	3.07679	0.280872
$121$	13.5911	1.23556
$122$	10.2837	0.931042
$123$	−2.72465	−0.245674
$124$	−6.04253	−0.542635
$125$	1.00000	0.0894427
$126$	5.52105	0.491854
$127$	17.1789	1.52438	0.762189	−	0.647354i	$-0.224125\pi$
$127$	17.1789	1.52438	0.762189	+	0.647354i	$0.224125\pi$
$128$	−1.73242	−0.153126
$129$	−1.74241	−0.153410
$130$	−2.51389	−0.220482
$131$	−6.57433	−0.574402	−0.287201	−	0.957870i	$-0.592725\pi$
$131$	−6.57433	−0.574402	−0.287201	+	0.957870i	$0.592725\pi$
$132$	−3.60553	−0.313821
$133$	37.9346	3.28934
$134$	11.6522	1.00659
$135$	1.00000	0.0860663
$136$	−7.63310	−0.654533
$137$	−9.17590	−0.783950	−0.391975	−	0.919976i	$-0.628208\pi$
$137$	−9.17590	−0.783950	−0.391975	+	0.919976i	$0.628208\pi$
$138$	0.174494	0.0148539
$139$	4.45641	0.377987	0.188994	−	0.981978i	$-0.439477\pi$
$139$	4.45641	0.377987	0.188994	+	0.981978i	$0.439477\pi$
$140$	3.55795	0.300702
$141$	−5.43472	−0.457686
$142$	−4.37401	−0.367059
$143$	11.0493	0.923986
$144$	−2.01721	−0.168101
$145$	−6.27073	−0.520756
$146$	3.45427	0.285878
$147$	16.9464	1.39772
$148$	−7.65734	−0.629429
$149$	−18.8360	−1.54310	−0.771551	−	0.636167i	$-0.780519\pi$
$149$	−18.8360	−1.54310	−0.771551	+	0.636167i	$0.780519\pi$
$150$	−1.12824	−0.0921204
$151$	−11.8802	−0.966795	−0.483397	−	0.875401i	$-0.660597\pi$
$151$	−11.8802	−0.966795	−0.483397	+	0.875401i	$0.660597\pi$
$152$	−23.8514	−1.93460
$153$	−2.48086	−0.200566
$154$	27.3786	2.20623
$155$	8.31073	0.667534
$156$	−1.62003	−0.129706
$157$	−16.3953	−1.30849	−0.654243	−	0.756284i	$-0.727013\pi$
$157$	−16.3953	−1.30849	−0.654243	+	0.756284i	$0.727013\pi$
$158$	2.07749	0.165276
$159$	2.70179	0.214266
$160$	−3.87769	−0.306558
$161$	0.756833	0.0596468
$162$	−1.12824	−0.0886429
$163$	24.5572	1.92347	0.961735	−	0.273981i	$-0.0883404\pi$
$163$	24.5572	1.92347	0.961735	+	0.273981i	$0.0883404\pi$
$164$	1.98103	0.154692
$165$	4.95894	0.386053
$166$	−9.85004	−0.764511
$167$	18.7805	1.45328	0.726641	−	0.687018i	$-0.241081\pi$
$167$	18.7805	1.45328	0.726641	+	0.687018i	$0.241081\pi$
$168$	−15.0563	−1.16162
$169$	−8.03535	−0.618104
$170$	2.79901	0.214674
$171$	−7.75202	−0.592812
$172$	1.26686	0.0965973
$173$	4.44434	0.337897	0.168948	−	0.985625i	$-0.445963\pi$
$173$	4.44434	0.337897	0.168948	+	0.985625i	$0.445963\pi$
$174$	7.07489	0.536346
$175$	−4.89351	−0.369914
$176$	−10.0032	−0.754022
$177$	−3.59171	−0.269969
$178$	8.87261	0.665030
$179$	17.4180	1.30188	0.650940	−	0.759130i	$-0.274375\pi$
$179$	17.4180	1.30188	0.650940	+	0.759130i	$0.274375\pi$
$180$	−0.727075	−0.0541930
$181$	−4.86910	−0.361918	−0.180959	−	0.983491i	$-0.557920\pi$
$181$	−4.86910	−0.361918	−0.180959	+	0.983491i	$0.557920\pi$
$182$	12.3017	0.911864
$183$	−9.11481	−0.673786
$184$	−0.475859	−0.0350808
$185$	10.5317	0.774306
$186$	−9.37650	−0.687518
$187$	−12.3025	−0.899645
$188$	3.95145	0.288189
$189$	−4.89351	−0.355950
$190$	8.74614	0.634511
$191$	15.5426	1.12463	0.562313	−	0.826924i	$-0.309912\pi$
$191$	15.5426	1.12463	0.562313	+	0.826924i	$0.309912\pi$
$192$	8.40939	0.606895
$193$	9.80599	0.705850	0.352925	−	0.935652i	$-0.385187\pi$
$193$	9.80599	0.705850	0.352925	+	0.935652i	$0.385187\pi$
$194$	−9.85032	−0.707212
$195$	2.22815	0.159561
$196$	−12.3213	−0.880094
$197$	−3.21642	−0.229161	−0.114580	−	0.993414i	$-0.536552\pi$
$197$	−3.21642	−0.229161	−0.114580	+	0.993414i	$0.536552\pi$
$198$	−5.59488	−0.397611
$199$	−20.3145	−1.44005	−0.720027	−	0.693946i	$-0.755871\pi$
$199$	−20.3145	−1.44005	−0.720027	+	0.693946i	$0.755871\pi$
$200$	3.07679	0.217562
$201$	−10.3277	−0.728463
$202$	11.5258	0.810956
$203$	30.6859	2.15373
$204$	1.80377	0.126289
$205$	−2.72465	−0.190298
$206$	7.38790	0.514740
$207$	−0.154661	−0.0107497
$208$	−4.49465	−0.311648
$209$	−38.4418	−2.65908
$210$	5.52105	0.380989
$211$	−14.5171	−0.999395	−0.499698	−	0.866200i	$-0.666556\pi$
$211$	−14.5171	−0.999395	−0.499698	+	0.866200i	$0.666556\pi$
$212$	−1.96440	−0.134916
$213$	3.87685	0.265637
$214$	−19.4546	−1.32989
$215$	−1.74241	−0.118831
$216$	3.07679	0.209349
$217$	−40.6686	−2.76077
$218$	9.51063	0.644141
$219$	−3.06165	−0.206887
$220$	−3.60553	−0.243085
$221$	−5.52773	−0.371836
$222$	−11.8823	−0.797486
$223$	−20.8994	−1.39953	−0.699765	−	0.714373i	$-0.746712\pi$
$223$	−20.8994	−1.39953	−0.699765	+	0.714373i	$0.746712\pi$
$224$	18.9755	1.26785
$225$	1.00000	0.0666667
$226$	7.87494	0.523833
$227$	−27.2084	−1.80588	−0.902942	−	0.429763i	$-0.858597\pi$
$227$	−27.2084	−1.80588	−0.902942	+	0.429763i	$0.858597\pi$
$228$	5.63630	0.373273
$229$	20.9816	1.38650	0.693252	−	0.720695i	$-0.256177\pi$
$229$	20.9816	1.38650	0.693252	+	0.720695i	$0.256177\pi$
$230$	0.174494	0.0115058
$231$	−24.2666	−1.59663
$232$	−19.2938	−1.26670
$233$	−26.7088	−1.74975	−0.874876	−	0.484347i	$-0.839057\pi$
$233$	−26.7088	−1.74975	−0.874876	+	0.484347i	$0.839057\pi$
$234$	−2.51389	−0.164338
$235$	−5.43472	−0.354522
$236$	2.61144	0.169990
$237$	−1.84135	−0.119609
$238$	−13.6970	−0.887842
$239$	−6.35709	−0.411206	−0.205603	−	0.978635i	$-0.565916\pi$
$239$	−6.35709	−0.411206	−0.205603	+	0.978635i	$0.565916\pi$
$240$	−2.01721	−0.130210
$241$	−21.4883	−1.38418	−0.692092	−	0.721809i	$-0.743311\pi$
$241$	−21.4883	−1.38418	−0.692092	+	0.721809i	$0.743311\pi$
$242$	−15.3340	−0.985710
$243$	1.00000	0.0641500
$244$	6.62716	0.424260
$245$	16.9464	1.08267
$246$	3.07406	0.195995
$247$	−17.2727	−1.09903
$248$	25.5704	1.62372
$249$	8.73045	0.553270
$250$	−1.12824	−0.0713561
$251$	−17.6797	−1.11594	−0.557968	−	0.829863i	$-0.688419\pi$
$251$	−17.6797	−1.11594	−0.557968	+	0.829863i	$0.688419\pi$
$252$	3.55795	0.224130
$253$	−0.766954	−0.0482180
$254$	−19.3819	−1.21613
$255$	−2.48086	−0.155358
$256$	−14.8642	−0.929012
$257$	4.89117	0.305103	0.152551	−	0.988296i	$-0.451251\pi$
$257$	4.89117	0.305103	0.152551	+	0.988296i	$0.451251\pi$
$258$	1.96585	0.122389
$259$	−51.5369	−3.20235
$260$	−1.62003	−0.100470
$261$	−6.27073	−0.388149
$262$	7.41742	0.458250
$263$	5.84613	0.360488	0.180244	−	0.983622i	$-0.442311\pi$
$263$	5.84613	0.360488	0.180244	+	0.983622i	$0.442311\pi$
$264$	15.2577	0.939044
$265$	2.70179	0.165969
$266$	−42.7993	−2.62419
$267$	−7.86411	−0.481276
$268$	7.50905	0.458688
$269$	17.2620	1.05248	0.526241	−	0.850336i	$-0.323601\pi$
$269$	17.2620	1.05248	0.526241	+	0.850336i	$0.323601\pi$
$270$	−1.12824	−0.0686625
$271$	10.8289	0.657807	0.328903	−	0.944364i	$-0.393321\pi$
$271$	10.8289	0.657807	0.328903	+	0.944364i	$0.393321\pi$
$272$	5.00442	0.303438
$273$	−10.9035	−0.659908
$274$	10.3526	0.625425
$275$	4.95894	0.299036
$276$	0.112450	0.00676869
$277$	−24.7192	−1.48523	−0.742616	−	0.669717i	$-0.766415\pi$
$277$	−24.7192	−1.48523	−0.742616	+	0.669717i	$0.766415\pi$
$278$	−5.02789	−0.301553
$279$	8.31073	0.497551
$280$	−15.0563	−0.899787
$281$	−6.25825	−0.373336	−0.186668	−	0.982423i	$-0.559769\pi$
$281$	−6.25825	−0.373336	−0.186668	+	0.982423i	$0.559769\pi$
$282$	6.13167	0.365136
$283$	24.7861	1.47338	0.736691	−	0.676229i	$-0.236387\pi$
$283$	24.7861	1.47338	0.736691	+	0.676229i	$0.236387\pi$
$284$	−2.81876	−0.167263
$285$	−7.75202	−0.459190
$286$	−12.4662	−0.737143
$287$	13.3331	0.787028
$288$	−3.87769	−0.228495
$289$	−10.8453	−0.637960
$290$	7.07489	0.415452
$291$	8.73070	0.511803
$292$	2.22605	0.130270
$293$	0.163827	0.00957089	0.00478544	−	0.999989i	$-0.498477\pi$
$293$	0.163827	0.00957089	0.00478544	+	0.999989i	$0.498477\pi$
$294$	−19.1196	−1.11508
$295$	−3.59171	−0.209117
$296$	32.4039	1.88344
$297$	4.95894	0.287747
$298$	21.2515	1.23107
$299$	−0.344607	−0.0199291
$300$	−0.727075	−0.0419777
$301$	8.52649	0.491459
$302$	13.4037	0.771295
$303$	−10.2158	−0.586881
$304$	15.6375	0.896870
$305$	−9.11481	−0.521913
$306$	2.79901	0.160009
$307$	−31.6279	−1.80510	−0.902550	−	0.430585i	$-0.858307\pi$
$307$	−31.6279	−1.80510	−0.902550	+	0.430585i	$0.858307\pi$
$308$	17.6437	1.00534
$309$	−6.54817	−0.372512
$310$	−9.37650	−0.532549
$311$	−34.6720	−1.96607	−0.983034	−	0.183424i	$-0.941282\pi$
$311$	−34.6720	−1.96607	−0.983034	+	0.183424i	$0.941282\pi$
$312$	6.85556	0.388120
$313$	−3.91609	−0.221350	−0.110675	−	0.993857i	$-0.535301\pi$
$313$	−3.91609	−0.221350	−0.110675	+	0.993857i	$0.535301\pi$
$314$	18.4978	1.04389
$315$	−4.89351	−0.275718
$316$	1.33880	0.0753134
$317$	11.3100	0.635232	0.317616	−	0.948219i	$-0.397118\pi$
$317$	11.3100	0.635232	0.317616	+	0.948219i	$0.397118\pi$
$318$	−3.04826	−0.170938
$319$	−31.0962	−1.74105
$320$	8.40939	0.470099
$321$	17.2433	0.962426
$322$	−0.853889	−0.0475854
$323$	19.2317	1.07008
$324$	−0.727075	−0.0403931
$325$	2.22815	0.123596
$326$	−27.7064	−1.53452
$327$	−8.42962	−0.466159
$328$	−8.38319	−0.462884
$329$	26.5949	1.46622
$330$	−5.59488	−0.307988
$331$	18.2344	1.00226	0.501128	−	0.865373i	$-0.332919\pi$
$331$	18.2344	1.00226	0.501128	+	0.865373i	$0.332919\pi$
$332$	−6.34770	−0.348375
$333$	10.5317	0.577133
$334$	−21.1889	−1.15941
$335$	−10.3277	−0.564265
$336$	9.87123	0.538520
$337$	−28.1213	−1.53186	−0.765932	−	0.642922i	$-0.777722\pi$
$337$	−28.1213	−1.53186	−0.765932	+	0.642922i	$0.777722\pi$
$338$	9.06580	0.493114
$339$	−6.97985	−0.379093
$340$	1.80377	0.0978234
$341$	41.2125	2.23178
$342$	8.74614	0.472937
$343$	−48.6729	−2.62809
$344$	−5.36103	−0.289047
$345$	−0.154661	−0.00832665
$346$	−5.01428	−0.269569
$347$	3.92677	0.210800	0.105400	−	0.994430i	$-0.466388\pi$
$347$	3.92677	0.210800	0.105400	+	0.994430i	$0.466388\pi$
$348$	4.55930	0.244404
$349$	−15.9248	−0.852437	−0.426218	−	0.904620i	$-0.640154\pi$
$349$	−15.9248	−0.852437	−0.426218	+	0.904620i	$0.640154\pi$
$350$	5.52105	0.295112
$351$	2.22815	0.118930
$352$	−19.2293	−1.02492
$353$	−33.4646	−1.78114	−0.890570	−	0.454846i	$-0.849694\pi$
$353$	−33.4646	−1.78114	−0.890570	+	0.454846i	$0.849694\pi$
$354$	4.05231	0.215378
$355$	3.87685	0.205762
$356$	5.71780	0.303043
$357$	12.1401	0.642523
$358$	−19.6516	−1.03862
$359$	−11.5509	−0.609634	−0.304817	−	0.952411i	$-0.598595\pi$
$359$	−11.5509	−0.609634	−0.304817	+	0.952411i	$0.598595\pi$
$360$	3.07679	0.162161
$361$	41.0938	2.16283
$362$	5.49351	0.288733
$363$	13.5911	0.713349
$364$	7.92764	0.415521
$365$	−3.06165	−0.160254
$366$	10.2837	0.537537
$367$	19.6550	1.02598	0.512990	−	0.858394i	$-0.328538\pi$
$367$	19.6550	1.02598	0.512990	+	0.858394i	$0.328538\pi$
$368$	0.311983	0.0162632
$369$	−2.72465	−0.141840
$370$	−11.8823	−0.617730
$371$	−13.2212	−0.686411
$372$	−6.04253	−0.313291
$373$	14.5159	0.751607	0.375804	−	0.926699i	$-0.377367\pi$
$373$	14.5159	0.751607	0.375804	+	0.926699i	$0.377367\pi$
$374$	13.8801	0.717724
$375$	1.00000	0.0516398
$376$	−16.7215	−0.862347
$377$	−13.9721	−0.719602
$378$	5.52105	0.283972
$379$	17.1697	0.881946	0.440973	−	0.897520i	$-0.354633\pi$
$379$	17.1697	0.881946	0.440973	+	0.897520i	$0.354633\pi$
$380$	5.63630	0.289136
$381$	17.1789	0.880100
$382$	−17.5358	−0.897211
$383$	−25.4365	−1.29975	−0.649873	−	0.760042i	$-0.725178\pi$
$383$	−25.4365	−1.29975	−0.649873	+	0.760042i	$0.725178\pi$
$384$	−1.73242	−0.0884071
$385$	−24.2666	−1.23674
$386$	−11.0635	−0.563117
$387$	−1.74241	−0.0885716
$388$	−6.34788	−0.322265
$389$	−1.29111	−0.0654621	−0.0327310	−	0.999464i	$-0.510420\pi$
$389$	−1.29111	−0.0654621	−0.0327310	+	0.999464i	$0.510420\pi$
$390$	−2.51389	−0.127296
$391$	0.383692	0.0194041
$392$	52.1406	2.63350
$393$	−6.57433	−0.331631
$394$	3.62890	0.182821
$395$	−1.84135	−0.0926484
$396$	−3.60553	−0.181185
$397$	−27.2387	−1.36707	−0.683537	−	0.729916i	$-0.739559\pi$
$397$	−27.2387	−1.36707	−0.683537	+	0.729916i	$0.739559\pi$
$398$	22.9196	1.14886
$399$	37.9346	1.89910
$400$	−2.01721	−0.100861
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	11.6522	0.581157
$403$	18.5176	0.922425
$404$	7.42764	0.369539
$405$	1.00000	0.0496904
$406$	−34.6210	−1.71821
$407$	52.2261	2.58875
$408$	−7.63310	−0.377895
$409$	2.96138	0.146431	0.0732155	−	0.997316i	$-0.476674\pi$
$409$	2.96138	0.146431	0.0732155	+	0.997316i	$0.476674\pi$
$410$	3.07406	0.151817
$411$	−9.17590	−0.452614
$412$	4.76101	0.234558
$413$	17.5761	0.864861
$414$	0.174494	0.00857593
$415$	8.73045	0.428561
$416$	−8.64008	−0.423615
$417$	4.45641	0.218231
$418$	43.3716	2.12137
$419$	−36.9679	−1.80600	−0.903000	−	0.429641i	$-0.858640\pi$
$419$	−36.9679	−1.80600	−0.903000	+	0.429641i	$0.858640\pi$
$420$	3.55795	0.173610
$421$	−12.2782	−0.598401	−0.299200	−	0.954190i	$-0.596720\pi$
$421$	−12.2782	−0.598401	−0.299200	+	0.954190i	$0.596720\pi$
$422$	16.3787	0.797304
$423$	−5.43472	−0.264245
$424$	8.31284	0.403707
$425$	−2.48086	−0.120339
$426$	−4.37401	−0.211922
$427$	44.6034	2.15851
$428$	−12.5372	−0.606007
$429$	11.0493	0.533464
$430$	1.96585	0.0948019
$431$	−2.93377	−0.141315	−0.0706574	−	0.997501i	$-0.522510\pi$
$431$	−2.93377	−0.141315	−0.0706574	+	0.997501i	$0.522510\pi$
$432$	−2.01721	−0.0970531
$433$	−18.6261	−0.895113	−0.447557	−	0.894256i	$-0.647706\pi$
$433$	−18.6261	−0.895113	−0.447557	+	0.894256i	$0.647706\pi$
$434$	45.8840	2.20250
$435$	−6.27073	−0.300659
$436$	6.12897	0.293524
$437$	1.19893	0.0573527
$438$	3.45427	0.165052
$439$	22.8880	1.09239	0.546193	−	0.837659i	$-0.316076\pi$
$439$	22.8880	1.09239	0.546193	+	0.837659i	$0.316076\pi$
$440$	15.2577	0.727380
$441$	16.9464	0.806972
$442$	6.23661	0.296645
$443$	−11.3645	−0.539944	−0.269972	−	0.962868i	$-0.587014\pi$
$443$	−11.3645	−0.539944	−0.269972	+	0.962868i	$0.587014\pi$
$444$	−7.65734	−0.363401
$445$	−7.86411	−0.372795
$446$	23.5796	1.11653
$447$	−18.8360	−0.890911
$448$	−41.1514	−1.94422
$449$	−14.0732	−0.664155	−0.332078	−	0.943252i	$-0.607750\pi$
$449$	−14.0732	−0.664155	−0.332078	+	0.943252i	$0.607750\pi$
$450$	−1.12824	−0.0531857
$451$	−13.5114	−0.636227
$452$	5.07488	0.238702
$453$	−11.8802	−0.558179
$454$	30.6976	1.44071
$455$	−10.9035	−0.511162
$456$	−23.8514	−1.11694
$457$	−24.9922	−1.16909	−0.584544	−	0.811362i	$-0.698726\pi$
$457$	−24.9922	−1.16909	−0.584544	+	0.811362i	$0.698726\pi$
$458$	−23.6723	−1.10613
$459$	−2.48086	−0.115797
$460$	0.112450	0.00524301
$461$	16.8195	0.783363	0.391681	−	0.920101i	$-0.371894\pi$
$461$	16.8195	0.783363	0.391681	+	0.920101i	$0.371894\pi$
$462$	27.3786	1.27377
$463$	15.0085	0.697505	0.348753	−	0.937215i	$-0.386605\pi$
$463$	15.0085	0.697505	0.348753	+	0.937215i	$0.386605\pi$
$464$	12.6494	0.587233
$465$	8.31073	0.385401
$466$	30.1339	1.39593
$467$	40.8667	1.89109	0.945543	−	0.325498i	$-0.105532\pi$
$467$	40.8667	1.89109	0.945543	+	0.325498i	$0.105532\pi$
$468$	−1.62003	−0.0748860
$469$	50.5389	2.33367
$470$	6.13167	0.282833
$471$	−16.3953	−0.755455
$472$	−11.0509	−0.508661
$473$	−8.64050	−0.397291
$474$	2.07749	0.0954220
$475$	−7.75202	−0.355687
$476$	−8.82678	−0.404575
$477$	2.70179	0.123706
$478$	7.17233	0.328055
$479$	−30.4737	−1.39238	−0.696189	−	0.717858i	$-0.745123\pi$
$479$	−30.4737	−1.39238	−0.696189	+	0.717858i	$0.745123\pi$
$480$	−3.87769	−0.176992
$481$	23.4662	1.06997
$482$	24.2440	1.10428
$483$	0.756833	0.0344371
$484$	−9.88177	−0.449172
$485$	8.73070	0.396441
$486$	−1.12824	−0.0511780
$487$	−9.15826	−0.415000	−0.207500	−	0.978235i	$-0.566533\pi$
$487$	−9.15826	−0.415000	−0.207500	+	0.978235i	$0.566533\pi$
$488$	−28.0444	−1.26951
$489$	24.5572	1.11052
$490$	−19.1196	−0.863736
$491$	−4.03451	−0.182075	−0.0910373	−	0.995847i	$-0.529018\pi$
$491$	−4.03451	−0.182075	−0.0910373	+	0.995847i	$0.529018\pi$
$492$	1.98103	0.0893116
$493$	15.5568	0.700644
$494$	19.4877	0.876793
$495$	4.95894	0.222888
$496$	−16.7645	−0.752748
$497$	−18.9714	−0.850982
$498$	−9.85004	−0.441391
$499$	12.4907	0.559162	0.279581	−	0.960122i	$-0.409804\pi$
$499$	12.4907	0.559162	0.279581	+	0.960122i	$0.409804\pi$
$500$	−0.727075	−0.0325158
$501$	18.7805	0.839052
$502$	19.9470	0.890278
$503$	−1.43036	−0.0637766	−0.0318883	−	0.999491i	$-0.510152\pi$
$503$	−1.43036	−0.0637766	−0.0318883	+	0.999491i	$0.510152\pi$
$504$	−15.0563	−0.670662
$505$	−10.2158	−0.454596
$506$	0.865307	0.0384676
$507$	−8.03535	−0.356862
$508$	−12.4903	−0.554169
$509$	−20.7562	−0.920003	−0.460002	−	0.887918i	$-0.652151\pi$
$509$	−20.7562	−0.920003	−0.460002	+	0.887918i	$0.652151\pi$
$510$	2.79901	0.123942
$511$	14.9822	0.662774
$512$	20.2352	0.894278
$513$	−7.75202	−0.342260
$514$	−5.51841	−0.243407
$515$	−6.54817	−0.288547
$516$	1.26686	0.0557705
$517$	−26.9505	−1.18528
$518$	58.1460	2.55479
$519$	4.44434	0.195085
$520$	6.85556	0.300636
$521$	0.927940	0.0406538	0.0203269	−	0.999793i	$-0.493529\pi$
$521$	0.927940	0.0406538	0.0203269	+	0.999793i	$0.493529\pi$
$522$	7.07489	0.309660
$523$	−6.33468	−0.276996	−0.138498	−	0.990363i	$-0.544228\pi$
$523$	−6.33468	−0.276996	−0.138498	+	0.990363i	$0.544228\pi$
$524$	4.78004	0.208817
$525$	−4.89351	−0.213570
$526$	−6.59583	−0.287592
$527$	−20.6178	−0.898125
$528$	−10.0032	−0.435335
$529$	−22.9761	−0.998960
$530$	−3.04826	−0.132408
$531$	−3.59171	−0.155867
$532$	−27.5813	−1.19580
$533$	−6.07093	−0.262961
$534$	8.87261	0.383955
$535$	17.2433	0.745492
$536$	−31.7763	−1.37253
$537$	17.4180	0.751640
$538$	−19.4757	−0.839655
$539$	84.0363	3.61970
$540$	−0.727075	−0.0312883
$541$	6.52670	0.280605	0.140302	−	0.990109i	$-0.455193\pi$
$541$	6.52670	0.280605	0.140302	+	0.990109i	$0.455193\pi$
$542$	−12.2176	−0.524789
$543$	−4.86910	−0.208953
$544$	9.62002	0.412455
$545$	−8.42962	−0.361085
$546$	12.3017	0.526465
$547$	−36.5487	−1.56271	−0.781355	−	0.624087i	$-0.785471\pi$
$547$	−36.5487	−1.56271	−0.781355	+	0.624087i	$0.785471\pi$
$548$	6.67157	0.284996
$549$	−9.11481	−0.389011
$550$	−5.59488	−0.238566
$551$	48.6109	2.07089
$552$	−0.475859	−0.0202539
$553$	9.01066	0.383172
$554$	27.8892	1.18490
$555$	10.5317	0.447046
$556$	−3.24014	−0.137413
$557$	−33.3407	−1.41269	−0.706345	−	0.707868i	$-0.749657\pi$
$557$	−33.3407	−1.41269	−0.706345	+	0.707868i	$0.749657\pi$
$558$	−9.37650	−0.396939
$559$	−3.88235	−0.164206
$560$	9.87123	0.417136
$561$	−12.3025	−0.519410
$562$	7.06081	0.297842
$563$	10.3115	0.434578	0.217289	−	0.976107i	$-0.430279\pi$
$563$	10.3115	0.434578	0.217289	+	0.976107i	$0.430279\pi$
$564$	3.95145	0.166386
$565$	−6.97985	−0.293644
$566$	−27.9647	−1.17544
$567$	−4.89351	−0.205508
$568$	11.9283	0.500499
$569$	−29.6200	−1.24174	−0.620868	−	0.783915i	$-0.713220\pi$
$569$	−29.6200	−1.24174	−0.620868	+	0.783915i	$0.713220\pi$
$570$	8.74614	0.366335
$571$	−14.4818	−0.606046	−0.303023	−	0.952983i	$-0.597996\pi$
$571$	−14.4818	−0.606046	−0.303023	+	0.952983i	$0.597996\pi$
$572$	−8.03365	−0.335904
$573$	15.5426	0.649303
$574$	−15.0429	−0.627880
$575$	−0.154661	−0.00644980
$576$	8.40939	0.350391
$577$	44.2305	1.84134	0.920671	−	0.390341i	$-0.127643\pi$
$577$	44.2305	1.84134	0.920671	+	0.390341i	$0.127643\pi$
$578$	12.2361	0.508956
$579$	9.80599	0.407523
$580$	4.55930	0.189314
$581$	−42.7225	−1.77243
$582$	−9.85032	−0.408309
$583$	13.3980	0.554889
$584$	−9.42006	−0.389805
$585$	2.22815	0.0921227
$586$	−0.184836	−0.00763552
$587$	−30.6309	−1.26427	−0.632137	−	0.774857i	$-0.717822\pi$
$587$	−30.6309	−1.26427	−0.632137	+	0.774857i	$0.717822\pi$
$588$	−12.3213	−0.508123
$589$	−64.4250	−2.65458
$590$	4.05231	0.166831
$591$	−3.21642	−0.132306
$592$	−21.2447	−0.873150
$593$	−9.59499	−0.394019	−0.197010	−	0.980402i	$-0.563123\pi$
$593$	−9.59499	−0.394019	−0.197010	+	0.980402i	$0.563123\pi$
$594$	−5.59488	−0.229561
$595$	12.1401	0.497696
$596$	13.6952	0.560976
$597$	−20.3145	−0.831416
$598$	0.388799	0.0158992
$599$	14.4345	0.589779	0.294889	−	0.955531i	$-0.404717\pi$
$599$	14.4345	0.589779	0.294889	+	0.955531i	$0.404717\pi$
$600$	3.07679	0.125610
$601$	12.6836	0.517374	0.258687	−	0.965961i	$-0.416710\pi$
$601$	12.6836	0.517374	0.258687	+	0.965961i	$0.416710\pi$
$602$	−9.61992	−0.392079
$603$	−10.3277	−0.420578
$604$	8.63778	0.351466
$605$	13.5911	0.552558
$606$	11.5258	0.468205
$607$	38.2043	1.55066	0.775332	−	0.631554i	$-0.217582\pi$
$607$	38.2043	1.55066	0.775332	+	0.631554i	$0.217582\pi$
$608$	30.0599	1.21909
$609$	30.6859	1.24345
$610$	10.2837	0.416374
$611$	−12.1094	−0.489893
$612$	1.80377	0.0729132
$613$	9.05911	0.365894	0.182947	−	0.983123i	$-0.441436\pi$
$613$	9.05911	0.365894	0.182947	+	0.983123i	$0.441436\pi$
$614$	35.6839	1.44008
$615$	−2.72465	−0.109869
$616$	−74.6634	−3.00828
$617$	29.4416	1.18527	0.592636	−	0.805470i	$-0.298087\pi$
$617$	29.4416	1.18527	0.592636	+	0.805470i	$0.298087\pi$
$618$	7.38790	0.297185
$619$	22.0611	0.886711	0.443355	−	0.896346i	$-0.353788\pi$
$619$	22.0611	0.886711	0.443355	+	0.896346i	$0.353788\pi$
$620$	−6.04253	−0.242674
$621$	−0.154661	−0.00620632
$622$	39.1183	1.56850
$623$	38.4831	1.54179
$624$	−4.49465	−0.179930
$625$	1.00000	0.0400000
$626$	4.41829	0.176590
$627$	−38.4418	−1.53522
$628$	11.9206	0.475684
$629$	−26.1277	−1.04178
$630$	5.52105	0.219964
$631$	−35.4117	−1.40972	−0.704859	−	0.709347i	$-0.748990\pi$
$631$	−35.4117	−1.40972	−0.704859	+	0.709347i	$0.748990\pi$
$632$	−5.66546	−0.225360
$633$	−14.5171	−0.577001
$634$	−12.7604	−0.506779
$635$	17.1789	0.681723
$636$	−1.96440	−0.0778936
$637$	37.7591	1.49607
$638$	35.0840	1.38899
$639$	3.87685	0.153366
$640$	−1.73242	−0.0684799
$641$	−20.0860	−0.793349	−0.396675	−	0.917959i	$-0.629836\pi$
$641$	−20.0860	−0.793349	−0.396675	+	0.917959i	$0.629836\pi$
$642$	−19.4546	−0.767810
$643$	−37.9059	−1.49486	−0.747431	−	0.664339i	$-0.768713\pi$
$643$	−37.9059	−1.49486	−0.747431	+	0.664339i	$0.768713\pi$
$644$	−0.550275	−0.0216839
$645$	−1.74241	−0.0686073
$646$	−21.6980	−0.853695
$647$	−40.8091	−1.60437	−0.802185	−	0.597075i	$-0.796329\pi$
$647$	−40.8091	−1.60437	−0.802185	+	0.597075i	$0.796329\pi$
$648$	3.07679	0.120868
$649$	−17.8111	−0.699146
$650$	−2.51389	−0.0986028
$651$	−40.6686	−1.59393
$652$	−17.8550	−0.699254
$653$	−24.8919	−0.974097	−0.487049	−	0.873375i	$-0.661927\pi$
$653$	−24.8919	−0.974097	−0.487049	+	0.873375i	$0.661927\pi$
$654$	9.51063	0.371895
$655$	−6.57433	−0.256881
$656$	5.49620	0.214590
$657$	−3.06165	−0.119446
$658$	−30.0054	−1.16973
$659$	−42.5738	−1.65844	−0.829220	−	0.558922i	$-0.811215\pi$
$659$	−42.5738	−1.65844	−0.829220	+	0.558922i	$0.811215\pi$
$660$	−3.60553	−0.140345
$661$	−47.2182	−1.83658	−0.918288	−	0.395912i	$-0.870428\pi$
$661$	−47.2182	−1.83658	−0.918288	+	0.395912i	$0.870428\pi$
$662$	−20.5728	−0.799586
$663$	−5.52773	−0.214679
$664$	26.8618	1.04244
$665$	37.9346	1.47104
$666$	−11.8823	−0.460429
$667$	0.969836	0.0375522
$668$	−13.6549	−0.528323
$669$	−20.8994	−0.808019
$670$	11.6522	0.450162
$671$	−45.1998	−1.74492
$672$	18.9755	0.731996
$673$	38.4147	1.48078	0.740389	−	0.672178i	$-0.234641\pi$
$673$	38.4147	1.48078	0.740389	+	0.672178i	$0.234641\pi$
$674$	31.7275	1.22210
$675$	1.00000	0.0384900
$676$	5.84230	0.224704
$677$	22.8892	0.879704	0.439852	−	0.898070i	$-0.355031\pi$
$677$	22.8892	0.879704	0.439852	+	0.898070i	$0.355031\pi$
$678$	7.87494	0.302435
$679$	−42.7237	−1.63959
$680$	−7.63310	−0.292716
$681$	−27.2084	−1.04263
$682$	−46.4975	−1.78048
$683$	21.9686	0.840604	0.420302	−	0.907384i	$-0.361924\pi$
$683$	21.9686	0.840604	0.420302	+	0.907384i	$0.361924\pi$
$684$	5.63630	0.215509
$685$	−9.17590	−0.350593
$686$	54.9146	2.09665
$687$	20.9816	0.800499
$688$	3.51480	0.134001
$689$	6.01998	0.229343
$690$	0.174494	0.00664288
$691$	−15.7764	−0.600162	−0.300081	−	0.953914i	$-0.597014\pi$
$691$	−15.7764	−0.600162	−0.300081	+	0.953914i	$0.597014\pi$
$692$	−3.23137	−0.122838
$693$	−24.2666	−0.921813
$694$	−4.43034	−0.168173
$695$	4.45641	0.169041
$696$	−19.2938	−0.731328
$697$	6.75948	0.256034
$698$	17.9670	0.680062
$699$	−26.7088	−1.01022
$700$	3.55795	0.134478
$701$	−4.86956	−0.183921	−0.0919603	−	0.995763i	$-0.529313\pi$
$701$	−4.86956	−0.183921	−0.0919603	+	0.995763i	$0.529313\pi$
$702$	−2.51389	−0.0948805
$703$	−81.6419	−3.07918
$704$	41.7017	1.57169
$705$	−5.43472	−0.204684
$706$	37.7561	1.42097
$707$	49.9910	1.88010
$708$	2.61144	0.0981440
$709$	7.43749	0.279321	0.139660	−	0.990199i	$-0.455399\pi$
$709$	7.43749	0.279321	0.139660	+	0.990199i	$0.455399\pi$
$710$	−4.37401	−0.164154
$711$	−1.84135	−0.0690561
$712$	−24.1963	−0.906793
$713$	−1.28534	−0.0481365
$714$	−13.6970	−0.512596
$715$	11.0493	0.413219
$716$	−12.6642	−0.473282
$717$	−6.35709	−0.237410
$718$	13.0322	0.486357
$719$	28.6458	1.06831	0.534155	−	0.845387i	$-0.320630\pi$
$719$	28.6458	1.06831	0.534155	+	0.845387i	$0.320630\pi$
$720$	−2.01721	−0.0751770
$721$	32.0435	1.19336
$722$	−46.3637	−1.72548
$723$	−21.4883	−0.799160
$724$	3.54021	0.131571
$725$	−6.27073	−0.232889
$726$	−15.3340	−0.569100
$727$	36.1433	1.34048	0.670241	−	0.742143i	$-0.266191\pi$
$727$	36.1433	1.34048	0.670241	+	0.742143i	$0.266191\pi$
$728$	−33.5477	−1.24336
$729$	1.00000	0.0370370
$730$	3.45427	0.127848
$731$	4.32267	0.159880
$732$	6.62716	0.244947
$733$	−20.0915	−0.742095	−0.371048	−	0.928614i	$-0.621001\pi$
$733$	−20.0915	−0.742095	−0.371048	+	0.928614i	$0.621001\pi$
$734$	−22.1755	−0.818513
$735$	16.9464	0.625078
$736$	0.599726	0.0221062
$737$	−51.2147	−1.88652
$738$	3.07406	0.113158
$739$	12.8774	0.473704	0.236852	−	0.971546i	$-0.423884\pi$
$739$	12.8774	0.473704	0.236852	+	0.971546i	$0.423884\pi$
$740$	−7.65734	−0.281489
$741$	−17.2727	−0.634527
$742$	14.9167	0.547609
$743$	18.0008	0.660387	0.330193	−	0.943913i	$-0.392886\pi$
$743$	18.0008	0.660387	0.330193	+	0.943913i	$0.392886\pi$
$744$	25.5704	0.937457
$745$	−18.8360	−0.690097
$746$	−16.3775	−0.599622
$747$	8.73045	0.319430
$748$	8.94481	0.327055
$749$	−84.3801	−3.08318
$750$	−1.12824	−0.0411975
$751$	−41.6415	−1.51952	−0.759760	−	0.650203i	$-0.774684\pi$
$751$	−41.6415	−1.51952	−0.759760	+	0.650203i	$0.774684\pi$
$752$	10.9630	0.399779
$753$	−17.6797	−0.644286
$754$	15.7639	0.574088
$755$	−11.8802	−0.432364
$756$	3.55795	0.129401
$757$	25.8820	0.940699	0.470349	−	0.882480i	$-0.344128\pi$
$757$	25.8820	0.940699	0.470349	+	0.882480i	$0.344128\pi$
$758$	−19.3715	−0.703604
$759$	−0.766954	−0.0278387
$760$	−23.8514	−0.865180
$761$	28.6080	1.03704	0.518519	−	0.855066i	$-0.326484\pi$
$761$	28.6080	1.03704	0.518519	+	0.855066i	$0.326484\pi$
$762$	−19.3819	−0.702132
$763$	41.2504	1.49336
$764$	−11.3007	−0.408844
$765$	−2.48086	−0.0896958
$766$	28.6985	1.03692
$767$	−8.00287	−0.288967
$768$	−14.8642	−0.536365
$769$	−8.99257	−0.324280	−0.162140	−	0.986768i	$-0.551840\pi$
$769$	−8.99257	−0.324280	−0.162140	+	0.986768i	$0.551840\pi$
$770$	27.3786	0.986655
$771$	4.89117	0.176151
$772$	−7.12969	−0.256603
$773$	38.1005	1.37038	0.685189	−	0.728365i	$-0.259720\pi$
$773$	38.1005	1.37038	0.685189	+	0.728365i	$0.259720\pi$
$774$	1.96585	0.0706612
$775$	8.31073	0.298530
$776$	26.8626	0.964310
$777$	−51.5369	−1.84888
$778$	1.45669	0.0522247
$779$	21.1216	0.756758
$780$	−1.62003	−0.0580065
$781$	19.2251	0.687927
$782$	−0.432896	−0.0154803
$783$	−6.27073	−0.224098
$784$	−34.1845	−1.22087
$785$	−16.3953	−0.585173
$786$	7.41742	0.264571
$787$	29.9590	1.06792	0.533962	−	0.845509i	$-0.320703\pi$
$787$	29.9590	1.06792	0.533962	+	0.845509i	$0.320703\pi$
$788$	2.33858	0.0833085
$789$	5.84613	0.208128
$790$	2.07749	0.0739136
$791$	34.1559	1.21445
$792$	15.2577	0.542157
$793$	−20.3092	−0.721200
$794$	30.7318	1.09063
$795$	2.70179	0.0958225
$796$	14.7702	0.523514
$797$	25.6671	0.909174	0.454587	−	0.890702i	$-0.349787\pi$
$797$	25.6671	0.909174	0.454587	+	0.890702i	$0.349787\pi$
$798$	−42.7993	−1.51508
$799$	13.4828	0.476987
$800$	−3.87769	−0.137097
$801$	−7.86411	−0.277865
$802$	1.12824	0.0398395
$803$	−15.1825	−0.535780
$804$	7.50905	0.264824
$805$	0.756833	0.0266749
$806$	−20.8922	−0.735898
$807$	17.2620	0.607651
$808$	−31.4319	−1.10577
$809$	−18.6184	−0.654586	−0.327293	−	0.944923i	$-0.606136\pi$
$809$	−18.6184	−0.654586	−0.327293	+	0.944923i	$0.606136\pi$
$810$	−1.12824	−0.0396423
$811$	−26.8860	−0.944095	−0.472048	−	0.881573i	$-0.656485\pi$
$811$	−26.8860	−0.944095	−0.472048	+	0.881573i	$0.656485\pi$
$812$	−22.3110	−0.782961
$813$	10.8289	0.379785
$814$	−58.9236	−2.06527
$815$	24.5572	0.860202
$816$	5.00442	0.175190
$817$	13.5072	0.472557
$818$	−3.34115	−0.116821
$819$	−10.9035	−0.380998
$820$	1.98103	0.0691805
$821$	42.5646	1.48551	0.742757	−	0.669561i	$-0.233518\pi$
$821$	42.5646	1.48551	0.742757	+	0.669561i	$0.233518\pi$
$822$	10.3526	0.361089
$823$	−10.5261	−0.366918	−0.183459	−	0.983027i	$-0.558729\pi$
$823$	−10.5261	−0.366918	−0.183459	+	0.983027i	$0.558729\pi$
$824$	−20.1474	−0.701867
$825$	4.95894	0.172648
$826$	−19.8300	−0.689974
$827$	35.2890	1.22712	0.613559	−	0.789649i	$-0.289737\pi$
$827$	35.2890	1.22712	0.613559	+	0.789649i	$0.289737\pi$
$828$	0.112450	0.00390791
$829$	−37.6394	−1.30727	−0.653636	−	0.756809i	$-0.726757\pi$
$829$	−37.6394	−1.30727	−0.653636	+	0.756809i	$0.726757\pi$
$830$	−9.85004	−0.341900
$831$	−24.7192	−0.857499
$832$	18.7374	0.649602
$833$	−42.0417	−1.45666
$834$	−5.02789	−0.174102
$835$	18.7805	0.649927
$836$	27.9501	0.966675
$837$	8.31073	0.287261
$838$	41.7086	1.44080
$839$	28.2700	0.975989	0.487995	−	0.872847i	$-0.337729\pi$
$839$	28.2700	0.975989	0.487995	+	0.872847i	$0.337729\pi$
$840$	−15.0563	−0.519492
$841$	10.3221	0.355935
$842$	13.8527	0.477396
$843$	−6.25825	−0.215546
$844$	10.5550	0.363318
$845$	−8.03535	−0.276424
$846$	6.13167	0.210811
$847$	−66.5083	−2.28525
$848$	−5.45007	−0.187156
$849$	24.7861	0.850658
$850$	2.79901	0.0960051
$851$	−1.62884	−0.0558359
$852$	−2.81876	−0.0965691
$853$	−43.3087	−1.48286	−0.741432	−	0.671028i	$-0.765853\pi$
$853$	−43.3087	−1.48286	−0.741432	+	0.671028i	$0.765853\pi$
$854$	−50.3233	−1.72203
$855$	−7.75202	−0.265114
$856$	53.0540	1.81335
$857$	20.0808	0.685949	0.342974	−	0.939345i	$-0.388566\pi$
$857$	20.0808	0.685949	0.342974	+	0.939345i	$0.388566\pi$
$858$	−12.4662	−0.425590
$859$	−1.08335	−0.0369633	−0.0184817	−	0.999829i	$-0.505883\pi$
$859$	−1.08335	−0.0369633	−0.0184817	+	0.999829i	$0.505883\pi$
$860$	1.26686	0.0431996
$861$	13.3331	0.454391
$862$	3.31000	0.112739
$863$	5.53413	0.188384	0.0941920	−	0.995554i	$-0.469973\pi$
$863$	5.53413	0.188384	0.0941920	+	0.995554i	$0.469973\pi$
$864$	−3.87769	−0.131922
$865$	4.44434	0.151112
$866$	21.0147	0.714109
$867$	−10.8453	−0.368326
$868$	29.5692	1.00364
$869$	−9.13116	−0.309753
$870$	7.07489	0.239861
$871$	−23.0118	−0.779723
$872$	−25.9362	−0.878311
$873$	8.73070	0.295489
$874$	−1.35268	−0.0457552
$875$	−4.89351	−0.165431
$876$	2.22605	0.0752112
$877$	42.8148	1.44575	0.722876	−	0.690978i	$-0.242820\pi$
$877$	42.8148	1.44575	0.722876	+	0.690978i	$0.242820\pi$
$878$	−25.8232	−0.871490
$879$	0.163827	0.00552575
$880$	−10.0032	−0.337209
$881$	53.1266	1.78988	0.894940	−	0.446186i	$-0.147218\pi$
$881$	53.1266	1.78988	0.894940	+	0.446186i	$0.147218\pi$
$882$	−19.1196	−0.643791
$883$	−40.3852	−1.35907	−0.679535	−	0.733643i	$-0.737818\pi$
$883$	−40.3852	−1.35907	−0.679535	+	0.733643i	$0.737818\pi$
$884$	4.01908	0.135176
$885$	−3.59171	−0.120734
$886$	12.8219	0.430759
$887$	16.8656	0.566290	0.283145	−	0.959077i	$-0.408622\pi$
$887$	16.8656	0.566290	0.283145	+	0.959077i	$0.408622\pi$
$888$	32.4039	1.08740
$889$	−84.0649	−2.81945
$890$	8.87261	0.297410
$891$	4.95894	0.166131
$892$	15.1955	0.508782
$893$	42.1301	1.40983
$894$	21.2515	0.710756
$895$	17.4180	0.582218
$896$	8.47761	0.283217
$897$	−0.344607	−0.0115061
$898$	15.8779	0.529854
$899$	−52.1144	−1.73811
$900$	−0.727075	−0.0242358
$901$	−6.70276	−0.223301
$902$	15.2441	0.507573
$903$	8.52649	0.283744
$904$	−21.4756	−0.714266
$905$	−4.86910	−0.161854
$906$	13.4037	0.445307
$907$	26.8644	0.892016	0.446008	−	0.895029i	$-0.352845\pi$
$907$	26.8644	0.892016	0.446008	+	0.895029i	$0.352845\pi$
$908$	19.7825	0.656507
$909$	−10.2158	−0.338836
$910$	12.3017	0.407798
$911$	38.6977	1.28211	0.641056	−	0.767494i	$-0.278497\pi$
$911$	38.6977	1.28211	0.641056	+	0.767494i	$0.278497\pi$
$912$	15.6375	0.517808
$913$	43.2938	1.43282
$914$	28.1972	0.932682
$915$	−9.11481	−0.301326
$916$	−15.2552	−0.504047
$917$	32.1716	1.06240
$918$	2.79901	0.0923810
$919$	4.71926	0.155674	0.0778370	−	0.996966i	$-0.475199\pi$
$919$	4.71926	0.155674	0.0778370	+	0.996966i	$0.475199\pi$
$920$	−0.475859	−0.0156886
$921$	−31.6279	−1.04217
$922$	−18.9764	−0.624956
$923$	8.63819	0.284330
$924$	17.6437	0.580434
$925$	10.5317	0.346280
$926$	−16.9332	−0.556460
$927$	−6.54817	−0.215070
$928$	24.3160	0.798211
$929$	−26.5222	−0.870164	−0.435082	−	0.900391i	$-0.643281\pi$
$929$	−26.5222	−0.870164	−0.435082	+	0.900391i	$0.643281\pi$
$930$	−9.37650	−0.307467
$931$	−131.369	−4.30544
$932$	19.4193	0.636101
$933$	−34.6720	−1.13511
$934$	−46.1074	−1.50868
$935$	−12.3025	−0.402333
$936$	6.85556	0.224081
$937$	−6.76319	−0.220944	−0.110472	−	0.993879i	$-0.535236\pi$
$937$	−6.76319	−0.220944	−0.110472	+	0.993879i	$0.535236\pi$
$938$	−57.0200	−1.86177
$939$	−3.91609	−0.127797
$940$	3.95145	0.128882
$941$	−11.7184	−0.382009	−0.191004	−	0.981589i	$-0.561174\pi$
$941$	−11.7184	−0.382009	−0.191004	+	0.981589i	$0.561174\pi$
$942$	18.4978	0.602691
$943$	0.421396	0.0137226
$944$	7.24523	0.235812
$945$	−4.89351	−0.159186
$946$	9.74856	0.316953
$947$	51.7048	1.68018	0.840090	−	0.542447i	$-0.182502\pi$
$947$	51.7048	1.68018	0.840090	+	0.542447i	$0.182502\pi$
$948$	1.33880	0.0434822
$949$	−6.82181	−0.221445
$950$	8.74614	0.283762
$951$	11.3100	0.366751
$952$	37.3526	1.21061
$953$	−6.17161	−0.199918	−0.0999591	−	0.994992i	$-0.531871\pi$
$953$	−6.17161	−0.199918	−0.0999591	+	0.994992i	$0.531871\pi$
$954$	−3.04826	−0.0986911
$955$	15.5426	0.502948
$956$	4.62209	0.149489
$957$	−31.0962	−1.00520
$958$	34.3816	1.11082
$959$	44.9024	1.44997
$960$	8.40939	0.271412
$961$	38.0683	1.22801
$962$	−26.4755	−0.853604
$963$	17.2433	0.555657
$964$	15.6236	0.503203
$965$	9.80599	0.315666
$966$	−0.853889	−0.0274734
$967$	14.4905	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$967$	14.4905	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$968$	41.8171	1.34405
$969$	19.2317	0.617811
$970$	−9.85032	−0.316275
$971$	33.5690	1.07728	0.538640	−	0.842536i	$-0.318938\pi$
$971$	33.5690	1.07728	0.538640	+	0.842536i	$0.318938\pi$
$972$	−0.727075	−0.0233210
$973$	−21.8075	−0.699115
$974$	10.3327	0.331081
$975$	2.22815	0.0713579
$976$	18.3865	0.588537
$977$	5.80043	0.185572	0.0927862	−	0.995686i	$-0.470423\pi$
$977$	5.80043	0.185572	0.0927862	+	0.995686i	$0.470423\pi$
$978$	−27.7064	−0.885954
$979$	−38.9977	−1.24637
$980$	−12.3213	−0.393590
$981$	−8.42962	−0.269137
$982$	4.55189	0.145257
$983$	14.5801	0.465034	0.232517	−	0.972592i	$-0.425304\pi$
$983$	14.5801	0.465034	0.232517	+	0.972592i	$0.425304\pi$
$984$	−8.38319	−0.267246
$985$	−3.21642	−0.102484
$986$	−17.5518	−0.558964
$987$	26.5949	0.846524
$988$	12.5585	0.399540
$989$	0.269482	0.00856903
$990$	−5.59488	−0.177817
$991$	−34.0751	−1.08243	−0.541216	−	0.840884i	$-0.682036\pi$
$991$	−34.0751	−1.08243	−0.541216	+	0.840884i	$0.682036\pi$
$992$	−32.2265	−1.02319
$993$	18.2344	0.578653
$994$	21.4043	0.678902
$995$	−20.3145	−0.644012
$996$	−6.34770	−0.201134
$997$	−28.6449	−0.907193	−0.453597	−	0.891207i	$-0.649859\pi$
$997$	−28.6449	−0.907193	−0.453597	+	0.891207i	$0.649859\pi$
$998$	−14.0925	−0.446091
$999$	10.5317	0.333208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.b.1.9	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.b.1.9	✓	23	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:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.12824 q^{2} +1.00000 q^{3} -0.727075 q^{4} +1.00000 q^{5} -1.12824 q^{6} -4.89351 q^{7} +3.07679 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.12824 q^{2} +1.00000 q^{3} -0.727075 q^{4} +1.00000 q^{5} -1.12824 q^{6} -4.89351 q^{7} +3.07679 q^{8} +1.00000 q^{9} -1.12824 q^{10} +4.95894 q^{11} -0.727075 q^{12} +2.22815 q^{13} +5.52105 q^{14} +1.00000 q^{15} -2.01721 q^{16} -2.48086 q^{17} -1.12824 q^{18} -7.75202 q^{19} -0.727075 q^{20} -4.89351 q^{21} -5.59488 q^{22} -0.154661 q^{23} +3.07679 q^{24} +1.00000 q^{25} -2.51389 q^{26} +1.00000 q^{27} +3.55795 q^{28} -6.27073 q^{29} -1.12824 q^{30} +8.31073 q^{31} -3.87769 q^{32} +4.95894 q^{33} +2.79901 q^{34} -4.89351 q^{35} -0.727075 q^{36} +10.5317 q^{37} +8.74614 q^{38} +2.22815 q^{39} +3.07679 q^{40} -2.72465 q^{41} +5.52105 q^{42} -1.74241 q^{43} -3.60553 q^{44} +1.00000 q^{45} +0.174494 q^{46} -5.43472 q^{47} -2.01721 q^{48} +16.9464 q^{49} -1.12824 q^{50} -2.48086 q^{51} -1.62003 q^{52} +2.70179 q^{53} -1.12824 q^{54} +4.95894 q^{55} -15.0563 q^{56} -7.75202 q^{57} +7.07489 q^{58} -3.59171 q^{59} -0.727075 q^{60} -9.11481 q^{61} -9.37650 q^{62} -4.89351 q^{63} +8.40939 q^{64} +2.22815 q^{65} -5.59488 q^{66} -10.3277 q^{67} +1.80377 q^{68} -0.154661 q^{69} +5.52105 q^{70} +3.87685 q^{71} +3.07679 q^{72} -3.06165 q^{73} -11.8823 q^{74} +1.00000 q^{75} +5.63630 q^{76} -24.2666 q^{77} -2.51389 q^{78} -1.84135 q^{79} -2.01721 q^{80} +1.00000 q^{81} +3.07406 q^{82} +8.73045 q^{83} +3.55795 q^{84} -2.48086 q^{85} +1.96585 q^{86} -6.27073 q^{87} +15.2577 q^{88} -7.86411 q^{89} -1.12824 q^{90} -10.9035 q^{91} +0.112450 q^{92} +8.31073 q^{93} +6.13167 q^{94} -7.75202 q^{95} -3.87769 q^{96} +8.73070 q^{97} -19.1196 q^{98} +4.95894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9	\( 23 q - 5 q^{2} + 23 q^{3} + 9 q^{4} + 23 q^{5} - 5 q^{6} - 16 q^{7} - 12 q^{8} + 23 q^{9} - 5 q^{10} - 13 q^{11} + 9 q^{12} - 18 q^{13} - 6 q^{14} + 23 q^{15} - 11 q^{16} - 34 q^{17} - 5 q^{18} - 35 q^{19} + 9 q^{20} - 16 q^{21} - 11 q^{22} - 14 q^{23} - 12 q^{24} + 23 q^{25} - 6 q^{26} + 23 q^{27} - 26 q^{28} - 43 q^{29} - 5 q^{30} - 21 q^{31} - 14 q^{32} - 13 q^{33} - 12 q^{34} - 16 q^{35} + 9 q^{36} - 18 q^{37} + 6 q^{38} - 18 q^{39} - 12 q^{40} - 45 q^{41} - 6 q^{42} - 43 q^{43} - 11 q^{44} + 23 q^{45} - 29 q^{46} - 14 q^{47} - 11 q^{48} - 25 q^{49} - 5 q^{50} - 34 q^{51} - 20 q^{52} - 3 q^{53} - 5 q^{54} - 13 q^{55} + 3 q^{56} - 35 q^{57} + 10 q^{58} - 9 q^{59} + 9 q^{60} - 67 q^{61} - 7 q^{62} - 16 q^{63} - 8 q^{64} - 18 q^{65} - 11 q^{66} - 32 q^{67} - 24 q^{68} - 14 q^{69} - 6 q^{70} - 8 q^{71} - 12 q^{72} - 39 q^{73} - 16 q^{74} + 23 q^{75} - 48 q^{76} - 26 q^{77} - 6 q^{78} - 59 q^{79} - 11 q^{80} + 23 q^{81} - q^{82} - 23 q^{83} - 26 q^{84} - 34 q^{85} - 7 q^{86} - 43 q^{87} + 17 q^{88} - 51 q^{89} - 5 q^{90} - 37 q^{91} + 11 q^{92} - 21 q^{93} + 8 q^{94} - 35 q^{95} - 14 q^{96} - 29 q^{97} + 32 q^{98} - 13 q^{99}+O(q^{100}) \) 23 * q - 5 * q^2 + 23 * q^3 + 9 * q^4 + 23 * q^5 - 5 * q^6 - 16 * q^7 - 12 * q^8 + 23 * q^9 - 5 * q^10 - 13 * q^11 + 9 * q^12 - 18 * q^13 - 6 * q^14 + 23 * q^15 - 11 * q^16 - 34 * q^17 - 5 * q^18 - 35 * q^19 + 9 * q^20 - 16 * q^21 - 11 * q^22 - 14 * q^23 - 12 * q^24 + 23 * q^25 - 6 * q^26 + 23 * q^27 - 26 * q^28 - 43 * q^29 - 5 * q^30 - 21 * q^31 - 14 * q^32 - 13 * q^33 - 12 * q^34 - 16 * q^35 + 9 * q^36 - 18 * q^37 + 6 * q^38 - 18 * q^39 - 12 * q^40 - 45 * q^41 - 6 * q^42 - 43 * q^43 - 11 * q^44 + 23 * q^45 - 29 * q^46 - 14 * q^47 - 11 * q^48 - 25 * q^49 - 5 * q^50 - 34 * q^51 - 20 * q^52 - 3 * q^53 - 5 * q^54 - 13 * q^55 + 3 * q^56 - 35 * q^57 + 10 * q^58 - 9 * q^59 + 9 * q^60 - 67 * q^61 - 7 * q^62 - 16 * q^63 - 8 * q^64 - 18 * q^65 - 11 * q^66 - 32 * q^67 - 24 * q^68 - 14 * q^69 - 6 * q^70 - 8 * q^71 - 12 * q^72 - 39 * q^73 - 16 * q^74 + 23 * q^75 - 48 * q^76 - 26 * q^77 - 6 * q^78 - 59 * q^79 - 11 * q^80 + 23 * q^81 - q^82 - 23 * q^83 - 26 * q^84 - 34 * q^85 - 7 * q^86 - 43 * q^87 + 17 * q^88 - 51 * q^89 - 5 * q^90 - 37 * q^91 + 11 * q^92 - 21 * q^93 + 8 * q^94 - 35 * q^95 - 14 * q^96 - 29 * q^97 + 32 * q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more