LMFDB - Embedded newform 4029.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26690 q^{2} +1.00000 q^{3} -0.394972 q^{4} +3.60603 q^{5} -1.26690 q^{6} -2.46109 q^{7} +3.03418 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.26690 q^{2} +1.00000 q^{3} -0.394972 q^{4} +3.60603 q^{5} -1.26690 q^{6} -2.46109 q^{7} +3.03418 q^{8} +1.00000 q^{9} -4.56847 q^{10} +4.32670 q^{11} -0.394972 q^{12} +6.87794 q^{13} +3.11794 q^{14} +3.60603 q^{15} -3.05405 q^{16} +1.00000 q^{17} -1.26690 q^{18} +1.02429 q^{19} -1.42428 q^{20} -2.46109 q^{21} -5.48148 q^{22} +3.18429 q^{23} +3.03418 q^{24} +8.00349 q^{25} -8.71364 q^{26} +1.00000 q^{27} +0.972059 q^{28} +2.11000 q^{29} -4.56847 q^{30} +11.0198 q^{31} -2.19919 q^{32} +4.32670 q^{33} -1.26690 q^{34} -8.87476 q^{35} -0.394972 q^{36} -3.84497 q^{37} -1.29767 q^{38} +6.87794 q^{39} +10.9414 q^{40} -3.71450 q^{41} +3.11794 q^{42} +5.79832 q^{43} -1.70892 q^{44} +3.60603 q^{45} -4.03416 q^{46} -10.7508 q^{47} -3.05405 q^{48} -0.943060 q^{49} -10.1396 q^{50} +1.00000 q^{51} -2.71659 q^{52} -1.67641 q^{53} -1.26690 q^{54} +15.6022 q^{55} -7.46738 q^{56} +1.02429 q^{57} -2.67315 q^{58} +1.63966 q^{59} -1.42428 q^{60} -7.36202 q^{61} -13.9609 q^{62} -2.46109 q^{63} +8.89426 q^{64} +24.8021 q^{65} -5.48148 q^{66} +2.03979 q^{67} -0.394972 q^{68} +3.18429 q^{69} +11.2434 q^{70} -3.93293 q^{71} +3.03418 q^{72} -7.02232 q^{73} +4.87118 q^{74} +8.00349 q^{75} -0.404566 q^{76} -10.6484 q^{77} -8.71364 q^{78} +1.00000 q^{79} -11.0130 q^{80} +1.00000 q^{81} +4.70588 q^{82} +2.00730 q^{83} +0.972059 q^{84} +3.60603 q^{85} -7.34587 q^{86} +2.11000 q^{87} +13.1280 q^{88} +3.75596 q^{89} -4.56847 q^{90} -16.9272 q^{91} -1.25770 q^{92} +11.0198 q^{93} +13.6201 q^{94} +3.69363 q^{95} -2.19919 q^{96} -0.409835 q^{97} +1.19476 q^{98} +4.32670 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * 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 $

$2$	−1.26690	−0.895831	−0.447916	−	0.894076i	$-0.647834\pi$
$2$	−1.26690	−0.895831	−0.447916	+	0.894076i	$0.647834\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.394972	−0.197486
$5$	3.60603	1.61267	0.806334	−	0.591461i	$-0.201448\pi$
$5$	3.60603	1.61267	0.806334	+	0.591461i	$0.201448\pi$
$6$	−1.26690	−0.517209
$7$	−2.46109	−0.930203	−0.465101	−	0.885257i	$-0.653982\pi$
$7$	−2.46109	−0.930203	−0.465101	+	0.885257i	$0.653982\pi$
$8$	3.03418	1.07275
$9$	1.00000	0.333333
$10$	−4.56847	−1.44468
$11$	4.32670	1.30455	0.652274	−	0.757983i	$-0.273815\pi$
$11$	4.32670	1.30455	0.652274	+	0.757983i	$0.273815\pi$
$12$	−0.394972	−0.114019
$13$	6.87794	1.90760	0.953799	−	0.300446i	$-0.0971355\pi$
$13$	6.87794	1.90760	0.953799	+	0.300446i	$0.0971355\pi$
$14$	3.11794	0.833305
$15$	3.60603	0.931074
$16$	−3.05405	−0.763513
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.26690	−0.298610
$19$	1.02429	0.234988	0.117494	−	0.993074i	$-0.462514\pi$
$19$	1.02429	0.234988	0.117494	+	0.993074i	$0.462514\pi$
$20$	−1.42428	−0.318479
$21$	−2.46109	−0.537053
$22$	−5.48148	−1.16866
$23$	3.18429	0.663969	0.331985	−	0.943285i	$-0.392282\pi$
$23$	3.18429	0.663969	0.331985	+	0.943285i	$0.392282\pi$
$24$	3.03418	0.619350
$25$	8.00349	1.60070
$26$	−8.71364	−1.70889
$27$	1.00000	0.192450
$28$	0.972059	0.183702
$29$	2.11000	0.391816	0.195908	−	0.980622i	$-0.437235\pi$
$29$	2.11000	0.391816	0.195908	+	0.980622i	$0.437235\pi$
$30$	−4.56847	−0.834086
$31$	11.0198	1.97921	0.989607	−	0.143801i	$-0.0459326\pi$
$31$	11.0198	1.97921	0.989607	+	0.143801i	$0.0459326\pi$
$32$	−2.19919	−0.388766
$33$	4.32670	0.753181
$34$	−1.26690	−0.217271
$35$	−8.87476	−1.50011
$36$	−0.394972	−0.0658286
$37$	−3.84497	−0.632108	−0.316054	−	0.948741i	$-0.602358\pi$
$37$	−3.84497	−0.632108	−0.316054	+	0.948741i	$0.602358\pi$
$38$	−1.29767	−0.210510
$39$	6.87794	1.10135
$40$	10.9414	1.72998
$41$	−3.71450	−0.580107	−0.290053	−	0.957010i	$-0.593673\pi$
$41$	−3.71450	−0.580107	−0.290053	+	0.957010i	$0.593673\pi$
$42$	3.11794	0.481109
$43$	5.79832	0.884235	0.442118	−	0.896957i	$-0.354227\pi$
$43$	5.79832	0.884235	0.442118	+	0.896957i	$0.354227\pi$
$44$	−1.70892	−0.257630
$45$	3.60603	0.537556
$46$	−4.03416	−0.594805
$47$	−10.7508	−1.56816	−0.784079	−	0.620661i	$-0.786864\pi$
$47$	−10.7508	−1.56816	−0.784079	+	0.620661i	$0.786864\pi$
$48$	−3.05405	−0.440815
$49$	−0.943060	−0.134723
$50$	−10.1396	−1.43395
$51$	1.00000	0.140028
$52$	−2.71659	−0.376724
$53$	−1.67641	−0.230272	−0.115136	−	0.993350i	$-0.536730\pi$
$53$	−1.67641	−0.230272	−0.115136	+	0.993350i	$0.536730\pi$
$54$	−1.26690	−0.172403
$55$	15.6022	2.10380
$56$	−7.46738	−0.997871
$57$	1.02429	0.135671
$58$	−2.67315	−0.351002
$59$	1.63966	0.213465	0.106733	−	0.994288i	$-0.465961\pi$
$59$	1.63966	0.213465	0.106733	+	0.994288i	$0.465961\pi$
$60$	−1.42428	−0.183874
$61$	−7.36202	−0.942611	−0.471305	−	0.881970i	$-0.656217\pi$
$61$	−7.36202	−0.942611	−0.471305	+	0.881970i	$0.656217\pi$
$62$	−13.9609	−1.77304
$63$	−2.46109	−0.310068
$64$	8.89426	1.11178
$65$	24.8021	3.07632
$66$	−5.48148	−0.674724
$67$	2.03979	0.249200	0.124600	−	0.992207i	$-0.460235\pi$
$67$	2.03979	0.249200	0.124600	+	0.992207i	$0.460235\pi$
$68$	−0.394972	−0.0478974
$69$	3.18429	0.383343
$70$	11.2434	1.34384
$71$	−3.93293	−0.466753	−0.233376	−	0.972386i	$-0.574977\pi$
$71$	−3.93293	−0.466753	−0.233376	+	0.972386i	$0.574977\pi$
$72$	3.03418	0.357582
$73$	−7.02232	−0.821901	−0.410950	−	0.911658i	$-0.634803\pi$
$73$	−7.02232	−0.821901	−0.410950	+	0.911658i	$0.634803\pi$
$74$	4.87118	0.566263
$75$	8.00349	0.924163
$76$	−0.404566	−0.0464069
$77$	−10.6484	−1.21349
$78$	−8.71364	−0.986626
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−11.0130	−1.23129
$81$	1.00000	0.111111
$82$	4.70588	0.519678
$83$	2.00730	0.220330	0.110165	−	0.993913i	$-0.464862\pi$
$83$	2.00730	0.220330	0.110165	+	0.993913i	$0.464862\pi$
$84$	0.972059	0.106060
$85$	3.60603	0.391129
$86$	−7.34587	−0.792126
$87$	2.11000	0.226215
$88$	13.1280	1.39945
$89$	3.75596	0.398131	0.199066	−	0.979986i	$-0.436209\pi$
$89$	3.75596	0.398131	0.199066	+	0.979986i	$0.436209\pi$
$90$	−4.56847	−0.481560
$91$	−16.9272	−1.77445
$92$	−1.25770	−0.131125
$93$	11.0198	1.14270
$94$	13.6201	1.40481
$95$	3.69363	0.378958
$96$	−2.19919	−0.224454
$97$	−0.409835	−0.0416124	−0.0208062	−	0.999784i	$-0.506623\pi$
$97$	−0.409835	−0.0416124	−0.0208062	+	0.999784i	$0.506623\pi$
$98$	1.19476	0.120689
$99$	4.32670	0.434849
$100$	−3.16115	−0.316115
$101$	−13.6638	−1.35960	−0.679802	−	0.733396i	$-0.737934\pi$
$101$	−13.6638	−1.35960	−0.679802	+	0.733396i	$0.737934\pi$
$102$	−1.26690	−0.125442
$103$	−12.4933	−1.23100	−0.615499	−	0.788137i	$-0.711046\pi$
$103$	−12.4933	−1.23100	−0.615499	+	0.788137i	$0.711046\pi$
$104$	20.8689	2.04637
$105$	−8.87476	−0.866088
$106$	2.12383	0.206285
$107$	−2.33864	−0.226085	−0.113043	−	0.993590i	$-0.536060\pi$
$107$	−2.33864	−0.226085	−0.113043	+	0.993590i	$0.536060\pi$
$108$	−0.394972	−0.0380062
$109$	−9.08357	−0.870049	−0.435024	−	0.900419i	$-0.643260\pi$
$109$	−9.08357	−0.870049	−0.435024	+	0.900419i	$0.643260\pi$
$110$	−19.7664	−1.88465
$111$	−3.84497	−0.364948
$112$	7.51629	0.710222
$113$	−13.4114	−1.26164	−0.630820	−	0.775929i	$-0.717282\pi$
$113$	−13.4114	−1.26164	−0.630820	+	0.775929i	$0.717282\pi$
$114$	−1.29767	−0.121538
$115$	11.4826	1.07076
$116$	−0.833389	−0.0773782
$117$	6.87794	0.635866
$118$	−2.07728	−0.191229
$119$	−2.46109	−0.225607
$120$	10.9414	0.998806
$121$	7.72031	0.701847
$122$	9.32693	0.844420
$123$	−3.71450	−0.334925
$124$	−4.35251	−0.390867
$125$	10.8307	0.968725
$126$	3.11794	0.277768
$127$	−2.05867	−0.182678	−0.0913388	−	0.995820i	$-0.529115\pi$
$127$	−2.05867	−0.182678	−0.0913388	+	0.995820i	$0.529115\pi$
$128$	−6.86972	−0.607203
$129$	5.79832	0.510513
$130$	−31.4217	−2.75587
$131$	12.4435	1.08719	0.543596	−	0.839347i	$-0.317062\pi$
$131$	12.4435	1.08719	0.543596	+	0.839347i	$0.317062\pi$
$132$	−1.70892	−0.148743
$133$	−2.52087	−0.218587
$134$	−2.58420	−0.223241
$135$	3.60603	0.310358
$136$	3.03418	0.260179
$137$	−6.62722	−0.566201	−0.283101	−	0.959090i	$-0.591363\pi$
$137$	−6.62722	−0.566201	−0.283101	+	0.959090i	$0.591363\pi$
$138$	−4.03416	−0.343411
$139$	8.97590	0.761326	0.380663	−	0.924714i	$-0.375696\pi$
$139$	8.97590	0.761326	0.380663	+	0.924714i	$0.375696\pi$
$140$	3.50528	0.296250
$141$	−10.7508	−0.905377
$142$	4.98262	0.418132
$143$	29.7588	2.48855
$144$	−3.05405	−0.254504
$145$	7.60872	0.631870
$146$	8.89656	0.736285
$147$	−0.943060	−0.0777823
$148$	1.51865	0.124833
$149$	1.81441	0.148642	0.0743210	−	0.997234i	$-0.476321\pi$
$149$	1.81441	0.148642	0.0743210	+	0.997234i	$0.476321\pi$
$150$	−10.1396	−0.827894
$151$	−16.4110	−1.33551	−0.667753	−	0.744383i	$-0.732744\pi$
$151$	−16.4110	−1.33551	−0.667753	+	0.744383i	$0.732744\pi$
$152$	3.10788	0.252083
$153$	1.00000	0.0808452
$154$	13.4904	1.08709
$155$	39.7378	3.19181
$156$	−2.71659	−0.217502
$157$	8.32711	0.664576	0.332288	−	0.943178i	$-0.392179\pi$
$157$	8.32711	0.664576	0.332288	+	0.943178i	$0.392179\pi$
$158$	−1.26690	−0.100789
$159$	−1.67641	−0.132948
$160$	−7.93037	−0.626951
$161$	−7.83680	−0.617626
$162$	−1.26690	−0.0995368
$163$	−7.54315	−0.590825	−0.295412	−	0.955370i	$-0.595457\pi$
$163$	−7.54315	−0.590825	−0.295412	+	0.955370i	$0.595457\pi$
$164$	1.46712	0.114563
$165$	15.6022	1.21463
$166$	−2.54304	−0.197378
$167$	−0.0281472	−0.00217810	−0.00108905	−	0.999999i	$-0.500347\pi$
$167$	−0.0281472	−0.00217810	−0.00108905	+	0.999999i	$0.500347\pi$
$168$	−7.46738	−0.576121
$169$	34.3061	2.63893
$170$	−4.56847	−0.350386
$171$	1.02429	0.0783294
$172$	−2.29017	−0.174624
$173$	23.8128	1.81045	0.905225	−	0.424932i	$-0.139702\pi$
$173$	23.8128	1.81045	0.905225	+	0.424932i	$0.139702\pi$
$174$	−2.67315	−0.202651
$175$	−19.6973	−1.48897
$176$	−13.2140	−0.996040
$177$	1.63966	0.123244
$178$	−4.75842	−0.356659
$179$	9.65959	0.721992	0.360996	−	0.932567i	$-0.382437\pi$
$179$	9.65959	0.721992	0.360996	+	0.932567i	$0.382437\pi$
$180$	−1.42428	−0.106160
$181$	−13.6390	−1.01378	−0.506888	−	0.862012i	$-0.669204\pi$
$181$	−13.6390	−1.01378	−0.506888	+	0.862012i	$0.669204\pi$
$182$	21.4450	1.58961
$183$	−7.36202	−0.544216
$184$	9.66170	0.712270
$185$	−13.8651	−1.01938
$186$	−13.9609	−1.02367
$187$	4.32670	0.316399
$188$	4.24624	0.309689
$189$	−2.46109	−0.179018
$190$	−4.67944	−0.339483
$191$	6.60493	0.477916	0.238958	−	0.971030i	$-0.423194\pi$
$191$	6.60493	0.477916	0.238958	+	0.971030i	$0.423194\pi$
$192$	8.89426	0.641888
$193$	−4.38819	−0.315869	−0.157935	−	0.987450i	$-0.550483\pi$
$193$	−4.38819	−0.315869	−0.157935	+	0.987450i	$0.550483\pi$
$194$	0.519219	0.0372777
$195$	24.8021	1.77611
$196$	0.372482	0.0266059
$197$	−4.55343	−0.324418	−0.162209	−	0.986756i	$-0.551862\pi$
$197$	−4.55343	−0.324418	−0.162209	+	0.986756i	$0.551862\pi$
$198$	−5.48148	−0.389552
$199$	−20.1322	−1.42714	−0.713568	−	0.700586i	$-0.752922\pi$
$199$	−20.1322	−1.42714	−0.713568	+	0.700586i	$0.752922\pi$
$200$	24.2840	1.71714
$201$	2.03979	0.143876
$202$	17.3107	1.21798
$203$	−5.19288	−0.364469
$204$	−0.394972	−0.0276536
$205$	−13.3946	−0.935520
$206$	15.8277	1.10277
$207$	3.18429	0.221323
$208$	−21.0056	−1.45648
$209$	4.43180	0.306554
$210$	11.2434	0.775869
$211$	23.4148	1.61194	0.805970	−	0.591957i	$-0.201644\pi$
$211$	23.4148	1.61194	0.805970	+	0.591957i	$0.201644\pi$
$212$	0.662133	0.0454755
$213$	−3.93293	−0.269480
$214$	2.96282	0.202534
$215$	20.9089	1.42598
$216$	3.03418	0.206450
$217$	−27.1206	−1.84107
$218$	11.5080	0.779417
$219$	−7.02232	−0.474525
$220$	−6.16244	−0.415472
$221$	6.87794	0.462660
$222$	4.87118	0.326932
$223$	−8.99029	−0.602034	−0.301017	−	0.953619i	$-0.597326\pi$
$223$	−8.99029	−0.602034	−0.301017	+	0.953619i	$0.597326\pi$
$224$	5.41240	0.361631
$225$	8.00349	0.533566
$226$	16.9909	1.13022
$227$	−13.9699	−0.927212	−0.463606	−	0.886041i	$-0.653445\pi$
$227$	−13.9699	−0.927212	−0.463606	+	0.886041i	$0.653445\pi$
$228$	−0.404566	−0.0267930
$229$	−7.43085	−0.491044	−0.245522	−	0.969391i	$-0.578959\pi$
$229$	−7.43085	−0.491044	−0.245522	+	0.969391i	$0.578959\pi$
$230$	−14.5473	−0.959222
$231$	−10.6484	−0.700611
$232$	6.40211	0.420319
$233$	−29.6762	−1.94415	−0.972075	−	0.234671i	$-0.924599\pi$
$233$	−29.6762	−1.94415	−0.972075	+	0.234671i	$0.924599\pi$
$234$	−8.71364	−0.569629
$235$	−38.7676	−2.52892
$236$	−0.647619	−0.0421564
$237$	1.00000	0.0649570
$238$	3.11794	0.202106
$239$	−20.5697	−1.33054	−0.665272	−	0.746601i	$-0.731684\pi$
$239$	−20.5697	−1.33054	−0.665272	+	0.746601i	$0.731684\pi$
$240$	−11.0130	−0.710888
$241$	−25.5963	−1.64880	−0.824402	−	0.566005i	$-0.808488\pi$
$241$	−25.5963	−1.64880	−0.824402	+	0.566005i	$0.808488\pi$
$242$	−9.78084	−0.628736
$243$	1.00000	0.0641500
$244$	2.90779	0.186152
$245$	−3.40071	−0.217263
$246$	4.70588	0.300036
$247$	7.04501	0.448263
$248$	33.4361	2.12319
$249$	2.00730	0.127207
$250$	−13.7213	−0.867814
$251$	−6.78324	−0.428154	−0.214077	−	0.976817i	$-0.568674\pi$
$251$	−6.78324	−0.428154	−0.214077	+	0.976817i	$0.568674\pi$
$252$	0.972059	0.0612340
$253$	13.7774	0.866180
$254$	2.60813	0.163648
$255$	3.60603	0.225819
$256$	−9.08529	−0.567830
$257$	19.9073	1.24178	0.620890	−	0.783898i	$-0.286771\pi$
$257$	19.9073	1.24178	0.620890	+	0.783898i	$0.286771\pi$
$258$	−7.34587	−0.457334
$259$	9.46279	0.587989
$260$	−9.79613	−0.607530
$261$	2.11000	0.130605
$262$	−15.7646	−0.973940
$263$	17.3086	1.06730	0.533649	−	0.845706i	$-0.320821\pi$
$263$	17.3086	1.06730	0.533649	+	0.845706i	$0.320821\pi$
$264$	13.1280	0.807972
$265$	−6.04517	−0.371352
$266$	3.19368	0.195817
$267$	3.75596	0.229861
$268$	−0.805660	−0.0492135
$269$	−20.7982	−1.26809	−0.634046	−	0.773296i	$-0.718607\pi$
$269$	−20.7982	−1.26809	−0.634046	+	0.773296i	$0.718607\pi$
$270$	−4.56847	−0.278029
$271$	12.1348	0.737136	0.368568	−	0.929601i	$-0.379848\pi$
$271$	12.1348	0.737136	0.368568	+	0.929601i	$0.379848\pi$
$272$	−3.05405	−0.185179
$273$	−16.9272	−1.02448
$274$	8.39600	0.507221
$275$	34.6287	2.08819
$276$	−1.25770	−0.0757048
$277$	26.2300	1.57601	0.788004	−	0.615671i	$-0.211115\pi$
$277$	26.2300	1.57601	0.788004	+	0.615671i	$0.211115\pi$
$278$	−11.3715	−0.682020
$279$	11.0198	0.659738
$280$	−26.9276	−1.60923
$281$	0.603688	0.0360130	0.0180065	−	0.999838i	$-0.494268\pi$
$281$	0.603688	0.0360130	0.0180065	+	0.999838i	$0.494268\pi$
$282$	13.6201	0.811065
$283$	12.5103	0.743659	0.371830	−	0.928301i	$-0.378731\pi$
$283$	12.5103	0.743659	0.371830	+	0.928301i	$0.378731\pi$
$284$	1.55340	0.0921771
$285$	3.69363	0.218792
$286$	−37.7013	−2.22932
$287$	9.14169	0.539617
$288$	−2.19919	−0.129589
$289$	1.00000	0.0588235
$290$	−9.63946	−0.566049
$291$	−0.409835	−0.0240250
$292$	2.77362	0.162314
$293$	−20.1930	−1.17969	−0.589843	−	0.807518i	$-0.700810\pi$
$293$	−20.1930	−1.17969	−0.589843	+	0.807518i	$0.700810\pi$
$294$	1.19476	0.0696798
$295$	5.91266	0.344249
$296$	−11.6663	−0.678092
$297$	4.32670	0.251060
$298$	−2.29867	−0.133158
$299$	21.9013	1.26659
$300$	−3.16115	−0.182509
$301$	−14.2702	−0.822518
$302$	20.7910	1.19639
$303$	−13.6638	−0.784968
$304$	−3.12824	−0.179417
$305$	−26.5477	−1.52012
$306$	−1.26690	−0.0724237
$307$	−1.30351	−0.0743955	−0.0371977	−	0.999308i	$-0.511843\pi$
$307$	−1.30351	−0.0743955	−0.0371977	+	0.999308i	$0.511843\pi$
$308$	4.20581	0.239648
$309$	−12.4933	−0.710718
$310$	−50.3436	−2.85933
$311$	−1.43428	−0.0813306	−0.0406653	−	0.999173i	$-0.512948\pi$
$311$	−1.43428	−0.0813306	−0.0406653	+	0.999173i	$0.512948\pi$
$312$	20.8689	1.18147
$313$	−1.80053	−0.101772	−0.0508860	−	0.998704i	$-0.516205\pi$
$313$	−1.80053	−0.101772	−0.0508860	+	0.998704i	$0.516205\pi$
$314$	−10.5496	−0.595348
$315$	−8.87476	−0.500036
$316$	−0.394972	−0.0222189
$317$	19.0499	1.06995	0.534974	−	0.844869i	$-0.320321\pi$
$317$	19.0499	1.06995	0.534974	+	0.844869i	$0.320321\pi$
$318$	2.12383	0.119099
$319$	9.12932	0.511144
$320$	32.0730	1.79294
$321$	−2.33864	−0.130530
$322$	9.92841	0.553289
$323$	1.02429	0.0569930
$324$	−0.394972	−0.0219429
$325$	55.0475	3.05349
$326$	9.55639	0.529279
$327$	−9.08357	−0.502323
$328$	−11.2705	−0.622307
$329$	26.4585	1.45871
$330$	−19.7664	−1.08811
$331$	−4.54377	−0.249748	−0.124874	−	0.992173i	$-0.539853\pi$
$331$	−4.54377	−0.249748	−0.124874	+	0.992173i	$0.539853\pi$
$332$	−0.792826	−0.0435120
$333$	−3.84497	−0.210703
$334$	0.0356596	0.00195121
$335$	7.35555	0.401877
$336$	7.51629	0.410047
$337$	−3.10873	−0.169343	−0.0846717	−	0.996409i	$-0.526984\pi$
$337$	−3.10873	−0.169343	−0.0846717	+	0.996409i	$0.526984\pi$
$338$	−43.4623	−2.36404
$339$	−13.4114	−0.728409
$340$	−1.42428	−0.0772425
$341$	47.6793	2.58198
$342$	−1.29767	−0.0701700
$343$	19.5485	1.05552
$344$	17.5932	0.948559
$345$	11.4826	0.618205
$346$	−30.1683	−1.62186
$347$	−5.27170	−0.283000	−0.141500	−	0.989938i	$-0.545192\pi$
$347$	−5.27170	−0.283000	−0.141500	+	0.989938i	$0.545192\pi$
$348$	−0.833389	−0.0446743
$349$	34.5823	1.85115	0.925573	−	0.378569i	$-0.123584\pi$
$349$	34.5823	1.85115	0.925573	+	0.378569i	$0.123584\pi$
$350$	24.9544	1.33387
$351$	6.87794	0.367117
$352$	−9.51525	−0.507164
$353$	−20.9297	−1.11398	−0.556989	−	0.830520i	$-0.688043\pi$
$353$	−20.9297	−1.11398	−0.556989	+	0.830520i	$0.688043\pi$
$354$	−2.07728	−0.110406
$355$	−14.1823	−0.752717
$356$	−1.48350	−0.0786254
$357$	−2.46109	−0.130254
$358$	−12.2377	−0.646783
$359$	7.75726	0.409412	0.204706	−	0.978823i	$-0.434376\pi$
$359$	7.75726	0.409412	0.204706	+	0.978823i	$0.434376\pi$
$360$	10.9414	0.576661
$361$	−17.9508	−0.944780
$362$	17.2792	0.908172
$363$	7.72031	0.405211
$364$	6.68577	0.350429
$365$	−25.3227	−1.32545
$366$	9.32693	0.487526
$367$	−20.1465	−1.05164	−0.525819	−	0.850596i	$-0.676241\pi$
$367$	−20.1465	−1.05164	−0.525819	+	0.850596i	$0.676241\pi$
$368$	−9.72498	−0.506949
$369$	−3.71450	−0.193369
$370$	17.5656	0.913193
$371$	4.12578	0.214200
$372$	−4.35251	−0.225667
$373$	−18.3044	−0.947768	−0.473884	−	0.880587i	$-0.657148\pi$
$373$	−18.3044	−0.947768	−0.473884	+	0.880587i	$0.657148\pi$
$374$	−5.48148	−0.283441
$375$	10.8307	0.559294
$376$	−32.6197	−1.68223
$377$	14.5124	0.747428
$378$	3.11794	0.160370
$379$	−6.98642	−0.358868	−0.179434	−	0.983770i	$-0.557427\pi$
$379$	−6.98642	−0.358868	−0.179434	+	0.983770i	$0.557427\pi$
$380$	−1.45888	−0.0748389
$381$	−2.05867	−0.105469
$382$	−8.36777	−0.428132
$383$	36.0036	1.83970	0.919849	−	0.392272i	$-0.128311\pi$
$383$	36.0036	1.83970	0.919849	+	0.392272i	$0.128311\pi$
$384$	−6.86972	−0.350569
$385$	−38.3984	−1.95696
$386$	5.55939	0.282965
$387$	5.79832	0.294745
$388$	0.161873	0.00821787
$389$	−17.4498	−0.884742	−0.442371	−	0.896832i	$-0.645863\pi$
$389$	−17.4498	−0.884742	−0.442371	+	0.896832i	$0.645863\pi$
$390$	−31.4217	−1.59110
$391$	3.18429	0.161036
$392$	−2.86142	−0.144523
$393$	12.4435	0.627690
$394$	5.76872	0.290624
$395$	3.60603	0.181439
$396$	−1.70892	−0.0858767
$397$	12.3394	0.619296	0.309648	−	0.950851i	$-0.399789\pi$
$397$	12.3394	0.619296	0.309648	+	0.950851i	$0.399789\pi$
$398$	25.5055	1.27847
$399$	−2.52087	−0.126201
$400$	−24.4431	−1.22215
$401$	−3.69472	−0.184505	−0.0922527	−	0.995736i	$-0.529407\pi$
$401$	−3.69472	−0.184505	−0.0922527	+	0.995736i	$0.529407\pi$
$402$	−2.58420	−0.128888
$403$	75.7935	3.77554
$404$	5.39684	0.268503
$405$	3.60603	0.179185
$406$	6.57884	0.326503
$407$	−16.6360	−0.824616
$408$	3.03418	0.150214
$409$	−27.6938	−1.36937	−0.684686	−	0.728838i	$-0.740061\pi$
$409$	−27.6938	−1.36937	−0.684686	+	0.728838i	$0.740061\pi$
$410$	16.9696	0.838068
$411$	−6.62722	−0.326896
$412$	4.93449	0.243105
$413$	−4.03534	−0.198566
$414$	−4.03416	−0.198268
$415$	7.23839	0.355318
$416$	−15.1259	−0.741610
$417$	8.97590	0.439552
$418$	−5.61463	−0.274620
$419$	27.1998	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$419$	27.1998	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$420$	3.50528	0.171040
$421$	18.4806	0.900687	0.450344	−	0.892855i	$-0.351301\pi$
$421$	18.4806	0.900687	0.450344	+	0.892855i	$0.351301\pi$
$422$	−29.6641	−1.44403
$423$	−10.7508	−0.522719
$424$	−5.08652	−0.247023
$425$	8.00349	0.388226
$426$	4.98262	0.241409
$427$	18.1186	0.876819
$428$	0.923699	0.0446487
$429$	29.7588	1.43677
$430$	−26.4895	−1.27744
$431$	32.6876	1.57451	0.787254	−	0.616629i	$-0.211502\pi$
$431$	32.6876	1.57451	0.787254	+	0.616629i	$0.211502\pi$
$432$	−3.05405	−0.146938
$433$	23.9317	1.15008	0.575041	−	0.818125i	$-0.304986\pi$
$433$	23.9317	1.15008	0.575041	+	0.818125i	$0.304986\pi$
$434$	34.3591	1.64929
$435$	7.60872	0.364810
$436$	3.58776	0.171822
$437$	3.26163	0.156025
$438$	8.89656	0.425094
$439$	36.2900	1.73203	0.866013	−	0.500022i	$-0.166675\pi$
$439$	36.2900	1.73203	0.866013	+	0.500022i	$0.166675\pi$
$440$	47.3400	2.25685
$441$	−0.943060	−0.0449076
$442$	−8.71364	−0.414466
$443$	1.85387	0.0880798	0.0440399	−	0.999030i	$-0.485977\pi$
$443$	1.85387	0.0880798	0.0440399	+	0.999030i	$0.485977\pi$
$444$	1.51865	0.0720721
$445$	13.5441	0.642054
$446$	11.3898	0.539321
$447$	1.81441	0.0858185
$448$	−21.8895	−1.03418
$449$	14.0063	0.660996	0.330498	−	0.943807i	$-0.392783\pi$
$449$	14.0063	0.660996	0.330498	+	0.943807i	$0.392783\pi$
$450$	−10.1396	−0.477985
$451$	−16.0715	−0.756778
$452$	5.29714	0.249156
$453$	−16.4110	−0.771055
$454$	17.6984	0.830626
$455$	−61.0401	−2.86160
$456$	3.10788	0.145540
$457$	−5.32504	−0.249095	−0.124548	−	0.992214i	$-0.539748\pi$
$457$	−5.32504	−0.249095	−0.124548	+	0.992214i	$0.539748\pi$
$458$	9.41412	0.439893
$459$	1.00000	0.0466760
$460$	−4.53532	−0.211460
$461$	13.0886	0.609595	0.304798	−	0.952417i	$-0.401411\pi$
$461$	13.0886	0.609595	0.304798	+	0.952417i	$0.401411\pi$
$462$	13.4904	0.627630
$463$	4.15712	0.193198	0.0965988	−	0.995323i	$-0.469204\pi$
$463$	4.15712	0.193198	0.0965988	+	0.995323i	$0.469204\pi$
$464$	−6.44404	−0.299157
$465$	39.7378	1.84279
$466$	37.5966	1.74163
$467$	−13.7026	−0.634081	−0.317040	−	0.948412i	$-0.602689\pi$
$467$	−13.7026	−0.634081	−0.317040	+	0.948412i	$0.602689\pi$
$468$	−2.71659	−0.125575
$469$	−5.02010	−0.231807
$470$	49.1145	2.26548
$471$	8.32711	0.383693
$472$	4.97502	0.228994
$473$	25.0876	1.15353
$474$	−1.26690	−0.0581905
$475$	8.19789	0.376145
$476$	0.972059	0.0445543
$477$	−1.67641	−0.0767573
$478$	26.0597	1.19194
$479$	11.6636	0.532923	0.266462	−	0.963846i	$-0.414145\pi$
$479$	11.6636	0.532923	0.266462	+	0.963846i	$0.414145\pi$
$480$	−7.93037	−0.361970
$481$	−26.4454	−1.20581
$482$	32.4279	1.47705
$483$	−7.83680	−0.356587
$484$	−3.04931	−0.138605
$485$	−1.47788	−0.0671071
$486$	−1.26690	−0.0574676
$487$	33.5884	1.52203	0.761017	−	0.648732i	$-0.224700\pi$
$487$	33.5884	1.52203	0.761017	+	0.648732i	$0.224700\pi$
$488$	−22.3377	−1.01118
$489$	−7.54315	−0.341113
$490$	4.30835	0.194631
$491$	−11.3079	−0.510318	−0.255159	−	0.966899i	$-0.582128\pi$
$491$	−11.3079	−0.510318	−0.255159	+	0.966899i	$0.582128\pi$
$492$	1.46712	0.0661430
$493$	2.11000	0.0950294
$494$	−8.92530	−0.401568
$495$	15.6022	0.701268
$496$	−33.6550	−1.51116
$497$	9.67928	0.434175
$498$	−2.54304	−0.113956
$499$	−23.0773	−1.03308	−0.516541	−	0.856262i	$-0.672781\pi$
$499$	−23.0773	−1.03308	−0.516541	+	0.856262i	$0.672781\pi$
$500$	−4.27781	−0.191310
$501$	−0.0281472	−0.00125752
$502$	8.59367	0.383554
$503$	36.0050	1.60538	0.802692	−	0.596393i	$-0.203400\pi$
$503$	36.0050	1.60538	0.802692	+	0.596393i	$0.203400\pi$
$504$	−7.46738	−0.332624
$505$	−49.2723	−2.19259
$506$	−17.4546	−0.775952
$507$	34.3061	1.52359
$508$	0.813118	0.0360763
$509$	39.6681	1.75826	0.879129	−	0.476583i	$-0.158125\pi$
$509$	39.6681	1.75826	0.879129	+	0.476583i	$0.158125\pi$
$510$	−4.56847	−0.202295
$511$	17.2825	0.764535
$512$	25.2496	1.11588
$513$	1.02429	0.0452235
$514$	−25.2204	−1.11243
$515$	−45.0512	−1.98519
$516$	−2.29017	−0.100819
$517$	−46.5152	−2.04574
$518$	−11.9884	−0.526739
$519$	23.8128	1.04526
$520$	75.2541	3.30011
$521$	28.9891	1.27003	0.635017	−	0.772498i	$-0.280993\pi$
$521$	28.9891	1.27003	0.635017	+	0.772498i	$0.280993\pi$
$522$	−2.67315	−0.117001
$523$	−15.3834	−0.672669	−0.336334	−	0.941743i	$-0.609187\pi$
$523$	−15.3834	−0.672669	−0.336334	+	0.941743i	$0.609187\pi$
$524$	−4.91482	−0.214705
$525$	−19.6973	−0.859659
$526$	−21.9283	−0.956118
$527$	11.0198	0.480030
$528$	−13.2140	−0.575064
$529$	−12.8603	−0.559145
$530$	7.65861	0.332669
$531$	1.63966	0.0711551
$532$	0.995671	0.0431678
$533$	−25.5481	−1.10661
$534$	−4.75842	−0.205917
$535$	−8.43323	−0.364601
$536$	6.18910	0.267328
$537$	9.65959	0.416842
$538$	26.3492	1.13600
$539$	−4.08034	−0.175752
$540$	−1.42428	−0.0612913
$541$	−11.9733	−0.514774	−0.257387	−	0.966308i	$-0.582861\pi$
$541$	−11.9733	−0.514774	−0.257387	+	0.966308i	$0.582861\pi$
$542$	−15.3735	−0.660350
$543$	−13.6390	−0.585304
$544$	−2.19919	−0.0942897
$545$	−32.7557	−1.40310
$546$	21.4450	0.917762
$547$	−9.94841	−0.425363	−0.212682	−	0.977122i	$-0.568220\pi$
$547$	−9.94841	−0.425363	−0.212682	+	0.977122i	$0.568220\pi$
$548$	2.61756	0.111817
$549$	−7.36202	−0.314204
$550$	−43.8710	−1.87066
$551$	2.16125	0.0920723
$552$	9.66170	0.411229
$553$	−2.46109	−0.104656
$554$	−33.2307	−1.41184
$555$	−13.8651	−0.588540
$556$	−3.54523	−0.150351
$557$	−16.1566	−0.684575	−0.342288	−	0.939595i	$-0.611202\pi$
$557$	−16.1566	−0.684575	−0.342288	+	0.939595i	$0.611202\pi$
$558$	−13.9609	−0.591014
$559$	39.8805	1.68676
$560$	27.1040	1.14535
$561$	4.32670	0.182673
$562$	−0.764810	−0.0322616
$563$	41.9107	1.76633	0.883163	−	0.469065i	$-0.155409\pi$
$563$	41.9107	1.76633	0.883163	+	0.469065i	$0.155409\pi$
$564$	4.24624	0.178799
$565$	−48.3621	−2.03461
$566$	−15.8492	−0.666193
$567$	−2.46109	−0.103356
$568$	−11.9332	−0.500707
$569$	−14.2557	−0.597630	−0.298815	−	0.954311i	$-0.596591\pi$
$569$	−14.2557	−0.597630	−0.298815	+	0.954311i	$0.596591\pi$
$570$	−4.67944	−0.196000
$571$	27.0724	1.13294	0.566472	−	0.824081i	$-0.308308\pi$
$571$	27.0724	1.13294	0.566472	+	0.824081i	$0.308308\pi$
$572$	−11.7539	−0.491454
$573$	6.60493	0.275925
$574$	−11.5816	−0.483406
$575$	25.4854	1.06281
$576$	8.89426	0.370594
$577$	−32.2946	−1.34444	−0.672221	−	0.740351i	$-0.734660\pi$
$577$	−32.2946	−1.34444	−0.672221	+	0.740351i	$0.734660\pi$
$578$	−1.26690	−0.0526960
$579$	−4.38819	−0.182367
$580$	−3.00523	−0.124785
$581$	−4.94013	−0.204951
$582$	0.519219	0.0215223
$583$	−7.25330	−0.300401
$584$	−21.3070	−0.881691
$585$	24.8021	1.02544
$586$	25.5824	1.05680
$587$	12.9438	0.534248	0.267124	−	0.963662i	$-0.413927\pi$
$587$	12.9438	0.534248	0.267124	+	0.963662i	$0.413927\pi$
$588$	0.372482	0.0153609
$589$	11.2875	0.465092
$590$	−7.49074	−0.308389
$591$	−4.55343	−0.187303
$592$	11.7427	0.482623
$593$	−26.5076	−1.08854	−0.544268	−	0.838911i	$-0.683193\pi$
$593$	−26.5076	−1.08854	−0.544268	+	0.838911i	$0.683193\pi$
$594$	−5.48148	−0.224908
$595$	−8.87476	−0.363830
$596$	−0.716640	−0.0293547
$597$	−20.1322	−0.823957
$598$	−27.7467	−1.13465
$599$	−22.6635	−0.926004	−0.463002	−	0.886357i	$-0.653228\pi$
$599$	−22.6635	−0.926004	−0.463002	+	0.886357i	$0.653228\pi$
$600$	24.2840	0.991392
$601$	19.2843	0.786623	0.393312	−	0.919405i	$-0.371329\pi$
$601$	19.2843	0.786623	0.393312	+	0.919405i	$0.371329\pi$
$602$	18.0788	0.736837
$603$	2.03979	0.0830667
$604$	6.48187	0.263744
$605$	27.8397	1.13185
$606$	17.3107	0.703199
$607$	−33.4003	−1.35568	−0.677839	−	0.735211i	$-0.737083\pi$
$607$	−33.4003	−1.35568	−0.677839	+	0.735211i	$0.737083\pi$
$608$	−2.25261	−0.0913555
$609$	−5.19288	−0.210426
$610$	33.6332	1.36177
$611$	−73.9430	−2.99141
$612$	−0.394972	−0.0159658
$613$	28.2913	1.14267	0.571337	−	0.820716i	$-0.306425\pi$
$613$	28.2913	1.14267	0.571337	+	0.820716i	$0.306425\pi$
$614$	1.65142	0.0666458
$615$	−13.3946	−0.540123
$616$	−32.3091	−1.30177
$617$	41.5676	1.67345	0.836725	−	0.547624i	$-0.184468\pi$
$617$	41.5676	1.67345	0.836725	+	0.547624i	$0.184468\pi$
$618$	15.8277	0.636683
$619$	45.5246	1.82979	0.914893	−	0.403696i	$-0.132275\pi$
$619$	45.5246	1.82979	0.914893	+	0.403696i	$0.132275\pi$
$620$	−15.6953	−0.630338
$621$	3.18429	0.127781
$622$	1.81709	0.0728585
$623$	−9.24375	−0.370343
$624$	−21.0056	−0.840897
$625$	−0.961643	−0.0384657
$626$	2.28109	0.0911705
$627$	4.43180	0.176989
$628$	−3.28897	−0.131244
$629$	−3.84497	−0.153309
$630$	11.2434	0.447948
$631$	−26.0677	−1.03774	−0.518868	−	0.854854i	$-0.673647\pi$
$631$	−26.0677	−1.03774	−0.518868	+	0.854854i	$0.673647\pi$
$632$	3.03418	0.120693
$633$	23.4148	0.930654
$634$	−24.1342	−0.958493
$635$	−7.42364	−0.294598
$636$	0.662133	0.0262553
$637$	−6.48631	−0.256997
$638$	−11.5659	−0.457898
$639$	−3.93293	−0.155584
$640$	−24.7725	−0.979217
$641$	13.6719	0.540009	0.270004	−	0.962859i	$-0.412975\pi$
$641$	13.6719	0.540009	0.270004	+	0.962859i	$0.412975\pi$
$642$	2.96282	0.116933
$643$	−24.6783	−0.973218	−0.486609	−	0.873620i	$-0.661766\pi$
$643$	−24.6783	−0.973218	−0.486609	+	0.873620i	$0.661766\pi$
$644$	3.09531	0.121972
$645$	20.9089	0.823288
$646$	−1.29767	−0.0510562
$647$	−2.12641	−0.0835979	−0.0417989	−	0.999126i	$-0.513309\pi$
$647$	−2.12641	−0.0835979	−0.0417989	+	0.999126i	$0.513309\pi$
$648$	3.03418	0.119194
$649$	7.09431	0.278476
$650$	−69.7395	−2.73541
$651$	−27.1206	−1.06294
$652$	2.97933	0.116680
$653$	−11.3322	−0.443463	−0.221732	−	0.975108i	$-0.571171\pi$
$653$	−11.3322	−0.443463	−0.221732	+	0.975108i	$0.571171\pi$
$654$	11.5080	0.449997
$655$	44.8716	1.75328
$656$	11.3443	0.442919
$657$	−7.02232	−0.273967
$658$	−33.5202	−1.30675
$659$	35.2566	1.37340	0.686701	−	0.726940i	$-0.259058\pi$
$659$	35.2566	1.37340	0.686701	+	0.726940i	$0.259058\pi$
$660$	−6.16244	−0.239873
$661$	−17.7745	−0.691349	−0.345674	−	0.938355i	$-0.612350\pi$
$661$	−17.7745	−0.691349	−0.345674	+	0.938355i	$0.612350\pi$
$662$	5.75649	0.223732
$663$	6.87794	0.267117
$664$	6.09051	0.236358
$665$	−9.09033	−0.352508
$666$	4.87118	0.188754
$667$	6.71883	0.260154
$668$	0.0111174	0.000430143 0
$669$	−8.99029	−0.347585
$670$	−9.31873	−0.360014
$671$	−31.8533	−1.22968
$672$	5.41240	0.208788
$673$	−22.9469	−0.884538	−0.442269	−	0.896883i	$-0.645826\pi$
$673$	−22.9469	−0.884538	−0.442269	+	0.896883i	$0.645826\pi$
$674$	3.93844	0.151703
$675$	8.00349	0.308054
$676$	−13.5499	−0.521151
$677$	−24.0602	−0.924710	−0.462355	−	0.886695i	$-0.652995\pi$
$677$	−24.0602	−0.924710	−0.462355	+	0.886695i	$0.652995\pi$
$678$	16.9909	0.652531
$679$	1.00864	0.0387080
$680$	10.9414	0.419582
$681$	−13.9699	−0.535326
$682$	−60.4048	−2.31302
$683$	−13.5985	−0.520334	−0.260167	−	0.965564i	$-0.583778\pi$
$683$	−13.5985	−0.520334	−0.260167	+	0.965564i	$0.583778\pi$
$684$	−0.404566	−0.0154690
$685$	−23.8980	−0.913094
$686$	−24.7660	−0.945570
$687$	−7.43085	−0.283504
$688$	−17.7084	−0.675125
$689$	−11.5302	−0.439266
$690$	−14.5473	−0.553807
$691$	−33.1769	−1.26211	−0.631055	−	0.775738i	$-0.717378\pi$
$691$	−33.1769	−1.26211	−0.631055	+	0.775738i	$0.717378\pi$
$692$	−9.40537	−0.357539
$693$	−10.6484	−0.404498
$694$	6.67870	0.253520
$695$	32.3674	1.22777
$696$	6.40211	0.242671
$697$	−3.71450	−0.140697
$698$	−43.8122	−1.65832
$699$	−29.6762	−1.12246
$700$	7.77986	0.294051
$701$	25.0473	0.946022	0.473011	−	0.881056i	$-0.343167\pi$
$701$	25.0473	0.946022	0.473011	+	0.881056i	$0.343167\pi$
$702$	−8.71364	−0.328875
$703$	−3.93836	−0.148538
$704$	38.4828	1.45037
$705$	−38.7676	−1.46007
$706$	26.5158	0.997936
$707$	33.6279	1.26471
$708$	−0.647619	−0.0243390
$709$	19.7256	0.740809	0.370404	−	0.928871i	$-0.379219\pi$
$709$	19.7256	0.740809	0.370404	+	0.928871i	$0.379219\pi$
$710$	17.9675	0.674308
$711$	1.00000	0.0375029
$712$	11.3963	0.427094
$713$	35.0902	1.31414
$714$	3.11794	0.116686
$715$	107.311	4.01321
$716$	−3.81527	−0.142583
$717$	−20.5697	−0.768190
$718$	−9.82765	−0.366764
$719$	−43.5503	−1.62415	−0.812076	−	0.583551i	$-0.801663\pi$
$719$	−43.5503	−1.62415	−0.812076	+	0.583551i	$0.801663\pi$
$720$	−11.0130	−0.410431
$721$	30.7470	1.14508
$722$	22.7419	0.846364
$723$	−25.5963	−0.951937
$724$	5.38700	0.200206
$725$	16.8873	0.627179
$726$	−9.78084	−0.363001
$727$	−25.7710	−0.955792	−0.477896	−	0.878416i	$-0.658600\pi$
$727$	−25.7710	−0.955792	−0.477896	+	0.878416i	$0.658600\pi$
$728$	−51.3602	−1.90354
$729$	1.00000	0.0370370
$730$	32.0813	1.18738
$731$	5.79832	0.214458
$732$	2.90779	0.107475
$733$	−18.6401	−0.688488	−0.344244	−	0.938880i	$-0.611865\pi$
$733$	−18.6401	−0.688488	−0.344244	+	0.938880i	$0.611865\pi$
$734$	25.5235	0.942091
$735$	−3.40071	−0.125437
$736$	−7.00286	−0.258129
$737$	8.82556	0.325094
$738$	4.70588	0.173226
$739$	−7.27703	−0.267690	−0.133845	−	0.991002i	$-0.542732\pi$
$739$	−7.27703	−0.267690	−0.133845	+	0.991002i	$0.542732\pi$
$740$	5.47632	0.201313
$741$	7.04501	0.258805
$742$	−5.22693	−0.191887
$743$	20.8378	0.764464	0.382232	−	0.924066i	$-0.375156\pi$
$743$	20.8378	0.764464	0.382232	+	0.924066i	$0.375156\pi$
$744$	33.4361	1.22583
$745$	6.54281	0.239710
$746$	23.1899	0.849041
$747$	2.00730	0.0734432
$748$	−1.70892	−0.0624844
$749$	5.75560	0.210305
$750$	−13.7213	−0.501033
$751$	−52.3594	−1.91062	−0.955311	−	0.295603i	$-0.904480\pi$
$751$	−52.3594	−1.91062	−0.955311	+	0.295603i	$0.904480\pi$
$752$	32.8334	1.19731
$753$	−6.78324	−0.247195
$754$	−18.3858	−0.669570
$755$	−59.1785	−2.15373
$756$	0.972059	0.0353535
$757$	−30.3943	−1.10470	−0.552350	−	0.833612i	$-0.686269\pi$
$757$	−30.3943	−1.10470	−0.552350	+	0.833612i	$0.686269\pi$
$758$	8.85107	0.321486
$759$	13.7774	0.500089
$760$	11.2071	0.406526
$761$	−37.2863	−1.35163	−0.675814	−	0.737073i	$-0.736208\pi$
$761$	−37.2863	−1.35163	−0.675814	+	0.737073i	$0.736208\pi$
$762$	2.60813	0.0944824
$763$	22.3554	0.809322
$764$	−2.60876	−0.0943817
$765$	3.60603	0.130376
$766$	−45.6129	−1.64806
$767$	11.2775	0.407206
$768$	−9.08529	−0.327837
$769$	7.59774	0.273981	0.136991	−	0.990572i	$-0.456257\pi$
$769$	7.59774	0.273981	0.136991	+	0.990572i	$0.456257\pi$
$770$	48.6468	1.75311
$771$	19.9073	0.716942
$772$	1.73321	0.0623797
$773$	5.98933	0.215421	0.107711	−	0.994182i	$-0.465648\pi$
$773$	5.98933	0.215421	0.107711	+	0.994182i	$0.465648\pi$
$774$	−7.34587	−0.264042
$775$	88.1968	3.16812
$776$	−1.24351	−0.0446396
$777$	9.46279	0.339476
$778$	22.1071	0.792580
$779$	−3.80472	−0.136318
$780$	−9.79613	−0.350758
$781$	−17.0166	−0.608902
$782$	−4.03416	−0.144261
$783$	2.11000	0.0754051
$784$	2.88016	0.102863
$785$	30.0279	1.07174
$786$	−15.7646	−0.562305
$787$	−9.32781	−0.332501	−0.166250	−	0.986084i	$-0.553166\pi$
$787$	−9.32781	−0.332501	−0.166250	+	0.986084i	$0.553166\pi$
$788$	1.79848	0.0640680
$789$	17.3086	0.616204
$790$	−4.56847	−0.162539
$791$	33.0067	1.17358
$792$	13.1280	0.466483
$793$	−50.6356	−1.79812
$794$	−15.6327	−0.554785
$795$	−6.04517	−0.214400
$796$	7.95166	0.281839
$797$	−2.97478	−0.105372	−0.0526861	−	0.998611i	$-0.516778\pi$
$797$	−2.97478	−0.105372	−0.0526861	+	0.998611i	$0.516778\pi$
$798$	3.19368	0.113055
$799$	−10.7508	−0.380334
$800$	−17.6012	−0.622297
$801$	3.75596	0.132710
$802$	4.68083	0.165286
$803$	−30.3835	−1.07221
$804$	−0.805660	−0.0284134
$805$	−28.2598	−0.996026
$806$	−96.0225	−3.38225
$807$	−20.7982	−0.732133
$808$	−41.4586	−1.45851
$809$	26.3095	0.924991	0.462496	−	0.886622i	$-0.346954\pi$
$809$	26.3095	0.924991	0.462496	+	0.886622i	$0.346954\pi$
$810$	−4.56847	−0.160520
$811$	33.1979	1.16574	0.582868	−	0.812567i	$-0.301930\pi$
$811$	33.1979	1.16574	0.582868	+	0.812567i	$0.301930\pi$
$812$	2.05104	0.0719774
$813$	12.1348	0.425586
$814$	21.0761	0.738717
$815$	−27.2008	−0.952804
$816$	−3.05405	−0.106913
$817$	5.93916	0.207785
$818$	35.0852	1.22673
$819$	−16.9272	−0.591484
$820$	5.29049	0.184752
$821$	−3.06087	−0.106825	−0.0534126	−	0.998573i	$-0.517010\pi$
$821$	−3.06087	−0.106825	−0.0534126	+	0.998573i	$0.517010\pi$
$822$	8.39600	0.292844
$823$	47.0026	1.63841	0.819204	−	0.573502i	$-0.194416\pi$
$823$	47.0026	1.63841	0.819204	+	0.573502i	$0.194416\pi$
$824$	−37.9069	−1.32055
$825$	34.6287	1.20562
$826$	5.11236	0.177882
$827$	−54.9477	−1.91072	−0.955360	−	0.295445i	$-0.904532\pi$
$827$	−54.9477	−1.91072	−0.955360	+	0.295445i	$0.904532\pi$
$828$	−1.25770	−0.0437082
$829$	−11.7361	−0.407612	−0.203806	−	0.979011i	$-0.565331\pi$
$829$	−11.7361	−0.407612	−0.203806	+	0.979011i	$0.565331\pi$
$830$	−9.17029	−0.318305
$831$	26.2300	0.909908
$832$	61.1742	2.12083
$833$	−0.943060	−0.0326751
$834$	−11.3715	−0.393764
$835$	−0.101500	−0.00351254
$836$	−1.75043	−0.0605400
$837$	11.0198	0.380900
$838$	−34.4593	−1.19038
$839$	−32.6383	−1.12680	−0.563399	−	0.826185i	$-0.690507\pi$
$839$	−32.6383	−1.12680	−0.563399	+	0.826185i	$0.690507\pi$
$840$	−26.9276	−0.929092
$841$	−24.5479	−0.846480
$842$	−23.4130	−0.806864
$843$	0.603688	0.0207921
$844$	−9.24818	−0.318335
$845$	123.709	4.25572
$846$	13.6201	0.468268
$847$	−19.0004	−0.652860
$848$	5.11983	0.175816
$849$	12.5103	0.429352
$850$	−10.1396	−0.347785
$851$	−12.2435	−0.419701
$852$	1.55340	0.0532185
$853$	−41.1697	−1.40962	−0.704812	−	0.709395i	$-0.748968\pi$
$853$	−41.1697	−1.40962	−0.704812	+	0.709395i	$0.748968\pi$
$854$	−22.9544	−0.785482
$855$	3.69363	0.126319
$856$	−7.09588	−0.242532
$857$	−37.4206	−1.27826	−0.639132	−	0.769097i	$-0.720706\pi$
$857$	−37.4206	−1.27826	−0.639132	+	0.769097i	$0.720706\pi$
$858$	−37.7013	−1.28710
$859$	52.4913	1.79098	0.895490	−	0.445081i	$-0.146825\pi$
$859$	52.4913	1.79098	0.895490	+	0.445081i	$0.146825\pi$
$860$	−8.25844	−0.281610
$861$	9.14169	0.311548
$862$	−41.4119	−1.41049
$863$	−32.3238	−1.10032	−0.550158	−	0.835061i	$-0.685432\pi$
$863$	−32.3238	−1.10032	−0.550158	+	0.835061i	$0.685432\pi$
$864$	−2.19919	−0.0748181
$865$	85.8696	2.91966
$866$	−30.3189	−1.03028
$867$	1.00000	0.0339618
$868$	10.7119	0.363585
$869$	4.32670	0.146773
$870$	−9.63946	−0.326808
$871$	14.0296	0.475373
$872$	−27.5612	−0.933341
$873$	−0.409835	−0.0138708
$874$	−4.13215	−0.139772
$875$	−26.6552	−0.901111
$876$	2.77362	0.0937120
$877$	−44.3456	−1.49744	−0.748722	−	0.662884i	$-0.769332\pi$
$877$	−44.3456	−1.49744	−0.748722	+	0.662884i	$0.769332\pi$
$878$	−45.9757	−1.55160
$879$	−20.1930	−0.681092
$880$	−47.6500	−1.60628
$881$	21.0812	0.710244	0.355122	−	0.934820i	$-0.384439\pi$
$881$	21.0812	0.710244	0.355122	+	0.934820i	$0.384439\pi$
$882$	1.19476	0.0402297
$883$	14.3239	0.482036	0.241018	−	0.970521i	$-0.422519\pi$
$883$	14.3239	0.482036	0.241018	+	0.970521i	$0.422519\pi$
$884$	−2.71659	−0.0913689
$885$	5.91266	0.198752
$886$	−2.34866	−0.0789047
$887$	−13.5909	−0.456336	−0.228168	−	0.973622i	$-0.573274\pi$
$887$	−13.5909	−0.456336	−0.228168	+	0.973622i	$0.573274\pi$
$888$	−11.6663	−0.391496
$889$	5.06657	0.169927
$890$	−17.1590	−0.575172
$891$	4.32670	0.144950
$892$	3.55091	0.118893
$893$	−11.0119	−0.368499
$894$	−2.29867	−0.0768789
$895$	34.8328	1.16433
$896$	16.9070	0.564822
$897$	21.9013	0.731264
$898$	−17.7445	−0.592141
$899$	23.2517	0.775488
$900$	−3.16115	−0.105372
$901$	−1.67641	−0.0558491
$902$	20.3609	0.677945
$903$	−14.2702	−0.474881
$904$	−40.6927	−1.35342
$905$	−49.1826	−1.63488
$906$	20.7910	0.690735
$907$	24.5040	0.813641	0.406821	−	0.913508i	$-0.366637\pi$
$907$	24.5040	0.813641	0.406821	+	0.913508i	$0.366637\pi$
$908$	5.51770	0.183111
$909$	−13.6638	−0.453201
$910$	77.3315	2.56351
$911$	−49.6960	−1.64650	−0.823250	−	0.567678i	$-0.807842\pi$
$911$	−49.6960	−1.64650	−0.823250	+	0.567678i	$0.807842\pi$
$912$	−3.12824	−0.103586
$913$	8.68497	0.287431
$914$	6.74628	0.223147
$915$	−26.5477	−0.877640
$916$	2.93497	0.0969743
$917$	−30.6245	−1.01131
$918$	−1.26690	−0.0418138
$919$	59.2114	1.95321	0.976603	−	0.215051i	$-0.0689919\pi$
$919$	59.2114	1.95321	0.976603	+	0.215051i	$0.0689919\pi$
$920$	34.8404	1.14866
$921$	−1.30351	−0.0429522
$922$	−16.5819	−0.546094
$923$	−27.0505	−0.890377
$924$	4.20581	0.138361
$925$	−30.7731	−1.01181
$926$	−5.26664	−0.173072
$927$	−12.4933	−0.410333
$928$	−4.64029	−0.152325
$929$	−32.4081	−1.06327	−0.531637	−	0.846972i	$-0.678423\pi$
$929$	−32.4081	−1.06327	−0.531637	+	0.846972i	$0.678423\pi$
$930$	−50.3436	−1.65083
$931$	−0.965967	−0.0316583
$932$	11.7212	0.383942
$933$	−1.43428	−0.0469562
$934$	17.3598	0.568030
$935$	15.6022	0.510247
$936$	20.8689	0.682122
$937$	12.6115	0.411999	0.205999	−	0.978552i	$-0.433956\pi$
$937$	12.6115	0.411999	0.205999	+	0.978552i	$0.433956\pi$
$938$	6.35995	0.207660
$939$	−1.80053	−0.0587581
$940$	15.3121	0.499426
$941$	−4.84264	−0.157865	−0.0789327	−	0.996880i	$-0.525151\pi$
$941$	−4.84264	−0.157865	−0.0789327	+	0.996880i	$0.525151\pi$
$942$	−10.5496	−0.343724
$943$	−11.8280	−0.385173
$944$	−5.00760	−0.162984
$945$	−8.87476	−0.288696
$946$	−31.7834	−1.03337
$947$	−53.6510	−1.74342	−0.871711	−	0.490021i	$-0.836989\pi$
$947$	−53.6510	−1.74342	−0.871711	+	0.490021i	$0.836989\pi$
$948$	−0.394972	−0.0128281
$949$	−48.2991	−1.56786
$950$	−10.3859	−0.336963
$951$	19.0499	0.617735
$952$	−7.46738	−0.242019
$953$	23.2884	0.754386	0.377193	−	0.926135i	$-0.376889\pi$
$953$	23.2884	0.754386	0.377193	+	0.926135i	$0.376889\pi$
$954$	2.12383	0.0687616
$955$	23.8176	0.770720
$956$	8.12446	0.262764
$957$	9.12932	0.295109
$958$	−14.7766	−0.477409
$959$	16.3101	0.526682
$960$	32.0730	1.03515
$961$	90.4358	2.91728
$962$	33.5037	1.08020
$963$	−2.33864	−0.0753618
$964$	10.1098	0.325616
$965$	−15.8240	−0.509392
$966$	9.92841	0.319442
$967$	21.8121	0.701429	0.350714	−	0.936482i	$-0.385939\pi$
$967$	21.8121	0.701429	0.350714	+	0.936482i	$0.385939\pi$
$968$	23.4248	0.752903
$969$	1.02429	0.0329049
$970$	1.87232	0.0601166
$971$	−13.3978	−0.429954	−0.214977	−	0.976619i	$-0.568968\pi$
$971$	−13.3978	−0.429954	−0.214977	+	0.976619i	$0.568968\pi$
$972$	−0.394972	−0.0126687
$973$	−22.0905	−0.708187
$974$	−42.5530	−1.36349
$975$	55.0475	1.76293
$976$	22.4840	0.719696
$977$	34.2518	1.09581	0.547906	−	0.836540i	$-0.315425\pi$
$977$	34.2518	1.09581	0.547906	+	0.836540i	$0.315425\pi$
$978$	9.55639	0.305580
$979$	16.2509	0.519382
$980$	1.34318	0.0429064
$981$	−9.08357	−0.290016
$982$	14.3259	0.457159
$983$	48.6071	1.55032	0.775162	−	0.631762i	$-0.217668\pi$
$983$	48.6071	1.55032	0.775162	+	0.631762i	$0.217668\pi$
$984$	−11.2705	−0.359289
$985$	−16.4198	−0.523179
$986$	−2.67315	−0.0851304
$987$	26.4585	0.842184
$988$	−2.78258	−0.0885257
$989$	18.4635	0.587105
$990$	−19.7664	−0.628218
$991$	−18.3700	−0.583542	−0.291771	−	0.956488i	$-0.594244\pi$
$991$	−18.3700	−0.583542	−0.291771	+	0.956488i	$0.594244\pi$
$992$	−24.2347	−0.769451
$993$	−4.54377	−0.144192
$994$	−12.2626	−0.388948
$995$	−72.5975	−2.30150
$996$	−0.792826	−0.0251217
$997$	−40.3551	−1.27806	−0.639029	−	0.769183i	$-0.720664\pi$
$997$	−40.3551	−1.27806	−0.639029	+	0.769183i	$0.720664\pi$
$998$	29.2366	0.925468
$999$	−3.84497	−0.121649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.10	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.10	✓	31	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26690 q^{2} +1.00000 q^{3} -0.394972 q^{4} +3.60603 q^{5} -1.26690 q^{6} -2.46109 q^{7} +3.03418 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.26690 q^{2} +1.00000 q^{3} -0.394972 q^{4} +3.60603 q^{5} -1.26690 q^{6} -2.46109 q^{7} +3.03418 q^{8} +1.00000 q^{9} -4.56847 q^{10} +4.32670 q^{11} -0.394972 q^{12} +6.87794 q^{13} +3.11794 q^{14} +3.60603 q^{15} -3.05405 q^{16} +1.00000 q^{17} -1.26690 q^{18} +1.02429 q^{19} -1.42428 q^{20} -2.46109 q^{21} -5.48148 q^{22} +3.18429 q^{23} +3.03418 q^{24} +8.00349 q^{25} -8.71364 q^{26} +1.00000 q^{27} +0.972059 q^{28} +2.11000 q^{29} -4.56847 q^{30} +11.0198 q^{31} -2.19919 q^{32} +4.32670 q^{33} -1.26690 q^{34} -8.87476 q^{35} -0.394972 q^{36} -3.84497 q^{37} -1.29767 q^{38} +6.87794 q^{39} +10.9414 q^{40} -3.71450 q^{41} +3.11794 q^{42} +5.79832 q^{43} -1.70892 q^{44} +3.60603 q^{45} -4.03416 q^{46} -10.7508 q^{47} -3.05405 q^{48} -0.943060 q^{49} -10.1396 q^{50} +1.00000 q^{51} -2.71659 q^{52} -1.67641 q^{53} -1.26690 q^{54} +15.6022 q^{55} -7.46738 q^{56} +1.02429 q^{57} -2.67315 q^{58} +1.63966 q^{59} -1.42428 q^{60} -7.36202 q^{61} -13.9609 q^{62} -2.46109 q^{63} +8.89426 q^{64} +24.8021 q^{65} -5.48148 q^{66} +2.03979 q^{67} -0.394972 q^{68} +3.18429 q^{69} +11.2434 q^{70} -3.93293 q^{71} +3.03418 q^{72} -7.02232 q^{73} +4.87118 q^{74} +8.00349 q^{75} -0.404566 q^{76} -10.6484 q^{77} -8.71364 q^{78} +1.00000 q^{79} -11.0130 q^{80} +1.00000 q^{81} +4.70588 q^{82} +2.00730 q^{83} +0.972059 q^{84} +3.60603 q^{85} -7.34587 q^{86} +2.11000 q^{87} +13.1280 q^{88} +3.75596 q^{89} -4.56847 q^{90} -16.9272 q^{91} -1.25770 q^{92} +11.0198 q^{93} +13.6201 q^{94} +3.69363 q^{95} -2.19919 q^{96} -0.409835 q^{97} +1.19476 q^{98} +4.32670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more