LMFDB - Embedded newform 4015.2.a.c.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.418197 q^{2} +2.15176 q^{3} -1.82511 q^{4} +1.00000 q^{5} +0.899862 q^{6} +0.524806 q^{7} -1.59965 q^{8} +1.63009 q^{9} +O(q^{10})$	\(q+0.418197 q^{2} +2.15176 q^{3} -1.82511 q^{4} +1.00000 q^{5} +0.899862 q^{6} +0.524806 q^{7} -1.59965 q^{8} +1.63009 q^{9} +0.418197 q^{10} -1.00000 q^{11} -3.92721 q^{12} -4.80974 q^{13} +0.219473 q^{14} +2.15176 q^{15} +2.98125 q^{16} -0.838646 q^{17} +0.681698 q^{18} +1.62168 q^{19} -1.82511 q^{20} +1.12926 q^{21} -0.418197 q^{22} -5.42611 q^{23} -3.44207 q^{24} +1.00000 q^{25} -2.01142 q^{26} -2.94773 q^{27} -0.957830 q^{28} +6.61468 q^{29} +0.899862 q^{30} -2.78526 q^{31} +4.44606 q^{32} -2.15176 q^{33} -0.350720 q^{34} +0.524806 q^{35} -2.97509 q^{36} -7.82759 q^{37} +0.678183 q^{38} -10.3494 q^{39} -1.59965 q^{40} -8.92449 q^{41} +0.472253 q^{42} -4.68898 q^{43} +1.82511 q^{44} +1.63009 q^{45} -2.26919 q^{46} +5.24365 q^{47} +6.41495 q^{48} -6.72458 q^{49} +0.418197 q^{50} -1.80457 q^{51} +8.77830 q^{52} +10.2945 q^{53} -1.23273 q^{54} -1.00000 q^{55} -0.839507 q^{56} +3.48947 q^{57} +2.76624 q^{58} -1.22859 q^{59} -3.92721 q^{60} -4.46331 q^{61} -1.16479 q^{62} +0.855480 q^{63} -4.10317 q^{64} -4.80974 q^{65} -0.899862 q^{66} -4.44529 q^{67} +1.53062 q^{68} -11.6757 q^{69} +0.219473 q^{70} -6.39629 q^{71} -2.60757 q^{72} +1.00000 q^{73} -3.27348 q^{74} +2.15176 q^{75} -2.95975 q^{76} -0.524806 q^{77} -4.32810 q^{78} -6.71627 q^{79} +2.98125 q^{80} -11.2331 q^{81} -3.73220 q^{82} +5.14201 q^{83} -2.06102 q^{84} -0.838646 q^{85} -1.96092 q^{86} +14.2332 q^{87} +1.59965 q^{88} +3.41612 q^{89} +0.681698 q^{90} -2.52418 q^{91} +9.90325 q^{92} -5.99322 q^{93} +2.19288 q^{94} +1.62168 q^{95} +9.56686 q^{96} +12.7106 q^{97} -2.81220 q^{98} -1.63009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * q^98 - 8 * 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 $

$2$	0.418197	0.295710	0.147855	−	0.989009i	$-0.452763\pi$
$2$	0.418197	0.295710	0.147855	+	0.989009i	$0.452763\pi$
$3$	2.15176	1.24232	0.621161	−	0.783683i	$-0.286661\pi$
$3$	2.15176	1.24232	0.621161	+	0.783683i	$0.286661\pi$
$4$	−1.82511	−0.912555
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.899862	0.367367
$7$	0.524806	0.198358	0.0991791	−	0.995070i	$-0.468378\pi$
$7$	0.524806	0.198358	0.0991791	+	0.995070i	$0.468378\pi$
$8$	−1.59965	−0.565562
$9$	1.63009	0.543362
$10$	0.418197	0.132246
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.92721	−1.13369
$13$	−4.80974	−1.33398	−0.666990	−	0.745066i	$-0.732418\pi$
$13$	−4.80974	−1.33398	−0.666990	+	0.745066i	$0.732418\pi$
$14$	0.219473	0.0586565
$15$	2.15176	0.555583
$16$	2.98125	0.745313
$17$	−0.838646	−0.203402	−0.101701	−	0.994815i	$-0.532428\pi$
$17$	−0.838646	−0.203402	−0.101701	+	0.994815i	$0.532428\pi$
$18$	0.681698	0.160678
$19$	1.62168	0.372039	0.186020	−	0.982546i	$-0.440441\pi$
$19$	1.62168	0.372039	0.186020	+	0.982546i	$0.440441\pi$
$20$	−1.82511	−0.408107
$21$	1.12926	0.246425
$22$	−0.418197	−0.0891600
$23$	−5.42611	−1.13142	−0.565711	−	0.824603i	$-0.691398\pi$
$23$	−5.42611	−1.13142	−0.565711	+	0.824603i	$0.691398\pi$
$24$	−3.44207	−0.702610
$25$	1.00000	0.200000
$26$	−2.01142	−0.394472
$27$	−2.94773	−0.567291
$28$	−0.957830	−0.181013
$29$	6.61468	1.22832	0.614158	−	0.789183i	$-0.289496\pi$
$29$	6.61468	1.22832	0.614158	+	0.789183i	$0.289496\pi$
$30$	0.899862	0.164292
$31$	−2.78526	−0.500247	−0.250124	−	0.968214i	$-0.580471\pi$
$31$	−2.78526	−0.500247	−0.250124	+	0.968214i	$0.580471\pi$
$32$	4.44606	0.785959
$33$	−2.15176	−0.374574
$34$	−0.350720	−0.0601479
$35$	0.524806	0.0887084
$36$	−2.97509	−0.495848
$37$	−7.82759	−1.28685	−0.643424	−	0.765510i	$-0.722487\pi$
$37$	−7.82759	−1.28685	−0.643424	+	0.765510i	$0.722487\pi$
$38$	0.678183	0.110016
$39$	−10.3494	−1.65723
$40$	−1.59965	−0.252927
$41$	−8.92449	−1.39377	−0.696886	−	0.717182i	$-0.745431\pi$
$41$	−8.92449	−1.39377	−0.696886	+	0.717182i	$0.745431\pi$
$42$	0.472253	0.0728703
$43$	−4.68898	−0.715062	−0.357531	−	0.933901i	$-0.616381\pi$
$43$	−4.68898	−0.715062	−0.357531	+	0.933901i	$0.616381\pi$
$44$	1.82511	0.275146
$45$	1.63009	0.242999
$46$	−2.26919	−0.334573
$47$	5.24365	0.764865	0.382432	−	0.923983i	$-0.375086\pi$
$47$	5.24365	0.764865	0.382432	+	0.923983i	$0.375086\pi$
$48$	6.41495	0.925918
$49$	−6.72458	−0.960654
$50$	0.418197	0.0591421
$51$	−1.80457	−0.252690
$52$	8.77830	1.21733
$53$	10.2945	1.41406	0.707030	−	0.707183i	$-0.250034\pi$
$53$	10.2945	1.41406	0.707030	+	0.707183i	$0.250034\pi$
$54$	−1.23273	−0.167754
$55$	−1.00000	−0.134840
$56$	−0.839507	−0.112184
$57$	3.48947	0.462192
$58$	2.76624	0.363225
$59$	−1.22859	−0.159949	−0.0799746	−	0.996797i	$-0.525484\pi$
$59$	−1.22859	−0.159949	−0.0799746	+	0.996797i	$0.525484\pi$
$60$	−3.92721	−0.507000
$61$	−4.46331	−0.571468	−0.285734	−	0.958309i	$-0.592237\pi$
$61$	−4.46331	−0.571468	−0.285734	+	0.958309i	$0.592237\pi$
$62$	−1.16479	−0.147928
$63$	0.855480	0.107780
$64$	−4.10317	−0.512897
$65$	−4.80974	−0.596574
$66$	−0.899862	−0.110765
$67$	−4.44529	−0.543079	−0.271539	−	0.962427i	$-0.587533\pi$
$67$	−4.44529	−0.543079	−0.271539	+	0.962427i	$0.587533\pi$
$68$	1.53062	0.185615
$69$	−11.6757	−1.40559
$70$	0.219473	0.0262320
$71$	−6.39629	−0.759100	−0.379550	−	0.925171i	$-0.623921\pi$
$71$	−6.39629	−0.759100	−0.379550	+	0.925171i	$0.623921\pi$
$72$	−2.60757	−0.307305
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	−3.27348	−0.380534
$75$	2.15176	0.248464
$76$	−2.95975	−0.339506
$77$	−0.524806	−0.0598072
$78$	−4.32810	−0.490061
$79$	−6.71627	−0.755639	−0.377820	−	0.925879i	$-0.623326\pi$
$79$	−6.71627	−0.755639	−0.377820	+	0.925879i	$0.623326\pi$
$80$	2.98125	0.333314
$81$	−11.2331	−1.24812
$82$	−3.73220	−0.412152
$83$	5.14201	0.564409	0.282205	−	0.959354i	$-0.408934\pi$
$83$	5.14201	0.564409	0.282205	+	0.959354i	$0.408934\pi$
$84$	−2.06102	−0.224876
$85$	−0.838646	−0.0909640
$86$	−1.96092	−0.211451
$87$	14.2332	1.52596
$88$	1.59965	0.170523
$89$	3.41612	0.362108	0.181054	−	0.983473i	$-0.442049\pi$
$89$	3.41612	0.362108	0.181054	+	0.983473i	$0.442049\pi$
$90$	0.681698	0.0718573
$91$	−2.52418	−0.264606
$92$	9.90325	1.03249
$93$	−5.99322	−0.621468
$94$	2.19288	0.226178
$95$	1.62168	0.166381
$96$	9.56686	0.976414
$97$	12.7106	1.29057	0.645283	−	0.763943i	$-0.276739\pi$
$97$	12.7106	1.29057	0.645283	+	0.763943i	$0.276739\pi$
$98$	−2.81220	−0.284075
$99$	−1.63009	−0.163830
$100$	−1.82511	−0.182511
$101$	−12.9278	−1.28636	−0.643182	−	0.765713i	$-0.722386\pi$
$101$	−12.9278	−1.28636	−0.643182	+	0.765713i	$0.722386\pi$
$102$	−0.754666	−0.0747231
$103$	−0.383109	−0.0377488	−0.0188744	−	0.999822i	$-0.506008\pi$
$103$	−0.383109	−0.0377488	−0.0188744	+	0.999822i	$0.506008\pi$
$104$	7.69390	0.754449
$105$	1.12926	0.110204
$106$	4.30514	0.418152
$107$	−15.3475	−1.48370	−0.741850	−	0.670565i	$-0.766052\pi$
$107$	−15.3475	−1.48370	−0.741850	+	0.670565i	$0.766052\pi$
$108$	5.37993	0.517684
$109$	−2.56778	−0.245949	−0.122975	−	0.992410i	$-0.539243\pi$
$109$	−2.56778	−0.245949	−0.122975	+	0.992410i	$0.539243\pi$
$110$	−0.418197	−0.0398736
$111$	−16.8431	−1.59868
$112$	1.56458	0.147839
$113$	−17.0139	−1.60054	−0.800268	−	0.599643i	$-0.795309\pi$
$113$	−17.0139	−1.60054	−0.800268	+	0.599643i	$0.795309\pi$
$114$	1.45929	0.136675
$115$	−5.42611	−0.505987
$116$	−12.0725	−1.12091
$117$	−7.84029	−0.724835
$118$	−0.513795	−0.0472986
$119$	−0.440127	−0.0403464
$120$	−3.44207	−0.314217
$121$	1.00000	0.0909091
$122$	−1.86654	−0.168989
$123$	−19.2034	−1.73151
$124$	5.08341	0.456503
$125$	1.00000	0.0894427
$126$	0.357760	0.0318718
$127$	−1.85178	−0.164319	−0.0821593	−	0.996619i	$-0.526182\pi$
$127$	−1.85178	−0.164319	−0.0821593	+	0.996619i	$0.526182\pi$
$128$	−10.6080	−0.937628
$129$	−10.0896	−0.888337
$130$	−2.01142	−0.176413
$131$	10.8823	0.950792	0.475396	−	0.879772i	$-0.342305\pi$
$131$	10.8823	0.950792	0.475396	+	0.879772i	$0.342305\pi$
$132$	3.92721	0.341820
$133$	0.851068	0.0737970
$134$	−1.85901	−0.160594
$135$	−2.94773	−0.253700
$136$	1.34154	0.115036
$137$	−13.6166	−1.16335	−0.581674	−	0.813422i	$-0.697602\pi$
$137$	−13.6166	−1.16335	−0.581674	+	0.813422i	$0.697602\pi$
$138$	−4.88275	−0.415647
$139$	13.4913	1.14432	0.572159	−	0.820143i	$-0.306106\pi$
$139$	13.4913	1.14432	0.572159	+	0.820143i	$0.306106\pi$
$140$	−0.957830	−0.0809514
$141$	11.2831	0.950208
$142$	−2.67491	−0.224474
$143$	4.80974	0.402210
$144$	4.85970	0.404975
$145$	6.61468	0.549319
$146$	0.418197	0.0346103
$147$	−14.4697	−1.19344
$148$	14.2862	1.17432
$149$	3.89259	0.318894	0.159447	−	0.987207i	$-0.449029\pi$
$149$	3.89259	0.318894	0.159447	+	0.987207i	$0.449029\pi$
$150$	0.899862	0.0734734
$151$	−1.47092	−0.119702	−0.0598509	−	0.998207i	$-0.519063\pi$
$151$	−1.47092	−0.119702	−0.0598509	+	0.998207i	$0.519063\pi$
$152$	−2.59412	−0.210411
$153$	−1.36707	−0.110521
$154$	−0.219473	−0.0176856
$155$	−2.78526	−0.223717
$156$	18.8888	1.51232
$157$	−8.11822	−0.647904	−0.323952	−	0.946073i	$-0.605012\pi$
$157$	−8.11822	−0.647904	−0.323952	+	0.946073i	$0.605012\pi$
$158$	−2.80873	−0.223450
$159$	22.1514	1.75672
$160$	4.44606	0.351492
$161$	−2.84766	−0.224427
$162$	−4.69764	−0.369082
$163$	15.9478	1.24913	0.624564	−	0.780974i	$-0.285277\pi$
$163$	15.9478	1.24913	0.624564	+	0.780974i	$0.285277\pi$
$164$	16.2882	1.27189
$165$	−2.15176	−0.167515
$166$	2.15038	0.166902
$167$	14.7812	1.14380	0.571902	−	0.820322i	$-0.306206\pi$
$167$	14.7812	1.14380	0.571902	+	0.820322i	$0.306206\pi$
$168$	−1.80642	−0.139368
$169$	10.1336	0.779504
$170$	−0.350720	−0.0268990
$171$	2.64348	0.202152
$172$	8.55790	0.652534
$173$	−2.99146	−0.227437	−0.113718	−	0.993513i	$-0.536276\pi$
$173$	−2.99146	−0.227437	−0.113718	+	0.993513i	$0.536276\pi$
$174$	5.95230	0.451243
$175$	0.524806	0.0396716
$176$	−2.98125	−0.224720
$177$	−2.64364	−0.198708
$178$	1.42861	0.107079
$179$	12.4212	0.928402	0.464201	−	0.885730i	$-0.346342\pi$
$179$	12.4212	0.928402	0.464201	+	0.885730i	$0.346342\pi$
$180$	−2.97509	−0.221750
$181$	−17.3425	−1.28906	−0.644529	−	0.764580i	$-0.722946\pi$
$181$	−17.3425	−1.28906	−0.644529	+	0.764580i	$0.722946\pi$
$182$	−1.05561	−0.0782467
$183$	−9.60398	−0.709947
$184$	8.67989	0.639890
$185$	−7.82759	−0.575496
$186$	−2.50635	−0.183774
$187$	0.838646	0.0613279
$188$	−9.57024	−0.697982
$189$	−1.54699	−0.112527
$190$	0.678183	0.0492006
$191$	−7.94784	−0.575086	−0.287543	−	0.957768i	$-0.592838\pi$
$191$	−7.94784	−0.575086	−0.287543	+	0.957768i	$0.592838\pi$
$192$	−8.82906	−0.637183
$193$	18.3433	1.32038	0.660190	−	0.751099i	$-0.270476\pi$
$193$	18.3433	1.32038	0.660190	+	0.751099i	$0.270476\pi$
$194$	5.31554	0.381634
$195$	−10.3494	−0.741137
$196$	12.2731	0.876650
$197$	17.0013	1.21129	0.605647	−	0.795734i	$-0.292914\pi$
$197$	17.0013	1.21129	0.605647	+	0.795734i	$0.292914\pi$
$198$	−0.681698	−0.0484462
$199$	−10.6301	−0.753546	−0.376773	−	0.926306i	$-0.622966\pi$
$199$	−10.6301	−0.753546	−0.376773	+	0.926306i	$0.622966\pi$
$200$	−1.59965	−0.113112
$201$	−9.56522	−0.674678
$202$	−5.40637	−0.380391
$203$	3.47143	0.243646
$204$	3.29354	0.230594
$205$	−8.92449	−0.623313
$206$	−0.160215	−0.0111627
$207$	−8.84503	−0.614772
$208$	−14.3390	−0.994233
$209$	−1.62168	−0.112174
$210$	0.472253	0.0325886
$211$	−16.4754	−1.13421	−0.567107	−	0.823644i	$-0.691937\pi$
$211$	−16.4754	−1.13421	−0.567107	+	0.823644i	$0.691937\pi$
$212$	−18.7886	−1.29041
$213$	−13.7633	−0.943046
$214$	−6.41829	−0.438746
$215$	−4.68898	−0.319786
$216$	4.71534	0.320838
$217$	−1.46172	−0.0992281
$218$	−1.07384	−0.0727297
$219$	2.15176	0.145403
$220$	1.82511	0.123049
$221$	4.03367	0.271334
$222$	−7.04375	−0.472746
$223$	−0.282405	−0.0189112	−0.00945562	−	0.999955i	$-0.503010\pi$
$223$	−0.282405	−0.0189112	−0.00945562	+	0.999955i	$0.503010\pi$
$224$	2.33332	0.155901
$225$	1.63009	0.108672
$226$	−7.11518	−0.473295
$227$	5.44697	0.361528	0.180764	−	0.983526i	$-0.442143\pi$
$227$	5.44697	0.361528	0.180764	+	0.983526i	$0.442143\pi$
$228$	−6.36868	−0.421776
$229$	10.5201	0.695187	0.347593	−	0.937645i	$-0.386999\pi$
$229$	10.5201	0.695187	0.347593	+	0.937645i	$0.386999\pi$
$230$	−2.26919	−0.149626
$231$	−1.12926	−0.0742998
$232$	−10.5812	−0.694689
$233$	8.64765	0.566526	0.283263	−	0.959042i	$-0.408583\pi$
$233$	8.64765	0.566526	0.283263	+	0.959042i	$0.408583\pi$
$234$	−3.27879	−0.214341
$235$	5.24365	0.342058
$236$	2.24232	0.145963
$237$	−14.4518	−0.938747
$238$	−0.184060	−0.0119308
$239$	4.79465	0.310140	0.155070	−	0.987903i	$-0.450440\pi$
$239$	4.79465	0.310140	0.155070	+	0.987903i	$0.450440\pi$
$240$	6.41495	0.414083
$241$	16.5327	1.06496	0.532482	−	0.846442i	$-0.321260\pi$
$241$	16.5327	1.06496	0.532482	+	0.846442i	$0.321260\pi$
$242$	0.418197	0.0268828
$243$	−15.3277	−0.983275
$244$	8.14603	0.521496
$245$	−6.72458	−0.429618
$246$	−8.03081	−0.512026
$247$	−7.79986	−0.496293
$248$	4.45544	0.282921
$249$	11.0644	0.701178
$250$	0.418197	0.0264491
$251$	−28.5539	−1.80231	−0.901153	−	0.433502i	$-0.857278\pi$
$251$	−28.5539	−1.80231	−0.901153	+	0.433502i	$0.857278\pi$
$252$	−1.56135	−0.0983555
$253$	5.42611	0.341137
$254$	−0.774408	−0.0485907
$255$	−1.80457	−0.113006
$256$	3.77009	0.235631
$257$	12.9729	0.809225	0.404613	−	0.914488i	$-0.367406\pi$
$257$	12.9729	0.809225	0.404613	+	0.914488i	$0.367406\pi$
$258$	−4.21943	−0.262690
$259$	−4.10797	−0.255257
$260$	8.77830	0.544407
$261$	10.7825	0.667420
$262$	4.55096	0.281159
$263$	9.62173	0.593301	0.296651	−	0.954986i	$-0.404130\pi$
$263$	9.62173	0.593301	0.296651	+	0.954986i	$0.404130\pi$
$264$	3.44207	0.211845
$265$	10.2945	0.632387
$266$	0.355915	0.0218225
$267$	7.35069	0.449855
$268$	8.11315	0.495589
$269$	10.1618	0.619574	0.309787	−	0.950806i	$-0.399742\pi$
$269$	10.1618	0.619574	0.309787	+	0.950806i	$0.399742\pi$
$270$	−1.23273	−0.0750217
$271$	−6.30487	−0.382994	−0.191497	−	0.981493i	$-0.561334\pi$
$271$	−6.30487	−0.382994	−0.191497	+	0.981493i	$0.561334\pi$
$272$	−2.50022	−0.151598
$273$	−5.43144	−0.328726
$274$	−5.69444	−0.344014
$275$	−1.00000	−0.0603023
$276$	21.3095	1.28268
$277$	13.8380	0.831443	0.415721	−	0.909492i	$-0.363529\pi$
$277$	13.8380	0.831443	0.415721	+	0.909492i	$0.363529\pi$
$278$	5.64203	0.338387
$279$	−4.54022	−0.271816
$280$	−0.839507	−0.0501702
$281$	4.17785	0.249230	0.124615	−	0.992205i	$-0.460230\pi$
$281$	4.17785	0.249230	0.124615	+	0.992205i	$0.460230\pi$
$282$	4.71856	0.280986
$283$	−13.2528	−0.787799	−0.393900	−	0.919153i	$-0.628874\pi$
$283$	−13.2528	−0.787799	−0.393900	+	0.919153i	$0.628874\pi$
$284$	11.6739	0.692721
$285$	3.48947	0.206699
$286$	2.01142	0.118938
$287$	−4.68363	−0.276466
$288$	7.24746	0.427061
$289$	−16.2967	−0.958628
$290$	2.76624	0.162439
$291$	27.3502	1.60330
$292$	−1.82511	−0.106807
$293$	−17.0398	−0.995477	−0.497738	−	0.867327i	$-0.665836\pi$
$293$	−17.0398	−0.995477	−0.497738	+	0.867327i	$0.665836\pi$
$294$	−6.05119	−0.352913
$295$	−1.22859	−0.0715315
$296$	12.5214	0.727793
$297$	2.94773	0.171045
$298$	1.62787	0.0943001
$299$	26.0982	1.50930
$300$	−3.92721	−0.226737
$301$	−2.46081	−0.141838
$302$	−0.615135	−0.0353970
$303$	−27.8176	−1.59808
$304$	4.83464	0.277286
$305$	−4.46331	−0.255568
$306$	−0.571704	−0.0326821
$307$	−21.7996	−1.24417	−0.622084	−	0.782951i	$-0.713714\pi$
$307$	−21.7996	−1.24417	−0.622084	+	0.782951i	$0.713714\pi$
$308$	0.957830	0.0545774
$309$	−0.824359	−0.0468962
$310$	−1.16479	−0.0661555
$311$	5.97812	0.338988	0.169494	−	0.985531i	$-0.445787\pi$
$311$	5.97812	0.338988	0.169494	+	0.985531i	$0.445787\pi$
$312$	16.5555	0.937268
$313$	−9.01455	−0.509532	−0.254766	−	0.967003i	$-0.581999\pi$
$313$	−9.01455	−0.509532	−0.254766	+	0.967003i	$0.581999\pi$
$314$	−3.39502	−0.191592
$315$	0.855480	0.0482008
$316$	12.2579	0.689563
$317$	−3.34633	−0.187949	−0.0939743	−	0.995575i	$-0.529957\pi$
$317$	−3.34633	−0.187949	−0.0939743	+	0.995575i	$0.529957\pi$
$318$	9.26365	0.519480
$319$	−6.61468	−0.370351
$320$	−4.10317	−0.229374
$321$	−33.0242	−1.84323
$322$	−1.19088	−0.0663653
$323$	−1.36002	−0.0756734
$324$	20.5016	1.13898
$325$	−4.80974	−0.266796
$326$	6.66933	0.369380
$327$	−5.52526	−0.305548
$328$	14.2761	0.788264
$329$	2.75190	0.151717
$330$	−0.899862	−0.0495358
$331$	−13.2414	−0.727812	−0.363906	−	0.931436i	$-0.618557\pi$
$331$	−13.2414	−0.727812	−0.363906	+	0.931436i	$0.618557\pi$
$332$	−9.38475	−0.515055
$333$	−12.7597	−0.699225
$334$	6.18146	0.338234
$335$	−4.44529	−0.242872
$336$	3.36661	0.183663
$337$	−12.7926	−0.696855	−0.348427	−	0.937336i	$-0.613284\pi$
$337$	−12.7926	−0.696855	−0.348427	+	0.937336i	$0.613284\pi$
$338$	4.23783	0.230507
$339$	−36.6099	−1.98838
$340$	1.53062	0.0830097
$341$	2.78526	0.150830
$342$	1.10550	0.0597785
$343$	−7.20274	−0.388912
$344$	7.50073	0.404412
$345$	−11.6757	−0.628599
$346$	−1.25102	−0.0672553
$347$	14.2743	0.766285	0.383143	−	0.923689i	$-0.374842\pi$
$347$	14.2743	0.766285	0.383143	+	0.923689i	$0.374842\pi$
$348$	−25.9772	−1.39253
$349$	−3.22183	−0.172461	−0.0862304	−	0.996275i	$-0.527482\pi$
$349$	−3.22183	−0.172461	−0.0862304	+	0.996275i	$0.527482\pi$
$350$	0.219473	0.0117313
$351$	14.1778	0.756755
$352$	−4.44606	−0.236976
$353$	31.9748	1.70185	0.850924	−	0.525289i	$-0.176043\pi$
$353$	31.9748	1.70185	0.850924	+	0.525289i	$0.176043\pi$
$354$	−1.10556	−0.0587601
$355$	−6.39629	−0.339480
$356$	−6.23480	−0.330444
$357$	−0.947049	−0.0501231
$358$	5.19450	0.274538
$359$	5.26550	0.277902	0.138951	−	0.990299i	$-0.455627\pi$
$359$	5.26550	0.277902	0.138951	+	0.990299i	$0.455627\pi$
$360$	−2.60757	−0.137431
$361$	−16.3702	−0.861587
$362$	−7.25259	−0.381188
$363$	2.15176	0.112938
$364$	4.60691	0.241468
$365$	1.00000	0.0523424
$366$	−4.01636	−0.209939
$367$	20.1694	1.05283	0.526417	−	0.850227i	$-0.323535\pi$
$367$	20.1694	1.05283	0.526417	+	0.850227i	$0.323535\pi$
$368$	−16.1766	−0.843264
$369$	−14.5477	−0.757323
$370$	−3.27348	−0.170180
$371$	5.40263	0.280490
$372$	10.9383	0.567124
$373$	9.70129	0.502314	0.251157	−	0.967946i	$-0.419189\pi$
$373$	9.70129	0.502314	0.251157	+	0.967946i	$0.419189\pi$
$374$	0.350720	0.0181353
$375$	2.15176	0.111117
$376$	−8.38801	−0.432579
$377$	−31.8149	−1.63855
$378$	−0.646946	−0.0332753
$379$	13.8404	0.710931	0.355466	−	0.934689i	$-0.384322\pi$
$379$	13.8404	0.710931	0.355466	+	0.934689i	$0.384322\pi$
$380$	−2.95975	−0.151832
$381$	−3.98458	−0.204136
$382$	−3.32377	−0.170059
$383$	−22.8534	−1.16775	−0.583876	−	0.811843i	$-0.698465\pi$
$383$	−22.8534	−1.16775	−0.583876	+	0.811843i	$0.698465\pi$
$384$	−22.8260	−1.16484
$385$	−0.524806	−0.0267466
$386$	7.67112	0.390450
$387$	−7.64344	−0.388538
$388$	−23.1983	−1.17771
$389$	17.3644	0.880410	0.440205	−	0.897897i	$-0.354906\pi$
$389$	17.3644	0.880410	0.440205	+	0.897897i	$0.354906\pi$
$390$	−4.32810	−0.219162
$391$	4.55059	0.230133
$392$	10.7570	0.543310
$393$	23.4162	1.18119
$394$	7.10991	0.358192
$395$	−6.71627	−0.337932
$396$	2.97509	0.149504
$397$	−25.1579	−1.26264	−0.631320	−	0.775522i	$-0.717487\pi$
$397$	−25.1579	−1.26264	−0.631320	+	0.775522i	$0.717487\pi$
$398$	−4.44547	−0.222831
$399$	1.83130	0.0916796
$400$	2.98125	0.149063
$401$	−6.64962	−0.332066	−0.166033	−	0.986120i	$-0.553096\pi$
$401$	−6.64962	−0.332066	−0.166033	+	0.986120i	$0.553096\pi$
$402$	−4.00015	−0.199509
$403$	13.3964	0.667320
$404$	23.5947	1.17388
$405$	−11.2331	−0.558176
$406$	1.45174	0.0720487
$407$	7.82759	0.387999
$408$	2.88668	0.142912
$409$	4.93944	0.244239	0.122120	−	0.992515i	$-0.461031\pi$
$409$	4.93944	0.244239	0.122120	+	0.992515i	$0.461031\pi$
$410$	−3.73220	−0.184320
$411$	−29.2998	−1.44525
$412$	0.699216	0.0344479
$413$	−0.644774	−0.0317272
$414$	−3.69897	−0.181794
$415$	5.14201	0.252412
$416$	−21.3844	−1.04845
$417$	29.0301	1.42161
$418$	−0.678183	−0.0331710
$419$	18.0476	0.881684	0.440842	−	0.897585i	$-0.354680\pi$
$419$	18.0476	0.881684	0.440842	+	0.897585i	$0.354680\pi$
$420$	−2.06102	−0.100568
$421$	−31.3969	−1.53019	−0.765096	−	0.643916i	$-0.777308\pi$
$421$	−31.3969	−1.53019	−0.765096	+	0.643916i	$0.777308\pi$
$422$	−6.88997	−0.335399
$423$	8.54761	0.415599
$424$	−16.4676	−0.799739
$425$	−0.838646	−0.0406803
$426$	−5.75578	−0.278868
$427$	−2.34237	−0.113355
$428$	28.0109	1.35396
$429$	10.3494	0.499674
$430$	−1.96092	−0.0945639
$431$	−16.6020	−0.799688	−0.399844	−	0.916583i	$-0.630936\pi$
$431$	−16.6020	−0.799688	−0.399844	+	0.916583i	$0.630936\pi$
$432$	−8.78792	−0.422809
$433$	18.3964	0.884077	0.442038	−	0.896996i	$-0.354256\pi$
$433$	18.3964	0.884077	0.442038	+	0.896996i	$0.354256\pi$
$434$	−0.611288	−0.0293428
$435$	14.2332	0.682431
$436$	4.68649	0.224442
$437$	−8.79942	−0.420933
$438$	0.899862	0.0429971
$439$	17.4468	0.832690	0.416345	−	0.909207i	$-0.363311\pi$
$439$	17.4468	0.832690	0.416345	+	0.909207i	$0.363311\pi$
$440$	1.59965	0.0762604
$441$	−10.9616	−0.521983
$442$	1.68687	0.0802362
$443$	−22.6298	−1.07517	−0.537586	−	0.843209i	$-0.680664\pi$
$443$	−22.6298	−1.07517	−0.537586	+	0.843209i	$0.680664\pi$
$444$	30.7406	1.45888
$445$	3.41612	0.161940
$446$	−0.118101	−0.00559225
$447$	8.37594	0.396168
$448$	−2.15337	−0.101737
$449$	−19.3877	−0.914961	−0.457481	−	0.889220i	$-0.651248\pi$
$449$	−19.3877	−0.914961	−0.457481	+	0.889220i	$0.651248\pi$
$450$	0.681698	0.0321356
$451$	8.92449	0.420238
$452$	31.0523	1.46058
$453$	−3.16507	−0.148708
$454$	2.27791	0.106908
$455$	−2.52418	−0.118335
$456$	−5.58194	−0.261398
$457$	29.0408	1.35847	0.679237	−	0.733919i	$-0.262311\pi$
$457$	29.0408	1.35847	0.679237	+	0.733919i	$0.262311\pi$
$458$	4.39947	0.205574
$459$	2.47210	0.115388
$460$	9.90325	0.461742
$461$	21.9620	1.02287	0.511437	−	0.859321i	$-0.329113\pi$
$461$	21.9620	1.02287	0.511437	+	0.859321i	$0.329113\pi$
$462$	−0.472253	−0.0219712
$463$	16.3916	0.761783	0.380892	−	0.924620i	$-0.375617\pi$
$463$	16.3916	0.761783	0.380892	+	0.924620i	$0.375617\pi$
$464$	19.7200	0.915479
$465$	−5.99322	−0.277929
$466$	3.61642	0.167528
$467$	−40.0620	−1.85385	−0.926924	−	0.375248i	$-0.877557\pi$
$467$	−40.0620	−1.85385	−0.926924	+	0.375248i	$0.877557\pi$
$468$	14.3094	0.661452
$469$	−2.33292	−0.107724
$470$	2.19288	0.101150
$471$	−17.4685	−0.804906
$472$	1.96532	0.0904613
$473$	4.68898	0.215599
$474$	−6.04372	−0.277597
$475$	1.62168	0.0744078
$476$	0.803280	0.0368183
$477$	16.7810	0.768347
$478$	2.00511	0.0917116
$479$	30.6883	1.40218	0.701091	−	0.713072i	$-0.252697\pi$
$479$	30.6883	1.40218	0.701091	+	0.713072i	$0.252697\pi$
$480$	9.56686	0.436665
$481$	37.6486	1.71663
$482$	6.91392	0.314921
$483$	−6.12748	−0.278810
$484$	−1.82511	−0.0829596
$485$	12.7106	0.577159
$486$	−6.41002	−0.290765
$487$	4.87988	0.221128	0.110564	−	0.993869i	$-0.464734\pi$
$487$	4.87988	0.221128	0.110564	+	0.993869i	$0.464734\pi$
$488$	7.13974	0.323201
$489$	34.3159	1.55182
$490$	−2.81220	−0.127042
$491$	−19.5065	−0.880316	−0.440158	−	0.897920i	$-0.645078\pi$
$491$	−19.5065	−0.880316	−0.440158	+	0.897920i	$0.645078\pi$
$492$	35.0483	1.58010
$493$	−5.54738	−0.249841
$494$	−3.26188	−0.146759
$495$	−1.63009	−0.0732670
$496$	−8.30356	−0.372841
$497$	−3.35681	−0.150574
$498$	4.62710	0.207346
$499$	8.05135	0.360428	0.180214	−	0.983627i	$-0.442321\pi$
$499$	8.05135	0.360428	0.180214	+	0.983627i	$0.442321\pi$
$500$	−1.82511	−0.0816214
$501$	31.8056	1.42097
$502$	−11.9412	−0.532960
$503$	40.4198	1.80223	0.901115	−	0.433581i	$-0.142750\pi$
$503$	40.4198	1.80223	0.901115	+	0.433581i	$0.142750\pi$
$504$	−1.36847	−0.0609565
$505$	−12.9278	−0.575279
$506$	2.26919	0.100878
$507$	21.8050	0.968395
$508$	3.37970	0.149950
$509$	33.1334	1.46861	0.734306	−	0.678818i	$-0.237508\pi$
$509$	33.1334	1.46861	0.734306	+	0.678818i	$0.237508\pi$
$510$	−0.754666	−0.0334172
$511$	0.524806	0.0232161
$512$	22.7927	1.00731
$513$	−4.78028	−0.211054
$514$	5.42522	0.239296
$515$	−0.383109	−0.0168818
$516$	18.4146	0.810657
$517$	−5.24365	−0.230615
$518$	−1.71794	−0.0754820
$519$	−6.43692	−0.282549
$520$	7.69390	0.337400
$521$	14.9837	0.656449	0.328224	−	0.944600i	$-0.393550\pi$
$521$	14.9837	0.656449	0.328224	+	0.944600i	$0.393550\pi$
$522$	4.50922	0.197363
$523$	−25.0824	−1.09678	−0.548389	−	0.836223i	$-0.684759\pi$
$523$	−25.0824	−1.09678	−0.548389	+	0.836223i	$0.684759\pi$
$524$	−19.8614	−0.867651
$525$	1.12926	0.0492849
$526$	4.02378	0.175445
$527$	2.33585	0.101751
$528$	−6.41495	−0.279175
$529$	6.44267	0.280116
$530$	4.30514	0.187003
$531$	−2.00271	−0.0869104
$532$	−1.55329	−0.0673438
$533$	42.9244	1.85926
$534$	3.07404	0.133027
$535$	−15.3475	−0.663531
$536$	7.11092	0.307145
$537$	26.7274	1.15337
$538$	4.24963	0.183214
$539$	6.72458	0.289648
$540$	5.37993	0.231515
$541$	−19.5097	−0.838788	−0.419394	−	0.907804i	$-0.637757\pi$
$541$	−19.5097	−0.838788	−0.419394	+	0.907804i	$0.637757\pi$
$542$	−2.63668	−0.113255
$543$	−37.3170	−1.60142
$544$	−3.72867	−0.159865
$545$	−2.56778	−0.109992
$546$	−2.27141	−0.0972075
$547$	16.3651	0.699721	0.349860	−	0.936802i	$-0.386229\pi$
$547$	16.3651	0.699721	0.349860	+	0.936802i	$0.386229\pi$
$548$	24.8519	1.06162
$549$	−7.27558	−0.310514
$550$	−0.418197	−0.0178320
$551$	10.7269	0.456981
$552$	18.6771	0.794949
$553$	−3.52474	−0.149887
$554$	5.78700	0.245866
$555$	−16.8431	−0.714951
$556$	−24.6231	−1.04425
$557$	6.89981	0.292354	0.146177	−	0.989258i	$-0.453303\pi$
$557$	6.89981	0.292354	0.146177	+	0.989258i	$0.453303\pi$
$558$	−1.89871	−0.0803787
$559$	22.5527	0.953880
$560$	1.56458	0.0661155
$561$	1.80457	0.0761890
$562$	1.74717	0.0736998
$563$	−39.0243	−1.64468	−0.822339	−	0.568998i	$-0.807331\pi$
$563$	−39.0243	−1.64468	−0.822339	+	0.568998i	$0.807331\pi$
$564$	−20.5929	−0.867117
$565$	−17.0139	−0.715781
$566$	−5.54230	−0.232960
$567$	−5.89519	−0.247575
$568$	10.2318	0.429318
$569$	12.3612	0.518209	0.259104	−	0.965849i	$-0.416573\pi$
$569$	12.3612	0.518209	0.259104	+	0.965849i	$0.416573\pi$
$570$	1.45929	0.0611229
$571$	28.1100	1.17636	0.588182	−	0.808728i	$-0.299844\pi$
$571$	28.1100	1.17636	0.588182	+	0.808728i	$0.299844\pi$
$572$	−8.77830	−0.367039
$573$	−17.1019	−0.714441
$574$	−1.95868	−0.0817538
$575$	−5.42611	−0.226284
$576$	−6.68853	−0.278689
$577$	−42.7127	−1.77815	−0.889077	−	0.457758i	$-0.848653\pi$
$577$	−42.7127	−1.77815	−0.889077	+	0.457758i	$0.848653\pi$
$578$	−6.81523	−0.283476
$579$	39.4704	1.64034
$580$	−12.0725	−0.501284
$581$	2.69856	0.111955
$582$	11.4378	0.474112
$583$	−10.2945	−0.426355
$584$	−1.59965	−0.0661941
$585$	−7.84029	−0.324156
$586$	−7.12601	−0.294373
$587$	−39.1923	−1.61764	−0.808820	−	0.588056i	$-0.799893\pi$
$587$	−39.1923	−1.61764	−0.808820	+	0.588056i	$0.799893\pi$
$588$	26.4088	1.08908
$589$	−4.51680	−0.186112
$590$	−0.513795	−0.0211526
$591$	36.5828	1.50482
$592$	−23.3360	−0.959104
$593$	−12.8275	−0.526762	−0.263381	−	0.964692i	$-0.584838\pi$
$593$	−12.8275	−0.526762	−0.263381	+	0.964692i	$0.584838\pi$
$594$	1.23273	0.0505796
$595$	−0.440127	−0.0180434
$596$	−7.10441	−0.291008
$597$	−22.8734	−0.936146
$598$	10.9142	0.446314
$599$	−23.8027	−0.972550	−0.486275	−	0.873806i	$-0.661645\pi$
$599$	−23.8027	−0.972550	−0.486275	+	0.873806i	$0.661645\pi$
$600$	−3.44207	−0.140522
$601$	32.1587	1.31178	0.655890	−	0.754856i	$-0.272293\pi$
$601$	32.1587	1.31178	0.655890	+	0.754856i	$0.272293\pi$
$602$	−1.02910	−0.0419431
$603$	−7.24621	−0.295089
$604$	2.68459	0.109234
$605$	1.00000	0.0406558
$606$	−11.6332	−0.472568
$607$	38.8908	1.57853	0.789265	−	0.614053i	$-0.210462\pi$
$607$	38.8908	1.57853	0.789265	+	0.614053i	$0.210462\pi$
$608$	7.21008	0.292408
$609$	7.46969	0.302687
$610$	−1.86654	−0.0755741
$611$	−25.2206	−1.02031
$612$	2.49505	0.100856
$613$	13.2562	0.535413	0.267707	−	0.963500i	$-0.413734\pi$
$613$	13.2562	0.535413	0.267707	+	0.963500i	$0.413734\pi$
$614$	−9.11653	−0.367913
$615$	−19.2034	−0.774356
$616$	0.839507	0.0338247
$617$	35.2110	1.41754	0.708771	−	0.705439i	$-0.249250\pi$
$617$	35.2110	1.41754	0.708771	+	0.705439i	$0.249250\pi$
$618$	−0.344745	−0.0138677
$619$	−6.45530	−0.259460	−0.129730	−	0.991549i	$-0.541411\pi$
$619$	−6.45530	−0.259460	−0.129730	+	0.991549i	$0.541411\pi$
$620$	5.08341	0.204155
$621$	15.9947	0.641845
$622$	2.50003	0.100242
$623$	1.79280	0.0718271
$624$	−30.8542	−1.23516
$625$	1.00000	0.0400000
$626$	−3.76986	−0.150674
$627$	−3.48947	−0.139356
$628$	14.8167	0.591249
$629$	6.56458	0.261747
$630$	0.357760	0.0142535
$631$	−15.0621	−0.599613	−0.299806	−	0.954000i	$-0.596922\pi$
$631$	−15.0621	−0.599613	−0.299806	+	0.954000i	$0.596922\pi$
$632$	10.7437	0.427361
$633$	−35.4512	−1.40906
$634$	−1.39943	−0.0555783
$635$	−1.85178	−0.0734855
$636$	−40.4287	−1.60310
$637$	32.3434	1.28149
$638$	−2.76624	−0.109517
$639$	−10.4265	−0.412466
$640$	−10.6080	−0.419320
$641$	2.30981	0.0912320	0.0456160	−	0.998959i	$-0.485475\pi$
$641$	2.30981	0.0912320	0.0456160	+	0.998959i	$0.485475\pi$
$642$	−13.8107	−0.545063
$643$	0.200576	0.00790994	0.00395497	−	0.999992i	$-0.498741\pi$
$643$	0.200576	0.00790994	0.00395497	+	0.999992i	$0.498741\pi$
$644$	5.19729	0.204802
$645$	−10.0896	−0.397277
$646$	−0.568756	−0.0223774
$647$	−20.5087	−0.806282	−0.403141	−	0.915138i	$-0.632082\pi$
$647$	−20.5087	−0.806282	−0.403141	+	0.915138i	$0.632082\pi$
$648$	17.9690	0.705889
$649$	1.22859	0.0482265
$650$	−2.01142	−0.0788944
$651$	−3.14528	−0.123273
$652$	−29.1065	−1.13990
$653$	46.4111	1.81621	0.908104	−	0.418746i	$-0.137530\pi$
$653$	46.4111	1.81621	0.908104	+	0.418746i	$0.137530\pi$
$654$	−2.31065	−0.0903536
$655$	10.8823	0.425207
$656$	−26.6061	−1.03880
$657$	1.63009	0.0635958
$658$	1.15084	0.0448643
$659$	28.4206	1.10711	0.553555	−	0.832812i	$-0.313271\pi$
$659$	28.4206	1.10711	0.553555	+	0.832812i	$0.313271\pi$
$660$	3.92721	0.152866
$661$	−22.7061	−0.883165	−0.441583	−	0.897221i	$-0.645583\pi$
$661$	−22.7061	−0.883165	−0.441583	+	0.897221i	$0.645583\pi$
$662$	−5.53751	−0.215222
$663$	8.67950	0.337084
$664$	−8.22543	−0.319209
$665$	0.851068	0.0330030
$666$	−5.33606	−0.206768
$667$	−35.8920	−1.38974
$668$	−26.9773	−1.04378
$669$	−0.607669	−0.0234938
$670$	−1.85901	−0.0718198
$671$	4.46331	0.172304
$672$	5.02075	0.193680
$673$	−30.7903	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$673$	−30.7903	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$674$	−5.34981	−0.206067
$675$	−2.94773	−0.113458
$676$	−18.4949	−0.711341
$677$	−6.38793	−0.245508	−0.122754	−	0.992437i	$-0.539173\pi$
$677$	−6.38793	−0.245508	−0.122754	+	0.992437i	$0.539173\pi$
$678$	−15.3102	−0.587984
$679$	6.67061	0.255994
$680$	1.34154	0.0514458
$681$	11.7206	0.449134
$682$	1.16479	0.0446021
$683$	0.381435	0.0145952	0.00729760	−	0.999973i	$-0.497677\pi$
$683$	0.381435	0.0145952	0.00729760	+	0.999973i	$0.497677\pi$
$684$	−4.82465	−0.184475
$685$	−13.6166	−0.520265
$686$	−3.01217	−0.115005
$687$	22.6367	0.863645
$688$	−13.9790	−0.532945
$689$	−49.5139	−1.88633
$690$	−4.88275	−0.185883
$691$	24.9952	0.950861	0.475430	−	0.879753i	$-0.342292\pi$
$691$	24.9952	0.950861	0.475430	+	0.879753i	$0.342292\pi$
$692$	5.45975	0.207548
$693$	−0.855480	−0.0324970
$694$	5.96948	0.226598
$695$	13.4913	0.511755
$696$	−22.7682	−0.863027
$697$	7.48449	0.283495
$698$	−1.34736	−0.0509984
$699$	18.6077	0.703808
$700$	−0.957830	−0.0362026
$701$	−6.74912	−0.254911	−0.127455	−	0.991844i	$-0.540681\pi$
$701$	−6.74912	−0.254911	−0.127455	+	0.991844i	$0.540681\pi$
$702$	5.92912	0.223780
$703$	−12.6939	−0.478758
$704$	4.10317	0.154644
$705$	11.2831	0.424946
$706$	13.3718	0.503254
$707$	−6.78459	−0.255161
$708$	4.82494	0.181332
$709$	2.91140	0.109340	0.0546700	−	0.998504i	$-0.482589\pi$
$709$	2.91140	0.109340	0.0546700	+	0.998504i	$0.482589\pi$
$710$	−2.67491	−0.100388
$711$	−10.9481	−0.410586
$712$	−5.46460	−0.204795
$713$	15.1131	0.565991
$714$	−0.396053	−0.0148219
$715$	4.80974	0.179874
$716$	−22.6700	−0.847218
$717$	10.3169	0.385293
$718$	2.20202	0.0821786
$719$	−3.23444	−0.120624	−0.0603122	−	0.998180i	$-0.519210\pi$
$719$	−3.23444	−0.120624	−0.0603122	+	0.998180i	$0.519210\pi$
$720$	4.85970	0.181110
$721$	−0.201058	−0.00748778
$722$	−6.84596	−0.254780
$723$	35.5744	1.32303
$724$	31.6520	1.17634
$725$	6.61468	0.245663
$726$	0.899862	0.0333970
$727$	48.5955	1.80231	0.901154	−	0.433500i	$-0.142722\pi$
$727$	48.5955	1.80231	0.901154	+	0.433500i	$0.142722\pi$
$728$	4.03781	0.149651
$729$	0.717550	0.0265759
$730$	0.418197	0.0154782
$731$	3.93239	0.145445
$732$	17.5283	0.647866
$733$	14.0623	0.519403	0.259701	−	0.965689i	$-0.416376\pi$
$733$	14.0623	0.519403	0.259701	+	0.965689i	$0.416376\pi$
$734$	8.43478	0.311334
$735$	−14.4697	−0.533723
$736$	−24.1248	−0.889251
$737$	4.44529	0.163744
$738$	−6.08381	−0.223948
$739$	2.20559	0.0811340	0.0405670	−	0.999177i	$-0.487084\pi$
$739$	2.20559	0.0811340	0.0405670	+	0.999177i	$0.487084\pi$
$740$	14.2862	0.525172
$741$	−16.7835	−0.616555
$742$	2.25937	0.0829439
$743$	1.65004	0.0605342	0.0302671	−	0.999542i	$-0.490364\pi$
$743$	1.65004	0.0605342	0.0302671	+	0.999542i	$0.490364\pi$
$744$	9.58706	0.351479
$745$	3.89259	0.142614
$746$	4.05706	0.148539
$747$	8.38193	0.306679
$748$	−1.53062	−0.0559651
$749$	−8.05447	−0.294304
$750$	0.899862	0.0328583
$751$	−18.0324	−0.658011	−0.329005	−	0.944328i	$-0.606713\pi$
$751$	−18.0324	−0.658011	−0.329005	+	0.944328i	$0.606713\pi$
$752$	15.6326	0.570064
$753$	−61.4412	−2.23904
$754$	−13.3049	−0.484536
$755$	−1.47092	−0.0535322
$756$	2.82342	0.102687
$757$	17.7891	0.646555	0.323278	−	0.946304i	$-0.395215\pi$
$757$	17.7891	0.646555	0.323278	+	0.946304i	$0.395215\pi$
$758$	5.78800	0.210230
$759$	11.6757	0.423801
$760$	−2.59412	−0.0940988
$761$	−39.9710	−1.44895	−0.724473	−	0.689303i	$-0.757917\pi$
$761$	−39.9710	−1.44895	−0.724473	+	0.689303i	$0.757917\pi$
$762$	−1.66634	−0.0603652
$763$	−1.34759	−0.0487860
$764$	14.5057	0.524798
$765$	−1.36707	−0.0494264
$766$	−9.55722	−0.345316
$767$	5.90921	0.213369
$768$	8.11234	0.292729
$769$	−22.4799	−0.810647	−0.405323	−	0.914173i	$-0.632841\pi$
$769$	−22.4799	−0.810647	−0.405323	+	0.914173i	$0.632841\pi$
$770$	−0.219473	−0.00790925
$771$	27.9145	1.00532
$772$	−33.4785	−1.20492
$773$	4.48883	0.161452	0.0807260	−	0.996736i	$-0.474276\pi$
$773$	4.48883	0.161452	0.0807260	+	0.996736i	$0.474276\pi$
$774$	−3.19647	−0.114895
$775$	−2.78526	−0.100049
$776$	−20.3325	−0.729896
$777$	−8.83938	−0.317111
$778$	7.26175	0.260346
$779$	−14.4727	−0.518537
$780$	18.8888	0.676329
$781$	6.39629	0.228877
$782$	1.90304	0.0680527
$783$	−19.4983	−0.696812
$784$	−20.0477	−0.715988
$785$	−8.11822	−0.289752
$786$	9.79258	0.349290
$787$	−1.52167	−0.0542418	−0.0271209	−	0.999632i	$-0.508634\pi$
$787$	−1.52167	−0.0542418	−0.0271209	+	0.999632i	$0.508634\pi$
$788$	−31.0293	−1.10537
$789$	20.7037	0.737071
$790$	−2.80873	−0.0999300
$791$	−8.92901	−0.317479
$792$	2.60757	0.0926560
$793$	21.4673	0.762327
$794$	−10.5210	−0.373376
$795$	22.1514	0.785628
$796$	19.4011	0.687652
$797$	19.8846	0.704350	0.352175	−	0.935934i	$-0.385442\pi$
$797$	19.8846	0.704350	0.352175	+	0.935934i	$0.385442\pi$
$798$	0.765844	0.0271106
$799$	−4.39757	−0.155575
$800$	4.44606	0.157192
$801$	5.56858	0.196756
$802$	−2.78085	−0.0981954
$803$	−1.00000	−0.0352892
$804$	17.4576	0.615681
$805$	−2.84766	−0.100367
$806$	5.60232	0.197333
$807$	21.8657	0.769710
$808$	20.6800	0.727519
$809$	−21.0595	−0.740412	−0.370206	−	0.928950i	$-0.620713\pi$
$809$	−21.0595	−0.740412	−0.370206	+	0.928950i	$0.620713\pi$
$810$	−4.69764	−0.165058
$811$	7.80039	0.273909	0.136954	−	0.990577i	$-0.456269\pi$
$811$	7.80039	0.273909	0.136954	+	0.990577i	$0.456269\pi$
$812$	−6.33574	−0.222341
$813$	−13.5666	−0.475801
$814$	3.27348	0.114735
$815$	15.9478	0.558627
$816$	−5.37987	−0.188333
$817$	−7.60403	−0.266031
$818$	2.06566	0.0722241
$819$	−4.11463	−0.143777
$820$	16.2882	0.568808
$821$	−42.7821	−1.49310	−0.746552	−	0.665327i	$-0.768292\pi$
$821$	−42.7821	−1.49310	−0.746552	+	0.665327i	$0.768292\pi$
$822$	−12.2531	−0.427376
$823$	18.2967	0.637784	0.318892	−	0.947791i	$-0.396689\pi$
$823$	18.2967	0.637784	0.318892	+	0.947791i	$0.396689\pi$
$824$	0.612840	0.0213493
$825$	−2.15176	−0.0749148
$826$	−0.269643	−0.00938207
$827$	−46.1825	−1.60592	−0.802962	−	0.596030i	$-0.796744\pi$
$827$	−46.1825	−1.60592	−0.802962	+	0.596030i	$0.796744\pi$
$828$	16.1432	0.561014
$829$	−32.0025	−1.11149	−0.555746	−	0.831352i	$-0.687567\pi$
$829$	−32.0025	−1.11149	−0.555746	+	0.831352i	$0.687567\pi$
$830$	2.15038	0.0746407
$831$	29.7760	1.03292
$832$	19.7352	0.684194
$833$	5.63954	0.195399
$834$	12.1403	0.420385
$835$	14.7812	0.511524
$836$	2.95975	0.102365
$837$	8.21019	0.283786
$838$	7.54747	0.260723
$839$	−54.8320	−1.89301	−0.946506	−	0.322686i	$-0.895414\pi$
$839$	−54.8320	−1.89301	−0.946506	+	0.322686i	$0.895414\pi$
$840$	−1.80642	−0.0623275
$841$	14.7540	0.508758
$842$	−13.1301	−0.452493
$843$	8.98975	0.309623
$844$	30.0694	1.03503
$845$	10.1336	0.348605
$846$	3.57459	0.122897
$847$	0.524806	0.0180326
$848$	30.6905	1.05392
$849$	−28.5170	−0.978700
$850$	−0.350720	−0.0120296
$851$	42.4734	1.45597
$852$	25.1196	0.860582
$853$	2.37091	0.0811783	0.0405892	−	0.999176i	$-0.487077\pi$
$853$	2.37091	0.0811783	0.0405892	+	0.999176i	$0.487077\pi$
$854$	−0.979574	−0.0335203
$855$	2.64348	0.0904052
$856$	24.5507	0.839125
$857$	15.4690	0.528412	0.264206	−	0.964466i	$-0.414890\pi$
$857$	15.4690	0.528412	0.264206	+	0.964466i	$0.414890\pi$
$858$	4.32810	0.147759
$859$	30.9204	1.05499	0.527495	−	0.849558i	$-0.323131\pi$
$859$	30.9204	1.05499	0.527495	+	0.849558i	$0.323131\pi$
$860$	8.55790	0.291822
$861$	−10.0781	−0.343459
$862$	−6.94290	−0.236476
$863$	−28.0729	−0.955614	−0.477807	−	0.878465i	$-0.658568\pi$
$863$	−28.0729	−0.955614	−0.477807	+	0.878465i	$0.658568\pi$
$864$	−13.1058	−0.445867
$865$	−2.99146	−0.101713
$866$	7.69335	0.261431
$867$	−35.0666	−1.19092
$868$	2.66780	0.0905512
$869$	6.71627	0.227834
$870$	5.95230	0.201802
$871$	21.3807	0.724457
$872$	4.10756	0.139100
$873$	20.7194	0.701245
$874$	−3.67990	−0.124474
$875$	0.524806	0.0177417
$876$	−3.92721	−0.132688
$877$	−2.36786	−0.0799570	−0.0399785	−	0.999201i	$-0.512729\pi$
$877$	−2.36786	−0.0799570	−0.0399785	+	0.999201i	$0.512729\pi$
$878$	7.29621	0.246235
$879$	−36.6657	−1.23670
$880$	−2.98125	−0.100498
$881$	−1.80380	−0.0607715	−0.0303858	−	0.999538i	$-0.509674\pi$
$881$	−1.80380	−0.0607715	−0.0303858	+	0.999538i	$0.509674\pi$
$882$	−4.58413	−0.154356
$883$	30.9042	1.04001	0.520005	−	0.854163i	$-0.325930\pi$
$883$	30.9042	1.04001	0.520005	+	0.854163i	$0.325930\pi$
$884$	−7.36189	−0.247607
$885$	−2.64364	−0.0888651
$886$	−9.46371	−0.317939
$887$	30.8936	1.03731	0.518653	−	0.854985i	$-0.326434\pi$
$887$	30.8936	1.03731	0.518653	+	0.854985i	$0.326434\pi$
$888$	26.9431	0.904153
$889$	−0.971824	−0.0325939
$890$	1.42861	0.0478872
$891$	11.2331	0.376322
$892$	0.515421	0.0172576
$893$	8.50353	0.284560
$894$	3.50280	0.117151
$895$	12.4212	0.415194
$896$	−5.56717	−0.185986
$897$	56.1571	1.87503
$898$	−8.10788	−0.270563
$899$	−18.4236	−0.614461
$900$	−2.97509	−0.0991697
$901$	−8.63346	−0.287622
$902$	3.73220	0.124269
$903$	−5.29507	−0.176209
$904$	27.2164	0.905202
$905$	−17.3425	−0.576484
$906$	−1.32362	−0.0439745
$907$	−34.3455	−1.14042	−0.570212	−	0.821498i	$-0.693139\pi$
$907$	−34.3455	−1.14042	−0.570212	+	0.821498i	$0.693139\pi$
$908$	−9.94133	−0.329914
$909$	−21.0734	−0.698962
$910$	−1.05561	−0.0349930
$911$	−12.5911	−0.417162	−0.208581	−	0.978005i	$-0.566885\pi$
$911$	−12.5911	−0.417162	−0.208581	+	0.978005i	$0.566885\pi$
$912$	10.4030	0.344478
$913$	−5.14201	−0.170176
$914$	12.1448	0.401715
$915$	−9.60398	−0.317498
$916$	−19.2003	−0.634396
$917$	5.71111	0.188597
$918$	1.03383	0.0341214
$919$	−28.6797	−0.946057	−0.473029	−	0.881047i	$-0.656839\pi$
$919$	−28.6797	−0.946057	−0.473029	+	0.881047i	$0.656839\pi$
$920$	8.67989	0.286167
$921$	−46.9075	−1.54566
$922$	9.18447	0.302474
$923$	30.7645	1.01262
$924$	2.06102	0.0678027
$925$	−7.82759	−0.257370
$926$	6.85493	0.225267
$927$	−0.624500	−0.0205113
$928$	29.4092	0.965405
$929$	−43.6276	−1.43138	−0.715688	−	0.698420i	$-0.753887\pi$
$929$	−43.6276	−1.43138	−0.715688	+	0.698420i	$0.753887\pi$
$930$	−2.50635	−0.0821864
$931$	−10.9051	−0.357401
$932$	−15.7829	−0.516987
$933$	12.8635	0.421132
$934$	−16.7538	−0.548202
$935$	0.838646	0.0274267
$936$	12.5417	0.409939
$937$	42.3887	1.38478	0.692389	−	0.721524i	$-0.256558\pi$
$937$	42.3887	1.38478	0.692389	+	0.721524i	$0.256558\pi$
$938$	−0.975620	−0.0318551
$939$	−19.3972	−0.633003
$940$	−9.57024	−0.312147
$941$	−54.3286	−1.77106	−0.885530	−	0.464582i	$-0.846205\pi$
$941$	−54.3286	−1.77106	−0.885530	+	0.464582i	$0.846205\pi$
$942$	−7.30528	−0.238019
$943$	48.4253	1.57694
$944$	−3.66275	−0.119212
$945$	−1.54699	−0.0503235
$946$	1.96092	0.0637550
$947$	20.8072	0.676144	0.338072	−	0.941120i	$-0.390225\pi$
$947$	20.8072	0.676144	0.338072	+	0.941120i	$0.390225\pi$
$948$	26.3762	0.856658
$949$	−4.80974	−0.156131
$950$	0.678183	0.0220032
$951$	−7.20051	−0.233493
$952$	0.704050	0.0228184
$953$	0.865352	0.0280315	0.0140157	−	0.999902i	$-0.495539\pi$
$953$	0.865352	0.0280315	0.0140157	+	0.999902i	$0.495539\pi$
$954$	7.01776	0.227208
$955$	−7.94784	−0.257186
$956$	−8.75076	−0.283020
$957$	−14.2332	−0.460095
$958$	12.8338	0.414640
$959$	−7.14609	−0.230759
$960$	−8.82906	−0.284957
$961$	−23.2423	−0.749753
$962$	15.7446	0.507625
$963$	−25.0178	−0.806187
$964$	−30.1740	−0.971838
$965$	18.3433	0.590492
$966$	−2.56250	−0.0824470
$967$	0.381308	0.0122620	0.00613102	−	0.999981i	$-0.498048\pi$
$967$	0.381308	0.0122620	0.00613102	+	0.999981i	$0.498048\pi$
$968$	−1.59965	−0.0514148
$969$	−2.92643	−0.0940106
$970$	5.31554	0.170672
$971$	21.9763	0.705254	0.352627	−	0.935764i	$-0.385289\pi$
$971$	21.9763	0.705254	0.352627	+	0.935764i	$0.385289\pi$
$972$	27.9748	0.897293
$973$	7.08032	0.226985
$974$	2.04075	0.0653899
$975$	−10.3494	−0.331447
$976$	−13.3062	−0.425922
$977$	−1.15141	−0.0368369	−0.0184185	−	0.999830i	$-0.505863\pi$
$977$	−1.15141	−0.0368369	−0.0184185	+	0.999830i	$0.505863\pi$
$978$	14.3508	0.458889
$979$	−3.41612	−0.109180
$980$	12.2731	0.392050
$981$	−4.18571	−0.133639
$982$	−8.15757	−0.260318
$983$	−4.58195	−0.146142	−0.0730708	−	0.997327i	$-0.523280\pi$
$983$	−4.58195	−0.146142	−0.0730708	+	0.997327i	$0.523280\pi$
$984$	30.7187	0.979278
$985$	17.0013	0.541707
$986$	−2.31990	−0.0738806
$987$	5.92144	0.188481
$988$	14.2356	0.452895
$989$	25.4429	0.809038
$990$	−0.681698	−0.0216658
$991$	49.7858	1.58150	0.790749	−	0.612140i	$-0.209691\pi$
$991$	49.7858	1.58150	0.790749	+	0.612140i	$0.209691\pi$
$992$	−12.3834	−0.393174
$993$	−28.4923	−0.904177
$994$	−1.40381	−0.0445262
$995$	−10.6301	−0.336996
$996$	−20.1938	−0.639864
$997$	45.4434	1.43921	0.719604	−	0.694385i	$-0.244324\pi$
$997$	45.4434	1.43921	0.719604	+	0.694385i	$0.244324\pi$
$998$	3.36705	0.106582
$999$	23.0736	0.730017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.c.1.13	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.c.1.13	✓	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.418197 q^{2} +2.15176 q^{3} -1.82511 q^{4} +1.00000 q^{5} +0.899862 q^{6} +0.524806 q^{7} -1.59965 q^{8} +1.63009 q^{9} +O(q^{10})\)	\(q+0.418197 q^{2} +2.15176 q^{3} -1.82511 q^{4} +1.00000 q^{5} +0.899862 q^{6} +0.524806 q^{7} -1.59965 q^{8} +1.63009 q^{9} +0.418197 q^{10} -1.00000 q^{11} -3.92721 q^{12} -4.80974 q^{13} +0.219473 q^{14} +2.15176 q^{15} +2.98125 q^{16} -0.838646 q^{17} +0.681698 q^{18} +1.62168 q^{19} -1.82511 q^{20} +1.12926 q^{21} -0.418197 q^{22} -5.42611 q^{23} -3.44207 q^{24} +1.00000 q^{25} -2.01142 q^{26} -2.94773 q^{27} -0.957830 q^{28} +6.61468 q^{29} +0.899862 q^{30} -2.78526 q^{31} +4.44606 q^{32} -2.15176 q^{33} -0.350720 q^{34} +0.524806 q^{35} -2.97509 q^{36} -7.82759 q^{37} +0.678183 q^{38} -10.3494 q^{39} -1.59965 q^{40} -8.92449 q^{41} +0.472253 q^{42} -4.68898 q^{43} +1.82511 q^{44} +1.63009 q^{45} -2.26919 q^{46} +5.24365 q^{47} +6.41495 q^{48} -6.72458 q^{49} +0.418197 q^{50} -1.80457 q^{51} +8.77830 q^{52} +10.2945 q^{53} -1.23273 q^{54} -1.00000 q^{55} -0.839507 q^{56} +3.48947 q^{57} +2.76624 q^{58} -1.22859 q^{59} -3.92721 q^{60} -4.46331 q^{61} -1.16479 q^{62} +0.855480 q^{63} -4.10317 q^{64} -4.80974 q^{65} -0.899862 q^{66} -4.44529 q^{67} +1.53062 q^{68} -11.6757 q^{69} +0.219473 q^{70} -6.39629 q^{71} -2.60757 q^{72} +1.00000 q^{73} -3.27348 q^{74} +2.15176 q^{75} -2.95975 q^{76} -0.524806 q^{77} -4.32810 q^{78} -6.71627 q^{79} +2.98125 q^{80} -11.2331 q^{81} -3.73220 q^{82} +5.14201 q^{83} -2.06102 q^{84} -0.838646 q^{85} -1.96092 q^{86} +14.2332 q^{87} +1.59965 q^{88} +3.41612 q^{89} +0.681698 q^{90} -2.52418 q^{91} +9.90325 q^{92} -5.99322 q^{93} +2.19288 q^{94} +1.62168 q^{95} +9.56686 q^{96} +12.7106 q^{97} -2.81220 q^{98} -1.63009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9	\( 23 q - 3 q^{2} - 5 q^{3} + 15 q^{4} + 23 q^{5} - 5 q^{6} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 23 q^{11} - 18 q^{12} - 5 q^{13} - 17 q^{14} - 5 q^{15} - q^{16} - 36 q^{17} + q^{18} + 6 q^{19} + 15 q^{20} - 18 q^{21} + 3 q^{22} - 14 q^{23} - 7 q^{24} + 23 q^{25} - 21 q^{26} - 29 q^{27} + 28 q^{28} - 36 q^{29} - 5 q^{30} - 16 q^{31} - 5 q^{32} + 5 q^{33} - 28 q^{34} - 14 q^{36} - 24 q^{37} + q^{38} - 10 q^{39} - 6 q^{40} - 36 q^{41} - 5 q^{42} + 17 q^{43} - 15 q^{44} + 8 q^{45} - 25 q^{46} - 21 q^{47} - 17 q^{48} - 27 q^{49} - 3 q^{50} + 19 q^{51} - 21 q^{52} - 28 q^{53} - 15 q^{54} - 23 q^{55} - 46 q^{56} - 23 q^{57} - 16 q^{58} - 61 q^{59} - 18 q^{60} - 17 q^{61} - 22 q^{62} - 9 q^{63} - 18 q^{64} - 5 q^{65} + 5 q^{66} + 2 q^{67} - 39 q^{68} - 36 q^{69} - 17 q^{70} - 50 q^{71} + 15 q^{72} + 23 q^{73} + 17 q^{74} - 5 q^{75} - 21 q^{76} + 49 q^{78} - 18 q^{79} - q^{80} - 57 q^{81} + 14 q^{82} - 20 q^{83} - 38 q^{84} - 36 q^{85} - 45 q^{86} + 37 q^{87} + 6 q^{88} - 93 q^{89} + q^{90} - 42 q^{91} - 39 q^{92} - 18 q^{93} - 6 q^{94} + 6 q^{95} - 9 q^{96} - 31 q^{97} - 31 q^{98} - 8 q^{99}+O(q^{100}) \) 23 * q - 3 * q^2 - 5 * q^3 + 15 * q^4 + 23 * q^5 - 5 * q^6 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 23 * q^11 - 18 * q^12 - 5 * q^13 - 17 * q^14 - 5 * q^15 - q^16 - 36 * q^17 + q^18 + 6 * q^19 + 15 * q^20 - 18 * q^21 + 3 * q^22 - 14 * q^23 - 7 * q^24 + 23 * q^25 - 21 * q^26 - 29 * q^27 + 28 * q^28 - 36 * q^29 - 5 * q^30 - 16 * q^31 - 5 * q^32 + 5 * q^33 - 28 * q^34 - 14 * q^36 - 24 * q^37 + q^38 - 10 * q^39 - 6 * q^40 - 36 * q^41 - 5 * q^42 + 17 * q^43 - 15 * q^44 + 8 * q^45 - 25 * q^46 - 21 * q^47 - 17 * q^48 - 27 * q^49 - 3 * q^50 + 19 * q^51 - 21 * q^52 - 28 * q^53 - 15 * q^54 - 23 * q^55 - 46 * q^56 - 23 * q^57 - 16 * q^58 - 61 * q^59 - 18 * q^60 - 17 * q^61 - 22 * q^62 - 9 * q^63 - 18 * q^64 - 5 * q^65 + 5 * q^66 + 2 * q^67 - 39 * q^68 - 36 * q^69 - 17 * q^70 - 50 * q^71 + 15 * q^72 + 23 * q^73 + 17 * q^74 - 5 * q^75 - 21 * q^76 + 49 * q^78 - 18 * q^79 - q^80 - 57 * q^81 + 14 * q^82 - 20 * q^83 - 38 * q^84 - 36 * q^85 - 45 * q^86 + 37 * q^87 + 6 * q^88 - 93 * q^89 + q^90 - 42 * q^91 - 39 * q^92 - 18 * q^93 - 6 * q^94 + 6 * q^95 - 9 * q^96 - 31 * q^97 - 31 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more