LMFDB - Embedded newform 6045.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 20x^{5} - 38x^{4} - 27x^{3} + 13x^{2} + 6x - 1 $ x^9 - 3x^8 - 7x^7 + 20x^6 + 20x^5 - 38x^4 - 27x^3 + 13x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.89284$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89284 q^{2} +1.00000 q^{3} +1.58283 q^{4} +1.00000 q^{5} -1.89284 q^{6} -4.37246 q^{7} +0.789632 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.89284 q^{2} +1.00000 q^{3} +1.58283 q^{4} +1.00000 q^{5} -1.89284 q^{6} -4.37246 q^{7} +0.789632 q^{8} +1.00000 q^{9} -1.89284 q^{10} +6.12628 q^{11} +1.58283 q^{12} -1.00000 q^{13} +8.27636 q^{14} +1.00000 q^{15} -4.66031 q^{16} -2.63346 q^{17} -1.89284 q^{18} -3.65166 q^{19} +1.58283 q^{20} -4.37246 q^{21} -11.5961 q^{22} -3.43477 q^{23} +0.789632 q^{24} +1.00000 q^{25} +1.89284 q^{26} +1.00000 q^{27} -6.92087 q^{28} +9.43934 q^{29} -1.89284 q^{30} -1.00000 q^{31} +7.24194 q^{32} +6.12628 q^{33} +4.98470 q^{34} -4.37246 q^{35} +1.58283 q^{36} -4.45352 q^{37} +6.91199 q^{38} -1.00000 q^{39} +0.789632 q^{40} +11.1894 q^{41} +8.27636 q^{42} -10.1255 q^{43} +9.69688 q^{44} +1.00000 q^{45} +6.50147 q^{46} -6.76351 q^{47} -4.66031 q^{48} +12.1184 q^{49} -1.89284 q^{50} -2.63346 q^{51} -1.58283 q^{52} +12.4465 q^{53} -1.89284 q^{54} +6.12628 q^{55} -3.45264 q^{56} -3.65166 q^{57} -17.8671 q^{58} -9.15041 q^{59} +1.58283 q^{60} -8.76708 q^{61} +1.89284 q^{62} -4.37246 q^{63} -4.38719 q^{64} -1.00000 q^{65} -11.5961 q^{66} -9.84253 q^{67} -4.16832 q^{68} -3.43477 q^{69} +8.27636 q^{70} -3.22454 q^{71} +0.789632 q^{72} +2.69524 q^{73} +8.42978 q^{74} +1.00000 q^{75} -5.77996 q^{76} -26.7870 q^{77} +1.89284 q^{78} -4.29920 q^{79} -4.66031 q^{80} +1.00000 q^{81} -21.1797 q^{82} +8.73838 q^{83} -6.92087 q^{84} -2.63346 q^{85} +19.1659 q^{86} +9.43934 q^{87} +4.83751 q^{88} -2.21335 q^{89} -1.89284 q^{90} +4.37246 q^{91} -5.43667 q^{92} -1.00000 q^{93} +12.8022 q^{94} -3.65166 q^{95} +7.24194 q^{96} +4.29194 q^{97} -22.9382 q^{98} +6.12628 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q - 3 * q^2 + 9 * q^3 + 5 * q^4 + 9 * q^5 - 3 * q^6 - 5 * q^7 - 18 * q^8 + 9 * q^9	\( 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9} - 3 q^{10} + 5 q^{12} - 9 q^{13} + 11 q^{14} + 9 q^{15} + 9 q^{16} - 5 q^{17} - 3 q^{18} - 12 q^{19} + 5 q^{20} - 5 q^{21} - 17 q^{22} - 21 q^{23} - 18 q^{24} + 9 q^{25} + 3 q^{26} + 9 q^{27} - 8 q^{28} - 15 q^{29} - 3 q^{30} - 9 q^{31} - 9 q^{32} + 9 q^{34} - 5 q^{35} + 5 q^{36} - 13 q^{37} - 24 q^{38} - 9 q^{39} - 18 q^{40} - 11 q^{41} + 11 q^{42} - 26 q^{43} + 5 q^{44} + 9 q^{45} + 14 q^{46} - 21 q^{47} + 9 q^{48} - 26 q^{49} - 3 q^{50} - 5 q^{51} - 5 q^{52} + 18 q^{53} - 3 q^{54} - 19 q^{56} - 12 q^{57} - 16 q^{58} - 24 q^{59} + 5 q^{60} - 4 q^{61} + 3 q^{62} - 5 q^{63} + 2 q^{64} - 9 q^{65} - 17 q^{66} - 20 q^{67} - 19 q^{68} - 21 q^{69} + 11 q^{70} - 13 q^{71} - 18 q^{72} + 15 q^{73} + 12 q^{74} + 9 q^{75} + 54 q^{76} - 28 q^{77} + 3 q^{78} - 15 q^{79} + 9 q^{80} + 9 q^{81} - 21 q^{82} - 5 q^{83} - 8 q^{84} - 5 q^{85} + 36 q^{86} - 15 q^{87} + 23 q^{88} + 31 q^{89} - 3 q^{90} + 5 q^{91} - 7 q^{92} - 9 q^{93} - 12 q^{95} - 9 q^{96} - 11 q^{97} - 23 q^{98}+O(q^{100}) \) 9 * q - 3 * q^2 + 9 * q^3 + 5 * q^4 + 9 * q^5 - 3 * q^6 - 5 * q^7 - 18 * q^8 + 9 * q^9 - 3 * q^10 + 5 * q^12 - 9 * q^13 + 11 * q^14 + 9 * q^15 + 9 * q^16 - 5 * q^17 - 3 * q^18 - 12 * q^19 + 5 * q^20 - 5 * q^21 - 17 * q^22 - 21 * q^23 - 18 * q^24 + 9 * q^25 + 3 * q^26 + 9 * q^27 - 8 * q^28 - 15 * q^29 - 3 * q^30 - 9 * q^31 - 9 * q^32 + 9 * q^34 - 5 * q^35 + 5 * q^36 - 13 * q^37 - 24 * q^38 - 9 * q^39 - 18 * q^40 - 11 * q^41 + 11 * q^42 - 26 * q^43 + 5 * q^44 + 9 * q^45 + 14 * q^46 - 21 * q^47 + 9 * q^48 - 26 * q^49 - 3 * q^50 - 5 * q^51 - 5 * q^52 + 18 * q^53 - 3 * q^54 - 19 * q^56 - 12 * q^57 - 16 * q^58 - 24 * q^59 + 5 * q^60 - 4 * q^61 + 3 * q^62 - 5 * q^63 + 2 * q^64 - 9 * q^65 - 17 * q^66 - 20 * q^67 - 19 * q^68 - 21 * q^69 + 11 * q^70 - 13 * q^71 - 18 * q^72 + 15 * q^73 + 12 * q^74 + 9 * q^75 + 54 * q^76 - 28 * q^77 + 3 * q^78 - 15 * q^79 + 9 * q^80 + 9 * q^81 - 21 * q^82 - 5 * q^83 - 8 * q^84 - 5 * q^85 + 36 * q^86 - 15 * q^87 + 23 * q^88 + 31 * q^89 - 3 * q^90 + 5 * q^91 - 7 * q^92 - 9 * q^93 - 12 * q^95 - 9 * q^96 - 11 * q^97 - 23 * q^98

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 $

$2$	−1.89284	−1.33844	−0.669219	−	0.743065i	$-0.733371\pi$
$2$	−1.89284	−1.33844	−0.669219	+	0.743065i	$0.733371\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.58283	0.791416
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.89284	−0.772747
$7$	−4.37246	−1.65264	−0.826318	−	0.563204i	$-0.809569\pi$
$7$	−4.37246	−1.65264	−0.826318	+	0.563204i	$0.809569\pi$
$8$	0.789632	0.279177
$9$	1.00000	0.333333
$10$	−1.89284	−0.598568
$11$	6.12628	1.84714	0.923572	−	0.383425i	$-0.125255\pi$
$11$	6.12628	1.84714	0.923572	+	0.383425i	$0.125255\pi$
$12$	1.58283	0.456924
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	8.27636	2.21195
$15$	1.00000	0.258199
$16$	−4.66031	−1.16508
$17$	−2.63346	−0.638707	−0.319353	−	0.947636i	$-0.603466\pi$
$17$	−2.63346	−0.638707	−0.319353	+	0.947636i	$0.603466\pi$
$18$	−1.89284	−0.446146
$19$	−3.65166	−0.837748	−0.418874	−	0.908044i	$-0.637575\pi$
$19$	−3.65166	−0.837748	−0.418874	+	0.908044i	$0.637575\pi$
$20$	1.58283	0.353932
$21$	−4.37246	−0.954150
$22$	−11.5961	−2.47229
$23$	−3.43477	−0.716200	−0.358100	−	0.933683i	$-0.616575\pi$
$23$	−3.43477	−0.716200	−0.358100	+	0.933683i	$0.616575\pi$
$24$	0.789632	0.161183
$25$	1.00000	0.200000
$26$	1.89284	0.371216
$27$	1.00000	0.192450
$28$	−6.92087	−1.30792
$29$	9.43934	1.75284	0.876421	−	0.481546i	$-0.159925\pi$
$29$	9.43934	1.75284	0.876421	+	0.481546i	$0.159925\pi$
$30$	−1.89284	−0.345583
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	7.24194	1.28021
$33$	6.12628	1.06645
$34$	4.98470	0.854870
$35$	−4.37246	−0.739081
$36$	1.58283	0.263805
$37$	−4.45352	−0.732154	−0.366077	−	0.930585i	$-0.619299\pi$
$37$	−4.45352	−0.732154	−0.366077	+	0.930585i	$0.619299\pi$
$38$	6.91199	1.12127
$39$	−1.00000	−0.160128
$40$	0.789632	0.124852
$41$	11.1894	1.74749	0.873745	−	0.486384i	$-0.161684\pi$
$41$	11.1894	1.74749	0.873745	+	0.486384i	$0.161684\pi$
$42$	8.27636	1.27707
$43$	−10.1255	−1.54412	−0.772061	−	0.635548i	$-0.780774\pi$
$43$	−10.1255	−1.54412	−0.772061	+	0.635548i	$0.780774\pi$
$44$	9.69688	1.46186
$45$	1.00000	0.149071
$46$	6.50147	0.958589
$47$	−6.76351	−0.986560	−0.493280	−	0.869871i	$-0.664202\pi$
$47$	−6.76351	−0.986560	−0.493280	+	0.869871i	$0.664202\pi$
$48$	−4.66031	−0.672657
$49$	12.1184	1.73121
$50$	−1.89284	−0.267688
$51$	−2.63346	−0.368758
$52$	−1.58283	−0.219499
$53$	12.4465	1.70966	0.854828	−	0.518912i	$-0.173663\pi$
$53$	12.4465	1.70966	0.854828	+	0.518912i	$0.173663\pi$
$54$	−1.89284	−0.257582
$55$	6.12628	0.826068
$56$	−3.45264	−0.461378
$57$	−3.65166	−0.483674
$58$	−17.8671	−2.34607
$59$	−9.15041	−1.19128	−0.595641	−	0.803251i	$-0.703102\pi$
$59$	−9.15041	−1.19128	−0.595641	+	0.803251i	$0.703102\pi$
$60$	1.58283	0.204343
$61$	−8.76708	−1.12251	−0.561255	−	0.827643i	$-0.689681\pi$
$61$	−8.76708	−1.12251	−0.561255	+	0.827643i	$0.689681\pi$
$62$	1.89284	0.240391
$63$	−4.37246	−0.550879
$64$	−4.38719	−0.548399
$65$	−1.00000	−0.124035
$66$	−11.5961	−1.42738
$67$	−9.84253	−1.20246	−0.601228	−	0.799077i	$-0.705322\pi$
$67$	−9.84253	−1.20246	−0.601228	+	0.799077i	$0.705322\pi$
$68$	−4.16832	−0.505483
$69$	−3.43477	−0.413498
$70$	8.27636	0.989214
$71$	−3.22454	−0.382683	−0.191341	−	0.981524i	$-0.561284\pi$
$71$	−3.22454	−0.382683	−0.191341	+	0.981524i	$0.561284\pi$
$72$	0.789632	0.0930590
$73$	2.69524	0.315453	0.157727	−	0.987483i	$-0.449583\pi$
$73$	2.69524	0.315453	0.157727	+	0.987483i	$0.449583\pi$
$74$	8.42978	0.979943
$75$	1.00000	0.115470
$76$	−5.77996	−0.663007
$77$	−26.7870	−3.05266
$78$	1.89284	0.214322
$79$	−4.29920	−0.483697	−0.241849	−	0.970314i	$-0.577754\pi$
$79$	−4.29920	−0.483697	−0.241849	+	0.970314i	$0.577754\pi$
$80$	−4.66031	−0.521038
$81$	1.00000	0.111111
$82$	−21.1797	−2.33891
$83$	8.73838	0.959161	0.479581	−	0.877498i	$-0.340789\pi$
$83$	8.73838	0.959161	0.479581	+	0.877498i	$0.340789\pi$
$84$	−6.92087	−0.755129
$85$	−2.63346	−0.285638
$86$	19.1659	2.06671
$87$	9.43934	1.01200
$88$	4.83751	0.515680
$89$	−2.21335	−0.234615	−0.117307	−	0.993096i	$-0.537426\pi$
$89$	−2.21335	−0.234615	−0.117307	+	0.993096i	$0.537426\pi$
$90$	−1.89284	−0.199523
$91$	4.37246	0.458359
$92$	−5.43667	−0.566812
$93$	−1.00000	−0.103695
$94$	12.8022	1.32045
$95$	−3.65166	−0.374652
$96$	7.24194	0.739127
$97$	4.29194	0.435780	0.217890	−	0.975973i	$-0.430083\pi$
$97$	4.29194	0.435780	0.217890	+	0.975973i	$0.430083\pi$
$98$	−22.9382	−2.31711
$99$	6.12628	0.615715
$100$	1.58283	0.158283
$101$	−10.7430	−1.06897	−0.534485	−	0.845178i	$-0.679494\pi$
$101$	−10.7430	−1.06897	−0.534485	+	0.845178i	$0.679494\pi$
$102$	4.98470	0.493559
$103$	2.89126	0.284884	0.142442	−	0.989803i	$-0.454505\pi$
$103$	2.89126	0.284884	0.142442	+	0.989803i	$0.454505\pi$
$104$	−0.789632	−0.0774298
$105$	−4.37246	−0.426709
$106$	−23.5592	−2.28827
$107$	0.0632155	0.00611128	0.00305564	−	0.999995i	$-0.499027\pi$
$107$	0.0632155	0.00611128	0.00305564	+	0.999995i	$0.499027\pi$
$108$	1.58283	0.152308
$109$	13.1309	1.25771	0.628855	−	0.777523i	$-0.283524\pi$
$109$	13.1309	1.25771	0.628855	+	0.777523i	$0.283524\pi$
$110$	−11.5961	−1.10564
$111$	−4.45352	−0.422709
$112$	20.3770	1.92545
$113$	−8.84392	−0.831966	−0.415983	−	0.909372i	$-0.636562\pi$
$113$	−8.84392	−0.831966	−0.415983	+	0.909372i	$0.636562\pi$
$114$	6.91199	0.647368
$115$	−3.43477	−0.320294
$116$	14.9409	1.38723
$117$	−1.00000	−0.0924500
$118$	17.3202	1.59446
$119$	11.5147	1.05555
$120$	0.789632	0.0720832
$121$	26.5314	2.41194
$122$	16.5947	1.50241
$123$	11.1894	1.00891
$124$	−1.58283	−0.142142
$125$	1.00000	0.0894427
$126$	8.27636	0.737317
$127$	−18.5243	−1.64377	−0.821883	−	0.569656i	$-0.807076\pi$
$127$	−18.5243	−1.64377	−0.821883	+	0.569656i	$0.807076\pi$
$128$	−6.17963	−0.546208
$129$	−10.1255	−0.891499
$130$	1.89284	0.166013
$131$	−12.1331	−1.06008	−0.530038	−	0.847974i	$-0.677822\pi$
$131$	−12.1331	−1.06008	−0.530038	+	0.847974i	$0.677822\pi$
$132$	9.69688	0.844005
$133$	15.9667	1.38449
$134$	18.6303	1.60941
$135$	1.00000	0.0860663
$136$	−2.07946	−0.178312
$137$	−4.71061	−0.402454	−0.201227	−	0.979545i	$-0.564493\pi$
$137$	−4.71061	−0.402454	−0.201227	+	0.979545i	$0.564493\pi$
$138$	6.50147	0.553441
$139$	−17.9055	−1.51873	−0.759363	−	0.650667i	$-0.774489\pi$
$139$	−17.9055	−1.51873	−0.759363	+	0.650667i	$0.774489\pi$
$140$	−6.92087	−0.584921
$141$	−6.76351	−0.569591
$142$	6.10353	0.512197
$143$	−6.12628	−0.512306
$144$	−4.66031	−0.388359
$145$	9.43934	0.783895
$146$	−5.10164	−0.422215
$147$	12.1184	0.999512
$148$	−7.04917	−0.579438
$149$	11.3954	0.933544	0.466772	−	0.884378i	$-0.345417\pi$
$149$	11.3954	0.933544	0.466772	+	0.884378i	$0.345417\pi$
$150$	−1.89284	−0.154549
$151$	5.00986	0.407697	0.203848	−	0.979002i	$-0.434655\pi$
$151$	5.00986	0.407697	0.203848	+	0.979002i	$0.434655\pi$
$152$	−2.88347	−0.233880
$153$	−2.63346	−0.212902
$154$	50.7033	4.08579
$155$	−1.00000	−0.0803219
$156$	−1.58283	−0.126728
$157$	15.7684	1.25846	0.629228	−	0.777221i	$-0.283371\pi$
$157$	15.7684	1.25846	0.629228	+	0.777221i	$0.283371\pi$
$158$	8.13768	0.647399
$159$	12.4465	0.987070
$160$	7.24194	0.572525
$161$	15.0184	1.18362
$162$	−1.89284	−0.148715
$163$	−19.9248	−1.56063	−0.780315	−	0.625387i	$-0.784941\pi$
$163$	−19.9248	−1.56063	−0.780315	+	0.625387i	$0.784941\pi$
$164$	17.7109	1.38299
$165$	6.12628	0.476931
$166$	−16.5403	−1.28378
$167$	11.0736	0.856900	0.428450	−	0.903565i	$-0.359060\pi$
$167$	11.0736	0.856900	0.428450	+	0.903565i	$0.359060\pi$
$168$	−3.45264	−0.266377
$169$	1.00000	0.0769231
$170$	4.98470	0.382309
$171$	−3.65166	−0.279249
$172$	−16.0269	−1.22204
$173$	7.82170	0.594673	0.297336	−	0.954773i	$-0.403902\pi$
$173$	7.82170	0.594673	0.297336	+	0.954773i	$0.403902\pi$
$174$	−17.8671	−1.35450
$175$	−4.37246	−0.330527
$176$	−28.5504	−2.15206
$177$	−9.15041	−0.687787
$178$	4.18951	0.314017
$179$	1.84650	0.138014	0.0690068	−	0.997616i	$-0.478017\pi$
$179$	1.84650	0.138014	0.0690068	+	0.997616i	$0.478017\pi$
$180$	1.58283	0.117977
$181$	−12.8828	−0.957574	−0.478787	−	0.877931i	$-0.658923\pi$
$181$	−12.8828	−0.957574	−0.478787	+	0.877931i	$0.658923\pi$
$182$	−8.27636	−0.613485
$183$	−8.76708	−0.648081
$184$	−2.71221	−0.199946
$185$	−4.45352	−0.327429
$186$	1.89284	0.138790
$187$	−16.1333	−1.17978
$188$	−10.7055	−0.780779
$189$	−4.37246	−0.318050
$190$	6.91199	0.501449
$191$	6.48173	0.469001	0.234501	−	0.972116i	$-0.424655\pi$
$191$	6.48173	0.469001	0.234501	+	0.972116i	$0.424655\pi$
$192$	−4.38719	−0.316618
$193$	−2.24614	−0.161681	−0.0808404	−	0.996727i	$-0.525760\pi$
$193$	−2.24614	−0.161681	−0.0808404	+	0.996727i	$0.525760\pi$
$194$	−8.12394	−0.583265
$195$	−1.00000	−0.0716115
$196$	19.1814	1.37010
$197$	11.6309	0.828665	0.414333	−	0.910126i	$-0.364015\pi$
$197$	11.6309	0.828665	0.414333	+	0.910126i	$0.364015\pi$
$198$	−11.5961	−0.824096
$199$	21.0403	1.49151	0.745753	−	0.666222i	$-0.232090\pi$
$199$	21.0403	1.49151	0.745753	+	0.666222i	$0.232090\pi$
$200$	0.789632	0.0558354
$201$	−9.84253	−0.694238
$202$	20.3348	1.43075
$203$	−41.2732	−2.89681
$204$	−4.16832	−0.291841
$205$	11.1894	0.781501
$206$	−5.47268	−0.381300
$207$	−3.43477	−0.238733
$208$	4.66031	0.323134
$209$	−22.3711	−1.54744
$210$	8.27636	0.571123
$211$	−17.2842	−1.18990	−0.594948	−	0.803764i	$-0.702827\pi$
$211$	−17.2842	−1.18990	−0.594948	+	0.803764i	$0.702827\pi$
$212$	19.7007	1.35305
$213$	−3.22454	−0.220942
$214$	−0.119657	−0.00817956
$215$	−10.1255	−0.690552
$216$	0.789632	0.0537276
$217$	4.37246	0.296822
$218$	−24.8546	−1.68337
$219$	2.69524	0.182127
$220$	9.69688	0.653763
$221$	2.63346	0.177145
$222$	8.42978	0.565770
$223$	3.64370	0.244000	0.122000	−	0.992530i	$-0.461069\pi$
$223$	3.64370	0.244000	0.122000	+	0.992530i	$0.461069\pi$
$224$	−31.6651	−2.11571
$225$	1.00000	0.0666667
$226$	16.7401	1.11354
$227$	−18.6493	−1.23780	−0.618900	−	0.785470i	$-0.712421\pi$
$227$	−18.6493	−1.23780	−0.618900	+	0.785470i	$0.712421\pi$
$228$	−5.77996	−0.382787
$229$	−23.8119	−1.57353	−0.786767	−	0.617250i	$-0.788247\pi$
$229$	−23.8119	−1.57353	−0.786767	+	0.617250i	$0.788247\pi$
$230$	6.50147	0.428694
$231$	−26.7870	−1.76245
$232$	7.45360	0.489353
$233$	−14.5787	−0.955084	−0.477542	−	0.878609i	$-0.658472\pi$
$233$	−14.5787	−0.955084	−0.477542	+	0.878609i	$0.658472\pi$
$234$	1.89284	0.123739
$235$	−6.76351	−0.441203
$236$	−14.4836	−0.942800
$237$	−4.29920	−0.279263
$238$	−21.7954	−1.41279
$239$	17.5006	1.13202	0.566009	−	0.824399i	$-0.308487\pi$
$239$	17.5006	1.13202	0.566009	+	0.824399i	$0.308487\pi$
$240$	−4.66031	−0.300822
$241$	3.83059	0.246750	0.123375	−	0.992360i	$-0.460628\pi$
$241$	3.83059	0.246750	0.123375	+	0.992360i	$0.460628\pi$
$242$	−50.2195	−3.22823
$243$	1.00000	0.0641500
$244$	−13.8768	−0.888372
$245$	12.1184	0.774218
$246$	−21.1797	−1.35037
$247$	3.65166	0.232349
$248$	−0.789632	−0.0501417
$249$	8.73838	0.553772
$250$	−1.89284	−0.119714
$251$	−21.8423	−1.37867	−0.689337	−	0.724440i	$-0.742098\pi$
$251$	−21.8423	−1.37867	−0.689337	+	0.724440i	$0.742098\pi$
$252$	−6.92087	−0.435974
$253$	−21.0424	−1.32292
$254$	35.0635	2.20008
$255$	−2.63346	−0.164913
$256$	20.4714	1.27946
$257$	−11.2931	−0.704442	−0.352221	−	0.935917i	$-0.614573\pi$
$257$	−11.2931	−0.704442	−0.352221	+	0.935917i	$0.614573\pi$
$258$	19.1659	1.19322
$259$	19.4728	1.20998
$260$	−1.58283	−0.0981631
$261$	9.43934	0.584281
$262$	22.9660	1.41885
$263$	24.8531	1.53251	0.766255	−	0.642537i	$-0.222118\pi$
$263$	24.8531	1.53251	0.766255	+	0.642537i	$0.222118\pi$
$264$	4.83751	0.297728
$265$	12.4465	0.764581
$266$	−30.2224	−1.85306
$267$	−2.21335	−0.135455
$268$	−15.5791	−0.951643
$269$	−20.6142	−1.25687	−0.628436	−	0.777861i	$-0.716305\pi$
$269$	−20.6142	−1.25687	−0.628436	+	0.777861i	$0.716305\pi$
$270$	−1.89284	−0.115194
$271$	−10.1536	−0.616785	−0.308393	−	0.951259i	$-0.599791\pi$
$271$	−10.1536	−0.616785	−0.308393	+	0.951259i	$0.599791\pi$
$272$	12.2727	0.744143
$273$	4.37246	0.264634
$274$	8.91641	0.538660
$275$	6.12628	0.369429
$276$	−5.43667	−0.327249
$277$	−25.7637	−1.54799	−0.773995	−	0.633192i	$-0.781744\pi$
$277$	−25.7637	−1.54799	−0.773995	+	0.633192i	$0.781744\pi$
$278$	33.8922	2.03272
$279$	−1.00000	−0.0598684
$280$	−3.45264	−0.206334
$281$	−21.6943	−1.29418	−0.647088	−	0.762415i	$-0.724013\pi$
$281$	−21.6943	−1.29418	−0.647088	+	0.762415i	$0.724013\pi$
$282$	12.8022	0.762361
$283$	24.6176	1.46336	0.731682	−	0.681646i	$-0.238736\pi$
$283$	24.6176	1.46336	0.731682	+	0.681646i	$0.238736\pi$
$284$	−5.10391	−0.302861
$285$	−3.65166	−0.216306
$286$	11.5961	0.685689
$287$	−48.9252	−2.88796
$288$	7.24194	0.426735
$289$	−10.0649	−0.592053
$290$	−17.8671	−1.04919
$291$	4.29194	0.251598
$292$	4.26610	0.249655
$293$	−0.239981	−0.0140198	−0.00700991	−	0.999975i	$-0.502231\pi$
$293$	−0.239981	−0.0140198	−0.00700991	+	0.999975i	$0.502231\pi$
$294$	−22.9382	−1.33778
$295$	−9.15041	−0.532758
$296$	−3.51664	−0.204401
$297$	6.12628	0.355483
$298$	−21.5696	−1.24949
$299$	3.43477	0.198638
$300$	1.58283	0.0913848
$301$	44.2733	2.55187
$302$	−9.48285	−0.545677
$303$	−10.7430	−0.617170
$304$	17.0179	0.976041
$305$	−8.76708	−0.502002
$306$	4.98470	0.284957
$307$	7.21498	0.411781	0.205890	−	0.978575i	$-0.433991\pi$
$307$	7.21498	0.411781	0.205890	+	0.978575i	$0.433991\pi$
$308$	−42.3992	−2.41592
$309$	2.89126	0.164478
$310$	1.89284	0.107506
$311$	−0.515701	−0.0292427	−0.0146214	−	0.999893i	$-0.504654\pi$
$311$	−0.515701	−0.0292427	−0.0146214	+	0.999893i	$0.504654\pi$
$312$	−0.789632	−0.0447041
$313$	−13.3780	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$313$	−13.3780	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$314$	−29.8470	−1.68436
$315$	−4.37246	−0.246360
$316$	−6.80490	−0.382806
$317$	16.4976	0.926597	0.463299	−	0.886202i	$-0.346666\pi$
$317$	16.4976	0.926597	0.463299	+	0.886202i	$0.346666\pi$
$318$	−23.5592	−1.32113
$319$	57.8281	3.23775
$320$	−4.38719	−0.245252
$321$	0.0632155	0.00352835
$322$	−28.4274	−1.58420
$323$	9.61648	0.535075
$324$	1.58283	0.0879351
$325$	−1.00000	−0.0554700
$326$	37.7144	2.08881
$327$	13.1309	0.726139
$328$	8.83550	0.487859
$329$	29.5732	1.63042
$330$	−11.5961	−0.638342
$331$	−22.3078	−1.22615	−0.613074	−	0.790026i	$-0.710067\pi$
$331$	−22.3078	−1.22615	−0.613074	+	0.790026i	$0.710067\pi$
$332$	13.8314	0.759095
$333$	−4.45352	−0.244051
$334$	−20.9605	−1.14691
$335$	−9.84253	−0.537755
$336$	20.3770	1.11166
$337$	34.4010	1.87394	0.936971	−	0.349408i	$-0.113617\pi$
$337$	34.4010	1.87394	0.936971	+	0.349408i	$0.113617\pi$
$338$	−1.89284	−0.102957
$339$	−8.84392	−0.480336
$340$	−4.16832	−0.226059
$341$	−6.12628	−0.331757
$342$	6.91199	0.373758
$343$	−22.3802	−1.20842
$344$	−7.99540	−0.431083
$345$	−3.43477	−0.184922
$346$	−14.8052	−0.795932
$347$	−20.3623	−1.09310	−0.546552	−	0.837425i	$-0.684060\pi$
$347$	−20.3623	−1.09310	−0.546552	+	0.837425i	$0.684060\pi$
$348$	14.9409	0.800916
$349$	−3.09881	−0.165875	−0.0829377	−	0.996555i	$-0.526430\pi$
$349$	−3.09881	−0.165875	−0.0829377	+	0.996555i	$0.526430\pi$
$350$	8.27636	0.442390
$351$	−1.00000	−0.0533761
$352$	44.3662	2.36472
$353$	−6.86557	−0.365417	−0.182709	−	0.983167i	$-0.558487\pi$
$353$	−6.86557	−0.365417	−0.182709	+	0.983167i	$0.558487\pi$
$354$	17.3202	0.920560
$355$	−3.22454	−0.171141
$356$	−3.50336	−0.185678
$357$	11.5147	0.609422
$358$	−3.49512	−0.184723
$359$	28.9847	1.52975	0.764877	−	0.644176i	$-0.222800\pi$
$359$	28.9847	1.52975	0.764877	+	0.644176i	$0.222800\pi$
$360$	0.789632	0.0416172
$361$	−5.66539	−0.298178
$362$	24.3851	1.28165
$363$	26.5314	1.39254
$364$	6.92087	0.362752
$365$	2.69524	0.141075
$366$	16.5947	0.867417
$367$	−0.557978	−0.0291262	−0.0145631	−	0.999894i	$-0.504636\pi$
$367$	−0.557978	−0.0291262	−0.0145631	+	0.999894i	$0.504636\pi$
$368$	16.0071	0.834428
$369$	11.1894	0.582497
$370$	8.42978	0.438244
$371$	−54.4218	−2.82544
$372$	−1.58283	−0.0820660
$373$	−6.17871	−0.319922	−0.159961	−	0.987123i	$-0.551137\pi$
$373$	−6.17871	−0.319922	−0.159961	+	0.987123i	$0.551137\pi$
$374$	30.5377	1.57907
$375$	1.00000	0.0516398
$376$	−5.34068	−0.275425
$377$	−9.43934	−0.486151
$378$	8.27636	0.425690
$379$	−22.7752	−1.16989	−0.584943	−	0.811074i	$-0.698883\pi$
$379$	−22.7752	−1.16989	−0.584943	+	0.811074i	$0.698883\pi$
$380$	−5.77996	−0.296506
$381$	−18.5243	−0.949029
$382$	−12.2688	−0.627729
$383$	18.1663	0.928256	0.464128	−	0.885768i	$-0.346368\pi$
$383$	18.1663	0.928256	0.464128	+	0.885768i	$0.346368\pi$
$384$	−6.17963	−0.315353
$385$	−26.7870	−1.36519
$386$	4.25158	0.216400
$387$	−10.1255	−0.514707
$388$	6.79342	0.344883
$389$	−36.8173	−1.86671	−0.933356	−	0.358951i	$-0.883135\pi$
$389$	−36.8173	−1.86671	−0.933356	+	0.358951i	$0.883135\pi$
$390$	1.89284	0.0958475
$391$	9.04532	0.457442
$392$	9.56910	0.483313
$393$	−12.1331	−0.612035
$394$	−22.0153	−1.10912
$395$	−4.29920	−0.216316
$396$	9.69688	0.487286
$397$	−19.5735	−0.982367	−0.491184	−	0.871056i	$-0.663436\pi$
$397$	−19.5735	−0.982367	−0.491184	+	0.871056i	$0.663436\pi$
$398$	−39.8258	−1.99629
$399$	15.9667	0.799337
$400$	−4.66031	−0.233015
$401$	−10.9947	−0.549050	−0.274525	−	0.961580i	$-0.588521\pi$
$401$	−10.9947	−0.549050	−0.274525	+	0.961580i	$0.588521\pi$
$402$	18.6303	0.929195
$403$	1.00000	0.0498135
$404$	−17.0044	−0.845999
$405$	1.00000	0.0496904
$406$	78.1234	3.87720
$407$	−27.2835	−1.35239
$408$	−2.07946	−0.102949
$409$	15.3007	0.756571	0.378286	−	0.925689i	$-0.376514\pi$
$409$	15.3007	0.756571	0.378286	+	0.925689i	$0.376514\pi$
$410$	−21.1797	−1.04599
$411$	−4.71061	−0.232357
$412$	4.57637	0.225462
$413$	40.0098	1.96876
$414$	6.50147	0.319530
$415$	8.73838	0.428950
$416$	−7.24194	−0.355065
$417$	−17.9055	−0.876837
$418$	42.3448	2.07115
$419$	−19.1286	−0.934492	−0.467246	−	0.884127i	$-0.654754\pi$
$419$	−19.1286	−0.934492	−0.467246	+	0.884127i	$0.654754\pi$
$420$	−6.92087	−0.337704
$421$	14.8550	0.723990	0.361995	−	0.932180i	$-0.382096\pi$
$421$	14.8550	0.723990	0.361995	+	0.932180i	$0.382096\pi$
$422$	32.7162	1.59260
$423$	−6.76351	−0.328853
$424$	9.82813	0.477296
$425$	−2.63346	−0.127741
$426$	6.10353	0.295717
$427$	38.3337	1.85510
$428$	0.100060	0.00483656
$429$	−6.12628	−0.295780
$430$	19.1659	0.924261
$431$	−7.60622	−0.366379	−0.183189	−	0.983078i	$-0.558642\pi$
$431$	−7.60622	−0.366379	−0.183189	+	0.983078i	$0.558642\pi$
$432$	−4.66031	−0.224219
$433$	−16.2081	−0.778912	−0.389456	−	0.921045i	$-0.627337\pi$
$433$	−16.2081	−0.778912	−0.389456	+	0.921045i	$0.627337\pi$
$434$	−8.27636	−0.397278
$435$	9.43934	0.452582
$436$	20.7840	0.995371
$437$	12.5426	0.599995
$438$	−5.10164	−0.243766
$439$	7.81038	0.372769	0.186385	−	0.982477i	$-0.440323\pi$
$439$	7.81038	0.372769	0.186385	+	0.982477i	$0.440323\pi$
$440$	4.83751	0.230619
$441$	12.1184	0.577068
$442$	−4.98470	−0.237098
$443$	30.4815	1.44822	0.724110	−	0.689684i	$-0.242251\pi$
$443$	30.4815	1.44822	0.724110	+	0.689684i	$0.242251\pi$
$444$	−7.04917	−0.334539
$445$	−2.21335	−0.104923
$446$	−6.89692	−0.326579
$447$	11.3954	0.538982
$448$	19.1828	0.906304
$449$	25.0915	1.18414	0.592072	−	0.805885i	$-0.298310\pi$
$449$	25.0915	1.18414	0.592072	+	0.805885i	$0.298310\pi$
$450$	−1.89284	−0.0892292
$451$	68.5494	3.22787
$452$	−13.9984	−0.658431
$453$	5.00986	0.235384
$454$	35.3002	1.65672
$455$	4.37246	0.204984
$456$	−2.88347	−0.135031
$457$	−16.0324	−0.749966	−0.374983	−	0.927032i	$-0.622351\pi$
$457$	−16.0324	−0.749966	−0.374983	+	0.927032i	$0.622351\pi$
$458$	45.0720	2.10608
$459$	−2.63346	−0.122919
$460$	−5.43667	−0.253486
$461$	16.6042	0.773337	0.386668	−	0.922219i	$-0.373626\pi$
$461$	16.6042	0.773337	0.386668	+	0.922219i	$0.373626\pi$
$462$	50.7033	2.35893
$463$	−0.558986	−0.0259783	−0.0129891	−	0.999916i	$-0.504135\pi$
$463$	−0.558986	−0.0259783	−0.0129891	+	0.999916i	$0.504135\pi$
$464$	−43.9902	−2.04220
$465$	−1.00000	−0.0463739
$466$	27.5952	1.27832
$467$	−15.3472	−0.710185	−0.355092	−	0.934831i	$-0.615551\pi$
$467$	−15.3472	−0.710185	−0.355092	+	0.934831i	$0.615551\pi$
$468$	−1.58283	−0.0731664
$469$	43.0361	1.98722
$470$	12.8022	0.590523
$471$	15.7684	0.726570
$472$	−7.22546	−0.332579
$473$	−62.0316	−2.85222
$474$	8.13768	0.373776
$475$	−3.65166	−0.167550
$476$	18.2258	0.835379
$477$	12.4465	0.569885
$478$	−33.1257	−1.51514
$479$	21.5123	0.982924	0.491462	−	0.870899i	$-0.336463\pi$
$479$	21.5123	0.982924	0.491462	+	0.870899i	$0.336463\pi$
$480$	7.24194	0.330548
$481$	4.45352	0.203063
$482$	−7.25069	−0.330260
$483$	15.0184	0.683362
$484$	41.9947	1.90885
$485$	4.29194	0.194887
$486$	−1.89284	−0.0858608
$487$	−5.82876	−0.264127	−0.132063	−	0.991241i	$-0.542160\pi$
$487$	−5.82876	−0.264127	−0.132063	+	0.991241i	$0.542160\pi$
$488$	−6.92277	−0.313379
$489$	−19.9248	−0.901030
$490$	−22.9382	−1.03624
$491$	−1.38499	−0.0625038	−0.0312519	−	0.999512i	$-0.509949\pi$
$491$	−1.38499	−0.0625038	−0.0312519	+	0.999512i	$0.509949\pi$
$492$	17.7109	0.798470
$493$	−24.8581	−1.11955
$494$	−6.91199	−0.310985
$495$	6.12628	0.275356
$496$	4.66031	0.209254
$497$	14.0992	0.632435
$498$	−16.5403	−0.741189
$499$	29.6010	1.32512	0.662561	−	0.749008i	$-0.269470\pi$
$499$	29.6010	1.32512	0.662561	+	0.749008i	$0.269470\pi$
$500$	1.58283	0.0707864
$501$	11.0736	0.494732
$502$	41.3439	1.84527
$503$	1.71678	0.0765474	0.0382737	−	0.999267i	$-0.487814\pi$
$503$	1.71678	0.0765474	0.0382737	+	0.999267i	$0.487814\pi$
$504$	−3.45264	−0.153793
$505$	−10.7430	−0.478058
$506$	39.8298	1.77065
$507$	1.00000	0.0444116
$508$	−29.3209	−1.30090
$509$	−34.1290	−1.51274	−0.756372	−	0.654142i	$-0.773030\pi$
$509$	−34.1290	−1.51274	−0.756372	+	0.654142i	$0.773030\pi$
$510$	4.98470	0.220726
$511$	−11.7848	−0.521330
$512$	−26.3898	−1.16628
$513$	−3.65166	−0.161225
$514$	21.3759	0.942852
$515$	2.89126	0.127404
$516$	−16.0269	−0.705547
$517$	−41.4352	−1.82232
$518$	−36.8589	−1.61949
$519$	7.82170	0.343334
$520$	−0.789632	−0.0346276
$521$	−31.4561	−1.37812	−0.689058	−	0.724707i	$-0.741975\pi$
$521$	−31.4561	−1.37812	−0.689058	+	0.724707i	$0.741975\pi$
$522$	−17.8671	−0.782023
$523$	−30.1047	−1.31639	−0.658194	−	0.752849i	$-0.728679\pi$
$523$	−30.1047	−1.31639	−0.658194	+	0.752849i	$0.728679\pi$
$524$	−19.2047	−0.838961
$525$	−4.37246	−0.190830
$526$	−47.0429	−2.05117
$527$	2.63346	0.114715
$528$	−28.5504	−1.24250
$529$	−11.2023	−0.487058
$530$	−23.5592	−1.02334
$531$	−9.15041	−0.397094
$532$	25.2727	1.09571
$533$	−11.1894	−0.484667
$534$	4.18951	0.181298
$535$	0.0632155	0.00273305
$536$	−7.77197	−0.335698
$537$	1.84650	0.0796822
$538$	39.0194	1.68225
$539$	74.2410	3.19779
$540$	1.58283	0.0681142
$541$	−34.7678	−1.49479	−0.747393	−	0.664382i	$-0.768695\pi$
$541$	−34.7678	−1.49479	−0.747393	+	0.664382i	$0.768695\pi$
$542$	19.2191	0.825529
$543$	−12.8828	−0.552856
$544$	−19.0713	−0.817676
$545$	13.1309	0.562465
$546$	−8.27636	−0.354196
$547$	−35.8161	−1.53139	−0.765694	−	0.643205i	$-0.777604\pi$
$547$	−35.8161	−1.53139	−0.765694	+	0.643205i	$0.777604\pi$
$548$	−7.45610	−0.318509
$549$	−8.76708	−0.374170
$550$	−11.5961	−0.494458
$551$	−34.4693	−1.46844
$552$	−2.71221	−0.115439
$553$	18.7981	0.799375
$554$	48.7664	2.07189
$555$	−4.45352	−0.189041
$556$	−28.3414	−1.20194
$557$	18.4628	0.782294	0.391147	−	0.920328i	$-0.372078\pi$
$557$	18.4628	0.782294	0.391147	+	0.920328i	$0.372078\pi$
$558$	1.89284	0.0801302
$559$	10.1255	0.428262
$560$	20.3770	0.861086
$561$	−16.1333	−0.681148
$562$	41.0639	1.73217
$563$	0.503952	0.0212390	0.0106195	−	0.999944i	$-0.496620\pi$
$563$	0.503952	0.0212390	0.0106195	+	0.999944i	$0.496620\pi$
$564$	−10.7055	−0.450783
$565$	−8.84392	−0.372067
$566$	−46.5971	−1.95862
$567$	−4.37246	−0.183626
$568$	−2.54620	−0.106836
$569$	−42.8234	−1.79525	−0.897624	−	0.440761i	$-0.854709\pi$
$569$	−42.8234	−1.79525	−0.897624	+	0.440761i	$0.854709\pi$
$570$	6.91199	0.289512
$571$	−44.9231	−1.87997	−0.939986	−	0.341212i	$-0.889163\pi$
$571$	−44.9231	−1.87997	−0.939986	+	0.341212i	$0.889163\pi$
$572$	−9.69688	−0.405447
$573$	6.48173	0.270778
$574$	92.6075	3.86536
$575$	−3.43477	−0.143240
$576$	−4.38719	−0.182800
$577$	−24.8759	−1.03560	−0.517798	−	0.855503i	$-0.673248\pi$
$577$	−24.8759	−1.03560	−0.517798	+	0.855503i	$0.673248\pi$
$578$	19.0512	0.792427
$579$	−2.24614	−0.0933465
$580$	14.9409	0.620387
$581$	−38.2082	−1.58514
$582$	−8.12394	−0.336748
$583$	76.2507	3.15798
$584$	2.12824	0.0880673
$585$	−1.00000	−0.0413449
$586$	0.454244	0.0187647
$587$	−16.5350	−0.682470	−0.341235	−	0.939978i	$-0.610845\pi$
$587$	−16.5350	−0.682470	−0.341235	+	0.939978i	$0.610845\pi$
$588$	19.1814	0.791029
$589$	3.65166	0.150464
$590$	17.3202	0.713063
$591$	11.6309	0.478430
$592$	20.7548	0.853016
$593$	−5.21088	−0.213985	−0.106993	−	0.994260i	$-0.534122\pi$
$593$	−5.21088	−0.213985	−0.106993	+	0.994260i	$0.534122\pi$
$594$	−11.5961	−0.475792
$595$	11.5147	0.472056
$596$	18.0369	0.738822
$597$	21.0403	0.861122
$598$	−6.50147	−0.265865
$599$	11.7408	0.479716	0.239858	−	0.970808i	$-0.422899\pi$
$599$	11.7408	0.479716	0.239858	+	0.970808i	$0.422899\pi$
$600$	0.789632	0.0322366
$601$	−23.6114	−0.963128	−0.481564	−	0.876411i	$-0.659931\pi$
$601$	−23.6114	−0.963128	−0.481564	+	0.876411i	$0.659931\pi$
$602$	−83.8022	−3.41552
$603$	−9.84253	−0.400819
$604$	7.92977	0.322658
$605$	26.5314	1.07865
$606$	20.3348	0.826043
$607$	37.7224	1.53110	0.765552	−	0.643374i	$-0.222466\pi$
$607$	37.7224	1.53110	0.765552	+	0.643374i	$0.222466\pi$
$608$	−26.4451	−1.07249
$609$	−41.2732	−1.67247
$610$	16.5947	0.671898
$611$	6.76351	0.273622
$612$	−4.16832	−0.168494
$613$	3.96939	0.160322	0.0801610	−	0.996782i	$-0.474457\pi$
$613$	3.96939	0.160322	0.0801610	+	0.996782i	$0.474457\pi$
$614$	−13.6568	−0.551143
$615$	11.1894	0.451200
$616$	−21.1518	−0.852231
$617$	18.1089	0.729038	0.364519	−	0.931196i	$-0.381233\pi$
$617$	18.1089	0.729038	0.364519	+	0.931196i	$0.381233\pi$
$618$	−5.47268	−0.220143
$619$	4.28953	0.172411	0.0862053	−	0.996277i	$-0.472526\pi$
$619$	4.28953	0.172411	0.0862053	+	0.996277i	$0.472526\pi$
$620$	−1.58283	−0.0635680
$621$	−3.43477	−0.137833
$622$	0.976138	0.0391396
$623$	9.67780	0.387733
$624$	4.66031	0.186562
$625$	1.00000	0.0400000
$626$	25.3223	1.01208
$627$	−22.3711	−0.893416
$628$	24.9587	0.995962
$629$	11.7281	0.467632
$630$	8.27636	0.329738
$631$	22.8877	0.911145	0.455573	−	0.890199i	$-0.349435\pi$
$631$	22.8877	0.911145	0.455573	+	0.890199i	$0.349435\pi$
$632$	−3.39478	−0.135037
$633$	−17.2842	−0.686986
$634$	−31.2273	−1.24019
$635$	−18.5243	−0.735115
$636$	19.7007	0.781183
$637$	−12.1184	−0.480150
$638$	−109.459	−4.33353
$639$	−3.22454	−0.127561
$640$	−6.17963	−0.244271
$641$	9.52929	0.376384	0.188192	−	0.982132i	$-0.439737\pi$
$641$	9.52929	0.376384	0.188192	+	0.982132i	$0.439737\pi$
$642$	−0.119657	−0.00472247
$643$	−36.6741	−1.44628	−0.723142	−	0.690699i	$-0.757303\pi$
$643$	−36.6741	−1.44628	−0.723142	+	0.690699i	$0.757303\pi$
$644$	23.7716	0.936733
$645$	−10.1255	−0.398691
$646$	−18.2024	−0.716165
$647$	−19.5869	−0.770040	−0.385020	−	0.922908i	$-0.625805\pi$
$647$	−19.5869	−0.770040	−0.385020	+	0.922908i	$0.625805\pi$
$648$	0.789632	0.0310197
$649$	−56.0580	−2.20047
$650$	1.89284	0.0742432
$651$	4.37246	0.171370
$652$	−31.5376	−1.23511
$653$	−16.1136	−0.630573	−0.315287	−	0.948996i	$-0.602101\pi$
$653$	−16.1136	−0.630573	−0.315287	+	0.948996i	$0.602101\pi$
$654$	−24.8546	−0.971892
$655$	−12.1331	−0.474080
$656$	−52.1460	−2.03596
$657$	2.69524	0.105151
$658$	−55.9773	−2.18222
$659$	4.39641	0.171260	0.0856299	−	0.996327i	$-0.472710\pi$
$659$	4.39641	0.171260	0.0856299	+	0.996327i	$0.472710\pi$
$660$	9.69688	0.377450
$661$	3.74624	0.145712	0.0728560	−	0.997342i	$-0.476789\pi$
$661$	3.74624	0.145712	0.0728560	+	0.997342i	$0.476789\pi$
$662$	42.2250	1.64112
$663$	2.63346	0.102275
$664$	6.90010	0.267776
$665$	15.9667	0.619164
$666$	8.42978	0.326648
$667$	−32.4220	−1.25538
$668$	17.5276	0.678164
$669$	3.64370	0.140873
$670$	18.6303	0.719751
$671$	−53.7096	−2.07344
$672$	−31.6651	−1.22151
$673$	−33.7537	−1.30111	−0.650555	−	0.759459i	$-0.725464\pi$
$673$	−33.7537	−1.30111	−0.650555	+	0.759459i	$0.725464\pi$
$674$	−65.1155	−2.50815
$675$	1.00000	0.0384900
$676$	1.58283	0.0608781
$677$	−19.7598	−0.759431	−0.379715	−	0.925103i	$-0.623978\pi$
$677$	−19.7598	−0.759431	−0.379715	+	0.925103i	$0.623978\pi$
$678$	16.7401	0.642900
$679$	−18.7663	−0.720186
$680$	−2.07946	−0.0797437
$681$	−18.6493	−0.714644
$682$	11.5961	0.444036
$683$	−26.9134	−1.02981	−0.514907	−	0.857246i	$-0.672173\pi$
$683$	−26.9134	−1.02981	−0.514907	+	0.857246i	$0.672173\pi$
$684$	−5.77996	−0.221002
$685$	−4.71061	−0.179983
$686$	42.3620	1.61739
$687$	−23.8119	−0.908480
$688$	47.1879	1.79902
$689$	−12.4465	−0.474173
$690$	6.50147	0.247507
$691$	13.0933	0.498092	0.249046	−	0.968492i	$-0.419883\pi$
$691$	13.0933	0.498092	0.249046	+	0.968492i	$0.419883\pi$
$692$	12.3804	0.470633
$693$	−26.7870	−1.01755
$694$	38.5424	1.46305
$695$	−17.9055	−0.679195
$696$	7.45360	0.282528
$697$	−29.4668	−1.11613
$698$	5.86554	0.222014
$699$	−14.5787	−0.551418
$700$	−6.92087	−0.261584
$701$	−21.8438	−0.825030	−0.412515	−	0.910951i	$-0.635350\pi$
$701$	−21.8438	−0.825030	−0.412515	+	0.910951i	$0.635350\pi$
$702$	1.89284	0.0714405
$703$	16.2627	0.613360
$704$	−26.8772	−1.01297
$705$	−6.76351	−0.254729
$706$	12.9954	0.489088
$707$	46.9734	1.76662
$708$	−14.4836	−0.544326
$709$	−19.3431	−0.726445	−0.363223	−	0.931702i	$-0.618324\pi$
$709$	−19.3431	−0.726445	−0.363223	+	0.931702i	$0.618324\pi$
$710$	6.10353	0.229061
$711$	−4.29920	−0.161232
$712$	−1.74773	−0.0654990
$713$	3.43477	0.128633
$714$	−21.7954	−0.815674
$715$	−6.12628	−0.229110
$716$	2.92269	0.109226
$717$	17.5006	0.653571
$718$	−54.8633	−2.04748
$719$	3.52867	0.131597	0.0657986	−	0.997833i	$-0.479041\pi$
$719$	3.52867	0.131597	0.0657986	+	0.997833i	$0.479041\pi$
$720$	−4.66031	−0.173679
$721$	−12.6419	−0.470810
$722$	10.7237	0.399093
$723$	3.83059	0.142461
$724$	−20.3914	−0.757839
$725$	9.43934	0.350568
$726$	−50.2195	−1.86382
$727$	−6.43162	−0.238536	−0.119268	−	0.992862i	$-0.538055\pi$
$727$	−6.43162	−0.238536	−0.119268	+	0.992862i	$0.538055\pi$
$728$	3.45264	0.127963
$729$	1.00000	0.0370370
$730$	−5.10164	−0.188820
$731$	26.6650	0.986241
$732$	−13.8768	−0.512902
$733$	9.53471	0.352173	0.176086	−	0.984375i	$-0.443656\pi$
$733$	9.53471	0.352173	0.176086	+	0.984375i	$0.443656\pi$
$734$	1.05616	0.0389836
$735$	12.1184	0.446995
$736$	−24.8744	−0.916883
$737$	−60.2981	−2.22111
$738$	−21.1797	−0.779636
$739$	−29.8673	−1.09869	−0.549343	−	0.835597i	$-0.685122\pi$
$739$	−29.8673	−1.09869	−0.549343	+	0.835597i	$0.685122\pi$
$740$	−7.04917	−0.259133
$741$	3.65166	0.134147
$742$	103.012	3.78167
$743$	−1.37031	−0.0502717	−0.0251358	−	0.999684i	$-0.508002\pi$
$743$	−1.37031	−0.0502717	−0.0251358	+	0.999684i	$0.508002\pi$
$744$	−0.789632	−0.0289493
$745$	11.3954	0.417494
$746$	11.6953	0.428195
$747$	8.73838	0.319720
$748$	−25.5363	−0.933700
$749$	−0.276408	−0.0100997
$750$	−1.89284	−0.0691166
$751$	−48.4010	−1.76618	−0.883089	−	0.469206i	$-0.844540\pi$
$751$	−48.4010	−1.76618	−0.883089	+	0.469206i	$0.844540\pi$
$752$	31.5200	1.14942
$753$	−21.8423	−0.795978
$754$	17.8671	0.650683
$755$	5.00986	0.182328
$756$	−6.92087	−0.251710
$757$	−43.3782	−1.57661	−0.788303	−	0.615287i	$-0.789040\pi$
$757$	−43.3782	−1.57661	−0.788303	+	0.615287i	$0.789040\pi$
$758$	43.1098	1.56582
$759$	−21.0424	−0.763791
$760$	−2.88347	−0.104594
$761$	48.8564	1.77104	0.885522	−	0.464597i	$-0.153801\pi$
$761$	48.8564	1.77104	0.885522	+	0.464597i	$0.153801\pi$
$762$	35.0635	1.27022
$763$	−57.4143	−2.07854
$764$	10.2595	0.371175
$765$	−2.63346	−0.0952128
$766$	−34.3859	−1.24241
$767$	9.15041	0.330402
$768$	20.4714	0.738699
$769$	14.9350	0.538570	0.269285	−	0.963061i	$-0.413213\pi$
$769$	14.9350	0.538570	0.269285	+	0.963061i	$0.413213\pi$
$770$	50.7033	1.82722
$771$	−11.2931	−0.406710
$772$	−3.55527	−0.127957
$773$	15.2037	0.546840	0.273420	−	0.961895i	$-0.411845\pi$
$773$	15.2037	0.546840	0.273420	+	0.961895i	$0.411845\pi$
$774$	19.1659	0.688904
$775$	−1.00000	−0.0359211
$776$	3.38905	0.121660
$777$	19.4728	0.698584
$778$	69.6892	2.49848
$779$	−40.8599	−1.46396
$780$	−1.58283	−0.0566745
$781$	−19.7545	−0.706870
$782$	−17.1213	−0.612257
$783$	9.43934	0.337335
$784$	−56.4756	−2.01699
$785$	15.7684	0.562799
$786$	22.9660	0.819171
$787$	−36.2768	−1.29313	−0.646565	−	0.762859i	$-0.723795\pi$
$787$	−36.2768	−1.29313	−0.646565	+	0.762859i	$0.723795\pi$
$788$	18.4097	0.655819
$789$	24.8531	0.884795
$790$	8.13768	0.289525
$791$	38.6697	1.37494
$792$	4.83751	0.171893
$793$	8.76708	0.311328
$794$	37.0495	1.31484
$795$	12.4465	0.441431
$796$	33.3032	1.18040
$797$	17.3251	0.613686	0.306843	−	0.951760i	$-0.400727\pi$
$797$	17.3251	0.613686	0.306843	+	0.951760i	$0.400727\pi$
$798$	−30.2224	−1.06986
$799$	17.8114	0.630123
$800$	7.24194	0.256041
$801$	−2.21335	−0.0782049
$802$	20.8112	0.734869
$803$	16.5118	0.582688
$804$	−15.5791	−0.549431
$805$	15.0184	0.529330
$806$	−1.89284	−0.0666723
$807$	−20.6142	−0.725655
$808$	−8.48302	−0.298432
$809$	−3.24621	−0.114131	−0.0570654	−	0.998370i	$-0.518174\pi$
$809$	−3.24621	−0.114131	−0.0570654	+	0.998370i	$0.518174\pi$
$810$	−1.89284	−0.0665075
$811$	11.8546	0.416271	0.208135	−	0.978100i	$-0.433261\pi$
$811$	11.8546	0.416271	0.208135	+	0.978100i	$0.433261\pi$
$812$	−65.3285	−2.29258
$813$	−10.1536	−0.356101
$814$	51.6433	1.81010
$815$	−19.9248	−0.697935
$816$	12.2727	0.429631
$817$	36.9748	1.29358
$818$	−28.9618	−1.01262
$819$	4.37246	0.152786
$820$	17.7109	0.618493
$821$	14.6438	0.511071	0.255535	−	0.966800i	$-0.417748\pi$
$821$	14.6438	0.511071	0.255535	+	0.966800i	$0.417748\pi$
$822$	8.91641	0.310996
$823$	51.5296	1.79621	0.898104	−	0.439783i	$-0.144945\pi$
$823$	51.5296	1.79621	0.898104	+	0.439783i	$0.144945\pi$
$824$	2.28303	0.0795331
$825$	6.12628	0.213290
$826$	−75.7321	−2.63506
$827$	11.1956	0.389309	0.194655	−	0.980872i	$-0.437641\pi$
$827$	11.1956	0.389309	0.194655	+	0.980872i	$0.437641\pi$
$828$	−5.43667	−0.188937
$829$	41.9907	1.45840	0.729198	−	0.684303i	$-0.239893\pi$
$829$	41.9907	1.45840	0.729198	+	0.684303i	$0.239893\pi$
$830$	−16.5403	−0.574123
$831$	−25.7637	−0.893732
$832$	4.38719	0.152099
$833$	−31.9134	−1.10573
$834$	33.8922	1.17359
$835$	11.0736	0.383217
$836$	−35.4097	−1.22467
$837$	−1.00000	−0.0345651
$838$	36.2073	1.25076
$839$	8.68967	0.300001	0.150000	−	0.988686i	$-0.452072\pi$
$839$	8.68967	0.300001	0.150000	+	0.988686i	$0.452072\pi$
$840$	−3.45264	−0.119127
$841$	60.1012	2.07245
$842$	−28.1182	−0.969016
$843$	−21.6943	−0.747193
$844$	−27.3580	−0.941702
$845$	1.00000	0.0344010
$846$	12.8022	0.440150
$847$	−116.007	−3.98606
$848$	−58.0044	−1.99188
$849$	24.6176	0.844874
$850$	4.98470	0.170974
$851$	15.2968	0.524368
$852$	−5.10391	−0.174857
$853$	−3.19389	−0.109357	−0.0546784	−	0.998504i	$-0.517413\pi$
$853$	−3.19389	−0.109357	−0.0546784	+	0.998504i	$0.517413\pi$
$854$	−72.5595	−2.48294
$855$	−3.65166	−0.124884
$856$	0.0499170	0.00170613
$857$	27.5104	0.939739	0.469869	−	0.882736i	$-0.344301\pi$
$857$	27.5104	0.939739	0.469869	+	0.882736i	$0.344301\pi$
$858$	11.5961	0.395883
$859$	−53.3521	−1.82035	−0.910175	−	0.414225i	$-0.864053\pi$
$859$	−53.3521	−1.82035	−0.910175	+	0.414225i	$0.864053\pi$
$860$	−16.0269	−0.546514
$861$	−48.9252	−1.66737
$862$	14.3973	0.490375
$863$	−16.0480	−0.546281	−0.273141	−	0.961974i	$-0.588062\pi$
$863$	−16.0480	−0.546281	−0.273141	+	0.961974i	$0.588062\pi$
$864$	7.24194	0.246376
$865$	7.82170	0.265946
$866$	30.6793	1.04252
$867$	−10.0649	−0.341822
$868$	6.92087	0.234910
$869$	−26.3381	−0.893459
$870$	−17.8671	−0.605753
$871$	9.84253	0.333501
$872$	10.3686	0.351123
$873$	4.29194	0.145260
$874$	−23.7411	−0.803056
$875$	−4.37246	−0.147816
$876$	4.26610	0.144138
$877$	−23.8523	−0.805434	−0.402717	−	0.915325i	$-0.631934\pi$
$877$	−23.8523	−0.805434	−0.402717	+	0.915325i	$0.631934\pi$
$878$	−14.7838	−0.498929
$879$	−0.239981	−0.00809434
$880$	−28.5504	−0.962433
$881$	16.8726	0.568452	0.284226	−	0.958757i	$-0.408263\pi$
$881$	16.8726	0.568452	0.284226	+	0.958757i	$0.408263\pi$
$882$	−22.9382	−0.772370
$883$	28.8564	0.971094	0.485547	−	0.874211i	$-0.338620\pi$
$883$	28.8564	0.971094	0.485547	+	0.874211i	$0.338620\pi$
$884$	4.16832	0.140196
$885$	−9.15041	−0.307588
$886$	−57.6966	−1.93835
$887$	−17.5865	−0.590495	−0.295248	−	0.955421i	$-0.595402\pi$
$887$	−17.5865	−0.590495	−0.295248	+	0.955421i	$0.595402\pi$
$888$	−3.51664	−0.118011
$889$	80.9969	2.71655
$890$	4.18951	0.140433
$891$	6.12628	0.205238
$892$	5.76736	0.193105
$893$	24.6980	0.826488
$894$	−21.5696	−0.721394
$895$	1.84650	0.0617216
$896$	27.0202	0.902682
$897$	3.43477	0.114684
$898$	−47.4942	−1.58490
$899$	−9.43934	−0.314820
$900$	1.58283	0.0527611
$901$	−32.7773	−1.09197
$902$	−129.753	−4.32030
$903$	44.2733	1.47332
$904$	−6.98344	−0.232266
$905$	−12.8828	−0.428240
$906$	−9.48285	−0.315047
$907$	−12.3167	−0.408970	−0.204485	−	0.978870i	$-0.565552\pi$
$907$	−12.3167	−0.408970	−0.204485	+	0.978870i	$0.565552\pi$
$908$	−29.5188	−0.979615
$909$	−10.7430	−0.356323
$910$	−8.27636	−0.274359
$911$	29.0320	0.961872	0.480936	−	0.876756i	$-0.340297\pi$
$911$	29.0320	0.961872	0.480936	+	0.876756i	$0.340297\pi$
$912$	17.0179	0.563517
$913$	53.5338	1.77171
$914$	30.3468	1.00378
$915$	−8.76708	−0.289831
$916$	−37.6902	−1.24532
$917$	53.0516	1.75192
$918$	4.98470	0.164520
$919$	8.21532	0.270998	0.135499	−	0.990777i	$-0.456736\pi$
$919$	8.21532	0.270998	0.135499	+	0.990777i	$0.456736\pi$
$920$	−2.71221	−0.0894188
$921$	7.21498	0.237742
$922$	−31.4291	−1.03506
$923$	3.22454	0.106137
$924$	−42.3992	−1.39483
$925$	−4.45352	−0.146431
$926$	1.05807	0.0347703
$927$	2.89126	0.0949614
$928$	68.3591	2.24400
$929$	40.2132	1.31935	0.659676	−	0.751550i	$-0.270694\pi$
$929$	40.2132	1.31935	0.659676	+	0.751550i	$0.270694\pi$
$930$	1.89284	0.0620686
$931$	−44.2524	−1.45031
$932$	−23.0757	−0.755869
$933$	−0.515701	−0.0168833
$934$	29.0498	0.950538
$935$	−16.1333	−0.527615
$936$	−0.789632	−0.0258099
$937$	−29.5677	−0.965935	−0.482967	−	0.875638i	$-0.660441\pi$
$937$	−29.5677	−0.965935	−0.482967	+	0.875638i	$0.660441\pi$
$938$	−81.4603	−2.65977
$939$	−13.3780	−0.436574
$940$	−10.7055	−0.349175
$941$	18.6193	0.606973	0.303486	−	0.952836i	$-0.401849\pi$
$941$	18.6193	0.606973	0.303486	+	0.952836i	$0.401849\pi$
$942$	−29.8470	−0.972468
$943$	−38.4330	−1.25155
$944$	42.6437	1.38794
$945$	−4.37246	−0.142236
$946$	117.416	3.81751
$947$	55.0313	1.78828	0.894139	−	0.447789i	$-0.147789\pi$
$947$	55.0313	1.78828	0.894139	+	0.447789i	$0.147789\pi$
$948$	−6.80490	−0.221013
$949$	−2.69524	−0.0874911
$950$	6.91199	0.224255
$951$	16.4976	0.534971
$952$	9.09236	0.294685
$953$	−28.8959	−0.936030	−0.468015	−	0.883720i	$-0.655031\pi$
$953$	−28.8959	−0.936030	−0.468015	+	0.883720i	$0.655031\pi$
$954$	−23.5592	−0.762756
$955$	6.48173	0.209744
$956$	27.7005	0.895897
$957$	57.8281	1.86932
$958$	−40.7194	−1.31558
$959$	20.5970	0.665110
$960$	−4.38719	−0.141596
$961$	1.00000	0.0322581
$962$	−8.42978	−0.271787
$963$	0.0632155	0.00203709
$964$	6.06318	0.195282
$965$	−2.24614	−0.0723059
$966$	−28.4274	−0.914637
$967$	22.7407	0.731291	0.365645	−	0.930754i	$-0.380848\pi$
$967$	22.7407	0.731291	0.365645	+	0.930754i	$0.380848\pi$
$968$	20.9500	0.673359
$969$	9.61648	0.308926
$970$	−8.12394	−0.260844
$971$	−5.92972	−0.190294	−0.0951468	−	0.995463i	$-0.530332\pi$
$971$	−5.92972	−0.190294	−0.0951468	+	0.995463i	$0.530332\pi$
$972$	1.58283	0.0507693
$973$	78.2912	2.50990
$974$	11.0329	0.353517
$975$	−1.00000	−0.0320256
$976$	40.8573	1.30781
$977$	1.80731	0.0578210	0.0289105	−	0.999582i	$-0.490796\pi$
$977$	1.80731	0.0578210	0.0289105	+	0.999582i	$0.490796\pi$
$978$	37.7144	1.20597
$979$	−13.5596	−0.433367
$980$	19.1814	0.612729
$981$	13.1309	0.419236
$982$	2.62156	0.0836575
$983$	−36.0446	−1.14964	−0.574821	−	0.818279i	$-0.694929\pi$
$983$	−36.0446	−1.14964	−0.574821	+	0.818279i	$0.694929\pi$
$984$	8.83550	0.281666
$985$	11.6309	0.370590
$986$	47.0523	1.49845
$987$	29.5732	0.941326
$988$	5.77996	0.183885
$989$	34.7787	1.10590
$990$	−11.5961	−0.368547
$991$	−48.4142	−1.53793	−0.768963	−	0.639293i	$-0.779227\pi$
$991$	−48.4142	−1.53793	−0.768963	+	0.639293i	$0.779227\pi$
$992$	−7.24194	−0.229932
$993$	−22.3078	−0.707916
$994$	−26.6875	−0.846475
$995$	21.0403	0.667022
$996$	13.8314	0.438264
$997$	19.3759	0.613640	0.306820	−	0.951768i	$-0.400735\pi$
$997$	19.3759	0.613640	0.306820	+	0.951768i	$0.400735\pi$
$998$	−56.0298	−1.77359
$999$	−4.45352	−0.140903

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.u.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.u.1.3	✓	9	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89284 q^{2} +1.00000 q^{3} +1.58283 q^{4} +1.00000 q^{5} -1.89284 q^{6} -4.37246 q^{7} +0.789632 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.89284 q^{2} +1.00000 q^{3} +1.58283 q^{4} +1.00000 q^{5} -1.89284 q^{6} -4.37246 q^{7} +0.789632 q^{8} +1.00000 q^{9} -1.89284 q^{10} +6.12628 q^{11} +1.58283 q^{12} -1.00000 q^{13} +8.27636 q^{14} +1.00000 q^{15} -4.66031 q^{16} -2.63346 q^{17} -1.89284 q^{18} -3.65166 q^{19} +1.58283 q^{20} -4.37246 q^{21} -11.5961 q^{22} -3.43477 q^{23} +0.789632 q^{24} +1.00000 q^{25} +1.89284 q^{26} +1.00000 q^{27} -6.92087 q^{28} +9.43934 q^{29} -1.89284 q^{30} -1.00000 q^{31} +7.24194 q^{32} +6.12628 q^{33} +4.98470 q^{34} -4.37246 q^{35} +1.58283 q^{36} -4.45352 q^{37} +6.91199 q^{38} -1.00000 q^{39} +0.789632 q^{40} +11.1894 q^{41} +8.27636 q^{42} -10.1255 q^{43} +9.69688 q^{44} +1.00000 q^{45} +6.50147 q^{46} -6.76351 q^{47} -4.66031 q^{48} +12.1184 q^{49} -1.89284 q^{50} -2.63346 q^{51} -1.58283 q^{52} +12.4465 q^{53} -1.89284 q^{54} +6.12628 q^{55} -3.45264 q^{56} -3.65166 q^{57} -17.8671 q^{58} -9.15041 q^{59} +1.58283 q^{60} -8.76708 q^{61} +1.89284 q^{62} -4.37246 q^{63} -4.38719 q^{64} -1.00000 q^{65} -11.5961 q^{66} -9.84253 q^{67} -4.16832 q^{68} -3.43477 q^{69} +8.27636 q^{70} -3.22454 q^{71} +0.789632 q^{72} +2.69524 q^{73} +8.42978 q^{74} +1.00000 q^{75} -5.77996 q^{76} -26.7870 q^{77} +1.89284 q^{78} -4.29920 q^{79} -4.66031 q^{80} +1.00000 q^{81} -21.1797 q^{82} +8.73838 q^{83} -6.92087 q^{84} -2.63346 q^{85} +19.1659 q^{86} +9.43934 q^{87} +4.83751 q^{88} -2.21335 q^{89} -1.89284 q^{90} +4.37246 q^{91} -5.43667 q^{92} -1.00000 q^{93} +12.8022 q^{94} -3.65166 q^{95} +7.24194 q^{96} +4.29194 q^{97} -22.9382 q^{98} +6.12628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q - 3 * q^2 + 9 * q^3 + 5 * q^4 + 9 * q^5 - 3 * q^6 - 5 * q^7 - 18 * q^8 + 9 * q^9	\( 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9} - 3 q^{10} + 5 q^{12} - 9 q^{13} + 11 q^{14} + 9 q^{15} + 9 q^{16} - 5 q^{17} - 3 q^{18} - 12 q^{19} + 5 q^{20} - 5 q^{21} - 17 q^{22} - 21 q^{23} - 18 q^{24} + 9 q^{25} + 3 q^{26} + 9 q^{27} - 8 q^{28} - 15 q^{29} - 3 q^{30} - 9 q^{31} - 9 q^{32} + 9 q^{34} - 5 q^{35} + 5 q^{36} - 13 q^{37} - 24 q^{38} - 9 q^{39} - 18 q^{40} - 11 q^{41} + 11 q^{42} - 26 q^{43} + 5 q^{44} + 9 q^{45} + 14 q^{46} - 21 q^{47} + 9 q^{48} - 26 q^{49} - 3 q^{50} - 5 q^{51} - 5 q^{52} + 18 q^{53} - 3 q^{54} - 19 q^{56} - 12 q^{57} - 16 q^{58} - 24 q^{59} + 5 q^{60} - 4 q^{61} + 3 q^{62} - 5 q^{63} + 2 q^{64} - 9 q^{65} - 17 q^{66} - 20 q^{67} - 19 q^{68} - 21 q^{69} + 11 q^{70} - 13 q^{71} - 18 q^{72} + 15 q^{73} + 12 q^{74} + 9 q^{75} + 54 q^{76} - 28 q^{77} + 3 q^{78} - 15 q^{79} + 9 q^{80} + 9 q^{81} - 21 q^{82} - 5 q^{83} - 8 q^{84} - 5 q^{85} + 36 q^{86} - 15 q^{87} + 23 q^{88} + 31 q^{89} - 3 q^{90} + 5 q^{91} - 7 q^{92} - 9 q^{93} - 12 q^{95} - 9 q^{96} - 11 q^{97} - 23 q^{98}+O(q^{100}) \) 9 * q - 3 * q^2 + 9 * q^3 + 5 * q^4 + 9 * q^5 - 3 * q^6 - 5 * q^7 - 18 * q^8 + 9 * q^9 - 3 * q^10 + 5 * q^12 - 9 * q^13 + 11 * q^14 + 9 * q^15 + 9 * q^16 - 5 * q^17 - 3 * q^18 - 12 * q^19 + 5 * q^20 - 5 * q^21 - 17 * q^22 - 21 * q^23 - 18 * q^24 + 9 * q^25 + 3 * q^26 + 9 * q^27 - 8 * q^28 - 15 * q^29 - 3 * q^30 - 9 * q^31 - 9 * q^32 + 9 * q^34 - 5 * q^35 + 5 * q^36 - 13 * q^37 - 24 * q^38 - 9 * q^39 - 18 * q^40 - 11 * q^41 + 11 * q^42 - 26 * q^43 + 5 * q^44 + 9 * q^45 + 14 * q^46 - 21 * q^47 + 9 * q^48 - 26 * q^49 - 3 * q^50 - 5 * q^51 - 5 * q^52 + 18 * q^53 - 3 * q^54 - 19 * q^56 - 12 * q^57 - 16 * q^58 - 24 * q^59 + 5 * q^60 - 4 * q^61 + 3 * q^62 - 5 * q^63 + 2 * q^64 - 9 * q^65 - 17 * q^66 - 20 * q^67 - 19 * q^68 - 21 * q^69 + 11 * q^70 - 13 * q^71 - 18 * q^72 + 15 * q^73 + 12 * q^74 + 9 * q^75 + 54 * q^76 - 28 * q^77 + 3 * q^78 - 15 * q^79 + 9 * q^80 + 9 * q^81 - 21 * q^82 - 5 * q^83 - 8 * q^84 - 5 * q^85 + 36 * q^86 - 15 * q^87 + 23 * q^88 + 31 * q^89 - 3 * q^90 + 5 * q^91 - 7 * q^92 - 9 * q^93 - 12 * q^95 - 9 * q^96 - 11 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more