LMFDB - Embedded newform 2645.2.a.v.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2645,2,Mod(1,2645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2645, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2645 = 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2645.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:	$21.1204313346$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 88 x^{13} + 93 x^{12} - 728 x^{11} + 58 x^{10} + 2760 x^{9} - 1764 x^{8} + \cdots - 18 $ x^16 - 4x^15 - 18x^14 + 88x^13 + 93x^12 - 728x^11 + 58x^10 + 2760x^9 - 1764x^8 - 4568x^7 + 4414x^6 + 2440x^5 - 2759x^4 - 684x^3 + 558x^2 + 108*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.53726$ of defining polynomial
Character	$\chi$	$=$	2645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53726 q^{2} -3.14832 q^{3} +4.43771 q^{4} -1.00000 q^{5} -7.98812 q^{6} -1.76258 q^{7} +6.18511 q^{8} +6.91193 q^{9} +O(q^{10})$	\(q+2.53726 q^{2} -3.14832 q^{3} +4.43771 q^{4} -1.00000 q^{5} -7.98812 q^{6} -1.76258 q^{7} +6.18511 q^{8} +6.91193 q^{9} -2.53726 q^{10} -4.19265 q^{11} -13.9713 q^{12} +3.43262 q^{13} -4.47213 q^{14} +3.14832 q^{15} +6.81785 q^{16} +2.90813 q^{17} +17.5374 q^{18} +0.741196 q^{19} -4.43771 q^{20} +5.54917 q^{21} -10.6379 q^{22} -19.4727 q^{24} +1.00000 q^{25} +8.70947 q^{26} -12.3160 q^{27} -7.82182 q^{28} -8.52090 q^{29} +7.98812 q^{30} +1.49553 q^{31} +4.92845 q^{32} +13.1998 q^{33} +7.37870 q^{34} +1.76258 q^{35} +30.6731 q^{36} -6.06172 q^{37} +1.88061 q^{38} -10.8070 q^{39} -6.18511 q^{40} -4.27402 q^{41} +14.0797 q^{42} -8.47750 q^{43} -18.6058 q^{44} -6.91193 q^{45} +0.572948 q^{47} -21.4648 q^{48} -3.89331 q^{49} +2.53726 q^{50} -9.15573 q^{51} +15.2330 q^{52} -6.78301 q^{53} -31.2490 q^{54} +4.19265 q^{55} -10.9018 q^{56} -2.33352 q^{57} -21.6198 q^{58} -6.52183 q^{59} +13.9713 q^{60} -4.75548 q^{61} +3.79456 q^{62} -12.1828 q^{63} -1.13091 q^{64} -3.43262 q^{65} +33.4914 q^{66} -5.71824 q^{67} +12.9054 q^{68} +4.47213 q^{70} +6.85182 q^{71} +42.7511 q^{72} +7.53697 q^{73} -15.3802 q^{74} -3.14832 q^{75} +3.28921 q^{76} +7.38989 q^{77} -27.4202 q^{78} -9.97670 q^{79} -6.81785 q^{80} +18.0390 q^{81} -10.8443 q^{82} +17.7349 q^{83} +24.6256 q^{84} -2.90813 q^{85} -21.5097 q^{86} +26.8265 q^{87} -25.9320 q^{88} +4.12129 q^{89} -17.5374 q^{90} -6.05027 q^{91} -4.70842 q^{93} +1.45372 q^{94} -0.741196 q^{95} -15.5163 q^{96} -1.07943 q^{97} -9.87835 q^{98} -28.9793 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9}+O(q^{10}) $ 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 4 q^{12} + 8 q^{13} - 8 q^{14} - 4 q^{15} + 28 q^{16} + 8 q^{17} + 4 q^{18} - 32 q^{19} - 20 q^{20} - 16 q^{21} - 44 q^{22} - 28 q^{24} + 16 q^{25} - 20 q^{26} - 8 q^{27} - 52 q^{28} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 4 q^{32} - 20 q^{33} - 24 q^{34} + 12 q^{35} - 4 q^{36} - 28 q^{37} + 20 q^{38} - 8 q^{39} - 4 q^{41} + 8 q^{42} - 48 q^{43} - 32 q^{44} - 8 q^{45} + 48 q^{47} + 24 q^{48} + 12 q^{49} + 4 q^{50} - 24 q^{51} + 12 q^{52} + 4 q^{53} - 44 q^{54} + 16 q^{55} - 64 q^{56} - 52 q^{57} - 28 q^{58} - 40 q^{59} + 4 q^{60} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{65} - 8 q^{66} - 68 q^{67} + 4 q^{68} + 8 q^{70} - 24 q^{71} - 12 q^{72} + 52 q^{73} - 40 q^{74} + 4 q^{75} - 24 q^{76} + 44 q^{77} + 12 q^{78} - 72 q^{79} - 28 q^{80} + 20 q^{81} - 20 q^{82} - 12 q^{83} - 32 q^{84} - 8 q^{85} + 56 q^{86} - 104 q^{88} - 48 q^{89} - 4 q^{90} - 48 q^{91} - 48 q^{93} - 60 q^{94} + 32 q^{95} - 108 q^{96} - 4 q^{97} + 8 q^{98} - 24 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 4 * q^12 + 8 * q^13 - 8 * q^14 - 4 * q^15 + 28 * q^16 + 8 * q^17 + 4 * q^18 - 32 * q^19 - 20 * q^20 - 16 * q^21 - 44 * q^22 - 28 * q^24 + 16 * q^25 - 20 * q^26 - 8 * q^27 - 52 * q^28 - 12 * q^29 + 12 * q^30 - 12 * q^31 + 4 * q^32 - 20 * q^33 - 24 * q^34 + 12 * q^35 - 4 * q^36 - 28 * q^37 + 20 * q^38 - 8 * q^39 - 4 * q^41 + 8 * q^42 - 48 * q^43 - 32 * q^44 - 8 * q^45 + 48 * q^47 + 24 * q^48 + 12 * q^49 + 4 * q^50 - 24 * q^51 + 12 * q^52 + 4 * q^53 - 44 * q^54 + 16 * q^55 - 64 * q^56 - 52 * q^57 - 28 * q^58 - 40 * q^59 + 4 * q^60 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^65 - 8 * q^66 - 68 * q^67 + 4 * q^68 + 8 * q^70 - 24 * q^71 - 12 * q^72 + 52 * q^73 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 44 * q^77 + 12 * q^78 - 72 * q^79 - 28 * q^80 + 20 * q^81 - 20 * q^82 - 12 * q^83 - 32 * q^84 - 8 * q^85 + 56 * q^86 - 104 * q^88 - 48 * q^89 - 4 * q^90 - 48 * q^91 - 48 * q^93 - 60 * q^94 + 32 * q^95 - 108 * q^96 - 4 * q^97 + 8 * q^98 - 24 * 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 $

$2$	2.53726	1.79412	0.897058	−	0.441912i	$-0.145700\pi$
$2$	2.53726	1.79412	0.897058	+	0.441912i	$0.145700\pi$
$3$	−3.14832	−1.81768	−0.908842	−	0.417140i	$-0.863032\pi$
$3$	−3.14832	−1.81768	−0.908842	+	0.417140i	$0.863032\pi$
$4$	4.43771	2.21885
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.98812	−3.26114
$7$	−1.76258	−0.666193	−0.333096	−	0.942893i	$-0.608093\pi$
$7$	−1.76258	−0.666193	−0.333096	+	0.942893i	$0.608093\pi$
$8$	6.18511	2.18677
$9$	6.91193	2.30398
$10$	−2.53726	−0.802353
$11$	−4.19265	−1.26413	−0.632066	−	0.774914i	$-0.717793\pi$
$11$	−4.19265	−1.26413	−0.632066	+	0.774914i	$0.717793\pi$
$12$	−13.9713	−4.03318
$13$	3.43262	0.952038	0.476019	−	0.879435i	$-0.342079\pi$
$13$	3.43262	0.952038	0.476019	+	0.879435i	$0.342079\pi$
$14$	−4.47213	−1.19523
$15$	3.14832	0.812893
$16$	6.81785	1.70446
$17$	2.90813	0.705325	0.352663	−	0.935751i	$-0.385276\pi$
$17$	2.90813	0.705325	0.352663	+	0.935751i	$0.385276\pi$
$18$	17.5374	4.13360
$19$	0.741196	0.170042	0.0850210	−	0.996379i	$-0.472904\pi$
$19$	0.741196	0.170042	0.0850210	+	0.996379i	$0.472904\pi$
$20$	−4.43771	−0.992302
$21$	5.54917	1.21093
$22$	−10.6379	−2.26800
$23$	0	0
$23$	0	0
$24$	−19.4727	−3.97485
$25$	1.00000	0.200000
$26$	8.70947	1.70807
$27$	−12.3160	−2.37022
$28$	−7.82182	−1.47819
$29$	−8.52090	−1.58229	−0.791145	−	0.611628i	$-0.790515\pi$
$29$	−8.52090	−1.58229	−0.791145	+	0.611628i	$0.790515\pi$
$30$	7.98812	1.45843
$31$	1.49553	0.268606	0.134303	−	0.990940i	$-0.457120\pi$
$31$	1.49553	0.268606	0.134303	+	0.990940i	$0.457120\pi$
$32$	4.92845	0.871235
$33$	13.1998	2.29779
$34$	7.37870	1.26544
$35$	1.76258	0.297931
$36$	30.6731	5.11219
$37$	−6.06172	−0.996540	−0.498270	−	0.867022i	$-0.666031\pi$
$37$	−6.06172	−0.996540	−0.498270	+	0.867022i	$0.666031\pi$
$38$	1.88061	0.305075
$39$	−10.8070	−1.73050
$40$	−6.18511	−0.977952
$41$	−4.27402	−0.667489	−0.333745	−	0.942664i	$-0.608312\pi$
$41$	−4.27402	−0.667489	−0.333745	+	0.942664i	$0.608312\pi$
$42$	14.0797	2.17255
$43$	−8.47750	−1.29281	−0.646404	−	0.762996i	$-0.723728\pi$
$43$	−8.47750	−1.29281	−0.646404	+	0.762996i	$0.723728\pi$
$44$	−18.6058	−2.80493
$45$	−6.91193	−1.03037
$46$	0	0
$47$	0.572948	0.0835731	0.0417866	−	0.999127i	$-0.486695\pi$
$47$	0.572948	0.0835731	0.0417866	+	0.999127i	$0.486695\pi$
$48$	−21.4648	−3.09817
$49$	−3.89331	−0.556187
$50$	2.53726	0.358823
$51$	−9.15573	−1.28206
$52$	15.2330	2.11243
$53$	−6.78301	−0.931718	−0.465859	−	0.884859i	$-0.654255\pi$
$53$	−6.78301	−0.931718	−0.465859	+	0.884859i	$0.654255\pi$
$54$	−31.2490	−4.25244
$55$	4.19265	0.565337
$56$	−10.9018	−1.45681
$57$	−2.33352	−0.309083
$58$	−21.6198	−2.83881
$59$	−6.52183	−0.849070	−0.424535	−	0.905412i	$-0.639562\pi$
$59$	−6.52183	−0.849070	−0.424535	+	0.905412i	$0.639562\pi$
$60$	13.9713	1.80369
$61$	−4.75548	−0.608877	−0.304438	−	0.952532i	$-0.598469\pi$
$61$	−4.75548	−0.608877	−0.304438	+	0.952532i	$0.598469\pi$
$62$	3.79456	0.481910
$63$	−12.1828	−1.53489
$64$	−1.13091	−0.141364
$65$	−3.43262	−0.425764
$66$	33.4914	4.12251
$67$	−5.71824	−0.698594	−0.349297	−	0.937012i	$-0.613580\pi$
$67$	−5.71824	−0.698594	−0.349297	+	0.937012i	$0.613580\pi$
$68$	12.9054	1.56501
$69$	0	0
$70$	4.47213	0.534522
$71$	6.85182	0.813161	0.406581	−	0.913615i	$-0.366721\pi$
$71$	6.85182	0.813161	0.406581	+	0.913615i	$0.366721\pi$
$72$	42.7511	5.03826
$73$	7.53697	0.882136	0.441068	−	0.897474i	$-0.354600\pi$
$73$	7.53697	0.882136	0.441068	+	0.897474i	$0.354600\pi$
$74$	−15.3802	−1.78791
$75$	−3.14832	−0.363537
$76$	3.28921	0.377298
$77$	7.38989	0.842156
$78$	−27.4202	−3.10473
$79$	−9.97670	−1.12247	−0.561233	−	0.827658i	$-0.689673\pi$
$79$	−9.97670	−1.12247	−0.561233	+	0.827658i	$0.689673\pi$
$80$	−6.81785	−0.762258
$81$	18.0390	2.00433
$82$	−10.8443	−1.19755
$83$	17.7349	1.94666	0.973331	−	0.229406i	$-0.0736785\pi$
$83$	17.7349	1.94666	0.973331	+	0.229406i	$0.0736785\pi$
$84$	24.6256	2.68687
$85$	−2.90813	−0.315431
$86$	−21.5097	−2.31945
$87$	26.8265	2.87610
$88$	−25.9320	−2.76436
$89$	4.12129	0.436856	0.218428	−	0.975853i	$-0.429907\pi$
$89$	4.12129	0.436856	0.218428	+	0.975853i	$0.429907\pi$
$90$	−17.5374	−1.84860
$91$	−6.05027	−0.634241
$92$	0	0
$93$	−4.70842	−0.488241
$94$	1.45372	0.149940
$95$	−0.741196	−0.0760451
$96$	−15.5163	−1.58363
$97$	−1.07943	−0.109599	−0.0547996	−	0.998497i	$-0.517452\pi$
$97$	−1.07943	−0.109599	−0.0547996	+	0.998497i	$0.517452\pi$
$98$	−9.87835	−0.997865
$99$	−28.9793	−2.91253
$100$	4.43771	0.443771
$101$	−4.53646	−0.451395	−0.225697	−	0.974197i	$-0.572466\pi$
$101$	−4.53646	−0.451395	−0.225697	+	0.974197i	$0.572466\pi$
$102$	−23.2305	−2.30016
$103$	−17.4274	−1.71717	−0.858587	−	0.512668i	$-0.828657\pi$
$103$	−17.4274	−1.71717	−0.858587	+	0.512668i	$0.828657\pi$
$104$	21.2312	2.08189
$105$	−5.54917	−0.541544
$106$	−17.2103	−1.67161
$107$	−18.3016	−1.76928	−0.884639	−	0.466276i	$-0.845595\pi$
$107$	−18.3016	−1.76928	−0.884639	+	0.466276i	$0.845595\pi$
$108$	−54.6549	−5.25917
$109$	9.56427	0.916091	0.458046	−	0.888929i	$-0.348550\pi$
$109$	9.56427	0.916091	0.458046	+	0.888929i	$0.348550\pi$
$110$	10.6379	1.01428
$111$	19.0842	1.81139
$112$	−12.0170	−1.13550
$113$	4.31293	0.405726	0.202863	−	0.979207i	$-0.434975\pi$
$113$	4.31293	0.405726	0.202863	+	0.979207i	$0.434975\pi$
$114$	−5.92076	−0.554530
$115$	0	0
$116$	−37.8133	−3.51087
$117$	23.7260	2.19347
$118$	−16.5476	−1.52333
$119$	−5.12582	−0.469883
$120$	19.4727	1.77761
$121$	6.57834	0.598031
$122$	−12.0659	−1.09240
$123$	13.4560	1.21328
$124$	6.63674	0.595997
$125$	−1.00000	−0.0894427
$126$	−30.9111	−2.75378
$127$	13.4451	1.19306	0.596528	−	0.802592i	$-0.296546\pi$
$127$	13.4451	1.19306	0.596528	+	0.802592i	$0.296546\pi$
$128$	−12.7263	−1.12486
$129$	26.6899	2.34992
$130$	−8.70947	−0.763871
$131$	−4.04258	−0.353202	−0.176601	−	0.984283i	$-0.556510\pi$
$131$	−4.04258	−0.353202	−0.176601	+	0.984283i	$0.556510\pi$
$132$	58.5770	5.09847
$133$	−1.30642	−0.113281
$134$	−14.5087	−1.25336
$135$	12.3160	1.05999
$136$	17.9871	1.54238
$137$	16.6090	1.41901	0.709503	−	0.704702i	$-0.248919\pi$
$137$	16.6090	1.41901	0.709503	+	0.704702i	$0.248919\pi$
$138$	0	0
$139$	−8.54236	−0.724553	−0.362277	−	0.932071i	$-0.618000\pi$
$139$	−8.54236	−0.724553	−0.362277	+	0.932071i	$0.618000\pi$
$140$	7.82182	0.661064
$141$	−1.80383	−0.151910
$142$	17.3849	1.45891
$143$	−14.3918	−1.20350
$144$	47.1245	3.92704
$145$	8.52090	0.707622
$146$	19.1233	1.58265
$147$	12.2574	1.01097
$148$	−26.9001	−2.21118
$149$	14.5471	1.19174	0.595871	−	0.803080i	$-0.296807\pi$
$149$	14.5471	1.19174	0.595871	+	0.803080i	$0.296807\pi$
$150$	−7.98812	−0.652228
$151$	−18.5705	−1.51124	−0.755622	−	0.655008i	$-0.772665\pi$
$151$	−18.5705	−1.51124	−0.755622	+	0.655008i	$0.772665\pi$
$152$	4.58438	0.371842
$153$	20.1008	1.62505
$154$	18.7501	1.51093
$155$	−1.49553	−0.120124
$156$	−47.9583	−3.83974
$157$	6.93072	0.553131	0.276566	−	0.960995i	$-0.410804\pi$
$157$	6.93072	0.553131	0.276566	+	0.960995i	$0.410804\pi$
$158$	−25.3135	−2.01384
$159$	21.3551	1.69357
$160$	−4.92845	−0.389628
$161$	0	0
$162$	45.7696	3.59600
$163$	−20.3775	−1.59609	−0.798044	−	0.602599i	$-0.794132\pi$
$163$	−20.3775	−1.59609	−0.798044	+	0.602599i	$0.794132\pi$
$164$	−18.9668	−1.48106
$165$	−13.1998	−1.02760
$166$	44.9982	3.49254
$167$	9.15807	0.708673	0.354336	−	0.935118i	$-0.384707\pi$
$167$	9.15807	0.708673	0.354336	+	0.935118i	$0.384707\pi$
$168$	34.3222	2.64802
$169$	−1.21710	−0.0936234
$170$	−7.37870	−0.565920
$171$	5.12309	0.391773
$172$	−37.6207	−2.86855
$173$	23.0704	1.75401	0.877006	−	0.480479i	$-0.159537\pi$
$173$	23.0704	1.75401	0.877006	+	0.480479i	$0.159537\pi$
$174$	68.0660	5.16007
$175$	−1.76258	−0.133239
$176$	−28.5849	−2.15467
$177$	20.5328	1.54334
$178$	10.4568	0.783770
$179$	−10.9679	−0.819780	−0.409890	−	0.912135i	$-0.634433\pi$
$179$	−10.9679	−0.819780	−0.409890	+	0.912135i	$0.634433\pi$
$180$	−30.6731	−2.28624
$181$	−5.94274	−0.441720	−0.220860	−	0.975306i	$-0.570886\pi$
$181$	−5.94274	−0.441720	−0.220860	+	0.975306i	$0.570886\pi$
$182$	−15.3511	−1.13790
$183$	14.9718	1.10675
$184$	0	0
$185$	6.06172	0.445666
$186$	−11.9465	−0.875960
$187$	−12.1928	−0.891625
$188$	2.54258	0.185437
$189$	21.7080	1.57902
$190$	−1.88061	−0.136434
$191$	−6.97458	−0.504663	−0.252332	−	0.967641i	$-0.581197\pi$
$191$	−6.97458	−0.504663	−0.252332	+	0.967641i	$0.581197\pi$
$192$	3.56047	0.256955
$193$	−2.20358	−0.158617	−0.0793085	−	0.996850i	$-0.525271\pi$
$193$	−2.20358	−0.158617	−0.0793085	+	0.996850i	$0.525271\pi$
$194$	−2.73879	−0.196634
$195$	10.8070	0.773905
$196$	−17.2774	−1.23410
$197$	1.60734	0.114518	0.0572590	−	0.998359i	$-0.481764\pi$
$197$	1.60734	0.114518	0.0572590	+	0.998359i	$0.481764\pi$
$198$	−73.5282	−5.22542
$199$	−21.8671	−1.55012	−0.775059	−	0.631889i	$-0.782280\pi$
$199$	−21.8671	−1.55012	−0.775059	+	0.631889i	$0.782280\pi$
$200$	6.18511	0.437354
$201$	18.0029	1.26982
$202$	−11.5102	−0.809855
$203$	15.0188	1.05411
$204$	−40.6305	−2.84470
$205$	4.27402	0.298510
$206$	−44.2179	−3.08081
$207$	0	0
$208$	23.4031	1.62271
$209$	−3.10758	−0.214956
$210$	−14.0797	−0.971592
$211$	8.59525	0.591721	0.295861	−	0.955231i	$-0.404394\pi$
$211$	8.59525	0.591721	0.295861	+	0.955231i	$0.404394\pi$
$212$	−30.1010	−2.06735
$213$	−21.5717	−1.47807
$214$	−46.4359	−3.17429
$215$	8.47750	0.578161
$216$	−76.1759	−5.18311
$217$	−2.63600	−0.178943
$218$	24.2671	1.64357
$219$	−23.7288	−1.60344
$220$	18.6058	1.25440
$221$	9.98251	0.671497
$222$	48.4217	3.24985
$223$	−12.4279	−0.832232	−0.416116	−	0.909311i	$-0.636609\pi$
$223$	−12.4279	−0.832232	−0.416116	+	0.909311i	$0.636609\pi$
$224$	−8.68679	−0.580411
$225$	6.91193	0.460795
$226$	10.9430	0.727921
$227$	12.4715	0.827760	0.413880	−	0.910331i	$-0.364173\pi$
$227$	12.4715	0.827760	0.413880	+	0.910331i	$0.364173\pi$
$228$	−10.3555	−0.685809
$229$	−3.57413	−0.236185	−0.118093	−	0.993003i	$-0.537678\pi$
$229$	−3.57413	−0.236185	−0.118093	+	0.993003i	$0.537678\pi$
$230$	0	0
$231$	−23.2657	−1.53077
$232$	−52.7027	−3.46010
$233$	22.9938	1.50637	0.753187	−	0.657807i	$-0.228516\pi$
$233$	22.9938	1.50637	0.753187	+	0.657807i	$0.228516\pi$
$234$	60.1992	3.93535
$235$	−0.572948	−0.0373750
$236$	−28.9420	−1.88396
$237$	31.4099	2.04029
$238$	−13.0055	−0.843024
$239$	5.42347	0.350815	0.175407	−	0.984496i	$-0.443876\pi$
$239$	5.42347	0.350815	0.175407	+	0.984496i	$0.443876\pi$
$240$	21.4648	1.38554
$241$	6.45618	0.415879	0.207940	−	0.978142i	$-0.433324\pi$
$241$	6.45618	0.415879	0.207940	+	0.978142i	$0.433324\pi$
$242$	16.6910	1.07294
$243$	−19.8444	−1.27302
$244$	−21.1034	−1.35101
$245$	3.89331	0.248734
$246$	34.1414	2.17677
$247$	2.54425	0.161886
$248$	9.25004	0.587378
$249$	−55.8353	−3.53842
$250$	−2.53726	−0.160471
$251$	−6.37112	−0.402141	−0.201071	−	0.979577i	$-0.564442\pi$
$251$	−6.37112	−0.402141	−0.201071	+	0.979577i	$0.564442\pi$
$252$	−54.0639	−3.40570
$253$	0	0
$254$	34.1137	2.14048
$255$	9.15573	0.573354
$256$	−30.0282	−1.87676
$257$	−21.5126	−1.34192	−0.670959	−	0.741494i	$-0.734118\pi$
$257$	−21.5126	−1.34192	−0.670959	+	0.741494i	$0.734118\pi$
$258$	67.7193	4.21602
$259$	10.6843	0.663888
$260$	−15.2330	−0.944709
$261$	−58.8958	−3.64556
$262$	−10.2571	−0.633685
$263$	−0.633987	−0.0390933	−0.0195467	−	0.999809i	$-0.506222\pi$
$263$	−0.633987	−0.0390933	−0.0195467	+	0.999809i	$0.506222\pi$
$264$	81.6424	5.02474
$265$	6.78301	0.416677
$266$	−3.31473	−0.203239
$267$	−12.9751	−0.794066
$268$	−25.3759	−1.55008
$269$	−17.2484	−1.05165	−0.525827	−	0.850591i	$-0.676244\pi$
$269$	−17.2484	−1.05165	−0.525827	+	0.850591i	$0.676244\pi$
$270$	31.2490	1.90175
$271$	26.6773	1.62053	0.810265	−	0.586064i	$-0.199323\pi$
$271$	26.6773	1.62053	0.810265	+	0.586064i	$0.199323\pi$
$272$	19.8272	1.20220
$273$	19.0482	1.15285
$274$	42.1415	2.54586
$275$	−4.19265	−0.252827
$276$	0	0
$277$	−7.09311	−0.426184	−0.213092	−	0.977032i	$-0.568353\pi$
$277$	−7.09311	−0.426184	−0.213092	+	0.977032i	$0.568353\pi$
$278$	−21.6742	−1.29993
$279$	10.3370	0.618861
$280$	10.9018	0.651505
$281$	22.4721	1.34057	0.670287	−	0.742102i	$-0.266171\pi$
$281$	22.4721	1.34057	0.670287	+	0.742102i	$0.266171\pi$
$282$	−4.57678	−0.272543
$283$	14.8194	0.880919	0.440460	−	0.897772i	$-0.354816\pi$
$283$	14.8194	0.880919	0.440460	+	0.897772i	$0.354816\pi$
$284$	30.4064	1.80429
$285$	2.33352	0.138226
$286$	−36.5158	−2.15922
$287$	7.53330	0.444676
$288$	34.0651	2.00730
$289$	−8.54277	−0.502516
$290$	21.6198	1.26956
$291$	3.39839	0.199217
$292$	33.4469	1.95733
$293$	8.23291	0.480972	0.240486	−	0.970653i	$-0.422693\pi$
$293$	8.23291	0.480972	0.240486	+	0.970653i	$0.422693\pi$
$294$	31.1002	1.81380
$295$	6.52183	0.379716
$296$	−37.4924	−2.17920
$297$	51.6367	2.99627
$298$	36.9098	2.13813
$299$	0	0
$300$	−13.9713	−0.806635
$301$	14.9423	0.861259
$302$	−47.1182	−2.71135
$303$	14.2822	0.820494
$304$	5.05336	0.289830
$305$	4.75548	0.272298
$306$	51.0010	2.91553
$307$	−6.54333	−0.373448	−0.186724	−	0.982412i	$-0.559787\pi$
$307$	−6.54333	−0.373448	−0.186724	+	0.982412i	$0.559787\pi$
$308$	32.7942	1.86862
$309$	54.8671	3.12128
$310$	−3.79456	−0.215517
$311$	−3.20074	−0.181497	−0.0907487	−	0.995874i	$-0.528926\pi$
$311$	−3.20074	−0.181497	−0.0907487	+	0.995874i	$0.528926\pi$
$312$	−66.8425	−3.78421
$313$	17.8318	1.00791	0.503957	−	0.863729i	$-0.331877\pi$
$313$	17.8318	1.00791	0.503957	+	0.863729i	$0.331877\pi$
$314$	17.5851	0.992382
$315$	12.1828	0.686425
$316$	−44.2737	−2.49059
$317$	23.4455	1.31683	0.658417	−	0.752654i	$-0.271227\pi$
$317$	23.4455	1.31683	0.658417	+	0.752654i	$0.271227\pi$
$318$	54.1835	3.03846
$319$	35.7252	2.00023
$320$	1.13091	0.0632198
$321$	57.6192	3.21599
$322$	0	0
$323$	2.15549	0.119935
$324$	80.0517	4.44731
$325$	3.43262	0.190408
$326$	−51.7031	−2.86357
$327$	−30.1114	−1.66516
$328$	−26.4353	−1.45964
$329$	−1.00987	−0.0556758
$330$	−33.4914	−1.84364
$331$	1.65752	0.0911054	0.0455527	−	0.998962i	$-0.485495\pi$
$331$	1.65752	0.0911054	0.0455527	+	0.998962i	$0.485495\pi$
$332$	78.7025	4.31936
$333$	−41.8981	−2.29600
$334$	23.2365	1.27144
$335$	5.71824	0.312421
$336$	37.8334	2.06398
$337$	28.8290	1.57041	0.785207	−	0.619233i	$-0.212556\pi$
$337$	28.8290	1.57041	0.785207	+	0.619233i	$0.212556\pi$
$338$	−3.08812	−0.167971
$339$	−13.5785	−0.737483
$340$	−12.9054	−0.699896
$341$	−6.27025	−0.339553
$342$	12.9986	0.702886
$343$	19.2003	1.03672
$344$	−52.4343	−2.82707
$345$	0	0
$346$	58.5358	3.14690
$347$	9.87931	0.530349	0.265174	−	0.964200i	$-0.414570\pi$
$347$	9.87931	0.530349	0.265174	+	0.964200i	$0.414570\pi$
$348$	119.048	6.38166
$349$	2.61080	0.139753	0.0698764	−	0.997556i	$-0.477740\pi$
$349$	2.61080	0.139753	0.0698764	+	0.997556i	$0.477740\pi$
$350$	−4.47213	−0.239046
$351$	−42.2762	−2.25654
$352$	−20.6633	−1.10136
$353$	−15.2459	−0.811458	−0.405729	−	0.913993i	$-0.632982\pi$
$353$	−15.2459	−0.811458	−0.405729	+	0.913993i	$0.632982\pi$
$354$	52.0972	2.76893
$355$	−6.85182	−0.363657
$356$	18.2891	0.969319
$357$	16.1377	0.854098
$358$	−27.8285	−1.47078
$359$	13.3327	0.703675	0.351838	−	0.936061i	$-0.385557\pi$
$359$	13.3327	0.703675	0.351838	+	0.936061i	$0.385557\pi$
$360$	−42.7511	−2.25318
$361$	−18.4506	−0.971086
$362$	−15.0783	−0.792497
$363$	−20.7107	−1.08703
$364$	−26.8494	−1.40729
$365$	−7.53697	−0.394503
$366$	37.9873	1.98563
$367$	−0.346651	−0.0180950	−0.00904751	−	0.999959i	$-0.502880\pi$
$367$	−0.346651	−0.0180950	−0.00904751	+	0.999959i	$0.502880\pi$
$368$	0	0
$369$	−29.5417	−1.53788
$370$	15.3802	0.799577
$371$	11.9556	0.620704
$372$	−20.8946	−1.08333
$373$	14.7472	0.763584	0.381792	−	0.924248i	$-0.375307\pi$
$373$	14.7472	0.763584	0.381792	+	0.924248i	$0.375307\pi$
$374$	−30.9363	−1.59968
$375$	3.14832	0.162579
$376$	3.54375	0.182755
$377$	−29.2490	−1.50640
$378$	55.0788	2.83295
$379$	−24.8681	−1.27739	−0.638694	−	0.769461i	$-0.720525\pi$
$379$	−24.8681	−1.27739	−0.638694	+	0.769461i	$0.720525\pi$
$380$	−3.28921	−0.168733
$381$	−42.3294	−2.16860
$382$	−17.6964	−0.905425
$383$	8.12847	0.415345	0.207673	−	0.978198i	$-0.433411\pi$
$383$	8.12847	0.415345	0.207673	+	0.978198i	$0.433411\pi$
$384$	40.0665	2.04464
$385$	−7.38989	−0.376624
$386$	−5.59106	−0.284577
$387$	−58.5959	−2.97860
$388$	−4.79019	−0.243185
$389$	−24.0733	−1.22057	−0.610283	−	0.792183i	$-0.708944\pi$
$389$	−24.0733	−1.22057	−0.610283	+	0.792183i	$0.708944\pi$
$390$	27.4202	1.38848
$391$	0	0
$392$	−24.0806	−1.21625
$393$	12.7273	0.642009
$394$	4.07824	0.205459
$395$	9.97670	0.501982
$396$	−128.602	−6.46248
$397$	24.7176	1.24054	0.620271	−	0.784387i	$-0.287023\pi$
$397$	24.7176	1.24054	0.620271	+	0.784387i	$0.287023\pi$
$398$	−55.4826	−2.78109
$399$	4.11302	0.205909
$400$	6.81785	0.340892
$401$	−36.0089	−1.79820	−0.899098	−	0.437747i	$-0.855777\pi$
$401$	−36.0089	−1.79820	−0.899098	+	0.437747i	$0.855777\pi$
$402$	45.6780	2.27821
$403$	5.13360	0.255723
$404$	−20.1315	−1.00158
$405$	−18.0390	−0.896363
$406$	38.1066	1.89120
$407$	25.4147	1.25976
$408$	−56.6292	−2.80356
$409$	−23.9401	−1.18376	−0.591882	−	0.806025i	$-0.701615\pi$
$409$	−23.9401	−1.18376	−0.591882	+	0.806025i	$0.701615\pi$
$410$	10.8443	0.535562
$411$	−52.2906	−2.57931
$412$	−77.3378	−3.81016
$413$	11.4953	0.565644
$414$	0	0
$415$	−17.7349	−0.870573
$416$	16.9175	0.829449
$417$	26.8941	1.31701
$418$	−7.88474	−0.385655
$419$	28.7028	1.40222	0.701111	−	0.713052i	$-0.252688\pi$
$419$	28.7028	1.40222	0.701111	+	0.713052i	$0.252688\pi$
$420$	−24.6256	−1.20161
$421$	5.01656	0.244492	0.122246	−	0.992500i	$-0.460990\pi$
$421$	5.01656	0.244492	0.122246	+	0.992500i	$0.460990\pi$
$422$	21.8084	1.06162
$423$	3.96018	0.192550
$424$	−41.9537	−2.03745
$425$	2.90813	0.141065
$426$	−54.7332	−2.65183
$427$	8.38191	0.405629
$428$	−81.2170	−3.92577
$429$	45.3100	2.18759
$430$	21.5097	1.03729
$431$	−13.7674	−0.663152	−0.331576	−	0.943429i	$-0.607580\pi$
$431$	−13.7674	−0.663152	−0.331576	+	0.943429i	$0.607580\pi$
$432$	−83.9686	−4.03994
$433$	−31.0124	−1.49036	−0.745179	−	0.666864i	$-0.767636\pi$
$433$	−31.0124	−1.49036	−0.745179	+	0.666864i	$0.767636\pi$
$434$	−6.68823	−0.321045
$435$	−26.8265	−1.28623
$436$	42.4435	2.03267
$437$	0	0
$438$	−60.2063	−2.87677
$439$	−15.4993	−0.739741	−0.369870	−	0.929083i	$-0.620598\pi$
$439$	−15.4993	−0.739741	−0.369870	+	0.929083i	$0.620598\pi$
$440$	25.9320	1.23626
$441$	−26.9103	−1.28144
$442$	25.3283	1.20474
$443$	9.03538	0.429284	0.214642	−	0.976693i	$-0.431142\pi$
$443$	9.03538	0.429284	0.214642	+	0.976693i	$0.431142\pi$
$444$	84.6903	4.01922
$445$	−4.12129	−0.195368
$446$	−31.5328	−1.49312
$447$	−45.7989	−2.16621
$448$	1.99332	0.0941756
$449$	−33.6306	−1.58713	−0.793564	−	0.608487i	$-0.791777\pi$
$449$	−33.6306	−1.58713	−0.793564	+	0.608487i	$0.791777\pi$
$450$	17.5374	0.826720
$451$	17.9195	0.843795
$452$	19.1395	0.900248
$453$	58.4658	2.74696
$454$	31.6434	1.48510
$455$	6.05027	0.283641
$456$	−14.4331	−0.675892
$457$	23.5859	1.10330	0.551652	−	0.834074i	$-0.313998\pi$
$457$	23.5859	1.10330	0.551652	+	0.834074i	$0.313998\pi$
$458$	−9.06851	−0.423744
$459$	−35.8166	−1.67177
$460$	0	0
$461$	19.3167	0.899670	0.449835	−	0.893112i	$-0.351483\pi$
$461$	19.3167	0.899670	0.449835	+	0.893112i	$0.351483\pi$
$462$	−59.0314	−2.74639
$463$	−13.9931	−0.650312	−0.325156	−	0.945660i	$-0.605417\pi$
$463$	−13.9931	−0.650312	−0.325156	+	0.945660i	$0.605417\pi$
$464$	−58.0942	−2.69695
$465$	4.70842	0.218348
$466$	58.3413	2.70261
$467$	−15.7795	−0.730190	−0.365095	−	0.930970i	$-0.618964\pi$
$467$	−15.7795	−0.730190	−0.365095	+	0.930970i	$0.618964\pi$
$468$	105.289	4.86700
$469$	10.0789	0.465398
$470$	−1.45372	−0.0670552
$471$	−21.8201	−1.00542
$472$	−40.3383	−1.85672
$473$	35.5432	1.63428
$474$	79.6951	3.66052
$475$	0.741196	0.0340084
$476$	−22.7469	−1.04260
$477$	−46.8837	−2.14666
$478$	13.7608	0.629403
$479$	42.0197	1.91993	0.959964	−	0.280125i	$-0.0903758\pi$
$479$	42.0197	1.91993	0.959964	+	0.280125i	$0.0903758\pi$
$480$	15.5163	0.708221
$481$	−20.8076	−0.948744
$482$	16.3810	0.746135
$483$	0	0
$484$	29.1928	1.32694
$485$	1.07943	0.0490143
$486$	−50.3506	−2.28395
$487$	19.7589	0.895361	0.447680	−	0.894194i	$-0.352250\pi$
$487$	19.7589	0.895361	0.447680	+	0.894194i	$0.352250\pi$
$488$	−29.4132	−1.33147
$489$	64.1549	2.90118
$490$	9.87835	0.446259
$491$	18.5203	0.835808	0.417904	−	0.908491i	$-0.362765\pi$
$491$	18.5203	0.835808	0.417904	+	0.908491i	$0.362765\pi$
$492$	59.7137	2.69210
$493$	−24.7799	−1.11603
$494$	6.45542	0.290443
$495$	28.9793	1.30252
$496$	10.1963	0.457828
$497$	−12.0769	−0.541722
$498$	−141.669	−6.34833
$499$	19.8163	0.887101	0.443551	−	0.896249i	$-0.353719\pi$
$499$	19.8163	0.887101	0.443551	+	0.896249i	$0.353719\pi$
$500$	−4.43771	−0.198460
$501$	−28.8326	−1.28814
$502$	−16.1652	−0.721489
$503$	31.0814	1.38585	0.692924	−	0.721010i	$-0.256322\pi$
$503$	31.0814	1.38585	0.692924	+	0.721010i	$0.256322\pi$
$504$	−75.3522	−3.35645
$505$	4.53646	0.201870
$506$	0	0
$507$	3.83184	0.170178
$508$	59.6653	2.64722
$509$	−21.2096	−0.940101	−0.470050	−	0.882640i	$-0.655764\pi$
$509$	−21.2096	−0.940101	−0.470050	+	0.882640i	$0.655764\pi$
$510$	23.2305	1.02866
$511$	−13.2845	−0.587673
$512$	−50.7369	−2.24227
$513$	−9.12857	−0.403036
$514$	−54.5831	−2.40756
$515$	17.4274	0.767943
$516$	118.442	5.21412
$517$	−2.40217	−0.105647
$518$	27.1088	1.19109
$519$	−72.6331	−3.18824
$520$	−21.2312	−0.931048
$521$	−10.0160	−0.438809	−0.219405	−	0.975634i	$-0.570412\pi$
$521$	−10.0160	−0.438809	−0.219405	+	0.975634i	$0.570412\pi$
$522$	−149.434	−6.54056
$523$	−7.23971	−0.316571	−0.158285	−	0.987393i	$-0.550597\pi$
$523$	−7.23971	−0.316571	−0.158285	+	0.987393i	$0.550597\pi$
$524$	−17.9398	−0.783703
$525$	5.54917	0.242186
$526$	−1.60859	−0.0701380
$527$	4.34921	0.189454
$528$	89.9943	3.91650
$529$	0	0
$530$	17.2103	0.747567
$531$	−45.0784	−1.95624
$532$	−5.79750	−0.251354
$533$	−14.6711	−0.635475
$534$	−32.9214	−1.42465
$535$	18.3016	0.791246
$536$	−35.3680	−1.52766
$537$	34.5305	1.49010
$538$	−43.7638	−1.88679
$539$	16.3233	0.703094
$540$	54.6549	2.35197
$541$	−34.8277	−1.49736	−0.748680	−	0.662931i	$-0.769312\pi$
$541$	−34.8277	−1.49736	−0.748680	+	0.662931i	$0.769312\pi$
$542$	67.6873	2.90742
$543$	18.7096	0.802908
$544$	14.3326	0.614504
$545$	−9.56427	−0.409688
$546$	48.3303	2.06835
$547$	8.19202	0.350266	0.175133	−	0.984545i	$-0.443965\pi$
$547$	8.19202	0.350266	0.175133	+	0.984545i	$0.443965\pi$
$548$	73.7061	3.14857
$549$	−32.8695	−1.40284
$550$	−10.6379	−0.453600
$551$	−6.31565	−0.269056
$552$	0	0
$553$	17.5847	0.747779
$554$	−17.9971	−0.764623
$555$	−19.0842	−0.810080
$556$	−37.9085	−1.60768
$557$	−22.5277	−0.954531	−0.477265	−	0.878759i	$-0.658372\pi$
$557$	−22.5277	−0.954531	−0.477265	+	0.878759i	$0.658372\pi$
$558$	26.2278	1.11031
$559$	−29.1001	−1.23080
$560$	12.0170	0.507811
$561$	38.3868	1.62069
$562$	57.0177	2.40515
$563$	22.7467	0.958659	0.479330	−	0.877635i	$-0.340880\pi$
$563$	22.7467	0.958659	0.479330	+	0.877635i	$0.340880\pi$
$564$	−8.00485	−0.337065
$565$	−4.31293	−0.181446
$566$	37.6006	1.58047
$567$	−31.7951	−1.33527
$568$	42.3793	1.77820
$569$	6.22842	0.261109	0.130555	−	0.991441i	$-0.458324\pi$
$569$	6.22842	0.261109	0.130555	+	0.991441i	$0.458324\pi$
$570$	5.92076	0.247993
$571$	−11.0590	−0.462805	−0.231403	−	0.972858i	$-0.574331\pi$
$571$	−11.0590	−0.462805	−0.231403	+	0.972858i	$0.574331\pi$
$572$	−63.8666	−2.67040
$573$	21.9582	0.917318
$574$	19.1140	0.797801
$575$	0	0
$576$	−7.81677	−0.325699
$577$	−17.1131	−0.712429	−0.356214	−	0.934404i	$-0.615933\pi$
$577$	−17.1131	−0.712429	−0.356214	+	0.934404i	$0.615933\pi$
$578$	−21.6753	−0.901573
$579$	6.93757	0.288315
$580$	37.8133	1.57011
$581$	−31.2592	−1.29685
$582$	8.62260	0.357418
$583$	28.4388	1.17782
$584$	46.6170	1.92903
$585$	−23.7260	−0.980951
$586$	20.8891	0.862919
$587$	14.1443	0.583798	0.291899	−	0.956449i	$-0.405713\pi$
$587$	14.1443	0.583798	0.291899	+	0.956449i	$0.405713\pi$
$588$	54.3947	2.24320
$589$	1.10848	0.0456743
$590$	16.5476	0.681254
$591$	−5.06041	−0.208158
$592$	−41.3278	−1.69856
$593$	−15.0292	−0.617177	−0.308588	−	0.951196i	$-0.599857\pi$
$593$	−15.0292	−0.617177	−0.308588	+	0.951196i	$0.599857\pi$
$594$	131.016	5.37565
$595$	5.12582	0.210138
$596$	64.5557	2.64430
$597$	68.8446	2.81762
$598$	0	0
$599$	1.76515	0.0721223	0.0360611	−	0.999350i	$-0.488519\pi$
$599$	1.76515	0.0721223	0.0360611	+	0.999350i	$0.488519\pi$
$600$	−19.4727	−0.794971
$601$	−16.6997	−0.681196	−0.340598	−	0.940209i	$-0.610629\pi$
$601$	−16.6997	−0.681196	−0.340598	+	0.940209i	$0.610629\pi$
$602$	37.9125	1.54520
$603$	−39.5241	−1.60954
$604$	−82.4103	−3.35323
$605$	−6.57834	−0.267448
$606$	36.2378	1.47206
$607$	−14.7335	−0.598013	−0.299006	−	0.954251i	$-0.596655\pi$
$607$	−14.7335	−0.598013	−0.299006	+	0.954251i	$0.596655\pi$
$608$	3.65295	0.148147
$609$	−47.2839	−1.91604
$610$	12.0659	0.488534
$611$	1.96672	0.0795648
$612$	89.2015	3.60576
$613$	−9.11293	−0.368068	−0.184034	−	0.982920i	$-0.558916\pi$
$613$	−9.11293	−0.368068	−0.184034	+	0.982920i	$0.558916\pi$
$614$	−16.6022	−0.670008
$615$	−13.4560	−0.542597
$616$	45.7073	1.84160
$617$	46.9667	1.89081	0.945404	−	0.325899i	$-0.105667\pi$
$617$	46.9667	1.89081	0.945404	+	0.325899i	$0.105667\pi$
$618$	139.212	5.59994
$619$	−43.8908	−1.76412	−0.882061	−	0.471136i	$-0.843844\pi$
$619$	−43.8908	−1.76412	−0.882061	+	0.471136i	$0.843844\pi$
$620$	−6.63674	−0.266538
$621$	0	0
$622$	−8.12112	−0.325627
$623$	−7.26410	−0.291030
$624$	−73.6804	−2.94958
$625$	1.00000	0.0400000
$626$	45.2440	1.80831
$627$	9.78365	0.390721
$628$	30.7565	1.22732
$629$	−17.6283	−0.702885
$630$	30.9111	1.23153
$631$	−27.1670	−1.08150	−0.540750	−	0.841184i	$-0.681859\pi$
$631$	−27.1670	−1.08150	−0.540750	+	0.841184i	$0.681859\pi$
$632$	−61.7070	−2.45457
$633$	−27.0606	−1.07556
$634$	59.4876	2.36255
$635$	−13.4451	−0.533551
$636$	94.7677	3.75779
$637$	−13.3643	−0.529511
$638$	90.6442	3.58864
$639$	47.3593	1.87350
$640$	12.7263	0.503052
$641$	37.7554	1.49125	0.745624	−	0.666367i	$-0.232152\pi$
$641$	37.7554	1.49125	0.745624	+	0.666367i	$0.232152\pi$
$642$	146.195	5.76986
$643$	−1.12503	−0.0443670	−0.0221835	−	0.999754i	$-0.507062\pi$
$643$	−1.12503	−0.0443670	−0.0221835	+	0.999754i	$0.507062\pi$
$644$	0	0
$645$	−26.6899	−1.05091
$646$	5.46906	0.215177
$647$	−28.6689	−1.12709	−0.563545	−	0.826085i	$-0.690563\pi$
$647$	−28.6689	−1.12709	−0.563545	+	0.826085i	$0.690563\pi$
$648$	111.573	4.38300
$649$	27.3438	1.07334
$650$	8.70947	0.341613
$651$	8.29897	0.325262
$652$	−90.4294	−3.54149
$653$	27.6420	1.08172	0.540858	−	0.841114i	$-0.318099\pi$
$653$	27.6420	1.08172	0.540858	+	0.841114i	$0.318099\pi$
$654$	−76.4006	−2.98750
$655$	4.04258	0.157957
$656$	−29.1396	−1.13771
$657$	52.0950	2.03242
$658$	−2.56230	−0.0998889
$659$	4.74091	0.184680	0.0923399	−	0.995728i	$-0.470565\pi$
$659$	4.74091	0.184680	0.0923399	+	0.995728i	$0.470565\pi$
$660$	−58.5770	−2.28011
$661$	15.7116	0.611109	0.305555	−	0.952175i	$-0.401158\pi$
$661$	15.7116	0.611109	0.305555	+	0.952175i	$0.401158\pi$
$662$	4.20556	0.163454
$663$	−31.4282	−1.22057
$664$	109.693	4.25690
$665$	1.30642	0.0506607
$666$	−106.307	−4.11930
$667$	0	0
$668$	40.6409	1.57244
$669$	39.1270	1.51274
$670$	14.5087	0.560519
$671$	19.9381	0.769701
$672$	27.3488	1.05500
$673$	−17.4053	−0.670925	−0.335463	−	0.942054i	$-0.608893\pi$
$673$	−17.4053	−0.670925	−0.335463	+	0.942054i	$0.608893\pi$
$674$	73.1467	2.81751
$675$	−12.3160	−0.474043
$676$	−5.40116	−0.207737
$677$	2.29090	0.0880466	0.0440233	−	0.999031i	$-0.485982\pi$
$677$	2.29090	0.0880466	0.0440233	+	0.999031i	$0.485982\pi$
$678$	−34.4522	−1.32313
$679$	1.90258	0.0730143
$680$	−17.9871	−0.689774
$681$	−39.2642	−1.50461
$682$	−15.9093	−0.609198
$683$	−39.6339	−1.51655	−0.758275	−	0.651935i	$-0.773958\pi$
$683$	−39.6339	−1.51655	−0.758275	+	0.651935i	$0.773958\pi$
$684$	22.7348	0.869287
$685$	−16.6090	−0.634599
$686$	48.7163	1.86000
$687$	11.2525	0.429310
$688$	−57.7983	−2.20354
$689$	−23.2835	−0.887031
$690$	0	0
$691$	5.68837	0.216396	0.108198	−	0.994129i	$-0.465492\pi$
$691$	5.68837	0.216396	0.108198	+	0.994129i	$0.465492\pi$
$692$	102.380	3.89190
$693$	51.0784	1.94031
$694$	25.0664	0.951508
$695$	8.54236	0.324030
$696$	165.925	6.28937
$697$	−12.4294	−0.470797
$698$	6.62428	0.250733
$699$	−72.3919	−2.73811
$700$	−7.82182	−0.295637
$701$	−19.5886	−0.739851	−0.369925	−	0.929061i	$-0.620617\pi$
$701$	−19.5886	−0.739851	−0.369925	+	0.929061i	$0.620617\pi$
$702$	−107.266	−4.04849
$703$	−4.49292	−0.169454
$704$	4.74152	0.178703
$705$	1.80383	0.0679360
$706$	−38.6829	−1.45585
$707$	7.99588	0.300716
$708$	91.1187	3.42445
$709$	−13.1199	−0.492727	−0.246364	−	0.969177i	$-0.579236\pi$
$709$	−13.1199	−0.492727	−0.246364	+	0.969177i	$0.579236\pi$
$710$	−17.3849	−0.652443
$711$	−68.9582	−2.58614
$712$	25.4906	0.955302
$713$	0	0
$714$	40.9456	1.53235
$715$	14.3918	0.538223
$716$	−48.6724	−1.81897
$717$	−17.0748	−0.637671
$718$	33.8287	1.26248
$719$	−34.4442	−1.28455	−0.642276	−	0.766473i	$-0.722010\pi$
$719$	−34.4442	−1.28455	−0.642276	+	0.766473i	$0.722010\pi$
$720$	−47.1245	−1.75622
$721$	30.7172	1.14397
$722$	−46.8141	−1.74224
$723$	−20.3261	−0.755937
$724$	−26.3721	−0.980113
$725$	−8.52090	−0.316458
$726$	−52.5486	−1.95026
$727$	30.6919	1.13830	0.569149	−	0.822234i	$-0.307273\pi$
$727$	30.6919	1.13830	0.569149	+	0.822234i	$0.307273\pi$
$728$	−37.4216	−1.38694
$729$	8.35976	0.309621
$730$	−19.1233	−0.707785
$731$	−24.6537	−0.911850
$732$	66.4404	2.45571
$733$	−26.8777	−0.992752	−0.496376	−	0.868108i	$-0.665336\pi$
$733$	−26.8777	−0.992752	−0.496376	+	0.868108i	$0.665336\pi$
$734$	−0.879544	−0.0324646
$735$	−12.2574	−0.452121
$736$	0	0
$737$	23.9746	0.883116
$738$	−74.9551	−2.75913
$739$	24.2563	0.892283	0.446141	−	0.894962i	$-0.352798\pi$
$739$	24.2563	0.892283	0.446141	+	0.894962i	$0.352798\pi$
$740$	26.9001	0.988869
$741$	−8.01010	−0.294258
$742$	30.3345	1.11362
$743$	−49.9756	−1.83343	−0.916713	−	0.399546i	$-0.869168\pi$
$743$	−49.9756	−1.83343	−0.916713	+	0.399546i	$0.869168\pi$
$744$	−29.1221	−1.06767
$745$	−14.5471	−0.532964
$746$	37.4177	1.36996
$747$	122.583	4.48506
$748$	−54.1080	−1.97839
$749$	32.2580	1.17868
$750$	7.98812	0.291685
$751$	40.6068	1.48176	0.740882	−	0.671635i	$-0.234408\pi$
$751$	40.6068	1.48176	0.740882	+	0.671635i	$0.234408\pi$
$752$	3.90627	0.142447
$753$	20.0583	0.730966
$754$	−74.2125	−2.70266
$755$	18.5705	0.675849
$756$	96.3336	3.50362
$757$	−8.26472	−0.300386	−0.150193	−	0.988657i	$-0.547990\pi$
$757$	−8.26472	−0.300386	−0.150193	+	0.988657i	$0.547990\pi$
$758$	−63.0969	−2.29178
$759$	0	0
$760$	−4.58438	−0.166293
$761$	34.2198	1.24047	0.620233	−	0.784417i	$-0.287038\pi$
$761$	34.2198	1.24047	0.620233	+	0.784417i	$0.287038\pi$
$762$	−107.401	−3.89072
$763$	−16.8578	−0.610293
$764$	−30.9512	−1.11977
$765$	−20.1008	−0.726746
$766$	20.6241	0.745178
$767$	−22.3870	−0.808347
$768$	94.5385	3.41136
$769$	44.0019	1.58675	0.793374	−	0.608734i	$-0.208322\pi$
$769$	44.0019	1.58675	0.793374	+	0.608734i	$0.208322\pi$
$770$	−18.7501	−0.675707
$771$	67.7286	2.43918
$772$	−9.77883	−0.351948
$773$	−15.6410	−0.562569	−0.281285	−	0.959624i	$-0.590761\pi$
$773$	−15.6410	−0.562569	−0.281285	+	0.959624i	$0.590761\pi$
$774$	−148.673	−5.34395
$775$	1.49553	0.0537212
$776$	−6.67638	−0.239668
$777$	−33.6375	−1.20674
$778$	−61.0804	−2.18984
$779$	−3.16788	−0.113501
$780$	47.9583	1.71718
$781$	−28.7273	−1.02794
$782$	0	0
$783$	104.943	3.75037
$784$	−26.5440	−0.947999
$785$	−6.93072	−0.247368
$786$	32.2926	1.15184
$787$	−28.3616	−1.01098	−0.505491	−	0.862832i	$-0.668688\pi$
$787$	−28.3616	−1.01098	−0.505491	+	0.862832i	$0.668688\pi$
$788$	7.13290	0.254099
$789$	1.99600	0.0710593
$790$	25.3135	0.900615
$791$	−7.60189	−0.270292
$792$	−179.240	−6.36903
$793$	−16.3238	−0.579674
$794$	62.7152	2.22568
$795$	−21.3551	−0.757387
$796$	−97.0398	−3.43948
$797$	−2.34830	−0.0831809	−0.0415905	−	0.999135i	$-0.513242\pi$
$797$	−2.34830	−0.0831809	−0.0415905	+	0.999135i	$0.513242\pi$
$798$	10.4358	0.369424
$799$	1.66621	0.0589462
$800$	4.92845	0.174247
$801$	28.4860	1.00650
$802$	−91.3640	−3.22617
$803$	−31.5999	−1.11514
$804$	79.8914	2.81755
$805$	0	0
$806$	13.0253	0.458797
$807$	54.3036	1.91158
$808$	−28.0585	−0.987096
$809$	−9.10313	−0.320049	−0.160025	−	0.987113i	$-0.551157\pi$
$809$	−9.10313	−0.320049	−0.160025	+	0.987113i	$0.551157\pi$
$810$	−45.7696	−1.60818
$811$	25.8112	0.906354	0.453177	−	0.891421i	$-0.350291\pi$
$811$	25.8112	0.906354	0.453177	+	0.891421i	$0.350291\pi$
$812$	66.6489	2.33892
$813$	−83.9886	−2.94561
$814$	64.4837	2.26015
$815$	20.3775	0.713792
$816$	−62.4224	−2.18522
$817$	−6.28349	−0.219831
$818$	−60.7425	−2.12381
$819$	−41.8191	−1.46128
$820$	18.9668	0.662351
$821$	−22.7062	−0.792451	−0.396225	−	0.918153i	$-0.629680\pi$
$821$	−22.7062	−0.792451	−0.396225	+	0.918153i	$0.629680\pi$
$822$	−132.675	−4.62758
$823$	−28.9011	−1.00743	−0.503714	−	0.863870i	$-0.668034\pi$
$823$	−28.9011	−1.00743	−0.503714	+	0.863870i	$0.668034\pi$
$824$	−107.791	−3.75506
$825$	13.1998	0.459559
$826$	29.1665	1.01483
$827$	−23.8936	−0.830860	−0.415430	−	0.909625i	$-0.636369\pi$
$827$	−23.8936	−0.830860	−0.415430	+	0.909625i	$0.636369\pi$
$828$	0	0
$829$	−15.3244	−0.532237	−0.266119	−	0.963940i	$-0.585741\pi$
$829$	−15.3244	−0.532237	−0.266119	+	0.963940i	$0.585741\pi$
$830$	−44.9982	−1.56191
$831$	22.3314	0.774667
$832$	−3.88199	−0.134584
$833$	−11.3223	−0.392293
$834$	68.2374	2.36287
$835$	−9.15807	−0.316928
$836$	−13.7905	−0.476955
$837$	−18.4190	−0.636654
$838$	72.8265	2.51575
$839$	−1.24156	−0.0428633	−0.0214317	−	0.999770i	$-0.506822\pi$
$839$	−1.24156	−0.0428633	−0.0214317	+	0.999770i	$0.506822\pi$
$840$	−34.3222	−1.18423
$841$	43.6057	1.50364
$842$	12.7283	0.438647
$843$	−70.7494	−2.43674
$844$	38.1432	1.31294
$845$	1.21710	0.0418697
$846$	10.0480	0.345458
$847$	−11.5949	−0.398404
$848$	−46.2455	−1.58808
$849$	−46.6561	−1.60123
$850$	7.37870	0.253087
$851$	0	0
$852$	−95.7291	−3.27962
$853$	−22.6675	−0.776121	−0.388060	−	0.921634i	$-0.626855\pi$
$853$	−22.6675	−0.776121	−0.388060	+	0.921634i	$0.626855\pi$
$854$	21.2671	0.727746
$855$	−5.12309	−0.175206
$856$	−113.197	−3.86900
$857$	32.8090	1.12074	0.560368	−	0.828244i	$-0.310660\pi$
$857$	32.8090	1.12074	0.560368	+	0.828244i	$0.310660\pi$
$858$	114.963	3.92479
$859$	45.4626	1.55116	0.775581	−	0.631248i	$-0.217457\pi$
$859$	45.4626	1.55116	0.775581	+	0.631248i	$0.217457\pi$
$860$	37.6207	1.28286
$861$	−23.7172	−0.808281
$862$	−34.9315	−1.18977
$863$	−7.68968	−0.261760	−0.130880	−	0.991398i	$-0.541780\pi$
$863$	−7.68968	−0.261760	−0.130880	+	0.991398i	$0.541780\pi$
$864$	−60.6988	−2.06502
$865$	−23.0704	−0.784418
$866$	−78.6865	−2.67388
$867$	26.8954	0.913416
$868$	−11.6978	−0.397049
$869$	41.8288	1.41895
$870$	−68.0660	−2.30765
$871$	−19.6286	−0.665088
$872$	59.1561	2.00328
$873$	−7.46093	−0.252514
$874$	0	0
$875$	1.76258	0.0595861
$876$	−105.302	−3.55781
$877$	29.4117	0.993164	0.496582	−	0.867990i	$-0.334588\pi$
$877$	29.4117	0.993164	0.496582	+	0.867990i	$0.334588\pi$
$878$	−39.3258	−1.32718
$879$	−25.9198	−0.874254
$880$	28.5849	0.963596
$881$	38.4550	1.29558	0.647790	−	0.761819i	$-0.275693\pi$
$881$	38.4550	1.29558	0.647790	+	0.761819i	$0.275693\pi$
$882$	−68.2785	−2.29906
$883$	−33.5311	−1.12841	−0.564206	−	0.825634i	$-0.690818\pi$
$883$	−33.5311	−1.12841	−0.564206	+	0.825634i	$0.690818\pi$
$884$	44.2995	1.48995
$885$	−20.5328	−0.690203
$886$	22.9251	0.770185
$887$	24.2411	0.813938	0.406969	−	0.913442i	$-0.366586\pi$
$887$	24.2411	0.813938	0.406969	+	0.913442i	$0.366586\pi$
$888$	118.038	3.96110
$889$	−23.6980	−0.794805
$890$	−10.4568	−0.350513
$891$	−75.6311	−2.53374
$892$	−55.1513	−1.84660
$893$	0.424667	0.0142109
$894$	−116.204	−3.88644
$895$	10.9679	0.366617
$896$	22.4312	0.749373
$897$	0	0
$898$	−85.3298	−2.84749
$899$	−12.7433	−0.425012
$900$	30.6731	1.02244
$901$	−19.7259	−0.657165
$902$	45.4664	1.51387
$903$	−47.0431	−1.56550
$904$	26.6760	0.887230
$905$	5.94274	0.197543
$906$	148.343	4.92837
$907$	26.3131	0.873712	0.436856	−	0.899531i	$-0.356092\pi$
$907$	26.3131	0.873712	0.436856	+	0.899531i	$0.356092\pi$
$908$	55.3447	1.83668
$909$	−31.3557	−1.04000
$910$	15.3511	0.508885
$911$	30.0615	0.995983	0.497991	−	0.867182i	$-0.334071\pi$
$911$	30.0615	0.995983	0.497991	+	0.867182i	$0.334071\pi$
$912$	−15.9096	−0.526819
$913$	−74.3564	−2.46084
$914$	59.8438	1.97946
$915$	−14.9718	−0.494952
$916$	−15.8609	−0.524061
$917$	7.12537	0.235300
$918$	−90.8761	−2.99936
$919$	30.8016	1.01605	0.508025	−	0.861342i	$-0.330376\pi$
$919$	30.8016	1.01605	0.508025	+	0.861342i	$0.330376\pi$
$920$	0	0
$921$	20.6005	0.678810
$922$	49.0117	1.61411
$923$	23.5197	0.774161
$924$	−103.247	−3.39656
$925$	−6.06172	−0.199308
$926$	−35.5041	−1.16674
$927$	−120.457	−3.95633
$928$	−41.9948	−1.37855
$929$	−20.6652	−0.678003	−0.339001	−	0.940786i	$-0.610089\pi$
$929$	−20.6652	−0.678003	−0.339001	+	0.940786i	$0.610089\pi$
$930$	11.9465	0.391741
$931$	−2.88570	−0.0945752
$932$	102.040	3.34242
$933$	10.0770	0.329905
$934$	−40.0369	−1.31005
$935$	12.1928	0.398747
$936$	146.748	4.79662
$937$	3.35287	0.109533	0.0547667	−	0.998499i	$-0.482558\pi$
$937$	3.35287	0.109533	0.0547667	+	0.998499i	$0.482558\pi$
$938$	25.5727	0.834979
$939$	−56.1403	−1.83207
$940$	−2.54258	−0.0829298
$941$	−28.2427	−0.920685	−0.460343	−	0.887741i	$-0.652273\pi$
$941$	−28.2427	−0.920685	−0.460343	+	0.887741i	$0.652273\pi$
$942$	−55.3634	−1.80384
$943$	0	0
$944$	−44.4648	−1.44721
$945$	−21.7080	−0.706160
$946$	90.1826	2.93209
$947$	−7.07101	−0.229777	−0.114889	−	0.993378i	$-0.536651\pi$
$947$	−7.07101	−0.229777	−0.114889	+	0.993378i	$0.536651\pi$
$948$	139.388	4.52711
$949$	25.8716	0.839827
$950$	1.88061	0.0610150
$951$	−73.8141	−2.39359
$952$	−31.7037	−1.02752
$953$	−5.80298	−0.187977	−0.0939885	−	0.995573i	$-0.529962\pi$
$953$	−5.80298	−0.187977	−0.0939885	+	0.995573i	$0.529962\pi$
$954$	−118.956	−3.85135
$955$	6.97458	0.225692
$956$	24.0678	0.778407
$957$	−112.474	−3.63578
$958$	106.615	3.44457
$959$	−29.2748	−0.945332
$960$	−3.56047	−0.114914
$961$	−28.7634	−0.927851
$962$	−52.7943	−1.70216
$963$	−126.499	−4.07638
$964$	28.6506	0.922775
$965$	2.20358	0.0709356
$966$	0	0
$967$	−22.3545	−0.718872	−0.359436	−	0.933170i	$-0.617031\pi$
$967$	−22.3545	−0.718872	−0.359436	+	0.933170i	$0.617031\pi$
$968$	40.6878	1.30776
$969$	−6.78619	−0.218004
$970$	2.73879	0.0879374
$971$	0.277038	0.00889058	0.00444529	−	0.999990i	$-0.498585\pi$
$971$	0.277038	0.00889058	0.00444529	+	0.999990i	$0.498585\pi$
$972$	−88.0638	−2.82465
$973$	15.0566	0.482692
$974$	50.1335	1.60638
$975$	−10.8070	−0.346101
$976$	−32.4221	−1.03781
$977$	−60.6182	−1.93935	−0.969674	−	0.244403i	$-0.921408\pi$
$977$	−60.6182	−1.93935	−0.969674	+	0.244403i	$0.921408\pi$
$978$	162.778	5.20506
$979$	−17.2791	−0.552243
$980$	17.2774	0.551906
$981$	66.1076	2.11065
$982$	46.9908	1.49954
$983$	−12.6984	−0.405018	−0.202509	−	0.979280i	$-0.564909\pi$
$983$	−12.6984	−0.405018	−0.202509	+	0.979280i	$0.564909\pi$
$984$	83.2267	2.65317
$985$	−1.60734	−0.0512140
$986$	−62.8731	−2.00229
$987$	3.17939	0.101201
$988$	11.2906	0.359203
$989$	0	0
$990$	73.5282	2.33688
$991$	34.1145	1.08368	0.541842	−	0.840480i	$-0.317727\pi$
$991$	34.1145	1.08368	0.541842	+	0.840480i	$0.317727\pi$
$992$	7.37066	0.234019
$993$	−5.21840	−0.165601
$994$	−30.6423	−0.971913
$995$	21.8671	0.693233
$996$	−247.781	−7.85123
$997$	44.2099	1.40014	0.700070	−	0.714074i	$-0.253152\pi$
$997$	44.2099	1.40014	0.700070	+	0.714074i	$0.253152\pi$
$998$	50.2793	1.59156
$999$	74.6561	2.36202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2645.2.a.v.1.15	✓	16
23.22	odd	2		2645.2.a.w.1.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2645.2.a.v.1.15	✓	16	1.1	even	1	trivial
2645.2.a.w.1.15	yes	16	23.22	odd	2

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:	\( N \)	\(=\)	\( 2645 = 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2645.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53726 q^{2} -3.14832 q^{3} +4.43771 q^{4} -1.00000 q^{5} -7.98812 q^{6} -1.76258 q^{7} +6.18511 q^{8} +6.91193 q^{9} +O(q^{10})\)	\(q+2.53726 q^{2} -3.14832 q^{3} +4.43771 q^{4} -1.00000 q^{5} -7.98812 q^{6} -1.76258 q^{7} +6.18511 q^{8} +6.91193 q^{9} -2.53726 q^{10} -4.19265 q^{11} -13.9713 q^{12} +3.43262 q^{13} -4.47213 q^{14} +3.14832 q^{15} +6.81785 q^{16} +2.90813 q^{17} +17.5374 q^{18} +0.741196 q^{19} -4.43771 q^{20} +5.54917 q^{21} -10.6379 q^{22} -19.4727 q^{24} +1.00000 q^{25} +8.70947 q^{26} -12.3160 q^{27} -7.82182 q^{28} -8.52090 q^{29} +7.98812 q^{30} +1.49553 q^{31} +4.92845 q^{32} +13.1998 q^{33} +7.37870 q^{34} +1.76258 q^{35} +30.6731 q^{36} -6.06172 q^{37} +1.88061 q^{38} -10.8070 q^{39} -6.18511 q^{40} -4.27402 q^{41} +14.0797 q^{42} -8.47750 q^{43} -18.6058 q^{44} -6.91193 q^{45} +0.572948 q^{47} -21.4648 q^{48} -3.89331 q^{49} +2.53726 q^{50} -9.15573 q^{51} +15.2330 q^{52} -6.78301 q^{53} -31.2490 q^{54} +4.19265 q^{55} -10.9018 q^{56} -2.33352 q^{57} -21.6198 q^{58} -6.52183 q^{59} +13.9713 q^{60} -4.75548 q^{61} +3.79456 q^{62} -12.1828 q^{63} -1.13091 q^{64} -3.43262 q^{65} +33.4914 q^{66} -5.71824 q^{67} +12.9054 q^{68} +4.47213 q^{70} +6.85182 q^{71} +42.7511 q^{72} +7.53697 q^{73} -15.3802 q^{74} -3.14832 q^{75} +3.28921 q^{76} +7.38989 q^{77} -27.4202 q^{78} -9.97670 q^{79} -6.81785 q^{80} +18.0390 q^{81} -10.8443 q^{82} +17.7349 q^{83} +24.6256 q^{84} -2.90813 q^{85} -21.5097 q^{86} +26.8265 q^{87} -25.9320 q^{88} +4.12129 q^{89} -17.5374 q^{90} -6.05027 q^{91} -4.70842 q^{93} +1.45372 q^{94} -0.741196 q^{95} -15.5163 q^{96} -1.07943 q^{97} -9.87835 q^{98} -28.9793 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9}+O(q^{10}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9	\( 16 q + 4 q^{2} + 4 q^{3} + 20 q^{4} - 16 q^{5} - 12 q^{6} - 12 q^{7} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 4 q^{12} + 8 q^{13} - 8 q^{14} - 4 q^{15} + 28 q^{16} + 8 q^{17} + 4 q^{18} - 32 q^{19} - 20 q^{20} - 16 q^{21} - 44 q^{22} - 28 q^{24} + 16 q^{25} - 20 q^{26} - 8 q^{27} - 52 q^{28} - 12 q^{29} + 12 q^{30} - 12 q^{31} + 4 q^{32} - 20 q^{33} - 24 q^{34} + 12 q^{35} - 4 q^{36} - 28 q^{37} + 20 q^{38} - 8 q^{39} - 4 q^{41} + 8 q^{42} - 48 q^{43} - 32 q^{44} - 8 q^{45} + 48 q^{47} + 24 q^{48} + 12 q^{49} + 4 q^{50} - 24 q^{51} + 12 q^{52} + 4 q^{53} - 44 q^{54} + 16 q^{55} - 64 q^{56} - 52 q^{57} - 28 q^{58} - 40 q^{59} + 4 q^{60} - 16 q^{61} - 4 q^{63} - 16 q^{64} - 8 q^{65} - 8 q^{66} - 68 q^{67} + 4 q^{68} + 8 q^{70} - 24 q^{71} - 12 q^{72} + 52 q^{73} - 40 q^{74} + 4 q^{75} - 24 q^{76} + 44 q^{77} + 12 q^{78} - 72 q^{79} - 28 q^{80} + 20 q^{81} - 20 q^{82} - 12 q^{83} - 32 q^{84} - 8 q^{85} + 56 q^{86} - 104 q^{88} - 48 q^{89} - 4 q^{90} - 48 q^{91} - 48 q^{93} - 60 q^{94} + 32 q^{95} - 108 q^{96} - 4 q^{97} + 8 q^{98} - 24 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 + 4 * q^3 + 20 * q^4 - 16 * q^5 - 12 * q^6 - 12 * q^7 + 8 * q^9 - 4 * q^10 - 16 * q^11 - 4 * q^12 + 8 * q^13 - 8 * q^14 - 4 * q^15 + 28 * q^16 + 8 * q^17 + 4 * q^18 - 32 * q^19 - 20 * q^20 - 16 * q^21 - 44 * q^22 - 28 * q^24 + 16 * q^25 - 20 * q^26 - 8 * q^27 - 52 * q^28 - 12 * q^29 + 12 * q^30 - 12 * q^31 + 4 * q^32 - 20 * q^33 - 24 * q^34 + 12 * q^35 - 4 * q^36 - 28 * q^37 + 20 * q^38 - 8 * q^39 - 4 * q^41 + 8 * q^42 - 48 * q^43 - 32 * q^44 - 8 * q^45 + 48 * q^47 + 24 * q^48 + 12 * q^49 + 4 * q^50 - 24 * q^51 + 12 * q^52 + 4 * q^53 - 44 * q^54 + 16 * q^55 - 64 * q^56 - 52 * q^57 - 28 * q^58 - 40 * q^59 + 4 * q^60 - 16 * q^61 - 4 * q^63 - 16 * q^64 - 8 * q^65 - 8 * q^66 - 68 * q^67 + 4 * q^68 + 8 * q^70 - 24 * q^71 - 12 * q^72 + 52 * q^73 - 40 * q^74 + 4 * q^75 - 24 * q^76 + 44 * q^77 + 12 * q^78 - 72 * q^79 - 28 * q^80 + 20 * q^81 - 20 * q^82 - 12 * q^83 - 32 * q^84 - 8 * q^85 + 56 * q^86 - 104 * q^88 - 48 * q^89 - 4 * q^90 - 48 * q^91 - 48 * q^93 - 60 * q^94 + 32 * q^95 - 108 * q^96 - 4 * q^97 + 8 * q^98 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more