LMFDB - Embedded newform 1445.2.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(1,1445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, 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("1445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.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:	$11.5383830921$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.55555$ of defining polynomial
Character	$\chi$	$=$	1445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55555 q^{2} +3.00797 q^{3} +0.419729 q^{4} -1.00000 q^{5} -4.67904 q^{6} +3.45467 q^{7} +2.45819 q^{8} +6.04787 q^{9} +O(q^{10})$	\(q-1.55555 q^{2} +3.00797 q^{3} +0.419729 q^{4} -1.00000 q^{5} -4.67904 q^{6} +3.45467 q^{7} +2.45819 q^{8} +6.04787 q^{9} +1.55555 q^{10} +4.24326 q^{11} +1.26253 q^{12} -0.127392 q^{13} -5.37391 q^{14} -3.00797 q^{15} -4.66329 q^{16} -9.40776 q^{18} +2.57338 q^{19} -0.419729 q^{20} +10.3916 q^{21} -6.60059 q^{22} +3.25221 q^{23} +7.39415 q^{24} +1.00000 q^{25} +0.198164 q^{26} +9.16791 q^{27} +1.45003 q^{28} -4.90337 q^{29} +4.67904 q^{30} -5.36372 q^{31} +2.33759 q^{32} +12.7636 q^{33} -3.45467 q^{35} +2.53847 q^{36} -1.77167 q^{37} -4.00302 q^{38} -0.383191 q^{39} -2.45819 q^{40} -10.0623 q^{41} -16.1646 q^{42} +2.53465 q^{43} +1.78102 q^{44} -6.04787 q^{45} -5.05897 q^{46} -4.59479 q^{47} -14.0270 q^{48} +4.93477 q^{49} -1.55555 q^{50} -0.0534701 q^{52} +1.63860 q^{53} -14.2611 q^{54} -4.24326 q^{55} +8.49224 q^{56} +7.74066 q^{57} +7.62743 q^{58} +6.14174 q^{59} -1.26253 q^{60} -4.04195 q^{61} +8.34352 q^{62} +20.8934 q^{63} +5.69034 q^{64} +0.127392 q^{65} -19.8544 q^{66} +6.88856 q^{67} +9.78255 q^{69} +5.37391 q^{70} -7.21741 q^{71} +14.8668 q^{72} +14.6252 q^{73} +2.75592 q^{74} +3.00797 q^{75} +1.08012 q^{76} +14.6591 q^{77} +0.596072 q^{78} +5.15061 q^{79} +4.66329 q^{80} +9.43315 q^{81} +15.6525 q^{82} -14.4462 q^{83} +4.36164 q^{84} -3.94276 q^{86} -14.7492 q^{87} +10.4307 q^{88} -0.600876 q^{89} +9.40776 q^{90} -0.440098 q^{91} +1.36505 q^{92} -16.1339 q^{93} +7.14741 q^{94} -2.57338 q^{95} +7.03140 q^{96} -7.67639 q^{97} -7.67628 q^{98} +25.6627 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 8 q^{3} + 12 q^{4} - 12 q^{5} + 8 q^{6} + 16 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 4 * q^2 + 8 * q^3 + 12 * q^4 - 12 * q^5 + 8 * q^6 + 16 * q^7 - 12 * q^8 + 12 * q^9	$ 12 q - 4 q^{2} + 8 q^{3} + 12 q^{4} - 12 q^{5} + 8 q^{6} + 16 q^{7} - 12 q^{8} + 12 q^{9} + 4 q^{10} + 16 q^{11} + 16 q^{12} - 8 q^{13} - 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} - 12 q^{20} + 16 q^{21} + 16 q^{22} + 16 q^{23} + 12 q^{25} + 16 q^{26} + 32 q^{27} + 40 q^{28} + 16 q^{29} - 8 q^{30} + 24 q^{31} - 28 q^{32} - 16 q^{35} + 12 q^{36} + 24 q^{37} - 24 q^{38} + 8 q^{39} + 12 q^{40} + 8 q^{41} - 16 q^{43} + 8 q^{44} - 12 q^{45} + 40 q^{46} - 32 q^{47} - 24 q^{48} + 20 q^{49} - 4 q^{50} - 24 q^{52} + 8 q^{54} - 16 q^{55} - 24 q^{56} + 32 q^{57} - 16 q^{58} + 8 q^{59} - 16 q^{60} + 24 q^{61} + 8 q^{62} + 48 q^{63} + 36 q^{64} + 8 q^{65} + 40 q^{66} - 8 q^{67} + 48 q^{69} + 16 q^{70} - 16 q^{71} + 12 q^{72} + 16 q^{73} + 8 q^{75} + 16 q^{76} + 24 q^{77} - 24 q^{78} + 40 q^{79} - 12 q^{80} + 36 q^{81} - 16 q^{82} - 40 q^{83} + 32 q^{84} - 8 q^{86} - 32 q^{87} + 48 q^{88} + 8 q^{89} - 4 q^{90} + 72 q^{91} - 8 q^{92} + 24 q^{93} - 16 q^{94} + 8 q^{96} + 32 q^{97} - 60 q^{98} + 56 q^{99}+O(q^{100}) $ 12 * q - 4 * q^2 + 8 * q^3 + 12 * q^4 - 12 * q^5 + 8 * q^6 + 16 * q^7 - 12 * q^8 + 12 * q^9 + 4 * q^10 + 16 * q^11 + 16 * q^12 - 8 * q^13 - 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 - 12 * q^20 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 12 * q^25 + 16 * q^26 + 32 * q^27 + 40 * q^28 + 16 * q^29 - 8 * q^30 + 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 + 24 * q^37 - 24 * q^38 + 8 * q^39 + 12 * q^40 + 8 * q^41 - 16 * q^43 + 8 * q^44 - 12 * q^45 + 40 * q^46 - 32 * q^47 - 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 + 8 * q^54 - 16 * q^55 - 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 16 * q^60 + 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 - 16 * q^71 + 12 * q^72 + 16 * q^73 + 8 * q^75 + 16 * q^76 + 24 * q^77 - 24 * q^78 + 40 * q^79 - 12 * q^80 + 36 * q^81 - 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 + 48 * q^88 + 8 * q^89 - 4 * q^90 + 72 * q^91 - 8 * q^92 + 24 * q^93 - 16 * q^94 + 8 * q^96 + 32 * q^97 - 60 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.55555	−1.09994	−0.549969	−	0.835185i	$-0.685361\pi$
$2$	−1.55555	−1.09994	−0.549969	+	0.835185i	$0.685361\pi$
$3$	3.00797	1.73665	0.868326	−	0.495995i	$-0.165196\pi$
$3$	3.00797	1.73665	0.868326	+	0.495995i	$0.165196\pi$
$4$	0.419729	0.209865
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.67904	−1.91021
$7$	3.45467	1.30574	0.652872	−	0.757468i	$-0.273564\pi$
$7$	3.45467	1.30574	0.652872	+	0.757468i	$0.273564\pi$
$8$	2.45819	0.869100
$9$	6.04787	2.01596
$10$	1.55555	0.491907
$11$	4.24326	1.27939	0.639695	−	0.768629i	$-0.279061\pi$
$11$	4.24326	1.27939	0.639695	+	0.768629i	$0.279061\pi$
$12$	1.26253	0.364461
$13$	−0.127392	−0.0353322	−0.0176661	−	0.999844i	$-0.505624\pi$
$13$	−0.127392	−0.0353322	−0.0176661	+	0.999844i	$0.505624\pi$
$14$	−5.37391	−1.43624
$15$	−3.00797	−0.776654
$16$	−4.66329	−1.16582
$17$	0	0
$17$	0	0
$18$	−9.40776	−2.21743
$19$	2.57338	0.590375	0.295187	−	0.955439i	$-0.404618\pi$
$19$	2.57338	0.590375	0.295187	+	0.955439i	$0.404618\pi$
$20$	−0.419729	−0.0938543
$21$	10.3916	2.26762
$22$	−6.60059	−1.40725
$23$	3.25221	0.678133	0.339066	−	0.940762i	$-0.389889\pi$
$23$	3.25221	0.678133	0.339066	+	0.940762i	$0.389889\pi$
$24$	7.39415	1.50932
$25$	1.00000	0.200000
$26$	0.198164	0.0388632
$27$	9.16791	1.76436
$28$	1.45003	0.274029
$29$	−4.90337	−0.910534	−0.455267	−	0.890355i	$-0.650456\pi$
$29$	−4.90337	−0.910534	−0.455267	+	0.890355i	$0.650456\pi$
$30$	4.67904	0.854272
$31$	−5.36372	−0.963352	−0.481676	−	0.876349i	$-0.659972\pi$
$31$	−5.36372	−0.963352	−0.481676	+	0.876349i	$0.659972\pi$
$32$	2.33759	0.413231
$33$	12.7636	2.22185
$34$	0	0
$35$	−3.45467	−0.583947
$36$	2.53847	0.423078
$37$	−1.77167	−0.291261	−0.145631	−	0.989339i	$-0.546521\pi$
$37$	−1.77167	−0.291261	−0.145631	+	0.989339i	$0.546521\pi$
$38$	−4.00302	−0.649376
$39$	−0.383191	−0.0613597
$40$	−2.45819	−0.388674
$41$	−10.0623	−1.57147	−0.785737	−	0.618561i	$-0.787716\pi$
$41$	−10.0623	−1.57147	−0.785737	+	0.618561i	$0.787716\pi$
$42$	−16.1646	−2.49424
$43$	2.53465	0.386530	0.193265	−	0.981147i	$-0.438092\pi$
$43$	2.53465	0.386530	0.193265	+	0.981147i	$0.438092\pi$
$44$	1.78102	0.268499
$45$	−6.04787	−0.901564
$46$	−5.05897	−0.745905
$47$	−4.59479	−0.670219	−0.335109	−	0.942179i	$-0.608773\pi$
$47$	−4.59479	−0.670219	−0.335109	+	0.942179i	$0.608773\pi$
$48$	−14.0270	−2.02463
$49$	4.93477	0.704968
$50$	−1.55555	−0.219988
$51$	0	0
$52$	−0.0534701	−0.00741498
$53$	1.63860	0.225078	0.112539	−	0.993647i	$-0.464102\pi$
$53$	1.63860	0.225078	0.112539	+	0.993647i	$0.464102\pi$
$54$	−14.2611	−1.94069
$55$	−4.24326	−0.572161
$56$	8.49224	1.13482
$57$	7.74066	1.02528
$58$	7.62743	1.00153
$59$	6.14174	0.799587	0.399793	−	0.916605i	$-0.369082\pi$
$59$	6.14174	0.799587	0.399793	+	0.916605i	$0.369082\pi$
$60$	−1.26253	−0.162992
$61$	−4.04195	−0.517519	−0.258760	−	0.965942i	$-0.583314\pi$
$61$	−4.04195	−0.517519	−0.258760	+	0.965942i	$0.583314\pi$
$62$	8.34352	1.05963
$63$	20.8934	2.63232
$64$	5.69034	0.711292
$65$	0.127392	0.0158010
$66$	−19.8544	−2.44390
$67$	6.88856	0.841571	0.420786	−	0.907160i	$-0.361754\pi$
$67$	6.88856	0.841571	0.420786	+	0.907160i	$0.361754\pi$
$68$	0	0
$69$	9.78255	1.17768
$70$	5.37391	0.642305
$71$	−7.21741	−0.856549	−0.428274	−	0.903649i	$-0.640878\pi$
$71$	−7.21741	−0.856549	−0.428274	+	0.903649i	$0.640878\pi$
$72$	14.8668	1.75207
$73$	14.6252	1.71175	0.855874	−	0.517184i	$-0.173020\pi$
$73$	14.6252	1.71175	0.855874	+	0.517184i	$0.173020\pi$
$74$	2.75592	0.320369
$75$	3.00797	0.347330
$76$	1.08012	0.123899
$77$	14.6591	1.67056
$78$	0.596072	0.0674919
$79$	5.15061	0.579489	0.289744	−	0.957104i	$-0.406430\pi$
$79$	5.15061	0.579489	0.289744	+	0.957104i	$0.406430\pi$
$80$	4.66329	0.521371
$81$	9.43315	1.04813
$82$	15.6525	1.72852
$83$	−14.4462	−1.58567	−0.792837	−	0.609434i	$-0.791397\pi$
$83$	−14.4462	−1.58567	−0.792837	+	0.609434i	$0.791397\pi$
$84$	4.36164	0.475893
$85$	0	0
$86$	−3.94276	−0.425159
$87$	−14.7492	−1.58128
$88$	10.4307	1.11192
$89$	−0.600876	−0.0636927	−0.0318463	−	0.999493i	$-0.510139\pi$
$89$	−0.600876	−0.0636927	−0.0318463	+	0.999493i	$0.510139\pi$
$90$	9.40776	0.991665
$91$	−0.440098	−0.0461348
$92$	1.36505	0.142316
$93$	−16.1339	−1.67301
$94$	7.14741	0.737199
$95$	−2.57338	−0.264024
$96$	7.03140	0.717639
$97$	−7.67639	−0.779420	−0.389710	−	0.920938i	$-0.627425\pi$
$97$	−7.67639	−0.779420	−0.389710	+	0.920938i	$0.627425\pi$
$98$	−7.67628	−0.775421
$99$	25.6627	2.57920
$100$	0.419729	0.0419729
$101$	−16.8485	−1.67649	−0.838243	−	0.545297i	$-0.816417\pi$
$101$	−16.8485	−1.67649	−0.838243	+	0.545297i	$0.816417\pi$
$102$	0	0
$103$	4.32255	0.425914	0.212957	−	0.977062i	$-0.431691\pi$
$103$	4.32255	0.425914	0.212957	+	0.977062i	$0.431691\pi$
$104$	−0.313154	−0.0307072
$105$	−10.3916	−1.01411
$106$	−2.54891	−0.247572
$107$	−4.72053	−0.456351	−0.228175	−	0.973620i	$-0.573276\pi$
$107$	−4.72053	−0.456351	−0.228175	+	0.973620i	$0.573276\pi$
$108$	3.84804	0.370277
$109$	15.3281	1.46816	0.734082	−	0.679061i	$-0.237613\pi$
$109$	15.3281	1.46816	0.734082	+	0.679061i	$0.237613\pi$
$110$	6.60059	0.629342
$111$	−5.32914	−0.505819
$112$	−16.1101	−1.52226
$113$	7.37161	0.693463	0.346732	−	0.937964i	$-0.387291\pi$
$113$	7.37161	0.693463	0.346732	+	0.937964i	$0.387291\pi$
$114$	−12.0410	−1.12774
$115$	−3.25221	−0.303270
$116$	−2.05809	−0.191089
$117$	−0.770451	−0.0712282
$118$	−9.55378	−0.879496
$119$	0	0
$120$	−7.39415	−0.674990
$121$	7.00523	0.636839
$122$	6.28745	0.569239
$123$	−30.2672	−2.72910
$124$	−2.25131	−0.202173
$125$	−1.00000	−0.0894427
$126$	−32.5007	−2.89540
$127$	8.02005	0.711664	0.355832	−	0.934550i	$-0.384197\pi$
$127$	8.02005	0.711664	0.355832	+	0.934550i	$0.384197\pi$
$128$	−13.5268	−1.19561
$129$	7.62413	0.671268
$130$	−0.198164	−0.0173802
$131$	14.2092	1.24146	0.620731	−	0.784023i	$-0.286836\pi$
$131$	14.2092	1.24146	0.620731	+	0.784023i	$0.286836\pi$
$132$	5.35725	0.466288
$133$	8.89020	0.770878
$134$	−10.7155	−0.925677
$135$	−9.16791	−0.789048
$136$	0	0
$137$	8.53083	0.728838	0.364419	−	0.931235i	$-0.381268\pi$
$137$	8.53083	0.728838	0.364419	+	0.931235i	$0.381268\pi$
$138$	−15.2172	−1.29538
$139$	6.83721	0.579925	0.289962	−	0.957038i	$-0.406357\pi$
$139$	6.83721	0.579925	0.289962	+	0.957038i	$0.406357\pi$
$140$	−1.45003	−0.122550
$141$	−13.8210	−1.16394
$142$	11.2270	0.942151
$143$	−0.540557	−0.0452037
$144$	−28.2030	−2.35025
$145$	4.90337	0.407203
$146$	−22.7502	−1.88282
$147$	14.8436	1.22428
$148$	−0.743623	−0.0611254
$149$	7.04072	0.576798	0.288399	−	0.957510i	$-0.406877\pi$
$149$	7.04072	0.576798	0.288399	+	0.957510i	$0.406877\pi$
$150$	−4.67904	−0.382042
$151$	−6.35689	−0.517316	−0.258658	−	0.965969i	$-0.583280\pi$
$151$	−6.35689	−0.517316	−0.258658	+	0.965969i	$0.583280\pi$
$152$	6.32586	0.513095
$153$	0	0
$154$	−22.8029	−1.83751
$155$	5.36372	0.430824
$156$	−0.160837	−0.0128772
$157$	−16.7739	−1.33870	−0.669351	−	0.742947i	$-0.733428\pi$
$157$	−16.7739	−1.33870	−0.669351	+	0.742947i	$0.733428\pi$
$158$	−8.01202	−0.637402
$159$	4.92885	0.390883
$160$	−2.33759	−0.184803
$161$	11.2353	0.885468
$162$	−14.6737	−1.15288
$163$	−4.59441	−0.359862	−0.179931	−	0.983679i	$-0.557587\pi$
$163$	−4.59441	−0.359862	−0.179931	+	0.983679i	$0.557587\pi$
$164$	−4.22346	−0.329797
$165$	−12.7636	−0.993644
$166$	22.4717	1.74414
$167$	−21.0265	−1.62708	−0.813540	−	0.581509i	$-0.802463\pi$
$167$	−21.0265	−1.62708	−0.813540	+	0.581509i	$0.802463\pi$
$168$	25.5444	1.97079
$169$	−12.9838	−0.998752
$170$	0	0
$171$	15.5635	1.19017
$172$	1.06386	0.0811189
$173$	−12.5524	−0.954343	−0.477171	−	0.878810i	$-0.658338\pi$
$173$	−12.5524	−0.954343	−0.477171	+	0.878810i	$0.658338\pi$
$174$	22.9431	1.73931
$175$	3.45467	0.261149
$176$	−19.7875	−1.49154
$177$	18.4742	1.38860
$178$	0.934691	0.0700580
$179$	−12.1536	−0.908404	−0.454202	−	0.890899i	$-0.650075\pi$
$179$	−12.1536	−0.908404	−0.454202	+	0.890899i	$0.650075\pi$
$180$	−2.53847	−0.189206
$181$	15.2624	1.13445	0.567224	−	0.823563i	$-0.308017\pi$
$181$	15.2624	1.13445	0.567224	+	0.823563i	$0.308017\pi$
$182$	0.684594	0.0507455
$183$	−12.1581	−0.898750
$184$	7.99454	0.589366
$185$	1.77167	0.130256
$186$	25.0970	1.84020
$187$	0	0
$188$	−1.92857	−0.140655
$189$	31.6721	2.30381
$190$	4.00302	0.290410
$191$	8.89972	0.643961	0.321981	−	0.946746i	$-0.395651\pi$
$191$	8.89972	0.643961	0.321981	+	0.946746i	$0.395651\pi$
$192$	17.1164	1.23527
$193$	18.3838	1.32330	0.661648	−	0.749814i	$-0.269857\pi$
$193$	18.3838	1.32330	0.661648	+	0.749814i	$0.269857\pi$
$194$	11.9410	0.857314
$195$	0.383191	0.0274409
$196$	2.07127	0.147948
$197$	10.6896	0.761603	0.380801	−	0.924657i	$-0.375648\pi$
$197$	10.6896	0.761603	0.380801	+	0.924657i	$0.375648\pi$
$198$	−39.9195	−2.83696
$199$	15.5191	1.10012	0.550058	−	0.835126i	$-0.314605\pi$
$199$	15.5191	1.10012	0.550058	+	0.835126i	$0.314605\pi$
$200$	2.45819	0.173820
$201$	20.7206	1.46152
$202$	26.2086	1.84403
$203$	−16.9396	−1.18892
$204$	0	0
$205$	10.0623	0.702785
$206$	−6.72393	−0.468479
$207$	19.6690	1.36709
$208$	0.594066	0.0411910
$209$	10.9195	0.755320
$210$	16.1646	1.11546
$211$	−8.75387	−0.602641	−0.301321	−	0.953523i	$-0.597427\pi$
$211$	−8.75387	−0.602641	−0.301321	+	0.953523i	$0.597427\pi$
$212$	0.687766	0.0472360
$213$	−21.7097	−1.48753
$214$	7.34301	0.501958
$215$	−2.53465	−0.172861
$216$	22.5364	1.53341
$217$	−18.5299	−1.25789
$218$	−23.8436	−1.61489
$219$	43.9921	2.97271
$220$	−1.78102	−0.120076
$221$	0	0
$222$	8.28973	0.556370
$223$	−21.1596	−1.41695	−0.708475	−	0.705736i	$-0.750616\pi$
$223$	−21.1596	−1.41695	−0.708475	+	0.705736i	$0.750616\pi$
$224$	8.07561	0.539574
$225$	6.04787	0.403192
$226$	−11.4669	−0.762767
$227$	−25.6333	−1.70134	−0.850671	−	0.525699i	$-0.823804\pi$
$227$	−25.6333	−1.70134	−0.850671	+	0.525699i	$0.823804\pi$
$228$	3.24898	0.215169
$229$	7.10038	0.469206	0.234603	−	0.972091i	$-0.424621\pi$
$229$	7.10038	0.469206	0.234603	+	0.972091i	$0.424621\pi$
$230$	5.05897	0.333579
$231$	44.0940	2.90117
$232$	−12.0534	−0.791345
$233$	−23.7142	−1.55357	−0.776785	−	0.629766i	$-0.783151\pi$
$233$	−23.7142	−1.55357	−0.776785	+	0.629766i	$0.783151\pi$
$234$	1.19847	0.0783467
$235$	4.59479	0.299731
$236$	2.57787	0.167805
$237$	15.4929	1.00637
$238$	0	0
$239$	0.109315	0.00707103	0.00353551	−	0.999994i	$-0.498875\pi$
$239$	0.109315	0.00707103	0.00353551	+	0.999994i	$0.498875\pi$
$240$	14.0270	0.905440
$241$	17.4869	1.12643	0.563215	−	0.826311i	$-0.309564\pi$
$241$	17.4869	1.12643	0.563215	+	0.826311i	$0.309564\pi$
$242$	−10.8970	−0.700484
$243$	0.870897	0.0558681
$244$	−1.69653	−0.108609
$245$	−4.93477	−0.315271
$246$	47.0821	3.00184
$247$	−0.327829	−0.0208592
$248$	−13.1850	−0.837249
$249$	−43.4536	−2.75376
$250$	1.55555	0.0983815
$251$	−3.28782	−0.207525	−0.103763	−	0.994602i	$-0.533088\pi$
$251$	−3.28782	−0.207525	−0.103763	+	0.994602i	$0.533088\pi$
$252$	8.76958	0.552432
$253$	13.8000	0.867597
$254$	−12.4756	−0.782787
$255$	0	0
$256$	9.66087	0.603804
$257$	21.3981	1.33478	0.667388	−	0.744710i	$-0.267412\pi$
$257$	21.3981	1.33478	0.667388	+	0.744710i	$0.267412\pi$
$258$	−11.8597	−0.738353
$259$	−6.12055	−0.380313
$260$	0.0534701	0.00331608
$261$	−29.6550	−1.83560
$262$	−22.1031	−1.36553
$263$	−10.1371	−0.625079	−0.312540	−	0.949905i	$-0.601180\pi$
$263$	−10.1371	−0.625079	−0.312540	+	0.949905i	$0.601180\pi$
$264$	31.3753	1.93101
$265$	−1.63860	−0.100658
$266$	−13.8291	−0.847919
$267$	−1.80741	−0.110612
$268$	2.89133	0.176616
$269$	15.0776	0.919299	0.459649	−	0.888100i	$-0.347975\pi$
$269$	15.0776	0.919299	0.459649	+	0.888100i	$0.347975\pi$
$270$	14.2611	0.867904
$271$	6.80304	0.413255	0.206628	−	0.978420i	$-0.433751\pi$
$271$	6.80304	0.413255	0.206628	+	0.978420i	$0.433751\pi$
$272$	0	0
$273$	−1.32380	−0.0801201
$274$	−13.2701	−0.801676
$275$	4.24326	0.255878
$276$	4.10602	0.247153
$277$	−17.3218	−1.04077	−0.520383	−	0.853933i	$-0.674211\pi$
$277$	−17.3218	−1.04077	−0.520383	+	0.853933i	$0.674211\pi$
$278$	−10.6356	−0.637882
$279$	−32.4391	−1.94208
$280$	−8.49224	−0.507508
$281$	18.0221	1.07511	0.537555	−	0.843228i	$-0.319348\pi$
$281$	18.0221	1.07511	0.537555	+	0.843228i	$0.319348\pi$
$282$	21.4992	1.28026
$283$	18.9013	1.12357	0.561784	−	0.827284i	$-0.310115\pi$
$283$	18.9013	1.12357	0.561784	+	0.827284i	$0.310115\pi$
$284$	−3.02936	−0.179759
$285$	−7.74066	−0.458517
$286$	0.840863	0.0497213
$287$	−34.7621	−2.05194
$288$	14.1374	0.833057
$289$	0	0
$290$	−7.62743	−0.447898
$291$	−23.0903	−1.35358
$292$	6.13862	0.359235
$293$	7.53184	0.440015	0.220007	−	0.975498i	$-0.429392\pi$
$293$	7.53184	0.440015	0.220007	+	0.975498i	$0.429392\pi$
$294$	−23.0900	−1.34664
$295$	−6.14174	−0.357586
$296$	−4.35510	−0.253135
$297$	38.9018	2.25731
$298$	−10.9522	−0.634442
$299$	−0.414306	−0.0239599
$300$	1.26253	0.0728923
$301$	8.75638	0.504709
$302$	9.88844	0.569016
$303$	−50.6797	−2.91147
$304$	−12.0004	−0.688272
$305$	4.04195	0.231442
$306$	0	0
$307$	−7.77074	−0.443499	−0.221750	−	0.975104i	$-0.571177\pi$
$307$	−7.77074	−0.443499	−0.221750	+	0.975104i	$0.571177\pi$
$308$	6.15284	0.350590
$309$	13.0021	0.739663
$310$	−8.34352	−0.473880
$311$	−8.64484	−0.490204	−0.245102	−	0.969497i	$-0.578821\pi$
$311$	−8.64484	−0.490204	−0.245102	+	0.969497i	$0.578821\pi$
$312$	−0.941956	−0.0533278
$313$	12.3380	0.697383	0.348691	−	0.937238i	$-0.386626\pi$
$313$	12.3380	0.697383	0.348691	+	0.937238i	$0.386626\pi$
$314$	26.0926	1.47249
$315$	−20.8934	−1.17721
$316$	2.16186	0.121614
$317$	8.17256	0.459016	0.229508	−	0.973307i	$-0.426288\pi$
$317$	8.17256	0.459016	0.229508	+	0.973307i	$0.426288\pi$
$318$	−7.66705	−0.429947
$319$	−20.8063	−1.16493
$320$	−5.69034	−0.318100
$321$	−14.1992	−0.792522
$322$	−17.4771	−0.973960
$323$	0	0
$324$	3.95937	0.219965
$325$	−0.127392	−0.00706644
$326$	7.14682	0.395826
$327$	46.1064	2.54969
$328$	−24.7351	−1.36577
$329$	−15.8735	−0.875134
$330$	19.8544	1.09295
$331$	−23.4664	−1.28983	−0.644916	−	0.764254i	$-0.723108\pi$
$331$	−23.4664	−1.28983	−0.644916	+	0.764254i	$0.723108\pi$
$332$	−6.06348	−0.332777
$333$	−10.7149	−0.587170
$334$	32.7077	1.78969
$335$	−6.88856	−0.376362
$336$	−48.4588	−2.64364
$337$	23.7182	1.29201	0.646005	−	0.763333i	$-0.276438\pi$
$337$	23.7182	1.29201	0.646005	+	0.763333i	$0.276438\pi$
$338$	20.1969	1.09857
$339$	22.1736	1.20430
$340$	0	0
$341$	−22.7596	−1.23250
$342$	−24.2098	−1.30911
$343$	−7.13468	−0.385237
$344$	6.23063	0.335933
$345$	−9.78255	−0.526675
$346$	19.5259	1.04972
$347$	−7.01884	−0.376791	−0.188395	−	0.982093i	$-0.560329\pi$
$347$	−7.01884	−0.376791	−0.188395	+	0.982093i	$0.560329\pi$
$348$	−6.19067	−0.331854
$349$	−29.3836	−1.57287	−0.786434	−	0.617674i	$-0.788075\pi$
$349$	−29.3836	−1.57287	−0.786434	+	0.617674i	$0.788075\pi$
$350$	−5.37391	−0.287248
$351$	−1.16792	−0.0623389
$352$	9.91899	0.528684
$353$	9.09110	0.483870	0.241935	−	0.970292i	$-0.422218\pi$
$353$	9.09110	0.483870	0.241935	+	0.970292i	$0.422218\pi$
$354$	−28.7375	−1.52738
$355$	7.21741	0.383060
$356$	−0.252205	−0.0133668
$357$	0	0
$358$	18.9055	0.999188
$359$	5.24008	0.276561	0.138280	−	0.990393i	$-0.455843\pi$
$359$	5.24008	0.276561	0.138280	+	0.990393i	$0.455843\pi$
$360$	−14.8668	−0.783549
$361$	−12.3777	−0.651458
$362$	−23.7415	−1.24782
$363$	21.0715	1.10597
$364$	−0.184722	−0.00968206
$365$	−14.6252	−0.765517
$366$	18.9125	0.988570
$367$	−18.2327	−0.951740	−0.475870	−	0.879516i	$-0.657867\pi$
$367$	−18.2327	−0.951740	−0.475870	+	0.879516i	$0.657867\pi$
$368$	−15.1660	−0.790582
$369$	−60.8558	−3.16803
$370$	−2.75592	−0.143274
$371$	5.66082	0.293895
$372$	−6.77186	−0.351105
$373$	−18.5489	−0.960428	−0.480214	−	0.877151i	$-0.659441\pi$
$373$	−18.5489	−0.960428	−0.480214	+	0.877151i	$0.659441\pi$
$374$	0	0
$375$	−3.00797	−0.155331
$376$	−11.2948	−0.582487
$377$	0.624651	0.0321712
$378$	−49.2675	−2.53405
$379$	20.0619	1.03051	0.515255	−	0.857037i	$-0.327697\pi$
$379$	20.0619	1.03051	0.515255	+	0.857037i	$0.327697\pi$
$380$	−1.08012	−0.0554092
$381$	24.1241	1.23591
$382$	−13.8439	−0.708317
$383$	−10.9283	−0.558412	−0.279206	−	0.960231i	$-0.590071\pi$
$383$	−10.9283	−0.558412	−0.279206	+	0.960231i	$0.590071\pi$
$384$	−40.6881	−2.07636
$385$	−14.6591	−0.747095
$386$	−28.5969	−1.45555
$387$	15.3292	0.779228
$388$	−3.22200	−0.163573
$389$	−23.7513	−1.20424	−0.602120	−	0.798405i	$-0.705677\pi$
$389$	−23.7513	−1.20424	−0.602120	+	0.798405i	$0.705677\pi$
$390$	−0.596072	−0.0301833
$391$	0	0
$392$	12.1306	0.612688
$393$	42.7408	2.15599
$394$	−16.6282	−0.837716
$395$	−5.15061	−0.259155
$396$	10.7714	0.541282
$397$	−28.3867	−1.42469	−0.712344	−	0.701830i	$-0.752367\pi$
$397$	−28.3867	−1.42469	−0.712344	+	0.701830i	$0.752367\pi$
$398$	−24.1406	−1.21006
$399$	26.7415	1.33875
$400$	−4.66329	−0.233164
$401$	−8.72614	−0.435763	−0.217881	−	0.975975i	$-0.569915\pi$
$401$	−8.72614	−0.435763	−0.217881	+	0.975975i	$0.569915\pi$
$402$	−32.2318	−1.60758
$403$	0.683295	0.0340373
$404$	−7.07179	−0.351835
$405$	−9.43315	−0.468737
$406$	26.3503	1.30774
$407$	−7.51766	−0.372637
$408$	0	0
$409$	10.5572	0.522020	0.261010	−	0.965336i	$-0.415944\pi$
$409$	10.5572	0.522020	0.261010	+	0.965336i	$0.415944\pi$
$410$	−15.6525	−0.773020
$411$	25.6605	1.26574
$412$	1.81430	0.0893841
$413$	21.2177	1.04406
$414$	−30.5960	−1.50371
$415$	14.4462	0.709135
$416$	−0.297790	−0.0146004
$417$	20.5661	1.00713
$418$	−16.9859	−0.830805
$419$	−14.2887	−0.698051	−0.349025	−	0.937113i	$-0.613487\pi$
$419$	−14.2887	−0.698051	−0.349025	+	0.937113i	$0.613487\pi$
$420$	−4.36164	−0.212826
$421$	18.2694	0.890398	0.445199	−	0.895432i	$-0.353133\pi$
$421$	18.2694	0.890398	0.445199	+	0.895432i	$0.353133\pi$
$422$	13.6171	0.662868
$423$	−27.7887	−1.35113
$424$	4.02798	0.195616
$425$	0	0
$426$	33.7705	1.63619
$427$	−13.9636	−0.675748
$428$	−1.98134	−0.0957718
$429$	−1.62598	−0.0785030
$430$	3.94276	0.190137
$431$	21.0425	1.01358	0.506791	−	0.862069i	$-0.330832\pi$
$431$	21.0425	1.01358	0.506791	+	0.862069i	$0.330832\pi$
$432$	−42.7526	−2.05693
$433$	−3.57657	−0.171879	−0.0859395	−	0.996300i	$-0.527389\pi$
$433$	−3.57657	−0.171879	−0.0859395	+	0.996300i	$0.527389\pi$
$434$	28.8241	1.38360
$435$	14.7492	0.707170
$436$	6.43364	0.308115
$437$	8.36919	0.400353
$438$	−68.4318	−3.26980
$439$	−3.28791	−0.156923	−0.0784617	−	0.996917i	$-0.525001\pi$
$439$	−3.28791	−0.156923	−0.0784617	+	0.996917i	$0.525001\pi$
$440$	−10.4307	−0.497265
$441$	29.8449	1.42119
$442$	0	0
$443$	−24.9391	−1.18489	−0.592447	−	0.805609i	$-0.701838\pi$
$443$	−24.9391	−1.18489	−0.592447	+	0.805609i	$0.701838\pi$
$444$	−2.23679	−0.106154
$445$	0.600876	0.0284842
$446$	32.9147	1.55856
$447$	21.1783	1.00170
$448$	19.6583	0.928766
$449$	−9.65637	−0.455712	−0.227856	−	0.973695i	$-0.573172\pi$
$449$	−9.65637	−0.455712	−0.227856	+	0.973695i	$0.573172\pi$
$450$	−9.40776	−0.443486
$451$	−42.6971	−2.01053
$452$	3.09408	0.145533
$453$	−19.1213	−0.898397
$454$	39.8738	1.87137
$455$	0.440098	0.0206321
$456$	19.0280	0.891067
$457$	−20.4776	−0.957900	−0.478950	−	0.877842i	$-0.658982\pi$
$457$	−20.4776	−0.957900	−0.478950	+	0.877842i	$0.658982\pi$
$458$	−11.0450	−0.516098
$459$	0	0
$460$	−1.36505	−0.0636457
$461$	−18.8498	−0.877923	−0.438961	−	0.898506i	$-0.644654\pi$
$461$	−18.8498	−0.877923	−0.438961	+	0.898506i	$0.644654\pi$
$462$	−68.5904	−3.19111
$463$	2.13430	0.0991892	0.0495946	−	0.998769i	$-0.484207\pi$
$463$	2.13430	0.0991892	0.0495946	+	0.998769i	$0.484207\pi$
$464$	22.8658	1.06152
$465$	16.1339	0.748191
$466$	36.8886	1.70883
$467$	−23.2636	−1.07651	−0.538255	−	0.842782i	$-0.680916\pi$
$467$	−23.2636	−1.07651	−0.538255	+	0.842782i	$0.680916\pi$
$468$	−0.323381	−0.0149483
$469$	23.7977	1.09888
$470$	−7.14741	−0.329686
$471$	−50.4553	−2.32486
$472$	15.0976	0.694921
$473$	10.7552	0.494523
$474$	−24.0999	−1.10694
$475$	2.57338	0.118075
$476$	0	0
$477$	9.91002	0.453749
$478$	−0.170045	−0.00777770
$479$	−31.2128	−1.42615	−0.713074	−	0.701088i	$-0.752698\pi$
$479$	−31.2128	−1.42615	−0.713074	+	0.701088i	$0.752698\pi$
$480$	−7.03140	−0.320938
$481$	0.225697	0.0102909
$482$	−27.2017	−1.23900
$483$	33.7955	1.53775
$484$	2.94030	0.133650
$485$	7.67639	0.348567
$486$	−1.35472	−0.0614514
$487$	−14.6528	−0.663980	−0.331990	−	0.943283i	$-0.607720\pi$
$487$	−14.6528	−0.663980	−0.331990	+	0.943283i	$0.607720\pi$
$488$	−9.93588	−0.449776
$489$	−13.8198	−0.624954
$490$	7.67628	0.346779
$491$	−13.8614	−0.625555	−0.312778	−	0.949826i	$-0.601259\pi$
$491$	−13.8614	−0.625555	−0.312778	+	0.949826i	$0.601259\pi$
$492$	−12.7040	−0.572742
$493$	0	0
$494$	0.509953	0.0229439
$495$	−25.6627	−1.15345
$496$	25.0125	1.12310
$497$	−24.9338	−1.11843
$498$	67.5942	3.02897
$499$	40.0320	1.79208	0.896039	−	0.443976i	$-0.146433\pi$
$499$	40.0320	1.79208	0.896039	+	0.443976i	$0.146433\pi$
$500$	−0.419729	−0.0187709
$501$	−63.2471	−2.82567
$502$	5.11435	0.228265
$503$	39.2423	1.74973	0.874863	−	0.484371i	$-0.160951\pi$
$503$	39.2423	1.74973	0.874863	+	0.484371i	$0.160951\pi$
$504$	51.3600	2.28775
$505$	16.8485	0.749747
$506$	−21.4665	−0.954303
$507$	−39.0548	−1.73448
$508$	3.36625	0.149353
$509$	26.7005	1.18348	0.591740	−	0.806129i	$-0.298441\pi$
$509$	26.7005	1.18348	0.591740	+	0.806129i	$0.298441\pi$
$510$	0	0
$511$	50.5253	2.23511
$512$	12.0256	0.531462
$513$	23.5925	1.04164
$514$	−33.2857	−1.46817
$515$	−4.32255	−0.190474
$516$	3.20007	0.140875
$517$	−19.4969	−0.857471
$518$	9.52081	0.418321
$519$	−37.7573	−1.65736
$520$	0.313154	0.0137327
$521$	−4.64101	−0.203326	−0.101663	−	0.994819i	$-0.532416\pi$
$521$	−4.64101	−0.203326	−0.101663	+	0.994819i	$0.532416\pi$
$522$	46.1298	2.01904
$523$	9.21401	0.402900	0.201450	−	0.979499i	$-0.435435\pi$
$523$	9.21401	0.402900	0.201450	+	0.979499i	$0.435435\pi$
$524$	5.96401	0.260539
$525$	10.3916	0.453524
$526$	15.7687	0.687549
$527$	0	0
$528$	−59.5202	−2.59029
$529$	−12.4231	−0.540136
$530$	2.54891	0.110718
$531$	37.1445	1.61193
$532$	3.73148	0.161780
$533$	1.28186	0.0555236
$534$	2.81152	0.121666
$535$	4.72053	0.204086
$536$	16.9334	0.731410
$537$	−36.5577	−1.57758
$538$	−23.4540	−1.01117
$539$	20.9395	0.901929
$540$	−3.84804	−0.165593
$541$	31.9717	1.37457	0.687285	−	0.726388i	$-0.258803\pi$
$541$	31.9717	1.37457	0.687285	+	0.726388i	$0.258803\pi$
$542$	−10.5824	−0.454555
$543$	45.9089	1.97014
$544$	0	0
$545$	−15.3281	−0.656583
$546$	2.05924	0.0881272
$547$	8.40850	0.359522	0.179761	−	0.983710i	$-0.442468\pi$
$547$	8.40850	0.359522	0.179761	+	0.983710i	$0.442468\pi$
$548$	3.58064	0.152957
$549$	−24.4452	−1.04330
$550$	−6.60059	−0.281450
$551$	−12.6183	−0.537556
$552$	24.0473	1.02352
$553$	17.7937	0.756664
$554$	26.9449	1.14478
$555$	5.32914	0.226209
$556$	2.86978	0.121706
$557$	1.01642	0.0430670	0.0215335	−	0.999768i	$-0.493145\pi$
$557$	1.01642	0.0430670	0.0215335	+	0.999768i	$0.493145\pi$
$558$	50.4605	2.13616
$559$	−0.322894	−0.0136570
$560$	16.1101	0.680777
$561$	0	0
$562$	−28.0343	−1.18256
$563$	10.3869	0.437757	0.218878	−	0.975752i	$-0.429760\pi$
$563$	10.3869	0.437757	0.218878	+	0.975752i	$0.429760\pi$
$564$	−5.80107	−0.244269
$565$	−7.37161	−0.310126
$566$	−29.4019	−1.23585
$567$	32.5885	1.36859
$568$	−17.7417	−0.744427
$569$	23.9733	1.00501	0.502506	−	0.864574i	$-0.332411\pi$
$569$	23.9733	1.00501	0.502506	+	0.864574i	$0.332411\pi$
$570$	12.0410	0.504340
$571$	−11.8509	−0.495944	−0.247972	−	0.968767i	$-0.579764\pi$
$571$	−11.8509	−0.495944	−0.247972	+	0.968767i	$0.579764\pi$
$572$	−0.226888	−0.00948665
$573$	26.7701	1.11834
$574$	54.0741	2.25701
$575$	3.25221	0.135627
$576$	34.4144	1.43394
$577$	−22.5581	−0.939107	−0.469553	−	0.882904i	$-0.655585\pi$
$577$	−22.5581	−0.939107	−0.469553	+	0.882904i	$0.655585\pi$
$578$	0	0
$579$	55.2980	2.29811
$580$	2.05809	0.0854575
$581$	−49.9068	−2.07048
$582$	35.9181	1.48885
$583$	6.95298	0.287963
$584$	35.9514	1.48768
$585$	0.770451	0.0318542
$586$	−11.7161	−0.483989
$587$	−42.1734	−1.74068	−0.870341	−	0.492449i	$-0.836102\pi$
$587$	−42.1734	−1.74068	−0.870341	+	0.492449i	$0.836102\pi$
$588$	6.23031	0.256934
$589$	−13.8029	−0.568739
$590$	9.55378	0.393323
$591$	32.1540	1.32264
$592$	8.26182	0.339559
$593$	8.48766	0.348547	0.174273	−	0.984697i	$-0.444242\pi$
$593$	8.48766	0.348547	0.174273	+	0.984697i	$0.444242\pi$
$594$	−60.5136	−2.48290
$595$	0	0
$596$	2.95519	0.121049
$597$	46.6808	1.91052
$598$	0.644473	0.0263545
$599$	−17.0226	−0.695523	−0.347762	−	0.937583i	$-0.613058\pi$
$599$	−17.0226	−0.695523	−0.347762	+	0.937583i	$0.613058\pi$
$600$	7.39415	0.301865
$601$	−20.2981	−0.827975	−0.413988	−	0.910282i	$-0.635864\pi$
$601$	−20.2981	−0.827975	−0.413988	+	0.910282i	$0.635864\pi$
$602$	−13.6210	−0.555149
$603$	41.6611	1.69657
$604$	−2.66817	−0.108566
$605$	−7.00523	−0.284803
$606$	78.8346	3.20244
$607$	−48.8334	−1.98209	−0.991043	−	0.133542i	$-0.957365\pi$
$607$	−48.8334	−1.98209	−0.991043	+	0.133542i	$0.957365\pi$
$608$	6.01552	0.243961
$609$	−50.9537	−2.06475
$610$	−6.28745	−0.254572
$611$	0.585340	0.0236803
$612$	0	0
$613$	−21.5320	−0.869669	−0.434834	−	0.900510i	$-0.643193\pi$
$613$	−21.5320	−0.869669	−0.434834	+	0.900510i	$0.643193\pi$
$614$	12.0878	0.487822
$615$	30.2672	1.22049
$616$	36.0347	1.45188
$617$	4.62599	0.186235	0.0931177	−	0.995655i	$-0.470317\pi$
$617$	4.62599	0.186235	0.0931177	+	0.995655i	$0.470317\pi$
$618$	−20.2254	−0.813584
$619$	−46.6252	−1.87402	−0.937012	−	0.349298i	$-0.886420\pi$
$619$	−46.6252	−1.87402	−0.937012	+	0.349298i	$0.886420\pi$
$620$	2.25131	0.0904147
$621$	29.8160	1.19647
$622$	13.4475	0.539194
$623$	−2.07583	−0.0831664
$624$	1.78693	0.0715345
$625$	1.00000	0.0400000
$626$	−19.1923	−0.767078
$627$	32.8456	1.31173
$628$	−7.04048	−0.280946
$629$	0	0
$630$	32.5007	1.29486
$631$	26.2970	1.04687	0.523433	−	0.852067i	$-0.324651\pi$
$631$	26.2970	1.04687	0.523433	+	0.852067i	$0.324651\pi$
$632$	12.6612	0.503634
$633$	−26.3314	−1.04658
$634$	−12.7128	−0.504890
$635$	−8.02005	−0.318266
$636$	2.06878	0.0820324
$637$	−0.628651	−0.0249081
$638$	32.3652	1.28135
$639$	−43.6500	−1.72677
$640$	13.5268	0.534693
$641$	6.56445	0.259280	0.129640	−	0.991561i	$-0.458618\pi$
$641$	6.56445	0.259280	0.129640	+	0.991561i	$0.458618\pi$
$642$	22.0875	0.871726
$643$	6.40706	0.252670	0.126335	−	0.991988i	$-0.459679\pi$
$643$	6.40706	0.252670	0.126335	+	0.991988i	$0.459679\pi$
$644$	4.71579	0.185828
$645$	−7.62413	−0.300200
$646$	0	0
$647$	−8.03230	−0.315782	−0.157891	−	0.987457i	$-0.550470\pi$
$647$	−8.03230	−0.315782	−0.157891	+	0.987457i	$0.550470\pi$
$648$	23.1884	0.910928
$649$	26.0610	1.02298
$650$	0.198164	0.00777265
$651$	−55.7373	−2.18452
$652$	−1.92841	−0.0755222
$653$	46.2055	1.80816	0.904080	−	0.427363i	$-0.140557\pi$
$653$	46.2055	1.80816	0.904080	+	0.427363i	$0.140557\pi$
$654$	−71.7207	−2.80450
$655$	−14.2092	−0.555199
$656$	46.9236	1.83206
$657$	88.4513	3.45081
$658$	24.6920	0.962594
$659$	47.9496	1.86785	0.933926	−	0.357465i	$-0.116359\pi$
$659$	47.9496	1.86785	0.933926	+	0.357465i	$0.116359\pi$
$660$	−5.35725	−0.208531
$661$	37.8757	1.47319	0.736596	−	0.676333i	$-0.236432\pi$
$661$	37.8757	1.47319	0.736596	+	0.676333i	$0.236432\pi$
$662$	36.5032	1.41874
$663$	0	0
$664$	−35.5114	−1.37811
$665$	−8.89020	−0.344747
$666$	16.6675	0.645851
$667$	−15.9468	−0.617463
$668$	−8.82544	−0.341466
$669$	−63.6473	−2.46075
$670$	10.7155	0.413975
$671$	−17.1511	−0.662109
$672$	24.2912	0.937053
$673$	4.06145	0.156557	0.0782786	−	0.996932i	$-0.475058\pi$
$673$	4.06145	0.156557	0.0782786	+	0.996932i	$0.475058\pi$
$674$	−36.8947	−1.42113
$675$	9.16791	0.352873
$676$	−5.44967	−0.209603
$677$	−3.48544	−0.133956	−0.0669782	−	0.997754i	$-0.521336\pi$
$677$	−3.48544	−0.133956	−0.0669782	+	0.997754i	$0.521336\pi$
$678$	−34.4921	−1.32466
$679$	−26.5194	−1.01772
$680$	0	0
$681$	−77.1041	−2.95464
$682$	35.4037	1.35568
$683$	−14.0816	−0.538818	−0.269409	−	0.963026i	$-0.586828\pi$
$683$	−14.0816	−0.538818	−0.269409	+	0.963026i	$0.586828\pi$
$684$	6.53245	0.249775
$685$	−8.53083	−0.325946
$686$	11.0983	0.423737
$687$	21.3577	0.814847
$688$	−11.8198	−0.450625
$689$	−0.208744	−0.00795252
$690$	15.2172	0.579310
$691$	44.6596	1.69893	0.849465	−	0.527644i	$-0.176925\pi$
$691$	44.6596	1.69893	0.849465	+	0.527644i	$0.176925\pi$
$692$	−5.26861	−0.200283
$693$	88.6562	3.36777
$694$	10.9181	0.414447
$695$	−6.83721	−0.259350
$696$	−36.2563	−1.37429
$697$	0	0
$698$	45.7076	1.73006
$699$	−71.3316	−2.69801
$700$	1.45003	0.0548059
$701$	1.84833	0.0698103	0.0349052	−	0.999391i	$-0.488887\pi$
$701$	1.84833	0.0698103	0.0349052	+	0.999391i	$0.488887\pi$
$702$	1.81675	0.0685689
$703$	−4.55920	−0.171953
$704$	24.1456	0.910020
$705$	13.8210	0.520528
$706$	−14.1416	−0.532228
$707$	−58.2060	−2.18906
$708$	7.75415	0.291419
$709$	30.0075	1.12695	0.563477	−	0.826132i	$-0.309463\pi$
$709$	30.0075	1.12695	0.563477	+	0.826132i	$0.309463\pi$
$710$	−11.2270	−0.421343
$711$	31.1502	1.16822
$712$	−1.47706	−0.0553553
$713$	−17.4439	−0.653281
$714$	0	0
$715$	0.540557	0.0202157
$716$	−5.10122	−0.190642
$717$	0.328817	0.0122799
$718$	−8.15119	−0.304200
$719$	−38.3910	−1.43174	−0.715872	−	0.698232i	$-0.753970\pi$
$719$	−38.3910	−1.43174	−0.715872	+	0.698232i	$0.753970\pi$
$720$	28.2030	1.05106
$721$	14.9330	0.556134
$722$	19.2541	0.716563
$723$	52.6000	1.95622
$724$	6.40609	0.238080
$725$	−4.90337	−0.182107
$726$	−32.7777	−1.21650
$727$	0.0992434	0.00368073	0.00184037	−	0.999998i	$-0.499414\pi$
$727$	0.0992434	0.00368073	0.00184037	+	0.999998i	$0.499414\pi$
$728$	−1.08184	−0.0400958
$729$	−25.6798	−0.951104
$730$	22.7502	0.842022
$731$	0	0
$732$	−5.10310	−0.188616
$733$	14.4487	0.533674	0.266837	−	0.963742i	$-0.414021\pi$
$733$	14.4487	0.533674	0.266837	+	0.963742i	$0.414021\pi$
$734$	28.3619	1.04686
$735$	−14.8436	−0.547516
$736$	7.60234	0.280226
$737$	29.2299	1.07670
$738$	94.6641	3.48463
$739$	4.56053	0.167762	0.0838808	−	0.996476i	$-0.473268\pi$
$739$	4.56053	0.167762	0.0838808	+	0.996476i	$0.473268\pi$
$740$	0.743623	0.0273361
$741$	−0.986098	−0.0362252
$742$	−8.80567	−0.323266
$743$	33.2731	1.22067	0.610337	−	0.792142i	$-0.291034\pi$
$743$	33.2731	1.22067	0.610337	+	0.792142i	$0.291034\pi$
$744$	−39.6601	−1.45401
$745$	−7.04072	−0.257952
$746$	28.8538	1.05641
$747$	−87.3686	−3.19665
$748$	0	0
$749$	−16.3079	−0.595877
$750$	4.67904	0.170854
$751$	−26.4160	−0.963933	−0.481967	−	0.876190i	$-0.660077\pi$
$751$	−26.4160	−0.963933	−0.481967	+	0.876190i	$0.660077\pi$
$752$	21.4268	0.781355
$753$	−9.88964	−0.360399
$754$	−0.971675	−0.0353863
$755$	6.35689	0.231351
$756$	13.2937	0.483488
$757$	22.1805	0.806163	0.403082	−	0.915164i	$-0.367939\pi$
$757$	22.1805	0.806163	0.403082	+	0.915164i	$0.367939\pi$
$758$	−31.2072	−1.13350
$759$	41.5099	1.50671
$760$	−6.32586	−0.229463
$761$	−23.1817	−0.840337	−0.420168	−	0.907446i	$-0.638029\pi$
$761$	−23.1817	−0.840337	−0.420168	+	0.907446i	$0.638029\pi$
$762$	−37.5261	−1.35943
$763$	52.9535	1.91705
$764$	3.73547	0.135145
$765$	0	0
$766$	16.9996	0.614219
$767$	−0.782409	−0.0282512
$768$	29.0596	1.04860
$769$	16.7701	0.604747	0.302373	−	0.953190i	$-0.402221\pi$
$769$	16.7701	0.604747	0.302373	+	0.953190i	$0.402221\pi$
$770$	22.8029	0.821759
$771$	64.3648	2.31804
$772$	7.71623	0.277713
$773$	−13.6907	−0.492419	−0.246210	−	0.969217i	$-0.579185\pi$
$773$	−13.6907	−0.492419	−0.246210	+	0.969217i	$0.579185\pi$
$774$	−23.8453	−0.857103
$775$	−5.36372	−0.192670
$776$	−18.8700	−0.677394
$777$	−18.4104	−0.660470
$778$	36.9463	1.32459
$779$	−25.8943	−0.927759
$780$	0.160837	0.00575887
$781$	−30.6253	−1.09586
$782$	0	0
$783$	−44.9537	−1.60651
$784$	−23.0123	−0.821867
$785$	16.7739	0.598685
$786$	−66.4854	−2.37145
$787$	29.2308	1.04197	0.520983	−	0.853567i	$-0.325565\pi$
$787$	29.2308	1.04197	0.520983	+	0.853567i	$0.325565\pi$
$788$	4.48674	0.159833
$789$	−30.4920	−1.08554
$790$	8.01202	0.285055
$791$	25.4665	0.905485
$792$	63.0837	2.24158
$793$	0.514913	0.0182851
$794$	44.1569	1.56707
$795$	−4.92885	−0.174808
$796$	6.51380	0.230875
$797$	−14.1482	−0.501153	−0.250577	−	0.968097i	$-0.580620\pi$
$797$	−14.1482	−0.501153	−0.250577	+	0.968097i	$0.580620\pi$
$798$	−41.5976	−1.47254
$799$	0	0
$800$	2.33759	0.0826463
$801$	−3.63402	−0.128402
$802$	13.5739	0.479312
$803$	62.0584	2.18999
$804$	8.69702	0.306720
$805$	−11.2353	−0.395993
$806$	−1.06290	−0.0374390
$807$	45.3530	1.59650
$808$	−41.4167	−1.45703
$809$	17.1630	0.603418	0.301709	−	0.953400i	$-0.402443\pi$
$809$	17.1630	0.603418	0.301709	+	0.953400i	$0.402443\pi$
$810$	14.6737	0.515582
$811$	45.2749	1.58982	0.794908	−	0.606730i	$-0.207519\pi$
$811$	45.2749	1.58982	0.794908	+	0.606730i	$0.207519\pi$
$812$	−7.11003	−0.249513
$813$	20.4633	0.717680
$814$	11.6941	0.409878
$815$	4.59441	0.160935
$816$	0	0
$817$	6.52262	0.228197
$818$	−16.4222	−0.574190
$819$	−2.66166	−0.0930058
$820$	4.22346	0.147490
$821$	23.9915	0.837309	0.418654	−	0.908146i	$-0.362502\pi$
$821$	23.9915	0.837309	0.418654	+	0.908146i	$0.362502\pi$
$822$	−39.9161	−1.39223
$823$	−43.3681	−1.51172	−0.755858	−	0.654735i	$-0.772780\pi$
$823$	−43.3681	−1.51172	−0.755858	+	0.654735i	$0.772780\pi$
$824$	10.6256	0.370162
$825$	12.7636	0.444371
$826$	−33.0052	−1.14840
$827$	38.4929	1.33853	0.669265	−	0.743024i	$-0.266609\pi$
$827$	38.4929	1.33853	0.669265	+	0.743024i	$0.266609\pi$
$828$	8.25563	0.286903
$829$	−54.4489	−1.89109	−0.945545	−	0.325492i	$-0.894470\pi$
$829$	−54.4489	−1.89109	−0.945545	+	0.325492i	$0.894470\pi$
$830$	−22.4717	−0.780004
$831$	−52.1034	−1.80745
$832$	−0.724904	−0.0251315
$833$	0	0
$834$	−31.9916	−1.10778
$835$	21.0265	0.727652
$836$	4.58324	0.158515
$837$	−49.1740	−1.69970
$838$	22.2268	0.767813
$839$	−11.8856	−0.410335	−0.205167	−	0.978727i	$-0.565774\pi$
$839$	−11.8856	−0.410335	−0.205167	+	0.978727i	$0.565774\pi$
$840$	−25.5444	−0.881365
$841$	−4.95692	−0.170928
$842$	−28.4190	−0.979383
$843$	54.2100	1.86709
$844$	−3.67425	−0.126473
$845$	12.9838	0.446655
$846$	43.2266	1.48616
$847$	24.2008	0.831549
$848$	−7.64124	−0.262401
$849$	56.8546	1.95124
$850$	0	0
$851$	−5.76186	−0.197514
$852$	−9.11221	−0.312179
$853$	−34.3786	−1.17710	−0.588550	−	0.808461i	$-0.700301\pi$
$853$	−34.3786	−1.17710	−0.588550	+	0.808461i	$0.700301\pi$
$854$	21.7211	0.743281
$855$	−15.5635	−0.532260
$856$	−11.6039	−0.396615
$857$	−18.4424	−0.629980	−0.314990	−	0.949095i	$-0.602001\pi$
$857$	−18.4424	−0.629980	−0.314990	+	0.949095i	$0.602001\pi$
$858$	2.52929	0.0863485
$859$	24.4156	0.833049	0.416524	−	0.909125i	$-0.363248\pi$
$859$	24.4156	0.833049	0.416524	+	0.909125i	$0.363248\pi$
$860$	−1.06386	−0.0362775
$861$	−104.563	−3.56351
$862$	−32.7326	−1.11488
$863$	−22.5253	−0.766770	−0.383385	−	0.923589i	$-0.625242\pi$
$863$	−22.5253	−0.766770	−0.383385	+	0.923589i	$0.625242\pi$
$864$	21.4308	0.729091
$865$	12.5524	0.426795
$866$	5.56353	0.189056
$867$	0	0
$868$	−7.77753	−0.263987
$869$	21.8554	0.741392
$870$	−22.9431	−0.777843
$871$	−0.877548	−0.0297346
$872$	37.6793	1.27598
$873$	−46.4258	−1.57128
$874$	−13.0187	−0.440363
$875$	−3.45467	−0.116789
$876$	18.4648	0.623866
$877$	56.5640	1.91003	0.955015	−	0.296557i	$-0.0958385\pi$
$877$	56.5640	1.91003	0.955015	+	0.296557i	$0.0958385\pi$
$878$	5.11450	0.172606
$879$	22.6555	0.764152
$880$	19.7875	0.667037
$881$	−33.2018	−1.11860	−0.559298	−	0.828967i	$-0.688929\pi$
$881$	−33.2018	−1.11860	−0.559298	+	0.828967i	$0.688929\pi$
$882$	−46.4252	−1.56322
$883$	11.2453	0.378433	0.189217	−	0.981935i	$-0.439405\pi$
$883$	11.2453	0.378433	0.189217	+	0.981935i	$0.439405\pi$
$884$	0	0
$885$	−18.4742	−0.621002
$886$	38.7940	1.30331
$887$	−1.01177	−0.0339718	−0.0169859	−	0.999856i	$-0.505407\pi$
$887$	−1.01177	−0.0339718	−0.0169859	+	0.999856i	$0.505407\pi$
$888$	−13.1000	−0.439608
$889$	27.7067	0.929252
$890$	−0.934691	−0.0313309
$891$	40.0273	1.34096
$892$	−8.88128	−0.297367
$893$	−11.8242	−0.395680
$894$	−32.9438	−1.10180
$895$	12.1536	0.406250
$896$	−46.7306	−1.56116
$897$	−1.24622	−0.0416100
$898$	15.0209	0.501255
$899$	26.3003	0.877164
$900$	2.53847	0.0846156
$901$	0	0
$902$	66.4174	2.21146
$903$	26.3389	0.876504
$904$	18.1208	0.602689
$905$	−15.2624	−0.507341
$906$	29.7441	0.988182
$907$	18.3308	0.608663	0.304332	−	0.952566i	$-0.401567\pi$
$907$	18.3308	0.608663	0.304332	+	0.952566i	$0.401567\pi$
$908$	−10.7590	−0.357051
$909$	−101.897	−3.37972
$910$	−0.684594	−0.0226941
$911$	27.7944	0.920870	0.460435	−	0.887694i	$-0.347694\pi$
$911$	27.7944	0.920870	0.460435	+	0.887694i	$0.347694\pi$
$912$	−36.0969	−1.19529
$913$	−61.2988	−2.02869
$914$	31.8538	1.05363
$915$	12.1581	0.401933
$916$	2.98023	0.0984697
$917$	49.0881	1.62103
$918$	0	0
$919$	−13.6372	−0.449850	−0.224925	−	0.974376i	$-0.572214\pi$
$919$	−13.6372	−0.449850	−0.224925	+	0.974376i	$0.572214\pi$
$920$	−7.99454	−0.263572
$921$	−23.3741	−0.770204
$922$	29.3218	0.965661
$923$	0.919441	0.0302638
$924$	18.5075	0.608853
$925$	−1.77167	−0.0582522
$926$	−3.32000	−0.109102
$927$	26.1422	0.858624
$928$	−11.4621	−0.376261
$929$	52.4121	1.71959	0.859793	−	0.510642i	$-0.170592\pi$
$929$	52.4121	1.71959	0.859793	+	0.510642i	$0.170592\pi$
$930$	−25.0970	−0.822964
$931$	12.6991	0.416195
$932$	−9.95354	−0.326039
$933$	−26.0034	−0.851313
$934$	36.1876	1.18409
$935$	0	0
$936$	−1.89391	−0.0619045
$937$	32.5221	1.06245	0.531225	−	0.847231i	$-0.321732\pi$
$937$	32.5221	1.06245	0.531225	+	0.847231i	$0.321732\pi$
$938$	−37.0185	−1.20870
$939$	37.1122	1.21111
$940$	1.92857	0.0629029
$941$	17.3587	0.565879	0.282939	−	0.959138i	$-0.408690\pi$
$941$	17.3587	0.565879	0.282939	+	0.959138i	$0.408690\pi$
$942$	78.4856	2.55720
$943$	−32.7249	−1.06567
$944$	−28.6407	−0.932176
$945$	−31.6721	−1.03029
$946$	−16.7302	−0.543944
$947$	14.2696	0.463699	0.231849	−	0.972752i	$-0.425522\pi$
$947$	14.2696	0.463699	0.231849	+	0.972752i	$0.425522\pi$
$948$	6.50280	0.211201
$949$	−1.86313	−0.0604799
$950$	−4.00302	−0.129875
$951$	24.5828	0.797152
$952$	0	0
$953$	30.1090	0.975326	0.487663	−	0.873032i	$-0.337850\pi$
$953$	30.1090	0.975326	0.487663	+	0.873032i	$0.337850\pi$
$954$	−15.4155	−0.499096
$955$	−8.89972	−0.287988
$956$	0.0458829	0.00148396
$957$	−62.5846	−2.02307
$958$	48.5530	1.56868
$959$	29.4712	0.951675
$960$	−17.1164	−0.552428
$961$	−2.23055	−0.0719533
$962$	−0.351083	−0.0113194
$963$	−28.5492	−0.919984
$964$	7.33976	0.236398
$965$	−18.3838	−0.591796
$966$	−52.5705	−1.69143
$967$	17.5196	0.563392	0.281696	−	0.959504i	$-0.409103\pi$
$967$	17.5196	0.563392	0.281696	+	0.959504i	$0.409103\pi$
$968$	17.2202	0.553477
$969$	0	0
$970$	−11.9410	−0.383402
$971$	−11.6359	−0.373414	−0.186707	−	0.982416i	$-0.559782\pi$
$971$	−11.6359	−0.373414	−0.186707	+	0.982416i	$0.559782\pi$
$972$	0.365541	0.0117247
$973$	23.6203	0.757234
$974$	22.7931	0.730337
$975$	−0.383191	−0.0122719
$976$	18.8488	0.603335
$977$	−43.3421	−1.38664	−0.693319	−	0.720631i	$-0.743852\pi$
$977$	−43.3421	−1.38664	−0.693319	+	0.720631i	$0.743852\pi$
$978$	21.4974	0.687411
$979$	−2.54967	−0.0814878
$980$	−2.07127	−0.0661642
$981$	92.7023	2.95976
$982$	21.5620	0.688072
$983$	−44.3919	−1.41588	−0.707941	−	0.706271i	$-0.750376\pi$
$983$	−44.3919	−1.41588	−0.707941	+	0.706271i	$0.750376\pi$
$984$	−74.4025	−2.37186
$985$	−10.6896	−0.340599
$986$	0	0
$987$	−47.7470	−1.51980
$988$	−0.137599	−0.00437761
$989$	8.24320	0.262119
$990$	39.9195	1.26873
$991$	57.7133	1.83332	0.916662	−	0.399664i	$-0.130873\pi$
$991$	57.7133	1.83332	0.916662	+	0.399664i	$0.130873\pi$
$992$	−12.5382	−0.398087
$993$	−70.5863	−2.23999
$994$	38.7857	1.23021
$995$	−15.5191	−0.491987
$996$	−18.2387	−0.577917
$997$	−11.6661	−0.369471	−0.184735	−	0.982788i	$-0.559143\pi$
$997$	−11.6661	−0.369471	−0.184735	+	0.982788i	$0.559143\pi$
$998$	−62.2716	−1.97117
$999$	−16.2425	−0.513891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.a.q.1.4		12
5.4	even	2		7225.2.a.bq.1.9		12
17.3	odd	16		85.2.l.a.26.3	✓	24
17.4	even	4		1445.2.d.j.866.17		24
17.6	odd	16		85.2.l.a.36.3	yes	24
17.13	even	4		1445.2.d.j.866.18		24
17.16	even	2		1445.2.a.p.1.4		12
51.20	even	16		765.2.be.b.451.4		24
51.23	even	16		765.2.be.b.631.4		24
85.3	even	16		425.2.n.c.349.4		24
85.23	even	16		425.2.n.f.274.3		24
85.37	even	16		425.2.n.f.349.3		24
85.54	odd	16		425.2.m.b.26.4		24
85.57	even	16		425.2.n.c.274.4		24
85.74	odd	16		425.2.m.b.376.4		24
85.84	even	2		7225.2.a.bs.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.3	✓	24	17.3	odd	16
85.2.l.a.36.3	yes	24	17.6	odd	16
425.2.m.b.26.4		24	85.54	odd	16
425.2.m.b.376.4		24	85.74	odd	16
425.2.n.c.274.4		24	85.57	even	16
425.2.n.c.349.4		24	85.3	even	16
425.2.n.f.274.3		24	85.23	even	16
425.2.n.f.349.3		24	85.37	even	16
765.2.be.b.451.4		24	51.20	even	16
765.2.be.b.631.4		24	51.23	even	16
1445.2.a.p.1.4		12	17.16	even	2
1445.2.a.q.1.4		12	1.1	even	1	trivial
1445.2.d.j.866.17		24	17.4	even	4
1445.2.d.j.866.18		24	17.13	even	4
7225.2.a.bq.1.9		12	5.4	even	2
7225.2.a.bs.1.9		12	85.84	even	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:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

\(f(q)\)	\(=\)	\(q-1.55555 q^{2} +3.00797 q^{3} +0.419729 q^{4} -1.00000 q^{5} -4.67904 q^{6} +3.45467 q^{7} +2.45819 q^{8} +6.04787 q^{9} +O(q^{10})\)	\(q-1.55555 q^{2} +3.00797 q^{3} +0.419729 q^{4} -1.00000 q^{5} -4.67904 q^{6} +3.45467 q^{7} +2.45819 q^{8} +6.04787 q^{9} +1.55555 q^{10} +4.24326 q^{11} +1.26253 q^{12} -0.127392 q^{13} -5.37391 q^{14} -3.00797 q^{15} -4.66329 q^{16} -9.40776 q^{18} +2.57338 q^{19} -0.419729 q^{20} +10.3916 q^{21} -6.60059 q^{22} +3.25221 q^{23} +7.39415 q^{24} +1.00000 q^{25} +0.198164 q^{26} +9.16791 q^{27} +1.45003 q^{28} -4.90337 q^{29} +4.67904 q^{30} -5.36372 q^{31} +2.33759 q^{32} +12.7636 q^{33} -3.45467 q^{35} +2.53847 q^{36} -1.77167 q^{37} -4.00302 q^{38} -0.383191 q^{39} -2.45819 q^{40} -10.0623 q^{41} -16.1646 q^{42} +2.53465 q^{43} +1.78102 q^{44} -6.04787 q^{45} -5.05897 q^{46} -4.59479 q^{47} -14.0270 q^{48} +4.93477 q^{49} -1.55555 q^{50} -0.0534701 q^{52} +1.63860 q^{53} -14.2611 q^{54} -4.24326 q^{55} +8.49224 q^{56} +7.74066 q^{57} +7.62743 q^{58} +6.14174 q^{59} -1.26253 q^{60} -4.04195 q^{61} +8.34352 q^{62} +20.8934 q^{63} +5.69034 q^{64} +0.127392 q^{65} -19.8544 q^{66} +6.88856 q^{67} +9.78255 q^{69} +5.37391 q^{70} -7.21741 q^{71} +14.8668 q^{72} +14.6252 q^{73} +2.75592 q^{74} +3.00797 q^{75} +1.08012 q^{76} +14.6591 q^{77} +0.596072 q^{78} +5.15061 q^{79} +4.66329 q^{80} +9.43315 q^{81} +15.6525 q^{82} -14.4462 q^{83} +4.36164 q^{84} -3.94276 q^{86} -14.7492 q^{87} +10.4307 q^{88} -0.600876 q^{89} +9.40776 q^{90} -0.440098 q^{91} +1.36505 q^{92} -16.1339 q^{93} +7.14741 q^{94} -2.57338 q^{95} +7.03140 q^{96} -7.67639 q^{97} -7.67628 q^{98} +25.6627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 8 q^{3} + 12 q^{4} - 12 q^{5} + 8 q^{6} + 16 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 4 * q^2 + 8 * q^3 + 12 * q^4 - 12 * q^5 + 8 * q^6 + 16 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} + 8 q^{3} + 12 q^{4} - 12 q^{5} + 8 q^{6} + 16 q^{7} - 12 q^{8} + 12 q^{9} + 4 q^{10} + 16 q^{11} + 16 q^{12} - 8 q^{13} - 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} - 12 q^{20} + 16 q^{21} + 16 q^{22} + 16 q^{23} + 12 q^{25} + 16 q^{26} + 32 q^{27} + 40 q^{28} + 16 q^{29} - 8 q^{30} + 24 q^{31} - 28 q^{32} - 16 q^{35} + 12 q^{36} + 24 q^{37} - 24 q^{38} + 8 q^{39} + 12 q^{40} + 8 q^{41} - 16 q^{43} + 8 q^{44} - 12 q^{45} + 40 q^{46} - 32 q^{47} - 24 q^{48} + 20 q^{49} - 4 q^{50} - 24 q^{52} + 8 q^{54} - 16 q^{55} - 24 q^{56} + 32 q^{57} - 16 q^{58} + 8 q^{59} - 16 q^{60} + 24 q^{61} + 8 q^{62} + 48 q^{63} + 36 q^{64} + 8 q^{65} + 40 q^{66} - 8 q^{67} + 48 q^{69} + 16 q^{70} - 16 q^{71} + 12 q^{72} + 16 q^{73} + 8 q^{75} + 16 q^{76} + 24 q^{77} - 24 q^{78} + 40 q^{79} - 12 q^{80} + 36 q^{81} - 16 q^{82} - 40 q^{83} + 32 q^{84} - 8 q^{86} - 32 q^{87} + 48 q^{88} + 8 q^{89} - 4 q^{90} + 72 q^{91} - 8 q^{92} + 24 q^{93} - 16 q^{94} + 8 q^{96} + 32 q^{97} - 60 q^{98} + 56 q^{99}+O(q^{100}) \) 12 * q - 4 * q^2 + 8 * q^3 + 12 * q^4 - 12 * q^5 + 8 * q^6 + 16 * q^7 - 12 * q^8 + 12 * q^9 + 4 * q^10 + 16 * q^11 + 16 * q^12 - 8 * q^13 - 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 - 12 * q^20 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 12 * q^25 + 16 * q^26 + 32 * q^27 + 40 * q^28 + 16 * q^29 - 8 * q^30 + 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 + 24 * q^37 - 24 * q^38 + 8 * q^39 + 12 * q^40 + 8 * q^41 - 16 * q^43 + 8 * q^44 - 12 * q^45 + 40 * q^46 - 32 * q^47 - 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 + 8 * q^54 - 16 * q^55 - 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 16 * q^60 + 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 - 16 * q^71 + 12 * q^72 + 16 * q^73 + 8 * q^75 + 16 * q^76 + 24 * q^77 - 24 * q^78 + 40 * q^79 - 12 * q^80 + 36 * q^81 - 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 + 48 * q^88 + 8 * q^89 - 4 * q^90 + 72 * q^91 - 8 * q^92 + 24 * q^93 - 16 * q^94 + 8 * q^96 + 32 * q^97 - 60 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more