LMFDB - Embedded newform 1445.2.a.p.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1445,2,Mod(1,1445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,-4,-8,12,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.5383830921$
Analytic rank:	$1$
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)

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

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	$-0.834804\pi$
$3$	−3.00797	−1.73665	−0.868326	+	0.495995i	$0.834804\pi$
$4$	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	$-0.726436\pi$
$7$	−3.45467	−1.30574	−0.652872	+	0.757468i	$0.726436\pi$
$8$	2.45819	0.869100
$9$	6.04787	2.01596
$10$	−1.55555	−0.491907
$11$	−4.24326	−1.27939	−0.639695	−	0.768629i	$-0.720939\pi$
$11$	−4.24326	−1.27939	−0.639695	+	0.768629i	$0.720939\pi$
$12$	−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	$-0.610111\pi$
$23$	−3.25221	−0.678133	−0.339066	+	0.940762i	$0.610111\pi$
$24$	−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	$-0.349544\pi$
$29$	4.90337	0.910534	0.455267	+	0.890355i	$0.349544\pi$
$30$	4.67904	0.854272
$31$	5.36372	0.963352	0.481676	−	0.876349i	$-0.340028\pi$
$31$	5.36372	0.963352	0.481676	+	0.876349i	$0.340028\pi$
$32$	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	$-0.453479\pi$
$37$	1.77167	0.291261	0.145631	+	0.989339i	$0.453479\pi$
$38$	−4.00302	−0.649376
$39$	0.383191	0.0613597
$40$	2.45819	0.388674
$41$	10.0623	1.57147	0.785737	−	0.618561i	$-0.212284\pi$
$41$	10.0623	1.57147	0.785737	+	0.618561i	$0.212284\pi$
$42$	−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	$-0.416686\pi$
$61$	4.04195	0.517519	0.258760	+	0.965942i	$0.416686\pi$
$62$	−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	$-0.359122\pi$
$71$	7.21741	0.856549	0.428274	+	0.903649i	$0.359122\pi$
$72$	14.8668	1.75207
$73$	−14.6252	−1.71175	−0.855874	−	0.517184i	$-0.826980\pi$
$73$	−14.6252	−1.71175	−0.855874	+	0.517184i	$0.826980\pi$
$74$	−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	$-0.593570\pi$
$79$	−5.15061	−0.579489	−0.289744	+	0.957104i	$0.593570\pi$
$80$	−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	$-0.372575\pi$
$97$	7.67639	0.779420	0.389710	+	0.920938i	$0.372575\pi$
$98$	−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	$-0.426724\pi$
$107$	4.72053	0.456351	0.228175	+	0.973620i	$0.426724\pi$
$108$	−3.84804	−0.370277
$109$	−15.3281	−1.46816	−0.734082	−	0.679061i	$-0.762387\pi$
$109$	−15.3281	−1.46816	−0.734082	+	0.679061i	$0.762387\pi$
$110$	6.60059	0.629342
$111$	−5.32914	−0.505819
$112$	16.1101	1.52226
$113$	−7.37161	−0.693463	−0.346732	−	0.937964i	$-0.612709\pi$
$113$	−7.37161	−0.693463	−0.346732	+	0.937964i	$0.612709\pi$
$114$	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	$-0.713164\pi$
$131$	−14.2092	−1.24146	−0.620731	+	0.784023i	$0.713164\pi$
$132$	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	$-0.593643\pi$
$139$	−6.83721	−0.579925	−0.289962	+	0.957038i	$0.593643\pi$
$140$	−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	$-0.442413\pi$
$163$	4.59441	0.359862	0.179931	+	0.983679i	$0.442413\pi$
$164$	4.22346	0.329797
$165$	12.7636	0.993644
$166$	22.4717	1.74414
$167$	21.0265	1.62708	0.813540	−	0.581509i	$-0.197537\pi$
$167$	21.0265	1.62708	0.813540	+	0.581509i	$0.197537\pi$
$168$	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	$-0.341662\pi$
$173$	12.5524	0.954343	0.477171	+	0.878810i	$0.341662\pi$
$174$	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	$-0.691983\pi$
$181$	−15.2624	−1.13445	−0.567224	+	0.823563i	$0.691983\pi$
$182$	−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	$-0.730143\pi$
$193$	−18.3838	−1.32330	−0.661648	+	0.749814i	$0.730143\pi$
$194$	−11.9410	−0.857314
$195$	0.383191	0.0274409
$196$	2.07127	0.147948
$197$	−10.6896	−0.761603	−0.380801	−	0.924657i	$-0.624352\pi$
$197$	−10.6896	−0.761603	−0.380801	+	0.924657i	$0.624352\pi$
$198$	39.9195	2.83696
$199$	−15.5191	−1.10012	−0.550058	−	0.835126i	$-0.685395\pi$
$199$	−15.5191	−1.10012	−0.550058	+	0.835126i	$0.685395\pi$
$200$	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	$-0.402573\pi$
$211$	8.75387	0.602641	0.301321	+	0.953523i	$0.402573\pi$
$212$	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	$-0.176196\pi$
$227$	25.6333	1.70134	0.850671	+	0.525699i	$0.176196\pi$
$228$	−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	$-0.216849\pi$
$233$	23.7142	1.55357	0.776785	+	0.629766i	$0.216849\pi$
$234$	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	$-0.690436\pi$
$241$	−17.4869	−1.12643	−0.563215	+	0.826311i	$0.690436\pi$
$242$	−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	$-0.652025\pi$
$269$	−15.0776	−0.919299	−0.459649	+	0.888100i	$0.652025\pi$
$270$	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	$-0.325789\pi$
$277$	17.3218	1.04077	0.520383	+	0.853933i	$0.325789\pi$
$278$	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	$-0.689885\pi$
$283$	−18.9013	−1.12357	−0.561784	+	0.827284i	$0.689885\pi$
$284$	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	$-0.421179\pi$
$311$	8.64484	0.490204	0.245102	+	0.969497i	$0.421179\pi$
$312$	0.941956	0.0533278
$313$	−12.3380	−0.697383	−0.348691	−	0.937238i	$-0.613374\pi$
$313$	−12.3380	−0.697383	−0.348691	+	0.937238i	$0.613374\pi$
$314$	26.0926	1.47249
$315$	−20.8934	−1.17721
$316$	−2.16186	−0.121614
$317$	−8.17256	−0.459016	−0.229508	−	0.973307i	$-0.573712\pi$
$317$	−8.17256	−0.459016	−0.229508	+	0.973307i	$0.573712\pi$
$318$	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	$-0.723562\pi$
$337$	−23.7182	−1.29201	−0.646005	+	0.763333i	$0.723562\pi$
$338$	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	$-0.439671\pi$
$347$	7.01884	0.376791	0.188395	+	0.982093i	$0.439671\pi$
$348$	−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	$-0.342133\pi$
$367$	18.2327	0.951740	0.475870	+	0.879516i	$0.342133\pi$
$368$	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	$-0.672303\pi$
$379$	−20.0619	−1.03051	−0.515255	+	0.857037i	$0.672303\pi$
$380$	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	$-0.247633\pi$
$397$	28.3867	1.42469	0.712344	+	0.701830i	$0.247633\pi$
$398$	24.1406	1.21006
$399$	26.7415	1.33875
$400$	−4.66329	−0.233164
$401$	8.72614	0.435763	0.217881	−	0.975975i	$-0.430085\pi$
$401$	8.72614	0.435763	0.217881	+	0.975975i	$0.430085\pi$
$402$	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	$-0.386513\pi$
$419$	14.2887	0.698051	0.349025	+	0.937113i	$0.386513\pi$
$420$	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	$-0.669168\pi$
$431$	−21.0425	−1.01358	−0.506791	+	0.862069i	$0.669168\pi$
$432$	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	$-0.474999\pi$
$439$	3.28791	0.156923	0.0784617	+	0.996917i	$0.474999\pi$
$440$	−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	$-0.426828\pi$
$449$	9.65637	0.455712	0.227856	+	0.973695i	$0.426828\pi$
$450$	−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	$-0.247302\pi$
$479$	31.2128	1.42615	0.713074	+	0.701088i	$0.247302\pi$
$480$	−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	$-0.392280\pi$
$487$	14.6528	0.663980	0.331990	+	0.943283i	$0.392280\pi$
$488$	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	$-0.853567\pi$
$499$	−40.0320	−1.79208	−0.896039	+	0.443976i	$0.853567\pi$
$500$	0.419729	0.0187709
$501$	−63.2471	−2.82567
$502$	5.11435	0.228265
$503$	−39.2423	−1.74973	−0.874863	−	0.484371i	$-0.839049\pi$
$503$	−39.2423	−1.74973	−0.874863	+	0.484371i	$0.839049\pi$
$504$	−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	$-0.467584\pi$
$521$	4.64101	0.203326	0.101663	+	0.994819i	$0.467584\pi$
$522$	−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	$-0.741197\pi$
$541$	−31.9717	−1.37457	−0.687285	+	0.726388i	$0.741197\pi$
$542$	−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	$-0.557532\pi$
$547$	−8.40850	−0.359522	−0.179761	+	0.983710i	$0.557532\pi$
$548$	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	$-0.420236\pi$
$571$	11.8509	0.495944	0.247972	+	0.968767i	$0.420236\pi$
$572$	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	$-0.364136\pi$
$601$	20.2981	0.827975	0.413988	+	0.910282i	$0.364136\pi$
$602$	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	$-0.0426352\pi$
$607$	48.8334	1.98209	0.991043	+	0.133542i	$0.0426352\pi$
$608$	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	$-0.529683\pi$
$617$	−4.62599	−0.186235	−0.0931177	+	0.995655i	$0.529683\pi$
$618$	20.2254	0.813584
$619$	46.6252	1.87402	0.937012	−	0.349298i	$-0.113580\pi$
$619$	46.6252	1.87402	0.937012	+	0.349298i	$0.113580\pi$
$620$	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	$-0.541382\pi$
$641$	−6.56445	−0.259280	−0.129640	+	0.991561i	$0.541382\pi$
$642$	22.0875	0.871726
$643$	−6.40706	−0.252670	−0.126335	−	0.991988i	$-0.540321\pi$
$643$	−6.40706	−0.252670	−0.126335	+	0.991988i	$0.540321\pi$
$644$	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	$-0.859443\pi$
$653$	−46.2055	−1.80816	−0.904080	+	0.427363i	$0.859443\pi$
$654$	−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	$-0.524942\pi$
$673$	−4.06145	−0.156557	−0.0782786	+	0.996932i	$0.524942\pi$
$674$	36.8947	1.42113
$675$	−9.16791	−0.352873
$676$	−5.44967	−0.209603
$677$	3.48544	0.133956	0.0669782	−	0.997754i	$-0.478664\pi$
$677$	3.48544	0.133956	0.0669782	+	0.997754i	$0.478664\pi$
$678$	−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	$-0.413172\pi$
$683$	14.0816	0.538818	0.269409	+	0.963026i	$0.413172\pi$
$684$	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	$-0.823075\pi$
$691$	−44.6596	−1.69893	−0.849465	+	0.527644i	$0.823075\pi$
$692$	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	$-0.690537\pi$
$709$	−30.0075	−1.12695	−0.563477	+	0.826132i	$0.690537\pi$
$710$	−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	$-0.246030\pi$
$719$	38.3910	1.43174	0.715872	+	0.698232i	$0.246030\pi$
$720$	−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	$-0.708966\pi$
$743$	−33.2731	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$744$	−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	$-0.339923\pi$
$751$	26.4160	0.963933	0.481967	+	0.876190i	$0.339923\pi$
$752$	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	$-0.674435\pi$
$787$	−29.2308	−1.04197	−0.520983	+	0.853567i	$0.674435\pi$
$788$	−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	$-0.597557\pi$
$809$	−17.1630	−0.603418	−0.301709	+	0.953400i	$0.597557\pi$
$810$	−14.6737	−0.515582
$811$	−45.2749	−1.58982	−0.794908	−	0.606730i	$-0.792481\pi$
$811$	−45.2749	−1.58982	−0.794908	+	0.606730i	$0.792481\pi$
$812$	−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	$-0.637498\pi$
$821$	−23.9915	−0.837309	−0.418654	+	0.908146i	$0.637498\pi$
$822$	39.9161	1.39223
$823$	43.3681	1.51172	0.755858	−	0.654735i	$-0.227220\pi$
$823$	43.3681	1.51172	0.755858	+	0.654735i	$0.227220\pi$
$824$	10.6256	0.370162
$825$	12.7636	0.444371
$826$	33.0052	1.14840
$827$	−38.4929	−1.33853	−0.669265	−	0.743024i	$-0.733391\pi$
$827$	−38.4929	−1.33853	−0.669265	+	0.743024i	$0.733391\pi$
$828$	−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	$-0.434226\pi$
$839$	11.8856	0.410335	0.205167	+	0.978727i	$0.434226\pi$
$840$	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	$-0.299699\pi$
$853$	34.3786	1.17710	0.588550	+	0.808461i	$0.299699\pi$
$854$	21.7211	0.743281
$855$	15.5635	0.532260
$856$	11.6039	0.396615
$857$	18.4424	0.629980	0.314990	−	0.949095i	$-0.397999\pi$
$857$	18.4424	0.629980	0.314990	+	0.949095i	$0.397999\pi$
$858$	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	$-0.904162\pi$
$877$	−56.5640	−1.91003	−0.955015	+	0.296557i	$0.904162\pi$
$878$	−5.11450	−0.172606
$879$	−22.6555	−0.764152
$880$	19.7875	0.667037
$881$	33.2018	1.11860	0.559298	−	0.828967i	$-0.311071\pi$
$881$	33.2018	1.11860	0.559298	+	0.828967i	$0.311071\pi$
$882$	−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	$-0.494593\pi$
$887$	1.01177	0.0339718	0.0169859	+	0.999856i	$0.494593\pi$
$888$	−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	$-0.598433\pi$
$907$	−18.3308	−0.608663	−0.304332	+	0.952566i	$0.598433\pi$
$908$	10.7590	0.357051
$909$	−101.897	−3.37972
$910$	−0.684594	−0.0226941
$911$	−27.7944	−0.920870	−0.460435	−	0.887694i	$-0.652306\pi$
$911$	−27.7944	−0.920870	−0.460435	+	0.887694i	$0.652306\pi$
$912$	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	$-0.829408\pi$
$929$	−52.4121	−1.71959	−0.859793	+	0.510642i	$0.829408\pi$
$930$	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	$-0.591310\pi$
$941$	−17.3587	−0.565879	−0.282939	+	0.959138i	$0.591310\pi$
$942$	−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	$-0.574478\pi$
$947$	−14.2696	−0.463699	−0.231849	+	0.972752i	$0.574478\pi$
$948$	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	$-0.249624\pi$
$983$	44.3919	1.41588	0.707941	+	0.706271i	$0.249624\pi$
$984$	−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	$-0.869127\pi$
$991$	−57.7133	−1.83332	−0.916662	+	0.399664i	$0.869127\pi$
$992$	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	$-0.440857\pi$
$997$	11.6661	0.369471	0.184735	+	0.982788i	$0.440857\pi$
$998$	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.p.1.4		12
5.4	even	2		7225.2.a.bs.1.9		12
17.4	even	4		1445.2.d.j.866.18		24
17.11	odd	16		85.2.l.a.36.3	yes	24
17.13	even	4		1445.2.d.j.866.17		24
17.14	odd	16		85.2.l.a.26.3	✓	24
17.16	even	2		1445.2.a.q.1.4		12
51.11	even	16		765.2.be.b.631.4		24
51.14	even	16		765.2.be.b.451.4		24
85.14	odd	16		425.2.m.b.26.4		24
85.28	even	16		425.2.n.f.274.3		24
85.48	even	16		425.2.n.c.349.4		24
85.62	even	16		425.2.n.c.274.4		24
85.79	odd	16		425.2.m.b.376.4		24
85.82	even	16		425.2.n.f.349.3		24
85.84	even	2		7225.2.a.bq.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.3	✓	24	17.14	odd	16
85.2.l.a.36.3	yes	24	17.11	odd	16
425.2.m.b.26.4		24	85.14	odd	16
425.2.m.b.376.4		24	85.79	odd	16
425.2.n.c.274.4		24	85.62	even	16
425.2.n.c.349.4		24	85.48	even	16
425.2.n.f.274.3		24	85.28	even	16
425.2.n.f.349.3		24	85.82	even	16
765.2.be.b.451.4		24	51.14	even	16
765.2.be.b.631.4		24	51.11	even	16
1445.2.a.p.1.4		12	1.1	even	1	trivial
1445.2.a.q.1.4		12	17.16	even	2
1445.2.d.j.866.17		24	17.13	even	4
1445.2.d.j.866.18		24	17.4	even	4
7225.2.a.bq.1.9		12	85.84	even	2
7225.2.a.bs.1.9		12	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

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

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