LMFDB - Embedded newform 1445.2.a.p.1.2

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:	$1$
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			1.2
Root			$2.63994$ 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)

$f(q)$	$=$	\(q-2.63994 q^{2} -2.12055 q^{3} +4.96928 q^{4} +1.00000 q^{5} +5.59814 q^{6} -4.06194 q^{7} -7.83873 q^{8} +1.49675 q^{9} -2.63994 q^{10} +1.74901 q^{11} -10.5376 q^{12} +1.67715 q^{13} +10.7233 q^{14} -2.12055 q^{15} +10.7552 q^{16} -3.95133 q^{18} +0.249037 q^{19} +4.96928 q^{20} +8.61356 q^{21} -4.61727 q^{22} -0.519715 q^{23} +16.6224 q^{24} +1.00000 q^{25} -4.42758 q^{26} +3.18772 q^{27} -20.1849 q^{28} -0.486878 q^{29} +5.59814 q^{30} -9.53040 q^{31} -12.7156 q^{32} -3.70886 q^{33} -4.06194 q^{35} +7.43778 q^{36} +4.26971 q^{37} -0.657443 q^{38} -3.55649 q^{39} -7.83873 q^{40} -5.28883 q^{41} -22.7393 q^{42} +3.17951 q^{43} +8.69131 q^{44} +1.49675 q^{45} +1.37202 q^{46} +6.26212 q^{47} -22.8070 q^{48} +9.49933 q^{49} -2.63994 q^{50} +8.33425 q^{52} +10.2436 q^{53} -8.41540 q^{54} +1.74901 q^{55} +31.8404 q^{56} -0.528097 q^{57} +1.28533 q^{58} +11.5986 q^{59} -10.5376 q^{60} +4.35334 q^{61} +25.1597 q^{62} -6.07971 q^{63} +12.0581 q^{64} +1.67715 q^{65} +9.79117 q^{66} -9.73489 q^{67} +1.10208 q^{69} +10.7233 q^{70} -1.00602 q^{71} -11.7326 q^{72} +2.91508 q^{73} -11.2718 q^{74} -2.12055 q^{75} +1.23754 q^{76} -7.10435 q^{77} +9.38893 q^{78} -6.81821 q^{79} +10.7552 q^{80} -11.2500 q^{81} +13.9622 q^{82} -3.19678 q^{83} +42.8032 q^{84} -8.39371 q^{86} +1.03245 q^{87} -13.7100 q^{88} +3.30525 q^{89} -3.95133 q^{90} -6.81249 q^{91} -2.58261 q^{92} +20.2097 q^{93} -16.5316 q^{94} +0.249037 q^{95} +26.9642 q^{96} -1.70094 q^{97} -25.0777 q^{98} +2.61783 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.63994	−1.86672	−0.933360	−	0.358942i	$-0.883137\pi$
$2$	−2.63994	−1.86672	−0.933360	+	0.358942i	$0.883137\pi$
$3$	−2.12055	−1.22430	−0.612151	−	0.790741i	$-0.709696\pi$
$3$	−2.12055	−1.22430	−0.612151	+	0.790741i	$0.709696\pi$
$4$	4.96928	2.48464
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	5.59814	2.28543
$7$	−4.06194	−1.53527	−0.767634	−	0.640889i	$-0.778566\pi$
$7$	−4.06194	−1.53527	−0.767634	+	0.640889i	$0.778566\pi$
$8$	−7.83873	−2.77141
$9$	1.49675	0.498917
$10$	−2.63994	−0.834822
$11$	1.74901	0.527345	0.263673	−	0.964612i	$-0.415066\pi$
$11$	1.74901	0.527345	0.263673	+	0.964612i	$0.415066\pi$
$12$	−10.5376	−3.04195
$13$	1.67715	0.465159	0.232579	−	0.972577i	$-0.425283\pi$
$13$	1.67715	0.465159	0.232579	+	0.972577i	$0.425283\pi$
$14$	10.7233	2.86591
$15$	−2.12055	−0.547525
$16$	10.7552	2.68880
$17$	0	0
$17$	0	0
$18$	−3.95133	−0.931338
$19$	0.249037	0.0571330	0.0285665	−	0.999592i	$-0.490906\pi$
$19$	0.249037	0.0571330	0.0285665	+	0.999592i	$0.490906\pi$
$20$	4.96928	1.11117
$21$	8.61356	1.87963
$22$	−4.61727	−0.984405
$23$	−0.519715	−0.108368	−0.0541841	−	0.998531i	$-0.517256\pi$
$23$	−0.519715	−0.108368	−0.0541841	+	0.998531i	$0.517256\pi$
$24$	16.6224	3.39304
$25$	1.00000	0.200000
$26$	−4.42758	−0.868320
$27$	3.18772	0.613477
$28$	−20.1849	−3.81459
$29$	−0.486878	−0.0904109	−0.0452055	−	0.998978i	$-0.514394\pi$
$29$	−0.486878	−0.0904109	−0.0452055	+	0.998978i	$0.514394\pi$
$30$	5.59814	1.02208
$31$	−9.53040	−1.71171	−0.855856	−	0.517215i	$-0.826969\pi$
$31$	−9.53040	−1.71171	−0.855856	+	0.517215i	$0.826969\pi$
$32$	−12.7156	−2.24783
$33$	−3.70886	−0.645630
$34$	0	0
$35$	−4.06194	−0.686593
$36$	7.43778	1.23963
$37$	4.26971	0.701937	0.350968	−	0.936387i	$-0.385852\pi$
$37$	4.26971	0.701937	0.350968	+	0.936387i	$0.385852\pi$
$38$	−0.657443	−0.106651
$39$	−3.55649	−0.569495
$40$	−7.83873	−1.23941
$41$	−5.28883	−0.825976	−0.412988	−	0.910736i	$-0.635515\pi$
$41$	−5.28883	−0.825976	−0.412988	+	0.910736i	$0.635515\pi$
$42$	−22.7393	−3.50875
$43$	3.17951	0.484871	0.242435	−	0.970168i	$-0.422054\pi$
$43$	3.17951	0.484871	0.242435	+	0.970168i	$0.422054\pi$
$44$	8.69131	1.31026
$45$	1.49675	0.223122
$46$	1.37202	0.202293
$47$	6.26212	0.913423	0.456712	−	0.889615i	$-0.349027\pi$
$47$	6.26212	0.913423	0.456712	+	0.889615i	$0.349027\pi$
$48$	−22.8070	−3.29191
$49$	9.49933	1.35705
$50$	−2.63994	−0.373344
$51$	0	0
$52$	8.33425	1.15575
$53$	10.2436	1.40707	0.703535	−	0.710661i	$-0.251604\pi$
$53$	10.2436	1.40707	0.703535	+	0.710661i	$0.251604\pi$
$54$	−8.41540	−1.14519
$55$	1.74901	0.235836
$56$	31.8404	4.25485
$57$	−0.528097	−0.0699481
$58$	1.28533	0.168772
$59$	11.5986	1.51001	0.755003	−	0.655722i	$-0.227636\pi$
$59$	11.5986	1.51001	0.755003	+	0.655722i	$0.227636\pi$
$60$	−10.5376	−1.36040
$61$	4.35334	0.557388	0.278694	−	0.960380i	$-0.410099\pi$
$61$	4.35334	0.557388	0.278694	+	0.960380i	$0.410099\pi$
$62$	25.1597	3.19528
$63$	−6.07971	−0.765971
$64$	12.0581	1.50726
$65$	1.67715	0.208025
$66$	9.79117	1.20521
$67$	−9.73489	−1.18931	−0.594653	−	0.803982i	$-0.702711\pi$
$67$	−9.73489	−1.18931	−0.594653	+	0.803982i	$0.702711\pi$
$68$	0	0
$69$	1.10208	0.132675
$70$	10.7233	1.28168
$71$	−1.00602	−0.119392	−0.0596960	−	0.998217i	$-0.519013\pi$
$71$	−1.00602	−0.119392	−0.0596960	+	0.998217i	$0.519013\pi$
$72$	−11.7326	−1.38270
$73$	2.91508	0.341184	0.170592	−	0.985342i	$-0.445432\pi$
$73$	2.91508	0.341184	0.170592	+	0.985342i	$0.445432\pi$
$74$	−11.2718	−1.31032
$75$	−2.12055	−0.244861
$76$	1.23754	0.141955
$77$	−7.10435	−0.809616
$78$	9.38893	1.06309
$79$	−6.81821	−0.767108	−0.383554	−	0.923518i	$-0.625300\pi$
$79$	−6.81821	−0.767108	−0.383554	+	0.923518i	$0.625300\pi$
$80$	10.7552	1.20247
$81$	−11.2500	−1.25000
$82$	13.9622	1.54187
$83$	−3.19678	−0.350892	−0.175446	−	0.984489i	$-0.556137\pi$
$83$	−3.19678	−0.350892	−0.175446	+	0.984489i	$0.556137\pi$
$84$	42.8032	4.67021
$85$	0	0
$86$	−8.39371	−0.905118
$87$	1.03245	0.110690
$88$	−13.7100	−1.46149
$89$	3.30525	0.350356	0.175178	−	0.984537i	$-0.443950\pi$
$89$	3.30525	0.350356	0.175178	+	0.984537i	$0.443950\pi$
$90$	−3.95133	−0.416507
$91$	−6.81249	−0.714143
$92$	−2.58261	−0.269256
$93$	20.2097	2.09565
$94$	−16.5316	−1.70511
$95$	0.249037	0.0255507
$96$	26.9642	2.75202
$97$	−1.70094	−0.172705	−0.0863523	−	0.996265i	$-0.527521\pi$
$97$	−1.70094	−0.172705	−0.0863523	+	0.996265i	$0.527521\pi$
$98$	−25.0777	−2.53323
$99$	2.61783	0.263101
$100$	4.96928	0.496928
$101$	9.86464	0.981569	0.490784	−	0.871281i	$-0.336710\pi$
$101$	9.86464	0.981569	0.490784	+	0.871281i	$0.336710\pi$
$102$	0	0
$103$	−14.0029	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$103$	−14.0029	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$104$	−13.1467	−1.28914
$105$	8.61356	0.840597
$106$	−27.0425	−2.62660
$107$	13.5272	1.30773	0.653863	−	0.756613i	$-0.273147\pi$
$107$	13.5272	1.30773	0.653863	+	0.756613i	$0.273147\pi$
$108$	15.8407	1.52427
$109$	−6.36995	−0.610131	−0.305065	−	0.952331i	$-0.598678\pi$
$109$	−6.36995	−0.610131	−0.305065	+	0.952331i	$0.598678\pi$
$110$	−4.61727	−0.440239
$111$	−9.05416	−0.859383
$112$	−43.6870	−4.12803
$113$	13.0583	1.22843	0.614213	−	0.789141i	$-0.289474\pi$
$113$	13.0583	1.22843	0.614213	+	0.789141i	$0.289474\pi$
$114$	1.39414	0.130574
$115$	−0.519715	−0.0484637
$116$	−2.41943	−0.224639
$117$	2.51028	0.232075
$118$	−30.6195	−2.81876
$119$	0	0
$120$	16.6224	1.51741
$121$	−7.94098	−0.721907
$122$	−11.4925	−1.04049
$123$	11.2152	1.01124
$124$	−47.3593	−4.25299
$125$	1.00000	0.0894427
$126$	16.0501	1.42985
$127$	−16.1404	−1.43223	−0.716114	−	0.697983i	$-0.754081\pi$
$127$	−16.1404	−1.43223	−0.716114	+	0.697983i	$0.754081\pi$
$128$	−6.40141	−0.565810
$129$	−6.74232	−0.593628
$130$	−4.42758	−0.388325
$131$	−15.7448	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$131$	−15.7448	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$132$	−18.4304	−1.60416
$133$	−1.01157	−0.0877145
$134$	25.6995	2.22010
$135$	3.18772	0.274355
$136$	0	0
$137$	−15.5752	−1.33068	−0.665339	−	0.746541i	$-0.731713\pi$
$137$	−15.5752	−1.33068	−0.665339	+	0.746541i	$0.731713\pi$
$138$	−2.90944	−0.247668
$139$	−10.5208	−0.892365	−0.446183	−	0.894942i	$-0.647217\pi$
$139$	−10.5208	−0.892365	−0.446183	+	0.894942i	$0.647217\pi$
$140$	−20.1849	−1.70594
$141$	−13.2792	−1.11831
$142$	2.65582	0.222871
$143$	2.93335	0.245299
$144$	16.0979	1.34149
$145$	−0.486878	−0.0404330
$146$	−7.69563	−0.636895
$147$	−20.1439	−1.66144
$148$	21.2174	1.74406
$149$	7.79902	0.638920	0.319460	−	0.947600i	$-0.396498\pi$
$149$	7.79902	0.638920	0.319460	+	0.947600i	$0.396498\pi$
$150$	5.59814	0.457086
$151$	−3.76216	−0.306161	−0.153080	−	0.988214i	$-0.548919\pi$
$151$	−3.76216	−0.306161	−0.153080	+	0.988214i	$0.548919\pi$
$152$	−1.95213	−0.158339
$153$	0	0
$154$	18.7551	1.51133
$155$	−9.53040	−0.765500
$156$	−17.6732	−1.41499
$157$	17.5226	1.39846	0.699229	−	0.714898i	$-0.253527\pi$
$157$	17.5226	1.39846	0.699229	+	0.714898i	$0.253527\pi$
$158$	17.9997	1.43198
$159$	−21.7221	−1.72268
$160$	−12.7156	−1.00526
$161$	2.11105	0.166374
$162$	29.6993	2.33340
$163$	−16.1671	−1.26630	−0.633152	−	0.774027i	$-0.718239\pi$
$163$	−16.1671	−1.26630	−0.633152	+	0.774027i	$0.718239\pi$
$164$	−26.2817	−2.05225
$165$	−3.70886	−0.288735
$166$	8.43930	0.655017
$167$	−14.3560	−1.11090	−0.555452	−	0.831549i	$-0.687455\pi$
$167$	−14.3560	−1.11090	−0.555452	+	0.831549i	$0.687455\pi$
$168$	−67.5193	−5.20923
$169$	−10.1872	−0.783628
$170$	0	0
$171$	0.372746	0.0285046
$172$	15.7999	1.20473
$173$	−18.0796	−1.37457	−0.687283	−	0.726390i	$-0.741197\pi$
$173$	−18.0796	−1.37457	−0.687283	+	0.726390i	$0.741197\pi$
$174$	−2.72561	−0.206628
$175$	−4.06194	−0.307054
$176$	18.8109	1.41793
$177$	−24.5954	−1.84870
$178$	−8.72567	−0.654017
$179$	7.51774	0.561902	0.280951	−	0.959722i	$-0.409350\pi$
$179$	7.51774	0.561902	0.280951	+	0.959722i	$0.409350\pi$
$180$	7.43778	0.554379
$181$	−4.39392	−0.326597	−0.163299	−	0.986577i	$-0.552213\pi$
$181$	−4.39392	−0.326597	−0.163299	+	0.986577i	$0.552213\pi$
$182$	17.9846	1.33310
$183$	−9.23149	−0.682411
$184$	4.07391	0.300332
$185$	4.26971	0.313916
$186$	−53.3525	−3.91199
$187$	0	0
$188$	31.1182	2.26953
$189$	−12.9483	−0.941852
$190$	−0.657443	−0.0476959
$191$	−5.03076	−0.364013	−0.182007	−	0.983297i	$-0.558259\pi$
$191$	−5.03076	−0.364013	−0.182007	+	0.983297i	$0.558259\pi$
$192$	−25.5699	−1.84535
$193$	−26.8878	−1.93543	−0.967713	−	0.252053i	$-0.918894\pi$
$193$	−26.8878	−1.93543	−0.967713	+	0.252053i	$0.918894\pi$
$194$	4.49039	0.322391
$195$	−3.55649	−0.254686
$196$	47.2049	3.37178
$197$	−8.51380	−0.606583	−0.303292	−	0.952898i	$-0.598086\pi$
$197$	−8.51380	−0.606583	−0.303292	+	0.952898i	$0.598086\pi$
$198$	−6.91090	−0.491136
$199$	16.7681	1.18866	0.594330	−	0.804221i	$-0.297417\pi$
$199$	16.7681	1.18866	0.594330	+	0.804221i	$0.297417\pi$
$200$	−7.83873	−0.554282
$201$	20.6434	1.45607
$202$	−26.0421	−1.83231
$203$	1.97767	0.138805
$204$	0	0
$205$	−5.28883	−0.369388
$206$	36.9669	2.57561
$207$	−0.777884	−0.0540667
$208$	18.0381	1.25072
$209$	0.435567	0.0301288
$210$	−22.7393	−1.56916
$211$	4.75316	0.327221	0.163611	−	0.986525i	$-0.447686\pi$
$211$	4.75316	0.327221	0.163611	+	0.986525i	$0.447686\pi$
$212$	50.9034	3.49606
$213$	2.13331	0.146172
$214$	−35.7111	−2.44116
$215$	3.17951	0.216841
$216$	−24.9877	−1.70020
$217$	38.7119	2.62794
$218$	16.8163	1.13894
$219$	−6.18158	−0.417712
$220$	8.69131	0.585968
$221$	0	0
$222$	23.9024	1.60423
$223$	−21.7349	−1.45548	−0.727738	−	0.685855i	$-0.759429\pi$
$223$	−21.7349	−1.45548	−0.727738	+	0.685855i	$0.759429\pi$
$224$	51.6501	3.45102
$225$	1.49675	0.0997834
$226$	−34.4732	−2.29313
$227$	2.81906	0.187108	0.0935539	−	0.995614i	$-0.470177\pi$
$227$	2.81906	0.187108	0.0935539	+	0.995614i	$0.470177\pi$
$228$	−2.62426	−0.173796
$229$	23.2096	1.53373	0.766867	−	0.641806i	$-0.221814\pi$
$229$	23.2096	1.53373	0.766867	+	0.641806i	$0.221814\pi$
$230$	1.37202	0.0904681
$231$	15.0652	0.991215
$232$	3.81650	0.250566
$233$	−9.69742	−0.635299	−0.317650	−	0.948208i	$-0.602894\pi$
$233$	−9.69742	−0.635299	−0.317650	+	0.948208i	$0.602894\pi$
$234$	−6.62699	−0.433220
$235$	6.26212	0.408495
$236$	57.6366	3.75182
$237$	14.4584	0.939173
$238$	0	0
$239$	6.98497	0.451820	0.225910	−	0.974148i	$-0.427464\pi$
$239$	6.98497	0.451820	0.225910	+	0.974148i	$0.427464\pi$
$240$	−22.8070	−1.47219
$241$	−3.83215	−0.246850	−0.123425	−	0.992354i	$-0.539388\pi$
$241$	−3.83215	−0.246850	−0.123425	+	0.992354i	$0.539388\pi$
$242$	20.9637	1.34760
$243$	14.2930	0.916899
$244$	21.6330	1.38491
$245$	9.49933	0.606890
$246$	−29.6076	−1.88771
$247$	0.417673	0.0265759
$248$	74.7062	4.74385
$249$	6.77894	0.429598
$250$	−2.63994	−0.166964
$251$	−12.5111	−0.789693	−0.394846	−	0.918747i	$-0.629202\pi$
$251$	−12.5111	−0.789693	−0.394846	+	0.918747i	$0.629202\pi$
$252$	−30.2118	−1.90316
$253$	−0.908985	−0.0571474
$254$	42.6097	2.67357
$255$	0	0
$256$	−7.21688	−0.451055
$257$	−13.8873	−0.866263	−0.433132	−	0.901331i	$-0.642591\pi$
$257$	−13.8873	−0.866263	−0.433132	+	0.901331i	$0.642591\pi$
$258$	17.7993	1.10814
$259$	−17.3433	−1.07766
$260$	8.33425	0.516868
$261$	−0.728735	−0.0451075
$262$	41.5652	2.56791
$263$	−30.0042	−1.85014	−0.925069	−	0.379798i	$-0.875993\pi$
$263$	−30.0042	−1.85014	−0.925069	+	0.379798i	$0.875993\pi$
$264$	29.0728	1.78930
$265$	10.2436	0.629260
$266$	2.67049	0.163738
$267$	−7.00897	−0.428942
$268$	−48.3754	−2.95500
$269$	7.27599	0.443625	0.221812	−	0.975089i	$-0.428803\pi$
$269$	7.27599	0.443625	0.221812	+	0.975089i	$0.428803\pi$
$270$	−8.41540	−0.512145
$271$	−6.39386	−0.388399	−0.194200	−	0.980962i	$-0.562211\pi$
$271$	−6.39386	−0.388399	−0.194200	+	0.980962i	$0.562211\pi$
$272$	0	0
$273$	14.4463	0.874327
$274$	41.1176	2.48400
$275$	1.74901	0.105469
$276$	5.47657	0.329651
$277$	7.78887	0.467988	0.233994	−	0.972238i	$-0.424820\pi$
$277$	7.78887	0.467988	0.233994	+	0.972238i	$0.424820\pi$
$278$	27.7744	1.66580
$279$	−14.2646	−0.854001
$280$	31.8404	1.90283
$281$	−21.4059	−1.27697	−0.638485	−	0.769634i	$-0.720439\pi$
$281$	−21.4059	−1.27697	−0.638485	+	0.769634i	$0.720439\pi$
$282$	35.0562	2.08756
$283$	−4.42397	−0.262978	−0.131489	−	0.991318i	$-0.541976\pi$
$283$	−4.42397	−0.262978	−0.131489	+	0.991318i	$0.541976\pi$
$284$	−4.99917	−0.296646
$285$	−0.528097	−0.0312817
$286$	−7.74387	−0.457905
$287$	21.4829	1.26809
$288$	−19.0321	−1.12148
$289$	0	0
$290$	1.28533	0.0754771
$291$	3.60694	0.211443
$292$	14.4858	0.847720
$293$	−19.6200	−1.14621	−0.573107	−	0.819481i	$-0.694262\pi$
$293$	−19.6200	−1.14621	−0.573107	+	0.819481i	$0.694262\pi$
$294$	53.1786	3.10144
$295$	11.5986	0.675295
$296$	−33.4691	−1.94535
$297$	5.57535	0.323514
$298$	−20.5889	−1.19268
$299$	−0.871642	−0.0504084
$300$	−10.5376	−0.608391
$301$	−12.9150	−0.744406
$302$	9.93189	0.571516
$303$	−20.9185	−1.20174
$304$	2.67844	0.153619
$305$	4.35334	0.249271
$306$	0	0
$307$	−13.3437	−0.761567	−0.380784	−	0.924664i	$-0.624346\pi$
$307$	−13.3437	−0.761567	−0.380784	+	0.924664i	$0.624346\pi$
$308$	−35.3035	−2.01161
$309$	29.6940	1.68923
$310$	25.1597	1.42897
$311$	5.35095	0.303424	0.151712	−	0.988425i	$-0.451521\pi$
$311$	5.35095	0.303424	0.151712	+	0.988425i	$0.451521\pi$
$312$	27.8784	1.57830
$313$	−15.9154	−0.899589	−0.449794	−	0.893132i	$-0.648503\pi$
$313$	−15.9154	−0.899589	−0.449794	+	0.893132i	$0.648503\pi$
$314$	−46.2587	−2.61053
$315$	−6.07971	−0.342553
$316$	−33.8816	−1.90599
$317$	2.37447	0.133363	0.0666817	−	0.997774i	$-0.478759\pi$
$317$	2.37447	0.133363	0.0666817	+	0.997774i	$0.478759\pi$
$318$	57.3452	3.21576
$319$	−0.851552	−0.0476778
$320$	12.0581	0.674069
$321$	−28.6852	−1.60105
$322$	−5.57305	−0.310574
$323$	0	0
$324$	−55.9044	−3.10580
$325$	1.67715	0.0930317
$326$	42.6802	2.36384
$327$	13.5078	0.746985
$328$	41.4577	2.28912
$329$	−25.4363	−1.40235
$330$	9.79117	0.538986
$331$	2.75697	0.151537	0.0757683	−	0.997125i	$-0.475859\pi$
$331$	2.75697	0.151537	0.0757683	+	0.997125i	$0.475859\pi$
$332$	−15.8857	−0.871840
$333$	6.39070	0.350208
$334$	37.8991	2.07374
$335$	−9.73489	−0.531874
$336$	92.6406	5.05396
$337$	25.2323	1.37449	0.687246	−	0.726425i	$-0.258820\pi$
$337$	25.2323	1.37449	0.687246	+	0.726425i	$0.258820\pi$
$338$	26.8935	1.46281
$339$	−27.6909	−1.50396
$340$	0	0
$341$	−16.6687	−0.902663
$342$	−0.984028	−0.0532101
$343$	−10.1521	−0.548164
$344$	−24.9233	−1.34377
$345$	1.10208	0.0593342
$346$	47.7290	2.56593
$347$	9.61842	0.516344	0.258172	−	0.966099i	$-0.416880\pi$
$347$	9.61842	0.516344	0.258172	+	0.966099i	$0.416880\pi$
$348$	5.13054	0.275026
$349$	23.1581	1.23962	0.619812	−	0.784750i	$-0.287209\pi$
$349$	23.1581	1.23962	0.619812	+	0.784750i	$0.287209\pi$
$350$	10.7233	0.573183
$351$	5.34630	0.285364
$352$	−22.2397	−1.18538
$353$	15.4317	0.821349	0.410674	−	0.911782i	$-0.365293\pi$
$353$	15.4317	0.821349	0.410674	+	0.911782i	$0.365293\pi$
$354$	64.9304	3.45101
$355$	−1.00602	−0.0533938
$356$	16.4247	0.870510
$357$	0	0
$358$	−19.8464	−1.04891
$359$	8.02467	0.423526	0.211763	−	0.977321i	$-0.432080\pi$
$359$	8.02467	0.423526	0.211763	+	0.977321i	$0.432080\pi$
$360$	−11.7326	−0.618363
$361$	−18.9380	−0.996736
$362$	11.5997	0.609666
$363$	16.8393	0.883833
$364$	−33.8532	−1.77439
$365$	2.91508	0.152582
$366$	24.3706	1.27387
$367$	20.7792	1.08466	0.542332	−	0.840164i	$-0.317541\pi$
$367$	20.7792	1.08466	0.542332	+	0.840164i	$0.317541\pi$
$368$	−5.58964	−0.291380
$369$	−7.91606	−0.412093
$370$	−11.2718	−0.585992
$371$	−41.6089	−2.16023
$372$	100.428	5.20694
$373$	12.6951	0.657330	0.328665	−	0.944447i	$-0.393401\pi$
$373$	12.6951	0.657330	0.328665	+	0.944447i	$0.393401\pi$
$374$	0	0
$375$	−2.12055	−0.109505
$376$	−49.0870	−2.53147
$377$	−0.816569	−0.0420554
$378$	34.1828	1.75817
$379$	−17.6102	−0.904577	−0.452288	−	0.891872i	$-0.649392\pi$
$379$	−17.6102	−0.904577	−0.452288	+	0.891872i	$0.649392\pi$
$380$	1.23754	0.0634842
$381$	34.2266	1.75348
$382$	13.2809	0.679510
$383$	−36.7734	−1.87903	−0.939516	−	0.342505i	$-0.888725\pi$
$383$	−36.7734	−1.87903	−0.939516	+	0.342505i	$0.888725\pi$
$384$	13.5745	0.692723
$385$	−7.10435	−0.362071
$386$	70.9822	3.61290
$387$	4.75893	0.241910
$388$	−8.45247	−0.429109
$389$	−9.34410	−0.473764	−0.236882	−	0.971538i	$-0.576126\pi$
$389$	−9.34410	−0.473764	−0.236882	+	0.971538i	$0.576126\pi$
$390$	9.38893	0.475427
$391$	0	0
$392$	−74.4627	−3.76093
$393$	33.3876	1.68418
$394$	22.4759	1.13232
$395$	−6.81821	−0.343061
$396$	13.0087	0.653712
$397$	−3.99973	−0.200740	−0.100370	−	0.994950i	$-0.532003\pi$
$397$	−3.99973	−0.200740	−0.100370	+	0.994950i	$0.532003\pi$
$398$	−44.2668	−2.21890
$399$	2.14510	0.107389
$400$	10.7552	0.537760
$401$	−21.4879	−1.07305	−0.536526	−	0.843884i	$-0.680264\pi$
$401$	−21.4879	−1.07305	−0.536526	+	0.843884i	$0.680264\pi$
$402$	−54.4972	−2.71808
$403$	−15.9839	−0.796217
$404$	49.0202	2.43885
$405$	−11.2500	−0.559016
$406$	−5.22092	−0.259110
$407$	7.46776	0.370163
$408$	0	0
$409$	18.4461	0.912103	0.456051	−	0.889954i	$-0.349263\pi$
$409$	18.4461	0.912103	0.456051	+	0.889954i	$0.349263\pi$
$410$	13.9622	0.689543
$411$	33.0280	1.62915
$412$	−69.5846	−3.42819
$413$	−47.1127	−2.31826
$414$	2.05357	0.100927
$415$	−3.19678	−0.156924
$416$	−21.3261	−1.04560
$417$	22.3100	1.09253
$418$	−1.14987	−0.0562421
$419$	16.6584	0.813817	0.406908	−	0.913469i	$-0.366607\pi$
$419$	16.6584	0.813817	0.406908	+	0.913469i	$0.366607\pi$
$420$	42.8032	2.08858
$421$	−34.9909	−1.70535	−0.852676	−	0.522440i	$-0.825022\pi$
$421$	−34.9909	−1.70535	−0.852676	+	0.522440i	$0.825022\pi$
$422$	−12.5481	−0.610830
$423$	9.37282	0.455722
$424$	−80.2969	−3.89956
$425$	0	0
$426$	−5.63181	−0.272862
$427$	−17.6830	−0.855739
$428$	67.2206	3.24923
$429$	−6.22033	−0.300320
$430$	−8.39371	−0.404781
$431$	4.19331	0.201985	0.100992	−	0.994887i	$-0.467798\pi$
$431$	4.19331	0.201985	0.100992	+	0.994887i	$0.467798\pi$
$432$	34.2846	1.64952
$433$	−3.66597	−0.176175	−0.0880877	−	0.996113i	$-0.528076\pi$
$433$	−3.66597	−0.176175	−0.0880877	+	0.996113i	$0.528076\pi$
$434$	−102.197	−4.90562
$435$	1.03245	0.0495022
$436$	−31.6541	−1.51596
$437$	−0.129428	−0.00619140
$438$	16.3190	0.779752
$439$	−25.1999	−1.20273	−0.601363	−	0.798976i	$-0.705375\pi$
$439$	−25.1999	−1.20273	−0.601363	+	0.798976i	$0.705375\pi$
$440$	−13.7100	−0.653598
$441$	14.2181	0.677054
$442$	0	0
$443$	35.8208	1.70190	0.850949	−	0.525248i	$-0.176027\pi$
$443$	35.8208	1.70190	0.850949	+	0.525248i	$0.176027\pi$
$444$	−44.9927	−2.13526
$445$	3.30525	0.156684
$446$	57.3788	2.71697
$447$	−16.5382	−0.782232
$448$	−48.9793	−2.31405
$449$	4.85930	0.229325	0.114662	−	0.993405i	$-0.463421\pi$
$449$	4.85930	0.229325	0.114662	+	0.993405i	$0.463421\pi$
$450$	−3.95133	−0.186268
$451$	−9.25019	−0.435575
$452$	64.8906	3.05220
$453$	7.97787	0.374833
$454$	−7.44216	−0.349278
$455$	−6.81249	−0.319374
$456$	4.13961	0.193855
$457$	−8.16176	−0.381791	−0.190895	−	0.981610i	$-0.561139\pi$
$457$	−8.16176	−0.381791	−0.190895	+	0.981610i	$0.561139\pi$
$458$	−61.2720	−2.86305
$459$	0	0
$460$	−2.58261	−0.120415
$461$	25.8811	1.20540	0.602702	−	0.797967i	$-0.294091\pi$
$461$	25.8811	1.20540	0.602702	+	0.797967i	$0.294091\pi$
$462$	−39.7711	−1.85032
$463$	−2.13987	−0.0994481	−0.0497240	−	0.998763i	$-0.515834\pi$
$463$	−2.13987	−0.0994481	−0.0497240	+	0.998763i	$0.515834\pi$
$464$	−5.23647	−0.243097
$465$	20.2097	0.937204
$466$	25.6006	1.18593
$467$	−13.2163	−0.611578	−0.305789	−	0.952099i	$-0.598920\pi$
$467$	−13.2163	−0.611578	−0.305789	+	0.952099i	$0.598920\pi$
$468$	12.4743	0.576624
$469$	39.5425	1.82590
$470$	−16.5316	−0.762546
$471$	−37.1577	−1.71214
$472$	−90.9180	−4.18484
$473$	5.56098	0.255694
$474$	−38.1693	−1.75317
$475$	0.249037	0.0114266
$476$	0	0
$477$	15.3321	0.702010
$478$	−18.4399	−0.843421
$479$	−23.5864	−1.07769	−0.538845	−	0.842405i	$-0.681139\pi$
$479$	−23.5864	−1.07769	−0.538845	+	0.842405i	$0.681139\pi$
$480$	26.9642	1.23074
$481$	7.16096	0.326512
$482$	10.1166	0.460800
$483$	−4.47660	−0.203692
$484$	−39.4610	−1.79368
$485$	−1.70094	−0.0772358
$486$	−37.7328	−1.71159
$487$	28.9667	1.31261	0.656304	−	0.754497i	$-0.272119\pi$
$487$	28.9667	1.31261	0.656304	+	0.754497i	$0.272119\pi$
$488$	−34.1246	−1.54475
$489$	34.2832	1.55034
$490$	−25.0777	−1.13289
$491$	−35.5262	−1.60328	−0.801638	−	0.597810i	$-0.796038\pi$
$491$	−35.5262	−1.60328	−0.801638	+	0.597810i	$0.796038\pi$
$492$	55.7317	2.51258
$493$	0	0
$494$	−1.10263	−0.0496098
$495$	2.61783	0.117663
$496$	−102.501	−4.60245
$497$	4.08637	0.183299
$498$	−17.8960	−0.801939
$499$	33.9551	1.52004	0.760019	−	0.649901i	$-0.225190\pi$
$499$	33.9551	1.52004	0.760019	+	0.649901i	$0.225190\pi$
$500$	4.96928	0.222233
$501$	30.4428	1.36008
$502$	33.0285	1.47413
$503$	19.6084	0.874295	0.437147	−	0.899390i	$-0.355989\pi$
$503$	19.6084	0.874295	0.437147	+	0.899390i	$0.355989\pi$
$504$	47.6572	2.12282
$505$	9.86464	0.438971
$506$	2.39967	0.106678
$507$	21.6024	0.959397
$508$	−80.2062	−3.55857
$509$	−1.06322	−0.0471265	−0.0235632	−	0.999722i	$-0.507501\pi$
$509$	−1.06322	−0.0471265	−0.0235632	+	0.999722i	$0.507501\pi$
$510$	0	0
$511$	−11.8409	−0.523809
$512$	31.8550	1.40780
$513$	0.793861	0.0350498
$514$	36.6615	1.61707
$515$	−14.0029	−0.617043
$516$	−33.5045	−1.47495
$517$	10.9525	0.481689
$518$	45.7853	2.01169
$519$	38.3387	1.68288
$520$	−13.1467	−0.576523
$521$	20.3011	0.889409	0.444705	−	0.895677i	$-0.353309\pi$
$521$	20.3011	0.889409	0.444705	+	0.895677i	$0.353309\pi$
$522$	1.92382	0.0842031
$523$	27.3245	1.19482	0.597408	−	0.801937i	$-0.296197\pi$
$523$	27.3245	1.19482	0.597408	+	0.801937i	$0.296197\pi$
$524$	−78.2402	−3.41794
$525$	8.61356	0.375926
$526$	79.2093	3.45369
$527$	0	0
$528$	−39.8896	−1.73597
$529$	−22.7299	−0.988256
$530$	−27.0425	−1.17465
$531$	17.3602	0.753367
$532$	−5.02679	−0.217939
$533$	−8.87017	−0.384210
$534$	18.5033	0.800715
$535$	13.5272	0.584833
$536$	76.3092	3.29605
$537$	−15.9418	−0.687938
$538$	−19.2082	−0.828123
$539$	16.6144	0.715632
$540$	15.8407	0.681675
$541$	25.8819	1.11275	0.556375	−	0.830931i	$-0.312192\pi$
$541$	25.8819	1.11275	0.556375	+	0.830931i	$0.312192\pi$
$542$	16.8794	0.725033
$543$	9.31754	0.399854
$544$	0	0
$545$	−6.36995	−0.272859
$546$	−38.1372	−1.63212
$547$	−35.5746	−1.52106	−0.760531	−	0.649302i	$-0.775061\pi$
$547$	−35.5746	−1.52106	−0.760531	+	0.649302i	$0.775061\pi$
$548$	−77.3975	−3.30626
$549$	6.51586	0.278090
$550$	−4.61727	−0.196881
$551$	−0.121251	−0.00516545
$552$	−8.63894	−0.367698
$553$	27.6951	1.17772
$554$	−20.5622	−0.873602
$555$	−9.05416	−0.384328
$556$	−52.2810	−2.21721
$557$	−13.1772	−0.558337	−0.279168	−	0.960242i	$-0.590059\pi$
$557$	−13.1772	−0.558337	−0.279168	+	0.960242i	$0.590059\pi$
$558$	37.6578	1.59418
$559$	5.33252	0.225542
$560$	−43.6870	−1.84611
$561$	0	0
$562$	56.5103	2.38374
$563$	−4.35207	−0.183418	−0.0917089	−	0.995786i	$-0.529233\pi$
$563$	−4.35207	−0.183418	−0.0917089	+	0.995786i	$0.529233\pi$
$564$	−65.9879	−2.77859
$565$	13.0583	0.549369
$566$	11.6790	0.490906
$567$	45.6967	1.91908
$568$	7.88588	0.330884
$569$	−19.4637	−0.815960	−0.407980	−	0.912991i	$-0.633767\pi$
$569$	−19.4637	−0.815960	−0.407980	+	0.912991i	$0.633767\pi$
$570$	1.39414	0.0583942
$571$	18.3772	0.769060	0.384530	−	0.923112i	$-0.374363\pi$
$571$	18.3772	0.769060	0.384530	+	0.923112i	$0.374363\pi$
$572$	14.5766	0.609480
$573$	10.6680	0.445662
$574$	−56.7135	−2.36718
$575$	−0.519715	−0.0216736
$576$	18.0480	0.751999
$577$	−23.8985	−0.994906	−0.497453	−	0.867491i	$-0.665731\pi$
$577$	−23.8985	−0.994906	−0.497453	+	0.867491i	$0.665731\pi$
$578$	0	0
$579$	57.0171	2.36955
$580$	−2.41943	−0.100462
$581$	12.9851	0.538713
$582$	−9.52211	−0.394704
$583$	17.9161	0.742011
$584$	−22.8505	−0.945560
$585$	2.51028	0.103787
$586$	51.7957	2.13966
$587$	−0.749317	−0.0309276	−0.0154638	−	0.999880i	$-0.504922\pi$
$587$	−0.749317	−0.0309276	−0.0154638	+	0.999880i	$0.504922\pi$
$588$	−100.100	−4.12807
$589$	−2.37342	−0.0977952
$590$	−30.6195	−1.26059
$591$	18.0540	0.742642
$592$	45.9216	1.88737
$593$	−20.0311	−0.822579	−0.411289	−	0.911505i	$-0.634921\pi$
$593$	−20.0311	−0.822579	−0.411289	+	0.911505i	$0.634921\pi$
$594$	−14.7186	−0.603911
$595$	0	0
$596$	38.7555	1.58749
$597$	−35.5577	−1.45528
$598$	2.30108	0.0940983
$599$	−12.9934	−0.530897	−0.265449	−	0.964125i	$-0.585520\pi$
$599$	−12.9934	−0.530897	−0.265449	+	0.964125i	$0.585520\pi$
$600$	16.6224	0.678609
$601$	−7.62279	−0.310940	−0.155470	−	0.987841i	$-0.549689\pi$
$601$	−7.62279	−0.310940	−0.155470	+	0.987841i	$0.549689\pi$
$602$	34.0947	1.38960
$603$	−14.5707	−0.593365
$604$	−18.6953	−0.760699
$605$	−7.94098	−0.322847
$606$	55.2236	2.24331
$607$	48.5958	1.97244	0.986222	−	0.165426i	$-0.0528998\pi$
$607$	48.5958	1.97244	0.986222	+	0.165426i	$0.0528998\pi$
$608$	−3.16667	−0.128425
$609$	−4.19375	−0.169939
$610$	−11.4925	−0.465320
$611$	10.5025	0.424887
$612$	0	0
$613$	5.92605	0.239351	0.119675	−	0.992813i	$-0.461815\pi$
$613$	5.92605	0.239351	0.119675	+	0.992813i	$0.461815\pi$
$614$	35.2267	1.42163
$615$	11.2152	0.452242
$616$	55.6891	2.24378
$617$	39.5840	1.59359	0.796796	−	0.604249i	$-0.206527\pi$
$617$	39.5840	1.59359	0.796796	+	0.604249i	$0.206527\pi$
$618$	−78.3904	−3.15332
$619$	−44.1102	−1.77294	−0.886470	−	0.462787i	$-0.846850\pi$
$619$	−44.1102	−1.77294	−0.886470	+	0.462787i	$0.846850\pi$
$620$	−47.3593	−1.90199
$621$	−1.65671	−0.0664814
$622$	−14.1262	−0.566408
$623$	−13.4257	−0.537891
$624$	−38.2508	−1.53126
$625$	1.00000	0.0400000
$626$	42.0156	1.67928
$627$	−0.923644	−0.0368868
$628$	87.0749	3.47467
$629$	0	0
$630$	16.0501	0.639450
$631$	2.25974	0.0899590	0.0449795	−	0.998988i	$-0.485678\pi$
$631$	2.25974	0.0899590	0.0449795	+	0.998988i	$0.485678\pi$
$632$	53.4461	2.12597
$633$	−10.0793	−0.400618
$634$	−6.26845	−0.248952
$635$	−16.1404	−0.640512
$636$	−107.943	−4.28024
$637$	15.9318	0.631242
$638$	2.24805	0.0890010
$639$	−1.50575	−0.0595667
$640$	−6.40141	−0.253038
$641$	36.7897	1.45311	0.726553	−	0.687110i	$-0.241121\pi$
$641$	36.7897	1.45311	0.726553	+	0.687110i	$0.241121\pi$
$642$	75.7272	2.98872
$643$	−24.5736	−0.969090	−0.484545	−	0.874766i	$-0.661015\pi$
$643$	−24.5736	−0.969090	−0.484545	+	0.874766i	$0.661015\pi$
$644$	10.4904	0.413380
$645$	−6.74232	−0.265479
$646$	0	0
$647$	39.0064	1.53350	0.766750	−	0.641946i	$-0.221873\pi$
$647$	39.0064	1.53350	0.766750	+	0.641946i	$0.221873\pi$
$648$	88.1856	3.46426
$649$	20.2860	0.796294
$650$	−4.42758	−0.173664
$651$	−82.0907	−3.21739
$652$	−80.3389	−3.14631
$653$	27.9254	1.09281	0.546403	−	0.837522i	$-0.315997\pi$
$653$	27.9254	1.09281	0.546403	+	0.837522i	$0.315997\pi$
$654$	−35.6598	−1.39441
$655$	−15.7448	−0.615199
$656$	−56.8824	−2.22089
$657$	4.36314	0.170222
$658$	67.1504	2.61779
$659$	−19.5830	−0.762847	−0.381423	−	0.924400i	$-0.624566\pi$
$659$	−19.5830	−0.762847	−0.381423	+	0.924400i	$0.624566\pi$
$660$	−18.4304	−0.717402
$661$	27.0356	1.05156	0.525782	−	0.850620i	$-0.323773\pi$
$661$	27.0356	1.05156	0.525782	+	0.850620i	$0.323773\pi$
$662$	−7.27823	−0.282876
$663$	0	0
$664$	25.0587	0.972465
$665$	−1.01157	−0.0392271
$666$	−16.8711	−0.653740
$667$	0.253038	0.00979766
$668$	−71.3392	−2.76020
$669$	46.0900	1.78194
$670$	25.6995	0.992859
$671$	7.61401	0.293936
$672$	−109.527	−4.22509
$673$	−50.1210	−1.93202	−0.966011	−	0.258499i	$-0.916772\pi$
$673$	−50.1210	−1.93202	−0.966011	+	0.258499i	$0.916772\pi$
$674$	−66.6118	−2.56579
$675$	3.18772	0.122695
$676$	−50.6229	−1.94703
$677$	2.14778	0.0825458	0.0412729	−	0.999148i	$-0.486859\pi$
$677$	2.14778	0.0825458	0.0412729	+	0.999148i	$0.486859\pi$
$678$	73.1024	2.80748
$679$	6.90912	0.265148
$680$	0	0
$681$	−5.97798	−0.229077
$682$	44.0045	1.68502
$683$	−3.40533	−0.130301	−0.0651507	−	0.997875i	$-0.520753\pi$
$683$	−3.40533	−0.130301	−0.0651507	+	0.997875i	$0.520753\pi$
$684$	1.85228	0.0708238
$685$	−15.5752	−0.595097
$686$	26.8010	1.02327
$687$	−49.2172	−1.87776
$688$	34.1963	1.30372
$689$	17.1801	0.654510
$690$	−2.90944	−0.110760
$691$	1.51390	0.0575916	0.0287958	−	0.999585i	$-0.490833\pi$
$691$	1.51390	0.0575916	0.0287958	+	0.999585i	$0.490833\pi$
$692$	−89.8426	−3.41530
$693$	−10.6334	−0.403931
$694$	−25.3920	−0.963869
$695$	−10.5208	−0.399078
$696$	−8.09310	−0.306768
$697$	0	0
$698$	−61.1360	−2.31403
$699$	20.5639	0.777798
$700$	−20.1849	−0.762918
$701$	−46.4213	−1.75331	−0.876654	−	0.481121i	$-0.840230\pi$
$701$	−46.4213	−1.75331	−0.876654	+	0.481121i	$0.840230\pi$
$702$	−14.1139	−0.532695
$703$	1.06332	0.0401038
$704$	21.0897	0.794848
$705$	−13.2792	−0.500122
$706$	−40.7389	−1.53323
$707$	−40.0696	−1.50697
$708$	−122.221	−4.59336
$709$	−13.4799	−0.506248	−0.253124	−	0.967434i	$-0.581458\pi$
$709$	−13.4799	−0.506248	−0.253124	+	0.967434i	$0.581458\pi$
$710$	2.65582	0.0996712
$711$	−10.2052	−0.382723
$712$	−25.9090	−0.970981
$713$	4.95310	0.185495
$714$	0	0
$715$	2.93335	0.109701
$716$	37.3578	1.39612
$717$	−14.8120	−0.553165
$718$	−21.1846	−0.790604
$719$	35.0421	1.30685	0.653426	−	0.756991i	$-0.273331\pi$
$719$	35.0421	1.30685	0.653426	+	0.756991i	$0.273331\pi$
$720$	16.0979	0.599932
$721$	56.8791	2.11829
$722$	49.9951	1.86063
$723$	8.12628	0.302219
$724$	−21.8346	−0.811477
$725$	−0.486878	−0.0180822
$726$	−44.4547	−1.64987
$727$	−15.6299	−0.579682	−0.289841	−	0.957075i	$-0.593602\pi$
$727$	−15.6299	−0.579682	−0.289841	+	0.957075i	$0.593602\pi$
$728$	53.4012	1.97918
$729$	3.44079	0.127437
$730$	−7.69563	−0.284828
$731$	0	0
$732$	−45.8739	−1.69555
$733$	−9.63543	−0.355893	−0.177946	−	0.984040i	$-0.556945\pi$
$733$	−9.63543	−0.355893	−0.177946	+	0.984040i	$0.556945\pi$
$734$	−54.8558	−2.02476
$735$	−20.1439	−0.743017
$736$	6.60851	0.243593
$737$	−17.0264	−0.627175
$738$	20.8979	0.769263
$739$	−0.106025	−0.00390019	−0.00195010	−	0.999998i	$-0.500621\pi$
$739$	−0.106025	−0.00390019	−0.00195010	+	0.999998i	$0.500621\pi$
$740$	21.2174	0.779968
$741$	−0.885699	−0.0325370
$742$	109.845	4.03254
$743$	3.59689	0.131957	0.0659786	−	0.997821i	$-0.478983\pi$
$743$	3.59689	0.131957	0.0659786	+	0.997821i	$0.478983\pi$
$744$	−158.419	−5.80791
$745$	7.79902	0.285734
$746$	−33.5144	−1.22705
$747$	−4.78478	−0.175066
$748$	0	0
$749$	−54.9467	−2.00771
$750$	5.59814	0.204415
$751$	17.0397	0.621788	0.310894	−	0.950445i	$-0.399372\pi$
$751$	17.0397	0.621788	0.310894	+	0.950445i	$0.399372\pi$
$752$	67.3503	2.45601
$753$	26.5304	0.966823
$754$	2.15569	0.0785057
$755$	−3.76216	−0.136919
$756$	−64.3439	−2.34017
$757$	−45.5016	−1.65378	−0.826891	−	0.562362i	$-0.809893\pi$
$757$	−45.5016	−1.65378	−0.826891	+	0.562362i	$0.809893\pi$
$758$	46.4899	1.68859
$759$	1.92755	0.0699657
$760$	−1.95213	−0.0708113
$761$	−11.3019	−0.409693	−0.204846	−	0.978794i	$-0.565670\pi$
$761$	−11.3019	−0.409693	−0.204846	+	0.978794i	$0.565670\pi$
$762$	−90.3561	−3.27326
$763$	25.8743	0.936714
$764$	−24.9993	−0.904442
$765$	0	0
$766$	97.0795	3.50763
$767$	19.4526	0.702392
$768$	15.3038	0.552228
$769$	9.75206	0.351668	0.175834	−	0.984420i	$-0.443738\pi$
$769$	9.75206	0.351668	0.175834	+	0.984420i	$0.443738\pi$
$770$	18.7551	0.675886
$771$	29.4487	1.06057
$772$	−133.613	−4.80884
$773$	−2.26624	−0.0815111	−0.0407555	−	0.999169i	$-0.512976\pi$
$773$	−2.26624	−0.0815111	−0.0407555	+	0.999169i	$0.512976\pi$
$774$	−12.5633	−0.451578
$775$	−9.53040	−0.342342
$776$	13.3332	0.478635
$777$	36.7774	1.31938
$778$	24.6679	0.884385
$779$	−1.31711	−0.0471905
$780$	−17.6732	−0.632803
$781$	−1.75953	−0.0629608
$782$	0	0
$783$	−1.55203	−0.0554651
$784$	102.167	3.64883
$785$	17.5226	0.625409
$786$	−88.1413	−3.14390
$787$	19.6778	0.701437	0.350718	−	0.936481i	$-0.385937\pi$
$787$	19.6778	0.701437	0.350718	+	0.936481i	$0.385937\pi$
$788$	−42.3075	−1.50714
$789$	63.6255	2.26513
$790$	17.9997	0.640399
$791$	−53.0422	−1.88596
$792$	−20.5204	−0.729161
$793$	7.30121	0.259274
$794$	10.5590	0.374726
$795$	−21.7221	−0.770405
$796$	83.3255	2.95340
$797$	−46.7746	−1.65684	−0.828422	−	0.560105i	$-0.810761\pi$
$797$	−46.7746	−1.65684	−0.828422	+	0.560105i	$0.810761\pi$
$798$	−5.66292	−0.200465
$799$	0	0
$800$	−12.7156	−0.449566
$801$	4.94714	0.174799
$802$	56.7266	2.00309
$803$	5.09849	0.179922
$804$	102.583	3.61781
$805$	2.11105	0.0744048
$806$	42.1967	1.48631
$807$	−15.4291	−0.543131
$808$	−77.3263	−2.72033
$809$	6.87875	0.241844	0.120922	−	0.992662i	$-0.461415\pi$
$809$	6.87875	0.241844	0.120922	+	0.992662i	$0.461415\pi$
$810$	29.6993	1.04353
$811$	−9.88871	−0.347240	−0.173620	−	0.984813i	$-0.555546\pi$
$811$	−9.88871	−0.347240	−0.173620	+	0.984813i	$0.555546\pi$
$812$	9.82759	0.344881
$813$	13.5585	0.475518
$814$	−19.7144	−0.690990
$815$	−16.1671	−0.566309
$816$	0	0
$817$	0.791816	0.0277021
$818$	−48.6967	−1.70264
$819$	−10.1966	−0.356298
$820$	−26.2817	−0.917796
$821$	3.99072	0.139277	0.0696386	−	0.997572i	$-0.477815\pi$
$821$	3.99072	0.139277	0.0696386	+	0.997572i	$0.477815\pi$
$822$	−87.1920	−3.04117
$823$	34.4402	1.20051	0.600254	−	0.799809i	$-0.295066\pi$
$823$	34.4402	1.20051	0.600254	+	0.799809i	$0.295066\pi$
$824$	109.765	3.82385
$825$	−3.70886	−0.129126
$826$	124.375	4.32755
$827$	−42.8760	−1.49095	−0.745473	−	0.666536i	$-0.767776\pi$
$827$	−42.8760	−1.49095	−0.745473	+	0.666536i	$0.767776\pi$
$828$	−3.86553	−0.134336
$829$	26.1065	0.906716	0.453358	−	0.891328i	$-0.350226\pi$
$829$	26.1065	0.906716	0.453358	+	0.891328i	$0.350226\pi$
$830$	8.43930	0.292932
$831$	−16.5167	−0.572959
$832$	20.2233	0.701117
$833$	0	0
$834$	−58.8970	−2.03944
$835$	−14.3560	−0.496811
$836$	2.16446	0.0748593
$837$	−30.3803	−1.05010
$838$	−43.9772	−1.51917
$839$	11.1671	0.385530	0.192765	−	0.981245i	$-0.438254\pi$
$839$	11.1671	0.385530	0.192765	+	0.981245i	$0.438254\pi$
$840$	−67.5193	−2.32964
$841$	−28.7629	−0.991826
$842$	92.3739	3.18341
$843$	45.3924	1.56340
$844$	23.6198	0.813027
$845$	−10.1872	−0.350449
$846$	−24.7437	−0.850706
$847$	32.2558	1.10832
$848$	110.172	3.78333
$849$	9.38128	0.321965
$850$	0	0
$851$	−2.21904	−0.0760676
$852$	10.6010	0.363185
$853$	21.1150	0.722965	0.361483	−	0.932379i	$-0.382271\pi$
$853$	21.1150	0.722965	0.361483	+	0.932379i	$0.382271\pi$
$854$	46.6820	1.59743
$855$	0.372746	0.0127477
$856$	−106.036	−3.62424
$857$	−55.5913	−1.89896	−0.949481	−	0.313826i	$-0.898389\pi$
$857$	−55.5913	−1.89896	−0.949481	+	0.313826i	$0.898389\pi$
$858$	16.4213	0.560614
$859$	1.84599	0.0629843	0.0314921	−	0.999504i	$-0.489974\pi$
$859$	1.84599	0.0629843	0.0314921	+	0.999504i	$0.489974\pi$
$860$	15.7999	0.538772
$861$	−45.5556	−1.55253
$862$	−11.0701	−0.377049
$863$	13.2346	0.450512	0.225256	−	0.974300i	$-0.427678\pi$
$863$	13.2346	0.450512	0.225256	+	0.974300i	$0.427678\pi$
$864$	−40.5339	−1.37899
$865$	−18.0796	−0.614724
$866$	9.67794	0.328870
$867$	0	0
$868$	192.370	6.52948
$869$	−11.9251	−0.404531
$870$	−2.72561	−0.0924068
$871$	−16.3269	−0.553216
$872$	49.9323	1.69092
$873$	−2.54589	−0.0861652
$874$	0.341683	0.0115576
$875$	−4.06194	−0.137319
$876$	−30.7180	−1.03787
$877$	−36.5039	−1.23265	−0.616325	−	0.787492i	$-0.711379\pi$
$877$	−36.5039	−1.23265	−0.616325	+	0.787492i	$0.711379\pi$
$878$	66.5262	2.24515
$879$	41.6053	1.40331
$880$	18.8109	0.634116
$881$	−24.0208	−0.809282	−0.404641	−	0.914476i	$-0.632603\pi$
$881$	−24.0208	−0.809282	−0.404641	+	0.914476i	$0.632603\pi$
$882$	−37.5350	−1.26387
$883$	42.2659	1.42236	0.711181	−	0.703009i	$-0.248161\pi$
$883$	42.2659	1.42236	0.711181	+	0.703009i	$0.248161\pi$
$884$	0	0
$885$	−24.5954	−0.826765
$886$	−94.5648	−3.17697
$887$	−7.41066	−0.248826	−0.124413	−	0.992231i	$-0.539705\pi$
$887$	−7.41066	−0.248826	−0.124413	+	0.992231i	$0.539705\pi$
$888$	70.9731	2.38170
$889$	65.5613	2.19885
$890$	−8.72567	−0.292485
$891$	−19.6763	−0.659181
$892$	−108.007	−3.61634
$893$	1.55950	0.0521866
$894$	43.6600	1.46021
$895$	7.51774	0.251290
$896$	26.0021	0.868670
$897$	1.84836	0.0617151
$898$	−12.8283	−0.428085
$899$	4.64014	0.154757
$900$	7.43778	0.247926
$901$	0	0
$902$	24.4200	0.813096
$903$	27.3869	0.911379
$904$	−102.361	−3.40447
$905$	−4.39392	−0.146059
$906$	−21.0611	−0.699708
$907$	10.6440	0.353429	0.176714	−	0.984262i	$-0.443453\pi$
$907$	10.6440	0.353429	0.176714	+	0.984262i	$0.443453\pi$
$908$	14.0087	0.464896
$909$	14.7649	0.489721
$910$	17.9846	0.596182
$911$	−6.49745	−0.215270	−0.107635	−	0.994190i	$-0.534328\pi$
$911$	−6.49745	−0.215270	−0.107635	+	0.994190i	$0.534328\pi$
$912$	−5.67979	−0.188077
$913$	−5.59118	−0.185041
$914$	21.5466	0.712697
$915$	−9.23149	−0.305184
$916$	115.335	3.81078
$917$	63.9542	2.11195
$918$	0	0
$919$	−9.77664	−0.322502	−0.161251	−	0.986913i	$-0.551553\pi$
$919$	−9.77664	−0.322502	−0.161251	+	0.986913i	$0.551553\pi$
$920$	4.07391	0.134313
$921$	28.2961	0.932388
$922$	−68.3246	−2.25015
$923$	−1.68724	−0.0555362
$924$	74.8631	2.46281
$925$	4.26971	0.140387
$926$	5.64912	0.185642
$927$	−20.9589	−0.688381
$928$	6.19096	0.203228
$929$	22.6724	0.743857	0.371929	−	0.928261i	$-0.378697\pi$
$929$	22.6724	0.743857	0.371929	+	0.928261i	$0.378697\pi$
$930$	−53.3525	−1.74950
$931$	2.36569	0.0775322
$932$	−48.1892	−1.57849
$933$	−11.3470	−0.371483
$934$	34.8903	1.14164
$935$	0	0
$936$	−19.6774	−0.643176
$937$	−22.7132	−0.742007	−0.371003	−	0.928632i	$-0.620986\pi$
$937$	−22.7132	−0.742007	−0.371003	+	0.928632i	$0.620986\pi$
$938$	−104.390	−3.40845
$939$	33.7494	1.10137
$940$	31.1182	1.01496
$941$	−37.4675	−1.22141	−0.610703	−	0.791860i	$-0.709113\pi$
$941$	−37.4675	−1.22141	−0.610703	+	0.791860i	$0.709113\pi$
$942$	98.0940	3.19608
$943$	2.74869	0.0895095
$944$	124.745	4.06010
$945$	−12.9483	−0.421209
$946$	−14.6807	−0.477309
$947$	27.1585	0.882533	0.441267	−	0.897376i	$-0.354529\pi$
$947$	27.1585	0.882533	0.441267	+	0.897376i	$0.354529\pi$
$948$	71.8478	2.33351
$949$	4.88903	0.158705
$950$	−0.657443	−0.0213303
$951$	−5.03519	−0.163277
$952$	0	0
$953$	−0.783845	−0.0253912	−0.0126956	−	0.999919i	$-0.504041\pi$
$953$	−0.783845	−0.0253912	−0.0126956	+	0.999919i	$0.504041\pi$
$954$	−40.4759	−1.31046
$955$	−5.03076	−0.162792
$956$	34.7103	1.12261
$957$	1.80576	0.0583720
$958$	62.2667	2.01175
$959$	63.2654	2.04295
$960$	−25.5699	−0.825264
$961$	59.8286	1.92995
$962$	−18.9045	−0.609506
$963$	20.2469	0.652446
$964$	−19.0430	−0.613335
$965$	−26.8878	−0.865549
$966$	11.8179	0.380236
$967$	52.9690	1.70337	0.851684	−	0.524056i	$-0.175582\pi$
$967$	52.9690	1.70337	0.851684	+	0.524056i	$0.175582\pi$
$968$	62.2472	2.00070
$969$	0	0
$970$	4.49039	0.144178
$971$	−7.21188	−0.231440	−0.115720	−	0.993282i	$-0.536918\pi$
$971$	−7.21188	−0.231440	−0.115720	+	0.993282i	$0.536918\pi$
$972$	71.0262	2.27817
$973$	42.7350	1.37002
$974$	−76.4704	−2.45027
$975$	−3.55649	−0.113899
$976$	46.8210	1.49870
$977$	10.0232	0.320669	0.160335	−	0.987063i	$-0.448743\pi$
$977$	10.0232	0.320669	0.160335	+	0.987063i	$0.448743\pi$
$978$	−90.5056	−2.89405
$979$	5.78091	0.184759
$980$	47.2049	1.50790
$981$	−9.53423	−0.304404
$982$	93.7871	2.99287
$983$	2.65289	0.0846141	0.0423071	−	0.999105i	$-0.486529\pi$
$983$	2.65289	0.0846141	0.0423071	+	0.999105i	$0.486529\pi$
$984$	−87.9133	−2.80257
$985$	−8.51380	−0.271272
$986$	0	0
$987$	53.9391	1.71690
$988$	2.07554	0.0660316
$989$	−1.65244	−0.0525445
$990$	−6.91090	−0.219643
$991$	−27.9374	−0.887460	−0.443730	−	0.896160i	$-0.646345\pi$
$991$	−27.9374	−0.887460	−0.443730	+	0.896160i	$0.646345\pi$
$992$	121.185	3.84763
$993$	−5.84630	−0.185527
$994$	−10.7878	−0.342167
$995$	16.7681	0.531585
$996$	33.6865	1.06740
$997$	25.9891	0.823082	0.411541	−	0.911391i	$-0.364991\pi$
$997$	25.9891	0.823082	0.411541	+	0.911391i	$0.364991\pi$
$998$	−89.6394	−2.83748
$999$	13.6107	0.430622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.a.p.1.2		12
5.4	even	2		7225.2.a.bs.1.11		12
17.4	even	4		1445.2.d.j.866.22		24
17.10	odd	16		85.2.l.a.66.6	✓	24
17.12	odd	16		85.2.l.a.76.6	yes	24
17.13	even	4		1445.2.d.j.866.21		24
17.16	even	2		1445.2.a.q.1.2		12
51.29	even	16		765.2.be.b.586.1		24
51.44	even	16		765.2.be.b.406.1		24
85.12	even	16		425.2.n.f.399.6		24
85.27	even	16		425.2.n.c.49.1		24
85.29	odd	16		425.2.m.b.76.1		24
85.44	odd	16		425.2.m.b.151.1		24
85.63	even	16		425.2.n.c.399.1		24
85.78	even	16		425.2.n.f.49.6		24
85.84	even	2		7225.2.a.bq.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.66.6	✓	24	17.10	odd	16
85.2.l.a.76.6	yes	24	17.12	odd	16
425.2.m.b.76.1		24	85.29	odd	16
425.2.m.b.151.1		24	85.44	odd	16
425.2.n.c.49.1		24	85.27	even	16
425.2.n.c.399.1		24	85.63	even	16
425.2.n.f.49.6		24	85.78	even	16
425.2.n.f.399.6		24	85.12	even	16
765.2.be.b.406.1		24	51.44	even	16
765.2.be.b.586.1		24	51.29	even	16
1445.2.a.p.1.2		12	1.1	even	1	trivial
1445.2.a.q.1.2		12	17.16	even	2
1445.2.d.j.866.21		24	17.13	even	4
1445.2.d.j.866.22		24	17.4	even	4
7225.2.a.bq.1.11		12	85.84	even	2
7225.2.a.bs.1.11		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.63994 q^{2} -2.12055 q^{3} +4.96928 q^{4} +1.00000 q^{5} +5.59814 q^{6} -4.06194 q^{7} -7.83873 q^{8} +1.49675 q^{9} -2.63994 q^{10} +1.74901 q^{11} -10.5376 q^{12} +1.67715 q^{13} +10.7233 q^{14} -2.12055 q^{15} +10.7552 q^{16} -3.95133 q^{18} +0.249037 q^{19} +4.96928 q^{20} +8.61356 q^{21} -4.61727 q^{22} -0.519715 q^{23} +16.6224 q^{24} +1.00000 q^{25} -4.42758 q^{26} +3.18772 q^{27} -20.1849 q^{28} -0.486878 q^{29} +5.59814 q^{30} -9.53040 q^{31} -12.7156 q^{32} -3.70886 q^{33} -4.06194 q^{35} +7.43778 q^{36} +4.26971 q^{37} -0.657443 q^{38} -3.55649 q^{39} -7.83873 q^{40} -5.28883 q^{41} -22.7393 q^{42} +3.17951 q^{43} +8.69131 q^{44} +1.49675 q^{45} +1.37202 q^{46} +6.26212 q^{47} -22.8070 q^{48} +9.49933 q^{49} -2.63994 q^{50} +8.33425 q^{52} +10.2436 q^{53} -8.41540 q^{54} +1.74901 q^{55} +31.8404 q^{56} -0.528097 q^{57} +1.28533 q^{58} +11.5986 q^{59} -10.5376 q^{60} +4.35334 q^{61} +25.1597 q^{62} -6.07971 q^{63} +12.0581 q^{64} +1.67715 q^{65} +9.79117 q^{66} -9.73489 q^{67} +1.10208 q^{69} +10.7233 q^{70} -1.00602 q^{71} -11.7326 q^{72} +2.91508 q^{73} -11.2718 q^{74} -2.12055 q^{75} +1.23754 q^{76} -7.10435 q^{77} +9.38893 q^{78} -6.81821 q^{79} +10.7552 q^{80} -11.2500 q^{81} +13.9622 q^{82} -3.19678 q^{83} +42.8032 q^{84} -8.39371 q^{86} +1.03245 q^{87} -13.7100 q^{88} +3.30525 q^{89} -3.95133 q^{90} -6.81249 q^{91} -2.58261 q^{92} +20.2097 q^{93} -16.5316 q^{94} +0.249037 q^{95} +26.9642 q^{96} -1.70094 q^{97} -25.0777 q^{98} +2.61783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21}+ \cdots - 56 q^{99}+O(q^{100}) \) 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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