LMFDB - Embedded newform 1445.2.a.p.1.10

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			1.10
Root			$-1.43840$ 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)

$f(q)$	$=$	\(q+1.43840 q^{2} +0.109907 q^{3} +0.0689897 q^{4} +1.00000 q^{5} +0.158090 q^{6} -0.695085 q^{7} -2.77756 q^{8} -2.98792 q^{9} +1.43840 q^{10} -4.85089 q^{11} +0.00758244 q^{12} +5.63906 q^{13} -0.999809 q^{14} +0.109907 q^{15} -4.13322 q^{16} -4.29782 q^{18} -2.32272 q^{19} +0.0689897 q^{20} -0.0763945 q^{21} -6.97750 q^{22} -4.63686 q^{23} -0.305273 q^{24} +1.00000 q^{25} +8.11121 q^{26} -0.658113 q^{27} -0.0479537 q^{28} -6.50618 q^{29} +0.158090 q^{30} -6.63194 q^{31} -0.390093 q^{32} -0.533145 q^{33} -0.695085 q^{35} -0.206136 q^{36} -0.118625 q^{37} -3.34100 q^{38} +0.619770 q^{39} -2.77756 q^{40} -1.07877 q^{41} -0.109886 q^{42} -0.641108 q^{43} -0.334661 q^{44} -2.98792 q^{45} -6.66965 q^{46} +4.93703 q^{47} -0.454269 q^{48} -6.51686 q^{49} +1.43840 q^{50} +0.389037 q^{52} +11.9864 q^{53} -0.946629 q^{54} -4.85089 q^{55} +1.93064 q^{56} -0.255283 q^{57} -9.35848 q^{58} -9.91829 q^{59} +0.00758244 q^{60} -1.60292 q^{61} -9.53937 q^{62} +2.07686 q^{63} +7.70533 q^{64} +5.63906 q^{65} -0.766875 q^{66} -2.99411 q^{67} -0.509622 q^{69} -0.999809 q^{70} +4.68852 q^{71} +8.29913 q^{72} -5.49911 q^{73} -0.170630 q^{74} +0.109907 q^{75} -0.160244 q^{76} +3.37178 q^{77} +0.891477 q^{78} -14.8439 q^{79} -4.13322 q^{80} +8.89143 q^{81} -1.55170 q^{82} +5.03506 q^{83} -0.00527044 q^{84} -0.922169 q^{86} -0.715073 q^{87} +13.4736 q^{88} -2.35657 q^{89} -4.29782 q^{90} -3.91962 q^{91} -0.319896 q^{92} -0.728895 q^{93} +7.10141 q^{94} -2.32272 q^{95} -0.0428738 q^{96} +2.70080 q^{97} -9.37384 q^{98} +14.4941 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.43840	1.01710	0.508551	−	0.861032i	$-0.330181\pi$
$2$	1.43840	1.01710	0.508551	+	0.861032i	$0.330181\pi$
$3$	0.109907	0.0634547	0.0317274	−	0.999497i	$-0.489899\pi$
$3$	0.109907	0.0634547	0.0317274	+	0.999497i	$0.489899\pi$
$4$	0.0689897	0.0344949
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.158090	0.0645399
$7$	−0.695085	−0.262717	−0.131359	−	0.991335i	$-0.541934\pi$
$7$	−0.695085	−0.262717	−0.131359	+	0.991335i	$0.541934\pi$
$8$	−2.77756	−0.982016
$9$	−2.98792	−0.995974
$10$	1.43840	0.454861
$11$	−4.85089	−1.46260	−0.731298	−	0.682058i	$-0.761085\pi$
$11$	−4.85089	−1.46260	−0.731298	+	0.682058i	$0.761085\pi$
$12$	0.00758244	0.00218886
$13$	5.63906	1.56399	0.781996	−	0.623283i	$-0.214202\pi$
$13$	5.63906	1.56399	0.781996	+	0.623283i	$0.214202\pi$
$14$	−0.999809	−0.267210
$15$	0.109907	0.0283778
$16$	−4.13322	−1.03330
$17$	0	0
$17$	0	0
$18$	−4.29782	−1.01301
$19$	−2.32272	−0.532869	−0.266434	−	0.963853i	$-0.585846\pi$
$19$	−2.32272	−0.532869	−0.266434	+	0.963853i	$0.585846\pi$
$20$	0.0689897	0.0154266
$21$	−0.0763945	−0.0166706
$22$	−6.97750	−1.48761
$23$	−4.63686	−0.966852	−0.483426	−	0.875385i	$-0.660608\pi$
$23$	−4.63686	−0.966852	−0.483426	+	0.875385i	$0.660608\pi$
$24$	−0.305273	−0.0623136
$25$	1.00000	0.200000
$26$	8.11121	1.59074
$27$	−0.658113	−0.126654
$28$	−0.0479537	−0.00906240
$29$	−6.50618	−1.20817	−0.604084	−	0.796921i	$-0.706461\pi$
$29$	−6.50618	−1.20817	−0.604084	+	0.796921i	$0.706461\pi$
$30$	0.158090	0.0288631
$31$	−6.63194	−1.19113	−0.595565	−	0.803307i	$-0.703072\pi$
$31$	−6.63194	−1.19113	−0.595565	+	0.803307i	$0.703072\pi$
$32$	−0.390093	−0.0689593
$33$	−0.533145	−0.0928087
$34$	0	0
$35$	−0.695085	−0.117491
$36$	−0.206136	−0.0343560
$37$	−0.118625	−0.0195018	−0.00975091	−	0.999952i	$-0.503104\pi$
$37$	−0.118625	−0.0195018	−0.00975091	+	0.999952i	$0.503104\pi$
$38$	−3.34100	−0.541981
$39$	0.619770	0.0992427
$40$	−2.77756	−0.439171
$41$	−1.07877	−0.168475	−0.0842375	−	0.996446i	$-0.526845\pi$
$41$	−1.07877	−0.168475	−0.0842375	+	0.996446i	$0.526845\pi$
$42$	−0.109886	−0.0169557
$43$	−0.641108	−0.0977681	−0.0488840	−	0.998804i	$-0.515566\pi$
$43$	−0.641108	−0.0977681	−0.0488840	+	0.998804i	$0.515566\pi$
$44$	−0.334661	−0.0504521
$45$	−2.98792	−0.445413
$46$	−6.66965	−0.983386
$47$	4.93703	0.720139	0.360070	−	0.932925i	$-0.382753\pi$
$47$	4.93703	0.720139	0.360070	+	0.932925i	$0.382753\pi$
$48$	−0.454269	−0.0655681
$49$	−6.51686	−0.930980
$50$	1.43840	0.203420
$51$	0	0
$52$	0.389037	0.0539497
$53$	11.9864	1.64646	0.823228	−	0.567711i	$-0.192171\pi$
$53$	11.9864	1.64646	0.823228	+	0.567711i	$0.192171\pi$
$54$	−0.946629	−0.128820
$55$	−4.85089	−0.654093
$56$	1.93064	0.257993
$57$	−0.255283	−0.0338130
$58$	−9.35848	−1.22883
$59$	−9.91829	−1.29125	−0.645626	−	0.763654i	$-0.723403\pi$
$59$	−9.91829	−1.29125	−0.645626	+	0.763654i	$0.723403\pi$
$60$	0.00758244	0.000978888 0
$61$	−1.60292	−0.205233	−0.102617	−	0.994721i	$-0.532722\pi$
$61$	−1.60292	−0.205233	−0.102617	+	0.994721i	$0.532722\pi$
$62$	−9.53937	−1.21150
$63$	2.07686	0.261659
$64$	7.70533	0.963166
$65$	5.63906	0.699439
$66$	−0.766875	−0.0943958
$67$	−2.99411	−0.365789	−0.182894	−	0.983133i	$-0.558547\pi$
$67$	−2.99411	−0.365789	−0.182894	+	0.983133i	$0.558547\pi$
$68$	0	0
$69$	−0.509622	−0.0613513
$70$	−0.999809	−0.119500
$71$	4.68852	0.556425	0.278213	−	0.960520i	$-0.410258\pi$
$71$	4.68852	0.556425	0.278213	+	0.960520i	$0.410258\pi$
$72$	8.29913	0.978062
$73$	−5.49911	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$73$	−5.49911	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$74$	−0.170630	−0.0198353
$75$	0.109907	0.0126909
$76$	−0.160244	−0.0183812
$77$	3.37178	0.384250
$78$	0.891477	0.100940
$79$	−14.8439	−1.67007	−0.835037	−	0.550193i	$-0.814554\pi$
$79$	−14.8439	−1.67007	−0.835037	+	0.550193i	$0.814554\pi$
$80$	−4.13322	−0.462108
$81$	8.89143	0.987937
$82$	−1.55170	−0.171356
$83$	5.03506	0.552670	0.276335	−	0.961061i	$-0.410880\pi$
$83$	5.03506	0.552670	0.276335	+	0.961061i	$0.410880\pi$
$84$	−0.00527044	−0.000575052 0
$85$	0	0
$86$	−0.922169	−0.0994400
$87$	−0.715073	−0.0766639
$88$	13.4736	1.43629
$89$	−2.35657	−0.249796	−0.124898	−	0.992170i	$-0.539860\pi$
$89$	−2.35657	−0.249796	−0.124898	+	0.992170i	$0.539860\pi$
$90$	−4.29782	−0.453030
$91$	−3.91962	−0.410888
$92$	−0.319896	−0.0333514
$93$	−0.728895	−0.0755829
$94$	7.10141	0.732454
$95$	−2.32272	−0.238306
$96$	−0.0428738	−0.00437579
$97$	2.70080	0.274225	0.137113	−	0.990555i	$-0.456218\pi$
$97$	2.70080	0.274225	0.137113	+	0.990555i	$0.456218\pi$
$98$	−9.37384	−0.946900
$99$	14.4941	1.45671
$100$	0.0689897	0.00689897
$101$	14.3025	1.42315	0.711575	−	0.702610i	$-0.247982\pi$
$101$	14.3025	1.42315	0.711575	+	0.702610i	$0.247982\pi$
$102$	0	0
$103$	−10.8963	−1.07365	−0.536824	−	0.843694i	$-0.680376\pi$
$103$	−10.8963	−1.07365	−0.536824	+	0.843694i	$0.680376\pi$
$104$	−15.6628	−1.53587
$105$	−0.0763945	−0.00745534
$106$	17.2412	1.67461
$107$	16.6273	1.60742	0.803712	−	0.595019i	$-0.202855\pi$
$107$	16.6273	1.60742	0.803712	+	0.595019i	$0.202855\pi$
$108$	−0.0454030	−0.00436891
$109$	10.5999	1.01529	0.507643	−	0.861568i	$-0.330517\pi$
$109$	10.5999	1.01529	0.507643	+	0.861568i	$0.330517\pi$
$110$	−6.97750	−0.665279
$111$	−0.0130377	−0.00123748
$112$	2.87294	0.271467
$113$	11.5397	1.08556	0.542780	−	0.839875i	$-0.317372\pi$
$113$	11.5397	1.08556	0.542780	+	0.839875i	$0.317372\pi$
$114$	−0.367198	−0.0343913
$115$	−4.63686	−0.432389
$116$	−0.448860	−0.0416756
$117$	−16.8490	−1.55770
$118$	−14.2665	−1.31333
$119$	0	0
$120$	−0.305273	−0.0278675
$121$	12.5311	1.13919
$122$	−2.30564	−0.208743
$123$	−0.118564	−0.0106905
$124$	−0.457535	−0.0410879
$125$	1.00000	0.0894427
$126$	2.98735	0.266134
$127$	−0.498539	−0.0442381	−0.0221191	−	0.999755i	$-0.507041\pi$
$127$	−0.498539	−0.0442381	−0.0221191	+	0.999755i	$0.507041\pi$
$128$	11.8635	1.04860
$129$	−0.0704621	−0.00620384
$130$	8.11121	0.711400
$131$	19.5341	1.70670	0.853350	−	0.521338i	$-0.174567\pi$
$131$	19.5341	1.70670	0.853350	+	0.521338i	$0.174567\pi$
$132$	−0.0367815	−0.00320142
$133$	1.61449	0.139994
$134$	−4.30672	−0.372044
$135$	−0.658113	−0.0566414
$136$	0	0
$137$	−3.81724	−0.326129	−0.163065	−	0.986615i	$-0.552138\pi$
$137$	−3.81724	−0.326129	−0.163065	+	0.986615i	$0.552138\pi$
$138$	−0.733040	−0.0624005
$139$	−13.2525	−1.12406	−0.562030	−	0.827117i	$-0.689980\pi$
$139$	−13.2525	−1.12406	−0.562030	+	0.827117i	$0.689980\pi$
$140$	−0.0479537	−0.00405283
$141$	0.542613	0.0456962
$142$	6.74396	0.565941
$143$	−27.3544	−2.28749
$144$	12.3497	1.02914
$145$	−6.50618	−0.540309
$146$	−7.90991	−0.654629
$147$	−0.716247	−0.0590750
$148$	−0.00818390	−0.000672712 0
$149$	−17.4543	−1.42991	−0.714955	−	0.699171i	$-0.753553\pi$
$149$	−17.4543	−1.42991	−0.714955	+	0.699171i	$0.753553\pi$
$150$	0.158090	0.0129080
$151$	−11.2715	−0.917262	−0.458631	−	0.888627i	$-0.651660\pi$
$151$	−11.2715	−0.917262	−0.458631	+	0.888627i	$0.651660\pi$
$152$	6.45150	0.523286
$153$	0	0
$154$	4.84996	0.390821
$155$	−6.63194	−0.532690
$156$	0.0427578	0.00342336
$157$	−6.14541	−0.490457	−0.245229	−	0.969465i	$-0.578863\pi$
$157$	−6.14541	−0.490457	−0.245229	+	0.969465i	$0.578863\pi$
$158$	−21.3515	−1.69863
$159$	1.31738	0.104475
$160$	−0.390093	−0.0308395
$161$	3.22301	0.254009
$162$	12.7894	1.00483
$163$	−11.2012	−0.877344	−0.438672	−	0.898647i	$-0.644551\pi$
$163$	−11.2012	−0.877344	−0.438672	+	0.898647i	$0.644551\pi$
$164$	−0.0744238	−0.00581152
$165$	−0.533145	−0.0415053
$166$	7.24243	0.562121
$167$	−5.70603	−0.441546	−0.220773	−	0.975325i	$-0.570858\pi$
$167$	−5.70603	−0.441546	−0.220773	+	0.975325i	$0.570858\pi$
$168$	0.212190	0.0163709
$169$	18.7989	1.44607
$170$	0	0
$171$	6.94010	0.530723
$172$	−0.0442299	−0.00337250
$173$	4.53754	0.344983	0.172491	−	0.985011i	$-0.444818\pi$
$173$	4.53754	0.344983	0.172491	+	0.985011i	$0.444818\pi$
$174$	−1.02856	−0.0779750
$175$	−0.695085	−0.0525435
$176$	20.0498	1.51131
$177$	−1.09009	−0.0819360
$178$	−3.38969	−0.254068
$179$	−18.3756	−1.37345	−0.686727	−	0.726915i	$-0.740953\pi$
$179$	−18.3756	−1.37345	−0.686727	+	0.726915i	$0.740953\pi$
$180$	−0.206136	−0.0153645
$181$	−10.7182	−0.796679	−0.398339	−	0.917238i	$-0.630413\pi$
$181$	−10.7182	−0.796679	−0.398339	+	0.917238i	$0.630413\pi$
$182$	−5.63798	−0.417915
$183$	−0.176172	−0.0130230
$184$	12.8792	0.949465
$185$	−0.118625	−0.00872148
$186$	−1.04844	−0.0768754
$187$	0	0
$188$	0.340604	0.0248411
$189$	0.457444	0.0332742
$190$	−3.34100	−0.242381
$191$	18.5397	1.34149	0.670743	−	0.741690i	$-0.265976\pi$
$191$	18.5397	1.34149	0.670743	+	0.741690i	$0.265976\pi$
$192$	0.846868	0.0611174
$193$	−9.88591	−0.711603	−0.355802	−	0.934562i	$-0.615792\pi$
$193$	−9.88591	−0.711603	−0.355802	+	0.934562i	$0.615792\pi$
$194$	3.88483	0.278915
$195$	0.619770	0.0443827
$196$	−0.449596	−0.0321140
$197$	−13.9534	−0.994138	−0.497069	−	0.867711i	$-0.665590\pi$
$197$	−13.9534	−0.994138	−0.497069	+	0.867711i	$0.665590\pi$
$198$	20.8482	1.48162
$199$	−12.7366	−0.902876	−0.451438	−	0.892303i	$-0.649089\pi$
$199$	−12.7366	−0.902876	−0.451438	+	0.892303i	$0.649089\pi$
$200$	−2.77756	−0.196403
$201$	−0.329073	−0.0232110
$202$	20.5727	1.44749
$203$	4.52235	0.317407
$204$	0	0
$205$	−1.07877	−0.0753443
$206$	−15.6733	−1.09201
$207$	13.8546	0.962959
$208$	−23.3075	−1.61608
$209$	11.2672	0.779372
$210$	−0.109886	−0.00758284
$211$	12.8778	0.886545	0.443273	−	0.896387i	$-0.353817\pi$
$211$	12.8778	0.886545	0.443273	+	0.896387i	$0.353817\pi$
$212$	0.826937	0.0567943
$213$	0.515300	0.0353078
$214$	23.9167	1.63491
$215$	−0.641108	−0.0437232
$216$	1.82795	0.124376
$217$	4.60976	0.312931
$218$	15.2469	1.03265
$219$	−0.604389	−0.0408409
$220$	−0.334661	−0.0225629
$221$	0	0
$222$	−0.0187534	−0.00125864
$223$	20.4208	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$223$	20.4208	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$224$	0.271147	0.0181168
$225$	−2.98792	−0.199195
$226$	16.5986	1.10412
$227$	−1.22029	−0.0809935	−0.0404968	−	0.999180i	$-0.512894\pi$
$227$	−1.22029	−0.0809935	−0.0404968	+	0.999180i	$0.512894\pi$
$228$	−0.0176119	−0.00116638
$229$	3.51944	0.232571	0.116286	−	0.993216i	$-0.462901\pi$
$229$	3.51944	0.232571	0.116286	+	0.993216i	$0.462901\pi$
$230$	−6.66965	−0.439784
$231$	0.370581	0.0243824
$232$	18.0713	1.18644
$233$	29.0026	1.90002	0.950011	−	0.312218i	$-0.101072\pi$
$233$	29.0026	1.90002	0.950011	+	0.312218i	$0.101072\pi$
$234$	−24.2356	−1.58433
$235$	4.93703	0.322056
$236$	−0.684260	−0.0445415
$237$	−1.63145	−0.105974
$238$	0	0
$239$	7.87133	0.509154	0.254577	−	0.967053i	$-0.418064\pi$
$239$	7.87133	0.509154	0.254577	+	0.967053i	$0.418064\pi$
$240$	−0.454269	−0.0293229
$241$	−11.6270	−0.748959	−0.374479	−	0.927235i	$-0.622179\pi$
$241$	−11.6270	−0.748959	−0.374479	+	0.927235i	$0.622179\pi$
$242$	18.0247	1.15867
$243$	2.95157	0.189343
$244$	−0.110585	−0.00707950
$245$	−6.51686	−0.416347
$246$	−0.170542	−0.0108734
$247$	−13.0979	−0.833403
$248$	18.4206	1.16971
$249$	0.553388	0.0350695
$250$	1.43840	0.0909723
$251$	−7.94692	−0.501605	−0.250803	−	0.968038i	$-0.580694\pi$
$251$	−7.94692	−0.501605	−0.250803	+	0.968038i	$0.580694\pi$
$252$	0.143282	0.00902591
$253$	22.4929	1.41411
$254$	−0.717097	−0.0449947
$255$	0	0
$256$	1.65381	0.103363
$257$	−7.92727	−0.494489	−0.247245	−	0.968953i	$-0.579525\pi$
$257$	−7.92727	−0.494489	−0.247245	+	0.968953i	$0.579525\pi$
$258$	−0.101353	−0.00630994
$259$	0.0824544	0.00512347
$260$	0.389037	0.0241270
$261$	19.4400	1.20330
$262$	28.0978	1.73589
$263$	8.53622	0.526366	0.263183	−	0.964746i	$-0.415228\pi$
$263$	8.53622	0.526366	0.263183	+	0.964746i	$0.415228\pi$
$264$	1.48084	0.0911396
$265$	11.9864	0.736318
$266$	2.32228	0.142388
$267$	−0.259003	−0.0158507
$268$	−0.206563	−0.0126178
$269$	−15.9828	−0.974490	−0.487245	−	0.873265i	$-0.661998\pi$
$269$	−15.9828	−0.974490	−0.487245	+	0.873265i	$0.661998\pi$
$270$	−0.946629	−0.0576100
$271$	−22.5289	−1.36853	−0.684266	−	0.729232i	$-0.739877\pi$
$271$	−22.5289	−1.36853	−0.684266	+	0.729232i	$0.739877\pi$
$272$	0	0
$273$	−0.430793	−0.0260728
$274$	−5.49072	−0.331706
$275$	−4.85089	−0.292519
$276$	−0.0351587	−0.00211630
$277$	6.90211	0.414708	0.207354	−	0.978266i	$-0.433515\pi$
$277$	6.90211	0.414708	0.207354	+	0.978266i	$0.433515\pi$
$278$	−19.0623	−1.14328
$279$	19.8157	1.18633
$280$	1.93064	0.115378
$281$	9.38062	0.559601	0.279801	−	0.960058i	$-0.409732\pi$
$281$	9.38062	0.559601	0.279801	+	0.960058i	$0.409732\pi$
$282$	0.780493	0.0464777
$283$	20.5769	1.22317	0.611584	−	0.791179i	$-0.290532\pi$
$283$	20.5769	1.22317	0.611584	+	0.791179i	$0.290532\pi$
$284$	0.323460	0.0191938
$285$	−0.255283	−0.0151216
$286$	−39.3465	−2.32661
$287$	0.749834	0.0442613
$288$	1.16557	0.0686816
$289$	0	0
$290$	−9.35848	−0.549549
$291$	0.296837	0.0174009
$292$	−0.379382	−0.0222017
$293$	−23.4539	−1.37019	−0.685097	−	0.728452i	$-0.740240\pi$
$293$	−23.4539	−1.37019	−0.685097	+	0.728452i	$0.740240\pi$
$294$	−1.03025	−0.0600853
$295$	−9.91829	−0.577465
$296$	0.329488	0.0191511
$297$	3.19243	0.185244
$298$	−25.1062	−1.45436
$299$	−26.1475	−1.51215
$300$	0.00758244	0.000437772 0
$301$	0.445624	0.0256854
$302$	−16.2129	−0.932948
$303$	1.57194	0.0903056
$304$	9.60031	0.550616
$305$	−1.60292	−0.0917832
$306$	0	0
$307$	30.4260	1.73650	0.868251	−	0.496126i	$-0.165244\pi$
$307$	30.4260	1.73650	0.868251	+	0.496126i	$0.165244\pi$
$308$	0.232618	0.0132546
$309$	−1.19758	−0.0681280
$310$	−9.53937	−0.541800
$311$	1.07989	0.0612348	0.0306174	−	0.999531i	$-0.490253\pi$
$311$	1.07989	0.0612348	0.0306174	+	0.999531i	$0.490253\pi$
$312$	−1.72145	−0.0974580
$313$	3.62554	0.204928	0.102464	−	0.994737i	$-0.467327\pi$
$313$	3.62554	0.204928	0.102464	+	0.994737i	$0.467327\pi$
$314$	−8.83955	−0.498845
$315$	2.07686	0.117018
$316$	−1.02408	−0.0576090
$317$	−8.79483	−0.493967	−0.246984	−	0.969020i	$-0.579439\pi$
$317$	−8.79483	−0.493967	−0.246984	+	0.969020i	$0.579439\pi$
$318$	1.89492	0.106262
$319$	31.5607	1.76706
$320$	7.70533	0.430741
$321$	1.82746	0.101999
$322$	4.63597	0.258353
$323$	0	0
$324$	0.613417	0.0340787
$325$	5.63906	0.312799
$326$	−16.1117	−0.892347
$327$	1.16500	0.0644246
$328$	2.99634	0.165445
$329$	−3.43165	−0.189193
$330$	−0.766875	−0.0422151
$331$	35.1634	1.93275	0.966377	−	0.257128i	$-0.0827760\pi$
$331$	35.1634	1.93275	0.966377	+	0.257128i	$0.0827760\pi$
$332$	0.347368	0.0190643
$333$	0.354442	0.0194233
$334$	−8.20754	−0.449097
$335$	−2.99411	−0.163586
$336$	0.315755	0.0172259
$337$	−31.5152	−1.71674	−0.858370	−	0.513031i	$-0.828523\pi$
$337$	−31.5152	−1.71674	−0.858370	+	0.513031i	$0.828523\pi$
$338$	27.0404	1.47080
$339$	1.26829	0.0688839
$340$	0	0
$341$	32.1708	1.74214
$342$	9.98263	0.539799
$343$	9.39536	0.507302
$344$	1.78072	0.0960098
$345$	−0.509622	−0.0274371
$346$	6.52679	0.350882
$347$	−0.440312	−0.0236372	−0.0118186	−	0.999930i	$-0.503762\pi$
$347$	−0.440312	−0.0236372	−0.0118186	+	0.999930i	$0.503762\pi$
$348$	−0.0493327	−0.00264451
$349$	−12.4222	−0.664944	−0.332472	−	0.943113i	$-0.607883\pi$
$349$	−12.4222	−0.664944	−0.332472	+	0.943113i	$0.607883\pi$
$350$	−0.999809	−0.0534420
$351$	−3.71114	−0.198086
$352$	1.89229	0.100860
$353$	−7.38055	−0.392827	−0.196414	−	0.980521i	$-0.562929\pi$
$353$	−7.38055	−0.392827	−0.196414	+	0.980521i	$0.562929\pi$
$354$	−1.56798	−0.0833372
$355$	4.68852	0.248841
$356$	−0.162579	−0.00861668
$357$	0	0
$358$	−26.4314	−1.39694
$359$	−7.05177	−0.372178	−0.186089	−	0.982533i	$-0.559581\pi$
$359$	−7.05177	−0.372178	−0.186089	+	0.982533i	$0.559581\pi$
$360$	8.29913	0.437403
$361$	−13.6050	−0.716051
$362$	−15.4171	−0.810303
$363$	1.37725	0.0722869
$364$	−0.270414	−0.0141735
$365$	−5.49911	−0.287837
$366$	−0.253406	−0.0132457
$367$	16.3167	0.851725	0.425862	−	0.904788i	$-0.359971\pi$
$367$	16.3167	0.851725	0.425862	+	0.904788i	$0.359971\pi$
$368$	19.1652	0.999053
$369$	3.22327	0.167797
$370$	−0.170630	−0.00887063
$371$	−8.33155	−0.432552
$372$	−0.0502862	−0.00260722
$373$	−18.3821	−0.951787	−0.475893	−	0.879503i	$-0.657875\pi$
$373$	−18.3821	−0.951787	−0.475893	+	0.879503i	$0.657875\pi$
$374$	0	0
$375$	0.109907	0.00567556
$376$	−13.7129	−0.707189
$377$	−36.6887	−1.88957
$378$	0.657987	0.0338432
$379$	−16.4955	−0.847315	−0.423657	−	0.905822i	$-0.639254\pi$
$379$	−16.4955	−0.847315	−0.423657	+	0.905822i	$0.639254\pi$
$380$	−0.160244	−0.00822033
$381$	−0.0547928	−0.00280712
$382$	26.6675	1.36443
$383$	−36.3426	−1.85702	−0.928510	−	0.371307i	$-0.878910\pi$
$383$	−36.3426	−1.85702	−0.928510	+	0.371307i	$0.878910\pi$
$384$	1.30388	0.0665384
$385$	3.37178	0.171842
$386$	−14.2199	−0.723772
$387$	1.91558	0.0973744
$388$	0.186328	0.00945935
$389$	6.36837	0.322889	0.161445	−	0.986882i	$-0.448385\pi$
$389$	6.36837	0.322889	0.161445	+	0.986882i	$0.448385\pi$
$390$	0.891477	0.0451417
$391$	0	0
$392$	18.1010	0.914237
$393$	2.14693	0.108298
$394$	−20.0705	−1.01114
$395$	−14.8439	−0.746880
$396$	0.999941	0.0502489
$397$	33.3778	1.67518	0.837591	−	0.546298i	$-0.183963\pi$
$397$	33.3778	1.67518	0.837591	+	0.546298i	$0.183963\pi$
$398$	−18.3203	−0.918316
$399$	0.177443	0.00888327
$400$	−4.13322	−0.206661
$401$	11.7500	0.586769	0.293384	−	0.955995i	$-0.405218\pi$
$401$	11.7500	0.586769	0.293384	+	0.955995i	$0.405218\pi$
$402$	−0.473338	−0.0236080
$403$	−37.3979	−1.86292
$404$	0.986724	0.0490914
$405$	8.89143	0.441819
$406$	6.50494	0.322835
$407$	0.575436	0.0285233
$408$	0	0
$409$	12.6834	0.627154	0.313577	−	0.949563i	$-0.398473\pi$
$409$	12.6834	0.627154	0.313577	+	0.949563i	$0.398473\pi$
$410$	−1.55170	−0.0766328
$411$	−0.419541	−0.0206944
$412$	−0.751735	−0.0370353
$413$	6.89405	0.339234
$414$	19.9284	0.979427
$415$	5.03506	0.247162
$416$	−2.19975	−0.107852
$417$	−1.45654	−0.0713269
$418$	16.2068	0.792700
$419$	35.0224	1.71095	0.855477	−	0.517840i	$-0.173264\pi$
$419$	35.0224	1.71095	0.855477	+	0.517840i	$0.173264\pi$
$420$	−0.00527044	−0.000257171 0
$421$	−10.0231	−0.488497	−0.244248	−	0.969713i	$-0.578541\pi$
$421$	−10.0231	−0.488497	−0.244248	+	0.969713i	$0.578541\pi$
$422$	18.5234	0.901706
$423$	−14.7514	−0.717240
$424$	−33.2929	−1.61685
$425$	0	0
$426$	0.741207	0.0359116
$427$	1.11417	0.0539184
$428$	1.14711	0.0554479
$429$	−3.00643	−0.145152
$430$	−0.922169	−0.0444709
$431$	−23.7461	−1.14381	−0.571905	−	0.820320i	$-0.693795\pi$
$431$	−23.7461	−1.14381	−0.571905	+	0.820320i	$0.693795\pi$
$432$	2.72013	0.130872
$433$	−32.2917	−1.55184	−0.775919	−	0.630832i	$-0.782714\pi$
$433$	−32.2917	−1.55184	−0.775919	+	0.630832i	$0.782714\pi$
$434$	6.63067	0.318282
$435$	−0.715073	−0.0342851
$436$	0.731283	0.0350221
$437$	10.7701	0.515205
$438$	−0.869353	−0.0415393
$439$	−21.3269	−1.01788	−0.508939	−	0.860803i	$-0.669962\pi$
$439$	−21.3269	−1.01788	−0.508939	+	0.860803i	$0.669962\pi$
$440$	13.4736	0.642330
$441$	19.4719	0.927231
$442$	0	0
$443$	−13.8187	−0.656546	−0.328273	−	0.944583i	$-0.606467\pi$
$443$	−13.8187	−0.656546	−0.328273	+	0.944583i	$0.606467\pi$
$444$	−0.000899466 0	−4.26868e−5 0
$445$	−2.35657	−0.111712
$446$	29.3732	1.39086
$447$	−1.91834	−0.0907345
$448$	−5.35586	−0.253040
$449$	−6.61380	−0.312125	−0.156062	−	0.987747i	$-0.549880\pi$
$449$	−6.61380	−0.312125	−0.156062	+	0.987747i	$0.549880\pi$
$450$	−4.29782	−0.202601
$451$	5.23297	0.246411
$452$	0.796118	0.0374462
$453$	−1.23881	−0.0582046
$454$	−1.75526	−0.0823786
$455$	−3.91962	−0.183755
$456$	0.709063	0.0332049
$457$	5.01039	0.234376	0.117188	−	0.993110i	$-0.462612\pi$
$457$	5.01039	0.234376	0.117188	+	0.993110i	$0.462612\pi$
$458$	5.06236	0.236548
$459$	0	0
$460$	−0.319896	−0.0149152
$461$	16.3822	0.762994	0.381497	−	0.924370i	$-0.375409\pi$
$461$	16.3822	0.762994	0.381497	+	0.924370i	$0.375409\pi$
$462$	0.533043	0.0247994
$463$	−27.3768	−1.27231	−0.636153	−	0.771563i	$-0.719475\pi$
$463$	−27.3768	−1.27231	−0.636153	+	0.771563i	$0.719475\pi$
$464$	26.8915	1.24841
$465$	−0.728895	−0.0338017
$466$	41.7172	1.93251
$467$	−0.646341	−0.0299091	−0.0149545	−	0.999888i	$-0.504760\pi$
$467$	−0.646341	−0.0299091	−0.0149545	+	0.999888i	$0.504760\pi$
$468$	−1.16241	−0.0537325
$469$	2.08116	0.0960991
$470$	7.10141	0.327564
$471$	−0.675423	−0.0311218
$472$	27.5487	1.26803
$473$	3.10994	0.142995
$474$	−2.34668	−0.107786
$475$	−2.32272	−0.106574
$476$	0	0
$477$	−35.8144	−1.63983
$478$	11.3221	0.517861
$479$	13.5649	0.619795	0.309898	−	0.950770i	$-0.399705\pi$
$479$	13.5649	0.619795	0.309898	+	0.950770i	$0.399705\pi$
$480$	−0.0428738	−0.00195691
$481$	−0.668933	−0.0305007
$482$	−16.7242	−0.761767
$483$	0.354231	0.0161181
$484$	0.864516	0.0392962
$485$	2.70080	0.122637
$486$	4.24553	0.192581
$487$	−29.0339	−1.31565	−0.657826	−	0.753170i	$-0.728524\pi$
$487$	−29.0339	−1.31565	−0.657826	+	0.753170i	$0.728524\pi$
$488$	4.45222	0.201543
$489$	−1.23108	−0.0556716
$490$	−9.37384	−0.423467
$491$	13.1298	0.592541	0.296271	−	0.955104i	$-0.404257\pi$
$491$	13.1298	0.592541	0.296271	+	0.955104i	$0.404257\pi$
$492$	−0.00817968	−0.000368768 0
$493$	0	0
$494$	−18.8401	−0.847655
$495$	14.4941	0.651460
$496$	27.4113	1.23080
$497$	−3.25892	−0.146183
$498$	0.795992	0.0356692
$499$	−11.3410	−0.507691	−0.253846	−	0.967245i	$-0.581695\pi$
$499$	−11.3410	−0.507691	−0.253846	+	0.967245i	$0.581695\pi$
$500$	0.0689897	0.00308531
$501$	−0.627131	−0.0280182
$502$	−11.4308	−0.510183
$503$	−20.5436	−0.915995	−0.457998	−	0.888953i	$-0.651433\pi$
$503$	−20.5436	−0.915995	−0.457998	+	0.888953i	$0.651433\pi$
$504$	−5.76860	−0.256954
$505$	14.3025	0.636452
$506$	32.3537	1.43830
$507$	2.06613	0.0917601
$508$	−0.0343940	−0.00152599
$509$	8.40900	0.372722	0.186361	−	0.982481i	$-0.440331\pi$
$509$	8.40900	0.372722	0.186361	+	0.982481i	$0.440331\pi$
$510$	0	0
$511$	3.82235	0.169091
$512$	−21.3482	−0.943467
$513$	1.52861	0.0674899
$514$	−11.4026	−0.502946
$515$	−10.8963	−0.480150
$516$	−0.00486116	−0.000214001 0
$517$	−23.9489	−1.05327
$518$	0.118602	0.00521108
$519$	0.498706	0.0218908
$520$	−15.6628	−0.686860
$521$	−21.3561	−0.935627	−0.467813	−	0.883827i	$-0.654958\pi$
$521$	−21.3561	−0.935627	−0.467813	+	0.883827i	$0.654958\pi$
$522$	27.9624	1.22388
$523$	−24.4644	−1.06975	−0.534877	−	0.844930i	$-0.679642\pi$
$523$	−24.4644	−1.06975	−0.534877	+	0.844930i	$0.679642\pi$
$524$	1.34765	0.0588724
$525$	−0.0763945	−0.00333413
$526$	12.2785	0.535367
$527$	0	0
$528$	2.20361	0.0958996
$529$	−1.49953	−0.0651969
$530$	17.2412	0.748909
$531$	29.6351	1.28605
$532$	0.111383	0.00482907
$533$	−6.08323	−0.263494
$534$	−0.372550	−0.0161218
$535$	16.6273	0.718862
$536$	8.31633	0.359211
$537$	−2.01960	−0.0871521
$538$	−22.9897	−0.991155
$539$	31.6125	1.36165
$540$	−0.0454030	−0.00195384
$541$	1.92406	0.0827219	0.0413610	−	0.999144i	$-0.486831\pi$
$541$	1.92406	0.0827219	0.0413610	+	0.999144i	$0.486831\pi$
$542$	−32.4055	−1.39194
$543$	−1.17800	−0.0505530
$544$	0	0
$545$	10.5999	0.454049
$546$	−0.619652	−0.0265186
$547$	−14.7222	−0.629475	−0.314737	−	0.949179i	$-0.601916\pi$
$547$	−14.7222	−0.629475	−0.314737	+	0.949179i	$0.601916\pi$
$548$	−0.263350	−0.0112498
$549$	4.78941	0.204407
$550$	−6.97750	−0.297522
$551$	15.1120	0.643794
$552$	1.41551	0.0602480
$553$	10.3178	0.438757
$554$	9.92798	0.421800
$555$	−0.0130377	−0.000553419 0
$556$	−0.914285	−0.0387743
$557$	−24.3617	−1.03224	−0.516119	−	0.856517i	$-0.672624\pi$
$557$	−24.3617	−1.03224	−0.516119	+	0.856517i	$0.672624\pi$
$558$	28.5029	1.20662
$559$	−3.61524	−0.152909
$560$	2.87294	0.121404
$561$	0	0
$562$	13.4931	0.569171
$563$	−30.7200	−1.29470	−0.647348	−	0.762195i	$-0.724122\pi$
$563$	−30.7200	−1.29470	−0.647348	+	0.762195i	$0.724122\pi$
$564$	0.0374347	0.00157628
$565$	11.5397	0.485477
$566$	29.5978	1.24409
$567$	−6.18030	−0.259548
$568$	−13.0227	−0.546419
$569$	5.35732	0.224590	0.112295	−	0.993675i	$-0.464180\pi$
$569$	5.35732	0.224590	0.112295	+	0.993675i	$0.464180\pi$
$570$	−0.367198	−0.0153802
$571$	−36.0360	−1.50806	−0.754031	−	0.656839i	$-0.771893\pi$
$571$	−36.0360	−1.50806	−0.754031	+	0.656839i	$0.771893\pi$
$572$	−1.88717	−0.0789067
$573$	2.03764	0.0851235
$574$	1.07856	0.0450182
$575$	−4.63686	−0.193370
$576$	−23.0229	−0.959288
$577$	−39.5472	−1.64637	−0.823186	−	0.567772i	$-0.807806\pi$
$577$	−39.5472	−1.64637	−0.823186	+	0.567772i	$0.807806\pi$
$578$	0	0
$579$	−1.08653	−0.0451546
$580$	−0.448860	−0.0186379
$581$	−3.49979	−0.145196
$582$	0.426969	0.0176984
$583$	−58.1446	−2.40810
$584$	15.2741	0.632048
$585$	−16.8490	−0.696622
$586$	−33.7361	−1.39363
$587$	17.9282	0.739975	0.369988	−	0.929037i	$-0.379362\pi$
$587$	17.9282	0.739975	0.369988	+	0.929037i	$0.379362\pi$
$588$	−0.0494137	−0.00203778
$589$	15.4041	0.634716
$590$	−14.2665	−0.587341
$591$	−1.53357	−0.0630827
$592$	0.490303	0.0201513
$593$	−15.2975	−0.628192	−0.314096	−	0.949391i	$-0.601701\pi$
$593$	−15.2975	−0.628192	−0.314096	+	0.949391i	$0.601701\pi$
$594$	4.59199	0.188412
$595$	0	0
$596$	−1.20416	−0.0493245
$597$	−1.39984	−0.0572917
$598$	−37.6105	−1.53801
$599$	34.6498	1.41575	0.707877	−	0.706336i	$-0.249653\pi$
$599$	34.6498	1.41575	0.707877	+	0.706336i	$0.249653\pi$
$600$	−0.305273	−0.0124627
$601$	16.0361	0.654126	0.327063	−	0.945003i	$-0.393941\pi$
$601$	16.0361	0.654126	0.327063	+	0.945003i	$0.393941\pi$
$602$	0.640985	0.0261246
$603$	8.94617	0.364316
$604$	−0.777618	−0.0316408
$605$	12.5311	0.509461
$606$	2.26108	0.0918499
$607$	−36.5119	−1.48197	−0.740986	−	0.671521i	$-0.765641\pi$
$607$	−36.5119	−1.48197	−0.740986	+	0.671521i	$0.765641\pi$
$608$	0.906076	0.0367462
$609$	0.497037	0.0201409
$610$	−2.30564	−0.0933528
$611$	27.8402	1.12629
$612$	0	0
$613$	33.2758	1.34400	0.671998	−	0.740553i	$-0.265436\pi$
$613$	33.2758	1.34400	0.671998	+	0.740553i	$0.265436\pi$
$614$	43.7647	1.76620
$615$	−0.118564	−0.00478095
$616$	−9.36532	−0.377339
$617$	31.7084	1.27653	0.638267	−	0.769815i	$-0.279652\pi$
$617$	31.7084	1.27653	0.638267	+	0.769815i	$0.279652\pi$
$618$	−1.72260	−0.0692931
$619$	−40.8302	−1.64110	−0.820552	−	0.571572i	$-0.806334\pi$
$619$	−40.8302	−1.64110	−0.820552	+	0.571572i	$0.806334\pi$
$620$	−0.457535	−0.0183751
$621$	3.05158	0.122456
$622$	1.55331	0.0622820
$623$	1.63802	0.0656258
$624$	−2.56165	−0.102548
$625$	1.00000	0.0400000
$626$	5.21497	0.208432
$627$	1.23835	0.0494548
$628$	−0.423970	−0.0169183
$629$	0	0
$630$	2.98735	0.119019
$631$	−15.8001	−0.628992	−0.314496	−	0.949259i	$-0.601835\pi$
$631$	−15.8001	−0.628992	−0.314496	+	0.949259i	$0.601835\pi$
$632$	41.2300	1.64004
$633$	1.41536	0.0562555
$634$	−12.6505	−0.502415
$635$	−0.498539	−0.0197839
$636$	0.0908860	0.00360386
$637$	−36.7489	−1.45605
$638$	45.3969	1.79728
$639$	−14.0089	−0.554185
$640$	11.8635	0.468947
$641$	−24.9339	−0.984831	−0.492415	−	0.870360i	$-0.663886\pi$
$641$	−24.9339	−0.984831	−0.492415	+	0.870360i	$0.663886\pi$
$642$	2.62861	0.103743
$643$	24.0161	0.947103	0.473552	−	0.880766i	$-0.342972\pi$
$643$	24.0161	0.947103	0.473552	+	0.880766i	$0.342972\pi$
$644$	0.222355	0.00876200
$645$	−0.0704621	−0.00277444
$646$	0	0
$647$	−23.8307	−0.936882	−0.468441	−	0.883495i	$-0.655184\pi$
$647$	−23.8307	−0.936882	−0.468441	+	0.883495i	$0.655184\pi$
$648$	−24.6965	−0.970170
$649$	48.1125	1.88858
$650$	8.11121	0.318148
$651$	0.506644	0.0198569
$652$	−0.772766	−0.0302638
$653$	16.5191	0.646441	0.323221	−	0.946324i	$-0.395234\pi$
$653$	16.5191	0.646441	0.323221	+	0.946324i	$0.395234\pi$
$654$	1.67573	0.0655264
$655$	19.5341	0.763260
$656$	4.45878	0.174086
$657$	16.4309	0.641031
$658$	−4.93608	−0.192428
$659$	10.2510	0.399324	0.199662	−	0.979865i	$-0.436016\pi$
$659$	10.2510	0.399324	0.199662	+	0.979865i	$0.436016\pi$
$660$	−0.0367815	−0.00143172
$661$	−21.3416	−0.830090	−0.415045	−	0.909801i	$-0.636234\pi$
$661$	−21.3416	−0.830090	−0.415045	+	0.909801i	$0.636234\pi$
$662$	50.5790	1.96581
$663$	0	0
$664$	−13.9852	−0.542731
$665$	1.61449	0.0626071
$666$	0.509829	0.0197555
$667$	30.1683	1.16812
$668$	−0.393657	−0.0152311
$669$	2.24438	0.0867728
$670$	−4.30672	−0.166383
$671$	7.77560	0.300174
$672$	0.0298009	0.00114960
$673$	27.5159	1.06066	0.530331	−	0.847791i	$-0.322068\pi$
$673$	27.5159	1.06066	0.530331	+	0.847791i	$0.322068\pi$
$674$	−45.3314	−1.74610
$675$	−0.658113	−0.0253308
$676$	1.29693	0.0498821
$677$	17.0437	0.655042	0.327521	−	0.944844i	$-0.393787\pi$
$677$	17.0437	0.655042	0.327521	+	0.944844i	$0.393787\pi$
$678$	1.82430	0.0700619
$679$	−1.87729	−0.0720437
$680$	0	0
$681$	−0.134118	−0.00513942
$682$	46.2744	1.77194
$683$	3.12281	0.119491	0.0597456	−	0.998214i	$-0.480971\pi$
$683$	3.12281	0.119491	0.0597456	+	0.998214i	$0.480971\pi$
$684$	0.478796	0.0183072
$685$	−3.81724	−0.145849
$686$	13.5143	0.515977
$687$	0.386810	0.0147577
$688$	2.64984	0.101024
$689$	67.5919	2.57505
$690$	−0.733040	−0.0279064
$691$	−36.1278	−1.37437	−0.687184	−	0.726484i	$-0.741153\pi$
$691$	−36.1278	−1.37437	−0.687184	+	0.726484i	$0.741153\pi$
$692$	0.313044	0.0119001
$693$	−10.0746	−0.382702
$694$	−0.633344	−0.0240414
$695$	−13.2525	−0.502695
$696$	1.98616	0.0752852
$697$	0	0
$698$	−17.8680	−0.676315
$699$	3.18758	0.120565
$700$	−0.0479537	−0.00181248
$701$	−18.1677	−0.686185	−0.343093	−	0.939302i	$-0.611474\pi$
$701$	−18.1677	−0.686185	−0.343093	+	0.939302i	$0.611474\pi$
$702$	−5.33809	−0.201473
$703$	0.275533	0.0103919
$704$	−37.3777	−1.40872
$705$	0.542613	0.0204360
$706$	−10.6162	−0.399545
$707$	−9.94144	−0.373886
$708$	−0.0752048	−0.00282637
$709$	−11.6045	−0.435817	−0.217909	−	0.975969i	$-0.569923\pi$
$709$	−11.6045	−0.435817	−0.217909	+	0.975969i	$0.569923\pi$
$710$	6.74396	0.253096
$711$	44.3525	1.66335
$712$	6.54553	0.245304
$713$	30.7514	1.15165
$714$	0	0
$715$	−27.3544	−1.02300
$716$	−1.26772	−0.0473771
$717$	0.865112	0.0323082
$718$	−10.1433	−0.378543
$719$	−25.7861	−0.961658	−0.480829	−	0.876814i	$-0.659664\pi$
$719$	−25.7861	−0.961658	−0.480829	+	0.876814i	$0.659664\pi$
$720$	12.3497	0.460247
$721$	7.57388	0.282066
$722$	−19.5694	−0.728296
$723$	−1.27788	−0.0475249
$724$	−0.739447	−0.0274813
$725$	−6.50618	−0.241634
$726$	1.98104	0.0735231
$727$	17.7430	0.658051	0.329025	−	0.944321i	$-0.393280\pi$
$727$	17.7430	0.658051	0.329025	+	0.944321i	$0.393280\pi$
$728$	10.8870	0.403499
$729$	−26.3499	−0.975922
$730$	−7.90991	−0.292759
$731$	0	0
$732$	−0.0121541	−0.000449227 0
$733$	−26.8221	−0.990695	−0.495348	−	0.868695i	$-0.664959\pi$
$733$	−26.8221	−0.990695	−0.495348	+	0.868695i	$0.664959\pi$
$734$	23.4699	0.866290
$735$	−0.716247	−0.0264192
$736$	1.80880	0.0666734
$737$	14.5241	0.535002
$738$	4.63634	0.170666
$739$	0.451458	0.0166071	0.00830357	−	0.999966i	$-0.497357\pi$
$739$	0.451458	0.0166071	0.00830357	+	0.999966i	$0.497357\pi$
$740$	−0.00818390	−0.000300846 0
$741$	−1.43955	−0.0528833
$742$	−11.9841	−0.439950
$743$	−20.6466	−0.757450	−0.378725	−	0.925509i	$-0.623637\pi$
$743$	−20.6466	−0.757450	−0.378725	+	0.925509i	$0.623637\pi$
$744$	2.02455	0.0742236
$745$	−17.4543	−0.639475
$746$	−26.4407	−0.968063
$747$	−15.0444	−0.550445
$748$	0	0
$749$	−11.5574	−0.422298
$750$	0.158090	0.00577262
$751$	−26.7059	−0.974514	−0.487257	−	0.873259i	$-0.662002\pi$
$751$	−26.7059	−0.974514	−0.487257	+	0.873259i	$0.662002\pi$
$752$	−20.4058	−0.744123
$753$	−0.873420	−0.0318292
$754$	−52.7730	−1.92188
$755$	−11.2715	−0.410212
$756$	0.0315589	0.00114779
$757$	19.3283	0.702499	0.351250	−	0.936282i	$-0.385757\pi$
$757$	19.3283	0.702499	0.351250	+	0.936282i	$0.385757\pi$
$758$	−23.7270	−0.861805
$759$	2.47212	0.0897322
$760$	6.45150	0.234020
$761$	−9.52382	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$761$	−9.52382	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$762$	−0.0788138	−0.00285512
$763$	−7.36782	−0.266733
$764$	1.27905	0.0462743
$765$	0	0
$766$	−52.2751	−1.88878
$767$	−55.9298	−2.01951
$768$	0.181764	0.00655886
$769$	21.8393	0.787544	0.393772	−	0.919208i	$-0.371170\pi$
$769$	21.8393	0.787544	0.393772	+	0.919208i	$0.371170\pi$
$770$	4.84996	0.174780
$771$	−0.871260	−0.0313777
$772$	−0.682026	−0.0245466
$773$	−18.2320	−0.655760	−0.327880	−	0.944719i	$-0.606334\pi$
$773$	−18.2320	−0.655760	−0.327880	+	0.944719i	$0.606334\pi$
$774$	2.75537	0.0990396
$775$	−6.63194	−0.238226
$776$	−7.50165	−0.269294
$777$	0.00906229	0.000325108 0
$778$	9.16025	0.328411
$779$	2.50567	0.0897751
$780$	0.0427578	0.00153097
$781$	−22.7435	−0.813826
$782$	0	0
$783$	4.28180	0.153019
$784$	26.9356	0.961986
$785$	−6.14541	−0.219339
$786$	3.08814	0.110150
$787$	10.0983	0.359967	0.179983	−	0.983670i	$-0.442396\pi$
$787$	10.0983	0.359967	0.179983	+	0.983670i	$0.442396\pi$
$788$	−0.962641	−0.0342927
$789$	0.938188	0.0334004
$790$	−21.3515	−0.759653
$791$	−8.02104	−0.285195
$792$	−40.2581	−1.43051
$793$	−9.03898	−0.320984
$794$	48.0105	1.70383
$795$	1.31738	0.0467228
$796$	−0.878696	−0.0311446
$797$	−30.2149	−1.07027	−0.535133	−	0.844768i	$-0.679738\pi$
$797$	−30.2149	−1.07027	−0.535133	+	0.844768i	$0.679738\pi$
$798$	0.255234	0.00903518
$799$	0	0
$800$	−0.390093	−0.0137919
$801$	7.04125	0.248790
$802$	16.9012	0.596803
$803$	26.6756	0.941360
$804$	−0.0227027	−0.000800661 0
$805$	3.22301	0.113596
$806$	−53.7930	−1.89478
$807$	−1.75662	−0.0618360
$808$	−39.7260	−1.39756
$809$	−25.8073	−0.907337	−0.453669	−	0.891170i	$-0.649885\pi$
$809$	−25.8073	−0.907337	−0.453669	+	0.891170i	$0.649885\pi$
$810$	12.7894	0.449374
$811$	47.1759	1.65657	0.828284	−	0.560308i	$-0.189317\pi$
$811$	47.1759	1.65657	0.828284	+	0.560308i	$0.189317\pi$
$812$	0.311995	0.0109489
$813$	−2.47608	−0.0868399
$814$	0.827706	0.0290111
$815$	−11.2012	−0.392360
$816$	0	0
$817$	1.48911	0.0520975
$818$	18.2438	0.637880
$819$	11.7115	0.409233
$820$	−0.0744238	−0.00259899
$821$	−34.8152	−1.21506	−0.607530	−	0.794297i	$-0.707839\pi$
$821$	−34.8152	−1.21506	−0.607530	+	0.794297i	$0.707839\pi$
$822$	−0.603467	−0.0210483
$823$	11.9291	0.415823	0.207912	−	0.978148i	$-0.433333\pi$
$823$	11.9291	0.415823	0.207912	+	0.978148i	$0.433333\pi$
$824$	30.2652	1.05434
$825$	−0.533145	−0.0185617
$826$	9.91639	0.345035
$827$	−7.58265	−0.263674	−0.131837	−	0.991271i	$-0.542088\pi$
$827$	−7.58265	−0.263674	−0.131837	+	0.991271i	$0.542088\pi$
$828$	0.955823	0.0332171
$829$	24.5064	0.851141	0.425570	−	0.904925i	$-0.360073\pi$
$829$	24.5064	0.851141	0.425570	+	0.904925i	$0.360073\pi$
$830$	7.24243	0.251388
$831$	0.758589	0.0263152
$832$	43.4508	1.50639
$833$	0	0
$834$	−2.09508	−0.0725467
$835$	−5.70603	−0.197465
$836$	0.777324	0.0268843
$837$	4.36456	0.150861
$838$	50.3761	1.74021
$839$	13.9747	0.482462	0.241231	−	0.970468i	$-0.422449\pi$
$839$	13.9747	0.482462	0.241231	+	0.970468i	$0.422449\pi$
$840$	0.212190	0.00732127
$841$	13.3304	0.459669
$842$	−14.4172	−0.496850
$843$	1.03099	0.0355093
$844$	0.888437	0.0305813
$845$	18.7989	0.646703
$846$	−21.2184	−0.729505
$847$	−8.71017	−0.299285
$848$	−49.5423	−1.70129
$849$	2.26154	0.0776158
$850$	0	0
$851$	0.550047	0.0188554
$852$	0.0355504	0.00121794
$853$	7.40009	0.253374	0.126687	−	0.991943i	$-0.459566\pi$
$853$	7.40009	0.253374	0.126687	+	0.991943i	$0.459566\pi$
$854$	1.60262	0.0548404
$855$	6.94010	0.237347
$856$	−46.1834	−1.57852
$857$	48.3185	1.65053	0.825264	−	0.564748i	$-0.191027\pi$
$857$	48.3185	1.65053	0.825264	+	0.564748i	$0.191027\pi$
$858$	−4.32445	−0.147634
$859$	−5.21085	−0.177792	−0.0888959	−	0.996041i	$-0.528334\pi$
$859$	−5.21085	−0.177792	−0.0888959	+	0.996041i	$0.528334\pi$
$860$	−0.0442299	−0.00150823
$861$	0.0824119	0.00280859
$862$	−34.1563	−1.16337
$863$	7.90307	0.269024	0.134512	−	0.990912i	$-0.457053\pi$
$863$	7.90307	0.269024	0.134512	+	0.990912i	$0.457053\pi$
$864$	0.256725	0.00873396
$865$	4.53754	0.154281
$866$	−46.4483	−1.57838
$867$	0	0
$868$	0.318026	0.0107945
$869$	72.0063	2.44265
$870$	−1.02856	−0.0348715
$871$	−16.8840	−0.572091
$872$	−29.4419	−0.997027
$873$	−8.06979	−0.273121
$874$	15.4917	0.524016
$875$	−0.695085	−0.0234982
$876$	−0.0416967	−0.00140880
$877$	27.5064	0.928824	0.464412	−	0.885619i	$-0.346266\pi$
$877$	27.5064	0.928824	0.464412	+	0.885619i	$0.346266\pi$
$878$	−30.6766	−1.03528
$879$	−2.57775	−0.0869452
$880$	20.0498	0.675878
$881$	19.7314	0.664769	0.332384	−	0.943144i	$-0.392147\pi$
$881$	19.7314	0.664769	0.332384	+	0.943144i	$0.392147\pi$
$882$	28.0083	0.943088
$883$	21.8566	0.735534	0.367767	−	0.929918i	$-0.380122\pi$
$883$	21.8566	0.735534	0.367767	+	0.929918i	$0.380122\pi$
$884$	0	0
$885$	−1.09009	−0.0366429
$886$	−19.8768	−0.667774
$887$	−2.77874	−0.0933011	−0.0466505	−	0.998911i	$-0.514855\pi$
$887$	−2.77874	−0.0933011	−0.0466505	+	0.998911i	$0.514855\pi$
$888$	0.0362130	0.00121523
$889$	0.346526	0.0116221
$890$	−3.38969	−0.113623
$891$	−43.1313	−1.44495
$892$	1.40882	0.0471709
$893$	−11.4673	−0.383740
$894$	−2.75934	−0.0922861
$895$	−18.3756	−0.614227
$896$	−8.24615	−0.275485
$897$	−2.87379	−0.0959530
$898$	−9.51328	−0.317462
$899$	43.1486	1.43909
$900$	−0.206136	−0.00687119
$901$	0	0
$902$	7.52710	0.250625
$903$	0.0489771	0.00162986
$904$	−32.0521	−1.06604
$905$	−10.7182	−0.356286
$906$	−1.78191	−0.0592000
$907$	−42.4769	−1.41042	−0.705210	−	0.708998i	$-0.749147\pi$
$907$	−42.4769	−1.41042	−0.705210	+	0.708998i	$0.749147\pi$
$908$	−0.0841875	−0.00279386
$909$	−42.7347	−1.41742
$910$	−5.63798	−0.186897
$911$	59.0680	1.95701	0.978505	−	0.206223i	$-0.0661171\pi$
$911$	59.0680	1.95701	0.978505	+	0.206223i	$0.0661171\pi$
$912$	1.05514	0.0349392
$913$	−24.4245	−0.808333
$914$	7.20694	0.238384
$915$	−0.176172	−0.00582407
$916$	0.242805	0.00802251
$917$	−13.5778	−0.448380
$918$	0	0
$919$	26.5657	0.876323	0.438161	−	0.898896i	$-0.355630\pi$
$919$	26.5657	0.876323	0.438161	+	0.898896i	$0.355630\pi$
$920$	12.8792	0.424614
$921$	3.34402	0.110189
$922$	23.5641	0.776042
$923$	26.4388	0.870245
$924$	0.0255663	0.000841069 0
$925$	−0.118625	−0.00390036
$926$	−39.3787	−1.29406
$927$	32.5574	1.06932
$928$	2.53801	0.0833144
$929$	38.9968	1.27944	0.639722	−	0.768606i	$-0.279049\pi$
$929$	38.9968	1.27944	0.639722	+	0.768606i	$0.279049\pi$
$930$	−1.04844	−0.0343797
$931$	15.1368	0.496090
$932$	2.00088	0.0655410
$933$	0.118687	0.00388563
$934$	−0.929695	−0.0304206
$935$	0	0
$936$	46.7993	1.52968
$937$	9.80514	0.320320	0.160160	−	0.987091i	$-0.448799\pi$
$937$	9.80514	0.320320	0.160160	+	0.987091i	$0.448799\pi$
$938$	2.99354	0.0977425
$939$	0.398471	0.0130036
$940$	0.340604	0.0111093
$941$	49.2240	1.60466	0.802328	−	0.596884i	$-0.203595\pi$
$941$	49.2240	1.60466	0.802328	+	0.596884i	$0.203595\pi$
$942$	−0.971527	−0.0316540
$943$	5.00209	0.162890
$944$	40.9945	1.33426
$945$	0.457444	0.0148807
$946$	4.47333	0.145441
$947$	−18.1975	−0.591339	−0.295669	−	0.955290i	$-0.595543\pi$
$947$	−18.1975	−0.591339	−0.295669	+	0.955290i	$0.595543\pi$
$948$	−0.112553	−0.00365556
$949$	−31.0098	−1.00662
$950$	−3.34100	−0.108396
$951$	−0.966612	−0.0313445
$952$	0	0
$953$	−5.23591	−0.169608	−0.0848039	−	0.996398i	$-0.527026\pi$
$953$	−5.23591	−0.169608	−0.0848039	+	0.996398i	$0.527026\pi$
$954$	−51.5153	−1.66787
$955$	18.5397	0.599930
$956$	0.543041	0.0175632
$957$	3.46874	0.112128
$958$	19.5117	0.630395
$959$	2.65331	0.0856797
$960$	0.846868	0.0273326
$961$	12.9826	0.418793
$962$	−0.962191	−0.0310223
$963$	−49.6811	−1.60095
$964$	−0.802141	−0.0258352
$965$	−9.88591	−0.318239
$966$	0.509525	0.0163937
$967$	3.19462	0.102732	0.0513661	−	0.998680i	$-0.483642\pi$
$967$	3.19462	0.102732	0.0513661	+	0.998680i	$0.483642\pi$
$968$	−34.8059	−1.11870
$969$	0	0
$970$	3.88483	0.124734
$971$	−27.2613	−0.874856	−0.437428	−	0.899253i	$-0.644110\pi$
$971$	−27.2613	−0.874856	−0.437428	+	0.899253i	$0.644110\pi$
$972$	0.203628	0.00653136
$973$	9.21160	0.295310
$974$	−41.7623	−1.33815
$975$	0.619770	0.0198485
$976$	6.62524	0.212069
$977$	8.00610	0.256138	0.128069	−	0.991765i	$-0.459122\pi$
$977$	8.00610	0.256138	0.128069	+	0.991765i	$0.459122\pi$
$978$	−1.77079	−0.0566236
$979$	11.4315	0.365351
$980$	−0.449596	−0.0143618
$981$	−31.6716	−1.01120
$982$	18.8859	0.602674
$983$	10.9464	0.349137	0.174569	−	0.984645i	$-0.444147\pi$
$983$	10.9464	0.349137	0.174569	+	0.984645i	$0.444147\pi$
$984$	0.329318	0.0104983
$985$	−13.9534	−0.444592
$986$	0	0
$987$	−0.377162	−0.0120052
$988$	−0.903624	−0.0287481
$989$	2.97273	0.0945273
$990$	20.8482	0.662600
$991$	−29.0834	−0.923863	−0.461932	−	0.886915i	$-0.652843\pi$
$991$	−29.0834	−0.923863	−0.461932	+	0.886915i	$0.652843\pi$
$992$	2.58707	0.0821395
$993$	3.86470	0.122642
$994$	−4.68762	−0.148682
$995$	−12.7366	−0.403778
$996$	0.0381780	0.00120972
$997$	13.6956	0.433745	0.216873	−	0.976200i	$-0.430414\pi$
$997$	13.6956	0.433745	0.216873	+	0.976200i	$0.430414\pi$
$998$	−16.3128	−0.516373
$999$	0.0780686	0.00246998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.a.p.1.10		12
5.4	even	2		7225.2.a.bs.1.3		12
17.4	even	4		1445.2.d.j.866.5		24
17.11	odd	16		85.2.l.a.36.5	yes	24
17.13	even	4		1445.2.d.j.866.6		24
17.14	odd	16		85.2.l.a.26.5	✓	24
17.16	even	2		1445.2.a.q.1.10		12
51.11	even	16		765.2.be.b.631.2		24
51.14	even	16		765.2.be.b.451.2		24
85.14	odd	16		425.2.m.b.26.2		24
85.28	even	16		425.2.n.f.274.5		24
85.48	even	16		425.2.n.c.349.2		24
85.62	even	16		425.2.n.c.274.2		24
85.79	odd	16		425.2.m.b.376.2		24
85.82	even	16		425.2.n.f.349.5		24
85.84	even	2		7225.2.a.bq.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.5	✓	24	17.14	odd	16
85.2.l.a.36.5	yes	24	17.11	odd	16
425.2.m.b.26.2		24	85.14	odd	16
425.2.m.b.376.2		24	85.79	odd	16
425.2.n.c.274.2		24	85.62	even	16
425.2.n.c.349.2		24	85.48	even	16
425.2.n.f.274.5		24	85.28	even	16
425.2.n.f.349.5		24	85.82	even	16
765.2.be.b.451.2		24	51.14	even	16
765.2.be.b.631.2		24	51.11	even	16
1445.2.a.p.1.10		12	1.1	even	1	trivial
1445.2.a.q.1.10		12	17.16	even	2
1445.2.d.j.866.5		24	17.4	even	4
1445.2.d.j.866.6		24	17.13	even	4
7225.2.a.bq.1.3		12	85.84	even	2
7225.2.a.bs.1.3		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)

\(f(q)\)	\(=\)	\(q+1.43840 q^{2} +0.109907 q^{3} +0.0689897 q^{4} +1.00000 q^{5} +0.158090 q^{6} -0.695085 q^{7} -2.77756 q^{8} -2.98792 q^{9} +1.43840 q^{10} -4.85089 q^{11} +0.00758244 q^{12} +5.63906 q^{13} -0.999809 q^{14} +0.109907 q^{15} -4.13322 q^{16} -4.29782 q^{18} -2.32272 q^{19} +0.0689897 q^{20} -0.0763945 q^{21} -6.97750 q^{22} -4.63686 q^{23} -0.305273 q^{24} +1.00000 q^{25} +8.11121 q^{26} -0.658113 q^{27} -0.0479537 q^{28} -6.50618 q^{29} +0.158090 q^{30} -6.63194 q^{31} -0.390093 q^{32} -0.533145 q^{33} -0.695085 q^{35} -0.206136 q^{36} -0.118625 q^{37} -3.34100 q^{38} +0.619770 q^{39} -2.77756 q^{40} -1.07877 q^{41} -0.109886 q^{42} -0.641108 q^{43} -0.334661 q^{44} -2.98792 q^{45} -6.66965 q^{46} +4.93703 q^{47} -0.454269 q^{48} -6.51686 q^{49} +1.43840 q^{50} +0.389037 q^{52} +11.9864 q^{53} -0.946629 q^{54} -4.85089 q^{55} +1.93064 q^{56} -0.255283 q^{57} -9.35848 q^{58} -9.91829 q^{59} +0.00758244 q^{60} -1.60292 q^{61} -9.53937 q^{62} +2.07686 q^{63} +7.70533 q^{64} +5.63906 q^{65} -0.766875 q^{66} -2.99411 q^{67} -0.509622 q^{69} -0.999809 q^{70} +4.68852 q^{71} +8.29913 q^{72} -5.49911 q^{73} -0.170630 q^{74} +0.109907 q^{75} -0.160244 q^{76} +3.37178 q^{77} +0.891477 q^{78} -14.8439 q^{79} -4.13322 q^{80} +8.89143 q^{81} -1.55170 q^{82} +5.03506 q^{83} -0.00527044 q^{84} -0.922169 q^{86} -0.715073 q^{87} +13.4736 q^{88} -2.35657 q^{89} -4.29782 q^{90} -3.91962 q^{91} -0.319896 q^{92} -0.728895 q^{93} +7.10141 q^{94} -2.32272 q^{95} -0.0428738 q^{96} +2.70080 q^{97} -9.37384 q^{98} +14.4941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21}+ \cdots - 56 q^{99}+O(q^{100}) \) 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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