LMFDB - Embedded newform 242.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [242,2,Mod(1,242)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("242.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	242.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.93237972891$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	242.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +3.23607 q^{5} +0.381966 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +3.23607 q^{10} +0.381966 q^{12} +1.23607 q^{13} -2.00000 q^{14} +1.23607 q^{15} +1.00000 q^{16} -0.618034 q^{17} -2.85410 q^{18} -5.85410 q^{19} +3.23607 q^{20} -0.763932 q^{21} +1.23607 q^{23} +0.381966 q^{24} +5.47214 q^{25} +1.23607 q^{26} -2.23607 q^{27} -2.00000 q^{28} +4.47214 q^{29} +1.23607 q^{30} +2.00000 q^{31} +1.00000 q^{32} -0.618034 q^{34} -6.47214 q^{35} -2.85410 q^{36} -3.70820 q^{37} -5.85410 q^{38} +0.472136 q^{39} +3.23607 q^{40} +5.61803 q^{41} -0.763932 q^{42} -8.56231 q^{43} -9.23607 q^{45} +1.23607 q^{46} -6.47214 q^{47} +0.381966 q^{48} -3.00000 q^{49} +5.47214 q^{50} -0.236068 q^{51} +1.23607 q^{52} -1.52786 q^{53} -2.23607 q^{54} -2.00000 q^{56} -2.23607 q^{57} +4.47214 q^{58} -8.61803 q^{59} +1.23607 q^{60} -2.47214 q^{61} +2.00000 q^{62} +5.70820 q^{63} +1.00000 q^{64} +4.00000 q^{65} +11.0902 q^{67} -0.618034 q^{68} +0.472136 q^{69} -6.47214 q^{70} -5.23607 q^{71} -2.85410 q^{72} +10.3820 q^{73} -3.70820 q^{74} +2.09017 q^{75} -5.85410 q^{76} +0.472136 q^{78} +13.4164 q^{79} +3.23607 q^{80} +7.70820 q^{81} +5.61803 q^{82} +9.32624 q^{83} -0.763932 q^{84} -2.00000 q^{85} -8.56231 q^{86} +1.70820 q^{87} -8.09017 q^{89} -9.23607 q^{90} -2.47214 q^{91} +1.23607 q^{92} +0.763932 q^{93} -6.47214 q^{94} -18.9443 q^{95} +0.381966 q^{96} +7.14590 q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{8} + q^{9} + 2 q^{10} + 3 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{15} + 2 q^{16} + q^{17} + q^{18} - 5 q^{19} + 2 q^{20} - 6 q^{21}+ \cdots - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 2 * q^8 + q^9 + 2 * q^10 + 3 * q^12 - 2 * q^13 - 4 * q^14 - 2 * q^15 + 2 * q^16 + q^17 + q^18 - 5 * q^19 + 2 * q^20 - 6 * q^21 - 2 * q^23 + 3 * q^24 + 2 * q^25 - 2 * q^26 - 4 * q^28 - 2 * q^30 + 4 * q^31 + 2 * q^32 + q^34 - 4 * q^35 + q^36 + 6 * q^37 - 5 * q^38 - 8 * q^39 + 2 * q^40 + 9 * q^41 - 6 * q^42 + 3 * q^43 - 14 * q^45 - 2 * q^46 - 4 * q^47 + 3 * q^48 - 6 * q^49 + 2 * q^50 + 4 * q^51 - 2 * q^52 - 12 * q^53 - 4 * q^56 - 15 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 + 8 * q^65 + 11 * q^67 + q^68 - 8 * q^69 - 4 * q^70 - 6 * q^71 + q^72 + 23 * q^73 + 6 * q^74 - 7 * q^75 - 5 * q^76 - 8 * q^78 + 2 * q^80 + 2 * q^81 + 9 * q^82 + 3 * q^83 - 6 * q^84 - 4 * q^85 + 3 * q^86 - 10 * q^87 - 5 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 6 * q^93 - 4 * q^94 - 20 * q^95 + 3 * q^96 + 21 * q^97 - 6 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	1.00000	0.500000
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0.381966	0.155937
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	−2.85410	−0.951367
$10$	3.23607	1.02333
$11$	0	0
$11$	0	0
$12$	0.381966	0.110264
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	−2.00000	−0.534522
$15$	1.23607	0.319151
$16$	1.00000	0.250000
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	−2.85410	−0.672718
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	3.23607	0.723607
$21$	−0.763932	−0.166704
$22$	0	0
$23$	1.23607	0.257738	0.128869	−	0.991662i	$-0.458865\pi$
$23$	1.23607	0.257738	0.128869	+	0.991662i	$0.458865\pi$
$24$	0.381966	0.0779685
$25$	5.47214	1.09443
$26$	1.23607	0.242413
$27$	−2.23607	−0.430331
$28$	−2.00000	−0.377964
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	1.23607	0.225674
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.618034	−0.105992
$35$	−6.47214	−1.09399
$36$	−2.85410	−0.475684
$37$	−3.70820	−0.609625	−0.304812	−	0.952412i	$-0.598594\pi$
$37$	−3.70820	−0.609625	−0.304812	+	0.952412i	$0.598594\pi$
$38$	−5.85410	−0.949661
$39$	0.472136	0.0756023
$40$	3.23607	0.511667
$41$	5.61803	0.877390	0.438695	−	0.898636i	$-0.355441\pi$
$41$	5.61803	0.877390	0.438695	+	0.898636i	$0.355441\pi$
$42$	−0.763932	−0.117877
$43$	−8.56231	−1.30574	−0.652870	−	0.757470i	$-0.726435\pi$
$43$	−8.56231	−1.30574	−0.652870	+	0.757470i	$0.726435\pi$
$44$	0	0
$45$	−9.23607	−1.37683
$46$	1.23607	0.182248
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0.381966	0.0551320
$49$	−3.00000	−0.428571
$50$	5.47214	0.773877
$51$	−0.236068	−0.0330561
$52$	1.23607	0.171412
$53$	−1.52786	−0.209868	−0.104934	−	0.994479i	$-0.533463\pi$
$53$	−1.52786	−0.209868	−0.104934	+	0.994479i	$0.533463\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−2.23607	−0.296174
$58$	4.47214	0.587220
$59$	−8.61803	−1.12197	−0.560986	−	0.827825i	$-0.689578\pi$
$59$	−8.61803	−1.12197	−0.560986	+	0.827825i	$0.689578\pi$
$60$	1.23607	0.159576
$61$	−2.47214	−0.316525	−0.158262	−	0.987397i	$-0.550589\pi$
$61$	−2.47214	−0.316525	−0.158262	+	0.987397i	$0.550589\pi$
$62$	2.00000	0.254000
$63$	5.70820	0.719166
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	11.0902	1.35488	0.677440	−	0.735578i	$-0.263089\pi$
$67$	11.0902	1.35488	0.677440	+	0.735578i	$0.263089\pi$
$68$	−0.618034	−0.0749476
$69$	0.472136	0.0568385
$70$	−6.47214	−0.773568
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	−2.85410	−0.336359
$73$	10.3820	1.21512	0.607559	−	0.794275i	$-0.292149\pi$
$73$	10.3820	1.21512	0.607559	+	0.794275i	$0.292149\pi$
$74$	−3.70820	−0.431070
$75$	2.09017	0.241352
$76$	−5.85410	−0.671512
$77$	0	0
$78$	0.472136	0.0534589
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	3.23607	0.361803
$81$	7.70820	0.856467
$82$	5.61803	0.620408
$83$	9.32624	1.02369	0.511844	−	0.859079i	$-0.328963\pi$
$83$	9.32624	1.02369	0.511844	+	0.859079i	$0.328963\pi$
$84$	−0.763932	−0.0833518
$85$	−2.00000	−0.216930
$86$	−8.56231	−0.923297
$87$	1.70820	0.183139
$88$	0	0
$89$	−8.09017	−0.857556	−0.428778	−	0.903410i	$-0.641056\pi$
$89$	−8.09017	−0.857556	−0.428778	+	0.903410i	$0.641056\pi$
$90$	−9.23607	−0.973567
$91$	−2.47214	−0.259150
$92$	1.23607	0.128869
$93$	0.763932	0.0792161
$94$	−6.47214	−0.667550
$95$	−18.9443	−1.94364
$96$	0.381966	0.0389842
$97$	7.14590	0.725556	0.362778	−	0.931876i	$-0.381828\pi$
$97$	7.14590	0.725556	0.362778	+	0.931876i	$0.381828\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	5.47214	0.547214
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	−0.236068	−0.0233742
$103$	15.7082	1.54778	0.773888	−	0.633323i	$-0.218309\pi$
$103$	15.7082	1.54778	0.773888	+	0.633323i	$0.218309\pi$
$104$	1.23607	0.121206
$105$	−2.47214	−0.241256
$106$	−1.52786	−0.148399
$107$	−1.14590	−0.110778	−0.0553891	−	0.998465i	$-0.517640\pi$
$107$	−1.14590	−0.110778	−0.0553891	+	0.998465i	$0.517640\pi$
$108$	−2.23607	−0.215166
$109$	18.9443	1.81453	0.907266	−	0.420557i	$-0.138165\pi$
$109$	18.9443	1.81453	0.907266	+	0.420557i	$0.138165\pi$
$110$	0	0
$111$	−1.41641	−0.134439
$112$	−2.00000	−0.188982
$113$	−1.85410	−0.174419	−0.0872096	−	0.996190i	$-0.527795\pi$
$113$	−1.85410	−0.174419	−0.0872096	+	0.996190i	$0.527795\pi$
$114$	−2.23607	−0.209427
$115$	4.00000	0.373002
$116$	4.47214	0.415227
$117$	−3.52786	−0.326151
$118$	−8.61803	−0.793354
$119$	1.23607	0.113310
$120$	1.23607	0.112837
$121$	0	0
$122$	−2.47214	−0.223817
$123$	2.14590	0.193489
$124$	2.00000	0.179605
$125$	1.52786	0.136656
$126$	5.70820	0.508527
$127$	−10.9443	−0.971147	−0.485574	−	0.874196i	$-0.661389\pi$
$127$	−10.9443	−0.971147	−0.485574	+	0.874196i	$0.661389\pi$
$128$	1.00000	0.0883883
$129$	−3.27051	−0.287952
$130$	4.00000	0.350823
$131$	6.79837	0.593977	0.296988	−	0.954881i	$-0.404018\pi$
$131$	6.79837	0.593977	0.296988	+	0.954881i	$0.404018\pi$
$132$	0	0
$133$	11.7082	1.01523
$134$	11.0902	0.958045
$135$	−7.23607	−0.622782
$136$	−0.618034	−0.0529960
$137$	16.0902	1.37468	0.687338	−	0.726338i	$-0.258779\pi$
$137$	16.0902	1.37468	0.687338	+	0.726338i	$0.258779\pi$
$138$	0.472136	0.0401909
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−6.47214	−0.546995
$141$	−2.47214	−0.208191
$142$	−5.23607	−0.439401
$143$	0	0
$144$	−2.85410	−0.237842
$145$	14.4721	1.20185
$146$	10.3820	0.859218
$147$	−1.14590	−0.0945121
$148$	−3.70820	−0.304812
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	2.09017	0.170662
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−5.85410	−0.474830
$153$	1.76393	0.142605
$154$	0	0
$155$	6.47214	0.519854
$156$	0.472136	0.0378011
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	13.4164	1.06735
$159$	−0.583592	−0.0462819
$160$	3.23607	0.255834
$161$	−2.47214	−0.194832
$162$	7.70820	0.605614
$163$	0.909830	0.0712634	0.0356317	−	0.999365i	$-0.488656\pi$
$163$	0.909830	0.0712634	0.0356317	+	0.999365i	$0.488656\pi$
$164$	5.61803	0.438695
$165$	0	0
$166$	9.32624	0.723856
$167$	−14.7639	−1.14247	−0.571234	−	0.820787i	$-0.693535\pi$
$167$	−14.7639	−1.14247	−0.571234	+	0.820787i	$0.693535\pi$
$168$	−0.763932	−0.0589386
$169$	−11.4721	−0.882472
$170$	−2.00000	−0.153393
$171$	16.7082	1.27771
$172$	−8.56231	−0.652870
$173$	21.8885	1.66416	0.832078	−	0.554659i	$-0.187151\pi$
$173$	21.8885	1.66416	0.832078	+	0.554659i	$0.187151\pi$
$174$	1.70820	0.129499
$175$	−10.9443	−0.827309
$176$	0	0
$177$	−3.29180	−0.247427
$178$	−8.09017	−0.606384
$179$	−8.61803	−0.644142	−0.322071	−	0.946715i	$-0.604379\pi$
$179$	−8.61803	−0.644142	−0.322071	+	0.946715i	$0.604379\pi$
$180$	−9.23607	−0.688416
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	−2.47214	−0.183247
$183$	−0.944272	−0.0698026
$184$	1.23607	0.0911241
$185$	−12.0000	−0.882258
$186$	0.763932	0.0560142
$187$	0	0
$188$	−6.47214	−0.472029
$189$	4.47214	0.325300
$190$	−18.9443	−1.37436
$191$	4.76393	0.344706	0.172353	−	0.985035i	$-0.444863\pi$
$191$	4.76393	0.344706	0.172353	+	0.985035i	$0.444863\pi$
$192$	0.381966	0.0275660
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	7.14590	0.513046
$195$	1.52786	0.109413
$196$	−3.00000	−0.214286
$197$	−20.9443	−1.49222	−0.746109	−	0.665824i	$-0.768080\pi$
$197$	−20.9443	−1.49222	−0.746109	+	0.665824i	$0.768080\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	5.47214	0.386938
$201$	4.23607	0.298789
$202$	−4.18034	−0.294128
$203$	−8.94427	−0.627765
$204$	−0.236068	−0.0165281
$205$	18.1803	1.26977
$206$	15.7082	1.09444
$207$	−3.52786	−0.245204
$208$	1.23607	0.0857059
$209$	0	0
$210$	−2.47214	−0.170594
$211$	−6.09017	−0.419265	−0.209632	−	0.977780i	$-0.567227\pi$
$211$	−6.09017	−0.419265	−0.209632	+	0.977780i	$0.567227\pi$
$212$	−1.52786	−0.104934
$213$	−2.00000	−0.137038
$214$	−1.14590	−0.0783320
$215$	−27.7082	−1.88968
$216$	−2.23607	−0.152145
$217$	−4.00000	−0.271538
$218$	18.9443	1.28307
$219$	3.96556	0.267968
$220$	0	0
$221$	−0.763932	−0.0513876
$222$	−1.41641	−0.0950631
$223$	25.7082	1.72155	0.860774	−	0.508987i	$-0.169980\pi$
$223$	25.7082	1.72155	0.860774	+	0.508987i	$0.169980\pi$
$224$	−2.00000	−0.133631
$225$	−15.6180	−1.04120
$226$	−1.85410	−0.123333
$227$	14.5066	0.962835	0.481418	−	0.876491i	$-0.340122\pi$
$227$	14.5066	0.962835	0.481418	+	0.876491i	$0.340122\pi$
$228$	−2.23607	−0.148087
$229$	1.70820	0.112881	0.0564406	−	0.998406i	$-0.482025\pi$
$229$	1.70820	0.112881	0.0564406	+	0.998406i	$0.482025\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	4.47214	0.293610
$233$	−9.61803	−0.630098	−0.315049	−	0.949075i	$-0.602021\pi$
$233$	−9.61803	−0.630098	−0.315049	+	0.949075i	$0.602021\pi$
$234$	−3.52786	−0.230624
$235$	−20.9443	−1.36625
$236$	−8.61803	−0.560986
$237$	5.12461	0.332879
$238$	1.23607	0.0801224
$239$	−21.7082	−1.40419	−0.702093	−	0.712085i	$-0.747751\pi$
$239$	−21.7082	−1.40419	−0.702093	+	0.712085i	$0.747751\pi$
$240$	1.23607	0.0797878
$241$	−11.0902	−0.714381	−0.357190	−	0.934032i	$-0.616265\pi$
$241$	−11.0902	−0.714381	−0.357190	+	0.934032i	$0.616265\pi$
$242$	0	0
$243$	9.65248	0.619207
$244$	−2.47214	−0.158262
$245$	−9.70820	−0.620234
$246$	2.14590	0.136817
$247$	−7.23607	−0.460420
$248$	2.00000	0.127000
$249$	3.56231	0.225752
$250$	1.52786	0.0966306
$251$	20.9443	1.32199	0.660995	−	0.750390i	$-0.270134\pi$
$251$	20.9443	1.32199	0.660995	+	0.750390i	$0.270134\pi$
$252$	5.70820	0.359583
$253$	0	0
$254$	−10.9443	−0.686705
$255$	−0.763932	−0.0478393
$256$	1.00000	0.0625000
$257$	−5.61803	−0.350443	−0.175222	−	0.984529i	$-0.556064\pi$
$257$	−5.61803	−0.350443	−0.175222	+	0.984529i	$0.556064\pi$
$258$	−3.27051	−0.203613
$259$	7.41641	0.460833
$260$	4.00000	0.248069
$261$	−12.7639	−0.790068
$262$	6.79837	0.420005
$263$	−23.2361	−1.43280	−0.716399	−	0.697691i	$-0.754211\pi$
$263$	−23.2361	−1.43280	−0.716399	+	0.697691i	$0.754211\pi$
$264$	0	0
$265$	−4.94427	−0.303724
$266$	11.7082	0.717876
$267$	−3.09017	−0.189115
$268$	11.0902	0.677440
$269$	6.18034	0.376822	0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034	0.376822	0.188411	+	0.982090i	$0.439666\pi$
$270$	−7.23607	−0.440373
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−0.618034	−0.0374738
$273$	−0.944272	−0.0571499
$274$	16.0902	0.972043
$275$	0	0
$276$	0.472136	0.0284192
$277$	−23.7082	−1.42449	−0.712244	−	0.701932i	$-0.752321\pi$
$277$	−23.7082	−1.42449	−0.712244	+	0.701932i	$0.752321\pi$
$278$	0	0
$279$	−5.70820	−0.341741
$280$	−6.47214	−0.386784
$281$	−16.0902	−0.959859	−0.479930	−	0.877307i	$-0.659338\pi$
$281$	−16.0902	−0.959859	−0.479930	+	0.877307i	$0.659338\pi$
$282$	−2.47214	−0.147214
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−5.23607	−0.310703
$285$	−7.23607	−0.428628
$286$	0	0
$287$	−11.2361	−0.663244
$288$	−2.85410	−0.168180
$289$	−16.6180	−0.977531
$290$	14.4721	0.849833
$291$	2.72949	0.160006
$292$	10.3820	0.607559
$293$	−28.3607	−1.65685	−0.828424	−	0.560101i	$-0.810762\pi$
$293$	−28.3607	−1.65685	−0.828424	+	0.560101i	$0.810762\pi$
$294$	−1.14590	−0.0668301
$295$	−27.8885	−1.62373
$296$	−3.70820	−0.215535
$297$	0	0
$298$	−6.18034	−0.358017
$299$	1.52786	0.0883587
$300$	2.09017	0.120676
$301$	17.1246	0.987046
$302$	−8.00000	−0.460348
$303$	−1.59675	−0.0917308
$304$	−5.85410	−0.335756
$305$	−8.00000	−0.458079
$306$	1.76393	0.100837
$307$	27.7984	1.58654	0.793268	−	0.608872i	$-0.208378\pi$
$307$	27.7984	1.58654	0.793268	+	0.608872i	$0.208378\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	6.47214	0.367593
$311$	26.0689	1.47823	0.739115	−	0.673579i	$-0.235244\pi$
$311$	26.0689	1.47823	0.739115	+	0.673579i	$0.235244\pi$
$312$	0.472136	0.0267294
$313$	4.32624	0.244533	0.122267	−	0.992497i	$-0.460984\pi$
$313$	4.32624	0.244533	0.122267	+	0.992497i	$0.460984\pi$
$314$	9.70820	0.547866
$315$	18.4721	1.04079
$316$	13.4164	0.754732
$317$	−3.70820	−0.208273	−0.104137	−	0.994563i	$-0.533208\pi$
$317$	−3.70820	−0.208273	−0.104137	+	0.994563i	$0.533208\pi$
$318$	−0.583592	−0.0327262
$319$	0	0
$320$	3.23607	0.180902
$321$	−0.437694	−0.0244297
$322$	−2.47214	−0.137767
$323$	3.61803	0.201313
$324$	7.70820	0.428234
$325$	6.76393	0.375195
$326$	0.909830	0.0503908
$327$	7.23607	0.400155
$328$	5.61803	0.310204
$329$	12.9443	0.713641
$330$	0	0
$331$	6.27051	0.344658	0.172329	−	0.985039i	$-0.444871\pi$
$331$	6.27051	0.344658	0.172329	+	0.985039i	$0.444871\pi$
$332$	9.32624	0.511844
$333$	10.5836	0.579977
$334$	−14.7639	−0.807846
$335$	35.8885	1.96080
$336$	−0.763932	−0.0416759
$337$	−26.7984	−1.45980	−0.729900	−	0.683554i	$-0.760433\pi$
$337$	−26.7984	−1.45980	−0.729900	+	0.683554i	$0.760433\pi$
$338$	−11.4721	−0.624002
$339$	−0.708204	−0.0384644
$340$	−2.00000	−0.108465
$341$	0	0
$342$	16.7082	0.903476
$343$	20.0000	1.07990
$344$	−8.56231	−0.461649
$345$	1.52786	0.0822574
$346$	21.8885	1.17674
$347$	16.6180	0.892103	0.446051	−	0.895007i	$-0.352830\pi$
$347$	16.6180	0.892103	0.446051	+	0.895007i	$0.352830\pi$
$348$	1.70820	0.0915693
$349$	−19.5967	−1.04899	−0.524495	−	0.851414i	$-0.675746\pi$
$349$	−19.5967	−1.04899	−0.524495	+	0.851414i	$0.675746\pi$
$350$	−10.9443	−0.584996
$351$	−2.76393	−0.147528
$352$	0	0
$353$	32.6180	1.73608	0.868041	−	0.496492i	$-0.165379\pi$
$353$	32.6180	1.73608	0.868041	+	0.496492i	$0.165379\pi$
$354$	−3.29180	−0.174957
$355$	−16.9443	−0.899309
$356$	−8.09017	−0.428778
$357$	0.472136	0.0249881
$358$	−8.61803	−0.455477
$359$	5.12461	0.270467	0.135233	−	0.990814i	$-0.456822\pi$
$359$	5.12461	0.270467	0.135233	+	0.990814i	$0.456822\pi$
$360$	−9.23607	−0.486784
$361$	15.2705	0.803711
$362$	−4.18034	−0.219714
$363$	0	0
$364$	−2.47214	−0.129575
$365$	33.5967	1.75853
$366$	−0.944272	−0.0493579
$367$	19.7082	1.02876	0.514380	−	0.857562i	$-0.328022\pi$
$367$	19.7082	1.02876	0.514380	+	0.857562i	$0.328022\pi$
$368$	1.23607	0.0644345
$369$	−16.0344	−0.834720
$370$	−12.0000	−0.623850
$371$	3.05573	0.158645
$372$	0.763932	0.0396080
$373$	−4.29180	−0.222221	−0.111110	−	0.993808i	$-0.535441\pi$
$373$	−4.29180	−0.222221	−0.111110	+	0.993808i	$0.535441\pi$
$374$	0	0
$375$	0.583592	0.0301366
$376$	−6.47214	−0.333775
$377$	5.52786	0.284699
$378$	4.47214	0.230022
$379$	14.2705	0.733027	0.366513	−	0.930413i	$-0.380551\pi$
$379$	14.2705	0.733027	0.366513	+	0.930413i	$0.380551\pi$
$380$	−18.9443	−0.971821
$381$	−4.18034	−0.214165
$382$	4.76393	0.243744
$383$	−28.3607	−1.44916	−0.724582	−	0.689189i	$-0.757967\pi$
$383$	−28.3607	−1.44916	−0.724582	+	0.689189i	$0.757967\pi$
$384$	0.381966	0.0194921
$385$	0	0
$386$	17.4164	0.886472
$387$	24.4377	1.24224
$388$	7.14590	0.362778
$389$	−10.6525	−0.540102	−0.270051	−	0.962846i	$-0.587041\pi$
$389$	−10.6525	−0.540102	−0.270051	+	0.962846i	$0.587041\pi$
$390$	1.52786	0.0773664
$391$	−0.763932	−0.0386337
$392$	−3.00000	−0.151523
$393$	2.59675	0.130989
$394$	−20.9443	−1.05516
$395$	43.4164	2.18452
$396$	0	0
$397$	−17.1246	−0.859460	−0.429730	−	0.902958i	$-0.641391\pi$
$397$	−17.1246	−0.859460	−0.429730	+	0.902958i	$0.641391\pi$
$398$	−18.9443	−0.949591
$399$	4.47214	0.223887
$400$	5.47214	0.273607
$401$	−17.7984	−0.888808	−0.444404	−	0.895826i	$-0.646585\pi$
$401$	−17.7984	−0.888808	−0.444404	+	0.895826i	$0.646585\pi$
$402$	4.23607	0.211276
$403$	2.47214	0.123146
$404$	−4.18034	−0.207980
$405$	24.9443	1.23949
$406$	−8.94427	−0.443897
$407$	0	0
$408$	−0.236068	−0.0116871
$409$	−13.4164	−0.663399	−0.331699	−	0.943385i	$-0.607622\pi$
$409$	−13.4164	−0.663399	−0.331699	+	0.943385i	$0.607622\pi$
$410$	18.1803	0.897863
$411$	6.14590	0.303155
$412$	15.7082	0.773888
$413$	17.2361	0.848131
$414$	−3.52786	−0.173385
$415$	30.1803	1.48149
$416$	1.23607	0.0606032
$417$	0	0
$418$	0	0
$419$	−10.8541	−0.530258	−0.265129	−	0.964213i	$-0.585414\pi$
$419$	−10.8541	−0.530258	−0.265129	+	0.964213i	$0.585414\pi$
$420$	−2.47214	−0.120628
$421$	−31.4164	−1.53114	−0.765571	−	0.643351i	$-0.777543\pi$
$421$	−31.4164	−1.53114	−0.765571	+	0.643351i	$0.777543\pi$
$422$	−6.09017	−0.296465
$423$	18.4721	0.898146
$424$	−1.52786	−0.0741996
$425$	−3.38197	−0.164049
$426$	−2.00000	−0.0969003
$427$	4.94427	0.239270
$428$	−1.14590	−0.0553891
$429$	0	0
$430$	−27.7082	−1.33621
$431$	−16.9443	−0.816177	−0.408088	−	0.912942i	$-0.633804\pi$
$431$	−16.9443	−0.816177	−0.408088	+	0.912942i	$0.633804\pi$
$432$	−2.23607	−0.107583
$433$	27.7426	1.33323	0.666613	−	0.745404i	$-0.267743\pi$
$433$	27.7426	1.33323	0.666613	+	0.745404i	$0.267743\pi$
$434$	−4.00000	−0.192006
$435$	5.52786	0.265041
$436$	18.9443	0.907266
$437$	−7.23607	−0.346148
$438$	3.96556	0.189482
$439$	−6.58359	−0.314218	−0.157109	−	0.987581i	$-0.550217\pi$
$439$	−6.58359	−0.314218	−0.157109	+	0.987581i	$0.550217\pi$
$440$	0	0
$441$	8.56231	0.407729
$442$	−0.763932	−0.0363365
$443$	−8.56231	−0.406807	−0.203404	−	0.979095i	$-0.565200\pi$
$443$	−8.56231	−0.406807	−0.203404	+	0.979095i	$0.565200\pi$
$444$	−1.41641	−0.0672197
$445$	−26.1803	−1.24107
$446$	25.7082	1.21732
$447$	−2.36068	−0.111656
$448$	−2.00000	−0.0944911
$449$	9.27051	0.437502	0.218751	−	0.975781i	$-0.429802\pi$
$449$	9.27051	0.437502	0.218751	+	0.975781i	$0.429802\pi$
$450$	−15.6180	−0.736241
$451$	0	0
$452$	−1.85410	−0.0872096
$453$	−3.05573	−0.143571
$454$	14.5066	0.680827
$455$	−8.00000	−0.375046
$456$	−2.23607	−0.104713
$457$	−9.43769	−0.441477	−0.220738	−	0.975333i	$-0.570847\pi$
$457$	−9.43769	−0.441477	−0.220738	+	0.975333i	$0.570847\pi$
$458$	1.70820	0.0798191
$459$	1.38197	0.0645046
$460$	4.00000	0.186501
$461$	1.34752	0.0627605	0.0313802	−	0.999508i	$-0.490010\pi$
$461$	1.34752	0.0627605	0.0313802	+	0.999508i	$0.490010\pi$
$462$	0	0
$463$	−19.4164	−0.902357	−0.451178	−	0.892434i	$-0.648996\pi$
$463$	−19.4164	−0.902357	−0.451178	+	0.892434i	$0.648996\pi$
$464$	4.47214	0.207614
$465$	2.47214	0.114643
$466$	−9.61803	−0.445547
$467$	−33.3050	−1.54117	−0.770585	−	0.637338i	$-0.780036\pi$
$467$	−33.3050	−1.54117	−0.770585	+	0.637338i	$0.780036\pi$
$468$	−3.52786	−0.163076
$469$	−22.1803	−1.02419
$470$	−20.9443	−0.966087
$471$	3.70820	0.170865
$472$	−8.61803	−0.396677
$473$	0	0
$474$	5.12461	0.235381
$475$	−32.0344	−1.46984
$476$	1.23607	0.0566551
$477$	4.36068	0.199662
$478$	−21.7082	−0.992910
$479$	−34.4721	−1.57507	−0.787536	−	0.616269i	$-0.788644\pi$
$479$	−34.4721	−1.57507	−0.787536	+	0.616269i	$0.788644\pi$
$480$	1.23607	0.0564185
$481$	−4.58359	−0.208994
$482$	−11.0902	−0.505143
$483$	−0.944272	−0.0429659
$484$	0	0
$485$	23.1246	1.05003
$486$	9.65248	0.437845
$487$	−1.34752	−0.0610621	−0.0305311	−	0.999534i	$-0.509720\pi$
$487$	−1.34752	−0.0610621	−0.0305311	+	0.999534i	$0.509720\pi$
$488$	−2.47214	−0.111908
$489$	0.347524	0.0157156
$490$	−9.70820	−0.438572
$491$	22.8541	1.03139	0.515696	−	0.856772i	$-0.327534\pi$
$491$	22.8541	1.03139	0.515696	+	0.856772i	$0.327534\pi$
$492$	2.14590	0.0967446
$493$	−2.76393	−0.124481
$494$	−7.23607	−0.325566
$495$	0	0
$496$	2.00000	0.0898027
$497$	10.4721	0.469739
$498$	3.56231	0.159631
$499$	−30.8541	−1.38122	−0.690610	−	0.723228i	$-0.742658\pi$
$499$	−30.8541	−1.38122	−0.690610	+	0.723228i	$0.742658\pi$
$500$	1.52786	0.0683282
$501$	−5.63932	−0.251946
$502$	20.9443	0.934789
$503$	−3.23607	−0.144289	−0.0721446	−	0.997394i	$-0.522984\pi$
$503$	−3.23607	−0.144289	−0.0721446	+	0.997394i	$0.522984\pi$
$504$	5.70820	0.254264
$505$	−13.5279	−0.601982
$506$	0	0
$507$	−4.38197	−0.194610
$508$	−10.9443	−0.485574
$509$	−8.29180	−0.367527	−0.183764	−	0.982970i	$-0.558828\pi$
$509$	−8.29180	−0.367527	−0.183764	+	0.982970i	$0.558828\pi$
$510$	−0.763932	−0.0338275
$511$	−20.7639	−0.918542
$512$	1.00000	0.0441942
$513$	13.0902	0.577945
$514$	−5.61803	−0.247801
$515$	50.8328	2.23996
$516$	−3.27051	−0.143976
$517$	0	0
$518$	7.41641	0.325858
$519$	8.36068	0.366993
$520$	4.00000	0.175412
$521$	−20.0344	−0.877725	−0.438862	−	0.898554i	$-0.644618\pi$
$521$	−20.0344	−0.877725	−0.438862	+	0.898554i	$0.644618\pi$
$522$	−12.7639	−0.558662
$523$	9.20163	0.402359	0.201180	−	0.979554i	$-0.435523\pi$
$523$	9.20163	0.402359	0.201180	+	0.979554i	$0.435523\pi$
$524$	6.79837	0.296988
$525$	−4.18034	−0.182445
$526$	−23.2361	−1.01314
$527$	−1.23607	−0.0538440
$528$	0	0
$529$	−21.4721	−0.933571
$530$	−4.94427	−0.214765
$531$	24.5967	1.06741
$532$	11.7082	0.507615
$533$	6.94427	0.300790
$534$	−3.09017	−0.133725
$535$	−3.70820	−0.160320
$536$	11.0902	0.479022
$537$	−3.29180	−0.142051
$538$	6.18034	0.266453
$539$	0	0
$540$	−7.23607	−0.311391
$541$	−3.12461	−0.134338	−0.0671688	−	0.997742i	$-0.521397\pi$
$541$	−3.12461	−0.134338	−0.0671688	+	0.997742i	$0.521397\pi$
$542$	2.00000	0.0859074
$543$	−1.59675	−0.0685230
$544$	−0.618034	−0.0264980
$545$	61.3050	2.62602
$546$	−0.944272	−0.0404111
$547$	8.32624	0.356004	0.178002	−	0.984030i	$-0.443037\pi$
$547$	8.32624	0.356004	0.178002	+	0.984030i	$0.443037\pi$
$548$	16.0902	0.687338
$549$	7.05573	0.301131
$550$	0	0
$551$	−26.1803	−1.11532
$552$	0.472136	0.0200954
$553$	−26.8328	−1.14105
$554$	−23.7082	−1.00727
$555$	−4.58359	−0.194563
$556$	0	0
$557$	35.2361	1.49300	0.746500	−	0.665385i	$-0.231733\pi$
$557$	35.2361	1.49300	0.746500	+	0.665385i	$0.231733\pi$
$558$	−5.70820	−0.241648
$559$	−10.5836	−0.447638
$560$	−6.47214	−0.273498
$561$	0	0
$562$	−16.0902	−0.678723
$563$	0.381966	0.0160979	0.00804897	−	0.999968i	$-0.497438\pi$
$563$	0.381966	0.0160979	0.00804897	+	0.999968i	$0.497438\pi$
$564$	−2.47214	−0.104096
$565$	−6.00000	−0.252422
$566$	24.0000	1.00880
$567$	−15.4164	−0.647428
$568$	−5.23607	−0.219701
$569$	38.2148	1.60205	0.801023	−	0.598633i	$-0.204289\pi$
$569$	38.2148	1.60205	0.801023	+	0.598633i	$0.204289\pi$
$570$	−7.23607	−0.303086
$571$	−2.47214	−0.103456	−0.0517278	−	0.998661i	$-0.516473\pi$
$571$	−2.47214	−0.103456	−0.0517278	+	0.998661i	$0.516473\pi$
$572$	0	0
$573$	1.81966	0.0760174
$574$	−11.2361	−0.468984
$575$	6.76393	0.282075
$576$	−2.85410	−0.118921
$577$	−6.79837	−0.283020	−0.141510	−	0.989937i	$-0.545196\pi$
$577$	−6.79837	−0.283020	−0.141510	+	0.989937i	$0.545196\pi$
$578$	−16.6180	−0.691219
$579$	6.65248	0.276467
$580$	14.4721	0.600923
$581$	−18.6525	−0.773835
$582$	2.72949	0.113141
$583$	0	0
$584$	10.3820	0.429609
$585$	−11.4164	−0.472010
$586$	−28.3607	−1.17157
$587$	22.1459	0.914059	0.457030	−	0.889452i	$-0.348913\pi$
$587$	22.1459	0.914059	0.457030	+	0.889452i	$0.348913\pi$
$588$	−1.14590	−0.0472560
$589$	−11.7082	−0.482428
$590$	−27.8885	−1.14815
$591$	−8.00000	−0.329076
$592$	−3.70820	−0.152406
$593$	−28.6869	−1.17803	−0.589015	−	0.808122i	$-0.700484\pi$
$593$	−28.6869	−1.17803	−0.589015	+	0.808122i	$0.700484\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−6.18034	−0.253157
$597$	−7.23607	−0.296153
$598$	1.52786	0.0624790
$599$	−13.4164	−0.548180	−0.274090	−	0.961704i	$-0.588377\pi$
$599$	−13.4164	−0.548180	−0.274090	+	0.961704i	$0.588377\pi$
$600$	2.09017	0.0853308
$601$	−27.1459	−1.10730	−0.553652	−	0.832748i	$-0.686766\pi$
$601$	−27.1459	−1.10730	−0.553652	+	0.832748i	$0.686766\pi$
$602$	17.1246	0.697947
$603$	−31.6525	−1.28899
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−1.59675	−0.0648634
$607$	23.1246	0.938599	0.469300	−	0.883039i	$-0.344506\pi$
$607$	23.1246	0.938599	0.469300	+	0.883039i	$0.344506\pi$
$608$	−5.85410	−0.237415
$609$	−3.41641	−0.138440
$610$	−8.00000	−0.323911
$611$	−8.00000	−0.323645
$612$	1.76393	0.0713027
$613$	35.7082	1.44224	0.721120	−	0.692810i	$-0.243627\pi$
$613$	35.7082	1.44224	0.721120	+	0.692810i	$0.243627\pi$
$614$	27.7984	1.12185
$615$	6.94427	0.280020
$616$	0	0
$617$	23.4508	0.944096	0.472048	−	0.881573i	$-0.343515\pi$
$617$	23.4508	0.944096	0.472048	+	0.881573i	$0.343515\pi$
$618$	6.00000	0.241355
$619$	5.20163	0.209071	0.104536	−	0.994521i	$-0.466664\pi$
$619$	5.20163	0.209071	0.104536	+	0.994521i	$0.466664\pi$
$620$	6.47214	0.259927
$621$	−2.76393	−0.110913
$622$	26.0689	1.04527
$623$	16.1803	0.648252
$624$	0.472136	0.0189006
$625$	−22.4164	−0.896656
$626$	4.32624	0.172911
$627$	0	0
$628$	9.70820	0.387400
$629$	2.29180	0.0913799
$630$	18.4721	0.735948
$631$	23.7082	0.943809	0.471904	−	0.881650i	$-0.343567\pi$
$631$	23.7082	0.943809	0.471904	+	0.881650i	$0.343567\pi$
$632$	13.4164	0.533676
$633$	−2.32624	−0.0924597
$634$	−3.70820	−0.147272
$635$	−35.4164	−1.40546
$636$	−0.583592	−0.0231409
$637$	−3.70820	−0.146924
$638$	0	0
$639$	14.9443	0.591186
$640$	3.23607	0.127917
$641$	16.6738	0.658574	0.329287	−	0.944230i	$-0.393192\pi$
$641$	16.6738	0.658574	0.329287	+	0.944230i	$0.393192\pi$
$642$	−0.437694	−0.0172744
$643$	39.9787	1.57661	0.788303	−	0.615287i	$-0.210960\pi$
$643$	39.9787	1.57661	0.788303	+	0.615287i	$0.210960\pi$
$644$	−2.47214	−0.0974158
$645$	−10.5836	−0.416729
$646$	3.61803	0.142350
$647$	29.0557	1.14230	0.571149	−	0.820846i	$-0.306498\pi$
$647$	29.0557	1.14230	0.571149	+	0.820846i	$0.306498\pi$
$648$	7.70820	0.302807
$649$	0	0
$650$	6.76393	0.265303
$651$	−1.52786	−0.0598817
$652$	0.909830	0.0356317
$653$	−15.3475	−0.600595	−0.300298	−	0.953846i	$-0.597086\pi$
$653$	−15.3475	−0.600595	−0.300298	+	0.953846i	$0.597086\pi$
$654$	7.23607	0.282953
$655$	22.0000	0.859611
$656$	5.61803	0.219347
$657$	−29.6312	−1.15602
$658$	12.9443	0.504620
$659$	−16.9098	−0.658713	−0.329357	−	0.944206i	$-0.606832\pi$
$659$	−16.9098	−0.658713	−0.329357	+	0.944206i	$0.606832\pi$
$660$	0	0
$661$	−3.52786	−0.137218	−0.0686090	−	0.997644i	$-0.521856\pi$
$661$	−3.52786	−0.137218	−0.0686090	+	0.997644i	$0.521856\pi$
$662$	6.27051	0.243710
$663$	−0.291796	−0.0113324
$664$	9.32624	0.361928
$665$	37.8885	1.46925
$666$	10.5836	0.410106
$667$	5.52786	0.214040
$668$	−14.7639	−0.571234
$669$	9.81966	0.379650
$670$	35.8885	1.38650
$671$	0	0
$672$	−0.763932	−0.0294693
$673$	9.85410	0.379848	0.189924	−	0.981799i	$-0.439176\pi$
$673$	9.85410	0.379848	0.189924	+	0.981799i	$0.439176\pi$
$674$	−26.7984	−1.03223
$675$	−12.2361	−0.470966
$676$	−11.4721	−0.441236
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	−0.708204	−0.0271984
$679$	−14.2918	−0.548469
$680$	−2.00000	−0.0766965
$681$	5.54102	0.212332
$682$	0	0
$683$	18.4721	0.706817	0.353408	−	0.935469i	$-0.385023\pi$
$683$	18.4721	0.706817	0.353408	+	0.935469i	$0.385023\pi$
$684$	16.7082	0.638854
$685$	52.0689	1.98945
$686$	20.0000	0.763604
$687$	0.652476	0.0248935
$688$	−8.56231	−0.326435
$689$	−1.88854	−0.0719478
$690$	1.52786	0.0581648
$691$	−26.2148	−0.997257	−0.498629	−	0.866816i	$-0.666163\pi$
$691$	−26.2148	−0.997257	−0.498629	+	0.866816i	$0.666163\pi$
$692$	21.8885	0.832078
$693$	0	0
$694$	16.6180	0.630812
$695$	0	0
$696$	1.70820	0.0647493
$697$	−3.47214	−0.131517
$698$	−19.5967	−0.741748
$699$	−3.67376	−0.138954
$700$	−10.9443	−0.413655
$701$	−14.8328	−0.560228	−0.280114	−	0.959967i	$-0.590372\pi$
$701$	−14.8328	−0.560228	−0.280114	+	0.959967i	$0.590372\pi$
$702$	−2.76393	−0.104318
$703$	21.7082	0.818740
$704$	0	0
$705$	−8.00000	−0.301297
$706$	32.6180	1.22760
$707$	8.36068	0.314436
$708$	−3.29180	−0.123713
$709$	24.4721	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$709$	24.4721	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$710$	−16.9443	−0.635907
$711$	−38.2918	−1.43605
$712$	−8.09017	−0.303192
$713$	2.47214	0.0925822
$714$	0.472136	0.0176692
$715$	0	0
$716$	−8.61803	−0.322071
$717$	−8.29180	−0.309663
$718$	5.12461	0.191249
$719$	26.8328	1.00070	0.500348	−	0.865825i	$-0.333206\pi$
$719$	26.8328	1.00070	0.500348	+	0.865825i	$0.333206\pi$
$720$	−9.23607	−0.344208
$721$	−31.4164	−1.17001
$722$	15.2705	0.568310
$723$	−4.23607	−0.157541
$724$	−4.18034	−0.155361
$725$	24.4721	0.908872
$726$	0	0
$727$	2.87539	0.106642	0.0533211	−	0.998577i	$-0.483019\pi$
$727$	2.87539	0.106642	0.0533211	+	0.998577i	$0.483019\pi$
$728$	−2.47214	−0.0916235
$729$	−19.4377	−0.719915
$730$	33.5967	1.24347
$731$	5.29180	0.195724
$732$	−0.944272	−0.0349013
$733$	−9.41641	−0.347803	−0.173901	−	0.984763i	$-0.555637\pi$
$733$	−9.41641	−0.347803	−0.173901	+	0.984763i	$0.555637\pi$
$734$	19.7082	0.727443
$735$	−3.70820	−0.136779
$736$	1.23607	0.0455621
$737$	0	0
$738$	−16.0344	−0.590236
$739$	−37.5623	−1.38175	−0.690876	−	0.722973i	$-0.742775\pi$
$739$	−37.5623	−1.38175	−0.690876	+	0.722973i	$0.742775\pi$
$740$	−12.0000	−0.441129
$741$	−2.76393	−0.101536
$742$	3.05573	0.112179
$743$	−8.76393	−0.321517	−0.160759	−	0.986994i	$-0.551394\pi$
$743$	−8.76393	−0.321517	−0.160759	+	0.986994i	$0.551394\pi$
$744$	0.763932	0.0280071
$745$	−20.0000	−0.732743
$746$	−4.29180	−0.157134
$747$	−26.6180	−0.973903
$748$	0	0
$749$	2.29180	0.0837404
$750$	0.583592	0.0213098
$751$	−33.7771	−1.23254	−0.616272	−	0.787534i	$-0.711358\pi$
$751$	−33.7771	−1.23254	−0.616272	+	0.787534i	$0.711358\pi$
$752$	−6.47214	−0.236015
$753$	8.00000	0.291536
$754$	5.52786	0.201313
$755$	−25.8885	−0.942181
$756$	4.47214	0.162650
$757$	−2.65248	−0.0964059	−0.0482029	−	0.998838i	$-0.515349\pi$
$757$	−2.65248	−0.0964059	−0.0482029	+	0.998838i	$0.515349\pi$
$758$	14.2705	0.518328
$759$	0	0
$760$	−18.9443	−0.687181
$761$	−4.25735	−0.154329	−0.0771645	−	0.997018i	$-0.524587\pi$
$761$	−4.25735	−0.154329	−0.0771645	+	0.997018i	$0.524587\pi$
$762$	−4.18034	−0.151438
$763$	−37.8885	−1.37166
$764$	4.76393	0.172353
$765$	5.70820	0.206381
$766$	−28.3607	−1.02471
$767$	−10.6525	−0.384639
$768$	0.381966	0.0137830
$769$	13.4164	0.483808	0.241904	−	0.970300i	$-0.422228\pi$
$769$	13.4164	0.483808	0.241904	+	0.970300i	$0.422228\pi$
$770$	0	0
$771$	−2.14590	−0.0772826
$772$	17.4164	0.626830
$773$	−3.88854	−0.139861	−0.0699306	−	0.997552i	$-0.522278\pi$
$773$	−3.88854	−0.139861	−0.0699306	+	0.997552i	$0.522278\pi$
$774$	24.4377	0.878395
$775$	10.9443	0.393130
$776$	7.14590	0.256523
$777$	2.83282	0.101627
$778$	−10.6525	−0.381910
$779$	−32.8885	−1.17835
$780$	1.52786	0.0547063
$781$	0	0
$782$	−0.763932	−0.0273182
$783$	−10.0000	−0.357371
$784$	−3.00000	−0.107143
$785$	31.4164	1.12130
$786$	2.59675	0.0926229
$787$	26.2148	0.934456	0.467228	−	0.884137i	$-0.345253\pi$
$787$	26.2148	0.934456	0.467228	+	0.884137i	$0.345253\pi$
$788$	−20.9443	−0.746109
$789$	−8.87539	−0.315972
$790$	43.4164	1.54469
$791$	3.70820	0.131849
$792$	0	0
$793$	−3.05573	−0.108512
$794$	−17.1246	−0.607730
$795$	−1.88854	−0.0669797
$796$	−18.9443	−0.671462
$797$	−24.7639	−0.877183	−0.438592	−	0.898686i	$-0.644523\pi$
$797$	−24.7639	−0.877183	−0.438592	+	0.898686i	$0.644523\pi$
$798$	4.47214	0.158312
$799$	4.00000	0.141510
$800$	5.47214	0.193469
$801$	23.0902	0.815851
$802$	−17.7984	−0.628482
$803$	0	0
$804$	4.23607	0.149395
$805$	−8.00000	−0.281963
$806$	2.47214	0.0870773
$807$	2.36068	0.0830999
$808$	−4.18034	−0.147064
$809$	16.9098	0.594518	0.297259	−	0.954797i	$-0.403928\pi$
$809$	16.9098	0.594518	0.297259	+	0.954797i	$0.403928\pi$
$810$	24.9443	0.876452
$811$	8.38197	0.294331	0.147165	−	0.989112i	$-0.452985\pi$
$811$	8.38197	0.294331	0.147165	+	0.989112i	$0.452985\pi$
$812$	−8.94427	−0.313882
$813$	0.763932	0.0267923
$814$	0	0
$815$	2.94427	0.103133
$816$	−0.236068	−0.00826403
$817$	50.1246	1.75364
$818$	−13.4164	−0.469094
$819$	7.05573	0.246547
$820$	18.1803	0.634885
$821$	30.5410	1.06589	0.532944	−	0.846150i	$-0.321085\pi$
$821$	30.5410	1.06589	0.532944	+	0.846150i	$0.321085\pi$
$822$	6.14590	0.214363
$823$	23.5967	0.822531	0.411265	−	0.911516i	$-0.365087\pi$
$823$	23.5967	0.822531	0.411265	+	0.911516i	$0.365087\pi$
$824$	15.7082	0.547221
$825$	0	0
$826$	17.2361	0.599720
$827$	39.3820	1.36945	0.684723	−	0.728804i	$-0.259923\pi$
$827$	39.3820	1.36945	0.684723	+	0.728804i	$0.259923\pi$
$828$	−3.52786	−0.122602
$829$	31.0557	1.07861	0.539305	−	0.842111i	$-0.318687\pi$
$829$	31.0557	1.07861	0.539305	+	0.842111i	$0.318687\pi$
$830$	30.1803	1.04757
$831$	−9.05573	−0.314140
$832$	1.23607	0.0428529
$833$	1.85410	0.0642408
$834$	0	0
$835$	−47.7771	−1.65339
$836$	0	0
$837$	−4.47214	−0.154580
$838$	−10.8541	−0.374949
$839$	12.1115	0.418134	0.209067	−	0.977901i	$-0.432957\pi$
$839$	12.1115	0.418134	0.209067	+	0.977901i	$0.432957\pi$
$840$	−2.47214	−0.0852968
$841$	−9.00000	−0.310345
$842$	−31.4164	−1.08268
$843$	−6.14590	−0.211676
$844$	−6.09017	−0.209632
$845$	−37.1246	−1.27713
$846$	18.4721	0.635085
$847$	0	0
$848$	−1.52786	−0.0524671
$849$	9.16718	0.314617
$850$	−3.38197	−0.116000
$851$	−4.58359	−0.157124
$852$	−2.00000	−0.0685189
$853$	−30.4721	−1.04335	−0.521673	−	0.853146i	$-0.674692\pi$
$853$	−30.4721	−1.04335	−0.521673	+	0.853146i	$0.674692\pi$
$854$	4.94427	0.169190
$855$	54.0689	1.84912
$856$	−1.14590	−0.0391660
$857$	−10.7426	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$857$	−10.7426	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$858$	0	0
$859$	46.5066	1.58678	0.793392	−	0.608711i	$-0.208313\pi$
$859$	46.5066	1.58678	0.793392	+	0.608711i	$0.208313\pi$
$860$	−27.7082	−0.944842
$861$	−4.29180	−0.146264
$862$	−16.9443	−0.577124
$863$	−33.2361	−1.13137	−0.565684	−	0.824622i	$-0.691388\pi$
$863$	−33.2361	−1.13137	−0.565684	+	0.824622i	$0.691388\pi$
$864$	−2.23607	−0.0760726
$865$	70.8328	2.40839
$866$	27.7426	0.942733
$867$	−6.34752	−0.215573
$868$	−4.00000	−0.135769
$869$	0	0
$870$	5.52786	0.187412
$871$	13.7082	0.464485
$872$	18.9443	0.641534
$873$	−20.3951	−0.690270
$874$	−7.23607	−0.244764
$875$	−3.05573	−0.103302
$876$	3.96556	0.133984
$877$	−28.1803	−0.951582	−0.475791	−	0.879558i	$-0.657838\pi$
$877$	−28.1803	−0.951582	−0.475791	+	0.879558i	$0.657838\pi$
$878$	−6.58359	−0.222185
$879$	−10.8328	−0.365382
$880$	0	0
$881$	−46.3394	−1.56121	−0.780607	−	0.625022i	$-0.785090\pi$
$881$	−46.3394	−1.56121	−0.780607	+	0.625022i	$0.785090\pi$
$882$	8.56231	0.288308
$883$	−3.43769	−0.115688	−0.0578438	−	0.998326i	$-0.518423\pi$
$883$	−3.43769	−0.115688	−0.0578438	+	0.998326i	$0.518423\pi$
$884$	−0.763932	−0.0256938
$885$	−10.6525	−0.358079
$886$	−8.56231	−0.287656
$887$	−37.7771	−1.26843	−0.634215	−	0.773157i	$-0.718677\pi$
$887$	−37.7771	−1.26843	−0.634215	+	0.773157i	$0.718677\pi$
$888$	−1.41641	−0.0475315
$889$	21.8885	0.734118
$890$	−26.1803	−0.877567
$891$	0	0
$892$	25.7082	0.860774
$893$	37.8885	1.26789
$894$	−2.36068	−0.0789529
$895$	−27.8885	−0.932211
$896$	−2.00000	−0.0668153
$897$	0.583592	0.0194856
$898$	9.27051	0.309361
$899$	8.94427	0.298308
$900$	−15.6180	−0.520601
$901$	0.944272	0.0314583
$902$	0	0
$903$	6.54102	0.217672
$904$	−1.85410	−0.0616665
$905$	−13.5279	−0.449681
$906$	−3.05573	−0.101520
$907$	10.4377	0.346578	0.173289	−	0.984871i	$-0.444561\pi$
$907$	10.4377	0.346578	0.173289	+	0.984871i	$0.444561\pi$
$908$	14.5066	0.481418
$909$	11.9311	0.395730
$910$	−8.00000	−0.265197
$911$	18.1803	0.602342	0.301171	−	0.953570i	$-0.402623\pi$
$911$	18.1803	0.602342	0.301171	+	0.953570i	$0.402623\pi$
$912$	−2.23607	−0.0740436
$913$	0	0
$914$	−9.43769	−0.312171
$915$	−3.05573	−0.101019
$916$	1.70820	0.0564406
$917$	−13.5967	−0.449004
$918$	1.38197	0.0456117
$919$	43.4164	1.43218	0.716088	−	0.698010i	$-0.245931\pi$
$919$	43.4164	1.43218	0.716088	+	0.698010i	$0.245931\pi$
$920$	4.00000	0.131876
$921$	10.6180	0.349876
$922$	1.34752	0.0443783
$923$	−6.47214	−0.213033
$924$	0	0
$925$	−20.2918	−0.667190
$926$	−19.4164	−0.638063
$927$	−44.8328	−1.47250
$928$	4.47214	0.146805
$929$	−16.2574	−0.533386	−0.266693	−	0.963781i	$-0.585931\pi$
$929$	−16.2574	−0.533386	−0.266693	+	0.963781i	$0.585931\pi$
$930$	2.47214	0.0810645
$931$	17.5623	0.575581
$932$	−9.61803	−0.315049
$933$	9.95743	0.325992
$934$	−33.3050	−1.08977
$935$	0	0
$936$	−3.52786	−0.115312
$937$	−37.8541	−1.23664	−0.618320	−	0.785927i	$-0.712186\pi$
$937$	−37.8541	−1.23664	−0.618320	+	0.785927i	$0.712186\pi$
$938$	−22.1803	−0.724214
$939$	1.65248	0.0539265
$940$	−20.9443	−0.683127
$941$	48.1803	1.57063	0.785317	−	0.619094i	$-0.212500\pi$
$941$	48.1803	1.57063	0.785317	+	0.619094i	$0.212500\pi$
$942$	3.70820	0.120820
$943$	6.94427	0.226137
$944$	−8.61803	−0.280493
$945$	14.4721	0.470779
$946$	0	0
$947$	−30.2148	−0.981848	−0.490924	−	0.871202i	$-0.663341\pi$
$947$	−30.2148	−0.981848	−0.490924	+	0.871202i	$0.663341\pi$
$948$	5.12461	0.166440
$949$	12.8328	0.416571
$950$	−32.0344	−1.03933
$951$	−1.41641	−0.0459302
$952$	1.23607	0.0400612
$953$	−11.3262	−0.366893	−0.183446	−	0.983030i	$-0.558725\pi$
$953$	−11.3262	−0.366893	−0.183446	+	0.983030i	$0.558725\pi$
$954$	4.36068	0.141182
$955$	15.4164	0.498863
$956$	−21.7082	−0.702093
$957$	0	0
$958$	−34.4721	−1.11374
$959$	−32.1803	−1.03916
$960$	1.23607	0.0398939
$961$	−27.0000	−0.870968
$962$	−4.58359	−0.147781
$963$	3.27051	0.105391
$964$	−11.0902	−0.357190
$965$	56.3607	1.81431
$966$	−0.944272	−0.0303815
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	0	0
$969$	1.38197	0.0443951
$970$	23.1246	0.742487
$971$	−10.1115	−0.324492	−0.162246	−	0.986750i	$-0.551874\pi$
$971$	−10.1115	−0.324492	−0.162246	+	0.986750i	$0.551874\pi$
$972$	9.65248	0.309603
$973$	0	0
$974$	−1.34752	−0.0431775
$975$	2.58359	0.0827412
$976$	−2.47214	−0.0791311
$977$	−43.3050	−1.38545	−0.692724	−	0.721203i	$-0.743590\pi$
$977$	−43.3050	−1.38545	−0.692724	+	0.721203i	$0.743590\pi$
$978$	0.347524	0.0111126
$979$	0	0
$980$	−9.70820	−0.310117
$981$	−54.0689	−1.72629
$982$	22.8541	0.729304
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	2.14590	0.0684087
$985$	−67.7771	−2.15956
$986$	−2.76393	−0.0880215
$987$	4.94427	0.157378
$988$	−7.23607	−0.230210
$989$	−10.5836	−0.336539
$990$	0	0
$991$	30.5410	0.970167	0.485084	−	0.874468i	$-0.338789\pi$
$991$	30.5410	0.970167	0.485084	+	0.874468i	$0.338789\pi$
$992$	2.00000	0.0635001
$993$	2.39512	0.0760069
$994$	10.4721	0.332156
$995$	−61.3050	−1.94350
$996$	3.56231	0.112876
$997$	8.65248	0.274027	0.137013	−	0.990569i	$-0.456250\pi$
$997$	8.65248	0.274027	0.137013	+	0.990569i	$0.456250\pi$
$998$	−30.8541	−0.976670
$999$	8.29180	0.262341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.2.a.f.1.1		2
3.2	odd	2		2178.2.a.p.1.1		2
4.3	odd	2		1936.2.a.o.1.2		2
5.4	even	2		6050.2.a.bs.1.2		2
8.3	odd	2		7744.2.a.cz.1.1		2
8.5	even	2		7744.2.a.bm.1.2		2
11.2	odd	10		242.2.c.c.81.1		4
11.3	even	5		242.2.c.a.9.1		4
11.4	even	5		242.2.c.a.27.1		4
11.5	even	5		22.2.c.a.3.1	✓	4
11.6	odd	10		242.2.c.c.3.1		4
11.7	odd	10		242.2.c.d.27.1		4
11.8	odd	10		242.2.c.d.9.1		4
11.9	even	5		22.2.c.a.15.1	yes	4
11.10	odd	2		242.2.a.d.1.1		2
33.5	odd	10		198.2.f.e.91.1		4
33.20	odd	10		198.2.f.e.37.1		4
33.32	even	2		2178.2.a.x.1.1		2
44.27	odd	10		176.2.m.c.113.1		4
44.31	odd	10		176.2.m.c.81.1		4
44.43	even	2		1936.2.a.n.1.2		2
55.9	even	10		550.2.h.h.301.1		4
55.27	odd	20		550.2.ba.c.399.1		8
55.38	odd	20		550.2.ba.c.399.2		8
55.42	odd	20		550.2.ba.c.499.2		8
55.49	even	10		550.2.h.h.201.1		4
55.53	odd	20		550.2.ba.c.499.1		8
55.54	odd	2		6050.2.a.ci.1.2		2
88.5	even	10		704.2.m.h.641.1		4
88.21	odd	2		7744.2.a.bn.1.2		2
88.27	odd	10		704.2.m.a.641.1		4
88.43	even	2		7744.2.a.cy.1.1		2
88.53	even	10		704.2.m.h.257.1		4
88.75	odd	10		704.2.m.a.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.2.c.a.3.1	✓	4	11.5	even	5
22.2.c.a.15.1	yes	4	11.9	even	5
176.2.m.c.81.1		4	44.31	odd	10
176.2.m.c.113.1		4	44.27	odd	10
198.2.f.e.37.1		4	33.20	odd	10
198.2.f.e.91.1		4	33.5	odd	10
242.2.a.d.1.1		2	11.10	odd	2
242.2.a.f.1.1		2	1.1	even	1	trivial
242.2.c.a.9.1		4	11.3	even	5
242.2.c.a.27.1		4	11.4	even	5
242.2.c.c.3.1		4	11.6	odd	10
242.2.c.c.81.1		4	11.2	odd	10
242.2.c.d.9.1		4	11.8	odd	10
242.2.c.d.27.1		4	11.7	odd	10
550.2.h.h.201.1		4	55.49	even	10
550.2.h.h.301.1		4	55.9	even	10
550.2.ba.c.399.1		8	55.27	odd	20
550.2.ba.c.399.2		8	55.38	odd	20
550.2.ba.c.499.1		8	55.53	odd	20
550.2.ba.c.499.2		8	55.42	odd	20
704.2.m.a.257.1		4	88.75	odd	10
704.2.m.a.641.1		4	88.27	odd	10
704.2.m.h.257.1		4	88.53	even	10
704.2.m.h.641.1		4	88.5	even	10
1936.2.a.n.1.2		2	44.43	even	2
1936.2.a.o.1.2		2	4.3	odd	2
2178.2.a.p.1.1		2	3.2	odd	2
2178.2.a.x.1.1		2	33.32	even	2
6050.2.a.bs.1.2		2	5.4	even	2
6050.2.a.ci.1.2		2	55.54	odd	2
7744.2.a.bm.1.2		2	8.5	even	2
7744.2.a.bn.1.2		2	88.21	odd	2
7744.2.a.cy.1.1		2	88.43	even	2
7744.2.a.cz.1.1		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} +3.23607 q^{5} +0.381966 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.85410 q^{9} +3.23607 q^{10} +0.381966 q^{12} +1.23607 q^{13} -2.00000 q^{14} +1.23607 q^{15} +1.00000 q^{16} -0.618034 q^{17} -2.85410 q^{18} -5.85410 q^{19} +3.23607 q^{20} -0.763932 q^{21} +1.23607 q^{23} +0.381966 q^{24} +5.47214 q^{25} +1.23607 q^{26} -2.23607 q^{27} -2.00000 q^{28} +4.47214 q^{29} +1.23607 q^{30} +2.00000 q^{31} +1.00000 q^{32} -0.618034 q^{34} -6.47214 q^{35} -2.85410 q^{36} -3.70820 q^{37} -5.85410 q^{38} +0.472136 q^{39} +3.23607 q^{40} +5.61803 q^{41} -0.763932 q^{42} -8.56231 q^{43} -9.23607 q^{45} +1.23607 q^{46} -6.47214 q^{47} +0.381966 q^{48} -3.00000 q^{49} +5.47214 q^{50} -0.236068 q^{51} +1.23607 q^{52} -1.52786 q^{53} -2.23607 q^{54} -2.00000 q^{56} -2.23607 q^{57} +4.47214 q^{58} -8.61803 q^{59} +1.23607 q^{60} -2.47214 q^{61} +2.00000 q^{62} +5.70820 q^{63} +1.00000 q^{64} +4.00000 q^{65} +11.0902 q^{67} -0.618034 q^{68} +0.472136 q^{69} -6.47214 q^{70} -5.23607 q^{71} -2.85410 q^{72} +10.3820 q^{73} -3.70820 q^{74} +2.09017 q^{75} -5.85410 q^{76} +0.472136 q^{78} +13.4164 q^{79} +3.23607 q^{80} +7.70820 q^{81} +5.61803 q^{82} +9.32624 q^{83} -0.763932 q^{84} -2.00000 q^{85} -8.56231 q^{86} +1.70820 q^{87} -8.09017 q^{89} -9.23607 q^{90} -2.47214 q^{91} +1.23607 q^{92} +0.763932 q^{93} -6.47214 q^{94} -18.9443 q^{95} +0.381966 q^{96} +7.14590 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{8} + q^{9} + 2 q^{10} + 3 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{15} + 2 q^{16} + q^{17} + q^{18} - 5 q^{19} + 2 q^{20} - 6 q^{21}+ \cdots - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 2 * q^8 + q^9 + 2 * q^10 + 3 * q^12 - 2 * q^13 - 4 * q^14 - 2 * q^15 + 2 * q^16 + q^17 + q^18 - 5 * q^19 + 2 * q^20 - 6 * q^21 - 2 * q^23 + 3 * q^24 + 2 * q^25 - 2 * q^26 - 4 * q^28 - 2 * q^30 + 4 * q^31 + 2 * q^32 + q^34 - 4 * q^35 + q^36 + 6 * q^37 - 5 * q^38 - 8 * q^39 + 2 * q^40 + 9 * q^41 - 6 * q^42 + 3 * q^43 - 14 * q^45 - 2 * q^46 - 4 * q^47 + 3 * q^48 - 6 * q^49 + 2 * q^50 + 4 * q^51 - 2 * q^52 - 12 * q^53 - 4 * q^56 - 15 * q^59 - 2 * q^60 + 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 + 8 * q^65 + 11 * q^67 + q^68 - 8 * q^69 - 4 * q^70 - 6 * q^71 + q^72 + 23 * q^73 + 6 * q^74 - 7 * q^75 - 5 * q^76 - 8 * q^78 + 2 * q^80 + 2 * q^81 + 9 * q^82 + 3 * q^83 - 6 * q^84 - 4 * q^85 + 3 * q^86 - 10 * q^87 - 5 * q^89 - 14 * q^90 + 4 * q^91 - 2 * q^92 + 6 * q^93 - 4 * q^94 - 20 * q^95 + 3 * q^96 + 21 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more