LMFDB - Embedded newform 409.2.a.b.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [409,2,Mod(1,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.26588144267$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.31596$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.31596 q^{2} +1.87410 q^{3} -0.268240 q^{4} -1.28432 q^{5} -2.46624 q^{6} -2.16419 q^{7} +2.98492 q^{8} +0.512236 q^{9} +1.69012 q^{10} +1.34988 q^{11} -0.502707 q^{12} +5.76832 q^{13} +2.84799 q^{14} -2.40695 q^{15} -3.39157 q^{16} +3.88611 q^{17} -0.674084 q^{18} +8.19658 q^{19} +0.344507 q^{20} -4.05589 q^{21} -1.77640 q^{22} +5.98263 q^{23} +5.59403 q^{24} -3.35051 q^{25} -7.59090 q^{26} -4.66231 q^{27} +0.580520 q^{28} +4.67769 q^{29} +3.16745 q^{30} -4.78270 q^{31} -1.50666 q^{32} +2.52981 q^{33} -5.11398 q^{34} +2.77952 q^{35} -0.137402 q^{36} +1.37721 q^{37} -10.7864 q^{38} +10.8104 q^{39} -3.83361 q^{40} -6.84895 q^{41} +5.33741 q^{42} -2.99849 q^{43} -0.362092 q^{44} -0.657877 q^{45} -7.87292 q^{46} +11.2068 q^{47} -6.35612 q^{48} -2.31630 q^{49} +4.40915 q^{50} +7.28294 q^{51} -1.54729 q^{52} +3.69504 q^{53} +6.13543 q^{54} -1.73369 q^{55} -6.45993 q^{56} +15.3612 q^{57} -6.15567 q^{58} -2.10575 q^{59} +0.645638 q^{60} +9.36043 q^{61} +6.29385 q^{62} -1.10857 q^{63} +8.76585 q^{64} -7.40840 q^{65} -3.32914 q^{66} -4.89108 q^{67} -1.04241 q^{68} +11.2120 q^{69} -3.65774 q^{70} +8.93427 q^{71} +1.52898 q^{72} -2.75468 q^{73} -1.81236 q^{74} -6.27918 q^{75} -2.19865 q^{76} -2.92140 q^{77} -14.2261 q^{78} -16.4012 q^{79} +4.35587 q^{80} -10.2743 q^{81} +9.01298 q^{82} +15.1179 q^{83} +1.08795 q^{84} -4.99102 q^{85} +3.94590 q^{86} +8.76644 q^{87} +4.02929 q^{88} -6.38372 q^{89} +0.865742 q^{90} -12.4837 q^{91} -1.60478 q^{92} -8.96323 q^{93} -14.7478 q^{94} -10.5271 q^{95} -2.82363 q^{96} -1.98117 q^{97} +3.04816 q^{98} +0.691458 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * 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 $

$2$	−1.31596	−0.930527	−0.465263	−	0.885172i	$-0.654040\pi$
$2$	−1.31596	−0.930527	−0.465263	+	0.885172i	$0.654040\pi$
$3$	1.87410	1.08201	0.541005	−	0.841019i	$-0.318044\pi$
$3$	1.87410	1.08201	0.541005	+	0.841019i	$0.318044\pi$
$4$	−0.268240	−0.134120
$5$	−1.28432	−0.574367	−0.287184	−	0.957876i	$-0.592719\pi$
$5$	−1.28432	−0.574367	−0.287184	+	0.957876i	$0.592719\pi$
$6$	−2.46624	−1.00684
$7$	−2.16419	−0.817986	−0.408993	−	0.912538i	$-0.634120\pi$
$7$	−2.16419	−0.817986	−0.408993	+	0.912538i	$0.634120\pi$
$8$	2.98492	1.05533
$9$	0.512236	0.170745
$10$	1.69012	0.534464
$11$	1.34988	0.407005	0.203503	−	0.979074i	$-0.434767\pi$
$11$	1.34988	0.407005	0.203503	+	0.979074i	$0.434767\pi$
$12$	−0.502707	−0.145119
$13$	5.76832	1.59984	0.799922	−	0.600103i	$-0.204874\pi$
$13$	5.76832	1.59984	0.799922	+	0.600103i	$0.204874\pi$
$14$	2.84799	0.761158
$15$	−2.40695	−0.621471
$16$	−3.39157	−0.847892
$17$	3.88611	0.942519	0.471260	−	0.881995i	$-0.343799\pi$
$17$	3.88611	0.942519	0.471260	+	0.881995i	$0.343799\pi$
$18$	−0.674084	−0.158883
$19$	8.19658	1.88042	0.940212	−	0.340588i	$-0.110626\pi$
$19$	8.19658	1.88042	0.940212	+	0.340588i	$0.110626\pi$
$20$	0.344507	0.0770340
$21$	−4.05589	−0.885069
$22$	−1.77640	−0.378729
$23$	5.98263	1.24746	0.623732	−	0.781638i	$-0.285616\pi$
$23$	5.98263	1.24746	0.623732	+	0.781638i	$0.285616\pi$
$24$	5.59403	1.14188
$25$	−3.35051	−0.670102
$26$	−7.59090	−1.48870
$27$	−4.66231	−0.897262
$28$	0.580520	0.109708
$29$	4.67769	0.868625	0.434312	−	0.900762i	$-0.356991\pi$
$29$	4.67769	0.868625	0.434312	+	0.900762i	$0.356991\pi$
$30$	3.16745	0.578295
$31$	−4.78270	−0.858997	−0.429499	−	0.903067i	$-0.641310\pi$
$31$	−4.78270	−0.858997	−0.429499	+	0.903067i	$0.641310\pi$
$32$	−1.50666	−0.266343
$33$	2.52981	0.440384
$34$	−5.11398	−0.877040
$35$	2.77952	0.469824
$36$	−0.137402	−0.0229003
$37$	1.37721	0.226413	0.113206	−	0.993572i	$-0.463888\pi$
$37$	1.37721	0.226413	0.113206	+	0.993572i	$0.463888\pi$
$38$	−10.7864	−1.74979
$39$	10.8104	1.73105
$40$	−3.83361	−0.606146
$41$	−6.84895	−1.06963	−0.534813	−	0.844970i	$-0.679618\pi$
$41$	−6.84895	−1.06963	−0.534813	+	0.844970i	$0.679618\pi$
$42$	5.33741	0.823580
$43$	−2.99849	−0.457265	−0.228632	−	0.973513i	$-0.573425\pi$
$43$	−2.99849	−0.457265	−0.228632	+	0.973513i	$0.573425\pi$
$44$	−0.362092	−0.0545874
$45$	−0.657877	−0.0980705
$46$	−7.87292	−1.16080
$47$	11.2068	1.63468	0.817342	−	0.576153i	$-0.195447\pi$
$47$	11.2068	1.63468	0.817342	+	0.576153i	$0.195447\pi$
$48$	−6.35612	−0.917428
$49$	−2.31630	−0.330899
$50$	4.40915	0.623548
$51$	7.28294	1.01982
$52$	−1.54729	−0.214571
$53$	3.69504	0.507552	0.253776	−	0.967263i	$-0.418327\pi$
$53$	3.69504	0.507552	0.253776	+	0.967263i	$0.418327\pi$
$54$	6.13543	0.834926
$55$	−1.73369	−0.233770
$56$	−6.45993	−0.863244
$57$	15.3612	2.03464
$58$	−6.15567	−0.808279
$59$	−2.10575	−0.274146	−0.137073	−	0.990561i	$-0.543769\pi$
$59$	−2.10575	−0.274146	−0.137073	+	0.990561i	$0.543769\pi$
$60$	0.645638	0.0833515
$61$	9.36043	1.19848	0.599240	−	0.800569i	$-0.295470\pi$
$61$	9.36043	1.19848	0.599240	+	0.800569i	$0.295470\pi$
$62$	6.29385	0.799320
$63$	−1.10857	−0.139667
$64$	8.76585	1.09573
$65$	−7.40840	−0.918898
$66$	−3.32914	−0.409789
$67$	−4.89108	−0.597540	−0.298770	−	0.954325i	$-0.596576\pi$
$67$	−4.89108	−0.597540	−0.298770	+	0.954325i	$0.596576\pi$
$68$	−1.04241	−0.126410
$69$	11.2120	1.34977
$70$	−3.65774	−0.437184
$71$	8.93427	1.06030	0.530152	−	0.847903i	$-0.322135\pi$
$71$	8.93427	1.06030	0.530152	+	0.847903i	$0.322135\pi$
$72$	1.52898	0.180192
$73$	−2.75468	−0.322410	−0.161205	−	0.986921i	$-0.551538\pi$
$73$	−2.75468	−0.322410	−0.161205	+	0.986921i	$0.551538\pi$
$74$	−1.81236	−0.210683
$75$	−6.27918	−0.725057
$76$	−2.19865	−0.252202
$77$	−2.92140	−0.332924
$78$	−14.2261	−1.61079
$79$	−16.4012	−1.84528	−0.922640	−	0.385662i	$-0.873973\pi$
$79$	−16.4012	−1.84528	−0.922640	+	0.385662i	$0.873973\pi$
$80$	4.35587	0.487001
$81$	−10.2743	−1.14159
$82$	9.01298	0.995317
$83$	15.1179	1.65941	0.829703	−	0.558204i	$-0.188510\pi$
$83$	15.1179	1.65941	0.829703	+	0.558204i	$0.188510\pi$
$84$	1.08795	0.118705
$85$	−4.99102	−0.541352
$86$	3.94590	0.425497
$87$	8.76644	0.939861
$88$	4.02929	0.429524
$89$	−6.38372	−0.676673	−0.338337	−	0.941025i	$-0.609864\pi$
$89$	−6.38372	−0.676673	−0.338337	+	0.941025i	$0.609864\pi$
$90$	0.865742	0.0912572
$91$	−12.4837	−1.30865
$92$	−1.60478	−0.167310
$93$	−8.96323	−0.929444
$94$	−14.7478	−1.52112
$95$	−10.5271	−1.08005
$96$	−2.82363	−0.288185
$97$	−1.98117	−0.201157	−0.100579	−	0.994929i	$-0.532069\pi$
$97$	−1.98117	−0.201157	−0.100579	+	0.994929i	$0.532069\pi$
$98$	3.04816	0.307911
$99$	0.691458	0.0694942
$100$	0.898740	0.0898740
$101$	14.7149	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$101$	14.7149	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$102$	−9.58408	−0.948965
$103$	−15.5971	−1.53683	−0.768415	−	0.639952i	$-0.778954\pi$
$103$	−15.5971	−1.53683	−0.768415	+	0.639952i	$0.778954\pi$
$104$	17.2180	1.68836
$105$	5.20908	0.508354
$106$	−4.86253	−0.472291
$107$	−17.4886	−1.69068	−0.845342	−	0.534226i	$-0.820603\pi$
$107$	−17.4886	−1.69068	−0.845342	+	0.534226i	$0.820603\pi$
$108$	1.25062	0.120341
$109$	0.420552	0.0402816	0.0201408	−	0.999797i	$-0.493589\pi$
$109$	0.420552	0.0402816	0.0201408	+	0.999797i	$0.493589\pi$
$110$	2.28147	0.217530
$111$	2.58103	0.244981
$112$	7.33999	0.693564
$113$	−13.5198	−1.27183	−0.635916	−	0.771758i	$-0.719377\pi$
$113$	−13.5198	−1.27183	−0.635916	+	0.771758i	$0.719377\pi$
$114$	−20.2148	−1.89329
$115$	−7.68364	−0.716503
$116$	−1.25474	−0.116500
$117$	2.95474	0.273166
$118$	2.77109	0.255100
$119$	−8.41026	−0.770967
$120$	−7.18455	−0.655856
$121$	−9.17782	−0.834347
$122$	−12.3180	−1.11522
$123$	−12.8356	−1.15735
$124$	1.28291	0.115209
$125$	10.7248	0.959252
$126$	1.45884	0.129964
$127$	19.9804	1.77297	0.886486	−	0.462756i	$-0.153139\pi$
$127$	19.9804	1.77297	0.886486	+	0.462756i	$0.153139\pi$
$128$	−8.52222	−0.753265
$129$	−5.61945	−0.494765
$130$	9.74918	0.855060
$131$	9.82761	0.858643	0.429321	−	0.903152i	$-0.358753\pi$
$131$	9.82761	0.858643	0.429321	+	0.903152i	$0.358753\pi$
$132$	−0.678595	−0.0590641
$133$	−17.7389	−1.53816
$134$	6.43648	0.556027
$135$	5.98792	0.515358
$136$	11.5997	0.994668
$137$	−1.20878	−0.103273	−0.0516367	−	0.998666i	$-0.516444\pi$
$137$	−1.20878	−0.103273	−0.0516367	+	0.998666i	$0.516444\pi$
$138$	−14.7546	−1.25600
$139$	−5.41003	−0.458873	−0.229436	−	0.973324i	$-0.573688\pi$
$139$	−5.41003	−0.458873	−0.229436	+	0.973324i	$0.573688\pi$
$140$	−0.745576	−0.0630127
$141$	21.0027	1.76874
$142$	−11.7572	−0.986641
$143$	7.78656	0.651145
$144$	−1.73728	−0.144774
$145$	−6.00767	−0.498910
$146$	3.62505	0.300012
$147$	−4.34096	−0.358036
$148$	−0.369423	−0.0303664
$149$	19.5581	1.60226	0.801131	−	0.598489i	$-0.204232\pi$
$149$	19.5581	1.60226	0.801131	+	0.598489i	$0.204232\pi$
$150$	8.26317	0.674685
$151$	−18.5723	−1.51139	−0.755696	−	0.654922i	$-0.772701\pi$
$151$	−18.5723	−1.51139	−0.755696	+	0.654922i	$0.772701\pi$
$152$	24.4661	1.98447
$153$	1.99060	0.160931
$154$	3.84446	0.309795
$155$	6.14253	0.493380
$156$	−2.89977	−0.232168
$157$	−6.01569	−0.480105	−0.240052	−	0.970760i	$-0.577165\pi$
$157$	−6.01569	−0.480105	−0.240052	+	0.970760i	$0.577165\pi$
$158$	21.5834	1.71708
$159$	6.92485	0.549177
$160$	1.93504	0.152978
$161$	−12.9475	−1.02041
$162$	13.5206	1.06228
$163$	16.2591	1.27351	0.636756	−	0.771065i	$-0.280276\pi$
$163$	16.2591	1.27351	0.636756	+	0.771065i	$0.280276\pi$
$164$	1.83716	0.143458
$165$	−3.24910	−0.252942
$166$	−19.8946	−1.54412
$167$	−3.01050	−0.232960	−0.116480	−	0.993193i	$-0.537161\pi$
$167$	−3.01050	−0.232960	−0.116480	+	0.993193i	$0.537161\pi$
$168$	−12.1065	−0.934038
$169$	20.2736	1.55950
$170$	6.56800	0.503743
$171$	4.19858	0.321074
$172$	0.804313	0.0613283
$173$	−1.41746	−0.107767	−0.0538836	−	0.998547i	$-0.517160\pi$
$173$	−1.41746	−0.107767	−0.0538836	+	0.998547i	$0.517160\pi$
$174$	−11.5363	−0.874566
$175$	7.25113	0.548134
$176$	−4.57822	−0.345096
$177$	−3.94638	−0.296628
$178$	8.40075	0.629663
$179$	−2.97671	−0.222490	−0.111245	−	0.993793i	$-0.535484\pi$
$179$	−2.97671	−0.222490	−0.111245	+	0.993793i	$0.535484\pi$
$180$	0.176469	0.0131532
$181$	−7.53706	−0.560225	−0.280113	−	0.959967i	$-0.590372\pi$
$181$	−7.53706	−0.560225	−0.280113	+	0.959967i	$0.590372\pi$
$182$	16.4281	1.21773
$183$	17.5423	1.29677
$184$	17.8577	1.31649
$185$	−1.76879	−0.130044
$186$	11.7953	0.864872
$187$	5.24579	0.383610
$188$	−3.00611	−0.219243
$189$	10.0901	0.733947
$190$	13.8532	1.00502
$191$	−13.3784	−0.968026	−0.484013	−	0.875061i	$-0.660821\pi$
$191$	−13.3784	−0.968026	−0.484013	+	0.875061i	$0.660821\pi$
$192$	16.4280	1.18559
$193$	−20.3104	−1.46197	−0.730986	−	0.682393i	$-0.760939\pi$
$193$	−20.3104	−1.46197	−0.730986	+	0.682393i	$0.760939\pi$
$194$	2.60715	0.187182
$195$	−13.8840	−0.994257
$196$	0.621322	0.0443801
$197$	−4.47740	−0.319002	−0.159501	−	0.987198i	$-0.550988\pi$
$197$	−4.47740	−0.319002	−0.159501	+	0.987198i	$0.550988\pi$
$198$	−0.909934	−0.0646662
$199$	2.12654	0.150747	0.0753733	−	0.997155i	$-0.475985\pi$
$199$	2.12654	0.150747	0.0753733	+	0.997155i	$0.475985\pi$
$200$	−10.0010	−0.707178
$201$	−9.16635	−0.646545
$202$	−19.3643	−1.36247
$203$	−10.1234	−0.710523
$204$	−1.95357	−0.136777
$205$	8.79628	0.614359
$206$	20.5252	1.43006
$207$	3.06452	0.212999
$208$	−19.5637	−1.35650
$209$	11.0644	0.765343
$210$	−6.85496	−0.473037
$211$	18.7398	1.29010	0.645051	−	0.764140i	$-0.276836\pi$
$211$	18.7398	1.29010	0.645051	+	0.764140i	$0.276836\pi$
$212$	−0.991155	−0.0680728
$213$	16.7437	1.14726
$214$	23.0143	1.57323
$215$	3.85103	0.262638
$216$	−13.9166	−0.946906
$217$	10.3506	0.702648
$218$	−0.553432	−0.0374831
$219$	−5.16253	−0.348851
$220$	0.465044	0.0313532
$221$	22.4163	1.50788
$222$	−3.39654	−0.227961
$223$	11.1622	0.747474	0.373737	−	0.927535i	$-0.378076\pi$
$223$	11.1622	0.747474	0.373737	+	0.927535i	$0.378076\pi$
$224$	3.26070	0.217864
$225$	−1.71625	−0.114417
$226$	17.7915	1.18347
$227$	1.67851	0.111407	0.0557033	−	0.998447i	$-0.482260\pi$
$227$	1.67851	0.111407	0.0557033	+	0.998447i	$0.482260\pi$
$228$	−4.12048	−0.272885
$229$	14.2489	0.941593	0.470797	−	0.882242i	$-0.343967\pi$
$229$	14.2489	0.941593	0.470797	+	0.882242i	$0.343967\pi$
$230$	10.1114	0.666725
$231$	−5.47498	−0.360227
$232$	13.9625	0.916685
$233$	0.364166	0.0238573	0.0119287	−	0.999929i	$-0.496203\pi$
$233$	0.364166	0.0238573	0.0119287	+	0.999929i	$0.496203\pi$
$234$	−3.88833	−0.254188
$235$	−14.3932	−0.938909
$236$	0.564846	0.0367683
$237$	−30.7374	−1.99661
$238$	11.0676	0.717406
$239$	12.1754	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$239$	12.1754	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$240$	8.16332	0.526940
$241$	−3.54661	−0.228457	−0.114229	−	0.993454i	$-0.536440\pi$
$241$	−3.54661	−0.228457	−0.114229	+	0.993454i	$0.536440\pi$
$242$	12.0777	0.776382
$243$	−5.26814	−0.337951
$244$	−2.51084	−0.160740
$245$	2.97487	0.190058
$246$	16.8912	1.07694
$247$	47.2805	3.00839
$248$	−14.2760	−0.906525
$249$	28.3324	1.79549
$250$	−14.1134	−0.892610
$251$	20.7483	1.30962	0.654810	−	0.755793i	$-0.272749\pi$
$251$	20.7483	1.30962	0.654810	+	0.755793i	$0.272749\pi$
$252$	0.297363	0.0187321
$253$	8.07585	0.507724
$254$	−26.2934	−1.64980
$255$	−9.35365	−0.585748
$256$	−6.31677	−0.394798
$257$	−20.8786	−1.30237	−0.651186	−	0.758918i	$-0.725728\pi$
$257$	−20.8786	−1.30237	−0.651186	+	0.758918i	$0.725728\pi$
$258$	7.39499	0.460392
$259$	−2.98055	−0.185202
$260$	1.98722	0.123242
$261$	2.39608	0.148314
$262$	−12.9328	−0.798990
$263$	−26.9177	−1.65982	−0.829908	−	0.557901i	$-0.811607\pi$
$263$	−26.9177	−1.65982	−0.829908	+	0.557901i	$0.811607\pi$
$264$	7.55128	0.464749
$265$	−4.74562	−0.291521
$266$	23.3438	1.43130
$267$	−11.9637	−0.732167
$268$	1.31198	0.0801420
$269$	−21.5860	−1.31612	−0.658062	−	0.752964i	$-0.728623\pi$
$269$	−21.5860	−1.31612	−0.658062	+	0.752964i	$0.728623\pi$
$270$	−7.87988	−0.479554
$271$	−15.1588	−0.920831	−0.460415	−	0.887704i	$-0.652299\pi$
$271$	−15.1588	−0.920831	−0.460415	+	0.887704i	$0.652299\pi$
$272$	−13.1800	−0.799155
$273$	−23.3957	−1.41597
$274$	1.59072	0.0960987
$275$	−4.52280	−0.272735
$276$	−3.00751	−0.181031
$277$	−27.5176	−1.65337	−0.826687	−	0.562663i	$-0.809777\pi$
$277$	−27.5176	−1.65337	−0.826687	+	0.562663i	$0.809777\pi$
$278$	7.11940	0.426993
$279$	−2.44987	−0.146670
$280$	8.29664	0.495819
$281$	−18.2568	−1.08911	−0.544555	−	0.838725i	$-0.683301\pi$
$281$	−18.2568	−1.08911	−0.544555	+	0.838725i	$0.683301\pi$
$282$	−27.6388	−1.64586
$283$	2.83505	0.168526	0.0842632	−	0.996444i	$-0.473146\pi$
$283$	2.83505	0.168526	0.0842632	+	0.996444i	$0.473146\pi$
$284$	−2.39653	−0.142208
$285$	−19.7287	−1.16863
$286$	−10.2468	−0.605908
$287$	14.8224	0.874940
$288$	−0.771765	−0.0454767
$289$	−1.89818	−0.111657
$290$	7.90587	0.464249
$291$	−3.71290	−0.217654
$292$	0.738913	0.0432416
$293$	−9.74209	−0.569139	−0.284570	−	0.958655i	$-0.591851\pi$
$293$	−9.74209	−0.569139	−0.284570	+	0.958655i	$0.591851\pi$
$294$	5.71255	0.333162
$295$	2.70447	0.157460
$296$	4.11087	0.238940
$297$	−6.29357	−0.365190
$298$	−25.7378	−1.49095
$299$	34.5097	1.99575
$300$	1.68432	0.0972445
$301$	6.48928	0.374036
$302$	24.4405	1.40639
$303$	27.5771	1.58427
$304$	−27.7993	−1.59440
$305$	−12.0218	−0.688368
$306$	−2.61956	−0.149750
$307$	10.6309	0.606740	0.303370	−	0.952873i	$-0.401888\pi$
$307$	10.6309	0.606740	0.303370	+	0.952873i	$0.401888\pi$
$308$	0.783635	0.0446517
$309$	−29.2305	−1.66286
$310$	−8.08335	−0.459103
$311$	23.1040	1.31011	0.655054	−	0.755582i	$-0.272646\pi$
$311$	23.1040	1.31011	0.655054	+	0.755582i	$0.272646\pi$
$312$	32.2682	1.82682
$313$	22.3419	1.26284	0.631420	−	0.775441i	$-0.282472\pi$
$313$	22.3419	1.26284	0.631420	+	0.775441i	$0.282472\pi$
$314$	7.91643	0.446750
$315$	1.42377	0.0802202
$316$	4.39945	0.247489
$317$	−34.0695	−1.91353	−0.956766	−	0.290858i	$-0.906059\pi$
$317$	−34.0695	−1.91353	−0.956766	+	0.290858i	$0.906059\pi$
$318$	−9.11286	−0.511024
$319$	6.31433	0.353535
$320$	−11.2582	−0.629352
$321$	−32.7753	−1.82934
$322$	17.0385	0.949517
$323$	31.8528	1.77234
$324$	2.75598	0.153110
$325$	−19.3268	−1.07206
$326$	−21.3964	−1.18504
$327$	0.788155	0.0435851
$328$	−20.4436	−1.12881
$329$	−24.2537	−1.33715
$330$	4.27569	0.235369
$331$	17.2304	0.947067	0.473534	−	0.880776i	$-0.342978\pi$
$331$	17.2304	0.947067	0.473534	+	0.880776i	$0.342978\pi$
$332$	−4.05522	−0.222559
$333$	0.705458	0.0386589
$334$	3.96171	0.216775
$335$	6.28173	0.343208
$336$	13.7558	0.750443
$337$	−0.449575	−0.0244899	−0.0122450	−	0.999925i	$-0.503898\pi$
$337$	−0.449575	−0.0244899	−0.0122450	+	0.999925i	$0.503898\pi$
$338$	−26.6793	−1.45116
$339$	−25.3373	−1.37613
$340$	1.33879	0.0726060
$341$	−6.45608	−0.349616
$342$	−5.52518	−0.298768
$343$	20.1622	1.08866
$344$	−8.95024	−0.482565
$345$	−14.3999	−0.775263
$346$	1.86532	0.100280
$347$	−14.6995	−0.789111	−0.394556	−	0.918872i	$-0.629101\pi$
$347$	−14.6995	−0.789111	−0.394556	+	0.918872i	$0.629101\pi$
$348$	−2.35150	−0.126054
$349$	−8.34547	−0.446723	−0.223361	−	0.974736i	$-0.571703\pi$
$349$	−8.34547	−0.446723	−0.223361	+	0.974736i	$0.571703\pi$
$350$	−9.54223	−0.510054
$351$	−26.8937	−1.43548
$352$	−2.03382	−0.108403
$353$	−16.7434	−0.891162	−0.445581	−	0.895242i	$-0.647003\pi$
$353$	−16.7434	−0.891162	−0.445581	+	0.895242i	$0.647003\pi$
$354$	5.19329	0.276021
$355$	−11.4745	−0.609003
$356$	1.71237	0.0907553
$357$	−15.7616	−0.834194
$358$	3.91725	0.207033
$359$	7.36108	0.388503	0.194251	−	0.980952i	$-0.437772\pi$
$359$	7.36108	0.388503	0.194251	+	0.980952i	$0.437772\pi$
$360$	−1.96371	−0.103497
$361$	48.1840	2.53600
$362$	9.91850	0.521305
$363$	−17.2001	−0.902771
$364$	3.34863	0.175516
$365$	3.53790	0.185182
$366$	−23.0851	−1.20668
$367$	−9.58293	−0.500225	−0.250112	−	0.968217i	$-0.580468\pi$
$367$	−9.58293	−0.500225	−0.250112	+	0.968217i	$0.580468\pi$
$368$	−20.2905	−1.05772
$369$	−3.50828	−0.182634
$370$	2.32766	0.121009
$371$	−7.99675	−0.415171
$372$	2.40429	0.124657
$373$	−20.9025	−1.08229	−0.541146	−	0.840929i	$-0.682009\pi$
$373$	−20.9025	−1.08229	−0.541146	+	0.840929i	$0.682009\pi$
$374$	−6.90327	−0.356960
$375$	20.0992	1.03792
$376$	33.4515	1.72513
$377$	26.9824	1.38967
$378$	−13.2782	−0.682958
$379$	0.709146	0.0364264	0.0182132	−	0.999834i	$-0.494202\pi$
$379$	0.709146	0.0364264	0.0182132	+	0.999834i	$0.494202\pi$
$380$	2.82378	0.144857
$381$	37.4451	1.91837
$382$	17.6055	0.900774
$383$	−7.16079	−0.365899	−0.182950	−	0.983122i	$-0.558564\pi$
$383$	−7.16079	−0.365899	−0.182950	+	0.983122i	$0.558564\pi$
$384$	−15.9715	−0.815040
$385$	3.75202	0.191221
$386$	26.7277	1.36040
$387$	−1.53593	−0.0780758
$388$	0.531428	0.0269792
$389$	9.67348	0.490465	0.245233	−	0.969464i	$-0.421136\pi$
$389$	9.67348	0.490465	0.245233	+	0.969464i	$0.421136\pi$
$390$	18.2709	0.925183
$391$	23.2491	1.17576
$392$	−6.91396	−0.349208
$393$	18.4179	0.929060
$394$	5.89210	0.296840
$395$	21.0645	1.05987
$396$	−0.185476	−0.00932054
$397$	−0.0784589	−0.00393774	−0.00196887	−	0.999998i	$-0.500627\pi$
$397$	−0.0784589	−0.00393774	−0.00196887	+	0.999998i	$0.500627\pi$
$398$	−2.79845	−0.140274
$399$	−33.2445	−1.66430
$400$	11.3635	0.568174
$401$	6.22825	0.311024	0.155512	−	0.987834i	$-0.450297\pi$
$401$	6.22825	0.311024	0.155512	+	0.987834i	$0.450297\pi$
$402$	12.0626	0.601627
$403$	−27.5881	−1.37426
$404$	−3.94712	−0.196376
$405$	13.1956	0.655693
$406$	13.3220	0.661160
$407$	1.85908	0.0921510
$408$	21.7390	1.07624
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	−11.5756	−0.571677
$411$	−2.26538	−0.111743
$412$	4.18376	0.206119
$413$	4.55724	0.224247
$414$	−4.03279	−0.198201
$415$	−19.4163	−0.953109
$416$	−8.69091	−0.426107
$417$	−10.1389	−0.496505
$418$	−14.5604	−0.712172
$419$	29.5594	1.44407	0.722035	−	0.691856i	$-0.243207\pi$
$419$	29.5594	1.44407	0.722035	+	0.691856i	$0.243207\pi$
$420$	−1.39728	−0.0681804
$421$	1.75282	0.0854270	0.0427135	−	0.999087i	$-0.486400\pi$
$421$	1.75282	0.0854270	0.0427135	+	0.999087i	$0.486400\pi$
$422$	−24.6609	−1.20047
$423$	5.74054	0.279115
$424$	11.0294	0.535635
$425$	−13.0204	−0.631584
$426$	−22.0341	−1.06755
$427$	−20.2577	−0.980340
$428$	4.69113	0.226754
$429$	14.5928	0.704545
$430$	−5.06781	−0.244392
$431$	3.20558	0.154408	0.0772038	−	0.997015i	$-0.475401\pi$
$431$	3.20558	0.154408	0.0772038	+	0.997015i	$0.475401\pi$
$432$	15.8125	0.760781
$433$	−23.6552	−1.13680	−0.568398	−	0.822753i	$-0.692437\pi$
$433$	−23.6552	−1.13680	−0.568398	+	0.822753i	$0.692437\pi$
$434$	−13.6211	−0.653832
$435$	−11.2589	−0.539825
$436$	−0.112809	−0.00540256
$437$	49.0371	2.34576
$438$	6.79370	0.324615
$439$	0.923939	0.0440972	0.0220486	−	0.999757i	$-0.492981\pi$
$439$	0.923939	0.0440972	0.0220486	+	0.999757i	$0.492981\pi$
$440$	−5.17492	−0.246705
$441$	−1.18649	−0.0564995
$442$	−29.4991	−1.40313
$443$	11.4658	0.544756	0.272378	−	0.962190i	$-0.412190\pi$
$443$	11.4658	0.544756	0.272378	+	0.962190i	$0.412190\pi$
$444$	−0.692334	−0.0328567
$445$	8.19877	0.388659
$446$	−14.6890	−0.695545
$447$	36.6538	1.73366
$448$	−18.9709	−0.896292
$449$	12.3343	0.582091	0.291045	−	0.956709i	$-0.405997\pi$
$449$	12.3343	0.582091	0.291045	+	0.956709i	$0.405997\pi$
$450$	2.25852	0.106468
$451$	−9.24529	−0.435344
$452$	3.62653	0.170578
$453$	−34.8063	−1.63534
$454$	−2.20886	−0.103667
$455$	16.0332	0.751646
$456$	45.8519	2.14721
$457$	−9.31360	−0.435672	−0.217836	−	0.975985i	$-0.569900\pi$
$457$	−9.31360	−0.435672	−0.217836	+	0.975985i	$0.569900\pi$
$458$	−18.7510	−0.876178
$459$	−18.1182	−0.845687
$460$	2.06105	0.0960972
$461$	−14.3508	−0.668385	−0.334192	−	0.942505i	$-0.608464\pi$
$461$	−14.3508	−0.668385	−0.334192	+	0.942505i	$0.608464\pi$
$462$	7.20488	0.335201
$463$	21.3247	0.991042	0.495521	−	0.868596i	$-0.334977\pi$
$463$	21.3247	0.991042	0.495521	+	0.868596i	$0.334977\pi$
$464$	−15.8647	−0.736500
$465$	11.5117	0.533842
$466$	−0.479229	−0.0221999
$467$	1.87580	0.0868017	0.0434009	−	0.999058i	$-0.486181\pi$
$467$	1.87580	0.0868017	0.0434009	+	0.999058i	$0.486181\pi$
$468$	−0.792578	−0.0366369
$469$	10.5852	0.488780
$470$	18.9409	0.873680
$471$	−11.2740	−0.519478
$472$	−6.28550	−0.289314
$473$	−4.04761	−0.186109
$474$	40.4494	1.85790
$475$	−27.4627	−1.26008
$476$	2.25596	0.103402
$477$	1.89273	0.0866621
$478$	−16.0224	−0.732847
$479$	32.8340	1.50022	0.750111	−	0.661312i	$-0.230000\pi$
$479$	32.8340	1.50022	0.750111	+	0.661312i	$0.230000\pi$
$480$	3.62645	0.165524
$481$	7.94421	0.362225
$482$	4.66721	0.212586
$483$	−24.2649	−1.10409
$484$	2.46185	0.111902
$485$	2.54446	0.115538
$486$	6.93268	0.314473
$487$	−32.8387	−1.48806	−0.744032	−	0.668144i	$-0.767089\pi$
$487$	−32.8387	−1.48806	−0.744032	+	0.668144i	$0.767089\pi$
$488$	27.9401	1.26479
$489$	30.4711	1.37795
$490$	−3.91483	−0.176854
$491$	−14.9471	−0.674553	−0.337276	−	0.941406i	$-0.609506\pi$
$491$	−14.9471	−0.674553	−0.337276	+	0.941406i	$0.609506\pi$
$492$	3.44301	0.155223
$493$	18.1780	0.818696
$494$	−62.2195	−2.79939
$495$	−0.888057	−0.0399152
$496$	16.2208	0.728337
$497$	−19.3354	−0.867313
$498$	−37.2844	−1.67076
$499$	15.6763	0.701767	0.350883	−	0.936419i	$-0.385881\pi$
$499$	15.6763	0.701767	0.350883	+	0.936419i	$0.385881\pi$
$500$	−2.87681	−0.128655
$501$	−5.64197	−0.252065
$502$	−27.3040	−1.21864
$503$	−21.6618	−0.965850	−0.482925	−	0.875662i	$-0.660426\pi$
$503$	−21.6618	−0.965850	−0.482925	+	0.875662i	$0.660426\pi$
$504$	−3.30900	−0.147395
$505$	−18.8987	−0.840981
$506$	−10.6275	−0.472451
$507$	37.9946	1.68740
$508$	−5.35953	−0.237791
$509$	20.1682	0.893938	0.446969	−	0.894549i	$-0.352503\pi$
$509$	20.1682	0.893938	0.446969	+	0.894549i	$0.352503\pi$
$510$	12.3091	0.545055
$511$	5.96163	0.263727
$512$	25.3571	1.12063
$513$	−38.2150	−1.68723
$514$	27.4755	1.21189
$515$	20.0318	0.882705
$516$	1.50736	0.0663578
$517$	15.1279	0.665325
$518$	3.92229	0.172336
$519$	−2.65645	−0.116605
$520$	−22.1135	−0.969740
$521$	−31.8810	−1.39673	−0.698365	−	0.715742i	$-0.746089\pi$
$521$	−31.8810	−1.39673	−0.698365	+	0.715742i	$0.746089\pi$
$522$	−3.15315	−0.138010
$523$	−25.0861	−1.09694	−0.548468	−	0.836171i	$-0.684789\pi$
$523$	−25.0861	−1.09694	−0.548468	+	0.836171i	$0.684789\pi$
$524$	−2.63615	−0.115161
$525$	13.5893	0.593086
$526$	35.4227	1.54450
$527$	−18.5861	−0.809622
$528$	−8.58003	−0.373398
$529$	12.7919	0.556168
$530$	6.24507	0.271269
$531$	−1.07864	−0.0468090
$532$	4.75828	0.206298
$533$	−39.5070	−1.71124
$534$	15.7438	0.681301
$535$	22.4610	0.971073
$536$	−14.5995	−0.630602
$537$	−5.57865	−0.240736
$538$	28.4064	1.22469
$539$	−3.12673	−0.134678
$540$	−1.60620	−0.0691197
$541$	−25.0444	−1.07674	−0.538372	−	0.842707i	$-0.680960\pi$
$541$	−25.0444	−1.07674	−0.538372	+	0.842707i	$0.680960\pi$
$542$	19.9484	0.856858
$543$	−14.1252	−0.606169
$544$	−5.85504	−0.251033
$545$	−0.540125	−0.0231364
$546$	30.7879	1.31760
$547$	−41.0332	−1.75445	−0.877227	−	0.480077i	$-0.840609\pi$
$547$	−41.0332	−1.75445	−0.877227	+	0.480077i	$0.840609\pi$
$548$	0.324244	0.0138510
$549$	4.79475	0.204635
$550$	5.95184	0.253787
$551$	38.3411	1.63338
$552$	33.4670	1.42445
$553$	35.4953	1.50941
$554$	36.2122	1.53851
$555$	−3.31488	−0.140709
$556$	1.45118	0.0615439
$557$	−16.7903	−0.711427	−0.355713	−	0.934595i	$-0.615762\pi$
$557$	−16.7903	−0.711427	−0.355713	+	0.934595i	$0.615762\pi$
$558$	3.22394	0.136480
$559$	−17.2962	−0.731553
$560$	−9.42692	−0.398360
$561$	9.83111	0.415070
$562$	24.0253	1.01345
$563$	−1.17601	−0.0495628	−0.0247814	−	0.999693i	$-0.507889\pi$
$563$	−1.17601	−0.0495628	−0.0247814	+	0.999693i	$0.507889\pi$
$564$	−5.63375	−0.237224
$565$	17.3638	0.730498
$566$	−3.73083	−0.156818
$567$	22.2356	0.933805
$568$	26.6681	1.11897
$569$	−23.8619	−1.00034	−0.500171	−	0.865927i	$-0.666729\pi$
$569$	−23.8619	−1.00034	−0.500171	+	0.865927i	$0.666729\pi$
$570$	25.9623	1.08744
$571$	5.19234	0.217293	0.108646	−	0.994080i	$-0.465348\pi$
$571$	5.19234	0.217293	0.108646	+	0.994080i	$0.465348\pi$
$572$	−2.08866	−0.0873314
$573$	−25.0724	−1.04741
$574$	−19.5058	−0.814155
$575$	−20.0449	−0.835929
$576$	4.49018	0.187091
$577$	37.5371	1.56269	0.781345	−	0.624099i	$-0.214534\pi$
$577$	37.5371	1.56269	0.781345	+	0.624099i	$0.214534\pi$
$578$	2.49793	0.103900
$579$	−38.0636	−1.58187
$580$	1.61149	0.0669136
$581$	−32.7180	−1.35737
$582$	4.88604	0.202533
$583$	4.98787	0.206576
$584$	−8.22249	−0.340249
$585$	−3.79485	−0.156898
$586$	12.8202	0.529599
$587$	32.7244	1.35068	0.675341	−	0.737506i	$-0.263996\pi$
$587$	32.7244	1.35068	0.675341	+	0.737506i	$0.263996\pi$
$588$	1.16442	0.0480198
$589$	−39.2018	−1.61528
$590$	−3.55898	−0.146521
$591$	−8.39108	−0.345163
$592$	−4.67091	−0.191973
$593$	−14.1286	−0.580194	−0.290097	−	0.956997i	$-0.593687\pi$
$593$	−14.1286	−0.580194	−0.290097	+	0.956997i	$0.593687\pi$
$594$	8.28211	0.339819
$595$	10.8015	0.442818
$596$	−5.24626	−0.214895
$597$	3.98535	0.163109
$598$	−45.4136	−1.85710
$599$	0.927088	0.0378798	0.0189399	−	0.999821i	$-0.493971\pi$
$599$	0.927088	0.0378798	0.0189399	+	0.999821i	$0.493971\pi$
$600$	−18.7429	−0.765174
$601$	−27.5669	−1.12448	−0.562238	−	0.826975i	$-0.690060\pi$
$601$	−27.5669	−1.12448	−0.562238	+	0.826975i	$0.690060\pi$
$602$	−8.53966	−0.348051
$603$	−2.50539	−0.102027
$604$	4.98182	0.202708
$605$	11.7873	0.479221
$606$	−36.2905	−1.47420
$607$	21.9170	0.889583	0.444791	−	0.895634i	$-0.353278\pi$
$607$	21.9170	0.889583	0.444791	+	0.895634i	$0.353278\pi$
$608$	−12.3495	−0.500837
$609$	−18.9722	−0.768793
$610$	15.8203	0.640545
$611$	64.6446	2.61524
$612$	−0.533958	−0.0215840
$613$	21.8913	0.884183	0.442091	−	0.896970i	$-0.354237\pi$
$613$	21.8913	0.884183	0.442091	+	0.896970i	$0.354237\pi$
$614$	−13.9899	−0.564588
$615$	16.4851	0.664742
$616$	−8.72015	−0.351345
$617$	−31.8798	−1.28343	−0.641717	−	0.766941i	$-0.721778\pi$
$617$	−31.8798	−1.28343	−0.641717	+	0.766941i	$0.721778\pi$
$618$	38.4663	1.54734
$619$	−0.651683	−0.0261933	−0.0130967	−	0.999914i	$-0.504169\pi$
$619$	−0.651683	−0.0261933	−0.0130967	+	0.999914i	$0.504169\pi$
$620$	−1.64767	−0.0661720
$621$	−27.8929	−1.11930
$622$	−30.4040	−1.21909
$623$	13.8156	0.553509
$624$	−36.6642	−1.46774
$625$	2.97849	0.119139
$626$	−29.4012	−1.17511
$627$	20.7358	0.828108
$628$	1.61365	0.0643915
$629$	5.35200	0.213398
$630$	−1.87363	−0.0746471
$631$	36.8625	1.46747	0.733736	−	0.679435i	$-0.237775\pi$
$631$	36.8625	1.46747	0.733736	+	0.679435i	$0.237775\pi$
$632$	−48.9563	−1.94738
$633$	35.1202	1.39590
$634$	44.8342	1.78059
$635$	−25.6613	−1.01834
$636$	−1.85752	−0.0736554
$637$	−13.3611	−0.529388
$638$	−8.30943	−0.328974
$639$	4.57645	0.181042
$640$	10.9453	0.432650
$641$	−1.50475	−0.0594341	−0.0297170	−	0.999558i	$-0.509461\pi$
$641$	−1.50475	−0.0594341	−0.0297170	+	0.999558i	$0.509461\pi$
$642$	43.1310	1.70225
$643$	−9.03610	−0.356349	−0.178175	−	0.983999i	$-0.557019\pi$
$643$	−9.03610	−0.356349	−0.178175	+	0.983999i	$0.557019\pi$
$644$	3.47304	0.136857
$645$	7.21720	0.284177
$646$	−41.9171	−1.64921
$647$	−23.5467	−0.925717	−0.462858	−	0.886432i	$-0.653176\pi$
$647$	−23.5467	−0.925717	−0.462858	+	0.886432i	$0.653176\pi$
$648$	−30.6680	−1.20475
$649$	−2.84252	−0.111579
$650$	25.4334	0.997580
$651$	19.3981	0.760272
$652$	−4.36134	−0.170803
$653$	−10.3739	−0.405961	−0.202980	−	0.979183i	$-0.565063\pi$
$653$	−10.3739	−0.405961	−0.202980	+	0.979183i	$0.565063\pi$
$654$	−1.03718	−0.0405571
$655$	−12.6218	−0.493176
$656$	23.2287	0.906928
$657$	−1.41104	−0.0550500
$658$	31.9170	1.24425
$659$	7.72597	0.300961	0.150481	−	0.988613i	$-0.451918\pi$
$659$	7.72597	0.300961	0.150481	+	0.988613i	$0.451918\pi$
$660$	0.871536	0.0339245
$661$	−11.5379	−0.448774	−0.224387	−	0.974500i	$-0.572038\pi$
$661$	−11.5379	−0.448774	−0.224387	+	0.974500i	$0.572038\pi$
$662$	−22.6746	−0.881272
$663$	42.0103	1.63155
$664$	45.1258	1.75122
$665$	22.7825	0.883469
$666$	−0.928357	−0.0359731
$667$	27.9849	1.08358
$668$	0.807535	0.0312445
$669$	20.9190	0.808774
$670$	−8.26653	−0.319364
$671$	12.6355	0.487788
$672$	6.11086	0.235731
$673$	−11.7815	−0.454142	−0.227071	−	0.973878i	$-0.572915\pi$
$673$	−11.7815	−0.454142	−0.227071	+	0.973878i	$0.572915\pi$
$674$	0.591625	0.0227885
$675$	15.6211	0.601257
$676$	−5.43817	−0.209160
$677$	6.08020	0.233681	0.116841	−	0.993151i	$-0.462723\pi$
$677$	6.08020	0.233681	0.116841	+	0.993151i	$0.462723\pi$
$678$	33.3430	1.28053
$679$	4.28762	0.164544
$680$	−14.8978	−0.571305
$681$	3.14569	0.120543
$682$	8.49597	0.325327
$683$	−18.0981	−0.692504	−0.346252	−	0.938141i	$-0.612546\pi$
$683$	−18.0981	−0.692504	−0.346252	+	0.938141i	$0.612546\pi$
$684$	−1.12623	−0.0430623
$685$	1.55247	0.0593169
$686$	−26.5327	−1.01302
$687$	26.7038	1.01881
$688$	10.1696	0.387711
$689$	21.3142	0.812005
$690$	18.9497	0.721403
$691$	9.80210	0.372890	0.186445	−	0.982465i	$-0.440303\pi$
$691$	9.80210	0.372890	0.186445	+	0.982465i	$0.440303\pi$
$692$	0.380218	0.0144537
$693$	−1.49645	−0.0568452
$694$	19.3440	0.734289
$695$	6.94823	0.263562
$696$	26.1671	0.991862
$697$	−26.6158	−1.00814
$698$	10.9823	0.415688
$699$	0.682482	0.0258138
$700$	−1.94504	−0.0735156
$701$	10.9662	0.414186	0.207093	−	0.978321i	$-0.433600\pi$
$701$	10.9662	0.414186	0.207093	+	0.978321i	$0.433600\pi$
$702$	35.3911	1.33575
$703$	11.2884	0.425752
$704$	11.8329	0.445968
$705$	−26.9742	−1.01591
$706$	22.0337	0.829250
$707$	−31.8458	−1.19768
$708$	1.05858	0.0397837
$709$	−24.8464	−0.933127	−0.466564	−	0.884488i	$-0.654508\pi$
$709$	−24.8464	−0.933127	−0.466564	+	0.884488i	$0.654508\pi$
$710$	15.1000	0.566694
$711$	−8.40128	−0.315073
$712$	−19.0549	−0.714113
$713$	−28.6131	−1.07157
$714$	20.7417	0.776240
$715$	−10.0005	−0.373996
$716$	0.798472	0.0298403
$717$	22.8179	0.852150
$718$	−9.68691	−0.361512
$719$	11.9979	0.447444	0.223722	−	0.974653i	$-0.428179\pi$
$719$	11.9979	0.447444	0.223722	+	0.974653i	$0.428179\pi$
$720$	2.23123	0.0831532
$721$	33.7551	1.25710
$722$	−63.4083	−2.35981
$723$	−6.64669	−0.247193
$724$	2.02174	0.0751373
$725$	−15.6726	−0.582068
$726$	22.6347	0.840053
$727$	−29.9238	−1.10981	−0.554906	−	0.831913i	$-0.687246\pi$
$727$	−29.9238	−1.10981	−0.554906	+	0.831913i	$0.687246\pi$
$728$	−37.2629	−1.38106
$729$	20.9500	0.775925
$730$	−4.65574	−0.172317
$731$	−11.6524	−0.430981
$732$	−4.70555	−0.173922
$733$	11.7907	0.435499	0.217750	−	0.976005i	$-0.430128\pi$
$733$	11.7907	0.435499	0.217750	+	0.976005i	$0.430128\pi$
$734$	12.6108	0.465473
$735$	5.57520	0.205644
$736$	−9.01379	−0.332253
$737$	−6.60239	−0.243202
$738$	4.61677	0.169946
$739$	−45.1729	−1.66171	−0.830856	−	0.556488i	$-0.812149\pi$
$739$	−45.1729	−1.66171	−0.830856	+	0.556488i	$0.812149\pi$
$740$	0.474459	0.0174415
$741$	88.6083	3.25511
$742$	10.5234	0.386327
$743$	9.18420	0.336936	0.168468	−	0.985707i	$-0.446118\pi$
$743$	9.18420	0.336936	0.168468	+	0.985707i	$0.446118\pi$
$744$	−26.7545	−0.980869
$745$	−25.1189	−0.920287
$746$	27.5070	1.00710
$747$	7.74394	0.283336
$748$	−1.40713	−0.0514497
$749$	37.8485	1.38296
$750$	−26.4499	−0.965813
$751$	50.4640	1.84146	0.920728	−	0.390204i	$-0.127596\pi$
$751$	50.4640	1.84146	0.920728	+	0.390204i	$0.127596\pi$
$752$	−38.0087	−1.38604
$753$	38.8843	1.41702
$754$	−35.5079	−1.29312
$755$	23.8529	0.868094
$756$	−2.70657	−0.0984368
$757$	8.18656	0.297546	0.148773	−	0.988871i	$-0.452468\pi$
$757$	8.18656	0.297546	0.148773	+	0.988871i	$0.452468\pi$
$758$	−0.933211	−0.0338957
$759$	15.1349	0.549363
$760$	−31.4225	−1.13981
$761$	−24.8915	−0.902317	−0.451158	−	0.892444i	$-0.648989\pi$
$761$	−24.8915	−0.902317	−0.451158	+	0.892444i	$0.648989\pi$
$762$	−49.2764	−1.78510
$763$	−0.910154	−0.0329498
$764$	3.58861	0.129831
$765$	−2.55658	−0.0924333
$766$	9.42333	0.340479
$767$	−12.1467	−0.438590
$768$	−11.8382	−0.427175
$769$	32.8369	1.18413	0.592064	−	0.805891i	$-0.298313\pi$
$769$	32.8369	1.18413	0.592064	+	0.805891i	$0.298313\pi$
$770$	−4.93753	−0.177936
$771$	−39.1285	−1.40918
$772$	5.44804	0.196079
$773$	−21.3470	−0.767796	−0.383898	−	0.923375i	$-0.625419\pi$
$773$	−21.3470	−0.767796	−0.383898	+	0.923375i	$0.625419\pi$
$774$	2.02123	0.0726516
$775$	16.0245	0.575616
$776$	−5.91363	−0.212287
$777$	−5.58583	−0.200391
$778$	−12.7300	−0.456391
$779$	−56.1380	−2.01135
$780$	3.72425	0.133350
$781$	12.0602	0.431549
$782$	−30.5950	−1.09408
$783$	−21.8088	−0.779384
$784$	7.85588	0.280567
$785$	7.72610	0.275756
$786$	−24.2373	−0.864515
$787$	14.5081	0.517158	0.258579	−	0.965990i	$-0.416746\pi$
$787$	14.5081	0.517158	0.258579	+	0.965990i	$0.416746\pi$
$788$	1.20102	0.0427844
$789$	−50.4463	−1.79594
$790$	−27.7201	−0.986236
$791$	29.2593	1.04034
$792$	2.06395	0.0733392
$793$	53.9940	1.91738
$794$	0.103249	0.00366417
$795$	−8.89376	−0.315429
$796$	−0.570423	−0.0202181
$797$	−3.77526	−0.133727	−0.0668633	−	0.997762i	$-0.521299\pi$
$797$	−3.77526	−0.133727	−0.0668633	+	0.997762i	$0.521299\pi$
$798$	43.7485	1.54868
$799$	43.5509	1.54072
$800$	5.04808	0.178477
$801$	−3.26997	−0.115539
$802$	−8.19616	−0.289416
$803$	−3.71849	−0.131223
$804$	2.45878	0.0867144
$805$	16.6288	0.586089
$806$	36.3050	1.27879
$807$	−40.4543	−1.42406
$808$	43.9228	1.54520
$809$	7.44684	0.261817	0.130909	−	0.991394i	$-0.458211\pi$
$809$	7.44684	0.261817	0.130909	+	0.991394i	$0.458211\pi$
$810$	−17.3649	−0.610140
$811$	41.5744	1.45988	0.729938	−	0.683514i	$-0.239549\pi$
$811$	41.5744	1.45988	0.729938	+	0.683514i	$0.239549\pi$
$812$	2.71549	0.0952951
$813$	−28.4090	−0.996348
$814$	−2.44648	−0.0857490
$815$	−20.8820	−0.731464
$816$	−24.7006	−0.864693
$817$	−24.5773	−0.859852
$818$	−1.31596	−0.0460116
$819$	−6.39461	−0.223446
$820$	−2.35951	−0.0823976
$821$	11.3450	0.395942	0.197971	−	0.980208i	$-0.436565\pi$
$821$	11.3450	0.395942	0.197971	+	0.980208i	$0.436565\pi$
$822$	2.98116	0.103980
$823$	−7.27405	−0.253557	−0.126779	−	0.991931i	$-0.540464\pi$
$823$	−7.27405	−0.253557	−0.126779	+	0.991931i	$0.540464\pi$
$824$	−46.5562	−1.62186
$825$	−8.47616	−0.295102
$826$	−5.99716	−0.208668
$827$	−19.6791	−0.684309	−0.342155	−	0.939644i	$-0.611157\pi$
$827$	−19.6791	−0.684309	−0.342155	+	0.939644i	$0.611157\pi$
$828$	−0.822024	−0.0285673
$829$	14.3752	0.499271	0.249636	−	0.968340i	$-0.419689\pi$
$829$	14.3752	0.499271	0.249636	+	0.968340i	$0.419689\pi$
$830$	25.5512	0.886893
$831$	−51.5707	−1.78897
$832$	50.5642	1.75300
$833$	−9.00137	−0.311879
$834$	13.3424	0.462011
$835$	3.86646	0.133804
$836$	−2.96792	−0.102648
$837$	22.2984	0.770746
$838$	−38.8991	−1.34375
$839$	−11.7440	−0.405448	−0.202724	−	0.979236i	$-0.564979\pi$
$839$	−11.7440	−0.405448	−0.202724	+	0.979236i	$0.564979\pi$
$840$	15.5487	0.536481
$841$	−7.11924	−0.245491
$842$	−2.30664	−0.0794921
$843$	−34.2150	−1.17843
$844$	−5.02676	−0.173028
$845$	−26.0378	−0.895728
$846$	−7.55434	−0.259724
$847$	19.8625	0.682484
$848$	−12.5320	−0.430350
$849$	5.31316	0.182347
$850$	17.1344	0.587706
$851$	8.23936	0.282442
$852$	−4.49132	−0.153870
$853$	22.8005	0.780673	0.390336	−	0.920672i	$-0.372359\pi$
$853$	22.8005	0.780673	0.390336	+	0.920672i	$0.372359\pi$
$854$	26.6584	0.912232
$855$	−5.39234	−0.184414
$856$	−52.2020	−1.78423
$857$	−4.10100	−0.140088	−0.0700438	−	0.997544i	$-0.522314\pi$
$857$	−4.10100	−0.140088	−0.0700438	+	0.997544i	$0.522314\pi$
$858$	−19.2036	−0.655598
$859$	−45.1593	−1.54081	−0.770407	−	0.637552i	$-0.779947\pi$
$859$	−45.1593	−1.54081	−0.770407	+	0.637552i	$0.779947\pi$
$860$	−1.03300	−0.0352249
$861$	27.7786	0.946693
$862$	−4.21843	−0.143680
$863$	54.0362	1.83941	0.919706	−	0.392607i	$-0.128427\pi$
$863$	54.0362	1.83941	0.919706	+	0.392607i	$0.128427\pi$
$864$	7.02452	0.238979
$865$	1.82047	0.0618979
$866$	31.1294	1.05782
$867$	−3.55736	−0.120814
$868$	−2.77645	−0.0942389
$869$	−22.1397	−0.751038
$870$	14.8164	0.502322
$871$	−28.2133	−0.955972
$872$	1.25532	0.0425103
$873$	−1.01483	−0.0343466
$874$	−64.5311	−2.18280
$875$	−23.2104	−0.784654
$876$	1.38479	0.0467878
$877$	14.0802	0.475456	0.237728	−	0.971332i	$-0.423597\pi$
$877$	14.0802	0.475456	0.237728	+	0.971332i	$0.423597\pi$
$878$	−1.21587	−0.0410336
$879$	−18.2576	−0.615814
$880$	5.87992	0.198212
$881$	3.78281	0.127446	0.0637231	−	0.997968i	$-0.479703\pi$
$881$	3.78281	0.127446	0.0637231	+	0.997968i	$0.479703\pi$
$882$	1.56138	0.0525743
$883$	−8.33617	−0.280534	−0.140267	−	0.990114i	$-0.544796\pi$
$883$	−8.33617	−0.280534	−0.140267	+	0.990114i	$0.544796\pi$
$884$	−6.01294	−0.202237
$885$	5.06843	0.170374
$886$	−15.0886	−0.506910
$887$	−39.7938	−1.33614	−0.668072	−	0.744096i	$-0.732880\pi$
$887$	−39.7938	−1.33614	−0.668072	+	0.744096i	$0.732880\pi$
$888$	7.70417	0.258535
$889$	−43.2413	−1.45027
$890$	−10.7893	−0.361658
$891$	−13.8691	−0.464633
$892$	−2.99414	−0.100251
$893$	91.8577	3.07390
$894$	−48.2350	−1.61322
$895$	3.82306	0.127791
$896$	18.4437	0.616160
$897$	64.6746	2.15942
$898$	−16.2315	−0.541651
$899$	−22.3720	−0.746146
$900$	0.460367	0.0153456
$901$	14.3593	0.478378
$902$	12.1665	0.405099
$903$	12.1615	0.404711
$904$	−40.3554	−1.34220
$905$	9.68003	0.321775
$906$	45.8038	1.52173
$907$	10.0289	0.333005	0.166502	−	0.986041i	$-0.446753\pi$
$907$	10.0289	0.333005	0.166502	+	0.986041i	$0.446753\pi$
$908$	−0.450243	−0.0149418
$909$	7.53750	0.250003
$910$	−21.0990	−0.699427
$911$	−7.98810	−0.264658	−0.132329	−	0.991206i	$-0.542245\pi$
$911$	−7.98810	−0.264658	−0.132329	+	0.991206i	$0.542245\pi$
$912$	−52.0985	−1.72515
$913$	20.4074	0.675387
$914$	12.2564	0.405404
$915$	−22.5301	−0.744821
$916$	−3.82212	−0.126286
$917$	−21.2688	−0.702357
$918$	23.8429	0.786934
$919$	−3.92914	−0.129610	−0.0648052	−	0.997898i	$-0.520643\pi$
$919$	−3.92914	−0.129610	−0.0648052	+	0.997898i	$0.520643\pi$
$920$	−22.9350	−0.756146
$921$	19.9234	0.656499
$922$	18.8852	0.621950
$923$	51.5358	1.69632
$924$	1.46861	0.0483136
$925$	−4.61437	−0.151720
$926$	−28.0625	−0.922191
$927$	−7.98940	−0.262406
$928$	−7.04769	−0.231352
$929$	−41.7813	−1.37080	−0.685400	−	0.728167i	$-0.740373\pi$
$929$	−41.7813	−1.37080	−0.685400	+	0.728167i	$0.740373\pi$
$930$	−15.1490	−0.496754
$931$	−18.9857	−0.622231
$932$	−0.0976837	−0.00319974
$933$	43.2991	1.41755
$934$	−2.46849	−0.0807713
$935$	−6.73729	−0.220333
$936$	8.81967	0.288280
$937$	−5.51739	−0.180245	−0.0901227	−	0.995931i	$-0.528726\pi$
$937$	−5.51739	−0.180245	−0.0901227	+	0.995931i	$0.528726\pi$
$938$	−13.9297	−0.454822
$939$	41.8709	1.36641
$940$	3.86083	0.125926
$941$	6.40287	0.208728	0.104364	−	0.994539i	$-0.466719\pi$
$941$	6.40287	0.208728	0.104364	+	0.994539i	$0.466719\pi$
$942$	14.8362	0.483388
$943$	−40.9748	−1.33432
$944$	7.14180	0.232446
$945$	−12.9590	−0.421555
$946$	5.32650	0.173180
$947$	10.7582	0.349596	0.174798	−	0.984604i	$-0.444073\pi$
$947$	10.7582	0.349596	0.174798	+	0.984604i	$0.444073\pi$
$948$	8.24500	0.267785
$949$	−15.8899	−0.515807
$950$	36.1400	1.17254
$951$	−63.8495	−2.07046
$952$	−25.1040	−0.813624
$953$	15.5384	0.503338	0.251669	−	0.967813i	$-0.419021\pi$
$953$	15.5384	0.503338	0.251669	+	0.967813i	$0.419021\pi$
$954$	−2.49076	−0.0806415
$955$	17.1822	0.556002
$956$	−3.26593	−0.105628
$957$	11.8337	0.382528
$958$	−43.2083	−1.39600
$959$	2.61604	0.0844762
$960$	−21.0989	−0.680965
$961$	−8.12583	−0.262123
$962$	−10.4543	−0.337060
$963$	−8.95827	−0.288676
$964$	0.951341	0.0306406
$965$	26.0851	0.839709
$966$	31.9317	1.02739
$967$	50.8279	1.63451	0.817257	−	0.576274i	$-0.195494\pi$
$967$	50.8279	1.63451	0.817257	+	0.576274i	$0.195494\pi$
$968$	−27.3951	−0.880510
$969$	59.6952	1.91769
$970$	−3.34842	−0.107511
$971$	26.4505	0.848837	0.424418	−	0.905466i	$-0.360479\pi$
$971$	26.4505	0.848837	0.424418	+	0.905466i	$0.360479\pi$
$972$	1.41312	0.0453259
$973$	11.7083	0.375351
$974$	43.2145	1.38468
$975$	−36.2203	−1.15998
$976$	−31.7465	−1.01618
$977$	38.0875	1.21853	0.609263	−	0.792968i	$-0.291465\pi$
$977$	38.0875	1.21853	0.609263	+	0.792968i	$0.291465\pi$
$978$	−40.0989	−1.28222
$979$	−8.61728	−0.275410
$980$	−0.797979	−0.0254905
$981$	0.215422	0.00687789
$982$	19.6698	0.627690
$983$	37.9226	1.20954	0.604771	−	0.796399i	$-0.293264\pi$
$983$	37.9226	1.20954	0.604771	+	0.796399i	$0.293264\pi$
$984$	−38.3132	−1.22138
$985$	5.75043	0.183224
$986$	−23.9216	−0.761818
$987$	−45.4537	−1.44681
$988$	−12.6825	−0.403484
$989$	−17.9388	−0.570422
$990$	1.16865	0.0371421
$991$	−30.2061	−0.959530	−0.479765	−	0.877397i	$-0.659278\pi$
$991$	−30.2061	−0.959530	−0.479765	+	0.877397i	$0.659278\pi$
$992$	7.20590	0.228788
$993$	32.2914	1.02474
$994$	25.4447	0.807058
$995$	−2.73117	−0.0865839
$996$	−7.59988	−0.240811
$997$	−57.2175	−1.81210	−0.906048	−	0.423175i	$-0.860916\pi$
$997$	−57.2175	−1.81210	−0.906048	+	0.423175i	$0.860916\pi$
$998$	−20.6294	−0.653013
$999$	−6.42100	−0.203151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.6	✓	20
3.2	odd	2		3681.2.a.i.1.15		20
4.3	odd	2		6544.2.a.i.1.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.6	✓	20	1.1	even	1	trivial
3681.2.a.i.1.15		20	3.2	odd	2
6544.2.a.i.1.6		20	4.3	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:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31596 q^{2} +1.87410 q^{3} -0.268240 q^{4} -1.28432 q^{5} -2.46624 q^{6} -2.16419 q^{7} +2.98492 q^{8} +0.512236 q^{9} +1.69012 q^{10} +1.34988 q^{11} -0.502707 q^{12} +5.76832 q^{13} +2.84799 q^{14} -2.40695 q^{15} -3.39157 q^{16} +3.88611 q^{17} -0.674084 q^{18} +8.19658 q^{19} +0.344507 q^{20} -4.05589 q^{21} -1.77640 q^{22} +5.98263 q^{23} +5.59403 q^{24} -3.35051 q^{25} -7.59090 q^{26} -4.66231 q^{27} +0.580520 q^{28} +4.67769 q^{29} +3.16745 q^{30} -4.78270 q^{31} -1.50666 q^{32} +2.52981 q^{33} -5.11398 q^{34} +2.77952 q^{35} -0.137402 q^{36} +1.37721 q^{37} -10.7864 q^{38} +10.8104 q^{39} -3.83361 q^{40} -6.84895 q^{41} +5.33741 q^{42} -2.99849 q^{43} -0.362092 q^{44} -0.657877 q^{45} -7.87292 q^{46} +11.2068 q^{47} -6.35612 q^{48} -2.31630 q^{49} +4.40915 q^{50} +7.28294 q^{51} -1.54729 q^{52} +3.69504 q^{53} +6.13543 q^{54} -1.73369 q^{55} -6.45993 q^{56} +15.3612 q^{57} -6.15567 q^{58} -2.10575 q^{59} +0.645638 q^{60} +9.36043 q^{61} +6.29385 q^{62} -1.10857 q^{63} +8.76585 q^{64} -7.40840 q^{65} -3.32914 q^{66} -4.89108 q^{67} -1.04241 q^{68} +11.2120 q^{69} -3.65774 q^{70} +8.93427 q^{71} +1.52898 q^{72} -2.75468 q^{73} -1.81236 q^{74} -6.27918 q^{75} -2.19865 q^{76} -2.92140 q^{77} -14.2261 q^{78} -16.4012 q^{79} +4.35587 q^{80} -10.2743 q^{81} +9.01298 q^{82} +15.1179 q^{83} +1.08795 q^{84} -4.99102 q^{85} +3.94590 q^{86} +8.76644 q^{87} +4.02929 q^{88} -6.38372 q^{89} +0.865742 q^{90} -12.4837 q^{91} -1.60478 q^{92} -8.96323 q^{93} -14.7478 q^{94} -10.5271 q^{95} -2.82363 q^{96} -1.98117 q^{97} +3.04816 q^{98} +0.691458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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