LMFDB - Embedded newform 465.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [465,2,Mod(1,465)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$3.71304369399$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.31743$ of defining polynomial
Character	$\chi$	$=$	465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} +1.00000 q^{5} -2.58181 q^{6} +0.946946 q^{7} +6.88240 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} +1.00000 q^{5} -2.58181 q^{6} +0.946946 q^{7} +6.88240 q^{8} +1.00000 q^{9} +2.58181 q^{10} -0.528753 q^{11} -4.66573 q^{12} -7.01937 q^{13} +2.44483 q^{14} -1.00000 q^{15} +8.43756 q^{16} +2.30059 q^{17} +2.58181 q^{18} +4.00000 q^{19} +4.66573 q^{20} -0.946946 q^{21} -1.36514 q^{22} -4.46843 q^{23} -6.88240 q^{24} +1.00000 q^{25} -18.1227 q^{26} -1.00000 q^{27} +4.41819 q^{28} +2.71878 q^{29} -2.58181 q^{30} -1.00000 q^{31} +8.01937 q^{32} +0.528753 q^{33} +5.93968 q^{34} +0.946946 q^{35} +4.66573 q^{36} -2.11056 q^{37} +10.3272 q^{38} +7.01937 q^{39} +6.88240 q^{40} -7.79848 q^{41} -2.44483 q^{42} +8.07243 q^{43} -2.46702 q^{44} +1.00000 q^{45} -11.5366 q^{46} -4.13275 q^{47} -8.43756 q^{48} -6.10329 q^{49} +2.58181 q^{50} -2.30059 q^{51} -32.7505 q^{52} -0.863025 q^{53} -2.58181 q^{54} -0.528753 q^{55} +6.51726 q^{56} -4.00000 q^{57} +7.01937 q^{58} -11.2139 q^{59} -4.66573 q^{60} +5.54367 q^{61} -2.58181 q^{62} +0.946946 q^{63} +3.82934 q^{64} -7.01937 q^{65} +1.36514 q^{66} -6.74542 q^{67} +10.7339 q^{68} +4.46843 q^{69} +2.44483 q^{70} +11.5215 q^{71} +6.88240 q^{72} -12.2167 q^{73} -5.44906 q^{74} -1.00000 q^{75} +18.6629 q^{76} -0.500701 q^{77} +18.1227 q^{78} +15.8251 q^{79} +8.43756 q^{80} +1.00000 q^{81} -20.1342 q^{82} -6.19448 q^{83} -4.41819 q^{84} +2.30059 q^{85} +20.8414 q^{86} -2.71878 q^{87} -3.63909 q^{88} +6.57899 q^{89} +2.58181 q^{90} -6.64697 q^{91} -20.8485 q^{92} +1.00000 q^{93} -10.6700 q^{94} +4.00000 q^{95} -8.01937 q^{96} -1.47125 q^{97} -15.7575 q^{98} -0.528753 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9} + 2 q^{10} + 6 q^{11} - 8 q^{12} + 2 q^{13} + 4 q^{14} - 4 q^{15} + 12 q^{16} - 10 q^{17} + 2 q^{18} + 16 q^{19} + 8 q^{20} - 4 q^{21} - 14 q^{22} + 6 q^{23} + 4 q^{25} - 10 q^{26} - 4 q^{27} + 26 q^{28} - 2 q^{30} - 4 q^{31} + 2 q^{32} - 6 q^{33} + 8 q^{34} + 4 q^{35} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 2 q^{39} - 6 q^{41} - 4 q^{42} + 2 q^{43} - 6 q^{44} + 4 q^{45} + 20 q^{46} - 2 q^{47} - 12 q^{48} + 8 q^{49} + 2 q^{50} + 10 q^{51} - 16 q^{52} - 6 q^{53} - 2 q^{54} + 6 q^{55} - 10 q^{56} - 16 q^{57} - 2 q^{58} + 4 q^{59} - 8 q^{60} - 2 q^{62} + 4 q^{63} - 12 q^{64} + 2 q^{65} + 14 q^{66} - 2 q^{67} - 10 q^{68} - 6 q^{69} + 4 q^{70} + 22 q^{71} - 32 q^{73} - 28 q^{74} - 4 q^{75} + 32 q^{76} - 28 q^{77} + 10 q^{78} + 24 q^{79} + 12 q^{80} + 4 q^{81} - 46 q^{82} - 6 q^{83} - 26 q^{84} - 10 q^{85} + 10 q^{86} - 18 q^{88} - 14 q^{89} + 2 q^{90} + 28 q^{91} - 54 q^{92} + 4 q^{93} - 44 q^{94} + 16 q^{95} - 2 q^{96} - 14 q^{97} + 8 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^9 + 2 * q^10 + 6 * q^11 - 8 * q^12 + 2 * q^13 + 4 * q^14 - 4 * q^15 + 12 * q^16 - 10 * q^17 + 2 * q^18 + 16 * q^19 + 8 * q^20 - 4 * q^21 - 14 * q^22 + 6 * q^23 + 4 * q^25 - 10 * q^26 - 4 * q^27 + 26 * q^28 - 2 * q^30 - 4 * q^31 + 2 * q^32 - 6 * q^33 + 8 * q^34 + 4 * q^35 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 2 * q^39 - 6 * q^41 - 4 * q^42 + 2 * q^43 - 6 * q^44 + 4 * q^45 + 20 * q^46 - 2 * q^47 - 12 * q^48 + 8 * q^49 + 2 * q^50 + 10 * q^51 - 16 * q^52 - 6 * q^53 - 2 * q^54 + 6 * q^55 - 10 * q^56 - 16 * q^57 - 2 * q^58 + 4 * q^59 - 8 * q^60 - 2 * q^62 + 4 * q^63 - 12 * q^64 + 2 * q^65 + 14 * q^66 - 2 * q^67 - 10 * q^68 - 6 * q^69 + 4 * q^70 + 22 * q^71 - 32 * q^73 - 28 * q^74 - 4 * q^75 + 32 * q^76 - 28 * q^77 + 10 * q^78 + 24 * q^79 + 12 * q^80 + 4 * q^81 - 46 * q^82 - 6 * q^83 - 26 * q^84 - 10 * q^85 + 10 * q^86 - 18 * q^88 - 14 * q^89 + 2 * q^90 + 28 * q^91 - 54 * q^92 + 4 * q^93 - 44 * q^94 + 16 * q^95 - 2 * q^96 - 14 * q^97 + 8 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.58181	1.82561	0.912807	−	0.408392i	$-0.133910\pi$
$2$	2.58181	1.82561	0.912807	+	0.408392i	$0.133910\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.66573	2.33286
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.58181	−1.05402
$7$	0.946946	0.357912	0.178956	−	0.983857i	$-0.442728\pi$
$7$	0.946946	0.357912	0.178956	+	0.983857i	$0.442728\pi$
$8$	6.88240	2.43329
$9$	1.00000	0.333333
$10$	2.58181	0.816439
$11$	−0.528753	−0.159425	−0.0797125	−	0.996818i	$-0.525400\pi$
$11$	−0.528753	−0.159425	−0.0797125	+	0.996818i	$0.525400\pi$
$12$	−4.66573	−1.34688
$13$	−7.01937	−1.94682	−0.973412	−	0.229062i	$-0.926434\pi$
$13$	−7.01937	−1.94682	−0.973412	+	0.229062i	$0.926434\pi$
$14$	2.44483	0.653409
$15$	−1.00000	−0.258199
$16$	8.43756	2.10939
$17$	2.30059	0.557975	0.278987	−	0.960295i	$-0.410001\pi$
$17$	2.30059	0.557975	0.278987	+	0.960295i	$0.410001\pi$
$18$	2.58181	0.608538
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	4.66573	1.04329
$21$	−0.946946	−0.206641
$22$	−1.36514	−0.291049
$23$	−4.46843	−0.931732	−0.465866	−	0.884855i	$-0.654257\pi$
$23$	−4.46843	−0.931732	−0.465866	+	0.884855i	$0.654257\pi$
$24$	−6.88240	−1.40486
$25$	1.00000	0.200000
$26$	−18.1227	−3.55415
$27$	−1.00000	−0.192450
$28$	4.41819	0.834960
$29$	2.71878	0.504865	0.252433	−	0.967614i	$-0.418769\pi$
$29$	2.71878	0.504865	0.252433	+	0.967614i	$0.418769\pi$
$30$	−2.58181	−0.471371
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	8.01937	1.41764
$33$	0.528753	0.0920441
$34$	5.93968	1.01865
$35$	0.946946	0.160063
$36$	4.66573	0.777621
$37$	−2.11056	−0.346974	−0.173487	−	0.984836i	$-0.555503\pi$
$37$	−2.11056	−0.346974	−0.173487	+	0.984836i	$0.555503\pi$
$38$	10.3272	1.67530
$39$	7.01937	1.12400
$40$	6.88240	1.08820
$41$	−7.79848	−1.21792	−0.608959	−	0.793202i	$-0.708412\pi$
$41$	−7.79848	−1.21792	−0.608959	+	0.793202i	$0.708412\pi$
$42$	−2.44483	−0.377246
$43$	8.07243	1.23103	0.615517	−	0.788124i	$-0.288947\pi$
$43$	8.07243	1.23103	0.615517	+	0.788124i	$0.288947\pi$
$44$	−2.46702	−0.371917
$45$	1.00000	0.149071
$46$	−11.5366	−1.70098
$47$	−4.13275	−0.602823	−0.301412	−	0.953494i	$-0.597458\pi$
$47$	−4.13275	−0.602823	−0.301412	+	0.953494i	$0.597458\pi$
$48$	−8.43756	−1.21786
$49$	−6.10329	−0.871899
$50$	2.58181	0.365123
$51$	−2.30059	−0.322147
$52$	−32.7505	−4.54167
$53$	−0.863025	−0.118546	−0.0592728	−	0.998242i	$-0.518878\pi$
$53$	−0.863025	−0.118546	−0.0592728	+	0.998242i	$0.518878\pi$
$54$	−2.58181	−0.351339
$55$	−0.528753	−0.0712971
$56$	6.51726	0.870905
$57$	−4.00000	−0.529813
$58$	7.01937	0.921689
$59$	−11.2139	−1.45992	−0.729960	−	0.683490i	$-0.760461\pi$
$59$	−11.2139	−1.45992	−0.729960	+	0.683490i	$0.760461\pi$
$60$	−4.66573	−0.602343
$61$	5.54367	0.709795	0.354897	−	0.934905i	$-0.384516\pi$
$61$	5.54367	0.709795	0.354897	+	0.934905i	$0.384516\pi$
$62$	−2.58181	−0.327890
$63$	0.946946	0.119304
$64$	3.82934	0.478668
$65$	−7.01937	−0.870646
$66$	1.36514	0.168037
$67$	−6.74542	−0.824084	−0.412042	−	0.911165i	$-0.635184\pi$
$67$	−6.74542	−0.824084	−0.412042	+	0.911165i	$0.635184\pi$
$68$	10.7339	1.30168
$69$	4.46843	0.537936
$70$	2.44483	0.292213
$71$	11.5215	1.36735	0.683674	−	0.729787i	$-0.260381\pi$
$71$	11.5215	1.36735	0.683674	+	0.729787i	$0.260381\pi$
$72$	6.88240	0.811098
$73$	−12.2167	−1.42985	−0.714926	−	0.699200i	$-0.753540\pi$
$73$	−12.2167	−1.42985	−0.714926	+	0.699200i	$0.753540\pi$
$74$	−5.44906	−0.633440
$75$	−1.00000	−0.115470
$76$	18.6629	2.14078
$77$	−0.500701	−0.0570601
$78$	18.1227	2.05199
$79$	15.8251	1.78046	0.890232	−	0.455507i	$-0.150542\pi$
$79$	15.8251	1.78046	0.890232	+	0.455507i	$0.150542\pi$
$80$	8.43756	0.943348
$81$	1.00000	0.111111
$82$	−20.1342	−2.22345
$83$	−6.19448	−0.679933	−0.339966	−	0.940438i	$-0.610416\pi$
$83$	−6.19448	−0.679933	−0.339966	+	0.940438i	$0.610416\pi$
$84$	−4.41819	−0.482064
$85$	2.30059	0.249534
$86$	20.8414	2.24739
$87$	−2.71878	−0.291484
$88$	−3.63909	−0.387928
$89$	6.57899	0.697372	0.348686	−	0.937240i	$-0.386628\pi$
$89$	6.57899	0.697372	0.348686	+	0.937240i	$0.386628\pi$
$90$	2.58181	0.272146
$91$	−6.64697	−0.696791
$92$	−20.8485	−2.17361
$93$	1.00000	0.103695
$94$	−10.6700	−1.10052
$95$	4.00000	0.410391
$96$	−8.01937	−0.818474
$97$	−1.47125	−0.149382	−0.0746912	−	0.997207i	$-0.523797\pi$
$97$	−1.47125	−0.149382	−0.0746912	+	0.997207i	$0.523797\pi$
$98$	−15.7575	−1.59175
$99$	−0.528753	−0.0531417
$100$	4.66573	0.466573
$101$	−8.80270	−0.875902	−0.437951	−	0.898999i	$-0.644296\pi$
$101$	−8.80270	−0.875902	−0.437951	+	0.898999i	$0.644296\pi$
$102$	−5.93968	−0.588116
$103$	18.6248	1.83515	0.917577	−	0.397558i	$-0.130142\pi$
$103$	18.6248	1.83515	0.917577	+	0.397558i	$0.130142\pi$
$104$	−48.3101	−4.73720
$105$	−0.946946	−0.0924125
$106$	−2.22816	−0.216418
$107$	−1.57031	−0.151808	−0.0759039	−	0.997115i	$-0.524184\pi$
$107$	−1.57031	−0.151808	−0.0759039	+	0.997115i	$0.524184\pi$
$108$	−4.66573	−0.448960
$109$	6.66150	0.638056	0.319028	−	0.947745i	$-0.396644\pi$
$109$	6.66150	0.638056	0.319028	+	0.947745i	$0.396644\pi$
$110$	−1.36514	−0.130161
$111$	2.11056	0.200326
$112$	7.98992	0.754976
$113$	18.8223	1.77065	0.885327	−	0.464970i	$-0.153935\pi$
$113$	18.8223	1.77065	0.885327	+	0.464970i	$0.153935\pi$
$114$	−10.3272	−0.967234
$115$	−4.46843	−0.416683
$116$	12.6851	1.17778
$117$	−7.01937	−0.648941
$118$	−28.9520	−2.66525
$119$	2.17853	0.199706
$120$	−6.88240	−0.628274
$121$	−10.7204	−0.974584
$122$	14.3127	1.29581
$123$	7.79848	0.703165
$124$	−4.66573	−0.418995
$125$	1.00000	0.0894427
$126$	2.44483	0.217803
$127$	19.7115	1.74911	0.874557	−	0.484923i	$-0.161152\pi$
$127$	19.7115	1.74911	0.874557	+	0.484923i	$0.161152\pi$
$128$	−6.15212	−0.543776
$129$	−8.07243	−0.710737
$130$	−18.1227	−1.58946
$131$	6.44483	0.563088	0.281544	−	0.959548i	$-0.409154\pi$
$131$	6.44483	0.563088	0.281544	+	0.959548i	$0.409154\pi$
$132$	2.46702	0.214726
$133$	3.78778	0.328443
$134$	−17.4154	−1.50446
$135$	−1.00000	−0.0860663
$136$	15.8336	1.35772
$137$	−6.30059	−0.538296	−0.269148	−	0.963099i	$-0.586742\pi$
$137$	−6.30059	−0.538296	−0.269148	+	0.963099i	$0.586742\pi$
$138$	11.5366	0.982063
$139$	20.5568	1.74361	0.871803	−	0.489857i	$-0.162951\pi$
$139$	20.5568	1.74361	0.871803	+	0.489857i	$0.162951\pi$
$140$	4.41819	0.373405
$141$	4.13275	0.348040
$142$	29.7463	2.49625
$143$	3.71152	0.310373
$144$	8.43756	0.703130
$145$	2.71878	0.225783
$146$	−31.5411	−2.61036
$147$	6.10329	0.503391
$148$	−9.84730	−0.809443
$149$	−18.4614	−1.51242	−0.756208	−	0.654331i	$-0.772950\pi$
$149$	−18.4614	−1.51242	−0.756208	+	0.654331i	$0.772950\pi$
$150$	−2.58181	−0.210804
$151$	16.6615	1.35589	0.677947	−	0.735111i	$-0.262870\pi$
$151$	16.6615	1.35589	0.677947	+	0.735111i	$0.262870\pi$
$152$	27.5296	2.23294
$153$	2.30059	0.185992
$154$	−1.29271	−0.104170
$155$	−1.00000	−0.0803219
$156$	32.7505	2.62214
$157$	9.73674	0.777077	0.388538	−	0.921433i	$-0.372980\pi$
$157$	9.73674	0.777077	0.388538	+	0.921433i	$0.372980\pi$
$158$	40.8574	3.25044
$159$	0.863025	0.0684423
$160$	8.01937	0.633987
$161$	−4.23136	−0.333478
$162$	2.58181	0.202846
$163$	−1.63464	−0.128035	−0.0640173	−	0.997949i	$-0.520391\pi$
$163$	−1.63464	−0.128035	−0.0640173	+	0.997949i	$0.520391\pi$
$164$	−36.3856	−2.84124
$165$	0.528753	0.0411634
$166$	−15.9930	−1.24129
$167$	−2.49507	−0.193074	−0.0965372	−	0.995329i	$-0.530777\pi$
$167$	−2.49507	−0.193074	−0.0965372	+	0.995329i	$0.530777\pi$
$168$	−6.51726	−0.502817
$169$	36.2716	2.79012
$170$	5.93968	0.455553
$171$	4.00000	0.305888
$172$	37.6637	2.87183
$173$	−18.1005	−1.37615	−0.688077	−	0.725638i	$-0.741545\pi$
$173$	−18.1005	−1.37615	−0.688077	+	0.725638i	$0.741545\pi$
$174$	−7.01937	−0.532137
$175$	0.946946	0.0715824
$176$	−4.46139	−0.336290
$177$	11.2139	0.842885
$178$	16.9857	1.27313
$179$	−0.668543	−0.0499693	−0.0249846	−	0.999688i	$-0.507954\pi$
$179$	−0.668543	−0.0499693	−0.0249846	+	0.999688i	$0.507954\pi$
$180$	4.66573	0.347763
$181$	5.87936	0.437009	0.218505	−	0.975836i	$-0.429882\pi$
$181$	5.87936	0.437009	0.218505	+	0.975836i	$0.429882\pi$
$182$	−17.1612	−1.27207
$183$	−5.54367	−0.409800
$184$	−30.7535	−2.26718
$185$	−2.11056	−0.155171
$186$	2.58181	0.189307
$187$	−1.21644	−0.0889552
$188$	−19.2823	−1.40630
$189$	−0.946946	−0.0688802
$190$	10.3272	0.749216
$191$	20.8193	1.50643	0.753214	−	0.657775i	$-0.228502\pi$
$191$	20.8193	1.50643	0.753214	+	0.657775i	$0.228502\pi$
$192$	−3.82934	−0.276359
$193$	−4.46702	−0.321543	−0.160772	−	0.986992i	$-0.551398\pi$
$193$	−4.46702	−0.321543	−0.160772	+	0.986992i	$0.551398\pi$
$194$	−3.79848	−0.272715
$195$	7.01937	0.502668
$196$	−28.4763	−2.03402
$197$	−22.3539	−1.59265	−0.796324	−	0.604871i	$-0.793225\pi$
$197$	−22.3539	−1.59265	−0.796324	+	0.604871i	$0.793225\pi$
$198$	−1.36514	−0.0970162
$199$	24.8064	1.75848	0.879238	−	0.476383i	$-0.158052\pi$
$199$	24.8064	1.75848	0.879238	+	0.476383i	$0.158052\pi$
$200$	6.88240	0.486659
$201$	6.74542	0.475785
$202$	−22.7269	−1.59906
$203$	2.57454	0.180697
$204$	−10.7339	−0.751525
$205$	−7.79848	−0.544669
$206$	48.0856	3.35028
$207$	−4.46843	−0.310577
$208$	−59.2264	−4.10661
$209$	−2.11501	−0.146298
$210$	−2.44483	−0.168709
$211$	−7.55258	−0.519941	−0.259970	−	0.965617i	$-0.583713\pi$
$211$	−7.55258	−0.519941	−0.259970	+	0.965617i	$0.583713\pi$
$212$	−4.02664	−0.276551
$213$	−11.5215	−0.789439
$214$	−4.05424	−0.277142
$215$	8.07243	0.550535
$216$	−6.88240	−0.468288
$217$	−0.946946	−0.0642829
$218$	17.1987	1.16484
$219$	12.2167	0.825526
$220$	−2.46702	−0.166326
$221$	−16.1487	−1.08628
$222$	5.44906	0.365717
$223$	−8.69660	−0.582367	−0.291184	−	0.956667i	$-0.594049\pi$
$223$	−8.69660	−0.582367	−0.291184	+	0.956667i	$0.594049\pi$
$224$	7.59391	0.507390
$225$	1.00000	0.0666667
$226$	48.5955	3.23253
$227$	−5.81442	−0.385917	−0.192958	−	0.981207i	$-0.561808\pi$
$227$	−5.81442	−0.385917	−0.192958	+	0.981207i	$0.561808\pi$
$228$	−18.6629	−1.23598
$229$	7.10188	0.469305	0.234653	−	0.972079i	$-0.424605\pi$
$229$	7.10188	0.469305	0.234653	+	0.972079i	$0.424605\pi$
$230$	−11.5366	−0.760703
$231$	0.500701	0.0329437
$232$	18.7117	1.22849
$233$	11.0926	0.726700	0.363350	−	0.931653i	$-0.381633\pi$
$233$	11.0926	0.726700	0.363350	+	0.931653i	$0.381633\pi$
$234$	−18.1227	−1.18472
$235$	−4.13275	−0.269591
$236$	−52.3208	−3.40579
$237$	−15.8251	−1.02795
$238$	5.62456	0.364586
$239$	−7.44826	−0.481788	−0.240894	−	0.970551i	$-0.577441\pi$
$239$	−7.44826	−0.481788	−0.240894	+	0.970551i	$0.577441\pi$
$240$	−8.43756	−0.544642
$241$	−17.0963	−1.10127	−0.550633	−	0.834747i	$-0.685614\pi$
$241$	−17.0963	−1.10127	−0.550633	+	0.834747i	$0.685614\pi$
$242$	−27.6781	−1.77921
$243$	−1.00000	−0.0641500
$244$	25.8653	1.65585
$245$	−6.10329	−0.389925
$246$	20.1342	1.28371
$247$	−28.0775	−1.78653
$248$	−6.88240	−0.437033
$249$	6.19448	0.392559
$250$	2.58181	0.163288
$251$	−10.8219	−0.683069	−0.341535	−	0.939869i	$-0.610947\pi$
$251$	−10.8219	−0.683069	−0.341535	+	0.939869i	$0.610947\pi$
$252$	4.41819	0.278320
$253$	2.36270	0.148542
$254$	50.8913	3.19321
$255$	−2.30059	−0.144069
$256$	−23.5423	−1.47139
$257$	12.8400	0.800939	0.400470	−	0.916310i	$-0.368847\pi$
$257$	12.8400	0.800939	0.400470	+	0.916310i	$0.368847\pi$
$258$	−20.8414	−1.29753
$259$	−1.99859	−0.124186
$260$	−32.7505	−2.03110
$261$	2.71878	0.168288
$262$	16.6393	1.02798
$263$	−27.4319	−1.69153	−0.845763	−	0.533559i	$-0.820854\pi$
$263$	−27.4319	−1.69153	−0.845763	+	0.533559i	$0.820854\pi$
$264$	3.63909	0.223970
$265$	−0.863025	−0.0530152
$266$	9.77933	0.599609
$267$	−6.57899	−0.402628
$268$	−31.4723	−1.92248
$269$	15.3732	0.937323	0.468661	−	0.883378i	$-0.344736\pi$
$269$	15.3732	0.937323	0.468661	+	0.883378i	$0.344736\pi$
$270$	−2.58181	−0.157124
$271$	−22.2510	−1.35165	−0.675825	−	0.737062i	$-0.736212\pi$
$271$	−22.2510	−1.35165	−0.675825	+	0.737062i	$0.736212\pi$
$272$	19.4114	1.17699
$273$	6.64697	0.402293
$274$	−16.2669	−0.982720
$275$	−0.528753	−0.0318850
$276$	20.8485	1.25493
$277$	12.4378	0.747314	0.373657	−	0.927567i	$-0.378104\pi$
$277$	12.4378	0.747314	0.373657	+	0.927567i	$0.378104\pi$
$278$	53.0737	3.18315
$279$	−1.00000	−0.0598684
$280$	6.51726	0.389481
$281$	13.3758	0.797935	0.398968	−	0.916965i	$-0.369369\pi$
$281$	13.3758	0.797935	0.398968	+	0.916965i	$0.369369\pi$
$282$	10.6700	0.635387
$283$	−10.8941	−0.647588	−0.323794	−	0.946128i	$-0.604958\pi$
$283$	−10.8941	−0.647588	−0.323794	+	0.946128i	$0.604958\pi$
$284$	53.7561	3.18984
$285$	−4.00000	−0.236940
$286$	9.58242	0.566620
$287$	−7.38474	−0.435907
$288$	8.01937	0.472546
$289$	−11.7073	−0.688664
$290$	7.01937	0.412192
$291$	1.47125	0.0862460
$292$	−56.9997	−3.33565
$293$	−11.0926	−0.648037	−0.324018	−	0.946051i	$-0.605034\pi$
$293$	−11.0926	−0.648037	−0.324018	+	0.946051i	$0.605034\pi$
$294$	15.7575	0.918998
$295$	−11.2139	−0.652896
$296$	−14.5257	−0.844290
$297$	0.528753	0.0306814
$298$	−47.6637	−2.76109
$299$	31.3656	1.81392
$300$	−4.66573	−0.269376
$301$	7.64415	0.440602
$302$	43.0168	2.47534
$303$	8.80270	0.505702
$304$	33.7503	1.93571
$305$	5.54367	0.317430
$306$	5.93968	0.339549
$307$	6.98569	0.398694	0.199347	−	0.979929i	$-0.436118\pi$
$307$	6.98569	0.398694	0.199347	+	0.979929i	$0.436118\pi$
$308$	−2.33613	−0.133114
$309$	−18.6248	−1.05953
$310$	−2.58181	−0.146637
$311$	29.8576	1.69307	0.846535	−	0.532333i	$-0.178685\pi$
$311$	29.8576	1.69307	0.846535	+	0.532333i	$0.178685\pi$
$312$	48.3101	2.73502
$313$	14.7398	0.833142	0.416571	−	0.909103i	$-0.363232\pi$
$313$	14.7398	0.833142	0.416571	+	0.909103i	$0.363232\pi$
$314$	25.1384	1.41864
$315$	0.946946	0.0533544
$316$	73.8357	4.15358
$317$	19.8919	1.11724	0.558621	−	0.829423i	$-0.311331\pi$
$317$	19.8919	1.11724	0.558621	+	0.829423i	$0.311331\pi$
$318$	2.22816	0.124949
$319$	−1.43756	−0.0804882
$320$	3.82934	0.214067
$321$	1.57031	0.0876463
$322$	−10.9246	−0.608802
$323$	9.20236	0.512033
$324$	4.66573	0.259207
$325$	−7.01937	−0.389365
$326$	−4.22032	−0.233742
$327$	−6.66150	−0.368382
$328$	−53.6722	−2.96355
$329$	−3.91349	−0.215758
$330$	1.36514	0.0751484
$331$	10.9444	0.601556	0.300778	−	0.953694i	$-0.402754\pi$
$331$	10.9444	0.601556	0.300778	+	0.953694i	$0.402754\pi$
$332$	−28.9018	−1.58619
$333$	−2.11056	−0.115658
$334$	−6.44179	−0.352479
$335$	−6.74542	−0.368542
$336$	−7.98992	−0.435886
$337$	−16.6500	−0.906984	−0.453492	−	0.891260i	$-0.649822\pi$
$337$	−16.6500	−0.906984	−0.453492	+	0.891260i	$0.649822\pi$
$338$	93.6462	5.09368
$339$	−18.8223	−1.02229
$340$	10.7339	0.582129
$341$	0.528753	0.0286336
$342$	10.3272	0.558433
$343$	−12.4081	−0.669975
$344$	55.5576	2.99547
$345$	4.46843	0.240572
$346$	−46.7319	−2.51232
$347$	−9.98547	−0.536048	−0.268024	−	0.963412i	$-0.586371\pi$
$347$	−9.98547	−0.536048	−0.268024	+	0.963412i	$0.586371\pi$
$348$	−12.6851	−0.679993
$349$	29.2038	1.56324	0.781621	−	0.623754i	$-0.214393\pi$
$349$	29.2038	1.56324	0.781621	+	0.623754i	$0.214393\pi$
$350$	2.44483	0.130682
$351$	7.01937	0.374666
$352$	−4.24027	−0.226007
$353$	−37.3738	−1.98921	−0.994605	−	0.103738i	$-0.966920\pi$
$353$	−37.3738	−1.98921	−0.994605	+	0.103738i	$0.966920\pi$
$354$	28.9520	1.53878
$355$	11.5215	0.611497
$356$	30.6958	1.62687
$357$	−2.17853	−0.115300
$358$	−1.72605	−0.0912246
$359$	16.0059	0.844757	0.422379	−	0.906419i	$-0.361195\pi$
$359$	16.0059	0.844757	0.422379	+	0.906419i	$0.361195\pi$
$360$	6.88240	0.362734
$361$	−3.00000	−0.157895
$362$	15.1794	0.797810
$363$	10.7204	0.562676
$364$	−31.0129	−1.62552
$365$	−12.2167	−0.639450
$366$	−14.3127	−0.748137
$367$	4.65446	0.242961	0.121480	−	0.992594i	$-0.461236\pi$
$367$	4.65446	0.242961	0.121480	+	0.992594i	$0.461236\pi$
$368$	−37.7027	−1.96539
$369$	−7.79848	−0.405972
$370$	−5.44906	−0.283283
$371$	−0.817238	−0.0424289
$372$	4.66573	0.241907
$373$	8.49507	0.439858	0.219929	−	0.975516i	$-0.429417\pi$
$373$	8.49507	0.439858	0.219929	+	0.975516i	$0.429417\pi$
$374$	−3.14062	−0.162398
$375$	−1.00000	−0.0516398
$376$	−28.4432	−1.46685
$377$	−19.0841	−0.982884
$378$	−2.44483	−0.125749
$379$	−18.8078	−0.966090	−0.483045	−	0.875596i	$-0.660469\pi$
$379$	−18.8078	−0.966090	−0.483045	+	0.875596i	$0.660469\pi$
$380$	18.6629	0.957387
$381$	−19.7115	−1.00985
$382$	53.7513	2.75016
$383$	−10.9546	−0.559754	−0.279877	−	0.960036i	$-0.590294\pi$
$383$	−10.9546	−0.559754	−0.279877	+	0.960036i	$0.590294\pi$
$384$	6.15212	0.313949
$385$	−0.500701	−0.0255181
$386$	−11.5330	−0.587013
$387$	8.07243	0.410344
$388$	−6.86444	−0.348489
$389$	−23.5602	−1.19455	−0.597276	−	0.802036i	$-0.703750\pi$
$389$	−23.5602	−1.19455	−0.597276	+	0.802036i	$0.703750\pi$
$390$	18.1227	0.917677
$391$	−10.2800	−0.519883
$392$	−42.0053	−2.12159
$393$	−6.44483	−0.325099
$394$	−57.7134	−2.90756
$395$	15.8251	0.796248
$396$	−2.46702	−0.123972
$397$	1.94672	0.0977032	0.0488516	−	0.998806i	$-0.484444\pi$
$397$	1.94672	0.0977032	0.0488516	+	0.998806i	$0.484444\pi$
$398$	64.0452	3.21030
$399$	−3.78778	−0.189626
$400$	8.43756	0.421878
$401$	−1.07924	−0.0538949	−0.0269475	−	0.999637i	$-0.508579\pi$
$401$	−1.07924	−0.0538949	−0.0269475	+	0.999637i	$0.508579\pi$
$402$	17.4154	0.868600
$403$	7.01937	0.349660
$404$	−41.0710	−2.04336
$405$	1.00000	0.0496904
$406$	6.64697	0.329883
$407$	1.11597	0.0553164
$408$	−15.8336	−0.783879
$409$	8.48617	0.419614	0.209807	−	0.977743i	$-0.432716\pi$
$409$	8.48617	0.419614	0.209807	+	0.977743i	$0.432716\pi$
$410$	−20.1342	−0.994355
$411$	6.30059	0.310785
$412$	86.8982	4.28117
$413$	−10.6189	−0.522523
$414$	−11.5366	−0.566994
$415$	−6.19448	−0.304075
$416$	−56.2910	−2.75989
$417$	−20.5568	−1.00667
$418$	−5.46056	−0.267084
$419$	−10.3724	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$419$	−10.3724	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$420$	−4.41819	−0.215586
$421$	−33.8812	−1.65127	−0.825635	−	0.564205i	$-0.809183\pi$
$421$	−33.8812	−1.65127	−0.825635	+	0.564205i	$0.809183\pi$
$422$	−19.4993	−0.949211
$423$	−4.13275	−0.200941
$424$	−5.93968	−0.288456
$425$	2.30059	0.111595
$426$	−29.7463	−1.44121
$427$	5.24956	0.254044
$428$	−7.32665	−0.354147
$429$	−3.71152	−0.179194
$430$	20.8414	1.00506
$431$	21.2245	1.02235	0.511175	−	0.859477i	$-0.329210\pi$
$431$	21.2245	1.02235	0.511175	+	0.859477i	$0.329210\pi$
$432$	−8.43756	−0.405953
$433$	33.2157	1.59625	0.798123	−	0.602495i	$-0.205827\pi$
$433$	33.2157	1.59625	0.798123	+	0.602495i	$0.205827\pi$
$434$	−2.44483	−0.117356
$435$	−2.71878	−0.130356
$436$	31.0808	1.48850
$437$	−17.8737	−0.855016
$438$	31.5411	1.50709
$439$	−15.8653	−0.757208	−0.378604	−	0.925559i	$-0.623596\pi$
$439$	−15.8653	−0.757208	−0.378604	+	0.925559i	$0.623596\pi$
$440$	−3.63909	−0.173487
$441$	−6.10329	−0.290633
$442$	−41.6928	−1.98313
$443$	−20.6363	−0.980459	−0.490229	−	0.871593i	$-0.663087\pi$
$443$	−20.6363	−0.980459	−0.490229	+	0.871593i	$0.663087\pi$
$444$	9.84730	0.467332
$445$	6.57899	0.311874
$446$	−22.4529	−1.06318
$447$	18.4614	0.873194
$448$	3.62618	0.171321
$449$	9.68933	0.457268	0.228634	−	0.973512i	$-0.426574\pi$
$449$	9.68933	0.457268	0.228634	+	0.973512i	$0.426574\pi$
$450$	2.58181	0.121708
$451$	4.12347	0.194167
$452$	87.8197	4.13069
$453$	−16.6615	−0.782826
$454$	−15.0117	−0.704535
$455$	−6.64697	−0.311615
$456$	−27.5296	−1.28919
$457$	−17.9988	−0.841949	−0.420975	−	0.907072i	$-0.638312\pi$
$457$	−17.9988	−0.841949	−0.420975	+	0.907072i	$0.638312\pi$
$458$	18.3357	0.856770
$459$	−2.30059	−0.107382
$460$	−20.8485	−0.972066
$461$	−25.1134	−1.16965	−0.584823	−	0.811161i	$-0.698836\pi$
$461$	−25.1134	−1.16965	−0.584823	+	0.811161i	$0.698836\pi$
$462$	1.29271	0.0601424
$463$	−6.73251	−0.312886	−0.156443	−	0.987687i	$-0.550003\pi$
$463$	−6.73251	−0.312886	−0.156443	+	0.987687i	$0.550003\pi$
$464$	22.9399	1.06496
$465$	1.00000	0.0463739
$466$	28.6390	1.32667
$467$	−3.35765	−0.155373	−0.0776867	−	0.996978i	$-0.524753\pi$
$467$	−3.35765	−0.155373	−0.0776867	+	0.996978i	$0.524753\pi$
$468$	−32.7505	−1.51389
$469$	−6.38755	−0.294950
$470$	−10.6700	−0.492168
$471$	−9.73674	−0.448645
$472$	−77.1782	−3.55241
$473$	−4.26832	−0.196258
$474$	−40.8574	−1.87664
$475$	4.00000	0.183533
$476$	10.1644	0.465887
$477$	−0.863025	−0.0395152
$478$	−19.2300	−0.879558
$479$	23.7196	1.08378	0.541888	−	0.840450i	$-0.317710\pi$
$479$	23.7196	1.08378	0.541888	+	0.840450i	$0.317710\pi$
$480$	−8.01937	−0.366033
$481$	14.8148	0.675497
$482$	−44.1392	−2.01049
$483$	4.23136	0.192534
$484$	−50.0186	−2.27357
$485$	−1.47125	−0.0668059
$486$	−2.58181	−0.117113
$487$	31.1495	1.41152	0.705760	−	0.708451i	$-0.250606\pi$
$487$	31.1495	1.41152	0.705760	+	0.708451i	$0.250606\pi$
$488$	38.1538	1.72714
$489$	1.63464	0.0739208
$490$	−15.7575	−0.711852
$491$	−21.8237	−0.984890	−0.492445	−	0.870344i	$-0.663897\pi$
$491$	−21.8237	−0.984890	−0.492445	+	0.870344i	$0.663897\pi$
$492$	36.3856	1.64039
$493$	6.25480	0.281702
$494$	−72.4907	−3.26151
$495$	−0.528753	−0.0237657
$496$	−8.43756	−0.378858
$497$	10.9102	0.489390
$498$	15.9930	0.716662
$499$	−23.6531	−1.05886	−0.529428	−	0.848355i	$-0.677593\pi$
$499$	−23.6531	−1.05886	−0.529428	+	0.848355i	$0.677593\pi$
$500$	4.66573	0.208658
$501$	2.49507	0.111472
$502$	−27.9399	−1.24702
$503$	−8.34496	−0.372084	−0.186042	−	0.982542i	$-0.559566\pi$
$503$	−8.34496	−0.372084	−0.186042	+	0.982542i	$0.559566\pi$
$504$	6.51726	0.290302
$505$	−8.80270	−0.391715
$506$	6.10003	0.271179
$507$	−36.2716	−1.61088
$508$	91.9686	4.08045
$509$	13.6752	0.606144	0.303072	−	0.952968i	$-0.401988\pi$
$509$	13.6752	0.606144	0.303072	+	0.952968i	$0.401988\pi$
$510$	−5.93968	−0.263013
$511$	−11.5685	−0.511761
$512$	−48.4774	−2.14242
$513$	−4.00000	−0.176604
$514$	33.1505	1.46221
$515$	18.6248	0.820706
$516$	−37.6637	−1.65805
$517$	2.18520	0.0961051
$518$	−5.15997	−0.226716
$519$	18.1005	0.794523
$520$	−48.3101	−2.11854
$521$	−13.7900	−0.604152	−0.302076	−	0.953284i	$-0.597680\pi$
$521$	−13.7900	−0.604152	−0.302076	+	0.953284i	$0.597680\pi$
$522$	7.01937	0.307230
$523$	−11.6199	−0.508105	−0.254052	−	0.967190i	$-0.581764\pi$
$523$	−11.6199	−0.508105	−0.254052	+	0.967190i	$0.581764\pi$
$524$	30.0698	1.31361
$525$	−0.946946	−0.0413281
$526$	−70.8240	−3.08807
$527$	−2.30059	−0.100215
$528$	4.46139	0.194157
$529$	−3.03311	−0.131875
$530$	−2.22816	−0.0967852
$531$	−11.2139	−0.486640
$532$	17.6728	0.766212
$533$	54.7404	2.37107
$534$	−16.9857	−0.735043
$535$	−1.57031	−0.0678905
$536$	−46.4247	−2.00524
$537$	0.668543	0.0288498
$538$	39.6907	1.71119
$539$	3.22714	0.139003
$540$	−4.66573	−0.200781
$541$	−14.7974	−0.636192	−0.318096	−	0.948059i	$-0.603043\pi$
$541$	−14.7974	−0.636192	−0.318096	+	0.948059i	$0.603043\pi$
$542$	−57.4477	−2.46759
$543$	−5.87936	−0.252307
$544$	18.4493	0.791007
$545$	6.66150	0.285347
$546$	17.1612	0.734431
$547$	−11.1344	−0.476072	−0.238036	−	0.971256i	$-0.576504\pi$
$547$	−11.1344	−0.476072	−0.238036	+	0.971256i	$0.576504\pi$
$548$	−29.3968	−1.25577
$549$	5.54367	0.236598
$550$	−1.36514	−0.0582097
$551$	10.8751	0.463296
$552$	30.7535	1.30896
$553$	14.9855	0.637250
$554$	32.1120	1.36431
$555$	2.11056	0.0895883
$556$	95.9125	4.06759
$557$	15.8527	0.671701	0.335851	−	0.941915i	$-0.390976\pi$
$557$	15.8527	0.671701	0.335851	+	0.941915i	$0.390976\pi$
$558$	−2.58181	−0.109297
$559$	−56.6634	−2.39660
$560$	7.98992	0.337636
$561$	1.21644	0.0513583
$562$	34.5338	1.45672
$563$	19.0523	0.802957	0.401478	−	0.915869i	$-0.368497\pi$
$563$	19.0523	0.802957	0.401478	+	0.915869i	$0.368497\pi$
$564$	19.2823	0.811930
$565$	18.8223	0.791860
$566$	−28.1265	−1.18225
$567$	0.946946	0.0397680
$568$	79.2954	3.32716
$569$	33.6242	1.40960	0.704800	−	0.709406i	$-0.251037\pi$
$569$	33.6242	1.40960	0.704800	+	0.709406i	$0.251037\pi$
$570$	−10.3272	−0.432560
$571$	−2.29694	−0.0961240	−0.0480620	−	0.998844i	$-0.515305\pi$
$571$	−2.29694	−0.0961240	−0.0480620	+	0.998844i	$0.515305\pi$
$572$	17.3169	0.724057
$573$	−20.8193	−0.869737
$574$	−19.0660	−0.795798
$575$	−4.46843	−0.186346
$576$	3.82934	0.159556
$577$	−16.5001	−0.686910	−0.343455	−	0.939169i	$-0.611597\pi$
$577$	−16.5001	−0.686910	−0.343455	+	0.939169i	$0.611597\pi$
$578$	−30.2260	−1.25723
$579$	4.46702	0.185643
$580$	12.6851	0.526720
$581$	−5.86584	−0.243356
$582$	3.79848	0.157452
$583$	0.456327	0.0188991
$584$	−84.0800	−3.47925
$585$	−7.01937	−0.290215
$586$	−28.6390	−1.18306
$587$	16.3446	0.674613	0.337307	−	0.941395i	$-0.390484\pi$
$587$	16.3446	0.674613	0.337307	+	0.941395i	$0.390484\pi$
$588$	28.4763	1.17434
$589$	−4.00000	−0.164817
$590$	−28.9520	−1.19194
$591$	22.3539	0.919515
$592$	−17.8080	−0.731904
$593$	−22.5568	−0.926297	−0.463148	−	0.886281i	$-0.653280\pi$
$593$	−22.5568	−0.926297	−0.463148	+	0.886281i	$0.653280\pi$
$594$	1.36514	0.0560123
$595$	2.17853	0.0893112
$596$	−86.1358	−3.52826
$597$	−24.8064	−1.01526
$598$	80.9799	3.31151
$599$	−20.3882	−0.833038	−0.416519	−	0.909127i	$-0.636750\pi$
$599$	−20.3882	−0.833038	−0.416519	+	0.909127i	$0.636750\pi$
$600$	−6.88240	−0.280973
$601$	10.2211	0.416928	0.208464	−	0.978030i	$-0.433154\pi$
$601$	10.2211	0.416928	0.208464	+	0.978030i	$0.433154\pi$
$602$	19.7357	0.804368
$603$	−6.74542	−0.274695
$604$	77.7380	3.16312
$605$	−10.7204	−0.435847
$606$	22.7269	0.923217
$607$	25.6738	1.04207	0.521034	−	0.853536i	$-0.325546\pi$
$607$	25.6738	1.04207	0.521034	+	0.853536i	$0.325546\pi$
$608$	32.0775	1.30091
$609$	−2.57454	−0.104326
$610$	14.3127	0.579504
$611$	29.0093	1.17359
$612$	10.7339	0.433893
$613$	−34.8374	−1.40707	−0.703536	−	0.710660i	$-0.748396\pi$
$613$	−34.8374	−1.40707	−0.703536	+	0.710660i	$0.748396\pi$
$614$	18.0357	0.727862
$615$	7.79848	0.314465
$616$	−3.44602	−0.138844
$617$	−19.1350	−0.770346	−0.385173	−	0.922844i	$-0.625858\pi$
$617$	−19.1350	−0.770346	−0.385173	+	0.922844i	$0.625858\pi$
$618$	−48.0856	−1.93429
$619$	43.8352	1.76189	0.880944	−	0.473221i	$-0.156909\pi$
$619$	43.8352	1.76189	0.880944	+	0.473221i	$0.156909\pi$
$620$	−4.66573	−0.187380
$621$	4.46843	0.179312
$622$	77.0866	3.09089
$623$	6.22995	0.249598
$624$	59.2264	2.37095
$625$	1.00000	0.0400000
$626$	38.0553	1.52100
$627$	2.11501	0.0844655
$628$	45.4290	1.81281
$629$	−4.85553	−0.193603
$630$	2.44483	0.0974044
$631$	−37.8568	−1.50706	−0.753528	−	0.657416i	$-0.771650\pi$
$631$	−37.8568	−1.50706	−0.753528	+	0.657416i	$0.771650\pi$
$632$	108.915	4.33239
$633$	7.55258	0.300188
$634$	51.3571	2.03965
$635$	19.7115	0.782228
$636$	4.02664	0.159667
$637$	42.8413	1.69743
$638$	−3.71152	−0.146940
$639$	11.5215	0.455783
$640$	−6.15212	−0.243184
$641$	2.67158	0.105521	0.0527606	−	0.998607i	$-0.483198\pi$
$641$	2.67158	0.105521	0.0527606	+	0.998607i	$0.483198\pi$
$642$	4.05424	0.160008
$643$	3.28426	0.129518	0.0647592	−	0.997901i	$-0.479372\pi$
$643$	3.28426	0.129518	0.0647592	+	0.997901i	$0.479372\pi$
$644$	−19.7424	−0.777959
$645$	−8.07243	−0.317851
$646$	23.7587	0.934774
$647$	−22.5040	−0.884723	−0.442361	−	0.896837i	$-0.645859\pi$
$647$	−22.5040	−0.884723	−0.442361	+	0.896837i	$0.645859\pi$
$648$	6.88240	0.270366
$649$	5.92936	0.232748
$650$	−18.1227	−0.710829
$651$	0.946946	0.0371137
$652$	−7.62677	−0.298687
$653$	47.3823	1.85421	0.927106	−	0.374798i	$-0.122288\pi$
$653$	47.3823	1.85421	0.927106	+	0.374798i	$0.122288\pi$
$654$	−17.1987	−0.672523
$655$	6.44483	0.251820
$656$	−65.8001	−2.56906
$657$	−12.2167	−0.476618
$658$	−10.1039	−0.393890
$659$	−10.0609	−0.391918	−0.195959	−	0.980612i	$-0.562782\pi$
$659$	−10.0609	−0.391918	−0.195959	+	0.980612i	$0.562782\pi$
$660$	2.46702	0.0960286
$661$	−31.1794	−1.21274	−0.606368	−	0.795184i	$-0.707374\pi$
$661$	−31.1794	−1.21274	−0.606368	+	0.795184i	$0.707374\pi$
$662$	28.2562	1.09821
$663$	16.1487	0.627163
$664$	−42.6329	−1.65448
$665$	3.78778	0.146884
$666$	−5.44906	−0.211147
$667$	−12.1487	−0.470399
$668$	−11.6413	−0.450416
$669$	8.69660	0.336230
$670$	−17.4154	−0.672815
$671$	−2.93123	−0.113159
$672$	−7.59391	−0.292942
$673$	−27.6767	−1.06686	−0.533428	−	0.845845i	$-0.679097\pi$
$673$	−27.6767	−1.06686	−0.533428	+	0.845845i	$0.679097\pi$
$674$	−42.9871	−1.65580
$675$	−1.00000	−0.0384900
$676$	169.233	6.50898
$677$	−33.1527	−1.27416	−0.637081	−	0.770796i	$-0.719859\pi$
$677$	−33.1527	−1.27416	−0.637081	+	0.770796i	$0.719859\pi$
$678$	−48.5955	−1.86630
$679$	−1.39319	−0.0534658
$680$	15.8336	0.607190
$681$	5.81442	0.222809
$682$	1.36514	0.0522739
$683$	−16.1771	−0.619000	−0.309500	−	0.950899i	$-0.600162\pi$
$683$	−16.1771	−0.619000	−0.309500	+	0.950899i	$0.600162\pi$
$684$	18.6629	0.713594
$685$	−6.30059	−0.240733
$686$	−32.0354	−1.22312
$687$	−7.10188	−0.270954
$688$	68.1116	2.59673
$689$	6.05789	0.230787
$690$	11.5366	0.439192
$691$	−40.7549	−1.55039	−0.775195	−	0.631722i	$-0.782349\pi$
$691$	−40.7549	−1.55039	−0.775195	+	0.631722i	$0.782349\pi$
$692$	−84.4519	−3.21038
$693$	−0.500701	−0.0190200
$694$	−25.7805	−0.978616
$695$	20.5568	0.779764
$696$	−18.7117	−0.709267
$697$	−17.9411	−0.679567
$698$	75.3985	2.85388
$699$	−11.0926	−0.419561
$700$	4.41819	0.166992
$701$	−45.1154	−1.70399	−0.851993	−	0.523554i	$-0.824606\pi$
$701$	−45.1154	−1.70399	−0.851993	+	0.523554i	$0.824606\pi$
$702$	18.1227	0.683996
$703$	−8.44224	−0.318405
$704$	−2.02478	−0.0763117
$705$	4.13275	0.155648
$706$	−96.4921	−3.63153
$707$	−8.33568	−0.313496
$708$	52.3208	1.96634
$709$	9.78778	0.367588	0.183794	−	0.982965i	$-0.441162\pi$
$709$	9.78778	0.367588	0.183794	+	0.982965i	$0.441162\pi$
$710$	29.7463	1.11636
$711$	15.8251	0.593488
$712$	45.2792	1.69691
$713$	4.46843	0.167344
$714$	−5.62456	−0.210494
$715$	3.71152	0.138803
$716$	−3.11924	−0.116572
$717$	7.44826	0.278160
$718$	41.3241	1.54220
$719$	−47.8844	−1.78579	−0.892894	−	0.450267i	$-0.851329\pi$
$719$	−47.8844	−1.78579	−0.892894	+	0.450267i	$0.851329\pi$
$720$	8.43756	0.314449
$721$	17.6367	0.656824
$722$	−7.74542	−0.288255
$723$	17.0963	0.635816
$724$	27.4315	1.01948
$725$	2.71878	0.100973
$726$	27.6781	1.02723
$727$	11.3910	0.422468	0.211234	−	0.977436i	$-0.432252\pi$
$727$	11.3910	0.422468	0.211234	+	0.977436i	$0.432252\pi$
$728$	−45.7471	−1.69550
$729$	1.00000	0.0370370
$730$	−31.5411	−1.16739
$731$	18.5713	0.686886
$732$	−25.8653	−0.956008
$733$	47.0115	1.73641	0.868205	−	0.496205i	$-0.165274\pi$
$733$	47.0115	1.73641	0.868205	+	0.496205i	$0.165274\pi$
$734$	12.0169	0.443552
$735$	6.10329	0.225123
$736$	−35.8340	−1.32086
$737$	3.56666	0.131380
$738$	−20.1342	−0.741149
$739$	−8.60822	−0.316659	−0.158329	−	0.987386i	$-0.550611\pi$
$739$	−8.60822	−0.316659	−0.158329	+	0.987386i	$0.550611\pi$
$740$	−9.84730	−0.361994
$741$	28.0775	1.03145
$742$	−2.10995	−0.0774587
$743$	4.56806	0.167586	0.0837930	−	0.996483i	$-0.473297\pi$
$743$	4.56806	0.167586	0.0837930	+	0.996483i	$0.473297\pi$
$744$	6.88240	0.252321
$745$	−18.4614	−0.676373
$746$	21.9326	0.803011
$747$	−6.19448	−0.226644
$748$	−5.67560	−0.207520
$749$	−1.48700	−0.0543338
$750$	−2.58181	−0.0942743
$751$	40.1751	1.46601	0.733006	−	0.680222i	$-0.238117\pi$
$751$	40.1751	1.46601	0.733006	+	0.680222i	$0.238117\pi$
$752$	−34.8703	−1.27159
$753$	10.8219	0.394370
$754$	−49.2716	−1.79437
$755$	16.6615	0.606374
$756$	−4.41819	−0.160688
$757$	−0.571501	−0.0207716	−0.0103858	−	0.999946i	$-0.503306\pi$
$757$	−0.571501	−0.0207716	−0.0103858	+	0.999946i	$0.503306\pi$
$758$	−48.5580	−1.76371
$759$	−2.36270	−0.0857605
$760$	27.5296	0.998603
$761$	51.3390	1.86104	0.930518	−	0.366245i	$-0.119357\pi$
$761$	51.3390	1.86104	0.930518	+	0.366245i	$0.119357\pi$
$762$	−50.8913	−1.84360
$763$	6.30808	0.228368
$764$	97.1370	3.51429
$765$	2.30059	0.0831780
$766$	−28.2827	−1.02189
$767$	78.7142	2.84221
$768$	23.5423	0.849509
$769$	−23.8320	−0.859403	−0.429701	−	0.902971i	$-0.641381\pi$
$769$	−23.8320	−0.859403	−0.429701	+	0.902971i	$0.641381\pi$
$770$	−1.29271	−0.0465861
$771$	−12.8400	−0.462423
$772$	−20.8419	−0.750116
$773$	−1.27498	−0.0458578	−0.0229289	−	0.999737i	$-0.507299\pi$
$773$	−1.27498	−0.0458578	−0.0229289	+	0.999737i	$0.507299\pi$
$774$	20.8414	0.749130
$775$	−1.00000	−0.0359211
$776$	−10.1257	−0.363492
$777$	1.99859	0.0716989
$778$	−60.8280	−2.18079
$779$	−31.1939	−1.11764
$780$	32.7505	1.17266
$781$	−6.09202	−0.217990
$782$	−26.5411	−0.949106
$783$	−2.71878	−0.0971614
$784$	−51.4969	−1.83918
$785$	9.73674	0.347519
$786$	−16.6393	−0.593505
$787$	1.41559	0.0504604	0.0252302	−	0.999682i	$-0.491968\pi$
$787$	1.41559	0.0504604	0.0252302	+	0.999682i	$0.491968\pi$
$788$	−104.297	−3.71543
$789$	27.4319	0.976603
$790$	40.8574	1.45364
$791$	17.8237	0.633738
$792$	−3.63909	−0.129309
$793$	−38.9131	−1.38184
$794$	5.02606	0.178368
$795$	0.863025	0.0306083
$796$	115.740	4.10229
$797$	46.4314	1.64468	0.822342	−	0.568994i	$-0.192667\pi$
$797$	46.4314	1.64468	0.822342	+	0.568994i	$0.192667\pi$
$798$	−9.77933	−0.346184
$799$	−9.50776	−0.336360
$800$	8.01937	0.283528
$801$	6.57899	0.232457
$802$	−2.78640	−0.0983913
$803$	6.45960	0.227954
$804$	31.4723	1.10994
$805$	−4.23136	−0.149136
$806$	18.1227	0.638344
$807$	−15.3732	−0.541164
$808$	−60.5837	−2.13133
$809$	30.2736	1.06436	0.532181	−	0.846630i	$-0.321372\pi$
$809$	30.2736	1.06436	0.532181	+	0.846630i	$0.321372\pi$
$810$	2.58181	0.0907155
$811$	−35.0289	−1.23003	−0.615015	−	0.788515i	$-0.710850\pi$
$811$	−35.0289	−1.23003	−0.615015	+	0.788515i	$0.710850\pi$
$812$	12.0121	0.421542
$813$	22.2510	0.780375
$814$	2.88121	0.100986
$815$	−1.63464	−0.0572588
$816$	−19.4114	−0.679534
$817$	32.2897	1.12967
$818$	21.9096	0.766053
$819$	−6.64697	−0.232264
$820$	−36.3856	−1.27064
$821$	−41.9548	−1.46423	−0.732117	−	0.681179i	$-0.761467\pi$
$821$	−41.9548	−1.46423	−0.732117	+	0.681179i	$0.761467\pi$
$822$	16.2669	0.567374
$823$	4.42220	0.154148	0.0770740	−	0.997025i	$-0.475442\pi$
$823$	4.42220	0.154148	0.0770740	+	0.997025i	$0.475442\pi$
$824$	128.183	4.46547
$825$	0.528753	0.0184088
$826$	−27.4160	−0.953924
$827$	32.5245	1.13099	0.565494	−	0.824752i	$-0.308685\pi$
$827$	32.5245	1.13099	0.565494	+	0.824752i	$0.308685\pi$
$828$	−20.8485	−0.724535
$829$	10.7831	0.374513	0.187256	−	0.982311i	$-0.440040\pi$
$829$	10.7831	0.374513	0.187256	+	0.982311i	$0.440040\pi$
$830$	−15.9930	−0.555124
$831$	−12.4378	−0.431462
$832$	−26.8796	−0.931882
$833$	−14.0412	−0.486498
$834$	−53.0737	−1.83779
$835$	−2.49507	−0.0863455
$836$	−9.86808	−0.341295
$837$	1.00000	0.0345651
$838$	−26.7796	−0.925084
$839$	4.07323	0.140624	0.0703118	−	0.997525i	$-0.477601\pi$
$839$	4.07323	0.140624	0.0703118	+	0.997525i	$0.477601\pi$
$840$	−6.51726	−0.224867
$841$	−21.6082	−0.745111
$842$	−87.4748	−3.01458
$843$	−13.3758	−0.460688
$844$	−35.2383	−1.21295
$845$	36.2716	1.24778
$846$	−10.6700	−0.366841
$847$	−10.1517	−0.348815
$848$	−7.28183	−0.250059
$849$	10.8941	0.373885
$850$	5.93968	0.203729
$851$	9.43090	0.323287
$852$	−53.7561	−1.84165
$853$	9.57173	0.327730	0.163865	−	0.986483i	$-0.447604\pi$
$853$	9.57173	0.327730	0.163865	+	0.986483i	$0.447604\pi$
$854$	13.5533	0.463786
$855$	4.00000	0.136797
$856$	−10.8075	−0.369393
$857$	37.3617	1.27625	0.638126	−	0.769932i	$-0.279710\pi$
$857$	37.3617	1.27625	0.638126	+	0.769932i	$0.279710\pi$
$858$	−9.58242	−0.327138
$859$	−2.32723	−0.0794040	−0.0397020	−	0.999212i	$-0.512641\pi$
$859$	−2.32723	−0.0794040	−0.0397020	+	0.999212i	$0.512641\pi$
$860$	37.6637	1.28432
$861$	7.38474	0.251671
$862$	54.7977	1.86642
$863$	−3.93264	−0.133868	−0.0669342	−	0.997757i	$-0.521322\pi$
$863$	−3.93264	−0.133868	−0.0669342	+	0.997757i	$0.521322\pi$
$864$	−8.01937	−0.272825
$865$	−18.1005	−0.615435
$866$	85.7566	2.91413
$867$	11.7073	0.397600
$868$	−4.41819	−0.149963
$869$	−8.36758	−0.283851
$870$	−7.01937	−0.237979
$871$	47.3486	1.60435
$872$	45.8471	1.55258
$873$	−1.47125	−0.0497942
$874$	−46.1465	−1.56093
$875$	0.946946	0.0320126
$876$	56.9997	1.92584
$877$	−12.5461	−0.423652	−0.211826	−	0.977307i	$-0.567941\pi$
$877$	−12.5461	−0.423652	−0.211826	+	0.977307i	$0.567941\pi$
$878$	−40.9611	−1.38237
$879$	11.0926	0.374144
$880$	−4.46139	−0.150393
$881$	−55.4929	−1.86960	−0.934801	−	0.355172i	$-0.884422\pi$
$881$	−55.4929	−1.86960	−0.934801	+	0.355172i	$0.884422\pi$
$882$	−15.7575	−0.530584
$883$	−33.9522	−1.14258	−0.571292	−	0.820747i	$-0.693558\pi$
$883$	−33.9522	−1.14258	−0.571292	+	0.820747i	$0.693558\pi$
$884$	−75.3454	−2.53414
$885$	11.2139	0.376950
$886$	−53.2789	−1.78994
$887$	46.6395	1.56600	0.783001	−	0.622020i	$-0.213688\pi$
$887$	46.6395	1.56600	0.783001	+	0.622020i	$0.213688\pi$
$888$	14.5257	0.487451
$889$	18.6657	0.626029
$890$	16.9857	0.569362
$891$	−0.528753	−0.0177139
$892$	−40.5760	−1.35858
$893$	−16.5310	−0.553188
$894$	47.6637	1.59411
$895$	−0.668543	−0.0223469
$896$	−5.82572	−0.194624
$897$	−31.3656	−1.04727
$898$	25.0160	0.834794
$899$	−2.71878	−0.0906765
$900$	4.66573	0.155524
$901$	−1.98547	−0.0661455
$902$	10.6460	0.354473
$903$	−7.64415	−0.254381
$904$	129.543	4.30852
$905$	5.87936	0.195436
$906$	−43.0168	−1.42914
$907$	20.0965	0.667292	0.333646	−	0.942698i	$-0.391721\pi$
$907$	20.0965	0.667292	0.333646	+	0.942698i	$0.391721\pi$
$908$	−27.1285	−0.900292
$909$	−8.80270	−0.291967
$910$	−17.1612	−0.568888
$911$	23.5386	0.779869	0.389934	−	0.920843i	$-0.372498\pi$
$911$	23.5386	0.779869	0.389934	+	0.920843i	$0.372498\pi$
$912$	−33.7503	−1.11758
$913$	3.27535	0.108398
$914$	−46.4695	−1.53707
$915$	−5.54367	−0.183268
$916$	33.1354	1.09483
$917$	6.10291	0.201536
$918$	−5.93968	−0.196039
$919$	45.3500	1.49596	0.747980	−	0.663721i	$-0.231024\pi$
$919$	45.3500	1.49596	0.747980	+	0.663721i	$0.231024\pi$
$920$	−30.7535	−1.01391
$921$	−6.98569	−0.230186
$922$	−64.8379	−2.13532
$923$	−80.8736	−2.66199
$924$	2.33613	0.0768532
$925$	−2.11056	−0.0693948
$926$	−17.3821	−0.571210
$927$	18.6248	0.611718
$928$	21.8029	0.715716
$929$	−7.36716	−0.241709	−0.120854	−	0.992670i	$-0.538563\pi$
$929$	−7.36716	−0.241709	−0.120854	+	0.992670i	$0.538563\pi$
$930$	2.58181	0.0846608
$931$	−24.4132	−0.800109
$932$	51.7551	1.69529
$933$	−29.8576	−0.977495
$934$	−8.66880	−0.283652
$935$	−1.21644	−0.0397820
$936$	−48.3101	−1.57907
$937$	33.5487	1.09599	0.547995	−	0.836482i	$-0.315391\pi$
$937$	33.5487	1.09599	0.547995	+	0.836482i	$0.315391\pi$
$938$	−16.4914	−0.538464
$939$	−14.7398	−0.481015
$940$	−19.2823	−0.628918
$941$	44.9454	1.46518	0.732589	−	0.680671i	$-0.238312\pi$
$941$	44.9454	1.46518	0.732589	+	0.680671i	$0.238312\pi$
$942$	−25.1384	−0.819053
$943$	34.8470	1.13477
$944$	−94.6176	−3.07954
$945$	−0.946946	−0.0308042
$946$	−11.0200	−0.358291
$947$	14.7638	0.479758	0.239879	−	0.970803i	$-0.422892\pi$
$947$	14.7638	0.479758	0.239879	+	0.970803i	$0.422892\pi$
$948$	−73.8357	−2.39807
$949$	85.7533	2.78367
$950$	10.3272	0.335060
$951$	−19.8919	−0.645039
$952$	14.9935	0.485943
$953$	−5.73771	−0.185863	−0.0929313	−	0.995673i	$-0.529624\pi$
$953$	−5.73771	−0.185863	−0.0929313	+	0.995673i	$0.529624\pi$
$954$	−2.22816	−0.0721395
$955$	20.8193	0.673695
$956$	−34.7515	−1.12395
$957$	1.43756	0.0464699
$958$	61.2395	1.97856
$959$	−5.96632	−0.192662
$960$	−3.82934	−0.123592
$961$	1.00000	0.0322581
$962$	38.2490	1.23320
$963$	−1.57031	−0.0506026
$964$	−79.7665	−2.56910
$965$	−4.46702	−0.143798
$966$	10.9246	0.351492
$967$	−10.9463	−0.352010	−0.176005	−	0.984389i	$-0.556318\pi$
$967$	−10.9463	−0.352010	−0.176005	+	0.984389i	$0.556318\pi$
$968$	−73.7822	−2.37145
$969$	−9.20236	−0.295622
$970$	−3.79848	−0.121962
$971$	−8.87638	−0.284857	−0.142428	−	0.989805i	$-0.545491\pi$
$971$	−8.87638	−0.284857	−0.142428	+	0.989805i	$0.545491\pi$
$972$	−4.66573	−0.149653
$973$	19.4662	0.624057
$974$	80.4221	2.57689
$975$	7.01937	0.224800
$976$	46.7751	1.49723
$977$	8.99295	0.287710	0.143855	−	0.989599i	$-0.454050\pi$
$977$	8.99295	0.287710	0.143855	+	0.989599i	$0.454050\pi$
$978$	4.22032	0.134951
$979$	−3.47866	−0.111179
$980$	−28.4763	−0.909642
$981$	6.66150	0.212685
$982$	−56.3446	−1.79803
$983$	48.7638	1.55532	0.777662	−	0.628682i	$-0.216405\pi$
$983$	48.7638	1.55532	0.777662	+	0.628682i	$0.216405\pi$
$984$	53.6722	1.71101
$985$	−22.3539	−0.712253
$986$	16.1487	0.514279
$987$	3.91349	0.124568
$988$	−131.002	−4.16773
$989$	−36.0711	−1.14699
$990$	−1.36514	−0.0433870
$991$	7.83498	0.248886	0.124443	−	0.992227i	$-0.460286\pi$
$991$	7.83498	0.248886	0.124443	+	0.992227i	$0.460286\pi$
$992$	−8.01937	−0.254615
$993$	−10.9444	−0.347309
$994$	28.1681	0.893438
$995$	24.8064	0.786414
$996$	28.9018	0.915788
$997$	58.1397	1.84130	0.920651	−	0.390387i	$-0.127659\pi$
$997$	58.1397	1.84130	0.920651	+	0.390387i	$0.127659\pi$
$998$	−61.0676	−1.93306
$999$	2.11056	0.0667752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.h.1.4	✓	4
3.2	odd	2		1395.2.a.k.1.1		4
4.3	odd	2		7440.2.a.bz.1.3		4
5.2	odd	4		2325.2.c.p.1024.8		8
5.3	odd	4		2325.2.c.p.1024.1		8
5.4	even	2		2325.2.a.v.1.1		4
15.14	odd	2		6975.2.a.bo.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.h.1.4	✓	4	1.1	even	1	trivial
1395.2.a.k.1.1		4	3.2	odd	2
2325.2.a.v.1.1		4	5.4	even	2
2325.2.c.p.1024.1		8	5.3	odd	4
2325.2.c.p.1024.8		8	5.2	odd	4
6975.2.a.bo.1.4		4	15.14	odd	2
7440.2.a.bz.1.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} +1.00000 q^{5} -2.58181 q^{6} +0.946946 q^{7} +6.88240 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} +1.00000 q^{5} -2.58181 q^{6} +0.946946 q^{7} +6.88240 q^{8} +1.00000 q^{9} +2.58181 q^{10} -0.528753 q^{11} -4.66573 q^{12} -7.01937 q^{13} +2.44483 q^{14} -1.00000 q^{15} +8.43756 q^{16} +2.30059 q^{17} +2.58181 q^{18} +4.00000 q^{19} +4.66573 q^{20} -0.946946 q^{21} -1.36514 q^{22} -4.46843 q^{23} -6.88240 q^{24} +1.00000 q^{25} -18.1227 q^{26} -1.00000 q^{27} +4.41819 q^{28} +2.71878 q^{29} -2.58181 q^{30} -1.00000 q^{31} +8.01937 q^{32} +0.528753 q^{33} +5.93968 q^{34} +0.946946 q^{35} +4.66573 q^{36} -2.11056 q^{37} +10.3272 q^{38} +7.01937 q^{39} +6.88240 q^{40} -7.79848 q^{41} -2.44483 q^{42} +8.07243 q^{43} -2.46702 q^{44} +1.00000 q^{45} -11.5366 q^{46} -4.13275 q^{47} -8.43756 q^{48} -6.10329 q^{49} +2.58181 q^{50} -2.30059 q^{51} -32.7505 q^{52} -0.863025 q^{53} -2.58181 q^{54} -0.528753 q^{55} +6.51726 q^{56} -4.00000 q^{57} +7.01937 q^{58} -11.2139 q^{59} -4.66573 q^{60} +5.54367 q^{61} -2.58181 q^{62} +0.946946 q^{63} +3.82934 q^{64} -7.01937 q^{65} +1.36514 q^{66} -6.74542 q^{67} +10.7339 q^{68} +4.46843 q^{69} +2.44483 q^{70} +11.5215 q^{71} +6.88240 q^{72} -12.2167 q^{73} -5.44906 q^{74} -1.00000 q^{75} +18.6629 q^{76} -0.500701 q^{77} +18.1227 q^{78} +15.8251 q^{79} +8.43756 q^{80} +1.00000 q^{81} -20.1342 q^{82} -6.19448 q^{83} -4.41819 q^{84} +2.30059 q^{85} +20.8414 q^{86} -2.71878 q^{87} -3.63909 q^{88} +6.57899 q^{89} +2.58181 q^{90} -6.64697 q^{91} -20.8485 q^{92} +1.00000 q^{93} -10.6700 q^{94} +4.00000 q^{95} -8.01937 q^{96} -1.47125 q^{97} -15.7575 q^{98} -0.528753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9} + 2 q^{10} + 6 q^{11} - 8 q^{12} + 2 q^{13} + 4 q^{14} - 4 q^{15} + 12 q^{16} - 10 q^{17} + 2 q^{18} + 16 q^{19} + 8 q^{20} - 4 q^{21} - 14 q^{22} + 6 q^{23} + 4 q^{25} - 10 q^{26} - 4 q^{27} + 26 q^{28} - 2 q^{30} - 4 q^{31} + 2 q^{32} - 6 q^{33} + 8 q^{34} + 4 q^{35} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 2 q^{39} - 6 q^{41} - 4 q^{42} + 2 q^{43} - 6 q^{44} + 4 q^{45} + 20 q^{46} - 2 q^{47} - 12 q^{48} + 8 q^{49} + 2 q^{50} + 10 q^{51} - 16 q^{52} - 6 q^{53} - 2 q^{54} + 6 q^{55} - 10 q^{56} - 16 q^{57} - 2 q^{58} + 4 q^{59} - 8 q^{60} - 2 q^{62} + 4 q^{63} - 12 q^{64} + 2 q^{65} + 14 q^{66} - 2 q^{67} - 10 q^{68} - 6 q^{69} + 4 q^{70} + 22 q^{71} - 32 q^{73} - 28 q^{74} - 4 q^{75} + 32 q^{76} - 28 q^{77} + 10 q^{78} + 24 q^{79} + 12 q^{80} + 4 q^{81} - 46 q^{82} - 6 q^{83} - 26 q^{84} - 10 q^{85} + 10 q^{86} - 18 q^{88} - 14 q^{89} + 2 q^{90} + 28 q^{91} - 54 q^{92} + 4 q^{93} - 44 q^{94} + 16 q^{95} - 2 q^{96} - 14 q^{97} + 8 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 4 * q^9 + 2 * q^10 + 6 * q^11 - 8 * q^12 + 2 * q^13 + 4 * q^14 - 4 * q^15 + 12 * q^16 - 10 * q^17 + 2 * q^18 + 16 * q^19 + 8 * q^20 - 4 * q^21 - 14 * q^22 + 6 * q^23 + 4 * q^25 - 10 * q^26 - 4 * q^27 + 26 * q^28 - 2 * q^30 - 4 * q^31 + 2 * q^32 - 6 * q^33 + 8 * q^34 + 4 * q^35 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 2 * q^39 - 6 * q^41 - 4 * q^42 + 2 * q^43 - 6 * q^44 + 4 * q^45 + 20 * q^46 - 2 * q^47 - 12 * q^48 + 8 * q^49 + 2 * q^50 + 10 * q^51 - 16 * q^52 - 6 * q^53 - 2 * q^54 + 6 * q^55 - 10 * q^56 - 16 * q^57 - 2 * q^58 + 4 * q^59 - 8 * q^60 - 2 * q^62 + 4 * q^63 - 12 * q^64 + 2 * q^65 + 14 * q^66 - 2 * q^67 - 10 * q^68 - 6 * q^69 + 4 * q^70 + 22 * q^71 - 32 * q^73 - 28 * q^74 - 4 * q^75 + 32 * q^76 - 28 * q^77 + 10 * q^78 + 24 * q^79 + 12 * q^80 + 4 * q^81 - 46 * q^82 - 6 * q^83 - 26 * q^84 - 10 * q^85 + 10 * q^86 - 18 * q^88 - 14 * q^89 + 2 * q^90 + 28 * q^91 - 54 * q^92 + 4 * q^93 - 44 * q^94 + 16 * q^95 - 2 * q^96 - 14 * q^97 + 8 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more