LMFDB - Embedded newform 465.2.a.f.1.2

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	$q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.311108 q^{6} +0.688892 q^{7} -1.21432 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.311108 q^{6} +0.688892 q^{7} -1.21432 q^{8} +1.00000 q^{9} +0.311108 q^{10} +2.00000 q^{11} -1.90321 q^{12} +3.73975 q^{13} +0.214320 q^{14} +1.00000 q^{15} +3.42864 q^{16} +2.28100 q^{17} +0.311108 q^{18} -1.90321 q^{20} +0.688892 q^{21} +0.622216 q^{22} +2.90321 q^{23} -1.21432 q^{24} +1.00000 q^{25} +1.16346 q^{26} +1.00000 q^{27} -1.31111 q^{28} -4.02074 q^{29} +0.311108 q^{30} +1.00000 q^{31} +3.49532 q^{32} +2.00000 q^{33} +0.709636 q^{34} +0.688892 q^{35} -1.90321 q^{36} -8.79060 q^{37} +3.73975 q^{39} -1.21432 q^{40} +2.75557 q^{41} +0.214320 q^{42} +1.05086 q^{43} -3.80642 q^{44} +1.00000 q^{45} +0.903212 q^{46} +4.70964 q^{47} +3.42864 q^{48} -6.52543 q^{49} +0.311108 q^{50} +2.28100 q^{51} -7.11753 q^{52} +2.14764 q^{53} +0.311108 q^{54} +2.00000 q^{55} -0.836535 q^{56} -1.25088 q^{58} -6.44938 q^{59} -1.90321 q^{60} -5.05086 q^{61} +0.311108 q^{62} +0.688892 q^{63} -5.76986 q^{64} +3.73975 q^{65} +0.622216 q^{66} -5.44446 q^{67} -4.34122 q^{68} +2.90321 q^{69} +0.214320 q^{70} +8.96989 q^{71} -1.21432 q^{72} -8.92396 q^{73} -2.73483 q^{74} +1.00000 q^{75} +1.37778 q^{77} +1.16346 q^{78} -8.38271 q^{79} +3.42864 q^{80} +1.00000 q^{81} +0.857279 q^{82} -10.5161 q^{83} -1.31111 q^{84} +2.28100 q^{85} +0.326929 q^{86} -4.02074 q^{87} -2.42864 q^{88} +13.5002 q^{89} +0.311108 q^{90} +2.57628 q^{91} -5.52543 q^{92} +1.00000 q^{93} +1.46520 q^{94} +3.49532 q^{96} -12.0000 q^{97} -2.03011 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 3 * q^3 + q^4 + 3 * q^5 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 6 q^{11} + q^{12} - 2 q^{13} - 6 q^{14} + 3 q^{15} - 3 q^{16} + q^{18} + q^{20} + 2 q^{21} + 2 q^{22} + 2 q^{23} + 3 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} - 4 q^{28} + 8 q^{29} + q^{30} + 3 q^{31} - 3 q^{32} + 6 q^{33} - 18 q^{34} + 2 q^{35} + q^{36} - 2 q^{39} + 3 q^{40} + 8 q^{41} - 6 q^{42} - 10 q^{43} + 2 q^{44} + 3 q^{45} - 4 q^{46} - 6 q^{47} - 3 q^{48} - 13 q^{49} + q^{50} - 8 q^{52} + q^{54} + 6 q^{55} + 4 q^{56} + 10 q^{58} + 14 q^{59} + q^{60} - 2 q^{61} + q^{62} + 2 q^{63} - 11 q^{64} - 2 q^{65} + 2 q^{66} - 16 q^{67} - 20 q^{68} + 2 q^{69} - 6 q^{70} + 20 q^{71} + 3 q^{72} - 28 q^{74} + 3 q^{75} + 4 q^{77} + 10 q^{78} + 8 q^{79} - 3 q^{80} + 3 q^{81} - 24 q^{82} + 2 q^{83} - 4 q^{84} + 14 q^{86} + 8 q^{87} + 6 q^{88} - 6 q^{89} + q^{90} - 12 q^{91} - 10 q^{92} + 3 q^{93} - 16 q^{94} - 3 q^{96} - 36 q^{97} - 13 q^{98} + 6 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + q^4 + 3 * q^5 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 6 * q^11 + q^12 - 2 * q^13 - 6 * q^14 + 3 * q^15 - 3 * q^16 + q^18 + q^20 + 2 * q^21 + 2 * q^22 + 2 * q^23 + 3 * q^24 + 3 * q^25 + 10 * q^26 + 3 * q^27 - 4 * q^28 + 8 * q^29 + q^30 + 3 * q^31 - 3 * q^32 + 6 * q^33 - 18 * q^34 + 2 * q^35 + q^36 - 2 * q^39 + 3 * q^40 + 8 * q^41 - 6 * q^42 - 10 * q^43 + 2 * q^44 + 3 * q^45 - 4 * q^46 - 6 * q^47 - 3 * q^48 - 13 * q^49 + q^50 - 8 * q^52 + q^54 + 6 * q^55 + 4 * q^56 + 10 * q^58 + 14 * q^59 + q^60 - 2 * q^61 + q^62 + 2 * q^63 - 11 * q^64 - 2 * q^65 + 2 * q^66 - 16 * q^67 - 20 * q^68 + 2 * q^69 - 6 * q^70 + 20 * q^71 + 3 * q^72 - 28 * q^74 + 3 * q^75 + 4 * q^77 + 10 * q^78 + 8 * q^79 - 3 * q^80 + 3 * q^81 - 24 * q^82 + 2 * q^83 - 4 * q^84 + 14 * q^86 + 8 * q^87 + 6 * q^88 - 6 * q^89 + q^90 - 12 * q^91 - 10 * q^92 + 3 * q^93 - 16 * q^94 - 3 * q^96 - 36 * q^97 - 13 * 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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.90321	−0.951606
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.311108	0.127009
$7$	0.688892	0.260377	0.130188	−	0.991489i	$-0.458442\pi$
$7$	0.688892	0.260377	0.130188	+	0.991489i	$0.458442\pi$
$8$	−1.21432	−0.429327
$9$	1.00000	0.333333
$10$	0.311108	0.0983809
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−1.90321	−0.549410
$13$	3.73975	1.03722	0.518610	−	0.855011i	$-0.326450\pi$
$13$	3.73975	1.03722	0.518610	+	0.855011i	$0.326450\pi$
$14$	0.214320	0.0572794
$15$	1.00000	0.258199
$16$	3.42864	0.857160
$17$	2.28100	0.553223	0.276611	−	0.960982i	$-0.410789\pi$
$17$	2.28100	0.553223	0.276611	+	0.960982i	$0.410789\pi$
$18$	0.311108	0.0733288
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.90321	−0.425571
$21$	0.688892	0.150329
$22$	0.622216	0.132657
$23$	2.90321	0.605362	0.302681	−	0.953092i	$-0.402118\pi$
$23$	2.90321	0.605362	0.302681	+	0.953092i	$0.402118\pi$
$24$	−1.21432	−0.247872
$25$	1.00000	0.200000
$26$	1.16346	0.228174
$27$	1.00000	0.192450
$28$	−1.31111	−0.247776
$29$	−4.02074	−0.746633	−0.373317	−	0.927704i	$-0.621780\pi$
$29$	−4.02074	−0.746633	−0.373317	+	0.927704i	$0.621780\pi$
$30$	0.311108	0.0568003
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.49532	0.617890
$33$	2.00000	0.348155
$34$	0.709636	0.121702
$35$	0.688892	0.116444
$36$	−1.90321	−0.317202
$37$	−8.79060	−1.44517	−0.722583	−	0.691284i	$-0.757045\pi$
$37$	−8.79060	−1.44517	−0.722583	+	0.691284i	$0.757045\pi$
$38$	0	0
$39$	3.73975	0.598839
$40$	−1.21432	−0.192001
$41$	2.75557	0.430348	0.215174	−	0.976576i	$-0.430968\pi$
$41$	2.75557	0.430348	0.215174	+	0.976576i	$0.430968\pi$
$42$	0.214320	0.0330703
$43$	1.05086	0.160254	0.0801270	−	0.996785i	$-0.474467\pi$
$43$	1.05086	0.160254	0.0801270	+	0.996785i	$0.474467\pi$
$44$	−3.80642	−0.573840
$45$	1.00000	0.149071
$46$	0.903212	0.133171
$47$	4.70964	0.686971	0.343485	−	0.939158i	$-0.388392\pi$
$47$	4.70964	0.686971	0.343485	+	0.939158i	$0.388392\pi$
$48$	3.42864	0.494881
$49$	−6.52543	−0.932204
$50$	0.311108	0.0439973
$51$	2.28100	0.319403
$52$	−7.11753	−0.987024
$53$	2.14764	0.295001	0.147501	−	0.989062i	$-0.452877\pi$
$53$	2.14764	0.295001	0.147501	+	0.989062i	$0.452877\pi$
$54$	0.311108	0.0423364
$55$	2.00000	0.269680
$56$	−0.836535	−0.111787
$57$	0	0
$58$	−1.25088	−0.164249
$59$	−6.44938	−0.839638	−0.419819	−	0.907608i	$-0.637906\pi$
$59$	−6.44938	−0.839638	−0.419819	+	0.907608i	$0.637906\pi$
$60$	−1.90321	−0.245704
$61$	−5.05086	−0.646696	−0.323348	−	0.946280i	$-0.604808\pi$
$61$	−5.05086	−0.646696	−0.323348	+	0.946280i	$0.604808\pi$
$62$	0.311108	0.0395107
$63$	0.688892	0.0867923
$64$	−5.76986	−0.721232
$65$	3.73975	0.463859
$66$	0.622216	0.0765895
$67$	−5.44446	−0.665147	−0.332573	−	0.943077i	$-0.607917\pi$
$67$	−5.44446	−0.665147	−0.332573	+	0.943077i	$0.607917\pi$
$68$	−4.34122	−0.526450
$69$	2.90321	0.349506
$70$	0.214320	0.0256161
$71$	8.96989	1.06453	0.532265	−	0.846578i	$-0.321341\pi$
$71$	8.96989	1.06453	0.532265	+	0.846578i	$0.321341\pi$
$72$	−1.21432	−0.143109
$73$	−8.92396	−1.04447	−0.522235	−	0.852802i	$-0.674902\pi$
$73$	−8.92396	−1.04447	−0.522235	+	0.852802i	$0.674902\pi$
$74$	−2.73483	−0.317917
$75$	1.00000	0.115470
$76$	0	0
$77$	1.37778	0.157013
$78$	1.16346	0.131736
$79$	−8.38271	−0.943128	−0.471564	−	0.881832i	$-0.656310\pi$
$79$	−8.38271	−0.943128	−0.471564	+	0.881832i	$0.656310\pi$
$80$	3.42864	0.383334
$81$	1.00000	0.111111
$82$	0.857279	0.0946706
$83$	−10.5161	−1.15429	−0.577144	−	0.816643i	$-0.695833\pi$
$83$	−10.5161	−1.15429	−0.577144	+	0.816643i	$0.695833\pi$
$84$	−1.31111	−0.143054
$85$	2.28100	0.247409
$86$	0.326929	0.0352537
$87$	−4.02074	−0.431069
$88$	−2.42864	−0.258894
$89$	13.5002	1.43102	0.715511	−	0.698601i	$-0.246194\pi$
$89$	13.5002	1.43102	0.715511	+	0.698601i	$0.246194\pi$
$90$	0.311108	0.0327936
$91$	2.57628	0.270068
$92$	−5.52543	−0.576066
$93$	1.00000	0.103695
$94$	1.46520	0.151124
$95$	0	0
$96$	3.49532	0.356739
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	−2.03011	−0.205072
$99$	2.00000	0.201008
$100$	−1.90321	−0.190321
$101$	8.29529	0.825412	0.412706	−	0.910864i	$-0.364584\pi$
$101$	8.29529	0.825412	0.412706	+	0.910864i	$0.364584\pi$
$102$	0.709636	0.0702644
$103$	2.98418	0.294040	0.147020	−	0.989134i	$-0.453032\pi$
$103$	2.98418	0.294040	0.147020	+	0.989134i	$0.453032\pi$
$104$	−4.54125	−0.445306
$105$	0.688892	0.0672290
$106$	0.668149	0.0648963
$107$	1.52543	0.147469	0.0737343	−	0.997278i	$-0.476508\pi$
$107$	1.52543	0.147469	0.0737343	+	0.997278i	$0.476508\pi$
$108$	−1.90321	−0.183137
$109$	3.76049	0.360190	0.180095	−	0.983649i	$-0.442360\pi$
$109$	3.76049	0.360190	0.180095	+	0.983649i	$0.442360\pi$
$110$	0.622216	0.0593259
$111$	−8.79060	−0.834367
$112$	2.36196	0.223185
$113$	−9.18421	−0.863978	−0.431989	−	0.901879i	$-0.642188\pi$
$113$	−9.18421	−0.863978	−0.431989	+	0.901879i	$0.642188\pi$
$114$	0	0
$115$	2.90321	0.270726
$116$	7.65233	0.710501
$117$	3.73975	0.345740
$118$	−2.00645	−0.184709
$119$	1.57136	0.144046
$120$	−1.21432	−0.110852
$121$	−7.00000	−0.636364
$122$	−1.57136	−0.142264
$123$	2.75557	0.248461
$124$	−1.90321	−0.170913
$125$	1.00000	0.0894427
$126$	0.214320	0.0190931
$127$	−11.4795	−1.01864	−0.509320	−	0.860577i	$-0.670103\pi$
$127$	−11.4795	−1.01864	−0.509320	+	0.860577i	$0.670103\pi$
$128$	−8.78568	−0.776552
$129$	1.05086	0.0925226
$130$	1.16346	0.102043
$131$	13.0716	1.14207	0.571035	−	0.820925i	$-0.306542\pi$
$131$	13.0716	1.14207	0.571035	+	0.820925i	$0.306542\pi$
$132$	−3.80642	−0.331307
$133$	0	0
$134$	−1.69381	−0.146323
$135$	1.00000	0.0860663
$136$	−2.76986	−0.237513
$137$	−11.1383	−0.951607	−0.475804	−	0.879552i	$-0.657843\pi$
$137$	−11.1383	−0.951607	−0.475804	+	0.879552i	$0.657843\pi$
$138$	0.903212	0.0768865
$139$	10.2351	0.868127	0.434063	−	0.900882i	$-0.357079\pi$
$139$	10.2351	0.868127	0.434063	+	0.900882i	$0.357079\pi$
$140$	−1.31111	−0.110809
$141$	4.70964	0.396623
$142$	2.79060	0.234182
$143$	7.47949	0.625467
$144$	3.42864	0.285720
$145$	−4.02074	−0.333905
$146$	−2.77631	−0.229769
$147$	−6.52543	−0.538208
$148$	16.7304	1.37523
$149$	−1.51114	−0.123797	−0.0618986	−	0.998082i	$-0.519716\pi$
$149$	−1.51114	−0.123797	−0.0618986	+	0.998082i	$0.519716\pi$
$150$	0.311108	0.0254018
$151$	−1.76049	−0.143267	−0.0716334	−	0.997431i	$-0.522821\pi$
$151$	−1.76049	−0.143267	−0.0716334	+	0.997431i	$0.522821\pi$
$152$	0	0
$153$	2.28100	0.184408
$154$	0.428639	0.0345408
$155$	1.00000	0.0803219
$156$	−7.11753	−0.569859
$157$	−3.43801	−0.274383	−0.137191	−	0.990545i	$-0.543808\pi$
$157$	−3.43801	−0.274383	−0.137191	+	0.990545i	$0.543808\pi$
$158$	−2.60793	−0.207475
$159$	2.14764	0.170319
$160$	3.49532	0.276329
$161$	2.00000	0.157622
$162$	0.311108	0.0244429
$163$	−10.5970	−0.830023	−0.415012	−	0.909816i	$-0.636222\pi$
$163$	−10.5970	−0.830023	−0.415012	+	0.909816i	$0.636222\pi$
$164$	−5.24443	−0.409521
$165$	2.00000	0.155700
$166$	−3.27163	−0.253928
$167$	−20.0415	−1.55086	−0.775428	−	0.631435i	$-0.782466\pi$
$167$	−20.0415	−1.55086	−0.775428	+	0.631435i	$0.782466\pi$
$168$	−0.836535	−0.0645401
$169$	0.985710	0.0758238
$170$	0.709636	0.0544266
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−18.8988	−1.43685	−0.718423	−	0.695606i	$-0.755136\pi$
$173$	−18.8988	−1.43685	−0.718423	+	0.695606i	$0.755136\pi$
$174$	−1.25088	−0.0948293
$175$	0.688892	0.0520754
$176$	6.85728	0.516887
$177$	−6.44938	−0.484765
$178$	4.20003	0.314806
$179$	16.2953	1.21797	0.608983	−	0.793183i	$-0.291578\pi$
$179$	16.2953	1.21797	0.608983	+	0.793183i	$0.291578\pi$
$180$	−1.90321	−0.141857
$181$	−1.70471	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$181$	−1.70471	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$182$	0.801502	0.0594113
$183$	−5.05086	−0.373370
$184$	−3.52543	−0.259898
$185$	−8.79060	−0.646298
$186$	0.311108	0.0228115
$187$	4.56199	0.333606
$188$	−8.96343	−0.653726
$189$	0.688892	0.0501095
$190$	0	0
$191$	−3.53188	−0.255558	−0.127779	−	0.991803i	$-0.540785\pi$
$191$	−3.53188	−0.255558	−0.127779	+	0.991803i	$0.540785\pi$
$192$	−5.76986	−0.416404
$193$	17.4193	1.25387	0.626933	−	0.779073i	$-0.284310\pi$
$193$	17.4193	1.25387	0.626933	+	0.779073i	$0.284310\pi$
$194$	−3.73329	−0.268035
$195$	3.73975	0.267809
$196$	12.4193	0.887091
$197$	10.0143	0.713489	0.356744	−	0.934202i	$-0.383887\pi$
$197$	10.0143	0.713489	0.356744	+	0.934202i	$0.383887\pi$
$198$	0.622216	0.0442189
$199$	27.1798	1.92672	0.963361	−	0.268208i	$-0.0864313\pi$
$199$	27.1798	1.92672	0.963361	+	0.268208i	$0.0864313\pi$
$200$	−1.21432	−0.0858654
$201$	−5.44446	−0.384023
$202$	2.58073	0.181579
$203$	−2.76986	−0.194406
$204$	−4.34122	−0.303946
$205$	2.75557	0.192457
$206$	0.928401	0.0646848
$207$	2.90321	0.201787
$208$	12.8222	0.889063
$209$	0	0
$210$	0.214320	0.0147895
$211$	−1.93978	−0.133540	−0.0667699	−	0.997768i	$-0.521269\pi$
$211$	−1.93978	−0.133540	−0.0667699	+	0.997768i	$0.521269\pi$
$212$	−4.08742	−0.280725
$213$	8.96989	0.614607
$214$	0.474572	0.0324411
$215$	1.05086	0.0716677
$216$	−1.21432	−0.0826240
$217$	0.688892	0.0467650
$218$	1.16992	0.0792369
$219$	−8.92396	−0.603025
$220$	−3.80642	−0.256629
$221$	8.53035	0.573813
$222$	−2.73483	−0.183549
$223$	23.5526	1.57720	0.788600	−	0.614906i	$-0.210806\pi$
$223$	23.5526	1.57720	0.788600	+	0.614906i	$0.210806\pi$
$224$	2.40790	0.160884
$225$	1.00000	0.0666667
$226$	−2.85728	−0.190063
$227$	19.8020	1.31430	0.657152	−	0.753758i	$-0.271761\pi$
$227$	19.8020	1.31430	0.657152	+	0.753758i	$0.271761\pi$
$228$	0	0
$229$	−24.4099	−1.61305	−0.806526	−	0.591199i	$-0.798655\pi$
$229$	−24.4099	−1.61305	−0.806526	+	0.591199i	$0.798655\pi$
$230$	0.903212	0.0595560
$231$	1.37778	0.0906516
$232$	4.88247	0.320550
$233$	3.13828	0.205595	0.102798	−	0.994702i	$-0.467221\pi$
$233$	3.13828	0.205595	0.102798	+	0.994702i	$0.467221\pi$
$234$	1.16346	0.0760581
$235$	4.70964	0.307223
$236$	12.2745	0.799005
$237$	−8.38271	−0.544515
$238$	0.488863	0.0316883
$239$	6.81579	0.440877	0.220438	−	0.975401i	$-0.429251\pi$
$239$	6.81579	0.440877	0.220438	+	0.975401i	$0.429251\pi$
$240$	3.42864	0.221318
$241$	17.0923	1.10101	0.550507	−	0.834830i	$-0.314434\pi$
$241$	17.0923	1.10101	0.550507	+	0.834830i	$0.314434\pi$
$242$	−2.17775	−0.139991
$243$	1.00000	0.0641500
$244$	9.61285	0.615400
$245$	−6.52543	−0.416894
$246$	0.857279	0.0546581
$247$	0	0
$248$	−1.21432	−0.0771094
$249$	−10.5161	−0.666428
$250$	0.311108	0.0196762
$251$	16.1748	1.02095	0.510473	−	0.859894i	$-0.329470\pi$
$251$	16.1748	1.02095	0.510473	+	0.859894i	$0.329470\pi$
$252$	−1.31111	−0.0825920
$253$	5.80642	0.365047
$254$	−3.57136	−0.224087
$255$	2.28100	0.142842
$256$	8.80642	0.550401
$257$	19.0464	1.18808	0.594041	−	0.804435i	$-0.297532\pi$
$257$	19.0464	1.18808	0.594041	+	0.804435i	$0.297532\pi$
$258$	0.326929	0.0203537
$259$	−6.05578	−0.376288
$260$	−7.11753	−0.441411
$261$	−4.02074	−0.248878
$262$	4.06668	0.251240
$263$	−6.23506	−0.384470	−0.192235	−	0.981349i	$-0.561574\pi$
$263$	−6.23506	−0.384470	−0.192235	+	0.981349i	$0.561574\pi$
$264$	−2.42864	−0.149472
$265$	2.14764	0.131929
$266$	0	0
$267$	13.5002	0.826201
$268$	10.3620	0.632958
$269$	20.8365	1.27043	0.635213	−	0.772337i	$-0.280912\pi$
$269$	20.8365	1.27043	0.635213	+	0.772337i	$0.280912\pi$
$270$	0.311108	0.0189334
$271$	6.62222	0.402271	0.201135	−	0.979563i	$-0.435537\pi$
$271$	6.62222	0.402271	0.201135	+	0.979563i	$0.435537\pi$
$272$	7.82071	0.474200
$273$	2.57628	0.155924
$274$	−3.46520	−0.209341
$275$	2.00000	0.120605
$276$	−5.52543	−0.332592
$277$	4.79060	0.287839	0.143920	−	0.989589i	$-0.454029\pi$
$277$	4.79060	0.287839	0.143920	+	0.989589i	$0.454029\pi$
$278$	3.18421	0.190976
$279$	1.00000	0.0598684
$280$	−0.836535	−0.0499926
$281$	−20.8256	−1.24235	−0.621177	−	0.783671i	$-0.713345\pi$
$281$	−20.8256	−1.24235	−0.621177	+	0.783671i	$0.713345\pi$
$282$	1.46520	0.0872517
$283$	−11.8731	−0.705783	−0.352891	−	0.935664i	$-0.614801\pi$
$283$	−11.8731	−0.705783	−0.352891	+	0.935664i	$0.614801\pi$
$284$	−17.0716	−1.01301
$285$	0	0
$286$	2.32693	0.137594
$287$	1.89829	0.112053
$288$	3.49532	0.205963
$289$	−11.7971	−0.693944
$290$	−1.25088	−0.0734545
$291$	−12.0000	−0.703452
$292$	16.9842	0.993924
$293$	−18.0143	−1.05241	−0.526203	−	0.850359i	$-0.676385\pi$
$293$	−18.0143	−1.05241	−0.526203	+	0.850359i	$0.676385\pi$
$294$	−2.03011	−0.118399
$295$	−6.44938	−0.375498
$296$	10.6746	0.620449
$297$	2.00000	0.116052
$298$	−0.470127	−0.0272337
$299$	10.8573	0.627893
$300$	−1.90321	−0.109882
$301$	0.723926	0.0417264
$302$	−0.547702	−0.0315167
$303$	8.29529	0.476552
$304$	0	0
$305$	−5.05086	−0.289211
$306$	0.709636	0.0405672
$307$	10.9240	0.623463	0.311732	−	0.950170i	$-0.399091\pi$
$307$	10.9240	0.623463	0.311732	+	0.950170i	$0.399091\pi$
$308$	−2.62222	−0.149415
$309$	2.98418	0.169764
$310$	0.311108	0.0176697
$311$	5.10324	0.289378	0.144689	−	0.989477i	$-0.453782\pi$
$311$	5.10324	0.289378	0.144689	+	0.989477i	$0.453782\pi$
$312$	−4.54125	−0.257098
$313$	−22.1082	−1.24963	−0.624814	−	0.780774i	$-0.714825\pi$
$313$	−22.1082	−1.24963	−0.624814	+	0.780774i	$0.714825\pi$
$314$	−1.06959	−0.0603605
$315$	0.688892	0.0388147
$316$	15.9541	0.897486
$317$	−1.06515	−0.0598245	−0.0299123	−	0.999553i	$-0.509523\pi$
$317$	−1.06515	−0.0598245	−0.0299123	+	0.999553i	$0.509523\pi$
$318$	0.668149	0.0374679
$319$	−8.04149	−0.450237
$320$	−5.76986	−0.322545
$321$	1.52543	0.0851411
$322$	0.622216	0.0346747
$323$	0	0
$324$	−1.90321	−0.105734
$325$	3.73975	0.207444
$326$	−3.29682	−0.182594
$327$	3.76049	0.207956
$328$	−3.34614	−0.184760
$329$	3.24443	0.178871
$330$	0.622216	0.0342518
$331$	−24.9447	−1.37108	−0.685542	−	0.728033i	$-0.740435\pi$
$331$	−24.9447	−1.37108	−0.685542	+	0.728033i	$0.740435\pi$
$332$	20.0143	1.09843
$333$	−8.79060	−0.481722
$334$	−6.23506	−0.341167
$335$	−5.44446	−0.297463
$336$	2.36196	0.128856
$337$	0.453829	0.0247216	0.0123608	−	0.999924i	$-0.496065\pi$
$337$	0.453829	0.0247216	0.0123608	+	0.999924i	$0.496065\pi$
$338$	0.306662	0.0166802
$339$	−9.18421	−0.498818
$340$	−4.34122	−0.235436
$341$	2.00000	0.108306
$342$	0	0
$343$	−9.31756	−0.503101
$344$	−1.27607	−0.0688013
$345$	2.90321	0.156304
$346$	−5.87955	−0.316087
$347$	−29.7560	−1.59739	−0.798694	−	0.601737i	$-0.794475\pi$
$347$	−29.7560	−1.59739	−0.798694	+	0.601737i	$0.794475\pi$
$348$	7.65233	0.410208
$349$	−9.65878	−0.517023	−0.258511	−	0.966008i	$-0.583232\pi$
$349$	−9.65878	−0.517023	−0.258511	+	0.966008i	$0.583232\pi$
$350$	0.214320	0.0114559
$351$	3.73975	0.199613
$352$	6.99063	0.372602
$353$	−16.8617	−0.897459	−0.448730	−	0.893668i	$-0.648123\pi$
$353$	−16.8617	−0.897459	−0.448730	+	0.893668i	$0.648123\pi$
$354$	−2.00645	−0.106642
$355$	8.96989	0.476072
$356$	−25.6938	−1.36177
$357$	1.57136	0.0831652
$358$	5.06959	0.267936
$359$	8.64296	0.456158	0.228079	−	0.973643i	$-0.426756\pi$
$359$	8.64296	0.456158	0.228079	+	0.973643i	$0.426756\pi$
$360$	−1.21432	−0.0640003
$361$	−19.0000	−1.00000
$362$	−0.530350	−0.0278746
$363$	−7.00000	−0.367405
$364$	−4.90321	−0.256998
$365$	−8.92396	−0.467101
$366$	−1.57136	−0.0821363
$367$	6.95899	0.363256	0.181628	−	0.983367i	$-0.441863\pi$
$367$	6.95899	0.363256	0.181628	+	0.983367i	$0.441863\pi$
$368$	9.95407	0.518892
$369$	2.75557	0.143449
$370$	−2.73483	−0.142177
$371$	1.47949	0.0768115
$372$	−1.90321	−0.0986769
$373$	21.0005	1.08736	0.543682	−	0.839291i	$-0.317030\pi$
$373$	21.0005	1.08736	0.543682	+	0.839291i	$0.317030\pi$
$374$	1.41927	0.0733888
$375$	1.00000	0.0516398
$376$	−5.71900	−0.294935
$377$	−15.0366	−0.774422
$378$	0.214320	0.0110234
$379$	29.7146	1.52633	0.763167	−	0.646201i	$-0.223643\pi$
$379$	29.7146	1.52633	0.763167	+	0.646201i	$0.223643\pi$
$380$	0	0
$381$	−11.4795	−0.588112
$382$	−1.09880	−0.0562193
$383$	4.05578	0.207241	0.103620	−	0.994617i	$-0.466957\pi$
$383$	4.05578	0.207241	0.103620	+	0.994617i	$0.466957\pi$
$384$	−8.78568	−0.448342
$385$	1.37778	0.0702184
$386$	5.41927	0.275834
$387$	1.05086	0.0534180
$388$	22.8385	1.15945
$389$	−10.9985	−0.557644	−0.278822	−	0.960343i	$-0.589944\pi$
$389$	−10.9985	−0.557644	−0.278822	+	0.960343i	$0.589944\pi$
$390$	1.16346	0.0589143
$391$	6.62222	0.334900
$392$	7.92396	0.400220
$393$	13.0716	0.659375
$394$	3.11552	0.156958
$395$	−8.38271	−0.421780
$396$	−3.80642	−0.191280
$397$	0.193576	0.00971531	0.00485765	−	0.999988i	$-0.498454\pi$
$397$	0.193576	0.00971531	0.00485765	+	0.999988i	$0.498454\pi$
$398$	8.45584	0.423853
$399$	0	0
$400$	3.42864	0.171432
$401$	−38.6242	−1.92880	−0.964401	−	0.264445i	$-0.914811\pi$
$401$	−38.6242	−1.92880	−0.964401	+	0.264445i	$0.914811\pi$
$402$	−1.69381	−0.0844798
$403$	3.73975	0.186290
$404$	−15.7877	−0.785467
$405$	1.00000	0.0496904
$406$	−0.861725	−0.0427667
$407$	−17.5812	−0.871468
$408$	−2.76986	−0.137128
$409$	2.94914	0.145826	0.0729129	−	0.997338i	$-0.476770\pi$
$409$	2.94914	0.145826	0.0729129	+	0.997338i	$0.476770\pi$
$410$	0.857279	0.0423380
$411$	−11.1383	−0.549411
$412$	−5.67952	−0.279810
$413$	−4.44293	−0.218622
$414$	0.903212	0.0443904
$415$	−10.5161	−0.516213
$416$	13.0716	0.640888
$417$	10.2351	0.501213
$418$	0	0
$419$	−26.0020	−1.27028	−0.635141	−	0.772397i	$-0.719058\pi$
$419$	−26.0020	−1.27028	−0.635141	+	0.772397i	$0.719058\pi$
$420$	−1.31111	−0.0639755
$421$	−20.0143	−0.975437	−0.487718	−	0.873001i	$-0.662171\pi$
$421$	−20.0143	−0.975437	−0.487718	+	0.873001i	$0.662171\pi$
$422$	−0.603480	−0.0293769
$423$	4.70964	0.228990
$424$	−2.60793	−0.126652
$425$	2.28100	0.110645
$426$	2.79060	0.135205
$427$	−3.47949	−0.168385
$428$	−2.90321	−0.140332
$429$	7.47949	0.361113
$430$	0.326929	0.0157659
$431$	−2.77631	−0.133730	−0.0668651	−	0.997762i	$-0.521300\pi$
$431$	−2.77631	−0.133730	−0.0668651	+	0.997762i	$0.521300\pi$
$432$	3.42864	0.164960
$433$	−30.7590	−1.47818	−0.739091	−	0.673606i	$-0.764744\pi$
$433$	−30.7590	−1.47818	−0.739091	+	0.673606i	$0.764744\pi$
$434$	0.214320	0.0102877
$435$	−4.02074	−0.192780
$436$	−7.15701	−0.342759
$437$	0	0
$438$	−2.77631	−0.132657
$439$	26.8385	1.28093	0.640467	−	0.767986i	$-0.278741\pi$
$439$	26.8385	1.28093	0.640467	+	0.767986i	$0.278741\pi$
$440$	−2.42864	−0.115781
$441$	−6.52543	−0.310735
$442$	2.65386	0.126231
$443$	−11.7190	−0.556787	−0.278393	−	0.960467i	$-0.589802\pi$
$443$	−11.7190	−0.556787	−0.278393	+	0.960467i	$0.589802\pi$
$444$	16.7304	0.793989
$445$	13.5002	0.639973
$446$	7.32741	0.346963
$447$	−1.51114	−0.0714743
$448$	−3.97481	−0.187792
$449$	33.8084	1.59552	0.797759	−	0.602976i	$-0.206019\pi$
$449$	33.8084	1.59552	0.797759	+	0.602976i	$0.206019\pi$
$450$	0.311108	0.0146658
$451$	5.51114	0.259509
$452$	17.4795	0.822166
$453$	−1.76049	−0.0827151
$454$	6.16055	0.289129
$455$	2.57628	0.120778
$456$	0	0
$457$	5.61930	0.262860	0.131430	−	0.991325i	$-0.458043\pi$
$457$	5.61930	0.262860	0.131430	+	0.991325i	$0.458043\pi$
$458$	−7.59411	−0.354850
$459$	2.28100	0.106468
$460$	−5.52543	−0.257624
$461$	−2.93825	−0.136848	−0.0684239	−	0.997656i	$-0.521797\pi$
$461$	−2.93825	−0.136848	−0.0684239	+	0.997656i	$0.521797\pi$
$462$	0.428639	0.0199421
$463$	−1.61285	−0.0749554	−0.0374777	−	0.999297i	$-0.511932\pi$
$463$	−1.61285	−0.0749554	−0.0374777	+	0.999297i	$0.511932\pi$
$464$	−13.7857	−0.639984
$465$	1.00000	0.0463739
$466$	0.976342	0.0452282
$467$	−11.5857	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$467$	−11.5857	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$468$	−7.11753	−0.329008
$469$	−3.75065	−0.173189
$470$	1.46520	0.0675848
$471$	−3.43801	−0.158415
$472$	7.83161	0.360479
$473$	2.10171	0.0966367
$474$	−2.60793	−0.119786
$475$	0	0
$476$	−2.99063	−0.137075
$477$	2.14764	0.0983338
$478$	2.12045	0.0969869
$479$	26.0020	1.18806	0.594031	−	0.804442i	$-0.297536\pi$
$479$	26.0020	1.18806	0.594031	+	0.804442i	$0.297536\pi$
$480$	3.49532	0.159539
$481$	−32.8746	−1.49895
$482$	5.31756	0.242208
$483$	2.00000	0.0910032
$484$	13.3225	0.605567
$485$	−12.0000	−0.544892
$486$	0.311108	0.0141121
$487$	40.8988	1.85330	0.926650	−	0.375925i	$-0.122675\pi$
$487$	40.8988	1.85330	0.926650	+	0.375925i	$0.122675\pi$
$488$	6.13335	0.277644
$489$	−10.5970	−0.479214
$490$	−2.03011	−0.0917111
$491$	16.6953	0.753450	0.376725	−	0.926325i	$-0.377050\pi$
$491$	16.6953	0.753450	0.376725	+	0.926325i	$0.377050\pi$
$492$	−5.24443	−0.236437
$493$	−9.17130	−0.413055
$494$	0	0
$495$	2.00000	0.0898933
$496$	3.42864	0.153950
$497$	6.17929	0.277179
$498$	−3.27163	−0.146605
$499$	−26.3684	−1.18041	−0.590206	−	0.807253i	$-0.700954\pi$
$499$	−26.3684	−1.18041	−0.590206	+	0.807253i	$0.700954\pi$
$500$	−1.90321	−0.0851142
$501$	−20.0415	−0.895388
$502$	5.03212	0.224594
$503$	−9.47505	−0.422472	−0.211236	−	0.977435i	$-0.567749\pi$
$503$	−9.47505	−0.422472	−0.211236	+	0.977435i	$0.567749\pi$
$504$	−0.836535	−0.0372622
$505$	8.29529	0.369135
$506$	1.80642	0.0803053
$507$	0.985710	0.0437769
$508$	21.8479	0.969344
$509$	21.9003	0.970714	0.485357	−	0.874316i	$-0.338690\pi$
$509$	21.9003	0.970714	0.485357	+	0.874316i	$0.338690\pi$
$510$	0.709636	0.0314232
$511$	−6.14764	−0.271956
$512$	20.3111	0.897633
$513$	0	0
$514$	5.92549	0.261362
$515$	2.98418	0.131499
$516$	−2.00000	−0.0880451
$517$	9.41927	0.414259
$518$	−1.88400	−0.0827782
$519$	−18.8988	−0.829564
$520$	−4.54125	−0.199147
$521$	−36.2034	−1.58610	−0.793050	−	0.609156i	$-0.791508\pi$
$521$	−36.2034	−1.58610	−0.793050	+	0.609156i	$0.791508\pi$
$522$	−1.25088	−0.0547497
$523$	−14.7971	−0.647030	−0.323515	−	0.946223i	$-0.604865\pi$
$523$	−14.7971	−0.647030	−0.323515	+	0.946223i	$0.604865\pi$
$524$	−24.8780	−1.08680
$525$	0.688892	0.0300657
$526$	−1.93978	−0.0845783
$527$	2.28100	0.0993618
$528$	6.85728	0.298425
$529$	−14.5714	−0.633537
$530$	0.668149	0.0290225
$531$	−6.44938	−0.279879
$532$	0	0
$533$	10.3051	0.446365
$534$	4.20003	0.181753
$535$	1.52543	0.0659500
$536$	6.61132	0.285565
$537$	16.2953	0.703194
$538$	6.48241	0.279476
$539$	−13.0509	−0.562140
$540$	−1.90321	−0.0819012
$541$	−41.9956	−1.80553	−0.902765	−	0.430134i	$-0.858466\pi$
$541$	−41.9956	−1.80553	−0.902765	+	0.430134i	$0.858466\pi$
$542$	2.06022	0.0884942
$543$	−1.70471	−0.0731563
$544$	7.97280	0.341831
$545$	3.76049	0.161082
$546$	0.801502	0.0343011
$547$	−34.8004	−1.48796	−0.743980	−	0.668202i	$-0.767064\pi$
$547$	−34.8004	−1.48796	−0.743980	+	0.668202i	$0.767064\pi$
$548$	21.1985	0.905555
$549$	−5.05086	−0.215565
$550$	0.622216	0.0265314
$551$	0	0
$552$	−3.52543	−0.150052
$553$	−5.77478	−0.245569
$554$	1.49039	0.0633208
$555$	−8.79060	−0.373140
$556$	−19.4795	−0.826115
$557$	26.3096	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$557$	26.3096	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$558$	0.311108	0.0131702
$559$	3.92993	0.166218
$560$	2.36196	0.0998112
$561$	4.56199	0.192607
$562$	−6.47902	−0.273301
$563$	−31.8306	−1.34150	−0.670749	−	0.741684i	$-0.734027\pi$
$563$	−31.8306	−1.34150	−0.670749	+	0.741684i	$0.734027\pi$
$564$	−8.96343	−0.377429
$565$	−9.18421	−0.386383
$566$	−3.69381	−0.155263
$567$	0.688892	0.0289308
$568$	−10.8923	−0.457031
$569$	23.6622	0.991970	0.495985	−	0.868331i	$-0.334807\pi$
$569$	23.6622	0.991970	0.495985	+	0.868331i	$0.334807\pi$
$570$	0	0
$571$	18.8889	0.790477	0.395238	−	0.918579i	$-0.370662\pi$
$571$	18.8889	0.790477	0.395238	+	0.918579i	$0.370662\pi$
$572$	−14.2351	−0.595198
$573$	−3.53188	−0.147546
$574$	0.590573	0.0246500
$575$	2.90321	0.121072
$576$	−5.76986	−0.240411
$577$	1.11108	0.0462548	0.0231274	−	0.999733i	$-0.492638\pi$
$577$	1.11108	0.0462548	0.0231274	+	0.999733i	$0.492638\pi$
$578$	−3.67016	−0.152658
$579$	17.4193	0.723920
$580$	7.65233	0.317746
$581$	−7.24443	−0.300550
$582$	−3.73329	−0.154750
$583$	4.29529	0.177893
$584$	10.8365	0.448419
$585$	3.73975	0.154620
$586$	−5.60439	−0.231515
$587$	−17.3778	−0.717258	−0.358629	−	0.933480i	$-0.616756\pi$
$587$	−17.3778	−0.717258	−0.358629	+	0.933480i	$0.616756\pi$
$588$	12.4193	0.512162
$589$	0	0
$590$	−2.00645	−0.0826044
$591$	10.0143	0.411933
$592$	−30.1398	−1.23874
$593$	−37.1753	−1.52661	−0.763304	−	0.646040i	$-0.776424\pi$
$593$	−37.1753	−1.52661	−0.763304	+	0.646040i	$0.776424\pi$
$594$	0.622216	0.0255298
$595$	1.57136	0.0644195
$596$	2.87601	0.117806
$597$	27.1798	1.11239
$598$	3.37778	0.138128
$599$	15.6207	0.638244	0.319122	−	0.947714i	$-0.396612\pi$
$599$	15.6207	0.638244	0.319122	+	0.947714i	$0.396612\pi$
$600$	−1.21432	−0.0495744
$601$	−26.7239	−1.09009	−0.545046	−	0.838406i	$-0.683488\pi$
$601$	−26.7239	−1.09009	−0.545046	+	0.838406i	$0.683488\pi$
$602$	0.225219	0.00917924
$603$	−5.44446	−0.221716
$604$	3.35059	0.136333
$605$	−7.00000	−0.284590
$606$	2.58073	0.104835
$607$	14.3936	0.584218	0.292109	−	0.956385i	$-0.405643\pi$
$607$	14.3936	0.584218	0.292109	+	0.956385i	$0.405643\pi$
$608$	0	0
$609$	−2.76986	−0.112240
$610$	−1.57136	−0.0636225
$611$	17.6128	0.712540
$612$	−4.34122	−0.175483
$613$	6.09526	0.246185	0.123093	−	0.992395i	$-0.460719\pi$
$613$	6.09526	0.246185	0.123093	+	0.992395i	$0.460719\pi$
$614$	3.39853	0.137153
$615$	2.75557	0.111115
$616$	−1.67307	−0.0674099
$617$	1.34614	0.0541936	0.0270968	−	0.999633i	$-0.491374\pi$
$617$	1.34614	0.0541936	0.0270968	+	0.999633i	$0.491374\pi$
$618$	0.928401	0.0373458
$619$	5.71900	0.229866	0.114933	−	0.993373i	$-0.463335\pi$
$619$	5.71900	0.229866	0.114933	+	0.993373i	$0.463335\pi$
$620$	−1.90321	−0.0764348
$621$	2.90321	0.116502
$622$	1.58766	0.0636593
$623$	9.30021	0.372605
$624$	12.8222	0.513301
$625$	1.00000	0.0400000
$626$	−6.87802	−0.274901
$627$	0	0
$628$	6.54326	0.261104
$629$	−20.0513	−0.799499
$630$	0.214320	0.00853870
$631$	37.3274	1.48598	0.742990	−	0.669302i	$-0.233407\pi$
$631$	37.3274	1.48598	0.742990	+	0.669302i	$0.233407\pi$
$632$	10.1793	0.404910
$633$	−1.93978	−0.0770992
$634$	−0.331375	−0.0131606
$635$	−11.4795	−0.455550
$636$	−4.08742	−0.162077
$637$	−24.4035	−0.966900
$638$	−2.50177	−0.0990460
$639$	8.96989	0.354843
$640$	−8.78568	−0.347285
$641$	36.1323	1.42714	0.713570	−	0.700584i	$-0.247077\pi$
$641$	36.1323	1.42714	0.713570	+	0.700584i	$0.247077\pi$
$642$	0.474572	0.0187299
$643$	−3.10123	−0.122301	−0.0611504	−	0.998129i	$-0.519477\pi$
$643$	−3.10123	−0.122301	−0.0611504	+	0.998129i	$0.519477\pi$
$644$	−3.80642	−0.149994
$645$	1.05086	0.0413774
$646$	0	0
$647$	29.6227	1.16459	0.582294	−	0.812978i	$-0.302155\pi$
$647$	29.6227	1.16459	0.582294	+	0.812978i	$0.302155\pi$
$648$	−1.21432	−0.0477030
$649$	−12.8988	−0.506321
$650$	1.16346	0.0456348
$651$	0.688892	0.0269998
$652$	20.1684	0.789855
$653$	−25.2114	−0.986599	−0.493299	−	0.869860i	$-0.664209\pi$
$653$	−25.2114	−0.986599	−0.493299	+	0.869860i	$0.664209\pi$
$654$	1.16992	0.0457474
$655$	13.0716	0.510750
$656$	9.44785	0.368877
$657$	−8.92396	−0.348157
$658$	1.00937	0.0393493
$659$	40.6242	1.58250	0.791248	−	0.611496i	$-0.209432\pi$
$659$	40.6242	1.58250	0.791248	+	0.611496i	$0.209432\pi$
$660$	−3.80642	−0.148165
$661$	−38.7783	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$661$	−38.7783	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$662$	−7.76049	−0.301620
$663$	8.53035	0.331291
$664$	12.7699	0.495567
$665$	0	0
$666$	−2.73483	−0.105972
$667$	−11.6731	−0.451983
$668$	38.1432	1.47580
$669$	23.5526	0.910597
$670$	−1.69381	−0.0654378
$671$	−10.1017	−0.389972
$672$	2.40790	0.0928866
$673$	40.1813	1.54888	0.774438	−	0.632650i	$-0.218033\pi$
$673$	40.1813	1.54888	0.774438	+	0.632650i	$0.218033\pi$
$674$	0.141190	0.00543842
$675$	1.00000	0.0384900
$676$	−1.87601	−0.0721544
$677$	10.6811	0.410506	0.205253	−	0.978709i	$-0.434198\pi$
$677$	10.6811	0.410506	0.205253	+	0.978709i	$0.434198\pi$
$678$	−2.85728	−0.109733
$679$	−8.26671	−0.317247
$680$	−2.76986	−0.106219
$681$	19.8020	0.758813
$682$	0.622216	0.0238259
$683$	31.6271	1.21018	0.605089	−	0.796158i	$-0.293137\pi$
$683$	31.6271	1.21018	0.605089	+	0.796158i	$0.293137\pi$
$684$	0	0
$685$	−11.1383	−0.425572
$686$	−2.89877	−0.110675
$687$	−24.4099	−0.931296
$688$	3.60300	0.137363
$689$	8.03164	0.305981
$690$	0.903212	0.0343847
$691$	10.6508	0.405175	0.202588	−	0.979264i	$-0.435065\pi$
$691$	10.6508	0.405175	0.202588	+	0.979264i	$0.435065\pi$
$692$	35.9684	1.36731
$693$	1.37778	0.0523377
$694$	−9.25734	−0.351404
$695$	10.2351	0.388238
$696$	4.88247	0.185069
$697$	6.28544	0.238078
$698$	−3.00492	−0.113738
$699$	3.13828	0.118700
$700$	−1.31111	−0.0495552
$701$	39.3590	1.48657	0.743285	−	0.668974i	$-0.233266\pi$
$701$	39.3590	1.48657	0.743285	+	0.668974i	$0.233266\pi$
$702$	1.16346	0.0439121
$703$	0	0
$704$	−11.5397	−0.434919
$705$	4.70964	0.177375
$706$	−5.24581	−0.197429
$707$	5.71456	0.214918
$708$	12.2745	0.461306
$709$	20.1847	0.758052	0.379026	−	0.925386i	$-0.376259\pi$
$709$	20.1847	0.758052	0.379026	+	0.925386i	$0.376259\pi$
$710$	2.79060	0.104729
$711$	−8.38271	−0.314376
$712$	−16.3936	−0.614376
$713$	2.90321	0.108726
$714$	0.488863	0.0182952
$715$	7.47949	0.279717
$716$	−31.0134	−1.15902
$717$	6.81579	0.254540
$718$	2.68889	0.100349
$719$	24.1017	0.898842	0.449421	−	0.893320i	$-0.351630\pi$
$719$	24.1017	0.898842	0.449421	+	0.893320i	$0.351630\pi$
$720$	3.42864	0.127778
$721$	2.05578	0.0765611
$722$	−5.91105	−0.219986
$723$	17.0923	0.635671
$724$	3.24443	0.120578
$725$	−4.02074	−0.149327
$726$	−2.17775	−0.0808241
$727$	−14.1180	−0.523608	−0.261804	−	0.965121i	$-0.584317\pi$
$727$	−14.1180	−0.523608	−0.261804	+	0.965121i	$0.584317\pi$
$728$	−3.12843	−0.115947
$729$	1.00000	0.0370370
$730$	−2.77631	−0.102756
$731$	2.39700	0.0886561
$732$	9.61285	0.355301
$733$	−23.1240	−0.854104	−0.427052	−	0.904227i	$-0.640448\pi$
$733$	−23.1240	−0.854104	−0.427052	+	0.904227i	$0.640448\pi$
$734$	2.16500	0.0799115
$735$	−6.52543	−0.240694
$736$	10.1476	0.374047
$737$	−10.8889	−0.401099
$738$	0.857279	0.0315569
$739$	−0.128907	−0.00474193	−0.00237097	−	0.999997i	$-0.500755\pi$
$739$	−0.128907	−0.00474193	−0.00237097	+	0.999997i	$0.500755\pi$
$740$	16.7304	0.615021
$741$	0	0
$742$	0.460282	0.0168975
$743$	28.0129	1.02769	0.513847	−	0.857882i	$-0.328220\pi$
$743$	28.0129	1.02769	0.513847	+	0.857882i	$0.328220\pi$
$744$	−1.21432	−0.0445191
$745$	−1.51114	−0.0553638
$746$	6.53341	0.239205
$747$	−10.5161	−0.384763
$748$	−8.68244	−0.317461
$749$	1.05086	0.0383974
$750$	0.311108	0.0113601
$751$	−3.56247	−0.129996	−0.0649982	−	0.997885i	$-0.520704\pi$
$751$	−3.56247	−0.129996	−0.0649982	+	0.997885i	$0.520704\pi$
$752$	16.1476	0.588844
$753$	16.1748	0.589444
$754$	−4.67799	−0.170362
$755$	−1.76049	−0.0640708
$756$	−1.31111	−0.0476845
$757$	27.0573	0.983415	0.491707	−	0.870761i	$-0.336373\pi$
$757$	27.0573	0.983415	0.491707	+	0.870761i	$0.336373\pi$
$758$	9.24443	0.335773
$759$	5.80642	0.210760
$760$	0	0
$761$	13.1447	0.476496	0.238248	−	0.971204i	$-0.423427\pi$
$761$	13.1447	0.476496	0.238248	+	0.971204i	$0.423427\pi$
$762$	−3.57136	−0.129377
$763$	2.59057	0.0937850
$764$	6.72192	0.243190
$765$	2.28100	0.0824696
$766$	1.26178	0.0455901
$767$	−24.1191	−0.870889
$768$	8.80642	0.317774
$769$	25.7418	0.928271	0.464136	−	0.885764i	$-0.346365\pi$
$769$	25.7418	0.928271	0.464136	+	0.885764i	$0.346365\pi$
$770$	0.428639	0.0154471
$771$	19.0464	0.685940
$772$	−33.1526	−1.19319
$773$	50.3007	1.80919	0.904595	−	0.426272i	$-0.140173\pi$
$773$	50.3007	1.80919	0.904595	+	0.426272i	$0.140173\pi$
$774$	0.326929	0.0117512
$775$	1.00000	0.0359211
$776$	14.5718	0.523098
$777$	−6.05578	−0.217250
$778$	−3.42171	−0.122674
$779$	0	0
$780$	−7.11753	−0.254849
$781$	17.9398	0.641936
$782$	2.06022	0.0736734
$783$	−4.02074	−0.143690
$784$	−22.3733	−0.799048
$785$	−3.43801	−0.122708
$786$	4.06668	0.145054
$787$	8.70519	0.310307	0.155153	−	0.987890i	$-0.450413\pi$
$787$	8.70519	0.310307	0.155153	+	0.987890i	$0.450413\pi$
$788$	−19.0593	−0.678960
$789$	−6.23506	−0.221974
$790$	−2.60793	−0.0927858
$791$	−6.32693	−0.224960
$792$	−2.42864	−0.0862979
$793$	−18.8889	−0.670765
$794$	0.0602231	0.00213724
$795$	2.14764	0.0761691
$796$	−51.7288	−1.83348
$797$	39.0464	1.38309	0.691547	−	0.722331i	$-0.256929\pi$
$797$	39.0464	1.38309	0.691547	+	0.722331i	$0.256929\pi$
$798$	0	0
$799$	10.7427	0.380048
$800$	3.49532	0.123578
$801$	13.5002	0.477007
$802$	−12.0163	−0.424310
$803$	−17.8479	−0.629839
$804$	10.3620	0.365438
$805$	2.00000	0.0704907
$806$	1.16346	0.0409813
$807$	20.8365	0.733481
$808$	−10.0731	−0.354371
$809$	−24.0435	−0.845324	−0.422662	−	0.906287i	$-0.638904\pi$
$809$	−24.0435	−0.845324	−0.422662	+	0.906287i	$0.638904\pi$
$810$	0.311108	0.0109312
$811$	45.5812	1.60057	0.800286	−	0.599618i	$-0.204681\pi$
$811$	45.5812	1.60057	0.800286	+	0.599618i	$0.204681\pi$
$812$	5.27163	0.184998
$813$	6.62222	0.232251
$814$	−5.46965	−0.191711
$815$	−10.5970	−0.371198
$816$	7.82071	0.273780
$817$	0	0
$818$	0.917502	0.0320797
$819$	2.57628	0.0900226
$820$	−5.24443	−0.183143
$821$	−32.2054	−1.12398	−0.561989	−	0.827145i	$-0.689964\pi$
$821$	−32.2054	−1.12398	−0.561989	+	0.827145i	$0.689964\pi$
$822$	−3.46520	−0.120863
$823$	33.1052	1.15398	0.576988	−	0.816752i	$-0.304228\pi$
$823$	33.1052	1.15398	0.576988	+	0.816752i	$0.304228\pi$
$824$	−3.62375	−0.126239
$825$	2.00000	0.0696311
$826$	−1.38223	−0.0480939
$827$	−0.147643	−0.00513406	−0.00256703	−	0.999997i	$-0.500817\pi$
$827$	−0.147643	−0.00513406	−0.00256703	+	0.999997i	$0.500817\pi$
$828$	−5.52543	−0.192022
$829$	−20.9175	−0.726495	−0.363247	−	0.931693i	$-0.618332\pi$
$829$	−20.9175	−0.726495	−0.363247	+	0.931693i	$0.618332\pi$
$830$	−3.27163	−0.113560
$831$	4.79060	0.166184
$832$	−21.5778	−0.748076
$833$	−14.8845	−0.515717
$834$	3.18421	0.110260
$835$	−20.0415	−0.693564
$836$	0	0
$837$	1.00000	0.0345651
$838$	−8.08943	−0.279445
$839$	−42.9066	−1.48130	−0.740650	−	0.671891i	$-0.765482\pi$
$839$	−42.9066	−1.48130	−0.740650	+	0.671891i	$0.765482\pi$
$840$	−0.836535	−0.0288632
$841$	−12.8336	−0.442539
$842$	−6.22660	−0.214583
$843$	−20.8256	−0.717273
$844$	3.69181	0.127077
$845$	0.985710	0.0339095
$846$	1.46520	0.0503748
$847$	−4.82225	−0.165694
$848$	7.36349	0.252863
$849$	−11.8731	−0.407484
$850$	0.709636	0.0243403
$851$	−25.5210	−0.874848
$852$	−17.0716	−0.584863
$853$	−49.2226	−1.68535	−0.842675	−	0.538422i	$-0.819021\pi$
$853$	−49.2226	−1.68535	−0.842675	+	0.538422i	$0.819021\pi$
$854$	−1.08250	−0.0370423
$855$	0	0
$856$	−1.85236	−0.0633123
$857$	34.2449	1.16978	0.584892	−	0.811111i	$-0.301137\pi$
$857$	34.2449	1.16978	0.584892	+	0.811111i	$0.301137\pi$
$858$	2.32693	0.0794401
$859$	14.5718	0.497185	0.248592	−	0.968608i	$-0.420032\pi$
$859$	14.5718	0.497185	0.248592	+	0.968608i	$0.420032\pi$
$860$	−2.00000	−0.0681994
$861$	1.89829	0.0646935
$862$	−0.863732	−0.0294188
$863$	44.6164	1.51876	0.759380	−	0.650648i	$-0.225503\pi$
$863$	44.6164	1.51876	0.759380	+	0.650648i	$0.225503\pi$
$864$	3.49532	0.118913
$865$	−18.8988	−0.642577
$866$	−9.56935	−0.325180
$867$	−11.7971	−0.400649
$868$	−1.31111	−0.0445019
$869$	−16.7654	−0.568728
$870$	−1.25088	−0.0424090
$871$	−20.3609	−0.689903
$872$	−4.56644	−0.154639
$873$	−12.0000	−0.406138
$874$	0	0
$875$	0.688892	0.0232888
$876$	16.9842	0.573842
$877$	8.05038	0.271842	0.135921	−	0.990720i	$-0.456601\pi$
$877$	8.05038	0.271842	0.135921	+	0.990720i	$0.456601\pi$
$878$	8.34968	0.281788
$879$	−18.0143	−0.607607
$880$	6.85728	0.231159
$881$	−28.7130	−0.967366	−0.483683	−	0.875243i	$-0.660701\pi$
$881$	−28.7130	−0.967366	−0.483683	+	0.875243i	$0.660701\pi$
$882$	−2.03011	−0.0683574
$883$	−26.4197	−0.889095	−0.444548	−	0.895755i	$-0.646636\pi$
$883$	−26.4197	−0.889095	−0.444548	+	0.895755i	$0.646636\pi$
$884$	−16.2351	−0.546044
$885$	−6.44938	−0.216794
$886$	−3.64587	−0.122486
$887$	−44.7926	−1.50399	−0.751994	−	0.659170i	$-0.770908\pi$
$887$	−44.7926	−1.50399	−0.751994	+	0.659170i	$0.770908\pi$
$888$	10.6746	0.358216
$889$	−7.90813	−0.265230
$890$	4.20003	0.140785
$891$	2.00000	0.0670025
$892$	−44.8256	−1.50087
$893$	0	0
$894$	−0.470127	−0.0157234
$895$	16.2953	0.544691
$896$	−6.05239	−0.202196
$897$	10.8573	0.362514
$898$	10.5181	0.350992
$899$	−4.02074	−0.134099
$900$	−1.90321	−0.0634404
$901$	4.89877	0.163202
$902$	1.71456	0.0570885
$903$	0.723926	0.0240907
$904$	11.1526	0.370929
$905$	−1.70471	−0.0566666
$906$	−0.547702	−0.0181962
$907$	7.21924	0.239711	0.119855	−	0.992791i	$-0.461757\pi$
$907$	7.21924	0.239711	0.119855	+	0.992791i	$0.461757\pi$
$908$	−37.6874	−1.25070
$909$	8.29529	0.275137
$910$	0.801502	0.0265695
$911$	21.1052	0.699248	0.349624	−	0.936890i	$-0.386309\pi$
$911$	21.1052	0.699248	0.349624	+	0.936890i	$0.386309\pi$
$912$	0	0
$913$	−21.0321	−0.696062
$914$	1.74821	0.0578256
$915$	−5.05086	−0.166976
$916$	46.4572	1.53499
$917$	9.00492	0.297369
$918$	0.709636	0.0234215
$919$	16.4415	0.542357	0.271178	−	0.962529i	$-0.412587\pi$
$919$	16.4415	0.542357	0.271178	+	0.962529i	$0.412587\pi$
$920$	−3.52543	−0.116230
$921$	10.9240	0.359957
$922$	−0.914111	−0.0301046
$923$	33.5451	1.10415
$924$	−2.62222	−0.0862646
$925$	−8.79060	−0.289033
$926$	−0.501770	−0.0164892
$927$	2.98418	0.0980133
$928$	−14.0538	−0.461338
$929$	6.26470	0.205538	0.102769	−	0.994705i	$-0.467230\pi$
$929$	6.26470	0.205538	0.102769	+	0.994705i	$0.467230\pi$
$930$	0.311108	0.0102016
$931$	0	0
$932$	−5.97280	−0.195646
$933$	5.10324	0.167073
$934$	−3.60439	−0.117939
$935$	4.56199	0.149193
$936$	−4.54125	−0.148435
$937$	37.1437	1.21343	0.606715	−	0.794919i	$-0.292487\pi$
$937$	37.1437	1.21343	0.606715	+	0.794919i	$0.292487\pi$
$938$	−1.16686	−0.0380992
$939$	−22.1082	−0.721473
$940$	−8.96343	−0.292355
$941$	39.1318	1.27566	0.637830	−	0.770177i	$-0.279832\pi$
$941$	39.1318	1.27566	0.637830	+	0.770177i	$0.279832\pi$
$942$	−1.06959	−0.0348492
$943$	8.00000	0.260516
$944$	−22.1126	−0.719704
$945$	0.688892	0.0224097
$946$	0.653858	0.0212588
$947$	−53.0968	−1.72541	−0.862707	−	0.505704i	$-0.831233\pi$
$947$	−53.0968	−1.72541	−0.862707	+	0.505704i	$0.831233\pi$
$948$	15.9541	0.518164
$949$	−33.3733	−1.08334
$950$	0	0
$951$	−1.06515	−0.0345397
$952$	−1.90813	−0.0618430
$953$	−6.57628	−0.213027	−0.106513	−	0.994311i	$-0.533969\pi$
$953$	−6.57628	−0.213027	−0.106513	+	0.994311i	$0.533969\pi$
$954$	0.668149	0.0216321
$955$	−3.53188	−0.114289
$956$	−12.9719	−0.419541
$957$	−8.04149	−0.259944
$958$	8.08943	0.261358
$959$	−7.67307	−0.247776
$960$	−5.76986	−0.186221
$961$	1.00000	0.0322581
$962$	−10.2276	−0.329750
$963$	1.52543	0.0491562
$964$	−32.5303	−1.04773
$965$	17.4193	0.560746
$966$	0.622216	0.0200195
$967$	54.2449	1.74440	0.872199	−	0.489151i	$-0.162693\pi$
$967$	54.2449	1.74440	0.872199	+	0.489151i	$0.162693\pi$
$968$	8.50024	0.273208
$969$	0	0
$970$	−3.73329	−0.119869
$971$	−42.8879	−1.37634	−0.688169	−	0.725551i	$-0.741585\pi$
$971$	−42.8879	−1.37634	−0.688169	+	0.725551i	$0.741585\pi$
$972$	−1.90321	−0.0610456
$973$	7.05086	0.226040
$974$	12.7239	0.407701
$975$	3.73975	0.119768
$976$	−17.3176	−0.554322
$977$	11.6316	0.372127	0.186064	−	0.982538i	$-0.440427\pi$
$977$	11.6316	0.372127	0.186064	+	0.982538i	$0.440427\pi$
$978$	−3.29682	−0.105421
$979$	27.0005	0.862939
$980$	12.4193	0.396719
$981$	3.76049	0.120063
$982$	5.19405	0.165749
$983$	−30.3595	−0.968318	−0.484159	−	0.874980i	$-0.660874\pi$
$983$	−30.3595	−0.968318	−0.484159	+	0.874980i	$0.660874\pi$
$984$	−3.34614	−0.106671
$985$	10.0143	0.319082
$986$	−2.85326	−0.0908664
$987$	3.24443	0.103271
$988$	0	0
$989$	3.05086	0.0970115
$990$	0.622216	0.0197753
$991$	42.5589	1.35193	0.675964	−	0.736934i	$-0.263727\pi$
$991$	42.5589	1.35193	0.675964	+	0.736934i	$0.263727\pi$
$992$	3.49532	0.110976
$993$	−24.9447	−0.791596
$994$	1.92242	0.0609756
$995$	27.1798	0.861656
$996$	20.0143	0.634177
$997$	−33.5339	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$997$	−33.5339	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$998$	−8.20342	−0.259675
$999$	−8.79060	−0.278122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.f.1.2	✓	3
3.2	odd	2		1395.2.a.i.1.2		3
4.3	odd	2		7440.2.a.bp.1.2		3
5.2	odd	4		2325.2.c.o.1024.4		6
5.3	odd	4		2325.2.c.o.1024.3		6
5.4	even	2		2325.2.a.q.1.2		3
15.14	odd	2		6975.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.2	✓	3	1.1	even	1	trivial
1395.2.a.i.1.2		3	3.2	odd	2
2325.2.a.q.1.2		3	5.4	even	2
2325.2.c.o.1024.3		6	5.3	odd	4
2325.2.c.o.1024.4		6	5.2	odd	4
6975.2.a.be.1.2		3	15.14	odd	2
7440.2.a.bp.1.2		3	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)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.311108 q^{6} +0.688892 q^{7} -1.21432 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +1.00000 q^{5} +0.311108 q^{6} +0.688892 q^{7} -1.21432 q^{8} +1.00000 q^{9} +0.311108 q^{10} +2.00000 q^{11} -1.90321 q^{12} +3.73975 q^{13} +0.214320 q^{14} +1.00000 q^{15} +3.42864 q^{16} +2.28100 q^{17} +0.311108 q^{18} -1.90321 q^{20} +0.688892 q^{21} +0.622216 q^{22} +2.90321 q^{23} -1.21432 q^{24} +1.00000 q^{25} +1.16346 q^{26} +1.00000 q^{27} -1.31111 q^{28} -4.02074 q^{29} +0.311108 q^{30} +1.00000 q^{31} +3.49532 q^{32} +2.00000 q^{33} +0.709636 q^{34} +0.688892 q^{35} -1.90321 q^{36} -8.79060 q^{37} +3.73975 q^{39} -1.21432 q^{40} +2.75557 q^{41} +0.214320 q^{42} +1.05086 q^{43} -3.80642 q^{44} +1.00000 q^{45} +0.903212 q^{46} +4.70964 q^{47} +3.42864 q^{48} -6.52543 q^{49} +0.311108 q^{50} +2.28100 q^{51} -7.11753 q^{52} +2.14764 q^{53} +0.311108 q^{54} +2.00000 q^{55} -0.836535 q^{56} -1.25088 q^{58} -6.44938 q^{59} -1.90321 q^{60} -5.05086 q^{61} +0.311108 q^{62} +0.688892 q^{63} -5.76986 q^{64} +3.73975 q^{65} +0.622216 q^{66} -5.44446 q^{67} -4.34122 q^{68} +2.90321 q^{69} +0.214320 q^{70} +8.96989 q^{71} -1.21432 q^{72} -8.92396 q^{73} -2.73483 q^{74} +1.00000 q^{75} +1.37778 q^{77} +1.16346 q^{78} -8.38271 q^{79} +3.42864 q^{80} +1.00000 q^{81} +0.857279 q^{82} -10.5161 q^{83} -1.31111 q^{84} +2.28100 q^{85} +0.326929 q^{86} -4.02074 q^{87} -2.42864 q^{88} +13.5002 q^{89} +0.311108 q^{90} +2.57628 q^{91} -5.52543 q^{92} +1.00000 q^{93} +1.46520 q^{94} +3.49532 q^{96} -12.0000 q^{97} -2.03011 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 3 * q^3 + q^4 + 3 * q^5 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 3 q^{3} + q^{4} + 3 q^{5} + q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 6 q^{11} + q^{12} - 2 q^{13} - 6 q^{14} + 3 q^{15} - 3 q^{16} + q^{18} + q^{20} + 2 q^{21} + 2 q^{22} + 2 q^{23} + 3 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} - 4 q^{28} + 8 q^{29} + q^{30} + 3 q^{31} - 3 q^{32} + 6 q^{33} - 18 q^{34} + 2 q^{35} + q^{36} - 2 q^{39} + 3 q^{40} + 8 q^{41} - 6 q^{42} - 10 q^{43} + 2 q^{44} + 3 q^{45} - 4 q^{46} - 6 q^{47} - 3 q^{48} - 13 q^{49} + q^{50} - 8 q^{52} + q^{54} + 6 q^{55} + 4 q^{56} + 10 q^{58} + 14 q^{59} + q^{60} - 2 q^{61} + q^{62} + 2 q^{63} - 11 q^{64} - 2 q^{65} + 2 q^{66} - 16 q^{67} - 20 q^{68} + 2 q^{69} - 6 q^{70} + 20 q^{71} + 3 q^{72} - 28 q^{74} + 3 q^{75} + 4 q^{77} + 10 q^{78} + 8 q^{79} - 3 q^{80} + 3 q^{81} - 24 q^{82} + 2 q^{83} - 4 q^{84} + 14 q^{86} + 8 q^{87} + 6 q^{88} - 6 q^{89} + q^{90} - 12 q^{91} - 10 q^{92} + 3 q^{93} - 16 q^{94} - 3 q^{96} - 36 q^{97} - 13 q^{98} + 6 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + q^4 + 3 * q^5 + q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 + q^10 + 6 * q^11 + q^12 - 2 * q^13 - 6 * q^14 + 3 * q^15 - 3 * q^16 + q^18 + q^20 + 2 * q^21 + 2 * q^22 + 2 * q^23 + 3 * q^24 + 3 * q^25 + 10 * q^26 + 3 * q^27 - 4 * q^28 + 8 * q^29 + q^30 + 3 * q^31 - 3 * q^32 + 6 * q^33 - 18 * q^34 + 2 * q^35 + q^36 - 2 * q^39 + 3 * q^40 + 8 * q^41 - 6 * q^42 - 10 * q^43 + 2 * q^44 + 3 * q^45 - 4 * q^46 - 6 * q^47 - 3 * q^48 - 13 * q^49 + q^50 - 8 * q^52 + q^54 + 6 * q^55 + 4 * q^56 + 10 * q^58 + 14 * q^59 + q^60 - 2 * q^61 + q^62 + 2 * q^63 - 11 * q^64 - 2 * q^65 + 2 * q^66 - 16 * q^67 - 20 * q^68 + 2 * q^69 - 6 * q^70 + 20 * q^71 + 3 * q^72 - 28 * q^74 + 3 * q^75 + 4 * q^77 + 10 * q^78 + 8 * q^79 - 3 * q^80 + 3 * q^81 - 24 * q^82 + 2 * q^83 - 4 * q^84 + 14 * q^86 + 8 * q^87 + 6 * q^88 - 6 * q^89 + q^90 - 12 * q^91 - 10 * q^92 + 3 * q^93 - 16 * q^94 - 3 * q^96 - 36 * q^97 - 13 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more