LMFDB - Embedded newform 431.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [431,2,Mod(1,431)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(431, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("431.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	431.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:	$3.44155232712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + 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			1.4
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35567 q^{2} -0.262360 q^{3} -0.162147 q^{4} -1.73764 q^{5} -0.355674 q^{6} -1.19353 q^{7} -2.93117 q^{8} -2.93117 q^{9} +O(q^{10})$	\(q+1.35567 q^{2} -0.262360 q^{3} -0.162147 q^{4} -1.73764 q^{5} -0.355674 q^{6} -1.19353 q^{7} -2.93117 q^{8} -2.93117 q^{9} -2.35567 q^{10} -4.16724 q^{11} +0.0425409 q^{12} +1.07392 q^{13} -1.61803 q^{14} +0.455887 q^{15} -3.64941 q^{16} +1.25120 q^{17} -3.97371 q^{18} +3.00509 q^{19} +0.281754 q^{20} +0.313133 q^{21} -5.64941 q^{22} +6.62312 q^{23} +0.769020 q^{24} -1.98061 q^{25} +1.45589 q^{26} +1.55610 q^{27} +0.193527 q^{28} +1.87350 q^{29} +0.618034 q^{30} -6.73764 q^{31} +0.914918 q^{32} +1.09331 q^{33} +1.69622 q^{34} +2.07392 q^{35} +0.475281 q^{36} -9.31508 q^{37} +4.07392 q^{38} -0.281754 q^{39} +5.09331 q^{40} +6.74699 q^{41} +0.424507 q^{42} -5.29374 q^{43} +0.675706 q^{44} +5.09331 q^{45} +8.97880 q^{46} -10.1529 q^{47} +0.957459 q^{48} -5.57549 q^{49} -2.68506 q^{50} -0.328264 q^{51} -0.174133 q^{52} -7.00194 q^{53} +2.10956 q^{54} +7.24116 q^{55} +3.49843 q^{56} -0.788414 q^{57} +2.53985 q^{58} +3.60687 q^{59} -0.0739208 q^{60} +12.0458 q^{61} -9.13405 q^{62} +3.49843 q^{63} +8.53916 q^{64} -1.86609 q^{65} +1.48218 q^{66} +7.35253 q^{67} -0.202878 q^{68} -1.73764 q^{69} +2.81156 q^{70} +13.5705 q^{71} +8.59174 q^{72} -14.1935 q^{73} -12.6282 q^{74} +0.519631 q^{75} -0.487267 q^{76} +4.97371 q^{77} -0.381966 q^{78} -11.8491 q^{79} +6.34137 q^{80} +8.38524 q^{81} +9.14672 q^{82} -0.740785 q^{83} -0.0507737 q^{84} -2.17413 q^{85} -7.17659 q^{86} -0.491530 q^{87} +12.2149 q^{88} -10.7225 q^{89} +6.90488 q^{90} -1.28175 q^{91} -1.07392 q^{92} +1.76769 q^{93} -13.7641 q^{94} -5.22176 q^{95} -0.240038 q^{96} +6.16806 q^{97} -7.55855 q^{98} +12.2149 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20} - 3 q^{21} - 11 q^{22} + 4 q^{23} - 6 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 6 q^{28} - 5 q^{29} - 2 q^{30} - 25 q^{31} + 8 q^{32} - 4 q^{33} - 12 q^{34} - q^{35} - 2 q^{36} - q^{37} + 7 q^{38} - 4 q^{39} + 12 q^{40} + 2 q^{41} + 4 q^{42} - 16 q^{43} + 2 q^{44} + 12 q^{45} + 7 q^{46} - 4 q^{47} + 6 q^{48} - 20 q^{49} + 13 q^{50} - 3 q^{51} + 7 q^{52} + 4 q^{53} - 13 q^{54} + 2 q^{55} + 7 q^{56} + 5 q^{57} + 20 q^{58} + 5 q^{59} + 9 q^{60} + 2 q^{62} + 7 q^{63} - 3 q^{64} + 14 q^{65} + 12 q^{66} + 9 q^{67} + 29 q^{68} - 5 q^{69} + 10 q^{71} + 19 q^{72} - 50 q^{73} - 10 q^{74} + 24 q^{75} + 10 q^{76} + 9 q^{77} - 6 q^{78} + 8 q^{79} - 8 q^{81} + 37 q^{82} - 15 q^{83} + 6 q^{84} - q^{85} + 12 q^{86} + 15 q^{87} + 11 q^{88} - 35 q^{89} + 8 q^{90} - 8 q^{91} + 5 q^{92} + 12 q^{93} - 4 q^{94} + 7 q^{95} + 7 q^{96} - 6 q^{97} + 6 q^{98} + 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * 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 $

$2$	1.35567	0.958606	0.479303	−	0.877649i	$-0.340889\pi$
$2$	1.35567	0.958606	0.479303	+	0.877649i	$0.340889\pi$
$3$	−0.262360	−0.151473	−0.0757367	−	0.997128i	$-0.524131\pi$
$3$	−0.262360	−0.151473	−0.0757367	+	0.997128i	$0.524131\pi$
$4$	−0.162147	−0.0810736
$5$	−1.73764	−0.777096	−0.388548	−	0.921428i	$-0.627023\pi$
$5$	−1.73764	−0.777096	−0.388548	+	0.921428i	$0.627023\pi$
$6$	−0.355674	−0.145203
$7$	−1.19353	−0.451111	−0.225555	−	0.974230i	$-0.572420\pi$
$7$	−1.19353	−0.451111	−0.225555	+	0.974230i	$0.572420\pi$
$8$	−2.93117	−1.03632
$9$	−2.93117	−0.977056
$10$	−2.35567	−0.744930
$11$	−4.16724	−1.25647	−0.628234	−	0.778024i	$-0.716222\pi$
$11$	−4.16724	−1.25647	−0.628234	+	0.778024i	$0.716222\pi$
$12$	0.0425409	0.0122805
$13$	1.07392	0.297852	0.148926	−	0.988848i	$-0.452418\pi$
$13$	1.07392	0.297852	0.148926	+	0.988848i	$0.452418\pi$
$14$	−1.61803	−0.432438
$15$	0.455887	0.117709
$16$	−3.64941	−0.912353
$17$	1.25120	0.303460	0.151730	−	0.988422i	$-0.451516\pi$
$17$	1.25120	0.303460	0.151730	+	0.988422i	$0.451516\pi$
$18$	−3.97371	−0.936612
$19$	3.00509	0.689415	0.344707	−	0.938710i	$-0.387978\pi$
$19$	3.00509	0.689415	0.344707	+	0.938710i	$0.387978\pi$
$20$	0.281754	0.0630020
$21$	0.313133	0.0683313
$22$	−5.64941	−1.20446
$23$	6.62312	1.38102	0.690508	−	0.723325i	$-0.257387\pi$
$23$	6.62312	1.38102	0.690508	+	0.723325i	$0.257387\pi$
$24$	0.769020	0.156976
$25$	−1.98061	−0.396121
$26$	1.45589	0.285523
$27$	1.55610	0.299471
$28$	0.193527	0.0365732
$29$	1.87350	0.347899	0.173950	−	0.984755i	$-0.444347\pi$
$29$	1.87350	0.347899	0.173950	+	0.984755i	$0.444347\pi$
$30$	0.618034	0.112837
$31$	−6.73764	−1.21012	−0.605058	−	0.796181i	$-0.706850\pi$
$31$	−6.73764	−1.21012	−0.605058	+	0.796181i	$0.706850\pi$
$32$	0.914918	0.161736
$33$	1.09331	0.190322
$34$	1.69622	0.290899
$35$	2.07392	0.350557
$36$	0.475281	0.0792134
$37$	−9.31508	−1.53139	−0.765695	−	0.643204i	$-0.777605\pi$
$37$	−9.31508	−1.53139	−0.765695	+	0.643204i	$0.777605\pi$
$38$	4.07392	0.660877
$39$	−0.281754	−0.0451167
$40$	5.09331	0.805324
$41$	6.74699	1.05370	0.526852	−	0.849957i	$-0.323372\pi$
$41$	6.74699	1.05370	0.526852	+	0.849957i	$0.323372\pi$
$42$	0.424507	0.0655028
$43$	−5.29374	−0.807288	−0.403644	−	0.914916i	$-0.632257\pi$
$43$	−5.29374	−0.807288	−0.403644	+	0.914916i	$0.632257\pi$
$44$	0.675706	0.101866
$45$	5.09331	0.759267
$46$	8.97880	1.32385
$47$	−10.1529	−1.48096	−0.740479	−	0.672080i	$-0.765401\pi$
$47$	−10.1529	−1.48096	−0.740479	+	0.672080i	$0.765401\pi$
$48$	0.957459	0.138197
$49$	−5.57549	−0.796499
$50$	−2.68506	−0.379724
$51$	−0.328264	−0.0459662
$52$	−0.174133	−0.0241479
$53$	−7.00194	−0.961791	−0.480895	−	0.876778i	$-0.659688\pi$
$53$	−7.00194	−0.961791	−0.480895	+	0.876778i	$0.659688\pi$
$54$	2.10956	0.287075
$55$	7.24116	0.976397
$56$	3.49843	0.467497
$57$	−0.788414	−0.104428
$58$	2.53985	0.333499
$59$	3.60687	0.469575	0.234787	−	0.972047i	$-0.424561\pi$
$59$	3.60687	0.469575	0.234787	+	0.972047i	$0.424561\pi$
$60$	−0.0739208	−0.00954313
$61$	12.0458	1.54231	0.771155	−	0.636648i	$-0.219680\pi$
$61$	12.0458	1.54231	0.771155	+	0.636648i	$0.219680\pi$
$62$	−9.13405	−1.16002
$63$	3.49843	0.440760
$64$	8.53916	1.06739
$65$	−1.86609	−0.231460
$66$	1.48218	0.182444
$67$	7.35253	0.898254	0.449127	−	0.893468i	$-0.351735\pi$
$67$	7.35253	0.898254	0.449127	+	0.893468i	$0.351735\pi$
$68$	−0.202878	−0.0246026
$69$	−1.73764	−0.209187
$70$	2.81156	0.336046
$71$	13.5705	1.61053	0.805263	−	0.592917i	$-0.202024\pi$
$71$	13.5705	1.61053	0.805263	+	0.592917i	$0.202024\pi$
$72$	8.59174	1.01255
$73$	−14.1935	−1.66123	−0.830613	−	0.556850i	$-0.812010\pi$
$73$	−14.1935	−1.66123	−0.830613	+	0.556850i	$0.812010\pi$
$74$	−12.6282	−1.46800
$75$	0.519631	0.0600018
$76$	−0.487267	−0.0558933
$77$	4.97371	0.566807
$78$	−0.381966	−0.0432491
$79$	−11.8491	−1.33313	−0.666567	−	0.745445i	$-0.732237\pi$
$79$	−11.8491	−1.33313	−0.666567	+	0.745445i	$0.732237\pi$
$80$	6.34137	0.708987
$81$	8.38524	0.931694
$82$	9.14672	1.01009
$83$	−0.740785	−0.0813117	−0.0406559	−	0.999173i	$-0.512945\pi$
$83$	−0.740785	−0.0813117	−0.0406559	+	0.999173i	$0.512945\pi$
$84$	−0.0507737	−0.00553987
$85$	−2.17413	−0.235818
$86$	−7.17659	−0.773871
$87$	−0.491530	−0.0526975
$88$	12.2149	1.30211
$89$	−10.7225	−1.13658	−0.568292	−	0.822827i	$-0.692396\pi$
$89$	−10.7225	−1.13658	−0.568292	+	0.822827i	$0.692396\pi$
$90$	6.90488	0.727838
$91$	−1.28175	−0.134364
$92$	−1.07392	−0.111964
$93$	1.76769	0.183300
$94$	−13.7641	−1.41965
$95$	−5.22176	−0.535742
$96$	−0.240038	−0.0244987
$97$	6.16806	0.626272	0.313136	−	0.949708i	$-0.398620\pi$
$97$	6.16806	0.626272	0.313136	+	0.949708i	$0.398620\pi$
$98$	−7.55855	−0.763529
$99$	12.2149	1.22764
$100$	0.321150	0.0321150
$101$	−7.07211	−0.703701	−0.351851	−	0.936056i	$-0.614448\pi$
$101$	−7.07211	−0.703701	−0.351851	+	0.936056i	$0.614448\pi$
$102$	−0.445019	−0.0440635
$103$	5.59571	0.551362	0.275681	−	0.961249i	$-0.411097\pi$
$103$	5.59571	0.551362	0.275681	+	0.961249i	$0.411097\pi$
$104$	−3.14784	−0.308671
$105$	−0.544113	−0.0531000
$106$	−9.49235	−0.921979
$107$	6.61489	0.639485	0.319743	−	0.947504i	$-0.396404\pi$
$107$	6.61489	0.639485	0.319743	+	0.947504i	$0.396404\pi$
$108$	−0.252317	−0.0242792
$109$	−15.9868	−1.53126	−0.765629	−	0.643282i	$-0.777572\pi$
$109$	−15.9868	−1.53126	−0.765629	+	0.643282i	$0.777572\pi$
$110$	9.81665	0.935981
$111$	2.44390	0.231965
$112$	4.35567	0.411573
$113$	−1.27374	−0.119823	−0.0599116	−	0.998204i	$-0.519082\pi$
$113$	−1.27374	−0.119823	−0.0599116	+	0.998204i	$0.519082\pi$
$114$	−1.06883	−0.100105
$115$	−11.5086	−1.07318
$116$	−0.303782	−0.0282055
$117$	−3.14784	−0.291018
$118$	4.88974	0.450138
$119$	−1.49334	−0.136894
$120$	−1.33628	−0.121985
$121$	6.36585	0.578714
$122$	16.3302	1.47847
$123$	−1.77014	−0.159608
$124$	1.09249	0.0981085
$125$	12.1298	1.08492
$126$	4.74273	0.422516
$127$	−15.5200	−1.37718	−0.688588	−	0.725153i	$-0.741769\pi$
$127$	−15.5200	−1.37718	−0.688588	+	0.725153i	$0.741769\pi$
$128$	9.74648	0.861475
$129$	1.38886	0.122283
$130$	−2.52981	−0.221879
$131$	−0.938890	−0.0820312	−0.0410156	−	0.999159i	$-0.513059\pi$
$131$	−0.938890	−0.0820312	−0.0410156	+	0.999159i	$0.513059\pi$
$132$	−0.177278	−0.0154301
$133$	−3.58665	−0.311002
$134$	9.96764	0.861073
$135$	−2.70394	−0.232718
$136$	−3.66747	−0.314483
$137$	−5.58141	−0.476852	−0.238426	−	0.971161i	$-0.576631\pi$
$137$	−5.58141	−0.476852	−0.238426	+	0.971161i	$0.576631\pi$
$138$	−2.35567	−0.200528
$139$	−10.9757	−0.930943	−0.465471	−	0.885063i	$-0.654115\pi$
$139$	−10.9757	−0.930943	−0.465471	+	0.885063i	$0.654115\pi$
$140$	−0.336280	−0.0284209
$141$	2.66372	0.224326
$142$	18.3972	1.54386
$143$	−4.47528	−0.374242
$144$	10.6970	0.891420
$145$	−3.25546	−0.270351
$146$	−19.2418	−1.59246
$147$	1.46278	0.120648
$148$	1.51041	0.124155
$149$	−4.51092	−0.369549	−0.184775	−	0.982781i	$-0.559156\pi$
$149$	−4.51092	−0.369549	−0.184775	+	0.982781i	$0.559156\pi$
$150$	0.704451	0.0575182
$151$	16.5293	1.34514	0.672569	−	0.740034i	$-0.265191\pi$
$151$	16.5293	1.34514	0.672569	+	0.740034i	$0.265191\pi$
$152$	−8.80842	−0.714457
$153$	−3.66747	−0.296498
$154$	6.74273	0.543345
$155$	11.7076	0.940377
$156$	0.0456855	0.00365777
$157$	15.3721	1.22682	0.613412	−	0.789763i	$-0.289797\pi$
$157$	15.3721	1.22682	0.613412	+	0.789763i	$0.289797\pi$
$158$	−16.0636	−1.27795
$159$	1.83703	0.145686
$160$	−1.58980	−0.125685
$161$	−7.90488	−0.622991
$162$	11.3677	0.893128
$163$	−5.18167	−0.405860	−0.202930	−	0.979193i	$-0.565046\pi$
$163$	−5.18167	−0.405860	−0.202930	+	0.979193i	$0.565046\pi$
$164$	−1.09401	−0.0854275
$165$	−1.89979	−0.147898
$166$	−1.00426	−0.0779459
$167$	−21.7077	−1.67979	−0.839897	−	0.542745i	$-0.817385\pi$
$167$	−21.7077	−1.67979	−0.839897	+	0.542745i	$0.817385\pi$
$168$	−0.917846	−0.0708134
$169$	−11.8467	−0.911284
$170$	−2.94742	−0.226057
$171$	−8.80842	−0.673596
$172$	0.858365	0.0654497
$173$	4.63053	0.352053	0.176026	−	0.984385i	$-0.443676\pi$
$173$	4.63053	0.352053	0.176026	+	0.984385i	$0.443676\pi$
$174$	−0.666354	−0.0505162
$175$	2.36391	0.178695
$176$	15.2080	1.14634
$177$	−0.946298	−0.0711281
$178$	−14.5362	−1.08954
$179$	16.1894	1.21005	0.605026	−	0.796206i	$-0.293163\pi$
$179$	16.1894	1.21005	0.605026	+	0.796206i	$0.293163\pi$
$180$	−0.825867	−0.0615565
$181$	−10.1978	−0.757996	−0.378998	−	0.925398i	$-0.623731\pi$
$181$	−10.1978	−0.757996	−0.378998	+	0.925398i	$0.623731\pi$
$182$	−1.73764	−0.128802
$183$	−3.16034	−0.233619
$184$	−19.4135	−1.43118
$185$	16.1863	1.19004
$186$	2.39641	0.175713
$187$	−5.21404	−0.381288
$188$	1.64627	0.120067
$189$	−1.85725	−0.135095
$190$	−7.07901	−0.513565
$191$	17.8994	1.29516	0.647578	−	0.761999i	$-0.275782\pi$
$191$	17.8994	1.29516	0.647578	+	0.761999i	$0.275782\pi$
$192$	−2.24033	−0.161682
$193$	6.09443	0.438687	0.219343	−	0.975648i	$-0.429608\pi$
$193$	6.09443	0.438687	0.219343	+	0.975648i	$0.429608\pi$
$194$	8.36188	0.600348
$195$	0.489586	0.0350600
$196$	0.904051	0.0645750
$197$	7.63053	0.543653	0.271826	−	0.962346i	$-0.412372\pi$
$197$	7.63053	0.543653	0.271826	+	0.962346i	$0.412372\pi$
$198$	16.5594	1.17682
$199$	−10.9777	−0.778187	−0.389093	−	0.921198i	$-0.627212\pi$
$199$	−10.9777	−0.778187	−0.389093	+	0.921198i	$0.627212\pi$
$200$	5.80549	0.410510
$201$	−1.92901	−0.136062
$202$	−9.58748	−0.674573
$203$	−2.23607	−0.156941
$204$	0.0532271	0.00372664
$205$	−11.7238	−0.818829
$206$	7.58596	0.528539
$207$	−19.4135	−1.34933
$208$	−3.91918	−0.271746
$209$	−12.5229	−0.866228
$210$	−0.737640	−0.0509020
$211$	16.4090	1.12964	0.564821	−	0.825213i	$-0.308945\pi$
$211$	16.4090	1.12964	0.564821	+	0.825213i	$0.308945\pi$
$212$	1.13535	0.0779759
$213$	−3.56036	−0.243952
$214$	8.96764	0.613015
$215$	9.19862	0.627340
$216$	−4.56119	−0.310349
$217$	8.04156	0.545896
$218$	−21.6729	−1.46787
$219$	3.72381	0.251632
$220$	−1.17413	−0.0791600
$221$	1.34369	0.0903863
$222$	3.31313	0.222363
$223$	23.3580	1.56417	0.782083	−	0.623175i	$-0.214157\pi$
$223$	23.3580	1.56417	0.782083	+	0.623175i	$0.214157\pi$
$224$	−1.09198	−0.0729610
$225$	5.80549	0.387033
$226$	−1.72677	−0.114863
$227$	−2.47722	−0.164419	−0.0822096	−	0.996615i	$-0.526198\pi$
$227$	−2.47722	−0.164419	−0.0822096	+	0.996615i	$0.526198\pi$
$228$	0.127839	0.00846635
$229$	−22.9422	−1.51606	−0.758031	−	0.652219i	$-0.773838\pi$
$229$	−22.9422	−1.51606	−0.758031	+	0.652219i	$0.773838\pi$
$230$	−15.6019	−1.02876
$231$	−1.30490	−0.0858562
$232$	−5.49153	−0.360537
$233$	−8.48037	−0.555567	−0.277784	−	0.960644i	$-0.589600\pi$
$233$	−8.48037	−0.555567	−0.277784	+	0.960644i	$0.589600\pi$
$234$	−4.26745	−0.278972
$235$	17.6421	1.15085
$236$	−0.584844	−0.0380701
$237$	3.10874	0.201934
$238$	−2.02448	−0.131228
$239$	2.33650	0.151135	0.0755677	−	0.997141i	$-0.475923\pi$
$239$	2.33650	0.151135	0.0755677	+	0.997141i	$0.475923\pi$
$240$	−1.66372	−0.107393
$241$	−27.8433	−1.79354	−0.896772	−	0.442493i	$-0.854094\pi$
$241$	−27.8433	−1.79354	−0.896772	+	0.442493i	$0.854094\pi$
$242$	8.63002	0.554759
$243$	−6.86825	−0.440598
$244$	−1.95320	−0.125041
$245$	9.68820	0.618956
$246$	−2.39973	−0.153001
$247$	3.22723	0.205344
$248$	19.7492	1.25407
$249$	0.194352	0.0123166
$250$	16.4440	1.04001
$251$	−2.47903	−0.156475	−0.0782376	−	0.996935i	$-0.524929\pi$
$251$	−2.47903	−0.156475	−0.0782376	+	0.996935i	$0.524929\pi$
$252$	−0.567260	−0.0357340
$253$	−27.6001	−1.73520
$254$	−21.0400	−1.32017
$255$	0.570405	0.0357201
$256$	−3.86526	−0.241579
$257$	4.22033	0.263257	0.131628	−	0.991299i	$-0.457979\pi$
$257$	4.22033	0.263257	0.131628	+	0.991299i	$0.457979\pi$
$258$	1.88285	0.117221
$259$	11.1178	0.690826
$260$	0.302581	0.0187653
$261$	−5.49153	−0.339917
$262$	−1.27283	−0.0786357
$263$	26.3555	1.62515	0.812575	−	0.582856i	$-0.198065\pi$
$263$	26.3555	1.62515	0.812575	+	0.582856i	$0.198065\pi$
$264$	−3.20469	−0.197235
$265$	12.1669	0.747404
$266$	−4.86233	−0.298129
$267$	2.81315	0.172162
$268$	−1.19219	−0.0728247
$269$	−17.2119	−1.04943	−0.524713	−	0.851279i	$-0.675827\pi$
$269$	−17.2119	−1.04943	−0.524713	+	0.851279i	$0.675827\pi$
$270$	−3.66566	−0.223085
$271$	9.45701	0.574472	0.287236	−	0.957860i	$-0.407264\pi$
$271$	9.45701	0.574472	0.287236	+	0.957860i	$0.407264\pi$
$272$	−4.56614	−0.276863
$273$	0.336280	0.0203526
$274$	−7.56657	−0.457113
$275$	8.25365	0.497714
$276$	0.281754	0.0169596
$277$	10.5426	0.633445	0.316722	−	0.948518i	$-0.397418\pi$
$277$	10.5426	0.633445	0.316722	+	0.948518i	$0.397418\pi$
$278$	−14.8794	−0.892408
$279$	19.7492	1.18235
$280$	−6.07901	−0.363290
$281$	−13.1965	−0.787235	−0.393617	−	0.919274i	$-0.628776\pi$
$281$	−13.1965	−0.787235	−0.393617	+	0.919274i	$0.628776\pi$
$282$	3.61114	0.215040
$283$	3.57714	0.212639	0.106320	−	0.994332i	$-0.466093\pi$
$283$	3.57714	0.212639	0.106320	+	0.994332i	$0.466093\pi$
$284$	−2.20042	−0.130571
$285$	1.36998	0.0811506
$286$	−6.06702	−0.358751
$287$	−8.05272	−0.475337
$288$	−2.68178	−0.158025
$289$	−15.4345	−0.907912
$290$	−4.41335	−0.259161
$291$	−1.61825	−0.0948635
$292$	2.30144	0.134682
$293$	−2.22430	−0.129945	−0.0649724	−	0.997887i	$-0.520696\pi$
$293$	−2.22430	−0.129945	−0.0649724	+	0.997887i	$0.520696\pi$
$294$	1.98306	0.115654
$295$	−6.26745	−0.364905
$296$	27.3040	1.58702
$297$	−6.48463	−0.376277
$298$	−6.11534	−0.354252
$299$	7.11271	0.411339
$300$	−0.0842568	−0.00486457
$301$	6.31822	0.364176
$302$	22.4084	1.28946
$303$	1.85544	0.106592
$304$	−10.9668	−0.628990
$305$	−20.9313	−1.19852
$306$	−4.97190	−0.284225
$307$	−9.59308	−0.547506	−0.273753	−	0.961800i	$-0.588265\pi$
$307$	−9.59308	−0.547506	−0.273753	+	0.961800i	$0.588265\pi$
$308$	−0.806473	−0.0459531
$309$	−1.46809	−0.0835167
$310$	15.8717	0.901451
$311$	−27.6506	−1.56792	−0.783961	−	0.620811i	$-0.786804\pi$
$311$	−27.6506	−1.56792	−0.783961	+	0.620811i	$0.786804\pi$
$312$	0.825867	0.0467555
$313$	3.01250	0.170276	0.0851382	−	0.996369i	$-0.472867\pi$
$313$	3.01250	0.170276	0.0851382	+	0.996369i	$0.472867\pi$
$314$	20.8395	1.17604
$315$	−6.07901	−0.342513
$316$	1.92131	0.108082
$317$	15.4751	0.869166	0.434583	−	0.900632i	$-0.356896\pi$
$317$	15.4751	0.869166	0.434583	+	0.900632i	$0.356896\pi$
$318$	2.49041	0.139655
$319$	−7.80730	−0.437125
$320$	−14.8380	−0.829469
$321$	−1.73548	−0.0968651
$322$	−10.7164	−0.597204
$323$	3.75996	0.209210
$324$	−1.35964	−0.0755358
$325$	−2.12701	−0.117986
$326$	−7.02466	−0.389060
$327$	4.19430	0.231945
$328$	−19.7766	−1.09198
$329$	12.1178	0.668076
$330$	−2.57549	−0.141776
$331$	7.61743	0.418692	0.209346	−	0.977842i	$-0.432867\pi$
$331$	7.61743	0.418692	0.209346	+	0.977842i	$0.432867\pi$
$332$	0.120116	0.00659223
$333$	27.3040	1.49625
$334$	−29.4286	−1.61026
$335$	−12.7761	−0.698030
$336$	−1.14275	−0.0623423
$337$	−23.7791	−1.29533	−0.647665	−	0.761925i	$-0.724255\pi$
$337$	−23.7791	−1.29533	−0.647665	+	0.761925i	$0.724255\pi$
$338$	−16.0603	−0.873563
$339$	0.334177	0.0181500
$340$	0.352530	0.0191186
$341$	28.0773	1.52047
$342$	−11.9413	−0.645714
$343$	15.0092	0.810420
$344$	15.5168	0.836612
$345$	3.01939	0.162559
$346$	6.27749	0.337480
$347$	10.5441	0.566038	0.283019	−	0.959114i	$-0.408664\pi$
$347$	10.5441	0.566038	0.283019	+	0.959114i	$0.408664\pi$
$348$	0.0797002	0.00427238
$349$	−9.31628	−0.498689	−0.249344	−	0.968415i	$-0.580215\pi$
$349$	−9.31628	−0.498689	−0.249344	+	0.968415i	$0.580215\pi$
$350$	3.20469	0.171298
$351$	1.67113	0.0891982
$352$	−3.81268	−0.203217
$353$	−23.3598	−1.24332	−0.621658	−	0.783289i	$-0.713541\pi$
$353$	−23.3598	−1.24332	−0.621658	+	0.783289i	$0.713541\pi$
$354$	−1.28287	−0.0681839
$355$	−23.5807	−1.25153
$356$	1.73863	0.0921469
$357$	0.391792	0.0207358
$358$	21.9476	1.15996
$359$	7.45805	0.393621	0.196810	−	0.980442i	$-0.436942\pi$
$359$	7.45805	0.393621	0.196810	+	0.980442i	$0.436942\pi$
$360$	−14.9294	−0.786846
$361$	−9.96945	−0.524708
$362$	−13.8249	−0.726620
$363$	−1.67014	−0.0876598
$364$	0.207833	0.0108934
$365$	24.6632	1.29093
$366$	−4.28439	−0.223949
$367$	21.4831	1.12141	0.560705	−	0.828015i	$-0.310530\pi$
$367$	21.4831	1.12141	0.560705	+	0.828015i	$0.310530\pi$
$368$	−24.1705	−1.25998
$369$	−19.7766	−1.02953
$370$	21.9433	1.14078
$371$	8.35701	0.433874
$372$	−0.286625	−0.0148608
$373$	8.95021	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$373$	8.95021	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$374$	−7.06854	−0.365505
$375$	−3.18237	−0.164337
$376$	29.7599	1.53475
$377$	2.01199	0.103623
$378$	−2.51782	−0.129503
$379$	1.74350	0.0895574	0.0447787	−	0.998997i	$-0.485742\pi$
$379$	1.74350	0.0895574	0.0447787	+	0.998997i	$0.485742\pi$
$380$	0.846694	0.0434345
$381$	4.07182	0.208606
$382$	24.2658	1.24154
$383$	18.2505	0.932557	0.466279	−	0.884638i	$-0.345594\pi$
$383$	18.2505	0.932557	0.466279	+	0.884638i	$0.345594\pi$
$384$	−2.55708	−0.130491
$385$	−8.64252	−0.440463
$386$	8.26207	0.420528
$387$	15.5168	0.788765
$388$	−1.00013	−0.0507741
$389$	33.3456	1.69069	0.845345	−	0.534221i	$-0.179395\pi$
$389$	33.3456	1.69069	0.845345	+	0.534221i	$0.179395\pi$
$390$	0.663720	0.0336087
$391$	8.28684	0.419084
$392$	16.3427	0.825431
$393$	0.246327	0.0124256
$394$	10.3445	0.521149
$395$	20.5896	1.03597
$396$	−1.98061	−0.0995292
$397$	−25.1539	−1.26244	−0.631219	−	0.775605i	$-0.717445\pi$
$397$	−25.1539	−1.26244	−0.631219	+	0.775605i	$0.717445\pi$
$398$	−14.8822	−0.745975
$399$	0.940993	0.0471086
$400$	7.22805	0.361403
$401$	5.14059	0.256709	0.128354	−	0.991728i	$-0.459030\pi$
$401$	5.14059	0.256709	0.128354	+	0.991728i	$0.459030\pi$
$402$	−2.61511	−0.130430
$403$	−7.23569	−0.360435
$404$	1.14672	0.0570516
$405$	−14.5705	−0.724016
$406$	−3.03138	−0.150445
$407$	38.8181	1.92414
$408$	0.962197	0.0476359
$409$	−24.8149	−1.22702	−0.613508	−	0.789688i	$-0.710242\pi$
$409$	−24.8149	−1.22702	−0.613508	+	0.789688i	$0.710242\pi$
$410$	−15.8937	−0.784935
$411$	1.46434	0.0722304
$412$	−0.907329	−0.0447009
$413$	−4.30490	−0.211830
$414$	−26.3184	−1.29348
$415$	1.28722	0.0631870
$416$	0.982550	0.0481735
$417$	2.87957	0.141013
$418$	−16.9770	−0.830372
$419$	−8.58783	−0.419543	−0.209771	−	0.977750i	$-0.567272\pi$
$419$	−8.58783	−0.419543	−0.209771	+	0.977750i	$0.567272\pi$
$420$	0.0882264	0.00430501
$421$	33.6854	1.64173	0.820863	−	0.571125i	$-0.193493\pi$
$421$	33.6854	1.64173	0.820863	+	0.571125i	$0.193493\pi$
$422$	22.2453	1.08288
$423$	29.7599	1.44698
$424$	20.5239	0.996727
$425$	−2.47813	−0.120207
$426$	−4.82669	−0.233854
$427$	−14.3770	−0.695752
$428$	−1.07259	−0.0518454
$429$	1.17413	0.0566877
$430$	12.4703	0.601373
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	−5.67885	−0.273224
$433$	3.27981	0.157618	0.0788088	−	0.996890i	$-0.474888\pi$
$433$	3.27981	0.157618	0.0788088	+	0.996890i	$0.474888\pi$
$434$	10.9017	0.523300
$435$	0.854102	0.0409511
$436$	2.59222	0.124145
$437$	19.9031	0.952093
$438$	5.04827	0.241216
$439$	24.8952	1.18818	0.594091	−	0.804398i	$-0.297512\pi$
$439$	24.8952	1.18818	0.594091	+	0.804398i	$0.297512\pi$
$440$	−21.2250	−1.01186
$441$	16.3427	0.778224
$442$	1.82160	0.0866448
$443$	10.3348	0.491019	0.245510	−	0.969394i	$-0.421045\pi$
$443$	10.3348	0.491019	0.245510	+	0.969394i	$0.421045\pi$
$444$	−0.396272	−0.0188062
$445$	18.6319	0.883235
$446$	31.6658	1.49942
$447$	1.18348	0.0559769
$448$	−10.1917	−0.481513
$449$	25.5135	1.20406	0.602028	−	0.798475i	$-0.294360\pi$
$449$	25.5135	1.20406	0.602028	+	0.798475i	$0.294360\pi$
$450$	7.87035	0.371012
$451$	−28.1163	−1.32394
$452$	0.206533	0.00971449
$453$	−4.33663	−0.203753
$454$	−3.35831	−0.157613
$455$	2.22723	0.104414
$456$	2.31097	0.108221
$457$	−14.0777	−0.658527	−0.329264	−	0.944238i	$-0.606801\pi$
$457$	−14.0777	−0.658527	−0.329264	+	0.944238i	$0.606801\pi$
$458$	−31.1021	−1.45331
$459$	1.94699	0.0908777
$460$	1.86609	0.0870068
$461$	35.1296	1.63615	0.818075	−	0.575112i	$-0.195041\pi$
$461$	35.1296	1.63615	0.818075	+	0.575112i	$0.195041\pi$
$462$	−1.76902	−0.0823023
$463$	31.8706	1.48115	0.740577	−	0.671972i	$-0.234552\pi$
$463$	31.8706	1.48115	0.740577	+	0.671972i	$0.234552\pi$
$464$	−6.83716	−0.317407
$465$	−3.07160	−0.142442
$466$	−11.4966	−0.532571
$467$	24.0856	1.11455	0.557275	−	0.830328i	$-0.311847\pi$
$467$	24.0856	1.11455	0.557275	+	0.830328i	$0.311847\pi$
$468$	0.510414	0.0235939
$469$	−8.77544	−0.405212
$470$	23.9170	1.10321
$471$	−4.03301	−0.185831
$472$	−10.5723	−0.486632
$473$	22.0603	1.01433
$474$	4.21444	0.193576
$475$	−5.95190	−0.273092
$476$	0.242141	0.0110985
$477$	20.5239	0.939723
$478$	3.16753	0.144879
$479$	−24.9418	−1.13962	−0.569810	−	0.821776i	$-0.692983\pi$
$479$	−24.9418	−1.13962	−0.569810	+	0.821776i	$0.692983\pi$
$480$	0.417099	0.0190379
$481$	−10.0037	−0.456127
$482$	−37.7464	−1.71930
$483$	2.07392	0.0943667
$484$	−1.03220	−0.0469184
$485$	−10.7179	−0.486673
$486$	−9.31111	−0.422360
$487$	−29.4008	−1.33228	−0.666139	−	0.745828i	$-0.732054\pi$
$487$	−29.4008	−1.33228	−0.666139	+	0.745828i	$0.732054\pi$
$488$	−35.3083	−1.59833
$489$	1.35946	0.0614770
$490$	13.1340	0.593336
$491$	26.2392	1.18416	0.592079	−	0.805880i	$-0.298307\pi$
$491$	26.2392	1.18416	0.592079	+	0.805880i	$0.298307\pi$
$492$	0.287023	0.0129400
$493$	2.34412	0.105574
$494$	4.37507	0.196844
$495$	−21.2250	−0.953995
$496$	24.5884	1.10405
$497$	−16.1968	−0.726526
$498$	0.263478	0.0118067
$499$	−34.6770	−1.55236	−0.776178	−	0.630514i	$-0.782844\pi$
$499$	−34.6770	−1.55236	−0.776178	+	0.630514i	$0.782844\pi$
$500$	−1.96681	−0.0879584
$501$	5.69523	0.254444
$502$	−3.36076	−0.149998
$503$	−1.48835	−0.0663622	−0.0331811	−	0.999449i	$-0.510564\pi$
$503$	−1.48835	−0.0663622	−0.0331811	+	0.999449i	$0.510564\pi$
$504$	−10.2545	−0.456771
$505$	12.2888	0.546844
$506$	−37.4168	−1.66338
$507$	3.10810	0.138035
$508$	2.51652	0.111653
$509$	13.6742	0.606100	0.303050	−	0.952975i	$-0.401995\pi$
$509$	13.6742	0.606100	0.303050	+	0.952975i	$0.401995\pi$
$510$	0.773283	0.0342416
$511$	16.9404	0.749397
$512$	−24.7330	−1.09305
$513$	4.67622	0.206460
$514$	5.72139	0.252360
$515$	−9.72333	−0.428461
$516$	−0.225200	−0.00991390
$517$	42.3096	1.86078
$518$	15.0721	0.662231
$519$	−1.21486	−0.0533266
$520$	5.46982	0.239867
$521$	−28.3688	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$521$	−28.3688	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$522$	−7.44473	−0.325847
$523$	12.7900	0.559270	0.279635	−	0.960106i	$-0.409787\pi$
$523$	12.7900	0.559270	0.279635	+	0.960106i	$0.409787\pi$
$524$	0.152238	0.00665057
$525$	−0.620194	−0.0270675
$526$	35.7295	1.55788
$527$	−8.43013	−0.367222
$528$	−3.98996	−0.173641
$529$	20.8657	0.907206
$530$	16.4943	0.716467
$531$	−10.5723	−0.458801
$532$	0.581566	0.0252141
$533$	7.24573	0.313848
$534$	3.81372	0.165036
$535$	−11.4943	−0.496942
$536$	−21.5515	−0.930883
$537$	−4.24745	−0.183291
$538$	−23.3337	−1.00599
$539$	23.2344	1.00078
$540$	0.438436	0.0188673
$541$	−27.1129	−1.16568	−0.582838	−	0.812588i	$-0.698058\pi$
$541$	−27.1129	−1.16568	−0.582838	+	0.812588i	$0.698058\pi$
$542$	12.8206	0.550693
$543$	2.67549	0.114816
$544$	1.14474	0.0490805
$545$	27.7793	1.18994
$546$	0.455887	0.0195102
$547$	29.4892	1.26087	0.630435	−	0.776242i	$-0.282877\pi$
$547$	29.4892	1.26087	0.630435	+	0.776242i	$0.282877\pi$
$548$	0.905009	0.0386601
$549$	−35.3083	−1.50692
$550$	11.1893	0.477112
$551$	5.63002	0.239847
$552$	5.09331	0.216786
$553$	14.1423	0.601391
$554$	14.2924	0.607224
$555$	−4.24662	−0.180259
$556$	1.77967	0.0754749
$557$	−26.1585	−1.10837	−0.554186	−	0.832393i	$-0.686970\pi$
$557$	−26.1585	−1.10837	−0.554186	+	0.832393i	$0.686970\pi$
$558$	26.7734	1.13341
$559$	−5.68506	−0.240452
$560$	−7.56860	−0.319832
$561$	1.36795	0.0577551
$562$	−17.8901	−0.754648
$563$	−39.0707	−1.64663	−0.823317	−	0.567581i	$-0.807879\pi$
$563$	−39.0707	−1.64663	−0.823317	+	0.567581i	$0.807879\pi$
$564$	−0.431915	−0.0181869
$565$	2.21330	0.0931141
$566$	4.84944	0.203837
$567$	−10.0080	−0.420297
$568$	−39.7775	−1.66903
$569$	9.14582	0.383413	0.191706	−	0.981452i	$-0.438598\pi$
$569$	9.14582	0.383413	0.191706	+	0.981452i	$0.438598\pi$
$570$	1.85725	0.0777915
$571$	−25.4577	−1.06537	−0.532687	−	0.846312i	$-0.678818\pi$
$571$	−25.4577	−1.06537	−0.532687	+	0.846312i	$0.678818\pi$
$572$	0.725654	0.0303411
$573$	−4.69608	−0.196182
$574$	−10.9169	−0.455661
$575$	−13.1178	−0.547050
$576$	−25.0297	−1.04290
$577$	−25.2383	−1.05068	−0.525341	−	0.850892i	$-0.676062\pi$
$577$	−25.2383	−1.05068	−0.525341	+	0.850892i	$0.676062\pi$
$578$	−20.9242	−0.870330
$579$	−1.59893	−0.0664494
$580$	0.527864	0.0219184
$581$	0.884147	0.0366806
$582$	−2.19382	−0.0909368
$583$	29.1787	1.20846
$584$	41.6036	1.72157
$585$	5.46982	0.226149
$586$	−3.01542	−0.124566
$587$	−1.15874	−0.0478261	−0.0239131	−	0.999714i	$-0.507612\pi$
$587$	−1.15874	−0.0478261	−0.0239131	+	0.999714i	$0.507612\pi$
$588$	−0.237186	−0.00978140
$589$	−20.2472	−0.834271
$590$	−8.49662	−0.349800
$591$	−2.00194	−0.0823490
$592$	33.9946	1.39717
$593$	−23.5426	−0.966779	−0.483390	−	0.875405i	$-0.660595\pi$
$593$	−23.5426	−0.966779	−0.483390	+	0.875405i	$0.660595\pi$
$594$	−8.79105	−0.360701
$595$	2.59489	0.106380
$596$	0.731434	0.0299607
$597$	2.88010	0.117875
$598$	9.64252	0.394312
$599$	−10.6446	−0.434926	−0.217463	−	0.976069i	$-0.569778\pi$
$599$	−10.6446	−0.434926	−0.217463	+	0.976069i	$0.569778\pi$
$600$	−1.52313	−0.0621814
$601$	−15.7427	−0.642158	−0.321079	−	0.947052i	$-0.604045\pi$
$601$	−15.7427	−0.642158	−0.321079	+	0.947052i	$0.604045\pi$
$602$	8.56545	0.349102
$603$	−21.5515	−0.877645
$604$	−2.68019	−0.109055
$605$	−11.0616	−0.449716
$606$	2.51537	0.102180
$607$	−0.362041	−0.0146948	−0.00734739	−	0.999973i	$-0.502339\pi$
$607$	−0.362041	−0.0146948	−0.00734739	+	0.999973i	$0.502339\pi$
$608$	2.74941	0.111503
$609$	0.586654	0.0237724
$610$	−28.3760	−1.14891
$611$	−10.9034	−0.441106
$612$	0.594670	0.0240381
$613$	−30.1080	−1.21605	−0.608026	−	0.793917i	$-0.708038\pi$
$613$	−30.1080	−1.21605	−0.608026	+	0.793917i	$0.708038\pi$
$614$	−13.0051	−0.524843
$615$	3.07586	0.124031
$616$	−14.5788	−0.587395
$617$	17.8643	0.719189	0.359594	−	0.933109i	$-0.382915\pi$
$617$	17.8643	0.719189	0.359594	+	0.933109i	$0.382915\pi$
$618$	−1.99025	−0.0800596
$619$	27.9226	1.12230	0.561152	−	0.827713i	$-0.310358\pi$
$619$	27.9226	1.12230	0.561152	+	0.827713i	$0.310358\pi$
$620$	−1.89835	−0.0762397
$621$	10.3062	0.413575
$622$	−37.4852	−1.50302
$623$	12.7976	0.512725
$624$	1.02824	0.0411624
$625$	−11.1742	−0.446967
$626$	4.08396	0.163228
$627$	3.28551	0.131211
$628$	−2.49254	−0.0994630
$629$	−11.6550	−0.464716
$630$	−8.24116	−0.328336
$631$	19.7889	0.787785	0.393892	−	0.919157i	$-0.371128\pi$
$631$	19.7889	0.787785	0.393892	+	0.919157i	$0.371128\pi$
$632$	34.7318	1.38156
$633$	−4.30506	−0.171111
$634$	20.9791	0.833188
$635$	26.9681	1.07020
$636$	−0.297869	−0.0118113
$637$	−5.98764	−0.237239
$638$	−10.5842	−0.419031
$639$	−39.7775	−1.57357
$640$	−16.9359	−0.669449
$641$	−24.9654	−0.986074	−0.493037	−	0.870008i	$-0.664113\pi$
$641$	−24.9654	−0.986074	−0.493037	+	0.870008i	$0.664113\pi$
$642$	−2.35275	−0.0928555
$643$	−22.8349	−0.900522	−0.450261	−	0.892897i	$-0.648669\pi$
$643$	−22.8349	−0.900522	−0.450261	+	0.892897i	$0.648669\pi$
$644$	1.28175	0.0505082
$645$	−2.41335	−0.0950254
$646$	5.09728	0.200550
$647$	−44.2508	−1.73968	−0.869839	−	0.493336i	$-0.835777\pi$
$647$	−44.2508	−1.73968	−0.869839	+	0.493336i	$0.835777\pi$
$648$	−24.5786	−0.965537
$649$	−15.0307	−0.590006
$650$	−2.88354	−0.113102
$651$	−2.10978	−0.0826888
$652$	0.840194	0.0329045
$653$	21.6883	0.848727	0.424364	−	0.905492i	$-0.360498\pi$
$653$	21.6883	0.848727	0.424364	+	0.905492i	$0.360498\pi$
$654$	5.68610	0.222344
$655$	1.63145	0.0637462
$656$	−24.6226	−0.961350
$657$	41.6036	1.62311
$658$	16.4278	0.640422
$659$	−31.4985	−1.22701	−0.613503	−	0.789692i	$-0.710240\pi$
$659$	−31.4985	−1.22701	−0.613503	+	0.789692i	$0.710240\pi$
$660$	0.308045	0.0119906
$661$	35.9230	1.39724	0.698621	−	0.715492i	$-0.253797\pi$
$661$	35.9230	1.39724	0.698621	+	0.715492i	$0.253797\pi$
$662$	10.3267	0.401360
$663$	−0.352530	−0.0136911
$664$	2.17136	0.0842653
$665$	6.23231	0.241679
$666$	37.0154	1.43432
$667$	12.4084	0.480455
$668$	3.51985	0.136187
$669$	−6.12819	−0.236929
$670$	−17.3202	−0.669136
$671$	−50.1978	−1.93786
$672$	0.286491	0.0110516
$673$	−30.6461	−1.18132	−0.590661	−	0.806920i	$-0.701133\pi$
$673$	−30.6461	−1.18132	−0.590661	+	0.806920i	$0.701133\pi$
$674$	−32.2367	−1.24171
$675$	−3.08202	−0.118627
$676$	1.92091	0.0738811
$677$	27.4146	1.05363	0.526815	−	0.849980i	$-0.323386\pi$
$677$	27.4146	1.05363	0.526815	+	0.849980i	$0.323386\pi$
$678$	0.453036	0.0173987
$679$	−7.36175	−0.282518
$680$	6.37275	0.244384
$681$	0.649924	0.0249051
$682$	38.0637	1.45754
$683$	15.7151	0.601322	0.300661	−	0.953731i	$-0.402793\pi$
$683$	15.7151	0.601322	0.300661	+	0.953731i	$0.402793\pi$
$684$	1.42826	0.0546109
$685$	9.69848	0.370560
$686$	20.3476	0.776874
$687$	6.01910	0.229643
$688$	19.3190	0.736532
$689$	−7.51953	−0.286471
$690$	4.09331	0.155830
$691$	5.49051	0.208869	0.104434	−	0.994532i	$-0.466697\pi$
$691$	5.49051	0.208869	0.104434	+	0.994532i	$0.466697\pi$
$692$	−0.750828	−0.0285422
$693$	−14.5788	−0.553802
$694$	14.2944	0.542607
$695$	19.0717	0.723432
$696$	1.44076	0.0546117
$697$	8.44183	0.319757
$698$	−12.6298	−0.478046
$699$	2.22491	0.0841537
$700$	−0.383301	−0.0144874
$701$	−25.4997	−0.963112	−0.481556	−	0.876415i	$-0.659928\pi$
$701$	−25.4997	−0.963112	−0.481556	+	0.876415i	$0.659928\pi$
$702$	2.26550	0.0855060
$703$	−27.9926	−1.05576
$704$	−35.5847	−1.34115
$705$	−4.62859	−0.174323
$706$	−31.6683	−1.19185
$707$	8.44076	0.317447
$708$	0.153440	0.00576661
$709$	−22.2696	−0.836353	−0.418176	−	0.908366i	$-0.637331\pi$
$709$	−22.2696	−0.836353	−0.418176	+	0.908366i	$0.637331\pi$
$710$	−31.9678	−1.19973
$711$	34.7318	1.30255
$712$	31.4295	1.17787
$713$	−44.6242	−1.67119
$714$	0.531142	0.0198775
$715$	7.77643	0.290822
$716$	−2.62507	−0.0981033
$717$	−0.613003	−0.0228930
$718$	10.1107	0.377327
$719$	−5.03417	−0.187743	−0.0938715	−	0.995584i	$-0.529924\pi$
$719$	−5.03417	−0.187743	−0.0938715	+	0.995584i	$0.529924\pi$
$720$	−18.5876	−0.692719
$721$	−6.67863	−0.248725
$722$	−13.5153	−0.502988
$723$	7.30496	0.271674
$724$	1.65354	0.0614534
$725$	−3.71066	−0.137810
$726$	−2.26417	−0.0840312
$727$	6.36687	0.236134	0.118067	−	0.993006i	$-0.462330\pi$
$727$	6.36687	0.236134	0.118067	+	0.993006i	$0.462330\pi$
$728$	3.75703	0.139245
$729$	−23.3538	−0.864955
$730$	33.4353	1.23750
$731$	−6.62352	−0.244980
$732$	0.512440	0.0189403
$733$	33.8114	1.24885	0.624425	−	0.781084i	$-0.285333\pi$
$733$	33.8114	1.24885	0.624425	+	0.781084i	$0.285333\pi$
$734$	29.1241	1.07499
$735$	−2.54179	−0.0937555
$736$	6.05962	0.223360
$737$	−30.6397	−1.12863
$738$	−26.8106	−0.986911
$739$	−44.1712	−1.62486	−0.812432	−	0.583056i	$-0.801857\pi$
$739$	−44.1712	−1.62486	−0.812432	+	0.583056i	$0.801857\pi$
$740$	−2.62456	−0.0964806
$741$	−0.846694	−0.0311041
$742$	11.3294	0.415915
$743$	−8.58951	−0.315118	−0.157559	−	0.987510i	$-0.550363\pi$
$743$	−8.58951	−0.315118	−0.157559	+	0.987510i	$0.550363\pi$
$744$	−5.18138	−0.189959
$745$	7.83836	0.287175
$746$	12.1336	0.444242
$747$	2.17136	0.0794461
$748$	0.845442	0.0309124
$749$	−7.89505	−0.288479
$750$	−4.31425	−0.157534
$751$	−1.02401	−0.0373666	−0.0186833	−	0.999825i	$-0.505947\pi$
$751$	−1.02401	−0.0373666	−0.0186833	+	0.999825i	$0.505947\pi$
$752$	37.0522	1.35116
$753$	0.650399	0.0237018
$754$	2.72760	0.0993333
$755$	−28.7220	−1.04530
$756$	0.301147	0.0109526
$757$	31.0815	1.12968	0.564838	−	0.825202i	$-0.308939\pi$
$757$	31.0815	1.12968	0.564838	+	0.825202i	$0.308939\pi$
$758$	2.36361	0.0858503
$759$	7.24116	0.262837
$760$	15.3059	0.555202
$761$	−4.27576	−0.154996	−0.0774980	−	0.996993i	$-0.524693\pi$
$761$	−4.27576	−0.154996	−0.0774980	+	0.996993i	$0.524693\pi$
$762$	5.52006	0.199971
$763$	19.0807	0.690768
$764$	−2.90234	−0.105003
$765$	6.37275	0.230407
$766$	24.7417	0.893956
$767$	3.87350	0.139864
$768$	1.01409	0.0365928
$769$	−19.2004	−0.692385	−0.346193	−	0.938163i	$-0.612526\pi$
$769$	−19.2004	−0.692385	−0.346193	+	0.938163i	$0.612526\pi$
$770$	−11.7164	−0.422231
$771$	−1.10724	−0.0398764
$772$	−0.988195	−0.0355659
$773$	−15.6584	−0.563194	−0.281597	−	0.959533i	$-0.590864\pi$
$773$	−15.6584	−0.563194	−0.281597	+	0.959533i	$0.590864\pi$
$774$	21.0358	0.756115
$775$	13.3446	0.479353
$776$	−18.0796	−0.649020
$777$	−2.91686	−0.104642
$778$	45.2058	1.62071
$779$	20.2753	0.726438
$780$	−0.0793850	−0.00284244
$781$	−56.5516	−2.02358
$782$	11.2343	0.401736
$783$	2.91535	0.104186
$784$	20.3473	0.726689
$785$	−26.7111	−0.953360
$786$	0.333939	0.0119112
$787$	1.57485	0.0561373	0.0280687	−	0.999606i	$-0.491064\pi$
$787$	1.57485	0.0561373	0.0280687	+	0.999606i	$0.491064\pi$
$788$	−1.23727	−0.0440759
$789$	−6.91462	−0.246167
$790$	27.9127	0.993090
$791$	1.52024	0.0540535
$792$	−35.8038	−1.27223
$793$	12.9363	0.459380
$794$	−34.1005	−1.21018
$795$	−3.19209	−0.113212
$796$	1.78000	0.0630904
$797$	35.5914	1.26071	0.630356	−	0.776306i	$-0.282909\pi$
$797$	35.5914	1.26071	0.630356	+	0.776306i	$0.282909\pi$
$798$	1.27568	0.0451586
$799$	−12.7033	−0.449412
$800$	−1.81209	−0.0640672
$801$	31.4295	1.11051
$802$	6.96897	0.246083
$803$	59.1478	2.08728
$804$	0.312783	0.0110310
$805$	13.7358	0.484124
$806$	−9.80924	−0.345516
$807$	4.51570	0.158960
$808$	20.7295	0.729263
$809$	5.92535	0.208324	0.104162	−	0.994560i	$-0.466784\pi$
$809$	5.92535	0.208324	0.104162	+	0.994560i	$0.466784\pi$
$810$	−19.7529	−0.694046
$811$	−31.1802	−1.09488	−0.547442	−	0.836844i	$-0.684398\pi$
$811$	−31.1802	−1.09488	−0.547442	+	0.836844i	$0.684398\pi$
$812$	0.362572	0.0127238
$813$	−2.48114	−0.0870173
$814$	52.6247	1.84450
$815$	9.00389	0.315392
$816$	1.19797	0.0419374
$817$	−15.9082	−0.556556
$818$	−33.6409	−1.17623
$819$	3.75703	0.131281
$820$	1.90099	0.0663854
$821$	−26.2296	−0.915419	−0.457710	−	0.889102i	$-0.651330\pi$
$821$	−26.2296	−0.915419	−0.457710	+	0.889102i	$0.651330\pi$
$822$	1.98516	0.0692405
$823$	−25.1236	−0.875753	−0.437877	−	0.899035i	$-0.644269\pi$
$823$	−25.1236	−0.875753	−0.437877	+	0.899035i	$0.644269\pi$
$824$	−16.4020	−0.571390
$825$	−2.16543	−0.0753904
$826$	−5.83604	−0.203062
$827$	4.86271	0.169093	0.0845465	−	0.996420i	$-0.473056\pi$
$827$	4.86271	0.169093	0.0845465	+	0.996420i	$0.473056\pi$
$828$	3.14784	0.109395
$829$	3.56491	0.123814	0.0619071	−	0.998082i	$-0.480282\pi$
$829$	3.56491	0.123814	0.0619071	+	0.998082i	$0.480282\pi$
$830$	1.74505	0.0605715
$831$	−2.76596	−0.0959500
$832$	9.17038	0.317926
$833$	−6.97605	−0.241706
$834$	3.90376	0.135176
$835$	37.7202	1.30536
$836$	2.03055	0.0702282
$837$	−10.4844	−0.362395
$838$	−11.6423	−0.402177
$839$	−41.0987	−1.41888	−0.709442	−	0.704764i	$-0.751053\pi$
$839$	−41.0987	−1.41888	−0.709442	+	0.704764i	$0.751053\pi$
$840$	1.59489	0.0550288
$841$	−25.4900	−0.878966
$842$	45.6664	1.57377
$843$	3.46222	0.119245
$844$	−2.66067	−0.0915842
$845$	20.5853	0.708156
$846$	40.3448	1.38708
$847$	−7.59782	−0.261064
$848$	25.5530	0.877493
$849$	−0.938498	−0.0322092
$850$	−3.35954	−0.115231
$851$	−61.6949	−2.11487
$852$	0.577303	0.0197781
$853$	52.7052	1.80459	0.902295	−	0.431119i	$-0.141881\pi$
$853$	52.7052	1.80459	0.902295	+	0.431119i	$0.141881\pi$
$854$	−19.4905	−0.666953
$855$	15.3059	0.523449
$856$	−19.3893	−0.662714
$857$	51.0437	1.74362	0.871810	−	0.489845i	$-0.162947\pi$
$857$	51.0437	1.74362	0.871810	+	0.489845i	$0.162947\pi$
$858$	1.59174	0.0543412
$859$	38.7732	1.32292	0.661462	−	0.749979i	$-0.269936\pi$
$859$	38.7732	1.32292	0.661462	+	0.749979i	$0.269936\pi$
$860$	−1.49153	−0.0508607
$861$	2.11271	0.0720009
$862$	−1.35567	−0.0461744
$863$	−4.49476	−0.153003	−0.0765017	−	0.997069i	$-0.524375\pi$
$863$	−4.49476	−0.153003	−0.0765017	+	0.997069i	$0.524375\pi$
$864$	1.42370	0.0484354
$865$	−8.04620	−0.273579
$866$	4.44635	0.151093
$867$	4.04939	0.137525
$868$	−1.30392	−0.0442578
$869$	49.3782	1.67504
$870$	1.15788	0.0392559
$871$	7.89603	0.267547
$872$	46.8600	1.58688
$873$	−18.0796	−0.611902
$874$	26.9821	0.912682
$875$	−14.4772	−0.489420
$876$	−0.603805	−0.0204007
$877$	2.19934	0.0742665	0.0371332	−	0.999310i	$-0.488177\pi$
$877$	2.19934	0.0742665	0.0371332	+	0.999310i	$0.488177\pi$
$878$	33.7498	1.13900
$879$	0.583566	0.0196832
$880$	−26.4260	−0.890819
$881$	6.55549	0.220860	0.110430	−	0.993884i	$-0.464777\pi$
$881$	6.55549	0.220860	0.110430	+	0.993884i	$0.464777\pi$
$882$	22.1554	0.746011
$883$	39.0467	1.31403	0.657013	−	0.753880i	$-0.271820\pi$
$883$	39.0467	1.31403	0.657013	+	0.753880i	$0.271820\pi$
$884$	−0.217875	−0.00732794
$885$	1.64433	0.0552734
$886$	14.0106	0.470694
$887$	39.8540	1.33817	0.669084	−	0.743187i	$-0.266687\pi$
$887$	39.8540	1.33817	0.669084	+	0.743187i	$0.266687\pi$
$888$	−7.16348	−0.240391
$889$	18.5235	0.621259
$890$	25.2587	0.846675
$891$	−34.9433	−1.17064
$892$	−3.78743	−0.126813
$893$	−30.5104	−1.02099
$894$	1.60442	0.0536598
$895$	−28.1314	−0.940327
$896$	−11.6327	−0.388621
$897$	−1.86609	−0.0623069
$898$	34.5880	1.15422
$899$	−12.6229	−0.420999
$900$	−0.941344	−0.0313781
$901$	−8.76082	−0.291865
$902$	−38.1165	−1.26914
$903$	−1.65765	−0.0551630
$904$	3.73354	0.124176
$905$	17.7201	0.589036
$906$	−5.87906	−0.195319
$907$	−35.8748	−1.19120	−0.595601	−	0.803280i	$-0.703086\pi$
$907$	−35.8748	−1.19120	−0.595601	+	0.803280i	$0.703086\pi$
$908$	0.401675	0.0133301
$909$	20.7295	0.687555
$910$	3.01939	0.100092
$911$	−37.4401	−1.24044	−0.620222	−	0.784426i	$-0.712958\pi$
$911$	−37.4401	−1.24044	−0.620222	+	0.784426i	$0.712958\pi$
$912$	2.87725	0.0952752
$913$	3.08703	0.102166
$914$	−19.0848	−0.631269
$915$	5.49153	0.181544
$916$	3.72001	0.122913
$917$	1.12059	0.0370052
$918$	2.63948	0.0871159
$919$	23.9715	0.790748	0.395374	−	0.918520i	$-0.370615\pi$
$919$	23.9715	0.790748	0.395374	+	0.918520i	$0.370615\pi$
$920$	33.7336	1.11217
$921$	2.51684	0.0829326
$922$	47.6243	1.56842
$923$	14.5737	0.479699
$924$	0.211586	0.00696067
$925$	18.4495	0.606616
$926$	43.2062	1.41984
$927$	−16.4020	−0.538711
$928$	1.71410	0.0562679
$929$	−17.2626	−0.566368	−0.283184	−	0.959066i	$-0.591391\pi$
$929$	−17.2626	−0.566368	−0.283184	+	0.959066i	$0.591391\pi$
$930$	−4.16409	−0.136546
$931$	−16.7548	−0.549118
$932$	1.37507	0.0450419
$933$	7.25440	0.237498
$934$	32.6523	1.06842
$935$	9.06012	0.296298
$936$	9.22685	0.301589
$937$	−16.6606	−0.544279	−0.272140	−	0.962258i	$-0.587731\pi$
$937$	−16.6606	−0.544279	−0.272140	+	0.962258i	$0.587731\pi$
$938$	−11.8966	−0.388439
$939$	−0.790358	−0.0257923
$940$	−2.86062	−0.0933033
$941$	16.6046	0.541294	0.270647	−	0.962679i	$-0.412762\pi$
$941$	16.6046	0.541294	0.270647	+	0.962679i	$0.412762\pi$
$942$	−5.46745	−0.178139
$943$	44.6861	1.45518
$944$	−13.1630	−0.428418
$945$	3.22723	0.104982
$946$	29.9065	0.972345
$947$	−19.3858	−0.629955	−0.314977	−	0.949099i	$-0.601997\pi$
$947$	−19.3858	−0.629955	−0.314977	+	0.949099i	$0.601997\pi$
$948$	−0.504073	−0.0163715
$949$	−15.2427	−0.494800
$950$	−8.06883	−0.261787
$951$	−4.06003	−0.131656
$952$	4.37723	0.141867
$953$	40.5403	1.31323	0.656614	−	0.754227i	$-0.271988\pi$
$953$	40.5403	1.31323	0.656614	+	0.754227i	$0.271988\pi$
$954$	27.8237	0.900825
$955$	−31.1027	−1.00646
$956$	−0.378856	−0.0122531
$957$	2.04832	0.0662128
$958$	−33.8130	−1.09245
$959$	6.66156	0.215113
$960$	3.89289	0.125642
$961$	14.3958	0.464381
$962$	−13.5617	−0.437247
$963$	−19.3893	−0.624813
$964$	4.51471	0.145409
$965$	−10.5899	−0.340902
$966$	2.81156	0.0904605
$967$	−7.25693	−0.233367	−0.116684	−	0.993169i	$-0.537226\pi$
$967$	−7.25693	−0.233367	−0.116684	+	0.993169i	$0.537226\pi$
$968$	−18.6594	−0.599735
$969$	−0.986463	−0.0316897
$970$	−14.5299	−0.466528
$971$	−33.1994	−1.06542	−0.532710	−	0.846298i	$-0.678826\pi$
$971$	−33.1994	−1.06542	−0.532710	+	0.846298i	$0.678826\pi$
$972$	1.11367	0.0357209
$973$	13.0997	0.419958
$974$	−39.8579	−1.27713
$975$	0.558043	0.0178717
$976$	−43.9602	−1.40713
$977$	−34.3678	−1.09952	−0.549761	−	0.835322i	$-0.685281\pi$
$977$	−34.3678	−1.09952	−0.549761	+	0.835322i	$0.685281\pi$
$978$	1.84299	0.0589323
$979$	44.6832	1.42808
$980$	−1.57091	−0.0501810
$981$	46.8600	1.49613
$982$	35.5718	1.13514
$983$	−41.0953	−1.31074	−0.655368	−	0.755310i	$-0.727486\pi$
$983$	−41.0953	−1.31074	−0.655368	+	0.755310i	$0.727486\pi$
$984$	5.18857	0.165406
$985$	−13.2591	−0.422471
$986$	3.17786	0.101204
$987$	−3.17922	−0.101196
$988$	−0.523286	−0.0166479
$989$	−35.0611	−1.11488
$990$	−28.7742	−0.914505
$991$	−29.2140	−0.928015	−0.464007	−	0.885831i	$-0.653589\pi$
$991$	−29.2140	−0.928015	−0.464007	+	0.885831i	$0.653589\pi$
$992$	−6.16439	−0.195720
$993$	−1.99851	−0.0634207
$994$	−21.9576	−0.696453
$995$	19.0753	0.604726
$996$	−0.0315137	−0.000998548 0
$997$	−2.36702	−0.0749642	−0.0374821	−	0.999297i	$-0.511934\pi$
$997$	−2.36702	−0.0749642	−0.0374821	+	0.999297i	$0.511934\pi$
$998$	−47.0107	−1.48810
$999$	−14.4952	−0.458607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.e.1.4	✓	4
3.2	odd	2		3879.2.a.m.1.1		4
4.3	odd	2		6896.2.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.e.1.4	✓	4	1.1	even	1	trivial
3879.2.a.m.1.1		4	3.2	odd	2
6896.2.a.o.1.2		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:	\( N \)	\(=\)	\( 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	431.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35567 q^{2} -0.262360 q^{3} -0.162147 q^{4} -1.73764 q^{5} -0.355674 q^{6} -1.19353 q^{7} -2.93117 q^{8} -2.93117 q^{9} +O(q^{10})\)	\(q+1.35567 q^{2} -0.262360 q^{3} -0.162147 q^{4} -1.73764 q^{5} -0.355674 q^{6} -1.19353 q^{7} -2.93117 q^{8} -2.93117 q^{9} -2.35567 q^{10} -4.16724 q^{11} +0.0425409 q^{12} +1.07392 q^{13} -1.61803 q^{14} +0.455887 q^{15} -3.64941 q^{16} +1.25120 q^{17} -3.97371 q^{18} +3.00509 q^{19} +0.281754 q^{20} +0.313133 q^{21} -5.64941 q^{22} +6.62312 q^{23} +0.769020 q^{24} -1.98061 q^{25} +1.45589 q^{26} +1.55610 q^{27} +0.193527 q^{28} +1.87350 q^{29} +0.618034 q^{30} -6.73764 q^{31} +0.914918 q^{32} +1.09331 q^{33} +1.69622 q^{34} +2.07392 q^{35} +0.475281 q^{36} -9.31508 q^{37} +4.07392 q^{38} -0.281754 q^{39} +5.09331 q^{40} +6.74699 q^{41} +0.424507 q^{42} -5.29374 q^{43} +0.675706 q^{44} +5.09331 q^{45} +8.97880 q^{46} -10.1529 q^{47} +0.957459 q^{48} -5.57549 q^{49} -2.68506 q^{50} -0.328264 q^{51} -0.174133 q^{52} -7.00194 q^{53} +2.10956 q^{54} +7.24116 q^{55} +3.49843 q^{56} -0.788414 q^{57} +2.53985 q^{58} +3.60687 q^{59} -0.0739208 q^{60} +12.0458 q^{61} -9.13405 q^{62} +3.49843 q^{63} +8.53916 q^{64} -1.86609 q^{65} +1.48218 q^{66} +7.35253 q^{67} -0.202878 q^{68} -1.73764 q^{69} +2.81156 q^{70} +13.5705 q^{71} +8.59174 q^{72} -14.1935 q^{73} -12.6282 q^{74} +0.519631 q^{75} -0.487267 q^{76} +4.97371 q^{77} -0.381966 q^{78} -11.8491 q^{79} +6.34137 q^{80} +8.38524 q^{81} +9.14672 q^{82} -0.740785 q^{83} -0.0507737 q^{84} -2.17413 q^{85} -7.17659 q^{86} -0.491530 q^{87} +12.2149 q^{88} -10.7225 q^{89} +6.90488 q^{90} -1.28175 q^{91} -1.07392 q^{92} +1.76769 q^{93} -13.7641 q^{94} -5.22176 q^{95} -0.240038 q^{96} +6.16806 q^{97} -7.55855 q^{98} +12.2149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20} - 3 q^{21} - 11 q^{22} + 4 q^{23} - 6 q^{24} - 7 q^{25} + q^{26} + 3 q^{27} - 6 q^{28} - 5 q^{29} - 2 q^{30} - 25 q^{31} + 8 q^{32} - 4 q^{33} - 12 q^{34} - q^{35} - 2 q^{36} - q^{37} + 7 q^{38} - 4 q^{39} + 12 q^{40} + 2 q^{41} + 4 q^{42} - 16 q^{43} + 2 q^{44} + 12 q^{45} + 7 q^{46} - 4 q^{47} + 6 q^{48} - 20 q^{49} + 13 q^{50} - 3 q^{51} + 7 q^{52} + 4 q^{53} - 13 q^{54} + 2 q^{55} + 7 q^{56} + 5 q^{57} + 20 q^{58} + 5 q^{59} + 9 q^{60} + 2 q^{62} + 7 q^{63} - 3 q^{64} + 14 q^{65} + 12 q^{66} + 9 q^{67} + 29 q^{68} - 5 q^{69} + 10 q^{71} + 19 q^{72} - 50 q^{73} - 10 q^{74} + 24 q^{75} + 10 q^{76} + 9 q^{77} - 6 q^{78} + 8 q^{79} - 8 q^{81} + 37 q^{82} - 15 q^{83} + 6 q^{84} - q^{85} + 12 q^{86} + 15 q^{87} + 11 q^{88} - 35 q^{89} + 8 q^{90} - 8 q^{91} + 5 q^{92} + 12 q^{93} - 4 q^{94} + 7 q^{95} + 7 q^{96} - 6 q^{97} + 6 q^{98} + 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more