LMFDB - Embedded newform 409.2.a.b.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.12
Root			$0.573477$ of defining polynomial
Character	$\chi$	$=$	409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.573477 q^{2} +1.77005 q^{3} -1.67112 q^{4} +1.27661 q^{5} +1.01508 q^{6} +2.77278 q^{7} -2.10531 q^{8} +0.133085 q^{9} +0.732108 q^{10} +3.42880 q^{11} -2.95798 q^{12} -0.246307 q^{13} +1.59012 q^{14} +2.25967 q^{15} +2.13490 q^{16} +1.18272 q^{17} +0.0763210 q^{18} +2.42449 q^{19} -2.13338 q^{20} +4.90796 q^{21} +1.96634 q^{22} -5.54747 q^{23} -3.72650 q^{24} -3.37026 q^{25} -0.141252 q^{26} -5.07459 q^{27} -4.63365 q^{28} +7.40574 q^{29} +1.29587 q^{30} -0.466653 q^{31} +5.43493 q^{32} +6.06916 q^{33} +0.678263 q^{34} +3.53976 q^{35} -0.222401 q^{36} +0.524788 q^{37} +1.39039 q^{38} -0.435977 q^{39} -2.68766 q^{40} +1.85544 q^{41} +2.81460 q^{42} -7.88904 q^{43} -5.72995 q^{44} +0.169898 q^{45} -3.18135 q^{46} -11.9084 q^{47} +3.77889 q^{48} +0.688284 q^{49} -1.93277 q^{50} +2.09348 q^{51} +0.411610 q^{52} -8.20771 q^{53} -2.91016 q^{54} +4.37725 q^{55} -5.83754 q^{56} +4.29148 q^{57} +4.24702 q^{58} -7.67709 q^{59} -3.77619 q^{60} +9.34587 q^{61} -0.267615 q^{62} +0.369014 q^{63} -1.15300 q^{64} -0.314439 q^{65} +3.48052 q^{66} +12.1402 q^{67} -1.97647 q^{68} -9.81931 q^{69} +2.02997 q^{70} -1.75984 q^{71} -0.280184 q^{72} -12.2331 q^{73} +0.300954 q^{74} -5.96553 q^{75} -4.05163 q^{76} +9.50730 q^{77} -0.250023 q^{78} -4.02729 q^{79} +2.72545 q^{80} -9.38154 q^{81} +1.06405 q^{82} +9.60080 q^{83} -8.20181 q^{84} +1.50988 q^{85} -4.52418 q^{86} +13.1085 q^{87} -7.21867 q^{88} +8.66515 q^{89} +0.0974324 q^{90} -0.682955 q^{91} +9.27051 q^{92} -0.826000 q^{93} -6.82921 q^{94} +3.09514 q^{95} +9.62011 q^{96} -18.2223 q^{97} +0.394715 q^{98} +0.456321 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) $ 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Coefficient data

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

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

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

$2$	0.573477	0.405509	0.202755	−	0.979230i	$-0.435011\pi$
$2$	0.573477	0.405509	0.202755	+	0.979230i	$0.435011\pi$
$3$	1.77005	1.02194	0.510970	−	0.859599i	$-0.329286\pi$
$3$	1.77005	1.02194	0.510970	+	0.859599i	$0.329286\pi$
$4$	−1.67112	−0.835562
$5$	1.27661	0.570919	0.285459	−	0.958391i	$-0.407854\pi$
$5$	1.27661	0.570919	0.285459	+	0.958391i	$0.407854\pi$
$6$	1.01508	0.414406
$7$	2.77278	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$7$	2.77278	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$8$	−2.10531	−0.744338
$9$	0.133085	0.0443616
$10$	0.732108	0.231513
$11$	3.42880	1.03382	0.516911	−	0.856039i	$-0.327082\pi$
$11$	3.42880	1.03382	0.516911	+	0.856039i	$0.327082\pi$
$12$	−2.95798	−0.853894
$13$	−0.246307	−0.0683134	−0.0341567	−	0.999416i	$-0.510875\pi$
$13$	−0.246307	−0.0683134	−0.0341567	+	0.999416i	$0.510875\pi$
$14$	1.59012	0.424978
$15$	2.25967	0.583445
$16$	2.13490	0.533726
$17$	1.18272	0.286852	0.143426	−	0.989661i	$-0.454188\pi$
$17$	1.18272	0.286852	0.143426	+	0.989661i	$0.454188\pi$
$18$	0.0763210	0.0179890
$19$	2.42449	0.556217	0.278109	−	0.960550i	$-0.410292\pi$
$19$	2.42449	0.556217	0.278109	+	0.960550i	$0.410292\pi$
$20$	−2.13338	−0.477038
$21$	4.90796	1.07100
$22$	1.96634	0.419225
$23$	−5.54747	−1.15673	−0.578364	−	0.815779i	$-0.696309\pi$
$23$	−5.54747	−1.15673	−0.578364	+	0.815779i	$0.696309\pi$
$24$	−3.72650	−0.760669
$25$	−3.37026	−0.674052
$26$	−0.141252	−0.0277017
$27$	−5.07459	−0.976605
$28$	−4.63365	−0.875678
$29$	7.40574	1.37521	0.687606	−	0.726084i	$-0.258662\pi$
$29$	7.40574	1.37521	0.687606	+	0.726084i	$0.258662\pi$
$30$	1.29587	0.236592
$31$	−0.466653	−0.0838133	−0.0419067	−	0.999122i	$-0.513343\pi$
$31$	−0.466653	−0.0838133	−0.0419067	+	0.999122i	$0.513343\pi$
$32$	5.43493	0.960769
$33$	6.06916	1.05650
$34$	0.678263	0.116321
$35$	3.53976	0.598329
$36$	−0.222401	−0.0370668
$37$	0.524788	0.0862747	0.0431373	−	0.999069i	$-0.486265\pi$
$37$	0.524788	0.0862747	0.0431373	+	0.999069i	$0.486265\pi$
$38$	1.39039	0.225551
$39$	−0.435977	−0.0698122
$40$	−2.68766	−0.424956
$41$	1.85544	0.289771	0.144885	−	0.989448i	$-0.453719\pi$
$41$	1.85544	0.289771	0.144885	+	0.989448i	$0.453719\pi$
$42$	2.81460	0.434302
$43$	−7.88904	−1.20307	−0.601534	−	0.798847i	$-0.705443\pi$
$43$	−7.88904	−1.20307	−0.601534	+	0.798847i	$0.705443\pi$
$44$	−5.72995	−0.863823
$45$	0.169898	0.0253269
$46$	−3.18135	−0.469064
$47$	−11.9084	−1.73702	−0.868511	−	0.495670i	$-0.834922\pi$
$47$	−11.9084	−1.73702	−0.868511	+	0.495670i	$0.834922\pi$
$48$	3.77889	0.545436
$49$	0.688284	0.0983263
$50$	−1.93277	−0.273334
$51$	2.09348	0.293145
$52$	0.411610	0.0570801
$53$	−8.20771	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$53$	−8.20771	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$54$	−2.91016	−0.396023
$55$	4.37725	0.590229
$56$	−5.83754	−0.780074
$57$	4.29148	0.568421
$58$	4.24702	0.557661
$59$	−7.67709	−0.999472	−0.499736	−	0.866178i	$-0.666570\pi$
$59$	−7.67709	−0.999472	−0.499736	+	0.866178i	$0.666570\pi$
$60$	−3.77619	−0.487504
$61$	9.34587	1.19662	0.598308	−	0.801266i	$-0.295840\pi$
$61$	9.34587	1.19662	0.598308	+	0.801266i	$0.295840\pi$
$62$	−0.267615	−0.0339871
$63$	0.369014	0.0464914
$64$	−1.15300	−0.144125
$65$	−0.314439	−0.0390014
$66$	3.48052	0.428423
$67$	12.1402	1.48316	0.741581	−	0.670863i	$-0.234076\pi$
$67$	12.1402	1.48316	0.741581	+	0.670863i	$0.234076\pi$
$68$	−1.97647	−0.239682
$69$	−9.81931	−1.18211
$70$	2.02997	0.242628
$71$	−1.75984	−0.208855	−0.104427	−	0.994533i	$-0.533301\pi$
$71$	−1.75984	−0.208855	−0.104427	+	0.994533i	$0.533301\pi$
$72$	−0.280184	−0.0330200
$73$	−12.2331	−1.43177	−0.715887	−	0.698216i	$-0.753977\pi$
$73$	−12.2331	−1.43177	−0.715887	+	0.698216i	$0.753977\pi$
$74$	0.300954	0.0349852
$75$	−5.96553	−0.688840
$76$	−4.05163	−0.464754
$77$	9.50730	1.08346
$78$	−0.250023	−0.0283095
$79$	−4.02729	−0.453106	−0.226553	−	0.973999i	$-0.572746\pi$
$79$	−4.02729	−0.453106	−0.226553	+	0.973999i	$0.572746\pi$
$80$	2.72545	0.304714
$81$	−9.38154	−1.04239
$82$	1.06405	0.117505
$83$	9.60080	1.05382	0.526912	−	0.849920i	$-0.323350\pi$
$83$	9.60080	1.05382	0.526912	+	0.849920i	$0.323350\pi$
$84$	−8.20181	−0.894890
$85$	1.50988	0.163769
$86$	−4.52418	−0.487855
$87$	13.1085	1.40538
$88$	−7.21867	−0.769513
$89$	8.66515	0.918504	0.459252	−	0.888306i	$-0.348117\pi$
$89$	8.66515	0.918504	0.459252	+	0.888306i	$0.348117\pi$
$90$	0.0974324	0.0102703
$91$	−0.682955	−0.0715932
$92$	9.27051	0.966517
$93$	−0.826000	−0.0856522
$94$	−6.82921	−0.704379
$95$	3.09514	0.317555
$96$	9.62011	0.981848
$97$	−18.2223	−1.85020	−0.925098	−	0.379727i	$-0.876018\pi$
$97$	−18.2223	−1.85020	−0.925098	+	0.379727i	$0.876018\pi$
$98$	0.394715	0.0398723
$99$	0.456321	0.0458620
$100$	5.63212	0.563212
$101$	2.89279	0.287844	0.143922	−	0.989589i	$-0.454029\pi$
$101$	2.89279	0.287844	0.143922	+	0.989589i	$0.454029\pi$
$102$	1.20056	0.118873
$103$	1.82539	0.179861	0.0899303	−	0.995948i	$-0.471336\pi$
$103$	1.82539	0.179861	0.0899303	+	0.995948i	$0.471336\pi$
$104$	0.518552	0.0508482
$105$	6.26556	0.611456
$106$	−4.70693	−0.457178
$107$	−3.79646	−0.367018	−0.183509	−	0.983018i	$-0.558746\pi$
$107$	−3.79646	−0.367018	−0.183509	+	0.983018i	$0.558746\pi$
$108$	8.48027	0.816014
$109$	−8.82032	−0.844834	−0.422417	−	0.906402i	$-0.638818\pi$
$109$	−8.82032	−0.844834	−0.422417	+	0.906402i	$0.638818\pi$
$110$	2.51025	0.239343
$111$	0.928903	0.0881676
$112$	5.91961	0.559351
$113$	−1.09202	−0.102729	−0.0513643	−	0.998680i	$-0.516357\pi$
$113$	−1.09202	−0.102729	−0.0513643	+	0.998680i	$0.516357\pi$
$114$	2.46107	0.230500
$115$	−7.08197	−0.660397
$116$	−12.3759	−1.14907
$117$	−0.0327798	−0.00303049
$118$	−4.40263	−0.405295
$119$	3.27942	0.300624
$120$	−4.75730	−0.434280
$121$	0.756678	0.0687889
$122$	5.35964	0.485239
$123$	3.28422	0.296128
$124$	0.779835	0.0700312
$125$	−10.6856	−0.955748
$126$	0.211621	0.0188527
$127$	3.47315	0.308192	0.154096	−	0.988056i	$-0.450754\pi$
$127$	3.47315	0.308192	0.154096	+	0.988056i	$0.450754\pi$
$128$	−11.5311	−1.01921
$129$	−13.9640	−1.22946
$130$	−0.180324	−0.0158154
$131$	9.51456	0.831291	0.415646	−	0.909527i	$-0.363556\pi$
$131$	9.51456	0.831291	0.415646	+	0.909527i	$0.363556\pi$
$132$	−10.1423	−0.882775
$133$	6.72258	0.582922
$134$	6.96213	0.601436
$135$	−6.47829	−0.557562
$136$	−2.48999	−0.213515
$137$	18.3320	1.56621	0.783105	−	0.621889i	$-0.213635\pi$
$137$	18.3320	1.56621	0.783105	+	0.621889i	$0.213635\pi$
$138$	−5.63115	−0.479355
$139$	17.1943	1.45840	0.729200	−	0.684300i	$-0.239892\pi$
$139$	17.1943	1.45840	0.729200	+	0.684300i	$0.239892\pi$
$140$	−5.91538	−0.499941
$141$	−21.0785	−1.77513
$142$	−1.00923	−0.0846926
$143$	−0.844539	−0.0706239
$144$	0.284123	0.0236769
$145$	9.45427	0.785134
$146$	−7.01539	−0.580598
$147$	1.21830	0.100484
$148$	−0.876987	−0.0720879
$149$	−5.03995	−0.412889	−0.206444	−	0.978458i	$-0.566189\pi$
$149$	−5.03995	−0.412889	−0.206444	+	0.978458i	$0.566189\pi$
$150$	−3.42110	−0.279331
$151$	−2.19479	−0.178610	−0.0893049	−	0.996004i	$-0.528465\pi$
$151$	−2.19479	−0.178610	−0.0893049	+	0.996004i	$0.528465\pi$
$152$	−5.10430	−0.414013
$153$	0.157402	0.0127252
$154$	5.45222	0.439352
$155$	−0.595735	−0.0478506
$156$	0.728572	0.0583324
$157$	−1.39949	−0.111692	−0.0558458	−	0.998439i	$-0.517786\pi$
$157$	−1.39949	−0.111692	−0.0558458	+	0.998439i	$0.517786\pi$
$158$	−2.30956	−0.183739
$159$	−14.5281	−1.15215
$160$	6.93830	0.548521
$161$	−15.3819	−1.21226
$162$	−5.38010	−0.422701
$163$	−16.7402	−1.31119	−0.655595	−	0.755113i	$-0.727582\pi$
$163$	−16.7402	−1.31119	−0.655595	+	0.755113i	$0.727582\pi$
$164$	−3.10067	−0.242122
$165$	7.74797	0.603178
$166$	5.50584	0.427336
$167$	20.9314	1.61972	0.809858	−	0.586625i	$-0.199544\pi$
$167$	20.9314	1.61972	0.809858	+	0.586625i	$0.199544\pi$
$168$	−10.3327	−0.797189
$169$	−12.9393	−0.995333
$170$	0.865880	0.0664099
$171$	0.322663	0.0246747
$172$	13.1836	1.00524
$173$	15.9066	1.20936	0.604679	−	0.796469i	$-0.293301\pi$
$173$	15.9066	1.20936	0.604679	+	0.796469i	$0.293301\pi$
$174$	7.51745	0.569897
$175$	−9.34497	−0.706413
$176$	7.32016	0.551778
$177$	−13.5888	−1.02140
$178$	4.96926	0.372462
$179$	11.5412	0.862631	0.431315	−	0.902201i	$-0.358050\pi$
$179$	11.5412	0.862631	0.431315	+	0.902201i	$0.358050\pi$
$180$	−0.283920	−0.0211622
$181$	1.78604	0.132756	0.0663778	−	0.997795i	$-0.478856\pi$
$181$	1.78604	0.132756	0.0663778	+	0.997795i	$0.478856\pi$
$182$	−0.391659	−0.0290317
$183$	16.5427	1.22287
$184$	11.6791	0.860996
$185$	0.669952	0.0492558
$186$	−0.473692	−0.0347328
$187$	4.05531	0.296554
$188$	19.9004	1.45139
$189$	−14.0707	−1.02349
$190$	1.77499	0.128772
$191$	12.8325	0.928530	0.464265	−	0.885696i	$-0.346319\pi$
$191$	12.8325	0.928530	0.464265	+	0.885696i	$0.346319\pi$
$192$	−2.04087	−0.147287
$193$	6.50538	0.468268	0.234134	−	0.972204i	$-0.424775\pi$
$193$	6.50538	0.468268	0.234134	+	0.972204i	$0.424775\pi$
$194$	−10.4501	−0.750272
$195$	−0.556574	−0.0398571
$196$	−1.15021	−0.0821577
$197$	11.7886	0.839904	0.419952	−	0.907546i	$-0.362047\pi$
$197$	11.7886	0.839904	0.419952	+	0.907546i	$0.362047\pi$
$198$	0.261690	0.0185975
$199$	7.24812	0.513806	0.256903	−	0.966437i	$-0.417298\pi$
$199$	7.24812	0.513806	0.256903	+	0.966437i	$0.417298\pi$
$200$	7.09542	0.501722
$201$	21.4888	1.51570
$202$	1.65895	0.116723
$203$	20.5345	1.44124
$204$	−3.49846	−0.244941
$205$	2.36868	0.165436
$206$	1.04682	0.0729352
$207$	−0.738283	−0.0513142
$208$	−0.525843	−0.0364606
$209$	8.31311	0.575030
$210$	3.59316	0.247951
$211$	1.10505	0.0760745	0.0380373	−	0.999276i	$-0.487889\pi$
$211$	1.10505	0.0760745	0.0380373	+	0.999276i	$0.487889\pi$
$212$	13.7161	0.942025
$213$	−3.11501	−0.213437
$214$	−2.17718	−0.148829
$215$	−10.0713	−0.686854
$216$	10.6836	0.726924
$217$	−1.29392	−0.0878373
$218$	−5.05825	−0.342588
$219$	−21.6532	−1.46319
$220$	−7.31493	−0.493173
$221$	−0.291313	−0.0195958
$222$	0.532704	0.0357528
$223$	7.06708	0.473246	0.236623	−	0.971601i	$-0.423959\pi$
$223$	7.06708	0.473246	0.236623	+	0.971601i	$0.423959\pi$
$224$	15.0698	1.00690
$225$	−0.448530	−0.0299020
$226$	−0.626248	−0.0416574
$227$	16.9741	1.12661	0.563305	−	0.826249i	$-0.309529\pi$
$227$	16.9741	1.12661	0.563305	+	0.826249i	$0.309529\pi$
$228$	−7.17160	−0.474951
$229$	18.0733	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$229$	18.0733	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$230$	−4.06135	−0.267797
$231$	16.8284	1.10723
$232$	−15.5913	−1.02362
$233$	−9.51914	−0.623619	−0.311810	−	0.950145i	$-0.600935\pi$
$233$	−9.51914	−0.623619	−0.311810	+	0.950145i	$0.600935\pi$
$234$	−0.0187984	−0.00122889
$235$	−15.2024	−0.991698
$236$	12.8294	0.835121
$237$	−7.12852	−0.463047
$238$	1.88067	0.121906
$239$	−1.41813	−0.0917309	−0.0458654	−	0.998948i	$-0.514605\pi$
$239$	−1.41813	−0.0917309	−0.0458654	+	0.998948i	$0.514605\pi$
$240$	4.82418	0.311400
$241$	−19.7371	−1.27138	−0.635688	−	0.771946i	$-0.719283\pi$
$241$	−19.7371	−1.27138	−0.635688	+	0.771946i	$0.719283\pi$
$242$	0.433938	0.0278946
$243$	−1.38205	−0.0886586
$244$	−15.6181	−0.999847
$245$	0.878673	0.0561363
$246$	1.88343	0.120083
$247$	−0.597171	−0.0379971
$248$	0.982447	0.0623854
$249$	16.9939	1.07695
$250$	−6.12794	−0.387565
$251$	−8.60203	−0.542955	−0.271478	−	0.962445i	$-0.587512\pi$
$251$	−8.60203	−0.542955	−0.271478	+	0.962445i	$0.587512\pi$
$252$	−0.616668	−0.0388464
$253$	−19.0212	−1.19585
$254$	1.99177	0.124975
$255$	2.67256	0.167362
$256$	−4.30681	−0.269175
$257$	20.5073	1.27921	0.639604	−	0.768705i	$-0.279098\pi$
$257$	20.5073	1.27921	0.639604	+	0.768705i	$0.279098\pi$
$258$	−8.00804	−0.498559
$259$	1.45512	0.0904168
$260$	0.525467	0.0325881
$261$	0.985591	0.0610065
$262$	5.45638	0.337096
$263$	−0.314295	−0.0193803	−0.00969013	−	0.999953i	$-0.503085\pi$
$263$	−0.314295	−0.0193803	−0.00969013	+	0.999953i	$0.503085\pi$
$264$	−12.7774	−0.786396
$265$	−10.4781	−0.643663
$266$	3.85524	0.236380
$267$	15.3378	0.938656
$268$	−20.2878	−1.23927
$269$	20.5769	1.25459	0.627296	−	0.778781i	$-0.284161\pi$
$269$	20.5769	1.25459	0.627296	+	0.778781i	$0.284161\pi$
$270$	−3.71515	−0.226097
$271$	8.32793	0.505886	0.252943	−	0.967481i	$-0.418602\pi$
$271$	8.32793	0.505886	0.252943	+	0.967481i	$0.418602\pi$
$272$	2.52499	0.153100
$273$	−1.20887	−0.0731639
$274$	10.5130	0.635113
$275$	−11.5559	−0.696850
$276$	16.4093	0.987723
$277$	19.2658	1.15757	0.578784	−	0.815481i	$-0.303527\pi$
$277$	19.2658	1.15757	0.578784	+	0.815481i	$0.303527\pi$
$278$	9.86053	0.591395
$279$	−0.0621043	−0.00371809
$280$	−7.45228	−0.445359
$281$	11.0940	0.661811	0.330905	−	0.943664i	$-0.392646\pi$
$281$	11.0940	0.661811	0.330905	+	0.943664i	$0.392646\pi$
$282$	−12.0880	−0.719833
$283$	−4.41826	−0.262638	−0.131319	−	0.991340i	$-0.541921\pi$
$283$	−4.41826	−0.262638	−0.131319	+	0.991340i	$0.541921\pi$
$284$	2.94092	0.174511
$285$	5.47856	0.324522
$286$	−0.484324	−0.0286387
$287$	5.14472	0.303683
$288$	0.723306	0.0426212
$289$	−15.6012	−0.917716
$290$	5.42181	0.318379
$291$	−32.2545	−1.89079
$292$	20.4430	1.19634
$293$	−13.7647	−0.804143	−0.402072	−	0.915608i	$-0.631710\pi$
$293$	−13.7647	−0.804143	−0.402072	+	0.915608i	$0.631710\pi$
$294$	0.698667	0.0407471
$295$	−9.80067	−0.570617
$296$	−1.10484	−0.0642175
$297$	−17.3998	−1.00964
$298$	−2.89029	−0.167430
$299$	1.36638	0.0790200
$300$	9.96915	0.575569
$301$	−21.8745	−1.26083
$302$	−1.25866	−0.0724280
$303$	5.12040	0.294159
$304$	5.17606	0.296868
$305$	11.9311	0.683170
$306$	0.0902664	0.00516019
$307$	−14.0406	−0.801337	−0.400669	−	0.916223i	$-0.631222\pi$
$307$	−14.0406	−0.801337	−0.400669	+	0.916223i	$0.631222\pi$
$308$	−15.8879	−0.905296
$309$	3.23103	0.183807
$310$	−0.341640	−0.0194039
$311$	26.9644	1.52901	0.764505	−	0.644618i	$-0.222984\pi$
$311$	26.9644	1.52901	0.764505	+	0.644618i	$0.222984\pi$
$312$	0.917865	0.0519639
$313$	−23.9253	−1.35234	−0.676170	−	0.736746i	$-0.736361\pi$
$313$	−23.9253	−1.35234	−0.676170	+	0.736746i	$0.736361\pi$
$314$	−0.802576	−0.0452920
$315$	0.471088	0.0265428
$316$	6.73010	0.378598
$317$	−6.91444	−0.388354	−0.194177	−	0.980967i	$-0.562204\pi$
$317$	−6.91444	−0.388354	−0.194177	+	0.980967i	$0.562204\pi$
$318$	−8.33151	−0.467208
$319$	25.3928	1.42172
$320$	−1.47194	−0.0822837
$321$	−6.71994	−0.375070
$322$	−8.82116	−0.491584
$323$	2.86750	0.159552
$324$	15.6777	0.870985
$325$	0.830120	0.0460468
$326$	−9.60009	−0.531700
$327$	−15.6124	−0.863369
$328$	−3.90627	−0.215687
$329$	−33.0194	−1.82042
$330$	4.44328	0.244595
$331$	−16.8176	−0.924376	−0.462188	−	0.886782i	$-0.652936\pi$
$331$	−16.8176	−0.924376	−0.462188	+	0.886782i	$0.652936\pi$
$332$	−16.0441	−0.880536
$333$	0.0698413	0.00382728
$334$	12.0036	0.656811
$335$	15.4983	0.846765
$336$	10.4780	0.571623
$337$	0.0242532	0.00132115	0.000660577	−	1.00000i	$-0.499790\pi$
$337$	0.0242532	0.00132115	0.000660577		1.00000i	$0.499790\pi$
$338$	−7.42041	−0.403617
$339$	−1.93293	−0.104982
$340$	−2.52319	−0.136839
$341$	−1.60006	−0.0866481
$342$	0.185040	0.0100058
$343$	−17.5010	−0.944964
$344$	16.6088	0.895489
$345$	−12.5355	−0.674887
$346$	9.12209	0.490406
$347$	−18.7892	−1.00866	−0.504328	−	0.863512i	$-0.668260\pi$
$347$	−18.7892	−1.00866	−0.504328	+	0.863512i	$0.668260\pi$
$348$	−21.9060	−1.17429
$349$	−17.2684	−0.924355	−0.462177	−	0.886788i	$-0.652932\pi$
$349$	−17.2684	−0.924355	−0.462177	+	0.886788i	$0.652932\pi$
$350$	−5.35913	−0.286457
$351$	1.24991	0.0667152
$352$	18.6353	0.993264
$353$	29.9434	1.59372	0.796862	−	0.604161i	$-0.206492\pi$
$353$	29.9434	1.59372	0.796862	+	0.604161i	$0.206492\pi$
$354$	−7.79289	−0.414188
$355$	−2.24664	−0.119239
$356$	−14.4805	−0.767467
$357$	5.80474	0.307219
$358$	6.61862	0.349805
$359$	2.59722	0.137076	0.0685380	−	0.997649i	$-0.478167\pi$
$359$	2.59722	0.137076	0.0685380	+	0.997649i	$0.478167\pi$
$360$	−0.357687	−0.0188517
$361$	−13.1218	−0.690622
$362$	1.02426	0.0538337
$363$	1.33936	0.0702982
$364$	1.14130	0.0598205
$365$	−15.6169	−0.817427
$366$	9.48684	0.495885
$367$	12.5717	0.656238	0.328119	−	0.944636i	$-0.393585\pi$
$367$	12.5717	0.656238	0.328119	+	0.944636i	$0.393585\pi$
$368$	−11.8433	−0.617375
$369$	0.246930	0.0128547
$370$	0.384202	0.0199737
$371$	−22.7581	−1.18154
$372$	1.38035	0.0715677
$373$	3.92511	0.203234	0.101617	−	0.994824i	$-0.467598\pi$
$373$	3.92511	0.203234	0.101617	+	0.994824i	$0.467598\pi$
$374$	2.32563	0.120255
$375$	−18.9140	−0.976717
$376$	25.0709	1.29293
$377$	−1.82409	−0.0939454
$378$	−8.06922	−0.415036
$379$	25.1999	1.29443	0.647216	−	0.762306i	$-0.275933\pi$
$379$	25.1999	1.29443	0.647216	+	0.762306i	$0.275933\pi$
$380$	−5.17237	−0.265337
$381$	6.14765	0.314954
$382$	7.35917	0.376528
$383$	−0.136582	−0.00697902	−0.00348951	−	0.999994i	$-0.501111\pi$
$383$	−0.136582	−0.00697902	−0.00348951	+	0.999994i	$0.501111\pi$
$384$	−20.4106	−1.04157
$385$	12.1371	0.618566
$386$	3.73069	0.189887
$387$	−1.04991	−0.0533700
$388$	30.4518	1.54595
$389$	−6.18694	−0.313690	−0.156845	−	0.987623i	$-0.550132\pi$
$389$	−6.18694	−0.313690	−0.156845	+	0.987623i	$0.550132\pi$
$390$	−0.319182	−0.0161624
$391$	−6.56111	−0.331809
$392$	−1.44905	−0.0731880
$393$	16.8413	0.849530
$394$	6.76050	0.340589
$395$	−5.14129	−0.258687
$396$	−0.762569	−0.0383205
$397$	−33.0741	−1.65994	−0.829971	−	0.557806i	$-0.811643\pi$
$397$	−33.0741	−1.65994	−0.829971	+	0.557806i	$0.811643\pi$
$398$	4.15663	0.208353
$399$	11.8993	0.595711
$400$	−7.19518	−0.359759
$401$	−24.8422	−1.24056	−0.620279	−	0.784381i	$-0.712981\pi$
$401$	−24.8422	−1.24056	−0.620279	+	0.784381i	$0.712981\pi$
$402$	12.3233	0.614632
$403$	0.114940	0.00572557
$404$	−4.83422	−0.240511
$405$	−11.9766	−0.595122
$406$	11.7760	0.584435
$407$	1.79940	0.0891927
$408$	−4.40741	−0.218199
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	1.35838	0.0670857
$411$	32.4486	1.60057
$412$	−3.05045	−0.150285
$413$	−21.2868	−1.04746
$414$	−0.423388	−0.0208084
$415$	12.2565	0.601648
$416$	−1.33866	−0.0656334
$417$	30.4348	1.49040
$418$	4.76738	0.233180
$419$	7.87576	0.384756	0.192378	−	0.981321i	$-0.438380\pi$
$419$	7.87576	0.384756	0.192378	+	0.981321i	$0.438380\pi$
$420$	−10.4705	−0.510910
$421$	16.6789	0.812880	0.406440	−	0.913677i	$-0.366770\pi$
$421$	16.6789	0.812880	0.406440	+	0.913677i	$0.366770\pi$
$422$	0.633719	0.0308489
$423$	−1.58483	−0.0770570
$424$	17.2797	0.839178
$425$	−3.98607	−0.193353
$426$	−1.78639	−0.0865508
$427$	25.9140	1.25407
$428$	6.34436	0.306666
$429$	−1.49488	−0.0721734
$430$	−5.77563	−0.278526
$431$	−10.3551	−0.498787	−0.249393	−	0.968402i	$-0.580231\pi$
$431$	−10.3551	−0.498787	−0.249393	+	0.968402i	$0.580231\pi$
$432$	−10.8338	−0.521240
$433$	−32.6854	−1.57076	−0.785379	−	0.619015i	$-0.787532\pi$
$433$	−32.6854	−1.57076	−0.785379	+	0.619015i	$0.787532\pi$
$434$	−0.742035	−0.0356188
$435$	16.7345	0.802360
$436$	14.7399	0.705911
$437$	−13.4498	−0.643392
$438$	−12.4176	−0.593336
$439$	−21.6730	−1.03440	−0.517198	−	0.855866i	$-0.673025\pi$
$439$	−21.6730	−1.03440	−0.517198	+	0.855866i	$0.673025\pi$
$440$	−9.21545	−0.439330
$441$	0.0916001	0.00436191
$442$	−0.167061	−0.00794629
$443$	38.9039	1.84838	0.924191	−	0.381931i	$-0.124741\pi$
$443$	38.9039	1.84838	0.924191	+	0.381931i	$0.124741\pi$
$444$	−1.55231	−0.0736695
$445$	11.0620	0.524391
$446$	4.05281	0.191906
$447$	−8.92097	−0.421947
$448$	−3.19701	−0.151045
$449$	4.26934	0.201483	0.100741	−	0.994913i	$-0.467879\pi$
$449$	4.26934	0.201483	0.100741	+	0.994913i	$0.467879\pi$
$450$	−0.257222	−0.0121255
$451$	6.36193	0.299572
$452$	1.82490	0.0858361
$453$	−3.88490	−0.182529
$454$	9.73426	0.456851
$455$	−0.871870	−0.0408739
$456$	−9.03488	−0.423097
$457$	−39.6438	−1.85446	−0.927230	−	0.374493i	$-0.877817\pi$
$457$	−39.6438	−1.85446	−0.927230	+	0.374493i	$0.877817\pi$
$458$	10.3646	0.484307
$459$	−6.00182	−0.280141
$460$	11.8349	0.551803
$461$	−11.8114	−0.550111	−0.275055	−	0.961428i	$-0.588696\pi$
$461$	−11.8114	−0.550111	−0.275055	+	0.961428i	$0.588696\pi$
$462$	9.65071	0.448992
$463$	−29.1995	−1.35702	−0.678509	−	0.734592i	$-0.737373\pi$
$463$	−29.1995	−1.35702	−0.678509	+	0.734592i	$0.737373\pi$
$464$	15.8105	0.733986
$465$	−1.05448	−0.0489004
$466$	−5.45901	−0.252884
$467$	−28.2659	−1.30799	−0.653995	−	0.756499i	$-0.726908\pi$
$467$	−28.2659	−1.30799	−0.653995	+	0.756499i	$0.726908\pi$
$468$	0.0547790	0.00253216
$469$	33.6621	1.55437
$470$	−8.71825	−0.402143
$471$	−2.47717	−0.114142
$472$	16.1626	0.743945
$473$	−27.0500	−1.24376
$474$	−4.08804	−0.187770
$475$	−8.17117	−0.374919
$476$	−5.48031	−0.251190
$477$	−1.09232	−0.0500139
$478$	−0.813262	−0.0371977
$479$	−22.6623	−1.03547	−0.517733	−	0.855542i	$-0.673224\pi$
$479$	−22.6623	−1.03547	−0.517733	+	0.855542i	$0.673224\pi$
$480$	12.2812	0.560556
$481$	−0.129259	−0.00589372
$482$	−11.3187	−0.515555
$483$	−27.2267	−1.23886
$484$	−1.26450	−0.0574774
$485$	−23.2629	−1.05631
$486$	−0.792574	−0.0359519
$487$	30.0041	1.35961	0.679807	−	0.733391i	$-0.262063\pi$
$487$	30.0041	1.35961	0.679807	+	0.733391i	$0.262063\pi$
$488$	−19.6759	−0.890686
$489$	−29.6309	−1.33996
$490$	0.503899	0.0227638
$491$	16.9366	0.764339	0.382169	−	0.924092i	$-0.375177\pi$
$491$	16.9366	0.764339	0.382169	+	0.924092i	$0.375177\pi$
$492$	−5.48834	−0.247434
$493$	8.75892	0.394482
$494$	−0.342464	−0.0154082
$495$	0.582545	0.0261835
$496$	−0.996259	−0.0447333
$497$	−4.87965	−0.218882
$498$	9.74562	0.436712
$499$	−5.75380	−0.257575	−0.128788	−	0.991672i	$-0.541109\pi$
$499$	−5.75380	−0.257575	−0.128788	+	0.991672i	$0.541109\pi$
$500$	17.8569	0.798586
$501$	37.0496	1.65525
$502$	−4.93307	−0.220174
$503$	6.56987	0.292936	0.146468	−	0.989215i	$-0.453209\pi$
$503$	6.56987	0.292936	0.146468	+	0.989215i	$0.453209\pi$
$504$	−0.776887	−0.0346053
$505$	3.69298	0.164335
$506$	−10.9082	−0.484929
$507$	−22.9033	−1.01717
$508$	−5.80406	−0.257513
$509$	−9.82750	−0.435596	−0.217798	−	0.975994i	$-0.569887\pi$
$509$	−9.82750	−0.435596	−0.217798	+	0.975994i	$0.569887\pi$
$510$	1.53265	0.0678670
$511$	−33.9196	−1.50051
$512$	20.5923	0.910060
$513$	−12.3033	−0.543205
$514$	11.7604	0.518731
$515$	2.33031	0.102686
$516$	23.3356	1.02729
$517$	−40.8316	−1.79577
$518$	0.834478	0.0366649
$519$	28.1556	1.23589
$520$	0.661991	0.0290302
$521$	16.7766	0.734995	0.367497	−	0.930025i	$-0.380215\pi$
$521$	16.7766	0.734995	0.367497	+	0.930025i	$0.380215\pi$
$522$	0.565214	0.0247387
$523$	−13.1326	−0.574251	−0.287125	−	0.957893i	$-0.592700\pi$
$523$	−13.1326	−0.574251	−0.287125	+	0.957893i	$0.592700\pi$
$524$	−15.9000	−0.694595
$525$	−16.5411	−0.721912
$526$	−0.180241	−0.00785888
$527$	−0.551920	−0.0240420
$528$	12.9571	0.563884
$529$	7.77442	0.338018
$530$	−6.00893	−0.261011
$531$	−1.02170	−0.0443381
$532$	−11.2343	−0.487067
$533$	−0.457008	−0.0197952
$534$	8.79586	0.380634
$535$	−4.84661	−0.209537
$536$	−25.5588	−1.10397
$537$	20.4285	0.881557
$538$	11.8004	0.508749
$539$	2.35999	0.101652
$540$	10.8260	0.465878
$541$	40.3606	1.73524	0.867619	−	0.497229i	$-0.165649\pi$
$541$	40.3606	1.73524	0.867619	+	0.497229i	$0.165649\pi$
$542$	4.77588	0.205142
$543$	3.16139	0.135668
$544$	6.42800	0.275598
$545$	−11.2601	−0.482331
$546$	−0.693257	−0.0296687
$547$	7.92586	0.338885	0.169443	−	0.985540i	$-0.445803\pi$
$547$	7.92586	0.338885	0.169443	+	0.985540i	$0.445803\pi$
$548$	−30.6351	−1.30867
$549$	1.24379	0.0530837
$550$	−6.62707	−0.282579
$551$	17.9552	0.764916
$552$	20.6726	0.879886
$553$	−11.1668	−0.474860
$554$	11.0485	0.469405
$555$	1.18585	0.0503365
$556$	−28.7338	−1.21858
$557$	−22.4202	−0.949973	−0.474986	−	0.879993i	$-0.657547\pi$
$557$	−22.4202	−0.949973	−0.474986	+	0.879993i	$0.657547\pi$
$558$	−0.0356154	−0.00150772
$559$	1.94313	0.0821856
$560$	7.55705	0.319344
$561$	7.17812	0.303060
$562$	6.36214	0.268371
$563$	12.6061	0.531285	0.265643	−	0.964072i	$-0.414416\pi$
$563$	12.6061	0.531285	0.265643	+	0.964072i	$0.414416\pi$
$564$	35.2248	1.48323
$565$	−1.39409	−0.0586497
$566$	−2.53377	−0.106502
$567$	−26.0129	−1.09244
$568$	3.70501	0.155459
$569$	−43.0625	−1.80528	−0.902638	−	0.430401i	$-0.858372\pi$
$569$	−43.0625	−1.80528	−0.902638	+	0.430401i	$0.858372\pi$
$570$	3.14183	0.131597
$571$	31.9454	1.33687	0.668436	−	0.743769i	$-0.266964\pi$
$571$	31.9454	1.33687	0.668436	+	0.743769i	$0.266964\pi$
$572$	1.41133	0.0590107
$573$	22.7143	0.948902
$574$	2.95038	0.123146
$575$	18.6964	0.779694
$576$	−0.153447	−0.00639362
$577$	−19.9680	−0.831279	−0.415639	−	0.909529i	$-0.636442\pi$
$577$	−19.9680	−0.831279	−0.415639	+	0.909529i	$0.636442\pi$
$578$	−8.94691	−0.372143
$579$	11.5149	0.478542
$580$	−15.7993	−0.656028
$581$	26.6209	1.10442
$582$	−18.4972	−0.766733
$583$	−28.1426	−1.16555
$584$	25.7544	1.06572
$585$	−0.0418471	−0.00173016
$586$	−7.89375	−0.326088
$587$	−15.3253	−0.632542	−0.316271	−	0.948669i	$-0.602431\pi$
$587$	−15.3253	−0.632542	−0.316271	+	0.948669i	$0.602431\pi$
$588$	−2.03593	−0.0839603
$589$	−1.13140	−0.0466184
$590$	−5.62046	−0.231391
$591$	20.8665	0.858331
$592$	1.12037	0.0460470
$593$	−3.23571	−0.132875	−0.0664374	−	0.997791i	$-0.521163\pi$
$593$	−3.23571	−0.132875	−0.0664374	+	0.997791i	$0.521163\pi$
$594$	−9.97836	−0.409417
$595$	4.18655	0.171632
$596$	8.42238	0.344994
$597$	12.8296	0.525079
$598$	0.783589	0.0320433
$599$	6.87340	0.280839	0.140420	−	0.990092i	$-0.455155\pi$
$599$	6.87340	0.280839	0.140420	+	0.990092i	$0.455155\pi$
$600$	12.5593	0.512730
$601$	30.6423	1.24992	0.624962	−	0.780655i	$-0.285115\pi$
$601$	30.6423	1.24992	0.624962	+	0.780655i	$0.285115\pi$
$602$	−12.5445	−0.511278
$603$	1.61568	0.0657954
$604$	3.66777	0.149240
$605$	0.965985	0.0392729
$606$	2.93643	0.119284
$607$	38.4923	1.56236	0.781178	−	0.624308i	$-0.214619\pi$
$607$	38.4923	1.56236	0.781178	+	0.624308i	$0.214619\pi$
$608$	13.1770	0.534396
$609$	36.3471	1.47286
$610$	6.84219	0.277032
$611$	2.93313	0.118662
$612$	−0.263038	−0.0106327
$613$	31.6313	1.27758	0.638789	−	0.769382i	$-0.279436\pi$
$613$	31.6313	1.27758	0.638789	+	0.769382i	$0.279436\pi$
$614$	−8.05194	−0.324950
$615$	4.19268	0.169065
$616$	−20.0158	−0.806458
$617$	−14.1816	−0.570930	−0.285465	−	0.958389i	$-0.592148\pi$
$617$	−14.1816	−0.570930	−0.285465	+	0.958389i	$0.592148\pi$
$618$	1.85292	0.0745354
$619$	−1.21737	−0.0489303	−0.0244652	−	0.999701i	$-0.507788\pi$
$619$	−1.21737	−0.0489303	−0.0244652	+	0.999701i	$0.507788\pi$
$620$	0.995547	0.0399821
$621$	28.1511	1.12967
$622$	15.4635	0.620028
$623$	24.0265	0.962602
$624$	−0.930769	−0.0372606
$625$	3.20993	0.128397
$626$	−13.7206	−0.548386
$627$	14.7146	0.587646
$628$	2.33872	0.0933252
$629$	0.620678	0.0247481
$630$	0.270158	0.0107634
$631$	46.0954	1.83503	0.917515	−	0.397700i	$-0.130192\pi$
$631$	46.0954	1.83503	0.917515	+	0.397700i	$0.130192\pi$
$632$	8.47868	0.337264
$633$	1.95599	0.0777436
$634$	−3.96527	−0.157481
$635$	4.43386	0.175953
$636$	24.2782	0.962693
$637$	−0.169530	−0.00671700
$638$	14.5622	0.576523
$639$	−0.234208	−0.00926513
$640$	−14.7207	−0.581888
$641$	16.1409	0.637526	0.318763	−	0.947834i	$-0.396733\pi$
$641$	16.1409	0.637526	0.318763	+	0.947834i	$0.396733\pi$
$642$	−3.85373	−0.152095
$643$	25.2497	0.995750	0.497875	−	0.867249i	$-0.334114\pi$
$643$	25.2497	0.995750	0.497875	+	0.867249i	$0.334114\pi$
$644$	25.7050	1.01292
$645$	−17.8266	−0.701924
$646$	1.64444	0.0646998
$647$	3.27943	0.128928	0.0644638	−	0.997920i	$-0.479466\pi$
$647$	3.27943	0.128928	0.0644638	+	0.997920i	$0.479466\pi$
$648$	19.7510	0.775893
$649$	−26.3232	−1.03328
$650$	0.476055	0.0186724
$651$	−2.29031	−0.0897644
$652$	27.9749	1.09558
$653$	23.6623	0.925977	0.462989	−	0.886364i	$-0.346777\pi$
$653$	23.6623	0.925977	0.462989	+	0.886364i	$0.346777\pi$
$654$	−8.95337	−0.350105
$655$	12.1464	0.474600
$656$	3.96118	0.154658
$657$	−1.62804	−0.0635157
$658$	−18.9359	−0.738196
$659$	−8.20730	−0.319711	−0.159855	−	0.987140i	$-0.551103\pi$
$659$	−8.20730	−0.319711	−0.159855	+	0.987140i	$0.551103\pi$
$660$	−12.9478	−0.503993
$661$	25.4872	0.991336	0.495668	−	0.868512i	$-0.334923\pi$
$661$	25.4872	0.991336	0.495668	+	0.868512i	$0.334923\pi$
$662$	−9.64448	−0.374843
$663$	−0.515639	−0.0200258
$664$	−20.2126	−0.784401
$665$	8.58213	0.332801
$666$	0.0400524	0.00155200
$667$	−41.0831	−1.59075
$668$	−34.9789	−1.35337
$669$	12.5091	0.483630
$670$	8.88795	0.343371
$671$	32.0451	1.23709
$672$	26.6744	1.02899
$673$	−7.88779	−0.304052	−0.152026	−	0.988376i	$-0.548580\pi$
$673$	−7.88779	−0.304052	−0.152026	+	0.988376i	$0.548580\pi$
$674$	0.0139086	0.000535741 0
$675$	17.1027	0.658282
$676$	21.6232	0.831663
$677$	−31.8220	−1.22302	−0.611510	−	0.791237i	$-0.709437\pi$
$677$	−31.8220	−1.22302	−0.611510	+	0.791237i	$0.709437\pi$
$678$	−1.10849	−0.0425714
$679$	−50.5264	−1.93903
$680$	−3.17875	−0.121900
$681$	30.0450	1.15133
$682$	−0.917598	−0.0351366
$683$	−32.9917	−1.26239	−0.631196	−	0.775623i	$-0.717436\pi$
$683$	−32.9917	−1.26239	−0.631196	+	0.775623i	$0.717436\pi$
$684$	−0.539210	−0.0206172
$685$	23.4029	0.894179
$686$	−10.0364	−0.383192
$687$	31.9907	1.22052
$688$	−16.8423	−0.642108
$689$	2.02162	0.0770176
$690$	−7.18880	−0.273673
$691$	33.4869	1.27390	0.636952	−	0.770904i	$-0.280195\pi$
$691$	33.4869	1.27390	0.636952	+	0.770904i	$0.280195\pi$
$692$	−26.5820	−1.01049
$693$	1.26528	0.0480638
$694$	−10.7752	−0.409020
$695$	21.9505	0.832629
$696$	−27.5975	−1.04608
$697$	2.19447	0.0831213
$698$	−9.90301	−0.374835
$699$	−16.8494	−0.637302
$700$	15.6166	0.590252
$701$	−3.20496	−0.121050	−0.0605249	−	0.998167i	$-0.519277\pi$
$701$	−3.20496	−0.121050	−0.0605249	+	0.998167i	$0.519277\pi$
$702$	0.716794	0.0270537
$703$	1.27235	0.0479875
$704$	−3.95341	−0.149000
$705$	−26.9091	−1.01346
$706$	17.1718	0.646271
$707$	8.02107	0.301663
$708$	22.7087	0.853443
$709$	−7.44076	−0.279444	−0.139722	−	0.990191i	$-0.544621\pi$
$709$	−7.44076	−0.279444	−0.139722	+	0.990191i	$0.544621\pi$
$710$	−1.28840	−0.0483526
$711$	−0.535971	−0.0201005
$712$	−18.2428	−0.683677
$713$	2.58874	0.0969492
$714$	3.32889	0.124580
$715$	−1.07815	−0.0403205
$716$	−19.2868	−0.720781
$717$	−2.51016	−0.0937435
$718$	1.48945	0.0555856
$719$	6.56216	0.244727	0.122364	−	0.992485i	$-0.460953\pi$
$719$	6.56216	0.244727	0.122364	+	0.992485i	$0.460953\pi$
$720$	0.362715	0.0135176
$721$	5.06139	0.188496
$722$	−7.52507	−0.280054
$723$	−34.9356	−1.29927
$724$	−2.98470	−0.110926
$725$	−24.9593	−0.926964
$726$	0.768092	0.0285066
$727$	11.4056	0.423009	0.211505	−	0.977377i	$-0.432164\pi$
$727$	11.4056	0.423009	0.211505	+	0.977377i	$0.432164\pi$
$728$	1.43783	0.0532895
$729$	25.6983	0.951790
$730$	−8.95594	−0.331474
$731$	−9.33053	−0.345102
$732$	−27.6449	−1.02178
$733$	−16.4234	−0.606610	−0.303305	−	0.952893i	$-0.598090\pi$
$733$	−16.4234	−0.606610	−0.303305	+	0.952893i	$0.598090\pi$
$734$	7.20959	0.266111
$735$	1.55530	0.0573680
$736$	−30.1501	−1.11135
$737$	41.6264	1.53333
$738$	0.141609	0.00521270
$739$	35.0610	1.28974	0.644870	−	0.764292i	$-0.276911\pi$
$739$	35.0610	1.28974	0.644870	+	0.764292i	$0.276911\pi$
$740$	−1.11957	−0.0411563
$741$	−1.05702	−0.0388307
$742$	−13.0513	−0.479127
$743$	52.7988	1.93700	0.968501	−	0.249011i	$-0.0801054\pi$
$743$	52.7988	1.93700	0.968501	+	0.249011i	$0.0801054\pi$
$744$	1.73898	0.0637542
$745$	−6.43406	−0.235726
$746$	2.25096	0.0824135
$747$	1.27772	0.0467493
$748$	−6.77693	−0.247789
$749$	−10.5267	−0.384639
$750$	−10.8468	−0.396068
$751$	−38.2376	−1.39531	−0.697655	−	0.716434i	$-0.745773\pi$
$751$	−38.2376	−1.39531	−0.697655	+	0.716434i	$0.745773\pi$
$752$	−25.4233	−0.927094
$753$	−15.2260	−0.554868
$754$	−1.04607	−0.0380957
$755$	−2.80190	−0.101972
$756$	23.5139	0.855192
$757$	−4.62426	−0.168072	−0.0840358	−	0.996463i	$-0.526781\pi$
$757$	−4.62426	−0.168072	−0.0840358	+	0.996463i	$0.526781\pi$
$758$	14.4516	0.524905
$759$	−33.6685	−1.22209
$760$	−6.51622	−0.236368
$761$	−8.86933	−0.321513	−0.160756	−	0.986994i	$-0.551393\pi$
$761$	−8.86933	−0.321513	−0.160756	+	0.986994i	$0.551393\pi$
$762$	3.52554	0.127717
$763$	−24.4568	−0.885395
$764$	−21.4448	−0.775844
$765$	0.200941	0.00726505
$766$	−0.0783267	−0.00283006
$767$	1.89092	0.0682773
$768$	−7.62327	−0.275081
$769$	−44.9504	−1.62095	−0.810477	−	0.585770i	$-0.800792\pi$
$769$	−44.9504	−1.62095	−0.810477	+	0.585770i	$0.800792\pi$
$770$	6.96037	0.250834
$771$	36.2989	1.30727
$772$	−10.8713	−0.391267
$773$	−11.9734	−0.430653	−0.215327	−	0.976542i	$-0.569082\pi$
$773$	−11.9734	−0.430653	−0.215327	+	0.976542i	$0.569082\pi$
$774$	−0.602100	−0.0216420
$775$	1.57274	0.0564945
$776$	38.3636	1.37717
$777$	2.57564	0.0924005
$778$	−3.54807	−0.127204
$779$	4.49850	0.161176
$780$	0.930104	0.0333031
$781$	−6.03415	−0.215919
$782$	−3.76264	−0.134552
$783$	−37.5811	−1.34304
$784$	1.46942	0.0524793
$785$	−1.78661	−0.0637668
$786$	9.65808	0.344492
$787$	−8.37109	−0.298397	−0.149199	−	0.988807i	$-0.547669\pi$
$787$	−8.37109	−0.298397	−0.149199	+	0.988807i	$0.547669\pi$
$788$	−19.7002	−0.701792
$789$	−0.556318	−0.0198055
$790$	−2.94841	−0.104900
$791$	−3.02793	−0.107661
$792$	−0.960695	−0.0341368
$793$	−2.30196	−0.0817449
$794$	−18.9673	−0.673123
$795$	−18.5467	−0.657785
$796$	−12.1125	−0.429317
$797$	33.6698	1.19265	0.596323	−	0.802745i	$-0.296628\pi$
$797$	33.6698	1.19265	0.596323	+	0.802745i	$0.296628\pi$
$798$	6.82398	0.241566
$799$	−14.0843	−0.498268
$800$	−18.3171	−0.647608
$801$	1.15320	0.0407463
$802$	−14.2464	−0.503058
$803$	−41.9448	−1.48020
$804$	−35.9105	−1.26646
$805$	−19.6367	−0.692104
$806$	0.0659155	0.00232177
$807$	36.4221	1.28212
$808$	−6.09021	−0.214253
$809$	29.2697	1.02907	0.514534	−	0.857470i	$-0.327965\pi$
$809$	29.2697	1.02907	0.514534	+	0.857470i	$0.327965\pi$
$810$	−6.86831	−0.241328
$811$	−3.54465	−0.124469	−0.0622347	−	0.998062i	$-0.519823\pi$
$811$	−3.54465	−0.124469	−0.0622347	+	0.998062i	$0.519823\pi$
$812$	−34.3156	−1.20424
$813$	14.7409	0.516985
$814$	1.03191	0.0361685
$815$	−21.3707	−0.748583
$816$	4.46937	0.156459
$817$	−19.1269	−0.669167
$818$	0.573477	0.0200512
$819$	−0.0908909	−0.00317598
$820$	−3.95835	−0.138232
$821$	31.0554	1.08384	0.541921	−	0.840429i	$-0.317697\pi$
$821$	31.0554	1.08384	0.541921	+	0.840429i	$0.317697\pi$
$822$	18.6086	0.649048
$823$	43.3865	1.51236	0.756179	−	0.654365i	$-0.227064\pi$
$823$	43.3865	1.51236	0.756179	+	0.654365i	$0.227064\pi$
$824$	−3.84300	−0.133877
$825$	−20.4546	−0.712139
$826$	−12.2075	−0.424754
$827$	33.7383	1.17319	0.586597	−	0.809879i	$-0.300467\pi$
$827$	33.7383	1.17319	0.586597	+	0.809879i	$0.300467\pi$
$828$	1.23376	0.0428762
$829$	−7.93092	−0.275452	−0.137726	−	0.990470i	$-0.543979\pi$
$829$	−7.93092	−0.275452	−0.137726	+	0.990470i	$0.543979\pi$
$830$	7.02882	0.243974
$831$	34.1014	1.18297
$832$	0.283993	0.00984568
$833$	0.814048	0.0282051
$834$	17.4537	0.604371
$835$	26.7212	0.924727
$836$	−13.8922	−0.480473
$837$	2.36807	0.0818525
$838$	4.51657	0.156022
$839$	−0.829060	−0.0286223	−0.0143112	−	0.999898i	$-0.504556\pi$
$839$	−0.829060	−0.0286223	−0.0143112	+	0.999898i	$0.504556\pi$
$840$	−13.1909	−0.455130
$841$	25.8450	0.891207
$842$	9.56497	0.329631
$843$	19.6369	0.676331
$844$	−1.84667	−0.0635650
$845$	−16.5185	−0.568255
$846$	−0.908863	−0.0312473
$847$	2.09810	0.0720915
$848$	−17.5227	−0.601731
$849$	−7.82055	−0.268401
$850$	−2.28592	−0.0784065
$851$	−2.91125	−0.0997963
$852$	5.20557	0.178340
$853$	48.1049	1.64708	0.823541	−	0.567257i	$-0.191995\pi$
$853$	48.1049	1.64708	0.823541	+	0.567257i	$0.191995\pi$
$854$	14.8611	0.508536
$855$	0.411916	0.0140872
$856$	7.99271	0.273185
$857$	−27.2391	−0.930470	−0.465235	−	0.885187i	$-0.654030\pi$
$857$	−27.2391	−0.930470	−0.465235	+	0.885187i	$0.654030\pi$
$858$	−0.857279	−0.0292670
$859$	−38.6029	−1.31711	−0.658557	−	0.752531i	$-0.728833\pi$
$859$	−38.6029	−1.31711	−0.658557	+	0.752531i	$0.728833\pi$
$860$	16.8303	0.573909
$861$	9.10641	0.310346
$862$	−5.93840	−0.202263
$863$	51.0584	1.73805	0.869025	−	0.494768i	$-0.164747\pi$
$863$	51.0584	1.73805	0.869025	+	0.494768i	$0.164747\pi$
$864$	−27.5800	−0.938292
$865$	20.3066	0.690446
$866$	−18.7443	−0.636957
$867$	−27.6149	−0.937851
$868$	2.16231	0.0733935
$869$	−13.8088	−0.468431
$870$	9.59688	0.325365
$871$	−2.99022	−0.101320
$872$	18.5695	0.628842
$873$	−2.42511	−0.0820776
$874$	−7.71316	−0.260901
$875$	−29.6287	−1.00163
$876$	36.1852	1.22258
$877$	8.73784	0.295056	0.147528	−	0.989058i	$-0.452868\pi$
$877$	8.73784	0.295056	0.147528	+	0.989058i	$0.452868\pi$
$878$	−12.4290	−0.419457
$879$	−24.3643	−0.821786
$880$	9.34502	0.315020
$881$	−51.3824	−1.73112	−0.865559	−	0.500808i	$-0.833036\pi$
$881$	−51.3824	−1.73112	−0.865559	+	0.500808i	$0.833036\pi$
$882$	0.0525306	0.00176880
$883$	−41.8244	−1.40750	−0.703752	−	0.710446i	$-0.748493\pi$
$883$	−41.8244	−1.40750	−0.703752	+	0.710446i	$0.748493\pi$
$884$	0.486820	0.0163735
$885$	−17.3477	−0.583137
$886$	22.3105	0.749536
$887$	−35.4061	−1.18882	−0.594410	−	0.804162i	$-0.702614\pi$
$887$	−35.4061	−1.18882	−0.594410	+	0.804162i	$0.702614\pi$
$888$	−1.95562	−0.0656265
$889$	9.63025	0.322988
$890$	6.34383	0.212646
$891$	−32.1674	−1.07765
$892$	−11.8100	−0.395427
$893$	−28.8719	−0.966161
$894$	−5.11597	−0.171104
$895$	14.7337	0.492492
$896$	−31.9731	−1.06815
$897$	2.41857	0.0807537
$898$	2.44837	0.0817032
$899$	−3.45591	−0.115261
$900$	0.749549	0.0249850
$901$	−9.70742	−0.323401
$902$	3.64842	0.121479
$903$	−38.7191	−1.28849
$904$	2.29903	0.0764648
$905$	2.28009	0.0757927
$906$	−2.22790	−0.0740170
$907$	−57.7182	−1.91650	−0.958251	−	0.285927i	$-0.907699\pi$
$907$	−57.7182	−1.91650	−0.958251	+	0.285927i	$0.907699\pi$
$908$	−28.3658	−0.941353
$909$	0.384987	0.0127692
$910$	−0.499997	−0.0165747
$911$	16.1671	0.535640	0.267820	−	0.963469i	$-0.413697\pi$
$911$	16.1671	0.535640	0.267820	+	0.963469i	$0.413697\pi$
$912$	9.16190	0.303381
$913$	32.9192	1.08947
$914$	−22.7348	−0.752001
$915$	21.1186	0.698159
$916$	−30.2027	−0.997926
$917$	26.3817	0.871202
$918$	−3.44191	−0.113600
$919$	47.8672	1.57899	0.789497	−	0.613754i	$-0.210342\pi$
$919$	47.8672	1.57899	0.789497	+	0.613754i	$0.210342\pi$
$920$	14.9097	0.491559
$921$	−24.8525	−0.818919
$922$	−6.77356	−0.223075
$923$	0.433462	0.0142676
$924$	−28.1224	−0.925158
$925$	−1.76867	−0.0581536
$926$	−16.7453	−0.550284
$927$	0.242931	0.00797890
$928$	40.2497	1.32126
$929$	−15.0151	−0.492629	−0.246315	−	0.969190i	$-0.579220\pi$
$929$	−15.0151	−0.492629	−0.246315	+	0.969190i	$0.579220\pi$
$930$	−0.604721	−0.0198296
$931$	1.66874	0.0546908
$932$	15.9077	0.521073
$933$	47.7284	1.56256
$934$	−16.2099	−0.530403
$935$	5.17707	0.169308
$936$	0.0690114	0.00225571
$937$	49.8602	1.62886	0.814431	−	0.580261i	$-0.197049\pi$
$937$	49.8602	1.62886	0.814431	+	0.580261i	$0.197049\pi$
$938$	19.3044	0.630312
$939$	−42.3491	−1.38201
$940$	25.4052	0.828625
$941$	−31.2000	−1.01709	−0.508546	−	0.861035i	$-0.669817\pi$
$941$	−31.2000	−1.01709	−0.508546	+	0.861035i	$0.669817\pi$
$942$	−1.42060	−0.0462857
$943$	−10.2930	−0.335186
$944$	−16.3898	−0.533444
$945$	−17.9628	−0.584331
$946$	−15.5125	−0.504356
$947$	−55.5752	−1.80595	−0.902975	−	0.429693i	$-0.858622\pi$
$947$	−55.5752	−1.80595	−0.902975	+	0.429693i	$0.858622\pi$
$948$	11.9126	0.386904
$949$	3.01310	0.0978093
$950$	−4.68598	−0.152033
$951$	−12.2389	−0.396874
$952$	−6.90418	−0.223766
$953$	39.9827	1.29517	0.647584	−	0.761994i	$-0.275780\pi$
$953$	39.9827	1.29517	0.647584	+	0.761994i	$0.275780\pi$
$954$	−0.626420	−0.0202811
$955$	16.3822	0.530115
$956$	2.36986	0.0766469
$957$	44.9466	1.45292
$958$	−12.9963	−0.419891
$959$	50.8306	1.64141
$960$	−2.60540	−0.0840891
$961$	−30.7822	−0.992975
$962$	−0.0741272	−0.00238996
$963$	−0.505251	−0.0162815
$964$	32.9831	1.06231
$965$	8.30486	0.267343
$966$	−15.6139	−0.502369
$967$	−40.8834	−1.31472	−0.657361	−	0.753576i	$-0.728327\pi$
$967$	−40.8834	−1.31472	−0.657361	+	0.753576i	$0.728327\pi$
$968$	−1.59304	−0.0512022
$969$	5.07562	0.163052
$970$	−13.3407	−0.428345
$971$	−32.1188	−1.03074	−0.515371	−	0.856967i	$-0.672346\pi$
$971$	−32.1188	−1.03074	−0.515371	+	0.856967i	$0.672346\pi$
$972$	2.30958	0.0740798
$973$	47.6759	1.52842
$974$	17.2066	0.551337
$975$	1.46936	0.0470570
$976$	19.9525	0.638665
$977$	4.87838	0.156073	0.0780366	−	0.996950i	$-0.475135\pi$
$977$	4.87838	0.156073	0.0780366	+	0.996950i	$0.475135\pi$
$978$	−16.9927	−0.543366
$979$	29.7111	0.949570
$980$	−1.46837	−0.0469054
$981$	−1.17385	−0.0374781
$982$	9.71276	0.309947
$983$	−4.93945	−0.157544	−0.0787721	−	0.996893i	$-0.525100\pi$
$983$	−4.93945	−0.157544	−0.0787721	+	0.996893i	$0.525100\pi$
$984$	−6.91429	−0.220420
$985$	15.0495	0.479517
$986$	5.02304	0.159966
$987$	−58.4460	−1.86036
$988$	0.997947	0.0317489
$989$	43.7642	1.39162
$990$	0.334076	0.0106176
$991$	61.3974	1.95035	0.975177	−	0.221428i	$-0.0710719\pi$
$991$	61.3974	1.95035	0.975177	+	0.221428i	$0.0710719\pi$
$992$	−2.53622	−0.0805252
$993$	−29.7680	−0.944657
$994$	−2.79837	−0.0887588
$995$	9.25305	0.293341
$996$	−28.3989	−0.899855
$997$	1.41985	0.0449672	0.0224836	−	0.999747i	$-0.492843\pi$
$997$	1.41985	0.0449672	0.0224836	+	0.999747i	$0.492843\pi$
$998$	−3.29967	−0.104449
$999$	−2.66309	−0.0842563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	409.2.a.b.1.12	✓	20
3.2	odd	2		3681.2.a.i.1.9		20
4.3	odd	2		6544.2.a.i.1.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.12	✓	20	1.1	even	1	trivial
3681.2.a.i.1.9		20	3.2	odd	2
6544.2.a.i.1.7		20	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.573477 q^{2} +1.77005 q^{3} -1.67112 q^{4} +1.27661 q^{5} +1.01508 q^{6} +2.77278 q^{7} -2.10531 q^{8} +0.133085 q^{9} +0.732108 q^{10} +3.42880 q^{11} -2.95798 q^{12} -0.246307 q^{13} +1.59012 q^{14} +2.25967 q^{15} +2.13490 q^{16} +1.18272 q^{17} +0.0763210 q^{18} +2.42449 q^{19} -2.13338 q^{20} +4.90796 q^{21} +1.96634 q^{22} -5.54747 q^{23} -3.72650 q^{24} -3.37026 q^{25} -0.141252 q^{26} -5.07459 q^{27} -4.63365 q^{28} +7.40574 q^{29} +1.29587 q^{30} -0.466653 q^{31} +5.43493 q^{32} +6.06916 q^{33} +0.678263 q^{34} +3.53976 q^{35} -0.222401 q^{36} +0.524788 q^{37} +1.39039 q^{38} -0.435977 q^{39} -2.68766 q^{40} +1.85544 q^{41} +2.81460 q^{42} -7.88904 q^{43} -5.72995 q^{44} +0.169898 q^{45} -3.18135 q^{46} -11.9084 q^{47} +3.77889 q^{48} +0.688284 q^{49} -1.93277 q^{50} +2.09348 q^{51} +0.411610 q^{52} -8.20771 q^{53} -2.91016 q^{54} +4.37725 q^{55} -5.83754 q^{56} +4.29148 q^{57} +4.24702 q^{58} -7.67709 q^{59} -3.77619 q^{60} +9.34587 q^{61} -0.267615 q^{62} +0.369014 q^{63} -1.15300 q^{64} -0.314439 q^{65} +3.48052 q^{66} +12.1402 q^{67} -1.97647 q^{68} -9.81931 q^{69} +2.02997 q^{70} -1.75984 q^{71} -0.280184 q^{72} -12.2331 q^{73} +0.300954 q^{74} -5.96553 q^{75} -4.05163 q^{76} +9.50730 q^{77} -0.250023 q^{78} -4.02729 q^{79} +2.72545 q^{80} -9.38154 q^{81} +1.06405 q^{82} +9.60080 q^{83} -8.20181 q^{84} +1.50988 q^{85} -4.52418 q^{86} +13.1085 q^{87} -7.21867 q^{88} +8.66515 q^{89} +0.0974324 q^{90} -0.682955 q^{91} +9.27051 q^{92} -0.826000 q^{93} -6.82921 q^{94} +3.09514 q^{95} +9.62011 q^{96} -18.2223 q^{97} +0.394715 q^{98} +0.456321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} + q^{3} + 23 q^{4} + 8 q^{5} - 4 q^{6} + 5 q^{7} + 12 q^{8} + 27 q^{9} - 5 q^{10} + 30 q^{11} - 5 q^{12} - 7 q^{13} + 17 q^{14} + 26 q^{15} + 29 q^{16} + 6 q^{17} - 4 q^{18} + q^{19} + 14 q^{20}+ \cdots + 43 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 + q^3 + 23 * q^4 + 8 * q^5 - 4 * q^6 + 5 * q^7 + 12 * q^8 + 27 * q^9 - 5 * q^10 + 30 * q^11 - 5 * q^12 - 7 * q^13 + 17 * q^14 + 26 * q^15 + 29 * q^16 + 6 * q^17 - 4 * q^18 + q^19 + 14 * q^20 - 3 * q^21 + q^22 + 37 * q^23 - 17 * q^24 + 14 * q^25 + 11 * q^26 - 5 * q^27 - 6 * q^28 + 12 * q^29 - 6 * q^30 + 10 * q^31 + 24 * q^32 - 22 * q^33 - 11 * q^34 + 12 * q^35 + 21 * q^36 - 4 * q^37 - 7 * q^38 + 31 * q^39 - 34 * q^40 - 7 * q^41 - 35 * q^42 - q^43 + 58 * q^44 - 10 * q^45 - 22 * q^46 + 51 * q^47 - 58 * q^48 + 5 * q^49 + q^50 + 6 * q^51 - 44 * q^52 + 10 * q^53 - 49 * q^54 - 5 * q^55 + 31 * q^56 - 30 * q^57 - 22 * q^58 + 43 * q^59 - 26 * q^60 - 21 * q^61 - 26 * q^62 + 8 * q^63 - 4 * q^64 - 10 * q^65 - 37 * q^66 + 4 * q^67 - 16 * q^68 - 19 * q^69 - 66 * q^70 + 97 * q^71 - 43 * q^72 - 36 * q^73 + 17 * q^74 - 19 * q^75 - 43 * q^76 - 20 * q^77 - 71 * q^78 + 16 * q^79 - 23 * q^80 + 16 * q^81 - 67 * q^82 + 28 * q^83 - 73 * q^84 - 30 * q^85 + 37 * q^86 - 14 * q^87 - 48 * q^88 + 2 * q^89 - 121 * q^90 - 25 * q^91 + 14 * q^92 - 27 * q^93 - 60 * q^94 + 61 * q^95 - 88 * q^96 - 36 * q^97 - 54 * q^98 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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