LMFDB - Embedded newform 4002.2.a.bk.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.95569$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.95569 q^{5} -1.00000 q^{6} +4.57890 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.95569 q^{5} -1.00000 q^{6} +4.57890 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.95569 q^{10} -4.22869 q^{11} -1.00000 q^{12} +5.48809 q^{13} +4.57890 q^{14} -3.95569 q^{15} +1.00000 q^{16} -0.273005 q^{17} +1.00000 q^{18} -6.48051 q^{19} +3.95569 q^{20} -4.57890 q^{21} -4.22869 q^{22} -1.00000 q^{23} -1.00000 q^{24} +10.6475 q^{25} +5.48809 q^{26} -1.00000 q^{27} +4.57890 q^{28} -1.00000 q^{29} -3.95569 q^{30} +5.86484 q^{31} +1.00000 q^{32} +4.22869 q^{33} -0.273005 q^{34} +18.1127 q^{35} +1.00000 q^{36} +1.93070 q^{37} -6.48051 q^{38} -5.48809 q^{39} +3.95569 q^{40} +0.433159 q^{41} -4.57890 q^{42} +5.69859 q^{43} -4.22869 q^{44} +3.95569 q^{45} -1.00000 q^{46} -1.89626 q^{47} -1.00000 q^{48} +13.9663 q^{49} +10.6475 q^{50} +0.273005 q^{51} +5.48809 q^{52} -7.90073 q^{53} -1.00000 q^{54} -16.7274 q^{55} +4.57890 q^{56} +6.48051 q^{57} -1.00000 q^{58} -9.96770 q^{59} -3.95569 q^{60} +6.22869 q^{61} +5.86484 q^{62} +4.57890 q^{63} +1.00000 q^{64} +21.7092 q^{65} +4.22869 q^{66} -8.50885 q^{67} -0.273005 q^{68} +1.00000 q^{69} +18.1127 q^{70} -4.98632 q^{71} +1.00000 q^{72} +8.19425 q^{73} +1.93070 q^{74} -10.6475 q^{75} -6.48051 q^{76} -19.3628 q^{77} -5.48809 q^{78} -8.86257 q^{79} +3.95569 q^{80} +1.00000 q^{81} +0.433159 q^{82} +12.7535 q^{83} -4.57890 q^{84} -1.07992 q^{85} +5.69859 q^{86} +1.00000 q^{87} -4.22869 q^{88} -6.20661 q^{89} +3.95569 q^{90} +25.1294 q^{91} -1.00000 q^{92} -5.86484 q^{93} -1.89626 q^{94} -25.6349 q^{95} -1.00000 q^{96} -6.36154 q^{97} +13.9663 q^{98} -4.22869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.95569	1.76904	0.884519	−	0.466504i	$-0.154487\pi$
$5$	3.95569	1.76904	0.884519	+	0.466504i	$0.154487\pi$
$6$	−1.00000	−0.408248
$7$	4.57890	1.73066	0.865330	−	0.501202i	$-0.167109\pi$
$7$	4.57890	1.73066	0.865330	+	0.501202i	$0.167109\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.95569	1.25090
$11$	−4.22869	−1.27500	−0.637500	−	0.770451i	$-0.720031\pi$
$11$	−4.22869	−1.27500	−0.637500	+	0.770451i	$0.720031\pi$
$12$	−1.00000	−0.288675
$13$	5.48809	1.52212	0.761061	−	0.648680i	$-0.224679\pi$
$13$	5.48809	1.52212	0.761061	+	0.648680i	$0.224679\pi$
$14$	4.57890	1.22376
$15$	−3.95569	−1.02135
$16$	1.00000	0.250000
$17$	−0.273005	−0.0662135	−0.0331067	−	0.999452i	$-0.510540\pi$
$17$	−0.273005	−0.0662135	−0.0331067	+	0.999452i	$0.510540\pi$
$18$	1.00000	0.235702
$19$	−6.48051	−1.48673	−0.743366	−	0.668885i	$-0.766772\pi$
$19$	−6.48051	−1.48673	−0.743366	+	0.668885i	$0.766772\pi$
$20$	3.95569	0.884519
$21$	−4.57890	−0.999198
$22$	−4.22869	−0.901561
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	10.6475	2.12950
$26$	5.48809	1.07630
$27$	−1.00000	−0.192450
$28$	4.57890	0.865330
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.95569	−0.722207
$31$	5.86484	1.05336	0.526678	−	0.850065i	$-0.323437\pi$
$31$	5.86484	1.05336	0.526678	+	0.850065i	$0.323437\pi$
$32$	1.00000	0.176777
$33$	4.22869	0.736121
$34$	−0.273005	−0.0468200
$35$	18.1127	3.06161
$36$	1.00000	0.166667
$37$	1.93070	0.317405	0.158702	−	0.987326i	$-0.449269\pi$
$37$	1.93070	0.317405	0.158702	+	0.987326i	$0.449269\pi$
$38$	−6.48051	−1.05128
$39$	−5.48809	−0.878798
$40$	3.95569	0.625449
$41$	0.433159	0.0676480	0.0338240	−	0.999428i	$-0.489231\pi$
$41$	0.433159	0.0676480	0.0338240	+	0.999428i	$0.489231\pi$
$42$	−4.57890	−0.706539
$43$	5.69859	0.869026	0.434513	−	0.900666i	$-0.356920\pi$
$43$	5.69859	0.869026	0.434513	+	0.900666i	$0.356920\pi$
$44$	−4.22869	−0.637500
$45$	3.95569	0.589679
$46$	−1.00000	−0.147442
$47$	−1.89626	−0.276597	−0.138299	−	0.990391i	$-0.544163\pi$
$47$	−1.89626	−0.276597	−0.138299	+	0.990391i	$0.544163\pi$
$48$	−1.00000	−0.144338
$49$	13.9663	1.99519
$50$	10.6475	1.50578
$51$	0.273005	0.0382284
$52$	5.48809	0.761061
$53$	−7.90073	−1.08525	−0.542625	−	0.839975i	$-0.682569\pi$
$53$	−7.90073	−1.08525	−0.542625	+	0.839975i	$0.682569\pi$
$54$	−1.00000	−0.136083
$55$	−16.7274	−2.25552
$56$	4.57890	0.611881
$57$	6.48051	0.858365
$58$	−1.00000	−0.131306
$59$	−9.96770	−1.29768	−0.648842	−	0.760923i	$-0.724747\pi$
$59$	−9.96770	−1.29768	−0.648842	+	0.760923i	$0.724747\pi$
$60$	−3.95569	−0.510677
$61$	6.22869	0.797503	0.398751	−	0.917059i	$-0.369444\pi$
$61$	6.22869	0.797503	0.398751	+	0.917059i	$0.369444\pi$
$62$	5.86484	0.744835
$63$	4.57890	0.576887
$64$	1.00000	0.125000
$65$	21.7092	2.69269
$66$	4.22869	0.520516
$67$	−8.50885	−1.03952	−0.519761	−	0.854312i	$-0.673979\pi$
$67$	−8.50885	−1.03952	−0.519761	+	0.854312i	$0.673979\pi$
$68$	−0.273005	−0.0331067
$69$	1.00000	0.120386
$70$	18.1127	2.16488
$71$	−4.98632	−0.591767	−0.295884	−	0.955224i	$-0.595614\pi$
$71$	−4.98632	−0.591767	−0.295884	+	0.955224i	$0.595614\pi$
$72$	1.00000	0.117851
$73$	8.19425	0.959065	0.479532	−	0.877524i	$-0.340806\pi$
$73$	8.19425	0.959065	0.479532	+	0.877524i	$0.340806\pi$
$74$	1.93070	0.224439
$75$	−10.6475	−1.22947
$76$	−6.48051	−0.743366
$77$	−19.3628	−2.20659
$78$	−5.48809	−0.621404
$79$	−8.86257	−0.997117	−0.498558	−	0.866856i	$-0.666137\pi$
$79$	−8.86257	−0.997117	−0.498558	+	0.866856i	$0.666137\pi$
$80$	3.95569	0.442260
$81$	1.00000	0.111111
$82$	0.433159	0.0478344
$83$	12.7535	1.39988	0.699940	−	0.714202i	$-0.253210\pi$
$83$	12.7535	1.39988	0.699940	+	0.714202i	$0.253210\pi$
$84$	−4.57890	−0.499599
$85$	−1.07992	−0.117134
$86$	5.69859	0.614494
$87$	1.00000	0.107211
$88$	−4.22869	−0.450780
$89$	−6.20661	−0.657899	−0.328950	−	0.944347i	$-0.606695\pi$
$89$	−6.20661	−0.657899	−0.328950	+	0.944347i	$0.606695\pi$
$90$	3.95569	0.416966
$91$	25.1294	2.63428
$92$	−1.00000	−0.104257
$93$	−5.86484	−0.608155
$94$	−1.89626	−0.195584
$95$	−25.6349	−2.63008
$96$	−1.00000	−0.102062
$97$	−6.36154	−0.645917	−0.322958	−	0.946413i	$-0.604677\pi$
$97$	−6.36154	−0.645917	−0.322958	+	0.946413i	$0.604677\pi$
$98$	13.9663	1.41081
$99$	−4.22869	−0.425000
$100$	10.6475	1.06475
$101$	−6.21580	−0.618495	−0.309248	−	0.950982i	$-0.600077\pi$
$101$	−6.21580	−0.618495	−0.309248	+	0.950982i	$0.600077\pi$
$102$	0.273005	0.0270315
$103$	−2.40509	−0.236981	−0.118490	−	0.992955i	$-0.537805\pi$
$103$	−2.40509	−0.236981	−0.118490	+	0.992955i	$0.537805\pi$
$104$	5.48809	0.538151
$105$	−18.1127	−1.76762
$106$	−7.90073	−0.767387
$107$	13.4034	1.29575	0.647876	−	0.761746i	$-0.275658\pi$
$107$	13.4034	1.29575	0.647876	+	0.761746i	$0.275658\pi$
$108$	−1.00000	−0.0962250
$109$	7.08602	0.678718	0.339359	−	0.940657i	$-0.389790\pi$
$109$	7.08602	0.678718	0.339359	+	0.940657i	$0.389790\pi$
$110$	−16.7274	−1.59490
$111$	−1.93070	−0.183254
$112$	4.57890	0.432665
$113$	−17.7916	−1.67369	−0.836846	−	0.547438i	$-0.815603\pi$
$113$	−17.7916	−1.67369	−0.836846	+	0.547438i	$0.815603\pi$
$114$	6.48051	0.606955
$115$	−3.95569	−0.368870
$116$	−1.00000	−0.0928477
$117$	5.48809	0.507374
$118$	−9.96770	−0.917602
$119$	−1.25006	−0.114593
$120$	−3.95569	−0.361103
$121$	6.88186	0.625624
$122$	6.22869	0.563920
$123$	−0.433159	−0.0390566
$124$	5.86484	0.526678
$125$	22.3397	1.99812
$126$	4.57890	0.407921
$127$	−19.6479	−1.74346	−0.871732	−	0.489982i	$-0.837003\pi$
$127$	−19.6479	−1.74346	−0.871732	+	0.489982i	$0.837003\pi$
$128$	1.00000	0.0883883
$129$	−5.69859	−0.501732
$130$	21.7092	1.90402
$131$	−10.0098	−0.874563	−0.437282	−	0.899325i	$-0.644059\pi$
$131$	−10.0098	−0.874563	−0.437282	+	0.899325i	$0.644059\pi$
$132$	4.22869	0.368061
$133$	−29.6736	−2.57303
$134$	−8.50885	−0.735053
$135$	−3.95569	−0.340452
$136$	−0.273005	−0.0234100
$137$	−6.86257	−0.586309	−0.293154	−	0.956065i	$-0.594705\pi$
$137$	−6.86257	−0.586309	−0.293154	+	0.956065i	$0.594705\pi$
$138$	1.00000	0.0851257
$139$	−3.84329	−0.325984	−0.162992	−	0.986627i	$-0.552114\pi$
$139$	−3.84329	−0.325984	−0.162992	+	0.986627i	$0.552114\pi$
$140$	18.1127	1.53080
$141$	1.89626	0.159694
$142$	−4.98632	−0.418443
$143$	−23.2075	−1.94070
$144$	1.00000	0.0833333
$145$	−3.95569	−0.328502
$146$	8.19425	0.678161
$147$	−13.9663	−1.15192
$148$	1.93070	0.158702
$149$	13.9976	1.14673	0.573365	−	0.819300i	$-0.305638\pi$
$149$	13.9976	1.14673	0.573365	+	0.819300i	$0.305638\pi$
$150$	−10.6475	−0.869363
$151$	−21.9867	−1.78925	−0.894627	−	0.446813i	$-0.852559\pi$
$151$	−21.9867	−1.78925	−0.894627	+	0.446813i	$0.852559\pi$
$152$	−6.48051	−0.525639
$153$	−0.273005	−0.0220712
$154$	−19.3628	−1.56030
$155$	23.1995	1.86343
$156$	−5.48809	−0.439399
$157$	16.9080	1.34940	0.674702	−	0.738090i	$-0.264272\pi$
$157$	16.9080	1.34940	0.674702	+	0.738090i	$0.264272\pi$
$158$	−8.86257	−0.705068
$159$	7.90073	0.626569
$160$	3.95569	0.312725
$161$	−4.57890	−0.360868
$162$	1.00000	0.0785674
$163$	17.7709	1.39193	0.695963	−	0.718077i	$-0.254978\pi$
$163$	17.7709	1.39193	0.695963	+	0.718077i	$0.254978\pi$
$164$	0.433159	0.0338240
$165$	16.7274	1.30223
$166$	12.7535	0.989865
$167$	11.4446	0.885609	0.442804	−	0.896618i	$-0.353984\pi$
$167$	11.4446	0.885609	0.442804	+	0.896618i	$0.353984\pi$
$168$	−4.57890	−0.353270
$169$	17.1191	1.31686
$170$	−1.07992	−0.0828263
$171$	−6.48051	−0.495577
$172$	5.69859	0.434513
$173$	−18.3555	−1.39554	−0.697772	−	0.716320i	$-0.745825\pi$
$173$	−18.3555	−1.39554	−0.697772	+	0.716320i	$0.745825\pi$
$174$	1.00000	0.0758098
$175$	48.7537	3.68544
$176$	−4.22869	−0.318750
$177$	9.96770	0.749219
$178$	−6.20661	−0.465205
$179$	4.22260	0.315612	0.157806	−	0.987470i	$-0.449558\pi$
$179$	4.22260	0.315612	0.157806	+	0.987470i	$0.449558\pi$
$180$	3.95569	0.294840
$181$	19.2109	1.42794	0.713968	−	0.700179i	$-0.246896\pi$
$181$	19.2109	1.42794	0.713968	+	0.700179i	$0.246896\pi$
$182$	25.1294	1.86272
$183$	−6.22869	−0.460438
$184$	−1.00000	−0.0737210
$185$	7.63724	0.561501
$186$	−5.86484	−0.430031
$187$	1.15446	0.0844221
$188$	−1.89626	−0.138299
$189$	−4.57890	−0.333066
$190$	−25.6349	−1.85975
$191$	16.9883	1.22923	0.614616	−	0.788826i	$-0.289311\pi$
$191$	16.9883	1.22923	0.614616	+	0.788826i	$0.289311\pi$
$192$	−1.00000	−0.0721688
$193$	−6.10105	−0.439163	−0.219582	−	0.975594i	$-0.570469\pi$
$193$	−6.10105	−0.439163	−0.219582	+	0.975594i	$0.570469\pi$
$194$	−6.36154	−0.456732
$195$	−21.7092	−1.55463
$196$	13.9663	0.997594
$197$	−14.9098	−1.06228	−0.531139	−	0.847285i	$-0.678236\pi$
$197$	−14.9098	−1.06228	−0.531139	+	0.847285i	$0.678236\pi$
$198$	−4.22869	−0.300520
$199$	−10.4805	−0.742945	−0.371472	−	0.928444i	$-0.621147\pi$
$199$	−10.4805	−0.742945	−0.371472	+	0.928444i	$0.621147\pi$
$200$	10.6475	0.752891
$201$	8.50885	0.600168
$202$	−6.21580	−0.437342
$203$	−4.57890	−0.321376
$204$	0.273005	0.0191142
$205$	1.71344	0.119672
$206$	−2.40509	−0.167571
$207$	−1.00000	−0.0695048
$208$	5.48809	0.380531
$209$	27.4041	1.89558
$210$	−18.1127	−1.24990
$211$	17.1514	1.18075	0.590375	−	0.807129i	$-0.298980\pi$
$211$	17.1514	1.18075	0.590375	+	0.807129i	$0.298980\pi$
$212$	−7.90073	−0.542625
$213$	4.98632	0.341657
$214$	13.4034	0.916234
$215$	22.5418	1.53734
$216$	−1.00000	−0.0680414
$217$	26.8545	1.82300
$218$	7.08602	0.479926
$219$	−8.19425	−0.553716
$220$	−16.7274	−1.12776
$221$	−1.49828	−0.100785
$222$	−1.93070	−0.129580
$223$	−4.61381	−0.308964	−0.154482	−	0.987996i	$-0.549371\pi$
$223$	−4.61381	−0.308964	−0.154482	+	0.987996i	$0.549371\pi$
$224$	4.57890	0.305941
$225$	10.6475	0.709832
$226$	−17.7916	−1.18348
$227$	−27.7867	−1.84427	−0.922136	−	0.386867i	$-0.873557\pi$
$227$	−27.7867	−1.84427	−0.922136	+	0.386867i	$0.873557\pi$
$228$	6.48051	0.429182
$229$	−2.27584	−0.150392	−0.0751959	−	0.997169i	$-0.523958\pi$
$229$	−2.27584	−0.150392	−0.0751959	+	0.997169i	$0.523958\pi$
$230$	−3.95569	−0.260830
$231$	19.3628	1.27398
$232$	−1.00000	−0.0656532
$233$	1.55620	0.101950	0.0509749	−	0.998700i	$-0.483767\pi$
$233$	1.55620	0.101950	0.0509749	+	0.998700i	$0.483767\pi$
$234$	5.48809	0.358768
$235$	−7.50100	−0.489311
$236$	−9.96770	−0.648842
$237$	8.86257	0.575686
$238$	−1.25006	−0.0810295
$239$	−1.42678	−0.0922909	−0.0461454	−	0.998935i	$-0.514694\pi$
$239$	−1.42678	−0.0922909	−0.0461454	+	0.998935i	$0.514694\pi$
$240$	−3.95569	−0.255339
$241$	−11.2979	−0.727759	−0.363879	−	0.931446i	$-0.618548\pi$
$241$	−11.2979	−0.727759	−0.363879	+	0.931446i	$0.618548\pi$
$242$	6.88186	0.442383
$243$	−1.00000	−0.0641500
$244$	6.22869	0.398751
$245$	55.2464	3.52956
$246$	−0.433159	−0.0276172
$247$	−35.5656	−2.26299
$248$	5.86484	0.372418
$249$	−12.7535	−0.808221
$250$	22.3397	1.41289
$251$	27.9300	1.76292	0.881462	−	0.472254i	$-0.156560\pi$
$251$	27.9300	1.76292	0.881462	+	0.472254i	$0.156560\pi$
$252$	4.57890	0.288443
$253$	4.22869	0.265856
$254$	−19.6479	−1.23282
$255$	1.07992	0.0676274
$256$	1.00000	0.0625000
$257$	11.1658	0.696501	0.348251	−	0.937401i	$-0.386776\pi$
$257$	11.1658	0.696501	0.348251	+	0.937401i	$0.386776\pi$
$258$	−5.69859	−0.354778
$259$	8.84047	0.549320
$260$	21.7092	1.34635
$261$	−1.00000	−0.0618984
$262$	−10.0098	−0.618410
$263$	16.4628	1.01514	0.507569	−	0.861611i	$-0.330544\pi$
$263$	16.4628	1.01514	0.507569	+	0.861611i	$0.330544\pi$
$264$	4.22869	0.260258
$265$	−31.2529	−1.91985
$266$	−29.6736	−1.81941
$267$	6.20661	0.379838
$268$	−8.50885	−0.519761
$269$	23.8852	1.45631	0.728153	−	0.685414i	$-0.240379\pi$
$269$	23.8852	1.45631	0.728153	+	0.685414i	$0.240379\pi$
$270$	−3.95569	−0.240736
$271$	−24.8321	−1.50844	−0.754221	−	0.656621i	$-0.771985\pi$
$271$	−24.8321	−1.50844	−0.754221	+	0.656621i	$0.771985\pi$
$272$	−0.273005	−0.0165534
$273$	−25.1294	−1.52090
$274$	−6.86257	−0.414583
$275$	−45.0249	−2.71511
$276$	1.00000	0.0601929
$277$	−19.0518	−1.14471	−0.572356	−	0.820005i	$-0.693970\pi$
$277$	−19.0518	−1.14471	−0.572356	+	0.820005i	$0.693970\pi$
$278$	−3.84329	−0.230505
$279$	5.86484	0.351119
$280$	18.1127	1.08244
$281$	8.99733	0.536736	0.268368	−	0.963316i	$-0.413516\pi$
$281$	8.99733	0.536736	0.268368	+	0.963316i	$0.413516\pi$
$282$	1.89626	0.112920
$283$	17.1658	1.02040	0.510199	−	0.860056i	$-0.329572\pi$
$283$	17.1658	1.02040	0.510199	+	0.860056i	$0.329572\pi$
$284$	−4.98632	−0.295884
$285$	25.6349	1.51848
$286$	−23.2075	−1.37229
$287$	1.98339	0.117076
$288$	1.00000	0.0589256
$289$	−16.9255	−0.995616
$290$	−3.95569	−0.232286
$291$	6.36154	0.372920
$292$	8.19425	0.479532
$293$	29.3370	1.71388	0.856941	−	0.515414i	$-0.172362\pi$
$293$	29.3370	1.71388	0.856941	+	0.515414i	$0.172362\pi$
$294$	−13.9663	−0.814532
$295$	−39.4291	−2.29565
$296$	1.93070	0.112219
$297$	4.22869	0.245374
$298$	13.9976	0.810860
$299$	−5.48809	−0.317384
$300$	−10.6475	−0.614733
$301$	26.0932	1.50399
$302$	−21.9867	−1.26519
$303$	6.21580	0.357088
$304$	−6.48051	−0.371683
$305$	24.6388	1.41081
$306$	−0.273005	−0.0156067
$307$	12.8019	0.730643	0.365322	−	0.930881i	$-0.380959\pi$
$307$	12.8019	0.730643	0.365322	+	0.930881i	$0.380959\pi$
$308$	−19.3628	−1.10330
$309$	2.40509	0.136821
$310$	23.1995	1.31764
$311$	2.05227	0.116374	0.0581869	−	0.998306i	$-0.481468\pi$
$311$	2.05227	0.116374	0.0581869	+	0.998306i	$0.481468\pi$
$312$	−5.48809	−0.310702
$313$	−6.99130	−0.395172	−0.197586	−	0.980286i	$-0.563310\pi$
$313$	−6.99130	−0.395172	−0.197586	+	0.980286i	$0.563310\pi$
$314$	16.9080	0.954173
$315$	18.1127	1.02054
$316$	−8.86257	−0.498558
$317$	−16.6476	−0.935022	−0.467511	−	0.883987i	$-0.654849\pi$
$317$	−16.6476	−0.935022	−0.467511	+	0.883987i	$0.654849\pi$
$318$	7.90073	0.443051
$319$	4.22869	0.236761
$320$	3.95569	0.221130
$321$	−13.4034	−0.748102
$322$	−4.57890	−0.255172
$323$	1.76921	0.0984416
$324$	1.00000	0.0555556
$325$	58.4343	3.24135
$326$	17.7709	0.984241
$327$	−7.08602	−0.391858
$328$	0.433159	0.0239172
$329$	−8.68276	−0.478696
$330$	16.7274	0.920813
$331$	−28.8743	−1.58707	−0.793536	−	0.608523i	$-0.791762\pi$
$331$	−28.8743	−1.58707	−0.793536	+	0.608523i	$0.791762\pi$
$332$	12.7535	0.699940
$333$	1.93070	0.105802
$334$	11.4446	0.626220
$335$	−33.6584	−1.83895
$336$	−4.57890	−0.249799
$337$	−20.0514	−1.09227	−0.546134	−	0.837698i	$-0.683901\pi$
$337$	−20.0514	−1.09227	−0.546134	+	0.837698i	$0.683901\pi$
$338$	17.1191	0.931158
$339$	17.7916	0.966307
$340$	−1.07992	−0.0585671
$341$	−24.8006	−1.34303
$342$	−6.48051	−0.350426
$343$	31.8980	1.72233
$344$	5.69859	0.307247
$345$	3.95569	0.212967
$346$	−18.3555	−0.986798
$347$	−8.13209	−0.436553	−0.218277	−	0.975887i	$-0.570044\pi$
$347$	−8.13209	−0.436553	−0.218277	+	0.975887i	$0.570044\pi$
$348$	1.00000	0.0536056
$349$	7.58819	0.406186	0.203093	−	0.979159i	$-0.434901\pi$
$349$	7.58819	0.406186	0.203093	+	0.979159i	$0.434901\pi$
$350$	48.7537	2.60600
$351$	−5.48809	−0.292933
$352$	−4.22869	−0.225390
$353$	−24.9873	−1.32994	−0.664970	−	0.746870i	$-0.731556\pi$
$353$	−24.9873	−1.32994	−0.664970	+	0.746870i	$0.731556\pi$
$354$	9.96770	0.529778
$355$	−19.7243	−1.04686
$356$	−6.20661	−0.328950
$357$	1.25006	0.0661603
$358$	4.22260	0.223171
$359$	8.15905	0.430618	0.215309	−	0.976546i	$-0.430924\pi$
$359$	8.15905	0.430618	0.215309	+	0.976546i	$0.430924\pi$
$360$	3.95569	0.208483
$361$	22.9970	1.21037
$362$	19.2109	1.00970
$363$	−6.88186	−0.361204
$364$	25.1294	1.31714
$365$	32.4139	1.69662
$366$	−6.22869	−0.325579
$367$	−15.8443	−0.827067	−0.413534	−	0.910489i	$-0.635706\pi$
$367$	−15.8443	−0.827067	−0.413534	+	0.910489i	$0.635706\pi$
$368$	−1.00000	−0.0521286
$369$	0.433159	0.0225493
$370$	7.63724	0.397041
$371$	−36.1767	−1.87820
$372$	−5.86484	−0.304078
$373$	−31.7270	−1.64276	−0.821380	−	0.570381i	$-0.806795\pi$
$373$	−31.7270	−1.64276	−0.821380	+	0.570381i	$0.806795\pi$
$374$	1.15446	0.0596955
$375$	−22.3397	−1.15362
$376$	−1.89626	−0.0977919
$377$	−5.48809	−0.282651
$378$	−4.57890	−0.235513
$379$	5.32609	0.273583	0.136791	−	0.990600i	$-0.456321\pi$
$379$	5.32609	0.273583	0.136791	+	0.990600i	$0.456321\pi$
$380$	−25.6349	−1.31504
$381$	19.6479	1.00659
$382$	16.9883	0.869198
$383$	15.1521	0.774238	0.387119	−	0.922030i	$-0.373470\pi$
$383$	15.1521	0.774238	0.387119	+	0.922030i	$0.373470\pi$
$384$	−1.00000	−0.0510310
$385$	−76.5931	−3.90355
$386$	−6.10105	−0.310535
$387$	5.69859	0.289675
$388$	−6.36154	−0.322958
$389$	17.8271	0.903869	0.451934	−	0.892051i	$-0.350734\pi$
$389$	17.8271	0.903869	0.451934	+	0.892051i	$0.350734\pi$
$390$	−21.7092	−1.09929
$391$	0.273005	0.0138065
$392$	13.9663	0.705405
$393$	10.0098	0.504929
$394$	−14.9098	−0.751144
$395$	−35.0576	−1.76394
$396$	−4.22869	−0.212500
$397$	24.4973	1.22948	0.614742	−	0.788728i	$-0.289260\pi$
$397$	24.4973	1.22948	0.614742	+	0.788728i	$0.289260\pi$
$398$	−10.4805	−0.525341
$399$	29.6736	1.48554
$400$	10.6475	0.532374
$401$	−25.8027	−1.28853	−0.644263	−	0.764804i	$-0.722836\pi$
$401$	−25.8027	−1.28853	−0.644263	+	0.764804i	$0.722836\pi$
$402$	8.50885	0.424383
$403$	32.1868	1.60334
$404$	−6.21580	−0.309248
$405$	3.95569	0.196560
$406$	−4.57890	−0.227247
$407$	−8.16433	−0.404691
$408$	0.273005	0.0135158
$409$	35.0530	1.73326	0.866630	−	0.498951i	$-0.166281\pi$
$409$	35.0530	1.73326	0.866630	+	0.498951i	$0.166281\pi$
$410$	1.71344	0.0846208
$411$	6.86257	0.338505
$412$	−2.40509	−0.118490
$413$	−45.6411	−2.24585
$414$	−1.00000	−0.0491473
$415$	50.4490	2.47644
$416$	5.48809	0.269076
$417$	3.84329	0.188207
$418$	27.4041	1.34038
$419$	33.6081	1.64186	0.820932	−	0.571025i	$-0.193454\pi$
$419$	33.6081	1.64186	0.820932	+	0.571025i	$0.193454\pi$
$420$	−18.1127	−0.883809
$421$	15.5020	0.755522	0.377761	−	0.925903i	$-0.376694\pi$
$421$	15.5020	0.755522	0.377761	+	0.925903i	$0.376694\pi$
$422$	17.1514	0.834916
$423$	−1.89626	−0.0921991
$424$	−7.90073	−0.383694
$425$	−2.90682	−0.141001
$426$	4.98632	0.241588
$427$	28.5206	1.38021
$428$	13.4034	0.647876
$429$	23.2075	1.12047
$430$	22.5418	1.08706
$431$	−7.05766	−0.339956	−0.169978	−	0.985448i	$-0.554370\pi$
$431$	−7.05766	−0.339956	−0.169978	+	0.985448i	$0.554370\pi$
$432$	−1.00000	−0.0481125
$433$	−16.1766	−0.777400	−0.388700	−	0.921364i	$-0.627076\pi$
$433$	−16.1766	−0.777400	−0.388700	+	0.921364i	$0.627076\pi$
$434$	26.8545	1.28906
$435$	3.95569	0.189661
$436$	7.08602	0.339359
$437$	6.48051	0.310005
$438$	−8.19425	−0.391537
$439$	13.9649	0.666510	0.333255	−	0.942837i	$-0.391853\pi$
$439$	13.9649	0.666510	0.333255	+	0.942837i	$0.391853\pi$
$440$	−16.7274	−0.797448
$441$	13.9663	0.665062
$442$	−1.49828	−0.0712657
$443$	−18.8457	−0.895386	−0.447693	−	0.894187i	$-0.647754\pi$
$443$	−18.8457	−0.895386	−0.447693	+	0.894187i	$0.647754\pi$
$444$	−1.93070	−0.0916268
$445$	−24.5514	−1.16385
$446$	−4.61381	−0.218470
$447$	−13.9976	−0.662065
$448$	4.57890	0.216333
$449$	19.9879	0.943286	0.471643	−	0.881790i	$-0.343661\pi$
$449$	19.9879	0.943286	0.471643	+	0.881790i	$0.343661\pi$
$450$	10.6475	0.501927
$451$	−1.83170	−0.0862512
$452$	−17.7916	−0.836846
$453$	21.9867	1.03303
$454$	−27.7867	−1.30410
$455$	99.4041	4.66014
$456$	6.48051	0.303478
$457$	−30.7310	−1.43754	−0.718768	−	0.695250i	$-0.755294\pi$
$457$	−30.7310	−1.43754	−0.718768	+	0.695250i	$0.755294\pi$
$458$	−2.27584	−0.106343
$459$	0.273005	0.0127428
$460$	−3.95569	−0.184435
$461$	19.4368	0.905261	0.452630	−	0.891698i	$-0.350486\pi$
$461$	19.4368	0.905261	0.452630	+	0.891698i	$0.350486\pi$
$462$	19.3628	0.900837
$463$	0.220124	0.0102300	0.00511502	−	0.999987i	$-0.498372\pi$
$463$	0.220124	0.0102300	0.00511502	+	0.999987i	$0.498372\pi$
$464$	−1.00000	−0.0464238
$465$	−23.1995	−1.07585
$466$	1.55620	0.0720894
$467$	20.6737	0.956666	0.478333	−	0.878179i	$-0.341241\pi$
$467$	20.6737	0.956666	0.478333	+	0.878179i	$0.341241\pi$
$468$	5.48809	0.253687
$469$	−38.9612	−1.79906
$470$	−7.50100	−0.345995
$471$	−16.9080	−0.779079
$472$	−9.96770	−0.458801
$473$	−24.0976	−1.10801
$474$	8.86257	0.407071
$475$	−69.0011	−3.16599
$476$	−1.25006	−0.0572965
$477$	−7.90073	−0.361750
$478$	−1.42678	−0.0652595
$479$	−7.32078	−0.334495	−0.167248	−	0.985915i	$-0.553488\pi$
$479$	−7.32078	−0.334495	−0.167248	+	0.985915i	$0.553488\pi$
$480$	−3.95569	−0.180552
$481$	10.5958	0.483129
$482$	−11.2979	−0.514603
$483$	4.57890	0.208347
$484$	6.88186	0.312812
$485$	−25.1643	−1.14265
$486$	−1.00000	−0.0453609
$487$	18.2058	0.824983	0.412491	−	0.910962i	$-0.364659\pi$
$487$	18.2058	0.824983	0.412491	+	0.910962i	$0.364659\pi$
$488$	6.22869	0.281960
$489$	−17.7709	−0.803629
$490$	55.2464	2.49578
$491$	−39.0601	−1.76276	−0.881378	−	0.472411i	$-0.843384\pi$
$491$	−39.0601	−1.76276	−0.881378	+	0.472411i	$0.843384\pi$
$492$	−0.433159	−0.0195283
$493$	0.273005	0.0122955
$494$	−35.5656	−1.60017
$495$	−16.7274	−0.751841
$496$	5.86484	0.263339
$497$	−22.8319	−1.02415
$498$	−12.7535	−0.571499
$499$	30.7227	1.37534	0.687668	−	0.726025i	$-0.258635\pi$
$499$	30.7227	1.37534	0.687668	+	0.726025i	$0.258635\pi$
$500$	22.3397	0.999061
$501$	−11.4446	−0.511307
$502$	27.9300	1.24658
$503$	−34.9237	−1.55717	−0.778585	−	0.627540i	$-0.784062\pi$
$503$	−34.9237	−1.55717	−0.778585	+	0.627540i	$0.784062\pi$
$504$	4.57890	0.203960
$505$	−24.5878	−1.09414
$506$	4.22869	0.187988
$507$	−17.1191	−0.760287
$508$	−19.6479	−0.871732
$509$	−7.76953	−0.344378	−0.172189	−	0.985064i	$-0.555084\pi$
$509$	−7.76953	−0.344378	−0.172189	+	0.985064i	$0.555084\pi$
$510$	1.07992	0.0478198
$511$	37.5207	1.65982
$512$	1.00000	0.0441942
$513$	6.48051	0.286122
$514$	11.1658	0.492501
$515$	−9.51379	−0.419228
$516$	−5.69859	−0.250866
$517$	8.01869	0.352661
$518$	8.84047	0.388428
$519$	18.3555	0.805717
$520$	21.7092	0.952010
$521$	−19.4781	−0.853350	−0.426675	−	0.904405i	$-0.640315\pi$
$521$	−19.4781	−0.853350	−0.426675	+	0.904405i	$0.640315\pi$
$522$	−1.00000	−0.0437688
$523$	23.6712	1.03507	0.517535	−	0.855662i	$-0.326850\pi$
$523$	23.6712	1.03507	0.517535	+	0.855662i	$0.326850\pi$
$524$	−10.0098	−0.437282
$525$	−48.7537	−2.12779
$526$	16.4628	0.717811
$527$	−1.60113	−0.0697464
$528$	4.22869	0.184030
$529$	1.00000	0.0434783
$530$	−31.2529	−1.35754
$531$	−9.96770	−0.432562
$532$	−29.6736	−1.28651
$533$	2.37721	0.102969
$534$	6.20661	0.268586
$535$	53.0195	2.29223
$536$	−8.50885	−0.367526
$537$	−4.22260	−0.182218
$538$	23.8852	1.02976
$539$	−59.0593	−2.54386
$540$	−3.95569	−0.170226
$541$	−43.1810	−1.85650	−0.928248	−	0.371962i	$-0.878685\pi$
$541$	−43.1810	−1.85650	−0.928248	+	0.371962i	$0.878685\pi$
$542$	−24.8321	−1.06663
$543$	−19.2109	−0.824419
$544$	−0.273005	−0.0117050
$545$	28.0301	1.20068
$546$	−25.1294	−1.07544
$547$	−4.10651	−0.175582	−0.0877909	−	0.996139i	$-0.527981\pi$
$547$	−4.10651	−0.175582	−0.0877909	+	0.996139i	$0.527981\pi$
$548$	−6.86257	−0.293154
$549$	6.22869	0.265834
$550$	−45.0249	−1.91987
$551$	6.48051	0.276079
$552$	1.00000	0.0425628
$553$	−40.5808	−1.72567
$554$	−19.0518	−0.809434
$555$	−7.63724	−0.324183
$556$	−3.84329	−0.162992
$557$	−36.1223	−1.53055	−0.765276	−	0.643702i	$-0.777398\pi$
$557$	−36.1223	−1.53055	−0.765276	+	0.643702i	$0.777398\pi$
$558$	5.86484	0.248278
$559$	31.2743	1.32276
$560$	18.1127	0.765401
$561$	−1.15446	−0.0487411
$562$	8.99733	0.379530
$563$	−24.3038	−1.02428	−0.512141	−	0.858901i	$-0.671148\pi$
$563$	−24.3038	−1.02428	−0.512141	+	0.858901i	$0.671148\pi$
$564$	1.89626	0.0798468
$565$	−70.3780	−2.96083
$566$	17.1658	0.721531
$567$	4.57890	0.192296
$568$	−4.98632	−0.209221
$569$	−13.7126	−0.574864	−0.287432	−	0.957801i	$-0.592801\pi$
$569$	−13.7126	−0.574864	−0.287432	+	0.957801i	$0.592801\pi$
$570$	25.6349	1.07373
$571$	24.6013	1.02953	0.514766	−	0.857331i	$-0.327879\pi$
$571$	24.6013	1.02953	0.514766	+	0.857331i	$0.327879\pi$
$572$	−23.2075	−0.970352
$573$	−16.9883	−0.709698
$574$	1.98339	0.0827851
$575$	−10.6475	−0.444031
$576$	1.00000	0.0416667
$577$	−8.78651	−0.365787	−0.182894	−	0.983133i	$-0.558546\pi$
$577$	−8.78651	−0.365787	−0.182894	+	0.983133i	$0.558546\pi$
$578$	−16.9255	−0.704007
$579$	6.10105	0.253551
$580$	−3.95569	−0.164251
$581$	58.3971	2.42272
$582$	6.36154	0.263694
$583$	33.4098	1.38369
$584$	8.19425	0.339081
$585$	21.7092	0.897564
$586$	29.3370	1.21190
$587$	−38.8961	−1.60542	−0.802708	−	0.596372i	$-0.796608\pi$
$587$	−38.8961	−1.60542	−0.802708	+	0.596372i	$0.796608\pi$
$588$	−13.9663	−0.575961
$589$	−38.0072	−1.56606
$590$	−39.4291	−1.62327
$591$	14.9098	0.613307
$592$	1.93070	0.0793512
$593$	−20.5300	−0.843065	−0.421532	−	0.906813i	$-0.638508\pi$
$593$	−20.5300	−0.843065	−0.421532	+	0.906813i	$0.638508\pi$
$594$	4.22869	0.173505
$595$	−4.94486	−0.202719
$596$	13.9976	0.573365
$597$	10.4805	0.428939
$598$	−5.48809	−0.224425
$599$	36.4319	1.48857	0.744284	−	0.667863i	$-0.232791\pi$
$599$	36.4319	1.48857	0.744284	+	0.667863i	$0.232791\pi$
$600$	−10.6475	−0.434682
$601$	−13.9225	−0.567910	−0.283955	−	0.958838i	$-0.591647\pi$
$601$	−13.9225	−0.567910	−0.283955	+	0.958838i	$0.591647\pi$
$602$	26.0932	1.06348
$603$	−8.50885	−0.346507
$604$	−21.9867	−0.894627
$605$	27.2225	1.10675
$606$	6.21580	0.252500
$607$	27.5968	1.12012	0.560059	−	0.828453i	$-0.310778\pi$
$607$	27.5968	1.12012	0.560059	+	0.828453i	$0.310778\pi$
$608$	−6.48051	−0.262819
$609$	4.57890	0.185546
$610$	24.6388	0.997595
$611$	−10.4068	−0.421015
$612$	−0.273005	−0.0110356
$613$	25.8997	1.04608	0.523040	−	0.852308i	$-0.324798\pi$
$613$	25.8997	1.04608	0.523040	+	0.852308i	$0.324798\pi$
$614$	12.8019	0.516643
$615$	−1.71344	−0.0690926
$616$	−19.3628	−0.780148
$617$	36.5726	1.47236	0.736179	−	0.676787i	$-0.236628\pi$
$617$	36.5726	1.47236	0.736179	+	0.676787i	$0.236628\pi$
$618$	2.40509	0.0967469
$619$	−23.8391	−0.958174	−0.479087	−	0.877767i	$-0.659032\pi$
$619$	−23.8391	−0.958174	−0.479087	+	0.877767i	$0.659032\pi$
$620$	23.1995	0.931714
$621$	1.00000	0.0401286
$622$	2.05227	0.0822886
$623$	−28.4194	−1.13860
$624$	−5.48809	−0.219699
$625$	35.1315	1.40526
$626$	−6.99130	−0.279429
$627$	−27.4041	−1.09441
$628$	16.9080	0.674702
$629$	−0.527090	−0.0210165
$630$	18.1127	0.721627
$631$	−13.6339	−0.542756	−0.271378	−	0.962473i	$-0.587479\pi$
$631$	−13.6339	−0.542756	−0.271378	+	0.962473i	$0.587479\pi$
$632$	−8.86257	−0.352534
$633$	−17.1514	−0.681706
$634$	−16.6476	−0.661160
$635$	−77.7208	−3.08426
$636$	7.90073	0.313284
$637$	76.6484	3.03692
$638$	4.22869	0.167416
$639$	−4.98632	−0.197256
$640$	3.95569	0.156362
$641$	−25.4410	−1.00486	−0.502430	−	0.864618i	$-0.667560\pi$
$641$	−25.4410	−1.00486	−0.502430	+	0.864618i	$0.667560\pi$
$642$	−13.4034	−0.528988
$643$	−0.399920	−0.0157713	−0.00788565	−	0.999969i	$-0.502510\pi$
$643$	−0.399920	−0.0157713	−0.00788565	+	0.999969i	$0.502510\pi$
$644$	−4.57890	−0.180434
$645$	−22.5418	−0.887584
$646$	1.76921	0.0696087
$647$	33.4638	1.31560	0.657798	−	0.753194i	$-0.271488\pi$
$647$	33.4638	1.31560	0.657798	+	0.753194i	$0.271488\pi$
$648$	1.00000	0.0392837
$649$	42.1504	1.65455
$650$	58.4343	2.29198
$651$	−26.8545	−1.05251
$652$	17.7709	0.695963
$653$	13.1734	0.515515	0.257757	−	0.966210i	$-0.417016\pi$
$653$	13.1734	0.515515	0.257757	+	0.966210i	$0.417016\pi$
$654$	−7.08602	−0.277085
$655$	−39.5958	−1.54714
$656$	0.433159	0.0169120
$657$	8.19425	0.319688
$658$	−8.68276	−0.338489
$659$	40.8003	1.58935	0.794677	−	0.607033i	$-0.207640\pi$
$659$	40.8003	1.58935	0.794677	+	0.607033i	$0.207640\pi$
$660$	16.7274	0.651113
$661$	2.48005	0.0964629	0.0482315	−	0.998836i	$-0.484641\pi$
$661$	2.48005	0.0964629	0.0482315	+	0.998836i	$0.484641\pi$
$662$	−28.8743	−1.12223
$663$	1.49828	0.0581882
$664$	12.7535	0.494932
$665$	−117.380	−4.55178
$666$	1.93070	0.0748130
$667$	1.00000	0.0387202
$668$	11.4446	0.442804
$669$	4.61381	0.178380
$670$	−33.6584	−1.30034
$671$	−26.3392	−1.01682
$672$	−4.57890	−0.176635
$673$	18.6781	0.719988	0.359994	−	0.932955i	$-0.382779\pi$
$673$	18.6781	0.719988	0.359994	+	0.932955i	$0.382779\pi$
$674$	−20.0514	−0.772351
$675$	−10.6475	−0.409822
$676$	17.1191	0.658428
$677$	−34.7636	−1.33607	−0.668036	−	0.744129i	$-0.732865\pi$
$677$	−34.7636	−1.33607	−0.668036	+	0.744129i	$0.732865\pi$
$678$	17.7916	0.683282
$679$	−29.1288	−1.11786
$680$	−1.07992	−0.0414132
$681$	27.7867	1.06479
$682$	−24.8006	−0.949665
$683$	−37.7600	−1.44485	−0.722424	−	0.691450i	$-0.756972\pi$
$683$	−37.7600	−1.44485	−0.722424	+	0.691450i	$0.756972\pi$
$684$	−6.48051	−0.247789
$685$	−27.1462	−1.03720
$686$	31.8980	1.21787
$687$	2.27584	0.0868287
$688$	5.69859	0.217257
$689$	−43.3599	−1.65188
$690$	3.95569	0.150591
$691$	−28.1811	−1.07206	−0.536030	−	0.844199i	$-0.680076\pi$
$691$	−28.1811	−1.07206	−0.536030	+	0.844199i	$0.680076\pi$
$692$	−18.3555	−0.697772
$693$	−19.3628	−0.735531
$694$	−8.13209	−0.308690
$695$	−15.2029	−0.576678
$696$	1.00000	0.0379049
$697$	−0.118254	−0.00447921
$698$	7.58819	0.287217
$699$	−1.55620	−0.0588608
$700$	48.7537	1.84272
$701$	4.12850	0.155931	0.0779657	−	0.996956i	$-0.475158\pi$
$701$	4.12850	0.155931	0.0779657	+	0.996956i	$0.475158\pi$
$702$	−5.48809	−0.207135
$703$	−12.5119	−0.471895
$704$	−4.22869	−0.159375
$705$	7.50100	0.282504
$706$	−24.9873	−0.940409
$707$	−28.4615	−1.07041
$708$	9.96770	0.374609
$709$	43.8998	1.64869	0.824345	−	0.566087i	$-0.191543\pi$
$709$	43.8998	1.64869	0.824345	+	0.566087i	$0.191543\pi$
$710$	−19.7243	−0.740241
$711$	−8.86257	−0.332372
$712$	−6.20661	−0.232603
$713$	−5.86484	−0.219640
$714$	1.25006	0.0467824
$715$	−91.8015	−3.43318
$716$	4.22260	0.157806
$717$	1.42678	0.0532842
$718$	8.15905	0.304493
$719$	−16.6930	−0.622543	−0.311271	−	0.950321i	$-0.600755\pi$
$719$	−16.6930	−0.622543	−0.311271	+	0.950321i	$0.600755\pi$
$720$	3.95569	0.147420
$721$	−11.0127	−0.410133
$722$	22.9970	0.855860
$723$	11.2979	0.420172
$724$	19.2109	0.713968
$725$	−10.6475	−0.395438
$726$	−6.88186	−0.255410
$727$	3.85764	0.143072	0.0715360	−	0.997438i	$-0.477210\pi$
$727$	3.85764	0.143072	0.0715360	+	0.997438i	$0.477210\pi$
$728$	25.1294	0.931358
$729$	1.00000	0.0370370
$730$	32.4139	1.19969
$731$	−1.55574	−0.0575412
$732$	−6.22869	−0.230219
$733$	25.9759	0.959444	0.479722	−	0.877421i	$-0.340738\pi$
$733$	25.9759	0.959444	0.479722	+	0.877421i	$0.340738\pi$
$734$	−15.8443	−0.584825
$735$	−55.2464	−2.03779
$736$	−1.00000	−0.0368605
$737$	35.9813	1.32539
$738$	0.433159	0.0159448
$739$	17.4632	0.642394	0.321197	−	0.947012i	$-0.395915\pi$
$739$	17.4632	0.642394	0.321197	+	0.947012i	$0.395915\pi$
$740$	7.63724	0.280750
$741$	35.5656	1.30654
$742$	−36.1767	−1.32809
$743$	−10.6893	−0.392151	−0.196076	−	0.980589i	$-0.562820\pi$
$743$	−10.6893	−0.392151	−0.196076	+	0.980589i	$0.562820\pi$
$744$	−5.86484	−0.215015
$745$	55.3702	2.02861
$746$	−31.7270	−1.16161
$747$	12.7535	0.466627
$748$	1.15446	0.0422111
$749$	61.3726	2.24251
$750$	−22.3397	−0.815730
$751$	−21.1889	−0.773195	−0.386597	−	0.922249i	$-0.626350\pi$
$751$	−21.1889	−0.773195	−0.386597	+	0.922249i	$0.626350\pi$
$752$	−1.89626	−0.0691493
$753$	−27.9300	−1.01783
$754$	−5.48809	−0.199864
$755$	−86.9727	−3.16526
$756$	−4.57890	−0.166533
$757$	24.6491	0.895886	0.447943	−	0.894062i	$-0.352157\pi$
$757$	24.6491	0.895886	0.447943	+	0.894062i	$0.352157\pi$
$758$	5.32609	0.193452
$759$	−4.22869	−0.153492
$760$	−25.6349	−0.929875
$761$	−3.14286	−0.113928	−0.0569642	−	0.998376i	$-0.518142\pi$
$761$	−3.14286	−0.113928	−0.0569642	+	0.998376i	$0.518142\pi$
$762$	19.6479	0.711767
$763$	32.4462	1.17463
$764$	16.9883	0.614616
$765$	−1.07992	−0.0390447
$766$	15.1521	0.547469
$767$	−54.7037	−1.97523
$768$	−1.00000	−0.0360844
$769$	10.4858	0.378126	0.189063	−	0.981965i	$-0.439455\pi$
$769$	10.4858	0.378126	0.189063	+	0.981965i	$0.439455\pi$
$770$	−76.5931	−2.76022
$771$	−11.1658	−0.402125
$772$	−6.10105	−0.219582
$773$	−18.6543	−0.670947	−0.335473	−	0.942050i	$-0.608896\pi$
$773$	−18.6543	−0.670947	−0.335473	+	0.942050i	$0.608896\pi$
$774$	5.69859	0.204831
$775$	62.4458	2.24312
$776$	−6.36154	−0.228366
$777$	−8.84047	−0.317150
$778$	17.8271	0.639132
$779$	−2.80709	−0.100574
$780$	−21.7092	−0.777313
$781$	21.0856	0.754503
$782$	0.273005	0.00976264
$783$	1.00000	0.0357371
$784$	13.9663	0.498797
$785$	66.8828	2.38715
$786$	10.0098	0.357039
$787$	−29.9529	−1.06771	−0.533854	−	0.845577i	$-0.679257\pi$
$787$	−29.9529	−1.06771	−0.533854	+	0.845577i	$0.679257\pi$
$788$	−14.9098	−0.531139
$789$	−16.4628	−0.586090
$790$	−35.0576	−1.24729
$791$	−81.4659	−2.89659
$792$	−4.22869	−0.150260
$793$	34.1836	1.21390
$794$	24.4973	0.869376
$795$	31.2529	1.10842
$796$	−10.4805	−0.371472
$797$	−27.3290	−0.968044	−0.484022	−	0.875056i	$-0.660825\pi$
$797$	−27.3290	−0.968044	−0.484022	+	0.875056i	$0.660825\pi$
$798$	29.6736	1.05043
$799$	0.517687	0.0183145
$800$	10.6475	0.376445
$801$	−6.20661	−0.219300
$802$	−25.8027	−0.911126
$803$	−34.6510	−1.22281
$804$	8.50885	0.300084
$805$	−18.1127	−0.638389
$806$	32.1868	1.13373
$807$	−23.8852	−0.840799
$808$	−6.21580	−0.218671
$809$	8.49222	0.298570	0.149285	−	0.988794i	$-0.452303\pi$
$809$	8.49222	0.298570	0.149285	+	0.988794i	$0.452303\pi$
$810$	3.95569	0.138989
$811$	47.8811	1.68133	0.840667	−	0.541552i	$-0.182163\pi$
$811$	47.8811	1.68133	0.840667	+	0.541552i	$0.182163\pi$
$812$	−4.57890	−0.160688
$813$	24.8321	0.870899
$814$	−8.16433	−0.286160
$815$	70.2963	2.46237
$816$	0.273005	0.00955709
$817$	−36.9297	−1.29201
$818$	35.0530	1.22560
$819$	25.1294	0.878092
$820$	1.71344	0.0598360
$821$	−41.0959	−1.43426	−0.717128	−	0.696942i	$-0.754544\pi$
$821$	−41.0959	−1.43426	−0.717128	+	0.696942i	$0.754544\pi$
$822$	6.86257	0.239359
$823$	26.7849	0.933662	0.466831	−	0.884347i	$-0.345396\pi$
$823$	26.7849	0.933662	0.466831	+	0.884347i	$0.345396\pi$
$824$	−2.40509	−0.0837853
$825$	45.0249	1.56757
$826$	−45.6411	−1.58806
$827$	21.5407	0.749044	0.374522	−	0.927218i	$-0.377807\pi$
$827$	21.5407	0.749044	0.374522	+	0.927218i	$0.377807\pi$
$828$	−1.00000	−0.0347524
$829$	36.4602	1.26631	0.633157	−	0.774023i	$-0.281759\pi$
$829$	36.4602	1.26631	0.633157	+	0.774023i	$0.281759\pi$
$830$	50.4490	1.75111
$831$	19.0518	0.660900
$832$	5.48809	0.190265
$833$	−3.81287	−0.132108
$834$	3.84329	0.133082
$835$	45.2712	1.56668
$836$	27.4041	0.947791
$837$	−5.86484	−0.202718
$838$	33.6081	1.16097
$839$	8.09937	0.279621	0.139811	−	0.990178i	$-0.455351\pi$
$839$	8.09937	0.279621	0.139811	+	0.990178i	$0.455351\pi$
$840$	−18.1127	−0.624948
$841$	1.00000	0.0344828
$842$	15.5020	0.534235
$843$	−8.99733	−0.309885
$844$	17.1514	0.590375
$845$	67.7179	2.32957
$846$	−1.89626	−0.0651946
$847$	31.5113	1.08274
$848$	−7.90073	−0.271312
$849$	−17.1658	−0.589127
$850$	−2.90682	−0.0997030
$851$	−1.93070	−0.0661834
$852$	4.98632	0.170829
$853$	25.7619	0.882070	0.441035	−	0.897490i	$-0.354611\pi$
$853$	25.7619	0.882070	0.441035	+	0.897490i	$0.354611\pi$
$854$	28.5206	0.975953
$855$	−25.6349	−0.876695
$856$	13.4034	0.458117
$857$	−1.73119	−0.0591365	−0.0295682	−	0.999563i	$-0.509413\pi$
$857$	−1.73119	−0.0591365	−0.0295682	+	0.999563i	$0.509413\pi$
$858$	23.2075	0.792289
$859$	−26.6334	−0.908720	−0.454360	−	0.890818i	$-0.650132\pi$
$859$	−26.6334	−0.908720	−0.454360	+	0.890818i	$0.650132\pi$
$860$	22.5418	0.768670
$861$	−1.98339	−0.0675937
$862$	−7.05766	−0.240385
$863$	−16.7930	−0.571640	−0.285820	−	0.958283i	$-0.592266\pi$
$863$	−16.7930	−0.571640	−0.285820	+	0.958283i	$0.592266\pi$
$864$	−1.00000	−0.0340207
$865$	−72.6087	−2.46877
$866$	−16.1766	−0.549705
$867$	16.9255	0.574819
$868$	26.8545	0.911501
$869$	37.4771	1.27132
$870$	3.95569	0.134110
$871$	−46.6973	−1.58228
$872$	7.08602	0.239963
$873$	−6.36154	−0.215306
$874$	6.48051	0.219207
$875$	102.291	3.45807
$876$	−8.19425	−0.276858
$877$	−0.545957	−0.0184357	−0.00921783	−	0.999958i	$-0.502934\pi$
$877$	−0.545957	−0.0184357	−0.00921783	+	0.999958i	$0.502934\pi$
$878$	13.9649	0.471293
$879$	−29.3370	−0.989511
$880$	−16.7274	−0.563881
$881$	25.9821	0.875360	0.437680	−	0.899131i	$-0.355800\pi$
$881$	25.9821	0.875360	0.437680	+	0.899131i	$0.355800\pi$
$882$	13.9663	0.470270
$883$	−35.4400	−1.19265	−0.596326	−	0.802742i	$-0.703374\pi$
$883$	−35.4400	−1.19265	−0.596326	+	0.802742i	$0.703374\pi$
$884$	−1.49828	−0.0503925
$885$	39.4291	1.32540
$886$	−18.8457	−0.633134
$887$	−51.8307	−1.74030	−0.870152	−	0.492783i	$-0.835980\pi$
$887$	−51.8307	−1.74030	−0.870152	+	0.492783i	$0.835980\pi$
$888$	−1.93070	−0.0647900
$889$	−89.9655	−3.01735
$890$	−24.5514	−0.822966
$891$	−4.22869	−0.141667
$892$	−4.61381	−0.154482
$893$	12.2887	0.411226
$894$	−13.9976	−0.468150
$895$	16.7033	0.558329
$896$	4.57890	0.152970
$897$	5.48809	0.183242
$898$	19.9879	0.667004
$899$	−5.86484	−0.195603
$900$	10.6475	0.354916
$901$	2.15694	0.0718581
$902$	−1.83170	−0.0609888
$903$	−26.0932	−0.868329
$904$	−17.7916	−0.591740
$905$	75.9924	2.52607
$906$	21.9867	0.730460
$907$	−22.8191	−0.757697	−0.378848	−	0.925459i	$-0.623680\pi$
$907$	−22.8191	−0.757697	−0.378848	+	0.925459i	$0.623680\pi$
$908$	−27.7867	−0.922136
$909$	−6.21580	−0.206165
$910$	99.4041	3.29521
$911$	−21.4245	−0.709825	−0.354913	−	0.934899i	$-0.615489\pi$
$911$	−21.4245	−0.709825	−0.354913	+	0.934899i	$0.615489\pi$
$912$	6.48051	0.214591
$913$	−53.9307	−1.78485
$914$	−30.7310	−1.01649
$915$	−24.6388	−0.814533
$916$	−2.27584	−0.0751959
$917$	−45.8340	−1.51357
$918$	0.273005	0.00901051
$919$	−13.7433	−0.453349	−0.226674	−	0.973971i	$-0.572785\pi$
$919$	−13.7433	−0.453349	−0.226674	+	0.973971i	$0.572785\pi$
$920$	−3.95569	−0.130415
$921$	−12.8019	−0.421837
$922$	19.4368	0.640116
$923$	−27.3654	−0.900742
$924$	19.3628	0.636988
$925$	20.5571	0.675912
$926$	0.220124	0.00723373
$927$	−2.40509	−0.0789936
$928$	−1.00000	−0.0328266
$929$	−38.8612	−1.27499	−0.637497	−	0.770453i	$-0.720030\pi$
$929$	−38.8612	−1.27499	−0.637497	+	0.770453i	$0.720030\pi$
$930$	−23.1995	−0.760741
$931$	−90.5088	−2.96631
$932$	1.55620	0.0509749
$933$	−2.05227	−0.0671884
$934$	20.6737	0.676465
$935$	4.56667	0.149346
$936$	5.48809	0.179384
$937$	−3.38126	−0.110461	−0.0552305	−	0.998474i	$-0.517589\pi$
$937$	−3.38126	−0.110461	−0.0552305	+	0.998474i	$0.517589\pi$
$938$	−38.9612	−1.27213
$939$	6.99130	0.228153
$940$	−7.50100	−0.244656
$941$	43.9953	1.43420	0.717102	−	0.696968i	$-0.245468\pi$
$941$	43.9953	1.43420	0.717102	+	0.696968i	$0.245468\pi$
$942$	−16.9080	−0.550892
$943$	−0.433159	−0.0141056
$944$	−9.96770	−0.324421
$945$	−18.1127	−0.589206
$946$	−24.0976	−0.783480
$947$	−22.4182	−0.728495	−0.364248	−	0.931302i	$-0.618674\pi$
$947$	−22.4182	−0.728495	−0.364248	+	0.931302i	$0.618674\pi$
$948$	8.86257	0.287843
$949$	44.9708	1.45981
$950$	−69.0011	−2.23869
$951$	16.6476	0.539835
$952$	−1.25006	−0.0405148
$953$	−23.3944	−0.757819	−0.378909	−	0.925434i	$-0.623701\pi$
$953$	−23.3944	−0.757819	−0.378909	+	0.925434i	$0.623701\pi$
$954$	−7.90073	−0.255796
$955$	67.2006	2.17456
$956$	−1.42678	−0.0461454
$957$	−4.22869	−0.136694
$958$	−7.32078	−0.236524
$959$	−31.4230	−1.01470
$960$	−3.95569	−0.127669
$961$	3.39634	0.109559
$962$	10.5958	0.341624
$963$	13.4034	0.431917
$964$	−11.2979	−0.363879
$965$	−24.1339	−0.776897
$966$	4.57890	0.147324
$967$	−37.5206	−1.20658	−0.603291	−	0.797521i	$-0.706144\pi$
$967$	−37.5206	−1.20658	−0.603291	+	0.797521i	$0.706144\pi$
$968$	6.88186	0.221191
$969$	−1.76921	−0.0568353
$970$	−25.1643	−0.807976
$971$	4.25547	0.136564	0.0682822	−	0.997666i	$-0.478248\pi$
$971$	4.25547	0.136564	0.0682822	+	0.997666i	$0.478248\pi$
$972$	−1.00000	−0.0320750
$973$	−17.5980	−0.564167
$974$	18.2058	0.583351
$975$	−58.4343	−1.87140
$976$	6.22869	0.199376
$977$	37.4690	1.19874	0.599370	−	0.800472i	$-0.295418\pi$
$977$	37.4690	1.19874	0.599370	+	0.800472i	$0.295418\pi$
$978$	−17.7709	−0.568252
$979$	26.2459	0.838821
$980$	55.2464	1.76478
$981$	7.08602	0.226239
$982$	−39.0601	−1.24646
$983$	15.5369	0.495549	0.247775	−	0.968818i	$-0.420301\pi$
$983$	15.5369	0.495549	0.247775	+	0.968818i	$0.420301\pi$
$984$	−0.433159	−0.0138086
$985$	−58.9785	−1.87921
$986$	0.273005	0.00869425
$987$	8.68276	0.276375
$988$	−35.5656	−1.13149
$989$	−5.69859	−0.181204
$990$	−16.7274	−0.531632
$991$	−34.5886	−1.09874	−0.549372	−	0.835578i	$-0.685133\pi$
$991$	−34.5886	−1.09874	−0.549372	+	0.835578i	$0.685133\pi$
$992$	5.86484	0.186209
$993$	28.8743	0.916297
$994$	−22.8319	−0.724183
$995$	−41.4577	−1.31430
$996$	−12.7535	−0.404111
$997$	−31.7107	−1.00429	−0.502143	−	0.864784i	$-0.667455\pi$
$997$	−31.7107	−1.00429	−0.502143	+	0.864784i	$0.667455\pi$
$998$	30.7227	0.972509
$999$	−1.93070	−0.0610846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.8	✓	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.95569 q^{5} -1.00000 q^{6} +4.57890 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.95569 q^{5} -1.00000 q^{6} +4.57890 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.95569 q^{10} -4.22869 q^{11} -1.00000 q^{12} +5.48809 q^{13} +4.57890 q^{14} -3.95569 q^{15} +1.00000 q^{16} -0.273005 q^{17} +1.00000 q^{18} -6.48051 q^{19} +3.95569 q^{20} -4.57890 q^{21} -4.22869 q^{22} -1.00000 q^{23} -1.00000 q^{24} +10.6475 q^{25} +5.48809 q^{26} -1.00000 q^{27} +4.57890 q^{28} -1.00000 q^{29} -3.95569 q^{30} +5.86484 q^{31} +1.00000 q^{32} +4.22869 q^{33} -0.273005 q^{34} +18.1127 q^{35} +1.00000 q^{36} +1.93070 q^{37} -6.48051 q^{38} -5.48809 q^{39} +3.95569 q^{40} +0.433159 q^{41} -4.57890 q^{42} +5.69859 q^{43} -4.22869 q^{44} +3.95569 q^{45} -1.00000 q^{46} -1.89626 q^{47} -1.00000 q^{48} +13.9663 q^{49} +10.6475 q^{50} +0.273005 q^{51} +5.48809 q^{52} -7.90073 q^{53} -1.00000 q^{54} -16.7274 q^{55} +4.57890 q^{56} +6.48051 q^{57} -1.00000 q^{58} -9.96770 q^{59} -3.95569 q^{60} +6.22869 q^{61} +5.86484 q^{62} +4.57890 q^{63} +1.00000 q^{64} +21.7092 q^{65} +4.22869 q^{66} -8.50885 q^{67} -0.273005 q^{68} +1.00000 q^{69} +18.1127 q^{70} -4.98632 q^{71} +1.00000 q^{72} +8.19425 q^{73} +1.93070 q^{74} -10.6475 q^{75} -6.48051 q^{76} -19.3628 q^{77} -5.48809 q^{78} -8.86257 q^{79} +3.95569 q^{80} +1.00000 q^{81} +0.433159 q^{82} +12.7535 q^{83} -4.57890 q^{84} -1.07992 q^{85} +5.69859 q^{86} +1.00000 q^{87} -4.22869 q^{88} -6.20661 q^{89} +3.95569 q^{90} +25.1294 q^{91} -1.00000 q^{92} -5.86484 q^{93} -1.89626 q^{94} -25.6349 q^{95} -1.00000 q^{96} -6.36154 q^{97} +13.9663 q^{98} -4.22869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more