LMFDB - Embedded newform 4002.2.a.bd.1.1

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:	$4$
Coefficient field:	4.4.19796.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 8 $ x^4 - x^3 - 7*x^2 + x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.28586$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.71045 q^{5} +1.00000 q^{6} +2.42459 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.71045 q^{5} +1.00000 q^{6} +2.42459 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.71045 q^{10} +5.28586 q^{11} +1.00000 q^{12} +1.77482 q^{13} +2.42459 q^{14} -3.71045 q^{15} +1.00000 q^{16} +2.42459 q^{17} +1.00000 q^{18} -2.42459 q^{19} -3.71045 q^{20} +2.42459 q^{21} +5.28586 q^{22} -1.00000 q^{23} +1.00000 q^{24} +8.76746 q^{25} +1.77482 q^{26} +1.00000 q^{27} +2.42459 q^{28} +1.00000 q^{29} -3.71045 q^{30} -7.19573 q^{31} +1.00000 q^{32} +5.28586 q^{33} +2.42459 q^{34} -8.99632 q^{35} +1.00000 q^{36} -8.82814 q^{37} -2.42459 q^{38} +1.77482 q^{39} -3.71045 q^{40} +11.7675 q^{41} +2.42459 q^{42} +4.45035 q^{43} +5.28586 q^{44} -3.71045 q^{45} -1.00000 q^{46} -10.5386 q^{47} +1.00000 q^{48} -1.12137 q^{49} +8.76746 q^{50} +2.42459 q^{51} +1.77482 q^{52} +8.75357 q^{53} +1.00000 q^{54} -19.6129 q^{55} +2.42459 q^{56} -2.42459 q^{57} +1.00000 q^{58} +12.3392 q^{59} -3.71045 q^{60} +8.37779 q^{61} -7.19573 q^{62} +2.42459 q^{63} +1.00000 q^{64} -6.58539 q^{65} +5.28586 q^{66} +5.55438 q^{67} +2.42459 q^{68} -1.00000 q^{69} -8.99632 q^{70} -1.49738 q^{71} +1.00000 q^{72} -4.66734 q^{73} -8.82814 q^{74} +8.76746 q^{75} -2.42459 q^{76} +12.8160 q^{77} +1.77482 q^{78} +6.41722 q^{79} -3.71045 q^{80} +1.00000 q^{81} +11.7675 q^{82} -1.75725 q^{83} +2.42459 q^{84} -8.99632 q^{85} +4.45035 q^{86} +1.00000 q^{87} +5.28586 q^{88} -4.75357 q^{89} -3.71045 q^{90} +4.30321 q^{91} -1.00000 q^{92} -7.19573 q^{93} -10.5386 q^{94} +8.99632 q^{95} +1.00000 q^{96} +10.1398 q^{97} -1.12137 q^{98} +5.28586 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * 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
$5$	−3.71045	−1.65936	−0.829682	−	0.558236i	$-0.811478\pi$
$5$	−3.71045	−1.65936	−0.829682	+	0.558236i	$0.811478\pi$
$6$	1.00000	0.408248
$7$	2.42459	0.916408	0.458204	−	0.888847i	$-0.348493\pi$
$7$	2.42459	0.916408	0.458204	+	0.888847i	$0.348493\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.71045	−1.17335
$11$	5.28586	1.59375	0.796874	−	0.604145i	$-0.206485\pi$
$11$	5.28586	1.59375	0.796874	+	0.604145i	$0.206485\pi$
$12$	1.00000	0.288675
$13$	1.77482	0.492247	0.246124	−	0.969238i	$-0.420843\pi$
$13$	1.77482	0.492247	0.246124	+	0.969238i	$0.420843\pi$
$14$	2.42459	0.647998
$15$	−3.71045	−0.958035
$16$	1.00000	0.250000
$17$	2.42459	0.588049	0.294024	−	0.955798i	$-0.405005\pi$
$17$	2.42459	0.588049	0.294024	+	0.955798i	$0.405005\pi$
$18$	1.00000	0.235702
$19$	−2.42459	−0.556239	−0.278119	−	0.960547i	$-0.589711\pi$
$19$	−2.42459	−0.556239	−0.278119	+	0.960547i	$0.589711\pi$
$20$	−3.71045	−0.829682
$21$	2.42459	0.529088
$22$	5.28586	1.12695
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	8.76746	1.75349
$26$	1.77482	0.348071
$27$	1.00000	0.192450
$28$	2.42459	0.458204
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.71045	−0.677433
$31$	−7.19573	−1.29239	−0.646195	−	0.763172i	$-0.723641\pi$
$31$	−7.19573	−1.29239	−0.646195	+	0.763172i	$0.723641\pi$
$32$	1.00000	0.176777
$33$	5.28586	0.920151
$34$	2.42459	0.415813
$35$	−8.99632	−1.52066
$36$	1.00000	0.166667
$37$	−8.82814	−1.45134	−0.725669	−	0.688044i	$-0.758470\pi$
$37$	−8.82814	−1.45134	−0.725669	+	0.688044i	$0.758470\pi$
$38$	−2.42459	−0.393320
$39$	1.77482	0.284199
$40$	−3.71045	−0.586674
$41$	11.7675	1.83777	0.918884	−	0.394528i	$-0.129092\pi$
$41$	11.7675	1.83777	0.918884	+	0.394528i	$0.129092\pi$
$42$	2.42459	0.374122
$43$	4.45035	0.678673	0.339336	−	0.940665i	$-0.389798\pi$
$43$	4.45035	0.678673	0.339336	+	0.940665i	$0.389798\pi$
$44$	5.28586	0.796874
$45$	−3.71045	−0.553122
$46$	−1.00000	−0.147442
$47$	−10.5386	−1.53721	−0.768606	−	0.639722i	$-0.779049\pi$
$47$	−10.5386	−1.53721	−0.768606	+	0.639722i	$0.779049\pi$
$48$	1.00000	0.144338
$49$	−1.12137	−0.160196
$50$	8.76746	1.23991
$51$	2.42459	0.339510
$52$	1.77482	0.246124
$53$	8.75357	1.20240	0.601198	−	0.799100i	$-0.294690\pi$
$53$	8.75357	1.20240	0.601198	+	0.799100i	$0.294690\pi$
$54$	1.00000	0.136083
$55$	−19.6129	−2.64461
$56$	2.42459	0.323999
$57$	−2.42459	−0.321144
$58$	1.00000	0.131306
$59$	12.3392	1.60643	0.803213	−	0.595692i	$-0.203122\pi$
$59$	12.3392	1.60643	0.803213	+	0.595692i	$0.203122\pi$
$60$	−3.71045	−0.479017
$61$	8.37779	1.07267	0.536333	−	0.844006i	$-0.319809\pi$
$61$	8.37779	1.07267	0.536333	+	0.844006i	$0.319809\pi$
$62$	−7.19573	−0.913858
$63$	2.42459	0.305469
$64$	1.00000	0.125000
$65$	−6.58539	−0.816818
$66$	5.28586	0.650645
$67$	5.55438	0.678576	0.339288	−	0.940683i	$-0.389814\pi$
$67$	5.55438	0.678576	0.339288	+	0.940683i	$0.389814\pi$
$68$	2.42459	0.294024
$69$	−1.00000	−0.120386
$70$	−8.99632	−1.07527
$71$	−1.49738	−0.177706	−0.0888530	−	0.996045i	$-0.528320\pi$
$71$	−1.49738	−0.177706	−0.0888530	+	0.996045i	$0.528320\pi$
$72$	1.00000	0.117851
$73$	−4.66734	−0.546270	−0.273135	−	0.961976i	$-0.588061\pi$
$73$	−4.66734	−0.546270	−0.273135	+	0.961976i	$0.588061\pi$
$74$	−8.82814	−1.02625
$75$	8.76746	1.01238
$76$	−2.42459	−0.278119
$77$	12.8160	1.46052
$78$	1.77482	0.200959
$79$	6.41722	0.721994	0.360997	−	0.932567i	$-0.382436\pi$
$79$	6.41722	0.721994	0.360997	+	0.932567i	$0.382436\pi$
$80$	−3.71045	−0.414841
$81$	1.00000	0.111111
$82$	11.7675	1.29950
$83$	−1.75725	−0.192883	−0.0964417	−	0.995339i	$-0.530746\pi$
$83$	−1.75725	−0.192883	−0.0964417	+	0.995339i	$0.530746\pi$
$84$	2.42459	0.264544
$85$	−8.99632	−0.975788
$86$	4.45035	0.479894
$87$	1.00000	0.107211
$88$	5.28586	0.563475
$89$	−4.75357	−0.503877	−0.251939	−	0.967743i	$-0.581068\pi$
$89$	−4.75357	−0.503877	−0.251939	+	0.967743i	$0.581068\pi$
$90$	−3.71045	−0.391116
$91$	4.30321	0.451099
$92$	−1.00000	−0.104257
$93$	−7.19573	−0.746162
$94$	−10.5386	−1.08697
$95$	8.99632	0.923003
$96$	1.00000	0.102062
$97$	10.1398	1.02954	0.514769	−	0.857329i	$-0.327878\pi$
$97$	10.1398	1.02954	0.514769	+	0.857329i	$0.327878\pi$
$98$	−1.12137	−0.113276
$99$	5.28586	0.531249
$100$	8.76746	0.876746
$101$	4.34655	0.432498	0.216249	−	0.976338i	$-0.430618\pi$
$101$	4.34655	0.432498	0.216249	+	0.976338i	$0.430618\pi$
$102$	2.42459	0.240070
$103$	−6.18657	−0.609581	−0.304791	−	0.952419i	$-0.598586\pi$
$103$	−6.18657	−0.609581	−0.304791	+	0.952419i	$0.598586\pi$
$104$	1.77482	0.174036
$105$	−8.99632	−0.877951
$106$	8.75357	0.850222
$107$	10.0184	0.968515	0.484258	−	0.874925i	$-0.339090\pi$
$107$	10.0184	0.968515	0.484258	+	0.874925i	$0.339090\pi$
$108$	1.00000	0.0962250
$109$	3.54965	0.339994	0.169997	−	0.985445i	$-0.445624\pi$
$109$	3.54965	0.339994	0.169997	+	0.985445i	$0.445624\pi$
$110$	−19.6129	−1.87002
$111$	−8.82814	−0.837930
$112$	2.42459	0.229102
$113$	20.5128	1.92968	0.964842	−	0.262829i	$-0.0846556\pi$
$113$	20.5128	1.92968	0.964842	+	0.262829i	$0.0846556\pi$
$114$	−2.42459	−0.227083
$115$	3.71045	0.346001
$116$	1.00000	0.0928477
$117$	1.77482	0.164082
$118$	12.3392	1.13591
$119$	5.87863	0.538893
$120$	−3.71045	−0.338716
$121$	16.9404	1.54003
$122$	8.37779	0.758489
$123$	11.7675	1.06104
$124$	−7.19573	−0.646195
$125$	−13.9790	−1.25032
$126$	2.42459	0.215999
$127$	1.94773	0.172833	0.0864166	−	0.996259i	$-0.472458\pi$
$127$	1.94773	0.172833	0.0864166	+	0.996259i	$0.472458\pi$
$128$	1.00000	0.0883883
$129$	4.45035	0.391832
$130$	−6.58539	−0.577577
$131$	−14.3825	−1.25661	−0.628304	−	0.777968i	$-0.716250\pi$
$131$	−14.3825	−1.25661	−0.628304	+	0.777968i	$0.716250\pi$
$132$	5.28586	0.460075
$133$	−5.87863	−0.509741
$134$	5.55438	0.479825
$135$	−3.71045	−0.319345
$136$	2.42459	0.207907
$137$	12.1229	1.03573	0.517866	−	0.855462i	$-0.326726\pi$
$137$	12.1229	1.03573	0.517866	+	0.855462i	$0.326726\pi$
$138$	−1.00000	−0.0851257
$139$	−7.81080	−0.662503	−0.331252	−	0.943542i	$-0.607471\pi$
$139$	−7.81080	−0.662503	−0.331252	+	0.943542i	$0.607471\pi$
$140$	−8.99632	−0.760328
$141$	−10.5386	−0.887510
$142$	−1.49738	−0.125657
$143$	9.38147	0.784518
$144$	1.00000	0.0833333
$145$	−3.71045	−0.308136
$146$	−4.66734	−0.386272
$147$	−1.12137	−0.0924894
$148$	−8.82814	−0.725669
$149$	−18.0930	−1.48223	−0.741117	−	0.671376i	$-0.765704\pi$
$149$	−18.0930	−1.48223	−0.741117	+	0.671376i	$0.765704\pi$
$150$	8.76746	0.715860
$151$	18.2869	1.48817	0.744084	−	0.668086i	$-0.232886\pi$
$151$	18.2869	1.48817	0.744084	+	0.668086i	$0.232886\pi$
$152$	−2.42459	−0.196660
$153$	2.42459	0.196016
$154$	12.8160	1.03275
$155$	26.6994	2.14455
$156$	1.77482	0.142100
$157$	−17.1450	−1.36832	−0.684161	−	0.729331i	$-0.739831\pi$
$157$	−17.1450	−1.36832	−0.684161	+	0.729331i	$0.739831\pi$
$158$	6.41722	0.510527
$159$	8.75357	0.694203
$160$	−3.71045	−0.293337
$161$	−2.42459	−0.191084
$162$	1.00000	0.0785674
$163$	−11.2473	−0.880954	−0.440477	−	0.897764i	$-0.645191\pi$
$163$	−11.2473	−0.880954	−0.440477	+	0.897764i	$0.645191\pi$
$164$	11.7675	0.918884
$165$	−19.6129	−1.52687
$166$	−1.75725	−0.136389
$167$	14.3134	1.10761	0.553803	−	0.832648i	$-0.313176\pi$
$167$	14.3134	1.10761	0.553803	+	0.832648i	$0.313176\pi$
$168$	2.42459	0.187061
$169$	−9.85000	−0.757693
$170$	−8.99632	−0.689986
$171$	−2.42459	−0.185413
$172$	4.45035	0.339336
$173$	4.26852	0.324529	0.162265	−	0.986747i	$-0.448120\pi$
$173$	4.26852	0.324529	0.162265	+	0.986747i	$0.448120\pi$
$174$	1.00000	0.0758098
$175$	21.2575	1.60691
$176$	5.28586	0.398437
$177$	12.3392	0.927470
$178$	−4.75357	−0.356295
$179$	−20.3568	−1.52154	−0.760768	−	0.649024i	$-0.775177\pi$
$179$	−20.3568	−1.52154	−0.760768	+	0.649024i	$0.775177\pi$
$180$	−3.71045	−0.276561
$181$	−24.2553	−1.80289	−0.901443	−	0.432898i	$-0.857491\pi$
$181$	−24.2553	−1.80289	−0.901443	+	0.432898i	$0.857491\pi$
$182$	4.30321	0.318975
$183$	8.37779	0.619304
$184$	−1.00000	−0.0737210
$185$	32.7564	2.40830
$186$	−7.19573	−0.527616
$187$	12.8160	0.937202
$188$	−10.5386	−0.768606
$189$	2.42459	0.176363
$190$	8.99632	0.652661
$191$	14.9584	1.08236	0.541178	−	0.840908i	$-0.317979\pi$
$191$	14.9584	1.08236	0.541178	+	0.840908i	$0.317979\pi$
$192$	1.00000	0.0721688
$193$	3.20392	0.230623	0.115312	−	0.993329i	$-0.463213\pi$
$193$	3.20392	0.230623	0.115312	+	0.993329i	$0.463213\pi$
$194$	10.1398	0.727993
$195$	−6.58539	−0.471590
$196$	−1.12137	−0.0800982
$197$	8.72780	0.621830	0.310915	−	0.950438i	$-0.399365\pi$
$197$	8.72780	0.621830	0.310915	+	0.950438i	$0.399365\pi$
$198$	5.28586	0.375650
$199$	−15.5212	−1.10027	−0.550136	−	0.835075i	$-0.685424\pi$
$199$	−15.5212	−1.10027	−0.550136	+	0.835075i	$0.685424\pi$
$200$	8.76746	0.619953
$201$	5.55438	0.391776
$202$	4.34655	0.305822
$203$	2.42459	0.170173
$204$	2.42459	0.169755
$205$	−43.6626	−3.04953
$206$	−6.18657	−0.431039
$207$	−1.00000	−0.0695048
$208$	1.77482	0.123062
$209$	−12.8160	−0.886504
$210$	−8.99632	−0.620805
$211$	1.74013	0.119795	0.0598976	−	0.998205i	$-0.480923\pi$
$211$	1.74013	0.119795	0.0598976	+	0.998205i	$0.480923\pi$
$212$	8.75357	0.601198
$213$	−1.49738	−0.102599
$214$	10.0184	0.684844
$215$	−16.5128	−1.12617
$216$	1.00000	0.0680414
$217$	−17.4467	−1.18436
$218$	3.54965	0.240412
$219$	−4.66734	−0.315389
$220$	−19.6129	−1.32230
$221$	4.30321	0.289465
$222$	−8.82814	−0.592506
$223$	4.81604	0.322506	0.161253	−	0.986913i	$-0.448446\pi$
$223$	4.81604	0.322506	0.161253	+	0.986913i	$0.448446\pi$
$224$	2.42459	0.162000
$225$	8.76746	0.584497
$226$	20.5128	1.36449
$227$	20.5297	1.36260	0.681301	−	0.732003i	$-0.261415\pi$
$227$	20.5297	1.36260	0.681301	+	0.732003i	$0.261415\pi$
$228$	−2.42459	−0.160572
$229$	19.5743	1.29351	0.646755	−	0.762698i	$-0.276126\pi$
$229$	19.5743	1.29351	0.646755	+	0.762698i	$0.276126\pi$
$230$	3.71045	0.244660
$231$	12.8160	0.843234
$232$	1.00000	0.0656532
$233$	0.700470	0.0458893	0.0229447	−	0.999737i	$-0.492696\pi$
$233$	0.700470	0.0458893	0.0229447	+	0.999737i	$0.492696\pi$
$234$	1.77482	0.116024
$235$	39.1030	2.55080
$236$	12.3392	0.803213
$237$	6.41722	0.416843
$238$	5.87863	0.381055
$239$	25.0686	1.62155	0.810775	−	0.585358i	$-0.199046\pi$
$239$	25.0686	1.62155	0.810775	+	0.585358i	$0.199046\pi$
$240$	−3.71045	−0.239509
$241$	−8.38253	−0.539966	−0.269983	−	0.962865i	$-0.587018\pi$
$241$	−8.38253	−0.539966	−0.269983	+	0.962865i	$0.587018\pi$
$242$	16.9404	1.08897
$243$	1.00000	0.0641500
$244$	8.37779	0.536333
$245$	4.16081	0.265824
$246$	11.7675	0.750266
$247$	−4.30321	−0.273807
$248$	−7.19573	−0.456929
$249$	−1.75725	−0.111361
$250$	−13.9790	−0.884108
$251$	14.5523	0.918531	0.459265	−	0.888299i	$-0.348113\pi$
$251$	14.5523	0.918531	0.459265	+	0.888299i	$0.348113\pi$
$252$	2.42459	0.152735
$253$	−5.28586	−0.332319
$254$	1.94773	0.122212
$255$	−8.99632	−0.563371
$256$	1.00000	0.0625000
$257$	−18.4446	−1.15054	−0.575270	−	0.817964i	$-0.695103\pi$
$257$	−18.4446	−1.15054	−0.575270	+	0.817964i	$0.695103\pi$
$258$	4.45035	0.277067
$259$	−21.4046	−1.33002
$260$	−6.58539	−0.408409
$261$	1.00000	0.0618984
$262$	−14.3825	−0.888555
$263$	−3.07352	−0.189522	−0.0947608	−	0.995500i	$-0.530209\pi$
$263$	−3.07352	−0.189522	−0.0947608	+	0.995500i	$0.530209\pi$
$264$	5.28586	0.325322
$265$	−32.4797	−1.99521
$266$	−5.87863	−0.360442
$267$	−4.75357	−0.290914
$268$	5.55438	0.339288
$269$	19.8483	1.21017	0.605087	−	0.796159i	$-0.293138\pi$
$269$	19.8483	1.21017	0.605087	+	0.796159i	$0.293138\pi$
$270$	−3.71045	−0.225811
$271$	5.02282	0.305114	0.152557	−	0.988295i	$-0.451249\pi$
$271$	5.02282	0.305114	0.152557	+	0.988295i	$0.451249\pi$
$272$	2.42459	0.147012
$273$	4.30321	0.260442
$274$	12.1229	0.732373
$275$	46.3436	2.79462
$276$	−1.00000	−0.0601929
$277$	−30.3392	−1.82291	−0.911453	−	0.411405i	$-0.865038\pi$
$277$	−30.3392	−1.82291	−0.911453	+	0.411405i	$0.865038\pi$
$278$	−7.81080	−0.468460
$279$	−7.19573	−0.430797
$280$	−8.99632	−0.537633
$281$	−10.2369	−0.610685	−0.305343	−	0.952243i	$-0.598771\pi$
$281$	−10.2369	−0.610685	−0.305343	+	0.952243i	$0.598771\pi$
$282$	−10.5386	−0.627564
$283$	27.4272	1.63038	0.815189	−	0.579195i	$-0.196633\pi$
$283$	27.4272	1.63038	0.815189	+	0.579195i	$0.196633\pi$
$284$	−1.49738	−0.0888530
$285$	8.99632	0.532896
$286$	9.38147	0.554738
$287$	28.5312	1.68415
$288$	1.00000	0.0589256
$289$	−11.1214	−0.654199
$290$	−3.71045	−0.217885
$291$	10.1398	0.594404
$292$	−4.66734	−0.273135
$293$	11.4798	0.670657	0.335329	−	0.942101i	$-0.391153\pi$
$293$	11.4798	0.670657	0.335329	+	0.942101i	$0.391153\pi$
$294$	−1.12137	−0.0653999
$295$	−45.7840	−2.66565
$296$	−8.82814	−0.513125
$297$	5.28586	0.306717
$298$	−18.0930	−1.04810
$299$	−1.77482	−0.102641
$300$	8.76746	0.506189
$301$	10.7903	0.621941
$302$	18.2869	1.05229
$303$	4.34655	0.249703
$304$	−2.42459	−0.139060
$305$	−31.0854	−1.77994
$306$	2.42459	0.138604
$307$	−26.0617	−1.48742	−0.743711	−	0.668501i	$-0.766936\pi$
$307$	−26.0617	−1.48742	−0.743711	+	0.668501i	$0.766936\pi$
$308$	12.8160	0.730262
$309$	−6.18657	−0.351942
$310$	26.6994	1.51642
$311$	−31.1450	−1.76607	−0.883036	−	0.469305i	$-0.844504\pi$
$311$	−31.1450	−1.76607	−0.883036	+	0.469305i	$0.844504\pi$
$312$	1.77482	0.100480
$313$	21.8623	1.23573	0.617866	−	0.786283i	$-0.287997\pi$
$313$	21.8623	1.23573	0.617866	+	0.786283i	$0.287997\pi$
$314$	−17.1450	−0.967550
$315$	−8.99632	−0.506885
$316$	6.41722	0.360997
$317$	−30.0722	−1.68902	−0.844512	−	0.535536i	$-0.820110\pi$
$317$	−30.0722	−1.68902	−0.844512	+	0.535536i	$0.820110\pi$
$318$	8.75357	0.490876
$319$	5.28586	0.295952
$320$	−3.71045	−0.207421
$321$	10.0184	0.559173
$322$	−2.42459	−0.135117
$323$	−5.87863	−0.327095
$324$	1.00000	0.0555556
$325$	15.5607	0.863151
$326$	−11.2473	−0.622928
$327$	3.54965	0.196296
$328$	11.7675	0.649749
$329$	−25.5518	−1.40871
$330$	−19.6129	−1.07966
$331$	−8.97792	−0.493471	−0.246735	−	0.969083i	$-0.579358\pi$
$331$	−8.97792	−0.493471	−0.246735	+	0.969083i	$0.579358\pi$
$332$	−1.75725	−0.0964417
$333$	−8.82814	−0.483779
$334$	14.3134	0.783195
$335$	−20.6093	−1.12600
$336$	2.42459	0.132272
$337$	−26.9842	−1.46992	−0.734962	−	0.678108i	$-0.762800\pi$
$337$	−26.9842	−1.46992	−0.734962	+	0.678108i	$0.762800\pi$
$338$	−9.85000	−0.535770
$339$	20.5128	1.11410
$340$	−8.99632	−0.487894
$341$	−38.0356	−2.05975
$342$	−2.42459	−0.131107
$343$	−19.6910	−1.06321
$344$	4.45035	0.239947
$345$	3.71045	0.199764
$346$	4.26852	0.229477
$347$	−21.1435	−1.13504	−0.567520	−	0.823359i	$-0.692097\pi$
$347$	−21.1435	−1.13504	−0.567520	+	0.823359i	$0.692097\pi$
$348$	1.00000	0.0536056
$349$	−27.0654	−1.44878	−0.724389	−	0.689391i	$-0.757878\pi$
$349$	−27.0654	−1.44878	−0.724389	+	0.689391i	$0.757878\pi$
$350$	21.2575	1.13626
$351$	1.77482	0.0947330
$352$	5.28586	0.281738
$353$	4.87494	0.259467	0.129734	−	0.991549i	$-0.458588\pi$
$353$	4.87494	0.259467	0.129734	+	0.991549i	$0.458588\pi$
$354$	12.3392	0.655820
$355$	5.55594	0.294879
$356$	−4.75357	−0.251939
$357$	5.87863	0.311130
$358$	−20.3568	−1.07589
$359$	−10.4414	−0.551077	−0.275539	−	0.961290i	$-0.588856\pi$
$359$	−10.4414	−0.551077	−0.275539	+	0.961290i	$0.588856\pi$
$360$	−3.71045	−0.195558
$361$	−13.1214	−0.690599
$362$	−24.2553	−1.27483
$363$	16.9404	0.889139
$364$	4.30321	0.225550
$365$	17.3179	0.906462
$366$	8.37779	0.437914
$367$	−21.3788	−1.11597	−0.557983	−	0.829852i	$-0.688425\pi$
$367$	−21.3788	−1.11597	−0.557983	+	0.829852i	$0.688425\pi$
$368$	−1.00000	−0.0521286
$369$	11.7675	0.612589
$370$	32.7564	1.70292
$371$	21.2238	1.10188
$372$	−7.19573	−0.373081
$373$	−1.66209	−0.0860598	−0.0430299	−	0.999074i	$-0.513701\pi$
$373$	−1.66209	−0.0860598	−0.0430299	+	0.999074i	$0.513701\pi$
$374$	12.8160	0.662702
$375$	−13.9790	−0.721871
$376$	−10.5386	−0.543487
$377$	1.77482	0.0914080
$378$	2.42459	0.124707
$379$	35.1030	1.80312	0.901559	−	0.432656i	$-0.142424\pi$
$379$	35.1030	1.80312	0.901559	+	0.432656i	$0.142424\pi$
$380$	8.99632	0.461501
$381$	1.94773	0.0997853
$382$	14.9584	0.765341
$383$	−20.3236	−1.03849	−0.519244	−	0.854626i	$-0.673787\pi$
$383$	−20.3236	−1.03849	−0.519244	+	0.854626i	$0.673787\pi$
$384$	1.00000	0.0510310
$385$	−47.5533	−2.42354
$386$	3.20392	0.163075
$387$	4.45035	0.226224
$388$	10.1398	0.514769
$389$	17.1802	0.871069	0.435535	−	0.900172i	$-0.356559\pi$
$389$	17.1802	0.871069	0.435535	+	0.900172i	$0.356559\pi$
$390$	−6.58539	−0.333464
$391$	−2.42459	−0.122617
$392$	−1.12137	−0.0566380
$393$	−14.3825	−0.725502
$394$	8.72780	0.439700
$395$	−23.8108	−1.19805
$396$	5.28586	0.265625
$397$	−11.6359	−0.583988	−0.291994	−	0.956420i	$-0.594319\pi$
$397$	−11.6359	−0.583988	−0.291994	+	0.956420i	$0.594319\pi$
$398$	−15.5212	−0.778010
$399$	−5.87863	−0.294299
$400$	8.76746	0.438373
$401$	−2.86284	−0.142963	−0.0714817	−	0.997442i	$-0.522773\pi$
$401$	−2.86284	−0.142963	−0.0714817	+	0.997442i	$0.522773\pi$
$402$	5.55438	0.277027
$403$	−12.7711	−0.636176
$404$	4.34655	0.216249
$405$	−3.71045	−0.184374
$406$	2.42459	0.120330
$407$	−46.6644	−2.31307
$408$	2.42459	0.120035
$409$	13.3783	0.661514	0.330757	−	0.943716i	$-0.392696\pi$
$409$	13.3783	0.661514	0.330757	+	0.943716i	$0.392696\pi$
$410$	−43.6626	−2.15634
$411$	12.1229	0.597980
$412$	−6.18657	−0.304791
$413$	29.9174	1.47214
$414$	−1.00000	−0.0491473
$415$	6.52020	0.320064
$416$	1.77482	0.0870178
$417$	−7.81080	−0.382496
$418$	−12.8160	−0.626853
$419$	−30.6510	−1.49740	−0.748701	−	0.662908i	$-0.769322\pi$
$419$	−30.6510	−1.49740	−0.748701	+	0.662908i	$0.769322\pi$
$420$	−8.99632	−0.438975
$421$	9.48812	0.462422	0.231211	−	0.972904i	$-0.425731\pi$
$421$	9.48812	0.462422	0.231211	+	0.972904i	$0.425731\pi$
$422$	1.74013	0.0847080
$423$	−10.5386	−0.512404
$424$	8.75357	0.425111
$425$	21.2575	1.03114
$426$	−1.49738	−0.0725481
$427$	20.3127	0.983000
$428$	10.0184	0.484258
$429$	9.38147	0.452942
$430$	−16.5128	−0.796319
$431$	−11.2496	−0.541872	−0.270936	−	0.962597i	$-0.587333\pi$
$431$	−11.2496	−0.541872	−0.270936	+	0.962597i	$0.587333\pi$
$432$	1.00000	0.0481125
$433$	−28.5218	−1.37067	−0.685334	−	0.728229i	$-0.740344\pi$
$433$	−28.5218	−1.37067	−0.685334	+	0.728229i	$0.740344\pi$
$434$	−17.4467	−0.837467
$435$	−3.71045	−0.177903
$436$	3.54965	0.169997
$437$	2.42459	0.115984
$438$	−4.66734	−0.223014
$439$	−7.96486	−0.380142	−0.190071	−	0.981770i	$-0.560872\pi$
$439$	−7.96486	−0.380142	−0.190071	+	0.981770i	$0.560872\pi$
$440$	−19.6129	−0.935011
$441$	−1.12137	−0.0533988
$442$	4.30321	0.204683
$443$	−25.7262	−1.22229	−0.611145	−	0.791519i	$-0.709291\pi$
$443$	−25.7262	−1.22229	−0.611145	+	0.791519i	$0.709291\pi$
$444$	−8.82814	−0.418965
$445$	17.6379	0.836116
$446$	4.81604	0.228046
$447$	−18.0930	−0.855768
$448$	2.42459	0.114551
$449$	−6.72854	−0.317539	−0.158770	−	0.987316i	$-0.550753\pi$
$449$	−6.72854	−0.317539	−0.158770	+	0.987316i	$0.550753\pi$
$450$	8.76746	0.413302
$451$	62.2012	2.92894
$452$	20.5128	0.964842
$453$	18.2869	0.859194
$454$	20.5297	0.963505
$455$	−15.9669	−0.748538
$456$	−2.42459	−0.113542
$457$	−23.3988	−1.09455	−0.547275	−	0.836953i	$-0.684335\pi$
$457$	−23.3988	−1.09455	−0.547275	+	0.836953i	$0.684335\pi$
$458$	19.5743	0.914649
$459$	2.42459	0.113170
$460$	3.71045	0.173001
$461$	−7.19729	−0.335211	−0.167606	−	0.985854i	$-0.553604\pi$
$461$	−7.19729	−0.335211	−0.167606	+	0.985854i	$0.553604\pi$
$462$	12.8160	0.596256
$463$	42.4997	1.97513	0.987563	−	0.157221i	$-0.0502536\pi$
$463$	42.4997	1.97513	0.987563	+	0.157221i	$0.0502536\pi$
$464$	1.00000	0.0464238
$465$	26.6994	1.23816
$466$	0.700470	0.0324486
$467$	22.3778	1.03552	0.517760	−	0.855526i	$-0.326766\pi$
$467$	22.3778	1.03552	0.517760	+	0.855526i	$0.326766\pi$
$468$	1.77482	0.0820412
$469$	13.4671	0.621852
$470$	39.1030	1.80369
$471$	−17.1450	−0.790001
$472$	12.3392	0.567957
$473$	23.5240	1.08163
$474$	6.41722	0.294753
$475$	−21.2575	−0.975360
$476$	5.87863	0.269446
$477$	8.75357	0.400798
$478$	25.0686	1.14661
$479$	−28.8203	−1.31683	−0.658417	−	0.752653i	$-0.728774\pi$
$479$	−28.8203	−1.31683	−0.658417	+	0.752653i	$0.728774\pi$
$480$	−3.71045	−0.169358
$481$	−15.6684	−0.714417
$482$	−8.38253	−0.381814
$483$	−2.42459	−0.110323
$484$	16.9404	0.770017
$485$	−37.6232	−1.70838
$486$	1.00000	0.0453609
$487$	24.8091	1.12421	0.562104	−	0.827066i	$-0.309992\pi$
$487$	24.8091	1.12421	0.562104	+	0.827066i	$0.309992\pi$
$488$	8.37779	0.379245
$489$	−11.2473	−0.508619
$490$	4.16081	0.187966
$491$	−14.8507	−0.670204	−0.335102	−	0.942182i	$-0.608771\pi$
$491$	−14.8507	−0.670204	−0.335102	+	0.942182i	$0.608771\pi$
$492$	11.7675	0.530518
$493$	2.42459	0.109198
$494$	−4.30321	−0.193611
$495$	−19.6129	−0.881537
$496$	−7.19573	−0.323098
$497$	−3.63052	−0.162851
$498$	−1.75725	−0.0787443
$499$	−10.7520	−0.481326	−0.240663	−	0.970609i	$-0.577365\pi$
$499$	−10.7520	−0.481326	−0.240663	+	0.970609i	$0.577365\pi$
$500$	−13.9790	−0.625158
$501$	14.3134	0.639476
$502$	14.5523	0.649499
$503$	−2.03313	−0.0906529	−0.0453265	−	0.998972i	$-0.514433\pi$
$503$	−2.03313	−0.0906529	−0.0453265	+	0.998972i	$0.514433\pi$
$504$	2.42459	0.108000
$505$	−16.1277	−0.717672
$506$	−5.28586	−0.234985
$507$	−9.85000	−0.437454
$508$	1.94773	0.0864166
$509$	29.5260	1.30872	0.654358	−	0.756185i	$-0.272939\pi$
$509$	29.5260	1.30872	0.654358	+	0.756185i	$0.272939\pi$
$510$	−8.99632	−0.398364
$511$	−11.3164	−0.500607
$512$	1.00000	0.0441942
$513$	−2.42459	−0.107048
$514$	−18.4446	−0.813554
$515$	22.9550	1.01152
$516$	4.45035	0.195916
$517$	−55.7056	−2.44993
$518$	−21.4046	−0.940465
$519$	4.26852	0.187367
$520$	−6.58539	−0.288789
$521$	7.63789	0.334622	0.167311	−	0.985904i	$-0.446492\pi$
$521$	7.63789	0.334622	0.167311	+	0.985904i	$0.446492\pi$
$522$	1.00000	0.0437688
$523$	−38.9789	−1.70443	−0.852213	−	0.523195i	$-0.824740\pi$
$523$	−38.9789	−1.70443	−0.852213	+	0.523195i	$0.824740\pi$
$524$	−14.3825	−0.628304
$525$	21.2575	0.927752
$526$	−3.07352	−0.134012
$527$	−17.4467	−0.759989
$528$	5.28586	0.230038
$529$	1.00000	0.0434783
$530$	−32.4797	−1.41083
$531$	12.3392	0.535475
$532$	−5.87863	−0.254871
$533$	20.8851	0.904636
$534$	−4.75357	−0.205707
$535$	−37.1728	−1.60712
$536$	5.55438	0.239913
$537$	−20.3568	−0.878459
$538$	19.8483	0.855722
$539$	−5.92743	−0.255313
$540$	−3.71045	−0.159672
$541$	14.4262	0.620229	0.310114	−	0.950699i	$-0.399633\pi$
$541$	14.4262	0.620229	0.310114	+	0.950699i	$0.399633\pi$
$542$	5.02282	0.215749
$543$	−24.2553	−1.04090
$544$	2.42459	0.103953
$545$	−13.1708	−0.564175
$546$	4.30321	0.184161
$547$	40.0221	1.71122	0.855610	−	0.517621i	$-0.173182\pi$
$547$	40.0221	1.71122	0.855610	+	0.517621i	$0.173182\pi$
$548$	12.1229	0.517866
$549$	8.37779	0.357555
$550$	46.3436	1.97610
$551$	−2.42459	−0.103291
$552$	−1.00000	−0.0425628
$553$	15.5591	0.661641
$554$	−30.3392	−1.28899
$555$	32.7564	1.39043
$556$	−7.81080	−0.331252
$557$	1.16293	0.0492748	0.0246374	−	0.999696i	$-0.492157\pi$
$557$	1.16293	0.0492748	0.0246374	+	0.999696i	$0.492157\pi$
$558$	−7.19573	−0.304619
$559$	7.89859	0.334075
$560$	−8.99632	−0.380164
$561$	12.8160	0.541094
$562$	−10.2369	−0.431820
$563$	11.6862	0.492517	0.246258	−	0.969204i	$-0.420799\pi$
$563$	11.6862	0.492517	0.246258	+	0.969204i	$0.420799\pi$
$564$	−10.5386	−0.443755
$565$	−76.1119	−3.20205
$566$	27.4272	1.15285
$567$	2.42459	0.101823
$568$	−1.49738	−0.0628285
$569$	−12.9721	−0.543819	−0.271910	−	0.962323i	$-0.587655\pi$
$569$	−12.9721	−0.543819	−0.271910	+	0.962323i	$0.587655\pi$
$570$	8.99632	0.376814
$571$	10.4824	0.438676	0.219338	−	0.975649i	$-0.429610\pi$
$571$	10.4824	0.438676	0.219338	+	0.975649i	$0.429610\pi$
$572$	9.38147	0.392259
$573$	14.9584	0.624898
$574$	28.5312	1.19087
$575$	−8.76746	−0.365628
$576$	1.00000	0.0416667
$577$	9.31057	0.387604	0.193802	−	0.981041i	$-0.437918\pi$
$577$	9.31057	0.387604	0.193802	+	0.981041i	$0.437918\pi$
$578$	−11.1214	−0.462588
$579$	3.20392	0.133150
$580$	−3.71045	−0.154068
$581$	−4.26061	−0.176760
$582$	10.1398	0.420307
$583$	46.2702	1.91631
$584$	−4.66734	−0.193136
$585$	−6.58539	−0.272273
$586$	11.4798	0.474226
$587$	−15.3936	−0.635361	−0.317680	−	0.948198i	$-0.602904\pi$
$587$	−15.3936	−0.635361	−0.317680	+	0.948198i	$0.602904\pi$
$588$	−1.12137	−0.0462447
$589$	17.4467	0.718878
$590$	−45.7840	−1.88490
$591$	8.72780	0.359014
$592$	−8.82814	−0.362834
$593$	−0.214971	−0.00882782	−0.00441391	−	0.999990i	$-0.501405\pi$
$593$	−0.214971	−0.00882782	−0.00441391	+	0.999990i	$0.501405\pi$
$594$	5.28586	0.216882
$595$	−21.8124	−0.894220
$596$	−18.0930	−0.741117
$597$	−15.5212	−0.635242
$598$	−1.77482	−0.0725779
$599$	−8.14714	−0.332883	−0.166442	−	0.986051i	$-0.553228\pi$
$599$	−8.14714	−0.332883	−0.166442	+	0.986051i	$0.553228\pi$
$600$	8.76746	0.357930
$601$	−30.4961	−1.24396	−0.621981	−	0.783033i	$-0.713672\pi$
$601$	−30.4961	−1.24396	−0.621981	+	0.783033i	$0.713672\pi$
$602$	10.7903	0.439779
$603$	5.55438	0.226192
$604$	18.2869	0.744084
$605$	−62.8564	−2.55548
$606$	4.34655	0.176567
$607$	−36.9663	−1.50042	−0.750208	−	0.661202i	$-0.770047\pi$
$607$	−36.9663	−1.50042	−0.750208	+	0.661202i	$0.770047\pi$
$608$	−2.42459	−0.0983300
$609$	2.42459	0.0982493
$610$	−31.0854	−1.25861
$611$	−18.7041	−0.756688
$612$	2.42459	0.0980081
$613$	8.98739	0.362997	0.181499	−	0.983391i	$-0.441905\pi$
$613$	8.98739	0.362997	0.181499	+	0.983391i	$0.441905\pi$
$614$	−26.0617	−1.05177
$615$	−43.6626	−1.76065
$616$	12.8160	0.516373
$617$	−34.6195	−1.39373	−0.696864	−	0.717204i	$-0.745422\pi$
$617$	−34.6195	−1.39373	−0.696864	+	0.717204i	$0.745422\pi$
$618$	−6.18657	−0.248861
$619$	5.91176	0.237614	0.118807	−	0.992917i	$-0.462093\pi$
$619$	5.91176	0.237614	0.118807	+	0.992917i	$0.462093\pi$
$620$	26.6994	1.07227
$621$	−1.00000	−0.0401286
$622$	−31.1450	−1.24880
$623$	−11.5254	−0.461757
$624$	1.77482	0.0710498
$625$	8.03101	0.321241
$626$	21.8623	0.873794
$627$	−12.8160	−0.511823
$628$	−17.1450	−0.684161
$629$	−21.4046	−0.853458
$630$	−8.99632	−0.358422
$631$	19.8939	0.791962	0.395981	−	0.918259i	$-0.370405\pi$
$631$	19.8939	0.791962	0.395981	+	0.918259i	$0.370405\pi$
$632$	6.41722	0.255263
$633$	1.74013	0.0691638
$634$	−30.0722	−1.19432
$635$	−7.22696	−0.286793
$636$	8.75357	0.347102
$637$	−1.99024	−0.0788562
$638$	5.28586	0.209269
$639$	−1.49738	−0.0592353
$640$	−3.71045	−0.146669
$641$	9.89336	0.390764	0.195382	−	0.980727i	$-0.437405\pi$
$641$	9.89336	0.390764	0.195382	+	0.980727i	$0.437405\pi$
$642$	10.0184	0.395395
$643$	26.6804	1.05217	0.526086	−	0.850431i	$-0.323659\pi$
$643$	26.6804	1.05217	0.526086	+	0.850431i	$0.323659\pi$
$644$	−2.42459	−0.0955421
$645$	−16.5128	−0.650192
$646$	−5.87863	−0.231291
$647$	−20.3130	−0.798585	−0.399293	−	0.916824i	$-0.630744\pi$
$647$	−20.3130	−0.798585	−0.399293	+	0.916824i	$0.630744\pi$
$648$	1.00000	0.0392837
$649$	65.2233	2.56024
$650$	15.5607	0.610340
$651$	−17.4467	−0.683789
$652$	−11.2473	−0.440477
$653$	43.7375	1.71158	0.855789	−	0.517324i	$-0.173072\pi$
$653$	43.7375	1.71158	0.855789	+	0.517324i	$0.173072\pi$
$654$	3.54965	0.138802
$655$	53.3657	2.08517
$656$	11.7675	0.459442
$657$	−4.66734	−0.182090
$658$	−25.5518	−0.996111
$659$	15.3184	0.596719	0.298360	−	0.954454i	$-0.403561\pi$
$659$	15.3184	0.596719	0.298360	+	0.954454i	$0.403561\pi$
$660$	−19.6129	−0.763433
$661$	29.2753	1.13868	0.569339	−	0.822103i	$-0.307199\pi$
$661$	29.2753	1.13868	0.569339	+	0.822103i	$0.307199\pi$
$662$	−8.97792	−0.348937
$663$	4.30321	0.167123
$664$	−1.75725	−0.0681946
$665$	21.8124	0.845847
$666$	−8.82814	−0.342084
$667$	−1.00000	−0.0387202
$668$	14.3134	0.553803
$669$	4.81604	0.186199
$670$	−20.6093	−0.796205
$671$	44.2839	1.70956
$672$	2.42459	0.0935305
$673$	−31.3893	−1.20997	−0.604985	−	0.796237i	$-0.706821\pi$
$673$	−31.3893	−1.20997	−0.604985	+	0.796237i	$0.706821\pi$
$674$	−26.9842	−1.03939
$675$	8.76746	0.337460
$676$	−9.85000	−0.378846
$677$	−22.9727	−0.882911	−0.441455	−	0.897283i	$-0.645538\pi$
$677$	−22.9727	−0.882911	−0.441455	+	0.897283i	$0.645538\pi$
$678$	20.5128	0.787790
$679$	24.5848	0.943477
$680$	−8.99632	−0.344993
$681$	20.5297	0.786699
$682$	−38.0356	−1.45646
$683$	8.89095	0.340203	0.170101	−	0.985427i	$-0.445590\pi$
$683$	8.89095	0.340203	0.170101	+	0.985427i	$0.445590\pi$
$684$	−2.42459	−0.0927064
$685$	−44.9816	−1.71866
$686$	−19.6910	−0.751805
$687$	19.5743	0.746808
$688$	4.45035	0.169668
$689$	15.5360	0.591876
$690$	3.71045	0.141255
$691$	−41.8990	−1.59391	−0.796957	−	0.604036i	$-0.793558\pi$
$691$	−41.8990	−1.59391	−0.796957	+	0.604036i	$0.793558\pi$
$692$	4.26852	0.162265
$693$	12.8160	0.486841
$694$	−21.1435	−0.802595
$695$	28.9816	1.09933
$696$	1.00000	0.0379049
$697$	28.5312	1.08070
$698$	−27.0654	−1.02444
$699$	0.700470	0.0264942
$700$	21.2575	0.803457
$701$	−15.5633	−0.587818	−0.293909	−	0.955833i	$-0.594956\pi$
$701$	−15.5633	−0.587818	−0.293909	+	0.955833i	$0.594956\pi$
$702$	1.77482	0.0669864
$703$	21.4046	0.807290
$704$	5.28586	0.199219
$705$	39.1030	1.47270
$706$	4.87494	0.183471
$707$	10.5386	0.396345
$708$	12.3392	0.463735
$709$	21.5638	0.809846	0.404923	−	0.914351i	$-0.367298\pi$
$709$	21.5638	0.809846	0.404923	+	0.914351i	$0.367298\pi$
$710$	5.55594	0.208511
$711$	6.41722	0.240665
$712$	−4.75357	−0.178147
$713$	7.19573	0.269482
$714$	5.87863	0.220002
$715$	−34.8095	−1.30180
$716$	−20.3568	−0.760768
$717$	25.0686	0.936202
$718$	−10.4414	−0.389671
$719$	7.05605	0.263146	0.131573	−	0.991306i	$-0.457997\pi$
$719$	7.05605	0.263146	0.131573	+	0.991306i	$0.457997\pi$
$720$	−3.71045	−0.138280
$721$	−14.9999	−0.558625
$722$	−13.1214	−0.488327
$723$	−8.38253	−0.311749
$724$	−24.2553	−0.901443
$725$	8.76746	0.325615
$726$	16.9404	0.628716
$727$	−30.0993	−1.11632	−0.558160	−	0.829733i	$-0.688493\pi$
$727$	−30.0993	−1.11632	−0.558160	+	0.829733i	$0.688493\pi$
$728$	4.30321	0.159488
$729$	1.00000	0.0370370
$730$	17.3179	0.640965
$731$	10.7903	0.399093
$732$	8.37779	0.309652
$733$	25.0893	0.926694	0.463347	−	0.886177i	$-0.346648\pi$
$733$	25.0893	0.926694	0.463347	+	0.886177i	$0.346648\pi$
$734$	−21.3788	−0.789107
$735$	4.16081	0.153474
$736$	−1.00000	−0.0368605
$737$	29.3597	1.08148
$738$	11.7675	0.433166
$739$	−13.4994	−0.496583	−0.248291	−	0.968685i	$-0.579869\pi$
$739$	−13.4994	−0.496583	−0.248291	+	0.968685i	$0.579869\pi$
$740$	32.7564	1.20415
$741$	−4.30321	−0.158082
$742$	21.2238	0.779150
$743$	17.4881	0.641577	0.320788	−	0.947151i	$-0.396052\pi$
$743$	17.4881	0.641577	0.320788	+	0.947151i	$0.396052\pi$
$744$	−7.19573	−0.263808
$745$	67.1331	2.45957
$746$	−1.66209	−0.0608534
$747$	−1.75725	−0.0642944
$748$	12.8160	0.468601
$749$	24.2905	0.887555
$750$	−13.9790	−0.510440
$751$	−39.3453	−1.43573	−0.717865	−	0.696183i	$-0.754880\pi$
$751$	−39.3453	−1.43573	−0.717865	+	0.696183i	$0.754880\pi$
$752$	−10.5386	−0.384303
$753$	14.5523	0.530314
$754$	1.77482	0.0646352
$755$	−67.8527	−2.46941
$756$	2.42459	0.0881814
$757$	39.6458	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$757$	39.6458	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$758$	35.1030	1.27500
$759$	−5.28586	−0.191865
$760$	8.99632	0.326331
$761$	−14.8235	−0.537352	−0.268676	−	0.963231i	$-0.586586\pi$
$761$	−14.8235	−0.537352	−0.268676	+	0.963231i	$0.586586\pi$
$762$	1.94773	0.0705589
$763$	8.60643	0.311574
$764$	14.9584	0.541178
$765$	−8.99632	−0.325263
$766$	−20.3236	−0.734322
$767$	21.8999	0.790758
$768$	1.00000	0.0360844
$769$	31.4881	1.13549	0.567745	−	0.823204i	$-0.307816\pi$
$769$	31.4881	1.13549	0.567745	+	0.823204i	$0.307816\pi$
$770$	−47.5533	−1.71370
$771$	−18.4446	−0.664264
$772$	3.20392	0.115312
$773$	−14.5612	−0.523731	−0.261866	−	0.965104i	$-0.584338\pi$
$773$	−14.5612	−0.523731	−0.261866	+	0.965104i	$0.584338\pi$
$774$	4.45035	0.159965
$775$	−63.0882	−2.26620
$776$	10.1398	0.363997
$777$	−21.4046	−0.767886
$778$	17.1802	0.615939
$779$	−28.5312	−1.02224
$780$	−6.58539	−0.235795
$781$	−7.91493	−0.283218
$782$	−2.42459	−0.0867031
$783$	1.00000	0.0357371
$784$	−1.12137	−0.0400491
$785$	63.6158	2.27054
$786$	−14.3825	−0.513008
$787$	23.9637	0.854213	0.427107	−	0.904201i	$-0.359533\pi$
$787$	23.9637	0.854213	0.427107	+	0.904201i	$0.359533\pi$
$788$	8.72780	0.310915
$789$	−3.07352	−0.109420
$790$	−23.8108	−0.847150
$791$	49.7352	1.76838
$792$	5.28586	0.187825
$793$	14.8691	0.528017
$794$	−11.6359	−0.412942
$795$	−32.4797	−1.15194
$796$	−15.5212	−0.550136
$797$	40.7073	1.44193	0.720963	−	0.692974i	$-0.243700\pi$
$797$	40.7073	1.44193	0.720963	+	0.692974i	$0.243700\pi$
$798$	−5.87863	−0.208101
$799$	−25.5518	−0.903956
$800$	8.76746	0.309976
$801$	−4.75357	−0.167959
$802$	−2.86284	−0.101090
$803$	−24.6709	−0.870618
$804$	5.55438	0.195888
$805$	8.99632	0.317079
$806$	−12.7711	−0.449844
$807$	19.8483	0.698694
$808$	4.34655	0.152911
$809$	−45.0543	−1.58402	−0.792012	−	0.610506i	$-0.790966\pi$
$809$	−45.0543	−1.58402	−0.792012	+	0.610506i	$0.790966\pi$
$810$	−3.71045	−0.130372
$811$	−0.293741	−0.0103147	−0.00515733	−	0.999987i	$-0.501642\pi$
$811$	−0.293741	−0.0103147	−0.00515733	+	0.999987i	$0.501642\pi$
$812$	2.42459	0.0850863
$813$	5.02282	0.176158
$814$	−46.6644	−1.63559
$815$	41.7324	1.46182
$816$	2.42459	0.0848775
$817$	−10.7903	−0.377504
$818$	13.3783	0.467761
$819$	4.30321	0.150366
$820$	−43.6626	−1.52476
$821$	23.8985	0.834063	0.417032	−	0.908892i	$-0.363070\pi$
$821$	23.8985	0.834063	0.417032	+	0.908892i	$0.363070\pi$
$822$	12.1229	0.422836
$823$	−13.3870	−0.466643	−0.233321	−	0.972400i	$-0.574959\pi$
$823$	−13.3870	−0.466643	−0.233321	+	0.972400i	$0.574959\pi$
$824$	−6.18657	−0.215520
$825$	46.3436	1.61348
$826$	29.9174	1.04096
$827$	23.3663	0.812526	0.406263	−	0.913756i	$-0.366832\pi$
$827$	23.3663	0.812526	0.406263	+	0.913756i	$0.366832\pi$
$828$	−1.00000	−0.0347524
$829$	−17.7851	−0.617703	−0.308851	−	0.951110i	$-0.599945\pi$
$829$	−17.7851	−0.617703	−0.308851	+	0.951110i	$0.599945\pi$
$830$	6.52020	0.226319
$831$	−30.3392	−1.05245
$832$	1.77482	0.0615309
$833$	−2.71887	−0.0942033
$834$	−7.81080	−0.270466
$835$	−53.1093	−1.83792
$836$	−12.8160	−0.443252
$837$	−7.19573	−0.248721
$838$	−30.6510	−1.05882
$839$	−1.07670	−0.0371717	−0.0185858	−	0.999827i	$-0.505916\pi$
$839$	−1.07670	−0.0371717	−0.0185858	+	0.999827i	$0.505916\pi$
$840$	−8.99632	−0.310402
$841$	1.00000	0.0344828
$842$	9.48812	0.326982
$843$	−10.2369	−0.352579
$844$	1.74013	0.0598976
$845$	36.5480	1.25729
$846$	−10.5386	−0.362324
$847$	41.0734	1.41130
$848$	8.75357	0.300599
$849$	27.4272	0.941299
$850$	21.2575	0.729125
$851$	8.82814	0.302625
$852$	−1.49738	−0.0512993
$853$	35.3547	1.21052	0.605262	−	0.796027i	$-0.293069\pi$
$853$	35.3547	1.21052	0.605262	+	0.796027i	$0.293069\pi$
$854$	20.3127	0.695086
$855$	8.99632	0.307668
$856$	10.0184	0.342422
$857$	29.6138	1.01159	0.505794	−	0.862654i	$-0.331200\pi$
$857$	29.6138	1.01159	0.505794	+	0.862654i	$0.331200\pi$
$858$	9.38147	0.320278
$859$	−13.9302	−0.475291	−0.237645	−	0.971352i	$-0.576376\pi$
$859$	−13.9302	−0.475291	−0.237645	+	0.971352i	$0.576376\pi$
$860$	−16.5128	−0.563083
$861$	28.5312	0.972342
$862$	−11.2496	−0.383162
$863$	−22.9256	−0.780398	−0.390199	−	0.920731i	$-0.627594\pi$
$863$	−22.9256	−0.780398	−0.390199	+	0.920731i	$0.627594\pi$
$864$	1.00000	0.0340207
$865$	−15.8381	−0.538512
$866$	−28.5218	−0.969209
$867$	−11.1214	−0.377702
$868$	−17.4467	−0.592179
$869$	33.9206	1.15068
$870$	−3.71045	−0.125796
$871$	9.85804	0.334027
$872$	3.54965	0.120206
$873$	10.1398	0.343179
$874$	2.42459	0.0820129
$875$	−33.8932	−1.14580
$876$	−4.66734	−0.157695
$877$	−13.5848	−0.458726	−0.229363	−	0.973341i	$-0.573664\pi$
$877$	−13.5848	−0.458726	−0.229363	+	0.973341i	$0.573664\pi$
$878$	−7.96486	−0.268801
$879$	11.4798	0.387204
$880$	−19.6129	−0.661152
$881$	39.1692	1.31964	0.659822	−	0.751422i	$-0.270632\pi$
$881$	39.1692	1.31964	0.659822	+	0.751422i	$0.270632\pi$
$882$	−1.12137	−0.0377586
$883$	−13.0479	−0.439095	−0.219548	−	0.975602i	$-0.570458\pi$
$883$	−13.0479	−0.439095	−0.219548	+	0.975602i	$0.570458\pi$
$884$	4.30321	0.144733
$885$	−45.7840	−1.53901
$886$	−25.7262	−0.864289
$887$	−8.23494	−0.276502	−0.138251	−	0.990397i	$-0.544148\pi$
$887$	−8.23494	−0.276502	−0.138251	+	0.990397i	$0.544148\pi$
$888$	−8.82814	−0.296253
$889$	4.72245	0.158386
$890$	17.6379	0.591223
$891$	5.28586	0.177083
$892$	4.81604	0.161253
$893$	25.5518	0.855057
$894$	−18.0930	−0.605120
$895$	75.5328	2.52478
$896$	2.42459	0.0809998
$897$	−1.77482	−0.0592596
$898$	−6.72854	−0.224534
$899$	−7.19573	−0.239991
$900$	8.76746	0.292249
$901$	21.2238	0.707067
$902$	62.2012	2.07107
$903$	10.7903	0.359078
$904$	20.5128	0.682247
$905$	89.9983	2.99165
$906$	18.2869	0.607542
$907$	4.57910	0.152046	0.0760232	−	0.997106i	$-0.475778\pi$
$907$	4.57910	0.152046	0.0760232	+	0.997106i	$0.475778\pi$
$908$	20.5297	0.681301
$909$	4.34655	0.144166
$910$	−15.9669	−0.529296
$911$	−46.5360	−1.54181	−0.770903	−	0.636953i	$-0.780195\pi$
$911$	−46.5360	−1.54181	−0.770903	+	0.636953i	$0.780195\pi$
$912$	−2.42459	−0.0802861
$913$	−9.28859	−0.307407
$914$	−23.3988	−0.773964
$915$	−31.0854	−1.02765
$916$	19.5743	0.646755
$917$	−34.8717	−1.15156
$918$	2.42459	0.0800233
$919$	44.0026	1.45151	0.725756	−	0.687952i	$-0.241490\pi$
$919$	44.0026	1.45151	0.725756	+	0.687952i	$0.241490\pi$
$920$	3.71045	0.122330
$921$	−26.0617	−0.858764
$922$	−7.19729	−0.237030
$923$	−2.65758	−0.0874752
$924$	12.8160	0.421617
$925$	−77.4004	−2.54491
$926$	42.4997	1.39663
$927$	−6.18657	−0.203194
$928$	1.00000	0.0328266
$929$	−20.6253	−0.676693	−0.338347	−	0.941022i	$-0.609868\pi$
$929$	−20.6253	−0.676693	−0.338347	+	0.941022i	$0.609868\pi$
$930$	26.6994	0.875508
$931$	2.71887	0.0891074
$932$	0.700470	0.0229447
$933$	−31.1450	−1.01964
$934$	22.3778	0.732224
$935$	−47.5533	−1.55516
$936$	1.77482	0.0580119
$937$	−32.8291	−1.07248	−0.536240	−	0.844066i	$-0.680156\pi$
$937$	−32.8291	−1.07248	−0.536240	+	0.844066i	$0.680156\pi$
$938$	13.4671	0.439716
$939$	21.8623	0.713450
$940$	39.1030	1.27540
$941$	32.7525	1.06770	0.533851	−	0.845579i	$-0.320745\pi$
$941$	32.7525	1.06770	0.533851	+	0.845579i	$0.320745\pi$
$942$	−17.1450	−0.558615
$943$	−11.7675	−0.383201
$944$	12.3392	0.401606
$945$	−8.99632	−0.292650
$946$	23.5240	0.764830
$947$	20.8676	0.678105	0.339053	−	0.940767i	$-0.389894\pi$
$947$	20.8676	0.678105	0.339053	+	0.940767i	$0.389894\pi$
$948$	6.41722	0.208422
$949$	−8.28370	−0.268900
$950$	−21.2575	−0.689683
$951$	−30.0722	−0.975159
$952$	5.87863	0.190527
$953$	−43.4351	−1.40700	−0.703500	−	0.710695i	$-0.748380\pi$
$953$	−43.4351	−1.40700	−0.703500	+	0.710695i	$0.748380\pi$
$954$	8.75357	0.283407
$955$	−55.5026	−1.79602
$956$	25.0686	0.810775
$957$	5.28586	0.170868
$958$	−28.8203	−0.931143
$959$	29.3931	0.949154
$960$	−3.71045	−0.119754
$961$	20.7785	0.670274
$962$	−15.6684	−0.505169
$963$	10.0184	0.322838
$964$	−8.38253	−0.269983
$965$	−11.8880	−0.382688
$966$	−2.42459	−0.0780098
$967$	−21.6914	−0.697549	−0.348775	−	0.937207i	$-0.613402\pi$
$967$	−21.6914	−0.697549	−0.348775	+	0.937207i	$0.613402\pi$
$968$	16.9404	0.544484
$969$	−5.87863	−0.188849
$970$	−37.6232	−1.20801
$971$	9.21593	0.295753	0.147877	−	0.989006i	$-0.452756\pi$
$971$	9.21593	0.295753	0.147877	+	0.989006i	$0.452756\pi$
$972$	1.00000	0.0320750
$973$	−18.9380	−0.607123
$974$	24.8091	0.794936
$975$	15.5607	0.498341
$976$	8.37779	0.268166
$977$	0.764553	0.0244602	0.0122301	−	0.999925i	$-0.496107\pi$
$977$	0.764553	0.0244602	0.0122301	+	0.999925i	$0.496107\pi$
$978$	−11.2473	−0.359648
$979$	−25.1267	−0.803053
$980$	4.16081	0.132912
$981$	3.54965	0.113331
$982$	−14.8507	−0.473906
$983$	−28.1739	−0.898606	−0.449303	−	0.893379i	$-0.648328\pi$
$983$	−28.1739	−0.898606	−0.449303	+	0.893379i	$0.648328\pi$
$984$	11.7675	0.375133
$985$	−32.3841	−1.03184
$986$	2.42459	0.0772146
$987$	−25.5518	−0.813321
$988$	−4.30321	−0.136903
$989$	−4.45035	−0.141513
$990$	−19.6129	−0.623340
$991$	17.4447	0.554148	0.277074	−	0.960849i	$-0.410635\pi$
$991$	17.4447	0.554148	0.277074	+	0.960849i	$0.410635\pi$
$992$	−7.19573	−0.228465
$993$	−8.97792	−0.284906
$994$	−3.63052	−0.115153
$995$	57.5909	1.82575
$996$	−1.75725	−0.0556806
$997$	31.0956	0.984807	0.492404	−	0.870367i	$-0.336118\pi$
$997$	31.0956	0.984807	0.492404	+	0.870367i	$0.336118\pi$
$998$	−10.7520	−0.340349
$999$	−8.82814	−0.279310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bd.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bd.1.1	✓	4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.71045 q^{5} +1.00000 q^{6} +2.42459 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.71045 q^{5} +1.00000 q^{6} +2.42459 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.71045 q^{10} +5.28586 q^{11} +1.00000 q^{12} +1.77482 q^{13} +2.42459 q^{14} -3.71045 q^{15} +1.00000 q^{16} +2.42459 q^{17} +1.00000 q^{18} -2.42459 q^{19} -3.71045 q^{20} +2.42459 q^{21} +5.28586 q^{22} -1.00000 q^{23} +1.00000 q^{24} +8.76746 q^{25} +1.77482 q^{26} +1.00000 q^{27} +2.42459 q^{28} +1.00000 q^{29} -3.71045 q^{30} -7.19573 q^{31} +1.00000 q^{32} +5.28586 q^{33} +2.42459 q^{34} -8.99632 q^{35} +1.00000 q^{36} -8.82814 q^{37} -2.42459 q^{38} +1.77482 q^{39} -3.71045 q^{40} +11.7675 q^{41} +2.42459 q^{42} +4.45035 q^{43} +5.28586 q^{44} -3.71045 q^{45} -1.00000 q^{46} -10.5386 q^{47} +1.00000 q^{48} -1.12137 q^{49} +8.76746 q^{50} +2.42459 q^{51} +1.77482 q^{52} +8.75357 q^{53} +1.00000 q^{54} -19.6129 q^{55} +2.42459 q^{56} -2.42459 q^{57} +1.00000 q^{58} +12.3392 q^{59} -3.71045 q^{60} +8.37779 q^{61} -7.19573 q^{62} +2.42459 q^{63} +1.00000 q^{64} -6.58539 q^{65} +5.28586 q^{66} +5.55438 q^{67} +2.42459 q^{68} -1.00000 q^{69} -8.99632 q^{70} -1.49738 q^{71} +1.00000 q^{72} -4.66734 q^{73} -8.82814 q^{74} +8.76746 q^{75} -2.42459 q^{76} +12.8160 q^{77} +1.77482 q^{78} +6.41722 q^{79} -3.71045 q^{80} +1.00000 q^{81} +11.7675 q^{82} -1.75725 q^{83} +2.42459 q^{84} -8.99632 q^{85} +4.45035 q^{86} +1.00000 q^{87} +5.28586 q^{88} -4.75357 q^{89} -3.71045 q^{90} +4.30321 q^{91} -1.00000 q^{92} -7.19573 q^{93} -10.5386 q^{94} +8.99632 q^{95} +1.00000 q^{96} +10.1398 q^{97} -1.12137 q^{98} +5.28586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - q^5 + 4 * q^6 + 2 * q^7 + 4 * q^8 + 4 * q^9 - q^10 + 15 * q^11 + 4 * q^12 + 11 * q^13 + 2 * q^14 - q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 - 2 * q^19 - q^20 + 2 * q^21 + 15 * q^22 - 4 * q^23 + 4 * q^24 - q^25 + 11 * q^26 + 4 * q^27 + 2 * q^28 + 4 * q^29 - q^30 - 5 * q^31 + 4 * q^32 + 15 * q^33 + 2 * q^34 - 16 * q^35 + 4 * q^36 + 3 * q^37 - 2 * q^38 + 11 * q^39 - q^40 + 11 * q^41 + 2 * q^42 + 10 * q^43 + 15 * q^44 - q^45 - 4 * q^46 + 10 * q^47 + 4 * q^48 - q^50 + 2 * q^51 + 11 * q^52 + 24 * q^53 + 4 * q^54 - 7 * q^55 + 2 * q^56 - 2 * q^57 + 4 * q^58 + q^59 - q^60 + 3 * q^61 - 5 * q^62 + 2 * q^63 + 4 * q^64 + 3 * q^65 + 15 * q^66 + 7 * q^67 + 2 * q^68 - 4 * q^69 - 16 * q^70 - 13 * q^71 + 4 * q^72 - 2 * q^73 + 3 * q^74 - q^75 - 2 * q^76 - 4 * q^77 + 11 * q^78 - 22 * q^79 - q^80 + 4 * q^81 + 11 * q^82 - 16 * q^83 + 2 * q^84 - 16 * q^85 + 10 * q^86 + 4 * q^87 + 15 * q^88 - 8 * q^89 - q^90 + 14 * q^91 - 4 * q^92 - 5 * q^93 + 10 * q^94 + 16 * q^95 + 4 * q^96 - 4 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more