LMFDB - Embedded newform 4002.2.a.t.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} -1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} -1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -1.00000 q^{12} +4.82843 q^{13} +1.41421 q^{14} +2.82843 q^{15} +1.00000 q^{16} +2.82843 q^{17} -1.00000 q^{18} -2.58579 q^{19} -2.82843 q^{20} +1.41421 q^{21} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.00000 q^{25} -4.82843 q^{26} -1.00000 q^{27} -1.41421 q^{28} +1.00000 q^{29} -2.82843 q^{30} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.82843 q^{34} +4.00000 q^{35} +1.00000 q^{36} -0.828427 q^{37} +2.58579 q^{38} -4.82843 q^{39} +2.82843 q^{40} -2.82843 q^{41} -1.41421 q^{42} +8.24264 q^{43} +4.00000 q^{44} -2.82843 q^{45} +1.00000 q^{46} -11.3137 q^{47} -1.00000 q^{48} -5.00000 q^{49} -3.00000 q^{50} -2.82843 q^{51} +4.82843 q^{52} +4.00000 q^{53} +1.00000 q^{54} -11.3137 q^{55} +1.41421 q^{56} +2.58579 q^{57} -1.00000 q^{58} -6.82843 q^{59} +2.82843 q^{60} -7.65685 q^{61} -6.00000 q^{62} -1.41421 q^{63} +1.00000 q^{64} -13.6569 q^{65} +4.00000 q^{66} +14.4853 q^{67} +2.82843 q^{68} +1.00000 q^{69} -4.00000 q^{70} +6.48528 q^{71} -1.00000 q^{72} -2.48528 q^{73} +0.828427 q^{74} -3.00000 q^{75} -2.58579 q^{76} -5.65685 q^{77} +4.82843 q^{78} +4.34315 q^{79} -2.82843 q^{80} +1.00000 q^{81} +2.82843 q^{82} -13.5563 q^{83} +1.41421 q^{84} -8.00000 q^{85} -8.24264 q^{86} -1.00000 q^{87} -4.00000 q^{88} -5.17157 q^{89} +2.82843 q^{90} -6.82843 q^{91} -1.00000 q^{92} -6.00000 q^{93} +11.3137 q^{94} +7.31371 q^{95} +1.00000 q^{96} +13.4142 q^{97} +5.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 8 q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{16} - 2 q^{18} - 8 q^{19} - 8 q^{22} - 2 q^{23} + 2 q^{24} + 6 q^{25} - 4 q^{26} - 2 q^{27} + 2 q^{29} + 12 q^{31} - 2 q^{32} - 8 q^{33} + 8 q^{35} + 2 q^{36} + 4 q^{37} + 8 q^{38} - 4 q^{39} + 8 q^{43} + 8 q^{44} + 2 q^{46} - 2 q^{48} - 10 q^{49} - 6 q^{50} + 4 q^{52} + 8 q^{53} + 2 q^{54} + 8 q^{57} - 2 q^{58} - 8 q^{59} - 4 q^{61} - 12 q^{62} + 2 q^{64} - 16 q^{65} + 8 q^{66} + 12 q^{67} + 2 q^{69} - 8 q^{70} - 4 q^{71} - 2 q^{72} + 12 q^{73} - 4 q^{74} - 6 q^{75} - 8 q^{76} + 4 q^{78} + 20 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{85} - 8 q^{86} - 2 q^{87} - 8 q^{88} - 16 q^{89} - 8 q^{91} - 2 q^{92} - 12 q^{93} - 8 q^{95} + 2 q^{96} + 24 q^{97} + 10 q^{98} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 8 * q^11 - 2 * q^12 + 4 * q^13 + 2 * q^16 - 2 * q^18 - 8 * q^19 - 8 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 4 * q^26 - 2 * q^27 + 2 * q^29 + 12 * q^31 - 2 * q^32 - 8 * q^33 + 8 * q^35 + 2 * q^36 + 4 * q^37 + 8 * q^38 - 4 * q^39 + 8 * q^43 + 8 * q^44 + 2 * q^46 - 2 * q^48 - 10 * q^49 - 6 * q^50 + 4 * q^52 + 8 * q^53 + 2 * q^54 + 8 * q^57 - 2 * q^58 - 8 * q^59 - 4 * q^61 - 12 * q^62 + 2 * q^64 - 16 * q^65 + 8 * q^66 + 12 * q^67 + 2 * q^69 - 8 * q^70 - 4 * q^71 - 2 * q^72 + 12 * q^73 - 4 * q^74 - 6 * q^75 - 8 * q^76 + 4 * q^78 + 20 * q^79 + 2 * q^81 + 4 * q^83 - 16 * q^85 - 8 * q^86 - 2 * q^87 - 8 * q^88 - 16 * q^89 - 8 * q^91 - 2 * q^92 - 12 * q^93 - 8 * q^95 + 2 * q^96 + 24 * q^97 + 10 * q^98 + 8 * 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
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	1.00000	0.408248
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.82843	0.894427
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	1.41421	0.377964
$15$	2.82843	0.730297
$16$	1.00000	0.250000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	−1.00000	−0.235702
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	−2.82843	−0.632456
$21$	1.41421	0.308607
$22$	−4.00000	−0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	3.00000	0.600000
$26$	−4.82843	−0.946932
$27$	−1.00000	−0.192450
$28$	−1.41421	−0.267261
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.82843	−0.516398
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−2.82843	−0.485071
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−0.828427	−0.136193	−0.0680963	−	0.997679i	$-0.521693\pi$
$37$	−0.828427	−0.136193	−0.0680963	+	0.997679i	$0.521693\pi$
$38$	2.58579	0.419470
$39$	−4.82843	−0.773167
$40$	2.82843	0.447214
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	−1.41421	−0.218218
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	4.00000	0.603023
$45$	−2.82843	−0.421637
$46$	1.00000	0.147442
$47$	−11.3137	−1.65027	−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137	−1.65027	−0.825137	+	0.564933i	$0.808902\pi$
$48$	−1.00000	−0.144338
$49$	−5.00000	−0.714286
$50$	−3.00000	−0.424264
$51$	−2.82843	−0.396059
$52$	4.82843	0.669582
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	1.00000	0.136083
$55$	−11.3137	−1.52554
$56$	1.41421	0.188982
$57$	2.58579	0.342496
$58$	−1.00000	−0.131306
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	2.82843	0.365148
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	−6.00000	−0.762001
$63$	−1.41421	−0.178174
$64$	1.00000	0.125000
$65$	−13.6569	−1.69392
$66$	4.00000	0.492366
$67$	14.4853	1.76966	0.884829	−	0.465915i	$-0.154275\pi$
$67$	14.4853	1.76966	0.884829	+	0.465915i	$0.154275\pi$
$68$	2.82843	0.342997
$69$	1.00000	0.120386
$70$	−4.00000	−0.478091
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	−1.00000	−0.117851
$73$	−2.48528	−0.290880	−0.145440	−	0.989367i	$-0.546460\pi$
$73$	−2.48528	−0.290880	−0.145440	+	0.989367i	$0.546460\pi$
$74$	0.828427	0.0963027
$75$	−3.00000	−0.346410
$76$	−2.58579	−0.296610
$77$	−5.65685	−0.644658
$78$	4.82843	0.546712
$79$	4.34315	0.488642	0.244321	−	0.969694i	$-0.421435\pi$
$79$	4.34315	0.488642	0.244321	+	0.969694i	$0.421435\pi$
$80$	−2.82843	−0.316228
$81$	1.00000	0.111111
$82$	2.82843	0.312348
$83$	−13.5563	−1.48800	−0.744001	−	0.668178i	$-0.767074\pi$
$83$	−13.5563	−1.48800	−0.744001	+	0.668178i	$0.767074\pi$
$84$	1.41421	0.154303
$85$	−8.00000	−0.867722
$86$	−8.24264	−0.888827
$87$	−1.00000	−0.107211
$88$	−4.00000	−0.426401
$89$	−5.17157	−0.548186	−0.274093	−	0.961703i	$-0.588378\pi$
$89$	−5.17157	−0.548186	−0.274093	+	0.961703i	$0.588378\pi$
$90$	2.82843	0.298142
$91$	−6.82843	−0.715814
$92$	−1.00000	−0.104257
$93$	−6.00000	−0.622171
$94$	11.3137	1.16692
$95$	7.31371	0.750371
$96$	1.00000	0.102062
$97$	13.4142	1.36201	0.681004	−	0.732280i	$-0.261544\pi$
$97$	13.4142	1.36201	0.681004	+	0.732280i	$0.261544\pi$
$98$	5.00000	0.505076
$99$	4.00000	0.402015
$100$	3.00000	0.300000
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	2.82843	0.280056
$103$	−1.41421	−0.139347	−0.0696733	−	0.997570i	$-0.522196\pi$
$103$	−1.41421	−0.139347	−0.0696733	+	0.997570i	$0.522196\pi$
$104$	−4.82843	−0.473466
$105$	−4.00000	−0.390360
$106$	−4.00000	−0.388514
$107$	−4.58579	−0.443325	−0.221662	−	0.975123i	$-0.571148\pi$
$107$	−4.58579	−0.443325	−0.221662	+	0.975123i	$0.571148\pi$
$108$	−1.00000	−0.0962250
$109$	−7.75736	−0.743020	−0.371510	−	0.928429i	$-0.621160\pi$
$109$	−7.75736	−0.743020	−0.371510	+	0.928429i	$0.621160\pi$
$110$	11.3137	1.07872
$111$	0.828427	0.0786308
$112$	−1.41421	−0.133631
$113$	−9.65685	−0.908440	−0.454220	−	0.890889i	$-0.650082\pi$
$113$	−9.65685	−0.908440	−0.454220	+	0.890889i	$0.650082\pi$
$114$	−2.58579	−0.242181
$115$	2.82843	0.263752
$116$	1.00000	0.0928477
$117$	4.82843	0.446388
$118$	6.82843	0.628608
$119$	−4.00000	−0.366679
$120$	−2.82843	−0.258199
$121$	5.00000	0.454545
$122$	7.65685	0.693219
$123$	2.82843	0.255031
$124$	6.00000	0.538816
$125$	5.65685	0.505964
$126$	1.41421	0.125988
$127$	15.3137	1.35887	0.679436	−	0.733735i	$-0.262225\pi$
$127$	15.3137	1.35887	0.679436	+	0.733735i	$0.262225\pi$
$128$	−1.00000	−0.0883883
$129$	−8.24264	−0.725724
$130$	13.6569	1.19779
$131$	−0.828427	−0.0723800	−0.0361900	−	0.999345i	$-0.511522\pi$
$131$	−0.828427	−0.0723800	−0.0361900	+	0.999345i	$0.511522\pi$
$132$	−4.00000	−0.348155
$133$	3.65685	0.317089
$134$	−14.4853	−1.25134
$135$	2.82843	0.243432
$136$	−2.82843	−0.242536
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	4.34315	0.368381	0.184190	−	0.982891i	$-0.441034\pi$
$139$	4.34315	0.368381	0.184190	+	0.982891i	$0.441034\pi$
$140$	4.00000	0.338062
$141$	11.3137	0.952786
$142$	−6.48528	−0.544233
$143$	19.3137	1.61509
$144$	1.00000	0.0833333
$145$	−2.82843	−0.234888
$146$	2.48528	0.205683
$147$	5.00000	0.412393
$148$	−0.828427	−0.0680963
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	3.00000	0.244949
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	2.58579	0.209735
$153$	2.82843	0.228665
$154$	5.65685	0.455842
$155$	−16.9706	−1.36311
$156$	−4.82843	−0.386584
$157$	14.4853	1.15605	0.578026	−	0.816019i	$-0.303823\pi$
$157$	14.4853	1.15605	0.578026	+	0.816019i	$0.303823\pi$
$158$	−4.34315	−0.345522
$159$	−4.00000	−0.317221
$160$	2.82843	0.223607
$161$	1.41421	0.111456
$162$	−1.00000	−0.0785674
$163$	−0.485281	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$163$	−0.485281	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$164$	−2.82843	−0.220863
$165$	11.3137	0.880771
$166$	13.5563	1.05218
$167$	19.3137	1.49454	0.747270	−	0.664521i	$-0.231364\pi$
$167$	19.3137	1.49454	0.747270	+	0.664521i	$0.231364\pi$
$168$	−1.41421	−0.109109
$169$	10.3137	0.793362
$170$	8.00000	0.613572
$171$	−2.58579	−0.197740
$172$	8.24264	0.628495
$173$	16.4853	1.25335	0.626676	−	0.779280i	$-0.284415\pi$
$173$	16.4853	1.25335	0.626676	+	0.779280i	$0.284415\pi$
$174$	1.00000	0.0758098
$175$	−4.24264	−0.320713
$176$	4.00000	0.301511
$177$	6.82843	0.513256
$178$	5.17157	0.387626
$179$	0.485281	0.0362716	0.0181358	−	0.999836i	$-0.494227\pi$
$179$	0.485281	0.0362716	0.0181358	+	0.999836i	$0.494227\pi$
$180$	−2.82843	−0.210819
$181$	15.0711	1.12022	0.560112	−	0.828417i	$-0.310758\pi$
$181$	15.0711	1.12022	0.560112	+	0.828417i	$0.310758\pi$
$182$	6.82843	0.506157
$183$	7.65685	0.566011
$184$	1.00000	0.0737210
$185$	2.34315	0.172272
$186$	6.00000	0.439941
$187$	11.3137	0.827340
$188$	−11.3137	−0.825137
$189$	1.41421	0.102869
$190$	−7.31371	−0.530592
$191$	−7.41421	−0.536474	−0.268237	−	0.963353i	$-0.586441\pi$
$191$	−7.41421	−0.536474	−0.268237	+	0.963353i	$0.586441\pi$
$192$	−1.00000	−0.0721688
$193$	7.65685	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$193$	7.65685	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$194$	−13.4142	−0.963084
$195$	13.6569	0.977988
$196$	−5.00000	−0.357143
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−4.00000	−0.284268
$199$	−8.24264	−0.584305	−0.292153	−	0.956372i	$-0.594372\pi$
$199$	−8.24264	−0.584305	−0.292153	+	0.956372i	$0.594372\pi$
$200$	−3.00000	−0.212132
$201$	−14.4853	−1.02171
$202$	4.82843	0.339727
$203$	−1.41421	−0.0992583
$204$	−2.82843	−0.198030
$205$	8.00000	0.558744
$206$	1.41421	0.0985329
$207$	−1.00000	−0.0695048
$208$	4.82843	0.334791
$209$	−10.3431	−0.715450
$210$	4.00000	0.276026
$211$	−2.82843	−0.194717	−0.0973585	−	0.995249i	$-0.531039\pi$
$211$	−2.82843	−0.194717	−0.0973585	+	0.995249i	$0.531039\pi$
$212$	4.00000	0.274721
$213$	−6.48528	−0.444364
$214$	4.58579	0.313478
$215$	−23.3137	−1.58998
$216$	1.00000	0.0680414
$217$	−8.48528	−0.576018
$218$	7.75736	0.525395
$219$	2.48528	0.167940
$220$	−11.3137	−0.762770
$221$	13.6569	0.918659
$222$	−0.828427	−0.0556004
$223$	10.3431	0.692628	0.346314	−	0.938119i	$-0.387433\pi$
$223$	10.3431	0.692628	0.346314	+	0.938119i	$0.387433\pi$
$224$	1.41421	0.0944911
$225$	3.00000	0.200000
$226$	9.65685	0.642364
$227$	6.24264	0.414339	0.207169	−	0.978305i	$-0.433575\pi$
$227$	6.24264	0.414339	0.207169	+	0.978305i	$0.433575\pi$
$228$	2.58579	0.171248
$229$	−11.6569	−0.770307	−0.385153	−	0.922853i	$-0.625851\pi$
$229$	−11.6569	−0.770307	−0.385153	+	0.922853i	$0.625851\pi$
$230$	−2.82843	−0.186501
$231$	5.65685	0.372194
$232$	−1.00000	−0.0656532
$233$	−4.82843	−0.316321	−0.158160	−	0.987413i	$-0.550556\pi$
$233$	−4.82843	−0.316321	−0.158160	+	0.987413i	$0.550556\pi$
$234$	−4.82843	−0.315644
$235$	32.0000	2.08745
$236$	−6.82843	−0.444493
$237$	−4.34315	−0.282118
$238$	4.00000	0.259281
$239$	−1.51472	−0.0979790	−0.0489895	−	0.998799i	$-0.515600\pi$
$239$	−1.51472	−0.0979790	−0.0489895	+	0.998799i	$0.515600\pi$
$240$	2.82843	0.182574
$241$	1.51472	0.0975716	0.0487858	−	0.998809i	$-0.484465\pi$
$241$	1.51472	0.0975716	0.0487858	+	0.998809i	$0.484465\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−7.65685	−0.490180
$245$	14.1421	0.903508
$246$	−2.82843	−0.180334
$247$	−12.4853	−0.794419
$248$	−6.00000	−0.381000
$249$	13.5563	0.859099
$250$	−5.65685	−0.357771
$251$	0.686292	0.0433183	0.0216592	−	0.999765i	$-0.493105\pi$
$251$	0.686292	0.0433183	0.0216592	+	0.999765i	$0.493105\pi$
$252$	−1.41421	−0.0890871
$253$	−4.00000	−0.251478
$254$	−15.3137	−0.960868
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	−19.1716	−1.19589	−0.597945	−	0.801537i	$-0.704016\pi$
$257$	−19.1716	−1.19589	−0.597945	+	0.801537i	$0.704016\pi$
$258$	8.24264	0.513164
$259$	1.17157	0.0727980
$260$	−13.6569	−0.846962
$261$	1.00000	0.0618984
$262$	0.828427	0.0511804
$263$	−17.5563	−1.08257	−0.541285	−	0.840839i	$-0.682062\pi$
$263$	−17.5563	−1.08257	−0.541285	+	0.840839i	$0.682062\pi$
$264$	4.00000	0.246183
$265$	−11.3137	−0.694996
$266$	−3.65685	−0.224216
$267$	5.17157	0.316495
$268$	14.4853	0.884829
$269$	−16.8284	−1.02605	−0.513024	−	0.858374i	$-0.671475\pi$
$269$	−16.8284	−1.02605	−0.513024	+	0.858374i	$0.671475\pi$
$270$	−2.82843	−0.172133
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	2.82843	0.171499
$273$	6.82843	0.413275
$274$	−8.00000	−0.483298
$275$	12.0000	0.723627
$276$	1.00000	0.0601929
$277$	8.82843	0.530449	0.265224	−	0.964187i	$-0.414554\pi$
$277$	8.82843	0.530449	0.265224	+	0.964187i	$0.414554\pi$
$278$	−4.34315	−0.260485
$279$	6.00000	0.359211
$280$	−4.00000	−0.239046
$281$	19.8995	1.18710	0.593552	−	0.804796i	$-0.297725\pi$
$281$	19.8995	1.18710	0.593552	+	0.804796i	$0.297725\pi$
$282$	−11.3137	−0.673722
$283$	1.51472	0.0900407	0.0450203	−	0.998986i	$-0.485665\pi$
$283$	1.51472	0.0900407	0.0450203	+	0.998986i	$0.485665\pi$
$284$	6.48528	0.384831
$285$	−7.31371	−0.433227
$286$	−19.3137	−1.14204
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−9.00000	−0.529412
$290$	2.82843	0.166091
$291$	−13.4142	−0.786355
$292$	−2.48528	−0.145440
$293$	10.7279	0.626732	0.313366	−	0.949632i	$-0.398543\pi$
$293$	10.7279	0.626732	0.313366	+	0.949632i	$0.398543\pi$
$294$	−5.00000	−0.291606
$295$	19.3137	1.12449
$296$	0.828427	0.0481513
$297$	−4.00000	−0.232104
$298$	4.00000	0.231714
$299$	−4.82843	−0.279235
$300$	−3.00000	−0.173205
$301$	−11.6569	−0.671890
$302$	−16.0000	−0.920697
$303$	4.82843	0.277386
$304$	−2.58579	−0.148305
$305$	21.6569	1.24007
$306$	−2.82843	−0.161690
$307$	22.8284	1.30289	0.651444	−	0.758697i	$-0.274164\pi$
$307$	22.8284	1.30289	0.651444	+	0.758697i	$0.274164\pi$
$308$	−5.65685	−0.322329
$309$	1.41421	0.0804518
$310$	16.9706	0.963863
$311$	17.4558	0.989830	0.494915	−	0.868941i	$-0.335199\pi$
$311$	17.4558	0.989830	0.494915	+	0.868941i	$0.335199\pi$
$312$	4.82843	0.273356
$313$	−24.6274	−1.39202	−0.696012	−	0.718030i	$-0.745044\pi$
$313$	−24.6274	−1.39202	−0.696012	+	0.718030i	$0.745044\pi$
$314$	−14.4853	−0.817452
$315$	4.00000	0.225374
$316$	4.34315	0.244321
$317$	20.1421	1.13130	0.565648	−	0.824647i	$-0.308626\pi$
$317$	20.1421	1.13130	0.565648	+	0.824647i	$0.308626\pi$
$318$	4.00000	0.224309
$319$	4.00000	0.223957
$320$	−2.82843	−0.158114
$321$	4.58579	0.255954
$322$	−1.41421	−0.0788110
$323$	−7.31371	−0.406946
$324$	1.00000	0.0555556
$325$	14.4853	0.803499
$326$	0.485281	0.0268772
$327$	7.75736	0.428983
$328$	2.82843	0.156174
$329$	16.0000	0.882109
$330$	−11.3137	−0.622799
$331$	11.7990	0.648531	0.324266	−	0.945966i	$-0.394883\pi$
$331$	11.7990	0.648531	0.324266	+	0.945966i	$0.394883\pi$
$332$	−13.5563	−0.744001
$333$	−0.828427	−0.0453975
$334$	−19.3137	−1.05680
$335$	−40.9706	−2.23846
$336$	1.41421	0.0771517
$337$	−4.24264	−0.231111	−0.115556	−	0.993301i	$-0.536865\pi$
$337$	−4.24264	−0.231111	−0.115556	+	0.993301i	$0.536865\pi$
$338$	−10.3137	−0.560992
$339$	9.65685	0.524488
$340$	−8.00000	−0.433861
$341$	24.0000	1.29967
$342$	2.58579	0.139823
$343$	16.9706	0.916324
$344$	−8.24264	−0.444413
$345$	−2.82843	−0.152277
$346$	−16.4853	−0.886254
$347$	−11.3137	−0.607352	−0.303676	−	0.952775i	$-0.598214\pi$
$347$	−11.3137	−0.607352	−0.303676	+	0.952775i	$0.598214\pi$
$348$	−1.00000	−0.0536056
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	4.24264	0.226779
$351$	−4.82843	−0.257722
$352$	−4.00000	−0.213201
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	−6.82843	−0.362927
$355$	−18.3431	−0.973553
$356$	−5.17157	−0.274093
$357$	4.00000	0.211702
$358$	−0.485281	−0.0256479
$359$	17.5563	0.926589	0.463294	−	0.886204i	$-0.346667\pi$
$359$	17.5563	0.926589	0.463294	+	0.886204i	$0.346667\pi$
$360$	2.82843	0.149071
$361$	−12.3137	−0.648090
$362$	−15.0711	−0.792118
$363$	−5.00000	−0.262432
$364$	−6.82843	−0.357907
$365$	7.02944	0.367938
$366$	−7.65685	−0.400230
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−1.00000	−0.0521286
$369$	−2.82843	−0.147242
$370$	−2.34315	−0.121814
$371$	−5.65685	−0.293689
$372$	−6.00000	−0.311086
$373$	15.7574	0.815885	0.407943	−	0.913008i	$-0.366246\pi$
$373$	15.7574	0.815885	0.407943	+	0.913008i	$0.366246\pi$
$374$	−11.3137	−0.585018
$375$	−5.65685	−0.292119
$376$	11.3137	0.583460
$377$	4.82843	0.248677
$378$	−1.41421	−0.0727393
$379$	−14.3848	−0.738896	−0.369448	−	0.929251i	$-0.620453\pi$
$379$	−14.3848	−0.738896	−0.369448	+	0.929251i	$0.620453\pi$
$380$	7.31371	0.375185
$381$	−15.3137	−0.784545
$382$	7.41421	0.379344
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	1.00000	0.0510310
$385$	16.0000	0.815436
$386$	−7.65685	−0.389724
$387$	8.24264	0.418997
$388$	13.4142	0.681004
$389$	25.5563	1.29576	0.647879	−	0.761743i	$-0.275656\pi$
$389$	25.5563	1.29576	0.647879	+	0.761743i	$0.275656\pi$
$390$	−13.6569	−0.691542
$391$	−2.82843	−0.143040
$392$	5.00000	0.252538
$393$	0.828427	0.0417886
$394$	−10.0000	−0.503793
$395$	−12.2843	−0.618089
$396$	4.00000	0.201008
$397$	−19.6569	−0.986549	−0.493275	−	0.869874i	$-0.664200\pi$
$397$	−19.6569	−0.986549	−0.493275	+	0.869874i	$0.664200\pi$
$398$	8.24264	0.413166
$399$	−3.65685	−0.183072
$400$	3.00000	0.150000
$401$	30.7279	1.53448	0.767240	−	0.641361i	$-0.221630\pi$
$401$	30.7279	1.53448	0.767240	+	0.641361i	$0.221630\pi$
$402$	14.4853	0.722460
$403$	28.9706	1.44313
$404$	−4.82843	−0.240223
$405$	−2.82843	−0.140546
$406$	1.41421	0.0701862
$407$	−3.31371	−0.164254
$408$	2.82843	0.140028
$409$	12.3431	0.610329	0.305165	−	0.952300i	$-0.401289\pi$
$409$	12.3431	0.610329	0.305165	+	0.952300i	$0.401289\pi$
$410$	−8.00000	−0.395092
$411$	−8.00000	−0.394611
$412$	−1.41421	−0.0696733
$413$	9.65685	0.475183
$414$	1.00000	0.0491473
$415$	38.3431	1.88219
$416$	−4.82843	−0.236733
$417$	−4.34315	−0.212685
$418$	10.3431	0.505900
$419$	18.0416	0.881391	0.440696	−	0.897657i	$-0.354732\pi$
$419$	18.0416	0.881391	0.440696	+	0.897657i	$0.354732\pi$
$420$	−4.00000	−0.195180
$421$	−0.343146	−0.0167239	−0.00836195	−	0.999965i	$-0.502662\pi$
$421$	−0.343146	−0.0167239	−0.00836195	+	0.999965i	$0.502662\pi$
$422$	2.82843	0.137686
$423$	−11.3137	−0.550091
$424$	−4.00000	−0.194257
$425$	8.48528	0.411597
$426$	6.48528	0.314213
$427$	10.8284	0.524024
$428$	−4.58579	−0.221662
$429$	−19.3137	−0.932475
$430$	23.3137	1.12429
$431$	16.4853	0.794068	0.397034	−	0.917804i	$-0.370039\pi$
$431$	16.4853	0.794068	0.397034	+	0.917804i	$0.370039\pi$
$432$	−1.00000	−0.0481125
$433$	15.5563	0.747590	0.373795	−	0.927511i	$-0.378056\pi$
$433$	15.5563	0.747590	0.373795	+	0.927511i	$0.378056\pi$
$434$	8.48528	0.407307
$435$	2.82843	0.135613
$436$	−7.75736	−0.371510
$437$	2.58579	0.123695
$438$	−2.48528	−0.118751
$439$	−17.6569	−0.842716	−0.421358	−	0.906894i	$-0.638446\pi$
$439$	−17.6569	−0.842716	−0.421358	+	0.906894i	$0.638446\pi$
$440$	11.3137	0.539360
$441$	−5.00000	−0.238095
$442$	−13.6569	−0.649590
$443$	−4.14214	−0.196799	−0.0983994	−	0.995147i	$-0.531372\pi$
$443$	−4.14214	−0.196799	−0.0983994	+	0.995147i	$0.531372\pi$
$444$	0.828427	0.0393154
$445$	14.6274	0.693406
$446$	−10.3431	−0.489762
$447$	4.00000	0.189194
$448$	−1.41421	−0.0668153
$449$	−3.79899	−0.179285	−0.0896427	−	0.995974i	$-0.528573\pi$
$449$	−3.79899	−0.179285	−0.0896427	+	0.995974i	$0.528573\pi$
$450$	−3.00000	−0.141421
$451$	−11.3137	−0.532742
$452$	−9.65685	−0.454220
$453$	−16.0000	−0.751746
$454$	−6.24264	−0.292982
$455$	19.3137	0.905441
$456$	−2.58579	−0.121091
$457$	16.6274	0.777798	0.388899	−	0.921280i	$-0.372856\pi$
$457$	16.6274	0.777798	0.388899	+	0.921280i	$0.372856\pi$
$458$	11.6569	0.544689
$459$	−2.82843	−0.132020
$460$	2.82843	0.131876
$461$	−10.4853	−0.488348	−0.244174	−	0.969731i	$-0.578517\pi$
$461$	−10.4853	−0.488348	−0.244174	+	0.969731i	$0.578517\pi$
$462$	−5.65685	−0.263181
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	1.00000	0.0464238
$465$	16.9706	0.786991
$466$	4.82843	0.223673
$467$	23.5147	1.08813	0.544066	−	0.839043i	$-0.316884\pi$
$467$	23.5147	1.08813	0.544066	+	0.839043i	$0.316884\pi$
$468$	4.82843	0.223194
$469$	−20.4853	−0.945922
$470$	−32.0000	−1.47605
$471$	−14.4853	−0.667447
$472$	6.82843	0.314304
$473$	32.9706	1.51599
$474$	4.34315	0.199487
$475$	−7.75736	−0.355932
$476$	−4.00000	−0.183340
$477$	4.00000	0.183147
$478$	1.51472	0.0692816
$479$	6.92893	0.316591	0.158295	−	0.987392i	$-0.449400\pi$
$479$	6.92893	0.316591	0.158295	+	0.987392i	$0.449400\pi$
$480$	−2.82843	−0.129099
$481$	−4.00000	−0.182384
$482$	−1.51472	−0.0689935
$483$	−1.41421	−0.0643489
$484$	5.00000	0.227273
$485$	−37.9411	−1.72282
$486$	1.00000	0.0453609
$487$	26.3431	1.19372	0.596861	−	0.802345i	$-0.296414\pi$
$487$	26.3431	1.19372	0.596861	+	0.802345i	$0.296414\pi$
$488$	7.65685	0.346610
$489$	0.485281	0.0219452
$490$	−14.1421	−0.638877
$491$	1.51472	0.0683583	0.0341791	−	0.999416i	$-0.489118\pi$
$491$	1.51472	0.0683583	0.0341791	+	0.999416i	$0.489118\pi$
$492$	2.82843	0.127515
$493$	2.82843	0.127386
$494$	12.4853	0.561739
$495$	−11.3137	−0.508513
$496$	6.00000	0.269408
$497$	−9.17157	−0.411401
$498$	−13.5563	−0.607475
$499$	−4.34315	−0.194426	−0.0972130	−	0.995264i	$-0.530993\pi$
$499$	−4.34315	−0.194426	−0.0972130	+	0.995264i	$0.530993\pi$
$500$	5.65685	0.252982
$501$	−19.3137	−0.862873
$502$	−0.686292	−0.0306307
$503$	−43.2132	−1.92678	−0.963391	−	0.268101i	$-0.913604\pi$
$503$	−43.2132	−1.92678	−0.963391	+	0.268101i	$0.913604\pi$
$504$	1.41421	0.0629941
$505$	13.6569	0.607722
$506$	4.00000	0.177822
$507$	−10.3137	−0.458048
$508$	15.3137	0.679436
$509$	9.45584	0.419123	0.209561	−	0.977795i	$-0.432796\pi$
$509$	9.45584	0.419123	0.209561	+	0.977795i	$0.432796\pi$
$510$	−8.00000	−0.354246
$511$	3.51472	0.155482
$512$	−1.00000	−0.0441942
$513$	2.58579	0.114165
$514$	19.1716	0.845622
$515$	4.00000	0.176261
$516$	−8.24264	−0.362862
$517$	−45.2548	−1.99031
$518$	−1.17157	−0.0514760
$519$	−16.4853	−0.723624
$520$	13.6569	0.598893
$521$	29.0711	1.27363	0.636813	−	0.771018i	$-0.280252\pi$
$521$	29.0711	1.27363	0.636813	+	0.771018i	$0.280252\pi$
$522$	−1.00000	−0.0437688
$523$	−7.17157	−0.313591	−0.156795	−	0.987631i	$-0.550116\pi$
$523$	−7.17157	−0.313591	−0.156795	+	0.987631i	$0.550116\pi$
$524$	−0.828427	−0.0361900
$525$	4.24264	0.185164
$526$	17.5563	0.765493
$527$	16.9706	0.739249
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	11.3137	0.491436
$531$	−6.82843	−0.296328
$532$	3.65685	0.158545
$533$	−13.6569	−0.591544
$534$	−5.17157	−0.223796
$535$	12.9706	0.560766
$536$	−14.4853	−0.625669
$537$	−0.485281	−0.0209414
$538$	16.8284	0.725525
$539$	−20.0000	−0.861461
$540$	2.82843	0.121716
$541$	−25.9411	−1.11530	−0.557648	−	0.830077i	$-0.688296\pi$
$541$	−25.9411	−1.11530	−0.557648	+	0.830077i	$0.688296\pi$
$542$	−4.00000	−0.171815
$543$	−15.0711	−0.646761
$544$	−2.82843	−0.121268
$545$	21.9411	0.939855
$546$	−6.82843	−0.292230
$547$	−2.97056	−0.127012	−0.0635060	−	0.997981i	$-0.520228\pi$
$547$	−2.97056	−0.127012	−0.0635060	+	0.997981i	$0.520228\pi$
$548$	8.00000	0.341743
$549$	−7.65685	−0.326787
$550$	−12.0000	−0.511682
$551$	−2.58579	−0.110158
$552$	−1.00000	−0.0425628
$553$	−6.14214	−0.261190
$554$	−8.82843	−0.375084
$555$	−2.34315	−0.0994610
$556$	4.34315	0.184190
$557$	−6.82843	−0.289330	−0.144665	−	0.989481i	$-0.546210\pi$
$557$	−6.82843	−0.289330	−0.144665	+	0.989481i	$0.546210\pi$
$558$	−6.00000	−0.254000
$559$	39.7990	1.68332
$560$	4.00000	0.169031
$561$	−11.3137	−0.477665
$562$	−19.8995	−0.839410
$563$	38.6274	1.62795	0.813976	−	0.580899i	$-0.197299\pi$
$563$	38.6274	1.62795	0.813976	+	0.580899i	$0.197299\pi$
$564$	11.3137	0.476393
$565$	27.3137	1.14910
$566$	−1.51472	−0.0636684
$567$	−1.41421	−0.0593914
$568$	−6.48528	−0.272116
$569$	36.9706	1.54989	0.774943	−	0.632031i	$-0.217778\pi$
$569$	36.9706	1.54989	0.774943	+	0.632031i	$0.217778\pi$
$570$	7.31371	0.306338
$571$	−17.3137	−0.724556	−0.362278	−	0.932070i	$-0.618001\pi$
$571$	−17.3137	−0.724556	−0.362278	+	0.932070i	$0.618001\pi$
$572$	19.3137	0.807547
$573$	7.41421	0.309733
$574$	−4.00000	−0.166957
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	1.31371	0.0546904	0.0273452	−	0.999626i	$-0.491295\pi$
$577$	1.31371	0.0546904	0.0273452	+	0.999626i	$0.491295\pi$
$578$	9.00000	0.374351
$579$	−7.65685	−0.318208
$580$	−2.82843	−0.117444
$581$	19.1716	0.795371
$582$	13.4142	0.556037
$583$	16.0000	0.662652
$584$	2.48528	0.102842
$585$	−13.6569	−0.564641
$586$	−10.7279	−0.443166
$587$	39.5980	1.63438	0.817192	−	0.576366i	$-0.195530\pi$
$587$	39.5980	1.63438	0.817192	+	0.576366i	$0.195530\pi$
$588$	5.00000	0.206197
$589$	−15.5147	−0.639273
$590$	−19.3137	−0.795133
$591$	−10.0000	−0.411345
$592$	−0.828427	−0.0340481
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	4.00000	0.164122
$595$	11.3137	0.463817
$596$	−4.00000	−0.163846
$597$	8.24264	0.337349
$598$	4.82843	0.197449
$599$	−31.7990	−1.29927	−0.649636	−	0.760246i	$-0.725079\pi$
$599$	−31.7990	−1.29927	−0.649636	+	0.760246i	$0.725079\pi$
$600$	3.00000	0.122474
$601$	−23.6569	−0.964983	−0.482492	−	0.875901i	$-0.660268\pi$
$601$	−23.6569	−0.964983	−0.482492	+	0.875901i	$0.660268\pi$
$602$	11.6569	0.475098
$603$	14.4853	0.589886
$604$	16.0000	0.651031
$605$	−14.1421	−0.574960
$606$	−4.82843	−0.196141
$607$	26.9706	1.09470	0.547351	−	0.836903i	$-0.315636\pi$
$607$	26.9706	1.09470	0.547351	+	0.836903i	$0.315636\pi$
$608$	2.58579	0.104867
$609$	1.41421	0.0573068
$610$	−21.6569	−0.876860
$611$	−54.6274	−2.20999
$612$	2.82843	0.114332
$613$	17.2132	0.695235	0.347617	−	0.937636i	$-0.386991\pi$
$613$	17.2132	0.695235	0.347617	+	0.937636i	$0.386991\pi$
$614$	−22.8284	−0.921280
$615$	−8.00000	−0.322591
$616$	5.65685	0.227921
$617$	32.2843	1.29972	0.649858	−	0.760056i	$-0.274828\pi$
$617$	32.2843	1.29972	0.649858	+	0.760056i	$0.274828\pi$
$618$	−1.41421	−0.0568880
$619$	−16.4437	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$619$	−16.4437	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$620$	−16.9706	−0.681554
$621$	1.00000	0.0401286
$622$	−17.4558	−0.699916
$623$	7.31371	0.293018
$624$	−4.82843	−0.193292
$625$	−31.0000	−1.24000
$626$	24.6274	0.984310
$627$	10.3431	0.413065
$628$	14.4853	0.578026
$629$	−2.34315	−0.0934273
$630$	−4.00000	−0.159364
$631$	8.92893	0.355455	0.177728	−	0.984080i	$-0.443125\pi$
$631$	8.92893	0.355455	0.177728	+	0.984080i	$0.443125\pi$
$632$	−4.34315	−0.172761
$633$	2.82843	0.112420
$634$	−20.1421	−0.799946
$635$	−43.3137	−1.71885
$636$	−4.00000	−0.158610
$637$	−24.1421	−0.956546
$638$	−4.00000	−0.158362
$639$	6.48528	0.256554
$640$	2.82843	0.111803
$641$	−2.34315	−0.0925487	−0.0462743	−	0.998929i	$-0.514735\pi$
$641$	−2.34315	−0.0925487	−0.0462743	+	0.998929i	$0.514735\pi$
$642$	−4.58579	−0.180987
$643$	−4.82843	−0.190415	−0.0952073	−	0.995457i	$-0.530351\pi$
$643$	−4.82843	−0.190415	−0.0952073	+	0.995457i	$0.530351\pi$
$644$	1.41421	0.0557278
$645$	23.3137	0.917976
$646$	7.31371	0.287754
$647$	29.7990	1.17152	0.585760	−	0.810485i	$-0.300796\pi$
$647$	29.7990	1.17152	0.585760	+	0.810485i	$0.300796\pi$
$648$	−1.00000	−0.0392837
$649$	−27.3137	−1.07216
$650$	−14.4853	−0.568159
$651$	8.48528	0.332564
$652$	−0.485281	−0.0190051
$653$	−0.343146	−0.0134283	−0.00671417	−	0.999977i	$-0.502137\pi$
$653$	−0.343146	−0.0134283	−0.00671417	+	0.999977i	$0.502137\pi$
$654$	−7.75736	−0.303337
$655$	2.34315	0.0915543
$656$	−2.82843	−0.110432
$657$	−2.48528	−0.0969601
$658$	−16.0000	−0.623745
$659$	−25.4558	−0.991619	−0.495809	−	0.868431i	$-0.665129\pi$
$659$	−25.4558	−0.991619	−0.495809	+	0.868431i	$0.665129\pi$
$660$	11.3137	0.440386
$661$	−4.44365	−0.172838	−0.0864190	−	0.996259i	$-0.527542\pi$
$661$	−4.44365	−0.172838	−0.0864190	+	0.996259i	$0.527542\pi$
$662$	−11.7990	−0.458581
$663$	−13.6569	−0.530388
$664$	13.5563	0.526088
$665$	−10.3431	−0.401090
$666$	0.828427	0.0321009
$667$	−1.00000	−0.0387202
$668$	19.3137	0.747270
$669$	−10.3431	−0.399889
$670$	40.9706	1.58283
$671$	−30.6274	−1.18236
$672$	−1.41421	−0.0545545
$673$	13.3137	0.513206	0.256603	−	0.966517i	$-0.417397\pi$
$673$	13.3137	0.513206	0.256603	+	0.966517i	$0.417397\pi$
$674$	4.24264	0.163420
$675$	−3.00000	−0.115470
$676$	10.3137	0.396681
$677$	−38.7279	−1.48843	−0.744217	−	0.667937i	$-0.767177\pi$
$677$	−38.7279	−1.48843	−0.744217	+	0.667937i	$0.767177\pi$
$678$	−9.65685	−0.370869
$679$	−18.9706	−0.728023
$680$	8.00000	0.306786
$681$	−6.24264	−0.239219
$682$	−24.0000	−0.919007
$683$	−48.9706	−1.87381	−0.936903	−	0.349589i	$-0.886321\pi$
$683$	−48.9706	−1.87381	−0.936903	+	0.349589i	$0.886321\pi$
$684$	−2.58579	−0.0988700
$685$	−22.6274	−0.864549
$686$	−16.9706	−0.647939
$687$	11.6569	0.444737
$688$	8.24264	0.314248
$689$	19.3137	0.735794
$690$	2.82843	0.107676
$691$	19.3137	0.734728	0.367364	−	0.930077i	$-0.380260\pi$
$691$	19.3137	0.734728	0.367364	+	0.930077i	$0.380260\pi$
$692$	16.4853	0.626676
$693$	−5.65685	−0.214886
$694$	11.3137	0.429463
$695$	−12.2843	−0.465969
$696$	1.00000	0.0379049
$697$	−8.00000	−0.303022
$698$	−2.00000	−0.0757011
$699$	4.82843	0.182628
$700$	−4.24264	−0.160357
$701$	−39.7990	−1.50319	−0.751594	−	0.659627i	$-0.770714\pi$
$701$	−39.7990	−1.50319	−0.751594	+	0.659627i	$0.770714\pi$
$702$	4.82843	0.182237
$703$	2.14214	0.0807922
$704$	4.00000	0.150756
$705$	−32.0000	−1.20519
$706$	10.0000	0.376355
$707$	6.82843	0.256809
$708$	6.82843	0.256628
$709$	−14.8701	−0.558457	−0.279228	−	0.960225i	$-0.590079\pi$
$709$	−14.8701	−0.558457	−0.279228	+	0.960225i	$0.590079\pi$
$710$	18.3431	0.688406
$711$	4.34315	0.162881
$712$	5.17157	0.193813
$713$	−6.00000	−0.224702
$714$	−4.00000	−0.149696
$715$	−54.6274	−2.04295
$716$	0.485281	0.0181358
$717$	1.51472	0.0565682
$718$	−17.5563	−0.655197
$719$	52.2843	1.94987	0.974937	−	0.222480i	$-0.0714154\pi$
$719$	52.2843	1.94987	0.974937	+	0.222480i	$0.0714154\pi$
$720$	−2.82843	−0.105409
$721$	2.00000	0.0744839
$722$	12.3137	0.458269
$723$	−1.51472	−0.0563330
$724$	15.0711	0.560112
$725$	3.00000	0.111417
$726$	5.00000	0.185567
$727$	−0.343146	−0.0127266	−0.00636329	−	0.999980i	$-0.502026\pi$
$727$	−0.343146	−0.0127266	−0.00636329	+	0.999980i	$0.502026\pi$
$728$	6.82843	0.253078
$729$	1.00000	0.0370370
$730$	−7.02944	−0.260171
$731$	23.3137	0.862289
$732$	7.65685	0.283005
$733$	20.8284	0.769316	0.384658	−	0.923059i	$-0.374319\pi$
$733$	20.8284	0.769316	0.384658	+	0.923059i	$0.374319\pi$
$734$	−10.0000	−0.369107
$735$	−14.1421	−0.521641
$736$	1.00000	0.0368605
$737$	57.9411	2.13429
$738$	2.82843	0.104116
$739$	11.0294	0.405724	0.202862	−	0.979207i	$-0.434976\pi$
$739$	11.0294	0.405724	0.202862	+	0.979207i	$0.434976\pi$
$740$	2.34315	0.0861358
$741$	12.4853	0.458658
$742$	5.65685	0.207670
$743$	24.5858	0.901965	0.450983	−	0.892533i	$-0.351074\pi$
$743$	24.5858	0.901965	0.450983	+	0.892533i	$0.351074\pi$
$744$	6.00000	0.219971
$745$	11.3137	0.414502
$746$	−15.7574	−0.576918
$747$	−13.5563	−0.496001
$748$	11.3137	0.413670
$749$	6.48528	0.236967
$750$	5.65685	0.206559
$751$	39.9411	1.45747	0.728736	−	0.684795i	$-0.240108\pi$
$751$	39.9411	1.45747	0.728736	+	0.684795i	$0.240108\pi$
$752$	−11.3137	−0.412568
$753$	−0.686292	−0.0250099
$754$	−4.82843	−0.175841
$755$	−45.2548	−1.64699
$756$	1.41421	0.0514344
$757$	−22.6863	−0.824547	−0.412274	−	0.911060i	$-0.635265\pi$
$757$	−22.6863	−0.824547	−0.412274	+	0.911060i	$0.635265\pi$
$758$	14.3848	0.522479
$759$	4.00000	0.145191
$760$	−7.31371	−0.265296
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	15.3137	0.554757
$763$	10.9706	0.397161
$764$	−7.41421	−0.268237
$765$	−8.00000	−0.289241
$766$	−8.00000	−0.289052
$767$	−32.9706	−1.19050
$768$	−1.00000	−0.0360844
$769$	18.8701	0.680472	0.340236	−	0.940340i	$-0.389493\pi$
$769$	18.8701	0.680472	0.340236	+	0.940340i	$0.389493\pi$
$770$	−16.0000	−0.576600
$771$	19.1716	0.690447
$772$	7.65685	0.275576
$773$	2.04163	0.0734323	0.0367162	−	0.999326i	$-0.488310\pi$
$773$	2.04163	0.0734323	0.0367162	+	0.999326i	$0.488310\pi$
$774$	−8.24264	−0.296276
$775$	18.0000	0.646579
$776$	−13.4142	−0.481542
$777$	−1.17157	−0.0420299
$778$	−25.5563	−0.916240
$779$	7.31371	0.262041
$780$	13.6569	0.488994
$781$	25.9411	0.928246
$782$	2.82843	0.101144
$783$	−1.00000	−0.0357371
$784$	−5.00000	−0.178571
$785$	−40.9706	−1.46230
$786$	−0.828427	−0.0295490
$787$	4.14214	0.147651	0.0738256	−	0.997271i	$-0.476479\pi$
$787$	4.14214	0.147651	0.0738256	+	0.997271i	$0.476479\pi$
$788$	10.0000	0.356235
$789$	17.5563	0.625023
$790$	12.2843	0.437055
$791$	13.6569	0.485582
$792$	−4.00000	−0.142134
$793$	−36.9706	−1.31286
$794$	19.6569	0.697596
$795$	11.3137	0.401256
$796$	−8.24264	−0.292153
$797$	26.5269	0.939631	0.469816	−	0.882765i	$-0.344320\pi$
$797$	26.5269	0.939631	0.469816	+	0.882765i	$0.344320\pi$
$798$	3.65685	0.129451
$799$	−32.0000	−1.13208
$800$	−3.00000	−0.106066
$801$	−5.17157	−0.182729
$802$	−30.7279	−1.08504
$803$	−9.94113	−0.350815
$804$	−14.4853	−0.510856
$805$	−4.00000	−0.140981
$806$	−28.9706	−1.02044
$807$	16.8284	0.592389
$808$	4.82843	0.169863
$809$	−51.7990	−1.82116	−0.910578	−	0.413338i	$-0.864363\pi$
$809$	−51.7990	−1.82116	−0.910578	+	0.413338i	$0.864363\pi$
$810$	2.82843	0.0993808
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	−1.41421	−0.0496292
$813$	−4.00000	−0.140286
$814$	3.31371	0.116145
$815$	1.37258	0.0480795
$816$	−2.82843	−0.0990148
$817$	−21.3137	−0.745672
$818$	−12.3431	−0.431568
$819$	−6.82843	−0.238605
$820$	8.00000	0.279372
$821$	−54.4264	−1.89949	−0.949747	−	0.313018i	$-0.898660\pi$
$821$	−54.4264	−1.89949	−0.949747	+	0.313018i	$0.898660\pi$
$822$	8.00000	0.279032
$823$	45.9411	1.60141	0.800703	−	0.599061i	$-0.204459\pi$
$823$	45.9411	1.60141	0.800703	+	0.599061i	$0.204459\pi$
$824$	1.41421	0.0492665
$825$	−12.0000	−0.417786
$826$	−9.65685	−0.336005
$827$	13.4558	0.467906	0.233953	−	0.972248i	$-0.424834\pi$
$827$	13.4558	0.467906	0.233953	+	0.972248i	$0.424834\pi$
$828$	−1.00000	−0.0347524
$829$	−17.3137	−0.601330	−0.300665	−	0.953730i	$-0.597209\pi$
$829$	−17.3137	−0.601330	−0.300665	+	0.953730i	$0.597209\pi$
$830$	−38.3431	−1.33091
$831$	−8.82843	−0.306255
$832$	4.82843	0.167396
$833$	−14.1421	−0.489996
$834$	4.34315	0.150391
$835$	−54.6274	−1.89046
$836$	−10.3431	−0.357725
$837$	−6.00000	−0.207390
$838$	−18.0416	−0.623238
$839$	49.3553	1.70394	0.851968	−	0.523594i	$-0.175409\pi$
$839$	49.3553	1.70394	0.851968	+	0.523594i	$0.175409\pi$
$840$	4.00000	0.138013
$841$	1.00000	0.0344828
$842$	0.343146	0.0118256
$843$	−19.8995	−0.685375
$844$	−2.82843	−0.0973585
$845$	−29.1716	−1.00353
$846$	11.3137	0.388973
$847$	−7.07107	−0.242965
$848$	4.00000	0.137361
$849$	−1.51472	−0.0519850
$850$	−8.48528	−0.291043
$851$	0.828427	0.0283981
$852$	−6.48528	−0.222182
$853$	19.9411	0.682771	0.341386	−	0.939923i	$-0.389104\pi$
$853$	19.9411	0.682771	0.341386	+	0.939923i	$0.389104\pi$
$854$	−10.8284	−0.370541
$855$	7.31371	0.250124
$856$	4.58579	0.156739
$857$	−28.1421	−0.961317	−0.480659	−	0.876908i	$-0.659602\pi$
$857$	−28.1421	−0.961317	−0.480659	+	0.876908i	$0.659602\pi$
$858$	19.3137	0.659359
$859$	−45.2548	−1.54408	−0.772038	−	0.635577i	$-0.780762\pi$
$859$	−45.2548	−1.54408	−0.772038	+	0.635577i	$0.780762\pi$
$860$	−23.3137	−0.794991
$861$	−4.00000	−0.136320
$862$	−16.4853	−0.561491
$863$	−2.34315	−0.0797616	−0.0398808	−	0.999204i	$-0.512698\pi$
$863$	−2.34315	−0.0797616	−0.0398808	+	0.999204i	$0.512698\pi$
$864$	1.00000	0.0340207
$865$	−46.6274	−1.58538
$866$	−15.5563	−0.528626
$867$	9.00000	0.305656
$868$	−8.48528	−0.288009
$869$	17.3726	0.589325
$870$	−2.82843	−0.0958927
$871$	69.9411	2.36986
$872$	7.75736	0.262697
$873$	13.4142	0.454002
$874$	−2.58579	−0.0874655
$875$	−8.00000	−0.270449
$876$	2.48528	0.0839699
$877$	−8.14214	−0.274940	−0.137470	−	0.990506i	$-0.543897\pi$
$877$	−8.14214	−0.274940	−0.137470	+	0.990506i	$0.543897\pi$
$878$	17.6569	0.595890
$879$	−10.7279	−0.361844
$880$	−11.3137	−0.381385
$881$	28.7696	0.969271	0.484635	−	0.874716i	$-0.338952\pi$
$881$	28.7696	0.969271	0.484635	+	0.874716i	$0.338952\pi$
$882$	5.00000	0.168359
$883$	12.6863	0.426928	0.213464	−	0.976951i	$-0.431525\pi$
$883$	12.6863	0.426928	0.213464	+	0.976951i	$0.431525\pi$
$884$	13.6569	0.459330
$885$	−19.3137	−0.649223
$886$	4.14214	0.139158
$887$	20.7696	0.697373	0.348687	−	0.937239i	$-0.386628\pi$
$887$	20.7696	0.697373	0.348687	+	0.937239i	$0.386628\pi$
$888$	−0.828427	−0.0278002
$889$	−21.6569	−0.726348
$890$	−14.6274	−0.490312
$891$	4.00000	0.134005
$892$	10.3431	0.346314
$893$	29.2548	0.978976
$894$	−4.00000	−0.133780
$895$	−1.37258	−0.0458804
$896$	1.41421	0.0472456
$897$	4.82843	0.161216
$898$	3.79899	0.126774
$899$	6.00000	0.200111
$900$	3.00000	0.100000
$901$	11.3137	0.376914
$902$	11.3137	0.376705
$903$	11.6569	0.387916
$904$	9.65685	0.321182
$905$	−42.6274	−1.41698
$906$	16.0000	0.531564
$907$	−15.7574	−0.523215	−0.261607	−	0.965174i	$-0.584253\pi$
$907$	−15.7574	−0.523215	−0.261607	+	0.965174i	$0.584253\pi$
$908$	6.24264	0.207169
$909$	−4.82843	−0.160149
$910$	−19.3137	−0.640243
$911$	24.1005	0.798485	0.399243	−	0.916845i	$-0.369273\pi$
$911$	24.1005	0.798485	0.399243	+	0.916845i	$0.369273\pi$
$912$	2.58579	0.0856239
$913$	−54.2254	−1.79460
$914$	−16.6274	−0.549986
$915$	−21.6569	−0.715954
$916$	−11.6569	−0.385153
$917$	1.17157	0.0386887
$918$	2.82843	0.0933520
$919$	36.2426	1.19553	0.597767	−	0.801670i	$-0.296055\pi$
$919$	36.2426	1.19553	0.597767	+	0.801670i	$0.296055\pi$
$920$	−2.82843	−0.0932505
$921$	−22.8284	−0.752222
$922$	10.4853	0.345314
$923$	31.3137	1.03070
$924$	5.65685	0.186097
$925$	−2.48528	−0.0817155
$926$	−28.0000	−0.920137
$927$	−1.41421	−0.0464489
$928$	−1.00000	−0.0328266
$929$	−8.62742	−0.283056	−0.141528	−	0.989934i	$-0.545202\pi$
$929$	−8.62742	−0.283056	−0.141528	+	0.989934i	$0.545202\pi$
$930$	−16.9706	−0.556487
$931$	12.9289	0.423729
$932$	−4.82843	−0.158160
$933$	−17.4558	−0.571479
$934$	−23.5147	−0.769425
$935$	−32.0000	−1.04651
$936$	−4.82843	−0.157822
$937$	−37.3137	−1.21899	−0.609493	−	0.792792i	$-0.708627\pi$
$937$	−37.3137	−1.21899	−0.609493	+	0.792792i	$0.708627\pi$
$938$	20.4853	0.668868
$939$	24.6274	0.803685
$940$	32.0000	1.04372
$941$	−31.5147	−1.02735	−0.513675	−	0.857985i	$-0.671716\pi$
$941$	−31.5147	−1.02735	−0.513675	+	0.857985i	$0.671716\pi$
$942$	14.4853	0.471956
$943$	2.82843	0.0921063
$944$	−6.82843	−0.222246
$945$	−4.00000	−0.130120
$946$	−32.9706	−1.07197
$947$	29.7990	0.968337	0.484169	−	0.874975i	$-0.339122\pi$
$947$	29.7990	0.968337	0.484169	+	0.874975i	$0.339122\pi$
$948$	−4.34315	−0.141059
$949$	−12.0000	−0.389536
$950$	7.75736	0.251682
$951$	−20.1421	−0.653153
$952$	4.00000	0.129641
$953$	8.10051	0.262401	0.131201	−	0.991356i	$-0.458117\pi$
$953$	8.10051	0.262401	0.131201	+	0.991356i	$0.458117\pi$
$954$	−4.00000	−0.129505
$955$	20.9706	0.678591
$956$	−1.51472	−0.0489895
$957$	−4.00000	−0.129302
$958$	−6.92893	−0.223864
$959$	−11.3137	−0.365339
$960$	2.82843	0.0912871
$961$	5.00000	0.161290
$962$	4.00000	0.128965
$963$	−4.58579	−0.147775
$964$	1.51472	0.0487858
$965$	−21.6569	−0.697159
$966$	1.41421	0.0455016
$967$	23.3137	0.749718	0.374859	−	0.927082i	$-0.377691\pi$
$967$	23.3137	0.749718	0.374859	+	0.927082i	$0.377691\pi$
$968$	−5.00000	−0.160706
$969$	7.31371	0.234950
$970$	37.9411	1.21822
$971$	39.5980	1.27076	0.635380	−	0.772200i	$-0.280844\pi$
$971$	39.5980	1.27076	0.635380	+	0.772200i	$0.280844\pi$
$972$	−1.00000	−0.0320750
$973$	−6.14214	−0.196908
$974$	−26.3431	−0.844089
$975$	−14.4853	−0.463900
$976$	−7.65685	−0.245090
$977$	45.0711	1.44195	0.720976	−	0.692960i	$-0.243694\pi$
$977$	45.0711	1.44195	0.720976	+	0.692960i	$0.243694\pi$
$978$	−0.485281	−0.0155176
$979$	−20.6863	−0.661137
$980$	14.1421	0.451754
$981$	−7.75736	−0.247673
$982$	−1.51472	−0.0483366
$983$	−14.7279	−0.469748	−0.234874	−	0.972026i	$-0.575468\pi$
$983$	−14.7279	−0.469748	−0.234874	+	0.972026i	$0.575468\pi$
$984$	−2.82843	−0.0901670
$985$	−28.2843	−0.901212
$986$	−2.82843	−0.0900755
$987$	−16.0000	−0.509286
$988$	−12.4853	−0.397210
$989$	−8.24264	−0.262101
$990$	11.3137	0.359573
$991$	−56.4853	−1.79431	−0.897157	−	0.441712i	$-0.854371\pi$
$991$	−56.4853	−1.79431	−0.897157	+	0.441712i	$0.854371\pi$
$992$	−6.00000	−0.190500
$993$	−11.7990	−0.374430
$994$	9.17157	0.290905
$995$	23.3137	0.739094
$996$	13.5563	0.429549
$997$	−48.6274	−1.54005	−0.770023	−	0.638016i	$-0.779755\pi$
$997$	−48.6274	−1.54005	−0.770023	+	0.638016i	$0.779755\pi$
$998$	4.34315	0.137480
$999$	0.828427	0.0262103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.t.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.t.1.1	✓	2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} -1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.00000 q^{6} -1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -1.00000 q^{12} +4.82843 q^{13} +1.41421 q^{14} +2.82843 q^{15} +1.00000 q^{16} +2.82843 q^{17} -1.00000 q^{18} -2.58579 q^{19} -2.82843 q^{20} +1.41421 q^{21} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.00000 q^{25} -4.82843 q^{26} -1.00000 q^{27} -1.41421 q^{28} +1.00000 q^{29} -2.82843 q^{30} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.82843 q^{34} +4.00000 q^{35} +1.00000 q^{36} -0.828427 q^{37} +2.58579 q^{38} -4.82843 q^{39} +2.82843 q^{40} -2.82843 q^{41} -1.41421 q^{42} +8.24264 q^{43} +4.00000 q^{44} -2.82843 q^{45} +1.00000 q^{46} -11.3137 q^{47} -1.00000 q^{48} -5.00000 q^{49} -3.00000 q^{50} -2.82843 q^{51} +4.82843 q^{52} +4.00000 q^{53} +1.00000 q^{54} -11.3137 q^{55} +1.41421 q^{56} +2.58579 q^{57} -1.00000 q^{58} -6.82843 q^{59} +2.82843 q^{60} -7.65685 q^{61} -6.00000 q^{62} -1.41421 q^{63} +1.00000 q^{64} -13.6569 q^{65} +4.00000 q^{66} +14.4853 q^{67} +2.82843 q^{68} +1.00000 q^{69} -4.00000 q^{70} +6.48528 q^{71} -1.00000 q^{72} -2.48528 q^{73} +0.828427 q^{74} -3.00000 q^{75} -2.58579 q^{76} -5.65685 q^{77} +4.82843 q^{78} +4.34315 q^{79} -2.82843 q^{80} +1.00000 q^{81} +2.82843 q^{82} -13.5563 q^{83} +1.41421 q^{84} -8.00000 q^{85} -8.24264 q^{86} -1.00000 q^{87} -4.00000 q^{88} -5.17157 q^{89} +2.82843 q^{90} -6.82843 q^{91} -1.00000 q^{92} -6.00000 q^{93} +11.3137 q^{94} +7.31371 q^{95} +1.00000 q^{96} +13.4142 q^{97} +5.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 8 q^{11} - 2 q^{12} + 4 q^{13} + 2 q^{16} - 2 q^{18} - 8 q^{19} - 8 q^{22} - 2 q^{23} + 2 q^{24} + 6 q^{25} - 4 q^{26} - 2 q^{27} + 2 q^{29} + 12 q^{31} - 2 q^{32} - 8 q^{33} + 8 q^{35} + 2 q^{36} + 4 q^{37} + 8 q^{38} - 4 q^{39} + 8 q^{43} + 8 q^{44} + 2 q^{46} - 2 q^{48} - 10 q^{49} - 6 q^{50} + 4 q^{52} + 8 q^{53} + 2 q^{54} + 8 q^{57} - 2 q^{58} - 8 q^{59} - 4 q^{61} - 12 q^{62} + 2 q^{64} - 16 q^{65} + 8 q^{66} + 12 q^{67} + 2 q^{69} - 8 q^{70} - 4 q^{71} - 2 q^{72} + 12 q^{73} - 4 q^{74} - 6 q^{75} - 8 q^{76} + 4 q^{78} + 20 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{85} - 8 q^{86} - 2 q^{87} - 8 q^{88} - 16 q^{89} - 8 q^{91} - 2 q^{92} - 12 q^{93} - 8 q^{95} + 2 q^{96} + 24 q^{97} + 10 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 2 * q^9 + 8 * q^11 - 2 * q^12 + 4 * q^13 + 2 * q^16 - 2 * q^18 - 8 * q^19 - 8 * q^22 - 2 * q^23 + 2 * q^24 + 6 * q^25 - 4 * q^26 - 2 * q^27 + 2 * q^29 + 12 * q^31 - 2 * q^32 - 8 * q^33 + 8 * q^35 + 2 * q^36 + 4 * q^37 + 8 * q^38 - 4 * q^39 + 8 * q^43 + 8 * q^44 + 2 * q^46 - 2 * q^48 - 10 * q^49 - 6 * q^50 + 4 * q^52 + 8 * q^53 + 2 * q^54 + 8 * q^57 - 2 * q^58 - 8 * q^59 - 4 * q^61 - 12 * q^62 + 2 * q^64 - 16 * q^65 + 8 * q^66 + 12 * q^67 + 2 * q^69 - 8 * q^70 - 4 * q^71 - 2 * q^72 + 12 * q^73 - 4 * q^74 - 6 * q^75 - 8 * q^76 + 4 * q^78 + 20 * q^79 + 2 * q^81 + 4 * q^83 - 16 * q^85 - 8 * q^86 - 2 * q^87 - 8 * q^88 - 16 * q^89 - 8 * q^91 - 2 * q^92 - 12 * q^93 - 8 * q^95 + 2 * q^96 + 24 * q^97 + 10 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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