LMFDB - Embedded newform 4002.2.a.t.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{6} +1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{6} +1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} +4.00000 q^{11} -1.00000 q^{12} -0.828427 q^{13} -1.41421 q^{14} -2.82843 q^{15} +1.00000 q^{16} -2.82843 q^{17} -1.00000 q^{18} -5.41421 q^{19} +2.82843 q^{20} -1.41421 q^{21} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.00000 q^{25} +0.828427 q^{26} -1.00000 q^{27} +1.41421 q^{28} +1.00000 q^{29} +2.82843 q^{30} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.82843 q^{34} +4.00000 q^{35} +1.00000 q^{36} +4.82843 q^{37} +5.41421 q^{38} +0.828427 q^{39} -2.82843 q^{40} +2.82843 q^{41} +1.41421 q^{42} -0.242641 q^{43} +4.00000 q^{44} +2.82843 q^{45} +1.00000 q^{46} +11.3137 q^{47} -1.00000 q^{48} -5.00000 q^{49} -3.00000 q^{50} +2.82843 q^{51} -0.828427 q^{52} +4.00000 q^{53} +1.00000 q^{54} +11.3137 q^{55} -1.41421 q^{56} +5.41421 q^{57} -1.00000 q^{58} -1.17157 q^{59} -2.82843 q^{60} +3.65685 q^{61} -6.00000 q^{62} +1.41421 q^{63} +1.00000 q^{64} -2.34315 q^{65} +4.00000 q^{66} -2.48528 q^{67} -2.82843 q^{68} +1.00000 q^{69} -4.00000 q^{70} -10.4853 q^{71} -1.00000 q^{72} +14.4853 q^{73} -4.82843 q^{74} -3.00000 q^{75} -5.41421 q^{76} +5.65685 q^{77} -0.828427 q^{78} +15.6569 q^{79} +2.82843 q^{80} +1.00000 q^{81} -2.82843 q^{82} +17.5563 q^{83} -1.41421 q^{84} -8.00000 q^{85} +0.242641 q^{86} -1.00000 q^{87} -4.00000 q^{88} -10.8284 q^{89} -2.82843 q^{90} -1.17157 q^{91} -1.00000 q^{92} -6.00000 q^{93} -11.3137 q^{94} -15.3137 q^{95} +1.00000 q^{96} +10.5858 q^{97} +5.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	1.00000	0.408248
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	−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
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	−1.41421	−0.377964
$15$	−2.82843	−0.730297
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	−1.00000	−0.235702
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	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
$26$	0.828427	0.162468
$27$	−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
$37$	4.82843	0.793789	0.396894	−	0.917864i	$-0.370088\pi$
$37$	4.82843	0.793789	0.396894	+	0.917864i	$0.370088\pi$
$38$	5.41421	0.878301
$39$	0.828427	0.132655
$40$	−2.82843	−0.447214
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	1.41421	0.218218
$43$	−0.242641	−0.0370024	−0.0185012	−	0.999829i	$-0.505889\pi$
$43$	−0.242641	−0.0370024	−0.0185012	+	0.999829i	$0.505889\pi$
$44$	4.00000	0.603023
$45$	2.82843	0.421637
$46$	1.00000	0.147442
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	−1.00000	−0.144338
$49$	−5.00000	−0.714286
$50$	−3.00000	−0.424264
$51$	2.82843	0.396059
$52$	−0.828427	−0.114882
$53$	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
$57$	5.41421	0.717130
$58$	−1.00000	−0.131306
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	−2.82843	−0.365148
$61$	3.65685	0.468212	0.234106	−	0.972211i	$-0.424784\pi$
$61$	3.65685	0.468212	0.234106	+	0.972211i	$0.424784\pi$
$62$	−6.00000	−0.762001
$63$	1.41421	0.178174
$64$	1.00000	0.125000
$65$	−2.34315	−0.290631
$66$	4.00000	0.492366
$67$	−2.48528	−0.303625	−0.151813	−	0.988409i	$-0.548511\pi$
$67$	−2.48528	−0.303625	−0.151813	+	0.988409i	$0.548511\pi$
$68$	−2.82843	−0.342997
$69$	1.00000	0.120386
$70$	−4.00000	−0.478091
$71$	−10.4853	−1.24437	−0.622187	−	0.782869i	$-0.713756\pi$
$71$	−10.4853	−1.24437	−0.622187	+	0.782869i	$0.713756\pi$
$72$	−1.00000	−0.117851
$73$	14.4853	1.69537	0.847687	−	0.530497i	$-0.177995\pi$
$73$	14.4853	1.69537	0.847687	+	0.530497i	$0.177995\pi$
$74$	−4.82843	−0.561293
$75$	−3.00000	−0.346410
$76$	−5.41421	−0.621053
$77$	5.65685	0.644658
$78$	−0.828427	−0.0938009
$79$	15.6569	1.76153	0.880767	−	0.473550i	$-0.157028\pi$
$79$	15.6569	1.76153	0.880767	+	0.473550i	$0.157028\pi$
$80$	2.82843	0.316228
$81$	1.00000	0.111111
$82$	−2.82843	−0.312348
$83$	17.5563	1.92706	0.963530	−	0.267601i	$-0.0862309\pi$
$83$	17.5563	1.92706	0.963530	+	0.267601i	$0.0862309\pi$
$84$	−1.41421	−0.154303
$85$	−8.00000	−0.867722
$86$	0.242641	0.0261646
$87$	−1.00000	−0.107211
$88$	−4.00000	−0.426401
$89$	−10.8284	−1.14781	−0.573905	−	0.818922i	$-0.694572\pi$
$89$	−10.8284	−1.14781	−0.573905	+	0.818922i	$0.694572\pi$
$90$	−2.82843	−0.298142
$91$	−1.17157	−0.122814
$92$	−1.00000	−0.104257
$93$	−6.00000	−0.622171
$94$	−11.3137	−1.16692
$95$	−15.3137	−1.57115
$96$	1.00000	0.102062
$97$	10.5858	1.07482	0.537412	−	0.843320i	$-0.319402\pi$
$97$	10.5858	1.07482	0.537412	+	0.843320i	$0.319402\pi$
$98$	5.00000	0.505076
$99$	4.00000	0.402015
$100$	3.00000	0.300000
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	−2.82843	−0.280056
$103$	1.41421	0.139347	0.0696733	−	0.997570i	$-0.477804\pi$
$103$	1.41421	0.139347	0.0696733	+	0.997570i	$0.477804\pi$
$104$	0.828427	0.0812340
$105$	−4.00000	−0.390360
$106$	−4.00000	−0.388514
$107$	−7.41421	−0.716759	−0.358380	−	0.933576i	$-0.616671\pi$
$107$	−7.41421	−0.716759	−0.358380	+	0.933576i	$0.616671\pi$
$108$	−1.00000	−0.0962250
$109$	−16.2426	−1.55576	−0.777881	−	0.628411i	$-0.783706\pi$
$109$	−16.2426	−1.55576	−0.777881	+	0.628411i	$0.783706\pi$
$110$	−11.3137	−1.07872
$111$	−4.82843	−0.458294
$112$	1.41421	0.133631
$113$	1.65685	0.155864	0.0779319	−	0.996959i	$-0.475168\pi$
$113$	1.65685	0.155864	0.0779319	+	0.996959i	$0.475168\pi$
$114$	−5.41421	−0.507088
$115$	−2.82843	−0.263752
$116$	1.00000	0.0928477
$117$	−0.828427	−0.0765881
$118$	1.17157	0.107852
$119$	−4.00000	−0.366679
$120$	2.82843	0.258199
$121$	5.00000	0.454545
$122$	−3.65685	−0.331076
$123$	−2.82843	−0.255031
$124$	6.00000	0.538816
$125$	−5.65685	−0.505964
$126$	−1.41421	−0.125988
$127$	−7.31371	−0.648987	−0.324493	−	0.945888i	$-0.605194\pi$
$127$	−7.31371	−0.648987	−0.324493	+	0.945888i	$0.605194\pi$
$128$	−1.00000	−0.0883883
$129$	0.242641	0.0213633
$130$	2.34315	0.205507
$131$	4.82843	0.421862	0.210931	−	0.977501i	$-0.432351\pi$
$131$	4.82843	0.421862	0.210931	+	0.977501i	$0.432351\pi$
$132$	−4.00000	−0.348155
$133$	−7.65685	−0.663933
$134$	2.48528	0.214696
$135$	−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
$139$	15.6569	1.32800	0.663999	−	0.747734i	$-0.268858\pi$
$139$	15.6569	1.32800	0.663999	+	0.747734i	$0.268858\pi$
$140$	4.00000	0.338062
$141$	−11.3137	−0.952786
$142$	10.4853	0.879905
$143$	−3.31371	−0.277106
$144$	1.00000	0.0833333
$145$	2.82843	0.234888
$146$	−14.4853	−1.19881
$147$	5.00000	0.412393
$148$	4.82843	0.396894
$149$	−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	$0.274337\pi$
$152$	5.41421	0.439151
$153$	−2.82843	−0.228665
$154$	−5.65685	−0.455842
$155$	16.9706	1.36311
$156$	0.828427	0.0663273
$157$	−2.48528	−0.198347	−0.0991735	−	0.995070i	$-0.531620\pi$
$157$	−2.48528	−0.198347	−0.0991735	+	0.995070i	$0.531620\pi$
$158$	−15.6569	−1.24559
$159$	−4.00000	−0.317221
$160$	−2.82843	−0.223607
$161$	−1.41421	−0.111456
$162$	−1.00000	−0.0785674
$163$	16.4853	1.29123	0.645613	−	0.763664i	$-0.276602\pi$
$163$	16.4853	1.29123	0.645613	+	0.763664i	$0.276602\pi$
$164$	2.82843	0.220863
$165$	−11.3137	−0.880771
$166$	−17.5563	−1.36264
$167$	−3.31371	−0.256422	−0.128211	−	0.991747i	$-0.540924\pi$
$167$	−3.31371	−0.256422	−0.128211	+	0.991747i	$0.540924\pi$
$168$	1.41421	0.109109
$169$	−12.3137	−0.947208
$170$	8.00000	0.613572
$171$	−5.41421	−0.414035
$172$	−0.242641	−0.0185012
$173$	−0.485281	−0.0368953	−0.0184476	−	0.999830i	$-0.505872\pi$
$173$	−0.485281	−0.0368953	−0.0184476	+	0.999830i	$0.505872\pi$
$174$	1.00000	0.0758098
$175$	4.24264	0.320713
$176$	4.00000	0.301511
$177$	1.17157	0.0880608
$178$	10.8284	0.811625
$179$	−16.4853	−1.23217	−0.616084	−	0.787681i	$-0.711282\pi$
$179$	−16.4853	−1.23217	−0.616084	+	0.787681i	$0.711282\pi$
$180$	2.82843	0.210819
$181$	0.928932	0.0690470	0.0345235	−	0.999404i	$-0.489009\pi$
$181$	0.928932	0.0690470	0.0345235	+	0.999404i	$0.489009\pi$
$182$	1.17157	0.0868428
$183$	−3.65685	−0.270322
$184$	1.00000	0.0737210
$185$	13.6569	1.00407
$186$	6.00000	0.439941
$187$	−11.3137	−0.827340
$188$	11.3137	0.825137
$189$	−1.41421	−0.102869
$190$	15.3137	1.11097
$191$	−4.58579	−0.331816	−0.165908	−	0.986141i	$-0.553055\pi$
$191$	−4.58579	−0.331816	−0.165908	+	0.986141i	$0.553055\pi$
$192$	−1.00000	−0.0721688
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	−10.5858	−0.760015
$195$	2.34315	0.167796
$196$	−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
$199$	0.242641	0.0172003	0.00860017	−	0.999963i	$-0.497262\pi$
$199$	0.242641	0.0172003	0.00860017	+	0.999963i	$0.497262\pi$
$200$	−3.00000	−0.212132
$201$	2.48528	0.175298
$202$	−0.828427	−0.0582879
$203$	1.41421	0.0992583
$204$	2.82843	0.198030
$205$	8.00000	0.558744
$206$	−1.41421	−0.0985329
$207$	−1.00000	−0.0695048
$208$	−0.828427	−0.0574411
$209$	−21.6569	−1.49804
$210$	4.00000	0.276026
$211$	2.82843	0.194717	0.0973585	−	0.995249i	$-0.468961\pi$
$211$	2.82843	0.194717	0.0973585	+	0.995249i	$0.468961\pi$
$212$	4.00000	0.274721
$213$	10.4853	0.718440
$214$	7.41421	0.506825
$215$	−0.686292	−0.0468047
$216$	1.00000	0.0680414
$217$	8.48528	0.576018
$218$	16.2426	1.10009
$219$	−14.4853	−0.978825
$220$	11.3137	0.762770
$221$	2.34315	0.157617
$222$	4.82843	0.324063
$223$	21.6569	1.45025	0.725125	−	0.688617i	$-0.241782\pi$
$223$	21.6569	1.45025	0.725125	+	0.688617i	$0.241782\pi$
$224$	−1.41421	−0.0944911
$225$	3.00000	0.200000
$226$	−1.65685	−0.110212
$227$	−2.24264	−0.148849	−0.0744246	−	0.997227i	$-0.523712\pi$
$227$	−2.24264	−0.148849	−0.0744246	+	0.997227i	$0.523712\pi$
$228$	5.41421	0.358565
$229$	−0.343146	−0.0226757	−0.0113379	−	0.999936i	$-0.503609\pi$
$229$	−0.343146	−0.0226757	−0.0113379	+	0.999936i	$0.503609\pi$
$230$	2.82843	0.186501
$231$	−5.65685	−0.372194
$232$	−1.00000	−0.0656532
$233$	0.828427	0.0542721	0.0271360	−	0.999632i	$-0.491361\pi$
$233$	0.828427	0.0542721	0.0271360	+	0.999632i	$0.491361\pi$
$234$	0.828427	0.0541560
$235$	32.0000	2.08745
$236$	−1.17157	−0.0762629
$237$	−15.6569	−1.01702
$238$	4.00000	0.259281
$239$	−18.4853	−1.19571	−0.597857	−	0.801603i	$-0.703981\pi$
$239$	−18.4853	−1.19571	−0.597857	+	0.801603i	$0.703981\pi$
$240$	−2.82843	−0.182574
$241$	18.4853	1.19074	0.595371	−	0.803451i	$-0.297005\pi$
$241$	18.4853	1.19074	0.595371	+	0.803451i	$0.297005\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	3.65685	0.234106
$245$	−14.1421	−0.903508
$246$	2.82843	0.180334
$247$	4.48528	0.285392
$248$	−6.00000	−0.381000
$249$	−17.5563	−1.11259
$250$	5.65685	0.357771
$251$	23.3137	1.47155	0.735774	−	0.677227i	$-0.236819\pi$
$251$	23.3137	1.47155	0.735774	+	0.677227i	$0.236819\pi$
$252$	1.41421	0.0890871
$253$	−4.00000	−0.251478
$254$	7.31371	0.458903
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	−24.8284	−1.54875	−0.774377	−	0.632724i	$-0.781937\pi$
$257$	−24.8284	−1.54875	−0.774377	+	0.632724i	$0.781937\pi$
$258$	−0.242641	−0.0151061
$259$	6.82843	0.424298
$260$	−2.34315	−0.145316
$261$	1.00000	0.0618984
$262$	−4.82843	−0.298301
$263$	13.5563	0.835920	0.417960	−	0.908465i	$-0.362745\pi$
$263$	13.5563	0.835920	0.417960	+	0.908465i	$0.362745\pi$
$264$	4.00000	0.246183
$265$	11.3137	0.694996
$266$	7.65685	0.469472
$267$	10.8284	0.662689
$268$	−2.48528	−0.151813
$269$	−11.1716	−0.681143	−0.340571	−	0.940219i	$-0.610621\pi$
$269$	−11.1716	−0.681143	−0.340571	+	0.940219i	$0.610621\pi$
$270$	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
$273$	1.17157	0.0709068
$274$	−8.00000	−0.483298
$275$	12.0000	0.723627
$276$	1.00000	0.0601929
$277$	3.17157	0.190561	0.0952807	−	0.995450i	$-0.469625\pi$
$277$	3.17157	0.190561	0.0952807	+	0.995450i	$0.469625\pi$
$278$	−15.6569	−0.939036
$279$	6.00000	0.359211
$280$	−4.00000	−0.239046
$281$	0.100505	0.00599563	0.00299781	−	0.999996i	$-0.499046\pi$
$281$	0.100505	0.00599563	0.00299781	+	0.999996i	$0.499046\pi$
$282$	11.3137	0.673722
$283$	18.4853	1.09884	0.549418	−	0.835548i	$-0.314849\pi$
$283$	18.4853	1.09884	0.549418	+	0.835548i	$0.314849\pi$
$284$	−10.4853	−0.622187
$285$	15.3137	0.907106
$286$	3.31371	0.195944
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	−9.00000	−0.529412
$290$	−2.82843	−0.166091
$291$	−10.5858	−0.620550
$292$	14.4853	0.847687
$293$	−14.7279	−0.860414	−0.430207	−	0.902730i	$-0.641560\pi$
$293$	−14.7279	−0.860414	−0.430207	+	0.902730i	$0.641560\pi$
$294$	−5.00000	−0.291606
$295$	−3.31371	−0.192932
$296$	−4.82843	−0.280647
$297$	−4.00000	−0.232104
$298$	4.00000	0.231714
$299$	0.828427	0.0479092
$300$	−3.00000	−0.173205
$301$	−0.343146	−0.0197786
$302$	−16.0000	−0.920697
$303$	−0.828427	−0.0475919
$304$	−5.41421	−0.310526
$305$	10.3431	0.592247
$306$	2.82843	0.161690
$307$	17.1716	0.980033	0.490017	−	0.871713i	$-0.336991\pi$
$307$	17.1716	0.980033	0.490017	+	0.871713i	$0.336991\pi$
$308$	5.65685	0.322329
$309$	−1.41421	−0.0804518
$310$	−16.9706	−0.963863
$311$	−33.4558	−1.89711	−0.948553	−	0.316617i	$-0.897453\pi$
$311$	−33.4558	−1.89711	−0.948553	+	0.316617i	$0.897453\pi$
$312$	−0.828427	−0.0469005
$313$	20.6274	1.16593	0.582965	−	0.812497i	$-0.301892\pi$
$313$	20.6274	1.16593	0.582965	+	0.812497i	$0.301892\pi$
$314$	2.48528	0.140253
$315$	4.00000	0.225374
$316$	15.6569	0.880767
$317$	−8.14214	−0.457308	−0.228654	−	0.973508i	$-0.573432\pi$
$317$	−8.14214	−0.457308	−0.228654	+	0.973508i	$0.573432\pi$
$318$	4.00000	0.224309
$319$	4.00000	0.223957
$320$	2.82843	0.158114
$321$	7.41421	0.413821
$322$	1.41421	0.0788110
$323$	15.3137	0.852078
$324$	1.00000	0.0555556
$325$	−2.48528	−0.137859
$326$	−16.4853	−0.913035
$327$	16.2426	0.898220
$328$	−2.82843	−0.156174
$329$	16.0000	0.882109
$330$	11.3137	0.622799
$331$	−27.7990	−1.52797	−0.763985	−	0.645234i	$-0.776760\pi$
$331$	−27.7990	−1.52797	−0.763985	+	0.645234i	$0.776760\pi$
$332$	17.5563	0.963530
$333$	4.82843	0.264596
$334$	3.31371	0.181318
$335$	−7.02944	−0.384059
$336$	−1.41421	−0.0771517
$337$	4.24264	0.231111	0.115556	−	0.993301i	$-0.463135\pi$
$337$	4.24264	0.231111	0.115556	+	0.993301i	$0.463135\pi$
$338$	12.3137	0.669777
$339$	−1.65685	−0.0899880
$340$	−8.00000	−0.433861
$341$	24.0000	1.29967
$342$	5.41421	0.292767
$343$	−16.9706	−0.916324
$344$	0.242641	0.0130823
$345$	2.82843	0.152277
$346$	0.485281	0.0260889
$347$	11.3137	0.607352	0.303676	−	0.952775i	$-0.401786\pi$
$347$	11.3137	0.607352	0.303676	+	0.952775i	$0.401786\pi$
$348$	−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
$351$	0.828427	0.0442182
$352$	−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	$0.585743\pi$
$354$	−1.17157	−0.0622684
$355$	−29.6569	−1.57402
$356$	−10.8284	−0.573905
$357$	4.00000	0.211702
$358$	16.4853	0.871274
$359$	−13.5563	−0.715477	−0.357738	−	0.933822i	$-0.616452\pi$
$359$	−13.5563	−0.715477	−0.357738	+	0.933822i	$0.616452\pi$
$360$	−2.82843	−0.149071
$361$	10.3137	0.542827
$362$	−0.928932	−0.0488236
$363$	−5.00000	−0.262432
$364$	−1.17157	−0.0614071
$365$	40.9706	2.14450
$366$	3.65685	0.191147
$367$	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
$370$	−13.6569	−0.709986
$371$	5.65685	0.293689
$372$	−6.00000	−0.311086
$373$	24.2426	1.25524	0.627618	−	0.778521i	$-0.284030\pi$
$373$	24.2426	1.25524	0.627618	+	0.778521i	$0.284030\pi$
$374$	11.3137	0.585018
$375$	5.65685	0.292119
$376$	−11.3137	−0.583460
$377$	−0.828427	−0.0426662
$378$	1.41421	0.0727393
$379$	22.3848	1.14983	0.574914	−	0.818214i	$-0.305035\pi$
$379$	22.3848	1.14983	0.574914	+	0.818214i	$0.305035\pi$
$380$	−15.3137	−0.785577
$381$	7.31371	0.374693
$382$	4.58579	0.234629
$383$	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
$386$	3.65685	0.186129
$387$	−0.242641	−0.0123341
$388$	10.5858	0.537412
$389$	−5.55635	−0.281718	−0.140859	−	0.990030i	$-0.544986\pi$
$389$	−5.55635	−0.281718	−0.140859	+	0.990030i	$0.544986\pi$
$390$	−2.34315	−0.118650
$391$	2.82843	0.143040
$392$	5.00000	0.252538
$393$	−4.82843	−0.243562
$394$	−10.0000	−0.503793
$395$	44.2843	2.22818
$396$	4.00000	0.201008
$397$	−8.34315	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$397$	−8.34315	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$398$	−0.242641	−0.0121625
$399$	7.65685	0.383322
$400$	3.00000	0.150000
$401$	5.27208	0.263275	0.131638	−	0.991298i	$-0.457977\pi$
$401$	5.27208	0.263275	0.131638	+	0.991298i	$0.457977\pi$
$402$	−2.48528	−0.123955
$403$	−4.97056	−0.247601
$404$	0.828427	0.0412158
$405$	2.82843	0.140546
$406$	−1.41421	−0.0701862
$407$	19.3137	0.957345
$408$	−2.82843	−0.140028
$409$	23.6569	1.16976	0.584878	−	0.811121i	$-0.301142\pi$
$409$	23.6569	1.16976	0.584878	+	0.811121i	$0.301142\pi$
$410$	−8.00000	−0.395092
$411$	−8.00000	−0.394611
$412$	1.41421	0.0696733
$413$	−1.65685	−0.0815285
$414$	1.00000	0.0491473
$415$	49.6569	2.43756
$416$	0.828427	0.0406170
$417$	−15.6569	−0.766719
$418$	21.6569	1.05927
$419$	−30.0416	−1.46763	−0.733815	−	0.679350i	$-0.762262\pi$
$419$	−30.0416	−1.46763	−0.733815	+	0.679350i	$0.762262\pi$
$420$	−4.00000	−0.195180
$421$	−11.6569	−0.568120	−0.284060	−	0.958806i	$-0.591682\pi$
$421$	−11.6569	−0.568120	−0.284060	+	0.958806i	$0.591682\pi$
$422$	−2.82843	−0.137686
$423$	11.3137	0.550091
$424$	−4.00000	−0.194257
$425$	−8.48528	−0.411597
$426$	−10.4853	−0.508014
$427$	5.17157	0.250270
$428$	−7.41421	−0.358380
$429$	3.31371	0.159987
$430$	0.686292	0.0330959
$431$	−0.485281	−0.0233752	−0.0116876	−	0.999932i	$-0.503720\pi$
$431$	−0.485281	−0.0233752	−0.0116876	+	0.999932i	$0.503720\pi$
$432$	−1.00000	−0.0481125
$433$	−15.5563	−0.747590	−0.373795	−	0.927511i	$-0.621944\pi$
$433$	−15.5563	−0.747590	−0.373795	+	0.927511i	$0.621944\pi$
$434$	−8.48528	−0.407307
$435$	−2.82843	−0.135613
$436$	−16.2426	−0.777881
$437$	5.41421	0.258997
$438$	14.4853	0.692134
$439$	−6.34315	−0.302742	−0.151371	−	0.988477i	$-0.548369\pi$
$439$	−6.34315	−0.302742	−0.151371	+	0.988477i	$0.548369\pi$
$440$	−11.3137	−0.539360
$441$	−5.00000	−0.238095
$442$	−2.34315	−0.111452
$443$	24.1421	1.14703	0.573514	−	0.819196i	$-0.305580\pi$
$443$	24.1421	1.14703	0.573514	+	0.819196i	$0.305580\pi$
$444$	−4.82843	−0.229147
$445$	−30.6274	−1.45188
$446$	−21.6569	−1.02548
$447$	4.00000	0.189194
$448$	1.41421	0.0668153
$449$	35.7990	1.68946	0.844729	−	0.535194i	$-0.179761\pi$
$449$	35.7990	1.68946	0.844729	+	0.535194i	$0.179761\pi$
$450$	−3.00000	−0.141421
$451$	11.3137	0.532742
$452$	1.65685	0.0779319
$453$	−16.0000	−0.751746
$454$	2.24264	0.105252
$455$	−3.31371	−0.155349
$456$	−5.41421	−0.253544
$457$	−28.6274	−1.33913	−0.669567	−	0.742752i	$-0.733520\pi$
$457$	−28.6274	−1.33913	−0.669567	+	0.742752i	$0.733520\pi$
$458$	0.343146	0.0160341
$459$	2.82843	0.132020
$460$	−2.82843	−0.131876
$461$	6.48528	0.302050	0.151025	−	0.988530i	$-0.451743\pi$
$461$	6.48528	0.302050	0.151025	+	0.988530i	$0.451743\pi$
$462$	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
$466$	−0.828427	−0.0383761
$467$	40.4853	1.87344	0.936718	−	0.350086i	$-0.113848\pi$
$467$	40.4853	1.87344	0.936718	+	0.350086i	$0.113848\pi$
$468$	−0.828427	−0.0382941
$469$	−3.51472	−0.162295
$470$	−32.0000	−1.47605
$471$	2.48528	0.114516
$472$	1.17157	0.0539260
$473$	−0.970563	−0.0446265
$474$	15.6569	0.719143
$475$	−16.2426	−0.745263
$476$	−4.00000	−0.183340
$477$	4.00000	0.183147
$478$	18.4853	0.845497
$479$	21.0711	0.962762	0.481381	−	0.876512i	$-0.340135\pi$
$479$	21.0711	0.962762	0.481381	+	0.876512i	$0.340135\pi$
$480$	2.82843	0.129099
$481$	−4.00000	−0.182384
$482$	−18.4853	−0.841981
$483$	1.41421	0.0643489
$484$	5.00000	0.227273
$485$	29.9411	1.35956
$486$	1.00000	0.0453609
$487$	37.6569	1.70639	0.853197	−	0.521588i	$-0.174660\pi$
$487$	37.6569	1.70639	0.853197	+	0.521588i	$0.174660\pi$
$488$	−3.65685	−0.165538
$489$	−16.4853	−0.745490
$490$	14.1421	0.638877
$491$	18.4853	0.834229	0.417115	−	0.908854i	$-0.363041\pi$
$491$	18.4853	0.834229	0.417115	+	0.908854i	$0.363041\pi$
$492$	−2.82843	−0.127515
$493$	−2.82843	−0.127386
$494$	−4.48528	−0.201802
$495$	11.3137	0.508513
$496$	6.00000	0.269408
$497$	−14.8284	−0.665146
$498$	17.5563	0.786719
$499$	−15.6569	−0.700897	−0.350449	−	0.936582i	$-0.613971\pi$
$499$	−15.6569	−0.700897	−0.350449	+	0.936582i	$0.613971\pi$
$500$	−5.65685	−0.252982
$501$	3.31371	0.148046
$502$	−23.3137	−1.04054
$503$	−0.786797	−0.0350815	−0.0175408	−	0.999846i	$-0.505584\pi$
$503$	−0.786797	−0.0350815	−0.0175408	+	0.999846i	$0.505584\pi$
$504$	−1.41421	−0.0629941
$505$	2.34315	0.104269
$506$	4.00000	0.177822
$507$	12.3137	0.546871
$508$	−7.31371	−0.324493
$509$	−41.4558	−1.83750	−0.918749	−	0.394842i	$-0.870799\pi$
$509$	−41.4558	−1.83750	−0.918749	+	0.394842i	$0.870799\pi$
$510$	−8.00000	−0.354246
$511$	20.4853	0.906215
$512$	−1.00000	−0.0441942
$513$	5.41421	0.239043
$514$	24.8284	1.09513
$515$	4.00000	0.176261
$516$	0.242641	0.0106817
$517$	45.2548	1.99031
$518$	−6.82843	−0.300024
$519$	0.485281	0.0213015
$520$	2.34315	0.102754
$521$	14.9289	0.654048	0.327024	−	0.945016i	$-0.393954\pi$
$521$	14.9289	0.654048	0.327024	+	0.945016i	$0.393954\pi$
$522$	−1.00000	−0.0437688
$523$	−12.8284	−0.560948	−0.280474	−	0.959862i	$-0.590492\pi$
$523$	−12.8284	−0.560948	−0.280474	+	0.959862i	$0.590492\pi$
$524$	4.82843	0.210931
$525$	−4.24264	−0.185164
$526$	−13.5563	−0.591085
$527$	−16.9706	−0.739249
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	−11.3137	−0.491436
$531$	−1.17157	−0.0508419
$532$	−7.65685	−0.331967
$533$	−2.34315	−0.101493
$534$	−10.8284	−0.468592
$535$	−20.9706	−0.906636
$536$	2.48528	0.107348
$537$	16.4853	0.711392
$538$	11.1716	0.481641
$539$	−20.0000	−0.861461
$540$	−2.82843	−0.121716
$541$	41.9411	1.80319	0.901595	−	0.432581i	$-0.142397\pi$
$541$	41.9411	1.80319	0.901595	+	0.432581i	$0.142397\pi$
$542$	−4.00000	−0.171815
$543$	−0.928932	−0.0398643
$544$	2.82843	0.121268
$545$	−45.9411	−1.96790
$546$	−1.17157	−0.0501387
$547$	30.9706	1.32421	0.662103	−	0.749413i	$-0.269664\pi$
$547$	30.9706	1.32421	0.662103	+	0.749413i	$0.269664\pi$
$548$	8.00000	0.341743
$549$	3.65685	0.156071
$550$	−12.0000	−0.511682
$551$	−5.41421	−0.230653
$552$	−1.00000	−0.0425628
$553$	22.1421	0.941579
$554$	−3.17157	−0.134747
$555$	−13.6569	−0.579701
$556$	15.6569	0.663999
$557$	−1.17157	−0.0496411	−0.0248206	−	0.999692i	$-0.507901\pi$
$557$	−1.17157	−0.0496411	−0.0248206	+	0.999692i	$0.507901\pi$
$558$	−6.00000	−0.254000
$559$	0.201010	0.00850182
$560$	4.00000	0.169031
$561$	11.3137	0.477665
$562$	−0.100505	−0.00423955
$563$	−6.62742	−0.279312	−0.139656	−	0.990200i	$-0.544600\pi$
$563$	−6.62742	−0.279312	−0.139656	+	0.990200i	$0.544600\pi$
$564$	−11.3137	−0.476393
$565$	4.68629	0.197154
$566$	−18.4853	−0.776994
$567$	1.41421	0.0593914
$568$	10.4853	0.439953
$569$	3.02944	0.127001	0.0635003	−	0.997982i	$-0.479774\pi$
$569$	3.02944	0.127001	0.0635003	+	0.997982i	$0.479774\pi$
$570$	−15.3137	−0.641421
$571$	5.31371	0.222372	0.111186	−	0.993800i	$-0.464535\pi$
$571$	5.31371	0.222372	0.111186	+	0.993800i	$0.464535\pi$
$572$	−3.31371	−0.138553
$573$	4.58579	0.191574
$574$	−4.00000	−0.166957
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	−21.3137	−0.887301	−0.443651	−	0.896200i	$-0.646317\pi$
$577$	−21.3137	−0.887301	−0.443651	+	0.896200i	$0.646317\pi$
$578$	9.00000	0.374351
$579$	3.65685	0.151974
$580$	2.82843	0.117444
$581$	24.8284	1.03006
$582$	10.5858	0.438795
$583$	16.0000	0.662652
$584$	−14.4853	−0.599405
$585$	−2.34315	−0.0968772
$586$	14.7279	0.608405
$587$	−39.5980	−1.63438	−0.817192	−	0.576366i	$-0.804470\pi$
$587$	−39.5980	−1.63438	−0.817192	+	0.576366i	$0.804470\pi$
$588$	5.00000	0.206197
$589$	−32.4853	−1.33853
$590$	3.31371	0.136423
$591$	−10.0000	−0.411345
$592$	4.82843	0.198447
$593$	−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
$597$	−0.242641	−0.00993062
$598$	−0.828427	−0.0338769
$599$	7.79899	0.318658	0.159329	−	0.987226i	$-0.449067\pi$
$599$	7.79899	0.318658	0.159329	+	0.987226i	$0.449067\pi$
$600$	3.00000	0.122474
$601$	−12.3431	−0.503487	−0.251744	−	0.967794i	$-0.581004\pi$
$601$	−12.3431	−0.503487	−0.251744	+	0.967794i	$0.581004\pi$
$602$	0.343146	0.0139856
$603$	−2.48528	−0.101208
$604$	16.0000	0.651031
$605$	14.1421	0.574960
$606$	0.828427	0.0336526
$607$	−6.97056	−0.282926	−0.141463	−	0.989944i	$-0.545181\pi$
$607$	−6.97056	−0.282926	−0.141463	+	0.989944i	$0.545181\pi$
$608$	5.41421	0.219575
$609$	−1.41421	−0.0573068
$610$	−10.3431	−0.418782
$611$	−9.37258	−0.379174
$612$	−2.82843	−0.114332
$613$	−25.2132	−1.01835	−0.509176	−	0.860663i	$-0.670050\pi$
$613$	−25.2132	−1.01835	−0.509176	+	0.860663i	$0.670050\pi$
$614$	−17.1716	−0.692988
$615$	−8.00000	−0.322591
$616$	−5.65685	−0.227921
$617$	−24.2843	−0.977648	−0.488824	−	0.872382i	$-0.662574\pi$
$617$	−24.2843	−0.977648	−0.488824	+	0.872382i	$0.662574\pi$
$618$	1.41421	0.0568880
$619$	−47.5563	−1.91145	−0.955726	−	0.294260i	$-0.904927\pi$
$619$	−47.5563	−1.91145	−0.955726	+	0.294260i	$0.904927\pi$
$620$	16.9706	0.681554
$621$	1.00000	0.0401286
$622$	33.4558	1.34146
$623$	−15.3137	−0.613531
$624$	0.828427	0.0331636
$625$	−31.0000	−1.24000
$626$	−20.6274	−0.824437
$627$	21.6569	0.864891
$628$	−2.48528	−0.0991735
$629$	−13.6569	−0.544534
$630$	−4.00000	−0.159364
$631$	23.0711	0.918445	0.459222	−	0.888321i	$-0.348128\pi$
$631$	23.0711	0.918445	0.459222	+	0.888321i	$0.348128\pi$
$632$	−15.6569	−0.622796
$633$	−2.82843	−0.112420
$634$	8.14214	0.323366
$635$	−20.6863	−0.820910
$636$	−4.00000	−0.158610
$637$	4.14214	0.164117
$638$	−4.00000	−0.158362
$639$	−10.4853	−0.414791
$640$	−2.82843	−0.111803
$641$	−13.6569	−0.539413	−0.269707	−	0.962943i	$-0.586927\pi$
$641$	−13.6569	−0.539413	−0.269707	+	0.962943i	$0.586927\pi$
$642$	−7.41421	−0.292616
$643$	0.828427	0.0326700	0.0163350	−	0.999867i	$-0.494800\pi$
$643$	0.828427	0.0326700	0.0163350	+	0.999867i	$0.494800\pi$
$644$	−1.41421	−0.0557278
$645$	0.686292	0.0270227
$646$	−15.3137	−0.602510
$647$	−9.79899	−0.385238	−0.192619	−	0.981274i	$-0.561698\pi$
$647$	−9.79899	−0.385238	−0.192619	+	0.981274i	$0.561698\pi$
$648$	−1.00000	−0.0392837
$649$	−4.68629	−0.183953
$650$	2.48528	0.0974808
$651$	−8.48528	−0.332564
$652$	16.4853	0.645613
$653$	−11.6569	−0.456168	−0.228084	−	0.973641i	$-0.573246\pi$
$653$	−11.6569	−0.456168	−0.228084	+	0.973641i	$0.573246\pi$
$654$	−16.2426	−0.635138
$655$	13.6569	0.533617
$656$	2.82843	0.110432
$657$	14.4853	0.565125
$658$	−16.0000	−0.623745
$659$	25.4558	0.991619	0.495809	−	0.868431i	$-0.334871\pi$
$659$	25.4558	0.991619	0.495809	+	0.868431i	$0.334871\pi$
$660$	−11.3137	−0.440386
$661$	−35.5563	−1.38298	−0.691491	−	0.722385i	$-0.743046\pi$
$661$	−35.5563	−1.38298	−0.691491	+	0.722385i	$0.743046\pi$
$662$	27.7990	1.08044
$663$	−2.34315	−0.0910002
$664$	−17.5563	−0.681318
$665$	−21.6569	−0.839817
$666$	−4.82843	−0.187098
$667$	−1.00000	−0.0387202
$668$	−3.31371	−0.128211
$669$	−21.6569	−0.837302
$670$	7.02944	0.271571
$671$	14.6274	0.564685
$672$	1.41421	0.0545545
$673$	−9.31371	−0.359017	−0.179509	−	0.983756i	$-0.557451\pi$
$673$	−9.31371	−0.359017	−0.179509	+	0.983756i	$0.557451\pi$
$674$	−4.24264	−0.163420
$675$	−3.00000	−0.115470
$676$	−12.3137	−0.473604
$677$	−13.2721	−0.510087	−0.255044	−	0.966930i	$-0.582090\pi$
$677$	−13.2721	−0.510087	−0.255044	+	0.966930i	$0.582090\pi$
$678$	1.65685	0.0636311
$679$	14.9706	0.574517
$680$	8.00000	0.306786
$681$	2.24264	0.0859382
$682$	−24.0000	−0.919007
$683$	−15.0294	−0.575085	−0.287543	−	0.957768i	$-0.592838\pi$
$683$	−15.0294	−0.575085	−0.287543	+	0.957768i	$0.592838\pi$
$684$	−5.41421	−0.207018
$685$	22.6274	0.864549
$686$	16.9706	0.647939
$687$	0.343146	0.0130918
$688$	−0.242641	−0.00925059
$689$	−3.31371	−0.126242
$690$	−2.82843	−0.107676
$691$	−3.31371	−0.126059	−0.0630297	−	0.998012i	$-0.520076\pi$
$691$	−3.31371	−0.126059	−0.0630297	+	0.998012i	$0.520076\pi$
$692$	−0.485281	−0.0184476
$693$	5.65685	0.214886
$694$	−11.3137	−0.429463
$695$	44.2843	1.67980
$696$	1.00000	0.0379049
$697$	−8.00000	−0.303022
$698$	−2.00000	−0.0757011
$699$	−0.828427	−0.0313340
$700$	4.24264	0.160357
$701$	−0.201010	−0.00759205	−0.00379602	−	0.999993i	$-0.501208\pi$
$701$	−0.201010	−0.00759205	−0.00379602	+	0.999993i	$0.501208\pi$
$702$	−0.828427	−0.0312670
$703$	−26.1421	−0.985969
$704$	4.00000	0.150756
$705$	−32.0000	−1.20519
$706$	10.0000	0.376355
$707$	1.17157	0.0440615
$708$	1.17157	0.0440304
$709$	38.8701	1.45980	0.729898	−	0.683556i	$-0.239568\pi$
$709$	38.8701	1.45980	0.729898	+	0.683556i	$0.239568\pi$
$710$	29.6569	1.11300
$711$	15.6569	0.587178
$712$	10.8284	0.405812
$713$	−6.00000	−0.224702
$714$	−4.00000	−0.149696
$715$	−9.37258	−0.350515
$716$	−16.4853	−0.616084
$717$	18.4853	0.690345
$718$	13.5563	0.505918
$719$	−4.28427	−0.159776	−0.0798882	−	0.996804i	$-0.525456\pi$
$719$	−4.28427	−0.159776	−0.0798882	+	0.996804i	$0.525456\pi$
$720$	2.82843	0.105409
$721$	2.00000	0.0744839
$722$	−10.3137	−0.383836
$723$	−18.4853	−0.687475
$724$	0.928932	0.0345235
$725$	3.00000	0.111417
$726$	5.00000	0.185567
$727$	−11.6569	−0.432329	−0.216164	−	0.976357i	$-0.569355\pi$
$727$	−11.6569	−0.432329	−0.216164	+	0.976357i	$0.569355\pi$
$728$	1.17157	0.0434214
$729$	1.00000	0.0370370
$730$	−40.9706	−1.51639
$731$	0.686292	0.0253834
$732$	−3.65685	−0.135161
$733$	15.1716	0.560375	0.280187	−	0.959945i	$-0.409603\pi$
$733$	15.1716	0.560375	0.280187	+	0.959945i	$0.409603\pi$
$734$	−10.0000	−0.369107
$735$	14.1421	0.521641
$736$	1.00000	0.0368605
$737$	−9.94113	−0.366186
$738$	−2.82843	−0.104116
$739$	44.9706	1.65427	0.827134	−	0.562004i	$-0.189969\pi$
$739$	44.9706	1.65427	0.827134	+	0.562004i	$0.189969\pi$
$740$	13.6569	0.502036
$741$	−4.48528	−0.164771
$742$	−5.65685	−0.207670
$743$	27.4142	1.00573	0.502865	−	0.864365i	$-0.332279\pi$
$743$	27.4142	1.00573	0.502865	+	0.864365i	$0.332279\pi$
$744$	6.00000	0.219971
$745$	−11.3137	−0.414502
$746$	−24.2426	−0.887586
$747$	17.5563	0.642353
$748$	−11.3137	−0.413670
$749$	−10.4853	−0.383124
$750$	−5.65685	−0.206559
$751$	−27.9411	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$751$	−27.9411	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$752$	11.3137	0.412568
$753$	−23.3137	−0.849599
$754$	0.828427	0.0301695
$755$	45.2548	1.64699
$756$	−1.41421	−0.0514344
$757$	−45.3137	−1.64695	−0.823477	−	0.567349i	$-0.807969\pi$
$757$	−45.3137	−1.64695	−0.823477	+	0.567349i	$0.807969\pi$
$758$	−22.3848	−0.813052
$759$	4.00000	0.145191
$760$	15.3137	0.555487
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	−7.31371	−0.264948
$763$	−22.9706	−0.831590
$764$	−4.58579	−0.165908
$765$	−8.00000	−0.289241
$766$	−8.00000	−0.289052
$767$	0.970563	0.0350450
$768$	−1.00000	−0.0360844
$769$	−34.8701	−1.25745	−0.628723	−	0.777629i	$-0.716422\pi$
$769$	−34.8701	−1.25745	−0.628723	+	0.777629i	$0.716422\pi$
$770$	−16.0000	−0.576600
$771$	24.8284	0.894174
$772$	−3.65685	−0.131613
$773$	−46.0416	−1.65600	−0.828001	−	0.560726i	$-0.810522\pi$
$773$	−46.0416	−1.65600	−0.828001	+	0.560726i	$0.810522\pi$
$774$	0.242641	0.00872154
$775$	18.0000	0.646579
$776$	−10.5858	−0.380008
$777$	−6.82843	−0.244968
$778$	5.55635	0.199205
$779$	−15.3137	−0.548671
$780$	2.34315	0.0838981
$781$	−41.9411	−1.50077
$782$	−2.82843	−0.101144
$783$	−1.00000	−0.0357371
$784$	−5.00000	−0.178571
$785$	−7.02944	−0.250891
$786$	4.82843	0.172224
$787$	−24.1421	−0.860574	−0.430287	−	0.902692i	$-0.641588\pi$
$787$	−24.1421	−0.860574	−0.430287	+	0.902692i	$0.641588\pi$
$788$	10.0000	0.356235
$789$	−13.5563	−0.482619
$790$	−44.2843	−1.57556
$791$	2.34315	0.0833127
$792$	−4.00000	−0.142134
$793$	−3.02944	−0.107578
$794$	8.34315	0.296087
$795$	−11.3137	−0.401256
$796$	0.242641	0.00860017
$797$	−38.5269	−1.36469	−0.682347	−	0.731029i	$-0.739040\pi$
$797$	−38.5269	−1.36469	−0.682347	+	0.731029i	$0.739040\pi$
$798$	−7.65685	−0.271050
$799$	−32.0000	−1.13208
$800$	−3.00000	−0.106066
$801$	−10.8284	−0.382604
$802$	−5.27208	−0.186164
$803$	57.9411	2.04470
$804$	2.48528	0.0876491
$805$	−4.00000	−0.140981
$806$	4.97056	0.175081
$807$	11.1716	0.393258
$808$	−0.828427	−0.0291440
$809$	−12.2010	−0.428965	−0.214482	−	0.976728i	$-0.568806\pi$
$809$	−12.2010	−0.428965	−0.214482	+	0.976728i	$0.568806\pi$
$810$	−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
$814$	−19.3137	−0.676945
$815$	46.6274	1.63329
$816$	2.82843	0.0990148
$817$	1.31371	0.0459608
$818$	−23.6569	−0.827143
$819$	−1.17157	−0.0409381
$820$	8.00000	0.279372
$821$	30.4264	1.06189	0.530944	−	0.847407i	$-0.321837\pi$
$821$	30.4264	1.06189	0.530944	+	0.847407i	$0.321837\pi$
$822$	8.00000	0.279032
$823$	−21.9411	−0.764820	−0.382410	−	0.923993i	$-0.624906\pi$
$823$	−21.9411	−0.764820	−0.382410	+	0.923993i	$0.624906\pi$
$824$	−1.41421	−0.0492665
$825$	−12.0000	−0.417786
$826$	1.65685	0.0576493
$827$	−37.4558	−1.30247	−0.651234	−	0.758877i	$-0.725748\pi$
$827$	−37.4558	−1.30247	−0.651234	+	0.758877i	$0.725748\pi$
$828$	−1.00000	−0.0347524
$829$	5.31371	0.184553	0.0922764	−	0.995733i	$-0.470586\pi$
$829$	5.31371	0.184553	0.0922764	+	0.995733i	$0.470586\pi$
$830$	−49.6569	−1.72361
$831$	−3.17157	−0.110021
$832$	−0.828427	−0.0287205
$833$	14.1421	0.489996
$834$	15.6569	0.542153
$835$	−9.37258	−0.324352
$836$	−21.6569	−0.749018
$837$	−6.00000	−0.207390
$838$	30.0416	1.03777
$839$	−21.3553	−0.737268	−0.368634	−	0.929575i	$-0.620174\pi$
$839$	−21.3553	−0.737268	−0.368634	+	0.929575i	$0.620174\pi$
$840$	4.00000	0.138013
$841$	1.00000	0.0344828
$842$	11.6569	0.401722
$843$	−0.100505	−0.00346158
$844$	2.82843	0.0973585
$845$	−34.8284	−1.19813
$846$	−11.3137	−0.388973
$847$	7.07107	0.242965
$848$	4.00000	0.137361
$849$	−18.4853	−0.634413
$850$	8.48528	0.291043
$851$	−4.82843	−0.165516
$852$	10.4853	0.359220
$853$	−47.9411	−1.64147	−0.820736	−	0.571307i	$-0.806437\pi$
$853$	−47.9411	−1.64147	−0.820736	+	0.571307i	$0.806437\pi$
$854$	−5.17157	−0.176968
$855$	−15.3137	−0.523718
$856$	7.41421	0.253413
$857$	0.142136	0.00485526	0.00242763	−	0.999997i	$-0.499227\pi$
$857$	0.142136	0.00485526	0.00242763	+	0.999997i	$0.499227\pi$
$858$	−3.31371	−0.113128
$859$	45.2548	1.54408	0.772038	−	0.635577i	$-0.219238\pi$
$859$	45.2548	1.54408	0.772038	+	0.635577i	$0.219238\pi$
$860$	−0.686292	−0.0234023
$861$	−4.00000	−0.136320
$862$	0.485281	0.0165287
$863$	−13.6569	−0.464885	−0.232442	−	0.972610i	$-0.574672\pi$
$863$	−13.6569	−0.464885	−0.232442	+	0.972610i	$0.574672\pi$
$864$	1.00000	0.0340207
$865$	−1.37258	−0.0466692
$866$	15.5563	0.528626
$867$	9.00000	0.305656
$868$	8.48528	0.288009
$869$	62.6274	2.12449
$870$	2.82843	0.0958927
$871$	2.05887	0.0697623
$872$	16.2426	0.550045
$873$	10.5858	0.358275
$874$	−5.41421	−0.183139
$875$	−8.00000	−0.270449
$876$	−14.4853	−0.489412
$877$	20.1421	0.680152	0.340076	−	0.940398i	$-0.389547\pi$
$877$	20.1421	0.680152	0.340076	+	0.940398i	$0.389547\pi$
$878$	6.34315	0.214071
$879$	14.7279	0.496760
$880$	11.3137	0.381385
$881$	−44.7696	−1.50832	−0.754162	−	0.656688i	$-0.771957\pi$
$881$	−44.7696	−1.50832	−0.754162	+	0.656688i	$0.771957\pi$
$882$	5.00000	0.168359
$883$	35.3137	1.18840	0.594200	−	0.804317i	$-0.297469\pi$
$883$	35.3137	1.18840	0.594200	+	0.804317i	$0.297469\pi$
$884$	2.34315	0.0788085
$885$	3.31371	0.111389
$886$	−24.1421	−0.811071
$887$	−52.7696	−1.77183	−0.885914	−	0.463849i	$-0.846468\pi$
$887$	−52.7696	−1.77183	−0.885914	+	0.463849i	$0.846468\pi$
$888$	4.82843	0.162031
$889$	−10.3431	−0.346898
$890$	30.6274	1.02663
$891$	4.00000	0.134005
$892$	21.6569	0.725125
$893$	−61.2548	−2.04981
$894$	−4.00000	−0.133780
$895$	−46.6274	−1.55858
$896$	−1.41421	−0.0472456
$897$	−0.828427	−0.0276604
$898$	−35.7990	−1.19463
$899$	6.00000	0.200111
$900$	3.00000	0.100000
$901$	−11.3137	−0.376914
$902$	−11.3137	−0.376705
$903$	0.343146	0.0114192
$904$	−1.65685	−0.0551062
$905$	2.62742	0.0873383
$906$	16.0000	0.531564
$907$	−24.2426	−0.804964	−0.402482	−	0.915428i	$-0.631852\pi$
$907$	−24.2426	−0.804964	−0.402482	+	0.915428i	$0.631852\pi$
$908$	−2.24264	−0.0744246
$909$	0.828427	0.0274772
$910$	3.31371	0.109848
$911$	43.8995	1.45446	0.727228	−	0.686396i	$-0.240808\pi$
$911$	43.8995	1.45446	0.727228	+	0.686396i	$0.240808\pi$
$912$	5.41421	0.179283
$913$	70.2254	2.32412
$914$	28.6274	0.946911
$915$	−10.3431	−0.341934
$916$	−0.343146	−0.0113379
$917$	6.82843	0.225495
$918$	−2.82843	−0.0933520
$919$	27.7574	0.915631	0.457815	−	0.889047i	$-0.348632\pi$
$919$	27.7574	0.915631	0.457815	+	0.889047i	$0.348632\pi$
$920$	2.82843	0.0932505
$921$	−17.1716	−0.565823
$922$	−6.48528	−0.213581
$923$	8.68629	0.285913
$924$	−5.65685	−0.186097
$925$	14.4853	0.476273
$926$	−28.0000	−0.920137
$927$	1.41421	0.0464489
$928$	−1.00000	−0.0328266
$929$	36.6274	1.20171	0.600853	−	0.799359i	$-0.294828\pi$
$929$	36.6274	1.20171	0.600853	+	0.799359i	$0.294828\pi$
$930$	16.9706	0.556487
$931$	27.0711	0.887218
$932$	0.828427	0.0271360
$933$	33.4558	1.09530
$934$	−40.4853	−1.32472
$935$	−32.0000	−1.04651
$936$	0.828427	0.0270780
$937$	−14.6863	−0.479780	−0.239890	−	0.970800i	$-0.577111\pi$
$937$	−14.6863	−0.479780	−0.239890	+	0.970800i	$0.577111\pi$
$938$	3.51472	0.114760
$939$	−20.6274	−0.673150
$940$	32.0000	1.04372
$941$	−48.4853	−1.58058	−0.790288	−	0.612736i	$-0.790069\pi$
$941$	−48.4853	−1.58058	−0.790288	+	0.612736i	$0.790069\pi$
$942$	−2.48528	−0.0809748
$943$	−2.82843	−0.0921063
$944$	−1.17157	−0.0381314
$945$	−4.00000	−0.130120
$946$	0.970563	0.0315557
$947$	−9.79899	−0.318424	−0.159212	−	0.987244i	$-0.550895\pi$
$947$	−9.79899	−0.318424	−0.159212	+	0.987244i	$0.550895\pi$
$948$	−15.6569	−0.508511
$949$	−12.0000	−0.389536
$950$	16.2426	0.526981
$951$	8.14214	0.264027
$952$	4.00000	0.129641
$953$	27.8995	0.903753	0.451877	−	0.892080i	$-0.350755\pi$
$953$	27.8995	0.903753	0.451877	+	0.892080i	$0.350755\pi$
$954$	−4.00000	−0.129505
$955$	−12.9706	−0.419718
$956$	−18.4853	−0.597857
$957$	−4.00000	−0.129302
$958$	−21.0711	−0.680775
$959$	11.3137	0.365339
$960$	−2.82843	−0.0912871
$961$	5.00000	0.161290
$962$	4.00000	0.128965
$963$	−7.41421	−0.238920
$964$	18.4853	0.595371
$965$	−10.3431	−0.332958
$966$	−1.41421	−0.0455016
$967$	0.686292	0.0220696	0.0110348	−	0.999939i	$-0.496487\pi$
$967$	0.686292	0.0220696	0.0110348	+	0.999939i	$0.496487\pi$
$968$	−5.00000	−0.160706
$969$	−15.3137	−0.491947
$970$	−29.9411	−0.961352
$971$	−39.5980	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$971$	−39.5980	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$972$	−1.00000	−0.0320750
$973$	22.1421	0.709844
$974$	−37.6569	−1.20660
$975$	2.48528	0.0795927
$976$	3.65685	0.117053
$977$	30.9289	0.989504	0.494752	−	0.869034i	$-0.335259\pi$
$977$	30.9289	0.989504	0.494752	+	0.869034i	$0.335259\pi$
$978$	16.4853	0.527141
$979$	−43.3137	−1.38431
$980$	−14.1421	−0.451754
$981$	−16.2426	−0.518588
$982$	−18.4853	−0.589889
$983$	10.7279	0.342168	0.171084	−	0.985256i	$-0.445273\pi$
$983$	10.7279	0.342168	0.171084	+	0.985256i	$0.445273\pi$
$984$	2.82843	0.0901670
$985$	28.2843	0.901212
$986$	2.82843	0.0900755
$987$	−16.0000	−0.509286
$988$	4.48528	0.142696
$989$	0.242641	0.00771552
$990$	−11.3137	−0.359573
$991$	−39.5147	−1.25523	−0.627613	−	0.778525i	$-0.715968\pi$
$991$	−39.5147	−1.25523	−0.627613	+	0.778525i	$0.715968\pi$
$992$	−6.00000	−0.190500
$993$	27.7990	0.882174
$994$	14.8284	0.470329
$995$	0.686292	0.0217569
$996$	−17.5563	−0.556294
$997$	−3.37258	−0.106811	−0.0534054	−	0.998573i	$-0.517008\pi$
$997$	−3.37258	−0.106811	−0.0534054	+	0.998573i	$0.517008\pi$
$998$	15.6569	0.495609
$999$	−4.82843	−0.152765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.t.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.t.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{6} +1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{6} +1.41421 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.82843 q^{10} +4.00000 q^{11} -1.00000 q^{12} -0.828427 q^{13} -1.41421 q^{14} -2.82843 q^{15} +1.00000 q^{16} -2.82843 q^{17} -1.00000 q^{18} -5.41421 q^{19} +2.82843 q^{20} -1.41421 q^{21} -4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.00000 q^{25} +0.828427 q^{26} -1.00000 q^{27} +1.41421 q^{28} +1.00000 q^{29} +2.82843 q^{30} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.82843 q^{34} +4.00000 q^{35} +1.00000 q^{36} +4.82843 q^{37} +5.41421 q^{38} +0.828427 q^{39} -2.82843 q^{40} +2.82843 q^{41} +1.41421 q^{42} -0.242641 q^{43} +4.00000 q^{44} +2.82843 q^{45} +1.00000 q^{46} +11.3137 q^{47} -1.00000 q^{48} -5.00000 q^{49} -3.00000 q^{50} +2.82843 q^{51} -0.828427 q^{52} +4.00000 q^{53} +1.00000 q^{54} +11.3137 q^{55} -1.41421 q^{56} +5.41421 q^{57} -1.00000 q^{58} -1.17157 q^{59} -2.82843 q^{60} +3.65685 q^{61} -6.00000 q^{62} +1.41421 q^{63} +1.00000 q^{64} -2.34315 q^{65} +4.00000 q^{66} -2.48528 q^{67} -2.82843 q^{68} +1.00000 q^{69} -4.00000 q^{70} -10.4853 q^{71} -1.00000 q^{72} +14.4853 q^{73} -4.82843 q^{74} -3.00000 q^{75} -5.41421 q^{76} +5.65685 q^{77} -0.828427 q^{78} +15.6569 q^{79} +2.82843 q^{80} +1.00000 q^{81} -2.82843 q^{82} +17.5563 q^{83} -1.41421 q^{84} -8.00000 q^{85} +0.242641 q^{86} -1.00000 q^{87} -4.00000 q^{88} -10.8284 q^{89} -2.82843 q^{90} -1.17157 q^{91} -1.00000 q^{92} -6.00000 q^{93} -11.3137 q^{94} -15.3137 q^{95} +1.00000 q^{96} +10.5858 q^{97} +5.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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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