LMFDB - Embedded newform 4002.2.a.s.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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} +3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} +3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} -2.00000 q^{11} -1.00000 q^{12} -5.12311 q^{13} -3.56155 q^{14} +3.56155 q^{15} +1.00000 q^{16} -2.43845 q^{17} -1.00000 q^{18} -1.56155 q^{19} -3.56155 q^{20} -3.56155 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +7.68466 q^{25} +5.12311 q^{26} -1.00000 q^{27} +3.56155 q^{28} +1.00000 q^{29} -3.56155 q^{30} -1.00000 q^{32} +2.00000 q^{33} +2.43845 q^{34} -12.6847 q^{35} +1.00000 q^{36} -1.56155 q^{37} +1.56155 q^{38} +5.12311 q^{39} +3.56155 q^{40} -3.56155 q^{41} +3.56155 q^{42} -0.684658 q^{43} -2.00000 q^{44} -3.56155 q^{45} +1.00000 q^{46} +2.43845 q^{47} -1.00000 q^{48} +5.68466 q^{49} -7.68466 q^{50} +2.43845 q^{51} -5.12311 q^{52} -2.00000 q^{53} +1.00000 q^{54} +7.12311 q^{55} -3.56155 q^{56} +1.56155 q^{57} -1.00000 q^{58} -1.56155 q^{59} +3.56155 q^{60} -3.12311 q^{61} +3.56155 q^{63} +1.00000 q^{64} +18.2462 q^{65} -2.00000 q^{66} -2.43845 q^{68} +1.00000 q^{69} +12.6847 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +1.56155 q^{74} -7.68466 q^{75} -1.56155 q^{76} -7.12311 q^{77} -5.12311 q^{78} +0.246211 q^{79} -3.56155 q^{80} +1.00000 q^{81} +3.56155 q^{82} -2.87689 q^{83} -3.56155 q^{84} +8.68466 q^{85} +0.684658 q^{86} -1.00000 q^{87} +2.00000 q^{88} -3.12311 q^{89} +3.56155 q^{90} -18.2462 q^{91} -1.00000 q^{92} -2.43845 q^{94} +5.56155 q^{95} +1.00000 q^{96} +13.1231 q^{97} -5.68466 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} + q^{19} - 3 q^{20} - 3 q^{21} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} - 2 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 2 q^{32} + 4 q^{33} + 9 q^{34} - 13 q^{35} + 2 q^{36} + q^{37} - q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 3 q^{42} + 11 q^{43} - 4 q^{44} - 3 q^{45} + 2 q^{46} + 9 q^{47} - 2 q^{48} - q^{49} - 3 q^{50} + 9 q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 6 q^{55} - 3 q^{56} - q^{57} - 2 q^{58} + q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{63} + 2 q^{64} + 20 q^{65} - 4 q^{66} - 9 q^{68} + 2 q^{69} + 13 q^{70} - 16 q^{71} - 2 q^{72} + 12 q^{73} - q^{74} - 3 q^{75} + q^{76} - 6 q^{77} - 2 q^{78} - 16 q^{79} - 3 q^{80} + 2 q^{81} + 3 q^{82} - 14 q^{83} - 3 q^{84} + 5 q^{85} - 11 q^{86} - 2 q^{87} + 4 q^{88} + 2 q^{89} + 3 q^{90} - 20 q^{91} - 2 q^{92} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 3 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 - 9 * q^17 - 2 * q^18 + q^19 - 3 * q^20 - 3 * q^21 + 4 * q^22 - 2 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 - 2 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 2 * q^32 + 4 * q^33 + 9 * q^34 - 13 * q^35 + 2 * q^36 + q^37 - q^38 + 2 * q^39 + 3 * q^40 - 3 * q^41 + 3 * q^42 + 11 * q^43 - 4 * q^44 - 3 * q^45 + 2 * q^46 + 9 * q^47 - 2 * q^48 - q^49 - 3 * q^50 + 9 * q^51 - 2 * q^52 - 4 * q^53 + 2 * q^54 + 6 * q^55 - 3 * q^56 - q^57 - 2 * q^58 + q^59 + 3 * q^60 + 2 * q^61 + 3 * q^63 + 2 * q^64 + 20 * q^65 - 4 * q^66 - 9 * q^68 + 2 * q^69 + 13 * q^70 - 16 * q^71 - 2 * q^72 + 12 * q^73 - q^74 - 3 * q^75 + q^76 - 6 * q^77 - 2 * q^78 - 16 * q^79 - 3 * q^80 + 2 * q^81 + 3 * q^82 - 14 * q^83 - 3 * q^84 + 5 * q^85 - 11 * q^86 - 2 * q^87 + 4 * q^88 + 2 * q^89 + 3 * q^90 - 20 * q^91 - 2 * q^92 - 9 * q^94 + 7 * q^95 + 2 * q^96 + 18 * q^97 + q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	1.00000	0.408248
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.56155	1.12626
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	−3.56155	−0.951865
$15$	3.56155	0.919589
$16$	1.00000	0.250000
$17$	−2.43845	−0.591410	−0.295705	−	0.955279i	$-0.595555\pi$
$17$	−2.43845	−0.591410	−0.295705	+	0.955279i	$0.595555\pi$
$18$	−1.00000	−0.235702
$19$	−1.56155	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$19$	−1.56155	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$20$	−3.56155	−0.796387
$21$	−3.56155	−0.777195
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	7.68466	1.53693
$26$	5.12311	1.00472
$27$	−1.00000	−0.192450
$28$	3.56155	0.673070
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.56155	−0.650248
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	2.43845	0.418190
$35$	−12.6847	−2.14410
$36$	1.00000	0.166667
$37$	−1.56155	−0.256718	−0.128359	−	0.991728i	$-0.540971\pi$
$37$	−1.56155	−0.256718	−0.128359	+	0.991728i	$0.540971\pi$
$38$	1.56155	0.253317
$39$	5.12311	0.820353
$40$	3.56155	0.563131
$41$	−3.56155	−0.556221	−0.278111	−	0.960549i	$-0.589708\pi$
$41$	−3.56155	−0.556221	−0.278111	+	0.960549i	$0.589708\pi$
$42$	3.56155	0.549560
$43$	−0.684658	−0.104409	−0.0522047	−	0.998636i	$-0.516625\pi$
$43$	−0.684658	−0.104409	−0.0522047	+	0.998636i	$0.516625\pi$
$44$	−2.00000	−0.301511
$45$	−3.56155	−0.530925
$46$	1.00000	0.147442
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	−1.00000	−0.144338
$49$	5.68466	0.812094
$50$	−7.68466	−1.08677
$51$	2.43845	0.341451
$52$	−5.12311	−0.710447
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	7.12311	0.960479
$56$	−3.56155	−0.475933
$57$	1.56155	0.206833
$58$	−1.00000	−0.131306
$59$	−1.56155	−0.203297	−0.101648	−	0.994820i	$-0.532412\pi$
$59$	−1.56155	−0.203297	−0.101648	+	0.994820i	$0.532412\pi$
$60$	3.56155	0.459794
$61$	−3.12311	−0.399873	−0.199936	−	0.979809i	$-0.564074\pi$
$61$	−3.12311	−0.399873	−0.199936	+	0.979809i	$0.564074\pi$
$62$	0	0
$63$	3.56155	0.448713
$64$	1.00000	0.125000
$65$	18.2462	2.26316
$66$	−2.00000	−0.246183
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−2.43845	−0.295705
$69$	1.00000	0.120386
$70$	12.6847	1.51611
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	1.56155	0.181527
$75$	−7.68466	−0.887348
$76$	−1.56155	−0.179122
$77$	−7.12311	−0.811753
$78$	−5.12311	−0.580077
$79$	0.246211	0.0277009	0.0138505	−	0.999904i	$-0.495591\pi$
$79$	0.246211	0.0277009	0.0138505	+	0.999904i	$0.495591\pi$
$80$	−3.56155	−0.398194
$81$	1.00000	0.111111
$82$	3.56155	0.393308
$83$	−2.87689	−0.315780	−0.157890	−	0.987457i	$-0.550469\pi$
$83$	−2.87689	−0.315780	−0.157890	+	0.987457i	$0.550469\pi$
$84$	−3.56155	−0.388597
$85$	8.68466	0.941983
$86$	0.684658	0.0738286
$87$	−1.00000	−0.107211
$88$	2.00000	0.213201
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	3.56155	0.375421
$91$	−18.2462	−1.91272
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−2.43845	−0.251507
$95$	5.56155	0.570603
$96$	1.00000	0.102062
$97$	13.1231	1.33245	0.666225	−	0.745751i	$-0.267909\pi$
$97$	13.1231	1.33245	0.666225	+	0.745751i	$0.267909\pi$
$98$	−5.68466	−0.574237
$99$	−2.00000	−0.201008
$100$	7.68466	0.768466
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	−2.43845	−0.241442
$103$	−5.80776	−0.572256	−0.286128	−	0.958191i	$-0.592368\pi$
$103$	−5.80776	−0.572256	−0.286128	+	0.958191i	$0.592368\pi$
$104$	5.12311	0.502362
$105$	12.6847	1.23790
$106$	2.00000	0.194257
$107$	−12.9309	−1.25007	−0.625037	−	0.780595i	$-0.714916\pi$
$107$	−12.9309	−1.25007	−0.625037	+	0.780595i	$0.714916\pi$
$108$	−1.00000	−0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	−7.12311	−0.679161
$111$	1.56155	0.148216
$112$	3.56155	0.336535
$113$	14.9309	1.40458	0.702289	−	0.711892i	$-0.252161\pi$
$113$	14.9309	1.40458	0.702289	+	0.711892i	$0.252161\pi$
$114$	−1.56155	−0.146253
$115$	3.56155	0.332117
$116$	1.00000	0.0928477
$117$	−5.12311	−0.473631
$118$	1.56155	0.143753
$119$	−8.68466	−0.796121
$120$	−3.56155	−0.325124
$121$	−7.00000	−0.636364
$122$	3.12311	0.282753
$123$	3.56155	0.321134
$124$	0	0
$125$	−9.56155	−0.855211
$126$	−3.56155	−0.317288
$127$	8.87689	0.787697	0.393849	−	0.919175i	$-0.371143\pi$
$127$	8.87689	0.787697	0.393849	+	0.919175i	$0.371143\pi$
$128$	−1.00000	−0.0883883
$129$	0.684658	0.0602808
$130$	−18.2462	−1.60030
$131$	−6.24621	−0.545734	−0.272867	−	0.962052i	$-0.587972\pi$
$131$	−6.24621	−0.545734	−0.272867	+	0.962052i	$0.587972\pi$
$132$	2.00000	0.174078
$133$	−5.56155	−0.482248
$134$	0	0
$135$	3.56155	0.306530
$136$	2.43845	0.209095
$137$	−13.3693	−1.14222	−0.571109	−	0.820874i	$-0.693487\pi$
$137$	−13.3693	−1.14222	−0.571109	+	0.820874i	$0.693487\pi$
$138$	−1.00000	−0.0851257
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	−12.6847	−1.07205
$141$	−2.43845	−0.205354
$142$	8.00000	0.671345
$143$	10.2462	0.856831
$144$	1.00000	0.0833333
$145$	−3.56155	−0.295771
$146$	−6.00000	−0.496564
$147$	−5.68466	−0.468863
$148$	−1.56155	−0.128359
$149$	−14.6847	−1.20301	−0.601507	−	0.798867i	$-0.705433\pi$
$149$	−14.6847	−1.20301	−0.601507	+	0.798867i	$0.705433\pi$
$150$	7.68466	0.627450
$151$	−3.31534	−0.269799	−0.134899	−	0.990859i	$-0.543071\pi$
$151$	−3.31534	−0.269799	−0.134899	+	0.990859i	$0.543071\pi$
$152$	1.56155	0.126659
$153$	−2.43845	−0.197137
$154$	7.12311	0.573996
$155$	0	0
$156$	5.12311	0.410177
$157$	4.68466	0.373876	0.186938	−	0.982372i	$-0.440144\pi$
$157$	4.68466	0.373876	0.186938	+	0.982372i	$0.440144\pi$
$158$	−0.246211	−0.0195875
$159$	2.00000	0.158610
$160$	3.56155	0.281565
$161$	−3.56155	−0.280690
$162$	−1.00000	−0.0785674
$163$	3.31534	0.259678	0.129839	−	0.991535i	$-0.458554\pi$
$163$	3.31534	0.259678	0.129839	+	0.991535i	$0.458554\pi$
$164$	−3.56155	−0.278111
$165$	−7.12311	−0.554533
$166$	2.87689	0.223290
$167$	−11.1231	−0.860732	−0.430366	−	0.902655i	$-0.641615\pi$
$167$	−11.1231	−0.860732	−0.430366	+	0.902655i	$0.641615\pi$
$168$	3.56155	0.274780
$169$	13.2462	1.01894
$170$	−8.68466	−0.666083
$171$	−1.56155	−0.119415
$172$	−0.684658	−0.0522047
$173$	16.0540	1.22056	0.610281	−	0.792185i	$-0.291057\pi$
$173$	16.0540	1.22056	0.610281	+	0.792185i	$0.291057\pi$
$174$	1.00000	0.0758098
$175$	27.3693	2.06893
$176$	−2.00000	−0.150756
$177$	1.56155	0.117373
$178$	3.12311	0.234087
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	−3.56155	−0.265462
$181$	10.2462	0.761595	0.380797	−	0.924658i	$-0.375650\pi$
$181$	10.2462	0.761595	0.380797	+	0.924658i	$0.375650\pi$
$182$	18.2462	1.35250
$183$	3.12311	0.230867
$184$	1.00000	0.0737210
$185$	5.56155	0.408893
$186$	0	0
$187$	4.87689	0.356634
$188$	2.43845	0.177842
$189$	−3.56155	−0.259065
$190$	−5.56155	−0.403477
$191$	21.5616	1.56014	0.780070	−	0.625693i	$-0.215184\pi$
$191$	21.5616	1.56014	0.780070	+	0.625693i	$0.215184\pi$
$192$	−1.00000	−0.0721688
$193$	9.12311	0.656696	0.328348	−	0.944557i	$-0.393508\pi$
$193$	9.12311	0.656696	0.328348	+	0.944557i	$0.393508\pi$
$194$	−13.1231	−0.942184
$195$	−18.2462	−1.30664
$196$	5.68466	0.406047
$197$	2.68466	0.191274	0.0956370	−	0.995416i	$-0.469511\pi$
$197$	2.68466	0.191274	0.0956370	+	0.995416i	$0.469511\pi$
$198$	2.00000	0.142134
$199$	−7.36932	−0.522397	−0.261199	−	0.965285i	$-0.584118\pi$
$199$	−7.36932	−0.522397	−0.261199	+	0.965285i	$0.584118\pi$
$200$	−7.68466	−0.543387
$201$	0	0
$202$	4.24621	0.298762
$203$	3.56155	0.249972
$204$	2.43845	0.170725
$205$	12.6847	0.885935
$206$	5.80776	0.404646
$207$	−1.00000	−0.0695048
$208$	−5.12311	−0.355223
$209$	3.12311	0.216030
$210$	−12.6847	−0.875325
$211$	14.4384	0.993984	0.496992	−	0.867755i	$-0.334438\pi$
$211$	14.4384	0.993984	0.496992	+	0.867755i	$0.334438\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	12.9309	0.883936
$215$	2.43845	0.166301
$216$	1.00000	0.0680414
$217$	0	0
$218$	−12.0000	−0.812743
$219$	−6.00000	−0.405442
$220$	7.12311	0.480240
$221$	12.4924	0.840331
$222$	−1.56155	−0.104805
$223$	18.2462	1.22186	0.610928	−	0.791686i	$-0.290796\pi$
$223$	18.2462	1.22186	0.610928	+	0.791686i	$0.290796\pi$
$224$	−3.56155	−0.237966
$225$	7.68466	0.512311
$226$	−14.9309	−0.993187
$227$	9.31534	0.618281	0.309140	−	0.951016i	$-0.399959\pi$
$227$	9.31534	0.618281	0.309140	+	0.951016i	$0.399959\pi$
$228$	1.56155	0.103416
$229$	9.56155	0.631845	0.315923	−	0.948785i	$-0.397686\pi$
$229$	9.56155	0.631845	0.315923	+	0.948785i	$0.397686\pi$
$230$	−3.56155	−0.234842
$231$	7.12311	0.468666
$232$	−1.00000	−0.0656532
$233$	5.12311	0.335626	0.167813	−	0.985819i	$-0.446330\pi$
$233$	5.12311	0.335626	0.167813	+	0.985819i	$0.446330\pi$
$234$	5.12311	0.334908
$235$	−8.68466	−0.566525
$236$	−1.56155	−0.101648
$237$	−0.246211	−0.0159931
$238$	8.68466	0.562943
$239$	17.3693	1.12353	0.561764	−	0.827298i	$-0.310123\pi$
$239$	17.3693	1.12353	0.561764	+	0.827298i	$0.310123\pi$
$240$	3.56155	0.229897
$241$	12.0540	0.776465	0.388232	−	0.921562i	$-0.373086\pi$
$241$	12.0540	0.776465	0.388232	+	0.921562i	$0.373086\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−3.12311	−0.199936
$245$	−20.2462	−1.29348
$246$	−3.56155	−0.227076
$247$	8.00000	0.509028
$248$	0	0
$249$	2.87689	0.182316
$250$	9.56155	0.604726
$251$	1.12311	0.0708898	0.0354449	−	0.999372i	$-0.488715\pi$
$251$	1.12311	0.0708898	0.0354449	+	0.999372i	$0.488715\pi$
$252$	3.56155	0.224357
$253$	2.00000	0.125739
$254$	−8.87689	−0.556986
$255$	−8.68466	−0.543854
$256$	1.00000	0.0625000
$257$	6.87689	0.428969	0.214484	−	0.976727i	$-0.431193\pi$
$257$	6.87689	0.428969	0.214484	+	0.976727i	$0.431193\pi$
$258$	−0.684658	−0.0426250
$259$	−5.56155	−0.345578
$260$	18.2462	1.13158
$261$	1.00000	0.0618984
$262$	6.24621	0.385892
$263$	25.1771	1.55249	0.776243	−	0.630434i	$-0.217123\pi$
$263$	25.1771	1.55249	0.776243	+	0.630434i	$0.217123\pi$
$264$	−2.00000	−0.123091
$265$	7.12311	0.437569
$266$	5.56155	0.341001
$267$	3.12311	0.191131
$268$	0	0
$269$	−4.24621	−0.258896	−0.129448	−	0.991586i	$-0.541321\pi$
$269$	−4.24621	−0.258896	−0.129448	+	0.991586i	$0.541321\pi$
$270$	−3.56155	−0.216749
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−2.43845	−0.147853
$273$	18.2462	1.10431
$274$	13.3693	0.807670
$275$	−15.3693	−0.926805
$276$	1.00000	0.0601929
$277$	−10.4924	−0.630429	−0.315214	−	0.949021i	$-0.602076\pi$
$277$	−10.4924	−0.630429	−0.315214	+	0.949021i	$0.602076\pi$
$278$	15.1231	0.907024
$279$	0	0
$280$	12.6847	0.758053
$281$	29.8617	1.78140	0.890701	−	0.454590i	$-0.150214\pi$
$281$	29.8617	1.78140	0.890701	+	0.454590i	$0.150214\pi$
$282$	2.43845	0.145207
$283$	−5.36932	−0.319173	−0.159586	−	0.987184i	$-0.551016\pi$
$283$	−5.36932	−0.319173	−0.159586	+	0.987184i	$0.551016\pi$
$284$	−8.00000	−0.474713
$285$	−5.56155	−0.329438
$286$	−10.2462	−0.605871
$287$	−12.6847	−0.748752
$288$	−1.00000	−0.0589256
$289$	−11.0540	−0.650234
$290$	3.56155	0.209142
$291$	−13.1231	−0.769290
$292$	6.00000	0.351123
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	5.68466	0.331536
$295$	5.56155	0.323806
$296$	1.56155	0.0907634
$297$	2.00000	0.116052
$298$	14.6847	0.850659
$299$	5.12311	0.296277
$300$	−7.68466	−0.443674
$301$	−2.43845	−0.140550
$302$	3.31534	0.190776
$303$	4.24621	0.243938
$304$	−1.56155	−0.0895612
$305$	11.1231	0.636907
$306$	2.43845	0.139397
$307$	−8.49242	−0.484688	−0.242344	−	0.970190i	$-0.577916\pi$
$307$	−8.49242	−0.484688	−0.242344	+	0.970190i	$0.577916\pi$
$308$	−7.12311	−0.405877
$309$	5.80776	0.330392
$310$	0	0
$311$	−10.0540	−0.570109	−0.285054	−	0.958511i	$-0.592012\pi$
$311$	−10.0540	−0.570109	−0.285054	+	0.958511i	$0.592012\pi$
$312$	−5.12311	−0.290039
$313$	10.1922	0.576099	0.288050	−	0.957615i	$-0.406993\pi$
$313$	10.1922	0.576099	0.288050	+	0.957615i	$0.406993\pi$
$314$	−4.68466	−0.264371
$315$	−12.6847	−0.714700
$316$	0.246211	0.0138505
$317$	25.1231	1.41105	0.705527	−	0.708683i	$-0.250710\pi$
$317$	25.1231	1.41105	0.705527	+	0.708683i	$0.250710\pi$
$318$	−2.00000	−0.112154
$319$	−2.00000	−0.111979
$320$	−3.56155	−0.199097
$321$	12.9309	0.721731
$322$	3.56155	0.198478
$323$	3.80776	0.211870
$324$	1.00000	0.0555556
$325$	−39.3693	−2.18382
$326$	−3.31534	−0.183620
$327$	−12.0000	−0.663602
$328$	3.56155	0.196654
$329$	8.68466	0.478801
$330$	7.12311	0.392114
$331$	25.5616	1.40499	0.702495	−	0.711689i	$-0.252069\pi$
$331$	25.5616	1.40499	0.702495	+	0.711689i	$0.252069\pi$
$332$	−2.87689	−0.157890
$333$	−1.56155	−0.0855726
$334$	11.1231	0.608629
$335$	0	0
$336$	−3.56155	−0.194299
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−13.2462	−0.720499
$339$	−14.9309	−0.810934
$340$	8.68466	0.470992
$341$	0	0
$342$	1.56155	0.0844391
$343$	−4.68466	−0.252948
$344$	0.684658	0.0369143
$345$	−3.56155	−0.191748
$346$	−16.0540	−0.863067
$347$	11.8078	0.633874	0.316937	−	0.948447i	$-0.397346\pi$
$347$	11.8078	0.633874	0.316937	+	0.948447i	$0.397346\pi$
$348$	−1.00000	−0.0536056
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−27.3693	−1.46295
$351$	5.12311	0.273451
$352$	2.00000	0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	−1.56155	−0.0829956
$355$	28.4924	1.51222
$356$	−3.12311	−0.165524
$357$	8.68466	0.459641
$358$	−16.0000	−0.845626
$359$	−3.80776	−0.200966	−0.100483	−	0.994939i	$-0.532039\pi$
$359$	−3.80776	−0.200966	−0.100483	+	0.994939i	$0.532039\pi$
$360$	3.56155	0.187710
$361$	−16.5616	−0.871661
$362$	−10.2462	−0.538529
$363$	7.00000	0.367405
$364$	−18.2462	−0.956361
$365$	−21.3693	−1.11852
$366$	−3.12311	−0.163247
$367$	−32.7386	−1.70894	−0.854471	−	0.519498i	$-0.826119\pi$
$367$	−32.7386	−1.70894	−0.854471	+	0.519498i	$0.826119\pi$
$368$	−1.00000	−0.0521286
$369$	−3.56155	−0.185407
$370$	−5.56155	−0.289131
$371$	−7.12311	−0.369813
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−4.87689	−0.252178
$375$	9.56155	0.493756
$376$	−2.43845	−0.125753
$377$	−5.12311	−0.263853
$378$	3.56155	0.183187
$379$	−0.492423	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$380$	5.56155	0.285302
$381$	−8.87689	−0.454777
$382$	−21.5616	−1.10319
$383$	−13.3693	−0.683140	−0.341570	−	0.939856i	$-0.610959\pi$
$383$	−13.3693	−0.683140	−0.341570	+	0.939856i	$0.610959\pi$
$384$	1.00000	0.0510310
$385$	25.3693	1.29294
$386$	−9.12311	−0.464354
$387$	−0.684658	−0.0348031
$388$	13.1231	0.666225
$389$	33.6155	1.70437	0.852187	−	0.523237i	$-0.175276\pi$
$389$	33.6155	1.70437	0.852187	+	0.523237i	$0.175276\pi$
$390$	18.2462	0.923933
$391$	2.43845	0.123318
$392$	−5.68466	−0.287119
$393$	6.24621	0.315080
$394$	−2.68466	−0.135251
$395$	−0.876894	−0.0441213
$396$	−2.00000	−0.100504
$397$	−6.49242	−0.325845	−0.162923	−	0.986639i	$-0.552092\pi$
$397$	−6.49242	−0.325845	−0.162923	+	0.986639i	$0.552092\pi$
$398$	7.36932	0.369390
$399$	5.56155	0.278426
$400$	7.68466	0.384233
$401$	−9.36932	−0.467881	−0.233941	−	0.972251i	$-0.575162\pi$
$401$	−9.36932	−0.467881	−0.233941	+	0.972251i	$0.575162\pi$
$402$	0	0
$403$	0	0
$404$	−4.24621	−0.211257
$405$	−3.56155	−0.176975
$406$	−3.56155	−0.176757
$407$	3.12311	0.154807
$408$	−2.43845	−0.120721
$409$	10.8769	0.537828	0.268914	−	0.963164i	$-0.413335\pi$
$409$	10.8769	0.537828	0.268914	+	0.963164i	$0.413335\pi$
$410$	−12.6847	−0.626451
$411$	13.3693	0.659460
$412$	−5.80776	−0.286128
$413$	−5.56155	−0.273666
$414$	1.00000	0.0491473
$415$	10.2462	0.502967
$416$	5.12311	0.251181
$417$	15.1231	0.740582
$418$	−3.12311	−0.152756
$419$	24.9309	1.21795	0.608976	−	0.793188i	$-0.291580\pi$
$419$	24.9309	1.21795	0.608976	+	0.793188i	$0.291580\pi$
$420$	12.6847	0.618948
$421$	−23.6155	−1.15095	−0.575475	−	0.817819i	$-0.695183\pi$
$421$	−23.6155	−1.15095	−0.575475	+	0.817819i	$0.695183\pi$
$422$	−14.4384	−0.702853
$423$	2.43845	0.118561
$424$	2.00000	0.0971286
$425$	−18.7386	−0.908957
$426$	−8.00000	−0.387601
$427$	−11.1231	−0.538285
$428$	−12.9309	−0.625037
$429$	−10.2462	−0.494692
$430$	−2.43845	−0.117592
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−1.00000	−0.0481125
$433$	−22.4924	−1.08092	−0.540458	−	0.841371i	$-0.681749\pi$
$433$	−22.4924	−1.08092	−0.540458	+	0.841371i	$0.681749\pi$
$434$	0	0
$435$	3.56155	0.170763
$436$	12.0000	0.574696
$437$	1.56155	0.0746992
$438$	6.00000	0.286691
$439$	−2.43845	−0.116381	−0.0581904	−	0.998306i	$-0.518533\pi$
$439$	−2.43845	−0.116381	−0.0581904	+	0.998306i	$0.518533\pi$
$440$	−7.12311	−0.339581
$441$	5.68466	0.270698
$442$	−12.4924	−0.594204
$443$	−5.75379	−0.273371	−0.136685	−	0.990615i	$-0.543645\pi$
$443$	−5.75379	−0.273371	−0.136685	+	0.990615i	$0.543645\pi$
$444$	1.56155	0.0741080
$445$	11.1231	0.527286
$446$	−18.2462	−0.863983
$447$	14.6847	0.694561
$448$	3.56155	0.168268
$449$	−11.5616	−0.545623	−0.272812	−	0.962067i	$-0.587954\pi$
$449$	−11.5616	−0.545623	−0.272812	+	0.962067i	$0.587954\pi$
$450$	−7.68466	−0.362258
$451$	7.12311	0.335414
$452$	14.9309	0.702289
$453$	3.31534	0.155768
$454$	−9.31534	−0.437191
$455$	64.9848	3.04654
$456$	−1.56155	−0.0731264
$457$	20.4384	0.956070	0.478035	−	0.878341i	$-0.341349\pi$
$457$	20.4384	0.956070	0.478035	+	0.878341i	$0.341349\pi$
$458$	−9.56155	−0.446782
$459$	2.43845	0.113817
$460$	3.56155	0.166058
$461$	38.1080	1.77486	0.887432	−	0.460938i	$-0.152487\pi$
$461$	38.1080	1.77486	0.887432	+	0.460938i	$0.152487\pi$
$462$	−7.12311	−0.331397
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−5.12311	−0.237323
$467$	−0.630683	−0.0291845	−0.0145923	−	0.999894i	$-0.504645\pi$
$467$	−0.630683	−0.0291845	−0.0145923	+	0.999894i	$0.504645\pi$
$468$	−5.12311	−0.236816
$469$	0	0
$470$	8.68466	0.400593
$471$	−4.68466	−0.215858
$472$	1.56155	0.0718763
$473$	1.36932	0.0629613
$474$	0.246211	0.0113089
$475$	−12.0000	−0.550598
$476$	−8.68466	−0.398061
$477$	−2.00000	−0.0915737
$478$	−17.3693	−0.794454
$479$	36.4924	1.66738	0.833691	−	0.552232i	$-0.186224\pi$
$479$	36.4924	1.66738	0.833691	+	0.552232i	$0.186224\pi$
$480$	−3.56155	−0.162562
$481$	8.00000	0.364769
$482$	−12.0540	−0.549044
$483$	3.56155	0.162056
$484$	−7.00000	−0.318182
$485$	−46.7386	−2.12229
$486$	1.00000	0.0453609
$487$	5.56155	0.252018	0.126009	−	0.992029i	$-0.459783\pi$
$487$	5.56155	0.252018	0.126009	+	0.992029i	$0.459783\pi$
$488$	3.12311	0.141376
$489$	−3.31534	−0.149925
$490$	20.2462	0.914631
$491$	25.3693	1.14490	0.572451	−	0.819939i	$-0.305993\pi$
$491$	25.3693	1.14490	0.572451	+	0.819939i	$0.305993\pi$
$492$	3.56155	0.160567
$493$	−2.43845	−0.109822
$494$	−8.00000	−0.359937
$495$	7.12311	0.320160
$496$	0	0
$497$	−28.4924	−1.27806
$498$	−2.87689	−0.128917
$499$	−3.61553	−0.161853	−0.0809266	−	0.996720i	$-0.525788\pi$
$499$	−3.61553	−0.161853	−0.0809266	+	0.996720i	$0.525788\pi$
$500$	−9.56155	−0.427606
$501$	11.1231	0.496944
$502$	−1.12311	−0.0501267
$503$	22.0540	0.983338	0.491669	−	0.870782i	$-0.336387\pi$
$503$	22.0540	0.983338	0.491669	+	0.870782i	$0.336387\pi$
$504$	−3.56155	−0.158644
$505$	15.1231	0.672969
$506$	−2.00000	−0.0889108
$507$	−13.2462	−0.588285
$508$	8.87689	0.393849
$509$	−7.94602	−0.352201	−0.176101	−	0.984372i	$-0.556348\pi$
$509$	−7.94602	−0.352201	−0.176101	+	0.984372i	$0.556348\pi$
$510$	8.68466	0.384563
$511$	21.3693	0.945323
$512$	−1.00000	−0.0441942
$513$	1.56155	0.0689442
$514$	−6.87689	−0.303327
$515$	20.6847	0.911475
$516$	0.684658	0.0301404
$517$	−4.87689	−0.214486
$518$	5.56155	0.244361
$519$	−16.0540	−0.704691
$520$	−18.2462	−0.800149
$521$	−33.8617	−1.48351	−0.741755	−	0.670671i	$-0.766006\pi$
$521$	−33.8617	−1.48351	−0.741755	+	0.670671i	$0.766006\pi$
$522$	−1.00000	−0.0437688
$523$	12.8769	0.563067	0.281534	−	0.959551i	$-0.409157\pi$
$523$	12.8769	0.563067	0.281534	+	0.959551i	$0.409157\pi$
$524$	−6.24621	−0.272867
$525$	−27.3693	−1.19449
$526$	−25.1771	−1.09777
$527$	0	0
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−7.12311	−0.309408
$531$	−1.56155	−0.0677656
$532$	−5.56155	−0.241124
$533$	18.2462	0.790331
$534$	−3.12311	−0.135150
$535$	46.0540	1.99109
$536$	0	0
$537$	−16.0000	−0.690451
$538$	4.24621	0.183067
$539$	−11.3693	−0.489711
$540$	3.56155	0.153265
$541$	−38.6847	−1.66318	−0.831592	−	0.555387i	$-0.812570\pi$
$541$	−38.6847	−1.66318	−0.831592	+	0.555387i	$0.812570\pi$
$542$	−4.00000	−0.171815
$543$	−10.2462	−0.439707
$544$	2.43845	0.104548
$545$	−42.7386	−1.83072
$546$	−18.2462	−0.780866
$547$	−13.3693	−0.571631	−0.285815	−	0.958285i	$-0.592264\pi$
$547$	−13.3693	−0.571631	−0.285815	+	0.958285i	$0.592264\pi$
$548$	−13.3693	−0.571109
$549$	−3.12311	−0.133291
$550$	15.3693	0.655350
$551$	−1.56155	−0.0665244
$552$	−1.00000	−0.0425628
$553$	0.876894	0.0372893
$554$	10.4924	0.445780
$555$	−5.56155	−0.236075
$556$	−15.1231	−0.641363
$557$	28.4384	1.20498	0.602488	−	0.798128i	$-0.294176\pi$
$557$	28.4384	1.20498	0.602488	+	0.798128i	$0.294176\pi$
$558$	0	0
$559$	3.50758	0.148355
$560$	−12.6847	−0.536025
$561$	−4.87689	−0.205903
$562$	−29.8617	−1.25964
$563$	−22.9848	−0.968696	−0.484348	−	0.874876i	$-0.660943\pi$
$563$	−22.9848	−0.968696	−0.484348	+	0.874876i	$0.660943\pi$
$564$	−2.43845	−0.102677
$565$	−53.1771	−2.23718
$566$	5.36932	0.225689
$567$	3.56155	0.149571
$568$	8.00000	0.335673
$569$	−23.3153	−0.977430	−0.488715	−	0.872444i	$-0.662534\pi$
$569$	−23.3153	−0.977430	−0.488715	+	0.872444i	$0.662534\pi$
$570$	5.56155	0.232948
$571$	43.1231	1.80465	0.902323	−	0.431061i	$-0.141861\pi$
$571$	43.1231	1.80465	0.902323	+	0.431061i	$0.141861\pi$
$572$	10.2462	0.428416
$573$	−21.5616	−0.900747
$574$	12.6847	0.529447
$575$	−7.68466	−0.320472
$576$	1.00000	0.0416667
$577$	−41.1231	−1.71198	−0.855989	−	0.516995i	$-0.827051\pi$
$577$	−41.1231	−1.71198	−0.855989	+	0.516995i	$0.827051\pi$
$578$	11.0540	0.459785
$579$	−9.12311	−0.379143
$580$	−3.56155	−0.147885
$581$	−10.2462	−0.425084
$582$	13.1231	0.543970
$583$	4.00000	0.165663
$584$	−6.00000	−0.248282
$585$	18.2462	0.754388
$586$	−22.0000	−0.908812
$587$	−12.1922	−0.503227	−0.251614	−	0.967828i	$-0.580961\pi$
$587$	−12.1922	−0.503227	−0.251614	+	0.967828i	$0.580961\pi$
$588$	−5.68466	−0.234431
$589$	0	0
$590$	−5.56155	−0.228966
$591$	−2.68466	−0.110432
$592$	−1.56155	−0.0641794
$593$	35.3693	1.45244	0.726222	−	0.687460i	$-0.241275\pi$
$593$	35.3693	1.45244	0.726222	+	0.687460i	$0.241275\pi$
$594$	−2.00000	−0.0820610
$595$	30.9309	1.26804
$596$	−14.6847	−0.601507
$597$	7.36932	0.301606
$598$	−5.12311	−0.209499
$599$	9.75379	0.398529	0.199265	−	0.979946i	$-0.436145\pi$
$599$	9.75379	0.398529	0.199265	+	0.979946i	$0.436145\pi$
$600$	7.68466	0.313725
$601$	−1.12311	−0.0458124	−0.0229062	−	0.999738i	$-0.507292\pi$
$601$	−1.12311	−0.0458124	−0.0229062	+	0.999738i	$0.507292\pi$
$602$	2.43845	0.0993837
$603$	0	0
$604$	−3.31534	−0.134899
$605$	24.9309	1.01358
$606$	−4.24621	−0.172491
$607$	22.7386	0.922933	0.461466	−	0.887158i	$-0.347324\pi$
$607$	22.7386	0.922933	0.461466	+	0.887158i	$0.347324\pi$
$608$	1.56155	0.0633293
$609$	−3.56155	−0.144321
$610$	−11.1231	−0.450361
$611$	−12.4924	−0.505389
$612$	−2.43845	−0.0985684
$613$	−6.63068	−0.267811	−0.133905	−	0.990994i	$-0.542752\pi$
$613$	−6.63068	−0.267811	−0.133905	+	0.990994i	$0.542752\pi$
$614$	8.49242	0.342726
$615$	−12.6847	−0.511495
$616$	7.12311	0.286998
$617$	4.19224	0.168773	0.0843865	−	0.996433i	$-0.473107\pi$
$617$	4.19224	0.168773	0.0843865	+	0.996433i	$0.473107\pi$
$618$	−5.80776	−0.233623
$619$	1.56155	0.0627641	0.0313821	−	0.999507i	$-0.490009\pi$
$619$	1.56155	0.0627641	0.0313821	+	0.999507i	$0.490009\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	10.0540	0.403128
$623$	−11.1231	−0.445638
$624$	5.12311	0.205088
$625$	−4.36932	−0.174773
$626$	−10.1922	−0.407364
$627$	−3.12311	−0.124725
$628$	4.68466	0.186938
$629$	3.80776	0.151825
$630$	12.6847	0.505369
$631$	19.1771	0.763428	0.381714	−	0.924281i	$-0.375334\pi$
$631$	19.1771	0.763428	0.381714	+	0.924281i	$0.375334\pi$
$632$	−0.246211	−0.00979376
$633$	−14.4384	−0.573877
$634$	−25.1231	−0.997766
$635$	−31.6155	−1.25462
$636$	2.00000	0.0793052
$637$	−29.1231	−1.15390
$638$	2.00000	0.0791808
$639$	−8.00000	−0.316475
$640$	3.56155	0.140783
$641$	18.4384	0.728275	0.364137	−	0.931345i	$-0.381364\pi$
$641$	18.4384	0.728275	0.364137	+	0.931345i	$0.381364\pi$
$642$	−12.9309	−0.510341
$643$	−3.50758	−0.138325	−0.0691627	−	0.997605i	$-0.522033\pi$
$643$	−3.50758	−0.138325	−0.0691627	+	0.997605i	$0.522033\pi$
$644$	−3.56155	−0.140345
$645$	−2.43845	−0.0960138
$646$	−3.80776	−0.149814
$647$	−3.12311	−0.122782	−0.0613910	−	0.998114i	$-0.519554\pi$
$647$	−3.12311	−0.122782	−0.0613910	+	0.998114i	$0.519554\pi$
$648$	−1.00000	−0.0392837
$649$	3.12311	0.122593
$650$	39.3693	1.54419
$651$	0	0
$652$	3.31534	0.129839
$653$	46.4924	1.81939	0.909695	−	0.415278i	$-0.136316\pi$
$653$	46.4924	1.81939	0.909695	+	0.415278i	$0.136316\pi$
$654$	12.0000	0.469237
$655$	22.2462	0.869231
$656$	−3.56155	−0.139055
$657$	6.00000	0.234082
$658$	−8.68466	−0.338563
$659$	−27.3693	−1.06616	−0.533079	−	0.846066i	$-0.678965\pi$
$659$	−27.3693	−1.06616	−0.533079	+	0.846066i	$0.678965\pi$
$660$	−7.12311	−0.277267
$661$	−17.7538	−0.690542	−0.345271	−	0.938503i	$-0.612213\pi$
$661$	−17.7538	−0.690542	−0.345271	+	0.938503i	$0.612213\pi$
$662$	−25.5616	−0.993478
$663$	−12.4924	−0.485165
$664$	2.87689	0.111645
$665$	19.8078	0.768112
$666$	1.56155	0.0605089
$667$	−1.00000	−0.0387202
$668$	−11.1231	−0.430366
$669$	−18.2462	−0.705439
$670$	0	0
$671$	6.24621	0.241132
$672$	3.56155	0.137390
$673$	−15.0691	−0.580872	−0.290436	−	0.956894i	$-0.593800\pi$
$673$	−15.0691	−0.580872	−0.290436	+	0.956894i	$0.593800\pi$
$674$	−6.00000	−0.231111
$675$	−7.68466	−0.295783
$676$	13.2462	0.509470
$677$	−32.7386	−1.25825	−0.629124	−	0.777305i	$-0.716586\pi$
$677$	−32.7386	−1.25825	−0.629124	+	0.777305i	$0.716586\pi$
$678$	14.9309	0.573417
$679$	46.7386	1.79366
$680$	−8.68466	−0.333041
$681$	−9.31534	−0.356965
$682$	0	0
$683$	51.4233	1.96766	0.983829	−	0.179111i	$-0.0573223\pi$
$683$	51.4233	1.96766	0.983829	+	0.179111i	$0.0573223\pi$
$684$	−1.56155	−0.0597075
$685$	47.6155	1.81930
$686$	4.68466	0.178861
$687$	−9.56155	−0.364796
$688$	−0.684658	−0.0261024
$689$	10.2462	0.390350
$690$	3.56155	0.135586
$691$	−8.49242	−0.323067	−0.161533	−	0.986867i	$-0.551644\pi$
$691$	−8.49242	−0.323067	−0.161533	+	0.986867i	$0.551644\pi$
$692$	16.0540	0.610281
$693$	−7.12311	−0.270584
$694$	−11.8078	−0.448217
$695$	53.8617	2.04309
$696$	1.00000	0.0379049
$697$	8.68466	0.328955
$698$	22.0000	0.832712
$699$	−5.12311	−0.193774
$700$	27.3693	1.03446
$701$	−42.3002	−1.59766	−0.798828	−	0.601559i	$-0.794546\pi$
$701$	−42.3002	−1.59766	−0.798828	+	0.601559i	$0.794546\pi$
$702$	−5.12311	−0.193359
$703$	2.43845	0.0919678
$704$	−2.00000	−0.0753778
$705$	8.68466	0.327083
$706$	−14.0000	−0.526897
$707$	−15.1231	−0.568763
$708$	1.56155	0.0586867
$709$	−0.384472	−0.0144391	−0.00721957	−	0.999974i	$-0.502298\pi$
$709$	−0.384472	−0.0144391	−0.00721957	+	0.999974i	$0.502298\pi$
$710$	−28.4924	−1.06930
$711$	0.246211	0.00923364
$712$	3.12311	0.117043
$713$	0	0
$714$	−8.68466	−0.325015
$715$	−36.4924	−1.36474
$716$	16.0000	0.597948
$717$	−17.3693	−0.648669
$718$	3.80776	0.142104
$719$	−9.75379	−0.363755	−0.181877	−	0.983321i	$-0.558217\pi$
$719$	−9.75379	−0.363755	−0.181877	+	0.983321i	$0.558217\pi$
$720$	−3.56155	−0.132731
$721$	−20.6847	−0.770337
$722$	16.5616	0.616357
$723$	−12.0540	−0.448292
$724$	10.2462	0.380797
$725$	7.68466	0.285401
$726$	−7.00000	−0.259794
$727$	29.2311	1.08412	0.542060	−	0.840340i	$-0.317645\pi$
$727$	29.2311	1.08412	0.542060	+	0.840340i	$0.317645\pi$
$728$	18.2462	0.676250
$729$	1.00000	0.0370370
$730$	21.3693	0.790914
$731$	1.66950	0.0617488
$732$	3.12311	0.115433
$733$	23.6155	0.872259	0.436130	−	0.899884i	$-0.356349\pi$
$733$	23.6155	0.872259	0.436130	+	0.899884i	$0.356349\pi$
$734$	32.7386	1.20841
$735$	20.2462	0.746793
$736$	1.00000	0.0368605
$737$	0	0
$738$	3.56155	0.131103
$739$	−1.26137	−0.0464001	−0.0232001	−	0.999731i	$-0.507385\pi$
$739$	−1.26137	−0.0464001	−0.0232001	+	0.999731i	$0.507385\pi$
$740$	5.56155	0.204447
$741$	−8.00000	−0.293887
$742$	7.12311	0.261497
$743$	−1.17708	−0.0431829	−0.0215915	−	0.999767i	$-0.506873\pi$
$743$	−1.17708	−0.0431829	−0.0215915	+	0.999767i	$0.506873\pi$
$744$	0	0
$745$	52.3002	1.91613
$746$	−20.0000	−0.732252
$747$	−2.87689	−0.105260
$748$	4.87689	0.178317
$749$	−46.0540	−1.68278
$750$	−9.56155	−0.349139
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	2.43845	0.0889210
$753$	−1.12311	−0.0409282
$754$	5.12311	0.186573
$755$	11.8078	0.429729
$756$	−3.56155	−0.129532
$757$	17.5616	0.638285	0.319143	−	0.947707i	$-0.396605\pi$
$757$	17.5616	0.638285	0.319143	+	0.947707i	$0.396605\pi$
$758$	0.492423	0.0178856
$759$	−2.00000	−0.0725954
$760$	−5.56155	−0.201739
$761$	−35.3693	−1.28214	−0.641068	−	0.767484i	$-0.721509\pi$
$761$	−35.3693	−1.28214	−0.641068	+	0.767484i	$0.721509\pi$
$762$	8.87689	0.321576
$763$	42.7386	1.54724
$764$	21.5616	0.780070
$765$	8.68466	0.313994
$766$	13.3693	0.483053
$767$	8.00000	0.288863
$768$	−1.00000	−0.0360844
$769$	26.4924	0.955341	0.477671	−	0.878539i	$-0.341481\pi$
$769$	26.4924	0.955341	0.477671	+	0.878539i	$0.341481\pi$
$770$	−25.3693	−0.914247
$771$	−6.87689	−0.247665
$772$	9.12311	0.328348
$773$	−19.7538	−0.710494	−0.355247	−	0.934772i	$-0.615603\pi$
$773$	−19.7538	−0.710494	−0.355247	+	0.934772i	$0.615603\pi$
$774$	0.684658	0.0246095
$775$	0	0
$776$	−13.1231	−0.471092
$777$	5.56155	0.199520
$778$	−33.6155	−1.20518
$779$	5.56155	0.199263
$780$	−18.2462	−0.653319
$781$	16.0000	0.572525
$782$	−2.43845	−0.0871987
$783$	−1.00000	−0.0357371
$784$	5.68466	0.203024
$785$	−16.6847	−0.595501
$786$	−6.24621	−0.222795
$787$	17.7538	0.632854	0.316427	−	0.948617i	$-0.397517\pi$
$787$	17.7538	0.632854	0.316427	+	0.948617i	$0.397517\pi$
$788$	2.68466	0.0956370
$789$	−25.1771	−0.896328
$790$	0.876894	0.0311985
$791$	53.1771	1.89076
$792$	2.00000	0.0710669
$793$	16.0000	0.568177
$794$	6.49242	0.230407
$795$	−7.12311	−0.252631
$796$	−7.36932	−0.261199
$797$	−25.1231	−0.889906	−0.444953	−	0.895554i	$-0.646780\pi$
$797$	−25.1231	−0.889906	−0.444953	+	0.895554i	$0.646780\pi$
$798$	−5.56155	−0.196877
$799$	−5.94602	−0.210355
$800$	−7.68466	−0.271694
$801$	−3.12311	−0.110350
$802$	9.36932	0.330842
$803$	−12.0000	−0.423471
$804$	0	0
$805$	12.6847	0.447075
$806$	0	0
$807$	4.24621	0.149474
$808$	4.24621	0.149381
$809$	2.49242	0.0876289	0.0438145	−	0.999040i	$-0.486049\pi$
$809$	2.49242	0.0876289	0.0438145	+	0.999040i	$0.486049\pi$
$810$	3.56155	0.125140
$811$	−30.7386	−1.07938	−0.539690	−	0.841864i	$-0.681458\pi$
$811$	−30.7386	−1.07938	−0.539690	+	0.841864i	$0.681458\pi$
$812$	3.56155	0.124986
$813$	−4.00000	−0.140286
$814$	−3.12311	−0.109465
$815$	−11.8078	−0.413608
$816$	2.43845	0.0853627
$817$	1.06913	0.0374041
$818$	−10.8769	−0.380302
$819$	−18.2462	−0.637574
$820$	12.6847	0.442967
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−13.3693	−0.466308
$823$	52.1080	1.81637	0.908184	−	0.418570i	$-0.137469\pi$
$823$	52.1080	1.81637	0.908184	+	0.418570i	$0.137469\pi$
$824$	5.80776	0.202323
$825$	15.3693	0.535091
$826$	5.56155	0.193511
$827$	15.3693	0.534444	0.267222	−	0.963635i	$-0.413894\pi$
$827$	15.3693	0.534444	0.267222	+	0.963635i	$0.413894\pi$
$828$	−1.00000	−0.0347524
$829$	5.80776	0.201712	0.100856	−	0.994901i	$-0.467842\pi$
$829$	5.80776	0.201712	0.100856	+	0.994901i	$0.467842\pi$
$830$	−10.2462	−0.355651
$831$	10.4924	0.363978
$832$	−5.12311	−0.177612
$833$	−13.8617	−0.480281
$834$	−15.1231	−0.523670
$835$	39.6155	1.37095
$836$	3.12311	0.108015
$837$	0	0
$838$	−24.9309	−0.861223
$839$	−44.3002	−1.52941	−0.764706	−	0.644379i	$-0.777116\pi$
$839$	−44.3002	−1.52941	−0.764706	+	0.644379i	$0.777116\pi$
$840$	−12.6847	−0.437662
$841$	1.00000	0.0344828
$842$	23.6155	0.813845
$843$	−29.8617	−1.02849
$844$	14.4384	0.496992
$845$	−47.1771	−1.62294
$846$	−2.43845	−0.0838355
$847$	−24.9309	−0.856635
$848$	−2.00000	−0.0686803
$849$	5.36932	0.184274
$850$	18.7386	0.642730
$851$	1.56155	0.0535293
$852$	8.00000	0.274075
$853$	−28.0540	−0.960550	−0.480275	−	0.877118i	$-0.659463\pi$
$853$	−28.0540	−0.960550	−0.480275	+	0.877118i	$0.659463\pi$
$854$	11.1231	0.380625
$855$	5.56155	0.190201
$856$	12.9309	0.441968
$857$	5.50758	0.188135	0.0940676	−	0.995566i	$-0.470013\pi$
$857$	5.50758	0.188135	0.0940676	+	0.995566i	$0.470013\pi$
$858$	10.2462	0.349800
$859$	−0.192236	−0.00655901	−0.00327950	−	0.999995i	$-0.501044\pi$
$859$	−0.192236	−0.00655901	−0.00327950	+	0.999995i	$0.501044\pi$
$860$	2.43845	0.0831504
$861$	12.6847	0.432292
$862$	4.00000	0.136241
$863$	44.4924	1.51454	0.757270	−	0.653102i	$-0.226533\pi$
$863$	44.4924	1.51454	0.757270	+	0.653102i	$0.226533\pi$
$864$	1.00000	0.0340207
$865$	−57.1771	−1.94408
$866$	22.4924	0.764324
$867$	11.0540	0.375413
$868$	0	0
$869$	−0.492423	−0.0167043
$870$	−3.56155	−0.120748
$871$	0	0
$872$	−12.0000	−0.406371
$873$	13.1231	0.444150
$874$	−1.56155	−0.0528203
$875$	−34.0540	−1.15123
$876$	−6.00000	−0.202721
$877$	−43.8617	−1.48111	−0.740553	−	0.671998i	$-0.765436\pi$
$877$	−43.8617	−1.48111	−0.740553	+	0.671998i	$0.765436\pi$
$878$	2.43845	0.0822936
$879$	−22.0000	−0.742042
$880$	7.12311	0.240120
$881$	−21.8617	−0.736541	−0.368270	−	0.929719i	$-0.620050\pi$
$881$	−21.8617	−0.736541	−0.368270	+	0.929719i	$0.620050\pi$
$882$	−5.68466	−0.191412
$883$	−7.50758	−0.252650	−0.126325	−	0.991989i	$-0.540318\pi$
$883$	−7.50758	−0.252650	−0.126325	+	0.991989i	$0.540318\pi$
$884$	12.4924	0.420166
$885$	−5.56155	−0.186950
$886$	5.75379	0.193302
$887$	16.9848	0.570295	0.285148	−	0.958484i	$-0.407957\pi$
$887$	16.9848	0.570295	0.285148	+	0.958484i	$0.407957\pi$
$888$	−1.56155	−0.0524023
$889$	31.6155	1.06035
$890$	−11.1231	−0.372847
$891$	−2.00000	−0.0670025
$892$	18.2462	0.610928
$893$	−3.80776	−0.127422
$894$	−14.6847	−0.491128
$895$	−56.9848	−1.90479
$896$	−3.56155	−0.118983
$897$	−5.12311	−0.171056
$898$	11.5616	0.385814
$899$	0	0
$900$	7.68466	0.256155
$901$	4.87689	0.162473
$902$	−7.12311	−0.237173
$903$	2.43845	0.0811464
$904$	−14.9309	−0.496593
$905$	−36.4924	−1.21305
$906$	−3.31534	−0.110145
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	9.31534	0.309140
$909$	−4.24621	−0.140838
$910$	−64.9848	−2.15423
$911$	−43.2311	−1.43231	−0.716154	−	0.697942i	$-0.754099\pi$
$911$	−43.2311	−1.43231	−0.716154	+	0.697942i	$0.754099\pi$
$912$	1.56155	0.0517082
$913$	5.75379	0.190423
$914$	−20.4384	−0.676044
$915$	−11.1231	−0.367719
$916$	9.56155	0.315923
$917$	−22.2462	−0.734635
$918$	−2.43845	−0.0804807
$919$	9.31534	0.307285	0.153642	−	0.988127i	$-0.450900\pi$
$919$	9.31534	0.307285	0.153642	+	0.988127i	$0.450900\pi$
$920$	−3.56155	−0.117421
$921$	8.49242	0.279835
$922$	−38.1080	−1.25502
$923$	40.9848	1.34903
$924$	7.12311	0.234333
$925$	−12.0000	−0.394558
$926$	8.00000	0.262896
$927$	−5.80776	−0.190752
$928$	−1.00000	−0.0328266
$929$	−26.4924	−0.869188	−0.434594	−	0.900626i	$-0.643108\pi$
$929$	−26.4924	−0.869188	−0.434594	+	0.900626i	$0.643108\pi$
$930$	0	0
$931$	−8.87689	−0.290928
$932$	5.12311	0.167813
$933$	10.0540	0.329152
$934$	0.630683	0.0206366
$935$	−17.3693	−0.568037
$936$	5.12311	0.167454
$937$	−30.3002	−0.989864	−0.494932	−	0.868932i	$-0.664807\pi$
$937$	−30.3002	−0.989864	−0.494932	+	0.868932i	$0.664807\pi$
$938$	0	0
$939$	−10.1922	−0.332611
$940$	−8.68466	−0.283262
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	4.68466	0.152634
$943$	3.56155	0.115980
$944$	−1.56155	−0.0508242
$945$	12.6847	0.412632
$946$	−1.36932	−0.0445203
$947$	−60.4924	−1.96574	−0.982870	−	0.184301i	$-0.940998\pi$
$947$	−60.4924	−1.96574	−0.982870	+	0.184301i	$0.940998\pi$
$948$	−0.246211	−0.00799657
$949$	−30.7386	−0.997818
$950$	12.0000	0.389331
$951$	−25.1231	−0.814673
$952$	8.68466	0.281471
$953$	−47.2311	−1.52996	−0.764982	−	0.644052i	$-0.777252\pi$
$953$	−47.2311	−1.52996	−0.764982	+	0.644052i	$0.777252\pi$
$954$	2.00000	0.0647524
$955$	−76.7926	−2.48495
$956$	17.3693	0.561764
$957$	2.00000	0.0646508
$958$	−36.4924	−1.17902
$959$	−47.6155	−1.53759
$960$	3.56155	0.114949
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	−12.9309	−0.416691
$964$	12.0540	0.388232
$965$	−32.4924	−1.04597
$966$	−3.56155	−0.114591
$967$	38.2462	1.22992	0.614958	−	0.788560i	$-0.289173\pi$
$967$	38.2462	1.22992	0.614958	+	0.788560i	$0.289173\pi$
$968$	7.00000	0.224989
$969$	−3.80776	−0.122323
$970$	46.7386	1.50069
$971$	3.75379	0.120465	0.0602324	−	0.998184i	$-0.480816\pi$
$971$	3.75379	0.120465	0.0602324	+	0.998184i	$0.480816\pi$
$972$	−1.00000	−0.0320750
$973$	−53.8617	−1.72673
$974$	−5.56155	−0.178204
$975$	39.3693	1.26083
$976$	−3.12311	−0.0999682
$977$	28.9848	0.927307	0.463654	−	0.886017i	$-0.346538\pi$
$977$	28.9848	0.927307	0.463654	+	0.886017i	$0.346538\pi$
$978$	3.31534	0.106013
$979$	6.24621	0.199630
$980$	−20.2462	−0.646741
$981$	12.0000	0.383131
$982$	−25.3693	−0.809567
$983$	−38.7386	−1.23557	−0.617785	−	0.786347i	$-0.711970\pi$
$983$	−38.7386	−1.23557	−0.617785	+	0.786347i	$0.711970\pi$
$984$	−3.56155	−0.113538
$985$	−9.56155	−0.304656
$986$	2.43845	0.0776560
$987$	−8.68466	−0.276436
$988$	8.00000	0.254514
$989$	0.684658	0.0217709
$990$	−7.12311	−0.226387
$991$	−11.4233	−0.362873	−0.181436	−	0.983403i	$-0.558075\pi$
$991$	−11.4233	−0.362873	−0.181436	+	0.983403i	$0.558075\pi$
$992$	0	0
$993$	−25.5616	−0.811171
$994$	28.4924	0.903725
$995$	26.2462	0.832061
$996$	2.87689	0.0911579
$997$	35.6695	1.12966	0.564832	−	0.825206i	$-0.308941\pi$
$997$	35.6695	1.12966	0.564832	+	0.825206i	$0.308941\pi$
$998$	3.61553	0.114448
$999$	1.56155	0.0494053

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.s.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} +3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.56155 q^{5} +1.00000 q^{6} +3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} -2.00000 q^{11} -1.00000 q^{12} -5.12311 q^{13} -3.56155 q^{14} +3.56155 q^{15} +1.00000 q^{16} -2.43845 q^{17} -1.00000 q^{18} -1.56155 q^{19} -3.56155 q^{20} -3.56155 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +7.68466 q^{25} +5.12311 q^{26} -1.00000 q^{27} +3.56155 q^{28} +1.00000 q^{29} -3.56155 q^{30} -1.00000 q^{32} +2.00000 q^{33} +2.43845 q^{34} -12.6847 q^{35} +1.00000 q^{36} -1.56155 q^{37} +1.56155 q^{38} +5.12311 q^{39} +3.56155 q^{40} -3.56155 q^{41} +3.56155 q^{42} -0.684658 q^{43} -2.00000 q^{44} -3.56155 q^{45} +1.00000 q^{46} +2.43845 q^{47} -1.00000 q^{48} +5.68466 q^{49} -7.68466 q^{50} +2.43845 q^{51} -5.12311 q^{52} -2.00000 q^{53} +1.00000 q^{54} +7.12311 q^{55} -3.56155 q^{56} +1.56155 q^{57} -1.00000 q^{58} -1.56155 q^{59} +3.56155 q^{60} -3.12311 q^{61} +3.56155 q^{63} +1.00000 q^{64} +18.2462 q^{65} -2.00000 q^{66} -2.43845 q^{68} +1.00000 q^{69} +12.6847 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +1.56155 q^{74} -7.68466 q^{75} -1.56155 q^{76} -7.12311 q^{77} -5.12311 q^{78} +0.246211 q^{79} -3.56155 q^{80} +1.00000 q^{81} +3.56155 q^{82} -2.87689 q^{83} -3.56155 q^{84} +8.68466 q^{85} +0.684658 q^{86} -1.00000 q^{87} +2.00000 q^{88} -3.12311 q^{89} +3.56155 q^{90} -18.2462 q^{91} -1.00000 q^{92} -2.43845 q^{94} +5.56155 q^{95} +1.00000 q^{96} +13.1231 q^{97} -5.68466 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} + q^{19} - 3 q^{20} - 3 q^{21} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} - 2 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 2 q^{32} + 4 q^{33} + 9 q^{34} - 13 q^{35} + 2 q^{36} + q^{37} - q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 3 q^{42} + 11 q^{43} - 4 q^{44} - 3 q^{45} + 2 q^{46} + 9 q^{47} - 2 q^{48} - q^{49} - 3 q^{50} + 9 q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 6 q^{55} - 3 q^{56} - q^{57} - 2 q^{58} + q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{63} + 2 q^{64} + 20 q^{65} - 4 q^{66} - 9 q^{68} + 2 q^{69} + 13 q^{70} - 16 q^{71} - 2 q^{72} + 12 q^{73} - q^{74} - 3 q^{75} + q^{76} - 6 q^{77} - 2 q^{78} - 16 q^{79} - 3 q^{80} + 2 q^{81} + 3 q^{82} - 14 q^{83} - 3 q^{84} + 5 q^{85} - 11 q^{86} - 2 q^{87} + 4 q^{88} + 2 q^{89} + 3 q^{90} - 20 q^{91} - 2 q^{92} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 3 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 - 9 * q^17 - 2 * q^18 + q^19 - 3 * q^20 - 3 * q^21 + 4 * q^22 - 2 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 - 2 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 2 * q^32 + 4 * q^33 + 9 * q^34 - 13 * q^35 + 2 * q^36 + q^37 - q^38 + 2 * q^39 + 3 * q^40 - 3 * q^41 + 3 * q^42 + 11 * q^43 - 4 * q^44 - 3 * q^45 + 2 * q^46 + 9 * q^47 - 2 * q^48 - q^49 - 3 * q^50 + 9 * q^51 - 2 * q^52 - 4 * q^53 + 2 * q^54 + 6 * q^55 - 3 * q^56 - q^57 - 2 * q^58 + q^59 + 3 * q^60 + 2 * q^61 + 3 * q^63 + 2 * q^64 + 20 * q^65 - 4 * q^66 - 9 * q^68 + 2 * q^69 + 13 * q^70 - 16 * q^71 - 2 * q^72 + 12 * q^73 - q^74 - 3 * q^75 + q^76 - 6 * q^77 - 2 * q^78 - 16 * q^79 - 3 * q^80 + 2 * q^81 + 3 * q^82 - 14 * q^83 - 3 * q^84 + 5 * q^85 - 11 * q^86 - 2 * q^87 + 4 * q^88 + 2 * q^89 + 3 * q^90 - 20 * q^91 - 2 * q^92 - 9 * q^94 + 7 * q^95 + 2 * q^96 + 18 * q^97 + q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more