LMFDB - Embedded newform 4002.2.a.bh.1.3

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 16x^{5} + 51x^{4} + 45x^{3} - 152x^{2} - 54x + 70 $ x^7 - 3x^6 - 16x^5 + 51x^4 + 45x^3 - 152x^2 - 54x + 70
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.56994$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.59086 q^{5} +1.00000 q^{6} +1.20696 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.59086 q^{5} +1.00000 q^{6} +1.20696 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.59086 q^{10} +4.80384 q^{11} +1.00000 q^{12} +1.05524 q^{13} +1.20696 q^{14} -1.59086 q^{15} +1.00000 q^{16} -2.39469 q^{17} +1.00000 q^{18} +6.34685 q^{19} -1.59086 q^{20} +1.20696 q^{21} +4.80384 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.46917 q^{25} +1.05524 q^{26} +1.00000 q^{27} +1.20696 q^{28} -1.00000 q^{29} -1.59086 q^{30} -5.20570 q^{31} +1.00000 q^{32} +4.80384 q^{33} -2.39469 q^{34} -1.92011 q^{35} +1.00000 q^{36} +7.70325 q^{37} +6.34685 q^{38} +1.05524 q^{39} -1.59086 q^{40} +1.69372 q^{41} +1.20696 q^{42} -7.44867 q^{43} +4.80384 q^{44} -1.59086 q^{45} +1.00000 q^{46} +8.28354 q^{47} +1.00000 q^{48} -5.54324 q^{49} -2.46917 q^{50} -2.39469 q^{51} +1.05524 q^{52} +1.92691 q^{53} +1.00000 q^{54} -7.64222 q^{55} +1.20696 q^{56} +6.34685 q^{57} -1.00000 q^{58} +2.23696 q^{59} -1.59086 q^{60} -5.98555 q^{61} -5.20570 q^{62} +1.20696 q^{63} +1.00000 q^{64} -1.67874 q^{65} +4.80384 q^{66} +1.48037 q^{67} -2.39469 q^{68} +1.00000 q^{69} -1.92011 q^{70} +7.48800 q^{71} +1.00000 q^{72} -3.40526 q^{73} +7.70325 q^{74} -2.46917 q^{75} +6.34685 q^{76} +5.79805 q^{77} +1.05524 q^{78} +17.0123 q^{79} -1.59086 q^{80} +1.00000 q^{81} +1.69372 q^{82} +4.46177 q^{83} +1.20696 q^{84} +3.80962 q^{85} -7.44867 q^{86} -1.00000 q^{87} +4.80384 q^{88} +13.3089 q^{89} -1.59086 q^{90} +1.27364 q^{91} +1.00000 q^{92} -5.20570 q^{93} +8.28354 q^{94} -10.0969 q^{95} +1.00000 q^{96} -17.6029 q^{97} -5.54324 q^{98} +4.80384 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{3} + 7 q^{4} + q^{5} + 7 q^{6} - 2 q^{7} + 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^3 + 7 * q^4 + q^5 + 7 * q^6 - 2 * q^7 + 7 * q^8 + 7 * q^9	\( 7 q + 7 q^{2} + 7 q^{3} + 7 q^{4} + q^{5} + 7 q^{6} - 2 q^{7} + 7 q^{8} + 7 q^{9} + q^{10} + 11 q^{11} + 7 q^{12} - q^{13} - 2 q^{14} + q^{15} + 7 q^{16} + 18 q^{17} + 7 q^{18} + 6 q^{19} + q^{20} - 2 q^{21} + 11 q^{22} + 7 q^{23} + 7 q^{24} + 12 q^{25} - q^{26} + 7 q^{27} - 2 q^{28} - 7 q^{29} + q^{30} + 3 q^{31} + 7 q^{32} + 11 q^{33} + 18 q^{34} - 4 q^{35} + 7 q^{36} + 5 q^{37} + 6 q^{38} - q^{39} + q^{40} + 25 q^{41} - 2 q^{42} + 20 q^{43} + 11 q^{44} + q^{45} + 7 q^{46} + 7 q^{48} + 7 q^{49} + 12 q^{50} + 18 q^{51} - q^{52} - 4 q^{53} + 7 q^{54} + 17 q^{55} - 2 q^{56} + 6 q^{57} - 7 q^{58} - 17 q^{59} + q^{60} + 5 q^{61} + 3 q^{62} - 2 q^{63} + 7 q^{64} + 7 q^{65} + 11 q^{66} + 15 q^{67} + 18 q^{68} + 7 q^{69} - 4 q^{70} + 15 q^{71} + 7 q^{72} + 14 q^{73} + 5 q^{74} + 12 q^{75} + 6 q^{76} - 12 q^{77} - q^{78} - 4 q^{79} + q^{80} + 7 q^{81} + 25 q^{82} + 14 q^{83} - 2 q^{84} + 34 q^{85} + 20 q^{86} - 7 q^{87} + 11 q^{88} + 8 q^{89} + q^{90} - 6 q^{91} + 7 q^{92} + 3 q^{93} - 18 q^{95} + 7 q^{96} - 18 q^{97} + 7 q^{98} + 11 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^3 + 7 * q^4 + q^5 + 7 * q^6 - 2 * q^7 + 7 * q^8 + 7 * q^9 + q^10 + 11 * q^11 + 7 * q^12 - q^13 - 2 * q^14 + q^15 + 7 * q^16 + 18 * q^17 + 7 * q^18 + 6 * q^19 + q^20 - 2 * q^21 + 11 * q^22 + 7 * q^23 + 7 * q^24 + 12 * q^25 - q^26 + 7 * q^27 - 2 * q^28 - 7 * q^29 + q^30 + 3 * q^31 + 7 * q^32 + 11 * q^33 + 18 * q^34 - 4 * q^35 + 7 * q^36 + 5 * q^37 + 6 * q^38 - q^39 + q^40 + 25 * q^41 - 2 * q^42 + 20 * q^43 + 11 * q^44 + q^45 + 7 * q^46 + 7 * q^48 + 7 * q^49 + 12 * q^50 + 18 * q^51 - q^52 - 4 * q^53 + 7 * q^54 + 17 * q^55 - 2 * q^56 + 6 * q^57 - 7 * q^58 - 17 * q^59 + q^60 + 5 * q^61 + 3 * q^62 - 2 * q^63 + 7 * q^64 + 7 * q^65 + 11 * q^66 + 15 * q^67 + 18 * q^68 + 7 * q^69 - 4 * q^70 + 15 * q^71 + 7 * q^72 + 14 * q^73 + 5 * q^74 + 12 * q^75 + 6 * q^76 - 12 * q^77 - q^78 - 4 * q^79 + q^80 + 7 * q^81 + 25 * q^82 + 14 * q^83 - 2 * q^84 + 34 * q^85 + 20 * q^86 - 7 * q^87 + 11 * q^88 + 8 * q^89 + q^90 - 6 * q^91 + 7 * q^92 + 3 * q^93 - 18 * q^95 + 7 * q^96 - 18 * q^97 + 7 * q^98 + 11 * 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
$5$	−1.59086	−0.711454	−0.355727	−	0.934590i	$-0.615767\pi$
$5$	−1.59086	−0.711454	−0.355727	+	0.934590i	$0.615767\pi$
$6$	1.00000	0.408248
$7$	1.20696	0.456189	0.228095	−	0.973639i	$-0.426750\pi$
$7$	1.20696	0.456189	0.228095	+	0.973639i	$0.426750\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.59086	−0.503074
$11$	4.80384	1.44841	0.724206	−	0.689584i	$-0.242207\pi$
$11$	4.80384	1.44841	0.724206	+	0.689584i	$0.242207\pi$
$12$	1.00000	0.288675
$13$	1.05524	0.292672	0.146336	−	0.989235i	$-0.453252\pi$
$13$	1.05524	0.292672	0.146336	+	0.989235i	$0.453252\pi$
$14$	1.20696	0.322574
$15$	−1.59086	−0.410758
$16$	1.00000	0.250000
$17$	−2.39469	−0.580799	−0.290399	−	0.956906i	$-0.593788\pi$
$17$	−2.39469	−0.580799	−0.290399	+	0.956906i	$0.593788\pi$
$18$	1.00000	0.235702
$19$	6.34685	1.45607	0.728033	−	0.685542i	$-0.240435\pi$
$19$	6.34685	1.45607	0.728033	+	0.685542i	$0.240435\pi$
$20$	−1.59086	−0.355727
$21$	1.20696	0.263381
$22$	4.80384	1.02418
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−2.46917	−0.493834
$26$	1.05524	0.206950
$27$	1.00000	0.192450
$28$	1.20696	0.228095
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.59086	−0.290450
$31$	−5.20570	−0.934971	−0.467485	−	0.884001i	$-0.654840\pi$
$31$	−5.20570	−0.934971	−0.467485	+	0.884001i	$0.654840\pi$
$32$	1.00000	0.176777
$33$	4.80384	0.836241
$34$	−2.39469	−0.410687
$35$	−1.92011	−0.324557
$36$	1.00000	0.166667
$37$	7.70325	1.26641	0.633203	−	0.773985i	$-0.281740\pi$
$37$	7.70325	1.26641	0.633203	+	0.773985i	$0.281740\pi$
$38$	6.34685	1.02959
$39$	1.05524	0.168974
$40$	−1.59086	−0.251537
$41$	1.69372	0.264515	0.132257	−	0.991215i	$-0.457777\pi$
$41$	1.69372	0.264515	0.132257	+	0.991215i	$0.457777\pi$
$42$	1.20696	0.186238
$43$	−7.44867	−1.13591	−0.567956	−	0.823059i	$-0.692266\pi$
$43$	−7.44867	−1.13591	−0.567956	+	0.823059i	$0.692266\pi$
$44$	4.80384	0.724206
$45$	−1.59086	−0.237151
$46$	1.00000	0.147442
$47$	8.28354	1.20828	0.604139	−	0.796879i	$-0.293517\pi$
$47$	8.28354	1.20828	0.604139	+	0.796879i	$0.293517\pi$
$48$	1.00000	0.144338
$49$	−5.54324	−0.791892
$50$	−2.46917	−0.349193
$51$	−2.39469	−0.335324
$52$	1.05524	0.146336
$53$	1.92691	0.264681	0.132340	−	0.991204i	$-0.457751\pi$
$53$	1.92691	0.264681	0.132340	+	0.991204i	$0.457751\pi$
$54$	1.00000	0.136083
$55$	−7.64222	−1.03048
$56$	1.20696	0.161287
$57$	6.34685	0.840660
$58$	−1.00000	−0.131306
$59$	2.23696	0.291227	0.145614	−	0.989342i	$-0.453484\pi$
$59$	2.23696	0.291227	0.145614	+	0.989342i	$0.453484\pi$
$60$	−1.59086	−0.205379
$61$	−5.98555	−0.766372	−0.383186	−	0.923671i	$-0.625173\pi$
$61$	−5.98555	−0.766372	−0.383186	+	0.923671i	$0.625173\pi$
$62$	−5.20570	−0.661124
$63$	1.20696	0.152063
$64$	1.00000	0.125000
$65$	−1.67874	−0.208222
$66$	4.80384	0.591311
$67$	1.48037	0.180856	0.0904281	−	0.995903i	$-0.471176\pi$
$67$	1.48037	0.180856	0.0904281	+	0.995903i	$0.471176\pi$
$68$	−2.39469	−0.290399
$69$	1.00000	0.120386
$70$	−1.92011	−0.229497
$71$	7.48800	0.888662	0.444331	−	0.895863i	$-0.353441\pi$
$71$	7.48800	0.888662	0.444331	+	0.895863i	$0.353441\pi$
$72$	1.00000	0.117851
$73$	−3.40526	−0.398556	−0.199278	−	0.979943i	$-0.563860\pi$
$73$	−3.40526	−0.398556	−0.199278	+	0.979943i	$0.563860\pi$
$74$	7.70325	0.895485
$75$	−2.46917	−0.285115
$76$	6.34685	0.728033
$77$	5.79805	0.660749
$78$	1.05524	0.119483
$79$	17.0123	1.91403	0.957015	−	0.290039i	$-0.0936682\pi$
$79$	17.0123	1.91403	0.957015	+	0.290039i	$0.0936682\pi$
$80$	−1.59086	−0.177863
$81$	1.00000	0.111111
$82$	1.69372	0.187040
$83$	4.46177	0.489743	0.244872	−	0.969556i	$-0.421254\pi$
$83$	4.46177	0.489743	0.244872	+	0.969556i	$0.421254\pi$
$84$	1.20696	0.131690
$85$	3.80962	0.413211
$86$	−7.44867	−0.803211
$87$	−1.00000	−0.107211
$88$	4.80384	0.512091
$89$	13.3089	1.41074	0.705370	−	0.708839i	$-0.250781\pi$
$89$	13.3089	1.41074	0.705370	+	0.708839i	$0.250781\pi$
$90$	−1.59086	−0.167691
$91$	1.27364	0.133514
$92$	1.00000	0.104257
$93$	−5.20570	−0.539806
$94$	8.28354	0.854382
$95$	−10.0969	−1.03592
$96$	1.00000	0.102062
$97$	−17.6029	−1.78730	−0.893652	−	0.448761i	$-0.851865\pi$
$97$	−17.6029	−1.78730	−0.893652	+	0.448761i	$0.851865\pi$
$98$	−5.54324	−0.559952
$99$	4.80384	0.482804
$100$	−2.46917	−0.246917
$101$	−9.38028	−0.933373	−0.466686	−	0.884423i	$-0.654552\pi$
$101$	−9.38028	−0.933373	−0.466686	+	0.884423i	$0.654552\pi$
$102$	−2.39469	−0.237110
$103$	7.72810	0.761472	0.380736	−	0.924684i	$-0.375671\pi$
$103$	7.72810	0.761472	0.380736	+	0.924684i	$0.375671\pi$
$104$	1.05524	0.103475
$105$	−1.92011	−0.187383
$106$	1.92691	0.187158
$107$	−1.73541	−0.167769	−0.0838843	−	0.996475i	$-0.526733\pi$
$107$	−1.73541	−0.167769	−0.0838843	+	0.996475i	$0.526733\pi$
$108$	1.00000	0.0962250
$109$	−12.9689	−1.24219	−0.621096	−	0.783735i	$-0.713312\pi$
$109$	−12.9689	−1.24219	−0.621096	+	0.783735i	$0.713312\pi$
$110$	−7.64222	−0.728657
$111$	7.70325	0.731160
$112$	1.20696	0.114047
$113$	6.19229	0.582521	0.291261	−	0.956644i	$-0.405925\pi$
$113$	6.19229	0.582521	0.291261	+	0.956644i	$0.405925\pi$
$114$	6.34685	0.594437
$115$	−1.59086	−0.148348
$116$	−1.00000	−0.0928477
$117$	1.05524	0.0975573
$118$	2.23696	0.205929
$119$	−2.89031	−0.264954
$120$	−1.59086	−0.145225
$121$	12.0768	1.09789
$122$	−5.98555	−0.541907
$123$	1.69372	0.152718
$124$	−5.20570	−0.467485
$125$	11.8824	1.06279
$126$	1.20696	0.107525
$127$	12.3449	1.09543	0.547717	−	0.836664i	$-0.315497\pi$
$127$	12.3449	1.09543	0.547717	+	0.836664i	$0.315497\pi$
$128$	1.00000	0.0883883
$129$	−7.44867	−0.655819
$130$	−1.67874	−0.147235
$131$	−3.25228	−0.284153	−0.142076	−	0.989856i	$-0.545378\pi$
$131$	−3.25228	−0.284153	−0.142076	+	0.989856i	$0.545378\pi$
$132$	4.80384	0.418120
$133$	7.66041	0.664242
$134$	1.48037	0.127885
$135$	−1.59086	−0.136919
$136$	−2.39469	−0.205343
$137$	0.424048	0.0362289	0.0181144	−	0.999836i	$-0.494234\pi$
$137$	0.424048	0.0362289	0.0181144	+	0.999836i	$0.494234\pi$
$138$	1.00000	0.0851257
$139$	14.4460	1.22530	0.612648	−	0.790356i	$-0.290104\pi$
$139$	14.4460	1.22530	0.612648	+	0.790356i	$0.290104\pi$
$140$	−1.92011	−0.162279
$141$	8.28354	0.697600
$142$	7.48800	0.628379
$143$	5.06922	0.423909
$144$	1.00000	0.0833333
$145$	1.59086	0.132114
$146$	−3.40526	−0.281822
$147$	−5.54324	−0.457199
$148$	7.70325	0.633203
$149$	−8.70747	−0.713344	−0.356672	−	0.934230i	$-0.616089\pi$
$149$	−8.70747	−0.713344	−0.356672	+	0.934230i	$0.616089\pi$
$150$	−2.46917	−0.201607
$151$	4.82785	0.392885	0.196442	−	0.980515i	$-0.437061\pi$
$151$	4.82785	0.392885	0.196442	+	0.980515i	$0.437061\pi$
$152$	6.34685	0.514797
$153$	−2.39469	−0.193600
$154$	5.79805	0.467220
$155$	8.28153	0.665188
$156$	1.05524	0.0844871
$157$	−11.1263	−0.887978	−0.443989	−	0.896032i	$-0.646437\pi$
$157$	−11.1263	−0.887978	−0.443989	+	0.896032i	$0.646437\pi$
$158$	17.0123	1.35342
$159$	1.92691	0.152814
$160$	−1.59086	−0.125768
$161$	1.20696	0.0951220
$162$	1.00000	0.0785674
$163$	12.1265	0.949819	0.474909	−	0.880035i	$-0.342481\pi$
$163$	12.1265	0.949819	0.474909	+	0.880035i	$0.342481\pi$
$164$	1.69372	0.132257
$165$	−7.64222	−0.594946
$166$	4.46177	0.346301
$167$	13.3094	1.02991	0.514957	−	0.857216i	$-0.327808\pi$
$167$	13.3094	1.02991	0.514957	+	0.857216i	$0.327808\pi$
$168$	1.20696	0.0931192
$169$	−11.8865	−0.914343
$170$	3.80962	0.292185
$171$	6.34685	0.485356
$172$	−7.44867	−0.567956
$173$	11.0714	0.841745	0.420872	−	0.907120i	$-0.361724\pi$
$173$	11.0714	0.841745	0.420872	+	0.907120i	$0.361724\pi$
$174$	−1.00000	−0.0758098
$175$	−2.98019	−0.225282
$176$	4.80384	0.362103
$177$	2.23696	0.168140
$178$	13.3089	0.997544
$179$	7.48516	0.559467	0.279733	−	0.960078i	$-0.409754\pi$
$179$	7.48516	0.559467	0.279733	+	0.960078i	$0.409754\pi$
$180$	−1.59086	−0.118576
$181$	−19.7635	−1.46901	−0.734504	−	0.678605i	$-0.762585\pi$
$181$	−19.7635	−1.46901	−0.734504	+	0.678605i	$0.762585\pi$
$182$	1.27364	0.0944084
$183$	−5.98555	−0.442465
$184$	1.00000	0.0737210
$185$	−12.2548	−0.900990
$186$	−5.20570	−0.381700
$187$	−11.5037	−0.841235
$188$	8.28354	0.604139
$189$	1.20696	0.0877936
$190$	−10.0969	−0.732509
$191$	−20.8384	−1.50781	−0.753905	−	0.656983i	$-0.771832\pi$
$191$	−20.8384	−1.50781	−0.753905	+	0.656983i	$0.771832\pi$
$192$	1.00000	0.0721688
$193$	10.1999	0.734203	0.367102	−	0.930181i	$-0.380350\pi$
$193$	10.1999	0.734203	0.367102	+	0.930181i	$0.380350\pi$
$194$	−17.6029	−1.26381
$195$	−1.67874	−0.120217
$196$	−5.54324	−0.395946
$197$	−13.5343	−0.964280	−0.482140	−	0.876094i	$-0.660140\pi$
$197$	−13.5343	−0.964280	−0.482140	+	0.876094i	$0.660140\pi$
$198$	4.80384	0.341394
$199$	2.66897	0.189198	0.0945990	−	0.995515i	$-0.469843\pi$
$199$	2.66897	0.189198	0.0945990	+	0.995515i	$0.469843\pi$
$200$	−2.46917	−0.174597
$201$	1.48037	0.104417
$202$	−9.38028	−0.659994
$203$	−1.20696	−0.0847122
$204$	−2.39469	−0.167662
$205$	−2.69447	−0.188190
$206$	7.72810	0.538442
$207$	1.00000	0.0695048
$208$	1.05524	0.0731680
$209$	30.4892	2.10898
$210$	−1.92011	−0.132500
$211$	27.2127	1.87340	0.936699	−	0.350136i	$-0.113865\pi$
$211$	27.2127	1.87340	0.936699	+	0.350136i	$0.113865\pi$
$212$	1.92691	0.132340
$213$	7.48800	0.513069
$214$	−1.73541	−0.118630
$215$	11.8498	0.808148
$216$	1.00000	0.0680414
$217$	−6.28308	−0.426523
$218$	−12.9689	−0.878362
$219$	−3.40526	−0.230106
$220$	−7.64222	−0.515239
$221$	−2.52699	−0.169983
$222$	7.70325	0.517008
$223$	−15.2739	−1.02281	−0.511407	−	0.859339i	$-0.670875\pi$
$223$	−15.2739	−1.02281	−0.511407	+	0.859339i	$0.670875\pi$
$224$	1.20696	0.0806436
$225$	−2.46917	−0.164611
$226$	6.19229	0.411905
$227$	−14.4356	−0.958122	−0.479061	−	0.877782i	$-0.659023\pi$
$227$	−14.4356	−0.958122	−0.479061	+	0.877782i	$0.659023\pi$
$228$	6.34685	0.420330
$229$	−18.9412	−1.25167	−0.625835	−	0.779956i	$-0.715241\pi$
$229$	−18.9412	−1.25167	−0.625835	+	0.779956i	$0.715241\pi$
$230$	−1.59086	−0.104898
$231$	5.79805	0.381484
$232$	−1.00000	−0.0656532
$233$	−9.39059	−0.615198	−0.307599	−	0.951516i	$-0.599526\pi$
$233$	−9.39059	−0.615198	−0.307599	+	0.951516i	$0.599526\pi$
$234$	1.05524	0.0689834
$235$	−13.1779	−0.859634
$236$	2.23696	0.145614
$237$	17.0123	1.10507
$238$	−2.89031	−0.187351
$239$	7.67262	0.496301	0.248150	−	0.968722i	$-0.420177\pi$
$239$	7.67262	0.496301	0.248150	+	0.968722i	$0.420177\pi$
$240$	−1.59086	−0.102689
$241$	−14.7332	−0.949047	−0.474523	−	0.880243i	$-0.657380\pi$
$241$	−14.7332	−0.949047	−0.474523	+	0.880243i	$0.657380\pi$
$242$	12.0768	0.776329
$243$	1.00000	0.0641500
$244$	−5.98555	−0.383186
$245$	8.81851	0.563394
$246$	1.69372	0.107988
$247$	6.69747	0.426150
$248$	−5.20570	−0.330562
$249$	4.46177	0.282753
$250$	11.8824	0.751508
$251$	23.7959	1.50198	0.750991	−	0.660313i	$-0.229576\pi$
$251$	23.7959	1.50198	0.750991	+	0.660313i	$0.229576\pi$
$252$	1.20696	0.0760315
$253$	4.80384	0.302015
$254$	12.3449	0.774589
$255$	3.80962	0.238568
$256$	1.00000	0.0625000
$257$	16.0774	1.00288	0.501439	−	0.865193i	$-0.332804\pi$
$257$	16.0774	1.00288	0.501439	+	0.865193i	$0.332804\pi$
$258$	−7.44867	−0.463734
$259$	9.29754	0.577721
$260$	−1.67874	−0.104111
$261$	−1.00000	−0.0618984
$262$	−3.25228	−0.200926
$263$	1.38492	0.0853977	0.0426989	−	0.999088i	$-0.486404\pi$
$263$	1.38492	0.0853977	0.0426989	+	0.999088i	$0.486404\pi$
$264$	4.80384	0.295656
$265$	−3.06544	−0.188308
$266$	7.66041	0.469690
$267$	13.3089	0.814492
$268$	1.48037	0.0904281
$269$	9.11781	0.555923	0.277961	−	0.960592i	$-0.410341\pi$
$269$	9.11781	0.555923	0.277961	+	0.960592i	$0.410341\pi$
$270$	−1.59086	−0.0968166
$271$	7.45184	0.452667	0.226334	−	0.974050i	$-0.427326\pi$
$271$	7.45184	0.452667	0.226334	+	0.974050i	$0.427326\pi$
$272$	−2.39469	−0.145200
$273$	1.27364	0.0770842
$274$	0.424048	0.0256177
$275$	−11.8615	−0.715274
$276$	1.00000	0.0601929
$277$	−16.9050	−1.01572	−0.507862	−	0.861438i	$-0.669564\pi$
$277$	−16.9050	−1.01572	−0.507862	+	0.861438i	$0.669564\pi$
$278$	14.4460	0.866415
$279$	−5.20570	−0.311657
$280$	−1.92011	−0.114748
$281$	10.4939	0.626012	0.313006	−	0.949751i	$-0.398664\pi$
$281$	10.4939	0.626012	0.313006	+	0.949751i	$0.398664\pi$
$282$	8.28354	0.493278
$283$	−24.5520	−1.45947	−0.729733	−	0.683732i	$-0.760356\pi$
$283$	−24.5520	−1.45947	−0.729733	+	0.683732i	$0.760356\pi$
$284$	7.48800	0.444331
$285$	−10.0969	−0.598091
$286$	5.06922	0.299749
$287$	2.04426	0.120669
$288$	1.00000	0.0589256
$289$	−11.2654	−0.662673
$290$	1.59086	0.0934184
$291$	−17.6029	−1.03190
$292$	−3.40526	−0.199278
$293$	−26.7409	−1.56222	−0.781109	−	0.624394i	$-0.785346\pi$
$293$	−26.7409	−1.56222	−0.781109	+	0.624394i	$0.785346\pi$
$294$	−5.54324	−0.323288
$295$	−3.55869	−0.207195
$296$	7.70325	0.447742
$297$	4.80384	0.278747
$298$	−8.70747	−0.504410
$299$	1.05524	0.0610263
$300$	−2.46917	−0.142558
$301$	−8.99027	−0.518190
$302$	4.82785	0.277812
$303$	−9.38028	−0.538883
$304$	6.34685	0.364017
$305$	9.52217	0.545238
$306$	−2.39469	−0.136896
$307$	−1.51966	−0.0867316	−0.0433658	−	0.999059i	$-0.513808\pi$
$307$	−1.51966	−0.0867316	−0.0433658	+	0.999059i	$0.513808\pi$
$308$	5.79805	0.330375
$309$	7.72810	0.439636
$310$	8.28153	0.470359
$311$	3.00344	0.170309	0.0851547	−	0.996368i	$-0.472862\pi$
$311$	3.00344	0.170309	0.0851547	+	0.996368i	$0.472862\pi$
$312$	1.05524	0.0597414
$313$	−17.2575	−0.975452	−0.487726	−	0.872997i	$-0.662174\pi$
$313$	−17.2575	−0.975452	−0.487726	+	0.872997i	$0.662174\pi$
$314$	−11.1263	−0.627895
$315$	−1.92011	−0.108186
$316$	17.0123	0.957015
$317$	−13.2800	−0.745881	−0.372940	−	0.927855i	$-0.621650\pi$
$317$	−13.2800	−0.745881	−0.372940	+	0.927855i	$0.621650\pi$
$318$	1.92691	0.108056
$319$	−4.80384	−0.268963
$320$	−1.59086	−0.0889317
$321$	−1.73541	−0.0968613
$322$	1.20696	0.0672614
$323$	−15.1988	−0.845682
$324$	1.00000	0.0555556
$325$	−2.60557	−0.144531
$326$	12.1265	0.671623
$327$	−12.9689	−0.717180
$328$	1.69372	0.0935200
$329$	9.99793	0.551204
$330$	−7.64222	−0.420691
$331$	−9.94877	−0.546834	−0.273417	−	0.961896i	$-0.588154\pi$
$331$	−9.94877	−0.546834	−0.273417	+	0.961896i	$0.588154\pi$
$332$	4.46177	0.244872
$333$	7.70325	0.422136
$334$	13.3094	0.728260
$335$	−2.35506	−0.128671
$336$	1.20696	0.0658452
$337$	29.8659	1.62690	0.813449	−	0.581637i	$-0.197588\pi$
$337$	29.8659	1.62690	0.813449	+	0.581637i	$0.197588\pi$
$338$	−11.8865	−0.646538
$339$	6.19229	0.336319
$340$	3.80962	0.206606
$341$	−25.0073	−1.35422
$342$	6.34685	0.343198
$343$	−15.1392	−0.817441
$344$	−7.44867	−0.401605
$345$	−1.59086	−0.0856489
$346$	11.0714	0.595204
$347$	26.0711	1.39957	0.699784	−	0.714354i	$-0.253279\pi$
$347$	26.0711	1.39957	0.699784	+	0.714354i	$0.253279\pi$
$348$	−1.00000	−0.0536056
$349$	−17.4269	−0.932842	−0.466421	−	0.884563i	$-0.654457\pi$
$349$	−17.4269	−0.932842	−0.466421	+	0.884563i	$0.654457\pi$
$350$	−2.98019	−0.159298
$351$	1.05524	0.0563247
$352$	4.80384	0.256045
$353$	19.0929	1.01621	0.508107	−	0.861294i	$-0.330345\pi$
$353$	19.0929	1.01621	0.508107	+	0.861294i	$0.330345\pi$
$354$	2.23696	0.118893
$355$	−11.9123	−0.632242
$356$	13.3089	0.705370
$357$	−2.89031	−0.152971
$358$	7.48516	0.395603
$359$	−27.4665	−1.44963	−0.724814	−	0.688945i	$-0.758074\pi$
$359$	−27.4665	−1.44963	−0.724814	+	0.688945i	$0.758074\pi$
$360$	−1.59086	−0.0838456
$361$	21.2825	1.12013
$362$	−19.7635	−1.03875
$363$	12.0768	0.633870
$364$	1.27364	0.0667568
$365$	5.41729	0.283554
$366$	−5.98555	−0.312870
$367$	−11.8549	−0.618823	−0.309411	−	0.950928i	$-0.600132\pi$
$367$	−11.8549	−0.618823	−0.309411	+	0.950928i	$0.600132\pi$
$368$	1.00000	0.0521286
$369$	1.69372	0.0881715
$370$	−12.2548	−0.637096
$371$	2.32570	0.120745
$372$	−5.20570	−0.269903
$373$	12.0350	0.623148	0.311574	−	0.950222i	$-0.399144\pi$
$373$	12.0350	0.623148	0.311574	+	0.950222i	$0.399144\pi$
$374$	−11.5037	−0.594843
$375$	11.8824	0.613604
$376$	8.28354	0.427191
$377$	−1.05524	−0.0543478
$378$	1.20696	0.0620795
$379$	−21.3092	−1.09458	−0.547291	−	0.836942i	$-0.684341\pi$
$379$	−21.3092	−1.09458	−0.547291	+	0.836942i	$0.684341\pi$
$380$	−10.0969	−0.517962
$381$	12.3449	0.632449
$382$	−20.8384	−1.06618
$383$	−17.8671	−0.912965	−0.456483	−	0.889732i	$-0.650891\pi$
$383$	−17.8671	−0.912965	−0.456483	+	0.889732i	$0.650891\pi$
$384$	1.00000	0.0510310
$385$	−9.22388	−0.470092
$386$	10.1999	0.519160
$387$	−7.44867	−0.378637
$388$	−17.6029	−0.893652
$389$	−38.1275	−1.93314	−0.966570	−	0.256403i	$-0.917463\pi$
$389$	−38.1275	−1.93314	−0.966570	+	0.256403i	$0.917463\pi$
$390$	−1.67874	−0.0850064
$391$	−2.39469	−0.121105
$392$	−5.54324	−0.279976
$393$	−3.25228	−0.164056
$394$	−13.5343	−0.681849
$395$	−27.0641	−1.36174
$396$	4.80384	0.241402
$397$	−33.0210	−1.65728	−0.828638	−	0.559785i	$-0.810884\pi$
$397$	−33.0210	−1.65728	−0.828638	+	0.559785i	$0.810884\pi$
$398$	2.66897	0.133783
$399$	7.66041	0.383500
$400$	−2.46917	−0.123458
$401$	21.3461	1.06597	0.532986	−	0.846124i	$-0.321070\pi$
$401$	21.3461	1.06597	0.532986	+	0.846124i	$0.321070\pi$
$402$	1.48037	0.0738343
$403$	−5.49328	−0.273640
$404$	−9.38028	−0.466686
$405$	−1.59086	−0.0790504
$406$	−1.20696	−0.0599006
$407$	37.0052	1.83428
$408$	−2.39469	−0.118555
$409$	−29.6900	−1.46808	−0.734039	−	0.679107i	$-0.762367\pi$
$409$	−29.6900	−1.46808	−0.734039	+	0.679107i	$0.762367\pi$
$410$	−2.69447	−0.133070
$411$	0.424048	0.0209168
$412$	7.72810	0.380736
$413$	2.69993	0.132855
$414$	1.00000	0.0491473
$415$	−7.09805	−0.348430
$416$	1.05524	0.0517376
$417$	14.4460	0.707425
$418$	30.4892	1.49128
$419$	−39.5696	−1.93310	−0.966550	−	0.256478i	$-0.917438\pi$
$419$	−39.5696	−1.93310	−0.966550	+	0.256478i	$0.917438\pi$
$420$	−1.92011	−0.0936916
$421$	10.7814	0.525454	0.262727	−	0.964870i	$-0.415378\pi$
$421$	10.7814	0.525454	0.262727	+	0.964870i	$0.415378\pi$
$422$	27.2127	1.32469
$423$	8.28354	0.402760
$424$	1.92691	0.0935789
$425$	5.91291	0.286818
$426$	7.48800	0.362795
$427$	−7.22434	−0.349610
$428$	−1.73541	−0.0838843
$429$	5.06922	0.244744
$430$	11.8498	0.571447
$431$	−35.0561	−1.68859	−0.844297	−	0.535875i	$-0.819982\pi$
$431$	−35.0561	−1.68859	−0.844297	+	0.535875i	$0.819982\pi$
$432$	1.00000	0.0481125
$433$	33.9279	1.63047	0.815236	−	0.579129i	$-0.196607\pi$
$433$	33.9279	1.63047	0.815236	+	0.579129i	$0.196607\pi$
$434$	−6.28308	−0.301598
$435$	1.59086	0.0762758
$436$	−12.9689	−0.621096
$437$	6.34685	0.303611
$438$	−3.40526	−0.162710
$439$	−10.3682	−0.494847	−0.247424	−	0.968907i	$-0.579584\pi$
$439$	−10.3682	−0.494847	−0.247424	+	0.968907i	$0.579584\pi$
$440$	−7.64222	−0.364329
$441$	−5.54324	−0.263964
$442$	−2.52699	−0.120196
$443$	−1.43888	−0.0683634	−0.0341817	−	0.999416i	$-0.510882\pi$
$443$	−1.43888	−0.0683634	−0.0341817	+	0.999416i	$0.510882\pi$
$444$	7.70325	0.365580
$445$	−21.1726	−1.00368
$446$	−15.2739	−0.723239
$447$	−8.70747	−0.411849
$448$	1.20696	0.0570236
$449$	10.7425	0.506972	0.253486	−	0.967339i	$-0.418423\pi$
$449$	10.7425	0.506972	0.253486	+	0.967339i	$0.418423\pi$
$450$	−2.46917	−0.116398
$451$	8.13635	0.383126
$452$	6.19229	0.291261
$453$	4.82785	0.226832
$454$	−14.4356	−0.677495
$455$	−2.02618	−0.0949888
$456$	6.34685	0.297218
$457$	−23.7209	−1.10962	−0.554809	−	0.831978i	$-0.687209\pi$
$457$	−23.7209	−1.10962	−0.554809	+	0.831978i	$0.687209\pi$
$458$	−18.9412	−0.885064
$459$	−2.39469	−0.111775
$460$	−1.59086	−0.0741742
$461$	−16.6544	−0.775672	−0.387836	−	0.921728i	$-0.626777\pi$
$461$	−16.6544	−0.775672	−0.387836	+	0.921728i	$0.626777\pi$
$462$	5.79805	0.269750
$463$	−19.1352	−0.889289	−0.444645	−	0.895707i	$-0.646670\pi$
$463$	−19.1352	−0.889289	−0.444645	+	0.895707i	$0.646670\pi$
$464$	−1.00000	−0.0464238
$465$	8.28153	0.384047
$466$	−9.39059	−0.435011
$467$	8.81835	0.408064	0.204032	−	0.978964i	$-0.434595\pi$
$467$	8.81835	0.408064	0.204032	+	0.978964i	$0.434595\pi$
$468$	1.05524	0.0487786
$469$	1.78675	0.0825046
$470$	−13.1779	−0.607853
$471$	−11.1263	−0.512674
$472$	2.23696	0.102964
$473$	−35.7822	−1.64527
$474$	17.0123	0.781399
$475$	−15.6714	−0.719055
$476$	−2.89031	−0.132477
$477$	1.92691	0.0882270
$478$	7.67262	0.350938
$479$	19.8699	0.907878	0.453939	−	0.891033i	$-0.350018\pi$
$479$	19.8699	0.907878	0.453939	+	0.891033i	$0.350018\pi$
$480$	−1.59086	−0.0726124
$481$	8.12880	0.370642
$482$	−14.7332	−0.671077
$483$	1.20696	0.0549187
$484$	12.0768	0.548947
$485$	28.0037	1.27158
$486$	1.00000	0.0453609
$487$	−31.1420	−1.41118	−0.705590	−	0.708620i	$-0.749318\pi$
$487$	−31.1420	−1.41118	−0.705590	+	0.708620i	$0.749318\pi$
$488$	−5.98555	−0.270953
$489$	12.1265	0.548378
$490$	8.81851	0.398380
$491$	−3.52322	−0.159001	−0.0795003	−	0.996835i	$-0.525332\pi$
$491$	−3.52322	−0.159001	−0.0795003	+	0.996835i	$0.525332\pi$
$492$	1.69372	0.0763588
$493$	2.39469	0.107852
$494$	6.69747	0.301333
$495$	−7.64222	−0.343492
$496$	−5.20570	−0.233743
$497$	9.03773	0.405398
$498$	4.46177	0.199937
$499$	6.16824	0.276128	0.138064	−	0.990423i	$-0.455912\pi$
$499$	6.16824	0.276128	0.138064	+	0.990423i	$0.455912\pi$
$500$	11.8824	0.531397
$501$	13.3094	0.594622
$502$	23.7959	1.06206
$503$	2.30315	0.102692	0.0513461	−	0.998681i	$-0.483649\pi$
$503$	2.30315	0.102692	0.0513461	+	0.998681i	$0.483649\pi$
$504$	1.20696	0.0537624
$505$	14.9227	0.664051
$506$	4.80384	0.213557
$507$	−11.8865	−0.527896
$508$	12.3449	0.547717
$509$	8.68767	0.385074	0.192537	−	0.981290i	$-0.438328\pi$
$509$	8.68767	0.385074	0.192537	+	0.981290i	$0.438328\pi$
$510$	3.80962	0.168693
$511$	−4.11003	−0.181817
$512$	1.00000	0.0441942
$513$	6.34685	0.280220
$514$	16.0774	0.709142
$515$	−12.2943	−0.541752
$516$	−7.44867	−0.327909
$517$	39.7928	1.75008
$518$	9.29754	0.408510
$519$	11.0714	0.485982
$520$	−1.67874	−0.0736177
$521$	−42.2502	−1.85102	−0.925508	−	0.378728i	$-0.876362\pi$
$521$	−42.2502	−1.85102	−0.925508	+	0.378728i	$0.876362\pi$
$522$	−1.00000	−0.0437688
$523$	37.8442	1.65481	0.827406	−	0.561604i	$-0.189816\pi$
$523$	37.8442	1.65481	0.827406	+	0.561604i	$0.189816\pi$
$524$	−3.25228	−0.142076
$525$	−2.98019	−0.130066
$526$	1.38492	0.0603853
$527$	12.4661	0.543030
$528$	4.80384	0.209060
$529$	1.00000	0.0434783
$530$	−3.06544	−0.133154
$531$	2.23696	0.0970758
$532$	7.66041	0.332121
$533$	1.78729	0.0774160
$534$	13.3089	0.575932
$535$	2.76080	0.119360
$536$	1.48037	0.0639423
$537$	7.48516	0.323008
$538$	9.11781	0.393097
$539$	−26.6288	−1.14698
$540$	−1.59086	−0.0684597
$541$	−18.4794	−0.794491	−0.397245	−	0.917712i	$-0.630034\pi$
$541$	−18.4794	−0.794491	−0.397245	+	0.917712i	$0.630034\pi$
$542$	7.45184	0.320084
$543$	−19.7635	−0.848132
$544$	−2.39469	−0.102672
$545$	20.6316	0.883762
$546$	1.27364	0.0545067
$547$	5.54813	0.237221	0.118610	−	0.992941i	$-0.462156\pi$
$547$	5.54813	0.237221	0.118610	+	0.992941i	$0.462156\pi$
$548$	0.424048	0.0181144
$549$	−5.98555	−0.255457
$550$	−11.8615	−0.505775
$551$	−6.34685	−0.270385
$552$	1.00000	0.0425628
$553$	20.5332	0.873159
$554$	−16.9050	−0.718226
$555$	−12.2548	−0.520187
$556$	14.4460	0.612648
$557$	−39.2163	−1.66165	−0.830825	−	0.556534i	$-0.812131\pi$
$557$	−39.2163	−1.66165	−0.830825	+	0.556534i	$0.812131\pi$
$558$	−5.20570	−0.220375
$559$	−7.86016	−0.332449
$560$	−1.92011	−0.0811393
$561$	−11.5037	−0.485687
$562$	10.4939	0.442657
$563$	40.0754	1.68898	0.844489	−	0.535573i	$-0.179904\pi$
$563$	40.0754	1.68898	0.844489	+	0.535573i	$0.179904\pi$
$564$	8.28354	0.348800
$565$	−9.85105	−0.414437
$566$	−24.5520	−1.03200
$567$	1.20696	0.0506877
$568$	7.48800	0.314189
$569$	−12.8124	−0.537125	−0.268563	−	0.963262i	$-0.586549\pi$
$569$	−12.8124	−0.537125	−0.268563	+	0.963262i	$0.586549\pi$
$570$	−10.0969	−0.422914
$571$	32.4189	1.35669	0.678344	−	0.734744i	$-0.262698\pi$
$571$	32.4189	1.35669	0.678344	+	0.734744i	$0.262698\pi$
$572$	5.06922	0.211955
$573$	−20.8384	−0.870535
$574$	2.04426	0.0853256
$575$	−2.46917	−0.102971
$576$	1.00000	0.0416667
$577$	16.1546	0.672525	0.336263	−	0.941768i	$-0.390837\pi$
$577$	16.1546	0.672525	0.336263	+	0.941768i	$0.390837\pi$
$578$	−11.2654	−0.468580
$579$	10.1999	0.423892
$580$	1.59086	0.0660568
$581$	5.38519	0.223416
$582$	−17.6029	−0.729664
$583$	9.25654	0.383367
$584$	−3.40526	−0.140911
$585$	−1.67874	−0.0694075
$586$	−26.7409	−1.10466
$587$	35.2192	1.45365	0.726825	−	0.686823i	$-0.240995\pi$
$587$	35.2192	1.45365	0.726825	+	0.686823i	$0.240995\pi$
$588$	−5.54324	−0.228599
$589$	−33.0398	−1.36138
$590$	−3.55869	−0.146509
$591$	−13.5343	−0.556727
$592$	7.70325	0.316602
$593$	15.1010	0.620126	0.310063	−	0.950716i	$-0.399650\pi$
$593$	15.1010	0.620126	0.310063	+	0.950716i	$0.399650\pi$
$594$	4.80384	0.197104
$595$	4.59807	0.188503
$596$	−8.70747	−0.356672
$597$	2.66897	0.109234
$598$	1.05524	0.0431521
$599$	−9.54660	−0.390064	−0.195032	−	0.980797i	$-0.562481\pi$
$599$	−9.54660	−0.390064	−0.195032	+	0.980797i	$0.562481\pi$
$600$	−2.46917	−0.100803
$601$	−22.0556	−0.899665	−0.449832	−	0.893113i	$-0.648516\pi$
$601$	−22.0556	−0.899665	−0.449832	+	0.893113i	$0.648516\pi$
$602$	−8.99027	−0.366416
$603$	1.48037	0.0602854
$604$	4.82785	0.196442
$605$	−19.2125	−0.781101
$606$	−9.38028	−0.381048
$607$	−2.12482	−0.0862440	−0.0431220	−	0.999070i	$-0.513730\pi$
$607$	−2.12482	−0.0862440	−0.0431220	+	0.999070i	$0.513730\pi$
$608$	6.34685	0.257399
$609$	−1.20696	−0.0489086
$610$	9.52217	0.385541
$611$	8.74115	0.353629
$612$	−2.39469	−0.0967998
$613$	37.3325	1.50785	0.753923	−	0.656963i	$-0.228159\pi$
$613$	37.3325	1.50785	0.753923	+	0.656963i	$0.228159\pi$
$614$	−1.51966	−0.0613285
$615$	−2.69447	−0.108651
$616$	5.79805	0.233610
$617$	36.2681	1.46010	0.730048	−	0.683395i	$-0.239497\pi$
$617$	36.2681	1.46010	0.730048	+	0.683395i	$0.239497\pi$
$618$	7.72810	0.310870
$619$	−14.3193	−0.575541	−0.287770	−	0.957699i	$-0.592914\pi$
$619$	−14.3193	−0.575541	−0.287770	+	0.957699i	$0.592914\pi$
$620$	8.28153	0.332594
$621$	1.00000	0.0401286
$622$	3.00344	0.120427
$623$	16.0633	0.643564
$624$	1.05524	0.0422435
$625$	−6.55736	−0.262294
$626$	−17.2575	−0.689749
$627$	30.4892	1.21762
$628$	−11.1263	−0.443989
$629$	−18.4469	−0.735528
$630$	−1.92011	−0.0764989
$631$	−22.8332	−0.908976	−0.454488	−	0.890753i	$-0.650178\pi$
$631$	−22.8332	−0.908976	−0.454488	+	0.890753i	$0.650178\pi$
$632$	17.0123	0.676712
$633$	27.2127	1.08161
$634$	−13.2800	−0.527417
$635$	−19.6390	−0.779350
$636$	1.92691	0.0764068
$637$	−5.84947	−0.231764
$638$	−4.80384	−0.190186
$639$	7.48800	0.296221
$640$	−1.59086	−0.0628842
$641$	−7.41546	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$641$	−7.41546	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$642$	−1.73541	−0.0684913
$643$	−19.0694	−0.752022	−0.376011	−	0.926615i	$-0.622704\pi$
$643$	−19.0694	−0.752022	−0.376011	+	0.926615i	$0.622704\pi$
$644$	1.20696	0.0475610
$645$	11.8498	0.466585
$646$	−15.1988	−0.597987
$647$	−2.66801	−0.104890	−0.0524452	−	0.998624i	$-0.516701\pi$
$647$	−2.66801	−0.104890	−0.0524452	+	0.998624i	$0.516701\pi$
$648$	1.00000	0.0392837
$649$	10.7460	0.421817
$650$	−2.60557	−0.102199
$651$	−6.28308	−0.246253
$652$	12.1265	0.474909
$653$	20.5941	0.805909	0.402954	−	0.915220i	$-0.367983\pi$
$653$	20.5941	0.805909	0.402954	+	0.915220i	$0.367983\pi$
$654$	−12.9689	−0.507123
$655$	5.17391	0.202162
$656$	1.69372	0.0661286
$657$	−3.40526	−0.132852
$658$	9.99793	0.389760
$659$	5.84398	0.227649	0.113825	−	0.993501i	$-0.463690\pi$
$659$	5.84398	0.227649	0.113825	+	0.993501i	$0.463690\pi$
$660$	−7.64222	−0.297473
$661$	9.59935	0.373371	0.186686	−	0.982420i	$-0.440225\pi$
$661$	9.59935	0.373371	0.186686	+	0.982420i	$0.440225\pi$
$662$	−9.94877	−0.386670
$663$	−2.52699	−0.0981400
$664$	4.46177	0.173150
$665$	−12.1866	−0.472577
$666$	7.70325	0.298495
$667$	−1.00000	−0.0387202
$668$	13.3094	0.514957
$669$	−15.2739	−0.590522
$670$	−2.35506	−0.0909840
$671$	−28.7536	−1.11002
$672$	1.20696	0.0465596
$673$	−22.8303	−0.880045	−0.440022	−	0.897987i	$-0.645029\pi$
$673$	−22.8303	−0.880045	−0.440022	+	0.897987i	$0.645029\pi$
$674$	29.8659	1.15039
$675$	−2.46917	−0.0950384
$676$	−11.8865	−0.457172
$677$	−13.3605	−0.513487	−0.256743	−	0.966480i	$-0.582649\pi$
$677$	−13.3605	−0.513487	−0.256743	+	0.966480i	$0.582649\pi$
$678$	6.19229	0.237813
$679$	−21.2460	−0.815348
$680$	3.80962	0.146092
$681$	−14.4356	−0.553172
$682$	−25.0073	−0.957579
$683$	−20.7352	−0.793409	−0.396705	−	0.917946i	$-0.629846\pi$
$683$	−20.7352	−0.793409	−0.396705	+	0.917946i	$0.629846\pi$
$684$	6.34685	0.242678
$685$	−0.674601	−0.0257752
$686$	−15.1392	−0.578018
$687$	−18.9412	−0.722652
$688$	−7.44867	−0.283978
$689$	2.03336	0.0774647
$690$	−1.59086	−0.0605630
$691$	−1.57135	−0.0597769	−0.0298884	−	0.999553i	$-0.509515\pi$
$691$	−1.57135	−0.0597769	−0.0298884	+	0.999553i	$0.509515\pi$
$692$	11.0714	0.420872
$693$	5.79805	0.220250
$694$	26.0711	0.989645
$695$	−22.9816	−0.871741
$696$	−1.00000	−0.0379049
$697$	−4.05594	−0.153630
$698$	−17.4269	−0.659619
$699$	−9.39059	−0.355185
$700$	−2.98019	−0.112641
$701$	−33.5536	−1.26730	−0.633652	−	0.773618i	$-0.718445\pi$
$701$	−33.5536	−1.26730	−0.633652	+	0.773618i	$0.718445\pi$
$702$	1.05524	0.0398276
$703$	48.8914	1.84397
$704$	4.80384	0.181051
$705$	−13.1779	−0.496310
$706$	19.0929	0.718572
$707$	−11.3216	−0.425794
$708$	2.23696	0.0840701
$709$	9.47066	0.355678	0.177839	−	0.984060i	$-0.443089\pi$
$709$	9.47066	0.355678	0.177839	+	0.984060i	$0.443089\pi$
$710$	−11.9123	−0.447062
$711$	17.0123	0.638010
$712$	13.3089	0.498772
$713$	−5.20570	−0.194955
$714$	−2.89031	−0.108167
$715$	−8.06441	−0.301592
$716$	7.48516	0.279733
$717$	7.67262	0.286539
$718$	−27.4665	−1.02504
$719$	−30.9345	−1.15366	−0.576830	−	0.816864i	$-0.695711\pi$
$719$	−30.9345	−1.15366	−0.576830	+	0.816864i	$0.695711\pi$
$720$	−1.59086	−0.0592878
$721$	9.32752	0.347375
$722$	21.2825	0.792051
$723$	−14.7332	−0.547932
$724$	−19.7635	−0.734504
$725$	2.46917	0.0917026
$726$	12.0768	0.448214
$727$	−36.6587	−1.35959	−0.679797	−	0.733400i	$-0.737932\pi$
$727$	−36.6587	−1.35959	−0.679797	+	0.733400i	$0.737932\pi$
$728$	1.27364	0.0472042
$729$	1.00000	0.0370370
$730$	5.41729	0.200503
$731$	17.8373	0.659736
$732$	−5.98555	−0.221232
$733$	37.2240	1.37490	0.687449	−	0.726232i	$-0.258730\pi$
$733$	37.2240	1.37490	0.687449	+	0.726232i	$0.258730\pi$
$734$	−11.8549	−0.437574
$735$	8.81851	0.325276
$736$	1.00000	0.0368605
$737$	7.11146	0.261954
$738$	1.69372	0.0623467
$739$	10.9517	0.402864	0.201432	−	0.979502i	$-0.435440\pi$
$739$	10.9517	0.402864	0.201432	+	0.979502i	$0.435440\pi$
$740$	−12.2548	−0.450495
$741$	6.69747	0.246038
$742$	2.32570	0.0853793
$743$	18.3969	0.674916	0.337458	−	0.941341i	$-0.390433\pi$
$743$	18.3969	0.674916	0.337458	+	0.941341i	$0.390433\pi$
$744$	−5.20570	−0.190850
$745$	13.8524	0.507511
$746$	12.0350	0.440632
$747$	4.46177	0.163248
$748$	−11.5037	−0.420618
$749$	−2.09458	−0.0765342
$750$	11.8824	0.433884
$751$	38.3660	1.40000	0.699998	−	0.714145i	$-0.253184\pi$
$751$	38.3660	1.40000	0.699998	+	0.714145i	$0.253184\pi$
$752$	8.28354	0.302070
$753$	23.7959	0.867169
$754$	−1.05524	−0.0384297
$755$	−7.68043	−0.279519
$756$	1.20696	0.0438968
$757$	−33.1391	−1.20446	−0.602231	−	0.798322i	$-0.705721\pi$
$757$	−33.1391	−1.20446	−0.602231	+	0.798322i	$0.705721\pi$
$758$	−21.3092	−0.773987
$759$	4.80384	0.174368
$760$	−10.0969	−0.366254
$761$	30.8883	1.11970	0.559850	−	0.828594i	$-0.310859\pi$
$761$	30.8883	1.11970	0.559850	+	0.828594i	$0.310859\pi$
$762$	12.3449	0.447209
$763$	−15.6529	−0.566674
$764$	−20.8384	−0.753905
$765$	3.80962	0.137737
$766$	−17.8671	−0.645564
$767$	2.36054	0.0852341
$768$	1.00000	0.0360844
$769$	−31.6360	−1.14082	−0.570411	−	0.821359i	$-0.693216\pi$
$769$	−31.6360	−1.14082	−0.570411	+	0.821359i	$0.693216\pi$
$770$	−9.22388	−0.332406
$771$	16.0774	0.579012
$772$	10.1999	0.367102
$773$	−15.8636	−0.570573	−0.285286	−	0.958442i	$-0.592089\pi$
$773$	−15.8636	−0.570573	−0.285286	+	0.958442i	$0.592089\pi$
$774$	−7.44867	−0.267737
$775$	12.8537	0.461720
$776$	−17.6029	−0.631907
$777$	9.29754	0.333547
$778$	−38.1275	−1.36694
$779$	10.7498	0.385151
$780$	−1.67874	−0.0601086
$781$	35.9711	1.28715
$782$	−2.39469	−0.0856341
$783$	−1.00000	−0.0357371
$784$	−5.54324	−0.197973
$785$	17.7004	0.631755
$786$	−3.25228	−0.116005
$787$	−1.29570	−0.0461867	−0.0230934	−	0.999733i	$-0.507351\pi$
$787$	−1.29570	−0.0461867	−0.0230934	+	0.999733i	$0.507351\pi$
$788$	−13.5343	−0.482140
$789$	1.38492	0.0493044
$790$	−27.0641	−0.962898
$791$	7.47386	0.265740
$792$	4.80384	0.170697
$793$	−6.31621	−0.224295
$794$	−33.0210	−1.17187
$795$	−3.06544	−0.108720
$796$	2.66897	0.0945990
$797$	26.5264	0.939615	0.469807	−	0.882769i	$-0.344323\pi$
$797$	26.5264	0.939615	0.469807	+	0.882769i	$0.344323\pi$
$798$	7.66041	0.271176
$799$	−19.8366	−0.701767
$800$	−2.46917	−0.0872983
$801$	13.3089	0.470247
$802$	21.3461	0.753756
$803$	−16.3583	−0.577273
$804$	1.48037	0.0522087
$805$	−1.92011	−0.0676749
$806$	−5.49328	−0.193492
$807$	9.11781	0.320962
$808$	−9.38028	−0.329997
$809$	53.3644	1.87619	0.938097	−	0.346373i	$-0.112587\pi$
$809$	53.3644	1.87619	0.938097	+	0.346373i	$0.112587\pi$
$810$	−1.59086	−0.0558971
$811$	18.2669	0.641439	0.320720	−	0.947174i	$-0.396075\pi$
$811$	18.2669	0.641439	0.320720	+	0.947174i	$0.396075\pi$
$812$	−1.20696	−0.0423561
$813$	7.45184	0.261348
$814$	37.0052	1.29703
$815$	−19.2915	−0.675752
$816$	−2.39469	−0.0838311
$817$	−47.2756	−1.65396
$818$	−29.6900	−1.03809
$819$	1.27364	0.0445046
$820$	−2.69447	−0.0940949
$821$	−4.78816	−0.167108	−0.0835539	−	0.996503i	$-0.526627\pi$
$821$	−4.78816	−0.167108	−0.0835539	+	0.996503i	$0.526627\pi$
$822$	0.424048	0.0147904
$823$	33.7492	1.17642	0.588212	−	0.808707i	$-0.299832\pi$
$823$	33.7492	1.17642	0.588212	+	0.808707i	$0.299832\pi$
$824$	7.72810	0.269221
$825$	−11.8615	−0.412964
$826$	2.69993	0.0939425
$827$	6.84674	0.238084	0.119042	−	0.992889i	$-0.462018\pi$
$827$	6.84674	0.238084	0.119042	+	0.992889i	$0.462018\pi$
$828$	1.00000	0.0347524
$829$	−28.9719	−1.00624	−0.503119	−	0.864217i	$-0.667814\pi$
$829$	−28.9719	−1.00624	−0.503119	+	0.864217i	$0.667814\pi$
$830$	−7.09805	−0.246377
$831$	−16.9050	−0.586429
$832$	1.05524	0.0365840
$833$	13.2744	0.459930
$834$	14.4460	0.500225
$835$	−21.1734	−0.732737
$836$	30.4892	1.05449
$837$	−5.20570	−0.179935
$838$	−39.5696	−1.36691
$839$	−4.11403	−0.142032	−0.0710160	−	0.997475i	$-0.522624\pi$
$839$	−4.11403	−0.142032	−0.0710160	+	0.997475i	$0.522624\pi$
$840$	−1.92011	−0.0662500
$841$	1.00000	0.0344828
$842$	10.7814	0.371552
$843$	10.4939	0.361428
$844$	27.2127	0.936699
$845$	18.9097	0.650513
$846$	8.28354	0.284794
$847$	14.5763	0.500848
$848$	1.92691	0.0661702
$849$	−24.5520	−0.842624
$850$	5.91291	0.202811
$851$	7.70325	0.264064
$852$	7.48800	0.256535
$853$	−47.4368	−1.62420	−0.812102	−	0.583515i	$-0.801677\pi$
$853$	−47.4368	−1.62420	−0.812102	+	0.583515i	$0.801677\pi$
$854$	−7.22434	−0.247212
$855$	−10.0969	−0.345308
$856$	−1.73541	−0.0593152
$857$	−5.46401	−0.186647	−0.0933235	−	0.995636i	$-0.529749\pi$
$857$	−5.46401	−0.186647	−0.0933235	+	0.995636i	$0.529749\pi$
$858$	5.06922	0.173060
$859$	28.4437	0.970486	0.485243	−	0.874379i	$-0.338731\pi$
$859$	28.4437	0.970486	0.485243	+	0.874379i	$0.338731\pi$
$860$	11.8498	0.404074
$861$	2.04426	0.0696681
$862$	−35.0561	−1.19402
$863$	36.4237	1.23988	0.619939	−	0.784650i	$-0.287157\pi$
$863$	36.4237	1.23988	0.619939	+	0.784650i	$0.287157\pi$
$864$	1.00000	0.0340207
$865$	−17.6131	−0.598862
$866$	33.9279	1.15292
$867$	−11.2654	−0.382594
$868$	−6.28308	−0.213262
$869$	81.7242	2.77230
$870$	1.59086	0.0539352
$871$	1.56215	0.0529315
$872$	−12.9689	−0.439181
$873$	−17.6029	−0.595768
$874$	6.34685	0.214685
$875$	14.3416	0.484835
$876$	−3.40526	−0.115053
$877$	8.33781	0.281548	0.140774	−	0.990042i	$-0.455041\pi$
$877$	8.33781	0.281548	0.140774	+	0.990042i	$0.455041\pi$
$878$	−10.3682	−0.349910
$879$	−26.7409	−0.901947
$880$	−7.64222	−0.257619
$881$	−37.7328	−1.27125	−0.635625	−	0.771998i	$-0.719257\pi$
$881$	−37.7328	−1.27125	−0.635625	+	0.771998i	$0.719257\pi$
$882$	−5.54324	−0.186651
$883$	8.89138	0.299219	0.149609	−	0.988745i	$-0.452198\pi$
$883$	8.89138	0.299219	0.149609	+	0.988745i	$0.452198\pi$
$884$	−2.52699	−0.0849917
$885$	−3.55869	−0.119624
$886$	−1.43888	−0.0483402
$887$	−24.2707	−0.814931	−0.407466	−	0.913221i	$-0.633587\pi$
$887$	−24.2707	−0.814931	−0.407466	+	0.913221i	$0.633587\pi$
$888$	7.70325	0.258504
$889$	14.8999	0.499725
$890$	−21.1726	−0.709707
$891$	4.80384	0.160935
$892$	−15.2739	−0.511407
$893$	52.5744	1.75933
$894$	−8.70747	−0.291221
$895$	−11.9078	−0.398035
$896$	1.20696	0.0403218
$897$	1.05524	0.0352335
$898$	10.7425	0.358483
$899$	5.20570	0.173620
$900$	−2.46917	−0.0823056
$901$	−4.61435	−0.153726
$902$	8.13635	0.270911
$903$	−8.99027	−0.299177
$904$	6.19229	0.205952
$905$	31.4409	1.04513
$906$	4.82785	0.160395
$907$	10.7068	0.355514	0.177757	−	0.984074i	$-0.443116\pi$
$907$	10.7068	0.355514	0.177757	+	0.984074i	$0.443116\pi$
$908$	−14.4356	−0.479061
$909$	−9.38028	−0.311124
$910$	−2.02618	−0.0671672
$911$	56.3103	1.86564	0.932822	−	0.360338i	$-0.117339\pi$
$911$	56.3103	1.86564	0.932822	+	0.360338i	$0.117339\pi$
$912$	6.34685	0.210165
$913$	21.4336	0.709350
$914$	−23.7209	−0.784618
$915$	9.52217	0.314793
$916$	−18.9412	−0.625835
$917$	−3.92538	−0.129627
$918$	−2.39469	−0.0790367
$919$	−45.9712	−1.51645	−0.758226	−	0.651992i	$-0.773933\pi$
$919$	−45.9712	−1.51645	−0.758226	+	0.651992i	$0.773933\pi$
$920$	−1.59086	−0.0524491
$921$	−1.51966	−0.0500745
$922$	−16.6544	−0.548483
$923$	7.90166	0.260086
$924$	5.79805	0.190742
$925$	−19.0206	−0.625394
$926$	−19.1352	−0.628823
$927$	7.72810	0.253824
$928$	−1.00000	−0.0328266
$929$	−5.92869	−0.194514	−0.0972570	−	0.995259i	$-0.531007\pi$
$929$	−5.92869	−0.194514	−0.0972570	+	0.995259i	$0.531007\pi$
$930$	8.28153	0.271562
$931$	−35.1821	−1.15305
$932$	−9.39059	−0.307599
$933$	3.00344	0.0983282
$934$	8.81835	0.288545
$935$	18.3008	0.598500
$936$	1.05524	0.0344917
$937$	−44.4877	−1.45335	−0.726675	−	0.686981i	$-0.758936\pi$
$937$	−44.4877	−1.45335	−0.726675	+	0.686981i	$0.758936\pi$
$938$	1.78675	0.0583396
$939$	−17.2575	−0.563178
$940$	−13.1779	−0.429817
$941$	40.1759	1.30970	0.654848	−	0.755760i	$-0.272733\pi$
$941$	40.1759	1.30970	0.654848	+	0.755760i	$0.272733\pi$
$942$	−11.1263	−0.362516
$943$	1.69372	0.0551551
$944$	2.23696	0.0728069
$945$	−1.92011	−0.0624611
$946$	−35.7822	−1.16338
$947$	20.6464	0.670917	0.335459	−	0.942055i	$-0.391109\pi$
$947$	20.6464	0.670917	0.335459	+	0.942055i	$0.391109\pi$
$948$	17.0123	0.552533
$949$	−3.59338	−0.116646
$950$	−15.6714	−0.508449
$951$	−13.2800	−0.430634
$952$	−2.89031	−0.0936754
$953$	22.9847	0.744548	0.372274	−	0.928123i	$-0.378578\pi$
$953$	22.9847	0.744548	0.372274	+	0.928123i	$0.378578\pi$
$954$	1.92691	0.0623859
$955$	33.1509	1.07274
$956$	7.67262	0.248150
$957$	−4.80384	−0.155286
$958$	19.8699	0.641967
$959$	0.511810	0.0165272
$960$	−1.59086	−0.0513447
$961$	−3.90073	−0.125830
$962$	8.12880	0.262083
$963$	−1.73541	−0.0559229
$964$	−14.7332	−0.474523
$965$	−16.2266	−0.522351
$966$	1.20696	0.0388334
$967$	−4.40489	−0.141652	−0.0708259	−	0.997489i	$-0.522563\pi$
$967$	−4.40489	−0.141652	−0.0708259	+	0.997489i	$0.522563\pi$
$968$	12.0768	0.388164
$969$	−15.1988	−0.488255
$970$	28.0037	0.899145
$971$	3.56224	0.114318	0.0571588	−	0.998365i	$-0.481796\pi$
$971$	3.56224	0.114318	0.0571588	+	0.998365i	$0.481796\pi$
$972$	1.00000	0.0320750
$973$	17.4358	0.558967
$974$	−31.1420	−0.997855
$975$	−2.60557	−0.0834451
$976$	−5.98555	−0.191593
$977$	−3.80343	−0.121683	−0.0608413	−	0.998147i	$-0.519378\pi$
$977$	−3.80343	−0.121683	−0.0608413	+	0.998147i	$0.519378\pi$
$978$	12.1265	0.387762
$979$	63.9338	2.04333
$980$	8.81851	0.281697
$981$	−12.9689	−0.414064
$982$	−3.52322	−0.112430
$983$	−16.8805	−0.538404	−0.269202	−	0.963084i	$-0.586760\pi$
$983$	−16.8805	−0.538404	−0.269202	+	0.963084i	$0.586760\pi$
$984$	1.69372	0.0539938
$985$	21.5312	0.686040
$986$	2.39469	0.0762626
$987$	9.99793	0.318238
$988$	6.69747	0.213075
$989$	−7.44867	−0.236854
$990$	−7.64222	−0.242886
$991$	−30.0552	−0.954734	−0.477367	−	0.878704i	$-0.658409\pi$
$991$	−30.0552	−0.954734	−0.477367	+	0.878704i	$0.658409\pi$
$992$	−5.20570	−0.165281
$993$	−9.94877	−0.315715
$994$	9.03773	0.286660
$995$	−4.24595	−0.134606
$996$	4.46177	0.141377
$997$	−51.9382	−1.64490	−0.822450	−	0.568838i	$-0.807393\pi$
$997$	−51.9382	−1.64490	−0.822450	+	0.568838i	$0.807393\pi$
$998$	6.16824	0.195252
$999$	7.70325	0.243720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bh.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bh.1.3	✓	7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.59086 q^{5} +1.00000 q^{6} +1.20696 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.59086 q^{5} +1.00000 q^{6} +1.20696 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.59086 q^{10} +4.80384 q^{11} +1.00000 q^{12} +1.05524 q^{13} +1.20696 q^{14} -1.59086 q^{15} +1.00000 q^{16} -2.39469 q^{17} +1.00000 q^{18} +6.34685 q^{19} -1.59086 q^{20} +1.20696 q^{21} +4.80384 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.46917 q^{25} +1.05524 q^{26} +1.00000 q^{27} +1.20696 q^{28} -1.00000 q^{29} -1.59086 q^{30} -5.20570 q^{31} +1.00000 q^{32} +4.80384 q^{33} -2.39469 q^{34} -1.92011 q^{35} +1.00000 q^{36} +7.70325 q^{37} +6.34685 q^{38} +1.05524 q^{39} -1.59086 q^{40} +1.69372 q^{41} +1.20696 q^{42} -7.44867 q^{43} +4.80384 q^{44} -1.59086 q^{45} +1.00000 q^{46} +8.28354 q^{47} +1.00000 q^{48} -5.54324 q^{49} -2.46917 q^{50} -2.39469 q^{51} +1.05524 q^{52} +1.92691 q^{53} +1.00000 q^{54} -7.64222 q^{55} +1.20696 q^{56} +6.34685 q^{57} -1.00000 q^{58} +2.23696 q^{59} -1.59086 q^{60} -5.98555 q^{61} -5.20570 q^{62} +1.20696 q^{63} +1.00000 q^{64} -1.67874 q^{65} +4.80384 q^{66} +1.48037 q^{67} -2.39469 q^{68} +1.00000 q^{69} -1.92011 q^{70} +7.48800 q^{71} +1.00000 q^{72} -3.40526 q^{73} +7.70325 q^{74} -2.46917 q^{75} +6.34685 q^{76} +5.79805 q^{77} +1.05524 q^{78} +17.0123 q^{79} -1.59086 q^{80} +1.00000 q^{81} +1.69372 q^{82} +4.46177 q^{83} +1.20696 q^{84} +3.80962 q^{85} -7.44867 q^{86} -1.00000 q^{87} +4.80384 q^{88} +13.3089 q^{89} -1.59086 q^{90} +1.27364 q^{91} +1.00000 q^{92} -5.20570 q^{93} +8.28354 q^{94} -10.0969 q^{95} +1.00000 q^{96} -17.6029 q^{97} -5.54324 q^{98} +4.80384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{3} + 7 q^{4} + q^{5} + 7 q^{6} - 2 q^{7} + 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^3 + 7 * q^4 + q^5 + 7 * q^6 - 2 * q^7 + 7 * q^8 + 7 * q^9	\( 7 q + 7 q^{2} + 7 q^{3} + 7 q^{4} + q^{5} + 7 q^{6} - 2 q^{7} + 7 q^{8} + 7 q^{9} + q^{10} + 11 q^{11} + 7 q^{12} - q^{13} - 2 q^{14} + q^{15} + 7 q^{16} + 18 q^{17} + 7 q^{18} + 6 q^{19} + q^{20} - 2 q^{21} + 11 q^{22} + 7 q^{23} + 7 q^{24} + 12 q^{25} - q^{26} + 7 q^{27} - 2 q^{28} - 7 q^{29} + q^{30} + 3 q^{31} + 7 q^{32} + 11 q^{33} + 18 q^{34} - 4 q^{35} + 7 q^{36} + 5 q^{37} + 6 q^{38} - q^{39} + q^{40} + 25 q^{41} - 2 q^{42} + 20 q^{43} + 11 q^{44} + q^{45} + 7 q^{46} + 7 q^{48} + 7 q^{49} + 12 q^{50} + 18 q^{51} - q^{52} - 4 q^{53} + 7 q^{54} + 17 q^{55} - 2 q^{56} + 6 q^{57} - 7 q^{58} - 17 q^{59} + q^{60} + 5 q^{61} + 3 q^{62} - 2 q^{63} + 7 q^{64} + 7 q^{65} + 11 q^{66} + 15 q^{67} + 18 q^{68} + 7 q^{69} - 4 q^{70} + 15 q^{71} + 7 q^{72} + 14 q^{73} + 5 q^{74} + 12 q^{75} + 6 q^{76} - 12 q^{77} - q^{78} - 4 q^{79} + q^{80} + 7 q^{81} + 25 q^{82} + 14 q^{83} - 2 q^{84} + 34 q^{85} + 20 q^{86} - 7 q^{87} + 11 q^{88} + 8 q^{89} + q^{90} - 6 q^{91} + 7 q^{92} + 3 q^{93} - 18 q^{95} + 7 q^{96} - 18 q^{97} + 7 q^{98} + 11 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^3 + 7 * q^4 + q^5 + 7 * q^6 - 2 * q^7 + 7 * q^8 + 7 * q^9 + q^10 + 11 * q^11 + 7 * q^12 - q^13 - 2 * q^14 + q^15 + 7 * q^16 + 18 * q^17 + 7 * q^18 + 6 * q^19 + q^20 - 2 * q^21 + 11 * q^22 + 7 * q^23 + 7 * q^24 + 12 * q^25 - q^26 + 7 * q^27 - 2 * q^28 - 7 * q^29 + q^30 + 3 * q^31 + 7 * q^32 + 11 * q^33 + 18 * q^34 - 4 * q^35 + 7 * q^36 + 5 * q^37 + 6 * q^38 - q^39 + q^40 + 25 * q^41 - 2 * q^42 + 20 * q^43 + 11 * q^44 + q^45 + 7 * q^46 + 7 * q^48 + 7 * q^49 + 12 * q^50 + 18 * q^51 - q^52 - 4 * q^53 + 7 * q^54 + 17 * q^55 - 2 * q^56 + 6 * q^57 - 7 * q^58 - 17 * q^59 + q^60 + 5 * q^61 + 3 * q^62 - 2 * q^63 + 7 * q^64 + 7 * q^65 + 11 * q^66 + 15 * q^67 + 18 * q^68 + 7 * q^69 - 4 * q^70 + 15 * q^71 + 7 * q^72 + 14 * q^73 + 5 * q^74 + 12 * q^75 + 6 * q^76 - 12 * q^77 - q^78 - 4 * q^79 + q^80 + 7 * q^81 + 25 * q^82 + 14 * q^83 - 2 * q^84 + 34 * q^85 + 20 * q^86 - 7 * q^87 + 11 * q^88 + 8 * q^89 + q^90 - 6 * q^91 + 7 * q^92 + 3 * q^93 - 18 * q^95 + 7 * q^96 - 18 * q^97 + 7 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more