LMFDB - Embedded newform 5610.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +5.23607 q^{13} -3.23607 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -5.23607 q^{19} +1.00000 q^{20} -3.23607 q^{21} -1.00000 q^{22} +2.76393 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} +2.00000 q^{29} +1.00000 q^{30} -2.47214 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.23607 q^{35} +1.00000 q^{36} -4.47214 q^{37} +5.23607 q^{38} -5.23607 q^{39} -1.00000 q^{40} +8.94427 q^{41} +3.23607 q^{42} +7.23607 q^{43} +1.00000 q^{44} +1.00000 q^{45} -2.76393 q^{46} +8.94427 q^{47} -1.00000 q^{48} +3.47214 q^{49} -1.00000 q^{50} -1.00000 q^{51} +5.23607 q^{52} +5.70820 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.23607 q^{56} +5.23607 q^{57} -2.00000 q^{58} +8.76393 q^{59} -1.00000 q^{60} -14.9443 q^{61} +2.47214 q^{62} +3.23607 q^{63} +1.00000 q^{64} +5.23607 q^{65} +1.00000 q^{66} +0.472136 q^{67} +1.00000 q^{68} -2.76393 q^{69} -3.23607 q^{70} +0.763932 q^{71} -1.00000 q^{72} -11.7082 q^{73} +4.47214 q^{74} -1.00000 q^{75} -5.23607 q^{76} +3.23607 q^{77} +5.23607 q^{78} +9.23607 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +12.9443 q^{83} -3.23607 q^{84} +1.00000 q^{85} -7.23607 q^{86} -2.00000 q^{87} -1.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +16.9443 q^{91} +2.76393 q^{92} +2.47214 q^{93} -8.94427 q^{94} -5.23607 q^{95} +1.00000 q^{96} -12.9443 q^{97} -3.47214 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} + 6 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} + 2 q^{24} + 2 q^{25} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 4 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} + 6 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{42} + 10 q^{43} + 2 q^{44} + 2 q^{45} - 10 q^{46} - 2 q^{48} - 2 q^{49} - 2 q^{50} - 2 q^{51} + 6 q^{52} - 2 q^{53} + 2 q^{54} + 2 q^{55} - 2 q^{56} + 6 q^{57} - 4 q^{58} + 22 q^{59} - 2 q^{60} - 12 q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{65} + 2 q^{66} - 8 q^{67} + 2 q^{68} - 10 q^{69} - 2 q^{70} + 6 q^{71} - 2 q^{72} - 10 q^{73} - 2 q^{75} - 6 q^{76} + 2 q^{77} + 6 q^{78} + 14 q^{79} + 2 q^{80} + 2 q^{81} + 8 q^{83} - 2 q^{84} + 2 q^{85} - 10 q^{86} - 4 q^{87} - 2 q^{88} + 28 q^{89} - 2 q^{90} + 16 q^{91} + 10 q^{92} - 4 q^{93} - 6 q^{95} + 2 q^{96} - 8 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 - 2 * q^12 + 6 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 + 2 * q^24 + 2 * q^25 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 4 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^35 + 2 * q^36 + 6 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^42 + 10 * q^43 + 2 * q^44 + 2 * q^45 - 10 * q^46 - 2 * q^48 - 2 * q^49 - 2 * q^50 - 2 * q^51 + 6 * q^52 - 2 * q^53 + 2 * q^54 + 2 * q^55 - 2 * q^56 + 6 * q^57 - 4 * q^58 + 22 * q^59 - 2 * q^60 - 12 * q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^65 + 2 * q^66 - 8 * q^67 + 2 * q^68 - 10 * q^69 - 2 * q^70 + 6 * q^71 - 2 * q^72 - 10 * q^73 - 2 * q^75 - 6 * q^76 + 2 * q^77 + 6 * q^78 + 14 * q^79 + 2 * q^80 + 2 * q^81 + 8 * q^83 - 2 * q^84 + 2 * q^85 - 10 * q^86 - 4 * q^87 - 2 * q^88 + 28 * q^89 - 2 * q^90 + 16 * q^91 + 10 * q^92 - 4 * q^93 - 6 * q^95 + 2 * q^96 - 8 * q^97 + 2 * q^98 + 2 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	−3.23607	−0.864876
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−5.23607	−1.20124	−0.600618	−	0.799536i	$-0.705079\pi$
$19$	−5.23607	−1.20124	−0.600618	+	0.799536i	$0.705079\pi$
$20$	1.00000	0.223607
$21$	−3.23607	−0.706168
$22$	−1.00000	−0.213201
$23$	2.76393	0.576320	0.288160	−	0.957582i	$-0.406957\pi$
$23$	2.76393	0.576320	0.288160	+	0.957582i	$0.406957\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−5.23607	−1.02688
$27$	−1.00000	−0.192450
$28$	3.23607	0.611559
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000	0.182574
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−1.00000	−0.171499
$35$	3.23607	0.546995
$36$	1.00000	0.166667
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	5.23607	0.849402
$39$	−5.23607	−0.838442
$40$	−1.00000	−0.158114
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	3.23607	0.499336
$43$	7.23607	1.10349	0.551745	−	0.834013i	$-0.313962\pi$
$43$	7.23607	1.10349	0.551745	+	0.834013i	$0.313962\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	−2.76393	−0.407520
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	−1.00000	−0.144338
$49$	3.47214	0.496019
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	5.23607	0.726112
$53$	5.70820	0.784082	0.392041	−	0.919948i	$-0.371769\pi$
$53$	5.70820	0.784082	0.392041	+	0.919948i	$0.371769\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−3.23607	−0.432438
$57$	5.23607	0.693534
$58$	−2.00000	−0.262613
$59$	8.76393	1.14097	0.570483	−	0.821309i	$-0.306756\pi$
$59$	8.76393	1.14097	0.570483	+	0.821309i	$0.306756\pi$
$60$	−1.00000	−0.129099
$61$	−14.9443	−1.91342	−0.956709	−	0.291046i	$-0.905997\pi$
$61$	−14.9443	−1.91342	−0.956709	+	0.291046i	$0.905997\pi$
$62$	2.47214	0.313962
$63$	3.23607	0.407706
$64$	1.00000	0.125000
$65$	5.23607	0.649454
$66$	1.00000	0.123091
$67$	0.472136	0.0576806	0.0288403	−	0.999584i	$-0.490819\pi$
$67$	0.472136	0.0576806	0.0288403	+	0.999584i	$0.490819\pi$
$68$	1.00000	0.121268
$69$	−2.76393	−0.332738
$70$	−3.23607	−0.386784
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	−1.00000	−0.117851
$73$	−11.7082	−1.37034	−0.685171	−	0.728382i	$-0.740272\pi$
$73$	−11.7082	−1.37034	−0.685171	+	0.728382i	$0.740272\pi$
$74$	4.47214	0.519875
$75$	−1.00000	−0.115470
$76$	−5.23607	−0.600618
$77$	3.23607	0.368784
$78$	5.23607	0.592868
$79$	9.23607	1.03914	0.519569	−	0.854428i	$-0.326092\pi$
$79$	9.23607	1.03914	0.519569	+	0.854428i	$0.326092\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−8.94427	−0.987730
$83$	12.9443	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$83$	12.9443	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$84$	−3.23607	−0.353084
$85$	1.00000	0.108465
$86$	−7.23607	−0.780285
$87$	−2.00000	−0.214423
$88$	−1.00000	−0.106600
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	−1.00000	−0.105409
$91$	16.9443	1.77624
$92$	2.76393	0.288160
$93$	2.47214	0.256349
$94$	−8.94427	−0.922531
$95$	−5.23607	−0.537209
$96$	1.00000	0.102062
$97$	−12.9443	−1.31429	−0.657146	−	0.753763i	$-0.728236\pi$
$97$	−12.9443	−1.31429	−0.657146	+	0.753763i	$0.728236\pi$
$98$	−3.47214	−0.350739
$99$	1.00000	0.100504
$100$	1.00000	0.100000
$101$	−1.52786	−0.152028	−0.0760141	−	0.997107i	$-0.524219\pi$
$101$	−1.52786	−0.152028	−0.0760141	+	0.997107i	$0.524219\pi$
$102$	1.00000	0.0990148
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−5.23607	−0.513439
$105$	−3.23607	−0.315808
$106$	−5.70820	−0.554430
$107$	−12.9443	−1.25137	−0.625685	−	0.780076i	$-0.715180\pi$
$107$	−12.9443	−1.25137	−0.625685	+	0.780076i	$0.715180\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.00000	−0.0953463
$111$	4.47214	0.424476
$112$	3.23607	0.305780
$113$	−21.1246	−1.98724	−0.993618	−	0.112796i	$-0.964019\pi$
$113$	−21.1246	−1.98724	−0.993618	+	0.112796i	$0.964019\pi$
$114$	−5.23607	−0.490403
$115$	2.76393	0.257738
$116$	2.00000	0.185695
$117$	5.23607	0.484075
$118$	−8.76393	−0.806785
$119$	3.23607	0.296650
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	14.9443	1.35299
$123$	−8.94427	−0.806478
$124$	−2.47214	−0.222004
$125$	1.00000	0.0894427
$126$	−3.23607	−0.288292
$127$	11.4164	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$127$	11.4164	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$128$	−1.00000	−0.0883883
$129$	−7.23607	−0.637100
$130$	−5.23607	−0.459234
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−1.00000	−0.0870388
$133$	−16.9443	−1.46925
$134$	−0.472136	−0.0407863
$135$	−1.00000	−0.0860663
$136$	−1.00000	−0.0857493
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	2.76393	0.235282
$139$	17.8885	1.51729	0.758643	−	0.651506i	$-0.225863\pi$
$139$	17.8885	1.51729	0.758643	+	0.651506i	$0.225863\pi$
$140$	3.23607	0.273498
$141$	−8.94427	−0.753244
$142$	−0.763932	−0.0641078
$143$	5.23607	0.437862
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	11.7082	0.968978
$147$	−3.47214	−0.286377
$148$	−4.47214	−0.367607
$149$	−16.9443	−1.38813	−0.694064	−	0.719913i	$-0.744182\pi$
$149$	−16.9443	−1.38813	−0.694064	+	0.719913i	$0.744182\pi$
$150$	1.00000	0.0816497
$151$	−4.94427	−0.402359	−0.201180	−	0.979554i	$-0.564477\pi$
$151$	−4.94427	−0.402359	−0.201180	+	0.979554i	$0.564477\pi$
$152$	5.23607	0.424701
$153$	1.00000	0.0808452
$154$	−3.23607	−0.260770
$155$	−2.47214	−0.198567
$156$	−5.23607	−0.419221
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	−9.23607	−0.734782
$159$	−5.70820	−0.452690
$160$	−1.00000	−0.0790569
$161$	8.94427	0.704907
$162$	−1.00000	−0.0785674
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	8.94427	0.698430
$165$	−1.00000	−0.0778499
$166$	−12.9443	−1.00467
$167$	−10.4721	−0.810358	−0.405179	−	0.914237i	$-0.632791\pi$
$167$	−10.4721	−0.810358	−0.405179	+	0.914237i	$0.632791\pi$
$168$	3.23607	0.249668
$169$	14.4164	1.10895
$170$	−1.00000	−0.0766965
$171$	−5.23607	−0.400412
$172$	7.23607	0.551745
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	2.00000	0.151620
$175$	3.23607	0.244624
$176$	1.00000	0.0753778
$177$	−8.76393	−0.658737
$178$	−14.0000	−1.04934
$179$	10.2918	0.769245	0.384622	−	0.923074i	$-0.374332\pi$
$179$	10.2918	0.769245	0.384622	+	0.923074i	$0.374332\pi$
$180$	1.00000	0.0745356
$181$	23.8885	1.77562	0.887811	−	0.460209i	$-0.152225\pi$
$181$	23.8885	1.77562	0.887811	+	0.460209i	$0.152225\pi$
$182$	−16.9443	−1.25599
$183$	14.9443	1.10471
$184$	−2.76393	−0.203760
$185$	−4.47214	−0.328798
$186$	−2.47214	−0.181266
$187$	1.00000	0.0731272
$188$	8.94427	0.652328
$189$	−3.23607	−0.235389
$190$	5.23607	0.379864
$191$	−13.8885	−1.00494	−0.502470	−	0.864595i	$-0.667575\pi$
$191$	−13.8885	−1.00494	−0.502470	+	0.864595i	$0.667575\pi$
$192$	−1.00000	−0.0721688
$193$	−8.29180	−0.596857	−0.298428	−	0.954432i	$-0.596462\pi$
$193$	−8.29180	−0.596857	−0.298428	+	0.954432i	$0.596462\pi$
$194$	12.9443	0.929345
$195$	−5.23607	−0.374963
$196$	3.47214	0.248010
$197$	16.4721	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$197$	16.4721	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$198$	−1.00000	−0.0710669
$199$	5.52786	0.391860	0.195930	−	0.980618i	$-0.437227\pi$
$199$	5.52786	0.391860	0.195930	+	0.980618i	$0.437227\pi$
$200$	−1.00000	−0.0707107
$201$	−0.472136	−0.0333019
$202$	1.52786	0.107500
$203$	6.47214	0.454255
$204$	−1.00000	−0.0700140
$205$	8.94427	0.624695
$206$	12.0000	0.836080
$207$	2.76393	0.192107
$208$	5.23607	0.363056
$209$	−5.23607	−0.362186
$210$	3.23607	0.223310
$211$	−7.41641	−0.510567	−0.255283	−	0.966866i	$-0.582169\pi$
$211$	−7.41641	−0.510567	−0.255283	+	0.966866i	$0.582169\pi$
$212$	5.70820	0.392041
$213$	−0.763932	−0.0523438
$214$	12.9443	0.884852
$215$	7.23607	0.493496
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	−10.0000	−0.677285
$219$	11.7082	0.791167
$220$	1.00000	0.0674200
$221$	5.23607	0.352216
$222$	−4.47214	−0.300150
$223$	−18.4721	−1.23699	−0.618493	−	0.785790i	$-0.712256\pi$
$223$	−18.4721	−1.23699	−0.618493	+	0.785790i	$0.712256\pi$
$224$	−3.23607	−0.216219
$225$	1.00000	0.0666667
$226$	21.1246	1.40519
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	5.23607	0.346767
$229$	−22.3607	−1.47764	−0.738818	−	0.673905i	$-0.764616\pi$
$229$	−22.3607	−1.47764	−0.738818	+	0.673905i	$0.764616\pi$
$230$	−2.76393	−0.182248
$231$	−3.23607	−0.212918
$232$	−2.00000	−0.131306
$233$	−16.4721	−1.07913	−0.539563	−	0.841945i	$-0.681410\pi$
$233$	−16.4721	−1.07913	−0.539563	+	0.841945i	$0.681410\pi$
$234$	−5.23607	−0.342292
$235$	8.94427	0.583460
$236$	8.76393	0.570483
$237$	−9.23607	−0.599947
$238$	−3.23607	−0.209763
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	−1.00000	−0.0645497
$241$	6.65248	0.428524	0.214262	−	0.976776i	$-0.431265\pi$
$241$	6.65248	0.428524	0.214262	+	0.976776i	$0.431265\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−14.9443	−0.956709
$245$	3.47214	0.221827
$246$	8.94427	0.570266
$247$	−27.4164	−1.74446
$248$	2.47214	0.156981
$249$	−12.9443	−0.820310
$250$	−1.00000	−0.0632456
$251$	6.29180	0.397135	0.198567	−	0.980087i	$-0.436371\pi$
$251$	6.29180	0.397135	0.198567	+	0.980087i	$0.436371\pi$
$252$	3.23607	0.203853
$253$	2.76393	0.173767
$254$	−11.4164	−0.716329
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	12.4721	0.777990	0.388995	−	0.921240i	$-0.372822\pi$
$257$	12.4721	0.777990	0.388995	+	0.921240i	$0.372822\pi$
$258$	7.23607	0.450498
$259$	−14.4721	−0.899255
$260$	5.23607	0.324727
$261$	2.00000	0.123797
$262$	−8.00000	−0.494242
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	1.00000	0.0615457
$265$	5.70820	0.350652
$266$	16.9443	1.03892
$267$	−14.0000	−0.856786
$268$	0.472136	0.0288403
$269$	−4.47214	−0.272671	−0.136335	−	0.990663i	$-0.543533\pi$
$269$	−4.47214	−0.272671	−0.136335	+	0.990663i	$0.543533\pi$
$270$	1.00000	0.0608581
$271$	−20.3607	−1.23682	−0.618412	−	0.785854i	$-0.712224\pi$
$271$	−20.3607	−1.23682	−0.618412	+	0.785854i	$0.712224\pi$
$272$	1.00000	0.0606339
$273$	−16.9443	−1.02551
$274$	2.00000	0.120824
$275$	1.00000	0.0603023
$276$	−2.76393	−0.166369
$277$	18.9443	1.13825	0.569125	−	0.822251i	$-0.307282\pi$
$277$	18.9443	1.13825	0.569125	+	0.822251i	$0.307282\pi$
$278$	−17.8885	−1.07288
$279$	−2.47214	−0.148003
$280$	−3.23607	−0.193392
$281$	−17.4164	−1.03898	−0.519488	−	0.854478i	$-0.673877\pi$
$281$	−17.4164	−1.03898	−0.519488	+	0.854478i	$0.673877\pi$
$282$	8.94427	0.532624
$283$	−19.4164	−1.15419	−0.577093	−	0.816679i	$-0.695813\pi$
$283$	−19.4164	−1.15419	−0.577093	+	0.816679i	$0.695813\pi$
$284$	0.763932	0.0453310
$285$	5.23607	0.310158
$286$	−5.23607	−0.309615
$287$	28.9443	1.70853
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	−2.00000	−0.117444
$291$	12.9443	0.758807
$292$	−11.7082	−0.685171
$293$	−23.8885	−1.39558	−0.697792	−	0.716301i	$-0.745834\pi$
$293$	−23.8885	−1.39558	−0.697792	+	0.716301i	$0.745834\pi$
$294$	3.47214	0.202499
$295$	8.76393	0.510256
$296$	4.47214	0.259938
$297$	−1.00000	−0.0580259
$298$	16.9443	0.981555
$299$	14.4721	0.836945
$300$	−1.00000	−0.0577350
$301$	23.4164	1.34970
$302$	4.94427	0.284511
$303$	1.52786	0.0877735
$304$	−5.23607	−0.300309
$305$	−14.9443	−0.855707
$306$	−1.00000	−0.0571662
$307$	30.6525	1.74943	0.874715	−	0.484638i	$-0.161049\pi$
$307$	30.6525	1.74943	0.874715	+	0.484638i	$0.161049\pi$
$308$	3.23607	0.184392
$309$	12.0000	0.682656
$310$	2.47214	0.140408
$311$	−24.1803	−1.37114	−0.685571	−	0.728006i	$-0.740447\pi$
$311$	−24.1803	−1.37114	−0.685571	+	0.728006i	$0.740447\pi$
$312$	5.23607	0.296434
$313$	25.8885	1.46331	0.731654	−	0.681677i	$-0.238749\pi$
$313$	25.8885	1.46331	0.731654	+	0.681677i	$0.238749\pi$
$314$	12.0000	0.677199
$315$	3.23607	0.182332
$316$	9.23607	0.519569
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	5.70820	0.320100
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	12.9443	0.722479
$322$	−8.94427	−0.498445
$323$	−5.23607	−0.291343
$324$	1.00000	0.0555556
$325$	5.23607	0.290445
$326$	6.47214	0.358458
$327$	−10.0000	−0.553001
$328$	−8.94427	−0.493865
$329$	28.9443	1.59575
$330$	1.00000	0.0550482
$331$	9.88854	0.543524	0.271762	−	0.962365i	$-0.412394\pi$
$331$	9.88854	0.543524	0.271762	+	0.962365i	$0.412394\pi$
$332$	12.9443	0.710409
$333$	−4.47214	−0.245072
$334$	10.4721	0.573010
$335$	0.472136	0.0257955
$336$	−3.23607	−0.176542
$337$	9.23607	0.503121	0.251560	−	0.967842i	$-0.419056\pi$
$337$	9.23607	0.503121	0.251560	+	0.967842i	$0.419056\pi$
$338$	−14.4164	−0.784149
$339$	21.1246	1.14733
$340$	1.00000	0.0542326
$341$	−2.47214	−0.133874
$342$	5.23607	0.283134
$343$	−11.4164	−0.616428
$344$	−7.23607	−0.390143
$345$	−2.76393	−0.148805
$346$	−6.94427	−0.373326
$347$	13.8885	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$347$	13.8885	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$348$	−2.00000	−0.107211
$349$	18.6525	0.998444	0.499222	−	0.866474i	$-0.333619\pi$
$349$	18.6525	0.998444	0.499222	+	0.866474i	$0.333619\pi$
$350$	−3.23607	−0.172975
$351$	−5.23607	−0.279481
$352$	−1.00000	−0.0533002
$353$	−19.5279	−1.03936	−0.519682	−	0.854360i	$-0.673949\pi$
$353$	−19.5279	−1.03936	−0.519682	+	0.854360i	$0.673949\pi$
$354$	8.76393	0.465798
$355$	0.763932	0.0405453
$356$	14.0000	0.741999
$357$	−3.23607	−0.171271
$358$	−10.2918	−0.543938
$359$	−10.4721	−0.552698	−0.276349	−	0.961057i	$-0.589125\pi$
$359$	−10.4721	−0.552698	−0.276349	+	0.961057i	$0.589125\pi$
$360$	−1.00000	−0.0527046
$361$	8.41641	0.442969
$362$	−23.8885	−1.25555
$363$	−1.00000	−0.0524864
$364$	16.9443	0.888121
$365$	−11.7082	−0.612835
$366$	−14.9443	−0.781150
$367$	15.5279	0.810548	0.405274	−	0.914195i	$-0.367176\pi$
$367$	15.5279	0.810548	0.405274	+	0.914195i	$0.367176\pi$
$368$	2.76393	0.144080
$369$	8.94427	0.465620
$370$	4.47214	0.232495
$371$	18.4721	0.959026
$372$	2.47214	0.128174
$373$	−24.6525	−1.27646	−0.638228	−	0.769847i	$-0.720332\pi$
$373$	−24.6525	−1.27646	−0.638228	+	0.769847i	$0.720332\pi$
$374$	−1.00000	−0.0517088
$375$	−1.00000	−0.0516398
$376$	−8.94427	−0.461266
$377$	10.4721	0.539342
$378$	3.23607	0.166445
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	−5.23607	−0.268605
$381$	−11.4164	−0.584880
$382$	13.8885	0.710600
$383$	21.8885	1.11845	0.559226	−	0.829015i	$-0.311098\pi$
$383$	21.8885	1.11845	0.559226	+	0.829015i	$0.311098\pi$
$384$	1.00000	0.0510310
$385$	3.23607	0.164925
$386$	8.29180	0.422041
$387$	7.23607	0.367830
$388$	−12.9443	−0.657146
$389$	20.2918	1.02883	0.514417	−	0.857540i	$-0.328008\pi$
$389$	20.2918	1.02883	0.514417	+	0.857540i	$0.328008\pi$
$390$	5.23607	0.265139
$391$	2.76393	0.139778
$392$	−3.47214	−0.175369
$393$	−8.00000	−0.403547
$394$	−16.4721	−0.829854
$395$	9.23607	0.464717
$396$	1.00000	0.0502519
$397$	−10.3607	−0.519988	−0.259994	−	0.965610i	$-0.583721\pi$
$397$	−10.3607	−0.519988	−0.259994	+	0.965610i	$0.583721\pi$
$398$	−5.52786	−0.277087
$399$	16.9443	0.848275
$400$	1.00000	0.0500000
$401$	27.1246	1.35454	0.677269	−	0.735735i	$-0.263163\pi$
$401$	27.1246	1.35454	0.677269	+	0.735735i	$0.263163\pi$
$402$	0.472136	0.0235480
$403$	−12.9443	−0.644800
$404$	−1.52786	−0.0760141
$405$	1.00000	0.0496904
$406$	−6.47214	−0.321207
$407$	−4.47214	−0.221676
$408$	1.00000	0.0495074
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	−8.94427	−0.441726
$411$	2.00000	0.0986527
$412$	−12.0000	−0.591198
$413$	28.3607	1.39554
$414$	−2.76393	−0.135840
$415$	12.9443	0.635409
$416$	−5.23607	−0.256719
$417$	−17.8885	−0.876006
$418$	5.23607	0.256104
$419$	−10.4721	−0.511597	−0.255799	−	0.966730i	$-0.582338\pi$
$419$	−10.4721	−0.511597	−0.255799	+	0.966730i	$0.582338\pi$
$420$	−3.23607	−0.157904
$421$	31.8885	1.55415	0.777076	−	0.629406i	$-0.216702\pi$
$421$	31.8885	1.55415	0.777076	+	0.629406i	$0.216702\pi$
$422$	7.41641	0.361025
$423$	8.94427	0.434885
$424$	−5.70820	−0.277215
$425$	1.00000	0.0485071
$426$	0.763932	0.0370126
$427$	−48.3607	−2.34034
$428$	−12.9443	−0.625685
$429$	−5.23607	−0.252800
$430$	−7.23607	−0.348954
$431$	−9.41641	−0.453572	−0.226786	−	0.973945i	$-0.572822\pi$
$431$	−9.41641	−0.453572	−0.226786	+	0.973945i	$0.572822\pi$
$432$	−1.00000	−0.0481125
$433$	−20.8328	−1.00116	−0.500581	−	0.865690i	$-0.666880\pi$
$433$	−20.8328	−1.00116	−0.500581	+	0.865690i	$0.666880\pi$
$434$	8.00000	0.384012
$435$	−2.00000	−0.0958927
$436$	10.0000	0.478913
$437$	−14.4721	−0.692296
$438$	−11.7082	−0.559440
$439$	11.1246	0.530949	0.265474	−	0.964118i	$-0.414471\pi$
$439$	11.1246	0.530949	0.265474	+	0.964118i	$0.414471\pi$
$440$	−1.00000	−0.0476731
$441$	3.47214	0.165340
$442$	−5.23607	−0.249054
$443$	7.70820	0.366228	0.183114	−	0.983092i	$-0.441382\pi$
$443$	7.70820	0.366228	0.183114	+	0.983092i	$0.441382\pi$
$444$	4.47214	0.212238
$445$	14.0000	0.663664
$446$	18.4721	0.874681
$447$	16.9443	0.801437
$448$	3.23607	0.152890
$449$	16.6525	0.785879	0.392939	−	0.919564i	$-0.371458\pi$
$449$	16.6525	0.785879	0.392939	+	0.919564i	$0.371458\pi$
$450$	−1.00000	−0.0471405
$451$	8.94427	0.421169
$452$	−21.1246	−0.993618
$453$	4.94427	0.232302
$454$	−4.00000	−0.187729
$455$	16.9443	0.794360
$456$	−5.23607	−0.245201
$457$	−16.8328	−0.787406	−0.393703	−	0.919238i	$-0.628806\pi$
$457$	−16.8328	−0.787406	−0.393703	+	0.919238i	$0.628806\pi$
$458$	22.3607	1.04485
$459$	−1.00000	−0.0466760
$460$	2.76393	0.128869
$461$	4.94427	0.230278	0.115139	−	0.993349i	$-0.463269\pi$
$461$	4.94427	0.230278	0.115139	+	0.993349i	$0.463269\pi$
$462$	3.23607	0.150556
$463$	26.4721	1.23026	0.615132	−	0.788424i	$-0.289103\pi$
$463$	26.4721	1.23026	0.615132	+	0.788424i	$0.289103\pi$
$464$	2.00000	0.0928477
$465$	2.47214	0.114643
$466$	16.4721	0.763057
$467$	−24.0689	−1.11378	−0.556888	−	0.830588i	$-0.688005\pi$
$467$	−24.0689	−1.11378	−0.556888	+	0.830588i	$0.688005\pi$
$468$	5.23607	0.242037
$469$	1.52786	0.0705502
$470$	−8.94427	−0.412568
$471$	12.0000	0.552931
$472$	−8.76393	−0.403393
$473$	7.23607	0.332715
$474$	9.23607	0.424227
$475$	−5.23607	−0.240247
$476$	3.23607	0.148325
$477$	5.70820	0.261361
$478$	−1.52786	−0.0698829
$479$	21.0557	0.962061	0.481030	−	0.876704i	$-0.340263\pi$
$479$	21.0557	0.962061	0.481030	+	0.876704i	$0.340263\pi$
$480$	1.00000	0.0456435
$481$	−23.4164	−1.06770
$482$	−6.65248	−0.303012
$483$	−8.94427	−0.406978
$484$	1.00000	0.0454545
$485$	−12.9443	−0.587769
$486$	1.00000	0.0453609
$487$	29.4164	1.33298	0.666492	−	0.745512i	$-0.267795\pi$
$487$	29.4164	1.33298	0.666492	+	0.745512i	$0.267795\pi$
$488$	14.9443	0.676495
$489$	6.47214	0.292680
$490$	−3.47214	−0.156855
$491$	4.47214	0.201825	0.100912	−	0.994895i	$-0.467824\pi$
$491$	4.47214	0.201825	0.100912	+	0.994895i	$0.467824\pi$
$492$	−8.94427	−0.403239
$493$	2.00000	0.0900755
$494$	27.4164	1.23352
$495$	1.00000	0.0449467
$496$	−2.47214	−0.111002
$497$	2.47214	0.110890
$498$	12.9443	0.580047
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	1.00000	0.0447214
$501$	10.4721	0.467861
$502$	−6.29180	−0.280817
$503$	−13.8885	−0.619260	−0.309630	−	0.950857i	$-0.600205\pi$
$503$	−13.8885	−0.619260	−0.309630	+	0.950857i	$0.600205\pi$
$504$	−3.23607	−0.144146
$505$	−1.52786	−0.0679891
$506$	−2.76393	−0.122872
$507$	−14.4164	−0.640255
$508$	11.4164	0.506521
$509$	43.1246	1.91146	0.955732	−	0.294237i	$-0.0950656\pi$
$509$	43.1246	1.91146	0.955732	+	0.294237i	$0.0950656\pi$
$510$	1.00000	0.0442807
$511$	−37.8885	−1.67609
$512$	−1.00000	−0.0441942
$513$	5.23607	0.231178
$514$	−12.4721	−0.550122
$515$	−12.0000	−0.528783
$516$	−7.23607	−0.318550
$517$	8.94427	0.393369
$518$	14.4721	0.635869
$519$	−6.94427	−0.304820
$520$	−5.23607	−0.229617
$521$	−26.5410	−1.16278	−0.581392	−	0.813624i	$-0.697492\pi$
$521$	−26.5410	−1.16278	−0.581392	+	0.813624i	$0.697492\pi$
$522$	−2.00000	−0.0875376
$523$	−17.1246	−0.748807	−0.374403	−	0.927266i	$-0.622153\pi$
$523$	−17.1246	−0.748807	−0.374403	+	0.927266i	$0.622153\pi$
$524$	8.00000	0.349482
$525$	−3.23607	−0.141234
$526$	−28.0000	−1.22086
$527$	−2.47214	−0.107688
$528$	−1.00000	−0.0435194
$529$	−15.3607	−0.667856
$530$	−5.70820	−0.247949
$531$	8.76393	0.380322
$532$	−16.9443	−0.734627
$533$	46.8328	2.02855
$534$	14.0000	0.605839
$535$	−12.9443	−0.559630
$536$	−0.472136	−0.0203932
$537$	−10.2918	−0.444124
$538$	4.47214	0.192807
$539$	3.47214	0.149555
$540$	−1.00000	−0.0430331
$541$	33.7771	1.45219	0.726095	−	0.687594i	$-0.241333\pi$
$541$	33.7771	1.45219	0.726095	+	0.687594i	$0.241333\pi$
$542$	20.3607	0.874566
$543$	−23.8885	−1.02516
$544$	−1.00000	−0.0428746
$545$	10.0000	0.428353
$546$	16.9443	0.725148
$547$	11.4164	0.488130	0.244065	−	0.969759i	$-0.421519\pi$
$547$	11.4164	0.488130	0.244065	+	0.969759i	$0.421519\pi$
$548$	−2.00000	−0.0854358
$549$	−14.9443	−0.637806
$550$	−1.00000	−0.0426401
$551$	−10.4721	−0.446128
$552$	2.76393	0.117641
$553$	29.8885	1.27099
$554$	−18.9443	−0.804865
$555$	4.47214	0.189832
$556$	17.8885	0.758643
$557$	−32.8328	−1.39117	−0.695586	−	0.718443i	$-0.744855\pi$
$557$	−32.8328	−1.39117	−0.695586	+	0.718443i	$0.744855\pi$
$558$	2.47214	0.104654
$559$	37.8885	1.60251
$560$	3.23607	0.136749
$561$	−1.00000	−0.0422200
$562$	17.4164	0.734667
$563$	−22.8328	−0.962288	−0.481144	−	0.876641i	$-0.659779\pi$
$563$	−22.8328	−0.962288	−0.481144	+	0.876641i	$0.659779\pi$
$564$	−8.94427	−0.376622
$565$	−21.1246	−0.888719
$566$	19.4164	0.816132
$567$	3.23607	0.135902
$568$	−0.763932	−0.0320539
$569$	−12.4721	−0.522859	−0.261430	−	0.965223i	$-0.584194\pi$
$569$	−12.4721	−0.522859	−0.261430	+	0.965223i	$0.584194\pi$
$570$	−5.23607	−0.219315
$571$	−3.05573	−0.127878	−0.0639391	−	0.997954i	$-0.520366\pi$
$571$	−3.05573	−0.127878	−0.0639391	+	0.997954i	$0.520366\pi$
$572$	5.23607	0.218931
$573$	13.8885	0.580202
$574$	−28.9443	−1.20811
$575$	2.76393	0.115264
$576$	1.00000	0.0416667
$577$	−25.0557	−1.04308	−0.521542	−	0.853226i	$-0.674643\pi$
$577$	−25.0557	−1.04308	−0.521542	+	0.853226i	$0.674643\pi$
$578$	−1.00000	−0.0415945
$579$	8.29180	0.344595
$580$	2.00000	0.0830455
$581$	41.8885	1.73783
$582$	−12.9443	−0.536557
$583$	5.70820	0.236410
$584$	11.7082	0.484489
$585$	5.23607	0.216485
$586$	23.8885	0.986827
$587$	44.6525	1.84300	0.921502	−	0.388373i	$-0.126963\pi$
$587$	44.6525	1.84300	0.921502	+	0.388373i	$0.126963\pi$
$588$	−3.47214	−0.143188
$589$	12.9443	0.533359
$590$	−8.76393	−0.360805
$591$	−16.4721	−0.677573
$592$	−4.47214	−0.183804
$593$	−23.3050	−0.957020	−0.478510	−	0.878082i	$-0.658823\pi$
$593$	−23.3050	−0.957020	−0.478510	+	0.878082i	$0.658823\pi$
$594$	1.00000	0.0410305
$595$	3.23607	0.132666
$596$	−16.9443	−0.694064
$597$	−5.52786	−0.226240
$598$	−14.4721	−0.591810
$599$	37.8885	1.54808	0.774042	−	0.633134i	$-0.218232\pi$
$599$	37.8885	1.54808	0.774042	+	0.633134i	$0.218232\pi$
$600$	1.00000	0.0408248
$601$	16.1803	0.660010	0.330005	−	0.943979i	$-0.392950\pi$
$601$	16.1803	0.660010	0.330005	+	0.943979i	$0.392950\pi$
$602$	−23.4164	−0.954382
$603$	0.472136	0.0192269
$604$	−4.94427	−0.201180
$605$	1.00000	0.0406558
$606$	−1.52786	−0.0620652
$607$	−4.76393	−0.193362	−0.0966810	−	0.995315i	$-0.530823\pi$
$607$	−4.76393	−0.193362	−0.0966810	+	0.995315i	$0.530823\pi$
$608$	5.23607	0.212351
$609$	−6.47214	−0.262264
$610$	14.9443	0.605076
$611$	46.8328	1.89465
$612$	1.00000	0.0404226
$613$	32.0689	1.29525	0.647625	−	0.761959i	$-0.275762\pi$
$613$	32.0689	1.29525	0.647625	+	0.761959i	$0.275762\pi$
$614$	−30.6525	−1.23703
$615$	−8.94427	−0.360668
$616$	−3.23607	−0.130385
$617$	16.1803	0.651396	0.325698	−	0.945474i	$-0.394401\pi$
$617$	16.1803	0.651396	0.325698	+	0.945474i	$0.394401\pi$
$618$	−12.0000	−0.482711
$619$	−10.8328	−0.435408	−0.217704	−	0.976015i	$-0.569857\pi$
$619$	−10.8328	−0.435408	−0.217704	+	0.976015i	$0.569857\pi$
$620$	−2.47214	−0.0992834
$621$	−2.76393	−0.110913
$622$	24.1803	0.969543
$623$	45.3050	1.81510
$624$	−5.23607	−0.209610
$625$	1.00000	0.0400000
$626$	−25.8885	−1.03471
$627$	5.23607	0.209108
$628$	−12.0000	−0.478852
$629$	−4.47214	−0.178316
$630$	−3.23607	−0.128928
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−9.23607	−0.367391
$633$	7.41641	0.294776
$634$	−2.00000	−0.0794301
$635$	11.4164	0.453046
$636$	−5.70820	−0.226345
$637$	18.1803	0.720331
$638$	−2.00000	−0.0791808
$639$	0.763932	0.0302207
$640$	−1.00000	−0.0395285
$641$	−27.7082	−1.09441	−0.547204	−	0.836999i	$-0.684308\pi$
$641$	−27.7082	−1.09441	−0.547204	+	0.836999i	$0.684308\pi$
$642$	−12.9443	−0.510870
$643$	−12.9443	−0.510472	−0.255236	−	0.966879i	$-0.582153\pi$
$643$	−12.9443	−0.510472	−0.255236	+	0.966879i	$0.582153\pi$
$644$	8.94427	0.352454
$645$	−7.23607	−0.284920
$646$	5.23607	0.206010
$647$	30.4721	1.19798	0.598992	−	0.800755i	$-0.295568\pi$
$647$	30.4721	1.19798	0.598992	+	0.800755i	$0.295568\pi$
$648$	−1.00000	−0.0392837
$649$	8.76393	0.344014
$650$	−5.23607	−0.205375
$651$	8.00000	0.313545
$652$	−6.47214	−0.253468
$653$	−24.8328	−0.971783	−0.485892	−	0.874019i	$-0.661505\pi$
$653$	−24.8328	−0.971783	−0.485892	+	0.874019i	$0.661505\pi$
$654$	10.0000	0.391031
$655$	8.00000	0.312586
$656$	8.94427	0.349215
$657$	−11.7082	−0.456781
$658$	−28.9443	−1.12837
$659$	31.5279	1.22815	0.614076	−	0.789247i	$-0.289529\pi$
$659$	31.5279	1.22815	0.614076	+	0.789247i	$0.289529\pi$
$660$	−1.00000	−0.0389249
$661$	15.5279	0.603964	0.301982	−	0.953314i	$-0.402352\pi$
$661$	15.5279	0.603964	0.301982	+	0.953314i	$0.402352\pi$
$662$	−9.88854	−0.384329
$663$	−5.23607	−0.203352
$664$	−12.9443	−0.502335
$665$	−16.9443	−0.657071
$666$	4.47214	0.173292
$667$	5.52786	0.214040
$668$	−10.4721	−0.405179
$669$	18.4721	0.714174
$670$	−0.472136	−0.0182402
$671$	−14.9443	−0.576917
$672$	3.23607	0.124834
$673$	31.7082	1.22226	0.611131	−	0.791530i	$-0.290715\pi$
$673$	31.7082	1.22226	0.611131	+	0.791530i	$0.290715\pi$
$674$	−9.23607	−0.355760
$675$	−1.00000	−0.0384900
$676$	14.4164	0.554477
$677$	37.4164	1.43803	0.719015	−	0.694995i	$-0.244593\pi$
$677$	37.4164	1.43803	0.719015	+	0.694995i	$0.244593\pi$
$678$	−21.1246	−0.811286
$679$	−41.8885	−1.60753
$680$	−1.00000	−0.0383482
$681$	−4.00000	−0.153280
$682$	2.47214	0.0946630
$683$	16.9443	0.648355	0.324177	−	0.945996i	$-0.394913\pi$
$683$	16.9443	0.648355	0.324177	+	0.945996i	$0.394913\pi$
$684$	−5.23607	−0.200206
$685$	−2.00000	−0.0764161
$686$	11.4164	0.435880
$687$	22.3607	0.853113
$688$	7.23607	0.275873
$689$	29.8885	1.13866
$690$	2.76393	0.105221
$691$	−6.11146	−0.232491	−0.116245	−	0.993221i	$-0.537086\pi$
$691$	−6.11146	−0.232491	−0.116245	+	0.993221i	$0.537086\pi$
$692$	6.94427	0.263982
$693$	3.23607	0.122928
$694$	−13.8885	−0.527202
$695$	17.8885	0.678551
$696$	2.00000	0.0758098
$697$	8.94427	0.338788
$698$	−18.6525	−0.706007
$699$	16.4721	0.623033
$700$	3.23607	0.122312
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	5.23607	0.197623
$703$	23.4164	0.883167
$704$	1.00000	0.0376889
$705$	−8.94427	−0.336861
$706$	19.5279	0.734941
$707$	−4.94427	−0.185948
$708$	−8.76393	−0.329369
$709$	−40.2492	−1.51159	−0.755796	−	0.654808i	$-0.772750\pi$
$709$	−40.2492	−1.51159	−0.755796	+	0.654808i	$0.772750\pi$
$710$	−0.763932	−0.0286699
$711$	9.23607	0.346380
$712$	−14.0000	−0.524672
$713$	−6.83282	−0.255891
$714$	3.23607	0.121107
$715$	5.23607	0.195818
$716$	10.2918	0.384622
$717$	−1.52786	−0.0570591
$718$	10.4721	0.390817
$719$	22.2918	0.831344	0.415672	−	0.909515i	$-0.363546\pi$
$719$	22.2918	0.831344	0.415672	+	0.909515i	$0.363546\pi$
$720$	1.00000	0.0372678
$721$	−38.8328	−1.44621
$722$	−8.41641	−0.313226
$723$	−6.65248	−0.247408
$724$	23.8885	0.887811
$725$	2.00000	0.0742781
$726$	1.00000	0.0371135
$727$	−15.0557	−0.558386	−0.279193	−	0.960235i	$-0.590067\pi$
$727$	−15.0557	−0.558386	−0.279193	+	0.960235i	$0.590067\pi$
$728$	−16.9443	−0.627996
$729$	1.00000	0.0370370
$730$	11.7082	0.433340
$731$	7.23607	0.267636
$732$	14.9443	0.552356
$733$	8.06888	0.298031	0.149016	−	0.988835i	$-0.452390\pi$
$733$	8.06888	0.298031	0.149016	+	0.988835i	$0.452390\pi$
$734$	−15.5279	−0.573144
$735$	−3.47214	−0.128072
$736$	−2.76393	−0.101880
$737$	0.472136	0.0173914
$738$	−8.94427	−0.329243
$739$	−1.23607	−0.0454695	−0.0227347	−	0.999742i	$-0.507237\pi$
$739$	−1.23607	−0.0454695	−0.0227347	+	0.999742i	$0.507237\pi$
$740$	−4.47214	−0.164399
$741$	27.4164	1.00717
$742$	−18.4721	−0.678133
$743$	−38.8328	−1.42464	−0.712319	−	0.701856i	$-0.752355\pi$
$743$	−38.8328	−1.42464	−0.712319	+	0.701856i	$0.752355\pi$
$744$	−2.47214	−0.0906329
$745$	−16.9443	−0.620790
$746$	24.6525	0.902591
$747$	12.9443	0.473606
$748$	1.00000	0.0365636
$749$	−41.8885	−1.53057
$750$	1.00000	0.0365148
$751$	−0.944272	−0.0344570	−0.0172285	−	0.999852i	$-0.505484\pi$
$751$	−0.944272	−0.0344570	−0.0172285	+	0.999852i	$0.505484\pi$
$752$	8.94427	0.326164
$753$	−6.29180	−0.229286
$754$	−10.4721	−0.381373
$755$	−4.94427	−0.179940
$756$	−3.23607	−0.117695
$757$	−53.3050	−1.93740	−0.968701	−	0.248232i	$-0.920151\pi$
$757$	−53.3050	−1.93740	−0.968701	+	0.248232i	$0.920151\pi$
$758$	0	0
$759$	−2.76393	−0.100324
$760$	5.23607	0.189932
$761$	−13.4164	−0.486344	−0.243172	−	0.969983i	$-0.578188\pi$
$761$	−13.4164	−0.486344	−0.243172	+	0.969983i	$0.578188\pi$
$762$	11.4164	0.413573
$763$	32.3607	1.17154
$764$	−13.8885	−0.502470
$765$	1.00000	0.0361551
$766$	−21.8885	−0.790865
$767$	45.8885	1.65694
$768$	−1.00000	−0.0360844
$769$	−9.41641	−0.339564	−0.169782	−	0.985482i	$-0.554306\pi$
$769$	−9.41641	−0.339564	−0.169782	+	0.985482i	$0.554306\pi$
$770$	−3.23607	−0.116620
$771$	−12.4721	−0.449173
$772$	−8.29180	−0.298428
$773$	47.0132	1.69095	0.845473	−	0.534018i	$-0.179319\pi$
$773$	47.0132	1.69095	0.845473	+	0.534018i	$0.179319\pi$
$774$	−7.23607	−0.260095
$775$	−2.47214	−0.0888017
$776$	12.9443	0.464672
$777$	14.4721	0.519185
$778$	−20.2918	−0.727496
$779$	−46.8328	−1.67796
$780$	−5.23607	−0.187481
$781$	0.763932	0.0273356
$782$	−2.76393	−0.0988380
$783$	−2.00000	−0.0714742
$784$	3.47214	0.124005
$785$	−12.0000	−0.428298
$786$	8.00000	0.285351
$787$	−35.7771	−1.27532	−0.637658	−	0.770320i	$-0.720097\pi$
$787$	−35.7771	−1.27532	−0.637658	+	0.770320i	$0.720097\pi$
$788$	16.4721	0.586796
$789$	−28.0000	−0.996826
$790$	−9.23607	−0.328605
$791$	−68.3607	−2.43063
$792$	−1.00000	−0.0355335
$793$	−78.2492	−2.77871
$794$	10.3607	0.367687
$795$	−5.70820	−0.202449
$796$	5.52786	0.195930
$797$	10.8754	0.385226	0.192613	−	0.981275i	$-0.438304\pi$
$797$	10.8754	0.385226	0.192613	+	0.981275i	$0.438304\pi$
$798$	−16.9443	−0.599821
$799$	8.94427	0.316426
$800$	−1.00000	−0.0353553
$801$	14.0000	0.494666
$802$	−27.1246	−0.957803
$803$	−11.7082	−0.413174
$804$	−0.472136	−0.0166510
$805$	8.94427	0.315244
$806$	12.9443	0.455943
$807$	4.47214	0.157427
$808$	1.52786	0.0537501
$809$	14.4721	0.508813	0.254407	−	0.967097i	$-0.418120\pi$
$809$	14.4721	0.508813	0.254407	+	0.967097i	$0.418120\pi$
$810$	−1.00000	−0.0351364
$811$	−45.5279	−1.59870	−0.799350	−	0.600866i	$-0.794823\pi$
$811$	−45.5279	−1.59870	−0.799350	+	0.600866i	$0.794823\pi$
$812$	6.47214	0.227127
$813$	20.3607	0.714080
$814$	4.47214	0.156748
$815$	−6.47214	−0.226709
$816$	−1.00000	−0.0350070
$817$	−37.8885	−1.32555
$818$	−6.00000	−0.209785
$819$	16.9443	0.592081
$820$	8.94427	0.312348
$821$	−38.7214	−1.35138	−0.675692	−	0.737184i	$-0.736155\pi$
$821$	−38.7214	−1.35138	−0.675692	+	0.737184i	$0.736155\pi$
$822$	−2.00000	−0.0697580
$823$	32.8328	1.14448	0.572240	−	0.820086i	$-0.306075\pi$
$823$	32.8328	1.14448	0.572240	+	0.820086i	$0.306075\pi$
$824$	12.0000	0.418040
$825$	−1.00000	−0.0348155
$826$	−28.3607	−0.986794
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	2.76393	0.0960533
$829$	24.8328	0.862479	0.431240	−	0.902237i	$-0.358076\pi$
$829$	24.8328	0.862479	0.431240	+	0.902237i	$0.358076\pi$
$830$	−12.9443	−0.449302
$831$	−18.9443	−0.657170
$832$	5.23607	0.181528
$833$	3.47214	0.120302
$834$	17.8885	0.619430
$835$	−10.4721	−0.362403
$836$	−5.23607	−0.181093
$837$	2.47214	0.0854495
$838$	10.4721	0.361754
$839$	32.5410	1.12344	0.561720	−	0.827327i	$-0.310140\pi$
$839$	32.5410	1.12344	0.561720	+	0.827327i	$0.310140\pi$
$840$	3.23607	0.111655
$841$	−25.0000	−0.862069
$842$	−31.8885	−1.09895
$843$	17.4164	0.599853
$844$	−7.41641	−0.255283
$845$	14.4164	0.495940
$846$	−8.94427	−0.307510
$847$	3.23607	0.111193
$848$	5.70820	0.196021
$849$	19.4164	0.666369
$850$	−1.00000	−0.0342997
$851$	−12.3607	−0.423719
$852$	−0.763932	−0.0261719
$853$	−38.3607	−1.31344	−0.656722	−	0.754132i	$-0.728058\pi$
$853$	−38.3607	−1.31344	−0.656722	+	0.754132i	$0.728058\pi$
$854$	48.3607	1.65487
$855$	−5.23607	−0.179070
$856$	12.9443	0.442426
$857$	−4.11146	−0.140445	−0.0702223	−	0.997531i	$-0.522371\pi$
$857$	−4.11146	−0.140445	−0.0702223	+	0.997531i	$0.522371\pi$
$858$	5.23607	0.178756
$859$	14.8328	0.506089	0.253045	−	0.967455i	$-0.418568\pi$
$859$	14.8328	0.506089	0.253045	+	0.967455i	$0.418568\pi$
$860$	7.23607	0.246748
$861$	−28.9443	−0.986418
$862$	9.41641	0.320724
$863$	12.5836	0.428350	0.214175	−	0.976795i	$-0.431294\pi$
$863$	12.5836	0.428350	0.214175	+	0.976795i	$0.431294\pi$
$864$	1.00000	0.0340207
$865$	6.94427	0.236112
$866$	20.8328	0.707928
$867$	−1.00000	−0.0339618
$868$	−8.00000	−0.271538
$869$	9.23607	0.313312
$870$	2.00000	0.0678064
$871$	2.47214	0.0837651
$872$	−10.0000	−0.338643
$873$	−12.9443	−0.438097
$874$	14.4721	0.489527
$875$	3.23607	0.109399
$876$	11.7082	0.395584
$877$	−35.3050	−1.19216	−0.596082	−	0.802924i	$-0.703277\pi$
$877$	−35.3050	−1.19216	−0.596082	+	0.802924i	$0.703277\pi$
$878$	−11.1246	−0.375437
$879$	23.8885	0.805741
$880$	1.00000	0.0337100
$881$	16.6525	0.561036	0.280518	−	0.959849i	$-0.409494\pi$
$881$	16.6525	0.561036	0.280518	+	0.959849i	$0.409494\pi$
$882$	−3.47214	−0.116913
$883$	−2.94427	−0.0990826	−0.0495413	−	0.998772i	$-0.515776\pi$
$883$	−2.94427	−0.0990826	−0.0495413	+	0.998772i	$0.515776\pi$
$884$	5.23607	0.176108
$885$	−8.76393	−0.294596
$886$	−7.70820	−0.258962
$887$	−4.58359	−0.153902	−0.0769510	−	0.997035i	$-0.524518\pi$
$887$	−4.58359	−0.153902	−0.0769510	+	0.997035i	$0.524518\pi$
$888$	−4.47214	−0.150075
$889$	36.9443	1.23907
$890$	−14.0000	−0.469281
$891$	1.00000	0.0335013
$892$	−18.4721	−0.618493
$893$	−46.8328	−1.56720
$894$	−16.9443	−0.566701
$895$	10.2918	0.344017
$896$	−3.23607	−0.108109
$897$	−14.4721	−0.483211
$898$	−16.6525	−0.555700
$899$	−4.94427	−0.164901
$900$	1.00000	0.0333333
$901$	5.70820	0.190168
$902$	−8.94427	−0.297812
$903$	−23.4164	−0.779249
$904$	21.1246	0.702594
$905$	23.8885	0.794082
$906$	−4.94427	−0.164262
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	4.00000	0.132745
$909$	−1.52786	−0.0506761
$910$	−16.9443	−0.561697
$911$	−52.1803	−1.72881	−0.864406	−	0.502795i	$-0.832305\pi$
$911$	−52.1803	−1.72881	−0.864406	+	0.502795i	$0.832305\pi$
$912$	5.23607	0.173384
$913$	12.9443	0.428393
$914$	16.8328	0.556780
$915$	14.9443	0.494042
$916$	−22.3607	−0.738818
$917$	25.8885	0.854915
$918$	1.00000	0.0330049
$919$	35.7771	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$919$	35.7771	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$920$	−2.76393	−0.0911241
$921$	−30.6525	−1.01003
$922$	−4.94427	−0.162831
$923$	4.00000	0.131662
$924$	−3.23607	−0.106459
$925$	−4.47214	−0.147043
$926$	−26.4721	−0.869928
$927$	−12.0000	−0.394132
$928$	−2.00000	−0.0656532
$929$	4.65248	0.152643	0.0763214	−	0.997083i	$-0.475682\pi$
$929$	4.65248	0.152643	0.0763214	+	0.997083i	$0.475682\pi$
$930$	−2.47214	−0.0810645
$931$	−18.1803	−0.595837
$932$	−16.4721	−0.539563
$933$	24.1803	0.791629
$934$	24.0689	0.787558
$935$	1.00000	0.0327035
$936$	−5.23607	−0.171146
$937$	40.4721	1.32217	0.661084	−	0.750312i	$-0.270097\pi$
$937$	40.4721	1.32217	0.661084	+	0.750312i	$0.270097\pi$
$938$	−1.52786	−0.0498865
$939$	−25.8885	−0.844841
$940$	8.94427	0.291730
$941$	−12.1115	−0.394822	−0.197411	−	0.980321i	$-0.563253\pi$
$941$	−12.1115	−0.394822	−0.197411	+	0.980321i	$0.563253\pi$
$942$	−12.0000	−0.390981
$943$	24.7214	0.805038
$944$	8.76393	0.285242
$945$	−3.23607	−0.105269
$946$	−7.23607	−0.235265
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−9.23607	−0.299974
$949$	−61.3050	−1.99004
$950$	5.23607	0.169880
$951$	−2.00000	−0.0648544
$952$	−3.23607	−0.104882
$953$	18.3607	0.594761	0.297380	−	0.954759i	$-0.403887\pi$
$953$	18.3607	0.594761	0.297380	+	0.954759i	$0.403887\pi$
$954$	−5.70820	−0.184810
$955$	−13.8885	−0.449423
$956$	1.52786	0.0494147
$957$	−2.00000	−0.0646508
$958$	−21.0557	−0.680280
$959$	−6.47214	−0.208996
$960$	−1.00000	−0.0322749
$961$	−24.8885	−0.802856
$962$	23.4164	0.754975
$963$	−12.9443	−0.417123
$964$	6.65248	0.214262
$965$	−8.29180	−0.266922
$966$	8.94427	0.287777
$967$	−53.6656	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$967$	−53.6656	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$968$	−1.00000	−0.0321412
$969$	5.23607	0.168207
$970$	12.9443	0.415616
$971$	−32.1803	−1.03272	−0.516358	−	0.856373i	$-0.672713\pi$
$971$	−32.1803	−1.03272	−0.516358	+	0.856373i	$0.672713\pi$
$972$	−1.00000	−0.0320750
$973$	57.8885	1.85582
$974$	−29.4164	−0.942563
$975$	−5.23607	−0.167688
$976$	−14.9443	−0.478354
$977$	56.2492	1.79957	0.899786	−	0.436331i	$-0.143722\pi$
$977$	56.2492	1.79957	0.899786	+	0.436331i	$0.143722\pi$
$978$	−6.47214	−0.206956
$979$	14.0000	0.447442
$980$	3.47214	0.110913
$981$	10.0000	0.319275
$982$	−4.47214	−0.142712
$983$	37.5967	1.19915	0.599575	−	0.800319i	$-0.295336\pi$
$983$	37.5967	1.19915	0.599575	+	0.800319i	$0.295336\pi$
$984$	8.94427	0.285133
$985$	16.4721	0.524846
$986$	−2.00000	−0.0636930
$987$	−28.9443	−0.921306
$988$	−27.4164	−0.872232
$989$	20.0000	0.635963
$990$	−1.00000	−0.0317821
$991$	−57.8885	−1.83889	−0.919445	−	0.393218i	$-0.871362\pi$
$991$	−57.8885	−1.83889	−0.919445	+	0.393218i	$0.871362\pi$
$992$	2.47214	0.0784904
$993$	−9.88854	−0.313803
$994$	−2.47214	−0.0784114
$995$	5.52786	0.175245
$996$	−12.9443	−0.410155
$997$	−22.5836	−0.715230	−0.357615	−	0.933869i	$-0.616410\pi$
$997$	−22.5836	−0.715230	−0.357615	+	0.933869i	$0.616410\pi$
$998$	−36.0000	−1.13956
$999$	4.47214	0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.br.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.br.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +5.23607 q^{13} -3.23607 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -5.23607 q^{19} +1.00000 q^{20} -3.23607 q^{21} -1.00000 q^{22} +2.76393 q^{23} +1.00000 q^{24} +1.00000 q^{25} -5.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} +2.00000 q^{29} +1.00000 q^{30} -2.47214 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.23607 q^{35} +1.00000 q^{36} -4.47214 q^{37} +5.23607 q^{38} -5.23607 q^{39} -1.00000 q^{40} +8.94427 q^{41} +3.23607 q^{42} +7.23607 q^{43} +1.00000 q^{44} +1.00000 q^{45} -2.76393 q^{46} +8.94427 q^{47} -1.00000 q^{48} +3.47214 q^{49} -1.00000 q^{50} -1.00000 q^{51} +5.23607 q^{52} +5.70820 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.23607 q^{56} +5.23607 q^{57} -2.00000 q^{58} +8.76393 q^{59} -1.00000 q^{60} -14.9443 q^{61} +2.47214 q^{62} +3.23607 q^{63} +1.00000 q^{64} +5.23607 q^{65} +1.00000 q^{66} +0.472136 q^{67} +1.00000 q^{68} -2.76393 q^{69} -3.23607 q^{70} +0.763932 q^{71} -1.00000 q^{72} -11.7082 q^{73} +4.47214 q^{74} -1.00000 q^{75} -5.23607 q^{76} +3.23607 q^{77} +5.23607 q^{78} +9.23607 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.94427 q^{82} +12.9443 q^{83} -3.23607 q^{84} +1.00000 q^{85} -7.23607 q^{86} -2.00000 q^{87} -1.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +16.9443 q^{91} +2.76393 q^{92} +2.47214 q^{93} -8.94427 q^{94} -5.23607 q^{95} +1.00000 q^{96} -12.9443 q^{97} -3.47214 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} + 6 q^{13} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} + 2 q^{24} + 2 q^{25} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 4 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} + 6 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{42} + 10 q^{43} + 2 q^{44} + 2 q^{45} - 10 q^{46} - 2 q^{48} - 2 q^{49} - 2 q^{50} - 2 q^{51} + 6 q^{52} - 2 q^{53} + 2 q^{54} + 2 q^{55} - 2 q^{56} + 6 q^{57} - 4 q^{58} + 22 q^{59} - 2 q^{60} - 12 q^{61} - 4 q^{62} + 2 q^{63} + 2 q^{64} + 6 q^{65} + 2 q^{66} - 8 q^{67} + 2 q^{68} - 10 q^{69} - 2 q^{70} + 6 q^{71} - 2 q^{72} - 10 q^{73} - 2 q^{75} - 6 q^{76} + 2 q^{77} + 6 q^{78} + 14 q^{79} + 2 q^{80} + 2 q^{81} + 8 q^{83} - 2 q^{84} + 2 q^{85} - 10 q^{86} - 4 q^{87} - 2 q^{88} + 28 q^{89} - 2 q^{90} + 16 q^{91} + 10 q^{92} - 4 q^{93} - 6 q^{95} + 2 q^{96} - 8 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 - 2 * q^12 + 6 * q^13 - 2 * q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 + 2 * q^24 + 2 * q^25 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 4 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^35 + 2 * q^36 + 6 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^42 + 10 * q^43 + 2 * q^44 + 2 * q^45 - 10 * q^46 - 2 * q^48 - 2 * q^49 - 2 * q^50 - 2 * q^51 + 6 * q^52 - 2 * q^53 + 2 * q^54 + 2 * q^55 - 2 * q^56 + 6 * q^57 - 4 * q^58 + 22 * q^59 - 2 * q^60 - 12 * q^61 - 4 * q^62 + 2 * q^63 + 2 * q^64 + 6 * q^65 + 2 * q^66 - 8 * q^67 + 2 * q^68 - 10 * q^69 - 2 * q^70 + 6 * q^71 - 2 * q^72 - 10 * q^73 - 2 * q^75 - 6 * q^76 + 2 * q^77 + 6 * q^78 + 14 * q^79 + 2 * q^80 + 2 * q^81 + 8 * q^83 - 2 * q^84 + 2 * q^85 - 10 * q^86 - 4 * q^87 - 2 * q^88 + 28 * q^89 - 2 * q^90 + 16 * q^91 + 10 * q^92 - 4 * q^93 - 6 * q^95 + 2 * q^96 - 8 * q^97 + 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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