LMFDB - Embedded newform 1003.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.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:	$8.00899532273$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.68133 q^{3} -1.00000 q^{4} -4.18953 q^{5} +1.68133 q^{6} +1.68133 q^{7} +3.00000 q^{8} -0.173127 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.68133 q^{3} -1.00000 q^{4} -4.18953 q^{5} +1.68133 q^{6} +1.68133 q^{7} +3.00000 q^{8} -0.173127 q^{9} +4.18953 q^{10} +1.68133 q^{12} +3.87086 q^{13} -1.68133 q^{14} +7.04399 q^{15} -1.00000 q^{16} -1.00000 q^{17} +0.173127 q^{18} -0.318669 q^{19} +4.18953 q^{20} -2.82687 q^{21} +5.87086 q^{23} -5.04399 q^{24} +12.5522 q^{25} -3.87086 q^{26} +5.33508 q^{27} -1.68133 q^{28} -7.55220 q^{29} -7.04399 q^{30} -6.50820 q^{31} -5.00000 q^{32} +1.00000 q^{34} -7.04399 q^{35} +0.173127 q^{36} +11.7417 q^{37} +0.318669 q^{38} -6.50820 q^{39} -12.5686 q^{40} +10.5358 q^{41} +2.82687 q^{42} -1.49180 q^{43} +0.725323 q^{45} -5.87086 q^{46} -6.88727 q^{47} +1.68133 q^{48} -4.17313 q^{49} -12.5522 q^{50} +1.68133 q^{51} -3.87086 q^{52} -12.1895 q^{53} -5.33508 q^{54} +5.04399 q^{56} +0.535789 q^{57} +7.55220 q^{58} -1.00000 q^{59} -7.04399 q^{60} -4.85446 q^{61} +6.50820 q^{62} -0.291084 q^{63} +7.00000 q^{64} -16.2171 q^{65} -10.7253 q^{67} +1.00000 q^{68} -9.87086 q^{69} +7.04399 q^{70} -2.63734 q^{71} -0.519382 q^{72} +3.36266 q^{73} -11.7417 q^{74} -21.1044 q^{75} +0.318669 q^{76} +6.50820 q^{78} +11.0768 q^{79} +4.18953 q^{80} -8.45065 q^{81} -10.5358 q^{82} +1.01641 q^{83} +2.82687 q^{84} +4.18953 q^{85} +1.49180 q^{86} +12.6977 q^{87} -11.8709 q^{89} -0.725323 q^{90} +6.50820 q^{91} -5.87086 q^{92} +10.9424 q^{93} +6.88727 q^{94} +1.33508 q^{95} +8.40665 q^{96} +17.2335 q^{97} +4.17313 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 2 q^{3} - 3 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 2 * q^3 - 3 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 9 * q^8 + 5 * q^9	$ 3 q - 3 q^{2} + 2 q^{3} - 3 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 9 q^{8} + 5 q^{9} + 4 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 3 q^{16} - 3 q^{17} - 5 q^{18} - 8 q^{19} + 4 q^{20} - 14 q^{21} + 2 q^{23} + 6 q^{24} + 15 q^{25} + 4 q^{26} + 20 q^{27} + 2 q^{28} - 18 q^{31} - 15 q^{32} + 3 q^{34} - 5 q^{36} + 4 q^{37} + 8 q^{38} - 18 q^{39} - 12 q^{40} + 12 q^{41} + 14 q^{42} - 6 q^{43} - 26 q^{45} - 2 q^{46} - 2 q^{47} - 2 q^{48} - 7 q^{49} - 15 q^{50} - 2 q^{51} + 4 q^{52} - 28 q^{53} - 20 q^{54} - 6 q^{56} - 18 q^{57} - 3 q^{59} - 2 q^{61} + 18 q^{62} - 26 q^{63} + 21 q^{64} - 22 q^{65} - 4 q^{67} + 3 q^{68} - 14 q^{69} - 22 q^{71} + 15 q^{72} - 4 q^{73} - 4 q^{74} - 18 q^{75} + 8 q^{76} + 18 q^{78} + 6 q^{79} + 4 q^{80} + 31 q^{81} - 12 q^{82} + 14 q^{84} + 4 q^{85} + 6 q^{86} + 28 q^{87} - 20 q^{89} + 26 q^{90} + 18 q^{91} - 2 q^{92} - 22 q^{93} + 2 q^{94} + 8 q^{95} - 10 q^{96} + 22 q^{97} + 7 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 2 * q^3 - 3 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 9 * q^8 + 5 * q^9 + 4 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 3 * q^16 - 3 * q^17 - 5 * q^18 - 8 * q^19 + 4 * q^20 - 14 * q^21 + 2 * q^23 + 6 * q^24 + 15 * q^25 + 4 * q^26 + 20 * q^27 + 2 * q^28 - 18 * q^31 - 15 * q^32 + 3 * q^34 - 5 * q^36 + 4 * q^37 + 8 * q^38 - 18 * q^39 - 12 * q^40 + 12 * q^41 + 14 * q^42 - 6 * q^43 - 26 * q^45 - 2 * q^46 - 2 * q^47 - 2 * q^48 - 7 * q^49 - 15 * q^50 - 2 * q^51 + 4 * q^52 - 28 * q^53 - 20 * q^54 - 6 * q^56 - 18 * q^57 - 3 * q^59 - 2 * q^61 + 18 * q^62 - 26 * q^63 + 21 * q^64 - 22 * q^65 - 4 * q^67 + 3 * q^68 - 14 * q^69 - 22 * q^71 + 15 * q^72 - 4 * q^73 - 4 * q^74 - 18 * q^75 + 8 * q^76 + 18 * q^78 + 6 * q^79 + 4 * q^80 + 31 * q^81 - 12 * q^82 + 14 * q^84 + 4 * q^85 + 6 * q^86 + 28 * q^87 - 20 * q^89 + 26 * q^90 + 18 * q^91 - 2 * q^92 - 22 * q^93 + 2 * q^94 + 8 * q^95 - 10 * q^96 + 22 * q^97 + 7 * q^98

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.68133	−0.970717	−0.485358	−	0.874315i	$-0.661311\pi$
$3$	−1.68133	−0.970717	−0.485358	+	0.874315i	$0.661311\pi$
$4$	−1.00000	−0.500000
$5$	−4.18953	−1.87362	−0.936808	−	0.349843i	$-0.886235\pi$
$5$	−4.18953	−1.87362	−0.936808	+	0.349843i	$0.886235\pi$
$6$	1.68133	0.686400
$7$	1.68133	0.635483	0.317742	−	0.948177i	$-0.397076\pi$
$7$	1.68133	0.635483	0.317742	+	0.948177i	$0.397076\pi$
$8$	3.00000	1.06066
$9$	−0.173127	−0.0577091
$10$	4.18953	1.32485
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.68133	0.485358
$13$	3.87086	1.07358	0.536792	−	0.843714i	$-0.319636\pi$
$13$	3.87086	1.07358	0.536792	+	0.843714i	$0.319636\pi$
$14$	−1.68133	−0.449355
$15$	7.04399	1.81875
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.173127	0.0408065
$19$	−0.318669	−0.0731078	−0.0365539	−	0.999332i	$-0.511638\pi$
$19$	−0.318669	−0.0731078	−0.0365539	+	0.999332i	$0.511638\pi$
$20$	4.18953	0.936808
$21$	−2.82687	−0.616874
$22$	0	0
$23$	5.87086	1.22416	0.612080	−	0.790796i	$-0.290333\pi$
$23$	5.87086	1.22416	0.612080	+	0.790796i	$0.290333\pi$
$24$	−5.04399	−1.02960
$25$	12.5522	2.51044
$26$	−3.87086	−0.759139
$27$	5.33508	1.02674
$28$	−1.68133	−0.317742
$29$	−7.55220	−1.40241	−0.701204	−	0.712961i	$-0.747354\pi$
$29$	−7.55220	−1.40241	−0.701204	+	0.712961i	$0.747354\pi$
$30$	−7.04399	−1.28605
$31$	−6.50820	−1.16891	−0.584454	−	0.811427i	$-0.698691\pi$
$31$	−6.50820	−1.16891	−0.584454	+	0.811427i	$0.698691\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	1.00000	0.171499
$35$	−7.04399	−1.19065
$36$	0.173127	0.0288545
$37$	11.7417	1.93033	0.965164	−	0.261645i	$-0.0842649\pi$
$37$	11.7417	1.93033	0.965164	+	0.261645i	$0.0842649\pi$
$38$	0.318669	0.0516950
$39$	−6.50820	−1.04215
$40$	−12.5686	−1.98727
$41$	10.5358	1.64541	0.822707	−	0.568466i	$-0.192463\pi$
$41$	10.5358	1.64541	0.822707	+	0.568466i	$0.192463\pi$
$42$	2.82687	0.436196
$43$	−1.49180	−0.227497	−0.113748	−	0.993510i	$-0.536286\pi$
$43$	−1.49180	−0.227497	−0.113748	+	0.993510i	$0.536286\pi$
$44$	0	0
$45$	0.725323	0.108125
$46$	−5.87086	−0.865612
$47$	−6.88727	−1.00461	−0.502306	−	0.864690i	$-0.667515\pi$
$47$	−6.88727	−1.00461	−0.502306	+	0.864690i	$0.667515\pi$
$48$	1.68133	0.242679
$49$	−4.17313	−0.596161
$50$	−12.5522	−1.77515
$51$	1.68133	0.235433
$52$	−3.87086	−0.536792
$53$	−12.1895	−1.67436	−0.837181	−	0.546926i	$-0.815798\pi$
$53$	−12.1895	−1.67436	−0.837181	+	0.546926i	$0.815798\pi$
$54$	−5.33508	−0.726012
$55$	0	0
$56$	5.04399	0.674032
$57$	0.535789	0.0709669
$58$	7.55220	0.991652
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−7.04399	−0.909375
$61$	−4.85446	−0.621550	−0.310775	−	0.950484i	$-0.600589\pi$
$61$	−4.85446	−0.621550	−0.310775	+	0.950484i	$0.600589\pi$
$62$	6.50820	0.826543
$63$	−0.291084	−0.0366732
$64$	7.00000	0.875000
$65$	−16.2171	−2.01149
$66$	0	0
$67$	−10.7253	−1.31031	−0.655153	−	0.755496i	$-0.727396\pi$
$67$	−10.7253	−1.31031	−0.655153	+	0.755496i	$0.727396\pi$
$68$	1.00000	0.121268
$69$	−9.87086	−1.18831
$70$	7.04399	0.841918
$71$	−2.63734	−0.312995	−0.156497	−	0.987678i	$-0.550020\pi$
$71$	−2.63734	−0.312995	−0.156497	+	0.987678i	$0.550020\pi$
$72$	−0.519382	−0.0612097
$73$	3.36266	0.393570	0.196785	−	0.980447i	$-0.436950\pi$
$73$	3.36266	0.393570	0.196785	+	0.980447i	$0.436950\pi$
$74$	−11.7417	−1.36495
$75$	−21.1044	−2.43693
$76$	0.318669	0.0365539
$77$	0	0
$78$	6.50820	0.736909
$79$	11.0768	1.24624	0.623119	−	0.782127i	$-0.285865\pi$
$79$	11.0768	1.24624	0.623119	+	0.782127i	$0.285865\pi$
$80$	4.18953	0.468404
$81$	−8.45065	−0.938961
$82$	−10.5358	−1.16348
$83$	1.01641	0.111565	0.0557826	−	0.998443i	$-0.482235\pi$
$83$	1.01641	0.111565	0.0557826	+	0.998443i	$0.482235\pi$
$84$	2.82687	0.308437
$85$	4.18953	0.454419
$86$	1.49180	0.160865
$87$	12.6977	1.36134
$88$	0	0
$89$	−11.8709	−1.25831	−0.629155	−	0.777280i	$-0.716599\pi$
$89$	−11.8709	−1.25831	−0.629155	+	0.777280i	$0.716599\pi$
$90$	−0.725323	−0.0764557
$91$	6.50820	0.682245
$92$	−5.87086	−0.612080
$93$	10.9424	1.13468
$94$	6.88727	0.710368
$95$	1.33508	0.136976
$96$	8.40665	0.858000
$97$	17.2335	1.74980	0.874900	−	0.484304i	$-0.160927\pi$
$97$	17.2335	1.74980	0.874900	+	0.484304i	$0.160927\pi$
$98$	4.17313	0.421550
$99$	0	0
$100$	−12.5522	−1.25522
$101$	6.37907	0.634741	0.317370	−	0.948302i	$-0.397200\pi$
$101$	6.37907	0.634741	0.317370	+	0.948302i	$0.397200\pi$
$102$	−1.68133	−0.166477
$103$	−9.23353	−0.909806	−0.454903	−	0.890541i	$-0.650326\pi$
$103$	−9.23353	−0.909806	−0.454903	+	0.890541i	$0.650326\pi$
$104$	11.6126	1.13871
$105$	11.8433	1.15579
$106$	12.1895	1.18395
$107$	2.31867	0.224154	0.112077	−	0.993700i	$-0.464250\pi$
$107$	2.31867	0.224154	0.112077	+	0.993700i	$0.464250\pi$
$108$	−5.33508	−0.513368
$109$	8.37907	0.802569	0.401285	−	0.915953i	$-0.368564\pi$
$109$	8.37907	0.802569	0.401285	+	0.915953i	$0.368564\pi$
$110$	0	0
$111$	−19.7417	−1.87380
$112$	−1.68133	−0.158871
$113$	8.59619	0.808661	0.404331	−	0.914613i	$-0.367505\pi$
$113$	8.59619	0.808661	0.404331	+	0.914613i	$0.367505\pi$
$114$	−0.535789	−0.0501812
$115$	−24.5962	−2.29361
$116$	7.55220	0.701204
$117$	−0.670152	−0.0619556
$118$	1.00000	0.0920575
$119$	−1.68133	−0.154127
$120$	21.1320	1.92908
$121$	−11.0000	−1.00000
$122$	4.85446	0.439502
$123$	−17.7141	−1.59723
$124$	6.50820	0.584454
$125$	−31.6402	−2.82998
$126$	0.291084	0.0259318
$127$	−1.33508	−0.118469	−0.0592344	−	0.998244i	$-0.518866\pi$
$127$	−1.33508	−0.118469	−0.0592344	+	0.998244i	$0.518866\pi$
$128$	3.00000	0.265165
$129$	2.50820	0.220835
$130$	16.2171	1.42234
$131$	1.65375	0.144488	0.0722442	−	0.997387i	$-0.476984\pi$
$131$	1.65375	0.144488	0.0722442	+	0.997387i	$0.476984\pi$
$132$	0	0
$133$	−0.535789	−0.0464588
$134$	10.7253	0.926527
$135$	−22.3515	−1.92371
$136$	−3.00000	−0.257248
$137$	3.81047	0.325550	0.162775	−	0.986663i	$-0.447956\pi$
$137$	3.81047	0.325550	0.162775	+	0.986663i	$0.447956\pi$
$138$	9.87086	0.840264
$139$	−8.08798	−0.686014	−0.343007	−	0.939333i	$-0.611445\pi$
$139$	−8.08798	−0.686014	−0.343007	+	0.939333i	$0.611445\pi$
$140$	7.04399	0.595326
$141$	11.5798	0.975193
$142$	2.63734	0.221321
$143$	0	0
$144$	0.173127	0.0144273
$145$	31.6402	2.62757
$146$	−3.36266	−0.278296
$147$	7.01641	0.578703
$148$	−11.7417	−0.965164
$149$	−17.9588	−1.47125	−0.735623	−	0.677391i	$-0.763111\pi$
$149$	−17.9588	−1.47125	−0.735623	+	0.677391i	$0.763111\pi$
$150$	21.1044	1.72317
$151$	10.9836	0.893832	0.446916	−	0.894576i	$-0.352522\pi$
$151$	10.9836	0.893832	0.446916	+	0.894576i	$0.352522\pi$
$152$	−0.956008	−0.0775425
$153$	0.173127	0.0139965
$154$	0	0
$155$	27.2663	2.19009
$156$	6.50820	0.521073
$157$	−4.50820	−0.359794	−0.179897	−	0.983685i	$-0.557576\pi$
$157$	−4.50820	−0.359794	−0.179897	+	0.983685i	$0.557576\pi$
$158$	−11.0768	−0.881223
$159$	20.4946	1.62533
$160$	20.9477	1.65606
$161$	9.87086	0.777933
$162$	8.45065	0.663945
$163$	−1.36266	−0.106732	−0.0533659	−	0.998575i	$-0.516995\pi$
$163$	−1.36266	−0.106732	−0.0533659	+	0.998575i	$0.516995\pi$
$164$	−10.5358	−0.822707
$165$	0	0
$166$	−1.01641	−0.0788885
$167$	−14.0604	−1.08803	−0.544013	−	0.839077i	$-0.683096\pi$
$167$	−14.0604	−1.08803	−0.544013	+	0.839077i	$0.683096\pi$
$168$	−8.48062	−0.654294
$169$	1.98359	0.152584
$170$	−4.18953	−0.321323
$171$	0.0551704	0.00421898
$172$	1.49180	0.113748
$173$	−11.5246	−0.876200	−0.438100	−	0.898926i	$-0.644348\pi$
$173$	−11.5246	−0.876200	−0.438100	+	0.898926i	$0.644348\pi$
$174$	−12.6977	−0.962613
$175$	21.1044	1.59534
$176$	0	0
$177$	1.68133	0.126377
$178$	11.8709	0.889759
$179$	−7.14554	−0.534083	−0.267041	−	0.963685i	$-0.586046\pi$
$179$	−7.14554	−0.534083	−0.267041	+	0.963685i	$0.586046\pi$
$180$	−0.725323	−0.0540624
$181$	−7.55220	−0.561350	−0.280675	−	0.959803i	$-0.590558\pi$
$181$	−7.55220	−0.561350	−0.280675	+	0.959803i	$0.590558\pi$
$182$	−6.50820	−0.482420
$183$	8.16195	0.603349
$184$	17.6126	1.29842
$185$	−49.1924	−3.61670
$186$	−10.9424	−0.802339
$187$	0	0
$188$	6.88727	0.502306
$189$	8.97003	0.652473
$190$	−1.33508	−0.0968566
$191$	21.6126	1.56383	0.781916	−	0.623384i	$-0.214242\pi$
$191$	21.6126	1.56383	0.781916	+	0.623384i	$0.214242\pi$
$192$	−11.7693	−0.849377
$193$	−13.8984	−1.00043	−0.500216	−	0.865901i	$-0.666746\pi$
$193$	−13.8984	−1.00043	−0.500216	+	0.865901i	$0.666746\pi$
$194$	−17.2335	−1.23730
$195$	27.2663	1.95258
$196$	4.17313	0.298081
$197$	−16.7253	−1.19163	−0.595815	−	0.803122i	$-0.703171\pi$
$197$	−16.7253	−1.19163	−0.595815	+	0.803122i	$0.703171\pi$
$198$	0	0
$199$	−19.0768	−1.35232	−0.676159	−	0.736755i	$-0.736357\pi$
$199$	−19.0768	−1.35232	−0.676159	+	0.736755i	$0.736357\pi$
$200$	37.6566	2.66272
$201$	18.0328	1.27194
$202$	−6.37907	−0.448830
$203$	−12.6977	−0.891206
$204$	−1.68133	−0.117717
$205$	−44.1400	−3.08287
$206$	9.23353	0.643330
$207$	−1.01641	−0.0706452
$208$	−3.87086	−0.268396
$209$	0	0
$210$	−11.8433	−0.817264
$211$	21.9260	1.50945	0.754725	−	0.656041i	$-0.227770\pi$
$211$	21.9260	1.50945	0.754725	+	0.656041i	$0.227770\pi$
$212$	12.1895	0.837181
$213$	4.43424	0.303829
$214$	−2.31867	−0.158501
$215$	6.24993	0.426242
$216$	16.0052	1.08902
$217$	−10.9424	−0.742821
$218$	−8.37907	−0.567502
$219$	−5.65375	−0.382045
$220$	0	0
$221$	−3.87086	−0.260383
$222$	19.7417	1.32498
$223$	2.98359	0.199796	0.0998981	−	0.994998i	$-0.468148\pi$
$223$	2.98359	0.199796	0.0998981	+	0.994998i	$0.468148\pi$
$224$	−8.40665	−0.561693
$225$	−2.17313	−0.144875
$226$	−8.59619	−0.571810
$227$	−22.0880	−1.46603	−0.733015	−	0.680212i	$-0.761888\pi$
$227$	−22.0880	−1.46603	−0.733015	+	0.680212i	$0.761888\pi$
$228$	−0.535789	−0.0354835
$229$	−4.46705	−0.295191	−0.147596	−	0.989048i	$-0.547153\pi$
$229$	−4.46705	−0.295191	−0.147596	+	0.989048i	$0.547153\pi$
$230$	24.5962	1.62182
$231$	0	0
$232$	−22.6566	−1.48748
$233$	16.7581	1.09786	0.548931	−	0.835868i	$-0.315035\pi$
$233$	16.7581	1.09786	0.548931	+	0.835868i	$0.315035\pi$
$234$	0.670152	0.0438092
$235$	28.8545	1.88226
$236$	1.00000	0.0650945
$237$	−18.6238	−1.20974
$238$	1.68133	0.108984
$239$	29.7693	1.92562	0.962808	−	0.270185i	$-0.0870849\pi$
$239$	29.7693	1.92562	0.962808	+	0.270185i	$0.0870849\pi$
$240$	−7.04399	−0.454688
$241$	2.48062	0.159791	0.0798953	−	0.996803i	$-0.474541\pi$
$241$	2.48062	0.159791	0.0798953	+	0.996803i	$0.474541\pi$
$242$	11.0000	0.707107
$243$	−1.79690	−0.115271
$244$	4.85446	0.310775
$245$	17.4835	1.11698
$246$	17.7141	1.12941
$247$	−1.23353	−0.0784874
$248$	−19.5246	−1.23981
$249$	−1.70892	−0.108298
$250$	31.6402	2.00110
$251$	−21.4559	−1.35428	−0.677141	−	0.735853i	$-0.736781\pi$
$251$	−21.4559	−1.35428	−0.677141	+	0.735853i	$0.736781\pi$
$252$	0.291084	0.0183366
$253$	0	0
$254$	1.33508	0.0837701
$255$	−7.04399	−0.441112
$256$	−17.0000	−1.06250
$257$	−30.2775	−1.88866	−0.944330	−	0.329000i	$-0.893288\pi$
$257$	−30.2775	−1.88866	−0.944330	+	0.329000i	$0.893288\pi$
$258$	−2.50820	−0.156154
$259$	19.7417	1.22669
$260$	16.2171	1.00574
$261$	1.30749	0.0809317
$262$	−1.65375	−0.102169
$263$	15.0440	0.927652	0.463826	−	0.885926i	$-0.346476\pi$
$263$	15.0440	0.927652	0.463826	+	0.885926i	$0.346476\pi$
$264$	0	0
$265$	51.0685	3.13711
$266$	0.535789	0.0328513
$267$	19.9588	1.22146
$268$	10.7253	0.655153
$269$	−12.6373	−0.770512	−0.385256	−	0.922810i	$-0.625887\pi$
$269$	−12.6373	−0.770512	−0.385256	+	0.922810i	$0.625887\pi$
$270$	22.3515	1.36027
$271$	19.6813	1.19556	0.597778	−	0.801662i	$-0.296050\pi$
$271$	19.6813	1.19556	0.597778	+	0.801662i	$0.296050\pi$
$272$	1.00000	0.0606339
$273$	−10.9424	−0.662267
$274$	−3.81047	−0.230199
$275$	0	0
$276$	9.87086	0.594156
$277$	−13.8984	−0.835077	−0.417538	−	0.908659i	$-0.637107\pi$
$277$	−13.8984	−0.835077	−0.417538	+	0.908659i	$0.637107\pi$
$278$	8.08798	0.485085
$279$	1.12675	0.0674566
$280$	−21.1320	−1.26288
$281$	−11.2388	−0.670448	−0.335224	−	0.942138i	$-0.608812\pi$
$281$	−11.2388	−0.670448	−0.335224	+	0.942138i	$0.608812\pi$
$282$	−11.5798	−0.689566
$283$	−22.6290	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$283$	−22.6290	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$284$	2.63734	0.156497
$285$	−2.24470	−0.132965
$286$	0	0
$287$	17.7141	1.04563
$288$	0.865636	0.0510081
$289$	1.00000	0.0588235
$290$	−31.6402	−1.85798
$291$	−28.9753	−1.69856
$292$	−3.36266	−0.196785
$293$	14.6566	0.856247	0.428123	−	0.903720i	$-0.359175\pi$
$293$	14.6566	0.856247	0.428123	+	0.903720i	$0.359175\pi$
$294$	−7.01641	−0.409205
$295$	4.18953	0.243924
$296$	35.2252	2.04742
$297$	0	0
$298$	17.9588	1.04033
$299$	22.7253	1.31424
$300$	21.1044	1.21846
$301$	−2.50820	−0.144570
$302$	−10.9836	−0.632035
$303$	−10.7253	−0.616154
$304$	0.318669	0.0182769
$305$	20.3379	1.16455
$306$	−0.173127	−0.00989703
$307$	−33.1976	−1.89469	−0.947344	−	0.320219i	$-0.896244\pi$
$307$	−33.1976	−1.89469	−0.947344	+	0.320219i	$0.896244\pi$
$308$	0	0
$309$	15.5246	0.883164
$310$	−27.2663	−1.54862
$311$	13.6813	0.775797	0.387899	−	0.921702i	$-0.373201\pi$
$311$	13.6813	0.775797	0.387899	+	0.921702i	$0.373201\pi$
$312$	−19.5246	−1.10536
$313$	5.61259	0.317243	0.158621	−	0.987339i	$-0.449295\pi$
$313$	5.61259	0.317243	0.158621	+	0.987339i	$0.449295\pi$
$314$	4.50820	0.254413
$315$	1.21951	0.0687114
$316$	−11.0768	−0.623119
$317$	7.27468	0.408587	0.204293	−	0.978910i	$-0.434510\pi$
$317$	7.27468	0.408587	0.204293	+	0.978910i	$0.434510\pi$
$318$	−20.4946	−1.14928
$319$	0	0
$320$	−29.3267	−1.63941
$321$	−3.89845	−0.217590
$322$	−9.87086	−0.550082
$323$	0.318669	0.0177312
$324$	8.45065	0.469480
$325$	48.5878	2.69517
$326$	1.36266	0.0754708
$327$	−14.0880	−0.779067
$328$	31.6074	1.74522
$329$	−11.5798	−0.638414
$330$	0	0
$331$	16.4395	0.903595	0.451797	−	0.892121i	$-0.350783\pi$
$331$	16.4395	0.903595	0.451797	+	0.892121i	$0.350783\pi$
$332$	−1.01641	−0.0557826
$333$	−2.03281	−0.111397
$334$	14.0604	0.769351
$335$	44.9341	2.45501
$336$	2.82687	0.154219
$337$	−17.0716	−0.929948	−0.464974	−	0.885324i	$-0.653936\pi$
$337$	−17.0716	−0.929948	−0.464974	+	0.885324i	$0.653936\pi$
$338$	−1.98359	−0.107893
$339$	−14.4530	−0.784981
$340$	−4.18953	−0.227209
$341$	0	0
$342$	−0.0551704	−0.00298327
$343$	−18.7857	−1.01433
$344$	−4.47539	−0.241297
$345$	41.3543	2.22644
$346$	11.5246	0.619567
$347$	−1.77454	−0.0952625	−0.0476312	−	0.998865i	$-0.515167\pi$
$347$	−1.77454	−0.0952625	−0.0476312	+	0.998865i	$0.515167\pi$
$348$	−12.6977	−0.680670
$349$	−8.03281	−0.429987	−0.214993	−	0.976616i	$-0.568973\pi$
$349$	−8.03281	−0.429987	−0.214993	+	0.976616i	$0.568973\pi$
$350$	−21.1044	−1.12808
$351$	20.6514	1.10229
$352$	0	0
$353$	6.05517	0.322284	0.161142	−	0.986931i	$-0.448482\pi$
$353$	6.05517	0.322284	0.161142	+	0.986931i	$0.448482\pi$
$354$	−1.68133	−0.0893617
$355$	11.0492	0.586432
$356$	11.8709	0.629155
$357$	2.82687	0.149614
$358$	7.14554	0.377654
$359$	−32.4946	−1.71500	−0.857501	−	0.514483i	$-0.827984\pi$
$359$	−32.4946	−1.71500	−0.857501	+	0.514483i	$0.827984\pi$
$360$	2.17597	0.114684
$361$	−18.8984	−0.994655
$362$	7.55220	0.396935
$363$	18.4946	0.970717
$364$	−6.50820	−0.341123
$365$	−14.0880	−0.737399
$366$	−8.16195	−0.426632
$367$	14.4671	0.755174	0.377587	−	0.925974i	$-0.376754\pi$
$367$	14.4671	0.755174	0.377587	+	0.925974i	$0.376754\pi$
$368$	−5.87086	−0.306040
$369$	−1.82403	−0.0949553
$370$	49.1924	2.55739
$371$	−20.4946	−1.06403
$372$	−10.9424	−0.567339
$373$	−2.95078	−0.152786	−0.0763928	−	0.997078i	$-0.524340\pi$
$373$	−2.95078	−0.152786	−0.0763928	+	0.997078i	$0.524340\pi$
$374$	0	0
$375$	53.1976	2.74711
$376$	−20.6618	−1.06555
$377$	−29.2335	−1.50560
$378$	−8.97003	−0.461368
$379$	10.3738	0.532868	0.266434	−	0.963853i	$-0.414155\pi$
$379$	10.3738	0.532868	0.266434	+	0.963853i	$0.414155\pi$
$380$	−1.33508	−0.0684880
$381$	2.24470	0.115000
$382$	−21.6126	−1.10580
$383$	−29.1924	−1.49166	−0.745830	−	0.666136i	$-0.767947\pi$
$383$	−29.1924	−1.49166	−0.745830	+	0.666136i	$0.767947\pi$
$384$	−5.04399	−0.257400
$385$	0	0
$386$	13.8984	0.707412
$387$	0.258271	0.0131286
$388$	−17.2335	−0.874900
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−27.2663	−1.38068
$391$	−5.87086	−0.296902
$392$	−12.5194	−0.632324
$393$	−2.78049	−0.140257
$394$	16.7253	0.842610
$395$	−46.4067	−2.33497
$396$	0	0
$397$	−34.8873	−1.75094	−0.875471	−	0.483271i	$-0.839449\pi$
$397$	−34.8873	−1.75094	−0.875471	+	0.483271i	$0.839449\pi$
$398$	19.0768	0.956234
$399$	0.900838	0.0450983
$400$	−12.5522	−0.627610
$401$	0.733661	0.0366373	0.0183186	−	0.999832i	$-0.494169\pi$
$401$	0.733661	0.0366373	0.0183186	+	0.999832i	$0.494169\pi$
$402$	−18.0328	−0.899395
$403$	−25.1924	−1.25492
$404$	−6.37907	−0.317370
$405$	35.4043	1.75925
$406$	12.6977	0.630178
$407$	0	0
$408$	5.04399	0.249715
$409$	0.129135	0.00638533	0.00319267	−	0.999995i	$-0.498984\pi$
$409$	0.129135	0.00638533	0.00319267	+	0.999995i	$0.498984\pi$
$410$	44.1400	2.17992
$411$	−6.40665	−0.316017
$412$	9.23353	0.454903
$413$	−1.68133	−0.0827329
$414$	1.01641	0.0499537
$415$	−4.25827	−0.209030
$416$	−19.3543	−0.948924
$417$	13.5986	0.665925
$418$	0	0
$419$	−7.78288	−0.380219	−0.190109	−	0.981763i	$-0.560884\pi$
$419$	−7.78288	−0.380219	−0.190109	+	0.981763i	$0.560884\pi$
$420$	−11.8433	−0.577893
$421$	4.38741	0.213829	0.106915	−	0.994268i	$-0.465903\pi$
$421$	4.38741	0.213829	0.106915	+	0.994268i	$0.465903\pi$
$422$	−21.9260	−1.06734
$423$	1.19237	0.0579752
$424$	−36.5686	−1.77593
$425$	−12.5522	−0.608871
$426$	−4.43424	−0.214840
$427$	−8.16195	−0.394984
$428$	−2.31867	−0.112077
$429$	0	0
$430$	−6.24993	−0.301399
$431$	12.4999	0.602097	0.301049	−	0.953609i	$-0.402663\pi$
$431$	12.4999	0.602097	0.301049	+	0.953609i	$0.402663\pi$
$432$	−5.33508	−0.256684
$433$	−1.14031	−0.0548000	−0.0274000	−	0.999625i	$-0.508723\pi$
$433$	−1.14031	−0.0548000	−0.0274000	+	0.999625i	$0.508723\pi$
$434$	10.9424	0.525254
$435$	−53.1976	−2.55063
$436$	−8.37907	−0.401285
$437$	−1.87086	−0.0894956
$438$	5.65375	0.270146
$439$	1.62093	0.0773629	0.0386814	−	0.999252i	$-0.487684\pi$
$439$	1.62093	0.0773629	0.0386814	+	0.999252i	$0.487684\pi$
$440$	0	0
$441$	0.722482	0.0344039
$442$	3.87086	0.184118
$443$	30.7498	1.46097	0.730483	−	0.682930i	$-0.239295\pi$
$443$	30.7498	1.46097	0.730483	+	0.682930i	$0.239295\pi$
$444$	19.7417	0.936901
$445$	49.7334	2.35759
$446$	−2.98359	−0.141277
$447$	30.1948	1.42816
$448$	11.7693	0.556048
$449$	−31.2283	−1.47375	−0.736877	−	0.676027i	$-0.763700\pi$
$449$	−31.2283	−1.47375	−0.736877	+	0.676027i	$0.763700\pi$
$450$	2.17313	0.102442
$451$	0	0
$452$	−8.59619	−0.404331
$453$	−18.4671	−0.867658
$454$	22.0880	1.03664
$455$	−27.2663	−1.27827
$456$	1.60737	0.0752718
$457$	−3.32985	−0.155764	−0.0778819	−	0.996963i	$-0.524816\pi$
$457$	−3.32985	−0.155764	−0.0778819	+	0.996963i	$0.524816\pi$
$458$	4.46705	0.208732
$459$	−5.33508	−0.249020
$460$	24.5962	1.14680
$461$	1.93437	0.0900927	0.0450464	−	0.998985i	$-0.485656\pi$
$461$	1.93437	0.0900927	0.0450464	+	0.998985i	$0.485656\pi$
$462$	0	0
$463$	18.6290	0.865763	0.432882	−	0.901451i	$-0.357497\pi$
$463$	18.6290	0.865763	0.432882	+	0.901451i	$0.357497\pi$
$464$	7.55220	0.350602
$465$	−45.8437	−2.12595
$466$	−16.7581	−0.776306
$467$	26.0880	1.20721	0.603604	−	0.797284i	$-0.293731\pi$
$467$	26.0880	1.20721	0.603604	+	0.797284i	$0.293731\pi$
$468$	0.670152	0.0309778
$469$	−18.0328	−0.832678
$470$	−28.8545	−1.33096
$471$	7.57978	0.349258
$472$	−3.00000	−0.138086
$473$	0	0
$474$	18.6238	0.855418
$475$	−4.00000	−0.183533
$476$	1.68133	0.0770637
$477$	2.11034	0.0966259
$478$	−29.7693	−1.36162
$479$	2.89561	0.132304	0.0661519	−	0.997810i	$-0.478928\pi$
$479$	2.89561	0.132304	0.0661519	+	0.997810i	$0.478928\pi$
$480$	−35.2200	−1.60756
$481$	45.4506	2.07237
$482$	−2.48062	−0.112989
$483$	−16.5962	−0.755153
$484$	11.0000	0.500000
$485$	−72.2004	−3.27845
$486$	1.79690	0.0815090
$487$	9.42306	0.427000	0.213500	−	0.976943i	$-0.431514\pi$
$487$	9.42306	0.427000	0.213500	+	0.976943i	$0.431514\pi$
$488$	−14.5634	−0.659253
$489$	2.29108	0.103606
$490$	−17.4835	−0.789822
$491$	−35.1648	−1.58696	−0.793482	−	0.608593i	$-0.791734\pi$
$491$	−35.1648	−1.58696	−0.793482	+	0.608593i	$0.791734\pi$
$492$	17.7141	0.798615
$493$	7.55220	0.340134
$494$	1.23353	0.0554990
$495$	0	0
$496$	6.50820	0.292227
$497$	−4.43424	−0.198903
$498$	1.70892	0.0765783
$499$	−39.8245	−1.78279	−0.891394	−	0.453228i	$-0.850272\pi$
$499$	−39.8245	−1.78279	−0.891394	+	0.453228i	$0.850272\pi$
$500$	31.6402	1.41499
$501$	23.6402	1.05617
$502$	21.4559	0.957622
$503$	−29.7089	−1.32466	−0.662328	−	0.749214i	$-0.730431\pi$
$503$	−29.7089	−1.32466	−0.662328	+	0.749214i	$0.730431\pi$
$504$	−0.873253	−0.0388978
$505$	−26.7253	−1.18926
$506$	0	0
$507$	−3.33508	−0.148116
$508$	1.33508	0.0592344
$509$	−7.49180	−0.332068	−0.166034	−	0.986120i	$-0.553096\pi$
$509$	−7.49180	−0.332068	−0.166034	+	0.986120i	$0.553096\pi$
$510$	7.04399	0.311913
$511$	5.65375	0.250107
$512$	11.0000	0.486136
$513$	−1.70013	−0.0750624
$514$	30.2775	1.33548
$515$	38.6842	1.70463
$516$	−2.50820	−0.110417
$517$	0	0
$518$	−19.7417	−0.867402
$519$	19.3767	0.850542
$520$	−48.6514	−2.13350
$521$	−16.0328	−0.702410	−0.351205	−	0.936299i	$-0.614228\pi$
$521$	−16.0328	−0.702410	−0.351205	+	0.936299i	$0.614228\pi$
$522$	−1.30749	−0.0572273
$523$	36.4946	1.59580	0.797900	−	0.602790i	$-0.205945\pi$
$523$	36.4946	1.59580	0.797900	+	0.602790i	$0.205945\pi$
$524$	−1.65375	−0.0722442
$525$	−35.4835	−1.54863
$526$	−15.0440	−0.655949
$527$	6.50820	0.283502
$528$	0	0
$529$	11.4671	0.498567
$530$	−51.0685	−2.21827
$531$	0.173127	0.00751308
$532$	0.535789	0.0232294
$533$	40.7826	1.76649
$534$	−19.9588	−0.863704
$535$	−9.71414	−0.419979
$536$	−32.1760	−1.38979
$537$	12.0140	0.518443
$538$	12.6373	0.544834
$539$	0	0
$540$	22.3515	0.961855
$541$	−6.12914	−0.263512	−0.131756	−	0.991282i	$-0.542062\pi$
$541$	−6.12914	−0.263512	−0.131756	+	0.991282i	$0.542062\pi$
$542$	−19.6813	−0.845386
$543$	12.6977	0.544912
$544$	5.00000	0.214373
$545$	−35.1044	−1.50371
$546$	10.9424	0.468293
$547$	−28.4119	−1.21480	−0.607402	−	0.794394i	$-0.707788\pi$
$547$	−28.4119	−1.21480	−0.607402	+	0.794394i	$0.707788\pi$
$548$	−3.81047	−0.162775
$549$	0.840439	0.0358691
$550$	0	0
$551$	2.40665	0.102527
$552$	−29.6126	−1.26040
$553$	18.6238	0.791963
$554$	13.8984	0.590489
$555$	82.7086	3.51079
$556$	8.08798	0.343007
$557$	−7.93126	−0.336058	−0.168029	−	0.985782i	$-0.553740\pi$
$557$	−7.93126	−0.336058	−0.168029	+	0.985782i	$0.553740\pi$
$558$	−1.12675	−0.0476990
$559$	−5.77454	−0.244237
$560$	7.04399	0.297663
$561$	0	0
$562$	11.2388	0.474078
$563$	−0.379068	−0.0159758	−0.00798791	−	0.999968i	$-0.502543\pi$
$563$	−0.379068	−0.0159758	−0.00798791	+	0.999968i	$0.502543\pi$
$564$	−11.5798	−0.487597
$565$	−36.0140	−1.51512
$566$	22.6290	0.951168
$567$	−14.2083	−0.596694
$568$	−7.91202	−0.331981
$569$	−17.0492	−0.714740	−0.357370	−	0.933963i	$-0.616327\pi$
$569$	−17.0492	−0.714740	−0.357370	+	0.933963i	$0.616327\pi$
$570$	2.24470	0.0940203
$571$	44.9997	1.88318	0.941590	−	0.336761i	$-0.109332\pi$
$571$	44.9997	1.88318	0.941590	+	0.336761i	$0.109332\pi$
$572$	0	0
$573$	−36.3379	−1.51804
$574$	−17.7141	−0.739374
$575$	73.6922	3.07318
$576$	−1.21189	−0.0504955
$577$	−14.5358	−0.605133	−0.302566	−	0.953128i	$-0.597843\pi$
$577$	−14.5358	−0.605133	−0.302566	+	0.953128i	$0.597843\pi$
$578$	−1.00000	−0.0415945
$579$	23.3679	0.971136
$580$	−31.6402	−1.31379
$581$	1.70892	0.0708978
$582$	28.9753	1.20106
$583$	0	0
$584$	10.0880	0.417444
$585$	2.80763	0.116081
$586$	−14.6566	−0.605458
$587$	18.6842	0.771178	0.385589	−	0.922671i	$-0.373998\pi$
$587$	18.6842	0.771178	0.385589	+	0.922671i	$0.373998\pi$
$588$	−7.01641	−0.289352
$589$	2.07396	0.0854562
$590$	−4.18953	−0.172480
$591$	28.1208	1.15674
$592$	−11.7417	−0.482582
$593$	−6.10155	−0.250561	−0.125280	−	0.992121i	$-0.539983\pi$
$593$	−6.10155	−0.250561	−0.125280	+	0.992121i	$0.539983\pi$
$594$	0	0
$595$	7.04399	0.288776
$596$	17.9588	0.735623
$597$	32.0744	1.31272
$598$	−22.7253	−0.929308
$599$	11.9396	0.487839	0.243919	−	0.969795i	$-0.421567\pi$
$599$	11.9396	0.487839	0.243919	+	0.969795i	$0.421567\pi$
$600$	−63.3132	−2.58475
$601$	35.2252	1.43687	0.718433	−	0.695596i	$-0.244860\pi$
$601$	35.2252	1.43687	0.718433	+	0.695596i	$0.244860\pi$
$602$	2.50820	0.102227
$603$	1.85685	0.0756166
$604$	−10.9836	−0.446916
$605$	46.0849	1.87362
$606$	10.7253	0.435686
$607$	−19.0112	−0.771640	−0.385820	−	0.922574i	$-0.626081\pi$
$607$	−19.0112	−0.771640	−0.385820	+	0.922574i	$0.626081\pi$
$608$	1.59335	0.0646187
$609$	21.3491	0.865109
$610$	−20.3379	−0.823458
$611$	−26.6597	−1.07854
$612$	−0.173127	−0.00699825
$613$	3.61259	0.145911	0.0729556	−	0.997335i	$-0.476757\pi$
$613$	3.61259	0.145911	0.0729556	+	0.997335i	$0.476757\pi$
$614$	33.1976	1.33975
$615$	74.2140	2.99260
$616$	0	0
$617$	42.9149	1.72769	0.863844	−	0.503760i	$-0.168050\pi$
$617$	42.9149	1.72769	0.863844	+	0.503760i	$0.168050\pi$
$618$	−15.5246	−0.624491
$619$	−30.9945	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$619$	−30.9945	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$620$	−27.2663	−1.09504
$621$	31.3215	1.25689
$622$	−13.6813	−0.548571
$623$	−19.9588	−0.799634
$624$	6.50820	0.260537
$625$	69.7966	2.79187
$626$	−5.61259	−0.224324
$627$	0	0
$628$	4.50820	0.179897
$629$	−11.7417	−0.468173
$630$	−1.21951	−0.0485863
$631$	−26.2088	−1.04336	−0.521678	−	0.853143i	$-0.674694\pi$
$631$	−26.2088	−1.04336	−0.521678	+	0.853143i	$0.674694\pi$
$632$	33.2304	1.32183
$633$	−36.8649	−1.46525
$634$	−7.27468	−0.288914
$635$	5.59335	0.221965
$636$	−20.4946	−0.812665
$637$	−16.1536	−0.640029
$638$	0	0
$639$	0.456595	0.0180626
$640$	−12.5686	−0.496818
$641$	4.29108	0.169488	0.0847438	−	0.996403i	$-0.472993\pi$
$641$	4.29108	0.169488	0.0847438	+	0.996403i	$0.472993\pi$
$642$	3.89845	0.153860
$643$	34.7529	1.37052	0.685260	−	0.728298i	$-0.259688\pi$
$643$	34.7529	1.37052	0.685260	+	0.728298i	$0.259688\pi$
$644$	−9.87086	−0.388967
$645$	−10.5082	−0.413760
$646$	−0.318669	−0.0125379
$647$	−1.90679	−0.0749636	−0.0374818	−	0.999297i	$-0.511934\pi$
$647$	−1.90679	−0.0749636	−0.0374818	+	0.999297i	$0.511934\pi$
$648$	−25.3519	−0.995918
$649$	0	0
$650$	−48.5878	−1.90577
$651$	18.3979	0.721069
$652$	1.36266	0.0533659
$653$	−44.6238	−1.74626	−0.873132	−	0.487485i	$-0.837915\pi$
$653$	−44.6238	−1.74626	−0.873132	+	0.487485i	$0.837915\pi$
$654$	14.0880	0.550884
$655$	−6.92842	−0.270716
$656$	−10.5358	−0.411353
$657$	−0.582168	−0.0227126
$658$	11.5798	0.451427
$659$	41.2580	1.60718	0.803592	−	0.595181i	$-0.202920\pi$
$659$	41.2580	1.60718	0.803592	+	0.595181i	$0.202920\pi$
$660$	0	0
$661$	−40.6894	−1.58263	−0.791317	−	0.611406i	$-0.790604\pi$
$661$	−40.6894	−1.58263	−0.791317	+	0.611406i	$0.790604\pi$
$662$	−16.4395	−0.638938
$663$	6.50820	0.252758
$664$	3.04922	0.118333
$665$	2.24470	0.0870459
$666$	2.03281	0.0787699
$667$	−44.3379	−1.71677
$668$	14.0604	0.544013
$669$	−5.01641	−0.193945
$670$	−44.9341	−1.73596
$671$	0	0
$672$	14.1344	0.545245
$673$	−21.9917	−0.847716	−0.423858	−	0.905729i	$-0.639324\pi$
$673$	−21.9917	−0.847716	−0.423858	+	0.905729i	$0.639324\pi$
$674$	17.0716	0.657573
$675$	66.9669	2.57756
$676$	−1.98359	−0.0762920
$677$	−7.53295	−0.289515	−0.144757	−	0.989467i	$-0.546240\pi$
$677$	−7.53295	−0.289515	−0.144757	+	0.989467i	$0.546240\pi$
$678$	14.4530	0.555065
$679$	28.9753	1.11197
$680$	12.5686	0.481984
$681$	37.1372	1.42310
$682$	0	0
$683$	27.2663	1.04332	0.521659	−	0.853154i	$-0.325313\pi$
$683$	27.2663	1.04332	0.521659	+	0.853154i	$0.325313\pi$
$684$	−0.0551704	−0.00210949
$685$	−15.9641	−0.609956
$686$	18.7857	0.717242
$687$	7.51059	0.286547
$688$	1.49180	0.0568742
$689$	−47.1840	−1.79757
$690$	−41.3543	−1.57433
$691$	31.0632	1.18170	0.590851	−	0.806781i	$-0.298792\pi$
$691$	31.0632	1.18170	0.590851	+	0.806781i	$0.298792\pi$
$692$	11.5246	0.438100
$693$	0	0
$694$	1.77454	0.0673607
$695$	33.8849	1.28533
$696$	38.0932	1.44392
$697$	−10.5358	−0.399071
$698$	8.03281	0.304046
$699$	−28.1760	−1.06571
$700$	−21.1044	−0.797671
$701$	−2.42022	−0.0914104	−0.0457052	−	0.998955i	$-0.514553\pi$
$701$	−2.42022	−0.0914104	−0.0457052	+	0.998955i	$0.514553\pi$
$702$	−20.6514	−0.779435
$703$	−3.74173	−0.141122
$704$	0	0
$705$	−48.5139	−1.82714
$706$	−6.05517	−0.227889
$707$	10.7253	0.403367
$708$	−1.68133	−0.0631883
$709$	20.7061	0.777633	0.388816	−	0.921315i	$-0.372884\pi$
$709$	20.7061	0.777633	0.388816	+	0.921315i	$0.372884\pi$
$710$	−11.0492	−0.414670
$711$	−1.91770	−0.0719193
$712$	−35.6126	−1.33464
$713$	−38.2088	−1.43093
$714$	−2.82687	−0.105793
$715$	0	0
$716$	7.14554	0.267041
$717$	−50.0521	−1.86923
$718$	32.4946	1.21269
$719$	28.0245	1.04514	0.522568	−	0.852597i	$-0.324974\pi$
$719$	28.0245	1.04514	0.522568	+	0.852597i	$0.324974\pi$
$720$	−0.725323	−0.0270312
$721$	−15.5246	−0.578167
$722$	18.8984	0.703327
$723$	−4.17074	−0.155111
$724$	7.55220	0.280675
$725$	−94.7966	−3.52066
$726$	−18.4946	−0.686400
$727$	33.4506	1.24062	0.620308	−	0.784358i	$-0.287008\pi$
$727$	33.4506	1.24062	0.620308	+	0.784358i	$0.287008\pi$
$728$	19.5246	0.723630
$729$	28.3731	1.05086
$730$	14.0880	0.521420
$731$	1.49180	0.0551761
$732$	−8.16195	−0.301674
$733$	−40.2744	−1.48757	−0.743785	−	0.668419i	$-0.766971\pi$
$733$	−40.2744	−1.48757	−0.743785	+	0.668419i	$0.766971\pi$
$734$	−14.4671	−0.533989
$735$	−29.3955	−1.08427
$736$	−29.3543	−1.08201
$737$	0	0
$738$	1.82403	0.0671436
$739$	15.7006	0.577555	0.288778	−	0.957396i	$-0.406751\pi$
$739$	15.7006	0.577555	0.288778	+	0.957396i	$0.406751\pi$
$740$	49.1924	1.80835
$741$	2.07396	0.0761890
$742$	20.4946	0.752382
$743$	−31.2028	−1.14472	−0.572360	−	0.820002i	$-0.693972\pi$
$743$	−31.2028	−1.14472	−0.572360	+	0.820002i	$0.693972\pi$
$744$	32.8273	1.20351
$745$	75.2392	2.75655
$746$	2.95078	0.108036
$747$	−0.175968	−0.00643832
$748$	0	0
$749$	3.89845	0.142446
$750$	−53.1976	−1.94250
$751$	0.313441	0.0114376	0.00571881	−	0.999984i	$-0.498180\pi$
$751$	0.313441	0.0114376	0.00571881	+	0.999984i	$0.498180\pi$
$752$	6.88727	0.251153
$753$	36.0744	1.31462
$754$	29.2335	1.06462
$755$	−46.0161	−1.67470
$756$	−8.97003	−0.326237
$757$	−9.08514	−0.330205	−0.165103	−	0.986276i	$-0.552796\pi$
$757$	−9.08514	−0.330205	−0.165103	+	0.986276i	$0.552796\pi$
$758$	−10.3738	−0.376795
$759$	0	0
$760$	4.00523	0.145285
$761$	−19.3596	−0.701783	−0.350892	−	0.936416i	$-0.614121\pi$
$761$	−19.3596	−0.701783	−0.350892	+	0.936416i	$0.614121\pi$
$762$	−2.24470	−0.0813171
$763$	14.0880	0.510019
$764$	−21.6126	−0.781916
$765$	−0.725323	−0.0262241
$766$	29.1924	1.05476
$767$	−3.87086	−0.139769
$768$	28.5826	1.03139
$769$	−12.3156	−0.444110	−0.222055	−	0.975034i	$-0.571277\pi$
$769$	−12.3156	−0.444110	−0.222055	+	0.975034i	$0.571277\pi$
$770$	0	0
$771$	50.9065	1.83335
$772$	13.8984	0.500216
$773$	27.4506	0.987331	0.493666	−	0.869652i	$-0.335657\pi$
$773$	27.4506	0.987331	0.493666	+	0.869652i	$0.335657\pi$
$774$	−0.258271	−0.00928335
$775$	−81.6922	−2.93447
$776$	51.7006	1.85594
$777$	−33.1924	−1.19077
$778$	26.0000	0.932145
$779$	−3.35743	−0.120293
$780$	−27.2663	−0.976292
$781$	0	0
$782$	5.87086	0.209942
$783$	−40.2915	−1.43990
$784$	4.17313	0.149040
$785$	18.8873	0.674116
$786$	2.78049	0.0991769
$787$	−20.4119	−0.727605	−0.363802	−	0.931476i	$-0.618522\pi$
$787$	−20.4119	−0.727605	−0.363802	+	0.931476i	$0.618522\pi$
$788$	16.7253	0.595815
$789$	−25.2939	−0.900488
$790$	46.4067	1.65107
$791$	14.4530	0.513891
$792$	0	0
$793$	−18.7909	−0.667286
$794$	34.8873	1.23810
$795$	−85.8630	−3.04525
$796$	19.0768	0.676159
$797$	17.3215	0.613559	0.306780	−	0.951781i	$-0.400749\pi$
$797$	17.3215	0.613559	0.306780	+	0.951781i	$0.400749\pi$
$798$	−0.900838	−0.0318893
$799$	6.88727	0.243654
$800$	−62.7610	−2.21894
$801$	2.05517	0.0726159
$802$	−0.733661	−0.0259065
$803$	0	0
$804$	−18.0328	−0.635968
$805$	−41.3543	−1.45755
$806$	25.1924	0.887364
$807$	21.2475	0.747949
$808$	19.1372	0.673244
$809$	28.0552	0.986367	0.493184	−	0.869925i	$-0.335833\pi$
$809$	28.0552	0.986367	0.493184	+	0.869925i	$0.335833\pi$
$810$	−35.4043	−1.24398
$811$	17.2747	0.606596	0.303298	−	0.952896i	$-0.401912\pi$
$811$	17.2747	0.606596	0.303298	+	0.952896i	$0.401912\pi$
$812$	12.6977	0.445603
$813$	−33.0908	−1.16055
$814$	0	0
$815$	5.70892	0.199975
$816$	−1.68133	−0.0588583
$817$	0.475390	0.0166318
$818$	−0.129135	−0.00451511
$819$	−1.12675	−0.0393717
$820$	44.1400	1.54144
$821$	35.1044	1.22515	0.612576	−	0.790412i	$-0.290133\pi$
$821$	35.1044	1.22515	0.612576	+	0.790412i	$0.290133\pi$
$822$	6.40665	0.223458
$823$	−42.4364	−1.47924	−0.739619	−	0.673026i	$-0.764994\pi$
$823$	−42.4364	−1.47924	−0.739619	+	0.673026i	$0.764994\pi$
$824$	−27.7006	−0.964995
$825$	0	0
$826$	1.68133	0.0585010
$827$	11.5881	0.402958	0.201479	−	0.979493i	$-0.435425\pi$
$827$	11.5881	0.402958	0.201479	+	0.979493i	$0.435425\pi$
$828$	1.01641	0.0353226
$829$	16.5134	0.573535	0.286768	−	0.958000i	$-0.407419\pi$
$829$	16.5134	0.573535	0.286768	+	0.958000i	$0.407419\pi$
$830$	4.25827	0.147807
$831$	23.3679	0.810623
$832$	27.0961	0.939387
$833$	4.17313	0.144590
$834$	−13.5986	−0.470880
$835$	58.9065	2.03854
$836$	0	0
$837$	−34.7218	−1.20016
$838$	7.78288	0.268855
$839$	13.0716	0.451281	0.225640	−	0.974211i	$-0.427553\pi$
$839$	13.0716	0.451281	0.225640	+	0.974211i	$0.427553\pi$
$840$	35.5298	1.22590
$841$	28.0357	0.966747
$842$	−4.38741	−0.151200
$843$	18.8961	0.650815
$844$	−21.9260	−0.754725
$845$	−8.31033	−0.285884
$846$	−1.19237	−0.0409947
$847$	−18.4946	−0.635483
$848$	12.1895	0.418590
$849$	38.0468	1.30576
$850$	12.5522	0.430537
$851$	68.9341	2.36303
$852$	−4.43424	−0.151915
$853$	−16.8269	−0.576141	−0.288071	−	0.957609i	$-0.593014\pi$
$853$	−16.8269	−0.576141	−0.288071	+	0.957609i	$0.593014\pi$
$854$	8.16195	0.279296
$855$	−0.231138	−0.00790476
$856$	6.95601	0.237751
$857$	25.6782	0.877151	0.438576	−	0.898694i	$-0.355483\pi$
$857$	25.6782	0.877151	0.438576	+	0.898694i	$0.355483\pi$
$858$	0	0
$859$	−24.3379	−0.830399	−0.415199	−	0.909730i	$-0.636288\pi$
$859$	−24.3379	−0.830399	−0.415199	+	0.909730i	$0.636288\pi$
$860$	−6.24993	−0.213121
$861$	−29.7833	−1.01501
$862$	−12.4999	−0.425747
$863$	−50.7805	−1.72859	−0.864294	−	0.502987i	$-0.832234\pi$
$863$	−50.7805	−1.72859	−0.864294	+	0.502987i	$0.832234\pi$
$864$	−26.6754	−0.907515
$865$	48.2827	1.64166
$866$	1.14031	0.0387494
$867$	−1.68133	−0.0571010
$868$	10.9424	0.371411
$869$	0	0
$870$	53.1976	1.80357
$871$	−41.5163	−1.40673
$872$	25.1372	0.851253
$873$	−2.98359	−0.100979
$874$	1.87086	0.0632829
$875$	−53.1976	−1.79841
$876$	5.65375	0.191022
$877$	5.46421	0.184513	0.0922567	−	0.995735i	$-0.470592\pi$
$877$	5.46421	0.184513	0.0922567	+	0.995735i	$0.470592\pi$
$878$	−1.62093	−0.0547038
$879$	−24.6426	−0.831173
$880$	0	0
$881$	−10.7909	−0.363556	−0.181778	−	0.983340i	$-0.558185\pi$
$881$	−10.7909	−0.363556	−0.181778	+	0.983340i	$0.558185\pi$
$882$	−0.722482	−0.0243272
$883$	49.5767	1.66839	0.834194	−	0.551471i	$-0.185933\pi$
$883$	49.5767	1.66839	0.834194	+	0.551471i	$0.185933\pi$
$884$	3.87086	0.130191
$885$	−7.04399	−0.236781
$886$	−30.7498	−1.03306
$887$	−32.1208	−1.07851	−0.539255	−	0.842142i	$-0.681294\pi$
$887$	−32.1208	−1.07851	−0.539255	+	0.842142i	$0.681294\pi$
$888$	−59.2252	−1.98747
$889$	−2.24470	−0.0752850
$890$	−49.7334	−1.66707
$891$	0	0
$892$	−2.98359	−0.0998981
$893$	2.19476	0.0734449
$894$	−30.1948	−1.00986
$895$	29.9365	1.00067
$896$	5.04399	0.168508
$897$	−38.2088	−1.27575
$898$	31.2283	1.04210
$899$	49.1512	1.63929
$900$	2.17313	0.0724376
$901$	12.1895	0.406092
$902$	0	0
$903$	4.21712	0.140337
$904$	25.7886	0.857715
$905$	31.6402	1.05176
$906$	18.4671	0.613527
$907$	7.14243	0.237161	0.118580	−	0.992944i	$-0.462166\pi$
$907$	7.14243	0.237161	0.118580	+	0.992944i	$0.462166\pi$
$908$	22.0880	0.733015
$909$	−1.10439	−0.0366303
$910$	27.2663	0.903870
$911$	−20.5275	−0.680105	−0.340052	−	0.940406i	$-0.610445\pi$
$911$	−20.5275	−0.680105	−0.340052	+	0.940406i	$0.610445\pi$
$912$	−0.535789	−0.0177417
$913$	0	0
$914$	3.32985	0.110142
$915$	−34.1948	−1.13044
$916$	4.46705	0.147596
$917$	2.78049	0.0918200
$918$	5.33508	0.176084
$919$	5.96719	0.196839	0.0984197	−	0.995145i	$-0.468621\pi$
$919$	5.96719	0.196839	0.0984197	+	0.995145i	$0.468621\pi$
$920$	−73.7886	−2.43274
$921$	55.8161	1.83920
$922$	−1.93437	−0.0637052
$923$	−10.2088	−0.336026
$924$	0	0
$925$	147.384	4.84597
$926$	−18.6290	−0.612187
$927$	1.59858	0.0525041
$928$	37.7610	1.23956
$929$	3.80736	0.124915	0.0624577	−	0.998048i	$-0.480106\pi$
$929$	3.80736	0.124915	0.0624577	+	0.998048i	$0.480106\pi$
$930$	45.8437	1.50328
$931$	1.32985	0.0435840
$932$	−16.7581	−0.548931
$933$	−23.0028	−0.753079
$934$	−26.0880	−0.853625
$935$	0	0
$936$	−2.01046	−0.0657138
$937$	−20.5494	−0.671318	−0.335659	−	0.941984i	$-0.608959\pi$
$937$	−20.5494	−0.671318	−0.335659	+	0.941984i	$0.608959\pi$
$938$	18.0328	0.588792
$939$	−9.43663	−0.307953
$940$	−28.8545	−0.941129
$941$	36.4587	1.18852	0.594260	−	0.804273i	$-0.297445\pi$
$941$	36.4587	1.18852	0.594260	+	0.804273i	$0.297445\pi$
$942$	−7.57978	−0.246963
$943$	61.8542	2.01425
$944$	1.00000	0.0325472
$945$	−37.5802	−1.22249
$946$	0	0
$947$	33.4782	1.08790	0.543948	−	0.839119i	$-0.316929\pi$
$947$	33.4782	1.08790	0.543948	+	0.839119i	$0.316929\pi$
$948$	18.6238	0.604872
$949$	13.0164	0.422530
$950$	4.00000	0.129777
$951$	−12.2311	−0.396622
$952$	−5.04399	−0.163477
$953$	6.32390	0.204851	0.102426	−	0.994741i	$-0.467340\pi$
$953$	6.32390	0.204851	0.102426	+	0.994741i	$0.467340\pi$
$954$	−2.11034	−0.0683248
$955$	−90.5467	−2.93002
$956$	−29.7693	−0.962808
$957$	0	0
$958$	−2.89561	−0.0935529
$959$	6.40665	0.206882
$960$	49.3079	1.59141
$961$	11.3567	0.366346
$962$	−45.4506	−1.46539
$963$	−0.401425	−0.0129357
$964$	−2.48062	−0.0798953
$965$	58.2280	1.87443
$966$	16.5962	0.533974
$967$	44.2968	1.42449	0.712244	−	0.701932i	$-0.247679\pi$
$967$	44.2968	1.42449	0.712244	+	0.701932i	$0.247679\pi$
$968$	−33.0000	−1.06066
$969$	−0.535789	−0.0172120
$970$	72.2004	2.31822
$971$	−1.58289	−0.0507974	−0.0253987	−	0.999677i	$-0.508086\pi$
$971$	−1.58289	−0.0507974	−0.0253987	+	0.999677i	$0.508086\pi$
$972$	1.79690	0.0576356
$973$	−13.5986	−0.435950
$974$	−9.42306	−0.301934
$975$	−81.6922	−2.61625
$976$	4.85446	0.155387
$977$	47.4751	1.51886	0.759432	−	0.650587i	$-0.225477\pi$
$977$	47.4751	1.51886	0.759432	+	0.650587i	$0.225477\pi$
$978$	−2.29108	−0.0732608
$979$	0	0
$980$	−17.4835	−0.558489
$981$	−1.45065	−0.0463155
$982$	35.1648	1.12215
$983$	−42.0880	−1.34240	−0.671199	−	0.741277i	$-0.734220\pi$
$983$	−42.0880	−1.34240	−0.671199	+	0.741277i	$0.734220\pi$
$984$	−53.1424	−1.69412
$985$	70.0713	2.23266
$986$	−7.55220	−0.240511
$987$	19.4694	0.619719
$988$	1.23353	0.0392437
$989$	−8.75814	−0.278493
$990$	0	0
$991$	2.53268	0.0804532	0.0402266	−	0.999191i	$-0.487192\pi$
$991$	2.53268	0.0804532	0.0402266	+	0.999191i	$0.487192\pi$
$992$	32.5410	1.03318
$993$	−27.6402	−0.877135
$994$	4.43424	0.140646
$995$	79.9229	2.53373
$996$	1.70892	0.0541491
$997$	−26.0192	−0.824038	−0.412019	−	0.911175i	$-0.635176\pi$
$997$	−26.0192	−0.824038	−0.412019	+	0.911175i	$0.635176\pi$
$998$	39.8245	1.26062
$999$	62.6430	1.98194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.e.1.1	✓	3
3.2	odd	2		9027.2.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.e.1.1	✓	3	1.1	even	1	trivial
9027.2.a.h.1.3		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.68133 q^{3} -1.00000 q^{4} -4.18953 q^{5} +1.68133 q^{6} +1.68133 q^{7} +3.00000 q^{8} -0.173127 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.68133 q^{3} -1.00000 q^{4} -4.18953 q^{5} +1.68133 q^{6} +1.68133 q^{7} +3.00000 q^{8} -0.173127 q^{9} +4.18953 q^{10} +1.68133 q^{12} +3.87086 q^{13} -1.68133 q^{14} +7.04399 q^{15} -1.00000 q^{16} -1.00000 q^{17} +0.173127 q^{18} -0.318669 q^{19} +4.18953 q^{20} -2.82687 q^{21} +5.87086 q^{23} -5.04399 q^{24} +12.5522 q^{25} -3.87086 q^{26} +5.33508 q^{27} -1.68133 q^{28} -7.55220 q^{29} -7.04399 q^{30} -6.50820 q^{31} -5.00000 q^{32} +1.00000 q^{34} -7.04399 q^{35} +0.173127 q^{36} +11.7417 q^{37} +0.318669 q^{38} -6.50820 q^{39} -12.5686 q^{40} +10.5358 q^{41} +2.82687 q^{42} -1.49180 q^{43} +0.725323 q^{45} -5.87086 q^{46} -6.88727 q^{47} +1.68133 q^{48} -4.17313 q^{49} -12.5522 q^{50} +1.68133 q^{51} -3.87086 q^{52} -12.1895 q^{53} -5.33508 q^{54} +5.04399 q^{56} +0.535789 q^{57} +7.55220 q^{58} -1.00000 q^{59} -7.04399 q^{60} -4.85446 q^{61} +6.50820 q^{62} -0.291084 q^{63} +7.00000 q^{64} -16.2171 q^{65} -10.7253 q^{67} +1.00000 q^{68} -9.87086 q^{69} +7.04399 q^{70} -2.63734 q^{71} -0.519382 q^{72} +3.36266 q^{73} -11.7417 q^{74} -21.1044 q^{75} +0.318669 q^{76} +6.50820 q^{78} +11.0768 q^{79} +4.18953 q^{80} -8.45065 q^{81} -10.5358 q^{82} +1.01641 q^{83} +2.82687 q^{84} +4.18953 q^{85} +1.49180 q^{86} +12.6977 q^{87} -11.8709 q^{89} -0.725323 q^{90} +6.50820 q^{91} -5.87086 q^{92} +10.9424 q^{93} +6.88727 q^{94} +1.33508 q^{95} +8.40665 q^{96} +17.2335 q^{97} +4.17313 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} - 3 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 2 * q^3 - 3 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 9 * q^8 + 5 * q^9	\( 3 q - 3 q^{2} + 2 q^{3} - 3 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 9 q^{8} + 5 q^{9} + 4 q^{10} - 2 q^{12} - 4 q^{13} + 2 q^{14} - 3 q^{16} - 3 q^{17} - 5 q^{18} - 8 q^{19} + 4 q^{20} - 14 q^{21} + 2 q^{23} + 6 q^{24} + 15 q^{25} + 4 q^{26} + 20 q^{27} + 2 q^{28} - 18 q^{31} - 15 q^{32} + 3 q^{34} - 5 q^{36} + 4 q^{37} + 8 q^{38} - 18 q^{39} - 12 q^{40} + 12 q^{41} + 14 q^{42} - 6 q^{43} - 26 q^{45} - 2 q^{46} - 2 q^{47} - 2 q^{48} - 7 q^{49} - 15 q^{50} - 2 q^{51} + 4 q^{52} - 28 q^{53} - 20 q^{54} - 6 q^{56} - 18 q^{57} - 3 q^{59} - 2 q^{61} + 18 q^{62} - 26 q^{63} + 21 q^{64} - 22 q^{65} - 4 q^{67} + 3 q^{68} - 14 q^{69} - 22 q^{71} + 15 q^{72} - 4 q^{73} - 4 q^{74} - 18 q^{75} + 8 q^{76} + 18 q^{78} + 6 q^{79} + 4 q^{80} + 31 q^{81} - 12 q^{82} + 14 q^{84} + 4 q^{85} + 6 q^{86} + 28 q^{87} - 20 q^{89} + 26 q^{90} + 18 q^{91} - 2 q^{92} - 22 q^{93} + 2 q^{94} + 8 q^{95} - 10 q^{96} + 22 q^{97} + 7 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 2 * q^3 - 3 * q^4 - 4 * q^5 - 2 * q^6 - 2 * q^7 + 9 * q^8 + 5 * q^9 + 4 * q^10 - 2 * q^12 - 4 * q^13 + 2 * q^14 - 3 * q^16 - 3 * q^17 - 5 * q^18 - 8 * q^19 + 4 * q^20 - 14 * q^21 + 2 * q^23 + 6 * q^24 + 15 * q^25 + 4 * q^26 + 20 * q^27 + 2 * q^28 - 18 * q^31 - 15 * q^32 + 3 * q^34 - 5 * q^36 + 4 * q^37 + 8 * q^38 - 18 * q^39 - 12 * q^40 + 12 * q^41 + 14 * q^42 - 6 * q^43 - 26 * q^45 - 2 * q^46 - 2 * q^47 - 2 * q^48 - 7 * q^49 - 15 * q^50 - 2 * q^51 + 4 * q^52 - 28 * q^53 - 20 * q^54 - 6 * q^56 - 18 * q^57 - 3 * q^59 - 2 * q^61 + 18 * q^62 - 26 * q^63 + 21 * q^64 - 22 * q^65 - 4 * q^67 + 3 * q^68 - 14 * q^69 - 22 * q^71 + 15 * q^72 - 4 * q^73 - 4 * q^74 - 18 * q^75 + 8 * q^76 + 18 * q^78 + 6 * q^79 + 4 * q^80 + 31 * q^81 - 12 * q^82 + 14 * q^84 + 4 * q^85 + 6 * q^86 + 28 * q^87 - 20 * q^89 + 26 * q^90 + 18 * q^91 - 2 * q^92 - 22 * q^93 + 2 * q^94 + 8 * q^95 - 10 * q^96 + 22 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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