LMFDB - Embedded newform 1003.2.a.e.1.3

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			1.3
Root			$-1.86081$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +3.32340 q^{3} -1.00000 q^{4} -2.39821 q^{5} -3.32340 q^{6} -3.32340 q^{7} +3.00000 q^{8} +8.04502 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.32340 q^{3} -1.00000 q^{4} -2.39821 q^{5} -3.32340 q^{6} -3.32340 q^{7} +3.00000 q^{8} +8.04502 q^{9} +2.39821 q^{10} -3.32340 q^{12} -2.92520 q^{13} +3.32340 q^{14} -7.97021 q^{15} -1.00000 q^{16} -1.00000 q^{17} -8.04502 q^{18} -5.32340 q^{19} +2.39821 q^{20} -11.0450 q^{21} -0.925197 q^{23} +9.97021 q^{24} +0.751399 q^{25} +2.92520 q^{26} +16.7666 q^{27} +3.32340 q^{28} +4.24860 q^{29} +7.97021 q^{30} -9.72161 q^{31} -5.00000 q^{32} +1.00000 q^{34} +7.97021 q^{35} -8.04502 q^{36} -1.85039 q^{37} +5.32340 q^{38} -9.72161 q^{39} -7.19462 q^{40} -7.69182 q^{41} +11.0450 q^{42} +1.72161 q^{43} -19.2936 q^{45} +0.925197 q^{46} -6.51803 q^{47} -3.32340 q^{48} +4.04502 q^{49} -0.751399 q^{50} -3.32340 q^{51} +2.92520 q^{52} -10.3982 q^{53} -16.7666 q^{54} -9.97021 q^{56} -17.6918 q^{57} -4.24860 q^{58} -1.00000 q^{59} +7.97021 q^{60} +8.36842 q^{61} +9.72161 q^{62} -26.7368 q^{63} +7.00000 q^{64} +7.01523 q^{65} +9.29362 q^{67} +1.00000 q^{68} -3.07480 q^{69} -7.97021 q^{70} -12.6468 q^{71} +24.1350 q^{72} -6.64681 q^{73} +1.85039 q^{74} +2.49720 q^{75} +5.32340 q^{76} +9.72161 q^{78} +8.91623 q^{79} +2.39821 q^{80} +31.5872 q^{81} +7.69182 q^{82} +7.44322 q^{83} +11.0450 q^{84} +2.39821 q^{85} -1.72161 q^{86} +14.1198 q^{87} -5.07480 q^{89} +19.2936 q^{90} +9.72161 q^{91} +0.925197 q^{92} -32.3088 q^{93} +6.51803 q^{94} +12.7666 q^{95} -16.6170 q^{96} +0.427995 q^{97} -4.04502 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.615027\pi$
$3$	3.32340	1.91877	0.959384	−	0.282103i	$-0.0910319\pi$
$3$	3.32340	1.91877	0.959384	+	0.282103i	$0.0910319\pi$
$4$	−1.00000	−0.500000
$5$	−2.39821	−1.07251	−0.536255	−	0.844056i	$-0.680162\pi$
$5$	−2.39821	−1.07251	−0.536255	+	0.844056i	$0.680162\pi$
$6$	−3.32340	−1.35677
$7$	−3.32340	−1.25613	−0.628064	−	0.778161i	$-0.716153\pi$
$7$	−3.32340	−1.25613	−0.628064	+	0.778161i	$0.716153\pi$
$8$	3.00000	1.06066
$9$	8.04502	2.68167
$10$	2.39821	0.758380
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−3.32340	−0.959384
$13$	−2.92520	−0.811304	−0.405652	−	0.914028i	$-0.632955\pi$
$13$	−2.92520	−0.811304	−0.405652	+	0.914028i	$0.632955\pi$
$14$	3.32340	0.888217
$15$	−7.97021	−2.05790
$16$	−1.00000	−0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−8.04502	−1.89623
$19$	−5.32340	−1.22127	−0.610636	−	0.791911i	$-0.709086\pi$
$19$	−5.32340	−1.22127	−0.610636	+	0.791911i	$0.709086\pi$
$20$	2.39821	0.536255
$21$	−11.0450	−2.41022
$22$	0	0
$23$	−0.925197	−0.192917	−0.0964584	−	0.995337i	$-0.530751\pi$
$23$	−0.925197	−0.192917	−0.0964584	+	0.995337i	$0.530751\pi$
$24$	9.97021	2.03516
$25$	0.751399	0.150280
$26$	2.92520	0.573678
$27$	16.7666	3.22674
$28$	3.32340	0.628064
$29$	4.24860	0.788945	0.394473	−	0.918908i	$-0.370927\pi$
$29$	4.24860	0.788945	0.394473	+	0.918908i	$0.370927\pi$
$30$	7.97021	1.45516
$31$	−9.72161	−1.74605	−0.873027	−	0.487673i	$-0.837846\pi$
$31$	−9.72161	−1.74605	−0.873027	+	0.487673i	$0.837846\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	1.00000	0.171499
$35$	7.97021	1.34721
$36$	−8.04502	−1.34084
$37$	−1.85039	−0.304203	−0.152101	−	0.988365i	$-0.548604\pi$
$37$	−1.85039	−0.304203	−0.152101	+	0.988365i	$0.548604\pi$
$38$	5.32340	0.863570
$39$	−9.72161	−1.55670
$40$	−7.19462	−1.13757
$41$	−7.69182	−1.20126	−0.600631	−	0.799527i	$-0.705084\pi$
$41$	−7.69182	−1.20126	−0.600631	+	0.799527i	$0.705084\pi$
$42$	11.0450	1.70428
$43$	1.72161	0.262543	0.131272	−	0.991346i	$-0.458094\pi$
$43$	1.72161	0.262543	0.131272	+	0.991346i	$0.458094\pi$
$44$	0	0
$45$	−19.2936	−2.87612
$46$	0.925197	0.136413
$47$	−6.51803	−0.950752	−0.475376	−	0.879783i	$-0.657688\pi$
$47$	−6.51803	−0.950752	−0.475376	+	0.879783i	$0.657688\pi$
$48$	−3.32340	−0.479692
$49$	4.04502	0.577859
$50$	−0.751399	−0.106264
$51$	−3.32340	−0.465370
$52$	2.92520	0.405652
$53$	−10.3982	−1.42830	−0.714152	−	0.699991i	$-0.753187\pi$
$53$	−10.3982	−1.42830	−0.714152	+	0.699991i	$0.753187\pi$
$54$	−16.7666	−2.28165
$55$	0	0
$56$	−9.97021	−1.33233
$57$	−17.6918	−2.34334
$58$	−4.24860	−0.557869
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	7.97021	1.02895
$61$	8.36842	1.07147	0.535733	−	0.844387i	$-0.320035\pi$
$61$	8.36842	1.07147	0.535733	+	0.844387i	$0.320035\pi$
$62$	9.72161	1.23465
$63$	−26.7368	−3.36853
$64$	7.00000	0.875000
$65$	7.01523	0.870132
$66$	0	0
$67$	9.29362	1.13540	0.567698	−	0.823237i	$-0.307834\pi$
$67$	9.29362	1.13540	0.567698	+	0.823237i	$0.307834\pi$
$68$	1.00000	0.121268
$69$	−3.07480	−0.370163
$70$	−7.97021	−0.952623
$71$	−12.6468	−1.50090	−0.750450	−	0.660927i	$-0.770163\pi$
$71$	−12.6468	−1.50090	−0.750450	+	0.660927i	$0.770163\pi$
$72$	24.1350	2.84434
$73$	−6.64681	−0.777950	−0.388975	−	0.921248i	$-0.627171\pi$
$73$	−6.64681	−0.777950	−0.388975	+	0.921248i	$0.627171\pi$
$74$	1.85039	0.215104
$75$	2.49720	0.288352
$76$	5.32340	0.610636
$77$	0	0
$78$	9.72161	1.10076
$79$	8.91623	1.00315	0.501577	−	0.865113i	$-0.332753\pi$
$79$	8.91623	1.00315	0.501577	+	0.865113i	$0.332753\pi$
$80$	2.39821	0.268128
$81$	31.5872	3.50969
$82$	7.69182	0.849420
$83$	7.44322	0.817000	0.408500	−	0.912758i	$-0.366052\pi$
$83$	7.44322	0.817000	0.408500	+	0.912758i	$0.366052\pi$
$84$	11.0450	1.20511
$85$	2.39821	0.260122
$86$	−1.72161	−0.185646
$87$	14.1198	1.51380
$88$	0	0
$89$	−5.07480	−0.537928	−0.268964	−	0.963150i	$-0.586681\pi$
$89$	−5.07480	−0.537928	−0.268964	+	0.963150i	$0.586681\pi$
$90$	19.2936	2.03373
$91$	9.72161	1.01910
$92$	0.925197	0.0964584
$93$	−32.3088	−3.35027
$94$	6.51803	0.672283
$95$	12.7666	1.30983
$96$	−16.6170	−1.69597
$97$	0.427995	0.0434563	0.0217281	−	0.999764i	$-0.493083\pi$
$97$	0.427995	0.0434563	0.0217281	+	0.999764i	$0.493083\pi$
$98$	−4.04502	−0.408608
$99$	0	0
$100$	−0.751399	−0.0751399
$101$	2.79641	0.278254	0.139127	−	0.990275i	$-0.455570\pi$
$101$	2.79641	0.278254	0.139127	+	0.990275i	$0.455570\pi$
$102$	3.32340	0.329066
$103$	7.57201	0.746092	0.373046	−	0.927813i	$-0.378313\pi$
$103$	7.57201	0.746092	0.373046	+	0.927813i	$0.378313\pi$
$104$	−8.77559	−0.860517
$105$	26.4882	2.58499
$106$	10.3982	1.00996
$107$	7.32340	0.707980	0.353990	−	0.935249i	$-0.384825\pi$
$107$	7.32340	0.707980	0.353990	+	0.935249i	$0.384825\pi$
$108$	−16.7666	−1.61337
$109$	4.79641	0.459413	0.229707	−	0.973260i	$-0.426223\pi$
$109$	4.79641	0.459413	0.229707	+	0.973260i	$0.426223\pi$
$110$	0	0
$111$	−6.14961	−0.583695
$112$	3.32340	0.314032
$113$	−18.2188	−1.71388	−0.856941	−	0.515415i	$-0.827638\pi$
$113$	−18.2188	−1.71388	−0.856941	+	0.515415i	$0.827638\pi$
$114$	17.6918	1.65699
$115$	2.21881	0.206905
$116$	−4.24860	−0.394473
$117$	−23.5333	−2.17565
$118$	1.00000	0.0920575
$119$	3.32340	0.304656
$120$	−23.9106	−2.18273
$121$	−11.0000	−1.00000
$122$	−8.36842	−0.757641
$123$	−25.5630	−2.30494
$124$	9.72161	0.873027
$125$	10.1890	0.911334
$126$	26.7368	2.38191
$127$	−12.7666	−1.13286	−0.566428	−	0.824112i	$-0.691675\pi$
$127$	−12.7666	−1.13286	−0.566428	+	0.824112i	$0.691675\pi$
$128$	3.00000	0.265165
$129$	5.72161	0.503760
$130$	−7.01523	−0.615276
$131$	18.0900	1.58053	0.790267	−	0.612763i	$-0.209942\pi$
$131$	18.0900	1.58053	0.790267	+	0.612763i	$0.209942\pi$
$132$	0	0
$133$	17.6918	1.53408
$134$	−9.29362	−0.802846
$135$	−40.2099	−3.46071
$136$	−3.00000	−0.257248
$137$	5.60179	0.478593	0.239297	−	0.970946i	$-0.423083\pi$
$137$	5.60179	0.478593	0.239297	+	0.970946i	$0.423083\pi$
$138$	3.07480	0.261745
$139$	21.9404	1.86096	0.930481	−	0.366339i	$-0.119389\pi$
$139$	21.9404	1.86096	0.930481	+	0.366339i	$0.119389\pi$
$140$	−7.97021	−0.673606
$141$	−21.6620	−1.82427
$142$	12.6468	1.06130
$143$	0	0
$144$	−8.04502	−0.670418
$145$	−10.1890	−0.846153
$146$	6.64681	0.550094
$147$	13.4432	1.10878
$148$	1.85039	0.152101
$149$	18.8656	1.54553	0.772766	−	0.634691i	$-0.218873\pi$
$149$	18.8656	1.54553	0.772766	+	0.634691i	$0.218873\pi$
$150$	−2.49720	−0.203896
$151$	4.55678	0.370825	0.185413	−	0.982661i	$-0.440638\pi$
$151$	4.55678	0.370825	0.185413	+	0.982661i	$0.440638\pi$
$152$	−15.9702	−1.29536
$153$	−8.04502	−0.650401
$154$	0	0
$155$	23.3144	1.87266
$156$	9.72161	0.778352
$157$	−7.72161	−0.616252	−0.308126	−	0.951346i	$-0.599702\pi$
$157$	−7.72161	−0.616252	−0.308126	+	0.951346i	$0.599702\pi$
$158$	−8.91623	−0.709337
$159$	−34.5574	−2.74058
$160$	11.9910	0.947975
$161$	3.07480	0.242328
$162$	−31.5872	−2.48173
$163$	8.64681	0.677270	0.338635	−	0.940918i	$-0.390035\pi$
$163$	8.64681	0.677270	0.338635	+	0.940918i	$0.390035\pi$
$164$	7.69182	0.600631
$165$	0	0
$166$	−7.44322	−0.577706
$167$	−5.47301	−0.423514	−0.211757	−	0.977322i	$-0.567919\pi$
$167$	−5.47301	−0.423514	−0.211757	+	0.977322i	$0.567919\pi$
$168$	−33.1350	−2.55642
$169$	−4.44322	−0.341786
$170$	−2.39821	−0.183934
$171$	−42.8269	−3.27505
$172$	−1.72161	−0.131272
$173$	−21.1648	−1.60913	−0.804566	−	0.593863i	$-0.797602\pi$
$173$	−21.1648	−1.60913	−0.804566	+	0.593863i	$0.797602\pi$
$174$	−14.1198	−1.07042
$175$	−2.49720	−0.188771
$176$	0	0
$177$	−3.32340	−0.249802
$178$	5.07480	0.380373
$179$	−20.3684	−1.52241	−0.761204	−	0.648513i	$-0.775391\pi$
$179$	−20.3684	−1.52241	−0.761204	+	0.648513i	$0.775391\pi$
$180$	19.2936	1.43806
$181$	4.24860	0.315796	0.157898	−	0.987455i	$-0.449528\pi$
$181$	4.24860	0.315796	0.157898	+	0.987455i	$0.449528\pi$
$182$	−9.72161	−0.720614
$183$	27.8116	2.05590
$184$	−2.77559	−0.204619
$185$	4.43763	0.326261
$186$	32.3088	2.36900
$187$	0	0
$188$	6.51803	0.475376
$189$	−55.7223	−4.05320
$190$	−12.7666	−0.926189
$191$	1.22441	0.0885952	0.0442976	−	0.999018i	$-0.485895\pi$
$191$	1.22441	0.0885952	0.0442976	+	0.999018i	$0.485895\pi$
$192$	23.2638	1.67892
$193$	14.3386	1.03212	0.516059	−	0.856553i	$-0.327399\pi$
$193$	14.3386	1.03212	0.516059	+	0.856553i	$0.327399\pi$
$194$	−0.427995	−0.0307282
$195$	23.3144	1.66958
$196$	−4.04502	−0.288930
$197$	3.29362	0.234661	0.117330	−	0.993093i	$-0.462566\pi$
$197$	3.29362	0.234661	0.117330	+	0.993093i	$0.462566\pi$
$198$	0	0
$199$	−16.9162	−1.19916	−0.599580	−	0.800315i	$-0.704666\pi$
$199$	−16.9162	−1.19916	−0.599580	+	0.800315i	$0.704666\pi$
$200$	2.25420	0.159396
$201$	30.8864	2.17856
$202$	−2.79641	−0.196755
$203$	−14.1198	−0.991017
$204$	3.32340	0.232685
$205$	18.4466	1.28837
$206$	−7.57201	−0.527567
$207$	−7.44322	−0.517340
$208$	2.92520	0.202826
$209$	0	0
$210$	−26.4882	−1.82786
$211$	−27.7521	−1.91053	−0.955266	−	0.295749i	$-0.904431\pi$
$211$	−27.7521	−1.91053	−0.955266	+	0.295749i	$0.904431\pi$
$212$	10.3982	0.714152
$213$	−42.0305	−2.87988
$214$	−7.32340	−0.500618
$215$	−4.12878	−0.281581
$216$	50.2999	3.42247
$217$	32.3088	2.19327
$218$	−4.79641	−0.324854
$219$	−22.0900	−1.49271
$220$	0	0
$221$	2.92520	0.196770
$222$	6.14961	0.412735
$223$	−3.44322	−0.230575	−0.115288	−	0.993332i	$-0.536779\pi$
$223$	−3.44322	−0.230575	−0.115288	+	0.993332i	$0.536779\pi$
$224$	16.6170	1.11027
$225$	6.04502	0.403001
$226$	18.2188	1.21190
$227$	7.94043	0.527025	0.263512	−	0.964656i	$-0.415119\pi$
$227$	7.94043	0.527025	0.263512	+	0.964656i	$0.415119\pi$
$228$	17.6918	1.17167
$229$	29.1440	1.92589	0.962945	−	0.269697i	$-0.0869236\pi$
$229$	29.1440	1.92589	0.962945	+	0.269697i	$0.0869236\pi$
$230$	−2.21881	−0.146304
$231$	0	0
$232$	12.7458	0.836803
$233$	9.59283	0.628447	0.314224	−	0.949349i	$-0.398256\pi$
$233$	9.59283	0.628447	0.314224	+	0.949349i	$0.398256\pi$
$234$	23.5333	1.53842
$235$	15.6316	1.01969
$236$	1.00000	0.0650945
$237$	29.6323	1.92482
$238$	−3.32340	−0.215424
$239$	−5.26383	−0.340489	−0.170244	−	0.985402i	$-0.554456\pi$
$239$	−5.26383	−0.340489	−0.170244	+	0.985402i	$0.554456\pi$
$240$	7.97021	0.514475
$241$	27.1350	1.74792	0.873961	−	0.485996i	$-0.161543\pi$
$241$	27.1350	1.74792	0.873961	+	0.485996i	$0.161543\pi$
$242$	11.0000	0.707107
$243$	54.6773	3.50755
$244$	−8.36842	−0.535733
$245$	−9.70079	−0.619761
$246$	25.5630	1.62984
$247$	15.5720	0.990823
$248$	−29.1648	−1.85197
$249$	24.7368	1.56763
$250$	−10.1890	−0.644411
$251$	−15.7126	−0.991774	−0.495887	−	0.868387i	$-0.665157\pi$
$251$	−15.7126	−0.991774	−0.495887	+	0.868387i	$0.665157\pi$
$252$	26.7368	1.68426
$253$	0	0
$254$	12.7666	0.801049
$255$	7.97021	0.499114
$256$	−17.0000	−1.06250
$257$	1.54222	0.0962009	0.0481005	−	0.998843i	$-0.484683\pi$
$257$	1.54222	0.0962009	0.0481005	+	0.998843i	$0.484683\pi$
$258$	−5.72161	−0.356212
$259$	6.14961	0.382118
$260$	−7.01523	−0.435066
$261$	34.1801	2.11569
$262$	−18.0900	−1.11761
$263$	0.0297872	0.00183676	0.000918378	−	1.00000i	$-0.499708\pi$
$263$	0.0297872	0.00183676	0.000918378		1.00000i	$0.499708\pi$
$264$	0	0
$265$	24.9371	1.53187
$266$	−17.6918	−1.08476
$267$	−16.8656	−1.03216
$268$	−9.29362	−0.567698
$269$	−22.6468	−1.38080	−0.690400	−	0.723428i	$-0.742566\pi$
$269$	−22.6468	−1.38080	−0.690400	+	0.723428i	$0.742566\pi$
$270$	40.2099	2.44709
$271$	14.6766	0.891540	0.445770	−	0.895148i	$-0.352930\pi$
$271$	14.6766	0.891540	0.445770	+	0.895148i	$0.352930\pi$
$272$	1.00000	0.0606339
$273$	32.3088	1.95542
$274$	−5.60179	−0.338417
$275$	0	0
$276$	3.07480	0.185081
$277$	14.3386	0.861525	0.430762	−	0.902465i	$-0.358245\pi$
$277$	14.3386	0.861525	0.430762	+	0.902465i	$0.358245\pi$
$278$	−21.9404	−1.31590
$279$	−78.2105	−4.68234
$280$	23.9106	1.42893
$281$	−28.7279	−1.71376	−0.856881	−	0.515515i	$-0.827601\pi$
$281$	−28.7279	−1.71376	−0.856881	+	0.515515i	$0.827601\pi$
$282$	21.6620	1.28996
$283$	−8.66763	−0.515237	−0.257619	−	0.966247i	$-0.582938\pi$
$283$	−8.66763	−0.515237	−0.257619	+	0.966247i	$0.582938\pi$
$284$	12.6468	0.750450
$285$	42.4287	2.51326
$286$	0	0
$287$	25.5630	1.50894
$288$	−40.2251	−2.37029
$289$	1.00000	0.0588235
$290$	10.1890	0.598320
$291$	1.42240	0.0833825
$292$	6.64681	0.388975
$293$	−20.7458	−1.21198	−0.605991	−	0.795471i	$-0.707223\pi$
$293$	−20.7458	−1.21198	−0.605991	+	0.795471i	$0.707223\pi$
$294$	−13.4432	−0.784025
$295$	2.39821	0.139629
$296$	−5.55118	−0.322656
$297$	0	0
$298$	−18.8656	−1.09286
$299$	2.70638	0.156514
$300$	−2.49720	−0.144176
$301$	−5.72161	−0.329788
$302$	−4.55678	−0.262213
$303$	9.29362	0.533904
$304$	5.32340	0.305318
$305$	−20.0692	−1.14916
$306$	8.04502	0.459903
$307$	−13.8623	−0.791161	−0.395580	−	0.918431i	$-0.629457\pi$
$307$	−13.8623	−0.791161	−0.395580	+	0.918431i	$0.629457\pi$
$308$	0	0
$309$	25.1648	1.43158
$310$	−23.3144	−1.32417
$311$	8.67660	0.492005	0.246002	−	0.969269i	$-0.420883\pi$
$311$	8.67660	0.492005	0.246002	+	0.969269i	$0.420883\pi$
$312$	−29.1648	−1.65113
$313$	−14.7756	−0.835166	−0.417583	−	0.908639i	$-0.637123\pi$
$313$	−14.7756	−0.835166	−0.417583	+	0.908639i	$0.637123\pi$
$314$	7.72161	0.435756
$315$	64.1205	3.61278
$316$	−8.91623	−0.501577
$317$	27.2936	1.53296	0.766481	−	0.642267i	$-0.222006\pi$
$317$	27.2936	1.53296	0.766481	+	0.642267i	$0.222006\pi$
$318$	34.5574	1.93789
$319$	0	0
$320$	−16.7875	−0.938447
$321$	24.3386	1.35845
$322$	−3.07480	−0.171352
$323$	5.32340	0.296202
$324$	−31.5872	−1.75485
$325$	−2.19799	−0.121923
$326$	−8.64681	−0.478902
$327$	15.9404	0.881508
$328$	−23.0755	−1.27413
$329$	21.6620	1.19427
$330$	0	0
$331$	4.26943	0.234669	0.117334	−	0.993092i	$-0.462565\pi$
$331$	4.26943	0.234669	0.117334	+	0.993092i	$0.462565\pi$
$332$	−7.44322	−0.408500
$333$	−14.8864	−0.815772
$334$	5.47301	0.299470
$335$	−22.2880	−1.21772
$336$	11.0450	0.602555
$337$	19.3836	1.05589	0.527947	−	0.849277i	$-0.322962\pi$
$337$	19.3836	1.05589	0.527947	+	0.849277i	$0.322962\pi$
$338$	4.44322	0.241679
$339$	−60.5485	−3.28854
$340$	−2.39821	−0.130061
$341$	0	0
$342$	42.8269	2.31581
$343$	9.82061	0.530263
$344$	5.16484	0.278469
$345$	7.37402	0.397004
$346$	21.1648	1.13783
$347$	−1.03605	−0.0556182	−0.0278091	−	0.999613i	$-0.508853\pi$
$347$	−1.03605	−0.0556182	−0.0278091	+	0.999613i	$0.508853\pi$
$348$	−14.1198	−0.756902
$349$	−20.8864	−1.11803	−0.559013	−	0.829159i	$-0.688820\pi$
$349$	−20.8864	−1.11803	−0.559013	+	0.829159i	$0.688820\pi$
$350$	2.49720	0.133481
$351$	−49.0457	−2.61787
$352$	0	0
$353$	−36.8269	−1.96010	−0.980048	−	0.198759i	$-0.936309\pi$
$353$	−36.8269	−1.96010	−0.980048	+	0.198759i	$0.936309\pi$
$354$	3.32340	0.176637
$355$	30.3297	1.60973
$356$	5.07480	0.268964
$357$	11.0450	0.584564
$358$	20.3684	1.07650
$359$	22.5574	1.19054	0.595268	−	0.803527i	$-0.297046\pi$
$359$	22.5574	1.19054	0.595268	+	0.803527i	$0.297046\pi$
$360$	−57.8809	−3.05059
$361$	9.33863	0.491507
$362$	−4.24860	−0.223302
$363$	−36.5574	−1.91877
$364$	−9.72161	−0.509551
$365$	15.9404	0.834360
$366$	−27.8116	−1.45374
$367$	−19.1440	−0.999309	−0.499655	−	0.866225i	$-0.666540\pi$
$367$	−19.1440	−0.999309	−0.499655	+	0.866225i	$0.666540\pi$
$368$	0.925197	0.0482292
$369$	−61.8809	−3.22139
$370$	−4.43763	−0.230701
$371$	34.5574	1.79413
$372$	32.3088	1.67514
$373$	16.3297	0.845518	0.422759	−	0.906242i	$-0.361062\pi$
$373$	16.3297	0.845518	0.422759	+	0.906242i	$0.361062\pi$
$374$	0	0
$375$	33.8623	1.74864
$376$	−19.5541	−1.00842
$377$	−12.4280	−0.640074
$378$	55.7223	2.86604
$379$	−27.5035	−1.41276	−0.706379	−	0.707834i	$-0.749673\pi$
$379$	−27.5035	−1.41276	−0.706379	+	0.707834i	$0.749673\pi$
$380$	−12.7666	−0.654914
$381$	−42.4287	−2.17369
$382$	−1.22441	−0.0626462
$383$	24.4376	1.24870	0.624352	−	0.781143i	$-0.285363\pi$
$383$	24.4376	1.24870	0.624352	+	0.781143i	$0.285363\pi$
$384$	9.97021	0.508790
$385$	0	0
$386$	−14.3386	−0.729817
$387$	13.8504	0.704055
$388$	−0.427995	−0.0217281
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−23.3144	−1.18057
$391$	0.925197	0.0467892
$392$	12.1350	0.612912
$393$	60.1205	3.03268
$394$	−3.29362	−0.165930
$395$	−21.3830	−1.07589
$396$	0	0
$397$	−34.5180	−1.73241	−0.866205	−	0.499689i	$-0.833448\pi$
$397$	−34.5180	−1.73241	−0.866205	+	0.499689i	$0.833448\pi$
$398$	16.9162	0.847934
$399$	58.7971	2.94354
$400$	−0.751399	−0.0375699
$401$	4.68556	0.233986	0.116993	−	0.993133i	$-0.462675\pi$
$401$	4.68556	0.233986	0.116993	+	0.993133i	$0.462675\pi$
$402$	−30.8864	−1.54048
$403$	28.4376	1.41658
$404$	−2.79641	−0.139127
$405$	−75.7527	−3.76418
$406$	14.1198	0.700755
$407$	0	0
$408$	−9.97021	−0.493599
$409$	6.92520	0.342429	0.171214	−	0.985234i	$-0.445231\pi$
$409$	6.92520	0.342429	0.171214	+	0.985234i	$0.445231\pi$
$410$	−18.4466	−0.911012
$411$	18.6170	0.918310
$412$	−7.57201	−0.373046
$413$	3.32340	0.163534
$414$	7.44322	0.365814
$415$	−17.8504	−0.876241
$416$	14.6260	0.717098
$417$	72.9169	3.57076
$418$	0	0
$419$	−31.0152	−1.51519	−0.757597	−	0.652723i	$-0.773626\pi$
$419$	−31.0152	−1.51519	−0.757597	+	0.652723i	$0.773626\pi$
$420$	−26.4882	−1.29249
$421$	24.7756	1.20749	0.603744	−	0.797178i	$-0.293675\pi$
$421$	24.7756	1.20749	0.603744	+	0.797178i	$0.293675\pi$
$422$	27.7521	1.35095
$423$	−52.4376	−2.54960
$424$	−31.1946	−1.51494
$425$	−0.751399	−0.0364482
$426$	42.0305	2.03638
$427$	−27.8116	−1.34590
$428$	−7.32340	−0.353990
$429$	0	0
$430$	4.12878	0.199108
$431$	−8.25756	−0.397753	−0.198876	−	0.980025i	$-0.563729\pi$
$431$	−8.25756	−0.397753	−0.198876	+	0.980025i	$0.563729\pi$
$432$	−16.7666	−0.806685
$433$	19.9315	0.957845	0.478922	−	0.877857i	$-0.341027\pi$
$433$	19.9315	0.957845	0.478922	+	0.877857i	$0.341027\pi$
$434$	−32.3088	−1.55087
$435$	−33.8623	−1.62357
$436$	−4.79641	−0.229707
$437$	4.92520	0.235604
$438$	22.0900	1.05550
$439$	5.20359	0.248354	0.124177	−	0.992260i	$-0.460371\pi$
$439$	5.20359	0.248354	0.124177	+	0.992260i	$0.460371\pi$
$440$	0	0
$441$	32.5422	1.54963
$442$	−2.92520	−0.139137
$443$	−0.386346	−0.0183559	−0.00917793	−	0.999958i	$-0.502921\pi$
$443$	−0.386346	−0.0183559	−0.00917793	+	0.999958i	$0.502921\pi$
$444$	6.14961	0.291847
$445$	12.1704	0.576934
$446$	3.44322	0.163041
$447$	62.6981	2.96552
$448$	−23.2638	−1.09911
$449$	19.8719	0.937812	0.468906	−	0.883248i	$-0.344648\pi$
$449$	19.8719	0.937812	0.468906	+	0.883248i	$0.344648\pi$
$450$	−6.04502	−0.284965
$451$	0	0
$452$	18.2188	0.856941
$453$	15.1440	0.711528
$454$	−7.94043	−0.372663
$455$	−23.3144	−1.09300
$456$	−53.0755	−2.48549
$457$	19.5333	0.913727	0.456864	−	0.889537i	$-0.348973\pi$
$457$	19.5333	0.913727	0.456864	+	0.889537i	$0.348973\pi$
$458$	−29.1440	−1.36181
$459$	−16.7666	−0.782599
$460$	−2.21881	−0.103453
$461$	−23.7729	−1.10721	−0.553607	−	0.832778i	$-0.686749\pi$
$461$	−23.7729	−1.10721	−0.553607	+	0.832778i	$0.686749\pi$
$462$	0	0
$463$	4.66763	0.216923	0.108462	−	0.994101i	$-0.465408\pi$
$463$	4.66763	0.216923	0.108462	+	0.994101i	$0.465408\pi$
$464$	−4.24860	−0.197236
$465$	77.4833	3.59320
$466$	−9.59283	−0.444379
$467$	−3.94043	−0.182341	−0.0911706	−	0.995835i	$-0.529061\pi$
$467$	−3.94043	−0.182341	−0.0911706	+	0.995835i	$0.529061\pi$
$468$	23.5333	1.08783
$469$	−30.8864	−1.42620
$470$	−15.6316	−0.721031
$471$	−25.6620	−1.18244
$472$	−3.00000	−0.138086
$473$	0	0
$474$	−29.6323	−1.36105
$475$	−4.00000	−0.183533
$476$	−3.32340	−0.152328
$477$	−83.6537	−3.83024
$478$	5.26383	0.240762
$479$	26.4972	1.21069	0.605344	−	0.795964i	$-0.293036\pi$
$479$	26.4972	1.21069	0.605344	+	0.795964i	$0.293036\pi$
$480$	39.8511	1.81894
$481$	5.41277	0.246801
$482$	−27.1350	−1.23597
$483$	10.2188	0.464972
$484$	11.0000	0.500000
$485$	−1.02642	−0.0466073
$486$	−54.6773	−2.48021
$487$	−9.17380	−0.415704	−0.207852	−	0.978160i	$-0.566647\pi$
$487$	−9.17380	−0.415704	−0.207852	+	0.978160i	$0.566647\pi$
$488$	25.1053	1.13646
$489$	28.7368	1.29952
$490$	9.70079	0.438237
$491$	−2.97581	−0.134296	−0.0671482	−	0.997743i	$-0.521390\pi$
$491$	−2.97581	−0.134296	−0.0671482	+	0.997743i	$0.521390\pi$
$492$	25.5630	1.15247
$493$	−4.24860	−0.191347
$494$	−15.5720	−0.700618
$495$	0	0
$496$	9.72161	0.436513
$497$	42.0305	1.88532
$498$	−24.7368	−1.10848
$499$	38.0907	1.70517	0.852587	−	0.522585i	$-0.175032\pi$
$499$	38.0907	1.70517	0.852587	+	0.522585i	$0.175032\pi$
$500$	−10.1890	−0.455667
$501$	−18.1890	−0.812626
$502$	15.7126	0.701290
$503$	−3.26316	−0.145497	−0.0727485	−	0.997350i	$-0.523177\pi$
$503$	−3.26316	−0.145497	−0.0727485	+	0.997350i	$0.523177\pi$
$504$	−80.2105	−3.57286
$505$	−6.70638	−0.298430
$506$	0	0
$507$	−14.7666	−0.655809
$508$	12.7666	0.566428
$509$	−4.27839	−0.189636	−0.0948181	−	0.995495i	$-0.530227\pi$
$509$	−4.27839	−0.189636	−0.0948181	+	0.995495i	$0.530227\pi$
$510$	−7.97021	−0.352927
$511$	22.0900	0.977205
$512$	11.0000	0.486136
$513$	−89.2555	−3.94073
$514$	−1.54222	−0.0680243
$515$	−18.1592	−0.800192
$516$	−5.72161	−0.251880
$517$	0	0
$518$	−6.14961	−0.270198
$519$	−70.3393	−3.08755
$520$	21.0457	0.922914
$521$	−28.8864	−1.26554	−0.632769	−	0.774341i	$-0.718082\pi$
$521$	−28.8864	−1.26554	−0.632769	+	0.774341i	$0.718082\pi$
$522$	−34.1801	−1.49602
$523$	−18.5574	−0.811461	−0.405730	−	0.913993i	$-0.632983\pi$
$523$	−18.5574	−0.811461	−0.405730	+	0.913993i	$0.632983\pi$
$524$	−18.0900	−0.790267
$525$	−8.29921	−0.362207
$526$	−0.0297872	−0.00129878
$527$	9.72161	0.423480
$528$	0	0
$529$	−22.1440	−0.962783
$530$	−24.9371	−1.08320
$531$	−8.04502	−0.349124
$532$	−17.6918	−0.767038
$533$	22.5001	0.974588
$534$	16.8656	0.729847
$535$	−17.5630	−0.759317
$536$	27.8809	1.20427
$537$	−67.6925	−2.92115
$538$	22.6468	0.976373
$539$	0	0
$540$	40.2099	1.73036
$541$	−12.9252	−0.555698	−0.277849	−	0.960625i	$-0.589621\pi$
$541$	−12.9252	−0.555698	−0.277849	+	0.960625i	$0.589621\pi$
$542$	−14.6766	−0.630414
$543$	14.1198	0.605939
$544$	5.00000	0.214373
$545$	−11.5028	−0.492726
$546$	−32.3088	−1.38269
$547$	−37.6829	−1.61120	−0.805601	−	0.592458i	$-0.798158\pi$
$547$	−37.6829	−1.61120	−0.805601	+	0.592458i	$0.798158\pi$
$548$	−5.60179	−0.239297
$549$	67.3241	2.87332
$550$	0	0
$551$	−22.6170	−0.963518
$552$	−9.22441	−0.392617
$553$	−29.6323	−1.26009
$554$	−14.3386	−0.609190
$555$	14.7480	0.626019
$556$	−21.9404	−0.930481
$557$	7.45219	0.315759	0.157880	−	0.987458i	$-0.449534\pi$
$557$	7.45219	0.315759	0.157880	+	0.987458i	$0.449534\pi$
$558$	78.2105	3.31092
$559$	−5.03605	−0.213002
$560$	−7.97021	−0.336803
$561$	0	0
$562$	28.7279	1.21181
$563$	3.20359	0.135015	0.0675075	−	0.997719i	$-0.478495\pi$
$563$	3.20359	0.135015	0.0675075	+	0.997719i	$0.478495\pi$
$564$	21.6620	0.912136
$565$	43.6925	1.83816
$566$	8.66763	0.364328
$567$	−104.977	−4.40863
$568$	−37.9404	−1.59194
$569$	−36.3297	−1.52302	−0.761510	−	0.648154i	$-0.775541\pi$
$569$	−36.3297	−1.52302	−0.761510	+	0.648154i	$0.775541\pi$
$570$	−42.4287	−1.77714
$571$	3.48487	0.145837	0.0729187	−	0.997338i	$-0.476769\pi$
$571$	3.48487	0.145837	0.0729187	+	0.997338i	$0.476769\pi$
$572$	0	0
$573$	4.06921	0.169994
$574$	−25.5630	−1.06698
$575$	−0.695192	−0.0289915
$576$	56.3151	2.34646
$577$	3.69182	0.153693	0.0768463	−	0.997043i	$-0.475515\pi$
$577$	3.69182	0.153693	0.0768463	+	0.997043i	$0.475515\pi$
$578$	−1.00000	−0.0415945
$579$	47.6531	1.98039
$580$	10.1890	0.423076
$581$	−24.7368	−1.02626
$582$	−1.42240	−0.0589603
$583$	0	0
$584$	−19.9404	−0.825141
$585$	56.4376	2.33341
$586$	20.7458	0.857001
$587$	−38.1592	−1.57500	−0.787500	−	0.616314i	$-0.788625\pi$
$587$	−38.1592	−1.57500	−0.787500	+	0.616314i	$0.788625\pi$
$588$	−13.4432	−0.554389
$589$	51.7521	2.13241
$590$	−2.39821	−0.0987326
$591$	10.9460	0.450259
$592$	1.85039	0.0760507
$593$	−34.3386	−1.41012	−0.705059	−	0.709148i	$-0.749080\pi$
$593$	−34.3386	−1.41012	−0.705059	+	0.709148i	$0.749080\pi$
$594$	0	0
$595$	−7.97021	−0.326747
$596$	−18.8656	−0.772766
$597$	−56.2195	−2.30091
$598$	−2.70638	−0.110672
$599$	20.5270	0.838710	0.419355	−	0.907822i	$-0.362256\pi$
$599$	20.5270	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$600$	7.49161	0.305844
$601$	−5.55118	−0.226437	−0.113219	−	0.993570i	$-0.536116\pi$
$601$	−5.55118	−0.226437	−0.113219	+	0.993570i	$0.536116\pi$
$602$	5.72161	0.233195
$603$	74.7673	3.04476
$604$	−4.55678	−0.185413
$605$	26.3803	1.07251
$606$	−9.29362	−0.377527
$607$	8.85666	0.359481	0.179740	−	0.983714i	$-0.442474\pi$
$607$	8.85666	0.359481	0.179740	+	0.983714i	$0.442474\pi$
$608$	26.6170	1.07946
$609$	−46.9259	−1.90153
$610$	20.0692	0.812578
$611$	19.0665	0.771348
$612$	8.04502	0.325200
$613$	−16.7756	−0.677560	−0.338780	−	0.940866i	$-0.610014\pi$
$613$	−16.7756	−0.677560	−0.338780	+	0.940866i	$0.610014\pi$
$614$	13.8623	0.559435
$615$	61.3055	2.47208
$616$	0	0
$617$	21.1046	0.849639	0.424819	−	0.905278i	$-0.360338\pi$
$617$	21.1046	0.849639	0.424819	+	0.905278i	$0.360338\pi$
$618$	−25.1648	−1.01228
$619$	44.8150	1.80127	0.900634	−	0.434579i	$-0.143103\pi$
$619$	44.8150	1.80127	0.900634	+	0.434579i	$0.143103\pi$
$620$	−23.3144	−0.936330
$621$	−15.5124	−0.622492
$622$	−8.67660	−0.347900
$623$	16.8656	0.675707
$624$	9.72161	0.389176
$625$	−28.1924	−1.12770
$626$	14.7756	0.590551
$627$	0	0
$628$	7.72161	0.308126
$629$	1.85039	0.0737800
$630$	−64.1205	−2.55462
$631$	20.9944	0.835774	0.417887	−	0.908499i	$-0.362771\pi$
$631$	20.9944	0.835774	0.417887	+	0.908499i	$0.362771\pi$
$632$	26.7487	1.06401
$633$	−92.2313	−3.66587
$634$	−27.2936	−1.08397
$635$	30.6170	1.21500
$636$	34.5574	1.37029
$637$	−11.8325	−0.468819
$638$	0	0
$639$	−101.744	−4.02492
$640$	−7.19462	−0.284392
$641$	30.7368	1.21403	0.607016	−	0.794690i	$-0.292366\pi$
$641$	30.7368	1.21403	0.607016	+	0.794690i	$0.292366\pi$
$642$	−24.3386	−0.960569
$643$	−6.70705	−0.264500	−0.132250	−	0.991216i	$-0.542220\pi$
$643$	−6.70705	−0.264500	−0.132250	+	0.991216i	$0.542220\pi$
$644$	−3.07480	−0.121164
$645$	−13.7216	−0.540288
$646$	−5.32340	−0.209447
$647$	2.35946	0.0927598	0.0463799	−	0.998924i	$-0.485232\pi$
$647$	2.35946	0.0927598	0.0463799	+	0.998924i	$0.485232\pi$
$648$	94.7617	3.72259
$649$	0	0
$650$	2.19799	0.0862122
$651$	107.375	4.20837
$652$	−8.64681	−0.338635
$653$	3.63225	0.142141	0.0710705	−	0.997471i	$-0.477358\pi$
$653$	3.63225	0.142141	0.0710705	+	0.997471i	$0.477358\pi$
$654$	−15.9404	−0.623320
$655$	−43.3836	−1.69514
$656$	7.69182	0.300315
$657$	−53.4737	−2.08621
$658$	−21.6620	−0.844474
$659$	13.3353	0.519468	0.259734	−	0.965680i	$-0.416365\pi$
$659$	13.3353	0.519468	0.259734	+	0.965680i	$0.416365\pi$
$660$	0	0
$661$	−18.1406	−0.705589	−0.352795	−	0.935701i	$-0.614769\pi$
$661$	−18.1406	−0.705589	−0.352795	+	0.935701i	$0.614769\pi$
$662$	−4.26943	−0.165936
$663$	9.72161	0.377556
$664$	22.3297	0.866559
$665$	−42.4287	−1.64531
$666$	14.8864	0.576838
$667$	−3.93079	−0.152201
$668$	5.47301	0.211757
$669$	−11.4432	−0.442421
$670$	22.2880	0.861061
$671$	0	0
$672$	55.2251	2.13035
$673$	1.97918	0.0762916	0.0381458	−	0.999272i	$-0.487855\pi$
$673$	1.97918	0.0762916	0.0381458	+	0.999272i	$0.487855\pi$
$674$	−19.3836	−0.746630
$675$	12.5984	0.484914
$676$	4.44322	0.170893
$677$	−41.1440	−1.58129	−0.790646	−	0.612273i	$-0.790255\pi$
$677$	−41.1440	−1.58129	−0.790646	+	0.612273i	$0.790255\pi$
$678$	60.5485	2.32535
$679$	−1.42240	−0.0545867
$680$	7.19462	0.275901
$681$	26.3892	1.01124
$682$	0	0
$683$	23.3144	0.892102	0.446051	−	0.895007i	$-0.352830\pi$
$683$	23.3144	0.892102	0.446051	+	0.895007i	$0.352830\pi$
$684$	42.8269	1.63753
$685$	−13.4343	−0.513297
$686$	−9.82061	−0.374952
$687$	96.8573	3.69534
$688$	−1.72161	−0.0656358
$689$	30.4168	1.15879
$690$	−7.37402	−0.280724
$691$	−29.3628	−1.11701	−0.558507	−	0.829500i	$-0.688626\pi$
$691$	−29.3628	−1.11701	−0.558507	+	0.829500i	$0.688626\pi$
$692$	21.1648	0.804566
$693$	0	0
$694$	1.03605	0.0393280
$695$	−52.6177	−1.99590
$696$	42.3595	1.60563
$697$	7.69182	0.291349
$698$	20.8864	0.790564
$699$	31.8809	1.20584
$700$	2.49720	0.0943854
$701$	−35.6620	−1.34694	−0.673468	−	0.739216i	$-0.735196\pi$
$701$	−35.6620	−1.34694	−0.673468	+	0.739216i	$0.735196\pi$
$702$	49.0457	1.85111
$703$	9.85039	0.371515
$704$	0	0
$705$	51.9501	1.95655
$706$	36.8269	1.38600
$707$	−9.29362	−0.349522
$708$	3.32340	0.124901
$709$	46.0990	1.73128	0.865642	−	0.500663i	$-0.166911\pi$
$709$	46.0990	1.73128	0.865642	+	0.500663i	$0.166911\pi$
$710$	−30.3297	−1.13825
$711$	71.7312	2.69013
$712$	−15.2244	−0.570559
$713$	8.99440	0.336843
$714$	−11.0450	−0.413349
$715$	0	0
$716$	20.3684	0.761204
$717$	−17.4938	−0.653319
$718$	−22.5574	−0.841836
$719$	16.9073	0.630535	0.315267	−	0.949003i	$-0.397906\pi$
$719$	16.9073	0.630535	0.315267	+	0.949003i	$0.397906\pi$
$720$	19.2936	0.719031
$721$	−25.1648	−0.937187
$722$	−9.33863	−0.347548
$723$	90.1807	3.35386
$724$	−4.24860	−0.157898
$725$	3.19239	0.118563
$726$	36.5574	1.35677
$727$	−6.58723	−0.244307	−0.122153	−	0.992511i	$-0.538980\pi$
$727$	−6.58723	−0.244307	−0.122153	+	0.992511i	$0.538980\pi$
$728$	29.1648	1.08092
$729$	86.9530	3.22048
$730$	−15.9404	−0.589982
$731$	−1.72161	−0.0636761
$732$	−27.8116	−1.02795
$733$	−18.7785	−0.693599	−0.346800	−	0.937939i	$-0.612732\pi$
$733$	−18.7785	−0.693599	−0.346800	+	0.937939i	$0.612732\pi$
$734$	19.1440	0.706618
$735$	−32.2396	−1.18918
$736$	4.62598	0.170516
$737$	0	0
$738$	61.8809	2.27787
$739$	−34.7160	−1.27705	−0.638525	−	0.769601i	$-0.720455\pi$
$739$	−34.7160	−1.27705	−0.638525	+	0.769601i	$0.720455\pi$
$740$	−4.43763	−0.163130
$741$	51.7521	1.90116
$742$	−34.5574	−1.26864
$743$	−46.1621	−1.69352	−0.846762	−	0.531971i	$-0.821451\pi$
$743$	−46.1621	−1.69352	−0.846762	+	0.531971i	$0.821451\pi$
$744$	−96.9265	−3.55350
$745$	−45.2437	−1.65760
$746$	−16.3297	−0.597872
$747$	59.8809	2.19093
$748$	0	0
$749$	−24.3386	−0.889314
$750$	−33.8623	−1.23647
$751$	−28.9765	−1.05737	−0.528683	−	0.848819i	$-0.677314\pi$
$751$	−28.9765	−1.05737	−0.528683	+	0.848819i	$0.677314\pi$
$752$	6.51803	0.237688
$753$	−52.2195	−1.90298
$754$	12.4280	0.452601
$755$	−10.9281	−0.397714
$756$	55.7223	2.02660
$757$	−30.8954	−1.12291	−0.561456	−	0.827506i	$-0.689759\pi$
$757$	−30.8954	−1.12291	−0.561456	+	0.827506i	$0.689759\pi$
$758$	27.5035	0.998971
$759$	0	0
$760$	38.2999	1.38928
$761$	−19.6739	−0.713178	−0.356589	−	0.934261i	$-0.616060\pi$
$761$	−19.6739	−0.713178	−0.356589	+	0.934261i	$0.616060\pi$
$762$	42.4287	1.53703
$763$	−15.9404	−0.577082
$764$	−1.22441	−0.0442976
$765$	19.2936	0.697562
$766$	−24.4376	−0.882967
$767$	2.92520	0.105623
$768$	−56.4979	−2.03869
$769$	−27.6441	−0.996872	−0.498436	−	0.866926i	$-0.666092\pi$
$769$	−27.6441	−0.996872	−0.498436	+	0.866926i	$0.666092\pi$
$770$	0	0
$771$	5.12541	0.184587
$772$	−14.3386	−0.516059
$773$	−12.5872	−0.452731	−0.226366	−	0.974042i	$-0.572684\pi$
$773$	−12.5872	−0.452731	−0.226366	+	0.974042i	$0.572684\pi$
$774$	−13.8504	−0.497842
$775$	−7.30481	−0.262396
$776$	1.28398	0.0460923
$777$	20.4376	0.733196
$778$	26.0000	0.932145
$779$	40.9467	1.46707
$780$	−23.3144	−0.834791
$781$	0	0
$782$	−0.925197	−0.0330850
$783$	71.2347	2.54572
$784$	−4.04502	−0.144465
$785$	18.5180	0.660937
$786$	−60.1205	−2.14443
$787$	−29.6829	−1.05808	−0.529040	−	0.848597i	$-0.677448\pi$
$787$	−29.6829	−1.05808	−0.529040	+	0.848597i	$0.677448\pi$
$788$	−3.29362	−0.117330
$789$	0.0989948	0.00352431
$790$	21.3830	0.760772
$791$	60.5485	2.15286
$792$	0	0
$793$	−24.4793	−0.869285
$794$	34.5180	1.22500
$795$	82.8759	2.93931
$796$	16.9162	0.599580
$797$	−29.5124	−1.04538	−0.522692	−	0.852522i	$-0.675072\pi$
$797$	−29.5124	−1.04538	−0.522692	+	0.852522i	$0.675072\pi$
$798$	−58.7971	−2.08139
$799$	6.51803	0.230591
$800$	−3.75699	−0.132830
$801$	−40.8269	−1.44255
$802$	−4.68556	−0.165453
$803$	0	0
$804$	−30.8864	−1.08928
$805$	−7.37402	−0.259900
$806$	−28.4376	−1.00167
$807$	−75.2645	−2.64944
$808$	8.38924	0.295133
$809$	−14.8269	−0.521285	−0.260643	−	0.965435i	$-0.583934\pi$
$809$	−14.8269	−0.521285	−0.260643	+	0.965435i	$0.583934\pi$
$810$	75.7527	2.66168
$811$	37.2936	1.30956	0.654778	−	0.755821i	$-0.272762\pi$
$811$	37.2936	1.30956	0.654778	+	0.755821i	$0.272762\pi$
$812$	14.1198	0.495509
$813$	48.7763	1.71066
$814$	0	0
$815$	−20.7368	−0.726380
$816$	3.32340	0.116342
$817$	−9.16484	−0.320637
$818$	−6.92520	−0.242134
$819$	78.2105	2.73290
$820$	−18.4466	−0.644183
$821$	11.5028	0.401450	0.200725	−	0.979648i	$-0.435670\pi$
$821$	11.5028	0.401450	0.200725	+	0.979648i	$0.435670\pi$
$822$	−18.6170	−0.649343
$823$	−40.5901	−1.41488	−0.707442	−	0.706772i	$-0.750151\pi$
$823$	−40.5901	−1.41488	−0.707442	+	0.706772i	$0.750151\pi$
$824$	22.7160	0.791350
$825$	0	0
$826$	−3.32340	−0.115636
$827$	2.31714	0.0805748	0.0402874	−	0.999188i	$-0.487173\pi$
$827$	2.31714	0.0805748	0.0402874	+	0.999188i	$0.487173\pi$
$828$	7.44322	0.258670
$829$	54.0215	1.87624	0.938122	−	0.346305i	$-0.112564\pi$
$829$	54.0215	1.87624	0.938122	+	0.346305i	$0.112564\pi$
$830$	17.8504	0.619596
$831$	47.6531	1.65307
$832$	−20.4764	−0.709891
$833$	−4.04502	−0.140151
$834$	−72.9169	−2.52491
$835$	13.1254	0.454224
$836$	0	0
$837$	−162.999	−5.63406
$838$	31.0152	1.07140
$839$	−23.3836	−0.807293	−0.403647	−	0.914915i	$-0.632257\pi$
$839$	−23.3836	−0.807293	−0.403647	+	0.914915i	$0.632257\pi$
$840$	79.4647	2.74179
$841$	−10.9494	−0.377565
$842$	−24.7756	−0.853823
$843$	−95.4744	−3.28831
$844$	27.7521	0.955266
$845$	10.6558	0.366570
$846$	52.4376	1.80284
$847$	36.5574	1.25613
$848$	10.3982	0.357076
$849$	−28.8060	−0.988621
$850$	0.751399	0.0257728
$851$	1.71198	0.0586859
$852$	42.0305	1.43994
$853$	−25.0450	−0.857525	−0.428763	−	0.903417i	$-0.641050\pi$
$853$	−25.0450	−0.857525	−0.428763	+	0.903417i	$0.641050\pi$
$854$	27.8116	0.951695
$855$	102.708	3.51253
$856$	21.9702	0.750926
$857$	30.9973	1.05885	0.529424	−	0.848357i	$-0.322408\pi$
$857$	30.9973	1.05885	0.529424	+	0.848357i	$0.322408\pi$
$858$	0	0
$859$	16.0692	0.548274	0.274137	−	0.961691i	$-0.411608\pi$
$859$	16.0692	0.548274	0.274137	+	0.961691i	$0.411608\pi$
$860$	4.12878	0.140790
$861$	84.9563	2.89530
$862$	8.25756	0.281254
$863$	12.1205	0.412586	0.206293	−	0.978490i	$-0.433860\pi$
$863$	12.1205	0.412586	0.206293	+	0.978490i	$0.433860\pi$
$864$	−83.8331	−2.85206
$865$	50.7577	1.72581
$866$	−19.9315	−0.677299
$867$	3.32340	0.112869
$868$	−32.3088	−1.09663
$869$	0	0
$870$	33.8623	1.14804
$871$	−27.1857	−0.921151
$872$	14.3892	0.487281
$873$	3.44322	0.116535
$874$	−4.92520	−0.166597
$875$	−33.8623	−1.14475
$876$	22.0900	0.746353
$877$	23.6918	0.800016	0.400008	−	0.916512i	$-0.369007\pi$
$877$	23.6918	0.800016	0.400008	+	0.916512i	$0.369007\pi$
$878$	−5.20359	−0.175612
$879$	−68.9467	−2.32551
$880$	0	0
$881$	−16.4793	−0.555201	−0.277600	−	0.960697i	$-0.589539\pi$
$881$	−16.4793	−0.555201	−0.277600	+	0.960697i	$0.589539\pi$
$882$	−32.5422	−1.09575
$883$	26.6587	0.897136	0.448568	−	0.893749i	$-0.351934\pi$
$883$	26.6587	0.897136	0.448568	+	0.893749i	$0.351934\pi$
$884$	−2.92520	−0.0983850
$885$	7.97021	0.267916
$886$	0.386346	0.0129795
$887$	−14.9460	−0.501838	−0.250919	−	0.968008i	$-0.580733\pi$
$887$	−14.9460	−0.501838	−0.250919	+	0.968008i	$0.580733\pi$
$888$	−18.4488	−0.619102
$889$	42.4287	1.42301
$890$	−12.1704	−0.407954
$891$	0	0
$892$	3.44322	0.115288
$893$	34.6981	1.16113
$894$	−62.6981	−2.09694
$895$	48.8477	1.63280
$896$	−9.97021	−0.333081
$897$	8.99440	0.300314
$898$	−19.8719	−0.663134
$899$	−41.3033	−1.37754
$900$	−6.04502	−0.201501
$901$	10.3982	0.346415
$902$	0	0
$903$	−19.0152	−0.632787
$904$	−54.6564	−1.81785
$905$	−10.1890	−0.338695
$906$	−15.1440	−0.503126
$907$	30.6891	1.01902	0.509508	−	0.860466i	$-0.329827\pi$
$907$	30.6891	1.01902	0.509508	+	0.860466i	$0.329827\pi$
$908$	−7.94043	−0.263512
$909$	22.4972	0.746185
$910$	23.3144	0.772866
$911$	21.6710	0.717992	0.358996	−	0.933339i	$-0.383119\pi$
$911$	21.6710	0.717992	0.358996	+	0.933339i	$0.383119\pi$
$912$	17.6918	0.585835
$913$	0	0
$914$	−19.5333	−0.646103
$915$	−66.6981	−2.20497
$916$	−29.1440	−0.962945
$917$	−60.1205	−1.98535
$918$	16.7666	0.553381
$919$	−6.88645	−0.227163	−0.113581	−	0.993529i	$-0.536232\pi$
$919$	−6.88645	−0.227163	−0.113581	+	0.993529i	$0.536232\pi$
$920$	6.65644	0.219456
$921$	−46.0699	−1.51805
$922$	23.7729	0.782919
$923$	36.9944	1.21769
$924$	0	0
$925$	−1.39038	−0.0457155
$926$	−4.66763	−0.153388
$927$	60.9169	2.00077
$928$	−21.2430	−0.697336
$929$	15.9225	0.522400	0.261200	−	0.965285i	$-0.415882\pi$
$929$	15.9225	0.522400	0.261200	+	0.965285i	$0.415882\pi$
$930$	−77.4833	−2.54078
$931$	−21.5333	−0.705724
$932$	−9.59283	−0.314224
$933$	28.8358	0.944043
$934$	3.94043	0.128935
$935$	0	0
$936$	−70.5998	−2.30763
$937$	−60.5872	−1.97930	−0.989649	−	0.143507i	$-0.954162\pi$
$937$	−60.5872	−1.97930	−0.989649	+	0.143507i	$0.954162\pi$
$938$	30.8864	1.00848
$939$	−49.1053	−1.60249
$940$	−15.6316	−0.509846
$941$	−21.1232	−0.688596	−0.344298	−	0.938860i	$-0.611883\pi$
$941$	−21.1232	−0.688596	−0.344298	+	0.938860i	$0.611883\pi$
$942$	25.6620	0.836114
$943$	7.11645	0.231744
$944$	1.00000	0.0325472
$945$	133.634	4.34710
$946$	0	0
$947$	−28.0007	−0.909900	−0.454950	−	0.890517i	$-0.650343\pi$
$947$	−28.0007	−0.909900	−0.454950	+	0.890517i	$0.650343\pi$
$948$	−29.6323	−0.962411
$949$	19.4432	0.631154
$950$	4.00000	0.129777
$951$	90.7077	2.94140
$952$	9.97021	0.323136
$953$	45.6233	1.47788	0.738942	−	0.673769i	$-0.235326\pi$
$953$	45.6233	1.47788	0.738942	+	0.673769i	$0.235326\pi$
$954$	83.6537	2.70839
$955$	−2.93639	−0.0950193
$956$	5.26383	0.170244
$957$	0	0
$958$	−26.4972	−0.856086
$959$	−18.6170	−0.601175
$960$	−55.7915	−1.80066
$961$	63.5097	2.04870
$962$	−5.41277	−0.174515
$963$	58.9169	1.89857
$964$	−27.1350	−0.873961
$965$	−34.3870	−1.10696
$966$	−10.2188	−0.328785
$967$	−32.9348	−1.05911	−0.529556	−	0.848275i	$-0.677642\pi$
$967$	−32.9348	−1.05911	−0.529556	+	0.848275i	$0.677642\pi$
$968$	−33.0000	−1.06066
$969$	17.6918	0.568343
$970$	1.02642	0.0329564
$971$	41.9827	1.34729	0.673645	−	0.739055i	$-0.264728\pi$
$971$	41.9827	1.34729	0.673645	+	0.739055i	$0.264728\pi$
$972$	−54.6773	−1.75377
$973$	−72.9169	−2.33761
$974$	9.17380	0.293947
$975$	−7.30481	−0.233941
$976$	−8.36842	−0.267867
$977$	−3.67996	−0.117732	−0.0588662	−	0.998266i	$-0.518749\pi$
$977$	−3.67996	−0.117732	−0.0588662	+	0.998266i	$0.518749\pi$
$978$	−28.7368	−0.918903
$979$	0	0
$980$	9.70079	0.309880
$981$	38.5872	1.23200
$982$	2.97581	0.0949619
$983$	−12.0596	−0.384641	−0.192320	−	0.981332i	$-0.561601\pi$
$983$	−12.0596	−0.384641	−0.192320	+	0.981332i	$0.561601\pi$
$984$	−76.6891	−2.44476
$985$	−7.89878	−0.251676
$986$	4.24860	0.135303
$987$	71.9917	2.29152
$988$	−15.5720	−0.495411
$989$	−1.59283	−0.0506490
$990$	0	0
$991$	−5.37112	−0.170619	−0.0853096	−	0.996354i	$-0.527188\pi$
$991$	−5.37112	−0.170619	−0.0853096	+	0.996354i	$0.527188\pi$
$992$	48.6081	1.54331
$993$	14.1890	0.450275
$994$	−42.0305	−1.33313
$995$	40.5686	1.28611
$996$	−24.7368	−0.783817
$997$	19.3926	0.614170	0.307085	−	0.951682i	$-0.400646\pi$
$997$	19.3926	0.614170	0.307085	+	0.951682i	$0.400646\pi$
$998$	−38.0907	−1.20574
$999$	−31.0249	−0.981583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.e.1.3	✓	3
3.2	odd	2		9027.2.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.e.1.3	✓	3	1.1	even	1	trivial
9027.2.a.h.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.32340 q^{3} -1.00000 q^{4} -2.39821 q^{5} -3.32340 q^{6} -3.32340 q^{7} +3.00000 q^{8} +8.04502 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.32340 q^{3} -1.00000 q^{4} -2.39821 q^{5} -3.32340 q^{6} -3.32340 q^{7} +3.00000 q^{8} +8.04502 q^{9} +2.39821 q^{10} -3.32340 q^{12} -2.92520 q^{13} +3.32340 q^{14} -7.97021 q^{15} -1.00000 q^{16} -1.00000 q^{17} -8.04502 q^{18} -5.32340 q^{19} +2.39821 q^{20} -11.0450 q^{21} -0.925197 q^{23} +9.97021 q^{24} +0.751399 q^{25} +2.92520 q^{26} +16.7666 q^{27} +3.32340 q^{28} +4.24860 q^{29} +7.97021 q^{30} -9.72161 q^{31} -5.00000 q^{32} +1.00000 q^{34} +7.97021 q^{35} -8.04502 q^{36} -1.85039 q^{37} +5.32340 q^{38} -9.72161 q^{39} -7.19462 q^{40} -7.69182 q^{41} +11.0450 q^{42} +1.72161 q^{43} -19.2936 q^{45} +0.925197 q^{46} -6.51803 q^{47} -3.32340 q^{48} +4.04502 q^{49} -0.751399 q^{50} -3.32340 q^{51} +2.92520 q^{52} -10.3982 q^{53} -16.7666 q^{54} -9.97021 q^{56} -17.6918 q^{57} -4.24860 q^{58} -1.00000 q^{59} +7.97021 q^{60} +8.36842 q^{61} +9.72161 q^{62} -26.7368 q^{63} +7.00000 q^{64} +7.01523 q^{65} +9.29362 q^{67} +1.00000 q^{68} -3.07480 q^{69} -7.97021 q^{70} -12.6468 q^{71} +24.1350 q^{72} -6.64681 q^{73} +1.85039 q^{74} +2.49720 q^{75} +5.32340 q^{76} +9.72161 q^{78} +8.91623 q^{79} +2.39821 q^{80} +31.5872 q^{81} +7.69182 q^{82} +7.44322 q^{83} +11.0450 q^{84} +2.39821 q^{85} -1.72161 q^{86} +14.1198 q^{87} -5.07480 q^{89} +19.2936 q^{90} +9.72161 q^{91} +0.925197 q^{92} -32.3088 q^{93} +6.51803 q^{94} +12.7666 q^{95} -16.6170 q^{96} +0.427995 q^{97} -4.04502 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} - 3 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10}) \) 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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