LMFDB - Embedded newform 4009.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.18144 q^{2} -2.69143 q^{3} +2.75868 q^{4} +0.125286 q^{5} +5.87120 q^{6} -3.80798 q^{7} -1.65501 q^{8} +4.24381 q^{9} +O(q^{10})$	\(q-2.18144 q^{2} -2.69143 q^{3} +2.75868 q^{4} +0.125286 q^{5} +5.87120 q^{6} -3.80798 q^{7} -1.65501 q^{8} +4.24381 q^{9} -0.273304 q^{10} -6.52712 q^{11} -7.42480 q^{12} -4.53493 q^{13} +8.30687 q^{14} -0.337199 q^{15} -1.90705 q^{16} +6.34825 q^{17} -9.25762 q^{18} -1.00000 q^{19} +0.345624 q^{20} +10.2489 q^{21} +14.2385 q^{22} +0.121667 q^{23} +4.45435 q^{24} -4.98430 q^{25} +9.89268 q^{26} -3.34763 q^{27} -10.5050 q^{28} +1.86140 q^{29} +0.735579 q^{30} -6.88197 q^{31} +7.47014 q^{32} +17.5673 q^{33} -13.8483 q^{34} -0.477087 q^{35} +11.7073 q^{36} -5.41126 q^{37} +2.18144 q^{38} +12.2055 q^{39} -0.207350 q^{40} +9.51886 q^{41} -22.3574 q^{42} +11.8652 q^{43} -18.0062 q^{44} +0.531690 q^{45} -0.265408 q^{46} -3.97080 q^{47} +5.13269 q^{48} +7.50070 q^{49} +10.8730 q^{50} -17.0859 q^{51} -12.5104 q^{52} +7.55282 q^{53} +7.30266 q^{54} -0.817758 q^{55} +6.30225 q^{56} +2.69143 q^{57} -4.06053 q^{58} -5.38697 q^{59} -0.930224 q^{60} +3.10948 q^{61} +15.0126 q^{62} -16.1603 q^{63} -12.4816 q^{64} -0.568164 q^{65} -38.3220 q^{66} +3.65948 q^{67} +17.5128 q^{68} -0.327458 q^{69} +1.04074 q^{70} -13.9384 q^{71} -7.02356 q^{72} -1.62253 q^{73} +11.8043 q^{74} +13.4149 q^{75} -2.75868 q^{76} +24.8551 q^{77} -26.6255 q^{78} +1.04258 q^{79} -0.238927 q^{80} -3.72151 q^{81} -20.7648 q^{82} +2.27460 q^{83} +28.2735 q^{84} +0.795348 q^{85} -25.8832 q^{86} -5.00983 q^{87} +10.8025 q^{88} -5.69222 q^{89} -1.15985 q^{90} +17.2689 q^{91} +0.335639 q^{92} +18.5224 q^{93} +8.66207 q^{94} -0.125286 q^{95} -20.1054 q^{96} +15.5872 q^{97} -16.3623 q^{98} -27.6999 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.18144	−1.54251	−0.771255	−	0.636526i	$-0.780371\pi$
$2$	−2.18144	−1.54251	−0.771255	+	0.636526i	$0.780371\pi$
$3$	−2.69143	−1.55390	−0.776950	−	0.629563i	$-0.783234\pi$
$3$	−2.69143	−1.55390	−0.776950	+	0.629563i	$0.783234\pi$
$4$	2.75868	1.37934
$5$	0.125286	0.0560296	0.0280148	−	0.999608i	$-0.491081\pi$
$5$	0.125286	0.0560296	0.0280148	+	0.999608i	$0.491081\pi$
$6$	5.87120	2.39691
$7$	−3.80798	−1.43928	−0.719640	−	0.694347i	$-0.755693\pi$
$7$	−3.80798	−1.43928	−0.719640	+	0.694347i	$0.755693\pi$
$8$	−1.65501	−0.585135
$9$	4.24381	1.41460
$10$	−0.273304	−0.0864263
$11$	−6.52712	−1.96800	−0.984001	−	0.178164i	$-0.942984\pi$
$11$	−6.52712	−1.96800	−0.984001	+	0.178164i	$0.942984\pi$
$12$	−7.42480	−2.14335
$13$	−4.53493	−1.25776	−0.628882	−	0.777501i	$-0.716487\pi$
$13$	−4.53493	−1.25776	−0.628882	+	0.777501i	$0.716487\pi$
$14$	8.30687	2.22011
$15$	−0.337199	−0.0870644
$16$	−1.90705	−0.476762
$17$	6.34825	1.53968	0.769839	−	0.638239i	$-0.220337\pi$
$17$	6.34825	1.53968	0.769839	+	0.638239i	$0.220337\pi$
$18$	−9.25762	−2.18204
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.345624	0.0772839
$21$	10.2489	2.23650
$22$	14.2385	3.03566
$23$	0.121667	0.0253692	0.0126846	−	0.999920i	$-0.495962\pi$
$23$	0.121667	0.0253692	0.0126846	+	0.999920i	$0.495962\pi$
$24$	4.45435	0.909241
$25$	−4.98430	−0.996861
$26$	9.89268	1.94011
$27$	−3.34763	−0.644252
$28$	−10.5050	−1.98526
$29$	1.86140	0.345653	0.172827	−	0.984952i	$-0.444710\pi$
$29$	1.86140	0.345653	0.172827	+	0.984952i	$0.444710\pi$
$30$	0.735579	0.134298
$31$	−6.88197	−1.23604	−0.618019	−	0.786163i	$-0.712065\pi$
$31$	−6.88197	−1.23604	−0.618019	+	0.786163i	$0.712065\pi$
$32$	7.47014	1.32055
$33$	17.5673	3.05808
$34$	−13.8483	−2.37497
$35$	−0.477087	−0.0806424
$36$	11.7073	1.95122
$37$	−5.41126	−0.889606	−0.444803	−	0.895628i	$-0.646726\pi$
$37$	−5.41126	−0.889606	−0.444803	+	0.895628i	$0.646726\pi$
$38$	2.18144	0.353876
$39$	12.2055	1.95444
$40$	−0.207350	−0.0327849
$41$	9.51886	1.48660	0.743298	−	0.668960i	$-0.233260\pi$
$41$	9.51886	1.48660	0.743298	+	0.668960i	$0.233260\pi$
$42$	−22.3574	−3.44982
$43$	11.8652	1.80942	0.904711	−	0.426026i	$-0.140087\pi$
$43$	11.8652	1.80942	0.904711	+	0.426026i	$0.140087\pi$
$44$	−18.0062	−2.71454
$45$	0.531690	0.0792597
$46$	−0.265408	−0.0391323
$47$	−3.97080	−0.579201	−0.289601	−	0.957148i	$-0.593522\pi$
$47$	−3.97080	−0.579201	−0.289601	+	0.957148i	$0.593522\pi$
$48$	5.13269	0.740841
$49$	7.50070	1.07153
$50$	10.8730	1.53767
$51$	−17.0859	−2.39250
$52$	−12.5104	−1.73488
$53$	7.55282	1.03746	0.518730	−	0.854938i	$-0.326405\pi$
$53$	7.55282	1.03746	0.518730	+	0.854938i	$0.326405\pi$
$54$	7.30266	0.993766
$55$	−0.817758	−0.110266
$56$	6.30225	0.842173
$57$	2.69143	0.356489
$58$	−4.06053	−0.533174
$59$	−5.38697	−0.701324	−0.350662	−	0.936502i	$-0.614043\pi$
$59$	−5.38697	−0.701324	−0.350662	+	0.936502i	$0.614043\pi$
$60$	−0.930224	−0.120091
$61$	3.10948	0.398128	0.199064	−	0.979987i	$-0.436210\pi$
$61$	3.10948	0.398128	0.199064	+	0.979987i	$0.436210\pi$
$62$	15.0126	1.90660
$63$	−16.1603	−2.03601
$64$	−12.4816	−1.56019
$65$	−0.568164	−0.0704720
$66$	−38.3220	−4.71712
$67$	3.65948	0.447076	0.223538	−	0.974695i	$-0.428239\pi$
$67$	3.65948	0.447076	0.223538	+	0.974695i	$0.428239\pi$
$68$	17.5128	2.12374
$69$	−0.327458	−0.0394213
$70$	1.04074	0.124392
$71$	−13.9384	−1.65418	−0.827091	−	0.562068i	$-0.810006\pi$
$71$	−13.9384	−1.65418	−0.827091	+	0.562068i	$0.810006\pi$
$72$	−7.02356	−0.827734
$73$	−1.62253	−0.189902	−0.0949512	−	0.995482i	$-0.530269\pi$
$73$	−1.62253	−0.189902	−0.0949512	+	0.995482i	$0.530269\pi$
$74$	11.8043	1.37223
$75$	13.4149	1.54902
$76$	−2.75868	−0.316442
$77$	24.8551	2.83251
$78$	−26.6255	−3.01474
$79$	1.04258	0.117300	0.0586499	−	0.998279i	$-0.481320\pi$
$79$	1.04258	0.117300	0.0586499	+	0.998279i	$0.481320\pi$
$80$	−0.238927	−0.0267128
$81$	−3.72151	−0.413501
$82$	−20.7648	−2.29309
$83$	2.27460	0.249669	0.124835	−	0.992178i	$-0.460160\pi$
$83$	2.27460	0.249669	0.124835	+	0.992178i	$0.460160\pi$
$84$	28.2735	3.08489
$85$	0.795348	0.0862676
$86$	−25.8832	−2.79105
$87$	−5.00983	−0.537110
$88$	10.8025	1.15155
$89$	−5.69222	−0.603374	−0.301687	−	0.953407i	$-0.597550\pi$
$89$	−5.69222	−0.603374	−0.301687	+	0.953407i	$0.597550\pi$
$90$	−1.15985	−0.122259
$91$	17.2689	1.81027
$92$	0.335639	0.0349928
$93$	18.5224	1.92068
$94$	8.66207	0.893424
$95$	−0.125286	−0.0128541
$96$	−20.1054	−2.05200
$97$	15.5872	1.58264	0.791322	−	0.611400i	$-0.209393\pi$
$97$	15.5872	1.58264	0.791322	+	0.611400i	$0.209393\pi$
$98$	−16.3623	−1.65284
$99$	−27.6999	−2.78394
$100$	−13.7501	−1.37501
$101$	−11.9391	−1.18798	−0.593991	−	0.804472i	$-0.702448\pi$
$101$	−11.9391	−1.18798	−0.593991	+	0.804472i	$0.702448\pi$
$102$	37.2718	3.69046
$103$	1.83153	0.180466	0.0902332	−	0.995921i	$-0.471239\pi$
$103$	1.83153	0.180466	0.0902332	+	0.995921i	$0.471239\pi$
$104$	7.50537	0.735962
$105$	1.28405	0.125310
$106$	−16.4760	−1.60029
$107$	8.19654	0.792389	0.396195	−	0.918167i	$-0.370331\pi$
$107$	8.19654	0.792389	0.396195	+	0.918167i	$0.370331\pi$
$108$	−9.23504	−0.888642
$109$	15.0360	1.44018	0.720092	−	0.693879i	$-0.244100\pi$
$109$	15.0360	1.44018	0.720092	+	0.693879i	$0.244100\pi$
$110$	1.78389	0.170087
$111$	14.5640	1.38236
$112$	7.26200	0.686194
$113$	−10.1208	−0.952083	−0.476041	−	0.879423i	$-0.657929\pi$
$113$	−10.1208	−0.952083	−0.476041	+	0.879423i	$0.657929\pi$
$114$	−5.87120	−0.549888
$115$	0.0152431	0.00142143
$116$	5.13500	0.476773
$117$	−19.2454	−1.77924
$118$	11.7513	1.08180
$119$	−24.1740	−2.21603
$120$	0.558068	0.0509444
$121$	31.6033	2.87303
$122$	−6.78313	−0.614116
$123$	−25.6194	−2.31002
$124$	−18.9851	−1.70492
$125$	−1.25089	−0.111883
$126$	35.2528	3.14057
$127$	22.4323	1.99055	0.995273	−	0.0971122i	$-0.0309606\pi$
$127$	22.4323	1.99055	0.995273	+	0.0971122i	$0.0309606\pi$
$128$	12.2875	1.08607
$129$	−31.9343	−2.81166
$130$	1.23941	0.108704
$131$	4.10359	0.358532	0.179266	−	0.983801i	$-0.442628\pi$
$131$	4.10359	0.358532	0.179266	+	0.983801i	$0.442628\pi$
$132$	48.4626	4.21813
$133$	3.80798	0.330194
$134$	−7.98293	−0.689620
$135$	−0.419412	−0.0360972
$136$	−10.5064	−0.900919
$137$	−4.85420	−0.414723	−0.207361	−	0.978264i	$-0.566488\pi$
$137$	−4.85420	−0.414723	−0.207361	+	0.978264i	$0.566488\pi$
$138$	0.714329	0.0608077
$139$	−2.31623	−0.196460	−0.0982299	−	0.995164i	$-0.531318\pi$
$139$	−2.31623	−0.196460	−0.0982299	+	0.995164i	$0.531318\pi$
$140$	−1.31613	−0.111233
$141$	10.6871	0.900020
$142$	30.4058	2.55159
$143$	29.6001	2.47528
$144$	−8.09315	−0.674429
$145$	0.233207	0.0193668
$146$	3.53944	0.292926
$147$	−20.1876	−1.66505
$148$	−14.9279	−1.22707
$149$	8.04906	0.659405	0.329702	−	0.944085i	$-0.393052\pi$
$149$	8.04906	0.659405	0.329702	+	0.944085i	$0.393052\pi$
$150$	−29.2638	−2.38938
$151$	−7.23318	−0.588627	−0.294314	−	0.955709i	$-0.595091\pi$
$151$	−7.23318	−0.588627	−0.294314	+	0.955709i	$0.595091\pi$
$152$	1.65501	0.134239
$153$	26.9408	2.17803
$154$	−54.2200	−4.36917
$155$	−0.862215	−0.0692548
$156$	33.6710	2.69583
$157$	11.2729	0.899672	0.449836	−	0.893111i	$-0.351482\pi$
$157$	11.2729	0.899672	0.449836	+	0.893111i	$0.351482\pi$
$158$	−2.27433	−0.180936
$159$	−20.3279	−1.61211
$160$	0.935904	0.0739897
$161$	−0.463304	−0.0365135
$162$	8.11824	0.637829
$163$	3.82958	0.299956	0.149978	−	0.988689i	$-0.452080\pi$
$163$	3.82958	0.299956	0.149978	+	0.988689i	$0.452080\pi$
$164$	26.2595	2.05052
$165$	2.20094	0.171343
$166$	−4.96189	−0.385118
$167$	15.6046	1.20752	0.603758	−	0.797167i	$-0.293669\pi$
$167$	15.6046	1.20752	0.603758	+	0.797167i	$0.293669\pi$
$168$	−16.9621	−1.30865
$169$	7.56561	0.581970
$170$	−1.73500	−0.133069
$171$	−4.24381	−0.324532
$172$	32.7322	2.49581
$173$	7.50691	0.570739	0.285370	−	0.958418i	$-0.407884\pi$
$173$	7.50691	0.570739	0.285370	+	0.958418i	$0.407884\pi$
$174$	10.9286	0.828499
$175$	18.9801	1.43476
$176$	12.4475	0.938269
$177$	14.4987	1.08979
$178$	12.4172	0.930711
$179$	25.9287	1.93800	0.969002	−	0.247053i	$-0.0794623\pi$
$179$	25.9287	1.93800	0.969002	+	0.247053i	$0.0794623\pi$
$180$	1.46676	0.109326
$181$	21.4134	1.59164	0.795822	−	0.605530i	$-0.207039\pi$
$181$	21.4134	1.59164	0.795822	+	0.605530i	$0.207039\pi$
$182$	−37.6711	−2.79237
$183$	−8.36894	−0.618650
$184$	−0.201360	−0.0148444
$185$	−0.677956	−0.0498443
$186$	−40.4054	−2.96267
$187$	−41.4358	−3.03009
$188$	−10.9542	−0.798915
$189$	12.7477	0.927259
$190$	0.273304	0.0198276
$191$	−17.8083	−1.28856	−0.644281	−	0.764789i	$-0.722843\pi$
$191$	−17.8083	−1.28856	−0.644281	+	0.764789i	$0.722843\pi$
$192$	33.5933	2.42438
$193$	14.5940	1.05050	0.525251	−	0.850947i	$-0.323971\pi$
$193$	14.5940	1.05050	0.525251	+	0.850947i	$0.323971\pi$
$194$	−34.0026	−2.44124
$195$	1.52917	0.109506
$196$	20.6920	1.47800
$197$	0.179358	0.0127788	0.00638938	−	0.999980i	$-0.497966\pi$
$197$	0.179358	0.0127788	0.00638938	+	0.999980i	$0.497966\pi$
$198$	60.4256	4.29426
$199$	18.3856	1.30332	0.651660	−	0.758511i	$-0.274073\pi$
$199$	18.3856	1.30332	0.651660	+	0.758511i	$0.274073\pi$
$200$	8.24908	0.583298
$201$	−9.84924	−0.694711
$202$	26.0443	1.83247
$203$	−7.08817	−0.497492
$204$	−47.1345	−3.30007
$205$	1.19258	0.0832934
$206$	−3.99538	−0.278371
$207$	0.516330	0.0358874
$208$	8.64834	0.599654
$209$	6.52712	0.451491
$210$	−2.80107	−0.193292
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	20.8358	1.43101
$213$	37.5142	2.57043
$214$	−17.8802	−1.22227
$215$	1.48654	0.101381
$216$	5.54037	0.376974
$217$	26.2064	1.77901
$218$	−32.8000	−2.22150
$219$	4.36692	0.295089
$220$	−2.25593	−0.152095
$221$	−28.7889	−1.93655
$222$	−31.7706	−2.13230
$223$	−15.9455	−1.06779	−0.533895	−	0.845550i	$-0.679272\pi$
$223$	−15.9455	−1.06779	−0.533895	+	0.845550i	$0.679272\pi$
$224$	−28.4461	−1.90064
$225$	−21.1524	−1.41016
$226$	22.0779	1.46860
$227$	26.2405	1.74164	0.870821	−	0.491600i	$-0.163588\pi$
$227$	26.2405	1.74164	0.870821	+	0.491600i	$0.163588\pi$
$228$	7.42480	0.491719
$229$	−16.6981	−1.10344	−0.551722	−	0.834028i	$-0.686029\pi$
$229$	−16.6981	−1.10344	−0.551722	+	0.834028i	$0.686029\pi$
$230$	−0.0332520	−0.00219257
$231$	−66.8959	−4.40143
$232$	−3.08064	−0.202254
$233$	−30.0504	−1.96867	−0.984334	−	0.176316i	$-0.943582\pi$
$233$	−30.0504	−1.96867	−0.984334	+	0.176316i	$0.943582\pi$
$234$	41.9827	2.74449
$235$	−0.497486	−0.0324524
$236$	−14.8609	−0.967363
$237$	−2.80604	−0.182272
$238$	52.7341	3.41825
$239$	−19.5362	−1.26369	−0.631847	−	0.775093i	$-0.717703\pi$
$239$	−19.5362	−1.26369	−0.631847	+	0.775093i	$0.717703\pi$
$240$	0.643055	0.0415090
$241$	−28.1364	−1.81242	−0.906212	−	0.422824i	$-0.861039\pi$
$241$	−28.1364	−1.81242	−0.906212	+	0.422824i	$0.861039\pi$
$242$	−68.9408	−4.43168
$243$	20.0591	1.28679
$244$	8.57804	0.549153
$245$	0.939733	0.0600373
$246$	55.8871	3.56323
$247$	4.53493	0.288551
$248$	11.3897	0.723249
$249$	−6.12192	−0.387961
$250$	2.72875	0.172581
$251$	10.2621	0.647735	0.323868	−	0.946102i	$-0.395017\pi$
$251$	10.2621	0.647735	0.323868	+	0.946102i	$0.395017\pi$
$252$	−44.5812	−2.80835
$253$	−0.794133	−0.0499267
$254$	−48.9348	−3.07044
$255$	−2.14062	−0.134051
$256$	−1.84129	−0.115081
$257$	12.2246	0.762549	0.381274	−	0.924462i	$-0.375485\pi$
$257$	12.2246	0.762549	0.381274	+	0.924462i	$0.375485\pi$
$258$	69.6628	4.33701
$259$	20.6060	1.28039
$260$	−1.56738	−0.0972049
$261$	7.89943	0.488962
$262$	−8.95172	−0.553039
$263$	−21.8518	−1.34744	−0.673719	−	0.738987i	$-0.735304\pi$
$263$	−21.8518	−1.34744	−0.673719	+	0.738987i	$0.735304\pi$
$264$	−29.0741	−1.78939
$265$	0.946263	0.0581285
$266$	−8.30687	−0.509327
$267$	15.3202	0.937583
$268$	10.0953	0.616670
$269$	−5.90671	−0.360139	−0.180069	−	0.983654i	$-0.557632\pi$
$269$	−5.90671	−0.360139	−0.180069	+	0.983654i	$0.557632\pi$
$270$	0.914921	0.0556803
$271$	−13.6173	−0.827192	−0.413596	−	0.910460i	$-0.635727\pi$
$271$	−13.6173	−0.827192	−0.413596	+	0.910460i	$0.635727\pi$
$272$	−12.1064	−0.734060
$273$	−46.4781	−2.81298
$274$	10.5892	0.639714
$275$	32.5332	1.96182
$276$	−0.903350	−0.0543753
$277$	−3.53766	−0.212558	−0.106279	−	0.994336i	$-0.533894\pi$
$277$	−3.53766	−0.212558	−0.106279	+	0.994336i	$0.533894\pi$
$278$	5.05271	0.303041
$279$	−29.2058	−1.74850
$280$	0.789584	0.0471867
$281$	20.1246	1.20053	0.600267	−	0.799799i	$-0.295061\pi$
$281$	20.1246	1.20053	0.600267	+	0.799799i	$0.295061\pi$
$282$	−23.3134	−1.38829
$283$	−24.0511	−1.42969	−0.714845	−	0.699283i	$-0.753503\pi$
$283$	−24.0511	−1.42969	−0.714845	+	0.699283i	$0.753503\pi$
$284$	−38.4515	−2.28168
$285$	0.337199	0.0199739
$286$	−64.5707	−3.81815
$287$	−36.2476	−2.13963
$288$	31.7018	1.86805
$289$	23.3003	1.37061
$290$	−0.508728	−0.0298735
$291$	−41.9520	−2.45927
$292$	−4.47603	−0.261940
$293$	−15.9790	−0.933501	−0.466751	−	0.884389i	$-0.654575\pi$
$293$	−15.9790	−0.933501	−0.466751	+	0.884389i	$0.654575\pi$
$294$	44.0381	2.56835
$295$	−0.674912	−0.0392949
$296$	8.95570	0.520540
$297$	21.8504	1.26789
$298$	−17.5585	−1.01714
$299$	−0.551750	−0.0319085
$300$	37.0074	2.13663
$301$	−45.1823	−2.60427
$302$	15.7787	0.907964
$303$	32.1332	1.84600
$304$	1.90705	0.109377
$305$	0.389574	0.0223069
$306$	−58.7697	−3.35964
$307$	5.33684	0.304590	0.152295	−	0.988335i	$-0.451334\pi$
$307$	5.33684	0.304590	0.152295	+	0.988335i	$0.451334\pi$
$308$	68.5673	3.90699
$309$	−4.92945	−0.280427
$310$	1.88087	0.106826
$311$	−20.0682	−1.13796	−0.568981	−	0.822351i	$-0.692662\pi$
$311$	−20.0682	−1.13796	−0.568981	+	0.822351i	$0.692662\pi$
$312$	−20.2002	−1.14361
$313$	−33.0274	−1.86682	−0.933410	−	0.358812i	$-0.883182\pi$
$313$	−33.0274	−1.86682	−0.933410	+	0.358812i	$0.883182\pi$
$314$	−24.5911	−1.38775
$315$	−2.02467	−0.114077
$316$	2.87615	0.161796
$317$	−19.2620	−1.08186	−0.540930	−	0.841068i	$-0.681927\pi$
$317$	−19.2620	−1.08186	−0.540930	+	0.841068i	$0.681927\pi$
$318$	44.3441	2.48669
$319$	−12.1496	−0.680246
$320$	−1.56376	−0.0874171
$321$	−22.0604	−1.23129
$322$	1.01067	0.0563224
$323$	−6.34825	−0.353226
$324$	−10.2664	−0.570358
$325$	22.6035	1.25382
$326$	−8.35399	−0.462685
$327$	−40.4683	−2.23790
$328$	−15.7538	−0.869860
$329$	15.1207	0.833633
$330$	−4.80122	−0.264298
$331$	−35.5457	−1.95377	−0.976885	−	0.213767i	$-0.931427\pi$
$331$	−35.5457	−1.95377	−0.976885	+	0.213767i	$0.931427\pi$
$332$	6.27488	0.344379
$333$	−22.9644	−1.25844
$334$	−34.0404	−1.86261
$335$	0.458481	0.0250495
$336$	−19.5452	−1.06628
$337$	−13.7481	−0.748906	−0.374453	−	0.927246i	$-0.622169\pi$
$337$	−13.7481	−0.748906	−0.374453	+	0.927246i	$0.622169\pi$
$338$	−16.5039	−0.897694
$339$	27.2394	1.47944
$340$	2.19411	0.118992
$341$	44.9195	2.43252
$342$	9.25762	0.500595
$343$	−1.90664	−0.102949
$344$	−19.6370	−1.05876
$345$	−0.0410259	−0.00220876
$346$	−16.3759	−0.880372
$347$	24.8564	1.33436	0.667181	−	0.744896i	$-0.267501\pi$
$347$	24.8564	1.33436	0.667181	+	0.744896i	$0.267501\pi$
$348$	−13.8205	−0.740857
$349$	16.8693	0.902995	0.451497	−	0.892272i	$-0.350890\pi$
$349$	16.8693	0.902995	0.451497	+	0.892272i	$0.350890\pi$
$350$	−41.4040	−2.21314
$351$	15.1813	0.810317
$352$	−48.7585	−2.59884
$353$	−7.27129	−0.387012	−0.193506	−	0.981099i	$-0.561986\pi$
$353$	−7.27129	−0.387012	−0.193506	+	0.981099i	$0.561986\pi$
$354$	−31.6280	−1.68101
$355$	−1.74629	−0.0926832
$356$	−15.7030	−0.832258
$357$	65.0627	3.44348
$358$	−56.5619	−2.98939
$359$	2.02290	0.106765	0.0533823	−	0.998574i	$-0.483000\pi$
$359$	2.02290	0.106765	0.0533823	+	0.998574i	$0.483000\pi$
$360$	−0.879954	−0.0463776
$361$	1.00000	0.0526316
$362$	−46.7120	−2.45513
$363$	−85.0582	−4.46440
$364$	47.6394	2.49698
$365$	−0.203280	−0.0106402
$366$	18.2563	0.954274
$367$	−32.2397	−1.68290	−0.841450	−	0.540335i	$-0.818298\pi$
$367$	−32.2397	−1.68290	−0.841450	+	0.540335i	$0.818298\pi$
$368$	−0.232024	−0.0120951
$369$	40.3962	2.10294
$370$	1.47892	0.0768854
$371$	−28.7610	−1.49320
$372$	51.0972	2.64927
$373$	−1.16000	−0.0600625	−0.0300312	−	0.999549i	$-0.509561\pi$
$373$	−1.16000	−0.0600625	−0.0300312	+	0.999549i	$0.509561\pi$
$374$	90.3897	4.67394
$375$	3.36670	0.173856
$376$	6.57173	0.338911
$377$	−8.44132	−0.434750
$378$	−27.8084	−1.43031
$379$	30.9811	1.59139	0.795696	−	0.605696i	$-0.207105\pi$
$379$	30.9811	1.59139	0.795696	+	0.605696i	$0.207105\pi$
$380$	−0.345624	−0.0177301
$381$	−60.3751	−3.09311
$382$	38.8477	1.98762
$383$	23.2655	1.18881	0.594405	−	0.804166i	$-0.297388\pi$
$383$	23.2655	1.18881	0.594405	+	0.804166i	$0.297388\pi$
$384$	−33.0709	−1.68764
$385$	3.11400	0.158704
$386$	−31.8360	−1.62041
$387$	50.3535	2.55961
$388$	43.0002	2.18300
$389$	22.2580	1.12853	0.564263	−	0.825595i	$-0.309160\pi$
$389$	22.2580	1.12853	0.564263	+	0.825595i	$0.309160\pi$
$390$	−3.33580	−0.168915
$391$	0.772371	0.0390605
$392$	−12.4137	−0.626989
$393$	−11.0445	−0.557123
$394$	−0.391260	−0.0197114
$395$	0.130621	0.00657227
$396$	−76.4150	−3.84000
$397$	−11.9526	−0.599882	−0.299941	−	0.953958i	$-0.596967\pi$
$397$	−11.9526	−0.599882	−0.299941	+	0.953958i	$0.596967\pi$
$398$	−40.1070	−2.01038
$399$	−10.2489	−0.513088
$400$	9.50531	0.475265
$401$	−14.1510	−0.706667	−0.353333	−	0.935497i	$-0.614952\pi$
$401$	−14.1510	−0.706667	−0.353333	+	0.935497i	$0.614952\pi$
$402$	21.4855	1.07160
$403$	31.2093	1.55464
$404$	−32.9360	−1.63863
$405$	−0.466253	−0.0231683
$406$	15.4624	0.767387
$407$	35.3200	1.75075
$408$	28.2774	1.39994
$409$	35.2486	1.74293	0.871466	−	0.490457i	$-0.163170\pi$
$409$	35.2486	1.74293	0.871466	+	0.490457i	$0.163170\pi$
$410$	−2.60154	−0.128481
$411$	13.0648	0.644437
$412$	5.05261	0.248924
$413$	20.5135	1.00940
$414$	−1.12634	−0.0553567
$415$	0.284975	0.0139889
$416$	−33.8766	−1.66093
$417$	6.23397	0.305279
$418$	−14.2385	−0.696429
$419$	9.22415	0.450629	0.225315	−	0.974286i	$-0.427659\pi$
$419$	9.22415	0.450629	0.225315	+	0.974286i	$0.427659\pi$
$420$	3.54227	0.172845
$421$	−25.2217	−1.22923	−0.614615	−	0.788827i	$-0.710689\pi$
$421$	−25.2217	−1.22923	−0.614615	+	0.788827i	$0.710689\pi$
$422$	2.18144	0.106191
$423$	−16.8513	−0.819340
$424$	−12.5000	−0.607054
$425$	−31.6416	−1.53484
$426$	−81.8350	−3.96492
$427$	−11.8408	−0.573017
$428$	22.6116	1.09297
$429$	−79.6666	−3.84634
$430$	−3.24280	−0.156382
$431$	−26.2827	−1.26599	−0.632996	−	0.774155i	$-0.718175\pi$
$431$	−26.2827	−1.26599	−0.632996	+	0.774155i	$0.718175\pi$
$432$	6.38410	0.307155
$433$	16.9226	0.813246	0.406623	−	0.913596i	$-0.366706\pi$
$433$	16.9226	0.813246	0.406623	+	0.913596i	$0.366706\pi$
$434$	−57.1676	−2.74413
$435$	−0.627662	−0.0300941
$436$	41.4794	1.98650
$437$	−0.121667	−0.00582010
$438$	−9.52617	−0.455178
$439$	21.1353	1.00873	0.504365	−	0.863490i	$-0.331726\pi$
$439$	21.1353	1.00873	0.504365	+	0.863490i	$0.331726\pi$
$440$	1.35340	0.0645208
$441$	31.8315	1.51579
$442$	62.8012	2.98715
$443$	21.8860	1.03984	0.519918	−	0.854216i	$-0.325963\pi$
$443$	21.8860	1.03984	0.519918	+	0.854216i	$0.325963\pi$
$444$	40.1775	1.90674
$445$	−0.713156	−0.0338068
$446$	34.7842	1.64708
$447$	−21.6635	−1.02465
$448$	47.5295	2.24556
$449$	−2.45723	−0.115964	−0.0579820	−	0.998318i	$-0.518467\pi$
$449$	−2.45723	−0.115964	−0.0579820	+	0.998318i	$0.518467\pi$
$450$	46.1428	2.17519
$451$	−62.1308	−2.92562
$452$	−27.9200	−1.31325
$453$	19.4676	0.914668
$454$	−57.2420	−2.68650
$455$	2.16356	0.101429
$456$	−4.45435	−0.208594
$457$	33.2328	1.55457	0.777283	−	0.629151i	$-0.216597\pi$
$457$	33.2328	1.55457	0.777283	+	0.629151i	$0.216597\pi$
$458$	36.4260	1.70207
$459$	−21.2516	−0.991940
$460$	0.0420509	0.00196063
$461$	14.3684	0.669203	0.334602	−	0.942360i	$-0.391398\pi$
$461$	14.3684	0.669203	0.334602	+	0.942360i	$0.391398\pi$
$462$	145.929	6.78925
$463$	11.3867	0.529183	0.264591	−	0.964361i	$-0.414763\pi$
$463$	11.3867	0.529183	0.264591	+	0.964361i	$0.414763\pi$
$464$	−3.54978	−0.164794
$465$	2.32059	0.107615
$466$	65.5531	3.03669
$467$	−14.3216	−0.662724	−0.331362	−	0.943504i	$-0.607508\pi$
$467$	−14.3216	−0.662724	−0.331362	+	0.943504i	$0.607508\pi$
$468$	−53.0918	−2.45417
$469$	−13.9352	−0.643468
$470$	1.08524	0.0500582
$471$	−30.3401	−1.39800
$472$	8.91550	0.410369
$473$	−77.4454	−3.56094
$474$	6.12122	0.281157
$475$	4.98430	0.228696
$476$	−66.6883	−3.05665
$477$	32.0527	1.46759
$478$	42.6171	1.94926
$479$	−35.6621	−1.62944	−0.814720	−	0.579854i	$-0.803110\pi$
$479$	−35.6621	−1.62944	−0.814720	+	0.579854i	$0.803110\pi$
$480$	−2.51892	−0.114973
$481$	24.5397	1.11891
$482$	61.3778	2.79568
$483$	1.24695	0.0567382
$484$	87.1834	3.96288
$485$	1.95286	0.0886749
$486$	−43.7577	−1.98489
$487$	10.0498	0.455400	0.227700	−	0.973731i	$-0.426879\pi$
$487$	10.0498	0.455400	0.227700	+	0.973731i	$0.426879\pi$
$488$	−5.14622	−0.232958
$489$	−10.3070	−0.466101
$490$	−2.04997	−0.0926082
$491$	21.4043	0.965960	0.482980	−	0.875631i	$-0.339554\pi$
$491$	21.4043	0.965960	0.482980	+	0.875631i	$0.339554\pi$
$492$	−70.6756	−3.18630
$493$	11.8166	0.532194
$494$	−9.89268	−0.445093
$495$	−3.47041	−0.155983
$496$	13.1242	0.589296
$497$	53.0771	2.38083
$498$	13.3546	0.598434
$499$	−19.2698	−0.862633	−0.431316	−	0.902201i	$-0.641951\pi$
$499$	−19.2698	−0.862633	−0.431316	+	0.902201i	$0.641951\pi$
$500$	−3.45082	−0.154325
$501$	−41.9986	−1.87636
$502$	−22.3861	−0.999138
$503$	5.35886	0.238940	0.119470	−	0.992838i	$-0.461880\pi$
$503$	5.35886	0.238940	0.119470	+	0.992838i	$0.461880\pi$
$504$	26.7456	1.19134
$505$	−1.49580	−0.0665622
$506$	1.73235	0.0770125
$507$	−20.3623	−0.904322
$508$	61.8836	2.74564
$509$	−1.73991	−0.0771202	−0.0385601	−	0.999256i	$-0.512277\pi$
$509$	−1.73991	−0.0771202	−0.0385601	+	0.999256i	$0.512277\pi$
$510$	4.66964	0.206775
$511$	6.17854	0.273323
$512$	−20.5583	−0.908557
$513$	3.34763	0.147802
$514$	−26.6672	−1.17624
$515$	0.229466	0.0101115
$516$	−88.0965	−3.87823
$517$	25.9179	1.13987
$518$	−44.9507	−1.97502
$519$	−20.2043	−0.886872
$520$	0.940318	0.0412357
$521$	−1.25469	−0.0549691	−0.0274846	−	0.999622i	$-0.508750\pi$
$521$	−1.25469	−0.0549691	−0.0274846	+	0.999622i	$0.508750\pi$
$522$	−17.2321	−0.754230
$523$	−8.76691	−0.383350	−0.191675	−	0.981458i	$-0.561392\pi$
$523$	−8.76691	−0.383350	−0.191675	+	0.981458i	$0.561392\pi$
$524$	11.3205	0.494537
$525$	−51.0837	−2.22948
$526$	47.6683	2.07844
$527$	−43.6885	−1.90310
$528$	−33.5017	−1.45798
$529$	−22.9852	−0.999356
$530$	−2.06422	−0.0896638
$531$	−22.8613	−0.992095
$532$	10.5050	0.455449
$533$	−43.1674	−1.86979
$534$	−33.4202	−1.44623
$535$	1.02691	0.0443973
$536$	−6.05648	−0.261600
$537$	−69.7854	−3.01146
$538$	12.8851	0.555518
$539$	−48.9580	−2.10877
$540$	−1.15702	−0.0497903
$541$	−18.6751	−0.802903	−0.401452	−	0.915880i	$-0.631494\pi$
$541$	−18.6751	−0.802903	−0.401452	+	0.915880i	$0.631494\pi$
$542$	29.7053	1.27595
$543$	−57.6327	−2.47326
$544$	47.4223	2.03321
$545$	1.88380	0.0806929
$546$	101.389	4.33906
$547$	−44.8914	−1.91942	−0.959709	−	0.280996i	$-0.909335\pi$
$547$	−44.8914	−1.91942	−0.959709	+	0.280996i	$0.909335\pi$
$548$	−13.3912	−0.572043
$549$	13.1960	0.563193
$550$	−70.9691	−3.02613
$551$	−1.86140	−0.0792983
$552$	0.541946	0.0230668
$553$	−3.97014	−0.168827
$554$	7.71720	0.327872
$555$	1.82467	0.0774530
$556$	−6.38972	−0.270985
$557$	30.2392	1.28127	0.640637	−	0.767844i	$-0.278670\pi$
$557$	30.2392	1.28127	0.640637	+	0.767844i	$0.278670\pi$
$558$	63.7106	2.69709
$559$	−53.8077	−2.27582
$560$	0.909827	0.0384472
$561$	111.522	4.70845
$562$	−43.9007	−1.85184
$563$	−22.4489	−0.946110	−0.473055	−	0.881033i	$-0.656849\pi$
$563$	−22.4489	−0.946110	−0.473055	+	0.881033i	$0.656849\pi$
$564$	29.4824	1.24143
$565$	−1.26799	−0.0533449
$566$	52.4660	2.20531
$567$	14.1714	0.595143
$568$	23.0682	0.967920
$569$	−24.0847	−1.00968	−0.504842	−	0.863212i	$-0.668449\pi$
$569$	−24.0847	−1.00968	−0.504842	+	0.863212i	$0.668449\pi$
$570$	−0.735579	−0.0308100
$571$	43.7014	1.82885	0.914423	−	0.404759i	$-0.132645\pi$
$571$	43.7014	1.82885	0.914423	+	0.404759i	$0.132645\pi$
$572$	81.6570	3.41425
$573$	47.9298	2.00230
$574$	79.0720	3.30040
$575$	−0.606423	−0.0252896
$576$	−52.9693	−2.20706
$577$	−5.35255	−0.222829	−0.111415	−	0.993774i	$-0.535538\pi$
$577$	−5.35255	−0.222829	−0.111415	+	0.993774i	$0.535538\pi$
$578$	−50.8282	−2.11418
$579$	−39.2789	−1.63237
$580$	0.643344	0.0267134
$581$	−8.66161	−0.359344
$582$	91.5157	3.79345
$583$	−49.2982	−2.04172
$584$	2.68530	0.111119
$585$	−2.41118	−0.0996900
$586$	34.8571	1.43994
$587$	2.05589	0.0848559	0.0424279	−	0.999100i	$-0.486491\pi$
$587$	2.05589	0.0848559	0.0424279	+	0.999100i	$0.486491\pi$
$588$	−55.6912	−2.29666
$589$	6.88197	0.283567
$590$	1.47228	0.0606128
$591$	−0.482731	−0.0198569
$592$	10.3195	0.424131
$593$	−11.1657	−0.458522	−0.229261	−	0.973365i	$-0.573631\pi$
$593$	−11.1657	−0.458522	−0.229261	+	0.973365i	$0.573631\pi$
$594$	−47.6653	−1.95573
$595$	−3.02867	−0.124163
$596$	22.2048	0.909543
$597$	−49.4836	−2.02523
$598$	1.20361	0.0492192
$599$	−2.41405	−0.0986355	−0.0493177	−	0.998783i	$-0.515705\pi$
$599$	−2.41405	−0.0986355	−0.0493177	+	0.998783i	$0.515705\pi$
$600$	−22.2018	−0.906387
$601$	37.3558	1.52378	0.761888	−	0.647709i	$-0.224273\pi$
$601$	37.3558	1.52378	0.761888	+	0.647709i	$0.224273\pi$
$602$	98.5625	4.01711
$603$	15.5301	0.632436
$604$	−19.9540	−0.811917
$605$	3.95946	0.160975
$606$	−70.0966	−2.84748
$607$	0.298918	0.0121327	0.00606636	−	0.999982i	$-0.498069\pi$
$607$	0.298918	0.0121327	0.00606636	+	0.999982i	$0.498069\pi$
$608$	−7.47014	−0.302954
$609$	19.0773	0.773052
$610$	−0.849832	−0.0344087
$611$	18.0073	0.728498
$612$	74.3210	3.00425
$613$	37.0069	1.49469	0.747347	−	0.664434i	$-0.231327\pi$
$613$	37.0069	1.49469	0.747347	+	0.664434i	$0.231327\pi$
$614$	−11.6420	−0.469833
$615$	−3.20975	−0.129430
$616$	−41.1356	−1.65740
$617$	−23.2863	−0.937470	−0.468735	−	0.883339i	$-0.655290\pi$
$617$	−23.2863	−0.937470	−0.468735	+	0.883339i	$0.655290\pi$
$618$	10.7533	0.432561
$619$	−7.33738	−0.294914	−0.147457	−	0.989068i	$-0.547109\pi$
$619$	−7.33738	−0.294914	−0.147457	+	0.989068i	$0.547109\pi$
$620$	−2.37857	−0.0955258
$621$	−0.407295	−0.0163442
$622$	43.7775	1.75532
$623$	21.6758	0.868425
$624$	−23.2764	−0.931802
$625$	24.7648	0.990592
$626$	72.0473	2.87959
$627$	−17.5673	−0.701571
$628$	31.0982	1.24095
$629$	−34.3521	−1.36971
$630$	4.41668	0.175965
$631$	−8.51562	−0.339002	−0.169501	−	0.985530i	$-0.554216\pi$
$631$	−8.51562	−0.339002	−0.169501	+	0.985530i	$0.554216\pi$
$632$	−1.72549	−0.0686363
$633$	2.69143	0.106975
$634$	42.0188	1.66878
$635$	2.81046	0.111530
$636$	−56.0782	−2.22364
$637$	−34.0151	−1.34773
$638$	26.5036	1.04929
$639$	−59.1519	−2.34001
$640$	1.53945	0.0608521
$641$	5.87878	0.232198	0.116099	−	0.993238i	$-0.462961\pi$
$641$	5.87878	0.232198	0.116099	+	0.993238i	$0.462961\pi$
$642$	48.1235	1.89928
$643$	12.1165	0.477826	0.238913	−	0.971041i	$-0.423209\pi$
$643$	12.1165	0.477826	0.238913	+	0.971041i	$0.423209\pi$
$644$	−1.27811	−0.0503645
$645$	−4.00092	−0.157536
$646$	13.8483	0.544855
$647$	−25.3487	−0.996559	−0.498280	−	0.867016i	$-0.666035\pi$
$647$	−25.3487	−0.996559	−0.498280	+	0.867016i	$0.666035\pi$
$648$	6.15914	0.241954
$649$	35.1614	1.38021
$650$	−49.3081	−1.93402
$651$	−70.5327	−2.76440
$652$	10.5646	0.413741
$653$	−1.09214	−0.0427386	−0.0213693	−	0.999772i	$-0.506803\pi$
$653$	−1.09214	−0.0427386	−0.0213693	+	0.999772i	$0.506803\pi$
$654$	88.2791	3.45198
$655$	0.514122	0.0200884
$656$	−18.1529	−0.708753
$657$	−6.88569	−0.268637
$658$	−32.9850	−1.28589
$659$	41.9861	1.63555	0.817773	−	0.575541i	$-0.195208\pi$
$659$	41.9861	1.63555	0.817773	+	0.575541i	$0.195208\pi$
$660$	6.07169	0.236340
$661$	28.0510	1.09106	0.545529	−	0.838092i	$-0.316329\pi$
$661$	28.0510	1.09106	0.545529	+	0.838092i	$0.316329\pi$
$662$	77.5408	3.01371
$663$	77.4834	3.00920
$664$	−3.76448	−0.146090
$665$	0.477087	0.0185006
$666$	50.0954	1.94116
$667$	0.226470	0.00876896
$668$	43.0480	1.66558
$669$	42.9163	1.65924
$670$	−1.00015	−0.0386392
$671$	−20.2959	−0.783516
$672$	76.5608	2.95340
$673$	−9.05115	−0.348896	−0.174448	−	0.984666i	$-0.555814\pi$
$673$	−9.05115	−0.348896	−0.174448	+	0.984666i	$0.555814\pi$
$674$	29.9906	1.15520
$675$	16.6856	0.642229
$676$	20.8711	0.802734
$677$	−27.6985	−1.06454	−0.532270	−	0.846575i	$-0.678661\pi$
$677$	−27.6985	−1.06454	−0.532270	+	0.846575i	$0.678661\pi$
$678$	−59.4211	−2.28205
$679$	−59.3558	−2.27787
$680$	−1.31631	−0.0504782
$681$	−70.6245	−2.70634
$682$	−97.9891	−3.75220
$683$	12.7774	0.488913	0.244456	−	0.969660i	$-0.421391\pi$
$683$	12.7774	0.488913	0.244456	+	0.969660i	$0.421391\pi$
$684$	−11.7073	−0.447640
$685$	−0.608164	−0.0232368
$686$	4.15922	0.158800
$687$	44.9419	1.71464
$688$	−22.6275	−0.862664
$689$	−34.2515	−1.30488
$690$	0.0894955	0.00340703
$691$	34.7752	1.32291	0.661456	−	0.749984i	$-0.269939\pi$
$691$	34.7752	1.32291	0.661456	+	0.749984i	$0.269939\pi$
$692$	20.7091	0.787243
$693$	105.480	4.00687
$694$	−54.2228	−2.05827
$695$	−0.290191	−0.0110076
$696$	8.29133	0.314282
$697$	60.4281	2.28888
$698$	−36.7994	−1.39288
$699$	80.8786	3.05911
$700$	52.3600	1.97902
$701$	−31.1300	−1.17576	−0.587882	−	0.808947i	$-0.700038\pi$
$701$	−31.1300	−1.17576	−0.587882	+	0.808947i	$0.700038\pi$
$702$	−33.1170	−1.24992
$703$	5.41126	0.204090
$704$	81.4686	3.07046
$705$	1.33895	0.0504278
$706$	15.8619	0.596970
$707$	45.4637	1.70984
$708$	39.9972	1.50319
$709$	−21.6171	−0.811848	−0.405924	−	0.913907i	$-0.633050\pi$
$709$	−21.6171	−0.811848	−0.405924	+	0.913907i	$0.633050\pi$
$710$	3.80942	0.142965
$711$	4.42453	0.165933
$712$	9.42069	0.353055
$713$	−0.837306	−0.0313574
$714$	−141.930	−5.31161
$715$	3.70847	0.138689
$716$	71.5290	2.67317
$717$	52.5805	1.96365
$718$	−4.41283	−0.164685
$719$	−27.1757	−1.01348	−0.506741	−	0.862098i	$-0.669150\pi$
$719$	−27.1757	−1.01348	−0.506741	+	0.862098i	$0.669150\pi$
$720$	−1.01396	−0.0377880
$721$	−6.97444	−0.259742
$722$	−2.18144	−0.0811848
$723$	75.7272	2.81632
$724$	59.0727	2.19542
$725$	−9.27778	−0.344568
$726$	185.549	6.88639
$727$	−10.4266	−0.386703	−0.193351	−	0.981130i	$-0.561936\pi$
$727$	−10.4266	−0.386703	−0.193351	+	0.981130i	$0.561936\pi$
$728$	−28.5803	−1.05926
$729$	−42.8231	−1.58604
$730$	0.443443	0.0164126
$731$	75.3231	2.78593
$732$	−23.0872	−0.853329
$733$	−29.0939	−1.07461	−0.537304	−	0.843389i	$-0.680557\pi$
$733$	−29.0939	−1.07461	−0.537304	+	0.843389i	$0.680557\pi$
$734$	70.3290	2.59589
$735$	−2.52923	−0.0932920
$736$	0.908866	0.0335013
$737$	−23.8859	−0.879847
$738$	−88.1219	−3.24381
$739$	−23.6128	−0.868612	−0.434306	−	0.900765i	$-0.643006\pi$
$739$	−23.6128	−0.868612	−0.434306	+	0.900765i	$0.643006\pi$
$740$	−1.87026	−0.0687522
$741$	−12.2055	−0.448379
$742$	62.7403	2.30327
$743$	−3.28055	−0.120352	−0.0601759	−	0.998188i	$-0.519166\pi$
$743$	−3.28055	−0.120352	−0.0601759	+	0.998188i	$0.519166\pi$
$744$	−30.6547	−1.12386
$745$	1.00844	0.0369462
$746$	2.53047	0.0926470
$747$	9.65295	0.353183
$748$	−114.308	−4.17952
$749$	−31.2122	−1.14047
$750$	−7.34425	−0.268174
$751$	−4.54757	−0.165943	−0.0829717	−	0.996552i	$-0.526441\pi$
$751$	−4.54757	−0.165943	−0.0829717	+	0.996552i	$0.526441\pi$
$752$	7.57251	0.276141
$753$	−27.6196	−1.00652
$754$	18.4142	0.670607
$755$	−0.906216	−0.0329806
$756$	35.1668	1.27901
$757$	−6.63586	−0.241184	−0.120592	−	0.992702i	$-0.538479\pi$
$757$	−6.63586	−0.241184	−0.120592	+	0.992702i	$0.538479\pi$
$758$	−67.5834	−2.45474
$759$	2.13736	0.0775811
$760$	0.207350	0.00752137
$761$	−25.8917	−0.938574	−0.469287	−	0.883046i	$-0.655489\pi$
$761$	−25.8917	−0.938574	−0.469287	+	0.883046i	$0.655489\pi$
$762$	131.705	4.77115
$763$	−57.2566	−2.07283
$764$	−49.1273	−1.77736
$765$	3.37530	0.122034
$766$	−50.7522	−1.83375
$767$	24.4295	0.882100
$768$	4.95572	0.178824
$769$	−1.88647	−0.0680277	−0.0340139	−	0.999421i	$-0.510829\pi$
$769$	−1.88647	−0.0680277	−0.0340139	+	0.999421i	$0.510829\pi$
$770$	−6.79301	−0.244803
$771$	−32.9017	−1.18492
$772$	40.2603	1.44900
$773$	−4.07570	−0.146593	−0.0732963	−	0.997310i	$-0.523352\pi$
$773$	−4.07570	−0.146593	−0.0732963	+	0.997310i	$0.523352\pi$
$774$	−109.843	−3.94823
$775$	34.3018	1.23216
$776$	−25.7971	−0.926060
$777$	−55.4596	−1.98960
$778$	−48.5545	−1.74076
$779$	−9.51886	−0.341049
$780$	4.21850	0.151047
$781$	90.9776	3.25543
$782$	−1.68488	−0.0602512
$783$	−6.23128	−0.222688
$784$	−14.3042	−0.510864
$785$	1.41233	0.0504083
$786$	24.0930	0.859367
$787$	13.9674	0.497883	0.248941	−	0.968519i	$-0.419917\pi$
$787$	13.9674	0.497883	0.248941	+	0.968519i	$0.419917\pi$
$788$	0.494792	0.0176262
$789$	58.8126	2.09378
$790$	−0.284942	−0.0101378
$791$	38.5397	1.37031
$792$	45.8436	1.62898
$793$	−14.1013	−0.500750
$794$	26.0738	0.925325
$795$	−2.54680	−0.0903258
$796$	50.7199	1.79772
$797$	25.7912	0.913571	0.456786	−	0.889577i	$-0.349001\pi$
$797$	25.7912	0.913571	0.456786	+	0.889577i	$0.349001\pi$
$798$	22.3574	0.791443
$799$	−25.2077	−0.891783
$800$	−37.2334	−1.31640
$801$	−24.1567	−0.853535
$802$	30.8695	1.09004
$803$	10.5904	0.373728
$804$	−27.1709	−0.958243
$805$	−0.0580455	−0.00204584
$806$	−68.0811	−2.39805
$807$	15.8975	0.559619
$808$	19.7593	0.695130
$809$	−15.9953	−0.562364	−0.281182	−	0.959654i	$-0.590726\pi$
$809$	−15.9953	−0.562364	−0.281182	+	0.959654i	$0.590726\pi$
$810$	1.01710	0.0357373
$811$	−14.3363	−0.503415	−0.251708	−	0.967803i	$-0.580992\pi$
$811$	−14.3363	−0.503415	−0.251708	+	0.967803i	$0.580992\pi$
$812$	−19.5540	−0.686210
$813$	36.6501	1.28537
$814$	−77.0484	−2.70054
$815$	0.479793	0.0168064
$816$	32.5836	1.14066
$817$	−11.8652	−0.415110
$818$	−76.8927	−2.68849
$819$	73.2860	2.56082
$820$	3.28995	0.114890
$821$	12.0585	0.420845	0.210422	−	0.977611i	$-0.432516\pi$
$821$	12.0585	0.420845	0.210422	+	0.977611i	$0.432516\pi$
$822$	−28.5000	−0.994051
$823$	−27.3288	−0.952624	−0.476312	−	0.879276i	$-0.658027\pi$
$823$	−27.3288	−0.952624	−0.476312	+	0.879276i	$0.658027\pi$
$824$	−3.03121	−0.105597
$825$	−87.5608	−3.04848
$826$	−44.7489	−1.55701
$827$	−28.4213	−0.988306	−0.494153	−	0.869375i	$-0.664522\pi$
$827$	−28.4213	−0.988306	−0.494153	+	0.869375i	$0.664522\pi$
$828$	1.42439	0.0495009
$829$	17.3302	0.601902	0.300951	−	0.953640i	$-0.402696\pi$
$829$	17.3302	0.601902	0.300951	+	0.953640i	$0.402696\pi$
$830$	−0.621656	−0.0215780
$831$	9.52138	0.330293
$832$	56.6030	1.96236
$833$	47.6163	1.64981
$834$	−13.5990	−0.470896
$835$	1.95503	0.0676567
$836$	18.0062	0.622759
$837$	23.0383	0.796320
$838$	−20.1219	−0.695100
$839$	−6.46195	−0.223091	−0.111546	−	0.993759i	$-0.535580\pi$
$839$	−6.46195	−0.223091	−0.111546	+	0.993759i	$0.535580\pi$
$840$	−2.12511	−0.0733233
$841$	−25.5352	−0.880524
$842$	55.0196	1.89610
$843$	−54.1641	−1.86551
$844$	−2.75868	−0.0949576
$845$	0.947865	0.0326075
$846$	36.7602	1.26384
$847$	−120.345	−4.13510
$848$	−14.4036	−0.494621
$849$	64.7319	2.22159
$850$	69.0243	2.36751
$851$	−0.658370	−0.0225686
$852$	103.490	3.54550
$853$	−23.5622	−0.806754	−0.403377	−	0.915034i	$-0.632164\pi$
$853$	−23.5622	−0.806754	−0.403377	+	0.915034i	$0.632164\pi$
$854$	25.8300	0.883885
$855$	−0.531690	−0.0181834
$856$	−13.5654	−0.463655
$857$	7.85120	0.268192	0.134096	−	0.990968i	$-0.457187\pi$
$857$	7.85120	0.268192	0.134096	+	0.990968i	$0.457187\pi$
$858$	173.788	5.93302
$859$	−0.315746	−0.0107731	−0.00538655	−	0.999985i	$-0.501715\pi$
$859$	−0.315746	−0.0107731	−0.00538655	+	0.999985i	$0.501715\pi$
$860$	4.10089	0.139839
$861$	97.5580	3.32477
$862$	57.3340	1.95281
$863$	12.0812	0.411249	0.205625	−	0.978631i	$-0.434077\pi$
$863$	12.0812	0.411249	0.205625	+	0.978631i	$0.434077\pi$
$864$	−25.0073	−0.850764
$865$	0.940511	0.0319783
$866$	−36.9155	−1.25444
$867$	−62.7112	−2.12978
$868$	72.2950	2.45385
$869$	−6.80507	−0.230846
$870$	1.36921	0.0464205
$871$	−16.5955	−0.562316
$872$	−24.8847	−0.842702
$873$	66.1492	2.23881
$874$	0.265408	0.00897757
$875$	4.76338	0.161032
$876$	12.0469	0.407028
$877$	14.6605	0.495051	0.247526	−	0.968881i	$-0.420383\pi$
$877$	14.6605	0.495051	0.247526	+	0.968881i	$0.420383\pi$
$878$	−46.1053	−1.55598
$879$	43.0063	1.45057
$880$	1.55950	0.0525709
$881$	−26.0133	−0.876409	−0.438205	−	0.898875i	$-0.644385\pi$
$881$	−26.0133	−0.876409	−0.438205	+	0.898875i	$0.644385\pi$
$882$	−69.4386	−2.33812
$883$	35.5425	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$883$	35.5425	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$884$	−79.4193	−2.67116
$885$	1.81648	0.0610603
$886$	−47.7430	−1.60396
$887$	−14.8644	−0.499098	−0.249549	−	0.968362i	$-0.580282\pi$
$887$	−14.8644	−0.499098	−0.249549	+	0.968362i	$0.580282\pi$
$888$	−24.1037	−0.808866
$889$	−85.4218	−2.86496
$890$	1.55571	0.0521474
$891$	24.2907	0.813770
$892$	−43.9886	−1.47285
$893$	3.97080	0.132878
$894$	47.2576	1.58053
$895$	3.24851	0.108586
$896$	−46.7905	−1.56316
$897$	1.48500	0.0495826
$898$	5.36030	0.178876
$899$	−12.8101	−0.427241
$900$	−58.3528	−1.94509
$901$	47.9472	1.59735
$902$	135.535	4.51281
$903$	121.605	4.04677
$904$	16.7500	0.557097
$905$	2.68280	0.0891793
$906$	−42.4674	−1.41089
$907$	−28.5080	−0.946594	−0.473297	−	0.880903i	$-0.656936\pi$
$907$	−28.5080	−0.946594	−0.473297	+	0.880903i	$0.656936\pi$
$908$	72.3891	2.40232
$909$	−50.6671	−1.68052
$910$	−4.71966	−0.156455
$911$	13.6167	0.451140	0.225570	−	0.974227i	$-0.427576\pi$
$911$	13.6167	0.451140	0.225570	+	0.974227i	$0.427576\pi$
$912$	−5.13269	−0.169960
$913$	−14.8466	−0.491350
$914$	−72.4954	−2.39793
$915$	−1.04851	−0.0346627
$916$	−46.0648	−1.52202
$917$	−15.6264	−0.516028
$918$	46.3591	1.53008
$919$	7.28278	0.240237	0.120118	−	0.992760i	$-0.461673\pi$
$919$	7.28278	0.240237	0.120118	+	0.992760i	$0.461673\pi$
$920$	−0.0252276	−0.000831728 0
$921$	−14.3638	−0.473302
$922$	−31.3438	−1.03225
$923$	63.2096	2.08057
$924$	−184.544	−6.07107
$925$	26.9714	0.886813
$926$	−24.8393	−0.816270
$927$	7.77268	0.255288
$928$	13.9049	0.456451
$929$	32.9106	1.07976	0.539880	−	0.841742i	$-0.318470\pi$
$929$	32.9106	1.07976	0.539880	+	0.841742i	$0.318470\pi$
$930$	−5.06223	−0.165997
$931$	−7.50070	−0.245825
$932$	−82.8994	−2.71546
$933$	54.0121	1.76828
$934$	31.2417	1.02226
$935$	−5.19133	−0.169775
$936$	31.8514	1.04109
$937$	−10.6632	−0.348350	−0.174175	−	0.984715i	$-0.555726\pi$
$937$	−10.6632	−0.348350	−0.174175	+	0.984715i	$0.555726\pi$
$938$	30.3988	0.992556
$939$	88.8911	2.90085
$940$	−1.37240	−0.0447629
$941$	−46.6338	−1.52022	−0.760109	−	0.649796i	$-0.774854\pi$
$941$	−46.6338	−1.52022	−0.760109	+	0.649796i	$0.774854\pi$
$942$	66.1852	2.15643
$943$	1.15813	0.0377138
$944$	10.2732	0.334365
$945$	1.59711	0.0519540
$946$	168.943	5.49280
$947$	33.8155	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$947$	33.8155	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$948$	−7.74098	−0.251415
$949$	7.35805	0.238852
$950$	−10.8730	−0.352765
$951$	51.8423	1.68110
$952$	40.0083	1.29668
$953$	−14.1205	−0.457409	−0.228704	−	0.973496i	$-0.573449\pi$
$953$	−14.1205	−0.457409	−0.228704	+	0.973496i	$0.573449\pi$
$954$	−69.9211	−2.26378
$955$	−2.23113	−0.0721976
$956$	−53.8942	−1.74306
$957$	32.6998	1.05703
$958$	77.7946	2.51343
$959$	18.4847	0.596902
$960$	4.20877	0.135837
$961$	16.3615	0.527790
$962$	−53.5319	−1.72594
$963$	34.7845	1.12092
$964$	−77.6192	−2.49995
$965$	1.82843	0.0588592
$966$	−2.72015	−0.0875194
$967$	54.8014	1.76229	0.881147	−	0.472842i	$-0.156772\pi$
$967$	54.8014	1.76229	0.881147	+	0.472842i	$0.156772\pi$
$968$	−52.3039	−1.68111
$969$	17.0859	0.548878
$970$	−4.26005	−0.136782
$971$	−33.5254	−1.07588	−0.537941	−	0.842982i	$-0.680798\pi$
$971$	−33.5254	−1.07588	−0.537941	+	0.842982i	$0.680798\pi$
$972$	55.3365	1.77492
$973$	8.82014	0.282761
$974$	−21.9230	−0.702459
$975$	−60.8357	−1.94830
$976$	−5.92992	−0.189812
$977$	11.9739	0.383078	0.191539	−	0.981485i	$-0.438652\pi$
$977$	11.9739	0.383078	0.191539	+	0.981485i	$0.438652\pi$
$978$	22.4842	0.718965
$979$	37.1538	1.18744
$980$	2.59242	0.0828118
$981$	63.8097	2.03729
$982$	−46.6921	−1.49000
$983$	7.23705	0.230826	0.115413	−	0.993318i	$-0.463181\pi$
$983$	7.23705	0.230826	0.115413	+	0.993318i	$0.463181\pi$
$984$	42.4004	1.35167
$985$	0.0224711	0.000715989 0
$986$	−25.7773	−0.820916
$987$	−40.6964	−1.29538
$988$	12.5104	0.398009
$989$	1.44360	0.0459037
$990$	7.57049	0.240606
$991$	25.7837	0.819047	0.409524	−	0.912299i	$-0.365695\pi$
$991$	25.7837	0.819047	0.409524	+	0.912299i	$0.365695\pi$
$992$	−51.4092	−1.63224
$993$	95.6689	3.03596
$994$	−115.784	−3.67246
$995$	2.30346	0.0730245
$996$	−16.8884	−0.535130
$997$	36.4739	1.15514	0.577570	−	0.816341i	$-0.304001\pi$
$997$	36.4739	1.15514	0.577570	+	0.816341i	$0.304001\pi$
$998$	42.0358	1.33062
$999$	18.1149	0.573130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.12	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.12	✓	75	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18144 q^{2} -2.69143 q^{3} +2.75868 q^{4} +0.125286 q^{5} +5.87120 q^{6} -3.80798 q^{7} -1.65501 q^{8} +4.24381 q^{9} +O(q^{10})\)	\(q-2.18144 q^{2} -2.69143 q^{3} +2.75868 q^{4} +0.125286 q^{5} +5.87120 q^{6} -3.80798 q^{7} -1.65501 q^{8} +4.24381 q^{9} -0.273304 q^{10} -6.52712 q^{11} -7.42480 q^{12} -4.53493 q^{13} +8.30687 q^{14} -0.337199 q^{15} -1.90705 q^{16} +6.34825 q^{17} -9.25762 q^{18} -1.00000 q^{19} +0.345624 q^{20} +10.2489 q^{21} +14.2385 q^{22} +0.121667 q^{23} +4.45435 q^{24} -4.98430 q^{25} +9.89268 q^{26} -3.34763 q^{27} -10.5050 q^{28} +1.86140 q^{29} +0.735579 q^{30} -6.88197 q^{31} +7.47014 q^{32} +17.5673 q^{33} -13.8483 q^{34} -0.477087 q^{35} +11.7073 q^{36} -5.41126 q^{37} +2.18144 q^{38} +12.2055 q^{39} -0.207350 q^{40} +9.51886 q^{41} -22.3574 q^{42} +11.8652 q^{43} -18.0062 q^{44} +0.531690 q^{45} -0.265408 q^{46} -3.97080 q^{47} +5.13269 q^{48} +7.50070 q^{49} +10.8730 q^{50} -17.0859 q^{51} -12.5104 q^{52} +7.55282 q^{53} +7.30266 q^{54} -0.817758 q^{55} +6.30225 q^{56} +2.69143 q^{57} -4.06053 q^{58} -5.38697 q^{59} -0.930224 q^{60} +3.10948 q^{61} +15.0126 q^{62} -16.1603 q^{63} -12.4816 q^{64} -0.568164 q^{65} -38.3220 q^{66} +3.65948 q^{67} +17.5128 q^{68} -0.327458 q^{69} +1.04074 q^{70} -13.9384 q^{71} -7.02356 q^{72} -1.62253 q^{73} +11.8043 q^{74} +13.4149 q^{75} -2.75868 q^{76} +24.8551 q^{77} -26.6255 q^{78} +1.04258 q^{79} -0.238927 q^{80} -3.72151 q^{81} -20.7648 q^{82} +2.27460 q^{83} +28.2735 q^{84} +0.795348 q^{85} -25.8832 q^{86} -5.00983 q^{87} +10.8025 q^{88} -5.69222 q^{89} -1.15985 q^{90} +17.2689 q^{91} +0.335639 q^{92} +18.5224 q^{93} +8.66207 q^{94} -0.125286 q^{95} -20.1054 q^{96} +15.5872 q^{97} -16.3623 q^{98} -27.6999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more