LMFDB - Embedded newform 4009.2.a.d.1.17

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			1.17
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)

$f(q)$	$=$	$q-1.81267 q^{2} -3.27382 q^{3} +1.28578 q^{4} +1.53613 q^{5} +5.93435 q^{6} -3.38684 q^{7} +1.29465 q^{8} +7.71788 q^{9} +O(q^{10})$	\(q-1.81267 q^{2} -3.27382 q^{3} +1.28578 q^{4} +1.53613 q^{5} +5.93435 q^{6} -3.38684 q^{7} +1.29465 q^{8} +7.71788 q^{9} -2.78450 q^{10} +2.37631 q^{11} -4.20939 q^{12} +6.14452 q^{13} +6.13923 q^{14} -5.02901 q^{15} -4.91833 q^{16} +5.44507 q^{17} -13.9900 q^{18} -1.00000 q^{19} +1.97512 q^{20} +11.0879 q^{21} -4.30746 q^{22} -3.37872 q^{23} -4.23846 q^{24} -2.64031 q^{25} -11.1380 q^{26} -15.4455 q^{27} -4.35472 q^{28} -6.27899 q^{29} +9.11594 q^{30} -3.49385 q^{31} +6.32601 q^{32} -7.77959 q^{33} -9.87012 q^{34} -5.20263 q^{35} +9.92346 q^{36} +10.3280 q^{37} +1.81267 q^{38} -20.1160 q^{39} +1.98876 q^{40} -1.53433 q^{41} -20.0987 q^{42} -8.07740 q^{43} +3.05540 q^{44} +11.8557 q^{45} +6.12452 q^{46} +1.50142 q^{47} +16.1017 q^{48} +4.47070 q^{49} +4.78600 q^{50} -17.8262 q^{51} +7.90047 q^{52} +4.34422 q^{53} +27.9975 q^{54} +3.65031 q^{55} -4.38479 q^{56} +3.27382 q^{57} +11.3817 q^{58} +7.19628 q^{59} -6.46618 q^{60} -12.8639 q^{61} +6.33320 q^{62} -26.1392 q^{63} -1.63031 q^{64} +9.43878 q^{65} +14.1018 q^{66} -5.61633 q^{67} +7.00114 q^{68} +11.0613 q^{69} +9.43065 q^{70} +3.25684 q^{71} +9.99198 q^{72} -7.75551 q^{73} -18.7213 q^{74} +8.64388 q^{75} -1.28578 q^{76} -8.04817 q^{77} +36.4637 q^{78} -13.9660 q^{79} -7.55520 q^{80} +27.4120 q^{81} +2.78123 q^{82} +8.36055 q^{83} +14.2566 q^{84} +8.36433 q^{85} +14.6417 q^{86} +20.5563 q^{87} +3.07649 q^{88} +0.928807 q^{89} -21.4904 q^{90} -20.8105 q^{91} -4.34428 q^{92} +11.4382 q^{93} -2.72159 q^{94} -1.53613 q^{95} -20.7102 q^{96} -4.62904 q^{97} -8.10391 q^{98} +18.3400 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.81267	−1.28175	−0.640876	−	0.767644i	$-0.721429\pi$
$2$	−1.81267	−1.28175	−0.640876	+	0.767644i	$0.721429\pi$
$3$	−3.27382	−1.89014	−0.945070	−	0.326869i	$-0.894006\pi$
$3$	−3.27382	−1.89014	−0.945070	+	0.326869i	$0.894006\pi$
$4$	1.28578	0.642888
$5$	1.53613	0.686978	0.343489	−	0.939157i	$-0.388391\pi$
$5$	1.53613	0.686978	0.343489	+	0.939157i	$0.388391\pi$
$6$	5.93435	2.42269
$7$	−3.38684	−1.28011	−0.640053	−	0.768331i	$-0.721088\pi$
$7$	−3.38684	−1.28011	−0.640053	+	0.768331i	$0.721088\pi$
$8$	1.29465	0.457729
$9$	7.71788	2.57263
$10$	−2.78450	−0.880536
$11$	2.37631	0.716483	0.358242	−	0.933629i	$-0.383376\pi$
$11$	2.37631	0.716483	0.358242	+	0.933629i	$0.383376\pi$
$12$	−4.20939	−1.21515
$13$	6.14452	1.70418	0.852091	−	0.523393i	$-0.175334\pi$
$13$	6.14452	1.70418	0.852091	+	0.523393i	$0.175334\pi$
$14$	6.13923	1.64078
$15$	−5.02901	−1.29848
$16$	−4.91833	−1.22958
$17$	5.44507	1.32062	0.660312	−	0.750992i	$-0.270424\pi$
$17$	5.44507	1.32062	0.660312	+	0.750992i	$0.270424\pi$
$18$	−13.9900	−3.29747
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.97512	0.441650
$21$	11.0879	2.41958
$22$	−4.30746	−0.918354
$23$	−3.37872	−0.704513	−0.352256	−	0.935904i	$-0.614585\pi$
$23$	−3.37872	−0.704513	−0.352256	+	0.935904i	$0.614585\pi$
$24$	−4.23846	−0.865172
$25$	−2.64031	−0.528061
$26$	−11.1380	−2.18434
$27$	−15.4455	−2.97248
$28$	−4.35472	−0.822965
$29$	−6.27899	−1.16598	−0.582990	−	0.812479i	$-0.698117\pi$
$29$	−6.27899	−1.16598	−0.582990	+	0.812479i	$0.698117\pi$
$30$	9.11594	1.66433
$31$	−3.49385	−0.627514	−0.313757	−	0.949503i	$-0.601588\pi$
$31$	−3.49385	−0.627514	−0.313757	+	0.949503i	$0.601588\pi$
$32$	6.32601	1.11829
$33$	−7.77959	−1.35425
$34$	−9.87012	−1.69271
$35$	−5.20263	−0.879405
$36$	9.92346	1.65391
$37$	10.3280	1.69791	0.848957	−	0.528463i	$-0.177231\pi$
$37$	10.3280	1.69791	0.848957	+	0.528463i	$0.177231\pi$
$38$	1.81267	0.294054
$39$	−20.1160	−3.22114
$40$	1.98876	0.314450
$41$	−1.53433	−0.239621	−0.119811	−	0.992797i	$-0.538229\pi$
$41$	−1.53433	−0.239621	−0.119811	+	0.992797i	$0.538229\pi$
$42$	−20.0987	−3.10130
$43$	−8.07740	−1.23179	−0.615896	−	0.787827i	$-0.711206\pi$
$43$	−8.07740	−1.23179	−0.615896	+	0.787827i	$0.711206\pi$
$44$	3.05540	0.460618
$45$	11.8557	1.76734
$46$	6.12452	0.903011
$47$	1.50142	0.219005	0.109503	−	0.993987i	$-0.465074\pi$
$47$	1.50142	0.219005	0.109503	+	0.993987i	$0.465074\pi$
$48$	16.1017	2.32408
$49$	4.47070	0.638671
$50$	4.78600	0.676843
$51$	−17.8262	−2.49616
$52$	7.90047	1.09560
$53$	4.34422	0.596725	0.298363	−	0.954453i	$-0.403560\pi$
$53$	4.34422	0.596725	0.298363	+	0.954453i	$0.403560\pi$
$54$	27.9975	3.80998
$55$	3.65031	0.492208
$56$	−4.38479	−0.585942
$57$	3.27382	0.433628
$58$	11.3817	1.49450
$59$	7.19628	0.936876	0.468438	−	0.883496i	$-0.344817\pi$
$59$	7.19628	0.936876	0.468438	+	0.883496i	$0.344817\pi$
$60$	−6.46618	−0.834780
$61$	−12.8639	−1.64706	−0.823530	−	0.567273i	$-0.807998\pi$
$61$	−12.8639	−1.64706	−0.823530	+	0.567273i	$0.807998\pi$
$62$	6.33320	0.804317
$63$	−26.1392	−3.29323
$64$	−1.63031	−0.203789
$65$	9.43878	1.17074
$66$	14.1018	1.73582
$67$	−5.61633	−0.686145	−0.343072	−	0.939309i	$-0.611468\pi$
$67$	−5.61633	−0.686145	−0.343072	+	0.939309i	$0.611468\pi$
$68$	7.00114	0.849013
$69$	11.0613	1.33163
$70$	9.43065	1.12718
$71$	3.25684	0.386516	0.193258	−	0.981148i	$-0.438095\pi$
$71$	3.25684	0.386516	0.193258	+	0.981148i	$0.438095\pi$
$72$	9.99198	1.17757
$73$	−7.75551	−0.907714	−0.453857	−	0.891075i	$-0.649952\pi$
$73$	−7.75551	−0.907714	−0.453857	+	0.891075i	$0.649952\pi$
$74$	−18.7213	−2.17630
$75$	8.64388	0.998109
$76$	−1.28578	−0.147489
$77$	−8.04817	−0.917174
$78$	36.4637	4.12871
$79$	−13.9660	−1.57130	−0.785650	−	0.618671i	$-0.787672\pi$
$79$	−13.9660	−1.57130	−0.785650	+	0.618671i	$0.787672\pi$
$80$	−7.55520	−0.844697
$81$	27.4120	3.04578
$82$	2.78123	0.307135
$83$	8.36055	0.917689	0.458845	−	0.888516i	$-0.348263\pi$
$83$	8.36055	0.917689	0.458845	+	0.888516i	$0.348263\pi$
$84$	14.2566	1.55552
$85$	8.36433	0.907239
$86$	14.6417	1.57885
$87$	20.5563	2.20386
$88$	3.07649	0.327955
$89$	0.928807	0.0984533	0.0492266	−	0.998788i	$-0.484324\pi$
$89$	0.928807	0.0984533	0.0492266	+	0.998788i	$0.484324\pi$
$90$	−21.4904	−2.26529
$91$	−20.8105	−2.18153
$92$	−4.34428	−0.452923
$93$	11.4382	1.18609
$94$	−2.72159	−0.280711
$95$	−1.53613	−0.157604
$96$	−20.7102	−2.11373
$97$	−4.62904	−0.470007	−0.235004	−	0.971994i	$-0.575510\pi$
$97$	−4.62904	−0.470007	−0.235004	+	0.971994i	$0.575510\pi$
$98$	−8.10391	−0.818618
$99$	18.3400	1.84324
$100$	−3.39484	−0.339484
$101$	0.147636	0.0146903	0.00734516	−	0.999973i	$-0.497662\pi$
$101$	0.147636	0.0146903	0.00734516	+	0.999973i	$0.497662\pi$
$102$	32.3130	3.19946
$103$	7.68631	0.757354	0.378677	−	0.925529i	$-0.376379\pi$
$103$	7.68631	0.757354	0.378677	+	0.925529i	$0.376379\pi$
$104$	7.95502	0.780054
$105$	17.0325	1.66220
$106$	−7.87465	−0.764854
$107$	−6.24167	−0.603405	−0.301702	−	0.953402i	$-0.597555\pi$
$107$	−6.24167	−0.603405	−0.301702	+	0.953402i	$0.597555\pi$
$108$	−19.8594	−1.91097
$109$	−17.1349	−1.64123	−0.820613	−	0.571485i	$-0.806368\pi$
$109$	−17.1349	−1.64123	−0.820613	+	0.571485i	$0.806368\pi$
$110$	−6.61682	−0.630889
$111$	−33.8120	−3.20929
$112$	16.6576	1.57400
$113$	16.9374	1.59334	0.796668	−	0.604417i	$-0.206594\pi$
$113$	16.9374	1.59334	0.796668	+	0.604417i	$0.206594\pi$
$114$	−5.93435	−0.555803
$115$	−5.19016	−0.483985
$116$	−8.07338	−0.749594
$117$	47.4226	4.38422
$118$	−13.0445	−1.20084
$119$	−18.4416	−1.69054
$120$	−6.51082	−0.594354
$121$	−5.35317	−0.486652
$122$	23.3181	2.11112
$123$	5.02310	0.452918
$124$	−4.49231	−0.403421
$125$	−11.7365	−1.04974
$126$	47.3818	4.22111
$127$	20.9988	1.86334	0.931670	−	0.363306i	$-0.118352\pi$
$127$	20.9988	1.86334	0.931670	+	0.363306i	$0.118352\pi$
$128$	−9.69680	−0.857085
$129$	26.4439	2.32826
$130$	−17.1094	−1.50059
$131$	−7.65857	−0.669132	−0.334566	−	0.942372i	$-0.608590\pi$
$131$	−7.65857	−0.669132	−0.334566	+	0.942372i	$0.608590\pi$
$132$	−10.0028	−0.870633
$133$	3.38684	0.293676
$134$	10.1806	0.879467
$135$	−23.7262	−2.04203
$136$	7.04948	0.604488
$137$	−14.9136	−1.27415	−0.637076	−	0.770801i	$-0.719856\pi$
$137$	−14.9136	−1.27415	−0.637076	+	0.770801i	$0.719856\pi$
$138$	−20.0505	−1.70682
$139$	−17.8947	−1.51781	−0.758906	−	0.651200i	$-0.774266\pi$
$139$	−17.8947	−1.51781	−0.758906	+	0.651200i	$0.774266\pi$
$140$	−6.68941	−0.565359
$141$	−4.91539	−0.413951
$142$	−5.90358	−0.495418
$143$	14.6013	1.22102
$144$	−37.9591	−3.16326
$145$	−9.64535	−0.801003
$146$	14.0582	1.16346
$147$	−14.6363	−1.20718
$148$	13.2795	1.09157
$149$	10.0627	0.824365	0.412183	−	0.911101i	$-0.364767\pi$
$149$	10.0627	0.824365	0.412183	+	0.911101i	$0.364767\pi$
$150$	−15.6685	−1.27933
$151$	22.3083	1.81542	0.907711	−	0.419595i	$-0.137828\pi$
$151$	22.3083	1.81542	0.907711	+	0.419595i	$0.137828\pi$
$152$	−1.29465	−0.105010
$153$	42.0244	3.39747
$154$	14.5887	1.17559
$155$	−5.36701	−0.431088
$156$	−25.8647	−2.07083
$157$	7.22765	0.576829	0.288414	−	0.957506i	$-0.406872\pi$
$157$	7.22765	0.576829	0.288414	+	0.957506i	$0.406872\pi$
$158$	25.3158	2.01402
$159$	−14.2222	−1.12789
$160$	9.71757	0.768242
$161$	11.4432	0.901851
$162$	−49.6889	−3.90393
$163$	16.2317	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$163$	16.2317	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$164$	−1.97280	−0.154050
$165$	−11.9505	−0.930342
$166$	−15.1549	−1.17625
$167$	12.2735	0.949752	0.474876	−	0.880053i	$-0.342493\pi$
$167$	12.2735	0.949752	0.474876	+	0.880053i	$0.342493\pi$
$168$	14.3550	1.10751
$169$	24.7551	1.90424
$170$	−15.1618	−1.16286
$171$	−7.71788	−0.590201
$172$	−10.3857	−0.791904
$173$	−15.0735	−1.14602	−0.573008	−	0.819550i	$-0.694224\pi$
$173$	−15.0735	−1.14602	−0.573008	+	0.819550i	$0.694224\pi$
$174$	−37.2618	−2.82481
$175$	8.94230	0.675974
$176$	−11.6875	−0.880975
$177$	−23.5593	−1.77083
$178$	−1.68362	−0.126193
$179$	16.6117	1.24162	0.620809	−	0.783962i	$-0.286804\pi$
$179$	16.6117	1.24162	0.620809	+	0.783962i	$0.286804\pi$
$180$	15.2437	1.13620
$181$	−15.4452	−1.14804	−0.574018	−	0.818843i	$-0.694616\pi$
$181$	−15.4452	−1.14804	−0.574018	+	0.818843i	$0.694616\pi$
$182$	37.7226	2.79619
$183$	42.1142	3.11317
$184$	−4.37428	−0.322476
$185$	15.8652	1.16643
$186$	−20.7337	−1.52027
$187$	12.9391	0.946204
$188$	1.93050	0.140796
$189$	52.3114	3.80509
$190$	2.78450	0.202009
$191$	−5.68586	−0.411415	−0.205707	−	0.978614i	$-0.565950\pi$
$191$	−5.68586	−0.411415	−0.205707	+	0.978614i	$0.565950\pi$
$192$	5.33734	0.385189
$193$	12.6487	0.910473	0.455237	−	0.890371i	$-0.349555\pi$
$193$	12.6487	0.910473	0.455237	+	0.890371i	$0.349555\pi$
$194$	8.39092	0.602433
$195$	−30.9008	−2.21285
$196$	5.74832	0.410594
$197$	−12.8013	−0.912057	−0.456029	−	0.889965i	$-0.650729\pi$
$197$	−12.8013	−0.912057	−0.456029	+	0.889965i	$0.650729\pi$
$198$	−33.2444	−2.36258
$199$	−0.594676	−0.0421554	−0.0210777	−	0.999778i	$-0.506710\pi$
$199$	−0.594676	−0.0421554	−0.0210777	+	0.999778i	$0.506710\pi$
$200$	−3.41828	−0.241709
$201$	18.3869	1.29691
$202$	−0.267615	−0.0188293
$203$	21.2660	1.49258
$204$	−22.9204	−1.60475
$205$	−2.35692	−0.164615
$206$	−13.9327	−0.970740
$207$	−26.0766	−1.81245
$208$	−30.2208	−2.09543
$209$	−2.37631	−0.164372
$210$	−30.8742	−2.13052
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	5.58570	0.383627
$213$	−10.6623	−0.730569
$214$	11.3141	0.773415
$215$	−12.4079	−0.846214
$216$	−19.9965	−1.36059
$217$	11.8331	0.803284
$218$	31.0599	2.10364
$219$	25.3901	1.71571
$220$	4.69348	0.316435
$221$	33.4573	2.25058
$222$	61.2900	4.11352
$223$	−12.5668	−0.841537	−0.420768	−	0.907168i	$-0.638239\pi$
$223$	−12.5668	−0.841537	−0.420768	+	0.907168i	$0.638239\pi$
$224$	−21.4252	−1.43153
$225$	−20.3776	−1.35850
$226$	−30.7019	−2.04226
$227$	−22.3647	−1.48440	−0.742199	−	0.670179i	$-0.766217\pi$
$227$	−22.3647	−1.48440	−0.742199	+	0.670179i	$0.766217\pi$
$228$	4.20939	0.278774
$229$	−25.6346	−1.69398	−0.846992	−	0.531605i	$-0.821589\pi$
$229$	−25.6346	−1.69398	−0.846992	+	0.531605i	$0.821589\pi$
$230$	9.40805	0.620349
$231$	26.3482	1.73359
$232$	−8.12912	−0.533703
$233$	−14.2178	−0.931436	−0.465718	−	0.884933i	$-0.654204\pi$
$233$	−14.2178	−0.931436	−0.465718	+	0.884933i	$0.654204\pi$
$234$	−85.9616	−5.61949
$235$	2.30638	0.150452
$236$	9.25280	0.602306
$237$	45.7222	2.96998
$238$	33.4285	2.16685
$239$	−15.7409	−1.01819	−0.509096	−	0.860710i	$-0.670020\pi$
$239$	−15.7409	−1.01819	−0.509096	+	0.860710i	$0.670020\pi$
$240$	24.7343	1.59659
$241$	−13.8298	−0.890854	−0.445427	−	0.895318i	$-0.646948\pi$
$241$	−13.8298	−0.890854	−0.445427	+	0.895318i	$0.646948\pi$
$242$	9.70354	0.623767
$243$	−43.4055	−2.78446
$244$	−16.5401	−1.05887
$245$	6.86757	0.438753
$246$	−9.10523	−0.580528
$247$	−6.14452	−0.390966
$248$	−4.52332	−0.287231
$249$	−27.3709	−1.73456
$250$	21.2744	1.34551
$251$	17.3057	1.09233	0.546165	−	0.837678i	$-0.316087\pi$
$251$	17.3057	1.09233	0.546165	+	0.837678i	$0.316087\pi$
$252$	−33.6092	−2.11718
$253$	−8.02888	−0.504772
$254$	−38.0639	−2.38834
$255$	−27.3833	−1.71481
$256$	20.8377	1.30236
$257$	−17.1375	−1.06901	−0.534505	−	0.845165i	$-0.679502\pi$
$257$	−17.1375	−1.06901	−0.534505	+	0.845165i	$0.679502\pi$
$258$	−47.9341	−2.98425
$259$	−34.9793	−2.17351
$260$	12.1362	0.752652
$261$	−48.4605	−2.99963
$262$	13.8825	0.857661
$263$	−0.175120	−0.0107983	−0.00539917	−	0.999985i	$-0.501719\pi$
$263$	−0.175120	−0.0107983	−0.00539917	+	0.999985i	$0.501719\pi$
$264$	−10.0719	−0.619881
$265$	6.67329	0.409937
$266$	−6.13923	−0.376420
$267$	−3.04074	−0.186090
$268$	−7.22135	−0.441114
$269$	11.8791	0.724279	0.362140	−	0.932124i	$-0.382046\pi$
$269$	11.8791	0.724279	0.362140	+	0.932124i	$0.382046\pi$
$270$	43.0079	2.61738
$271$	−9.79292	−0.594877	−0.297439	−	0.954741i	$-0.596132\pi$
$271$	−9.79292	−0.594877	−0.297439	+	0.954741i	$0.596132\pi$
$272$	−26.7807	−1.62382
$273$	68.1298	4.12340
$274$	27.0334	1.63315
$275$	−6.27417	−0.378347
$276$	14.2224	0.856087
$277$	−8.14083	−0.489135	−0.244568	−	0.969632i	$-0.578646\pi$
$277$	−8.14083	−0.489135	−0.244568	+	0.969632i	$0.578646\pi$
$278$	32.4373	1.94546
$279$	−26.9651	−1.61436
$280$	−6.73560	−0.402529
$281$	8.44511	0.503793	0.251897	−	0.967754i	$-0.418946\pi$
$281$	8.44511	0.503793	0.251897	+	0.967754i	$0.418946\pi$
$282$	8.90999	0.530582
$283$	21.3241	1.26759	0.633794	−	0.773502i	$-0.281497\pi$
$283$	21.3241	1.26759	0.633794	+	0.773502i	$0.281497\pi$
$284$	4.18757	0.248486
$285$	5.02901	0.297893
$286$	−26.4673	−1.56504
$287$	5.19652	0.306741
$288$	48.8234	2.87694
$289$	12.6488	0.744045
$290$	17.4838	1.02669
$291$	15.1546	0.888379
$292$	−9.97184	−0.583558
$293$	18.8610	1.10187	0.550935	−	0.834548i	$-0.314271\pi$
$293$	18.8610	1.10187	0.550935	+	0.834548i	$0.314271\pi$
$294$	26.5307	1.54730
$295$	11.0544	0.643613
$296$	13.3712	0.777184
$297$	−36.7032	−2.12973
$298$	−18.2403	−1.05663
$299$	−20.7606	−1.20062
$300$	11.1141	0.641672
$301$	27.3569	1.57682
$302$	−40.4376	−2.32692
$303$	−0.483333	−0.0277667
$304$	4.91833	0.282086
$305$	−19.7607	−1.13149
$306$	−76.1764	−4.35471
$307$	4.34683	0.248087	0.124043	−	0.992277i	$-0.460414\pi$
$307$	4.34683	0.248087	0.124043	+	0.992277i	$0.460414\pi$
$308$	−10.3481	−0.589640
$309$	−25.1636	−1.43151
$310$	9.72861	0.552548
$311$	−5.82741	−0.330442	−0.165221	−	0.986257i	$-0.552834\pi$
$311$	−5.82741	−0.330442	−0.165221	+	0.986257i	$0.552834\pi$
$312$	−26.0433	−1.47441
$313$	16.5618	0.936127	0.468063	−	0.883695i	$-0.344952\pi$
$313$	16.5618	0.936127	0.468063	+	0.883695i	$0.344952\pi$
$314$	−13.1013	−0.739352
$315$	−40.1533	−2.26238
$316$	−17.9572	−1.01017
$317$	5.48763	0.308216	0.154108	−	0.988054i	$-0.450750\pi$
$317$	5.48763	0.308216	0.154108	+	0.988054i	$0.450750\pi$
$318$	25.7802	1.44568
$319$	−14.9208	−0.835405
$320$	−2.50437	−0.139998
$321$	20.4341	1.14052
$322$	−20.7428	−1.15595
$323$	−5.44507	−0.302972
$324$	35.2457	1.95809
$325$	−16.2234	−0.899913
$326$	−29.4227	−1.62957
$327$	56.0965	3.10214
$328$	−1.98642	−0.109682
$329$	−5.08509	−0.280350
$330$	21.6623	1.19247
$331$	9.84574	0.541171	0.270585	−	0.962696i	$-0.412783\pi$
$331$	9.84574	0.541171	0.270585	+	0.962696i	$0.412783\pi$
$332$	10.7498	0.589971
$333$	79.7103	4.36810
$334$	−22.2478	−1.21735
$335$	−8.62742	−0.471366
$336$	−54.5340	−2.97507
$337$	−30.0197	−1.63528	−0.817639	−	0.575731i	$-0.804718\pi$
$337$	−30.0197	−1.63528	−0.817639	+	0.575731i	$0.804718\pi$
$338$	−44.8729	−2.44076
$339$	−55.4499	−3.01163
$340$	10.7547	0.583253
$341$	−8.30245	−0.449603
$342$	13.9900	0.756491
$343$	8.56634	0.462539
$344$	−10.4574	−0.563827
$345$	16.9916	0.914799
$346$	27.3233	1.46891
$347$	−16.2339	−0.871480	−0.435740	−	0.900072i	$-0.643513\pi$
$347$	−16.2339	−0.871480	−0.435740	+	0.900072i	$0.643513\pi$
$348$	26.4308	1.41684
$349$	−3.33126	−0.178318	−0.0891591	−	0.996017i	$-0.528418\pi$
$349$	−3.33126	−0.178318	−0.0891591	+	0.996017i	$0.528418\pi$
$350$	−16.2094	−0.866431
$351$	−94.9050	−5.06565
$352$	15.0325	0.801237
$353$	−19.4499	−1.03521	−0.517607	−	0.855618i	$-0.673177\pi$
$353$	−19.4499	−1.03521	−0.517607	+	0.855618i	$0.673177\pi$
$354$	42.7053	2.26976
$355$	5.00293	0.265528
$356$	1.19424	0.0632944
$357$	60.3744	3.19535
$358$	−30.1116	−1.59145
$359$	6.20521	0.327498	0.163749	−	0.986502i	$-0.447641\pi$
$359$	6.20521	0.327498	0.163749	+	0.986502i	$0.447641\pi$
$360$	15.3490	0.808962
$361$	1.00000	0.0526316
$362$	27.9971	1.47150
$363$	17.5253	0.919840
$364$	−26.7577	−1.40248
$365$	−11.9135	−0.623579
$366$	−76.3392	−3.99031
$367$	30.6457	1.59969	0.799845	−	0.600206i	$-0.204915\pi$
$367$	30.6457	1.59969	0.799845	+	0.600206i	$0.204915\pi$
$368$	16.6177	0.866257
$369$	−11.8417	−0.616456
$370$	−28.7583	−1.49507
$371$	−14.7132	−0.763871
$372$	14.7070	0.762522
$373$	−24.5104	−1.26910	−0.634550	−	0.772881i	$-0.718815\pi$
$373$	−24.5104	−1.26910	−0.634550	+	0.772881i	$0.718815\pi$
$374$	−23.4544	−1.21280
$375$	38.4232	1.98416
$376$	1.94383	0.100245
$377$	−38.5814	−1.98704
$378$	−94.8233	−4.87718
$379$	18.8156	0.966491	0.483246	−	0.875485i	$-0.339458\pi$
$379$	18.8156	0.966491	0.483246	+	0.875485i	$0.339458\pi$
$380$	−1.97512	−0.101321
$381$	−68.7461	−3.52197
$382$	10.3066	0.527332
$383$	−31.4522	−1.60713	−0.803566	−	0.595216i	$-0.797067\pi$
$383$	−31.4522	−1.60713	−0.803566	+	0.595216i	$0.797067\pi$
$384$	31.7456	1.62001
$385$	−12.3630	−0.630079
$386$	−22.9279	−1.16700
$387$	−62.3404	−3.16894
$388$	−5.95190	−0.302162
$389$	−31.8989	−1.61734	−0.808668	−	0.588265i	$-0.799811\pi$
$389$	−31.8989	−1.61734	−0.808668	+	0.588265i	$0.799811\pi$
$390$	56.0130	2.83633
$391$	−18.3974	−0.930396
$392$	5.78801	0.292338
$393$	25.0728	1.26475
$394$	23.2046	1.16903
$395$	−21.4536	−1.07945
$396$	23.5812	1.18500
$397$	0.0590682	0.00296455	0.00148227	−	0.999999i	$-0.499528\pi$
$397$	0.0590682	0.00296455	0.00148227	+	0.999999i	$0.499528\pi$
$398$	1.07795	0.0540328
$399$	−11.0879	−0.555089
$400$	12.9859	0.649295
$401$	−28.2982	−1.41314	−0.706572	−	0.707641i	$-0.749759\pi$
$401$	−28.2982	−1.41314	−0.706572	+	0.707641i	$0.749759\pi$
$402$	−33.3293	−1.66232
$403$	−21.4680	−1.06940
$404$	0.189827	0.00944423
$405$	42.1084	2.09238
$406$	−38.5482	−1.91311
$407$	24.5425	1.21653
$408$	−23.0787	−1.14257
$409$	1.98997	0.0983979	0.0491989	−	0.998789i	$-0.484333\pi$
$409$	1.98997	0.0983979	0.0491989	+	0.998789i	$0.484333\pi$
$410$	4.27232	0.210995
$411$	48.8243	2.40832
$412$	9.88287	0.486894
$413$	−24.3727	−1.19930
$414$	47.2683	2.32311
$415$	12.8429	0.630433
$416$	38.8703	1.90577
$417$	58.5841	2.86888
$418$	4.30746	0.210685
$419$	4.37323	0.213646	0.106823	−	0.994278i	$-0.465932\pi$
$419$	4.37323	0.213646	0.106823	+	0.994278i	$0.465932\pi$
$420$	21.8999	1.06861
$421$	21.5589	1.05071	0.525357	−	0.850882i	$-0.323932\pi$
$421$	21.5589	1.05071	0.525357	+	0.850882i	$0.323932\pi$
$422$	1.81267	0.0882394
$423$	11.5878	0.563419
$424$	5.62427	0.273139
$425$	−14.3766	−0.697370
$426$	19.3272	0.936408
$427$	43.5681	2.10841
$428$	−8.02539	−0.387922
$429$	−47.8018	−2.30789
$430$	22.4915	1.08464
$431$	21.2256	1.02240	0.511200	−	0.859462i	$-0.329201\pi$
$431$	21.2256	1.02240	0.511200	+	0.859462i	$0.329201\pi$
$432$	75.9659	3.65491
$433$	−36.1825	−1.73882	−0.869410	−	0.494091i	$-0.835501\pi$
$433$	−36.1825	−1.73882	−0.869410	+	0.494091i	$0.835501\pi$
$434$	−21.4495	−1.02961
$435$	31.5771	1.51401
$436$	−22.0316	−1.05512
$437$	3.37872	0.161626
$438$	−46.0239	−2.19911
$439$	11.0133	0.525637	0.262818	−	0.964845i	$-0.415348\pi$
$439$	11.0133	0.525637	0.262818	+	0.964845i	$0.415348\pi$
$440$	4.72589	0.225298
$441$	34.5043	1.64306
$442$	−60.6471	−2.88469
$443$	−26.5695	−1.26236	−0.631178	−	0.775638i	$-0.717428\pi$
$443$	−26.5695	−1.26236	−0.631178	+	0.775638i	$0.717428\pi$
$444$	−43.4746	−2.06321
$445$	1.42677	0.0676353
$446$	22.7795	1.07864
$447$	−32.9433	−1.55816
$448$	5.52160	0.260871
$449$	12.2539	0.578299	0.289149	−	0.957284i	$-0.406628\pi$
$449$	12.2539	0.578299	0.289149	+	0.957284i	$0.406628\pi$
$450$	36.9378	1.74126
$451$	−3.64603	−0.171685
$452$	21.7777	1.02434
$453$	−73.0333	−3.43140
$454$	40.5399	1.90263
$455$	−31.9677	−1.49867
$456$	4.23846	0.198484
$457$	6.89717	0.322636	0.161318	−	0.986902i	$-0.448425\pi$
$457$	6.89717	0.322636	0.161318	+	0.986902i	$0.448425\pi$
$458$	46.4672	2.17127
$459$	−84.1016	−3.92553
$460$	−6.67338	−0.311148
$461$	26.9171	1.25365	0.626827	−	0.779158i	$-0.284353\pi$
$461$	26.9171	1.25365	0.626827	+	0.779158i	$0.284353\pi$
$462$	−47.7607	−2.22203
$463$	7.62676	0.354446	0.177223	−	0.984171i	$-0.443289\pi$
$463$	7.62676	0.354446	0.177223	+	0.984171i	$0.443289\pi$
$464$	30.8822	1.43367
$465$	17.5706	0.814817
$466$	25.7721	1.19387
$467$	−15.9322	−0.737254	−0.368627	−	0.929577i	$-0.620172\pi$
$467$	−15.9322	−0.737254	−0.368627	+	0.929577i	$0.620172\pi$
$468$	60.9749	2.81856
$469$	19.0216	0.878338
$470$	−4.18071	−0.192842
$471$	−23.6620	−1.09029
$472$	9.31669	0.428835
$473$	−19.1944	−0.882558
$474$	−82.8793	−3.80677
$475$	2.64031	0.121146
$476$	−23.7117	−1.08683
$477$	33.5282	1.53515
$478$	28.5330	1.30507
$479$	9.45379	0.431955	0.215977	−	0.976398i	$-0.430706\pi$
$479$	9.45379	0.431955	0.215977	+	0.976398i	$0.430706\pi$
$480$	−31.8136	−1.45208
$481$	63.4606	2.89355
$482$	25.0688	1.14185
$483$	−37.4630	−1.70462
$484$	−6.88298	−0.312863
$485$	−7.11080	−0.322885
$486$	78.6798	3.56899
$487$	36.9439	1.67409	0.837043	−	0.547136i	$-0.184282\pi$
$487$	36.9439	1.67409	0.837043	+	0.547136i	$0.184282\pi$
$488$	−16.6544	−0.753907
$489$	−53.1395	−2.40305
$490$	−12.4487	−0.562373
$491$	−37.3118	−1.68386	−0.841930	−	0.539587i	$-0.818580\pi$
$491$	−37.3118	−1.68386	−0.841930	+	0.539587i	$0.818580\pi$
$492$	6.45858	0.291175
$493$	−34.1896	−1.53982
$494$	11.1380	0.501122
$495$	28.1727	1.26627
$496$	17.1839	0.771580
$497$	−11.0304	−0.494781
$498$	49.6144	2.22328
$499$	−7.08713	−0.317263	−0.158632	−	0.987338i	$-0.550708\pi$
$499$	−7.08713	−0.317263	−0.158632	+	0.987338i	$0.550708\pi$
$500$	−15.0905	−0.674868
$501$	−40.1812	−1.79516
$502$	−31.3696	−1.40010
$503$	−25.4960	−1.13681	−0.568405	−	0.822749i	$-0.692439\pi$
$503$	−25.4960	−1.13681	−0.568405	+	0.822749i	$0.692439\pi$
$504$	−33.8413	−1.50741
$505$	0.226788	0.0100919
$506$	14.5537	0.646992
$507$	−81.0437	−3.59928
$508$	26.9997	1.19792
$509$	28.2801	1.25349	0.626747	−	0.779223i	$-0.284386\pi$
$509$	28.2801	1.25349	0.626747	+	0.779223i	$0.284386\pi$
$510$	49.6369	2.19796
$511$	26.2667	1.16197
$512$	−18.3783	−0.812216
$513$	15.4455	0.681934
$514$	31.0647	1.37021
$515$	11.8072	0.520286
$516$	34.0010	1.49681
$517$	3.56784	0.156914
$518$	63.4060	2.78590
$519$	49.3478	2.16613
$520$	12.2199	0.535880
$521$	−11.1888	−0.490190	−0.245095	−	0.969499i	$-0.578819\pi$
$521$	−11.1888	−0.490190	−0.245095	+	0.969499i	$0.578819\pi$
$522$	87.8430	3.84478
$523$	14.0018	0.612255	0.306128	−	0.951990i	$-0.400967\pi$
$523$	14.0018	0.612255	0.306128	+	0.951990i	$0.400967\pi$
$524$	−9.84720	−0.430177
$525$	−29.2754	−1.27769
$526$	0.317434	0.0138408
$527$	−19.0242	−0.828709
$528$	38.2626	1.66517
$529$	−11.5842	−0.503662
$530$	−12.0965	−0.525438
$531$	55.5400	2.41023
$532$	4.35472	0.188801
$533$	−9.42769	−0.408359
$534$	5.51187	0.238522
$535$	−9.58801	−0.414526
$536$	−7.27121	−0.314068
$537$	−54.3837	−2.34683
$538$	−21.5328	−0.928346
$539$	10.6237	0.457597
$540$	−30.5066	−1.31280
$541$	−35.8274	−1.54034	−0.770169	−	0.637840i	$-0.779828\pi$
$541$	−35.8274	−1.54034	−0.770169	+	0.637840i	$0.779828\pi$
$542$	17.7513	0.762485
$543$	50.5649	2.16995
$544$	34.4456	1.47684
$545$	−26.3214	−1.12749
$546$	−123.497	−5.28518
$547$	43.6486	1.86628	0.933140	−	0.359513i	$-0.117057\pi$
$547$	43.6486	1.86628	0.933140	+	0.359513i	$0.117057\pi$
$548$	−19.1755	−0.819137
$549$	−99.2824	−4.23727
$550$	11.3730	0.484947
$551$	6.27899	0.267494
$552$	14.3206	0.609525
$553$	47.3007	2.01143
$554$	14.7566	0.626950
$555$	−51.9396	−2.20471
$556$	−23.0086	−0.975783
$557$	−12.6645	−0.536612	−0.268306	−	0.963334i	$-0.586464\pi$
$557$	−12.6645	−0.536612	−0.268306	+	0.963334i	$0.586464\pi$
$558$	48.8788	2.06921
$559$	−49.6317	−2.09920
$560$	25.5883	1.08130
$561$	−42.3604	−1.78846
$562$	−15.3082	−0.645738
$563$	−28.4930	−1.20084	−0.600419	−	0.799686i	$-0.704999\pi$
$563$	−28.4930	−1.20084	−0.600419	+	0.799686i	$0.704999\pi$
$564$	−6.32009	−0.266124
$565$	26.0180	1.09459
$566$	−38.6536	−1.62473
$567$	−92.8401	−3.89892
$568$	4.21648	0.176920
$569$	19.3100	0.809516	0.404758	−	0.914424i	$-0.367356\pi$
$569$	19.3100	0.809516	0.404758	+	0.914424i	$0.367356\pi$
$570$	−9.11594	−0.381825
$571$	−16.7723	−0.701900	−0.350950	−	0.936394i	$-0.614141\pi$
$571$	−16.7723	−0.701900	−0.350950	+	0.936394i	$0.614141\pi$
$572$	18.7739	0.784978
$573$	18.6145	0.777631
$574$	−9.41957	−0.393165
$575$	8.92086	0.372026
$576$	−12.5825	−0.524272
$577$	−8.63289	−0.359392	−0.179696	−	0.983722i	$-0.557511\pi$
$577$	−8.63289	−0.359392	−0.179696	+	0.983722i	$0.557511\pi$
$578$	−22.9281	−0.953681
$579$	−41.4095	−1.72092
$580$	−12.4018	−0.514955
$581$	−28.3159	−1.17474
$582$	−27.4703	−1.13868
$583$	10.3232	0.427543
$584$	−10.0407	−0.415487
$585$	72.8473	3.01187
$586$	−34.1887	−1.41232
$587$	3.06939	0.126687	0.0633437	−	0.997992i	$-0.479824\pi$
$587$	3.06939	0.126687	0.0633437	+	0.997992i	$0.479824\pi$
$588$	−18.8189	−0.776080
$589$	3.49385	0.143962
$590$	−20.0380	−0.824952
$591$	41.9092	1.72391
$592$	−50.7965	−2.08773
$593$	−19.5361	−0.802250	−0.401125	−	0.916023i	$-0.631381\pi$
$593$	−19.5361	−0.802250	−0.401125	+	0.916023i	$0.631381\pi$
$594$	66.5307	2.72979
$595$	−28.3287	−1.16136
$596$	12.9383	0.529974
$597$	1.94686	0.0796797
$598$	37.6322	1.53890
$599$	−42.8069	−1.74904	−0.874520	−	0.484989i	$-0.838824\pi$
$599$	−42.8069	−1.74904	−0.874520	+	0.484989i	$0.838824\pi$
$600$	11.1908	0.456863
$601$	−0.746734	−0.0304599	−0.0152300	−	0.999884i	$-0.504848\pi$
$601$	−0.746734	−0.0304599	−0.0152300	+	0.999884i	$0.504848\pi$
$602$	−49.5890	−2.02110
$603$	−43.3462	−1.76519
$604$	28.6835	1.16711
$605$	−8.22317	−0.334319
$606$	0.876123	0.0355901
$607$	17.8943	0.726308	0.363154	−	0.931729i	$-0.381700\pi$
$607$	17.8943	0.726308	0.363154	+	0.931729i	$0.381700\pi$
$608$	−6.32601	−0.256554
$609$	−69.6209	−2.82118
$610$	35.8196	1.45029
$611$	9.22553	0.373225
$612$	54.0339	2.18419
$613$	48.0618	1.94120	0.970600	−	0.240700i	$-0.0773770\pi$
$613$	48.0618	1.94120	0.970600	+	0.240700i	$0.0773770\pi$
$614$	−7.87937	−0.317985
$615$	7.71613	0.311145
$616$	−10.4196	−0.419817
$617$	25.7472	1.03654	0.518271	−	0.855217i	$-0.326576\pi$
$617$	25.7472	1.03654	0.518271	+	0.855217i	$0.326576\pi$
$618$	45.6133	1.83483
$619$	−23.3571	−0.938801	−0.469400	−	0.882985i	$-0.655530\pi$
$619$	−23.3571	−0.938801	−0.469400	+	0.882985i	$0.655530\pi$
$620$	−6.90077	−0.277141
$621$	52.1860	2.09415
$622$	10.5632	0.423544
$623$	−3.14572	−0.126031
$624$	98.9373	3.96066
$625$	−4.82726	−0.193091
$626$	−30.0210	−1.19988
$627$	7.77959	0.310687
$628$	9.29313	0.370836
$629$	56.2367	2.24230
$630$	72.7846	2.89981
$631$	8.43836	0.335926	0.167963	−	0.985793i	$-0.446281\pi$
$631$	8.43836	0.335926	0.167963	+	0.985793i	$0.446281\pi$
$632$	−18.0812	−0.719230
$633$	3.27382	0.130123
$634$	−9.94726	−0.395056
$635$	32.2568	1.28007
$636$	−18.2866	−0.725109
$637$	27.4703	1.08841
$638$	27.0465	1.07078
$639$	25.1359	0.994361
$640$	−14.8956	−0.588798
$641$	30.7004	1.21259	0.606297	−	0.795238i	$-0.292654\pi$
$641$	30.7004	1.21259	0.606297	+	0.795238i	$0.292654\pi$
$642$	−37.0403	−1.46186
$643$	26.1122	1.02976	0.514882	−	0.857261i	$-0.327836\pi$
$643$	26.1122	1.02976	0.514882	+	0.857261i	$0.327836\pi$
$644$	14.7134	0.579789
$645$	40.6213	1.59946
$646$	9.87012	0.388335
$647$	38.5072	1.51387	0.756937	−	0.653488i	$-0.226695\pi$
$647$	38.5072	1.51387	0.756937	+	0.653488i	$0.226695\pi$
$648$	35.4890	1.39414
$649$	17.1006	0.671256
$650$	29.4077	1.15346
$651$	−38.7395	−1.51832
$652$	20.8703	0.817344
$653$	−9.15473	−0.358252	−0.179126	−	0.983826i	$-0.557327\pi$
$653$	−9.15473	−0.358252	−0.179126	+	0.983826i	$0.557327\pi$
$654$	−101.685	−3.97618
$655$	−11.7646	−0.459679
$656$	7.54632	0.294634
$657$	−59.8561	−2.33521
$658$	9.21759	0.359339
$659$	−7.61221	−0.296530	−0.148265	−	0.988948i	$-0.547369\pi$
$659$	−7.61221	−0.296530	−0.148265	+	0.988948i	$0.547369\pi$
$660$	−15.3656	−0.598106
$661$	9.65387	0.375492	0.187746	−	0.982218i	$-0.439882\pi$
$661$	9.65387	0.375492	0.187746	+	0.982218i	$0.439882\pi$
$662$	−17.8471	−0.693646
$663$	−109.533	−4.25392
$664$	10.8240	0.420053
$665$	5.20263	0.201749
$666$	−144.488	−5.59881
$667$	21.2150	0.821448
$668$	15.7810	0.610584
$669$	41.1415	1.59062
$670$	15.6387	0.604175
$671$	−30.5687	−1.18009
$672$	70.1422	2.70579
$673$	11.6751	0.450042	0.225021	−	0.974354i	$-0.427755\pi$
$673$	11.6751	0.450042	0.225021	+	0.974354i	$0.427755\pi$
$674$	54.4159	2.09602
$675$	40.7807	1.56965
$676$	31.8295	1.22421
$677$	−34.5661	−1.32848	−0.664242	−	0.747517i	$-0.731246\pi$
$677$	−34.5661	−1.32848	−0.664242	+	0.747517i	$0.731246\pi$
$678$	100.512	3.86016
$679$	15.6778	0.601659
$680$	10.8289	0.415270
$681$	73.2180	2.80572
$682$	15.0496	0.576280
$683$	1.20581	0.0461389	0.0230695	−	0.999734i	$-0.492656\pi$
$683$	1.20581	0.0461389	0.0230695	+	0.999734i	$0.492656\pi$
$684$	−9.92346	−0.379433
$685$	−22.9092	−0.875314
$686$	−15.5280	−0.592860
$687$	83.9231	3.20187
$688$	39.7273	1.51459
$689$	26.6932	1.01693
$690$	−30.8002	−1.17255
$691$	24.8468	0.945215	0.472608	−	0.881273i	$-0.343313\pi$
$691$	24.8468	0.945215	0.472608	+	0.881273i	$0.343313\pi$
$692$	−19.3811	−0.736760
$693$	−62.1148	−2.35955
$694$	29.4267	1.11702
$695$	−27.4886	−1.04270
$696$	26.6133	1.00877
$697$	−8.35450	−0.316449
$698$	6.03848	0.228560
$699$	46.5463	1.76054
$700$	11.4978	0.434576
$701$	19.8541	0.749880	0.374940	−	0.927049i	$-0.377663\pi$
$701$	19.8541	0.749880	0.374940	+	0.927049i	$0.377663\pi$
$702$	172.031	6.49291
$703$	−10.3280	−0.389528
$704$	−3.87412	−0.146011
$705$	−7.55068	−0.284375
$706$	35.2563	1.32689
$707$	−0.500019	−0.0188052
$708$	−30.2920	−1.13844
$709$	−20.6104	−0.774040	−0.387020	−	0.922071i	$-0.626496\pi$
$709$	−20.6104	−0.774040	−0.387020	+	0.922071i	$0.626496\pi$
$710$	−9.06867	−0.340341
$711$	−107.788	−4.04237
$712$	1.20248	0.0450649
$713$	11.8048	0.442092
$714$	−109.439	−4.09565
$715$	22.4294	0.838813
$716$	21.3589	0.798221
$717$	51.5328	1.92453
$718$	−11.2480	−0.419771
$719$	−38.7952	−1.44682	−0.723408	−	0.690421i	$-0.757425\pi$
$719$	−38.7952	−1.44682	−0.723408	+	0.690421i	$0.757425\pi$
$720$	−58.3101	−2.17309
$721$	−26.0323	−0.969494
$722$	−1.81267	−0.0674606
$723$	45.2762	1.68384
$724$	−19.8591	−0.738059
$725$	16.5785	0.615709
$726$	−31.7676	−1.17901
$727$	−8.04160	−0.298247	−0.149123	−	0.988819i	$-0.547645\pi$
$727$	−8.04160	−0.298247	−0.149123	+	0.988819i	$0.547645\pi$
$728$	−26.9424	−0.998552
$729$	59.8656	2.21724
$730$	21.5952	0.799274
$731$	−43.9820	−1.62673
$732$	54.1494	2.00142
$733$	35.0381	1.29416	0.647081	−	0.762422i	$-0.275990\pi$
$733$	35.0381	1.29416	0.647081	+	0.762422i	$0.275990\pi$
$734$	−55.5505	−2.05041
$735$	−22.4832	−0.829305
$736$	−21.3738	−0.787851
$737$	−13.3461	−0.491611
$738$	21.4652	0.790144
$739$	−4.18623	−0.153993	−0.0769964	−	0.997031i	$-0.524533\pi$
$739$	−4.18623	−0.153993	−0.0769964	+	0.997031i	$0.524533\pi$
$740$	20.3990	0.749883
$741$	20.1160	0.738981
$742$	26.6702	0.979094
$743$	5.72878	0.210168	0.105084	−	0.994463i	$-0.466489\pi$
$743$	5.72878	0.210168	0.105084	+	0.994463i	$0.466489\pi$
$744$	14.8085	0.542907
$745$	15.4576	0.566321
$746$	44.4293	1.62667
$747$	64.5257	2.36087
$748$	16.6368	0.608303
$749$	21.1395	0.772422
$750$	−69.6485	−2.54320
$751$	−22.8026	−0.832077	−0.416039	−	0.909347i	$-0.636582\pi$
$751$	−22.8026	−0.832077	−0.416039	+	0.909347i	$0.636582\pi$
$752$	−7.38451	−0.269285
$753$	−56.6559	−2.06465
$754$	69.9354	2.54690
$755$	34.2684	1.24716
$756$	67.2607	2.44625
$757$	−48.3192	−1.75619	−0.878096	−	0.478485i	$-0.841186\pi$
$757$	−48.3192	−1.75619	−0.878096	+	0.478485i	$0.841186\pi$
$758$	−34.1064	−1.23880
$759$	26.2851	0.954088
$760$	−1.98876	−0.0721398
$761$	31.1685	1.12986	0.564928	−	0.825140i	$-0.308904\pi$
$761$	31.1685	1.12986	0.564928	+	0.825140i	$0.308904\pi$
$762$	124.614	4.51429
$763$	58.0332	2.10094
$764$	−7.31075	−0.264494
$765$	64.5549	2.33399
$766$	57.0125	2.05994
$767$	44.2177	1.59661
$768$	−68.2189	−2.46164
$769$	10.2703	0.370355	0.185178	−	0.982705i	$-0.440714\pi$
$769$	10.2703	0.370355	0.185178	+	0.982705i	$0.440714\pi$
$770$	22.4101	0.807605
$771$	56.1052	2.02058
$772$	16.2634	0.585332
$773$	36.1245	1.29931	0.649655	−	0.760230i	$-0.274914\pi$
$773$	36.1245	1.29931	0.649655	+	0.760230i	$0.274914\pi$
$774$	113.003	4.06180
$775$	9.22483	0.331366
$776$	−5.99300	−0.215136
$777$	114.516	4.10823
$778$	57.8221	2.07302
$779$	1.53433	0.0549729
$780$	−39.7315	−1.42262
$781$	7.73925	0.276932
$782$	33.3484	1.19254
$783$	96.9820	3.46585
$784$	−21.9884	−0.785299
$785$	11.1026	0.396269
$786$	−45.4486	−1.62110
$787$	−48.4514	−1.72711	−0.863553	−	0.504257i	$-0.831766\pi$
$787$	−48.4514	−1.72711	−0.863553	+	0.504257i	$0.831766\pi$
$788$	−16.4596	−0.586350
$789$	0.573310	0.0204104
$790$	38.8884	1.38359
$791$	−57.3643	−2.03964
$792$	23.7440	0.843706
$793$	−79.0427	−2.80689
$794$	−0.107071	−0.00379981
$795$	−21.8471	−0.774838
$796$	−0.764620	−0.0271012
$797$	26.0167	0.921560	0.460780	−	0.887515i	$-0.347570\pi$
$797$	26.0167	0.921560	0.460780	+	0.887515i	$0.347570\pi$
$798$	20.0987	0.711487
$799$	8.17536	0.289224
$800$	−16.7026	−0.590526
$801$	7.16842	0.253283
$802$	51.2953	1.81130
$803$	−18.4295	−0.650361
$804$	23.6414	0.833767
$805$	17.5783	0.619552
$806$	38.9145	1.37070
$807$	−38.8899	−1.36899
$808$	0.191137	0.00672419
$809$	−17.1447	−0.602776	−0.301388	−	0.953502i	$-0.597450\pi$
$809$	−17.1447	−0.602776	−0.301388	+	0.953502i	$0.597450\pi$
$810$	−76.3287	−2.68192
$811$	−18.7983	−0.660099	−0.330050	−	0.943964i	$-0.607065\pi$
$811$	−18.7983	−0.660099	−0.330050	+	0.943964i	$0.607065\pi$
$812$	27.3433	0.959560
$813$	32.0602	1.12440
$814$	−44.4875	−1.55928
$815$	24.9340	0.873399
$816$	87.6750	3.06924
$817$	8.07740	0.282592
$818$	−3.60717	−0.126122
$819$	−160.613	−5.61227
$820$	−3.03047	−0.105829
$821$	−26.0170	−0.908001	−0.454000	−	0.891001i	$-0.650004\pi$
$821$	−26.0170	−0.908001	−0.454000	+	0.891001i	$0.650004\pi$
$822$	−88.5023	−3.08687
$823$	−21.9527	−0.765223	−0.382611	−	0.923909i	$-0.624975\pi$
$823$	−21.9527	−0.765223	−0.382611	+	0.923909i	$0.624975\pi$
$824$	9.95111	0.346663
$825$	20.5405	0.715128
$826$	44.1796	1.53721
$827$	−26.0314	−0.905200	−0.452600	−	0.891714i	$-0.649503\pi$
$827$	−26.0314	−0.905200	−0.452600	+	0.891714i	$0.649503\pi$
$828$	−33.5286	−1.16520
$829$	−15.2469	−0.529546	−0.264773	−	0.964311i	$-0.585297\pi$
$829$	−15.2469	−0.529546	−0.264773	+	0.964311i	$0.585297\pi$
$830$	−23.2799	−0.808058
$831$	26.6516	0.924533
$832$	−10.0175	−0.347293
$833$	24.3433	0.843444
$834$	−106.194	−3.67719
$835$	18.8537	0.652459
$836$	−3.05540	−0.105673
$837$	53.9641	1.86527
$838$	−7.92724	−0.273842
$839$	−42.9082	−1.48135	−0.740677	−	0.671861i	$-0.765495\pi$
$839$	−42.9082	−1.48135	−0.740677	+	0.671861i	$0.765495\pi$
$840$	22.0511	0.760836
$841$	10.4258	0.359509
$842$	−39.0791	−1.34676
$843$	−27.6478	−0.952239
$844$	−1.28578	−0.0442582
$845$	38.0271	1.30817
$846$	−21.0049	−0.722163
$847$	18.1303	0.622966
$848$	−21.3663	−0.733723
$849$	−69.8113	−2.39592
$850$	26.0601	0.893855
$851$	−34.8955	−1.19620
$852$	−13.7093	−0.469674
$853$	1.80047	0.0616471	0.0308235	−	0.999525i	$-0.490187\pi$
$853$	1.80047	0.0616471	0.0308235	+	0.999525i	$0.490187\pi$
$854$	−78.9747	−2.70246
$855$	−11.8557	−0.405455
$856$	−8.08080	−0.276196
$857$	−38.1872	−1.30445	−0.652225	−	0.758025i	$-0.726164\pi$
$857$	−38.1872	−1.30445	−0.652225	+	0.758025i	$0.726164\pi$
$858$	86.6490	2.95815
$859$	7.16493	0.244464	0.122232	−	0.992502i	$-0.460995\pi$
$859$	7.16493	0.244464	0.122232	+	0.992502i	$0.460995\pi$
$860$	−15.9538	−0.544021
$861$	−17.0124	−0.579783
$862$	−38.4750	−1.31046
$863$	42.9807	1.46308	0.731540	−	0.681798i	$-0.238802\pi$
$863$	42.9807	1.46308	0.731540	+	0.681798i	$0.238802\pi$
$864$	−97.7082	−3.32410
$865$	−23.1548	−0.787288
$866$	65.5870	2.22874
$867$	−41.4098	−1.40635
$868$	15.2147	0.516422
$869$	−33.1875	−1.12581
$870$	−57.2389	−1.94058
$871$	−34.5097	−1.16932
$872$	−22.1838	−0.751237
$873$	−35.7263	−1.20915
$874$	−6.12452	−0.207165
$875$	39.7497	1.34378
$876$	32.6460	1.10301
$877$	−36.8722	−1.24509	−0.622544	−	0.782585i	$-0.713901\pi$
$877$	−36.8722	−1.24509	−0.622544	+	0.782585i	$0.713901\pi$
$878$	−19.9635	−0.673736
$879$	−61.7474	−2.08269
$880$	−17.9535	−0.605211
$881$	30.8536	1.03948	0.519742	−	0.854323i	$-0.326028\pi$
$881$	30.8536	1.03948	0.519742	+	0.854323i	$0.326028\pi$
$882$	−62.5450	−2.10600
$883$	52.2873	1.75961	0.879804	−	0.475337i	$-0.157674\pi$
$883$	52.2873	1.75961	0.879804	+	0.475337i	$0.157674\pi$
$884$	43.0186	1.44687
$885$	−36.1901	−1.21652
$886$	48.1618	1.61803
$887$	41.6538	1.39860	0.699299	−	0.714829i	$-0.253495\pi$
$887$	41.6538	1.39860	0.699299	+	0.714829i	$0.253495\pi$
$888$	−43.7748	−1.46899
$889$	−71.1195	−2.38527
$890$	−2.58626	−0.0866916
$891$	65.1393	2.18225
$892$	−16.1581	−0.541014
$893$	−1.50142	−0.0502433
$894$	59.7154	1.99718
$895$	25.5177	0.852964
$896$	32.8415	1.09716
$897$	67.9665	2.26934
$898$	−22.2123	−0.741236
$899$	21.9379	0.731668
$900$	−26.2010	−0.873365
$901$	23.6546	0.788049
$902$	6.60904	0.220057
$903$	−89.5614	−2.98042
$904$	21.9281	0.729317
$905$	−23.7259	−0.788676
$906$	132.385	4.39821
$907$	2.47631	0.0822246	0.0411123	−	0.999155i	$-0.486910\pi$
$907$	2.47631	0.0822246	0.0411123	+	0.999155i	$0.486910\pi$
$908$	−28.7560	−0.954302
$909$	1.13944	0.0377927
$910$	57.9468	1.92092
$911$	−30.6483	−1.01542	−0.507711	−	0.861527i	$-0.669508\pi$
$911$	−30.6483	−1.01542	−0.507711	+	0.861527i	$0.669508\pi$
$912$	−16.1017	−0.533181
$913$	19.8672	0.657509
$914$	−12.5023	−0.413540
$915$	64.6929	2.13868
$916$	−32.9604	−1.08904
$917$	25.9384	0.856560
$918$	152.449	5.03155
$919$	−15.1289	−0.499057	−0.249528	−	0.968367i	$-0.580276\pi$
$919$	−15.1289	−0.499057	−0.249528	+	0.968367i	$0.580276\pi$
$920$	−6.71946	−0.221534
$921$	−14.2307	−0.468918
$922$	−48.7918	−1.60687
$923$	20.0117	0.658694
$924$	33.8779	1.11450
$925$	−27.2691	−0.896602
$926$	−13.8248	−0.454312
$927$	59.3220	1.94839
$928$	−39.7210	−1.30391
$929$	−37.4209	−1.22774	−0.613870	−	0.789407i	$-0.710388\pi$
$929$	−37.4209	−1.22774	−0.613870	+	0.789407i	$0.710388\pi$
$930$	−31.8497	−1.04439
$931$	−4.47070	−0.146521
$932$	−18.2808	−0.598809
$933$	19.0779	0.624581
$934$	28.8798	0.944977
$935$	19.8762	0.650022
$936$	61.3959	2.00679
$937$	28.4516	0.929472	0.464736	−	0.885449i	$-0.346149\pi$
$937$	28.4516	0.929472	0.464736	+	0.885449i	$0.346149\pi$
$938$	−34.4800	−1.12581
$939$	−54.2202	−1.76941
$940$	2.96549	0.0967237
$941$	15.9339	0.519430	0.259715	−	0.965685i	$-0.416371\pi$
$941$	15.9339	0.519430	0.259715	+	0.965685i	$0.416371\pi$
$942$	42.8914	1.39748
$943$	5.18406	0.168816
$944$	−35.3937	−1.15197
$945$	80.3570	2.61401
$946$	34.7931	1.13122
$947$	1.78501	0.0580049	0.0290024	−	0.999579i	$-0.490767\pi$
$947$	1.78501	0.0580049	0.0290024	+	0.999579i	$0.490767\pi$
$948$	58.7885	1.90936
$949$	−47.6539	−1.54691
$950$	−4.78600	−0.155278
$951$	−17.9655	−0.582571
$952$	−23.8755	−0.773808
$953$	−31.1978	−1.01060	−0.505298	−	0.862945i	$-0.668618\pi$
$953$	−31.1978	−1.01060	−0.505298	+	0.862945i	$0.668618\pi$
$954$	−60.7756	−1.96768
$955$	−8.73423	−0.282633
$956$	−20.2392	−0.654584
$957$	48.8480	1.57903
$958$	−17.1366	−0.553659
$959$	50.5099	1.63105
$960$	8.19884	0.264617
$961$	−18.7930	−0.606226
$962$	−115.033	−3.70882
$963$	−48.1724	−1.55233
$964$	−17.7820	−0.572719
$965$	19.4300	0.625475
$966$	67.9080	2.18490
$967$	−17.2564	−0.554930	−0.277465	−	0.960736i	$-0.589494\pi$
$967$	−17.2564	−0.554930	−0.277465	+	0.960736i	$0.589494\pi$
$968$	−6.93050	−0.222755
$969$	17.8262	0.572659
$970$	12.8895	0.413858
$971$	35.6886	1.14530	0.572651	−	0.819799i	$-0.305915\pi$
$971$	35.6886	1.14530	0.572651	+	0.819799i	$0.305915\pi$
$972$	−55.8097	−1.79010
$973$	60.6067	1.94296
$974$	−66.9671	−2.14576
$975$	53.1125	1.70096
$976$	63.2692	2.02520
$977$	−9.13340	−0.292203	−0.146102	−	0.989270i	$-0.546673\pi$
$977$	−9.13340	−0.292203	−0.146102	+	0.989270i	$0.546673\pi$
$978$	96.3245	3.08012
$979$	2.20713	0.0705401
$980$	8.83016	0.282069
$981$	−132.245	−4.22226
$982$	67.6341	2.15829
$983$	−7.83634	−0.249940	−0.124970	−	0.992160i	$-0.539884\pi$
$983$	−7.83634	−0.249940	−0.124970	+	0.992160i	$0.539884\pi$
$984$	6.50317	0.207314
$985$	−19.6645	−0.626563
$986$	61.9744	1.97367
$987$	16.6477	0.529901
$988$	−7.90047	−0.251348
$989$	27.2913	0.867813
$990$	−51.0678	−1.62304
$991$	−38.8917	−1.23543	−0.617717	−	0.786400i	$-0.711942\pi$
$991$	−38.8917	−1.23543	−0.617717	+	0.786400i	$0.711942\pi$
$992$	−22.1021	−0.701743
$993$	−32.2331	−1.02289
$994$	19.9945	0.634187
$995$	−0.913499	−0.0289599
$996$	−35.1928	−1.11513
$997$	−29.3556	−0.929702	−0.464851	−	0.885389i	$-0.653892\pi$
$997$	−29.3556	−0.929702	−0.464851	+	0.885389i	$0.653892\pi$
$998$	12.8466	0.406653
$999$	−159.521	−5.04702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.17	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.17	✓	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)

\(f(q)\)	\(=\)	\(q-1.81267 q^{2} -3.27382 q^{3} +1.28578 q^{4} +1.53613 q^{5} +5.93435 q^{6} -3.38684 q^{7} +1.29465 q^{8} +7.71788 q^{9} +O(q^{10})\)	\(q-1.81267 q^{2} -3.27382 q^{3} +1.28578 q^{4} +1.53613 q^{5} +5.93435 q^{6} -3.38684 q^{7} +1.29465 q^{8} +7.71788 q^{9} -2.78450 q^{10} +2.37631 q^{11} -4.20939 q^{12} +6.14452 q^{13} +6.13923 q^{14} -5.02901 q^{15} -4.91833 q^{16} +5.44507 q^{17} -13.9900 q^{18} -1.00000 q^{19} +1.97512 q^{20} +11.0879 q^{21} -4.30746 q^{22} -3.37872 q^{23} -4.23846 q^{24} -2.64031 q^{25} -11.1380 q^{26} -15.4455 q^{27} -4.35472 q^{28} -6.27899 q^{29} +9.11594 q^{30} -3.49385 q^{31} +6.32601 q^{32} -7.77959 q^{33} -9.87012 q^{34} -5.20263 q^{35} +9.92346 q^{36} +10.3280 q^{37} +1.81267 q^{38} -20.1160 q^{39} +1.98876 q^{40} -1.53433 q^{41} -20.0987 q^{42} -8.07740 q^{43} +3.05540 q^{44} +11.8557 q^{45} +6.12452 q^{46} +1.50142 q^{47} +16.1017 q^{48} +4.47070 q^{49} +4.78600 q^{50} -17.8262 q^{51} +7.90047 q^{52} +4.34422 q^{53} +27.9975 q^{54} +3.65031 q^{55} -4.38479 q^{56} +3.27382 q^{57} +11.3817 q^{58} +7.19628 q^{59} -6.46618 q^{60} -12.8639 q^{61} +6.33320 q^{62} -26.1392 q^{63} -1.63031 q^{64} +9.43878 q^{65} +14.1018 q^{66} -5.61633 q^{67} +7.00114 q^{68} +11.0613 q^{69} +9.43065 q^{70} +3.25684 q^{71} +9.99198 q^{72} -7.75551 q^{73} -18.7213 q^{74} +8.64388 q^{75} -1.28578 q^{76} -8.04817 q^{77} +36.4637 q^{78} -13.9660 q^{79} -7.55520 q^{80} +27.4120 q^{81} +2.78123 q^{82} +8.36055 q^{83} +14.2566 q^{84} +8.36433 q^{85} +14.6417 q^{86} +20.5563 q^{87} +3.07649 q^{88} +0.928807 q^{89} -21.4904 q^{90} -20.8105 q^{91} -4.34428 q^{92} +11.4382 q^{93} -2.72159 q^{94} -1.53613 q^{95} -20.7102 q^{96} -4.62904 q^{97} -8.10391 q^{98} +18.3400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 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