LMFDB - Embedded newform 4010.2.a.n.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, 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("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.103543 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.103543 q^{6} +2.39084 q^{7} +1.00000 q^{8} -2.98928 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.103543 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.103543 q^{6} +2.39084 q^{7} +1.00000 q^{8} -2.98928 q^{9} +1.00000 q^{10} +3.58736 q^{11} -0.103543 q^{12} -3.15486 q^{13} +2.39084 q^{14} -0.103543 q^{15} +1.00000 q^{16} +3.91846 q^{17} -2.98928 q^{18} +4.85015 q^{19} +1.00000 q^{20} -0.247555 q^{21} +3.58736 q^{22} -3.16219 q^{23} -0.103543 q^{24} +1.00000 q^{25} -3.15486 q^{26} +0.620148 q^{27} +2.39084 q^{28} -1.07235 q^{29} -0.103543 q^{30} +8.56460 q^{31} +1.00000 q^{32} -0.371446 q^{33} +3.91846 q^{34} +2.39084 q^{35} -2.98928 q^{36} -7.23809 q^{37} +4.85015 q^{38} +0.326663 q^{39} +1.00000 q^{40} +12.5691 q^{41} -0.247555 q^{42} -12.9209 q^{43} +3.58736 q^{44} -2.98928 q^{45} -3.16219 q^{46} +4.73294 q^{47} -0.103543 q^{48} -1.28388 q^{49} +1.00000 q^{50} -0.405729 q^{51} -3.15486 q^{52} -1.93024 q^{53} +0.620148 q^{54} +3.58736 q^{55} +2.39084 q^{56} -0.502199 q^{57} -1.07235 q^{58} +5.45283 q^{59} -0.103543 q^{60} -2.79150 q^{61} +8.56460 q^{62} -7.14689 q^{63} +1.00000 q^{64} -3.15486 q^{65} -0.371446 q^{66} +6.53368 q^{67} +3.91846 q^{68} +0.327422 q^{69} +2.39084 q^{70} +9.05407 q^{71} -2.98928 q^{72} -3.79835 q^{73} -7.23809 q^{74} -0.103543 q^{75} +4.85015 q^{76} +8.57681 q^{77} +0.326663 q^{78} +0.480807 q^{79} +1.00000 q^{80} +8.90362 q^{81} +12.5691 q^{82} +16.4106 q^{83} -0.247555 q^{84} +3.91846 q^{85} -12.9209 q^{86} +0.111034 q^{87} +3.58736 q^{88} +1.58419 q^{89} -2.98928 q^{90} -7.54276 q^{91} -3.16219 q^{92} -0.886804 q^{93} +4.73294 q^{94} +4.85015 q^{95} -0.103543 q^{96} +6.24326 q^{97} -1.28388 q^{98} -10.7236 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 22 q^{2} + q^{3} + 22 q^{4} + 22 q^{5} + q^{6} + 22 q^{8} + 43 q^{9}+O(q^{10}) $ 22 * q + 22 * q^2 + q^3 + 22 * q^4 + 22 * q^5 + q^6 + 22 * q^8 + 43 * q^9	\( 22 q + 22 q^{2} + q^{3} + 22 q^{4} + 22 q^{5} + q^{6} + 22 q^{8} + 43 q^{9} + 22 q^{10} + 12 q^{11} + q^{12} + 10 q^{13} + q^{15} + 22 q^{16} + 24 q^{17} + 43 q^{18} + 13 q^{19} + 22 q^{20} + 13 q^{21} + 12 q^{22} + 7 q^{23} + q^{24} + 22 q^{25} + 10 q^{26} - 5 q^{27} + 22 q^{29} + q^{30} + 14 q^{31} + 22 q^{32} + 31 q^{33} + 24 q^{34} + 43 q^{36} + 35 q^{37} + 13 q^{38} + 4 q^{39} + 22 q^{40} + 29 q^{41} + 13 q^{42} + 7 q^{43} + 12 q^{44} + 43 q^{45} + 7 q^{46} - 21 q^{47} + q^{48} + 32 q^{49} + 22 q^{50} - 6 q^{51} + 10 q^{52} + 29 q^{53} - 5 q^{54} + 12 q^{55} - 13 q^{57} + 22 q^{58} + 12 q^{59} + q^{60} + 24 q^{61} + 14 q^{62} - 8 q^{63} + 22 q^{64} + 10 q^{65} + 31 q^{66} + 25 q^{67} + 24 q^{68} + 3 q^{69} + 31 q^{71} + 43 q^{72} + 30 q^{73} + 35 q^{74} + q^{75} + 13 q^{76} + 10 q^{77} + 4 q^{78} + 35 q^{79} + 22 q^{80} + 74 q^{81} + 29 q^{82} - 33 q^{83} + 13 q^{84} + 24 q^{85} + 7 q^{86} - 24 q^{87} + 12 q^{88} + 38 q^{89} + 43 q^{90} - 32 q^{91} + 7 q^{92} + 3 q^{93} - 21 q^{94} + 13 q^{95} + q^{96} + 11 q^{97} + 32 q^{98} - 41 q^{99}+O(q^{100}) \) 22 * q + 22 * q^2 + q^3 + 22 * q^4 + 22 * q^5 + q^6 + 22 * q^8 + 43 * q^9 + 22 * q^10 + 12 * q^11 + q^12 + 10 * q^13 + q^15 + 22 * q^16 + 24 * q^17 + 43 * q^18 + 13 * q^19 + 22 * q^20 + 13 * q^21 + 12 * q^22 + 7 * q^23 + q^24 + 22 * q^25 + 10 * q^26 - 5 * q^27 + 22 * q^29 + q^30 + 14 * q^31 + 22 * q^32 + 31 * q^33 + 24 * q^34 + 43 * q^36 + 35 * q^37 + 13 * q^38 + 4 * q^39 + 22 * q^40 + 29 * q^41 + 13 * q^42 + 7 * q^43 + 12 * q^44 + 43 * q^45 + 7 * q^46 - 21 * q^47 + q^48 + 32 * q^49 + 22 * q^50 - 6 * q^51 + 10 * q^52 + 29 * q^53 - 5 * q^54 + 12 * q^55 - 13 * q^57 + 22 * q^58 + 12 * q^59 + q^60 + 24 * q^61 + 14 * q^62 - 8 * q^63 + 22 * q^64 + 10 * q^65 + 31 * q^66 + 25 * q^67 + 24 * q^68 + 3 * q^69 + 31 * q^71 + 43 * q^72 + 30 * q^73 + 35 * q^74 + q^75 + 13 * q^76 + 10 * q^77 + 4 * q^78 + 35 * q^79 + 22 * q^80 + 74 * q^81 + 29 * q^82 - 33 * q^83 + 13 * q^84 + 24 * q^85 + 7 * q^86 - 24 * q^87 + 12 * q^88 + 38 * q^89 + 43 * q^90 - 32 * q^91 + 7 * q^92 + 3 * q^93 - 21 * q^94 + 13 * q^95 + q^96 + 11 * q^97 + 32 * q^98 - 41 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.103543	−0.0597806	−0.0298903	−	0.999553i	$-0.509516\pi$
$3$	−0.103543	−0.0597806	−0.0298903	+	0.999553i	$0.509516\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.103543	−0.0422712
$7$	2.39084	0.903653	0.451826	−	0.892106i	$-0.350773\pi$
$7$	2.39084	0.903653	0.451826	+	0.892106i	$0.350773\pi$
$8$	1.00000	0.353553
$9$	−2.98928	−0.996426
$10$	1.00000	0.316228
$11$	3.58736	1.08163	0.540815	−	0.841141i	$-0.318116\pi$
$11$	3.58736	1.08163	0.540815	+	0.841141i	$0.318116\pi$
$12$	−0.103543	−0.0298903
$13$	−3.15486	−0.875000	−0.437500	−	0.899218i	$-0.644136\pi$
$13$	−3.15486	−0.875000	−0.437500	+	0.899218i	$0.644136\pi$
$14$	2.39084	0.638979
$15$	−0.103543	−0.0267347
$16$	1.00000	0.250000
$17$	3.91846	0.950367	0.475183	−	0.879887i	$-0.342382\pi$
$17$	3.91846	0.950367	0.475183	+	0.879887i	$0.342382\pi$
$18$	−2.98928	−0.704580
$19$	4.85015	1.11270	0.556351	−	0.830947i	$-0.312201\pi$
$19$	4.85015	1.11270	0.556351	+	0.830947i	$0.312201\pi$
$20$	1.00000	0.223607
$21$	−0.247555	−0.0540209
$22$	3.58736	0.764828
$23$	−3.16219	−0.659362	−0.329681	−	0.944092i	$-0.606941\pi$
$23$	−3.16219	−0.659362	−0.329681	+	0.944092i	$0.606941\pi$
$24$	−0.103543	−0.0211356
$25$	1.00000	0.200000
$26$	−3.15486	−0.618718
$27$	0.620148	0.119347
$28$	2.39084	0.451826
$29$	−1.07235	−0.199130	−0.0995650	−	0.995031i	$-0.531745\pi$
$29$	−1.07235	−0.199130	−0.0995650	+	0.995031i	$0.531745\pi$
$30$	−0.103543	−0.0189043
$31$	8.56460	1.53825	0.769124	−	0.639100i	$-0.220693\pi$
$31$	8.56460	1.53825	0.769124	+	0.639100i	$0.220693\pi$
$32$	1.00000	0.176777
$33$	−0.371446	−0.0646605
$34$	3.91846	0.672011
$35$	2.39084	0.404126
$36$	−2.98928	−0.498213
$37$	−7.23809	−1.18993	−0.594967	−	0.803750i	$-0.702835\pi$
$37$	−7.23809	−1.18993	−0.594967	+	0.803750i	$0.702835\pi$
$38$	4.85015	0.786799
$39$	0.326663	0.0523080
$40$	1.00000	0.158114
$41$	12.5691	1.96297	0.981485	−	0.191539i	$-0.0613478\pi$
$41$	12.5691	1.96297	0.981485	+	0.191539i	$0.0613478\pi$
$42$	−0.247555	−0.0381985
$43$	−12.9209	−1.97042	−0.985212	−	0.171343i	$-0.945190\pi$
$43$	−12.9209	−1.97042	−0.985212	+	0.171343i	$0.945190\pi$
$44$	3.58736	0.540815
$45$	−2.98928	−0.445615
$46$	−3.16219	−0.466239
$47$	4.73294	0.690371	0.345185	−	0.938535i	$-0.387816\pi$
$47$	4.73294	0.690371	0.345185	+	0.938535i	$0.387816\pi$
$48$	−0.103543	−0.0149451
$49$	−1.28388	−0.183412
$50$	1.00000	0.141421
$51$	−0.405729	−0.0568135
$52$	−3.15486	−0.437500
$53$	−1.93024	−0.265139	−0.132569	−	0.991174i	$-0.542323\pi$
$53$	−1.93024	−0.265139	−0.132569	+	0.991174i	$0.542323\pi$
$54$	0.620148	0.0843914
$55$	3.58736	0.483720
$56$	2.39084	0.319489
$57$	−0.502199	−0.0665179
$58$	−1.07235	−0.140806
$59$	5.45283	0.709899	0.354949	−	0.934886i	$-0.384498\pi$
$59$	5.45283	0.709899	0.354949	+	0.934886i	$0.384498\pi$
$60$	−0.103543	−0.0133673
$61$	−2.79150	−0.357415	−0.178708	−	0.983902i	$-0.557192\pi$
$61$	−2.79150	−0.357415	−0.178708	+	0.983902i	$0.557192\pi$
$62$	8.56460	1.08771
$63$	−7.14689	−0.900423
$64$	1.00000	0.125000
$65$	−3.15486	−0.391312
$66$	−0.371446	−0.0457219
$67$	6.53368	0.798217	0.399108	−	0.916904i	$-0.369320\pi$
$67$	6.53368	0.798217	0.399108	+	0.916904i	$0.369320\pi$
$68$	3.91846	0.475183
$69$	0.327422	0.0394170
$70$	2.39084	0.285760
$71$	9.05407	1.07452	0.537260	−	0.843416i	$-0.319459\pi$
$71$	9.05407	1.07452	0.537260	+	0.843416i	$0.319459\pi$
$72$	−2.98928	−0.352290
$73$	−3.79835	−0.444563	−0.222281	−	0.974983i	$-0.571350\pi$
$73$	−3.79835	−0.444563	−0.222281	+	0.974983i	$0.571350\pi$
$74$	−7.23809	−0.841411
$75$	−0.103543	−0.0119561
$76$	4.85015	0.556351
$77$	8.57681	0.977418
$78$	0.326663	0.0369873
$79$	0.480807	0.0540950	0.0270475	−	0.999634i	$-0.491389\pi$
$79$	0.480807	0.0540950	0.0270475	+	0.999634i	$0.491389\pi$
$80$	1.00000	0.111803
$81$	8.90362	0.989292
$82$	12.5691	1.38803
$83$	16.4106	1.80129	0.900647	−	0.434551i	$-0.143093\pi$
$83$	16.4106	1.80129	0.900647	+	0.434551i	$0.143093\pi$
$84$	−0.247555	−0.0270104
$85$	3.91846	0.425017
$86$	−12.9209	−1.39330
$87$	0.111034	0.0119041
$88$	3.58736	0.382414
$89$	1.58419	0.167924	0.0839621	−	0.996469i	$-0.473243\pi$
$89$	1.58419	0.167924	0.0839621	+	0.996469i	$0.473243\pi$
$90$	−2.98928	−0.315098
$91$	−7.54276	−0.790696
$92$	−3.16219	−0.329681
$93$	−0.886804	−0.0919573
$94$	4.73294	0.488166
$95$	4.85015	0.497615
$96$	−0.103543	−0.0105678
$97$	6.24326	0.633907	0.316953	−	0.948441i	$-0.397340\pi$
$97$	6.24326	0.633907	0.316953	+	0.948441i	$0.397340\pi$
$98$	−1.28388	−0.129692
$99$	−10.7236	−1.07777
$100$	1.00000	0.100000
$101$	−6.33364	−0.630221	−0.315110	−	0.949055i	$-0.602042\pi$
$101$	−6.33364	−0.630221	−0.315110	+	0.949055i	$0.602042\pi$
$102$	−0.405729	−0.0401732
$103$	15.5270	1.52992	0.764962	−	0.644076i	$-0.222758\pi$
$103$	15.5270	1.52992	0.764962	+	0.644076i	$0.222758\pi$
$104$	−3.15486	−0.309359
$105$	−0.247555	−0.0241589
$106$	−1.93024	−0.187481
$107$	−8.84625	−0.855199	−0.427600	−	0.903968i	$-0.640641\pi$
$107$	−8.84625	−0.855199	−0.427600	+	0.903968i	$0.640641\pi$
$108$	0.620148	0.0596737
$109$	−13.0461	−1.24959	−0.624795	−	0.780789i	$-0.714817\pi$
$109$	−13.0461	−1.24959	−0.624795	+	0.780789i	$0.714817\pi$
$110$	3.58736	0.342042
$111$	0.749453	0.0711349
$112$	2.39084	0.225913
$113$	−18.8854	−1.77659	−0.888293	−	0.459277i	$-0.848109\pi$
$113$	−18.8854	−1.77659	−0.888293	+	0.459277i	$0.848109\pi$
$114$	−0.502199	−0.0470353
$115$	−3.16219	−0.294876
$116$	−1.07235	−0.0995650
$117$	9.43075	0.871873
$118$	5.45283	0.501974
$119$	9.36842	0.858801
$120$	−0.103543	−0.00945214
$121$	1.86918	0.169925
$122$	−2.79150	−0.252731
$123$	−1.30145	−0.117347
$124$	8.56460	0.769124
$125$	1.00000	0.0894427
$126$	−7.14689	−0.636695
$127$	7.97822	0.707953	0.353976	−	0.935254i	$-0.384829\pi$
$127$	7.97822	0.707953	0.353976	+	0.935254i	$0.384829\pi$
$128$	1.00000	0.0883883
$129$	1.33787	0.117793
$130$	−3.15486	−0.276699
$131$	−4.83563	−0.422491	−0.211245	−	0.977433i	$-0.567752\pi$
$131$	−4.83563	−0.422491	−0.211245	+	0.977433i	$0.567752\pi$
$132$	−0.371446	−0.0323303
$133$	11.5959	1.00550
$134$	6.53368	0.564424
$135$	0.620148	0.0533738
$136$	3.91846	0.336005
$137$	1.75116	0.149612	0.0748058	−	0.997198i	$-0.476166\pi$
$137$	1.75116	0.149612	0.0748058	+	0.997198i	$0.476166\pi$
$138$	0.327422	0.0278720
$139$	−5.64685	−0.478960	−0.239480	−	0.970901i	$-0.576977\pi$
$139$	−5.64685	−0.478960	−0.239480	+	0.970901i	$0.576977\pi$
$140$	2.39084	0.202063
$141$	−0.490063	−0.0412708
$142$	9.05407	0.759801
$143$	−11.3176	−0.946427
$144$	−2.98928	−0.249107
$145$	−1.07235	−0.0890536
$146$	−3.79835	−0.314353
$147$	0.132937	0.0109645
$148$	−7.23809	−0.594967
$149$	2.06358	0.169055	0.0845275	−	0.996421i	$-0.473062\pi$
$149$	2.06358	0.169055	0.0845275	+	0.996421i	$0.473062\pi$
$150$	−0.103543	−0.00845425
$151$	3.48738	0.283799	0.141900	−	0.989881i	$-0.454679\pi$
$151$	3.48738	0.283799	0.141900	+	0.989881i	$0.454679\pi$
$152$	4.85015	0.393400
$153$	−11.7134	−0.946970
$154$	8.57681	0.691139
$155$	8.56460	0.687925
$156$	0.326663	0.0261540
$157$	5.94415	0.474395	0.237197	−	0.971461i	$-0.423771\pi$
$157$	5.94415	0.474395	0.237197	+	0.971461i	$0.423771\pi$
$158$	0.480807	0.0382510
$159$	0.199863	0.0158501
$160$	1.00000	0.0790569
$161$	−7.56028	−0.595834
$162$	8.90362	0.699535
$163$	18.3183	1.43480	0.717400	−	0.696661i	$-0.245332\pi$
$163$	18.3183	1.43480	0.717400	+	0.696661i	$0.245332\pi$
$164$	12.5691	0.981485
$165$	−0.371446	−0.0289171
$166$	16.4106	1.27371
$167$	4.02051	0.311116	0.155558	−	0.987827i	$-0.450282\pi$
$167$	4.02051	0.311116	0.155558	+	0.987827i	$0.450282\pi$
$168$	−0.247555	−0.0190993
$169$	−3.04687	−0.234375
$170$	3.91846	0.300532
$171$	−14.4985	−1.10873
$172$	−12.9209	−0.985212
$173$	−6.81384	−0.518047	−0.259023	−	0.965871i	$-0.583401\pi$
$173$	−6.81384	−0.518047	−0.259023	+	0.965871i	$0.583401\pi$
$174$	0.111034	0.00841747
$175$	2.39084	0.180731
$176$	3.58736	0.270408
$177$	−0.564603	−0.0424381
$178$	1.58419	0.118740
$179$	−7.68797	−0.574626	−0.287313	−	0.957837i	$-0.592762\pi$
$179$	−7.68797	−0.574626	−0.287313	+	0.957837i	$0.592762\pi$
$180$	−2.98928	−0.222808
$181$	2.18647	0.162519	0.0812595	−	0.996693i	$-0.474106\pi$
$181$	2.18647	0.162519	0.0812595	+	0.996693i	$0.474106\pi$
$182$	−7.54276	−0.559107
$183$	0.289041	0.0213665
$184$	−3.16219	−0.233120
$185$	−7.23809	−0.532155
$186$	−0.886804	−0.0650236
$187$	14.0570	1.02795
$188$	4.73294	0.345185
$189$	1.48267	0.107849
$190$	4.85015	0.351867
$191$	16.6168	1.20235	0.601175	−	0.799117i	$-0.294699\pi$
$191$	16.6168	1.20235	0.601175	+	0.799117i	$0.294699\pi$
$192$	−0.103543	−0.00747257
$193$	−19.0101	−1.36838	−0.684189	−	0.729304i	$-0.739844\pi$
$193$	−19.0101	−1.36838	−0.684189	+	0.729304i	$0.739844\pi$
$194$	6.24326	0.448240
$195$	0.326663	0.0233928
$196$	−1.28388	−0.0917061
$197$	−11.4808	−0.817970	−0.408985	−	0.912541i	$-0.634117\pi$
$197$	−11.4808	−0.817970	−0.408985	+	0.912541i	$0.634117\pi$
$198$	−10.7236	−0.762095
$199$	−14.5148	−1.02893	−0.514463	−	0.857513i	$-0.672009\pi$
$199$	−14.5148	−1.02893	−0.514463	+	0.857513i	$0.672009\pi$
$200$	1.00000	0.0707107
$201$	−0.676517	−0.0477178
$202$	−6.33364	−0.445634
$203$	−2.56381	−0.179944
$204$	−0.405729	−0.0284067
$205$	12.5691	0.877867
$206$	15.5270	1.08182
$207$	9.45266	0.657005
$208$	−3.15486	−0.218750
$209$	17.3993	1.20353
$210$	−0.247555	−0.0170829
$211$	−15.8897	−1.09389	−0.546945	−	0.837168i	$-0.684209\pi$
$211$	−15.8897	−1.09389	−0.546945	+	0.837168i	$0.684209\pi$
$212$	−1.93024	−0.132569
$213$	−0.937486	−0.0642355
$214$	−8.84625	−0.604717
$215$	−12.9209	−0.881200
$216$	0.620148	0.0421957
$217$	20.4766	1.39004
$218$	−13.0461	−0.883593
$219$	0.393292	0.0265762
$220$	3.58736	0.241860
$221$	−12.3622	−0.831571
$222$	0.749453	0.0503000
$223$	18.1323	1.21423	0.607113	−	0.794616i	$-0.292328\pi$
$223$	18.1323	1.21423	0.607113	+	0.794616i	$0.292328\pi$
$224$	2.39084	0.159745
$225$	−2.98928	−0.199285
$226$	−18.8854	−1.25624
$227$	−19.0813	−1.26647	−0.633234	−	0.773960i	$-0.718273\pi$
$227$	−19.0813	−1.26647	−0.633234	+	0.773960i	$0.718273\pi$
$228$	−0.502199	−0.0332590
$229$	27.8705	1.84173	0.920866	−	0.389878i	$-0.127483\pi$
$229$	27.8705	1.84173	0.920866	+	0.389878i	$0.127483\pi$
$230$	−3.16219	−0.208508
$231$	−0.888069	−0.0584306
$232$	−1.07235	−0.0704031
$233$	1.22179	0.0800421	0.0400210	−	0.999199i	$-0.487258\pi$
$233$	1.22179	0.0800421	0.0400210	+	0.999199i	$0.487258\pi$
$234$	9.43075	0.616507
$235$	4.73294	0.308743
$236$	5.45283	0.354949
$237$	−0.0497842	−0.00323383
$238$	9.36842	0.607264
$239$	−0.336310	−0.0217541	−0.0108770	−	0.999941i	$-0.503462\pi$
$239$	−0.336310	−0.0217541	−0.0108770	+	0.999941i	$0.503462\pi$
$240$	−0.103543	−0.00668367
$241$	17.5576	1.13099	0.565493	−	0.824753i	$-0.308686\pi$
$241$	17.5576	1.13099	0.565493	+	0.824753i	$0.308686\pi$
$242$	1.86918	0.120155
$243$	−2.78235	−0.178488
$244$	−2.79150	−0.178708
$245$	−1.28388	−0.0820244
$246$	−1.30145	−0.0829772
$247$	−15.3015	−0.973614
$248$	8.56460	0.543853
$249$	−1.69920	−0.107682
$250$	1.00000	0.0632456
$251$	−1.47659	−0.0932013	−0.0466007	−	0.998914i	$-0.514839\pi$
$251$	−1.47659	−0.0932013	−0.0466007	+	0.998914i	$0.514839\pi$
$252$	−7.14689	−0.450212
$253$	−11.3439	−0.713186
$254$	7.97822	0.500598
$255$	−0.405729	−0.0254078
$256$	1.00000	0.0625000
$257$	−14.9720	−0.933928	−0.466964	−	0.884276i	$-0.654652\pi$
$257$	−14.9720	−0.933928	−0.466964	+	0.884276i	$0.654652\pi$
$258$	1.33787	0.0832922
$259$	−17.3051	−1.07529
$260$	−3.15486	−0.195656
$261$	3.20555	0.198418
$262$	−4.83563	−0.298746
$263$	22.3328	1.37710	0.688551	−	0.725188i	$-0.258247\pi$
$263$	22.3328	1.37710	0.688551	+	0.725188i	$0.258247\pi$
$264$	−0.371446	−0.0228609
$265$	−1.93024	−0.118574
$266$	11.5959	0.710993
$267$	−0.164032	−0.0100386
$268$	6.53368	0.399108
$269$	11.9129	0.726344	0.363172	−	0.931722i	$-0.381694\pi$
$269$	11.9129	0.726344	0.363172	+	0.931722i	$0.381694\pi$
$270$	0.620148	0.0377410
$271$	19.1585	1.16379	0.581897	−	0.813262i	$-0.302311\pi$
$271$	19.1585	1.16379	0.581897	+	0.813262i	$0.302311\pi$
$272$	3.91846	0.237592
$273$	0.781000	0.0472683
$274$	1.75116	0.105791
$275$	3.58736	0.216326
$276$	0.327422	0.0197085
$277$	−9.81408	−0.589671	−0.294836	−	0.955548i	$-0.595265\pi$
$277$	−9.81408	−0.589671	−0.294836	+	0.955548i	$0.595265\pi$
$278$	−5.64685	−0.338676
$279$	−25.6020	−1.53275
$280$	2.39084	0.142880
$281$	−19.2782	−1.15004	−0.575020	−	0.818139i	$-0.695006\pi$
$281$	−19.2782	−1.15004	−0.575020	+	0.818139i	$0.695006\pi$
$282$	−0.490063	−0.0291828
$283$	−5.75699	−0.342218	−0.171109	−	0.985252i	$-0.554735\pi$
$283$	−5.75699	−0.342218	−0.171109	+	0.985252i	$0.554735\pi$
$284$	9.05407	0.537260
$285$	−0.502199	−0.0297477
$286$	−11.3176	−0.669225
$287$	30.0508	1.77384
$288$	−2.98928	−0.176145
$289$	−1.64565	−0.0968030
$290$	−1.07235	−0.0629704
$291$	−0.646445	−0.0378953
$292$	−3.79835	−0.222281
$293$	−18.3702	−1.07320	−0.536600	−	0.843837i	$-0.680292\pi$
$293$	−18.3702	−1.07320	−0.536600	+	0.843837i	$0.680292\pi$
$294$	0.132937	0.00775306
$295$	5.45283	0.317476
$296$	−7.23809	−0.420705
$297$	2.22470	0.129090
$298$	2.06358	0.119540
$299$	9.97625	0.576942
$300$	−0.103543	−0.00597806
$301$	−30.8919	−1.78058
$302$	3.48738	0.200676
$303$	0.655804	0.0376750
$304$	4.85015	0.278175
$305$	−2.79150	−0.159841
$306$	−11.7134	−0.669609
$307$	−13.5880	−0.775510	−0.387755	−	0.921762i	$-0.626749\pi$
$307$	−13.5880	−0.775510	−0.387755	+	0.921762i	$0.626749\pi$
$308$	8.57681	0.488709
$309$	−1.60772	−0.0914597
$310$	8.56460	0.486437
$311$	1.14835	0.0651168	0.0325584	−	0.999470i	$-0.489635\pi$
$311$	1.14835	0.0651168	0.0325584	+	0.999470i	$0.489635\pi$
$312$	0.326663	0.0184937
$313$	9.72942	0.549940	0.274970	−	0.961453i	$-0.411332\pi$
$313$	9.72942	0.549940	0.274970	+	0.961453i	$0.411332\pi$
$314$	5.94415	0.335448
$315$	−7.14689	−0.402681
$316$	0.480807	0.0270475
$317$	8.82275	0.495535	0.247768	−	0.968819i	$-0.420303\pi$
$317$	8.82275	0.495535	0.247768	+	0.968819i	$0.420303\pi$
$318$	0.199863	0.0112077
$319$	−3.84690	−0.215385
$320$	1.00000	0.0559017
$321$	0.915967	0.0511243
$322$	−7.56028	−0.421318
$323$	19.0052	1.05747
$324$	8.90362	0.494646
$325$	−3.15486	−0.175000
$326$	18.3183	1.01456
$327$	1.35083	0.0747012
$328$	12.5691	0.694015
$329$	11.3157	0.623855
$330$	−0.371446	−0.0204474
$331$	−14.9145	−0.819772	−0.409886	−	0.912137i	$-0.634432\pi$
$331$	−14.9145	−0.819772	−0.409886	+	0.912137i	$0.634432\pi$
$332$	16.4106	0.900647
$333$	21.6367	1.18568
$334$	4.02051	0.219992
$335$	6.53368	0.356973
$336$	−0.247555	−0.0135052
$337$	−0.335498	−0.0182757	−0.00913786	−	0.999958i	$-0.502909\pi$
$337$	−0.335498	−0.0182757	−0.00913786	+	0.999958i	$0.502909\pi$
$338$	−3.04687	−0.165728
$339$	1.95545	0.106205
$340$	3.91846	0.212508
$341$	30.7243	1.66382
$342$	−14.4985	−0.783987
$343$	−19.8054	−1.06939
$344$	−12.9209	−0.696650
$345$	0.327422	0.0176278
$346$	−6.81384	−0.366314
$347$	0.998566	0.0536058	0.0268029	−	0.999641i	$-0.491467\pi$
$347$	0.998566	0.0536058	0.0268029	+	0.999641i	$0.491467\pi$
$348$	0.111034	0.00595205
$349$	12.6314	0.676144	0.338072	−	0.941120i	$-0.390225\pi$
$349$	12.6314	0.676144	0.338072	+	0.941120i	$0.390225\pi$
$350$	2.39084	0.127796
$351$	−1.95648	−0.104429
$352$	3.58736	0.191207
$353$	25.3186	1.34757	0.673786	−	0.738927i	$-0.264667\pi$
$353$	25.3186	1.34757	0.673786	+	0.738927i	$0.264667\pi$
$354$	−0.564603	−0.0300083
$355$	9.05407	0.480540
$356$	1.58419	0.0839621
$357$	−0.970034	−0.0513396
$358$	−7.68797	−0.406322
$359$	32.0380	1.69090	0.845451	−	0.534052i	$-0.179331\pi$
$359$	32.0380	1.69090	0.845451	+	0.534052i	$0.179331\pi$
$360$	−2.98928	−0.157549
$361$	4.52400	0.238105
$362$	2.18647	0.114918
$363$	−0.193540	−0.0101582
$364$	−7.54276	−0.395348
$365$	−3.79835	−0.198814
$366$	0.289041	0.0151084
$367$	−27.4318	−1.43193	−0.715965	−	0.698137i	$-0.754013\pi$
$367$	−27.4318	−1.43193	−0.715965	+	0.698137i	$0.754013\pi$
$368$	−3.16219	−0.164840
$369$	−37.5727	−1.95595
$370$	−7.23809	−0.376290
$371$	−4.61489	−0.239593
$372$	−0.886804	−0.0459787
$373$	9.29220	0.481132	0.240566	−	0.970633i	$-0.422667\pi$
$373$	9.29220	0.481132	0.240566	+	0.970633i	$0.422667\pi$
$374$	14.0570	0.726868
$375$	−0.103543	−0.00534694
$376$	4.73294	0.244083
$377$	3.38310	0.174239
$378$	1.48267	0.0762605
$379$	3.45354	0.177396	0.0886982	−	0.996059i	$-0.471729\pi$
$379$	3.45354	0.177396	0.0886982	+	0.996059i	$0.471729\pi$
$380$	4.85015	0.248808
$381$	−0.826089	−0.0423218
$382$	16.6168	0.850190
$383$	−24.0180	−1.22726	−0.613630	−	0.789594i	$-0.710291\pi$
$383$	−24.0180	−1.22726	−0.613630	+	0.789594i	$0.710291\pi$
$384$	−0.103543	−0.00528391
$385$	8.57681	0.437115
$386$	−19.0101	−0.967590
$387$	38.6243	1.96338
$388$	6.24326	0.316953
$389$	−2.45579	−0.124513	−0.0622567	−	0.998060i	$-0.519830\pi$
$389$	−2.45579	−0.124513	−0.0622567	+	0.998060i	$0.519830\pi$
$390$	0.326663	0.0165412
$391$	−12.3909	−0.626635
$392$	−1.28388	−0.0648460
$393$	0.500695	0.0252567
$394$	−11.4808	−0.578392
$395$	0.480807	0.0241920
$396$	−10.7236	−0.538883
$397$	34.4051	1.72674	0.863371	−	0.504570i	$-0.168349\pi$
$397$	34.4051	1.72674	0.863371	+	0.504570i	$0.168349\pi$
$398$	−14.5148	−0.727560
$399$	−1.20068	−0.0601091
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−0.676517	−0.0337416
$403$	−27.0201	−1.34597
$404$	−6.33364	−0.315110
$405$	8.90362	0.442425
$406$	−2.56381	−0.127240
$407$	−25.9656	−1.28707
$408$	−0.405729	−0.0200866
$409$	−25.4710	−1.25946	−0.629730	−	0.776814i	$-0.716835\pi$
$409$	−25.4710	−1.25946	−0.629730	+	0.776814i	$0.716835\pi$
$410$	12.5691	0.620746
$411$	−0.181320	−0.00894387
$412$	15.5270	0.764962
$413$	13.0369	0.641502
$414$	9.45266	0.464573
$415$	16.4106	0.805563
$416$	−3.15486	−0.154680
$417$	0.584692	0.0286325
$418$	17.3993	0.851026
$419$	−27.8568	−1.36089	−0.680446	−	0.732798i	$-0.738214\pi$
$419$	−27.8568	−1.36089	−0.680446	+	0.732798i	$0.738214\pi$
$420$	−0.247555	−0.0120794
$421$	8.28299	0.403688	0.201844	−	0.979418i	$-0.435307\pi$
$421$	8.28299	0.403688	0.201844	+	0.979418i	$0.435307\pi$
$422$	−15.8897	−0.773498
$423$	−14.1481	−0.687904
$424$	−1.93024	−0.0937407
$425$	3.91846	0.190073
$426$	−0.937486	−0.0454213
$427$	−6.67404	−0.322979
$428$	−8.84625	−0.427600
$429$	1.17186	0.0565779
$430$	−12.9209	−0.623102
$431$	18.4328	0.887877	0.443938	−	0.896057i	$-0.353581\pi$
$431$	18.4328	0.887877	0.443938	+	0.896057i	$0.353581\pi$
$432$	0.620148	0.0298369
$433$	−12.4895	−0.600206	−0.300103	−	0.953907i	$-0.597021\pi$
$433$	−12.4895	−0.600206	−0.300103	+	0.953907i	$0.597021\pi$
$434$	20.4766	0.982908
$435$	0.111034	0.00532368
$436$	−13.0461	−0.624795
$437$	−15.3371	−0.733673
$438$	0.393292	0.0187922
$439$	−32.1011	−1.53210	−0.766052	−	0.642779i	$-0.777781\pi$
$439$	−32.1011	−1.53210	−0.766052	+	0.642779i	$0.777781\pi$
$440$	3.58736	0.171021
$441$	3.83789	0.182757
$442$	−12.3622	−0.588009
$443$	−13.8794	−0.659431	−0.329716	−	0.944080i	$-0.606953\pi$
$443$	−13.8794	−0.659431	−0.329716	+	0.944080i	$0.606953\pi$
$444$	0.749453	0.0355675
$445$	1.58419	0.0750980
$446$	18.1323	0.858587
$447$	−0.213669	−0.0101062
$448$	2.39084	0.112957
$449$	28.8811	1.36298	0.681491	−	0.731826i	$-0.261332\pi$
$449$	28.8811	1.36298	0.681491	+	0.731826i	$0.261332\pi$
$450$	−2.98928	−0.140916
$451$	45.0901	2.12321
$452$	−18.8854	−0.888293
$453$	−0.361094	−0.0169657
$454$	−19.0813	−0.895529
$455$	−7.54276	−0.353610
$456$	−0.502199	−0.0235176
$457$	−0.825716	−0.0386254	−0.0193127	−	0.999813i	$-0.506148\pi$
$457$	−0.825716	−0.0386254	−0.0193127	+	0.999813i	$0.506148\pi$
$458$	27.8705	1.30230
$459$	2.43003	0.113424
$460$	−3.16219	−0.147438
$461$	−17.8190	−0.829912	−0.414956	−	0.909842i	$-0.636203\pi$
$461$	−17.8190	−0.829912	−0.414956	+	0.909842i	$0.636203\pi$
$462$	−0.888069	−0.0413167
$463$	−23.6748	−1.10026	−0.550131	−	0.835078i	$-0.685422\pi$
$463$	−23.6748	−1.10026	−0.550131	+	0.835078i	$0.685422\pi$
$464$	−1.07235	−0.0497825
$465$	−0.886804	−0.0411246
$466$	1.22179	0.0565983
$467$	−21.6743	−1.00297	−0.501483	−	0.865168i	$-0.667212\pi$
$467$	−21.6743	−1.00297	−0.501483	+	0.865168i	$0.667212\pi$
$468$	9.43075	0.435937
$469$	15.6210	0.721310
$470$	4.73294	0.218314
$471$	−0.615475	−0.0283596
$472$	5.45283	0.250987
$473$	−46.3521	−2.13127
$474$	−0.0497842	−0.00228666
$475$	4.85015	0.222540
$476$	9.36842	0.429401
$477$	5.77003	0.264191
$478$	−0.336310	−0.0153824
$479$	−15.9043	−0.726685	−0.363342	−	0.931656i	$-0.618364\pi$
$479$	−15.9043	−0.726685	−0.363342	+	0.931656i	$0.618364\pi$
$480$	−0.103543	−0.00472607
$481$	22.8351	1.04119
$482$	17.5576	0.799728
$483$	0.782814	0.0356193
$484$	1.86918	0.0849626
$485$	6.24326	0.283492
$486$	−2.78235	−0.126210
$487$	−20.2457	−0.917420	−0.458710	−	0.888586i	$-0.651688\pi$
$487$	−20.2457	−0.917420	−0.458710	+	0.888586i	$0.651688\pi$
$488$	−2.79150	−0.126365
$489$	−1.89673	−0.0857732
$490$	−1.28388	−0.0580000
$491$	−22.0874	−0.996792	−0.498396	−	0.866950i	$-0.666077\pi$
$491$	−22.0874	−0.996792	−0.498396	+	0.866950i	$0.666077\pi$
$492$	−1.30145	−0.0586737
$493$	−4.20196	−0.189247
$494$	−15.3015	−0.688449
$495$	−10.7236	−0.481991
$496$	8.56460	0.384562
$497$	21.6468	0.970993
$498$	−1.69920	−0.0761429
$499$	−4.60664	−0.206221	−0.103111	−	0.994670i	$-0.532880\pi$
$499$	−4.60664	−0.206221	−0.103111	+	0.994670i	$0.532880\pi$
$500$	1.00000	0.0447214
$501$	−0.416295	−0.0185987
$502$	−1.47659	−0.0659033
$503$	−27.3092	−1.21766	−0.608829	−	0.793302i	$-0.708360\pi$
$503$	−27.3092	−1.21766	−0.608829	+	0.793302i	$0.708360\pi$
$504$	−7.14689	−0.318348
$505$	−6.33364	−0.281843
$506$	−11.3439	−0.504299
$507$	0.315482	0.0140111
$508$	7.97822	0.353976
$509$	−30.0392	−1.33146	−0.665731	−	0.746192i	$-0.731880\pi$
$509$	−30.0392	−1.33146	−0.665731	+	0.746192i	$0.731880\pi$
$510$	−0.405729	−0.0179660
$511$	−9.08123	−0.401730
$512$	1.00000	0.0441942
$513$	3.00781	0.132798
$514$	−14.9720	−0.660387
$515$	15.5270	0.684203
$516$	1.33787	0.0588965
$517$	16.9788	0.746726
$518$	−17.3051	−0.760343
$519$	0.705526	0.0309691
$520$	−3.15486	−0.138350
$521$	17.1734	0.752379	0.376190	−	0.926543i	$-0.377234\pi$
$521$	17.1734	0.752379	0.376190	+	0.926543i	$0.377234\pi$
$522$	3.20555	0.140303
$523$	−22.0883	−0.965852	−0.482926	−	0.875661i	$-0.660426\pi$
$523$	−22.0883	−0.965852	−0.482926	+	0.875661i	$0.660426\pi$
$524$	−4.83563	−0.211245
$525$	−0.247555	−0.0108042
$526$	22.3328	0.973758
$527$	33.5601	1.46190
$528$	−0.371446	−0.0161651
$529$	−13.0006	−0.565242
$530$	−1.93024	−0.0838443
$531$	−16.3000	−0.707362
$532$	11.5959	0.502748
$533$	−39.6538	−1.71760
$534$	−0.164032	−0.00709836
$535$	−8.84625	−0.382457
$536$	6.53368	0.282212
$537$	0.796036	0.0343515
$538$	11.9129	0.513603
$539$	−4.60576	−0.198384
$540$	0.620148	0.0266869
$541$	21.8844	0.940883	0.470441	−	0.882431i	$-0.344095\pi$
$541$	21.8844	0.940883	0.470441	+	0.882431i	$0.344095\pi$
$542$	19.1585	0.822927
$543$	−0.226393	−0.00971547
$544$	3.91846	0.168003
$545$	−13.0461	−0.558833
$546$	0.781000	0.0334237
$547$	27.2968	1.16713	0.583563	−	0.812068i	$-0.301658\pi$
$547$	27.2968	1.16713	0.583563	+	0.812068i	$0.301658\pi$
$548$	1.75116	0.0748058
$549$	8.34458	0.356138
$550$	3.58736	0.152966
$551$	−5.20105	−0.221572
$552$	0.327422	0.0139360
$553$	1.14953	0.0488831
$554$	−9.81408	−0.416960
$555$	0.749453	0.0318125
$556$	−5.64685	−0.239480
$557$	−26.9100	−1.14022	−0.570108	−	0.821570i	$-0.693098\pi$
$557$	−26.9100	−1.14022	−0.570108	+	0.821570i	$0.693098\pi$
$558$	−25.6020	−1.08382
$559$	40.7637	1.72412
$560$	2.39084	0.101031
$561$	−1.45550	−0.0614512
$562$	−19.2782	−0.813201
$563$	45.1851	1.90432	0.952162	−	0.305593i	$-0.0988548\pi$
$563$	45.1851	1.90432	0.952162	+	0.305593i	$0.0988548\pi$
$564$	−0.490063	−0.0206354
$565$	−18.8854	−0.794514
$566$	−5.75699	−0.241984
$567$	21.2871	0.893976
$568$	9.05407	0.379900
$569$	18.4084	0.771721	0.385860	−	0.922557i	$-0.373905\pi$
$569$	18.4084	0.771721	0.385860	+	0.922557i	$0.373905\pi$
$570$	−0.502199	−0.0210348
$571$	−11.9994	−0.502160	−0.251080	−	0.967966i	$-0.580786\pi$
$571$	−11.9994	−0.502160	−0.251080	+	0.967966i	$0.580786\pi$
$572$	−11.3176	−0.473214
$573$	−1.72055	−0.0718772
$574$	30.0508	1.25430
$575$	−3.16219	−0.131872
$576$	−2.98928	−0.124553
$577$	−45.7063	−1.90278	−0.951388	−	0.307994i	$-0.900342\pi$
$577$	−45.7063	−1.90278	−0.951388	+	0.307994i	$0.900342\pi$
$578$	−1.64565	−0.0684501
$579$	1.96836	0.0818025
$580$	−1.07235	−0.0445268
$581$	39.2350	1.62774
$582$	−0.646445	−0.0267960
$583$	−6.92447	−0.286782
$584$	−3.79835	−0.157177
$585$	9.43075	0.389913
$586$	−18.3702	−0.758867
$587$	−15.8510	−0.654242	−0.327121	−	0.944982i	$-0.606079\pi$
$587$	−15.8510	−0.654242	−0.327121	+	0.944982i	$0.606079\pi$
$588$	0.132937	0.00548224
$589$	41.5396	1.71161
$590$	5.45283	0.224490
$591$	1.18875	0.0488987
$592$	−7.23809	−0.297484
$593$	−30.5789	−1.25573	−0.627863	−	0.778324i	$-0.716070\pi$
$593$	−30.5789	−1.25573	−0.627863	+	0.778324i	$0.716070\pi$
$594$	2.22470	0.0912804
$595$	9.36842	0.384068
$596$	2.06358	0.0845275
$597$	1.50290	0.0615097
$598$	9.97625	0.407959
$599$	−35.0011	−1.43011	−0.715053	−	0.699070i	$-0.753597\pi$
$599$	−35.0011	−1.43011	−0.715053	+	0.699070i	$0.753597\pi$
$600$	−0.103543	−0.00422712
$601$	1.87965	0.0766724	0.0383362	−	0.999265i	$-0.487794\pi$
$601$	1.87965	0.0766724	0.0383362	+	0.999265i	$0.487794\pi$
$602$	−30.8919	−1.25906
$603$	−19.5310	−0.795364
$604$	3.48738	0.141900
$605$	1.86918	0.0759929
$606$	0.655804	0.0266402
$607$	−32.5228	−1.32006	−0.660030	−	0.751239i	$-0.729457\pi$
$607$	−32.5228	−1.32006	−0.660030	+	0.751239i	$0.729457\pi$
$608$	4.85015	0.196700
$609$	0.265465	0.0107572
$610$	−2.79150	−0.113025
$611$	−14.9318	−0.604074
$612$	−11.7134	−0.473485
$613$	−23.0123	−0.929459	−0.464729	−	0.885453i	$-0.653848\pi$
$613$	−23.0123	−0.929459	−0.464729	+	0.885453i	$0.653848\pi$
$614$	−13.5880	−0.548368
$615$	−1.30145	−0.0524794
$616$	8.57681	0.345570
$617$	10.2531	0.412776	0.206388	−	0.978470i	$-0.433829\pi$
$617$	10.2531	0.412776	0.206388	+	0.978470i	$0.433829\pi$
$618$	−1.60772	−0.0646718
$619$	16.6255	0.668234	0.334117	−	0.942532i	$-0.391562\pi$
$619$	16.6255	0.668234	0.334117	+	0.942532i	$0.391562\pi$
$620$	8.56460	0.343963
$621$	−1.96102	−0.0786932
$622$	1.14835	0.0460445
$623$	3.78755	0.151745
$624$	0.326663	0.0130770
$625$	1.00000	0.0400000
$626$	9.72942	0.388866
$627$	−1.80157	−0.0719479
$628$	5.94415	0.237197
$629$	−28.3622	−1.13087
$630$	−7.14689	−0.284739
$631$	−20.2824	−0.807429	−0.403715	−	0.914885i	$-0.632281\pi$
$631$	−20.2824	−0.807429	−0.403715	+	0.914885i	$0.632281\pi$
$632$	0.480807	0.0191255
$633$	1.64527	0.0653934
$634$	8.82275	0.350396
$635$	7.97822	0.316606
$636$	0.199863	0.00792507
$637$	4.05047	0.160486
$638$	−3.84690	−0.152300
$639$	−27.0652	−1.07068
$640$	1.00000	0.0395285
$641$	1.66724	0.0658521	0.0329261	−	0.999458i	$-0.489517\pi$
$641$	1.66724	0.0658521	0.0329261	+	0.999458i	$0.489517\pi$
$642$	0.915967	0.0361503
$643$	−28.0866	−1.10763	−0.553813	−	0.832641i	$-0.686828\pi$
$643$	−28.0866	−1.10763	−0.553813	+	0.832641i	$0.686828\pi$
$644$	−7.56028	−0.297917
$645$	1.33787	0.0526786
$646$	19.0052	0.747748
$647$	−23.6547	−0.929963	−0.464982	−	0.885320i	$-0.653939\pi$
$647$	−23.6547	−0.929963	−0.464982	+	0.885320i	$0.653939\pi$
$648$	8.90362	0.349767
$649$	19.5613	0.767848
$650$	−3.15486	−0.123744
$651$	−2.12021	−0.0830975
$652$	18.3183	0.717400
$653$	23.4614	0.918116	0.459058	−	0.888406i	$-0.348187\pi$
$653$	23.4614	0.918116	0.459058	+	0.888406i	$0.348187\pi$
$654$	1.35083	0.0528217
$655$	−4.83563	−0.188944
$656$	12.5691	0.490743
$657$	11.3543	0.442974
$658$	11.3157	0.441132
$659$	−23.2487	−0.905639	−0.452820	−	0.891602i	$-0.649582\pi$
$659$	−23.2487	−0.905639	−0.452820	+	0.891602i	$0.649582\pi$
$660$	−0.371446	−0.0144585
$661$	22.5685	0.877813	0.438906	−	0.898533i	$-0.355366\pi$
$661$	22.5685	0.877813	0.438906	+	0.898533i	$0.355366\pi$
$662$	−14.9145	−0.579667
$663$	1.28002	0.0497118
$664$	16.4106	0.636854
$665$	11.5959	0.449671
$666$	21.6367	0.838404
$667$	3.39097	0.131299
$668$	4.02051	0.155558
$669$	−1.87747	−0.0725871
$670$	6.53368	0.252418
$671$	−10.0141	−0.386592
$672$	−0.247555	−0.00954963
$673$	−8.47243	−0.326588	−0.163294	−	0.986577i	$-0.552212\pi$
$673$	−8.47243	−0.326588	−0.163294	+	0.986577i	$0.552212\pi$
$674$	−0.335498	−0.0129229
$675$	0.620148	0.0238695
$676$	−3.04687	−0.117187
$677$	−11.5354	−0.443340	−0.221670	−	0.975122i	$-0.571151\pi$
$677$	−11.5354	−0.443340	−0.221670	+	0.975122i	$0.571151\pi$
$678$	1.95545	0.0750985
$679$	14.9266	0.572831
$680$	3.91846	0.150266
$681$	1.97573	0.0757102
$682$	30.7243	1.17650
$683$	3.62278	0.138622	0.0693110	−	0.997595i	$-0.477920\pi$
$683$	3.62278	0.138622	0.0693110	+	0.997595i	$0.477920\pi$
$684$	−14.4985	−0.554363
$685$	1.75116	0.0669084
$686$	−19.8054	−0.756175
$687$	−2.88579	−0.110100
$688$	−12.9209	−0.492606
$689$	6.08963	0.231996
$690$	0.327422	0.0124648
$691$	−14.6011	−0.555454	−0.277727	−	0.960660i	$-0.589581\pi$
$691$	−14.6011	−0.555454	−0.277727	+	0.960660i	$0.589581\pi$
$692$	−6.81384	−0.259023
$693$	−25.6385	−0.973925
$694$	0.998566	0.0379051
$695$	−5.64685	−0.214197
$696$	0.111034	0.00420874
$697$	49.2517	1.86554
$698$	12.6314	0.478106
$699$	−0.126508	−0.00478496
$700$	2.39084	0.0903653
$701$	−3.50590	−0.132416	−0.0662080	−	0.997806i	$-0.521090\pi$
$701$	−3.50590	−0.132416	−0.0662080	+	0.997806i	$0.521090\pi$
$702$	−1.95648	−0.0738425
$703$	−35.1058	−1.32404
$704$	3.58736	0.135204
$705$	−0.490063	−0.0184568
$706$	25.3186	0.952877
$707$	−15.1427	−0.569501
$708$	−0.564603	−0.0212191
$709$	29.4476	1.10593	0.552965	−	0.833205i	$-0.313496\pi$
$709$	29.4476	1.10593	0.552965	+	0.833205i	$0.313496\pi$
$710$	9.05407	0.339793
$711$	−1.43727	−0.0539017
$712$	1.58419	0.0593701
$713$	−27.0829	−1.01426
$714$	−0.970034	−0.0363026
$715$	−11.3176	−0.423255
$716$	−7.68797	−0.287313
$717$	0.0348225	0.00130047
$718$	32.0380	1.19565
$719$	−19.9792	−0.745100	−0.372550	−	0.928012i	$-0.621516\pi$
$719$	−19.9792	−0.745100	−0.372550	+	0.928012i	$0.621516\pi$
$720$	−2.98928	−0.111404
$721$	37.1226	1.38252
$722$	4.52400	0.168366
$723$	−1.81797	−0.0676110
$724$	2.18647	0.0812595
$725$	−1.07235	−0.0398260
$726$	−0.193540	−0.00718295
$727$	46.9863	1.74263	0.871314	−	0.490727i	$-0.163269\pi$
$727$	46.9863	1.74263	0.871314	+	0.490727i	$0.163269\pi$
$728$	−7.54276	−0.279553
$729$	−26.4228	−0.978622
$730$	−3.79835	−0.140583
$731$	−50.6302	−1.87262
$732$	0.289041	0.0106832
$733$	−14.9263	−0.551315	−0.275658	−	0.961256i	$-0.588896\pi$
$733$	−14.9263	−0.551315	−0.275658	+	0.961256i	$0.588896\pi$
$734$	−27.4318	−1.01253
$735$	0.132937	0.00490346
$736$	−3.16219	−0.116560
$737$	23.4387	0.863376
$738$	−37.5727	−1.38307
$739$	32.0153	1.17770	0.588850	−	0.808242i	$-0.299581\pi$
$739$	32.0153	1.17770	0.588850	+	0.808242i	$0.299581\pi$
$740$	−7.23809	−0.266077
$741$	1.58437	0.0582032
$742$	−4.61489	−0.169418
$743$	−34.9584	−1.28250	−0.641250	−	0.767332i	$-0.721584\pi$
$743$	−34.9584	−1.28250	−0.641250	+	0.767332i	$0.721584\pi$
$744$	−0.886804	−0.0325118
$745$	2.06358	0.0756037
$746$	9.29220	0.340212
$747$	−49.0558	−1.79486
$748$	14.0570	0.513973
$749$	−21.1500	−0.772803
$750$	−0.103543	−0.00378086
$751$	−13.8984	−0.507160	−0.253580	−	0.967314i	$-0.581608\pi$
$751$	−13.8984	−0.507160	−0.253580	+	0.967314i	$0.581608\pi$
$752$	4.73294	0.172593
$753$	0.152890	0.00557163
$754$	3.38310	0.123205
$755$	3.48738	0.126919
$756$	1.48267	0.0539243
$757$	9.79725	0.356087	0.178044	−	0.984023i	$-0.443023\pi$
$757$	9.79725	0.356087	0.178044	+	0.984023i	$0.443023\pi$
$758$	3.45354	0.125438
$759$	1.17458	0.0426347
$760$	4.85015	0.175934
$761$	−14.0361	−0.508808	−0.254404	−	0.967098i	$-0.581879\pi$
$761$	−14.0361	−0.508808	−0.254404	+	0.967098i	$0.581879\pi$
$762$	−0.826089	−0.0299261
$763$	−31.1911	−1.12919
$764$	16.6168	0.601175
$765$	−11.7134	−0.423498
$766$	−24.0180	−0.867804
$767$	−17.2029	−0.621161
$768$	−0.103543	−0.00373629
$769$	12.3807	0.446458	0.223229	−	0.974766i	$-0.428340\pi$
$769$	12.3807	0.446458	0.223229	+	0.974766i	$0.428340\pi$
$770$	8.57681	0.309087
$771$	1.55025	0.0558307
$772$	−19.0101	−0.684189
$773$	−36.1190	−1.29911	−0.649556	−	0.760314i	$-0.725045\pi$
$773$	−36.1190	−1.29911	−0.649556	+	0.760314i	$0.725045\pi$
$774$	38.6243	1.38832
$775$	8.56460	0.307650
$776$	6.24326	0.224120
$777$	1.79182	0.0642813
$778$	−2.45579	−0.0880443
$779$	60.9623	2.18420
$780$	0.326663	0.0116964
$781$	32.4803	1.16223
$782$	−12.3909	−0.443098
$783$	−0.665014	−0.0237657
$784$	−1.28388	−0.0458530
$785$	5.94415	0.212156
$786$	0.500695	0.0178592
$787$	−38.0372	−1.35588	−0.677940	−	0.735117i	$-0.737127\pi$
$787$	−38.0372	−1.35588	−0.677940	+	0.735117i	$0.737127\pi$
$788$	−11.4808	−0.408985
$789$	−2.31241	−0.0823239
$790$	0.480807	0.0171064
$791$	−45.1519	−1.60542
$792$	−10.7236	−0.381048
$793$	8.80680	0.312739
$794$	34.4051	1.22099
$795$	0.199863	0.00708840
$796$	−14.5148	−0.514463
$797$	−46.7582	−1.65626	−0.828131	−	0.560535i	$-0.810596\pi$
$797$	−46.7582	−1.65626	−0.828131	+	0.560535i	$0.810596\pi$
$798$	−1.20068	−0.0425036
$799$	18.5459	0.656105
$800$	1.00000	0.0353553
$801$	−4.73559	−0.167324
$802$	1.00000	0.0353112
$803$	−13.6260	−0.480853
$804$	−0.676517	−0.0238589
$805$	−7.56028	−0.266465
$806$	−27.0201	−0.951742
$807$	−1.23350	−0.0434213
$808$	−6.33364	−0.222817
$809$	29.4606	1.03578	0.517889	−	0.855448i	$-0.326718\pi$
$809$	29.4606	1.03578	0.517889	+	0.855448i	$0.326718\pi$
$810$	8.90362	0.312841
$811$	−26.3992	−0.927003	−0.463501	−	0.886096i	$-0.653407\pi$
$811$	−26.3992	−0.927003	−0.463501	+	0.886096i	$0.653407\pi$
$812$	−2.56381	−0.0899722
$813$	−1.98372	−0.0695723
$814$	−25.9656	−0.910096
$815$	18.3183	0.641662
$816$	−0.405729	−0.0142034
$817$	−62.6685	−2.19249
$818$	−25.4710	−0.890573
$819$	22.5474	0.787870
$820$	12.5691	0.438933
$821$	−48.5301	−1.69371	−0.846855	−	0.531823i	$-0.821507\pi$
$821$	−48.5301	−1.69371	−0.846855	+	0.531823i	$0.821507\pi$
$822$	−0.181320	−0.00632427
$823$	−10.5724	−0.368531	−0.184265	−	0.982877i	$-0.558991\pi$
$823$	−10.5724	−0.368531	−0.184265	+	0.982877i	$0.558991\pi$
$824$	15.5270	0.540910
$825$	−0.371446	−0.0129321
$826$	13.0369	0.453610
$827$	1.41479	0.0491969	0.0245985	−	0.999697i	$-0.492169\pi$
$827$	1.41479	0.0491969	0.0245985	+	0.999697i	$0.492169\pi$
$828$	9.45266	0.328503
$829$	18.1844	0.631569	0.315785	−	0.948831i	$-0.397732\pi$
$829$	18.1844	0.631569	0.315785	+	0.948831i	$0.397732\pi$
$830$	16.4106	0.569619
$831$	1.01618	0.0352509
$832$	−3.15486	−0.109375
$833$	−5.03085	−0.174309
$834$	0.584692	0.0202462
$835$	4.02051	0.139135
$836$	17.3993	0.601766
$837$	5.31132	0.183586
$838$	−27.8568	−0.962296
$839$	55.9923	1.93307	0.966535	−	0.256536i	$-0.0825814\pi$
$839$	55.9923	1.93307	0.966535	+	0.256536i	$0.0825814\pi$
$840$	−0.247555	−0.00854145
$841$	−27.8501	−0.960347
$842$	8.28299	0.285451
$843$	1.99612	0.0687501
$844$	−15.8897	−0.546945
$845$	−3.04687	−0.104816
$846$	−14.1481	−0.486421
$847$	4.46890	0.153553
$848$	−1.93024	−0.0662847
$849$	0.596096	0.0204580
$850$	3.91846	0.134402
$851$	22.8882	0.784597
$852$	−0.937486	−0.0321177
$853$	55.0083	1.88345	0.941725	−	0.336384i	$-0.109204\pi$
$853$	55.0083	1.88345	0.941725	+	0.336384i	$0.109204\pi$
$854$	−6.67404	−0.228381
$855$	−14.4985	−0.495837
$856$	−8.84625	−0.302359
$857$	−34.9050	−1.19233	−0.596166	−	0.802861i	$-0.703310\pi$
$857$	−34.9050	−1.19233	−0.596166	+	0.802861i	$0.703310\pi$
$858$	1.17186	0.0400066
$859$	25.7905	0.879962	0.439981	−	0.898007i	$-0.354985\pi$
$859$	25.7905	0.879962	0.439981	+	0.898007i	$0.354985\pi$
$860$	−12.9209	−0.440600
$861$	−3.11155	−0.106041
$862$	18.4328	0.627824
$863$	−32.0219	−1.09004	−0.545018	−	0.838424i	$-0.683477\pi$
$863$	−32.0219	−1.09004	−0.545018	+	0.838424i	$0.683477\pi$
$864$	0.620148	0.0210979
$865$	−6.81384	−0.231678
$866$	−12.4895	−0.424410
$867$	0.170396	0.00578694
$868$	20.4766	0.695021
$869$	1.72483	0.0585109
$870$	0.111034	0.00376441
$871$	−20.6128	−0.698440
$872$	−13.0461	−0.441797
$873$	−18.6628	−0.631641
$874$	−15.3371	−0.518785
$875$	2.39084	0.0808251
$876$	0.393292	0.0132881
$877$	53.1807	1.79578	0.897892	−	0.440216i	$-0.145098\pi$
$877$	53.1807	1.79578	0.897892	+	0.440216i	$0.145098\pi$
$878$	−32.1011	−1.08336
$879$	1.90211	0.0641565
$880$	3.58736	0.120930
$881$	−39.8235	−1.34169	−0.670843	−	0.741599i	$-0.734068\pi$
$881$	−39.8235	−1.34169	−0.670843	+	0.741599i	$0.734068\pi$
$882$	3.83789	0.129228
$883$	−13.3816	−0.450328	−0.225164	−	0.974321i	$-0.572292\pi$
$883$	−13.3816	−0.450328	−0.225164	+	0.974321i	$0.572292\pi$
$884$	−12.3622	−0.415785
$885$	−0.564603	−0.0189789
$886$	−13.8794	−0.466288
$887$	43.5009	1.46062	0.730308	−	0.683118i	$-0.239377\pi$
$887$	43.5009	1.46062	0.730308	+	0.683118i	$0.239377\pi$
$888$	0.749453	0.0251500
$889$	19.0747	0.639743
$890$	1.58419	0.0531023
$891$	31.9405	1.07005
$892$	18.1323	0.607113
$893$	22.9555	0.768177
$894$	−0.213669	−0.00714616
$895$	−7.68797	−0.256981
$896$	2.39084	0.0798724
$897$	−1.03297	−0.0344899
$898$	28.8811	0.963774
$899$	−9.18423	−0.306311
$900$	−2.98928	−0.0996426
$901$	−7.56357	−0.251979
$902$	45.0901	1.50134
$903$	3.19864	0.106444
$904$	−18.8854	−0.628118
$905$	2.18647	0.0726807
$906$	−0.361094	−0.0119965
$907$	−4.33027	−0.143784	−0.0718921	−	0.997412i	$-0.522904\pi$
$907$	−4.33027	−0.143784	−0.0718921	+	0.997412i	$0.522904\pi$
$908$	−19.0813	−0.633234
$909$	18.9330	0.627969
$910$	−7.54276	−0.250040
$911$	−17.0764	−0.565766	−0.282883	−	0.959154i	$-0.591291\pi$
$911$	−17.0764	−0.565766	−0.282883	+	0.959154i	$0.591291\pi$
$912$	−0.502199	−0.0166295
$913$	58.8707	1.94834
$914$	−0.825716	−0.0273123
$915$	0.289041	0.00955539
$916$	27.8705	0.920866
$917$	−11.5612	−0.381785
$918$	2.43003	0.0802028
$919$	15.7584	0.519822	0.259911	−	0.965633i	$-0.416307\pi$
$919$	15.7584	0.519822	0.259911	+	0.965633i	$0.416307\pi$
$920$	−3.16219	−0.104254
$921$	1.40695	0.0463604
$922$	−17.8190	−0.586836
$923$	−28.5643	−0.940206
$924$	−0.888069	−0.0292153
$925$	−7.23809	−0.237987
$926$	−23.6748	−0.778003
$927$	−46.4146	−1.52446
$928$	−1.07235	−0.0352015
$929$	−16.4033	−0.538173	−0.269087	−	0.963116i	$-0.586722\pi$
$929$	−16.4033	−0.538173	−0.269087	+	0.963116i	$0.586722\pi$
$930$	−0.886804	−0.0290795
$931$	−6.22704	−0.204083
$932$	1.22179	0.0400210
$933$	−0.118903	−0.00389272
$934$	−21.6743	−0.709204
$935$	14.0570	0.459711
$936$	9.43075	0.308254
$937$	−7.34558	−0.239970	−0.119985	−	0.992776i	$-0.538285\pi$
$937$	−7.34558	−0.239970	−0.119985	+	0.992776i	$0.538285\pi$
$938$	15.6210	0.510043
$939$	−1.00741	−0.0328757
$940$	4.73294	0.154372
$941$	46.8497	1.52726	0.763629	−	0.645656i	$-0.223416\pi$
$941$	46.8497	1.52726	0.763629	+	0.645656i	$0.223416\pi$
$942$	−0.615475	−0.0200533
$943$	−39.7460	−1.29431
$944$	5.45283	0.177475
$945$	1.48267	0.0482314
$946$	−46.3521	−1.50704
$947$	−39.8777	−1.29585	−0.647925	−	0.761704i	$-0.724363\pi$
$947$	−39.8777	−1.29585	−0.647925	+	0.761704i	$0.724363\pi$
$948$	−0.0497842	−0.00161692
$949$	11.9832	0.388992
$950$	4.85015	0.157360
$951$	−0.913534	−0.0296234
$952$	9.36842	0.303632
$953$	25.1304	0.814053	0.407027	−	0.913416i	$-0.366566\pi$
$953$	25.1304	0.814053	0.407027	+	0.913416i	$0.366566\pi$
$954$	5.77003	0.186811
$955$	16.6168	0.537708
$956$	−0.336310	−0.0108770
$957$	0.398320	0.0128758
$958$	−15.9043	−0.513844
$959$	4.18674	0.135197
$960$	−0.103543	−0.00334184
$961$	42.3524	1.36621
$962$	22.8351	0.736234
$963$	26.4439	0.852143
$964$	17.5576	0.565493
$965$	−19.0101	−0.611958
$966$	0.782814	0.0251866
$967$	33.0811	1.06382	0.531908	−	0.846802i	$-0.321475\pi$
$967$	33.0811	1.06382	0.531908	+	0.846802i	$0.321475\pi$
$968$	1.86918	0.0600777
$969$	−1.96785	−0.0632164
$970$	6.24326	0.200459
$971$	−7.80184	−0.250373	−0.125187	−	0.992133i	$-0.539953\pi$
$971$	−7.80184	−0.250373	−0.125187	+	0.992133i	$0.539953\pi$
$972$	−2.78235	−0.0892440
$973$	−13.5007	−0.432813
$974$	−20.2457	−0.648714
$975$	0.326663	0.0104616
$976$	−2.79150	−0.0893539
$977$	26.5800	0.850370	0.425185	−	0.905107i	$-0.360209\pi$
$977$	26.5800	0.850370	0.425185	+	0.905107i	$0.360209\pi$
$978$	−1.89673	−0.0606508
$979$	5.68308	0.181632
$980$	−1.28388	−0.0410122
$981$	38.9984	1.24512
$982$	−22.0874	−0.704838
$983$	12.0699	0.384971	0.192485	−	0.981300i	$-0.438345\pi$
$983$	12.0699	0.384971	0.192485	+	0.981300i	$0.438345\pi$
$984$	−1.30145	−0.0414886
$985$	−11.4808	−0.365807
$986$	−4.20196	−0.133818
$987$	−1.17166	−0.0372944
$988$	−15.3015	−0.486807
$989$	40.8584	1.29922
$990$	−10.7236	−0.340819
$991$	−26.9428	−0.855866	−0.427933	−	0.903810i	$-0.640758\pi$
$991$	−26.9428	−0.855866	−0.427933	+	0.903810i	$0.640758\pi$
$992$	8.56460	0.271926
$993$	1.54429	0.0490065
$994$	21.6468	0.686596
$995$	−14.5148	−0.460149
$996$	−1.69920	−0.0538412
$997$	−10.5774	−0.334990	−0.167495	−	0.985873i	$-0.553568\pi$
$997$	−10.5774	−0.334990	−0.167495	+	0.985873i	$0.553568\pi$
$998$	−4.60664	−0.145821
$999$	−4.48868	−0.142016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.n.1.11	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.n.1.11	✓	22	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:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.103543 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.103543 q^{6} +2.39084 q^{7} +1.00000 q^{8} -2.98928 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.103543 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.103543 q^{6} +2.39084 q^{7} +1.00000 q^{8} -2.98928 q^{9} +1.00000 q^{10} +3.58736 q^{11} -0.103543 q^{12} -3.15486 q^{13} +2.39084 q^{14} -0.103543 q^{15} +1.00000 q^{16} +3.91846 q^{17} -2.98928 q^{18} +4.85015 q^{19} +1.00000 q^{20} -0.247555 q^{21} +3.58736 q^{22} -3.16219 q^{23} -0.103543 q^{24} +1.00000 q^{25} -3.15486 q^{26} +0.620148 q^{27} +2.39084 q^{28} -1.07235 q^{29} -0.103543 q^{30} +8.56460 q^{31} +1.00000 q^{32} -0.371446 q^{33} +3.91846 q^{34} +2.39084 q^{35} -2.98928 q^{36} -7.23809 q^{37} +4.85015 q^{38} +0.326663 q^{39} +1.00000 q^{40} +12.5691 q^{41} -0.247555 q^{42} -12.9209 q^{43} +3.58736 q^{44} -2.98928 q^{45} -3.16219 q^{46} +4.73294 q^{47} -0.103543 q^{48} -1.28388 q^{49} +1.00000 q^{50} -0.405729 q^{51} -3.15486 q^{52} -1.93024 q^{53} +0.620148 q^{54} +3.58736 q^{55} +2.39084 q^{56} -0.502199 q^{57} -1.07235 q^{58} +5.45283 q^{59} -0.103543 q^{60} -2.79150 q^{61} +8.56460 q^{62} -7.14689 q^{63} +1.00000 q^{64} -3.15486 q^{65} -0.371446 q^{66} +6.53368 q^{67} +3.91846 q^{68} +0.327422 q^{69} +2.39084 q^{70} +9.05407 q^{71} -2.98928 q^{72} -3.79835 q^{73} -7.23809 q^{74} -0.103543 q^{75} +4.85015 q^{76} +8.57681 q^{77} +0.326663 q^{78} +0.480807 q^{79} +1.00000 q^{80} +8.90362 q^{81} +12.5691 q^{82} +16.4106 q^{83} -0.247555 q^{84} +3.91846 q^{85} -12.9209 q^{86} +0.111034 q^{87} +3.58736 q^{88} +1.58419 q^{89} -2.98928 q^{90} -7.54276 q^{91} -3.16219 q^{92} -0.886804 q^{93} +4.73294 q^{94} +4.85015 q^{95} -0.103543 q^{96} +6.24326 q^{97} -1.28388 q^{98} -10.7236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 22 q^{2} + q^{3} + 22 q^{4} + 22 q^{5} + q^{6} + 22 q^{8} + 43 q^{9}+O(q^{10}) \) 22 * q + 22 * q^2 + q^3 + 22 * q^4 + 22 * q^5 + q^6 + 22 * q^8 + 43 * q^9	\( 22 q + 22 q^{2} + q^{3} + 22 q^{4} + 22 q^{5} + q^{6} + 22 q^{8} + 43 q^{9} + 22 q^{10} + 12 q^{11} + q^{12} + 10 q^{13} + q^{15} + 22 q^{16} + 24 q^{17} + 43 q^{18} + 13 q^{19} + 22 q^{20} + 13 q^{21} + 12 q^{22} + 7 q^{23} + q^{24} + 22 q^{25} + 10 q^{26} - 5 q^{27} + 22 q^{29} + q^{30} + 14 q^{31} + 22 q^{32} + 31 q^{33} + 24 q^{34} + 43 q^{36} + 35 q^{37} + 13 q^{38} + 4 q^{39} + 22 q^{40} + 29 q^{41} + 13 q^{42} + 7 q^{43} + 12 q^{44} + 43 q^{45} + 7 q^{46} - 21 q^{47} + q^{48} + 32 q^{49} + 22 q^{50} - 6 q^{51} + 10 q^{52} + 29 q^{53} - 5 q^{54} + 12 q^{55} - 13 q^{57} + 22 q^{58} + 12 q^{59} + q^{60} + 24 q^{61} + 14 q^{62} - 8 q^{63} + 22 q^{64} + 10 q^{65} + 31 q^{66} + 25 q^{67} + 24 q^{68} + 3 q^{69} + 31 q^{71} + 43 q^{72} + 30 q^{73} + 35 q^{74} + q^{75} + 13 q^{76} + 10 q^{77} + 4 q^{78} + 35 q^{79} + 22 q^{80} + 74 q^{81} + 29 q^{82} - 33 q^{83} + 13 q^{84} + 24 q^{85} + 7 q^{86} - 24 q^{87} + 12 q^{88} + 38 q^{89} + 43 q^{90} - 32 q^{91} + 7 q^{92} + 3 q^{93} - 21 q^{94} + 13 q^{95} + q^{96} + 11 q^{97} + 32 q^{98} - 41 q^{99}+O(q^{100}) \) 22 * q + 22 * q^2 + q^3 + 22 * q^4 + 22 * q^5 + q^6 + 22 * q^8 + 43 * q^9 + 22 * q^10 + 12 * q^11 + q^12 + 10 * q^13 + q^15 + 22 * q^16 + 24 * q^17 + 43 * q^18 + 13 * q^19 + 22 * q^20 + 13 * q^21 + 12 * q^22 + 7 * q^23 + q^24 + 22 * q^25 + 10 * q^26 - 5 * q^27 + 22 * q^29 + q^30 + 14 * q^31 + 22 * q^32 + 31 * q^33 + 24 * q^34 + 43 * q^36 + 35 * q^37 + 13 * q^38 + 4 * q^39 + 22 * q^40 + 29 * q^41 + 13 * q^42 + 7 * q^43 + 12 * q^44 + 43 * q^45 + 7 * q^46 - 21 * q^47 + q^48 + 32 * q^49 + 22 * q^50 - 6 * q^51 + 10 * q^52 + 29 * q^53 - 5 * q^54 + 12 * q^55 - 13 * q^57 + 22 * q^58 + 12 * q^59 + q^60 + 24 * q^61 + 14 * q^62 - 8 * q^63 + 22 * q^64 + 10 * q^65 + 31 * q^66 + 25 * q^67 + 24 * q^68 + 3 * q^69 + 31 * q^71 + 43 * q^72 + 30 * q^73 + 35 * q^74 + q^75 + 13 * q^76 + 10 * q^77 + 4 * q^78 + 35 * q^79 + 22 * q^80 + 74 * q^81 + 29 * q^82 - 33 * q^83 + 13 * q^84 + 24 * q^85 + 7 * q^86 - 24 * q^87 + 12 * q^88 + 38 * q^89 + 43 * q^90 - 32 * q^91 + 7 * q^92 + 3 * q^93 - 21 * q^94 + 13 * q^95 + q^96 + 11 * q^97 + 32 * q^98 - 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more