LMFDB - Embedded newform 2005.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89780 q^{2} +0.675950 q^{3} +1.60165 q^{4} -1.00000 q^{5} -1.28282 q^{6} -2.83198 q^{7} +0.755984 q^{8} -2.54309 q^{9} +O(q^{10})$	\(q-1.89780 q^{2} +0.675950 q^{3} +1.60165 q^{4} -1.00000 q^{5} -1.28282 q^{6} -2.83198 q^{7} +0.755984 q^{8} -2.54309 q^{9} +1.89780 q^{10} -4.44765 q^{11} +1.08264 q^{12} -4.91859 q^{13} +5.37454 q^{14} -0.675950 q^{15} -4.63801 q^{16} -1.03069 q^{17} +4.82629 q^{18} -2.32528 q^{19} -1.60165 q^{20} -1.91428 q^{21} +8.44077 q^{22} -3.44522 q^{23} +0.511007 q^{24} +1.00000 q^{25} +9.33451 q^{26} -3.74685 q^{27} -4.53585 q^{28} -5.81648 q^{29} +1.28282 q^{30} +7.09126 q^{31} +7.29006 q^{32} -3.00639 q^{33} +1.95604 q^{34} +2.83198 q^{35} -4.07315 q^{36} +4.87407 q^{37} +4.41293 q^{38} -3.32472 q^{39} -0.755984 q^{40} +6.57797 q^{41} +3.63292 q^{42} -6.72917 q^{43} -7.12360 q^{44} +2.54309 q^{45} +6.53834 q^{46} +9.31278 q^{47} -3.13506 q^{48} +1.02011 q^{49} -1.89780 q^{50} -0.696692 q^{51} -7.87787 q^{52} -4.77702 q^{53} +7.11078 q^{54} +4.44765 q^{55} -2.14093 q^{56} -1.57177 q^{57} +11.0385 q^{58} -2.96718 q^{59} -1.08264 q^{60} +10.8696 q^{61} -13.4578 q^{62} +7.20199 q^{63} -4.55907 q^{64} +4.91859 q^{65} +5.70553 q^{66} -9.28704 q^{67} -1.65080 q^{68} -2.32879 q^{69} -5.37454 q^{70} +7.86021 q^{71} -1.92254 q^{72} -2.85175 q^{73} -9.25002 q^{74} +0.675950 q^{75} -3.72430 q^{76} +12.5957 q^{77} +6.30966 q^{78} +8.86201 q^{79} +4.63801 q^{80} +5.09659 q^{81} -12.4837 q^{82} +4.67184 q^{83} -3.06600 q^{84} +1.03069 q^{85} +12.7706 q^{86} -3.93165 q^{87} -3.36236 q^{88} -11.3910 q^{89} -4.82629 q^{90} +13.9293 q^{91} -5.51804 q^{92} +4.79333 q^{93} -17.6738 q^{94} +2.32528 q^{95} +4.92772 q^{96} -13.9653 q^{97} -1.93597 q^{98} +11.3108 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) $ 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * 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 $

$2$	−1.89780	−1.34195	−0.670974	−	0.741481i	$-0.734124\pi$
$2$	−1.89780	−1.34195	−0.670974	+	0.741481i	$0.734124\pi$
$3$	0.675950	0.390260	0.195130	−	0.980777i	$-0.437487\pi$
$3$	0.675950	0.390260	0.195130	+	0.980777i	$0.437487\pi$
$4$	1.60165	0.800826
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.28282	−0.523708
$7$	−2.83198	−1.07039	−0.535194	−	0.844729i	$-0.679761\pi$
$7$	−2.83198	−1.07039	−0.535194	+	0.844729i	$0.679761\pi$
$8$	0.755984	0.267281
$9$	−2.54309	−0.847697
$10$	1.89780	0.600138
$11$	−4.44765	−1.34102	−0.670509	−	0.741901i	$-0.733924\pi$
$11$	−4.44765	−1.34102	−0.670509	+	0.741901i	$0.733924\pi$
$12$	1.08264	0.312530
$13$	−4.91859	−1.36417	−0.682086	−	0.731272i	$-0.738927\pi$
$13$	−4.91859	−1.36417	−0.682086	+	0.731272i	$0.738927\pi$
$14$	5.37454	1.43641
$15$	−0.675950	−0.174529
$16$	−4.63801	−1.15950
$17$	−1.03069	−0.249978	−0.124989	−	0.992158i	$-0.539890\pi$
$17$	−1.03069	−0.249978	−0.124989	+	0.992158i	$0.539890\pi$
$18$	4.82629	1.13757
$19$	−2.32528	−0.533457	−0.266728	−	0.963772i	$-0.585943\pi$
$19$	−2.32528	−0.533457	−0.266728	+	0.963772i	$0.585943\pi$
$20$	−1.60165	−0.358140
$21$	−1.91428	−0.417729
$22$	8.44077	1.79958
$23$	−3.44522	−0.718378	−0.359189	−	0.933265i	$-0.616947\pi$
$23$	−3.44522	−0.718378	−0.359189	+	0.933265i	$0.616947\pi$
$24$	0.511007	0.104309
$25$	1.00000	0.200000
$26$	9.33451	1.83065
$27$	−3.74685	−0.721082
$28$	−4.53585	−0.857195
$29$	−5.81648	−1.08009	−0.540047	−	0.841635i	$-0.681593\pi$
$29$	−5.81648	−1.08009	−0.540047	+	0.841635i	$0.681593\pi$
$30$	1.28282	0.234210
$31$	7.09126	1.27363	0.636814	−	0.771018i	$-0.280252\pi$
$31$	7.09126	1.27363	0.636814	+	0.771018i	$0.280252\pi$
$32$	7.29006	1.28871
$33$	−3.00639	−0.523345
$34$	1.95604	0.335458
$35$	2.83198	0.478692
$36$	−4.07315	−0.678858
$37$	4.87407	0.801292	0.400646	−	0.916233i	$-0.368786\pi$
$37$	4.87407	0.801292	0.400646	+	0.916233i	$0.368786\pi$
$38$	4.41293	0.715871
$39$	−3.32472	−0.532381
$40$	−0.755984	−0.119532
$41$	6.57797	1.02731	0.513653	−	0.857998i	$-0.328292\pi$
$41$	6.57797	1.02731	0.513653	+	0.857998i	$0.328292\pi$
$42$	3.63292	0.560571
$43$	−6.72917	−1.02619	−0.513094	−	0.858332i	$-0.671501\pi$
$43$	−6.72917	−1.02619	−0.513094	+	0.858332i	$0.671501\pi$
$44$	−7.12360	−1.07392
$45$	2.54309	0.379102
$46$	6.53834	0.964026
$47$	9.31278	1.35841	0.679204	−	0.733949i	$-0.262325\pi$
$47$	9.31278	1.35841	0.679204	+	0.733949i	$0.262325\pi$
$48$	−3.13506	−0.452507
$49$	1.02011	0.145730
$50$	−1.89780	−0.268390
$51$	−0.696692	−0.0975564
$52$	−7.87787	−1.09246
$53$	−4.77702	−0.656175	−0.328087	−	0.944647i	$-0.606404\pi$
$53$	−4.77702	−0.656175	−0.328087	+	0.944647i	$0.606404\pi$
$54$	7.11078	0.967655
$55$	4.44765	0.599722
$56$	−2.14093	−0.286094
$57$	−1.57177	−0.208187
$58$	11.0385	1.44943
$59$	−2.96718	−0.386294	−0.193147	−	0.981170i	$-0.561869\pi$
$59$	−2.96718	−0.386294	−0.193147	+	0.981170i	$0.561869\pi$
$60$	−1.08264	−0.139768
$61$	10.8696	1.39171	0.695855	−	0.718182i	$-0.255025\pi$
$61$	10.8696	1.39171	0.695855	+	0.718182i	$0.255025\pi$
$62$	−13.4578	−1.70914
$63$	7.20199	0.907365
$64$	−4.55907	−0.569884
$65$	4.91859	0.610076
$66$	5.70553	0.702303
$67$	−9.28704	−1.13459	−0.567296	−	0.823514i	$-0.692010\pi$
$67$	−9.28704	−1.13459	−0.567296	+	0.823514i	$0.692010\pi$
$68$	−1.65080	−0.200189
$69$	−2.32879	−0.280354
$70$	−5.37454	−0.642380
$71$	7.86021	0.932836	0.466418	−	0.884565i	$-0.345544\pi$
$71$	7.86021	0.932836	0.466418	+	0.884565i	$0.345544\pi$
$72$	−1.92254	−0.226573
$73$	−2.85175	−0.333772	−0.166886	−	0.985976i	$-0.553371\pi$
$73$	−2.85175	−0.333772	−0.166886	+	0.985976i	$0.553371\pi$
$74$	−9.25002	−1.07529
$75$	0.675950	0.0780519
$76$	−3.72430	−0.427206
$77$	12.5957	1.43541
$78$	6.30966	0.714428
$79$	8.86201	0.997054	0.498527	−	0.866874i	$-0.333874\pi$
$79$	8.86201	0.997054	0.498527	+	0.866874i	$0.333874\pi$
$80$	4.63801	0.518546
$81$	5.09659	0.566288
$82$	−12.4837	−1.37859
$83$	4.67184	0.512801	0.256401	−	0.966571i	$-0.417463\pi$
$83$	4.67184	0.512801	0.256401	+	0.966571i	$0.417463\pi$
$84$	−3.06600	−0.334529
$85$	1.03069	0.111794
$86$	12.7706	1.37709
$87$	−3.93165	−0.421517
$88$	−3.36236	−0.358428
$89$	−11.3910	−1.20744	−0.603722	−	0.797195i	$-0.706316\pi$
$89$	−11.3910	−1.20744	−0.603722	+	0.797195i	$0.706316\pi$
$90$	−4.82629	−0.508735
$91$	13.9293	1.46019
$92$	−5.51804	−0.575296
$93$	4.79333	0.497046
$94$	−17.6738	−1.82291
$95$	2.32528	0.238569
$96$	4.92772	0.502933
$97$	−13.9653	−1.41796	−0.708979	−	0.705229i	$-0.750844\pi$
$97$	−13.9653	−1.41796	−0.708979	+	0.705229i	$0.750844\pi$
$98$	−1.93597	−0.195562
$99$	11.3108	1.13678
$100$	1.60165	0.160165
$101$	2.48854	0.247619	0.123809	−	0.992306i	$-0.460489\pi$
$101$	2.48854	0.247619	0.123809	+	0.992306i	$0.460489\pi$
$102$	1.32218	0.130916
$103$	−17.5396	−1.72823	−0.864116	−	0.503292i	$-0.832122\pi$
$103$	−17.5396	−1.72823	−0.864116	+	0.503292i	$0.832122\pi$
$104$	−3.71838	−0.364617
$105$	1.91428	0.186814
$106$	9.06585	0.880553
$107$	19.1064	1.84708	0.923542	−	0.383496i	$-0.125280\pi$
$107$	19.1064	1.84708	0.923542	+	0.383496i	$0.125280\pi$
$108$	−6.00115	−0.577461
$109$	−6.25634	−0.599249	−0.299624	−	0.954057i	$-0.596861\pi$
$109$	−6.25634	−0.599249	−0.299624	+	0.954057i	$0.596861\pi$
$110$	−8.44077	−0.804796
$111$	3.29463	0.312712
$112$	13.1348	1.24112
$113$	−3.93380	−0.370061	−0.185030	−	0.982733i	$-0.559238\pi$
$113$	−3.93380	−0.370061	−0.185030	+	0.982733i	$0.559238\pi$
$114$	2.98292	0.279376
$115$	3.44522	0.321268
$116$	−9.31599	−0.864968
$117$	12.5084	1.15640
$118$	5.63112	0.518386
$119$	2.91888	0.267574
$120$	−0.511007	−0.0466484
$121$	8.78163	0.798330
$122$	−20.6284	−1.86760
$123$	4.44638	0.400916
$124$	11.3577	1.01995
$125$	−1.00000	−0.0894427
$126$	−13.6679	−1.21764
$127$	−20.0757	−1.78143	−0.890714	−	0.454563i	$-0.849795\pi$
$127$	−20.0757	−1.78143	−0.890714	+	0.454563i	$0.849795\pi$
$128$	−5.92791	−0.523958
$129$	−4.54858	−0.400480
$130$	−9.33451	−0.818691
$131$	−11.4034	−0.996318	−0.498159	−	0.867086i	$-0.665990\pi$
$131$	−11.4034	−0.996318	−0.498159	+	0.867086i	$0.665990\pi$
$132$	−4.81519	−0.419109
$133$	6.58516	0.571005
$134$	17.6250	1.52256
$135$	3.74685	0.322478
$136$	−0.779182	−0.0668143
$137$	−2.69197	−0.229991	−0.114995	−	0.993366i	$-0.536685\pi$
$137$	−2.69197	−0.229991	−0.114995	+	0.993366i	$0.536685\pi$
$138$	4.41959	0.376220
$139$	16.2106	1.37497	0.687484	−	0.726199i	$-0.258715\pi$
$139$	16.2106	1.37497	0.687484	+	0.726199i	$0.258715\pi$
$140$	4.53585	0.383349
$141$	6.29497	0.530132
$142$	−14.9171	−1.25182
$143$	21.8762	1.82938
$144$	11.7949	0.982908
$145$	5.81648	0.483033
$146$	5.41206	0.447905
$147$	0.689544	0.0568726
$148$	7.80657	0.641696
$149$	16.4884	1.35078	0.675392	−	0.737459i	$-0.263974\pi$
$149$	16.4884	1.35078	0.675392	+	0.737459i	$0.263974\pi$
$150$	−1.28282	−0.104742
$151$	−20.8143	−1.69384	−0.846922	−	0.531718i	$-0.821547\pi$
$151$	−20.8143	−1.69384	−0.846922	+	0.531718i	$0.821547\pi$
$152$	−1.75788	−0.142583
$153$	2.62113	0.211906
$154$	−23.9041	−1.92625
$155$	−7.09126	−0.569584
$156$	−5.32504	−0.426345
$157$	−4.85684	−0.387618	−0.193809	−	0.981039i	$-0.562084\pi$
$157$	−4.85684	−0.387618	−0.193809	+	0.981039i	$0.562084\pi$
$158$	−16.8183	−1.33800
$159$	−3.22903	−0.256079
$160$	−7.29006	−0.576330
$161$	9.75679	0.768943
$162$	−9.67233	−0.759930
$163$	2.60904	0.204356	0.102178	−	0.994766i	$-0.467419\pi$
$163$	2.60904	0.204356	0.102178	+	0.994766i	$0.467419\pi$
$164$	10.5356	0.822694
$165$	3.00639	0.234047
$166$	−8.86623	−0.688153
$167$	7.00993	0.542445	0.271222	−	0.962517i	$-0.412572\pi$
$167$	7.00993	0.542445	0.271222	+	0.962517i	$0.412572\pi$
$168$	−1.44716	−0.111651
$169$	11.1925	0.860963
$170$	−1.95604	−0.150021
$171$	5.91341	0.452210
$172$	−10.7778	−0.821799
$173$	9.09495	0.691476	0.345738	−	0.938331i	$-0.387629\pi$
$173$	9.09495	0.691476	0.345738	+	0.938331i	$0.387629\pi$
$174$	7.46149	0.565654
$175$	−2.83198	−0.214078
$176$	20.6283	1.55492
$177$	−2.00566	−0.150755
$178$	21.6179	1.62033
$179$	−2.45287	−0.183336	−0.0916680	−	0.995790i	$-0.529220\pi$
$179$	−2.45287	−0.183336	−0.0916680	+	0.995790i	$0.529220\pi$
$180$	4.07315	0.303595
$181$	21.0121	1.56182	0.780908	−	0.624646i	$-0.214757\pi$
$181$	21.0121	1.56182	0.780908	+	0.624646i	$0.214757\pi$
$182$	−26.4351	−1.95950
$183$	7.34731	0.543128
$184$	−2.60453	−0.192008
$185$	−4.87407	−0.358349
$186$	−9.09680	−0.667010
$187$	4.58414	0.335225
$188$	14.9158	1.08785
$189$	10.6110	0.771837
$190$	−4.41293	−0.320147
$191$	−10.2180	−0.739349	−0.369675	−	0.929161i	$-0.620531\pi$
$191$	−10.2180	−0.739349	−0.369675	+	0.929161i	$0.620531\pi$
$192$	−3.08170	−0.222403
$193$	−22.6469	−1.63016	−0.815081	−	0.579346i	$-0.803308\pi$
$193$	−22.6469	−1.63016	−0.815081	+	0.579346i	$0.803308\pi$
$194$	26.5033	1.90283
$195$	3.32472	0.238088
$196$	1.63386	0.116705
$197$	−1.17607	−0.0837917	−0.0418959	−	0.999122i	$-0.513340\pi$
$197$	−1.17607	−0.0837917	−0.0418959	+	0.999122i	$0.513340\pi$
$198$	−21.4657	−1.52550
$199$	−11.4441	−0.811253	−0.405627	−	0.914039i	$-0.632947\pi$
$199$	−11.4441	−0.811253	−0.405627	+	0.914039i	$0.632947\pi$
$200$	0.755984	0.0534562
$201$	−6.27757	−0.442786
$202$	−4.72275	−0.332292
$203$	16.4722	1.15612
$204$	−1.11586	−0.0781257
$205$	−6.57797	−0.459425
$206$	33.2868	2.31920
$207$	8.76151	0.608967
$208$	22.8125	1.58176
$209$	10.3421	0.715375
$210$	−3.63292	−0.250695
$211$	15.8915	1.09402	0.547009	−	0.837127i	$-0.315766\pi$
$211$	15.8915	1.09402	0.547009	+	0.837127i	$0.315766\pi$
$212$	−7.65113	−0.525482
$213$	5.31311	0.364048
$214$	−36.2602	−2.47869
$215$	6.72917	0.458926
$216$	−2.83256	−0.192731
$217$	−20.0823	−1.36328
$218$	11.8733	0.804161
$219$	−1.92764	−0.130258
$220$	7.12360	0.480273
$221$	5.06952	0.341013
$222$	−6.25255	−0.419643
$223$	−22.9804	−1.53888	−0.769440	−	0.638719i	$-0.779464\pi$
$223$	−22.9804	−1.53888	−0.769440	+	0.638719i	$0.779464\pi$
$224$	−20.6453	−1.37942
$225$	−2.54309	−0.169539
$226$	7.46558	0.496603
$227$	5.37658	0.356856	0.178428	−	0.983953i	$-0.442899\pi$
$227$	5.37658	0.356856	0.178428	+	0.983953i	$0.442899\pi$
$228$	−2.51744	−0.166721
$229$	−2.87172	−0.189769	−0.0948845	−	0.995488i	$-0.530248\pi$
$229$	−2.87172	−0.189769	−0.0948845	+	0.995488i	$0.530248\pi$
$230$	−6.53834	−0.431125
$231$	8.51404	0.560182
$232$	−4.39717	−0.288688
$233$	−3.11835	−0.204290	−0.102145	−	0.994770i	$-0.532571\pi$
$233$	−3.11835	−0.204290	−0.102145	+	0.994770i	$0.532571\pi$
$234$	−23.7385	−1.55184
$235$	−9.31278	−0.607499
$236$	−4.75239	−0.309354
$237$	5.99027	0.389110
$238$	−5.53946	−0.359070
$239$	25.9036	1.67556	0.837781	−	0.546006i	$-0.183852\pi$
$239$	25.9036	1.67556	0.837781	+	0.546006i	$0.183852\pi$
$240$	3.13506	0.202367
$241$	6.26351	0.403468	0.201734	−	0.979440i	$-0.435342\pi$
$241$	6.26351	0.403468	0.201734	+	0.979440i	$0.435342\pi$
$242$	−16.6658	−1.07132
$243$	14.6856	0.942081
$244$	17.4093	1.11452
$245$	−1.02011	−0.0651725
$246$	−8.43834	−0.538009
$247$	11.4371	0.727726
$248$	5.36088	0.340416
$249$	3.15793	0.200126
$250$	1.89780	0.120028
$251$	29.1639	1.84081	0.920403	−	0.390971i	$-0.127861\pi$
$251$	29.1639	1.84081	0.920403	+	0.390971i	$0.127861\pi$
$252$	11.5351	0.726642
$253$	15.3231	0.963357
$254$	38.0997	2.39059
$255$	0.696692	0.0436285
$256$	20.3681	1.27301
$257$	14.9139	0.930301	0.465151	−	0.885232i	$-0.346000\pi$
$257$	14.9139	0.930301	0.465151	+	0.885232i	$0.346000\pi$
$258$	8.63230	0.537424
$259$	−13.8033	−0.857693
$260$	7.87787	0.488565
$261$	14.7919	0.915593
$262$	21.6414	1.33701
$263$	−19.7395	−1.21719	−0.608595	−	0.793481i	$-0.708267\pi$
$263$	−19.7395	−1.21719	−0.608595	+	0.793481i	$0.708267\pi$
$264$	−2.27278	−0.139880
$265$	4.77702	0.293450
$266$	−12.4973	−0.766260
$267$	−7.69974	−0.471217
$268$	−14.8746	−0.908611
$269$	22.5932	1.37753	0.688765	−	0.724985i	$-0.258153\pi$
$269$	22.5932	1.37753	0.688765	+	0.724985i	$0.258153\pi$
$270$	−7.11078	−0.432748
$271$	6.65290	0.404135	0.202067	−	0.979372i	$-0.435234\pi$
$271$	6.65290	0.404135	0.202067	+	0.979372i	$0.435234\pi$
$272$	4.78034	0.289851
$273$	9.41554	0.569854
$274$	5.10883	0.308636
$275$	−4.44765	−0.268204
$276$	−3.72992	−0.224515
$277$	−26.9907	−1.62171	−0.810856	−	0.585246i	$-0.800998\pi$
$277$	−26.9907	−1.62171	−0.810856	+	0.585246i	$0.800998\pi$
$278$	−30.7646	−1.84514
$279$	−18.0337	−1.07965
$280$	2.14093	0.127945
$281$	1.29166	0.0770539	0.0385269	−	0.999258i	$-0.487733\pi$
$281$	1.29166	0.0770539	0.0385269	+	0.999258i	$0.487733\pi$
$282$	−11.9466	−0.711410
$283$	29.0358	1.72600	0.862999	−	0.505205i	$-0.168583\pi$
$283$	29.0358	1.72600	0.862999	+	0.505205i	$0.168583\pi$
$284$	12.5893	0.747039
$285$	1.57177	0.0931039
$286$	−41.5167	−2.45493
$287$	−18.6287	−1.09962
$288$	−18.5393	−1.09244
$289$	−15.9377	−0.937511
$290$	−11.0385	−0.648205
$291$	−9.43982	−0.553372
$292$	−4.56752	−0.267294
$293$	17.4658	1.02036	0.510182	−	0.860067i	$-0.329578\pi$
$293$	17.4658	1.02036	0.510182	+	0.860067i	$0.329578\pi$
$294$	−1.30862	−0.0763201
$295$	2.96718	0.172756
$296$	3.68472	0.214170
$297$	16.6647	0.966984
$298$	−31.2918	−1.81268
$299$	16.9456	0.979990
$300$	1.08264	0.0625060
$301$	19.0569	1.09842
$302$	39.5014	2.27305
$303$	1.68213	0.0966356
$304$	10.7847	0.618545
$305$	−10.8696	−0.622392
$306$	−4.97439	−0.284367
$307$	16.0640	0.916820	0.458410	−	0.888741i	$-0.348419\pi$
$307$	16.0640	0.916820	0.458410	+	0.888741i	$0.348419\pi$
$308$	20.1739	1.14951
$309$	−11.8559	−0.674459
$310$	13.4578	0.764352
$311$	0.658983	0.0373675	0.0186837	−	0.999825i	$-0.494052\pi$
$311$	0.658983	0.0373675	0.0186837	+	0.999825i	$0.494052\pi$
$312$	−2.51343	−0.142295
$313$	2.66090	0.150403	0.0752016	−	0.997168i	$-0.476040\pi$
$313$	2.66090	0.150403	0.0752016	+	0.997168i	$0.476040\pi$
$314$	9.21733	0.520164
$315$	−7.20199	−0.405786
$316$	14.1939	0.798467
$317$	−25.1631	−1.41330	−0.706650	−	0.707564i	$-0.749794\pi$
$317$	−25.1631	−1.41330	−0.706650	+	0.707564i	$0.749794\pi$
$318$	6.12805	0.343644
$319$	25.8697	1.44843
$320$	4.55907	0.254860
$321$	12.9150	0.720843
$322$	−18.5165	−1.03188
$323$	2.39664	0.133352
$324$	8.16298	0.453499
$325$	−4.91859	−0.272834
$326$	−4.95145	−0.274235
$327$	−4.22897	−0.233863
$328$	4.97284	0.274579
$329$	−26.3736	−1.45402
$330$	−5.70553	−0.314079
$331$	0.861202	0.0473359	0.0236680	−	0.999720i	$-0.492466\pi$
$331$	0.861202	0.0473359	0.0236680	+	0.999720i	$0.492466\pi$
$332$	7.48267	0.410665
$333$	−12.3952	−0.679253
$334$	−13.3035	−0.727933
$335$	9.28704	0.507405
$336$	8.87844	0.484358
$337$	12.6957	0.691576	0.345788	−	0.938313i	$-0.387612\pi$
$337$	12.6957	0.691576	0.345788	+	0.938313i	$0.387612\pi$
$338$	−21.2412	−1.15537
$339$	−2.65905	−0.144420
$340$	1.65080	0.0895273
$341$	−31.5395	−1.70796
$342$	−11.2225	−0.606842
$343$	16.9349	0.914400
$344$	−5.08715	−0.274281
$345$	2.32879	0.125378
$346$	−17.2604	−0.927926
$347$	25.9247	1.39171	0.695857	−	0.718181i	$-0.255025\pi$
$347$	25.9247	1.39171	0.695857	+	0.718181i	$0.255025\pi$
$348$	−6.29714	−0.337562
$349$	−31.0530	−1.66223	−0.831113	−	0.556103i	$-0.812296\pi$
$349$	−31.0530	−1.66223	−0.831113	+	0.556103i	$0.812296\pi$
$350$	5.37454	0.287281
$351$	18.4292	0.983679
$352$	−32.4237	−1.72819
$353$	4.93435	0.262629	0.131314	−	0.991341i	$-0.458080\pi$
$353$	4.93435	0.262629	0.131314	+	0.991341i	$0.458080\pi$
$354$	3.80635	0.202305
$355$	−7.86021	−0.417177
$356$	−18.2444	−0.966953
$357$	1.97302	0.104423
$358$	4.65506	0.246028
$359$	32.3490	1.70732	0.853658	−	0.520834i	$-0.174379\pi$
$359$	32.3490	1.70732	0.853658	+	0.520834i	$0.174379\pi$
$360$	1.92254	0.101327
$361$	−13.5931	−0.715424
$362$	−39.8768	−2.09588
$363$	5.93594	0.311556
$364$	22.3100	1.16936
$365$	2.85175	0.149268
$366$	−13.9437	−0.728851
$367$	−5.75071	−0.300185	−0.150092	−	0.988672i	$-0.547957\pi$
$367$	−5.75071	−0.300185	−0.150092	+	0.988672i	$0.547957\pi$
$368$	15.9790	0.832961
$369$	−16.7284	−0.870845
$370$	9.25002	0.480886
$371$	13.5284	0.702361
$372$	7.67726	0.398047
$373$	−13.1891	−0.682907	−0.341453	−	0.939899i	$-0.610919\pi$
$373$	−13.1891	−0.682907	−0.341453	+	0.939899i	$0.610919\pi$
$374$	−8.69978	−0.449855
$375$	−0.675950	−0.0349059
$376$	7.04032	0.363077
$377$	28.6089	1.47343
$378$	−20.1376	−1.03577
$379$	−11.2691	−0.578853	−0.289426	−	0.957200i	$-0.593465\pi$
$379$	−11.2691	−0.578853	−0.289426	+	0.957200i	$0.593465\pi$
$380$	3.72430	0.191052
$381$	−13.5701	−0.695220
$382$	19.3918	0.992169
$383$	−20.9653	−1.07128	−0.535639	−	0.844447i	$-0.679929\pi$
$383$	−20.9653	−1.07128	−0.535639	+	0.844447i	$0.679929\pi$
$384$	−4.00697	−0.204480
$385$	−12.5957	−0.641935
$386$	42.9794	2.18759
$387$	17.1129	0.869898
$388$	−22.3675	−1.13554
$389$	−22.2800	−1.12964	−0.564820	−	0.825214i	$-0.691054\pi$
$389$	−22.2800	−1.12964	−0.564820	+	0.825214i	$0.691054\pi$
$390$	−6.30966	−0.319502
$391$	3.55094	0.179579
$392$	0.771188	0.0389509
$393$	−7.70811	−0.388823
$394$	2.23195	0.112444
$395$	−8.86201	−0.445896
$396$	18.1160	0.910362
$397$	2.34144	0.117513	0.0587567	−	0.998272i	$-0.481286\pi$
$397$	2.34144	0.117513	0.0587567	+	0.998272i	$0.481286\pi$
$398$	21.7187	1.08866
$399$	4.45123	0.222840
$400$	−4.63801	−0.231901
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	11.9136	0.594195
$403$	−34.8790	−1.73745
$404$	3.98577	0.198300
$405$	−5.09659	−0.253252
$406$	−31.2609	−1.55145
$407$	−21.6782	−1.07455
$408$	−0.526688	−0.0260749
$409$	−5.86619	−0.290064	−0.145032	−	0.989427i	$-0.546329\pi$
$409$	−5.86619	−0.290064	−0.145032	+	0.989427i	$0.546329\pi$
$410$	12.4837	0.616525
$411$	−1.81964	−0.0897561
$412$	−28.0924	−1.38401
$413$	8.40299	0.413484
$414$	−16.6276	−0.817202
$415$	−4.67184	−0.229332
$416$	−35.8568	−1.75803
$417$	10.9576	0.536595
$418$	−19.6272	−0.959997
$419$	−7.03623	−0.343742	−0.171871	−	0.985119i	$-0.554981\pi$
$419$	−7.03623	−0.343742	−0.171871	+	0.985119i	$0.554981\pi$
$420$	3.06600	0.149606
$421$	17.7623	0.865683	0.432841	−	0.901470i	$-0.357511\pi$
$421$	17.7623	0.865683	0.432841	+	0.901470i	$0.357511\pi$
$422$	−30.1590	−1.46812
$423$	−23.6833	−1.15152
$424$	−3.61135	−0.175383
$425$	−1.03069	−0.0499956
$426$	−10.0832	−0.488534
$427$	−30.7825	−1.48967
$428$	30.6018	1.47919
$429$	14.7872	0.713933
$430$	−12.7706	−0.615855
$431$	−27.1943	−1.30991	−0.654953	−	0.755670i	$-0.727311\pi$
$431$	−27.1943	−1.30991	−0.654953	+	0.755670i	$0.727311\pi$
$432$	17.3779	0.836097
$433$	26.8174	1.28876	0.644380	−	0.764705i	$-0.277116\pi$
$433$	26.8174	1.28876	0.644380	+	0.764705i	$0.277116\pi$
$434$	38.1122	1.82945
$435$	3.93165	0.188508
$436$	−10.0205	−0.479894
$437$	8.01111	0.383223
$438$	3.65828	0.174799
$439$	−12.8756	−0.614520	−0.307260	−	0.951626i	$-0.599412\pi$
$439$	−12.8756	−0.614520	−0.307260	+	0.951626i	$0.599412\pi$
$440$	3.36236	0.160294
$441$	−2.59424	−0.123535
$442$	−9.62095	−0.457622
$443$	10.0867	0.479233	0.239616	−	0.970868i	$-0.422978\pi$
$443$	10.0867	0.479233	0.239616	+	0.970868i	$0.422978\pi$
$444$	5.27685	0.250428
$445$	11.3910	0.539985
$446$	43.6122	2.06510
$447$	11.1453	0.527157
$448$	12.9112	0.609997
$449$	11.5306	0.544164	0.272082	−	0.962274i	$-0.412288\pi$
$449$	11.5306	0.544164	0.272082	+	0.962274i	$0.412288\pi$
$450$	4.82629	0.227513
$451$	−29.2565	−1.37764
$452$	−6.30058	−0.296355
$453$	−14.0694	−0.661039
$454$	−10.2037	−0.478883
$455$	−13.9293	−0.653018
$456$	−1.18824	−0.0556443
$457$	−12.7761	−0.597640	−0.298820	−	0.954309i	$-0.596593\pi$
$457$	−12.7761	−0.597640	−0.298820	+	0.954309i	$0.596593\pi$
$458$	5.44997	0.254660
$459$	3.86183	0.180255
$460$	5.51804	0.257280
$461$	−41.6907	−1.94173	−0.970865	−	0.239627i	$-0.922975\pi$
$461$	−41.6907	−1.94173	−0.970865	+	0.239627i	$0.922975\pi$
$462$	−16.1580	−0.751736
$463$	−30.1100	−1.39933	−0.699666	−	0.714470i	$-0.746668\pi$
$463$	−30.1100	−1.39933	−0.699666	+	0.714470i	$0.746668\pi$
$464$	26.9769	1.25237
$465$	−4.79333	−0.222286
$466$	5.91801	0.274147
$467$	−19.4320	−0.899204	−0.449602	−	0.893229i	$-0.648434\pi$
$467$	−19.4320	−0.899204	−0.449602	+	0.893229i	$0.648434\pi$
$468$	20.0342	0.926079
$469$	26.3007	1.21445
$470$	17.6738	0.815232
$471$	−3.28298	−0.151272
$472$	−2.24314	−0.103249
$473$	29.9290	1.37614
$474$	−11.3684	−0.522166
$475$	−2.32528	−0.106691
$476$	4.67504	0.214280
$477$	12.1484	0.556238
$478$	−49.1598	−2.24852
$479$	−3.40359	−0.155514	−0.0777570	−	0.996972i	$-0.524776\pi$
$479$	−3.40359	−0.155514	−0.0777570	+	0.996972i	$0.524776\pi$
$480$	−4.92772	−0.224918
$481$	−23.9735	−1.09310
$482$	−11.8869	−0.541433
$483$	6.59510	0.300087
$484$	14.0651	0.639324
$485$	13.9653	0.634131
$486$	−27.8703	−1.26422
$487$	−7.78641	−0.352836	−0.176418	−	0.984315i	$-0.556451\pi$
$487$	−7.78641	−0.352836	−0.176418	+	0.984315i	$0.556451\pi$
$488$	8.21725	0.371977
$489$	1.76358	0.0797520
$490$	1.93597	0.0874582
$491$	24.4275	1.10240	0.551198	−	0.834374i	$-0.314171\pi$
$491$	24.4275	1.10240	0.551198	+	0.834374i	$0.314171\pi$
$492$	7.12155	0.321064
$493$	5.99497	0.270000
$494$	−21.7054	−0.976571
$495$	−11.3108	−0.508382
$496$	−32.8894	−1.47678
$497$	−22.2600	−0.998496
$498$	−5.99312	−0.268558
$499$	−24.5861	−1.10062	−0.550312	−	0.834959i	$-0.685491\pi$
$499$	−24.5861	−1.10062	−0.550312	+	0.834959i	$0.685491\pi$
$500$	−1.60165	−0.0716281
$501$	4.73836	0.211694
$502$	−55.3472	−2.47027
$503$	−42.5073	−1.89531	−0.947654	−	0.319299i	$-0.896553\pi$
$503$	−42.5073	−1.89531	−0.947654	+	0.319299i	$0.896553\pi$
$504$	5.44459	0.242521
$505$	−2.48854	−0.110738
$506$	−29.0803	−1.29278
$507$	7.56558	0.335999
$508$	−32.1543	−1.42662
$509$	−24.9064	−1.10396	−0.551978	−	0.833859i	$-0.686127\pi$
$509$	−24.9064	−1.10396	−0.551978	+	0.833859i	$0.686127\pi$
$510$	−1.32218	−0.0585473
$511$	8.07610	0.357266
$512$	−26.7989	−1.18435
$513$	8.71249	0.384666
$514$	−28.3036	−1.24842
$515$	17.5396	0.772889
$516$	−7.28524	−0.320715
$517$	−41.4200	−1.82165
$518$	26.1959	1.15098
$519$	6.14773	0.269855
$520$	3.71838	0.163062
$521$	41.0681	1.79922	0.899612	−	0.436690i	$-0.143849\pi$
$521$	41.0681	1.79922	0.899612	+	0.436690i	$0.143849\pi$
$522$	−28.0720	−1.22868
$523$	33.2801	1.45524	0.727620	−	0.685981i	$-0.240627\pi$
$523$	33.2801	1.45524	0.727620	+	0.685981i	$0.240627\pi$
$524$	−18.2643	−0.797878
$525$	−1.91428	−0.0835458
$526$	37.4617	1.63341
$527$	−7.30886	−0.318379
$528$	13.9437	0.606821
$529$	−11.1305	−0.483934
$530$	−9.06585	−0.393795
$531$	7.54581	0.327460
$532$	10.5471	0.457276
$533$	−32.3543	−1.40142
$534$	14.6126	0.632349
$535$	−19.1064	−0.826041
$536$	−7.02085	−0.303255
$537$	−1.65802	−0.0715487
$538$	−42.8774	−1.84857
$539$	−4.53710	−0.195427
$540$	6.00115	0.258249
$541$	−16.9604	−0.729186	−0.364593	−	0.931167i	$-0.618792\pi$
$541$	−16.9604	−0.729186	−0.364593	+	0.931167i	$0.618792\pi$
$542$	−12.6259	−0.542328
$543$	14.2031	0.609514
$544$	−7.51377	−0.322150
$545$	6.25634	0.267992
$546$	−17.8688	−0.764715
$547$	−24.1266	−1.03158	−0.515790	−	0.856715i	$-0.672502\pi$
$547$	−24.1266	−1.03158	−0.515790	+	0.856715i	$0.672502\pi$
$548$	−4.31160	−0.184183
$549$	−27.6424	−1.17975
$550$	8.44077	0.359916
$551$	13.5250	0.576183
$552$	−1.76053	−0.0749332
$553$	−25.0970	−1.06723
$554$	51.2229	2.17625
$555$	−3.29463	−0.139849
$556$	25.9638	1.10111
$557$	−32.3121	−1.36911	−0.684554	−	0.728962i	$-0.740003\pi$
$557$	−32.3121	−1.36911	−0.684554	+	0.728962i	$0.740003\pi$
$558$	34.2244	1.44884
$559$	33.0980	1.39990
$560$	−13.1348	−0.555045
$561$	3.09864	0.130825
$562$	−2.45131	−0.103402
$563$	−27.1205	−1.14299	−0.571496	−	0.820605i	$-0.693637\pi$
$563$	−27.1205	−1.14299	−0.571496	+	0.820605i	$0.693637\pi$
$564$	10.0824	0.424544
$565$	3.93380	0.165496
$566$	−55.1042	−2.31620
$567$	−14.4335	−0.606148
$568$	5.94220	0.249329
$569$	9.08554	0.380886	0.190443	−	0.981698i	$-0.439008\pi$
$569$	9.08554	0.380886	0.190443	+	0.981698i	$0.439008\pi$
$570$	−2.98292	−0.124941
$571$	27.7228	1.16016	0.580081	−	0.814559i	$-0.303021\pi$
$571$	27.7228	1.16016	0.580081	+	0.814559i	$0.303021\pi$
$572$	35.0381	1.46501
$573$	−6.90686	−0.288538
$574$	35.3535	1.47563
$575$	−3.44522	−0.143676
$576$	11.5941	0.483089
$577$	2.50332	0.104214	0.0521072	−	0.998641i	$-0.483406\pi$
$577$	2.50332	0.104214	0.0521072	+	0.998641i	$0.483406\pi$
$578$	30.2466	1.25809
$579$	−15.3082	−0.636187
$580$	9.31599	0.386825
$581$	−13.2306	−0.548896
$582$	17.9149	0.742597
$583$	21.2466	0.879942
$584$	−2.15588	−0.0892109
$585$	−12.5084	−0.517160
$586$	−33.1467	−1.36928
$587$	−24.7245	−1.02049	−0.510245	−	0.860029i	$-0.670445\pi$
$587$	−24.7245	−1.02049	−0.510245	+	0.860029i	$0.670445\pi$
$588$	1.10441	0.0455451
$589$	−16.4892	−0.679425
$590$	−5.63112	−0.231829
$591$	−0.794966	−0.0327005
$592$	−22.6060	−0.929101
$593$	−8.68584	−0.356685	−0.178342	−	0.983968i	$-0.557073\pi$
$593$	−8.68584	−0.356685	−0.178342	+	0.983968i	$0.557073\pi$
$594$	−31.6263	−1.29764
$595$	−2.91888	−0.119663
$596$	26.4087	1.08174
$597$	−7.73566	−0.316599
$598$	−32.1594	−1.31510
$599$	22.9709	0.938566	0.469283	−	0.883048i	$-0.344512\pi$
$599$	22.9709	0.938566	0.469283	+	0.883048i	$0.344512\pi$
$600$	0.511007	0.0208618
$601$	30.3273	1.23708	0.618538	−	0.785755i	$-0.287725\pi$
$601$	30.3273	1.23708	0.618538	+	0.785755i	$0.287725\pi$
$602$	−36.1662	−1.47402
$603$	23.6178	0.961791
$604$	−33.3373	−1.35647
$605$	−8.78163	−0.357024
$606$	−3.19234	−0.129680
$607$	25.3595	1.02931	0.514654	−	0.857398i	$-0.327920\pi$
$607$	25.3595	1.02931	0.514654	+	0.857398i	$0.327920\pi$
$608$	−16.9515	−0.687473
$609$	11.1344	0.451187
$610$	20.6284	0.835218
$611$	−45.8058	−1.85310
$612$	4.19814	0.169700
$613$	15.5258	0.627080	0.313540	−	0.949575i	$-0.398485\pi$
$613$	15.5258	0.627080	0.313540	+	0.949575i	$0.398485\pi$
$614$	−30.4863	−1.23033
$615$	−4.44638	−0.179295
$616$	9.52213	0.383657
$617$	34.0496	1.37079	0.685394	−	0.728173i	$-0.259630\pi$
$617$	34.0496	1.37079	0.685394	+	0.728173i	$0.259630\pi$
$618$	22.5002	0.905090
$619$	38.8662	1.56216	0.781082	−	0.624429i	$-0.214668\pi$
$619$	38.8662	1.56216	0.781082	+	0.624429i	$0.214668\pi$
$620$	−11.3577	−0.456138
$621$	12.9087	0.518009
$622$	−1.25062	−0.0501452
$623$	32.2591	1.29243
$624$	15.4201	0.617298
$625$	1.00000	0.0400000
$626$	−5.04987	−0.201833
$627$	6.99071	0.279182
$628$	−7.77898	−0.310415
$629$	−5.02364	−0.200305
$630$	13.6679	0.544544
$631$	−38.6989	−1.54058	−0.770289	−	0.637695i	$-0.779888\pi$
$631$	−38.6989	−1.54058	−0.770289	+	0.637695i	$0.779888\pi$
$632$	6.69954	0.266493
$633$	10.7419	0.426951
$634$	47.7545	1.89658
$635$	20.0757	0.796679
$636$	−5.17178	−0.205074
$637$	−5.01751	−0.198801
$638$	−49.0956	−1.94371
$639$	−19.9892	−0.790762
$640$	5.92791	0.234321
$641$	−23.7769	−0.939130	−0.469565	−	0.882898i	$-0.655589\pi$
$641$	−23.7769	−0.939130	−0.469565	+	0.882898i	$0.655589\pi$
$642$	−24.5100	−0.967334
$643$	46.1994	1.82193	0.910963	−	0.412488i	$-0.135340\pi$
$643$	46.1994	1.82193	0.910963	+	0.412488i	$0.135340\pi$
$644$	15.6270	0.615790
$645$	4.54858	0.179100
$646$	−4.54834	−0.178952
$647$	−14.8191	−0.582600	−0.291300	−	0.956632i	$-0.594088\pi$
$647$	−14.8191	−0.582600	−0.291300	+	0.956632i	$0.594088\pi$
$648$	3.85294	0.151358
$649$	13.1970	0.518027
$650$	9.33451	0.366130
$651$	−13.5746	−0.532032
$652$	4.17878	0.163654
$653$	22.7053	0.888525	0.444263	−	0.895897i	$-0.353466\pi$
$653$	22.7053	0.888525	0.444263	+	0.895897i	$0.353466\pi$
$654$	8.02575	0.313832
$655$	11.4034	0.445567
$656$	−30.5087	−1.19117
$657$	7.25227	0.282938
$658$	50.0519	1.95123
$659$	19.3392	0.753347	0.376674	−	0.926346i	$-0.377068\pi$
$659$	19.3392	0.753347	0.376674	+	0.926346i	$0.377068\pi$
$660$	4.81519	0.187431
$661$	−29.8052	−1.15929	−0.579644	−	0.814870i	$-0.696809\pi$
$661$	−29.8052	−1.15929	−0.579644	+	0.814870i	$0.696809\pi$
$662$	−1.63439	−0.0635224
$663$	3.42674	0.133084
$664$	3.53184	0.137062
$665$	−6.58516	−0.255361
$666$	23.5237	0.911523
$667$	20.0390	0.775915
$668$	11.2275	0.434404
$669$	−15.5336	−0.600563
$670$	−17.6250	−0.680912
$671$	−48.3443	−1.86631
$672$	−13.9552	−0.538333
$673$	−26.5757	−1.02442	−0.512209	−	0.858861i	$-0.671173\pi$
$673$	−26.5757	−1.02442	−0.512209	+	0.858861i	$0.671173\pi$
$674$	−24.0938	−0.928060
$675$	−3.74685	−0.144216
$676$	17.9265	0.689482
$677$	29.9035	1.14929	0.574643	−	0.818404i	$-0.305141\pi$
$677$	29.9035	1.14929	0.574643	+	0.818404i	$0.305141\pi$
$678$	5.04635	0.193804
$679$	39.5494	1.51777
$680$	0.779182	0.0298803
$681$	3.63430	0.139267
$682$	59.8557	2.29199
$683$	−0.207184	−0.00792769	−0.00396384	−	0.999992i	$-0.501262\pi$
$683$	−0.207184	−0.00792769	−0.00396384	+	0.999992i	$0.501262\pi$
$684$	9.47123	0.362142
$685$	2.69197	0.102855
$686$	−32.1391	−1.22708
$687$	−1.94114	−0.0740592
$688$	31.2100	1.18987
$689$	23.4962	0.895135
$690$	−4.41959	−0.168251
$691$	−31.3776	−1.19366	−0.596830	−	0.802367i	$-0.703573\pi$
$691$	−31.3776	−1.19366	−0.596830	+	0.802367i	$0.703573\pi$
$692$	14.5670	0.553753
$693$	−32.0319	−1.21679
$694$	−49.2000	−1.86761
$695$	−16.2106	−0.614904
$696$	−2.97226	−0.112663
$697$	−6.77982	−0.256804
$698$	58.9324	2.23062
$699$	−2.10785	−0.0797261
$700$	−4.53585	−0.171439
$701$	18.8940	0.713617	0.356808	−	0.934178i	$-0.383865\pi$
$701$	18.8940	0.713617	0.356808	+	0.934178i	$0.383865\pi$
$702$	−34.9750	−1.32005
$703$	−11.3336	−0.427455
$704$	20.2772	0.764225
$705$	−6.29497	−0.237082
$706$	−9.36442	−0.352434
$707$	−7.04749	−0.265048
$708$	−3.21238	−0.120728
$709$	32.8031	1.23194	0.615972	−	0.787768i	$-0.288763\pi$
$709$	32.8031	1.23194	0.615972	+	0.787768i	$0.288763\pi$
$710$	14.9171	0.559830
$711$	−22.5369	−0.845200
$712$	−8.61142	−0.322727
$713$	−24.4309	−0.914946
$714$	−3.74440	−0.140131
$715$	−21.8762	−0.818123
$716$	−3.92865	−0.146820
$717$	17.5095	0.653904
$718$	−61.3920	−2.29113
$719$	21.0894	0.786502	0.393251	−	0.919431i	$-0.371350\pi$
$719$	21.0894	0.786502	0.393251	+	0.919431i	$0.371350\pi$
$720$	−11.7949	−0.439570
$721$	49.6719	1.84988
$722$	25.7969	0.960062
$723$	4.23382	0.157457
$724$	33.6541	1.25074
$725$	−5.81648	−0.216019
$726$	−11.2652	−0.418092
$727$	8.74125	0.324195	0.162098	−	0.986775i	$-0.448174\pi$
$727$	8.74125	0.324195	0.162098	+	0.986775i	$0.448174\pi$
$728$	10.5304	0.390281
$729$	−5.36306	−0.198632
$730$	−5.41206	−0.200309
$731$	6.93566	0.256525
$732$	11.7678	0.434952
$733$	19.8119	0.731768	0.365884	−	0.930661i	$-0.380767\pi$
$733$	19.8119	0.731768	0.365884	+	0.930661i	$0.380767\pi$
$734$	10.9137	0.402833
$735$	−0.689544	−0.0254342
$736$	−25.1159	−0.925783
$737$	41.3055	1.52151
$738$	31.7472	1.16863
$739$	32.0112	1.17755	0.588776	−	0.808296i	$-0.299610\pi$
$739$	32.0112	1.17755	0.588776	+	0.808296i	$0.299610\pi$
$740$	−7.80657	−0.286975
$741$	7.73091	0.284002
$742$	−25.6743	−0.942533
$743$	23.6903	0.869114	0.434557	−	0.900644i	$-0.356905\pi$
$743$	23.6903	0.869114	0.434557	+	0.900644i	$0.356905\pi$
$744$	3.62368	0.132851
$745$	−16.4884	−0.604089
$746$	25.0303	0.916426
$747$	−11.8809	−0.434700
$748$	7.34219	0.268457
$749$	−54.1089	−1.97710
$750$	1.28282	0.0468419
$751$	−16.9011	−0.616729	−0.308364	−	0.951268i	$-0.599782\pi$
$751$	−16.9011	−0.616729	−0.308364	+	0.951268i	$0.599782\pi$
$752$	−43.1928	−1.57508
$753$	19.7133	0.718392
$754$	−54.2940	−1.97727
$755$	20.8143	0.757510
$756$	16.9951	0.618108
$757$	19.0048	0.690742	0.345371	−	0.938466i	$-0.387753\pi$
$757$	19.0048	0.690742	0.345371	+	0.938466i	$0.387753\pi$
$758$	21.3864	0.776790
$759$	10.3577	0.375960
$760$	1.75788	0.0637649
$761$	49.0185	1.77692	0.888459	−	0.458955i	$-0.151776\pi$
$761$	49.0185	1.77692	0.888459	+	0.458955i	$0.151776\pi$
$762$	25.7535	0.932949
$763$	17.7178	0.641428
$764$	−16.3657	−0.592090
$765$	−2.62113	−0.0947672
$766$	39.7880	1.43760
$767$	14.5943	0.526971
$768$	13.7678	0.496804
$769$	−33.2931	−1.20058	−0.600289	−	0.799783i	$-0.704948\pi$
$769$	−33.2931	−1.20058	−0.600289	+	0.799783i	$0.704948\pi$
$770$	23.9041	0.861443
$771$	10.0810	0.363059
$772$	−36.2725	−1.30548
$773$	−2.27055	−0.0816660	−0.0408330	−	0.999166i	$-0.513001\pi$
$773$	−2.27055	−0.0816660	−0.0408330	+	0.999166i	$0.513001\pi$
$774$	−32.4769	−1.16736
$775$	7.09126	0.254726
$776$	−10.5575	−0.378993
$777$	−9.33031	−0.334723
$778$	42.2830	1.51592
$779$	−15.2956	−0.548023
$780$	5.32504	0.190667
$781$	−34.9595	−1.25095
$782$	−6.73898	−0.240985
$783$	21.7935	0.778836
$784$	−4.73129	−0.168975
$785$	4.85684	0.173348
$786$	14.6285	0.521780
$787$	−20.0781	−0.715708	−0.357854	−	0.933778i	$-0.616491\pi$
$787$	−20.0781	−0.715708	−0.357854	+	0.933778i	$0.616491\pi$
$788$	−1.88366	−0.0671026
$789$	−13.3429	−0.475020
$790$	16.8183	0.598370
$791$	11.1404	0.396109
$792$	8.55078	0.303839
$793$	−53.4631	−1.89853
$794$	−4.44359	−0.157697
$795$	3.22903	0.114522
$796$	−18.3295	−0.649673
$797$	−23.6907	−0.839169	−0.419584	−	0.907716i	$-0.637824\pi$
$797$	−23.6907	−0.839169	−0.419584	+	0.907716i	$0.637824\pi$
$798$	−8.44756	−0.299040
$799$	−9.59856	−0.339572
$800$	7.29006	0.257743
$801$	28.9684	1.02355
$802$	−1.89780	−0.0670137
$803$	12.6836	0.447595
$804$	−10.0545	−0.354594
$805$	−9.75679	−0.343882
$806$	66.1934	2.33156
$807$	15.2718	0.537594
$808$	1.88129	0.0661837
$809$	−22.3088	−0.784335	−0.392168	−	0.919894i	$-0.628275\pi$
$809$	−22.3088	−0.784335	−0.392168	+	0.919894i	$0.628275\pi$
$810$	9.67233	0.339851
$811$	42.6341	1.49709	0.748543	−	0.663086i	$-0.230754\pi$
$811$	42.6341	1.49709	0.748543	+	0.663086i	$0.230754\pi$
$812$	26.3827	0.925851
$813$	4.49702	0.157718
$814$	41.1409	1.44199
$815$	−2.60904	−0.0913908
$816$	3.23127	0.113117
$817$	15.6472	0.547427
$818$	11.1329	0.389252
$819$	−35.4236	−1.23780
$820$	−10.5356	−0.367920
$821$	−10.5588	−0.368506	−0.184253	−	0.982879i	$-0.558987\pi$
$821$	−10.5588	−0.368506	−0.184253	+	0.982879i	$0.558987\pi$
$822$	3.45331	0.120448
$823$	37.3564	1.30216	0.651081	−	0.759009i	$-0.274316\pi$
$823$	37.3564	1.30216	0.651081	+	0.759009i	$0.274316\pi$
$824$	−13.2597	−0.461923
$825$	−3.00639	−0.104669
$826$	−15.9472	−0.554875
$827$	5.62178	0.195488	0.0977442	−	0.995212i	$-0.468837\pi$
$827$	5.62178	0.195488	0.0977442	+	0.995212i	$0.468837\pi$
$828$	14.0329	0.487677
$829$	42.1771	1.46487	0.732436	−	0.680836i	$-0.238383\pi$
$829$	42.1771	1.46487	0.732436	+	0.680836i	$0.238383\pi$
$830$	8.86623	0.307751
$831$	−18.2443	−0.632889
$832$	22.4242	0.777419
$833$	−1.05141	−0.0364294
$834$	−20.7953	−0.720082
$835$	−7.00993	−0.242589
$836$	16.5644	0.572891
$837$	−26.5699	−0.918390
$838$	13.3534	0.461284
$839$	18.7622	0.647742	0.323871	−	0.946101i	$-0.395016\pi$
$839$	18.7622	0.647742	0.323871	+	0.946101i	$0.395016\pi$
$840$	1.44716	0.0499318
$841$	4.83147	0.166602
$842$	−33.7094	−1.16170
$843$	0.873096	0.0300710
$844$	25.4527	0.876118
$845$	−11.1925	−0.385035
$846$	44.9461	1.54528
$847$	−24.8694	−0.854523
$848$	22.1559	0.760837
$849$	19.6267	0.673588
$850$	1.95604	0.0670916
$851$	−16.7922	−0.575630
$852$	8.50975	0.291539
$853$	13.8796	0.475229	0.237615	−	0.971360i	$-0.423634\pi$
$853$	13.8796	0.475229	0.237615	+	0.971360i	$0.423634\pi$
$854$	58.4191	1.99906
$855$	−5.91341	−0.202234
$856$	14.4441	0.493690
$857$	−5.36809	−0.183370	−0.0916852	−	0.995788i	$-0.529225\pi$
$857$	−5.36809	−0.183370	−0.0916852	+	0.995788i	$0.529225\pi$
$858$	−28.0632	−0.958061
$859$	−22.5090	−0.767998	−0.383999	−	0.923334i	$-0.625453\pi$
$859$	−22.5090	−0.767998	−0.383999	+	0.923334i	$0.625453\pi$
$860$	10.7778	0.367520
$861$	−12.5920	−0.429136
$862$	51.6095	1.75783
$863$	−9.30185	−0.316639	−0.158319	−	0.987388i	$-0.550608\pi$
$863$	−9.30185	−0.316639	−0.158319	+	0.987388i	$0.550608\pi$
$864$	−27.3148	−0.929268
$865$	−9.09495	−0.309238
$866$	−50.8940	−1.72945
$867$	−10.7731	−0.365873
$868$	−32.1649	−1.09175
$869$	−39.4152	−1.33707
$870$	−7.46149	−0.252968
$871$	45.6791	1.54778
$872$	−4.72969	−0.160168
$873$	35.5150	1.20200
$874$	−15.2035	−0.514266
$875$	2.83198	0.0957384
$876$	−3.08741	−0.104314
$877$	10.2108	0.344794	0.172397	−	0.985028i	$-0.444849\pi$
$877$	10.2108	0.344794	0.172397	+	0.985028i	$0.444849\pi$
$878$	24.4354	0.824654
$879$	11.8060	0.398207
$880$	−20.6283	−0.695379
$881$	−4.81815	−0.162327	−0.0811637	−	0.996701i	$-0.525864\pi$
$881$	−4.81815	−0.162327	−0.0811637	+	0.996701i	$0.525864\pi$
$882$	4.92335	0.165778
$883$	29.4743	0.991891	0.495945	−	0.868354i	$-0.334822\pi$
$883$	29.4743	0.991891	0.495945	+	0.868354i	$0.334822\pi$
$884$	8.11961	0.273092
$885$	2.00566	0.0674196
$886$	−19.1425	−0.643106
$887$	51.0769	1.71500	0.857498	−	0.514487i	$-0.172018\pi$
$887$	51.0769	1.71500	0.857498	+	0.514487i	$0.172018\pi$
$888$	2.49068	0.0835819
$889$	56.8539	1.90682
$890$	−21.6179	−0.724633
$891$	−22.6679	−0.759403
$892$	−36.8066	−1.23238
$893$	−21.6549	−0.724652
$894$	−21.1517	−0.707417
$895$	2.45287	0.0819904
$896$	16.7877	0.560839
$897$	11.4544	0.382451
$898$	−21.8828	−0.730240
$899$	−41.2462	−1.37564
$900$	−4.07315	−0.135772
$901$	4.92361	0.164029
$902$	55.5231	1.84872
$903$	12.8815	0.428669
$904$	−2.97389	−0.0989102
$905$	−21.0121	−0.698465
$906$	26.7010	0.887080
$907$	33.6744	1.11814	0.559070	−	0.829121i	$-0.311158\pi$
$907$	33.6744	1.11814	0.559070	+	0.829121i	$0.311158\pi$
$908$	8.61141	0.285780
$909$	−6.32858	−0.209906
$910$	26.4351	0.876317
$911$	9.84059	0.326033	0.163017	−	0.986623i	$-0.447878\pi$
$911$	9.84059	0.326033	0.163017	+	0.986623i	$0.447878\pi$
$912$	7.28991	0.241393
$913$	−20.7787	−0.687676
$914$	24.2465	0.802003
$915$	−7.34731	−0.242894
$916$	−4.59951	−0.151972
$917$	32.2941	1.06645
$918$	−7.32898	−0.241893
$919$	−37.0597	−1.22249	−0.611243	−	0.791443i	$-0.709330\pi$
$919$	−37.0597	−1.22249	−0.611243	+	0.791443i	$0.709330\pi$
$920$	2.60453	0.0858688
$921$	10.8584	0.357798
$922$	79.1207	2.60570
$923$	−38.6612	−1.27255
$924$	13.6365	0.448609
$925$	4.87407	0.160258
$926$	57.1429	1.87783
$927$	44.6049	1.46502
$928$	−42.4025	−1.39193
$929$	−22.2859	−0.731176	−0.365588	−	0.930777i	$-0.619132\pi$
$929$	−22.2859	−0.731176	−0.365588	+	0.930777i	$0.619132\pi$
$930$	9.09680	0.298296
$931$	−2.37205	−0.0777407
$932$	−4.99452	−0.163601
$933$	0.445439	0.0145830
$934$	36.8780	1.20669
$935$	−4.58414	−0.149917
$936$	9.45617	0.309085
$937$	2.43335	0.0794942	0.0397471	−	0.999210i	$-0.487345\pi$
$937$	2.43335	0.0794942	0.0397471	+	0.999210i	$0.487345\pi$
$938$	−49.9135	−1.62973
$939$	1.79864	0.0586963
$940$	−14.9158	−0.486501
$941$	−28.7019	−0.935656	−0.467828	−	0.883820i	$-0.654963\pi$
$941$	−28.7019	−0.935656	−0.467828	+	0.883820i	$0.654963\pi$
$942$	6.23045	0.202999
$943$	−22.6625	−0.737994
$944$	13.7618	0.447909
$945$	−10.6110	−0.345176
$946$	−56.7994	−1.84671
$947$	−28.4335	−0.923966	−0.461983	−	0.886889i	$-0.652862\pi$
$947$	−28.4335	−0.923966	−0.461983	+	0.886889i	$0.652862\pi$
$948$	9.59434	0.311610
$949$	14.0266	0.455323
$950$	4.41293	0.143174
$951$	−17.0090	−0.551554
$952$	2.20663	0.0715173
$953$	4.91678	0.159270	0.0796351	−	0.996824i	$-0.474624\pi$
$953$	4.91678	0.159270	0.0796351	+	0.996824i	$0.474624\pi$
$954$	−23.0553	−0.746442
$955$	10.2180	0.330647
$956$	41.4885	1.34183
$957$	17.4866	0.565262
$958$	6.45934	0.208692
$959$	7.62361	0.246179
$960$	3.08170	0.0994615
$961$	19.2860	0.622128
$962$	45.4970	1.46688
$963$	−48.5893	−1.56577
$964$	10.0320	0.323108
$965$	22.6469	0.729031
$966$	−12.5162	−0.402702
$967$	46.6949	1.50161	0.750803	−	0.660526i	$-0.229667\pi$
$967$	46.6949	1.50161	0.750803	+	0.660526i	$0.229667\pi$
$968$	6.63877	0.213378
$969$	1.62001	0.0520421
$970$	−26.5033	−0.850971
$971$	16.3604	0.525031	0.262515	−	0.964928i	$-0.415448\pi$
$971$	16.3604	0.525031	0.262515	+	0.964928i	$0.415448\pi$
$972$	23.5212	0.754444
$973$	−45.9082	−1.47175
$974$	14.7771	0.473488
$975$	−3.32472	−0.106476
$976$	−50.4134	−1.61369
$977$	−49.8298	−1.59420	−0.797098	−	0.603850i	$-0.793633\pi$
$977$	−49.8298	−1.59420	−0.797098	+	0.603850i	$0.793633\pi$
$978$	−3.34693	−0.107023
$979$	50.6632	1.61920
$980$	−1.63386	−0.0521919
$981$	15.9104	0.507981
$982$	−46.3585	−1.47936
$983$	−5.83599	−0.186139	−0.0930696	−	0.995660i	$-0.529668\pi$
$983$	−5.83599	−0.186139	−0.0930696	+	0.995660i	$0.529668\pi$
$984$	3.36139	0.107157
$985$	1.17607	0.0374728
$986$	−11.3773	−0.362326
$987$	−17.8272	−0.567447
$988$	18.3183	0.582782
$989$	23.1835	0.737191
$990$	21.4657	0.682223
$991$	−35.8751	−1.13961	−0.569804	−	0.821780i	$-0.692981\pi$
$991$	−35.8751	−1.13961	−0.569804	+	0.821780i	$0.692981\pi$
$992$	51.6957	1.64134
$993$	0.582129	0.0184733
$994$	42.2450	1.33993
$995$	11.4441	0.362803
$996$	5.05791	0.160266
$997$	−22.2570	−0.704886	−0.352443	−	0.935833i	$-0.614649\pi$
$997$	−22.2570	−0.704886	−0.352443	+	0.935833i	$0.614649\pi$
$998$	46.6595	1.47698
$999$	−18.2624	−0.577797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.f.1.7	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.f.1.7	✓	37	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89780 q^{2} +0.675950 q^{3} +1.60165 q^{4} -1.00000 q^{5} -1.28282 q^{6} -2.83198 q^{7} +0.755984 q^{8} -2.54309 q^{9} +O(q^{10})\)	\(q-1.89780 q^{2} +0.675950 q^{3} +1.60165 q^{4} -1.00000 q^{5} -1.28282 q^{6} -2.83198 q^{7} +0.755984 q^{8} -2.54309 q^{9} +1.89780 q^{10} -4.44765 q^{11} +1.08264 q^{12} -4.91859 q^{13} +5.37454 q^{14} -0.675950 q^{15} -4.63801 q^{16} -1.03069 q^{17} +4.82629 q^{18} -2.32528 q^{19} -1.60165 q^{20} -1.91428 q^{21} +8.44077 q^{22} -3.44522 q^{23} +0.511007 q^{24} +1.00000 q^{25} +9.33451 q^{26} -3.74685 q^{27} -4.53585 q^{28} -5.81648 q^{29} +1.28282 q^{30} +7.09126 q^{31} +7.29006 q^{32} -3.00639 q^{33} +1.95604 q^{34} +2.83198 q^{35} -4.07315 q^{36} +4.87407 q^{37} +4.41293 q^{38} -3.32472 q^{39} -0.755984 q^{40} +6.57797 q^{41} +3.63292 q^{42} -6.72917 q^{43} -7.12360 q^{44} +2.54309 q^{45} +6.53834 q^{46} +9.31278 q^{47} -3.13506 q^{48} +1.02011 q^{49} -1.89780 q^{50} -0.696692 q^{51} -7.87787 q^{52} -4.77702 q^{53} +7.11078 q^{54} +4.44765 q^{55} -2.14093 q^{56} -1.57177 q^{57} +11.0385 q^{58} -2.96718 q^{59} -1.08264 q^{60} +10.8696 q^{61} -13.4578 q^{62} +7.20199 q^{63} -4.55907 q^{64} +4.91859 q^{65} +5.70553 q^{66} -9.28704 q^{67} -1.65080 q^{68} -2.32879 q^{69} -5.37454 q^{70} +7.86021 q^{71} -1.92254 q^{72} -2.85175 q^{73} -9.25002 q^{74} +0.675950 q^{75} -3.72430 q^{76} +12.5957 q^{77} +6.30966 q^{78} +8.86201 q^{79} +4.63801 q^{80} +5.09659 q^{81} -12.4837 q^{82} +4.67184 q^{83} -3.06600 q^{84} +1.03069 q^{85} +12.7706 q^{86} -3.93165 q^{87} -3.36236 q^{88} -11.3910 q^{89} -4.82629 q^{90} +13.9293 q^{91} -5.51804 q^{92} +4.79333 q^{93} -17.6738 q^{94} +2.32528 q^{95} +4.92772 q^{96} -13.9653 q^{97} -1.93597 q^{98} +11.3108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more