LMFDB - Embedded newform 6047.2.a.a.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6047,2,Mod(1,6047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6047, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6047 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6047.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:	$48.2855381023$
Analytic rank:	$1$
Dimension:	$217$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44753 q^{2} -2.68992 q^{3} +3.99041 q^{4} -1.41647 q^{5} +6.58366 q^{6} -0.890122 q^{7} -4.87158 q^{8} +4.23566 q^{9} +O(q^{10})$	\(q-2.44753 q^{2} -2.68992 q^{3} +3.99041 q^{4} -1.41647 q^{5} +6.58366 q^{6} -0.890122 q^{7} -4.87158 q^{8} +4.23566 q^{9} +3.46686 q^{10} +5.63194 q^{11} -10.7339 q^{12} +6.51825 q^{13} +2.17860 q^{14} +3.81019 q^{15} +3.94253 q^{16} +0.509200 q^{17} -10.3669 q^{18} +2.70183 q^{19} -5.65229 q^{20} +2.39435 q^{21} -13.7843 q^{22} +0.817422 q^{23} +13.1041 q^{24} -2.99361 q^{25} -15.9536 q^{26} -3.32382 q^{27} -3.55195 q^{28} +8.47541 q^{29} -9.32556 q^{30} +6.30475 q^{31} +0.0937034 q^{32} -15.1494 q^{33} -1.24628 q^{34} +1.26083 q^{35} +16.9020 q^{36} -9.26639 q^{37} -6.61280 q^{38} -17.5335 q^{39} +6.90045 q^{40} -0.0960551 q^{41} -5.86026 q^{42} -1.68517 q^{43} +22.4737 q^{44} -5.99969 q^{45} -2.00066 q^{46} -9.10936 q^{47} -10.6051 q^{48} -6.20768 q^{49} +7.32695 q^{50} -1.36971 q^{51} +26.0105 q^{52} -1.10602 q^{53} +8.13514 q^{54} -7.97748 q^{55} +4.33630 q^{56} -7.26769 q^{57} -20.7438 q^{58} -6.44584 q^{59} +15.2042 q^{60} -7.59288 q^{61} -15.4311 q^{62} -3.77025 q^{63} -8.11440 q^{64} -9.23291 q^{65} +37.0787 q^{66} -5.28086 q^{67} +2.03191 q^{68} -2.19880 q^{69} -3.08592 q^{70} +14.7116 q^{71} -20.6343 q^{72} +3.10261 q^{73} +22.6798 q^{74} +8.05256 q^{75} +10.7814 q^{76} -5.01311 q^{77} +42.9139 q^{78} -5.94542 q^{79} -5.58448 q^{80} -3.76618 q^{81} +0.235098 q^{82} -2.53055 q^{83} +9.55445 q^{84} -0.721267 q^{85} +4.12451 q^{86} -22.7982 q^{87} -27.4364 q^{88} +7.64966 q^{89} +14.6844 q^{90} -5.80203 q^{91} +3.26184 q^{92} -16.9592 q^{93} +22.2954 q^{94} -3.82706 q^{95} -0.252055 q^{96} +6.26189 q^{97} +15.1935 q^{98} +23.8550 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) $ 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.44753	−1.73067	−0.865333	−	0.501198i	$-0.832893\pi$
$2$	−2.44753	−1.73067	−0.865333	+	0.501198i	$0.832893\pi$
$3$	−2.68992	−1.55302	−0.776512	−	0.630102i	$-0.783013\pi$
$3$	−2.68992	−1.55302	−0.776512	+	0.630102i	$0.783013\pi$
$4$	3.99041	1.99520
$5$	−1.41647	−0.633465	−0.316733	−	0.948515i	$-0.602586\pi$
$5$	−1.41647	−0.633465	−0.316733	+	0.948515i	$0.602586\pi$
$6$	6.58366	2.68777
$7$	−0.890122	−0.336434	−0.168217	−	0.985750i	$-0.553801\pi$
$7$	−0.890122	−0.336434	−0.168217	+	0.985750i	$0.553801\pi$
$8$	−4.87158	−1.72236
$9$	4.23566	1.41189
$10$	3.46686	1.09632
$11$	5.63194	1.69809	0.849046	−	0.528318i	$-0.177177\pi$
$11$	5.63194	1.69809	0.849046	+	0.528318i	$0.177177\pi$
$12$	−10.7339	−3.09860
$13$	6.51825	1.80784	0.903918	−	0.427705i	$-0.140678\pi$
$13$	6.51825	1.80784	0.903918	+	0.427705i	$0.140678\pi$
$14$	2.17860	0.582256
$15$	3.81019	0.983787
$16$	3.94253	0.985632
$17$	0.509200	0.123499	0.0617495	−	0.998092i	$-0.480332\pi$
$17$	0.509200	0.123499	0.0617495	+	0.998092i	$0.480332\pi$
$18$	−10.3669	−2.44350
$19$	2.70183	0.619842	0.309921	−	0.950762i	$-0.399697\pi$
$19$	2.70183	0.619842	0.309921	+	0.950762i	$0.399697\pi$
$20$	−5.65229	−1.26389
$21$	2.39435	0.522491
$22$	−13.7843	−2.93883
$23$	0.817422	0.170444	0.0852221	−	0.996362i	$-0.472840\pi$
$23$	0.817422	0.170444	0.0852221	+	0.996362i	$0.472840\pi$
$24$	13.1041	2.67487
$25$	−2.99361	−0.598722
$26$	−15.9536	−3.12876
$27$	−3.32382	−0.639669
$28$	−3.55195	−0.671255
$29$	8.47541	1.57384	0.786922	−	0.617052i	$-0.211673\pi$
$29$	8.47541	1.57384	0.786922	+	0.617052i	$0.211673\pi$
$30$	−9.32556	−1.70261
$31$	6.30475	1.13237	0.566183	−	0.824280i	$-0.308420\pi$
$31$	6.30475	1.13237	0.566183	+	0.824280i	$0.308420\pi$
$32$	0.0937034	0.0165646
$33$	−15.1494	−2.63718
$34$	−1.24628	−0.213736
$35$	1.26083	0.213120
$36$	16.9020	2.81700
$37$	−9.26639	−1.52339	−0.761693	−	0.647938i	$-0.775631\pi$
$37$	−9.26639	−1.52339	−0.761693	+	0.647938i	$0.775631\pi$
$38$	−6.61280	−1.07274
$39$	−17.5335	−2.80761
$40$	6.90045	1.09106
$41$	−0.0960551	−0.0150013	−0.00750064	−	0.999972i	$-0.502388\pi$
$41$	−0.0960551	−0.0150013	−0.00750064	+	0.999972i	$0.502388\pi$
$42$	−5.86026	−0.904257
$43$	−1.68517	−0.256986	−0.128493	−	0.991710i	$-0.541014\pi$
$43$	−1.68517	−0.256986	−0.128493	+	0.991710i	$0.541014\pi$
$44$	22.4737	3.38804
$45$	−5.99969	−0.894380
$46$	−2.00066	−0.294982
$47$	−9.10936	−1.32874	−0.664369	−	0.747405i	$-0.731300\pi$
$47$	−9.10936	−1.32874	−0.664369	+	0.747405i	$0.731300\pi$
$48$	−10.6051	−1.53071
$49$	−6.20768	−0.886812
$50$	7.32695	1.03619
$51$	−1.36971	−0.191797
$52$	26.0105	3.60700
$53$	−1.10602	−0.151923	−0.0759616	−	0.997111i	$-0.524203\pi$
$53$	−1.10602	−0.151923	−0.0759616	+	0.997111i	$0.524203\pi$
$54$	8.13514	1.10705
$55$	−7.97748	−1.07568
$56$	4.33630	0.579462
$57$	−7.26769	−0.962629
$58$	−20.7438	−2.72380
$59$	−6.44584	−0.839177	−0.419589	−	0.907714i	$-0.637826\pi$
$59$	−6.44584	−0.839177	−0.419589	+	0.907714i	$0.637826\pi$
$60$	15.2042	1.96285
$61$	−7.59288	−0.972169	−0.486084	−	0.873912i	$-0.661575\pi$
$61$	−7.59288	−0.972169	−0.486084	+	0.873912i	$0.661575\pi$
$62$	−15.4311	−1.95975
$63$	−3.77025	−0.475007
$64$	−8.11440	−1.01430
$65$	−9.23291	−1.14520
$66$	37.0787	4.56408
$67$	−5.28086	−0.645160	−0.322580	−	0.946542i	$-0.604550\pi$
$67$	−5.28086	−0.645160	−0.322580	+	0.946542i	$0.604550\pi$
$68$	2.03191	0.246406
$69$	−2.19880	−0.264704
$70$	−3.08592	−0.368839
$71$	14.7116	1.74595	0.872975	−	0.487766i	$-0.162188\pi$
$71$	14.7116	1.74595	0.872975	+	0.487766i	$0.162188\pi$
$72$	−20.6343	−2.43178
$73$	3.10261	0.363133	0.181567	−	0.983379i	$-0.441883\pi$
$73$	3.10261	0.363133	0.181567	+	0.983379i	$0.441883\pi$
$74$	22.6798	2.63647
$75$	8.05256	0.929830
$76$	10.7814	1.23671
$77$	−5.01311	−0.571297
$78$	42.9139	4.85904
$79$	−5.94542	−0.668912	−0.334456	−	0.942411i	$-0.608553\pi$
$79$	−5.94542	−0.668912	−0.334456	+	0.942411i	$0.608553\pi$
$80$	−5.58448	−0.624363
$81$	−3.76618	−0.418464
$82$	0.235098	0.0259622
$83$	−2.53055	−0.277763	−0.138882	−	0.990309i	$-0.544351\pi$
$83$	−2.53055	−0.277763	−0.138882	+	0.990309i	$0.544351\pi$
$84$	9.55445	1.04248
$85$	−0.721267	−0.0782323
$86$	4.12451	0.444757
$87$	−22.7982	−2.44422
$88$	−27.4364	−2.92473
$89$	7.64966	0.810862	0.405431	−	0.914126i	$-0.367121\pi$
$89$	7.64966	0.810862	0.405431	+	0.914126i	$0.367121\pi$
$90$	14.6844	1.54787
$91$	−5.80203	−0.608219
$92$	3.26184	0.340071
$93$	−16.9592	−1.75859
$94$	22.2954	2.29960
$95$	−3.82706	−0.392648
$96$	−0.252055	−0.0257252
$97$	6.26189	0.635799	0.317899	−	0.948124i	$-0.397023\pi$
$97$	6.26189	0.635799	0.317899	+	0.948124i	$0.397023\pi$
$98$	15.1935	1.53477
$99$	23.8550	2.39751
$100$	−11.9457	−1.19457
$101$	−16.5310	−1.64489	−0.822446	−	0.568844i	$-0.807391\pi$
$101$	−16.5310	−1.64489	−0.822446	+	0.568844i	$0.807391\pi$
$102$	3.35240	0.331937
$103$	−9.78801	−0.964441	−0.482221	−	0.876050i	$-0.660170\pi$
$103$	−9.78801	−0.964441	−0.482221	+	0.876050i	$0.660170\pi$
$104$	−31.7542	−3.11375
$105$	−3.39153	−0.330980
$106$	2.70701	0.262928
$107$	−17.9574	−1.73601	−0.868006	−	0.496554i	$-0.834598\pi$
$107$	−17.9574	−1.73601	−0.868006	+	0.496554i	$0.834598\pi$
$108$	−13.2634	−1.27627
$109$	4.19032	0.401359	0.200680	−	0.979657i	$-0.435685\pi$
$109$	4.19032	0.401359	0.200680	+	0.979657i	$0.435685\pi$
$110$	19.5251	1.86165
$111$	24.9258	2.36586
$112$	−3.50933	−0.331600
$113$	2.11039	0.198529	0.0992645	−	0.995061i	$-0.468351\pi$
$113$	2.11039	0.198529	0.0992645	+	0.995061i	$0.468351\pi$
$114$	17.7879	1.66599
$115$	−1.15785	−0.107970
$116$	33.8203	3.14014
$117$	27.6091	2.55246
$118$	15.7764	1.45234
$119$	−0.453250	−0.0415493
$120$	−18.5616	−1.69444
$121$	20.7187	1.88352
$122$	18.5838	1.68250
$123$	0.258380	0.0232974
$124$	25.1585	2.25930
$125$	11.3227	1.01273
$126$	9.22781	0.822078
$127$	−10.3827	−0.921318	−0.460659	−	0.887577i	$-0.652387\pi$
$127$	−10.3827	−0.921318	−0.460659	+	0.887577i	$0.652387\pi$
$128$	19.6728	1.73885
$129$	4.53297	0.399106
$130$	22.5978	1.98196
$131$	−8.52841	−0.745131	−0.372565	−	0.928006i	$-0.621522\pi$
$131$	−8.52841	−0.745131	−0.372565	+	0.928006i	$0.621522\pi$
$132$	−60.4524	−5.26171
$133$	−2.40496	−0.208536
$134$	12.9251	1.11656
$135$	4.70809	0.405208
$136$	−2.48061	−0.212710
$137$	−13.8047	−1.17941	−0.589707	−	0.807617i	$-0.700757\pi$
$137$	−13.8047	−1.17941	−0.589707	+	0.807617i	$0.700757\pi$
$138$	5.38162	0.458114
$139$	−10.7081	−0.908253	−0.454126	−	0.890937i	$-0.650049\pi$
$139$	−10.7081	−0.908253	−0.454126	+	0.890937i	$0.650049\pi$
$140$	5.03123	0.425217
$141$	24.5034	2.06356
$142$	−36.0072	−3.02165
$143$	36.7104	3.06987
$144$	16.6992	1.39160
$145$	−12.0052	−0.996976
$146$	−7.59374	−0.628462
$147$	16.6982	1.37724
$148$	−36.9767	−3.03946
$149$	−12.2400	−1.00274	−0.501372	−	0.865232i	$-0.667171\pi$
$149$	−12.2400	−1.00274	−0.501372	+	0.865232i	$0.667171\pi$
$150$	−19.7089	−1.60922
$151$	−12.3544	−1.00539	−0.502694	−	0.864464i	$-0.667658\pi$
$151$	−12.3544	−1.00539	−0.502694	+	0.864464i	$0.667658\pi$
$152$	−13.1622	−1.06759
$153$	2.15680	0.174367
$154$	12.2697	0.988724
$155$	−8.93049	−0.717314
$156$	−69.9660	−5.60176
$157$	−12.5772	−1.00377	−0.501885	−	0.864934i	$-0.667360\pi$
$157$	−12.5772	−1.00377	−0.501885	+	0.864934i	$0.667360\pi$
$158$	14.5516	1.15766
$159$	2.97510	0.235941
$160$	−0.132728	−0.0104931
$161$	−0.727605	−0.0573433
$162$	9.21784	0.724222
$163$	−15.4289	−1.20848	−0.604241	−	0.796801i	$-0.706524\pi$
$163$	−15.4289	−1.20848	−0.604241	+	0.796801i	$0.706524\pi$
$164$	−0.383299	−0.0299306
$165$	21.4588	1.67056
$166$	6.19359	0.480716
$167$	17.7040	1.36997	0.684987	−	0.728556i	$-0.259808\pi$
$167$	17.7040	1.36997	0.684987	+	0.728556i	$0.259808\pi$
$168$	−11.6643	−0.899919
$169$	29.4875	2.26827
$170$	1.76532	0.135394
$171$	11.4440	0.875145
$172$	−6.72451	−0.512739
$173$	6.90724	0.525148	0.262574	−	0.964912i	$-0.415429\pi$
$173$	6.90724	0.525148	0.262574	+	0.964912i	$0.415429\pi$
$174$	55.7992	4.23013
$175$	2.66468	0.201431
$176$	22.2041	1.67369
$177$	17.3388	1.30326
$178$	−18.7228	−1.40333
$179$	6.94263	0.518916	0.259458	−	0.965754i	$-0.416456\pi$
$179$	6.94263	0.518916	0.259458	+	0.965754i	$0.416456\pi$
$180$	−23.9412	−1.78447
$181$	−15.5054	−1.15251	−0.576254	−	0.817271i	$-0.695486\pi$
$181$	−15.5054	−1.15251	−0.576254	+	0.817271i	$0.695486\pi$
$182$	14.2007	1.05262
$183$	20.4242	1.50980
$184$	−3.98213	−0.293567
$185$	13.1256	0.965012
$186$	41.5083	3.04353
$187$	2.86778	0.209713
$188$	−36.3501	−2.65110
$189$	2.95860	0.215207
$190$	9.36685	0.679542
$191$	−7.70328	−0.557390	−0.278695	−	0.960380i	$-0.589902\pi$
$191$	−7.70328	−0.557390	−0.278695	+	0.960380i	$0.589902\pi$
$192$	21.8271	1.57523
$193$	−17.1272	−1.23284	−0.616421	−	0.787417i	$-0.711418\pi$
$193$	−17.1272	−1.23284	−0.616421	+	0.787417i	$0.711418\pi$
$194$	−15.3262	−1.10035
$195$	24.8358	1.77853
$196$	−24.7712	−1.76937
$197$	18.1320	1.29185	0.645927	−	0.763399i	$-0.276471\pi$
$197$	18.1320	1.29185	0.645927	+	0.763399i	$0.276471\pi$
$198$	−58.3857	−4.14929
$199$	−23.2708	−1.64962	−0.824812	−	0.565407i	$-0.808719\pi$
$199$	−23.2708	−1.64962	−0.824812	+	0.565407i	$0.808719\pi$
$200$	14.5836	1.03122
$201$	14.2051	1.00195
$202$	40.4600	2.84676
$203$	−7.54415	−0.529496
$204$	−5.46568	−0.382674
$205$	0.136059	0.00950279
$206$	23.9565	1.66913
$207$	3.46232	0.240648
$208$	25.6984	1.78186
$209$	15.2165	1.05255
$210$	8.30088	0.572815
$211$	13.9098	0.957594	0.478797	−	0.877926i	$-0.341073\pi$
$211$	13.9098	0.957594	0.478797	+	0.877926i	$0.341073\pi$
$212$	−4.41346	−0.303118
$213$	−39.5731	−2.71150
$214$	43.9514	3.00445
$215$	2.38699	0.162792
$216$	16.1922	1.10174
$217$	−5.61199	−0.380967
$218$	−10.2559	−0.694619
$219$	−8.34577	−0.563955
$220$	−31.8334	−2.14621
$221$	3.31909	0.223266
$222$	−61.0067	−4.09450
$223$	2.12520	0.142314	0.0711570	−	0.997465i	$-0.477331\pi$
$223$	2.12520	0.142314	0.0711570	+	0.997465i	$0.477331\pi$
$224$	−0.0834075	−0.00557290
$225$	−12.6799	−0.845327
$226$	−5.16525	−0.343587
$227$	16.0941	1.06820	0.534102	−	0.845420i	$-0.320650\pi$
$227$	16.0941	1.06820	0.534102	+	0.845420i	$0.320650\pi$
$228$	−29.0010	−1.92064
$229$	11.4833	0.758837	0.379418	−	0.925225i	$-0.376124\pi$
$229$	11.4833	0.758837	0.379418	+	0.925225i	$0.376124\pi$
$230$	2.83388	0.186861
$231$	13.4849	0.887238
$232$	−41.2886	−2.71073
$233$	16.9021	1.10729	0.553645	−	0.832753i	$-0.313236\pi$
$233$	16.9021	1.10729	0.553645	+	0.832753i	$0.313236\pi$
$234$	−67.5740	−4.41745
$235$	12.9032	0.841709
$236$	−25.7215	−1.67433
$237$	15.9927	1.03884
$238$	1.10934	0.0719080
$239$	−23.1196	−1.49548	−0.747740	−	0.663992i	$-0.768861\pi$
$239$	−23.1196	−1.49548	−0.747740	+	0.663992i	$0.768861\pi$
$240$	15.0218	0.969652
$241$	−2.67252	−0.172152	−0.0860761	−	0.996289i	$-0.527433\pi$
$241$	−2.67252	−0.172152	−0.0860761	+	0.996289i	$0.527433\pi$
$242$	−50.7097	−3.25974
$243$	20.1022	1.28955
$244$	−30.2987	−1.93967
$245$	8.79300	0.561764
$246$	−0.632393	−0.0403199
$247$	17.6112	1.12057
$248$	−30.7141	−1.95035
$249$	6.80696	0.431373
$250$	−27.7127	−1.75270
$251$	4.30218	0.271551	0.135776	−	0.990740i	$-0.456647\pi$
$251$	4.30218	0.271551	0.135776	+	0.990740i	$0.456647\pi$
$252$	−15.0448	−0.947735
$253$	4.60367	0.289430
$254$	25.4120	1.59449
$255$	1.94015	0.121497
$256$	−31.9210	−1.99507
$257$	−22.4307	−1.39919	−0.699594	−	0.714541i	$-0.746636\pi$
$257$	−22.4307	−1.39919	−0.699594	+	0.714541i	$0.746636\pi$
$258$	−11.0946	−0.690718
$259$	8.24822	0.512519
$260$	−36.8431	−2.28491
$261$	35.8989	2.22209
$262$	20.8735	1.28957
$263$	19.7071	1.21519	0.607595	−	0.794247i	$-0.292134\pi$
$263$	19.7071	1.21519	0.607595	+	0.794247i	$0.292134\pi$
$264$	73.8017	4.54218
$265$	1.56664	0.0962381
$266$	5.88620	0.360906
$267$	−20.5769	−1.25929
$268$	−21.0728	−1.28723
$269$	−32.2781	−1.96803	−0.984017	−	0.178076i	$-0.943013\pi$
$269$	−32.2781	−1.96803	−0.984017	+	0.178076i	$0.943013\pi$
$270$	−11.5232	−0.701279
$271$	23.8537	1.44901	0.724503	−	0.689271i	$-0.242069\pi$
$271$	23.8537	1.44901	0.724503	+	0.689271i	$0.242069\pi$
$272$	2.00753	0.121725
$273$	15.6070	0.944578
$274$	33.7874	2.04117
$275$	−16.8598	−1.01669
$276$	−8.77409	−0.528138
$277$	−27.4984	−1.65222	−0.826109	−	0.563511i	$-0.809450\pi$
$277$	−27.4984	−1.65222	−0.826109	+	0.563511i	$0.809450\pi$
$278$	26.2085	1.57188
$279$	26.7047	1.59877
$280$	−6.14224	−0.367069
$281$	5.41702	0.323152	0.161576	−	0.986860i	$-0.448342\pi$
$281$	5.41702	0.323152	0.161576	+	0.986860i	$0.448342\pi$
$282$	−59.9729	−3.57134
$283$	−23.7937	−1.41439	−0.707195	−	0.707018i	$-0.750040\pi$
$283$	−23.7937	−1.41439	−0.707195	+	0.707018i	$0.750040\pi$
$284$	58.7054	3.48352
$285$	10.2945	0.609792
$286$	−89.8497	−5.31292
$287$	0.0855007	0.00504695
$288$	0.396896	0.0233873
$289$	−16.7407	−0.984748
$290$	29.3830	1.72543
$291$	−16.8440	−0.987411
$292$	12.3807	0.724524
$293$	5.34461	0.312235	0.156118	−	0.987738i	$-0.450102\pi$
$293$	5.34461	0.312235	0.156118	+	0.987738i	$0.450102\pi$
$294$	−40.8692	−2.38354
$295$	9.13035	0.531590
$296$	45.1420	2.62382
$297$	−18.7195	−1.08622
$298$	29.9579	1.73541
$299$	5.32816	0.308135
$300$	32.1330	1.85520
$301$	1.50001	0.0864590
$302$	30.2378	1.73999
$303$	44.4669	2.55456
$304$	10.6520	0.610935
$305$	10.7551	0.615835
$306$	−5.27882	−0.301770
$307$	−31.8833	−1.81967	−0.909837	−	0.414965i	$-0.863794\pi$
$307$	−31.8833	−1.81967	−0.909837	+	0.414965i	$0.863794\pi$
$308$	−20.0043	−1.13985
$309$	26.3289	1.49780
$310$	21.8576	1.24143
$311$	11.7871	0.668385	0.334193	−	0.942505i	$-0.391536\pi$
$311$	11.7871	0.668385	0.334193	+	0.942505i	$0.391536\pi$
$312$	85.4161	4.83573
$313$	17.5302	0.990867	0.495433	−	0.868646i	$-0.335009\pi$
$313$	17.5302	0.990867	0.495433	+	0.868646i	$0.335009\pi$
$314$	30.7831	1.73719
$315$	5.34045	0.300900
$316$	−23.7247	−1.33462
$317$	−12.3681	−0.694664	−0.347332	−	0.937742i	$-0.612912\pi$
$317$	−12.3681	−0.694664	−0.347332	+	0.937742i	$0.612912\pi$
$318$	−7.28164	−0.408334
$319$	47.7330	2.67253
$320$	11.4938	0.642523
$321$	48.3040	2.69607
$322$	1.78084	0.0992421
$323$	1.37577	0.0765498
$324$	−15.0286	−0.834921
$325$	−19.5131	−1.08239
$326$	37.7626	2.09148
$327$	−11.2716	−0.623321
$328$	0.467940	0.0258377
$329$	8.10845	0.447033
$330$	−52.5210	−2.89118
$331$	15.9544	0.876931	0.438466	−	0.898748i	$-0.355522\pi$
$331$	15.9544	0.876931	0.438466	+	0.898748i	$0.355522\pi$
$332$	−10.0979	−0.554194
$333$	−39.2493	−2.15085
$334$	−43.3310	−2.37097
$335$	7.48019	0.408686
$336$	9.43981	0.514984
$337$	−21.1110	−1.14999	−0.574995	−	0.818157i	$-0.694996\pi$
$337$	−21.1110	−1.14999	−0.574995	+	0.818157i	$0.694996\pi$
$338$	−72.1717	−3.92562
$339$	−5.67678	−0.308321
$340$	−2.87815	−0.156089
$341$	35.5079	1.92286
$342$	−28.0096	−1.51458
$343$	11.7564	0.634789
$344$	8.20944	0.442623
$345$	3.11453	0.167681
$346$	−16.9057	−0.908855
$347$	29.5371	1.58563	0.792817	−	0.609460i	$-0.208614\pi$
$347$	29.5371	1.58563	0.792817	+	0.609460i	$0.208614\pi$
$348$	−90.9739	−4.87671
$349$	7.76249	0.415516	0.207758	−	0.978180i	$-0.433383\pi$
$349$	7.76249	0.415516	0.207758	+	0.978180i	$0.433383\pi$
$350$	−6.52188	−0.348609
$351$	−21.6655	−1.15642
$352$	0.527732	0.0281282
$353$	−12.4518	−0.662745	−0.331372	−	0.943500i	$-0.607512\pi$
$353$	−12.4518	−0.662745	−0.331372	+	0.943500i	$0.607512\pi$
$354$	−42.4372	−2.25551
$355$	−20.8386	−1.10600
$356$	30.5252	1.61783
$357$	1.21920	0.0645272
$358$	−16.9923	−0.898071
$359$	11.8809	0.627051	0.313525	−	0.949580i	$-0.398490\pi$
$359$	11.8809	0.627051	0.313525	+	0.949580i	$0.398490\pi$
$360$	29.2279	1.54045
$361$	−11.7001	−0.615796
$362$	37.9500	1.99461
$363$	−55.7316	−2.92515
$364$	−23.1525	−1.21352
$365$	−4.39476	−0.230032
$366$	−49.9889	−2.61296
$367$	16.2758	0.849587	0.424794	−	0.905290i	$-0.360347\pi$
$367$	16.2758	0.849587	0.424794	+	0.905290i	$0.360347\pi$
$368$	3.22271	0.167995
$369$	−0.406856	−0.0211801
$370$	−32.1253	−1.67011
$371$	0.984491	0.0511122
$372$	−67.6743	−3.50875
$373$	−13.5560	−0.701906	−0.350953	−	0.936393i	$-0.614142\pi$
$373$	−13.5560	−0.701906	−0.350953	+	0.936393i	$0.614142\pi$
$374$	−7.01898	−0.362943
$375$	−30.4572	−1.57280
$376$	44.3770	2.28857
$377$	55.2448	2.84525
$378$	−7.24127	−0.372451
$379$	34.0902	1.75110	0.875549	−	0.483130i	$-0.160500\pi$
$379$	34.0902	1.75110	0.875549	+	0.483130i	$0.160500\pi$
$380$	−15.2715	−0.783412
$381$	27.9287	1.43083
$382$	18.8540	0.964656
$383$	3.42962	0.175245	0.0876226	−	0.996154i	$-0.472073\pi$
$383$	3.42962	0.175245	0.0876226	+	0.996154i	$0.472073\pi$
$384$	−52.9183	−2.70047
$385$	7.10093	0.361897
$386$	41.9193	2.13364
$387$	−7.13780	−0.362835
$388$	24.9875	1.26855
$389$	−5.83791	−0.295994	−0.147997	−	0.988988i	$-0.547283\pi$
$389$	−5.83791	−0.295994	−0.147997	+	0.988988i	$0.547283\pi$
$390$	−60.7863	−3.07803
$391$	0.416231	0.0210497
$392$	30.2412	1.52741
$393$	22.9407	1.15721
$394$	−44.3787	−2.23577
$395$	8.42152	0.423733
$396$	95.1909	4.78353
$397$	−23.9999	−1.20452	−0.602261	−	0.798299i	$-0.705733\pi$
$397$	−23.9999	−1.20452	−0.602261	+	0.798299i	$0.705733\pi$
$398$	56.9560	2.85495
$399$	6.46913	0.323862
$400$	−11.8024	−0.590119
$401$	−23.0233	−1.14973	−0.574865	−	0.818249i	$-0.694945\pi$
$401$	−23.0233	−1.14973	−0.574865	+	0.818249i	$0.694945\pi$
$402$	−34.7674	−1.73404
$403$	41.0959	2.04713
$404$	−65.9652	−3.28189
$405$	5.33468	0.265083
$406$	18.4645	0.916380
$407$	−52.1877	−2.58685
$408$	6.67263	0.330344
$409$	−34.0606	−1.68419	−0.842095	−	0.539329i	$-0.818678\pi$
$409$	−34.0606	−1.68419	−0.842095	+	0.539329i	$0.818678\pi$
$410$	−0.333009	−0.0164461
$411$	37.1335	1.83166
$412$	−39.0581	−1.92426
$413$	5.73759	0.282328
$414$	−8.47413	−0.416481
$415$	3.58444	0.175953
$416$	0.610782	0.0299461
$417$	28.8040	1.41054
$418$	−37.2429	−1.82161
$419$	14.9589	0.730790	0.365395	−	0.930853i	$-0.380934\pi$
$419$	14.9589	0.730790	0.365395	+	0.930853i	$0.380934\pi$
$420$	−13.5336	−0.660372
$421$	5.09761	0.248442	0.124221	−	0.992255i	$-0.460357\pi$
$421$	5.09761	0.248442	0.124221	+	0.992255i	$0.460357\pi$
$422$	−34.0448	−1.65727
$423$	−38.5841	−1.87603
$424$	5.38805	0.261667
$425$	−1.52434	−0.0739416
$426$	96.8563	4.69270
$427$	6.75859	0.327071
$428$	−71.6575	−3.46369
$429$	−98.7478	−4.76759
$430$	−5.84224	−0.281738
$431$	4.41808	0.212811	0.106406	−	0.994323i	$-0.466066\pi$
$431$	4.41808	0.212811	0.106406	+	0.994323i	$0.466066\pi$
$432$	−13.1042	−0.630478
$433$	−18.6392	−0.895744	−0.447872	−	0.894098i	$-0.647818\pi$
$433$	−18.6392	−0.895744	−0.447872	+	0.894098i	$0.647818\pi$
$434$	13.7355	0.659326
$435$	32.2929	1.54833
$436$	16.7211	0.800793
$437$	2.20853	0.105648
$438$	20.4265	0.976017
$439$	18.9840	0.906058	0.453029	−	0.891496i	$-0.350343\pi$
$439$	18.9840	0.906058	0.453029	+	0.891496i	$0.350343\pi$
$440$	38.8629	1.85272
$441$	−26.2936	−1.25208
$442$	−8.12357	−0.386399
$443$	25.4883	1.21098	0.605492	−	0.795852i	$-0.292976\pi$
$443$	25.4883	1.21098	0.605492	+	0.795852i	$0.292976\pi$
$444$	99.4642	4.72036
$445$	−10.8355	−0.513653
$446$	−5.20149	−0.246298
$447$	32.9247	1.55729
$448$	7.22280	0.341245
$449$	−31.4551	−1.48446	−0.742230	−	0.670145i	$-0.766232\pi$
$449$	−31.4551	−1.48446	−0.742230	+	0.670145i	$0.766232\pi$
$450$	31.0345	1.46298
$451$	−0.540976	−0.0254736
$452$	8.42132	0.396106
$453$	33.2324	1.56139
$454$	−39.3909	−1.84871
$455$	8.21841	0.385285
$456$	35.4051	1.65800
$457$	−12.0040	−0.561526	−0.280763	−	0.959777i	$-0.590587\pi$
$457$	−12.0040	−0.561526	−0.280763	+	0.959777i	$0.590587\pi$
$458$	−28.1057	−1.31329
$459$	−1.69249	−0.0789985
$460$	−4.62031	−0.215423
$461$	−9.10238	−0.423940	−0.211970	−	0.977276i	$-0.567988\pi$
$461$	−9.10238	−0.423940	−0.211970	+	0.977276i	$0.567988\pi$
$462$	−33.0046	−1.53551
$463$	27.2522	1.26652	0.633259	−	0.773940i	$-0.281717\pi$
$463$	27.2522	1.26652	0.633259	+	0.773940i	$0.281717\pi$
$464$	33.4145	1.55123
$465$	24.0223	1.11401
$466$	−41.3683	−1.91635
$467$	−3.39628	−0.157161	−0.0785805	−	0.996908i	$-0.525039\pi$
$467$	−3.39628	−0.157161	−0.0785805	+	0.996908i	$0.525039\pi$
$468$	110.171	5.09267
$469$	4.70061	0.217054
$470$	−31.5809	−1.45672
$471$	33.8316	1.55888
$472$	31.4014	1.44537
$473$	−9.49077	−0.436386
$474$	−39.1426	−1.79788
$475$	−8.08821	−0.371113
$476$	−1.80865	−0.0828994
$477$	−4.68471	−0.214498
$478$	56.5858	2.58817
$479$	−23.7043	−1.08308	−0.541538	−	0.840676i	$-0.682158\pi$
$479$	−23.7043	−1.08308	−0.541538	+	0.840676i	$0.682158\pi$
$480$	0.357028	0.0162960
$481$	−60.4006	−2.75403
$482$	6.54107	0.297938
$483$	1.95720	0.0890556
$484$	82.6761	3.75800
$485$	−8.86979	−0.402756
$486$	−49.2007	−2.23179
$487$	−35.7815	−1.62141	−0.810706	−	0.585453i	$-0.800917\pi$
$487$	−35.7815	−1.62141	−0.810706	+	0.585453i	$0.800917\pi$
$488$	36.9893	1.67443
$489$	41.5024	1.87680
$490$	−21.5211	−0.972226
$491$	−31.7010	−1.43065	−0.715324	−	0.698793i	$-0.753721\pi$
$491$	−31.7010	−1.43065	−0.715324	+	0.698793i	$0.753721\pi$
$492$	1.03104	0.0464830
$493$	4.31568	0.194368
$494$	−43.1039	−1.93934
$495$	−33.7899	−1.51874
$496$	24.8566	1.11610
$497$	−13.0951	−0.587398
$498$	−16.6602	−0.746563
$499$	−8.97807	−0.401914	−0.200957	−	0.979600i	$-0.564405\pi$
$499$	−8.97807	−0.401914	−0.200957	+	0.979600i	$0.564405\pi$
$500$	45.1822	2.02061
$501$	−47.6222	−2.12760
$502$	−10.5297	−0.469965
$503$	−29.0742	−1.29635	−0.648177	−	0.761490i	$-0.724468\pi$
$503$	−29.0742	−1.29635	−0.648177	+	0.761490i	$0.724468\pi$
$504$	18.3671	0.818135
$505$	23.4156	1.04198
$506$	−11.2676	−0.500907
$507$	−79.3191	−3.52268
$508$	−41.4313	−1.83822
$509$	−18.2555	−0.809159	−0.404580	−	0.914503i	$-0.632582\pi$
$509$	−18.2555	−0.809159	−0.404580	+	0.914503i	$0.632582\pi$
$510$	−4.74857	−0.210270
$511$	−2.76170	−0.122171
$512$	38.7821	1.71394
$513$	−8.98038	−0.396493
$514$	54.8998	2.42152
$515$	13.8644	0.610940
$516$	18.0884	0.796297
$517$	−51.3034	−2.25632
$518$	−20.1878	−0.887000
$519$	−18.5799	−0.815568
$520$	44.9788	1.97245
$521$	9.18330	0.402328	0.201164	−	0.979558i	$-0.435528\pi$
$521$	9.18330	0.402328	0.201164	+	0.979558i	$0.435528\pi$
$522$	−87.8638	−3.84569
$523$	−13.6056	−0.594932	−0.297466	−	0.954732i	$-0.596141\pi$
$523$	−13.6056	−0.594932	−0.297466	+	0.954732i	$0.596141\pi$
$524$	−34.0318	−1.48669
$525$	−7.16776	−0.312827
$526$	−48.2337	−2.10309
$527$	3.21037	0.139846
$528$	−59.7271	−2.59929
$529$	−22.3318	−0.970949
$530$	−3.83441	−0.166556
$531$	−27.3024	−1.18482
$532$	−9.59675	−0.416072
$533$	−0.626111	−0.0271199
$534$	50.3627	2.17941
$535$	25.4362	1.09970
$536$	25.7261	1.11120
$537$	−18.6751	−0.805890
$538$	79.0018	3.40601
$539$	−34.9613	−1.50589
$540$	18.7872	0.808472
$541$	15.2985	0.657734	0.328867	−	0.944376i	$-0.393333\pi$
$541$	15.2985	0.657734	0.328867	+	0.944376i	$0.393333\pi$
$542$	−58.3826	−2.50775
$543$	41.7083	1.78987
$544$	0.0477138	0.00204571
$545$	−5.93546	−0.254247
$546$	−38.1986	−1.63475
$547$	−11.1899	−0.478447	−0.239224	−	0.970965i	$-0.576893\pi$
$547$	−11.1899	−0.478447	−0.239224	+	0.970965i	$0.576893\pi$
$548$	−55.0863	−2.35317
$549$	−32.1608	−1.37259
$550$	41.2649	1.75954
$551$	22.8991	0.975534
$552$	10.7116	0.455917
$553$	5.29215	0.225045
$554$	67.3031	2.85944
$555$	−35.3067	−1.49869
$556$	−42.7298	−1.81215
$557$	−31.1512	−1.31992	−0.659960	−	0.751301i	$-0.729427\pi$
$557$	−31.1512	−1.31992	−0.659960	+	0.751301i	$0.729427\pi$
$558$	−65.3607	−2.76694
$559$	−10.9844	−0.464589
$560$	4.97086	0.210057
$561$	−7.71409	−0.325689
$562$	−13.2583	−0.559268
$563$	−27.2331	−1.14774	−0.573868	−	0.818948i	$-0.694558\pi$
$563$	−27.2331	−1.14774	−0.573868	+	0.818948i	$0.694558\pi$
$564$	97.7787	4.11722
$565$	−2.98931	−0.125761
$566$	58.2359	2.44784
$567$	3.35236	0.140786
$568$	−71.6689	−3.00716
$569$	21.4062	0.897393	0.448696	−	0.893684i	$-0.351888\pi$
$569$	21.4062	0.897393	0.448696	+	0.893684i	$0.351888\pi$
$570$	−25.1960	−1.05535
$571$	−4.12253	−0.172522	−0.0862612	−	0.996273i	$-0.527492\pi$
$571$	−4.12253	−0.172522	−0.0862612	+	0.996273i	$0.527492\pi$
$572$	146.489	6.12502
$573$	20.7212	0.865640
$574$	−0.209266	−0.00873458
$575$	−2.44704	−0.102049
$576$	−34.3698	−1.43207
$577$	24.4685	1.01864	0.509319	−	0.860578i	$-0.329897\pi$
$577$	24.4685	1.01864	0.509319	+	0.860578i	$0.329897\pi$
$578$	40.9734	1.70427
$579$	46.0707	1.91463
$580$	−47.9055	−1.98917
$581$	2.25249	0.0934492
$582$	41.2261	1.70888
$583$	−6.22902	−0.257980
$584$	−15.1146	−0.625447
$585$	−39.1074	−1.61689
$586$	−13.0811	−0.540375
$587$	33.1711	1.36912	0.684560	−	0.728957i	$-0.259994\pi$
$587$	33.1711	1.36912	0.684560	+	0.728957i	$0.259994\pi$
$588$	66.6324	2.74787
$589$	17.0343	0.701887
$590$	−22.3468	−0.920004
$591$	−48.7737	−2.00628
$592$	−36.5330	−1.50150
$593$	−30.1606	−1.23855	−0.619273	−	0.785175i	$-0.712573\pi$
$593$	−30.1606	−1.23855	−0.619273	+	0.785175i	$0.712573\pi$
$594$	45.8166	1.87988
$595$	0.642015	0.0263201
$596$	−48.8427	−2.00068
$597$	62.5966	2.56191
$598$	−13.0408	−0.533279
$599$	−16.7701	−0.685208	−0.342604	−	0.939480i	$-0.611309\pi$
$599$	−16.7701	−0.685208	−0.342604	+	0.939480i	$0.611309\pi$
$600$	−39.2287	−1.60150
$601$	32.8379	1.33949	0.669743	−	0.742593i	$-0.266404\pi$
$601$	32.8379	1.33949	0.669743	+	0.742593i	$0.266404\pi$
$602$	−3.67131	−0.149632
$603$	−22.3679	−0.910892
$604$	−49.2992	−2.00595
$605$	−29.3475	−1.19314
$606$	−108.834	−4.42108
$607$	10.7068	0.434577	0.217289	−	0.976107i	$-0.430279\pi$
$607$	10.7068	0.434577	0.217289	+	0.976107i	$0.430279\pi$
$608$	0.253170	0.0102674
$609$	20.2931	0.822320
$610$	−26.3234	−1.06580
$611$	−59.3771	−2.40214
$612$	8.60649	0.347897
$613$	−35.1182	−1.41841	−0.709206	−	0.705001i	$-0.750946\pi$
$613$	−35.1182	−1.41841	−0.709206	+	0.705001i	$0.750946\pi$
$614$	78.0353	3.14925
$615$	−0.365988	−0.0147581
$616$	24.4218	0.983981
$617$	−27.6583	−1.11348	−0.556740	−	0.830687i	$-0.687948\pi$
$617$	−27.6583	−1.11348	−0.556740	+	0.830687i	$0.687948\pi$
$618$	−64.4409	−2.59219
$619$	46.1219	1.85380	0.926898	−	0.375313i	$-0.122465\pi$
$619$	46.1219	1.85380	0.926898	+	0.375313i	$0.122465\pi$
$620$	−35.6363	−1.43119
$621$	−2.71696	−0.109028
$622$	−28.8493	−1.15675
$623$	−6.80913	−0.272802
$624$	−69.1265	−2.76727
$625$	−1.07025	−0.0428102
$626$	−42.9058	−1.71486
$627$	−40.9312	−1.63463
$628$	−50.1881	−2.00272
$629$	−4.71844	−0.188137
$630$	−13.0709	−0.520758
$631$	−18.5482	−0.738392	−0.369196	−	0.929352i	$-0.620367\pi$
$631$	−18.5482	−0.738392	−0.369196	+	0.929352i	$0.620367\pi$
$632$	28.9636	1.15211
$633$	−37.4164	−1.48717
$634$	30.2714	1.20223
$635$	14.7068	0.583623
$636$	11.8718	0.470749
$637$	−40.4632	−1.60321
$638$	−116.828	−4.62526
$639$	62.3134	2.46508
$640$	−27.8660	−1.10150
$641$	16.4954	0.651528	0.325764	−	0.945451i	$-0.394379\pi$
$641$	16.4954	0.651528	0.325764	+	0.945451i	$0.394379\pi$
$642$	−118.226	−4.66599
$643$	27.6160	1.08907	0.544535	−	0.838738i	$-0.316706\pi$
$643$	27.6160	1.08907	0.544535	+	0.838738i	$0.316706\pi$
$644$	−2.90344	−0.114412
$645$	−6.42082	−0.252820
$646$	−3.36724	−0.132482
$647$	−6.07994	−0.239027	−0.119514	−	0.992833i	$-0.538134\pi$
$647$	−6.07994	−0.239027	−0.119514	+	0.992833i	$0.538134\pi$
$648$	18.3472	0.720748
$649$	−36.3026	−1.42500
$650$	47.7589	1.87326
$651$	15.0958	0.591651
$652$	−61.5675	−2.41117
$653$	42.9964	1.68258	0.841289	−	0.540586i	$-0.181797\pi$
$653$	42.9964	1.68258	0.841289	+	0.540586i	$0.181797\pi$
$654$	27.5876	1.07876
$655$	12.0802	0.472014
$656$	−0.378700	−0.0147857
$657$	13.1416	0.512703
$658$	−19.8457	−0.773665
$659$	−4.40998	−0.171788	−0.0858942	−	0.996304i	$-0.527375\pi$
$659$	−4.40998	−0.171788	−0.0858942	+	0.996304i	$0.527375\pi$
$660$	85.6291	3.33311
$661$	−5.15605	−0.200547	−0.100274	−	0.994960i	$-0.531972\pi$
$661$	−5.15605	−0.200547	−0.100274	+	0.994960i	$0.531972\pi$
$662$	−39.0488	−1.51767
$663$	−8.92808	−0.346738
$664$	12.3277	0.478409
$665$	3.40655	0.132100
$666$	96.0638	3.72240
$667$	6.92799	0.268253
$668$	70.6460	2.73337
$669$	−5.71662	−0.221017
$670$	−18.3080	−0.707299
$671$	−42.7626	−1.65083
$672$	0.224359	0.00865485
$673$	−23.9965	−0.924997	−0.462499	−	0.886620i	$-0.653047\pi$
$673$	−23.9965	−0.924997	−0.462499	+	0.886620i	$0.653047\pi$
$674$	51.6699	1.99025
$675$	9.95021	0.382984
$676$	117.667	4.52566
$677$	−45.6796	−1.75561	−0.877805	−	0.479019i	$-0.840993\pi$
$677$	−45.6796	−1.75561	−0.877805	+	0.479019i	$0.840993\pi$
$678$	13.8941	0.533600
$679$	−5.57385	−0.213905
$680$	3.51371	0.134745
$681$	−43.2919	−1.65895
$682$	−86.9067	−3.32783
$683$	7.75642	0.296791	0.148396	−	0.988928i	$-0.452589\pi$
$683$	7.75642	0.296791	0.148396	+	0.988928i	$0.452589\pi$
$684$	45.6663	1.74609
$685$	19.5540	0.747118
$686$	−28.7743	−1.09861
$687$	−30.8891	−1.17849
$688$	−6.64383	−0.253294
$689$	−7.20930	−0.274652
$690$	−7.62291	−0.290199
$691$	−32.6197	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$691$	−32.6197	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$692$	27.5627	1.04778
$693$	−21.2338	−0.806606
$694$	−72.2929	−2.74420
$695$	15.1678	0.575346
$696$	111.063	4.20983
$697$	−0.0489112	−0.00185264
$698$	−18.9989	−0.719120
$699$	−45.4651	−1.71965
$700$	10.6331	0.401895
$701$	22.0897	0.834315	0.417157	−	0.908834i	$-0.363026\pi$
$701$	22.0897	0.834315	0.417157	+	0.908834i	$0.363026\pi$
$702$	53.0269	2.00137
$703$	−25.0362	−0.944258
$704$	−45.6998	−1.72237
$705$	−34.7084	−1.30719
$706$	30.4763	1.14699
$707$	14.7146	0.553398
$708$	69.1888	2.60027
$709$	7.85616	0.295044	0.147522	−	0.989059i	$-0.452870\pi$
$709$	7.85616	0.295044	0.147522	+	0.989059i	$0.452870\pi$
$710$	51.0031	1.91411
$711$	−25.1828	−0.944428
$712$	−37.2659	−1.39660
$713$	5.15364	0.193005
$714$	−2.98404	−0.111675
$715$	−51.9992	−1.94466
$716$	27.7039	1.03534
$717$	62.1897	2.32252
$718$	−29.0789	−1.08522
$719$	10.0478	0.374718	0.187359	−	0.982291i	$-0.440007\pi$
$719$	10.0478	0.374718	0.187359	+	0.982291i	$0.440007\pi$
$720$	−23.6539	−0.881530
$721$	8.71252	0.324471
$722$	28.6364	1.06574
$723$	7.18886	0.267356
$724$	−61.8729	−2.29949
$725$	−25.3721	−0.942295
$726$	136.405	5.06246
$727$	−5.68745	−0.210936	−0.105468	−	0.994423i	$-0.533634\pi$
$727$	−5.68745	−0.210936	−0.105468	+	0.994423i	$0.533634\pi$
$728$	28.2651	1.04757
$729$	−42.7746	−1.58425
$730$	10.7563	0.398109
$731$	−0.858088	−0.0317375
$732$	81.5009	3.01236
$733$	10.6157	0.392101	0.196050	−	0.980594i	$-0.437188\pi$
$733$	10.6157	0.392101	0.196050	+	0.980594i	$0.437188\pi$
$734$	−39.8354	−1.47035
$735$	−23.6525	−0.872434
$736$	0.0765952	0.00282334
$737$	−29.7415	−1.09554
$738$	0.995793	0.0366557
$739$	35.3411	1.30004	0.650022	−	0.759916i	$-0.274760\pi$
$739$	35.3411	1.30004	0.650022	+	0.759916i	$0.274760\pi$
$740$	52.3764	1.92539
$741$	−47.3726	−1.74028
$742$	−2.40957	−0.0884582
$743$	−49.6459	−1.82133	−0.910665	−	0.413146i	$-0.864430\pi$
$743$	−49.6459	−1.82133	−0.910665	+	0.413146i	$0.864430\pi$
$744$	82.6183	3.02893
$745$	17.3377	0.635203
$746$	33.1788	1.21476
$747$	−10.7185	−0.392170
$748$	11.4436	0.418420
$749$	15.9843	0.584054
$750$	74.5449	2.72199
$751$	−1.74195	−0.0635646	−0.0317823	−	0.999495i	$-0.510118\pi$
$751$	−1.74195	−0.0635646	−0.0317823	+	0.999495i	$0.510118\pi$
$752$	−35.9139	−1.30965
$753$	−11.5725	−0.421726
$754$	−135.213	−4.92418
$755$	17.4997	0.636879
$756$	11.8060	0.429381
$757$	−21.0742	−0.765956	−0.382978	−	0.923757i	$-0.625102\pi$
$757$	−21.0742	−0.765956	−0.382978	+	0.923757i	$0.625102\pi$
$758$	−83.4369	−3.03056
$759$	−12.3835	−0.449492
$760$	18.6438	0.676283
$761$	17.9383	0.650263	0.325132	−	0.945669i	$-0.394591\pi$
$761$	17.9383	0.650263	0.325132	+	0.945669i	$0.394591\pi$
$762$	−68.3563	−2.47629
$763$	−3.72989	−0.135031
$764$	−30.7392	−1.11211
$765$	−3.05504	−0.110455
$766$	−8.39409	−0.303291
$767$	−42.0156	−1.51710
$768$	85.8650	3.09839
$769$	−53.9528	−1.94559	−0.972794	−	0.231672i	$-0.925580\pi$
$769$	−53.9528	−1.94559	−0.972794	+	0.231672i	$0.925580\pi$
$770$	−17.3797	−0.626322
$771$	60.3367	2.17297
$772$	−68.3445	−2.45977
$773$	42.4103	1.52539	0.762697	−	0.646756i	$-0.223875\pi$
$773$	42.4103	1.52539	0.762697	+	0.646756i	$0.223875\pi$
$774$	17.4700	0.627946
$775$	−18.8739	−0.677972
$776$	−30.5053	−1.09508
$777$	−22.1870	−0.795955
$778$	14.2885	0.512266
$779$	−0.259524	−0.00929842
$780$	99.1048	3.54852
$781$	82.8550	2.96478
$782$	−1.01874	−0.0364300
$783$	−28.1707	−1.00674
$784$	−24.4740	−0.874070
$785$	17.8152	0.635853
$786$	−56.1481	−2.00274
$787$	5.50093	0.196087	0.0980434	−	0.995182i	$-0.468742\pi$
$787$	5.50093	0.196087	0.0980434	+	0.995182i	$0.468742\pi$
$788$	72.3542	2.57751
$789$	−53.0104	−1.88722
$790$	−20.6119	−0.733340
$791$	−1.87851	−0.0667920
$792$	−116.211	−4.12939
$793$	−49.4923	−1.75752
$794$	58.7406	2.08463
$795$	−4.21414	−0.149460
$796$	−92.8600	−3.29134
$797$	13.7990	0.488787	0.244393	−	0.969676i	$-0.421411\pi$
$797$	13.7990	0.488787	0.244393	+	0.969676i	$0.421411\pi$
$798$	−15.8334	−0.560496
$799$	−4.63849	−0.164098
$800$	−0.280512	−0.00991758
$801$	32.4013	1.14484
$802$	56.3502	1.98980
$803$	17.4737	0.616634
$804$	56.6841	1.99909
$805$	1.03063	0.0363250
$806$	−100.583	−3.54290
$807$	86.8256	3.05640
$808$	80.5318	2.83310
$809$	8.46772	0.297709	0.148855	−	0.988859i	$-0.452441\pi$
$809$	8.46772	0.297709	0.148855	+	0.988859i	$0.452441\pi$
$810$	−13.0568	−0.458769
$811$	−2.76648	−0.0971444	−0.0485722	−	0.998820i	$-0.515467\pi$
$811$	−2.76648	−0.0971444	−0.0485722	+	0.998820i	$0.515467\pi$
$812$	−30.1042	−1.05645
$813$	−64.1644	−2.25034
$814$	127.731	4.47697
$815$	21.8546	0.765532
$816$	−5.40010	−0.189041
$817$	−4.55304	−0.159291
$818$	83.3645	2.91477
$819$	−24.5754	−0.858735
$820$	0.542932	0.0189600
$821$	−40.3137	−1.40696	−0.703479	−	0.710716i	$-0.748371\pi$
$821$	−40.3137	−1.40696	−0.703479	+	0.710716i	$0.748371\pi$
$822$	−90.8854	−3.16999
$823$	−1.95277	−0.0680692	−0.0340346	−	0.999421i	$-0.510836\pi$
$823$	−1.95277	−0.0680692	−0.0340346	+	0.999421i	$0.510836\pi$
$824$	47.6831	1.66112
$825$	45.3515	1.57894
$826$	−14.0429	−0.488616
$827$	−25.3430	−0.881262	−0.440631	−	0.897688i	$-0.645245\pi$
$827$	−25.3430	−0.881262	−0.440631	+	0.897688i	$0.645245\pi$
$828$	13.8161	0.480141
$829$	−11.2952	−0.392298	−0.196149	−	0.980574i	$-0.562844\pi$
$829$	−11.2952	−0.392298	−0.196149	+	0.980574i	$0.562844\pi$
$830$	−8.77304	−0.304517
$831$	73.9684	2.56593
$832$	−52.8916	−1.83369
$833$	−3.16095	−0.109520
$834$	−70.4987	−2.44117
$835$	−25.0771	−0.867830
$836$	60.7201	2.10005
$837$	−20.9558	−0.724339
$838$	−36.6123	−1.26475
$839$	50.8320	1.75492	0.877458	−	0.479654i	$-0.159238\pi$
$839$	50.8320	1.75492	0.877458	+	0.479654i	$0.159238\pi$
$840$	16.5221	0.570068
$841$	42.8326	1.47699
$842$	−12.4765	−0.429970
$843$	−14.5713	−0.501863
$844$	55.5059	1.91059
$845$	−41.7683	−1.43687
$846$	94.4359	3.24677
$847$	−18.4422	−0.633681
$848$	−4.36051	−0.149740
$849$	64.0032	2.19658
$850$	3.73088	0.127968
$851$	−7.57455	−0.259652
$852$	−157.913	−5.41000
$853$	−4.37117	−0.149666	−0.0748331	−	0.997196i	$-0.523842\pi$
$853$	−4.37117	−0.149666	−0.0748331	+	0.997196i	$0.523842\pi$
$854$	−16.5419	−0.566051
$855$	−16.2101	−0.554374
$856$	87.4811	2.99004
$857$	−20.5532	−0.702083	−0.351041	−	0.936360i	$-0.614172\pi$
$857$	−20.5532	−0.702083	−0.351041	+	0.936360i	$0.614172\pi$
$858$	241.688	8.25110
$859$	29.4768	1.00573	0.502867	−	0.864364i	$-0.332278\pi$
$859$	29.4768	1.00573	0.502867	+	0.864364i	$0.332278\pi$
$860$	9.52508	0.324802
$861$	−0.229990	−0.00783803
$862$	−10.8134	−0.368305
$863$	−9.45370	−0.321808	−0.160904	−	0.986970i	$-0.551441\pi$
$863$	−9.45370	−0.321808	−0.160904	+	0.986970i	$0.551441\pi$
$864$	−0.311453	−0.0105958
$865$	−9.78391	−0.332663
$866$	45.6201	1.55023
$867$	45.0311	1.52934
$868$	−22.3941	−0.760106
$869$	−33.4843	−1.13588
$870$	−79.0380	−2.67964
$871$	−34.4220	−1.16634
$872$	−20.4135	−0.691287
$873$	26.5232	0.897675
$874$	−5.40545	−0.182842
$875$	−10.0786	−0.340719
$876$	−33.3030	−1.12520
$877$	−38.5399	−1.30140	−0.650700	−	0.759335i	$-0.725525\pi$
$877$	−38.5399	−1.30140	−0.650700	+	0.759335i	$0.725525\pi$
$878$	−46.4640	−1.56808
$879$	−14.3766	−0.484909
$880$	−31.4514	−1.06023
$881$	−16.0186	−0.539681	−0.269841	−	0.962905i	$-0.586971\pi$
$881$	−16.0186	−0.539681	−0.269841	+	0.962905i	$0.586971\pi$
$882$	64.3544	2.16693
$883$	4.96567	0.167108	0.0835540	−	0.996503i	$-0.473373\pi$
$883$	4.96567	0.167108	0.0835540	+	0.996503i	$0.473373\pi$
$884$	13.2445	0.445461
$885$	−24.5599	−0.825572
$886$	−62.3833	−2.09581
$887$	41.4801	1.39277	0.696383	−	0.717670i	$-0.254791\pi$
$887$	41.4801	1.39277	0.696383	+	0.717670i	$0.254791\pi$
$888$	−121.428	−4.07486
$889$	9.24190	0.309963
$890$	26.5203	0.888961
$891$	−21.2109	−0.710591
$892$	8.48041	0.283945
$893$	−24.6119	−0.823607
$894$	−80.5842	−2.69514
$895$	−9.83403	−0.328715
$896$	−17.5112	−0.585009
$897$	−14.3323	−0.478542
$898$	76.9874	2.56910
$899$	53.4353	1.78217
$900$	−50.5980	−1.68660
$901$	−0.563184	−0.0187624
$902$	1.32406	0.0440862
$903$	−4.03490	−0.134273
$904$	−10.2809	−0.341939
$905$	21.9630	0.730074
$906$	−81.3373	−2.70225
$907$	28.9150	0.960105	0.480053	−	0.877240i	$-0.340618\pi$
$907$	28.9150	0.960105	0.480053	+	0.877240i	$0.340618\pi$
$908$	64.2221	2.13129
$909$	−70.0194	−2.32240
$910$	−20.1148	−0.666800
$911$	6.43776	0.213293	0.106646	−	0.994297i	$-0.465989\pi$
$911$	6.43776	0.213293	0.106646	+	0.994297i	$0.465989\pi$
$912$	−28.6531	−0.948798
$913$	−14.2519	−0.471668
$914$	29.3803	0.971813
$915$	−28.9303	−0.956407
$916$	45.8230	1.51403
$917$	7.59133	0.250688
$918$	4.14241	0.136720
$919$	44.3753	1.46381	0.731904	−	0.681408i	$-0.238632\pi$
$919$	44.3753	1.46381	0.731904	+	0.681408i	$0.238632\pi$
$920$	5.64058	0.185964
$921$	85.7634	2.82600
$922$	22.2783	0.733698
$923$	95.8940	3.15639
$924$	53.8100	1.77022
$925$	27.7400	0.912084
$926$	−66.7007	−2.19192
$927$	−41.4587	−1.36168
$928$	0.794175	0.0260701
$929$	51.0452	1.67474	0.837369	−	0.546638i	$-0.184093\pi$
$929$	51.0452	1.67474	0.837369	+	0.546638i	$0.184093\pi$
$930$	−58.7953	−1.92797
$931$	−16.7721	−0.549683
$932$	67.4461	2.20927
$933$	−31.7063	−1.03802
$934$	8.31250	0.271993
$935$	−4.06213	−0.132846
$936$	−134.500	−4.39626
$937$	30.4957	0.996251	0.498125	−	0.867105i	$-0.334022\pi$
$937$	30.4957	0.996251	0.498125	+	0.867105i	$0.334022\pi$
$938$	−11.5049	−0.375648
$939$	−47.1549	−1.53884
$940$	51.4888	1.67938
$941$	1.47064	0.0479414	0.0239707	−	0.999713i	$-0.492369\pi$
$941$	1.47064	0.0479414	0.0239707	+	0.999713i	$0.492369\pi$
$942$	−82.8039	−2.69790
$943$	−0.0785175	−0.00255688
$944$	−25.4129	−0.827120
$945$	−4.19077	−0.136326
$946$	23.2290	0.755238
$947$	34.8514	1.13252	0.566259	−	0.824228i	$-0.308390\pi$
$947$	34.8514	1.13252	0.566259	+	0.824228i	$0.308390\pi$
$948$	63.8174	2.07269
$949$	20.2236	0.656485
$950$	19.7962	0.642272
$951$	33.2693	1.07883
$952$	2.20804	0.0715631
$953$	12.2115	0.395570	0.197785	−	0.980245i	$-0.436625\pi$
$953$	12.2115	0.395570	0.197785	+	0.980245i	$0.436625\pi$
$954$	11.4660	0.371225
$955$	10.9115	0.353087
$956$	−92.2564	−2.98379
$957$	−128.398	−4.15051
$958$	58.0170	1.87444
$959$	12.2879	0.396796
$960$	−30.9174	−0.997855
$961$	8.74981	0.282252
$962$	147.832	4.76631
$963$	−76.0616	−2.45105
$964$	−10.6644	−0.343478
$965$	24.2602	0.780963
$966$	−4.79030	−0.154125
$967$	−5.92843	−0.190645	−0.0953227	−	0.995446i	$-0.530388\pi$
$967$	−5.92843	−0.190645	−0.0953227	+	0.995446i	$0.530388\pi$
$968$	−100.933	−3.24410
$969$	−3.70071	−0.118884
$970$	21.7091	0.697036
$971$	−23.5661	−0.756273	−0.378136	−	0.925750i	$-0.623435\pi$
$971$	−23.5661	−0.756273	−0.378136	+	0.925750i	$0.623435\pi$
$972$	80.2158	2.57292
$973$	9.53155	0.305568
$974$	87.5762	2.80612
$975$	52.4886	1.68098
$976$	−29.9351	−0.958200
$977$	44.0856	1.41042	0.705211	−	0.708997i	$-0.250852\pi$
$977$	44.0856	1.41042	0.705211	+	0.708997i	$0.250852\pi$
$978$	−101.578	−3.24812
$979$	43.0824	1.37692
$980$	35.0877	1.12083
$981$	17.7487	0.566674
$982$	77.5893	2.47597
$983$	2.18152	0.0695796	0.0347898	−	0.999395i	$-0.488924\pi$
$983$	2.18152	0.0695796	0.0347898	+	0.999395i	$0.488924\pi$
$984$	−1.25872	−0.0401265
$985$	−25.6835	−0.818344
$986$	−10.5628	−0.336387
$987$	−21.8111	−0.694253
$988$	70.2757	2.23577
$989$	−1.37749	−0.0438018
$990$	82.7017	2.62843
$991$	−9.22446	−0.293025	−0.146512	−	0.989209i	$-0.546805\pi$
$991$	−9.22446	−0.293025	−0.146512	+	0.989209i	$0.546805\pi$
$992$	0.590776	0.0187572
$993$	−42.9159	−1.36190
$994$	32.0508	1.01659
$995$	32.9624	1.04498
$996$	27.1625	0.860678
$997$	−42.0745	−1.33251	−0.666257	−	0.745722i	$-0.732105\pi$
$997$	−42.0745	−1.33251	−0.666257	+	0.745722i	$0.732105\pi$
$998$	21.9741	0.695578
$999$	30.7998	0.974462

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6047.2.a.a.1.20	✓	217

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6047.2.a.a.1.20	✓	217	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:	\( N \)	\(=\)	\( 6047 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44753 q^{2} -2.68992 q^{3} +3.99041 q^{4} -1.41647 q^{5} +6.58366 q^{6} -0.890122 q^{7} -4.87158 q^{8} +4.23566 q^{9} +O(q^{10})\)	\(q-2.44753 q^{2} -2.68992 q^{3} +3.99041 q^{4} -1.41647 q^{5} +6.58366 q^{6} -0.890122 q^{7} -4.87158 q^{8} +4.23566 q^{9} +3.46686 q^{10} +5.63194 q^{11} -10.7339 q^{12} +6.51825 q^{13} +2.17860 q^{14} +3.81019 q^{15} +3.94253 q^{16} +0.509200 q^{17} -10.3669 q^{18} +2.70183 q^{19} -5.65229 q^{20} +2.39435 q^{21} -13.7843 q^{22} +0.817422 q^{23} +13.1041 q^{24} -2.99361 q^{25} -15.9536 q^{26} -3.32382 q^{27} -3.55195 q^{28} +8.47541 q^{29} -9.32556 q^{30} +6.30475 q^{31} +0.0937034 q^{32} -15.1494 q^{33} -1.24628 q^{34} +1.26083 q^{35} +16.9020 q^{36} -9.26639 q^{37} -6.61280 q^{38} -17.5335 q^{39} +6.90045 q^{40} -0.0960551 q^{41} -5.86026 q^{42} -1.68517 q^{43} +22.4737 q^{44} -5.99969 q^{45} -2.00066 q^{46} -9.10936 q^{47} -10.6051 q^{48} -6.20768 q^{49} +7.32695 q^{50} -1.36971 q^{51} +26.0105 q^{52} -1.10602 q^{53} +8.13514 q^{54} -7.97748 q^{55} +4.33630 q^{56} -7.26769 q^{57} -20.7438 q^{58} -6.44584 q^{59} +15.2042 q^{60} -7.59288 q^{61} -15.4311 q^{62} -3.77025 q^{63} -8.11440 q^{64} -9.23291 q^{65} +37.0787 q^{66} -5.28086 q^{67} +2.03191 q^{68} -2.19880 q^{69} -3.08592 q^{70} +14.7116 q^{71} -20.6343 q^{72} +3.10261 q^{73} +22.6798 q^{74} +8.05256 q^{75} +10.7814 q^{76} -5.01311 q^{77} +42.9139 q^{78} -5.94542 q^{79} -5.58448 q^{80} -3.76618 q^{81} +0.235098 q^{82} -2.53055 q^{83} +9.55445 q^{84} -0.721267 q^{85} +4.12451 q^{86} -22.7982 q^{87} -27.4364 q^{88} +7.64966 q^{89} +14.6844 q^{90} -5.80203 q^{91} +3.26184 q^{92} -16.9592 q^{93} +22.2954 q^{94} -3.82706 q^{95} -0.252055 q^{96} +6.26189 q^{97} +15.1935 q^{98} +23.8550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more