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Properties

Label 8410.2.a.e.1.1
Level $8410$
Weight $2$
Character 8410.1
Self dual yes
Analytic conductor $67.154$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8410,2,Mod(1,8410)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8410.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8410, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8410 = 2 \cdot 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8410.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,0,1,1,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.1541880999\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 290)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8410.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +3.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{28} +10.0000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} -3.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -12.0000 q^{41} -4.00000 q^{43} -2.00000 q^{44} -3.00000 q^{45} +8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -2.00000 q^{53} -2.00000 q^{55} -4.00000 q^{56} -12.0000 q^{59} +4.00000 q^{61} -10.0000 q^{62} -12.0000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} -2.00000 q^{76} -8.00000 q^{77} +10.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} +12.0000 q^{82} -4.00000 q^{83} -6.00000 q^{85} +4.00000 q^{86} +2.00000 q^{88} +12.0000 q^{89} +3.00000 q^{90} +8.00000 q^{91} -8.00000 q^{94} -2.00000 q^{95} +2.00000 q^{97} -9.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 3.00000 0.707107
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 0 0
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 4.00000 0.676123
\(36\) −3.00000 −0.500000
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −10.0000 −1.27000
\(63\) −12.0000 −1.51186
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) 12.0000 1.32518
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 3.00000 0.316228
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −9.00000 −0.909137
\(99\) 6.00000 0.603023
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8410.2.a.e.1.1 1
29.12 odd 4 290.2.c.a.231.1 ✓ 2
29.17 odd 4 290.2.c.a.231.2 yes 2
29.28 even 2 8410.2.a.l.1.1 1
87.17 even 4 2610.2.f.c.811.1 2
87.41 even 4 2610.2.f.c.811.2 2
116.75 even 4 2320.2.g.a.1681.1 2
116.99 even 4 2320.2.g.a.1681.2 2
145.12 even 4 1450.2.d.d.1449.2 2
145.17 even 4 1450.2.d.a.1449.2 2
145.99 odd 4 1450.2.c.b.1101.2 2
145.104 odd 4 1450.2.c.b.1101.1 2
145.128 even 4 1450.2.d.a.1449.1 2
145.133 even 4 1450.2.d.d.1449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.c.a.231.1 ✓ 2 29.12 odd 4
290.2.c.a.231.2 yes 2 29.17 odd 4
1450.2.c.b.1101.1 2 145.104 odd 4
1450.2.c.b.1101.2 2 145.99 odd 4
1450.2.d.a.1449.1 2 145.128 even 4
1450.2.d.a.1449.2 2 145.17 even 4
1450.2.d.d.1449.1 2 145.133 even 4
1450.2.d.d.1449.2 2 145.12 even 4
2320.2.g.a.1681.1 2 116.75 even 4
2320.2.g.a.1681.2 2 116.99 even 4
2610.2.f.c.811.1 2 87.17 even 4
2610.2.f.c.811.2 2 87.41 even 4
8410.2.a.e.1.1 1 1.1 even 1 trivial
8410.2.a.l.1.1 1 29.28 even 2