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Properties

Label 5808.2.a.cj.1.1
Level $5808$
Weight $2$
Character 5808.1
Self dual yes
Analytic conductor $46.377$
Analytic rank $1$
Dimension $2$
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] = [5808,2,Mod(1,5808)] 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("5808.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5808.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,1,0,2,0,2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.3771134940\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5808.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.236068 q^{13} -0.618034 q^{15} -1.14590 q^{17} -5.85410 q^{19} +1.00000 q^{21} -0.236068 q^{23} -4.61803 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.09017 q^{31} -0.618034 q^{35} -6.23607 q^{37} +0.236068 q^{39} +0.236068 q^{41} +6.70820 q^{43} -0.618034 q^{45} +10.0902 q^{47} -6.00000 q^{49} -1.14590 q^{51} -0.381966 q^{53} -5.85410 q^{57} -7.38197 q^{59} -11.5623 q^{61} +1.00000 q^{63} -0.145898 q^{65} -1.85410 q^{67} -0.236068 q^{69} -10.3262 q^{71} -5.70820 q^{73} -4.61803 q^{75} -11.0000 q^{79} +1.00000 q^{81} -1.47214 q^{83} +0.708204 q^{85} -6.00000 q^{87} -8.23607 q^{89} +0.236068 q^{91} +6.09017 q^{93} +3.61803 q^{95} +7.85410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} + q^{15} - 9 q^{17} - 5 q^{19} + 2 q^{21} + 4 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{31} + q^{35} - 8 q^{37} - 4 q^{39} - 4 q^{41} + q^{45}+ \cdots + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.618034 −0.276393 −0.138197 − 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0.236068 0.0654735 0.0327367 − 0.999464i \(-0.489578\pi\)
0.0327367 + 0.999464i \(0.489578\pi\)
\(14\) 0 0
\(15\) −0.618034 −0.159576
\(16\) 0 0
\(17\) −1.14590 −0.277921 −0.138961 − 0.990298i \(-0.544376\pi\)
−0.138961 + 0.990298i \(0.544376\pi\)
\(18\) 0 0
\(19\) −5.85410 −1.34302 −0.671512 − 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −0.236068 −0.0492236 −0.0246118 − 0.999697i \(-0.507835\pi\)
−0.0246118 + 0.999697i \(0.507835\pi\)
\(24\) 0 0
\(25\) −4.61803 −0.923607
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.09017 1.09383 0.546913 − 0.837189i \(-0.315803\pi\)
0.546913 + 0.837189i \(0.315803\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.618034 −0.104467
\(36\) 0 0
\(37\) −6.23607 −1.02520 −0.512602 − 0.858627i \(-0.671318\pi\)
−0.512602 + 0.858627i \(0.671318\pi\)
\(38\) 0 0
\(39\) 0.236068 0.0378011
\(40\) 0 0
\(41\) 0.236068 0.0368676 0.0184338 − 0.999830i \(-0.494132\pi\)
0.0184338 + 0.999830i \(0.494132\pi\)
\(42\) 0 0
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0 0
\(47\) 10.0902 1.47180 0.735901 − 0.677089i \(-0.236759\pi\)
0.735901 + 0.677089i \(0.236759\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −1.14590 −0.160458
\(52\) 0 0
\(53\) −0.381966 −0.0524671 −0.0262335 − 0.999656i \(-0.508351\pi\)
−0.0262335 + 0.999656i \(0.508351\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.85410 −0.775395
\(58\) 0 0
\(59\) −7.38197 −0.961050 −0.480525 − 0.876981i \(-0.659554\pi\)
−0.480525 + 0.876981i \(0.659554\pi\)
\(60\) 0 0
\(61\) −11.5623 −1.48040 −0.740201 − 0.672386i \(-0.765270\pi\)
−0.740201 + 0.672386i \(0.765270\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −0.145898 −0.0180964
\(66\) 0 0
\(67\) −1.85410 −0.226515 −0.113257 − 0.993566i \(-0.536128\pi\)
−0.113257 + 0.993566i \(0.536128\pi\)
\(68\) 0 0
\(69\) −0.236068 −0.0284192
\(70\) 0 0
\(71\) −10.3262 −1.22550 −0.612749 − 0.790277i \(-0.709937\pi\)
−0.612749 + 0.790277i \(0.709937\pi\)
\(72\) 0 0
\(73\) −5.70820 −0.668095 −0.334047 − 0.942556i \(-0.608415\pi\)
−0.334047 + 0.942556i \(0.608415\pi\)
\(74\) 0 0
\(75\) −4.61803 −0.533245
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.47214 −0.161588 −0.0807940 − 0.996731i \(-0.525746\pi\)
−0.0807940 + 0.996731i \(0.525746\pi\)
\(84\) 0 0
\(85\) 0.708204 0.0768155
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −8.23607 −0.873021 −0.436511 − 0.899699i \(-0.643786\pi\)
−0.436511 + 0.899699i \(0.643786\pi\)
\(90\) 0 0
\(91\) 0.236068 0.0247466
\(92\) 0 0
\(93\) 6.09017 0.631521
\(94\) 0 0
\(95\) 3.61803 0.371202
\(96\) 0 0
\(97\) 7.85410 0.797463 0.398732 − 0.917068i \(-0.369451\pi\)
0.398732 + 0.917068i \(0.369451\pi\)
\(98\) 0 0
\(99\) 0 0
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 5808.2.a.cj.1.1 2
4.3 odd 2 363.2.a.d.1.1 2
11.3 even 5 528.2.y.b.97.1 4
11.4 even 5 528.2.y.b.49.1 4
11.10 odd 2 5808.2.a.ci.1.1 2
12.11 even 2 1089.2.a.t.1.2 2
20.19 odd 2 9075.2.a.cb.1.2 2
44.3 odd 10 33.2.e.b.31.1 yes 4
44.7 even 10 363.2.e.f.148.1 4
44.15 odd 10 33.2.e.b.16.1 ✓ 4
44.19 even 10 363.2.e.f.130.1 4
44.27 odd 10 363.2.e.k.124.1 4
44.31 odd 10 363.2.e.k.202.1 4
44.35 even 10 363.2.e.b.202.1 4
44.39 even 10 363.2.e.b.124.1 4
44.43 even 2 363.2.a.i.1.2 2
132.47 even 10 99.2.f.a.64.1 4
132.59 even 10 99.2.f.a.82.1 4
132.131 odd 2 1089.2.a.l.1.1 2
220.3 even 20 825.2.bx.d.724.2 8
220.47 even 20 825.2.bx.d.724.1 8
220.59 odd 10 825.2.n.c.676.1 4
220.103 even 20 825.2.bx.d.49.1 8
220.147 even 20 825.2.bx.d.49.2 8
220.179 odd 10 825.2.n.c.526.1 4
220.219 even 2 9075.2.a.u.1.1 2
396.47 even 30 891.2.n.b.757.1 8
396.59 even 30 891.2.n.b.379.1 8
396.103 odd 30 891.2.n.c.379.1 8
396.191 even 30 891.2.n.b.676.1 8
396.223 odd 30 891.2.n.c.757.1 8
396.311 even 30 891.2.n.b.460.1 8
396.355 odd 30 891.2.n.c.460.1 8
396.367 odd 30 891.2.n.c.676.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.16.1 ✓ 4 44.15 odd 10
33.2.e.b.31.1 yes 4 44.3 odd 10
99.2.f.a.64.1 4 132.47 even 10
99.2.f.a.82.1 4 132.59 even 10
363.2.a.d.1.1 2 4.3 odd 2
363.2.a.i.1.2 2 44.43 even 2
363.2.e.b.124.1 4 44.39 even 10
363.2.e.b.202.1 4 44.35 even 10
363.2.e.f.130.1 4 44.19 even 10
363.2.e.f.148.1 4 44.7 even 10
363.2.e.k.124.1 4 44.27 odd 10
363.2.e.k.202.1 4 44.31 odd 10
528.2.y.b.49.1 4 11.4 even 5
528.2.y.b.97.1 4 11.3 even 5
825.2.n.c.526.1 4 220.179 odd 10
825.2.n.c.676.1 4 220.59 odd 10
825.2.bx.d.49.1 8 220.103 even 20
825.2.bx.d.49.2 8 220.147 even 20
825.2.bx.d.724.1 8 220.47 even 20
825.2.bx.d.724.2 8 220.3 even 20
891.2.n.b.379.1 8 396.59 even 30
891.2.n.b.460.1 8 396.311 even 30
891.2.n.b.676.1 8 396.191 even 30
891.2.n.b.757.1 8 396.47 even 30
891.2.n.c.379.1 8 396.103 odd 30
891.2.n.c.460.1 8 396.355 odd 30
891.2.n.c.676.1 8 396.367 odd 30
891.2.n.c.757.1 8 396.223 odd 30
1089.2.a.l.1.1 2 132.131 odd 2
1089.2.a.t.1.2 2 12.11 even 2
5808.2.a.ci.1.1 2 11.10 odd 2
5808.2.a.cj.1.1 2 1.1 even 1 trivial
9075.2.a.u.1.1 2 220.219 even 2
9075.2.a.cb.1.2 2 20.19 odd 2