Properties

Label 8004.2.a.f.1.8
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
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] = [8004,2,Mod(1,8004)] 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("8004.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,9,0,-1,0,-5] 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: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.91513\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.91513 q^{5} -2.97719 q^{7} +1.00000 q^{9} +2.04500 q^{11} +1.36883 q^{13} +1.91513 q^{15} -6.26690 q^{17} -2.09894 q^{19} -2.97719 q^{21} -1.00000 q^{23} -1.33227 q^{25} +1.00000 q^{27} -1.00000 q^{29} +1.34340 q^{31} +2.04500 q^{33} -5.70171 q^{35} -5.21257 q^{37} +1.36883 q^{39} +7.87339 q^{41} -1.58152 q^{43} +1.91513 q^{45} +3.90675 q^{47} +1.86366 q^{49} -6.26690 q^{51} -9.86566 q^{53} +3.91645 q^{55} -2.09894 q^{57} -2.00323 q^{59} -9.80143 q^{61} -2.97719 q^{63} +2.62148 q^{65} +2.45384 q^{67} -1.00000 q^{69} -8.65971 q^{71} +14.2737 q^{73} -1.33227 q^{75} -6.08837 q^{77} +0.878902 q^{79} +1.00000 q^{81} -13.3255 q^{83} -12.0019 q^{85} -1.00000 q^{87} -5.49647 q^{89} -4.07526 q^{91} +1.34340 q^{93} -4.01975 q^{95} -8.26837 q^{97} +2.04500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(5\) 1.91513 0.856473 0.428236 − 0.903667i \(-0.359135\pi\)
0.428236 + 0.903667i \(0.359135\pi\)
\(6\) 0 0
\(7\) −2.97719 −1.12527 −0.562636 − 0.826705i \(-0.690213\pi\)
−0.562636 + 0.826705i \(0.690213\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.04500 0.616592 0.308296 − 0.951290i \(-0.400241\pi\)
0.308296 + 0.951290i \(0.400241\pi\)
\(12\) 0 0
\(13\) 1.36883 0.379645 0.189822 − 0.981818i \(-0.439209\pi\)
0.189822 + 0.981818i \(0.439209\pi\)
\(14\) 0 0
\(15\) 1.91513 0.494485
\(16\) 0 0
\(17\) −6.26690 −1.51995 −0.759974 − 0.649954i \(-0.774788\pi\)
−0.759974 + 0.649954i \(0.774788\pi\)
\(18\) 0 0
\(19\) −2.09894 −0.481530 −0.240765 − 0.970583i \(-0.577398\pi\)
−0.240765 + 0.970583i \(0.577398\pi\)
\(20\) 0 0
\(21\) −2.97719 −0.649676
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −1.33227 −0.266455
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.34340 0.241282 0.120641 − 0.992696i \(-0.461505\pi\)
0.120641 + 0.992696i \(0.461505\pi\)
\(32\) 0 0
\(33\) 2.04500 0.355990
\(34\) 0 0
\(35\) −5.70171 −0.963765
\(36\) 0 0
\(37\) −5.21257 −0.856942 −0.428471 − 0.903556i \(-0.640948\pi\)
−0.428471 + 0.903556i \(0.640948\pi\)
\(38\) 0 0
\(39\) 1.36883 0.219188
\(40\) 0 0
\(41\) 7.87339 1.22962 0.614808 − 0.788677i \(-0.289233\pi\)
0.614808 + 0.788677i \(0.289233\pi\)
\(42\) 0 0
\(43\) −1.58152 −0.241180 −0.120590 − 0.992702i \(-0.538479\pi\)
−0.120590 + 0.992702i \(0.538479\pi\)
\(44\) 0 0
\(45\) 1.91513 0.285491
\(46\) 0 0
\(47\) 3.90675 0.569858 0.284929 − 0.958549i \(-0.408030\pi\)
0.284929 + 0.958549i \(0.408030\pi\)
\(48\) 0 0
\(49\) 1.86366 0.266237
\(50\) 0 0
\(51\) −6.26690 −0.877542
\(52\) 0 0
\(53\) −9.86566 −1.35515 −0.677576 − 0.735453i \(-0.736970\pi\)
−0.677576 + 0.735453i \(0.736970\pi\)
\(54\) 0 0
\(55\) 3.91645 0.528094
\(56\) 0 0
\(57\) −2.09894 −0.278012
\(58\) 0 0
\(59\) −2.00323 −0.260799 −0.130399 − 0.991462i \(-0.541626\pi\)
−0.130399 + 0.991462i \(0.541626\pi\)
\(60\) 0 0
\(61\) −9.80143 −1.25494 −0.627472 − 0.778639i \(-0.715910\pi\)
−0.627472 + 0.778639i \(0.715910\pi\)
\(62\) 0 0
\(63\) −2.97719 −0.375091
\(64\) 0 0
\(65\) 2.62148 0.325155
\(66\) 0 0
\(67\) 2.45384 0.299784 0.149892 − 0.988702i \(-0.452107\pi\)
0.149892 + 0.988702i \(0.452107\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.65971 −1.02772 −0.513859 − 0.857875i \(-0.671785\pi\)
−0.513859 + 0.857875i \(0.671785\pi\)
\(72\) 0 0
\(73\) 14.2737 1.67061 0.835303 − 0.549790i \(-0.185292\pi\)
0.835303 + 0.549790i \(0.185292\pi\)
\(74\) 0 0
\(75\) −1.33227 −0.153838
\(76\) 0 0
\(77\) −6.08837 −0.693834
\(78\) 0 0
\(79\) 0.878902 0.0988842 0.0494421 − 0.998777i \(-0.484256\pi\)
0.0494421 + 0.998777i \(0.484256\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.3255 −1.46266 −0.731330 − 0.682024i \(-0.761100\pi\)
−0.731330 + 0.682024i \(0.761100\pi\)
\(84\) 0 0
\(85\) −12.0019 −1.30179
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −5.49647 −0.582624 −0.291312 − 0.956628i \(-0.594092\pi\)
−0.291312 + 0.956628i \(0.594092\pi\)
\(90\) 0 0
\(91\) −4.07526 −0.427203
\(92\) 0 0
\(93\) 1.34340 0.139304
\(94\) 0 0
\(95\) −4.01975 −0.412418
\(96\) 0 0
\(97\) −8.26837 −0.839525 −0.419763 − 0.907634i \(-0.637887\pi\)
−0.419763 + 0.907634i \(0.637887\pi\)
\(98\) 0 0
\(99\) 2.04500 0.205531
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 8004.2.a.f.1.8 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.f.1.8 ✓ 9 1.1 even 1 trivial