Properties

Label 8004.2.a.f.1.7
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 1.7
Root \(-1.09926\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.09926 q^{5} -1.62400 q^{7} +1.00000 q^{9} -4.16402 q^{11} +3.95343 q^{13} +1.09926 q^{15} -2.36393 q^{17} +4.37074 q^{19} -1.62400 q^{21} -1.00000 q^{23} -3.79163 q^{25} +1.00000 q^{27} -1.00000 q^{29} -8.09478 q^{31} -4.16402 q^{33} -1.78520 q^{35} +3.52968 q^{37} +3.95343 q^{39} -3.81475 q^{41} +1.25079 q^{43} +1.09926 q^{45} +6.41448 q^{47} -4.36263 q^{49} -2.36393 q^{51} -3.55412 q^{53} -4.57734 q^{55} +4.37074 q^{57} +5.96232 q^{59} -12.3425 q^{61} -1.62400 q^{63} +4.34585 q^{65} -12.8312 q^{67} -1.00000 q^{69} +8.30445 q^{71} -2.26071 q^{73} -3.79163 q^{75} +6.76236 q^{77} -14.7456 q^{79} +1.00000 q^{81} +9.49232 q^{83} -2.59857 q^{85} -1.00000 q^{87} -4.52312 q^{89} -6.42036 q^{91} -8.09478 q^{93} +4.80459 q^{95} -2.79546 q^{97} -4.16402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(5\) 1.09926 0.491604 0.245802 − 0.969320i \(-0.420949\pi\)
0.245802 + 0.969320i \(0.420949\pi\)
\(6\) 0 0
\(7\) −1.62400 −0.613814 −0.306907 − 0.951740i \(-0.599294\pi\)
−0.306907 + 0.951740i \(0.599294\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.16402 −1.25550 −0.627750 − 0.778415i \(-0.716024\pi\)
−0.627750 + 0.778415i \(0.716024\pi\)
\(12\) 0 0
\(13\) 3.95343 1.09648 0.548242 − 0.836320i \(-0.315297\pi\)
0.548242 + 0.836320i \(0.315297\pi\)
\(14\) 0 0
\(15\) 1.09926 0.283828
\(16\) 0 0
\(17\) −2.36393 −0.573337 −0.286668 − 0.958030i \(-0.592548\pi\)
−0.286668 + 0.958030i \(0.592548\pi\)
\(18\) 0 0
\(19\) 4.37074 1.00272 0.501359 − 0.865240i \(-0.332834\pi\)
0.501359 + 0.865240i \(0.332834\pi\)
\(20\) 0 0
\(21\) −1.62400 −0.354386
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.79163 −0.758325
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.09478 −1.45387 −0.726933 − 0.686708i \(-0.759055\pi\)
−0.726933 + 0.686708i \(0.759055\pi\)
\(32\) 0 0
\(33\) −4.16402 −0.724863
\(34\) 0 0
\(35\) −1.78520 −0.301754
\(36\) 0 0
\(37\) 3.52968 0.580276 0.290138 − 0.956985i \(-0.406299\pi\)
0.290138 + 0.956985i \(0.406299\pi\)
\(38\) 0 0
\(39\) 3.95343 0.633055
\(40\) 0 0
\(41\) −3.81475 −0.595764 −0.297882 − 0.954603i \(-0.596280\pi\)
−0.297882 + 0.954603i \(0.596280\pi\)
\(42\) 0 0
\(43\) 1.25079 0.190744 0.0953721 − 0.995442i \(-0.469596\pi\)
0.0953721 + 0.995442i \(0.469596\pi\)
\(44\) 0 0
\(45\) 1.09926 0.163868
\(46\) 0 0
\(47\) 6.41448 0.935649 0.467824 − 0.883821i \(-0.345038\pi\)
0.467824 + 0.883821i \(0.345038\pi\)
\(48\) 0 0
\(49\) −4.36263 −0.623233
\(50\) 0 0
\(51\) −2.36393 −0.331016
\(52\) 0 0
\(53\) −3.55412 −0.488196 −0.244098 − 0.969751i \(-0.578492\pi\)
−0.244098 + 0.969751i \(0.578492\pi\)
\(54\) 0 0
\(55\) −4.57734 −0.617209
\(56\) 0 0
\(57\) 4.37074 0.578919
\(58\) 0 0
\(59\) 5.96232 0.776228 0.388114 − 0.921611i \(-0.373127\pi\)
0.388114 + 0.921611i \(0.373127\pi\)
\(60\) 0 0
\(61\) −12.3425 −1.58030 −0.790149 − 0.612915i \(-0.789997\pi\)
−0.790149 + 0.612915i \(0.789997\pi\)
\(62\) 0 0
\(63\) −1.62400 −0.204605
\(64\) 0 0
\(65\) 4.34585 0.539036
\(66\) 0 0
\(67\) −12.8312 −1.56758 −0.783791 − 0.621024i \(-0.786717\pi\)
−0.783791 + 0.621024i \(0.786717\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.30445 0.985557 0.492778 − 0.870155i \(-0.335981\pi\)
0.492778 + 0.870155i \(0.335981\pi\)
\(72\) 0 0
\(73\) −2.26071 −0.264596 −0.132298 − 0.991210i \(-0.542236\pi\)
−0.132298 + 0.991210i \(0.542236\pi\)
\(74\) 0 0
\(75\) −3.79163 −0.437819
\(76\) 0 0
\(77\) 6.76236 0.770643
\(78\) 0 0
\(79\) −14.7456 −1.65900 −0.829502 − 0.558504i \(-0.811376\pi\)
−0.829502 + 0.558504i \(0.811376\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.49232 1.04192 0.520959 − 0.853582i \(-0.325575\pi\)
0.520959 + 0.853582i \(0.325575\pi\)
\(84\) 0 0
\(85\) −2.59857 −0.281855
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −4.52312 −0.479450 −0.239725 − 0.970841i \(-0.577057\pi\)
−0.239725 + 0.970841i \(0.577057\pi\)
\(90\) 0 0
\(91\) −6.42036 −0.673036
\(92\) 0 0
\(93\) −8.09478 −0.839390
\(94\) 0 0
\(95\) 4.80459 0.492940
\(96\) 0 0
\(97\) −2.79546 −0.283836 −0.141918 − 0.989878i \(-0.545327\pi\)
−0.141918 + 0.989878i \(0.545327\pi\)
\(98\) 0 0
\(99\) −4.16402 −0.418500
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.1.7 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.f.1.7 ✓ 9 1.1 even 1 trivial