Properties

Label 8004.2.a.e.1.5
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,-3,0,7] 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: \( x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.761439\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.461945 q^{5} +0.732054 q^{7} +1.00000 q^{9} +0.317021 q^{11} +1.73156 q^{13} -0.461945 q^{15} -1.85694 q^{17} -0.400818 q^{19} -0.732054 q^{21} -1.00000 q^{23} -4.78661 q^{25} -1.00000 q^{27} +1.00000 q^{29} -2.76393 q^{31} -0.317021 q^{33} +0.338169 q^{35} +6.86987 q^{37} -1.73156 q^{39} +2.88726 q^{41} +7.54434 q^{43} +0.461945 q^{45} -11.3820 q^{47} -6.46410 q^{49} +1.85694 q^{51} -10.8018 q^{53} +0.146446 q^{55} +0.400818 q^{57} +3.88177 q^{59} -4.01686 q^{61} +0.732054 q^{63} +0.799886 q^{65} -6.53778 q^{67} +1.00000 q^{69} -0.411185 q^{71} -13.9499 q^{73} +4.78661 q^{75} +0.232077 q^{77} +11.1974 q^{79} +1.00000 q^{81} +6.15780 q^{83} -0.857805 q^{85} -1.00000 q^{87} -4.04181 q^{89} +1.26760 q^{91} +2.76393 q^{93} -0.185156 q^{95} +10.3181 q^{97} +0.317021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(5\) 0.461945 0.206588 0.103294 − 0.994651i \(-0.467062\pi\)
0.103294 + 0.994651i \(0.467062\pi\)
\(6\) 0 0
\(7\) 0.732054 0.276691 0.138345 − 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.317021 0.0955854 0.0477927 − 0.998857i \(-0.484781\pi\)
0.0477927 + 0.998857i \(0.484781\pi\)
\(12\) 0 0
\(13\) 1.73156 0.480249 0.240124 − 0.970742i \(-0.422812\pi\)
0.240124 + 0.970742i \(0.422812\pi\)
\(14\) 0 0
\(15\) −0.461945 −0.119274
\(16\) 0 0
\(17\) −1.85694 −0.450375 −0.225187 − 0.974315i \(-0.572299\pi\)
−0.225187 + 0.974315i \(0.572299\pi\)
\(18\) 0 0
\(19\) −0.400818 −0.0919540 −0.0459770 − 0.998942i \(-0.514640\pi\)
−0.0459770 + 0.998942i \(0.514640\pi\)
\(20\) 0 0
\(21\) −0.732054 −0.159747
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.78661 −0.957321
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.76393 −0.496417 −0.248208 − 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) 0 0
\(33\) −0.317021 −0.0551863
\(34\) 0 0
\(35\) 0.338169 0.0571610
\(36\) 0 0
\(37\) 6.86987 1.12940 0.564700 − 0.825296i \(-0.308992\pi\)
0.564700 + 0.825296i \(0.308992\pi\)
\(38\) 0 0
\(39\) −1.73156 −0.277272
\(40\) 0 0
\(41\) 2.88726 0.450915 0.225457 − 0.974253i \(-0.427612\pi\)
0.225457 + 0.974253i \(0.427612\pi\)
\(42\) 0 0
\(43\) 7.54434 1.15050 0.575250 − 0.817977i \(-0.304905\pi\)
0.575250 + 0.817977i \(0.304905\pi\)
\(44\) 0 0
\(45\) 0.461945 0.0688627
\(46\) 0 0
\(47\) −11.3820 −1.66024 −0.830118 − 0.557588i \(-0.811727\pi\)
−0.830118 + 0.557588i \(0.811727\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923442
\(50\) 0 0
\(51\) 1.85694 0.260024
\(52\) 0 0
\(53\) −10.8018 −1.48374 −0.741872 − 0.670541i \(-0.766062\pi\)
−0.741872 + 0.670541i \(0.766062\pi\)
\(54\) 0 0
\(55\) 0.146446 0.0197468
\(56\) 0 0
\(57\) 0.400818 0.0530896
\(58\) 0 0
\(59\) 3.88177 0.505364 0.252682 − 0.967549i \(-0.418687\pi\)
0.252682 + 0.967549i \(0.418687\pi\)
\(60\) 0 0
\(61\) −4.01686 −0.514306 −0.257153 − 0.966371i \(-0.582784\pi\)
−0.257153 + 0.966371i \(0.582784\pi\)
\(62\) 0 0
\(63\) 0.732054 0.0922302
\(64\) 0 0
\(65\) 0.799886 0.0992137
\(66\) 0 0
\(67\) −6.53778 −0.798716 −0.399358 − 0.916795i \(-0.630767\pi\)
−0.399358 + 0.916795i \(0.630767\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.411185 −0.0487987 −0.0243994 − 0.999702i \(-0.507767\pi\)
−0.0243994 + 0.999702i \(0.507767\pi\)
\(72\) 0 0
\(73\) −13.9499 −1.63271 −0.816354 − 0.577552i \(-0.804008\pi\)
−0.816354 + 0.577552i \(0.804008\pi\)
\(74\) 0 0
\(75\) 4.78661 0.552710
\(76\) 0 0
\(77\) 0.232077 0.0264476
\(78\) 0 0
\(79\) 11.1974 1.25980 0.629902 − 0.776675i \(-0.283095\pi\)
0.629902 + 0.776675i \(0.283095\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.15780 0.675906 0.337953 − 0.941163i \(-0.390265\pi\)
0.337953 + 0.941163i \(0.390265\pi\)
\(84\) 0 0
\(85\) −0.857805 −0.0930420
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −4.04181 −0.428431 −0.214216 − 0.976786i \(-0.568720\pi\)
−0.214216 + 0.976786i \(0.568720\pi\)
\(90\) 0 0
\(91\) 1.26760 0.132880
\(92\) 0 0
\(93\) 2.76393 0.286606
\(94\) 0 0
\(95\) −0.185156 −0.0189966
\(96\) 0 0
\(97\) 10.3181 1.04765 0.523823 − 0.851827i \(-0.324505\pi\)
0.523823 + 0.851827i \(0.324505\pi\)
\(98\) 0 0
\(99\) 0.317021 0.0318618
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.e.1.5 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.5 ✓ 9 1.1 even 1 trivial