Properties

Label 8004.2.a.e.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,-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 1.8
Root \(0.511914\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.34732 q^{5} +2.91839 q^{7} +1.00000 q^{9} +3.69746 q^{11} -6.03680 q^{13} -1.34732 q^{15} +5.75034 q^{17} -6.30556 q^{19} -2.91839 q^{21} -1.00000 q^{23} -3.18473 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.671792 q^{31} -3.69746 q^{33} +3.93200 q^{35} -4.42860 q^{37} +6.03680 q^{39} -2.20091 q^{41} -8.58949 q^{43} +1.34732 q^{45} +0.549485 q^{47} +1.51697 q^{49} -5.75034 q^{51} -2.93706 q^{53} +4.98166 q^{55} +6.30556 q^{57} -4.20516 q^{59} -10.6176 q^{61} +2.91839 q^{63} -8.13349 q^{65} -3.98884 q^{67} +1.00000 q^{69} -9.08197 q^{71} -2.11062 q^{73} +3.18473 q^{75} +10.7906 q^{77} -12.8630 q^{79} +1.00000 q^{81} -6.50563 q^{83} +7.74754 q^{85} -1.00000 q^{87} +4.64297 q^{89} -17.6177 q^{91} +0.671792 q^{93} -8.49560 q^{95} -1.50834 q^{97} +3.69746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(5\) 1.34732 0.602539 0.301270 − 0.953539i \(-0.402590\pi\)
0.301270 + 0.953539i \(0.402590\pi\)
\(6\) 0 0
\(7\) 2.91839 1.10305 0.551523 − 0.834160i \(-0.314047\pi\)
0.551523 + 0.834160i \(0.314047\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.69746 1.11483 0.557413 − 0.830235i \(-0.311794\pi\)
0.557413 + 0.830235i \(0.311794\pi\)
\(12\) 0 0
\(13\) −6.03680 −1.67431 −0.837153 − 0.546969i \(-0.815782\pi\)
−0.837153 + 0.546969i \(0.815782\pi\)
\(14\) 0 0
\(15\) −1.34732 −0.347876
\(16\) 0 0
\(17\) 5.75034 1.39466 0.697331 − 0.716750i \(-0.254371\pi\)
0.697331 + 0.716750i \(0.254371\pi\)
\(18\) 0 0
\(19\) −6.30556 −1.44659 −0.723297 − 0.690537i \(-0.757374\pi\)
−0.723297 + 0.690537i \(0.757374\pi\)
\(20\) 0 0
\(21\) −2.91839 −0.636844
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.18473 −0.636946
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −0.671792 −0.120657 −0.0603287 − 0.998179i \(-0.519215\pi\)
−0.0603287 + 0.998179i \(0.519215\pi\)
\(32\) 0 0
\(33\) −3.69746 −0.643645
\(34\) 0 0
\(35\) 3.93200 0.664629
\(36\) 0 0
\(37\) −4.42860 −0.728058 −0.364029 − 0.931388i \(-0.618599\pi\)
−0.364029 + 0.931388i \(0.618599\pi\)
\(38\) 0 0
\(39\) 6.03680 0.966661
\(40\) 0 0
\(41\) −2.20091 −0.343725 −0.171862 − 0.985121i \(-0.554978\pi\)
−0.171862 + 0.985121i \(0.554978\pi\)
\(42\) 0 0
\(43\) −8.58949 −1.30988 −0.654942 − 0.755679i \(-0.727307\pi\)
−0.654942 + 0.755679i \(0.727307\pi\)
\(44\) 0 0
\(45\) 1.34732 0.200846
\(46\) 0 0
\(47\) 0.549485 0.0801506 0.0400753 − 0.999197i \(-0.487240\pi\)
0.0400753 + 0.999197i \(0.487240\pi\)
\(48\) 0 0
\(49\) 1.51697 0.216711
\(50\) 0 0
\(51\) −5.75034 −0.805208
\(52\) 0 0
\(53\) −2.93706 −0.403436 −0.201718 − 0.979444i \(-0.564652\pi\)
−0.201718 + 0.979444i \(0.564652\pi\)
\(54\) 0 0
\(55\) 4.98166 0.671726
\(56\) 0 0
\(57\) 6.30556 0.835191
\(58\) 0 0
\(59\) −4.20516 −0.547465 −0.273733 − 0.961806i \(-0.588258\pi\)
−0.273733 + 0.961806i \(0.588258\pi\)
\(60\) 0 0
\(61\) −10.6176 −1.35945 −0.679724 − 0.733468i \(-0.737900\pi\)
−0.679724 + 0.733468i \(0.737900\pi\)
\(62\) 0 0
\(63\) 2.91839 0.367682
\(64\) 0 0
\(65\) −8.13349 −1.00884
\(66\) 0 0
\(67\) −3.98884 −0.487314 −0.243657 − 0.969861i \(-0.578347\pi\)
−0.243657 + 0.969861i \(0.578347\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.08197 −1.07783 −0.538916 − 0.842359i \(-0.681166\pi\)
−0.538916 + 0.842359i \(0.681166\pi\)
\(72\) 0 0
\(73\) −2.11062 −0.247029 −0.123515 − 0.992343i \(-0.539417\pi\)
−0.123515 + 0.992343i \(0.539417\pi\)
\(74\) 0 0
\(75\) 3.18473 0.367741
\(76\) 0 0
\(77\) 10.7906 1.22970
\(78\) 0 0
\(79\) −12.8630 −1.44720 −0.723598 − 0.690222i \(-0.757513\pi\)
−0.723598 + 0.690222i \(0.757513\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.50563 −0.714085 −0.357043 − 0.934088i \(-0.616215\pi\)
−0.357043 + 0.934088i \(0.616215\pi\)
\(84\) 0 0
\(85\) 7.74754 0.840338
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 4.64297 0.492154 0.246077 − 0.969250i \(-0.420858\pi\)
0.246077 + 0.969250i \(0.420858\pi\)
\(90\) 0 0
\(91\) −17.6177 −1.84684
\(92\) 0 0
\(93\) 0.671792 0.0696616
\(94\) 0 0
\(95\) −8.49560 −0.871630
\(96\) 0 0
\(97\) −1.50834 −0.153148 −0.0765742 − 0.997064i \(-0.524398\pi\)
−0.0765742 + 0.997064i \(0.524398\pi\)
\(98\) 0 0
\(99\) 3.69746 0.371609
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.1.8 ✓ 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.8 ✓ 9 1.1 even 1 trivial