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Properties

Label 8464.2.a.cf.1.7
Level $8464$
Weight $2$
Character 8464.1
Self dual yes
Analytic conductor $67.585$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-8,0,0,0,0,0,8,0,0,0,16] 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: \(67.5853802708\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 4232)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.90564\) of defining polynomial
Character \(\chi\) \(=\) 8464.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.368525 q^{3} -1.72190 q^{5} +0.634565 q^{7} -2.86419 q^{9} +6.09935 q^{11} +6.49865 q^{13} +0.634565 q^{15} -2.10696 q^{17} -4.13463 q^{19} -0.233853 q^{21} -2.03505 q^{25} +2.16110 q^{27} +0.137641 q^{29} -6.17402 q^{31} -2.24777 q^{33} -1.09266 q^{35} -9.22178 q^{37} -2.39492 q^{39} -3.58758 q^{41} +5.59880 q^{43} +4.93186 q^{45} -7.61453 q^{47} -6.59733 q^{49} +0.776469 q^{51} +7.23472 q^{53} -10.5025 q^{55} +1.52371 q^{57} +9.94415 q^{59} +1.63915 q^{61} -1.81751 q^{63} -11.1900 q^{65} -13.6926 q^{67} -4.13695 q^{71} +4.17219 q^{73} +0.749968 q^{75} +3.87043 q^{77} +2.88084 q^{79} +7.79615 q^{81} +10.3441 q^{83} +3.62799 q^{85} -0.0507241 q^{87} -1.91162 q^{89} +4.12381 q^{91} +2.27528 q^{93} +7.11942 q^{95} +6.18953 q^{97} -17.4697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} - 8 q^{27} - 8 q^{31} - 56 q^{35} - 64 q^{39} - 40 q^{41} - 32 q^{47} + 28 q^{49} - 64 q^{55} - 60 q^{59} + 32 q^{71} + 28 q^{73} - 16 q^{75} + 24 q^{77}+ \cdots - 16 q^{95}+O(q^{100}) \) 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
\(3\) −0.368525 −0.212768 −0.106384 − 0.994325i \(-0.533927\pi\)
−0.106384 + 0.994325i \(0.533927\pi\)
\(4\) 0 0
\(5\) −1.72190 −0.770058 −0.385029 − 0.922904i \(-0.625809\pi\)
−0.385029 + 0.922904i \(0.625809\pi\)
\(6\) 0 0
\(7\) 0.634565 0.239843 0.119921 − 0.992783i \(-0.461736\pi\)
0.119921 + 0.992783i \(0.461736\pi\)
\(8\) 0 0
\(9\) −2.86419 −0.954730
\(10\) 0 0
\(11\) 6.09935 1.83902 0.919512 − 0.393062i \(-0.128584\pi\)
0.919512 + 0.393062i \(0.128584\pi\)
\(12\) 0 0
\(13\) 6.49865 1.80240 0.901201 − 0.433402i \(-0.142687\pi\)
0.901201 + 0.433402i \(0.142687\pi\)
\(14\) 0 0
\(15\) 0.634565 0.163844
\(16\) 0 0
\(17\) −2.10696 −0.511014 −0.255507 − 0.966807i \(-0.582242\pi\)
−0.255507 + 0.966807i \(0.582242\pi\)
\(18\) 0 0
\(19\) −4.13463 −0.948548 −0.474274 − 0.880377i \(-0.657289\pi\)
−0.474274 + 0.880377i \(0.657289\pi\)
\(20\) 0 0
\(21\) −0.233853 −0.0510309
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −2.03505 −0.407010
\(26\) 0 0
\(27\) 2.16110 0.415904
\(28\) 0 0
\(29\) 0.137641 0.0255593 0.0127796 − 0.999918i \(-0.495932\pi\)
0.0127796 + 0.999918i \(0.495932\pi\)
\(30\) 0 0
\(31\) −6.17402 −1.10889 −0.554443 − 0.832222i \(-0.687069\pi\)
−0.554443 + 0.832222i \(0.687069\pi\)
\(32\) 0 0
\(33\) −2.24777 −0.391286
\(34\) 0 0
\(35\) −1.09266 −0.184693
\(36\) 0 0
\(37\) −9.22178 −1.51605 −0.758026 − 0.652225i \(-0.773836\pi\)
−0.758026 + 0.652225i \(0.773836\pi\)
\(38\) 0 0
\(39\) −2.39492 −0.383494
\(40\) 0 0
\(41\) −3.58758 −0.560286 −0.280143 − 0.959958i \(-0.590382\pi\)
−0.280143 + 0.959958i \(0.590382\pi\)
\(42\) 0 0
\(43\) 5.59880 0.853809 0.426904 − 0.904297i \(-0.359604\pi\)
0.426904 + 0.904297i \(0.359604\pi\)
\(44\) 0 0
\(45\) 4.93186 0.735198
\(46\) 0 0
\(47\) −7.61453 −1.11069 −0.555347 − 0.831619i \(-0.687414\pi\)
−0.555347 + 0.831619i \(0.687414\pi\)
\(48\) 0 0
\(49\) −6.59733 −0.942475
\(50\) 0 0
\(51\) 0.776469 0.108727
\(52\) 0 0
\(53\) 7.23472 0.993766 0.496883 − 0.867818i \(-0.334478\pi\)
0.496883 + 0.867818i \(0.334478\pi\)
\(54\) 0 0
\(55\) −10.5025 −1.41616
\(56\) 0 0
\(57\) 1.52371 0.201821
\(58\) 0 0
\(59\) 9.94415 1.29462 0.647309 − 0.762228i \(-0.275894\pi\)
0.647309 + 0.762228i \(0.275894\pi\)
\(60\) 0 0
\(61\) 1.63915 0.209872 0.104936 − 0.994479i \(-0.466536\pi\)
0.104936 + 0.994479i \(0.466536\pi\)
\(62\) 0 0
\(63\) −1.81751 −0.228985
\(64\) 0 0
\(65\) −11.1900 −1.38795
\(66\) 0 0
\(67\) −13.6926 −1.67282 −0.836411 − 0.548103i \(-0.815350\pi\)
−0.836411 + 0.548103i \(0.815350\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.13695 −0.490966 −0.245483 − 0.969401i \(-0.578947\pi\)
−0.245483 + 0.969401i \(0.578947\pi\)
\(72\) 0 0
\(73\) 4.17219 0.488318 0.244159 − 0.969735i \(-0.421488\pi\)
0.244159 + 0.969735i \(0.421488\pi\)
\(74\) 0 0
\(75\) 0.749968 0.0865988
\(76\) 0 0
\(77\) 3.87043 0.441077
\(78\) 0 0
\(79\) 2.88084 0.324120 0.162060 − 0.986781i \(-0.448186\pi\)
0.162060 + 0.986781i \(0.448186\pi\)
\(80\) 0 0
\(81\) 7.79615 0.866239
\(82\) 0 0
\(83\) 10.3441 1.13541 0.567705 − 0.823232i \(-0.307831\pi\)
0.567705 + 0.823232i \(0.307831\pi\)
\(84\) 0 0
\(85\) 3.62799 0.393510
\(86\) 0 0
\(87\) −0.0507241 −0.00543820
\(88\) 0 0
\(89\) −1.91162 −0.202632 −0.101316 − 0.994854i \(-0.532305\pi\)
−0.101316 + 0.994854i \(0.532305\pi\)
\(90\) 0 0
\(91\) 4.12381 0.432293
\(92\) 0 0
\(93\) 2.27528 0.235936
\(94\) 0 0
\(95\) 7.11942 0.730437
\(96\) 0 0
\(97\) 6.18953 0.628451 0.314226 − 0.949348i \(-0.398255\pi\)
0.314226 + 0.949348i \(0.398255\pi\)
\(98\) 0 0
\(99\) −17.4697 −1.75577
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 8464.2.a.cf.1.7 12
4.3 odd 2 4232.2.a.x.1.5 ✓ 12
23.22 odd 2 inner 8464.2.a.cf.1.8 12
92.91 even 2 4232.2.a.x.1.6 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.x.1.5 ✓ 12 4.3 odd 2
4232.2.a.x.1.6 yes 12 92.91 even 2
8464.2.a.cf.1.7 12 1.1 even 1 trivial
8464.2.a.cf.1.8 12 23.22 odd 2 inner