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Properties

Label 8464.2.a.cf.1.8
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 1.8
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 \(-0.374191\pi\)
0.385029 + 0.922904i \(0.374191\pi\)
\(6\) 0 0
\(7\) −0.634565 −0.239843 −0.119921 − 0.992783i \(-0.538264\pi\)
−0.119921 + 0.992783i \(0.538264\pi\)
\(8\) 0 0
\(9\) −2.86419 −0.954730
\(10\) 0 0
\(11\) −6.09935 −1.83902 −0.919512 − 0.393062i \(-0.871416\pi\)
−0.919512 + 0.393062i \(0.871416\pi\)
\(12\) 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 \(-0.417758\pi\)
0.255507 + 0.966807i \(0.417758\pi\)
\(18\) 0 0
\(19\) 4.13463 0.948548 0.474274 − 0.880377i \(-0.342711\pi\)
0.474274 + 0.880377i \(0.342711\pi\)
\(20\) 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 \(-0.226164\pi\)
0.758026 + 0.652225i \(0.226164\pi\)
\(38\) 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 \(-0.640396\pi\)
−0.426904 + 0.904297i \(0.640396\pi\)
\(44\) 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 \(-0.665522\pi\)
−0.496883 + 0.867818i \(0.665522\pi\)
\(54\) 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 \(-0.533464\pi\)
−0.104936 + 0.994479i \(0.533464\pi\)
\(62\) 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 \(-0.184650\pi\)
0.836411 + 0.548103i \(0.184650\pi\)
\(68\) 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 \(-0.551814\pi\)
−0.162060 + 0.986781i \(0.551814\pi\)
\(80\) 0 0
\(81\) 7.79615 0.866239
\(82\) 0 0
\(83\) −10.3441 −1.13541 −0.567705 − 0.823232i \(-0.692169\pi\)
−0.567705 + 0.823232i \(0.692169\pi\)
\(84\) 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 \(-0.467695\pi\)
0.101316 + 0.994854i \(0.467695\pi\)
\(90\) 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 \(-0.601745\pi\)
−0.314226 + 0.949348i \(0.601745\pi\)
\(98\) 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.1.8 12
4.3 odd 2 4232.2.a.x.1.6 yes 12
23.22 odd 2 inner 8464.2.a.cf.1.7 12
92.91 even 2 4232.2.a.x.1.5 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.x.1.5 ✓ 12 92.91 even 2
4232.2.a.x.1.6 yes 12 4.3 odd 2
8464.2.a.cf.1.7 12 23.22 odd 2 inner
8464.2.a.cf.1.8 12 1.1 even 1 trivial