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Properties

Label 4232.2.a.x.1.2
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
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] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4232 = 2^{3} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,8,0,0,0,0] 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: \(33.7926901354\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.52595\) of defining polynomial
Character \(\chi\) \(=\) 4232.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.38043 q^{3} +0.0992850 q^{5} -0.236341 q^{7} +2.66645 q^{9} +2.12038 q^{11} -5.09308 q^{13} -0.236341 q^{15} +6.44776 q^{17} +0.582013 q^{19} +0.562594 q^{21} -4.99014 q^{25} +0.793990 q^{27} +3.65591 q^{29} -3.91182 q^{31} -5.04741 q^{33} -0.0234651 q^{35} -7.09213 q^{37} +12.1237 q^{39} -7.57960 q^{41} +10.7311 q^{43} +0.264739 q^{45} +10.8700 q^{47} -6.94414 q^{49} -15.3485 q^{51} -6.54082 q^{53} +0.210522 q^{55} -1.38544 q^{57} +8.89024 q^{59} +4.79354 q^{61} -0.630192 q^{63} -0.505666 q^{65} +3.29978 q^{67} -12.2161 q^{71} -3.90127 q^{73} +11.8787 q^{75} -0.501132 q^{77} -11.7440 q^{79} -9.88939 q^{81} +7.31907 q^{83} +0.640167 q^{85} -8.70263 q^{87} +3.04701 q^{89} +1.20370 q^{91} +9.31181 q^{93} +0.0577851 q^{95} -4.49485 q^{97} +5.65388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) \) 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
\(3\) −2.38043 −1.37434 −0.687171 − 0.726496i \(-0.741148\pi\)
−0.687171 + 0.726496i \(0.741148\pi\)
\(4\) 0 0
\(5\) 0.0992850 0.0444016 0.0222008 − 0.999754i \(-0.492933\pi\)
0.0222008 + 0.999754i \(0.492933\pi\)
\(6\) 0 0
\(7\) −0.236341 −0.0893286 −0.0446643 − 0.999002i \(-0.514222\pi\)
−0.0446643 + 0.999002i \(0.514222\pi\)
\(8\) 0 0
\(9\) 2.66645 0.888817
\(10\) 0 0
\(11\) 2.12038 0.639318 0.319659 − 0.947533i \(-0.396432\pi\)
0.319659 + 0.947533i \(0.396432\pi\)
\(12\) 0 0
\(13\) −5.09308 −1.41257 −0.706283 − 0.707930i \(-0.749629\pi\)
−0.706283 + 0.707930i \(0.749629\pi\)
\(14\) 0 0
\(15\) −0.236341 −0.0610230
\(16\) 0 0
\(17\) 6.44776 1.56381 0.781906 − 0.623396i \(-0.214248\pi\)
0.781906 + 0.623396i \(0.214248\pi\)
\(18\) 0 0
\(19\) 0.582013 0.133523 0.0667614 − 0.997769i \(-0.478733\pi\)
0.0667614 + 0.997769i \(0.478733\pi\)
\(20\) 0 0
\(21\) 0.562594 0.122768
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.99014 −0.998028
\(26\) 0 0
\(27\) 0.793990 0.152803
\(28\) 0 0
\(29\) 3.65591 0.678885 0.339442 − 0.940627i \(-0.389762\pi\)
0.339442 + 0.940627i \(0.389762\pi\)
\(30\) 0 0
\(31\) −3.91182 −0.702583 −0.351292 − 0.936266i \(-0.614257\pi\)
−0.351292 + 0.936266i \(0.614257\pi\)
\(32\) 0 0
\(33\) −5.04741 −0.878641
\(34\) 0 0
\(35\) −0.0234651 −0.00396633
\(36\) 0 0
\(37\) −7.09213 −1.16594 −0.582970 − 0.812494i \(-0.698109\pi\)
−0.582970 + 0.812494i \(0.698109\pi\)
\(38\) 0 0
\(39\) 12.1237 1.94135
\(40\) 0 0
\(41\) −7.57960 −1.18374 −0.591868 − 0.806035i \(-0.701609\pi\)
−0.591868 + 0.806035i \(0.701609\pi\)
\(42\) 0 0
\(43\) 10.7311 1.63648 0.818242 − 0.574873i \(-0.194949\pi\)
0.818242 + 0.574873i \(0.194949\pi\)
\(44\) 0 0
\(45\) 0.264739 0.0394649
\(46\) 0 0
\(47\) 10.8700 1.58555 0.792774 − 0.609515i \(-0.208636\pi\)
0.792774 + 0.609515i \(0.208636\pi\)
\(48\) 0 0
\(49\) −6.94414 −0.992020
\(50\) 0 0
\(51\) −15.3485 −2.14921
\(52\) 0 0
\(53\) −6.54082 −0.898450 −0.449225 − 0.893419i \(-0.648300\pi\)
−0.449225 + 0.893419i \(0.648300\pi\)
\(54\) 0 0
\(55\) 0.210522 0.0283867
\(56\) 0 0
\(57\) −1.38544 −0.183506
\(58\) 0 0
\(59\) 8.89024 1.15741 0.578706 − 0.815537i \(-0.303558\pi\)
0.578706 + 0.815537i \(0.303558\pi\)
\(60\) 0 0
\(61\) 4.79354 0.613751 0.306875 − 0.951750i \(-0.400717\pi\)
0.306875 + 0.951750i \(0.400717\pi\)
\(62\) 0 0
\(63\) −0.630192 −0.0793968
\(64\) 0 0
\(65\) −0.505666 −0.0627202
\(66\) 0 0
\(67\) 3.29978 0.403132 0.201566 − 0.979475i \(-0.435397\pi\)
0.201566 + 0.979475i \(0.435397\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.2161 −1.44978 −0.724892 − 0.688862i \(-0.758111\pi\)
−0.724892 + 0.688862i \(0.758111\pi\)
\(72\) 0 0
\(73\) −3.90127 −0.456609 −0.228305 − 0.973590i \(-0.573318\pi\)
−0.228305 + 0.973590i \(0.573318\pi\)
\(74\) 0 0
\(75\) 11.8787 1.37163
\(76\) 0 0
\(77\) −0.501132 −0.0571093
\(78\) 0 0
\(79\) −11.7440 −1.32131 −0.660653 − 0.750691i \(-0.729721\pi\)
−0.660653 + 0.750691i \(0.729721\pi\)
\(80\) 0 0
\(81\) −9.88939 −1.09882
\(82\) 0 0
\(83\) 7.31907 0.803372 0.401686 − 0.915778i \(-0.368424\pi\)
0.401686 + 0.915778i \(0.368424\pi\)
\(84\) 0 0
\(85\) 0.640167 0.0694358
\(86\) 0 0
\(87\) −8.70263 −0.933020
\(88\) 0 0
\(89\) 3.04701 0.322983 0.161491 − 0.986874i \(-0.448370\pi\)
0.161491 + 0.986874i \(0.448370\pi\)
\(90\) 0 0
\(91\) 1.20370 0.126182
\(92\) 0 0
\(93\) 9.31181 0.965590
\(94\) 0 0
\(95\) 0.0577851 0.00592863
\(96\) 0 0
\(97\) −4.49485 −0.456383 −0.228191 − 0.973616i \(-0.573281\pi\)
−0.228191 + 0.973616i \(0.573281\pi\)
\(98\) 0 0
\(99\) 5.65388 0.568236
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 4232.2.a.x.1.2 yes 12
4.3 odd 2 8464.2.a.cf.1.12 12
23.22 odd 2 inner 4232.2.a.x.1.1 ✓ 12
92.91 even 2 8464.2.a.cf.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.x.1.1 ✓ 12 23.22 odd 2 inner
4232.2.a.x.1.2 yes 12 1.1 even 1 trivial
8464.2.a.cf.1.11 12 92.91 even 2
8464.2.a.cf.1.12 12 4.3 odd 2