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Properties

Label 4232.2.a.x.1.9
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 1.9
Root \(1.31154\) 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)
 
\(f(q)\) \(=\) \(q+2.27985 q^{3} -1.62262 q^{5} -3.69933 q^{7} +2.19774 q^{9} -5.39319 q^{11} -0.869675 q^{13} -3.69933 q^{15} +3.44423 q^{17} +5.99215 q^{19} -8.43394 q^{21} -2.36711 q^{25} -1.82904 q^{27} +6.59876 q^{29} +6.66600 q^{31} -12.2957 q^{33} +6.00260 q^{35} -1.89437 q^{37} -1.98273 q^{39} +8.09539 q^{41} +12.3421 q^{43} -3.56609 q^{45} -3.55630 q^{47} +6.68506 q^{49} +7.85233 q^{51} -8.82665 q^{53} +8.75109 q^{55} +13.6612 q^{57} +12.5898 q^{59} -7.43203 q^{61} -8.13016 q^{63} +1.41115 q^{65} -7.08638 q^{67} +0.0791492 q^{71} +3.26498 q^{73} -5.39667 q^{75} +19.9512 q^{77} +9.31522 q^{79} -10.7632 q^{81} +14.6390 q^{83} -5.58866 q^{85} +15.0442 q^{87} -2.62678 q^{89} +3.21722 q^{91} +15.1975 q^{93} -9.72297 q^{95} +3.58937 q^{97} -11.8528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) \) 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
\(3\) 2.27985 1.31627 0.658137 − 0.752898i \(-0.271345\pi\)
0.658137 + 0.752898i \(0.271345\pi\)
\(4\) 0 0
\(5\) −1.62262 −0.725657 −0.362828 − 0.931856i \(-0.618189\pi\)
−0.362828 + 0.931856i \(0.618189\pi\)
\(6\) 0 0
\(7\) −3.69933 −1.39822 −0.699108 − 0.715016i \(-0.746419\pi\)
−0.699108 + 0.715016i \(0.746419\pi\)
\(8\) 0 0
\(9\) 2.19774 0.732579
\(10\) 0 0
\(11\) −5.39319 −1.62611 −0.813054 − 0.582188i \(-0.802197\pi\)
−0.813054 + 0.582188i \(0.802197\pi\)
\(12\) 0 0
\(13\) −0.869675 −0.241204 −0.120602 − 0.992701i \(-0.538483\pi\)
−0.120602 + 0.992701i \(0.538483\pi\)
\(14\) 0 0
\(15\) −3.69933 −0.955164
\(16\) 0 0
\(17\) 3.44423 0.835347 0.417674 − 0.908597i \(-0.362846\pi\)
0.417674 + 0.908597i \(0.362846\pi\)
\(18\) 0 0
\(19\) 5.99215 1.37469 0.687347 − 0.726329i \(-0.258775\pi\)
0.687347 + 0.726329i \(0.258775\pi\)
\(20\) 0 0
\(21\) −8.43394 −1.84044
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −2.36711 −0.473422
\(26\) 0 0
\(27\) −1.82904 −0.351999
\(28\) 0 0
\(29\) 6.59876 1.22536 0.612679 − 0.790332i \(-0.290092\pi\)
0.612679 + 0.790332i \(0.290092\pi\)
\(30\) 0 0
\(31\) 6.66600 1.19725 0.598625 − 0.801030i \(-0.295714\pi\)
0.598625 + 0.801030i \(0.295714\pi\)
\(32\) 0 0
\(33\) −12.2957 −2.14041
\(34\) 0 0
\(35\) 6.00260 1.01463
\(36\) 0 0
\(37\) −1.89437 −0.311433 −0.155717 − 0.987802i \(-0.549769\pi\)
−0.155717 + 0.987802i \(0.549769\pi\)
\(38\) 0 0
\(39\) −1.98273 −0.317491
\(40\) 0 0
\(41\) 8.09539 1.26429 0.632144 − 0.774851i \(-0.282175\pi\)
0.632144 + 0.774851i \(0.282175\pi\)
\(42\) 0 0
\(43\) 12.3421 1.88216 0.941078 − 0.338190i \(-0.109815\pi\)
0.941078 + 0.338190i \(0.109815\pi\)
\(44\) 0 0
\(45\) −3.56609 −0.531601
\(46\) 0 0
\(47\) −3.55630 −0.518739 −0.259370 − 0.965778i \(-0.583515\pi\)
−0.259370 + 0.965778i \(0.583515\pi\)
\(48\) 0 0
\(49\) 6.68506 0.955009
\(50\) 0 0
\(51\) 7.85233 1.09955
\(52\) 0 0
\(53\) −8.82665 −1.21243 −0.606217 − 0.795299i \(-0.707314\pi\)
−0.606217 + 0.795299i \(0.707314\pi\)
\(54\) 0 0
\(55\) 8.75109 1.18000
\(56\) 0 0
\(57\) 13.6612 1.80947
\(58\) 0 0
\(59\) 12.5898 1.63905 0.819526 − 0.573041i \(-0.194237\pi\)
0.819526 + 0.573041i \(0.194237\pi\)
\(60\) 0 0
\(61\) −7.43203 −0.951575 −0.475787 − 0.879560i \(-0.657837\pi\)
−0.475787 + 0.879560i \(0.657837\pi\)
\(62\) 0 0
\(63\) −8.13016 −1.02430
\(64\) 0 0
\(65\) 1.41115 0.175032
\(66\) 0 0
\(67\) −7.08638 −0.865739 −0.432869 − 0.901457i \(-0.642499\pi\)
−0.432869 + 0.901457i \(0.642499\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.0791492 0.00939328 0.00469664 − 0.999989i \(-0.498505\pi\)
0.00469664 + 0.999989i \(0.498505\pi\)
\(72\) 0 0
\(73\) 3.26498 0.382137 0.191069 − 0.981577i \(-0.438805\pi\)
0.191069 + 0.981577i \(0.438805\pi\)
\(74\) 0 0
\(75\) −5.39667 −0.623154
\(76\) 0 0
\(77\) 19.9512 2.27365
\(78\) 0 0
\(79\) 9.31522 1.04804 0.524022 − 0.851705i \(-0.324431\pi\)
0.524022 + 0.851705i \(0.324431\pi\)
\(80\) 0 0
\(81\) −10.7632 −1.19591
\(82\) 0 0
\(83\) 14.6390 1.60684 0.803422 − 0.595410i \(-0.203010\pi\)
0.803422 + 0.595410i \(0.203010\pi\)
\(84\) 0 0
\(85\) −5.58866 −0.606175
\(86\) 0 0
\(87\) 15.0442 1.61291
\(88\) 0 0
\(89\) −2.62678 −0.278438 −0.139219 − 0.990262i \(-0.544459\pi\)
−0.139219 + 0.990262i \(0.544459\pi\)
\(90\) 0 0
\(91\) 3.21722 0.337256
\(92\) 0 0
\(93\) 15.1975 1.57591
\(94\) 0 0
\(95\) −9.72297 −0.997556
\(96\) 0 0
\(97\) 3.58937 0.364445 0.182222 − 0.983257i \(-0.441671\pi\)
0.182222 + 0.983257i \(0.441671\pi\)
\(98\) 0 0
\(99\) −11.8528 −1.19125
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.1.9 ✓ 12
4.3 odd 2 8464.2.a.cf.1.3 12
23.22 odd 2 inner 4232.2.a.x.1.10 yes 12
92.91 even 2 8464.2.a.cf.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.x.1.9 ✓ 12 1.1 even 1 trivial
4232.2.a.x.1.10 yes 12 23.22 odd 2 inner
8464.2.a.cf.1.3 12 4.3 odd 2
8464.2.a.cf.1.4 12 92.91 even 2