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Properties

Label 4232.2.a.t.1.3
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $1$
Dimension $4$
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 = [4,0,0,0,0,0,0] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.28825\) 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)
 
\(f(q)\) \(=\) \(q+2.23607 q^{3} -2.28825 q^{5} +3.70246 q^{7} +2.00000 q^{9} +0.540182 q^{11} -6.23607 q^{13} -5.11667 q^{15} -3.70246 q^{17} +0.333851 q^{19} +8.27895 q^{21} +0.236068 q^{25} -2.23607 q^{27} -8.70820 q^{29} -7.94427 q^{31} +1.20788 q^{33} -8.47214 q^{35} +9.69316 q^{37} -13.9443 q^{39} +2.70820 q^{41} -8.48528 q^{43} -4.57649 q^{45} -9.47214 q^{47} +6.70820 q^{49} -8.27895 q^{51} +6.65841 q^{53} -1.23607 q^{55} +0.746512 q^{57} +13.7082 q^{59} -9.48683 q^{61} +7.40492 q^{63} +14.2697 q^{65} -2.95595 q^{67} +3.76393 q^{71} -8.23607 q^{73} +0.527864 q^{75} +2.00000 q^{77} -13.3956 q^{79} -11.0000 q^{81} +7.19859 q^{83} +8.47214 q^{85} -19.4721 q^{87} -3.90879 q^{89} -23.0888 q^{91} -17.7639 q^{93} -0.763932 q^{95} +12.1877 q^{97} +1.08036 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9} - 16 q^{13} - 8 q^{25} - 8 q^{29} + 4 q^{31} - 16 q^{35} - 20 q^{39} - 16 q^{41} - 20 q^{47} + 4 q^{55} + 28 q^{59} + 24 q^{71} - 24 q^{73} + 20 q^{75} + 8 q^{77} - 44 q^{81} + 16 q^{85} - 60 q^{87}+ \cdots - 12 q^{95}+O(q^{100}) \) 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
\(3\) 2.23607 1.29099 0.645497 − 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 0 0
\(5\) −2.28825 −1.02333 −0.511667 − 0.859184i \(-0.670972\pi\)
−0.511667 + 0.859184i \(0.670972\pi\)
\(6\) 0 0
\(7\) 3.70246 1.39940 0.699699 − 0.714438i \(-0.253317\pi\)
0.699699 + 0.714438i \(0.253317\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 0.540182 0.162871 0.0814354 − 0.996679i \(-0.474050\pi\)
0.0814354 + 0.996679i \(0.474050\pi\)
\(12\) 0 0
\(13\) −6.23607 −1.72957 −0.864787 − 0.502139i \(-0.832547\pi\)
−0.864787 + 0.502139i \(0.832547\pi\)
\(14\) 0 0
\(15\) −5.11667 −1.32112
\(16\) 0 0
\(17\) −3.70246 −0.897978 −0.448989 − 0.893537i \(-0.648216\pi\)
−0.448989 + 0.893537i \(0.648216\pi\)
\(18\) 0 0
\(19\) 0.333851 0.0765906 0.0382953 − 0.999266i \(-0.487807\pi\)
0.0382953 + 0.999266i \(0.487807\pi\)
\(20\) 0 0
\(21\) 8.27895 1.80662
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 0.236068 0.0472136
\(26\) 0 0
\(27\) −2.23607 −0.430331
\(28\) 0 0
\(29\) −8.70820 −1.61707 −0.808536 − 0.588446i \(-0.799740\pi\)
−0.808536 + 0.588446i \(0.799740\pi\)
\(30\) 0 0
\(31\) −7.94427 −1.42683 −0.713417 − 0.700740i \(-0.752853\pi\)
−0.713417 + 0.700740i \(0.752853\pi\)
\(32\) 0 0
\(33\) 1.20788 0.210265
\(34\) 0 0
\(35\) −8.47214 −1.43205
\(36\) 0 0
\(37\) 9.69316 1.59355 0.796773 − 0.604279i \(-0.206539\pi\)
0.796773 + 0.604279i \(0.206539\pi\)
\(38\) 0 0
\(39\) −13.9443 −2.23287
\(40\) 0 0
\(41\) 2.70820 0.422950 0.211475 − 0.977383i \(-0.432173\pi\)
0.211475 + 0.977383i \(0.432173\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 − 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) −4.57649 −0.682223
\(46\) 0 0
\(47\) −9.47214 −1.38165 −0.690827 − 0.723021i \(-0.742753\pi\)
−0.690827 + 0.723021i \(0.742753\pi\)
\(48\) 0 0
\(49\) 6.70820 0.958315
\(50\) 0 0
\(51\) −8.27895 −1.15928
\(52\) 0 0
\(53\) 6.65841 0.914602 0.457301 − 0.889312i \(-0.348816\pi\)
0.457301 + 0.889312i \(0.348816\pi\)
\(54\) 0 0
\(55\) −1.23607 −0.166671
\(56\) 0 0
\(57\) 0.746512 0.0988780
\(58\) 0 0
\(59\) 13.7082 1.78466 0.892328 − 0.451387i \(-0.149071\pi\)
0.892328 + 0.451387i \(0.149071\pi\)
\(60\) 0 0
\(61\) −9.48683 −1.21466 −0.607332 − 0.794448i \(-0.707760\pi\)
−0.607332 + 0.794448i \(0.707760\pi\)
\(62\) 0 0
\(63\) 7.40492 0.932932
\(64\) 0 0
\(65\) 14.2697 1.76993
\(66\) 0 0
\(67\) −2.95595 −0.361126 −0.180563 − 0.983563i \(-0.557792\pi\)
−0.180563 + 0.983563i \(0.557792\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.76393 0.446697 0.223348 − 0.974739i \(-0.428301\pi\)
0.223348 + 0.974739i \(0.428301\pi\)
\(72\) 0 0
\(73\) −8.23607 −0.963959 −0.481979 − 0.876183i \(-0.660082\pi\)
−0.481979 + 0.876183i \(0.660082\pi\)
\(74\) 0 0
\(75\) 0.527864 0.0609525
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −13.3956 −1.50713 −0.753563 − 0.657376i \(-0.771666\pi\)
−0.753563 + 0.657376i \(0.771666\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 7.19859 0.790148 0.395074 − 0.918649i \(-0.370719\pi\)
0.395074 + 0.918649i \(0.370719\pi\)
\(84\) 0 0
\(85\) 8.47214 0.918932
\(86\) 0 0
\(87\) −19.4721 −2.08763
\(88\) 0 0
\(89\) −3.90879 −0.414331 −0.207165 − 0.978306i \(-0.566424\pi\)
−0.207165 + 0.978306i \(0.566424\pi\)
\(90\) 0 0
\(91\) −23.0888 −2.42036
\(92\) 0 0
\(93\) −17.7639 −1.84203
\(94\) 0 0
\(95\) −0.763932 −0.0783778
\(96\) 0 0
\(97\) 12.1877 1.23748 0.618739 − 0.785597i \(-0.287644\pi\)
0.618739 + 0.785597i \(0.287644\pi\)
\(98\) 0 0
\(99\) 1.08036 0.108581
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.t.1.3 ✓ 4
4.3 odd 2 8464.2.a.bm.1.1 4
23.22 odd 2 inner 4232.2.a.t.1.4 yes 4
92.91 even 2 8464.2.a.bm.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.t.1.3 ✓ 4 1.1 even 1 trivial
4232.2.a.t.1.4 yes 4 23.22 odd 2 inner
8464.2.a.bm.1.1 4 4.3 odd 2
8464.2.a.bm.1.2 4 92.91 even 2