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Properties

Label 4232.2.a.t.1.1
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 1.1
Root \(0.874032\) 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)
 
\(f(q)\) \(=\) \(q-2.23607 q^{3} -0.874032 q^{5} -0.540182 q^{7} +2.00000 q^{9} -3.70246 q^{11} -1.76393 q^{13} +1.95440 q^{15} +0.540182 q^{17} +5.99070 q^{19} +1.20788 q^{21} -4.23607 q^{25} +2.23607 q^{27} +4.70820 q^{29} +9.94427 q^{31} +8.27895 q^{33} +0.472136 q^{35} -0.206331 q^{37} +3.94427 q^{39} -10.7082 q^{41} +8.48528 q^{43} -1.74806 q^{45} -0.527864 q^{47} -6.70820 q^{49} -1.20788 q^{51} +12.3153 q^{53} +3.23607 q^{55} -13.3956 q^{57} +0.291796 q^{59} -9.48683 q^{61} -1.08036 q^{63} +1.54173 q^{65} -12.8554 q^{67} +8.23607 q^{71} -3.76393 q^{73} +9.47214 q^{75} +2.00000 q^{77} +0.746512 q^{79} -11.0000 q^{81} +8.61280 q^{83} -0.472136 q^{85} -10.5279 q^{87} +10.2333 q^{89} +0.952843 q^{91} -22.2361 q^{93} -5.23607 q^{95} -9.02546 q^{97} -7.40492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9} - 16 q^{13} - 8 q^{25} - 8 q^{29} + 4 q^{31} - 16 q^{35} - 20 q^{39} - 16 q^{41} - 20 q^{47} + 4 q^{55} + 28 q^{59} + 24 q^{71} - 24 q^{73} + 20 q^{75} + 8 q^{77} - 44 q^{81} + 16 q^{85} - 60 q^{87}+ \cdots - 12 q^{95}+O(q^{100}) \) 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 \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 0 0
\(5\) −0.874032 −0.390879 −0.195440 − 0.980716i \(-0.562613\pi\)
−0.195440 + 0.980716i \(0.562613\pi\)
\(6\) 0 0
\(7\) −0.540182 −0.204169 −0.102085 − 0.994776i \(-0.532551\pi\)
−0.102085 + 0.994776i \(0.532551\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.70246 −1.11633 −0.558167 − 0.829729i \(-0.688495\pi\)
−0.558167 + 0.829729i \(0.688495\pi\)
\(12\) 0 0
\(13\) −1.76393 −0.489227 −0.244613 − 0.969621i \(-0.578661\pi\)
−0.244613 + 0.969621i \(0.578661\pi\)
\(14\) 0 0
\(15\) 1.95440 0.504623
\(16\) 0 0
\(17\) 0.540182 0.131013 0.0655066 − 0.997852i \(-0.479134\pi\)
0.0655066 + 0.997852i \(0.479134\pi\)
\(18\) 0 0
\(19\) 5.99070 1.37436 0.687181 − 0.726486i \(-0.258848\pi\)
0.687181 + 0.726486i \(0.258848\pi\)
\(20\) 0 0
\(21\) 1.20788 0.263582
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.23607 −0.847214
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) 4.70820 0.874292 0.437146 − 0.899391i \(-0.355989\pi\)
0.437146 + 0.899391i \(0.355989\pi\)
\(30\) 0 0
\(31\) 9.94427 1.78604 0.893022 − 0.450013i \(-0.148581\pi\)
0.893022 + 0.450013i \(0.148581\pi\)
\(32\) 0 0
\(33\) 8.27895 1.44118
\(34\) 0 0
\(35\) 0.472136 0.0798055
\(36\) 0 0
\(37\) −0.206331 −0.0339206 −0.0169603 − 0.999856i \(-0.505399\pi\)
−0.0169603 + 0.999856i \(0.505399\pi\)
\(38\) 0 0
\(39\) 3.94427 0.631589
\(40\) 0 0
\(41\) −10.7082 −1.67234 −0.836170 − 0.548470i \(-0.815210\pi\)
−0.836170 + 0.548470i \(0.815210\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 − 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) −1.74806 −0.260586
\(46\) 0 0
\(47\) −0.527864 −0.0769969 −0.0384984 − 0.999259i \(-0.512257\pi\)
−0.0384984 + 0.999259i \(0.512257\pi\)
\(48\) 0 0
\(49\) −6.70820 −0.958315
\(50\) 0 0
\(51\) −1.20788 −0.169137
\(52\) 0 0
\(53\) 12.3153 1.69163 0.845816 − 0.533475i \(-0.179114\pi\)
0.845816 + 0.533475i \(0.179114\pi\)
\(54\) 0 0
\(55\) 3.23607 0.436351
\(56\) 0 0
\(57\) −13.3956 −1.77429
\(58\) 0 0
\(59\) 0.291796 0.0379886 0.0189943 − 0.999820i \(-0.493954\pi\)
0.0189943 + 0.999820i \(0.493954\pi\)
\(60\) 0 0
\(61\) −9.48683 −1.21466 −0.607332 − 0.794448i \(-0.707760\pi\)
−0.607332 + 0.794448i \(0.707760\pi\)
\(62\) 0 0
\(63\) −1.08036 −0.136113
\(64\) 0 0
\(65\) 1.54173 0.191228
\(66\) 0 0
\(67\) −12.8554 −1.57054 −0.785271 − 0.619152i \(-0.787476\pi\)
−0.785271 + 0.619152i \(0.787476\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.23607 0.977441 0.488721 − 0.872440i \(-0.337464\pi\)
0.488721 + 0.872440i \(0.337464\pi\)
\(72\) 0 0
\(73\) −3.76393 −0.440535 −0.220267 − 0.975440i \(-0.570693\pi\)
−0.220267 + 0.975440i \(0.570693\pi\)
\(74\) 0 0
\(75\) 9.47214 1.09375
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 0.746512 0.0839892 0.0419946 − 0.999118i \(-0.486629\pi\)
0.0419946 + 0.999118i \(0.486629\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 8.61280 0.945378 0.472689 − 0.881229i \(-0.343283\pi\)
0.472689 + 0.881229i \(0.343283\pi\)
\(84\) 0 0
\(85\) −0.472136 −0.0512103
\(86\) 0 0
\(87\) −10.5279 −1.12871
\(88\) 0 0
\(89\) 10.2333 1.08473 0.542366 − 0.840142i \(-0.317529\pi\)
0.542366 + 0.840142i \(0.317529\pi\)
\(90\) 0 0
\(91\) 0.952843 0.0998851
\(92\) 0 0
\(93\) −22.2361 −2.30577
\(94\) 0 0
\(95\) −5.23607 −0.537209
\(96\) 0 0
\(97\) −9.02546 −0.916397 −0.458198 − 0.888850i \(-0.651505\pi\)
−0.458198 + 0.888850i \(0.651505\pi\)
\(98\) 0 0
\(99\) −7.40492 −0.744222
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.1.1 ✓ 4
4.3 odd 2 8464.2.a.bm.1.3 4
23.22 odd 2 inner 4232.2.a.t.1.2 yes 4
92.91 even 2 8464.2.a.bm.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.t.1.1 ✓ 4 1.1 even 1 trivial
4232.2.a.t.1.2 yes 4 23.22 odd 2 inner
8464.2.a.bm.1.3 4 4.3 odd 2
8464.2.a.bm.1.4 4 92.91 even 2