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Properties

Label 4232.2.a.y.1.9
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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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 = [15,0,-1,0,-10,0,-10] 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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.322533\) 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)
 
\(f(q)\) \(=\) \(q+0.322533 q^{3} +2.48772 q^{5} +0.274819 q^{7} -2.89597 q^{9} -2.55684 q^{11} -3.41303 q^{13} +0.802372 q^{15} +3.52774 q^{17} -3.46313 q^{19} +0.0886380 q^{21} +1.18877 q^{25} -1.90164 q^{27} -1.16074 q^{29} +8.51726 q^{31} -0.824665 q^{33} +0.683673 q^{35} -0.463759 q^{37} -1.10082 q^{39} -11.4991 q^{41} +3.94854 q^{43} -7.20438 q^{45} +0.258587 q^{47} -6.92447 q^{49} +1.13781 q^{51} +1.04859 q^{53} -6.36071 q^{55} -1.11697 q^{57} +10.0383 q^{59} -5.79233 q^{61} -0.795867 q^{63} -8.49068 q^{65} -3.94712 q^{67} -6.19780 q^{71} -4.03700 q^{73} +0.383416 q^{75} -0.702667 q^{77} -17.4853 q^{79} +8.07457 q^{81} -15.5326 q^{83} +8.77604 q^{85} -0.374375 q^{87} -7.96709 q^{89} -0.937965 q^{91} +2.74710 q^{93} -8.61532 q^{95} -7.27807 q^{97} +7.40454 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} - 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} - 20 q^{17} + 3 q^{19} - 21 q^{21} + 11 q^{25} - 7 q^{27} - 4 q^{29} - 14 q^{31} - 34 q^{33} + 18 q^{35} - 26 q^{37} + 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) 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
\(3\) 0.322533 0.186214 0.0931072 − 0.995656i \(-0.470320\pi\)
0.0931072 + 0.995656i \(0.470320\pi\)
\(4\) 0 0
\(5\) 2.48772 1.11254 0.556272 − 0.831000i \(-0.312231\pi\)
0.556272 + 0.831000i \(0.312231\pi\)
\(6\) 0 0
\(7\) 0.274819 0.103872 0.0519358 − 0.998650i \(-0.483461\pi\)
0.0519358 + 0.998650i \(0.483461\pi\)
\(8\) 0 0
\(9\) −2.89597 −0.965324
\(10\) 0 0
\(11\) −2.55684 −0.770917 −0.385458 − 0.922725i \(-0.625957\pi\)
−0.385458 + 0.922725i \(0.625957\pi\)
\(12\) 0 0
\(13\) −3.41303 −0.946605 −0.473303 − 0.880900i \(-0.656938\pi\)
−0.473303 + 0.880900i \(0.656938\pi\)
\(14\) 0 0
\(15\) 0.802372 0.207172
\(16\) 0 0
\(17\) 3.52774 0.855603 0.427801 − 0.903873i \(-0.359288\pi\)
0.427801 + 0.903873i \(0.359288\pi\)
\(18\) 0 0
\(19\) −3.46313 −0.794497 −0.397249 − 0.917711i \(-0.630035\pi\)
−0.397249 + 0.917711i \(0.630035\pi\)
\(20\) 0 0
\(21\) 0.0886380 0.0193424
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 1.18877 0.237753
\(26\) 0 0
\(27\) −1.90164 −0.365972
\(28\) 0 0
\(29\) −1.16074 −0.215543 −0.107772 − 0.994176i \(-0.534372\pi\)
−0.107772 + 0.994176i \(0.534372\pi\)
\(30\) 0 0
\(31\) 8.51726 1.52975 0.764873 − 0.644181i \(-0.222802\pi\)
0.764873 + 0.644181i \(0.222802\pi\)
\(32\) 0 0
\(33\) −0.824665 −0.143556
\(34\) 0 0
\(35\) 0.683673 0.115562
\(36\) 0 0
\(37\) −0.463759 −0.0762414 −0.0381207 − 0.999273i \(-0.512137\pi\)
−0.0381207 + 0.999273i \(0.512137\pi\)
\(38\) 0 0
\(39\) −1.10082 −0.176272
\(40\) 0 0
\(41\) −11.4991 −1.79585 −0.897926 − 0.440146i \(-0.854927\pi\)
−0.897926 + 0.440146i \(0.854927\pi\)
\(42\) 0 0
\(43\) 3.94854 0.602146 0.301073 − 0.953601i \(-0.402655\pi\)
0.301073 + 0.953601i \(0.402655\pi\)
\(44\) 0 0
\(45\) −7.20438 −1.07397
\(46\) 0 0
\(47\) 0.258587 0.0377188 0.0188594 − 0.999822i \(-0.493997\pi\)
0.0188594 + 0.999822i \(0.493997\pi\)
\(48\) 0 0
\(49\) −6.92447 −0.989211
\(50\) 0 0
\(51\) 1.13781 0.159326
\(52\) 0 0
\(53\) 1.04859 0.144035 0.0720175 − 0.997403i \(-0.477056\pi\)
0.0720175 + 0.997403i \(0.477056\pi\)
\(54\) 0 0
\(55\) −6.36071 −0.857678
\(56\) 0 0
\(57\) −1.11697 −0.147947
\(58\) 0 0
\(59\) 10.0383 1.30688 0.653439 − 0.756979i \(-0.273326\pi\)
0.653439 + 0.756979i \(0.273326\pi\)
\(60\) 0 0
\(61\) −5.79233 −0.741631 −0.370816 − 0.928706i \(-0.620922\pi\)
−0.370816 + 0.928706i \(0.620922\pi\)
\(62\) 0 0
\(63\) −0.795867 −0.100270
\(64\) 0 0
\(65\) −8.49068 −1.05314
\(66\) 0 0
\(67\) −3.94712 −0.482218 −0.241109 − 0.970498i \(-0.577511\pi\)
−0.241109 + 0.970498i \(0.577511\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.19780 −0.735543 −0.367771 − 0.929916i \(-0.619879\pi\)
−0.367771 + 0.929916i \(0.619879\pi\)
\(72\) 0 0
\(73\) −4.03700 −0.472495 −0.236248 − 0.971693i \(-0.575918\pi\)
−0.236248 + 0.971693i \(0.575918\pi\)
\(74\) 0 0
\(75\) 0.383416 0.0442731
\(76\) 0 0
\(77\) −0.702667 −0.0800764
\(78\) 0 0
\(79\) −17.4853 −1.96724 −0.983622 − 0.180241i \(-0.942312\pi\)
−0.983622 + 0.180241i \(0.942312\pi\)
\(80\) 0 0
\(81\) 8.07457 0.897175
\(82\) 0 0
\(83\) −15.5326 −1.70493 −0.852464 − 0.522785i \(-0.824893\pi\)
−0.852464 + 0.522785i \(0.824893\pi\)
\(84\) 0 0
\(85\) 8.77604 0.951895
\(86\) 0 0
\(87\) −0.374375 −0.0401372
\(88\) 0 0
\(89\) −7.96709 −0.844510 −0.422255 − 0.906477i \(-0.638761\pi\)
−0.422255 + 0.906477i \(0.638761\pi\)
\(90\) 0 0
\(91\) −0.937965 −0.0983255
\(92\) 0 0
\(93\) 2.74710 0.284861
\(94\) 0 0
\(95\) −8.61532 −0.883913
\(96\) 0 0
\(97\) −7.27807 −0.738976 −0.369488 − 0.929235i \(-0.620467\pi\)
−0.369488 + 0.929235i \(0.620467\pi\)
\(98\) 0 0
\(99\) 7.40454 0.744184
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.y.1.9 15
4.3 odd 2 8464.2.a.ci.1.7 15
23.3 even 11 184.2.i.a.9.2 ✓ 30
23.8 even 11 184.2.i.a.41.2 yes 30
23.22 odd 2 4232.2.a.z.1.9 15
92.3 odd 22 368.2.m.f.193.2 30
92.31 odd 22 368.2.m.f.225.2 30
92.91 even 2 8464.2.a.cj.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.9.2 ✓ 30 23.3 even 11
184.2.i.a.41.2 yes 30 23.8 even 11
368.2.m.f.193.2 30 92.3 odd 22
368.2.m.f.225.2 30 92.31 odd 22
4232.2.a.y.1.9 15 1.1 even 1 trivial
4232.2.a.z.1.9 15 23.22 odd 2
8464.2.a.ci.1.7 15 4.3 odd 2
8464.2.a.cj.1.7 15 92.91 even 2