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Properties

Label 4232.2.a.z.1.2
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
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: \(0\)
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 1.2
Root \(2.80169\) 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)
 
\(f(q)\) \(=\) \(q-2.80169 q^{3} +0.737681 q^{5} -3.85057 q^{7} +4.84949 q^{9} -3.18997 q^{11} +3.22799 q^{13} -2.06676 q^{15} +4.11959 q^{17} +5.88860 q^{19} +10.7881 q^{21} -4.45583 q^{25} -5.18169 q^{27} +0.0202068 q^{29} -10.0832 q^{31} +8.93731 q^{33} -2.84049 q^{35} -2.39698 q^{37} -9.04385 q^{39} -8.31372 q^{41} -7.88990 q^{43} +3.57738 q^{45} -1.14678 q^{47} +7.82686 q^{49} -11.5418 q^{51} -1.56158 q^{53} -2.35318 q^{55} -16.4981 q^{57} +3.01520 q^{59} -2.81671 q^{61} -18.6733 q^{63} +2.38123 q^{65} +13.5043 q^{67} +14.9848 q^{71} -10.4551 q^{73} +12.4839 q^{75} +12.2832 q^{77} -3.04318 q^{79} -0.0309418 q^{81} -15.0810 q^{83} +3.03895 q^{85} -0.0566132 q^{87} +4.95749 q^{89} -12.4296 q^{91} +28.2501 q^{93} +4.34391 q^{95} +9.93096 q^{97} -15.4697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 10 q^{5} + 10 q^{7} + 16 q^{9} - 9 q^{11} + 12 q^{15} + 20 q^{17} - 3 q^{19} + 21 q^{21} + 11 q^{25} - 7 q^{27} - 4 q^{29} - 14 q^{31} + 34 q^{33} + 18 q^{35} + 26 q^{37} + 33 q^{39} - 11 q^{41}+ \cdots - 29 q^{99}+O(q^{100}) \) 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
\(3\) −2.80169 −1.61756 −0.808779 − 0.588112i \(-0.799871\pi\)
−0.808779 + 0.588112i \(0.799871\pi\)
\(4\) 0 0
\(5\) 0.737681 0.329901 0.164951 − 0.986302i \(-0.447254\pi\)
0.164951 + 0.986302i \(0.447254\pi\)
\(6\) 0 0
\(7\) −3.85057 −1.45538 −0.727688 − 0.685908i \(-0.759405\pi\)
−0.727688 + 0.685908i \(0.759405\pi\)
\(8\) 0 0
\(9\) 4.84949 1.61650
\(10\) 0 0
\(11\) −3.18997 −0.961811 −0.480906 − 0.876772i \(-0.659692\pi\)
−0.480906 + 0.876772i \(0.659692\pi\)
\(12\) 0 0
\(13\) 3.22799 0.895284 0.447642 − 0.894213i \(-0.352264\pi\)
0.447642 + 0.894213i \(0.352264\pi\)
\(14\) 0 0
\(15\) −2.06676 −0.533634
\(16\) 0 0
\(17\) 4.11959 0.999148 0.499574 − 0.866271i \(-0.333490\pi\)
0.499574 + 0.866271i \(0.333490\pi\)
\(18\) 0 0
\(19\) 5.88860 1.35094 0.675469 − 0.737388i \(-0.263941\pi\)
0.675469 + 0.737388i \(0.263941\pi\)
\(20\) 0 0
\(21\) 10.7881 2.35416
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.45583 −0.891165
\(26\) 0 0
\(27\) −5.18169 −0.997217
\(28\) 0 0
\(29\) 0.0202068 0.00375231 0.00187615 − 0.999998i \(-0.499403\pi\)
0.00187615 + 0.999998i \(0.499403\pi\)
\(30\) 0 0
\(31\) −10.0832 −1.81100 −0.905500 − 0.424346i \(-0.860504\pi\)
−0.905500 + 0.424346i \(0.860504\pi\)
\(32\) 0 0
\(33\) 8.93731 1.55579
\(34\) 0 0
\(35\) −2.84049 −0.480130
\(36\) 0 0
\(37\) −2.39698 −0.394061 −0.197031 − 0.980397i \(-0.563130\pi\)
−0.197031 + 0.980397i \(0.563130\pi\)
\(38\) 0 0
\(39\) −9.04385 −1.44817
\(40\) 0 0
\(41\) −8.31372 −1.29838 −0.649192 − 0.760624i \(-0.724893\pi\)
−0.649192 + 0.760624i \(0.724893\pi\)
\(42\) 0 0
\(43\) −7.88990 −1.20320 −0.601599 − 0.798798i \(-0.705470\pi\)
−0.601599 + 0.798798i \(0.705470\pi\)
\(44\) 0 0
\(45\) 3.57738 0.533284
\(46\) 0 0
\(47\) −1.14678 −0.167275 −0.0836375 − 0.996496i \(-0.526654\pi\)
−0.0836375 + 0.996496i \(0.526654\pi\)
\(48\) 0 0
\(49\) 7.82686 1.11812
\(50\) 0 0
\(51\) −11.5418 −1.61618
\(52\) 0 0
\(53\) −1.56158 −0.214499 −0.107250 − 0.994232i \(-0.534204\pi\)
−0.107250 + 0.994232i \(0.534204\pi\)
\(54\) 0 0
\(55\) −2.35318 −0.317303
\(56\) 0 0
\(57\) −16.4981 −2.18522
\(58\) 0 0
\(59\) 3.01520 0.392546 0.196273 − 0.980549i \(-0.437116\pi\)
0.196273 + 0.980549i \(0.437116\pi\)
\(60\) 0 0
\(61\) −2.81671 −0.360643 −0.180322 − 0.983608i \(-0.557714\pi\)
−0.180322 + 0.983608i \(0.557714\pi\)
\(62\) 0 0
\(63\) −18.6733 −2.35261
\(64\) 0 0
\(65\) 2.38123 0.295355
\(66\) 0 0
\(67\) 13.5043 1.64982 0.824909 − 0.565265i \(-0.191226\pi\)
0.824909 + 0.565265i \(0.191226\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.9848 1.77836 0.889182 − 0.457554i \(-0.151275\pi\)
0.889182 + 0.457554i \(0.151275\pi\)
\(72\) 0 0
\(73\) −10.4551 −1.22368 −0.611840 − 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(74\) 0 0
\(75\) 12.4839 1.44151
\(76\) 0 0
\(77\) 12.2832 1.39980
\(78\) 0 0
\(79\) −3.04318 −0.342385 −0.171192 − 0.985238i \(-0.554762\pi\)
−0.171192 + 0.985238i \(0.554762\pi\)
\(80\) 0 0
\(81\) −0.0309418 −0.00343798
\(82\) 0 0
\(83\) −15.0810 −1.65535 −0.827676 − 0.561207i \(-0.810337\pi\)
−0.827676 + 0.561207i \(0.810337\pi\)
\(84\) 0 0
\(85\) 3.03895 0.329620
\(86\) 0 0
\(87\) −0.0566132 −0.00606958
\(88\) 0 0
\(89\) 4.95749 0.525493 0.262747 − 0.964865i \(-0.415372\pi\)
0.262747 + 0.964865i \(0.415372\pi\)
\(90\) 0 0
\(91\) −12.4296 −1.30298
\(92\) 0 0
\(93\) 28.2501 2.92940
\(94\) 0 0
\(95\) 4.34391 0.445676
\(96\) 0 0
\(97\) 9.93096 1.00834 0.504168 − 0.863606i \(-0.331799\pi\)
0.504168 + 0.863606i \(0.331799\pi\)
\(98\) 0 0
\(99\) −15.4697 −1.55476
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.z.1.2 15
4.3 odd 2 8464.2.a.cj.1.14 15
23.15 odd 22 184.2.i.a.41.3 yes 30
23.20 odd 22 184.2.i.a.9.3 ✓ 30
23.22 odd 2 4232.2.a.y.1.2 15
92.15 even 22 368.2.m.f.225.1 30
92.43 even 22 368.2.m.f.193.1 30
92.91 even 2 8464.2.a.ci.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.9.3 ✓ 30 23.20 odd 22
184.2.i.a.41.3 yes 30 23.15 odd 22
368.2.m.f.193.1 30 92.43 even 22
368.2.m.f.225.1 30 92.15 even 22
4232.2.a.y.1.2 15 23.22 odd 2
4232.2.a.z.1.2 15 1.1 even 1 trivial
8464.2.a.ci.1.14 15 92.91 even 2
8464.2.a.cj.1.14 15 4.3 odd 2