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Properties

Label 4232.2.a.y.1.6
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 1.6
Root \(0.881524\) 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)
 
\(f(q)\) \(=\) \(q-0.881524 q^{3} +2.46842 q^{5} +1.39730 q^{7} -2.22292 q^{9} -0.760030 q^{11} -3.31151 q^{13} -2.17597 q^{15} +2.14059 q^{17} +0.416417 q^{19} -1.23176 q^{21} +1.09309 q^{25} +4.60412 q^{27} -7.29582 q^{29} -7.39924 q^{31} +0.669985 q^{33} +3.44913 q^{35} +7.55976 q^{37} +2.91918 q^{39} +0.0737081 q^{41} +1.34201 q^{43} -5.48709 q^{45} +5.43776 q^{47} -5.04755 q^{49} -1.88698 q^{51} -7.32667 q^{53} -1.87607 q^{55} -0.367081 q^{57} -14.9188 q^{59} -11.5155 q^{61} -3.10609 q^{63} -8.17420 q^{65} +11.2996 q^{67} -4.20758 q^{71} -6.28419 q^{73} -0.963582 q^{75} -1.06199 q^{77} -4.57589 q^{79} +2.61011 q^{81} +2.48083 q^{83} +5.28386 q^{85} +6.43144 q^{87} -6.81924 q^{89} -4.62718 q^{91} +6.52260 q^{93} +1.02789 q^{95} +6.48057 q^{97} +1.68948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} - 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} - 20 q^{17} + 3 q^{19} - 21 q^{21} + 11 q^{25} - 7 q^{27} - 4 q^{29} - 14 q^{31} - 34 q^{33} + 18 q^{35} - 26 q^{37} + 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) 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
\(3\) −0.881524 −0.508948 −0.254474 − 0.967080i \(-0.581902\pi\)
−0.254474 + 0.967080i \(0.581902\pi\)
\(4\) 0 0
\(5\) 2.46842 1.10391 0.551955 − 0.833874i \(-0.313882\pi\)
0.551955 + 0.833874i \(0.313882\pi\)
\(6\) 0 0
\(7\) 1.39730 0.528131 0.264065 − 0.964505i \(-0.414937\pi\)
0.264065 + 0.964505i \(0.414937\pi\)
\(8\) 0 0
\(9\) −2.22292 −0.740972
\(10\) 0 0
\(11\) −0.760030 −0.229158 −0.114579 − 0.993414i \(-0.536552\pi\)
−0.114579 + 0.993414i \(0.536552\pi\)
\(12\) 0 0
\(13\) −3.31151 −0.918448 −0.459224 − 0.888320i \(-0.651873\pi\)
−0.459224 + 0.888320i \(0.651873\pi\)
\(14\) 0 0
\(15\) −2.17597 −0.561833
\(16\) 0 0
\(17\) 2.14059 0.519168 0.259584 − 0.965721i \(-0.416415\pi\)
0.259584 + 0.965721i \(0.416415\pi\)
\(18\) 0 0
\(19\) 0.416417 0.0955325 0.0477663 − 0.998859i \(-0.484790\pi\)
0.0477663 + 0.998859i \(0.484790\pi\)
\(20\) 0 0
\(21\) −1.23176 −0.268791
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 1.09309 0.218617
\(26\) 0 0
\(27\) 4.60412 0.886064
\(28\) 0 0
\(29\) −7.29582 −1.35480 −0.677400 − 0.735615i \(-0.736893\pi\)
−0.677400 + 0.735615i \(0.736893\pi\)
\(30\) 0 0
\(31\) −7.39924 −1.32894 −0.664471 − 0.747314i \(-0.731343\pi\)
−0.664471 + 0.747314i \(0.731343\pi\)
\(32\) 0 0
\(33\) 0.669985 0.116629
\(34\) 0 0
\(35\) 3.44913 0.583009
\(36\) 0 0
\(37\) 7.55976 1.24282 0.621408 − 0.783487i \(-0.286561\pi\)
0.621408 + 0.783487i \(0.286561\pi\)
\(38\) 0 0
\(39\) 2.91918 0.467442
\(40\) 0 0
\(41\) 0.0737081 0.0115113 0.00575563 − 0.999983i \(-0.498168\pi\)
0.00575563 + 0.999983i \(0.498168\pi\)
\(42\) 0 0
\(43\) 1.34201 0.204655 0.102327 − 0.994751i \(-0.467371\pi\)
0.102327 + 0.994751i \(0.467371\pi\)
\(44\) 0 0
\(45\) −5.48709 −0.817966
\(46\) 0 0
\(47\) 5.43776 0.793179 0.396590 − 0.917996i \(-0.370194\pi\)
0.396590 + 0.917996i \(0.370194\pi\)
\(48\) 0 0
\(49\) −5.04755 −0.721078
\(50\) 0 0
\(51\) −1.88698 −0.264230
\(52\) 0 0
\(53\) −7.32667 −1.00640 −0.503198 − 0.864171i \(-0.667843\pi\)
−0.503198 + 0.864171i \(0.667843\pi\)
\(54\) 0 0
\(55\) −1.87607 −0.252970
\(56\) 0 0
\(57\) −0.367081 −0.0486211
\(58\) 0 0
\(59\) −14.9188 −1.94226 −0.971130 − 0.238550i \(-0.923328\pi\)
−0.971130 + 0.238550i \(0.923328\pi\)
\(60\) 0 0
\(61\) −11.5155 −1.47441 −0.737204 − 0.675670i \(-0.763854\pi\)
−0.737204 + 0.675670i \(0.763854\pi\)
\(62\) 0 0
\(63\) −3.10609 −0.391330
\(64\) 0 0
\(65\) −8.17420 −1.01388
\(66\) 0 0
\(67\) 11.2996 1.38046 0.690231 − 0.723589i \(-0.257509\pi\)
0.690231 + 0.723589i \(0.257509\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.20758 −0.499347 −0.249674 − 0.968330i \(-0.580323\pi\)
−0.249674 + 0.968330i \(0.580323\pi\)
\(72\) 0 0
\(73\) −6.28419 −0.735509 −0.367754 − 0.929923i \(-0.619873\pi\)
−0.367754 + 0.929923i \(0.619873\pi\)
\(74\) 0 0
\(75\) −0.963582 −0.111265
\(76\) 0 0
\(77\) −1.06199 −0.121025
\(78\) 0 0
\(79\) −4.57589 −0.514827 −0.257414 − 0.966301i \(-0.582870\pi\)
−0.257414 + 0.966301i \(0.582870\pi\)
\(80\) 0 0
\(81\) 2.61011 0.290012
\(82\) 0 0
\(83\) 2.48083 0.272307 0.136153 − 0.990688i \(-0.456526\pi\)
0.136153 + 0.990688i \(0.456526\pi\)
\(84\) 0 0
\(85\) 5.28386 0.573115
\(86\) 0 0
\(87\) 6.43144 0.689523
\(88\) 0 0
\(89\) −6.81924 −0.722838 −0.361419 − 0.932404i \(-0.617708\pi\)
−0.361419 + 0.932404i \(0.617708\pi\)
\(90\) 0 0
\(91\) −4.62718 −0.485061
\(92\) 0 0
\(93\) 6.52260 0.676362
\(94\) 0 0
\(95\) 1.02789 0.105459
\(96\) 0 0
\(97\) 6.48057 0.658003 0.329001 − 0.944329i \(-0.393288\pi\)
0.329001 + 0.944329i \(0.393288\pi\)
\(98\) 0 0
\(99\) 1.68948 0.169800
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.1.6 15
4.3 odd 2 8464.2.a.ci.1.10 15
23.4 even 11 184.2.i.a.177.2 yes 30
23.6 even 11 184.2.i.a.105.2 ✓ 30
23.22 odd 2 4232.2.a.z.1.6 15
92.27 odd 22 368.2.m.f.177.2 30
92.75 odd 22 368.2.m.f.289.2 30
92.91 even 2 8464.2.a.cj.1.10 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.105.2 ✓ 30 23.6 even 11
184.2.i.a.177.2 yes 30 23.4 even 11
368.2.m.f.177.2 30 92.27 odd 22
368.2.m.f.289.2 30 92.75 odd 22
4232.2.a.y.1.6 15 1.1 even 1 trivial
4232.2.a.z.1.6 15 23.22 odd 2
8464.2.a.ci.1.10 15 4.3 odd 2
8464.2.a.cj.1.10 15 92.91 even 2