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Properties

Label 4232.2.a.bb.1.11
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 = [15,0,1,0,0,0,10] 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: \( x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 \) 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 1.11
Root \(1.44882\) 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)
 
\(f(q)\) \(=\) \(q+1.44882 q^{3} -2.99678 q^{5} -1.69224 q^{7} -0.900926 q^{9} +5.69500 q^{11} -6.47165 q^{13} -4.34179 q^{15} +5.94707 q^{17} +0.280108 q^{19} -2.45174 q^{21} +3.98068 q^{25} -5.65173 q^{27} +1.20529 q^{29} -2.99338 q^{31} +8.25101 q^{33} +5.07126 q^{35} +0.265368 q^{37} -9.37624 q^{39} +5.81542 q^{41} -5.66327 q^{43} +2.69988 q^{45} -1.51532 q^{47} -4.13634 q^{49} +8.61622 q^{51} +6.60567 q^{53} -17.0666 q^{55} +0.405826 q^{57} +3.21061 q^{59} +8.76330 q^{61} +1.52458 q^{63} +19.3941 q^{65} -3.11516 q^{67} -2.19420 q^{71} +12.6635 q^{73} +5.76729 q^{75} -9.63728 q^{77} +0.653751 q^{79} -5.48555 q^{81} +13.8264 q^{83} -17.8221 q^{85} +1.74624 q^{87} -9.80353 q^{89} +10.9516 q^{91} -4.33686 q^{93} -0.839423 q^{95} +3.82589 q^{97} -5.13077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) 1.44882 0.836476 0.418238 − 0.908338i \(-0.362648\pi\)
0.418238 + 0.908338i \(0.362648\pi\)
\(4\) 0 0
\(5\) −2.99678 −1.34020 −0.670100 − 0.742271i \(-0.733749\pi\)
−0.670100 + 0.742271i \(0.733749\pi\)
\(6\) 0 0
\(7\) −1.69224 −0.639605 −0.319803 − 0.947484i \(-0.603617\pi\)
−0.319803 + 0.947484i \(0.603617\pi\)
\(8\) 0 0
\(9\) −0.900926 −0.300309
\(10\) 0 0
\(11\) 5.69500 1.71711 0.858553 − 0.512725i \(-0.171364\pi\)
0.858553 + 0.512725i \(0.171364\pi\)
\(12\) 0 0
\(13\) −6.47165 −1.79491 −0.897456 − 0.441104i \(-0.854587\pi\)
−0.897456 + 0.441104i \(0.854587\pi\)
\(14\) 0 0
\(15\) −4.34179 −1.12104
\(16\) 0 0
\(17\) 5.94707 1.44238 0.721188 − 0.692739i \(-0.243596\pi\)
0.721188 + 0.692739i \(0.243596\pi\)
\(18\) 0 0
\(19\) 0.280108 0.0642613 0.0321306 − 0.999484i \(-0.489771\pi\)
0.0321306 + 0.999484i \(0.489771\pi\)
\(20\) 0 0
\(21\) −2.45174 −0.535014
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 3.98068 0.796136
\(26\) 0 0
\(27\) −5.65173 −1.08768
\(28\) 0 0
\(29\) 1.20529 0.223816 0.111908 − 0.993719i \(-0.464304\pi\)
0.111908 + 0.993719i \(0.464304\pi\)
\(30\) 0 0
\(31\) −2.99338 −0.537626 −0.268813 − 0.963192i \(-0.586631\pi\)
−0.268813 + 0.963192i \(0.586631\pi\)
\(32\) 0 0
\(33\) 8.25101 1.43632
\(34\) 0 0
\(35\) 5.07126 0.857199
\(36\) 0 0
\(37\) 0.265368 0.0436262 0.0218131 − 0.999762i \(-0.493056\pi\)
0.0218131 + 0.999762i \(0.493056\pi\)
\(38\) 0 0
\(39\) −9.37624 −1.50140
\(40\) 0 0
\(41\) 5.81542 0.908216 0.454108 − 0.890947i \(-0.349958\pi\)
0.454108 + 0.890947i \(0.349958\pi\)
\(42\) 0 0
\(43\) −5.66327 −0.863640 −0.431820 − 0.901960i \(-0.642129\pi\)
−0.431820 + 0.901960i \(0.642129\pi\)
\(44\) 0 0
\(45\) 2.69988 0.402474
\(46\) 0 0
\(47\) −1.51532 −0.221032 −0.110516 − 0.993874i \(-0.535250\pi\)
−0.110516 + 0.993874i \(0.535250\pi\)
\(48\) 0 0
\(49\) −4.13634 −0.590905
\(50\) 0 0
\(51\) 8.61622 1.20651
\(52\) 0 0
\(53\) 6.60567 0.907358 0.453679 − 0.891165i \(-0.350111\pi\)
0.453679 + 0.891165i \(0.350111\pi\)
\(54\) 0 0
\(55\) −17.0666 −2.30127
\(56\) 0 0
\(57\) 0.405826 0.0537530
\(58\) 0 0
\(59\) 3.21061 0.417985 0.208993 − 0.977917i \(-0.432982\pi\)
0.208993 + 0.977917i \(0.432982\pi\)
\(60\) 0 0
\(61\) 8.76330 1.12203 0.561013 − 0.827807i \(-0.310412\pi\)
0.561013 + 0.827807i \(0.310412\pi\)
\(62\) 0 0
\(63\) 1.52458 0.192079
\(64\) 0 0
\(65\) 19.3941 2.40554
\(66\) 0 0
\(67\) −3.11516 −0.380577 −0.190289 − 0.981728i \(-0.560942\pi\)
−0.190289 + 0.981728i \(0.560942\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.19420 −0.260403 −0.130202 − 0.991488i \(-0.541562\pi\)
−0.130202 + 0.991488i \(0.541562\pi\)
\(72\) 0 0
\(73\) 12.6635 1.48216 0.741078 − 0.671419i \(-0.234315\pi\)
0.741078 + 0.671419i \(0.234315\pi\)
\(74\) 0 0
\(75\) 5.76729 0.665949
\(76\) 0 0
\(77\) −9.63728 −1.09827
\(78\) 0 0
\(79\) 0.653751 0.0735528 0.0367764 − 0.999324i \(-0.488291\pi\)
0.0367764 + 0.999324i \(0.488291\pi\)
\(80\) 0 0
\(81\) −5.48555 −0.609506
\(82\) 0 0
\(83\) 13.8264 1.51765 0.758823 − 0.651297i \(-0.225775\pi\)
0.758823 + 0.651297i \(0.225775\pi\)
\(84\) 0 0
\(85\) −17.8221 −1.93307
\(86\) 0 0
\(87\) 1.74624 0.187217
\(88\) 0 0
\(89\) −9.80353 −1.03917 −0.519586 − 0.854418i \(-0.673914\pi\)
−0.519586 + 0.854418i \(0.673914\pi\)
\(90\) 0 0
\(91\) 10.9516 1.14804
\(92\) 0 0
\(93\) −4.33686 −0.449711
\(94\) 0 0
\(95\) −0.839423 −0.0861230
\(96\) 0 0
\(97\) 3.82589 0.388460 0.194230 − 0.980956i \(-0.437779\pi\)
0.194230 + 0.980956i \(0.437779\pi\)
\(98\) 0 0
\(99\) −5.13077 −0.515662
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.bb.1.11 15
4.3 odd 2 8464.2.a.cg.1.5 15
23.4 even 11 184.2.i.b.177.2 yes 30
23.6 even 11 184.2.i.b.105.2 ✓ 30
23.22 odd 2 4232.2.a.ba.1.11 15
92.27 odd 22 368.2.m.e.177.2 30
92.75 odd 22 368.2.m.e.289.2 30
92.91 even 2 8464.2.a.ch.1.5 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.105.2 ✓ 30 23.6 even 11
184.2.i.b.177.2 yes 30 23.4 even 11
368.2.m.e.177.2 30 92.27 odd 22
368.2.m.e.289.2 30 92.75 odd 22
4232.2.a.ba.1.11 15 23.22 odd 2
4232.2.a.bb.1.11 15 1.1 even 1 trivial
8464.2.a.cg.1.5 15 4.3 odd 2
8464.2.a.ch.1.5 15 92.91 even 2