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Properties

Label 4232.2.a.bb.1.1
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 1.1
Root \(-3.13479\) 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)
 
\(f(q)\) \(=\) \(q-3.13479 q^{3} -3.00137 q^{5} -0.714288 q^{7} +6.82690 q^{9} +1.17733 q^{11} +4.39302 q^{13} +9.40866 q^{15} -2.25057 q^{17} +8.02507 q^{19} +2.23914 q^{21} +4.00822 q^{25} -11.9965 q^{27} -8.34446 q^{29} +3.19860 q^{31} -3.69067 q^{33} +2.14384 q^{35} +4.74615 q^{37} -13.7712 q^{39} -10.8973 q^{41} +8.08417 q^{43} -20.4900 q^{45} +1.20581 q^{47} -6.48979 q^{49} +7.05508 q^{51} +0.460183 q^{53} -3.53359 q^{55} -25.1569 q^{57} +4.53574 q^{59} +2.56511 q^{61} -4.87637 q^{63} -13.1851 q^{65} -3.89321 q^{67} -8.18786 q^{71} -0.0978311 q^{73} -12.5649 q^{75} -0.840950 q^{77} -10.9981 q^{79} +17.1258 q^{81} +1.88740 q^{83} +6.75481 q^{85} +26.1581 q^{87} -5.90204 q^{89} -3.13788 q^{91} -10.0269 q^{93} -24.0862 q^{95} +17.4863 q^{97} +8.03748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) −3.13479 −1.80987 −0.904935 − 0.425549i \(-0.860081\pi\)
−0.904935 + 0.425549i \(0.860081\pi\)
\(4\) 0 0
\(5\) −3.00137 −1.34225 −0.671127 − 0.741343i \(-0.734189\pi\)
−0.671127 + 0.741343i \(0.734189\pi\)
\(6\) 0 0
\(7\) −0.714288 −0.269975 −0.134988 − 0.990847i \(-0.543100\pi\)
−0.134988 + 0.990847i \(0.543100\pi\)
\(8\) 0 0
\(9\) 6.82690 2.27563
\(10\) 0 0
\(11\) 1.17733 0.354977 0.177489 − 0.984123i \(-0.443203\pi\)
0.177489 + 0.984123i \(0.443203\pi\)
\(12\) 0 0
\(13\) 4.39302 1.21840 0.609202 − 0.793015i \(-0.291490\pi\)
0.609202 + 0.793015i \(0.291490\pi\)
\(14\) 0 0
\(15\) 9.40866 2.42931
\(16\) 0 0
\(17\) −2.25057 −0.545845 −0.272922 − 0.962036i \(-0.587990\pi\)
−0.272922 + 0.962036i \(0.587990\pi\)
\(18\) 0 0
\(19\) 8.02507 1.84108 0.920538 − 0.390652i \(-0.127750\pi\)
0.920538 + 0.390652i \(0.127750\pi\)
\(20\) 0 0
\(21\) 2.23914 0.488621
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 4.00822 0.801644
\(26\) 0 0
\(27\) −11.9965 −2.30873
\(28\) 0 0
\(29\) −8.34446 −1.54953 −0.774764 − 0.632251i \(-0.782131\pi\)
−0.774764 + 0.632251i \(0.782131\pi\)
\(30\) 0 0
\(31\) 3.19860 0.574485 0.287243 − 0.957858i \(-0.407261\pi\)
0.287243 + 0.957858i \(0.407261\pi\)
\(32\) 0 0
\(33\) −3.69067 −0.642463
\(34\) 0 0
\(35\) 2.14384 0.362376
\(36\) 0 0
\(37\) 4.74615 0.780262 0.390131 − 0.920759i \(-0.372430\pi\)
0.390131 + 0.920759i \(0.372430\pi\)
\(38\) 0 0
\(39\) −13.7712 −2.20515
\(40\) 0 0
\(41\) −10.8973 −1.70187 −0.850937 − 0.525268i \(-0.823965\pi\)
−0.850937 + 0.525268i \(0.823965\pi\)
\(42\) 0 0
\(43\) 8.08417 1.23282 0.616412 − 0.787423i \(-0.288585\pi\)
0.616412 + 0.787423i \(0.288585\pi\)
\(44\) 0 0
\(45\) −20.4900 −3.05448
\(46\) 0 0
\(47\) 1.20581 0.175886 0.0879428 − 0.996126i \(-0.471971\pi\)
0.0879428 + 0.996126i \(0.471971\pi\)
\(48\) 0 0
\(49\) −6.48979 −0.927113
\(50\) 0 0
\(51\) 7.05508 0.987908
\(52\) 0 0
\(53\) 0.460183 0.0632110 0.0316055 − 0.999500i \(-0.489938\pi\)
0.0316055 + 0.999500i \(0.489938\pi\)
\(54\) 0 0
\(55\) −3.53359 −0.476469
\(56\) 0 0
\(57\) −25.1569 −3.33211
\(58\) 0 0
\(59\) 4.53574 0.590503 0.295252 − 0.955420i \(-0.404597\pi\)
0.295252 + 0.955420i \(0.404597\pi\)
\(60\) 0 0
\(61\) 2.56511 0.328429 0.164214 − 0.986425i \(-0.447491\pi\)
0.164214 + 0.986425i \(0.447491\pi\)
\(62\) 0 0
\(63\) −4.87637 −0.614365
\(64\) 0 0
\(65\) −13.1851 −1.63541
\(66\) 0 0
\(67\) −3.89321 −0.475632 −0.237816 − 0.971310i \(-0.576431\pi\)
−0.237816 + 0.971310i \(0.576431\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.18786 −0.971720 −0.485860 − 0.874037i \(-0.661493\pi\)
−0.485860 + 0.874037i \(0.661493\pi\)
\(72\) 0 0
\(73\) −0.0978311 −0.0114503 −0.00572513 − 0.999984i \(-0.501822\pi\)
−0.00572513 + 0.999984i \(0.501822\pi\)
\(74\) 0 0
\(75\) −12.5649 −1.45087
\(76\) 0 0
\(77\) −0.840950 −0.0958351
\(78\) 0 0
\(79\) −10.9981 −1.23738 −0.618691 − 0.785635i \(-0.712337\pi\)
−0.618691 + 0.785635i \(0.712337\pi\)
\(80\) 0 0
\(81\) 17.1258 1.90287
\(82\) 0 0
\(83\) 1.88740 0.207169 0.103584 − 0.994621i \(-0.466969\pi\)
0.103584 + 0.994621i \(0.466969\pi\)
\(84\) 0 0
\(85\) 6.75481 0.732662
\(86\) 0 0
\(87\) 26.1581 2.80444
\(88\) 0 0
\(89\) −5.90204 −0.625615 −0.312808 − 0.949816i \(-0.601270\pi\)
−0.312808 + 0.949816i \(0.601270\pi\)
\(90\) 0 0
\(91\) −3.13788 −0.328939
\(92\) 0 0
\(93\) −10.0269 −1.03974
\(94\) 0 0
\(95\) −24.0862 −2.47119
\(96\) 0 0
\(97\) 17.4863 1.77547 0.887734 − 0.460356i \(-0.152278\pi\)
0.887734 + 0.460356i \(0.152278\pi\)
\(98\) 0 0
\(99\) 8.03748 0.807798
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.1.1 15
4.3 odd 2 8464.2.a.cg.1.15 15
23.2 even 11 184.2.i.b.73.3 ✓ 30
23.12 even 11 184.2.i.b.121.3 yes 30
23.22 odd 2 4232.2.a.ba.1.1 15
92.35 odd 22 368.2.m.e.305.1 30
92.71 odd 22 368.2.m.e.257.1 30
92.91 even 2 8464.2.a.ch.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.73.3 ✓ 30 23.2 even 11
184.2.i.b.121.3 yes 30 23.12 even 11
368.2.m.e.257.1 30 92.71 odd 22
368.2.m.e.305.1 30 92.35 odd 22
4232.2.a.ba.1.1 15 23.22 odd 2
4232.2.a.bb.1.1 15 1.1 even 1 trivial
8464.2.a.cg.1.15 15 4.3 odd 2
8464.2.a.ch.1.15 15 92.91 even 2