↑

Properties

Label 4232.2.a.bb.1.4
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.4
Root \(-2.01779\) 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)
 
\(f(q)\) \(=\) \(q-2.01779 q^{3} -0.684021 q^{5} -2.16510 q^{7} +1.07147 q^{9} -1.89186 q^{11} -2.22346 q^{13} +1.38021 q^{15} -2.80474 q^{17} +2.23273 q^{19} +4.36872 q^{21} -4.53211 q^{25} +3.89136 q^{27} -8.59972 q^{29} -2.31109 q^{31} +3.81737 q^{33} +1.48098 q^{35} -5.55335 q^{37} +4.48647 q^{39} +3.44767 q^{41} +1.37494 q^{43} -0.732910 q^{45} -11.4999 q^{47} -2.31234 q^{49} +5.65938 q^{51} -8.37106 q^{53} +1.29407 q^{55} -4.50517 q^{57} -8.40725 q^{59} -5.71825 q^{61} -2.31985 q^{63} +1.52089 q^{65} +15.4601 q^{67} -1.24569 q^{71} +13.4084 q^{73} +9.14485 q^{75} +4.09607 q^{77} +8.14191 q^{79} -11.0664 q^{81} +9.82785 q^{83} +1.91850 q^{85} +17.3524 q^{87} -11.0489 q^{89} +4.81402 q^{91} +4.66329 q^{93} -1.52723 q^{95} -16.8611 q^{97} -2.02707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) −2.01779 −1.16497 −0.582485 − 0.812841i \(-0.697920\pi\)
−0.582485 + 0.812841i \(0.697920\pi\)
\(4\) 0 0
\(5\) −0.684021 −0.305904 −0.152952 − 0.988234i \(-0.548878\pi\)
−0.152952 + 0.988234i \(0.548878\pi\)
\(6\) 0 0
\(7\) −2.16510 −0.818331 −0.409166 − 0.912460i \(-0.634180\pi\)
−0.409166 + 0.912460i \(0.634180\pi\)
\(8\) 0 0
\(9\) 1.07147 0.357157
\(10\) 0 0
\(11\) −1.89186 −0.570417 −0.285208 − 0.958466i \(-0.592063\pi\)
−0.285208 + 0.958466i \(0.592063\pi\)
\(12\) 0 0
\(13\) −2.22346 −0.616677 −0.308339 − 0.951277i \(-0.599773\pi\)
−0.308339 + 0.951277i \(0.599773\pi\)
\(14\) 0 0
\(15\) 1.38021 0.356369
\(16\) 0 0
\(17\) −2.80474 −0.680250 −0.340125 − 0.940380i \(-0.610469\pi\)
−0.340125 + 0.940380i \(0.610469\pi\)
\(18\) 0 0
\(19\) 2.23273 0.512222 0.256111 − 0.966647i \(-0.417559\pi\)
0.256111 + 0.966647i \(0.417559\pi\)
\(20\) 0 0
\(21\) 4.36872 0.953332
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.53211 −0.906423
\(26\) 0 0
\(27\) 3.89136 0.748893
\(28\) 0 0
\(29\) −8.59972 −1.59693 −0.798464 − 0.602042i \(-0.794354\pi\)
−0.798464 + 0.602042i \(0.794354\pi\)
\(30\) 0 0
\(31\) −2.31109 −0.415084 −0.207542 − 0.978226i \(-0.566546\pi\)
−0.207542 + 0.978226i \(0.566546\pi\)
\(32\) 0 0
\(33\) 3.81737 0.664519
\(34\) 0 0
\(35\) 1.48098 0.250331
\(36\) 0 0
\(37\) −5.55335 −0.912966 −0.456483 − 0.889732i \(-0.650891\pi\)
−0.456483 + 0.889732i \(0.650891\pi\)
\(38\) 0 0
\(39\) 4.48647 0.718411
\(40\) 0 0
\(41\) 3.44767 0.538436 0.269218 − 0.963079i \(-0.413235\pi\)
0.269218 + 0.963079i \(0.413235\pi\)
\(42\) 0 0
\(43\) 1.37494 0.209676 0.104838 − 0.994489i \(-0.466568\pi\)
0.104838 + 0.994489i \(0.466568\pi\)
\(44\) 0 0
\(45\) −0.732910 −0.109256
\(46\) 0 0
\(47\) −11.4999 −1.67743 −0.838717 − 0.544568i \(-0.816694\pi\)
−0.838717 + 0.544568i \(0.816694\pi\)
\(48\) 0 0
\(49\) −2.31234 −0.330334
\(50\) 0 0
\(51\) 5.65938 0.792472
\(52\) 0 0
\(53\) −8.37106 −1.14985 −0.574927 − 0.818205i \(-0.694970\pi\)
−0.574927 + 0.818205i \(0.694970\pi\)
\(54\) 0 0
\(55\) 1.29407 0.174493
\(56\) 0 0
\(57\) −4.50517 −0.596724
\(58\) 0 0
\(59\) −8.40725 −1.09453 −0.547265 − 0.836959i \(-0.684331\pi\)
−0.547265 + 0.836959i \(0.684331\pi\)
\(60\) 0 0
\(61\) −5.71825 −0.732147 −0.366073 − 0.930586i \(-0.619298\pi\)
−0.366073 + 0.930586i \(0.619298\pi\)
\(62\) 0 0
\(63\) −2.31985 −0.292273
\(64\) 0 0
\(65\) 1.52089 0.188644
\(66\) 0 0
\(67\) 15.4601 1.88875 0.944376 − 0.328868i \(-0.106667\pi\)
0.944376 + 0.328868i \(0.106667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.24569 −0.147836 −0.0739180 − 0.997264i \(-0.523550\pi\)
−0.0739180 + 0.997264i \(0.523550\pi\)
\(72\) 0 0
\(73\) 13.4084 1.56934 0.784668 − 0.619916i \(-0.212833\pi\)
0.784668 + 0.619916i \(0.212833\pi\)
\(74\) 0 0
\(75\) 9.14485 1.05596
\(76\) 0 0
\(77\) 4.09607 0.466790
\(78\) 0 0
\(79\) 8.14191 0.916036 0.458018 − 0.888943i \(-0.348560\pi\)
0.458018 + 0.888943i \(0.348560\pi\)
\(80\) 0 0
\(81\) −11.0664 −1.22960
\(82\) 0 0
\(83\) 9.82785 1.07875 0.539374 − 0.842067i \(-0.318661\pi\)
0.539374 + 0.842067i \(0.318661\pi\)
\(84\) 0 0
\(85\) 1.91850 0.208091
\(86\) 0 0
\(87\) 17.3524 1.86038
\(88\) 0 0
\(89\) −11.0489 −1.17118 −0.585590 − 0.810607i \(-0.699137\pi\)
−0.585590 + 0.810607i \(0.699137\pi\)
\(90\) 0 0
\(91\) 4.81402 0.504646
\(92\) 0 0
\(93\) 4.66329 0.483560
\(94\) 0 0
\(95\) −1.52723 −0.156691
\(96\) 0 0
\(97\) −16.8611 −1.71199 −0.855993 − 0.516988i \(-0.827053\pi\)
−0.855993 + 0.516988i \(0.827053\pi\)
\(98\) 0 0
\(99\) −2.02707 −0.203729
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.1.4 15
4.3 odd 2 8464.2.a.cg.1.12 15
23.3 even 11 184.2.i.b.9.3 ✓ 30
23.8 even 11 184.2.i.b.41.3 yes 30
23.22 odd 2 4232.2.a.ba.1.4 15
92.3 odd 22 368.2.m.e.193.1 30
92.31 odd 22 368.2.m.e.225.1 30
92.91 even 2 8464.2.a.ch.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.9.3 ✓ 30 23.3 even 11
184.2.i.b.41.3 yes 30 23.8 even 11
368.2.m.e.193.1 30 92.3 odd 22
368.2.m.e.225.1 30 92.31 odd 22
4232.2.a.ba.1.4 15 23.22 odd 2
4232.2.a.bb.1.4 15 1.1 even 1 trivial
8464.2.a.cg.1.12 15 4.3 odd 2
8464.2.a.ch.1.12 15 92.91 even 2