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Properties

Label 4232.2.a.z.1.5
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 1.5
Root \(1.43553\) 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)
 
\(f(q)\) \(=\) \(q-1.43553 q^{3} -0.827591 q^{5} +5.07987 q^{7} -0.939239 q^{9} -4.86639 q^{11} -5.30671 q^{13} +1.18804 q^{15} -4.09247 q^{17} -4.91886 q^{19} -7.29233 q^{21} -4.31509 q^{25} +5.65492 q^{27} +1.28740 q^{29} +5.71147 q^{31} +6.98588 q^{33} -4.20405 q^{35} +0.628840 q^{37} +7.61796 q^{39} +1.31109 q^{41} +2.33680 q^{43} +0.777306 q^{45} +4.18373 q^{47} +18.8051 q^{49} +5.87488 q^{51} +3.36194 q^{53} +4.02738 q^{55} +7.06120 q^{57} -0.797365 q^{59} +4.20032 q^{61} -4.77121 q^{63} +4.39178 q^{65} +8.28829 q^{67} -5.45066 q^{71} +4.84059 q^{73} +6.19447 q^{75} -24.7206 q^{77} -9.74767 q^{79} -5.30011 q^{81} -3.37620 q^{83} +3.38689 q^{85} -1.84810 q^{87} +17.4367 q^{89} -26.9574 q^{91} -8.19901 q^{93} +4.07080 q^{95} -0.181183 q^{97} +4.57071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - q^{3} + 10 q^{5} + 10 q^{7} + 16 q^{9} - 9 q^{11} + 12 q^{15} + 20 q^{17} - 3 q^{19} + 21 q^{21} + 11 q^{25} - 7 q^{27} - 4 q^{29} - 14 q^{31} + 34 q^{33} + 18 q^{35} + 26 q^{37} + 33 q^{39} - 11 q^{41}+ \cdots - 29 q^{99}+O(q^{100}) \) 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
\(3\) −1.43553 −0.828807 −0.414403 − 0.910093i \(-0.636010\pi\)
−0.414403 + 0.910093i \(0.636010\pi\)
\(4\) 0 0
\(5\) −0.827591 −0.370110 −0.185055 − 0.982728i \(-0.559246\pi\)
−0.185055 + 0.982728i \(0.559246\pi\)
\(6\) 0 0
\(7\) 5.07987 1.92001 0.960005 − 0.279984i \(-0.0903291\pi\)
0.960005 + 0.279984i \(0.0903291\pi\)
\(8\) 0 0
\(9\) −0.939239 −0.313080
\(10\) 0 0
\(11\) −4.86639 −1.46727 −0.733637 − 0.679542i \(-0.762179\pi\)
−0.733637 + 0.679542i \(0.762179\pi\)
\(12\) 0 0
\(13\) −5.30671 −1.47182 −0.735908 − 0.677082i \(-0.763244\pi\)
−0.735908 + 0.677082i \(0.763244\pi\)
\(14\) 0 0
\(15\) 1.18804 0.306749
\(16\) 0 0
\(17\) −4.09247 −0.992569 −0.496284 − 0.868160i \(-0.665303\pi\)
−0.496284 + 0.868160i \(0.665303\pi\)
\(18\) 0 0
\(19\) −4.91886 −1.12846 −0.564232 − 0.825616i \(-0.690828\pi\)
−0.564232 + 0.825616i \(0.690828\pi\)
\(20\) 0 0
\(21\) −7.29233 −1.59132
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −4.31509 −0.863019
\(26\) 0 0
\(27\) 5.65492 1.08829
\(28\) 0 0
\(29\) 1.28740 0.239064 0.119532 − 0.992830i \(-0.461861\pi\)
0.119532 + 0.992830i \(0.461861\pi\)
\(30\) 0 0
\(31\) 5.71147 1.02581 0.512905 − 0.858445i \(-0.328569\pi\)
0.512905 + 0.858445i \(0.328569\pi\)
\(32\) 0 0
\(33\) 6.98588 1.21609
\(34\) 0 0
\(35\) −4.20405 −0.710614
\(36\) 0 0
\(37\) 0.628840 0.103381 0.0516904 − 0.998663i \(-0.483539\pi\)
0.0516904 + 0.998663i \(0.483539\pi\)
\(38\) 0 0
\(39\) 7.61796 1.21985
\(40\) 0 0
\(41\) 1.31109 0.204758 0.102379 − 0.994745i \(-0.467355\pi\)
0.102379 + 0.994745i \(0.467355\pi\)
\(42\) 0 0
\(43\) 2.33680 0.356359 0.178179 − 0.983998i \(-0.442979\pi\)
0.178179 + 0.983998i \(0.442979\pi\)
\(44\) 0 0
\(45\) 0.777306 0.115874
\(46\) 0 0
\(47\) 4.18373 0.610260 0.305130 − 0.952311i \(-0.401300\pi\)
0.305130 + 0.952311i \(0.401300\pi\)
\(48\) 0 0
\(49\) 18.8051 2.68644
\(50\) 0 0
\(51\) 5.87488 0.822647
\(52\) 0 0
\(53\) 3.36194 0.461799 0.230899 − 0.972978i \(-0.425833\pi\)
0.230899 + 0.972978i \(0.425833\pi\)
\(54\) 0 0
\(55\) 4.02738 0.543052
\(56\) 0 0
\(57\) 7.06120 0.935278
\(58\) 0 0
\(59\) −0.797365 −0.103808 −0.0519040 − 0.998652i \(-0.516529\pi\)
−0.0519040 + 0.998652i \(0.516529\pi\)
\(60\) 0 0
\(61\) 4.20032 0.537796 0.268898 − 0.963169i \(-0.413341\pi\)
0.268898 + 0.963169i \(0.413341\pi\)
\(62\) 0 0
\(63\) −4.77121 −0.601116
\(64\) 0 0
\(65\) 4.39178 0.544733
\(66\) 0 0
\(67\) 8.28829 1.01258 0.506288 − 0.862364i \(-0.331017\pi\)
0.506288 + 0.862364i \(0.331017\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.45066 −0.646875 −0.323437 − 0.946250i \(-0.604839\pi\)
−0.323437 + 0.946250i \(0.604839\pi\)
\(72\) 0 0
\(73\) 4.84059 0.566548 0.283274 − 0.959039i \(-0.408579\pi\)
0.283274 + 0.959039i \(0.408579\pi\)
\(74\) 0 0
\(75\) 6.19447 0.715276
\(76\) 0 0
\(77\) −24.7206 −2.81718
\(78\) 0 0
\(79\) −9.74767 −1.09670 −0.548349 − 0.836249i \(-0.684744\pi\)
−0.548349 + 0.836249i \(0.684744\pi\)
\(80\) 0 0
\(81\) −5.30011 −0.588901
\(82\) 0 0
\(83\) −3.37620 −0.370587 −0.185293 − 0.982683i \(-0.559324\pi\)
−0.185293 + 0.982683i \(0.559324\pi\)
\(84\) 0 0
\(85\) 3.38689 0.367359
\(86\) 0 0
\(87\) −1.84810 −0.198138
\(88\) 0 0
\(89\) 17.4367 1.84829 0.924144 − 0.382044i \(-0.124780\pi\)
0.924144 + 0.382044i \(0.124780\pi\)
\(90\) 0 0
\(91\) −26.9574 −2.82590
\(92\) 0 0
\(93\) −8.19901 −0.850198
\(94\) 0 0
\(95\) 4.07080 0.417656
\(96\) 0 0
\(97\) −0.181183 −0.0183963 −0.00919817 − 0.999958i \(-0.502928\pi\)
−0.00919817 + 0.999958i \(0.502928\pi\)
\(98\) 0 0
\(99\) 4.57071 0.459374
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.1.5 15
4.3 odd 2 8464.2.a.cj.1.11 15
23.11 odd 22 184.2.i.a.121.3 yes 30
23.21 odd 22 184.2.i.a.73.3 ✓ 30
23.22 odd 2 4232.2.a.y.1.5 15
92.11 even 22 368.2.m.f.305.1 30
92.67 even 22 368.2.m.f.257.1 30
92.91 even 2 8464.2.a.ci.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.73.3 ✓ 30 23.21 odd 22
184.2.i.a.121.3 yes 30 23.11 odd 22
368.2.m.f.257.1 30 92.67 even 22
368.2.m.f.305.1 30 92.11 even 22
4232.2.a.y.1.5 15 23.22 odd 2
4232.2.a.z.1.5 15 1.1 even 1 trivial
8464.2.a.ci.1.11 15 92.91 even 2
8464.2.a.cj.1.11 15 4.3 odd 2