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Properties

Label 4232.2.a.bb.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 = [15,0,1,0,0,0,10] 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 1.5
Root \(-1.67059\) 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)
 
\(f(q)\) \(=\) \(q-1.67059 q^{3} -3.30085 q^{5} +1.48378 q^{7} -0.209129 q^{9} +2.77429 q^{11} +3.41846 q^{13} +5.51436 q^{15} +3.76631 q^{17} -0.514351 q^{19} -2.47878 q^{21} +5.89559 q^{25} +5.36114 q^{27} +7.25504 q^{29} -7.65695 q^{31} -4.63470 q^{33} -4.89772 q^{35} -3.28450 q^{37} -5.71085 q^{39} -10.7718 q^{41} -6.13046 q^{43} +0.690301 q^{45} -8.93322 q^{47} -4.79840 q^{49} -6.29196 q^{51} -11.3758 q^{53} -9.15751 q^{55} +0.859270 q^{57} -2.03410 q^{59} +10.6262 q^{61} -0.310300 q^{63} -11.2838 q^{65} +5.97360 q^{67} +13.0664 q^{71} +10.0288 q^{73} -9.84911 q^{75} +4.11643 q^{77} +7.58909 q^{79} -8.32888 q^{81} +8.04453 q^{83} -12.4320 q^{85} -12.1202 q^{87} +15.7531 q^{89} +5.07224 q^{91} +12.7916 q^{93} +1.69779 q^{95} +7.14396 q^{97} -0.580184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 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
\(3\) −1.67059 −0.964516 −0.482258 − 0.876029i \(-0.660183\pi\)
−0.482258 + 0.876029i \(0.660183\pi\)
\(4\) 0 0
\(5\) −3.30085 −1.47618 −0.738092 − 0.674700i \(-0.764273\pi\)
−0.738092 + 0.674700i \(0.764273\pi\)
\(6\) 0 0
\(7\) 1.48378 0.560815 0.280408 − 0.959881i \(-0.409530\pi\)
0.280408 + 0.959881i \(0.409530\pi\)
\(8\) 0 0
\(9\) −0.209129 −0.0697095
\(10\) 0 0
\(11\) 2.77429 0.836480 0.418240 − 0.908336i \(-0.362647\pi\)
0.418240 + 0.908336i \(0.362647\pi\)
\(12\) 0 0
\(13\) 3.41846 0.948112 0.474056 − 0.880495i \(-0.342789\pi\)
0.474056 + 0.880495i \(0.342789\pi\)
\(14\) 0 0
\(15\) 5.51436 1.42380
\(16\) 0 0
\(17\) 3.76631 0.913464 0.456732 − 0.889604i \(-0.349020\pi\)
0.456732 + 0.889604i \(0.349020\pi\)
\(18\) 0 0
\(19\) −0.514351 −0.118000 −0.0590001 − 0.998258i \(-0.518791\pi\)
−0.0590001 + 0.998258i \(0.518791\pi\)
\(20\) 0 0
\(21\) −2.47878 −0.540915
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) 5.89559 1.17912
\(26\) 0 0
\(27\) 5.36114 1.03175
\(28\) 0 0
\(29\) 7.25504 1.34723 0.673614 − 0.739083i \(-0.264741\pi\)
0.673614 + 0.739083i \(0.264741\pi\)
\(30\) 0 0
\(31\) −7.65695 −1.37523 −0.687614 − 0.726076i \(-0.741342\pi\)
−0.687614 + 0.726076i \(0.741342\pi\)
\(32\) 0 0
\(33\) −4.63470 −0.806798
\(34\) 0 0
\(35\) −4.89772 −0.827866
\(36\) 0 0
\(37\) −3.28450 −0.539968 −0.269984 − 0.962865i \(-0.587018\pi\)
−0.269984 + 0.962865i \(0.587018\pi\)
\(38\) 0 0
\(39\) −5.71085 −0.914468
\(40\) 0 0
\(41\) −10.7718 −1.68227 −0.841137 − 0.540821i \(-0.818113\pi\)
−0.841137 + 0.540821i \(0.818113\pi\)
\(42\) 0 0
\(43\) −6.13046 −0.934887 −0.467443 − 0.884023i \(-0.654825\pi\)
−0.467443 + 0.884023i \(0.654825\pi\)
\(44\) 0 0
\(45\) 0.690301 0.102904
\(46\) 0 0
\(47\) −8.93322 −1.30304 −0.651522 − 0.758630i \(-0.725869\pi\)
−0.651522 + 0.758630i \(0.725869\pi\)
\(48\) 0 0
\(49\) −4.79840 −0.685486
\(50\) 0 0
\(51\) −6.29196 −0.881050
\(52\) 0 0
\(53\) −11.3758 −1.56259 −0.781294 − 0.624164i \(-0.785440\pi\)
−0.781294 + 0.624164i \(0.785440\pi\)
\(54\) 0 0
\(55\) −9.15751 −1.23480
\(56\) 0 0
\(57\) 0.859270 0.113813
\(58\) 0 0
\(59\) −2.03410 −0.264817 −0.132409 − 0.991195i \(-0.542271\pi\)
−0.132409 + 0.991195i \(0.542271\pi\)
\(60\) 0 0
\(61\) 10.6262 1.36055 0.680273 − 0.732959i \(-0.261861\pi\)
0.680273 + 0.732959i \(0.261861\pi\)
\(62\) 0 0
\(63\) −0.310300 −0.0390942
\(64\) 0 0
\(65\) −11.2838 −1.39959
\(66\) 0 0
\(67\) 5.97360 0.729791 0.364896 − 0.931048i \(-0.381105\pi\)
0.364896 + 0.931048i \(0.381105\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0664 1.55070 0.775349 − 0.631533i \(-0.217574\pi\)
0.775349 + 0.631533i \(0.217574\pi\)
\(72\) 0 0
\(73\) 10.0288 1.17378 0.586892 − 0.809665i \(-0.300351\pi\)
0.586892 + 0.809665i \(0.300351\pi\)
\(74\) 0 0
\(75\) −9.84911 −1.13728
\(76\) 0 0
\(77\) 4.11643 0.469111
\(78\) 0 0
\(79\) 7.58909 0.853840 0.426920 − 0.904289i \(-0.359599\pi\)
0.426920 + 0.904289i \(0.359599\pi\)
\(80\) 0 0
\(81\) −8.32888 −0.925431
\(82\) 0 0
\(83\) 8.04453 0.883002 0.441501 − 0.897261i \(-0.354446\pi\)
0.441501 + 0.897261i \(0.354446\pi\)
\(84\) 0 0
\(85\) −12.4320 −1.34844
\(86\) 0 0
\(87\) −12.1202 −1.29942
\(88\) 0 0
\(89\) 15.7531 1.66983 0.834915 − 0.550379i \(-0.185517\pi\)
0.834915 + 0.550379i \(0.185517\pi\)
\(90\) 0 0
\(91\) 5.07224 0.531716
\(92\) 0 0
\(93\) 12.7916 1.32643
\(94\) 0 0
\(95\) 1.69779 0.174190
\(96\) 0 0
\(97\) 7.14396 0.725359 0.362679 − 0.931914i \(-0.381862\pi\)
0.362679 + 0.931914i \(0.381862\pi\)
\(98\) 0 0
\(99\) −0.580184 −0.0583107
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.1.5 15
4.3 odd 2 8464.2.a.cg.1.11 15
23.13 even 11 184.2.i.b.169.1 yes 30
23.16 even 11 184.2.i.b.49.1 ✓ 30
23.22 odd 2 4232.2.a.ba.1.5 15
92.39 odd 22 368.2.m.e.49.3 30
92.59 odd 22 368.2.m.e.353.3 30
92.91 even 2 8464.2.a.ch.1.11 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.b.49.1 ✓ 30 23.16 even 11
184.2.i.b.169.1 yes 30 23.13 even 11
368.2.m.e.49.3 30 92.39 odd 22
368.2.m.e.353.3 30 92.59 odd 22
4232.2.a.ba.1.5 15 23.22 odd 2
4232.2.a.bb.1.5 15 1.1 even 1 trivial
8464.2.a.cg.1.11 15 4.3 odd 2
8464.2.a.ch.1.11 15 92.91 even 2