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Properties

Label 4232.2.a.v.1.3
Level $4232$
Weight $2$
Character 4232.1
Self dual yes
Analytic conductor $33.793$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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 = [6,0,4,0,0,0,0] 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: \(6\)
Coefficient field: 6.6.26849792.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 14x^{4} - 2x^{3} + 28x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.52912\) 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)
 
\(f(q)\) \(=\) \(q+0.357926 q^{3} -1.57672 q^{5} -2.08290 q^{7} -2.87189 q^{9} +0.668688 q^{11} -0.357926 q^{13} -0.564349 q^{15} -2.99093 q^{17} -5.58002 q^{19} -0.745525 q^{21} -2.51396 q^{25} -2.10170 q^{27} +5.53341 q^{29} -3.87189 q^{31} +0.239341 q^{33} +3.28415 q^{35} -2.58909 q^{37} -0.128111 q^{39} +10.8176 q^{41} +9.71832 q^{43} +4.52816 q^{45} +8.58774 q^{47} -2.66152 q^{49} -1.07053 q^{51} -3.33461 q^{53} -1.05433 q^{55} -1.99724 q^{57} +1.77018 q^{59} +4.13830 q^{61} +5.98186 q^{63} +0.564349 q^{65} -7.66292 q^{67} +5.64207 q^{71} +10.2493 q^{73} -0.899813 q^{75} -1.39281 q^{77} +1.99724 q^{79} +7.86341 q^{81} -10.7889 q^{83} +4.71585 q^{85} +1.98055 q^{87} +2.72409 q^{89} +0.745525 q^{91} -1.38585 q^{93} +8.79811 q^{95} +10.3102 q^{97} -1.92040 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + 10 q^{9} - 4 q^{13} + 14 q^{25} + 40 q^{27} - 12 q^{29} + 4 q^{31} + 16 q^{35} - 28 q^{39} + 16 q^{41} + 28 q^{47} + 2 q^{49} - 28 q^{55} + 36 q^{59} + 32 q^{71} + 20 q^{73} + 80 q^{75}+ \cdots + 20 q^{95}+O(q^{100}) \) 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
\(3\) 0.357926 0.206649 0.103324 − 0.994648i \(-0.467052\pi\)
0.103324 + 0.994648i \(0.467052\pi\)
\(4\) 0 0
\(5\) −1.57672 −0.705129 −0.352565 − 0.935787i \(-0.614690\pi\)
−0.352565 + 0.935787i \(0.614690\pi\)
\(6\) 0 0
\(7\) −2.08290 −0.787263 −0.393631 − 0.919268i \(-0.628781\pi\)
−0.393631 + 0.919268i \(0.628781\pi\)
\(8\) 0 0
\(9\) −2.87189 −0.957296
\(10\) 0 0
\(11\) 0.668688 0.201617 0.100809 − 0.994906i \(-0.467857\pi\)
0.100809 + 0.994906i \(0.467857\pi\)
\(12\) 0 0
\(13\) −0.357926 −0.0992709 −0.0496355 − 0.998767i \(-0.515806\pi\)
−0.0496355 + 0.998767i \(0.515806\pi\)
\(14\) 0 0
\(15\) −0.564349 −0.145714
\(16\) 0 0
\(17\) −2.99093 −0.725407 −0.362704 − 0.931905i \(-0.618146\pi\)
−0.362704 + 0.931905i \(0.618146\pi\)
\(18\) 0 0
\(19\) −5.58002 −1.28014 −0.640072 − 0.768315i \(-0.721095\pi\)
−0.640072 + 0.768315i \(0.721095\pi\)
\(20\) 0 0
\(21\) −0.745525 −0.162687
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −2.51396 −0.502792
\(26\) 0 0
\(27\) −2.10170 −0.404473
\(28\) 0 0
\(29\) 5.53341 1.02753 0.513764 − 0.857931i \(-0.328251\pi\)
0.513764 + 0.857931i \(0.328251\pi\)
\(30\) 0 0
\(31\) −3.87189 −0.695412 −0.347706 − 0.937604i \(-0.613039\pi\)
−0.347706 + 0.937604i \(0.613039\pi\)
\(32\) 0 0
\(33\) 0.239341 0.0416639
\(34\) 0 0
\(35\) 3.28415 0.555122
\(36\) 0 0
\(37\) −2.58909 −0.425643 −0.212822 − 0.977091i \(-0.568265\pi\)
−0.212822 + 0.977091i \(0.568265\pi\)
\(38\) 0 0
\(39\) −0.128111 −0.0205142
\(40\) 0 0
\(41\) 10.8176 1.68942 0.844709 − 0.535225i \(-0.179773\pi\)
0.844709 + 0.535225i \(0.179773\pi\)
\(42\) 0 0
\(43\) 9.71832 1.48203 0.741015 − 0.671489i \(-0.234345\pi\)
0.741015 + 0.671489i \(0.234345\pi\)
\(44\) 0 0
\(45\) 4.52816 0.675018
\(46\) 0 0
\(47\) 8.58774 1.25265 0.626325 − 0.779562i \(-0.284558\pi\)
0.626325 + 0.779562i \(0.284558\pi\)
\(48\) 0 0
\(49\) −2.66152 −0.380217
\(50\) 0 0
\(51\) −1.07053 −0.149905
\(52\) 0 0
\(53\) −3.33461 −0.458044 −0.229022 − 0.973421i \(-0.573553\pi\)
−0.229022 + 0.973421i \(0.573553\pi\)
\(54\) 0 0
\(55\) −1.05433 −0.142166
\(56\) 0 0
\(57\) −1.99724 −0.264540
\(58\) 0 0
\(59\) 1.77018 0.230458 0.115229 − 0.993339i \(-0.463240\pi\)
0.115229 + 0.993339i \(0.463240\pi\)
\(60\) 0 0
\(61\) 4.13830 0.529855 0.264928 − 0.964268i \(-0.414652\pi\)
0.264928 + 0.964268i \(0.414652\pi\)
\(62\) 0 0
\(63\) 5.98186 0.753644
\(64\) 0 0
\(65\) 0.564349 0.0699988
\(66\) 0 0
\(67\) −7.66292 −0.936175 −0.468087 − 0.883682i \(-0.655057\pi\)
−0.468087 + 0.883682i \(0.655057\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.64207 0.669591 0.334795 − 0.942291i \(-0.391333\pi\)
0.334795 + 0.942291i \(0.391333\pi\)
\(72\) 0 0
\(73\) 10.2493 1.19959 0.599793 − 0.800155i \(-0.295250\pi\)
0.599793 + 0.800155i \(0.295250\pi\)
\(74\) 0 0
\(75\) −0.899813 −0.103902
\(76\) 0 0
\(77\) −1.39281 −0.158726
\(78\) 0 0
\(79\) 1.99724 0.224707 0.112353 − 0.993668i \(-0.464161\pi\)
0.112353 + 0.993668i \(0.464161\pi\)
\(80\) 0 0
\(81\) 7.86341 0.873712
\(82\) 0 0
\(83\) −10.7889 −1.18423 −0.592115 − 0.805853i \(-0.701707\pi\)
−0.592115 + 0.805853i \(0.701707\pi\)
\(84\) 0 0
\(85\) 4.71585 0.511506
\(86\) 0 0
\(87\) 1.98055 0.212338
\(88\) 0 0
\(89\) 2.72409 0.288753 0.144376 − 0.989523i \(-0.453882\pi\)
0.144376 + 0.989523i \(0.453882\pi\)
\(90\) 0 0
\(91\) 0.745525 0.0781523
\(92\) 0 0
\(93\) −1.38585 −0.143706
\(94\) 0 0
\(95\) 8.79811 0.902667
\(96\) 0 0
\(97\) 10.3102 1.04684 0.523420 − 0.852075i \(-0.324656\pi\)
0.523420 + 0.852075i \(0.324656\pi\)
\(98\) 0 0
\(99\) −1.92040 −0.193007
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.v.1.3 ✓ 6
4.3 odd 2 8464.2.a.by.1.3 6
23.22 odd 2 inner 4232.2.a.v.1.4 yes 6
92.91 even 2 8464.2.a.by.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4232.2.a.v.1.3 ✓ 6 1.1 even 1 trivial
4232.2.a.v.1.4 yes 6 23.22 odd 2 inner
8464.2.a.by.1.3 6 4.3 odd 2
8464.2.a.by.1.4 6 92.91 even 2