Properties

Label 1232.2.e.d
Level $1232$
Weight $2$
Character orbit 1232.e
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 308)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -2 \beta_{2} q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -2 \beta_{2} q^{5} -\beta_{3} q^{7} + q^{9} + ( -2 - \beta_{3} ) q^{11} -\beta_{1} q^{13} -4 q^{15} -\beta_{1} q^{17} -2 \beta_{1} q^{19} + \beta_{1} q^{21} -4 q^{23} -3 q^{25} -4 \beta_{2} q^{27} -\beta_{2} q^{31} + ( \beta_{1} + 2 \beta_{2} ) q^{33} + 2 \beta_{1} q^{35} -4 q^{37} -2 \beta_{3} q^{39} + \beta_{1} q^{41} + 4 \beta_{3} q^{43} -2 \beta_{2} q^{45} + 7 \beta_{2} q^{47} -7 q^{49} -2 \beta_{3} q^{51} -4 q^{53} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{55} -4 \beta_{3} q^{57} + \beta_{2} q^{59} -3 \beta_{1} q^{61} -\beta_{3} q^{63} -4 \beta_{3} q^{65} + 12 q^{67} + 4 \beta_{2} q^{69} -8 q^{71} + 3 \beta_{1} q^{73} + 3 \beta_{2} q^{75} + ( -7 + 2 \beta_{3} ) q^{77} + 4 \beta_{3} q^{79} -5 q^{81} -2 \beta_{1} q^{83} -4 \beta_{3} q^{85} + 12 \beta_{2} q^{89} -7 \beta_{2} q^{91} -2 q^{93} -8 \beta_{3} q^{95} -4 \beta_{2} q^{97} + ( -2 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{9} - 8 q^{11} - 16 q^{15} - 16 q^{23} - 12 q^{25} - 16 q^{37} - 28 q^{49} - 16 q^{53} + 48 q^{67} - 32 q^{71} - 28 q^{77} - 20 q^{81} - 8 q^{93} - 8 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 4 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 3 \nu^{2} + 17 \nu - 8 \)
\(\beta_{3}\)\(=\)\( -4 \nu^{3} + 6 \nu^{2} - 32 \nu + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{3} - 13 \beta_{2} + 3 \beta_{1} - 11\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
0.500000 + 2.73709i
0.500000 + 0.0913379i
0.500000 0.0913379i
0.500000 2.73709i
0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.2 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.3 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
769.4 0 1.41421i 0 2.82843i 0 2.64575i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.e.d 4
4.b odd 2 1 308.2.c.b 4
7.b odd 2 1 inner 1232.2.e.d 4
11.b odd 2 1 inner 1232.2.e.d 4
12.b even 2 1 2772.2.i.a 4
28.d even 2 1 308.2.c.b 4
28.f even 6 2 2156.2.q.a 8
28.g odd 6 2 2156.2.q.a 8
44.c even 2 1 308.2.c.b 4
77.b even 2 1 inner 1232.2.e.d 4
84.h odd 2 1 2772.2.i.a 4
132.d odd 2 1 2772.2.i.a 4
308.g odd 2 1 308.2.c.b 4
308.m odd 6 2 2156.2.q.a 8
308.n even 6 2 2156.2.q.a 8
924.n even 2 1 2772.2.i.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.b 4 4.b odd 2 1
308.2.c.b 4 28.d even 2 1
308.2.c.b 4 44.c even 2 1
308.2.c.b 4 308.g odd 2 1
1232.2.e.d 4 1.a even 1 1 trivial
1232.2.e.d 4 7.b odd 2 1 inner
1232.2.e.d 4 11.b odd 2 1 inner
1232.2.e.d 4 77.b even 2 1 inner
2156.2.q.a 8 28.f even 6 2
2156.2.q.a 8 28.g odd 6 2
2156.2.q.a 8 308.m odd 6 2
2156.2.q.a 8 308.n even 6 2
2772.2.i.a 4 12.b even 2 1
2772.2.i.a 4 84.h odd 2 1
2772.2.i.a 4 132.d odd 2 1
2772.2.i.a 4 924.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{2} + 2 \)
\( T_{13}^{2} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 2 + T^{2} )^{2} \)
$5$ \( ( 8 + T^{2} )^{2} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( 11 + 4 T + T^{2} )^{2} \)
$13$ \( ( -14 + T^{2} )^{2} \)
$17$ \( ( -14 + T^{2} )^{2} \)
$19$ \( ( -56 + T^{2} )^{2} \)
$23$ \( ( 4 + T )^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 2 + T^{2} )^{2} \)
$37$ \( ( 4 + T )^{4} \)
$41$ \( ( -14 + T^{2} )^{2} \)
$43$ \( ( 112 + T^{2} )^{2} \)
$47$ \( ( 98 + T^{2} )^{2} \)
$53$ \( ( 4 + T )^{4} \)
$59$ \( ( 2 + T^{2} )^{2} \)
$61$ \( ( -126 + T^{2} )^{2} \)
$67$ \( ( -12 + T )^{4} \)
$71$ \( ( 8 + T )^{4} \)
$73$ \( ( -126 + T^{2} )^{2} \)
$79$ \( ( 112 + T^{2} )^{2} \)
$83$ \( ( -56 + T^{2} )^{2} \)
$89$ \( ( 288 + T^{2} )^{2} \)
$97$ \( ( 32 + T^{2} )^{2} \)
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