Properties

Label 1232.2.e.b
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: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} -2 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{3} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} -2 q^{9} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{2} q^{13} + 5 q^{15} -4 \beta_{2} q^{17} + \beta_{2} q^{19} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{21} + 5 q^{23} + \beta_{3} q^{27} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{29} + \beta_{3} q^{31} + ( -5 \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{35} + 11 q^{37} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{39} + 7 \beta_{2} q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} + 2 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( 2 + 3 \beta_{3} ) q^{49} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{51} + 8 q^{53} + ( 5 \beta_{2} - \beta_{3} ) q^{55} + ( -2 \beta_{1} - \beta_{2} ) q^{57} -\beta_{3} q^{59} -3 \beta_{2} q^{61} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{63} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{65} -9 q^{67} + 5 \beta_{3} q^{69} + q^{71} -3 \beta_{2} q^{73} + ( 5 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{77} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{79} -11 q^{81} + \beta_{2} q^{83} + ( -8 \beta_{1} - 4 \beta_{2} ) q^{85} -15 \beta_{2} q^{87} + 3 \beta_{3} q^{89} + ( -6 - 2 \beta_{3} ) q^{91} -5 q^{93} + ( 2 \beta_{1} + \beta_{2} ) q^{95} + 7 \beta_{3} q^{97} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{9} + 4 q^{11} + 20 q^{15} + 20 q^{23} + 44 q^{37} + 8 q^{49} + 32 q^{53} - 36 q^{67} + 4 q^{71} + 20 q^{77} - 44 q^{81} - 24 q^{91} - 20 q^{93} - 8 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 2\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - \beta_{1}\)

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
−0.707107 + 1.58114i
0.707107 1.58114i
−0.707107 1.58114i
0.707107 + 1.58114i
0 2.23607i 0 2.23607i 0 −2.12132 + 1.58114i 0 −2.00000 0
769.2 0 2.23607i 0 2.23607i 0 2.12132 1.58114i 0 −2.00000 0
769.3 0 2.23607i 0 2.23607i 0 −2.12132 1.58114i 0 −2.00000 0
769.4 0 2.23607i 0 2.23607i 0 2.12132 + 1.58114i 0 −2.00000 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.b 4
4.b odd 2 1 308.2.c.a 4
7.b odd 2 1 inner 1232.2.e.b 4
11.b odd 2 1 inner 1232.2.e.b 4
12.b even 2 1 2772.2.i.c 4
28.d even 2 1 308.2.c.a 4
28.f even 6 2 2156.2.q.b 8
28.g odd 6 2 2156.2.q.b 8
44.c even 2 1 308.2.c.a 4
77.b even 2 1 inner 1232.2.e.b 4
84.h odd 2 1 2772.2.i.c 4
132.d odd 2 1 2772.2.i.c 4
308.g odd 2 1 308.2.c.a 4
308.m odd 6 2 2156.2.q.b 8
308.n even 6 2 2156.2.q.b 8
924.n even 2 1 2772.2.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.a 4 4.b odd 2 1
308.2.c.a 4 28.d even 2 1
308.2.c.a 4 44.c even 2 1
308.2.c.a 4 308.g odd 2 1
1232.2.e.b 4 1.a even 1 1 trivial
1232.2.e.b 4 7.b odd 2 1 inner
1232.2.e.b 4 11.b odd 2 1 inner
1232.2.e.b 4 77.b even 2 1 inner
2156.2.q.b 8 28.f even 6 2
2156.2.q.b 8 28.g odd 6 2
2156.2.q.b 8 308.m odd 6 2
2156.2.q.b 8 308.n even 6 2
2772.2.i.c 4 12.b even 2 1
2772.2.i.c 4 84.h odd 2 1
2772.2.i.c 4 132.d odd 2 1
2772.2.i.c 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 \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{2} + 5 \)
\( T_{13}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 5 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( 49 - 4 T^{2} + T^{4} \)
$11$ \( ( 11 - 2 T + T^{2} )^{2} \)
$13$ \( ( -8 + T^{2} )^{2} \)
$17$ \( ( -32 + T^{2} )^{2} \)
$19$ \( ( -2 + T^{2} )^{2} \)
$23$ \( ( -5 + T )^{4} \)
$29$ \( ( 90 + T^{2} )^{2} \)
$31$ \( ( 5 + T^{2} )^{2} \)
$37$ \( ( -11 + T )^{4} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( 10 + T^{2} )^{2} \)
$47$ \( ( 20 + T^{2} )^{2} \)
$53$ \( ( -8 + T )^{4} \)
$59$ \( ( 5 + T^{2} )^{2} \)
$61$ \( ( -18 + T^{2} )^{2} \)
$67$ \( ( 9 + T )^{4} \)
$71$ \( ( -1 + T )^{4} \)
$73$ \( ( -18 + T^{2} )^{2} \)
$79$ \( ( 40 + T^{2} )^{2} \)
$83$ \( ( -2 + T^{2} )^{2} \)
$89$ \( ( 45 + T^{2} )^{2} \)
$97$ \( ( 245 + T^{2} )^{2} \)
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