Properties

Label 1232.2.e.c
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 77)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} -2 q^{9} + ( 3 - \beta_{2} ) q^{11} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{13} -5 q^{15} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( 3 \beta_{1} + \beta_{2} ) q^{21} + 3 q^{23} + \beta_{3} q^{27} -\beta_{2} q^{29} -3 \beta_{3} q^{31} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{33} + ( 3 \beta_{1} + \beta_{2} ) q^{35} - q^{37} + 10 \beta_{2} q^{39} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{41} -3 \beta_{2} q^{43} -2 \beta_{3} q^{45} + 2 \beta_{3} q^{47} + ( -2 - 3 \beta_{3} ) q^{49} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{55} + 5 \beta_{2} q^{57} -\beta_{3} q^{59} + ( -2 \beta_{1} + \beta_{2} ) q^{61} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{63} + 10 \beta_{2} q^{65} -11 q^{67} + 3 \beta_{3} q^{69} -9 q^{71} + ( -2 \beta_{1} + \beta_{2} ) q^{73} + ( -3 + 3 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{77} + 6 \beta_{2} q^{79} -11 q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( 2 \beta_{1} - \beta_{2} ) q^{87} + \beta_{3} q^{89} + ( 10 - 6 \beta_{3} ) q^{91} + 15 q^{93} + 5 \beta_{2} q^{95} -3 \beta_{3} q^{97} + ( -6 + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9} + O(q^{10}) \) \( 4 q - 8 q^{9} + 12 q^{11} - 20 q^{15} + 12 q^{23} - 4 q^{37} - 8 q^{49} - 44 q^{67} - 36 q^{71} - 12 q^{77} - 44 q^{81} + 40 q^{91} + 60 q^{93} - 24 q^{99} + O(q^{100}) \)

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
−1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.58114 + 0.707107i
0 2.23607i 0 2.23607i 0 −1.58114 2.12132i 0 −2.00000 0
769.2 0 2.23607i 0 2.23607i 0 1.58114 + 2.12132i 0 −2.00000 0
769.3 0 2.23607i 0 2.23607i 0 −1.58114 + 2.12132i 0 −2.00000 0
769.4 0 2.23607i 0 2.23607i 0 1.58114 2.12132i 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.c 4
4.b odd 2 1 77.2.b.b 4
7.b odd 2 1 inner 1232.2.e.c 4
11.b odd 2 1 inner 1232.2.e.c 4
12.b even 2 1 693.2.c.b 4
28.d even 2 1 77.2.b.b 4
28.f even 6 2 539.2.i.b 8
28.g odd 6 2 539.2.i.b 8
44.c even 2 1 77.2.b.b 4
44.g even 10 4 847.2.l.g 16
44.h odd 10 4 847.2.l.g 16
77.b even 2 1 inner 1232.2.e.c 4
84.h odd 2 1 693.2.c.b 4
132.d odd 2 1 693.2.c.b 4
308.g odd 2 1 77.2.b.b 4
308.m odd 6 2 539.2.i.b 8
308.n even 6 2 539.2.i.b 8
308.s odd 10 4 847.2.l.g 16
308.t even 10 4 847.2.l.g 16
924.n even 2 1 693.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.b.b 4 4.b odd 2 1
77.2.b.b 4 28.d even 2 1
77.2.b.b 4 44.c even 2 1
77.2.b.b 4 308.g odd 2 1
539.2.i.b 8 28.f even 6 2
539.2.i.b 8 28.g odd 6 2
539.2.i.b 8 308.m odd 6 2
539.2.i.b 8 308.n even 6 2
693.2.c.b 4 12.b even 2 1
693.2.c.b 4 84.h odd 2 1
693.2.c.b 4 132.d odd 2 1
693.2.c.b 4 924.n even 2 1
847.2.l.g 16 44.g even 10 4
847.2.l.g 16 44.h odd 10 4
847.2.l.g 16 308.s odd 10 4
847.2.l.g 16 308.t even 10 4
1232.2.e.c 4 1.a even 1 1 trivial
1232.2.e.c 4 7.b odd 2 1 inner
1232.2.e.c 4 11.b odd 2 1 inner
1232.2.e.c 4 77.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} + 5 \)
\( T_{13}^{2} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 5 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( 49 + 4 T^{2} + T^{4} \)
$11$ \( ( 11 - 6 T + T^{2} )^{2} \)
$13$ \( ( -40 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -10 + T^{2} )^{2} \)
$23$ \( ( -3 + T )^{4} \)
$29$ \( ( 2 + T^{2} )^{2} \)
$31$ \( ( 45 + T^{2} )^{2} \)
$37$ \( ( 1 + T )^{4} \)
$41$ \( ( -90 + T^{2} )^{2} \)
$43$ \( ( 18 + T^{2} )^{2} \)
$47$ \( ( 20 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 5 + T^{2} )^{2} \)
$61$ \( ( -10 + T^{2} )^{2} \)
$67$ \( ( 11 + T )^{4} \)
$71$ \( ( 9 + T )^{4} \)
$73$ \( ( -10 + T^{2} )^{2} \)
$79$ \( ( 72 + T^{2} )^{2} \)
$83$ \( ( -90 + T^{2} )^{2} \)
$89$ \( ( 5 + T^{2} )^{2} \)
$97$ \( ( 45 + T^{2} )^{2} \)
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