Properties

Label 1232.3.m.a
Level $1232$
Weight $3$
Character orbit 1232.m
Self dual yes
Analytic conductor $33.570$
Analytic rank $0$
Dimension $4$
CM discriminant -308
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1232.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(33.5695685692\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{7}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -7 q^{7} + ( 9 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -7 q^{7} + ( 9 + \beta_{2} ) q^{9} -11 q^{11} + ( 2 \beta_{1} - \beta_{3} ) q^{13} + ( 4 \beta_{1} + \beta_{3} ) q^{17} -7 \beta_{1} q^{21} + 25 q^{25} + ( 17 \beta_{1} + \beta_{3} ) q^{27} + ( 7 \beta_{1} - 2 \beta_{3} ) q^{31} -11 \beta_{1} q^{33} -3 \beta_{2} q^{37} + ( 34 + \beta_{2} ) q^{39} + ( 11 \beta_{1} + 2 \beta_{3} ) q^{41} -3 \beta_{2} q^{43} + ( \beta_{1} + 4 \beta_{3} ) q^{47} + 49 q^{49} + ( 74 + 5 \beta_{2} ) q^{51} -3 \beta_{2} q^{53} + ( -2 \beta_{1} - 5 \beta_{3} ) q^{59} + ( 19 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -63 - 7 \beta_{2} ) q^{63} + ( -14 \beta_{1} - 5 \beta_{3} ) q^{73} + 25 \beta_{1} q^{75} + 77 q^{77} + 9 \beta_{2} q^{79} + ( 227 + 9 \beta_{2} ) q^{81} + ( -14 \beta_{1} + 7 \beta_{3} ) q^{91} + ( 122 + 5 \beta_{2} ) q^{93} + ( -99 - 11 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{7} + 36 q^{9} + O(q^{10}) \) \( 4 q - 28 q^{7} + 36 q^{9} - 44 q^{11} + 100 q^{25} + 136 q^{39} + 196 q^{49} + 296 q^{51} - 252 q^{63} + 308 q^{77} + 908 q^{81} + 488 q^{93} - 396 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 18 \)
\(\beta_{3}\)\(=\)\( 8 \nu^{3} - 70 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 18\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 35 \beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1231.1
−2.98119
−0.335437
0.335437
2.98119
0 −5.96238 0 0 0 −7.00000 0 26.5499 0
1231.2 0 −0.670873 0 0 0 −7.00000 0 −8.54993 0
1231.3 0 0.670873 0 0 0 −7.00000 0 −8.54993 0
1231.4 0 5.96238 0 0 0 −7.00000 0 26.5499 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
308.g odd 2 1 CM by \(\Q(\sqrt{-77}) \)
7.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.3.m.a 4
4.b odd 2 1 1232.3.m.b yes 4
7.b odd 2 1 inner 1232.3.m.a 4
11.b odd 2 1 1232.3.m.b yes 4
28.d even 2 1 1232.3.m.b yes 4
44.c even 2 1 inner 1232.3.m.a 4
77.b even 2 1 1232.3.m.b yes 4
308.g odd 2 1 CM 1232.3.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1232.3.m.a 4 1.a even 1 1 trivial
1232.3.m.a 4 7.b odd 2 1 inner
1232.3.m.a 4 44.c even 2 1 inner
1232.3.m.a 4 308.g odd 2 1 CM
1232.3.m.b yes 4 4.b odd 2 1
1232.3.m.b yes 4 11.b odd 2 1
1232.3.m.b yes 4 28.d even 2 1
1232.3.m.b yes 4 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{4} - 36 T_{3}^{2} + 16 \)
\( T_{107} + 38 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 - 36 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 + T )^{4} \)
$11$ \( ( 11 + T )^{4} \)
$13$ \( 44944 - 676 T^{2} + T^{4} \)
$17$ \( 309136 - 1156 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 3225616 - 3844 T^{2} + T^{4} \)
$37$ \( ( -2772 + T^{2} )^{2} \)
$41$ \( 7907344 - 6724 T^{2} + T^{4} \)
$43$ \( ( -2772 + T^{2} )^{2} \)
$47$ \( 3083536 - 8836 T^{2} + T^{4} \)
$53$ \( ( -2772 + T^{2} )^{2} \)
$59$ \( 10523536 - 13924 T^{2} + T^{4} \)
$61$ \( 39790864 - 14884 T^{2} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 113124496 - 21316 T^{2} + T^{4} \)
$79$ \( ( -24948 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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