Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( 1 + \beta_{2} ) q^{11} + ( -1 + 2 \beta_{3} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( 2 + 2 \beta_{2} ) q^{17} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{19} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{21} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{23} + ( 3 + 3 \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{27} + q^{29} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{35} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{37} + ( -5 - 3 \beta_{1} - 5 \beta_{2} ) q^{39} + ( 2 + 3 \beta_{3} ) q^{41} + 8 q^{43} + ( 4 + 4 \beta_{2} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( -6 + 5 \beta_{3} ) q^{57} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{59} + ( -2 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -4 - 2 \beta_{1} + 4 \beta_{2} ) q^{63} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{65} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( 2 - 3 \beta_{3} ) q^{69} + ( 10 - 3 \beta_{3} ) q^{71} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{79} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( 2 + 6 \beta_{3} ) q^{83} -2 \beta_{3} q^{85} + ( 1 + \beta_{1} + \beta_{2} ) q^{87} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -8 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{91} + 4 \beta_{2} q^{93} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -13 - 2 \beta_{3} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{7} + O(q^{10}) \) \( 4 q + 2 q^{3} - 2 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{17} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{33} + 12 q^{35} + 4 q^{37} - 10 q^{39} + 8 q^{41} + 32 q^{43} + 8 q^{45} + 4 q^{47} + 10 q^{49} - 4 q^{51} - 8 q^{53} - 24 q^{57} + 6 q^{59} + 22 q^{61} - 24 q^{63} - 8 q^{65} + 2 q^{67} + 8 q^{69} + 40 q^{71} + 4 q^{73} - 6 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} + 8 q^{83} + 2 q^{87} + 24 q^{89} - 22 q^{91} - 8 q^{93} + 4 q^{95} - 52 q^{97} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.207107 0.358719i 0 −0.707107 + 1.22474i 0 −2.62132 0.358719i 0 1.41421 2.44949i 0
177.2 0 1.20711 + 2.09077i 0 0.707107 1.22474i 0 1.62132 + 2.09077i 0 −1.41421 + 2.44949i 0
529.1 0 −0.207107 + 0.358719i 0 −0.707107 1.22474i 0 −2.62132 + 0.358719i 0 1.41421 + 2.44949i 0
529.2 0 1.20711 2.09077i 0 0.707107 + 1.22474i 0 1.62132 2.09077i 0 −1.41421 2.44949i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.h 4
4.b odd 2 1 616.2.q.b 4
7.c even 3 1 inner 1232.2.q.h 4
7.c even 3 1 8624.2.a.bg 2
7.d odd 6 1 8624.2.a.cd 2
28.f even 6 1 4312.2.a.m 2
28.g odd 6 1 616.2.q.b 4
28.g odd 6 1 4312.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.b 4 4.b odd 2 1
616.2.q.b 4 28.g odd 6 1
1232.2.q.h 4 1.a even 1 1 trivial
1232.2.q.h 4 7.c even 3 1 inner
4312.2.a.m 2 28.f even 6 1
4312.2.a.u 2 28.g odd 6 1
8624.2.a.bg 2 7.c even 3 1
8624.2.a.cd 2 7.d odd 6 1

Hecke kernels

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

\( T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1 \)
\( T_{13}^{2} + 2 T_{13} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( 4 + 2 T^{2} + T^{4} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -7 + 2 T + T^{2} )^{2} \)
$17$ \( ( 4 - 2 T + T^{2} )^{2} \)
$19$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$23$ \( 196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( 256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( -14 - 4 T + T^{2} )^{2} \)
$43$ \( ( -8 + T )^{4} \)
$47$ \( 16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$53$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( 49 - 42 T + 29 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( 12769 - 2486 T + 371 T^{2} - 22 T^{3} + T^{4} \)
$67$ \( 2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( 82 - 20 T + T^{2} )^{2} \)
$73$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 2401 + 98 T + 53 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( ( -68 - 4 T + T^{2} )^{2} \)
$89$ \( 18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4} \)
$97$ \( ( 161 + 26 T + T^{2} )^{2} \)
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