Properties

Label 1232.2.q.f
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 154)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \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} -\beta_{3} q^{15} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{19} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{21} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + ( -3 + 4 \beta_{3} ) q^{29} + ( -4 - 4 \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{35} + ( \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{37} + ( 5 - 3 \beta_{1} + 5 \beta_{2} ) q^{39} + ( -4 - \beta_{3} ) q^{41} + 4 \beta_{3} q^{43} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{45} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 5 - 2 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{51} + ( 2 + 7 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 2 - \beta_{3} ) q^{55} + \beta_{3} q^{57} + ( -7 - \beta_{1} - 7 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{63} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 7 + 3 \beta_{1} + 7 \beta_{2} ) q^{67} + ( 8 + 5 \beta_{3} ) q^{69} + ( 4 + 5 \beta_{3} ) q^{71} + ( 8 - \beta_{1} + 8 \beta_{2} ) q^{73} + ( 3 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} + ( -2 + 10 \beta_{3} ) q^{83} + ( 12 - 10 \beta_{3} ) q^{85} + ( -5 + \beta_{1} - 5 \beta_{2} ) q^{87} + ( -6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{89} + ( 8 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{91} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{93} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{95} + ( -1 - 2 \beta_{3} ) q^{97} -2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 4q^{5} + 2q^{7} + O(q^{10}) \) \( 4q - 2q^{3} + 4q^{5} + 2q^{7} + 2q^{11} - 4q^{13} + 4q^{17} - 4q^{19} - 4q^{21} - 4q^{23} - 2q^{25} + 4q^{27} - 12q^{29} - 8q^{31} + 2q^{33} + 8q^{35} + 16q^{37} + 10q^{39} - 16q^{41} - 8q^{45} - 4q^{47} + 10q^{49} - 12q^{51} + 4q^{53} + 8q^{55} - 14q^{59} - 18q^{61} + 24q^{63} + 4q^{65} + 14q^{67} + 32q^{69} + 16q^{71} + 16q^{73} + 14q^{75} + 4q^{77} - 18q^{79} + 2q^{81} - 8q^{83} + 48q^{85} - 10q^{87} - 8q^{89} + 22q^{91} - 8q^{93} + 12q^{95} - 4q^{97} + O(q^{100}) \)

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 −1.20711 2.09077i 0 0.292893 0.507306i 0 −1.62132 2.09077i 0 −1.41421 + 2.44949i 0
177.2 0 0.207107 + 0.358719i 0 1.70711 2.95680i 0 2.62132 + 0.358719i 0 1.41421 2.44949i 0
529.1 0 −1.20711 + 2.09077i 0 0.292893 + 0.507306i 0 −1.62132 + 2.09077i 0 −1.41421 2.44949i 0
529.2 0 0.207107 0.358719i 0 1.70711 + 2.95680i 0 2.62132 0.358719i 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.f 4
4.b odd 2 1 154.2.e.e 4
7.c even 3 1 inner 1232.2.q.f 4
7.c even 3 1 8624.2.a.cc 2
7.d odd 6 1 8624.2.a.bh 2
12.b even 2 1 1386.2.k.t 4
28.d even 2 1 1078.2.e.m 4
28.f even 6 1 1078.2.a.x 2
28.f even 6 1 1078.2.e.m 4
28.g odd 6 1 154.2.e.e 4
28.g odd 6 1 1078.2.a.t 2
84.j odd 6 1 9702.2.a.ch 2
84.n even 6 1 1386.2.k.t 4
84.n even 6 1 9702.2.a.cx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.e 4 4.b odd 2 1
154.2.e.e 4 28.g odd 6 1
1078.2.a.t 2 28.g odd 6 1
1078.2.a.x 2 28.f even 6 1
1078.2.e.m 4 28.d even 2 1
1078.2.e.m 4 28.f even 6 1
1232.2.q.f 4 1.a even 1 1 trivial
1232.2.q.f 4 7.c even 3 1 inner
1386.2.k.t 4 12.b even 2 1
1386.2.k.t 4 84.n even 6 1
8624.2.a.bh 2 7.d odd 6 1
8624.2.a.cc 2 7.c even 3 1
9702.2.a.ch 2 84.j odd 6 1
9702.2.a.cx 2 84.n even 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

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( ( -7 + 2 T + T^{2} )^{2} \)
$17$ \( 784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$29$ \( ( -23 + 6 T + T^{2} )^{2} \)
$31$ \( ( 16 + 4 T + T^{2} )^{2} \)
$37$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( ( 14 + 8 T + T^{2} )^{2} \)
$43$ \( ( -32 + T^{2} )^{2} \)
$47$ \( 4624 - 272 T + 84 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 8836 + 376 T + 110 T^{2} - 4 T^{3} + T^{4} \)
$59$ \( 2209 + 658 T + 149 T^{2} + 14 T^{3} + T^{4} \)
$61$ \( 5329 + 1314 T + 251 T^{2} + 18 T^{3} + T^{4} \)
$67$ \( 961 - 434 T + 165 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( ( -34 - 8 T + T^{2} )^{2} \)
$73$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$79$ \( 3969 + 1134 T + 261 T^{2} + 18 T^{3} + T^{4} \)
$83$ \( ( -196 + 4 T + T^{2} )^{2} \)
$89$ \( 3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( -7 + 2 T + T^{2} )^{2} \)
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