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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{3} - 2 \beta_1) q^{9} + (\beta_{2} + 1) q^{11} + ( - 2 \beta_{3} - 1) q^{13} - \beta_{3} q^{15} + (2 \beta_{2} + 4 \beta_1 + 2) q^{17} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{21} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{23} + ( - \beta_{2} - 4 \beta_1 - 1) q^{25} + ( - \beta_{3} + 1) q^{27} + (4 \beta_{3} - 3) q^{29} + ( - 4 \beta_{2} - 4) q^{31} + (\beta_{3} - \beta_{2} + \beta_1) q^{33} + ( - 4 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{35} + (\beta_{3} - 8 \beta_{2} + \beta_1) q^{37} + (5 \beta_{2} - 3 \beta_1 + 5) q^{39} + ( - \beta_{3} - 4) q^{41} + 4 \beta_{3} q^{43} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{45} + ( - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{47} + ( - 4 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 5) q^{49} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{51} + (2 \beta_{2} + 7 \beta_1 + 2) q^{53} + ( - \beta_{3} + 2) q^{55} + \beta_{3} q^{57} + ( - 7 \beta_{2} - \beta_1 - 7) q^{59} + ( - 2 \beta_{3} + 9 \beta_{2} - 2 \beta_1) q^{61} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{65} + (7 \beta_{2} + 3 \beta_1 + 7) q^{67} + (5 \beta_{3} + 8) q^{69} + (5 \beta_{3} + 4) q^{71} + (8 \beta_{2} - \beta_1 + 8) q^{73} + (3 \beta_{3} - 7 \beta_{2} + 3 \beta_1) q^{75} + (\beta_{3} + 2 \beta_1 + 1) q^{77} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{79} + (\beta_{2} - 6 \beta_1 + 1) q^{81} + (10 \beta_{3} - 2) q^{83} + ( - 10 \beta_{3} + 12) q^{85} + ( - 5 \beta_{2} + \beta_1 - 5) q^{87} + ( - 6 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{89} + ( - \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 8) q^{91} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{93} + (6 \beta_{2} + 4 \beta_1 + 6) q^{95} + ( - 2 \beta_{3} - 1) q^{97} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 4 q^{19} - 4 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} - 12 q^{29} - 8 q^{31} + 2 q^{33} + 8 q^{35} + 16 q^{37} + 10 q^{39} - 16 q^{41} - 8 q^{45} - 4 q^{47} + 10 q^{49} - 12 q^{51} + 4 q^{53} + 8 q^{55} - 14 q^{59} - 18 q^{61} + 24 q^{63} + 4 q^{65} + 14 q^{67} + 32 q^{69} + 16 q^{71} + 16 q^{73} + 14 q^{75} + 4 q^{77} - 18 q^{79} + 2 q^{81} - 8 q^{83} + 48 q^{85} - 10 q^{87} - 8 q^{89} + 22 q^{91} - 8 q^{93} + 12 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

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