Properties

Label 1232.2.q.g
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + 4 \beta_{2} q^{9} + (\beta_{2} + 1) q^{11} + 5 q^{13} + (\beta_{3} + 7) q^{15} + ( - 6 \beta_{2} - 6) q^{17} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{19} + 7 q^{21} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{23} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{25} + \beta_{3} q^{27} + ( - 2 \beta_{3} + 1) q^{29} + ( - 4 \beta_{2} - 4) q^{31} + (\beta_{3} + \beta_1) q^{33} + ( - 7 \beta_{2} + \beta_1 - 7) q^{35} + (\beta_{3} + \beta_{2} + \beta_1) q^{37} + 5 \beta_1 q^{39} + ( - 3 \beta_{3} + 3) q^{41} + 4 q^{43} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{45} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{47} + ( - 7 \beta_{2} - 7) q^{49} + ( - 6 \beta_{3} - 6 \beta_1) q^{51} + (\beta_{2} - \beta_1 + 1) q^{53} + ( - \beta_{3} - 1) q^{55} + (3 \beta_{3} - 7) q^{57} + (2 \beta_{2} + \beta_1 + 2) q^{59} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1) q^{61} + 4 \beta_1 q^{63} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{65} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{67} + (\beta_{3} + 7) q^{69} + ( - \beta_{3} - 7) q^{71} + ( - 3 \beta_{2} + \beta_1 - 3) q^{73} + ( - 3 \beta_{3} + 14 \beta_{2} - 3 \beta_1) q^{75} - \beta_{3} q^{77} + (\beta_{3} + \beta_1) q^{79} + (5 \beta_{2} + 5) q^{81} + ( - 2 \beta_{3} - 8) q^{83} + (6 \beta_{3} + 6) q^{85} + (14 \beta_{2} + \beta_1 + 14) q^{87} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{89} + ( - 5 \beta_{3} - 5 \beta_1) q^{91} + ( - 4 \beta_{3} - 4 \beta_1) q^{93} + (4 \beta_{2} + 2 \beta_1 + 4) q^{95} + (2 \beta_{3} - 11) q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 8 q^{9} + 2 q^{11} + 20 q^{13} + 28 q^{15} - 12 q^{17} - 6 q^{19} + 28 q^{21} - 2 q^{23} - 6 q^{25} + 4 q^{29} - 8 q^{31} - 14 q^{35} - 2 q^{37} + 12 q^{41} + 16 q^{43} - 8 q^{45} + 16 q^{47} - 14 q^{49} + 2 q^{53} - 4 q^{55} - 28 q^{57} + 4 q^{59} - 18 q^{61} - 10 q^{65} - 8 q^{67} + 28 q^{69} - 28 q^{71} - 6 q^{73} - 28 q^{75} + 10 q^{81} - 32 q^{83} + 24 q^{85} + 28 q^{87} + 8 q^{89} + 8 q^{95} - 44 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) 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
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 2.29129i
0 −1.32288 2.29129i 0 −1.82288 + 3.15731i 0 −1.32288 + 2.29129i 0 −2.00000 + 3.46410i 0
177.2 0 1.32288 + 2.29129i 0 0.822876 1.42526i 0 1.32288 2.29129i 0 −2.00000 + 3.46410i 0
529.1 0 −1.32288 + 2.29129i 0 −1.82288 3.15731i 0 −1.32288 2.29129i 0 −2.00000 3.46410i 0
529.2 0 1.32288 2.29129i 0 0.822876 + 1.42526i 0 1.32288 + 2.29129i 0 −2.00000 3.46410i 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.g 4
4.b odd 2 1 154.2.e.f 4
7.c even 3 1 inner 1232.2.q.g 4
7.c even 3 1 8624.2.a.ca 2
7.d odd 6 1 8624.2.a.bk 2
12.b even 2 1 1386.2.k.s 4
28.d even 2 1 1078.2.e.v 4
28.f even 6 1 1078.2.a.n 2
28.f even 6 1 1078.2.e.v 4
28.g odd 6 1 154.2.e.f 4
28.g odd 6 1 1078.2.a.s 2
84.j odd 6 1 9702.2.a.dr 2
84.n even 6 1 1386.2.k.s 4
84.n even 6 1 9702.2.a.cz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 4.b odd 2 1
154.2.e.f 4 28.g odd 6 1
1078.2.a.n 2 28.f even 6 1
1078.2.a.s 2 28.g odd 6 1
1078.2.e.v 4 28.d even 2 1
1078.2.e.v 4 28.f even 6 1
1232.2.q.g 4 1.a even 1 1 trivial
1232.2.q.g 4 7.c even 3 1 inner
1386.2.k.s 4 12.b even 2 1
1386.2.k.s 4 84.n even 6 1
8624.2.a.bk 2 7.d odd 6 1
8624.2.a.ca 2 7.c even 3 1
9702.2.a.cz 2 84.n even 6 1
9702.2.a.dr 2 84.j odd 6 1

Hecke kernels

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

\( T_{3}^{4} + 7T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 5)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 34 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 16 T^{3} + 220 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + 271 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 111 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$71$ \( (T^{2} + 14 T + 42)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} + 34 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$79$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$83$ \( (T^{2} + 16 T + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + 160 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$97$ \( (T^{2} + 22 T + 93)^{2} \) Copy content Toggle raw display
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