Properties

Label 1232.2.q.a
Level $1232$
Weight $2$
Character orbit 1232.q
Analytic conductor $9.838$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -3 + 3 \zeta_{6} ) q^{3} + 4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} -6 \zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} - q^{13} -12 q^{15} + ( -2 + 2 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} + ( -9 + 6 \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 9 q^{27} + q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -3 \zeta_{6} q^{33} + ( -4 + 12 \zeta_{6} ) q^{35} + 2 \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} -2 q^{41} -4 q^{43} + ( 24 - 24 \zeta_{6} ) q^{45} + 2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} -4 q^{55} -18 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + 5 \zeta_{6} q^{61} + ( 6 - 18 \zeta_{6} ) q^{63} -4 \zeta_{6} q^{65} + ( -9 + 9 \zeta_{6} ) q^{67} + 6 q^{69} -4 q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} -33 \zeta_{6} q^{75} + ( -3 + 2 \zeta_{6} ) q^{77} -15 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 6 q^{83} -8 q^{85} + ( -3 + 3 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} + 12 \zeta_{6} q^{93} + ( -24 + 24 \zeta_{6} ) q^{95} -5 q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 4q^{5} + 5q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 4q^{5} + 5q^{7} - 6q^{9} - q^{11} - 2q^{13} - 24q^{15} - 2q^{17} + 6q^{19} - 12q^{21} - 2q^{23} - 11q^{25} + 18q^{27} + 2q^{29} + 4q^{31} - 3q^{33} + 4q^{35} + 2q^{37} + 3q^{39} - 4q^{41} - 8q^{43} + 24q^{45} + 2q^{47} + 11q^{49} - 6q^{51} + 12q^{53} - 8q^{55} - 36q^{57} + 9q^{59} + 5q^{61} - 6q^{63} - 4q^{65} - 9q^{67} + 12q^{69} - 8q^{71} + 2q^{73} - 33q^{75} - 4q^{77} - 15q^{79} - 9q^{81} + 12q^{83} - 16q^{85} - 3q^{87} - 6q^{89} - 5q^{91} + 12q^{93} - 24q^{95} - 10q^{97} + 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 2.59808i 0 2.00000 3.46410i 0 2.50000 0.866025i 0 −3.00000 + 5.19615i 0
529.1 0 −1.50000 + 2.59808i 0 2.00000 + 3.46410i 0 2.50000 + 0.866025i 0 −3.00000 5.19615i 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.a 2
4.b odd 2 1 154.2.e.d 2
7.c even 3 1 inner 1232.2.q.a 2
7.c even 3 1 8624.2.a.bd 1
7.d odd 6 1 8624.2.a.d 1
12.b even 2 1 1386.2.k.a 2
28.d even 2 1 1078.2.e.g 2
28.f even 6 1 1078.2.a.f 1
28.f even 6 1 1078.2.e.g 2
28.g odd 6 1 154.2.e.d 2
28.g odd 6 1 1078.2.a.a 1
84.j odd 6 1 9702.2.a.bb 1
84.n even 6 1 1386.2.k.a 2
84.n even 6 1 9702.2.a.cg 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.d 2 4.b odd 2 1
154.2.e.d 2 28.g odd 6 1
1078.2.a.a 1 28.g odd 6 1
1078.2.a.f 1 28.f even 6 1
1078.2.e.g 2 28.d even 2 1
1078.2.e.g 2 28.f even 6 1
1232.2.q.a 2 1.a even 1 1 trivial
1232.2.q.a 2 7.c even 3 1 inner
1386.2.k.a 2 12.b even 2 1
1386.2.k.a 2 84.n even 6 1
8624.2.a.d 1 7.d odd 6 1
8624.2.a.bd 1 7.c even 3 1
9702.2.a.bb 1 84.j odd 6 1
9702.2.a.cg 1 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 \(S_{2}^{\mathrm{new}}(1232, [\chi])\):

\( T_{3}^{2} + 3 T_{3} + 9 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 3 T + T^{2} \)
$5$ \( 16 - 4 T + T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 4 + 2 T + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( 16 - 4 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 4 - 2 T + T^{2} \)
$53$ \( 144 - 12 T + T^{2} \)
$59$ \( 81 - 9 T + T^{2} \)
$61$ \( 25 - 5 T + T^{2} \)
$67$ \( 81 + 9 T + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 225 + 15 T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 5 + T )^{2} \)
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