Properties

Label 1232.2.a.p
Level $1232$
Weight $2$
Character orbit 1232.a
Self dual yes
Analytic conductor $9.838$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 1 + \beta ) q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 1 + \beta ) q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} - q^{11} + ( -1 + \beta ) q^{13} + ( 6 + 2 \beta ) q^{15} + ( -2 - 2 \beta ) q^{17} + ( 5 - \beta ) q^{19} + ( -1 - \beta ) q^{21} -4 q^{23} + ( 1 + 2 \beta ) q^{25} + ( 10 + 2 \beta ) q^{27} -2 \beta q^{29} -2 q^{31} + ( -1 - \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( -2 - 4 \beta ) q^{37} + 4 q^{39} + ( 2 + 2 \beta ) q^{41} + ( 6 - 2 \beta ) q^{43} + ( 13 + 5 \beta ) q^{45} + 2 q^{47} + q^{49} + ( -12 - 4 \beta ) q^{51} + ( 4 - 2 \beta ) q^{53} + ( -1 - \beta ) q^{55} + 4 \beta q^{57} + ( -5 - \beta ) q^{59} + ( -3 - \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + 4 q^{65} + ( 2 + 6 \beta ) q^{67} + ( -4 - 4 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + ( 4 - 4 \beta ) q^{73} + ( 11 + 3 \beta ) q^{75} + q^{77} + ( 11 + 6 \beta ) q^{81} + ( 1 - 5 \beta ) q^{83} + ( -12 - 4 \beta ) q^{85} + ( -10 - 2 \beta ) q^{87} + 10 q^{89} + ( 1 - \beta ) q^{91} + ( -2 - 2 \beta ) q^{93} + 4 \beta q^{95} + ( 8 - 2 \beta ) q^{97} + ( -3 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} + 12 q^{15} - 4 q^{17} + 10 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 4 q^{41} + 12 q^{43} + 26 q^{45} + 4 q^{47} + 2 q^{49} - 24 q^{51} + 8 q^{53} - 2 q^{55} - 10 q^{59} - 6 q^{61} - 6 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 4 q^{71} + 8 q^{73} + 22 q^{75} + 2 q^{77} + 22 q^{81} + 2 q^{83} - 24 q^{85} - 20 q^{87} + 20 q^{89} + 2 q^{91} - 4 q^{93} + 16 q^{97} - 6 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −1.23607 0 −1.23607 0 −1.00000 0 −1.47214 0
1.2 0 3.23607 0 3.23607 0 −1.00000 0 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.a.p 2
4.b odd 2 1 154.2.a.d 2
7.b odd 2 1 8624.2.a.bf 2
8.b even 2 1 4928.2.a.bk 2
8.d odd 2 1 4928.2.a.bt 2
12.b even 2 1 1386.2.a.m 2
20.d odd 2 1 3850.2.a.bj 2
20.e even 4 2 3850.2.c.q 4
28.d even 2 1 1078.2.a.w 2
28.f even 6 2 1078.2.e.n 4
28.g odd 6 2 1078.2.e.q 4
44.c even 2 1 1694.2.a.l 2
84.h odd 2 1 9702.2.a.cu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 4.b odd 2 1
1078.2.a.w 2 28.d even 2 1
1078.2.e.n 4 28.f even 6 2
1078.2.e.q 4 28.g odd 6 2
1232.2.a.p 2 1.a even 1 1 trivial
1386.2.a.m 2 12.b even 2 1
1694.2.a.l 2 44.c even 2 1
3850.2.a.bj 2 20.d odd 2 1
3850.2.c.q 4 20.e even 4 2
4928.2.a.bk 2 8.b even 2 1
4928.2.a.bt 2 8.d odd 2 1
8624.2.a.bf 2 7.b odd 2 1
9702.2.a.cu 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1232))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -4 + 2 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( 20 - 10 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( ( 2 + T )^{2} \)
$37$ \( -76 + 4 T + T^{2} \)
$41$ \( -16 - 4 T + T^{2} \)
$43$ \( 16 - 12 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( 20 + 10 T + T^{2} \)
$61$ \( 4 + 6 T + T^{2} \)
$67$ \( -176 - 4 T + T^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( -64 - 8 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -124 - 2 T + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 44 - 16 T + T^{2} \)
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