Properties

Label 1232.2.a.m
Level $1232$
Weight $2$
Character orbit 1232.a
Self dual yes
Analytic conductor $9.838$
Analytic rank $1$
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: \(1\)
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 77)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{3} -2 q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{3} -2 q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} + q^{11} + ( 1 - \beta ) q^{13} + ( 2 + 2 \beta ) q^{15} + ( -1 + \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( 2 + 2 \beta ) q^{23} - q^{25} + ( -10 - 2 \beta ) q^{27} + ( 4 - 2 \beta ) q^{29} + ( 5 + \beta ) q^{31} + ( -1 - \beta ) q^{33} + 2 q^{35} + ( -4 + 2 \beta ) q^{37} + 4 q^{39} + ( -9 + \beta ) q^{41} -8 q^{43} + ( -6 - 4 \beta ) q^{45} + ( -5 - \beta ) q^{47} + q^{49} -4 q^{51} + ( 4 + 2 \beta ) q^{53} -2 q^{55} -8 q^{57} + ( -1 - \beta ) q^{59} + ( -5 + \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + ( -2 + 2 \beta ) q^{65} + ( -10 + 2 \beta ) q^{67} + ( -12 - 4 \beta ) q^{69} + ( 6 - 2 \beta ) q^{71} + ( -3 - \beta ) q^{73} + ( 1 + \beta ) q^{75} - q^{77} -4 \beta q^{79} + ( 11 + 6 \beta ) q^{81} + ( -2 - 6 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + ( 6 - 2 \beta ) q^{87} + 2 q^{89} + ( -1 + \beta ) q^{91} + ( -10 - 6 \beta ) q^{93} + ( 4 - 4 \beta ) q^{95} + ( 4 - 6 \beta ) q^{97} + ( 3 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 2 q^{13} + 4 q^{15} - 2 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{23} - 2 q^{25} - 20 q^{27} + 8 q^{29} + 10 q^{31} - 2 q^{33} + 4 q^{35} - 8 q^{37} + 8 q^{39} - 18 q^{41} - 16 q^{43} - 12 q^{45} - 10 q^{47} + 2 q^{49} - 8 q^{51} + 8 q^{53} - 4 q^{55} - 16 q^{57} - 2 q^{59} - 10 q^{61} - 6 q^{63} - 4 q^{65} - 20 q^{67} - 24 q^{69} + 12 q^{71} - 6 q^{73} + 2 q^{75} - 2 q^{77} + 22 q^{81} - 4 q^{83} + 4 q^{85} + 12 q^{87} + 4 q^{89} - 2 q^{91} - 20 q^{93} + 8 q^{95} + 8 q^{97} + 6 q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
0 −3.23607 0 −2.00000 0 −1.00000 0 7.47214 0
1.2 0 1.23607 0 −2.00000 0 −1.00000 0 −1.47214 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.m 2
4.b odd 2 1 77.2.a.d 2
7.b odd 2 1 8624.2.a.ce 2
8.b even 2 1 4928.2.a.bv 2
8.d odd 2 1 4928.2.a.bm 2
12.b even 2 1 693.2.a.h 2
20.d odd 2 1 1925.2.a.r 2
20.e even 4 2 1925.2.b.h 4
28.d even 2 1 539.2.a.f 2
28.f even 6 2 539.2.e.j 4
28.g odd 6 2 539.2.e.i 4
44.c even 2 1 847.2.a.f 2
44.g even 10 2 847.2.f.b 4
44.g even 10 2 847.2.f.m 4
44.h odd 10 2 847.2.f.a 4
44.h odd 10 2 847.2.f.n 4
84.h odd 2 1 4851.2.a.y 2
132.d odd 2 1 7623.2.a.bl 2
308.g odd 2 1 5929.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 4.b odd 2 1
539.2.a.f 2 28.d even 2 1
539.2.e.i 4 28.g odd 6 2
539.2.e.j 4 28.f even 6 2
693.2.a.h 2 12.b even 2 1
847.2.a.f 2 44.c even 2 1
847.2.f.a 4 44.h odd 10 2
847.2.f.b 4 44.g even 10 2
847.2.f.m 4 44.g even 10 2
847.2.f.n 4 44.h odd 10 2
1232.2.a.m 2 1.a even 1 1 trivial
1925.2.a.r 2 20.d odd 2 1
1925.2.b.h 4 20.e even 4 2
4851.2.a.y 2 84.h odd 2 1
4928.2.a.bm 2 8.d odd 2 1
4928.2.a.bv 2 8.b even 2 1
5929.2.a.m 2 308.g odd 2 1
7623.2.a.bl 2 132.d odd 2 1
8624.2.a.ce 2 7.b odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 + 2 T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -4 + 2 T + T^{2} \)
$19$ \( -16 + 4 T + T^{2} \)
$23$ \( -16 - 4 T + T^{2} \)
$29$ \( -4 - 8 T + T^{2} \)
$31$ \( 20 - 10 T + T^{2} \)
$37$ \( -4 + 8 T + T^{2} \)
$41$ \( 76 + 18 T + T^{2} \)
$43$ \( ( 8 + T )^{2} \)
$47$ \( 20 + 10 T + T^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( -4 + 2 T + T^{2} \)
$61$ \( 20 + 10 T + T^{2} \)
$67$ \( 80 + 20 T + T^{2} \)
$71$ \( 16 - 12 T + T^{2} \)
$73$ \( 4 + 6 T + T^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -176 + 4 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( -164 - 8 T + T^{2} \)
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