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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1232))\):

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

Hecke characteristic polynomials

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