Properties

Label 2496.2.a.bi
Level $2496$
Weight $2$
Character orbit 2496.a
Self dual yes
Analytic conductor $19.931$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} - \beta q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta q^{5} - \beta q^{7} + q^{9} - 2 q^{11} + q^{13} + \beta q^{15} + (2 \beta + 2) q^{17} - \beta q^{19} - \beta q^{21} + 4 q^{23} + 3 q^{25} + q^{27} - 2 q^{29} + ( - \beta + 4) q^{31} - 2 q^{33} - 8 q^{35} + (2 \beta + 2) q^{37} + q^{39} + ( - \beta + 8) q^{41} + ( - 2 \beta + 4) q^{43} + \beta q^{45} + (2 \beta + 6) q^{47} + q^{49} + (2 \beta + 2) q^{51} + 2 q^{53} - 2 \beta q^{55} - \beta q^{57} + (2 \beta + 2) q^{59} + ( - 4 \beta - 2) q^{61} - \beta q^{63} + \beta q^{65} + (\beta + 4) q^{67} + 4 q^{69} - 2 q^{71} + ( - 2 \beta + 6) q^{73} + 3 q^{75} + 2 \beta q^{77} + 4 \beta q^{79} + q^{81} + (2 \beta - 2) q^{83} + (2 \beta + 16) q^{85} - 2 q^{87} + (\beta + 12) q^{89} - \beta q^{91} + ( - \beta + 4) q^{93} - 8 q^{95} + (2 \beta - 2) q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{17} + 8 q^{23} + 6 q^{25} + 2 q^{27} - 4 q^{29} + 8 q^{31} - 4 q^{33} - 16 q^{35} + 4 q^{37} + 2 q^{39} + 16 q^{41} + 8 q^{43} + 12 q^{47} + 2 q^{49} + 4 q^{51} + 4 q^{53} + 4 q^{59} - 4 q^{61} + 8 q^{67} + 8 q^{69} - 4 q^{71} + 12 q^{73} + 6 q^{75} + 2 q^{81} - 4 q^{83} + 32 q^{85} - 4 q^{87} + 24 q^{89} + 8 q^{93} - 16 q^{95} - 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
0 1.00000 0 −2.82843 0 2.82843 0 1.00000 0
1.2 0 1.00000 0 2.82843 0 −2.82843 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

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 2496.2.a.bi 2
3.b odd 2 1 7488.2.a.co 2
4.b odd 2 1 2496.2.a.bf 2
8.b even 2 1 624.2.a.k 2
8.d odd 2 1 39.2.a.b 2
12.b even 2 1 7488.2.a.cl 2
24.f even 2 1 117.2.a.c 2
24.h odd 2 1 1872.2.a.w 2
40.e odd 2 1 975.2.a.l 2
40.k even 4 2 975.2.c.h 4
56.e even 2 1 1911.2.a.h 2
72.l even 6 2 1053.2.e.e 4
72.p odd 6 2 1053.2.e.m 4
88.g even 2 1 4719.2.a.p 2
104.e even 2 1 8112.2.a.bm 2
104.h odd 2 1 507.2.a.h 2
104.m even 4 2 507.2.b.e 4
104.n odd 6 2 507.2.e.h 4
104.p odd 6 2 507.2.e.d 4
104.u even 12 4 507.2.j.f 8
120.m even 2 1 2925.2.a.v 2
120.q odd 4 2 2925.2.c.u 4
168.e odd 2 1 5733.2.a.u 2
312.h even 2 1 1521.2.a.f 2
312.w odd 4 2 1521.2.b.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 8.d odd 2 1
117.2.a.c 2 24.f even 2 1
507.2.a.h 2 104.h odd 2 1
507.2.b.e 4 104.m even 4 2
507.2.e.d 4 104.p odd 6 2
507.2.e.h 4 104.n odd 6 2
507.2.j.f 8 104.u even 12 4
624.2.a.k 2 8.b even 2 1
975.2.a.l 2 40.e odd 2 1
975.2.c.h 4 40.k even 4 2
1053.2.e.e 4 72.l even 6 2
1053.2.e.m 4 72.p odd 6 2
1521.2.a.f 2 312.h even 2 1
1521.2.b.j 4 312.w odd 4 2
1872.2.a.w 2 24.h odd 2 1
1911.2.a.h 2 56.e even 2 1
2496.2.a.bf 2 4.b odd 2 1
2496.2.a.bi 2 1.a even 1 1 trivial
2925.2.a.v 2 120.m even 2 1
2925.2.c.u 4 120.q odd 4 2
4719.2.a.p 2 88.g even 2 1
5733.2.a.u 2 168.e odd 2 1
7488.2.a.cl 2 12.b even 2 1
7488.2.a.co 2 3.b odd 2 1
8112.2.a.bm 2 104.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2496))\):

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{19}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
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