Properties

Label 2496.2.c.f
Level $2496$
Weight $2$
Character orbit 2496.c
Analytic conductor $19.931$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). 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 \( q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + ( - \beta + 3) q^{13} - \beta q^{15} - 2 q^{17} - 3 \beta q^{19} - \beta q^{21} + 4 q^{23} + q^{25} - q^{27} + 10 q^{29} - 5 \beta q^{31} - 4 q^{35} + 4 \beta q^{37} + (\beta - 3) q^{39} - 5 \beta q^{41} - 4 q^{43} + \beta q^{45} + 6 \beta q^{47} + 3 q^{49} + 2 q^{51} + 6 q^{53} + 3 \beta q^{57} + 2 \beta q^{59} - 2 q^{61} + \beta q^{63} + (3 \beta + 4) q^{65} - \beta q^{67} - 4 q^{69} + 2 \beta q^{73} - q^{75} + q^{81} - 2 \beta q^{83} - 2 \beta q^{85} - 10 q^{87} + 3 \beta q^{89} + (3 \beta + 4) q^{91} + 5 \beta q^{93} + 12 q^{95} + 6 \beta q^{97} +O(q^{100}) \) 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 \( 2 q - 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 2 q^{27} + 20 q^{29} - 8 q^{35} - 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} + 12 q^{53} - 4 q^{61} + 8 q^{65} - 8 q^{69} - 2 q^{75} + 2 q^{81} - 20 q^{87} + 8 q^{91} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.00000i
1.00000i
0 −1.00000 0 2.00000i 0 2.00000i 0 1.00000 0
961.2 0 −1.00000 0 2.00000i 0 2.00000i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.2.c.f 2
4.b odd 2 1 2496.2.c.m 2
8.b even 2 1 78.2.b.a 2
8.d odd 2 1 624.2.c.a 2
13.b even 2 1 inner 2496.2.c.f 2
24.f even 2 1 1872.2.c.b 2
24.h odd 2 1 234.2.b.a 2
40.f even 2 1 1950.2.b.c 2
40.i odd 4 1 1950.2.f.d 2
40.i odd 4 1 1950.2.f.g 2
52.b odd 2 1 2496.2.c.m 2
56.h odd 2 1 3822.2.c.d 2
104.e even 2 1 78.2.b.a 2
104.h odd 2 1 624.2.c.a 2
104.j odd 4 1 1014.2.a.b 1
104.j odd 4 1 1014.2.a.g 1
104.m even 4 1 8112.2.a.g 1
104.m even 4 1 8112.2.a.j 1
104.r even 6 2 1014.2.i.c 4
104.s even 6 2 1014.2.i.c 4
104.x odd 12 2 1014.2.e.b 2
104.x odd 12 2 1014.2.e.e 2
312.b odd 2 1 234.2.b.a 2
312.h even 2 1 1872.2.c.b 2
312.y even 4 1 3042.2.a.c 1
312.y even 4 1 3042.2.a.n 1
520.p even 2 1 1950.2.b.c 2
520.bg odd 4 1 1950.2.f.d 2
520.bg odd 4 1 1950.2.f.g 2
728.l odd 2 1 3822.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 8.b even 2 1
78.2.b.a 2 104.e even 2 1
234.2.b.a 2 24.h odd 2 1
234.2.b.a 2 312.b odd 2 1
624.2.c.a 2 8.d odd 2 1
624.2.c.a 2 104.h odd 2 1
1014.2.a.b 1 104.j odd 4 1
1014.2.a.g 1 104.j odd 4 1
1014.2.e.b 2 104.x odd 12 2
1014.2.e.e 2 104.x odd 12 2
1014.2.i.c 4 104.r even 6 2
1014.2.i.c 4 104.s even 6 2
1872.2.c.b 2 24.f even 2 1
1872.2.c.b 2 312.h even 2 1
1950.2.b.c 2 40.f even 2 1
1950.2.b.c 2 520.p even 2 1
1950.2.f.d 2 40.i odd 4 1
1950.2.f.d 2 520.bg odd 4 1
1950.2.f.g 2 40.i odd 4 1
1950.2.f.g 2 520.bg odd 4 1
2496.2.c.f 2 1.a even 1 1 trivial
2496.2.c.f 2 13.b even 2 1 inner
2496.2.c.m 2 4.b odd 2 1
2496.2.c.m 2 52.b odd 2 1
3042.2.a.c 1 312.y even 4 1
3042.2.a.n 1 312.y even 4 1
3822.2.c.d 2 56.h odd 2 1
3822.2.c.d 2 728.l odd 2 1
8112.2.a.g 1 104.m even 4 1
8112.2.a.j 1 104.m even 4 1

Hecke kernels

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

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{23} - 4 \) Copy content Toggle raw display
\( T_{43} + 4 \) 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
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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