Properties

Label 1300.2.a.d
Level $1300$
Weight $2$
Character orbit 1300.a
Self dual yes
Analytic conductor $10.381$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3805522628\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{7} - 3 q^{9} - 2 q^{11} + q^{13} - 6 q^{17} - 6 q^{19} - 8 q^{23} + 2 q^{29} + 10 q^{31} + 6 q^{37} - 6 q^{41} - 4 q^{43} + 2 q^{47} - 3 q^{49} - 6 q^{53} - 10 q^{59} - 2 q^{61} - 6 q^{63} - 10 q^{67} + 10 q^{71} - 2 q^{73} - 4 q^{77} - 4 q^{79} + 9 q^{81} + 6 q^{83} - 6 q^{89} + 2 q^{91} - 2 q^{97} + 6 q^{99}+O(q^{100}) \) 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
0
0 0 0 0 0 2.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

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 1300.2.a.d 1
4.b odd 2 1 5200.2.a.q 1
5.b even 2 1 52.2.a.a 1
5.c odd 4 2 1300.2.c.c 2
15.d odd 2 1 468.2.a.b 1
20.d odd 2 1 208.2.a.c 1
35.c odd 2 1 2548.2.a.e 1
35.i odd 6 2 2548.2.j.f 2
35.j even 6 2 2548.2.j.e 2
40.e odd 2 1 832.2.a.f 1
40.f even 2 1 832.2.a.e 1
45.h odd 6 2 4212.2.i.i 2
45.j even 6 2 4212.2.i.d 2
55.d odd 2 1 6292.2.a.g 1
60.h even 2 1 1872.2.a.f 1
65.d even 2 1 676.2.a.c 1
65.g odd 4 2 676.2.d.c 2
65.l even 6 2 676.2.e.b 2
65.n even 6 2 676.2.e.c 2
65.s odd 12 4 676.2.h.c 4
80.k odd 4 2 3328.2.b.e 2
80.q even 4 2 3328.2.b.q 2
120.i odd 2 1 7488.2.a.bn 1
120.m even 2 1 7488.2.a.bw 1
195.e odd 2 1 6084.2.a.m 1
195.n even 4 2 6084.2.b.m 2
260.g odd 2 1 2704.2.a.g 1
260.u even 4 2 2704.2.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 5.b even 2 1
208.2.a.c 1 20.d odd 2 1
468.2.a.b 1 15.d odd 2 1
676.2.a.c 1 65.d even 2 1
676.2.d.c 2 65.g odd 4 2
676.2.e.b 2 65.l even 6 2
676.2.e.c 2 65.n even 6 2
676.2.h.c 4 65.s odd 12 4
832.2.a.e 1 40.f even 2 1
832.2.a.f 1 40.e odd 2 1
1300.2.a.d 1 1.a even 1 1 trivial
1300.2.c.c 2 5.c odd 4 2
1872.2.a.f 1 60.h even 2 1
2548.2.a.e 1 35.c odd 2 1
2548.2.j.e 2 35.j even 6 2
2548.2.j.f 2 35.i odd 6 2
2704.2.a.g 1 260.g odd 2 1
2704.2.f.f 2 260.u even 4 2
3328.2.b.e 2 80.k odd 4 2
3328.2.b.q 2 80.q even 4 2
4212.2.i.d 2 45.j even 6 2
4212.2.i.i 2 45.h odd 6 2
5200.2.a.q 1 4.b odd 2 1
6084.2.a.m 1 195.e odd 2 1
6084.2.b.m 2 195.n even 4 2
6292.2.a.g 1 55.d odd 2 1
7488.2.a.bn 1 120.i odd 2 1
7488.2.a.bw 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1300))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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