Properties

Label 6300.2.a.a
Level $6300$
Weight $2$
Character orbit 6300.a
Self dual yes
Analytic conductor $50.306$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.3057532734\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{7} + O(q^{10}) \) \( q - q^{7} - 6q^{11} + 4q^{13} + 6q^{17} + 2q^{19} - 6q^{29} - 10q^{31} - 2q^{37} + 6q^{41} + 4q^{43} + q^{49} - 12q^{53} + 14q^{61} + 4q^{67} - 6q^{71} + 4q^{73} + 6q^{77} - 16q^{79} - 12q^{83} - 6q^{89} - 4q^{91} + 16q^{97} + O(q^{100}) \)

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
0
0 0 0 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 6300.2.a.a 1
3.b odd 2 1 2100.2.a.d 1
5.b even 2 1 1260.2.a.i 1
5.c odd 4 2 6300.2.k.a 2
12.b even 2 1 8400.2.a.cj 1
15.d odd 2 1 420.2.a.c 1
15.e even 4 2 2100.2.k.j 2
20.d odd 2 1 5040.2.a.bc 1
35.c odd 2 1 8820.2.a.b 1
60.h even 2 1 1680.2.a.a 1
105.g even 2 1 2940.2.a.f 1
105.o odd 6 2 2940.2.q.e 2
105.p even 6 2 2940.2.q.i 2
120.i odd 2 1 6720.2.a.x 1
120.m even 2 1 6720.2.a.ch 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 15.d odd 2 1
1260.2.a.i 1 5.b even 2 1
1680.2.a.a 1 60.h even 2 1
2100.2.a.d 1 3.b odd 2 1
2100.2.k.j 2 15.e even 4 2
2940.2.a.f 1 105.g even 2 1
2940.2.q.e 2 105.o odd 6 2
2940.2.q.i 2 105.p even 6 2
5040.2.a.bc 1 20.d odd 2 1
6300.2.a.a 1 1.a even 1 1 trivial
6300.2.k.a 2 5.c odd 4 2
6720.2.a.x 1 120.i odd 2 1
6720.2.a.ch 1 120.m even 2 1
8400.2.a.cj 1 12.b even 2 1
8820.2.a.b 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6300))\):

\( T_{11} + 6 \)
\( T_{13} - 4 \)
\( T_{17} - 6 \)
\( T_{37} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( 6 + T \)
$13$ \( -4 + T \)
$17$ \( -6 + T \)
$19$ \( -2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( 2 + T \)
$41$ \( -6 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( 12 + T \)
$59$ \( T \)
$61$ \( -14 + T \)
$67$ \( -4 + T \)
$71$ \( 6 + T \)
$73$ \( -4 + T \)
$79$ \( 16 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( -16 + T \)
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