Properties

Label 6300.2.a.f
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 2100)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{7} + O(q^{10}) \) \( q - q^{7} - q^{11} - 4q^{13} + 2q^{17} - 4q^{19} + 7q^{23} + 9q^{29} - 2q^{31} - q^{37} - 8q^{41} + 9q^{43} + 4q^{47} + q^{49} + 6q^{53} - 4q^{59} + 4q^{61} - 9q^{67} - 5q^{71} - 10q^{73} + q^{77} - 15q^{79} - 6q^{83} - 8q^{89} + 4q^{91} - 10q^{97} + O(q^{100}) \)

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.f 1
3.b odd 2 1 2100.2.a.m yes 1
5.b even 2 1 6300.2.a.x 1
5.c odd 4 2 6300.2.k.h 2
12.b even 2 1 8400.2.a.x 1
15.d odd 2 1 2100.2.a.g 1
15.e even 4 2 2100.2.k.f 2
60.h even 2 1 8400.2.a.bv 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.g 1 15.d odd 2 1
2100.2.a.m yes 1 3.b odd 2 1
2100.2.k.f 2 15.e even 4 2
6300.2.a.f 1 1.a even 1 1 trivial
6300.2.a.x 1 5.b even 2 1
6300.2.k.h 2 5.c odd 4 2
8400.2.a.x 1 12.b even 2 1
8400.2.a.bv 1 60.h even 2 1

Hecke kernels

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

\( T_{11} + 1 \)
\( T_{13} + 4 \)
\( T_{17} - 2 \)
\( T_{37} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 + T \)
$11$ \( 1 + T + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 7 T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 + 2 T + 31 T^{2} \)
$37$ \( 1 + T + 37 T^{2} \)
$41$ \( 1 + 8 T + 41 T^{2} \)
$43$ \( 1 - 9 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 - 4 T + 61 T^{2} \)
$67$ \( 1 + 9 T + 67 T^{2} \)
$71$ \( 1 + 5 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 + 15 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 + 8 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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