Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{7} + \beta q^{11} - 2 \beta q^{17} + 3 \beta q^{23} - 3 \beta q^{29} - 10 q^{31} - 11 q^{37} - 4 \beta q^{41} - q^{43} + 4 \beta q^{47} + q^{49} - 4 \beta q^{53} + 4 \beta q^{59} - 8 q^{61} + 3 q^{67} - \beta q^{71} + 10 q^{73} + \beta q^{77} - 11 q^{79} - 2 \beta q^{83} + 4 \beta q^{89} - 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} - 20 q^{31} - 22 q^{37} - 2 q^{43} + 2 q^{49} - 16 q^{61} + 6 q^{67} + 20 q^{73} - 22 q^{79} - 4 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
0 0 0 0 0 1.00000 0 0 0
1.2 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6300.2.a.bi yes 2
3.b odd 2 1 inner 6300.2.a.bi yes 2
5.b even 2 1 6300.2.a.bh 2
5.c odd 4 2 6300.2.k.t 4
15.d odd 2 1 6300.2.a.bh 2
15.e even 4 2 6300.2.k.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6300.2.a.bh 2 5.b even 2 1
6300.2.a.bh 2 15.d odd 2 1
6300.2.a.bi yes 2 1.a even 1 1 trivial
6300.2.a.bi yes 2 3.b odd 2 1 inner
6300.2.k.t 4 5.c odd 4 2
6300.2.k.t 4 15.e even 4 2

Hecke kernels

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

\( T_{11}^{2} - 7 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} - 28 \) Copy content Toggle raw display
\( T_{37} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 28 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 63 \) Copy content Toggle raw display
$29$ \( T^{2} - 63 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( (T + 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 112 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 112 \) Copy content Toggle raw display
$53$ \( T^{2} - 112 \) Copy content Toggle raw display
$59$ \( T^{2} - 112 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T - 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 7 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 112 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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