Properties

Label 6300.2.k.a
Level $6300$
Weight $2$
Character orbit 6300.k
Analytic conductor $50.306$
Analytic rank $0$
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.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(50.3057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q -i q^{7} +O(q^{10})\) \( q -i q^{7} -6 q^{11} -4 i q^{13} + 6 i q^{17} -2 q^{19} + 6 q^{29} -10 q^{31} -2 i q^{37} + 6 q^{41} -4 i q^{43} - q^{49} + 12 i q^{53} + 14 q^{61} + 4 i q^{67} -6 q^{71} -4 i q^{73} + 6 i q^{77} + 16 q^{79} + 12 i q^{83} + 6 q^{89} -4 q^{91} + 16 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 12q^{11} - 4q^{19} + 12q^{29} - 20q^{31} + 12q^{41} - 2q^{49} + 28q^{61} - 12q^{71} + 32q^{79} + 12q^{89} - 8q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times\).

\(n\) \(2801\) \(3151\) \(3277\) \(3601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6049.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
6049.2 0 0 0 0 0 1.00000i 0 0 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
5.b even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{11} + 6 \)
\( T_{13}^{2} + 16 \)
\( T_{17}^{2} + 36 \)
\( T_{41} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 6 + T )^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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