Properties

Label 6300.2.k.k
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 2100)
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.

\(f(q)\) \(=\) \( q -i q^{7} +O(q^{10})\) \( q -i q^{7} + q^{11} + 2 i q^{13} -6 q^{19} + i q^{23} + q^{29} -2 q^{31} -7 i q^{37} + 8 q^{41} + i q^{43} + 2 i q^{47} - q^{49} -14 i q^{53} + 10 q^{59} -3 i q^{67} + 9 q^{71} -i q^{77} - q^{79} + 2 i q^{83} + 2 q^{89} + 2 q^{91} -10 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{11} - 12q^{19} + 2q^{29} - 4q^{31} + 16q^{41} - 2q^{49} + 20q^{59} + 18q^{71} - 2q^{79} + 4q^{89} + 4q^{91} + O(q^{100}) \)

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.k 2
3.b odd 2 1 2100.2.k.d 2
5.b even 2 1 inner 6300.2.k.k 2
5.c odd 4 1 6300.2.a.k 1
5.c odd 4 1 6300.2.a.z 1
15.d odd 2 1 2100.2.k.d 2
15.e even 4 1 2100.2.a.c 1
15.e even 4 1 2100.2.a.p yes 1
60.l odd 4 1 8400.2.a.h 1
60.l odd 4 1 8400.2.a.cp 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.c 1 15.e even 4 1
2100.2.a.p yes 1 15.e even 4 1
2100.2.k.d 2 3.b odd 2 1
2100.2.k.d 2 15.d odd 2 1
6300.2.a.k 1 5.c odd 4 1
6300.2.a.z 1 5.c odd 4 1
6300.2.k.k 2 1.a even 1 1 trivial
6300.2.k.k 2 5.b even 2 1 inner
8400.2.a.h 1 60.l odd 4 1
8400.2.a.cp 1 60.l odd 4 1

Hecke kernels

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

\( T_{11} - 1 \)
\( T_{13}^{2} + 4 \)
\( T_{17} \)
\( T_{41} - 8 \)