Properties

Label 6300.2.k.c
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 140)
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.

\(f(q)\) \(=\) \( q -i q^{7} +O(q^{10})\) \( q -i q^{7} -3 q^{11} -i q^{13} -3 i q^{17} -2 q^{19} + 6 i q^{23} -9 q^{29} + 8 q^{31} + 10 i q^{37} + 2 i q^{43} -3 i q^{47} - q^{49} + 12 q^{59} + 8 q^{61} -8 i q^{67} + 14 i q^{73} + 3 i q^{77} -5 q^{79} + 12 i q^{83} + 12 q^{89} - q^{91} -17 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 6q^{11} - 4q^{19} - 18q^{29} + 16q^{31} - 2q^{49} + 24q^{59} + 16q^{61} - 10q^{79} + 24q^{89} - 2q^{91} + O(q^{100}) \)

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.c 2
3.b odd 2 1 700.2.e.c 2
5.b even 2 1 inner 6300.2.k.c 2
5.c odd 4 1 1260.2.a.c 1
5.c odd 4 1 6300.2.a.d 1
12.b even 2 1 2800.2.g.j 2
15.d odd 2 1 700.2.e.c 2
15.e even 4 1 140.2.a.a 1
15.e even 4 1 700.2.a.d 1
20.e even 4 1 5040.2.a.h 1
21.c even 2 1 4900.2.e.l 2
35.f even 4 1 8820.2.a.r 1
60.h even 2 1 2800.2.g.j 2
60.l odd 4 1 560.2.a.c 1
60.l odd 4 1 2800.2.a.y 1
105.g even 2 1 4900.2.e.l 2
105.k odd 4 1 980.2.a.c 1
105.k odd 4 1 4900.2.a.p 1
105.w odd 12 2 980.2.i.h 2
105.x even 12 2 980.2.i.d 2
120.q odd 4 1 2240.2.a.r 1
120.w even 4 1 2240.2.a.g 1
420.w even 4 1 3920.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 15.e even 4 1
560.2.a.c 1 60.l odd 4 1
700.2.a.d 1 15.e even 4 1
700.2.e.c 2 3.b odd 2 1
700.2.e.c 2 15.d odd 2 1
980.2.a.c 1 105.k odd 4 1
980.2.i.d 2 105.x even 12 2
980.2.i.h 2 105.w odd 12 2
1260.2.a.c 1 5.c odd 4 1
2240.2.a.g 1 120.w even 4 1
2240.2.a.r 1 120.q odd 4 1
2800.2.a.y 1 60.l odd 4 1
2800.2.g.j 2 12.b even 2 1
2800.2.g.j 2 60.h even 2 1
3920.2.a.u 1 420.w even 4 1
4900.2.a.p 1 105.k odd 4 1
4900.2.e.l 2 21.c even 2 1
4900.2.e.l 2 105.g even 2 1
5040.2.a.h 1 20.e even 4 1
6300.2.a.d 1 5.c odd 4 1
6300.2.k.c 2 1.a even 1 1 trivial
6300.2.k.c 2 5.b even 2 1 inner
8820.2.a.r 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 \(S_{2}^{\mathrm{new}}(6300, [\chi])\):

\( T_{11} + 3 \)
\( T_{13}^{2} + 1 \)
\( T_{17}^{2} + 9 \)
\( T_{41} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( ( 5 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( 289 + T^{2} \)
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