Properties

Label 6300.2.a.p
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 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{7} + O(q^{10}) \) \( q - q^{7} + 6q^{11} - 2q^{13} - 4q^{19} - 6q^{23} - 6q^{29} + 8q^{31} - 2q^{37} - 12q^{41} + 4q^{43} + 12q^{47} + q^{49} - 6q^{53} - 10q^{61} - 8q^{67} - 6q^{71} + 10q^{73} - 6q^{77} - 4q^{79} - 12q^{83} - 12q^{89} + 2q^{91} + 10q^{97} + O(q^{100}) \)

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:

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.p 1
3.b odd 2 1 2100.2.a.a 1
5.b even 2 1 252.2.a.b 1
5.c odd 4 2 6300.2.k.r 2
12.b even 2 1 8400.2.a.ct 1
15.d odd 2 1 84.2.a.b 1
15.e even 4 2 2100.2.k.a 2
20.d odd 2 1 1008.2.a.g 1
35.c odd 2 1 1764.2.a.g 1
35.i odd 6 2 1764.2.k.d 2
35.j even 6 2 1764.2.k.e 2
40.e odd 2 1 4032.2.a.t 1
40.f even 2 1 4032.2.a.u 1
45.h odd 6 2 2268.2.j.i 2
45.j even 6 2 2268.2.j.f 2
60.h even 2 1 336.2.a.b 1
105.g even 2 1 588.2.a.c 1
105.o odd 6 2 588.2.i.c 2
105.p even 6 2 588.2.i.f 2
120.i odd 2 1 1344.2.a.f 1
120.m even 2 1 1344.2.a.o 1
140.c even 2 1 7056.2.a.x 1
240.t even 4 2 5376.2.c.x 2
240.bm odd 4 2 5376.2.c.i 2
420.o odd 2 1 2352.2.a.s 1
420.ba even 6 2 2352.2.q.s 2
420.be odd 6 2 2352.2.q.g 2
840.b odd 2 1 9408.2.a.r 1
840.u even 2 1 9408.2.a.co 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 15.d odd 2 1
252.2.a.b 1 5.b even 2 1
336.2.a.b 1 60.h even 2 1
588.2.a.c 1 105.g even 2 1
588.2.i.c 2 105.o odd 6 2
588.2.i.f 2 105.p even 6 2
1008.2.a.g 1 20.d odd 2 1
1344.2.a.f 1 120.i odd 2 1
1344.2.a.o 1 120.m even 2 1
1764.2.a.g 1 35.c odd 2 1
1764.2.k.d 2 35.i odd 6 2
1764.2.k.e 2 35.j even 6 2
2100.2.a.a 1 3.b odd 2 1
2100.2.k.a 2 15.e even 4 2
2268.2.j.f 2 45.j even 6 2
2268.2.j.i 2 45.h odd 6 2
2352.2.a.s 1 420.o odd 2 1
2352.2.q.g 2 420.be odd 6 2
2352.2.q.s 2 420.ba even 6 2
4032.2.a.t 1 40.e odd 2 1
4032.2.a.u 1 40.f even 2 1
5376.2.c.i 2 240.bm odd 4 2
5376.2.c.x 2 240.t even 4 2
6300.2.a.p 1 1.a even 1 1 trivial
6300.2.k.r 2 5.c odd 4 2
7056.2.a.x 1 140.c even 2 1
8400.2.a.ct 1 12.b even 2 1
9408.2.a.r 1 840.b odd 2 1
9408.2.a.co 1 840.u even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

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

\( T_{11} - 6 \)
\( T_{13} + 2 \)
\( T_{17} \)
\( T_{37} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 + T \)
$11$ \( 1 - 6 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 6 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 12 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 12 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 6 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 + 4 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 12 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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