Properties

Label 8400.2.a.k
Level $8400$
Weight $2$
Character orbit 8400.a
Self dual yes
Analytic conductor $67.074$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{7} + q^{9} + 4 q^{11} - 6 q^{13} - 2 q^{17} + 4 q^{19} + q^{21} + 8 q^{23} - q^{27} - 2 q^{29} - 4 q^{33} + 10 q^{37} + 6 q^{39} - 6 q^{41} - 4 q^{43} + q^{49} + 2 q^{51} - 6 q^{53} - 4 q^{57} - 4 q^{59} + 6 q^{61} - q^{63} + 4 q^{67} - 8 q^{69} - 8 q^{71} - 10 q^{73} - 4 q^{77} + q^{81} - 4 q^{83} + 2 q^{87} - 6 q^{89} + 6 q^{91} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 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
0
0 −1.00000 0 0 0 −1.00000 0 1.00000 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

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 8400.2.a.k 1
4.b odd 2 1 1050.2.a.i 1
5.b even 2 1 336.2.a.d 1
12.b even 2 1 3150.2.a.bo 1
15.d odd 2 1 1008.2.a.j 1
20.d odd 2 1 42.2.a.a 1
20.e even 4 2 1050.2.g.a 2
28.d even 2 1 7350.2.a.f 1
35.c odd 2 1 2352.2.a.l 1
35.i odd 6 2 2352.2.q.n 2
35.j even 6 2 2352.2.q.i 2
40.e odd 2 1 1344.2.a.q 1
40.f even 2 1 1344.2.a.i 1
60.h even 2 1 126.2.a.a 1
60.l odd 4 2 3150.2.g.r 2
80.k odd 4 2 5376.2.c.bc 2
80.q even 4 2 5376.2.c.e 2
105.g even 2 1 7056.2.a.k 1
120.i odd 2 1 4032.2.a.m 1
120.m even 2 1 4032.2.a.e 1
140.c even 2 1 294.2.a.g 1
140.p odd 6 2 294.2.e.c 2
140.s even 6 2 294.2.e.a 2
180.n even 6 2 1134.2.f.j 2
180.p odd 6 2 1134.2.f.g 2
220.g even 2 1 5082.2.a.d 1
260.g odd 2 1 7098.2.a.f 1
280.c odd 2 1 9408.2.a.bw 1
280.n even 2 1 9408.2.a.n 1
420.o odd 2 1 882.2.a.b 1
420.ba even 6 2 882.2.g.h 2
420.be odd 6 2 882.2.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 20.d odd 2 1
126.2.a.a 1 60.h even 2 1
294.2.a.g 1 140.c even 2 1
294.2.e.a 2 140.s even 6 2
294.2.e.c 2 140.p odd 6 2
336.2.a.d 1 5.b even 2 1
882.2.a.b 1 420.o odd 2 1
882.2.g.h 2 420.ba even 6 2
882.2.g.j 2 420.be odd 6 2
1008.2.a.j 1 15.d odd 2 1
1050.2.a.i 1 4.b odd 2 1
1050.2.g.a 2 20.e even 4 2
1134.2.f.g 2 180.p odd 6 2
1134.2.f.j 2 180.n even 6 2
1344.2.a.i 1 40.f even 2 1
1344.2.a.q 1 40.e odd 2 1
2352.2.a.l 1 35.c odd 2 1
2352.2.q.i 2 35.j even 6 2
2352.2.q.n 2 35.i odd 6 2
3150.2.a.bo 1 12.b even 2 1
3150.2.g.r 2 60.l odd 4 2
4032.2.a.e 1 120.m even 2 1
4032.2.a.m 1 120.i odd 2 1
5082.2.a.d 1 220.g even 2 1
5376.2.c.e 2 80.q even 4 2
5376.2.c.bc 2 80.k odd 4 2
7056.2.a.k 1 105.g even 2 1
7098.2.a.f 1 260.g odd 2 1
7350.2.a.f 1 28.d even 2 1
8400.2.a.k 1 1.a even 1 1 trivial
9408.2.a.n 1 280.n even 2 1
9408.2.a.bw 1 280.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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