Properties

Label 6800.2.a.n
Level $6800$
Weight $2$
Character orbit 6800.a
Self dual yes
Analytic conductor $54.298$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.2982733745\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{7} - 3 q^{9} + 2 q^{13} - q^{17} + 4 q^{19} + 4 q^{23} + 6 q^{29} - 4 q^{31} + 2 q^{37} - 6 q^{41} + 4 q^{43} + 9 q^{49} - 6 q^{53} + 12 q^{59} - 10 q^{61} - 12 q^{63} + 4 q^{67} + 4 q^{71} + 6 q^{73} - 12 q^{79} + 9 q^{81} - 4 q^{83} + 10 q^{89} + 8 q^{91} - 2 q^{97}+O(q^{100}) \) 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 0 0 0 0 4.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(17\) \(1\)

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 6800.2.a.n 1
4.b odd 2 1 425.2.a.d 1
5.b even 2 1 272.2.a.b 1
12.b even 2 1 3825.2.a.d 1
15.d odd 2 1 2448.2.a.o 1
20.d odd 2 1 17.2.a.a 1
20.e even 4 2 425.2.b.b 2
40.e odd 2 1 1088.2.a.i 1
40.f even 2 1 1088.2.a.h 1
60.h even 2 1 153.2.a.c 1
68.d odd 2 1 7225.2.a.g 1
85.c even 2 1 4624.2.a.d 1
120.i odd 2 1 9792.2.a.i 1
120.m even 2 1 9792.2.a.n 1
140.c even 2 1 833.2.a.a 1
140.p odd 6 2 833.2.e.b 2
140.s even 6 2 833.2.e.a 2
220.g even 2 1 2057.2.a.e 1
260.g odd 2 1 2873.2.a.c 1
340.d odd 2 1 289.2.a.a 1
340.n odd 4 2 289.2.b.a 2
340.ba odd 8 4 289.2.c.a 4
340.bg even 16 8 289.2.d.d 8
380.d even 2 1 6137.2.a.b 1
420.o odd 2 1 7497.2.a.l 1
460.g even 2 1 8993.2.a.a 1
1020.b even 2 1 2601.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 20.d odd 2 1
153.2.a.c 1 60.h even 2 1
272.2.a.b 1 5.b even 2 1
289.2.a.a 1 340.d odd 2 1
289.2.b.a 2 340.n odd 4 2
289.2.c.a 4 340.ba odd 8 4
289.2.d.d 8 340.bg even 16 8
425.2.a.d 1 4.b odd 2 1
425.2.b.b 2 20.e even 4 2
833.2.a.a 1 140.c even 2 1
833.2.e.a 2 140.s even 6 2
833.2.e.b 2 140.p odd 6 2
1088.2.a.h 1 40.f even 2 1
1088.2.a.i 1 40.e odd 2 1
2057.2.a.e 1 220.g even 2 1
2448.2.a.o 1 15.d odd 2 1
2601.2.a.g 1 1020.b even 2 1
2873.2.a.c 1 260.g odd 2 1
3825.2.a.d 1 12.b even 2 1
4624.2.a.d 1 85.c even 2 1
6137.2.a.b 1 380.d even 2 1
6800.2.a.n 1 1.a even 1 1 trivial
7225.2.a.g 1 68.d odd 2 1
7497.2.a.l 1 420.o odd 2 1
8993.2.a.a 1 460.g even 2 1
9792.2.a.i 1 120.i odd 2 1
9792.2.a.n 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) 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
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T - 4 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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