Properties

Label 8450.2.a.w
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + q^{7} + q^{8} + q^{9} + 3 q^{11} + 2 q^{12} + q^{14} + q^{16} + 6 q^{17} + q^{18} + 5 q^{19} + 2 q^{21} + 3 q^{22} + 2 q^{24} - 4 q^{27} + q^{28} - 4 q^{31} + q^{32} + 6 q^{33} + 6 q^{34} + q^{36} - 11 q^{37} + 5 q^{38} + 6 q^{41} + 2 q^{42} - 2 q^{43} + 3 q^{44} + 3 q^{47} + 2 q^{48} - 6 q^{49} + 12 q^{51} + 9 q^{53} - 4 q^{54} + q^{56} + 10 q^{57} + 8 q^{61} - 4 q^{62} + q^{63} + q^{64} + 6 q^{66} + 16 q^{67} + 6 q^{68} + 6 q^{71} + q^{72} - 14 q^{73} - 11 q^{74} + 5 q^{76} + 3 q^{77} - 16 q^{79} - 11 q^{81} + 6 q^{82} + 6 q^{83} + 2 q^{84} - 2 q^{86} + 3 q^{88} + 9 q^{89} - 8 q^{93} + 3 q^{94} + 2 q^{96} + 10 q^{97} - 6 q^{98} + 3 q^{99}+O(q^{100}) \) 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
1.00000 2.00000 1.00000 0 2.00000 1.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 8450.2.a.w 1
5.b even 2 1 1690.2.a.a 1
13.b even 2 1 8450.2.a.k 1
13.c even 3 2 650.2.e.a 2
65.d even 2 1 1690.2.a.g 1
65.g odd 4 2 1690.2.d.a 2
65.l even 6 2 1690.2.e.e 2
65.n even 6 2 130.2.e.b 2
65.q odd 12 4 650.2.o.b 4
65.s odd 12 4 1690.2.l.i 4
195.x odd 6 2 1170.2.i.f 2
260.v odd 6 2 1040.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 65.n even 6 2
650.2.e.a 2 13.c even 3 2
650.2.o.b 4 65.q odd 12 4
1040.2.q.c 2 260.v odd 6 2
1170.2.i.f 2 195.x odd 6 2
1690.2.a.a 1 5.b even 2 1
1690.2.a.g 1 65.d even 2 1
1690.2.d.a 2 65.g odd 4 2
1690.2.e.e 2 65.l even 6 2
1690.2.l.i 4 65.s odd 12 4
8450.2.a.k 1 13.b even 2 1
8450.2.a.w 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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