Properties

Label 4050.2.a.bg
Level $4050$
Weight $2$
Character orbit 4050.a
Self dual yes
Analytic conductor $32.339$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.3394128186\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} - 4q^{13} + 2q^{14} + q^{16} - 6q^{17} - 7q^{19} - 4q^{26} + 2q^{28} - 6q^{29} - 10q^{31} + q^{32} - 6q^{34} + 2q^{37} - 7q^{38} + 9q^{41} - q^{43} + 6q^{47} - 3q^{49} - 4q^{52} - 12q^{53} + 2q^{56} - 6q^{58} - 9q^{59} - 4q^{61} - 10q^{62} + q^{64} - 13q^{67} - 6q^{68} + 6q^{71} - q^{73} + 2q^{74} - 7q^{76} + 2q^{79} + 9q^{82} + 9q^{83} - q^{86} + 15q^{89} - 8q^{91} + 6q^{94} + 17q^{97} - 3q^{98} + O(q^{100}) \)

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
1.00000 0 1.00000 0 0 2.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4050.2.a.bg 1
3.b odd 2 1 4050.2.a.o 1
5.b even 2 1 4050.2.a.d 1
5.c odd 4 2 4050.2.c.h 2
9.c even 3 2 450.2.e.c 2
9.d odd 6 2 1350.2.e.h 2
15.d odd 2 1 4050.2.a.u 1
15.e even 4 2 4050.2.c.m 2
45.h odd 6 2 1350.2.e.d 2
45.j even 6 2 450.2.e.f yes 2
45.k odd 12 4 450.2.j.b 4
45.l even 12 4 1350.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.c 2 9.c even 3 2
450.2.e.f yes 2 45.j even 6 2
450.2.j.b 4 45.k odd 12 4
1350.2.e.d 2 45.h odd 6 2
1350.2.e.h 2 9.d odd 6 2
1350.2.j.b 4 45.l even 12 4
4050.2.a.d 1 5.b even 2 1
4050.2.a.o 1 3.b odd 2 1
4050.2.a.u 1 15.d odd 2 1
4050.2.a.bg 1 1.a even 1 1 trivial
4050.2.c.h 2 5.c odd 4 2
4050.2.c.m 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\):

\( T_{7} - 2 \)
\( T_{11} \)
\( T_{13} + 4 \)
\( T_{17} + 6 \)
\( T_{23} \)
\( T_{41} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( 4 + T \)
$17$ \( 6 + T \)
$19$ \( 7 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( -2 + T \)
$41$ \( -9 + T \)
$43$ \( 1 + T \)
$47$ \( -6 + T \)
$53$ \( 12 + T \)
$59$ \( 9 + T \)
$61$ \( 4 + T \)
$67$ \( 13 + T \)
$71$ \( -6 + T \)
$73$ \( 1 + T \)
$79$ \( -2 + T \)
$83$ \( -9 + T \)
$89$ \( -15 + T \)
$97$ \( -17 + T \)
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