Properties

Label 2850.2.a.s
Level $2850$
Weight $2$
Character orbit 2850.a
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - 4q^{11} - q^{12} - 4q^{13} + q^{16} + 6q^{17} + q^{18} - q^{19} - 4q^{22} - q^{24} - 4q^{26} - q^{27} - 2q^{29} + q^{32} + 4q^{33} + 6q^{34} + q^{36} - 4q^{37} - q^{38} + 4q^{39} - 12q^{41} + 6q^{43} - 4q^{44} - q^{48} - 7q^{49} - 6q^{51} - 4q^{52} - 14q^{53} - q^{54} + q^{57} - 2q^{58} - 10q^{59} - 6q^{61} + q^{64} + 4q^{66} - 4q^{67} + 6q^{68} + 12q^{71} + q^{72} + 8q^{73} - 4q^{74} - q^{76} + 4q^{78} - 8q^{79} + q^{81} - 12q^{82} + 12q^{83} + 6q^{86} + 2q^{87} - 4q^{88} - 8q^{89} - q^{96} - 10q^{97} - 7q^{98} - 4q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 2850.2.a.s 1
3.b odd 2 1 8550.2.a.j 1
5.b even 2 1 2850.2.a.k 1
5.c odd 4 2 570.2.d.b 2
15.d odd 2 1 8550.2.a.bc 1
15.e even 4 2 1710.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.b 2 5.c odd 4 2
1710.2.d.a 2 15.e even 4 2
2850.2.a.k 1 5.b even 2 1
2850.2.a.s 1 1.a even 1 1 trivial
8550.2.a.j 1 3.b odd 2 1
8550.2.a.bc 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2850))\):

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13} + 4 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 4 + T \)
$17$ \( -6 + T \)
$19$ \( 1 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( T \)
$37$ \( 4 + T \)
$41$ \( 12 + T \)
$43$ \( -6 + T \)
$47$ \( T \)
$53$ \( 14 + T \)
$59$ \( 10 + T \)
$61$ \( 6 + T \)
$67$ \( 4 + T \)
$71$ \( -12 + T \)
$73$ \( -8 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( 8 + T \)
$97$ \( 10 + T \)
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