Properties

Label 5070.2.a.i
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 5q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 5q^{7} - q^{8} + q^{9} + q^{10} - 3q^{11} + q^{12} + 5q^{14} - q^{15} + q^{16} - 8q^{17} - q^{18} - 5q^{19} - q^{20} - 5q^{21} + 3q^{22} - 4q^{23} - q^{24} + q^{25} + q^{27} - 5q^{28} - 4q^{29} + q^{30} - 2q^{31} - q^{32} - 3q^{33} + 8q^{34} + 5q^{35} + q^{36} - 7q^{37} + 5q^{38} + q^{40} + 6q^{41} + 5q^{42} + 6q^{43} - 3q^{44} - q^{45} + 4q^{46} - 3q^{47} + q^{48} + 18q^{49} - q^{50} - 8q^{51} + q^{53} - q^{54} + 3q^{55} + 5q^{56} - 5q^{57} + 4q^{58} + 12q^{59} - q^{60} + 2q^{61} + 2q^{62} - 5q^{63} + q^{64} + 3q^{66} + 8q^{67} - 8q^{68} - 4q^{69} - 5q^{70} + 2q^{71} - q^{72} + 7q^{74} + q^{75} - 5q^{76} + 15q^{77} - 2q^{79} - q^{80} + q^{81} - 6q^{82} + 8q^{83} - 5q^{84} + 8q^{85} - 6q^{86} - 4q^{87} + 3q^{88} - 11q^{89} + q^{90} - 4q^{92} - 2q^{93} + 3q^{94} + 5q^{95} - q^{96} - 18q^{98} - 3q^{99} + O(q^{100}) \)

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 −1.00000 −1.00000 −5.00000 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 5070.2.a.i 1
13.b even 2 1 5070.2.a.x 1
13.c even 3 2 390.2.i.d 2
13.d odd 4 2 5070.2.b.l 2
39.i odd 6 2 1170.2.i.g 2
65.n even 6 2 1950.2.i.i 2
65.q odd 12 4 1950.2.z.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 13.c even 3 2
1170.2.i.g 2 39.i odd 6 2
1950.2.i.i 2 65.n even 6 2
1950.2.z.j 4 65.q odd 12 4
5070.2.a.i 1 1.a even 1 1 trivial
5070.2.a.x 1 13.b even 2 1
5070.2.b.l 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7} + 5 \)
\( T_{11} + 3 \)
\( T_{17} + 8 \)
\( T_{31} + 2 \)

Hecke characteristic polynomials

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