Properties

Label 5070.2.a.f
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} - q^{11} - q^{12} + 2q^{14} - q^{15} + q^{16} + 2q^{17} - q^{18} - 6q^{19} + q^{20} + 2q^{21} + q^{22} - 3q^{23} + q^{24} + q^{25} - q^{27} - 2q^{28} - q^{29} + q^{30} + 3q^{31} - q^{32} + q^{33} - 2q^{34} - 2q^{35} + q^{36} + 5q^{37} + 6q^{38} - q^{40} - 10q^{41} - 2q^{42} + 5q^{43} - q^{44} + q^{45} + 3q^{46} - 3q^{47} - q^{48} - 3q^{49} - q^{50} - 2q^{51} + 14q^{53} + q^{54} - q^{55} + 2q^{56} + 6q^{57} + q^{58} + 5q^{59} - q^{60} - 10q^{61} - 3q^{62} - 2q^{63} + q^{64} - q^{66} + 2q^{68} + 3q^{69} + 2q^{70} - 4q^{71} - q^{72} + 2q^{73} - 5q^{74} - q^{75} - 6q^{76} + 2q^{77} + 5q^{79} + q^{80} + q^{81} + 10q^{82} + 6q^{83} + 2q^{84} + 2q^{85} - 5q^{86} + q^{87} + q^{88} - 10q^{89} - q^{90} - 3q^{92} - 3q^{93} + 3q^{94} - 6q^{95} + q^{96} + 10q^{97} + 3q^{98} - q^{99} + O(q^{100}) \)

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 −2.00000 −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.f 1
13.b even 2 1 5070.2.a.o 1
13.d odd 4 2 5070.2.b.g 2
13.e even 6 2 390.2.i.a 2
39.h odd 6 2 1170.2.i.k 2
65.l even 6 2 1950.2.i.s 2
65.r odd 12 4 1950.2.z.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.a 2 13.e even 6 2
1170.2.i.k 2 39.h odd 6 2
1950.2.i.s 2 65.l even 6 2
1950.2.z.h 4 65.r odd 12 4
5070.2.a.f 1 1.a even 1 1 trivial
5070.2.a.o 1 13.b even 2 1
5070.2.b.g 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7} + 2 \)
\( T_{11} + 1 \)
\( T_{17} - 2 \)
\( T_{31} - 3 \)

Hecke characteristic polynomials

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