Properties

Label 770.2.a.g
Level $770$
Weight $2$
Character orbit 770.a
Self dual yes
Analytic conductor $6.148$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} - 2q^{12} + 2q^{13} + q^{14} - 2q^{15} + q^{16} + 6q^{17} + q^{18} + 2q^{19} + q^{20} - 2q^{21} - q^{22} - 6q^{23} - 2q^{24} + q^{25} + 2q^{26} + 4q^{27} + q^{28} - 2q^{30} + 8q^{31} + q^{32} + 2q^{33} + 6q^{34} + q^{35} + q^{36} + 8q^{37} + 2q^{38} - 4q^{39} + q^{40} - 2q^{42} - 4q^{43} - q^{44} + q^{45} - 6q^{46} - 2q^{48} + q^{49} + q^{50} - 12q^{51} + 2q^{52} + 12q^{53} + 4q^{54} - q^{55} + q^{56} - 4q^{57} + 12q^{59} - 2q^{60} - 10q^{61} + 8q^{62} + q^{63} + q^{64} + 2q^{65} + 2q^{66} - 4q^{67} + 6q^{68} + 12q^{69} + q^{70} - 12q^{71} + q^{72} + 14q^{73} + 8q^{74} - 2q^{75} + 2q^{76} - q^{77} - 4q^{78} - 10q^{79} + q^{80} - 11q^{81} - 2q^{84} + 6q^{85} - 4q^{86} - q^{88} - 18q^{89} + q^{90} + 2q^{91} - 6q^{92} - 16q^{93} + 2q^{95} - 2q^{96} + 8q^{97} + q^{98} - q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 770.2.a.g 1
3.b odd 2 1 6930.2.a.f 1
4.b odd 2 1 6160.2.a.n 1
5.b even 2 1 3850.2.a.j 1
5.c odd 4 2 3850.2.c.a 2
7.b odd 2 1 5390.2.a.bh 1
11.b odd 2 1 8470.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.g 1 1.a even 1 1 trivial
3850.2.a.j 1 5.b even 2 1
3850.2.c.a 2 5.c odd 4 2
5390.2.a.bh 1 7.b odd 2 1
6160.2.a.n 1 4.b odd 2 1
6930.2.a.f 1 3.b odd 2 1
8470.2.a.e 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\):

\( T_{3} + 2 \)
\( T_{13} - 2 \)
\( T_{17} - 6 \)
\( T_{19} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( -6 + T \)
$19$ \( -2 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( -8 + T \)
$37$ \( -8 + T \)
$41$ \( T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( -12 + T \)
$59$ \( -12 + T \)
$61$ \( 10 + T \)
$67$ \( 4 + T \)
$71$ \( 12 + T \)
$73$ \( -14 + T \)
$79$ \( 10 + T \)
$83$ \( T \)
$89$ \( 18 + T \)
$97$ \( -8 + T \)
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