Properties

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

Related objects

Downloads

Learn more

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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - 3q^{9} - q^{10} - q^{11} - 6q^{13} - q^{14} + q^{16} - 2q^{17} + 3q^{18} - 4q^{19} + q^{20} + q^{22} - 4q^{23} + q^{25} + 6q^{26} + q^{28} + 6q^{29} - q^{32} + 2q^{34} + q^{35} - 3q^{36} - 2q^{37} + 4q^{38} - q^{40} - 6q^{41} - 4q^{43} - q^{44} - 3q^{45} + 4q^{46} + 4q^{47} + q^{49} - q^{50} - 6q^{52} - 2q^{53} - q^{55} - q^{56} - 6q^{58} + 12q^{59} - 2q^{61} - 3q^{63} + q^{64} - 6q^{65} - 8q^{67} - 2q^{68} - q^{70} - 8q^{71} + 3q^{72} - 10q^{73} + 2q^{74} - 4q^{76} - q^{77} - 8q^{79} + q^{80} + 9q^{81} + 6q^{82} - 12q^{83} - 2q^{85} + 4q^{86} + q^{88} + 10q^{89} + 3q^{90} - 6q^{91} - 4q^{92} - 4q^{94} - 4q^{95} - 6q^{97} - q^{98} + 3q^{99} + O(q^{100}) \)

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 0 1.00000 1.00000 0 1.00000 −1.00000 −3.00000 −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.d 1
3.b odd 2 1 6930.2.a.x 1
4.b odd 2 1 6160.2.a.e 1
5.b even 2 1 3850.2.a.s 1
5.c odd 4 2 3850.2.c.m 2
7.b odd 2 1 5390.2.a.j 1
11.b odd 2 1 8470.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.d 1 1.a even 1 1 trivial
3850.2.a.s 1 5.b even 2 1
3850.2.c.m 2 5.c odd 4 2
5390.2.a.j 1 7.b odd 2 1
6160.2.a.e 1 4.b odd 2 1
6930.2.a.x 1 3.b odd 2 1
8470.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(770))\):

\( T_{3} \)
\( T_{13} + 6 \)
\( T_{17} + 2 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

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