Properties

Label 775.2.a.b
Level $775$
Weight $2$
Character orbit 775.a
Self dual yes
Analytic conductor $6.188$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + 4 q^{11} + 2 q^{12} - 4 q^{14} - q^{16} + 8 q^{17} + q^{18} + 4 q^{19} + 8 q^{21} + 4 q^{22} - 2 q^{23} + 6 q^{24} + 4 q^{27} + 4 q^{28} - 6 q^{29} + q^{31} + 5 q^{32} - 8 q^{33} + 8 q^{34} - q^{36} + 4 q^{37} + 4 q^{38} - 6 q^{41} + 8 q^{42} + 6 q^{43} - 4 q^{44} - 2 q^{46} - 8 q^{47} + 2 q^{48} + 9 q^{49} - 16 q^{51} + 12 q^{53} + 4 q^{54} + 12 q^{56} - 8 q^{57} - 6 q^{58} - 4 q^{59} + 10 q^{61} + q^{62} - 4 q^{63} + 7 q^{64} - 8 q^{66} - 8 q^{67} - 8 q^{68} + 4 q^{69} - 3 q^{72} + 4 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{77} - 11 q^{81} - 6 q^{82} - 2 q^{83} - 8 q^{84} + 6 q^{86} + 12 q^{87} - 12 q^{88} + 14 q^{89} + 2 q^{92} - 2 q^{93} - 8 q^{94} - 10 q^{96} + 18 q^{97} + 9 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0 −2.00000 −4.00000 −3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(31\) \(-1\)

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 775.2.a.b 1
3.b odd 2 1 6975.2.a.d 1
5.b even 2 1 155.2.a.b 1
5.c odd 4 2 775.2.b.b 2
15.d odd 2 1 1395.2.a.d 1
20.d odd 2 1 2480.2.a.b 1
35.c odd 2 1 7595.2.a.c 1
40.e odd 2 1 9920.2.a.bd 1
40.f even 2 1 9920.2.a.g 1
155.c odd 2 1 4805.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.b 1 5.b even 2 1
775.2.a.b 1 1.a even 1 1 trivial
775.2.b.b 2 5.c odd 4 2
1395.2.a.d 1 15.d odd 2 1
2480.2.a.b 1 20.d odd 2 1
4805.2.a.d 1 155.c odd 2 1
6975.2.a.d 1 3.b odd 2 1
7595.2.a.c 1 35.c odd 2 1
9920.2.a.g 1 40.f even 2 1
9920.2.a.bd 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\). Copy content Toggle raw display

Hecke characteristic polynomials

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