Properties

Label 363.2.a.c
Level $363$
Weight $2$
Character orbit 363.a
Self dual yes
Analytic conductor $2.899$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89856959337\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} - q^{7} + q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{13} - 2 q^{14} - 4 q^{15} - 4 q^{16} - 4 q^{17} + 2 q^{18} + 3 q^{19} + 8 q^{20} + q^{21} + 2 q^{23} + 11 q^{25} + 4 q^{26} - q^{27} - 2 q^{28} - 6 q^{29} - 8 q^{30} - 5 q^{31} - 8 q^{32} - 8 q^{34} - 4 q^{35} + 2 q^{36} + 3 q^{37} + 6 q^{38} - 2 q^{39} + 2 q^{41} + 2 q^{42} - 12 q^{43} + 4 q^{45} + 4 q^{46} + 2 q^{47} + 4 q^{48} - 6 q^{49} + 22 q^{50} + 4 q^{51} + 4 q^{52} + 6 q^{53} - 2 q^{54} - 3 q^{57} - 12 q^{58} - 10 q^{59} - 8 q^{60} - 3 q^{61} - 10 q^{62} - q^{63} - 8 q^{64} + 8 q^{65} - q^{67} - 8 q^{68} - 2 q^{69} - 8 q^{70} + 11 q^{73} + 6 q^{74} - 11 q^{75} + 6 q^{76} - 4 q^{78} - 11 q^{79} - 16 q^{80} + q^{81} + 4 q^{82} - 6 q^{83} + 2 q^{84} - 16 q^{85} - 24 q^{86} + 6 q^{87} + 12 q^{89} + 8 q^{90} - 2 q^{91} + 4 q^{92} + 5 q^{93} + 4 q^{94} + 12 q^{95} + 8 q^{96} + 5 q^{97} - 12 q^{98}+O(q^{100}) \) 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
2.00000 −1.00000 2.00000 4.00000 −2.00000 −1.00000 0 1.00000 8.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

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 363.2.a.c yes 1
3.b odd 2 1 1089.2.a.a 1
4.b odd 2 1 5808.2.a.bi 1
5.b even 2 1 9075.2.a.b 1
11.b odd 2 1 363.2.a.a 1
11.c even 5 4 363.2.e.d 4
11.d odd 10 4 363.2.e.i 4
33.d even 2 1 1089.2.a.k 1
44.c even 2 1 5808.2.a.bh 1
55.d odd 2 1 9075.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 11.b odd 2 1
363.2.a.c yes 1 1.a even 1 1 trivial
363.2.e.d 4 11.c even 5 4
363.2.e.i 4 11.d odd 10 4
1089.2.a.a 1 3.b odd 2 1
1089.2.a.k 1 33.d even 2 1
5808.2.a.bh 1 44.c even 2 1
5808.2.a.bi 1 4.b odd 2 1
9075.2.a.b 1 5.b even 2 1
9075.2.a.t 1 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

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