Properties

Label 363.3.b.b
Level $363$
Weight $3$
Character orbit 363.b
Self dual yes
Analytic conductor $9.891$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} + 4 q^{4} + 11 q^{7} + 9 q^{9} + O(q^{10}) \) \( q - 3 q^{3} + 4 q^{4} + 11 q^{7} + 9 q^{9} - 12 q^{12} - 22 q^{13} + 16 q^{16} + 11 q^{19} - 33 q^{21} + 25 q^{25} - 27 q^{27} + 44 q^{28} + 59 q^{31} + 36 q^{36} + 47 q^{37} + 66 q^{39} - 22 q^{43} - 48 q^{48} + 72 q^{49} - 88 q^{52} - 33 q^{57} - 121 q^{61} + 99 q^{63} + 64 q^{64} - 13 q^{67} + 143 q^{73} - 75 q^{75} + 44 q^{76} + 11 q^{79} + 81 q^{81} - 132 q^{84} - 242 q^{91} - 177 q^{93} - 169 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
0
0 −3.00000 4.00000 0 0 11.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.b yes 1
3.b odd 2 1 CM 363.3.b.b yes 1
11.b odd 2 1 363.3.b.a 1
11.c even 5 4 363.3.h.a 4
11.d odd 10 4 363.3.h.b 4
33.d even 2 1 363.3.b.a 1
33.f even 10 4 363.3.h.b 4
33.h odd 10 4 363.3.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.a 1 11.b odd 2 1
363.3.b.a 1 33.d even 2 1
363.3.b.b yes 1 1.a even 1 1 trivial
363.3.b.b yes 1 3.b odd 2 1 CM
363.3.h.a 4 11.c even 5 4
363.3.h.a 4 33.h odd 10 4
363.3.h.b 4 11.d odd 10 4
363.3.h.b 4 33.f even 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \)
\( T_{5} \)
\( T_{7} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -11 + T \)
$11$ \( T \)
$13$ \( 22 + T \)
$17$ \( T \)
$19$ \( -11 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -59 + T \)
$37$ \( -47 + T \)
$41$ \( T \)
$43$ \( 22 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 121 + T \)
$67$ \( 13 + T \)
$71$ \( T \)
$73$ \( -143 + T \)
$79$ \( -11 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 169 + T \)
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