Properties

Label 363.3.b.e
Level $363$
Weight $3$
Character orbit 363.b
Self dual yes
Analytic conductor $9.891$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + 4 q^{4} -5 \beta q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + 4 q^{4} -5 \beta q^{7} + 9 q^{9} + 12 q^{12} + 8 \beta q^{13} + 16 q^{16} + 21 \beta q^{19} -15 \beta q^{21} + 25 q^{25} + 27 q^{27} -20 \beta q^{28} -59 q^{31} + 36 q^{36} -47 q^{37} + 24 \beta q^{39} -48 \beta q^{43} + 48 q^{48} + 26 q^{49} + 32 \beta q^{52} + 63 \beta q^{57} -9 \beta q^{61} -45 \beta q^{63} + 64 q^{64} + 13 q^{67} + 17 \beta q^{73} + 75 q^{75} + 84 \beta q^{76} -91 \beta q^{79} + 81 q^{81} -60 \beta q^{84} -120 q^{91} -177 q^{93} -169 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 8 q^{4} + 18 q^{9} + O(q^{10}) \) \( 2 q + 6 q^{3} + 8 q^{4} + 18 q^{9} + 24 q^{12} + 32 q^{16} + 50 q^{25} + 54 q^{27} - 118 q^{31} + 72 q^{36} - 94 q^{37} + 96 q^{48} + 52 q^{49} + 128 q^{64} + 26 q^{67} + 150 q^{75} + 162 q^{81} - 240 q^{91} - 354 q^{93} - 338 q^{97} + O(q^{100}) \)

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
1.73205
−1.73205
0 3.00000 4.00000 0 0 −8.66025 0 9.00000 0
122.2 0 3.00000 4.00000 0 0 8.66025 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 \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.e 2
3.b odd 2 1 CM 363.3.b.e 2
11.b odd 2 1 inner 363.3.b.e 2
11.c even 5 4 363.3.h.c 8
11.d odd 10 4 363.3.h.c 8
33.d even 2 1 inner 363.3.b.e 2
33.f even 10 4 363.3.h.c 8
33.h odd 10 4 363.3.h.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.e 2 1.a even 1 1 trivial
363.3.b.e 2 3.b odd 2 1 CM
363.3.b.e 2 11.b odd 2 1 inner
363.3.b.e 2 33.d even 2 1 inner
363.3.h.c 8 11.c even 5 4
363.3.h.c 8 11.d odd 10 4
363.3.h.c 8 33.f even 10 4
363.3.h.c 8 33.h odd 10 4

Hecke kernels

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

\( T_{2} \)
\( T_{5} \)
\( T_{7}^{2} - 75 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -75 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -192 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( -1323 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 59 + T )^{2} \)
$37$ \( ( 47 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( -6912 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -243 + T^{2} \)
$67$ \( ( -13 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( -867 + T^{2} \)
$79$ \( -24843 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 169 + T )^{2} \)
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