Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + q^{4} - 3 q^{5} - \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + q^{4} - 3 q^{5} - \beta q^{6} - 2 \beta q^{7} - \beta q^{8} + q^{9} - 3 \beta q^{10} - q^{12} - \beta q^{13} - 6 q^{14} + 3 q^{15} - 5 q^{16} + \beta q^{17} + \beta q^{18} + 4 \beta q^{19} - 3 q^{20} + 2 \beta q^{21} - 6 q^{23} + \beta q^{24} + 4 q^{25} - 3 q^{26} - q^{27} - 2 \beta q^{28} - \beta q^{29} + 3 \beta q^{30} + 4 q^{31} - 3 \beta q^{32} + 3 q^{34} + 6 \beta q^{35} + q^{36} - 11 q^{37} + 12 q^{38} + \beta q^{39} + 3 \beta q^{40} - \beta q^{41} + 6 q^{42} + 2 \beta q^{43} - 3 q^{45} - 6 \beta q^{46} + 5 q^{48} + 5 q^{49} + 4 \beta q^{50} - \beta q^{51} - \beta q^{52} - 9 q^{53} - \beta q^{54} + 6 q^{56} - 4 \beta q^{57} - 3 q^{58} - 6 q^{59} + 3 q^{60} + 4 \beta q^{62} - 2 \beta q^{63} + q^{64} + 3 \beta q^{65} - 2 q^{67} + \beta q^{68} + 6 q^{69} + 18 q^{70} - 6 q^{71} - \beta q^{72} + 4 \beta q^{73} - 11 \beta q^{74} - 4 q^{75} + 4 \beta q^{76} + 3 q^{78} + 15 q^{80} + q^{81} - 3 q^{82} + 2 \beta q^{84} - 3 \beta q^{85} + 6 q^{86} + \beta q^{87} + 9 q^{89} - 3 \beta q^{90} + 6 q^{91} - 6 q^{92} - 4 q^{93} - 12 \beta q^{95} + 3 \beta q^{96} - 7 q^{97} + 5 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{5} + 2 q^{9} - 2 q^{12} - 12 q^{14} + 6 q^{15} - 10 q^{16} - 6 q^{20} - 12 q^{23} + 8 q^{25} - 6 q^{26} - 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} - 22 q^{37} + 24 q^{38} + 12 q^{42} - 6 q^{45} + 10 q^{48} + 10 q^{49} - 18 q^{53} + 12 q^{56} - 6 q^{58} - 12 q^{59} + 6 q^{60} + 2 q^{64} - 4 q^{67} + 12 q^{69} + 36 q^{70} - 12 q^{71} - 8 q^{75} + 6 q^{78} + 30 q^{80} + 2 q^{81} - 6 q^{82} + 12 q^{86} + 18 q^{89} + 12 q^{91} - 12 q^{92} - 8 q^{93} - 14 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 −1.00000 1.00000 −3.00000 1.73205 3.46410 1.73205 1.00000 5.19615
1.2 1.73205 −1.00000 1.00000 −3.00000 −1.73205 −3.46410 −1.73205 1.00000 −5.19615
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.a.f 2
3.b odd 2 1 1089.2.a.o 2
4.b odd 2 1 5808.2.a.ca 2
5.b even 2 1 9075.2.a.bo 2
11.b odd 2 1 inner 363.2.a.f 2
11.c even 5 4 363.2.e.m 8
11.d odd 10 4 363.2.e.m 8
33.d even 2 1 1089.2.a.o 2
44.c even 2 1 5808.2.a.ca 2
55.d odd 2 1 9075.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 1.a even 1 1 trivial
363.2.a.f 2 11.b odd 2 1 inner
363.2.e.m 8 11.c even 5 4
363.2.e.m 8 11.d odd 10 4
1089.2.a.o 2 3.b odd 2 1
1089.2.a.o 2 33.d even 2 1
5808.2.a.ca 2 4.b odd 2 1
5808.2.a.ca 2 44.c even 2 1
9075.2.a.bo 2 5.b even 2 1
9075.2.a.bo 2 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 48 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 3 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 11)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3 \) Copy content Toggle raw display
$43$ \( T^{2} - 12 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 48 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( (T + 7)^{2} \) Copy content Toggle raw display
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