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: |
\( x^{2} - 3 \)
|
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.
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 | −1.00000 | 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 | |||||||||||||||||||||||||
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(363))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3 \)
$3$
\( (T + 1)^{2} \)
$5$
\( (T + 3)^{2} \)
$7$
\( T^{2} - 12 \)
$11$
\( T^{2} \)
$13$
\( T^{2} - 3 \)
$17$
\( T^{2} - 3 \)
$19$
\( T^{2} - 48 \)
$23$
\( (T + 6)^{2} \)
$29$
\( T^{2} - 3 \)
$31$
\( (T - 4)^{2} \)
$37$
\( (T + 11)^{2} \)
$41$
\( T^{2} - 3 \)
$43$
\( T^{2} - 12 \)
$47$
\( T^{2} \)
$53$
\( (T + 9)^{2} \)
$59$
\( (T + 6)^{2} \)
$61$
\( T^{2} \)
$67$
\( (T + 2)^{2} \)
$71$
\( (T + 6)^{2} \)
$73$
\( T^{2} - 48 \)
$79$
\( T^{2} \)
$83$
\( T^{2} \)
$89$
\( (T - 9)^{2} \)
$97$
\( (T + 7)^{2} \)
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