Newspace parameters
Level: | \( N \) | \(=\) | \( 82 = 2 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 82.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.654773296574\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - 2 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
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 \(\beta = \sqrt{2}\). 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 |
|
1.00000 | −1.41421 | 1.00000 | 2.82843 | −1.41421 | −0.585786 | 1.00000 | −1.00000 | 2.82843 | ||||||||||||||||||||||||
1.2 | 1.00000 | 1.41421 | 1.00000 | −2.82843 | 1.41421 | −3.41421 | 1.00000 | −1.00000 | −2.82843 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(41\) | \(1\) |
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 | 82.2.a.b | ✓ | 2 |
3.b | odd | 2 | 1 | 738.2.a.k | 2 | ||
4.b | odd | 2 | 1 | 656.2.a.e | 2 | ||
5.b | even | 2 | 1 | 2050.2.a.h | 2 | ||
5.c | odd | 4 | 2 | 2050.2.c.l | 4 | ||
7.b | odd | 2 | 1 | 4018.2.a.ba | 2 | ||
8.b | even | 2 | 1 | 2624.2.a.j | 2 | ||
8.d | odd | 2 | 1 | 2624.2.a.l | 2 | ||
11.b | odd | 2 | 1 | 9922.2.a.i | 2 | ||
12.b | even | 2 | 1 | 5904.2.a.z | 2 | ||
41.b | even | 2 | 1 | 3362.2.a.m | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
82.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
656.2.a.e | 2 | 4.b | odd | 2 | 1 | ||
738.2.a.k | 2 | 3.b | odd | 2 | 1 | ||
2050.2.a.h | 2 | 5.b | even | 2 | 1 | ||
2050.2.c.l | 4 | 5.c | odd | 4 | 2 | ||
2624.2.a.j | 2 | 8.b | even | 2 | 1 | ||
2624.2.a.l | 2 | 8.d | odd | 2 | 1 | ||
3362.2.a.m | 2 | 41.b | even | 2 | 1 | ||
4018.2.a.ba | 2 | 7.b | odd | 2 | 1 | ||
5904.2.a.z | 2 | 12.b | even | 2 | 1 | ||
9922.2.a.i | 2 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(82))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} - 2 \)
$5$
\( T^{2} - 8 \)
$7$
\( T^{2} + 4T + 2 \)
$11$
\( T^{2} - 18 \)
$13$
\( T^{2} \)
$17$
\( T^{2} - 4T - 28 \)
$19$
\( T^{2} + 8T + 14 \)
$23$
\( T^{2} - 8T + 8 \)
$29$
\( T^{2} - 8T - 16 \)
$31$
\( T^{2} + 8T + 8 \)
$37$
\( T^{2} - 72 \)
$41$
\( (T + 1)^{2} \)
$43$
\( T^{2} - 8T - 16 \)
$47$
\( T^{2} + 4T - 46 \)
$53$
\( (T - 12)^{2} \)
$59$
\( T^{2} + 8T + 8 \)
$61$
\( (T - 6)^{2} \)
$67$
\( T^{2} + 8T - 2 \)
$71$
\( T^{2} + 4T + 2 \)
$73$
\( T^{2} + 16T + 32 \)
$79$
\( T^{2} + 12T + 18 \)
$83$
\( T^{2} - 24T + 112 \)
$89$
\( T^{2} + 12T + 4 \)
$97$
\( T^{2} + 4T - 28 \)
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