Newspace parameters
Level: | \( N \) | \(=\) | \( 145 = 5 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 145.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.15783082931\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - 2 \)
|
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 \(\beta = \sqrt{2}\). 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 |
|
−2.41421 | −2.00000 | 3.82843 | 1.00000 | 4.82843 | 0.828427 | −4.41421 | 1.00000 | −2.41421 | ||||||||||||||||||||||||
1.2 | 0.414214 | −2.00000 | −1.82843 | 1.00000 | −0.828427 | −4.82843 | −1.58579 | 1.00000 | 0.414214 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(29\) | \(-1\) |
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 | 145.2.a.b | ✓ | 2 |
3.b | odd | 2 | 1 | 1305.2.a.n | 2 | ||
4.b | odd | 2 | 1 | 2320.2.a.k | 2 | ||
5.b | even | 2 | 1 | 725.2.a.c | 2 | ||
5.c | odd | 4 | 2 | 725.2.b.c | 4 | ||
7.b | odd | 2 | 1 | 7105.2.a.e | 2 | ||
8.b | even | 2 | 1 | 9280.2.a.be | 2 | ||
8.d | odd | 2 | 1 | 9280.2.a.w | 2 | ||
15.d | odd | 2 | 1 | 6525.2.a.p | 2 | ||
29.b | even | 2 | 1 | 4205.2.a.d | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
145.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
725.2.a.c | 2 | 5.b | even | 2 | 1 | ||
725.2.b.c | 4 | 5.c | odd | 4 | 2 | ||
1305.2.a.n | 2 | 3.b | odd | 2 | 1 | ||
2320.2.a.k | 2 | 4.b | odd | 2 | 1 | ||
4205.2.a.d | 2 | 29.b | even | 2 | 1 | ||
6525.2.a.p | 2 | 15.d | odd | 2 | 1 | ||
7105.2.a.e | 2 | 7.b | odd | 2 | 1 | ||
9280.2.a.w | 2 | 8.d | odd | 2 | 1 | ||
9280.2.a.be | 2 | 8.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(145))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T - 1 \)
$3$
\( (T + 2)^{2} \)
$5$
\( (T - 1)^{2} \)
$7$
\( T^{2} + 4T - 4 \)
$11$
\( T^{2} + 4T - 4 \)
$13$
\( (T + 2)^{2} \)
$17$
\( T^{2} - 8 \)
$19$
\( T^{2} + 4T - 4 \)
$23$
\( T^{2} + 12T + 28 \)
$29$
\( (T - 1)^{2} \)
$31$
\( T^{2} + 4T - 68 \)
$37$
\( T^{2} - 72 \)
$41$
\( (T + 6)^{2} \)
$43$
\( (T + 6)^{2} \)
$47$
\( T^{2} + 12T + 4 \)
$53$
\( T^{2} - 4T - 28 \)
$59$
\( T^{2} \)
$61$
\( T^{2} - 4T - 28 \)
$67$
\( T^{2} + 4T - 68 \)
$71$
\( T^{2} + 8T - 112 \)
$73$
\( T^{2} - 72 \)
$79$
\( T^{2} - 12T - 36 \)
$83$
\( T^{2} - 20T + 92 \)
$89$
\( T^{2} + 4T - 28 \)
$97$
\( T^{2} + 8T - 56 \)
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