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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