Newspace parameters
| Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 46.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.71408786026\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{41}) \) |
| Defining polynomial: |
\( x^{2} - x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). 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 |
|
−2.00000 | −10.1047 | 4.00000 | 11.4031 | 20.2094 | 9.40312 | −8.00000 | 75.1047 | −22.8062 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 9.10469 | 4.00000 | −1.40312 | −18.2094 | −3.40312 | −8.00000 | 55.8953 | 2.80625 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(23\) | \(1\) |
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 | 46.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 414.4.a.j | 2 | ||
| 4.b | odd | 2 | 1 | 368.4.a.g | 2 | ||
| 5.b | even | 2 | 1 | 1150.4.a.k | 2 | ||
| 5.c | odd | 4 | 2 | 1150.4.b.i | 4 | ||
| 7.b | odd | 2 | 1 | 2254.4.a.d | 2 | ||
| 8.b | even | 2 | 1 | 1472.4.a.m | 2 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.l | 2 | ||
| 23.b | odd | 2 | 1 | 1058.4.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 368.4.a.g | 2 | 4.b | odd | 2 | 1 | ||
| 414.4.a.j | 2 | 3.b | odd | 2 | 1 | ||
| 1058.4.a.f | 2 | 23.b | odd | 2 | 1 | ||
| 1150.4.a.k | 2 | 5.b | even | 2 | 1 | ||
| 1150.4.b.i | 4 | 5.c | odd | 4 | 2 | ||
| 1472.4.a.l | 2 | 8.d | odd | 2 | 1 | ||
| 1472.4.a.m | 2 | 8.b | even | 2 | 1 | ||
| 2254.4.a.d | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + T_{3} - 92 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(46))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 2)^{2} \)
$3$
\( T^{2} + T - 92 \)
$5$
\( T^{2} - 10T - 16 \)
$7$
\( T^{2} - 6T - 32 \)
$11$
\( T^{2} - 72T + 640 \)
$13$
\( T^{2} + 111T + 2578 \)
$17$
\( T^{2} - 124T - 2060 \)
$19$
\( T^{2} - 22T - 3200 \)
$23$
\( (T + 23)^{2} \)
$29$
\( T^{2} - 15T - 43250 \)
$31$
\( T^{2} - 67T - 1840 \)
$37$
\( T^{2} - 18T - 50144 \)
$41$
\( T^{2} - 485T + 54286 \)
$43$
\( T^{2} + 440T + 16256 \)
$47$
\( T^{2} - 215T - 77096 \)
$53$
\( T^{2} - 240T - 143204 \)
$59$
\( T^{2} - 792T + 146320 \)
$61$
\( T^{2} - 456T - 408692 \)
$67$
\( T^{2} - 240T - 9216 \)
$71$
\( T^{2} + 705T + 39376 \)
$73$
\( T^{2} - 27T - 8438 \)
$79$
\( T^{2} - 594T + 32080 \)
$83$
\( T^{2} - 394T - 285952 \)
$89$
\( T^{2} + 486T - 31520 \)
$97$
\( T^{2} - 2152 T + 882092 \)
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