Newspace parameters
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.18307067021\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.21208.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 13x - 3 \)
|
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 8x^{2} + 13x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 6\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 7\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 7\beta _1 - 5 \)
|
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 |
|
−5.64104 | 2.22256 | 23.8213 | −12.1011 | −12.5375 | −9.22618 | −89.2487 | −22.0602 | 68.2629 | ||||||||||||||||||||||||||||||
| 1.2 | −2.06923 | 0.265551 | −3.71830 | −5.08678 | −0.549486 | −27.2773 | 24.2478 | −26.9295 | 10.5257 | |||||||||||||||||||||||||||||||
| 1.3 | 0.389055 | −4.19207 | −7.84864 | −19.1145 | −1.63095 | 34.7993 | −6.16599 | −9.42651 | −7.43658 | |||||||||||||||||||||||||||||||
| 1.4 | 1.32121 | −9.29603 | −6.25440 | 7.30237 | −12.2820 | −30.2958 | −18.8331 | 59.4162 | 9.64798 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(-1\) |
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 | 37.4.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 333.4.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 592.4.a.f | 4 | ||
| 5.b | even | 2 | 1 | 925.4.a.a | 4 | ||
| 7.b | odd | 2 | 1 | 1813.4.a.b | 4 | ||
| 8.b | even | 2 | 1 | 2368.4.a.l | 4 | ||
| 8.d | odd | 2 | 1 | 2368.4.a.g | 4 | ||
| 37.b | even | 2 | 1 | 1369.4.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.4.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 333.4.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 592.4.a.f | 4 | 4.b | odd | 2 | 1 | ||
| 925.4.a.a | 4 | 5.b | even | 2 | 1 | ||
| 1369.4.a.c | 4 | 37.b | even | 2 | 1 | ||
| 1813.4.a.b | 4 | 7.b | odd | 2 | 1 | ||
| 2368.4.a.g | 4 | 8.d | odd | 2 | 1 | ||
| 2368.4.a.l | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 6T_{2}^{3} - T_{2}^{2} - 16T_{2} + 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(37))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 6 T^{3} - T^{2} - 16 T + 6 \)
$3$
\( T^{4} + 11 T^{3} + 6 T^{2} - 89 T + 23 \)
$5$
\( T^{4} + 29 T^{3} + 125 T^{2} + \cdots - 8592 \)
$7$
\( T^{4} + 32 T^{3} - 967 T^{2} + \cdots - 265324 \)
$11$
\( T^{4} + 11 T^{3} - 1432 T^{2} + \cdots + 169317 \)
$13$
\( T^{4} + 45 T^{3} - 4635 T^{2} + \cdots - 4630592 \)
$17$
\( T^{4} + 56 T^{3} + 344 T^{2} + \cdots - 1776 \)
$19$
\( T^{4} + 144 T^{3} + \cdots - 115380208 \)
$23$
\( T^{4} + 275 T^{3} + \cdots - 93600684 \)
$29$
\( T^{4} + 29 T^{3} - 24515 T^{2} + \cdots - 20921868 \)
$31$
\( T^{4} - 111 T^{3} + \cdots + 443578864 \)
$37$
\( (T - 37)^{4} \)
$41$
\( T^{4} + 9 T^{3} - 55962 T^{2} + \cdots + 239475447 \)
$43$
\( T^{4} + 190 T^{3} + \cdots + 118152128 \)
$47$
\( T^{4} + 472 T^{3} + \cdots - 4884453612 \)
$53$
\( T^{4} - 318 T^{3} - 4891 T^{2} + \cdots + 1515396 \)
$59$
\( T^{4} + 1530 T^{3} + \cdots - 19839940416 \)
$61$
\( T^{4} + 31 T^{3} + \cdots - 1974966496 \)
$67$
\( T^{4} + 5 T^{3} + \cdots + 3584880464 \)
$71$
\( T^{4} - 390 T^{3} + \cdots - 21095220732 \)
$73$
\( T^{4} + 967 T^{3} + \cdots - 10092864749 \)
$79$
\( T^{4} - 17 T^{3} + \cdots - 29390715884 \)
$83$
\( T^{4} + 3138 T^{3} + \cdots - 146330370096 \)
$89$
\( T^{4} - 224 T^{3} + \cdots - 212866753728 \)
$97$
\( T^{4} + 2532 T^{3} + \cdots + 3402009472 \)
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