Newspace parameters
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.1041806482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{37})\) |
| Defining polynomial: |
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 14 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−2.54138 | + | 4.40180i | 0 | 3.08276 | + | 5.33950i | 20.9138 | − | 36.2238i | 0 | −127.159 | − | 25.2522i | −193.986 | 0 | 106.300 | + | 184.117i | ||||||||||||||||||||
| 37.2 | 3.54138 | − | 6.13385i | 0 | −9.08276 | − | 15.7318i | −39.9138 | + | 69.1328i | 0 | 43.1587 | + | 122.247i | 97.9863 | 0 | 282.700 | + | 489.651i | |||||||||||||||||||||
| 46.1 | −2.54138 | − | 4.40180i | 0 | 3.08276 | − | 5.33950i | 20.9138 | + | 36.2238i | 0 | −127.159 | + | 25.2522i | −193.986 | 0 | 106.300 | − | 184.117i | |||||||||||||||||||||
| 46.2 | 3.54138 | + | 6.13385i | 0 | −9.08276 | + | 15.7318i | −39.9138 | − | 69.1328i | 0 | 43.1587 | − | 122.247i | 97.9863 | 0 | 282.700 | − | 489.651i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.6.e.d | 4 | |
| 3.b | odd | 2 | 1 | 7.6.c.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 63.6.e.d | 4 | |
| 7.c | even | 3 | 1 | 441.6.a.n | 2 | ||
| 7.d | odd | 6 | 1 | 441.6.a.m | 2 | ||
| 12.b | even | 2 | 1 | 112.6.i.c | 4 | ||
| 21.c | even | 2 | 1 | 49.6.c.f | 4 | ||
| 21.g | even | 6 | 1 | 49.6.a.e | 2 | ||
| 21.g | even | 6 | 1 | 49.6.c.f | 4 | ||
| 21.h | odd | 6 | 1 | 7.6.c.a | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 49.6.a.d | 2 | ||
| 84.j | odd | 6 | 1 | 784.6.a.t | 2 | ||
| 84.n | even | 6 | 1 | 112.6.i.c | 4 | ||
| 84.n | even | 6 | 1 | 784.6.a.ba | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.6.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 7.6.c.a | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 49.6.a.d | 2 | 21.h | odd | 6 | 1 | ||
| 49.6.a.e | 2 | 21.g | even | 6 | 1 | ||
| 49.6.c.f | 4 | 21.c | even | 2 | 1 | ||
| 49.6.c.f | 4 | 21.g | even | 6 | 1 | ||
| 63.6.e.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 63.6.e.d | 4 | 7.c | even | 3 | 1 | inner | |
| 112.6.i.c | 4 | 12.b | even | 2 | 1 | ||
| 112.6.i.c | 4 | 84.n | even | 6 | 1 | ||
| 441.6.a.m | 2 | 7.d | odd | 6 | 1 | ||
| 441.6.a.n | 2 | 7.c | even | 3 | 1 | ||
| 784.6.a.t | 2 | 84.j | odd | 6 | 1 | ||
| 784.6.a.ba | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 40T_{2}^{2} + 72T_{2} + 1296 \)
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2 T^{3} + 40 T^{2} + \cdots + 1296 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 38 T^{3} + 4783 T^{2} + \cdots + 11148921 \)
$7$
\( T^{4} + 168 T^{3} + \cdots + 282475249 \)
$11$
\( T^{4} - 424 T^{3} + \cdots + 643687641 \)
$13$
\( (T^{2} + 924 T + 184436)^{2} \)
$17$
\( T^{4} + 2346 T^{3} + \cdots + 534713400081 \)
$19$
\( T^{4} - 360 T^{3} + \cdots + 7876852004329 \)
$23$
\( T^{4} + 12 T^{3} + \cdots + 31018606641 \)
$29$
\( (T^{2} - 7052 T - 5697324)^{2} \)
$31$
\( T^{4} + \cdots + 248637676526001 \)
$37$
\( T^{4} + 11090 T^{3} + \cdots + 58604000855625 \)
$41$
\( (T^{2} + 3500 T - 24814188)^{2} \)
$43$
\( (T^{2} + 12680 T - 26638832)^{2} \)
$47$
\( T^{4} + 22956 T^{3} + \cdots + 10\!\cdots\!81 \)
$53$
\( T^{4} - 3042 T^{3} + \cdots + 14\!\cdots\!09 \)
$59$
\( T^{4} + 65808 T^{3} + \cdots + 11\!\cdots\!81 \)
$61$
\( T^{4} - 42486 T^{3} + \cdots + 35\!\cdots\!49 \)
$67$
\( T^{4} + 42312 T^{3} + \cdots + 17\!\cdots\!41 \)
$71$
\( (T^{2} - 2208 T - 175265856)^{2} \)
$73$
\( T^{4} - 50506 T^{3} + \cdots + 18\!\cdots\!01 \)
$79$
\( T^{4} + 9004 T^{3} + \cdots + 95\!\cdots\!81 \)
$83$
\( (T^{2} - 104328 T + 1959796944)^{2} \)
$89$
\( T^{4} + 26666 T^{3} + \cdots + 93\!\cdots\!49 \)
$97$
\( (T^{2} - 209132 T + 10932626964)^{2} \)
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