Newspace parameters
Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 27.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.21146025774\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-10}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 10 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
|
− | 9.48683i | 0 | −26.0000 | − | 132.816i | 0 | −403.000 | − | 360.500i | 0 | −1260.00 | |||||||||||||||||||||
26.2 | 9.48683i | 0 | −26.0000 | 132.816i | 0 | −403.000 | 360.500i | 0 | −1260.00 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 27.7.b.a | ✓ | 2 |
3.b | odd | 2 | 1 | inner | 27.7.b.a | ✓ | 2 |
4.b | odd | 2 | 1 | 432.7.e.g | 2 | ||
9.c | even | 3 | 2 | 81.7.d.c | 4 | ||
9.d | odd | 6 | 2 | 81.7.d.c | 4 | ||
12.b | even | 2 | 1 | 432.7.e.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.7.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
27.7.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
81.7.d.c | 4 | 9.c | even | 3 | 2 | ||
81.7.d.c | 4 | 9.d | odd | 6 | 2 | ||
432.7.e.g | 2 | 4.b | odd | 2 | 1 | ||
432.7.e.g | 2 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 90 \)
acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 90 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 17640 \)
$7$
\( (T + 403)^{2} \)
$11$
\( T^{2} + 2246760 \)
$13$
\( (T + 961)^{2} \)
$17$
\( T^{2} + 92537640 \)
$19$
\( (T - 8021)^{2} \)
$23$
\( T^{2} + 112493160 \)
$29$
\( T^{2} + 6084000 \)
$31$
\( (T - 48854)^{2} \)
$37$
\( (T - 24167)^{2} \)
$41$
\( T^{2} + 5008644000 \)
$43$
\( (T + 60802)^{2} \)
$47$
\( T^{2} + 728974440 \)
$53$
\( T^{2} + 18943385760 \)
$59$
\( T^{2} + 9514757160 \)
$61$
\( (T - 272999)^{2} \)
$67$
\( (T + 85579)^{2} \)
$71$
\( T^{2} + 116795571840 \)
$73$
\( (T + 152737)^{2} \)
$79$
\( (T + 74059)^{2} \)
$83$
\( T^{2} + 9326916000 \)
$89$
\( T^{2} + 1424896404840 \)
$97$
\( (T + 1197313)^{2} \)
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