Newspace parameters
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.42608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 371.1 |
|
− | 1.00000i | −1.22474 | − | 1.22474i | −1.00000 | 1.00000i | −1.22474 | + | 1.22474i | −4.00000 | 1.00000i | 3.00000i | 1.00000 | |||||||||||||||||||||||||
| 371.2 | − | 1.00000i | 1.22474 | + | 1.22474i | −1.00000 | 1.00000i | 1.22474 | − | 1.22474i | −4.00000 | 1.00000i | 3.00000i | 1.00000 | ||||||||||||||||||||||||||
| 371.3 | 1.00000i | −1.22474 | + | 1.22474i | −1.00000 | − | 1.00000i | −1.22474 | − | 1.22474i | −4.00000 | − | 1.00000i | − | 3.00000i | 1.00000 | ||||||||||||||||||||||||
| 371.4 | 1.00000i | 1.22474 | − | 1.22474i | −1.00000 | − | 1.00000i | 1.22474 | + | 1.22474i | −4.00000 | − | 1.00000i | − | 3.00000i | 1.00000 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 31.b | odd | 2 | 1 | inner |
| 93.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 930.2.h.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 930.2.h.a | ✓ | 4 |
| 31.b | odd | 2 | 1 | inner | 930.2.h.a | ✓ | 4 |
| 93.c | even | 2 | 1 | inner | 930.2.h.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.h.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 930.2.h.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 930.2.h.a | ✓ | 4 | 31.b | odd | 2 | 1 | inner |
| 930.2.h.a | ✓ | 4 | 93.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( T^{4} + 9 \)
$5$
\( (T^{2} + 1)^{2} \)
$7$
\( (T + 4)^{4} \)
$11$
\( (T^{2} - 24)^{2} \)
$13$
\( (T^{2} + 6)^{2} \)
$17$
\( (T^{2} - 54)^{2} \)
$19$
\( (T + 2)^{4} \)
$23$
\( (T^{2} - 54)^{2} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} - 10 T + 31)^{2} \)
$37$
\( (T^{2} + 6)^{2} \)
$41$
\( (T^{2} + 144)^{2} \)
$43$
\( (T^{2} + 6)^{2} \)
$47$
\( (T^{2} + 144)^{2} \)
$53$
\( (T^{2} - 6)^{2} \)
$59$
\( (T^{2} + 36)^{2} \)
$61$
\( T^{4} \)
$67$
\( (T - 4)^{4} \)
$71$
\( T^{4} \)
$73$
\( (T^{2} + 6)^{2} \)
$79$
\( (T^{2} + 24)^{2} \)
$83$
\( (T^{2} - 150)^{2} \)
$89$
\( (T^{2} - 24)^{2} \)
$97$
\( (T - 14)^{4} \)
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