Newspace parameters
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.f (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.90162250860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
4.00000 | + | 4.00000i | −11.0227 | + | 11.0227i | 32.0000i | −123.371 | − | 20.1135i | −88.1816 | −152.287 | − | 152.287i | −128.000 | + | 128.000i | − | 243.000i | −413.031 | − | 573.939i | |||||||||||||||||
| 7.2 | 4.00000 | + | 4.00000i | 11.0227 | − | 11.0227i | 32.0000i | −86.6288 | + | 90.1135i | 88.1816 | 450.287 | + | 450.287i | −128.000 | + | 128.000i | − | 243.000i | −706.969 | + | 13.9388i | ||||||||||||||||||
| 13.1 | 4.00000 | − | 4.00000i | −11.0227 | − | 11.0227i | − | 32.0000i | −123.371 | + | 20.1135i | −88.1816 | −152.287 | + | 152.287i | −128.000 | − | 128.000i | 243.000i | −413.031 | + | 573.939i | ||||||||||||||||||
| 13.2 | 4.00000 | − | 4.00000i | 11.0227 | + | 11.0227i | − | 32.0000i | −86.6288 | − | 90.1135i | 88.1816 | 450.287 | − | 450.287i | −128.000 | − | 128.000i | 243.000i | −706.969 | − | 13.9388i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 30.7.f.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 90.7.g.c | 4 | ||
| 4.b | odd | 2 | 1 | 240.7.bg.a | 4 | ||
| 5.b | even | 2 | 1 | 150.7.f.a | 4 | ||
| 5.c | odd | 4 | 1 | inner | 30.7.f.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 150.7.f.a | 4 | ||
| 15.d | odd | 2 | 1 | 450.7.g.i | 4 | ||
| 15.e | even | 4 | 1 | 90.7.g.c | 4 | ||
| 15.e | even | 4 | 1 | 450.7.g.i | 4 | ||
| 20.e | even | 4 | 1 | 240.7.bg.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.7.f.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 30.7.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 90.7.g.c | 4 | 3.b | odd | 2 | 1 | ||
| 90.7.g.c | 4 | 15.e | even | 4 | 1 | ||
| 150.7.f.a | 4 | 5.b | even | 2 | 1 | ||
| 150.7.f.a | 4 | 5.c | odd | 4 | 1 | ||
| 240.7.bg.a | 4 | 4.b | odd | 2 | 1 | ||
| 240.7.bg.a | 4 | 20.e | even | 4 | 1 | ||
| 450.7.g.i | 4 | 15.d | odd | 2 | 1 | ||
| 450.7.g.i | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 596T_{7}^{3} + 177608T_{7}^{2} + 81739016T_{7} + 18809025316 \)
acting on \(S_{7}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 32)^{2} \)
|
| $3$ |
\( T^{4} + 59049 \)
|
| $5$ |
\( T^{4} + 420 T^{3} + \cdots + 244140625 \)
|
| $7$ |
\( T^{4} + \cdots + 18809025316 \)
|
| $11$ |
\( (T^{2} - 472 T - 1450310)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 6188332868496 \)
|
| $17$ |
\( T^{4} + \cdots + 86477426856976 \)
|
| $19$ |
\( T^{4} + \cdots + 24771524410000 \)
|
| $23$ |
\( T^{4} + \cdots + 75\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 93\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} - 68360 T + 712251256)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!44 \)
|
| $41$ |
\( (T^{2} + 195164 T + 9270770668)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 48\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 71\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 92\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} - 225252 T - 16225204500)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} - 388472 T - 134580660080)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 90\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 18\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 15\!\cdots\!44 \)
|
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