Newspace parameters
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(23.0502954168\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
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|
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| Defining polynomial: |
\( x^{6} + 350078x^{4} + 30638651521x^{2} + 173683668788100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 350078x^{4} + 30638651521x^{2} + 173683668788100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 175039\nu ) / 13178910 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} + 1198\nu^{4} - 875195\nu^{3} + 156981082\nu^{2} - 46925470052\nu - 6151994261460 ) / 1396964460 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{5} - 1089\nu^{4} + 1166962\nu^{3} - 230154201\nu^{2} + 97497589614\nu - 4613646454980 ) / 1396964460 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 588\nu^{4} + 875195\nu^{3} - 181996392\nu^{2} + 46925470052\nu - 9227292909960 ) / 1396964460 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6\nu^{5} + 1089\nu^{4} + 1750390\nu^{3} + 230154201\nu^{2} + 114805407004\nu + 4613646454980 ) / 1396964460 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{4} + 6\beta_{3} + 3\beta_{2} - 2\beta _1 + 1 ) / 150 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 305\beta_{5} - 1590\beta_{4} - 183\beta_{3} - 1224\beta_{2} + 61\beta _1 - 17504308 ) / 150 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 875195\beta_{4} - 1050234\beta_{3} - 525117\beta_{2} + 1977186578\beta _1 - 175039 ) / 150 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12507655 \beta_{5} + 278312010 \beta_{4} - 7504593 \beta_{3} + 293321196 \beta_{2} + \cdots + 3063962925832 ) / 150 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 20954466900 \beta_{5} - 149173690055 \beta_{4} + 199962894966 \beta_{3} + 89504214033 \beta_{2} + \cdots + 29834738011 ) / 150 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 32.0000i | − | 243.000i | −1024.00 | −6709.81 | − | 1951.05i | −7776.00 | 52907.3i | 32768.0i | −59049.0 | −62433.8 | + | 214714.i | ||||||||||||||||||||||||||||||
| 19.2 | − | 32.0000i | − | 243.000i | −1024.00 | −5008.33 | + | 4872.86i | −7776.00 | − | 78763.7i | 32768.0i | −59049.0 | 155931. | + | 160267.i | ||||||||||||||||||||||||||||||
| 19.3 | − | 32.0000i | − | 243.000i | −1024.00 | 6755.14 | − | 1787.80i | −7776.00 | − | 41527.6i | 32768.0i | −59049.0 | −57209.7 | − | 216164.i | ||||||||||||||||||||||||||||||
| 19.4 | 32.0000i | 243.000i | −1024.00 | −6709.81 | + | 1951.05i | −7776.00 | − | 52907.3i | − | 32768.0i | −59049.0 | −62433.8 | − | 214714.i | |||||||||||||||||||||||||||||||
| 19.5 | 32.0000i | 243.000i | −1024.00 | −5008.33 | − | 4872.86i | −7776.00 | 78763.7i | − | 32768.0i | −59049.0 | 155931. | − | 160267.i | ||||||||||||||||||||||||||||||||
| 19.6 | 32.0000i | 243.000i | −1024.00 | 6755.14 | + | 1787.80i | −7776.00 | 41527.6i | − | 32768.0i | −59049.0 | −57209.7 | + | 216164.i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 30.12.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.12.c.c | 6 | ||
| 4.b | odd | 2 | 1 | 240.12.f.b | 6 | ||
| 5.b | even | 2 | 1 | inner | 30.12.c.b | ✓ | 6 |
| 5.c | odd | 4 | 1 | 150.12.a.t | 3 | ||
| 5.c | odd | 4 | 1 | 150.12.a.u | 3 | ||
| 15.d | odd | 2 | 1 | 90.12.c.c | 6 | ||
| 20.d | odd | 2 | 1 | 240.12.f.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.12.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 30.12.c.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 90.12.c.c | 6 | 3.b | odd | 2 | 1 | ||
| 90.12.c.c | 6 | 15.d | odd | 2 | 1 | ||
| 150.12.a.t | 3 | 5.c | odd | 4 | 1 | ||
| 150.12.a.u | 3 | 5.c | odd | 4 | 1 | ||
| 240.12.f.b | 6 | 4.b | odd | 2 | 1 | ||
| 240.12.f.b | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 10727432952T_{7}^{4} + 32891155395360292368T_{7}^{2} + 29947171023497902154506242304 \)
acting on \(S_{12}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1024)^{3} \)
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| $3$ |
\( (T^{2} + 59049)^{3} \)
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| $5$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
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| $7$ |
\( T^{6} + \cdots + 29\!\cdots\!04 \)
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| $11$ |
\( (T^{3} + \cdots + 22\!\cdots\!00)^{2} \)
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| $13$ |
\( T^{6} + \cdots + 21\!\cdots\!36 \)
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| $17$ |
\( T^{6} + \cdots + 48\!\cdots\!44 \)
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| $19$ |
\( (T^{3} + \cdots - 29\!\cdots\!36)^{2} \)
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| $23$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
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| $29$ |
\( (T^{3} + \cdots + 12\!\cdots\!16)^{2} \)
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| $31$ |
\( (T^{3} + \cdots + 26\!\cdots\!88)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 45\!\cdots\!36 \)
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| $41$ |
\( (T^{3} + \cdots + 12\!\cdots\!16)^{2} \)
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| $43$ |
\( T^{6} + \cdots + 57\!\cdots\!04 \)
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| $47$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
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| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!44 \)
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| $59$ |
\( (T^{3} + \cdots - 58\!\cdots\!00)^{2} \)
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| $61$ |
\( (T^{3} + \cdots - 34\!\cdots\!00)^{2} \)
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| $67$ |
\( T^{6} + \cdots + 25\!\cdots\!04 \)
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| $71$ |
\( (T^{3} + \cdots + 78\!\cdots\!00)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 37\!\cdots\!36 \)
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| $79$ |
\( (T^{3} + \cdots - 29\!\cdots\!12)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 59\!\cdots\!16 \)
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| $89$ |
\( (T^{3} + \cdots - 39\!\cdots\!32)^{2} \)
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| $97$ |
\( T^{6} + \cdots + 60\!\cdots\!36 \)
|
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