Newspace parameters
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(20.6506685020\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} + 67x^{4} - 378x^{3} + 4489x^{2} - 12663x + 35721 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 67x^{4} - 378x^{3} + 4489x^{2} - 12663x + 35721 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 67\nu^{3} - 189\nu^{2} + 4489\nu ) / 12663 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 189 ) / 67 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} + 21\nu^{4} + 335\nu^{3} - 945\nu^{2} + 12848\nu ) / 1407 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} - 251\nu^{3} + 2352\nu^{2} - 16817\nu + 47439 ) / 1407 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 45\beta_{2} - 4\beta _1 - 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( 67\beta_{3} + 189 \)
|
| \(\nu^{4}\) | \(=\) |
\( 67\beta_{4} - 3015\beta_{2} + 457\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 189\beta_{5} - 5245\beta_{3} + 21168\beta_{2} - 5245\beta _1 - 21168 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 51.1 |
|
−1.00000 | + | 1.73205i | −3.46216 | − | 5.99664i | −2.00000 | − | 3.46410i | 0 | 13.8487 | 13.4353 | + | 12.7473i | 8.00000 | −10.4731 | + | 18.1400i | 0 | ||||||||||||||||||||||||||
| 51.2 | −1.00000 | + | 1.73205i | −2.20784 | − | 3.82408i | −2.00000 | − | 3.46410i | 0 | 8.83134 | −2.04309 | − | 18.4072i | 8.00000 | 3.75092 | − | 6.49679i | 0 | |||||||||||||||||||||||||||
| 51.3 | −1.00000 | + | 1.73205i | 4.17000 | + | 7.22265i | −2.00000 | − | 3.46410i | 0 | −16.6800 | 16.6078 | − | 8.19644i | 8.00000 | −21.2778 | + | 36.8542i | 0 | |||||||||||||||||||||||||||
| 151.1 | −1.00000 | − | 1.73205i | −3.46216 | + | 5.99664i | −2.00000 | + | 3.46410i | 0 | 13.8487 | 13.4353 | − | 12.7473i | 8.00000 | −10.4731 | − | 18.1400i | 0 | |||||||||||||||||||||||||||
| 151.2 | −1.00000 | − | 1.73205i | −2.20784 | + | 3.82408i | −2.00000 | + | 3.46410i | 0 | 8.83134 | −2.04309 | + | 18.4072i | 8.00000 | 3.75092 | + | 6.49679i | 0 | |||||||||||||||||||||||||||
| 151.3 | −1.00000 | − | 1.73205i | 4.17000 | − | 7.22265i | −2.00000 | + | 3.46410i | 0 | −16.6800 | 16.6078 | + | 8.19644i | 8.00000 | −21.2778 | − | 36.8542i | 0 | |||||||||||||||||||||||||||
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 | 350.4.e.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 350.4.e.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 350.4.j.h | 12 | ||
| 7.c | even | 3 | 1 | inner | 350.4.e.i | ✓ | 6 |
| 7.c | even | 3 | 1 | 2450.4.a.ci | 3 | ||
| 7.d | odd | 6 | 1 | 2450.4.a.ch | 3 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.cd | 3 | ||
| 35.j | even | 6 | 1 | 350.4.e.j | yes | 6 | |
| 35.j | even | 6 | 1 | 2450.4.a.cc | 3 | ||
| 35.l | odd | 12 | 2 | 350.4.j.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.4.e.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 350.4.e.i | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 350.4.e.j | yes | 6 | 5.b | even | 2 | 1 | |
| 350.4.e.j | yes | 6 | 35.j | even | 6 | 1 | |
| 350.4.j.h | 12 | 5.c | odd | 4 | 2 | ||
| 350.4.j.h | 12 | 35.l | odd | 12 | 2 | ||
| 2450.4.a.cc | 3 | 35.j | even | 6 | 1 | ||
| 2450.4.a.cd | 3 | 35.i | odd | 6 | 1 | ||
| 2450.4.a.ch | 3 | 7.d | odd | 6 | 1 | ||
| 2450.4.a.ci | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):
|
\( T_{3}^{6} + 3T_{3}^{5} + 73T_{3}^{4} + 318T_{3}^{3} + 4861T_{3}^{2} + 16320T_{3} + 65025 \)
|
|
\( T_{11}^{6} - 26T_{11}^{5} + 2069T_{11}^{4} - 36596T_{11}^{3} + 2887031T_{11}^{2} - 50714951T_{11} + 1325469649 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots + 65025 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 56 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + \cdots + 1325469649 \)
|
| $13$ |
\( (T^{3} + 80 T^{2} + \cdots - 160725)^{2} \)
|
| $17$ |
\( T^{6} - 30 T^{5} + \cdots + 40208281 \)
|
| $19$ |
\( T^{6} + \cdots + 9063992025 \)
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| $23$ |
\( T^{6} + \cdots + 8977373001 \)
|
| $29$ |
\( (T^{3} + 404 T^{2} + \cdots + 1808541)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 57192570880929 \)
|
| $37$ |
\( T^{6} + \cdots + 128685360417849 \)
|
| $41$ |
\( (T^{3} + 101 T^{2} + \cdots - 7167)^{2} \)
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| $43$ |
\( (T^{3} + 249 T^{2} + \cdots - 27999735)^{2} \)
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| $47$ |
\( T^{6} + \cdots + 27\!\cdots\!61 \)
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| $53$ |
\( T^{6} + \cdots + 87\!\cdots\!89 \)
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| $59$ |
\( T^{6} + \cdots + 86\!\cdots\!25 \)
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| $61$ |
\( T^{6} + \cdots + 863302997881 \)
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| $67$ |
\( T^{6} + \cdots + 7149548690496 \)
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| $71$ |
\( (T^{3} - 533 T^{2} + \cdots + 74350109)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 43\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots + 30\!\cdots\!25 \)
|
| $83$ |
\( (T^{3} - 520 T^{2} + \cdots - 294469047)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 71\!\cdots\!69 \)
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| $97$ |
\( (T^{3} - 1027 T^{2} + \cdots + 883021)^{2} \)
|
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