Newspace parameters
| Level: | \( N \) | \(=\) | \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1400.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(82.6026740080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{57})\) |
|
|
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| Defining polynomial: |
\( x^{4} + 29x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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
\( x^{4} + 29x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 15\nu ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 43\nu ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 29 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 29 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -15\beta_{2} + 43\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).
| \(n\) | \(351\) | \(701\) | \(801\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 8.54983i | 0 | 0 | 0 | − | 7.00000i | 0 | −46.0997 | 0 | ||||||||||||||||||||||||||||
| 449.2 | 0 | − | 6.54983i | 0 | 0 | 0 | 7.00000i | 0 | −15.9003 | 0 | ||||||||||||||||||||||||||||||
| 449.3 | 0 | 6.54983i | 0 | 0 | 0 | − | 7.00000i | 0 | −15.9003 | 0 | ||||||||||||||||||||||||||||||
| 449.4 | 0 | 8.54983i | 0 | 0 | 0 | 7.00000i | 0 | −46.0997 | 0 | |||||||||||||||||||||||||||||||
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 | 1400.4.g.h | 4 | |
| 5.b | even | 2 | 1 | inner | 1400.4.g.h | 4 | |
| 5.c | odd | 4 | 1 | 56.4.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1400.4.a.i | 2 | ||
| 15.e | even | 4 | 1 | 504.4.a.i | 2 | ||
| 20.e | even | 4 | 1 | 112.4.a.h | 2 | ||
| 35.f | even | 4 | 1 | 392.4.a.h | 2 | ||
| 35.k | even | 12 | 2 | 392.4.i.i | 4 | ||
| 35.l | odd | 12 | 2 | 392.4.i.l | 4 | ||
| 40.i | odd | 4 | 1 | 448.4.a.s | 2 | ||
| 40.k | even | 4 | 1 | 448.4.a.r | 2 | ||
| 60.l | odd | 4 | 1 | 1008.4.a.x | 2 | ||
| 140.j | odd | 4 | 1 | 784.4.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 112.4.a.h | 2 | 20.e | even | 4 | 1 | ||
| 392.4.a.h | 2 | 35.f | even | 4 | 1 | ||
| 392.4.i.i | 4 | 35.k | even | 12 | 2 | ||
| 392.4.i.l | 4 | 35.l | odd | 12 | 2 | ||
| 448.4.a.r | 2 | 40.k | even | 4 | 1 | ||
| 448.4.a.s | 2 | 40.i | odd | 4 | 1 | ||
| 504.4.a.i | 2 | 15.e | even | 4 | 1 | ||
| 784.4.a.t | 2 | 140.j | odd | 4 | 1 | ||
| 1008.4.a.x | 2 | 60.l | odd | 4 | 1 | ||
| 1400.4.a.i | 2 | 5.c | odd | 4 | 1 | ||
| 1400.4.g.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 1400.4.g.h | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1400, [\chi])\):
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\( T_{3}^{4} + 116T_{3}^{2} + 3136 \)
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\( T_{11}^{2} - 36T_{11} - 1728 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 116T^{2} + 3136 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
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| $11$ |
\( (T^{2} - 36 T - 1728)^{2} \)
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| $13$ |
\( T^{4} + 3732 T^{2} + 968256 \)
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| $17$ |
\( T^{4} + 4136 T^{2} + 4145296 \)
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| $19$ |
\( (T^{2} - 118 T - 6152)^{2} \)
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| $23$ |
\( T^{4} + 7232 T^{2} + 3211264 \)
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| $29$ |
\( (T^{2} - 56 T - 26804)^{2} \)
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| $31$ |
\( (T^{2} - 20 T - 11072)^{2} \)
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| $37$ |
\( T^{4} + \cdots + 3341302416 \)
|
| $41$ |
\( (T^{2} + 544 T + 55516)^{2} \)
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| $43$ |
\( T^{4} + 107216 T^{2} + 977437696 \)
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| $47$ |
\( T^{4} + \cdots + 2634563584 \)
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| $53$ |
\( T^{4} + \cdots + 77024011024 \)
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| $59$ |
\( (T^{2} - 198 T + 168)^{2} \)
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| $61$ |
\( (T^{2} + 346 T - 24848)^{2} \)
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| $67$ |
\( T^{4} + 1035168 T^{2} + 91240704 \)
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| $71$ |
\( (T^{2} + 1224 T + 264192)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 2947621264 \)
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| $79$ |
\( (T^{2} + 584 T + 11392)^{2} \)
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| $83$ |
\( T^{4} + \cdots + 63661345344 \)
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| $89$ |
\( (T^{2} + 596 T - 1370396)^{2} \)
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| $97$ |
\( T^{4} + \cdots + 9238285456 \)
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