Newspace parameters
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.8992105744\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
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|
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| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 351.1 |
|
0 | − | 1.73205i | 0 | 0 | 0 | − | 3.46410i | 0 | 6.00000 | 0 | ||||||||||||||||||||||
| 351.2 | 0 | 1.73205i | 0 | 0 | 0 | 3.46410i | 0 | 6.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.3.b.e | ✓ | 2 |
| 3.b | odd | 2 | 1 | 3600.3.e.j | 2 | ||
| 4.b | odd | 2 | 1 | inner | 400.3.b.e | ✓ | 2 |
| 5.b | even | 2 | 1 | 400.3.b.f | yes | 2 | |
| 5.c | odd | 4 | 2 | 400.3.h.b | 4 | ||
| 8.b | even | 2 | 1 | 1600.3.b.j | 2 | ||
| 8.d | odd | 2 | 1 | 1600.3.b.j | 2 | ||
| 12.b | even | 2 | 1 | 3600.3.e.j | 2 | ||
| 15.d | odd | 2 | 1 | 3600.3.e.u | 2 | ||
| 15.e | even | 4 | 2 | 3600.3.j.f | 4 | ||
| 20.d | odd | 2 | 1 | 400.3.b.f | yes | 2 | |
| 20.e | even | 4 | 2 | 400.3.h.b | 4 | ||
| 40.e | odd | 2 | 1 | 1600.3.b.i | 2 | ||
| 40.f | even | 2 | 1 | 1600.3.b.i | 2 | ||
| 40.i | odd | 4 | 2 | 1600.3.h.h | 4 | ||
| 40.k | even | 4 | 2 | 1600.3.h.h | 4 | ||
| 60.h | even | 2 | 1 | 3600.3.e.u | 2 | ||
| 60.l | odd | 4 | 2 | 3600.3.j.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 400.3.b.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 400.3.b.e | ✓ | 2 | 4.b | odd | 2 | 1 | inner |
| 400.3.b.f | yes | 2 | 5.b | even | 2 | 1 | |
| 400.3.b.f | yes | 2 | 20.d | odd | 2 | 1 | |
| 400.3.h.b | 4 | 5.c | odd | 4 | 2 | ||
| 400.3.h.b | 4 | 20.e | even | 4 | 2 | ||
| 1600.3.b.i | 2 | 40.e | odd | 2 | 1 | ||
| 1600.3.b.i | 2 | 40.f | even | 2 | 1 | ||
| 1600.3.b.j | 2 | 8.b | even | 2 | 1 | ||
| 1600.3.b.j | 2 | 8.d | odd | 2 | 1 | ||
| 1600.3.h.h | 4 | 40.i | odd | 4 | 2 | ||
| 1600.3.h.h | 4 | 40.k | even | 4 | 2 | ||
| 3600.3.e.j | 2 | 3.b | odd | 2 | 1 | ||
| 3600.3.e.j | 2 | 12.b | even | 2 | 1 | ||
| 3600.3.e.u | 2 | 15.d | odd | 2 | 1 | ||
| 3600.3.e.u | 2 | 60.h | even | 2 | 1 | ||
| 3600.3.j.f | 4 | 15.e | even | 4 | 2 | ||
| 3600.3.j.f | 4 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\):
|
\( T_{3}^{2} + 3 \)
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\( T_{13} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} + 3 \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} + 12 \)
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| $11$ |
\( T^{2} + 147 \)
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| $13$ |
\( (T + 16)^{2} \)
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| $17$ |
\( (T - 3)^{2} \)
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| $19$ |
\( T^{2} + 507 \)
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| $23$ |
\( T^{2} + 2028 \)
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| $29$ |
\( (T - 12)^{2} \)
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| $31$ |
\( T^{2} + 972 \)
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| $37$ |
\( (T + 50)^{2} \)
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| $41$ |
\( (T + 63)^{2} \)
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| $43$ |
\( T^{2} + 3888 \)
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| $47$ |
\( T^{2} + 192 \)
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| $53$ |
\( (T + 18)^{2} \)
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| $59$ |
\( T^{2} + 3072 \)
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| $61$ |
\( (T - 26)^{2} \)
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| $67$ |
\( T^{2} + 1083 \)
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| $71$ |
\( T^{2} + 768 \)
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| $73$ |
\( (T - 17)^{2} \)
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| $79$ |
\( T^{2} + 7500 \)
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| $83$ |
\( T^{2} + 9747 \)
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| $89$ |
\( (T - 99)^{2} \)
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| $97$ |
\( (T - 134)^{2} \)
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