Newspace parameters
| Level: | \( N \) | \(=\) | \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1100.k (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.78354422234\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
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|
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| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 220) |
| 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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 8\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(551\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 593.1 |
|
0 | 1.00000 | + | 1.00000i | 0 | 0 | 0 | −3.31662 | − | 3.31662i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||
| 593.2 | 0 | 1.00000 | + | 1.00000i | 0 | 0 | 0 | 3.31662 | + | 3.31662i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||
| 857.1 | 0 | 1.00000 | − | 1.00000i | 0 | 0 | 0 | −3.31662 | + | 3.31662i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||
| 857.2 | 0 | 1.00000 | − | 1.00000i | 0 | 0 | 0 | 3.31662 | − | 3.31662i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 55.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1100.2.k.a | 4 | |
| 5.b | even | 2 | 1 | 220.2.k.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 220.2.k.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 1100.2.k.a | 4 | |
| 11.b | odd | 2 | 1 | inner | 1100.2.k.a | 4 | |
| 15.d | odd | 2 | 1 | 1980.2.y.a | 4 | ||
| 15.e | even | 4 | 1 | 1980.2.y.a | 4 | ||
| 20.d | odd | 2 | 1 | 880.2.bd.f | 4 | ||
| 20.e | even | 4 | 1 | 880.2.bd.f | 4 | ||
| 55.d | odd | 2 | 1 | 220.2.k.a | ✓ | 4 | |
| 55.e | even | 4 | 1 | 220.2.k.a | ✓ | 4 | |
| 55.e | even | 4 | 1 | inner | 1100.2.k.a | 4 | |
| 165.d | even | 2 | 1 | 1980.2.y.a | 4 | ||
| 165.l | odd | 4 | 1 | 1980.2.y.a | 4 | ||
| 220.g | even | 2 | 1 | 880.2.bd.f | 4 | ||
| 220.i | odd | 4 | 1 | 880.2.bd.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 220.2.k.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 220.2.k.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 220.2.k.a | ✓ | 4 | 55.d | odd | 2 | 1 | |
| 220.2.k.a | ✓ | 4 | 55.e | even | 4 | 1 | |
| 880.2.bd.f | 4 | 20.d | odd | 2 | 1 | ||
| 880.2.bd.f | 4 | 20.e | even | 4 | 1 | ||
| 880.2.bd.f | 4 | 220.g | even | 2 | 1 | ||
| 880.2.bd.f | 4 | 220.i | odd | 4 | 1 | ||
| 1100.2.k.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1100.2.k.a | 4 | 5.c | odd | 4 | 1 | inner | |
| 1100.2.k.a | 4 | 11.b | odd | 2 | 1 | inner | |
| 1100.2.k.a | 4 | 55.e | even | 4 | 1 | inner | |
| 1980.2.y.a | 4 | 15.d | odd | 2 | 1 | ||
| 1980.2.y.a | 4 | 15.e | even | 4 | 1 | ||
| 1980.2.y.a | 4 | 165.d | even | 2 | 1 | ||
| 1980.2.y.a | 4 | 165.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\):
|
\( T_{3}^{2} - 2T_{3} + 2 \)
|
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\( T_{19} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 484 \)
|
| $11$ |
\( (T^{2} + 11)^{2} \)
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| $13$ |
\( T^{4} + 484 \)
|
| $17$ |
\( T^{4} + 484 \)
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| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 6 T + 18)^{2} \)
|
| $29$ |
\( (T^{2} - 44)^{2} \)
|
| $31$ |
\( (T + 4)^{4} \)
|
| $37$ |
\( (T^{2} + 10 T + 50)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} + 484 \)
|
| $47$ |
\( (T^{2} + 10 T + 50)^{2} \)
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| $53$ |
\( (T^{2} - 6 T + 18)^{2} \)
|
| $59$ |
\( (T^{2} + 100)^{2} \)
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| $61$ |
\( (T^{2} + 176)^{2} \)
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| $67$ |
\( (T^{2} - 6 T + 18)^{2} \)
|
| $71$ |
\( (T + 4)^{4} \)
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| $73$ |
\( T^{4} + 484 \)
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| $79$ |
\( (T^{2} - 176)^{2} \)
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| $83$ |
\( T^{4} + 484 \)
|
| $89$ |
\( (T^{2} + 144)^{2} \)
|
| $97$ |
\( (T^{2} + 10 T + 50)^{2} \)
|
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