Newspace parameters
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(15.2594118263\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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 \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).
| \(n\) | \(638\) | \(1471\) | \(1522\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 883.1 |
|
− | 1.73205i | −1.00000 | −1.00000 | − | 3.46410i | 1.73205i | 0 | − | 1.73205i | 1.00000 | −6.00000 | |||||||||||||||||||||
| 883.2 | 1.73205i | −1.00000 | −1.00000 | 3.46410i | − | 1.73205i | 0 | 1.73205i | 1.00000 | −6.00000 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1911.2.c.b | 2 | |
| 7.b | odd | 2 | 1 | 1911.2.c.e | 2 | ||
| 7.c | even | 3 | 1 | 273.2.bj.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | 273.2.bj.b | yes | 2 | |
| 13.b | even | 2 | 1 | inner | 1911.2.c.b | 2 | |
| 21.h | odd | 6 | 1 | 819.2.dl.a | 2 | ||
| 21.h | odd | 6 | 1 | 819.2.dl.d | 2 | ||
| 91.b | odd | 2 | 1 | 1911.2.c.e | 2 | ||
| 91.r | even | 6 | 1 | 273.2.bj.a | ✓ | 2 | |
| 91.r | even | 6 | 1 | 273.2.bj.b | yes | 2 | |
| 273.w | odd | 6 | 1 | 819.2.dl.a | 2 | ||
| 273.w | odd | 6 | 1 | 819.2.dl.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.bj.a | ✓ | 2 | 7.c | even | 3 | 1 | |
| 273.2.bj.a | ✓ | 2 | 91.r | even | 6 | 1 | |
| 273.2.bj.b | yes | 2 | 7.c | even | 3 | 1 | |
| 273.2.bj.b | yes | 2 | 91.r | even | 6 | 1 | |
| 819.2.dl.a | 2 | 21.h | odd | 6 | 1 | ||
| 819.2.dl.a | 2 | 273.w | odd | 6 | 1 | ||
| 819.2.dl.d | 2 | 21.h | odd | 6 | 1 | ||
| 819.2.dl.d | 2 | 273.w | odd | 6 | 1 | ||
| 1911.2.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 1911.2.c.b | 2 | 13.b | even | 2 | 1 | inner | |
| 1911.2.c.e | 2 | 7.b | odd | 2 | 1 | ||
| 1911.2.c.e | 2 | 91.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1911, [\chi])\):
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\( T_{2}^{2} + 3 \)
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\( T_{5}^{2} + 12 \)
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\( T_{17} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 3 \)
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| $3$ |
\( (T + 1)^{2} \)
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| $5$ |
\( T^{2} + 12 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} + 27 \)
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| $13$ |
\( T^{2} + 2T + 13 \)
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| $17$ |
\( (T + 3)^{2} \)
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| $19$ |
\( T^{2} + 27 \)
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| $23$ |
\( (T - 6)^{2} \)
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| $29$ |
\( (T - 3)^{2} \)
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| $31$ |
\( T^{2} + 12 \)
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| $37$ |
\( T^{2} + 12 \)
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| $41$ |
\( T^{2} + 108 \)
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| $43$ |
\( (T - 8)^{2} \)
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| $47$ |
\( T^{2} + 3 \)
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| $53$ |
\( (T - 3)^{2} \)
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| $59$ |
\( T^{2} + 75 \)
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| $61$ |
\( (T - 1)^{2} \)
|
| $67$ |
\( T^{2} + 3 \)
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| $71$ |
\( T^{2} + 147 \)
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| $73$ |
\( T^{2} + 108 \)
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| $79$ |
\( (T - 4)^{2} \)
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| $83$ |
\( T^{2} + 12 \)
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| $89$ |
\( T^{2} + 12 \)
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| $97$ |
\( T^{2} + 300 \)
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