Newspace parameters
| Level: | \( N \) | \(=\) | \( 1055 = 5 \cdot 211 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1055.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.526513588328\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{36})^+\) |
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| Defining polynomial: |
\( x^{6} - 6x^{4} + 9x^{2} - 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{18}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
$q$-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 \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 4 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 4\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 6 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 11\beta_1 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1055\mathbb{Z}\right)^\times\).
| \(n\) | \(212\) | \(846\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1054.1 |
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−1.73205 | −1.28558 | 2.00000 | −1.00000 | 2.22668 | 1.96962 | −1.73205 | 0.652704 | 1.73205 | ||||||||||||||||||||||||||||||||||||
| 1054.2 | −1.73205 | −0.684040 | 2.00000 | −1.00000 | 1.18479 | −1.28558 | −1.73205 | −0.532089 | 1.73205 | |||||||||||||||||||||||||||||||||||||
| 1054.3 | −1.73205 | 1.96962 | 2.00000 | −1.00000 | −3.41147 | −0.684040 | −1.73205 | 2.87939 | 1.73205 | |||||||||||||||||||||||||||||||||||||
| 1054.4 | 1.73205 | −1.96962 | 2.00000 | −1.00000 | −3.41147 | 0.684040 | 1.73205 | 2.87939 | −1.73205 | |||||||||||||||||||||||||||||||||||||
| 1054.5 | 1.73205 | 0.684040 | 2.00000 | −1.00000 | 1.18479 | 1.28558 | 1.73205 | −0.532089 | −1.73205 | |||||||||||||||||||||||||||||||||||||
| 1054.6 | 1.73205 | 1.28558 | 2.00000 | −1.00000 | 2.22668 | −1.96962 | 1.73205 | 0.652704 | −1.73205 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 1055.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-1055}) \) |
| 5.b | even | 2 | 1 | inner |
| 211.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1055.1.d.h | ✓ | 6 |
| 5.b | even | 2 | 1 | inner | 1055.1.d.h | ✓ | 6 |
| 211.b | odd | 2 | 1 | inner | 1055.1.d.h | ✓ | 6 |
| 1055.d | odd | 2 | 1 | CM | 1055.1.d.h | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1055.1.d.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1055.1.d.h | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 1055.1.d.h | ✓ | 6 | 211.b | odd | 2 | 1 | inner |
| 1055.1.d.h | ✓ | 6 | 1055.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1055, [\chi])\):
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\( T_{2}^{2} - 3 \)
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\( T_{3}^{6} - 6T_{3}^{4} + 9T_{3}^{2} - 3 \)
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\( T_{11}^{3} - 3T_{11} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 3)^{3} \)
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| $3$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
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| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
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| $11$ |
\( (T^{3} - 3 T + 1)^{2} \)
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| $13$ |
\( T^{6} \)
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| $17$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
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| $19$ |
\( (T^{3} - 3 T + 1)^{2} \)
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| $23$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} \)
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| $37$ |
\( T^{6} \)
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| $41$ |
\( T^{6} \)
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| $43$ |
\( T^{6} \)
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| $47$ |
\( T^{6} \)
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| $53$ |
\( T^{6} \)
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| $59$ |
\( (T^{3} - 3 T + 1)^{2} \)
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| $61$ |
\( T^{6} \)
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| $67$ |
\( T^{6} \)
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| $71$ |
\( (T - 1)^{6} \)
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| $73$ |
\( T^{6} \)
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| $79$ |
\( (T^{3} - 3 T + 1)^{2} \)
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| $83$ |
\( T^{6} \)
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| $89$ |
\( T^{6} \)
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| $97$ |
\( T^{6} - 6 T^{4} + \cdots - 3 \)
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