Newspace parameters
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.j (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.42608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
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| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{3} \)
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| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
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| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
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| \(\beta_{4}\) | \(=\) |
\( 2\zeta_{24}^{4} - 1 \)
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| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24} \)
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| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
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| \(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} - \zeta_{24}^{3} \)
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| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
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| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
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| \(\zeta_{24}^{3}\) | \(=\) |
\( \beta_1 \)
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| \(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{4} + 1 ) / 2 \)
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| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} ) / 2 \)
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| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
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| \(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_1 ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 497.1 |
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−0.707107 | + | 0.707107i | −1.22474 | + | 1.22474i | − | 1.00000i | −2.12132 | − | 0.707107i | − | 1.73205i | −3.44949 | − | 3.44949i | 0.707107 | + | 0.707107i | − | 3.00000i | 2.00000 | − | 1.00000i | |||||||||||||||||||||||||||
| 497.2 | −0.707107 | + | 0.707107i | 1.22474 | − | 1.22474i | − | 1.00000i | −2.12132 | − | 0.707107i | 1.73205i | 1.44949 | + | 1.44949i | 0.707107 | + | 0.707107i | − | 3.00000i | 2.00000 | − | 1.00000i | |||||||||||||||||||||||||||||
| 497.3 | 0.707107 | − | 0.707107i | −1.22474 | + | 1.22474i | − | 1.00000i | 2.12132 | + | 0.707107i | 1.73205i | −3.44949 | − | 3.44949i | −0.707107 | − | 0.707107i | − | 3.00000i | 2.00000 | − | 1.00000i | |||||||||||||||||||||||||||||
| 497.4 | 0.707107 | − | 0.707107i | 1.22474 | − | 1.22474i | − | 1.00000i | 2.12132 | + | 0.707107i | − | 1.73205i | 1.44949 | + | 1.44949i | −0.707107 | − | 0.707107i | − | 3.00000i | 2.00000 | − | 1.00000i | ||||||||||||||||||||||||||||
| 683.1 | −0.707107 | − | 0.707107i | −1.22474 | − | 1.22474i | 1.00000i | −2.12132 | + | 0.707107i | 1.73205i | −3.44949 | + | 3.44949i | 0.707107 | − | 0.707107i | 3.00000i | 2.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||
| 683.2 | −0.707107 | − | 0.707107i | 1.22474 | + | 1.22474i | 1.00000i | −2.12132 | + | 0.707107i | − | 1.73205i | 1.44949 | − | 1.44949i | 0.707107 | − | 0.707107i | 3.00000i | 2.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||
| 683.3 | 0.707107 | + | 0.707107i | −1.22474 | − | 1.22474i | 1.00000i | 2.12132 | − | 0.707107i | − | 1.73205i | −3.44949 | + | 3.44949i | −0.707107 | + | 0.707107i | 3.00000i | 2.00000 | + | 1.00000i | ||||||||||||||||||||||||||||||
| 683.4 | 0.707107 | + | 0.707107i | 1.22474 | + | 1.22474i | 1.00000i | 2.12132 | − | 0.707107i | 1.73205i | 1.44949 | − | 1.44949i | −0.707107 | + | 0.707107i | 3.00000i | 2.00000 | + | 1.00000i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 930.2.j.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 930.2.j.d | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 930.2.j.d | ✓ | 8 |
| 15.e | even | 4 | 1 | inner | 930.2.j.d | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.j.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 930.2.j.d | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 930.2.j.d | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 930.2.j.d | ✓ | 8 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\):
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\( T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 40T_{7} + 100 \)
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\( T_{11}^{4} + 40T_{11}^{2} + 16 \)
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\( T_{17}^{8} + 392T_{17}^{4} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
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| $3$ |
\( (T^{4} + 9)^{2} \)
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| $5$ |
\( (T^{4} - 8 T^{2} + 25)^{2} \)
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| $7$ |
\( (T^{4} + 4 T^{3} + \cdots + 100)^{2} \)
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| $11$ |
\( (T^{4} + 40 T^{2} + 16)^{2} \)
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| $13$ |
\( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
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| $17$ |
\( T^{8} + 392T^{4} + 16 \)
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| $19$ |
\( (T^{2} + 24)^{4} \)
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| $23$ |
\( (T^{4} + 36)^{2} \)
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| $29$ |
\( (T^{2} - 8)^{4} \)
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| $31$ |
\( (T + 1)^{8} \)
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| $37$ |
\( T^{8} \)
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| $41$ |
\( (T^{2} + 8)^{4} \)
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| $43$ |
\( (T^{4} + 144)^{2} \)
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| $47$ |
\( (T^{4} + 4096)^{2} \)
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| $53$ |
\( (T^{4} + 1296)^{2} \)
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| $59$ |
\( (T^{4} - 124 T^{2} + 1444)^{2} \)
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| $61$ |
\( (T^{2} + 4 T - 2)^{4} \)
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| $67$ |
\( (T^{4} + 24 T^{3} + \cdots + 3600)^{2} \)
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| $71$ |
\( (T^{4} + 88 T^{2} + 400)^{2} \)
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| $73$ |
\( (T^{4} + 2304)^{2} \)
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| $79$ |
\( (T^{4} + 80 T^{2} + 64)^{2} \)
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| $83$ |
\( T^{8} + 6272 T^{4} + 4096 \)
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| $89$ |
\( (T^{4} - 352 T^{2} + 6400)^{2} \)
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| $97$ |
\( (T^{2} + 6 T + 18)^{4} \)
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