Newspace parameters
| Level: | \( N \) | \(=\) | \( 104 = 2^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 104.s (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.830444181021\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(53\) | \(79\) |
| \(\chi(n)\) | \(\zeta_{12}^{2}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.1 |
|
−1.00000 | − | 1.00000i | −2.36603 | − | 1.36603i | 2.00000i | −0.267949 | 1.00000 | + | 3.73205i | −3.00000 | + | 1.73205i | 2.00000 | − | 2.00000i | 2.23205 | + | 3.86603i | 0.267949 | + | 0.267949i | ||||||||||||||||
| 69.2 | −1.00000 | + | 1.00000i | −0.633975 | − | 0.366025i | − | 2.00000i | −3.73205 | 1.00000 | − | 0.267949i | −3.00000 | + | 1.73205i | 2.00000 | + | 2.00000i | −1.23205 | − | 2.13397i | 3.73205 | − | 3.73205i | ||||||||||||||||
| 101.1 | −1.00000 | − | 1.00000i | −0.633975 | + | 0.366025i | 2.00000i | −3.73205 | 1.00000 | + | 0.267949i | −3.00000 | − | 1.73205i | 2.00000 | − | 2.00000i | −1.23205 | + | 2.13397i | 3.73205 | + | 3.73205i | |||||||||||||||||
| 101.2 | −1.00000 | + | 1.00000i | −2.36603 | + | 1.36603i | − | 2.00000i | −0.267949 | 1.00000 | − | 3.73205i | −3.00000 | − | 1.73205i | 2.00000 | + | 2.00000i | 2.23205 | − | 3.86603i | 0.267949 | − | 0.267949i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 104.s | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 104.2.s.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 936.2.dg.b | 4 | ||
| 4.b | odd | 2 | 1 | 416.2.ba.b | 4 | ||
| 8.b | even | 2 | 1 | 104.2.s.b | yes | 4 | |
| 8.d | odd | 2 | 1 | 416.2.ba.a | 4 | ||
| 13.e | even | 6 | 1 | 104.2.s.b | yes | 4 | |
| 24.h | odd | 2 | 1 | 936.2.dg.a | 4 | ||
| 39.h | odd | 6 | 1 | 936.2.dg.a | 4 | ||
| 52.i | odd | 6 | 1 | 416.2.ba.a | 4 | ||
| 104.p | odd | 6 | 1 | 416.2.ba.b | 4 | ||
| 104.s | even | 6 | 1 | inner | 104.2.s.a | ✓ | 4 |
| 312.bg | odd | 6 | 1 | 936.2.dg.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.s.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 104.2.s.a | ✓ | 4 | 104.s | even | 6 | 1 | inner |
| 104.2.s.b | yes | 4 | 8.b | even | 2 | 1 | |
| 104.2.s.b | yes | 4 | 13.e | even | 6 | 1 | |
| 416.2.ba.a | 4 | 8.d | odd | 2 | 1 | ||
| 416.2.ba.a | 4 | 52.i | odd | 6 | 1 | ||
| 416.2.ba.b | 4 | 4.b | odd | 2 | 1 | ||
| 416.2.ba.b | 4 | 104.p | odd | 6 | 1 | ||
| 936.2.dg.a | 4 | 24.h | odd | 2 | 1 | ||
| 936.2.dg.a | 4 | 39.h | odd | 6 | 1 | ||
| 936.2.dg.b | 4 | 3.b | odd | 2 | 1 | ||
| 936.2.dg.b | 4 | 312.bg | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6T_{3}^{3} + 14T_{3}^{2} + 12T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 4 \)
|
| $5$ |
\( (T^{2} + 4 T + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $11$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $13$ |
\( T^{4} - T^{2} + 169 \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 9 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
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| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 324 \)
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| $29$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $31$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $37$ |
\( T^{4} + 12 T^{3} + \cdots + 1089 \)
|
| $41$ |
\( T^{4} + 18 T^{3} + \cdots + 121 \)
|
| $43$ |
\( T^{4} - 12 T^{3} + \cdots + 576 \)
|
| $47$ |
\( T^{4} + 56T^{2} + 484 \)
|
| $53$ |
\( T^{4} + 114T^{2} + 1521 \)
|
| $59$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
|
| $61$ |
\( T^{4} - T^{2} + 1 \)
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| $67$ |
\( T^{4} + 18 T^{3} + \cdots + 6084 \)
|
| $71$ |
\( T^{4} + 6 T^{3} + \cdots + 13924 \)
|
| $73$ |
\( (T^{2} + 3)^{2} \)
|
| $79$ |
\( (T^{2} - 108)^{2} \)
|
| $83$ |
\( (T^{2} - 4 T - 8)^{2} \)
|
| $89$ |
\( T^{4} - 12 T^{3} + \cdots + 16 \)
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| $97$ |
\( T^{4} - 36T^{2} + 1296 \)
|
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