Newspace parameters
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.n (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.8538461238\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 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
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.43649 | − | 2.48808i | 4.50000 | + | 2.59808i | 3.87298 | − | 6.70820i | −9.68246 | + | 5.59017i | − | 14.9285i | −33.6663 | + | 35.6031i | −68.2218 | 13.5000 | + | 23.3827i | 27.8175 | + | 16.0605i | |||||||||||||||
| 31.2 | 2.43649 | + | 4.22013i | 4.50000 | + | 2.59808i | −3.87298 | + | 6.70820i | 9.68246 | − | 5.59017i | 25.3208i | 47.6663 | − | 11.3544i | 40.2218 | 13.5000 | + | 23.3827i | 47.1825 | + | 27.2408i | |||||||||||||||||
| 61.1 | −1.43649 | + | 2.48808i | 4.50000 | − | 2.59808i | 3.87298 | + | 6.70820i | −9.68246 | − | 5.59017i | 14.9285i | −33.6663 | − | 35.6031i | −68.2218 | 13.5000 | − | 23.3827i | 27.8175 | − | 16.0605i | |||||||||||||||||
| 61.2 | 2.43649 | − | 4.22013i | 4.50000 | − | 2.59808i | −3.87298 | − | 6.70820i | 9.68246 | + | 5.59017i | − | 25.3208i | 47.6663 | + | 11.3544i | 40.2218 | 13.5000 | − | 23.3827i | 47.1825 | − | 27.2408i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.5.n.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | inner | 105.5.n.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.5.n.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 105.5.n.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 18T_{2}^{2} + 28T_{2} + 196 \)
acting on \(S_{5}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 196 \)
|
| $3$ |
\( (T^{2} - 9 T + 27)^{2} \)
|
| $5$ |
\( T^{4} - 125 T^{2} + 15625 \)
|
| $7$ |
\( T^{4} - 28 T^{3} + \cdots + 5764801 \)
|
| $11$ |
\( T^{4} + 190 T^{3} + \cdots + 21996100 \)
|
| $13$ |
\( T^{4} + 33786 T^{2} + 83119689 \)
|
| $17$ |
\( T^{4} + 402 T^{3} + \cdots + 34363044 \)
|
| $19$ |
\( T^{4} + \cdots + 15846529689 \)
|
| $23$ |
\( T^{4} + \cdots + 218004948100 \)
|
| $29$ |
\( (T^{2} + 1442 T + 516466)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 577215504009 \)
|
| $37$ |
\( T^{4} + \cdots + 3258697225 \)
|
| $41$ |
\( T^{4} + \cdots + 38228852484 \)
|
| $43$ |
\( (T^{2} + 56 T - 427631)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 31051418864400 \)
|
| $53$ |
\( T^{4} + \cdots + 102637485192256 \)
|
| $59$ |
\( T^{4} + \cdots + 348440387567364 \)
|
| $61$ |
\( T^{4} + \cdots + 340588944322704 \)
|
| $67$ |
\( T^{4} + \cdots + 577518231128881 \)
|
| $71$ |
\( (T^{2} - 1558 T - 14124374)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 697493751751449 \)
|
| $79$ |
\( T^{4} + \cdots + 27\!\cdots\!21 \)
|
| $83$ |
\( T^{4} + \cdots + 160308568335204 \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 15\!\cdots\!64 \)
|
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