Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(54.2097310968\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{849})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 425x^{2} + 44944 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 425x^{2} + 44944 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 213\nu ) / 106 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 213 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 213 \)
|
| \(\nu^{3}\) | \(=\) |
\( 106\beta_{2} - 213\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 4.00000i | −10.0688 | −16.0000 | − | 80.2064i | 40.2752i | 208.619i | 64.0000i | −141.619 | −320.826 | ||||||||||||||||||||||||||||
| 337.2 | − | 4.00000i | 19.0688 | −16.0000 | 7.20641i | − | 76.2752i | − | 53.6192i | 64.0000i | 120.619 | 28.8256 | ||||||||||||||||||||||||||||
| 337.3 | 4.00000i | −10.0688 | −16.0000 | 80.2064i | − | 40.2752i | − | 208.619i | − | 64.0000i | −141.619 | −320.826 | ||||||||||||||||||||||||||||
| 337.4 | 4.00000i | 19.0688 | −16.0000 | − | 7.20641i | 76.2752i | 53.6192i | − | 64.0000i | 120.619 | 28.8256 | |||||||||||||||||||||||||||||
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 | 338.6.b.b | 4 | |
| 13.b | even | 2 | 1 | inner | 338.6.b.b | 4 | |
| 13.d | odd | 4 | 1 | 26.6.a.c | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 338.6.a.f | 2 | ||
| 39.f | even | 4 | 1 | 234.6.a.h | 2 | ||
| 52.f | even | 4 | 1 | 208.6.a.g | 2 | ||
| 65.f | even | 4 | 1 | 650.6.b.h | 4 | ||
| 65.g | odd | 4 | 1 | 650.6.a.b | 2 | ||
| 65.k | even | 4 | 1 | 650.6.b.h | 4 | ||
| 104.j | odd | 4 | 1 | 832.6.a.k | 2 | ||
| 104.m | even | 4 | 1 | 832.6.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.6.a.c | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 208.6.a.g | 2 | 52.f | even | 4 | 1 | ||
| 234.6.a.h | 2 | 39.f | even | 4 | 1 | ||
| 338.6.a.f | 2 | 13.d | odd | 4 | 1 | ||
| 338.6.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 338.6.b.b | 4 | 13.b | even | 2 | 1 | inner | |
| 650.6.a.b | 2 | 65.g | odd | 4 | 1 | ||
| 650.6.b.h | 4 | 65.f | even | 4 | 1 | ||
| 650.6.b.h | 4 | 65.k | even | 4 | 1 | ||
| 832.6.a.k | 2 | 104.j | odd | 4 | 1 | ||
| 832.6.a.m | 2 | 104.m | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 9T_{3} - 192 \)
acting on \(S_{6}^{\mathrm{new}}(338, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{2} \)
|
| $3$ |
\( (T^{2} - 9 T - 192)^{2} \)
|
| $5$ |
\( T^{4} + 6485 T^{2} + 334084 \)
|
| $7$ |
\( T^{4} + 46397 T^{2} + 125126596 \)
|
| $11$ |
\( T^{4} + \cdots + 12134344336 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 189 T - 3523122)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 629489907216 \)
|
| $23$ |
\( (T^{2} - 3044 T - 159200)^{2} \)
|
| $29$ |
\( (T^{2} - 1900 T - 13890476)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 804852814725184 \)
|
| $37$ |
\( T^{4} + \cdots + 52\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots + 131469156000000 \)
|
| $43$ |
\( (T^{2} - 4069 T - 6040532)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!76 \)
|
| $53$ |
\( (T^{2} + 4614 T - 129839400)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 86\!\cdots\!56 \)
|
| $61$ |
\( (T^{2} - 96830 T + 2309711776)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots + 34\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} - 52024 T + 439936528)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 69\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $97$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
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