Newspace parameters
| Level: | \( N \) | \(=\) | \( 704 = 2^{6} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 704.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.1826106110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 44) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).
| \(n\) | \(133\) | \(321\) | \(639\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −0.372281 | 0 | −9.11684 | 0 | 0 | 0 | −8.86141 | 0 | ||||||||||||||||||||||||
| 65.2 | 0 | 5.37228 | 0 | 8.11684 | 0 | 0 | 0 | 19.8614 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 704.3.h.f | 2 | |
| 4.b | odd | 2 | 1 | 704.3.h.c | 2 | ||
| 8.b | even | 2 | 1 | 176.3.h.b | 2 | ||
| 8.d | odd | 2 | 1 | 44.3.d.a | ✓ | 2 | |
| 11.b | odd | 2 | 1 | CM | 704.3.h.f | 2 | |
| 24.f | even | 2 | 1 | 396.3.f.a | 2 | ||
| 24.h | odd | 2 | 1 | 1584.3.j.c | 2 | ||
| 40.e | odd | 2 | 1 | 1100.3.f.a | 2 | ||
| 40.k | even | 4 | 2 | 1100.3.e.a | 4 | ||
| 44.c | even | 2 | 1 | 704.3.h.c | 2 | ||
| 56.e | even | 2 | 1 | 2156.3.h.a | 2 | ||
| 88.b | odd | 2 | 1 | 176.3.h.b | 2 | ||
| 88.g | even | 2 | 1 | 44.3.d.a | ✓ | 2 | |
| 88.k | even | 10 | 4 | 484.3.f.b | 8 | ||
| 88.l | odd | 10 | 4 | 484.3.f.b | 8 | ||
| 264.m | even | 2 | 1 | 1584.3.j.c | 2 | ||
| 264.p | odd | 2 | 1 | 396.3.f.a | 2 | ||
| 440.c | even | 2 | 1 | 1100.3.f.a | 2 | ||
| 440.w | odd | 4 | 2 | 1100.3.e.a | 4 | ||
| 616.g | odd | 2 | 1 | 2156.3.h.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.3.d.a | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 44.3.d.a | ✓ | 2 | 88.g | even | 2 | 1 | |
| 176.3.h.b | 2 | 8.b | even | 2 | 1 | ||
| 176.3.h.b | 2 | 88.b | odd | 2 | 1 | ||
| 396.3.f.a | 2 | 24.f | even | 2 | 1 | ||
| 396.3.f.a | 2 | 264.p | odd | 2 | 1 | ||
| 484.3.f.b | 8 | 88.k | even | 10 | 4 | ||
| 484.3.f.b | 8 | 88.l | odd | 10 | 4 | ||
| 704.3.h.c | 2 | 4.b | odd | 2 | 1 | ||
| 704.3.h.c | 2 | 44.c | even | 2 | 1 | ||
| 704.3.h.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 704.3.h.f | 2 | 11.b | odd | 2 | 1 | CM | |
| 1100.3.e.a | 4 | 40.k | even | 4 | 2 | ||
| 1100.3.e.a | 4 | 440.w | odd | 4 | 2 | ||
| 1100.3.f.a | 2 | 40.e | odd | 2 | 1 | ||
| 1100.3.f.a | 2 | 440.c | even | 2 | 1 | ||
| 1584.3.j.c | 2 | 24.h | odd | 2 | 1 | ||
| 1584.3.j.c | 2 | 264.m | even | 2 | 1 | ||
| 2156.3.h.a | 2 | 56.e | even | 2 | 1 | ||
| 2156.3.h.a | 2 | 616.g | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 5T_{3} - 2 \)
acting on \(S_{3}^{\mathrm{new}}(704, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 5T - 2 \)
|
| $5$ |
\( T^{2} + T - 74 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T + 11)^{2} \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} - 35T - 362 \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} + 37T - 1514 \)
|
| $37$ |
\( T^{2} + 25T - 3482 \)
|
| $41$ |
\( T^{2} \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( (T + 50)^{2} \)
|
| $53$ |
\( (T - 70)^{2} \)
|
| $59$ |
\( T^{2} + 107T + 1006 \)
|
| $61$ |
\( T^{2} \)
|
| $67$ |
\( T^{2} + 35T - 12242 \)
|
| $71$ |
\( T^{2} + 133T + 2566 \)
|
| $73$ |
\( T^{2} \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} - 97T - 14354 \)
|
| $97$ |
\( T^{2} + 95T - 19202 \)
|
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