Newspace parameters
| Level: | \( N \) | \(=\) | \( 275 = 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 275.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(16.2255252516\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2301792529.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - 22x^{4} + 101x^{2} - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 55) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 22x^{4} + 101x^{2} - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 13\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 17\nu^{2} + 32 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 21\nu^{3} + 84\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 13\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 17\beta_{2} + 87 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 42\beta_{3} + 189\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−3.94784 | 3.56498 | 7.58548 | 0 | −14.0740 | 2.30429 | 1.63647 | −14.2909 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.50006 | −2.61871 | −1.74972 | 0 | 6.54691 | 4.41889 | 24.3748 | −20.1424 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.405276 | 8.56932 | −7.83575 | 0 | −3.47294 | −27.4984 | 6.41784 | 46.4333 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.405276 | −8.56932 | −7.83575 | 0 | −3.47294 | 27.4984 | −6.41784 | 46.4333 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.50006 | 2.61871 | −1.74972 | 0 | 6.54691 | −4.41889 | −24.3748 | −20.1424 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 3.94784 | −3.56498 | 7.58548 | 0 | −14.0740 | −2.30429 | −1.63647 | −14.2909 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.4.a.j | 6 | |
| 3.b | odd | 2 | 1 | 2475.4.a.bn | 6 | ||
| 5.b | even | 2 | 1 | inner | 275.4.a.j | 6 | |
| 5.c | odd | 4 | 2 | 55.4.b.a | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 2475.4.a.bn | 6 | ||
| 15.e | even | 4 | 2 | 495.4.c.a | 6 | ||
| 20.e | even | 4 | 2 | 880.4.b.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.b.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 275.4.a.j | 6 | 1.a | even | 1 | 1 | trivial | |
| 275.4.a.j | 6 | 5.b | even | 2 | 1 | inner | |
| 495.4.c.a | 6 | 15.e | even | 4 | 2 | ||
| 880.4.b.f | 6 | 20.e | even | 4 | 2 | ||
| 2475.4.a.bn | 6 | 3.b | odd | 2 | 1 | ||
| 2475.4.a.bn | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 22T_{2}^{4} + 101T_{2}^{2} - 16 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(275))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 22 T^{4} + \cdots - 16 \)
|
| $3$ |
\( T^{6} - 93 T^{4} + \cdots - 6400 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 781 T^{4} + \cdots - 78400 \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} - 3096 T^{4} + \cdots - 595360000 \)
|
| $17$ |
\( T^{6} - 5441 T^{4} + \cdots - 545502736 \)
|
| $19$ |
\( (T^{3} + 129 T^{2} + \cdots - 493744)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 994439750656 \)
|
| $29$ |
\( (T^{3} + 247 T^{2} + \cdots - 2047004)^{2} \)
|
| $31$ |
\( (T^{3} + 257 T^{2} + \cdots - 484864)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 294700670255104 \)
|
| $41$ |
\( (T^{3} + 412 T^{2} + \cdots + 143488)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 23\!\cdots\!96 \)
|
| $47$ |
\( T^{6} + \cdots - 5233772609536 \)
|
| $53$ |
\( T^{6} + \cdots - 14\!\cdots\!44 \)
|
| $59$ |
\( (T^{3} + 100 T^{2} + \cdots - 20068352)^{2} \)
|
| $61$ |
\( (T^{3} + 1105 T^{2} + \cdots - 114682660)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 10\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} + 135 T^{2} + \cdots - 68294144)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 27232283023936 \)
|
| $79$ |
\( (T^{3} + 412 T^{2} + \cdots - 287752192)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 35\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} - 93 T^{2} + \cdots - 155639300)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| show more | |
| show less |