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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 38x^{3} + 61x^{2} + 304x - 148 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 38x^{3} + 61x^{2} + 304x - 148 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 22\nu^{2} - 87\nu - 18 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 4\nu^{3} + 26\nu^{2} + 87\nu - 46 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} - 8\nu^{3} + 74\nu^{2} + 165\nu - 82 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + \beta_{2} + 24\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{4} + 30\beta_{3} + 22\beta_{2} - 9\beta _1 + 362 \)
|
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 |
|
−5.19955 | 7.06563 | 19.0353 | 0 | −36.7381 | 32.4859 | −57.3785 | 22.9231 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.75920 | −2.30980 | −0.386826 | 0 | 6.37319 | 13.2555 | 23.1409 | −21.6648 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.457079 | −2.45289 | −7.79108 | 0 | −1.12116 | −23.1894 | −7.21777 | −20.9833 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.72095 | −8.29546 | 14.2874 | 0 | −39.1625 | 5.48389 | 29.6825 | 41.8147 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.78071 | 5.99252 | 14.8552 | 0 | 28.6485 | 11.9641 | 32.7728 | 8.91034 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 275.4.a.h | yes | 5 |
| 3.b | odd | 2 | 1 | 2475.4.a.bh | 5 | ||
| 5.b | even | 2 | 1 | 275.4.a.g | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 275.4.b.f | 10 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.bl | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 275.4.a.g | ✓ | 5 | 5.b | even | 2 | 1 | |
| 275.4.a.h | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 275.4.b.f | 10 | 5.c | odd | 4 | 2 | ||
| 2475.4.a.bh | 5 | 3.b | odd | 2 | 1 | ||
| 2475.4.a.bl | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 38T_{2}^{3} + 61T_{2}^{2} + 304T_{2} - 148 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(275))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 148 \)
|
| $3$ |
\( T^{5} - 83 T^{3} + \cdots + 1990 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 40 T^{4} + \cdots + 655160 \)
|
| $11$ |
\( (T + 11)^{5} \)
|
| $13$ |
\( T^{5} - 211 T^{4} + \cdots + 259208125 \)
|
| $17$ |
\( T^{5} + \cdots - 2036840548 \)
|
| $19$ |
\( T^{5} + \cdots - 2305690119 \)
|
| $23$ |
\( T^{5} - 57 T^{4} + \cdots + 1037717 \)
|
| $29$ |
\( T^{5} + \cdots - 16865962259 \)
|
| $31$ |
\( T^{5} + \cdots + 94367463777 \)
|
| $37$ |
\( T^{5} + \cdots - 32982950164 \)
|
| $41$ |
\( T^{5} + \cdots + 29753829648 \)
|
| $43$ |
\( T^{5} + \cdots - 1411617547728 \)
|
| $47$ |
\( T^{5} + \cdots - 566636405656 \)
|
| $53$ |
\( T^{5} + \cdots + 983599875694 \)
|
| $59$ |
\( T^{5} + \cdots + 13620640524856 \)
|
| $61$ |
\( T^{5} + \cdots + 2812375273130 \)
|
| $67$ |
\( T^{5} + \cdots - 13838126809088 \)
|
| $71$ |
\( T^{5} + \cdots + 38185247888 \)
|
| $73$ |
\( T^{5} + \cdots + 17706351601732 \)
|
| $79$ |
\( T^{5} + \cdots - 718883827912 \)
|
| $83$ |
\( T^{5} + \cdots - 17345803708953 \)
|
| $89$ |
\( T^{5} + \cdots - 28452224444275 \)
|
| $97$ |
\( T^{5} + \cdots + 46751646619135 \)
|
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