Newspace parameters
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.n (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.95259490005\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.819390625.1 |
| Defining polynomial: |
\( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 165) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -256\nu^{7} + 341\nu^{6} + 3310\nu^{5} - 16865\nu^{4} + 32996\nu^{3} - 59433\nu^{2} + 33270\nu - 118459 ) / 171589 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -620\nu^{7} + 2532\nu^{6} - 9045\nu^{5} + 18870\nu^{4} - 41955\nu^{3} + 81515\nu^{2} - 102225\nu - 16104 ) / 171589 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -672\nu^{7} + 2845\nu^{6} - 10810\nu^{5} + 23975\nu^{4} - 77175\nu^{3} + 52625\nu^{2} - 72556\nu - 75020 ) / 171589 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1187 \nu^{7} + 4445 \nu^{6} - 17191 \nu^{5} + 14238 \nu^{4} - 12015 \nu^{3} - 32510 \nu^{2} - 25003 \nu - 143748 ) / 171589 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1188 \nu^{7} - 4751 \nu^{6} + 16325 \nu^{5} - 32635 \nu^{4} + 48690 \nu^{3} - 20331 \nu^{2} + 62530 \nu + 144881 ) / 171589 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1432 \nu^{7} + 1420 \nu^{6} - 7808 \nu^{5} - 2207 \nu^{4} - 27965 \nu^{3} - 33635 \nu^{2} - 187297 \nu - 249744 ) / 171589 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{4} + 3\beta_{3} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} - 5\beta_{4} + 2\beta_{3} - \beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 15\beta_{6} - 8\beta_{5} - 8\beta_{4} - 5\beta_{3} - 5\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{7} - 25\beta_{6} - 28\beta_{5} + 25\beta_{2} - 2\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -55\beta_{7} + 55\beta_{4} + 22\beta_{3} + 110\beta_{2} - 45\beta _1 + 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( -165\beta_{7} + 175\beta_{6} + 165\beta_{5} + 188\beta_{4} - 80\beta_{3} - 188\beta _1 - 175 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) | \(397\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−2.00643 | + | 1.45776i | 0 | 1.28267 | − | 3.94765i | −0.809017 | − | 0.587785i | 0 | −1.35808 | + | 4.17973i | 1.64835 | + | 5.07311i | 0 | 2.48008 | ||||||||||||||||||||||||||||||||
| 91.2 | 1.50643 | − | 1.09448i | 0 | 0.453397 | − | 1.39541i | −0.809017 | − | 0.587785i | 0 | 0.812990 | − | 2.50213i | 0.306561 | + | 0.943499i | 0 | −1.86205 | |||||||||||||||||||||||||||||||||
| 136.1 | −2.00643 | − | 1.45776i | 0 | 1.28267 | + | 3.94765i | −0.809017 | + | 0.587785i | 0 | −1.35808 | − | 4.17973i | 1.64835 | − | 5.07311i | 0 | 2.48008 | |||||||||||||||||||||||||||||||||
| 136.2 | 1.50643 | + | 1.09448i | 0 | 0.453397 | + | 1.39541i | −0.809017 | + | 0.587785i | 0 | 0.812990 | + | 2.50213i | 0.306561 | − | 0.943499i | 0 | −1.86205 | |||||||||||||||||||||||||||||||||
| 181.1 | −0.788477 | − | 2.42668i | 0 | −3.64906 | + | 2.65120i | 0.309017 | − | 0.951057i | 0 | 3.39382 | − | 2.46575i | 5.18229 | + | 3.76516i | 0 | −2.55157 | |||||||||||||||||||||||||||||||||
| 181.2 | 0.288477 | + | 0.887841i | 0 | 0.912991 | − | 0.663327i | 0.309017 | − | 0.951057i | 0 | 1.65127 | − | 1.19972i | 2.36279 | + | 1.71667i | 0 | 0.933531 | |||||||||||||||||||||||||||||||||
| 361.1 | −0.788477 | + | 2.42668i | 0 | −3.64906 | − | 2.65120i | 0.309017 | + | 0.951057i | 0 | 3.39382 | + | 2.46575i | 5.18229 | − | 3.76516i | 0 | −2.55157 | |||||||||||||||||||||||||||||||||
| 361.2 | 0.288477 | − | 0.887841i | 0 | 0.912991 | + | 0.663327i | 0.309017 | + | 0.951057i | 0 | 1.65127 | + | 1.19972i | 2.36279 | − | 1.71667i | 0 | 0.933531 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 495.2.n.b | 8 | |
| 3.b | odd | 2 | 1 | 165.2.m.b | ✓ | 8 | |
| 11.c | even | 5 | 1 | inner | 495.2.n.b | 8 | |
| 11.c | even | 5 | 1 | 5445.2.a.bk | 4 | ||
| 11.d | odd | 10 | 1 | 5445.2.a.br | 4 | ||
| 15.d | odd | 2 | 1 | 825.2.n.i | 8 | ||
| 15.e | even | 4 | 2 | 825.2.bx.g | 16 | ||
| 33.f | even | 10 | 1 | 1815.2.a.r | 4 | ||
| 33.h | odd | 10 | 1 | 165.2.m.b | ✓ | 8 | |
| 33.h | odd | 10 | 1 | 1815.2.a.v | 4 | ||
| 165.o | odd | 10 | 1 | 825.2.n.i | 8 | ||
| 165.o | odd | 10 | 1 | 9075.2.a.cq | 4 | ||
| 165.r | even | 10 | 1 | 9075.2.a.dg | 4 | ||
| 165.v | even | 20 | 2 | 825.2.bx.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.m.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 165.2.m.b | ✓ | 8 | 33.h | odd | 10 | 1 | |
| 495.2.n.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 495.2.n.b | 8 | 11.c | even | 5 | 1 | inner | |
| 825.2.n.i | 8 | 15.d | odd | 2 | 1 | ||
| 825.2.n.i | 8 | 165.o | odd | 10 | 1 | ||
| 825.2.bx.g | 16 | 15.e | even | 4 | 2 | ||
| 825.2.bx.g | 16 | 165.v | even | 20 | 2 | ||
| 1815.2.a.r | 4 | 33.f | even | 10 | 1 | ||
| 1815.2.a.v | 4 | 33.h | odd | 10 | 1 | ||
| 5445.2.a.bk | 4 | 11.c | even | 5 | 1 | ||
| 5445.2.a.br | 4 | 11.d | odd | 10 | 1 | ||
| 9075.2.a.cq | 4 | 165.o | odd | 10 | 1 | ||
| 9075.2.a.dg | 4 | 165.r | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} - 3T_{2}^{5} + 4T_{2}^{4} + 3T_{2}^{3} + 135T_{2}^{2} - 77T_{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{7} + 5 T^{6} - 3 T^{5} + \cdots + 121 \)
$3$
\( T^{8} \)
$5$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$7$
\( T^{8} - 9 T^{7} + 55 T^{6} + \cdots + 9801 \)
$11$
\( T^{8} - 3 T^{7} + 8 T^{6} + \cdots + 14641 \)
$13$
\( T^{8} - 10 T^{7} + 57 T^{6} + \cdots + 9801 \)
$17$
\( T^{8} - 2 T^{7} + 21 T^{6} + \cdots + 9801 \)
$19$
\( T^{8} + 2 T^{7} + T^{6} - 9 T^{5} + \cdots + 81 \)
$23$
\( (T^{4} - T^{3} - 39 T^{2} + 29 T + 341)^{2} \)
$29$
\( T^{8} + 14 T^{7} + 235 T^{6} + \cdots + 383161 \)
$31$
\( T^{8} + 5 T^{7} + 95 T^{6} + 1025 T^{5} + \cdots + 625 \)
$37$
\( T^{8} + 27 T^{7} + 415 T^{6} + \cdots + 1324801 \)
$41$
\( T^{8} + T^{7} + 45 T^{6} + 279 T^{5} + \cdots + 121 \)
$43$
\( (T^{4} + 14 T^{3} + 50 T^{2} + 57 T + 9)^{2} \)
$47$
\( T^{8} - 27 T^{7} + 423 T^{6} + \cdots + 12313081 \)
$53$
\( T^{8} - T^{7} + 50 T^{6} + \cdots + 2927521 \)
$59$
\( T^{8} + 13 T^{7} + 103 T^{6} + \cdots + 9801 \)
$61$
\( T^{8} + 3 T^{7} + 110 T^{6} + \cdots + 1234321 \)
$67$
\( (T^{4} - 5 T^{3} - 74 T^{2} + 180 T + 1049)^{2} \)
$71$
\( T^{8} + 9 T^{7} + 269 T^{6} + \cdots + 674041 \)
$73$
\( T^{8} - 5 T^{7} + 318 T^{6} + \cdots + 22801 \)
$79$
\( T^{8} + 10 T^{7} + 407 T^{6} + \cdots + 707281 \)
$83$
\( T^{8} - 25 T^{7} + 335 T^{6} + \cdots + 3900625 \)
$89$
\( (T^{4} + 2 T^{3} - 55 T^{2} - 156 T + 99)^{2} \)
$97$
\( T^{8} - 13 T^{7} + 160 T^{6} + \cdots + 793881 \)
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