Newspace parameters
Level: | \( N \) | \(=\) | \( 1375 = 5^{3} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1375.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.9794302779\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.6988960000.10 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 9x^{6} + 22x^{4} + 11x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 9x^{6} + 22x^{4} + 11x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 2 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 5 ) / 2 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} + 10\nu^{4} + 26\nu^{2} + 9 ) / 2 \) |
\(\beta_{5}\) | \(=\) | \( -\nu^{7} - 9\nu^{5} - 21\nu^{3} - 6\nu \) |
\(\beta_{6}\) | \(=\) | \( \nu^{7} + 9\nu^{5} + 22\nu^{3} + 11\nu \) |
\(\beta_{7}\) | \(=\) | \( ( 3\nu^{7} + 26\nu^{5} + 58\nu^{3} + 17\nu ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{6} + \beta_{5} - 5\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} - \beta_{3} - 5\beta_{2} + 8 \) |
\(\nu^{5}\) | \(=\) | \( -2\beta_{7} - 5\beta_{6} - 8\beta_{5} + 24\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -8\beta_{4} + 10\beta_{3} + 24\beta_{2} - 37 \) |
\(\nu^{7}\) | \(=\) | \( 18\beta_{7} + 24\beta_{6} + 50\beta_{5} - 117\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).
\(n\) | \(376\) | \(1002\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
749.1 |
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− | 2.25947i | − | 0.716111i | −3.10522 | 0 | −1.61803 | 3.36025i | 2.49721i | 2.48718 | 0 | ||||||||||||||||||||||||||||||||||||||||
749.2 | − | 1.80692i | 0.342036i | −1.26498 | 0 | 0.618034 | 3.40246i | − | 1.32813i | 2.88301 | 0 | |||||||||||||||||||||||||||||||||||||||||
749.3 | − | 0.716111i | − | 2.25947i | 1.48718 | 0 | −1.61803 | − | 2.22368i | − | 2.49721i | −2.10522 | 0 | |||||||||||||||||||||||||||||||||||||||
749.4 | − | 0.342036i | 1.80692i | 1.88301 | 0 | 0.618034 | − | 0.432668i | − | 1.32813i | −0.264977 | 0 | ||||||||||||||||||||||||||||||||||||||||
749.5 | 0.342036i | − | 1.80692i | 1.88301 | 0 | 0.618034 | 0.432668i | 1.32813i | −0.264977 | 0 | ||||||||||||||||||||||||||||||||||||||||||
749.6 | 0.716111i | 2.25947i | 1.48718 | 0 | −1.61803 | 2.22368i | 2.49721i | −2.10522 | 0 | |||||||||||||||||||||||||||||||||||||||||||
749.7 | 1.80692i | − | 0.342036i | −1.26498 | 0 | 0.618034 | − | 3.40246i | 1.32813i | 2.88301 | 0 | |||||||||||||||||||||||||||||||||||||||||
749.8 | 2.25947i | 0.716111i | −3.10522 | 0 | −1.61803 | − | 3.36025i | − | 2.49721i | 2.48718 | 0 | |||||||||||||||||||||||||||||||||||||||||
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 | 1375.2.b.b | 8 | |
5.b | even | 2 | 1 | inner | 1375.2.b.b | 8 | |
5.c | odd | 4 | 2 | 1375.2.a.c | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1375.2.a.c | ✓ | 8 | 5.c | odd | 4 | 2 | |
1375.2.b.b | 8 | 1.a | even | 1 | 1 | trivial | |
1375.2.b.b | 8 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 9T_{2}^{6} + 22T_{2}^{4} + 11T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1375, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 9 T^{6} + 22 T^{4} + 11 T^{2} + \cdots + 1 \)
$3$
\( T^{8} + 9 T^{6} + 22 T^{4} + 11 T^{2} + \cdots + 1 \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 28 T^{6} + 249 T^{4} + \cdots + 121 \)
$11$
\( (T + 1)^{8} \)
$13$
\( T^{8} + 60 T^{6} + 1075 T^{4} + \cdots + 10000 \)
$17$
\( T^{8} + 56 T^{6} + 987 T^{4} + \cdots + 15376 \)
$19$
\( (T^{4} - 9 T^{3} - 9 T^{2} + 116 T + 176)^{2} \)
$23$
\( T^{8} + 59 T^{6} + 1062 T^{4} + \cdots + 10201 \)
$29$
\( (T^{4} - 18 T^{3} + 113 T^{2} - 288 T + 251)^{2} \)
$31$
\( (T^{4} + 8 T^{3} - 55 T^{2} - 374 T + 484)^{2} \)
$37$
\( T^{8} + 121 T^{6} + 4807 T^{4} + \cdots + 364816 \)
$41$
\( (T^{4} + 23 T^{3} + 168 T^{2} + 483 T + 461)^{2} \)
$43$
\( T^{8} + 93 T^{6} + 2974 T^{4} + \cdots + 128881 \)
$47$
\( T^{8} + 181 T^{6} + 6846 T^{4} + \cdots + 109561 \)
$53$
\( T^{8} + 344 T^{6} + \cdots + 16128256 \)
$59$
\( (T^{4} + 7 T^{3} - 115 T^{2} - 316 T + 844)^{2} \)
$61$
\( (T^{4} + 10 T^{3} - 191 T^{2} - 1840 T + 359)^{2} \)
$67$
\( T^{8} + 404 T^{6} + \cdots + 15335056 \)
$71$
\( (T^{4} - 9 T^{3} - 65 T^{2} + 828 T - 1936)^{2} \)
$73$
\( T^{8} + 327 T^{6} + 20579 T^{4} + \cdots + 55696 \)
$79$
\( (T^{4} - 17 T^{3} + 13 T^{2} + 948 T - 3764)^{2} \)
$83$
\( T^{8} + 331 T^{6} + 10311 T^{4} + \cdots + 274576 \)
$89$
\( (T^{4} - 36 T^{3} + 417 T^{2} - 1444 T - 1159)^{2} \)
$97$
\( T^{8} + 732 T^{6} + \cdots + 939790336 \)
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