Newspace parameters
Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 350.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.53680925261\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.205520896.4 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \( x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( 6\nu^{7} - 14\nu^{6} + 26\nu^{5} + 79\nu^{4} - 262\nu^{3} + 134\nu^{2} + 182\nu + 44 ) / 51 \) |
\(\beta_{2}\) | \(=\) | \( ( 32\nu^{7} - 137\nu^{6} + 422\nu^{5} - 508\nu^{4} - 94\nu^{3} + 335\nu^{2} + 336\nu + 76 ) / 51 \) |
\(\beta_{3}\) | \(=\) | \( ( -2\nu^{7} + 9\nu^{6} - 28\nu^{5} + 36\nu^{4} + 4\nu^{3} - 33\nu^{2} - 10\nu ) / 3 \) |
\(\beta_{4}\) | \(=\) | \( ( -35\nu^{7} + 161\nu^{6} - 503\nu^{5} + 664\nu^{4} + 55\nu^{3} - 623\nu^{2} - 189\nu + 140 ) / 51 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{7} + 11\nu^{6} - 31\nu^{5} + 20\nu^{4} + 49\nu^{3} - 43\nu^{2} - 51\nu - 12 ) / 3 \) |
\(\beta_{6}\) | \(=\) | \( ( 3\nu^{7} - 13\nu^{6} + 41\nu^{5} - 52\nu^{4} + \nu^{3} + 29\nu^{2} + 27\nu + 6 ) / 3 \) |
\(\beta_{7}\) | \(=\) | \( ( 61\nu^{7} - 267\nu^{6} + 831\nu^{5} - 1030\nu^{4} - 159\nu^{3} + 909\nu^{2} + 275\nu + 28 ) / 51 \) |
\(\nu\) | \(=\) | \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{2} - 2\beta _1 + 2 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} - 2\beta _1 - 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -5\beta_{7} + \beta_{6} + 5\beta_{5} + 9\beta_{4} - 18\beta_{3} + 4\beta_{2} + 14\beta _1 - 22 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{6} + 8\beta_{5} + 12\beta_{2} + 21\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( ( 37\beta_{7} - 7\beta_{6} + 37\beta_{5} - 81\beta_{4} + 150\beta_{3} + 52\beta_{2} + 98\beta _1 + 202 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 43\beta_{7} + 43\beta_{6} - 91\beta_{5} - 91\beta_{4} + 171\beta_{3} - 171\beta_{2} - 226\beta _1 + 226 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -103\beta_{7} + 305\beta_{6} - 713\beta_{5} + 305\beta_{4} - 504\beta_{3} - 1286\beta_{2} - 1790\beta _1 - 782 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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349.1 |
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− | 1.41421i | −2.97127 | −2.00000 | 0 | 4.20201i | −5.43275 | − | 4.41421i | 2.82843i | −0.171573 | 0 | |||||||||||||||||||||||||||||||||||||||
349.2 | − | 1.41421i | −1.78089 | −2.00000 | 0 | 2.51856i | 6.81801 | + | 1.58579i | 2.82843i | −5.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
349.3 | − | 1.41421i | 1.78089 | −2.00000 | 0 | − | 2.51856i | −6.81801 | + | 1.58579i | 2.82843i | −5.82843 | 0 | |||||||||||||||||||||||||||||||||||||||
349.4 | − | 1.41421i | 2.97127 | −2.00000 | 0 | − | 4.20201i | 5.43275 | − | 4.41421i | 2.82843i | −0.171573 | 0 | |||||||||||||||||||||||||||||||||||||||
349.5 | 1.41421i | −2.97127 | −2.00000 | 0 | − | 4.20201i | −5.43275 | + | 4.41421i | − | 2.82843i | −0.171573 | 0 | |||||||||||||||||||||||||||||||||||||||
349.6 | 1.41421i | −1.78089 | −2.00000 | 0 | − | 2.51856i | 6.81801 | − | 1.58579i | − | 2.82843i | −5.82843 | 0 | |||||||||||||||||||||||||||||||||||||||
349.7 | 1.41421i | 1.78089 | −2.00000 | 0 | 2.51856i | −6.81801 | − | 1.58579i | − | 2.82843i | −5.82843 | 0 | ||||||||||||||||||||||||||||||||||||||||
349.8 | 1.41421i | 2.97127 | −2.00000 | 0 | 4.20201i | 5.43275 | + | 4.41421i | − | 2.82843i | −0.171573 | 0 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 350.3.d.a | 8 | |
5.b | even | 2 | 1 | inner | 350.3.d.a | 8 | |
5.c | odd | 4 | 1 | 350.3.b.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 350.3.b.b | yes | 4 | |
7.b | odd | 2 | 1 | inner | 350.3.d.a | 8 | |
35.c | odd | 2 | 1 | inner | 350.3.d.a | 8 | |
35.f | even | 4 | 1 | 350.3.b.a | ✓ | 4 | |
35.f | even | 4 | 1 | 350.3.b.b | yes | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
350.3.b.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
350.3.b.a | ✓ | 4 | 35.f | even | 4 | 1 | |
350.3.b.b | yes | 4 | 5.c | odd | 4 | 1 | |
350.3.b.b | yes | 4 | 35.f | even | 4 | 1 | |
350.3.d.a | 8 | 1.a | even | 1 | 1 | trivial | |
350.3.d.a | 8 | 5.b | even | 2 | 1 | inner | |
350.3.d.a | 8 | 7.b | odd | 2 | 1 | inner | |
350.3.d.a | 8 | 35.c | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 12T_{3}^{2} + 28 \)
acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{4} \)
$3$
\( (T^{4} - 12 T^{2} + 28)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 108 T^{6} + 6566 T^{4} + \cdots + 5764801 \)
$11$
\( (T^{2} - 10 T + 7)^{4} \)
$13$
\( (T^{4} - 276 T^{2} + 8092)^{2} \)
$17$
\( (T^{4} - 944 T^{2} + 129472)^{2} \)
$19$
\( (T^{4} + 768 T^{2} + 114688)^{2} \)
$23$
\( (T^{4} + 166 T^{2} + 6241)^{2} \)
$29$
\( (T^{2} - 42 T - 71)^{4} \)
$31$
\( (T^{4} + 1356 T^{2} + 149212)^{2} \)
$37$
\( (T^{4} + 3394 T^{2} + 57121)^{2} \)
$41$
\( (T^{4} + 5556 T^{2} + 3489052)^{2} \)
$43$
\( (T^{4} + 822 T^{2} + 96721)^{2} \)
$47$
\( (T^{4} - 1872 T^{2} + 430528)^{2} \)
$53$
\( (T^{4} + 5808 T^{2} + 8248384)^{2} \)
$59$
\( (T^{4} + 8300 T^{2} + 16817500)^{2} \)
$61$
\( (T^{4} + 7668 T^{2} + 4502428)^{2} \)
$67$
\( (T^{4} + 13878 T^{2} + 36687249)^{2} \)
$71$
\( (T^{2} - 46 T - 2833)^{4} \)
$73$
\( (T^{4} - 12340 T^{2} + 14812)^{2} \)
$79$
\( (T^{2} + 102 T - 3449)^{4} \)
$83$
\( (T^{4} - 36588 T^{2} + \cdots + 285109468)^{2} \)
$89$
\( (T^{4} + 9684 T^{2} + 1849372)^{2} \)
$97$
\( (T^{4} - 34996 T^{2} + \cdots + 276596572)^{2} \)
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