Newspace parameters
Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 630.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(17.1662566547\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.98344960000.4 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{6} \) |
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 \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{6} - \nu^{4} + 11\nu^{2} + 270 ) / 90 \) |
\(\beta_{2}\) | \(=\) | \( ( 4\nu^{7} + 19\nu^{5} + 151\nu^{3} - 1155\nu ) / 450 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + 7\nu^{4} + 55\nu^{2} - 108 ) / 18 \) |
\(\beta_{4}\) | \(=\) | \( ( 7\nu^{7} + 52\nu^{5} + 358\nu^{3} - 615\nu ) / 450 \) |
\(\beta_{5}\) | \(=\) | \( ( -4\nu^{6} - 34\nu^{4} - 196\nu^{2} + 405 ) / 45 \) |
\(\beta_{6}\) | \(=\) | \( ( 2\nu^{7} + 17\nu^{5} + 113\nu^{3} - 195\nu ) / 90 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{7} - 6\nu^{5} - 24\nu^{3} + 235\nu ) / 30 \) |
\(\nu\) | \(=\) | \( ( -\beta_{6} + 2\beta_{4} - \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 2\beta_{3} + 2\beta _1 - 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 3\beta_{7} + 2\beta_{6} - \beta_{4} + 8\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -11\beta_{5} - 16\beta_{3} + 8\beta _1 - 21 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -15\beta_{7} + 65\beta_{6} - 97\beta_{4} - 49\beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 11\beta_{5} + 19\beta_{3} - 83\beta _1 + 264 \) |
\(\nu^{7}\) | \(=\) | \( ( -42\beta_{7} - 673\beta_{6} + 1076\beta_{4} - 133\beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(281\) | \(451\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
71.1 |
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− | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −2.64575 | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||
71.2 | − | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 2.64575 | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||||||||
71.3 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −2.64575 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
71.4 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 2.64575 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
71.5 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −2.64575 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
71.6 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 2.64575 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
71.7 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −2.64575 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
71.8 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 2.64575 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 630.3.e.a | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 630.3.e.a | ✓ | 8 |
5.b | even | 2 | 1 | 3150.3.e.i | 8 | ||
5.c | odd | 4 | 2 | 3150.3.c.d | 16 | ||
15.d | odd | 2 | 1 | 3150.3.e.i | 8 | ||
15.e | even | 4 | 2 | 3150.3.c.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
630.3.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
630.3.e.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
3150.3.c.d | 16 | 5.c | odd | 4 | 2 | ||
3150.3.c.d | 16 | 15.e | even | 4 | 2 | ||
3150.3.e.i | 8 | 5.b | even | 2 | 1 | ||
3150.3.e.i | 8 | 15.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 32T_{11}^{2} + 144 \)
acting on \(S_{3}^{\mathrm{new}}(630, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 2)^{4} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 5)^{4} \)
$7$
\( (T^{2} - 7)^{4} \)
$11$
\( (T^{4} + 32 T^{2} + 144)^{2} \)
$13$
\( (T^{4} + 16 T^{3} - 124 T^{2} - 1504 T - 2364)^{2} \)
$17$
\( T^{8} + 896 T^{6} + \cdots + 26873856 \)
$19$
\( (T^{4} - 40 T^{3} + 196 T^{2} + \cdots - 29916)^{2} \)
$23$
\( T^{8} + 2408 T^{6} + \cdots + 75666805776 \)
$29$
\( T^{8} + 2792 T^{6} + \cdots + 19925016336 \)
$31$
\( (T^{4} + 56 T^{3} - 300 T^{2} + \cdots - 111676)^{2} \)
$37$
\( (T^{4} + 16 T^{3} - 1128 T^{2} + \cdots - 7664)^{2} \)
$41$
\( T^{8} + 5776 T^{6} + \cdots + 1274875842816 \)
$43$
\( (T^{4} - 64 T^{3} - 744 T^{2} + \cdots + 87056)^{2} \)
$47$
\( T^{8} + 11488 T^{6} + \cdots + 3582449565696 \)
$53$
\( T^{8} + 12112 T^{6} + \cdots + 58107995136 \)
$59$
\( T^{8} + 9328 T^{6} + \cdots + 21234991124736 \)
$61$
\( (T^{4} + 40 T^{3} - 3736 T^{2} + \cdots - 42096)^{2} \)
$67$
\( (T^{4} - 64 T^{3} - 5000 T^{2} + \cdots + 8709264)^{2} \)
$71$
\( T^{8} + \cdots + 842087039037696 \)
$73$
\( (T^{4} - 11100 T^{2} - 560000 T - 6997500)^{2} \)
$79$
\( (T^{4} - 7376 T^{2} + 7795264)^{2} \)
$83$
\( T^{8} + 5728 T^{6} + \cdots + 2702420361216 \)
$89$
\( T^{8} + 26224 T^{6} + \cdots + 70331973325056 \)
$97$
\( (T^{4} + 112 T^{3} - 16572 T^{2} + \cdots + 18658756)^{2} \)
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