Newspace parameters
Level: | \( N \) | \(=\) | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 330.m (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.63506326670\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 3x^{7} + 14x^{6} - 12x^{5} + 121x^{4} + 120x^{3} + 1400x^{2} + 3000x + 10000 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \( x^{8} - 3x^{7} + 14x^{6} - 12x^{5} + 121x^{4} + 120x^{3} + 1400x^{2} + 3000x + 10000 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} - 13\nu^{6} + 144\nu^{5} - 1452\nu^{4} + 14641\nu^{3} - 146290\nu^{2} + 133300\nu + 1000 ) / 1331000 \) |
\(\beta_{3}\) | \(=\) | \( ( 41\nu^{7} - 423\nu^{6} + 3374\nu^{5} - 18392\nu^{4} + 29161\nu^{3} - 7180\nu^{2} + 9300\nu + 52000 ) / 1331000 \) |
\(\beta_{4}\) | \(=\) | \( ( -289\nu^{7} + 1557\nu^{6} - 3116\nu^{5} - 3872\nu^{4} + 3751\nu^{3} - 72190\nu^{2} - 79800\nu - 630000 ) / 1331000 \) |
\(\beta_{5}\) | \(=\) | \( ( 3\nu^{7} - 28\nu^{6} + 179\nu^{5} - 242\nu^{4} + 121\nu^{3} + 481\nu^{2} + 710\nu + 4100 ) / 13310 \) |
\(\beta_{6}\) | \(=\) | \( ( 324\nu^{7} + 683\nu^{6} - 5429\nu^{5} + 29282\nu^{4} - 40656\nu^{3} + 239135\nu^{2} + 47200\nu + 2084000 ) / 665500 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} - 2\nu^{6} + 12\nu^{5} + 121\nu^{3} + 241\nu^{2} + 1641\nu + 3310 ) / 1331 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 11\beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( 12\beta_{7} - 12\beta_{6} - 11\beta_{5} - 22\beta_{3} - 22\beta_{2} - 11\beta _1 + 12 \) |
\(\nu^{4}\) | \(=\) | \( 23\beta_{7} - 22\beta_{6} + 10\beta_{4} - 142\beta_{3} - 32\beta_{2} - 22\beta _1 + 22 \) |
\(\nu^{5}\) | \(=\) | \( 33\beta_{7} + 132\beta_{5} + 220\beta_{4} - 220\beta_{3} - 33\beta _1 - 10 \) |
\(\nu^{6}\) | \(=\) | \( 385\beta_{6} + 385\beta_{5} + 1320\beta_{4} + 385\beta_{3} + 715\beta_{2} - 43\beta _1 - 715 \) |
\(\nu^{7}\) | \(=\) | \( -758\beta_{7} + 2463\beta_{6} + 758\beta_{5} + 6313\beta_{3} + 6743\beta_{2} - 6313 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).
\(n\) | \(67\) | \(211\) | \(221\) |
\(\chi(n)\) | \(1\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
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0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | −3.29524 | − | 2.39413i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 1.00000 | ||||||||||||||||||||||||||
31.2 | 0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | + | 0.951057i | 1.98622 | + | 1.44308i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
91.1 | −0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | −1.07734 | + | 3.31572i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
91.2 | −0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | − | 0.587785i | 0.886361 | − | 2.72794i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
181.1 | 0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | −3.29524 | + | 2.39413i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
181.2 | 0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | − | 0.951057i | 1.98622 | − | 1.44308i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 1.00000 | |||||||||||||||||||||||||||
301.1 | −0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | −1.07734 | − | 3.31572i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
301.2 | −0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | + | 0.587785i | 0.886361 | + | 2.72794i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 1.00000 | |||||||||||||||||||||||||||
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 | 330.2.m.e | ✓ | 8 |
3.b | odd | 2 | 1 | 990.2.n.k | 8 | ||
11.c | even | 5 | 1 | inner | 330.2.m.e | ✓ | 8 |
11.c | even | 5 | 1 | 3630.2.a.bt | 4 | ||
11.d | odd | 10 | 1 | 3630.2.a.br | 4 | ||
33.h | odd | 10 | 1 | 990.2.n.k | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
330.2.m.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
330.2.m.e | ✓ | 8 | 11.c | even | 5 | 1 | inner |
990.2.n.k | 8 | 3.b | odd | 2 | 1 | ||
990.2.n.k | 8 | 33.h | odd | 10 | 1 | ||
3630.2.a.br | 4 | 11.d | odd | 10 | 1 | ||
3630.2.a.bt | 4 | 11.c | even | 5 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 3T_{7}^{7} + 14T_{7}^{6} + 12T_{7}^{5} + 121T_{7}^{4} - 120T_{7}^{3} + 1400T_{7}^{2} - 3000T_{7} + 10000 \)
acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$3$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$5$
\( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \)
$7$
\( T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 10000 \)
$11$
\( T^{8} + 11 T^{6} - 50 T^{5} + \cdots + 14641 \)
$13$
\( T^{8} - 3 T^{7} + 2 T^{6} + 45 T^{5} + \cdots + 121 \)
$17$
\( T^{8} - 5 T^{7} + 54 T^{6} - 280 T^{5} + \cdots + 256 \)
$19$
\( T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 48400 \)
$23$
\( (T^{4} - 8 T^{3} - 29 T^{2} + 330 T - 605)^{2} \)
$29$
\( T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 10000 \)
$31$
\( T^{8} + 22 T^{7} + 234 T^{6} + \cdots + 48400 \)
$37$
\( (T^{4} - 2 T^{3} + 64 T^{2} + 247 T + 361)^{2} \)
$41$
\( T^{8} - T^{7} + 52 T^{6} + \cdots + 6739216 \)
$43$
\( (T^{4} + T^{3} - 107 T^{2} + 242 T + 404)^{2} \)
$47$
\( T^{8} + T^{7} + 202 T^{6} + \cdots + 7027801 \)
$53$
\( T^{8} + 30 T^{7} + 466 T^{6} + \cdots + 355216 \)
$59$
\( T^{8} + 8 T^{7} + 4 T^{6} - 448 T^{5} + \cdots + 3025 \)
$61$
\( T^{8} - 12 T^{7} + 112 T^{6} + \cdots + 495616 \)
$67$
\( (T^{4} - 29 T^{3} + 83 T^{2} + 3662 T - 26476)^{2} \)
$71$
\( T^{8} + 2 T^{7} + 184 T^{6} + \cdots + 2560000 \)
$73$
\( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \)
$79$
\( T^{8} - 41 T^{7} + 1006 T^{6} + \cdots + 31584400 \)
$83$
\( (T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256)^{2} \)
$89$
\( (T^{4} - 5 T^{3} - 145 T^{2} + 350 T + 500)^{2} \)
$97$
\( T^{8} + 40 T^{7} + 744 T^{6} + \cdots + 26050816 \)
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