Newspace parameters
Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 231.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.84454428669\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} + 15x^{8} + 72x^{6} + 120x^{4} + 72x^{2} + 12 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{4} + 7\nu^{2} + 5 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{9} + 15\nu^{7} + 70\nu^{5} + 102\nu^{3} + 36\nu - 4 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{9} + 2\nu^{8} + 13\nu^{7} + 28\nu^{6} + 44\nu^{5} + 118\nu^{4} + 6\nu^{3} + 140\nu^{2} - 36\nu + 40 ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{9} + \nu^{8} + 14\nu^{7} + 15\nu^{6} + 59\nu^{5} + 70\nu^{4} + 70\nu^{3} + 100\nu^{2} + 18\nu + 32 ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{9} - \nu^{8} - 14\nu^{7} - 13\nu^{6} - 59\nu^{5} - 48\nu^{4} - 70\nu^{3} - 36\nu^{2} - 18\nu + 4 ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{8} - 14\nu^{6} - 59\nu^{4} - 70\nu^{2} - 20 ) / 2 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{9} - 2\nu^{8} + 15\nu^{7} - 30\nu^{6} + 74\nu^{5} - 140\nu^{4} + 130\nu^{3} - 204\nu^{2} + 60\nu - 76 ) / 8 \) |
\(\beta_{8}\) | \(=\) | \( ( -3\nu^{9} - 43\nu^{7} - 188\nu^{5} - 4\nu^{4} - 246\nu^{3} - 28\nu^{2} - 96\nu - 20 ) / 8 \) |
\(\beta_{9}\) | \(=\) | \( ( -\nu^{9} + \nu^{8} - 14\nu^{7} + 15\nu^{6} - 59\nu^{5} + 70\nu^{4} - 70\nu^{3} + 100\nu^{2} - 18\nu + 32 ) / 4 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 1 ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{9} - \beta_{6} + \beta_{5} - 3 \) |
\(\nu^{3}\) | \(=\) | \( -3\beta_{9} + 2\beta_{8} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - 4\beta_{3} + 2\beta_{2} + \beta _1 + 1 \) |
\(\nu^{4}\) | \(=\) | \( 7\beta_{9} + 7\beta_{6} - 7\beta_{5} + \beta _1 + 16 \) |
\(\nu^{5}\) | \(=\) | \( 17 \beta_{9} - 10 \beta_{8} + 26 \beta_{7} - \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 24 \beta_{3} - 12 \beta_{2} - 5 \beta _1 - 6 \) |
\(\nu^{6}\) | \(=\) | \( -45\beta_{9} - 45\beta_{6} + 47\beta_{5} + 2\beta_{4} - 11\beta _1 - 98 \) |
\(\nu^{7}\) | \(=\) | \( - 101 \beta_{9} + 58 \beta_{8} - 170 \beta_{7} + 11 \beta_{6} - 85 \beta_{5} + 16 \beta_{4} - 148 \beta_{3} + 88 \beta_{2} + 29 \beta _1 + 44 \) |
\(\nu^{8}\) | \(=\) | \( 287\beta_{9} + 285\beta_{6} - 315\beta_{5} - 28\beta_{4} + 95\beta _1 + 618 \) |
\(\nu^{9}\) | \(=\) | \( 607 \beta_{9} - 350 \beta_{8} + 1114 \beta_{7} - 95 \beta_{6} + 557 \beta_{5} - 50 \beta_{4} + 924 \beta_{3} - 652 \beta_{2} - 175 \beta _1 - 326 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).
\(n\) | \(155\) | \(199\) | \(211\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 |
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−1.29725 | − | 2.24690i | 0.500000 | − | 0.866025i | −2.36571 | + | 4.09752i | −1.09601 | − | 1.89835i | −2.59450 | −2.61329 | + | 0.413178i | 7.08664 | −0.500000 | − | 0.866025i | −2.84359 | + | 4.92525i | ||||||||||||||||||||||||||||||||||
67.2 | −1.17616 | − | 2.03717i | 0.500000 | − | 0.866025i | −1.76671 | + | 3.06003i | 2.05761 | + | 3.56389i | −2.35232 | 2.51585 | − | 0.818848i | 3.60708 | −0.500000 | − | 0.866025i | 4.84017 | − | 8.38342i | |||||||||||||||||||||||||||||||||||
67.3 | −0.307468 | − | 0.532550i | 0.500000 | − | 0.866025i | 0.810927 | − | 1.40457i | −0.747986 | − | 1.29555i | −0.614936 | 2.59895 | + | 0.495442i | −2.22721 | −0.500000 | − | 0.866025i | −0.459963 | + | 0.796680i | |||||||||||||||||||||||||||||||||||
67.4 | 0.534421 | + | 0.925645i | 0.500000 | − | 0.866025i | 0.428788 | − | 0.742682i | 1.34592 | + | 2.33120i | 1.06884 | −0.855706 | − | 2.50355i | 3.05430 | −0.500000 | − | 0.866025i | −1.43858 | + | 2.49169i | |||||||||||||||||||||||||||||||||||
67.5 | 1.24646 | + | 2.15892i | 0.500000 | − | 0.866025i | −2.10730 | + | 3.64995i | 0.440463 | + | 0.762904i | 2.49291 | −2.14580 | + | 1.54775i | −5.52081 | −0.500000 | − | 0.866025i | −1.09804 | + | 1.90185i | |||||||||||||||||||||||||||||||||||
100.1 | −1.29725 | + | 2.24690i | 0.500000 | + | 0.866025i | −2.36571 | − | 4.09752i | −1.09601 | + | 1.89835i | −2.59450 | −2.61329 | − | 0.413178i | 7.08664 | −0.500000 | + | 0.866025i | −2.84359 | − | 4.92525i | |||||||||||||||||||||||||||||||||||
100.2 | −1.17616 | + | 2.03717i | 0.500000 | + | 0.866025i | −1.76671 | − | 3.06003i | 2.05761 | − | 3.56389i | −2.35232 | 2.51585 | + | 0.818848i | 3.60708 | −0.500000 | + | 0.866025i | 4.84017 | + | 8.38342i | |||||||||||||||||||||||||||||||||||
100.3 | −0.307468 | + | 0.532550i | 0.500000 | + | 0.866025i | 0.810927 | + | 1.40457i | −0.747986 | + | 1.29555i | −0.614936 | 2.59895 | − | 0.495442i | −2.22721 | −0.500000 | + | 0.866025i | −0.459963 | − | 0.796680i | |||||||||||||||||||||||||||||||||||
100.4 | 0.534421 | − | 0.925645i | 0.500000 | + | 0.866025i | 0.428788 | + | 0.742682i | 1.34592 | − | 2.33120i | 1.06884 | −0.855706 | + | 2.50355i | 3.05430 | −0.500000 | + | 0.866025i | −1.43858 | − | 2.49169i | |||||||||||||||||||||||||||||||||||
100.5 | 1.24646 | − | 2.15892i | 0.500000 | + | 0.866025i | −2.10730 | − | 3.64995i | 0.440463 | − | 0.762904i | 2.49291 | −2.14580 | − | 1.54775i | −5.52081 | −0.500000 | + | 0.866025i | −1.09804 | − | 1.90185i | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 231.2.i.f | ✓ | 10 |
3.b | odd | 2 | 1 | 693.2.i.j | 10 | ||
7.c | even | 3 | 1 | inner | 231.2.i.f | ✓ | 10 |
7.c | even | 3 | 1 | 1617.2.a.ba | 5 | ||
7.d | odd | 6 | 1 | 1617.2.a.bb | 5 | ||
21.g | even | 6 | 1 | 4851.2.a.bz | 5 | ||
21.h | odd | 6 | 1 | 693.2.i.j | 10 | ||
21.h | odd | 6 | 1 | 4851.2.a.ca | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
231.2.i.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
231.2.i.f | ✓ | 10 | 7.c | even | 3 | 1 | inner |
693.2.i.j | 10 | 3.b | odd | 2 | 1 | ||
693.2.i.j | 10 | 21.h | odd | 6 | 1 | ||
1617.2.a.ba | 5 | 7.c | even | 3 | 1 | ||
1617.2.a.bb | 5 | 7.d | odd | 6 | 1 | ||
4851.2.a.bz | 5 | 21.g | even | 6 | 1 | ||
4851.2.a.ca | 5 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 2T_{2}^{9} + 12T_{2}^{8} + 12T_{2}^{7} + 81T_{2}^{6} + 78T_{2}^{5} + 264T_{2}^{4} + 6T_{2}^{3} + 261T_{2}^{2} + 110T_{2} + 100 \)
acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 2 T^{9} + 12 T^{8} + 12 T^{7} + \cdots + 100 \)
$3$
\( (T^{2} - T + 1)^{5} \)
$5$
\( T^{10} - 4 T^{9} + 24 T^{8} + \cdots + 1024 \)
$7$
\( T^{10} + T^{9} - 15 T^{8} - 36 T^{7} + \cdots + 16807 \)
$11$
\( (T^{2} + T + 1)^{5} \)
$13$
\( (T^{5} - 5 T^{4} - 38 T^{3} + 80 T^{2} + \cdots + 476)^{2} \)
$17$
\( T^{10} + 2 T^{9} + 12 T^{8} + 12 T^{7} + \cdots + 100 \)
$19$
\( T^{10} - 3 T^{9} + 51 T^{8} + \cdots + 363609 \)
$23$
\( T^{10} + 16 T^{9} + 186 T^{8} + \cdots + 38416 \)
$29$
\( (T^{5} - 96 T^{3} + 24 T^{2} + 1707 T - 1290)^{2} \)
$31$
\( T^{10} + 5 T^{9} + 33 T^{8} + 24 T^{7} + \cdots + 400 \)
$37$
\( T^{10} + 15 T^{9} + 195 T^{8} + \cdots + 40401 \)
$41$
\( (T^{5} + 22 T^{4} + 124 T^{3} + \cdots - 11536)^{2} \)
$43$
\( (T^{5} - 3 T^{4} - 108 T^{3} + 402 T^{2} + \cdots - 2973)^{2} \)
$47$
\( T^{10} - 2 T^{9} + 198 T^{8} + \cdots + 783664036 \)
$53$
\( T^{10} + 6 T^{9} + 132 T^{8} + \cdots + 42302016 \)
$59$
\( T^{10} + 16 T^{9} + 186 T^{8} + \cdots + 38416 \)
$61$
\( T^{10} + 12 T^{9} + \cdots + 2361960000 \)
$67$
\( T^{10} + 7 T^{9} + 141 T^{8} + \cdots + 9265936 \)
$71$
\( (T^{5} - 24 T^{4} + 66 T^{3} + 1380 T^{2} + \cdots + 5232)^{2} \)
$73$
\( T^{10} + 17 T^{9} + 387 T^{8} + \cdots + 9265936 \)
$79$
\( T^{10} + 7 T^{9} + \cdots + 8999937424 \)
$83$
\( (T^{5} + 12 T^{4} - 120 T^{3} - 816 T^{2} + \cdots + 2688)^{2} \)
$89$
\( T^{10} - 6 T^{9} + 204 T^{8} + \cdots + 315417600 \)
$97$
\( (T^{5} + 14 T^{4} - 326 T^{3} + \cdots - 38906)^{2} \)
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