Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + 10\nu^{4} + 22\nu^{2} + 4\nu + 6 ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{11} + 20\nu^{9} + 144\nu^{7} + 446\nu^{5} + 544\nu^{3} + 180\nu + 24 ) / 48 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{11} - 20\nu^{9} - 142\nu^{7} - 426\nu^{5} - 500\nu^{3} + 8\nu^{2} - 168\nu + 24 ) / 16 \) |
\(\beta_{6}\) | \(=\) | \( ( - 5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} - 6 \nu^{6} - 1510 \nu^{5} - 84 \nu^{4} - 1064 \nu^{3} - 300 \nu^{2} + 72 \nu - 156 ) / 48 \) |
\(\beta_{7}\) | \(=\) | \( ( - 5 \nu^{11} - 94 \nu^{9} - 606 \nu^{7} + 6 \nu^{6} - 1510 \nu^{5} + 84 \nu^{4} - 1064 \nu^{3} + 300 \nu^{2} + 72 \nu + 156 ) / 48 \) |
\(\beta_{8}\) | \(=\) | \( ( - 5 \nu^{11} + 6 \nu^{10} - 94 \nu^{9} + 114 \nu^{8} - 612 \nu^{7} + 756 \nu^{6} - 1594 \nu^{5} + 2040 \nu^{4} - 1412 \nu^{3} + 1932 \nu^{2} - 324 \nu + 384 ) / 48 \) |
\(\beta_{9}\) | \(=\) | \( ( 4 \nu^{11} + 77 \nu^{9} + 516 \nu^{7} - 3 \nu^{6} + 1394 \nu^{5} - 42 \nu^{4} + 1306 \nu^{3} - 162 \nu^{2} + 360 \nu - 114 ) / 24 \) |
\(\beta_{10}\) | \(=\) | \( ( - 5 \nu^{11} - 6 \nu^{10} - 94 \nu^{9} - 108 \nu^{8} - 612 \nu^{7} - 660 \nu^{6} - 1594 \nu^{5} - 1548 \nu^{4} - 1388 \nu^{3} - 1056 \nu^{2} - 180 \nu - 24 ) / 48 \) |
\(\beta_{11}\) | \(=\) | \( ( 5 \nu^{11} + 6 \nu^{10} + 94 \nu^{9} + 114 \nu^{8} + 612 \nu^{7} + 756 \nu^{6} + 1594 \nu^{5} + 2040 \nu^{4} + 1412 \nu^{3} + 1932 \nu^{2} + 324 \nu + 384 ) / 48 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 3 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{9} + \beta_{7} + \beta_{5} - 6\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{7} - \beta_{6} - 2\beta_{3} - 7\beta_{2} + \beta _1 + 16 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{11} - 9\beta_{9} - \beta_{8} - 8\beta_{7} + \beta_{6} - 11\beta_{5} - 6\beta_{4} + \beta_{2} + 39\beta _1 + 3 \) |
\(\nu^{6}\) | \(=\) | \( -10\beta_{7} + 10\beta_{6} + 28\beta_{3} + 48\beta_{2} - 14\beta _1 - 100 \) |
\(\nu^{7}\) | \(=\) | \( - 10 \beta_{11} + 68 \beta_{9} + 10 \beta_{8} + 58 \beta_{7} - 10 \beta_{6} + 96 \beta_{5} + 84 \beta_{4} - 14 \beta_{2} - 264 \beta _1 - 42 \) |
\(\nu^{8}\) | \(=\) | \( 8 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + 74 \beta_{7} - 78 \beta_{6} - 4 \beta_{5} - 284 \beta_{3} - 340 \beta_{2} + 142 \beta _1 + 666 \) |
\(\nu^{9}\) | \(=\) | \( 70 \beta_{11} - 488 \beta_{9} - 70 \beta_{8} - 414 \beta_{7} + 74 \beta_{6} - 780 \beta_{5} - 836 \beta_{4} + 146 \beta_{2} + 1830 \beta _1 + 418 \) |
\(\nu^{10}\) | \(=\) | \( - 148 \beta_{11} - 152 \beta_{10} + 76 \beta_{9} + 4 \beta_{8} - 486 \beta_{7} + 562 \beta_{6} + 76 \beta_{5} + 2548 \beta_{3} + 2470 \beta_{2} - 1274 \beta _1 - 4592 \) |
\(\nu^{11}\) | \(=\) | \( - 406 \beta_{11} + 3438 \beta_{9} + 406 \beta_{8} + 2952 \beta_{7} - 486 \beta_{6} + 6138 \beta_{5} + 7348 \beta_{4} - 1350 \beta_{2} - 12894 \beta _1 - 3674 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(1\) | \(\beta_{4}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 |
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−2.13631 | − | 1.23340i | 0 | 2.04255 | + | 3.53780i | 0.500000 | − | 0.866025i | 0 | −1.85267 | + | 1.88881i | − | 5.14351i | 0 | −2.13631 | + | 1.23340i | |||||||||||||||||||||||||||||||||||||||||||
26.2 | −1.72830 | − | 0.997835i | 0 | 0.991350 | + | 1.71707i | 0.500000 | − | 0.866025i | 0 | 1.78020 | + | 1.95727i | 0.0345244i | 0 | −1.72830 | + | 0.997835i | |||||||||||||||||||||||||||||||||||||||||||||
26.3 | −0.344861 | − | 0.199105i | 0 | −0.920714 | − | 1.59472i | 0.500000 | − | 0.866025i | 0 | −2.30243 | − | 1.30338i | 1.52970i | 0 | −0.344861 | + | 0.199105i | |||||||||||||||||||||||||||||||||||||||||||||
26.4 | 0.848417 | + | 0.489834i | 0 | −0.520126 | − | 0.900884i | 0.500000 | − | 0.866025i | 0 | 1.24698 | + | 2.33346i | − | 2.97844i | 0 | 0.848417 | − | 0.489834i | ||||||||||||||||||||||||||||||||||||||||||||
26.5 | 0.986962 | + | 0.569823i | 0 | −0.350603 | − | 0.607263i | 0.500000 | − | 0.866025i | 0 | 2.18931 | − | 1.48558i | − | 3.07842i | 0 | 0.986962 | − | 0.569823i | ||||||||||||||||||||||||||||||||||||||||||||
26.6 | 2.37409 | + | 1.37068i | 0 | 2.75754 | + | 4.77621i | 0.500000 | − | 0.866025i | 0 | −2.06138 | − | 1.65853i | 9.63615i | 0 | 2.37409 | − | 1.37068i | |||||||||||||||||||||||||||||||||||||||||||||
206.1 | −2.13631 | + | 1.23340i | 0 | 2.04255 | − | 3.53780i | 0.500000 | + | 0.866025i | 0 | −1.85267 | − | 1.88881i | 5.14351i | 0 | −2.13631 | − | 1.23340i | |||||||||||||||||||||||||||||||||||||||||||||
206.2 | −1.72830 | + | 0.997835i | 0 | 0.991350 | − | 1.71707i | 0.500000 | + | 0.866025i | 0 | 1.78020 | − | 1.95727i | − | 0.0345244i | 0 | −1.72830 | − | 0.997835i | ||||||||||||||||||||||||||||||||||||||||||||
206.3 | −0.344861 | + | 0.199105i | 0 | −0.920714 | + | 1.59472i | 0.500000 | + | 0.866025i | 0 | −2.30243 | + | 1.30338i | − | 1.52970i | 0 | −0.344861 | − | 0.199105i | ||||||||||||||||||||||||||||||||||||||||||||
206.4 | 0.848417 | − | 0.489834i | 0 | −0.520126 | + | 0.900884i | 0.500000 | + | 0.866025i | 0 | 1.24698 | − | 2.33346i | 2.97844i | 0 | 0.848417 | + | 0.489834i | |||||||||||||||||||||||||||||||||||||||||||||
206.5 | 0.986962 | − | 0.569823i | 0 | −0.350603 | + | 0.607263i | 0.500000 | + | 0.866025i | 0 | 2.18931 | + | 1.48558i | 3.07842i | 0 | 0.986962 | + | 0.569823i | |||||||||||||||||||||||||||||||||||||||||||||
206.6 | 2.37409 | − | 1.37068i | 0 | 2.75754 | − | 4.77621i | 0.500000 | + | 0.866025i | 0 | −2.06138 | + | 1.65853i | − | 9.63615i | 0 | 2.37409 | + | 1.37068i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.bj.b | yes | 12 |
3.b | odd | 2 | 1 | 315.2.bj.a | ✓ | 12 | |
5.b | even | 2 | 1 | 1575.2.bk.e | 12 | ||
5.c | odd | 4 | 2 | 1575.2.bc.c | 24 | ||
7.c | even | 3 | 1 | 2205.2.b.a | 12 | ||
7.d | odd | 6 | 1 | 315.2.bj.a | ✓ | 12 | |
7.d | odd | 6 | 1 | 2205.2.b.b | 12 | ||
15.d | odd | 2 | 1 | 1575.2.bk.f | 12 | ||
15.e | even | 4 | 2 | 1575.2.bc.d | 24 | ||
21.g | even | 6 | 1 | inner | 315.2.bj.b | yes | 12 |
21.g | even | 6 | 1 | 2205.2.b.a | 12 | ||
21.h | odd | 6 | 1 | 2205.2.b.b | 12 | ||
35.i | odd | 6 | 1 | 1575.2.bk.f | 12 | ||
35.k | even | 12 | 2 | 1575.2.bc.d | 24 | ||
105.p | even | 6 | 1 | 1575.2.bk.e | 12 | ||
105.w | odd | 12 | 2 | 1575.2.bc.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bj.a | ✓ | 12 | 3.b | odd | 2 | 1 | |
315.2.bj.a | ✓ | 12 | 7.d | odd | 6 | 1 | |
315.2.bj.b | yes | 12 | 1.a | even | 1 | 1 | trivial |
315.2.bj.b | yes | 12 | 21.g | even | 6 | 1 | inner |
1575.2.bc.c | 24 | 5.c | odd | 4 | 2 | ||
1575.2.bc.c | 24 | 105.w | odd | 12 | 2 | ||
1575.2.bc.d | 24 | 15.e | even | 4 | 2 | ||
1575.2.bc.d | 24 | 35.k | even | 12 | 2 | ||
1575.2.bk.e | 12 | 5.b | even | 2 | 1 | ||
1575.2.bk.e | 12 | 105.p | even | 6 | 1 | ||
1575.2.bk.f | 12 | 15.d | odd | 2 | 1 | ||
1575.2.bk.f | 12 | 35.i | odd | 6 | 1 | ||
2205.2.b.a | 12 | 7.c | even | 3 | 1 | ||
2205.2.b.a | 12 | 21.g | even | 6 | 1 | ||
2205.2.b.b | 12 | 7.d | odd | 6 | 1 | ||
2205.2.b.b | 12 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 10T_{2}^{10} + 78T_{2}^{8} + 12T_{2}^{7} - 208T_{2}^{6} + 424T_{2}^{4} - 264T_{2}^{3} - 84T_{2}^{2} + 72T_{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} - 10 T^{10} + 78 T^{8} + 12 T^{7} + \cdots + 36 \)
$3$
\( T^{12} \)
$5$
\( (T^{2} - T + 1)^{6} \)
$7$
\( T^{12} + 2 T^{11} - T^{10} + 6 T^{9} + \cdots + 117649 \)
$11$
\( T^{12} + 12 T^{11} + 38 T^{10} + \cdots + 19044 \)
$13$
\( T^{12} + 102 T^{10} + 3831 T^{8} + \cdots + 2025 \)
$17$
\( T^{12} + 60 T^{10} - 120 T^{9} + \cdots + 2125764 \)
$19$
\( T^{12} - 6 T^{11} - 39 T^{10} + \cdots + 227529 \)
$23$
\( T^{12} + 12 T^{11} + \cdots + 113422500 \)
$29$
\( T^{12} + 164 T^{10} + 8988 T^{8} + \cdots + 4088484 \)
$31$
\( T^{12} - 6 T^{11} - 111 T^{10} + \cdots + 1750329 \)
$37$
\( T^{12} + 10 T^{11} + \cdots + 12130158769 \)
$41$
\( (T^{6} + 12 T^{5} - 54 T^{4} - 756 T^{3} + \cdots - 36450)^{2} \)
$43$
\( (T^{6} + 2 T^{5} - 193 T^{4} + \cdots - 114167)^{2} \)
$47$
\( T^{12} + 120 T^{10} + \cdots + 466560000 \)
$53$
\( T^{12} + 12 T^{11} - 28 T^{10} + \cdots + 43877376 \)
$59$
\( T^{12} - 24 T^{11} + 438 T^{10} + \cdots + 72900 \)
$61$
\( T^{12} - 228 T^{10} + \cdots + 593994384 \)
$67$
\( T^{12} - 6 T^{11} + 183 T^{10} + \cdots + 235898881 \)
$71$
\( T^{12} + 392 T^{10} + \cdots + 491508900 \)
$73$
\( T^{12} + 42 T^{11} + 681 T^{10} + \cdots + 1565001 \)
$79$
\( T^{12} - 18 T^{11} + \cdots + 1973580625 \)
$83$
\( (T^{6} - 12 T^{5} - 198 T^{4} + \cdots - 287874)^{2} \)
$89$
\( T^{12} + 12 T^{11} + \cdots + 872493444 \)
$97$
\( T^{12} + 732 T^{10} + \cdots + 71571600 \)
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