Newspace parameters
Level: | \( N \) | \(=\) | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1456.cc (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(11.6262185343\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 12.0.58891012706304.1 |
Defining polynomial: |
\( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 91) |
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
\( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{11} - 5\nu^{9} - 2\nu^{8} + 15\nu^{7} + 2\nu^{6} - 30\nu^{5} + 4\nu^{4} + 60\nu^{3} - 16\nu^{2} - 80\nu ) / 32 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{11} + 4 \nu^{10} + 7 \nu^{9} - 6 \nu^{8} - 13 \nu^{7} + 30 \nu^{6} - 6 \nu^{5} - 28 \nu^{4} - 4 \nu^{3} + 16 \nu^{2} - 48 \nu + 96 ) / 32 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 3 \nu^{11} + 2 \nu^{10} - 11 \nu^{9} - 8 \nu^{8} + 21 \nu^{7} + 4 \nu^{6} - 42 \nu^{5} + 84 \nu^{3} + 24 \nu^{2} - 64 \nu - 64 ) / 32 \)
|
\(\beta_{5}\) | \(=\) |
\( ( \nu^{11} + 3 \nu^{10} - 3 \nu^{9} - 13 \nu^{8} + 7 \nu^{7} + 23 \nu^{6} - 26 \nu^{5} - 38 \nu^{4} + 60 \nu^{3} + 68 \nu^{2} - 56 \nu - 64 ) / 16 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 5 \nu^{11} + 4 \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 42 \nu^{6} + 10 \nu^{5} - 68 \nu^{4} - 20 \nu^{3} + 48 \nu^{2} - 32 \nu + 64 ) / 32 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 5 \nu^{11} + 2 \nu^{10} + 17 \nu^{9} - 8 \nu^{8} - 39 \nu^{7} + 44 \nu^{6} + 50 \nu^{5} - 88 \nu^{4} - 68 \nu^{3} + 120 \nu^{2} + 16 \nu - 32 ) / 32 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - \nu^{11} + \nu^{10} + 5 \nu^{9} - 3 \nu^{8} - 13 \nu^{7} + 13 \nu^{6} + 20 \nu^{5} - 34 \nu^{4} - 28 \nu^{3} + 52 \nu^{2} + 24 \nu - 32 ) / 8 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - \nu^{11} + 3 \nu^{10} + 5 \nu^{9} - 13 \nu^{8} - 13 \nu^{7} + 35 \nu^{6} + 12 \nu^{5} - 70 \nu^{4} + 8 \nu^{3} + 108 \nu^{2} - 16 \nu - 80 ) / 16 \)
|
\(\beta_{10}\) | \(=\) |
\( ( - 3 \nu^{11} + 15 \nu^{9} - 2 \nu^{8} - 37 \nu^{7} + 18 \nu^{6} + 66 \nu^{5} - 68 \nu^{4} - 92 \nu^{3} + 96 \nu^{2} + 96 \nu - 64 ) / 16 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - \nu^{11} + 4 \nu^{10} + 9 \nu^{9} - 18 \nu^{8} - 27 \nu^{7} + 50 \nu^{6} + 34 \nu^{5} - 116 \nu^{4} - 20 \nu^{3} + 192 \nu^{2} + 16 \nu - 160 ) / 16 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 1 \)
|
\(\nu^{3}\) | \(=\) |
\( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 \)
|
\(\nu^{4}\) | \(=\) |
\( -\beta_{11} + \beta_{9} + \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1 \)
|
\(\nu^{5}\) | \(=\) |
\( -2\beta_{11} + 4\beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta _1 + 1 \)
|
\(\nu^{6}\) | \(=\) |
\( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta _1 - 2 \)
|
\(\nu^{7}\) | \(=\) |
\( - 5 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} + 7 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1 \)
|
\(\nu^{8}\) | \(=\) |
\( -3\beta_{11} + 7\beta_{8} + \beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{4} + 4\beta_{3} + 10\beta_{2} + 8\beta _1 - 3 \)
|
\(\nu^{9}\) | \(=\) |
\( - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 4 \beta_{7} - 11 \beta_{6} + 12 \beta_{5} - 17 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + \beta _1 - 8 \)
|
\(\nu^{10}\) | \(=\) |
\( - 17 \beta_{11} + 4 \beta_{10} + 13 \beta_{9} + 18 \beta_{8} - 9 \beta_{7} - 8 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 6 \beta _1 - 11 \)
|
\(\nu^{11}\) | \(=\) |
\( 18 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 21 \beta_{7} - 11 \beta_{6} - \beta_{5} - 26 \beta_{4} + 22 \beta_{3} + 14 \beta_{2} + 3 \beta _1 - 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).
\(n\) | \(561\) | \(911\) | \(1093\) | \(1249\) |
\(\chi(n)\) | \(1 - \beta_{9}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
225.1 |
|
0 | −1.41289 | + | 2.44719i | 0 | 0.518957i | 0 | 0.866025 | − | 0.500000i | 0 | −2.49250 | − | 4.31714i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
225.2 | 0 | −0.583963 | + | 1.01145i | 0 | 1.81487i | 0 | 0.866025 | − | 0.500000i | 0 | 0.817975 | + | 1.41677i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
225.3 | 0 | −0.291146 | + | 0.504280i | 0 | − | 1.68817i | 0 | −0.866025 | + | 0.500000i | 0 | 1.33047 | + | 2.30444i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
225.4 | 0 | −0.172975 | + | 0.299601i | 0 | 3.25812i | 0 | −0.866025 | + | 0.500000i | 0 | 1.44016 | + | 2.49443i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
225.5 | 0 | 1.13082 | − | 1.95864i | 0 | − | 3.60178i | 0 | 0.866025 | − | 0.500000i | 0 | −1.05753 | − | 1.83169i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
225.6 | 0 | 1.33015 | − | 2.30388i | 0 | 3.16209i | 0 | −0.866025 | + | 0.500000i | 0 | −2.03858 | − | 3.53092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
673.1 | 0 | −1.41289 | − | 2.44719i | 0 | − | 0.518957i | 0 | 0.866025 | + | 0.500000i | 0 | −2.49250 | + | 4.31714i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
673.2 | 0 | −0.583963 | − | 1.01145i | 0 | − | 1.81487i | 0 | 0.866025 | + | 0.500000i | 0 | 0.817975 | − | 1.41677i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
673.3 | 0 | −0.291146 | − | 0.504280i | 0 | 1.68817i | 0 | −0.866025 | − | 0.500000i | 0 | 1.33047 | − | 2.30444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
673.4 | 0 | −0.172975 | − | 0.299601i | 0 | − | 3.25812i | 0 | −0.866025 | − | 0.500000i | 0 | 1.44016 | − | 2.49443i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
673.5 | 0 | 1.13082 | + | 1.95864i | 0 | 3.60178i | 0 | 0.866025 | + | 0.500000i | 0 | −1.05753 | + | 1.83169i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
673.6 | 0 | 1.33015 | + | 2.30388i | 0 | − | 3.16209i | 0 | −0.866025 | − | 0.500000i | 0 | −2.03858 | + | 3.53092i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1456.2.cc.c | 12 | |
4.b | odd | 2 | 1 | 91.2.q.a | ✓ | 12 | |
12.b | even | 2 | 1 | 819.2.ct.a | 12 | ||
13.e | even | 6 | 1 | inner | 1456.2.cc.c | 12 | |
28.d | even | 2 | 1 | 637.2.q.h | 12 | ||
28.f | even | 6 | 1 | 637.2.k.g | 12 | ||
28.f | even | 6 | 1 | 637.2.u.i | 12 | ||
28.g | odd | 6 | 1 | 637.2.k.h | 12 | ||
28.g | odd | 6 | 1 | 637.2.u.h | 12 | ||
52.i | odd | 6 | 1 | 91.2.q.a | ✓ | 12 | |
52.i | odd | 6 | 1 | 1183.2.c.i | 12 | ||
52.j | odd | 6 | 1 | 1183.2.c.i | 12 | ||
52.l | even | 12 | 1 | 1183.2.a.m | 6 | ||
52.l | even | 12 | 1 | 1183.2.a.p | 6 | ||
156.r | even | 6 | 1 | 819.2.ct.a | 12 | ||
364.s | odd | 6 | 1 | 637.2.k.h | 12 | ||
364.w | even | 6 | 1 | 637.2.u.i | 12 | ||
364.bc | even | 6 | 1 | 637.2.q.h | 12 | ||
364.bk | odd | 6 | 1 | 637.2.u.h | 12 | ||
364.bp | even | 6 | 1 | 637.2.k.g | 12 | ||
364.bv | odd | 12 | 1 | 8281.2.a.by | 6 | ||
364.bv | odd | 12 | 1 | 8281.2.a.ch | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.2.q.a | ✓ | 12 | 4.b | odd | 2 | 1 | |
91.2.q.a | ✓ | 12 | 52.i | odd | 6 | 1 | |
637.2.k.g | 12 | 28.f | even | 6 | 1 | ||
637.2.k.g | 12 | 364.bp | even | 6 | 1 | ||
637.2.k.h | 12 | 28.g | odd | 6 | 1 | ||
637.2.k.h | 12 | 364.s | odd | 6 | 1 | ||
637.2.q.h | 12 | 28.d | even | 2 | 1 | ||
637.2.q.h | 12 | 364.bc | even | 6 | 1 | ||
637.2.u.h | 12 | 28.g | odd | 6 | 1 | ||
637.2.u.h | 12 | 364.bk | odd | 6 | 1 | ||
637.2.u.i | 12 | 28.f | even | 6 | 1 | ||
637.2.u.i | 12 | 364.w | even | 6 | 1 | ||
819.2.ct.a | 12 | 12.b | even | 2 | 1 | ||
819.2.ct.a | 12 | 156.r | even | 6 | 1 | ||
1183.2.a.m | 6 | 52.l | even | 12 | 1 | ||
1183.2.a.p | 6 | 52.l | even | 12 | 1 | ||
1183.2.c.i | 12 | 52.i | odd | 6 | 1 | ||
1183.2.c.i | 12 | 52.j | odd | 6 | 1 | ||
1456.2.cc.c | 12 | 1.a | even | 1 | 1 | trivial | |
1456.2.cc.c | 12 | 13.e | even | 6 | 1 | inner | |
8281.2.a.by | 6 | 364.bv | odd | 12 | 1 | ||
8281.2.a.ch | 6 | 364.bv | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 11 T_{3}^{10} + 4 T_{3}^{9} + 96 T_{3}^{8} + 42 T_{3}^{7} + 287 T_{3}^{6} + 390 T_{3}^{5} + 709 T_{3}^{4} + 516 T_{3}^{3} + 300 T_{3}^{2} + 80 T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( T^{12} + 11 T^{10} + 4 T^{9} + 96 T^{8} + \cdots + 16 \)
$5$
\( T^{12} + 40 T^{10} + 600 T^{8} + \cdots + 3481 \)
$7$
\( (T^{4} - T^{2} + 1)^{3} \)
$11$
\( T^{12} + 6 T^{11} - 7 T^{10} - 114 T^{9} + \cdots + 256 \)
$13$
\( T^{12} - 4 T^{11} + 21 T^{10} + \cdots + 4826809 \)
$17$
\( T^{12} + 4 T^{11} + 37 T^{10} + \cdots + 241081 \)
$19$
\( T^{12} - 29 T^{10} + 748 T^{8} + \cdots + 55696 \)
$23$
\( T^{12} - 12 T^{11} + 164 T^{10} + \cdots + 38539264 \)
$29$
\( T^{12} - 8 T^{11} + 108 T^{10} + \cdots + 10042561 \)
$31$
\( T^{12} + 136 T^{10} + 5854 T^{8} + \cdots + 913936 \)
$37$
\( T^{12} + 42 T^{11} + \cdots + 1755945216 \)
$41$
\( T^{12} - 30 T^{11} + \cdots + 884705536 \)
$43$
\( T^{12} + 2 T^{11} + 113 T^{10} + \cdots + 2408704 \)
$47$
\( T^{12} + 272 T^{10} + 21782 T^{8} + \cdots + 9461776 \)
$53$
\( (T^{6} + 22 T^{5} + 91 T^{4} - 700 T^{3} + \cdots - 2339)^{2} \)
$59$
\( T^{12} + 18 T^{11} + \cdots + 4571923456 \)
$61$
\( T^{12} - 14 T^{11} + 283 T^{10} + \cdots + 5607424 \)
$67$
\( T^{12} - 24 T^{11} + \cdots + 613651984 \)
$71$
\( T^{12} - 24 T^{11} + 212 T^{10} + \cdots + 46895104 \)
$73$
\( T^{12} + 334 T^{10} + \cdots + 1386221824 \)
$79$
\( (T^{6} - 28 T^{5} + 212 T^{4} - 192 T^{3} + \cdots - 512)^{2} \)
$83$
\( T^{12} + 304 T^{10} + \cdots + 141324544 \)
$89$
\( T^{12} + 12 T^{11} + \cdots + 1834580224 \)
$97$
\( T^{12} - 6 T^{11} - 173 T^{10} + \cdots + 53465344 \)
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