Newspace parameters
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.q (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(53.8889634572\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
Defining polynomial: |
\( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7^{2} \) |
Twist minimal: | no (minimal twist has level 21) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \)
:
\(\beta_{1}\) | \(=\) |
\( ( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 141627885 \nu - 307508418 ) / 9888988410 \)
|
\(\beta_{2}\) | \(=\) |
\( ( 8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 39697581525 \nu - 10196496828 ) / 4944494205 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 65603 \nu^{7} - 25662916 \nu^{6} + 83553680 \nu^{5} - 2627557456 \nu^{4} + 3363632432 \nu^{3} - 228707310580 \nu^{2} + \cdots - 714835153872 ) / 9888988410 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 731039 \nu^{7} - 7994302 \nu^{6} + 26027960 \nu^{5} - 628584958 \nu^{4} + 1047811304 \nu^{3} - 71245033510 \nu^{2} - 451136527737 \nu - 222679608984 ) / 9888988410 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 1437521 \nu^{7} - 1642579 \nu^{6} + 136763060 \nu^{5} + 99854873 \nu^{4} + 13093975028 \nu^{3} - 2811264295 \nu^{2} + 41000921157 \nu + 2771341902 ) / 4944494205 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 484709 \nu^{7} - 433105 \nu^{6} + 45997385 \nu^{5} + 45127907 \nu^{4} + 4285048727 \nu^{3} + 153033125 \nu^{2} + 442204518 \nu - 5171325642 ) / 282542526 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 2434409 \nu^{7} - 2482177 \nu^{6} + 231017885 \nu^{5} + 226651007 \nu^{4} + 21466747139 \nu^{3} + 768595625 \nu^{2} + 2220933918 \nu + 102356221716 ) / 1412712630 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 2 ) / 8 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 98\beta _1 + 1 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -16\beta_{6} + 95\beta_{5} - 95\beta_{2} - 542 ) / 8 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -101\beta_{4} - 97\beta_{3} - 22\beta_{2} + 9050\beta _1 - 9050 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( -360\beta_{7} + 1928\beta_{6} - 9009\beta_{5} - 1928\beta_{4} - 360\beta_{3} + 85442\beta _1 - 360 ) / 8 \)
|
\(\nu^{6}\) | \(=\) |
\( ( -9205\beta_{7} + 9957\beta_{6} - 4220\beta_{5} + 4220\beta_{2} + 860245 ) / 2 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 219312\beta_{4} + 69024\beta_{3} + 862207\beta_{2} - 11352926\beta _1 + 11352926 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
\(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
193.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | −29.1836 | − | 50.5475i | 0 | 21.4366 | + | 127.857i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||||||||||||||
193.2 | 0 | 4.50000 | − | 7.79423i | 0 | −11.8764 | − | 20.5705i | 0 | −30.6840 | − | 125.958i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
193.3 | 0 | 4.50000 | − | 7.79423i | 0 | −11.0358 | − | 19.1146i | 0 | −126.882 | − | 26.6059i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
193.4 | 0 | 4.50000 | − | 7.79423i | 0 | 52.0958 | + | 90.2327i | 0 | 7.12980 | − | 129.446i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
289.1 | 0 | 4.50000 | + | 7.79423i | 0 | −29.1836 | + | 50.5475i | 0 | 21.4366 | − | 127.857i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
289.2 | 0 | 4.50000 | + | 7.79423i | 0 | −11.8764 | + | 20.5705i | 0 | −30.6840 | + | 125.958i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
289.3 | 0 | 4.50000 | + | 7.79423i | 0 | −11.0358 | + | 19.1146i | 0 | −126.882 | + | 26.6059i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
289.4 | 0 | 4.50000 | + | 7.79423i | 0 | 52.0958 | − | 90.2327i | 0 | 7.12980 | + | 129.446i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||||||||||||
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 | 336.6.q.j | 8 | |
4.b | odd | 2 | 1 | 21.6.e.c | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 336.6.q.j | 8 | |
12.b | even | 2 | 1 | 63.6.e.e | 8 | ||
28.d | even | 2 | 1 | 147.6.e.o | 8 | ||
28.f | even | 6 | 1 | 147.6.a.l | 4 | ||
28.f | even | 6 | 1 | 147.6.e.o | 8 | ||
28.g | odd | 6 | 1 | 21.6.e.c | ✓ | 8 | |
28.g | odd | 6 | 1 | 147.6.a.m | 4 | ||
84.j | odd | 6 | 1 | 441.6.a.v | 4 | ||
84.n | even | 6 | 1 | 63.6.e.e | 8 | ||
84.n | even | 6 | 1 | 441.6.a.w | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.6.e.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
21.6.e.c | ✓ | 8 | 28.g | odd | 6 | 1 | |
63.6.e.e | 8 | 12.b | even | 2 | 1 | ||
63.6.e.e | 8 | 84.n | even | 6 | 1 | ||
147.6.a.l | 4 | 28.f | even | 6 | 1 | ||
147.6.a.m | 4 | 28.g | odd | 6 | 1 | ||
147.6.e.o | 8 | 28.d | even | 2 | 1 | ||
147.6.e.o | 8 | 28.f | even | 6 | 1 | ||
336.6.q.j | 8 | 1.a | even | 1 | 1 | trivial | |
336.6.q.j | 8 | 7.c | even | 3 | 1 | inner | |
441.6.a.v | 4 | 84.j | odd | 6 | 1 | ||
441.6.a.w | 4 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + 67214905692 T_{5}^{2} + 965081458800 T_{5} + 10164899803536 \)
acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} - 9 T + 81)^{4} \)
$5$
\( T^{8} + 7657 T^{6} + \cdots + 10164899803536 \)
$7$
\( T^{8} + 258 T^{7} + \cdots + 79\!\cdots\!01 \)
$11$
\( T^{8} - 402 T^{7} + \cdots + 28\!\cdots\!76 \)
$13$
\( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \)
$17$
\( T^{8} + 276 T^{7} + \cdots + 25\!\cdots\!24 \)
$19$
\( T^{8} - 510 T^{7} + \cdots + 54\!\cdots\!76 \)
$23$
\( T^{8} - 6900 T^{7} + \cdots + 90\!\cdots\!76 \)
$29$
\( (T^{4} - 540 T^{3} + \cdots + 408027025117872)^{2} \)
$31$
\( T^{8} + 6410 T^{7} + \cdots + 75\!\cdots\!09 \)
$37$
\( T^{8} + 15250 T^{7} + \cdots + 25\!\cdots\!36 \)
$41$
\( (T^{4} - 4308 T^{3} + \cdots - 18\!\cdots\!52)^{2} \)
$43$
\( (T^{4} + 29198 T^{3} + \cdots - 991662745581932)^{2} \)
$47$
\( T^{8} + 15060 T^{7} + \cdots + 73\!\cdots\!36 \)
$53$
\( T^{8} + 13692 T^{7} + \cdots + 72\!\cdots\!84 \)
$59$
\( T^{8} - 34830 T^{7} + \cdots + 66\!\cdots\!96 \)
$61$
\( T^{8} - 5364 T^{7} + \cdots + 32\!\cdots\!76 \)
$67$
\( T^{8} + 5994 T^{7} + \cdots + 30\!\cdots\!56 \)
$71$
\( (T^{4} + 89268 T^{3} + \cdots + 21\!\cdots\!68)^{2} \)
$73$
\( T^{8} + 59638 T^{7} + \cdots + 14\!\cdots\!00 \)
$79$
\( T^{8} + 44062 T^{7} + \cdots + 27\!\cdots\!81 \)
$83$
\( (T^{4} - 208446 T^{3} + \cdots - 41\!\cdots\!32)^{2} \)
$89$
\( T^{8} - 77520 T^{7} + \cdots + 22\!\cdots\!24 \)
$97$
\( (T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44)^{2} \)
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