Newspace parameters
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(10.1041806482\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3\cdot 7^{2} \) |
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
\( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 8905 \nu^{7} + 366059 \nu^{6} - 1191820 \nu^{5} + 33153695 \nu^{4} - 47979268 \nu^{3} + 3262309295 \nu^{2} - 141627885 \nu + 307508418 ) / 9888988410 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 1257 \nu^{7} - 279647 \nu^{6} + 388990 \nu^{5} - 26314019 \nu^{4} - 14452616 \nu^{3} - 2382173465 \nu^{2} - 51985031 \nu - 251390874 ) / 156968070 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 22795 \nu^{7} + 56368 \nu^{6} + 2163175 \nu^{5} + 2122285 \nu^{4} + 222588334 \nu^{3} + 7196875 \nu^{2} + 20796090 \nu - 31645054854 ) / 706356315 \)
|
\(\beta_{5}\) | \(=\) |
\( ( - 341623 \nu^{7} - 8468248 \nu^{6} + 27571040 \nu^{5} - 966332554 \nu^{4} + 1109931296 \nu^{3} - 75468829240 \nu^{2} + \cdots - 235881275616 ) / 9888988410 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 25511 \nu^{7} + 22795 \nu^{6} - 2420915 \nu^{5} - 2375153 \nu^{4} - 232964210 \nu^{3} - 8054375 \nu^{2} - 23273922 \nu - 54979470 ) / 706356315 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 15589754 \nu^{7} - 23777815 \nu^{6} + 1539409445 \nu^{5} + 516927917 \nu^{4} + 141447402185 \nu^{3} - 70438948615 \nu^{2} + \cdots + 24613669710 ) / 9888988410 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{3} + 49\beta_{2} + \beta_1 \)
|
\(\nu^{3}\) | \(=\) |
\( -2\beta_{7} - 91\beta_{6} + 2\beta_{5} - 2\beta_{4} - 44 \)
|
\(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 99\beta_{4} - 99\beta_{3} - 4505\beta_{2} - 181\beta _1 - 4406 \)
|
\(\nu^{5}\) | \(=\) |
\( 196\beta_{7} + 8707\beta_{6} - 286\beta_{3} - 8624\beta_{2} - 8707\beta_1 \)
|
\(\nu^{6}\) | \(=\) |
\( 376\beta_{7} + 25333\beta_{6} - 376\beta_{5} + 9581\beta_{4} + 421300 \)
|
\(\nu^{7}\) | \(=\) |
\( -18786\beta_{5} + 36042\beta_{4} + 36042\beta_{3} + 1222350\beta_{2} + 841891\beta _1 + 1186308 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
\(n\) | \(10\) | \(29\) |
\(\chi(n)\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
−4.59061 | + | 7.95118i | 0 | −26.1475 | − | 45.2888i | 11.0358 | − | 19.1146i | 0 | 126.882 | − | 26.6059i | 186.333 | 0 | 101.322 | + | 175.495i | ||||||||||||||||||||||||||||||||
37.2 | −0.395402 | + | 0.684857i | 0 | 15.6873 | + | 27.1712i | −52.0958 | + | 90.2327i | 0 | −7.12980 | − | 129.446i | −50.1170 | 0 | −41.1977 | − | 71.3564i | |||||||||||||||||||||||||||||||||
37.3 | 1.37409 | − | 2.37999i | 0 | 12.2237 | + | 21.1722i | 29.1836 | − | 50.5475i | 0 | −21.4366 | + | 127.857i | 155.128 | 0 | −80.2019 | − | 138.914i | |||||||||||||||||||||||||||||||||
37.4 | 5.11193 | − | 8.85412i | 0 | −36.2636 | − | 62.8104i | 11.8764 | − | 20.5705i | 0 | 30.6840 | − | 125.958i | −414.344 | 0 | −121.423 | − | 210.310i | |||||||||||||||||||||||||||||||||
46.1 | −4.59061 | − | 7.95118i | 0 | −26.1475 | + | 45.2888i | 11.0358 | + | 19.1146i | 0 | 126.882 | + | 26.6059i | 186.333 | 0 | 101.322 | − | 175.495i | |||||||||||||||||||||||||||||||||
46.2 | −0.395402 | − | 0.684857i | 0 | 15.6873 | − | 27.1712i | −52.0958 | − | 90.2327i | 0 | −7.12980 | + | 129.446i | −50.1170 | 0 | −41.1977 | + | 71.3564i | |||||||||||||||||||||||||||||||||
46.3 | 1.37409 | + | 2.37999i | 0 | 12.2237 | − | 21.1722i | 29.1836 | + | 50.5475i | 0 | −21.4366 | − | 127.857i | 155.128 | 0 | −80.2019 | + | 138.914i | |||||||||||||||||||||||||||||||||
46.4 | 5.11193 | + | 8.85412i | 0 | −36.2636 | + | 62.8104i | 11.8764 | + | 20.5705i | 0 | 30.6840 | + | 125.958i | −414.344 | 0 | −121.423 | + | 210.310i | |||||||||||||||||||||||||||||||||
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 | 63.6.e.e | 8 | |
3.b | odd | 2 | 1 | 21.6.e.c | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 63.6.e.e | 8 | |
7.c | even | 3 | 1 | 441.6.a.w | 4 | ||
7.d | odd | 6 | 1 | 441.6.a.v | 4 | ||
12.b | even | 2 | 1 | 336.6.q.j | 8 | ||
21.c | even | 2 | 1 | 147.6.e.o | 8 | ||
21.g | even | 6 | 1 | 147.6.a.l | 4 | ||
21.g | even | 6 | 1 | 147.6.e.o | 8 | ||
21.h | odd | 6 | 1 | 21.6.e.c | ✓ | 8 | |
21.h | odd | 6 | 1 | 147.6.a.m | 4 | ||
84.n | even | 6 | 1 | 336.6.q.j | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.6.e.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
21.6.e.c | ✓ | 8 | 21.h | odd | 6 | 1 | |
63.6.e.e | 8 | 1.a | even | 1 | 1 | trivial | |
63.6.e.e | 8 | 7.c | even | 3 | 1 | inner | |
147.6.a.l | 4 | 21.g | even | 6 | 1 | ||
147.6.a.m | 4 | 21.h | odd | 6 | 1 | ||
147.6.e.o | 8 | 21.c | even | 2 | 1 | ||
147.6.e.o | 8 | 21.g | even | 6 | 1 | ||
336.6.q.j | 8 | 12.b | even | 2 | 1 | ||
336.6.q.j | 8 | 84.n | even | 6 | 1 | ||
441.6.a.v | 4 | 7.d | odd | 6 | 1 | ||
441.6.a.w | 4 | 7.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 3T_{2}^{7} + 103T_{2}^{6} - 90T_{2}^{5} + 9190T_{2}^{4} - 16260T_{2}^{3} + 53772T_{2}^{2} + 37944T_{2} + 41616 \)
acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 3 T^{7} + 103 T^{6} + \cdots + 41616 \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 7657 T^{6} + \cdots + 10164899803536 \)
$7$
\( T^{8} - 258 T^{7} + \cdots + 79\!\cdots\!01 \)
$11$
\( T^{8} - 402 T^{7} + \cdots + 28\!\cdots\!76 \)
$13$
\( (T^{4} - 462 T^{3} + \cdots + 149501563456)^{2} \)
$17$
\( T^{8} - 276 T^{7} + \cdots + 25\!\cdots\!24 \)
$19$
\( T^{8} + 510 T^{7} + \cdots + 54\!\cdots\!76 \)
$23$
\( T^{8} - 6900 T^{7} + \cdots + 90\!\cdots\!76 \)
$29$
\( (T^{4} + 540 T^{3} + \cdots + 408027025117872)^{2} \)
$31$
\( T^{8} - 6410 T^{7} + \cdots + 75\!\cdots\!09 \)
$37$
\( T^{8} + 15250 T^{7} + \cdots + 25\!\cdots\!36 \)
$41$
\( (T^{4} + 4308 T^{3} + \cdots - 18\!\cdots\!52)^{2} \)
$43$
\( (T^{4} - 29198 T^{3} + \cdots - 991662745581932)^{2} \)
$47$
\( T^{8} + 15060 T^{7} + \cdots + 73\!\cdots\!36 \)
$53$
\( T^{8} - 13692 T^{7} + \cdots + 72\!\cdots\!84 \)
$59$
\( T^{8} - 34830 T^{7} + \cdots + 66\!\cdots\!96 \)
$61$
\( T^{8} - 5364 T^{7} + \cdots + 32\!\cdots\!76 \)
$67$
\( T^{8} - 5994 T^{7} + \cdots + 30\!\cdots\!56 \)
$71$
\( (T^{4} + 89268 T^{3} + \cdots + 21\!\cdots\!68)^{2} \)
$73$
\( T^{8} + 59638 T^{7} + \cdots + 14\!\cdots\!00 \)
$79$
\( T^{8} - 44062 T^{7} + \cdots + 27\!\cdots\!81 \)
$83$
\( (T^{4} - 208446 T^{3} + \cdots - 41\!\cdots\!32)^{2} \)
$89$
\( T^{8} + 77520 T^{7} + \cdots + 22\!\cdots\!24 \)
$97$
\( (T^{4} + 188630 T^{3} + \cdots - 11\!\cdots\!44)^{2} \)
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