Newspace parameters
Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 200.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(32.0767639626\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.12220785438976.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{15} \) |
Twist minimal: | no (minimal twist has level 8) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) :
\(\beta_{1}\) | \(=\) | \( ( 5\nu^{7} - 7\nu^{5} + 164\nu^{3} - 832\nu ) / 2304 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} + 5\nu^{5} + 116\nu^{3} + 320\nu ) / 256 \) |
\(\beta_{3}\) | \(=\) | \( ( -5\nu^{6} + 7\nu^{4} - 164\nu^{2} + 976 ) / 144 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} - 11\nu^{4} + 100\nu^{2} - 608 ) / 48 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} - 21\nu^{5} - 68\nu^{3} - 1280\nu ) / 128 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{6} + 13\nu^{4} + 124\nu^{2} + 664 ) / 24 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{7} - 13\nu^{5} - 76\nu^{3} + 368\nu ) / 72 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} - \beta_{5} + \beta_{2} + \beta_1 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + 3\beta_{4} + 3\beta_{3} - 10 ) / 8 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{7} + 9\beta_{5} + 55\beta_{2} - 73\beta_1 ) / 16 \) |
\(\nu^{4}\) | \(=\) | \( ( 7\beta_{6} - 19\beta_{4} - 3\beta_{3} - 414 ) / 8 \) |
\(\nu^{5}\) | \(=\) | \( ( -63\beta_{7} - 41\beta_{5} - 151\beta_{2} - 279\beta_1 ) / 16 \) |
\(\nu^{6}\) | \(=\) | \( ( -23\beta_{6} - 125\beta_{4} - 333\beta_{3} + 1310 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( 111\beta_{7} - 519\beta_{5} - 1849\beta_{2} + 9543\beta_1 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(151\) | \(177\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 |
|
−4.21569 | − | 3.77200i | −3.25452 | 3.54400 | + | 31.8031i | 0 | 13.7200 | + | 12.2760i | 112.704i | 105.021 | − | 147.440i | −232.408 | 0 | ||||||||||||||||||||||||||||||||||
149.2 | −4.21569 | + | 3.77200i | −3.25452 | 3.54400 | − | 31.8031i | 0 | 13.7200 | − | 12.2760i | − | 112.704i | 105.021 | + | 147.440i | −232.408 | 0 | ||||||||||||||||||||||||||||||||||
149.3 | −3.03776 | − | 4.77200i | 23.6095 | −13.5440 | + | 28.9924i | 0 | −71.7200 | − | 112.665i | 160.704i | 179.495 | − | 23.4400i | 314.408 | 0 | |||||||||||||||||||||||||||||||||||
149.4 | −3.03776 | + | 4.77200i | 23.6095 | −13.5440 | − | 28.9924i | 0 | −71.7200 | + | 112.665i | − | 160.704i | 179.495 | + | 23.4400i | 314.408 | 0 | ||||||||||||||||||||||||||||||||||
149.5 | 3.03776 | − | 4.77200i | −23.6095 | −13.5440 | − | 28.9924i | 0 | −71.7200 | + | 112.665i | 160.704i | −179.495 | − | 23.4400i | 314.408 | 0 | |||||||||||||||||||||||||||||||||||
149.6 | 3.03776 | + | 4.77200i | −23.6095 | −13.5440 | + | 28.9924i | 0 | −71.7200 | − | 112.665i | − | 160.704i | −179.495 | + | 23.4400i | 314.408 | 0 | ||||||||||||||||||||||||||||||||||
149.7 | 4.21569 | − | 3.77200i | 3.25452 | 3.54400 | − | 31.8031i | 0 | 13.7200 | − | 12.2760i | 112.704i | −105.021 | − | 147.440i | −232.408 | 0 | |||||||||||||||||||||||||||||||||||
149.8 | 4.21569 | + | 3.77200i | 3.25452 | 3.54400 | + | 31.8031i | 0 | 13.7200 | + | 12.2760i | − | 112.704i | −105.021 | + | 147.440i | −232.408 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 200.6.f.a | 8 | |
4.b | odd | 2 | 1 | 800.6.f.a | 8 | ||
5.b | even | 2 | 1 | inner | 200.6.f.a | 8 | |
5.c | odd | 4 | 1 | 8.6.b.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 200.6.d.a | 4 | ||
8.b | even | 2 | 1 | inner | 200.6.f.a | 8 | |
8.d | odd | 2 | 1 | 800.6.f.a | 8 | ||
15.e | even | 4 | 1 | 72.6.d.b | 4 | ||
20.d | odd | 2 | 1 | 800.6.f.a | 8 | ||
20.e | even | 4 | 1 | 32.6.b.a | 4 | ||
20.e | even | 4 | 1 | 800.6.d.a | 4 | ||
40.e | odd | 2 | 1 | 800.6.f.a | 8 | ||
40.f | even | 2 | 1 | inner | 200.6.f.a | 8 | |
40.i | odd | 4 | 1 | 8.6.b.a | ✓ | 4 | |
40.i | odd | 4 | 1 | 200.6.d.a | 4 | ||
40.k | even | 4 | 1 | 32.6.b.a | 4 | ||
40.k | even | 4 | 1 | 800.6.d.a | 4 | ||
60.l | odd | 4 | 1 | 288.6.d.b | 4 | ||
80.i | odd | 4 | 1 | 256.6.a.k | 4 | ||
80.j | even | 4 | 1 | 256.6.a.n | 4 | ||
80.s | even | 4 | 1 | 256.6.a.n | 4 | ||
80.t | odd | 4 | 1 | 256.6.a.k | 4 | ||
120.q | odd | 4 | 1 | 288.6.d.b | 4 | ||
120.w | even | 4 | 1 | 72.6.d.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.6.b.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
8.6.b.a | ✓ | 4 | 40.i | odd | 4 | 1 | |
32.6.b.a | 4 | 20.e | even | 4 | 1 | ||
32.6.b.a | 4 | 40.k | even | 4 | 1 | ||
72.6.d.b | 4 | 15.e | even | 4 | 1 | ||
72.6.d.b | 4 | 120.w | even | 4 | 1 | ||
200.6.d.a | 4 | 5.c | odd | 4 | 1 | ||
200.6.d.a | 4 | 40.i | odd | 4 | 1 | ||
200.6.f.a | 8 | 1.a | even | 1 | 1 | trivial | |
200.6.f.a | 8 | 5.b | even | 2 | 1 | inner | |
200.6.f.a | 8 | 8.b | even | 2 | 1 | inner | |
200.6.f.a | 8 | 40.f | even | 2 | 1 | inner | |
256.6.a.k | 4 | 80.i | odd | 4 | 1 | ||
256.6.a.k | 4 | 80.t | odd | 4 | 1 | ||
256.6.a.n | 4 | 80.j | even | 4 | 1 | ||
256.6.a.n | 4 | 80.s | even | 4 | 1 | ||
288.6.d.b | 4 | 60.l | odd | 4 | 1 | ||
288.6.d.b | 4 | 120.q | odd | 4 | 1 | ||
800.6.d.a | 4 | 20.e | even | 4 | 1 | ||
800.6.d.a | 4 | 40.k | even | 4 | 1 | ||
800.6.f.a | 8 | 4.b | odd | 2 | 1 | ||
800.6.f.a | 8 | 8.d | odd | 2 | 1 | ||
800.6.f.a | 8 | 20.d | odd | 2 | 1 | ||
800.6.f.a | 8 | 40.e | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 568T_{3}^{2} + 5904 \)
acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 20 T^{6} + 1856 T^{4} + \cdots + 1048576 \)
$3$
\( (T^{4} - 568 T^{2} + 5904)^{2} \)
$5$
\( T^{8} \)
$7$
\( (T^{4} + 38528 T^{2} + \cdots + 328044544)^{2} \)
$11$
\( (T^{4} + 347768 T^{2} + \cdots + 5520765456)^{2} \)
$13$
\( (T^{4} - 590944 T^{2} + \cdots + 7999305984)^{2} \)
$17$
\( (T^{4} + 154504 T^{2} + \cdots + 5220351504)^{2} \)
$19$
\( (T^{4} + 3109816 T^{2} + \cdots + 120994976016)^{2} \)
$23$
\( (T^{4} + 6998656 T^{2} + \cdots + 7936705990656)^{2} \)
$29$
\( (T^{4} + 50789216 T^{2} + \cdots + 535633608132864)^{2} \)
$31$
\( (T^{2} + 6464 T + 7754752)^{4} \)
$37$
\( (T^{4} - 36113248 T^{2} + \cdots + 306881230162176)^{2} \)
$41$
\( (T^{2} + 2284 T - 85109148)^{4} \)
$43$
\( (T^{4} - 121686904 T^{2} + \cdots + 23\!\cdots\!56)^{2} \)
$47$
\( (T^{4} + 483273216 T^{2} + \cdots + 17\!\cdots\!64)^{2} \)
$53$
\( (T^{4} - 633629792 T^{2} + \cdots + 76\!\cdots\!44)^{2} \)
$59$
\( (T^{4} + 1322273016 T^{2} + \cdots + 22\!\cdots\!36)^{2} \)
$61$
\( (T^{4} + 4119483744 T^{2} + \cdots + 41\!\cdots\!16)^{2} \)
$67$
\( (T^{4} - 4033664568 T^{2} + \cdots + 32\!\cdots\!84)^{2} \)
$71$
\( (T^{2} - 103344 T + 2609278272)^{4} \)
$73$
\( (T^{4} + 3608600776 T^{2} + \cdots + 25\!\cdots\!56)^{2} \)
$79$
\( (T^{2} - 123936 T + 3701816576)^{4} \)
$83$
\( (T^{4} - 5708307384 T^{2} + \cdots + 72\!\cdots\!56)^{2} \)
$89$
\( (T^{2} - 42316 T - 6875717724)^{4} \)
$97$
\( (T^{4} + 4045463048 T^{2} + \cdots + 61\!\cdots\!04)^{2} \)
show more
show less