Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,3,Mod(65,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.65");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.r (of order \(6\), degree \(2\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(8.28340003655\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 917x^{4} - 1582x^{3} + 4294x^{2} - 3528x + 4971 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 76) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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
\( x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 917x^{4} - 1582x^{3} + 4294x^{2} - 3528x + 4971 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu + 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{7} - 28\nu^{6} + 434\nu^{5} - 1015\nu^{4} + 6314\nu^{3} - 8470\nu^{2} + 16740\nu - 7506 ) / 1029 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} + 14\nu^{6} - 217\nu^{5} + 1022\nu^{4} - 4186\nu^{3} + 17612\nu^{2} - 21747\nu + 39768 ) / 1029 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -47\nu^{7} - 7\nu^{6} - 1778\nu^{5} - 2569\nu^{4} - 14714\nu^{3} - 57169\nu^{2} + 21531\nu - 187170 ) / 2058 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} - 46\nu^{4} + 87\nu^{3} - 562\nu^{2} + 519\nu - 1251 ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 34\nu^{7} - 119\nu^{6} + 1673\nu^{5} - 3885\nu^{4} + 22204\nu^{3} - 29309\nu^{2} + 56739\nu - 21439 ) / 343 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 - 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + 4\beta_{5} - \beta_{3} + \beta_{2} - 17\beta _1 - 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 4\beta_{6} + 8\beta_{5} + 2\beta_{4} - \beta_{3} - 24\beta_{2} - 34\beta _1 + 246 \)
|
| \(\nu^{5}\) | \(=\) |
\( -24\beta_{7} + 44\beta_{6} - 88\beta_{5} + 5\beta_{4} + 50\beta_{3} - 48\beta_{2} + 329\beta _1 + 466 \)
|
| \(\nu^{6}\) | \(=\) |
\( -77\beta_{7} + 136\beta_{6} - 284\beta_{5} - 77\beta_{4} + 109\beta_{3} + 485\beta_{2} + 1029\beta _1 - 4820 \)
|
| \(\nu^{7}\) | \(=\) |
\( 497\beta_{7} - 840\beta_{6} + 1638\beta_{5} - 287\beta_{4} - 1540\beta_{3} + 1526\beta_{2} - 6177\beta _1 - 15083 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(191\) | \(229\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(1\) | \(1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −4.72527 | − | 2.72814i | 0 | 3.36497 | − | 5.82829i | 0 | 10.3204 | 0 | 10.3855 | + | 17.9882i | 0 | ||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −2.28889 | − | 1.32149i | 0 | −4.81674 | + | 8.34284i | 0 | −7.59243 | 0 | −1.00732 | − | 1.74474i | 0 | |||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | 0.707846 | + | 0.408675i | 0 | 2.50973 | − | 4.34699i | 0 | −7.91625 | 0 | −4.16597 | − | 7.21567i | 0 | |||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | 3.30632 | + | 1.90890i | 0 | −1.55796 | + | 2.69846i | 0 | 11.1883 | 0 | 2.78782 | + | 4.82865i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.1 | 0 | −4.72527 | + | 2.72814i | 0 | 3.36497 | + | 5.82829i | 0 | 10.3204 | 0 | 10.3855 | − | 17.9882i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.2 | 0 | −2.28889 | + | 1.32149i | 0 | −4.81674 | − | 8.34284i | 0 | −7.59243 | 0 | −1.00732 | + | 1.74474i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.3 | 0 | 0.707846 | − | 0.408675i | 0 | 2.50973 | + | 4.34699i | 0 | −7.91625 | 0 | −4.16597 | + | 7.21567i | 0 | |||||||||||||||||||||||||||||||||||||
| 145.4 | 0 | 3.30632 | − | 1.90890i | 0 | −1.55796 | − | 2.69846i | 0 | 11.1883 | 0 | 2.78782 | − | 4.82865i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 304.3.r.c | 8 | |
| 4.b | odd | 2 | 1 | 76.3.h.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 684.3.y.h | 8 | ||
| 19.d | odd | 6 | 1 | inner | 304.3.r.c | 8 | |
| 76.f | even | 6 | 1 | 76.3.h.a | ✓ | 8 | |
| 76.f | even | 6 | 1 | 1444.3.c.b | 8 | ||
| 76.g | odd | 6 | 1 | 1444.3.c.b | 8 | ||
| 228.n | odd | 6 | 1 | 684.3.y.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.3.h.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 76.3.h.a | ✓ | 8 | 76.f | even | 6 | 1 | |
| 304.3.r.c | 8 | 1.a | even | 1 | 1 | trivial | |
| 304.3.r.c | 8 | 19.d | odd | 6 | 1 | inner | |
| 684.3.y.h | 8 | 12.b | even | 2 | 1 | ||
| 684.3.y.h | 8 | 228.n | odd | 6 | 1 | ||
| 1444.3.c.b | 8 | 76.f | even | 6 | 1 | ||
| 1444.3.c.b | 8 | 76.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 6T_{3}^{7} - 8T_{3}^{6} - 120T_{3}^{5} + 211T_{3}^{4} + 1440T_{3}^{3} + 828T_{3}^{2} - 3240T_{3} + 2025 \)
acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 6 T^{7} + \cdots + 2025 \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots + 1028196 \)
|
| $7$ |
\( (T^{4} - 6 T^{3} + \cdots + 6940)^{2} \)
|
| $11$ |
\( (T^{4} - 5 T^{3} + \cdots - 4746)^{2} \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots + 46785600 \)
|
| $17$ |
\( T^{8} - 23 T^{7} + \cdots + 56250000 \)
|
| $19$ |
\( T^{8} + \cdots + 16983563041 \)
|
| $23$ |
\( T^{8} + \cdots + 21126622500 \)
|
| $29$ |
\( T^{8} + 105 T^{7} + \cdots + 21068100 \)
|
| $31$ |
\( T^{8} + \cdots + 2425365504 \)
|
| $37$ |
\( T^{8} + \cdots + 409702406400 \)
|
| $41$ |
\( T^{8} + \cdots + 74733890625 \)
|
| $43$ |
\( T^{8} + \cdots + 386933761600 \)
|
| $47$ |
\( T^{8} + \cdots + 105819282922500 \)
|
| $53$ |
\( T^{8} + \cdots + 143608586342400 \)
|
| $59$ |
\( T^{8} + \cdots + 8199855604521 \)
|
| $61$ |
\( T^{8} + \cdots + 4850565316 \)
|
| $67$ |
\( T^{8} + \cdots + 669597133730625 \)
|
| $71$ |
\( T^{8} + \cdots + 918515225664 \)
|
| $73$ |
\( T^{8} + \cdots + 11\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 771333601536 \)
|
| $83$ |
\( (T^{4} - 41 T^{3} + \cdots - 897750)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 498714472231056 \)
|
| $97$ |
\( T^{8} + \cdots + 258895984160025 \)
|
| show more | |
| show less |