Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,6,Mod(49,304)]
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, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.i (of order \(3\), 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: | \(48.7566812231\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 38) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 2x^{7} + 386x^{6} + 3436x^{5} + 128708x^{4} + 568528x^{3} + 7340704x^{2} - 19430784x + 211527936 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 390471533 \nu^{7} + 3142267604 \nu^{6} + 128794858006 \nu^{5} + 2734091869688 \nu^{4} + \cdots + 21\!\cdots\!64 ) / 27\!\cdots\!56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6473945 \nu^{7} + 36183422 \nu^{6} - 2297742050 \nu^{5} - 13997685940 \nu^{4} + \cdots + 136296431422848 ) / 44885936230176 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18756773 \nu^{7} + 22066664 \nu^{6} - 6657181370 \nu^{5} - 40555089316 \nu^{4} + \cdots + 20\!\cdots\!88 ) / 22442968115088 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14831978953 \nu^{7} - 117204668308 \nu^{6} - 4803969780662 \nu^{5} + \cdots - 79\!\cdots\!28 ) / 13\!\cdots\!28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 35647275847 \nu^{7} - 1347014053248 \nu^{6} + 26507480732490 \nu^{5} + \cdots - 12\!\cdots\!44 ) / 90\!\cdots\!52 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 71425529135 \nu^{7} + 280447752060 \nu^{6} + 11494956177090 \nu^{5} + 561804417935112 \nu^{4} + \cdots + 19\!\cdots\!60 ) / 90\!\cdots\!52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} - 8\beta_{3} + 188\beta_{2} - 188 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} + 28\beta_{4} - 326\beta_{3} - 326\beta _1 - 1414 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{7} - 820\beta_{5} - 60100\beta_{2} - 5044\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 772\beta_{6} - 15008\beta_{5} - 15008\beta_{4} + 130764\beta_{3} - 911516\beta_{2} + 911516 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5744\beta_{7} + 5744\beta_{6} - 351576\beta_{4} + 2507520\beta_{3} + 2507520\beta _1 + 23936064 \)
|
| \(\nu^{7}\) | \(=\) |
\( 282648\beta_{7} + 7124496\beta_{5} + 455799624\beta_{2} + 56964328\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(191\) | \(229\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −4.84386 | + | 8.38982i | 0 | −23.1201 | + | 40.0451i | 0 | −177.800 | 0 | 74.5740 | + | 129.166i | 0 | ||||||||||||||||||||||||||||||||||||
| 49.2 | 0 | −4.18703 | + | 7.25215i | 0 | −12.5904 | + | 21.8073i | 0 | 187.086 | 0 | 86.4375 | + | 149.714i | 0 | |||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | 3.86148 | − | 6.68827i | 0 | 46.3693 | − | 80.3140i | 0 | −13.5472 | 0 | 91.6780 | + | 158.791i | 0 | |||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 12.1694 | − | 21.0781i | 0 | −28.6588 | + | 49.6385i | 0 | −33.7394 | 0 | −174.689 | − | 302.571i | 0 | |||||||||||||||||||||||||||||||||||||
| 273.1 | 0 | −4.84386 | − | 8.38982i | 0 | −23.1201 | − | 40.0451i | 0 | −177.800 | 0 | 74.5740 | − | 129.166i | 0 | |||||||||||||||||||||||||||||||||||||
| 273.2 | 0 | −4.18703 | − | 7.25215i | 0 | −12.5904 | − | 21.8073i | 0 | 187.086 | 0 | 86.4375 | − | 149.714i | 0 | |||||||||||||||||||||||||||||||||||||
| 273.3 | 0 | 3.86148 | + | 6.68827i | 0 | 46.3693 | + | 80.3140i | 0 | −13.5472 | 0 | 91.6780 | − | 158.791i | 0 | |||||||||||||||||||||||||||||||||||||
| 273.4 | 0 | 12.1694 | + | 21.0781i | 0 | −28.6588 | − | 49.6385i | 0 | −33.7394 | 0 | −174.689 | + | 302.571i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 304.6.i.b | 8 | |
| 4.b | odd | 2 | 1 | 38.6.c.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 342.6.g.d | 8 | ||
| 19.c | even | 3 | 1 | inner | 304.6.i.b | 8 | |
| 76.f | even | 6 | 1 | 722.6.a.g | 4 | ||
| 76.g | odd | 6 | 1 | 38.6.c.b | ✓ | 8 | |
| 76.g | odd | 6 | 1 | 722.6.a.j | 4 | ||
| 228.m | even | 6 | 1 | 342.6.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.c.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 38.6.c.b | ✓ | 8 | 76.g | odd | 6 | 1 | |
| 304.6.i.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 304.6.i.b | 8 | 19.c | even | 3 | 1 | inner | |
| 342.6.g.d | 8 | 12.b | even | 2 | 1 | ||
| 342.6.g.d | 8 | 228.m | even | 6 | 1 | ||
| 722.6.a.g | 4 | 76.f | even | 6 | 1 | ||
| 722.6.a.j | 4 | 76.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 14 T_{3}^{7} + 506 T_{3}^{6} + 2752 T_{3}^{5} + 91967 T_{3}^{4} + 180832 T_{3}^{3} + \cdots + 232532001 \)
acting on \(S_{6}^{\mathrm{new}}(304, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 14 T^{7} + \cdots + 232532001 \)
|
| $5$ |
\( T^{8} + \cdots + 38307063132900 \)
|
| $7$ |
\( (T^{4} + 38 T^{3} + \cdots - 15204096)^{2} \)
|
| $11$ |
\( (T^{4} + 72 T^{3} + \cdots - 423612144)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 10\!\cdots\!44 \)
|
| $17$ |
\( T^{8} + \cdots + 23\!\cdots\!24 \)
|
| $19$ |
\( T^{8} + \cdots + 37\!\cdots\!01 \)
|
| $23$ |
\( T^{8} + \cdots + 42\!\cdots\!44 \)
|
| $29$ |
\( T^{8} + \cdots + 41\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots - 9213535294816)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 27\!\cdots\!80)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 51\!\cdots\!25 \)
|
| $43$ |
\( T^{8} + \cdots + 78\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 88\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 59\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!16 \)
|
| $67$ |
\( T^{8} + \cdots + 52\!\cdots\!25 \)
|
| $71$ |
\( T^{8} + \cdots + 31\!\cdots\!96 \)
|
| $73$ |
\( T^{8} + \cdots + 28\!\cdots\!09 \)
|
| $79$ |
\( T^{8} + \cdots + 64\!\cdots\!16 \)
|
| $83$ |
\( (T^{4} + \cdots - 61\!\cdots\!52)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 61\!\cdots\!16 \)
|
| $97$ |
\( T^{8} + \cdots + 53\!\cdots\!25 \)
|
| show more | |
| show less |