Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,4,Mod(64,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.64");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.d (of order \(2\), degree \(1\), 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: | \(18.5856016518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.84052224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 36x^{2} - 36x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 105) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 6x^{3} + 36x^{2} - 36x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{5} + 3\nu^{4} + 53\nu^{3} - 130\nu^{2} + 42\nu - 9 ) / 327 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{5} - 19\nu^{4} + 64\nu^{3} + 24\nu^{2} - 48\nu + 57 ) / 327 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15\nu^{5} - 22\nu^{4} + 11\nu^{3} + 154\nu^{2} + 564\nu - 261 ) / 327 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -72\nu^{5} + 62\nu^{4} - 31\nu^{3} - 434\nu^{2} - 2838\nu - 186 ) / 327 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -78\nu^{5} + 158\nu^{4} - 79\nu^{3} - 452\nu^{2} - 2802\nu + 2796 ) / 327 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 10\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 3\beta_{2} + 3\beta _1 - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 3\beta_{2} - 37 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{4} - 52\beta_{3} + 21\beta_{2} - 21\beta _1 - 52 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(136\) | \(281\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
− | 4.10645i | 0 | −8.86293 | 6.58150 | − | 9.03792i | 0 | − | 7.00000i | 3.54358i | 0 | −37.1137 | − | 27.0266i | ||||||||||||||||||||||||||||||
| 64.2 | − | 3.18296i | 0 | −2.13122 | 11.0627 | − | 1.61735i | 0 | 7.00000i | − | 18.6801i | 0 | −5.14795 | − | 35.2122i | |||||||||||||||||||||||||||||||
| 64.3 | − | 0.0765073i | 0 | 7.99415 | −10.6442 | + | 3.42057i | 0 | − | 7.00000i | − | 1.22367i | 0 | 0.261698 | + | 0.814361i | ||||||||||||||||||||||||||||||
| 64.4 | 0.0765073i | 0 | 7.99415 | −10.6442 | − | 3.42057i | 0 | 7.00000i | 1.22367i | 0 | 0.261698 | − | 0.814361i | |||||||||||||||||||||||||||||||||
| 64.5 | 3.18296i | 0 | −2.13122 | 11.0627 | + | 1.61735i | 0 | − | 7.00000i | 18.6801i | 0 | −5.14795 | + | 35.2122i | ||||||||||||||||||||||||||||||||
| 64.6 | 4.10645i | 0 | −8.86293 | 6.58150 | + | 9.03792i | 0 | 7.00000i | − | 3.54358i | 0 | −37.1137 | + | 27.0266i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 315.4.d.a | 6 | |
| 3.b | odd | 2 | 1 | 105.4.d.a | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 315.4.d.a | 6 | |
| 5.c | odd | 4 | 1 | 1575.4.a.bc | 3 | ||
| 5.c | odd | 4 | 1 | 1575.4.a.bd | 3 | ||
| 15.d | odd | 2 | 1 | 105.4.d.a | ✓ | 6 | |
| 15.e | even | 4 | 1 | 525.4.a.q | 3 | ||
| 15.e | even | 4 | 1 | 525.4.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.d.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 105.4.d.a | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 315.4.d.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 315.4.d.a | 6 | 5.b | even | 2 | 1 | inner | |
| 525.4.a.q | 3 | 15.e | even | 4 | 1 | ||
| 525.4.a.r | 3 | 15.e | even | 4 | 1 | ||
| 1575.4.a.bc | 3 | 5.c | odd | 4 | 1 | ||
| 1575.4.a.bd | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 27T_{2}^{4} + 171T_{2}^{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 27 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 14 T^{5} + \cdots + 1953125 \)
|
| $7$ |
\( (T^{2} + 49)^{3} \)
|
| $11$ |
\( (T^{3} - 66 T^{2} + \cdots + 6120)^{2} \)
|
| $13$ |
\( T^{6} + 4236 T^{4} + \cdots + 823460416 \)
|
| $17$ |
\( T^{6} + \cdots + 4661065984 \)
|
| $19$ |
\( (T^{3} - 138 T^{2} + \cdots - 70632)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 52356761856 \)
|
| $29$ |
\( (T^{3} - 170 T^{2} + \cdots + 1680584)^{2} \)
|
| $31$ |
\( (T^{3} + 366 T^{2} + \cdots - 3510632)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 67599442284544 \)
|
| $41$ |
\( (T^{3} - 206 T^{2} + \cdots + 18222200)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 187722443001856 \)
|
| $47$ |
\( T^{6} + \cdots + 436551881342976 \)
|
| $53$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + 880 T^{2} + \cdots - 22878976)^{2} \)
|
| $61$ |
\( (T^{3} + 870 T^{2} + \cdots - 112468792)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} - 1018 T^{2} + \cdots + 133243912)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 39\!\cdots\!96 \)
|
| $79$ |
\( (T^{3} - 1620 T^{2} + \cdots + 258624448)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 47\!\cdots\!24 \)
|
| $89$ |
\( (T^{3} + 1938 T^{2} + \cdots + 181473048)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 19\!\cdots\!64 \)
|
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