Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(249,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.b (of order \(2\), degree \(1\), 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: | \(6.18840615665\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 155) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -215\nu^{7} + 864\nu^{6} + 144\nu^{5} + 454\nu^{4} - 7744\nu^{3} + 30084\nu^{2} - 6214\nu - 956 ) / 10798 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} - 5590\nu - 860 ) / 5399 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -216\nu^{7} - 36\nu^{6} - 6\nu^{5} + 431\nu^{4} - 6876\nu^{3} + 1446\nu^{2} + 5208\nu - 860 ) / 5399 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -655\nu^{7} - 1009\nu^{6} - 1068\nu^{5} + 1132\nu^{4} - 14552\nu^{3} - 12562\nu^{2} - 12402\nu - 1908 ) / 10798 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 714\nu^{7} + 119\nu^{6} - 880\nu^{5} - 5174\nu^{4} + 17330\nu^{3} - 3880\nu^{2} - 13616\nu - 36150 ) / 10798 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -858\nu^{7} - 143\nu^{6} + 876\nu^{5} + 1862\nu^{4} - 21914\nu^{3} + 4844\nu^{2} + 17088\nu - 14814 ) / 10798 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1955\nu^{7} - 574\nu^{6} + 1704\nu^{5} - 3626\nu^{4} + 52336\nu^{3} - 43532\nu^{2} + 43446\nu + 6684 ) / 21596 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{4} + 3\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} + 2\beta_{6} - 4\beta_{3} - 4\beta_{2} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{6} - 3\beta_{5} + \beta_{3} - 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14\beta_{7} + 13\beta_{6} + \beta_{5} + 2\beta_{4} - 20\beta_{3} + 20\beta_{2} + 9\beta _1 + 18 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -34\beta_{7} - 32\beta_{4} - 10\beta_{2} - 37\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 44\beta_{7} - 38\beta_{6} - 6\beta_{5} + 12\beta_{4} + 53\beta_{3} + 53\beta_{2} + 32\beta _1 - 64 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(652\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
− | 2.80027i | 0.342376i | −5.84153 | 0 | 0.958747 | 1.04125i | 10.7573i | 2.88278 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 249.2 | − | 2.27244i | − | 0.632112i | −3.16400 | 0 | −1.43644 | − | 3.43644i | 2.64511i | 2.60043 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 249.3 | − | 1.62946i | 3.05273i | −0.655151 | 0 | 4.97431 | − | 2.97431i | − | 2.19138i | −6.31916 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 249.4 | − | 1.15729i | 3.02722i | 0.660672 | 0 | 3.50338 | 1.50338i | − | 3.07918i | −6.16405 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 249.5 | 1.15729i | − | 3.02722i | 0.660672 | 0 | 3.50338 | − | 1.50338i | 3.07918i | −6.16405 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 249.6 | 1.62946i | − | 3.05273i | −0.655151 | 0 | 4.97431 | 2.97431i | 2.19138i | −6.31916 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 249.7 | 2.27244i | 0.632112i | −3.16400 | 0 | −1.43644 | 3.43644i | − | 2.64511i | 2.60043 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 249.8 | 2.80027i | − | 0.342376i | −5.84153 | 0 | 0.958747 | − | 1.04125i | − | 10.7573i | 2.88278 | 0 | ||||||||||||||||||||||||||||||||||||||||
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 | 775.2.b.e | 8 | |
| 5.b | even | 2 | 1 | inner | 775.2.b.e | 8 | |
| 5.c | odd | 4 | 1 | 155.2.a.d | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 775.2.a.g | 4 | ||
| 15.e | even | 4 | 1 | 1395.2.a.m | 4 | ||
| 15.e | even | 4 | 1 | 6975.2.a.bj | 4 | ||
| 20.e | even | 4 | 1 | 2480.2.a.z | 4 | ||
| 35.f | even | 4 | 1 | 7595.2.a.q | 4 | ||
| 40.i | odd | 4 | 1 | 9920.2.a.ch | 4 | ||
| 40.k | even | 4 | 1 | 9920.2.a.cd | 4 | ||
| 155.f | even | 4 | 1 | 4805.2.a.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 155.2.a.d | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 775.2.a.g | 4 | 5.c | odd | 4 | 1 | ||
| 775.2.b.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 775.2.b.e | 8 | 5.b | even | 2 | 1 | inner | |
| 1395.2.a.m | 4 | 15.e | even | 4 | 1 | ||
| 2480.2.a.z | 4 | 20.e | even | 4 | 1 | ||
| 4805.2.a.j | 4 | 155.f | even | 4 | 1 | ||
| 6975.2.a.bj | 4 | 15.e | even | 4 | 1 | ||
| 7595.2.a.q | 4 | 35.f | even | 4 | 1 | ||
| 9920.2.a.cd | 4 | 40.k | even | 4 | 1 | ||
| 9920.2.a.ch | 4 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 17T_{2}^{6} + 96T_{2}^{4} + 208T_{2}^{2} + 144 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 17 T^{6} + \cdots + 144 \)
|
| $3$ |
\( T^{8} + 19 T^{6} + \cdots + 4 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 24 T^{6} + \cdots + 256 \)
|
| $11$ |
\( (T^{4} + 6 T^{3} + \cdots - 144)^{2} \)
|
| $13$ |
\( T^{8} + 88 T^{6} + \cdots + 4096 \)
|
| $17$ |
\( T^{8} + 51 T^{6} + \cdots + 576 \)
|
| $19$ |
\( (T^{4} + 5 T^{3} + \cdots + 108)^{2} \)
|
| $23$ |
\( T^{8} + 128 T^{6} + \cdots + 576 \)
|
| $29$ |
\( (T^{4} + 6 T^{3} + \cdots - 456)^{2} \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} + 67 T^{6} + \cdots + 16 \)
|
| $41$ |
\( (T^{4} - 13 T^{3} + \cdots - 294)^{2} \)
|
| $43$ |
\( T^{8} + 63 T^{6} + \cdots + 45796 \)
|
| $47$ |
\( T^{8} + 124 T^{6} + \cdots + 36864 \)
|
| $53$ |
\( T^{8} + 271 T^{6} + \cdots + 8363664 \)
|
| $59$ |
\( (T^{4} + 13 T^{3} + \cdots + 2484)^{2} \)
|
| $61$ |
\( (T^{4} - 22 T^{3} + \cdots - 32)^{2} \)
|
| $67$ |
\( T^{8} + 84 T^{6} + \cdots + 1024 \)
|
| $71$ |
\( (T^{4} - 3 T^{3} + \cdots + 384)^{2} \)
|
| $73$ |
\( T^{8} + 271 T^{6} + \cdots + 204304 \)
|
| $79$ |
\( (T^{4} - 16 T^{3} + \cdots + 256)^{2} \)
|
| $83$ |
\( T^{8} + 295 T^{6} + \cdots + 544644 \)
|
| $89$ |
\( (T^{4} - 12 T^{3} + \cdots + 1656)^{2} \)
|
| $97$ |
\( T^{8} + 144 T^{6} + \cdots + 256 \)
|
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