Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1175,2,Mod(424,1175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1175, 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("1175.424");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1175 = 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1175.c (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: | \(9.38242223750\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.980441344.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 47) |
| 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
\( x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{7} - 7\nu^{5} - 12\nu^{3} - \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} - 7\nu^{4} - 12\nu^{2} - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} + 8\nu^{4} + 17\nu^{2} + 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} + 8\nu^{5} + 17\nu^{3} + 5\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} + 8\nu^{5} + 18\nu^{3} + 9\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} - 4\beta_{4} + 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{7} + 6\beta_{6} + \beta_{3} + 16\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16\beta_{5} + 15\beta_{4} - 23\beta_{2} - 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23\beta_{7} - 30\beta_{6} - 8\beta_{3} - 65\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times\).
| \(n\) | \(377\) | \(851\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 424.1 |
|
− | 2.30887i | − | 2.87576i | −3.33090 | 0 | −6.63978 | − | 1.65227i | 3.07289i | −5.27002 | 0 | |||||||||||||||||||||||||||||||||||||||
| 424.2 | − | 2.26608i | − | 0.824788i | −3.13511 | 0 | −1.86903 | 3.21243i | 2.57226i | 2.31972 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 424.3 | − | 0.673363i | − | 0.188279i | 1.54658 | 0 | −0.126780 | 3.31128i | − | 2.38814i | 2.96455 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 424.4 | − | 0.283841i | 2.23925i | 1.91943 | 0 | 0.635593 | − | 2.44658i | − | 1.11250i | −2.01426 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 424.5 | 0.283841i | − | 2.23925i | 1.91943 | 0 | 0.635593 | 2.44658i | 1.11250i | −2.01426 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 424.6 | 0.673363i | 0.188279i | 1.54658 | 0 | −0.126780 | − | 3.31128i | 2.38814i | 2.96455 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 424.7 | 2.26608i | 0.824788i | −3.13511 | 0 | −1.86903 | − | 3.21243i | − | 2.57226i | 2.31972 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 424.8 | 2.30887i | 2.87576i | −3.33090 | 0 | −6.63978 | 1.65227i | − | 3.07289i | −5.27002 | 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 | 1175.2.c.e | 8 | |
| 5.b | even | 2 | 1 | inner | 1175.2.c.e | 8 | |
| 5.c | odd | 4 | 1 | 47.2.a.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1175.2.a.f | 4 | ||
| 15.e | even | 4 | 1 | 423.2.a.k | 4 | ||
| 20.e | even | 4 | 1 | 752.2.a.h | 4 | ||
| 35.f | even | 4 | 1 | 2303.2.a.h | 4 | ||
| 40.i | odd | 4 | 1 | 3008.2.a.q | 4 | ||
| 40.k | even | 4 | 1 | 3008.2.a.p | 4 | ||
| 55.e | even | 4 | 1 | 5687.2.a.s | 4 | ||
| 60.l | odd | 4 | 1 | 6768.2.a.bv | 4 | ||
| 65.h | odd | 4 | 1 | 7943.2.a.h | 4 | ||
| 235.e | even | 4 | 1 | 2209.2.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.2.a.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 423.2.a.k | 4 | 15.e | even | 4 | 1 | ||
| 752.2.a.h | 4 | 20.e | even | 4 | 1 | ||
| 1175.2.a.f | 4 | 5.c | odd | 4 | 1 | ||
| 1175.2.c.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1175.2.c.e | 8 | 5.b | even | 2 | 1 | inner | |
| 2209.2.a.e | 4 | 235.e | even | 4 | 1 | ||
| 2303.2.a.h | 4 | 35.f | even | 4 | 1 | ||
| 3008.2.a.p | 4 | 40.k | even | 4 | 1 | ||
| 3008.2.a.q | 4 | 40.i | odd | 4 | 1 | ||
| 5687.2.a.s | 4 | 55.e | even | 4 | 1 | ||
| 6768.2.a.bv | 4 | 60.l | odd | 4 | 1 | ||
| 7943.2.a.h | 4 | 65.h | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1175, [\chi])\):
|
\( T_{2}^{8} + 11T_{2}^{6} + 33T_{2}^{4} + 15T_{2}^{2} + 1 \)
|
|
\( T_{11}^{4} + 6T_{11}^{3} - 4T_{11}^{2} - 56T_{11} - 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 11 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{8} + 14 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 30 T^{6} + \cdots + 1849 \)
|
| $11$ |
\( (T^{4} + 6 T^{3} - 4 T^{2} + \cdots - 48)^{2} \)
|
| $13$ |
\( T^{8} + 64 T^{6} + \cdots + 2304 \)
|
| $17$ |
\( T^{8} + 78 T^{6} + \cdots + 19881 \)
|
| $19$ |
\( (T^{4} - 16 T^{2} + \cdots + 16)^{2} \)
|
| $23$ |
\( T^{8} + 76 T^{6} + \cdots + 256 \)
|
| $29$ |
\( (T^{4} - 10 T^{3} + \cdots - 16)^{2} \)
|
| $31$ |
\( (T^{4} + 8 T^{3} - 56 T + 48)^{2} \)
|
| $37$ |
\( T^{8} + 70 T^{6} + \cdots + 81 \)
|
| $41$ |
\( (T^{4} - 6 T^{3} - 8 T^{2} + \cdots - 16)^{2} \)
|
| $43$ |
\( T^{8} + 164 T^{6} + \cdots + 186624 \)
|
| $47$ |
\( (T^{2} + 1)^{4} \)
|
| $53$ |
\( T^{8} + 238 T^{6} + \cdots + 5900041 \)
|
| $59$ |
\( (T^{4} + 4 T^{3} + \cdots - 519)^{2} \)
|
| $61$ |
\( (T^{4} + 6 T^{3} + \cdots + 337)^{2} \)
|
| $67$ |
\( T^{8} + 340 T^{6} + \cdots + 10137856 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots + 657)^{2} \)
|
| $73$ |
\( T^{8} + 364 T^{6} + \cdots + 58736896 \)
|
| $79$ |
\( (T^{4} + 20 T^{3} + \cdots - 47)^{2} \)
|
| $83$ |
\( T^{8} + 240 T^{6} + \cdots + 65536 \)
|
| $89$ |
\( (T^{4} - 6 T^{3} + \cdots + 4841)^{2} \)
|
| $97$ |
\( T^{8} + 542 T^{6} + \cdots + 204690249 \)
|
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