Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1075,2,Mod(1,1075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1075 = 5^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1075.a (trivial) |
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: | yes |
| Analytic conductor: | \(8.58391821729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 6x^{5} + 13x^{4} + 15x^{3} - 7x^{2} - 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 215) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 6x^{5} + 13x^{4} + 15x^{3} - 7x^{2} - 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + 4\nu^{5} + 2\nu^{4} - 14\nu^{3} - 3\nu^{2} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 4\nu^{5} + 2\nu^{4} - 15\nu^{3} - \nu^{2} + 10\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 9\nu^{3} + 14\nu^{2} + 4\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 13\nu^{3} + 14\nu^{2} - 5\nu - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 13\nu^{3} + 15\nu^{2} - 7\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} - 2\beta_{5} - \beta_{3} + \beta_{2} + 8\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{6} - 9\beta_{5} + \beta_{4} - 4\beta_{3} + 4\beta_{2} + 23\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 23\beta_{6} - 26\beta_{5} + 4\beta_{4} - 17\beta_{3} + 18\beta_{2} + 77\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 77\beta_{6} - 91\beta_{5} + 18\beta_{4} - 62\beta_{3} + 65\beta_{2} + 242\beta _1 + 70 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.25270 | 2.38208 | 3.07465 | 0 | −5.36611 | 3.07793 | −2.42085 | 2.67432 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.38203 | −0.670186 | −0.0899964 | 0 | 0.926217 | 3.14172 | 2.88844 | −2.55085 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0.449707 | −0.980548 | −1.79776 | 0 | −0.440960 | −3.80803 | −1.70788 | −2.03853 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.787169 | 2.31820 | −1.38037 | 0 | 1.82481 | 1.65206 | −2.66092 | 2.37403 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.54717 | −2.43386 | 0.393729 | 0 | −3.76559 | 2.19857 | −2.48517 | 2.92369 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.29269 | 1.48286 | 3.25644 | 0 | 3.39974 | 1.39135 | 2.88062 | −0.801134 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.55799 | 2.90146 | 4.54331 | 0 | 7.42190 | −4.65360 | 6.50577 | 5.41847 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(43\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1075.2.a.t | 7 | |
| 3.b | odd | 2 | 1 | 9675.2.a.cm | 7 | ||
| 5.b | even | 2 | 1 | 1075.2.a.s | 7 | ||
| 5.c | odd | 4 | 2 | 215.2.b.b | ✓ | 14 | |
| 15.d | odd | 2 | 1 | 9675.2.a.cn | 7 | ||
| 15.e | even | 4 | 2 | 1935.2.b.d | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.b.b | ✓ | 14 | 5.c | odd | 4 | 2 | |
| 1075.2.a.s | 7 | 5.b | even | 2 | 1 | ||
| 1075.2.a.t | 7 | 1.a | even | 1 | 1 | trivial | |
| 1935.2.b.d | 14 | 15.e | even | 4 | 2 | ||
| 9675.2.a.cm | 7 | 3.b | odd | 2 | 1 | ||
| 9675.2.a.cn | 7 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1075))\):
|
\( T_{2}^{7} - 4T_{2}^{6} - 3T_{2}^{5} + 27T_{2}^{4} - 18T_{2}^{3} - 32T_{2}^{2} + 38T_{2} - 10 \)
|
|
\( T_{3}^{7} - 5T_{3}^{6} - 2T_{3}^{5} + 39T_{3}^{4} - 30T_{3}^{3} - 62T_{3}^{2} + 40T_{3} + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots - 10 \)
|
| $3$ |
\( T^{7} - 5 T^{6} + \cdots + 38 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 3 T^{6} + \cdots - 866 \)
|
| $11$ |
\( T^{7} - T^{6} + \cdots + 719 \)
|
| $13$ |
\( T^{7} - 14 T^{6} + \cdots - 200 \)
|
| $17$ |
\( T^{7} - 12 T^{6} + \cdots + 152 \)
|
| $19$ |
\( T^{7} - 40 T^{5} + \cdots + 2560 \)
|
| $23$ |
\( T^{7} - 18 T^{6} + \cdots + 6680 \)
|
| $29$ |
\( T^{7} - 10 T^{6} + \cdots + 304 \)
|
| $31$ |
\( T^{7} + 15 T^{6} + \cdots - 5149 \)
|
| $37$ |
\( T^{7} - 5 T^{6} + \cdots + 3800 \)
|
| $41$ |
\( T^{7} + 21 T^{6} + \cdots + 60091 \)
|
| $43$ |
\( (T + 1)^{7} \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots - 1504 \)
|
| $53$ |
\( T^{7} - 42 T^{6} + \cdots + 2712440 \)
|
| $59$ |
\( T^{7} - 9 T^{6} + \cdots + 26380 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots + 89984 \)
|
| $67$ |
\( T^{7} + 28 T^{6} + \cdots + 611648 \)
|
| $71$ |
\( T^{7} - 2 T^{6} + \cdots + 4105856 \)
|
| $73$ |
\( T^{7} - 11 T^{6} + \cdots + 5582 \)
|
| $79$ |
\( T^{7} + 9 T^{6} + \cdots + 78308 \)
|
| $83$ |
\( T^{7} - 12 T^{6} + \cdots - 12736 \)
|
| $89$ |
\( T^{7} - 6 T^{6} + \cdots + 251248 \)
|
| $97$ |
\( T^{7} - 26 T^{6} + \cdots + 14691064 \)
|
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