Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,2,Mod(349,725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, 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("725.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.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: | \(5.78915414654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 145) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 7\nu^{2} - 32\nu + 13 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 21\nu^{5} - 30\nu^{4} + 15\nu^{3} + 67\nu^{2} + 96\nu - 39 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 3\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{4} + \beta_{3} + 4\beta_{2} - 3\beta _1 - 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} - 3\beta _1 - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{5} + 6\beta_{4} + 5\beta_{3} - 17\beta_{2} - 11\beta _1 - 18 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).
| \(n\) | \(176\) | \(552\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
− | 2.67513i | − | 0.806063i | −5.15633 | 0 | −2.15633 | 4.15633i | 8.44358i | 2.35026 | 0 | ||||||||||||||||||||||||||||||||||
| 349.2 | − | 1.53919i | 1.70928i | −0.369102 | 0 | 2.63090 | − | 0.630898i | − | 2.51026i | 0.0783777 | 0 | ||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.21432i | 2.90321i | 0.525428 | 0 | 3.52543 | 1.52543i | − | 3.06668i | −5.42864 | 0 | |||||||||||||||||||||||||||||||||||
| 349.4 | 1.21432i | − | 2.90321i | 0.525428 | 0 | 3.52543 | − | 1.52543i | 3.06668i | −5.42864 | 0 | |||||||||||||||||||||||||||||||||||
| 349.5 | 1.53919i | − | 1.70928i | −0.369102 | 0 | 2.63090 | 0.630898i | 2.51026i | 0.0783777 | 0 | ||||||||||||||||||||||||||||||||||||
| 349.6 | 2.67513i | 0.806063i | −5.15633 | 0 | −2.15633 | − | 4.15633i | − | 8.44358i | 2.35026 | 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 | 725.2.b.d | 6 | |
| 5.b | even | 2 | 1 | inner | 725.2.b.d | 6 | |
| 5.c | odd | 4 | 1 | 145.2.a.d | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 725.2.a.d | 3 | ||
| 15.e | even | 4 | 1 | 1305.2.a.o | 3 | ||
| 15.e | even | 4 | 1 | 6525.2.a.bh | 3 | ||
| 20.e | even | 4 | 1 | 2320.2.a.s | 3 | ||
| 35.f | even | 4 | 1 | 7105.2.a.p | 3 | ||
| 40.i | odd | 4 | 1 | 9280.2.a.bu | 3 | ||
| 40.k | even | 4 | 1 | 9280.2.a.bm | 3 | ||
| 145.h | odd | 4 | 1 | 4205.2.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.a.d | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 725.2.a.d | 3 | 5.c | odd | 4 | 1 | ||
| 725.2.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 725.2.b.d | 6 | 5.b | even | 2 | 1 | inner | |
| 1305.2.a.o | 3 | 15.e | even | 4 | 1 | ||
| 2320.2.a.s | 3 | 20.e | even | 4 | 1 | ||
| 4205.2.a.e | 3 | 145.h | odd | 4 | 1 | ||
| 6525.2.a.bh | 3 | 15.e | even | 4 | 1 | ||
| 7105.2.a.p | 3 | 35.f | even | 4 | 1 | ||
| 9280.2.a.bm | 3 | 40.k | even | 4 | 1 | ||
| 9280.2.a.bu | 3 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\):
|
\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25 \)
|
|
\( T_{3}^{6} + 12T_{3}^{4} + 32T_{3}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 11 T^{4} + 31 T^{2} + 25 \)
|
| $3$ |
\( T^{6} + 12 T^{4} + 32 T^{2} + 16 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 20 T^{4} + 48 T^{2} + 16 \)
|
| $11$ |
\( (T^{3} - 8 T^{2} + 16 T - 4)^{2} \)
|
| $13$ |
\( T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 5776 \)
|
| $19$ |
\( (T^{3} - 28 T - 52)^{2} \)
|
| $23$ |
\( T^{6} + 76 T^{4} + 1472 T^{2} + \cdots + 5776 \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( (T^{3} - 12 T^{2} + 20 T - 4)^{2} \)
|
| $37$ |
\( T^{6} + 96 T^{4} + 2144 T^{2} + \cdots + 4624 \)
|
| $41$ |
\( (T^{3} + 10 T^{2} + 20 T - 8)^{2} \)
|
| $43$ |
\( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \)
|
| $47$ |
\( T^{6} + 204 T^{4} + 6912 T^{2} + \cdots + 8464 \)
|
| $53$ |
\( T^{6} + 60 T^{4} + 560 T^{2} + \cdots + 64 \)
|
| $59$ |
\( (T^{3} + 4 T^{2} - 48 T + 80)^{2} \)
|
| $61$ |
\( (T^{3} - 6 T^{2} - 4 T + 40)^{2} \)
|
| $67$ |
\( T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400 \)
|
| $71$ |
\( (T^{3} - 24 T^{2} + 176 T - 368)^{2} \)
|
| $73$ |
\( T^{6} + 376 T^{4} + 41264 T^{2} + \cdots + 1227664 \)
|
| $79$ |
\( (T^{3} + 8 T^{2} - 56 T - 20)^{2} \)
|
| $83$ |
\( T^{6} + 68 T^{4} + 816 T^{2} + \cdots + 2704 \)
|
| $89$ |
\( (T^{3} + 22 T^{2} + 124 T + 200)^{2} \)
|
| $97$ |
\( T^{6} + 600 T^{4} + 88560 T^{2} + \cdots + 204304 \)
|
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