Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [425,4,Mod(324,425)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(425, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("425.324");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 425 = 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 425.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: | \(25.0758117524\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 44x^{8} + 690x^{6} + 4708x^{4} + 14337x^{2} + 15876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 44x^{8} + 690x^{6} + 4708x^{4} + 14337x^{2} + 15876 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 13\nu^{9} + 509\nu^{7} + 6576\nu^{5} + 31153\nu^{3} + 47403\nu ) / 1890 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{9} - 509\nu^{7} - 6576\nu^{5} - 31153\nu^{3} - 43623\nu ) / 1890 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} + 35\nu^{6} + 375\nu^{4} + 1189\nu^{2} + 720 ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{9} + 281\nu^{7} + 3669\nu^{5} + 17107\nu^{3} + 25812\nu ) / 540 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -59\nu^{9} - 2407\nu^{7} - 32583\nu^{5} - 163049\nu^{3} - 259794\nu ) / 3780 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -107\nu^{9} - 4141\nu^{7} - 51339\nu^{5} - 216287\nu^{3} - 265302\nu ) / 3780 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -7\nu^{8} - 281\nu^{6} - 3669\nu^{4} - 16927\nu^{2} - 23832 ) / 180 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7\nu^{8} + 281\nu^{6} + 3669\nu^{4} + 17107\nu^{2} + 25452 ) / 180 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{8} + 38\nu^{6} + 477\nu^{4} + 2206\nu^{2} + 3261 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{6} + 4\beta_{5} + 6\beta_{4} - 11\beta_{2} - 5\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - 20\beta_{8} - 19\beta_{7} - \beta_{3} + 115 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -36\beta_{6} - 80\beta_{5} - 124\beta_{4} + 155\beta_{2} + 59\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -29\beta_{9} + 341\beta_{8} + 307\beta_{7} + 22\beta_{3} - 1706 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 570\beta_{6} + 1288\beta_{5} + 2162\beta_{4} - 2381\beta_{2} - 1187\beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( 640\beta_{9} - 5624\beta_{8} - 4809\beta_{7} - 359\beta_{3} + 26566 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -8900\beta_{6} - 19548\beta_{5} - 36304\beta_{4} + 37533\beta_{2} + 25257\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 324.1 |
|
− | 5.46013i | − | 0.820806i | −21.8130 | 0 | −4.48171 | 22.0083i | 75.4209i | 26.3263 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 324.2 | − | 5.14668i | 1.13153i | −18.4883 | 0 | 5.82364 | − | 26.5778i | 53.9797i | 25.7196 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 324.3 | − | 3.58234i | 2.57070i | −4.83317 | 0 | 9.20914 | − | 25.2782i | − | 11.3447i | 20.3915 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 324.4 | − | 2.18655i | − | 6.08748i | 3.21900 | 0 | −13.3106 | 10.8418i | − | 24.5309i | −10.0574 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 324.5 | − | 0.290753i | − | 7.70584i | 7.91546 | 0 | −2.24049 | − | 22.4661i | − | 4.62746i | −32.3800 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 324.6 | 0.290753i | 7.70584i | 7.91546 | 0 | −2.24049 | 22.4661i | 4.62746i | −32.3800 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 324.7 | 2.18655i | 6.08748i | 3.21900 | 0 | −13.3106 | − | 10.8418i | 24.5309i | −10.0574 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 324.8 | 3.58234i | − | 2.57070i | −4.83317 | 0 | 9.20914 | 25.2782i | 11.3447i | 20.3915 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 324.9 | 5.14668i | − | 1.13153i | −18.4883 | 0 | 5.82364 | 26.5778i | − | 53.9797i | 25.7196 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 324.10 | 5.46013i | 0.820806i | −21.8130 | 0 | −4.48171 | − | 22.0083i | − | 75.4209i | 26.3263 | 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 | 425.4.b.i | 10 | |
| 5.b | even | 2 | 1 | inner | 425.4.b.i | 10 | |
| 5.c | odd | 4 | 1 | 85.4.a.g | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 425.4.a.i | 5 | ||
| 15.e | even | 4 | 1 | 765.4.a.m | 5 | ||
| 20.e | even | 4 | 1 | 1360.4.a.w | 5 | ||
| 85.g | odd | 4 | 1 | 1445.4.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.4.a.g | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 425.4.a.i | 5 | 5.c | odd | 4 | 1 | ||
| 425.4.b.i | 10 | 1.a | even | 1 | 1 | trivial | |
| 425.4.b.i | 10 | 5.b | even | 2 | 1 | inner | |
| 765.4.a.m | 5 | 15.e | even | 4 | 1 | ||
| 1360.4.a.w | 5 | 20.e | even | 4 | 1 | ||
| 1445.4.a.l | 5 | 85.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):
|
\( T_{2}^{10} + 74T_{2}^{8} + 1849T_{2}^{6} + 17520T_{2}^{4} + 49920T_{2}^{2} + 4096 \)
|
|
\( T_{3}^{10} + 105T_{3}^{8} + 3040T_{3}^{6} + 20176T_{3}^{4} + 30864T_{3}^{2} + 12544 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 74 T^{8} + \cdots + 4096 \)
|
| $3$ |
\( T^{10} + 105 T^{8} + \cdots + 12544 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots + 12970600566784 \)
|
| $11$ |
\( (T^{5} - 126 T^{4} + \cdots - 167488)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 58426088540416 \)
|
| $17$ |
\( (T^{2} + 289)^{5} \)
|
| $19$ |
\( (T^{5} + 55 T^{4} + \cdots + 2274428800)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 51\!\cdots\!36 \)
|
| $29$ |
\( (T^{5} + 195 T^{4} + \cdots + 2844978640)^{2} \)
|
| $31$ |
\( (T^{5} - 97 T^{4} + \cdots - 73509549120)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 48\!\cdots\!04 \)
|
| $41$ |
\( (T^{5} - 298 T^{4} + \cdots - 478624610720)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 22\!\cdots\!84 \)
|
| $47$ |
\( T^{10} + \cdots + 70\!\cdots\!04 \)
|
| $53$ |
\( T^{10} + \cdots + 22\!\cdots\!00 \)
|
| $59$ |
\( (T^{5} + \cdots - 1194050514560)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 37656238184848)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 15\!\cdots\!84 \)
|
| $71$ |
\( (T^{5} + \cdots + 26612610106464)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 38\!\cdots\!44 \)
|
| $79$ |
\( (T^{5} + \cdots - 10071329239040)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 93\!\cdots\!16 \)
|
| $89$ |
\( (T^{5} + \cdots + 30709087651760)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 29\!\cdots\!00 \)
|
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