Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,10,Mod(193,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.193");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.d (of order \(2\), degree \(1\), 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: | \(197.773761087\) |
| 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: |
\( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{2} \) |
| Twist minimal: | yes |
| 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
\( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4766\nu^{6} + 51851374\nu^{4} + 88099997114\nu^{2} - 43229725983072 ) / 34840779813 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 169786\nu^{6} + 1579828010\nu^{4} + 1426285970446\nu^{2} - 1771419698208576 ) / 243885458691 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 199366\nu^{6} + 2241395438\nu^{4} + 5124570485866\nu^{2} + 1439961094183104 ) / 243885458691 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 28496\nu^{7} + 372522607\nu^{5} + 1264029013547\nu^{3} + 1137344409469551\nu ) / 4843958573246310 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 797877302 \nu^{7} - 9758638510954 \nu^{5} + \cdots - 22\!\cdots\!32 \nu ) / 75\!\cdots\!05 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1012635742 \nu^{7} - 15512580206174 \nu^{5} + \cdots - 96\!\cdots\!32 \nu ) / 75\!\cdots\!05 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 575969794 \nu^{7} + 7049643741998 \nu^{5} + \cdots + 18\!\cdots\!44 \nu ) / 10\!\cdots\!15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 5\beta_{5} - 96\beta_{4} ) / 192 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 48\beta_{3} + 13\beta_{2} - 353\beta _1 - 626976 ) / 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5251\beta_{7} - 2988\beta_{6} - 36371\beta_{5} - 24620160\beta_{4} ) / 192 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -307416\beta_{3} - 258409\beta_{2} + 3152165\beta _1 + 3849294240 ) / 192 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 17617919\beta_{7} + 13863726\beta_{6} + 145754053\beta_{5} + 134259999792\beta_{4} ) / 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1228613316\beta_{3} + 1285518437\beta_{2} - 13182475621\beta _1 - 14273447764896 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -267619114645\beta_{7} - 229933464768\beta_{6} - 2397041072585\beta_{5} - 2381735984318112\beta_{4} ) / 192 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | − | 81.0000i | 0 | − | 1783.68i | 0 | −4965.03 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | − | 81.0000i | 0 | − | 1428.10i | 0 | 6382.13 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | − | 81.0000i | 0 | 473.533i | 0 | −574.939 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | − | 81.0000i | 0 | 2178.24i | 0 | −7658.16 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | 81.0000i | 0 | − | 2178.24i | 0 | −7658.16 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.6 | 0 | 81.0000i | 0 | − | 473.533i | 0 | −574.939 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 193.7 | 0 | 81.0000i | 0 | 1428.10i | 0 | 6382.13 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 193.8 | 0 | 81.0000i | 0 | 1783.68i | 0 | −4965.03 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 384.10.d.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 384.10.d.b | yes | 8 | |
| 8.b | even | 2 | 1 | inner | 384.10.d.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | 384.10.d.b | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.10.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 384.10.d.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 384.10.d.b | yes | 8 | 4.b | odd | 2 | 1 | |
| 384.10.d.b | yes | 8 | 8.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 \(S_{10}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{8} + 10189952T_{5}^{6} + 33495402213376T_{5}^{4} + 37796298115527475200T_{5}^{2} + 6903383511575235133440000 \)
|
|
\( T_{7}^{4} + 6816T_{7}^{3} - 38951584T_{7}^{2} - 267125408256T_{7} - 139519152670464 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 6561)^{4} \)
|
| $5$ |
\( T^{8} + \cdots + 69\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 139519152670464)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 95\!\cdots\!76 \)
|
| $13$ |
\( T^{8} + \cdots + 12\!\cdots\!64 \)
|
| $17$ |
\( (T^{4} + \cdots - 30\!\cdots\!80)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( (T^{4} + \cdots - 24\!\cdots\!20)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 21\!\cdots\!36 \)
|
| $31$ |
\( (T^{4} + \cdots - 12\!\cdots\!76)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $41$ |
\( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $47$ |
\( (T^{4} + \cdots - 77\!\cdots\!80)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 46\!\cdots\!64 \)
|
| $59$ |
\( T^{8} + \cdots + 94\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 45\!\cdots\!96 \)
|
| $67$ |
\( T^{8} + \cdots + 14\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + \cdots - 21\!\cdots\!16)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots - 50\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 11\!\cdots\!08)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 34\!\cdots\!96 \)
|
| $89$ |
\( (T^{4} + \cdots + 15\!\cdots\!60)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 71\!\cdots\!20)^{2} \)
|
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