Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [272,10,Mod(33,272)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(272, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("272.33");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 272.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: | \(140.089747437\) |
| 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: |
\( x^{8} + 117950x^{6} + 4041468216x^{4} + 33029034367104x^{2} + 61354193073357312 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 34) |
| 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
\( x^{8} + 117950x^{6} + 4041468216x^{4} + 33029034367104x^{2} + 61354193073357312 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4957\nu^{7} + 467912294\nu^{5} + 11265376287960\nu^{3} + 26641045932694848\nu ) / 1241367006600000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -37\nu^{7} - 1120654\nu^{5} + 42941215640\nu^{3} - 156091916011968\nu ) / 6568079400000 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} + 29488 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 68342\nu^{4} - 833408280\nu^{2} - 2468684412864 ) / 36450000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 58481\nu^{7} + 7058387002\nu^{5} + 236681330791680\nu^{3} + 1404342177198427584\nu ) / 620683503300000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} + 122342\nu^{4} + 3983444280\nu^{2} + 16660303344864 ) / 85050000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 29488 \)
|
| \(\nu^{3}\) | \(=\) |
\( 29\beta_{6} - 199\beta_{3} - 965\beta_{2} - 49634\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1575\beta_{7} + 675\beta_{5} - 58334\beta_{4} + 1457345234 \)
|
| \(\nu^{5}\) | \(=\) |
\( -1553086\beta_{6} + 13426466\beta_{3} + 55586710\beta_{2} + 2640105148\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -107638650\beta_{7} - 82580850\beta_{5} + 3153253948\beta_{4} - 77491029034252 \)
|
| \(\nu^{7}\) | \(=\) |
\( 80696413292\beta_{6} - 815129841652\beta_{3} - 2803560594620\beta_{2} - 141785962367960\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(239\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | − | 235.374i | 0 | 563.763i | 0 | 10271.0i | 0 | −35718.1 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | − | 227.701i | 0 | − | 1970.96i | 0 | − | 11819.1i | 0 | −32164.7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | − | 89.7032i | 0 | 2532.93i | 0 | − | 5673.64i | 0 | 11636.3 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | − | 51.5217i | 0 | − | 49.0991i | 0 | − | 2584.83i | 0 | 17028.5 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | 51.5217i | 0 | 49.0991i | 0 | 2584.83i | 0 | 17028.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | 89.7032i | 0 | − | 2532.93i | 0 | 5673.64i | 0 | 11636.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 227.701i | 0 | 1970.96i | 0 | 11819.1i | 0 | −32164.7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 235.374i | 0 | − | 563.763i | 0 | − | 10271.0i | 0 | −35718.1 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 272.10.b.b | 8 | |
| 4.b | odd | 2 | 1 | 34.10.b.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 306.10.b.b | 8 | ||
| 17.b | even | 2 | 1 | inner | 272.10.b.b | 8 | |
| 68.d | odd | 2 | 1 | 34.10.b.b | ✓ | 8 | |
| 204.h | even | 2 | 1 | 306.10.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.10.b.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 34.10.b.b | ✓ | 8 | 68.d | odd | 2 | 1 | |
| 272.10.b.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 272.10.b.b | 8 | 17.b | even | 2 | 1 | inner | |
| 306.10.b.b | 8 | 12.b | even | 2 | 1 | ||
| 306.10.b.b | 8 | 204.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 117950T_{3}^{6} + 4041468216T_{3}^{4} + 33029034367104T_{3}^{2} + 61354193073357312 \)
acting on \(S_{10}^{\mathrm{new}}(272, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 61\!\cdots\!12 \)
|
| $5$ |
\( T^{8} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 31\!\cdots\!00 \)
|
| $11$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( (T^{4} + \cdots + 71\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 19\!\cdots\!81 \)
|
| $19$ |
\( (T^{4} + \cdots - 34\!\cdots\!76)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 22\!\cdots\!92 \)
|
| $29$ |
\( T^{8} + \cdots + 55\!\cdots\!72 \)
|
| $31$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 64\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots + 13\!\cdots\!12 \)
|
| $43$ |
\( (T^{4} + \cdots + 22\!\cdots\!56)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 89\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots - 10\!\cdots\!64)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots - 12\!\cdots\!32)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 82\!\cdots\!88 \)
|
| $67$ |
\( (T^{4} + \cdots + 43\!\cdots\!40)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 44\!\cdots\!92 \)
|
| $73$ |
\( T^{8} + \cdots + 92\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 15\!\cdots\!28 \)
|
| $83$ |
\( (T^{4} + \cdots - 88\!\cdots\!52)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 12\!\cdots\!72)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
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