Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,7,Mod(21,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.21");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.b (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: | \(5.06118983964\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2707\nu^{5} - 22767\nu^{4} + 4050251\nu^{3} + 3625173\nu^{2} - 657163750\nu + 10557684654 ) / 963515190 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5414\nu^{5} + 45534\nu^{4} - 8100502\nu^{3} - 7250346\nu^{2} + 3241357880\nu - 21115369308 ) / 481757595 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 44423 \nu^{5} - 907518 \nu^{4} + 67178179 \nu^{3} + 1011971817 \nu^{2} - 24365375270 \nu - 111796059144 ) / 3372303165 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 95153 \nu^{5} - 18774978 \nu^{4} - 154836221 \nu^{3} + 16296294372 \nu^{2} + \cdots - 1601025981984 ) / 6744606330 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 46261 \nu^{5} - 187378 \nu^{4} + 90358871 \nu^{3} - 137288348 \nu^{2} - 45368513086 \nu + 315774403296 ) / 449640422 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + 14\beta_{3} - 24\beta _1 + 1192 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 99\beta_{5} + 15\beta_{4} + 28\beta_{3} + 1834\beta_{2} + 3504\beta _1 - 23048 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -859\beta_{5} - 887\beta_{4} + 6181\beta_{3} - 5748\beta_{2} - 16062\beta _1 + 521678 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 174345\beta_{5} + 49605\beta_{4} - 128548\beta_{3} + 2694664\beta_{2} + 3324008\beta _1 - 33241672 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.1 |
|
− | 5.65685i | −41.5131 | −32.0000 | 155.573 | 234.833i | 562.567i | 181.019i | 994.336 | − | 880.056i | ||||||||||||||||||||||||||||||||||
| 21.2 | − | 5.65685i | −1.47212 | −32.0000 | −147.340 | 8.32754i | 44.7193i | 181.019i | −726.833 | 833.480i | ||||||||||||||||||||||||||||||||||||
| 21.3 | − | 5.65685i | 16.9852 | −32.0000 | 175.767 | − | 96.0828i | − | 412.125i | 181.019i | −440.503 | − | 994.286i | |||||||||||||||||||||||||||||||||
| 21.4 | 5.65685i | −41.5131 | −32.0000 | 155.573 | − | 234.833i | − | 562.567i | − | 181.019i | 994.336 | 880.056i | ||||||||||||||||||||||||||||||||||
| 21.5 | 5.65685i | −1.47212 | −32.0000 | −147.340 | − | 8.32754i | − | 44.7193i | − | 181.019i | −726.833 | − | 833.480i | |||||||||||||||||||||||||||||||||
| 21.6 | 5.65685i | 16.9852 | −32.0000 | 175.767 | 96.0828i | 412.125i | − | 181.019i | −440.503 | 994.286i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 22.7.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 198.7.d.a | 6 | ||
| 4.b | odd | 2 | 1 | 176.7.h.e | 6 | ||
| 11.b | odd | 2 | 1 | inner | 22.7.b.a | ✓ | 6 |
| 33.d | even | 2 | 1 | 198.7.d.a | 6 | ||
| 44.c | even | 2 | 1 | 176.7.h.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.7.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 22.7.b.a | ✓ | 6 | 11.b | odd | 2 | 1 | inner |
| 176.7.h.e | 6 | 4.b | odd | 2 | 1 | ||
| 176.7.h.e | 6 | 44.c | even | 2 | 1 | ||
| 198.7.d.a | 6 | 3.b | odd | 2 | 1 | ||
| 198.7.d.a | 6 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(22, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32)^{3} \)
|
| $3$ |
\( (T^{3} + 26 T^{2} + \cdots - 1038)^{2} \)
|
| $5$ |
\( (T^{3} - 184 T^{2} + \cdots + 4028950)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 107496740487168 \)
|
| $11$ |
\( T^{6} + \cdots + 55\!\cdots\!81 \)
|
| $13$ |
\( T^{6} + \cdots + 91\!\cdots\!32 \)
|
| $17$ |
\( T^{6} + \cdots + 29\!\cdots\!72 \)
|
| $19$ |
\( T^{6} + \cdots + 61\!\cdots\!88 \)
|
| $23$ |
\( (T^{3} - 1078 T^{2} + \cdots + 6239437498)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 56\!\cdots\!92 \)
|
| $31$ |
\( (T^{3} + \cdots - 1132011557222)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 26364256495174)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 22\!\cdots\!08 \)
|
| $43$ |
\( T^{6} + \cdots + 19\!\cdots\!08 \)
|
| $47$ |
\( (T^{3} + \cdots - 432230445439256)^{2} \)
|
| $53$ |
\( (T^{3} + \cdots - 31\!\cdots\!92)^{2} \)
|
| $59$ |
\( (T^{3} + \cdots - 90\!\cdots\!90)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots + 18\!\cdots\!74)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots + 97\!\cdots\!82)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 15\!\cdots\!68 \)
|
| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!72 \)
|
| $89$ |
\( (T^{3} + \cdots - 22\!\cdots\!14)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 17\!\cdots\!14)^{2} \)
|
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