Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,12,Mod(1,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.a (trivial) |
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: | yes |
| Analytic conductor: | \(159.047038376\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 23) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{8} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 60015645 \nu^{7} - 645905638 \nu^{6} - 140469464484 \nu^{5} + 1167187182454 \nu^{4} + \cdots + 15\!\cdots\!32 ) / 39027522759904 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 45014461 \nu^{7} + 1873557490 \nu^{6} + 83490232948 \nu^{5} - 2952828856806 \nu^{4} + \cdots - 10\!\cdots\!48 ) / 29270642069928 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 45014461 \nu^{7} - 1873557490 \nu^{6} - 83490232948 \nu^{5} + 2952828856806 \nu^{4} + \cdots + 30\!\cdots\!32 ) / 29270642069928 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 51174359 \nu^{7} - 314582090 \nu^{6} - 117621421388 \nu^{5} + 1155900187554 \nu^{4} + \cdots + 11\!\cdots\!92 ) / 16726081182816 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 229993795 \nu^{7} - 673359386 \nu^{6} + 449139676420 \nu^{5} + 2731402695342 \nu^{4} + \cdots + 24\!\cdots\!52 ) / 58541284139856 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 405250443 \nu^{7} + 10607611714 \nu^{6} + 633216494716 \nu^{5} - 16692616293018 \nu^{4} + \cdots - 36\!\cdots\!32 ) / 39027522759904 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 2\beta _1 + 2672 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{7} + 6\beta_{6} - 7\beta_{5} - 22\beta_{4} - 3\beta_{3} + 28\beta_{2} + 4077\beta _1 + 3704 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 51 \beta_{7} + 134 \beta_{6} + 97 \beta_{5} + 2272 \beta_{4} + 2567 \beta_{3} + 100 \beta_{2} + \cdots + 5467404 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2993 \beta_{7} + 4586 \beta_{6} - 5041 \beta_{5} - 16976 \beta_{4} - 185 \beta_{3} + 18328 \beta_{2} + \cdots + 2278446 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 50961 \beta_{7} + 103810 \beta_{6} + 73187 \beta_{5} + 1181571 \beta_{4} + 1533596 \beta_{3} + \cdots + 3051054240 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 9150313 \beta_{7} + 11165394 \beta_{6} - 12322917 \beta_{5} - 45803970 \beta_{4} + 3189615 \beta_{3} + \cdots + 6436740960 ) / 8 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−63.8784 | 0 | 2032.45 | 10644.5 | 0 | −35785.2 | 993.065 | 0 | −679953. | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −63.0482 | 0 | 1927.07 | −5906.93 | 0 | −36688.5 | 7624.20 | 0 | 372421. | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −28.6126 | 0 | −1229.32 | 2645.71 | 0 | 76456.7 | 93772.7 | 0 | −75700.7 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −14.8702 | 0 | −1826.88 | −2118.93 | 0 | −68580.9 | 57620.3 | 0 | 31509.0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 15.9125 | 0 | −1794.79 | −8501.73 | 0 | 38766.9 | −61148.6 | 0 | −135284. | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 43.0298 | 0 | −196.437 | 1082.06 | 0 | 38616.0 | −96577.7 | 0 | 46560.9 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 69.2357 | 0 | 2745.59 | 10567.4 | 0 | −39263.4 | 48297.9 | 0 | 731642. | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 74.2314 | 0 | 3462.31 | 3053.92 | 0 | −27639.7 | 104986. | 0 | 226697. | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 207.12.a.a | 8 | |
| 3.b | odd | 2 | 1 | 23.12.a.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.12.a.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 207.12.a.a | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 32 T_{2}^{7} - 10240 T_{2}^{6} + 243056 T_{2}^{5} + 32897712 T_{2}^{4} + \cdots + 6030172585984 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 6030172585984 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots - 11\!\cdots\!36 \)
|
| $11$ |
\( T^{8} + \cdots - 66\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 24\!\cdots\!64 \)
|
| $17$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots - 45\!\cdots\!20 \)
|
| $23$ |
\( (T + 6436343)^{8} \)
|
| $29$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 20\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 26\!\cdots\!32 \)
|
| $41$ |
\( T^{8} + \cdots + 99\!\cdots\!80 \)
|
| $43$ |
\( T^{8} + \cdots - 46\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots - 10\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots + 17\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots - 74\!\cdots\!76 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!08 \)
|
| $67$ |
\( T^{8} + \cdots + 12\!\cdots\!40 \)
|
| $71$ |
\( T^{8} + \cdots + 31\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots - 39\!\cdots\!08 \)
|
| $79$ |
\( T^{8} + \cdots + 17\!\cdots\!76 \)
|
| $83$ |
\( T^{8} + \cdots + 89\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 22\!\cdots\!20 \)
|
| $97$ |
\( T^{8} + \cdots + 33\!\cdots\!04 \)
|
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