Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,30,Mod(1,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 30);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.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: | \(37.2946296681\) |
| 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: |
\( x^{8} - 3 x^{7} - 3348071721 x^{6} + 1059446710053 x^{5} + \cdots + 67\!\cdots\!22 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{12}\cdot 5\cdot 7^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{8} - 3 x^{7} - 3348071721 x^{6} + 1059446710053 x^{5} + \cdots + 67\!\cdots\!22 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 37\!\cdots\!99 \nu^{7} + \cdots + 21\!\cdots\!10 ) / 56\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 70\!\cdots\!81 \nu^{7} + \cdots - 51\!\cdots\!42 ) / 56\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 57\!\cdots\!97 \nu^{7} + \cdots + 18\!\cdots\!18 ) / 14\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 45\!\cdots\!23 \nu^{7} + \cdots + 18\!\cdots\!14 ) / 41\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 75\!\cdots\!41 \nu^{7} + \cdots + 20\!\cdots\!78 ) / 40\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31\!\cdots\!69 \nu^{7} + \cdots + 19\!\cdots\!26 ) / 28\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 19\beta_{2} - 479\beta _1 + 837018109 \)
|
| \(\nu^{3}\) | \(=\) |
\( 41 \beta_{7} + \beta_{6} - 25 \beta_{5} - 497 \beta_{4} - 3270 \beta_{3} + 609683 \beta_{2} + \cdots - 394025271258 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1244786 \beta_{7} - 249441 \beta_{6} - 26848 \beta_{5} + 5663316 \beta_{4} + 1606633555 \beta_{3} + \cdots + 11\!\cdots\!50 \)
|
| \(\nu^{5}\) | \(=\) |
\( 100972064592 \beta_{7} + 2323207500 \beta_{6} - 61220180088 \beta_{5} - 918019732296 \beta_{4} + \cdots - 86\!\cdots\!36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32\!\cdots\!06 \beta_{7} - 809493890677038 \beta_{6} - 518724977219466 \beta_{5} + \cdots + 16\!\cdots\!21 \)
|
| \(\nu^{7}\) | \(=\) |
\( 20\!\cdots\!33 \beta_{7} + \cdots + 48\!\cdots\!28 \)
|
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 |
|
−42967.5 | 1.54958e7 | 1.30934e9 | −5.88828e9 | −6.65817e11 | 6.78223e11 | −3.31910e13 | 1.71490e14 | 2.53005e14 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −30785.7 | −1.88236e6 | 4.10890e8 | −5.78649e9 | 5.79497e10 | 6.78223e11 | 3.87842e12 | −6.50871e13 | 1.78141e14 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −27331.8 | −1.58855e7 | 2.10156e8 | 1.83311e10 | 4.34181e11 | 6.78223e11 | 8.92970e12 | 1.83720e14 | −5.01021e14 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2948.36 | 1.06725e7 | −5.28178e8 | 1.95261e10 | −3.14663e10 | 6.78223e11 | 3.14015e12 | 4.52715e13 | −5.75700e13 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2718.39 | −4.63979e6 | −5.29481e8 | 9.12402e8 | −1.26127e10 | 6.78223e11 | −2.89876e12 | −4.71027e13 | 2.48026e12 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 18170.8 | 2.24754e6 | −2.06692e8 | −2.42208e10 | 4.08397e10 | 6.78223e11 | −1.35112e13 | −6.35789e13 | −4.40112e14 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 36988.5 | −6.25657e6 | 8.31275e8 | 1.74387e10 | −2.31421e11 | 6.78223e11 | 1.08896e13 | −2.94857e13 | 6.45029e14 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 38064.7 | 1.26067e7 | 9.12053e8 | 3.88769e9 | 4.79871e11 | 6.78223e11 | 1.42812e13 | 9.02990e13 | 1.47984e14 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
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 | 7.30.a.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.30.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 8091 T_{2}^{7} - 3319431102 T_{2}^{6} - 21310916973480 T_{2}^{5} + \cdots + 74\!\cdots\!40 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 74\!\cdots\!40 \)
|
| $3$ |
\( T^{8} + \cdots + 40\!\cdots\!84 \)
|
| $5$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
|
| $7$ |
\( (T - 678223072849)^{8} \)
|
| $11$ |
\( T^{8} + \cdots - 24\!\cdots\!04 \)
|
| $13$ |
\( T^{8} + \cdots + 33\!\cdots\!20 \)
|
| $17$ |
\( T^{8} + \cdots - 18\!\cdots\!96 \)
|
| $19$ |
\( T^{8} + \cdots + 47\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + \cdots - 86\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 12\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 45\!\cdots\!56 \)
|
| $41$ |
\( T^{8} + \cdots - 31\!\cdots\!16 \)
|
| $43$ |
\( T^{8} + \cdots + 43\!\cdots\!60 \)
|
| $47$ |
\( T^{8} + \cdots - 15\!\cdots\!68 \)
|
| $53$ |
\( T^{8} + \cdots - 21\!\cdots\!52 \)
|
| $59$ |
\( T^{8} + \cdots + 77\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 58\!\cdots\!84 \)
|
| $67$ |
\( T^{8} + \cdots - 14\!\cdots\!08 \)
|
| $71$ |
\( T^{8} + \cdots - 26\!\cdots\!40 \)
|
| $73$ |
\( T^{8} + \cdots + 86\!\cdots\!52 \)
|
| $79$ |
\( T^{8} + \cdots - 17\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 30\!\cdots\!92 \)
|
| $89$ |
\( T^{8} + \cdots + 37\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 30\!\cdots\!72 \)
|
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