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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2 x^{6} - 818624691 x^{5} - 543736213254 x^{4} + \cdots - 58\!\cdots\!46 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{7}\cdot 5\cdot 7^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2 x^{6} - 818624691 x^{5} - 543736213254 x^{4} + \cdots - 58\!\cdots\!46 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 185212798681 \nu^{6} + \cdots - 93\!\cdots\!66 ) / 14\!\cdots\!32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 185212798681 \nu^{6} + \cdots - 84\!\cdots\!10 ) / 80\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 20\!\cdots\!49 \nu^{6} + \cdots - 11\!\cdots\!26 ) / 72\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 252502215366499 \nu^{6} + \cdots - 15\!\cdots\!86 ) / 80\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 6782676204739 \nu^{6} + \cdots - 15\!\cdots\!66 ) / 99\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 18\beta_{2} + 1997\beta _1 + 935571928 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -50\beta_{6} - 189\beta_{5} + 216\beta_{4} - 260\beta_{3} + 268713\beta_{2} + 404819939\beta _1 + 467636539131 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5193980 \beta_{6} - 4271570 \beta_{5} - 3531220 \beta_{4} + 527856219 \beta_{3} - 19121355208 \beta_{2} + \cdots + 37\!\cdots\!68 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 32957133416 \beta_{6} - 121362711236 \beta_{5} + 166889512448 \beta_{4} - 321693323865 \beta_{3} + \cdots + 21\!\cdots\!45 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 44\!\cdots\!48 \beta_{6} + \cdots + 17\!\cdots\!16 ) / 4 \)
|
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 |
|
−45400.3 | −8.89933e6 | 1.52431e9 | −1.41045e10 | 4.04032e11 | −6.78223e11 | −4.48302e13 | 1.05677e13 | 6.40350e14 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −34162.6 | 2.89147e6 | 6.30213e8 | 1.97373e10 | −9.87801e10 | −6.78223e11 | −3.18881e12 | −6.02698e13 | −6.74276e14 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −13105.5 | −1.02032e7 | −3.65117e8 | −1.60363e10 | 1.33718e11 | −6.78223e11 | 1.18210e13 | 3.54748e13 | 2.10164e14 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1638.46 | −6.25098e6 | −5.34186e8 | 1.56937e10 | −1.02420e10 | −6.78223e11 | −1.75489e12 | −2.95557e13 | 2.57135e13 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 16588.8 | 1.16803e7 | −2.61682e8 | −5.22505e9 | 1.93763e11 | −6.78223e11 | −1.32471e13 | 6.77993e13 | −8.66775e13 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 28543.3 | 1.74260e6 | 2.77849e8 | 8.17459e9 | 4.97395e10 | −6.78223e11 | −7.39334e12 | −6.55937e13 | 2.33330e14 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 45347.8 | −2.94982e6 | 1.51955e9 | −2.18051e10 | −1.33768e11 | −6.78223e11 | 4.45624e13 | −5.99289e13 | −9.88815e14 | ||||||||||||||||||||||||||||||||||||||||
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.a | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.30.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 550 T_{2}^{6} - 3274369128 T_{2}^{5} + 3072852162432 T_{2}^{4} + \cdots + 71\!\cdots\!00 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 71\!\cdots\!00 \)
|
| $3$ |
\( T^{7} + \cdots - 98\!\cdots\!64 \)
|
| $5$ |
\( T^{7} + \cdots - 65\!\cdots\!00 \)
|
| $7$ |
\( (T + 678223072849)^{7} \)
|
| $11$ |
\( T^{7} + \cdots - 41\!\cdots\!52 \)
|
| $13$ |
\( T^{7} + \cdots - 17\!\cdots\!00 \)
|
| $17$ |
\( T^{7} + \cdots + 19\!\cdots\!04 \)
|
| $19$ |
\( T^{7} + \cdots + 13\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 59\!\cdots\!76 \)
|
| $29$ |
\( T^{7} + \cdots - 28\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 53\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 23\!\cdots\!32 \)
|
| $41$ |
\( T^{7} + \cdots - 20\!\cdots\!68 \)
|
| $43$ |
\( T^{7} + \cdots + 12\!\cdots\!00 \)
|
| $47$ |
\( T^{7} + \cdots - 35\!\cdots\!96 \)
|
| $53$ |
\( T^{7} + \cdots + 91\!\cdots\!16 \)
|
| $59$ |
\( T^{7} + \cdots + 16\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots - 73\!\cdots\!12 \)
|
| $67$ |
\( T^{7} + \cdots - 12\!\cdots\!32 \)
|
| $71$ |
\( T^{7} + \cdots + 43\!\cdots\!00 \)
|
| $73$ |
\( T^{7} + \cdots + 19\!\cdots\!72 \)
|
| $79$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 92\!\cdots\!96 \)
|
| $89$ |
\( T^{7} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 10\!\cdots\!08 \)
|
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