Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,Mod(1,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(180.262542657\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 19610x^{4} + 704364x^{3} + 22509888x^{2} + 142780860x + 152654544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2}\cdot 5^{5} \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 19610x^{4} + 704364x^{3} + 22509888x^{2} + 142780860x + 152654544 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 30253357 \nu^{5} + 118583794 \nu^{4} + 595084857083 \nu^{3} - 22908245479098 \nu^{2} + \cdots - 24\!\cdots\!60 ) / 7048620009396 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 424746695 \nu^{5} + 9861530420 \nu^{4} + 8295933771943 \nu^{3} - 476612542972146 \nu^{2} + \cdots + 32\!\cdots\!40 ) / 7048620009396 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 113590357 \nu^{5} + 396214819 \nu^{4} + 2235931504958 \nu^{3} - 85321813931718 \nu^{2} + \cdots - 77\!\cdots\!84 ) / 1762155002349 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 162608450 \nu^{5} - 1431567095 \nu^{4} - 3172527158575 \nu^{3} + 139011545028720 \nu^{2} + \cdots + 81\!\cdots\!24 ) / 2349540003132 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 251124229 \nu^{5} - 2562909013 \nu^{4} - 4887969075476 \nu^{3} + 222955837312236 \nu^{2} + \cdots + 21\!\cdots\!36 ) / 1762155002349 \)
|
| \(\nu\) | \(=\) |
\( ( -4\beta_{5} + 12\beta_{4} - 17\beta_{3} + 316\beta _1 + 260 ) / 1200 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 824\beta_{5} - 2172\beta_{4} - 143\beta_{3} - 5516\beta _1 + 3919820 ) / 600 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11864\beta_{5} + 46932\beta_{4} - 19335\beta_{3} + 4040\beta_{2} + 596508\beta _1 - 40957708 ) / 120 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4224201\beta_{5} - 11152668\beta_{4} + 1001243\beta_{3} + 165250\beta_{2} - 56934934\beta _1 + 16891694810 ) / 150 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 841194334 \beta_{5} + 2979562092 \beta_{4} - 799238117 \beta_{3} + 199962900 \beta_{2} + \cdots - 3391823435680 ) / 300 \)
|
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 |
|
16.0000 | −201.043 | 256.000 | 0 | −3216.69 | −2401.00 | 4096.00 | 20735.3 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −126.454 | 256.000 | 0 | −2023.27 | −2401.00 | 4096.00 | −3692.35 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | −103.598 | 256.000 | 0 | −1657.57 | −2401.00 | 4096.00 | −8950.46 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 18.7845 | 256.000 | 0 | 300.552 | −2401.00 | 4096.00 | −19330.1 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 16.0000 | 110.036 | 256.000 | 0 | 1760.57 | −2401.00 | 4096.00 | −7575.15 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 16.0000 | 225.275 | 256.000 | 0 | 3604.40 | −2401.00 | 4096.00 | 31065.8 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 350.10.a.v | 6 | |
| 5.b | even | 2 | 1 | 350.10.a.u | 6 | ||
| 5.c | odd | 4 | 2 | 70.10.c.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.10.c.a | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 350.10.a.u | 6 | 5.b | even | 2 | 1 | ||
| 350.10.a.v | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 77T_{3}^{5} - 62211T_{3}^{4} - 5446287T_{3}^{3} + 711704286T_{3}^{2} + 54239134950T_{3} - 1226363846400 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 1226363846400 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T + 2401)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots - 22\!\cdots\!24 \)
|
| $17$ |
\( T^{6} + \cdots + 70\!\cdots\!64 \)
|
| $19$ |
\( T^{6} + \cdots - 39\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 59\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 23\!\cdots\!64 \)
|
| $31$ |
\( T^{6} + \cdots - 77\!\cdots\!36 \)
|
| $37$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 70\!\cdots\!24 \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!36 \)
|
| $47$ |
\( T^{6} + \cdots - 38\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots - 62\!\cdots\!96 \)
|
| $59$ |
\( T^{6} + \cdots - 38\!\cdots\!16 \)
|
| $61$ |
\( T^{6} + \cdots - 84\!\cdots\!44 \)
|
| $67$ |
\( T^{6} + \cdots - 10\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 24\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots - 88\!\cdots\!36 \)
|
| $83$ |
\( T^{6} + \cdots + 61\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
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