Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,10,Mod(1,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, 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("342.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.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: | \(176.142255968\) |
| 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: |
\( x^{6} - 2x^{5} - 625764x^{4} + 19915656x^{3} + 50419290137x^{2} - 293829896134x - 170483716854 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 625764x^{4} + 19915656x^{3} + 50419290137x^{2} - 293829896134x - 170483716854 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 30446374007 \nu^{5} - 440817033759 \nu^{4} + \cdots + 14\!\cdots\!78 ) / 30\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1057965228989 \nu^{5} + 7295939607357 \nu^{4} + \cdots + 28\!\cdots\!46 ) / 82\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 969408695993 \nu^{5} - 1636190558739 \nu^{4} + \cdots - 32\!\cdots\!02 ) / 55\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1043143392601 \nu^{5} - 2973783866661 \nu^{4} + \cdots + 10\!\cdots\!78 ) / 33\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 221\beta_{5} + 287\beta_{4} + 2\beta_{3} - 196\beta_{2} + 112\beta _1 + 1877371 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12973\beta_{5} + 84719\beta_{4} + 105464\beta_{3} + 55283\beta_{2} + 1380487\beta _1 - 84028043 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 91428155 \beta_{5} + 136454645 \beta_{4} + 52211120 \beta_{3} - 115867060 \beta_{2} + \cdots + 872015113099 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 4139461315 \beta_{5} + 15915259805 \beta_{4} + 21936076610 \beta_{3} + 12750595505 \beta_{2} + \cdots - 28724748682069 ) / 3 \)
|
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 | 0 | 256.000 | −2182.33 | 0 | −3091.61 | −4096.00 | 0 | 34917.3 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −16.0000 | 0 | 256.000 | −808.558 | 0 | −4284.75 | −4096.00 | 0 | 12936.9 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 0 | 256.000 | 61.4049 | 0 | 8741.02 | −4096.00 | 0 | −982.478 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −16.0000 | 0 | 256.000 | 82.0430 | 0 | −8193.86 | −4096.00 | 0 | −1312.69 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −16.0000 | 0 | 256.000 | 1049.86 | 0 | 3988.64 | −4096.00 | 0 | −16797.8 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −16.0000 | 0 | 256.000 | 2181.58 | 0 | −7109.44 | −4096.00 | 0 | −34905.3 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 342.10.a.n | ✓ | 6 |
| 3.b | odd | 2 | 1 | 342.10.a.o | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 342.10.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 342.10.a.o | yes | 6 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 384 T_{5}^{5} - 5570451 T_{5}^{4} + 1951716384 T_{5}^{3} + 3848466710700 T_{5}^{2} + \cdots + 20\!\cdots\!00 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 26\!\cdots\!04 \)
|
| $11$ |
\( T^{6} + \cdots + 61\!\cdots\!68 \)
|
| $13$ |
\( T^{6} + \cdots + 24\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots - 15\!\cdots\!72 \)
|
| $19$ |
\( (T - 130321)^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 67\!\cdots\!72 \)
|
| $29$ |
\( T^{6} + \cdots + 27\!\cdots\!08 \)
|
| $31$ |
\( T^{6} + \cdots - 76\!\cdots\!84 \)
|
| $37$ |
\( T^{6} + \cdots - 43\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 54\!\cdots\!60 \)
|
| $47$ |
\( T^{6} + \cdots - 23\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots - 17\!\cdots\!40 \)
|
| $59$ |
\( T^{6} + \cdots + 42\!\cdots\!08 \)
|
| $61$ |
\( T^{6} + \cdots - 21\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots - 36\!\cdots\!68 \)
|
| $71$ |
\( T^{6} + \cdots - 70\!\cdots\!52 \)
|
| $73$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 20\!\cdots\!24 \)
|
| $83$ |
\( T^{6} + \cdots + 66\!\cdots\!92 \)
|
| $89$ |
\( T^{6} + \cdots + 57\!\cdots\!92 \)
|
| $97$ |
\( T^{6} + \cdots + 38\!\cdots\!44 \)
|
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