Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,6,Mod(1,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 195.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: | \(31.2748448635\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - 2x^{6} - 201x^{5} + 480x^{4} + 11274x^{3} - 29276x^{2} - 162072x + 455280 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{7} - 2x^{6} - 201x^{5} + 480x^{4} + 11274x^{3} - 29276x^{2} - 162072x + 455280 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 58 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -33\nu^{6} - 96\nu^{5} + 4465\nu^{4} + 12866\nu^{3} - 113758\nu^{2} - 440704\nu + 721272 ) / 37328 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{6} + 32\nu^{5} - 2266\nu^{4} - 5844\nu^{3} + 132017\nu^{2} + 257330\nu - 1882856 ) / 4666 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 121\nu^{6} + 352\nu^{5} - 22593\nu^{4} - 40954\nu^{3} + 1151230\nu^{2} + 856912\nu - 13656424 ) / 18664 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -443\nu^{6} + 408\nu^{5} + 84259\nu^{4} - 72178\nu^{3} - 4181362\nu^{2} + 1537040\nu + 43030008 ) / 37328 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 58 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} + 87\beta _1 - 56 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} - 2\beta_{4} - 20\beta_{3} + 119\beta_{2} - 153\beta _1 + 5018 \)
|
| \(\nu^{5}\) | \(=\) |
\( 22\beta_{6} + 173\beta_{5} - 260\beta_{4} + 280\beta_{3} + 3\beta_{2} + 8793\beta _1 - 8930 \)
|
| \(\nu^{6}\) | \(=\) |
\( -64\beta_{6} - 384\beta_{5} - 294\beta_{4} - 3872\beta_{3} + 13035\beta_{2} - 22269\beta _1 + 505014 \)
|
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 |
|
−11.0349 | −9.00000 | 89.7686 | −25.0000 | 99.3139 | −171.189 | −637.470 | 81.0000 | 275.872 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.90544 | −9.00000 | 30.4961 | −25.0000 | 71.1490 | 176.088 | 11.8893 | 81.0000 | 197.636 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.78396 | −9.00000 | −9.11374 | −25.0000 | 43.0556 | 73.5517 | 196.686 | 81.0000 | 119.599 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 3.01237 | −9.00000 | −22.9256 | −25.0000 | −27.1114 | −242.606 | −165.456 | 81.0000 | −75.3093 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.36956 | −9.00000 | −12.9069 | −25.0000 | −39.3261 | 51.6597 | −196.224 | 81.0000 | −109.239 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 8.06197 | −9.00000 | 32.9954 | −25.0000 | −72.5577 | −68.7454 | 8.02471 | 81.0000 | −201.549 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.2804 | −9.00000 | 73.6862 | −25.0000 | −92.5234 | 213.240 | 428.550 | 81.0000 | −257.009 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 195.6.a.i | ✓ | 7 |
| 3.b | odd | 2 | 1 | 585.6.a.n | 7 | ||
| 5.b | even | 2 | 1 | 975.6.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.6.a.i | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 585.6.a.n | 7 | 3.b | odd | 2 | 1 | ||
| 975.6.a.m | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2T_{2}^{6} - 201T_{2}^{5} + 480T_{2}^{4} + 11274T_{2}^{3} - 29276T_{2}^{2} - 162072T_{2} + 455280 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(195))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 455280 \)
|
| $3$ |
\( (T + 9)^{7} \)
|
| $5$ |
\( (T + 25)^{7} \)
|
| $7$ |
\( T^{7} + \cdots + 407345549010432 \)
|
| $11$ |
\( T^{7} + \cdots + 32\!\cdots\!56 \)
|
| $13$ |
\( (T - 169)^{7} \)
|
| $17$ |
\( T^{7} + \cdots + 62\!\cdots\!44 \)
|
| $19$ |
\( T^{7} + \cdots - 28\!\cdots\!40 \)
|
| $23$ |
\( T^{7} + \cdots + 29\!\cdots\!20 \)
|
| $29$ |
\( T^{7} + \cdots + 35\!\cdots\!20 \)
|
| $31$ |
\( T^{7} + \cdots - 17\!\cdots\!16 \)
|
| $37$ |
\( T^{7} + \cdots - 48\!\cdots\!32 \)
|
| $41$ |
\( T^{7} + \cdots - 52\!\cdots\!04 \)
|
| $43$ |
\( T^{7} + \cdots + 84\!\cdots\!08 \)
|
| $47$ |
\( T^{7} + \cdots + 24\!\cdots\!00 \)
|
| $53$ |
\( T^{7} + \cdots + 72\!\cdots\!64 \)
|
| $59$ |
\( T^{7} + \cdots - 11\!\cdots\!28 \)
|
| $61$ |
\( T^{7} + \cdots + 44\!\cdots\!08 \)
|
| $67$ |
\( T^{7} + \cdots - 81\!\cdots\!12 \)
|
| $71$ |
\( T^{7} + \cdots + 57\!\cdots\!20 \)
|
| $73$ |
\( T^{7} + \cdots + 34\!\cdots\!52 \)
|
| $79$ |
\( T^{7} + \cdots + 26\!\cdots\!96 \)
|
| $83$ |
\( T^{7} + \cdots - 33\!\cdots\!36 \)
|
| $89$ |
\( T^{7} + \cdots + 20\!\cdots\!40 \)
|
| $97$ |
\( T^{7} + \cdots - 28\!\cdots\!00 \)
|
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