Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,4,Mod(1,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("95.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 95.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: | \(5.60518145055\) |
| 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: |
\( x^{6} - x^{5} - 36x^{4} + 30x^{3} + 241x^{2} - 347x + 76 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - x^{5} - 36x^{4} + 30x^{3} + 241x^{2} - 347x + 76 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} - 28\nu^{4} - 268\nu^{3} + 1166\nu^{2} + 3321\nu - 7364 ) / 542 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\nu^{5} + 2\nu^{4} - 639\nu^{3} - 122\nu^{2} + 3460\nu - 2726 ) / 271 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -43\nu^{5} + 24\nu^{4} + 1546\nu^{3} - 380\nu^{2} - 10241\nu + 6312 ) / 542 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47\nu^{5} + 62\nu^{4} - 1652\nu^{3} - 2156\nu^{2} + 8887\nu + 46 ) / 542 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + 5\beta_{3} - 2\beta_{2} + 19\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 35\beta_{4} + 27\beta_{3} + 30\beta_{2} + 18\beta _1 + 271 \)
|
| \(\nu^{5}\) | \(=\) |
\( -68\beta_{5} + 70\beta_{4} + 186\beta_{3} - 64\beta_{2} + 455\beta _1 + 192 \)
|
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 |
|
−5.47638 | −4.48105 | 21.9908 | −5.00000 | 24.5399 | −26.3667 | −76.6187 | −6.92019 | 27.3819 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.27701 | 9.52774 | −2.81523 | −5.00000 | −21.6948 | 21.5257 | 24.6264 | 63.7779 | 11.3850 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.32917 | −8.73981 | −6.23331 | −5.00000 | 11.6167 | −25.8183 | 18.9185 | 49.3843 | 6.64584 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.271302 | −3.45171 | −7.92640 | −5.00000 | 0.936455 | 31.6242 | 4.32086 | −15.0857 | 1.35651 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.43846 | 9.54338 | 3.82299 | −5.00000 | 32.8145 | −6.80350 | −14.3625 | 64.0762 | −17.1923 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.91540 | 2.60145 | 16.1612 | −5.00000 | 12.7872 | 10.8387 | 40.1155 | −20.2325 | −24.5770 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(1\) |
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 | 95.4.a.g | ✓ | 6 |
| 3.b | odd | 2 | 1 | 855.4.a.r | 6 | ||
| 4.b | odd | 2 | 1 | 1520.4.a.y | 6 | ||
| 5.b | even | 2 | 1 | 475.4.a.i | 6 | ||
| 5.c | odd | 4 | 2 | 475.4.b.i | 12 | ||
| 19.b | odd | 2 | 1 | 1805.4.a.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.4.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 475.4.a.i | 6 | 5.b | even | 2 | 1 | ||
| 475.4.b.i | 12 | 5.c | odd | 4 | 2 | ||
| 855.4.a.r | 6 | 3.b | odd | 2 | 1 | ||
| 1520.4.a.y | 6 | 4.b | odd | 2 | 1 | ||
| 1805.4.a.m | 6 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(95))\):
|
\( T_{2}^{6} + T_{2}^{5} - 36T_{2}^{4} - 30T_{2}^{3} + 241T_{2}^{2} + 347T_{2} + 76 \)
|
|
\( T_{3}^{6} - 5T_{3}^{5} - 136T_{3}^{4} + 404T_{3}^{3} + 5044T_{3}^{2} - 1060T_{3} - 31976 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 76 \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots - 31976 \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots - 34171968 \)
|
| $11$ |
\( T^{6} + \cdots - 1164746752 \)
|
| $13$ |
\( T^{6} + \cdots + 32296714744 \)
|
| $17$ |
\( T^{6} + \cdots - 22420426912 \)
|
| $19$ |
\( (T + 19)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 19519388352 \)
|
| $29$ |
\( T^{6} + \cdots + 5164992142240 \)
|
| $31$ |
\( T^{6} + \cdots + 3563293473792 \)
|
| $37$ |
\( T^{6} + \cdots + 81154435293744 \)
|
| $41$ |
\( T^{6} + \cdots + 151734187110336 \)
|
| $43$ |
\( T^{6} + \cdots + 14416604266624 \)
|
| $47$ |
\( T^{6} + \cdots + 544317707716608 \)
|
| $53$ |
\( T^{6} + \cdots - 289016191472744 \)
|
| $59$ |
\( T^{6} + \cdots + 49133724670720 \)
|
| $61$ |
\( T^{6} + \cdots + 56277540209408 \)
|
| $67$ |
\( T^{6} + \cdots - 80297782152504 \)
|
| $71$ |
\( T^{6} + \cdots + 15461733531648 \)
|
| $73$ |
\( T^{6} + \cdots + 92\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots - 16\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots - 81\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 83\!\cdots\!08 \)
|
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