Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,4,Mod(1,361)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.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: | \(21.2996895121\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3144.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 16x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 19) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 16x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
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 |
|
−5.07177 | 8.66998 | 17.7229 | −2.61710 | −43.9722 | −15.1058 | −49.3121 | 48.1686 | 13.2733 | |||||||||||||||||||||||||||
| 1.2 | −1.89080 | −2.95388 | −4.42486 | −1.51710 | 5.58521 | 5.94196 | 23.4930 | −18.2746 | 2.86853 | ||||||||||||||||||||||||||||
| 1.3 | 3.96257 | −6.71610 | 7.70200 | 18.1342 | −26.6130 | −25.8362 | −1.18085 | 18.1060 | 71.8581 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(-1\) |
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 | 361.4.a.i | 3 | |
| 19.b | odd | 2 | 1 | 19.4.a.b | ✓ | 3 | |
| 57.d | even | 2 | 1 | 171.4.a.f | 3 | ||
| 76.d | even | 2 | 1 | 304.4.a.i | 3 | ||
| 95.d | odd | 2 | 1 | 475.4.a.f | 3 | ||
| 95.g | even | 4 | 2 | 475.4.b.f | 6 | ||
| 133.c | even | 2 | 1 | 931.4.a.c | 3 | ||
| 152.b | even | 2 | 1 | 1216.4.a.u | 3 | ||
| 152.g | odd | 2 | 1 | 1216.4.a.s | 3 | ||
| 209.d | even | 2 | 1 | 2299.4.a.h | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.4.a.b | ✓ | 3 | 19.b | odd | 2 | 1 | |
| 171.4.a.f | 3 | 57.d | even | 2 | 1 | ||
| 304.4.a.i | 3 | 76.d | even | 2 | 1 | ||
| 361.4.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 475.4.a.f | 3 | 95.d | odd | 2 | 1 | ||
| 475.4.b.f | 6 | 95.g | even | 4 | 2 | ||
| 931.4.a.c | 3 | 133.c | even | 2 | 1 | ||
| 1216.4.a.s | 3 | 152.g | odd | 2 | 1 | ||
| 1216.4.a.u | 3 | 152.b | even | 2 | 1 | ||
| 2299.4.a.h | 3 | 209.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 3T_{2}^{2} - 18T_{2} - 38 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(361))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3 T^{2} + \cdots - 38 \)
|
| $3$ |
\( T^{3} + T^{2} + \cdots - 172 \)
|
| $5$ |
\( T^{3} - 14 T^{2} + \cdots - 72 \)
|
| $7$ |
\( T^{3} + 35 T^{2} + \cdots - 2319 \)
|
| $11$ |
\( T^{3} - 16 T^{2} + \cdots + 1182 \)
|
| $13$ |
\( T^{3} + 65 T^{2} + \cdots - 4848 \)
|
| $17$ |
\( T^{3} - 29 T^{2} + \cdots - 218619 \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( T^{3} + 101 T^{2} + \cdots - 378176 \)
|
| $29$ |
\( T^{3} + 377 T^{2} + \cdots - 4544396 \)
|
| $31$ |
\( T^{3} - 140 T^{2} + \cdots + 2444352 \)
|
| $37$ |
\( T^{3} - 290 T^{2} + \cdots + 10001448 \)
|
| $41$ |
\( T^{3} + 956 T^{2} + \cdots + 31578144 \)
|
| $43$ |
\( T^{3} + 570 T^{2} + \cdots - 65963504 \)
|
| $47$ |
\( T^{3} - 66 T^{2} + \cdots + 2940624 \)
|
| $53$ |
\( T^{3} + 817 T^{2} + \cdots + 16824816 \)
|
| $59$ |
\( T^{3} + 265 T^{2} + \cdots - 31557612 \)
|
| $61$ |
\( T^{3} - 988 T^{2} + \cdots + 76875874 \)
|
| $67$ |
\( T^{3} - 207 T^{2} + \cdots + 7515248 \)
|
| $71$ |
\( T^{3} + 846 T^{2} + \cdots - 1727928 \)
|
| $73$ |
\( T^{3} - 627 T^{2} + \cdots + 145581839 \)
|
| $79$ |
\( T^{3} + 382 T^{2} + \cdots + 56023488 \)
|
| $83$ |
\( T^{3} + 766 T^{2} + \cdots - 78728352 \)
|
| $89$ |
\( T^{3} - 172 T^{2} + \cdots + 76923456 \)
|
| $97$ |
\( T^{3} - 2450 T^{2} + \cdots - 196438912 \)
|
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