Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(57.8985589525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 84x^{2} - 154x + 484 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 84x^{2} - 154x + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 62\nu - 132 ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 13\nu^{2} + 102\nu - 220 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 2\beta _1 + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 26\beta_{2} + 64\beta _1 + 176 \)
|
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 |
|
−9.02832 | −7.95930 | 49.5106 | −73.8370 | 71.8591 | 41.7769 | −158.091 | −179.649 | 666.624 | ||||||||||||||||||||||||||||||
| 1.2 | −7.37689 | 3.41393 | 22.4185 | 108.514 | −25.1842 | −80.6465 | 70.6815 | −231.345 | −800.499 | |||||||||||||||||||||||||||||||
| 1.3 | −1.84953 | −30.2395 | −28.5793 | 10.2772 | 55.9287 | 19.2784 | 112.043 | 671.426 | −19.0079 | |||||||||||||||||||||||||||||||
| 1.4 | 9.25474 | 28.7849 | 53.6502 | 45.0454 | 266.396 | −170.409 | 200.367 | 585.568 | 416.883 | |||||||||||||||||||||||||||||||
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 | 361.6.a.e | 4 | |
| 19.b | odd | 2 | 1 | 19.6.a.d | ✓ | 4 | |
| 57.d | even | 2 | 1 | 171.6.a.i | 4 | ||
| 76.d | even | 2 | 1 | 304.6.a.l | 4 | ||
| 95.d | odd | 2 | 1 | 475.6.a.e | 4 | ||
| 133.c | even | 2 | 1 | 931.6.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.6.a.d | ✓ | 4 | 19.b | odd | 2 | 1 | |
| 171.6.a.i | 4 | 57.d | even | 2 | 1 | ||
| 304.6.a.l | 4 | 76.d | even | 2 | 1 | ||
| 361.6.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 475.6.a.e | 4 | 95.d | odd | 2 | 1 | ||
| 931.6.a.d | 4 | 133.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 9T_{2}^{3} - 72T_{2}^{2} - 774T_{2} - 1140 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 9 T^{3} + \cdots - 1140 \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 23652 \)
|
| $5$ |
\( T^{4} - 90 T^{3} + \cdots - 3709248 \)
|
| $7$ |
\( T^{4} + 190 T^{3} + \cdots + 11068411 \)
|
| $11$ |
\( T^{4} + \cdots + 23023022796 \)
|
| $13$ |
\( T^{4} + \cdots + 19147784368 \)
|
| $17$ |
\( T^{4} + \cdots + 597258958449 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 6908113340160 \)
|
| $29$ |
\( T^{4} + \cdots + 56401950031212 \)
|
| $31$ |
\( T^{4} + \cdots - 5096621276672 \)
|
| $37$ |
\( T^{4} + \cdots + 63\!\cdots\!52 \)
|
| $41$ |
\( T^{4} + \cdots + 813179406508032 \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots - 942340690593024 \)
|
| $53$ |
\( T^{4} + \cdots + 13\!\cdots\!52 \)
|
| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots - 10\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!12 \)
|
| $71$ |
\( T^{4} + \cdots + 16\!\cdots\!16 \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!37 \)
|
| $79$ |
\( T^{4} + \cdots - 20\!\cdots\!56 \)
|
| $83$ |
\( T^{4} + \cdots - 13\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 38\!\cdots\!72 \)
|
| $97$ |
\( T^{4} + \cdots + 42\!\cdots\!12 \)
|
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