Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,6,Mod(1,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("164.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.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: | \(26.3029464493\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 1807 x^{8} - 1186 x^{7} + 1075622 x^{6} + 1575146 x^{5} - 242812142 x^{4} - 535064182 x^{3} + \cdots - 425549129499 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 1807 x^{8} - 1186 x^{7} + 1075622 x^{6} + 1575146 x^{5} - 242812142 x^{4} - 535064182 x^{3} + \cdots - 425549129499 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 20\!\cdots\!42 \nu^{9} + \cdots + 61\!\cdots\!49 ) / 17\!\cdots\!30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 32\!\cdots\!33 \nu^{9} + \cdots - 27\!\cdots\!88 ) / 14\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25\!\cdots\!67 \nu^{9} + \cdots + 86\!\cdots\!89 ) / 47\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 77\!\cdots\!89 \nu^{9} + \cdots + 49\!\cdots\!63 ) / 14\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\!\cdots\!99 \nu^{9} + \cdots - 94\!\cdots\!83 ) / 14\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 97\!\cdots\!67 \nu^{9} + \cdots + 57\!\cdots\!09 ) / 71\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 10\!\cdots\!53 \nu^{9} + \cdots - 49\!\cdots\!81 ) / 35\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15\!\cdots\!31 \nu^{9} + \cdots + 42\!\cdots\!85 ) / 47\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} - 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 361 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{9} + 9 \beta_{8} + 21 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 12 \beta_{4} - 6 \beta_{3} + \cdots + 374 \)
|
| \(\nu^{4}\) | \(=\) |
\( 23 \beta_{9} + 42 \beta_{8} + 878 \beta_{7} - 1003 \beta_{6} - 1717 \beta_{5} - 1152 \beta_{4} + \cdots + 222496 \)
|
| \(\nu^{5}\) | \(=\) |
\( 328 \beta_{9} + 7233 \beta_{8} + 16896 \beta_{7} + 2063 \beta_{6} + 2946 \beta_{5} - 10931 \beta_{4} + \cdots + 300472 \)
|
| \(\nu^{6}\) | \(=\) |
\( 71724 \beta_{9} + 126864 \beta_{8} + 749781 \beta_{7} - 913260 \beta_{6} - 1353764 \beta_{5} + \cdots + 159794874 \)
|
| \(\nu^{7}\) | \(=\) |
\( 956626 \beta_{9} + 5462538 \beta_{8} + 12524557 \beta_{7} + 3701767 \beta_{6} + 3823815 \beta_{5} + \cdots + 213006620 \)
|
| \(\nu^{8}\) | \(=\) |
\( 99607642 \beta_{9} + 174250386 \beta_{8} + 630748328 \beta_{7} - 809100144 \beta_{6} + \cdots + 120949229163 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1187764244 \beta_{9} + 4262654670 \beta_{8} + 9326592460 \beta_{7} + 4037378436 \beta_{6} + \cdots + 147038296506 \)
|
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 |
|
0 | −27.8044 | 0 | 33.0128 | 0 | −225.112 | 0 | 530.085 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −22.8246 | 0 | 105.662 | 0 | 182.914 | 0 | 277.961 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −12.6020 | 0 | −99.4633 | 0 | 15.8113 | 0 | −84.1902 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.60967 | 0 | −7.92999 | 0 | 170.769 | 0 | −150.654 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −7.13065 | 0 | −43.0883 | 0 | −181.850 | 0 | −192.154 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 5.25838 | 0 | −76.6066 | 0 | −175.490 | 0 | −215.349 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 11.5737 | 0 | 47.3907 | 0 | 183.989 | 0 | −109.050 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 17.2169 | 0 | 68.9224 | 0 | −99.1523 | 0 | 53.4233 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 26.1896 | 0 | −66.7543 | 0 | 117.962 | 0 | 442.894 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 29.7327 | 0 | 70.8542 | 0 | 98.1583 | 0 | 641.034 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(1\) |
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 | 164.6.a.b | ✓ | 10 |
| 4.b | odd | 2 | 1 | 656.6.a.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.6.a.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 656.6.a.h | 10 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 10 T_{3}^{9} - 1762 T_{3}^{8} + 13150 T_{3}^{7} + 1033538 T_{3}^{6} - 4802552 T_{3}^{5} + \cdots - 447130693728 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(164))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} + \cdots - 447130693728 \)
|
| $5$ |
\( T^{10} + \cdots - 14\!\cdots\!96 \)
|
| $7$ |
\( T^{10} + \cdots + 74\!\cdots\!72 \)
|
| $11$ |
\( T^{10} + \cdots - 53\!\cdots\!96 \)
|
| $13$ |
\( T^{10} + \cdots - 16\!\cdots\!04 \)
|
| $17$ |
\( T^{10} + \cdots - 12\!\cdots\!92 \)
|
| $19$ |
\( T^{10} + \cdots + 33\!\cdots\!84 \)
|
| $23$ |
\( T^{10} + \cdots + 14\!\cdots\!12 \)
|
| $29$ |
\( T^{10} + \cdots + 77\!\cdots\!44 \)
|
| $31$ |
\( T^{10} + \cdots - 70\!\cdots\!64 \)
|
| $37$ |
\( T^{10} + \cdots - 24\!\cdots\!40 \)
|
| $41$ |
\( (T + 1681)^{10} \)
|
| $43$ |
\( T^{10} + \cdots - 22\!\cdots\!28 \)
|
| $47$ |
\( T^{10} + \cdots + 45\!\cdots\!12 \)
|
| $53$ |
\( T^{10} + \cdots - 18\!\cdots\!60 \)
|
| $59$ |
\( T^{10} + \cdots + 88\!\cdots\!80 \)
|
| $61$ |
\( T^{10} + \cdots - 26\!\cdots\!72 \)
|
| $67$ |
\( T^{10} + \cdots - 47\!\cdots\!84 \)
|
| $71$ |
\( T^{10} + \cdots + 70\!\cdots\!64 \)
|
| $73$ |
\( T^{10} + \cdots - 79\!\cdots\!60 \)
|
| $79$ |
\( T^{10} + \cdots + 30\!\cdots\!48 \)
|
| $83$ |
\( T^{10} + \cdots - 54\!\cdots\!64 \)
|
| $89$ |
\( T^{10} + \cdots - 10\!\cdots\!32 \)
|
| $97$ |
\( T^{10} + \cdots + 59\!\cdots\!44 \)
|
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