Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1152,2,Mod(385,1152)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1152.385");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1152 = 2^{7} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1152.i (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(9.19876631285\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{11} - 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 22 \nu^{7} - 42 \nu^{6} + 51 \nu^{5} - 126 \nu^{4} + \cdots - 486 ) / 243 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{11} + 2 \nu^{10} - 3 \nu^{9} + 8 \nu^{8} - 13 \nu^{7} + 24 \nu^{6} - 51 \nu^{5} + 108 \nu^{4} + \cdots + 162 ) / 162 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 5 \nu^{8} + 40 \nu^{7} + 15 \nu^{5} - 279 \nu^{4} + 36 \nu^{3} + \cdots - 1701 ) / 486 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2 \nu^{11} + \nu^{10} - 3 \nu^{9} + 13 \nu^{8} - 38 \nu^{7} + 33 \nu^{6} - 69 \nu^{5} + 243 \nu^{4} + \cdots + 1215 ) / 162 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11 \nu^{11} + 4 \nu^{10} - 33 \nu^{9} + 52 \nu^{8} - 179 \nu^{7} + 192 \nu^{6} - 327 \nu^{5} + \cdots + 4374 ) / 486 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16 \nu^{11} + \nu^{10} + 36 \nu^{9} - 29 \nu^{8} + 196 \nu^{7} - 135 \nu^{6} + 240 \nu^{5} + \cdots - 6075 ) / 486 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{11} - 4 \nu^{9} + 7 \nu^{8} - 30 \nu^{7} + 26 \nu^{6} - 48 \nu^{5} + 165 \nu^{4} - 108 \nu^{3} + \cdots + 891 ) / 54 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 17 \nu^{11} - 2 \nu^{10} - 42 \nu^{9} + 46 \nu^{8} - 275 \nu^{7} + 210 \nu^{6} - 444 \nu^{5} + \cdots + 9234 ) / 486 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25 \nu^{11} - 2 \nu^{10} + 78 \nu^{9} - 92 \nu^{8} + 463 \nu^{7} - 462 \nu^{6} + 870 \nu^{5} + \cdots - 13122 ) / 486 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{11} - 4 \nu^{10} + 24 \nu^{9} - 34 \nu^{8} + 161 \nu^{7} - 138 \nu^{6} + 210 \nu^{5} + \cdots - 4050 ) / 162 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2 \nu^{11} - 4 \nu^{9} + 7 \nu^{8} - 30 \nu^{7} + 26 \nu^{6} - 48 \nu^{5} + 165 \nu^{4} - 108 \nu^{3} + \cdots + 918 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - 2\beta_{7} - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{11} - \beta_{7} + 3\beta_{6} - 3\beta_{4} - 3\beta_{3} - 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} - 3\beta_{9} - 6\beta_{8} + \beta_{7} - 3\beta_{5} - 3\beta_{3} + 3\beta _1 + 5 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + \cdots - 14 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 5 \beta_{11} + 3 \beta_{10} - 9 \beta_{8} + 7 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + \cdots - 7 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} - 21 \beta_{8} - 7 \beta_{7} - 30 \beta_{6} - 6 \beta_{5} + \cdots + 13 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13 \beta_{11} + 18 \beta_{10} + 21 \beta_{9} + 24 \beta_{8} - 23 \beta_{7} - 24 \beta_{6} + 21 \beta_{5} + \cdots + 17 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 41 \beta_{11} + 3 \beta_{10} + 27 \beta_{9} - 25 \beta_{7} + 69 \beta_{6} + 99 \beta_{5} - 33 \beta_{4} + \cdots + 55 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 130 \beta_{11} - 21 \beta_{10} - 30 \beta_{9} - 69 \beta_{8} - 95 \beta_{7} + 87 \beta_{6} - 165 \beta_{5} + \cdots - 94 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 16 \beta_{11} - 66 \beta_{10} - 66 \beta_{9} - 69 \beta_{8} + 92 \beta_{7} + 165 \beta_{6} + \cdots + 241 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 176 \beta_{11} + 36 \beta_{10} - 324 \beta_{9} - 450 \beta_{8} + 70 \beta_{7} - 138 \beta_{6} + \cdots + 248 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 385.1 |
|
0 | −1.72908 | − | 0.101475i | 0 | 1.24278 | + | 2.15256i | 0 | 0.909142 | − | 1.57468i | 0 | 2.97941 | + | 0.350917i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 385.2 | 0 | −1.23541 | + | 1.21398i | 0 | −2.22043 | − | 3.84590i | 0 | 1.45488 | − | 2.51992i | 0 | 0.0524919 | − | 2.99954i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 385.3 | 0 | −0.857134 | − | 1.50510i | 0 | −0.551563 | − | 0.955334i | 0 | −1.62490 | + | 2.81442i | 0 | −1.53064 | + | 2.58014i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 385.4 | 0 | −0.366879 | + | 1.69275i | 0 | 1.05471 | + | 1.82681i | 0 | −1.43914 | + | 2.49267i | 0 | −2.73080 | − | 1.24207i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 385.5 | 0 | 0.494250 | − | 1.66004i | 0 | −0.268104 | − | 0.464369i | 0 | 2.35014 | − | 4.07056i | 0 | −2.51143 | − | 1.64095i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 385.6 | 0 | 1.69425 | + | 0.359877i | 0 | 1.74260 | + | 3.01828i | 0 | 1.34988 | − | 2.33807i | 0 | 2.74098 | + | 1.21944i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.1 | 0 | −1.72908 | + | 0.101475i | 0 | 1.24278 | − | 2.15256i | 0 | 0.909142 | + | 1.57468i | 0 | 2.97941 | − | 0.350917i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.2 | 0 | −1.23541 | − | 1.21398i | 0 | −2.22043 | + | 3.84590i | 0 | 1.45488 | + | 2.51992i | 0 | 0.0524919 | + | 2.99954i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.3 | 0 | −0.857134 | + | 1.50510i | 0 | −0.551563 | + | 0.955334i | 0 | −1.62490 | − | 2.81442i | 0 | −1.53064 | − | 2.58014i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.4 | 0 | −0.366879 | − | 1.69275i | 0 | 1.05471 | − | 1.82681i | 0 | −1.43914 | − | 2.49267i | 0 | −2.73080 | + | 1.24207i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.5 | 0 | 0.494250 | + | 1.66004i | 0 | −0.268104 | + | 0.464369i | 0 | 2.35014 | + | 4.07056i | 0 | −2.51143 | + | 1.64095i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 769.6 | 0 | 1.69425 | − | 0.359877i | 0 | 1.74260 | − | 3.01828i | 0 | 1.34988 | + | 2.33807i | 0 | 2.74098 | − | 1.21944i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1152.2.i.j | yes | 12 |
| 3.b | odd | 2 | 1 | 3456.2.i.j | 12 | ||
| 4.b | odd | 2 | 1 | 1152.2.i.l | yes | 12 | |
| 8.b | even | 2 | 1 | 1152.2.i.k | yes | 12 | |
| 8.d | odd | 2 | 1 | 1152.2.i.i | ✓ | 12 | |
| 9.c | even | 3 | 1 | inner | 1152.2.i.j | yes | 12 |
| 9.d | odd | 6 | 1 | 3456.2.i.j | 12 | ||
| 12.b | even | 2 | 1 | 3456.2.i.i | 12 | ||
| 24.f | even | 2 | 1 | 3456.2.i.k | 12 | ||
| 24.h | odd | 2 | 1 | 3456.2.i.l | 12 | ||
| 36.f | odd | 6 | 1 | 1152.2.i.l | yes | 12 | |
| 36.h | even | 6 | 1 | 3456.2.i.i | 12 | ||
| 72.j | odd | 6 | 1 | 3456.2.i.l | 12 | ||
| 72.l | even | 6 | 1 | 3456.2.i.k | 12 | ||
| 72.n | even | 6 | 1 | 1152.2.i.k | yes | 12 | |
| 72.p | odd | 6 | 1 | 1152.2.i.i | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1152.2.i.i | ✓ | 12 | 8.d | odd | 2 | 1 | |
| 1152.2.i.i | ✓ | 12 | 72.p | odd | 6 | 1 | |
| 1152.2.i.j | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 1152.2.i.j | yes | 12 | 9.c | even | 3 | 1 | inner |
| 1152.2.i.k | yes | 12 | 8.b | even | 2 | 1 | |
| 1152.2.i.k | yes | 12 | 72.n | even | 6 | 1 | |
| 1152.2.i.l | yes | 12 | 4.b | odd | 2 | 1 | |
| 1152.2.i.l | yes | 12 | 36.f | odd | 6 | 1 | |
| 3456.2.i.i | 12 | 12.b | even | 2 | 1 | ||
| 3456.2.i.i | 12 | 36.h | even | 6 | 1 | ||
| 3456.2.i.j | 12 | 3.b | odd | 2 | 1 | ||
| 3456.2.i.j | 12 | 9.d | odd | 6 | 1 | ||
| 3456.2.i.k | 12 | 24.f | even | 2 | 1 | ||
| 3456.2.i.k | 12 | 72.l | even | 6 | 1 | ||
| 3456.2.i.l | 12 | 24.h | odd | 2 | 1 | ||
| 3456.2.i.l | 12 | 72.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):
|
\( T_{5}^{12} - 2 T_{5}^{11} + 24 T_{5}^{10} - 60 T_{5}^{9} + 465 T_{5}^{8} - 948 T_{5}^{7} + 2928 T_{5}^{6} + \cdots + 2304 \)
|
|
\( T_{7}^{12} - 6 T_{7}^{11} + 48 T_{7}^{10} - 152 T_{7}^{9} + 861 T_{7}^{8} - 2400 T_{7}^{7} + \cdots + 394384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 4 T^{11} + \cdots + 729 \)
|
| $5$ |
\( T^{12} - 2 T^{11} + \cdots + 2304 \)
|
| $7$ |
\( T^{12} - 6 T^{11} + \cdots + 394384 \)
|
| $11$ |
\( T^{12} + 4 T^{11} + \cdots + 229441 \)
|
| $13$ |
\( T^{12} + 10 T^{11} + \cdots + 6533136 \)
|
| $17$ |
\( (T^{6} - 2 T^{5} + \cdots + 1812)^{2} \)
|
| $19$ |
\( (T^{6} + 2 T^{5} + \cdots - 3408)^{2} \)
|
| $23$ |
\( T^{12} - 8 T^{11} + \cdots + 204304 \)
|
| $29$ |
\( T^{12} + \cdots + 229704336 \)
|
| $31$ |
\( T^{12} + \cdots + 1021953024 \)
|
| $37$ |
\( (T^{6} - 60 T^{4} + \cdots - 128)^{2} \)
|
| $41$ |
\( T^{12} + 2 T^{11} + \cdots + 2259009 \)
|
| $43$ |
\( T^{12} - 2 T^{11} + \cdots + 16621929 \)
|
| $47$ |
\( T^{12} + 14 T^{11} + \cdots + 2178576 \)
|
| $53$ |
\( (T^{6} + 12 T^{5} + \cdots + 1728)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 4100737369 \)
|
| $61$ |
\( T^{12} + 14 T^{11} + \cdots + 60715264 \)
|
| $67$ |
\( T^{12} + \cdots + 3020711521 \)
|
| $71$ |
\( (T^{6} + 14 T^{5} + \cdots + 1728)^{2} \)
|
| $73$ |
\( (T^{6} - 30 T^{5} + \cdots + 39892)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 674337024 \)
|
| $83$ |
\( T^{12} + \cdots + 734843664 \)
|
| $89$ |
\( (T^{6} + 24 T^{5} + \cdots - 2864)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 78140934369 \)
|
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