Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1472,2,Mod(369,1472)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1472, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1472.369");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1472 = 2^{6} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1472.j (of order \(4\), degree \(2\), not 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: | \(11.7539791775\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | 12.0.221124989353984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 12 x^{8} - 8 x^{7} - 14 x^{6} - 16 x^{5} + 48 x^{4} + 16 x^{3} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 368) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{12} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 12 x^{8} - 8 x^{7} - 14 x^{6} - 16 x^{5} + 48 x^{4} + 16 x^{3} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{11} + 2\nu^{10} - 2\nu^{9} + 2\nu^{8} - 12\nu^{7} + 8\nu^{6} + 2\nu^{5} + 24\nu^{4} - 32\nu^{3} - 64\nu + 96 ) / 32 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + 12 \nu^{7} - 8 \nu^{6} - 14 \nu^{5} - 16 \nu^{4} + \cdots - 64 ) / 32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{11} - 2 \nu^{10} + 10 \nu^{9} - 2 \nu^{8} - 4 \nu^{7} - 24 \nu^{6} + 30 \nu^{5} + 8 \nu^{4} + \cdots + 64 \nu ) / 32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3 \nu^{11} + \nu^{10} + 6 \nu^{9} + 12 \nu^{8} - 26 \nu^{7} - 16 \nu^{6} + 2 \nu^{5} + 86 \nu^{4} + \cdots + 144 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{11} + \nu^{10} + 14 \nu^{9} - 26 \nu^{7} - 24 \nu^{6} + 42 \nu^{5} + 70 \nu^{4} - 20 \nu^{3} + \cdots + 128 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4 \nu^{11} - \nu^{10} - 12 \nu^{9} - 10 \nu^{8} + 34 \nu^{7} + 24 \nu^{6} - 24 \nu^{5} - 98 \nu^{4} + \cdots - 160 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{11} + 2 \nu^{10} + 5 \nu^{9} + 2 \nu^{8} - 18 \nu^{7} - 6 \nu^{6} + 16 \nu^{5} + 44 \nu^{4} + \cdots + 72 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 9 \nu^{11} + 6 \nu^{10} + 26 \nu^{9} + 22 \nu^{8} - 100 \nu^{7} - 40 \nu^{6} + 62 \nu^{5} + \cdots + 576 ) / 32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9 \nu^{11} - 2 \nu^{10} - 34 \nu^{9} - 22 \nu^{8} + 92 \nu^{7} + 72 \nu^{6} - 78 \nu^{5} - 272 \nu^{4} + \cdots - 512 ) / 32 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 13 \nu^{11} + 10 \nu^{10} + 30 \nu^{9} + 30 \nu^{8} - 116 \nu^{7} - 48 \nu^{6} + 70 \nu^{5} + \cdots + 544 ) / 32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 7 \nu^{11} + 3 \nu^{10} + 24 \nu^{9} + 12 \nu^{8} - 66 \nu^{7} - 44 \nu^{6} + 66 \nu^{5} + \cdots + 336 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{2} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{8} - \beta_{7} - 2\beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} + 3\beta_{9} + \beta_{8} - 3\beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{2} + \beta _1 - 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{10} + \beta_{9} - \beta_{8} - 4\beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{3} - \beta_{2} + 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{11} + 3 \beta_{10} + 7 \beta_{9} - \beta_{8} - 4 \beta_{7} - \beta_{6} + \beta_{5} + 5 \beta_{4} + \cdots + 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 14 \beta_{11} + 8 \beta_{10} + 6 \beta_{9} + 10 \beta_{8} - 6 \beta_{7} - 4 \beta_{6} + 10 \beta_{5} + \cdots - 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 10 \beta_{11} + 8 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + \cdots + 18 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 14 \beta_{11} + 8 \beta_{10} + 12 \beta_{9} + 18 \beta_{8} - 6 \beta_{7} + 10 \beta_{5} + 28 \beta_{3} + \cdots - 22 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times\).
| \(n\) | \(645\) | \(833\) | \(1151\) |
| \(\chi(n)\) | \(\beta_{7}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 369.1 |
|
0 | −1.49351 | + | 1.49351i | 0 | −1.17219 | − | 1.17219i | 0 | 0.836699i | 0 | − | 1.46112i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 369.2 | 0 | −1.19673 | + | 1.19673i | 0 | 0.672033 | + | 0.672033i | 0 | − | 1.79918i | 0 | 0.135652i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 369.3 | 0 | −0.631541 | + | 0.631541i | 0 | −2.74053 | − | 2.74053i | 0 | − | 1.52921i | 0 | 2.20231i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 369.4 | 0 | −0.339667 | + | 0.339667i | 0 | 1.46929 | + | 1.46929i | 0 | 2.63128i | 0 | 2.76925i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 369.5 | 0 | 0.955624 | − | 0.955624i | 0 | 2.38359 | + | 2.38359i | 0 | 0.198699i | 0 | 1.17357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 369.6 | 0 | 1.70582 | − | 1.70582i | 0 | −2.61219 | − | 2.61219i | 0 | 1.66171i | 0 | − | 2.81967i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.1 | 0 | −1.49351 | − | 1.49351i | 0 | −1.17219 | + | 1.17219i | 0 | − | 0.836699i | 0 | 1.46112i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.2 | 0 | −1.19673 | − | 1.19673i | 0 | 0.672033 | − | 0.672033i | 0 | 1.79918i | 0 | − | 0.135652i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.3 | 0 | −0.631541 | − | 0.631541i | 0 | −2.74053 | + | 2.74053i | 0 | 1.52921i | 0 | − | 2.20231i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.4 | 0 | −0.339667 | − | 0.339667i | 0 | 1.46929 | − | 1.46929i | 0 | − | 2.63128i | 0 | − | 2.76925i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.5 | 0 | 0.955624 | + | 0.955624i | 0 | 2.38359 | − | 2.38359i | 0 | − | 0.198699i | 0 | − | 1.17357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1105.6 | 0 | 1.70582 | + | 1.70582i | 0 | −2.61219 | + | 2.61219i | 0 | − | 1.66171i | 0 | 2.81967i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1472.2.j.c | 12 | |
| 4.b | odd | 2 | 1 | 368.2.j.c | ✓ | 12 | |
| 16.e | even | 4 | 1 | inner | 1472.2.j.c | 12 | |
| 16.f | odd | 4 | 1 | 368.2.j.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 368.2.j.c | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 368.2.j.c | ✓ | 12 | 16.f | odd | 4 | 1 | |
| 1472.2.j.c | 12 | 1.a | even | 1 | 1 | trivial | |
| 1472.2.j.c | 12 | 16.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 2 T_{3}^{11} + 2 T_{3}^{10} + 2 T_{3}^{9} + 35 T_{3}^{8} + 76 T_{3}^{7} + 84 T_{3}^{6} + \cdots + 25 \)
acting on \(S_{2}^{\mathrm{new}}(1472, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 2 T^{11} + \cdots + 25 \)
|
| $5$ |
\( T^{12} + 4 T^{11} + \cdots + 24964 \)
|
| $7$ |
\( T^{12} + 16 T^{10} + \cdots + 4 \)
|
| $11$ |
\( T^{12} - 4 T^{11} + \cdots + 14884 \)
|
| $13$ |
\( T^{12} - 18 T^{11} + \cdots + 25 \)
|
| $17$ |
\( (T^{6} + 4 T^{5} + \cdots + 202)^{2} \)
|
| $19$ |
\( T^{12} - 8 T^{11} + \cdots + 576 \)
|
| $23$ |
\( (T^{2} + 1)^{6} \)
|
| $29$ |
\( T^{12} - 2 T^{11} + \cdots + 16641 \)
|
| $31$ |
\( (T^{6} + 10 T^{5} + \cdots - 3727)^{2} \)
|
| $37$ |
\( T^{12} + 4 T^{11} + \cdots + 30140100 \)
|
| $41$ |
\( T^{12} + 238 T^{10} + \cdots + 4995225 \)
|
| $43$ |
\( T^{12} + \cdots + 2437989376 \)
|
| $47$ |
\( (T^{6} - 8 T^{5} + \cdots - 51211)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 10123579456 \)
|
| $59$ |
\( T^{12} + \cdots + 17761425984 \)
|
| $61$ |
\( T^{12} - 12 T^{11} + \cdots + 2304 \)
|
| $67$ |
\( T^{12} - 4 T^{11} + \cdots + 438244 \)
|
| $71$ |
\( T^{12} + \cdots + 138180025 \)
|
| $73$ |
\( T^{12} + \cdots + 1176284209 \)
|
| $79$ |
\( (T^{6} - 2 T^{5} + \cdots - 120)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 1492276900 \)
|
| $89$ |
\( T^{12} + \cdots + 3846976576 \)
|
| $97$ |
\( (T^{6} - 18 T^{5} + \cdots - 123118)^{2} \)
|
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