Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9328,2,Mod(1,9328)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9328, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9328.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9328 = 2^{4} \cdot 11 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9328.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: | \(74.4844550055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 4664) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 18 \nu^{10} + 107 \nu^{9} + 30 \nu^{8} - 991 \nu^{7} + 603 \nu^{6} + 2930 \nu^{5} - 1835 \nu^{4} + \cdots + 30 ) / 130 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{10} - 7 \nu^{9} + 5 \nu^{8} + 56 \nu^{7} - 99 \nu^{6} - 115 \nu^{5} + 311 \nu^{4} + 28 \nu^{3} + \cdots + 46 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18 \nu^{10} + 107 \nu^{9} + 30 \nu^{8} - 991 \nu^{7} + 603 \nu^{6} + 2930 \nu^{5} - 1835 \nu^{4} + \cdots - 360 ) / 130 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 9 \nu^{10} - 8 \nu^{9} - 249 \nu^{8} + 242 \nu^{7} + 1980 \nu^{6} - 1361 \nu^{5} - 6109 \nu^{4} + \cdots - 1302 ) / 130 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 35 \nu^{10} + 203 \nu^{9} + 67 \nu^{8} - 1828 \nu^{7} + 1075 \nu^{6} + 5053 \nu^{5} - 3407 \nu^{4} + \cdots - 258 ) / 130 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 28 \nu^{10} + 178 \nu^{9} - 14 \nu^{8} - 1621 \nu^{7} + 1640 \nu^{6} + 4604 \nu^{5} - 5635 \nu^{4} + \cdots - 1418 ) / 130 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 33 \nu^{10} - 168 \nu^{9} - 211 \nu^{8} + 1741 \nu^{7} + 240 \nu^{6} - 5909 \nu^{5} - 70 \nu^{4} + \cdots - 562 ) / 130 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 37 \nu^{10} + 173 \nu^{9} + 313 \nu^{8} - 1850 \nu^{7} - 1055 \nu^{6} + 6537 \nu^{5} + 2541 \nu^{4} + \cdots + 482 ) / 130 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 34 \nu^{10} - 127 \nu^{9} - 451 \nu^{8} + 1518 \nu^{7} + 2579 \nu^{6} - 5949 \nu^{5} - 7360 \nu^{4} + \cdots - 1712 ) / 130 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} - \beta_{8} + \beta_{6} + 2\beta_{4} + \beta_{3} - 3\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{10} - 3\beta_{8} - 2\beta_{7} + 3\beta_{6} + 10\beta_{4} + 2\beta_{3} - 13\beta_{2} + 14\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{10} - 4 \beta_{9} - 15 \beta_{8} - 6 \beta_{7} + 18 \beta_{6} + 26 \beta_{4} + 15 \beta_{3} + \cdots + 39 \)
|
| \(\nu^{6}\) | \(=\) |
\( 46 \beta_{10} + 7 \beta_{9} - 55 \beta_{8} - 33 \beta_{7} + 64 \beta_{6} - 5 \beta_{5} + 101 \beta_{4} + \cdots + 162 \)
|
| \(\nu^{7}\) | \(=\) |
\( 186 \beta_{10} + 18 \beta_{9} - 219 \beta_{8} - 114 \beta_{7} + 279 \beta_{6} - 26 \beta_{5} + \cdots + 447 \)
|
| \(\nu^{8}\) | \(=\) |
\( 810 \beta_{10} + 194 \beta_{9} - 823 \beta_{8} - 472 \beta_{7} + 1032 \beta_{6} - 152 \beta_{5} + \cdots + 1633 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3186 \beta_{10} + 763 \beta_{9} - 3117 \beta_{8} - 1703 \beta_{7} + 4068 \beta_{6} - 673 \beta_{5} + \cdots + 5152 \)
|
| \(\nu^{10}\) | \(=\) |
\( 12688 \beta_{10} + 3624 \beta_{9} - 11649 \beta_{8} - 6512 \beta_{7} + 15137 \beta_{6} - 2990 \beta_{5} + \cdots + 18203 \)
|
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 | −2.66270 | 0 | −0.240194 | 0 | 3.16150 | 0 | 4.08996 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.92319 | 0 | −2.25184 | 0 | −2.05673 | 0 | 0.698658 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.15382 | 0 | 4.11573 | 0 | 1.30816 | 0 | −1.66869 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.492404 | 0 | 0.923502 | 0 | −1.84390 | 0 | −2.75754 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.418978 | 0 | −0.101827 | 0 | 3.67256 | 0 | −2.82446 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.532294 | 0 | −1.86524 | 0 | −3.07135 | 0 | −2.71666 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.51312 | 0 | −3.62351 | 0 | 0.892845 | 0 | −0.710478 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.84134 | 0 | 3.45804 | 0 | 1.90801 | 0 | 0.390530 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.09994 | 0 | −1.01201 | 0 | −2.31863 | 0 | 1.40977 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.67321 | 0 | 1.73724 | 0 | −1.26953 | 0 | 4.14608 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.15323 | 0 | 1.86010 | 0 | 4.61706 | 0 | 6.94284 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(1\) |
| \(53\) | \(-1\) |
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 | 9328.2.a.bm | 11 | |
| 4.b | odd | 2 | 1 | 4664.2.a.k | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4664.2.a.k | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 9328.2.a.bm | 11 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9328))\):
|
\( T_{3}^{11} - 6 T_{3}^{10} - 2 T_{3}^{9} + 70 T_{3}^{8} - 85 T_{3}^{7} - 210 T_{3}^{6} + 411 T_{3}^{5} + \cdots - 32 \)
|
|
\( T_{5}^{11} - 3 T_{5}^{10} - 25 T_{5}^{9} + 65 T_{5}^{8} + 197 T_{5}^{7} - 404 T_{5}^{6} - 599 T_{5}^{5} + \cdots - 16 \)
|
|
\( T_{7}^{11} - 5 T_{7}^{10} - 25 T_{7}^{9} + 123 T_{7}^{8} + 253 T_{7}^{7} - 1082 T_{7}^{6} - 1310 T_{7}^{5} + \cdots + 4096 \)
|
|
\( T_{13}^{11} + 13 T_{13}^{10} + 28 T_{13}^{9} - 186 T_{13}^{8} - 640 T_{13}^{7} + 65 T_{13}^{6} + \cdots + 32 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 6 T^{10} + \cdots - 32 \)
|
| $5$ |
\( T^{11} - 3 T^{10} + \cdots - 16 \)
|
| $7$ |
\( T^{11} - 5 T^{10} + \cdots + 4096 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} + 13 T^{10} + \cdots + 32 \)
|
| $17$ |
\( T^{11} + 7 T^{10} + \cdots + 4640 \)
|
| $19$ |
\( T^{11} - 21 T^{10} + \cdots + 295936 \)
|
| $23$ |
\( T^{11} + 11 T^{10} + \cdots - 731708 \)
|
| $29$ |
\( T^{11} - 112 T^{9} + \cdots - 152008 \)
|
| $31$ |
\( T^{11} - 5 T^{10} + \cdots + 18608 \)
|
| $37$ |
\( T^{11} + 4 T^{10} + \cdots + 1917278 \)
|
| $41$ |
\( T^{11} + 11 T^{10} + \cdots + 534512 \)
|
| $43$ |
\( T^{11} - 296 T^{9} + \cdots - 5291680 \)
|
| $47$ |
\( T^{11} + \cdots + 307232768 \)
|
| $53$ |
\( (T - 1)^{11} \)
|
| $59$ |
\( T^{11} - 19 T^{10} + \cdots - 36528344 \)
|
| $61$ |
\( T^{11} + 2 T^{10} + \cdots - 35035136 \)
|
| $67$ |
\( T^{11} + \cdots - 163195904 \)
|
| $71$ |
\( T^{11} + \cdots - 254186564 \)
|
| $73$ |
\( T^{11} - 5 T^{10} + \cdots + 14568256 \)
|
| $79$ |
\( T^{11} + \cdots - 320973344 \)
|
| $83$ |
\( T^{11} + \cdots - 77177586608 \)
|
| $89$ |
\( T^{11} - 6 T^{10} + \cdots - 1090808 \)
|
| $97$ |
\( T^{11} + \cdots - 375661550 \)
|
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