Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8304,2,Mod(1,8304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8304, 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("8304.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8304 = 2^{4} \cdot 3 \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8304.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: | \(66.3077738385\) |
| 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: |
\( x^{11} - 3x^{10} - 12x^{9} + 39x^{8} + 43x^{7} - 165x^{6} - 45x^{5} + 271x^{4} - 29x^{3} - 134x^{2} + 60x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 519) |
| 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
\( x^{11} - 3x^{10} - 12x^{9} + 39x^{8} + 43x^{7} - 165x^{6} - 45x^{5} + 271x^{4} - 29x^{3} - 134x^{2} + 60x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 4 \nu^{10} + 3 \nu^{9} + 59 \nu^{8} - 36 \nu^{7} - 287 \nu^{6} + 146 \nu^{5} + 500 \nu^{4} + \cdots - 105 ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9 \nu^{10} - 11 \nu^{9} - 120 \nu^{8} + 115 \nu^{7} + 514 \nu^{6} - 320 \nu^{5} - 768 \nu^{4} + \cdots + 96 ) / 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10 \nu^{10} - 16 \nu^{9} - 139 \nu^{8} + 192 \nu^{7} + 658 \nu^{6} - 705 \nu^{5} - 1301 \nu^{4} + \cdots - 35 ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{10} + 15 \nu^{9} + 40 \nu^{8} - 214 \nu^{7} - 194 \nu^{6} + 1019 \nu^{5} + 477 \nu^{4} + \cdots - 134 ) / 17 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10 \nu^{10} - \nu^{9} + 156 \nu^{8} + 29 \nu^{7} - 811 \nu^{6} - 213 \nu^{5} + 1539 \nu^{4} + \cdots + 1 ) / 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5 \nu^{10} - 9 \nu^{9} + 95 \nu^{8} + 125 \nu^{7} - 601 \nu^{6} - 574 \nu^{5} + 1424 \nu^{4} + \cdots + 196 ) / 17 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5 \nu^{10} - 25 \nu^{9} - 61 \nu^{8} + 351 \nu^{7} + 244 \nu^{6} - 1636 \nu^{5} - 438 \nu^{4} + \cdots + 263 ) / 17 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 18 \nu^{10} + 39 \nu^{9} + 240 \nu^{8} - 502 \nu^{7} - 1062 \nu^{6} + 2102 \nu^{5} + 1876 \nu^{4} + \cdots - 345 ) / 17 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 30 \nu^{10} - 65 \nu^{9} - 400 \nu^{8} + 831 \nu^{7} + 1770 \nu^{6} - 3424 \nu^{5} - 3138 \nu^{4} + \cdots + 524 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{10} + \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} + 5\beta _1 + 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{10} - 6\beta_{9} + 5\beta_{8} - \beta_{7} + 5\beta_{5} + 3\beta_{4} + \beta_{3} + 3\beta_{2} + 4\beta _1 + 29 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 7 \beta_{10} + 2 \beta_{9} - \beta_{8} + 9 \beta_{7} - 10 \beta_{6} - 19 \beta_{5} + 7 \beta_{4} + \cdots + 27 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 27 \beta_{10} - 38 \beta_{9} + 24 \beta_{8} - 9 \beta_{7} - \beta_{6} + 24 \beta_{5} + 7 \beta_{4} + \cdots + 159 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 45 \beta_{10} + 19 \beta_{9} - 13 \beta_{8} + 68 \beta_{7} - 80 \beta_{6} - 145 \beta_{5} + \cdots + 148 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 158 \beta_{10} - 250 \beta_{9} + 118 \beta_{8} - 60 \beta_{7} - 18 \beta_{6} + 120 \beta_{5} + \cdots + 939 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 144 \beta_{10} + 66 \beta_{9} - 64 \beta_{8} + 243 \beta_{7} - 296 \beta_{6} - 516 \beta_{5} + \cdots + 425 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 976 \beta_{10} - 1676 \beta_{9} + 594 \beta_{8} - 358 \beta_{7} - 208 \beta_{6} + 626 \beta_{5} + \cdots + 5815 ) / 2 \)
|
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 | −1.00000 | 0 | −3.95340 | 0 | 4.85037 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −1.74767 | 0 | −2.24842 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −0.724904 | 0 | −2.94502 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | −0.0483161 | 0 | 3.16101 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | −0.0400868 | 0 | −0.678956 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 0.476766 | 0 | −1.82350 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −1.00000 | 0 | 2.18366 | 0 | 0.682789 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | −1.00000 | 0 | 2.46712 | 0 | −1.34068 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | −1.00000 | 0 | 2.76161 | 0 | 5.18186 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | −1.00000 | 0 | 3.75758 | 0 | 2.48323 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | −1.00000 | 0 | 3.86764 | 0 | −4.32268 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(173\) | \( +1 \) |
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 | 8304.2.a.bj | 11 | |
| 4.b | odd | 2 | 1 | 519.2.a.c | ✓ | 11 | |
| 12.b | even | 2 | 1 | 1557.2.a.e | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 519.2.a.c | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 1557.2.a.e | 11 | 12.b | even | 2 | 1 | ||
| 8304.2.a.bj | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8304))\):
|
\( T_{5}^{11} - 9 T_{5}^{10} + 7 T_{5}^{9} + 150 T_{5}^{8} - 451 T_{5}^{7} - 80 T_{5}^{6} + 1582 T_{5}^{5} + \cdots + 1 \)
|
|
\( T_{7}^{11} - 3 T_{7}^{10} - 48 T_{7}^{9} + 95 T_{7}^{8} + 850 T_{7}^{7} - 712 T_{7}^{6} - 6576 T_{7}^{5} + \cdots - 6400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 9 T^{10} + \cdots + 1 \)
|
| $7$ |
\( T^{11} - 3 T^{10} + \cdots - 6400 \)
|
| $11$ |
\( T^{11} + 13 T^{10} + \cdots - 243 \)
|
| $13$ |
\( T^{11} + T^{10} + \cdots + 30524 \)
|
| $17$ |
\( T^{11} - 18 T^{10} + \cdots + 732564 \)
|
| $19$ |
\( T^{11} + T^{10} + \cdots + 4782848 \)
|
| $23$ |
\( T^{11} + 15 T^{10} + \cdots + 164096 \)
|
| $29$ |
\( T^{11} - 6 T^{10} + \cdots + 2179072 \)
|
| $31$ |
\( T^{11} + T^{10} + \cdots + 1552763 \)
|
| $37$ |
\( T^{11} + 13 T^{10} + \cdots + 63731983 \)
|
| $41$ |
\( T^{11} - 17 T^{10} + \cdots - 7439616 \)
|
| $43$ |
\( T^{11} - 22 T^{10} + \cdots - 12852472 \)
|
| $47$ |
\( T^{11} + 14 T^{10} + \cdots - 379136 \)
|
| $53$ |
\( T^{11} - 27 T^{10} + \cdots - 4817921 \)
|
| $59$ |
\( T^{11} + \cdots - 3367690816 \)
|
| $61$ |
\( T^{11} - 6 T^{10} + \cdots - 26858752 \)
|
| $67$ |
\( T^{11} - 21 T^{10} + \cdots - 492443 \)
|
| $71$ |
\( T^{11} + 9 T^{10} + \cdots + 65166829 \)
|
| $73$ |
\( T^{11} - T^{10} + \cdots + 1925631 \)
|
| $79$ |
\( T^{11} + \cdots - 197045504 \)
|
| $83$ |
\( T^{11} + \cdots + 210432256 \)
|
| $89$ |
\( T^{11} + \cdots + 559299328 \)
|
| $97$ |
\( T^{11} + T^{10} + \cdots + 34048 \)
|
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