Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2205,4,Mod(1,2205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2205.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2205.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: | \(130.099211563\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 45x^{6} + 134x^{5} + 641x^{4} - 1130x^{3} - 2877x^{2} + 2584x + 3696 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 315) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 45x^{6} + 134x^{5} + 641x^{4} - 1130x^{3} - 2877x^{2} + 2584x + 3696 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} - 28\nu^{6} - 117\nu^{5} + 794\nu^{4} - 467\nu^{3} - 4650\nu^{2} + 12331\nu + 2228 ) / 632 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 27\nu^{6} + 31\nu^{5} - 1076\nu^{4} + 408\nu^{3} + 11577\nu^{2} - 3621\nu - 23264 ) / 316 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -21\nu^{7} + 86\nu^{6} + 839\nu^{5} - 2450\nu^{4} - 9857\nu^{3} + 13368\nu^{2} + 25535\nu - 9484 ) / 632 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + \nu^{6} + 165\nu^{5} + 124\nu^{4} - 2627\nu^{3} - 4083\nu^{2} + 8433\nu + 13610 ) / 158 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 14\nu^{7} - 31\nu^{6} - 691\nu^{5} + 896\nu^{4} + 9784\nu^{3} - 4725\nu^{2} - 29479\nu + 4216 ) / 316 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} + 21\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} + 33\beta_{2} + 49\beta _1 + 293 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 42\beta_{6} - 36\beta_{5} + 42\beta_{4} - 28\beta_{3} + 132\beta_{2} + 559\beta _1 + 722 \)
|
| \(\nu^{6}\) | \(=\) |
\( 50\beta_{7} + 172\beta_{6} - 70\beta_{5} + 144\beta_{4} - 46\beta_{3} + 1059\beta_{2} + 1919\beta _1 + 7997 \)
|
| \(\nu^{7}\) | \(=\) |
\( 168\beta_{7} + 1563\beta_{6} - 1169\beta_{5} + 1565\beta_{4} - 721\beta_{3} + 4989\beta_{2} + 16471\beta _1 + 28195 \)
|
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 |
|
−4.95030 | 0 | 16.5055 | 5.00000 | 0 | 0 | −42.1048 | 0 | −24.7515 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.92080 | 0 | 7.37264 | 5.00000 | 0 | 0 | 2.45977 | 0 | −19.6040 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.30262 | 0 | −6.30317 | 5.00000 | 0 | 0 | 18.6317 | 0 | −6.51312 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.813699 | 0 | −7.33789 | 5.00000 | 0 | 0 | 12.4804 | 0 | −4.06849 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.95066 | 0 | −4.19492 | 5.00000 | 0 | 0 | −23.7882 | 0 | 9.75331 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.02408 | 0 | 1.14504 | 5.00000 | 0 | 0 | −20.7299 | 0 | 15.1204 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.42072 | 0 | 11.5428 | 5.00000 | 0 | 0 | 15.6616 | 0 | 22.1036 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.59196 | 0 | 23.2700 | 5.00000 | 0 | 0 | 85.3894 | 0 | 27.9598 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 2205.4.a.cg | 8 | |
| 3.b | odd | 2 | 1 | 2205.4.a.cb | 8 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.cf | 8 | ||
| 7.d | odd | 6 | 2 | 315.4.j.i | ✓ | 16 | |
| 21.c | even | 2 | 1 | 2205.4.a.cc | 8 | ||
| 21.g | even | 6 | 2 | 315.4.j.j | yes | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.4.j.i | ✓ | 16 | 7.d | odd | 6 | 2 | |
| 315.4.j.j | yes | 16 | 21.g | even | 6 | 2 | |
| 2205.4.a.cb | 8 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.cc | 8 | 21.c | even | 2 | 1 | ||
| 2205.4.a.cf | 8 | 7.b | odd | 2 | 1 | ||
| 2205.4.a.cg | 8 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):
|
\( T_{2}^{8} - 4T_{2}^{7} - 45T_{2}^{6} + 164T_{2}^{5} + 566T_{2}^{4} - 1790T_{2}^{3} - 1812T_{2}^{2} + 3616T_{2} + 3000 \)
|
|
\( T_{11}^{8} - 100 T_{11}^{7} + 169 T_{11}^{6} + 181044 T_{11}^{5} - 816344 T_{11}^{4} + \cdots + 199776500820 \)
|
|
\( T_{13}^{8} - 102 T_{13}^{7} - 6020 T_{13}^{6} + 866738 T_{13}^{5} - 9517198 T_{13}^{4} + \cdots - 1519211972487 \)
|
|
\( T_{17}^{8} + 56 T_{17}^{7} - 20312 T_{17}^{6} - 508468 T_{17}^{5} + 148726728 T_{17}^{4} + \cdots - 130894197809760 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots + 3000 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T - 5)^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 199776500820 \)
|
| $13$ |
\( T^{8} + \cdots - 1519211972487 \)
|
| $17$ |
\( T^{8} + \cdots - 130894197809760 \)
|
| $19$ |
\( T^{8} + \cdots - 37338980824211 \)
|
| $23$ |
\( T^{8} + \cdots + 36927470728260 \)
|
| $29$ |
\( T^{8} + \cdots - 32\!\cdots\!40 \)
|
| $31$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 13\!\cdots\!71 \)
|
| $41$ |
\( T^{8} + \cdots + 54\!\cdots\!80 \)
|
| $43$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 44\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots - 37\!\cdots\!20 \)
|
| $59$ |
\( T^{8} + \cdots + 26\!\cdots\!20 \)
|
| $61$ |
\( T^{8} + \cdots - 43\!\cdots\!72 \)
|
| $67$ |
\( T^{8} + \cdots + 68\!\cdots\!64 \)
|
| $71$ |
\( T^{8} + \cdots + 24\!\cdots\!80 \)
|
| $73$ |
\( T^{8} + \cdots - 10\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots + 95\!\cdots\!64 \)
|
| $83$ |
\( T^{8} + \cdots + 31\!\cdots\!60 \)
|
| $89$ |
\( T^{8} + \cdots + 19\!\cdots\!20 \)
|
| $97$ |
\( T^{8} + \cdots - 69\!\cdots\!88 \)
|
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