Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7448,2,Mod(1,7448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, 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("7448.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7448.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: | \(59.4725794254\) |
| 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: |
\( x^{11} - 25x^{9} - 7x^{8} + 212x^{7} + 112x^{6} - 694x^{5} - 480x^{4} + 740x^{3} + 632x^{2} + 48x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1064) |
| 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
\( x^{11} - 25x^{9} - 7x^{8} + 212x^{7} + 112x^{6} - 694x^{5} - 480x^{4} + 740x^{3} + 632x^{2} + 48x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 287 \nu^{10} + 438 \nu^{9} - 6680 \nu^{8} - 12589 \nu^{7} + 48473 \nu^{6} + 112345 \nu^{5} + \cdots + 76402 ) / 11062 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 765 \nu^{10} + 69 \nu^{9} - 18403 \nu^{8} - 4976 \nu^{7} + 144661 \nu^{6} + 56150 \nu^{5} + \cdots - 15316 ) / 22124 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1122 \nu^{10} - 1005 \nu^{9} - 28466 \nu^{8} + 17407 \nu^{7} + 250149 \nu^{6} - 87264 \nu^{5} + \cdots - 165532 ) / 22124 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1171 \nu^{10} + 149 \nu^{9} - 30281 \nu^{8} - 10104 \nu^{7} + 270701 \nu^{6} + 133596 \nu^{5} + \cdots - 48 ) / 22124 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1270 \nu^{10} - 223 \nu^{9} + 32142 \nu^{8} + 12635 \nu^{7} - 271137 \nu^{6} - 155178 \nu^{5} + \cdots - 81000 ) / 22124 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1696 \nu^{10} + 129 \nu^{9} + 41826 \nu^{8} + 7771 \nu^{7} - 347095 \nu^{6} - 137426 \nu^{5} + \cdots - 31520 ) / 22124 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1973 \nu^{10} - 525 \nu^{9} + 49603 \nu^{8} + 25356 \nu^{7} - 420609 \nu^{6} - 297370 \nu^{5} + \cdots + 14572 ) / 22124 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3004 \nu^{10} - 1833 \nu^{9} - 73118 \nu^{8} + 21809 \nu^{7} + 604935 \nu^{6} + 2214 \nu^{5} + \cdots - 114484 ) / 22124 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{5} + \beta_{3} + 8\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + 11\beta_{2} + 2\beta _1 + 41 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 16\beta_{8} - \beta_{7} + \beta_{6} + 12\beta_{5} + \beta_{4} + 13\beta_{3} + \beta_{2} + 73\beta _1 + 30 \)
|
| \(\nu^{6}\) | \(=\) |
\( 29\beta_{9} - 15\beta_{8} - 11\beta_{7} + 17\beta_{6} - 14\beta_{4} + 15\beta_{3} + 116\beta_{2} + 34\beta _1 + 386 \)
|
| \(\nu^{7}\) | \(=\) |
\( 17 \beta_{10} + 6 \beta_{9} + 195 \beta_{8} - 18 \beta_{7} + 24 \beta_{6} + 124 \beta_{5} + 26 \beta_{4} + \cdots + 381 \)
|
| \(\nu^{8}\) | \(=\) |
\( 7 \beta_{10} + 343 \beta_{9} - 167 \beta_{8} - 98 \beta_{7} + 224 \beta_{6} - 8 \beta_{5} - 145 \beta_{4} + \cdots + 3848 \)
|
| \(\nu^{9}\) | \(=\) |
\( 217 \beta_{10} + 143 \beta_{9} + 2186 \beta_{8} - 231 \beta_{7} + 403 \beta_{6} + 1232 \beta_{5} + \cdots + 4521 \)
|
| \(\nu^{10}\) | \(=\) |
\( 186 \beta_{10} + 3853 \beta_{9} - 1636 \beta_{8} - 830 \beta_{7} + 2750 \beta_{6} - 200 \beta_{5} + \cdots + 39345 \)
|
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 | −3.16715 | 0 | −4.36423 | 0 | 0 | 0 | 7.03082 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.46711 | 0 | 3.45917 | 0 | 0 | 0 | 3.08665 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.02819 | 0 | −1.45328 | 0 | 0 | 0 | 1.11356 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.41808 | 0 | −1.57417 | 0 | 0 | 0 | −0.989040 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.728917 | 0 | 3.18420 | 0 | 0 | 0 | −2.46868 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.261012 | 0 | −0.284735 | 0 | 0 | 0 | −2.93187 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.119898 | 0 | −0.190070 | 0 | 0 | 0 | −2.98562 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.61712 | 0 | −4.18709 | 0 | 0 | 0 | −0.384920 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.82026 | 0 | 3.85544 | 0 | 0 | 0 | 0.313338 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 3.20667 | 0 | −3.33994 | 0 | 0 | 0 | 7.28271 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.30652 | 0 | 1.89470 | 0 | 0 | 0 | 7.93307 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 7448.2.a.bs | 11 | |
| 7.b | odd | 2 | 1 | 7448.2.a.bt | 11 | ||
| 7.d | odd | 6 | 2 | 1064.2.q.o | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1064.2.q.o | ✓ | 22 | 7.d | odd | 6 | 2 | |
| 7448.2.a.bs | 11 | 1.a | even | 1 | 1 | trivial | |
| 7448.2.a.bt | 11 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7448))\):
|
\( T_{3}^{11} - 25 T_{3}^{9} - 7 T_{3}^{8} + 212 T_{3}^{7} + 112 T_{3}^{6} - 694 T_{3}^{5} - 480 T_{3}^{4} + \cdots - 16 \)
|
|
\( T_{5}^{11} + 3 T_{5}^{10} - 42 T_{5}^{9} - 114 T_{5}^{8} + 617 T_{5}^{7} + 1511 T_{5}^{6} - 3508 T_{5}^{5} + \cdots + 608 \)
|
|
\( T_{11}^{11} + 6 T_{11}^{10} - 84 T_{11}^{9} - 454 T_{11}^{8} + 2850 T_{11}^{7} + 12598 T_{11}^{6} + \cdots - 1228088 \)
|
|
\( T_{13}^{11} - 8 T_{13}^{10} - 87 T_{13}^{9} + 903 T_{13}^{8} + 1328 T_{13}^{7} - 32896 T_{13}^{6} + \cdots - 2873232 \)
|
|
\( T_{17}^{11} + 13 T_{17}^{10} - 30 T_{17}^{9} - 1025 T_{17}^{8} - 3174 T_{17}^{7} + 11786 T_{17}^{6} + \cdots + 191104 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 25 T^{9} + \cdots - 16 \)
|
| $5$ |
\( T^{11} + 3 T^{10} + \cdots + 608 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} + 6 T^{10} + \cdots - 1228088 \)
|
| $13$ |
\( T^{11} - 8 T^{10} + \cdots - 2873232 \)
|
| $17$ |
\( T^{11} + 13 T^{10} + \cdots + 191104 \)
|
| $19$ |
\( (T - 1)^{11} \)
|
| $23$ |
\( T^{11} + 2 T^{10} + \cdots - 840211 \)
|
| $29$ |
\( T^{11} - 16 T^{10} + \cdots + 140032 \)
|
| $31$ |
\( T^{11} - 6 T^{10} + \cdots - 53453952 \)
|
| $37$ |
\( T^{11} - 9 T^{10} + \cdots - 2133760 \)
|
| $41$ |
\( T^{11} - 4 T^{10} + \cdots - 14745600 \)
|
| $43$ |
\( T^{11} - 31 T^{10} + \cdots - 79136 \)
|
| $47$ |
\( T^{11} + \cdots + 674499103 \)
|
| $53$ |
\( T^{11} - 329 T^{9} + \cdots + 62176112 \)
|
| $59$ |
\( T^{11} + \cdots - 139677696 \)
|
| $61$ |
\( T^{11} + \cdots + 1682545288 \)
|
| $67$ |
\( T^{11} + 14 T^{10} + \cdots + 5574656 \)
|
| $71$ |
\( T^{11} + \cdots - 109996928 \)
|
| $73$ |
\( T^{11} + \cdots - 298611091 \)
|
| $79$ |
\( T^{11} + 10 T^{10} + \cdots - 7497856 \)
|
| $83$ |
\( T^{11} - 16 T^{10} + \cdots - 4664104 \)
|
| $89$ |
\( T^{11} - 20 T^{10} + \cdots + 2771456 \)
|
| $97$ |
\( T^{11} + \cdots + 15192225792 \)
|
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