Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2254,4,Mod(1,2254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2254.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2254.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: | \(132.990305153\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 178 x^{9} - 29 x^{8} + 11409 x^{7} + 5153 x^{6} - 327598 x^{5} - 270393 x^{4} + 4277164 x^{3} + \cdots - 33055073 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 322) |
| 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
\( x^{11} - 178 x^{9} - 29 x^{8} + 11409 x^{7} + 5153 x^{6} - 327598 x^{5} - 270393 x^{4} + 4277164 x^{3} + \cdots - 33055073 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 894730436 \nu^{10} + 22340866396 \nu^{9} - 113873204817 \nu^{8} - 3696821733796 \nu^{7} + \cdots + 79\!\cdots\!13 ) / 247483648506465 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3367525582 \nu^{10} + 84689834653 \nu^{9} + 521736289584 \nu^{8} + \cdots + 18\!\cdots\!84 ) / 742450945519395 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1210343378 \nu^{10} + 3533447093 \nu^{9} + 172671180807 \nu^{8} - 516669798044 \nu^{7} + \cdots - 16\!\cdots\!18 ) / 148490189103879 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 9699812414 \nu^{10} + 17970313301 \nu^{9} + 1626206213568 \nu^{8} + \cdots - 14\!\cdots\!52 ) / 742450945519395 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1401849014 \nu^{10} + 10498739203 \nu^{9} + 235670829176 \nu^{8} - 1686231730597 \nu^{7} + \cdots + 29\!\cdots\!84 ) / 49496729701293 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 26898920738 \nu^{10} + 97647142948 \nu^{9} - 4247574688626 \nu^{8} - 16789859715403 \nu^{7} + \cdots + 26\!\cdots\!59 ) / 742450945519395 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 9871258094 \nu^{10} + 16438387241 \nu^{9} + 1617580714560 \nu^{8} + \cdots + 26\!\cdots\!65 ) / 148490189103879 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 19755070309 \nu^{10} - 71185717561 \nu^{9} - 3281784074643 \nu^{8} + 11219304530896 \nu^{7} + \cdots - 19\!\cdots\!13 ) / 247483648506465 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 83140076393 \nu^{10} - 43653383927 \nu^{9} - 13865417081346 \nu^{8} + \cdots - 34\!\cdots\!21 ) / 742450945519395 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 33 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{9} + \beta_{8} + \beta_{7} + 5\beta_{6} - \beta_{5} - 3\beta_{3} - 4\beta_{2} + 47\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{10} - 7 \beta_{9} - 8 \beta_{8} - 13 \beta_{7} + 12 \beta_{6} + 6 \beta_{5} + 76 \beta_{4} + \cdots + 1662 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 13 \beta_{10} + 199 \beta_{9} + 146 \beta_{8} + 124 \beta_{7} + 387 \beta_{6} - 148 \beta_{5} + \cdots + 144 \)
|
| \(\nu^{6}\) | \(=\) |
\( 866 \beta_{10} - 1012 \beta_{9} - 890 \beta_{8} - 1395 \beta_{7} + 1111 \beta_{6} + 665 \beta_{5} + \cdots + 100201 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1480 \beta_{10} + 16418 \beta_{9} + 14377 \beta_{8} + 10743 \beta_{7} + 26176 \beta_{6} + \cdots - 30330 \)
|
| \(\nu^{8}\) | \(=\) |
\( 79682 \beta_{10} - 103552 \beta_{9} - 77990 \beta_{8} - 116535 \beta_{7} + 78277 \beta_{6} + \cdots + 6513465 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 131365 \beta_{10} + 1280149 \beta_{9} + 1239167 \beta_{8} + 854206 \beta_{7} + 1739427 \beta_{6} + \cdots - 4834334 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6598968 \beta_{10} - 9225737 \beta_{9} - 6355162 \beta_{8} - 9066898 \beta_{7} + 5039062 \beta_{6} + \cdots + 439414551 \)
|
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 |
|
−2.00000 | −8.65181 | 4.00000 | −8.28906 | 17.3036 | 0 | −8.00000 | 47.8538 | 16.5781 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −6.92791 | 4.00000 | 18.3858 | 13.8558 | 0 | −8.00000 | 20.9959 | −36.7717 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −4.68259 | 4.00000 | −19.8302 | 9.36518 | 0 | −8.00000 | −5.07336 | 39.6604 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −3.62203 | 4.00000 | 11.6858 | 7.24405 | 0 | −8.00000 | −13.8809 | −23.3717 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −3.17667 | 4.00000 | 0.0260643 | 6.35334 | 0 | −8.00000 | −16.9088 | −0.0521286 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | −1.96532 | 4.00000 | −4.37870 | 3.93063 | 0 | −8.00000 | −23.1375 | 8.75740 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | 3.85892 | 4.00000 | 15.2739 | −7.71785 | 0 | −8.00000 | −12.1087 | −30.5477 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | 4.04898 | 4.00000 | −5.73149 | −8.09796 | 0 | −8.00000 | −10.6058 | 11.4630 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.00000 | 5.44340 | 4.00000 | −7.06685 | −10.8868 | 0 | −8.00000 | 2.63057 | 14.1337 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −2.00000 | 7.40070 | 4.00000 | 3.86030 | −14.8014 | 0 | −8.00000 | 27.7703 | −7.72059 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −2.00000 | 8.27433 | 4.00000 | −20.9356 | −16.5487 | 0 | −8.00000 | 41.4645 | 41.8712 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 2254.4.a.s | 11 | |
| 7.b | odd | 2 | 1 | 2254.4.a.t | 11 | ||
| 7.d | odd | 6 | 2 | 322.4.e.c | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.e.c | ✓ | 22 | 7.d | odd | 6 | 2 | |
| 2254.4.a.s | 11 | 1.a | even | 1 | 1 | trivial | |
| 2254.4.a.t | 11 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{11} - 178 T_{3}^{9} - 29 T_{3}^{8} + 11409 T_{3}^{7} + 5153 T_{3}^{6} - 327598 T_{3}^{5} + \cdots - 33055073 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{11} \)
|
| $3$ |
\( T^{11} - 178 T^{9} + \cdots - 33055073 \)
|
| $5$ |
\( T^{11} + \cdots - 201518793 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} + \cdots + 11\!\cdots\!25 \)
|
| $13$ |
\( T^{11} + \cdots - 10\!\cdots\!67 \)
|
| $17$ |
\( T^{11} + \cdots - 62\!\cdots\!61 \)
|
| $19$ |
\( T^{11} + \cdots + 24\!\cdots\!29 \)
|
| $23$ |
\( (T + 23)^{11} \)
|
| $29$ |
\( T^{11} + \cdots + 80\!\cdots\!55 \)
|
| $31$ |
\( T^{11} + \cdots - 22\!\cdots\!25 \)
|
| $37$ |
\( T^{11} + \cdots - 15\!\cdots\!23 \)
|
| $41$ |
\( T^{11} + \cdots - 14\!\cdots\!51 \)
|
| $43$ |
\( T^{11} + \cdots + 22\!\cdots\!55 \)
|
| $47$ |
\( T^{11} + \cdots - 54\!\cdots\!27 \)
|
| $53$ |
\( T^{11} + \cdots + 16\!\cdots\!23 \)
|
| $59$ |
\( T^{11} + \cdots - 79\!\cdots\!99 \)
|
| $61$ |
\( T^{11} + \cdots - 94\!\cdots\!93 \)
|
| $67$ |
\( T^{11} + \cdots - 12\!\cdots\!91 \)
|
| $71$ |
\( T^{11} + \cdots + 31\!\cdots\!13 \)
|
| $73$ |
\( T^{11} + \cdots - 90\!\cdots\!45 \)
|
| $79$ |
\( T^{11} + \cdots - 24\!\cdots\!75 \)
|
| $83$ |
\( T^{11} + \cdots - 53\!\cdots\!33 \)
|
| $89$ |
\( T^{11} + \cdots - 93\!\cdots\!05 \)
|
| $97$ |
\( T^{11} + \cdots - 85\!\cdots\!95 \)
|
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