Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1334,2,Mod(1,1334)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1334, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1334.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1334 = 2 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1334.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: | \(10.6520436296\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5x^{9} - 8x^{8} + 60x^{7} + 13x^{6} - 241x^{5} + 6x^{4} + 346x^{3} + 16x^{2} - 64x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5x^{9} - 8x^{8} + 60x^{7} + 13x^{6} - 241x^{5} + 6x^{4} + 346x^{3} + 16x^{2} - 64x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 6\nu^{8} - 4\nu^{7} + 70\nu^{6} - 29\nu^{5} - 276\nu^{4} + 132\nu^{3} + 392\nu^{2} - 80\nu - 52 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + 5\nu^{8} + 8\nu^{7} - 61\nu^{6} - 8\nu^{5} + 246\nu^{4} - 51\nu^{3} - 342\nu^{2} + 84\nu + 48 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{9} + 11\nu^{8} + 11\nu^{7} - 127\nu^{6} + 29\nu^{5} + 488\nu^{4} - 206\nu^{3} - 664\nu^{2} + 196\nu + 88 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{9} + 16\nu^{8} + 20\nu^{7} - 192\nu^{6} + 13\nu^{5} + 768\nu^{4} - 230\nu^{3} - 1080\nu^{2} + 228\nu + 148 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{9} - 21\nu^{8} - 25\nu^{7} + 239\nu^{6} - 23\nu^{5} - 890\nu^{4} + 296\nu^{3} + 1148\nu^{2} - 308\nu - 148 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -2\nu^{9} + 11\nu^{8} + 11\nu^{7} - 126\nu^{6} + 26\nu^{5} + 479\nu^{4} - 181\nu^{3} - 646\nu^{2} + 150\nu + 90 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11 \nu^{9} - 57 \nu^{8} - 75 \nu^{7} + 665 \nu^{6} - 10 \nu^{5} - 2560 \nu^{4} + 682 \nu^{3} + \cdots - 448 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 2\beta_{6} + \beta_{5} - 2\beta_{4} + 4\beta_{3} + 8\beta_{2} + 11\beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + 2\beta_{8} + 14\beta_{6} + \beta_{5} - 12\beta_{4} + 17\beta_{3} + 16\beta_{2} + 47\beta _1 + 55 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3 \beta_{9} + 17 \beta_{8} + 9 \beta_{7} + 35 \beta_{6} + 8 \beta_{5} - 29 \beta_{4} + 62 \beta_{3} + \cdots + 243 \)
|
| \(\nu^{7}\) | \(=\) |
\( 20 \beta_{9} + 50 \beta_{8} + 2 \beta_{7} + 161 \beta_{6} + 2 \beta_{5} - 119 \beta_{4} + 221 \beta_{3} + \cdots + 636 \)
|
| \(\nu^{8}\) | \(=\) |
\( 70 \beta_{9} + 249 \beta_{8} + 59 \beta_{7} + 462 \beta_{6} + 13 \beta_{5} - 316 \beta_{4} + 770 \beta_{3} + \cdots + 2376 \)
|
| \(\nu^{9}\) | \(=\) |
\( 319 \beta_{9} + 838 \beta_{8} + 8 \beta_{7} + 1792 \beta_{6} - 169 \beta_{5} - 1110 \beta_{4} + \cdots + 7069 \)
|
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 |
|
1.00000 | −2.37864 | 1.00000 | 0.983407 | −2.37864 | −3.11298 | 1.00000 | 2.65792 | 0.983407 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.05371 | 1.00000 | −4.01245 | −2.05371 | −3.53542 | 1.00000 | 1.21772 | −4.01245 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.20998 | 1.00000 | −1.11783 | −1.20998 | −0.129988 | 1.00000 | −1.53596 | −1.11783 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.02950 | 1.00000 | 2.59754 | −1.02950 | 4.22406 | 1.00000 | −1.94012 | 2.59754 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.478690 | 1.00000 | −3.82976 | 0.478690 | 4.18279 | 1.00000 | −2.77086 | −3.82976 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.26719 | 1.00000 | 1.36043 | 1.26719 | 1.67775 | 1.00000 | −1.39423 | 1.36043 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 1.29016 | 1.00000 | 4.10547 | 1.29016 | −1.09974 | 1.00000 | −1.33548 | 4.10547 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.61750 | 1.00000 | 0.316103 | 2.61750 | −4.23734 | 1.00000 | 3.85129 | 0.316103 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 2.74474 | 1.00000 | 1.89466 | 2.74474 | 2.43465 | 1.00000 | 4.53361 | 1.89466 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 3.27355 | 1.00000 | −3.29757 | 3.27355 | 1.59623 | 1.00000 | 7.71610 | −3.29757 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(23\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 1334.2.a.k | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1334.2.a.k | ✓ | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):
|
\( T_{3}^{10} - 5T_{3}^{9} - 8T_{3}^{8} + 64T_{3}^{7} - T_{3}^{6} - 271T_{3}^{5} + 116T_{3}^{4} + 428T_{3}^{3} - 224T_{3}^{2} - 224T_{3} + 112 \)
|
|
\( T_{5}^{10} + T_{5}^{9} - 36 T_{5}^{8} - 16 T_{5}^{7} + 433 T_{5}^{6} - 119 T_{5}^{5} - 1942 T_{5}^{4} + \cdots + 484 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{10} \)
|
| $3$ |
\( T^{10} - 5 T^{9} + \cdots + 112 \)
|
| $5$ |
\( T^{10} + T^{9} + \cdots + 484 \)
|
| $7$ |
\( T^{10} - 2 T^{9} + \cdots - 768 \)
|
| $11$ |
\( T^{10} - T^{9} + \cdots - 288 \)
|
| $13$ |
\( T^{10} - 9 T^{9} + \cdots + 1568 \)
|
| $17$ |
\( T^{10} - 62 T^{8} + \cdots + 11944 \)
|
| $19$ |
\( T^{10} - 20 T^{9} + \cdots - 3816320 \)
|
| $23$ |
\( (T - 1)^{10} \)
|
| $29$ |
\( (T - 1)^{10} \)
|
| $31$ |
\( T^{10} - 21 T^{9} + \cdots + 665088 \)
|
| $37$ |
\( T^{10} + 8 T^{9} + \cdots + 13526272 \)
|
| $41$ |
\( T^{10} + 4 T^{9} + \cdots + 29042496 \)
|
| $43$ |
\( T^{10} + 3 T^{9} + \cdots - 3872 \)
|
| $47$ |
\( T^{10} + T^{9} + \cdots - 384 \)
|
| $53$ |
\( T^{10} + 13 T^{9} + \cdots + 24284932 \)
|
| $59$ |
\( T^{10} - 14 T^{9} + \cdots - 61239296 \)
|
| $61$ |
\( T^{10} - 12 T^{9} + \cdots - 4864512 \)
|
| $67$ |
\( T^{10} + \cdots + 209133512 \)
|
| $71$ |
\( T^{10} - 8 T^{9} + \cdots + 1443840 \)
|
| $73$ |
\( T^{10} - 16 T^{9} + \cdots - 7603392 \)
|
| $79$ |
\( T^{10} - 7 T^{9} + \cdots + 9788 \)
|
| $83$ |
\( T^{10} - 18 T^{9} + \cdots - 16748704 \)
|
| $89$ |
\( T^{10} + \cdots - 541360056 \)
|
| $97$ |
\( T^{10} + \cdots + 158407200 \)
|
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