Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,4,Mod(1,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, 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("435.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 435.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: | \(25.6658308525\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 43x^{6} + 117x^{5} + 586x^{4} - 701x^{3} - 1792x^{2} + 924x + 1424 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 5\nu^{6} + 85\nu^{5} - 98\nu^{4} - 1081\nu^{3} - 143\nu^{2} + 2426\nu + 1128 ) / 136 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{7} + 43\nu^{6} + 204\nu^{5} - 1193\nu^{4} - 1837\nu^{3} + 7838\nu^{2} + 3544\nu - 9040 ) / 272 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 21\nu^{6} + 204\nu^{5} - 619\nu^{4} - 2371\nu^{3} + 3578\nu^{2} + 1968\nu - 3232 ) / 136 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} + 21\nu^{6} + 204\nu^{5} - 619\nu^{4} - 2507\nu^{3} + 3850\nu^{2} + 4824\nu - 4320 ) / 136 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 9\nu^{6} - 374\nu^{5} - 31\nu^{4} + 5883\nu^{3} + 6952\nu^{2} - 14220\nu - 15440 ) / 272 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta_{2} + 25\beta _1 + 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - 3\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + 28\beta_{2} + 86\beta _1 + 273 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{7} - 26\beta_{6} + 28\beta_{5} - 12\beta_{3} + 89\beta_{2} + 700\beta _1 + 780 \)
|
| \(\nu^{6}\) | \(=\) |
\( -48\beta_{7} - 119\beta_{6} + 93\beta_{5} + 44\beta_{4} - 96\beta_{3} + 780\beta_{2} + 3065\beta _1 + 7324 \)
|
| \(\nu^{7}\) | \(=\) |
\( -241\beta_{7} - 715\beta_{6} + 784\beta_{5} + 61\beta_{4} - 769\beta_{3} + 3208\beta_{2} + 20756\beta _1 + 29141 \)
|
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 |
|
−5.17466 | −3.00000 | 18.7771 | −5.00000 | 15.5240 | −11.1931 | −55.7681 | 9.00000 | 25.8733 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.76263 | −3.00000 | 14.6826 | −5.00000 | 14.2879 | 35.2196 | −31.8267 | 9.00000 | 23.8131 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.46965 | −3.00000 | −1.90084 | −5.00000 | 7.40894 | −36.5630 | 24.4516 | 9.00000 | 12.3482 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.85976 | −3.00000 | −4.54128 | −5.00000 | 5.57929 | −0.560700 | 23.3238 | 9.00000 | 9.29882 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.262226 | −3.00000 | −7.93124 | −5.00000 | −0.786679 | 12.1027 | −4.17759 | 9.00000 | −1.31113 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.818792 | −3.00000 | −7.32958 | −5.00000 | −2.45638 | 13.8632 | −12.5517 | 9.00000 | −4.09396 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.42731 | −3.00000 | 11.6011 | −5.00000 | −13.2819 | −2.85439 | 15.9430 | 9.00000 | −22.1365 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.75837 | −3.00000 | 14.6421 | −5.00000 | −14.2751 | −11.0144 | 31.6057 | 9.00000 | −23.7919 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(29\) | \( -1 \) |
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 | 435.4.a.k | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1305.4.a.p | 8 | ||
| 5.b | even | 2 | 1 | 2175.4.a.o | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.k | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1305.4.a.p | 8 | 3.b | odd | 2 | 1 | ||
| 2175.4.a.o | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 4T_{2}^{7} - 43T_{2}^{6} - 169T_{2}^{5} + 456T_{2}^{4} + 1869T_{2}^{3} + 90T_{2}^{2} - 2112T_{2} + 512 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(435))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 4 T^{7} + \cdots + 512 \)
|
| $3$ |
\( (T + 3)^{8} \)
|
| $5$ |
\( (T + 5)^{8} \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots - 42630912 \)
|
| $11$ |
\( T^{8} + \cdots - 109041848480 \)
|
| $13$ |
\( T^{8} + \cdots + 791998533632 \)
|
| $17$ |
\( T^{8} + \cdots + 13781129317504 \)
|
| $19$ |
\( T^{8} + \cdots + 66835834880 \)
|
| $23$ |
\( T^{8} + \cdots - 22\!\cdots\!16 \)
|
| $29$ |
\( (T - 29)^{8} \)
|
| $31$ |
\( T^{8} + \cdots + 33\!\cdots\!52 \)
|
| $37$ |
\( T^{8} + \cdots - 22\!\cdots\!04 \)
|
| $41$ |
\( T^{8} + \cdots - 10\!\cdots\!20 \)
|
| $43$ |
\( T^{8} + \cdots - 11\!\cdots\!40 \)
|
| $47$ |
\( T^{8} + \cdots + 48\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots - 79\!\cdots\!92 \)
|
| $59$ |
\( T^{8} + \cdots + 17\!\cdots\!88 \)
|
| $61$ |
\( T^{8} + \cdots - 19\!\cdots\!08 \)
|
| $67$ |
\( T^{8} + \cdots - 21\!\cdots\!04 \)
|
| $71$ |
\( T^{8} + \cdots + 43\!\cdots\!24 \)
|
| $73$ |
\( T^{8} + \cdots - 15\!\cdots\!24 \)
|
| $79$ |
\( T^{8} + \cdots + 89\!\cdots\!28 \)
|
| $83$ |
\( T^{8} + \cdots + 14\!\cdots\!44 \)
|
| $89$ |
\( T^{8} + \cdots + 73\!\cdots\!60 \)
|
| $97$ |
\( T^{8} + \cdots + 70\!\cdots\!80 \)
|
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