Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,8,Mod(1,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 29.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: | \(9.05916573904\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 616x^{5} + 1864x^{4} + 96785x^{3} - 257817x^{2} - 3929114x + 2682946 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 616x^{5} + 1864x^{4} + 96785x^{3} - 257817x^{2} - 3929114x + 2682946 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{6} + 8\nu^{5} + 4336\nu^{4} - 6904\nu^{3} - 520799\nu^{2} + 364840\nu - 1397186 ) / 275968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 68\nu^{5} + 500\nu^{4} + 31580\nu^{3} - 112485\nu^{2} - 2531648\nu + 498330 ) / 137984 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\nu^{6} + 384\nu^{5} - 11080\nu^{4} - 119488\nu^{3} + 951099\nu^{2} + 7649544\nu - 3144918 ) / 275968 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 15\nu^{5} - 1209\nu^{4} - 6169\nu^{3} + 192817\nu^{2} + 645190\nu - 5881078 ) / 34496 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15\nu^{6} - 80\nu^{5} - 8952\nu^{4} + 73968\nu^{3} + 1365855\nu^{2} - 9863224\nu - 53289950 ) / 275968 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 2\beta_{2} - 3\beta _1 + 177 \)
|
| \(\nu^{3}\) | \(=\) |
\( 11\beta_{6} - 16\beta_{5} - \beta_{4} - 24\beta_{3} - 10\beta_{2} + 293\beta _1 - 579 \)
|
| \(\nu^{4}\) | \(=\) |
\( -110\beta_{6} + 421\beta_{5} + 66\beta_{4} + 669\beta_{3} + 790\beta_{2} - 2405\beta _1 + 52869 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5371\beta_{6} - 8814\beta_{5} - 585\beta_{4} - 15006\beta_{3} - 6606\beta_{2} + 106441\beta _1 - 451429 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 72848 \beta_{6} + 192087 \beta_{5} + 41200 \beta_{4} + 346519 \beta_{3} + 303438 \beta_{2} + \cdots + 19435337 \)
|
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 |
|
−17.4977 | −74.2053 | 178.170 | 273.079 | 1298.42 | 59.8847 | −877.860 | 3319.43 | −4778.26 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −15.3447 | −11.2656 | 107.460 | 20.3214 | 172.867 | 1114.59 | 315.178 | −2060.09 | −311.826 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −10.8581 | 69.4472 | −10.1010 | −257.927 | −754.067 | −96.3589 | 1499.52 | 2635.91 | 2800.60 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.66133 | −6.32051 | −125.240 | 488.146 | 10.5005 | −620.024 | 420.716 | −2147.05 | −810.974 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.63803 | 43.0759 | −96.2126 | −368.912 | 242.863 | −945.236 | −1264.12 | −331.464 | −2079.94 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 11.1332 | −21.8586 | −4.05193 | −88.9174 | −243.356 | −61.2811 | −1470.16 | −1709.20 | −989.935 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 20.5907 | −80.8732 | 295.975 | −385.790 | −1665.23 | −1155.57 | 3458.72 | 4353.47 | −7943.68 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(-1\) |
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 | 29.8.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 261.8.a.b | 7 | ||
| 4.b | odd | 2 | 1 | 464.8.a.f | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.8.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 261.8.a.b | 7 | 3.b | odd | 2 | 1 | ||
| 464.8.a.f | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 8T_{2}^{6} - 589T_{2}^{5} - 4894T_{2}^{4} + 83224T_{2}^{3} + 530864T_{2}^{2} - 3133648T_{2} - 6259936 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 8 T^{6} + \cdots - 6259936 \)
|
| $3$ |
\( T^{7} + \cdots + 27941895432 \)
|
| $5$ |
\( T^{7} + \cdots - 88\!\cdots\!50 \)
|
| $7$ |
\( T^{7} + \cdots + 26\!\cdots\!52 \)
|
| $11$ |
\( T^{7} + \cdots + 39\!\cdots\!12 \)
|
| $13$ |
\( T^{7} + \cdots - 21\!\cdots\!46 \)
|
| $17$ |
\( T^{7} + \cdots - 11\!\cdots\!16 \)
|
| $19$ |
\( T^{7} + \cdots + 19\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 23\!\cdots\!16 \)
|
| $29$ |
\( (T - 24389)^{7} \)
|
| $31$ |
\( T^{7} + \cdots - 11\!\cdots\!28 \)
|
| $37$ |
\( T^{7} + \cdots + 19\!\cdots\!16 \)
|
| $41$ |
\( T^{7} + \cdots + 78\!\cdots\!48 \)
|
| $43$ |
\( T^{7} + \cdots + 64\!\cdots\!16 \)
|
| $47$ |
\( T^{7} + \cdots - 51\!\cdots\!68 \)
|
| $53$ |
\( T^{7} + \cdots - 34\!\cdots\!58 \)
|
| $59$ |
\( T^{7} + \cdots + 35\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots - 29\!\cdots\!68 \)
|
| $67$ |
\( T^{7} + \cdots + 68\!\cdots\!12 \)
|
| $71$ |
\( T^{7} + \cdots + 79\!\cdots\!76 \)
|
| $73$ |
\( T^{7} + \cdots - 98\!\cdots\!88 \)
|
| $79$ |
\( T^{7} + \cdots + 42\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 44\!\cdots\!76 \)
|
| $89$ |
\( T^{7} + \cdots - 39\!\cdots\!40 \)
|
| $97$ |
\( T^{7} + \cdots + 22\!\cdots\!24 \)
|
| show more | |
| show less |