Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,4,Mod(1,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, 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("177.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(10.4433380710\) |
| 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: |
\( x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 27\nu^{5} - 154\nu^{4} - 783\nu^{3} + 2919\nu^{2} + 4790\nu - 7944 ) / 584 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{6} - 30\nu^{5} - 202\nu^{4} + 797\nu^{3} + 1380\nu^{2} - 3992\nu - 1296 ) / 584 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{6} + 49\nu^{5} + 218\nu^{4} - 1129\nu^{3} - 867\nu^{2} + 3610\nu - 920 ) / 584 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 24\nu^{6} - 9\nu^{5} - 776\nu^{4} + 42\nu^{3} + 6473\nu^{2} + 2102\nu - 10784 ) / 584 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -15\nu^{6} + 33\nu^{5} + 412\nu^{4} - 811\nu^{3} - 2175\nu^{2} + 3340\nu - 852 ) / 292 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + 4\beta_{3} - \beta_{2} + 17\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 22\beta_{6} + 24\beta_{5} + 16\beta_{3} - 28\beta_{2} + \beta _1 + 162 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 9\beta_{5} + 86\beta_{4} + 89\beta_{3} - 35\beta_{2} + 320\beta _1 + 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( 442\beta_{6} + 534\beta_{5} + 27\beta_{4} + 274\beta_{3} - 647\beta_{2} + 35\beta _1 + 3000 \)
|
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 |
|
−5.60512 | 3.00000 | 23.4174 | 12.9748 | −16.8154 | −18.2988 | −86.4162 | 9.00000 | −72.7251 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.58179 | 3.00000 | 12.9928 | −12.2965 | −13.7454 | 15.3493 | −22.8759 | 9.00000 | 56.3401 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.46227 | 3.00000 | −1.93724 | −11.2702 | −7.38680 | 18.6588 | 24.4681 | 9.00000 | 27.7502 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.77500 | 3.00000 | −4.84937 | 6.23028 | −5.32500 | −18.0779 | 22.8076 | 9.00000 | −11.0588 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.29817 | 3.00000 | −6.31476 | 7.05496 | 3.89451 | −33.4497 | −18.5830 | 9.00000 | 9.15853 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.68175 | 3.00000 | −5.17172 | −9.78761 | 5.04525 | 6.87552 | −22.1515 | 9.00000 | −16.4603 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.44426 | 3.00000 | 3.86292 | −20.9057 | 10.3328 | −30.0572 | −14.2492 | 9.00000 | −72.0046 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(1\) |
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 | 177.4.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 531.4.a.d | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.4.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 531.4.a.d | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 8T_{2}^{6} - 7T_{2}^{5} - 145T_{2}^{4} - 70T_{2}^{3} + 637T_{2}^{2} + 244T_{2} - 844 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 8 T^{6} + \cdots - 844 \)
|
| $3$ |
\( (T - 3)^{7} \)
|
| $5$ |
\( T^{7} + 28 T^{6} + \cdots - 16171712 \)
|
| $7$ |
\( T^{7} + 59 T^{6} + \cdots - 654921472 \)
|
| $11$ |
\( T^{7} + \cdots + 35342635856 \)
|
| $13$ |
\( T^{7} + \cdots + 17075623568 \)
|
| $17$ |
\( T^{7} + \cdots + 83907218653452 \)
|
| $19$ |
\( T^{7} + \cdots - 83214285665216 \)
|
| $23$ |
\( T^{7} + \cdots - 498329696312416 \)
|
| $29$ |
\( T^{7} + \cdots + 102487884628576 \)
|
| $31$ |
\( T^{7} + \cdots + 71281602727072 \)
|
| $37$ |
\( T^{7} + \cdots - 21\!\cdots\!32 \)
|
| $41$ |
\( T^{7} + \cdots - 15\!\cdots\!84 \)
|
| $43$ |
\( T^{7} + \cdots - 67\!\cdots\!12 \)
|
| $47$ |
\( T^{7} + \cdots - 290351607809472 \)
|
| $53$ |
\( T^{7} + \cdots + 15\!\cdots\!64 \)
|
| $59$ |
\( (T + 59)^{7} \)
|
| $61$ |
\( T^{7} + \cdots + 479182635497312 \)
|
| $67$ |
\( T^{7} + \cdots + 94\!\cdots\!28 \)
|
| $71$ |
\( T^{7} + \cdots - 11\!\cdots\!48 \)
|
| $73$ |
\( T^{7} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{7} + \cdots + 11\!\cdots\!68 \)
|
| $83$ |
\( T^{7} + \cdots + 31\!\cdots\!56 \)
|
| $89$ |
\( T^{7} + \cdots - 35\!\cdots\!48 \)
|
| $97$ |
\( T^{7} + \cdots - 14\!\cdots\!96 \)
|
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