Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6171,2,Mod(1,6171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6171, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6171.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6171 = 3 \cdot 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6171.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: | \(49.2756830873\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.14034713125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 16x^{6} + 20x^{5} + 74x^{4} - 97x^{3} - 110x^{2} + 135x + 19 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 561) |
| 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
\( x^{8} - x^{7} - 16x^{6} + 20x^{5} + 74x^{4} - 97x^{3} - 110x^{2} + 135x + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} + 18\nu^{5} + 31\nu^{4} - 125\nu^{3} - 67\nu^{2} + 255\nu + 17 ) / 35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} + 25\nu^{5} - 39\nu^{4} - 132\nu^{3} + 87\nu^{2} + 171\nu + 10 ) / 35 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{7} + 5\nu^{6} - 44\nu^{5} - 19\nu^{4} + 122\nu^{3} - 61\nu^{2} - 61\nu + 149 ) / 35 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -12\nu^{7} - 15\nu^{6} + 167\nu^{5} + 127\nu^{4} - 716\nu^{3} - 307\nu^{2} + 953\nu + 148 ) / 35 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -13\nu^{7} - 11\nu^{6} + 192\nu^{5} + 88\nu^{4} - 848\nu^{3} - 185\nu^{2} + 1124\nu + 18 ) / 35 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -17\nu^{7} - 9\nu^{6} + 243\nu^{5} + 37\nu^{4} - 1012\nu^{3} - 40\nu^{2} + 1276\nu + 37 ) / 35 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} - \beta_{2} + 5\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 10\beta_{6} - 10\beta_{5} + 3\beta_{4} - 13\beta_{3} - 2\beta_{2} - 2\beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{7} - 16\beta_{6} + 27\beta_{5} - 13\beta_{4} + 42\beta_{3} - 6\beta_{2} + 32\beta _1 - 47 \)
|
| \(\nu^{6}\) | \(=\) |
\( 21\beta_{7} + 93\beta_{6} - 103\beta_{5} + 42\beta_{4} - 142\beta_{3} - 20\beta_{2} - 35\beta _1 + 237 \)
|
| \(\nu^{7}\) | \(=\) |
\( -123\beta_{7} - 199\beta_{6} + 302\beta_{5} - 142\beta_{4} + 471\beta_{3} - 20\beta_{2} + 249\beta _1 - 565 \)
|
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 |
|
−2.19353 | −1.00000 | 2.81156 | −2.46655 | 2.19353 | 2.20625 | −1.78018 | 1.00000 | 5.41044 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.19353 | −1.00000 | 2.81156 | 0.0106593 | 2.19353 | −0.750362 | −1.78018 | 1.00000 | −0.0233815 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.29496 | −1.00000 | −0.323071 | −4.14920 | 1.29496 | 1.70112 | 3.00829 | 1.00000 | 5.37305 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.29496 | −1.00000 | −0.323071 | 1.37697 | 1.29496 | 0.0711042 | 3.00829 | 1.00000 | −1.78313 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.294963 | −1.00000 | −1.91300 | −1.20855 | −0.294963 | −3.66032 | −1.15419 | 1.00000 | −0.356476 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.294963 | −1.00000 | −1.91300 | 2.59880 | −0.294963 | 1.27006 | −1.15419 | 1.00000 | 0.766551 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.19353 | −1.00000 | −0.575493 | −2.14864 | −1.19353 | −3.35742 | −3.07392 | 1.00000 | −2.56446 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.19353 | −1.00000 | −0.575493 | 0.986489 | −1.19353 | 3.51956 | −3.07392 | 1.00000 | 1.17740 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 6171.2.a.bf | 8 | |
| 11.b | odd | 2 | 1 | 6171.2.a.bi | 8 | ||
| 11.d | odd | 10 | 2 | 561.2.m.c | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 561.2.m.c | ✓ | 16 | 11.d | odd | 10 | 2 | |
| 6171.2.a.bf | 8 | 1.a | even | 1 | 1 | trivial | |
| 6171.2.a.bi | 8 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 2T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{5}^{8} + 5T_{5}^{7} - 7T_{5}^{6} - 51T_{5}^{5} - T_{5}^{4} + 131T_{5}^{3} + 16T_{5}^{2} - 94T_{5} + 1 \)
|
|
\( T_{7}^{8} - T_{7}^{7} - 23T_{7}^{6} + 32T_{7}^{5} + 132T_{7}^{4} - 253T_{7}^{3} - 15T_{7}^{2} + 157T_{7} - 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} - 2 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + 5 T^{7} + \cdots + 1 \)
|
| $7$ |
\( T^{8} - T^{7} + \cdots - 11 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 10 T^{7} + \cdots + 1831 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} + 2 T^{7} + \cdots + 891 \)
|
| $23$ |
\( T^{8} - 19 T^{7} + \cdots - 1331 \)
|
| $29$ |
\( T^{8} + 6 T^{7} + \cdots + 4279 \)
|
| $31$ |
\( T^{8} + 12 T^{7} + \cdots - 492725 \)
|
| $37$ |
\( T^{8} - 5 T^{7} + \cdots + 509 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots - 149581 \)
|
| $43$ |
\( T^{8} - 18 T^{7} + \cdots - 161051 \)
|
| $47$ |
\( T^{8} - T^{7} + \cdots - 521 \)
|
| $53$ |
\( T^{8} + 26 T^{7} + \cdots + 47179 \)
|
| $59$ |
\( T^{8} + 22 T^{7} + \cdots + 175351 \)
|
| $61$ |
\( T^{8} - 4 T^{7} + \cdots + 306641 \)
|
| $67$ |
\( T^{8} - 4 T^{7} + \cdots + 17260771 \)
|
| $71$ |
\( T^{8} - 24 T^{7} + \cdots + 5035469 \)
|
| $73$ |
\( T^{8} - 17 T^{7} + \cdots + 176779 \)
|
| $79$ |
\( T^{8} + 13 T^{7} + \cdots - 325249 \)
|
| $83$ |
\( T^{8} - 5 T^{7} + \cdots - 55016269 \)
|
| $89$ |
\( T^{8} + 12 T^{7} + \cdots + 97261 \)
|
| $97$ |
\( T^{8} + 27 T^{7} + \cdots - 16982971 \)
|
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