Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [961,6,Mod(1,961)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("961.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 961 = 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 961.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: | \(154.128850840\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5\cdot 13 \) |
| Twist minimal: | no (minimal twist has level 31) |
| 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
\( x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 375 \nu^{7} + 7594 \nu^{6} + 28965 \nu^{5} - 752733 \nu^{4} - 7619468 \nu^{3} + 10748347 \nu^{2} + \cdots + 108553724 ) / 12432384 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 617 \nu^{7} - 19862 \nu^{6} + 62853 \nu^{5} + 3039811 \nu^{4} - 1759756 \nu^{3} + \cdots + 643617404 ) / 12432384 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 161 \nu^{7} + 1114 \nu^{6} + 17373 \nu^{5} - 180629 \nu^{4} + 97556 \nu^{3} + 6416003 \nu^{2} + \cdots + 7452188 ) / 1308672 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4621 \nu^{7} + 10926 \nu^{6} - 734361 \nu^{5} - 1111055 \nu^{4} + 35939452 \nu^{3} + \cdots + 260217876 ) / 24864768 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1891 \nu^{7} + 9170 \nu^{6} - 323607 \nu^{5} - 1272353 \nu^{4} + 16630116 \nu^{3} + \cdots - 35821076 ) / 6216192 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{7} + 7\beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} - \beta_{2} + 75\beta _1 - 31 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{7} - 7\beta_{6} + 15\beta_{5} + 29\beta_{4} + 31\beta_{3} + 106\beta_{2} - 124\beta _1 + 3585 \)
|
| \(\nu^{5}\) | \(=\) |
\( -560\beta_{7} + 897\beta_{6} + 47\beta_{5} - 339\beta_{4} - 245\beta_{3} - 221\beta_{2} + 6060\beta _1 - 4243 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2296 \beta_{7} - 1840 \beta_{6} + 2432 \beta_{5} + 4199 \beta_{4} + 5009 \beta_{3} + 10467 \beta_{2} + \cdots + 280872 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 60428 \beta_{7} + 96156 \beta_{6} - 2452 \beta_{5} - 41274 \beta_{4} - 27770 \beta_{3} + \cdots - 516454 \)
|
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 |
|
−8.19708 | −28.0551 | 35.1921 | −15.0267 | 229.970 | −14.0824 | −26.1663 | 544.091 | 123.175 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.08260 | 28.6238 | 18.1632 | −77.0065 | −202.731 | −166.782 | 98.0006 | 576.322 | 545.406 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.36806 | 5.61682 | −3.18388 | −13.6922 | −30.1515 | 106.967 | 188.869 | −211.451 | 73.5006 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.21681 | 24.0001 | −30.5194 | 80.9514 | −29.2035 | 175.330 | 76.0739 | 333.006 | −98.5022 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.51317 | −22.4697 | −29.7103 | 77.4836 | −34.0005 | 20.7014 | −93.3783 | 261.888 | 117.246 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.55453 | −5.46345 | 25.0709 | 20.6359 | −41.2738 | 223.462 | −52.3459 | −213.151 | 155.894 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.89102 | −18.9313 | 47.0502 | −33.8630 | −168.319 | −134.850 | 133.811 | 115.395 | −301.077 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.9058 | 18.6789 | 86.9372 | 88.5175 | 203.709 | −122.747 | 599.135 | 105.900 | 965.357 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(31\) | \( -1 \) |
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 | 961.6.a.c | 8 | |
| 31.b | odd | 2 | 1 | 31.6.a.b | ✓ | 8 | |
| 93.c | even | 2 | 1 | 279.6.a.f | 8 | ||
| 124.d | even | 2 | 1 | 496.6.a.h | 8 | ||
| 155.c | odd | 2 | 1 | 775.6.a.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.6.a.b | ✓ | 8 | 31.b | odd | 2 | 1 | |
| 279.6.a.f | 8 | 93.c | even | 2 | 1 | ||
| 496.6.a.h | 8 | 124.d | even | 2 | 1 | ||
| 775.6.a.b | 8 | 155.c | odd | 2 | 1 | ||
| 961.6.a.c | 8 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(961))\):
|
\( T_{2}^{8} - 7T_{2}^{7} - 178T_{2}^{6} + 903T_{2}^{5} + 10963T_{2}^{4} - 30550T_{2}^{3} - 240688T_{2}^{2} + 115128T_{2} + 420336 \)
|
|
\( T_{3}^{8} - 2 T_{3}^{7} - 1726 T_{3}^{6} + 2256 T_{3}^{5} + 959544 T_{3}^{4} - 632448 T_{3}^{3} + \cdots + 4699378944 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 7 T^{7} + \cdots + 420336 \)
|
| $3$ |
\( T^{8} + \cdots + 4699378944 \)
|
| $5$ |
\( T^{8} + \cdots + 6147184721424 \)
|
| $7$ |
\( T^{8} + \cdots + 33\!\cdots\!08 \)
|
| $11$ |
\( T^{8} + \cdots - 29\!\cdots\!08 \)
|
| $13$ |
\( T^{8} + \cdots + 18\!\cdots\!20 \)
|
| $17$ |
\( T^{8} + \cdots + 16\!\cdots\!84 \)
|
| $19$ |
\( T^{8} + \cdots - 44\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 18\!\cdots\!88 \)
|
| $29$ |
\( T^{8} + \cdots - 40\!\cdots\!00 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + \cdots - 35\!\cdots\!88 \)
|
| $41$ |
\( T^{8} + \cdots + 29\!\cdots\!52 \)
|
| $43$ |
\( T^{8} + \cdots + 33\!\cdots\!36 \)
|
| $47$ |
\( T^{8} + \cdots - 36\!\cdots\!72 \)
|
| $53$ |
\( T^{8} + \cdots + 85\!\cdots\!56 \)
|
| $59$ |
\( T^{8} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 52\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 96\!\cdots\!48 \)
|
| $71$ |
\( T^{8} + \cdots - 54\!\cdots\!80 \)
|
| $73$ |
\( T^{8} + \cdots - 28\!\cdots\!92 \)
|
| $79$ |
\( T^{8} + \cdots - 89\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 35\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots - 22\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 34\!\cdots\!36 \)
|
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