Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [31,10,Mod(1,31)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.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: | \(15.9661109211\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3 x^{9} - 3696 x^{8} + 756 x^{7} + 4729174 x^{6} + 8503590 x^{5} - 2447393116 x^{4} + \cdots - 14818099999228 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 23 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3 x^{9} - 3696 x^{8} + 756 x^{7} + 4729174 x^{6} + 8503590 x^{5} - 2447393116 x^{4} + \cdots - 14818099999228 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 739 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 95347897226425 \nu^{9} + \cdots + 38\!\cdots\!28 ) / 51\!\cdots\!96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!97 \nu^{9} + \cdots + 28\!\cdots\!76 ) / 28\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 976309082175181 \nu^{9} + \cdots + 35\!\cdots\!52 ) / 14\!\cdots\!64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 633171766216563 \nu^{9} + \cdots - 51\!\cdots\!12 ) / 47\!\cdots\!88 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 31\!\cdots\!05 \nu^{9} + \cdots - 39\!\cdots\!92 ) / 18\!\cdots\!52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 158100092582003 \nu^{9} - 1623885479930 \nu^{8} + \cdots - 72\!\cdots\!92 ) / 85\!\cdots\!16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 328688198239295 \nu^{9} + \cdots + 40\!\cdots\!72 ) / 17\!\cdots\!08 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 739 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{9} - 4\beta_{8} + 5\beta_{7} + \beta_{5} - \beta_{4} - 39\beta_{3} + 8\beta_{2} + 1149\beta _1 + 2759 \)
|
| \(\nu^{4}\) | \(=\) |
\( 120 \beta_{9} - 4 \beta_{8} - 23 \beta_{7} - 12 \beta_{6} + 49 \beta_{5} - 61 \beta_{4} - 739 \beta_{3} + \cdots + 849998 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6488 \beta_{9} - 5368 \beta_{8} + 9274 \beta_{7} - 1616 \beta_{6} + 4690 \beta_{5} - 4258 \beta_{4} + \cdots + 7338662 \)
|
| \(\nu^{6}\) | \(=\) |
\( 250768 \beta_{9} + 37816 \beta_{8} - 47306 \beta_{7} - 78696 \beta_{6} + 114326 \beta_{5} + \cdots + 1132959845 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9487452 \beta_{9} - 5025900 \beta_{8} + 14491779 \beta_{7} - 5203488 \beta_{6} + 11018615 \beta_{5} + \cdots + 15559027265 \)
|
| \(\nu^{8}\) | \(=\) |
\( 421955080 \beta_{9} + 134138100 \beta_{8} - 66146905 \beta_{7} - 246936660 \beta_{6} + \cdots + 1634280887240 \)
|
| \(\nu^{9}\) | \(=\) |
\( 13907262992 \beta_{9} - 2544274912 \beta_{8} + 21581705200 \beta_{7} - 12301996208 \beta_{6} + \cdots + 30327051957256 \)
|
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 |
|
−44.8360 | 245.176 | 1498.27 | −1315.71 | −10992.7 | −892.479 | −44220.3 | 40428.1 | 58991.3 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −37.9842 | −159.973 | 930.801 | −2781.75 | 6076.43 | 5528.23 | −15907.8 | 5908.21 | 105663. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −28.4831 | 124.497 | 299.286 | 80.3284 | −3546.07 | −5048.52 | 6058.75 | −4183.42 | −2288.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −20.2746 | −215.536 | −100.939 | 617.887 | 4369.92 | 1464.39 | 12427.1 | 26772.8 | −12527.4 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −7.84470 | −24.3740 | −450.461 | 461.457 | 191.207 | 12121.0 | 7550.21 | −19088.9 | −3619.99 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.70467 | 105.830 | −479.457 | 432.188 | 603.723 | −1120.62 | −5655.93 | −8483.10 | 2465.49 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 14.5425 | 180.349 | −300.515 | −689.673 | 2622.73 | −10283.2 | −11816.0 | 12842.9 | −10029.6 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 23.4522 | −191.817 | 38.0068 | 2312.28 | −4498.54 | 1476.60 | −11116.2 | 17110.8 | 54228.2 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 29.0262 | 16.1429 | 330.521 | −2277.55 | 468.566 | 6354.33 | −5267.65 | −19422.4 | −66108.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 33.6970 | −88.2948 | 623.490 | −136.454 | −2975.27 | −10094.7 | 3756.86 | −11887.0 | −4598.10 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 31.10.a.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 279.10.a.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.10.a.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 279.10.a.c | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 33 T_{2}^{9} - 3210 T_{2}^{8} - 85248 T_{2}^{7} + 3805720 T_{2}^{6} + 70982160 T_{2}^{5} + \cdots - 14681900400640 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(31))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 14681900400640 \)
|
| $3$ |
\( T^{10} + \cdots - 13\!\cdots\!96 \)
|
| $5$ |
\( T^{10} + \cdots - 17\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 48\!\cdots\!80 \)
|
| $11$ |
\( T^{10} + \cdots - 11\!\cdots\!72 \)
|
| $13$ |
\( T^{10} + \cdots - 13\!\cdots\!48 \)
|
| $17$ |
\( T^{10} + \cdots + 10\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots + 93\!\cdots\!36 \)
|
| $23$ |
\( T^{10} + \cdots - 15\!\cdots\!92 \)
|
| $29$ |
\( T^{10} + \cdots + 84\!\cdots\!80 \)
|
| $31$ |
\( (T + 923521)^{10} \)
|
| $37$ |
\( T^{10} + \cdots + 78\!\cdots\!32 \)
|
| $41$ |
\( T^{10} + \cdots - 51\!\cdots\!56 \)
|
| $43$ |
\( T^{10} + \cdots + 90\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots + 37\!\cdots\!12 \)
|
| $53$ |
\( T^{10} + \cdots - 72\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots - 44\!\cdots\!04 \)
|
| $61$ |
\( T^{10} + \cdots + 27\!\cdots\!08 \)
|
| $67$ |
\( T^{10} + \cdots - 61\!\cdots\!52 \)
|
| $71$ |
\( T^{10} + \cdots - 37\!\cdots\!64 \)
|
| $73$ |
\( T^{10} + \cdots - 78\!\cdots\!04 \)
|
| $79$ |
\( T^{10} + \cdots - 47\!\cdots\!20 \)
|
| $83$ |
\( T^{10} + \cdots + 21\!\cdots\!92 \)
|
| $89$ |
\( T^{10} + \cdots + 38\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 27\!\cdots\!96 \)
|
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