Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [131,2,Mod(1,131)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(131, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("131.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 131 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 131.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: | \(1.04604026648\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{10} - 18x^{8} - 2x^{7} + 111x^{6} + 18x^{5} - 270x^{4} - 28x^{3} + 232x^{2} - 16x - 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{10} - 18x^{8} - 2x^{7} + 111x^{6} + 18x^{5} - 270x^{4} - 28x^{3} + 232x^{2} - 16x - 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 9\nu^{4} + 9\nu^{3} + 14\nu^{2} - 18\nu + 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 11\nu^{4} + 9\nu^{3} + 32\nu^{2} - 18\nu - 20 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} - 18\nu^{7} - 2\nu^{6} + 111\nu^{5} + 18\nu^{4} - 270\nu^{3} - 28\nu^{2} + 216\nu - 16 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} - 14\nu^{7} + 26\nu^{6} + 59\nu^{5} - 108\nu^{4} - 54\nu^{3} + 152\nu^{2} - 72\nu - 16 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 16\nu^{6} + 81\nu^{4} - 134\nu^{2} + 40 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{9} - 2\nu^{8} - 14\nu^{7} + 30\nu^{6} + 55\nu^{5} - 144\nu^{4} - 34\nu^{3} + 224\nu^{2} - 64\nu - 32 ) / 16 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{9} + \nu^{8} + 16\nu^{7} - 12\nu^{6} - 81\nu^{5} + 41\nu^{4} + 126\nu^{3} - 34\nu^{2} - 16\nu - 16 ) / 8 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} - \nu^{8} - 16\nu^{7} + 12\nu^{6} + 85\nu^{5} - 41\nu^{4} - 162\nu^{3} + 34\nu^{2} + 72\nu + 8 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{9} - 9\beta_{5} - 9\beta_{4} + 7\beta_{3} - 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{9} + 2\beta_{8} - 9\beta_{7} - 9\beta_{4} + 9\beta_{3} + 31\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 69\beta_{9} + 2\beta_{8} - 67\beta_{5} - 67\beta_{4} + 49\beta_{3} - 45\beta_{2} + 4\beta _1 + 91 \)
|
| \(\nu^{7}\) | \(=\) |
\( 99 \beta_{9} + 28 \beta_{8} - 63 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 75 \beta_{4} + 71 \beta_{3} + \cdots + 91 \)
|
| \(\nu^{8}\) | \(=\) |
\( 509 \beta_{9} + 32 \beta_{8} + 8 \beta_{6} - 477 \beta_{5} - 477 \beta_{4} + 351 \beta_{3} - 287 \beta_{2} + \cdots + 603 \)
|
| \(\nu^{9}\) | \(=\) |
\( 835 \beta_{9} + 286 \beta_{8} - 405 \beta_{7} + 72 \beta_{6} - 72 \beta_{5} - 605 \beta_{4} + 549 \beta_{3} + \cdots + 699 \)
|
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 |
|
−2.77595 | 2.75245 | 5.70589 | 2.13268 | −7.64065 | −0.507406 | −10.2874 | 4.57597 | −5.92020 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.61816 | −2.26482 | 4.85474 | −3.21931 | 5.92966 | −1.61608 | −7.47417 | 2.12942 | 8.42865 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.64897 | 0.939890 | 0.719087 | −0.908601 | −1.54985 | 4.24019 | 2.11218 | −2.11661 | 1.49825 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.960983 | −3.31776 | −1.07651 | 2.09792 | 3.18831 | 0.358420 | 2.95648 | 8.00754 | −2.01606 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.501920 | 1.39139 | −1.74808 | 4.24361 | −0.698367 | −2.27110 | 1.88124 | −1.06404 | −2.12995 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.355001 | 3.04591 | −1.87397 | −1.37468 | 1.08130 | 2.58803 | −1.37526 | 6.27756 | −0.488013 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.49686 | −1.04612 | 0.240604 | 2.58763 | −1.56590 | 4.53364 | −2.63358 | −1.90564 | 3.87333 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.70157 | 1.62192 | 0.895331 | 0.00430284 | 2.75981 | −4.05652 | −1.87967 | −0.369366 | 0.00732156 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.37860 | 0.205831 | 3.65772 | −3.94759 | 0.489590 | 2.07349 | 3.94304 | −2.95763 | −9.38972 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.57394 | −2.32869 | 4.62519 | 2.38404 | −5.99391 | −4.34267 | 6.75709 | 2.42278 | 6.13639 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(131\) | \(-1\) |
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 | 131.2.a.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1179.2.a.g | 10 | ||
| 4.b | odd | 2 | 1 | 2096.2.a.r | 10 | ||
| 5.b | even | 2 | 1 | 3275.2.a.f | 10 | ||
| 7.b | odd | 2 | 1 | 6419.2.a.d | 10 | ||
| 8.b | even | 2 | 1 | 8384.2.a.bt | 10 | ||
| 8.d | odd | 2 | 1 | 8384.2.a.bu | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 131.2.a.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1179.2.a.g | 10 | 3.b | odd | 2 | 1 | ||
| 2096.2.a.r | 10 | 4.b | odd | 2 | 1 | ||
| 3275.2.a.f | 10 | 5.b | even | 2 | 1 | ||
| 6419.2.a.d | 10 | 7.b | odd | 2 | 1 | ||
| 8384.2.a.bt | 10 | 8.b | even | 2 | 1 | ||
| 8384.2.a.bu | 10 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 18T_{2}^{8} + 2T_{2}^{7} + 111T_{2}^{6} - 18T_{2}^{5} - 270T_{2}^{4} + 28T_{2}^{3} + 232T_{2}^{2} + 16T_{2} - 32 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(131))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 18 T^{8} + \cdots - 32 \)
|
| $3$ |
\( T^{10} - T^{9} + \cdots + 67 \)
|
| $5$ |
\( T^{10} - 4 T^{9} + \cdots + 8 \)
|
| $7$ |
\( T^{10} - T^{9} + \cdots - 1213 \)
|
| $11$ |
\( T^{10} - 2 T^{9} + \cdots - 17852 \)
|
| $13$ |
\( T^{10} - 11 T^{9} + \cdots - 31 \)
|
| $17$ |
\( T^{10} + 2 T^{9} + \cdots + 2048 \)
|
| $19$ |
\( T^{10} - 110 T^{8} + \cdots + 64000 \)
|
| $23$ |
\( T^{10} + 10 T^{9} + \cdots + 512 \)
|
| $29$ |
\( T^{10} - 16 T^{9} + \cdots + 40960 \)
|
| $31$ |
\( T^{10} - 6 T^{9} + \cdots - 2020864 \)
|
| $37$ |
\( T^{10} - 34 T^{9} + \cdots + 55889408 \)
|
| $41$ |
\( T^{10} + 13 T^{9} + \cdots - 544027 \)
|
| $43$ |
\( T^{10} - 9 T^{9} + \cdots - 13498661 \)
|
| $47$ |
\( T^{10} + 6 T^{9} + \cdots + 25248256 \)
|
| $53$ |
\( T^{10} - 30 T^{9} + \cdots - 57328 \)
|
| $59$ |
\( T^{10} + 5 T^{9} + \cdots - 272185 \)
|
| $61$ |
\( T^{10} - 51 T^{9} + \cdots - 32394611 \)
|
| $67$ |
\( T^{10} + 10 T^{9} + \cdots + 217088 \)
|
| $71$ |
\( T^{10} - 324 T^{8} + \cdots + 43725824 \)
|
| $73$ |
\( T^{10} + 14 T^{9} + \cdots - 45719552 \)
|
| $79$ |
\( T^{10} + \cdots - 467968000 \)
|
| $83$ |
\( T^{10} + 22 T^{9} + \cdots + 5208064 \)
|
| $89$ |
\( T^{10} - 14 T^{9} + \cdots - 2616560 \)
|
| $97$ |
\( T^{10} - 4 T^{9} + \cdots + 1846784 \)
|
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