Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [167,2,Mod(1,167)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(167, 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("167.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 167.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: | \(1.33350171376\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 2 x^{11} - 17 x^{10} + 33 x^{9} + 103 x^{8} - 189 x^{7} - 277 x^{6} + 447 x^{5} + 363 x^{4} + \cdots + 9 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 2 x^{11} - 17 x^{10} + 33 x^{9} + 103 x^{8} - 189 x^{7} - 277 x^{6} + 447 x^{5} + 363 x^{4} + \cdots + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 67 \nu^{11} - 220 \nu^{10} + 1397 \nu^{9} + 3708 \nu^{8} - 10537 \nu^{7} - 21384 \nu^{6} + \cdots - 384 ) / 933 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 172 \nu^{11} - 410 \nu^{10} - 2528 \nu^{9} + 6147 \nu^{8} + 11788 \nu^{7} - 29589 \nu^{6} + \cdots + 1320 ) / 933 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 190 \nu^{11} - 415 \nu^{10} + 4268 \nu^{9} + 6825 \nu^{8} - 34741 \nu^{7} - 38472 \nu^{6} + \cdots + 2838 ) / 933 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 227 \nu^{11} - 118 \nu^{10} - 4009 \nu^{9} + 1446 \nu^{8} + 25061 \nu^{7} - 3819 \nu^{6} - 66263 \nu^{5} + \cdots - 5787 ) / 933 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 290 \nu^{11} + 496 \nu^{10} + 4501 \nu^{9} - 7359 \nu^{8} - 22826 \nu^{7} + 34776 \nu^{6} + \cdots - 2052 ) / 933 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 160 \nu^{11} - 27 \nu^{10} - 2923 \nu^{9} + 178 \nu^{8} + 19189 \nu^{7} + 1232 \nu^{6} - 54224 \nu^{5} + \cdots - 1195 ) / 311 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 544 \nu^{11} - 157 \nu^{10} + 10187 \nu^{9} + 3189 \nu^{8} - 68788 \nu^{7} - 22911 \nu^{6} + \cdots + 3441 ) / 933 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 230 \nu^{11} + 136 \nu^{10} + 3988 \nu^{9} - 1772 \nu^{8} - 24377 \nu^{7} + 6004 \nu^{6} + \cdots + 1815 ) / 311 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 719 \nu^{11} - 271 \nu^{10} - 12694 \nu^{9} + 2973 \nu^{8} + 79892 \nu^{7} - 4509 \nu^{6} + \cdots - 7479 ) / 933 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 715 \nu^{11} - 869 \nu^{10} - 11789 \nu^{9} + 12594 \nu^{8} + 67120 \nu^{7} - 56850 \nu^{6} + \cdots + 1842 ) / 933 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - 2\beta_{3} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 8\beta_{7} + 8\beta_{6} - \beta_{5} + 7\beta_{4} - \beta_{3} - \beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{11} - 9 \beta_{9} + 9 \beta_{8} + 2 \beta_{7} + 9 \beta_{6} - 10 \beta_{5} + \beta_{4} + \cdots - 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{11} - 10 \beta_{10} - 13 \beta_{9} + 11 \beta_{8} + 54 \beta_{7} + 57 \beta_{6} - 11 \beta_{5} + \cdots + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 54 \beta_{11} + \beta_{10} - 67 \beta_{9} + 66 \beta_{8} + 23 \beta_{7} + 68 \beta_{6} - 80 \beta_{5} + \cdots - 77 \)
|
| \(\nu^{8}\) | \(=\) |
\( 78 \beta_{11} - 80 \beta_{10} - 121 \beta_{9} + 89 \beta_{8} + 349 \beta_{7} + 392 \beta_{6} + \cdots + 428 \)
|
| \(\nu^{9}\) | \(=\) |
\( 347 \beta_{11} + 17 \beta_{10} - 475 \beta_{9} + 455 \beta_{8} + 198 \beta_{7} + 489 \beta_{6} + \cdots - 541 \)
|
| \(\nu^{10}\) | \(=\) |
\( 562 \beta_{11} - 595 \beta_{10} - 999 \beta_{9} + 652 \beta_{8} + 2228 \beta_{7} + 2665 \beta_{6} + \cdots + 2513 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2195 \beta_{11} + 187 \beta_{10} - 3316 \beta_{9} + 3063 \beta_{8} + 1547 \beta_{7} + 3460 \beta_{6} + \cdots - 3663 \)
|
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 |
|
−2.54892 | 1.77962 | 4.49697 | 4.18324 | −4.53611 | −0.0397534 | −6.36457 | 0.167056 | −10.6627 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43035 | 0.147197 | 3.90659 | −2.63845 | −0.357741 | 3.42804 | −4.63367 | −2.97833 | 6.41235 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.32632 | 3.36459 | −0.240885 | −1.38185 | −4.46251 | 1.92318 | 2.97212 | 8.32045 | 1.83277 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.18929 | −3.09492 | −0.585595 | −2.93022 | 3.68076 | 1.83981 | 3.07502 | 6.57856 | 3.48488 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.977398 | −1.54181 | −1.04469 | 2.68216 | 1.50696 | −1.27122 | 2.97588 | −0.622820 | −2.62154 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.0682496 | 1.03644 | −1.99534 | 0.779119 | −0.0707366 | 4.67370 | 0.272680 | −1.92579 | −0.0531746 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.466291 | 2.35499 | −1.78257 | 3.59549 | 1.09811 | −3.04979 | −1.76378 | 2.54600 | 1.67654 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.55935 | 1.36773 | 0.431581 | 0.367289 | 2.13277 | 1.25543 | −2.44572 | −1.12932 | 0.572732 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.60424 | −3.10595 | 0.573578 | 4.20693 | −4.98269 | 3.68943 | −2.28832 | 6.64695 | 6.74892 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.85971 | 2.55841 | 1.45851 | −3.62823 | 4.75790 | −1.79477 | −1.00701 | 3.54548 | −6.74745 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.41160 | −0.698370 | 3.81581 | 1.67948 | −1.68419 | −4.46027 | 4.37901 | −2.51228 | 4.05022 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.63933 | −1.16792 | 4.96605 | −2.91495 | −3.08253 | 4.80620 | 7.82836 | −1.63595 | −7.69351 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(167\) | \(-1\) |
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 | 167.2.a.b | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1503.2.a.g | 12 | ||
| 4.b | odd | 2 | 1 | 2672.2.a.o | 12 | ||
| 5.b | even | 2 | 1 | 4175.2.a.d | 12 | ||
| 7.b | odd | 2 | 1 | 8183.2.a.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 167.2.a.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 1503.2.a.g | 12 | 3.b | odd | 2 | 1 | ||
| 2672.2.a.o | 12 | 4.b | odd | 2 | 1 | ||
| 4175.2.a.d | 12 | 5.b | even | 2 | 1 | ||
| 8183.2.a.k | 12 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 2 T_{2}^{11} - 17 T_{2}^{10} + 33 T_{2}^{9} + 103 T_{2}^{8} - 189 T_{2}^{7} - 277 T_{2}^{6} + \cdots + 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 2 T^{11} + \cdots + 9 \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots - 91 \)
|
| $5$ |
\( T^{12} - 4 T^{11} + \cdots - 9216 \)
|
| $7$ |
\( T^{12} - 11 T^{11} + \cdots - 1557 \)
|
| $11$ |
\( T^{12} - 77 T^{10} + \cdots + 86192 \)
|
| $13$ |
\( T^{12} - 9 T^{11} + \cdots - 37888 \)
|
| $17$ |
\( T^{12} - 3 T^{11} + \cdots - 37888 \)
|
| $19$ |
\( T^{12} - 105 T^{10} + \cdots + 53116 \)
|
| $23$ |
\( T^{12} - T^{11} + \cdots - 846848 \)
|
| $29$ |
\( T^{12} + 6 T^{11} + \cdots + 1467907 \)
|
| $31$ |
\( T^{12} - 2 T^{11} + \cdots - 9529468 \)
|
| $37$ |
\( T^{12} - 34 T^{11} + \cdots - 84354048 \)
|
| $41$ |
\( T^{12} + 14 T^{11} + \cdots - 65094656 \)
|
| $43$ |
\( T^{12} - 6 T^{11} + \cdots + 2249728 \)
|
| $47$ |
\( T^{12} - 2 T^{11} + \cdots + 5029119 \)
|
| $53$ |
\( T^{12} + \cdots + 363016192 \)
|
| $59$ |
\( T^{12} + \cdots + 16274108416 \)
|
| $61$ |
\( T^{12} - 2 T^{11} + \cdots + 52291179 \)
|
| $67$ |
\( T^{12} + \cdots + 742796288 \)
|
| $71$ |
\( T^{12} + 9 T^{11} + \cdots + 30477312 \)
|
| $73$ |
\( T^{12} + \cdots + 4539186176 \)
|
| $79$ |
\( T^{12} + \cdots + 16330576896 \)
|
| $83$ |
\( T^{12} + \cdots - 23933785088 \)
|
| $89$ |
\( T^{12} + \cdots - 347261236 \)
|
| $97$ |
\( T^{12} + \cdots - 1132990973381 \)
|
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