Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [154,4,Mod(23,154)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(154, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("154.23");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 154 = 2 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 154.e (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(9.08629414088\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} + 4 x^{8} - 99 x^{7} + 51 x^{6} + 738 x^{5} + 1071 x^{4} - 43659 x^{3} + 37044 x^{2} + \cdots + 4084101 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{10} - 2 x^{9} + 4 x^{8} - 99 x^{7} + 51 x^{6} + 738 x^{5} + 1071 x^{4} - 43659 x^{3} + 37044 x^{2} + \cdots + 4084101 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 610 \nu^{9} - 2581 \nu^{8} + 11336 \nu^{7} + 23577 \nu^{6} - 555 \nu^{5} - 708858 \nu^{4} + \cdots - 200509911 ) / 252047376 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 620 \nu^{9} + 10585 \nu^{8} - 220502 \nu^{7} - 160107 \nu^{6} + 338883 \nu^{5} + \cdots + 8858804031 ) / 252047376 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} - 2 \nu^{8} + 4 \nu^{7} - 99 \nu^{6} + 51 \nu^{5} + 738 \nu^{4} + 1071 \nu^{3} + \cdots - 388962 ) / 194481 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 737 \nu^{9} + 1844 \nu^{8} + 40853 \nu^{7} - 19119 \nu^{6} - 177558 \nu^{5} - 844047 \nu^{4} + \cdots - 1897745598 ) / 126023688 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 188 \nu^{9} + 145 \nu^{8} - 23222 \nu^{7} - 25971 \nu^{6} + 187035 \nu^{5} + 54540 \nu^{4} + \cdots + 1003327479 ) / 28005264 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 134 \nu^{9} + 541 \nu^{8} - 1964 \nu^{7} - 2400 \nu^{6} - 345 \nu^{5} + 317223 \nu^{4} + \cdots + 52898832 ) / 15752961 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 58 \nu^{9} - 517 \nu^{8} + 2672 \nu^{7} + 1257 \nu^{6} - 28995 \nu^{5} - 167130 \nu^{4} + \cdots - 99824319 ) / 6001128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1348 \nu^{9} + 12503 \nu^{8} + 26150 \nu^{7} - 12981 \nu^{6} - 409011 \nu^{5} + \cdots - 802623087 ) / 84015792 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5620 \nu^{9} - 19787 \nu^{8} + 51922 \nu^{7} + 171921 \nu^{6} + 385719 \nu^{5} + \cdots - 3521856429 ) / 252047376 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{9} - \beta_{8} + 2\beta_{4} - 2\beta_{3} - 2\beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{9} + \beta_{8} + 3\beta_{7} + 3\beta_{5} - 2\beta_{4} - 4\beta_{3} - 3\beta_{2} - 28\beta _1 + 11 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 23 \beta_{9} + 5 \beta_{8} + 6 \beta_{7} + 36 \beta_{6} + 6 \beta_{5} + 26 \beta_{4} - 2 \beta_{3} + \cdots + 145 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 41 \beta_{9} - 50 \beta_{8} + 3 \beta_{7} + 72 \beta_{6} - 15 \beta_{5} + \beta_{4} - 43 \beta_{3} + \cdots + 341 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 25 \beta_{9} + 11 \beta_{8} - 606 \beta_{7} - 450 \beta_{6} + 708 \beta_{5} + 482 \beta_{4} + \cdots - 4199 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1564 \beta_{9} + 556 \beta_{8} - 285 \beta_{7} + 477 \beta_{6} - 411 \beta_{5} - 1643 \beta_{4} + \cdots + 1661 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 4663 \beta_{9} - 1765 \beta_{8} - 5502 \beta_{7} - 2394 \beta_{6} - 1596 \beta_{5} + 9704 \beta_{4} + \cdots + 210463 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 22931 \beta_{9} - 7118 \beta_{8} - 12156 \beta_{7} - 15750 \beta_{6} + 2649 \beta_{5} + 39247 \beta_{4} + \cdots - 27835 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 203639 \beta_{9} + 86441 \beta_{8} + 198960 \beta_{7} - 80856 \beta_{6} + 177162 \beta_{5} + \cdots + 2382931 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−1.00000 | + | 1.73205i | −3.65723 | − | 6.33450i | −2.00000 | − | 3.46410i | 7.18631 | − | 12.4471i | 14.6289 | 8.74618 | − | 16.3250i | 8.00000 | −13.2506 | + | 22.9507i | 14.3726 | + | 24.8941i | ||||||||||||||||||||||||||||||||||
| 23.2 | −1.00000 | + | 1.73205i | −0.926283 | − | 1.60437i | −2.00000 | − | 3.46410i | −0.430075 | + | 0.744911i | 3.70513 | −15.1973 | + | 10.5850i | 8.00000 | 11.7840 | − | 20.4105i | −0.860149 | − | 1.48982i | |||||||||||||||||||||||||||||||||||
| 23.3 | −1.00000 | + | 1.73205i | −0.644420 | − | 1.11617i | −2.00000 | − | 3.46410i | −3.59556 | + | 6.22770i | 2.57768 | 15.6956 | + | 9.83093i | 8.00000 | 12.6694 | − | 21.9441i | −7.19113 | − | 12.4554i | |||||||||||||||||||||||||||||||||||
| 23.4 | −1.00000 | + | 1.73205i | 3.65226 | + | 6.32591i | −2.00000 | − | 3.46410i | 8.79533 | − | 15.2340i | −14.6091 | 7.83281 | + | 16.7823i | 8.00000 | −13.1781 | + | 22.8251i | 17.5907 | + | 30.4679i | |||||||||||||||||||||||||||||||||||
| 23.5 | −1.00000 | + | 1.73205i | 5.07567 | + | 8.79131i | −2.00000 | − | 3.46410i | −6.95600 | + | 12.0482i | −20.3027 | 13.9227 | − | 12.2131i | 8.00000 | −38.0248 | + | 65.8608i | −13.9120 | − | 24.0963i | |||||||||||||||||||||||||||||||||||
| 67.1 | −1.00000 | − | 1.73205i | −3.65723 | + | 6.33450i | −2.00000 | + | 3.46410i | 7.18631 | + | 12.4471i | 14.6289 | 8.74618 | + | 16.3250i | 8.00000 | −13.2506 | − | 22.9507i | 14.3726 | − | 24.8941i | |||||||||||||||||||||||||||||||||||
| 67.2 | −1.00000 | − | 1.73205i | −0.926283 | + | 1.60437i | −2.00000 | + | 3.46410i | −0.430075 | − | 0.744911i | 3.70513 | −15.1973 | − | 10.5850i | 8.00000 | 11.7840 | + | 20.4105i | −0.860149 | + | 1.48982i | |||||||||||||||||||||||||||||||||||
| 67.3 | −1.00000 | − | 1.73205i | −0.644420 | + | 1.11617i | −2.00000 | + | 3.46410i | −3.59556 | − | 6.22770i | 2.57768 | 15.6956 | − | 9.83093i | 8.00000 | 12.6694 | + | 21.9441i | −7.19113 | + | 12.4554i | |||||||||||||||||||||||||||||||||||
| 67.4 | −1.00000 | − | 1.73205i | 3.65226 | − | 6.32591i | −2.00000 | + | 3.46410i | 8.79533 | + | 15.2340i | −14.6091 | 7.83281 | − | 16.7823i | 8.00000 | −13.1781 | − | 22.8251i | 17.5907 | − | 30.4679i | |||||||||||||||||||||||||||||||||||
| 67.5 | −1.00000 | − | 1.73205i | 5.07567 | − | 8.79131i | −2.00000 | + | 3.46410i | −6.95600 | − | 12.0482i | −20.3027 | 13.9227 | + | 12.2131i | 8.00000 | −38.0248 | − | 65.8608i | −13.9120 | + | 24.0963i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 154.4.e.b | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 154.4.e.b | ✓ | 10 |
| 7.c | even | 3 | 1 | 1078.4.a.y | 5 | ||
| 7.d | odd | 6 | 1 | 1078.4.a.z | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.e.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 154.4.e.b | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 1078.4.a.y | 5 | 7.c | even | 3 | 1 | ||
| 1078.4.a.z | 5 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 7 T_{3}^{9} + 132 T_{3}^{8} - 119 T_{3}^{7} + 7763 T_{3}^{6} - 5691 T_{3}^{5} + \cdots + 1677025 \)
acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{5} \)
|
| $3$ |
\( T^{10} - 7 T^{9} + \cdots + 1677025 \)
|
| $5$ |
\( T^{10} + \cdots + 473323536 \)
|
| $7$ |
\( T^{10} + \cdots + 4747561509943 \)
|
| $11$ |
\( (T^{2} - 11 T + 121)^{5} \)
|
| $13$ |
\( (T^{5} + 5 T^{4} + \cdots + 130487245)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 38\!\cdots\!04 \)
|
| $19$ |
\( T^{10} + \cdots + 53\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 37\!\cdots\!16 \)
|
| $29$ |
\( (T^{5} - 15 T^{4} + \cdots + 285272513829)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 19\!\cdots\!64 \)
|
| $37$ |
\( T^{10} + \cdots + 57\!\cdots\!36 \)
|
| $41$ |
\( (T^{5} + 333 T^{4} + \cdots + 17939058768)^{2} \)
|
| $43$ |
\( (T^{5} - 92 T^{4} + \cdots + 105037305792)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 17\!\cdots\!56 \)
|
| $53$ |
\( T^{10} + \cdots + 10\!\cdots\!84 \)
|
| $59$ |
\( T^{10} + \cdots + 24\!\cdots\!21 \)
|
| $61$ |
\( T^{10} + \cdots + 25\!\cdots\!24 \)
|
| $67$ |
\( T^{10} + \cdots + 24\!\cdots\!44 \)
|
| $71$ |
\( (T^{5} + \cdots - 417540873289068)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 11\!\cdots\!96 \)
|
| $79$ |
\( T^{10} + \cdots + 39\!\cdots\!09 \)
|
| $83$ |
\( (T^{5} + \cdots + 1296970438428)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 10\!\cdots\!44 \)
|
| $97$ |
\( (T^{5} + 193 T^{4} + \cdots + 236004933927)^{2} \)
|
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