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: |
\( x^{10} - 2 x^{9} + 77 x^{8} + 92 x^{7} + 4681 x^{6} + 945 x^{5} + 52191 x^{4} + 34722 x^{3} + \cdots + 11664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{10} - 2 x^{9} + 77 x^{8} + 92 x^{7} + 4681 x^{6} + 945 x^{5} + 52191 x^{4} + 34722 x^{3} + \cdots + 11664 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8821 \nu^{9} + 18484 \nu^{8} - 87799 \nu^{7} + 4454858 \nu^{6} - 1714391 \nu^{5} + \cdots - 4922773704 ) / 31930695720 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 592343 \nu^{9} + 4162372 \nu^{8} - 19771267 \nu^{7} + 121238858 \nu^{6} + \cdots + 2142928038672 ) / 325693096344 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 34022909 \nu^{9} + 79883704 \nu^{8} - 3474434929 \nu^{7} + 2252985578 \nu^{6} + \cdots + 69833751056856 ) / 9770792890320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37567553 \nu^{9} + 241953752 \nu^{8} + 1945707013 \nu^{7} + 24877144654 \nu^{6} + \cdots + 75850436982168 ) / 9770792890320 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 844097 \nu^{9} + 1670552 \nu^{8} - 65032437 \nu^{7} - 77481326 \nu^{6} - 3960127773 \nu^{5} + \cdots - 64265554872 ) / 63861391440 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 51404525 \nu^{9} - 104900032 \nu^{8} + 4014687601 \nu^{7} + 4460655214 \nu^{6} + \cdots + 4309456060440 ) / 651386192688 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1032613531 \nu^{9} + 1784716136 \nu^{8} - 78254231351 \nu^{7} - 116570544218 \nu^{6} + \cdots - 70387043394456 ) / 9770792890320 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1034939203 \nu^{9} + 2312899208 \nu^{8} - 80763100943 \nu^{7} - 73195985114 \nu^{6} + \cdots - 6379727884488 ) / 9770792890320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{9} - \beta_{8} + \beta_{7} + 30\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{9} + 2\beta_{8} + 7\beta_{5} + 7\beta_{4} - 7\beta_{3} - 57\beta_{2} - 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( 56 \beta_{9} + 142 \beta_{8} - 23 \beta_{7} - 1731 \beta_{6} + 56 \beta_{5} + 86 \beta_{4} + 23 \beta_{3} + \cdots - 1731 \)
|
| \(\nu^{5}\) | \(=\) |
\( 741 \beta_{9} + 450 \beta_{8} + 492 \beta_{7} - 6486 \beta_{6} - 291 \beta_{5} - 450 \beta_{4} + \cdots - 3770 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 6104\beta_{9} - 6104\beta_{8} - 10255\beta_{5} - 10255\beta_{4} - 182\beta_{3} + 17528\beta_{2} + 116817 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 36022 \beta_{9} - 65639 \beta_{8} - 31367 \beta_{7} + 619227 \beta_{6} - 36022 \beta_{5} + \cdots + 619227 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 759446 \beta_{9} - 323065 \beta_{8} - 51554 \beta_{7} + 8369979 \beta_{6} + 436381 \beta_{5} + \cdots + 1584122 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2656425 \beta_{9} + 2656425 \beta_{8} + 5498136 \beta_{5} + 5498136 \beta_{4} - 2054913 \beta_{3} + \cdots - 54049350 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-1 - \beta_{6}\) | \(1\) |
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 | −4.93282 | − | 8.54389i | −2.00000 | − | 3.46410i | −5.44610 | + | 9.43293i | −19.7313 | −4.68224 | − | 17.9186i | −8.00000 | −35.1654 | + | 60.9082i | 10.8922 | + | 18.8659i | ||||||||||||||||||||||||||||||||||
| 23.2 | 1.00000 | − | 1.73205i | −2.24673 | − | 3.89145i | −2.00000 | − | 3.46410i | 5.50258 | − | 9.53075i | −8.98692 | −18.4932 | + | 1.00015i | −8.00000 | 3.40440 | − | 5.89659i | −11.0052 | − | 19.0615i | |||||||||||||||||||||||||||||||||||
| 23.3 | 1.00000 | − | 1.73205i | −0.422421 | − | 0.731654i | −2.00000 | − | 3.46410i | −8.52709 | + | 14.7693i | −1.68968 | 18.4904 | − | 1.05055i | −8.00000 | 13.1431 | − | 22.7646i | 17.0542 | + | 29.5387i | |||||||||||||||||||||||||||||||||||
| 23.4 | 1.00000 | − | 1.73205i | 1.10811 | + | 1.91931i | −2.00000 | − | 3.46410i | 6.77456 | − | 11.7339i | 4.43246 | 12.0234 | + | 14.0868i | −8.00000 | 11.0442 | − | 19.1291i | −13.5491 | − | 23.4678i | |||||||||||||||||||||||||||||||||||
| 23.5 | 1.00000 | − | 1.73205i | 2.99385 | + | 5.18551i | −2.00000 | − | 3.46410i | −8.30395 | + | 14.3829i | 11.9754 | −17.3384 | − | 6.51010i | −8.00000 | −4.42633 | + | 7.66662i | 16.6079 | + | 28.7657i | |||||||||||||||||||||||||||||||||||
| 67.1 | 1.00000 | + | 1.73205i | −4.93282 | + | 8.54389i | −2.00000 | + | 3.46410i | −5.44610 | − | 9.43293i | −19.7313 | −4.68224 | + | 17.9186i | −8.00000 | −35.1654 | − | 60.9082i | 10.8922 | − | 18.8659i | |||||||||||||||||||||||||||||||||||
| 67.2 | 1.00000 | + | 1.73205i | −2.24673 | + | 3.89145i | −2.00000 | + | 3.46410i | 5.50258 | + | 9.53075i | −8.98692 | −18.4932 | − | 1.00015i | −8.00000 | 3.40440 | + | 5.89659i | −11.0052 | + | 19.0615i | |||||||||||||||||||||||||||||||||||
| 67.3 | 1.00000 | + | 1.73205i | −0.422421 | + | 0.731654i | −2.00000 | + | 3.46410i | −8.52709 | − | 14.7693i | −1.68968 | 18.4904 | + | 1.05055i | −8.00000 | 13.1431 | + | 22.7646i | 17.0542 | − | 29.5387i | |||||||||||||||||||||||||||||||||||
| 67.4 | 1.00000 | + | 1.73205i | 1.10811 | − | 1.91931i | −2.00000 | + | 3.46410i | 6.77456 | + | 11.7339i | 4.43246 | 12.0234 | − | 14.0868i | −8.00000 | 11.0442 | + | 19.1291i | −13.5491 | + | 23.4678i | |||||||||||||||||||||||||||||||||||
| 67.5 | 1.00000 | + | 1.73205i | 2.99385 | − | 5.18551i | −2.00000 | + | 3.46410i | −8.30395 | − | 14.3829i | 11.9754 | −17.3384 | + | 6.51010i | −8.00000 | −4.42633 | − | 7.66662i | 16.6079 | − | 28.7657i | |||||||||||||||||||||||||||||||||||
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.c | ✓ | 10 |
| 7.c | even | 3 | 1 | inner | 154.4.e.c | ✓ | 10 |
| 7.c | even | 3 | 1 | 1078.4.a.w | 5 | ||
| 7.d | odd | 6 | 1 | 1078.4.a.t | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.e.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 154.4.e.c | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 1078.4.a.t | 5 | 7.d | odd | 6 | 1 | ||
| 1078.4.a.w | 5 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} + 7 T_{3}^{9} + 104 T_{3}^{8} + 63 T_{3}^{7} + 4151 T_{3}^{6} + 6629 T_{3}^{5} + \cdots + 247009 \)
acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{5} \)
|
| $3$ |
\( T^{10} + 7 T^{9} + \cdots + 247009 \)
|
| $5$ |
\( T^{10} + \cdots + 211611040144 \)
|
| $7$ |
\( T^{10} + \cdots + 4747561509943 \)
|
| $11$ |
\( (T^{2} - 11 T + 121)^{5} \)
|
| $13$ |
\( (T^{5} + 5 T^{4} + \cdots + 95294869)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 68\!\cdots\!36 \)
|
| $19$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots + 52\!\cdots\!04 \)
|
| $29$ |
\( (T^{5} + 381 T^{4} + \cdots - 4870964167)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 15\!\cdots\!16 \)
|
| $37$ |
\( T^{10} + \cdots + 69\!\cdots\!76 \)
|
| $41$ |
\( (T^{5} - 529 T^{4} + \cdots - 425891823268)^{2} \)
|
| $43$ |
\( (T^{5} + 624 T^{4} + \cdots - 4275494800)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 20\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 65\!\cdots\!04 \)
|
| $59$ |
\( T^{10} + \cdots + 18\!\cdots\!25 \)
|
| $61$ |
\( T^{10} + \cdots + 39\!\cdots\!44 \)
|
| $67$ |
\( T^{10} + \cdots + 15\!\cdots\!96 \)
|
| $71$ |
\( (T^{5} + 581 T^{4} + \cdots - 117826341388)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 20\!\cdots\!04 \)
|
| $79$ |
\( T^{10} + \cdots + 14\!\cdots\!81 \)
|
| $83$ |
\( (T^{5} + \cdots - 40031950056932)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 63\!\cdots\!84 \)
|
| $97$ |
\( (T^{5} + \cdots + 288050282075731)^{2} \)
|
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