Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,4,Mod(337,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.337");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.c (of order \(2\), degree \(1\), 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: | \(32.2150428631\) |
| 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: |
\( x^{10} + 472x^{8} + 80178x^{6} + 5943740x^{4} + 189932577x^{2} + 2157416704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{10} + 472x^{8} + 80178x^{6} + 5943740x^{4} + 189932577x^{2} + 2157416704 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -84\nu^{8} - 27863\nu^{6} - 2969246\nu^{4} - 116826777\nu^{2} - 24092302\nu - 1386658592 ) / 24092302 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -84\nu^{8} - 27863\nu^{6} - 2969246\nu^{4} - 116826777\nu^{2} + 24092302\nu - 1386658592 ) / 24092302 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9647\nu^{8} + 4534717\nu^{6} + 716133261\nu^{4} + 42635302303\nu^{2} + 758614306800 ) / 819138268 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25327\nu^{8} - 10971313\nu^{6} - 1631159293\nu^{4} - 94243291663\nu^{2} - 1663421280700 ) / 819138268 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 55317\nu^{8} + 24206387\nu^{6} + 3621893195\nu^{4} + 209699570501\nu^{2} + 3719108841108 ) / 819138268 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14927\nu^{9} + 5094728\nu^{7} + 549726694\nu^{5} + 19764437876\nu^{3} + 121938507831\nu ) / 559519621648 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5610282 \nu^{9} + 2499985589 \nu^{7} + 380115547329 \nu^{5} + 22226874649543 \nu^{3} + 392472644268417 \nu ) / 2377958392004 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 49611357 \nu^{9} + 22042872516 \nu^{7} + 3334936788934 \nu^{5} + 193880830566944 \nu^{3} + 34\!\cdots\!61 \nu ) / 9511833568016 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 112607111 \nu^{9} + 49450497100 \nu^{7} + 7425335199354 \nu^{5} + 431284783249712 \nu^{3} + 76\!\cdots\!63 \nu ) / 9511833568016 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} + 3\beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 - 95 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{9} + 16\beta_{8} - 82\beta_{7} - 314\beta_{6} - 111\beta_{2} + 111\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -407\beta_{5} - 671\beta_{4} + 494\beta_{3} - 132\beta_{2} - 132\beta _1 + 12598 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1424\beta_{9} - 4876\beta_{8} + 17348\beta_{7} + 51028\beta_{6} + 15279\beta_{2} - 15279\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 70604\beta_{5} + 122909\beta_{4} - 72551\beta_{3} + 16244\beta_{2} + 16244\beta _1 - 1910051 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 169330\beta_{9} + 1041148\beta_{8} - 3061958\beta_{7} - 7908038\beta_{6} - 2315589\beta_{2} + 2315589\beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( -11814401\beta_{5} - 21222973\beta_{4} + 10775730\beta_{3} - 2256417\beta_{2} - 2256417\beta _1 + 303869942 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 18592072 \beta_{9} - 196967208 \beta_{8} + 514763072 \beta_{7} + 1310572896 \beta_{6} + \cdots - 366445773 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
| \(n\) | \(157\) | \(365\) | \(379\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 2.00000i | 3.00000 | −4.00000 | − | 19.0430i | − | 6.00000i | − | 7.00000i | 8.00000i | 9.00000 | −38.0859 | ||||||||||||||||||||||||||||||||||||||||||||
| 337.2 | − | 2.00000i | 3.00000 | −4.00000 | − | 9.08035i | − | 6.00000i | − | 7.00000i | 8.00000i | 9.00000 | −18.1607 | |||||||||||||||||||||||||||||||||||||||||||||
| 337.3 | − | 2.00000i | 3.00000 | −4.00000 | 2.31954i | − | 6.00000i | − | 7.00000i | 8.00000i | 9.00000 | 4.63908 | ||||||||||||||||||||||||||||||||||||||||||||||
| 337.4 | − | 2.00000i | 3.00000 | −4.00000 | 8.59071i | − | 6.00000i | − | 7.00000i | 8.00000i | 9.00000 | 17.1814 | ||||||||||||||||||||||||||||||||||||||||||||||
| 337.5 | − | 2.00000i | 3.00000 | −4.00000 | 11.2131i | − | 6.00000i | − | 7.00000i | 8.00000i | 9.00000 | 22.4261 | ||||||||||||||||||||||||||||||||||||||||||||||
| 337.6 | 2.00000i | 3.00000 | −4.00000 | − | 11.2131i | 6.00000i | 7.00000i | − | 8.00000i | 9.00000 | 22.4261 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.7 | 2.00000i | 3.00000 | −4.00000 | − | 8.59071i | 6.00000i | 7.00000i | − | 8.00000i | 9.00000 | 17.1814 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.8 | 2.00000i | 3.00000 | −4.00000 | − | 2.31954i | 6.00000i | 7.00000i | − | 8.00000i | 9.00000 | 4.63908 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.9 | 2.00000i | 3.00000 | −4.00000 | 9.08035i | 6.00000i | 7.00000i | − | 8.00000i | 9.00000 | −18.1607 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.10 | 2.00000i | 3.00000 | −4.00000 | 19.0430i | 6.00000i | 7.00000i | − | 8.00000i | 9.00000 | −38.0859 | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 546.4.c.b | ✓ | 10 |
| 13.b | even | 2 | 1 | inner | 546.4.c.b | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.c.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 546.4.c.b | ✓ | 10 | 13.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 650T_{5}^{8} + 131457T_{5}^{6} + 10784696T_{5}^{4} + 331766800T_{5}^{2} + 1492740496 \)
acting on \(S_{4}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{5} \)
|
| $3$ |
\( (T - 3)^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 1492740496 \)
|
| $7$ |
\( (T^{2} + 49)^{5} \)
|
| $11$ |
\( T^{10} + \cdots + 39\!\cdots\!36 \)
|
| $13$ |
\( T^{10} + \cdots + 51\!\cdots\!57 \)
|
| $17$ |
\( (T^{5} - 173 T^{4} + \cdots - 118083392)^{2} \)
|
| $19$ |
\( T^{10} + \cdots + 106252225168384 \)
|
| $23$ |
\( (T^{5} + 66 T^{4} + \cdots - 87921192)^{2} \)
|
| $29$ |
\( (T^{5} + 382 T^{4} + \cdots - 124472088)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 29\!\cdots\!24 \)
|
| $37$ |
\( T^{10} + \cdots + 92\!\cdots\!16 \)
|
| $41$ |
\( T^{10} + \cdots + 21\!\cdots\!16 \)
|
| $43$ |
\( (T^{5} + 68 T^{4} + \cdots - 105663767296)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 13\!\cdots\!16 \)
|
| $53$ |
\( (T^{5} + \cdots + 4886981196336)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 36\!\cdots\!00 \)
|
| $61$ |
\( (T^{5} + \cdots + 84036731674400)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 14\!\cdots\!76 \)
|
| $71$ |
\( T^{10} + \cdots + 17\!\cdots\!04 \)
|
| $73$ |
\( T^{10} + \cdots + 81\!\cdots\!76 \)
|
| $79$ |
\( (T^{5} + \cdots - 24942754724160)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 26\!\cdots\!84 \)
|
| $89$ |
\( T^{10} + \cdots + 26\!\cdots\!16 \)
|
| $97$ |
\( T^{10} + \cdots + 17\!\cdots\!24 \)
|
| show more | |
| show less |