Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,4,Mod(1,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.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: | \(19.9426455819\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6681389953.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 13^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 7\nu^{4} - 77\nu^{3} - 531\nu^{2} + 1431\nu + 9815 ) / 39 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -22\nu^{5} - 102\nu^{4} + 1655\nu^{3} + 7574\nu^{2} - 30520\nu - 139594 ) / 273 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 50\nu^{5} + 207\nu^{4} - 3902\nu^{3} - 16163\nu^{2} + 73955\nu + 314496 ) / 273 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 131\nu^{5} + 657\nu^{4} - 10347\nu^{3} - 51114\nu^{2} + 200123\nu + 982345 ) / 273 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 172\nu^{5} + 814\nu^{4} - 13543\nu^{3} - 62888\nu^{2} + 260211\nu + 1204567 ) / 273 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} - 2\beta_{3} - 3\beta_{2} - 8\beta _1 - 1 ) / 13 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{5} - 5\beta_{4} - 8\beta_{3} - 2\beta_{2} - 3\beta _1 + 466 ) / 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -66\beta_{5} + 92\beta_{4} - 40\beta_{3} - 145\beta_{2} - 270\beta _1 + 54 ) / 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 443\beta_{5} - 363\beta_{4} - 625\beta_{3} - 143\beta_{2} - 77\beta _1 + 17789 ) / 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3566\beta_{5} + 4108\beta_{4} - 91\beta_{3} - 6933\beta_{2} - 9889\beta _1 + 917 ) / 13 \)
|
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 |
|
−2.00000 | −8.14801 | 4.00000 | 12.6646 | 16.2960 | 28.7705 | −8.00000 | 39.3901 | −25.3292 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.92743 | 4.00000 | 0.152827 | 7.85487 | −33.9913 | −8.00000 | −11.5753 | −0.305653 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −1.24585 | 4.00000 | −20.5002 | 2.49171 | −21.2545 | −8.00000 | −25.4478 | 41.0004 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.35815 | 4.00000 | −6.80407 | −6.71631 | −15.5380 | −8.00000 | −15.7228 | 13.6081 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 8.95458 | 4.00000 | −17.3267 | −17.9092 | 16.7510 | −8.00000 | 53.1845 | 34.6535 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 10.0086 | 4.00000 | 13.8136 | −20.0171 | 0.262285 | −8.00000 | 73.1713 | −27.6271 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 338.4.a.n | ✓ | 6 |
| 13.b | even | 2 | 1 | 338.4.a.o | yes | 6 | |
| 13.c | even | 3 | 2 | 338.4.c.p | 12 | ||
| 13.d | odd | 4 | 2 | 338.4.b.h | 12 | ||
| 13.e | even | 6 | 2 | 338.4.c.o | 12 | ||
| 13.f | odd | 12 | 4 | 338.4.e.i | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.4.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 338.4.a.o | yes | 6 | 13.b | even | 2 | 1 | |
| 338.4.b.h | 12 | 13.d | odd | 4 | 2 | ||
| 338.4.c.o | 12 | 13.e | even | 6 | 2 | ||
| 338.4.c.p | 12 | 13.c | even | 3 | 2 | ||
| 338.4.e.i | 24 | 13.f | odd | 12 | 4 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3}^{6} - 9T_{3}^{5} - 97T_{3}^{4} + 731T_{3}^{3} + 2313T_{3}^{2} - 8047T_{3} - 11999 \)
|
|
\( T_{5}^{6} + 18T_{5}^{5} - 397T_{5}^{4} - 5935T_{5}^{3} + 44090T_{5}^{2} + 416208T_{5} - 64616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 9 T^{5} + \cdots - 11999 \)
|
| $5$ |
\( T^{6} + 18 T^{5} + \cdots - 64616 \)
|
| $7$ |
\( T^{6} + 25 T^{5} + \cdots - 1418984 \)
|
| $11$ |
\( T^{6} + 37 T^{5} + \cdots - 284930113 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 137132126111 \)
|
| $19$ |
\( T^{6} + \cdots + 98183802067 \)
|
| $23$ |
\( T^{6} + \cdots - 920633860328 \)
|
| $29$ |
\( T^{6} + \cdots + 83240818856 \)
|
| $31$ |
\( T^{6} + \cdots + 254759240632 \)
|
| $37$ |
\( T^{6} + \cdots + 19945258052936 \)
|
| $41$ |
\( T^{6} + \cdots - 6202922352809 \)
|
| $43$ |
\( T^{6} + \cdots - 12\!\cdots\!89 \)
|
| $47$ |
\( T^{6} + \cdots + 549423790361944 \)
|
| $53$ |
\( T^{6} + \cdots + 14\!\cdots\!88 \)
|
| $59$ |
\( T^{6} + \cdots + 35\!\cdots\!77 \)
|
| $61$ |
\( T^{6} + \cdots + 833240079095872 \)
|
| $67$ |
\( T^{6} + \cdots - 2729196191063 \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $73$ |
\( T^{6} + \cdots - 52281997669273 \)
|
| $79$ |
\( T^{6} + \cdots + 79\!\cdots\!96 \)
|
| $83$ |
\( T^{6} + \cdots - 99\!\cdots\!91 \)
|
| $89$ |
\( T^{6} + \cdots - 95\!\cdots\!09 \)
|
| $97$ |
\( T^{6} + \cdots + 15227749501399 \)
|
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