Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,7,Mod(79,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 0, 2]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.79");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.be (of order \(6\), 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: | \(77.2981720963\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{15} + 12577 x^{14} + 7962770 x^{13} - 439315715 x^{12} - 44007154545 x^{11} + \cdots + 16\!\cdots\!12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{2}\cdot 7^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - x^{15} + 12577 x^{14} + 7962770 x^{13} - 439315715 x^{12} - 44007154545 x^{11} + \cdots + 16\!\cdots\!12 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 33\!\cdots\!99 \nu^{15} + \cdots + 10\!\cdots\!12 ) / 38\!\cdots\!32 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 64\!\cdots\!28 \nu^{15} + \cdots - 10\!\cdots\!80 ) / 60\!\cdots\!12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25\!\cdots\!49 \nu^{15} + \cdots - 28\!\cdots\!76 ) / 25\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 29\!\cdots\!17 \nu^{15} + \cdots + 13\!\cdots\!72 ) / 23\!\cdots\!32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 32\!\cdots\!63 \nu^{15} + \cdots - 72\!\cdots\!48 ) / 23\!\cdots\!32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19\!\cdots\!89 \nu^{15} + \cdots - 25\!\cdots\!00 ) / 11\!\cdots\!16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 60\!\cdots\!01 \nu^{15} + \cdots + 11\!\cdots\!52 ) / 23\!\cdots\!84 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 36\!\cdots\!85 \nu^{15} + \cdots + 44\!\cdots\!36 ) / 11\!\cdots\!16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 90\!\cdots\!95 \nu^{15} + \cdots + 19\!\cdots\!52 ) / 23\!\cdots\!32 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15\!\cdots\!85 \nu^{15} + \cdots + 79\!\cdots\!24 ) / 23\!\cdots\!32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 79\!\cdots\!33 \nu^{15} + \cdots - 15\!\cdots\!16 ) / 11\!\cdots\!16 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 16\!\cdots\!89 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 23\!\cdots\!32 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 43\!\cdots\!13 \nu^{15} + \cdots + 15\!\cdots\!60 ) / 47\!\cdots\!68 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 24\!\cdots\!59 \nu^{15} + \cdots + 42\!\cdots\!36 ) / 23\!\cdots\!32 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 61\!\cdots\!17 \nu^{15} + \cdots - 12\!\cdots\!56 ) / 23\!\cdots\!32 \)
|
| \(\nu\) | \(=\) |
\( ( 2 \beta_{15} + 14 \beta_{14} + 7 \beta_{13} - 12 \beta_{12} - 19 \beta_{11} - 12 \beta_{10} + \cdots + 52 ) / 336 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 278 \beta_{15} + 182 \beta_{14} + 469 \beta_{13} - 2748 \beta_{12} - 13 \beta_{11} - 2100 \beta_{10} + \cdots - 89570 ) / 336 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 165538 \beta_{15} - 13258 \beta_{14} - 150535 \beta_{13} + 439020 \beta_{12} + 887741 \beta_{11} + \cdots - 639929870 ) / 336 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 41955082 \beta_{15} - 70082782 \beta_{14} + 39737313 \beta_{13} - 3652932 \beta_{12} + \cdots + 50264413468 ) / 336 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2391826480 \beta_{15} + 5599988044 \beta_{14} - 8482873623 \beta_{13} + 4796712516 \beta_{12} + \cdots + 8999906443610 ) / 336 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1173100517024 \beta_{15} + 950313523460 \beta_{14} + 2534479154501 \beta_{13} - 2247589047708 \beta_{12} + \cdots + 15\!\cdots\!34 ) / 336 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 153976836898600 \beta_{15} + 247887270548176 \beta_{14} - 432372805187379 \beta_{13} + \cdots - 25\!\cdots\!96 ) / 336 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 13\!\cdots\!06 \beta_{15} + \cdots - 39\!\cdots\!06 ) / 112 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 70\!\cdots\!62 \beta_{15} + \cdots + 13\!\cdots\!54 ) / 336 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 39\!\cdots\!46 \beta_{15} + \cdots + 28\!\cdots\!80 ) / 336 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 28\!\cdots\!28 \beta_{15} + \cdots + 89\!\cdots\!50 ) / 336 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 26\!\cdots\!84 \beta_{15} + \cdots - 19\!\cdots\!14 ) / 336 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 10\!\cdots\!60 \beta_{15} + \cdots + 73\!\cdots\!32 ) / 336 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 12\!\cdots\!18 \beta_{15} + \cdots - 24\!\cdots\!74 ) / 336 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 55\!\cdots\!26 \beta_{15} + \cdots + 11\!\cdots\!50 ) / 336 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
0 | 13.5000 | − | 7.79423i | 0 | −122.025 | + | 211.354i | 0 | 292.124 | + | 179.757i | 0 | 121.500 | − | 210.444i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.2 | 0 | 13.5000 | − | 7.79423i | 0 | −96.4257 | + | 167.014i | 0 | −297.005 | + | 171.572i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.3 | 0 | 13.5000 | − | 7.79423i | 0 | −47.3833 | + | 82.0703i | 0 | 118.618 | − | 321.836i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.4 | 0 | 13.5000 | − | 7.79423i | 0 | −15.4579 | + | 26.7738i | 0 | −337.064 | − | 63.5355i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.5 | 0 | 13.5000 | − | 7.79423i | 0 | 9.66076 | − | 16.7329i | 0 | 318.080 | + | 128.351i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.6 | 0 | 13.5000 | − | 7.79423i | 0 | 46.3170 | − | 80.2234i | 0 | −172.181 | + | 296.653i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.7 | 0 | 13.5000 | − | 7.79423i | 0 | 70.0525 | − | 121.334i | 0 | 14.5983 | + | 342.689i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 79.8 | 0 | 13.5000 | − | 7.79423i | 0 | 99.2617 | − | 171.926i | 0 | −99.1709 | − | 328.351i | 0 | 121.500 | − | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.1 | 0 | 13.5000 | + | 7.79423i | 0 | −122.025 | − | 211.354i | 0 | 292.124 | − | 179.757i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | 13.5000 | + | 7.79423i | 0 | −96.4257 | − | 167.014i | 0 | −297.005 | − | 171.572i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | 13.5000 | + | 7.79423i | 0 | −47.3833 | − | 82.0703i | 0 | 118.618 | + | 321.836i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | 13.5000 | + | 7.79423i | 0 | −15.4579 | − | 26.7738i | 0 | −337.064 | + | 63.5355i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | 13.5000 | + | 7.79423i | 0 | 9.66076 | + | 16.7329i | 0 | 318.080 | − | 128.351i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 13.5000 | + | 7.79423i | 0 | 46.3170 | + | 80.2234i | 0 | −172.181 | − | 296.653i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 13.5000 | + | 7.79423i | 0 | 70.0525 | + | 121.334i | 0 | 14.5983 | − | 342.689i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 13.5000 | + | 7.79423i | 0 | 99.2617 | + | 171.926i | 0 | −99.1709 | + | 328.351i | 0 | 121.500 | + | 210.444i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 28.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.7.be.d | yes | 16 |
| 4.b | odd | 2 | 1 | 336.7.be.a | ✓ | 16 | |
| 7.c | even | 3 | 1 | 336.7.be.a | ✓ | 16 | |
| 28.g | odd | 6 | 1 | inner | 336.7.be.d | yes | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.7.be.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 336.7.be.a | ✓ | 16 | 7.c | even | 3 | 1 | |
| 336.7.be.d | yes | 16 | 1.a | even | 1 | 1 | trivial |
| 336.7.be.d | yes | 16 | 28.g | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\):
|
\( T_{5}^{16} + 112 T_{5}^{15} + 93614 T_{5}^{14} + 2541504 T_{5}^{13} + 5287633159 T_{5}^{12} + \cdots + 47\!\cdots\!00 \)
|
|
\( T_{11}^{16} + 252 T_{11}^{15} - 9893886 T_{11}^{14} - 2498593608 T_{11}^{13} + 69963978470163 T_{11}^{12} + \cdots + 66\!\cdots\!56 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( (T^{2} - 27 T + 243)^{8} \)
|
| $5$ |
\( T^{16} + \cdots + 47\!\cdots\!00 \)
|
| $7$ |
\( T^{16} + \cdots + 36\!\cdots\!01 \)
|
| $11$ |
\( T^{16} + \cdots + 66\!\cdots\!56 \)
|
| $13$ |
\( (T^{8} + \cdots - 30\!\cdots\!92)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 39\!\cdots\!04 \)
|
| $19$ |
\( T^{16} + \cdots + 43\!\cdots\!76 \)
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| $23$ |
\( T^{16} + \cdots + 19\!\cdots\!04 \)
|
| $29$ |
\( (T^{8} + \cdots + 26\!\cdots\!12)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 90\!\cdots\!81 \)
|
| $37$ |
\( T^{16} + \cdots + 65\!\cdots\!96 \)
|
| $41$ |
\( (T^{8} + \cdots - 61\!\cdots\!36)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 23\!\cdots\!64 \)
|
| $47$ |
\( T^{16} + \cdots + 56\!\cdots\!24 \)
|
| $53$ |
\( T^{16} + \cdots + 72\!\cdots\!36 \)
|
| $59$ |
\( T^{16} + \cdots + 31\!\cdots\!16 \)
|
| $61$ |
\( T^{16} + \cdots + 15\!\cdots\!44 \)
|
| $67$ |
\( T^{16} + \cdots + 73\!\cdots\!56 \)
|
| $71$ |
\( T^{16} + \cdots + 61\!\cdots\!36 \)
|
| $73$ |
\( T^{16} + \cdots + 32\!\cdots\!36 \)
|
| $79$ |
\( T^{16} + \cdots + 13\!\cdots\!69 \)
|
| $83$ |
\( T^{16} + \cdots + 65\!\cdots\!44 \)
|
| $89$ |
\( T^{16} + \cdots + 12\!\cdots\!84 \)
|
| $97$ |
\( (T^{8} + \cdots + 12\!\cdots\!72)^{2} \)
|
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