Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,3,Mod(99,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.99");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.d (of order \(4\), 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: | \(9.20983293538\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(6\) over \(\Q(i)\) |
| Coefficient field: | 12.0.4739148267126784.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 21x^{8} + 98x^{4} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 13^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 21x^{8} + 98x^{4} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{10} + 22\nu^{6} + 133\nu^{2} ) / 91 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{10} - \nu^{9} - \nu^{8} - 44\nu^{6} - 35\nu^{5} - 35\nu^{4} - 175\nu^{2} - 224\nu - 133 ) / 91 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{9} + 84\nu^{5} + 119\nu ) / 91 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{11} + 53\nu^{7} + 126\nu^{3} ) / 91 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{11} + 5\nu^{10} + 22\nu^{7} + 84\nu^{6} + 133\nu^{3} + 210\nu^{2} + 91\nu ) / 91 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{11} - \nu^{10} - \nu^{9} - 66\nu^{7} - 9\nu^{6} - 35\nu^{5} - 308\nu^{3} + 49\nu^{2} - 224\nu ) / 91 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{11} - \nu^{9} - 4\nu^{8} + 66\nu^{7} - 35\nu^{5} - 49\nu^{4} + 308\nu^{3} - 224\nu + 14 ) / 91 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -3\nu^{11} - 2\nu^{10} + \nu^{8} - 66\nu^{7} - 44\nu^{6} + 35\nu^{4} - 308\nu^{3} - 175\nu^{2} + 133 ) / 91 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 4\nu^{11} - 9\nu^{8} + 75\nu^{7} - 133\nu^{4} + 259\nu^{3} - 91\nu - 196 ) / 91 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 4\nu^{10} - 6\nu^{9} + 5\nu^{8} + 75\nu^{6} - 119\nu^{5} + 84\nu^{4} + 259\nu^{2} - 434\nu + 210 ) / 91 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 4\nu^{11} - 4\nu^{10} + 5\nu^{8} + 88\nu^{7} - 75\nu^{6} + 84\nu^{4} + 441\nu^{3} - 259\nu^{2} + 210 ) / 91 \)
|
| \(\nu\) | \(=\) |
\( ( -5\beta_{10} - 4\beta_{9} + 3\beta_{7} + 3\beta_{6} + 4\beta_{5} + 4\beta_{4} - 5\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 13 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{11} - \beta_{10} + 5\beta_{8} - 4\beta_{6} - \beta_{5} - \beta_{3} + 5\beta_{2} + 29\beta_1 ) / 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{11} + 2\beta_{10} - 2\beta_{9} + 23\beta_{8} + 21\beta_{7} + 2\beta_{4} + 2\beta_{3} - 23\beta_{2} - 84 ) / 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 45 \beta_{10} + 49 \beta_{9} - 53 \beta_{7} - 53 \beta_{6} - 49 \beta_{5} - 49 \beta_{4} + 32 \beta_{3} + \cdots + 4 ) / 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{8} + 7\beta_{6} - 7\beta_{2} - 21\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 168 \beta_{11} - 189 \beta_{9} - 21 \beta_{8} + 210 \beta_{7} - 210 \beta_{6} - 189 \beta_{5} + \cdots - 21 ) / 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 21 \beta_{11} + 21 \beta_{10} - 21 \beta_{9} - 350 \beta_{8} - 371 \beta_{7} + 21 \beta_{4} + \cdots + 938 ) / 13 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 49 \beta_{10} - 56 \beta_{9} + 63 \beta_{7} + 63 \beta_{6} + 56 \beta_{5} + 56 \beta_{4} - 14 \beta_{3} + \cdots - 7 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 133 \beta_{11} + 133 \beta_{10} + 1337 \beta_{8} - 1470 \beta_{6} + 133 \beta_{5} + \cdots + 3332 \beta_1 ) / 13 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 2422 \beta_{11} + 2793 \beta_{9} + 371 \beta_{8} - 3164 \beta_{7} + 3164 \beta_{6} + 2793 \beta_{5} + \cdots + 371 ) / 13 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
1.00000 | + | 1.00000i | −5.52382 | 2.00000i | 3.04055 | + | 3.04055i | −5.52382 | − | 5.52382i | −2.62254 | + | 2.62254i | −2.00000 | + | 2.00000i | 21.5126 | 6.08110i | ||||||||||||||||||||||||||||||||||||||||||||
| 99.2 | 1.00000 | + | 1.00000i | −3.23112 | 2.00000i | −0.725610 | − | 0.725610i | −3.23112 | − | 3.23112i | −6.96533 | + | 6.96533i | −2.00000 | + | 2.00000i | 1.44017 | − | 1.45122i | ||||||||||||||||||||||||||||||||||||||||||||
| 99.3 | 1.00000 | + | 1.00000i | −1.26283 | 2.00000i | 5.71353 | + | 5.71353i | −1.26283 | − | 1.26283i | −5.84630 | + | 5.84630i | −2.00000 | + | 2.00000i | −7.40525 | 11.4271i | |||||||||||||||||||||||||||||||||||||||||||||
| 99.4 | 1.00000 | + | 1.00000i | −0.728366 | 2.00000i | −0.573283 | − | 0.573283i | −0.728366 | − | 0.728366i | 1.71704 | − | 1.71704i | −2.00000 | + | 2.00000i | −8.46948 | − | 1.14657i | ||||||||||||||||||||||||||||||||||||||||||||
| 99.5 | 1.00000 | + | 1.00000i | 2.33224 | 2.00000i | −6.63447 | − | 6.63447i | 2.33224 | + | 2.33224i | −6.38729 | + | 6.38729i | −2.00000 | + | 2.00000i | −3.56065 | − | 13.2689i | ||||||||||||||||||||||||||||||||||||||||||||
| 99.6 | 1.00000 | + | 1.00000i | 4.41391 | 2.00000i | −4.82072 | − | 4.82072i | 4.41391 | + | 4.41391i | 8.10442 | − | 8.10442i | −2.00000 | + | 2.00000i | 10.4826 | − | 9.64144i | ||||||||||||||||||||||||||||||||||||||||||||
| 239.1 | 1.00000 | − | 1.00000i | −5.52382 | − | 2.00000i | 3.04055 | − | 3.04055i | −5.52382 | + | 5.52382i | −2.62254 | − | 2.62254i | −2.00000 | − | 2.00000i | 21.5126 | − | 6.08110i | |||||||||||||||||||||||||||||||||||||||||||
| 239.2 | 1.00000 | − | 1.00000i | −3.23112 | − | 2.00000i | −0.725610 | + | 0.725610i | −3.23112 | + | 3.23112i | −6.96533 | − | 6.96533i | −2.00000 | − | 2.00000i | 1.44017 | 1.45122i | ||||||||||||||||||||||||||||||||||||||||||||
| 239.3 | 1.00000 | − | 1.00000i | −1.26283 | − | 2.00000i | 5.71353 | − | 5.71353i | −1.26283 | + | 1.26283i | −5.84630 | − | 5.84630i | −2.00000 | − | 2.00000i | −7.40525 | − | 11.4271i | |||||||||||||||||||||||||||||||||||||||||||
| 239.4 | 1.00000 | − | 1.00000i | −0.728366 | − | 2.00000i | −0.573283 | + | 0.573283i | −0.728366 | + | 0.728366i | 1.71704 | + | 1.71704i | −2.00000 | − | 2.00000i | −8.46948 | 1.14657i | ||||||||||||||||||||||||||||||||||||||||||||
| 239.5 | 1.00000 | − | 1.00000i | 2.33224 | − | 2.00000i | −6.63447 | + | 6.63447i | 2.33224 | − | 2.33224i | −6.38729 | − | 6.38729i | −2.00000 | − | 2.00000i | −3.56065 | 13.2689i | ||||||||||||||||||||||||||||||||||||||||||||
| 239.6 | 1.00000 | − | 1.00000i | 4.41391 | − | 2.00000i | −4.82072 | + | 4.82072i | 4.41391 | − | 4.41391i | 8.10442 | + | 8.10442i | −2.00000 | − | 2.00000i | 10.4826 | 9.64144i | ||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 338.3.d.i | yes | 12 |
| 13.b | even | 2 | 1 | 338.3.d.h | ✓ | 12 | |
| 13.c | even | 3 | 2 | 338.3.f.k | 24 | ||
| 13.d | odd | 4 | 1 | 338.3.d.h | ✓ | 12 | |
| 13.d | odd | 4 | 1 | inner | 338.3.d.i | yes | 12 |
| 13.e | even | 6 | 2 | 338.3.f.l | 24 | ||
| 13.f | odd | 12 | 2 | 338.3.f.k | 24 | ||
| 13.f | odd | 12 | 2 | 338.3.f.l | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.3.d.h | ✓ | 12 | 13.b | even | 2 | 1 | |
| 338.3.d.h | ✓ | 12 | 13.d | odd | 4 | 1 | |
| 338.3.d.i | yes | 12 | 1.a | even | 1 | 1 | trivial |
| 338.3.d.i | yes | 12 | 13.d | odd | 4 | 1 | inner |
| 338.3.f.k | 24 | 13.c | even | 3 | 2 | ||
| 338.3.f.k | 24 | 13.f | odd | 12 | 2 | ||
| 338.3.f.l | 24 | 13.e | even | 6 | 2 | ||
| 338.3.f.l | 24 | 13.f | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):
|
\( T_{3}^{6} + 4T_{3}^{5} - 26T_{3}^{4} - 90T_{3}^{3} + 95T_{3}^{2} + 338T_{3} + 169 \)
|
|
\( T_{5}^{12} + 8 T_{5}^{11} + 32 T_{5}^{10} - 168 T_{5}^{9} + 5402 T_{5}^{8} + 35656 T_{5}^{7} + \cdots + 3418801 \)
|
|
\( T_{7}^{12} + 24 T_{7}^{11} + 288 T_{7}^{10} + 1932 T_{7}^{9} + 24009 T_{7}^{8} + 419528 T_{7}^{7} + \cdots + 5766427969 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{6} \)
|
| $3$ |
\( (T^{6} + 4 T^{5} + \cdots + 169)^{2} \)
|
| $5$ |
\( T^{12} + 8 T^{11} + \cdots + 3418801 \)
|
| $7$ |
\( T^{12} + \cdots + 5766427969 \)
|
| $11$ |
\( T^{12} + \cdots + 67458114529 \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} + \cdots + 249297494209 \)
|
| $19$ |
\( T^{12} + \cdots + 10872858529 \)
|
| $23$ |
\( T^{12} + \cdots + 131156022761569 \)
|
| $29$ |
\( (T^{6} + 14 T^{5} + \cdots - 1342159)^{2} \)
|
| $31$ |
\( T^{12} + \cdots + 65\!\cdots\!69 \)
|
| $37$ |
\( T^{12} + \cdots + 84\!\cdots\!49 \)
|
| $41$ |
\( T^{12} + \cdots + 29\!\cdots\!21 \)
|
| $43$ |
\( T^{12} + \cdots + 44\!\cdots\!29 \)
|
| $47$ |
\( T^{12} + \cdots + 820190087159521 \)
|
| $53$ |
\( (T^{6} - 94 T^{5} + \cdots - 4994639)^{2} \)
|
| $59$ |
\( T^{12} + \cdots + 26863727418529 \)
|
| $61$ |
\( (T^{6} - 152 T^{5} + \cdots + 5893577033)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 27588976855441 \)
|
| $71$ |
\( T^{12} + \cdots + 89\!\cdots\!41 \)
|
| $73$ |
\( T^{12} + \cdots + 27\!\cdots\!21 \)
|
| $79$ |
\( (T^{6} - 86 T^{5} + \cdots + 6816365417)^{2} \)
|
| $83$ |
\( T^{12} + \cdots + 51\!\cdots\!49 \)
|
| $89$ |
\( T^{12} + \cdots + 95\!\cdots\!89 \)
|
| $97$ |
\( T^{12} + \cdots + 47\!\cdots\!29 \)
|
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