Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,14,Mod(9,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.c (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: | \(15.0123300533\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + \cdots + 75\!\cdots\!25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 209077 x^{6} - 47718852 x^{5} + 40973427094 x^{4} - 4988457209802 x^{3} + \cdots + 75\!\cdots\!25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 36\!\cdots\!43 \nu^{7} + \cdots - 22\!\cdots\!25 ) / 44\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 19\!\cdots\!19 \nu^{7} + \cdots + 20\!\cdots\!25 ) / 65\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 15\!\cdots\!46 \nu^{7} + \cdots - 17\!\cdots\!19 ) / 12\!\cdots\!59 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\!\cdots\!77 \nu^{7} + \cdots + 16\!\cdots\!05 ) / 17\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 20\!\cdots\!81 \nu^{7} + \cdots - 85\!\cdots\!05 ) / 44\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 64\!\cdots\!91 \nu^{7} + \cdots - 12\!\cdots\!35 ) / 58\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 53\!\cdots\!25 \nu^{7} + \cdots - 92\!\cdots\!35 ) / 42\!\cdots\!24 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - 3\beta_{4} - \beta_{3} - 50\beta_{2} - 24\beta _1 + 1 ) / 336 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 890\beta_{7} - 1345\beta_{6} - 435\beta_{5} - 262\beta_{3} + 193\beta_{2} - 35125512\beta _1 - 35126402 ) / 336 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 7549 \beta_{7} + 30461 \beta_{6} + 38010 \beta_{5} + 38010 \beta_{4} - 329923 \beta_{3} + \cdots + 375640003 ) / 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 305067991 \beta_{7} + 71270609 \beta_{6} - 162526773 \beta_{4} + 71270609 \beta_{3} + \cdots + 305067991 ) / 336 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 117654471422 \beta_{7} - 105997118719 \beta_{6} - 129311824125 \beta_{5} + 985682722142 \beta_{3} + \cdots - 20\!\cdots\!02 ) / 336 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 96184064357 \beta_{7} + 518498001245 \beta_{6} + 422313936888 \beta_{5} + 422313936888 \beta_{4} + \cdots + 12\!\cdots\!70 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 28\!\cdots\!09 \beta_{7} - 405888753844057 \beta_{6} + \cdots + 28\!\cdots\!09 ) / 336 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
32.0000 | − | 55.4256i | −746.854 | − | 1293.59i | −2048.00 | − | 3547.24i | −7178.20 | + | 12433.0i | −95597.3 | −302505. | − | 73348.4i | −262144. | −318420. | + | 551520.i | 459405. | + | 795712.i | ||||||||||||||||||||||||||||
| 9.2 | 32.0000 | − | 55.4256i | −244.702 | − | 423.837i | −2048.00 | − | 3547.24i | 34096.3 | − | 59056.5i | −31321.9 | 100530. | + | 294589.i | −262144. | 677403. | − | 1.17330e6i | −2.18216e6 | − | 3.77962e6i | |||||||||||||||||||||||||||||
| 9.3 | 32.0000 | − | 55.4256i | 194.343 | + | 336.611i | −2048.00 | − | 3547.24i | −12951.1 | + | 22432.0i | 24875.9 | 286520. | − | 121636.i | −262144. | 721623. | − | 1.24989e6i | 828871. | + | 1.43565e6i | |||||||||||||||||||||||||||||
| 9.4 | 32.0000 | − | 55.4256i | 888.214 | + | 1538.43i | −2048.00 | − | 3547.24i | −13071.0 | + | 22639.6i | 113691. | −218721. | + | 221472.i | −262144. | −780686. | + | 1.35219e6i | 836543. | + | 1.44893e6i | |||||||||||||||||||||||||||||
| 11.1 | 32.0000 | + | 55.4256i | −746.854 | + | 1293.59i | −2048.00 | + | 3547.24i | −7178.20 | − | 12433.0i | −95597.3 | −302505. | + | 73348.4i | −262144. | −318420. | − | 551520.i | 459405. | − | 795712.i | |||||||||||||||||||||||||||||
| 11.2 | 32.0000 | + | 55.4256i | −244.702 | + | 423.837i | −2048.00 | + | 3547.24i | 34096.3 | + | 59056.5i | −31321.9 | 100530. | − | 294589.i | −262144. | 677403. | + | 1.17330e6i | −2.18216e6 | + | 3.77962e6i | |||||||||||||||||||||||||||||
| 11.3 | 32.0000 | + | 55.4256i | 194.343 | − | 336.611i | −2048.00 | + | 3547.24i | −12951.1 | − | 22432.0i | 24875.9 | 286520. | + | 121636.i | −262144. | 721623. | + | 1.24989e6i | 828871. | − | 1.43565e6i | |||||||||||||||||||||||||||||
| 11.4 | 32.0000 | + | 55.4256i | 888.214 | − | 1538.43i | −2048.00 | + | 3547.24i | −13071.0 | − | 22639.6i | 113691. | −218721. | − | 221472.i | −262144. | −780686. | − | 1.35219e6i | 836543. | − | 1.44893e6i | |||||||||||||||||||||||||||||
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 | 14.14.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 126.14.g.b | 8 | ||
| 7.b | odd | 2 | 1 | 98.14.c.o | 8 | ||
| 7.c | even | 3 | 1 | inner | 14.14.c.b | ✓ | 8 |
| 7.c | even | 3 | 1 | 98.14.a.h | 4 | ||
| 7.d | odd | 6 | 1 | 98.14.a.j | 4 | ||
| 7.d | odd | 6 | 1 | 98.14.c.o | 8 | ||
| 21.h | odd | 6 | 1 | 126.14.g.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.14.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 14.14.c.b | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 98.14.a.h | 4 | 7.c | even | 3 | 1 | ||
| 98.14.a.j | 4 | 7.d | odd | 6 | 1 | ||
| 98.14.c.o | 8 | 7.b | odd | 2 | 1 | ||
| 98.14.c.o | 8 | 7.d | odd | 6 | 1 | ||
| 126.14.g.b | 8 | 3.b | odd | 2 | 1 | ||
| 126.14.g.b | 8 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 182 T_{3}^{7} + 2905288 T_{3}^{6} + 949684092 T_{3}^{5} + 7705719718857 T_{3}^{4} + \cdots + 25\!\cdots\!25 \)
acting on \(S_{14}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 64 T + 4096)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 25\!\cdots\!25 \)
|
| $5$ |
\( T^{8} + \cdots + 43\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 88\!\cdots\!01 \)
|
| $11$ |
\( T^{8} + \cdots + 48\!\cdots\!25 \)
|
| $13$ |
\( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 61\!\cdots\!41 \)
|
| $19$ |
\( T^{8} + \cdots + 12\!\cdots\!25 \)
|
| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!09 \)
|
| $29$ |
\( (T^{4} + \cdots - 94\!\cdots\!96)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 15\!\cdots\!25 \)
|
| $37$ |
\( T^{8} + \cdots + 82\!\cdots\!69 \)
|
| $41$ |
\( (T^{4} + \cdots - 96\!\cdots\!16)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 22\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 26\!\cdots\!21 \)
|
| $53$ |
\( T^{8} + \cdots + 91\!\cdots\!21 \)
|
| $59$ |
\( T^{8} + \cdots + 11\!\cdots\!25 \)
|
| $61$ |
\( T^{8} + \cdots + 90\!\cdots\!41 \)
|
| $67$ |
\( T^{8} + \cdots + 52\!\cdots\!21 \)
|
| $71$ |
\( (T^{4} + \cdots + 19\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 76\!\cdots\!25 \)
|
| $79$ |
\( T^{8} + \cdots + 10\!\cdots\!25 \)
|
| $83$ |
\( (T^{4} + \cdots - 29\!\cdots\!64)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 25\!\cdots\!61 \)
|
| $97$ |
\( (T^{4} + \cdots - 20\!\cdots\!80)^{2} \)
|
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