Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,10,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), not 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: | \(25.2367559720\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 427x^{4} - 3606x^{3} + 183492x^{2} - 858816x + 4064256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 415\nu^{5} - 177205\nu^{4} - 1825733\nu^{3} - 76149180\nu^{2} + 356408640\nu - 18279850176 ) / 464953608 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -451\nu^{5} + 192577\nu^{4} - 4738111\nu^{3} + 82754892\nu^{2} - 387326016\nu + 30551662512 ) / 464953608 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4331\nu^{5} + 4283\nu^{4} - 1828841\nu^{3} + 6865794\nu^{2} - 785896236\nu - 41319936 ) / 3719628864 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -97909\nu^{5} - 629099\nu^{4} - 41343799\nu^{3} + 155211966\nu^{2} - 24361417764\nu - 934101504 ) / 1859814432 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 710\nu^{5} + 4339\nu^{4} + 299810\nu^{3} - 1125540\nu^{2} + 97141857\nu + 6773760 ) / 12915378 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} - 2\beta_{3} ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -25\beta_{5} + 11\beta_{4} - 1706\beta_{3} + 11\beta_{2} - 25\beta _1 - 1706 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -415\beta_{2} - 451\beta _1 + 9538 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9085\beta_{5} - 6287\beta_{4} + 713186\beta_{3} ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 150811\beta_{5} + 192679\beta_{4} - 6789298\beta_{3} + 192679\beta_{2} + 150811\beta _1 - 6789298 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−20.9010 | − | 36.2015i | −0.116170 | + | 0.201212i | −617.701 | + | 1069.89i | 895.944 | + | 1551.82i | 9.71222 | 0 | 30239.6 | 9841.47 | + | 17045.9i | 37452.2 | − | 64869.1i | ||||||||||||||||||||||||
| 18.2 | −6.68036 | − | 11.5707i | −81.7073 | + | 141.521i | 166.746 | − | 288.812i | −961.094 | − | 1664.66i | 2183.34 | 0 | −11296.4 | −3510.66 | − | 6080.64i | −12840.9 | + | 22241.1i | |||||||||||||||||||||||||
| 18.3 | 17.0813 | + | 29.5857i | 39.8234 | − | 68.9762i | −327.544 | + | 567.323i | −711.850 | − | 1232.96i | 2720.95 | 0 | −4888.28 | 6669.69 | + | 11552.2i | 24318.7 | − | 42121.2i | |||||||||||||||||||||||||
| 30.1 | −20.9010 | + | 36.2015i | −0.116170 | − | 0.201212i | −617.701 | − | 1069.89i | 895.944 | − | 1551.82i | 9.71222 | 0 | 30239.6 | 9841.47 | − | 17045.9i | 37452.2 | + | 64869.1i | |||||||||||||||||||||||||
| 30.2 | −6.68036 | + | 11.5707i | −81.7073 | − | 141.521i | 166.746 | + | 288.812i | −961.094 | + | 1664.66i | 2183.34 | 0 | −11296.4 | −3510.66 | + | 6080.64i | −12840.9 | − | 22241.1i | |||||||||||||||||||||||||
| 30.3 | 17.0813 | − | 29.5857i | 39.8234 | + | 68.9762i | −327.544 | − | 567.323i | −711.850 | + | 1232.96i | 2720.95 | 0 | −4888.28 | 6669.69 | − | 11552.2i | 24318.7 | + | 42121.2i | |||||||||||||||||||||||||
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 | 49.10.c.d | 6 | |
| 7.b | odd | 2 | 1 | 49.10.c.e | 6 | ||
| 7.c | even | 3 | 1 | 7.10.a.b | ✓ | 3 | |
| 7.c | even | 3 | 1 | inner | 49.10.c.d | 6 | |
| 7.d | odd | 6 | 1 | 49.10.a.c | 3 | ||
| 7.d | odd | 6 | 1 | 49.10.c.e | 6 | ||
| 21.h | odd | 6 | 1 | 63.10.a.e | 3 | ||
| 28.g | odd | 6 | 1 | 112.10.a.h | 3 | ||
| 35.j | even | 6 | 1 | 175.10.a.d | 3 | ||
| 35.l | odd | 12 | 2 | 175.10.b.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.a.b | ✓ | 3 | 7.c | even | 3 | 1 | |
| 49.10.a.c | 3 | 7.d | odd | 6 | 1 | ||
| 49.10.c.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 49.10.c.d | 6 | 7.c | even | 3 | 1 | inner | |
| 49.10.c.e | 6 | 7.b | odd | 2 | 1 | ||
| 49.10.c.e | 6 | 7.d | odd | 6 | 1 | ||
| 63.10.a.e | 3 | 21.h | odd | 6 | 1 | ||
| 112.10.a.h | 3 | 28.g | odd | 6 | 1 | ||
| 175.10.a.d | 3 | 35.j | even | 6 | 1 | ||
| 175.10.b.d | 6 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{6} + 21T_{2}^{5} + 1767T_{2}^{4} + 10314T_{2}^{3} + 2158956T_{2}^{2} + 25300080T_{2} + 364046400 \)
|
|
\( T_{3}^{6} + 84T_{3}^{5} + 20052T_{3}^{4} - 1085616T_{3}^{3} + 169150032T_{3}^{2} + 39299904T_{3} + 9144576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 21 T^{5} + \cdots + 364046400 \)
|
| $3$ |
\( T^{6} + 84 T^{5} + \cdots + 9144576 \)
|
| $5$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!04 \)
|
| $13$ |
\( (T^{3} + \cdots - 41548412541440)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 48\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 94\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots + 44\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 55\!\cdots\!56 \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots - 19\!\cdots\!12)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 68\!\cdots\!80)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 19\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 57\!\cdots\!84 \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 26\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 41\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 16\!\cdots\!80)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 38\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots + 18\!\cdots\!48)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 49\!\cdots\!16)^{2} \)
|
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