Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(245,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.245");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.h (of order \(10\), degree \(4\), 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: | \(9.89103359628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{20}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 245.1 |
|
−0.587785 | − | 0.809017i | 1.50859 | − | 2.59310i | 0.927051 | − | 2.85317i | 2.57565 | − | 3.54508i | −2.98459 | + | 0.303706i | −0.663119 | + | 2.04087i | −6.65740 | + | 2.16312i | −4.44829 | − | 7.82385i | −4.38197 | ||||||||||||||||||||||||||
| 245.2 | 0.587785 | + | 0.809017i | −2.74466 | − | 1.21113i | 0.927051 | − | 2.85317i | −2.57565 | + | 3.54508i | −0.633446 | − | 2.93236i | −0.663119 | + | 2.04087i | 6.65740 | − | 2.16312i | 6.06633 | + | 6.64828i | −4.38197 | |||||||||||||||||||||||||||
| 251.1 | −0.951057 | − | 0.309017i | 2.93236 | + | 0.633446i | −2.42705 | − | 1.76336i | 6.29412 | − | 2.04508i | −2.59310 | − | 1.50859i | 7.16312 | + | 5.20431i | 4.11450 | + | 5.66312i | 8.19749 | + | 3.71499i | −6.61803 | |||||||||||||||||||||||||||
| 251.2 | 0.951057 | + | 0.309017i | 0.303706 | − | 2.98459i | −2.42705 | − | 1.76336i | −6.29412 | + | 2.04508i | 1.21113 | − | 2.74466i | 7.16312 | + | 5.20431i | −4.11450 | − | 5.66312i | −8.81553 | − | 1.81288i | −6.61803 | |||||||||||||||||||||||||||
| 269.1 | −0.951057 | + | 0.309017i | 2.93236 | − | 0.633446i | −2.42705 | + | 1.76336i | 6.29412 | + | 2.04508i | −2.59310 | + | 1.50859i | 7.16312 | − | 5.20431i | 4.11450 | − | 5.66312i | 8.19749 | − | 3.71499i | −6.61803 | |||||||||||||||||||||||||||
| 269.2 | 0.951057 | − | 0.309017i | 0.303706 | + | 2.98459i | −2.42705 | + | 1.76336i | −6.29412 | − | 2.04508i | 1.21113 | + | 2.74466i | 7.16312 | − | 5.20431i | −4.11450 | + | 5.66312i | −8.81553 | + | 1.81288i | −6.61803 | |||||||||||||||||||||||||||
| 323.1 | −0.587785 | + | 0.809017i | 1.50859 | + | 2.59310i | 0.927051 | + | 2.85317i | 2.57565 | + | 3.54508i | −2.98459 | − | 0.303706i | −0.663119 | − | 2.04087i | −6.65740 | − | 2.16312i | −4.44829 | + | 7.82385i | −4.38197 | |||||||||||||||||||||||||||
| 323.2 | 0.587785 | − | 0.809017i | −2.74466 | + | 1.21113i | 0.927051 | + | 2.85317i | −2.57565 | − | 3.54508i | −0.633446 | + | 2.93236i | −0.663119 | − | 2.04087i | 6.65740 | + | 2.16312i | 6.06633 | − | 6.64828i | −4.38197 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 33.h | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.3.h.h | 8 | |
| 3.b | odd | 2 | 1 | inner | 363.3.h.h | 8 | |
| 11.b | odd | 2 | 1 | 363.3.h.i | 8 | ||
| 11.c | even | 5 | 2 | 33.3.h.a | ✓ | 8 | |
| 11.c | even | 5 | 1 | 363.3.b.f | 4 | ||
| 11.c | even | 5 | 1 | inner | 363.3.h.h | 8 | |
| 11.d | odd | 10 | 1 | 363.3.b.g | 4 | ||
| 11.d | odd | 10 | 2 | 363.3.h.g | 8 | ||
| 11.d | odd | 10 | 1 | 363.3.h.i | 8 | ||
| 33.d | even | 2 | 1 | 363.3.h.i | 8 | ||
| 33.f | even | 10 | 1 | 363.3.b.g | 4 | ||
| 33.f | even | 10 | 2 | 363.3.h.g | 8 | ||
| 33.f | even | 10 | 1 | 363.3.h.i | 8 | ||
| 33.h | odd | 10 | 2 | 33.3.h.a | ✓ | 8 | |
| 33.h | odd | 10 | 1 | 363.3.b.f | 4 | ||
| 33.h | odd | 10 | 1 | inner | 363.3.h.h | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.3.h.a | ✓ | 8 | 11.c | even | 5 | 2 | |
| 33.3.h.a | ✓ | 8 | 33.h | odd | 10 | 2 | |
| 363.3.b.f | 4 | 11.c | even | 5 | 1 | ||
| 363.3.b.f | 4 | 33.h | odd | 10 | 1 | ||
| 363.3.b.g | 4 | 11.d | odd | 10 | 1 | ||
| 363.3.b.g | 4 | 33.f | even | 10 | 1 | ||
| 363.3.h.g | 8 | 11.d | odd | 10 | 2 | ||
| 363.3.h.g | 8 | 33.f | even | 10 | 2 | ||
| 363.3.h.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 363.3.h.h | 8 | 3.b | odd | 2 | 1 | inner | |
| 363.3.h.h | 8 | 11.c | even | 5 | 1 | inner | |
| 363.3.h.h | 8 | 33.h | odd | 10 | 1 | inner | |
| 363.3.h.i | 8 | 11.b | odd | 2 | 1 | ||
| 363.3.h.i | 8 | 11.d | odd | 10 | 1 | ||
| 363.3.h.i | 8 | 33.d | even | 2 | 1 | ||
| 363.3.h.i | 8 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \)
|
|
\( T_{5}^{8} - 59T_{5}^{6} + 1446T_{5}^{4} - 3364T_{5}^{2} + 707281 \)
|
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\( T_{7}^{4} - 13T_{7}^{3} + 64T_{7}^{2} + 38T_{7} + 361 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 6561 \)
|
| $5$ |
\( T^{8} - 59 T^{6} + \cdots + 707281 \)
|
| $7$ |
\( (T^{4} - 13 T^{3} + \cdots + 361)^{2} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T^{4} - 6 T^{3} + \cdots + 13456)^{2} \)
|
| $17$ |
\( T^{8} - 36 T^{6} + \cdots + 1679616 \)
|
| $19$ |
\( (T^{4} + 34 T^{3} + \cdots + 80656)^{2} \)
|
| $23$ |
\( (T^{4} + 1008 T^{2} + 215296)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 2449228130001 \)
|
| $31$ |
\( (T^{4} - 46 T^{3} + \cdots + 36481)^{2} \)
|
| $37$ |
\( (T^{4} - 50 T^{3} + \cdots + 336400)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 3544535296 \)
|
| $43$ |
\( (T^{2} + 66 T + 1044)^{4} \)
|
| $47$ |
\( T^{8} + 164 T^{6} + \cdots + 33362176 \)
|
| $53$ |
\( T^{8} + \cdots + 3969126001 \)
|
| $59$ |
\( T^{8} + \cdots + 276281640625 \)
|
| $61$ |
\( (T^{4} + 102 T^{3} + \cdots + 6533136)^{2} \)
|
| $67$ |
\( (T^{2} + 6 T - 5436)^{4} \)
|
| $71$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $73$ |
\( (T^{4} - 14 T^{3} + \cdots + 866761)^{2} \)
|
| $79$ |
\( (T^{4} - 175 T^{3} + \cdots + 55428025)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 22\!\cdots\!01 \)
|
| $89$ |
\( (T^{4} + 9632 T^{2} + 891136)^{2} \)
|
| $97$ |
\( (T^{4} + 237 T^{3} + \cdots + 147403881)^{2} \)
|
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