Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [343,2,Mod(30,343)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(343, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([32]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("343.30");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 343.g (of order \(21\), degree \(12\), 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: | \(2.73886878933\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{21})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 49) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\zeta_{21}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 30.1 |
|
0.458528 | − | 0.312619i | −1.64715 | + | 1.52833i | −0.618165 | + | 1.57506i | −1.38084 | − | 0.425934i | −0.277479 | + | 1.21572i | 0 | 0.455927 | + | 1.99755i | 0.153116 | − | 2.04319i | −0.766310 | + | 0.236375i | ||||||||||||||||||||||||||||||||||||||
| 67.1 | 2.14715 | − | 0.662309i | −0.292981 | − | 0.746503i | 2.51913 | − | 1.71752i | −0.244221 | − | 0.0368104i | −1.12349 | − | 1.40881i | 0 | 1.46950 | − | 1.84270i | 1.72773 | − | 1.60310i | −0.548760 | + | 0.0827122i | |||||||||||||||||||||||||||||||||||||||
| 79.1 | 0.792981 | − | 0.119523i | 0.458528 | + | 0.312619i | −1.29661 | + | 0.399952i | −0.209389 | + | 2.79410i | 0.400969 | + | 0.193096i | 0 | −2.42543 | + | 1.16802i | −0.983506 | − | 2.50593i | 0.167917 | + | 2.24070i | |||||||||||||||||||||||||||||||||||||||
| 116.1 | 0.0414721 | + | 0.553406i | 2.14715 | + | 0.662309i | 1.67312 | − | 0.252183i | 1.05929 | − | 0.982878i | −0.277479 | + | 1.21572i | 0 | 0.455927 | + | 1.99755i | 1.69289 | + | 1.15420i | 0.587862 | + | 0.545456i | |||||||||||||||||||||||||||||||||||||||
| 128.1 | 2.14715 | + | 0.662309i | −0.292981 | + | 0.746503i | 2.51913 | + | 1.71752i | −0.244221 | + | 0.0368104i | −1.12349 | + | 1.40881i | 0 | 1.46950 | + | 1.84270i | 1.72773 | + | 1.60310i | −0.548760 | − | 0.0827122i | |||||||||||||||||||||||||||||||||||||||
| 165.1 | 0.792981 | + | 0.119523i | 0.458528 | − | 0.312619i | −1.29661 | − | 0.399952i | −0.209389 | − | 2.79410i | 0.400969 | − | 0.193096i | 0 | −2.42543 | − | 1.16802i | −0.983506 | + | 2.50593i | 0.167917 | − | 2.24070i | |||||||||||||||||||||||||||||||||||||||
| 177.1 | −1.64715 | − | 1.52833i | 0.792981 | + | 0.119523i | 0.227846 | + | 3.04039i | 0.0902318 | + | 0.229907i | −1.12349 | − | 1.40881i | 0 | 1.46950 | − | 1.84270i | −2.25219 | − | 0.694707i | 0.202749 | − | 0.516596i | |||||||||||||||||||||||||||||||||||||||
| 214.1 | −0.292981 | + | 0.746503i | 0.0414721 | − | 0.553406i | 0.994675 | + | 0.922924i | −2.31507 | − | 1.57839i | 0.400969 | + | 0.193096i | 0 | −2.42543 | + | 1.16802i | 2.66195 | + | 0.401225i | 1.85654 | − | 1.26577i | |||||||||||||||||||||||||||||||||||||||
| 226.1 | −0.292981 | − | 0.746503i | 0.0414721 | + | 0.553406i | 0.994675 | − | 0.922924i | −2.31507 | + | 1.57839i | 0.400969 | − | 0.193096i | 0 | −2.42543 | − | 1.16802i | 2.66195 | − | 0.401225i | 1.85654 | + | 1.26577i | |||||||||||||||||||||||||||||||||||||||
| 263.1 | 0.458528 | + | 0.312619i | −1.64715 | − | 1.52833i | −0.618165 | − | 1.57506i | −1.38084 | + | 0.425934i | −0.277479 | − | 1.21572i | 0 | 0.455927 | − | 1.99755i | 0.153116 | + | 2.04319i | −0.766310 | − | 0.236375i | |||||||||||||||||||||||||||||||||||||||
| 275.1 | 0.0414721 | − | 0.553406i | 2.14715 | − | 0.662309i | 1.67312 | + | 0.252183i | 1.05929 | + | 0.982878i | −0.277479 | − | 1.21572i | 0 | 0.455927 | − | 1.99755i | 1.69289 | − | 1.15420i | 0.587862 | − | 0.545456i | |||||||||||||||||||||||||||||||||||||||
| 312.1 | −1.64715 | + | 1.52833i | 0.792981 | − | 0.119523i | 0.227846 | − | 3.04039i | 0.0902318 | − | 0.229907i | −1.12349 | + | 1.40881i | 0 | 1.46950 | + | 1.84270i | −2.25219 | + | 0.694707i | 0.202749 | + | 0.516596i | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 49.e | even | 7 | 1 | inner |
| 49.g | even | 21 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 343.2.g.f | 12 | |
| 7.b | odd | 2 | 1 | 343.2.g.e | 12 | ||
| 7.c | even | 3 | 1 | 49.2.e.a | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 343.2.g.f | 12 | |
| 7.d | odd | 6 | 1 | 343.2.e.a | 6 | ||
| 7.d | odd | 6 | 1 | 343.2.g.e | 12 | ||
| 21.h | odd | 6 | 1 | 441.2.u.a | 6 | ||
| 28.g | odd | 6 | 1 | 784.2.u.a | 6 | ||
| 49.e | even | 7 | 1 | inner | 343.2.g.f | 12 | |
| 49.f | odd | 14 | 1 | 343.2.g.e | 12 | ||
| 49.g | even | 21 | 1 | 49.2.e.a | ✓ | 6 | |
| 49.g | even | 21 | 1 | inner | 343.2.g.f | 12 | |
| 49.g | even | 21 | 1 | 2401.2.a.b | 3 | ||
| 49.h | odd | 42 | 1 | 343.2.e.a | 6 | ||
| 49.h | odd | 42 | 1 | 343.2.g.e | 12 | ||
| 49.h | odd | 42 | 1 | 2401.2.a.a | 3 | ||
| 147.n | odd | 42 | 1 | 441.2.u.a | 6 | ||
| 196.o | odd | 42 | 1 | 784.2.u.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.2.e.a | ✓ | 6 | 7.c | even | 3 | 1 | |
| 49.2.e.a | ✓ | 6 | 49.g | even | 21 | 1 | |
| 343.2.e.a | 6 | 7.d | odd | 6 | 1 | ||
| 343.2.e.a | 6 | 49.h | odd | 42 | 1 | ||
| 343.2.g.e | 12 | 7.b | odd | 2 | 1 | ||
| 343.2.g.e | 12 | 7.d | odd | 6 | 1 | ||
| 343.2.g.e | 12 | 49.f | odd | 14 | 1 | ||
| 343.2.g.e | 12 | 49.h | odd | 42 | 1 | ||
| 343.2.g.f | 12 | 1.a | even | 1 | 1 | trivial | |
| 343.2.g.f | 12 | 7.c | even | 3 | 1 | inner | |
| 343.2.g.f | 12 | 49.e | even | 7 | 1 | inner | |
| 343.2.g.f | 12 | 49.g | even | 21 | 1 | inner | |
| 441.2.u.a | 6 | 21.h | odd | 6 | 1 | ||
| 441.2.u.a | 6 | 147.n | odd | 42 | 1 | ||
| 784.2.u.a | 6 | 28.g | odd | 6 | 1 | ||
| 784.2.u.a | 6 | 196.o | odd | 42 | 1 | ||
| 2401.2.a.a | 3 | 49.h | odd | 42 | 1 | ||
| 2401.2.a.b | 3 | 49.g | even | 21 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):
|
\( T_{2}^{12} - 3 T_{2}^{11} - T_{2}^{9} + 31 T_{2}^{8} - 56 T_{2}^{7} + 57 T_{2}^{6} - 56 T_{2}^{5} + \cdots + 1 \)
|
|
\( T_{3}^{12} - 3 T_{3}^{11} - T_{3}^{9} + 31 T_{3}^{8} - 56 T_{3}^{7} + 57 T_{3}^{6} - 56 T_{3}^{5} + \cdots + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 3 T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots + 1 \)
|
| $5$ |
\( T^{12} + 6 T^{11} + \cdots + 1 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} + 2 T^{11} + \cdots + 28561 \)
|
| $13$ |
\( (T^{6} + 14 T^{4} + \cdots + 41209)^{2} \)
|
| $17$ |
\( T^{12} + 4 T^{11} + \cdots + 2825761 \)
|
| $19$ |
\( (T^{6} - 4 T^{5} + 27 T^{4} + \cdots + 1)^{2} \)
|
| $23$ |
\( T^{12} - 10 T^{11} + \cdots + 28561 \)
|
| $29$ |
\( (T^{6} + 16 T^{5} + \cdots + 6889)^{2} \)
|
| $31$ |
\( (T^{6} - 5 T^{5} + 19 T^{4} + \cdots + 1)^{2} \)
|
| $37$ |
\( T^{12} + \cdots + 116985856 \)
|
| $41$ |
\( (T^{6} + 56 T^{4} + \cdots + 3136)^{2} \)
|
| $43$ |
\( (T^{6} - 12 T^{5} + \cdots + 841)^{2} \)
|
| $47$ |
\( T^{12} - 15 T^{11} + \cdots + 88529281 \)
|
| $53$ |
\( T^{12} + \cdots + 2897022976 \)
|
| $59$ |
\( T^{12} + \cdots + 3262808641 \)
|
| $61$ |
\( T^{12} - 8 T^{11} + \cdots + 3418801 \)
|
| $67$ |
\( (T^{6} - 6 T^{5} + \cdots + 169)^{2} \)
|
| $71$ |
\( (T^{6} - 5 T^{5} + \cdots + 85849)^{2} \)
|
| $73$ |
\( T^{12} + 4 T^{11} + \cdots + 25411681 \)
|
| $79$ |
\( (T^{6} + 30 T^{5} + \cdots + 877969)^{2} \)
|
| $83$ |
\( (T^{6} + 14 T^{5} + \cdots + 625681)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 35152125121 \)
|
| $97$ |
\( (T^{3} - 7 T - 7)^{4} \)
|
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