Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,5,Mod(43,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.43");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 84.g (of order \(2\), degree \(1\), 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: | \(8.68307689904\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −3.98722 | − | 0.319546i | − | 5.19615i | 15.7958 | + | 2.54820i | 40.3533 | −1.66041 | + | 20.7182i | 18.5203i | −62.1669 | − | 15.2077i | −27.0000 | −160.897 | − | 12.8947i | |||||||
43.2 | −3.98722 | + | 0.319546i | 5.19615i | 15.7958 | − | 2.54820i | 40.3533 | −1.66041 | − | 20.7182i | − | 18.5203i | −62.1669 | + | 15.2077i | −27.0000 | −160.897 | + | 12.8947i | |||||||
43.3 | −3.81725 | − | 1.19525i | − | 5.19615i | 13.1428 | + | 9.12512i | −7.49303 | −6.21069 | + | 19.8350i | − | 18.5203i | −39.2624 | − | 50.5417i | −27.0000 | 28.6027 | + | 8.95603i | ||||||
43.4 | −3.81725 | + | 1.19525i | 5.19615i | 13.1428 | − | 9.12512i | −7.49303 | −6.21069 | − | 19.8350i | 18.5203i | −39.2624 | + | 50.5417i | −27.0000 | 28.6027 | − | 8.95603i | ||||||||
43.5 | −3.63258 | − | 1.67461i | 5.19615i | 10.3913 | + | 12.1664i | −41.3815 | 8.70155 | − | 18.8755i | − | 18.5203i | −17.3735 | − | 61.5968i | −27.0000 | 150.322 | + | 69.2981i | |||||||
43.6 | −3.63258 | + | 1.67461i | − | 5.19615i | 10.3913 | − | 12.1664i | −41.3815 | 8.70155 | + | 18.8755i | 18.5203i | −17.3735 | + | 61.5968i | −27.0000 | 150.322 | − | 69.2981i | |||||||
43.7 | −3.26473 | − | 2.31117i | 5.19615i | 5.31694 | + | 15.0907i | 10.0790 | 12.0092 | − | 16.9640i | 18.5203i | 17.5189 | − | 61.5556i | −27.0000 | −32.9051 | − | 23.2943i | ||||||||
43.8 | −3.26473 | + | 2.31117i | − | 5.19615i | 5.31694 | − | 15.0907i | 10.0790 | 12.0092 | + | 16.9640i | − | 18.5203i | 17.5189 | + | 61.5556i | −27.0000 | −32.9051 | + | 23.2943i | ||||||
43.9 | −1.52999 | − | 3.69583i | − | 5.19615i | −11.3182 | + | 11.3092i | −22.1164 | −19.2041 | + | 7.95007i | − | 18.5203i | 59.1135 | + | 24.5273i | −27.0000 | 33.8379 | + | 81.7383i | ||||||
43.10 | −1.52999 | + | 3.69583i | 5.19615i | −11.3182 | − | 11.3092i | −22.1164 | −19.2041 | − | 7.95007i | 18.5203i | 59.1135 | − | 24.5273i | −27.0000 | 33.8379 | − | 81.7383i | ||||||||
43.11 | −1.25775 | − | 3.79711i | 5.19615i | −12.8361 | + | 9.55162i | 5.76271 | 19.7304 | − | 6.53545i | − | 18.5203i | 52.4132 | + | 36.7268i | −27.0000 | −7.24803 | − | 21.8817i | |||||||
43.12 | −1.25775 | + | 3.79711i | − | 5.19615i | −12.8361 | − | 9.55162i | 5.76271 | 19.7304 | + | 6.53545i | 18.5203i | 52.4132 | − | 36.7268i | −27.0000 | −7.24803 | + | 21.8817i | |||||||
43.13 | 0.113690 | − | 3.99838i | 5.19615i | −15.9741 | − | 0.909150i | 19.7185 | 20.7762 | + | 0.590749i | 18.5203i | −5.45122 | + | 63.7674i | −27.0000 | 2.24179 | − | 78.8421i | ||||||||
43.14 | 0.113690 | + | 3.99838i | − | 5.19615i | −15.9741 | + | 0.909150i | 19.7185 | 20.7762 | − | 0.590749i | − | 18.5203i | −5.45122 | − | 63.7674i | −27.0000 | 2.24179 | + | 78.8421i | ||||||
43.15 | 1.64066 | − | 3.64804i | 5.19615i | −10.6165 | − | 11.9704i | −17.1026 | 18.9558 | + | 8.52513i | − | 18.5203i | −61.0866 | + | 19.0899i | −27.0000 | −28.0596 | + | 62.3910i | |||||||
43.16 | 1.64066 | + | 3.64804i | − | 5.19615i | −10.6165 | + | 11.9704i | −17.1026 | 18.9558 | − | 8.52513i | 18.5203i | −61.0866 | − | 19.0899i | −27.0000 | −28.0596 | − | 62.3910i | |||||||
43.17 | 1.76690 | − | 3.58860i | − | 5.19615i | −9.75616 | − | 12.6814i | −22.5825 | −18.6469 | − | 9.18106i | 18.5203i | −62.7466 | + | 12.6043i | −27.0000 | −39.9010 | + | 81.0398i | |||||||
43.18 | 1.76690 | + | 3.58860i | 5.19615i | −9.75616 | + | 12.6814i | −22.5825 | −18.6469 | + | 9.18106i | − | 18.5203i | −62.7466 | − | 12.6043i | −27.0000 | −39.9010 | − | 81.0398i | |||||||
43.19 | 3.20050 | − | 2.39933i | − | 5.19615i | 4.48640 | − | 15.3581i | 28.1610 | −12.4673 | − | 16.6303i | − | 18.5203i | −22.4906 | − | 59.9181i | −27.0000 | 90.1292 | − | 67.5676i | ||||||
43.20 | 3.20050 | + | 2.39933i | 5.19615i | 4.48640 | + | 15.3581i | 28.1610 | −12.4673 | + | 16.6303i | 18.5203i | −22.4906 | + | 59.9181i | −27.0000 | 90.1292 | + | 67.5676i | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 84.5.g.a | ✓ | 24 |
3.b | odd | 2 | 1 | 252.5.g.d | 24 | ||
4.b | odd | 2 | 1 | inner | 84.5.g.a | ✓ | 24 |
8.b | even | 2 | 1 | 1344.5.m.e | 24 | ||
8.d | odd | 2 | 1 | 1344.5.m.e | 24 | ||
12.b | even | 2 | 1 | 252.5.g.d | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
84.5.g.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
84.5.g.a | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
252.5.g.d | 24 | 3.b | odd | 2 | 1 | ||
252.5.g.d | 24 | 12.b | even | 2 | 1 | ||
1344.5.m.e | 24 | 8.b | even | 2 | 1 | ||
1344.5.m.e | 24 | 8.d | odd | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(84, [\chi])\).