Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [270,4,Mod(17,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.17");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 270.m (of order \(12\), 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: | \(15.9305157015\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 90) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | 8.83318 | − | 6.85382i | 0 | −6.42552 | + | 23.9804i | −5.65685 | − | 5.65685i | 0 | −20.6122 | + | 8.66817i | ||||||
17.2 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | −3.66689 | + | 10.5619i | 0 | 4.59296 | − | 17.1411i | −5.65685 | − | 5.65685i | 0 | 12.5511 | − | 18.5059i | ||||||
17.3 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | 11.1581 | + | 0.704786i | 0 | −3.01094 | + | 11.2370i | −5.65685 | − | 5.65685i | 0 | −21.1910 | − | 7.13740i | ||||||
17.4 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | −10.4719 | − | 3.91650i | 0 | −1.06330 | + | 3.96829i | −5.65685 | − | 5.65685i | 0 | 18.2029 | + | 12.9868i | ||||||
17.5 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | 0.302381 | + | 11.1763i | 0 | 2.28208 | − | 8.51682i | −5.65685 | − | 5.65685i | 0 | 5.20110 | − | 21.7474i | ||||||
17.6 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | 0.744935 | − | 11.1555i | 0 | −3.80534 | + | 14.2017i | −5.65685 | − | 5.65685i | 0 | −7.21361 | + | 21.1652i | ||||||
17.7 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | −10.4745 | + | 3.90957i | 0 | −4.71196 | + | 17.5853i | −5.65685 | − | 5.65685i | 0 | 22.2589 | − | 2.13070i | ||||||
17.8 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | −4.14145 | − | 10.3850i | 0 | 8.50201 | − | 31.7299i | −5.65685 | − | 5.65685i | 0 | 2.62500 | + | 22.2061i | ||||||
17.9 | −1.93185 | − | 0.517638i | 0 | 3.46410 | + | 2.00000i | 11.1803 | − | 0.0416934i | 0 | 8.57118 | − | 31.9881i | −5.65685 | − | 5.65685i | 0 | −21.6202 | − | 5.70678i | ||||||
17.10 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | 11.1621 | + | 0.639246i | 0 | 6.82827 | − | 25.4835i | 5.65685 | + | 5.65685i | 0 | 21.2325 | + | 7.01283i | ||||||
17.11 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | −8.43508 | + | 7.33821i | 0 | 2.96302 | − | 11.0581i | 5.65685 | + | 5.65685i | 0 | −20.0939 | + | 9.81002i | ||||||
17.12 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | 4.23604 | + | 10.3468i | 0 | 2.69511 | − | 10.0583i | 5.65685 | + | 5.65685i | 0 | 2.82751 | + | 22.1812i | ||||||
17.13 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | −7.32550 | − | 8.44612i | 0 | 1.57990 | − | 5.89626i | 5.65685 | + | 5.65685i | 0 | −9.77975 | − | 20.1086i | ||||||
17.14 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | 5.40078 | − | 9.78936i | 0 | 0.447095 | − | 1.66858i | 5.65685 | + | 5.65685i | 0 | 15.5008 | − | 16.1159i | ||||||
17.15 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | −10.3242 | − | 4.29078i | 0 | 0.319128 | − | 1.19100i | 5.65685 | + | 5.65685i | 0 | −17.7238 | − | 13.6334i | ||||||
17.16 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | 10.7798 | − | 2.96568i | 0 | −5.36661 | + | 20.0285i | 5.65685 | + | 5.65685i | 0 | 22.3602 | − | 0.149197i | ||||||
17.17 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | −6.55934 | − | 9.05401i | 0 | −7.99699 | + | 29.8452i | 5.65685 | + | 5.65685i | 0 | −7.98497 | − | 20.8864i | ||||||
17.18 | 1.93185 | + | 0.517638i | 0 | 3.46410 | + | 2.00000i | 4.52953 | + | 10.2217i | 0 | −6.40009 | + | 23.8855i | 5.65685 | + | 5.65685i | 0 | 3.45924 | + | 22.0915i | ||||||
143.1 | −1.93185 | + | 0.517638i | 0 | 3.46410 | − | 2.00000i | 8.83318 | + | 6.85382i | 0 | −6.42552 | − | 23.9804i | −5.65685 | + | 5.65685i | 0 | −20.6122 | − | 8.66817i | ||||||
143.2 | −1.93185 | + | 0.517638i | 0 | 3.46410 | − | 2.00000i | −3.66689 | − | 10.5619i | 0 | 4.59296 | + | 17.1411i | −5.65685 | + | 5.65685i | 0 | 12.5511 | + | 18.5059i | ||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 270.4.m.a | 72 | |
3.b | odd | 2 | 1 | 90.4.l.a | ✓ | 72 | |
5.c | odd | 4 | 1 | inner | 270.4.m.a | 72 | |
9.c | even | 3 | 1 | 90.4.l.a | ✓ | 72 | |
9.d | odd | 6 | 1 | inner | 270.4.m.a | 72 | |
15.e | even | 4 | 1 | 90.4.l.a | ✓ | 72 | |
45.k | odd | 12 | 1 | 90.4.l.a | ✓ | 72 | |
45.l | even | 12 | 1 | inner | 270.4.m.a | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.4.l.a | ✓ | 72 | 3.b | odd | 2 | 1 | |
90.4.l.a | ✓ | 72 | 9.c | even | 3 | 1 | |
90.4.l.a | ✓ | 72 | 15.e | even | 4 | 1 | |
90.4.l.a | ✓ | 72 | 45.k | odd | 12 | 1 | |
270.4.m.a | 72 | 1.a | even | 1 | 1 | trivial | |
270.4.m.a | 72 | 5.c | odd | 4 | 1 | inner | |
270.4.m.a | 72 | 9.d | odd | 6 | 1 | inner | |
270.4.m.a | 72 | 45.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(270, [\chi])\).