Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,4,Mod(26,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.26");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bj (of order \(6\), degree \(2\), 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: | \(18.5856016518\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
26.1 | −4.79668 | − | 2.76937i | 0 | 11.3388 | + | 19.6393i | 2.50000 | − | 4.33013i | 0 | −3.51757 | − | 18.1831i | − | 81.2950i | 0 | −23.9834 | + | 13.8468i | |||||||
26.2 | −4.28337 | − | 2.47301i | 0 | 8.23152 | + | 14.2574i | 2.50000 | − | 4.33013i | 0 | −12.4677 | + | 13.6952i | − | 41.8583i | 0 | −21.4169 | + | 12.3650i | |||||||
26.3 | −3.10158 | − | 1.79070i | 0 | 2.41320 | + | 4.17978i | 2.50000 | − | 4.33013i | 0 | 18.5132 | − | 0.511080i | 11.3659i | 0 | −15.5079 | + | 8.95349i | ||||||||
26.4 | −2.82648 | − | 1.63187i | 0 | 1.32598 | + | 2.29666i | 2.50000 | − | 4.33013i | 0 | −18.2872 | + | 2.92918i | 17.4546i | 0 | −14.1324 | + | 8.15933i | ||||||||
26.5 | −2.71573 | − | 1.56793i | 0 | 0.916790 | + | 1.58793i | 2.50000 | − | 4.33013i | 0 | 15.0868 | − | 10.7418i | 19.3370i | 0 | −13.5786 | + | 7.83964i | ||||||||
26.6 | −1.57393 | − | 0.908710i | 0 | −2.34849 | − | 4.06771i | 2.50000 | − | 4.33013i | 0 | −16.6185 | − | 8.17469i | 23.0758i | 0 | −7.86966 | + | 4.54355i | ||||||||
26.7 | −1.45507 | − | 0.840086i | 0 | −2.58851 | − | 4.48343i | 2.50000 | − | 4.33013i | 0 | 2.62579 | + | 18.3332i | 22.1397i | 0 | −7.27536 | + | 4.20043i | ||||||||
26.8 | −0.452924 | − | 0.261496i | 0 | −3.86324 | − | 6.69133i | 2.50000 | − | 4.33013i | 0 | 1.23679 | + | 18.4789i | 8.22481i | 0 | −2.26462 | + | 1.30748i | ||||||||
26.9 | −0.0946322 | − | 0.0546359i | 0 | −3.99403 | − | 6.91786i | 2.50000 | − | 4.33013i | 0 | 2.69161 | − | 18.3236i | 1.74705i | 0 | −0.473161 | + | 0.273180i | ||||||||
26.10 | 1.23944 | + | 0.715594i | 0 | −2.97585 | − | 5.15433i | 2.50000 | − | 4.33013i | 0 | 16.7856 | − | 7.82585i | − | 19.9675i | 0 | 6.19722 | − | 3.57797i | |||||||
26.11 | 2.02851 | + | 1.17116i | 0 | −1.25677 | − | 2.17679i | 2.50000 | − | 4.33013i | 0 | −7.09670 | + | 17.1066i | − | 24.6261i | 0 | 10.1425 | − | 5.85580i | |||||||
26.12 | 2.89621 | + | 1.67213i | 0 | 1.59202 | + | 2.75747i | 2.50000 | − | 4.33013i | 0 | −11.0741 | − | 14.8447i | − | 16.1058i | 0 | 14.4811 | − | 8.36064i | |||||||
26.13 | 2.91070 | + | 1.68049i | 0 | 1.64813 | + | 2.85464i | 2.50000 | − | 4.33013i | 0 | −14.9406 | + | 10.9443i | − | 15.8092i | 0 | 14.5535 | − | 8.40247i | |||||||
26.14 | 3.50335 | + | 2.02266i | 0 | 4.18231 | + | 7.24398i | 2.50000 | − | 4.33013i | 0 | −5.54534 | − | 17.6706i | 1.47503i | 0 | 17.5168 | − | 10.1133i | ||||||||
26.15 | 4.24064 | + | 2.44833i | 0 | 7.98866 | + | 13.8368i | 2.50000 | − | 4.33013i | 0 | 16.8273 | + | 7.73571i | 39.0623i | 0 | 21.2032 | − | 12.2417i | ||||||||
26.16 | 4.48155 | + | 2.58742i | 0 | 9.38951 | + | 16.2631i | 2.50000 | − | 4.33013i | 0 | −6.21952 | + | 17.4447i | 55.7798i | 0 | 22.4077 | − | 12.9371i | ||||||||
206.1 | −4.79668 | + | 2.76937i | 0 | 11.3388 | − | 19.6393i | 2.50000 | + | 4.33013i | 0 | −3.51757 | + | 18.1831i | 81.2950i | 0 | −23.9834 | − | 13.8468i | ||||||||
206.2 | −4.28337 | + | 2.47301i | 0 | 8.23152 | − | 14.2574i | 2.50000 | + | 4.33013i | 0 | −12.4677 | − | 13.6952i | 41.8583i | 0 | −21.4169 | − | 12.3650i | ||||||||
206.3 | −3.10158 | + | 1.79070i | 0 | 2.41320 | − | 4.17978i | 2.50000 | + | 4.33013i | 0 | 18.5132 | + | 0.511080i | − | 11.3659i | 0 | −15.5079 | − | 8.95349i | |||||||
206.4 | −2.82648 | + | 1.63187i | 0 | 1.32598 | − | 2.29666i | 2.50000 | + | 4.33013i | 0 | −18.2872 | − | 2.92918i | − | 17.4546i | 0 | −14.1324 | − | 8.15933i | |||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.4.bj.b | yes | 32 |
3.b | odd | 2 | 1 | 315.4.bj.a | ✓ | 32 | |
7.d | odd | 6 | 1 | 315.4.bj.a | ✓ | 32 | |
21.g | even | 6 | 1 | inner | 315.4.bj.b | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.4.bj.a | ✓ | 32 | 3.b | odd | 2 | 1 | |
315.4.bj.a | ✓ | 32 | 7.d | odd | 6 | 1 | |
315.4.bj.b | yes | 32 | 1.a | even | 1 | 1 | trivial |
315.4.bj.b | yes | 32 | 21.g | even | 6 | 1 | inner |