Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,4,Mod(3,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 13]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.k (of order \(18\), degree \(6\), 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: | \(4.48414516044\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −2.82429 | + | 0.152912i | 5.54644 | − | 2.01874i | 7.95324 | − | 0.863738i | 2.56623 | − | 14.5538i | −15.3561 | + | 6.54962i | −1.22017 | + | 0.704465i | −22.3302 | + | 3.65560i | 6.00448 | − | 5.03835i | −5.02232 | + | 41.4966i |
3.2 | −2.77864 | + | 0.528343i | −2.57091 | + | 0.935735i | 7.44171 | − | 2.93615i | −1.56986 | + | 8.90313i | 6.64925 | − | 3.95839i | 16.8987 | − | 9.75650i | −19.1266 | + | 12.0903i | −14.9492 | + | 12.5439i | −0.341819 | − | 25.5680i |
3.3 | −2.67496 | + | 0.919012i | −7.61194 | + | 2.77052i | 6.31083 | − | 4.91664i | 2.53263 | − | 14.3633i | 17.8155 | − | 14.4065i | −10.1706 | + | 5.87199i | −12.3628 | + | 18.9516i | 29.5827 | − | 24.8228i | 6.42533 | + | 40.7487i |
3.4 | −2.66917 | − | 0.935695i | 2.28426 | − | 0.831401i | 6.24895 | + | 4.99506i | −1.38264 | + | 7.84134i | −6.87501 | + | 0.0817851i | −8.80596 | + | 5.08413i | −12.0057 | − | 19.1798i | −16.1566 | + | 13.5570i | 11.0276 | − | 19.6361i |
3.5 | −2.42254 | − | 1.45990i | −6.72563 | + | 2.44793i | 3.73740 | + | 7.07332i | −0.791289 | + | 4.48762i | 19.8668 | + | 3.88852i | −1.86219 | + | 1.07514i | 1.27231 | − | 22.5916i | 18.5585 | − | 15.5724i | 8.46839 | − | 9.71624i |
3.6 | −2.38811 | + | 1.51557i | 9.08416 | − | 3.30637i | 3.40609 | − | 7.23868i | −3.54781 | + | 20.1206i | −16.6829 | + | 21.6636i | 4.44622 | − | 2.56703i | 2.83662 | + | 22.4489i | 50.9068 | − | 42.7159i | −22.0217 | − | 53.4271i |
3.7 | −1.90584 | + | 2.08992i | 1.16156 | − | 0.422774i | −0.735541 | − | 7.96611i | 0.336107 | − | 1.90616i | −1.33019 | + | 3.23331i | −25.4625 | + | 14.7008i | 18.0504 | + | 13.6449i | −19.5127 | + | 16.3731i | 3.34316 | + | 4.33527i |
3.8 | −1.72722 | + | 2.23980i | −1.16156 | + | 0.422774i | −2.03339 | − | 7.73727i | 0.336107 | − | 1.90616i | 1.05935 | − | 3.33189i | 25.4625 | − | 14.7008i | 20.8420 | + | 8.80962i | −19.5127 | + | 16.3731i | 3.68888 | + | 4.04518i |
3.9 | −1.70042 | − | 2.26022i | 5.85727 | − | 2.13187i | −2.21716 | + | 7.68663i | 0.184577 | − | 1.04679i | −14.7783 | − | 9.61362i | 24.9926 | − | 14.4295i | 21.1435 | − | 8.05922i | 9.07949 | − | 7.61860i | −2.67983 | + | 1.36279i |
3.10 | −1.58740 | − | 2.34097i | −1.55643 | + | 0.566493i | −2.96030 | + | 7.43213i | 2.88655 | − | 16.3705i | 3.79682 | + | 2.74430i | −17.3301 | + | 10.0055i | 22.0976 | − | 4.86782i | −18.5816 | + | 15.5919i | −42.9049 | + | 19.2292i |
3.11 | −1.07786 | + | 2.61500i | −9.08416 | + | 3.30637i | −5.67646 | − | 5.63718i | −3.54781 | + | 20.1206i | 1.14527 | − | 27.3189i | −4.44622 | + | 2.56703i | 20.8596 | − | 8.76787i | 50.9068 | − | 42.7159i | −48.7914 | − | 30.9646i |
3.12 | −0.518452 | − | 2.78050i | 2.06434 | − | 0.751360i | −7.46241 | + | 2.88312i | −3.33172 | + | 18.8951i | −3.15942 | − | 5.35037i | −15.6472 | + | 9.03393i | 11.8854 | + | 19.2545i | −16.9862 | + | 14.2531i | 54.2653 | − | 0.532358i |
3.13 | −0.440548 | + | 2.79391i | 7.61194 | − | 2.77052i | −7.61183 | − | 2.46170i | 2.53263 | − | 14.3633i | 4.38715 | + | 22.4876i | 10.1706 | − | 5.87199i | 10.2311 | − | 20.1823i | 29.5827 | − | 24.8228i | 39.0139 | + | 13.4036i |
3.14 | −0.428429 | − | 2.79579i | −6.50066 | + | 2.36605i | −7.63290 | + | 2.39560i | −0.324438 | + | 1.83998i | 9.40005 | + | 17.1608i | 19.7617 | − | 11.4094i | 9.96774 | + | 20.3136i | 15.9773 | − | 13.4065i | 5.28320 | + | 0.118761i |
3.15 | −0.0378098 | + | 2.82817i | 2.57091 | − | 0.935735i | −7.99714 | − | 0.213866i | −1.56986 | + | 8.90313i | 2.54921 | + | 7.30636i | −16.8987 | + | 9.75650i | 0.907220 | − | 22.6092i | −14.9492 | + | 12.5439i | −25.1202 | − | 4.77647i |
3.16 | 0.311236 | − | 2.81125i | 8.99625 | − | 3.27437i | −7.80626 | − | 1.74992i | 1.82627 | − | 10.3573i | −6.40511 | − | 26.3098i | −21.9414 | + | 12.6679i | −7.34906 | + | 21.4007i | 49.5278 | − | 41.5588i | −28.5485 | − | 8.35765i |
3.17 | 0.339844 | + | 2.80794i | −5.54644 | + | 2.01874i | −7.76901 | + | 1.90852i | 2.56623 | − | 14.5538i | −7.55341 | − | 14.8880i | 1.22017 | − | 0.704465i | −7.99925 | − | 21.1663i | 6.00448 | − | 5.03835i | 41.7383 | + | 2.25979i |
3.18 | 0.844697 | − | 2.69935i | −1.12992 | + | 0.411256i | −6.57297 | − | 4.56026i | 2.09426 | − | 11.8771i | 0.155687 | + | 3.39743i | 6.44819 | − | 3.72286i | −17.8619 | + | 13.8907i | −19.5756 | + | 16.4259i | −30.2915 | − | 15.6857i |
3.19 | 1.38498 | + | 2.46614i | −2.28426 | + | 0.831401i | −4.16368 | + | 6.83109i | −1.38264 | + | 7.84134i | −5.21399 | − | 4.48182i | 8.80596 | − | 5.08413i | −22.6130 | − | 0.807318i | −16.1566 | + | 13.5570i | −21.2527 | + | 7.45029i |
3.20 | 1.71026 | − | 2.25278i | −5.58264 | + | 2.03191i | −2.15002 | − | 7.70567i | −1.53918 | + | 8.72914i | −4.97031 | + | 16.0515i | −18.0252 | + | 10.4069i | −21.0363 | − | 8.33517i | 6.35395 | − | 5.33160i | 17.0324 | + | 18.3965i |
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.f | odd | 18 | 1 | inner |
76.k | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.4.k.a | ✓ | 168 |
4.b | odd | 2 | 1 | inner | 76.4.k.a | ✓ | 168 |
19.f | odd | 18 | 1 | inner | 76.4.k.a | ✓ | 168 |
76.k | even | 18 | 1 | inner | 76.4.k.a | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.4.k.a | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
76.4.k.a | ✓ | 168 | 4.b | odd | 2 | 1 | inner |
76.4.k.a | ✓ | 168 | 19.f | odd | 18 | 1 | inner |
76.4.k.a | ✓ | 168 | 76.k | even | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).