Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,3,Mod(3,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([5, 24]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.3");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 77.p (of order \(30\), degree \(8\), 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: | \(2.09809803557\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{30})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{30}]$ |
$q$-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 | −3.47800 | − | 1.54850i | 0.405840 | − | 1.90933i | 7.02207 | + | 7.79880i | 4.28573 | + | 0.450449i | −4.36811 | + | 6.01219i | 2.91525 | + | 6.36407i | −7.64037 | − | 23.5147i | 4.74108 | + | 2.11087i | −14.2082 | − | 8.20313i |
3.2 | −3.14125 | − | 1.39857i | −0.904545 | + | 4.25555i | 5.23491 | + | 5.81396i | −2.97205 | − | 0.312375i | 8.79310 | − | 12.1027i | −0.982273 | − | 6.93074i | −4.06266 | − | 12.5036i | −9.06958 | − | 4.03804i | 8.89907 | + | 5.13788i |
3.3 | −2.13423 | − | 0.950219i | 0.866650 | − | 4.07727i | 0.975483 | + | 1.08338i | −7.57520 | − | 0.796185i | −5.72392 | + | 7.87830i | 6.99898 | − | 0.119278i | 1.83525 | + | 5.64833i | −7.65111 | − | 3.40649i | 15.4106 | + | 8.89733i |
3.4 | −2.11301 | − | 0.940772i | −0.286000 | + | 1.34552i | 0.903232 | + | 1.00314i | 0.485505 | + | 0.0510287i | 1.87015 | − | 2.57404i | −4.83539 | + | 5.06153i | 1.89419 | + | 5.82970i | 6.49328 | + | 2.89099i | −0.977871 | − | 0.564574i |
3.5 | −1.63381 | − | 0.727421i | 0.994951 | − | 4.68088i | −0.536317 | − | 0.595640i | 7.45896 | + | 0.783969i | −5.03053 | + | 6.92393i | −5.98810 | − | 3.62528i | 2.65358 | + | 8.16689i | −12.6988 | − | 5.65386i | −11.6163 | − | 6.70666i |
3.6 | −1.39802 | − | 0.622439i | −0.407108 | + | 1.91529i | −1.10949 | − | 1.23221i | 3.14966 | + | 0.331043i | 1.76130 | − | 2.42422i | 6.37630 | − | 2.88839i | 2.67570 | + | 8.23495i | 4.71931 | + | 2.10117i | −4.19724 | − | 2.42328i |
3.7 | −0.169983 | − | 0.0756813i | 0.0447743 | − | 0.210647i | −2.65336 | − | 2.94685i | −7.93821 | − | 0.834340i | −0.0235529 | + | 0.0324177i | −6.97723 | + | 0.564195i | 0.457998 | + | 1.40957i | 8.17954 | + | 3.64177i | 1.28622 | + | 0.742598i |
3.8 | 0.333205 | + | 0.148352i | −1.16799 | + | 5.49496i | −2.58751 | − | 2.87372i | −5.30093 | − | 0.557150i | −1.20437 | + | 1.65767i | 4.75117 | + | 5.14066i | −0.886687 | − | 2.72894i | −20.6085 | − | 9.17550i | −1.68364 | − | 0.972049i |
3.9 | 0.710777 | + | 0.316458i | 0.455331 | − | 2.14216i | −2.27146 | − | 2.52272i | 1.67507 | + | 0.176057i | 1.00154 | − | 1.37851i | 1.90441 | − | 6.73596i | −1.77788 | − | 5.47176i | 3.84038 | + | 1.70985i | 1.13489 | + | 0.655228i |
3.10 | 1.25182 | + | 0.557345i | −0.248510 | + | 1.16915i | −1.42011 | − | 1.57719i | 8.69060 | + | 0.913418i | −0.962708 | + | 1.32505i | −0.656163 | + | 6.96918i | −2.59244 | − | 7.97872i | 6.91676 | + | 3.07954i | 10.3699 | + | 5.98709i |
3.11 | 1.35964 | + | 0.605349i | 0.923126 | − | 4.34296i | −1.19436 | − | 1.32647i | −1.52806 | − | 0.160606i | 3.88412 | − | 5.34604i | 3.55163 | + | 6.03207i | −2.66057 | − | 8.18838i | −9.78727 | − | 4.35757i | −1.98039 | − | 1.14338i |
3.12 | 2.43731 | + | 1.08516i | −0.812940 | + | 3.82458i | 2.08639 | + | 2.31717i | 2.08180 | + | 0.218806i | −6.13168 | + | 8.43953i | −4.71907 | − | 5.17015i | −0.727112 | − | 2.23782i | −5.74466 | − | 2.55769i | 4.83655 | + | 2.79239i |
3.13 | 2.86109 | + | 1.27384i | −0.315131 | + | 1.48258i | 3.88664 | + | 4.31656i | −4.70921 | − | 0.494958i | −2.79018 | + | 3.84036i | 6.97032 | + | 0.643947i | 1.75026 | + | 5.38674i | 6.12319 | + | 2.72622i | −12.8430 | − | 7.41490i |
3.14 | 3.07171 | + | 1.36761i | 0.674115 | − | 3.17146i | 4.88852 | + | 5.42926i | −1.65544 | − | 0.173994i | 6.40802 | − | 8.81989i | −6.98372 | + | 0.477162i | 3.43485 | + | 10.5714i | −1.38183 | − | 0.615230i | −4.84708 | − | 2.79846i |
5.1 | −3.67332 | − | 0.780787i | 4.52797 | + | 0.475909i | 9.22944 | + | 4.10921i | −2.23369 | + | 2.01122i | −16.2611 | − | 5.28355i | −0.790956 | + | 6.95517i | −18.5416 | − | 13.4712i | 11.4727 | + | 2.43860i | 9.77538 | − | 5.64382i |
5.2 | −3.42306 | − | 0.727594i | −4.13871 | − | 0.434996i | 7.53377 | + | 3.35425i | −6.41701 | + | 5.77790i | 13.8506 | + | 4.50032i | −0.243758 | − | 6.99575i | −12.0233 | − | 8.73544i | 8.13640 | + | 1.72945i | 26.1698 | − | 15.1091i |
5.3 | −2.88156 | − | 0.612495i | −0.771458 | − | 0.0810835i | 4.27406 | + | 1.90294i | 3.48954 | − | 3.14199i | 2.17334 | + | 0.706161i | −6.97684 | + | 0.568896i | −1.61719 | − | 1.17496i | −8.21476 | − | 1.74610i | −11.9798 | + | 6.91652i |
5.4 | −2.08046 | − | 0.442214i | 2.46033 | + | 0.258591i | 0.478559 | + | 0.213068i | 1.04600 | − | 0.941822i | −5.00425 | − | 1.62598i | 5.96372 | − | 3.66525i | 5.98150 | + | 4.34582i | −2.81699 | − | 0.598769i | −2.59264 | + | 1.49686i |
5.5 | −1.85255 | − | 0.393773i | −4.21020 | − | 0.442510i | −0.377280 | − | 0.167976i | 1.58739 | − | 1.42930i | 7.62537 | + | 2.47763i | 4.15912 | + | 5.63043i | 6.76171 | + | 4.91267i | 8.72662 | + | 1.85490i | −3.50355 | + | 2.02278i |
5.6 | −0.979296 | − | 0.208156i | 0.761101 | + | 0.0799950i | −2.73849 | − | 1.21925i | −5.47727 | + | 4.93175i | −0.728692 | − | 0.236766i | −0.164579 | + | 6.99806i | 5.66787 | + | 4.11795i | −8.23045 | − | 1.74944i | 6.39044 | − | 3.68952i |
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
11.c | even | 5 | 1 | inner |
77.p | odd | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 77.3.p.a | ✓ | 112 |
7.d | odd | 6 | 1 | inner | 77.3.p.a | ✓ | 112 |
11.c | even | 5 | 1 | inner | 77.3.p.a | ✓ | 112 |
77.p | odd | 30 | 1 | inner | 77.3.p.a | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.3.p.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
77.3.p.a | ✓ | 112 | 7.d | odd | 6 | 1 | inner |
77.3.p.a | ✓ | 112 | 11.c | even | 5 | 1 | inner |
77.3.p.a | ✓ | 112 | 77.p | odd | 30 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(77, [\chi])\).