Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,4,Mod(5,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.5");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.g (of order \(22\), degree \(10\), 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: | \(4.07113179040\) |
Analytic rank: | \(0\) |
Dimension: | \(220\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.55934 | + | 5.31062i | 3.37983 | − | 3.94674i | −19.0411 | − | 12.2370i | 1.69511 | − | 11.7897i | 15.6894 | + | 24.1033i | 10.2590 | − | 4.68513i | 61.2142 | − | 53.0424i | −4.15356 | − | 26.6786i | 59.9675 | + | 27.3862i |
5.2 | −1.48758 | + | 5.06624i | −5.07448 | + | 1.11787i | −16.7239 | − | 10.7478i | −2.59342 | + | 18.0376i | 1.88530 | − | 27.3715i | 2.05975 | − | 0.940657i | 47.4055 | − | 41.0771i | 24.5007 | − | 11.3452i | −87.5251 | − | 39.9714i |
5.3 | −1.20747 | + | 4.11226i | −2.48738 | + | 4.56212i | −8.72270 | − | 5.60574i | 2.62405 | − | 18.2506i | −15.7572 | − | 15.7374i | −25.3847 | + | 11.5928i | 7.67232 | − | 6.64810i | −14.6259 | − | 22.6954i | 71.8830 | + | 32.8279i |
5.4 | −1.17465 | + | 4.00049i | 4.31812 | + | 2.89031i | −7.89411 | − | 5.07323i | −0.800687 | + | 5.56890i | −16.6349 | + | 13.8795i | −5.95353 | + | 2.71889i | 4.36018 | − | 3.77812i | 10.2923 | + | 24.9614i | −21.3378 | − | 9.74465i |
5.5 | −1.00406 | + | 3.41950i | −0.818289 | − | 5.13132i | −3.95481 | − | 2.54160i | −1.20932 | + | 8.41101i | 18.3681 | + | 2.35399i | −15.7004 | + | 7.17015i | −8.88523 | + | 7.69910i | −25.6608 | + | 8.39780i | −27.5472 | − | 12.5804i |
5.6 | −0.978191 | + | 3.33141i | −4.71623 | − | 2.18110i | −3.41142 | − | 2.19239i | 1.70238 | − | 11.8403i | 11.8795 | − | 13.5782i | 22.3835 | − | 10.2222i | −10.3513 | + | 8.96943i | 17.4856 | + | 20.5731i | 37.7798 | + | 17.2535i |
5.7 | −0.698369 | + | 2.37843i | 4.35121 | − | 2.84024i | 1.56084 | + | 1.00309i | −0.0326676 | + | 0.227208i | 3.71656 | + | 12.3326i | 16.1821 | − | 7.39011i | −18.4629 | + | 15.9982i | 10.8660 | − | 24.7170i | −0.517584 | − | 0.236373i |
5.8 | −0.579088 | + | 1.97219i | −1.85709 | + | 4.85296i | 3.17583 | + | 2.04098i | −1.18604 | + | 8.24906i | −8.49555 | − | 6.47283i | 14.0781 | − | 6.42924i | −18.2916 | + | 15.8497i | −20.1024 | − | 18.0248i | −15.5819 | − | 7.11602i |
5.9 | −0.222958 | + | 0.759325i | 4.12739 | + | 3.15668i | 6.20316 | + | 3.98653i | 2.75670 | − | 19.1733i | −3.31718 | + | 2.43022i | 15.1346 | − | 6.91176i | −9.19481 | + | 7.96735i | 7.07070 | + | 26.0577i | 13.9441 | + | 6.36807i |
5.10 | −0.201291 | + | 0.685535i | −5.17034 | − | 0.517272i | 6.30059 | + | 4.04914i | 0.256284 | − | 1.78249i | 1.39535 | − | 3.44033i | −23.2362 | + | 10.6116i | −8.36380 | + | 7.24728i | 26.4649 | + | 5.34895i | 1.17037 | + | 0.534491i |
5.11 | −0.0243207 | + | 0.0828288i | 0.514506 | − | 5.17062i | 6.72376 | + | 4.32110i | 2.06531 | − | 14.3646i | 0.415763 | + | 0.168369i | −11.4770 | + | 5.24135i | −1.04336 | + | 0.904078i | −26.4706 | − | 5.32063i | 1.13957 | + | 0.520424i |
5.12 | 0.0243207 | − | 0.0828288i | 5.04477 | − | 1.24512i | 6.72376 | + | 4.32110i | −2.06531 | + | 14.3646i | 0.0195602 | − | 0.448134i | −11.4770 | + | 5.24135i | 1.04336 | − | 0.904078i | 23.8993 | − | 12.5627i | 1.13957 | + | 0.520424i |
5.13 | 0.201291 | − | 0.685535i | 1.24782 | + | 5.04410i | 6.30059 | + | 4.04914i | −0.256284 | + | 1.78249i | 3.70908 | + | 0.159907i | −23.2362 | + | 10.6116i | 8.36380 | − | 7.24728i | −23.8859 | + | 12.5883i | 1.17037 | + | 0.534491i |
5.14 | 0.222958 | − | 0.759325i | −3.71194 | − | 3.63614i | 6.20316 | + | 3.98653i | −2.75670 | + | 19.1733i | −3.58862 | + | 2.00786i | 15.1346 | − | 6.91176i | 9.19481 | − | 7.96735i | 0.557026 | + | 26.9943i | 13.9441 | + | 6.36807i |
5.15 | 0.579088 | − | 1.97219i | −4.53927 | + | 2.52884i | 3.17583 | + | 2.04098i | 1.18604 | − | 8.24906i | 2.35872 | + | 10.4167i | 14.0781 | − | 6.42924i | 18.2916 | − | 15.8497i | 14.2100 | − | 22.9582i | −15.5819 | − | 7.11602i |
5.16 | 0.698369 | − | 2.37843i | 2.19209 | − | 4.71113i | 1.56084 | + | 1.00309i | 0.0326676 | − | 0.227208i | −9.67418 | − | 8.50383i | 16.1821 | − | 7.39011i | 18.4629 | − | 15.9982i | −17.3895 | − | 20.6545i | −0.517584 | − | 0.236373i |
5.17 | 0.978191 | − | 3.33141i | 2.83009 | + | 4.35782i | −3.41142 | − | 2.19239i | −1.70238 | + | 11.8403i | 17.2861 | − | 5.16540i | 22.3835 | − | 10.2222i | 10.3513 | − | 8.96943i | −10.9812 | + | 24.6660i | 37.7798 | + | 17.2535i |
5.18 | 1.00406 | − | 3.41950i | 5.19554 | + | 0.0796979i | −3.95481 | − | 2.54160i | 1.20932 | − | 8.41101i | 5.48914 | − | 17.6861i | −15.7004 | + | 7.17015i | 8.88523 | − | 7.69910i | 26.9873 | + | 0.828147i | −27.5472 | − | 12.5804i |
5.19 | 1.17465 | − | 4.00049i | −3.47542 | − | 3.86283i | −7.89411 | − | 5.07323i | 0.800687 | − | 5.56890i | −19.5356 | + | 9.36591i | −5.95353 | + | 2.71889i | −4.36018 | + | 3.77812i | −2.84293 | + | 26.8499i | −21.3378 | − | 9.74465i |
5.20 | 1.20747 | − | 4.11226i | −4.16170 | + | 3.11132i | −8.72270 | − | 5.60574i | −2.62405 | + | 18.2506i | 7.76943 | + | 20.8708i | −25.3847 | + | 11.5928i | −7.67232 | + | 6.64810i | 7.63942 | − | 25.8967i | 71.8830 | + | 32.8279i |
See next 80 embeddings (of 220 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.4.g.a | ✓ | 220 |
3.b | odd | 2 | 1 | inner | 69.4.g.a | ✓ | 220 |
23.d | odd | 22 | 1 | inner | 69.4.g.a | ✓ | 220 |
69.g | even | 22 | 1 | inner | 69.4.g.a | ✓ | 220 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.4.g.a | ✓ | 220 | 1.a | even | 1 | 1 | trivial |
69.4.g.a | ✓ | 220 | 3.b | odd | 2 | 1 | inner |
69.4.g.a | ✓ | 220 | 23.d | odd | 22 | 1 | inner |
69.4.g.a | ✓ | 220 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(69, [\chi])\).