Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [229,4,Mod(95,229)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(229, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("229.95");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 229 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 229.e (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: | \(13.5114373913\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(56\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
95.1 | − | 5.59890i | 2.13688 | + | 3.70118i | −23.3477 | 0.397054 | − | 0.687718i | 20.7225 | − | 11.9642i | −23.3838 | − | 13.5007i | 85.9303i | 4.36752 | − | 7.56476i | −3.85047 | − | 2.22307i | |||||
95.2 | − | 5.24809i | 1.35071 | + | 2.33949i | −19.5425 | 3.56085 | − | 6.16757i | 12.2779 | − | 7.08863i | 26.9437 | + | 15.5559i | 60.5760i | 9.85119 | − | 17.0628i | −32.3680 | − | 18.6877i | |||||
95.3 | − | 5.16927i | −3.57187 | − | 6.18665i | −18.7214 | −2.27892 | + | 3.94721i | −31.9805 | + | 18.4640i | −12.1333 | − | 7.00516i | 55.4219i | −12.0164 | + | 20.8131i | 20.4042 | + | 11.7804i | |||||
95.4 | − | 4.89891i | −2.60564 | − | 4.51310i | −15.9994 | 9.09963 | − | 15.7610i | −22.1093 | + | 12.7648i | 7.54611 | + | 4.35675i | 39.1882i | −0.0787278 | + | 0.136360i | −77.2119 | − | 44.5783i | |||||
95.5 | − | 4.88435i | −0.341672 | − | 0.591793i | −15.8569 | −9.80902 | + | 16.9897i | −2.89053 | + | 1.66885i | 9.50542 | + | 5.48796i | 38.3757i | 13.2665 | − | 22.9783i | 82.9838 | + | 47.9107i | |||||
95.6 | − | 4.57710i | 4.53130 | + | 7.84844i | −12.9499 | 9.29894 | − | 16.1062i | 35.9231 | − | 20.7402i | −9.47962 | − | 5.47306i | 22.6560i | −27.5653 | + | 47.7446i | −73.7199 | − | 42.5622i | |||||
95.7 | − | 4.49258i | 3.29550 | + | 5.70797i | −12.1833 | −4.76266 | + | 8.24917i | 25.6435 | − | 14.8053i | 2.69506 | + | 1.55599i | 18.7936i | −8.22062 | + | 14.2385i | 37.0601 | + | 21.3966i | |||||
95.8 | − | 4.45453i | −4.21061 | − | 7.29299i | −11.8429 | −3.05880 | + | 5.29800i | −32.4869 | + | 18.7563i | 27.0271 | + | 15.6041i | 17.1182i | −21.9585 | + | 38.0332i | 23.6001 | + | 13.6255i | |||||
95.9 | − | 4.25392i | −0.572421 | − | 0.991463i | −10.0958 | 2.83491 | − | 4.91021i | −4.21760 | + | 2.43503i | −9.49060 | − | 5.47940i | 8.91539i | 12.8447 | − | 22.2476i | −20.8876 | − | 12.0595i | |||||
95.10 | − | 3.93710i | 4.04600 | + | 7.00788i | −7.50078 | −0.327278 | + | 0.566862i | 27.5907 | − | 15.9295i | 8.49678 | + | 4.90562i | − | 1.96549i | −19.2403 | + | 33.3251i | 2.23179 | + | 1.28853i | ||||
95.11 | − | 3.88906i | −0.370198 | − | 0.641201i | −7.12477 | 8.76886 | − | 15.1881i | −2.49367 | + | 1.43972i | −13.2364 | − | 7.64205i | − | 3.40383i | 13.2259 | − | 22.9079i | −59.0674 | − | 34.1026i | ||||
95.12 | − | 3.71526i | 2.77922 | + | 4.81375i | −5.80315 | −8.74246 | + | 15.1424i | 17.8843 | − | 10.3255i | −30.0974 | − | 17.3767i | − | 8.16188i | −1.94812 | + | 3.37424i | 56.2579 | + | 32.4805i | ||||
95.13 | − | 3.12602i | −1.86331 | − | 3.22736i | −1.77198 | −0.838868 | + | 1.45296i | −10.0888 | + | 5.82475i | 15.9054 | + | 9.18298i | − | 19.4689i | 6.55612 | − | 11.3555i | 4.54199 | + | 2.62232i | ||||
95.14 | − | 3.11916i | −5.14483 | − | 8.91110i | −1.72913 | 7.94771 | − | 13.7658i | −27.7951 | + | 16.0475i | −8.25062 | − | 4.76350i | − | 19.5598i | −39.4385 | + | 68.3095i | −42.9378 | − | 24.7901i | ||||
95.15 | − | 3.03910i | −3.43070 | − | 5.94215i | −1.23613 | −4.75713 | + | 8.23959i | −18.0588 | + | 10.4262i | −24.7427 | − | 14.2852i | − | 20.5561i | −10.0394 | + | 17.3888i | 25.0410 | + | 14.4574i | ||||
95.16 | − | 2.71014i | 0.766356 | + | 1.32737i | 0.655139 | −2.06905 | + | 3.58371i | 3.59735 | − | 2.07693i | 11.5089 | + | 6.64469i | − | 23.4566i | 12.3254 | − | 21.3482i | 9.71235 | + | 5.60743i | ||||
95.17 | − | 2.67545i | −0.911074 | − | 1.57803i | 0.841950 | −7.26188 | + | 12.5779i | −4.22194 | + | 2.43754i | 0.0589830 | + | 0.0340538i | − | 23.6562i | 11.8399 | − | 20.5073i | 33.6517 | + | 19.4288i | ||||
95.18 | − | 2.65057i | −3.09449 | − | 5.35982i | 0.974499 | 3.56708 | − | 6.17836i | −14.2066 | + | 8.20216i | 21.8927 | + | 12.6398i | − | 23.7875i | −5.65180 | + | 9.78920i | −16.3761 | − | 9.45477i | ||||
95.19 | − | 2.34635i | 2.35926 | + | 4.08636i | 2.49466 | 9.80565 | − | 16.9839i | 9.58802 | − | 5.53565i | 22.4546 | + | 12.9642i | − | 24.6241i | 2.36776 | − | 4.10108i | −39.8501 | − | 23.0075i | ||||
95.20 | − | 2.09994i | 1.89908 | + | 3.28931i | 3.59026 | 5.51530 | − | 9.55279i | 6.90734 | − | 3.98796i | −26.8118 | − | 15.4798i | − | 24.3388i | 6.28697 | − | 10.8894i | −20.0603 | − | 11.5818i | ||||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
229.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 229.4.e.a | ✓ | 112 |
229.e | even | 6 | 1 | inner | 229.4.e.a | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
229.4.e.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
229.4.e.a | ✓ | 112 | 229.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(229, [\chi])\).