Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,2,Mod(27,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.27");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 260.o (of order \(4\), 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: | \(2.07611045255\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(36\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 | −1.41404 | + | 0.0221303i | 0.798638 | + | 0.798638i | 1.99902 | − | 0.0625863i | −1.57330 | − | 1.58893i | −1.14698 | − | 1.11163i | 1.30625 | − | 1.30625i | −2.82531 | + | 0.132739i | − | 1.72435i | 2.25988 | + | 2.21200i | |
27.2 | −1.41202 | − | 0.0788259i | −1.63309 | − | 1.63309i | 1.98757 | + | 0.222607i | −0.963620 | + | 2.01778i | 2.17721 | + | 2.43467i | 3.04756 | − | 3.04756i | −2.78894 | − | 0.470996i | 2.33394i | 1.51970 | − | 2.77318i | ||
27.3 | −1.39950 | + | 0.203483i | 0.0775283 | + | 0.0775283i | 1.91719 | − | 0.569548i | 0.871245 | + | 2.05935i | −0.124276 | − | 0.0927250i | −2.89635 | + | 2.89635i | −2.56721 | + | 1.18720i | − | 2.98798i | −1.63835 | − | 2.70478i | |
27.4 | −1.34983 | − | 0.421862i | −1.90910 | − | 1.90910i | 1.64407 | + | 1.13888i | −0.987103 | − | 2.00640i | 1.77158 | + | 3.38234i | −3.15102 | + | 3.15102i | −1.73875 | − | 2.23086i | 4.28935i | 0.485997 | + | 3.12471i | ||
27.5 | −1.32767 | − | 0.487116i | 0.396814 | + | 0.396814i | 1.52544 | + | 1.29346i | 2.02359 | − | 0.951360i | −0.333545 | − | 0.720134i | 0.477103 | − | 0.477103i | −1.39521 | − | 2.46036i | − | 2.68508i | −3.15009 | + | 0.277373i | |
27.6 | −1.26622 | + | 0.629837i | 2.21225 | + | 2.21225i | 1.20661 | − | 1.59502i | 1.36305 | − | 1.77259i | −4.19454 | − | 1.40783i | −2.47925 | + | 2.47925i | −0.523229 | + | 2.77961i | 6.78807i | −0.609477 | + | 3.10299i | ||
27.7 | −1.23527 | + | 0.688553i | −1.59413 | − | 1.59413i | 1.05179 | − | 1.70110i | 1.94514 | − | 1.10291i | 3.06682 | + | 0.871538i | 0.0942377 | − | 0.0942377i | −0.127946 | + | 2.82553i | 2.08249i | −1.64337 | + | 2.70173i | ||
27.8 | −1.18329 | − | 0.774482i | 2.05925 | + | 2.05925i | 0.800354 | + | 1.83288i | 1.34695 | + | 1.78486i | −0.841839 | − | 4.03155i | 0.845670 | − | 0.845670i | 0.472478 | − | 2.78869i | 5.48103i | −0.211491 | − | 3.15520i | ||
27.9 | −1.08734 | + | 0.904262i | 1.78782 | + | 1.78782i | 0.364619 | − | 1.96648i | −1.34049 | + | 1.78972i | −3.56063 | − | 0.327311i | 2.34459 | − | 2.34459i | 1.38175 | + | 2.46795i | 3.39260i | −0.160802 | − | 3.15819i | ||
27.10 | −0.942648 | + | 1.05424i | 0.158208 | + | 0.158208i | −0.222831 | − | 1.98755i | −2.23604 | + | 0.0111104i | −0.315922 | + | 0.0176543i | −2.20671 | + | 2.20671i | 2.30540 | + | 1.63864i | − | 2.94994i | 2.09608 | − | 2.36779i | |
27.11 | −0.774482 | − | 1.18329i | −2.05925 | − | 2.05925i | −0.800354 | + | 1.83288i | 1.34695 | + | 1.78486i | −0.841839 | + | 4.03155i | −0.845670 | + | 0.845670i | 2.78869 | − | 0.472478i | 5.48103i | 1.06882 | − | 2.97618i | ||
27.12 | −0.769019 | + | 1.18685i | −0.808783 | − | 0.808783i | −0.817219 | − | 1.82542i | 0.885949 | + | 2.05307i | 1.58187 | − | 0.337933i | 0.771151 | − | 0.771151i | 2.79495 | + | 0.433867i | − | 1.69174i | −3.11799 | − | 0.527363i | |
27.13 | −0.516756 | + | 1.31642i | −2.23963 | − | 2.23963i | −1.46593 | − | 1.36054i | −2.20168 | − | 0.390628i | 4.10564 | − | 1.79095i | 0.918543 | − | 0.918543i | 2.54856 | − | 1.22671i | 7.03189i | 1.65196 | − | 2.69648i | ||
27.14 | −0.505980 | + | 1.32060i | 0.658144 | + | 0.658144i | −1.48797 | − | 1.33639i | 0.820865 | − | 2.07995i | −1.20215 | + | 0.536137i | 1.89124 | − | 1.89124i | 2.51772 | − | 1.28882i | − | 2.13369i | 2.33144 | + | 2.13645i | |
27.15 | −0.487116 | − | 1.32767i | −0.396814 | − | 0.396814i | −1.52544 | + | 1.29346i | 2.02359 | − | 0.951360i | −0.333545 | + | 0.720134i | −0.477103 | + | 0.477103i | 2.46036 | + | 1.39521i | − | 2.68508i | −2.24882 | − | 2.22324i | |
27.16 | −0.421862 | − | 1.34983i | 1.90910 | + | 1.90910i | −1.64407 | + | 1.13888i | −0.987103 | − | 2.00640i | 1.77158 | − | 3.38234i | 3.15102 | − | 3.15102i | 2.23086 | + | 1.73875i | 4.28935i | −2.29187 | + | 2.17884i | ||
27.17 | −0.0788259 | − | 1.41202i | 1.63309 | + | 1.63309i | −1.98757 | + | 0.222607i | −0.963620 | + | 2.01778i | 2.17721 | − | 2.43467i | −3.04756 | + | 3.04756i | 0.470996 | + | 2.78894i | 2.33394i | 2.92509 | + | 1.20159i | ||
27.18 | −0.0447185 | + | 1.41351i | 1.60820 | + | 1.60820i | −1.99600 | − | 0.126420i | 1.86263 | + | 1.23718i | −2.34511 | + | 2.20128i | −0.275770 | + | 0.275770i | 0.267954 | − | 2.81571i | 2.17259i | −1.83206 | + | 2.57751i | ||
27.19 | 0.0221303 | − | 1.41404i | −0.798638 | − | 0.798638i | −1.99902 | − | 0.0625863i | −1.57330 | − | 1.58893i | −1.14698 | + | 1.11163i | −1.30625 | + | 1.30625i | −0.132739 | + | 2.82531i | − | 1.72435i | −2.28164 | + | 2.18955i | |
27.20 | 0.203483 | − | 1.39950i | −0.0775283 | − | 0.0775283i | −1.91719 | − | 0.569548i | 0.871245 | + | 2.05935i | −0.124276 | + | 0.0927250i | 2.89635 | − | 2.89635i | −1.18720 | + | 2.56721i | − | 2.98798i | 3.05934 | − | 0.800263i | |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 260.2.o.a | ✓ | 72 |
4.b | odd | 2 | 1 | inner | 260.2.o.a | ✓ | 72 |
5.c | odd | 4 | 1 | inner | 260.2.o.a | ✓ | 72 |
20.e | even | 4 | 1 | inner | 260.2.o.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
260.2.o.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
260.2.o.a | ✓ | 72 | 4.b | odd | 2 | 1 | inner |
260.2.o.a | ✓ | 72 | 5.c | odd | 4 | 1 | inner |
260.2.o.a | ✓ | 72 | 20.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(260, [\chi])\).