Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(30,833)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.30");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 833.o (of order \(12\), degree \(4\), not 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: | \(6.65153848837\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 119) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
30.1 | −2.17421 | − | 1.25528i | −0.677398 | + | 2.52808i | 2.15147 | + | 3.72645i | 2.89431 | − | 0.775527i | 4.64626 | − | 4.64626i | 0 | − | 5.78166i | −3.33426 | − | 1.92503i | −7.26634 | − | 1.94701i | |||
30.2 | −2.01236 | − | 1.16184i | 0.545242 | − | 2.03487i | 1.69972 | + | 2.94400i | 1.42180 | − | 0.380969i | −3.46141 | + | 3.46141i | 0 | − | 3.25185i | −1.24533 | − | 0.718992i | −3.30378 | − | 0.885246i | |||
30.3 | −1.36164 | − | 0.786146i | −0.526518 | + | 1.96499i | 0.236050 | + | 0.408850i | −3.27915 | + | 0.878646i | 2.26170 | − | 2.26170i | 0 | 2.40230i | −0.985896 | − | 0.569207i | 5.15578 | + | 1.38149i | ||||
30.4 | −1.21683 | − | 0.702538i | 0.166262 | − | 0.620497i | −0.0128813 | − | 0.0223111i | −0.286192 | + | 0.0766850i | −0.638235 | + | 0.638235i | 0 | 2.84635i | 2.24070 | + | 1.29367i | 0.402122 | + | 0.107748i | ||||
30.5 | −0.107682 | − | 0.0621703i | −0.445460 | + | 1.66248i | −0.992270 | − | 1.71866i | 3.11267 | − | 0.834039i | 0.151325 | − | 0.151325i | 0 | 0.495440i | 0.0326760 | + | 0.0188655i | −0.387032 | − | 0.103705i | ||||
30.6 | 0.504509 | + | 0.291278i | −0.223439 | + | 0.833886i | −0.830314 | − | 1.43815i | −1.93011 | + | 0.517173i | −0.355620 | + | 0.355620i | 0 | − | 2.13252i | 1.95264 | + | 1.12735i | −1.12440 | − | 0.301282i | |||
30.7 | 0.615152 | + | 0.355158i | 0.775643 | − | 2.89474i | −0.747726 | − | 1.29510i | 3.64268 | − | 0.976053i | 1.50523 | − | 1.50523i | 0 | − | 2.48287i | −5.17981 | − | 2.99057i | 2.58745 | + | 0.693306i | |||
30.8 | 1.63202 | + | 0.942246i | 0.342447 | − | 1.27803i | 0.775655 | + | 1.34347i | −1.33407 | + | 0.357462i | 1.76310 | − | 1.76310i | 0 | − | 0.845554i | 1.08198 | + | 0.624683i | −2.51404 | − | 0.673635i | |||
30.9 | 1.84967 | + | 1.06791i | −0.799375 | + | 2.98331i | 1.28084 | + | 2.21849i | −1.05290 | + | 0.282125i | −4.66447 | + | 4.66447i | 0 | 1.19966i | −5.66305 | − | 3.26956i | −2.24880 | − | 0.602565i | ||||
30.10 | 2.27138 | + | 1.31138i | 0.110545 | − | 0.412559i | 2.43945 | + | 4.22526i | 2.27508 | − | 0.609605i | 0.792113 | − | 0.792113i | 0 | 7.55071i | 2.44009 | + | 1.40879i | 5.96699 | + | 1.59885i | ||||
361.1 | −2.17421 | + | 1.25528i | −0.677398 | − | 2.52808i | 2.15147 | − | 3.72645i | 2.89431 | + | 0.775527i | 4.64626 | + | 4.64626i | 0 | 5.78166i | −3.33426 | + | 1.92503i | −7.26634 | + | 1.94701i | ||||
361.2 | −2.01236 | + | 1.16184i | 0.545242 | + | 2.03487i | 1.69972 | − | 2.94400i | 1.42180 | + | 0.380969i | −3.46141 | − | 3.46141i | 0 | 3.25185i | −1.24533 | + | 0.718992i | −3.30378 | + | 0.885246i | ||||
361.3 | −1.36164 | + | 0.786146i | −0.526518 | − | 1.96499i | 0.236050 | − | 0.408850i | −3.27915 | − | 0.878646i | 2.26170 | + | 2.26170i | 0 | − | 2.40230i | −0.985896 | + | 0.569207i | 5.15578 | − | 1.38149i | |||
361.4 | −1.21683 | + | 0.702538i | 0.166262 | + | 0.620497i | −0.0128813 | + | 0.0223111i | −0.286192 | − | 0.0766850i | −0.638235 | − | 0.638235i | 0 | − | 2.84635i | 2.24070 | − | 1.29367i | 0.402122 | − | 0.107748i | |||
361.5 | −0.107682 | + | 0.0621703i | −0.445460 | − | 1.66248i | −0.992270 | + | 1.71866i | 3.11267 | + | 0.834039i | 0.151325 | + | 0.151325i | 0 | − | 0.495440i | 0.0326760 | − | 0.0188655i | −0.387032 | + | 0.103705i | |||
361.6 | 0.504509 | − | 0.291278i | −0.223439 | − | 0.833886i | −0.830314 | + | 1.43815i | −1.93011 | − | 0.517173i | −0.355620 | − | 0.355620i | 0 | 2.13252i | 1.95264 | − | 1.12735i | −1.12440 | + | 0.301282i | ||||
361.7 | 0.615152 | − | 0.355158i | 0.775643 | + | 2.89474i | −0.747726 | + | 1.29510i | 3.64268 | + | 0.976053i | 1.50523 | + | 1.50523i | 0 | 2.48287i | −5.17981 | + | 2.99057i | 2.58745 | − | 0.693306i | ||||
361.8 | 1.63202 | − | 0.942246i | 0.342447 | + | 1.27803i | 0.775655 | − | 1.34347i | −1.33407 | − | 0.357462i | 1.76310 | + | 1.76310i | 0 | 0.845554i | 1.08198 | − | 0.624683i | −2.51404 | + | 0.673635i | ||||
361.9 | 1.84967 | − | 1.06791i | −0.799375 | − | 2.98331i | 1.28084 | − | 2.21849i | −1.05290 | − | 0.282125i | −4.66447 | − | 4.66447i | 0 | − | 1.19966i | −5.66305 | + | 3.26956i | −2.24880 | + | 0.602565i | |||
361.10 | 2.27138 | − | 1.31138i | 0.110545 | + | 0.412559i | 2.43945 | − | 4.22526i | 2.27508 | + | 0.609605i | 0.792113 | + | 0.792113i | 0 | − | 7.55071i | 2.44009 | − | 1.40879i | 5.96699 | − | 1.59885i | |||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
17.c | even | 4 | 1 | inner |
119.n | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 833.2.o.g | 40 | |
7.b | odd | 2 | 1 | 833.2.o.f | 40 | ||
7.c | even | 3 | 1 | 119.2.g.a | ✓ | 20 | |
7.c | even | 3 | 1 | inner | 833.2.o.g | 40 | |
7.d | odd | 6 | 1 | 833.2.g.h | 20 | ||
7.d | odd | 6 | 1 | 833.2.o.f | 40 | ||
17.c | even | 4 | 1 | inner | 833.2.o.g | 40 | |
21.h | odd | 6 | 1 | 1071.2.n.c | 20 | ||
119.f | odd | 4 | 1 | 833.2.o.f | 40 | ||
119.m | odd | 12 | 1 | 833.2.g.h | 20 | ||
119.m | odd | 12 | 1 | 833.2.o.f | 40 | ||
119.n | even | 12 | 1 | 119.2.g.a | ✓ | 20 | |
119.n | even | 12 | 1 | inner | 833.2.o.g | 40 | |
119.q | even | 24 | 1 | 2023.2.a.m | 10 | ||
119.q | even | 24 | 1 | 2023.2.a.n | 10 | ||
357.z | odd | 12 | 1 | 1071.2.n.c | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
119.2.g.a | ✓ | 20 | 7.c | even | 3 | 1 | |
119.2.g.a | ✓ | 20 | 119.n | even | 12 | 1 | |
833.2.g.h | 20 | 7.d | odd | 6 | 1 | ||
833.2.g.h | 20 | 119.m | odd | 12 | 1 | ||
833.2.o.f | 40 | 7.b | odd | 2 | 1 | ||
833.2.o.f | 40 | 7.d | odd | 6 | 1 | ||
833.2.o.f | 40 | 119.f | odd | 4 | 1 | ||
833.2.o.f | 40 | 119.m | odd | 12 | 1 | ||
833.2.o.g | 40 | 1.a | even | 1 | 1 | trivial | |
833.2.o.g | 40 | 7.c | even | 3 | 1 | inner | |
833.2.o.g | 40 | 17.c | even | 4 | 1 | inner | |
833.2.o.g | 40 | 119.n | even | 12 | 1 | inner | |
1071.2.n.c | 20 | 21.h | odd | 6 | 1 | ||
1071.2.n.c | 20 | 357.z | odd | 12 | 1 | ||
2023.2.a.m | 10 | 119.q | even | 24 | 1 | ||
2023.2.a.n | 10 | 119.q | even | 24 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):
\( T_{2}^{40} - 32 T_{2}^{38} + 592 T_{2}^{36} - 7424 T_{2}^{34} + 70064 T_{2}^{32} - 514322 T_{2}^{30} + 3023872 T_{2}^{28} - 14348720 T_{2}^{26} + 55422528 T_{2}^{24} - 173395264 T_{2}^{22} + 438238483 T_{2}^{20} + \cdots + 2401 \) |
\( T_{3}^{40} - 4 T_{3}^{39} + 8 T_{3}^{38} - 32 T_{3}^{37} - 38 T_{3}^{36} + 404 T_{3}^{35} - 800 T_{3}^{34} + 3252 T_{3}^{33} + 2137 T_{3}^{32} - 34108 T_{3}^{31} + 66480 T_{3}^{30} - 270980 T_{3}^{29} + 389802 T_{3}^{28} + \cdots + 9834496 \) |