Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1003,2,Mod(237,1003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1003, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1003.237");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1003 = 17 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1003.d (of order \(2\), degree \(1\), 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: | \(8.00899532273\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
237.1 | −2.79415 | 3.34993i | 5.80725 | 0.976926i | − | 9.36019i | − | 0.327980i | −10.6380 | −8.22203 | − | 2.72968i | |||||||||||||||
237.2 | −2.79415 | − | 3.34993i | 5.80725 | − | 0.976926i | 9.36019i | 0.327980i | −10.6380 | −8.22203 | 2.72968i | ||||||||||||||||
237.3 | −2.54791 | 1.18403i | 4.49184 | 1.53761i | − | 3.01681i | − | 1.67517i | −6.34897 | 1.59806 | − | 3.91769i | |||||||||||||||
237.4 | −2.54791 | − | 1.18403i | 4.49184 | − | 1.53761i | 3.01681i | 1.67517i | −6.34897 | 1.59806 | 3.91769i | ||||||||||||||||
237.5 | −2.49154 | − | 1.56691i | 4.20776 | 2.43371i | 3.90401i | − | 4.19592i | −5.50071 | 0.544794 | − | 6.06369i | |||||||||||||||
237.6 | −2.49154 | 1.56691i | 4.20776 | − | 2.43371i | − | 3.90401i | 4.19592i | −5.50071 | 0.544794 | 6.06369i | ||||||||||||||||
237.7 | −2.22677 | − | 2.55813i | 2.95850 | − | 0.0680526i | 5.69638i | 1.82148i | −2.13436 | −3.54405 | 0.151537i | ||||||||||||||||
237.8 | −2.22677 | 2.55813i | 2.95850 | 0.0680526i | − | 5.69638i | − | 1.82148i | −2.13436 | −3.54405 | − | 0.151537i | |||||||||||||||
237.9 | −2.14935 | − | 1.94376i | 2.61968 | − | 4.42011i | 4.17781i | 2.30552i | −1.33191 | −0.778194 | 9.50034i | ||||||||||||||||
237.10 | −2.14935 | 1.94376i | 2.61968 | 4.42011i | − | 4.17781i | − | 2.30552i | −1.33191 | −0.778194 | − | 9.50034i | |||||||||||||||
237.11 | −1.97698 | − | 2.79908i | 1.90846 | 3.92158i | 5.53373i | 4.82675i | 0.180982 | −4.83486 | − | 7.75288i | ||||||||||||||||
237.12 | −1.97698 | 2.79908i | 1.90846 | − | 3.92158i | − | 5.53373i | − | 4.82675i | 0.180982 | −4.83486 | 7.75288i | |||||||||||||||
237.13 | −1.70427 | − | 0.948035i | 0.904538 | 1.68981i | 1.61571i | − | 2.20232i | 1.86696 | 2.10123 | − | 2.87990i | |||||||||||||||
237.14 | −1.70427 | 0.948035i | 0.904538 | − | 1.68981i | − | 1.61571i | 2.20232i | 1.86696 | 2.10123 | 2.87990i | ||||||||||||||||
237.15 | −1.66727 | − | 1.18715i | 0.779800 | 2.91515i | 1.97930i | − | 0.647324i | 2.03441 | 1.59067 | − | 4.86036i | |||||||||||||||
237.16 | −1.66727 | 1.18715i | 0.779800 | − | 2.91515i | − | 1.97930i | 0.647324i | 2.03441 | 1.59067 | 4.86036i | ||||||||||||||||
237.17 | −0.869109 | − | 2.92010i | −1.24465 | − | 4.00486i | 2.53788i | − | 0.621239i | 2.81995 | −5.52697 | 3.48066i | |||||||||||||||
237.18 | −0.869109 | 2.92010i | −1.24465 | 4.00486i | − | 2.53788i | 0.621239i | 2.81995 | −5.52697 | − | 3.48066i | ||||||||||||||||
237.19 | −0.774747 | 1.02838i | −1.39977 | 1.34000i | − | 0.796735i | 3.95629i | 2.63396 | 1.94243 | − | 1.03816i | ||||||||||||||||
237.20 | −0.774747 | − | 1.02838i | −1.39977 | − | 1.34000i | 0.796735i | − | 3.95629i | 2.63396 | 1.94243 | 1.03816i | |||||||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1003.2.d.d | ✓ | 44 |
17.b | even | 2 | 1 | inner | 1003.2.d.d | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1003.2.d.d | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
1003.2.d.d | ✓ | 44 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 4 T_{2}^{21} - 29 T_{2}^{20} - 131 T_{2}^{19} + 325 T_{2}^{18} + 1789 T_{2}^{17} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(1003, [\chi])\).