Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8015,2,Mod(1,8015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8015, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8015 = 5 \cdot 7 \cdot 229 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8015.a (trivial) |
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: | yes |
Analytic conductor: | \(64.0000972201\) |
Analytic rank: | \(1\) |
Dimension: | \(45\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.78224 | −1.94372 | 5.74083 | −1.00000 | 5.40788 | 1.00000 | −10.4079 | 0.778035 | 2.78224 | ||||||||||||||||||
1.2 | −2.58820 | 1.46821 | 4.69879 | −1.00000 | −3.80002 | 1.00000 | −6.98503 | −0.844367 | 2.58820 | ||||||||||||||||||
1.3 | −2.51360 | 3.19000 | 4.31820 | −1.00000 | −8.01840 | 1.00000 | −5.82702 | 7.17611 | 2.51360 | ||||||||||||||||||
1.4 | −2.46499 | 0.580032 | 4.07617 | −1.00000 | −1.42977 | 1.00000 | −5.11774 | −2.66356 | 2.46499 | ||||||||||||||||||
1.5 | −2.28900 | 0.276088 | 3.23954 | −1.00000 | −0.631967 | 1.00000 | −2.83731 | −2.92378 | 2.28900 | ||||||||||||||||||
1.6 | −2.26339 | 2.16114 | 3.12292 | −1.00000 | −4.89149 | 1.00000 | −2.54160 | 1.67051 | 2.26339 | ||||||||||||||||||
1.7 | −2.24619 | −2.43741 | 3.04539 | −1.00000 | 5.47490 | 1.00000 | −2.34814 | 2.94098 | 2.24619 | ||||||||||||||||||
1.8 | −2.21589 | −3.14559 | 2.91017 | −1.00000 | 6.97028 | 1.00000 | −2.01683 | 6.89472 | 2.21589 | ||||||||||||||||||
1.9 | −1.81008 | −0.874461 | 1.27639 | −1.00000 | 1.58284 | 1.00000 | 1.30979 | −2.23532 | 1.81008 | ||||||||||||||||||
1.10 | −1.69745 | 0.851741 | 0.881321 | −1.00000 | −1.44578 | 1.00000 | 1.89890 | −2.27454 | 1.69745 | ||||||||||||||||||
1.11 | −1.67660 | −1.48567 | 0.810994 | −1.00000 | 2.49087 | 1.00000 | 1.99349 | −0.792790 | 1.67660 | ||||||||||||||||||
1.12 | −1.65958 | −1.94100 | 0.754194 | −1.00000 | 3.22125 | 1.00000 | 2.06751 | 0.767498 | 1.65958 | ||||||||||||||||||
1.13 | −1.60365 | 2.74735 | 0.571694 | −1.00000 | −4.40579 | 1.00000 | 2.29050 | 4.54792 | 1.60365 | ||||||||||||||||||
1.14 | −1.48089 | −0.908664 | 0.193023 | −1.00000 | 1.34563 | 1.00000 | 2.67593 | −2.17433 | 1.48089 | ||||||||||||||||||
1.15 | −1.27565 | −2.40768 | −0.372716 | −1.00000 | 3.07136 | 1.00000 | 3.02676 | 2.79693 | 1.27565 | ||||||||||||||||||
1.16 | −1.13794 | 1.06234 | −0.705083 | −1.00000 | −1.20889 | 1.00000 | 3.07823 | −1.87143 | 1.13794 | ||||||||||||||||||
1.17 | −0.903684 | 3.04554 | −1.18335 | −1.00000 | −2.75220 | 1.00000 | 2.87675 | 6.27529 | 0.903684 | ||||||||||||||||||
1.18 | −0.883459 | 2.68305 | −1.21950 | −1.00000 | −2.37036 | 1.00000 | 2.84430 | 4.19874 | 0.883459 | ||||||||||||||||||
1.19 | −0.647127 | −0.415940 | −1.58123 | −1.00000 | 0.269166 | 1.00000 | 2.31751 | −2.82699 | 0.647127 | ||||||||||||||||||
1.20 | −0.556743 | −0.855264 | −1.69004 | −1.00000 | 0.476163 | 1.00000 | 2.05440 | −2.26852 | 0.556743 | ||||||||||||||||||
See all 45 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(1\) |
\(7\) | \(-1\) |
\(229\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 8015.2.a.j | ✓ | 45 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.j | ✓ | 45 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{45} + 6 T_{2}^{44} - 44 T_{2}^{43} - 323 T_{2}^{42} + 795 T_{2}^{41} + 7988 T_{2}^{40} + \cdots + 104 \) |
\( T_{3}^{45} - 82 T_{3}^{43} - 4 T_{3}^{42} + 3099 T_{3}^{41} + 295 T_{3}^{40} - 71657 T_{3}^{39} + \cdots + 2816 \) |