Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6015,2,Mod(1,6015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6015, 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("6015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6015 = 3 \cdot 5 \cdot 401 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6015.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: | \(48.0300168158\) |
Analytic rank: | \(0\) |
Dimension: | \(31\) |
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 | −2.46359 | −1.00000 | 4.06929 | −1.00000 | 2.46359 | −1.56442 | −5.09790 | 1.00000 | 2.46359 | ||||||||||||||||||
1.2 | −2.32925 | −1.00000 | 3.42539 | −1.00000 | 2.32925 | −3.29161 | −3.32009 | 1.00000 | 2.32925 | ||||||||||||||||||
1.3 | −2.16689 | −1.00000 | 2.69543 | −1.00000 | 2.16689 | 0.00504296 | −1.50693 | 1.00000 | 2.16689 | ||||||||||||||||||
1.4 | −2.16128 | −1.00000 | 2.67112 | −1.00000 | 2.16128 | 1.73409 | −1.45047 | 1.00000 | 2.16128 | ||||||||||||||||||
1.5 | −2.02086 | −1.00000 | 2.08386 | −1.00000 | 2.02086 | 0.941480 | −0.169465 | 1.00000 | 2.02086 | ||||||||||||||||||
1.6 | −1.61585 | −1.00000 | 0.610967 | −1.00000 | 1.61585 | 5.09249 | 2.24447 | 1.00000 | 1.61585 | ||||||||||||||||||
1.7 | −1.60901 | −1.00000 | 0.588914 | −1.00000 | 1.60901 | −4.04417 | 2.27045 | 1.00000 | 1.60901 | ||||||||||||||||||
1.8 | −1.52097 | −1.00000 | 0.313357 | −1.00000 | 1.52097 | −2.49520 | 2.56534 | 1.00000 | 1.52097 | ||||||||||||||||||
1.9 | −1.22966 | −1.00000 | −0.487935 | −1.00000 | 1.22966 | −1.61954 | 3.05932 | 1.00000 | 1.22966 | ||||||||||||||||||
1.10 | −0.900767 | −1.00000 | −1.18862 | −1.00000 | 0.900767 | 3.65606 | 2.87220 | 1.00000 | 0.900767 | ||||||||||||||||||
1.11 | −0.707695 | −1.00000 | −1.49917 | −1.00000 | 0.707695 | −2.27723 | 2.47634 | 1.00000 | 0.707695 | ||||||||||||||||||
1.12 | −0.349021 | −1.00000 | −1.87818 | −1.00000 | 0.349021 | 0.242299 | 1.35357 | 1.00000 | 0.349021 | ||||||||||||||||||
1.13 | −0.325483 | −1.00000 | −1.89406 | −1.00000 | 0.325483 | −4.65914 | 1.26745 | 1.00000 | 0.325483 | ||||||||||||||||||
1.14 | −0.0812850 | −1.00000 | −1.99339 | −1.00000 | 0.0812850 | 2.03254 | 0.324603 | 1.00000 | 0.0812850 | ||||||||||||||||||
1.15 | 0.0495237 | −1.00000 | −1.99755 | −1.00000 | −0.0495237 | −1.50762 | −0.197974 | 1.00000 | −0.0495237 | ||||||||||||||||||
1.16 | 0.0951324 | −1.00000 | −1.99095 | −1.00000 | −0.0951324 | 0.303287 | −0.379669 | 1.00000 | −0.0951324 | ||||||||||||||||||
1.17 | 0.363417 | −1.00000 | −1.86793 | −1.00000 | −0.363417 | 3.29378 | −1.40567 | 1.00000 | −0.363417 | ||||||||||||||||||
1.18 | 0.698893 | −1.00000 | −1.51155 | −1.00000 | −0.698893 | 4.36738 | −2.45420 | 1.00000 | −0.698893 | ||||||||||||||||||
1.19 | 0.917882 | −1.00000 | −1.15749 | −1.00000 | −0.917882 | −0.377330 | −2.89821 | 1.00000 | −0.917882 | ||||||||||||||||||
1.20 | 1.04920 | −1.00000 | −0.899176 | −1.00000 | −1.04920 | 2.71724 | −3.04182 | 1.00000 | −1.04920 | ||||||||||||||||||
See all 31 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(5\) | \(1\) |
\(401\) | \(-1\) |
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 | 6015.2.a.e | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6015.2.a.e | ✓ | 31 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{31} - 6 T_{2}^{30} - 25 T_{2}^{29} + 209 T_{2}^{28} + 182 T_{2}^{27} - 3192 T_{2}^{26} + 818 T_{2}^{25} + \cdots - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6015))\).