Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4015,2,Mod(1,4015)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, 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("4015.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4015 = 5 \cdot 11 \cdot 73 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4015.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: | \(32.0599364115\) |
Analytic rank: | \(0\) |
Dimension: | \(38\) |
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 | −2.76981 | 1.77605 | 5.67182 | 1.00000 | −4.91932 | −0.953960 | −10.1702 | 0.154367 | −2.76981 | ||||||||||||||||||
1.2 | −2.64676 | −3.26170 | 5.00536 | 1.00000 | 8.63297 | 0.334745 | −7.95449 | 7.63872 | −2.64676 | ||||||||||||||||||
1.3 | −2.64612 | −1.80895 | 5.00197 | 1.00000 | 4.78670 | −4.64214 | −7.94360 | 0.272285 | −2.64612 | ||||||||||||||||||
1.4 | −2.40266 | 2.98645 | 3.77280 | 1.00000 | −7.17544 | 4.58733 | −4.25943 | 5.91889 | −2.40266 | ||||||||||||||||||
1.5 | −2.35033 | 0.502249 | 3.52405 | 1.00000 | −1.18045 | 0.984719 | −3.58202 | −2.74775 | −2.35033 | ||||||||||||||||||
1.6 | −2.26509 | 0.846807 | 3.13061 | 1.00000 | −1.91809 | −3.17432 | −2.56093 | −2.28292 | −2.26509 | ||||||||||||||||||
1.7 | −1.83787 | −3.02191 | 1.37778 | 1.00000 | 5.55388 | 4.06050 | 1.14356 | 6.13192 | −1.83787 | ||||||||||||||||||
1.8 | −1.73652 | −1.80706 | 1.01551 | 1.00000 | 3.13799 | −4.26912 | 1.70959 | 0.265453 | −1.73652 | ||||||||||||||||||
1.9 | −1.69384 | 0.407884 | 0.869088 | 1.00000 | −0.690890 | 2.45284 | 1.91558 | −2.83363 | −1.69384 | ||||||||||||||||||
1.10 | −1.61749 | 2.78278 | 0.616262 | 1.00000 | −4.50111 | −5.17171 | 2.23818 | 4.74386 | −1.61749 | ||||||||||||||||||
1.11 | −1.50911 | −0.619433 | 0.277405 | 1.00000 | 0.934791 | 1.62740 | 2.59958 | −2.61630 | −1.50911 | ||||||||||||||||||
1.12 | −1.49358 | 2.85677 | 0.230796 | 1.00000 | −4.26684 | 0.0973246 | 2.64246 | 5.16116 | −1.49358 | ||||||||||||||||||
1.13 | −1.16964 | −3.05105 | −0.631953 | 1.00000 | 3.56861 | −2.34807 | 3.07843 | 6.30888 | −1.16964 | ||||||||||||||||||
1.14 | −1.00909 | −1.30445 | −0.981746 | 1.00000 | 1.31630 | 1.28192 | 3.00884 | −1.29840 | −1.00909 | ||||||||||||||||||
1.15 | −0.454453 | 2.93224 | −1.79347 | 1.00000 | −1.33257 | 4.01407 | 1.72395 | 5.59806 | −0.454453 | ||||||||||||||||||
1.16 | −0.445505 | −0.895604 | −1.80153 | 1.00000 | 0.398996 | −1.53298 | 1.69360 | −2.19789 | −0.445505 | ||||||||||||||||||
1.17 | −0.364998 | 1.30973 | −1.86678 | 1.00000 | −0.478050 | −1.97253 | 1.41137 | −1.28460 | −0.364998 | ||||||||||||||||||
1.18 | −0.0798923 | 0.0600931 | −1.99362 | 1.00000 | −0.00480098 | −4.79075 | 0.319059 | −2.99639 | −0.0798923 | ||||||||||||||||||
1.19 | 0.0204713 | 2.32254 | −1.99958 | 1.00000 | 0.0475455 | 2.78561 | −0.0818767 | 2.39419 | 0.0204713 | ||||||||||||||||||
1.20 | 0.0631532 | 0.00106429 | −1.99601 | 1.00000 | 6.72133e−5 | 0 | 3.26238 | −0.252361 | −3.00000 | 0.0631532 | |||||||||||||||||
See all 38 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(5\) | \(-1\) |
\(11\) | \(1\) |
\(73\) | \(1\) |
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 | 4015.2.a.i | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4015.2.a.i | ✓ | 38 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} - 4 T_{2}^{37} - 55 T_{2}^{36} + 231 T_{2}^{35} + 1363 T_{2}^{34} - 6076 T_{2}^{33} + \cdots + 192 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4015))\).