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: | \(0\) |
Dimension: | \(73\) |
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 | −2.77376 | 0.883907 | 5.69373 | 1.00000 | −2.45174 | 1.00000 | −10.2455 | −2.21871 | −2.77376 | ||||||||||||||||||
1.2 | −2.76326 | 3.03127 | 5.63562 | 1.00000 | −8.37621 | 1.00000 | −10.0462 | 6.18862 | −2.76326 | ||||||||||||||||||
1.3 | −2.72942 | −2.22505 | 5.44974 | 1.00000 | 6.07311 | 1.00000 | −9.41580 | 1.95086 | −2.72942 | ||||||||||||||||||
1.4 | −2.69749 | −3.25136 | 5.27643 | 1.00000 | 8.77051 | 1.00000 | −8.83812 | 7.57137 | −2.69749 | ||||||||||||||||||
1.5 | −2.68177 | −0.319561 | 5.19187 | 1.00000 | 0.856987 | 1.00000 | −8.55985 | −2.89788 | −2.68177 | ||||||||||||||||||
1.6 | −2.45360 | 1.56371 | 4.02015 | 1.00000 | −3.83672 | 1.00000 | −4.95665 | −0.554809 | −2.45360 | ||||||||||||||||||
1.7 | −2.45327 | 3.30020 | 4.01854 | 1.00000 | −8.09629 | 1.00000 | −4.95203 | 7.89132 | −2.45327 | ||||||||||||||||||
1.8 | −2.45229 | −0.606235 | 4.01370 | 1.00000 | 1.48666 | 1.00000 | −4.93818 | −2.63248 | −2.45229 | ||||||||||||||||||
1.9 | −2.41178 | −1.36759 | 3.81666 | 1.00000 | 3.29833 | 1.00000 | −4.38139 | −1.12969 | −2.41178 | ||||||||||||||||||
1.10 | −2.29631 | 0.975796 | 3.27305 | 1.00000 | −2.24073 | 1.00000 | −2.92333 | −2.04782 | −2.29631 | ||||||||||||||||||
1.11 | −2.19612 | 2.45439 | 2.82293 | 1.00000 | −5.39013 | 1.00000 | −1.80725 | 3.02403 | −2.19612 | ||||||||||||||||||
1.12 | −2.19191 | −0.935723 | 2.80448 | 1.00000 | 2.05102 | 1.00000 | −1.76336 | −2.12442 | −2.19191 | ||||||||||||||||||
1.13 | −2.08852 | −0.140980 | 2.36194 | 1.00000 | 0.294440 | 1.00000 | −0.755913 | −2.98012 | −2.08852 | ||||||||||||||||||
1.14 | −2.01224 | 2.54989 | 2.04911 | 1.00000 | −5.13098 | 1.00000 | −0.0988175 | 3.50192 | −2.01224 | ||||||||||||||||||
1.15 | −1.86693 | −0.178602 | 1.48541 | 1.00000 | 0.333437 | 1.00000 | 0.960700 | −2.96810 | −1.86693 | ||||||||||||||||||
1.16 | −1.84781 | −2.80168 | 1.41441 | 1.00000 | 5.17697 | 1.00000 | 1.08207 | 4.84939 | −1.84781 | ||||||||||||||||||
1.17 | −1.63497 | 2.59836 | 0.673111 | 1.00000 | −4.24822 | 1.00000 | 2.16942 | 3.75146 | −1.63497 | ||||||||||||||||||
1.18 | −1.53607 | −1.95086 | 0.359523 | 1.00000 | 2.99667 | 1.00000 | 2.51989 | 0.805862 | −1.53607 | ||||||||||||||||||
1.19 | −1.50054 | 2.96680 | 0.251626 | 1.00000 | −4.45181 | 1.00000 | 2.62351 | 5.80192 | −1.50054 | ||||||||||||||||||
1.20 | −1.40112 | 2.16021 | −0.0368596 | 1.00000 | −3.02672 | 1.00000 | 2.85389 | 1.66651 | −1.40112 | ||||||||||||||||||
See all 73 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.o | ✓ | 73 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8015.2.a.o | ✓ | 73 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):
\( T_{2}^{73} - 7 T_{2}^{72} - 96 T_{2}^{71} + 771 T_{2}^{70} + 4193 T_{2}^{69} - 40402 T_{2}^{68} - 107553 T_{2}^{67} + 1340585 T_{2}^{66} + 1686113 T_{2}^{65} - 31625519 T_{2}^{64} - 12629340 T_{2}^{63} + \cdots + 31509048 \) |
\( T_{3}^{73} - 14 T_{3}^{72} - 67 T_{3}^{71} + 1810 T_{3}^{70} - 982 T_{3}^{69} - 108249 T_{3}^{68} + 294790 T_{3}^{67} + 3940592 T_{3}^{66} - 17099479 T_{3}^{65} - 96004886 T_{3}^{64} + 588675495 T_{3}^{63} + \cdots - 451619323904 \) |