Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(24,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.24");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.u (of order \(18\), degree \(6\), not 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: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 95) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24.1 | −0.883478 | − | 2.42733i | 0.243956 | + | 0.0430161i | −3.57933 | + | 3.00342i | 0 | −0.111115 | − | 0.630167i | −0.347830 | + | 0.200820i | 5.97847 | + | 3.45167i | −2.76141 | − | 1.00507i | 0 | ||||
24.2 | −0.429906 | − | 1.18116i | −2.96649 | − | 0.523072i | 0.321776 | − | 0.270002i | 0 | 0.657481 | + | 3.72876i | 3.22501 | − | 1.86196i | −2.63437 | − | 1.52095i | 5.70738 | + | 2.07732i | 0 | ||||
24.3 | −0.105338 | − | 0.289414i | −1.61894 | − | 0.285463i | 1.45942 | − | 1.22460i | 0 | 0.0879194 | + | 0.498616i | −0.0772459 | + | 0.0445979i | −1.04160 | − | 0.601369i | −0.279590 | − | 0.101762i | 0 | ||||
24.4 | 0.105338 | + | 0.289414i | 1.61894 | + | 0.285463i | 1.45942 | − | 1.22460i | 0 | 0.0879194 | + | 0.498616i | 0.0772459 | − | 0.0445979i | 1.04160 | + | 0.601369i | −0.279590 | − | 0.101762i | 0 | ||||
24.5 | 0.429906 | + | 1.18116i | 2.96649 | + | 0.523072i | 0.321776 | − | 0.270002i | 0 | 0.657481 | + | 3.72876i | −3.22501 | + | 1.86196i | 2.63437 | + | 1.52095i | 5.70738 | + | 2.07732i | 0 | ||||
24.6 | 0.883478 | + | 2.42733i | −0.243956 | − | 0.0430161i | −3.57933 | + | 3.00342i | 0 | −0.111115 | − | 0.630167i | 0.347830 | − | 0.200820i | −5.97847 | − | 3.45167i | −2.76141 | − | 1.00507i | 0 | ||||
74.1 | −2.18897 | − | 0.385975i | 0.666572 | − | 0.794389i | 2.76323 | + | 1.00573i | 0 | −1.76572 | + | 1.48162i | −1.75377 | + | 1.01254i | −1.81055 | − | 1.04532i | 0.334208 | + | 1.89539i | 0 | ||||
74.2 | −2.09998 | − | 0.370282i | −1.43367 | + | 1.70859i | 2.39340 | + | 0.871127i | 0 | 3.64334 | − | 3.05712i | −1.28659 | + | 0.742812i | −1.01015 | − | 0.583208i | −0.342900 | − | 1.94468i | 0 | ||||
74.3 | −0.207778 | − | 0.0366369i | −0.0513559 | + | 0.0612035i | −1.83756 | − | 0.668816i | 0 | 0.0129129 | − | 0.0108352i | 1.46118 | − | 0.843614i | 0.722735 | + | 0.417271i | 0.519836 | + | 2.94814i | 0 | ||||
74.4 | 0.207778 | + | 0.0366369i | 0.0513559 | − | 0.0612035i | −1.83756 | − | 0.668816i | 0 | 0.0129129 | − | 0.0108352i | −1.46118 | + | 0.843614i | −0.722735 | − | 0.417271i | 0.519836 | + | 2.94814i | 0 | ||||
74.5 | 2.09998 | + | 0.370282i | 1.43367 | − | 1.70859i | 2.39340 | + | 0.871127i | 0 | 3.64334 | − | 3.05712i | 1.28659 | − | 0.742812i | 1.01015 | + | 0.583208i | −0.342900 | − | 1.94468i | 0 | ||||
74.6 | 2.18897 | + | 0.385975i | −0.666572 | + | 0.794389i | 2.76323 | + | 1.00573i | 0 | −1.76572 | + | 1.48162i | 1.75377 | − | 1.01254i | 1.81055 | + | 1.04532i | 0.334208 | + | 1.89539i | 0 | ||||
99.1 | −0.883478 | + | 2.42733i | 0.243956 | − | 0.0430161i | −3.57933 | − | 3.00342i | 0 | −0.111115 | + | 0.630167i | −0.347830 | − | 0.200820i | 5.97847 | − | 3.45167i | −2.76141 | + | 1.00507i | 0 | ||||
99.2 | −0.429906 | + | 1.18116i | −2.96649 | + | 0.523072i | 0.321776 | + | 0.270002i | 0 | 0.657481 | − | 3.72876i | 3.22501 | + | 1.86196i | −2.63437 | + | 1.52095i | 5.70738 | − | 2.07732i | 0 | ||||
99.3 | −0.105338 | + | 0.289414i | −1.61894 | + | 0.285463i | 1.45942 | + | 1.22460i | 0 | 0.0879194 | − | 0.498616i | −0.0772459 | − | 0.0445979i | −1.04160 | + | 0.601369i | −0.279590 | + | 0.101762i | 0 | ||||
99.4 | 0.105338 | − | 0.289414i | 1.61894 | − | 0.285463i | 1.45942 | + | 1.22460i | 0 | 0.0879194 | − | 0.498616i | 0.0772459 | + | 0.0445979i | 1.04160 | − | 0.601369i | −0.279590 | + | 0.101762i | 0 | ||||
99.5 | 0.429906 | − | 1.18116i | 2.96649 | − | 0.523072i | 0.321776 | + | 0.270002i | 0 | 0.657481 | − | 3.72876i | −3.22501 | − | 1.86196i | 2.63437 | − | 1.52095i | 5.70738 | − | 2.07732i | 0 | ||||
99.6 | 0.883478 | − | 2.42733i | −0.243956 | + | 0.0430161i | −3.57933 | − | 3.00342i | 0 | −0.111115 | + | 0.630167i | 0.347830 | + | 0.200820i | −5.97847 | + | 3.45167i | −2.76141 | + | 1.00507i | 0 | ||||
149.1 | −1.47196 | + | 1.75422i | 1.13116 | + | 3.10785i | −0.563307 | − | 3.19467i | 0 | −7.11687 | − | 2.59033i | −2.54534 | + | 1.46955i | 2.46697 | + | 1.42431i | −6.08105 | + | 5.10261i | 0 | ||||
149.2 | −0.443593 | + | 0.528654i | −0.237653 | − | 0.652945i | 0.264596 | + | 1.50060i | 0 | 0.450603 | + | 0.164006i | −2.02186 | + | 1.16732i | −2.10598 | − | 1.21589i | 1.92827 | − | 1.61801i | 0 | ||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
95.p | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.u.b | 36 | |
5.b | even | 2 | 1 | inner | 475.2.u.b | 36 | |
5.c | odd | 4 | 1 | 95.2.k.a | ✓ | 18 | |
5.c | odd | 4 | 1 | 475.2.l.c | 18 | ||
15.e | even | 4 | 1 | 855.2.bs.c | 18 | ||
19.e | even | 9 | 1 | inner | 475.2.u.b | 36 | |
95.p | even | 18 | 1 | inner | 475.2.u.b | 36 | |
95.q | odd | 36 | 1 | 95.2.k.a | ✓ | 18 | |
95.q | odd | 36 | 1 | 475.2.l.c | 18 | ||
95.q | odd | 36 | 1 | 1805.2.a.v | 9 | ||
95.q | odd | 36 | 1 | 9025.2.a.cc | 9 | ||
95.r | even | 36 | 1 | 1805.2.a.s | 9 | ||
95.r | even | 36 | 1 | 9025.2.a.cf | 9 | ||
285.bi | even | 36 | 1 | 855.2.bs.c | 18 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.2.k.a | ✓ | 18 | 5.c | odd | 4 | 1 | |
95.2.k.a | ✓ | 18 | 95.q | odd | 36 | 1 | |
475.2.l.c | 18 | 5.c | odd | 4 | 1 | ||
475.2.l.c | 18 | 95.q | odd | 36 | 1 | ||
475.2.u.b | 36 | 1.a | even | 1 | 1 | trivial | |
475.2.u.b | 36 | 5.b | even | 2 | 1 | inner | |
475.2.u.b | 36 | 19.e | even | 9 | 1 | inner | |
475.2.u.b | 36 | 95.p | even | 18 | 1 | inner | |
855.2.bs.c | 18 | 15.e | even | 4 | 1 | ||
855.2.bs.c | 18 | 285.bi | even | 36 | 1 | ||
1805.2.a.s | 9 | 95.r | even | 36 | 1 | ||
1805.2.a.v | 9 | 95.q | odd | 36 | 1 | ||
9025.2.a.cc | 9 | 95.q | odd | 36 | 1 | ||
9025.2.a.cf | 9 | 95.r | even | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{34} - 12 T_{2}^{32} - 175 T_{2}^{30} + 1281 T_{2}^{28} - 2784 T_{2}^{26} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).