Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [375,3,Mod(74,375)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(375, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("375.74");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 375.h (of order \(10\), degree \(4\), 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: | \(10.2180099135\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{10})\) |
Twist minimal: | no (minimal twist has level 75) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
74.1 | −2.91455 | + | 2.11754i | 1.30764 | − | 2.70002i | 2.77453 | − | 8.53912i | 0 | 1.90623 | + | 10.6383i | − | 9.61929i | 5.54243 | + | 17.0578i | −5.58018 | − | 7.06128i | 0 | |||||
74.2 | −2.79240 | + | 2.02880i | 1.49624 | + | 2.60025i | 2.44541 | − | 7.52621i | 0 | −9.45346 | − | 4.22537i | − | 4.79411i | 4.17417 | + | 12.8468i | −4.52255 | + | 7.78116i | 0 | |||||
74.3 | −2.48516 | + | 1.80557i | −2.15485 | − | 2.08725i | 1.67984 | − | 5.17002i | 0 | 9.12382 | + | 1.29641i | 8.51768i | 1.36320 | + | 4.19549i | 0.286751 | + | 8.99543i | 0 | ||||||
74.4 | −2.05424 | + | 1.49250i | −2.06574 | + | 2.17548i | 0.756307 | − | 2.32767i | 0 | 0.996648 | − | 7.55208i | − | 6.97211i | −1.21820 | − | 3.74924i | −0.465411 | − | 8.98796i | 0 | |||||
74.5 | −1.88806 | + | 1.37175i | 2.67611 | + | 1.35589i | 0.446980 | − | 1.37566i | 0 | −6.91259 | + | 1.11096i | 12.5544i | −1.84155 | − | 5.66769i | 5.32311 | + | 7.25703i | 0 | ||||||
74.6 | −1.27392 | + | 0.925557i | 1.60087 | − | 2.53717i | −0.469852 | + | 1.44606i | 0 | 0.308913 | + | 4.71384i | 3.45468i | −2.68623 | − | 8.26736i | −3.87443 | − | 8.12335i | 0 | ||||||
74.7 | −0.949573 | + | 0.689905i | −1.65555 | + | 2.50183i | −0.810348 | + | 2.49399i | 0 | −0.153954 | − | 3.51784i | 4.52613i | −2.40195 | − | 7.39246i | −3.51828 | − | 8.28382i | 0 | ||||||
74.8 | −0.680034 | + | 0.494074i | 2.98817 | − | 0.266209i | −1.01773 | + | 3.13225i | 0 | −1.90053 | + | 1.65741i | − | 9.76040i | −1.89447 | − | 5.83059i | 8.85827 | − | 1.59096i | 0 | |||||
74.9 | −0.624639 | + | 0.453826i | −2.28023 | − | 1.94950i | −1.05185 | + | 3.23727i | 0 | 2.30905 | + | 0.182906i | − | 1.71117i | −1.76649 | − | 5.43671i | 1.39889 | + | 8.89062i | 0 | |||||
74.10 | 0.624639 | − | 0.453826i | −2.99063 | + | 0.236895i | −1.05185 | + | 3.23727i | 0 | −1.76055 | + | 1.50520i | − | 1.71117i | 1.76649 | + | 5.43671i | 8.88776 | − | 1.41693i | 0 | |||||
74.11 | 0.680034 | − | 0.494074i | 2.26100 | + | 1.97177i | −1.01773 | + | 3.13225i | 0 | 2.51176 | + | 0.223767i | − | 9.76040i | 1.89447 | + | 5.83059i | 1.22427 | + | 8.91634i | 0 | |||||
74.12 | 0.949573 | − | 0.689905i | 0.131165 | − | 2.99713i | −0.810348 | + | 2.49399i | 0 | −1.94319 | − | 2.93649i | 4.52613i | 2.40195 | + | 7.39246i | −8.96559 | − | 0.786239i | 0 | ||||||
74.13 | 1.27392 | − | 0.925557i | −0.196179 | + | 2.99358i | −0.469852 | + | 1.44606i | 0 | 2.52081 | + | 3.99515i | 3.45468i | 2.68623 | + | 8.26736i | −8.92303 | − | 1.17455i | 0 | ||||||
74.14 | 1.88806 | − | 1.37175i | 2.96199 | + | 0.476037i | 0.446980 | − | 1.37566i | 0 | 6.24541 | − | 3.16433i | 12.5544i | 1.84155 | + | 5.66769i | 8.54678 | + | 2.82003i | 0 | ||||||
74.15 | 2.05424 | − | 1.49250i | −0.392507 | − | 2.97421i | 0.756307 | − | 2.32767i | 0 | −5.24530 | − | 5.52394i | − | 6.97211i | 1.21820 | + | 3.74924i | −8.69188 | + | 2.33480i | 0 | |||||
74.16 | 2.48516 | − | 1.80557i | −2.97017 | + | 0.422034i | 1.67984 | − | 5.17002i | 0 | −6.61931 | + | 6.41167i | 8.51768i | −1.36320 | − | 4.19549i | 8.64377 | − | 2.50702i | 0 | ||||||
74.17 | 2.79240 | − | 2.02880i | 2.73887 | − | 1.22418i | 2.44541 | − | 7.52621i | 0 | 5.16441 | − | 8.97500i | − | 4.79411i | −4.17417 | − | 12.8468i | 6.00278 | − | 6.70572i | 0 | |||||
74.18 | 2.91455 | − | 2.11754i | −0.529130 | + | 2.95297i | 2.77453 | − | 8.53912i | 0 | 4.71086 | + | 9.72702i | − | 9.61929i | −5.54243 | − | 17.0578i | −8.44004 | − | 3.12501i | 0 | |||||
149.1 | −1.16846 | + | 3.59614i | −2.98146 | + | 0.333051i | −8.33088 | − | 6.05274i | 0 | 2.28600 | − | 11.1109i | 3.47419i | 19.2645 | − | 13.9965i | 8.77815 | − | 1.98595i | 0 | ||||||
149.2 | −1.10327 | + | 3.39551i | 2.43927 | − | 1.74641i | −7.07623 | − | 5.14118i | 0 | 3.23878 | + | 10.2093i | − | 0.132726i | 13.7104 | − | 9.96116i | 2.90011 | − | 8.51994i | 0 | |||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.e | even | 10 | 1 | inner |
75.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 375.3.h.a | 72 | |
3.b | odd | 2 | 1 | inner | 375.3.h.a | 72 | |
5.b | even | 2 | 1 | 75.3.h.a | ✓ | 72 | |
5.c | odd | 4 | 2 | 375.3.j.b | 144 | ||
15.d | odd | 2 | 1 | 75.3.h.a | ✓ | 72 | |
15.e | even | 4 | 2 | 375.3.j.b | 144 | ||
25.d | even | 5 | 1 | 75.3.h.a | ✓ | 72 | |
25.e | even | 10 | 1 | inner | 375.3.h.a | 72 | |
25.f | odd | 20 | 2 | 375.3.j.b | 144 | ||
75.h | odd | 10 | 1 | inner | 375.3.h.a | 72 | |
75.j | odd | 10 | 1 | 75.3.h.a | ✓ | 72 | |
75.l | even | 20 | 2 | 375.3.j.b | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.3.h.a | ✓ | 72 | 5.b | even | 2 | 1 | |
75.3.h.a | ✓ | 72 | 15.d | odd | 2 | 1 | |
75.3.h.a | ✓ | 72 | 25.d | even | 5 | 1 | |
75.3.h.a | ✓ | 72 | 75.j | odd | 10 | 1 | |
375.3.h.a | 72 | 1.a | even | 1 | 1 | trivial | |
375.3.h.a | 72 | 3.b | odd | 2 | 1 | inner | |
375.3.h.a | 72 | 25.e | even | 10 | 1 | inner | |
375.3.h.a | 72 | 75.h | odd | 10 | 1 | inner | |
375.3.j.b | 144 | 5.c | odd | 4 | 2 | ||
375.3.j.b | 144 | 15.e | even | 4 | 2 | ||
375.3.j.b | 144 | 25.f | odd | 20 | 2 | ||
375.3.j.b | 144 | 75.l | even | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{72} + 55 T_{2}^{70} + 1735 T_{2}^{68} + 41465 T_{2}^{66} + 847535 T_{2}^{64} + 14877956 T_{2}^{62} + 227671100 T_{2}^{60} + 3100547455 T_{2}^{58} + 38147838725 T_{2}^{56} + \cdots + 16\!\cdots\!25 \)
acting on \(S_{3}^{\mathrm{new}}(375, [\chi])\).