Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,4,Mod(46,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.46");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.h (of order \(5\), degree \(4\), 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: | \(13.2754297513\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{5})\) |
Twist minimal: | no (minimal twist has level 75) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 | −1.48860 | + | 4.58143i | 0 | −12.3014 | − | 8.93752i | −5.89885 | + | 9.49755i | 0 | 1.13492 | 28.0809 | − | 20.4020i | 0 | −34.7314 | − | 41.1632i | ||||||||
46.2 | −1.31310 | + | 4.04131i | 0 | −8.13578 | − | 5.91099i | 8.20823 | − | 7.59111i | 0 | −28.2853 | 7.06929 | − | 5.13614i | 0 | 19.8998 | + | 43.1398i | ||||||||
46.3 | −0.536885 | + | 1.65236i | 0 | 4.03008 | + | 2.92802i | −8.44668 | − | 7.32487i | 0 | 7.66213 | −18.2465 | + | 13.2569i | 0 | 16.6382 | − | 10.0244i | ||||||||
46.4 | 0.177563 | − | 0.546483i | 0 | 6.20502 | + | 4.50821i | 9.45304 | − | 5.96993i | 0 | −2.67744 | 7.28438 | − | 5.29241i | 0 | −1.58395 | − | 6.22597i | ||||||||
46.5 | 0.722966 | − | 2.22506i | 0 | 2.04392 | + | 1.48499i | 1.87995 | + | 11.0212i | 0 | −32.9322 | 19.9239 | − | 14.4756i | 0 | 25.8819 | + | 3.78491i | ||||||||
46.6 | 0.907834 | − | 2.79403i | 0 | −0.510282 | − | 0.370741i | −10.7234 | + | 3.16365i | 0 | 18.9115 | 17.5148 | − | 12.7253i | 0 | −0.895733 | + | 32.8335i | ||||||||
46.7 | 1.53022 | − | 4.70953i | 0 | −13.3660 | − | 9.71093i | 9.83672 | + | 5.31403i | 0 | 28.2766 | −34.1374 | + | 24.8023i | 0 | 40.0789 | − | 38.1947i | ||||||||
91.1 | −3.87763 | − | 2.81726i | 0 | 4.62691 | + | 14.2402i | −7.51887 | + | 8.27445i | 0 | 0.140520 | 10.3279 | − | 31.7859i | 0 | 52.4667 | − | 10.9026i | ||||||||
91.2 | −3.25026 | − | 2.36146i | 0 | 2.51561 | + | 7.74226i | 6.00716 | − | 9.42943i | 0 | 1.75849 | 0.174670 | − | 0.537580i | 0 | −41.7920 | + | 16.4625i | ||||||||
91.3 | −0.772797 | − | 0.561470i | 0 | −2.19017 | − | 6.74065i | −11.0970 | + | 1.36218i | 0 | 12.4836 | −4.45357 | + | 13.7067i | 0 | 9.34059 | + | 5.17797i | ||||||||
91.4 | −0.109191 | − | 0.0793317i | 0 | −2.46651 | − | 7.59113i | 6.22327 | + | 9.28822i | 0 | −17.3099 | −0.666555 | + | 2.05145i | 0 | 0.0573272 | − | 1.50789i | ||||||||
91.5 | 1.08389 | + | 0.787491i | 0 | −1.91746 | − | 5.90135i | 3.83217 | − | 10.5031i | 0 | −12.2101 | 5.88101 | − | 18.0999i | 0 | 12.4247 | − | 8.36635i | ||||||||
91.6 | 3.16070 | + | 2.29638i | 0 | 2.24451 | + | 6.90789i | 11.0852 | + | 1.45535i | 0 | 22.0918 | 0.889297 | − | 2.73697i | 0 | 31.6950 | + | 30.0558i | ||||||||
91.7 | 3.76530 | + | 2.73565i | 0 | 4.22155 | + | 12.9926i | −5.34090 | + | 9.82216i | 0 | −26.0445 | −8.14206 | + | 25.0587i | 0 | −46.9800 | + | 22.3725i | ||||||||
136.1 | −3.87763 | + | 2.81726i | 0 | 4.62691 | − | 14.2402i | −7.51887 | − | 8.27445i | 0 | 0.140520 | 10.3279 | + | 31.7859i | 0 | 52.4667 | + | 10.9026i | ||||||||
136.2 | −3.25026 | + | 2.36146i | 0 | 2.51561 | − | 7.74226i | 6.00716 | + | 9.42943i | 0 | 1.75849 | 0.174670 | + | 0.537580i | 0 | −41.7920 | − | 16.4625i | ||||||||
136.3 | −0.772797 | + | 0.561470i | 0 | −2.19017 | + | 6.74065i | −11.0970 | − | 1.36218i | 0 | 12.4836 | −4.45357 | − | 13.7067i | 0 | 9.34059 | − | 5.17797i | ||||||||
136.4 | −0.109191 | + | 0.0793317i | 0 | −2.46651 | + | 7.59113i | 6.22327 | − | 9.28822i | 0 | −17.3099 | −0.666555 | − | 2.05145i | 0 | 0.0573272 | + | 1.50789i | ||||||||
136.5 | 1.08389 | − | 0.787491i | 0 | −1.91746 | + | 5.90135i | 3.83217 | + | 10.5031i | 0 | −12.2101 | 5.88101 | + | 18.0999i | 0 | 12.4247 | + | 8.36635i | ||||||||
136.6 | 3.16070 | − | 2.29638i | 0 | 2.24451 | − | 6.90789i | 11.0852 | − | 1.45535i | 0 | 22.0918 | 0.889297 | + | 2.73697i | 0 | 31.6950 | − | 30.0558i | ||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.4.h.a | 28 | |
3.b | odd | 2 | 1 | 75.4.g.b | ✓ | 28 | |
25.d | even | 5 | 1 | inner | 225.4.h.a | 28 | |
75.h | odd | 10 | 1 | 1875.4.a.f | 14 | ||
75.j | odd | 10 | 1 | 75.4.g.b | ✓ | 28 | |
75.j | odd | 10 | 1 | 1875.4.a.g | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.4.g.b | ✓ | 28 | 3.b | odd | 2 | 1 | |
75.4.g.b | ✓ | 28 | 75.j | odd | 10 | 1 | |
225.4.h.a | 28 | 1.a | even | 1 | 1 | trivial | |
225.4.h.a | 28 | 25.d | even | 5 | 1 | inner | |
1875.4.a.f | 14 | 75.h | odd | 10 | 1 | ||
1875.4.a.g | 14 | 75.j | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 43 T_{2}^{26} - 21 T_{2}^{25} + 1285 T_{2}^{24} - 803 T_{2}^{23} + 33580 T_{2}^{22} + \cdots + 1769380096 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).