Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,4,Mod(16,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.16");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.g (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: | \(4.42514325043\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −3.76530 | − | 2.73565i | 0.927051 | + | 2.85317i | 4.22155 | + | 12.9926i | 5.34090 | − | 9.82216i | 4.31465 | − | 13.2791i | −26.0445 | 8.14206 | − | 25.0587i | −7.28115 | + | 5.29007i | −46.9800 | + | 22.3725i | ||
16.2 | −3.16070 | − | 2.29638i | 0.927051 | + | 2.85317i | 2.24451 | + | 6.90789i | −11.0852 | − | 1.45535i | 3.62184 | − | 11.1469i | 22.0918 | −0.889297 | + | 2.73697i | −7.28115 | + | 5.29007i | 31.6950 | + | 30.0558i | ||
16.3 | −1.08389 | − | 0.787491i | 0.927051 | + | 2.85317i | −1.91746 | − | 5.90135i | −3.83217 | + | 10.5031i | 1.24203 | − | 3.82256i | −12.2101 | −5.88101 | + | 18.0999i | −7.28115 | + | 5.29007i | 12.4247 | − | 8.36635i | ||
16.4 | 0.109191 | + | 0.0793317i | 0.927051 | + | 2.85317i | −2.46651 | − | 7.59113i | −6.22327 | − | 9.28822i | −0.125121 | + | 0.385084i | −17.3099 | 0.666555 | − | 2.05145i | −7.28115 | + | 5.29007i | 0.0573272 | − | 1.50789i | ||
16.5 | 0.772797 | + | 0.561470i | 0.927051 | + | 2.85317i | −2.19017 | − | 6.74065i | 11.0970 | − | 1.36218i | −0.885547 | + | 2.72543i | 12.4836 | 4.45357 | − | 13.7067i | −7.28115 | + | 5.29007i | 9.34059 | + | 5.17797i | ||
16.6 | 3.25026 | + | 2.36146i | 0.927051 | + | 2.85317i | 2.51561 | + | 7.74226i | −6.00716 | + | 9.42943i | −3.72447 | + | 11.4627i | 1.75849 | −0.174670 | + | 0.537580i | −7.28115 | + | 5.29007i | −41.7920 | + | 16.4625i | ||
16.7 | 3.87763 | + | 2.81726i | 0.927051 | + | 2.85317i | 4.62691 | + | 14.2402i | 7.51887 | − | 8.27445i | −4.44337 | + | 13.6753i | 0.140520 | −10.3279 | + | 31.7859i | −7.28115 | + | 5.29007i | 52.4667 | − | 10.9026i | ||
31.1 | −1.53022 | − | 4.70953i | −2.42705 | + | 1.76336i | −13.3660 | + | 9.71093i | −9.83672 | + | 5.31403i | 12.0185 | + | 8.73195i | 28.2766 | 34.1374 | + | 24.8023i | 2.78115 | − | 8.55951i | 40.0789 | + | 38.1947i | ||
31.2 | −0.907834 | − | 2.79403i | −2.42705 | + | 1.76336i | −0.510282 | + | 0.370741i | 10.7234 | + | 3.16365i | 7.13022 | + | 5.18041i | 18.9115 | −17.5148 | − | 12.7253i | 2.78115 | − | 8.55951i | −0.895733 | − | 32.8335i | ||
31.3 | −0.722966 | − | 2.22506i | −2.42705 | + | 1.76336i | 2.04392 | − | 1.48499i | −1.87995 | + | 11.0212i | 5.67825 | + | 4.12549i | −32.9322 | −19.9239 | − | 14.4756i | 2.78115 | − | 8.55951i | 25.8819 | − | 3.78491i | ||
31.4 | −0.177563 | − | 0.546483i | −2.42705 | + | 1.76336i | 6.20502 | − | 4.50821i | −9.45304 | − | 5.96993i | 1.39460 | + | 1.01324i | −2.67744 | −7.28438 | − | 5.29241i | 2.78115 | − | 8.55951i | −1.58395 | + | 6.22597i | ||
31.5 | 0.536885 | + | 1.65236i | −2.42705 | + | 1.76336i | 4.03008 | − | 2.92802i | 8.44668 | − | 7.32487i | −4.21675 | − | 3.06365i | 7.66213 | 18.2465 | + | 13.2569i | 2.78115 | − | 8.55951i | 16.6382 | + | 10.0244i | ||
31.6 | 1.31310 | + | 4.04131i | −2.42705 | + | 1.76336i | −8.13578 | + | 5.91099i | −8.20823 | − | 7.59111i | −10.3132 | − | 7.49299i | −28.2853 | −7.06929 | − | 5.13614i | 2.78115 | − | 8.55951i | 19.8998 | − | 43.1398i | ||
31.7 | 1.48860 | + | 4.58143i | −2.42705 | + | 1.76336i | −12.3014 | + | 8.93752i | 5.89885 | + | 9.49755i | −11.6916 | − | 8.49444i | 1.13492 | −28.0809 | − | 20.4020i | 2.78115 | − | 8.55951i | −34.7314 | + | 41.1632i | ||
46.1 | −1.53022 | + | 4.70953i | −2.42705 | − | 1.76336i | −13.3660 | − | 9.71093i | −9.83672 | − | 5.31403i | 12.0185 | − | 8.73195i | 28.2766 | 34.1374 | − | 24.8023i | 2.78115 | + | 8.55951i | 40.0789 | − | 38.1947i | ||
46.2 | −0.907834 | + | 2.79403i | −2.42705 | − | 1.76336i | −0.510282 | − | 0.370741i | 10.7234 | − | 3.16365i | 7.13022 | − | 5.18041i | 18.9115 | −17.5148 | + | 12.7253i | 2.78115 | + | 8.55951i | −0.895733 | + | 32.8335i | ||
46.3 | −0.722966 | + | 2.22506i | −2.42705 | − | 1.76336i | 2.04392 | + | 1.48499i | −1.87995 | − | 11.0212i | 5.67825 | − | 4.12549i | −32.9322 | −19.9239 | + | 14.4756i | 2.78115 | + | 8.55951i | 25.8819 | + | 3.78491i | ||
46.4 | −0.177563 | + | 0.546483i | −2.42705 | − | 1.76336i | 6.20502 | + | 4.50821i | −9.45304 | + | 5.96993i | 1.39460 | − | 1.01324i | −2.67744 | −7.28438 | + | 5.29241i | 2.78115 | + | 8.55951i | −1.58395 | − | 6.22597i | ||
46.5 | 0.536885 | − | 1.65236i | −2.42705 | − | 1.76336i | 4.03008 | + | 2.92802i | 8.44668 | + | 7.32487i | −4.21675 | + | 3.06365i | 7.66213 | 18.2465 | − | 13.2569i | 2.78115 | + | 8.55951i | 16.6382 | − | 10.0244i | ||
46.6 | 1.31310 | − | 4.04131i | −2.42705 | − | 1.76336i | −8.13578 | − | 5.91099i | −8.20823 | + | 7.59111i | −10.3132 | + | 7.49299i | −28.2853 | −7.06929 | + | 5.13614i | 2.78115 | + | 8.55951i | 19.8998 | + | 43.1398i | ||
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 | 75.4.g.b | ✓ | 28 |
3.b | odd | 2 | 1 | 225.4.h.a | 28 | ||
25.d | even | 5 | 1 | inner | 75.4.g.b | ✓ | 28 |
25.d | even | 5 | 1 | 1875.4.a.g | 14 | ||
25.e | even | 10 | 1 | 1875.4.a.f | 14 | ||
75.j | odd | 10 | 1 | 225.4.h.a | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.4.g.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
75.4.g.b | ✓ | 28 | 25.d | even | 5 | 1 | inner |
225.4.h.a | 28 | 3.b | odd | 2 | 1 | ||
225.4.h.a | 28 | 75.j | odd | 10 | 1 | ||
1875.4.a.f | 14 | 25.e | even | 10 | 1 | ||
1875.4.a.g | 14 | 25.d | even | 5 | 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 \(S_{4}^{\mathrm{new}}(75, [\chi])\).