Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [150,3,Mod(11,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 150.j (of order \(10\), 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: | \(4.08720396540\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −0.831254 | − | 1.14412i | −2.89929 | + | 0.770808i | −0.618034 | + | 1.90211i | 4.91294 | − | 0.928983i | 3.29194 | + | 2.67640i | −5.64982 | 2.68999 | − | 0.874032i | 7.81171 | − | 4.46959i | −5.14677 | − | 4.84879i | ||
11.2 | −0.831254 | − | 1.14412i | −2.74953 | − | 1.20004i | −0.618034 | + | 1.90211i | −3.75827 | − | 3.29779i | 0.912562 | + | 4.14334i | −3.79877 | 2.68999 | − | 0.874032i | 6.11980 | + | 6.59909i | −0.648993 | + | 7.04122i | ||
11.3 | −0.831254 | − | 1.14412i | −2.21254 | − | 2.02600i | −0.618034 | + | 1.90211i | 1.16748 | + | 4.86179i | −0.478809 | + | 4.21554i | 6.03242 | 2.68999 | − | 0.874032i | 0.790661 | + | 8.96520i | 4.59201 | − | 5.37712i | ||
11.4 | −0.831254 | − | 1.14412i | −1.90427 | + | 2.31814i | −0.618034 | + | 1.90211i | −4.93889 | + | 0.779331i | 4.23516 | + | 0.251761i | 8.16986 | 2.68999 | − | 0.874032i | −1.74750 | − | 8.82872i | 4.99712 | + | 5.00288i | ||
11.5 | −0.831254 | − | 1.14412i | −0.513013 | + | 2.95581i | −0.618034 | + | 1.90211i | 2.36498 | − | 4.40532i | 3.80825 | − | 1.87008i | 1.96623 | 2.68999 | − | 0.874032i | −8.47364 | − | 3.03274i | −7.00613 | + | 0.956120i | ||
11.6 | −0.831254 | − | 1.14412i | 0.571625 | − | 2.94504i | −0.618034 | + | 1.90211i | 0.970509 | − | 4.90491i | −3.84465 | + | 1.79406i | 12.2853 | 2.68999 | − | 0.874032i | −8.34649 | − | 3.36692i | −6.41856 | + | 2.96684i | ||
11.7 | −0.831254 | − | 1.14412i | 0.888659 | + | 2.86536i | −0.618034 | + | 1.90211i | 1.35845 | + | 4.81192i | 2.53962 | − | 3.39858i | −9.92920 | 2.68999 | − | 0.874032i | −7.42057 | + | 5.09266i | 4.37621 | − | 5.55417i | ||
11.8 | −0.831254 | − | 1.14412i | 2.70308 | − | 1.30129i | −0.618034 | + | 1.90211i | 4.40886 | + | 2.35838i | −3.73578 | − | 2.01095i | 0.752203 | 2.68999 | − | 0.874032i | 5.61328 | − | 7.03499i | −0.966599 | − | 7.00469i | ||
11.9 | −0.831254 | − | 1.14412i | 2.78863 | + | 1.10613i | −0.618034 | + | 1.90211i | −4.36319 | + | 2.44184i | −1.05251 | − | 4.11002i | 4.66363 | 2.68999 | − | 0.874032i | 6.55293 | + | 6.16920i | 6.42068 | + | 2.96224i | ||
11.10 | −0.831254 | − | 1.14412i | 2.82664 | − | 1.00504i | −0.618034 | + | 1.90211i | −1.09537 | − | 4.87854i | −3.49954 | − | 2.39858i | −13.2558 | 2.68999 | − | 0.874032i | 6.97980 | − | 5.68176i | −4.67112 | + | 5.30855i | ||
11.11 | 0.831254 | + | 1.14412i | −2.90622 | − | 0.744235i | −0.618034 | + | 1.90211i | 4.36319 | − | 2.44184i | −1.56431 | − | 3.94372i | 4.66363 | −2.68999 | + | 0.874032i | 7.89223 | + | 4.32582i | 6.42068 | + | 2.96224i | ||
11.12 | 0.831254 | + | 1.14412i | −2.40316 | + | 1.79578i | −0.618034 | + | 1.90211i | −1.35845 | − | 4.81192i | −4.05223 | − | 1.25675i | −9.92920 | −2.68999 | + | 0.874032i | 2.55032 | − | 8.63110i | 4.37621 | − | 5.55417i | ||
11.13 | 0.831254 | + | 1.14412i | −1.69605 | − | 2.47455i | −0.618034 | + | 1.90211i | 1.09537 | + | 4.87854i | 1.42134 | − | 3.99747i | −13.2558 | −2.68999 | + | 0.874032i | −3.24680 | + | 8.39394i | −4.67112 | + | 5.30855i | ||
11.14 | 0.831254 | + | 1.14412i | −1.42196 | − | 2.64160i | −0.618034 | + | 1.90211i | −4.40886 | − | 2.35838i | 1.84031 | − | 3.82273i | 0.752203 | −2.68999 | + | 0.874032i | −4.95608 | + | 7.51247i | −0.966599 | − | 7.00469i | ||
11.15 | 0.831254 | + | 1.14412i | −1.32235 | + | 2.69284i | −0.618034 | + | 1.90211i | −2.36498 | + | 4.40532i | −4.18015 | + | 0.725510i | 1.96623 | −2.68999 | + | 0.874032i | −5.50280 | − | 7.12174i | −7.00613 | + | 0.956120i | ||
11.16 | 0.831254 | + | 1.14412i | 0.178022 | + | 2.99471i | −0.618034 | + | 1.90211i | 4.93889 | − | 0.779331i | −3.27834 | + | 2.69305i | 8.16986 | −2.68999 | + | 0.874032i | −8.93662 | + | 1.06625i | 4.99712 | + | 5.00288i | ||
11.17 | 0.831254 | + | 1.14412i | 1.26859 | − | 2.71858i | −0.618034 | + | 1.90211i | −0.970509 | + | 4.90491i | 4.16491 | − | 0.808400i | 12.2853 | −2.68999 | + | 0.874032i | −5.78133 | − | 6.89755i | −6.41856 | + | 2.96684i | ||
11.18 | 0.831254 | + | 1.14412i | 1.89250 | + | 2.32775i | −0.618034 | + | 1.90211i | −4.91294 | + | 0.928983i | −1.09009 | + | 4.10021i | −5.64982 | −2.68999 | + | 0.874032i | −1.83688 | + | 8.81055i | −5.14677 | − | 4.84879i | ||
11.19 | 0.831254 | + | 1.14412i | 2.92978 | + | 0.645279i | −0.618034 | + | 1.90211i | 3.75827 | + | 3.29779i | 1.69711 | + | 3.88842i | −3.79877 | −2.68999 | + | 0.874032i | 8.16723 | + | 3.78105i | −0.648993 | + | 7.04122i | ||
11.20 | 0.831254 | + | 1.14412i | 2.98083 | − | 0.338569i | −0.618034 | + | 1.90211i | −1.16748 | − | 4.86179i | 2.86519 | + | 3.12900i | 6.03242 | −2.68999 | + | 0.874032i | 8.77074 | − | 2.01844i | 4.59201 | − | 5.37712i | ||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 150.3.j.a | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 150.3.j.a | ✓ | 80 |
25.d | even | 5 | 1 | inner | 150.3.j.a | ✓ | 80 |
75.j | odd | 10 | 1 | inner | 150.3.j.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
150.3.j.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
150.3.j.a | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
150.3.j.a | ✓ | 80 | 25.d | even | 5 | 1 | inner |
150.3.j.a | ✓ | 80 | 75.j | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(150, [\chi])\).