Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,3,Mod(67,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 310.o (of order \(12\), 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: | \(8.44688819517\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(15\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −1.00000 | − | 1.00000i | −1.44258 | − | 5.38379i | 2.00000i | 3.11675 | + | 3.90971i | −3.94121 | + | 6.82637i | 9.48540 | − | 2.54160i | 2.00000 | − | 2.00000i | −19.1099 | + | 11.0331i | 0.792965 | − | 7.02646i | ||
67.2 | −1.00000 | − | 1.00000i | −1.30026 | − | 4.85263i | 2.00000i | −4.14890 | + | 2.79045i | −3.55237 | + | 6.15289i | −5.56600 | + | 1.49140i | 2.00000 | − | 2.00000i | −14.0631 | + | 8.11934i | 6.93935 | + | 1.35845i | ||
67.3 | −1.00000 | − | 1.00000i | −1.20732 | − | 4.50577i | 2.00000i | 3.15820 | − | 3.87631i | −3.29845 | + | 5.71308i | −8.21042 | + | 2.19998i | 2.00000 | − | 2.00000i | −11.0501 | + | 6.37978i | −7.03451 | + | 0.718103i | ||
67.4 | −1.00000 | − | 1.00000i | −0.748521 | − | 2.79352i | 2.00000i | 3.71811 | − | 3.34300i | −2.04500 | + | 3.54204i | 3.99794 | − | 1.07125i | 2.00000 | − | 2.00000i | 0.550769 | − | 0.317987i | −7.06111 | − | 0.375114i | ||
67.5 | −1.00000 | − | 1.00000i | −0.684780 | − | 2.55563i | 2.00000i | −4.42728 | + | 2.32362i | −1.87085 | + | 3.24041i | 9.25624 | − | 2.48020i | 2.00000 | − | 2.00000i | 1.73188 | − | 0.999904i | 6.75090 | + | 2.10366i | ||
67.6 | −1.00000 | − | 1.00000i | −0.625156 | − | 2.33312i | 2.00000i | −4.41762 | − | 2.34192i | −1.70796 | + | 2.95827i | 2.12950 | − | 0.570598i | 2.00000 | − | 2.00000i | 2.74162 | − | 1.58288i | 2.07570 | + | 6.75955i | ||
67.7 | −1.00000 | − | 1.00000i | −0.587359 | − | 2.19206i | 2.00000i | 3.39124 | + | 3.67417i | −1.60470 | + | 2.77941i | −9.33925 | + | 2.50244i | 2.00000 | − | 2.00000i | 3.33411 | − | 1.92495i | 0.282931 | − | 7.06541i | ||
67.8 | −1.00000 | − | 1.00000i | −0.133096 | − | 0.496721i | 2.00000i | −1.38599 | − | 4.80406i | −0.363625 | + | 0.629817i | 3.18158 | − | 0.852501i | 2.00000 | − | 2.00000i | 7.56521 | − | 4.36778i | −3.41807 | + | 6.19006i | ||
67.9 | −1.00000 | − | 1.00000i | −0.00691904 | − | 0.0258222i | 2.00000i | −1.45893 | + | 4.78242i | −0.0189032 | + | 0.0327412i | −4.08615 | + | 1.09488i | 2.00000 | − | 2.00000i | 7.79361 | − | 4.49964i | 6.24135 | − | 3.32349i | ||
67.10 | −1.00000 | − | 1.00000i | 0.495067 | + | 1.84762i | 2.00000i | −1.47284 | − | 4.77815i | 1.35255 | − | 2.34268i | −8.53581 | + | 2.28716i | 2.00000 | − | 2.00000i | 4.62564 | − | 2.67061i | −3.30532 | + | 6.25099i | ||
67.11 | −1.00000 | − | 1.00000i | 0.547772 | + | 2.04431i | 2.00000i | 2.08818 | + | 4.54307i | 1.49654 | − | 2.59209i | 5.63559 | − | 1.51005i | 2.00000 | − | 2.00000i | 3.91506 | − | 2.26036i | 2.45489 | − | 6.63125i | ||
67.12 | −1.00000 | − | 1.00000i | 0.596728 | + | 2.22702i | 2.00000i | 4.61146 | − | 1.93247i | 1.63029 | − | 2.82375i | 7.04237 | − | 1.88700i | 2.00000 | − | 2.00000i | 3.19070 | − | 1.84215i | −6.54393 | − | 2.67900i | ||
67.13 | −1.00000 | − | 1.00000i | 0.758485 | + | 2.83070i | 2.00000i | −4.88288 | + | 1.07588i | 2.07222 | − | 3.58919i | −7.55379 | + | 2.02403i | 2.00000 | − | 2.00000i | 0.356638 | − | 0.205905i | 5.95876 | + | 3.80699i | ||
67.14 | −1.00000 | − | 1.00000i | 1.14761 | + | 4.28293i | 2.00000i | 4.96940 | + | 0.552337i | 3.13532 | − | 5.43054i | −10.4628 | + | 2.80349i | 2.00000 | − | 2.00000i | −9.23228 | + | 5.33026i | −4.41706 | − | 5.52174i | ||
67.15 | −1.00000 | − | 1.00000i | 1.36020 | + | 5.07634i | 2.00000i | −0.760826 | − | 4.94178i | 3.71614 | − | 6.43654i | −2.00070 | + | 0.536087i | 2.00000 | − | 2.00000i | −16.1248 | + | 9.30968i | −4.18095 | + | 5.70260i | ||
87.1 | −1.00000 | − | 1.00000i | −5.07634 | − | 1.36020i | 2.00000i | −3.89929 | + | 3.12978i | 3.71614 | + | 6.43654i | 0.536087 | − | 2.00070i | 2.00000 | − | 2.00000i | 16.1248 | + | 9.30968i | 7.02907 | + | 0.769507i | ||
87.2 | −1.00000 | − | 1.00000i | −4.28293 | − | 1.14761i | 2.00000i | −2.00636 | − | 4.57979i | 3.13532 | + | 5.43054i | 2.80349 | − | 10.4628i | 2.00000 | − | 2.00000i | 9.23228 | + | 5.33026i | −2.57343 | + | 6.58616i | ||
87.3 | −1.00000 | − | 1.00000i | −2.83070 | − | 0.758485i | 2.00000i | 3.37318 | + | 3.69075i | 2.07222 | + | 3.58919i | 2.02403 | − | 7.55379i | 2.00000 | − | 2.00000i | −0.356638 | − | 0.205905i | 0.317571 | − | 7.06393i | ||
87.4 | −1.00000 | − | 1.00000i | −2.22702 | − | 0.596728i | 2.00000i | −3.97929 | − | 3.02741i | 1.63029 | + | 2.82375i | −1.88700 | + | 7.04237i | 2.00000 | − | 2.00000i | −3.19070 | − | 1.84215i | 0.951885 | + | 7.00671i | ||
87.5 | −1.00000 | − | 1.00000i | −2.04431 | − | 0.547772i | 2.00000i | 2.89033 | − | 4.07995i | 1.49654 | + | 2.59209i | −1.51005 | + | 5.63559i | 2.00000 | − | 2.00000i | −3.91506 | − | 2.26036i | −6.97028 | + | 1.18963i | ||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
31.c | even | 3 | 1 | inner |
155.o | odd | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 310.3.o.c | ✓ | 60 |
5.c | odd | 4 | 1 | inner | 310.3.o.c | ✓ | 60 |
31.c | even | 3 | 1 | inner | 310.3.o.c | ✓ | 60 |
155.o | odd | 12 | 1 | inner | 310.3.o.c | ✓ | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
310.3.o.c | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
310.3.o.c | ✓ | 60 | 5.c | odd | 4 | 1 | inner |
310.3.o.c | ✓ | 60 | 31.c | even | 3 | 1 | inner |
310.3.o.c | ✓ | 60 | 155.o | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{60} - 10 T_{3}^{59} + 50 T_{3}^{58} - 340 T_{3}^{57} + 380 T_{3}^{56} + 9668 T_{3}^{55} - 57880 T_{3}^{54} + 419978 T_{3}^{53} - 955032 T_{3}^{52} - 7784772 T_{3}^{51} + 51536312 T_{3}^{50} + \cdots + 53\!\cdots\!00 \)
acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\).