Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,2,Mod(71,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.71");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.bu (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: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(14\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
71.1 | −0.965926 | + | 0.258819i | −1.68734 | + | 0.390990i | 0.866025 | − | 0.500000i | −2.95685 | − | 2.95685i | 1.52865 | − | 0.814384i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 2.69425 | − | 1.31947i | 3.62139 | + | 2.09081i |
71.2 | −0.965926 | + | 0.258819i | −1.35787 | − | 1.07526i | 0.866025 | − | 0.500000i | 0.489109 | + | 0.489109i | 1.58990 | + | 0.687181i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 0.687622 | + | 2.92013i | −0.599034 | − | 0.345852i |
71.3 | −0.965926 | + | 0.258819i | −0.847008 | + | 1.51082i | 0.866025 | − | 0.500000i | 1.36895 | + | 1.36895i | 0.427118 | − | 1.67856i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | −1.56515 | − | 2.55935i | −1.67662 | − | 0.967996i |
71.4 | −0.965926 | + | 0.258819i | 0.504239 | − | 1.65703i | 0.866025 | − | 0.500000i | −2.02216 | − | 2.02216i | −0.0581866 | + | 1.73107i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | −2.49149 | − | 1.67108i | 2.47663 | + | 1.42988i |
71.5 | −0.965926 | + | 0.258819i | 0.523037 | + | 1.65119i | 0.866025 | − | 0.500000i | −1.11112 | − | 1.11112i | −0.932575 | − | 1.45956i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | −2.45286 | + | 1.72727i | 1.36084 | + | 0.785680i |
71.6 | −0.965926 | + | 0.258819i | 1.41231 | − | 1.00269i | 0.866025 | − | 0.500000i | 2.61115 | + | 2.61115i | −1.10467 | + | 1.33405i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 0.989233 | − | 2.83221i | −3.19799 | − | 1.84636i |
71.7 | −0.965926 | + | 0.258819i | 1.71146 | − | 0.266310i | 0.866025 | − | 0.500000i | −0.828570 | − | 0.828570i | −1.58421 | + | 0.700193i | 0.258819 | − | 0.965926i | −0.707107 | + | 0.707107i | 2.85816 | − | 0.911557i | 1.01479 | + | 0.585888i |
71.8 | 0.965926 | − | 0.258819i | −1.67570 | + | 0.438205i | 0.866025 | − | 0.500000i | −1.24412 | − | 1.24412i | −1.50519 | + | 0.856977i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 2.61595 | − | 1.46860i | −1.52373 | − | 0.879726i |
71.9 | 0.965926 | − | 0.258819i | −1.39783 | + | 1.02278i | 0.866025 | − | 0.500000i | 2.04268 | + | 2.04268i | −1.08548 | + | 1.34972i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 0.907836 | − | 2.85934i | 2.50176 | + | 1.44439i |
71.10 | 0.965926 | − | 0.258819i | −1.38152 | − | 1.04470i | 0.866025 | − | 0.500000i | 1.28362 | + | 1.28362i | −1.60483 | − | 0.651543i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 0.817188 | + | 2.88656i | 1.57211 | + | 0.907660i |
71.11 | 0.965926 | − | 0.258819i | 0.442137 | + | 1.67467i | 0.866025 | − | 0.500000i | 0.359298 | + | 0.359298i | 0.860508 | + | 1.50317i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | −2.60903 | + | 1.48087i | 0.440048 | + | 0.254062i |
71.12 | 0.965926 | − | 0.258819i | 0.563184 | − | 1.63793i | 0.866025 | − | 0.500000i | −1.58301 | − | 1.58301i | 0.120065 | − | 1.72788i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | −2.36565 | − | 1.84491i | −1.93879 | − | 1.11936i |
71.13 | 0.965926 | − | 0.258819i | 1.59445 | − | 0.676548i | 0.866025 | − | 0.500000i | −0.0940280 | − | 0.0940280i | 1.36502 | − | 1.06617i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 2.08457 | − | 2.15745i | −0.115160 | − | 0.0664879i |
71.14 | 0.965926 | − | 0.258819i | 1.59645 | + | 0.671817i | 0.866025 | − | 0.500000i | 1.68505 | + | 1.68505i | 1.71593 | + | 0.235733i | −0.258819 | + | 0.965926i | 0.707107 | − | 0.707107i | 2.09732 | + | 2.14505i | 2.06376 | + | 1.19151i |
197.1 | −0.258819 | + | 0.965926i | −1.73063 | − | 0.0700467i | −0.866025 | − | 0.500000i | 2.88717 | + | 2.88717i | 0.515581 | − | 1.65353i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 2.99019 | + | 0.242450i | −3.53605 | + | 2.04154i |
197.2 | −0.258819 | + | 0.965926i | −1.60917 | − | 0.640767i | −0.866025 | − | 0.500000i | −2.11813 | − | 2.11813i | 1.03542 | − | 1.38849i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 2.17884 | + | 2.06220i | 2.59417 | − | 1.49774i |
197.3 | −0.258819 | + | 0.965926i | −0.703834 | + | 1.58260i | −0.866025 | − | 0.500000i | −2.95010 | − | 2.95010i | −1.34651 | − | 1.08946i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −2.00924 | − | 2.22777i | 3.61312 | − | 2.08604i |
197.4 | −0.258819 | + | 0.965926i | 0.684122 | + | 1.59122i | −0.866025 | − | 0.500000i | 0.803802 | + | 0.803802i | −1.71406 | + | 0.248973i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −2.06396 | + | 2.17717i | −0.984452 | + | 0.568374i |
197.5 | −0.258819 | + | 0.965926i | 0.957713 | − | 1.44319i | −0.866025 | − | 0.500000i | −0.0544192 | − | 0.0544192i | 1.14614 | + | 1.29860i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | −1.16557 | − | 2.76432i | 0.0666496 | − | 0.0384802i |
197.6 | −0.258819 | + | 0.965926i | 1.63803 | + | 0.562898i | −0.866025 | − | 0.500000i | 1.20435 | + | 1.20435i | −0.967672 | + | 1.43653i | 0.965926 | − | 0.258819i | 0.707107 | − | 0.707107i | 2.36629 | + | 1.84409i | −1.47502 | + | 0.851606i |
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
39.k | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.bu.a | ✓ | 56 |
3.b | odd | 2 | 1 | 546.2.bu.b | yes | 56 | |
13.f | odd | 12 | 1 | 546.2.bu.b | yes | 56 | |
39.k | even | 12 | 1 | inner | 546.2.bu.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.bu.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
546.2.bu.a | ✓ | 56 | 39.k | even | 12 | 1 | inner |
546.2.bu.b | yes | 56 | 3.b | odd | 2 | 1 | |
546.2.bu.b | yes | 56 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} - 16 T_{5}^{53} + 860 T_{5}^{52} - 224 T_{5}^{51} + 128 T_{5}^{50} - 10400 T_{5}^{49} + \cdots + 19932921856 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).