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 | −1.67137 | + | 0.454432i | 0.866025 | − | 0.500000i | −0.359298 | − | 0.359298i | 1.49681 | − | 0.871531i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | 2.58698 | − | 1.51905i | 0.440048 | + | 0.254062i |
71.2 | −0.965926 | + | 0.258819i | −1.38004 | − | 1.04666i | 0.866025 | − | 0.500000i | −1.68505 | − | 1.68505i | 1.60391 | + | 0.653816i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | 0.809005 | + | 2.88886i | 2.06376 | + | 1.19151i |
71.3 | −0.965926 | + | 0.258819i | −0.211319 | − | 1.71911i | 0.866025 | − | 0.500000i | 0.0940280 | + | 0.0940280i | 0.649057 | + | 1.60584i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | −2.91069 | + | 0.726562i | −0.115160 | − | 0.0664879i |
71.4 | −0.965926 | + | 0.258819i | −0.186842 | + | 1.72194i | 0.866025 | − | 0.500000i | −2.04268 | − | 2.04268i | −0.265197 | − | 1.71163i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | −2.93018 | − | 0.643461i | 2.50176 | + | 1.44439i |
71.5 | −0.965926 | + | 0.258819i | 0.458354 | + | 1.67030i | 0.866025 | − | 0.500000i | 1.24412 | + | 1.24412i | −0.875042 | − | 1.49476i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | −2.57982 | + | 1.53118i | −1.52373 | − | 0.879726i |
71.6 | −0.965926 | + | 0.258819i | 1.13690 | − | 1.30670i | 0.866025 | − | 0.500000i | 1.58301 | + | 1.58301i | −0.759962 | + | 1.55642i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | −0.414918 | − | 2.97117i | −1.93879 | − | 1.11936i |
71.7 | −0.965926 | + | 0.258819i | 1.59550 | + | 0.674078i | 0.866025 | − | 0.500000i | −1.28362 | − | 1.28362i | −1.71560 | − | 0.238164i | −0.258819 | + | 0.965926i | −0.707107 | + | 0.707107i | 2.09124 | + | 2.15098i | 1.57211 | + | 0.907660i |
71.8 | 0.965926 | − | 0.258819i | −1.69149 | + | 0.372632i | 0.866025 | − | 0.500000i | 1.11112 | + | 1.11112i | −1.53741 | + | 0.797725i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | 2.72229 | − | 1.26061i | 1.36084 | + | 0.785680i |
71.9 | 0.965926 | − | 0.258819i | −0.884905 | + | 1.48894i | 0.866025 | − | 0.500000i | −1.36895 | − | 1.36895i | −0.469386 | + | 1.66724i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | −1.43389 | − | 2.63514i | −1.67662 | − | 0.967996i |
71.10 | 0.965926 | − | 0.258819i | −0.625096 | − | 1.61532i | 0.866025 | − | 0.500000i | 0.828570 | + | 0.828570i | −1.02187 | − | 1.39849i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | −2.21851 | + | 2.01946i | 1.01479 | + | 0.585888i |
71.11 | 0.965926 | − | 0.258819i | 0.162199 | − | 1.72444i | 0.866025 | − | 0.500000i | −2.61115 | − | 2.61115i | −0.289646 | − | 1.70766i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | −2.94738 | − | 0.559404i | −3.19799 | − | 1.84636i |
71.12 | 0.965926 | − | 0.258819i | 0.505064 | + | 1.65678i | 0.866025 | − | 0.500000i | 2.95685 | + | 2.95685i | 0.916660 | + | 1.46960i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | −2.48982 | + | 1.67356i | 3.62139 | + | 2.09081i |
71.13 | 0.965926 | − | 0.258819i | 1.18291 | − | 1.26520i | 0.866025 | − | 0.500000i | 2.02216 | + | 2.02216i | 0.815146 | − | 1.52825i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | −0.201451 | − | 2.99323i | 2.47663 | + | 1.42988i |
71.14 | 0.965926 | − | 0.258819i | 1.61014 | + | 0.638319i | 0.866025 | − | 0.500000i | −0.489109 | − | 0.489109i | 1.72048 | + | 0.199834i | 0.258819 | − | 0.965926i | 0.707107 | − | 0.707107i | 2.18510 | + | 2.05556i | −0.599034 | − | 0.345852i |
197.1 | −0.258819 | + | 0.965926i | −1.64483 | + | 0.542708i | −0.866025 | − | 0.500000i | 1.01262 | + | 1.01262i | −0.0985019 | − | 1.72925i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | 2.41094 | − | 1.78532i | −1.24020 | + | 0.716030i |
197.2 | −0.258819 | + | 0.965926i | −1.34996 | − | 1.08518i | −0.866025 | − | 0.500000i | 0.429580 | + | 0.429580i | 1.39760 | − | 1.02310i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | 0.644780 | + | 2.92989i | −0.526126 | + | 0.303759i |
197.3 | −0.258819 | + | 0.965926i | −1.12111 | + | 1.32027i | −0.866025 | − | 0.500000i | −0.381662 | − | 0.381662i | −0.985117 | − | 1.42462i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | −0.486221 | − | 2.96034i | 0.467439 | − | 0.269876i |
197.4 | −0.258819 | + | 0.965926i | −0.413071 | − | 1.68207i | −0.866025 | − | 0.500000i | −2.27299 | − | 2.27299i | 1.73167 | + | 0.0363565i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | −2.65874 | + | 1.38963i | 2.78383 | − | 1.60724i |
197.5 | −0.258819 | + | 0.965926i | 0.984178 | − | 1.42527i | −0.866025 | − | 0.500000i | −1.05727 | − | 1.05727i | 1.12198 | + | 1.31953i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | −1.06279 | − | 2.80544i | 1.29489 | − | 0.747605i |
197.6 | −0.258819 | + | 0.965926i | 1.01587 | + | 1.40286i | −0.866025 | − | 0.500000i | −1.99319 | − | 1.99319i | −1.61798 | + | 0.618168i | −0.965926 | + | 0.258819i | 0.707107 | − | 0.707107i | −0.936020 | + | 2.85024i | 2.44115 | − | 1.40940i |
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.b | yes | 56 |
3.b | odd | 2 | 1 | 546.2.bu.a | ✓ | 56 | |
13.f | odd | 12 | 1 | 546.2.bu.a | ✓ | 56 | |
39.k | even | 12 | 1 | inner | 546.2.bu.b | yes | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.bu.a | ✓ | 56 | 3.b | odd | 2 | 1 | |
546.2.bu.a | ✓ | 56 | 13.f | odd | 12 | 1 | |
546.2.bu.b | yes | 56 | 1.a | even | 1 | 1 | trivial |
546.2.bu.b | yes | 56 | 39.k | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{56} + 16 T_{5}^{53} + 860 T_{5}^{52} + 224 T_{5}^{51} + 128 T_{5}^{50} + 10400 T_{5}^{49} + 287904 T_{5}^{48} + 140800 T_{5}^{47} + 81408 T_{5}^{46} + 2525712 T_{5}^{45} + 48653252 T_{5}^{44} + \cdots + 19932921856 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).