Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [252,2,Mod(31,252)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(252, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("252.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 252.n (of order \(6\), degree \(2\), 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: | \(2.01223013094\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.41347 | − | 0.0457258i | −0.962720 | − | 1.43985i | 1.99582 | + | 0.129265i | 0.724862i | 1.29494 | + | 2.07921i | −1.25991 | + | 2.32651i | −2.81513 | − | 0.273973i | −1.14634 | + | 2.77235i | 0.0331449 | − | 1.02457i | ||
31.2 | −1.41219 | + | 0.0755732i | −1.69164 | + | 0.371957i | 1.98858 | − | 0.213448i | 1.05993i | 2.36081 | − | 0.653118i | 0.349222 | − | 2.62260i | −2.79212 | + | 0.451713i | 2.72330 | − | 1.25843i | −0.0801021 | − | 1.49682i | ||
31.3 | −1.39808 | − | 0.213015i | −0.709178 | + | 1.58021i | 1.90925 | + | 0.595623i | − | 2.93241i | 1.32809 | − | 2.05819i | −2.24678 | + | 1.39713i | −2.54241 | − | 1.23943i | −1.99413 | − | 2.24130i | −0.624647 | + | 4.09975i | |
31.4 | −1.38558 | + | 0.283143i | 0.385748 | + | 1.68855i | 1.83966 | − | 0.784633i | 3.94623i | −1.01258 | − | 2.23040i | 2.37341 | + | 1.16916i | −2.32683 | + | 1.60806i | −2.70240 | + | 1.30271i | −1.11734 | − | 5.46781i | ||
31.5 | −1.33749 | + | 0.459470i | 1.58493 | − | 0.698557i | 1.57778 | − | 1.22907i | − | 2.69249i | −1.79887 | + | 1.66254i | 1.51663 | + | 2.16791i | −1.54554 | + | 2.36882i | 2.02404 | − | 2.21433i | 1.23712 | + | 3.60119i | |
31.6 | −1.32994 | − | 0.480902i | 1.67381 | + | 0.445360i | 1.53747 | + | 1.27914i | − | 0.815110i | −2.01189 | − | 1.39724i | −0.448419 | − | 2.60747i | −1.42959 | − | 2.44055i | 2.60331 | + | 1.49090i | −0.391988 | + | 1.08405i | |
31.7 | −1.26565 | − | 0.630973i | 1.11108 | − | 1.32872i | 1.20375 | + | 1.59718i | 2.00240i | −2.24463 | + | 0.980638i | 2.64523 | + | 0.0525529i | −0.515742 | − | 2.78101i | −0.531008 | − | 2.95263i | 1.26346 | − | 2.53433i | ||
31.8 | −1.18174 | + | 0.776846i | −0.238791 | − | 1.71551i | 0.793020 | − | 1.83606i | 0.121070i | 1.61488 | + | 1.84178i | 0.910048 | − | 2.48431i | 0.489193 | + | 2.78580i | −2.88596 | + | 0.819299i | −0.0940527 | − | 0.143073i | ||
31.9 | −1.16223 | + | 0.805738i | 1.39154 | + | 1.03131i | 0.701573 | − | 1.87291i | 0.0505362i | −2.44827 | − | 0.0774077i | −2.64573 | − | 0.0112150i | 0.693684 | + | 2.74204i | 0.872785 | + | 2.87023i | −0.0407190 | − | 0.0587349i | ||
31.10 | −1.13008 | − | 0.850244i | −1.49040 | − | 0.882451i | 0.554170 | + | 1.92169i | − | 4.05112i | 0.933971 | + | 2.26444i | 2.21649 | − | 1.44470i | 1.00765 | − | 2.64285i | 1.44256 | + | 2.63040i | −3.44444 | + | 4.57810i | |
31.11 | −0.998009 | + | 1.00199i | −1.26661 | + | 1.18139i | −0.00795539 | − | 1.99998i | − | 1.65619i | 0.0803514 | − | 2.44817i | 2.45603 | + | 0.983840i | 2.01190 | + | 1.98803i | 0.208620 | − | 2.99274i | 1.65948 | + | 1.65289i | |
31.12 | −0.989784 | − | 1.01011i | −0.187304 | + | 1.72189i | −0.0406542 | + | 1.99959i | 2.71053i | 1.92470 | − | 1.51511i | −1.53391 | − | 2.15572i | 2.06005 | − | 1.93809i | −2.92983 | − | 0.645035i | 2.73794 | − | 2.68284i | ||
31.13 | −0.964964 | − | 1.03385i | 0.422109 | − | 1.67983i | −0.137689 | + | 1.99525i | − | 0.594537i | −2.14401 | + | 1.18458i | −2.59555 | + | 0.512964i | 2.19566 | − | 1.78300i | −2.64365 | − | 1.41814i | −0.614662 | + | 0.573707i | |
31.14 | −0.836845 | + | 1.14004i | −1.73205 | − | 0.00306272i | −0.599381 | − | 1.90807i | 3.10969i | 1.45295 | − | 1.97204i | −2.27748 | + | 1.34650i | 2.67687 | + | 0.913443i | 2.99998 | + | 0.0106095i | −3.54517 | − | 2.60233i | ||
31.15 | −0.795200 | − | 1.16947i | 0.815267 | + | 1.52818i | −0.735314 | + | 1.85992i | − | 1.91789i | 1.13886 | − | 2.16864i | 1.85299 | + | 1.88850i | 2.75984 | − | 0.619084i | −1.67068 | + | 2.49175i | −2.24291 | + | 1.52511i | |
31.16 | −0.568881 | + | 1.29475i | 1.73205 | + | 0.00306272i | −1.35275 | − | 1.47312i | 3.10969i | −0.989294 | + | 2.24082i | 2.27748 | − | 1.34650i | 2.67687 | − | 0.913443i | 2.99998 | + | 0.0106095i | −4.02627 | − | 1.76904i | ||
31.17 | −0.368742 | + | 1.36529i | 1.26661 | − | 1.18139i | −1.72806 | − | 1.00688i | − | 1.65619i | 1.14590 | + | 2.16493i | −2.45603 | − | 0.983840i | 2.01190 | − | 1.98803i | 0.208620 | − | 2.99274i | 2.26119 | + | 0.610706i | |
31.18 | −0.290988 | − | 1.38395i | −1.20973 | − | 1.23958i | −1.83065 | + | 0.805427i | 2.88398i | −1.36351 | + | 2.03491i | 2.24777 | + | 1.39554i | 1.64737 | + | 2.29917i | −0.0731219 | + | 2.99911i | 3.99129 | − | 0.839202i | ||
31.19 | −0.121544 | − | 1.40898i | 1.71229 | − | 0.260902i | −1.97045 | + | 0.342507i | − | 2.56839i | −0.575725 | − | 2.38087i | −0.477445 | − | 2.60232i | 0.722083 | + | 2.73470i | 2.86386 | − | 0.893480i | −3.61882 | + | 0.312174i | |
31.20 | −0.116673 | + | 1.40939i | −1.39154 | − | 1.03131i | −1.97277 | − | 0.328876i | 0.0505362i | 1.61588 | − | 1.84090i | 2.64573 | + | 0.0112150i | 0.693684 | − | 2.74204i | 0.872785 | + | 2.87023i | −0.0712254 | − | 0.00589621i | ||
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.k | odd | 6 | 1 | inner |
252.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 252.2.n.b | ✓ | 84 |
3.b | odd | 2 | 1 | 756.2.n.b | 84 | ||
4.b | odd | 2 | 1 | inner | 252.2.n.b | ✓ | 84 |
7.d | odd | 6 | 1 | 252.2.bj.b | yes | 84 | |
9.c | even | 3 | 1 | 252.2.bj.b | yes | 84 | |
9.d | odd | 6 | 1 | 756.2.bj.b | 84 | ||
12.b | even | 2 | 1 | 756.2.n.b | 84 | ||
21.g | even | 6 | 1 | 756.2.bj.b | 84 | ||
28.f | even | 6 | 1 | 252.2.bj.b | yes | 84 | |
36.f | odd | 6 | 1 | 252.2.bj.b | yes | 84 | |
36.h | even | 6 | 1 | 756.2.bj.b | 84 | ||
63.k | odd | 6 | 1 | inner | 252.2.n.b | ✓ | 84 |
63.s | even | 6 | 1 | 756.2.n.b | 84 | ||
84.j | odd | 6 | 1 | 756.2.bj.b | 84 | ||
252.n | even | 6 | 1 | inner | 252.2.n.b | ✓ | 84 |
252.bn | odd | 6 | 1 | 756.2.n.b | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.n.b | ✓ | 84 | 1.a | even | 1 | 1 | trivial |
252.2.n.b | ✓ | 84 | 4.b | odd | 2 | 1 | inner |
252.2.n.b | ✓ | 84 | 63.k | odd | 6 | 1 | inner |
252.2.n.b | ✓ | 84 | 252.n | even | 6 | 1 | inner |
252.2.bj.b | yes | 84 | 7.d | odd | 6 | 1 | |
252.2.bj.b | yes | 84 | 9.c | even | 3 | 1 | |
252.2.bj.b | yes | 84 | 28.f | even | 6 | 1 | |
252.2.bj.b | yes | 84 | 36.f | odd | 6 | 1 | |
756.2.n.b | 84 | 3.b | odd | 2 | 1 | ||
756.2.n.b | 84 | 12.b | even | 2 | 1 | ||
756.2.n.b | 84 | 63.s | even | 6 | 1 | ||
756.2.n.b | 84 | 252.bn | odd | 6 | 1 | ||
756.2.bj.b | 84 | 9.d | odd | 6 | 1 | ||
756.2.bj.b | 84 | 21.g | even | 6 | 1 | ||
756.2.bj.b | 84 | 36.h | even | 6 | 1 | ||
756.2.bj.b | 84 | 84.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{42} + 113 T_{5}^{40} + 5811 T_{5}^{38} + 180273 T_{5}^{36} + 3769914 T_{5}^{34} + 56237622 T_{5}^{32} + \cdots + 2187 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).