↑

Properties

Label 252.2.ba.a
Level $252$
Weight $2$
Character orbit 252.ba
Analytic conductor $2.012$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(155,252)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.ba (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: \(72\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{6} + 4 q^{9} - 12 q^{12} - 34 q^{18} - 42 q^{20} - 2 q^{24} + 36 q^{25} + 28 q^{30} + 30 q^{32} - 44 q^{33} - 12 q^{34} + 20 q^{36} - 12 q^{40} - 60 q^{41} + 20 q^{42} - 24 q^{45} - 24 q^{46} - 28 q^{48} + 36 q^{49} - 78 q^{50} - 18 q^{52} - 10 q^{54} - 4 q^{57} - 18 q^{58} - 76 q^{60} - 60 q^{64} + 24 q^{65} + 54 q^{66} + 78 q^{68} + 24 q^{69} + 74 q^{72} - 24 q^{73} + 12 q^{76} - 20 q^{78} - 4 q^{81} - 36 q^{82} + 14 q^{84} + 30 q^{86} + 24 q^{88} + 8 q^{90} - 114 q^{92} - 96 q^{93} + 42 q^{94} + 112 q^{96} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.40524 + 0.159031i −1.66588 + 0.474188i 1.94942 − 0.446956i −0.233892 + 0.135037i 2.26555 − 0.931276i −0.866025 − 0.500000i −2.66833 + 0.938100i 2.55029 − 1.57988i 0.307199 − 0.226956i
155.2 −1.40378 + 0.171499i −1.10270 − 1.33569i 1.94118 − 0.481494i 3.64166 − 2.10251i 1.77701 + 1.68589i 0.866025 + 0.500000i −2.64240 + 1.00882i −0.568119 + 2.94572i −4.75149 + 3.57600i
155.3 −1.34115 + 0.448692i 1.72102 + 0.195202i 1.59735 − 1.20352i −1.04266 + 0.601980i −2.39572 + 0.510412i 0.866025 + 0.500000i −1.60227 + 2.33082i 2.92379 + 0.671890i 1.12826 − 1.27518i
155.4 −1.27695 − 0.607789i −1.63407 − 0.574282i 1.26118 + 1.55223i −3.32396 + 1.91909i 1.73758 + 1.72650i 0.866025 + 0.500000i −0.667037 − 2.74865i 2.34040 + 1.87684i 5.41092 − 0.430306i
155.5 −1.26027 − 0.641659i 0.953579 − 1.44592i 1.17655 + 1.61732i 0.898347 − 0.518661i −2.12955 + 1.21037i −0.866025 − 0.500000i −0.444992 − 2.79320i −1.18137 − 2.75760i −1.46496 + 0.0772186i
155.6 −1.24454 + 0.671660i −0.586217 − 1.62983i 1.09775 − 1.67181i −2.16781 + 1.25159i 1.82426 + 1.63465i −0.866025 − 0.500000i −0.243298 + 2.81794i −2.31270 + 1.91087i 1.85728 − 3.01367i
155.7 −1.18583 − 0.770594i −0.953579 + 1.44592i 0.812371 + 1.82758i 0.898347 − 0.518661i 2.24500 − 0.979790i 0.866025 + 0.500000i 0.444992 − 2.79320i −1.18137 − 2.75760i −1.46496 − 0.0772186i
155.8 −1.17848 + 0.781786i 1.48209 − 0.896336i 0.777621 − 1.84264i 2.57586 − 1.48718i −1.04586 + 2.21499i −0.866025 − 0.500000i 0.524137 + 2.77944i 1.39317 − 2.65689i −1.87295 + 3.76638i
155.9 −1.16483 − 0.801973i 1.63407 + 0.574282i 0.713677 + 1.86833i −3.32396 + 1.91909i −1.44287 − 1.97943i −0.866025 − 0.500000i 0.667037 − 2.74865i 2.34040 + 1.87684i 5.41092 + 0.430306i
155.10 −1.08267 + 0.909846i −0.483200 + 1.66329i 0.344359 − 1.97013i −3.28588 + 1.89710i −0.990187 − 2.24043i 0.866025 + 0.500000i 1.41969 + 2.44632i −2.53304 − 1.60740i 1.83146 − 5.04359i
155.11 −0.896785 + 1.09352i −1.24317 + 1.20604i −0.391553 − 1.96130i 2.06874 − 1.19439i −0.203962 − 2.44098i −0.866025 − 0.500000i 2.49585 + 1.33069i 0.0909512 − 2.99862i −0.549133 + 3.33330i
155.12 −0.687522 + 1.23585i −1.67512 − 0.440428i −1.05463 − 1.69934i 0.256612 − 0.148155i 1.69598 − 1.76738i 0.866025 + 0.500000i 2.82520 − 0.135024i 2.61205 + 1.47554i 0.00667042 + 0.418992i
155.13 −0.564896 − 1.29649i 1.66588 − 0.474188i −1.36178 + 1.46477i −0.233892 + 0.135037i −1.55583 − 1.89193i 0.866025 + 0.500000i 2.66833 + 0.938100i 2.55029 − 1.57988i 0.307199 + 0.226956i
155.14 −0.553365 − 1.30146i 1.10270 + 1.33569i −1.38757 + 1.44036i 3.64166 − 2.10251i 1.12814 − 2.17423i −0.866025 − 0.500000i 2.64240 + 1.00882i −0.568119 + 2.94572i −4.75149 − 3.57600i
155.15 −0.484025 + 1.32880i 1.33288 + 1.10609i −1.53144 − 1.28635i 2.24261 − 1.29477i −2.11492 + 1.23577i 0.866025 + 0.500000i 2.45056 − 1.41236i 0.553149 + 2.94856i 0.635018 + 3.60669i
155.16 −0.281995 − 1.38581i −1.72102 − 0.195202i −1.84096 + 0.781584i −1.04266 + 0.601980i 0.214804 + 2.44005i −0.866025 − 0.500000i 1.60227 + 2.33082i 2.92379 + 0.671890i 1.12826 + 1.27518i
155.17 −0.0405943 − 1.41363i 0.586217 + 1.62983i −1.99670 + 0.114771i −2.16781 + 1.25159i 2.28018 − 0.894856i 0.866025 + 0.500000i 0.243298 + 2.81794i −2.31270 + 1.91087i 1.85728 + 3.01367i
155.18 0.0344832 + 1.41379i 0.0354296 + 1.73169i −1.99762 + 0.0975041i −0.948457 + 0.547592i −2.44703 + 0.109804i −0.866025 − 0.500000i −0.206735 − 2.82086i −2.99749 + 0.122706i −0.806888 − 1.32204i
155.19 0.0439092 + 1.41353i −1.25879 − 1.18972i −1.99614 + 0.124134i −0.245511 + 0.141746i 1.62644 − 1.83159i −0.866025 − 0.500000i −0.263117 − 2.81616i 0.169128 + 2.99523i −0.211142 − 0.340813i
155.20 0.0878075 − 1.41148i −1.48209 + 0.896336i −1.98458 − 0.247878i 2.57586 − 1.48718i 1.13503 + 2.17065i 0.866025 + 0.500000i −0.524137 + 2.77944i 1.39317 − 2.65689i −1.87295 − 3.76638i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h 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.ba.a ✓ 72
3.b odd 2 1 756.2.ba.a 72
4.b odd 2 1 inner 252.2.ba.a ✓ 72
9.c even 3 1 756.2.ba.a 72
9.d odd 6 1 inner 252.2.ba.a ✓ 72
12.b even 2 1 756.2.ba.a 72
36.f odd 6 1 756.2.ba.a 72
36.h even 6 1 inner 252.2.ba.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.ba.a ✓ 72 1.a even 1 1 trivial
252.2.ba.a ✓ 72 4.b odd 2 1 inner
252.2.ba.a ✓ 72 9.d odd 6 1 inner
252.2.ba.a ✓ 72 36.h even 6 1 inner
756.2.ba.a 72 3.b odd 2 1
756.2.ba.a 72 9.c even 3 1
756.2.ba.a 72 12.b even 2 1
756.2.ba.a 72 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).