# 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

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None 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.

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

## 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.

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 239.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database