# Properties

 Label 504.2.br.a Level 504 Weight 2 Character orbit 504.br Analytic conductor 4.024 Analytic rank 0 Dimension 144 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.br (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$72$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 6q^{6} - 12q^{12} + 34q^{18} - 42q^{20} - 30q^{24} - 72q^{25} + 24q^{27} - 36q^{30} - 30q^{32} - 16q^{33} + 12q^{34} - 12q^{36} + 12q^{40} + 24q^{41} - 20q^{42} - 24q^{46} - 24q^{48} + 72q^{49} - 78q^{50} + 18q^{52} + 10q^{54} + 8q^{57} - 18q^{58} - 72q^{59} + 88q^{60} - 12q^{64} + 2q^{66} + 78q^{68} + 42q^{72} + 84q^{74} - 112q^{75} + 12q^{76} + 112q^{78} - 8q^{81} - 36q^{82} - 14q^{84} + 30q^{86} + 24q^{88} + 104q^{90} - 114q^{92} - 42q^{94} - 80q^{96} - 64q^{99} + O(q^{100})$$

## 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 −1.41306 + 0.0570983i 0.376059 + 1.69073i 1.99348 0.161367i 1.12986 + 1.95698i −0.627932 2.36764i 0.866025 + 0.500000i −2.80769 + 0.341845i −2.71716 + 1.27163i −1.70831 2.70082i
155.2 −1.41302 + 0.0581930i −0.927403 1.46285i 1.99323 0.164455i 0.0780272 + 0.135147i 1.39556 + 2.01306i −0.866025 0.500000i −2.80689 + 0.348370i −1.27985 + 2.71330i −0.118118 0.186424i
155.3 −1.41262 + 0.0671848i 1.61186 0.633969i 1.99097 0.189813i −0.683073 1.18312i −2.23434 + 1.00385i 0.866025 + 0.500000i −2.79973 + 0.401896i 2.19617 2.04374i 1.04441 + 1.62540i
155.4 −1.40981 + 0.111460i −1.64186 + 0.551643i 1.97515 0.314275i −1.45065 2.51260i 2.25323 0.960715i −0.866025 0.500000i −2.74957 + 0.663220i 2.39138 1.81144i 2.32520 + 3.38061i
155.5 −1.38704 0.275894i −0.942212 1.45335i 1.84776 + 0.765353i −1.23804 2.14435i 0.905915 + 2.27581i 0.866025 + 0.500000i −2.35177 1.57136i −1.22447 + 2.73873i 1.12560 + 3.31587i
155.6 −1.37639 0.324902i −0.139646 + 1.72641i 1.78888 + 0.894380i 0.851872 + 1.47549i 0.753121 2.33084i −0.866025 0.500000i −2.17160 1.81222i −2.96100 0.482174i −0.693117 2.30761i
155.7 −1.35659 0.399563i 1.37962 + 1.04721i 1.68070 + 1.08409i −1.36514 2.36449i −1.45315 1.97189i −0.866025 0.500000i −1.84686 2.14222i 0.806685 + 2.88951i 0.907175 + 3.75311i
155.8 −1.33761 + 0.459123i 0.847856 1.51034i 1.57841 1.22826i 0.551827 + 0.955792i −0.440668 + 2.40953i −0.866025 0.500000i −1.54738 + 2.36762i −1.56228 2.56111i −1.17696 1.02512i
155.9 −1.29748 + 0.562635i 1.54346 + 0.785952i 1.36688 1.46001i 0.481160 + 0.833393i −2.44481 0.151347i −0.866025 0.500000i −0.952046 + 2.66338i 1.76456 + 2.42618i −1.09319 0.810590i
155.10 −1.29192 0.575281i 0.410623 1.68267i 1.33810 + 1.48643i 1.67611 + 2.90312i −1.49850 + 1.93765i 0.866025 + 0.500000i −0.873604 2.69013i −2.66278 1.38189i −0.495295 4.71482i
155.11 −1.27467 0.612543i −1.30518 + 1.13865i 1.24958 + 1.56158i −1.17058 2.02750i 2.36114 0.651926i 0.866025 + 0.500000i −0.636271 2.75593i 0.406968 2.97227i 0.250172 + 3.30142i
155.12 −1.26542 + 0.631437i −1.22677 + 1.22272i 1.20258 1.59807i −0.127264 0.220428i 0.780309 2.32188i 0.866025 + 0.500000i −0.512685 + 2.78157i 0.00992315 2.99998i 0.300229 + 0.198575i
155.13 −1.25232 + 0.657031i 0.906679 + 1.47578i 1.13662 1.64563i −2.13367 3.69562i −2.10509 1.25244i 0.866025 + 0.500000i −0.342188 + 2.80765i −1.35587 + 2.67612i 5.10018 + 3.22622i
155.14 −1.16781 0.797628i −1.30518 + 1.13865i 0.727580 + 1.86296i 1.17058 + 2.02750i 2.43242 0.288684i −0.866025 0.500000i 0.636271 2.75593i 0.406968 2.97227i 0.250172 3.30142i
155.15 −1.14417 0.831193i 0.410623 1.68267i 0.618236 + 1.90205i −1.67611 2.90312i −1.86845 + 1.58395i −0.866025 0.500000i 0.873604 2.69013i −2.66278 1.38189i −0.495295 + 4.71482i
155.16 −1.06211 + 0.933765i 1.61727 0.620022i 0.256166 1.98353i 1.88595 + 3.26656i −1.13877 + 2.16869i 0.866025 + 0.500000i 1.58007 + 2.34593i 2.23114 2.00549i −5.05329 1.70842i
155.17 −1.04337 + 0.954667i −1.44707 0.951840i 0.177222 1.99213i −0.0190253 0.0329529i 2.41851 0.388350i 0.866025 + 0.500000i 1.71692 + 2.24771i 1.18800 + 2.75475i 0.0513094 + 0.0162190i
155.18 −1.02433 0.975064i 1.37962 + 1.04721i 0.0985016 + 1.99757i 1.36514 + 2.36449i −0.392082 2.41791i 0.866025 + 0.500000i 1.84686 2.14222i 0.806685 + 2.88951i 0.907175 3.75311i
155.19 −0.969566 1.02953i −0.139646 + 1.72641i −0.119883 + 1.99640i −0.851872 1.47549i 1.91280 1.53010i 0.866025 + 0.500000i 2.17160 1.81222i −2.96100 0.482174i −0.693117 + 2.30761i
155.20 −0.949957 + 1.04765i −1.73068 0.0689108i −0.195162 1.99046i −0.590671 1.02307i 1.71627 1.74769i −0.866025 0.500000i 2.27071 + 1.68639i 2.99050 + 0.238525i 1.63294 + 0.353056i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 491.72 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.d odd 6 1 inner
72.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.br.a 144
8.d odd 2 1 inner 504.2.br.a 144
9.d odd 6 1 inner 504.2.br.a 144
72.l even 6 1 inner 504.2.br.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.br.a 144 1.a even 1 1 trivial
504.2.br.a 144 8.d odd 2 1 inner
504.2.br.a 144 9.d odd 6 1 inner
504.2.br.a 144 72.l even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database