# Properties

 Label 504.2.y.a Level 504 Weight 2 Character orbit 504.y Analytic conductor 4.024 Analytic rank 0 Dimension 184 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.y (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$92$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 3q^{2} + q^{4} + 6q^{6} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 3q^{2} + q^{4} + 6q^{6} - 2q^{7} - 2q^{9} - 6q^{10} - 3q^{12} - 3q^{14} - 2q^{15} + q^{16} - 15q^{18} - 6q^{22} - 12q^{24} - 156q^{25} + 6q^{26} - 8q^{28} - 14q^{30} - 6q^{31} - 33q^{32} - 6q^{33} - 6q^{34} + 22q^{36} - 66q^{38} + 10q^{39} - 15q^{42} + 9q^{44} + 2q^{46} - 6q^{47} - 9q^{48} - 2q^{49} + 9q^{50} + 24q^{54} + 60q^{56} + 4q^{57} + 6q^{58} + 34q^{60} - 12q^{62} - 30q^{63} - 8q^{64} - 6q^{65} - 21q^{66} - 36q^{68} + 30q^{70} + 9q^{72} - 12q^{73} - 12q^{76} + 19q^{78} + 2q^{79} + 57q^{80} + 6q^{81} + 9q^{84} + 12q^{87} - 18q^{88} + 24q^{89} + 75q^{90} - 36q^{92} - 3q^{94} + 54q^{95} - 54q^{96} + 45q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
173.1 −1.41306 + 0.0570616i 0.0165996 + 1.73197i 1.99349 0.161263i 0.186906i −0.122285 2.44644i −0.0220103 + 2.64566i −2.80772 + 0.341626i −2.99945 + 0.0574999i −0.0106652 0.264110i
173.2 −1.41306 0.0571030i 0.216493 1.71847i 1.99348 + 0.161380i 1.61593i −0.404047 + 2.41594i 2.40829 1.09550i −2.80769 0.341873i −2.90626 0.744072i 0.0922743 2.28340i
173.3 −1.40941 0.116469i 1.69912 + 0.336136i 1.97287 + 0.328305i 3.14427i −2.35561 0.671648i 0.623342 + 2.57127i −2.74234 0.692495i 2.77403 + 1.14227i −0.366210 + 4.43156i
173.4 −1.40767 0.135848i −1.56804 + 0.735705i 1.96309 + 0.382459i 3.12398i 2.30723 0.822619i 2.22515 + 1.43133i −2.71143 0.805059i 1.91747 2.30723i 0.424387 4.39755i
173.5 −1.40645 0.147961i 1.68230 0.412165i 1.95621 + 0.416201i 0.157459i −2.42705 + 0.330775i −2.21308 1.44992i −2.68974 0.874811i 2.66024 1.38677i 0.0232979 0.221459i
173.6 −1.39680 0.221244i −1.40670 + 1.01054i 1.90210 + 0.618068i 4.36497i 2.18846 1.10030i 1.72321 2.00762i −2.52011 1.28415i 0.957621 2.84306i −0.965725 + 6.09699i
173.7 −1.39532 + 0.230384i −0.267989 + 1.71119i 1.89385 0.642919i 0.170174i −0.0203002 2.44941i −1.51202 2.17113i −2.49441 + 1.33339i −2.85636 0.917161i 0.0392054 + 0.237448i
173.8 −1.38113 + 0.304086i 1.31500 + 1.12729i 1.81506 0.839967i 3.74037i −2.15898 1.15707i 0.706907 2.54957i −2.25142 + 1.71204i 0.458435 + 2.96477i −1.13739 5.16595i
173.9 −1.34205 + 0.445987i −1.28072 1.16609i 1.60219 1.19707i 0.636978i 2.23885 + 0.993762i 2.14318 + 1.55138i −1.61634 + 2.32109i 0.280479 + 2.98686i 0.284084 + 0.854856i
173.10 −1.33822 0.457359i 0.789770 1.54151i 1.58165 + 1.22409i 2.01312i −1.76191 + 1.70167i −1.48443 + 2.19008i −1.55673 2.36148i −1.75253 2.43488i 0.920719 2.69399i
173.11 −1.33533 0.465720i −1.66375 0.481605i 1.56621 + 1.24378i 1.79202i 1.99736 + 1.41794i −1.87653 + 1.86511i −1.51215 2.39027i 2.53611 + 1.60254i −0.834578 + 2.39293i
173.12 −1.31489 + 0.520628i 1.30232 1.14191i 1.45789 1.36914i 1.77147i −1.11790 + 2.17952i 1.98439 1.74991i −1.20416 + 2.55930i 0.392079 2.97427i 0.922278 + 2.32930i
173.13 −1.30575 + 0.543166i −0.944647 1.45177i 1.40994 1.41847i 3.97106i 2.02202 + 1.38254i −2.57827 0.593742i −1.07056 + 2.61800i −1.21528 + 2.74282i −2.15695 5.18519i
173.14 −1.26838 + 0.625475i −1.71321 + 0.254787i 1.21756 1.58668i 1.07899i 2.01363 1.39473i −2.47543 + 0.933935i −0.551904 + 2.77406i 2.87017 0.873007i 0.674878 + 1.36856i
173.15 −1.26678 0.628698i 1.22962 + 1.21985i 1.20948 + 1.59285i 1.38607i −0.790746 2.31834i 1.97507 1.76042i −0.530723 2.77819i 0.0239358 + 2.99990i −0.871418 + 1.75585i
173.16 −1.23015 0.697666i −1.59170 0.683004i 1.02652 + 1.71646i 2.06988i 1.48151 + 1.95067i 0.440031 2.60890i −0.0652554 2.82767i 2.06701 + 2.17427i 1.44408 2.54625i
173.17 −1.17587 + 0.785710i 1.71321 0.254787i 0.765320 1.84778i 1.07899i −1.81431 + 1.64568i −2.47543 + 0.933935i 0.551904 + 2.77406i 2.87017 0.873007i −0.847769 1.26874i
173.18 −1.14306 0.832711i 1.10317 + 1.33530i 0.613184 + 1.90368i 4.07380i −0.149078 2.44495i −1.75571 + 1.97926i 0.884310 2.68663i −0.566029 + 2.94612i 3.39230 4.65661i
173.19 −1.14122 0.835238i −0.987509 + 1.42296i 0.604756 + 1.90638i 0.709981i 2.31548 0.799106i −2.47476 0.935707i 0.902119 2.68071i −1.04965 2.81038i 0.593003 0.810243i
173.20 −1.12327 + 0.859225i 0.944647 + 1.45177i 0.523464 1.93028i 3.97106i −2.30849 0.819065i −2.57827 0.593742i 1.07056 + 2.61800i −1.21528 + 2.74282i 3.41204 + 4.46057i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 437.92 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.b even 2 1 inner
63.s even 6 1 inner
504.y 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.y.a 184
7.d odd 6 1 504.2.ca.a yes 184
8.b even 2 1 inner 504.2.y.a 184
9.d odd 6 1 504.2.ca.a yes 184
56.j odd 6 1 504.2.ca.a yes 184
63.s even 6 1 inner 504.2.y.a 184
72.j odd 6 1 504.2.ca.a yes 184
504.y even 6 1 inner 504.2.y.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.y.a 184 1.a even 1 1 trivial
504.2.y.a 184 8.b even 2 1 inner
504.2.y.a 184 63.s even 6 1 inner
504.2.y.a 184 504.y even 6 1 inner
504.2.ca.a yes 184 7.d odd 6 1
504.2.ca.a yes 184 9.d odd 6 1
504.2.ca.a yes 184 56.j odd 6 1
504.2.ca.a yes 184 72.j odd 6 1

## 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