# Properties

 Label 504.2.cy.a Level 504 Weight 2 Character orbit 504.cy 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.cy (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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 3q^{2} - 2q^{3} + q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 3q^{2} - 2q^{3} + q^{4} - 2q^{6} - 2q^{9} - 6q^{10} - 5q^{12} - 3q^{14} + q^{16} + 5q^{18} - 4q^{19} - 6q^{20} + 2q^{22} - 8q^{24} + 148q^{25} + 6q^{26} - 8q^{27} + 24q^{30} - 33q^{32} + 22q^{33} - 4q^{34} - 30q^{35} - 38q^{36} - 12q^{40} - 12q^{41} + 7q^{42} - 4q^{43} - 9q^{44} - 6q^{46} - 5q^{48} - 2q^{49} - 21q^{50} + 26q^{51} - 18q^{52} - 40q^{54} + 18q^{56} + 4q^{57} + 6q^{58} - 6q^{59} - 2q^{60} - 8q^{64} - 6q^{65} + 43q^{66} + 2q^{67} - 18q^{70} - 11q^{72} - 4q^{73} - 36q^{75} + 2q^{76} - 29q^{78} - 87q^{80} - 10q^{81} - 4q^{82} - 72q^{83} - 65q^{84} + 14q^{88} + 24q^{89} - 49q^{90} - 36q^{91} - 36q^{92} + 9q^{94} - 88q^{96} - 4q^{97} - 57q^{98} + 10q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
347.1 −1.41373 + 0.0369625i 1.72290 + 0.177799i 1.99727 0.104510i 1.53459 −2.44229 0.187678i 2.64302 + 0.120259i −2.81973 + 0.221573i 2.93677 + 0.612662i −2.16949 + 0.0567221i
347.2 −1.41326 + 0.0519573i 0.179393 1.72274i 1.99460 0.146858i −2.18143 −0.164020 + 2.44399i 2.31224 + 1.28590i −2.81126 + 0.311183i −2.93564 0.618093i 3.08292 0.113341i
347.3 −1.40993 + 0.110035i 1.32847 + 1.11138i 1.97578 0.310283i −3.42656 −1.99534 1.42079i 1.12525 2.39454i −2.75157 + 0.654882i 0.529666 + 2.95287i 4.83120 0.377042i
347.4 −1.40917 0.119296i −1.11238 + 1.32763i 1.97154 + 0.336217i −4.38485 1.72592 1.73816i −2.42141 + 1.06619i −2.73813 0.708985i −0.525222 2.95367i 6.17901 + 0.523095i
347.5 −1.40410 0.168823i 1.00207 + 1.41275i 1.94300 + 0.474089i 3.02374 −1.16851 2.15281i −2.60675 0.452601i −2.64813 0.993691i −0.991703 + 2.83135i −4.24563 0.510476i
347.6 −1.39790 + 0.214156i 1.53386 0.804539i 1.90827 0.598739i 2.18793 −1.97189 + 1.45315i −2.62554 0.326387i −2.53936 + 1.24565i 1.70544 2.46809i −3.05852 + 0.468558i
347.7 −1.37793 0.318287i −0.613399 1.61980i 1.79739 + 0.877155i 3.53729 0.329661 + 2.42720i 1.58882 2.11557i −2.19749 1.78074i −2.24748 + 1.98716i −4.87414 1.12587i
347.8 −1.36752 + 0.360384i −0.541502 + 1.64523i 1.74025 0.985669i 0.493530 0.147602 2.44504i 1.32629 2.28931i −2.02461 + 1.97508i −2.41355 1.78179i −0.674914 + 0.177860i
347.9 −1.36622 0.365298i −1.64155 + 0.552549i 1.73312 + 0.998154i 0.188732 2.44456 0.155249i 2.31931 + 1.27310i −2.00319 1.99680i 2.38938 1.81407i −0.257849 0.0689432i
347.10 −1.36156 + 0.382313i −1.59594 0.673028i 1.70767 1.04108i −0.632847 2.43027 + 0.306217i −1.94512 + 1.79346i −1.92708 + 2.07036i 2.09407 + 2.14823i 0.861657 0.241945i
347.11 −1.34051 + 0.450600i −0.774779 1.54910i 1.59392 1.20807i −0.714950 1.73662 + 1.72747i −1.52030 2.16534i −1.59230 + 2.33764i −1.79943 + 2.40042i 0.958395 0.322157i
347.12 −1.29887 + 0.559401i −0.869153 + 1.49819i 1.37414 1.45318i 2.96535 0.290830 2.43216i −0.557344 + 2.58638i −0.971921 + 2.65619i −1.48915 2.60431i −3.85161 + 1.65882i
347.13 −1.29676 0.564292i 1.23687 1.21249i 1.36315 + 1.46350i −2.35082 −2.28812 + 0.874349i −1.09033 2.41064i −0.941831 2.66701i 0.0597186 2.99941i 3.04843 + 1.32655i
347.14 −1.28812 0.583745i 0.0908862 + 1.72966i 1.31848 + 1.50386i −0.177604 0.892611 2.28106i 1.51672 + 2.16785i −0.820487 2.70681i −2.98348 + 0.314405i 0.228775 + 0.103676i
347.15 −1.27658 0.608568i −1.44822 0.950089i 1.25929 + 1.55377i 2.40991 1.27056 + 2.09420i −1.71811 + 2.01199i −0.662008 2.74986i 1.19466 + 2.75187i −3.07644 1.46660i
347.16 −1.22414 + 0.708150i 1.63621 0.568159i 0.997046 1.73375i −2.54780 −1.60062 + 1.85419i −0.190034 + 2.63892i 0.00723210 + 2.82842i 2.35439 1.85926i 3.11887 1.80422i
347.17 −1.16532 0.801263i −1.44822 0.950089i 0.715955 + 1.86746i −2.40991 0.926369 + 2.26756i 1.71811 2.01199i 0.662008 2.74986i 1.19466 + 2.75187i 2.80833 + 1.93097i
347.18 −1.15683 + 0.813481i −1.71844 + 0.216724i 0.676496 1.88211i −2.66759 1.81163 1.64863i 2.37290 1.17020i 0.748476 + 2.72760i 2.90606 0.744855i 3.08594 2.17004i
347.19 −1.14960 0.823668i 0.0908862 + 1.72966i 0.643141 + 1.89377i 0.177604 1.32019 2.06328i −1.51672 2.16785i 0.820487 2.70681i −2.98348 + 0.314405i −0.204173 0.146287i
347.20 −1.13707 0.840877i 1.23687 1.21249i 0.585851 + 1.91227i 2.35082 −2.42597 + 0.338629i 1.09033 + 2.41064i 0.941831 2.66701i 0.0597186 2.99941i −2.67304 1.97675i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 443.92 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.d odd 2 1 inner
63.n odd 6 1 inner
504.cy 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.cy.a yes 184
7.c even 3 1 504.2.bt.a 184
8.d odd 2 1 inner 504.2.cy.a yes 184
9.d odd 6 1 504.2.bt.a 184
56.k odd 6 1 504.2.bt.a 184
63.n odd 6 1 inner 504.2.cy.a yes 184
72.l even 6 1 504.2.bt.a 184
504.cy even 6 1 inner 504.2.cy.a yes 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bt.a 184 7.c even 3 1
504.2.bt.a 184 9.d odd 6 1
504.2.bt.a 184 56.k odd 6 1
504.2.bt.a 184 72.l even 6 1
504.2.cy.a yes 184 1.a even 1 1 trivial
504.2.cy.a yes 184 8.d odd 2 1 inner
504.2.cy.a yes 184 63.n odd 6 1 inner
504.2.cy.a yes 184 504.cy 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