# Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 2q^{4} - 2q^{6} - 2q^{7} - 8q^{8} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 2q^{4} - 2q^{6} - 2q^{7} - 8q^{8} - 2q^{9} + 2q^{10} + 16q^{12} - 4q^{14} - 2q^{15} - 2q^{16} - 4q^{17} - 14q^{18} + 6q^{20} + 2q^{22} + 2q^{23} + 12q^{24} + 78q^{25} - 4q^{26} - 8q^{28} - 9q^{30} - 4q^{31} - 2q^{32} - 14q^{33} - 18q^{36} - 5q^{38} + 4q^{39} - 4q^{40} - 4q^{41} - 12q^{42} + 17q^{44} - 6q^{46} - 84q^{47} - 3q^{48} - 2q^{49} - 31q^{50} + 9q^{52} - q^{54} + 4q^{55} - 16q^{56} - 20q^{57} + 5q^{58} + 17q^{60} + 32q^{62} + 8q^{63} - 8q^{64} - 44q^{65} - 56q^{66} - 12q^{68} + 5q^{70} - 16q^{71} + 36q^{72} - 4q^{73} + 19q^{74} - 6q^{76} - 47q^{78} - 4q^{79} - 11q^{80} - 18q^{81} - 65q^{84} - 23q^{86} - 14q^{87} - 7q^{88} + 4q^{89} - 35q^{90} - 48q^{92} - 18q^{94} - 44q^{95} + 95q^{96} - 4q^{97} - 83q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1 −1.41195 0.0799747i 1.37829 + 1.04896i 1.98721 + 0.225841i 3.10331 + 1.79170i −1.86219 1.59130i −1.70224 2.02544i −2.78778 0.477802i 0.799378 + 2.89154i −4.23843 2.77797i
277.2 −1.41195 + 0.0799747i −1.37829 1.04896i 1.98721 0.225841i −3.10331 1.79170i 2.02997 + 1.37085i −1.70224 2.02544i −2.78778 + 0.477802i 0.799378 + 2.89154i 4.52501 + 2.28160i
277.3 −1.40384 0.171013i −1.37821 + 1.04907i 1.94151 + 0.480149i 0.213745 + 0.123406i 2.11418 1.23703i −2.18219 + 1.49601i −2.64345 1.00607i 0.798909 2.89167i −0.278959 0.209795i
277.4 −1.40384 + 0.171013i 1.37821 1.04907i 1.94151 0.480149i −0.213745 0.123406i −1.75537 + 1.70841i −2.18219 + 1.49601i −2.64345 + 1.00607i 0.798909 2.89167i 0.321167 + 0.136688i
277.5 −1.38793 0.271374i 1.73202 + 0.00998187i 1.85271 + 0.753297i −1.80782 1.04374i −2.40122 0.483879i 2.62626 0.320557i −2.36702 1.54830i 2.99980 + 0.0345777i 2.22588 + 1.93924i
277.6 −1.38793 + 0.271374i −1.73202 0.00998187i 1.85271 0.753297i 1.80782 + 1.04374i 2.40664 0.456171i 2.62626 0.320557i −2.36702 + 1.54830i 2.99980 + 0.0345777i −2.79237 0.958051i
277.7 −1.38729 0.274643i −0.170331 1.72366i 1.84914 + 0.762020i −2.08943 1.20633i −0.237093 + 2.43799i 0.664788 + 2.56087i −2.35601 1.56500i −2.94197 + 0.587183i 2.56733 + 2.24738i
277.8 −1.38729 + 0.274643i 0.170331 + 1.72366i 1.84914 0.762020i 2.08943 + 1.20633i −0.709689 2.34443i 0.664788 + 2.56087i −2.35601 + 1.56500i −2.94197 + 0.587183i −3.22995 1.09968i
277.9 −1.37280 0.339745i −0.0247783 + 1.73187i 1.76915 + 0.932801i −0.736012 0.424937i 0.622410 2.36909i 1.20653 2.35463i −2.11177 1.88161i −2.99877 0.0858258i 0.866026 + 0.833408i
277.10 −1.37280 + 0.339745i 0.0247783 1.73187i 1.76915 0.932801i 0.736012 + 0.424937i 0.554379 + 2.38593i 1.20653 2.35463i −2.11177 + 1.88161i −2.99877 0.0858258i −1.15477 0.333296i
277.11 −1.28938 0.580936i 1.21975 1.22972i 1.32503 + 1.49810i 3.55178 + 2.05062i −2.28712 + 0.876982i 2.40176 + 1.10974i −0.838168 2.70138i −0.0244139 2.99990i −3.38833 4.70739i
277.12 −1.28938 + 0.580936i −1.21975 + 1.22972i 1.32503 1.49810i −3.55178 2.05062i 0.858341 2.29418i 2.40176 + 1.10974i −0.838168 + 2.70138i −0.0244139 2.99990i 5.77089 + 0.580682i
277.13 −1.27676 0.608187i −1.55038 0.772217i 1.26022 + 1.55302i 2.14506 + 1.23845i 1.50981 + 1.92886i −2.51672 0.816158i −0.664465 2.74927i 1.80736 + 2.39446i −1.98551 2.88580i
277.14 −1.27676 + 0.608187i 1.55038 + 0.772217i 1.26022 1.55302i −2.14506 1.23845i −2.44911 0.0430112i −2.51672 0.816158i −0.664465 + 2.74927i 1.80736 + 2.39446i 3.49193 + 0.276602i
277.15 −1.23730 0.684906i 0.549172 + 1.64268i 1.06181 + 1.69486i −3.35113 1.93477i 0.445596 2.40862i −2.41512 + 1.08036i −0.152946 2.82429i −2.39682 + 1.80423i 2.82120 + 4.68910i
277.16 −1.23730 + 0.684906i −0.549172 1.64268i 1.06181 1.69486i 3.35113 + 1.93477i 1.80457 + 1.65636i −2.41512 + 1.08036i −0.152946 + 2.82429i −2.39682 + 1.80423i −5.47148 0.0986814i
277.17 −1.16914 0.795682i 0.775861 1.54856i 0.733781 + 1.86053i −0.241073 0.139184i −2.13925 + 1.19315i −2.22477 1.43192i 0.622496 2.75908i −1.79608 2.40294i 0.171103 + 0.354543i
277.18 −1.16914 + 0.795682i −0.775861 + 1.54856i 0.733781 1.86053i 0.241073 + 0.139184i −0.325070 2.42782i −2.22477 1.43192i 0.622496 + 2.75908i −1.79608 2.40294i −0.392595 + 0.0290923i
277.19 −1.15465 0.816568i −1.68787 + 0.388703i 0.666434 + 1.88570i −1.19505 0.689965i 2.26630 + 0.929446i 0.677727 + 2.55748i 0.770303 2.72151i 2.69782 1.31216i 0.816467 + 1.77251i
277.20 −1.15465 + 0.816568i 1.68787 0.388703i 0.666434 1.88570i 1.19505 + 0.689965i −1.63150 + 1.82708i 0.677727 + 2.55748i 0.770303 + 2.72151i 2.69782 1.31216i −1.94327 + 0.179175i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.92 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.h even 3 1 inner
504.cq 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.cq.a yes 184
7.c even 3 1 504.2.w.a 184
8.b even 2 1 inner 504.2.cq.a yes 184
9.c even 3 1 504.2.w.a 184
56.p even 6 1 504.2.w.a 184
63.h even 3 1 inner 504.2.cq.a yes 184
72.n even 6 1 504.2.w.a 184
504.cq even 6 1 inner 504.2.cq.a yes 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.w.a 184 7.c even 3 1
504.2.w.a 184 9.c even 3 1
504.2.w.a 184 56.p even 6 1
504.2.w.a 184 72.n even 6 1
504.2.cq.a yes 184 1.a even 1 1 trivial
504.2.cq.a yes 184 8.b even 2 1 inner
504.2.cq.a yes 184 63.h even 3 1 inner
504.2.cq.a yes 184 504.cq 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