# Properties

 Label 504.2.cs.a Level 504 Weight 2 Character orbit 504.cs Analytic conductor 4.024 Analytic rank 0 Dimension 72 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.cs (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 8q^{6} + 36q^{7} + 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 8q^{6} + 36q^{7} + 6q^{8} + 20q^{12} + 40q^{17} - q^{18} + 2q^{20} - 12q^{22} + 12q^{23} + 10q^{24} + 36q^{25} - 42q^{26} + 4q^{30} + 5q^{32} + 8q^{33} + 6q^{34} - 18q^{36} - 25q^{38} - 4q^{39} - 9q^{40} + 24q^{41} - 7q^{42} + 12q^{46} - 53q^{48} - 36q^{49} + 41q^{50} - 9q^{52} + 12q^{54} + 3q^{56} + 4q^{57} + 9q^{58} - 50q^{60} + 60q^{62} - 6q^{64} - 40q^{65} - 2q^{66} + 23q^{68} - 56q^{71} + 19q^{72} + 22q^{74} - 24q^{76} - 54q^{78} - 6q^{80} - 4q^{81} - 48q^{82} + 10q^{84} - 39q^{86} - 76q^{87} - 12q^{88} - 88q^{89} - 77q^{90} + 11q^{92} + 24q^{94} - 24q^{95} + 59q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1 −1.41413 + 0.0151305i −1.72706 + 0.131454i 1.99954 0.0427932i −1.26304 0.729219i 2.44030 0.212025i 0.500000 + 0.866025i −2.82697 + 0.0907694i 2.96544 0.454056i 1.79715 + 1.01210i
85.2 −1.39778 0.214989i 0.0497101 + 1.73134i 1.90756 + 0.601014i 0.261066 + 0.150726i 0.302736 2.43071i 0.500000 + 0.866025i −2.53713 1.25019i −2.99506 + 0.172130i −0.332507 0.266808i
85.3 −1.38589 0.281624i 1.31821 1.12353i 1.84138 + 0.780600i 2.74638 + 1.58562i −2.14331 + 1.18585i 0.500000 + 0.866025i −2.33211 1.60040i 0.475349 2.96210i −3.35962 2.97094i
85.4 −1.31153 + 0.529035i 0.349968 1.69633i 1.44024 1.38770i −2.84022 1.63980i 0.438422 + 2.40993i 0.500000 + 0.866025i −1.15479 + 2.58195i −2.75505 1.18732i 4.59255 + 0.648078i
85.5 −1.27427 0.613390i 1.34597 + 1.09012i 1.24750 + 1.56324i −2.33508 1.34816i −1.04645 2.21471i 0.500000 + 0.866025i −0.630773 2.75720i 0.623266 + 2.93454i 2.14857 + 3.15023i
85.6 −1.25626 + 0.649466i 1.65525 0.510053i 1.15639 1.63180i 1.14688 + 0.662153i −1.74816 + 1.71579i 0.500000 + 0.866025i −0.392930 + 2.80100i 2.47969 1.68853i −1.87083 0.0869767i
85.7 −1.20390 + 0.742040i −1.50659 + 0.854512i 0.898754 1.78668i 3.20115 + 1.84819i 1.17970 2.14670i 0.500000 + 0.866025i 0.243781 + 2.81790i 1.53962 2.57480i −5.22530 + 0.150350i
85.8 −1.14128 0.835149i 1.40449 1.01361i 0.605053 + 1.90628i −1.92732 1.11274i −2.44944 0.0161356i 0.500000 + 0.866025i 0.901492 2.68092i 0.945172 2.84722i 1.27031 + 2.87955i
85.9 −0.972917 1.02637i −0.120790 1.72783i −0.106865 + 1.99714i 2.23744 + 1.29179i −1.65588 + 1.80501i 0.500000 + 0.866025i 2.15378 1.83337i −2.97082 + 0.417410i −0.850996 3.55324i
85.10 −0.930716 + 1.06479i −1.06868 1.36306i −0.267536 1.98203i 0.220108 + 0.127079i 2.44600 + 0.130709i 0.500000 + 0.866025i 2.35943 + 1.55983i −0.715859 + 2.91334i −0.340170 + 0.116093i
85.11 −0.870602 + 1.11447i −0.119319 + 1.72794i −0.484104 1.94053i −0.525102 0.303168i −1.82186 1.63732i 0.500000 + 0.866025i 2.58413 + 1.14991i −2.97153 0.412353i 0.795028 0.321274i
85.12 −0.828761 1.14593i 1.26004 + 1.18840i −0.626309 + 1.89940i 3.33304 + 1.92433i 0.317546 2.42882i 0.500000 + 0.866025i 2.69564 0.856447i 0.175418 + 2.99487i −0.557147 5.41424i
85.13 −0.589905 + 1.28531i 1.55592 + 0.760985i −1.30402 1.51642i −3.48956 2.01470i −1.89595 + 1.55093i 0.500000 + 0.866025i 2.71831 0.781526i 1.84180 + 2.36807i 4.64802 3.29667i
85.14 −0.578023 1.29069i −1.26004 1.18840i −1.33178 + 1.49210i −3.33304 1.92433i −0.805523 + 2.31325i 0.500000 + 0.866025i 2.69564 + 0.856447i 0.175418 + 2.99487i −0.557147 + 5.41424i
85.15 −0.402403 1.35576i 0.120790 + 1.72783i −1.67614 + 1.09112i −2.23744 1.29179i 2.29391 0.859046i 0.500000 + 0.866025i 2.15378 + 1.83337i −2.97082 + 0.417410i −0.850996 + 3.55324i
85.16 −0.305084 + 1.38091i 1.64418 + 0.544667i −1.81385 0.842588i 2.23227 + 1.28880i −1.25375 + 2.10431i 0.500000 + 0.866025i 1.71692 2.24771i 2.40668 + 1.79106i −2.46075 + 2.68938i
85.17 −0.248810 + 1.39215i −1.73065 0.0695272i −1.87619 0.692764i 1.22717 + 0.708510i 0.527397 2.39204i 0.500000 + 0.866025i 1.43125 2.43958i 2.99033 + 0.240655i −1.29169 + 1.53213i
85.18 −0.152619 1.40595i −1.40449 + 1.01361i −1.95342 + 0.429150i 1.92732 + 1.11274i 1.63945 + 1.81995i 0.500000 + 0.866025i 0.901492 + 2.68092i 0.945172 2.84722i 1.27031 2.87955i
85.19 0.105921 1.41024i −1.34597 1.09012i −1.97756 0.298749i 2.33508 + 1.34816i −1.67990 + 1.78267i 0.500000 + 0.866025i −0.630773 + 2.75720i 0.623266 + 2.93454i 2.14857 3.15023i
85.20 0.109593 + 1.40996i 0.416131 1.68132i −1.97598 + 0.309044i −0.226907 0.131005i 2.41620 + 0.402467i 0.500000 + 0.866025i −0.652293 2.75218i −2.65367 1.39930i 0.159844 0.334288i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n 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.cs.a 72
8.b even 2 1 inner 504.2.cs.a 72
9.c even 3 1 inner 504.2.cs.a 72
72.n even 6 1 inner 504.2.cs.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cs.a 72 1.a even 1 1 trivial
504.2.cs.a 72 8.b even 2 1 inner
504.2.cs.a 72 9.c even 3 1 inner
504.2.cs.a 72 72.n even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{72} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database