# Properties

 Label 504.2.cs.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 2q^{6} - 36q^{7} + 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 2q^{6} - 36q^{7} + 6q^{8} - 8q^{12} - 40q^{17} - 21q^{18} + 12q^{20} + 12q^{22} + 12q^{23} - 12q^{24} + 36q^{25} - 14q^{26} - 60q^{30} - 15q^{32} + 8q^{33} + 6q^{34} + 18q^{36} - 3q^{38} - 20q^{39} + 21q^{40} - 32q^{41} - 13q^{42} - 64q^{44} + 12q^{46} + 29q^{48} - 36q^{49} + 5q^{50} - 9q^{52} + 30q^{54} - 3q^{56} + 4q^{57} + 9q^{58} + 34q^{60} - 12q^{62} - 54q^{64} + 40q^{65} + 120q^{66} + 55q^{68} - 56q^{71} + 15q^{72} - 22q^{74} + 12q^{76} + 62q^{78} + 94q^{80} - 4q^{81} + 12q^{82} + 4q^{84} - 3q^{86} - 28q^{87} - 12q^{88} + 88q^{89} - 83q^{90} + 55q^{92} - 18q^{94} - 40q^{95} - 83q^{96} + O(q^{100})$$

## 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 −1.40280 + 0.179343i −0.897491 + 1.48139i 1.93567 0.503164i 0.991981 + 0.572721i 0.993320 2.23904i −0.500000 0.866025i −2.62511 + 1.05299i −1.38902 2.65906i −1.49426 0.625505i
85.2 −1.40215 0.184326i 1.61889 + 0.615780i 1.93205 + 0.516905i 2.02215 + 1.16749i −2.15643 1.16182i −0.500000 0.866025i −2.61374 1.08091i 2.24163 + 1.99376i −2.62016 2.00973i
85.3 −1.35959 0.389240i 0.885523 1.48857i 1.69698 + 1.05841i −0.313644 0.181082i −1.78336 + 1.67917i −0.500000 0.866025i −1.89523 2.09955i −1.43170 2.63633i 0.355944 + 0.368281i
85.4 −1.35007 + 0.421096i 0.234769 + 1.71607i 1.64536 1.13701i −3.20237 1.84889i −1.03958 2.21794i −0.500000 0.866025i −1.74255 + 2.22790i −2.88977 + 0.805757i 5.10197 + 1.14762i
85.5 −1.30114 0.554117i −1.71503 0.242257i 1.38591 + 1.44196i 3.59330 + 2.07459i 2.09724 + 1.26553i −0.500000 0.866025i −1.00424 2.64415i 2.88262 + 0.830955i −3.52580 4.69043i
85.6 −1.28916 + 0.581429i −1.31449 1.12788i 1.32388 1.49911i 1.50625 + 0.869635i 2.35037 + 0.689740i −0.500000 0.866025i −0.835071 + 2.70234i 0.455768 + 2.96518i −2.44743 0.245323i
85.7 −1.14498 0.830079i −1.41639 + 0.996919i 0.621938 + 1.90084i −3.39378 1.95940i 2.44925 + 0.0342651i −0.500000 0.866025i 0.865744 2.69267i 1.01230 2.82405i 2.25934 + 5.06056i
85.8 −1.14188 + 0.834337i −1.71270 0.258185i 0.607763 1.90542i −1.96075 1.13204i 2.17110 1.13415i −0.500000 0.866025i 0.895772 + 2.68283i 2.86668 + 0.884386i 3.18343 0.343276i
85.9 −0.972690 + 1.02658i 0.821098 + 1.52506i −0.107748 1.99710i 2.72052 + 1.57069i −2.36427 0.640483i −0.500000 0.866025i 2.15499 + 1.83194i −1.65160 + 2.50444i −4.25867 + 1.26504i
85.10 −0.908808 + 1.08354i 1.55167 0.769621i −0.348135 1.96947i −1.05668 0.610074i −0.576254 + 2.38074i −0.500000 0.866025i 2.45039 + 1.41265i 1.81537 2.38840i 1.62136 0.590518i
85.11 −0.889632 1.09934i 1.72874 + 0.107029i −0.417109 + 1.95602i −0.436219 0.251851i −1.42028 1.99570i −0.500000 0.866025i 2.52141 1.28159i 2.97709 + 0.370051i 0.111204 + 0.703609i
85.12 −0.808719 1.16016i −0.339544 1.69844i −0.691947 + 1.87649i 0.642635 + 0.371025i −1.69587 + 1.76749i −0.500000 0.866025i 2.73662 0.714781i −2.76942 + 1.15339i −0.0892617 1.04561i
85.13 −0.600369 1.28045i 0.339544 + 1.69844i −1.27911 + 1.53749i −0.642635 0.371025i 1.97092 1.45446i −0.500000 0.866025i 2.73662 + 0.714781i −2.76942 + 1.15339i −0.0892617 + 1.04561i
85.14 −0.507243 1.32012i −1.72874 0.107029i −1.48541 + 1.33924i 0.436219 + 0.251851i 0.735600 + 2.33643i −0.500000 0.866025i 2.52141 + 1.28159i 2.97709 + 0.370051i 0.111204 0.703609i
85.15 −0.462760 + 1.33636i −1.16981 + 1.27732i −1.57171 1.23683i −0.563250 0.325193i −1.16562 2.15438i −0.500000 0.866025i 2.38016 1.52801i −0.263093 2.98844i 0.695224 0.602218i
85.16 −0.451845 + 1.34009i −0.169829 1.72370i −1.59167 1.21102i −2.48308 1.43361i 2.38665 + 0.551261i −0.500000 0.866025i 2.34207 1.58579i −2.94232 + 0.585469i 3.04313 2.67978i
85.17 −0.146382 1.40662i 1.41639 0.996919i −1.95714 + 0.411806i 3.39378 + 1.95940i −1.60962 1.84638i −0.500000 0.866025i 0.865744 + 2.69267i 1.01230 2.82405i 2.25934 5.06056i
85.18 0.0728331 + 1.41234i 1.64907 0.529693i −1.98939 + 0.205730i 1.76518 + 1.01913i 0.868211 + 2.29046i −0.500000 0.866025i −0.435453 2.79471i 2.43885 1.74700i −1.31079 + 2.56725i
85.19 0.106383 + 1.41021i −1.33989 1.09759i −1.97737 + 0.300044i 0.882573 + 0.509554i 1.40529 2.00628i −0.500000 0.866025i −0.633481 2.75657i 0.590596 + 2.94129i −0.624686 + 1.29882i
85.20 0.158116 + 1.40535i 0.757390 + 1.55768i −1.95000 + 0.444416i −2.00933 1.16009i −2.06932 + 1.31069i −0.500000 0.866025i −0.932885 2.67015i −1.85272 + 2.35954i 1.31262 3.00724i
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.b 72
8.b even 2 1 inner 504.2.cs.b 72
9.c even 3 1 inner 504.2.cs.b 72
72.n even 6 1 inner 504.2.cs.b 72

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