# Properties

 Label 504.2.cz.b Level 504 Weight 2 Character orbit 504.cz Analytic conductor 4.024 Analytic rank 0 Dimension 180 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.cz (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$180$$ Relative dimension: $$90$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$180q + 3q^{2} + q^{4} + 6q^{6} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$180q + 3q^{2} + q^{4} + 6q^{6} - 8q^{9} + 16q^{11} - 3q^{12} + 7q^{14} - 7q^{16} - 18q^{17} - 13q^{18} - 6q^{19} - 36q^{20} - 16q^{22} - 24q^{24} + 156q^{25} - 6q^{26} + 16q^{28} - 8q^{30} + 13q^{32} - 36q^{33} + 12q^{34} - 12q^{35} + 2q^{36} + 42q^{41} + 31q^{42} + 14q^{43} - 21q^{44} - 12q^{46} + 9q^{48} + 20q^{49} + 15q^{50} - 42q^{51} - 12q^{54} - 40q^{56} - 26q^{57} - 38q^{58} + 18q^{59} - 38q^{60} - 8q^{64} - 12q^{65} - 21q^{66} - 14q^{67} - 42q^{70} + 5q^{72} + 18q^{73} - 98q^{74} - 48q^{75} + 12q^{76} - 33q^{78} - 63q^{80} + 8q^{81} - 54q^{82} - 6q^{83} - 77q^{84} + 26q^{86} - 58q^{88} - 66q^{89} + 51q^{90} + 2q^{91} - 60q^{92} + 9q^{94} - 30q^{96} + 6q^{97} + 31q^{98} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
187.1 −1.41404 0.0223915i −0.178596 1.72282i 1.99900 + 0.0633248i −2.54012 0.213965 + 2.44013i −2.62252 + 0.349814i −2.82524 0.134304i −2.93621 + 0.615378i 3.59182 + 0.0568771i
187.2 −1.40798 0.132660i 1.55918 0.754290i 1.96480 + 0.373564i 0.987364 −2.29536 + 0.855183i 1.42351 + 2.23016i −2.71684 0.786619i 1.86209 2.35215i −1.39019 0.130983i
187.3 −1.40451 0.165416i 0.410054 1.68281i 1.94528 + 0.464656i 3.30804 −0.854288 + 2.29569i −0.351034 2.62236i −2.65529 0.974392i −2.66371 1.38009i −4.64616 0.547203i
187.4 −1.39379 + 0.239493i 0.682265 + 1.59202i 1.88529 0.667604i −3.62353 −1.33221 2.05553i 0.820118 + 2.51543i −2.46780 + 1.38201i −2.06903 + 2.17235i 5.05043 0.867809i
187.5 −1.38863 0.267785i −1.73191 0.0223461i 1.85658 + 0.743707i −3.43019 2.39899 + 0.494808i −0.366949 2.62018i −2.37895 1.52990i 2.99900 + 0.0774026i 4.76327 + 0.918553i
187.6 −1.37793 + 0.318310i −1.42741 0.981074i 1.79736 0.877216i 2.04097 2.27915 + 0.897487i 2.55041 0.703859i −2.19740 + 1.78086i 1.07499 + 2.80079i −2.81230 + 0.649662i
187.7 −1.37605 + 0.326323i −1.36395 + 1.06753i 1.78703 0.898073i −1.29544 1.52851 1.91407i 2.57211 + 0.619896i −2.16598 + 1.81894i 0.720742 2.91214i 1.78258 0.422730i
187.8 −1.37231 + 0.341717i 1.71120 + 0.267930i 1.76646 0.937883i −1.57990 −2.43985 + 0.217065i −2.61033 0.431470i −2.10363 + 1.89069i 2.85643 + 0.916964i 2.16811 0.539878i
187.9 −1.34927 0.423642i 1.23482 + 1.21459i 1.64105 + 1.14321i 1.33433 −1.15155 2.16193i −2.63615 0.225222i −1.72991 2.23772i 0.0495475 + 2.99959i −1.80038 0.565280i
187.10 −1.33390 + 0.469805i 0.0180886 + 1.73196i 1.55857 1.25334i 4.13478 −0.837810 2.30175i −1.26742 + 2.32242i −1.49014 + 2.40405i −2.99935 + 0.0626575i −5.51537 + 1.94254i
187.11 −1.33082 0.478466i −0.622274 + 1.61641i 1.54214 + 1.27350i 0.133301 1.60153 1.85340i −0.788476 2.52553i −1.44298 2.43266i −2.22555 2.01170i −0.177399 0.0637802i
187.12 −1.26757 0.627104i −1.24359 + 1.20561i 1.21348 + 1.58980i 3.37565 2.33238 0.748336i 2.41832 + 1.07319i −0.541207 2.77617i 0.0930234 2.99856i −4.27888 2.11688i
187.13 −1.26090 + 0.640415i −0.200514 + 1.72041i 1.17974 1.61500i 0.275961 −0.848945 2.29767i −0.484862 2.60094i −0.453262 + 2.79187i −2.91959 0.689930i −0.347959 + 0.176729i
187.14 −1.25542 0.651094i 1.72271 + 0.179625i 1.15215 + 1.63479i −3.71413 −2.04577 1.34715i 2.06154 1.65833i −0.382032 2.80251i 2.93547 + 0.618883i 4.66278 + 2.41824i
187.15 −1.25541 + 0.651116i 0.939159 1.45533i 1.15209 1.63483i −2.12129 −0.231438 + 2.43853i 2.08410 1.62989i −0.381881 + 2.80253i −1.23596 2.73357i 2.66308 1.38120i
187.16 −1.24319 + 0.674147i −1.71073 0.270952i 1.09105 1.67619i 0.373039 2.30942 0.816436i −2.63561 0.231432i −0.226389 + 2.81935i 2.85317 + 0.927050i −0.463759 + 0.251483i
187.17 −1.14303 0.832759i −0.192996 1.72126i 0.613025 + 1.90373i −1.85503 −1.21280 + 2.12817i 2.48224 + 0.915690i 0.884646 2.68652i −2.92551 + 0.664393i 2.12035 + 1.54479i
187.18 −1.10154 0.886915i −0.590800 + 1.62818i 0.426762 + 1.95394i −2.97526 2.09484 1.26950i −1.42958 + 2.22627i 1.26288 2.53083i −2.30191 1.92385i 3.27736 + 2.63881i
187.19 −1.06762 + 0.927460i 1.47718 0.904397i 0.279635 1.98035i 3.27868 −0.738280 + 2.33558i −2.42344 1.06158i 1.53815 + 2.37362i 1.36413 2.67192i −3.50039 + 3.04084i
187.20 −1.06331 0.932395i 1.19034 1.25821i 0.261277 + 1.98286i −1.28954 −2.43885 + 0.228006i −2.40873 + 1.09453i 1.57099 2.35202i −0.166184 2.99539i 1.37119 + 1.20236i
See next 80 embeddings (of 180 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 283.90 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.k odd 6 1 inner
504.cz 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.cz.b yes 180
7.d odd 6 1 504.2.bf.b 180
8.d odd 2 1 inner 504.2.cz.b yes 180
9.c even 3 1 504.2.bf.b 180
56.m even 6 1 504.2.bf.b 180
63.k odd 6 1 inner 504.2.cz.b yes 180
72.p odd 6 1 504.2.bf.b 180
504.cz even 6 1 inner 504.2.cz.b yes 180

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bf.b 180 7.d odd 6 1
504.2.bf.b 180 9.c even 3 1
504.2.bf.b 180 56.m even 6 1
504.2.bf.b 180 72.p odd 6 1
504.2.cz.b yes 180 1.a even 1 1 trivial
504.2.cz.b yes 180 8.d odd 2 1 inner
504.2.cz.b yes 180 63.k odd 6 1 inner
504.2.cz.b yes 180 504.cz even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{90} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database