# Properties

 Label 756.2.cf.a Level 756 Weight 2 Character orbit 756.cf Analytic conductor 6.037 Analytic rank 0 Dimension 648 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.cf (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$648$$ Relative dimension: $$108$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$648q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$648q + 18q^{12} + 42q^{18} + 42q^{20} - 48q^{30} - 30q^{32} + 60q^{33} + 60q^{41} - 30q^{42} - 144q^{44} - 18q^{48} - 156q^{50} + 54q^{52} - 12q^{57} + 54q^{58} - 186q^{60} - 24q^{65} - 186q^{66} - 78q^{68} - 42q^{72} - 24q^{81} - 30q^{86} - 36q^{89} + 198q^{90} + 114q^{92} - 144q^{93} - 54q^{94} + 156q^{96} + 36q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1 −1.41420 + 0.00498059i 1.72099 + 0.195426i 1.99995 0.0140872i −1.38437 + 0.244102i −2.43481 0.267801i −0.642788 + 0.766044i −2.82827 + 0.0298831i 2.92362 + 0.672654i 1.95657 0.352106i
155.2 −1.41412 0.0164052i 0.147426 + 1.72577i 1.99946 + 0.0463977i −3.09496 + 0.545724i −0.180167 2.44285i 0.642788 0.766044i −2.82671 0.0984134i −2.95653 + 0.508847i 4.38559 0.720946i
155.3 −1.41391 + 0.0291490i −0.812950 1.52942i 1.99830 0.0824282i 2.69171 0.474621i 1.19402 + 2.13876i 0.642788 0.766044i −2.82302 + 0.174795i −1.67823 + 2.48668i −3.79200 + 0.749533i
155.4 −1.41317 0.0543791i 0.596337 1.62616i 1.99409 + 0.153693i −1.26911 + 0.223778i −0.931153 + 2.26560i 0.642788 0.766044i −2.80962 0.325631i −2.28876 1.93947i 1.80563 0.247223i
155.5 −1.37502 0.330626i 0.740762 + 1.56565i 1.78137 + 0.909235i 3.54867 0.625726i −0.500919 2.39772i 0.642788 0.766044i −2.14881 1.83919i −1.90254 + 2.31955i −5.08638 0.312893i
155.6 −1.37092 + 0.347239i −1.36710 + 1.06350i 1.75885 0.952074i −0.934734 + 0.164819i 1.50490 1.93269i 0.642788 0.766044i −2.08065 + 1.91596i 0.737920 2.90783i 1.22422 0.550530i
155.7 −1.37029 + 0.349726i −0.0159569 + 1.73198i 1.75538 0.958451i 1.66542 0.293658i −0.583852 2.37889i −0.642788 + 0.766044i −2.07019 + 1.92726i −2.99949 0.0552741i −2.17940 + 0.984835i
155.8 −1.36715 + 0.361799i 0.921295 1.46670i 1.73820 0.989266i 2.55429 0.450391i −0.728898 + 2.33853i −0.642788 + 0.766044i −2.01847 + 1.98136i −1.30243 2.70253i −3.32915 + 1.53989i
155.9 −1.36600 0.366116i −1.45832 0.934508i 1.73192 + 1.00023i 2.77927 0.490060i 1.64993 + 1.81045i −0.642788 + 0.766044i −1.99960 2.00040i 1.25339 + 2.72562i −3.97591 0.348112i
155.10 −1.36570 + 0.367248i −1.73062 0.0704418i 1.73026 1.00310i 1.13893 0.200824i 2.38937 0.539363i −0.642788 + 0.766044i −1.99462 + 2.00536i 2.99008 + 0.243815i −1.48168 + 0.692535i
155.11 −1.36197 0.380829i −0.829350 + 1.52058i 1.70994 + 1.03736i 1.49277 0.263215i 1.70863 1.75515i −0.642788 + 0.766044i −1.93384 2.06404i −1.62436 2.52220i −2.13335 0.209997i
155.12 −1.35441 0.406892i 1.16480 1.28189i 1.66888 + 1.10220i −2.59226 + 0.457086i −2.09921 + 1.26227i −0.642788 + 0.766044i −1.81188 2.17189i −0.286497 2.98629i 3.69698 + 0.435688i
155.13 −1.33123 0.477307i 1.70586 0.300078i 1.54436 + 1.27081i 0.831135 0.146552i −2.41412 0.414745i 0.642788 0.766044i −1.44933 2.42888i 2.81991 1.02378i −1.17638 0.201613i
155.14 −1.32875 0.484174i −1.44056 + 0.961654i 1.53115 + 1.28669i −4.04339 + 0.712959i 2.37976 0.580314i −0.642788 + 0.766044i −1.41153 2.45103i 1.15044 2.77065i 5.71785 + 1.01036i
155.15 −1.30248 0.550953i −1.12266 1.31896i 1.39290 + 1.43521i −2.72975 + 0.481329i 0.735554 + 2.33644i 0.642788 0.766044i −1.02349 2.63675i −0.479285 + 2.96147i 3.82063 + 0.877043i
155.16 −1.28166 + 0.597796i 1.70881 0.282776i 1.28528 1.53234i 3.03124 0.534490i −2.02107 + 1.38394i 0.642788 0.766044i −0.731262 + 2.73226i 2.84008 0.966421i −3.56549 + 2.49710i
155.17 −1.24559 + 0.669697i 1.67938 0.423888i 1.10301 1.66834i −3.97359 + 0.700651i −1.80795 + 1.65267i 0.642788 0.766044i −0.256623 + 2.81676i 2.64064 1.42374i 4.48026 3.53383i
155.18 −1.19801 + 0.751520i 0.195130 + 1.72102i 0.870434 1.80065i −3.08937 + 0.544739i −1.52715 1.91515i −0.642788 + 0.766044i 0.310441 + 2.81134i −2.92385 + 0.671646i 3.29170 2.97433i
155.19 −1.18703 + 0.768741i −0.163462 1.72432i 0.818074 1.82504i −0.0623853 + 0.0110002i 1.51959 + 1.92116i −0.642788 + 0.766044i 0.431902 + 2.79526i −2.94656 + 0.563722i 0.0655969 0.0610157i
155.20 −1.16457 + 0.802363i −1.47459 0.908615i 0.712427 1.86881i 1.21839 0.214834i 2.44630 0.125015i 0.642788 0.766044i 0.669794 + 2.74798i 1.34884 + 2.67967i −1.24652 + 1.22778i
See next 80 embeddings (of 648 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 743.108 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.cf.a 648
4.b odd 2 1 inner 756.2.cf.a 648
27.f odd 18 1 inner 756.2.cf.a 648
108.l even 18 1 inner 756.2.cf.a 648

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.cf.a 648 1.a even 1 1 trivial
756.2.cf.a 648 4.b odd 2 1 inner
756.2.cf.a 648 27.f odd 18 1 inner
756.2.cf.a 648 108.l even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database