# Properties

 Label 504.2.be.a Level 504 Weight 2 Character orbit 504.be Analytic conductor 4.024 Analytic rank 0 Dimension 184 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.be (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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 2q^{4} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 2q^{2} - 2q^{4} - 8q^{8} - 8q^{9} - 4q^{11} - 10q^{14} - 2q^{16} - 10q^{18} + 6q^{22} - 80q^{25} - 12q^{28} + 4q^{30} - 12q^{32} + 12q^{35} - 46q^{36} + 10q^{42} - 4q^{43} - 36q^{44} - 16q^{46} - 2q^{49} + 16q^{50} - 24q^{51} + 40q^{56} + 16q^{57} - 10q^{58} + 40q^{60} - 8q^{64} + 36q^{65} - 4q^{67} - 76q^{72} - 40q^{74} + 60q^{78} + 8q^{81} + 40q^{84} + 32q^{86} - 18q^{88} + 20q^{91} - 26q^{92} - 132q^{98} + 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.41404 + 0.0218513i −1.61755 0.619288i 1.99905 0.0617973i 0.694615 1.20311i 2.30083 + 0.840355i −0.407789 2.61414i −2.82539 + 0.131066i 2.23297 + 2.00346i −0.955927 + 1.71643i
139.2 −1.41404 + 0.0218513i 1.61755 + 0.619288i 1.99905 0.0617973i −0.694615 + 1.20311i −2.30083 0.840355i 2.06001 + 1.66022i −2.82539 + 0.131066i 2.23297 + 2.00346i 0.955927 1.71643i
139.3 −1.41366 0.0394185i −1.54504 + 0.782839i 1.99689 + 0.111449i 1.11062 1.92366i 2.21503 1.04577i −2.08899 + 1.62361i −2.81854 0.236266i 1.77433 2.41904i −1.64588 + 2.67563i
139.4 −1.41366 0.0394185i 1.54504 0.782839i 1.99689 + 0.111449i −1.11062 + 1.92366i −2.21503 + 1.04577i −2.45058 + 0.997318i −2.81854 0.236266i 1.77433 2.41904i 1.64588 2.67563i
139.5 −1.38885 0.266643i −0.855137 + 1.50623i 1.85780 + 0.740655i −1.56998 + 2.71929i 1.58928 1.86392i 1.10903 2.40210i −2.38272 1.52403i −1.53748 2.57607i 2.90555 3.35806i
139.6 −1.38885 0.266643i 0.855137 1.50623i 1.85780 + 0.740655i 1.56998 2.71929i −1.58928 + 1.86392i 2.63479 + 0.240603i −2.38272 1.52403i −1.53748 2.57607i −2.90555 + 3.35806i
139.7 −1.38351 + 0.293103i −0.0384188 1.73162i 1.82818 0.811019i −1.45134 + 2.51379i 0.560696 + 2.38445i 0.907665 2.48519i −2.29159 + 1.65790i −2.99705 + 0.133054i 1.27114 3.90323i
139.8 −1.38351 + 0.293103i 0.0384188 + 1.73162i 1.82818 0.811019i 1.45134 2.51379i −0.560696 2.38445i 2.60607 + 0.456532i −2.29159 + 1.65790i −2.99705 + 0.133054i −1.27114 + 3.90323i
139.9 −1.37416 0.334195i −0.522528 1.65135i 1.77663 + 0.918475i −0.649625 + 1.12518i 0.166162 + 2.44385i −0.237312 + 2.63509i −2.13442 1.85587i −2.45393 + 1.72576i 1.26872 1.32908i
139.10 −1.37416 0.334195i 0.522528 + 1.65135i 1.77663 + 0.918475i 0.649625 1.12518i −0.166162 2.44385i −2.40071 1.11203i −2.13442 1.85587i −2.45393 + 1.72576i −1.26872 + 1.32908i
139.11 −1.29729 + 0.563073i −1.71842 0.216873i 1.36590 1.46093i −1.92861 + 3.34045i 2.35140 0.686250i 1.79440 + 1.94426i −0.949346 + 2.66435i 2.90593 + 0.745357i 0.621039 5.41946i
139.12 −1.29729 + 0.563073i 1.71842 + 0.216873i 1.36590 1.46093i 1.92861 3.34045i −2.35140 + 0.686250i −0.786580 2.52612i −0.949346 + 2.66435i 2.90593 + 0.745357i −0.621039 + 5.41946i
139.13 −1.28821 + 0.583540i −0.476817 + 1.66513i 1.31896 1.50344i −0.852648 + 1.47683i −0.357427 2.42327i −1.56588 + 2.13261i −0.821782 + 2.70641i −2.54529 1.58792i 0.236600 2.40002i
139.14 −1.28821 + 0.583540i 0.476817 1.66513i 1.31896 1.50344i 0.852648 1.47683i 0.357427 + 2.42327i −2.62983 + 0.289791i −0.821782 + 2.70641i −2.54529 1.58792i −0.236600 + 2.40002i
139.15 −1.20351 0.742667i −1.22607 1.22342i 0.896891 + 1.78762i 0.417659 0.723407i 0.566998 + 2.38296i 2.60856 0.442060i 0.248186 2.81752i 0.00649163 + 2.99999i −1.03991 + 0.560448i
139.16 −1.20351 0.742667i 1.22607 + 1.22342i 0.896891 + 1.78762i −0.417659 + 0.723407i −0.566998 2.38296i 1.68711 2.03805i 0.248186 2.81752i 0.00649163 + 2.99999i 1.03991 0.560448i
139.17 −1.19759 0.752180i −1.58643 + 0.695160i 0.868451 + 1.80161i 0.0673890 0.116721i 2.42278 + 0.360761i 1.08501 + 2.41304i 0.315085 2.81082i 2.03351 2.20564i −0.168500 + 0.0890957i
139.18 −1.19759 0.752180i 1.58643 0.695160i 0.868451 + 1.80161i −0.0673890 + 0.116721i −2.42278 0.360761i −1.54725 2.14617i 0.315085 2.81082i 2.03351 2.20564i 0.168500 0.0890957i
139.19 −1.12708 + 0.854224i −1.71594 + 0.235665i 0.540602 1.92555i 0.892228 1.54538i 1.73269 1.73141i 2.64346 0.110098i 1.03555 + 2.63204i 2.88892 0.808775i 0.314496 + 2.50393i
139.20 −1.12708 + 0.854224i 1.71594 0.235665i 0.540602 1.92555i −0.892228 + 1.54538i −1.73269 + 1.73141i 1.41708 2.23425i 1.03555 + 2.63204i 2.88892 0.808775i −0.314496 2.50393i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 475.92 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
56.e even 2 1 inner
63.l odd 6 1 inner
72.p odd 6 1 inner
504.be 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.be.a 184
7.b odd 2 1 inner 504.2.be.a 184
8.d odd 2 1 inner 504.2.be.a 184
9.c even 3 1 inner 504.2.be.a 184
56.e even 2 1 inner 504.2.be.a 184
63.l odd 6 1 inner 504.2.be.a 184
72.p odd 6 1 inner 504.2.be.a 184
504.be even 6 1 inner 504.2.be.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.be.a 184 1.a even 1 1 trivial
504.2.be.a 184 7.b odd 2 1 inner
504.2.be.a 184 8.d odd 2 1 inner
504.2.be.a 184 9.c even 3 1 inner
504.2.be.a 184 56.e even 2 1 inner
504.2.be.a 184 63.l odd 6 1 inner
504.2.be.a 184 72.p odd 6 1 inner
504.2.be.a 184 504.be 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