# Properties

 Label 504.2.bf.b Level 504 Weight 2 Character orbit 504.bf Analytic conductor 4.024 Analytic rank 0 Dimension 180 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.bf (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.

 $$\operatorname{Tr}(f)(q) =$$ $$180q - 6q^{2} - 6q^{3} - 2q^{4} - 6q^{6} + 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$180q - 6q^{2} - 6q^{3} - 2q^{4} - 6q^{6} + 10q^{9} - 8q^{11} + 12q^{12} + 10q^{14} + 14q^{16} - 18q^{17} + 8q^{18} - 6q^{19} + 36q^{20} - 16q^{22} - 12q^{24} - 78q^{25} - 6q^{26} + 16q^{28} + q^{30} - 26q^{32} - 36q^{33} - 12q^{34} - 12q^{35} + 2q^{36} - 27q^{38} + 24q^{40} - 42q^{41} + 22q^{42} + 14q^{43} - 21q^{44} - 12q^{46} - 9q^{48} + 2q^{49} + 15q^{50} + 9q^{52} - 51q^{54} + 14q^{56} - 26q^{57} + 19q^{58} + 37q^{60} - 8q^{64} + 24q^{65} + 6q^{66} + 28q^{67} + 12q^{68} + 27q^{70} - 28q^{72} + 18q^{73} + 49q^{74} - 18q^{75} - 12q^{76} - 33q^{78} - 63q^{80} - 22q^{81} - 54q^{82} + 6q^{83} + 31q^{84} - 13q^{86} + 29q^{88} - 66q^{89} - 51q^{90} + 2q^{91} - 60q^{92} - 45q^{96} - 6q^{97} + 31q^{98} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1 −1.41416 0.0128144i 1.65702 + 0.504266i 1.99967 + 0.0362430i 2.12331 3.67768i −2.33682 0.734345i 2.60163 + 0.481180i −2.82738 0.0768778i 2.49143 + 1.67116i −3.04982 + 5.17360i
115.2 −1.41416 + 0.0128144i 1.65702 + 0.504266i 1.99967 0.0362430i −2.12331 + 3.67768i −2.34975 0.691877i −2.60163 0.481180i −2.82738 + 0.0768778i 2.49143 + 1.67116i 2.95556 5.22802i
115.3 −1.41229 0.0736946i −1.11864 1.32236i 1.98914 + 0.208157i 1.36131 2.35786i 1.48240 + 1.95000i −1.25248 + 2.33051i −2.79390 0.440567i −0.497287 + 2.95850i −2.09633 + 3.22967i
115.4 −1.41229 + 0.0736946i −1.11864 1.32236i 1.98914 0.208157i −1.36131 + 2.35786i 1.67730 + 1.78512i 1.25248 2.33051i −2.79390 + 0.440567i −0.497287 + 2.95850i 1.74881 3.43031i
115.5 −1.40172 0.187532i 0.894629 1.48312i 1.92966 + 0.525737i −0.356494 + 0.617465i −1.53216 + 1.91115i −0.577640 + 2.58192i −2.60626 1.09881i −1.39928 2.65368i 0.615501 0.798662i
115.6 −1.40172 + 0.187532i 0.894629 1.48312i 1.92966 0.525737i 0.356494 0.617465i −0.975890 + 2.24669i 0.577640 2.58192i −2.60626 + 1.09881i −1.39928 2.65368i −0.383911 + 0.932370i
115.7 −1.37865 0.315175i −0.265334 + 1.71161i 1.80133 + 0.869029i −1.58477 + 2.74491i 0.905257 2.27607i 2.63076 + 0.281244i −2.20950 1.76582i −2.85920 0.908295i 3.04997 3.28478i
115.8 −1.37865 + 0.315175i −0.265334 + 1.71161i 1.80133 0.869029i 1.58477 2.74491i −0.173654 2.44333i −2.63076 0.281244i −2.20950 + 1.76582i −2.85920 0.908295i −1.31972 + 4.28374i
115.9 −1.37396 0.335011i −1.69738 + 0.344799i 1.77554 + 0.920584i 0.237272 0.410968i 2.44765 + 0.0949016i −2.23531 1.41542i −2.13111 1.85967i 2.76223 1.17051i −0.463682 + 0.485165i
115.10 −1.37396 + 0.335011i −1.69738 + 0.344799i 1.77554 0.920584i −0.237272 + 0.410968i 2.21663 1.04238i 2.23531 + 1.41542i −2.13111 + 1.85967i 2.76223 1.17051i 0.188324 0.644143i
115.11 −1.32350 0.498350i 0.336489 + 1.69905i 1.50330 + 1.31913i 0.402355 0.696900i 0.401379 2.41638i 0.321531 2.62614i −1.33222 2.49503i −2.77355 + 1.14342i −0.879816 + 0.721832i
115.12 −1.32350 + 0.498350i 0.336489 + 1.69905i 1.50330 1.31913i −0.402355 + 0.696900i −1.29206 2.08100i −0.321531 + 2.62614i −1.33222 + 2.49503i −2.77355 + 1.14342i 0.185217 1.12286i
115.13 −1.29841 0.560472i 1.69247 0.368153i 1.37174 + 1.45545i −0.863157 + 1.49503i −2.40386 0.470569i 2.59622 + 0.509573i −0.965350 2.65859i 2.72893 1.24618i 1.95866 1.45739i
115.14 −1.29841 + 0.560472i 1.69247 0.368153i 1.37174 1.45545i 0.863157 1.49503i −1.99119 + 1.42660i −2.59622 0.509573i −0.965350 + 2.65859i 2.72893 1.24618i −0.282810 + 2.42494i
115.15 −1.16893 0.795991i −0.801348 1.53553i 0.732796 + 1.86092i −1.06898 + 1.85154i −0.285545 + 2.43279i −2.52157 0.801064i 0.624685 2.75858i −1.71568 + 2.46098i 2.72337 1.31341i
115.16 −1.16893 + 0.795991i −0.801348 1.53553i 0.732796 1.86092i 1.06898 1.85154i 2.15899 + 1.15706i 2.52157 + 0.801064i 0.624685 + 2.75858i −1.71568 + 2.46098i 0.224237 + 3.01522i
115.17 −1.15708 0.813118i 1.26585 + 1.18221i 0.677678 + 1.88169i 0.495626 0.858449i −0.503410 2.39720i −1.90467 + 1.83636i 0.745907 2.72830i 0.204739 + 2.99301i −1.27150 + 0.590294i
115.18 −1.15708 + 0.813118i 1.26585 + 1.18221i 0.677678 1.88169i −0.495626 + 0.858449i −2.42597 0.338635i 1.90467 1.83636i 0.745907 + 2.72830i 0.204739 + 2.99301i −0.124541 1.39630i
115.19 −1.14308 0.832691i −1.17641 + 1.27125i 0.613252 + 1.90366i 1.57644 2.73047i 2.40328 0.473549i 1.25267 + 2.33041i 0.884165 2.68668i −0.232133 2.99101i −4.07562 + 1.80845i
115.20 −1.14308 + 0.832691i −1.17641 + 1.27125i 0.613252 1.90366i −1.57644 + 2.73047i 0.286170 2.43272i −1.25267 2.33041i 0.884165 + 2.68668i −0.232133 2.99101i −0.471646 4.43382i
See next 80 embeddings (of 180 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 355.90 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.t odd 6 1 inner
504.bf 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.bf.b 180
7.d odd 6 1 504.2.cz.b yes 180
8.d odd 2 1 inner 504.2.bf.b 180
9.c even 3 1 504.2.cz.b yes 180
56.m even 6 1 504.2.cz.b yes 180
63.t odd 6 1 inner 504.2.bf.b 180
72.p odd 6 1 504.2.cz.b yes 180
504.bf even 6 1 inner 504.2.bf.b 180

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{180} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database