# Properties

 Label 504.2.cc.b Level 504 Weight 2 Character orbit 504.cc Analytic conductor 4.024 Analytic rank 0 Dimension 168 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.cc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$168$$ Relative dimension: $$84$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$168q - 6q^{2} - 18q^{4} - 2q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$168q - 6q^{2} - 18q^{4} - 2q^{7} - 8q^{9} + 12q^{14} - 4q^{15} + 30q^{16} - 2q^{18} + 6q^{22} - 84q^{23} + 32q^{25} + 4q^{28} - 52q^{30} + 24q^{32} - 26q^{36} + 44q^{39} - 16q^{46} + 54q^{49} + 108q^{50} - 24q^{56} - 40q^{57} + 6q^{58} - 4q^{60} - 86q^{63} + 120q^{64} + 60q^{65} - 6q^{70} + 52q^{72} + 48q^{74} - 184q^{78} - 4q^{79} - 104q^{81} - 72q^{84} - 144q^{86} - 18q^{88} - 90q^{92} - 72q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1 −1.41305 + 0.0572344i −1.67485 0.441454i 1.99345 0.161751i −2.14271 1.23709i 2.39192 + 0.527939i −0.933167 + 2.47572i −2.80759 + 0.342656i 2.61024 + 1.47874i 3.09857 + 1.62545i
293.2 −1.41305 + 0.0572344i 1.67485 + 0.441454i 1.99345 0.161751i 2.14271 + 1.23709i −2.39192 0.527939i 2.61062 + 0.429715i −2.80759 + 0.342656i 2.61024 + 1.47874i −3.09857 1.62545i
293.3 −1.39399 0.238316i −1.00341 1.41180i 1.88641 + 0.664419i 1.68580 + 0.973297i 1.06229 + 2.20716i −2.24361 1.40222i −2.47129 1.37576i −0.986337 + 2.83322i −2.11804 1.75852i
293.4 −1.39399 0.238316i 1.00341 + 1.41180i 1.88641 + 0.664419i −1.68580 0.973297i −1.06229 2.20716i −0.0925508 2.64413i −2.47129 1.37576i −0.986337 + 2.83322i 2.11804 + 1.75852i
293.5 −1.38810 0.270536i −1.21338 + 1.23600i 1.85362 + 0.751060i 2.17541 + 1.25598i 2.01867 1.38743i −2.64254 0.130417i −2.36981 1.54401i −0.0554045 2.99949i −2.67990 2.33194i
293.6 −1.38810 0.270536i 1.21338 1.23600i 1.85362 + 0.751060i −2.17541 1.25598i −2.01867 + 1.38743i 1.20832 2.35371i −2.36981 1.54401i −0.0554045 2.99949i 2.67990 + 2.33194i
293.7 −1.37160 + 0.344550i −1.55842 + 0.755861i 1.76257 0.945170i −0.673579 0.388891i 1.87710 1.57369i 0.999298 2.44978i −2.09188 + 1.90369i 1.85735 2.35590i 1.05787 + 0.301321i
293.8 −1.37160 + 0.344550i 1.55842 0.755861i 1.76257 0.945170i 0.673579 + 0.388891i −1.87710 + 1.57369i −2.62122 0.359471i −2.09188 + 1.90369i 1.85735 2.35590i −1.05787 0.301321i
293.9 −1.36240 0.379297i −0.713494 1.57827i 1.71227 + 1.03351i −0.431334 0.249031i 0.373433 + 2.42086i 2.62075 + 0.362852i −1.94079 2.05751i −1.98185 + 2.25217i 0.493193 + 0.502883i
293.10 −1.36240 0.379297i 0.713494 + 1.57827i 1.71227 + 1.03351i 0.431334 + 0.249031i −0.373433 2.42086i −0.996137 + 2.45106i −1.94079 2.05751i −1.98185 + 2.25217i −0.493193 0.502883i
293.11 −1.30669 + 0.540891i −0.0157202 + 1.73198i 1.41487 1.41355i −1.10228 0.636400i −0.916271 2.27166i 2.59794 0.500682i −1.08422 + 2.61237i −2.99951 0.0544542i 1.78456 + 0.235365i
293.12 −1.30669 + 0.540891i 0.0157202 1.73198i 1.41487 1.41355i 1.10228 + 0.636400i 0.916271 + 2.27166i −1.73258 + 1.99955i −1.08422 + 2.61237i −2.99951 0.0544542i −1.78456 0.235365i
293.13 −1.21983 0.715553i −1.72909 + 0.101211i 0.975968 + 1.74571i 2.82704 + 1.63219i 2.18162 + 1.11380i 2.40792 1.09632i 0.0586299 2.82782i 2.97951 0.350006i −2.28058 4.01389i
293.14 −1.21983 0.715553i 1.72909 0.101211i 0.975968 + 1.74571i −2.82704 1.63219i −2.18162 1.11380i −2.15340 + 1.53716i 0.0586299 2.82782i 2.97951 0.350006i 2.28058 + 4.01389i
293.15 −1.21170 0.729232i −1.13327 + 1.30985i 0.936442 + 1.76722i −2.78508 1.60797i 2.32837 0.760729i 2.37927 + 1.15719i 0.154027 2.82423i −0.431402 2.96882i 2.20210 + 3.97934i
293.16 −1.21170 0.729232i 1.13327 1.30985i 0.936442 + 1.76722i 2.78508 + 1.60797i −2.32837 + 0.760729i −0.187474 + 2.63910i 0.154027 2.82423i −0.431402 2.96882i −2.20210 3.97934i
293.17 −1.12177 + 0.861180i −0.0157202 + 1.73198i 0.516736 1.93209i −1.10228 0.636400i −1.47391 1.95642i −1.73258 + 1.99955i 1.08422 + 2.61237i −2.99951 0.0544542i 1.78456 0.235365i
293.18 −1.12177 + 0.861180i 0.0157202 1.73198i 0.516736 1.93209i 1.10228 + 0.636400i 1.47391 + 1.95642i 2.59794 0.500682i 1.08422 + 2.61237i −2.99951 0.0544542i −1.78456 + 0.235365i
293.19 −0.995788 1.00419i −0.539289 + 1.64595i −0.0168127 + 1.99993i −0.194059 0.112040i 2.18988 1.09747i −1.79621 1.94258i 2.02506 1.97462i −2.41834 1.77529i 0.0807318 + 0.306441i
293.20 −0.995788 1.00419i 0.539289 1.64595i −0.0168127 + 1.99993i 0.194059 + 0.112040i −2.18988 + 1.09747i −0.784223 2.52685i 2.02506 1.97462i −2.41834 1.77529i −0.0807318 0.306441i
See next 80 embeddings (of 168 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.84 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.b even 2 1 inner
9.d odd 6 1 inner
56.h odd 2 1 inner
63.o even 6 1 inner
72.j odd 6 1 inner
504.cc 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.cc.b 168
7.b odd 2 1 inner 504.2.cc.b 168
8.b even 2 1 inner 504.2.cc.b 168
9.d odd 6 1 inner 504.2.cc.b 168
56.h odd 2 1 inner 504.2.cc.b 168
63.o even 6 1 inner 504.2.cc.b 168
72.j odd 6 1 inner 504.2.cc.b 168
504.cc even 6 1 inner 504.2.cc.b 168

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cc.b 168 1.a even 1 1 trivial
504.2.cc.b 168 7.b odd 2 1 inner
504.2.cc.b 168 8.b even 2 1 inner
504.2.cc.b 168 9.d odd 6 1 inner
504.2.cc.b 168 56.h odd 2 1 inner
504.2.cc.b 168 63.o even 6 1 inner
504.2.cc.b 168 72.j odd 6 1 inner
504.2.cc.b 168 504.cc even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{84} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database