# Properties

 Label 756.2.n.b Level 756 Weight 2 Character orbit 756.n Analytic conductor 6.037 Analytic rank 0 Dimension 84 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.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$42$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 252) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q + q^{2} + q^{4} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q + q^{2} + q^{4} + 16q^{8} - 18q^{10} - 18q^{13} + 25q^{14} - 7q^{16} - 6q^{17} - 24q^{20} + 6q^{22} - 32q^{25} + 30q^{26} - 4q^{28} - 10q^{29} - 9q^{32} + 24q^{34} + 2q^{37} + 6q^{41} + 13q^{44} + 10q^{46} + 2q^{49} + 17q^{50} + 2q^{53} + 32q^{56} + 26q^{58} - 24q^{61} - 8q^{64} - 50q^{65} - 4q^{70} + 30q^{73} - 46q^{74} - 46q^{77} - 3q^{80} - 18q^{82} - 50q^{85} - 18q^{86} - 2q^{88} + 102q^{89} - 28q^{92} + 3q^{94} - 6q^{97} - 21q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41039 + 0.103929i 0 1.97840 0.293161i 1.91789i 0 −1.85299 + 1.88850i −2.75984 + 0.619084i 0 0.199325 + 2.70497i
19.2 −1.37782 + 0.318759i 0 1.79679 0.878385i 0.594537i 0 2.59555 + 0.512964i −2.19566 + 1.78300i 0 0.189514 + 0.819166i
19.3 −1.36968 + 0.352122i 0 1.75202 0.964586i 2.71053i 0 1.53391 2.15572i −2.06005 + 1.93809i 0 −0.954438 3.71255i
19.4 −1.34403 0.439974i 0 1.61285 + 1.18268i 2.88398i 0 −2.24777 + 1.39554i −1.64737 2.29917i 0 1.26887 3.87616i
19.5 −1.30137 + 0.553558i 0 1.38715 1.44077i 4.05112i 0 −2.21649 1.44470i −1.00765 + 2.64285i 0 2.24253 + 5.27202i
19.6 −1.28099 0.599230i 0 1.28185 + 1.53521i 2.56839i 0 0.477445 2.60232i −0.722083 2.73470i 0 −1.53906 + 3.29008i
19.7 −1.27769 0.606229i 0 1.26497 + 1.54914i 0.139892i 0 −1.99326 1.73980i −0.677105 2.74618i 0 −0.0848066 + 0.178738i
19.8 −1.17926 + 0.780599i 0 0.781329 1.84107i 2.00240i 0 −2.64523 + 0.0525529i 0.515742 + 2.78101i 0 −1.56307 2.36135i
19.9 −1.08144 + 0.911308i 0 0.339034 1.97105i 0.815110i 0 0.448419 2.60747i 1.42959 + 2.44055i 0 0.742817 + 0.881495i
19.10 −1.07941 0.913714i 0 0.330255 + 1.97254i 3.48216i 0 −0.638585 + 2.56753i 1.44586 2.43094i 0 −3.18169 + 3.75868i
19.11 −0.883516 + 1.10426i 0 −0.438800 1.95127i 2.93241i 0 2.24678 + 1.39713i 2.54241 + 1.23943i 0 3.23816 + 2.59083i
19.12 −0.825493 1.14829i 0 −0.637123 + 1.89580i 0.299382i 0 2.33495 1.24419i 2.70287 0.833374i 0 −0.343777 + 0.247138i
19.13 −0.746337 + 1.20124i 0 −0.885963 1.79306i 0.724862i 0 1.25991 + 2.32651i 2.81513 + 0.273973i 0 −0.870735 0.540991i
19.14 −0.726200 1.21352i 0 −0.945266 + 1.76252i 2.80831i 0 −0.514387 2.59527i 2.82531 0.132842i 0 3.40794 2.03940i
19.15 −0.687840 1.23567i 0 −1.05375 + 1.69988i 2.80831i 0 0.514387 + 2.59527i 2.82531 + 0.132842i 0 3.47014 1.93167i
19.16 −0.640648 + 1.26078i 0 −1.17914 1.61543i 1.05993i 0 −0.349222 2.62260i 2.79212 0.451713i 0 −1.33634 0.679041i
19.17 −0.581699 1.28904i 0 −1.32325 + 1.49967i 0.299382i 0 −2.33495 + 1.24419i 2.70287 + 0.833374i 0 −0.385916 + 0.174150i
19.18 −0.447581 + 1.34152i 0 −1.59934 1.20088i 3.94623i 0 −2.37341 + 1.16916i 2.32683 1.60806i 0 −5.29393 1.76626i
19.19 −0.270834 + 1.38804i 0 −1.85330 0.751856i 2.69249i 0 −1.51663 + 2.16791i 1.54554 2.36882i 0 3.73728 + 0.729218i
19.20 −0.251594 1.39165i 0 −1.87340 + 0.700263i 3.48216i 0 0.638585 2.56753i 1.44586 + 2.43094i 0 −4.84596 + 0.876089i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.42 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
63.k odd 6 1 inner
252.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.n.b 84
3.b odd 2 1 252.2.n.b 84
4.b odd 2 1 inner 756.2.n.b 84
7.d odd 6 1 756.2.bj.b 84
9.c even 3 1 756.2.bj.b 84
9.d odd 6 1 252.2.bj.b yes 84
12.b even 2 1 252.2.n.b 84
21.g even 6 1 252.2.bj.b yes 84
28.f even 6 1 756.2.bj.b 84
36.f odd 6 1 756.2.bj.b 84
36.h even 6 1 252.2.bj.b yes 84
63.k odd 6 1 inner 756.2.n.b 84
63.s even 6 1 252.2.n.b 84
84.j odd 6 1 252.2.bj.b yes 84
252.n even 6 1 inner 756.2.n.b 84
252.bn odd 6 1 252.2.n.b 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.n.b 84 3.b odd 2 1
252.2.n.b 84 12.b even 2 1
252.2.n.b 84 63.s even 6 1
252.2.n.b 84 252.bn odd 6 1
252.2.bj.b yes 84 9.d odd 6 1
252.2.bj.b yes 84 21.g even 6 1
252.2.bj.b yes 84 36.h even 6 1
252.2.bj.b yes 84 84.j odd 6 1
756.2.n.b 84 1.a even 1 1 trivial
756.2.n.b 84 4.b odd 2 1 inner
756.2.n.b 84 63.k odd 6 1 inner
756.2.n.b 84 252.n even 6 1 inner
756.2.bj.b 84 7.d odd 6 1
756.2.bj.b 84 9.c even 3 1
756.2.bj.b 84 28.f even 6 1
756.2.bj.b 84 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{42} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database