# Properties

 Label 756.2.bj.b Level 756 Weight 2 Character orbit 756.bj 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.bj (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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q - 2q^{2} - 2q^{4} - 6q^{5} + 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q - 2q^{2} - 2q^{4} - 6q^{5} + 16q^{8} - 18q^{10} + 18q^{13} - 14q^{14} + 14q^{16} - 6q^{17} + 24q^{20} + 6q^{22} + 16q^{25} + 30q^{26} - 4q^{28} - 10q^{29} + 18q^{32} - 24q^{34} + 2q^{37} - 33q^{38} + 6q^{40} - 6q^{41} + 13q^{44} + 10q^{46} - 28q^{49} + 17q^{50} - 27q^{52} + 2q^{53} - 58q^{56} - 13q^{58} - 8q^{64} + 100q^{65} + 18q^{68} - 19q^{70} + 30q^{73} + 23q^{74} + 2q^{77} - 3q^{80} - 18q^{82} - 50q^{85} + 9q^{86} + q^{88} + 102q^{89} - 28q^{92} + 6q^{97} - 21q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1 −1.40573 0.154709i 0 1.95213 + 0.434957i 2.69307 + 1.55484i 0 2.30485 + 1.29911i −2.67687 0.913443i 0 −3.54517 2.60233i
451.2 −1.40573 + 0.154709i 0 1.95213 0.434957i 2.69307 + 1.55484i 0 −2.30485 1.29911i −2.67687 + 0.913443i 0 −4.02627 1.76904i
451.3 −1.36675 0.363308i 0 1.73601 + 0.993103i −1.43430 0.828095i 0 −0.375983 2.61890i −2.01190 1.98803i 0 1.65948 + 1.65289i
451.4 −1.36675 + 0.363308i 0 1.73601 0.993103i −1.43430 0.828095i 0 0.375983 + 2.61890i −2.01190 + 1.98803i 0 2.26119 + 0.610706i
451.5 −1.27891 0.603655i 0 1.27120 + 1.54404i 0.0437657 + 0.0252681i 0 1.31315 + 2.29687i −0.693684 2.74204i 0 −0.0407190 0.0587349i
451.6 −1.27891 + 0.603655i 0 1.27120 1.54404i 0.0437657 + 0.0252681i 0 −1.31315 2.29687i −0.693684 + 2.74204i 0 −0.0712254 0.00589621i
451.7 −1.26364 0.634994i 0 1.19357 + 1.60481i 0.104850 + 0.0605350i 0 −2.60650 + 0.454031i −0.489193 2.78580i 0 −0.0940527 0.143073i
451.8 −1.26364 + 0.634994i 0 1.19357 1.60481i 0.104850 + 0.0605350i 0 2.60650 0.454031i −0.489193 + 2.78580i 0 −0.170931 0.00991543i
451.9 −1.06666 0.928568i 0 0.275522 + 1.98093i −2.33177 1.34625i 0 1.11915 2.39740i 1.54554 2.36882i 0 1.23712 + 3.60119i
451.10 −1.06666 + 0.928568i 0 0.275522 1.98093i −2.33177 1.34625i 0 −1.11915 + 2.39740i 1.54554 + 2.36882i 0 3.73728 0.729218i
451.11 −0.937998 1.05838i 0 −0.240318 + 1.98551i 3.41753 + 1.97311i 0 −0.174185 2.64001i 2.32683 1.60806i 0 −1.11734 5.46781i
451.12 −0.937998 + 1.05838i 0 −0.240318 1.98551i 3.41753 + 1.97311i 0 0.174185 + 2.64001i 2.32683 + 1.60806i 0 −5.29393 + 1.76626i
451.13 −0.771545 1.18521i 0 −0.809438 + 1.82888i 0.917924 + 0.529964i 0 −2.44585 + 1.00887i 2.79212 0.451713i 0 −0.0801021 1.49682i
451.14 −0.771545 + 1.18521i 0 −0.809438 1.82888i 0.917924 + 0.529964i 0 2.44585 1.00887i 2.79212 + 0.451713i 0 −1.33634 + 0.679041i
451.15 −0.667137 1.24697i 0 −1.10986 + 1.66380i 0.627749 + 0.362431i 0 2.64477 0.0721413i 2.81513 + 0.273973i 0 0.0331449 1.02457i
451.16 −0.667137 + 1.24697i 0 −1.10986 1.66380i 0.627749 + 0.362431i 0 −2.64477 + 0.0721413i 2.81513 0.273973i 0 −0.870735 + 0.540991i
451.17 −0.514563 1.31728i 0 −1.47045 + 1.35565i −2.53954 1.46621i 0 2.33334 + 1.24720i 2.54241 + 1.23943i 0 −0.624647 + 4.09975i
451.18 −0.514563 + 1.31728i 0 −1.47045 1.35565i −2.53954 1.46621i 0 −2.33334 1.24720i 2.54241 1.23943i 0 3.23816 2.59083i
451.19 −0.248495 1.39221i 0 −1.87650 + 0.691915i −0.705906 0.407555i 0 −2.03393 + 1.69208i 1.42959 + 2.44055i 0 −0.391988 + 1.08405i
451.20 −0.248495 + 1.39221i 0 −1.87650 0.691915i −0.705906 0.407555i 0 2.03393 1.69208i 1.42959 2.44055i 0 0.742817 0.881495i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 523.42 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.t odd 6 1 inner
252.bj 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.bj.b 84
3.b odd 2 1 252.2.bj.b yes 84
4.b odd 2 1 inner 756.2.bj.b 84
7.d odd 6 1 756.2.n.b 84
9.c even 3 1 756.2.n.b 84
9.d odd 6 1 252.2.n.b 84
12.b even 2 1 252.2.bj.b yes 84
21.g even 6 1 252.2.n.b 84
28.f even 6 1 756.2.n.b 84
36.f odd 6 1 756.2.n.b 84
36.h even 6 1 252.2.n.b 84
63.i even 6 1 252.2.bj.b yes 84
63.t odd 6 1 inner 756.2.bj.b 84
84.j odd 6 1 252.2.n.b 84
252.r odd 6 1 252.2.bj.b yes 84
252.bj even 6 1 inner 756.2.bj.b 84

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

## 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