# Properties

 Label 756.2.o.a Level 756 Weight 2 Character orbit 756.o Analytic conductor 6.037 Analytic rank 0 Dimension 88 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.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q + 3q^{2} + q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q + 3q^{2} + q^{4} + 2q^{10} - 4q^{13} + 3q^{14} + q^{16} + 6q^{20} - 6q^{22} - 60q^{25} + 6q^{26} + 24q^{29} - 27q^{32} - 4q^{34} - 4q^{37} + 8q^{40} + 12q^{41} + 57q^{44} - 6q^{46} - 2q^{49} - 9q^{50} + 14q^{52} + 66q^{56} - 10q^{58} + 2q^{61} - 8q^{64} - 18q^{65} + 30q^{70} - 4q^{73} - 6q^{76} + 30q^{77} - 87q^{80} - 4q^{82} - 14q^{85} - 18q^{88} - 60q^{89} - 24q^{92} + 9q^{94} - 4q^{97} + 57q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1 −1.41419 0.00770275i 0 1.99988 + 0.0217863i 2.17323i 0 −0.191082 + 2.63884i −2.82805 0.0462147i 0 −0.0167398 + 3.07336i
179.2 −1.41030 0.105074i 0 1.97792 + 0.296374i 4.26346i 0 −1.55034 2.14393i −2.75833 0.625806i 0 0.447980 6.01278i
179.3 −1.40984 + 0.111168i 0 1.97528 0.313458i 3.13896i 0 0.228884 2.63583i −2.74998 + 0.661514i 0 0.348953 + 4.42542i
179.4 −1.37735 + 0.320803i 0 1.79417 0.883713i 1.64200i 0 −1.79804 + 1.94089i −2.18770 + 1.79275i 0 −0.526757 2.26160i
179.5 −1.34745 0.429408i 0 1.63122 + 1.15721i 0.501474i 0 −2.46962 0.949199i −1.70106 2.25973i 0 −0.215337 + 0.675709i
179.6 −1.32602 0.491613i 0 1.51663 + 1.30377i 0.834477i 0 2.47783 0.927568i −1.37012 2.47442i 0 0.410240 1.10653i
179.7 −1.20615 + 0.738385i 0 0.909574 1.78120i 1.66943i 0 2.08401 + 1.63000i 0.218133 + 2.82000i 0 −1.23268 2.01358i
179.8 −1.17812 + 0.782325i 0 0.775934 1.84335i 3.27841i 0 2.52907 + 0.777042i 0.527954 + 2.77872i 0 2.56478 + 3.86236i
179.9 −1.15931 + 0.809940i 0 0.687993 1.87794i 0.492858i 0 −1.41089 2.23817i 0.723425 + 2.73435i 0 0.399186 + 0.571375i
179.10 −1.08876 0.902556i 0 0.370785 + 1.96533i 0.834477i 0 −2.47783 + 0.927568i 1.37012 2.47442i 0 0.753162 0.908543i
179.11 −1.04560 0.952218i 0 0.186562 + 1.99128i 0.501474i 0 2.46962 + 0.949199i 1.70106 2.25973i 0 −0.477513 + 0.524342i
179.12 −0.798016 + 1.16755i 0 −0.726340 1.86345i 1.15027i 0 0.422424 2.61181i 2.75529 + 0.639023i 0 −1.34300 0.917936i
179.13 −0.796149 1.16882i 0 −0.732292 + 1.86111i 4.26346i 0 1.55034 + 2.14393i 2.75833 0.625806i 0 4.98323 3.39435i
179.14 −0.718821 + 1.21791i 0 −0.966593 1.75091i 3.37581i 0 2.01379 + 1.71600i 2.82726 + 0.0813731i 0 −4.11142 2.42660i
179.15 −0.713767 1.22088i 0 −0.981073 + 1.74284i 2.17323i 0 0.191082 2.63884i 2.82805 0.0462147i 0 −2.65324 + 1.55118i
179.16 −0.608644 1.27654i 0 −1.25910 + 1.55392i 3.13896i 0 −0.228884 + 2.63583i 2.74998 + 0.661514i 0 −4.00700 + 1.91051i
179.17 −0.571896 + 1.29342i 0 −1.34587 1.47940i 1.08703i 0 −2.19009 + 1.48443i 2.68319 0.894712i 0 −1.40598 0.621665i
179.18 −0.410851 1.35322i 0 −1.66240 + 1.11194i 1.64200i 0 1.79804 1.94089i 2.18770 + 1.79275i 0 2.22198 0.674616i
179.19 −0.343960 + 1.37175i 0 −1.76338 0.943653i 3.68619i 0 2.28903 1.32677i 1.90099 2.09434i 0 5.05653 + 1.26790i
179.20 −0.177841 + 1.40299i 0 −1.93675 0.499017i 1.62292i 0 −1.09603 2.40805i 1.04455 2.62848i 0 −2.27693 0.288621i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 359.44 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.n odd 6 1 inner
252.o 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.o.a 88
3.b odd 2 1 252.2.o.a 88
4.b odd 2 1 inner 756.2.o.a 88
7.c even 3 1 756.2.bb.a 88
9.c even 3 1 252.2.bb.a yes 88
9.d odd 6 1 756.2.bb.a 88
12.b even 2 1 252.2.o.a 88
21.h odd 6 1 252.2.bb.a yes 88
28.g odd 6 1 756.2.bb.a 88
36.f odd 6 1 252.2.bb.a yes 88
36.h even 6 1 756.2.bb.a 88
63.g even 3 1 252.2.o.a 88
63.n odd 6 1 inner 756.2.o.a 88
84.n even 6 1 252.2.bb.a yes 88
252.o even 6 1 inner 756.2.o.a 88
252.bl odd 6 1 252.2.o.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.o.a 88 3.b odd 2 1
252.2.o.a 88 12.b even 2 1
252.2.o.a 88 63.g even 3 1
252.2.o.a 88 252.bl odd 6 1
252.2.bb.a yes 88 9.c even 3 1
252.2.bb.a yes 88 21.h odd 6 1
252.2.bb.a yes 88 36.f odd 6 1
252.2.bb.a yes 88 84.n even 6 1
756.2.o.a 88 1.a even 1 1 trivial
756.2.o.a 88 4.b odd 2 1 inner
756.2.o.a 88 63.n odd 6 1 inner
756.2.o.a 88 252.o even 6 1 inner
756.2.bb.a 88 7.c even 3 1
756.2.bb.a 88 9.d odd 6 1
756.2.bb.a 88 28.g odd 6 1
756.2.bb.a 88 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database