# Properties

 Label 756.2.bi.c Level 756 Weight 2 Character orbit 756.bi Analytic conductor 6.037 Analytic rank 0 Dimension 80 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 756.bi (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 6q^{2} - 2q^{4} + 24q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 6q^{2} - 2q^{4} + 24q^{8} - 10q^{14} - 18q^{16} - 14q^{22} + 32q^{25} + 28q^{28} - 8q^{29} + 16q^{32} - 16q^{37} + 84q^{44} + 24q^{46} - 24q^{49} + 12q^{50} + 48q^{53} - 32q^{56} - 14q^{58} - 8q^{64} + 40q^{65} - 22q^{70} - 64q^{74} + 12q^{77} + 40q^{85} + 52q^{86} + 6q^{88} - 30q^{92} - 20q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1 −1.40431 0.167104i 0 1.94415 + 0.469331i −1.39056 0.802839i 0 0.686390 2.55517i −2.65176 0.983961i 0 1.81861 + 1.35980i
307.2 −1.40431 0.167104i 0 1.94415 + 0.469331i 1.39056 + 0.802839i 0 2.55603 + 0.683151i −2.65176 0.983961i 0 −1.81861 1.35980i
307.3 −1.33314 + 0.471949i 0 1.55453 1.25835i −0.0753642 0.0435115i 0 −0.0425387 + 2.64541i −1.47853 + 2.41122i 0 0.121006 + 0.0224389i
307.4 −1.33314 + 0.471949i 0 1.55453 1.25835i 0.0753642 + 0.0435115i 0 −2.31226 1.28587i −1.47853 + 2.41122i 0 −0.121006 0.0224389i
307.5 −1.07464 + 0.919317i 0 0.309712 1.97587i −3.16954 1.82994i 0 2.31955 1.27266i 1.48363 + 2.40808i 0 5.08842 0.947288i
307.6 −1.07464 + 0.919317i 0 0.309712 1.97587i 3.16954 + 1.82994i 0 2.26193 1.37246i 1.48363 + 2.40808i 0 −5.08842 + 0.947288i
307.7 −1.05092 0.946342i 0 0.208875 + 1.98906i −0.416111 0.240242i 0 2.63514 0.236713i 1.66282 2.28802i 0 0.209950 + 0.646259i
307.8 −1.05092 0.946342i 0 0.208875 + 1.98906i 0.416111 + 0.240242i 0 1.52257 2.16374i 1.66282 2.28802i 0 −0.209950 0.646259i
307.9 −1.02930 0.969818i 0 0.118905 + 1.99646i −3.45303 1.99361i 0 −2.03435 1.69157i 1.81382 2.17027i 0 1.62076 + 5.40083i
307.10 −1.02930 0.969818i 0 0.118905 + 1.99646i 3.45303 + 1.99361i 0 0.447766 + 2.60759i 1.81382 2.17027i 0 −1.62076 5.40083i
307.11 −0.955367 + 1.04272i 0 −0.174547 1.99237i −1.82425 1.05323i 0 −0.655553 + 2.56325i 2.24425 + 1.72144i 0 2.84106 0.895969i
307.12 −0.955367 + 1.04272i 0 −0.174547 1.99237i 1.82425 + 1.05323i 0 −2.54762 0.713899i 2.24425 + 1.72144i 0 −2.84106 + 0.895969i
307.13 −0.648812 + 1.25660i 0 −1.15809 1.63059i −2.66647 1.53948i 0 −1.15806 2.37885i 2.80038 0.397302i 0 3.66455 2.35184i
307.14 −0.648812 + 1.25660i 0 −1.15809 1.63059i 2.66647 + 1.53948i 0 1.48111 + 2.19233i 2.80038 0.397302i 0 −3.66455 + 2.35184i
307.15 −0.325239 1.37631i 0 −1.78844 + 0.895257i −3.45303 1.99361i 0 2.03435 + 1.69157i 1.81382 + 2.17027i 0 −1.62076 + 5.40083i
307.16 −0.325239 1.37631i 0 −1.78844 + 0.895257i 3.45303 + 1.99361i 0 −0.447766 2.60759i 1.81382 + 2.17027i 0 1.62076 5.40083i
307.17 −0.294095 1.38330i 0 −1.82702 + 0.813640i −0.416111 0.240242i 0 −2.63514 + 0.236713i 1.66282 + 2.28802i 0 −0.209950 + 0.646259i
307.18 −0.294095 1.38330i 0 −1.82702 + 0.813640i 0.416111 + 0.240242i 0 −1.52257 + 2.16374i 1.66282 + 2.28802i 0 0.209950 0.646259i
307.19 −0.139426 + 1.40732i 0 −1.96112 0.392436i −1.35495 0.782280i 0 0.314165 + 2.62703i 0.825716 2.70522i 0 1.28984 1.79778i
307.20 −0.139426 + 1.40732i 0 −1.96112 0.392436i 1.35495 + 0.782280i 0 −2.11799 1.58559i 0.825716 2.70522i 0 −1.28984 + 1.79778i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.40 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
7.b odd 2 1 inner
9.c even 3 1 inner
28.d even 2 1 inner
36.f odd 6 1 inner
63.l odd 6 1 inner
252.bi 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.bi.c 80
3.b odd 2 1 252.2.bi.c 80
4.b odd 2 1 inner 756.2.bi.c 80
7.b odd 2 1 inner 756.2.bi.c 80
9.c even 3 1 inner 756.2.bi.c 80
9.d odd 6 1 252.2.bi.c 80
12.b even 2 1 252.2.bi.c 80
21.c even 2 1 252.2.bi.c 80
28.d even 2 1 inner 756.2.bi.c 80
36.f odd 6 1 inner 756.2.bi.c 80
36.h even 6 1 252.2.bi.c 80
63.l odd 6 1 inner 756.2.bi.c 80
63.o even 6 1 252.2.bi.c 80
84.h odd 2 1 252.2.bi.c 80
252.s odd 6 1 252.2.bi.c 80
252.bi even 6 1 inner 756.2.bi.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bi.c 80 3.b odd 2 1
252.2.bi.c 80 9.d odd 6 1
252.2.bi.c 80 12.b even 2 1
252.2.bi.c 80 21.c even 2 1
252.2.bi.c 80 36.h even 6 1
252.2.bi.c 80 63.o even 6 1
252.2.bi.c 80 84.h odd 2 1
252.2.bi.c 80 252.s odd 6 1
756.2.bi.c 80 1.a even 1 1 trivial
756.2.bi.c 80 4.b odd 2 1 inner
756.2.bi.c 80 7.b odd 2 1 inner
756.2.bi.c 80 9.c even 3 1 inner
756.2.bi.c 80 28.d even 2 1 inner
756.2.bi.c 80 36.f odd 6 1 inner
756.2.bi.c 80 63.l odd 6 1 inner
756.2.bi.c 80 252.bi even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(756, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database