# Properties

 Label 756.2.bb.a Level 756 Weight 2 Character orbit 756.bb 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.bb (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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} + 6q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} + 6q^{5} + 2q^{10} - 4q^{13} + 18q^{14} - 2q^{16} + 6q^{20} - 6q^{22} + 30q^{25} - 6q^{26} + 24q^{29} - 4q^{34} - 4q^{37} + 45q^{38} - 4q^{40} + 12q^{41} - 57q^{44} - 6q^{46} - 2q^{49} - 9q^{50} - 7q^{52} + 24q^{56} + 5q^{58} - 4q^{61} - 8q^{64} + 12q^{68} - 27q^{70} - 4q^{73} - 51q^{74} - 6q^{76} + 30q^{77} + 87q^{80} - 4q^{82} - 14q^{85} - 81q^{86} + 9q^{88} + 60q^{89} - 24q^{92} - 18q^{94} - 4q^{97} - 57q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
611.1 −1.41415 + 0.0135638i 0 1.99963 0.0383626i 2.92354 1.68790i 0 0.479205 + 2.60199i −2.82726 + 0.0813731i 0 −4.11142 + 2.42660i
611.2 −1.41013 + 0.107328i 0 1.97696 0.302694i 0.996165 0.575136i 0 −2.47311 0.940076i −2.75529 + 0.639023i 0 −1.34300 + 0.917936i
611.3 −1.40608 0.151434i 0 1.95414 + 0.425857i 0.941391 0.543513i 0 2.38059 1.15446i −2.68319 0.894712i 0 −1.40598 + 0.621665i
611.4 −1.35995 0.387995i 0 1.69892 + 1.05531i −3.19234 + 1.84310i 0 −2.29354 + 1.31898i −1.90099 2.09434i 0 5.05653 1.26790i
611.5 −1.30394 0.547479i 0 1.40053 + 1.42776i 1.40549 0.811459i 0 −1.53742 2.15322i −1.04455 2.62848i 0 −2.27693 + 0.288621i
611.6 −1.28108 + 0.599020i 0 1.28235 1.53479i −0.426828 + 0.246429i 0 −1.23286 2.34095i −0.723425 + 2.73435i 0 0.399186 0.571375i
611.7 −1.26657 + 0.629119i 0 1.20842 1.59365i −2.83919 + 1.63921i 0 −0.591598 + 2.57876i −0.527954 + 2.77872i 0 2.56478 3.86236i
611.8 −1.24253 + 0.675360i 0 1.08778 1.67831i 1.44577 0.834716i 0 0.369619 + 2.61981i −0.218133 + 2.82000i 0 −1.23268 + 2.01358i
611.9 −1.12310 0.859445i 0 0.522707 + 1.93049i −2.12868 + 1.22899i 0 2.32866 1.25593i 1.07210 2.61737i 0 3.44698 + 0.449201i
611.10 −1.08038 0.912567i 0 0.334444 + 1.97184i −0.934343 + 0.539443i 0 −0.423599 + 2.61162i 1.43811 2.43554i 0 1.50172 + 0.269846i
611.11 −1.07292 0.921328i 0 0.302311 + 1.97702i 2.48793 1.43641i 0 −2.33229 + 1.24916i 1.49713 2.39971i 0 −3.99275 0.751051i
611.12 −0.966497 + 1.03242i 0 −0.131768 1.99565i 1.42201 0.820999i 0 2.57988 0.586703i 2.18770 + 1.79275i 0 −0.526757 + 2.26160i
611.13 −0.803010 1.16412i 0 −0.710349 + 1.86960i 1.33842 0.772734i 0 2.59837 + 0.498458i 2.74686 0.674378i 0 −1.97432 0.937562i
611.14 −0.801193 + 1.16537i 0 −0.716179 1.86737i −2.71842 + 1.56948i 0 −2.39714 1.11970i 2.74998 + 0.661514i 0 0.348953 4.42542i
611.15 −0.700426 + 1.22858i 0 −1.01881 1.72105i −1.88207 + 1.08661i 0 2.38085 + 1.15394i 2.82805 0.0462147i 0 −0.0167398 3.07336i
611.16 −0.692480 1.23307i 0 −1.04094 + 1.70776i −1.47575 + 0.852024i 0 0.540398 2.58997i 2.82662 + 0.100974i 0 2.07253 + 1.22970i
611.17 −0.614155 + 1.27390i 0 −1.24563 1.56474i 3.69227 2.13173i 0 −1.08153 2.41460i 2.75833 0.625806i 0 0.447980 + 6.01278i
611.18 −0.301845 + 1.38163i 0 −1.81778 0.834072i −0.434289 + 0.250737i 0 0.412779 2.61335i 1.70106 2.25973i 0 −0.215337 0.675709i
611.19 −0.298214 1.38241i 0 −1.82214 + 0.824510i −2.82413 + 1.63051i 0 −2.14986 1.54211i 1.68320 + 2.27307i 0 3.09624 + 3.41787i
611.20 −0.237258 + 1.39417i 0 −1.88742 0.661555i 0.722678 0.417239i 0 −2.04221 + 1.68208i 1.37012 2.47442i 0 0.410240 + 1.10653i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.44 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.j odd 6 1 inner
252.bb 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.bb.a 88
3.b odd 2 1 252.2.bb.a yes 88
4.b odd 2 1 inner 756.2.bb.a 88
7.c even 3 1 756.2.o.a 88
9.c even 3 1 252.2.o.a 88
9.d odd 6 1 756.2.o.a 88
12.b even 2 1 252.2.bb.a yes 88
21.h odd 6 1 252.2.o.a 88
28.g odd 6 1 756.2.o.a 88
36.f odd 6 1 252.2.o.a 88
36.h even 6 1 756.2.o.a 88
63.h even 3 1 252.2.bb.a yes 88
63.j odd 6 1 inner 756.2.bb.a 88
84.n even 6 1 252.2.o.a 88
252.u odd 6 1 252.2.bb.a yes 88
252.bb even 6 1 inner 756.2.bb.a 88

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

## 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