# Properties

 Label 252.2.bb.a Level 252 Weight 2 Character orbit 252.bb Analytic conductor 2.012 Analytic rank 0 Dimension 88 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.bb (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} - 6q^{5} - 6q^{6} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} - 6q^{5} - 6q^{6} - 2q^{9} + 2q^{10} + 6q^{12} - 4q^{13} - 18q^{14} - 2q^{16} + 2q^{18} - 6q^{20} - 6q^{21} - 6q^{22} - 14q^{24} + 30q^{25} + 6q^{26} - 24q^{29} - 29q^{30} + 10q^{33} - 4q^{34} + 2q^{36} - 4q^{37} - 45q^{38} - 4q^{40} - 12q^{41} - 46q^{42} + 57q^{44} - 18q^{45} - 6q^{46} - 43q^{48} - 2q^{49} + 9q^{50} - 7q^{52} + 23q^{54} - 24q^{56} - 28q^{57} + 5q^{58} - 19q^{60} - 4q^{61} - 8q^{64} + 60q^{66} - 12q^{68} - 6q^{69} - 27q^{70} - 10q^{72} - 4q^{73} + 51q^{74} - 6q^{76} - 30q^{77} + 55q^{78} - 87q^{80} - 34q^{81} - 4q^{82} - 55q^{84} - 14q^{85} + 81q^{86} + 9q^{88} - 60q^{89} + 41q^{90} + 24q^{92} + 30q^{93} - 18q^{94} - 29q^{96} - 4q^{97} + 57q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41415 + 0.0135638i 1.41411 + 1.00015i 1.99963 0.0383626i −2.92354 1.68790i −2.01332 1.39518i −0.479205 + 2.60199i −2.82726 + 0.0813731i 0.999400 + 2.82864i 4.15721 + 2.34729i
11.2 −1.41013 + 0.107328i −1.49957 0.866773i 1.97696 0.302694i −0.996165 0.575136i 2.20762 + 1.06132i 2.47311 0.940076i −2.75529 + 0.639023i 1.49741 + 2.59957i 1.46646 + 0.704103i
11.3 −1.40608 0.151434i 0.377313 1.69045i 1.95414 + 0.425857i −0.941391 0.543513i −0.786525 + 2.31978i −2.38059 1.15446i −2.68319 0.894712i −2.71527 1.27566i 1.24137 + 0.906782i
11.4 −1.35995 0.387995i 0.672325 + 1.59624i 1.69892 + 1.05531i 3.19234 + 1.84310i −0.294994 2.43166i 2.29354 + 1.31898i −1.90099 2.09434i −2.09596 + 2.14638i −3.62630 3.74513i
11.5 −1.30394 0.547479i −1.06571 + 1.36538i 1.40053 + 1.42776i −1.40549 0.811459i 2.13714 1.19692i 1.53742 2.15322i −1.04455 2.62848i −0.728526 2.91020i 1.38842 + 1.82757i
11.6 −1.28108 + 0.599020i 1.73101 + 0.0599198i 1.28235 1.53479i 0.426828 + 0.246429i −2.25347 + 0.960151i 1.23286 2.34095i −0.723425 + 2.73435i 2.99282 + 0.207444i −0.694418 0.0600176i
11.7 −1.26657 + 0.629119i 0.555129 1.64068i 1.20842 1.59365i 2.83919 + 1.63921i 0.329071 + 2.42728i 0.591598 + 2.57876i −0.527954 + 2.77872i −2.38366 1.82158i −4.62729 0.289988i
11.8 −1.24253 + 0.675360i −1.31771 + 1.12412i 1.08778 1.67831i −1.44577 0.834716i 0.878106 2.28669i −0.369619 + 2.61981i −0.218133 + 2.82000i 0.472697 2.96253i 2.36015 + 0.0607467i
11.9 −1.12310 0.859445i 1.72883 0.105547i 0.522707 + 1.93049i 2.12868 + 1.22899i −2.03236 1.36730i −2.32866 1.25593i 1.07210 2.61737i 2.97772 0.364945i −1.33447 3.20977i
11.10 −1.08038 0.912567i −1.05609 1.37284i 0.334444 + 1.97184i 0.934343 + 0.539443i −0.111830 + 2.44694i 0.423599 + 2.61162i 1.43811 2.43554i −0.769364 + 2.89967i −0.517168 1.43545i
11.11 −1.07292 0.921328i 1.31640 1.12565i 0.302311 + 1.97702i −2.48793 1.43641i −2.44948 0.00509913i 2.33229 + 1.24916i 1.49713 2.39971i 0.465809 2.96362i 1.34595 + 3.83335i
11.12 −0.966497 + 1.03242i −1.48378 0.893531i −0.131768 1.99565i −1.42201 0.820999i 2.35656 0.668284i −2.57988 0.586703i 2.18770 + 1.79275i 1.40320 + 2.65161i 2.22198 0.674616i
11.13 −0.803010 1.16412i 0.130605 + 1.72712i −0.710349 + 1.86960i −1.33842 0.772734i 1.90570 1.53893i −2.59837 + 0.498458i 2.74686 0.674378i −2.96588 + 0.451141i 0.175206 + 2.17859i
11.14 −0.801193 + 1.16537i −1.41297 + 1.00176i −0.716179 1.86737i 2.71842 + 1.56948i −0.0353630 2.44923i 2.39714 1.11970i 2.74998 + 0.661514i 0.992951 2.83091i −4.00700 + 1.91051i
11.15 −0.700426 + 1.22858i 1.17274 + 1.27463i −1.01881 1.72105i 1.88207 + 1.08661i −2.38740 + 0.548016i −2.38085 + 1.15394i 2.82805 0.0462147i −0.249371 + 2.98962i −2.65324 + 1.55118i
11.16 −0.692480 1.23307i −1.73156 + 0.0411748i −1.04094 + 1.70776i 1.47575 + 0.852024i 1.24984 + 2.10663i −0.540398 2.58997i 2.82662 + 0.100974i 2.99661 0.142593i 0.0286827 2.40972i
11.17 −0.614155 + 1.27390i 0.368715 + 1.69235i −1.24563 1.56474i −3.69227 2.13173i −2.38233 0.569660i 1.08153 2.41460i 2.75833 0.625806i −2.72810 + 1.24799i 4.98323 3.39435i
11.18 −0.301845 + 1.38163i 0.331433 1.70004i −1.81778 0.834072i 0.434289 + 0.250737i 2.24878 + 0.971066i −0.412779 2.61335i 1.70106 2.25973i −2.78030 1.12690i −0.477513 + 0.524342i
11.19 −0.298214 1.38241i 0.829150 1.52069i −1.82214 + 0.824510i 2.82413 + 1.63051i −2.34949 0.692737i 2.14986 1.54211i 1.68320 + 2.27307i −1.62502 2.52177i 1.41185 4.39036i
11.20 −0.237258 + 1.39417i 1.62755 0.592510i −1.88742 0.661555i −0.722678 0.417239i 0.439910 + 2.40966i 2.04221 + 1.68208i 1.37012 2.47442i 2.29786 1.92868i 0.753162 0.908543i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.44 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 252.2.bb.a yes 88
3.b odd 2 1 756.2.bb.a 88
4.b odd 2 1 inner 252.2.bb.a yes 88
7.c even 3 1 252.2.o.a 88
9.c even 3 1 756.2.o.a 88
9.d odd 6 1 252.2.o.a 88
12.b even 2 1 756.2.bb.a 88
21.h odd 6 1 756.2.o.a 88
28.g odd 6 1 252.2.o.a 88
36.f odd 6 1 756.2.o.a 88
36.h even 6 1 252.2.o.a 88
63.h even 3 1 756.2.bb.a 88
63.j odd 6 1 inner 252.2.bb.a yes 88
84.n even 6 1 756.2.o.a 88
252.u odd 6 1 756.2.bb.a 88
252.bb even 6 1 inner 252.2.bb.a yes 88

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database