# Properties

 Label 252.2.bj.b Level 252 Weight 2 Character orbit 252.bj Analytic conductor 2.012 Analytic rank 0 Dimension 84 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$42$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q + 2q^{2} - 2q^{4} + 6q^{5} - 16q^{8} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q + 2q^{2} - 2q^{4} + 6q^{5} - 16q^{8} + 4q^{9} - 18q^{10} - 6q^{12} + 18q^{13} + 14q^{14} + 14q^{16} + 6q^{17} - 10q^{18} - 24q^{20} + 4q^{21} + 6q^{22} - 6q^{24} + 16q^{25} - 30q^{26} - 4q^{28} + 10q^{29} + 11q^{30} - 18q^{32} - 18q^{33} - 24q^{34} - 38q^{36} + 2q^{37} + 33q^{38} + 6q^{40} + 6q^{41} - 38q^{42} - 13q^{44} - 54q^{45} + 10q^{46} + 9q^{48} - 28q^{49} - 17q^{50} - 27q^{52} - 2q^{53} - 39q^{54} + 58q^{56} + 6q^{57} - 13q^{58} - 31q^{60} - 8q^{64} - 100q^{65} + 30q^{66} - 18q^{68} - 18q^{69} - 19q^{70} + 26q^{72} + 30q^{73} - 23q^{74} - 2q^{77} + 15q^{78} + 3q^{80} - 32q^{81} - 18q^{82} + 23q^{84} - 50q^{85} - 9q^{86} + q^{88} - 102q^{89} - 39q^{90} + 28q^{92} - 36q^{93} + 63q^{96} + 6q^{97} + 21q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1 −1.41404 0.0221477i 0.799650 + 1.53641i 1.99902 + 0.0626353i −2.43207 + 1.40416i −1.09671 2.19026i −2.50476 0.852161i −2.82531 0.132842i −1.72112 + 2.45718i 3.47014 1.93167i
103.2 −1.41404 + 0.0221477i −0.799650 1.53641i 1.99902 0.0626353i −2.43207 + 1.40416i 1.16477 + 2.15484i 2.50476 + 0.852161i −2.82531 + 0.132842i −1.72112 + 2.45718i 3.40794 2.03940i
103.3 −1.40719 0.140755i 1.46735 0.920267i 1.96038 + 0.396138i 0.259273 0.149691i −2.19437 + 1.08846i 0.0899789 2.64422i −2.70287 0.833374i 1.30622 2.70070i −0.385916 + 0.174150i
103.4 −1.40719 + 0.140755i −1.46735 + 0.920267i 1.96038 0.396138i 0.259273 0.149691i 1.93531 1.50153i −0.0899789 + 2.64422i −2.70287 + 0.833374i 1.30622 2.70070i −0.343777 + 0.247138i
103.5 −1.33100 0.477940i 1.43852 + 0.964710i 1.54315 + 1.27228i 3.01564 1.74108i −1.45360 1.97156i 1.90425 + 1.83680i −1.44586 2.43094i 1.13867 + 2.77551i −4.84596 + 0.876089i
103.6 −1.33100 + 0.477940i −1.43852 0.964710i 1.54315 1.27228i 3.01564 1.74108i 2.37575 + 0.596507i −1.90425 1.83680i −1.44586 + 2.43094i 1.13867 + 2.77551i −3.18169 + 3.75868i
103.7 −1.16385 0.803396i −0.477968 1.66480i 0.709111 + 1.87007i 0.121150 0.0699460i −0.781205 + 2.32158i −2.50334 + 0.856310i 0.677105 2.74618i −2.54309 + 1.59144i −0.197195 0.0159245i
103.8 −1.16385 + 0.803396i 0.477968 + 1.66480i 0.709111 1.87007i 0.121150 0.0699460i −1.89378 1.55358i 2.50334 0.856310i 0.677105 + 2.74618i −2.54309 + 1.59144i −0.0848066 + 0.178738i
103.9 −1.15944 0.809751i −0.630196 + 1.61334i 0.688607 + 1.87772i 2.22429 1.28420i 2.03708 1.36027i −2.01495 1.71464i 0.722083 2.73470i −2.20571 2.03344i −3.61882 0.312174i
103.10 −1.15944 + 0.809751i 0.630196 1.61334i 0.688607 1.87772i 2.22429 1.28420i 0.575725 + 2.38087i 2.01495 + 1.71464i 0.722083 + 2.73470i −2.20571 2.03344i −1.53906 + 3.29008i
103.11 −1.05304 0.943979i 1.67837 0.427864i 0.217806 + 1.98810i −2.49760 + 1.44199i −2.17130 1.13379i 0.0846904 + 2.64440i 1.64737 2.29917i 2.63387 1.43623i 3.99129 + 0.839202i
103.12 −1.05304 + 0.943979i −1.67837 + 0.427864i 0.217806 1.98810i −2.49760 + 1.44199i 1.36351 2.03491i −0.0846904 2.64440i 1.64737 + 2.29917i 2.63387 1.43623i 1.26887 3.87616i
103.13 −0.615190 1.27340i −1.73108 0.0580485i −1.24308 + 1.56676i 1.66094 0.958945i 0.991022 + 2.24006i 0.708998 + 2.54898i 2.75984 + 0.619084i 2.99326 + 0.200973i −2.24291 1.52511i
103.14 −0.615190 + 1.27340i 1.73108 + 0.0580485i −1.24308 1.56676i 1.66094 0.958945i −1.13886 + 2.16864i −0.708998 2.54898i 2.75984 0.619084i 2.99326 + 0.200973i 0.199325 + 2.70497i
103.15 −0.412858 1.35261i 1.24372 + 1.20547i −1.65910 + 1.11687i 0.514884 0.297269i 1.11705 2.17995i 1.74201 1.99133i 2.19566 + 1.78300i 0.0936760 + 2.99854i −0.614662 0.573707i
103.16 −0.412858 + 1.35261i −1.24372 1.20547i −1.65910 1.11687i 0.514884 0.297269i 2.14401 1.18458i −1.74201 + 1.99133i 2.19566 1.78300i 0.0936760 + 2.99854i 0.189514 + 0.819166i
103.17 −0.379891 1.36223i −1.39755 1.02316i −1.71137 + 1.03500i −2.34739 + 1.35527i −0.862862 + 2.29248i −1.09995 2.40626i 2.06005 + 1.93809i 0.906301 + 2.85983i 2.73794 + 2.68284i
103.18 −0.379891 + 1.36223i 1.39755 + 1.02316i −1.71137 1.03500i −2.34739 + 1.35527i −1.92470 + 1.51511i 1.09995 + 2.40626i 2.06005 1.93809i 0.906301 + 2.85983i −0.954438 3.71255i
103.19 −0.171292 1.40380i 1.50942 0.849495i −1.94132 + 0.480920i 3.50837 2.02556i −1.45107 1.97342i −2.35939 + 1.19719i 1.00765 + 2.64285i 1.55672 2.56450i −3.44444 4.57810i
103.20 −0.171292 + 1.40380i −1.50942 + 0.849495i −1.94132 0.480920i 3.50837 2.02556i −0.933971 2.26444i 2.35939 1.19719i 1.00765 2.64285i 1.55672 2.56450i 2.24253 + 5.27202i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 115.42 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 252.2.bj.b yes 84
3.b odd 2 1 756.2.bj.b 84
4.b odd 2 1 inner 252.2.bj.b yes 84
7.d odd 6 1 252.2.n.b 84
9.c even 3 1 252.2.n.b 84
9.d odd 6 1 756.2.n.b 84
12.b even 2 1 756.2.bj.b 84
21.g even 6 1 756.2.n.b 84
28.f even 6 1 252.2.n.b 84
36.f odd 6 1 252.2.n.b 84
36.h even 6 1 756.2.n.b 84
63.i even 6 1 756.2.bj.b 84
63.t odd 6 1 inner 252.2.bj.b yes 84
84.j odd 6 1 756.2.n.b 84
252.r odd 6 1 756.2.bj.b 84
252.bj even 6 1 inner 252.2.bj.b yes 84

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{42} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database