# Properties

 Label 432.2.bj.a Level 432 Weight 2 Character orbit 432.bj Analytic conductor 3.450 Analytic rank 0 Dimension 840 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.bj (of order $$36$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$840$$ Relative dimension: $$70$$ over $$\Q(\zeta_{36})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$840q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 24q^{7} - 18q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$840q - 12q^{2} - 12q^{3} - 12q^{4} - 12q^{5} - 12q^{6} - 24q^{7} - 18q^{8} - 6q^{10} - 12q^{11} + 6q^{12} - 12q^{13} - 12q^{14} - 12q^{16} - 36q^{17} - 12q^{18} - 6q^{19} + 30q^{20} - 12q^{21} - 12q^{22} - 24q^{23} - 42q^{24} - 12q^{27} - 24q^{28} - 12q^{29} - 12q^{30} - 12q^{32} - 24q^{33} - 24q^{34} - 18q^{35} - 12q^{36} - 6q^{37} - 12q^{38} - 24q^{39} - 12q^{40} + 48q^{42} - 12q^{43} - 18q^{44} - 12q^{45} - 6q^{46} - 12q^{48} - 24q^{49} + 144q^{50} - 30q^{51} - 12q^{52} - 180q^{54} - 48q^{55} - 54q^{56} - 66q^{58} + 24q^{59} - 72q^{60} - 12q^{61} - 216q^{62} - 6q^{64} - 24q^{65} - 6q^{66} - 12q^{67} - 114q^{68} - 12q^{69} + 30q^{70} - 36q^{71} - 48q^{72} - 96q^{74} + 72q^{75} - 12q^{76} - 12q^{77} - 90q^{78} - 24q^{81} - 24q^{82} - 72q^{83} - 192q^{84} - 42q^{85} - 72q^{86} - 24q^{87} - 12q^{88} - 66q^{90} - 6q^{91} - 126q^{92} - 12q^{93} - 12q^{94} + 96q^{96} - 24q^{97} - 18q^{98} - 12q^{99} + O(q^{100})$$

## 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 −1.41347 0.0457693i 1.41081 + 1.00480i 1.99581 + 0.129387i −0.451047 + 0.210327i −1.94815 1.48483i 0.650509 3.68922i −2.81510 0.274232i 0.980759 + 2.83516i 0.647170 0.276647i
11.2 −1.41262 + 0.0671088i −1.64576 0.539886i 1.99099 0.189599i −2.14457 + 1.00003i 2.36106 + 0.652210i −0.446923 + 2.53462i −2.79979 + 0.401444i 2.41705 + 1.77705i 2.96236 1.55658i
11.3 −1.41099 0.0954482i 1.71137 0.266838i 1.98178 + 0.269353i −2.91453 + 1.35907i −2.44020 + 0.213158i −0.433865 + 2.46057i −2.77056 0.569211i 2.85760 0.913318i 4.24209 1.63944i
11.4 −1.40904 0.120812i −0.646554 + 1.60685i 1.97081 + 0.340458i 1.63314 0.761545i 1.10515 2.18601i −0.638985 + 3.62387i −2.73582 0.717817i −2.16394 2.07783i −2.39317 + 0.875748i
11.5 −1.40536 0.158024i −1.72785 + 0.120515i 1.95006 + 0.444159i 3.64812 1.70115i 2.44729 + 0.103674i 0.736424 4.17647i −2.67034 0.932357i 2.97095 0.416466i −5.39573 + 1.81423i
11.6 −1.36790 0.358938i 0.748873 1.56179i 1.74233 + 0.981986i 1.61065 0.751058i −1.58497 + 1.86758i −0.149029 + 0.845186i −2.03087 1.96865i −1.87838 2.33917i −2.47280 + 0.449253i
11.7 −1.36652 + 0.364177i −0.589257 1.62873i 1.73475 0.995311i −2.14350 + 0.999531i 1.39838 + 2.01110i 0.824762 4.67746i −2.00810 + 1.99187i −2.30555 + 1.91949i 2.56513 2.14649i
11.8 −1.34888 + 0.424869i 1.40837 + 1.00821i 1.63897 1.14620i 1.83680 0.856513i −2.32808 0.761591i −0.211132 + 1.19739i −1.72380 + 2.24243i 0.967010 + 2.83988i −2.11372 + 1.93573i
11.9 −1.33243 + 0.473947i 1.23188 1.21757i 1.55075 1.26300i 2.78882 1.30045i −1.06433 + 2.20617i 0.243644 1.38177i −1.46767 + 2.41784i 0.0350450 2.99980i −3.09957 + 3.05451i
11.10 −1.32255 0.500851i −1.23524 + 1.21416i 1.49830 + 1.32480i −2.38847 + 1.11376i 2.24179 0.987116i 0.330679 1.87537i −1.31805 2.50255i 0.0516452 2.99956i 3.71671 0.276743i
11.11 −1.25692 0.648191i −0.892926 1.48414i 1.15970 + 1.62945i 0.756449 0.352738i 0.160328 + 2.44424i −0.222648 + 1.26270i −0.401452 2.79979i −1.40537 + 2.65046i −1.17944 0.0469600i
11.12 −1.25580 + 0.650350i −0.416170 + 1.68131i 1.15409 1.63343i 0.706106 0.329263i −0.570812 2.38205i 0.241012 1.36684i −0.387011 + 2.80182i −2.65360 1.39942i −0.672595 + 0.872706i
11.13 −1.17422 + 0.788161i 0.237306 1.71572i 0.757603 1.85096i −0.814785 + 0.379940i 1.07361 + 2.20167i −0.728449 + 4.13124i 0.569257 + 2.77055i −2.88737 0.814300i 0.657285 1.08832i
11.14 −1.12355 0.858857i 0.837200 + 1.51628i 0.524731 + 1.92994i −1.42689 + 0.665372i 0.361629 2.42265i −0.255884 + 1.45119i 1.06798 2.61905i −1.59819 + 2.53885i 2.17465 + 0.477919i
11.15 −1.11141 + 0.874510i −1.49998 0.866058i 0.470465 1.94388i 2.31389 1.07899i 2.42447 0.349202i −0.254246 + 1.44190i 1.17706 + 2.57187i 1.49989 + 2.59814i −1.62810 + 3.22272i
11.16 −1.09183 0.898834i 0.592090 1.62771i 0.384194 + 1.96275i −2.62704 + 1.22501i −2.10950 + 1.24499i 0.576268 3.26818i 1.34471 2.48832i −2.29886 1.92750i 3.96937 + 1.02377i
11.17 −1.09036 + 0.900614i −1.63814 + 0.562587i 0.377790 1.96399i −1.73541 + 0.809237i 1.27949 2.08875i 0.446922 2.53462i 1.35687 + 2.48171i 2.36699 1.84319i 1.16342 2.44530i
11.18 −1.03330 + 0.965550i 1.61462 0.626890i 0.135428 1.99541i −2.00571 + 0.935279i −1.06310 + 2.20677i 0.295687 1.67692i 1.78673 + 2.19262i 2.21402 2.02438i 1.16945 2.90304i
11.19 −0.956562 1.04163i −1.71601 + 0.235149i −0.169980 + 1.99276i 1.98476 0.925510i 1.88641 + 1.56252i −0.864532 + 4.90300i 2.23832 1.72915i 2.88941 0.807036i −2.86258 1.18208i
11.20 −0.948532 1.04895i 1.73143 0.0462528i −0.200575 + 1.98992i 1.65781 0.773051i −1.69084 1.77231i 0.761547 4.31895i 2.27757 1.67711i 2.99572 0.160167i −2.38338 1.00569i
See next 80 embeddings (of 840 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.70 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
27.f odd 18 1 inner
432.bj even 36 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.bj.a 840
16.f odd 4 1 inner 432.2.bj.a 840
27.f odd 18 1 inner 432.2.bj.a 840
432.bj even 36 1 inner 432.2.bj.a 840

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.bj.a 840 1.a even 1 1 trivial
432.2.bj.a 840 16.f odd 4 1 inner
432.2.bj.a 840 27.f odd 18 1 inner
432.2.bj.a 840 432.bj even 36 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database