# Properties

 Label 441.2.bb.f Level $441$ Weight $2$ Character orbit 441.bb Analytic conductor $3.521$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.bb (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$6$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 14q^{4} - 28q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 14q^{4} - 28q^{7} + 42q^{13} - 26q^{16} + 28q^{19} + 8q^{22} - 24q^{25} - 28q^{28} + 28q^{31} + 28q^{34} - 106q^{37} + 70q^{40} - 2q^{43} + 60q^{46} + 28q^{49} + 70q^{52} - 42q^{55} + 56q^{58} + 112q^{61} + 38q^{64} - 36q^{67} - 14q^{70} - 28q^{73} + 56q^{76} + 62q^{79} - 70q^{82} - 100q^{85} - 262q^{88} - 154q^{91} - 294q^{94} + 56q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −2.14208 + 1.46044i 0 1.72492 4.39502i 0.478860 + 0.147709i 0 −1.61928 2.09235i 1.56997 + 6.87850i 0 −1.24148 + 0.382945i
37.2 −1.14923 + 0.783534i 0 −0.0238713 + 0.0608230i −3.53905 1.09165i 0 −2.60134 + 0.482718i −0.639241 2.80070i 0 4.92254 1.51840i
37.3 −0.715278 + 0.487668i 0 −0.456880 + 1.16411i 2.34510 + 0.723367i 0 2.39271 1.12913i −0.626178 2.74347i 0 −2.03016 + 0.626222i
37.4 0.715278 0.487668i 0 −0.456880 + 1.16411i −2.34510 0.723367i 0 2.39271 1.12913i 0.626178 + 2.74347i 0 −2.03016 + 0.626222i
37.5 1.14923 0.783534i 0 −0.0238713 + 0.0608230i 3.53905 + 1.09165i 0 −2.60134 + 0.482718i 0.639241 + 2.80070i 0 4.92254 1.51840i
37.6 2.14208 1.46044i 0 1.72492 4.39502i −0.478860 0.147709i 0 −1.61928 2.09235i −1.56997 6.87850i 0 −1.24148 + 0.382945i
46.1 −1.00262 + 2.55465i 0 −4.05486 3.76236i 1.62281 + 1.10641i 0 1.40782 2.24010i 8.73185 4.20504i 0 −4.45356 + 3.03638i
46.2 −0.576178 + 1.46808i 0 −0.357164 0.331400i −0.810468 0.552568i 0 −1.54377 2.14867i −2.14952 + 1.03515i 0 1.27819 0.871452i
46.3 −0.144768 + 0.368863i 0 1.35100 + 1.25355i −2.49239 1.69928i 0 2.63807 0.201526i −1.37200 + 0.660718i 0 0.987622 0.673350i
46.4 0.144768 0.368863i 0 1.35100 + 1.25355i 2.49239 + 1.69928i 0 2.63807 0.201526i 1.37200 0.660718i 0 0.987622 0.673350i
46.5 0.576178 1.46808i 0 −0.357164 0.331400i 0.810468 + 0.552568i 0 −1.54377 2.14867i 2.14952 1.03515i 0 1.27819 0.871452i
46.6 1.00262 2.55465i 0 −4.05486 3.76236i −1.62281 1.10641i 0 1.40782 2.24010i −8.73185 + 4.20504i 0 −4.45356 + 3.03638i
100.1 −2.49548 0.769754i 0 3.98243 + 2.71518i −1.69843 + 0.255997i 0 −2.62887 0.298395i −4.59158 5.75765i 0 4.43546 + 0.668538i
100.2 −2.02995 0.626155i 0 2.07613 + 1.41548i 0.321603 0.0484738i 0 2.57904 + 0.590375i −0.679131 0.851603i 0 −0.683188 0.102974i
100.3 −1.12270 0.346307i 0 −0.511951 0.349042i 1.86885 0.281684i 0 −0.330078 2.62508i 1.91896 + 2.40631i 0 −2.19571 0.330950i
100.4 1.12270 + 0.346307i 0 −0.511951 0.349042i −1.86885 + 0.281684i 0 −0.330078 2.62508i −1.91896 2.40631i 0 −2.19571 0.330950i
100.5 2.02995 + 0.626155i 0 2.07613 + 1.41548i −0.321603 + 0.0484738i 0 2.57904 + 0.590375i 0.679131 + 0.851603i 0 −0.683188 0.102974i
100.6 2.49548 + 0.769754i 0 3.98243 + 2.71518i 1.69843 0.255997i 0 −2.62887 0.298395i 4.59158 + 5.75765i 0 4.43546 + 0.668538i
109.1 −1.80148 + 1.67153i 0 0.301863 4.02808i −1.44677 + 3.68631i 0 −0.227334 + 2.63597i 3.12479 + 3.91837i 0 −3.55545 9.05915i
109.2 −0.970582 + 0.900568i 0 −0.0184545 + 0.246258i −0.843767 + 2.14988i 0 −0.921217 2.48019i −1.85490 2.32597i 0 −1.11717 2.84651i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.g even 21 1 inner
147.n odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.f 72
3.b odd 2 1 inner 441.2.bb.f 72
49.g even 21 1 inner 441.2.bb.f 72
147.n odd 42 1 inner 441.2.bb.f 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.bb.f 72 1.a even 1 1 trivial
441.2.bb.f 72 3.b odd 2 1 inner
441.2.bb.f 72 49.g even 21 1 inner
441.2.bb.f 72 147.n odd 42 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$85\!\cdots\!88$$$$T_{2}^{32} +$$$$27\!\cdots\!37$$$$T_{2}^{30} -$$$$27\!\cdots\!47$$$$T_{2}^{28} -$$$$76\!\cdots\!01$$$$T_{2}^{26} +$$$$38\!\cdots\!77$$$$T_{2}^{24} -$$$$87\!\cdots\!91$$$$T_{2}^{22} +$$$$12\!\cdots\!68$$$$T_{2}^{20} -$$$$13\!\cdots\!96$$$$T_{2}^{18} +$$$$78\!\cdots\!56$$$$T_{2}^{16} -$$$$67\!\cdots\!75$$$$T_{2}^{14} -$$$$28\!\cdots\!59$$$$T_{2}^{12} +$$$$20\!\cdots\!50$$$$T_{2}^{10} -$$$$33\!\cdots\!00$$$$T_{2}^{8} -$$$$35\!\cdots\!50$$$$T_{2}^{6} +$$$$16\!\cdots\!00$$$$T_{2}^{4} +$$$$50\!\cdots\!50$$$$T_{2}^{2} + 594248265625$$">$$T_{2}^{72} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.