# Properties

 Label 441.2.bb.e Level $441$ Weight $2$ Character orbit 441.bb Analytic conductor $3.521$ Analytic rank $0$ Dimension $60$ CM no Inner twists $2$

# 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: $$60$$ Relative dimension: $$5$$ over $$\Q(\zeta_{21})$$ Twist minimal: no (minimal twist has level 147) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q - q^{2} + 5q^{4} + 2q^{5} + 5q^{7} - 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q - q^{2} + 5q^{4} + 2q^{5} + 5q^{7} - 6q^{8} - 34q^{10} + 11q^{11} - 2q^{13} - 40q^{14} - 31q^{16} + 9q^{17} - 29q^{19} + 43q^{20} + 9q^{22} + 4q^{23} + 55q^{25} - 36q^{26} - 57q^{28} - 4q^{29} - 39q^{31} + 92q^{32} - 36q^{34} + 33q^{35} - 24q^{37} - 118q^{38} - 35q^{41} + 2q^{43} - 40q^{44} - 40q^{46} + 5q^{47} + 129q^{49} + 176q^{50} - 6q^{52} - 26q^{53} + 2q^{55} - 63q^{56} + 11q^{58} + 41q^{59} + 6q^{61} - 36q^{62} + 74q^{64} + 51q^{65} - 55q^{67} + 22q^{68} - 68q^{70} + 66q^{71} + 24q^{73} - 28q^{74} + 3q^{76} + 34q^{77} - 51q^{79} + 5q^{80} - 41q^{82} - 30q^{83} + 68q^{85} - 110q^{86} + 129q^{88} - 75q^{89} + 5q^{91} + 38q^{94} - 36q^{95} - 168q^{97} - 227q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −2.23975 + 1.52704i 0 1.95397 4.97862i 2.15929 + 0.666054i 0 1.59254 + 2.11277i 2.01973 + 8.84903i 0 −5.85337 + 1.80553i
37.2 −1.04710 + 0.713898i 0 −0.143921 + 0.366706i −1.23854 0.382039i 0 2.59083 + 0.536271i −0.675095 2.95778i 0 1.56961 0.484160i
37.3 −0.190250 + 0.129710i 0 −0.711312 + 1.81239i −1.13743 0.350851i 0 −2.33995 + 1.23477i −0.202234 0.886046i 0 0.261905 0.0807869i
37.4 0.788109 0.537324i 0 −0.398283 + 1.01481i 2.91381 + 0.898791i 0 −0.446646 2.60778i 0.655894 + 2.87366i 0 2.77934 0.857313i
37.5 1.86275 1.27000i 0 1.12626 2.86965i −3.53817 1.09138i 0 2.24157 1.40547i −0.543186 2.37985i 0 −7.97679 + 2.46051i
46.1 −0.820550 + 2.09073i 0 −2.23173 2.07075i 0.297708 + 0.202974i 0 2.33428 + 1.24544i 2.11349 1.01780i 0 −0.668648 + 0.455876i
46.2 −0.489645 + 1.24760i 0 0.149360 + 0.138586i 2.98135 + 2.03265i 0 −2.47781 0.927615i −2.66107 + 1.28150i 0 −3.99573 + 2.72425i
46.3 −0.317978 + 0.810194i 0 0.910799 + 0.845098i −2.75602 1.87902i 0 −2.42550 + 1.05686i −2.54264 + 1.22447i 0 2.39873 1.63543i
46.4 0.322730 0.822302i 0 0.894077 + 0.829582i −0.910959 0.621082i 0 1.47122 2.19898i 2.56248 1.23403i 0 −0.804711 + 0.548643i
46.5 0.940102 2.39534i 0 −3.38776 3.14338i 3.23329 + 2.20442i 0 2.49214 0.888404i −6.07754 + 2.92679i 0 8.31997 5.67246i
100.1 −2.27757 0.702538i 0 3.04130 + 2.07352i 3.42100 0.515632i 0 2.20464 1.46273i −2.49792 3.13230i 0 −8.15382 1.22899i
100.2 −1.64310 0.506829i 0 0.790427 + 0.538904i −1.86453 + 0.281032i 0 −2.42176 + 1.06541i 1.11855 + 1.40262i 0 3.20604 + 0.483233i
100.3 −0.380106 0.117247i 0 −1.52174 1.03751i −0.996754 + 0.150237i 0 2.64575 0.00303574i 0.952800 + 1.19477i 0 0.396487 + 0.0597608i
100.4 1.48554 + 0.458227i 0 0.344369 + 0.234787i −3.62814 + 0.546855i 0 −2.49763 0.872846i −1.53457 1.92429i 0 −5.64032 0.850142i
100.5 1.85967 + 0.573632i 0 1.47684 + 1.00689i 3.25272 0.490269i 0 −0.730570 + 2.54289i −0.257935 0.323440i 0 6.33022 + 0.954128i
109.1 −1.60276 + 1.48714i 0 0.207780 2.77264i −0.252155 + 0.642481i 0 −0.451208 2.60699i 1.06386 + 1.33404i 0 −0.551316 1.40473i
109.2 −0.502390 + 0.466150i 0 −0.114360 + 1.52603i 1.18268 3.01342i 0 −2.63978 + 0.177638i −1.50851 1.89161i 0 0.810538 + 2.06522i
109.3 −0.227517 + 0.211105i 0 −0.142261 + 1.89835i −1.53122 + 3.90148i 0 2.06200 + 1.65775i −0.755410 0.947254i 0 −0.475244 1.21090i
109.4 1.03880 0.963867i 0 0.000608887 0.00812503i 0.107830 0.274746i 0 −1.39076 + 2.25073i 1.75989 + 2.20683i 0 −0.152805 0.389340i
109.5 2.02691 1.88070i 0 0.421883 5.62963i −0.259960 + 0.662367i 0 2.63989 + 0.175969i −6.28460 7.88063i 0 0.718799 + 1.83147i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.e 60
3.b odd 2 1 147.2.m.b 60
49.g even 21 1 inner 441.2.bb.e 60
147.n odd 42 1 147.2.m.b 60
147.n odd 42 1 7203.2.a.n 30
147.o even 42 1 7203.2.a.m 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.b 60 3.b odd 2 1
147.2.m.b 60 147.n odd 42 1
441.2.bb.e 60 1.a even 1 1 trivial
441.2.bb.e 60 49.g even 21 1 inner
7203.2.a.m 30 147.o even 42 1
7203.2.a.n 30 147.n odd 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{60} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.