# Properties

 Label 441.2.bb.d Level $441$ Weight $2$ Character orbit 441.bb Analytic conductor $3.521$ Analytic rank $0$ Dimension $48$ 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: $$48$$ Relative dimension: $$4$$ over $$\Q(\zeta_{21})$$ Twist minimal: no (minimal twist has level 49) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 13q^{2} - 9q^{4} + 14q^{5} - 14q^{7} + 20q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 13q^{2} - 9q^{4} + 14q^{5} - 14q^{7} + 20q^{8} - 14q^{10} + 3q^{11} - 14q^{13} - 21q^{14} - 3q^{16} + 7q^{17} + 21q^{19} - 14q^{20} - 20q^{22} - 15q^{23} - 4q^{25} + 28q^{28} - 12q^{29} + 35q^{31} - 45q^{32} + 70q^{34} + 15q^{37} + 28q^{38} - 42q^{40} + 42q^{41} - 30q^{43} + 50q^{44} - 78q^{46} - 21q^{47} - 70q^{49} - 40q^{50} - 70q^{52} - 11q^{53} - 7q^{55} + 28q^{56} + 16q^{58} + 28q^{59} + 7q^{61} + 28q^{62} - 32q^{64} - 14q^{65} + 11q^{67} - 77q^{68} + 70q^{70} - 19q^{71} + 7q^{73} - 34q^{74} + 119q^{76} - 7q^{77} + 15q^{79} - 70q^{80} - 14q^{82} - 26q^{85} + 33q^{86} - 77q^{88} + 14q^{89} + 84q^{91} + 38q^{92} + 14q^{94} + 61q^{95} + 161q^{98} + O(q^{100})$$

## 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 −1.52543 + 1.04002i 0 0.514606 1.31120i 1.32986 + 0.410207i 0 −1.73764 1.99515i −0.242977 1.06455i 0 −2.45523 + 0.757337i
37.2 −0.174575 + 0.119023i 0 −0.714372 + 1.82019i 3.75050 + 1.15688i 0 1.85388 + 1.88763i −0.185965 0.814768i 0 −0.792437 + 0.244435i
37.3 0.968018 0.659983i 0 −0.229202 + 0.583997i −3.03447 0.936009i 0 −2.50722 0.844891i 0.684966 + 3.00103i 0 −3.55517 + 1.09662i
37.4 2.09732 1.42993i 0 1.62338 4.13631i 2.38971 + 0.737129i 0 −1.40100 + 2.24437i −1.38019 6.04701i 0 6.06605 1.87113i
46.1 −0.940144 + 2.39545i 0 −3.38820 3.14379i 0.392378 + 0.267519i 0 −1.60453 + 2.10369i 6.07919 2.92758i 0 −1.00972 + 0.688415i
46.2 −0.0341744 + 0.0870750i 0 1.45969 + 1.35439i 2.09852 + 1.43075i 0 −0.301609 2.62850i −0.336373 + 0.161989i 0 −0.196298 + 0.133834i
46.3 0.354207 0.902506i 0 0.777050 + 0.720997i 0.605902 + 0.413097i 0 −0.310399 + 2.62748i 2.67297 1.28723i 0 0.587437 0.400508i
46.4 0.887059 2.26019i 0 −2.85548 2.64950i −1.29033 0.879733i 0 −2.25101 1.39030i −4.14619 + 1.99670i 0 −3.13296 + 2.13602i
100.1 −1.16258 0.358609i 0 −0.429482 0.292816i 2.56601 0.386764i 0 −0.339612 + 2.62386i 1.91142 + 2.39684i 0 −3.12189 0.470550i
100.2 −0.681025 0.210069i 0 −1.23281 0.840516i −2.65874 + 0.400741i 0 −0.518691 2.59441i 1.55172 + 1.94579i 0 1.89485 + 0.285603i
100.3 1.16438 + 0.359164i 0 −0.425690 0.290231i 0.478452 0.0721150i 0 1.90196 1.83918i −1.91089 2.39618i 0 0.583002 + 0.0878735i
100.4 2.50546 + 0.772833i 0 4.02760 + 2.74597i −2.04183 + 0.307757i 0 0.189789 + 2.63894i 4.69930 + 5.89274i 0 −5.35358 0.806922i
109.1 −1.51353 + 1.40435i 0 0.169112 2.25665i 0.0830372 0.211575i 0 1.07894 + 2.41576i 0.338532 + 0.424506i 0 0.171447 + 0.436839i
109.2 −0.168237 + 0.156102i 0 −0.145524 + 1.94188i −0.711168 + 1.81203i 0 −2.04196 1.68237i −0.564834 0.708279i 0 −0.163215 0.415865i
109.3 1.02480 0.950878i 0 −0.00340854 + 0.0454838i 1.11243 2.83442i 0 2.44418 1.01290i 1.78303 + 2.23585i 0 −1.55517 3.96251i
109.4 1.73170 1.60678i 0 0.267571 3.57049i 0.567937 1.44708i 0 −2.62161 0.356599i −2.32789 2.91908i 0 −1.34164 3.41845i
163.1 −0.940144 2.39545i 0 −3.38820 + 3.14379i 0.392378 0.267519i 0 −1.60453 2.10369i 6.07919 + 2.92758i 0 −1.00972 0.688415i
163.2 −0.0341744 0.0870750i 0 1.45969 1.35439i 2.09852 1.43075i 0 −0.301609 + 2.62850i −0.336373 0.161989i 0 −0.196298 0.133834i
163.3 0.354207 + 0.902506i 0 0.777050 0.720997i 0.605902 0.413097i 0 −0.310399 2.62748i 2.67297 + 1.28723i 0 0.587437 + 0.400508i
163.4 0.887059 + 2.26019i 0 −2.85548 + 2.64950i −1.29033 + 0.879733i 0 −2.25101 + 1.39030i −4.14619 1.99670i 0 −3.13296 2.13602i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.4 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.d 48
3.b odd 2 1 49.2.g.a 48
12.b even 2 1 784.2.bg.c 48
21.c even 2 1 343.2.g.g 48
21.g even 6 1 343.2.e.c 48
21.g even 6 1 343.2.g.h 48
21.h odd 6 1 343.2.e.d 48
21.h odd 6 1 343.2.g.i 48
49.g even 21 1 inner 441.2.bb.d 48
147.k even 14 1 343.2.g.h 48
147.l odd 14 1 343.2.g.i 48
147.n odd 42 1 49.2.g.a 48
147.n odd 42 1 343.2.e.d 48
147.n odd 42 1 2401.2.a.h 24
147.o even 42 1 343.2.e.c 48
147.o even 42 1 343.2.g.g 48
147.o even 42 1 2401.2.a.i 24
588.bb even 42 1 784.2.bg.c 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.g.a 48 3.b odd 2 1
49.2.g.a 48 147.n odd 42 1
343.2.e.c 48 21.g even 6 1
343.2.e.c 48 147.o even 42 1
343.2.e.d 48 21.h odd 6 1
343.2.e.d 48 147.n odd 42 1
343.2.g.g 48 21.c even 2 1
343.2.g.g 48 147.o even 42 1
343.2.g.h 48 21.g even 6 1
343.2.g.h 48 147.k even 14 1
343.2.g.i 48 21.h odd 6 1
343.2.g.i 48 147.l odd 14 1
441.2.bb.d 48 1.a even 1 1 trivial
441.2.bb.d 48 49.g even 21 1 inner
784.2.bg.c 48 12.b even 2 1
784.2.bg.c 48 588.bb even 42 1
2401.2.a.h 24 147.n odd 42 1
2401.2.a.i 24 147.o even 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.