# Properties

 Label 441.2.bb.c 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 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + q^{2} + 3q^{4} - 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + q^{2} + 3q^{4} - 6q^{8} + 30q^{10} + 9q^{11} + 42q^{14} + 29q^{16} + 5q^{17} - 26q^{19} + 5q^{20} + q^{22} + 4q^{23} - 56q^{25} + 62q^{26} + 7q^{28} - 12q^{29} - 36q^{31} + 14q^{32} - 76q^{34} - 7q^{35} - 20q^{37} + 60q^{38} + 28q^{40} - 41q^{41} + 12q^{43} - 44q^{44} + 16q^{46} - 13q^{47} - 84q^{49} + 12q^{50} + 92q^{52} - 14q^{53} - 38q^{55} - 105q^{56} + 3q^{58} - 57q^{59} + 11q^{61} + 16q^{62} - 110q^{64} - 21q^{65} + 34q^{67} - 22q^{68} - 6q^{71} - 69q^{73} + 90q^{74} - 49q^{76} + 34q^{79} - 55q^{80} + 91q^{82} - 28q^{83} - 44q^{85} + 26q^{86} + 45q^{88} - 11q^{89} - 84q^{92} + 90q^{94} - 150q^{95} + 124q^{97} - 119q^{98} + O(q^{100})$$

## 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 −1.65175 + 1.12614i 0 0.729391 1.85846i 0.400180 + 0.123439i 0 −1.90889 + 1.83197i −0.00157168 0.00688598i 0 −0.800006 + 0.246769i
37.2 −0.449603 + 0.306534i 0 −0.622503 + 1.58611i −1.43317 0.442074i 0 0.881776 2.49449i −0.448490 1.96496i 0 0.779867 0.240557i
37.3 1.08067 0.736786i 0 −0.105696 + 0.269309i −1.01467 0.312983i 0 0.746631 + 2.53822i 0.666286 + 2.91919i 0 −1.32712 + 0.409362i
37.4 1.84692 1.25921i 0 1.09483 2.78958i 1.49226 + 0.460300i 0 1.29066 2.30959i −0.495786 2.17218i 0 3.33570 1.02893i
46.1 −0.687295 + 1.75120i 0 −1.12822 1.04683i −2.01668 1.37495i 0 −0.492953 2.59942i −0.781251 + 0.376230i 0 3.79385 2.58660i
46.2 −0.293772 + 0.748519i 0 0.992125 + 0.920558i 1.11822 + 0.762387i 0 1.80149 + 1.93768i −2.42946 + 1.16997i 0 −0.899162 + 0.613038i
46.3 0.633632 1.61447i 0 −0.738910 0.685609i −2.02525 1.38079i 0 2.55966 0.669438i 1.55011 0.746495i 0 −3.51251 + 2.39479i
46.4 0.712776 1.81612i 0 −1.32415 1.22863i 2.50104 + 1.70518i 0 −1.59223 + 2.11301i 0.340382 0.163920i 0 4.87950 3.32679i
100.1 −2.24916 0.693774i 0 2.92492 + 1.99418i 0.982858 0.148142i 0 −2.07146 1.64592i −2.26005 2.83402i 0 −2.31338 0.348686i
100.2 0.393224 + 0.121294i 0 −1.51256 1.03125i 1.73113 0.260925i 0 −2.49962 + 0.867115i −0.982833 1.23243i 0 0.712370 + 0.107372i
100.3 0.742928 + 0.229163i 0 −1.15305 0.786137i −1.96019 + 0.295451i 0 1.38027 + 2.25718i −1.64597 2.06398i 0 −1.52399 0.229704i
100.4 2.06858 + 0.638073i 0 2.21941 + 1.51317i 2.32995 0.351184i 0 0.990624 2.45330i 0.926114 + 1.16131i 0 5.04378 + 0.760227i
109.1 −1.90236 + 1.76513i 0 0.353823 4.72145i 1.29324 3.29513i 0 −2.54955 + 0.706959i 4.42482 + 5.54855i 0 3.35613 + 8.55127i
109.2 −0.615612 + 0.571205i 0 −0.0967566 + 1.29113i 0.568152 1.44763i 0 1.56778 2.13121i −1.72514 2.16325i 0 0.477130 + 1.21571i
109.3 0.492413 0.456893i 0 −0.115740 + 1.54445i 0.137164 0.349488i 0 0.0148404 + 2.64571i 1.48629 + 1.86375i 0 −0.0921371 0.234762i
109.4 1.29251 1.19927i 0 0.0828637 1.10574i −1.35494 + 3.45233i 0 1.47509 2.19638i 0.979680 + 1.22848i 0 2.38902 + 6.08712i
163.1 −0.687295 1.75120i 0 −1.12822 + 1.04683i −2.01668 + 1.37495i 0 −0.492953 + 2.59942i −0.781251 0.376230i 0 3.79385 + 2.58660i
163.2 −0.293772 0.748519i 0 0.992125 0.920558i 1.11822 0.762387i 0 1.80149 1.93768i −2.42946 1.16997i 0 −0.899162 0.613038i
163.3 0.633632 + 1.61447i 0 −0.738910 + 0.685609i −2.02525 + 1.38079i 0 2.55966 + 0.669438i 1.55011 + 0.746495i 0 −3.51251 2.39479i
163.4 0.712776 + 1.81612i 0 −1.32415 + 1.22863i 2.50104 1.70518i 0 −1.59223 2.11301i 0.340382 + 0.163920i 0 4.87950 + 3.32679i
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.c 48
3.b odd 2 1 147.2.m.a 48
49.g even 21 1 inner 441.2.bb.c 48
147.n odd 42 1 147.2.m.a 48
147.n odd 42 1 7203.2.a.i 24
147.o even 42 1 7203.2.a.k 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.a 48 3.b odd 2 1
147.2.m.a 48 147.n odd 42 1
441.2.bb.c 48 1.a even 1 1 trivial
441.2.bb.c 48 49.g even 21 1 inner
7203.2.a.i 24 147.n odd 42 1
7203.2.a.k 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 $$S_{2}^{\mathrm{new}}(441, [\chi])$$.