# Properties

 Label 441.2.i.d Level $441$ Weight $2$ Character orbit 441.i Analytic conductor $3.521$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{4} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 48q^{4} - 8q^{9} + 24q^{11} - 40q^{15} + 48q^{16} - 16q^{18} + 48q^{23} - 24q^{25} - 24q^{30} - 8q^{36} - 56q^{39} - 96q^{44} + 48q^{50} - 24q^{51} - 48q^{53} + 80q^{57} + 168q^{60} - 48q^{64} - 88q^{72} + 168q^{74} - 88q^{78} + 48q^{79} - 24q^{81} - 24q^{85} - 24q^{86} - 144q^{92} + 16q^{93} - 72q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1 2.70883i −0.572034 1.63486i −5.33776 −0.601464 1.04177i −4.42857 + 1.54954i 0 9.04141i −2.34556 + 1.87039i −2.82197 + 1.62926i
68.2 2.70883i 0.572034 + 1.63486i −5.33776 0.601464 + 1.04177i 4.42857 1.54954i 0 9.04141i −2.34556 + 1.87039i 2.82197 1.62926i
68.3 2.37274i −1.65098 0.523694i −3.62990 1.71774 + 2.97522i −1.24259 + 3.91735i 0 3.86732i 2.45149 + 1.72922i 7.05942 4.07576i
68.4 2.37274i 1.65098 + 0.523694i −3.62990 −1.71774 2.97522i 1.24259 3.91735i 0 3.86732i 2.45149 + 1.72922i −7.05942 + 4.07576i
68.5 1.48451i −0.995298 + 1.41753i −0.203760 −0.154215 0.267109i 2.10433 + 1.47753i 0 2.66653i −1.01876 2.82172i −0.396525 + 0.228934i
68.6 1.48451i 0.995298 1.41753i −0.203760 0.154215 + 0.267109i −2.10433 1.47753i 0 2.66653i −1.01876 2.82172i 0.396525 0.228934i
68.7 1.17820i −0.278055 1.70959i 0.611843 −2.16601 3.75164i −2.01424 + 0.327604i 0 3.07728i −2.84537 + 0.950717i −4.42019 + 2.55200i
68.8 1.17820i 0.278055 + 1.70959i 0.611843 2.16601 + 3.75164i 2.01424 0.327604i 0 3.07728i −2.84537 + 0.950717i 4.42019 2.55200i
68.9 0.981621i −1.73160 0.0395255i 1.03642 −0.940599 1.62916i −0.0387990 + 1.69978i 0 2.98061i 2.99688 + 0.136885i −1.59922 + 0.923312i
68.10 0.981621i 1.73160 + 0.0395255i 1.03642 0.940599 + 1.62916i 0.0387990 1.69978i 0 2.98061i 2.99688 + 0.136885i 1.59922 0.923312i
68.11 0.424201i −0.625041 + 1.61534i 1.82005 −1.80381 3.12430i 0.685229 + 0.265143i 0 1.62047i −2.21865 2.01931i −1.32533 + 0.765180i
68.12 0.424201i 0.625041 1.61534i 1.82005 1.80381 + 3.12430i −0.685229 0.265143i 0 1.62047i −2.21865 2.01931i 1.32533 0.765180i
68.13 0.122344i −0.937657 + 1.45630i 1.98503 0.264715 + 0.458500i −0.178170 0.114717i 0 0.487547i −1.24160 2.73101i −0.0560949 + 0.0323864i
68.14 0.122344i 0.937657 1.45630i 1.98503 −0.264715 0.458500i 0.178170 + 0.114717i 0 0.487547i −1.24160 2.73101i 0.0560949 0.0323864i
68.15 0.664297i −1.15747 1.28852i 1.55871 −0.0141520 0.0245119i 0.855956 0.768901i 0 2.36404i −0.320544 + 2.98283i 0.0162832 0.00940110i
68.16 0.664297i 1.15747 + 1.28852i 1.55871 0.0141520 + 0.0245119i −0.855956 + 0.768901i 0 2.36404i −0.320544 + 2.98283i −0.0162832 + 0.00940110i
68.17 1.83202i −1.55144 0.770089i −1.35631 −0.322784 0.559079i 1.41082 2.84227i 0 1.17925i 1.81393 + 2.38949i 1.02425 0.591348i
68.18 1.83202i 1.55144 + 0.770089i −1.35631 0.322784 + 0.559079i −1.41082 + 2.84227i 0 1.17925i 1.81393 + 2.38949i −1.02425 + 0.591348i
68.19 1.86894i −1.72324 0.174470i −1.49292 1.25287 + 2.17003i 0.326074 3.22063i 0 0.947692i 2.93912 + 0.601309i −4.05565 + 2.34153i
68.20 1.86894i 1.72324 + 0.174470i −1.49292 −1.25287 2.17003i −0.326074 + 3.22063i 0 0.947692i 2.93912 + 0.601309i 4.05565 2.34153i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 227.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.i.d 48
3.b odd 2 1 1323.2.i.d 48
7.b odd 2 1 inner 441.2.i.d 48
7.c even 3 1 441.2.o.e 48
7.c even 3 1 441.2.s.d 48
7.d odd 6 1 441.2.o.e 48
7.d odd 6 1 441.2.s.d 48
9.c even 3 1 1323.2.s.d 48
9.d odd 6 1 441.2.s.d 48
21.c even 2 1 1323.2.i.d 48
21.g even 6 1 1323.2.o.e 48
21.g even 6 1 1323.2.s.d 48
21.h odd 6 1 1323.2.o.e 48
21.h odd 6 1 1323.2.s.d 48
63.g even 3 1 1323.2.o.e 48
63.h even 3 1 1323.2.i.d 48
63.i even 6 1 inner 441.2.i.d 48
63.j odd 6 1 inner 441.2.i.d 48
63.k odd 6 1 1323.2.o.e 48
63.l odd 6 1 1323.2.s.d 48
63.n odd 6 1 441.2.o.e 48
63.o even 6 1 441.2.s.d 48
63.s even 6 1 441.2.o.e 48
63.t odd 6 1 1323.2.i.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.i.d 48 1.a even 1 1 trivial
441.2.i.d 48 7.b odd 2 1 inner
441.2.i.d 48 63.i even 6 1 inner
441.2.i.d 48 63.j odd 6 1 inner
441.2.o.e 48 7.c even 3 1
441.2.o.e 48 7.d odd 6 1
441.2.o.e 48 63.n odd 6 1
441.2.o.e 48 63.s even 6 1
441.2.s.d 48 7.c even 3 1
441.2.s.d 48 7.d odd 6 1
441.2.s.d 48 9.d odd 6 1
441.2.s.d 48 63.o even 6 1
1323.2.i.d 48 3.b odd 2 1
1323.2.i.d 48 21.c even 2 1
1323.2.i.d 48 63.h even 3 1
1323.2.i.d 48 63.t odd 6 1
1323.2.o.e 48 21.g even 6 1
1323.2.o.e 48 21.h odd 6 1
1323.2.o.e 48 63.g even 3 1
1323.2.o.e 48 63.k odd 6 1
1323.2.s.d 48 9.c even 3 1
1323.2.s.d 48 21.g even 6 1
1323.2.s.d 48 21.h odd 6 1
1323.2.s.d 48 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.