# Properties

 Label 441.2.g.h Level $441$ Weight $2$ Character orbit 441.g Analytic conductor $3.521$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} - 12q^{4} - 24q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} - 12q^{4} - 24q^{8} - 4q^{9} - 40q^{11} + 4q^{15} - 12q^{16} + 28q^{18} - 64q^{23} + 24q^{25} + 16q^{29} + 84q^{30} + 48q^{32} - 4q^{36} - 12q^{37} - 40q^{39} + 56q^{44} + 24q^{46} - 4q^{50} - 8q^{51} + 32q^{53} - 12q^{57} + 56q^{60} + 96q^{64} + 60q^{65} - 12q^{67} - 112q^{71} - 168q^{72} - 136q^{74} - 60q^{78} + 12q^{79} - 40q^{81} + 12q^{85} - 152q^{86} + 16q^{92} + 112q^{93} + 64q^{95} + 20q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −1.08816 + 1.88474i −1.18045 + 1.26749i −1.36816 2.36973i 1.26829 −1.10439 3.60407i 0 1.60248 −0.213085 2.99242i −1.38010 + 2.39040i
67.2 −1.08816 + 1.88474i 1.18045 1.26749i −1.36816 2.36973i −1.26829 1.10439 + 3.60407i 0 1.60248 −0.213085 2.99242i 1.38010 2.39040i
67.3 −0.649936 + 1.12572i −1.52504 0.821126i 0.155166 + 0.268756i 3.52584 1.91554 1.18309i 0 −3.00314 1.65150 + 2.50450i −2.29157 + 3.96912i
67.4 −0.649936 + 1.12572i 1.52504 + 0.821126i 0.155166 + 0.268756i −3.52584 −1.91554 + 1.18309i 0 −3.00314 1.65150 + 2.50450i 2.29157 3.96912i
67.5 −0.0341870 + 0.0592136i −1.15559 + 1.29020i 0.997662 + 1.72800i −2.66379 −0.0368912 0.112535i 0 −0.273176 −0.329225 2.98188i 0.0910670 0.157733i
67.6 −0.0341870 + 0.0592136i 1.15559 1.29020i 0.997662 + 1.72800i 2.66379 0.0368912 + 0.112535i 0 −0.273176 −0.329225 2.98188i −0.0910670 + 0.157733i
67.7 0.551407 0.955065i −0.454745 + 1.67129i 0.391901 + 0.678793i 0.105466 1.34544 + 1.35587i 0 3.07001 −2.58641 1.52002i 0.0581547 0.100727i
67.8 0.551407 0.955065i 0.454745 1.67129i 0.391901 + 0.678793i −0.105466 −1.34544 1.35587i 0 3.07001 −2.58641 1.52002i −0.0581547 + 0.100727i
67.9 0.863305 1.49529i −0.615283 1.61908i −0.490592 0.849731i −3.51231 −2.95217 0.477737i 0 1.75910 −2.24285 + 1.99239i −3.03220 + 5.25192i
67.10 0.863305 1.49529i 0.615283 + 1.61908i −0.490592 0.849731i 3.51231 2.95217 + 0.477737i 0 1.75910 −2.24285 + 1.99239i 3.03220 5.25192i
67.11 1.35757 2.35137i −1.69116 0.374116i −2.68597 4.65224i −1.58639 −3.17555 + 3.46867i 0 −9.15528 2.72007 + 1.26538i −2.15363 + 3.73020i
67.12 1.35757 2.35137i 1.69116 + 0.374116i −2.68597 4.65224i 1.58639 3.17555 3.46867i 0 −9.15528 2.72007 + 1.26538i 2.15363 3.73020i
79.1 −1.08816 1.88474i −1.18045 1.26749i −1.36816 + 2.36973i 1.26829 −1.10439 + 3.60407i 0 1.60248 −0.213085 + 2.99242i −1.38010 2.39040i
79.2 −1.08816 1.88474i 1.18045 + 1.26749i −1.36816 + 2.36973i −1.26829 1.10439 3.60407i 0 1.60248 −0.213085 + 2.99242i 1.38010 + 2.39040i
79.3 −0.649936 1.12572i −1.52504 + 0.821126i 0.155166 0.268756i 3.52584 1.91554 + 1.18309i 0 −3.00314 1.65150 2.50450i −2.29157 3.96912i
79.4 −0.649936 1.12572i 1.52504 0.821126i 0.155166 0.268756i −3.52584 −1.91554 1.18309i 0 −3.00314 1.65150 2.50450i 2.29157 + 3.96912i
79.5 −0.0341870 0.0592136i −1.15559 1.29020i 0.997662 1.72800i −2.66379 −0.0368912 + 0.112535i 0 −0.273176 −0.329225 + 2.98188i 0.0910670 + 0.157733i
79.6 −0.0341870 0.0592136i 1.15559 + 1.29020i 0.997662 1.72800i 2.66379 0.0368912 0.112535i 0 −0.273176 −0.329225 + 2.98188i −0.0910670 0.157733i
79.7 0.551407 + 0.955065i −0.454745 1.67129i 0.391901 0.678793i 0.105466 1.34544 1.35587i 0 3.07001 −2.58641 + 1.52002i 0.0581547 + 0.100727i
79.8 0.551407 + 0.955065i 0.454745 + 1.67129i 0.391901 0.678793i −0.105466 −1.34544 + 1.35587i 0 3.07001 −2.58641 + 1.52002i −0.0581547 0.100727i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.12 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.g even 3 1 inner
63.k 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.g.h 24
3.b odd 2 1 1323.2.g.h 24
7.b odd 2 1 inner 441.2.g.h 24
7.c even 3 1 441.2.f.h 24
7.c even 3 1 441.2.h.h 24
7.d odd 6 1 441.2.f.h 24
7.d odd 6 1 441.2.h.h 24
9.c even 3 1 441.2.h.h 24
9.d odd 6 1 1323.2.h.h 24
21.c even 2 1 1323.2.g.h 24
21.g even 6 1 1323.2.f.h 24
21.g even 6 1 1323.2.h.h 24
21.h odd 6 1 1323.2.f.h 24
21.h odd 6 1 1323.2.h.h 24
63.g even 3 1 inner 441.2.g.h 24
63.g even 3 1 3969.2.a.bh 12
63.h even 3 1 441.2.f.h 24
63.i even 6 1 1323.2.f.h 24
63.j odd 6 1 1323.2.f.h 24
63.k odd 6 1 inner 441.2.g.h 24
63.k odd 6 1 3969.2.a.bh 12
63.l odd 6 1 441.2.h.h 24
63.n odd 6 1 1323.2.g.h 24
63.n odd 6 1 3969.2.a.bi 12
63.o even 6 1 1323.2.h.h 24
63.s even 6 1 1323.2.g.h 24
63.s even 6 1 3969.2.a.bi 12
63.t odd 6 1 441.2.f.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.h 24 7.c even 3 1
441.2.f.h 24 7.d odd 6 1
441.2.f.h 24 63.h even 3 1
441.2.f.h 24 63.t odd 6 1
441.2.g.h 24 1.a even 1 1 trivial
441.2.g.h 24 7.b odd 2 1 inner
441.2.g.h 24 63.g even 3 1 inner
441.2.g.h 24 63.k odd 6 1 inner
441.2.h.h 24 7.c even 3 1
441.2.h.h 24 7.d odd 6 1
441.2.h.h 24 9.c even 3 1
441.2.h.h 24 63.l odd 6 1
1323.2.f.h 24 21.g even 6 1
1323.2.f.h 24 21.h odd 6 1
1323.2.f.h 24 63.i even 6 1
1323.2.f.h 24 63.j odd 6 1
1323.2.g.h 24 3.b odd 2 1
1323.2.g.h 24 21.c even 2 1
1323.2.g.h 24 63.n odd 6 1
1323.2.g.h 24 63.s even 6 1
1323.2.h.h 24 9.d odd 6 1
1323.2.h.h 24 21.g even 6 1
1323.2.h.h 24 21.h odd 6 1
1323.2.h.h 24 63.o even 6 1
3969.2.a.bh 12 63.g even 3 1
3969.2.a.bh 12 63.k odd 6 1
3969.2.a.bi 12 63.n odd 6 1
3969.2.a.bi 12 63.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{12} - \cdots$$ $$T_{5}^{12} - 36 T_{5}^{10} + 465 T_{5}^{8} - 2580 T_{5}^{6} + 5850 T_{5}^{4} - 4470 T_{5}^{2} + 49$$