# Properties

 Label 441.2.h.h Level $441$ Weight $2$ Character orbit 441.h 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.h (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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} + 24q^{4} - 24q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{2} + 24q^{4} - 24q^{8} - 4q^{9} + 20q^{11} + 4q^{15} + 24q^{16} - 32q^{18} + 32q^{23} - 12q^{25} + 16q^{29} - 84q^{30} - 96q^{32} - 4q^{36} - 12q^{37} + 8q^{39} + 56q^{44} + 24q^{46} - 4q^{50} + 64q^{51} + 32q^{53} - 12q^{57} + 32q^{60} + 96q^{64} - 120q^{65} + 24q^{67} - 112q^{71} + 68q^{74} - 60q^{78} - 24q^{79} - 40q^{81} + 12q^{85} + 76q^{86} + 16q^{92} - 32q^{93} - 128q^{95} + 20q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
214.1 −2.71513 −1.16958 1.27753i 5.37195 −0.793197 + 1.37386i 3.17555 + 3.46867i 0 −9.15528 −0.264183 + 2.98835i 2.15363 3.73020i
214.2 −2.71513 1.16958 + 1.27753i 5.37195 0.793197 1.37386i −3.17555 3.46867i 0 −9.15528 −0.264183 + 2.98835i −2.15363 + 3.73020i
214.3 −1.72661 −1.70981 + 0.276691i 0.981184 −1.75616 + 3.04175i 2.95217 0.477737i 0 1.75910 2.84688 0.946176i 3.03220 5.25192i
214.4 −1.72661 1.70981 0.276691i 0.981184 1.75616 3.04175i −2.95217 + 0.477737i 0 1.75910 2.84688 0.946176i −3.03220 + 5.25192i
214.5 −1.10281 −1.22001 + 1.22947i −0.783802 −0.0527330 + 0.0913363i 1.34544 1.35587i 0 3.07001 −0.0231690 2.99991i 0.0581547 0.100727i
214.6 −1.10281 1.22001 1.22947i −0.783802 0.0527330 0.0913363i −1.34544 + 1.35587i 0 3.07001 −0.0231690 2.99991i −0.0581547 + 0.100727i
214.7 0.0683740 −0.539550 + 1.64587i −1.99532 1.33190 2.30691i −0.0368912 + 0.112535i 0 −0.273176 −2.41777 1.77606i 0.0910670 0.157733i
214.8 0.0683740 0.539550 1.64587i −1.99532 −1.33190 + 2.30691i 0.0368912 0.112535i 0 −0.273176 −2.41777 1.77606i −0.0910670 + 0.157733i
214.9 1.29987 −1.47364 0.910162i −0.310333 1.76292 3.05347i −1.91554 1.18309i 0 −3.00314 1.34321 + 2.68250i 2.29157 3.96912i
214.10 1.29987 1.47364 + 0.910162i −0.310333 −1.76292 + 3.05347i 1.91554 + 1.18309i 0 −3.00314 1.34321 + 2.68250i −2.29157 + 3.96912i
214.11 2.17631 −0.507459 + 1.65605i 2.73633 −0.634145 + 1.09837i −1.10439 + 3.60407i 0 1.60248 −2.48497 1.68075i −1.38010 + 2.39040i
214.12 2.17631 0.507459 1.65605i 2.73633 0.634145 1.09837i 1.10439 3.60407i 0 1.60248 −2.48497 1.68075i 1.38010 2.39040i
373.1 −2.71513 −1.16958 + 1.27753i 5.37195 −0.793197 1.37386i 3.17555 3.46867i 0 −9.15528 −0.264183 2.98835i 2.15363 + 3.73020i
373.2 −2.71513 1.16958 1.27753i 5.37195 0.793197 + 1.37386i −3.17555 + 3.46867i 0 −9.15528 −0.264183 2.98835i −2.15363 3.73020i
373.3 −1.72661 −1.70981 0.276691i 0.981184 −1.75616 3.04175i 2.95217 + 0.477737i 0 1.75910 2.84688 + 0.946176i 3.03220 + 5.25192i
373.4 −1.72661 1.70981 + 0.276691i 0.981184 1.75616 + 3.04175i −2.95217 0.477737i 0 1.75910 2.84688 + 0.946176i −3.03220 5.25192i
373.5 −1.10281 −1.22001 1.22947i −0.783802 −0.0527330 0.0913363i 1.34544 + 1.35587i 0 3.07001 −0.0231690 + 2.99991i 0.0581547 + 0.100727i
373.6 −1.10281 1.22001 + 1.22947i −0.783802 0.0527330 + 0.0913363i −1.34544 1.35587i 0 3.07001 −0.0231690 + 2.99991i −0.0581547 0.100727i
373.7 0.0683740 −0.539550 1.64587i −1.99532 1.33190 + 2.30691i −0.0368912 0.112535i 0 −0.273176 −2.41777 + 1.77606i 0.0910670 + 0.157733i
373.8 0.0683740 0.539550 + 1.64587i −1.99532 −1.33190 2.30691i 0.0368912 + 0.112535i 0 −0.273176 −2.41777 + 1.77606i −0.0910670 0.157733i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.12 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.h even 3 1 inner
63.t 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.h.h 24
3.b odd 2 1 1323.2.h.h 24
7.b odd 2 1 inner 441.2.h.h 24
7.c even 3 1 441.2.f.h 24
7.c even 3 1 441.2.g.h 24
7.d odd 6 1 441.2.f.h 24
7.d odd 6 1 441.2.g.h 24
9.c even 3 1 441.2.g.h 24
9.d odd 6 1 1323.2.g.h 24
21.c even 2 1 1323.2.h.h 24
21.g even 6 1 1323.2.f.h 24
21.g even 6 1 1323.2.g.h 24
21.h odd 6 1 1323.2.f.h 24
21.h odd 6 1 1323.2.g.h 24
63.g even 3 1 441.2.f.h 24
63.h even 3 1 inner 441.2.h.h 24
63.h even 3 1 3969.2.a.bh 12
63.i even 6 1 1323.2.h.h 24
63.i even 6 1 3969.2.a.bi 12
63.j odd 6 1 1323.2.h.h 24
63.j odd 6 1 3969.2.a.bi 12
63.k odd 6 1 441.2.f.h 24
63.l odd 6 1 441.2.g.h 24
63.n odd 6 1 1323.2.f.h 24
63.o even 6 1 1323.2.g.h 24
63.s even 6 1 1323.2.f.h 24
63.t odd 6 1 inner 441.2.h.h 24
63.t odd 6 1 3969.2.a.bh 12

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.g even 3 1
441.2.f.h 24 63.k odd 6 1
441.2.g.h 24 7.c even 3 1
441.2.g.h 24 7.d odd 6 1
441.2.g.h 24 9.c even 3 1
441.2.g.h 24 63.l odd 6 1
441.2.h.h 24 1.a even 1 1 trivial
441.2.h.h 24 7.b odd 2 1 inner
441.2.h.h 24 63.h even 3 1 inner
441.2.h.h 24 63.t odd 6 1 inner
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.n odd 6 1
1323.2.f.h 24 63.s even 6 1
1323.2.g.h 24 9.d odd 6 1
1323.2.g.h 24 21.g even 6 1
1323.2.g.h 24 21.h odd 6 1
1323.2.g.h 24 63.o even 6 1
1323.2.h.h 24 3.b odd 2 1
1323.2.h.h 24 21.c even 2 1
1323.2.h.h 24 63.i even 6 1
1323.2.h.h 24 63.j odd 6 1
3969.2.a.bh 12 63.h even 3 1
3969.2.a.bh 12 63.t odd 6 1
3969.2.a.bi 12 63.i even 6 1
3969.2.a.bi 12 63.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{6} + 2 T_{2}^{5} - 7 T_{2}^{4} - 12 T_{2}^{3} + 10 T_{2}^{2} + 14 T_{2} - 1$$ $$T_{5}^{24} + \cdots$$