# Properties

 Label 637.2.i.a Level $637$ Weight $2$ Character orbit 637.i Analytic conductor $5.086$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} - 16q^{8} - 16q^{9} + 20q^{11} - 44q^{15} - 24q^{16} + 8q^{18} - 8q^{22} + 16q^{29} - 8q^{32} + 16q^{37} + 12q^{39} + 84q^{44} - 24q^{46} + 88q^{50} + 24q^{53} + 40q^{57} - 52q^{58} - 32q^{60} + 16q^{65} - 32q^{67} - 36q^{71} - 44q^{72} - 24q^{74} - 176q^{78} + 64q^{79} - 32q^{81} - 84q^{85} - 84q^{86} + 48q^{92} - 12q^{93} - 24q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
489.1 −1.74842 1.74842i 2.04432i 4.11394i −1.15810 + 1.15810i −3.57432 + 3.57432i 0 3.69606 3.69606i −1.17922 4.04969
489.2 −1.74842 1.74842i 2.04432i 4.11394i 1.15810 1.15810i 3.57432 3.57432i 0 3.69606 3.69606i −1.17922 −4.04969
489.3 −1.12364 1.12364i 0.503603i 0.525123i 0.0563066 0.0563066i −0.565867 + 0.565867i 0 −1.65723 + 1.65723i 2.74638 −0.126536
489.4 −1.12364 1.12364i 0.503603i 0.525123i −0.0563066 + 0.0563066i 0.565867 0.565867i 0 −1.65723 + 1.65723i 2.74638 0.126536
489.5 −0.577557 0.577557i 3.00778i 1.33286i 2.22392 2.22392i −1.73717 + 1.73717i 0 −1.92491 + 1.92491i −6.04677 −2.56888
489.6 −0.577557 0.577557i 3.00778i 1.33286i −2.22392 + 2.22392i 1.73717 1.73717i 0 −1.92491 + 1.92491i −6.04677 2.56888
489.7 −0.508388 0.508388i 1.66432i 1.48308i 1.36989 1.36989i −0.846120 + 0.846120i 0 −1.77076 + 1.77076i 0.230041 −1.39287
489.8 −0.508388 0.508388i 1.66432i 1.48308i −1.36989 + 1.36989i 0.846120 0.846120i 0 −1.77076 + 1.77076i 0.230041 1.39287
489.9 0.546480 + 0.546480i 0.487133i 1.40272i 1.29040 1.29040i 0.266208 0.266208i 0 1.85952 1.85952i 2.76270 1.41035
489.10 0.546480 + 0.546480i 0.487133i 1.40272i −1.29040 + 1.29040i −0.266208 + 0.266208i 0 1.85952 1.85952i 2.76270 −1.41035
489.11 1.15715 + 1.15715i 1.98136i 0.678010i −2.02979 + 2.02979i 2.29273 2.29273i 0 1.52975 1.52975i −0.925772 −4.69755
489.12 1.15715 + 1.15715i 1.98136i 0.678010i 2.02979 2.02979i −2.29273 + 2.29273i 0 1.52975 1.52975i −0.925772 4.69755
489.13 1.34850 + 1.34850i 2.64781i 1.63690i 2.41949 2.41949i 3.57057 3.57057i 0 0.489646 0.489646i −4.01091 6.52537
489.14 1.34850 + 1.34850i 2.64781i 1.63690i −2.41949 + 2.41949i −3.57057 + 3.57057i 0 0.489646 0.489646i −4.01091 −6.52537
489.15 1.90587 + 1.90587i 0.759247i 5.26469i 1.78720 1.78720i 1.44703 1.44703i 0 −6.22207 + 6.22207i 2.42354 6.81236
489.16 1.90587 + 1.90587i 0.759247i 5.26469i −1.78720 + 1.78720i −1.44703 + 1.44703i 0 −6.22207 + 6.22207i 2.42354 −6.81236
538.1 −1.74842 + 1.74842i 2.04432i 4.11394i 1.15810 + 1.15810i 3.57432 + 3.57432i 0 3.69606 + 3.69606i −1.17922 −4.04969
538.2 −1.74842 + 1.74842i 2.04432i 4.11394i −1.15810 1.15810i −3.57432 3.57432i 0 3.69606 + 3.69606i −1.17922 4.04969
538.3 −1.12364 + 1.12364i 0.503603i 0.525123i −0.0563066 0.0563066i 0.565867 + 0.565867i 0 −1.65723 1.65723i 2.74638 0.126536
538.4 −1.12364 + 1.12364i 0.503603i 0.525123i 0.0563066 + 0.0563066i −0.565867 0.565867i 0 −1.65723 1.65723i 2.74638 −0.126536
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 538.16 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
13.d odd 4 1 inner
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.i.a 32
7.b odd 2 1 inner 637.2.i.a 32
7.c even 3 1 91.2.bb.a 32
7.c even 3 1 637.2.bc.b 32
7.d odd 6 1 91.2.bb.a 32
7.d odd 6 1 637.2.bc.b 32
13.d odd 4 1 inner 637.2.i.a 32
21.g even 6 1 819.2.fn.e 32
21.h odd 6 1 819.2.fn.e 32
91.i even 4 1 inner 637.2.i.a 32
91.z odd 12 1 91.2.bb.a 32
91.z odd 12 1 637.2.bc.b 32
91.bb even 12 1 91.2.bb.a 32
91.bb even 12 1 637.2.bc.b 32
273.cb odd 12 1 819.2.fn.e 32
273.cd even 12 1 819.2.fn.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bb.a 32 7.c even 3 1
91.2.bb.a 32 7.d odd 6 1
91.2.bb.a 32 91.z odd 12 1
91.2.bb.a 32 91.bb even 12 1
637.2.i.a 32 1.a even 1 1 trivial
637.2.i.a 32 7.b odd 2 1 inner
637.2.i.a 32 13.d odd 4 1 inner
637.2.i.a 32 91.i even 4 1 inner
637.2.bc.b 32 7.c even 3 1
637.2.bc.b 32 7.d odd 6 1
637.2.bc.b 32 91.z odd 12 1
637.2.bc.b 32 91.bb even 12 1
819.2.fn.e 32 21.g even 6 1
819.2.fn.e 32 21.h odd 6 1
819.2.fn.e 32 273.cb odd 12 1
819.2.fn.e 32 273.cd even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.