# Properties

 Label 1148.2.k.b Level $1148$ Weight $2$ Character orbit 1148.k Analytic conductor $9.167$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(i)$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 12q^{11} - 16q^{17} - 4q^{19} - 36q^{23} - 64q^{25} + 12q^{27} + 16q^{29} - 28q^{31} + 12q^{35} + 48q^{37} + 4q^{41} + 36q^{45} + 12q^{47} - 12q^{51} - 12q^{53} + 12q^{55} + 76q^{57} + 20q^{59} - 4q^{65} - 44q^{67} + 72q^{69} - 20q^{71} + 72q^{75} - 8q^{79} - 100q^{81} - 40q^{83} - 8q^{85} - 16q^{89} + 20q^{93} + 76q^{95} - 16q^{97} + 56q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1 0 −2.33489 2.33489i 0 3.90568i 0 −0.707107 0.707107i 0 7.90342i 0
337.2 0 −2.32681 2.32681i 0 4.24622i 0 0.707107 + 0.707107i 0 7.82813i 0
337.3 0 −1.89621 1.89621i 0 1.39357i 0 −0.707107 0.707107i 0 4.19121i 0
337.4 0 −1.64273 1.64273i 0 1.39147i 0 −0.707107 0.707107i 0 2.39713i 0
337.5 0 −1.38781 1.38781i 0 2.11167i 0 0.707107 + 0.707107i 0 0.852014i 0
337.6 0 −0.898855 0.898855i 0 3.17533i 0 −0.707107 0.707107i 0 1.38412i 0
337.7 0 −0.796852 0.796852i 0 1.53973i 0 0.707107 + 0.707107i 0 1.73005i 0
337.8 0 −0.350714 0.350714i 0 3.08423i 0 −0.707107 0.707107i 0 2.75400i 0
337.9 0 −0.197445 0.197445i 0 3.12585i 0 0.707107 + 0.707107i 0 2.92203i 0
337.10 0 0.147004 + 0.147004i 0 2.61619i 0 0.707107 + 0.707107i 0 2.95678i 0
337.11 0 0.178034 + 0.178034i 0 0.542323i 0 −0.707107 0.707107i 0 2.93661i 0
337.12 0 0.805830 + 0.805830i 0 0.251332i 0 0.707107 + 0.707107i 0 1.70128i 0
337.13 0 1.46058 + 1.46058i 0 2.07990i 0 −0.707107 0.707107i 0 1.26660i 0
337.14 0 1.62776 + 1.62776i 0 3.63967i 0 0.707107 + 0.707107i 0 2.29923i 0
337.15 0 1.65216 + 1.65216i 0 2.95400i 0 −0.707107 0.707107i 0 2.45924i 0
337.16 0 1.71130 + 1.71130i 0 2.46933i 0 −0.707107 0.707107i 0 2.85712i 0
337.17 0 1.86437 + 1.86437i 0 2.96253i 0 0.707107 + 0.707107i 0 3.95177i 0
337.18 0 2.38527 + 2.38527i 0 0.515487i 0 0.707107 + 0.707107i 0 8.37900i 0
729.1 0 −2.33489 + 2.33489i 0 3.90568i 0 −0.707107 + 0.707107i 0 7.90342i 0
729.2 0 −2.32681 + 2.32681i 0 4.24622i 0 0.707107 0.707107i 0 7.82813i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 729.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.k.b 36
41.c even 4 1 inner 1148.2.k.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.k.b 36 1.a even 1 1 trivial
1148.2.k.b 36 41.c even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1148, [\chi])$$.