# Properties

 Label 1368.2.e.g Level $1368$ Weight $2$ Character orbit 1368.e Analytic conductor $10.924$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1368.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.9235349965$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: no (minimal twist has level 456) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40 q - 4 q^{4} + 4 q^{16} + 8 q^{19} - 32 q^{20} - 40 q^{25} - 40 q^{26} - 8 q^{28} + 48 q^{35} + 8 q^{44} - 56 q^{49} + 16 q^{58} - 40 q^{62} + 68 q^{64} + 88 q^{68} - 16 q^{73} + 40 q^{74} - 12 q^{76} + 32 q^{80} - 64 q^{82} - 80 q^{83} + 48 q^{92} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1 −1.41195 0.0800606i 0 1.98718 + 0.226082i 1.12499i 0 4.22432i −2.78769 0.478311i 0 −0.0900673 + 1.58842i
379.2 −1.41195 + 0.0800606i 0 1.98718 0.226082i 1.12499i 0 4.22432i −2.78769 + 0.478311i 0 −0.0900673 1.58842i
379.3 −1.38840 0.268989i 0 1.85529 + 0.746926i 3.66131i 0 0.477575i −2.37496 1.53608i 0 0.984850 5.08335i
379.4 −1.38840 + 0.268989i 0 1.85529 0.746926i 3.66131i 0 0.477575i −2.37496 + 1.53608i 0 0.984850 + 5.08335i
379.5 −1.33983 0.452615i 0 1.59028 + 1.21285i 0.393257i 0 1.35370i −1.58175 2.34480i 0 −0.177994 + 0.526897i
379.6 −1.33983 + 0.452615i 0 1.59028 1.21285i 0.393257i 0 1.35370i −1.58175 + 2.34480i 0 −0.177994 0.526897i
379.7 −1.10595 0.881408i 0 0.446239 + 1.94958i 0.946513i 0 1.89799i 1.22486 2.54945i 0 0.834264 1.04679i
379.8 −1.10595 + 0.881408i 0 0.446239 1.94958i 0.946513i 0 1.89799i 1.22486 + 2.54945i 0 0.834264 + 1.04679i
379.9 −0.939794 1.05678i 0 −0.233575 + 1.98631i 1.70737i 0 4.15645i 2.31861 1.61989i 0 1.80431 1.60457i
379.10 −0.939794 + 1.05678i 0 −0.233575 1.98631i 1.70737i 0 4.15645i 2.31861 + 1.61989i 0 1.80431 + 1.60457i
379.11 −0.906554 1.08543i 0 −0.356321 + 1.96800i 3.19828i 0 0.607439i 2.45916 1.39734i 0 −3.47151 + 2.89941i
379.12 −0.906554 + 1.08543i 0 −0.356321 1.96800i 3.19828i 0 0.607439i 2.45916 + 1.39734i 0 −3.47151 2.89941i
379.13 −0.612280 1.27480i 0 −1.25023 + 1.56107i 4.23483i 0 3.84958i 2.75554 + 0.637979i 0 5.39856 2.59290i
379.14 −0.612280 + 1.27480i 0 −1.25023 1.56107i 4.23483i 0 3.84958i 2.75554 0.637979i 0 5.39856 + 2.59290i
379.15 −0.558288 1.29935i 0 −1.37663 + 1.45082i 1.71991i 0 0.342554i 2.65369 + 0.978748i 0 −2.23477 + 0.960208i
379.16 −0.558288 + 1.29935i 0 −1.37663 1.45082i 1.71991i 0 0.342554i 2.65369 0.978748i 0 −2.23477 0.960208i
379.17 −0.388306 1.35986i 0 −1.69844 + 1.05608i 1.84435i 0 2.17959i 2.09564 + 1.89955i 0 2.50805 0.716171i
379.18 −0.388306 + 1.35986i 0 −1.69844 1.05608i 1.84435i 0 2.17959i 2.09564 1.89955i 0 2.50805 + 0.716171i
379.19 −0.134537 1.40780i 0 −1.96380 + 0.378802i 2.61555i 0 4.81248i 0.797481 + 2.71367i 0 3.68217 0.351888i
379.20 −0.134537 + 1.40780i 0 −1.96380 0.378802i 2.61555i 0 4.81248i 0.797481 2.71367i 0 3.68217 + 0.351888i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.e.g 40
3.b odd 2 1 456.2.e.a 40
4.b odd 2 1 5472.2.e.g 40
8.b even 2 1 5472.2.e.g 40
8.d odd 2 1 inner 1368.2.e.g 40
12.b even 2 1 1824.2.e.a 40
19.b odd 2 1 inner 1368.2.e.g 40
24.f even 2 1 456.2.e.a 40
24.h odd 2 1 1824.2.e.a 40
57.d even 2 1 456.2.e.a 40
76.d even 2 1 5472.2.e.g 40
152.b even 2 1 inner 1368.2.e.g 40
152.g odd 2 1 5472.2.e.g 40
228.b odd 2 1 1824.2.e.a 40
456.l odd 2 1 456.2.e.a 40
456.p even 2 1 1824.2.e.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.e.a 40 3.b odd 2 1
456.2.e.a 40 24.f even 2 1
456.2.e.a 40 57.d even 2 1
456.2.e.a 40 456.l odd 2 1
1368.2.e.g 40 1.a even 1 1 trivial
1368.2.e.g 40 8.d odd 2 1 inner
1368.2.e.g 40 19.b odd 2 1 inner
1368.2.e.g 40 152.b even 2 1 inner
1824.2.e.a 40 12.b even 2 1
1824.2.e.a 40 24.h odd 2 1
1824.2.e.a 40 228.b odd 2 1
1824.2.e.a 40 456.p even 2 1
5472.2.e.g 40 4.b odd 2 1
5472.2.e.g 40 8.b even 2 1
5472.2.e.g 40 76.d even 2 1
5472.2.e.g 40 152.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1368, [\chi])$$:

 $$T_{5}^{20} + \cdots$$ $$T_{7}^{20} + \cdots$$