# Properties

 Label 1368.2.e.f Level $1368$ Weight $2$ Character orbit 1368.e Analytic conductor $10.924$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

# 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: $$24$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 20 q^{4} - 12 q^{16} + 24 q^{19} - 40 q^{25} - 40 q^{28} - 72 q^{49} - 104 q^{58} + 20 q^{64} + 80 q^{73} - 12 q^{76} + 56 q^{82} + O(q^{100})$$

## 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 −1.39298 0.244153i 0 1.88078 + 0.680200i 3.10261i 0 4.34495i −2.45381 1.40670i 0 0.757512 4.32187i
379.2 −1.39298 0.244153i 0 1.88078 + 0.680200i 3.10261i 0 4.34495i −2.45381 1.40670i 0 −0.757512 + 4.32187i
379.3 −1.39298 + 0.244153i 0 1.88078 0.680200i 3.10261i 0 4.34495i −2.45381 + 1.40670i 0 0.757512 + 4.32187i
379.4 −1.39298 + 0.244153i 0 1.88078 0.680200i 3.10261i 0 4.34495i −2.45381 + 1.40670i 0 −0.757512 4.32187i
379.5 −1.26128 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i −0.458259 2.79106i 0 −1.25652 + 2.47760i
379.6 −1.26128 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i −0.458259 2.79106i 0 1.25652 2.47760i
379.7 −1.26128 + 0.639662i 0 1.18166 1.61359i 1.96435i 0 1.25539i −0.458259 + 2.79106i 0 −1.25652 2.47760i
379.8 −1.26128 + 0.639662i 0 1.18166 1.61359i 1.96435i 0 1.25539i −0.458259 + 2.79106i 0 1.25652 + 2.47760i
379.9 −0.847808 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i 2.64930 0.990551i 0 −2.88918 + 2.16401i
379.10 −0.847808 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i 2.64930 0.990551i 0 2.88918 2.16401i
379.11 −0.847808 + 1.13191i 0 −0.562443 1.91929i 2.55248i 0 3.08957i 2.64930 + 0.990551i 0 −2.88918 2.16401i
379.12 −0.847808 + 1.13191i 0 −0.562443 1.91929i 2.55248i 0 3.08957i 2.64930 + 0.990551i 0 2.88918 + 2.16401i
379.13 0.847808 1.13191i 0 −0.562443 1.91929i 2.55248i 0 3.08957i −2.64930 0.990551i 0 2.88918 + 2.16401i
379.14 0.847808 1.13191i 0 −0.562443 1.91929i 2.55248i 0 3.08957i −2.64930 0.990551i 0 −2.88918 2.16401i
379.15 0.847808 + 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i −2.64930 + 0.990551i 0 2.88918 2.16401i
379.16 0.847808 + 1.13191i 0 −0.562443 + 1.91929i 2.55248i 0 3.08957i −2.64930 + 0.990551i 0 −2.88918 + 2.16401i
379.17 1.26128 0.639662i 0 1.18166 1.61359i 1.96435i 0 1.25539i 0.458259 2.79106i 0 −1.25652 2.47760i
379.18 1.26128 0.639662i 0 1.18166 1.61359i 1.96435i 0 1.25539i 0.458259 2.79106i 0 1.25652 + 2.47760i
379.19 1.26128 + 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i 0.458259 + 2.79106i 0 −1.25652 + 2.47760i
379.20 1.26128 + 0.639662i 0 1.18166 + 1.61359i 1.96435i 0 1.25539i 0.458259 + 2.79106i 0 1.25652 2.47760i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
24.f even 2 1 inner
57.d even 2 1 inner
152.b even 2 1 inner
456.l odd 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.f 24
3.b odd 2 1 inner 1368.2.e.f 24
4.b odd 2 1 5472.2.e.f 24
8.b even 2 1 5472.2.e.f 24
8.d odd 2 1 inner 1368.2.e.f 24
12.b even 2 1 5472.2.e.f 24
19.b odd 2 1 inner 1368.2.e.f 24
24.f even 2 1 inner 1368.2.e.f 24
24.h odd 2 1 5472.2.e.f 24
57.d even 2 1 inner 1368.2.e.f 24
76.d even 2 1 5472.2.e.f 24
152.b even 2 1 inner 1368.2.e.f 24
152.g odd 2 1 5472.2.e.f 24
228.b odd 2 1 5472.2.e.f 24
456.l odd 2 1 inner 1368.2.e.f 24
456.p even 2 1 5472.2.e.f 24

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

## Hecke kernels

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

 $$T_{5}^{6} + 20 T_{5}^{4} + 125 T_{5}^{2} + 242$$ $$T_{7}^{6} + 30 T_{7}^{4} + 225 T_{7}^{2} + 284$$