# Properties

 Label 1400.2.x.c Level $1400$ Weight $2$ Character orbit 1400.x Analytic conductor $11.179$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{11} - 40q^{21} + 32q^{51} + 128q^{71} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
657.1 0 −2.09284 2.09284i 0 0 0 0.510946 2.59595i 0 5.75996i 0
657.2 0 −2.09284 2.09284i 0 0 0 2.59595 0.510946i 0 5.75996i 0
657.3 0 −1.26512 1.26512i 0 0 0 −1.75168 1.98283i 0 0.201074i 0
657.4 0 −1.26512 1.26512i 0 0 0 1.98283 + 1.75168i 0 0.201074i 0
657.5 0 −0.923076 0.923076i 0 0 0 −2.48246 + 0.915096i 0 1.29586i 0
657.6 0 −0.923076 0.923076i 0 0 0 −0.915096 + 2.48246i 0 1.29586i 0
657.7 0 −0.409160 0.409160i 0 0 0 0.738062 2.54072i 0 2.66518i 0
657.8 0 −0.409160 0.409160i 0 0 0 2.54072 0.738062i 0 2.66518i 0
657.9 0 0.409160 + 0.409160i 0 0 0 −2.54072 + 0.738062i 0 2.66518i 0
657.10 0 0.409160 + 0.409160i 0 0 0 −0.738062 + 2.54072i 0 2.66518i 0
657.11 0 0.923076 + 0.923076i 0 0 0 0.915096 2.48246i 0 1.29586i 0
657.12 0 0.923076 + 0.923076i 0 0 0 2.48246 0.915096i 0 1.29586i 0
657.13 0 1.26512 + 1.26512i 0 0 0 −1.98283 1.75168i 0 0.201074i 0
657.14 0 1.26512 + 1.26512i 0 0 0 1.75168 + 1.98283i 0 0.201074i 0
657.15 0 2.09284 + 2.09284i 0 0 0 −2.59595 + 0.510946i 0 5.75996i 0
657.16 0 2.09284 + 2.09284i 0 0 0 −0.510946 + 2.59595i 0 5.75996i 0
993.1 0 −2.09284 + 2.09284i 0 0 0 0.510946 + 2.59595i 0 5.75996i 0
993.2 0 −2.09284 + 2.09284i 0 0 0 2.59595 + 0.510946i 0 5.75996i 0
993.3 0 −1.26512 + 1.26512i 0 0 0 −1.75168 + 1.98283i 0 0.201074i 0
993.4 0 −1.26512 + 1.26512i 0 0 0 1.98283 1.75168i 0 0.201074i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 993.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.x.c 32
5.b even 2 1 inner 1400.2.x.c 32
5.c odd 4 2 inner 1400.2.x.c 32
7.b odd 2 1 inner 1400.2.x.c 32
35.c odd 2 1 inner 1400.2.x.c 32
35.f even 4 2 inner 1400.2.x.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.x.c 32 1.a even 1 1 trivial
1400.2.x.c 32 5.b even 2 1 inner
1400.2.x.c 32 5.c odd 4 2 inner
1400.2.x.c 32 7.b odd 2 1 inner
1400.2.x.c 32 35.c odd 2 1 inner
1400.2.x.c 32 35.f even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} + 90 T_{3}^{12} + 1049 T_{3}^{8} + 2400 T_{3}^{4} + 256$$ acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$.