# Properties

 Label 1078.2.i.d Level $1078$ Weight $2$ Character orbit 1078.i Analytic conductor $8.608$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 16 q^{4} - 16 q^{9}+O(q^{10})$$ 32 * q + 16 * q^4 - 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q + 16 q^{4} - 16 q^{9} - 16 q^{11} - 16 q^{16} - 16 q^{22} + 16 q^{23} + 64 q^{25} - 32 q^{36} + 80 q^{37} + 16 q^{44} - 32 q^{53} + 48 q^{58} - 32 q^{64} - 16 q^{67} - 96 q^{71} + 32 q^{78} + 48 q^{81} - 32 q^{86} - 8 q^{88} + 32 q^{92} + 48 q^{93} + 48 q^{99}+O(q^{100})$$ 32 * q + 16 * q^4 - 16 * q^9 - 16 * q^11 - 16 * q^16 - 16 * q^22 + 16 * q^23 + 64 * q^25 - 32 * q^36 + 80 * q^37 + 16 * q^44 - 32 * q^53 + 48 * q^58 - 32 * q^64 - 16 * q^67 - 96 * q^71 + 32 * q^78 + 48 * q^81 - 32 * q^86 - 8 * q^88 + 32 * q^92 + 48 * q^93 + 48 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1 −0.866025 0.500000i −0.662827 + 0.382683i 0.500000 + 0.866025i −3.37695 1.94969i 0.765367 0 1.00000i −1.20711 + 2.09077i 1.94969 + 3.37695i
901.2 −0.866025 0.500000i 0.662827 0.382683i 0.500000 + 0.866025i 3.37695 + 1.94969i −0.765367 0 1.00000i −1.20711 + 2.09077i −1.94969 3.37695i
901.3 −0.866025 0.500000i −1.60021 + 0.923880i 0.500000 + 0.866025i 2.80322 + 1.61844i 1.84776 0 1.00000i 0.207107 0.358719i −1.61844 2.80322i
901.4 −0.866025 0.500000i 1.60021 0.923880i 0.500000 + 0.866025i −2.80322 1.61844i −1.84776 0 1.00000i 0.207107 0.358719i 1.61844 + 2.80322i
901.5 −0.866025 0.500000i −1.60021 + 0.923880i 0.500000 + 0.866025i −2.14039 1.23576i 1.84776 0 1.00000i 0.207107 0.358719i 1.23576 + 2.14039i
901.6 −0.866025 0.500000i 1.60021 0.923880i 0.500000 + 0.866025i 2.14039 + 1.23576i −1.84776 0 1.00000i 0.207107 0.358719i −1.23576 2.14039i
901.7 −0.866025 0.500000i −0.662827 + 0.382683i 0.500000 + 0.866025i 1.77675 + 1.02581i 0.765367 0 1.00000i −1.20711 + 2.09077i −1.02581 1.77675i
901.8 −0.866025 0.500000i 0.662827 0.382683i 0.500000 + 0.866025i −1.77675 1.02581i −0.765367 0 1.00000i −1.20711 + 2.09077i 1.02581 + 1.77675i
901.9 0.866025 + 0.500000i −1.60021 + 0.923880i 0.500000 + 0.866025i −2.14039 1.23576i −1.84776 0 1.00000i 0.207107 0.358719i −1.23576 2.14039i
901.10 0.866025 + 0.500000i 1.60021 0.923880i 0.500000 + 0.866025i 2.14039 + 1.23576i 1.84776 0 1.00000i 0.207107 0.358719i 1.23576 + 2.14039i
901.11 0.866025 + 0.500000i −0.662827 + 0.382683i 0.500000 + 0.866025i −3.37695 1.94969i −0.765367 0 1.00000i −1.20711 + 2.09077i −1.94969 3.37695i
901.12 0.866025 + 0.500000i 0.662827 0.382683i 0.500000 + 0.866025i 3.37695 + 1.94969i 0.765367 0 1.00000i −1.20711 + 2.09077i 1.94969 + 3.37695i
901.13 0.866025 + 0.500000i −0.662827 + 0.382683i 0.500000 + 0.866025i 1.77675 + 1.02581i −0.765367 0 1.00000i −1.20711 + 2.09077i 1.02581 + 1.77675i
901.14 0.866025 + 0.500000i 0.662827 0.382683i 0.500000 + 0.866025i −1.77675 1.02581i 0.765367 0 1.00000i −1.20711 + 2.09077i −1.02581 1.77675i
901.15 0.866025 + 0.500000i −1.60021 + 0.923880i 0.500000 + 0.866025i 2.80322 + 1.61844i −1.84776 0 1.00000i 0.207107 0.358719i 1.61844 + 2.80322i
901.16 0.866025 + 0.500000i 1.60021 0.923880i 0.500000 + 0.866025i −2.80322 1.61844i 1.84776 0 1.00000i 0.207107 0.358719i −1.61844 2.80322i
1011.1 −0.866025 + 0.500000i −0.662827 0.382683i 0.500000 0.866025i −3.37695 + 1.94969i 0.765367 0 1.00000i −1.20711 2.09077i 1.94969 3.37695i
1011.2 −0.866025 + 0.500000i 0.662827 + 0.382683i 0.500000 0.866025i 3.37695 1.94969i −0.765367 0 1.00000i −1.20711 2.09077i −1.94969 + 3.37695i
1011.3 −0.866025 + 0.500000i −1.60021 0.923880i 0.500000 0.866025i 2.80322 1.61844i 1.84776 0 1.00000i 0.207107 + 0.358719i −1.61844 + 2.80322i
1011.4 −0.866025 + 0.500000i 1.60021 + 0.923880i 0.500000 0.866025i −2.80322 + 1.61844i −1.84776 0 1.00000i 0.207107 + 0.358719i 1.61844 2.80322i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1011.16 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
7.c even 3 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.i.d 32
7.b odd 2 1 inner 1078.2.i.d 32
7.c even 3 1 1078.2.c.c 16
7.c even 3 1 inner 1078.2.i.d 32
7.d odd 6 1 1078.2.c.c 16
7.d odd 6 1 inner 1078.2.i.d 32
11.b odd 2 1 inner 1078.2.i.d 32
77.b even 2 1 inner 1078.2.i.d 32
77.h odd 6 1 1078.2.c.c 16
77.h odd 6 1 inner 1078.2.i.d 32
77.i even 6 1 1078.2.c.c 16
77.i even 6 1 inner 1078.2.i.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.c 16 7.c even 3 1
1078.2.c.c 16 7.d odd 6 1
1078.2.c.c 16 77.h odd 6 1
1078.2.c.c 16 77.i even 6 1
1078.2.i.d 32 1.a even 1 1 trivial
1078.2.i.d 32 7.b odd 2 1 inner
1078.2.i.d 32 7.c even 3 1 inner
1078.2.i.d 32 7.d odd 6 1 inner
1078.2.i.d 32 11.b odd 2 1 inner
1078.2.i.d 32 77.b even 2 1 inner
1078.2.i.d 32 77.h odd 6 1 inner
1078.2.i.d 32 77.i even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 4T_{3}^{6} + 14T_{3}^{4} - 8T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$.