# Properties

 Label 668.2.b.a Level $668$ Weight $2$ Character orbit 668.b Analytic conductor $5.334$ Analytic rank $0$ Dimension $22$ CM discriminant -167 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$668 = 2^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 668.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.33400685502$$ Analytic rank: $$0$$ Dimension: $$22$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 66q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 66q^{9} + 110q^{25} - 11q^{42} - 33q^{44} + 55q^{48} - 154q^{49} + 77q^{54} - 99q^{62} - 121q^{72} + 198q^{81} + 143q^{84} + 165q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
667.1 −1.40042 0.197044i 2.07563i 1.92235 + 0.551888i 0 0.408990 2.90675i 4.81975i −2.58335 1.15166i −1.30824 0
667.2 −1.40042 + 0.197044i 2.07563i 1.92235 0.551888i 0 0.408990 + 2.90675i 4.81975i −2.58335 + 1.15166i −1.30824 0
667.3 −1.28464 0.591360i 0.736748i 1.30059 + 1.51937i 0 −0.435683 + 0.946454i 5.23983i −0.772291 2.72095i 2.45720 0
667.4 −1.28464 + 0.591360i 0.736748i 1.30059 1.51937i 0 −0.435683 0.946454i 5.23983i −0.772291 + 2.72095i 2.45720 0
667.5 −1.07158 0.922888i 3.45525i 0.296557 + 1.97789i 0 −3.18880 + 3.70256i 4.00920i 1.50759 2.39315i −8.93873 0
667.6 −1.07158 + 0.922888i 3.45525i 0.296557 1.97789i 0 −3.18880 3.70256i 4.00920i 1.50759 + 2.39315i −8.93873 0
667.7 −0.760993 1.19201i 3.04057i −0.841781 + 1.81422i 0 3.62439 2.31385i 5.23541i 2.80316 0.377198i −6.24504 0
667.8 −0.760993 + 1.19201i 3.04057i −0.841781 1.81422i 0 3.62439 + 2.31385i 5.23541i 2.80316 + 0.377198i −6.24504 0
667.9 −0.402518 1.35572i 2.44978i −1.67596 + 1.09140i 0 3.32122 0.986080i 2.87385i 2.15424 + 1.83282i −3.00142 0
667.10 −0.402518 + 1.35572i 2.44978i −1.67596 1.09140i 0 3.32122 + 0.986080i 2.87385i 2.15424 1.83282i −3.00142 0
667.11 0.00426214 1.41421i 3.24555i −1.99996 0.0120551i 0 −4.58987 0.0138330i 4.80685i −0.0255725 + 2.82831i −7.53357 0
667.12 0.00426214 + 1.41421i 3.24555i −1.99996 + 0.0120551i 0 −4.58987 + 0.0138330i 4.80685i −0.0255725 2.82831i −7.53357 0
667.13 0.394339 1.35812i 0.246717i −1.68899 1.07112i 0 0.335073 + 0.0972902i 1.50567i −2.12075 + 1.87148i 2.93913 0
667.14 0.394339 + 1.35812i 0.246717i −1.68899 + 1.07112i 0 0.335073 0.0972902i 1.50567i −2.12075 1.87148i 2.93913 0
667.15 0.768164 1.18740i 1.21020i −0.819849 1.82424i 0 1.43699 + 0.929628i 3.98887i −2.79588 0.427823i 1.53543 0
667.16 0.768164 + 1.18740i 1.21020i −0.819849 + 1.82424i 0 1.43699 0.929628i 3.98887i −2.79588 + 0.427823i 1.53543 0
667.17 1.06600 0.929330i 2.77291i 0.272692 1.98132i 0 −2.57695 2.95591i 0.0155204i −1.55061 2.36550i −4.68903 0
667.18 1.06600 + 0.929330i 2.77291i 0.272692 + 1.98132i 0 −2.57695 + 2.95591i 0.0155204i −1.55061 + 2.36550i −4.68903 0
667.19 1.28818 0.583606i 1.66053i 1.31881 1.50358i 0 0.969093 + 2.13906i 2.84773i 0.821364 2.70654i 0.242651 0
667.20 1.28818 + 0.583606i 1.66053i 1.31881 + 1.50358i 0 0.969093 2.13906i 2.84773i 0.821364 + 2.70654i 0.242651 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by $$\Q(\sqrt{-167})$$
4.b odd 2 1 inner
668.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.2.b.a 22
4.b odd 2 1 inner 668.2.b.a 22
167.b odd 2 1 CM 668.2.b.a 22
668.b even 2 1 inner 668.2.b.a 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.b.a 22 1.a even 1 1 trivial
668.2.b.a 22 4.b odd 2 1 inner
668.2.b.a 22 167.b odd 2 1 CM
668.2.b.a 22 668.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{22} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(668, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database