# Properties

 Label 668.2.b.b Level $668$ Weight $2$ Character orbit 668.b Analytic conductor $5.334$ Analytic rank $0$ Dimension $60$ CM no 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: $$60$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q - 2q^{2} + 2q^{4} - 8q^{6} - 8q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q - 2q^{2} + 2q^{4} - 8q^{6} - 8q^{8} - 16q^{9} + 18q^{12} + 10q^{14} + 10q^{16} - 20q^{18} + 20q^{22} + 10q^{24} - 188q^{25} + 4q^{29} + 18q^{32} + 8q^{33} + 30q^{36} - 36q^{38} + 14q^{42} + 28q^{44} - 28q^{48} + 72q^{49} - 40q^{50} - 74q^{54} + 50q^{56} + 8q^{57} - 22q^{58} + 36q^{61} + 104q^{62} + 8q^{64} + 24q^{65} + 24q^{66} + 90q^{72} - 36q^{76} - 84q^{81} - 110q^{84} - 16q^{85} - 20q^{88} + 28q^{89} + 72q^{93} + 90q^{94} + 2q^{96} - 4q^{97} - 114q^{98} + O(q^{100})$$

## 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 −1.37801 0.317929i 2.42728i 1.79784 + 0.876221i 2.41022i 0.771702 3.34482i 1.33262i −2.19887 1.77903i −2.89167 −0.766278 + 3.32131i
667.2 −1.37801 0.317929i 2.42728i 1.79784 + 0.876221i 2.41022i 0.771702 3.34482i 1.33262i −2.19887 1.77903i −2.89167 0.766278 3.32131i
667.3 −1.37801 + 0.317929i 2.42728i 1.79784 0.876221i 2.41022i 0.771702 + 3.34482i 1.33262i −2.19887 + 1.77903i −2.89167 0.766278 + 3.32131i
667.4 −1.37801 + 0.317929i 2.42728i 1.79784 0.876221i 2.41022i 0.771702 + 3.34482i 1.33262i −2.19887 + 1.77903i −2.89167 −0.766278 3.32131i
667.5 −1.37670 0.323577i 0.453678i 1.79060 + 0.890936i 4.09468i −0.146800 + 0.624578i 3.11488i −2.17682 1.80595i 2.79418 −1.32494 + 5.63714i
667.6 −1.37670 0.323577i 0.453678i 1.79060 + 0.890936i 4.09468i −0.146800 + 0.624578i 3.11488i −2.17682 1.80595i 2.79418 1.32494 5.63714i
667.7 −1.37670 + 0.323577i 0.453678i 1.79060 0.890936i 4.09468i −0.146800 0.624578i 3.11488i −2.17682 + 1.80595i 2.79418 1.32494 + 5.63714i
667.8 −1.37670 + 0.323577i 0.453678i 1.79060 0.890936i 4.09468i −0.146800 0.624578i 3.11488i −2.17682 + 1.80595i 2.79418 −1.32494 5.63714i
667.9 −1.30799 0.537726i 2.06566i 1.42170 + 1.40669i 1.78350i −1.11076 + 2.70187i 0.540270i −1.10317 2.60442i −1.26695 −0.959034 + 2.33281i
667.10 −1.30799 0.537726i 2.06566i 1.42170 + 1.40669i 1.78350i −1.11076 + 2.70187i 0.540270i −1.10317 2.60442i −1.26695 0.959034 2.33281i
667.11 −1.30799 + 0.537726i 2.06566i 1.42170 1.40669i 1.78350i −1.11076 2.70187i 0.540270i −1.10317 + 2.60442i −1.26695 0.959034 + 2.33281i
667.12 −1.30799 + 0.537726i 2.06566i 1.42170 1.40669i 1.78350i −1.11076 2.70187i 0.540270i −1.10317 + 2.60442i −1.26695 −0.959034 2.33281i
667.13 −1.12623 0.855338i 0.530251i 0.536794 + 1.92662i 0.936151i 0.453544 0.597186i 2.10467i 1.04335 2.62896i 2.71883 −0.800726 + 1.05432i
667.14 −1.12623 0.855338i 0.530251i 0.536794 + 1.92662i 0.936151i 0.453544 0.597186i 2.10467i 1.04335 2.62896i 2.71883 0.800726 1.05432i
667.15 −1.12623 + 0.855338i 0.530251i 0.536794 1.92662i 0.936151i 0.453544 + 0.597186i 2.10467i 1.04335 + 2.62896i 2.71883 0.800726 + 1.05432i
667.16 −1.12623 + 0.855338i 0.530251i 0.536794 1.92662i 0.936151i 0.453544 + 0.597186i 2.10467i 1.04335 + 2.62896i 2.71883 −0.800726 1.05432i
667.17 −0.844041 1.13472i 0.873998i −0.575190 + 1.91550i 2.77771i 0.991745 0.737690i 1.63351i 2.65905 0.964083i 2.23613 −3.15193 + 2.34450i
667.18 −0.844041 1.13472i 0.873998i −0.575190 + 1.91550i 2.77771i 0.991745 0.737690i 1.63351i 2.65905 0.964083i 2.23613 3.15193 2.34450i
667.19 −0.844041 + 1.13472i 0.873998i −0.575190 1.91550i 2.77771i 0.991745 + 0.737690i 1.63351i 2.65905 + 0.964083i 2.23613 3.15193 + 2.34450i
667.20 −0.844041 + 1.13472i 0.873998i −0.575190 1.91550i 2.77771i 0.991745 + 0.737690i 1.63351i 2.65905 + 0.964083i 2.23613 −3.15193 2.34450i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 667.60 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
167.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.b 60
4.b odd 2 1 inner 668.2.b.b 60
167.b odd 2 1 inner 668.2.b.b 60
668.b even 2 1 inner 668.2.b.b 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.b.b 60 1.a even 1 1 trivial
668.2.b.b 60 4.b odd 2 1 inner
668.2.b.b 60 167.b odd 2 1 inner
668.2.b.b 60 668.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(668, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database