# Properties

 Label 170.2.o.a Level $170$ Weight $2$ Character orbit 170.o Analytic conductor $1.357$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.o (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 8q^{10} - 40q^{15} + 16q^{18} + 8q^{20} - 8q^{25} + 8q^{26} - 72q^{27} + 8q^{28} + 8q^{29} - 16q^{31} - 64q^{33} - 24q^{34} + 32q^{35} + 16q^{37} + 32q^{39} - 8q^{40} + 16q^{41} - 40q^{42} + 48q^{43} + 16q^{44} + 24q^{45} - 64q^{47} + 16q^{49} + 32q^{50} + 32q^{51} - 16q^{52} - 24q^{54} + 8q^{55} + 8q^{56} - 8q^{57} - 16q^{58} + 64q^{59} - 48q^{60} - 24q^{61} - 24q^{62} - 24q^{63} - 16q^{65} - 16q^{67} - 16q^{68} + 24q^{70} + 8q^{71} + 16q^{73} - 8q^{74} - 8q^{75} + 40q^{77} + 48q^{78} - 72q^{79} + 8q^{80} + 48q^{81} + 16q^{82} + 16q^{83} - 8q^{85} - 64q^{86} + 24q^{87} + 16q^{88} - 16q^{89} + 48q^{90} + 48q^{91} + 8q^{92} + 8q^{93} - 8q^{94} + 40q^{95} + 16q^{97} + 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −0.382683 + 0.923880i −0.800867 1.19858i −0.707107 0.707107i −0.598768 + 2.15441i 1.41382 0.281227i 0.660909 + 3.32261i 0.923880 0.382683i 0.352840 0.851831i −1.76128 1.37765i
3.2 −0.382683 + 0.923880i −0.572498 0.856804i −0.707107 0.707107i −1.01937 1.99020i 1.01067 0.201035i 0.326988 + 1.64388i 0.923880 0.382683i 0.741692 1.79060i 2.22880 0.180163i
3.3 −0.382683 + 0.923880i −0.189672 0.283864i −0.707107 0.707107i 1.79797 1.32941i 0.334840 0.0666039i −0.701012 3.52423i 0.923880 0.382683i 1.10345 2.66396i 0.540159 + 2.16985i
3.4 −0.382683 + 0.923880i 1.56304 + 2.33925i −0.707107 0.707107i −2.19349 0.434263i −2.75933 + 0.548865i 0.254312 + 1.27851i 0.923880 0.382683i −1.88095 + 4.54102i 1.24062 1.86034i
7.1 −0.923880 + 0.382683i −0.518138 2.60485i 0.707107 0.707107i 0.404835 2.19912i 1.47553 + 2.20829i −1.33574 + 0.892510i −0.382683 + 0.923880i −3.74516 + 1.55130i 0.467546 + 2.18664i
7.2 −0.923880 + 0.382683i −0.241738 1.21530i 0.707107 0.707107i 1.17123 + 1.90479i 0.688411 + 1.03028i −0.590918 + 0.394839i −0.382683 + 0.923880i 1.35312 0.560483i −1.81101 1.31159i
7.3 −0.923880 + 0.382683i 0.268393 + 1.34931i 0.707107 0.707107i −2.19088 + 0.447284i −0.764320 1.14389i −3.27846 + 2.19060i −0.382683 + 0.923880i 1.02305 0.423761i 1.85294 1.25165i
7.4 −0.923880 + 0.382683i 0.491482 + 2.47085i 0.707107 0.707107i 0.780722 + 2.09535i −1.39962 2.09468i 3.89855 2.60493i −0.382683 + 0.923880i −3.09190 + 1.28071i −1.52315 1.63708i
27.1 0.923880 0.382683i −3.22346 + 0.641185i 0.707107 0.707107i 2.10251 + 0.761213i −2.73271 + 1.82594i 1.37046 + 2.05104i 0.382683 0.923880i 7.20792 2.98562i 2.23377 0.101327i
27.2 0.923880 0.382683i 0.0811867 0.0161490i 0.707107 0.707107i 1.07264 1.96200i 0.0688268 0.0459886i 0.143301 + 0.214466i 0.382683 0.923880i −2.76531 + 1.14543i 0.240162 2.22313i
27.3 0.923880 0.382683i 0.917782 0.182558i 0.707107 0.707107i 0.304091 + 2.21529i 0.778058 0.519882i 1.08548 + 1.62454i 0.382683 0.923880i −1.96264 + 0.812953i 1.12870 + 1.93029i
27.4 0.923880 0.382683i 2.22449 0.442478i 0.707107 0.707107i −2.23094 + 0.151404i 1.88583 1.26007i −1.29268 1.93464i 0.382683 0.923880i 1.98092 0.820524i −2.00318 + 0.993622i
57.1 −0.382683 0.923880i −0.800867 + 1.19858i −0.707107 + 0.707107i −0.598768 2.15441i 1.41382 + 0.281227i 0.660909 3.32261i 0.923880 + 0.382683i 0.352840 + 0.851831i −1.76128 + 1.37765i
57.2 −0.382683 0.923880i −0.572498 + 0.856804i −0.707107 + 0.707107i −1.01937 + 1.99020i 1.01067 + 0.201035i 0.326988 1.64388i 0.923880 + 0.382683i 0.741692 + 1.79060i 2.22880 + 0.180163i
57.3 −0.382683 0.923880i −0.189672 + 0.283864i −0.707107 + 0.707107i 1.79797 + 1.32941i 0.334840 + 0.0666039i −0.701012 + 3.52423i 0.923880 + 0.382683i 1.10345 + 2.66396i 0.540159 2.16985i
57.4 −0.382683 0.923880i 1.56304 2.33925i −0.707107 + 0.707107i −2.19349 + 0.434263i −2.75933 0.548865i 0.254312 1.27851i 0.923880 + 0.382683i −1.88095 4.54102i 1.24062 + 1.86034i
63.1 0.923880 + 0.382683i −3.22346 0.641185i 0.707107 + 0.707107i 2.10251 0.761213i −2.73271 1.82594i 1.37046 2.05104i 0.382683 + 0.923880i 7.20792 + 2.98562i 2.23377 + 0.101327i
63.2 0.923880 + 0.382683i 0.0811867 + 0.0161490i 0.707107 + 0.707107i 1.07264 + 1.96200i 0.0688268 + 0.0459886i 0.143301 0.214466i 0.382683 + 0.923880i −2.76531 1.14543i 0.240162 + 2.22313i
63.3 0.923880 + 0.382683i 0.917782 + 0.182558i 0.707107 + 0.707107i 0.304091 2.21529i 0.778058 + 0.519882i 1.08548 1.62454i 0.382683 + 0.923880i −1.96264 0.812953i 1.12870 1.93029i
63.4 0.923880 + 0.382683i 2.22449 + 0.442478i 0.707107 + 0.707107i −2.23094 0.151404i 1.88583 + 1.26007i −1.29268 + 1.93464i 0.382683 + 0.923880i 1.98092 + 0.820524i −2.00318 0.993622i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 147.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.o even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.o.a 32
5.b even 2 1 850.2.s.c 32
5.c odd 4 1 170.2.r.a yes 32
5.c odd 4 1 850.2.v.c 32
17.e odd 16 1 170.2.r.a yes 32
85.o even 16 1 inner 170.2.o.a 32
85.p odd 16 1 850.2.v.c 32
85.r even 16 1 850.2.s.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.a 32 1.a even 1 1 trivial
170.2.o.a 32 85.o even 16 1 inner
170.2.r.a yes 32 5.c odd 4 1
170.2.r.a yes 32 17.e odd 16 1
850.2.s.c 32 5.b even 2 1
850.2.s.c 32 85.r even 16 1
850.2.v.c 32 5.c odd 4 1
850.2.v.c 32 85.p odd 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(170, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database