# Properties

 Label 170.2.r.a Level $170$ Weight $2$ Character orbit 170.r 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.r (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 8q^{10} - 48q^{15} + 16q^{18} - 24q^{25} + 8q^{26} + 24q^{27} - 8q^{28} - 8q^{29} - 16q^{30} - 16q^{31} + 64q^{33} + 24q^{34} + 32q^{35} - 32q^{37} - 32q^{39} + 16q^{41} - 24q^{42} - 16q^{43} - 16q^{44} - 24q^{45} - 16q^{49} - 32q^{50} + 32q^{51} - 16q^{52} + 16q^{53} + 24q^{54} - 8q^{55} + 8q^{56} - 24q^{57} - 16q^{58} - 64q^{59} + 40q^{60} - 24q^{61} - 40q^{62} - 24q^{63} + 32q^{65} + 16q^{67} + 40q^{70} + 8q^{71} - 16q^{72} + 32q^{73} + 8q^{74} - 56q^{75} + 24q^{77} + 32q^{78} + 72q^{79} + 8q^{80} + 48q^{81} + 48q^{82} + 16q^{83} + 8q^{85} - 64q^{86} + 40q^{87} + 32q^{88} + 16q^{89} + 24q^{90} + 48q^{91} + 24q^{92} - 8q^{93} + 8q^{94} + 56q^{95} - 48q^{97} - 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −0.923880 + 0.382683i −1.19858 0.800867i 0.707107 0.707107i 2.21955 0.271267i 1.41382 + 0.281227i −3.32261 0.660909i −0.382683 + 0.923880i −0.352840 0.851831i −1.94679 + 1.10000i
23.2 −0.923880 + 0.382683i −0.856804 0.572498i 0.707107 0.707107i −1.44860 + 1.70339i 1.01067 + 0.201035i −1.64388 0.326988i −0.382683 + 0.923880i −0.741692 1.79060i 0.686475 2.12809i
23.3 −0.923880 + 0.382683i −0.283864 0.189672i 0.707107 0.707107i −1.91626 1.15236i 0.334840 + 0.0666039i 3.52423 + 0.701012i −0.382683 + 0.923880i −1.10345 2.66396i 2.21139 + 0.331322i
23.4 −0.923880 + 0.382683i 2.33925 + 1.56304i 0.707107 0.707107i 0.438207 + 2.19271i −2.75933 0.548865i −1.27851 0.254312i −0.382683 + 0.923880i 1.88095 + 4.54102i −1.24396 1.85810i
37.1 −0.923880 0.382683i −1.19858 + 0.800867i 0.707107 + 0.707107i 2.21955 + 0.271267i 1.41382 0.281227i −3.32261 + 0.660909i −0.382683 0.923880i −0.352840 + 0.851831i −1.94679 1.10000i
37.2 −0.923880 0.382683i −0.856804 + 0.572498i 0.707107 + 0.707107i −1.44860 1.70339i 1.01067 0.201035i −1.64388 + 0.326988i −0.382683 0.923880i −0.741692 + 1.79060i 0.686475 + 2.12809i
37.3 −0.923880 0.382683i −0.283864 + 0.189672i 0.707107 + 0.707107i −1.91626 + 1.15236i 0.334840 0.0666039i 3.52423 0.701012i −0.382683 0.923880i −1.10345 + 2.66396i 2.21139 0.331322i
37.4 −0.923880 0.382683i 2.33925 1.56304i 0.707107 + 0.707107i 0.438207 2.19271i −2.75933 + 0.548865i −1.27851 + 0.254312i −0.382683 0.923880i 1.88095 4.54102i −1.24396 + 1.85810i
97.1 −0.382683 + 0.923880i −0.641185 + 3.22346i −0.707107 0.707107i 1.65116 + 1.50787i −2.73271 1.82594i 2.05104 + 1.37046i 0.923880 0.382683i −7.20792 2.98562i −2.02496 + 0.948441i
97.2 −0.382683 + 0.923880i 0.0161490 0.0811867i −0.707107 0.707107i 1.74181 1.40217i 0.0688268 + 0.0459886i 0.214466 + 0.143301i 0.923880 0.382683i 2.76531 + 1.14543i 0.628876 + 2.14581i
97.3 −0.382683 + 0.923880i 0.182558 0.917782i −0.707107 0.707107i −0.566813 + 2.16304i 0.778058 + 0.519882i 1.62454 + 1.08548i 0.923880 0.382683i 1.96264 + 0.812953i −1.78147 1.35143i
97.4 −0.382683 + 0.923880i 0.442478 2.22449i −0.707107 0.707107i −2.11906 0.713863i 1.88583 + 1.26007i −1.93464 1.29268i 0.923880 0.382683i −1.98092 0.820524i 1.47045 1.68457i
107.1 0.382683 0.923880i −2.47085 0.491482i −0.707107 0.707107i 0.0805608 2.23462i −1.39962 + 2.09468i −2.60493 + 3.89855i −0.923880 + 0.382683i 3.09190 + 1.28071i −2.03369 0.929579i
107.2 0.382683 0.923880i −1.34931 0.268393i −0.707107 0.707107i 2.19527 + 0.425176i −0.764320 + 1.14389i 2.19060 3.27846i −0.923880 + 0.382683i −1.02305 0.423761i 1.23291 1.86546i
107.3 0.382683 0.923880i 1.21530 + 0.241738i −0.707107 0.707107i −0.353144 2.20801i 0.688411 1.03028i 0.394839 0.590918i −0.923880 + 0.382683i −1.35312 0.560483i −2.17507 0.518705i
107.4 0.382683 0.923880i 2.60485 + 0.518138i −0.707107 0.707107i −1.21558 + 1.87679i 1.47553 2.20829i 0.892510 1.33574i −0.923880 + 0.382683i 3.74516 + 1.55130i 1.26875 + 1.84127i
113.1 0.923880 0.382683i −1.13636 + 1.70069i 0.707107 0.707107i −0.0888762 + 2.23430i −0.399038 + 2.00610i −0.252993 + 1.27188i 0.382683 0.923880i −0.452969 1.09356i 0.772919 + 2.09824i
113.2 0.923880 0.382683i −0.683024 + 1.02222i 0.707107 0.707107i 1.78940 1.34092i −0.239846 + 1.20579i 0.0311808 0.156756i 0.382683 0.923880i 0.569643 + 1.37524i 1.14004 1.92362i
113.3 0.923880 0.382683i 0.776902 1.16272i 0.707107 0.707107i −1.16598 + 1.90801i 0.272812 1.37152i 0.749924 3.77012i 0.382683 0.923880i 0.399719 + 0.965007i −0.347062 + 2.20897i
113.4 0.923880 0.382683i 1.04249 1.56019i 0.707107 0.707107i −1.24165 1.85965i 0.366072 1.84037i −0.635763 + 3.19619i 0.382683 0.923880i −0.199368 0.481317i −1.85879 1.24294i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 167.4 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.r 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.r.a yes 32
5.b even 2 1 850.2.v.c 32
5.c odd 4 1 170.2.o.a 32
5.c odd 4 1 850.2.s.c 32
17.e odd 16 1 170.2.o.a 32
85.o even 16 1 850.2.v.c 32
85.p odd 16 1 850.2.s.c 32
85.r even 16 1 inner 170.2.r.a yes 32

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

## Hecke kernels

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