Properties

Label 170.2.r.b
Level $170$
Weight $2$
Character orbit 170.r
Analytic conductor $1.357$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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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: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 16q^{10} - 16q^{18} + 8q^{25} - 8q^{26} + 24q^{27} - 8q^{28} + 8q^{29} + 16q^{30} - 16q^{31} - 32q^{33} + 8q^{34} - 32q^{35} - 32q^{39} - 56q^{41} - 24q^{42} + 16q^{43} + 16q^{44} + 24q^{45} + 16q^{49} - 32q^{51} - 16q^{52} + 16q^{53} - 24q^{54} - 8q^{55} - 8q^{56} - 120q^{57} + 16q^{58} + 8q^{60} + 24q^{61} - 8q^{62} - 24q^{63} - 32q^{65} + 16q^{67} - 8q^{70} + 24q^{71} + 56q^{72} + 88q^{73} + 32q^{74} + 8q^{75} + 24q^{77} + 32q^{78} - 104q^{79} + 8q^{80} + 48q^{81} + 16q^{82} + 16q^{83} + 136q^{85} + 96q^{86} + 136q^{87} - 16q^{89} + 24q^{90} + 48q^{91} - 8q^{92} - 8q^{93} - 8q^{94} - 136q^{95} + 16q^{97} + 72q^{98} + 160q^{99} + O(q^{100}) \)

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 −2.47936 1.65666i 0.707107 0.707107i −2.20283 + 0.384106i −2.92460 0.581740i −2.11600 0.420899i 0.382683 0.923880i 2.25467 + 5.44325i −1.88816 + 1.19785i
23.2 0.923880 0.382683i −0.628452 0.419918i 0.707107 0.707107i −0.448594 2.19061i −0.741309 0.147456i 1.27154 + 0.252926i 0.382683 0.923880i −0.929430 2.24384i −1.25276 1.85219i
23.3 0.923880 0.382683i −0.432293 0.288849i 0.707107 0.707107i 0.601626 + 2.15361i −0.509925 0.101430i 3.10821 + 0.618262i 0.382683 0.923880i −1.04461 2.52190i 1.37998 + 1.75945i
23.4 0.923880 0.382683i 1.41027 + 0.942312i 0.707107 0.707107i 2.21673 0.293448i 1.66353 + 0.330896i −5.02568 0.999670i 0.382683 0.923880i −0.0471409 0.113808i 1.93569 1.11942i
23.5 0.923880 0.382683i 2.12983 + 1.42311i 0.707107 0.707107i −2.23146 0.143452i 2.51231 + 0.499730i 0.0411526 + 0.00818577i 0.382683 0.923880i 1.36290 + 3.29034i −2.11650 + 0.721411i
37.1 0.923880 + 0.382683i −2.47936 + 1.65666i 0.707107 + 0.707107i −2.20283 0.384106i −2.92460 + 0.581740i −2.11600 + 0.420899i 0.382683 + 0.923880i 2.25467 5.44325i −1.88816 1.19785i
37.2 0.923880 + 0.382683i −0.628452 + 0.419918i 0.707107 + 0.707107i −0.448594 + 2.19061i −0.741309 + 0.147456i 1.27154 0.252926i 0.382683 + 0.923880i −0.929430 + 2.24384i −1.25276 + 1.85219i
37.3 0.923880 + 0.382683i −0.432293 + 0.288849i 0.707107 + 0.707107i 0.601626 2.15361i −0.509925 + 0.101430i 3.10821 0.618262i 0.382683 + 0.923880i −1.04461 + 2.52190i 1.37998 1.75945i
37.4 0.923880 + 0.382683i 1.41027 0.942312i 0.707107 + 0.707107i 2.21673 + 0.293448i 1.66353 0.330896i −5.02568 + 0.999670i 0.382683 + 0.923880i −0.0471409 + 0.113808i 1.93569 + 1.11942i
37.5 0.923880 + 0.382683i 2.12983 1.42311i 0.707107 + 0.707107i −2.23146 + 0.143452i 2.51231 0.499730i 0.0411526 0.00818577i 0.382683 + 0.923880i 1.36290 3.29034i −2.11650 0.721411i
97.1 0.382683 0.923880i −0.592739 + 2.97990i −0.707107 0.707107i −1.33977 + 1.79026i 2.52624 + 1.68798i −0.842302 0.562808i −0.923880 + 0.382683i −5.75684 2.38456i 1.14127 + 1.92289i
97.2 0.382683 0.923880i −0.364682 + 1.83338i −0.707107 0.707107i 0.389869 2.20182i 1.55427 + 1.03853i 3.99085 + 2.66660i −0.923880 + 0.382683i −0.456659 0.189154i −1.88502 1.20279i
97.3 0.382683 0.923880i 0.0178485 0.0897304i −0.707107 0.707107i −0.936619 2.03045i −0.0760698 0.0508282i −3.86692 2.58379i −0.923880 + 0.382683i 2.76391 + 1.14485i −2.23432 + 0.0883019i
97.4 0.382683 0.923880i 0.372511 1.87274i −0.707107 0.707107i 2.21955 + 0.271292i −1.58763 1.06082i −0.930986 0.622065i −0.923880 + 0.382683i −0.596748 0.247181i 1.10003 1.94678i
97.5 0.382683 0.923880i 0.567062 2.85081i −0.707107 0.707107i −2.18819 0.460262i −2.41680 1.61486i 3.60477 + 2.40863i −0.923880 + 0.382683i −5.03395 2.08513i −1.26261 + 1.84549i
107.1 −0.382683 + 0.923880i −2.87914 0.572697i −0.707107 0.707107i 2.22112 0.258133i 1.63090 2.44082i −0.222469 + 0.332949i 0.923880 0.382683i 5.18983 + 2.14970i −0.611501 + 2.15083i
107.2 −0.382683 + 0.923880i −1.27477 0.253568i −0.707107 0.707107i −1.36286 + 1.77274i 0.722100 1.08070i −0.128721 + 0.192645i 0.923880 0.382683i −1.21089 0.501569i −1.11625 1.93752i
107.3 −0.382683 + 0.923880i −0.706293 0.140490i −0.707107 0.707107i −0.971395 2.01405i 0.400083 0.598766i 1.70679 2.55439i 0.923880 0.382683i −2.29253 0.949595i 2.23248 0.126709i
107.4 −0.382683 + 0.923880i 1.51925 + 0.302197i −0.707107 0.707107i 1.28645 + 1.82895i −0.860584 + 1.28795i 0.134383 0.201119i 0.923880 0.382683i −0.554854 0.229828i −2.18203 + 0.488612i
107.5 −0.382683 + 0.923880i 3.34096 + 0.664558i −0.707107 0.707107i −0.732362 2.11273i −1.89250 + 2.83233i −0.616963 + 0.923351i 0.923880 0.382683i 7.94873 + 3.29247i 2.23217 + 0.131894i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 167.5
Significant digits:
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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.b yes 40
5.b even 2 1 850.2.v.d 40
5.c odd 4 1 170.2.o.b 40
5.c odd 4 1 850.2.s.d 40
17.e odd 16 1 170.2.o.b 40
85.o even 16 1 850.2.v.d 40
85.p odd 16 1 850.2.s.d 40
85.r even 16 1 inner 170.2.r.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.o.b 40 5.c odd 4 1
170.2.o.b 40 17.e odd 16 1
170.2.r.b yes 40 1.a even 1 1 trivial
170.2.r.b yes 40 85.r even 16 1 inner
850.2.s.d 40 5.c odd 4 1
850.2.s.d 40 85.p odd 16 1
850.2.v.d 40 5.b even 2 1
850.2.v.d 40 85.o even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{40} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(170, [\chi])\).