# Properties

 Label 17.5.e.a Level 17 Weight 5 Character orbit 17.e Analytic conductor 1.757 Analytic rank 0 Dimension 40 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.75728937243$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{2} - 8q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 8q^{9} + 376q^{10} + 112q^{11} - 776q^{12} - 416q^{13} - 776q^{14} - 704q^{15} + 256q^{17} + 2032q^{18} + 688q^{19} + 2680q^{20} + 2032q^{21} + 760q^{22} - 176q^{23} + 1672q^{24} - 2600q^{26} - 2600q^{27} - 7448q^{28} - 3368q^{29} - 9800q^{30} - 3720q^{31} - 2400q^{32} + 4280q^{34} + 4208q^{35} + 11960q^{36} + 7416q^{37} + 16720q^{38} + 15624q^{39} + 20280q^{40} + 2656q^{41} - 6392q^{42} - 7512q^{43} - 31592q^{44} - 23368q^{45} - 25752q^{46} - 10208q^{47} - 14080q^{48} - 3112q^{49} + 3224q^{51} + 12784q^{52} + 24424q^{53} + 51672q^{54} + 26648q^{55} + 40432q^{56} + 10352q^{57} - 4336q^{58} - 3176q^{59} - 38896q^{60} - 24600q^{61} - 39248q^{62} - 55664q^{63} - 45560q^{64} - 37928q^{65} - 29376q^{66} + 34912q^{68} + 46592q^{69} + 59536q^{70} + 21736q^{71} + 59824q^{72} + 28592q^{73} + 15976q^{74} + 46168q^{75} - 9280q^{76} + 2392q^{77} - 6048q^{78} - 15912q^{79} - 47640q^{80} - 48696q^{81} - 58368q^{82} - 27296q^{83} + 18872q^{85} + 74336q^{86} + 38536q^{87} + 55608q^{88} - 1232q^{89} - 24136q^{90} - 7800q^{91} - 27032q^{92} - 32232q^{93} - 37096q^{94} - 49640q^{95} - 79696q^{96} - 12392q^{97} - 76304q^{98} + 6056q^{99} + O(q^{100})$$

## 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 −7.12052 + 2.94942i −0.829686 0.165035i 30.6890 30.6890i −14.8705 22.2553i 6.39455 1.27196i 16.8953 25.2857i −80.8164 + 195.108i −74.1731 30.7235i 171.526 + 114.610i
3.2 −2.92749 + 1.21261i 8.22545 + 1.63614i −4.21393 + 4.21393i 21.2448 + 31.7951i −26.0639 + 5.18443i 5.22138 7.81435i 26.6281 64.2859i −9.85322 4.08134i −100.749 67.3181i
3.3 −1.63167 + 0.675861i −11.4219 2.27196i −9.10814 + 9.10814i −6.20205 9.28203i 20.1724 4.01253i −12.6661 + 18.9562i 19.5194 47.1241i 50.4644 + 20.9030i 16.3931 + 10.9535i
3.4 3.43536 1.42297i 6.42184 + 1.27738i −1.53684 + 1.53684i −8.81924 13.1989i 23.8790 4.74983i 5.40438 8.08823i −25.8603 + 62.4323i −35.2259 14.5911i −49.0790 32.7936i
3.5 6.53721 2.70780i −10.8435 2.15691i 24.0892 24.0892i 16.7298 + 25.0379i −76.7267 + 15.2619i −31.8191 + 47.6207i 48.9226 118.110i 38.0949 + 15.7794i 177.164 + 118.377i
5.1 −2.55538 6.16923i −1.90204 2.84660i −20.2157 + 20.2157i −0.0751892 + 0.378002i −12.7009 + 19.0082i −11.8164 59.4051i 77.6662 + 32.1704i 26.5120 64.0055i 2.52411 0.502078i
5.2 −1.10044 2.65671i 8.44506 + 12.6389i 5.46660 5.46660i 3.31718 16.6766i 24.2846 36.3445i 6.66998 + 33.5323i −63.0461 26.1146i −57.4261 + 138.639i −47.9552 + 9.53888i
5.3 −0.210299 0.507707i −5.96248 8.92348i 11.1002 11.1002i 1.57331 7.90954i −3.27661 + 4.90379i 7.85546 + 39.4921i −16.0933 6.66606i −13.0800 + 31.5779i −4.34659 + 0.864591i
5.4 1.12676 + 2.72024i 2.50076 + 3.74265i 5.18357 5.18357i −7.34972 + 36.9495i −7.36316 + 11.0198i −15.5535 78.1927i 63.4651 + 26.2881i 23.2437 56.1153i −108.793 + 21.6403i
5.5 2.44646 + 5.90629i 0.660541 + 0.988570i −17.5853 + 17.5853i 5.91128 29.7180i −4.22279 + 6.31985i 9.51676 + 47.8440i −52.3853 21.6987i 30.4564 73.5283i 189.985 37.7903i
6.1 −7.12052 2.94942i −0.829686 + 0.165035i 30.6890 + 30.6890i −14.8705 + 22.2553i 6.39455 + 1.27196i 16.8953 + 25.2857i −80.8164 195.108i −74.1731 + 30.7235i 171.526 114.610i
6.2 −2.92749 1.21261i 8.22545 1.63614i −4.21393 4.21393i 21.2448 31.7951i −26.0639 5.18443i 5.22138 + 7.81435i 26.6281 + 64.2859i −9.85322 + 4.08134i −100.749 + 67.3181i
6.3 −1.63167 0.675861i −11.4219 + 2.27196i −9.10814 9.10814i −6.20205 + 9.28203i 20.1724 + 4.01253i −12.6661 18.9562i 19.5194 + 47.1241i 50.4644 20.9030i 16.3931 10.9535i
6.4 3.43536 + 1.42297i 6.42184 1.27738i −1.53684 1.53684i −8.81924 + 13.1989i 23.8790 + 4.74983i 5.40438 + 8.08823i −25.8603 62.4323i −35.2259 + 14.5911i −49.0790 + 32.7936i
6.5 6.53721 + 2.70780i −10.8435 + 2.15691i 24.0892 + 24.0892i 16.7298 25.0379i −76.7267 15.2619i −31.8191 47.6207i 48.9226 + 118.110i 38.0949 15.7794i 177.164 118.377i
7.1 −2.55538 + 6.16923i −1.90204 + 2.84660i −20.2157 20.2157i −0.0751892 0.378002i −12.7009 19.0082i −11.8164 + 59.4051i 77.6662 32.1704i 26.5120 + 64.0055i 2.52411 + 0.502078i
7.2 −1.10044 + 2.65671i 8.44506 12.6389i 5.46660 + 5.46660i 3.31718 + 16.6766i 24.2846 + 36.3445i 6.66998 33.5323i −63.0461 + 26.1146i −57.4261 138.639i −47.9552 9.53888i
7.3 −0.210299 + 0.507707i −5.96248 + 8.92348i 11.1002 + 11.1002i 1.57331 + 7.90954i −3.27661 4.90379i 7.85546 39.4921i −16.0933 + 6.66606i −13.0800 31.5779i −4.34659 0.864591i
7.4 1.12676 2.72024i 2.50076 3.74265i 5.18357 + 5.18357i −7.34972 36.9495i −7.36316 11.0198i −15.5535 + 78.1927i 63.4651 26.2881i 23.2437 + 56.1153i −108.793 21.6403i
7.5 2.44646 5.90629i 0.660541 0.988570i −17.5853 17.5853i 5.91128 + 29.7180i −4.22279 6.31985i 9.51676 47.8440i −52.3853 + 21.6987i 30.4564 + 73.5283i 189.985 + 37.7903i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.5.e.a 40
3.b odd 2 1 153.5.p.a 40
17.e odd 16 1 inner 17.5.e.a 40
51.i even 16 1 153.5.p.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.5.e.a 40 1.a even 1 1 trivial
17.5.e.a 40 17.e odd 16 1 inner
153.5.p.a 40 3.b odd 2 1
153.5.p.a 40 51.i even 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(17, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database