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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.75728937243\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
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.

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

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:
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Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(17, [\chi])\).