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Properties

Label 289.10.a.i
Level $289$
Weight $10$
Character orbit 289.a
Self dual yes
Analytic conductor $148.845$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + 156200q^{13} + 1207872q^{15} + 3407880q^{16} + 2193336q^{18} + 1185568q^{19} + 5198336q^{21} + 13827692q^{25} + 3618944q^{26} - 9167544q^{30} + 61884888q^{32} + 1635208q^{33} + 46992776q^{35} + 156027320q^{36} + 84813952q^{38} - 4635776q^{42} + 125448912q^{43} + 164193176q^{47} + 270850284q^{49} - 226223888q^{50} + 103553016q^{52} + 426167208q^{53} + 677761520q^{55} + 375214512q^{59} + 336918024q^{60} + 190014416q^{64} + 1377178928q^{66} + 311910088q^{67} + 533688136q^{69} + 1477690280q^{70} + 2757942680q^{72} + 4047975520q^{76} + 3440336432q^{77} + 3266558756q^{81} + 2072890608q^{83} + 2630025952q^{84} + 1538547296q^{86} - 1010436256q^{87} + 1873849184q^{89} - 1998451624q^{93} - 6880776704q^{94} - 4667454128q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −41.2664 −175.384 1190.92 699.952 7237.48 8848.92 −28016.5 11076.6 −28884.5
1.2 −41.2664 175.384 1190.92 −699.952 −7237.48 −8848.92 −28016.5 11076.6 28884.5
1.3 −40.6100 −102.962 1137.17 −769.524 4181.30 1567.27 −25388.3 −9081.79 31250.4
1.4 −40.6100 102.962 1137.17 769.524 −4181.30 −1567.27 −25388.3 −9081.79 −31250.4
1.5 −37.3725 −245.740 884.705 −1549.27 9183.93 −9379.03 −13928.9 40705.3 57900.2
1.6 −37.3725 245.740 884.705 1549.27 −9183.93 9379.03 −13928.9 40705.3 −57900.2
1.7 −35.0759 −118.571 718.321 −1635.44 4158.98 10785.0 −7236.92 −5623.96 57364.5
1.8 −35.0759 118.571 718.321 1635.44 −4158.98 −10785.0 −7236.92 −5623.96 −57364.5
1.9 −31.9341 −159.334 507.789 1397.46 5088.20 −9976.78 134.480 5704.34 −44626.6
1.10 −31.9341 159.334 507.789 −1397.46 −5088.20 9976.78 134.480 5704.34 44626.6
1.11 −26.9890 −111.659 216.409 −2126.40 3013.56 −8475.21 7977.73 −7215.34 57389.5
1.12 −26.9890 111.659 216.409 2126.40 −3013.56 8475.21 7977.73 −7215.34 −57389.5
1.13 −25.4513 −80.7952 135.770 783.444 2056.34 5744.00 9575.55 −13155.1 −19939.7
1.14 −25.4513 80.7952 135.770 −783.444 −2056.34 −5744.00 9575.55 −13155.1 19939.7
1.15 −24.0067 −194.286 64.3195 2543.89 4664.15 −1962.84 10747.3 18063.9 −61070.2
1.16 −24.0067 194.286 64.3195 −2543.89 −4664.15 1962.84 10747.3 18063.9 61070.2
1.17 −14.8865 −140.613 −290.392 1537.70 2093.24 2138.55 11944.8 89.0110 −22891.0
1.18 −14.8865 140.613 −290.392 −1537.70 −2093.24 −2138.55 11944.8 89.0110 22891.0
1.19 −8.91738 −33.9373 −432.480 34.8748 302.632 8539.90 8422.29 −18531.3 −310.992
1.20 −8.91738 33.9373 −432.480 −34.8748 −302.632 −8539.90 8422.29 −18531.3 310.992
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.10.a.i 52
17.b even 2 1 inner 289.10.a.i 52
17.e odd 16 2 17.10.d.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.d.a ✓ 52 17.e odd 16 2
289.10.a.i 52 1.a even 1 1 trivial
289.10.a.i 52 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):

\(26\!\cdots\!16\)\( T_{2}^{19} + \)\(13\!\cdots\!92\)\( T_{2}^{18} - \)\(38\!\cdots\!08\)\( T_{2}^{17} - \)\(13\!\cdots\!44\)\( T_{2}^{16} + \)\(36\!\cdots\!20\)\( T_{2}^{15} + \)\(84\!\cdots\!84\)\( T_{2}^{14} - \)\(22\!\cdots\!56\)\( T_{2}^{13} - \)\(35\!\cdots\!64\)\( T_{2}^{12} + \)\(88\!\cdots\!72\)\( T_{2}^{11} + \)\(95\!\cdots\!20\)\( T_{2}^{10} - \)\(21\!\cdots\!24\)\( T_{2}^{9} - \)\(16\!\cdots\!80\)\( T_{2}^{8} + \)\(30\!\cdots\!08\)\( T_{2}^{7} + \)\(17\!\cdots\!28\)\( T_{2}^{6} - \)\(20\!\cdots\!16\)\( T_{2}^{5} - \)\(11\!\cdots\!64\)\( T_{2}^{4} + \)\(43\!\cdots\!24\)\( T_{2}^{3} + \)\(30\!\cdots\!44\)\( T_{2}^{2} + \)\(19\!\cdots\!40\)\( T_{2} - \)\(60\!\cdots\!00\)\( \)">\(T_{2}^{26} - \cdots\)
\(42\!\cdots\!00\)\( T_{3}^{46} + \)\(58\!\cdots\!60\)\( T_{3}^{44} - \)\(58\!\cdots\!72\)\( T_{3}^{42} + \)\(45\!\cdots\!60\)\( T_{3}^{40} - \)\(27\!\cdots\!40\)\( T_{3}^{38} + \)\(13\!\cdots\!04\)\( T_{3}^{36} - \)\(53\!\cdots\!76\)\( T_{3}^{34} + \)\(17\!\cdots\!92\)\( T_{3}^{32} - \)\(44\!\cdots\!92\)\( T_{3}^{30} + \)\(95\!\cdots\!68\)\( T_{3}^{28} - \)\(16\!\cdots\!44\)\( T_{3}^{26} + \)\(24\!\cdots\!96\)\( T_{3}^{24} - \)\(27\!\cdots\!16\)\( T_{3}^{22} + \)\(26\!\cdots\!36\)\( T_{3}^{20} - \)\(19\!\cdots\!52\)\( T_{3}^{18} + \)\(11\!\cdots\!80\)\( T_{3}^{16} - \)\(51\!\cdots\!08\)\( T_{3}^{14} + \)\(17\!\cdots\!44\)\( T_{3}^{12} - \)\(43\!\cdots\!56\)\( T_{3}^{10} + \)\(73\!\cdots\!04\)\( T_{3}^{8} - \)\(82\!\cdots\!28\)\( T_{3}^{6} + \)\(57\!\cdots\!92\)\( T_{3}^{4} - \)\(21\!\cdots\!08\)\( T_{3}^{2} + \)\(32\!\cdots\!88\)\( \)">\(T_{3}^{52} - \cdots\)