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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.

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

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:
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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 \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2}^{26} - 32 T_{2}^{25} - 9472 T_{2}^{24} + 294912 T_{2}^{23} + 38993919 T_{2}^{22} - 1179754716 T_{2}^{21} - 91560086140 T_{2}^{20} + 2688404815616 T_{2}^{19} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{52} - 682344 T_{3}^{50} + 216870628408 T_{3}^{48} + \cdots + 32\!\cdots\!88 \) Copy content Toggle raw display