# Properties

 Label 289.10.a.f Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $1$ Dimension $24$ CM no Inner twists $2$

# Related objects

## 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: $$1$$ Dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 5124q^{4} + 108368q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 5124q^{4} + 108368q^{9} - 244832q^{13} - 127332q^{15} - 279932q^{16} + 888764q^{18} - 2280212q^{19} + 775748q^{21} - 2762628q^{25} - 3334452q^{26} - 39084792q^{30} + 2010240q^{32} - 30349992q^{33} - 25532364q^{35} + 31177320q^{36} - 13171392q^{38} - 86527192q^{42} + 61046960q^{43} - 153365328q^{47} + 169445876q^{49} - 236105676q^{50} - 209898380q^{52} + 85777812q^{53} - 255767540q^{55} - 767709024q^{59} + 429639656q^{60} - 1006108924q^{64} + 346830788q^{66} - 26076868q^{67} - 751973532q^{69} - 319504544q^{70} - 1171736028q^{72} - 1640047616q^{76} - 174401076q^{77} - 347156560q^{81} - 1649346672q^{83} - 935672904q^{84} + 690159588q^{86} - 257027500q^{87} + 191594460q^{89} + 1842509012q^{93} + 2877218432q^{94} + 4413444720q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −38.1697 −135.302 944.927 33.6875 5164.44 −4816.06 −16524.7 −1376.34 −1285.84
1.2 −38.1697 135.302 944.927 −33.6875 −5164.44 4816.06 −16524.7 −1376.34 1285.84
1.3 −36.7615 −213.730 839.411 −2187.52 7857.06 −1444.09 −12036.1 25997.7 80416.5
1.4 −36.7615 213.730 839.411 2187.52 −7857.06 1444.09 −12036.1 25997.7 −80416.5
1.5 −31.4159 −60.4230 474.957 1582.21 1898.24 −10488.5 1163.73 −16032.1 −49706.4
1.6 −31.4159 60.4230 474.957 −1582.21 −1898.24 10488.5 1163.73 −16032.1 49706.4
1.7 −17.1503 −137.714 −217.866 1247.89 2361.84 10551.1 12517.4 −717.820 −21401.7
1.8 −17.1503 137.714 −217.866 −1247.89 −2361.84 −10551.1 12517.4 −717.820 21401.7
1.9 −12.8587 −204.729 −346.654 120.661 2632.54 4525.78 11041.2 22230.9 −1551.54
1.10 −12.8587 204.729 −346.654 −120.661 −2632.54 −4525.78 11041.2 22230.9 1551.54
1.11 −9.57683 −3.76279 −420.284 1873.25 36.0355 −1553.61 8928.33 −19668.8 −17939.8
1.12 −9.57683 3.76279 −420.284 −1873.25 −36.0355 1553.61 8928.33 −19668.8 17939.8
1.13 7.42963 −213.375 −456.801 298.376 −1585.29 −7723.69 −7197.83 25845.7 2216.82
1.14 7.42963 213.375 −456.801 −298.376 1585.29 7723.69 −7197.83 25845.7 −2216.82
1.15 12.9147 −101.422 −345.210 −2078.59 −1309.84 −2161.61 −11070.6 −9396.49 −26844.4
1.16 12.9147 101.422 −345.210 2078.59 1309.84 2161.61 −11070.6 −9396.49 26844.4
1.17 21.4252 −32.0790 −52.9591 −1493.58 −687.300 −1220.42 −12104.4 −18653.9 −32000.3
1.18 21.4252 32.0790 −52.9591 1493.58 687.300 1220.42 −12104.4 −18653.9 32000.3
1.19 28.8329 −253.694 319.336 1602.54 −7314.73 −6961.05 −5555.06 44677.5 46205.9
1.20 28.8329 253.694 319.336 −1602.54 7314.73 6961.05 −5555.06 44677.5 −46205.9
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 24
17.b even 2 1 inner 289.10.a.f 24
17.d even 8 2 17.10.c.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.c.a 24 17.d even 8 2
289.10.a.f 24 1.a even 1 1 trivial
289.10.a.f 24 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$15\!\cdots\!24$$$$T_{2}^{4} +$$$$10\!\cdots\!08$$$$T_{2}^{3} -$$$$19\!\cdots\!68$$$$T_{2}^{2} -$$$$16\!\cdots\!72$$$$T_{2} +$$$$78\!\cdots\!12$$">$$T_{2}^{12} - \cdots$$ $$24\!\cdots\!36$$$$T_{3}^{18} +$$$$10\!\cdots\!76$$$$T_{3}^{16} -$$$$28\!\cdots\!12$$$$T_{3}^{14} +$$$$48\!\cdots\!24$$$$T_{3}^{12} -$$$$50\!\cdots\!08$$$$T_{3}^{10} +$$$$30\!\cdots\!12$$$$T_{3}^{8} -$$$$97\!\cdots\!72$$$$T_{3}^{6} +$$$$14\!\cdots\!80$$$$T_{3}^{4} -$$$$77\!\cdots\!48$$$$T_{3}^{2} +$$$$10\!\cdots\!00$$">$$T_{3}^{24} - \cdots$$