# Properties

 Label 289.10.a.g Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $1$ Dimension $36$ CM no Inner twists $1$

# 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: $$36$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 486q^{3} + 9216q^{4} - 3750q^{5} - 11061q^{6} - 29040q^{7} + 24837q^{8} + 236196q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 486q^{3} + 9216q^{4} - 3750q^{5} - 11061q^{6} - 29040q^{7} + 24837q^{8} + 236196q^{9} - 60000q^{10} - 76902q^{11} - 373248q^{12} + 54216q^{13} - 17373q^{14} - 34122q^{15} + 2359296q^{16} - 1779435q^{18} - 245058q^{19} - 6439479q^{20} - 138102q^{21} - 267324q^{22} - 4041462q^{23} - 7653888q^{24} + 16582356q^{25} + 15822744q^{26} - 13281612q^{27} - 18614784q^{28} - 4005936q^{29} + 22471686q^{30} - 21257064q^{31} - 30922641q^{32} + 35736474q^{33} - 9039642q^{35} + 39076761q^{36} - 22076682q^{37} - 27401376q^{38} - 62736162q^{39} + 12231630q^{40} - 59641782q^{41} + 150001536q^{42} - 47951586q^{43} + 49578936q^{44} - 129308238q^{45} - 140524827q^{46} - 118557912q^{47} - 407719119q^{48} + 99849138q^{49} + 435669051q^{50} - 105017607q^{52} + 13698846q^{53} - 209848575q^{54} - 365439924q^{55} - 203095059q^{56} + 4614108q^{57} + 179071413q^{58} + 343015128q^{59} + 427179186q^{60} - 175597116q^{61} - 720602571q^{62} - 587415936q^{63} + 853082511q^{64} - 393820182q^{65} - 494661978q^{66} + 502776528q^{67} - 469106598q^{69} - 1062525966q^{70} - 1308709542q^{71} - 275337849q^{72} - 494841342q^{73} - 1545361890q^{74} - 1824677616q^{75} + 242064891q^{76} - 792768144q^{77} - 2270624538q^{78} - 1980107868q^{79} - 2897000199q^{80} + 1598298840q^{81} - 898743654q^{82} + 275294520q^{83} - 2144532369q^{84} - 2880848046q^{86} + 1088458710q^{87} + 2705904618q^{88} + 148394658q^{89} - 117916215q^{90} - 636340896q^{91} + 3458472327q^{92} - 628345524q^{93} - 200245965q^{94} - 4878626298q^{95} + 8390096634q^{96} + 891786822q^{97} + 4285627647q^{98} + 1476187998q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −44.7383 −202.519 1489.51 −1685.49 9060.35 4589.24 −43732.3 21330.9 75405.9
1.2 −43.7427 −83.5916 1401.42 712.522 3656.52 −5542.36 −38905.7 −12695.4 −31167.6
1.3 −38.3451 247.035 958.344 −875.650 −9472.58 −7072.73 −17115.1 41343.4 33576.9
1.4 −37.5513 −70.9462 898.103 98.5382 2664.12 4159.85 −14498.7 −14649.6 −3700.24
1.5 −35.4923 64.4314 747.701 1779.33 −2286.82 −1603.15 −8365.56 −15531.6 −63152.6
1.6 −33.0540 199.787 580.568 −108.765 −6603.77 1478.44 −2266.45 20231.9 3595.11
1.7 −32.5443 213.342 547.129 −2406.57 −6943.04 −8347.75 −1143.25 25831.6 78320.2
1.8 −26.9492 −20.4039 214.258 −1220.88 549.868 4892.40 8023.91 −19266.7 32901.7
1.9 −22.6693 −265.602 1.89626 −368.101 6021.01 2877.49 11563.7 50861.5 8344.59
1.10 −22.4936 −161.121 −6.03889 −1284.66 3624.20 −9354.17 11652.5 6277.10 28896.7
1.11 −21.2323 −64.4551 −61.1895 −398.305 1368.53 3851.88 12170.1 −15528.5 8456.93
1.12 −20.0534 107.002 −109.862 2675.64 −2145.75 −1982.25 12470.4 −8233.60 −53655.7
1.13 −16.2320 −276.194 −248.524 1638.38 4483.17 −8498.12 12344.8 56600.2 −26594.2
1.14 −15.8730 −48.4836 −260.047 2428.05 769.582 −4466.03 12254.7 −17332.3 −38540.5
1.15 −10.3896 121.426 −404.056 −1455.35 −1261.57 5787.38 9517.45 −4938.65 15120.5
1.16 −8.02687 214.990 −447.569 357.217 −1725.70 11795.1 7702.34 26537.7 −2867.34
1.17 −7.00956 −248.596 −462.866 1624.49 1742.55 8782.15 6833.38 42117.0 −11386.9
1.18 −3.07469 18.6108 −502.546 −2540.06 −57.2223 −10949.3 3119.41 −19336.6 7809.90
1.19 −0.561663 191.593 −511.685 819.697 −107.611 1085.21 574.966 17024.9 −460.393
1.20 2.64758 −62.7112 −504.990 −1957.56 −166.033 −4153.19 −2692.56 −15750.3 −5182.80
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.10.a.g 36
17.b even 2 1 289.10.a.h yes 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.10.a.g 36 1.a even 1 1 trivial
289.10.a.h yes 36 17.b even 2 1

## Hecke kernels

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

 $$82\!\cdots\!91$$$$T_{2}^{28} +$$$$23\!\cdots\!35$$$$T_{2}^{27} -$$$$14\!\cdots\!46$$$$T_{2}^{26} -$$$$54\!\cdots\!52$$$$T_{2}^{25} +$$$$19\!\cdots\!00$$$$T_{2}^{24} +$$$$87\!\cdots\!44$$$$T_{2}^{23} -$$$$18\!\cdots\!40$$$$T_{2}^{22} -$$$$10\!\cdots\!36$$$$T_{2}^{21} +$$$$13\!\cdots\!92$$$$T_{2}^{20} +$$$$86\!\cdots\!48$$$$T_{2}^{19} -$$$$74\!\cdots\!20$$$$T_{2}^{18} -$$$$52\!\cdots\!28$$$$T_{2}^{17} +$$$$29\!\cdots\!20$$$$T_{2}^{16} +$$$$23\!\cdots\!04$$$$T_{2}^{15} -$$$$84\!\cdots\!28$$$$T_{2}^{14} -$$$$74\!\cdots\!92$$$$T_{2}^{13} +$$$$16\!\cdots\!56$$$$T_{2}^{12} +$$$$16\!\cdots\!56$$$$T_{2}^{11} -$$$$22\!\cdots\!44$$$$T_{2}^{10} -$$$$23\!\cdots\!84$$$$T_{2}^{9} +$$$$17\!\cdots\!60$$$$T_{2}^{8} +$$$$20\!\cdots\!64$$$$T_{2}^{7} -$$$$70\!\cdots\!84$$$$T_{2}^{6} -$$$$95\!\cdots\!16$$$$T_{2}^{5} +$$$$78\!\cdots\!28$$$$T_{2}^{4} +$$$$18\!\cdots\!04$$$$T_{2}^{3} +$$$$69\!\cdots\!32$$$$T_{2}^{2} -$$$$97\!\cdots\!64$$$$T_{2} -$$$$53\!\cdots\!16$$">$$T_{2}^{36} - \cdots$$ $$36\!\cdots\!48$$$$T_{3}^{31} -$$$$39\!\cdots\!08$$$$T_{3}^{30} -$$$$40\!\cdots\!24$$$$T_{3}^{29} +$$$$12\!\cdots\!97$$$$T_{3}^{28} +$$$$29\!\cdots\!60$$$$T_{3}^{27} +$$$$42\!\cdots\!72$$$$T_{3}^{26} -$$$$14\!\cdots\!18$$$$T_{3}^{25} -$$$$66\!\cdots\!42$$$$T_{3}^{24} +$$$$52\!\cdots\!96$$$$T_{3}^{23} +$$$$35\!\cdots\!40$$$$T_{3}^{22} -$$$$13\!\cdots\!84$$$$T_{3}^{21} -$$$$11\!\cdots\!16$$$$T_{3}^{20} +$$$$24\!\cdots\!18$$$$T_{3}^{19} +$$$$26\!\cdots\!94$$$$T_{3}^{18} -$$$$32\!\cdots\!94$$$$T_{3}^{17} -$$$$40\!\cdots\!93$$$$T_{3}^{16} +$$$$27\!\cdots\!84$$$$T_{3}^{15} +$$$$42\!\cdots\!64$$$$T_{3}^{14} -$$$$14\!\cdots\!36$$$$T_{3}^{13} -$$$$30\!\cdots\!61$$$$T_{3}^{12} +$$$$28\!\cdots\!94$$$$T_{3}^{11} +$$$$14\!\cdots\!18$$$$T_{3}^{10} +$$$$11\!\cdots\!44$$$$T_{3}^{9} -$$$$42\!\cdots\!83$$$$T_{3}^{8} -$$$$84\!\cdots\!20$$$$T_{3}^{7} +$$$$67\!\cdots\!16$$$$T_{3}^{6} +$$$$19\!\cdots\!44$$$$T_{3}^{5} -$$$$49\!\cdots\!70$$$$T_{3}^{4} -$$$$17\!\cdots\!50$$$$T_{3}^{3} +$$$$98\!\cdots\!22$$$$T_{3}^{2} +$$$$40\!\cdots\!30$$$$T_{3} +$$$$96\!\cdots\!49$$">$$T_{3}^{36} + \cdots$$