# Properties

 Label 289.10.a.h Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $0$ Dimension $36$ CM no Inner twists $1$

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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: $$36$$ 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.h yes 36
17.b even 2 1 289.10.a.g 36

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

## Hecke kernels

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

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