# Properties

 Label 637.2.a.j.1.1 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} -2.81361 q^{5} -5.62721 q^{6} +1.28917 q^{8} +6.62721 q^{9} +O(q^{10})$$ $$q-1.81361 q^{2} +3.10278 q^{3} +1.28917 q^{4} -2.81361 q^{5} -5.62721 q^{6} +1.28917 q^{8} +6.62721 q^{9} +5.10278 q^{10} +3.10278 q^{11} +4.00000 q^{12} -1.00000 q^{13} -8.72999 q^{15} -4.91638 q^{16} +0.524438 q^{17} -12.0192 q^{18} -0.813607 q^{19} -3.62721 q^{20} -5.62721 q^{22} +7.33804 q^{23} +4.00000 q^{24} +2.91638 q^{25} +1.81361 q^{26} +11.2544 q^{27} +8.28917 q^{29} +15.8328 q^{30} -1.39194 q^{31} +6.33804 q^{32} +9.62721 q^{33} -0.951124 q^{34} +8.54359 q^{36} -6.15165 q^{37} +1.47556 q^{38} -3.10278 q^{39} -3.62721 q^{40} +4.20555 q^{41} +6.75971 q^{43} +4.00000 q^{44} -18.6464 q^{45} -13.3083 q^{46} +5.97028 q^{47} -15.2544 q^{48} -5.28917 q^{50} +1.62721 q^{51} -1.28917 q^{52} -2.49472 q^{53} -20.4111 q^{54} -8.72999 q^{55} -2.52444 q^{57} -15.0333 q^{58} +4.47054 q^{59} -11.2544 q^{60} +2.00000 q^{61} +2.52444 q^{62} -1.66196 q^{64} +2.81361 q^{65} -17.4600 q^{66} +10.0383 q^{67} +0.676089 q^{68} +22.7683 q^{69} -8.72999 q^{71} +8.54359 q^{72} +2.34307 q^{73} +11.1567 q^{74} +9.04888 q^{75} -1.04888 q^{76} +5.62721 q^{78} -13.5436 q^{79} +13.8328 q^{80} +15.0383 q^{81} -7.62721 q^{82} -16.4791 q^{83} -1.47556 q^{85} -12.2594 q^{86} +25.7194 q^{87} +4.00000 q^{88} +10.6464 q^{89} +33.8172 q^{90} +9.45998 q^{92} -4.31889 q^{93} -10.8277 q^{94} +2.28917 q^{95} +19.6655 q^{96} +1.18639 q^{97} +20.5628 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9}+O(q^{10})$$ 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 $$3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23} + 12 q^{24} - 5 q^{25} - q^{26} + 8 q^{27} + 24 q^{29} + 20 q^{30} + 4 q^{31} + 7 q^{32} + 16 q^{33} - 14 q^{34} - q^{36} + 10 q^{38} - 2 q^{39} + 2 q^{40} - 2 q^{41} + 10 q^{43} + 12 q^{44} - 22 q^{45} - 18 q^{46} + 8 q^{47} - 20 q^{48} - 15 q^{50} - 8 q^{51} - 3 q^{52} + 8 q^{53} - 32 q^{54} - 6 q^{55} - 2 q^{57} + 12 q^{58} + 4 q^{59} - 8 q^{60} + 6 q^{61} + 2 q^{62} - 17 q^{64} + 2 q^{65} - 12 q^{66} - 12 q^{67} - 22 q^{68} + 6 q^{69} - 6 q^{71} - q^{72} + 10 q^{73} + 30 q^{74} + 16 q^{75} + 8 q^{76} + 4 q^{78} - 14 q^{79} + 14 q^{80} + 3 q^{81} - 10 q^{82} + 12 q^{83} - 10 q^{85} - 26 q^{86} + 26 q^{87} + 12 q^{88} - 2 q^{89} + 28 q^{90} - 12 q^{92} - 22 q^{93} + 10 q^{94} + 6 q^{95} + 4 q^{96} + 10 q^{97} + 14 q^{99}+O(q^{100})$$ 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 3 * q^13 - 6 * q^15 - q^16 - 4 * q^17 - 15 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 10 * q^23 + 12 * q^24 - 5 * q^25 - q^26 + 8 * q^27 + 24 * q^29 + 20 * q^30 + 4 * q^31 + 7 * q^32 + 16 * q^33 - 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 2 * q^41 + 10 * q^43 + 12 * q^44 - 22 * q^45 - 18 * q^46 + 8 * q^47 - 20 * q^48 - 15 * q^50 - 8 * q^51 - 3 * q^52 + 8 * q^53 - 32 * q^54 - 6 * q^55 - 2 * q^57 + 12 * q^58 + 4 * q^59 - 8 * q^60 + 6 * q^61 + 2 * q^62 - 17 * q^64 + 2 * q^65 - 12 * q^66 - 12 * q^67 - 22 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 10 * q^73 + 30 * q^74 + 16 * q^75 + 8 * q^76 + 4 * q^78 - 14 * q^79 + 14 * q^80 + 3 * q^81 - 10 * q^82 + 12 * q^83 - 10 * q^85 - 26 * q^86 + 26 * q^87 + 12 * q^88 - 2 * q^89 + 28 * q^90 - 12 * q^92 - 22 * q^93 + 10 * q^94 + 6 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.81361 −1.28241 −0.641207 0.767368i $$-0.721566\pi$$
−0.641207 + 0.767368i $$0.721566\pi$$
$$3$$ 3.10278 1.79139 0.895694 0.444671i $$-0.146679\pi$$
0.895694 + 0.444671i $$0.146679\pi$$
$$4$$ 1.28917 0.644584
$$5$$ −2.81361 −1.25828 −0.629142 0.777291i $$-0.716593\pi$$
−0.629142 + 0.777291i $$0.716593\pi$$
$$6$$ −5.62721 −2.29730
$$7$$ 0 0
$$8$$ 1.28917 0.455790
$$9$$ 6.62721 2.20907
$$10$$ 5.10278 1.61364
$$11$$ 3.10278 0.935522 0.467761 0.883855i $$-0.345061\pi$$
0.467761 + 0.883855i $$0.345061\pi$$
$$12$$ 4.00000 1.15470
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −8.72999 −2.25407
$$16$$ −4.91638 −1.22910
$$17$$ 0.524438 0.127195 0.0635974 0.997976i $$-0.479743\pi$$
0.0635974 + 0.997976i $$0.479743\pi$$
$$18$$ −12.0192 −2.83294
$$19$$ −0.813607 −0.186654 −0.0933271 0.995636i $$-0.529750\pi$$
−0.0933271 + 0.995636i $$0.529750\pi$$
$$20$$ −3.62721 −0.811069
$$21$$ 0 0
$$22$$ −5.62721 −1.19973
$$23$$ 7.33804 1.53009 0.765044 0.643978i $$-0.222717\pi$$
0.765044 + 0.643978i $$0.222717\pi$$
$$24$$ 4.00000 0.816497
$$25$$ 2.91638 0.583276
$$26$$ 1.81361 0.355677
$$27$$ 11.2544 2.16592
$$28$$ 0 0
$$29$$ 8.28917 1.53926 0.769630 0.638490i $$-0.220441\pi$$
0.769630 + 0.638490i $$0.220441\pi$$
$$30$$ 15.8328 2.89065
$$31$$ −1.39194 −0.250000 −0.125000 0.992157i $$-0.539893\pi$$
−0.125000 + 0.992157i $$0.539893\pi$$
$$32$$ 6.33804 1.12042
$$33$$ 9.62721 1.67588
$$34$$ −0.951124 −0.163116
$$35$$ 0 0
$$36$$ 8.54359 1.42393
$$37$$ −6.15165 −1.01133 −0.505663 0.862731i $$-0.668752\pi$$
−0.505663 + 0.862731i $$0.668752\pi$$
$$38$$ 1.47556 0.239368
$$39$$ −3.10278 −0.496842
$$40$$ −3.62721 −0.573513
$$41$$ 4.20555 0.656797 0.328398 0.944539i $$-0.393491\pi$$
0.328398 + 0.944539i $$0.393491\pi$$
$$42$$ 0 0
$$43$$ 6.75971 1.03085 0.515423 0.856936i $$-0.327635\pi$$
0.515423 + 0.856936i $$0.327635\pi$$
$$44$$ 4.00000 0.603023
$$45$$ −18.6464 −2.77964
$$46$$ −13.3083 −1.96221
$$47$$ 5.97028 0.870855 0.435427 0.900224i $$-0.356597\pi$$
0.435427 + 0.900224i $$0.356597\pi$$
$$48$$ −15.2544 −2.20179
$$49$$ 0 0
$$50$$ −5.28917 −0.748001
$$51$$ 1.62721 0.227855
$$52$$ −1.28917 −0.178776
$$53$$ −2.49472 −0.342676 −0.171338 0.985212i $$-0.554809\pi$$
−0.171338 + 0.985212i $$0.554809\pi$$
$$54$$ −20.4111 −2.77760
$$55$$ −8.72999 −1.17715
$$56$$ 0 0
$$57$$ −2.52444 −0.334370
$$58$$ −15.0333 −1.97397
$$59$$ 4.47054 0.582015 0.291007 0.956721i $$-0.406010\pi$$
0.291007 + 0.956721i $$0.406010\pi$$
$$60$$ −11.2544 −1.45294
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 2.52444 0.320604
$$63$$ 0 0
$$64$$ −1.66196 −0.207744
$$65$$ 2.81361 0.348985
$$66$$ −17.4600 −2.14917
$$67$$ 10.0383 1.22638 0.613188 0.789937i $$-0.289887\pi$$
0.613188 + 0.789937i $$0.289887\pi$$
$$68$$ 0.676089 0.0819878
$$69$$ 22.7683 2.74098
$$70$$ 0 0
$$71$$ −8.72999 −1.03606 −0.518029 0.855363i $$-0.673334\pi$$
−0.518029 + 0.855363i $$0.673334\pi$$
$$72$$ 8.54359 1.00687
$$73$$ 2.34307 0.274235 0.137118 0.990555i $$-0.456216\pi$$
0.137118 + 0.990555i $$0.456216\pi$$
$$74$$ 11.1567 1.29694
$$75$$ 9.04888 1.04487
$$76$$ −1.04888 −0.120314
$$77$$ 0 0
$$78$$ 5.62721 0.637156
$$79$$ −13.5436 −1.52377 −0.761887 0.647710i $$-0.775727\pi$$
−0.761887 + 0.647710i $$0.775727\pi$$
$$80$$ 13.8328 1.54655
$$81$$ 15.0383 1.67092
$$82$$ −7.62721 −0.842285
$$83$$ −16.4791 −1.80882 −0.904410 0.426665i $$-0.859688\pi$$
−0.904410 + 0.426665i $$0.859688\pi$$
$$84$$ 0 0
$$85$$ −1.47556 −0.160047
$$86$$ −12.2594 −1.32197
$$87$$ 25.7194 2.75741
$$88$$ 4.00000 0.426401
$$89$$ 10.6464 1.12851 0.564256 0.825600i $$-0.309163\pi$$
0.564256 + 0.825600i $$0.309163\pi$$
$$90$$ 33.8172 3.56464
$$91$$ 0 0
$$92$$ 9.45998 0.986271
$$93$$ −4.31889 −0.447848
$$94$$ −10.8277 −1.11680
$$95$$ 2.28917 0.234864
$$96$$ 19.6655 2.00710
$$97$$ 1.18639 0.120460 0.0602300 0.998185i $$-0.480817\pi$$
0.0602300 + 0.998185i $$0.480817\pi$$
$$98$$ 0 0
$$99$$ 20.5628 2.06663
$$100$$ 3.75971 0.375971
$$101$$ −13.1028 −1.30377 −0.651887 0.758316i $$-0.726023\pi$$
−0.651887 + 0.758316i $$0.726023\pi$$
$$102$$ −2.95112 −0.292205
$$103$$ −4.41110 −0.434639 −0.217319 0.976101i $$-0.569731\pi$$
−0.217319 + 0.976101i $$0.569731\pi$$
$$104$$ −1.28917 −0.126413
$$105$$ 0 0
$$106$$ 4.52444 0.439452
$$107$$ 0.578337 0.0559100 0.0279550 0.999609i $$-0.491100\pi$$
0.0279550 + 0.999609i $$0.491100\pi$$
$$108$$ 14.5089 1.39611
$$109$$ 5.57331 0.533827 0.266913 0.963721i $$-0.413996\pi$$
0.266913 + 0.963721i $$0.413996\pi$$
$$110$$ 15.8328 1.50959
$$111$$ −19.0872 −1.81168
$$112$$ 0 0
$$113$$ 5.44584 0.512302 0.256151 0.966637i $$-0.417546\pi$$
0.256151 + 0.966637i $$0.417546\pi$$
$$114$$ 4.57834 0.428801
$$115$$ −20.6464 −1.92528
$$116$$ 10.6861 0.992183
$$117$$ −6.62721 −0.612686
$$118$$ −8.10780 −0.746383
$$119$$ 0 0
$$120$$ −11.2544 −1.02738
$$121$$ −1.37279 −0.124799
$$122$$ −3.62721 −0.328392
$$123$$ 13.0489 1.17658
$$124$$ −1.79445 −0.161146
$$125$$ 5.86248 0.524356
$$126$$ 0 0
$$127$$ −12.8816 −1.14306 −0.571530 0.820581i $$-0.693650\pi$$
−0.571530 + 0.820581i $$0.693650\pi$$
$$128$$ −9.66196 −0.854004
$$129$$ 20.9739 1.84664
$$130$$ −5.10278 −0.447543
$$131$$ −9.04888 −0.790604 −0.395302 0.918551i $$-0.629360\pi$$
−0.395302 + 0.918551i $$0.629360\pi$$
$$132$$ 12.4111 1.08025
$$133$$ 0 0
$$134$$ −18.2056 −1.57272
$$135$$ −31.6655 −2.72533
$$136$$ 0.676089 0.0579741
$$137$$ −6.25945 −0.534781 −0.267390 0.963588i $$-0.586161\pi$$
−0.267390 + 0.963588i $$0.586161\pi$$
$$138$$ −41.2927 −3.51507
$$139$$ 11.5733 0.981636 0.490818 0.871262i $$-0.336698\pi$$
0.490818 + 0.871262i $$0.336698\pi$$
$$140$$ 0 0
$$141$$ 18.5244 1.56004
$$142$$ 15.8328 1.32866
$$143$$ −3.10278 −0.259467
$$144$$ −32.5819 −2.71516
$$145$$ −23.3225 −1.93682
$$146$$ −4.24940 −0.351683
$$147$$ 0 0
$$148$$ −7.93051 −0.651884
$$149$$ 8.52444 0.698349 0.349175 0.937058i $$-0.386462\pi$$
0.349175 + 0.937058i $$0.386462\pi$$
$$150$$ −16.4111 −1.33996
$$151$$ −11.9844 −0.975278 −0.487639 0.873045i $$-0.662142\pi$$
−0.487639 + 0.873045i $$0.662142\pi$$
$$152$$ −1.04888 −0.0850751
$$153$$ 3.47556 0.280983
$$154$$ 0 0
$$155$$ 3.91638 0.314571
$$156$$ −4.00000 −0.320256
$$157$$ 12.8277 1.02377 0.511883 0.859055i $$-0.328948\pi$$
0.511883 + 0.859055i $$0.328948\pi$$
$$158$$ 24.5628 1.95411
$$159$$ −7.74055 −0.613866
$$160$$ −17.8328 −1.40980
$$161$$ 0 0
$$162$$ −27.2736 −2.14282
$$163$$ −13.4600 −1.05427 −0.527133 0.849783i $$-0.676733\pi$$
−0.527133 + 0.849783i $$0.676733\pi$$
$$164$$ 5.42166 0.423361
$$165$$ −27.0872 −2.10873
$$166$$ 29.8867 2.31965
$$167$$ 2.02972 0.157064 0.0785322 0.996912i $$-0.474977\pi$$
0.0785322 + 0.996912i $$0.474977\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 2.67609 0.205247
$$171$$ −5.39194 −0.412332
$$172$$ 8.71440 0.664467
$$173$$ −20.2978 −1.54321 −0.771605 0.636102i $$-0.780546\pi$$
−0.771605 + 0.636102i $$0.780546\pi$$
$$174$$ −46.6449 −3.53614
$$175$$ 0 0
$$176$$ −15.2544 −1.14985
$$177$$ 13.8711 1.04261
$$178$$ −19.3083 −1.44722
$$179$$ −11.0036 −0.822445 −0.411223 0.911535i $$-0.634898\pi$$
−0.411223 + 0.911535i $$0.634898\pi$$
$$180$$ −24.0383 −1.79171
$$181$$ −0.691675 −0.0514118 −0.0257059 0.999670i $$-0.508183\pi$$
−0.0257059 + 0.999670i $$0.508183\pi$$
$$182$$ 0 0
$$183$$ 6.20555 0.458727
$$184$$ 9.45998 0.697399
$$185$$ 17.3083 1.27253
$$186$$ 7.83276 0.574326
$$187$$ 1.62721 0.118994
$$188$$ 7.69670 0.561339
$$189$$ 0 0
$$190$$ −4.15165 −0.301192
$$191$$ 7.83276 0.566759 0.283379 0.959008i $$-0.408544\pi$$
0.283379 + 0.959008i $$0.408544\pi$$
$$192$$ −5.15667 −0.372151
$$193$$ −12.2056 −0.878575 −0.439287 0.898347i $$-0.644769\pi$$
−0.439287 + 0.898347i $$0.644769\pi$$
$$194$$ −2.15165 −0.154480
$$195$$ 8.72999 0.625167
$$196$$ 0 0
$$197$$ −18.8222 −1.34103 −0.670513 0.741898i $$-0.733926\pi$$
−0.670513 + 0.741898i $$0.733926\pi$$
$$198$$ −37.2927 −2.65028
$$199$$ −21.6116 −1.53201 −0.766004 0.642836i $$-0.777758\pi$$
−0.766004 + 0.642836i $$0.777758\pi$$
$$200$$ 3.75971 0.265851
$$201$$ 31.1466 2.19691
$$202$$ 23.7633 1.67198
$$203$$ 0 0
$$204$$ 2.09775 0.146872
$$205$$ −11.8328 −0.826436
$$206$$ 8.00000 0.557386
$$207$$ 48.6308 3.38007
$$208$$ 4.91638 0.340890
$$209$$ −2.52444 −0.174619
$$210$$ 0 0
$$211$$ −17.3764 −1.19624 −0.598119 0.801407i $$-0.704085\pi$$
−0.598119 + 0.801407i $$0.704085\pi$$
$$212$$ −3.21611 −0.220884
$$213$$ −27.0872 −1.85598
$$214$$ −1.04888 −0.0716997
$$215$$ −19.0192 −1.29710
$$216$$ 14.5089 0.987202
$$217$$ 0 0
$$218$$ −10.1078 −0.684586
$$219$$ 7.27001 0.491262
$$220$$ −11.2544 −0.758773
$$221$$ −0.524438 −0.0352775
$$222$$ 34.6167 2.32332
$$223$$ 10.5486 0.706388 0.353194 0.935550i $$-0.385096\pi$$
0.353194 + 0.935550i $$0.385096\pi$$
$$224$$ 0 0
$$225$$ 19.3275 1.28850
$$226$$ −9.87662 −0.656983
$$227$$ 6.95112 0.461362 0.230681 0.973029i $$-0.425905\pi$$
0.230681 + 0.973029i $$0.425905\pi$$
$$228$$ −3.25443 −0.215530
$$229$$ 21.0872 1.39348 0.696740 0.717323i $$-0.254633\pi$$
0.696740 + 0.717323i $$0.254633\pi$$
$$230$$ 37.4444 2.46901
$$231$$ 0 0
$$232$$ 10.6861 0.701579
$$233$$ 6.08362 0.398551 0.199276 0.979943i $$-0.436141\pi$$
0.199276 + 0.979943i $$0.436141\pi$$
$$234$$ 12.0192 0.785717
$$235$$ −16.7980 −1.09578
$$236$$ 5.76328 0.375157
$$237$$ −42.0227 −2.72967
$$238$$ 0 0
$$239$$ −14.2056 −0.918881 −0.459440 0.888209i $$-0.651950\pi$$
−0.459440 + 0.888209i $$0.651950\pi$$
$$240$$ 42.9200 2.77047
$$241$$ −8.44082 −0.543721 −0.271860 0.962337i $$-0.587639\pi$$
−0.271860 + 0.962337i $$0.587639\pi$$
$$242$$ 2.48970 0.160044
$$243$$ 12.8972 0.827357
$$244$$ 2.57834 0.165061
$$245$$ 0 0
$$246$$ −23.6655 −1.50886
$$247$$ 0.813607 0.0517685
$$248$$ −1.79445 −0.113948
$$249$$ −51.1310 −3.24030
$$250$$ −10.6322 −0.672442
$$251$$ 23.9844 1.51388 0.756941 0.653483i $$-0.226693\pi$$
0.756941 + 0.653483i $$0.226693\pi$$
$$252$$ 0 0
$$253$$ 22.7683 1.43143
$$254$$ 23.3622 1.46588
$$255$$ −4.57834 −0.286707
$$256$$ 20.8469 1.30293
$$257$$ −15.6116 −0.973827 −0.486913 0.873450i $$-0.661877\pi$$
−0.486913 + 0.873450i $$0.661877\pi$$
$$258$$ −38.0383 −2.36816
$$259$$ 0 0
$$260$$ 3.62721 0.224950
$$261$$ 54.9341 3.40033
$$262$$ 16.4111 1.01388
$$263$$ −15.1708 −0.935472 −0.467736 0.883868i $$-0.654930\pi$$
−0.467736 + 0.883868i $$0.654930\pi$$
$$264$$ 12.4111 0.763850
$$265$$ 7.01916 0.431183
$$266$$ 0 0
$$267$$ 33.0333 2.02160
$$268$$ 12.9411 0.790502
$$269$$ −23.2927 −1.42018 −0.710092 0.704109i $$-0.751347\pi$$
−0.710092 + 0.704109i $$0.751347\pi$$
$$270$$ 57.4288 3.49501
$$271$$ −12.7456 −0.774238 −0.387119 0.922030i $$-0.626530\pi$$
−0.387119 + 0.922030i $$0.626530\pi$$
$$272$$ −2.57834 −0.156335
$$273$$ 0 0
$$274$$ 11.3522 0.685810
$$275$$ 9.04888 0.545668
$$276$$ 29.3522 1.76679
$$277$$ 8.12193 0.488000 0.244000 0.969775i $$-0.421540\pi$$
0.244000 + 0.969775i $$0.421540\pi$$
$$278$$ −20.9894 −1.25886
$$279$$ −9.22471 −0.552269
$$280$$ 0 0
$$281$$ 19.0333 1.13543 0.567715 0.823225i $$-0.307827\pi$$
0.567715 + 0.823225i $$0.307827\pi$$
$$282$$ −33.5960 −2.00062
$$283$$ −11.1466 −0.662598 −0.331299 0.943526i $$-0.607487\pi$$
−0.331299 + 0.943526i $$0.607487\pi$$
$$284$$ −11.2544 −0.667827
$$285$$ 7.10278 0.420732
$$286$$ 5.62721 0.332744
$$287$$ 0 0
$$288$$ 42.0036 2.47508
$$289$$ −16.7250 −0.983821
$$290$$ 42.2978 2.48381
$$291$$ 3.68111 0.215791
$$292$$ 3.02061 0.176768
$$293$$ 14.1758 0.828161 0.414080 0.910240i $$-0.364103\pi$$
0.414080 + 0.910240i $$0.364103\pi$$
$$294$$ 0 0
$$295$$ −12.5783 −0.732339
$$296$$ −7.93051 −0.460952
$$297$$ 34.9200 2.02626
$$298$$ −15.4600 −0.895572
$$299$$ −7.33804 −0.424370
$$300$$ 11.6655 0.673509
$$301$$ 0 0
$$302$$ 21.7350 1.25071
$$303$$ −40.6550 −2.33557
$$304$$ 4.00000 0.229416
$$305$$ −5.62721 −0.322213
$$306$$ −6.30330 −0.360336
$$307$$ −13.5592 −0.773863 −0.386932 0.922108i $$-0.626465\pi$$
−0.386932 + 0.922108i $$0.626465\pi$$
$$308$$ 0 0
$$309$$ −13.6867 −0.778606
$$310$$ −7.10278 −0.403411
$$311$$ −0.426686 −0.0241952 −0.0120976 0.999927i $$-0.503851\pi$$
−0.0120976 + 0.999927i $$0.503851\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 18.1517 1.02599 0.512996 0.858391i $$-0.328536\pi$$
0.512996 + 0.858391i $$0.328536\pi$$
$$314$$ −23.2645 −1.31289
$$315$$ 0 0
$$316$$ −17.4600 −0.982200
$$317$$ 9.42166 0.529173 0.264587 0.964362i $$-0.414764\pi$$
0.264587 + 0.964362i $$0.414764\pi$$
$$318$$ 14.0383 0.787230
$$319$$ 25.7194 1.44001
$$320$$ 4.67609 0.261401
$$321$$ 1.79445 0.100156
$$322$$ 0 0
$$323$$ −0.426686 −0.0237415
$$324$$ 19.3869 1.07705
$$325$$ −2.91638 −0.161772
$$326$$ 24.4111 1.35201
$$327$$ 17.2927 0.956291
$$328$$ 5.42166 0.299361
$$329$$ 0 0
$$330$$ 49.1255 2.70427
$$331$$ −17.4005 −0.956420 −0.478210 0.878246i $$-0.658714\pi$$
−0.478210 + 0.878246i $$0.658714\pi$$
$$332$$ −21.2444 −1.16594
$$333$$ −40.7683 −2.23409
$$334$$ −3.68111 −0.201421
$$335$$ −28.2439 −1.54313
$$336$$ 0 0
$$337$$ 22.0524 1.20127 0.600637 0.799522i $$-0.294914\pi$$
0.600637 + 0.799522i $$0.294914\pi$$
$$338$$ −1.81361 −0.0986472
$$339$$ 16.8972 0.917731
$$340$$ −1.90225 −0.103164
$$341$$ −4.31889 −0.233881
$$342$$ 9.77886 0.528780
$$343$$ 0 0
$$344$$ 8.71440 0.469849
$$345$$ −64.0610 −3.44893
$$346$$ 36.8122 1.97903
$$347$$ 25.3522 1.36098 0.680488 0.732759i $$-0.261768\pi$$
0.680488 + 0.732759i $$0.261768\pi$$
$$348$$ 33.1567 1.77738
$$349$$ 5.70529 0.305397 0.152699 0.988273i $$-0.451204\pi$$
0.152699 + 0.988273i $$0.451204\pi$$
$$350$$ 0 0
$$351$$ −11.2544 −0.600717
$$352$$ 19.6655 1.04818
$$353$$ 28.6761 1.52627 0.763137 0.646237i $$-0.223658\pi$$
0.763137 + 0.646237i $$0.223658\pi$$
$$354$$ −25.1567 −1.33706
$$355$$ 24.5628 1.30366
$$356$$ 13.7250 0.727422
$$357$$ 0 0
$$358$$ 19.9561 1.05472
$$359$$ 11.0433 0.582845 0.291423 0.956594i $$-0.405871\pi$$
0.291423 + 0.956594i $$0.405871\pi$$
$$360$$ −24.0383 −1.26693
$$361$$ −18.3380 −0.965160
$$362$$ 1.25443 0.0659312
$$363$$ −4.25945 −0.223563
$$364$$ 0 0
$$365$$ −6.59247 −0.345066
$$366$$ −11.2544 −0.588278
$$367$$ −27.3466 −1.42748 −0.713741 0.700409i $$-0.753001\pi$$
−0.713741 + 0.700409i $$0.753001\pi$$
$$368$$ −36.0766 −1.88062
$$369$$ 27.8711 1.45091
$$370$$ −31.3905 −1.63191
$$371$$ 0 0
$$372$$ −5.56777 −0.288676
$$373$$ 16.1461 0.836014 0.418007 0.908444i $$-0.362729\pi$$
0.418007 + 0.908444i $$0.362729\pi$$
$$374$$ −2.95112 −0.152599
$$375$$ 18.1900 0.939326
$$376$$ 7.69670 0.396927
$$377$$ −8.28917 −0.426914
$$378$$ 0 0
$$379$$ 26.1305 1.34223 0.671117 0.741351i $$-0.265815\pi$$
0.671117 + 0.741351i $$0.265815\pi$$
$$380$$ 2.95112 0.151389
$$381$$ −39.9688 −2.04767
$$382$$ −14.2056 −0.726819
$$383$$ 21.0489 1.07555 0.537774 0.843089i $$-0.319265\pi$$
0.537774 + 0.843089i $$0.319265\pi$$
$$384$$ −29.9789 −1.52985
$$385$$ 0 0
$$386$$ 22.1361 1.12670
$$387$$ 44.7980 2.27721
$$388$$ 1.52946 0.0776466
$$389$$ −21.6061 −1.09547 −0.547736 0.836651i $$-0.684510\pi$$
−0.547736 + 0.836651i $$0.684510\pi$$
$$390$$ −15.8328 −0.801723
$$391$$ 3.84835 0.194619
$$392$$ 0 0
$$393$$ −28.0766 −1.41628
$$394$$ 34.1361 1.71975
$$395$$ 38.1063 1.91734
$$396$$ 26.5089 1.33212
$$397$$ 27.6952 1.38998 0.694992 0.719017i $$-0.255408\pi$$
0.694992 + 0.719017i $$0.255408\pi$$
$$398$$ 39.1950 1.96467
$$399$$ 0 0
$$400$$ −14.3380 −0.716902
$$401$$ −2.57834 −0.128756 −0.0643780 0.997926i $$-0.520506\pi$$
−0.0643780 + 0.997926i $$0.520506\pi$$
$$402$$ −56.4877 −2.81735
$$403$$ 1.39194 0.0693376
$$404$$ −16.8917 −0.840393
$$405$$ −42.3119 −2.10250
$$406$$ 0 0
$$407$$ −19.0872 −0.946117
$$408$$ 2.09775 0.103854
$$409$$ −15.1169 −0.747483 −0.373742 0.927533i $$-0.621925\pi$$
−0.373742 + 0.927533i $$0.621925\pi$$
$$410$$ 21.4600 1.05983
$$411$$ −19.4217 −0.958000
$$412$$ −5.68665 −0.280161
$$413$$ 0 0
$$414$$ −88.1971 −4.33465
$$415$$ 46.3658 2.27601
$$416$$ −6.33804 −0.310748
$$417$$ 35.9094 1.75849
$$418$$ 4.57834 0.223934
$$419$$ 9.99446 0.488261 0.244131 0.969742i $$-0.421497\pi$$
0.244131 + 0.969742i $$0.421497\pi$$
$$420$$ 0 0
$$421$$ −25.9250 −1.26351 −0.631753 0.775170i $$-0.717664\pi$$
−0.631753 + 0.775170i $$0.717664\pi$$
$$422$$ 31.5139 1.53407
$$423$$ 39.5663 1.92378
$$424$$ −3.21611 −0.156188
$$425$$ 1.52946 0.0741898
$$426$$ 49.1255 2.38014
$$427$$ 0 0
$$428$$ 0.745574 0.0360387
$$429$$ −9.62721 −0.464806
$$430$$ 34.4933 1.66341
$$431$$ 30.6761 1.47762 0.738808 0.673916i $$-0.235389\pi$$
0.738808 + 0.673916i $$0.235389\pi$$
$$432$$ −55.3311 −2.66212
$$433$$ 3.51941 0.169132 0.0845661 0.996418i $$-0.473050\pi$$
0.0845661 + 0.996418i $$0.473050\pi$$
$$434$$ 0 0
$$435$$ −72.3643 −3.46960
$$436$$ 7.18494 0.344096
$$437$$ −5.97028 −0.285597
$$438$$ −13.1849 −0.630001
$$439$$ −32.3517 −1.54406 −0.772030 0.635586i $$-0.780759\pi$$
−0.772030 + 0.635586i $$0.780759\pi$$
$$440$$ −11.2544 −0.536534
$$441$$ 0 0
$$442$$ 0.951124 0.0452404
$$443$$ 15.4458 0.733854 0.366927 0.930250i $$-0.380410\pi$$
0.366927 + 0.930250i $$0.380410\pi$$
$$444$$ −24.6066 −1.16778
$$445$$ −29.9547 −1.41999
$$446$$ −19.1310 −0.905881
$$447$$ 26.4494 1.25101
$$448$$ 0 0
$$449$$ −14.4705 −0.682907 −0.341453 0.939899i $$-0.610919\pi$$
−0.341453 + 0.939899i $$0.610919\pi$$
$$450$$ −35.0524 −1.65239
$$451$$ 13.0489 0.614448
$$452$$ 7.02061 0.330222
$$453$$ −37.1849 −1.74710
$$454$$ −12.6066 −0.591657
$$455$$ 0 0
$$456$$ −3.25443 −0.152402
$$457$$ −34.6705 −1.62182 −0.810910 0.585171i $$-0.801027\pi$$
−0.810910 + 0.585171i $$0.801027\pi$$
$$458$$ −38.2439 −1.78702
$$459$$ 5.90225 0.275493
$$460$$ −26.6167 −1.24101
$$461$$ −12.5400 −0.584047 −0.292024 0.956411i $$-0.594329\pi$$
−0.292024 + 0.956411i $$0.594329\pi$$
$$462$$ 0 0
$$463$$ 12.1517 0.564735 0.282368 0.959306i $$-0.408880\pi$$
0.282368 + 0.959306i $$0.408880\pi$$
$$464$$ −40.7527 −1.89190
$$465$$ 12.1517 0.563519
$$466$$ −11.0333 −0.511107
$$467$$ 37.0333 1.71370 0.856848 0.515569i $$-0.172419\pi$$
0.856848 + 0.515569i $$0.172419\pi$$
$$468$$ −8.54359 −0.394928
$$469$$ 0 0
$$470$$ 30.4650 1.40525
$$471$$ 39.8016 1.83396
$$472$$ 5.76328 0.265276
$$473$$ 20.9739 0.964379
$$474$$ 76.2127 3.50056
$$475$$ −2.37279 −0.108871
$$476$$ 0 0
$$477$$ −16.5330 −0.756996
$$478$$ 25.7633 1.17838
$$479$$ −12.0086 −0.548687 −0.274343 0.961632i $$-0.588460\pi$$
−0.274343 + 0.961632i $$0.588460\pi$$
$$480$$ −55.3311 −2.52551
$$481$$ 6.15165 0.280491
$$482$$ 15.3083 0.697275
$$483$$ 0 0
$$484$$ −1.76975 −0.0804434
$$485$$ −3.33804 −0.151573
$$486$$ −23.3905 −1.06101
$$487$$ 11.1184 0.503821 0.251911 0.967751i $$-0.418941\pi$$
0.251911 + 0.967751i $$0.418941\pi$$
$$488$$ 2.57834 0.116716
$$489$$ −41.7633 −1.88860
$$490$$ 0 0
$$491$$ 0.0594386 0.00268243 0.00134121 0.999999i $$-0.499573\pi$$
0.00134121 + 0.999999i $$0.499573\pi$$
$$492$$ 16.8222 0.758403
$$493$$ 4.34715 0.195786
$$494$$ −1.47556 −0.0663887
$$495$$ −57.8555 −2.60041
$$496$$ 6.84333 0.307274
$$497$$ 0 0
$$498$$ 92.7316 4.15540
$$499$$ 10.2978 0.460991 0.230496 0.973073i $$-0.425965\pi$$
0.230496 + 0.973073i $$0.425965\pi$$
$$500$$ 7.55773 0.337992
$$501$$ 6.29776 0.281363
$$502$$ −43.4983 −1.94142
$$503$$ 9.32391 0.415733 0.207866 0.978157i $$-0.433348\pi$$
0.207866 + 0.978157i $$0.433348\pi$$
$$504$$ 0 0
$$505$$ 36.8661 1.64052
$$506$$ −41.2927 −1.83569
$$507$$ 3.10278 0.137799
$$508$$ −16.6066 −0.736799
$$509$$ −39.6952 −1.75946 −0.879730 0.475473i $$-0.842277\pi$$
−0.879730 + 0.475473i $$0.842277\pi$$
$$510$$ 8.30330 0.367676
$$511$$ 0 0
$$512$$ −18.4842 −0.816892
$$513$$ −9.15667 −0.404277
$$514$$ 28.3133 1.24885
$$515$$ 12.4111 0.546898
$$516$$ 27.0388 1.19032
$$517$$ 18.5244 0.814704
$$518$$ 0 0
$$519$$ −62.9794 −2.76449
$$520$$ 3.62721 0.159064
$$521$$ −22.3627 −0.979729 −0.489865 0.871798i $$-0.662954\pi$$
−0.489865 + 0.871798i $$0.662954\pi$$
$$522$$ −99.6288 −4.36063
$$523$$ 20.6550 0.903178 0.451589 0.892226i $$-0.350857\pi$$
0.451589 + 0.892226i $$0.350857\pi$$
$$524$$ −11.6655 −0.509611
$$525$$ 0 0
$$526$$ 27.5139 1.19966
$$527$$ −0.729988 −0.0317988
$$528$$ −47.3311 −2.05982
$$529$$ 30.8469 1.34117
$$530$$ −12.7300 −0.552955
$$531$$ 29.6272 1.28571
$$532$$ 0 0
$$533$$ −4.20555 −0.182163
$$534$$ −59.9094 −2.59253
$$535$$ −1.62721 −0.0703506
$$536$$ 12.9411 0.558969
$$537$$ −34.1416 −1.47332
$$538$$ 42.2439 1.82126
$$539$$ 0 0
$$540$$ −40.8222 −1.75671
$$541$$ 5.62167 0.241695 0.120847 0.992671i $$-0.461439\pi$$
0.120847 + 0.992671i $$0.461439\pi$$
$$542$$ 23.1155 0.992894
$$543$$ −2.14611 −0.0920985
$$544$$ 3.32391 0.142512
$$545$$ −15.6811 −0.671705
$$546$$ 0 0
$$547$$ −10.3970 −0.444542 −0.222271 0.974985i $$-0.571347\pi$$
−0.222271 + 0.974985i $$0.571347\pi$$
$$548$$ −8.06949 −0.344711
$$549$$ 13.2544 0.565685
$$550$$ −16.4111 −0.699772
$$551$$ −6.74412 −0.287309
$$552$$ 29.3522 1.24931
$$553$$ 0 0
$$554$$ −14.7300 −0.625817
$$555$$ 53.7038 2.27960
$$556$$ 14.9200 0.632747
$$557$$ 14.6550 0.620951 0.310475 0.950581i $$-0.399512\pi$$
0.310475 + 0.950581i $$0.399512\pi$$
$$558$$ 16.7300 0.708237
$$559$$ −6.75971 −0.285905
$$560$$ 0 0
$$561$$ 5.04888 0.213164
$$562$$ −34.5189 −1.45609
$$563$$ 24.7456 1.04290 0.521451 0.853281i $$-0.325391\pi$$
0.521451 + 0.853281i $$0.325391\pi$$
$$564$$ 23.8811 1.00558
$$565$$ −15.3225 −0.644621
$$566$$ 20.2156 0.849725
$$567$$ 0 0
$$568$$ −11.2544 −0.472225
$$569$$ −20.5330 −0.860789 −0.430395 0.902641i $$-0.641626\pi$$
−0.430395 + 0.902641i $$0.641626\pi$$
$$570$$ −12.8816 −0.539552
$$571$$ −41.8953 −1.75326 −0.876631 0.481163i $$-0.840214\pi$$
−0.876631 + 0.481163i $$0.840214\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 24.3033 1.01529
$$574$$ 0 0
$$575$$ 21.4005 0.892464
$$576$$ −11.0141 −0.458922
$$577$$ 20.1744 0.839870 0.419935 0.907554i $$-0.362053\pi$$
0.419935 + 0.907554i $$0.362053\pi$$
$$578$$ 30.3325 1.26167
$$579$$ −37.8711 −1.57387
$$580$$ −30.0666 −1.24845
$$581$$ 0 0
$$582$$ −6.67609 −0.276733
$$583$$ −7.74055 −0.320581
$$584$$ 3.02061 0.124994
$$585$$ 18.6464 0.770933
$$586$$ −25.7094 −1.06204
$$587$$ 18.7441 0.773653 0.386826 0.922153i $$-0.373571\pi$$
0.386826 + 0.922153i $$0.373571\pi$$
$$588$$ 0 0
$$589$$ 1.13249 0.0466636
$$590$$ 22.8122 0.939162
$$591$$ −58.4011 −2.40230
$$592$$ 30.2439 1.24302
$$593$$ 2.98084 0.122409 0.0612043 0.998125i $$-0.480506\pi$$
0.0612043 + 0.998125i $$0.480506\pi$$
$$594$$ −63.3311 −2.59850
$$595$$ 0 0
$$596$$ 10.9894 0.450145
$$597$$ −67.0560 −2.74442
$$598$$ 13.3083 0.544218
$$599$$ 7.47411 0.305384 0.152692 0.988274i $$-0.451206\pi$$
0.152692 + 0.988274i $$0.451206\pi$$
$$600$$ 11.6655 0.476243
$$601$$ −21.4700 −0.875780 −0.437890 0.899028i $$-0.644274\pi$$
−0.437890 + 0.899028i $$0.644274\pi$$
$$602$$ 0 0
$$603$$ 66.5260 2.70915
$$604$$ −15.4499 −0.628649
$$605$$ 3.86248 0.157032
$$606$$ 73.7321 2.99516
$$607$$ 22.9044 0.929660 0.464830 0.885400i $$-0.346116\pi$$
0.464830 + 0.885400i $$0.346116\pi$$
$$608$$ −5.15667 −0.209131
$$609$$ 0 0
$$610$$ 10.2056 0.413211
$$611$$ −5.97028 −0.241532
$$612$$ 4.48059 0.181117
$$613$$ −20.1461 −0.813694 −0.406847 0.913496i $$-0.633372\pi$$
−0.406847 + 0.913496i $$0.633372\pi$$
$$614$$ 24.5910 0.992413
$$615$$ −36.7144 −1.48047
$$616$$ 0 0
$$617$$ −13.7844 −0.554939 −0.277470 0.960734i $$-0.589496\pi$$
−0.277470 + 0.960734i $$0.589496\pi$$
$$618$$ 24.8222 0.998495
$$619$$ 19.6655 0.790424 0.395212 0.918590i $$-0.370671\pi$$
0.395212 + 0.918590i $$0.370671\pi$$
$$620$$ 5.04888 0.202768
$$621$$ 82.5855 3.31404
$$622$$ 0.773841 0.0310282
$$623$$ 0 0
$$624$$ 15.2544 0.610666
$$625$$ −31.0766 −1.24307
$$626$$ −32.9200 −1.31575
$$627$$ −7.83276 −0.312810
$$628$$ 16.5371 0.659903
$$629$$ −3.22616 −0.128635
$$630$$ 0 0
$$631$$ −16.1672 −0.643608 −0.321804 0.946806i $$-0.604289\pi$$
−0.321804 + 0.946806i $$0.604289\pi$$
$$632$$ −17.4600 −0.694521
$$633$$ −53.9149 −2.14293
$$634$$ −17.0872 −0.678619
$$635$$ 36.2439 1.43829
$$636$$ −9.97887 −0.395688
$$637$$ 0 0
$$638$$ −46.6449 −1.84669
$$639$$ −57.8555 −2.28873
$$640$$ 27.1849 1.07458
$$641$$ −29.0036 −1.14557 −0.572786 0.819705i $$-0.694137\pi$$
−0.572786 + 0.819705i $$0.694137\pi$$
$$642$$ −3.25443 −0.128442
$$643$$ 39.2233 1.54681 0.773407 0.633910i $$-0.218551\pi$$
0.773407 + 0.633910i $$0.218551\pi$$
$$644$$ 0 0
$$645$$ −59.0122 −2.32360
$$646$$ 0.773841 0.0304464
$$647$$ 11.9844 0.471156 0.235578 0.971855i $$-0.424302\pi$$
0.235578 + 0.971855i $$0.424302\pi$$
$$648$$ 19.3869 0.761590
$$649$$ 13.8711 0.544487
$$650$$ 5.28917 0.207458
$$651$$ 0 0
$$652$$ −17.3522 −0.679564
$$653$$ 45.3311 1.77394 0.886971 0.461826i $$-0.152806\pi$$
0.886971 + 0.461826i $$0.152806\pi$$
$$654$$ −31.3622 −1.22636
$$655$$ 25.4600 0.994804
$$656$$ −20.6761 −0.807266
$$657$$ 15.5280 0.605805
$$658$$ 0 0
$$659$$ 6.12193 0.238477 0.119238 0.992866i $$-0.461955\pi$$
0.119238 + 0.992866i $$0.461955\pi$$
$$660$$ −34.9200 −1.35926
$$661$$ 27.5280 1.07072 0.535358 0.844625i $$-0.320177\pi$$
0.535358 + 0.844625i $$0.320177\pi$$
$$662$$ 31.5577 1.22653
$$663$$ −1.62721 −0.0631957
$$664$$ −21.2444 −0.824442
$$665$$ 0 0
$$666$$ 73.9377 2.86503
$$667$$ 60.8263 2.35520
$$668$$ 2.61665 0.101241
$$669$$ 32.7300 1.26541
$$670$$ 51.2233 1.97893
$$671$$ 6.20555 0.239563
$$672$$ 0 0
$$673$$ 27.9547 1.07757 0.538787 0.842442i $$-0.318883\pi$$
0.538787 + 0.842442i $$0.318883\pi$$
$$674$$ −39.9945 −1.54053
$$675$$ 32.8222 1.26333
$$676$$ 1.28917 0.0495834
$$677$$ 12.6605 0.486583 0.243291 0.969953i $$-0.421773\pi$$
0.243291 + 0.969953i $$0.421773\pi$$
$$678$$ −30.6449 −1.17691
$$679$$ 0 0
$$680$$ −1.90225 −0.0729479
$$681$$ 21.5678 0.826479
$$682$$ 7.83276 0.299932
$$683$$ −28.3033 −1.08300 −0.541498 0.840702i $$-0.682143\pi$$
−0.541498 + 0.840702i $$0.682143\pi$$
$$684$$ −6.95112 −0.265783
$$685$$ 17.6116 0.672906
$$686$$ 0 0
$$687$$ 65.4288 2.49626
$$688$$ −33.2333 −1.26701
$$689$$ 2.49472 0.0950412
$$690$$ 116.182 4.42295
$$691$$ −12.2353 −0.465452 −0.232726 0.972542i $$-0.574764\pi$$
−0.232726 + 0.972542i $$0.574764\pi$$
$$692$$ −26.1672 −0.994729
$$693$$ 0 0
$$694$$ −45.9789 −1.74533
$$695$$ −32.5628 −1.23518
$$696$$ 33.1567 1.25680
$$697$$ 2.20555 0.0835412
$$698$$ −10.3472 −0.391646
$$699$$ 18.8761 0.713960
$$700$$ 0 0
$$701$$ 51.0419 1.92783 0.963913 0.266219i $$-0.0857743\pi$$
0.963913 + 0.266219i $$0.0857743\pi$$
$$702$$ 20.4111 0.770367
$$703$$ 5.00502 0.188768
$$704$$ −5.15667 −0.194349
$$705$$ −52.1205 −1.96297
$$706$$ −52.0071 −1.95731
$$707$$ 0 0
$$708$$ 17.8822 0.672053
$$709$$ 42.5910 1.59954 0.799770 0.600307i $$-0.204955\pi$$
0.799770 + 0.600307i $$0.204955\pi$$
$$710$$ −44.5472 −1.67183
$$711$$ −89.7563 −3.36612
$$712$$ 13.7250 0.514365
$$713$$ −10.2141 −0.382523
$$714$$ 0 0
$$715$$ 8.72999 0.326483
$$716$$ −14.1855 −0.530135
$$717$$ −44.0766 −1.64607
$$718$$ −20.0283 −0.747448
$$719$$ 42.4933 1.58473 0.792366 0.610046i $$-0.208849\pi$$
0.792366 + 0.610046i $$0.208849\pi$$
$$720$$ 91.6727 3.41644
$$721$$ 0 0
$$722$$ 33.2580 1.23773
$$723$$ −26.1900 −0.974015
$$724$$ −0.891685 −0.0331392
$$725$$ 24.1744 0.897814
$$726$$ 7.72496 0.286700
$$727$$ 3.75614 0.139307 0.0696537 0.997571i $$-0.477811\pi$$
0.0696537 + 0.997571i $$0.477811\pi$$
$$728$$ 0 0
$$729$$ −5.09775 −0.188806
$$730$$ 11.9561 0.442517
$$731$$ 3.54505 0.131118
$$732$$ 8.00000 0.295689
$$733$$ −45.7819 −1.69099 −0.845497 0.533980i $$-0.820696\pi$$
−0.845497 + 0.533980i $$0.820696\pi$$
$$734$$ 49.5960 1.83062
$$735$$ 0 0
$$736$$ 46.5089 1.71434
$$737$$ 31.1466 1.14730
$$738$$ −50.5472 −1.86067
$$739$$ −14.0539 −0.516981 −0.258491 0.966014i $$-0.583225\pi$$
−0.258491 + 0.966014i $$0.583225\pi$$
$$740$$ 22.3133 0.820255
$$741$$ 2.52444 0.0927375
$$742$$ 0 0
$$743$$ 4.74557 0.174098 0.0870491 0.996204i $$-0.472256\pi$$
0.0870491 + 0.996204i $$0.472256\pi$$
$$744$$ −5.56777 −0.204125
$$745$$ −23.9844 −0.878721
$$746$$ −29.2827 −1.07212
$$747$$ −109.211 −3.99581
$$748$$ 2.09775 0.0767014
$$749$$ 0 0
$$750$$ −32.9894 −1.20460
$$751$$ 36.1008 1.31734 0.658669 0.752433i $$-0.271120\pi$$
0.658669 + 0.752433i $$0.271120\pi$$
$$752$$ −29.3522 −1.07036
$$753$$ 74.4182 2.71195
$$754$$ 15.0333 0.547480
$$755$$ 33.7194 1.22718
$$756$$ 0 0
$$757$$ −1.03474 −0.0376084 −0.0188042 0.999823i $$-0.505986\pi$$
−0.0188042 + 0.999823i $$0.505986\pi$$
$$758$$ −47.3905 −1.72130
$$759$$ 70.6449 2.56425
$$760$$ 2.95112 0.107049
$$761$$ −29.8414 −1.08175 −0.540874 0.841104i $$-0.681906\pi$$
−0.540874 + 0.841104i $$0.681906\pi$$
$$762$$ 72.4877 2.62595
$$763$$ 0 0
$$764$$ 10.0978 0.365324
$$765$$ −9.77886 −0.353556
$$766$$ −38.1744 −1.37930
$$767$$ −4.47054 −0.161422
$$768$$ 64.6832 2.33406
$$769$$ −23.6358 −0.852329 −0.426164 0.904646i $$-0.640136\pi$$
−0.426164 + 0.904646i $$0.640136\pi$$
$$770$$ 0 0
$$771$$ −48.4394 −1.74450
$$772$$ −15.7350 −0.566315
$$773$$ −13.0278 −0.468576 −0.234288 0.972167i $$-0.575276\pi$$
−0.234288 + 0.972167i $$0.575276\pi$$
$$774$$ −81.2460 −2.92033
$$775$$ −4.05944 −0.145819
$$776$$ 1.52946 0.0549045
$$777$$ 0 0
$$778$$ 39.1849 1.40485
$$779$$ −3.42166 −0.122594
$$780$$ 11.2544 0.402973
$$781$$ −27.0872 −0.969256
$$782$$ −6.97939 −0.249583
$$783$$ 93.2898 3.33391
$$784$$ 0 0
$$785$$ −36.0922 −1.28819
$$786$$ 50.9200 1.81625
$$787$$ −46.2141 −1.64736 −0.823678 0.567058i $$-0.808082\pi$$
−0.823678 + 0.567058i $$0.808082\pi$$
$$788$$ −24.2650 −0.864404
$$789$$ −47.0716 −1.67579
$$790$$ −69.1099 −2.45882
$$791$$ 0 0
$$792$$ 26.5089 0.941951
$$793$$ −2.00000 −0.0710221
$$794$$ −50.2283 −1.78253
$$795$$ 21.7789 0.772417
$$796$$ −27.8610 −0.987508
$$797$$ −53.1155 −1.88145 −0.940723 0.339176i $$-0.889852\pi$$
−0.940723 + 0.339176i $$0.889852\pi$$
$$798$$ 0 0
$$799$$ 3.13104 0.110768
$$800$$ 18.4842 0.653514
$$801$$ 70.5558 2.49297
$$802$$ 4.67609 0.165118
$$803$$ 7.27001 0.256553
$$804$$ 40.1533 1.41610
$$805$$ 0 0
$$806$$ −2.52444 −0.0889195
$$807$$ −72.2721 −2.54410
$$808$$ −16.8917 −0.594247
$$809$$ −54.4635 −1.91484 −0.957418 0.288705i $$-0.906775\pi$$
−0.957418 + 0.288705i $$0.906775\pi$$
$$810$$ 76.7371 2.69627
$$811$$ −38.0978 −1.33779 −0.668897 0.743356i $$-0.733233\pi$$
−0.668897 + 0.743356i $$0.733233\pi$$
$$812$$ 0 0
$$813$$ −39.5466 −1.38696
$$814$$ 34.6167 1.21331
$$815$$ 37.8711 1.32657
$$816$$ −8.00000 −0.280056
$$817$$ −5.49974 −0.192412
$$818$$ 27.4161 0.958582
$$819$$ 0 0
$$820$$ −15.2544 −0.532708
$$821$$ −2.30330 −0.0803858 −0.0401929 0.999192i $$-0.512797\pi$$
−0.0401929 + 0.999192i $$0.512797\pi$$
$$822$$ 35.2233 1.22855
$$823$$ 23.6172 0.823243 0.411621 0.911355i $$-0.364963\pi$$
0.411621 + 0.911355i $$0.364963\pi$$
$$824$$ −5.68665 −0.198104
$$825$$ 28.0766 0.977503
$$826$$ 0 0
$$827$$ −48.1643 −1.67484 −0.837419 0.546562i $$-0.815936\pi$$
−0.837419 + 0.546562i $$0.815936\pi$$
$$828$$ 62.6933 2.17874
$$829$$ −13.0716 −0.453996 −0.226998 0.973895i $$-0.572891\pi$$
−0.226998 + 0.973895i $$0.572891\pi$$
$$830$$ −84.0893 −2.91878
$$831$$ 25.2005 0.874197
$$832$$ 1.66196 0.0576179
$$833$$ 0 0
$$834$$ −65.1255 −2.25511
$$835$$ −5.71083 −0.197631
$$836$$ −3.25443 −0.112557
$$837$$ −15.6655 −0.541480
$$838$$ −18.1260 −0.626153
$$839$$ 17.6756 0.610229 0.305114 0.952316i $$-0.401305\pi$$
0.305114 + 0.952316i $$0.401305\pi$$
$$840$$ 0 0
$$841$$ 39.7103 1.36932
$$842$$ 47.0177 1.62034
$$843$$ 59.0560 2.03400
$$844$$ −22.4011 −0.771076
$$845$$ −2.81361 −0.0967910
$$846$$ −71.7577 −2.46708
$$847$$ 0 0
$$848$$ 12.2650 0.421181
$$849$$ −34.5855 −1.18697
$$850$$ −2.77384 −0.0951420
$$851$$ −45.1411 −1.54742
$$852$$ −34.9200 −1.19634
$$853$$ 5.48970 0.187964 0.0939818 0.995574i $$-0.470040\pi$$
0.0939818 + 0.995574i $$0.470040\pi$$
$$854$$ 0 0
$$855$$ 15.1708 0.518831
$$856$$ 0.745574 0.0254832
$$857$$ 11.0489 0.377422 0.188711 0.982033i $$-0.439569\pi$$
0.188711 + 0.982033i $$0.439569\pi$$
$$858$$ 17.4600 0.596074
$$859$$ −45.2616 −1.54430 −0.772152 0.635437i $$-0.780820\pi$$
−0.772152 + 0.635437i $$0.780820\pi$$
$$860$$ −24.5189 −0.836087
$$861$$ 0 0
$$862$$ −55.6344 −1.89491
$$863$$ −5.90225 −0.200915 −0.100457 0.994941i $$-0.532031\pi$$
−0.100457 + 0.994941i $$0.532031\pi$$
$$864$$ 71.3311 2.42673
$$865$$ 57.1099 1.94180
$$866$$ −6.38283 −0.216898
$$867$$ −51.8938 −1.76241
$$868$$ 0 0
$$869$$ −42.0227 −1.42552
$$870$$ 131.240 4.44947
$$871$$ −10.0383 −0.340135
$$872$$ 7.18494 0.243313
$$873$$ 7.86248 0.266105
$$874$$ 10.8277 0.366254
$$875$$ 0 0
$$876$$ 9.37227 0.316660
$$877$$ −4.90727 −0.165707 −0.0828534 0.996562i $$-0.526403\pi$$
−0.0828534 + 0.996562i $$0.526403\pi$$
$$878$$ 58.6732 1.98012
$$879$$ 43.9844 1.48356
$$880$$ 42.9200 1.44683
$$881$$ −44.2822 −1.49190 −0.745952 0.665999i $$-0.768005\pi$$
−0.745952 + 0.665999i $$0.768005\pi$$
$$882$$ 0 0
$$883$$ −58.8605 −1.98081 −0.990407 0.138181i $$-0.955874\pi$$
−0.990407 + 0.138181i $$0.955874\pi$$
$$884$$ −0.676089 −0.0227393
$$885$$ −39.0278 −1.31190
$$886$$ −28.0127 −0.941104
$$887$$ 10.1289 0.340096 0.170048 0.985436i $$-0.445608\pi$$
0.170048 + 0.985436i $$0.445608\pi$$
$$888$$ −24.6066 −0.825744
$$889$$ 0 0
$$890$$ 54.3260 1.82101
$$891$$ 46.6605 1.56319
$$892$$ 13.5989 0.455326
$$893$$ −4.85746 −0.162549
$$894$$ −47.9688 −1.60432
$$895$$ 30.9597 1.03487
$$896$$ 0 0
$$897$$ −22.7683 −0.760211
$$898$$ 26.2439 0.875769
$$899$$ −11.5381 −0.384816
$$900$$ 24.9164 0.830546
$$901$$ −1.30833 −0.0435866
$$902$$ −23.6655 −0.787976
$$903$$ 0 0
$$904$$ 7.02061 0.233502
$$905$$ 1.94610 0.0646906
$$906$$ 67.4389 2.24051
$$907$$ 37.9547 1.26026 0.630132 0.776488i $$-0.283001\pi$$
0.630132 + 0.776488i $$0.283001\pi$$
$$908$$ 8.96117 0.297387
$$909$$ −86.8349 −2.88013
$$910$$ 0 0
$$911$$ 5.57477 0.184700 0.0923501 0.995727i $$-0.470562\pi$$
0.0923501 + 0.995727i $$0.470562\pi$$
$$912$$ 12.4111 0.410973
$$913$$ −51.1310 −1.69219
$$914$$ 62.8787 2.07984
$$915$$ −17.4600 −0.577209
$$916$$ 27.1849 0.898216
$$917$$ 0 0
$$918$$ −10.7044 −0.353296
$$919$$ 15.7844 0.520679 0.260340 0.965517i $$-0.416165\pi$$
0.260340 + 0.965517i $$0.416165\pi$$
$$920$$ −26.6167 −0.877525
$$921$$ −42.0711 −1.38629
$$922$$ 22.7427 0.748990
$$923$$ 8.72999 0.287351
$$924$$ 0 0
$$925$$ −17.9406 −0.589882
$$926$$ −22.0383 −0.724224
$$927$$ −29.2333 −0.960148
$$928$$ 52.5371 1.72462
$$929$$ 45.2630 1.48503 0.742516 0.669829i $$-0.233632\pi$$
0.742516 + 0.669829i $$0.233632\pi$$
$$930$$ −22.0383 −0.722665
$$931$$ 0 0
$$932$$ 7.84281 0.256900
$$933$$ −1.32391 −0.0433429
$$934$$ −67.1638 −2.19767
$$935$$ −4.57834 −0.149728
$$936$$ −8.54359 −0.279256
$$937$$ −53.6188 −1.75165 −0.875824 0.482630i $$-0.839682\pi$$
−0.875824 + 0.482630i $$0.839682\pi$$
$$938$$ 0 0
$$939$$ 56.3205 1.83795
$$940$$ −21.6555 −0.706324
$$941$$ −20.7753 −0.677255 −0.338628 0.940920i $$-0.609963\pi$$
−0.338628 + 0.940920i $$0.609963\pi$$
$$942$$ −72.1844 −2.35190
$$943$$ 30.8605 1.00496
$$944$$ −21.9789 −0.715351
$$945$$ 0 0
$$946$$ −38.0383 −1.23673
$$947$$ −10.8605 −0.352919 −0.176460 0.984308i $$-0.556465\pi$$
−0.176460 + 0.984308i $$0.556465\pi$$
$$948$$ −54.1744 −1.75950
$$949$$ −2.34307 −0.0760592
$$950$$ 4.30330 0.139618
$$951$$ 29.2333 0.947955
$$952$$ 0 0
$$953$$ −25.7180 −0.833087 −0.416543 0.909116i $$-0.636759\pi$$
−0.416543 + 0.909116i $$0.636759\pi$$
$$954$$ 29.9844 0.970781
$$955$$ −22.0383 −0.713143
$$956$$ −18.3133 −0.592296
$$957$$ 79.8016 2.57962
$$958$$ 21.7789 0.703643
$$959$$ 0 0
$$960$$ 14.5089 0.468271
$$961$$ −29.0625 −0.937500
$$962$$ −11.1567 −0.359706
$$963$$ 3.83276 0.123509
$$964$$ −10.8816 −0.350474
$$965$$ 34.3416 1.10550
$$966$$ 0 0
$$967$$ 33.5038 1.07741 0.538705 0.842494i $$-0.318914\pi$$
0.538705 + 0.842494i $$0.318914\pi$$
$$968$$ −1.76975 −0.0568820
$$969$$ −1.32391 −0.0425302
$$970$$ 6.05390 0.194379
$$971$$ −2.03831 −0.0654126 −0.0327063 0.999465i $$-0.510413\pi$$
−0.0327063 + 0.999465i $$0.510413\pi$$
$$972$$ 16.6267 0.533302
$$973$$ 0 0
$$974$$ −20.1643 −0.646107
$$975$$ −9.04888 −0.289796
$$976$$ −9.83276 −0.314739
$$977$$ 15.1411 0.484406 0.242203 0.970226i $$-0.422130\pi$$
0.242203 + 0.970226i $$0.422130\pi$$
$$978$$ 75.7422 2.42197
$$979$$ 33.0333 1.05575
$$980$$ 0 0
$$981$$ 36.9355 1.17926
$$982$$ −0.107798 −0.00343998
$$983$$ −49.3124 −1.57282 −0.786411 0.617704i $$-0.788063\pi$$
−0.786411 + 0.617704i $$0.788063\pi$$
$$984$$ 16.8222 0.536272
$$985$$ 52.9583 1.68739
$$986$$ −7.88403 −0.251079
$$987$$ 0 0
$$988$$ 1.04888 0.0333692
$$989$$ 49.6030 1.57728
$$990$$ 104.927 3.33480
$$991$$ 5.43171 0.172544 0.0862720 0.996272i $$-0.472505\pi$$
0.0862720 + 0.996272i $$0.472505\pi$$
$$992$$ −8.82220 −0.280105
$$993$$ −53.9900 −1.71332
$$994$$ 0 0
$$995$$ 60.8066 1.92770
$$996$$ −65.9165 −2.08865
$$997$$ 53.6061 1.69772 0.848861 0.528616i $$-0.177289\pi$$
0.848861 + 0.528616i $$0.177289\pi$$
$$998$$ −18.6761 −0.591181
$$999$$ −69.2333 −2.19044
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.a.j.1.1 3
3.2 odd 2 5733.2.a.x.1.3 3
7.2 even 3 637.2.e.i.508.3 6
7.3 odd 6 637.2.e.j.79.3 6
7.4 even 3 637.2.e.i.79.3 6
7.5 odd 6 637.2.e.j.508.3 6
7.6 odd 2 91.2.a.d.1.1 3
13.12 even 2 8281.2.a.bg.1.3 3
21.20 even 2 819.2.a.i.1.3 3
28.27 even 2 1456.2.a.t.1.3 3
35.34 odd 2 2275.2.a.m.1.3 3
56.13 odd 2 5824.2.a.by.1.3 3
56.27 even 2 5824.2.a.bs.1.1 3
91.34 even 4 1183.2.c.f.337.2 6
91.83 even 4 1183.2.c.f.337.5 6
91.90 odd 2 1183.2.a.i.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 7.6 odd 2
637.2.a.j.1.1 3 1.1 even 1 trivial
637.2.e.i.79.3 6 7.4 even 3
637.2.e.i.508.3 6 7.2 even 3
637.2.e.j.79.3 6 7.3 odd 6
637.2.e.j.508.3 6 7.5 odd 6
819.2.a.i.1.3 3 21.20 even 2
1183.2.a.i.1.3 3 91.90 odd 2
1183.2.c.f.337.2 6 91.34 even 4
1183.2.c.f.337.5 6 91.83 even 4
1456.2.a.t.1.3 3 28.27 even 2
2275.2.a.m.1.3 3 35.34 odd 2
5733.2.a.x.1.3 3 3.2 odd 2
5824.2.a.bs.1.1 3 56.27 even 2
5824.2.a.by.1.3 3 56.13 odd 2
8281.2.a.bg.1.3 3 13.12 even 2