# Properties

 Label 9702.2.a.ck.1.2 Level $9702$ Weight $2$ Character 9702.1 Self dual yes Analytic conductor $77.471$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$77.4708600410$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3234) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 9702.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{8} -1.23607 q^{10} +1.00000 q^{11} +1.00000 q^{16} -5.23607 q^{17} +7.70820 q^{19} +1.23607 q^{20} -1.00000 q^{22} +2.47214 q^{23} -3.47214 q^{25} -4.47214 q^{29} +2.76393 q^{31} -1.00000 q^{32} +5.23607 q^{34} -10.9443 q^{37} -7.70820 q^{38} -1.23607 q^{40} -5.23607 q^{41} +6.47214 q^{43} +1.00000 q^{44} -2.47214 q^{46} +3.70820 q^{47} +3.47214 q^{50} +6.00000 q^{53} +1.23607 q^{55} +4.47214 q^{58} -1.52786 q^{59} -2.76393 q^{62} +1.00000 q^{64} +15.4164 q^{67} -5.23607 q^{68} +2.47214 q^{71} +15.7082 q^{73} +10.9443 q^{74} +7.70820 q^{76} +11.4164 q^{79} +1.23607 q^{80} +5.23607 q^{82} +1.23607 q^{83} -6.47214 q^{85} -6.47214 q^{86} -1.00000 q^{88} -6.47214 q^{89} +2.47214 q^{92} -3.70820 q^{94} +9.52786 q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{16} - 6 q^{17} + 2 q^{19} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} + 10 q^{31} - 2 q^{32} + 6 q^{34} - 4 q^{37} - 2 q^{38} + 2 q^{40} - 6 q^{41} + 4 q^{43} + 2 q^{44} + 4 q^{46} - 6 q^{47} - 2 q^{50} + 12 q^{53} - 2 q^{55} - 12 q^{59} - 10 q^{62} + 2 q^{64} + 4 q^{67} - 6 q^{68} - 4 q^{71} + 18 q^{73} + 4 q^{74} + 2 q^{76} - 4 q^{79} - 2 q^{80} + 6 q^{82} - 2 q^{83} - 4 q^{85} - 4 q^{86} - 2 q^{88} - 4 q^{89} - 4 q^{92} + 6 q^{94} + 28 q^{95}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 + 2 * q^11 + 2 * q^16 - 6 * q^17 + 2 * q^19 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 + 10 * q^31 - 2 * q^32 + 6 * q^34 - 4 * q^37 - 2 * q^38 + 2 * q^40 - 6 * q^41 + 4 * q^43 + 2 * q^44 + 4 * q^46 - 6 * q^47 - 2 * q^50 + 12 * q^53 - 2 * q^55 - 12 * q^59 - 10 * q^62 + 2 * q^64 + 4 * q^67 - 6 * q^68 - 4 * q^71 + 18 * q^73 + 4 * q^74 + 2 * q^76 - 4 * q^79 - 2 * q^80 + 6 * q^82 - 2 * q^83 - 4 * q^85 - 4 * q^86 - 2 * q^88 - 4 * q^89 - 4 * q^92 + 6 * q^94 + 28 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.23607 0.552786 0.276393 0.961045i $$-0.410861\pi$$
0.276393 + 0.961045i $$0.410861\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.23607 −0.390879
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −5.23607 −1.26993 −0.634967 0.772540i $$-0.718986\pi$$
−0.634967 + 0.772540i $$0.718986\pi$$
$$18$$ 0 0
$$19$$ 7.70820 1.76838 0.884192 0.467124i $$-0.154710\pi$$
0.884192 + 0.467124i $$0.154710\pi$$
$$20$$ 1.23607 0.276393
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 2.47214 0.515476 0.257738 0.966215i $$-0.417023\pi$$
0.257738 + 0.966215i $$0.417023\pi$$
$$24$$ 0 0
$$25$$ −3.47214 −0.694427
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ 2.76393 0.496417 0.248208 0.968707i $$-0.420158\pi$$
0.248208 + 0.968707i $$0.420158\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 5.23607 0.897978
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.9443 −1.79923 −0.899614 0.436687i $$-0.856152\pi$$
−0.899614 + 0.436687i $$0.856152\pi$$
$$38$$ −7.70820 −1.25044
$$39$$ 0 0
$$40$$ −1.23607 −0.195440
$$41$$ −5.23607 −0.817736 −0.408868 0.912593i $$-0.634076\pi$$
−0.408868 + 0.912593i $$0.634076\pi$$
$$42$$ 0 0
$$43$$ 6.47214 0.986991 0.493496 0.869748i $$-0.335719\pi$$
0.493496 + 0.869748i $$0.335719\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −2.47214 −0.364497
$$47$$ 3.70820 0.540897 0.270449 0.962734i $$-0.412828\pi$$
0.270449 + 0.962734i $$0.412828\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 3.47214 0.491034
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 1.23607 0.166671
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 4.47214 0.587220
$$59$$ −1.52786 −0.198911 −0.0994555 0.995042i $$-0.531710\pi$$
−0.0994555 + 0.995042i $$0.531710\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −2.76393 −0.351020
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 15.4164 1.88341 0.941707 0.336434i $$-0.109221\pi$$
0.941707 + 0.336434i $$0.109221\pi$$
$$68$$ −5.23607 −0.634967
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.47214 0.293389 0.146694 0.989182i $$-0.453137\pi$$
0.146694 + 0.989182i $$0.453137\pi$$
$$72$$ 0 0
$$73$$ 15.7082 1.83851 0.919253 0.393667i $$-0.128794\pi$$
0.919253 + 0.393667i $$0.128794\pi$$
$$74$$ 10.9443 1.27225
$$75$$ 0 0
$$76$$ 7.70820 0.884192
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.4164 1.28445 0.642223 0.766518i $$-0.278012\pi$$
0.642223 + 0.766518i $$0.278012\pi$$
$$80$$ 1.23607 0.138197
$$81$$ 0 0
$$82$$ 5.23607 0.578227
$$83$$ 1.23607 0.135676 0.0678380 0.997696i $$-0.478390\pi$$
0.0678380 + 0.997696i $$0.478390\pi$$
$$84$$ 0 0
$$85$$ −6.47214 −0.702002
$$86$$ −6.47214 −0.697908
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ −6.47214 −0.686045 −0.343023 0.939327i $$-0.611451\pi$$
−0.343023 + 0.939327i $$0.611451\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.47214 0.257738
$$93$$ 0 0
$$94$$ −3.70820 −0.382472
$$95$$ 9.52786 0.977538
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −3.47214 −0.347214
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −10.7639 −1.06060 −0.530301 0.847810i $$-0.677921\pi$$
−0.530301 + 0.847810i $$0.677921\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −0.944272 −0.0912862 −0.0456431 0.998958i $$-0.514534\pi$$
−0.0456431 + 0.998958i $$0.514534\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −1.23607 −0.117854
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −13.4164 −1.26211 −0.631055 0.775738i $$-0.717378\pi$$
−0.631055 + 0.775738i $$0.717378\pi$$
$$114$$ 0 0
$$115$$ 3.05573 0.284948
$$116$$ −4.47214 −0.415227
$$117$$ 0 0
$$118$$ 1.52786 0.140651
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 2.76393 0.248208
$$125$$ −10.4721 −0.936656
$$126$$ 0 0
$$127$$ −6.47214 −0.574309 −0.287155 0.957884i $$-0.592709\pi$$
−0.287155 + 0.957884i $$0.592709\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.76393 0.590967 0.295484 0.955348i $$-0.404519\pi$$
0.295484 + 0.955348i $$0.404519\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −15.4164 −1.33177
$$135$$ 0 0
$$136$$ 5.23607 0.448989
$$137$$ 16.4721 1.40731 0.703655 0.710542i $$-0.251550\pi$$
0.703655 + 0.710542i $$0.251550\pi$$
$$138$$ 0 0
$$139$$ 10.1803 0.863485 0.431743 0.901997i $$-0.357899\pi$$
0.431743 + 0.901997i $$0.357899\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2.47214 −0.207457
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −5.52786 −0.459064
$$146$$ −15.7082 −1.30002
$$147$$ 0 0
$$148$$ −10.9443 −0.899614
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −7.70820 −0.625218
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.41641 0.274412
$$156$$ 0 0
$$157$$ 14.7639 1.17829 0.589145 0.808027i $$-0.299465\pi$$
0.589145 + 0.808027i $$0.299465\pi$$
$$158$$ −11.4164 −0.908241
$$159$$ 0 0
$$160$$ −1.23607 −0.0977198
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −5.23607 −0.408868
$$165$$ 0 0
$$166$$ −1.23607 −0.0959375
$$167$$ 18.4721 1.42942 0.714708 0.699423i $$-0.246559\pi$$
0.714708 + 0.699423i $$0.246559\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 6.47214 0.496390
$$171$$ 0 0
$$172$$ 6.47214 0.493496
$$173$$ 19.4164 1.47620 0.738101 0.674690i $$-0.235723\pi$$
0.738101 + 0.674690i $$0.235723\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 6.47214 0.485107
$$179$$ 5.52786 0.413172 0.206586 0.978428i $$-0.433765\pi$$
0.206586 + 0.978428i $$0.433765\pi$$
$$180$$ 0 0
$$181$$ −11.7082 −0.870264 −0.435132 0.900367i $$-0.643298\pi$$
−0.435132 + 0.900367i $$0.643298\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −2.47214 −0.182248
$$185$$ −13.5279 −0.994588
$$186$$ 0 0
$$187$$ −5.23607 −0.382899
$$188$$ 3.70820 0.270449
$$189$$ 0 0
$$190$$ −9.52786 −0.691224
$$191$$ 10.4721 0.757737 0.378869 0.925450i $$-0.376313\pi$$
0.378869 + 0.925450i $$0.376313\pi$$
$$192$$ 0 0
$$193$$ 5.05573 0.363919 0.181960 0.983306i $$-0.441756\pi$$
0.181960 + 0.983306i $$0.441756\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.3607 1.59313 0.796566 0.604551i $$-0.206648\pi$$
0.796566 + 0.604551i $$0.206648\pi$$
$$198$$ 0 0
$$199$$ 23.1246 1.63926 0.819630 0.572893i $$-0.194179\pi$$
0.819630 + 0.572893i $$0.194179\pi$$
$$200$$ 3.47214 0.245517
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.47214 −0.452034
$$206$$ 10.7639 0.749959
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 7.70820 0.533188
$$210$$ 0 0
$$211$$ 3.41641 0.235195 0.117598 0.993061i $$-0.462481\pi$$
0.117598 + 0.993061i $$0.462481\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 0.944272 0.0645491
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 0 0
$$220$$ 1.23607 0.0833357
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 15.7082 1.05190 0.525950 0.850516i $$-0.323710\pi$$
0.525950 + 0.850516i $$0.323710\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 13.4164 0.892446
$$227$$ 8.65248 0.574285 0.287142 0.957888i $$-0.407295\pi$$
0.287142 + 0.957888i $$0.407295\pi$$
$$228$$ 0 0
$$229$$ −19.7082 −1.30235 −0.651177 0.758926i $$-0.725725\pi$$
−0.651177 + 0.758926i $$0.725725\pi$$
$$230$$ −3.05573 −0.201489
$$231$$ 0 0
$$232$$ 4.47214 0.293610
$$233$$ 9.05573 0.593260 0.296630 0.954992i $$-0.404137\pi$$
0.296630 + 0.954992i $$0.404137\pi$$
$$234$$ 0 0
$$235$$ 4.58359 0.299001
$$236$$ −1.52786 −0.0994555
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −18.1803 −1.17110 −0.585549 0.810637i $$-0.699121\pi$$
−0.585549 + 0.810637i $$0.699121\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2.76393 −0.175510
$$249$$ 0 0
$$250$$ 10.4721 0.662316
$$251$$ −6.47214 −0.408518 −0.204259 0.978917i $$-0.565478\pi$$
−0.204259 + 0.978917i $$0.565478\pi$$
$$252$$ 0 0
$$253$$ 2.47214 0.155422
$$254$$ 6.47214 0.406098
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 22.4721 1.40177 0.700887 0.713273i $$-0.252788\pi$$
0.700887 + 0.713273i $$0.252788\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.76393 −0.417877
$$263$$ 9.52786 0.587513 0.293757 0.955880i $$-0.405094\pi$$
0.293757 + 0.955880i $$0.405094\pi$$
$$264$$ 0 0
$$265$$ 7.41641 0.455586
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 15.4164 0.941707
$$269$$ −27.7082 −1.68940 −0.844700 0.535241i $$-0.820221\pi$$
−0.844700 + 0.535241i $$0.820221\pi$$
$$270$$ 0 0
$$271$$ 2.47214 0.150172 0.0750858 0.997177i $$-0.476077\pi$$
0.0750858 + 0.997177i $$0.476077\pi$$
$$272$$ −5.23607 −0.317483
$$273$$ 0 0
$$274$$ −16.4721 −0.995118
$$275$$ −3.47214 −0.209378
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −10.1803 −0.610576
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −19.8885 −1.18645 −0.593226 0.805036i $$-0.702146\pi$$
−0.593226 + 0.805036i $$0.702146\pi$$
$$282$$ 0 0
$$283$$ −23.7082 −1.40931 −0.704653 0.709552i $$-0.748897\pi$$
−0.704653 + 0.709552i $$0.748897\pi$$
$$284$$ 2.47214 0.146694
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 10.4164 0.612730
$$290$$ 5.52786 0.324607
$$291$$ 0 0
$$292$$ 15.7082 0.919253
$$293$$ −4.58359 −0.267776 −0.133888 0.990996i $$-0.542746\pi$$
−0.133888 + 0.990996i $$0.542746\pi$$
$$294$$ 0 0
$$295$$ −1.88854 −0.109955
$$296$$ 10.9443 0.636123
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 7.70820 0.442096
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −25.5967 −1.46088 −0.730442 0.682975i $$-0.760686\pi$$
−0.730442 + 0.682975i $$0.760686\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −3.41641 −0.194039
$$311$$ −8.65248 −0.490637 −0.245318 0.969443i $$-0.578893\pi$$
−0.245318 + 0.969443i $$0.578893\pi$$
$$312$$ 0 0
$$313$$ −5.52786 −0.312453 −0.156227 0.987721i $$-0.549933\pi$$
−0.156227 + 0.987721i $$0.549933\pi$$
$$314$$ −14.7639 −0.833177
$$315$$ 0 0
$$316$$ 11.4164 0.642223
$$317$$ 31.8885 1.79104 0.895520 0.445022i $$-0.146804\pi$$
0.895520 + 0.445022i $$0.146804\pi$$
$$318$$ 0 0
$$319$$ −4.47214 −0.250392
$$320$$ 1.23607 0.0690983
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −40.3607 −2.24573
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 5.23607 0.289113
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.41641 −0.407643 −0.203821 0.979008i $$-0.565336\pi$$
−0.203821 + 0.979008i $$0.565336\pi$$
$$332$$ 1.23607 0.0678380
$$333$$ 0 0
$$334$$ −18.4721 −1.01075
$$335$$ 19.0557 1.04113
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ −6.47214 −0.351001
$$341$$ 2.76393 0.149675
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −6.47214 −0.348954
$$345$$ 0 0
$$346$$ −19.4164 −1.04383
$$347$$ 26.8328 1.44046 0.720231 0.693735i $$-0.244036\pi$$
0.720231 + 0.693735i $$0.244036\pi$$
$$348$$ 0 0
$$349$$ 0.583592 0.0312390 0.0156195 0.999878i $$-0.495028\pi$$
0.0156195 + 0.999878i $$0.495028\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 24.3607 1.29659 0.648294 0.761390i $$-0.275483\pi$$
0.648294 + 0.761390i $$0.275483\pi$$
$$354$$ 0 0
$$355$$ 3.05573 0.162181
$$356$$ −6.47214 −0.343023
$$357$$ 0 0
$$358$$ −5.52786 −0.292157
$$359$$ −1.52786 −0.0806376 −0.0403188 0.999187i $$-0.512837\pi$$
−0.0403188 + 0.999187i $$0.512837\pi$$
$$360$$ 0 0
$$361$$ 40.4164 2.12718
$$362$$ 11.7082 0.615370
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 19.4164 1.01630
$$366$$ 0 0
$$367$$ −8.29180 −0.432828 −0.216414 0.976302i $$-0.569436\pi$$
−0.216414 + 0.976302i $$0.569436\pi$$
$$368$$ 2.47214 0.128869
$$369$$ 0 0
$$370$$ 13.5279 0.703280
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.41641 −0.0733388 −0.0366694 0.999327i $$-0.511675\pi$$
−0.0366694 + 0.999327i $$0.511675\pi$$
$$374$$ 5.23607 0.270751
$$375$$ 0 0
$$376$$ −3.70820 −0.191236
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 9.52786 0.488769
$$381$$ 0 0
$$382$$ −10.4721 −0.535801
$$383$$ −0.652476 −0.0333400 −0.0166700 0.999861i $$-0.505306\pi$$
−0.0166700 + 0.999861i $$0.505306\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.05573 −0.257330
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 6.94427 0.352089 0.176044 0.984382i $$-0.443670\pi$$
0.176044 + 0.984382i $$0.443670\pi$$
$$390$$ 0 0
$$391$$ −12.9443 −0.654620
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −22.3607 −1.12651
$$395$$ 14.1115 0.710024
$$396$$ 0 0
$$397$$ 1.81966 0.0913261 0.0456631 0.998957i $$-0.485460\pi$$
0.0456631 + 0.998957i $$0.485460\pi$$
$$398$$ −23.1246 −1.15913
$$399$$ 0 0
$$400$$ −3.47214 −0.173607
$$401$$ 8.47214 0.423078 0.211539 0.977370i $$-0.432152\pi$$
0.211539 + 0.977370i $$0.432152\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.9443 −0.542487
$$408$$ 0 0
$$409$$ −2.76393 −0.136668 −0.0683338 0.997663i $$-0.521768\pi$$
−0.0683338 + 0.997663i $$0.521768\pi$$
$$410$$ 6.47214 0.319636
$$411$$ 0 0
$$412$$ −10.7639 −0.530301
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.52786 0.0749999
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −7.70820 −0.377021
$$419$$ 15.0557 0.735520 0.367760 0.929921i $$-0.380125\pi$$
0.367760 + 0.929921i $$0.380125\pi$$
$$420$$ 0 0
$$421$$ −5.05573 −0.246401 −0.123201 0.992382i $$-0.539316\pi$$
−0.123201 + 0.992382i $$0.539316\pi$$
$$422$$ −3.41641 −0.166308
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 18.1803 0.881876
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −0.944272 −0.0456431
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 22.4721 1.08244 0.541222 0.840880i $$-0.317962\pi$$
0.541222 + 0.840880i $$0.317962\pi$$
$$432$$ 0 0
$$433$$ −9.88854 −0.475213 −0.237607 0.971361i $$-0.576363\pi$$
−0.237607 + 0.971361i $$0.576363\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 19.0557 0.911559
$$438$$ 0 0
$$439$$ −31.4164 −1.49942 −0.749712 0.661765i $$-0.769808\pi$$
−0.749712 + 0.661765i $$0.769808\pi$$
$$440$$ −1.23607 −0.0589272
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.41641 −0.352364 −0.176182 0.984358i $$-0.556375\pi$$
−0.176182 + 0.984358i $$0.556375\pi$$
$$444$$ 0 0
$$445$$ −8.00000 −0.379236
$$446$$ −15.7082 −0.743805
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −19.5279 −0.921577 −0.460788 0.887510i $$-0.652433\pi$$
−0.460788 + 0.887510i $$0.652433\pi$$
$$450$$ 0 0
$$451$$ −5.23607 −0.246557
$$452$$ −13.4164 −0.631055
$$453$$ 0 0
$$454$$ −8.65248 −0.406081
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 19.7082 0.920904
$$459$$ 0 0
$$460$$ 3.05573 0.142474
$$461$$ −26.8328 −1.24973 −0.624864 0.780733i $$-0.714846\pi$$
−0.624864 + 0.780733i $$0.714846\pi$$
$$462$$ 0 0
$$463$$ −1.88854 −0.0877681 −0.0438840 0.999037i $$-0.513973\pi$$
−0.0438840 + 0.999037i $$0.513973\pi$$
$$464$$ −4.47214 −0.207614
$$465$$ 0 0
$$466$$ −9.05573 −0.419499
$$467$$ −29.8885 −1.38308 −0.691538 0.722340i $$-0.743067\pi$$
−0.691538 + 0.722340i $$0.743067\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4.58359 −0.211425
$$471$$ 0 0
$$472$$ 1.52786 0.0703256
$$473$$ 6.47214 0.297589
$$474$$ 0 0
$$475$$ −26.7639 −1.22801
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 35.7771 1.63470 0.817348 0.576144i $$-0.195443\pi$$
0.817348 + 0.576144i $$0.195443\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 18.1803 0.828092
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 36.9443 1.67410 0.837052 0.547123i $$-0.184277\pi$$
0.837052 + 0.547123i $$0.184277\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 29.8885 1.34885 0.674426 0.738343i $$-0.264391\pi$$
0.674426 + 0.738343i $$0.264391\pi$$
$$492$$ 0 0
$$493$$ 23.4164 1.05462
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 2.76393 0.124104
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 7.41641 0.332004 0.166002 0.986125i $$-0.446914\pi$$
0.166002 + 0.986125i $$0.446914\pi$$
$$500$$ −10.4721 −0.468328
$$501$$ 0 0
$$502$$ 6.47214 0.288866
$$503$$ −41.3050 −1.84170 −0.920848 0.389921i $$-0.872502\pi$$
−0.920848 + 0.389921i $$0.872502\pi$$
$$504$$ 0 0
$$505$$ −14.8328 −0.660052
$$506$$ −2.47214 −0.109900
$$507$$ 0 0
$$508$$ −6.47214 −0.287155
$$509$$ −6.76393 −0.299806 −0.149903 0.988701i $$-0.547896\pi$$
−0.149903 + 0.988701i $$0.547896\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −22.4721 −0.991203
$$515$$ −13.3050 −0.586286
$$516$$ 0 0
$$517$$ 3.70820 0.163087
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −21.3050 −0.933387 −0.466693 0.884419i $$-0.654555\pi$$
−0.466693 + 0.884419i $$0.654555\pi$$
$$522$$ 0 0
$$523$$ 24.2918 1.06221 0.531103 0.847307i $$-0.321778\pi$$
0.531103 + 0.847307i $$0.321778\pi$$
$$524$$ 6.76393 0.295484
$$525$$ 0 0
$$526$$ −9.52786 −0.415435
$$527$$ −14.4721 −0.630416
$$528$$ 0 0
$$529$$ −16.8885 −0.734285
$$530$$ −7.41641 −0.322148
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −1.16718 −0.0504618
$$536$$ −15.4164 −0.665887
$$537$$ 0 0
$$538$$ 27.7082 1.19459
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 21.4164 0.920763 0.460382 0.887721i $$-0.347713\pi$$
0.460382 + 0.887721i $$0.347713\pi$$
$$542$$ −2.47214 −0.106187
$$543$$ 0 0
$$544$$ 5.23607 0.224495
$$545$$ 7.41641 0.317684
$$546$$ 0 0
$$547$$ 38.4721 1.64495 0.822475 0.568801i $$-0.192593\pi$$
0.822475 + 0.568801i $$0.192593\pi$$
$$548$$ 16.4721 0.703655
$$549$$ 0 0
$$550$$ 3.47214 0.148052
$$551$$ −34.4721 −1.46856
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ 10.1803 0.431743
$$557$$ 10.5836 0.448441 0.224221 0.974538i $$-0.428016\pi$$
0.224221 + 0.974538i $$0.428016\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 19.8885 0.838948
$$563$$ 6.76393 0.285066 0.142533 0.989790i $$-0.454475\pi$$
0.142533 + 0.989790i $$0.454475\pi$$
$$564$$ 0 0
$$565$$ −16.5836 −0.697677
$$566$$ 23.7082 0.996530
$$567$$ 0 0
$$568$$ −2.47214 −0.103729
$$569$$ 34.9443 1.46494 0.732470 0.680799i $$-0.238367\pi$$
0.732470 + 0.680799i $$0.238367\pi$$
$$570$$ 0 0
$$571$$ −17.5279 −0.733518 −0.366759 0.930316i $$-0.619533\pi$$
−0.366759 + 0.930316i $$0.619533\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.58359 −0.357961
$$576$$ 0 0
$$577$$ 39.4164 1.64093 0.820463 0.571699i $$-0.193716\pi$$
0.820463 + 0.571699i $$0.193716\pi$$
$$578$$ −10.4164 −0.433265
$$579$$ 0 0
$$580$$ −5.52786 −0.229532
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.00000 0.248495
$$584$$ −15.7082 −0.650010
$$585$$ 0 0
$$586$$ 4.58359 0.189346
$$587$$ 17.5279 0.723452 0.361726 0.932284i $$-0.382188\pi$$
0.361726 + 0.932284i $$0.382188\pi$$
$$588$$ 0 0
$$589$$ 21.3050 0.877855
$$590$$ 1.88854 0.0777501
$$591$$ 0 0
$$592$$ −10.9443 −0.449807
$$593$$ 18.7639 0.770542 0.385271 0.922803i $$-0.374108\pi$$
0.385271 + 0.922803i $$0.374108\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −25.3050 −1.03393 −0.516966 0.856006i $$-0.672939\pi$$
−0.516966 + 0.856006i $$0.672939\pi$$
$$600$$ 0 0
$$601$$ 0.291796 0.0119026 0.00595130 0.999982i $$-0.498106\pi$$
0.00595130 + 0.999982i $$0.498106\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1.23607 0.0502533
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −7.70820 −0.312609
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12.4721 0.503745 0.251872 0.967760i $$-0.418954\pi$$
0.251872 + 0.967760i $$0.418954\pi$$
$$614$$ 25.5967 1.03300
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −23.8885 −0.961717 −0.480858 0.876798i $$-0.659675\pi$$
−0.480858 + 0.876798i $$0.659675\pi$$
$$618$$ 0 0
$$619$$ −18.8328 −0.756955 −0.378477 0.925611i $$-0.623552\pi$$
−0.378477 + 0.925611i $$0.623552\pi$$
$$620$$ 3.41641 0.137206
$$621$$ 0 0
$$622$$ 8.65248 0.346933
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 4.41641 0.176656
$$626$$ 5.52786 0.220938
$$627$$ 0 0
$$628$$ 14.7639 0.589145
$$629$$ 57.3050 2.28490
$$630$$ 0 0
$$631$$ 9.88854 0.393657 0.196828 0.980438i $$-0.436936\pi$$
0.196828 + 0.980438i $$0.436936\pi$$
$$632$$ −11.4164 −0.454120
$$633$$ 0 0
$$634$$ −31.8885 −1.26646
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 4.47214 0.177054
$$639$$ 0 0
$$640$$ −1.23607 −0.0488599
$$641$$ 35.8885 1.41751 0.708756 0.705454i $$-0.249257\pi$$
0.708756 + 0.705454i $$0.249257\pi$$
$$642$$ 0 0
$$643$$ −43.4164 −1.71218 −0.856088 0.516830i $$-0.827112\pi$$
−0.856088 + 0.516830i $$0.827112\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.3607 1.58797
$$647$$ 45.0132 1.76965 0.884825 0.465924i $$-0.154278\pi$$
0.884825 + 0.465924i $$0.154278\pi$$
$$648$$ 0 0
$$649$$ −1.52786 −0.0599739
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 14.9443 0.584815 0.292407 0.956294i $$-0.405544\pi$$
0.292407 + 0.956294i $$0.405544\pi$$
$$654$$ 0 0
$$655$$ 8.36068 0.326679
$$656$$ −5.23607 −0.204434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −13.8885 −0.541021 −0.270510 0.962717i $$-0.587192\pi$$
−0.270510 + 0.962717i $$0.587192\pi$$
$$660$$ 0 0
$$661$$ −5.59675 −0.217688 −0.108844 0.994059i $$-0.534715\pi$$
−0.108844 + 0.994059i $$0.534715\pi$$
$$662$$ 7.41641 0.288247
$$663$$ 0 0
$$664$$ −1.23607 −0.0479687
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −11.0557 −0.428080
$$668$$ 18.4721 0.714708
$$669$$ 0 0
$$670$$ −19.0557 −0.736187
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 34.9443 1.34700 0.673501 0.739186i $$-0.264790\pi$$
0.673501 + 0.739186i $$0.264790\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −46.4721 −1.78607 −0.893035 0.449988i $$-0.851428\pi$$
−0.893035 + 0.449988i $$0.851428\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 6.47214 0.248195
$$681$$ 0 0
$$682$$ −2.76393 −0.105836
$$683$$ 4.36068 0.166857 0.0834284 0.996514i $$-0.473413\pi$$
0.0834284 + 0.996514i $$0.473413\pi$$
$$684$$ 0 0
$$685$$ 20.3607 0.777942
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 6.47214 0.246748
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 17.5279 0.666791 0.333396 0.942787i $$-0.391806\pi$$
0.333396 + 0.942787i $$0.391806\pi$$
$$692$$ 19.4164 0.738101
$$693$$ 0 0
$$694$$ −26.8328 −1.01856
$$695$$ 12.5836 0.477323
$$696$$ 0 0
$$697$$ 27.4164 1.03847
$$698$$ −0.583592 −0.0220893
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.11146 0.306365 0.153183 0.988198i $$-0.451048\pi$$
0.153183 + 0.988198i $$0.451048\pi$$
$$702$$ 0 0
$$703$$ −84.3607 −3.18172
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −24.3607 −0.916826
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5.05573 0.189872 0.0949359 0.995483i $$-0.469735\pi$$
0.0949359 + 0.995483i $$0.469735\pi$$
$$710$$ −3.05573 −0.114679
$$711$$ 0 0
$$712$$ 6.47214 0.242554
$$713$$ 6.83282 0.255891
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 5.52786 0.206586
$$717$$ 0 0
$$718$$ 1.52786 0.0570194
$$719$$ 20.2918 0.756756 0.378378 0.925651i $$-0.376482\pi$$
0.378378 + 0.925651i $$0.376482\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −40.4164 −1.50414
$$723$$ 0 0
$$724$$ −11.7082 −0.435132
$$725$$ 15.5279 0.576690
$$726$$ 0 0
$$727$$ −45.2361 −1.67771 −0.838856 0.544353i $$-0.816775\pi$$
−0.838856 + 0.544353i $$0.816775\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −19.4164 −0.718633
$$731$$ −33.8885 −1.25341
$$732$$ 0 0
$$733$$ 22.8328 0.843349 0.421675 0.906747i $$-0.361442\pi$$
0.421675 + 0.906747i $$0.361442\pi$$
$$734$$ 8.29180 0.306056
$$735$$ 0 0
$$736$$ −2.47214 −0.0911241
$$737$$ 15.4164 0.567871
$$738$$ 0 0
$$739$$ 43.4164 1.59710 0.798549 0.601930i $$-0.205601\pi$$
0.798549 + 0.601930i $$0.205601\pi$$
$$740$$ −13.5279 −0.497294
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 25.8885 0.949759 0.474879 0.880051i $$-0.342492\pi$$
0.474879 + 0.880051i $$0.342492\pi$$
$$744$$ 0 0
$$745$$ 7.41641 0.271716
$$746$$ 1.41641 0.0518584
$$747$$ 0 0
$$748$$ −5.23607 −0.191450
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14.8328 −0.541257 −0.270629 0.962684i $$-0.587232\pi$$
−0.270629 + 0.962684i $$0.587232\pi$$
$$752$$ 3.70820 0.135224
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.9443 0.833924 0.416962 0.908924i $$-0.363095\pi$$
0.416962 + 0.908924i $$0.363095\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ −9.52786 −0.345612
$$761$$ 23.1246 0.838267 0.419133 0.907925i $$-0.362334\pi$$
0.419133 + 0.907925i $$0.362334\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 10.4721 0.378869
$$765$$ 0 0
$$766$$ 0.652476 0.0235749
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 12.0689 0.435215 0.217608 0.976036i $$-0.430175\pi$$
0.217608 + 0.976036i $$0.430175\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.05573 0.181960
$$773$$ −49.9574 −1.79684 −0.898422 0.439133i $$-0.855286\pi$$
−0.898422 + 0.439133i $$0.855286\pi$$
$$774$$ 0 0
$$775$$ −9.59675 −0.344725
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −6.94427 −0.248964
$$779$$ −40.3607 −1.44607
$$780$$ 0 0
$$781$$ 2.47214 0.0884600
$$782$$ 12.9443 0.462886
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 18.2492 0.651343
$$786$$ 0 0
$$787$$ 22.5410 0.803501 0.401750 0.915749i $$-0.368402\pi$$
0.401750 + 0.915749i $$0.368402\pi$$
$$788$$ 22.3607 0.796566
$$789$$ 0 0
$$790$$ −14.1115 −0.502063
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −1.81966 −0.0645773
$$795$$ 0 0
$$796$$ 23.1246 0.819630
$$797$$ 12.8754 0.456070 0.228035 0.973653i $$-0.426770\pi$$
0.228035 + 0.973653i $$0.426770\pi$$
$$798$$ 0 0
$$799$$ −19.4164 −0.686903
$$800$$ 3.47214 0.122759
$$801$$ 0 0
$$802$$ −8.47214 −0.299162
$$803$$ 15.7082 0.554330
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 12.0000 0.422159
$$809$$ −44.8328 −1.57624 −0.788119 0.615523i $$-0.788945\pi$$
−0.788119 + 0.615523i $$0.788945\pi$$
$$810$$ 0 0
$$811$$ −17.5967 −0.617905 −0.308953 0.951077i $$-0.599978\pi$$
−0.308953 + 0.951077i $$0.599978\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 10.9443 0.383597
$$815$$ −14.8328 −0.519571
$$816$$ 0 0
$$817$$ 49.8885 1.74538
$$818$$ 2.76393 0.0966386
$$819$$ 0 0
$$820$$ −6.47214 −0.226017
$$821$$ −29.7771 −1.03923 −0.519614 0.854401i $$-0.673924\pi$$
−0.519614 + 0.854401i $$0.673924\pi$$
$$822$$ 0 0
$$823$$ −30.8328 −1.07476 −0.537382 0.843339i $$-0.680587\pi$$
−0.537382 + 0.843339i $$0.680587\pi$$
$$824$$ 10.7639 0.374979
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.7214 −1.27693 −0.638463 0.769652i $$-0.720430\pi$$
−0.638463 + 0.769652i $$0.720430\pi$$
$$828$$ 0 0
$$829$$ −47.4853 −1.64923 −0.824616 0.565693i $$-0.808609\pi$$
−0.824616 + 0.565693i $$0.808609\pi$$
$$830$$ −1.52786 −0.0530329
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 22.8328 0.790162
$$836$$ 7.70820 0.266594
$$837$$ 0 0
$$838$$ −15.0557 −0.520091
$$839$$ 25.2361 0.871246 0.435623 0.900129i $$-0.356528\pi$$
0.435623 + 0.900129i $$0.356528\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 5.05573 0.174232
$$843$$ 0 0
$$844$$ 3.41641 0.117598
$$845$$ −16.0689 −0.552786
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ −18.1803 −0.623581
$$851$$ −27.0557 −0.927458
$$852$$ 0 0
$$853$$ 33.8885 1.16032 0.580161 0.814502i $$-0.302990\pi$$
0.580161 + 0.814502i $$0.302990\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0.944272 0.0322745
$$857$$ 47.1246 1.60975 0.804873 0.593447i $$-0.202233\pi$$
0.804873 + 0.593447i $$0.202233\pi$$
$$858$$ 0 0
$$859$$ 43.4164 1.48135 0.740674 0.671864i $$-0.234506\pi$$
0.740674 + 0.671864i $$0.234506\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ −22.4721 −0.765404
$$863$$ 44.3607 1.51006 0.755028 0.655693i $$-0.227623\pi$$
0.755028 + 0.655693i $$0.227623\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ 9.88854 0.336026
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 11.4164 0.387275
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −6.00000 −0.203186
$$873$$ 0 0
$$874$$ −19.0557 −0.644570
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ 31.4164 1.06025
$$879$$ 0 0
$$880$$ 1.23607 0.0416678
$$881$$ 26.8328 0.904021 0.452010 0.892013i $$-0.350707\pi$$
0.452010 + 0.892013i $$0.350707\pi$$
$$882$$ 0 0
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 7.41641 0.249159
$$887$$ −36.9443 −1.24047 −0.620234 0.784417i $$-0.712962\pi$$
−0.620234 + 0.784417i $$0.712962\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 8.00000 0.268161
$$891$$ 0 0
$$892$$ 15.7082 0.525950
$$893$$ 28.5836 0.956513
$$894$$ 0 0
$$895$$ 6.83282 0.228396
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 19.5279 0.651653
$$899$$ −12.3607 −0.412252
$$900$$ 0 0
$$901$$ −31.4164 −1.04663
$$902$$ 5.23607 0.174342
$$903$$ 0 0
$$904$$ 13.4164 0.446223
$$905$$ −14.4721 −0.481070
$$906$$ 0 0
$$907$$ −33.3050 −1.10587 −0.552936 0.833223i $$-0.686493\pi$$
−0.552936 + 0.833223i $$0.686493\pi$$
$$908$$ 8.65248 0.287142
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31.4164 1.04087 0.520436 0.853901i $$-0.325769\pi$$
0.520436 + 0.853901i $$0.325769\pi$$
$$912$$ 0 0
$$913$$ 1.23607 0.0409079
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −19.7082 −0.651177
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −53.6656 −1.77027 −0.885133 0.465338i $$-0.845933\pi$$
−0.885133 + 0.465338i $$0.845933\pi$$
$$920$$ −3.05573 −0.100744
$$921$$ 0 0
$$922$$ 26.8328 0.883692
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 38.0000 1.24943
$$926$$ 1.88854 0.0620614
$$927$$ 0 0
$$928$$ 4.47214 0.146805
$$929$$ 8.36068 0.274305 0.137153 0.990550i $$-0.456205\pi$$
0.137153 + 0.990550i $$0.456205\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9.05573 0.296630
$$933$$ 0 0
$$934$$ 29.8885 0.977983
$$935$$ −6.47214 −0.211661
$$936$$ 0 0
$$937$$ 1.59675 0.0521635 0.0260817 0.999660i $$-0.491697\pi$$
0.0260817 + 0.999660i $$0.491697\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 4.58359 0.149500
$$941$$ −7.63932 −0.249035 −0.124517 0.992217i $$-0.539738\pi$$
−0.124517 + 0.992217i $$0.539738\pi$$
$$942$$ 0 0
$$943$$ −12.9443 −0.421523
$$944$$ −1.52786 −0.0497277
$$945$$ 0 0
$$946$$ −6.47214 −0.210427
$$947$$ 37.8885 1.23121 0.615606 0.788054i $$-0.288911\pi$$
0.615606 + 0.788054i $$0.288911\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 26.7639 0.868337
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 30.9443 1.00238 0.501192 0.865336i $$-0.332895\pi$$
0.501192 + 0.865336i $$0.332895\pi$$
$$954$$ 0 0
$$955$$ 12.9443 0.418867
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −35.7771 −1.15591
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23.3607 −0.753570
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −18.1803 −0.585549
$$965$$ 6.24922 0.201170
$$966$$ 0 0
$$967$$ 54.8328 1.76330 0.881652 0.471900i $$-0.156432\pi$$
0.881652 + 0.471900i $$0.156432\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −39.0557 −1.25336 −0.626679 0.779278i $$-0.715586\pi$$
−0.626679 + 0.779278i $$0.715586\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −36.9443 −1.18377
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 11.8885 0.380348 0.190174 0.981750i $$-0.439095\pi$$
0.190174 + 0.981750i $$0.439095\pi$$
$$978$$ 0 0
$$979$$ −6.47214 −0.206850
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −29.8885 −0.953782
$$983$$ 51.7082 1.64924 0.824618 0.565690i $$-0.191390\pi$$
0.824618 + 0.565690i $$0.191390\pi$$
$$984$$ 0 0
$$985$$ 27.6393 0.880662
$$986$$ −23.4164 −0.745730
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −30.8328 −0.979437 −0.489718 0.871881i $$-0.662900\pi$$
−0.489718 + 0.871881i $$0.662900\pi$$
$$992$$ −2.76393 −0.0877549
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 28.5836 0.906161
$$996$$ 0 0
$$997$$ −54.2492 −1.71809 −0.859045 0.511900i $$-0.828942\pi$$
−0.859045 + 0.511900i $$0.828942\pi$$
$$998$$ −7.41641 −0.234762
$$999$$ 0 0
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 9702.2.a.ck.1.2 2
3.2 odd 2 3234.2.a.be.1.1 yes 2
7.6 odd 2 9702.2.a.cw.1.1 2
21.20 even 2 3234.2.a.bb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bb.1.2 2 21.20 even 2
3234.2.a.be.1.1 yes 2 3.2 odd 2
9702.2.a.ck.1.2 2 1.1 even 1 trivial
9702.2.a.cw.1.1 2 7.6 odd 2