# Properties

 Label 5733.2.a.be.1.3 Level $5733$ Weight $2$ Character 5733.1 Self dual yes Analytic conductor $45.778$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5733 = 3^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5733.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.7782354788$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.65544$$ of defining polynomial Character $$\chi$$ $$=$$ 5733.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.65544 q^{2} +5.05137 q^{4} +3.65544 q^{5} +8.10275 q^{8} +O(q^{10})$$ $$q+2.65544 q^{2} +5.05137 q^{4} +3.65544 q^{5} +8.10275 q^{8} +9.70682 q^{10} -0.655442 q^{11} +1.00000 q^{13} +11.4136 q^{16} +2.39593 q^{17} +2.70682 q^{19} +18.4650 q^{20} -1.74049 q^{22} -7.36226 q^{23} +8.36226 q^{25} +2.65544 q^{26} +0.208136 q^{29} +1.13642 q^{31} +14.1027 q^{32} +6.36226 q^{34} -7.44731 q^{37} +7.18780 q^{38} +29.6191 q^{40} -10.2055 q^{41} -3.10275 q^{43} -3.31088 q^{44} -19.5501 q^{46} +4.60407 q^{47} +22.2055 q^{50} +5.05137 q^{52} -5.25951 q^{53} -2.39593 q^{55} +0.552694 q^{58} +8.25951 q^{59} +1.89725 q^{61} +3.01770 q^{62} +14.6218 q^{64} +3.65544 q^{65} -12.8946 q^{67} +12.1027 q^{68} +6.75819 q^{71} -12.5367 q^{73} -19.7759 q^{74} +13.6731 q^{76} -1.51902 q^{79} +41.7219 q^{80} -27.1001 q^{82} +15.7582 q^{83} +8.75819 q^{85} -8.23917 q^{86} -5.31088 q^{88} -14.8096 q^{89} -37.1895 q^{92} +12.2258 q^{94} +9.89461 q^{95} +10.0177 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{8} + O(q^{10})$$ $$3 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{8} + 14 q^{10} + 4 q^{11} + 3 q^{13} + 4 q^{16} + 4 q^{17} - 7 q^{19} + 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} + 2 q^{26} + 7 q^{29} + 3 q^{31} + 24 q^{32} - 2 q^{34} - 10 q^{37} + 12 q^{38} + 22 q^{40} + 6 q^{41} + 9 q^{43} + 2 q^{44} - 28 q^{46} + 17 q^{47} + 30 q^{50} + 6 q^{52} - 13 q^{53} - 4 q^{55} + 14 q^{58} + 22 q^{59} + 24 q^{61} - 18 q^{62} + 20 q^{64} + 5 q^{65} - 14 q^{67} + 18 q^{68} - 4 q^{71} - 5 q^{73} - 8 q^{74} + 8 q^{76} + q^{79} + 40 q^{80} - 20 q^{82} + 23 q^{83} + 2 q^{85} - 6 q^{86} - 4 q^{88} - 11 q^{89} - 30 q^{92} + 16 q^{94} + 5 q^{95} + 3 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.65544 1.87768 0.938841 0.344352i $$-0.111901\pi$$
0.938841 + 0.344352i $$0.111901\pi$$
$$3$$ 0 0
$$4$$ 5.05137 2.52569
$$5$$ 3.65544 1.63476 0.817382 0.576096i $$-0.195425\pi$$
0.817382 + 0.576096i $$0.195425\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 8.10275 2.86475
$$9$$ 0 0
$$10$$ 9.70682 3.06956
$$11$$ −0.655442 −0.197623 −0.0988117 0.995106i $$-0.531504\pi$$
−0.0988117 + 0.995106i $$0.531504\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 11.4136 2.85341
$$17$$ 2.39593 0.581099 0.290549 0.956860i $$-0.406162\pi$$
0.290549 + 0.956860i $$0.406162\pi$$
$$18$$ 0 0
$$19$$ 2.70682 0.620986 0.310493 0.950576i $$-0.399506\pi$$
0.310493 + 0.950576i $$0.399506\pi$$
$$20$$ 18.4650 4.12890
$$21$$ 0 0
$$22$$ −1.74049 −0.371074
$$23$$ −7.36226 −1.53514 −0.767569 0.640967i $$-0.778534\pi$$
−0.767569 + 0.640967i $$0.778534\pi$$
$$24$$ 0 0
$$25$$ 8.36226 1.67245
$$26$$ 2.65544 0.520775
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.208136 0.0386499 0.0193250 0.999813i $$-0.493848\pi$$
0.0193250 + 0.999813i $$0.493848\pi$$
$$30$$ 0 0
$$31$$ 1.13642 0.204107 0.102054 0.994779i $$-0.467459\pi$$
0.102054 + 0.994779i $$0.467459\pi$$
$$32$$ 14.1027 2.49304
$$33$$ 0 0
$$34$$ 6.36226 1.09112
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.44731 −1.22433 −0.612165 0.790730i $$-0.709701\pi$$
−0.612165 + 0.790730i $$0.709701\pi$$
$$38$$ 7.18780 1.16601
$$39$$ 0 0
$$40$$ 29.6191 4.68320
$$41$$ −10.2055 −1.59383 −0.796915 0.604091i $$-0.793536\pi$$
−0.796915 + 0.604091i $$0.793536\pi$$
$$42$$ 0 0
$$43$$ −3.10275 −0.473165 −0.236582 0.971611i $$-0.576027\pi$$
−0.236582 + 0.971611i $$0.576027\pi$$
$$44$$ −3.31088 −0.499135
$$45$$ 0 0
$$46$$ −19.5501 −2.88250
$$47$$ 4.60407 0.671572 0.335786 0.941938i $$-0.390998\pi$$
0.335786 + 0.941938i $$0.390998\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 22.2055 3.14033
$$51$$ 0 0
$$52$$ 5.05137 0.700500
$$53$$ −5.25951 −0.722449 −0.361225 0.932479i $$-0.617641\pi$$
−0.361225 + 0.932479i $$0.617641\pi$$
$$54$$ 0 0
$$55$$ −2.39593 −0.323067
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0.552694 0.0725723
$$59$$ 8.25951 1.07530 0.537648 0.843169i $$-0.319313\pi$$
0.537648 + 0.843169i $$0.319313\pi$$
$$60$$ 0 0
$$61$$ 1.89725 0.242918 0.121459 0.992596i $$-0.461243\pi$$
0.121459 + 0.992596i $$0.461243\pi$$
$$62$$ 3.01770 0.383248
$$63$$ 0 0
$$64$$ 14.6218 1.82772
$$65$$ 3.65544 0.453402
$$66$$ 0 0
$$67$$ −12.8946 −1.57533 −0.787664 0.616105i $$-0.788710\pi$$
−0.787664 + 0.616105i $$0.788710\pi$$
$$68$$ 12.1027 1.46767
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.75819 0.802050 0.401025 0.916067i $$-0.368654\pi$$
0.401025 + 0.916067i $$0.368654\pi$$
$$72$$ 0 0
$$73$$ −12.5367 −1.46731 −0.733656 0.679521i $$-0.762188\pi$$
−0.733656 + 0.679521i $$0.762188\pi$$
$$74$$ −19.7759 −2.29890
$$75$$ 0 0
$$76$$ 13.6731 1.56842
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.51902 −0.170903 −0.0854516 0.996342i $$-0.527233\pi$$
−0.0854516 + 0.996342i $$0.527233\pi$$
$$80$$ 41.7219 4.66465
$$81$$ 0 0
$$82$$ −27.1001 −2.99271
$$83$$ 15.7582 1.72969 0.864843 0.502042i $$-0.167418\pi$$
0.864843 + 0.502042i $$0.167418\pi$$
$$84$$ 0 0
$$85$$ 8.75819 0.949959
$$86$$ −8.23917 −0.888453
$$87$$ 0 0
$$88$$ −5.31088 −0.566142
$$89$$ −14.8096 −1.56981 −0.784905 0.619616i $$-0.787288\pi$$
−0.784905 + 0.619616i $$0.787288\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −37.1895 −3.87728
$$93$$ 0 0
$$94$$ 12.2258 1.26100
$$95$$ 9.89461 1.01517
$$96$$ 0 0
$$97$$ 10.0177 1.01714 0.508572 0.861020i $$-0.330174\pi$$
0.508572 + 0.861020i $$0.330174\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 42.2409 4.22409
$$101$$ −3.01770 −0.300273 −0.150136 0.988665i $$-0.547971\pi$$
−0.150136 + 0.988665i $$0.547971\pi$$
$$102$$ 0 0
$$103$$ 5.03804 0.496413 0.248207 0.968707i $$-0.420159\pi$$
0.248207 + 0.968707i $$0.420159\pi$$
$$104$$ 8.10275 0.794540
$$105$$ 0 0
$$106$$ −13.9663 −1.35653
$$107$$ −11.8432 −1.14493 −0.572465 0.819929i $$-0.694013\pi$$
−0.572465 + 0.819929i $$0.694013\pi$$
$$108$$ 0 0
$$109$$ 3.55005 0.340034 0.170017 0.985441i $$-0.445618\pi$$
0.170017 + 0.985441i $$0.445618\pi$$
$$110$$ −6.36226 −0.606618
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 9.46501 0.890393 0.445197 0.895433i $$-0.353134\pi$$
0.445197 + 0.895433i $$0.353134\pi$$
$$114$$ 0 0
$$115$$ −26.9123 −2.50959
$$116$$ 1.05137 0.0976176
$$117$$ 0 0
$$118$$ 21.9327 2.01906
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.5704 −0.960945
$$122$$ 5.03804 0.456123
$$123$$ 0 0
$$124$$ 5.74049 0.515511
$$125$$ 12.2905 1.09930
$$126$$ 0 0
$$127$$ 5.46765 0.485175 0.242588 0.970130i $$-0.422004\pi$$
0.242588 + 0.970130i $$0.422004\pi$$
$$128$$ 10.6218 0.938841
$$129$$ 0 0
$$130$$ 9.70682 0.851344
$$131$$ 9.82991 0.858843 0.429421 0.903104i $$-0.358718\pi$$
0.429421 + 0.903104i $$0.358718\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −34.2409 −2.95796
$$135$$ 0 0
$$136$$ 19.4136 1.66471
$$137$$ −17.5501 −1.49940 −0.749701 0.661777i $$-0.769803\pi$$
−0.749701 + 0.661777i $$0.769803\pi$$
$$138$$ 0 0
$$139$$ 4.91495 0.416881 0.208440 0.978035i $$-0.433161\pi$$
0.208440 + 0.978035i $$0.433161\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 17.9460 1.50599
$$143$$ −0.655442 −0.0548108
$$144$$ 0 0
$$145$$ 0.760830 0.0631835
$$146$$ −33.2905 −2.75515
$$147$$ 0 0
$$148$$ −37.6191 −3.09227
$$149$$ 10.3419 0.847243 0.423621 0.905839i $$-0.360759\pi$$
0.423621 + 0.905839i $$0.360759\pi$$
$$150$$ 0 0
$$151$$ 5.07171 0.412730 0.206365 0.978475i $$-0.433837\pi$$
0.206365 + 0.978475i $$0.433837\pi$$
$$152$$ 21.9327 1.77897
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.15412 0.333667
$$156$$ 0 0
$$157$$ 12.6014 1.00570 0.502852 0.864373i $$-0.332284\pi$$
0.502852 + 0.864373i $$0.332284\pi$$
$$158$$ −4.03367 −0.320902
$$159$$ 0 0
$$160$$ 51.5518 4.07553
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3.20814 0.251281 0.125640 0.992076i $$-0.459901\pi$$
0.125640 + 0.992076i $$0.459901\pi$$
$$164$$ −51.5518 −4.02552
$$165$$ 0 0
$$166$$ 41.8450 3.24780
$$167$$ 8.12045 0.628379 0.314190 0.949360i $$-0.398267\pi$$
0.314190 + 0.949360i $$0.398267\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 23.2569 1.78372
$$171$$ 0 0
$$172$$ −15.6731 −1.19507
$$173$$ −10.3286 −0.785268 −0.392634 0.919695i $$-0.628436\pi$$
−0.392634 + 0.919695i $$0.628436\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −7.48098 −0.563900
$$177$$ 0 0
$$178$$ −39.3259 −2.94760
$$179$$ 2.37823 0.177757 0.0888786 0.996042i $$-0.471672\pi$$
0.0888786 + 0.996042i $$0.471672\pi$$
$$180$$ 0 0
$$181$$ −21.8096 −1.62109 −0.810546 0.585675i $$-0.800830\pi$$
−0.810546 + 0.585675i $$0.800830\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −59.6545 −4.39779
$$185$$ −27.2232 −2.00149
$$186$$ 0 0
$$187$$ −1.57040 −0.114839
$$188$$ 23.2569 1.69618
$$189$$ 0 0
$$190$$ 26.2746 1.90616
$$191$$ 25.0868 1.81522 0.907608 0.419819i $$-0.137907\pi$$
0.907608 + 0.419819i $$0.137907\pi$$
$$192$$ 0 0
$$193$$ 11.3109 0.814175 0.407088 0.913389i $$-0.366544\pi$$
0.407088 + 0.913389i $$0.366544\pi$$
$$194$$ 26.6014 1.90987
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.7919 1.19637 0.598185 0.801358i $$-0.295889\pi$$
0.598185 + 0.801358i $$0.295889\pi$$
$$198$$ 0 0
$$199$$ −20.5341 −1.45562 −0.727811 0.685777i $$-0.759462\pi$$
−0.727811 + 0.685777i $$0.759462\pi$$
$$200$$ 67.7573 4.79116
$$201$$ 0 0
$$202$$ −8.01333 −0.563816
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −37.3056 −2.60554
$$206$$ 13.3782 0.932105
$$207$$ 0 0
$$208$$ 11.4136 0.791393
$$209$$ −1.77416 −0.122721
$$210$$ 0 0
$$211$$ −15.7785 −1.08624 −0.543119 0.839655i $$-0.682757\pi$$
−0.543119 + 0.839655i $$0.682757\pi$$
$$212$$ −26.5678 −1.82468
$$213$$ 0 0
$$214$$ −31.4490 −2.14981
$$215$$ −11.3419 −0.773512
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 9.42697 0.638475
$$219$$ 0 0
$$220$$ −12.1027 −0.815967
$$221$$ 2.39593 0.161168
$$222$$ 0 0
$$223$$ 8.44731 0.565673 0.282837 0.959168i $$-0.408725\pi$$
0.282837 + 0.959168i $$0.408725\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 25.1338 1.67187
$$227$$ 6.68912 0.443972 0.221986 0.975050i $$-0.428746\pi$$
0.221986 + 0.975050i $$0.428746\pi$$
$$228$$ 0 0
$$229$$ −8.63510 −0.570624 −0.285312 0.958435i $$-0.592097\pi$$
−0.285312 + 0.958435i $$0.592097\pi$$
$$230$$ −71.4641 −4.71220
$$231$$ 0 0
$$232$$ 1.68648 0.110723
$$233$$ 4.16745 0.273019 0.136510 0.990639i $$-0.456412\pi$$
0.136510 + 0.990639i $$0.456412\pi$$
$$234$$ 0 0
$$235$$ 16.8299 1.09786
$$236$$ 41.7219 2.71586
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.79450 0.116077 0.0580384 0.998314i $$-0.481515\pi$$
0.0580384 + 0.998314i $$0.481515\pi$$
$$240$$ 0 0
$$241$$ −13.8609 −0.892862 −0.446431 0.894818i $$-0.647305\pi$$
−0.446431 + 0.894818i $$0.647305\pi$$
$$242$$ −28.0691 −1.80435
$$243$$ 0 0
$$244$$ 9.58373 0.613535
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.70682 0.172231
$$248$$ 9.20814 0.584717
$$249$$ 0 0
$$250$$ 32.6368 2.06413
$$251$$ 14.7449 0.930687 0.465344 0.885130i $$-0.345931\pi$$
0.465344 + 0.885130i $$0.345931\pi$$
$$252$$ 0 0
$$253$$ 4.82554 0.303379
$$254$$ 14.5190 0.911004
$$255$$ 0 0
$$256$$ −1.03804 −0.0648776
$$257$$ 23.7068 1.47879 0.739395 0.673272i $$-0.235112\pi$$
0.739395 + 0.673272i $$0.235112\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 18.4650 1.14515
$$261$$ 0 0
$$262$$ 26.1027 1.61263
$$263$$ −11.3756 −0.701449 −0.350725 0.936479i $$-0.614065\pi$$
−0.350725 + 0.936479i $$0.614065\pi$$
$$264$$ 0 0
$$265$$ −19.2258 −1.18103
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −65.1355 −3.97878
$$269$$ −11.1054 −0.677107 −0.338554 0.940947i $$-0.609938\pi$$
−0.338554 + 0.940947i $$0.609938\pi$$
$$270$$ 0 0
$$271$$ 12.7245 0.772959 0.386480 0.922298i $$-0.373691\pi$$
0.386480 + 0.922298i $$0.373691\pi$$
$$272$$ 27.3463 1.65811
$$273$$ 0 0
$$274$$ −46.6032 −2.81540
$$275$$ −5.48098 −0.330515
$$276$$ 0 0
$$277$$ −3.00000 −0.180253 −0.0901263 0.995930i $$-0.528727\pi$$
−0.0901263 + 0.995930i $$0.528727\pi$$
$$278$$ 13.0514 0.782769
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.44731 −0.205649 −0.102825 0.994700i $$-0.532788\pi$$
−0.102825 + 0.994700i $$0.532788\pi$$
$$282$$ 0 0
$$283$$ 12.4783 0.741760 0.370880 0.928681i $$-0.379056\pi$$
0.370880 + 0.928681i $$0.379056\pi$$
$$284$$ 34.1382 2.02573
$$285$$ 0 0
$$286$$ −1.74049 −0.102917
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −11.2595 −0.662324
$$290$$ 2.02034 0.118638
$$291$$ 0 0
$$292$$ −63.3277 −3.70597
$$293$$ −13.5341 −0.790670 −0.395335 0.918537i $$-0.629371\pi$$
−0.395335 + 0.918537i $$0.629371\pi$$
$$294$$ 0 0
$$295$$ 30.1922 1.75786
$$296$$ −60.3436 −3.50740
$$297$$ 0 0
$$298$$ 27.4624 1.59085
$$299$$ −7.36226 −0.425770
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 13.4676 0.774976
$$303$$ 0 0
$$304$$ 30.8946 1.77193
$$305$$ 6.93529 0.397114
$$306$$ 0 0
$$307$$ −28.2365 −1.61154 −0.805772 0.592226i $$-0.798249\pi$$
−0.805772 + 0.592226i $$0.798249\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 11.0310 0.626521
$$311$$ −24.7042 −1.40085 −0.700423 0.713728i $$-0.747005\pi$$
−0.700423 + 0.713728i $$0.747005\pi$$
$$312$$ 0 0
$$313$$ 20.8122 1.17638 0.588188 0.808724i $$-0.299842\pi$$
0.588188 + 0.808724i $$0.299842\pi$$
$$314$$ 33.4624 1.88839
$$315$$ 0 0
$$316$$ −7.67314 −0.431648
$$317$$ −26.1382 −1.46806 −0.734032 0.679114i $$-0.762364\pi$$
−0.734032 + 0.679114i $$0.762364\pi$$
$$318$$ 0 0
$$319$$ −0.136421 −0.00763813
$$320$$ 53.4490 2.98789
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6.48535 0.360854
$$324$$ 0 0
$$325$$ 8.36226 0.463855
$$326$$ 8.51902 0.471825
$$327$$ 0 0
$$328$$ −82.6926 −4.56593
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −24.1382 −1.32675 −0.663376 0.748286i $$-0.730877\pi$$
−0.663376 + 0.748286i $$0.730877\pi$$
$$332$$ 79.6005 4.36865
$$333$$ 0 0
$$334$$ 21.5634 1.17990
$$335$$ −47.1355 −2.57529
$$336$$ 0 0
$$337$$ −30.5297 −1.66306 −0.831530 0.555480i $$-0.812534\pi$$
−0.831530 + 0.555480i $$0.812534\pi$$
$$338$$ 2.65544 0.144437
$$339$$ 0 0
$$340$$ 44.2409 2.39930
$$341$$ −0.744859 −0.0403364
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −25.1408 −1.35550
$$345$$ 0 0
$$346$$ −27.4270 −1.47448
$$347$$ 24.9974 1.34193 0.670964 0.741490i $$-0.265880\pi$$
0.670964 + 0.741490i $$0.265880\pi$$
$$348$$ 0 0
$$349$$ 1.83887 0.0984324 0.0492162 0.998788i $$-0.484328\pi$$
0.0492162 + 0.998788i $$0.484328\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −9.24354 −0.492682
$$353$$ −23.2569 −1.23784 −0.618919 0.785455i $$-0.712429\pi$$
−0.618919 + 0.785455i $$0.712429\pi$$
$$354$$ 0 0
$$355$$ 24.7042 1.31116
$$356$$ −74.8087 −3.96485
$$357$$ 0 0
$$358$$ 6.31525 0.333772
$$359$$ 21.4473 1.13195 0.565973 0.824424i $$-0.308501\pi$$
0.565973 + 0.824424i $$0.308501\pi$$
$$360$$ 0 0
$$361$$ −11.6731 −0.614376
$$362$$ −57.9140 −3.04389
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −45.8273 −2.39871
$$366$$ 0 0
$$367$$ 1.12045 0.0584870 0.0292435 0.999572i $$-0.490690\pi$$
0.0292435 + 0.999572i $$0.490690\pi$$
$$368$$ −84.0301 −4.38037
$$369$$ 0 0
$$370$$ −72.2896 −3.75816
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 15.6058 0.808038 0.404019 0.914751i $$-0.367613\pi$$
0.404019 + 0.914751i $$0.367613\pi$$
$$374$$ −4.17009 −0.215630
$$375$$ 0 0
$$376$$ 37.3056 1.92389
$$377$$ 0.208136 0.0107196
$$378$$ 0 0
$$379$$ 12.7849 0.656714 0.328357 0.944554i $$-0.393505\pi$$
0.328357 + 0.944554i $$0.393505\pi$$
$$380$$ 49.9814 2.56399
$$381$$ 0 0
$$382$$ 66.6165 3.40840
$$383$$ 34.0354 1.73913 0.869564 0.493820i $$-0.164400\pi$$
0.869564 + 0.493820i $$0.164400\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 30.0354 1.52876
$$387$$ 0 0
$$388$$ 50.6032 2.56899
$$389$$ 24.6705 1.25084 0.625422 0.780287i $$-0.284927\pi$$
0.625422 + 0.780287i $$0.284927\pi$$
$$390$$ 0 0
$$391$$ −17.6395 −0.892066
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 44.5898 2.24640
$$395$$ −5.55269 −0.279386
$$396$$ 0 0
$$397$$ 4.97966 0.249922 0.124961 0.992162i $$-0.460119\pi$$
0.124961 + 0.992162i $$0.460119\pi$$
$$398$$ −54.5271 −2.73320
$$399$$ 0 0
$$400$$ 95.4438 4.77219
$$401$$ −0.689115 −0.0344128 −0.0172064 0.999852i $$-0.505477\pi$$
−0.0172064 + 0.999852i $$0.505477\pi$$
$$402$$ 0 0
$$403$$ 1.13642 0.0566092
$$404$$ −15.2435 −0.758394
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.88128 0.241956
$$408$$ 0 0
$$409$$ −19.9770 −0.987800 −0.493900 0.869519i $$-0.664429\pi$$
−0.493900 + 0.869519i $$0.664429\pi$$
$$410$$ −99.0629 −4.89237
$$411$$ 0 0
$$412$$ 25.4490 1.25378
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 57.6032 2.82763
$$416$$ 14.1027 0.691444
$$417$$ 0 0
$$418$$ −4.71119 −0.230432
$$419$$ 19.6661 0.960754 0.480377 0.877062i $$-0.340500\pi$$
0.480377 + 0.877062i $$0.340500\pi$$
$$420$$ 0 0
$$421$$ 14.9283 0.727560 0.363780 0.931485i $$-0.381486\pi$$
0.363780 + 0.931485i $$0.381486\pi$$
$$422$$ −41.8990 −2.03961
$$423$$ 0 0
$$424$$ −42.6165 −2.06964
$$425$$ 20.0354 0.971860
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −59.8246 −2.89173
$$429$$ 0 0
$$430$$ −30.1178 −1.45241
$$431$$ 32.6838 1.57433 0.787163 0.616746i $$-0.211549\pi$$
0.787163 + 0.616746i $$0.211549\pi$$
$$432$$ 0 0
$$433$$ 8.96196 0.430684 0.215342 0.976539i $$-0.430913\pi$$
0.215342 + 0.976539i $$0.430913\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 17.9327 0.858818
$$437$$ −19.9283 −0.953299
$$438$$ 0 0
$$439$$ 7.93265 0.378605 0.189302 0.981919i $$-0.439377\pi$$
0.189302 + 0.981919i $$0.439377\pi$$
$$440$$ −19.4136 −0.925509
$$441$$ 0 0
$$442$$ 6.36226 0.302622
$$443$$ 8.91058 0.423355 0.211677 0.977340i $$-0.432107\pi$$
0.211677 + 0.977340i $$0.432107\pi$$
$$444$$ 0 0
$$445$$ −54.1355 −2.56627
$$446$$ 22.4313 1.06215
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 8.45168 0.398859 0.199430 0.979912i $$-0.436091\pi$$
0.199430 + 0.979912i $$0.436091\pi$$
$$450$$ 0 0
$$451$$ 6.68912 0.314978
$$452$$ 47.8113 2.24885
$$453$$ 0 0
$$454$$ 17.7626 0.833638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.1692 1.08381 0.541904 0.840440i $$-0.317703\pi$$
0.541904 + 0.840440i $$0.317703\pi$$
$$458$$ −22.9300 −1.07145
$$459$$ 0 0
$$460$$ −135.944 −6.33843
$$461$$ 2.27284 0.105857 0.0529284 0.998598i $$-0.483144\pi$$
0.0529284 + 0.998598i $$0.483144\pi$$
$$462$$ 0 0
$$463$$ −4.10976 −0.190997 −0.0954983 0.995430i $$-0.530444\pi$$
−0.0954983 + 0.995430i $$0.530444\pi$$
$$464$$ 2.37559 0.110284
$$465$$ 0 0
$$466$$ 11.0664 0.512643
$$467$$ 32.9150 1.52312 0.761561 0.648093i $$-0.224433\pi$$
0.761561 + 0.648093i $$0.224433\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 44.6908 2.06143
$$471$$ 0 0
$$472$$ 66.9247 3.08046
$$473$$ 2.03367 0.0935084
$$474$$ 0 0
$$475$$ 22.6351 1.03857
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 4.76520 0.217955
$$479$$ −9.31525 −0.425625 −0.212812 0.977093i $$-0.568262\pi$$
−0.212812 + 0.977093i $$0.568262\pi$$
$$480$$ 0 0
$$481$$ −7.44731 −0.339568
$$482$$ −36.8069 −1.67651
$$483$$ 0 0
$$484$$ −53.3950 −2.42705
$$485$$ 36.6191 1.66279
$$486$$ 0 0
$$487$$ 20.2409 0.917203 0.458601 0.888642i $$-0.348351\pi$$
0.458601 + 0.888642i $$0.348351\pi$$
$$488$$ 15.3730 0.695901
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.36226 0.196866 0.0984330 0.995144i $$-0.468617\pi$$
0.0984330 + 0.995144i $$0.468617\pi$$
$$492$$ 0 0
$$493$$ 0.498680 0.0224594
$$494$$ 7.18780 0.323394
$$495$$ 0 0
$$496$$ 12.9707 0.582401
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9.69348 0.433940 0.216970 0.976178i $$-0.430383\pi$$
0.216970 + 0.976178i $$0.430383\pi$$
$$500$$ 62.0841 2.77649
$$501$$ 0 0
$$502$$ 39.1541 1.74753
$$503$$ 2.64843 0.118088 0.0590439 0.998255i $$-0.481195\pi$$
0.0590439 + 0.998255i $$0.481195\pi$$
$$504$$ 0 0
$$505$$ −11.0310 −0.490875
$$506$$ 12.8139 0.569649
$$507$$ 0 0
$$508$$ 27.6191 1.22540
$$509$$ −13.9416 −0.617951 −0.308976 0.951070i $$-0.599986\pi$$
−0.308976 + 0.951070i $$0.599986\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −24.0000 −1.06066
$$513$$ 0 0
$$514$$ 62.9521 2.77670
$$515$$ 18.4163 0.811518
$$516$$ 0 0
$$517$$ −3.01770 −0.132718
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 29.6191 1.29888
$$521$$ −14.6218 −0.640591 −0.320296 0.947318i $$-0.603782\pi$$
−0.320296 + 0.947318i $$0.603782\pi$$
$$522$$ 0 0
$$523$$ −16.5190 −0.722326 −0.361163 0.932503i $$-0.617620\pi$$
−0.361163 + 0.932503i $$0.617620\pi$$
$$524$$ 49.6545 2.16917
$$525$$ 0 0
$$526$$ −30.2072 −1.31710
$$527$$ 2.72279 0.118607
$$528$$ 0 0
$$529$$ 31.2029 1.35665
$$530$$ −51.0531 −2.21761
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10.2055 −0.442049
$$534$$ 0 0
$$535$$ −43.2923 −1.87169
$$536$$ −104.482 −4.51293
$$537$$ 0 0
$$538$$ −29.4897 −1.27139
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −43.1018 −1.85309 −0.926546 0.376181i $$-0.877237\pi$$
−0.926546 + 0.376181i $$0.877237\pi$$
$$542$$ 33.7892 1.45137
$$543$$ 0 0
$$544$$ 33.7892 1.44870
$$545$$ 12.9770 0.555874
$$546$$ 0 0
$$547$$ −13.5057 −0.577462 −0.288731 0.957410i $$-0.593233\pi$$
−0.288731 + 0.957410i $$0.593233\pi$$
$$548$$ −88.6519 −3.78702
$$549$$ 0 0
$$550$$ −14.5544 −0.620603
$$551$$ 0.563387 0.0240011
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −7.96633 −0.338457
$$555$$ 0 0
$$556$$ 24.8273 1.05291
$$557$$ −1.35157 −0.0572677 −0.0286338 0.999590i $$-0.509116\pi$$
−0.0286338 + 0.999590i $$0.509116\pi$$
$$558$$ 0 0
$$559$$ −3.10275 −0.131232
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −9.15412 −0.386143
$$563$$ 31.7626 1.33863 0.669316 0.742978i $$-0.266587\pi$$
0.669316 + 0.742978i $$0.266587\pi$$
$$564$$ 0 0
$$565$$ 34.5988 1.45558
$$566$$ 33.1355 1.39279
$$567$$ 0 0
$$568$$ 54.7599 2.29768
$$569$$ −30.3730 −1.27330 −0.636650 0.771153i $$-0.719680\pi$$
−0.636650 + 0.771153i $$0.719680\pi$$
$$570$$ 0 0
$$571$$ −0.432244 −0.0180888 −0.00904442 0.999959i $$-0.502879\pi$$
−0.00904442 + 0.999959i $$0.502879\pi$$
$$572$$ −3.31088 −0.138435
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −61.5651 −2.56744
$$576$$ 0 0
$$577$$ −18.1382 −0.755101 −0.377551 0.925989i $$-0.623234\pi$$
−0.377551 + 0.925989i $$0.623234\pi$$
$$578$$ −29.8990 −1.24363
$$579$$ 0 0
$$580$$ 3.84324 0.159582
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 3.44731 0.142773
$$584$$ −101.582 −4.20349
$$585$$ 0 0
$$586$$ −35.9390 −1.48463
$$587$$ 19.3065 0.796865 0.398433 0.917198i $$-0.369554\pi$$
0.398433 + 0.917198i $$0.369554\pi$$
$$588$$ 0 0
$$589$$ 3.07608 0.126748
$$590$$ 80.1736 3.30069
$$591$$ 0 0
$$592$$ −85.0008 −3.49351
$$593$$ 8.20113 0.336780 0.168390 0.985720i $$-0.446143\pi$$
0.168390 + 0.985720i $$0.446143\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 52.2409 2.13987
$$597$$ 0 0
$$598$$ −19.5501 −0.799461
$$599$$ −23.4783 −0.959299 −0.479649 0.877460i $$-0.659236\pi$$
−0.479649 + 0.877460i $$0.659236\pi$$
$$600$$ 0 0
$$601$$ 8.96196 0.365566 0.182783 0.983153i $$-0.441489\pi$$
0.182783 + 0.983153i $$0.441489\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 25.6191 1.04243
$$605$$ −38.6395 −1.57092
$$606$$ 0 0
$$607$$ −43.4641 −1.76415 −0.882077 0.471106i $$-0.843855\pi$$
−0.882077 + 0.471106i $$0.843855\pi$$
$$608$$ 38.1736 1.54814
$$609$$ 0 0
$$610$$ 18.4163 0.745653
$$611$$ 4.60407 0.186261
$$612$$ 0 0
$$613$$ −21.4490 −0.866318 −0.433159 0.901317i $$-0.642601\pi$$
−0.433159 + 0.901317i $$0.642601\pi$$
$$614$$ −74.9805 −3.02597
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.1294 0.488312 0.244156 0.969736i $$-0.421489\pi$$
0.244156 + 0.969736i $$0.421489\pi$$
$$618$$ 0 0
$$619$$ −12.7245 −0.511442 −0.255721 0.966751i $$-0.582313\pi$$
−0.255721 + 0.966751i $$0.582313\pi$$
$$620$$ 20.9840 0.842739
$$621$$ 0 0
$$622$$ −65.6005 −2.63034
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 3.11608 0.124643
$$626$$ 55.2656 2.20886
$$627$$ 0 0
$$628$$ 63.6545 2.54009
$$629$$ −17.8432 −0.711456
$$630$$ 0 0
$$631$$ −11.7538 −0.467912 −0.233956 0.972247i $$-0.575167\pi$$
−0.233956 + 0.972247i $$0.575167\pi$$
$$632$$ −12.3082 −0.489596
$$633$$ 0 0
$$634$$ −69.4084 −2.75656
$$635$$ 19.9867 0.793147
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −0.362259 −0.0143420
$$639$$ 0 0
$$640$$ 38.8273 1.53478
$$641$$ −4.68648 −0.185105 −0.0925523 0.995708i $$-0.529503\pi$$
−0.0925523 + 0.995708i $$0.529503\pi$$
$$642$$ 0 0
$$643$$ 0.751182 0.0296237 0.0148119 0.999890i $$-0.495285\pi$$
0.0148119 + 0.999890i $$0.495285\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 17.2215 0.677570
$$647$$ 40.8476 1.60589 0.802943 0.596056i $$-0.203267\pi$$
0.802943 + 0.596056i $$0.203267\pi$$
$$648$$ 0 0
$$649$$ −5.41363 −0.212504
$$650$$ 22.2055 0.870971
$$651$$ 0 0
$$652$$ 16.2055 0.634656
$$653$$ 46.4783 1.81884 0.909419 0.415881i $$-0.136527\pi$$
0.909419 + 0.415881i $$0.136527\pi$$
$$654$$ 0 0
$$655$$ 35.9327 1.40400
$$656$$ −116.482 −4.54785
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −30.3596 −1.18264 −0.591321 0.806436i $$-0.701394\pi$$
−0.591321 + 0.806436i $$0.701394\pi$$
$$660$$ 0 0
$$661$$ 0.107118 0.00416640 0.00208320 0.999998i $$-0.499337\pi$$
0.00208320 + 0.999998i $$0.499337\pi$$
$$662$$ −64.0975 −2.49122
$$663$$ 0 0
$$664$$ 127.685 4.95513
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.53235 −0.0593330
$$668$$ 41.0194 1.58709
$$669$$ 0 0
$$670$$ −125.166 −4.83557
$$671$$ −1.24354 −0.0480063
$$672$$ 0 0
$$673$$ 38.5385 1.48555 0.742774 0.669542i $$-0.233510\pi$$
0.742774 + 0.669542i $$0.233510\pi$$
$$674$$ −81.0699 −3.12270
$$675$$ 0 0
$$676$$ 5.05137 0.194284
$$677$$ −20.7803 −0.798650 −0.399325 0.916809i $$-0.630756\pi$$
−0.399325 + 0.916809i $$0.630756\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 70.9654 2.72140
$$681$$ 0 0
$$682$$ −1.97793 −0.0757388
$$683$$ −27.5837 −1.05546 −0.527731 0.849412i $$-0.676957\pi$$
−0.527731 + 0.849412i $$0.676957\pi$$
$$684$$ 0 0
$$685$$ −64.1532 −2.45117
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −35.4136 −1.35013
$$689$$ −5.25951 −0.200371
$$690$$ 0 0
$$691$$ −43.6775 −1.66157 −0.830785 0.556593i $$-0.812108\pi$$
−0.830785 + 0.556593i $$0.812108\pi$$
$$692$$ −52.1736 −1.98334
$$693$$ 0 0
$$694$$ 66.3791 2.51971
$$695$$ 17.9663 0.681502
$$696$$ 0 0
$$697$$ −24.4517 −0.926173
$$698$$ 4.88301 0.184825
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20.8973 0.789278 0.394639 0.918836i $$-0.370870\pi$$
0.394639 + 0.918836i $$0.370870\pi$$
$$702$$ 0 0
$$703$$ −20.1585 −0.760292
$$704$$ −9.58373 −0.361200
$$705$$ 0 0
$$706$$ −61.7573 −2.32427
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 9.93966 0.373292 0.186646 0.982427i $$-0.440238\pi$$
0.186646 + 0.982427i $$0.440238\pi$$
$$710$$ 65.6005 2.46194
$$711$$ 0 0
$$712$$ −119.998 −4.49712
$$713$$ −8.36663 −0.313333
$$714$$ 0 0
$$715$$ −2.39593 −0.0896028
$$716$$ 12.0133 0.448959
$$717$$ 0 0
$$718$$ 56.9521 2.12543
$$719$$ −11.9797 −0.446766 −0.223383 0.974731i $$-0.571710\pi$$
−0.223383 + 0.974731i $$0.571710\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −30.9974 −1.15360
$$723$$ 0 0
$$724$$ −110.168 −4.09437
$$725$$ 1.74049 0.0646402
$$726$$ 0 0
$$727$$ −24.1736 −0.896547 −0.448274 0.893896i $$-0.647961\pi$$
−0.448274 + 0.893896i $$0.647961\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −121.692 −4.50401
$$731$$ −7.43397 −0.274955
$$732$$ 0 0
$$733$$ 36.4473 1.34621 0.673106 0.739546i $$-0.264960\pi$$
0.673106 + 0.739546i $$0.264960\pi$$
$$734$$ 2.97529 0.109820
$$735$$ 0 0
$$736$$ −103.828 −3.82715
$$737$$ 8.45168 0.311321
$$738$$ 0 0
$$739$$ 43.2772 1.59198 0.795989 0.605311i $$-0.206951\pi$$
0.795989 + 0.605311i $$0.206951\pi$$
$$740$$ −137.515 −5.05514
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −22.6572 −0.831211 −0.415606 0.909545i $$-0.636430\pi$$
−0.415606 + 0.909545i $$0.636430\pi$$
$$744$$ 0 0
$$745$$ 37.8043 1.38504
$$746$$ 41.4403 1.51724
$$747$$ 0 0
$$748$$ −7.93265 −0.290047
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29.8679 1.08990 0.544948 0.838470i $$-0.316549\pi$$
0.544948 + 0.838470i $$0.316549\pi$$
$$752$$ 52.5491 1.91627
$$753$$ 0 0
$$754$$ 0.552694 0.0201279
$$755$$ 18.5394 0.674716
$$756$$ 0 0
$$757$$ 2.55706 0.0929380 0.0464690 0.998920i $$-0.485203\pi$$
0.0464690 + 0.998920i $$0.485203\pi$$
$$758$$ 33.9494 1.23310
$$759$$ 0 0
$$760$$ 80.1736 2.90820
$$761$$ 14.7289 0.533922 0.266961 0.963707i $$-0.413981\pi$$
0.266961 + 0.963707i $$0.413981\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 126.723 4.58467
$$765$$ 0 0
$$766$$ 90.3791 3.26553
$$767$$ 8.25951 0.298234
$$768$$ 0 0
$$769$$ 36.1692 1.30429 0.652147 0.758092i $$-0.273868\pi$$
0.652147 + 0.758092i $$0.273868\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 57.1355 2.05635
$$773$$ 38.9567 1.40117 0.700587 0.713567i $$-0.252921\pi$$
0.700587 + 0.713567i $$0.252921\pi$$
$$774$$ 0 0
$$775$$ 9.50305 0.341360
$$776$$ 81.1709 2.91387
$$777$$ 0 0
$$778$$ 65.5111 2.34869
$$779$$ −27.6244 −0.989747
$$780$$ 0 0
$$781$$ −4.42960 −0.158504
$$782$$ −46.8406 −1.67502
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 46.0638 1.64409
$$786$$ 0 0
$$787$$ −29.1045 −1.03746 −0.518731 0.854937i $$-0.673595\pi$$
−0.518731 + 0.854937i $$0.673595\pi$$
$$788$$ 84.8220 3.02166
$$789$$ 0 0
$$790$$ −14.7449 −0.524599
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 1.89725 0.0673734
$$794$$ 13.2232 0.469274
$$795$$ 0 0
$$796$$ −103.725 −3.67645
$$797$$ −17.8920 −0.633766 −0.316883 0.948465i $$-0.602636\pi$$
−0.316883 + 0.948465i $$0.602636\pi$$
$$798$$ 0 0
$$799$$ 11.0310 0.390250
$$800$$ 117.931 4.16948
$$801$$ 0 0
$$802$$ −1.82991 −0.0646162
$$803$$ 8.21710 0.289975
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3.01770 0.106294
$$807$$ 0 0
$$808$$ −24.4517 −0.860207
$$809$$ −20.4543 −0.719135 −0.359568 0.933119i $$-0.617076\pi$$
−0.359568 + 0.933119i $$0.617076\pi$$
$$810$$ 0 0
$$811$$ 31.6458 1.11123 0.555617 0.831438i $$-0.312482\pi$$
0.555617 + 0.831438i $$0.312482\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 12.9620 0.454316
$$815$$ 11.7272 0.410784
$$816$$ 0 0
$$817$$ −8.39857 −0.293829
$$818$$ −53.0478 −1.85477
$$819$$ 0 0
$$820$$ −188.445 −6.58077
$$821$$ −18.0761 −0.630860 −0.315430 0.948949i $$-0.602149\pi$$
−0.315430 + 0.948949i $$0.602149\pi$$
$$822$$ 0 0
$$823$$ −23.0514 −0.803520 −0.401760 0.915745i $$-0.631601\pi$$
−0.401760 + 0.915745i $$0.631601\pi$$
$$824$$ 40.8220 1.42210
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 26.3756 0.917169 0.458585 0.888651i $$-0.348357\pi$$
0.458585 + 0.888651i $$0.348357\pi$$
$$828$$ 0 0
$$829$$ −45.1152 −1.56691 −0.783457 0.621446i $$-0.786546\pi$$
−0.783457 + 0.621446i $$0.786546\pi$$
$$830$$ 152.962 5.30938
$$831$$ 0 0
$$832$$ 14.6218 0.506919
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 29.6838 1.02725
$$836$$ −8.96196 −0.309956
$$837$$ 0 0
$$838$$ 52.2223 1.80399
$$839$$ −30.4871 −1.05253 −0.526265 0.850320i $$-0.676408\pi$$
−0.526265 + 0.850320i $$0.676408\pi$$
$$840$$ 0 0
$$841$$ −28.9567 −0.998506
$$842$$ 39.6412 1.36613
$$843$$ 0 0
$$844$$ −79.7033 −2.74350
$$845$$ 3.65544 0.125751
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −60.0301 −2.06144
$$849$$ 0 0
$$850$$ 53.2029 1.82484
$$851$$ 54.8290 1.87951
$$852$$ 0 0
$$853$$ 33.2746 1.13930 0.569650 0.821888i $$-0.307079\pi$$
0.569650 + 0.821888i $$0.307079\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −95.9628 −3.27994
$$857$$ 45.6139 1.55814 0.779070 0.626937i $$-0.215692\pi$$
0.779070 + 0.626937i $$0.215692\pi$$
$$858$$ 0 0
$$859$$ −20.9353 −0.714303 −0.357151 0.934046i $$-0.616252\pi$$
−0.357151 + 0.934046i $$0.616252\pi$$
$$860$$ −57.2923 −1.95365
$$861$$ 0 0
$$862$$ 86.7900 2.95608
$$863$$ −2.96196 −0.100826 −0.0504131 0.998728i $$-0.516054\pi$$
−0.0504131 + 0.998728i $$0.516054\pi$$
$$864$$ 0 0
$$865$$ −37.7556 −1.28373
$$866$$ 23.7980 0.808688
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0.995631 0.0337745
$$870$$ 0 0
$$871$$ −12.8946 −0.436917
$$872$$ 28.7652 0.974113
$$873$$ 0 0
$$874$$ −52.9184 −1.78999
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36.9370 −1.24727 −0.623637 0.781714i $$-0.714346\pi$$
−0.623637 + 0.781714i $$0.714346\pi$$
$$878$$ 21.0647 0.710899
$$879$$ 0 0
$$880$$ −27.3463 −0.921843
$$881$$ 30.8946 1.04087 0.520433 0.853903i $$-0.325771\pi$$
0.520433 + 0.853903i $$0.325771\pi$$
$$882$$ 0 0
$$883$$ −9.64648 −0.324630 −0.162315 0.986739i $$-0.551896\pi$$
−0.162315 + 0.986739i $$0.551896\pi$$
$$884$$ 12.1027 0.407059
$$885$$ 0 0
$$886$$ 23.6615 0.794925
$$887$$ −51.3056 −1.72267 −0.861337 0.508034i $$-0.830372\pi$$
−0.861337 + 0.508034i $$0.830372\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −143.754 −4.81864
$$891$$ 0 0
$$892$$ 42.6705 1.42871
$$893$$ 12.4624 0.417037
$$894$$ 0 0
$$895$$ 8.69348 0.290591
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 22.4429 0.748931
$$899$$ 0.236531 0.00788873
$$900$$ 0 0
$$901$$ −12.6014 −0.419814
$$902$$ 17.7626 0.591429
$$903$$ 0 0
$$904$$ 76.6926 2.55076
$$905$$ −79.7236 −2.65010
$$906$$ 0 0
$$907$$ −18.5385 −0.615559 −0.307780 0.951458i $$-0.599586\pi$$
−0.307780 + 0.951458i $$0.599586\pi$$
$$908$$ 33.7892 1.12133
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −20.7272 −0.686721 −0.343361 0.939204i $$-0.611565\pi$$
−0.343361 + 0.939204i $$0.611565\pi$$
$$912$$ 0 0
$$913$$ −10.3286 −0.341826
$$914$$ 61.5244 2.03505
$$915$$ 0 0
$$916$$ −43.6191 −1.44122
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5.13205 −0.169291 −0.0846454 0.996411i $$-0.526976\pi$$
−0.0846454 + 0.996411i $$0.526976\pi$$
$$920$$ −218.064 −7.18935
$$921$$ 0 0
$$922$$ 6.03540 0.198765
$$923$$ 6.75819 0.222449
$$924$$ 0 0
$$925$$ −62.2763 −2.04763
$$926$$ −10.9132 −0.358631
$$927$$ 0 0
$$928$$ 2.93529 0.0963557
$$929$$ 19.9283 0.653826 0.326913 0.945054i $$-0.393992\pi$$
0.326913 + 0.945054i $$0.393992\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 21.0514 0.689561
$$933$$ 0 0
$$934$$ 87.4038 2.85994
$$935$$ −5.74049 −0.187734
$$936$$ 0 0
$$937$$ 1.64475 0.0537316 0.0268658 0.999639i $$-0.491447\pi$$
0.0268658 + 0.999639i $$0.491447\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 85.0142 2.77286
$$941$$ −17.7849 −0.579770 −0.289885 0.957062i $$-0.593617\pi$$
−0.289885 + 0.957062i $$0.593617\pi$$
$$942$$ 0 0
$$943$$ 75.1355 2.44675
$$944$$ 94.2710 3.06826
$$945$$ 0 0
$$946$$ 5.40030 0.175579
$$947$$ −10.6484 −0.346028 −0.173014 0.984919i $$-0.555351\pi$$
−0.173014 + 0.984919i $$0.555351\pi$$
$$948$$ 0 0
$$949$$ −12.5367 −0.406959
$$950$$ 60.1062 1.95010
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 41.5544 1.34608 0.673040 0.739606i $$-0.264988\pi$$
0.673040 + 0.739606i $$0.264988\pi$$
$$954$$ 0 0
$$955$$ 91.7033 2.96745
$$956$$ 9.06471 0.293174
$$957$$ 0 0
$$958$$ −24.7361 −0.799188
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29.7085 −0.958340
$$962$$ −19.7759 −0.637600
$$963$$ 0 0
$$964$$ −70.0168 −2.25509
$$965$$ 41.3463 1.33098
$$966$$ 0 0
$$967$$ −5.51465 −0.177339 −0.0886696 0.996061i $$-0.528262\pi$$
−0.0886696 + 0.996061i $$0.528262\pi$$
$$968$$ −85.6493 −2.75287
$$969$$ 0 0
$$970$$ 97.2400 3.12219
$$971$$ 39.8246 1.27803 0.639017 0.769193i $$-0.279342\pi$$
0.639017 + 0.769193i $$0.279342\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 53.7485 1.72221
$$975$$ 0 0
$$976$$ 21.6545 0.693145
$$977$$ −1.93092 −0.0617757 −0.0308879 0.999523i $$-0.509833\pi$$
−0.0308879 + 0.999523i $$0.509833\pi$$
$$978$$ 0 0
$$979$$ 9.70682 0.310231
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 11.5837 0.369652
$$983$$ −43.0238 −1.37225 −0.686123 0.727485i $$-0.740689\pi$$
−0.686123 + 0.727485i $$0.740689\pi$$
$$984$$ 0 0
$$985$$ 61.3817 1.95578
$$986$$ 1.32422 0.0421717
$$987$$ 0 0
$$988$$ 13.6731 0.435001
$$989$$ 22.8432 0.726373
$$990$$ 0 0
$$991$$ −17.6058 −0.559267 −0.279633 0.960107i $$-0.590213\pi$$
−0.279633 + 0.960107i $$0.590213\pi$$
$$992$$ 16.0267 0.508847
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −75.0612 −2.37960
$$996$$ 0 0
$$997$$ 12.9707 0.410786 0.205393 0.978680i $$-0.434153\pi$$
0.205393 + 0.978680i $$0.434153\pi$$
$$998$$ 25.7405 0.814801
$$999$$ 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 5733.2.a.be.1.3 3
3.2 odd 2 637.2.a.h.1.1 3
7.6 odd 2 5733.2.a.bd.1.3 3
21.2 odd 6 637.2.e.l.508.3 6
21.5 even 6 637.2.e.k.508.3 6
21.11 odd 6 637.2.e.l.79.3 6
21.17 even 6 637.2.e.k.79.3 6
21.20 even 2 637.2.a.i.1.1 yes 3
39.38 odd 2 8281.2.a.bh.1.3 3
273.272 even 2 8281.2.a.bk.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.1 3 3.2 odd 2
637.2.a.i.1.1 yes 3 21.20 even 2
637.2.e.k.79.3 6 21.17 even 6
637.2.e.k.508.3 6 21.5 even 6
637.2.e.l.79.3 6 21.11 odd 6
637.2.e.l.508.3 6 21.2 odd 6
5733.2.a.bd.1.3 3 7.6 odd 2
5733.2.a.be.1.3 3 1.1 even 1 trivial
8281.2.a.bh.1.3 3 39.38 odd 2
8281.2.a.bk.1.3 3 273.272 even 2