# Properties

 Label 3234.2.a.bb.1.1 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$1$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 3234.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} +3.23607 q^{15} +1.00000 q^{16} -0.763932 q^{17} +1.00000 q^{18} +5.70820 q^{19} -3.23607 q^{20} -1.00000 q^{22} +6.47214 q^{23} -1.00000 q^{24} +5.47214 q^{25} -1.00000 q^{27} -4.47214 q^{29} +3.23607 q^{30} -7.23607 q^{31} +1.00000 q^{32} +1.00000 q^{33} -0.763932 q^{34} +1.00000 q^{36} +6.94427 q^{37} +5.70820 q^{38} -3.23607 q^{40} -0.763932 q^{41} -2.47214 q^{43} -1.00000 q^{44} -3.23607 q^{45} +6.47214 q^{46} -9.70820 q^{47} -1.00000 q^{48} +5.47214 q^{50} +0.763932 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.23607 q^{55} -5.70820 q^{57} -4.47214 q^{58} -10.4721 q^{59} +3.23607 q^{60} -7.23607 q^{62} +1.00000 q^{64} +1.00000 q^{66} -11.4164 q^{67} -0.763932 q^{68} -6.47214 q^{69} +6.47214 q^{71} +1.00000 q^{72} -2.29180 q^{73} +6.94427 q^{74} -5.47214 q^{75} +5.70820 q^{76} -15.4164 q^{79} -3.23607 q^{80} +1.00000 q^{81} -0.763932 q^{82} -3.23607 q^{83} +2.47214 q^{85} -2.47214 q^{86} +4.47214 q^{87} -1.00000 q^{88} +2.47214 q^{89} -3.23607 q^{90} +6.47214 q^{92} +7.23607 q^{93} -9.70820 q^{94} -18.4721 q^{95} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{30} - 10 q^{31} + 2 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{46} - 6 q^{47} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{60} - 10 q^{62} + 2 q^{64} + 2 q^{66} + 4 q^{67} - 6 q^{68} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 18 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 4 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{92} + 10 q^{93} - 6 q^{94} - 28 q^{95} - 2 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 + 2 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 2 * q^30 - 10 * q^31 + 2 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^40 - 6 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^45 + 4 * q^46 - 6 * q^47 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^57 - 12 * q^59 + 2 * q^60 - 10 * q^62 + 2 * q^64 + 2 * q^66 + 4 * q^67 - 6 * q^68 - 4 * q^69 + 4 * q^71 + 2 * q^72 - 18 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 - 4 * q^79 - 2 * q^80 + 2 * q^81 - 6 * q^82 - 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 - 4 * q^89 - 2 * q^90 + 4 * q^92 + 10 * q^93 - 6 * q^94 - 28 * q^95 - 2 * q^96 - 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.23607 −1.44721 −0.723607 0.690212i $$-0.757517\pi$$
−0.723607 + 0.690212i $$0.757517\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −3.23607 −1.02333
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 3.23607 0.835549
$$16$$ 1.00000 0.250000
$$17$$ −0.763932 −0.185281 −0.0926404 0.995700i $$-0.529531\pi$$
−0.0926404 + 0.995700i $$0.529531\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 5.70820 1.30955 0.654776 0.755823i $$-0.272763\pi$$
0.654776 + 0.755823i $$0.272763\pi$$
$$20$$ −3.23607 −0.723607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 6.47214 1.34953 0.674767 0.738031i $$-0.264244\pi$$
0.674767 + 0.738031i $$0.264244\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 5.47214 1.09443
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 3.23607 0.590822
$$31$$ −7.23607 −1.29964 −0.649818 0.760090i $$-0.725155\pi$$
−0.649818 + 0.760090i $$0.725155\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ −0.763932 −0.131013
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.94427 1.14163 0.570816 0.821078i $$-0.306627\pi$$
0.570816 + 0.821078i $$0.306627\pi$$
$$38$$ 5.70820 0.925993
$$39$$ 0 0
$$40$$ −3.23607 −0.511667
$$41$$ −0.763932 −0.119306 −0.0596531 0.998219i $$-0.518999\pi$$
−0.0596531 + 0.998219i $$0.518999\pi$$
$$42$$ 0 0
$$43$$ −2.47214 −0.376997 −0.188499 0.982073i $$-0.560362\pi$$
−0.188499 + 0.982073i $$0.560362\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −3.23607 −0.482405
$$46$$ 6.47214 0.954264
$$47$$ −9.70820 −1.41609 −0.708044 0.706169i $$-0.750422\pi$$
−0.708044 + 0.706169i $$0.750422\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 5.47214 0.773877
$$51$$ 0.763932 0.106972
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.23607 0.436351
$$56$$ 0 0
$$57$$ −5.70820 −0.756070
$$58$$ −4.47214 −0.587220
$$59$$ −10.4721 −1.36336 −0.681678 0.731652i $$-0.738749\pi$$
−0.681678 + 0.731652i $$0.738749\pi$$
$$60$$ 3.23607 0.417775
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −7.23607 −0.918982
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ −11.4164 −1.39474 −0.697368 0.716713i $$-0.745646\pi$$
−0.697368 + 0.716713i $$0.745646\pi$$
$$68$$ −0.763932 −0.0926404
$$69$$ −6.47214 −0.779154
$$70$$ 0 0
$$71$$ 6.47214 0.768101 0.384051 0.923312i $$-0.374529\pi$$
0.384051 + 0.923312i $$0.374529\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −2.29180 −0.268234 −0.134117 0.990965i $$-0.542820\pi$$
−0.134117 + 0.990965i $$0.542820\pi$$
$$74$$ 6.94427 0.807255
$$75$$ −5.47214 −0.631868
$$76$$ 5.70820 0.654776
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −15.4164 −1.73448 −0.867241 0.497889i $$-0.834109\pi$$
−0.867241 + 0.497889i $$0.834109\pi$$
$$80$$ −3.23607 −0.361803
$$81$$ 1.00000 0.111111
$$82$$ −0.763932 −0.0843622
$$83$$ −3.23607 −0.355205 −0.177602 0.984102i $$-0.556834\pi$$
−0.177602 + 0.984102i $$0.556834\pi$$
$$84$$ 0 0
$$85$$ 2.47214 0.268141
$$86$$ −2.47214 −0.266577
$$87$$ 4.47214 0.479463
$$88$$ −1.00000 −0.106600
$$89$$ 2.47214 0.262046 0.131023 0.991379i $$-0.458174\pi$$
0.131023 + 0.991379i $$0.458174\pi$$
$$90$$ −3.23607 −0.341112
$$91$$ 0 0
$$92$$ 6.47214 0.674767
$$93$$ 7.23607 0.750345
$$94$$ −9.70820 −1.00132
$$95$$ −18.4721 −1.89520
$$96$$ −1.00000 −0.102062
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 5.47214 0.547214
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0.763932 0.0756405
$$103$$ 15.2361 1.50125 0.750627 0.660726i $$-0.229751\pi$$
0.750627 + 0.660726i $$0.229751\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −16.9443 −1.63806 −0.819032 0.573747i $$-0.805489\pi$$
−0.819032 + 0.573747i $$0.805489\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 3.23607 0.308547
$$111$$ −6.94427 −0.659121
$$112$$ 0 0
$$113$$ −13.4164 −1.26211 −0.631055 0.775738i $$-0.717378\pi$$
−0.631055 + 0.775738i $$0.717378\pi$$
$$114$$ −5.70820 −0.534622
$$115$$ −20.9443 −1.95306
$$116$$ −4.47214 −0.415227
$$117$$ 0 0
$$118$$ −10.4721 −0.964038
$$119$$ 0 0
$$120$$ 3.23607 0.295411
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0.763932 0.0688814
$$124$$ −7.23607 −0.649818
$$125$$ −1.52786 −0.136656
$$126$$ 0 0
$$127$$ 2.47214 0.219367 0.109683 0.993967i $$-0.465016\pi$$
0.109683 + 0.993967i $$0.465016\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 2.47214 0.217659
$$130$$ 0 0
$$131$$ 11.2361 0.981700 0.490850 0.871244i $$-0.336686\pi$$
0.490850 + 0.871244i $$0.336686\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ −11.4164 −0.986227
$$135$$ 3.23607 0.278516
$$136$$ −0.763932 −0.0655066
$$137$$ −7.52786 −0.643149 −0.321574 0.946884i $$-0.604212\pi$$
−0.321574 + 0.946884i $$0.604212\pi$$
$$138$$ −6.47214 −0.550945
$$139$$ 12.1803 1.03312 0.516561 0.856250i $$-0.327212\pi$$
0.516561 + 0.856250i $$0.327212\pi$$
$$140$$ 0 0
$$141$$ 9.70820 0.817578
$$142$$ 6.47214 0.543130
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 14.4721 1.20185
$$146$$ −2.29180 −0.189670
$$147$$ 0 0
$$148$$ 6.94427 0.570816
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −5.47214 −0.446798
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 5.70820 0.462996
$$153$$ −0.763932 −0.0617602
$$154$$ 0 0
$$155$$ 23.4164 1.88085
$$156$$ 0 0
$$157$$ −19.2361 −1.53521 −0.767603 0.640926i $$-0.778551\pi$$
−0.767603 + 0.640926i $$0.778551\pi$$
$$158$$ −15.4164 −1.22646
$$159$$ 6.00000 0.475831
$$160$$ −3.23607 −0.255834
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −0.763932 −0.0596531
$$165$$ −3.23607 −0.251928
$$166$$ −3.23607 −0.251168
$$167$$ 9.52786 0.737288 0.368644 0.929571i $$-0.379822\pi$$
0.368644 + 0.929571i $$0.379822\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 2.47214 0.189604
$$171$$ 5.70820 0.436517
$$172$$ −2.47214 −0.188499
$$173$$ −7.41641 −0.563859 −0.281930 0.959435i $$-0.590974\pi$$
−0.281930 + 0.959435i $$0.590974\pi$$
$$174$$ 4.47214 0.339032
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 10.4721 0.787134
$$178$$ 2.47214 0.185294
$$179$$ −14.4721 −1.08170 −0.540849 0.841120i $$-0.681897\pi$$
−0.540849 + 0.841120i $$0.681897\pi$$
$$180$$ −3.23607 −0.241202
$$181$$ −1.70820 −0.126970 −0.0634849 0.997983i $$-0.520221\pi$$
−0.0634849 + 0.997983i $$0.520221\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.47214 0.477132
$$185$$ −22.4721 −1.65218
$$186$$ 7.23607 0.530574
$$187$$ 0.763932 0.0558642
$$188$$ −9.70820 −0.708044
$$189$$ 0 0
$$190$$ −18.4721 −1.34011
$$191$$ −1.52786 −0.110552 −0.0552762 0.998471i $$-0.517604\pi$$
−0.0552762 + 0.998471i $$0.517604\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 22.9443 1.65156 0.825782 0.563989i $$-0.190734\pi$$
0.825782 + 0.563989i $$0.190734\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.3607 1.59313 0.796566 0.604551i $$-0.206648\pi$$
0.796566 + 0.604551i $$0.206648\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 17.1246 1.21393 0.606966 0.794728i $$-0.292386\pi$$
0.606966 + 0.794728i $$0.292386\pi$$
$$200$$ 5.47214 0.386938
$$201$$ 11.4164 0.805251
$$202$$ −12.0000 −0.844317
$$203$$ 0 0
$$204$$ 0.763932 0.0534859
$$205$$ 2.47214 0.172661
$$206$$ 15.2361 1.06155
$$207$$ 6.47214 0.449845
$$208$$ 0 0
$$209$$ −5.70820 −0.394845
$$210$$ 0 0
$$211$$ −23.4164 −1.61205 −0.806026 0.591880i $$-0.798386\pi$$
−0.806026 + 0.591880i $$0.798386\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −6.47214 −0.443463
$$214$$ −16.9443 −1.15829
$$215$$ 8.00000 0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 6.00000 0.406371
$$219$$ 2.29180 0.154865
$$220$$ 3.23607 0.218176
$$221$$ 0 0
$$222$$ −6.94427 −0.466069
$$223$$ −2.29180 −0.153470 −0.0767350 0.997052i $$-0.524450\pi$$
−0.0767350 + 0.997052i $$0.524450\pi$$
$$224$$ 0 0
$$225$$ 5.47214 0.364809
$$226$$ −13.4164 −0.892446
$$227$$ −22.6525 −1.50350 −0.751749 0.659450i $$-0.770789\pi$$
−0.751749 + 0.659450i $$0.770789\pi$$
$$228$$ −5.70820 −0.378035
$$229$$ 6.29180 0.415774 0.207887 0.978153i $$-0.433341\pi$$
0.207887 + 0.978153i $$0.433341\pi$$
$$230$$ −20.9443 −1.38102
$$231$$ 0 0
$$232$$ −4.47214 −0.293610
$$233$$ −26.9443 −1.76518 −0.882589 0.470145i $$-0.844201\pi$$
−0.882589 + 0.470145i $$0.844201\pi$$
$$234$$ 0 0
$$235$$ 31.4164 2.04938
$$236$$ −10.4721 −0.681678
$$237$$ 15.4164 1.00140
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 3.23607 0.208887
$$241$$ −4.18034 −0.269279 −0.134640 0.990895i $$-0.542988\pi$$
−0.134640 + 0.990895i $$0.542988\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0.763932 0.0487065
$$247$$ 0 0
$$248$$ −7.23607 −0.459491
$$249$$ 3.23607 0.205077
$$250$$ −1.52786 −0.0966306
$$251$$ 2.47214 0.156040 0.0780199 0.996952i $$-0.475140\pi$$
0.0780199 + 0.996952i $$0.475140\pi$$
$$252$$ 0 0
$$253$$ −6.47214 −0.406900
$$254$$ 2.47214 0.155116
$$255$$ −2.47214 −0.154811
$$256$$ 1.00000 0.0625000
$$257$$ 13.5279 0.843845 0.421922 0.906632i $$-0.361355\pi$$
0.421922 + 0.906632i $$0.361355\pi$$
$$258$$ 2.47214 0.153908
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.47214 −0.276818
$$262$$ 11.2361 0.694167
$$263$$ −18.4721 −1.13904 −0.569520 0.821977i $$-0.692871\pi$$
−0.569520 + 0.821977i $$0.692871\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 19.4164 1.19274
$$266$$ 0 0
$$267$$ −2.47214 −0.151292
$$268$$ −11.4164 −0.697368
$$269$$ −14.2918 −0.871386 −0.435693 0.900095i $$-0.643497\pi$$
−0.435693 + 0.900095i $$0.643497\pi$$
$$270$$ 3.23607 0.196941
$$271$$ 6.47214 0.393154 0.196577 0.980488i $$-0.437017\pi$$
0.196577 + 0.980488i $$0.437017\pi$$
$$272$$ −0.763932 −0.0463202
$$273$$ 0 0
$$274$$ −7.52786 −0.454775
$$275$$ −5.47214 −0.329982
$$276$$ −6.47214 −0.389577
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 12.1803 0.730528
$$279$$ −7.23607 −0.433212
$$280$$ 0 0
$$281$$ −15.8885 −0.947831 −0.473916 0.880570i $$-0.657160\pi$$
−0.473916 + 0.880570i $$0.657160\pi$$
$$282$$ 9.70820 0.578115
$$283$$ 10.2918 0.611784 0.305892 0.952066i $$-0.401045\pi$$
0.305892 + 0.952066i $$0.401045\pi$$
$$284$$ 6.47214 0.384051
$$285$$ 18.4721 1.09419
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −16.4164 −0.965671
$$290$$ 14.4721 0.849833
$$291$$ 0 0
$$292$$ −2.29180 −0.134117
$$293$$ −31.4164 −1.83537 −0.917683 0.397313i $$-0.869943\pi$$
−0.917683 + 0.397313i $$0.869943\pi$$
$$294$$ 0 0
$$295$$ 33.8885 1.97307
$$296$$ 6.94427 0.403628
$$297$$ 1.00000 0.0580259
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ −5.47214 −0.315934
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 12.0000 0.689382
$$304$$ 5.70820 0.327388
$$305$$ 0 0
$$306$$ −0.763932 −0.0436711
$$307$$ −23.5967 −1.34674 −0.673369 0.739307i $$-0.735153\pi$$
−0.673369 + 0.739307i $$0.735153\pi$$
$$308$$ 0 0
$$309$$ −15.2361 −0.866750
$$310$$ 23.4164 1.32996
$$311$$ 22.6525 1.28450 0.642252 0.766494i $$-0.278000\pi$$
0.642252 + 0.766494i $$0.278000\pi$$
$$312$$ 0 0
$$313$$ 14.4721 0.818013 0.409007 0.912531i $$-0.365875\pi$$
0.409007 + 0.912531i $$0.365875\pi$$
$$314$$ −19.2361 −1.08555
$$315$$ 0 0
$$316$$ −15.4164 −0.867241
$$317$$ 3.88854 0.218402 0.109201 0.994020i $$-0.465171\pi$$
0.109201 + 0.994020i $$0.465171\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 4.47214 0.250392
$$320$$ −3.23607 −0.180902
$$321$$ 16.9443 0.945737
$$322$$ 0 0
$$323$$ −4.36068 −0.242635
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ −6.00000 −0.331801
$$328$$ −0.763932 −0.0421811
$$329$$ 0 0
$$330$$ −3.23607 −0.178140
$$331$$ 19.4164 1.06722 0.533611 0.845730i $$-0.320835\pi$$
0.533611 + 0.845730i $$0.320835\pi$$
$$332$$ −3.23607 −0.177602
$$333$$ 6.94427 0.380544
$$334$$ 9.52786 0.521342
$$335$$ 36.9443 2.01848
$$336$$ 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
$$339$$ 13.4164 0.728679
$$340$$ 2.47214 0.134070
$$341$$ 7.23607 0.391855
$$342$$ 5.70820 0.308664
$$343$$ 0 0
$$344$$ −2.47214 −0.133289
$$345$$ 20.9443 1.12760
$$346$$ −7.41641 −0.398709
$$347$$ 26.8328 1.44046 0.720231 0.693735i $$-0.244036\pi$$
0.720231 + 0.693735i $$0.244036\pi$$
$$348$$ 4.47214 0.239732
$$349$$ −27.4164 −1.46757 −0.733783 0.679384i $$-0.762247\pi$$
−0.733783 + 0.679384i $$0.762247\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ −20.3607 −1.08369 −0.541845 0.840479i $$-0.682274\pi$$
−0.541845 + 0.840479i $$0.682274\pi$$
$$354$$ 10.4721 0.556588
$$355$$ −20.9443 −1.11161
$$356$$ 2.47214 0.131023
$$357$$ 0 0
$$358$$ −14.4721 −0.764876
$$359$$ 10.4721 0.552698 0.276349 0.961057i $$-0.410875\pi$$
0.276349 + 0.961057i $$0.410875\pi$$
$$360$$ −3.23607 −0.170556
$$361$$ 13.5836 0.714926
$$362$$ −1.70820 −0.0897812
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 7.41641 0.388193
$$366$$ 0 0
$$367$$ 21.7082 1.13316 0.566580 0.824007i $$-0.308266\pi$$
0.566580 + 0.824007i $$0.308266\pi$$
$$368$$ 6.47214 0.337383
$$369$$ −0.763932 −0.0397687
$$370$$ −22.4721 −1.16827
$$371$$ 0 0
$$372$$ 7.23607 0.375173
$$373$$ 25.4164 1.31601 0.658006 0.753013i $$-0.271400\pi$$
0.658006 + 0.753013i $$0.271400\pi$$
$$374$$ 0.763932 0.0395020
$$375$$ 1.52786 0.0788986
$$376$$ −9.70820 −0.500662
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ −18.4721 −0.947601
$$381$$ −2.47214 −0.126651
$$382$$ −1.52786 −0.0781723
$$383$$ 30.6525 1.56627 0.783134 0.621853i $$-0.213620\pi$$
0.783134 + 0.621853i $$0.213620\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 22.9443 1.16783
$$387$$ −2.47214 −0.125666
$$388$$ 0 0
$$389$$ 10.9443 0.554897 0.277448 0.960741i $$-0.410511\pi$$
0.277448 + 0.960741i $$0.410511\pi$$
$$390$$ 0 0
$$391$$ −4.94427 −0.250043
$$392$$ 0 0
$$393$$ −11.2361 −0.566785
$$394$$ 22.3607 1.12651
$$395$$ 49.8885 2.51017
$$396$$ −1.00000 −0.0502519
$$397$$ −24.1803 −1.21358 −0.606788 0.794864i $$-0.707542\pi$$
−0.606788 + 0.794864i $$0.707542\pi$$
$$398$$ 17.1246 0.858379
$$399$$ 0 0
$$400$$ 5.47214 0.273607
$$401$$ 0.472136 0.0235773 0.0117887 0.999931i $$-0.496247\pi$$
0.0117887 + 0.999931i $$0.496247\pi$$
$$402$$ 11.4164 0.569399
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ −3.23607 −0.160802
$$406$$ 0 0
$$407$$ −6.94427 −0.344215
$$408$$ 0.763932 0.0378203
$$409$$ 7.23607 0.357801 0.178900 0.983867i $$-0.442746\pi$$
0.178900 + 0.983867i $$0.442746\pi$$
$$410$$ 2.47214 0.122090
$$411$$ 7.52786 0.371322
$$412$$ 15.2361 0.750627
$$413$$ 0 0
$$414$$ 6.47214 0.318088
$$415$$ 10.4721 0.514057
$$416$$ 0 0
$$417$$ −12.1803 −0.596474
$$418$$ −5.70820 −0.279197
$$419$$ 32.9443 1.60943 0.804717 0.593659i $$-0.202317\pi$$
0.804717 + 0.593659i $$0.202317\pi$$
$$420$$ 0 0
$$421$$ −22.9443 −1.11824 −0.559118 0.829088i $$-0.688860\pi$$
−0.559118 + 0.829088i $$0.688860\pi$$
$$422$$ −23.4164 −1.13989
$$423$$ −9.70820 −0.472029
$$424$$ −6.00000 −0.291386
$$425$$ −4.18034 −0.202776
$$426$$ −6.47214 −0.313576
$$427$$ 0 0
$$428$$ −16.9443 −0.819032
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −13.5279 −0.651614 −0.325807 0.945436i $$-0.605636\pi$$
−0.325807 + 0.945436i $$0.605636\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −25.8885 −1.24412 −0.622062 0.782968i $$-0.713705\pi$$
−0.622062 + 0.782968i $$0.713705\pi$$
$$434$$ 0 0
$$435$$ −14.4721 −0.693886
$$436$$ 6.00000 0.287348
$$437$$ 36.9443 1.76728
$$438$$ 2.29180 0.109506
$$439$$ 4.58359 0.218763 0.109381 0.994000i $$-0.465113\pi$$
0.109381 + 0.994000i $$0.465113\pi$$
$$440$$ 3.23607 0.154273
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −19.4164 −0.922501 −0.461251 0.887270i $$-0.652599\pi$$
−0.461251 + 0.887270i $$0.652599\pi$$
$$444$$ −6.94427 −0.329561
$$445$$ −8.00000 −0.379236
$$446$$ −2.29180 −0.108520
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 28.4721 1.34368 0.671842 0.740695i $$-0.265504\pi$$
0.671842 + 0.740695i $$0.265504\pi$$
$$450$$ 5.47214 0.257959
$$451$$ 0.763932 0.0359722
$$452$$ −13.4164 −0.631055
$$453$$ 0 0
$$454$$ −22.6525 −1.06313
$$455$$ 0 0
$$456$$ −5.70820 −0.267311
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 6.29180 0.293996
$$459$$ 0.763932 0.0356573
$$460$$ −20.9443 −0.976532
$$461$$ 26.8328 1.24973 0.624864 0.780733i $$-0.285154\pi$$
0.624864 + 0.780733i $$0.285154\pi$$
$$462$$ 0 0
$$463$$ 33.8885 1.57493 0.787467 0.616357i $$-0.211392\pi$$
0.787467 + 0.616357i $$0.211392\pi$$
$$464$$ −4.47214 −0.207614
$$465$$ −23.4164 −1.08591
$$466$$ −26.9443 −1.24817
$$467$$ 5.88854 0.272489 0.136245 0.990675i $$-0.456497\pi$$
0.136245 + 0.990675i $$0.456497\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 31.4164 1.44913
$$471$$ 19.2361 0.886351
$$472$$ −10.4721 −0.482019
$$473$$ 2.47214 0.113669
$$474$$ 15.4164 0.708099
$$475$$ 31.2361 1.43321
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −35.7771 −1.63470 −0.817348 0.576144i $$-0.804557\pi$$
−0.817348 + 0.576144i $$0.804557\pi$$
$$480$$ 3.23607 0.147706
$$481$$ 0 0
$$482$$ −4.18034 −0.190409
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 19.0557 0.863497 0.431749 0.901994i $$-0.357897\pi$$
0.431749 + 0.901994i $$0.357897\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 5.88854 0.265746 0.132873 0.991133i $$-0.457580\pi$$
0.132873 + 0.991133i $$0.457580\pi$$
$$492$$ 0.763932 0.0344407
$$493$$ 3.41641 0.153867
$$494$$ 0 0
$$495$$ 3.23607 0.145450
$$496$$ −7.23607 −0.324909
$$497$$ 0 0
$$498$$ 3.23607 0.145012
$$499$$ −19.4164 −0.869198 −0.434599 0.900624i $$-0.643110\pi$$
−0.434599 + 0.900624i $$0.643110\pi$$
$$500$$ −1.52786 −0.0683282
$$501$$ −9.52786 −0.425674
$$502$$ 2.47214 0.110337
$$503$$ 21.3050 0.949941 0.474970 0.880002i $$-0.342459\pi$$
0.474970 + 0.880002i $$0.342459\pi$$
$$504$$ 0 0
$$505$$ 38.8328 1.72804
$$506$$ −6.47214 −0.287722
$$507$$ 13.0000 0.577350
$$508$$ 2.47214 0.109683
$$509$$ −11.2361 −0.498030 −0.249015 0.968500i $$-0.580107\pi$$
−0.249015 + 0.968500i $$0.580107\pi$$
$$510$$ −2.47214 −0.109468
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −5.70820 −0.252023
$$514$$ 13.5279 0.596689
$$515$$ −49.3050 −2.17264
$$516$$ 2.47214 0.108830
$$517$$ 9.70820 0.426966
$$518$$ 0 0
$$519$$ 7.41641 0.325544
$$520$$ 0 0
$$521$$ 41.3050 1.80960 0.904801 0.425834i $$-0.140019\pi$$
0.904801 + 0.425834i $$0.140019\pi$$
$$522$$ −4.47214 −0.195740
$$523$$ −37.7082 −1.64886 −0.824432 0.565961i $$-0.808505\pi$$
−0.824432 + 0.565961i $$0.808505\pi$$
$$524$$ 11.2361 0.490850
$$525$$ 0 0
$$526$$ −18.4721 −0.805423
$$527$$ 5.52786 0.240798
$$528$$ 1.00000 0.0435194
$$529$$ 18.8885 0.821241
$$530$$ 19.4164 0.843395
$$531$$ −10.4721 −0.454452
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −2.47214 −0.106980
$$535$$ 54.8328 2.37063
$$536$$ −11.4164 −0.493114
$$537$$ 14.4721 0.624519
$$538$$ −14.2918 −0.616163
$$539$$ 0 0
$$540$$ 3.23607 0.139258
$$541$$ −5.41641 −0.232870 −0.116435 0.993198i $$-0.537147\pi$$
−0.116435 + 0.993198i $$0.537147\pi$$
$$542$$ 6.47214 0.278002
$$543$$ 1.70820 0.0733060
$$544$$ −0.763932 −0.0327533
$$545$$ −19.4164 −0.831708
$$546$$ 0 0
$$547$$ 29.5279 1.26252 0.631260 0.775571i $$-0.282538\pi$$
0.631260 + 0.775571i $$0.282538\pi$$
$$548$$ −7.52786 −0.321574
$$549$$ 0 0
$$550$$ −5.47214 −0.233333
$$551$$ −25.5279 −1.08752
$$552$$ −6.47214 −0.275472
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 22.4721 0.953889
$$556$$ 12.1803 0.516561
$$557$$ −37.4164 −1.58538 −0.792692 0.609622i $$-0.791321\pi$$
−0.792692 + 0.609622i $$0.791321\pi$$
$$558$$ −7.23607 −0.306327
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −0.763932 −0.0322532
$$562$$ −15.8885 −0.670218
$$563$$ 11.2361 0.473544 0.236772 0.971565i $$-0.423911\pi$$
0.236772 + 0.971565i $$0.423911\pi$$
$$564$$ 9.70820 0.408789
$$565$$ 43.4164 1.82654
$$566$$ 10.2918 0.432596
$$567$$ 0 0
$$568$$ 6.47214 0.271565
$$569$$ −17.0557 −0.715013 −0.357507 0.933911i $$-0.616373\pi$$
−0.357507 + 0.933911i $$0.616373\pi$$
$$570$$ 18.4721 0.773713
$$571$$ −26.4721 −1.10782 −0.553912 0.832575i $$-0.686866\pi$$
−0.553912 + 0.832575i $$0.686866\pi$$
$$572$$ 0 0
$$573$$ 1.52786 0.0638274
$$574$$ 0 0
$$575$$ 35.4164 1.47697
$$576$$ 1.00000 0.0416667
$$577$$ −12.5836 −0.523862 −0.261931 0.965087i $$-0.584359\pi$$
−0.261931 + 0.965087i $$0.584359\pi$$
$$578$$ −16.4164 −0.682833
$$579$$ −22.9443 −0.953531
$$580$$ 14.4721 0.600923
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.00000 0.248495
$$584$$ −2.29180 −0.0948352
$$585$$ 0 0
$$586$$ −31.4164 −1.29780
$$587$$ 26.4721 1.09262 0.546311 0.837582i $$-0.316032\pi$$
0.546311 + 0.837582i $$0.316032\pi$$
$$588$$ 0 0
$$589$$ −41.3050 −1.70194
$$590$$ 33.8885 1.39517
$$591$$ −22.3607 −0.919795
$$592$$ 6.94427 0.285408
$$593$$ 23.2361 0.954191 0.477095 0.878851i $$-0.341690\pi$$
0.477095 + 0.878851i $$0.341690\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −17.1246 −0.700864
$$598$$ 0 0
$$599$$ −37.3050 −1.52424 −0.762120 0.647436i $$-0.775841\pi$$
−0.762120 + 0.647436i $$0.775841\pi$$
$$600$$ −5.47214 −0.223399
$$601$$ −13.7082 −0.559169 −0.279585 0.960121i $$-0.590197\pi$$
−0.279585 + 0.960121i $$0.590197\pi$$
$$602$$ 0 0
$$603$$ −11.4164 −0.464912
$$604$$ 0 0
$$605$$ −3.23607 −0.131565
$$606$$ 12.0000 0.487467
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ 5.70820 0.231498
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −0.763932 −0.0308801
$$613$$ 3.52786 0.142489 0.0712445 0.997459i $$-0.477303\pi$$
0.0712445 + 0.997459i $$0.477303\pi$$
$$614$$ −23.5967 −0.952287
$$615$$ −2.47214 −0.0996861
$$616$$ 0 0
$$617$$ −11.8885 −0.478615 −0.239307 0.970944i $$-0.576920\pi$$
−0.239307 + 0.970944i $$0.576920\pi$$
$$618$$ −15.2361 −0.612885
$$619$$ −34.8328 −1.40005 −0.700025 0.714119i $$-0.746828\pi$$
−0.700025 + 0.714119i $$0.746828\pi$$
$$620$$ 23.4164 0.940426
$$621$$ −6.47214 −0.259718
$$622$$ 22.6525 0.908282
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −22.4164 −0.896656
$$626$$ 14.4721 0.578423
$$627$$ 5.70820 0.227964
$$628$$ −19.2361 −0.767603
$$629$$ −5.30495 −0.211522
$$630$$ 0 0
$$631$$ −25.8885 −1.03061 −0.515303 0.857008i $$-0.672321\pi$$
−0.515303 + 0.857008i $$0.672321\pi$$
$$632$$ −15.4164 −0.613232
$$633$$ 23.4164 0.930719
$$634$$ 3.88854 0.154434
$$635$$ −8.00000 −0.317470
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 4.47214 0.177054
$$639$$ 6.47214 0.256034
$$640$$ −3.23607 −0.127917
$$641$$ −0.111456 −0.00440225 −0.00220113 0.999998i $$-0.500701\pi$$
−0.00220113 + 0.999998i $$0.500701\pi$$
$$642$$ 16.9443 0.668737
$$643$$ 16.5836 0.653993 0.326997 0.945026i $$-0.393963\pi$$
0.326997 + 0.945026i $$0.393963\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ −4.36068 −0.171569
$$647$$ −31.0132 −1.21925 −0.609626 0.792689i $$-0.708681\pi$$
−0.609626 + 0.792689i $$0.708681\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 10.4721 0.411067
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ 2.94427 0.115218 0.0576091 0.998339i $$-0.481652\pi$$
0.0576091 + 0.998339i $$0.481652\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ −36.3607 −1.42073
$$656$$ −0.763932 −0.0298265
$$657$$ −2.29180 −0.0894115
$$658$$ 0 0
$$659$$ −21.8885 −0.852657 −0.426328 0.904569i $$-0.640193\pi$$
−0.426328 + 0.904569i $$0.640193\pi$$
$$660$$ −3.23607 −0.125964
$$661$$ −43.5967 −1.69572 −0.847858 0.530223i $$-0.822108\pi$$
−0.847858 + 0.530223i $$0.822108\pi$$
$$662$$ 19.4164 0.754640
$$663$$ 0 0
$$664$$ −3.23607 −0.125584
$$665$$ 0 0
$$666$$ 6.94427 0.269085
$$667$$ −28.9443 −1.12073
$$668$$ 9.52786 0.368644
$$669$$ 2.29180 0.0886060
$$670$$ 36.9443 1.42728
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 17.0557 0.657450 0.328725 0.944426i $$-0.393381\pi$$
0.328725 + 0.944426i $$0.393381\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ −5.47214 −0.210623
$$676$$ −13.0000 −0.500000
$$677$$ −37.5279 −1.44231 −0.721156 0.692772i $$-0.756389\pi$$
−0.721156 + 0.692772i $$0.756389\pi$$
$$678$$ 13.4164 0.515254
$$679$$ 0 0
$$680$$ 2.47214 0.0948021
$$681$$ 22.6525 0.868045
$$682$$ 7.23607 0.277083
$$683$$ 40.3607 1.54436 0.772179 0.635405i $$-0.219167\pi$$
0.772179 + 0.635405i $$0.219167\pi$$
$$684$$ 5.70820 0.218259
$$685$$ 24.3607 0.930774
$$686$$ 0 0
$$687$$ −6.29180 −0.240047
$$688$$ −2.47214 −0.0942493
$$689$$ 0 0
$$690$$ 20.9443 0.797335
$$691$$ −26.4721 −1.00705 −0.503524 0.863981i $$-0.667963\pi$$
−0.503524 + 0.863981i $$0.667963\pi$$
$$692$$ −7.41641 −0.281930
$$693$$ 0 0
$$694$$ 26.8328 1.01856
$$695$$ −39.4164 −1.49515
$$696$$ 4.47214 0.169516
$$697$$ 0.583592 0.0221051
$$698$$ −27.4164 −1.03773
$$699$$ 26.9443 1.01913
$$700$$ 0 0
$$701$$ −43.8885 −1.65765 −0.828824 0.559510i $$-0.810989\pi$$
−0.828824 + 0.559510i $$0.810989\pi$$
$$702$$ 0 0
$$703$$ 39.6393 1.49503
$$704$$ −1.00000 −0.0376889
$$705$$ −31.4164 −1.18321
$$706$$ −20.3607 −0.766284
$$707$$ 0 0
$$708$$ 10.4721 0.393567
$$709$$ 22.9443 0.861690 0.430845 0.902426i $$-0.358216\pi$$
0.430845 + 0.902426i $$0.358216\pi$$
$$710$$ −20.9443 −0.786025
$$711$$ −15.4164 −0.578160
$$712$$ 2.47214 0.0926472
$$713$$ −46.8328 −1.75390
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −14.4721 −0.540849
$$717$$ 0 0
$$718$$ 10.4721 0.390817
$$719$$ 33.7082 1.25710 0.628552 0.777768i $$-0.283648\pi$$
0.628552 + 0.777768i $$0.283648\pi$$
$$720$$ −3.23607 −0.120601
$$721$$ 0 0
$$722$$ 13.5836 0.505529
$$723$$ 4.18034 0.155469
$$724$$ −1.70820 −0.0634849
$$725$$ −24.4721 −0.908872
$$726$$ −1.00000 −0.0371135
$$727$$ 40.7639 1.51185 0.755925 0.654658i $$-0.227187\pi$$
0.755925 + 0.654658i $$0.227187\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 7.41641 0.274494
$$731$$ 1.88854 0.0698503
$$732$$ 0 0
$$733$$ 30.8328 1.13884 0.569418 0.822048i $$-0.307169\pi$$
0.569418 + 0.822048i $$0.307169\pi$$
$$734$$ 21.7082 0.801264
$$735$$ 0 0
$$736$$ 6.47214 0.238566
$$737$$ 11.4164 0.420529
$$738$$ −0.763932 −0.0281207
$$739$$ 16.5836 0.610037 0.305019 0.952346i $$-0.401337\pi$$
0.305019 + 0.952346i $$0.401337\pi$$
$$740$$ −22.4721 −0.826092
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.88854 0.362775 0.181388 0.983412i $$-0.441941\pi$$
0.181388 + 0.983412i $$0.441941\pi$$
$$744$$ 7.23607 0.265287
$$745$$ 19.4164 0.711362
$$746$$ 25.4164 0.930561
$$747$$ −3.23607 −0.118402
$$748$$ 0.763932 0.0279321
$$749$$ 0 0
$$750$$ 1.52786 0.0557897
$$751$$ 38.8328 1.41703 0.708515 0.705696i $$-0.249366\pi$$
0.708515 + 0.705696i $$0.249366\pi$$
$$752$$ −9.70820 −0.354022
$$753$$ −2.47214 −0.0900896
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 5.05573 0.183754 0.0918768 0.995770i $$-0.470713\pi$$
0.0918768 + 0.995770i $$0.470713\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 6.47214 0.234924
$$760$$ −18.4721 −0.670055
$$761$$ −17.1246 −0.620767 −0.310383 0.950611i $$-0.600457\pi$$
−0.310383 + 0.950611i $$0.600457\pi$$
$$762$$ −2.47214 −0.0895560
$$763$$ 0 0
$$764$$ −1.52786 −0.0552762
$$765$$ 2.47214 0.0893803
$$766$$ 30.6525 1.10752
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 46.0689 1.66129 0.830643 0.556805i $$-0.187973\pi$$
0.830643 + 0.556805i $$0.187973\pi$$
$$770$$ 0 0
$$771$$ −13.5279 −0.487194
$$772$$ 22.9443 0.825782
$$773$$ 43.9574 1.58104 0.790519 0.612437i $$-0.209811\pi$$
0.790519 + 0.612437i $$0.209811\pi$$
$$774$$ −2.47214 −0.0888591
$$775$$ −39.5967 −1.42236
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 10.9443 0.392371
$$779$$ −4.36068 −0.156238
$$780$$ 0 0
$$781$$ −6.47214 −0.231591
$$782$$ −4.94427 −0.176807
$$783$$ 4.47214 0.159821
$$784$$ 0 0
$$785$$ 62.2492 2.22177
$$786$$ −11.2361 −0.400777
$$787$$ 44.5410 1.58772 0.793858 0.608103i $$-0.208069\pi$$
0.793858 + 0.608103i $$0.208069\pi$$
$$788$$ 22.3607 0.796566
$$789$$ 18.4721 0.657625
$$790$$ 49.8885 1.77495
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −24.1803 −0.858128
$$795$$ −19.4164 −0.688629
$$796$$ 17.1246 0.606966
$$797$$ 53.1246 1.88177 0.940885 0.338726i $$-0.109996\pi$$
0.940885 + 0.338726i $$0.109996\pi$$
$$798$$ 0 0
$$799$$ 7.41641 0.262374
$$800$$ 5.47214 0.193469
$$801$$ 2.47214 0.0873486
$$802$$ 0.472136 0.0166717
$$803$$ 2.29180 0.0808757
$$804$$ 11.4164 0.402626
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.2918 0.503095
$$808$$ −12.0000 −0.422159
$$809$$ −8.83282 −0.310545 −0.155273 0.987872i $$-0.549626\pi$$
−0.155273 + 0.987872i $$0.549626\pi$$
$$810$$ −3.23607 −0.113704
$$811$$ −31.5967 −1.10951 −0.554756 0.832013i $$-0.687188\pi$$
−0.554756 + 0.832013i $$0.687188\pi$$
$$812$$ 0 0
$$813$$ −6.47214 −0.226988
$$814$$ −6.94427 −0.243397
$$815$$ 38.8328 1.36025
$$816$$ 0.763932 0.0267430
$$817$$ −14.1115 −0.493697
$$818$$ 7.23607 0.253003
$$819$$ 0 0
$$820$$ 2.47214 0.0863307
$$821$$ −41.7771 −1.45803 −0.729015 0.684498i $$-0.760022\pi$$
−0.729015 + 0.684498i $$0.760022\pi$$
$$822$$ 7.52786 0.262564
$$823$$ 22.8328 0.795902 0.397951 0.917407i $$-0.369721\pi$$
0.397951 + 0.917407i $$0.369721\pi$$
$$824$$ 15.2361 0.530774
$$825$$ 5.47214 0.190515
$$826$$ 0 0
$$827$$ −52.7214 −1.83330 −0.916651 0.399689i $$-0.869118\pi$$
−0.916651 + 0.399689i $$0.869118\pi$$
$$828$$ 6.47214 0.224922
$$829$$ −37.4853 −1.30192 −0.650959 0.759113i $$-0.725633\pi$$
−0.650959 + 0.759113i $$0.725633\pi$$
$$830$$ 10.4721 0.363493
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −12.1803 −0.421771
$$835$$ −30.8328 −1.06701
$$836$$ −5.70820 −0.197422
$$837$$ 7.23607 0.250115
$$838$$ 32.9443 1.13804
$$839$$ 20.7639 0.716851 0.358425 0.933558i $$-0.383314\pi$$
0.358425 + 0.933558i $$0.383314\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ −22.9443 −0.790712
$$843$$ 15.8885 0.547231
$$844$$ −23.4164 −0.806026
$$845$$ 42.0689 1.44721
$$846$$ −9.70820 −0.333775
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −10.2918 −0.353214
$$850$$ −4.18034 −0.143384
$$851$$ 44.9443 1.54067
$$852$$ −6.47214 −0.221732
$$853$$ 1.88854 0.0646625 0.0323313 0.999477i $$-0.489707\pi$$
0.0323313 + 0.999477i $$0.489707\pi$$
$$854$$ 0 0
$$855$$ −18.4721 −0.631734
$$856$$ −16.9443 −0.579143
$$857$$ 6.87539 0.234859 0.117429 0.993081i $$-0.462535\pi$$
0.117429 + 0.993081i $$0.462535\pi$$
$$858$$ 0 0
$$859$$ −16.5836 −0.565825 −0.282912 0.959146i $$-0.591301\pi$$
−0.282912 + 0.959146i $$0.591301\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ −13.5279 −0.460761
$$863$$ 0.360680 0.0122777 0.00613884 0.999981i $$-0.498046\pi$$
0.00613884 + 0.999981i $$0.498046\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 24.0000 0.816024
$$866$$ −25.8885 −0.879729
$$867$$ 16.4164 0.557530
$$868$$ 0 0
$$869$$ 15.4164 0.522966
$$870$$ −14.4721 −0.490651
$$871$$ 0 0
$$872$$ 6.00000 0.203186
$$873$$ 0 0
$$874$$ 36.9443 1.24966
$$875$$ 0 0
$$876$$ 2.29180 0.0774326
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ 4.58359 0.154689
$$879$$ 31.4164 1.05965
$$880$$ 3.23607 0.109088
$$881$$ −26.8328 −0.904021 −0.452010 0.892013i $$-0.649293\pi$$
−0.452010 + 0.892013i $$0.649293\pi$$
$$882$$ 0 0
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ 0 0
$$885$$ −33.8885 −1.13915
$$886$$ −19.4164 −0.652307
$$887$$ −19.0557 −0.639829 −0.319914 0.947446i $$-0.603654\pi$$
−0.319914 + 0.947446i $$0.603654\pi$$
$$888$$ −6.94427 −0.233035
$$889$$ 0 0
$$890$$ −8.00000 −0.268161
$$891$$ −1.00000 −0.0335013
$$892$$ −2.29180 −0.0767350
$$893$$ −55.4164 −1.85444
$$894$$ 6.00000 0.200670
$$895$$ 46.8328 1.56545
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 28.4721 0.950127
$$899$$ 32.3607 1.07929
$$900$$ 5.47214 0.182405
$$901$$ 4.58359 0.152702
$$902$$ 0.763932 0.0254362
$$903$$ 0 0
$$904$$ −13.4164 −0.446223
$$905$$ 5.52786 0.183752
$$906$$ 0 0
$$907$$ 29.3050 0.973055 0.486527 0.873665i $$-0.338263\pi$$
0.486527 + 0.873665i $$0.338263\pi$$
$$908$$ −22.6525 −0.751749
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −4.58359 −0.151861 −0.0759306 0.997113i $$-0.524193\pi$$
−0.0759306 + 0.997113i $$0.524193\pi$$
$$912$$ −5.70820 −0.189018
$$913$$ 3.23607 0.107098
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ 6.29180 0.207887
$$917$$ 0 0
$$918$$ 0.763932 0.0252135
$$919$$ 53.6656 1.77027 0.885133 0.465338i $$-0.154067\pi$$
0.885133 + 0.465338i $$0.154067\pi$$
$$920$$ −20.9443 −0.690512
$$921$$ 23.5967 0.777539
$$922$$ 26.8328 0.883692
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 38.0000 1.24943
$$926$$ 33.8885 1.11365
$$927$$ 15.2361 0.500418
$$928$$ −4.47214 −0.146805
$$929$$ −36.3607 −1.19296 −0.596478 0.802630i $$-0.703434\pi$$
−0.596478 + 0.802630i $$0.703434\pi$$
$$930$$ −23.4164 −0.767854
$$931$$ 0 0
$$932$$ −26.9443 −0.882589
$$933$$ −22.6525 −0.741609
$$934$$ 5.88854 0.192679
$$935$$ −2.47214 −0.0808475
$$936$$ 0 0
$$937$$ 47.5967 1.55492 0.777459 0.628934i $$-0.216508\pi$$
0.777459 + 0.628934i $$0.216508\pi$$
$$938$$ 0 0
$$939$$ −14.4721 −0.472280
$$940$$ 31.4164 1.02469
$$941$$ −52.3607 −1.70691 −0.853455 0.521167i $$-0.825497\pi$$
−0.853455 + 0.521167i $$0.825497\pi$$
$$942$$ 19.2361 0.626745
$$943$$ −4.94427 −0.161008
$$944$$ −10.4721 −0.340839
$$945$$ 0 0
$$946$$ 2.47214 0.0803761
$$947$$ −2.11146 −0.0686131 −0.0343066 0.999411i $$-0.510922\pi$$
−0.0343066 + 0.999411i $$0.510922\pi$$
$$948$$ 15.4164 0.500702
$$949$$ 0 0
$$950$$ 31.2361 1.01343
$$951$$ −3.88854 −0.126095
$$952$$ 0 0
$$953$$ −13.0557 −0.422917 −0.211458 0.977387i $$-0.567821\pi$$
−0.211458 + 0.977387i $$0.567821\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 4.94427 0.159993
$$956$$ 0 0
$$957$$ −4.47214 −0.144564
$$958$$ −35.7771 −1.15591
$$959$$ 0 0
$$960$$ 3.23607 0.104444
$$961$$ 21.3607 0.689054
$$962$$ 0 0
$$963$$ −16.9443 −0.546022
$$964$$ −4.18034 −0.134640
$$965$$ −74.2492 −2.39017
$$966$$ 0 0
$$967$$ 1.16718 0.0375341 0.0187671 0.999824i $$-0.494026\pi$$
0.0187671 + 0.999824i $$0.494026\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 4.36068 0.140085
$$970$$ 0 0
$$971$$ −56.9443 −1.82743 −0.913714 0.406357i $$-0.866799\pi$$
−0.913714 + 0.406357i $$0.866799\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 19.0557 0.610585
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 23.8885 0.764262 0.382131 0.924108i $$-0.375190\pi$$
0.382131 + 0.924108i $$0.375190\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −2.47214 −0.0790098
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ 5.88854 0.187911
$$983$$ 38.2918 1.22132 0.610659 0.791893i $$-0.290904\pi$$
0.610659 + 0.791893i $$0.290904\pi$$
$$984$$ 0.763932 0.0243533
$$985$$ −72.3607 −2.30560
$$986$$ 3.41641 0.108801
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 3.23607 0.102849
$$991$$ 22.8328 0.725308 0.362654 0.931924i $$-0.381871\pi$$
0.362654 + 0.931924i $$0.381871\pi$$
$$992$$ −7.23607 −0.229745
$$993$$ −19.4164 −0.616161
$$994$$ 0 0
$$995$$ −55.4164 −1.75682
$$996$$ 3.23607 0.102539
$$997$$ −26.2492 −0.831321 −0.415661 0.909520i $$-0.636450\pi$$
−0.415661 + 0.909520i $$0.636450\pi$$
$$998$$ −19.4164 −0.614616
$$999$$ −6.94427 −0.219707
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 3234.2.a.bb.1.1 2
3.2 odd 2 9702.2.a.cw.1.2 2
7.6 odd 2 3234.2.a.be.1.2 yes 2
21.20 even 2 9702.2.a.ck.1.1 2

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