Properties

 Label 8400.2.a.dg.1.3 Level $8400$ Weight $2$ Character 8400.1 Self dual yes Analytic conductor $67.074$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8400.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$67.0743376979$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 8400.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +6.42864 q^{13} -4.42864 q^{17} +2.42864 q^{19} +1.00000 q^{21} +1.37778 q^{23} -1.00000 q^{27} +0.755569 q^{29} -5.18421 q^{31} +2.00000 q^{33} -7.61285 q^{37} -6.42864 q^{39} -8.23506 q^{41} +10.1017 q^{43} +2.75557 q^{47} +1.00000 q^{49} +4.42864 q^{51} -9.18421 q^{53} -2.42864 q^{57} -14.1017 q^{59} +6.85728 q^{61} -1.00000 q^{63} +2.75557 q^{67} -1.37778 q^{69} -2.00000 q^{71} +1.57136 q^{73} +2.00000 q^{77} +4.85728 q^{79} +1.00000 q^{81} +11.6128 q^{83} -0.755569 q^{87} +4.62222 q^{89} -6.42864 q^{91} +5.18421 q^{93} +11.9398 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53} + 6 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{63} + 8 q^{67} - 4 q^{69} - 6 q^{71} + 18 q^{73} + 6 q^{77} - 12 q^{79} + 3 q^{81} + 8 q^{83} - 2 q^{87} + 14 q^{89} - 6 q^{91} + 2 q^{93} + 22 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 6 * q^13 - 6 * q^19 + 3 * q^21 + 4 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 + 4 * q^37 - 6 * q^39 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 3 * q^49 - 14 * q^53 + 6 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^63 + 8 * q^67 - 4 * q^69 - 6 * q^71 + 18 * q^73 + 6 * q^77 - 12 * q^79 + 3 * q^81 + 8 * q^83 - 2 * q^87 + 14 * q^89 - 6 * q^91 + 2 * q^93 + 22 * q^97 - 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 6.42864 1.78298 0.891492 0.453037i $$-0.149659\pi$$
0.891492 + 0.453037i $$0.149659\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.42864 −1.07410 −0.537051 0.843550i $$-0.680462\pi$$
−0.537051 + 0.843550i $$0.680462\pi$$
$$18$$ 0 0
$$19$$ 2.42864 0.557168 0.278584 0.960412i $$-0.410135\pi$$
0.278584 + 0.960412i $$0.410135\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 1.37778 0.287288 0.143644 0.989629i $$-0.454118\pi$$
0.143644 + 0.989629i $$0.454118\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 0.755569 0.140306 0.0701528 0.997536i $$-0.477651\pi$$
0.0701528 + 0.997536i $$0.477651\pi$$
$$30$$ 0 0
$$31$$ −5.18421 −0.931111 −0.465556 0.885019i $$-0.654145\pi$$
−0.465556 + 0.885019i $$0.654145\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.61285 −1.25154 −0.625772 0.780006i $$-0.715216\pi$$
−0.625772 + 0.780006i $$0.715216\pi$$
$$38$$ 0 0
$$39$$ −6.42864 −1.02941
$$40$$ 0 0
$$41$$ −8.23506 −1.28610 −0.643050 0.765824i $$-0.722331\pi$$
−0.643050 + 0.765824i $$0.722331\pi$$
$$42$$ 0 0
$$43$$ 10.1017 1.54050 0.770248 0.637744i $$-0.220132\pi$$
0.770248 + 0.637744i $$0.220132\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.75557 0.401941 0.200971 0.979597i $$-0.435590\pi$$
0.200971 + 0.979597i $$0.435590\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 4.42864 0.620134
$$52$$ 0 0
$$53$$ −9.18421 −1.26155 −0.630774 0.775967i $$-0.717263\pi$$
−0.630774 + 0.775967i $$0.717263\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.42864 −0.321681
$$58$$ 0 0
$$59$$ −14.1017 −1.83589 −0.917943 0.396712i $$-0.870151\pi$$
−0.917943 + 0.396712i $$0.870151\pi$$
$$60$$ 0 0
$$61$$ 6.85728 0.877985 0.438992 0.898491i $$-0.355336\pi$$
0.438992 + 0.898491i $$0.355336\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.75557 0.336646 0.168323 0.985732i $$-0.446165\pi$$
0.168323 + 0.985732i $$0.446165\pi$$
$$68$$ 0 0
$$69$$ −1.37778 −0.165866
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 1.57136 0.183914 0.0919569 0.995763i $$-0.470688\pi$$
0.0919569 + 0.995763i $$0.470688\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ 4.85728 0.546487 0.273243 0.961945i $$-0.411904\pi$$
0.273243 + 0.961945i $$0.411904\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 11.6128 1.27468 0.637338 0.770585i $$-0.280036\pi$$
0.637338 + 0.770585i $$0.280036\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.755569 −0.0810055
$$88$$ 0 0
$$89$$ 4.62222 0.489954 0.244977 0.969529i $$-0.421220\pi$$
0.244977 + 0.969529i $$0.421220\pi$$
$$90$$ 0 0
$$91$$ −6.42864 −0.673905
$$92$$ 0 0
$$93$$ 5.18421 0.537577
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 11.9398 1.21230 0.606150 0.795350i $$-0.292713\pi$$
0.606150 + 0.795350i $$0.292713\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 1.47949 0.147215 0.0736076 0.997287i $$-0.476549\pi$$
0.0736076 + 0.997287i $$0.476549\pi$$
$$102$$ 0 0
$$103$$ −8.85728 −0.872734 −0.436367 0.899769i $$-0.643735\pi$$
−0.436367 + 0.899769i $$0.643735\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.76494 0.170623 0.0853114 0.996354i $$-0.472811\pi$$
0.0853114 + 0.996354i $$0.472811\pi$$
$$108$$ 0 0
$$109$$ 5.61285 0.537613 0.268807 0.963194i $$-0.413371\pi$$
0.268807 + 0.963194i $$0.413371\pi$$
$$110$$ 0 0
$$111$$ 7.61285 0.722580
$$112$$ 0 0
$$113$$ 11.2859 1.06169 0.530845 0.847469i $$-0.321875\pi$$
0.530845 + 0.847469i $$0.321875\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.42864 0.594328
$$118$$ 0 0
$$119$$ 4.42864 0.405973
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 8.23506 0.742531
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.8573 −1.14090 −0.570450 0.821333i $$-0.693231\pi$$
−0.570450 + 0.821333i $$0.693231\pi$$
$$128$$ 0 0
$$129$$ −10.1017 −0.889406
$$130$$ 0 0
$$131$$ 2.10171 0.183627 0.0918136 0.995776i $$-0.470734\pi$$
0.0918136 + 0.995776i $$0.470734\pi$$
$$132$$ 0 0
$$133$$ −2.42864 −0.210590
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.9398 −1.36183 −0.680914 0.732364i $$-0.738417\pi$$
−0.680914 + 0.732364i $$0.738417\pi$$
$$138$$ 0 0
$$139$$ −11.6731 −0.990097 −0.495048 0.868865i $$-0.664850\pi$$
−0.495048 + 0.868865i $$0.664850\pi$$
$$140$$ 0 0
$$141$$ −2.75557 −0.232061
$$142$$ 0 0
$$143$$ −12.8573 −1.07518
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ −21.2257 −1.73888 −0.869438 0.494041i $$-0.835519\pi$$
−0.869438 + 0.494041i $$0.835519\pi$$
$$150$$ 0 0
$$151$$ −16.8573 −1.37183 −0.685913 0.727684i $$-0.740597\pi$$
−0.685913 + 0.727684i $$0.740597\pi$$
$$152$$ 0 0
$$153$$ −4.42864 −0.358034
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.4286 0.832296 0.416148 0.909297i $$-0.363380\pi$$
0.416148 + 0.909297i $$0.363380\pi$$
$$158$$ 0 0
$$159$$ 9.18421 0.728355
$$160$$ 0 0
$$161$$ −1.37778 −0.108585
$$162$$ 0 0
$$163$$ 20.8573 1.63367 0.816834 0.576873i $$-0.195727\pi$$
0.816834 + 0.576873i $$0.195727\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 15.3461 1.18752 0.593760 0.804642i $$-0.297643\pi$$
0.593760 + 0.804642i $$0.297643\pi$$
$$168$$ 0 0
$$169$$ 28.3274 2.17903
$$170$$ 0 0
$$171$$ 2.42864 0.185723
$$172$$ 0 0
$$173$$ −2.06022 −0.156636 −0.0783179 0.996928i $$-0.524955\pi$$
−0.0783179 + 0.996928i $$0.524955\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.1017 1.05995
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −12.1017 −0.899513 −0.449757 0.893151i $$-0.648489\pi$$
−0.449757 + 0.893151i $$0.648489\pi$$
$$182$$ 0 0
$$183$$ −6.85728 −0.506905
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.85728 0.647708
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −0.488863 −0.0353729 −0.0176864 0.999844i $$-0.505630\pi$$
−0.0176864 + 0.999844i $$0.505630\pi$$
$$192$$ 0 0
$$193$$ 22.9590 1.65262 0.826312 0.563212i $$-0.190435\pi$$
0.826312 + 0.563212i $$0.190435\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.18421 0.0843713 0.0421857 0.999110i $$-0.486568\pi$$
0.0421857 + 0.999110i $$0.486568\pi$$
$$198$$ 0 0
$$199$$ −8.79706 −0.623607 −0.311803 0.950147i $$-0.600933\pi$$
−0.311803 + 0.950147i $$0.600933\pi$$
$$200$$ 0 0
$$201$$ −2.75557 −0.194363
$$202$$ 0 0
$$203$$ −0.755569 −0.0530305
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.37778 0.0957626
$$208$$ 0 0
$$209$$ −4.85728 −0.335985
$$210$$ 0 0
$$211$$ −23.2257 −1.59892 −0.799461 0.600717i $$-0.794882\pi$$
−0.799461 + 0.600717i $$0.794882\pi$$
$$212$$ 0 0
$$213$$ 2.00000 0.137038
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.18421 0.351927
$$218$$ 0 0
$$219$$ −1.57136 −0.106183
$$220$$ 0 0
$$221$$ −28.4701 −1.91511
$$222$$ 0 0
$$223$$ −15.2257 −1.01959 −0.509794 0.860297i $$-0.670278\pi$$
−0.509794 + 0.860297i $$0.670278\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −14.3684 −0.953665 −0.476833 0.878994i $$-0.658215\pi$$
−0.476833 + 0.878994i $$0.658215\pi$$
$$228$$ 0 0
$$229$$ −5.61285 −0.370907 −0.185454 0.982653i $$-0.559375\pi$$
−0.185454 + 0.982653i $$0.559375\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 0 0
$$233$$ −23.2859 −1.52551 −0.762756 0.646687i $$-0.776154\pi$$
−0.762756 + 0.646687i $$0.776154\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.85728 −0.315514
$$238$$ 0 0
$$239$$ 8.48886 0.549099 0.274549 0.961573i $$-0.411471\pi$$
0.274549 + 0.961573i $$0.411471\pi$$
$$240$$ 0 0
$$241$$ −7.24443 −0.466655 −0.233327 0.972398i $$-0.574961\pi$$
−0.233327 + 0.972398i $$0.574961\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 15.6128 0.993422
$$248$$ 0 0
$$249$$ −11.6128 −0.735934
$$250$$ 0 0
$$251$$ −27.6128 −1.74291 −0.871454 0.490478i $$-0.836822\pi$$
−0.871454 + 0.490478i $$0.836822\pi$$
$$252$$ 0 0
$$253$$ −2.75557 −0.173241
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0.428639 0.0267378 0.0133689 0.999911i $$-0.495744\pi$$
0.0133689 + 0.999911i $$0.495744\pi$$
$$258$$ 0 0
$$259$$ 7.61285 0.473039
$$260$$ 0 0
$$261$$ 0.755569 0.0467685
$$262$$ 0 0
$$263$$ −9.37778 −0.578259 −0.289129 0.957290i $$-0.593366\pi$$
−0.289129 + 0.957290i $$0.593366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −4.62222 −0.282875
$$268$$ 0 0
$$269$$ 1.74620 0.106468 0.0532339 0.998582i $$-0.483047\pi$$
0.0532339 + 0.998582i $$0.483047\pi$$
$$270$$ 0 0
$$271$$ 2.69535 0.163731 0.0818653 0.996643i $$-0.473912\pi$$
0.0818653 + 0.996643i $$0.473912\pi$$
$$272$$ 0 0
$$273$$ 6.42864 0.389079
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5.12399 0.307870 0.153935 0.988081i $$-0.450805\pi$$
0.153935 + 0.988081i $$0.450805\pi$$
$$278$$ 0 0
$$279$$ −5.18421 −0.310370
$$280$$ 0 0
$$281$$ 23.9813 1.43060 0.715301 0.698816i $$-0.246290\pi$$
0.715301 + 0.698816i $$0.246290\pi$$
$$282$$ 0 0
$$283$$ 2.36842 0.140788 0.0703939 0.997519i $$-0.477574\pi$$
0.0703939 + 0.997519i $$0.477574\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.23506 0.486100
$$288$$ 0 0
$$289$$ 2.61285 0.153697
$$290$$ 0 0
$$291$$ −11.9398 −0.699922
$$292$$ 0 0
$$293$$ −8.42864 −0.492406 −0.246203 0.969218i $$-0.579183\pi$$
−0.246203 + 0.969218i $$0.579183\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ 8.85728 0.512230
$$300$$ 0 0
$$301$$ −10.1017 −0.582253
$$302$$ 0 0
$$303$$ −1.47949 −0.0849947
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −22.5718 −1.28824 −0.644121 0.764923i $$-0.722777\pi$$
−0.644121 + 0.764923i $$0.722777\pi$$
$$308$$ 0 0
$$309$$ 8.85728 0.503873
$$310$$ 0 0
$$311$$ 24.0830 1.36562 0.682810 0.730596i $$-0.260758\pi$$
0.682810 + 0.730596i $$0.260758\pi$$
$$312$$ 0 0
$$313$$ −9.65433 −0.545695 −0.272848 0.962057i $$-0.587965\pi$$
−0.272848 + 0.962057i $$0.587965\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.04149 0.339324 0.169662 0.985502i $$-0.445732\pi$$
0.169662 + 0.985502i $$0.445732\pi$$
$$318$$ 0 0
$$319$$ −1.51114 −0.0846075
$$320$$ 0 0
$$321$$ −1.76494 −0.0985092
$$322$$ 0 0
$$323$$ −10.7556 −0.598456
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −5.61285 −0.310391
$$328$$ 0 0
$$329$$ −2.75557 −0.151919
$$330$$ 0 0
$$331$$ −13.5111 −0.742639 −0.371320 0.928505i $$-0.621095\pi$$
−0.371320 + 0.928505i $$0.621095\pi$$
$$332$$ 0 0
$$333$$ −7.61285 −0.417181
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −10.4889 −0.571365 −0.285682 0.958324i $$-0.592220\pi$$
−0.285682 + 0.958324i $$0.592220\pi$$
$$338$$ 0 0
$$339$$ −11.2859 −0.612967
$$340$$ 0 0
$$341$$ 10.3684 0.561481
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.7239 −0.897787 −0.448894 0.893585i $$-0.648182\pi$$
−0.448894 + 0.893585i $$0.648182\pi$$
$$348$$ 0 0
$$349$$ −16.3684 −0.876181 −0.438091 0.898931i $$-0.644345\pi$$
−0.438091 + 0.898931i $$0.644345\pi$$
$$350$$ 0 0
$$351$$ −6.42864 −0.343135
$$352$$ 0 0
$$353$$ −0.549086 −0.0292249 −0.0146124 0.999893i $$-0.504651\pi$$
−0.0146124 + 0.999893i $$0.504651\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.42864 −0.234388
$$358$$ 0 0
$$359$$ 0.285442 0.0150651 0.00753253 0.999972i $$-0.497602\pi$$
0.00753253 + 0.999972i $$0.497602\pi$$
$$360$$ 0 0
$$361$$ −13.1017 −0.689564
$$362$$ 0 0
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 1.71456 0.0894992 0.0447496 0.998998i $$-0.485751\pi$$
0.0447496 + 0.998998i $$0.485751\pi$$
$$368$$ 0 0
$$369$$ −8.23506 −0.428700
$$370$$ 0 0
$$371$$ 9.18421 0.476820
$$372$$ 0 0
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.85728 0.250163
$$378$$ 0 0
$$379$$ 4.85728 0.249502 0.124751 0.992188i $$-0.460187\pi$$
0.124751 + 0.992188i $$0.460187\pi$$
$$380$$ 0 0
$$381$$ 12.8573 0.658698
$$382$$ 0 0
$$383$$ −8.38715 −0.428563 −0.214282 0.976772i $$-0.568741\pi$$
−0.214282 + 0.976772i $$0.568741\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.1017 0.513499
$$388$$ 0 0
$$389$$ 8.95899 0.454239 0.227119 0.973867i $$-0.427069\pi$$
0.227119 + 0.973867i $$0.427069\pi$$
$$390$$ 0 0
$$391$$ −6.10171 −0.308577
$$392$$ 0 0
$$393$$ −2.10171 −0.106017
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.54909 0.127935 0.0639675 0.997952i $$-0.479625\pi$$
0.0639675 + 0.997952i $$0.479625\pi$$
$$398$$ 0 0
$$399$$ 2.42864 0.121584
$$400$$ 0 0
$$401$$ 0.958989 0.0478896 0.0239448 0.999713i $$-0.492377\pi$$
0.0239448 + 0.999713i $$0.492377\pi$$
$$402$$ 0 0
$$403$$ −33.3274 −1.66016
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 15.2257 0.754710
$$408$$ 0 0
$$409$$ −31.9813 −1.58137 −0.790686 0.612222i $$-0.790276\pi$$
−0.790686 + 0.612222i $$0.790276\pi$$
$$410$$ 0 0
$$411$$ 15.9398 0.786251
$$412$$ 0 0
$$413$$ 14.1017 0.693900
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 11.6731 0.571633
$$418$$ 0 0
$$419$$ 0.470127 0.0229672 0.0114836 0.999934i $$-0.496345\pi$$
0.0114836 + 0.999934i $$0.496345\pi$$
$$420$$ 0 0
$$421$$ −33.6128 −1.63819 −0.819095 0.573658i $$-0.805524\pi$$
−0.819095 + 0.573658i $$0.805524\pi$$
$$422$$ 0 0
$$423$$ 2.75557 0.133980
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.85728 −0.331847
$$428$$ 0 0
$$429$$ 12.8573 0.620755
$$430$$ 0 0
$$431$$ −11.7146 −0.564270 −0.282135 0.959375i $$-0.591043\pi$$
−0.282135 + 0.959375i $$0.591043\pi$$
$$432$$ 0 0
$$433$$ 0.0602231 0.00289414 0.00144707 0.999999i $$-0.499539\pi$$
0.00144707 + 0.999999i $$0.499539\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.34614 0.160068
$$438$$ 0 0
$$439$$ 22.4286 1.07046 0.535230 0.844706i $$-0.320225\pi$$
0.535230 + 0.844706i $$0.320225\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −23.9496 −1.13788 −0.568940 0.822379i $$-0.692647\pi$$
−0.568940 + 0.822379i $$0.692647\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 21.2257 1.00394
$$448$$ 0 0
$$449$$ 29.4291 1.38885 0.694423 0.719567i $$-0.255660\pi$$
0.694423 + 0.719567i $$0.255660\pi$$
$$450$$ 0 0
$$451$$ 16.4701 0.775548
$$452$$ 0 0
$$453$$ 16.8573 0.792024
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.14272 0.147010 0.0735051 0.997295i $$-0.476581\pi$$
0.0735051 + 0.997295i $$0.476581\pi$$
$$458$$ 0 0
$$459$$ 4.42864 0.206711
$$460$$ 0 0
$$461$$ −3.37778 −0.157319 −0.0786596 0.996902i $$-0.525064\pi$$
−0.0786596 + 0.996902i $$0.525064\pi$$
$$462$$ 0 0
$$463$$ 20.8573 0.969320 0.484660 0.874703i $$-0.338943\pi$$
0.484660 + 0.874703i $$0.338943\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −14.3684 −0.664891 −0.332446 0.943122i $$-0.607874\pi$$
−0.332446 + 0.943122i $$0.607874\pi$$
$$468$$ 0 0
$$469$$ −2.75557 −0.127240
$$470$$ 0 0
$$471$$ −10.4286 −0.480526
$$472$$ 0 0
$$473$$ −20.2034 −0.928954
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.18421 −0.420516
$$478$$ 0 0
$$479$$ −6.36842 −0.290980 −0.145490 0.989360i $$-0.546476\pi$$
−0.145490 + 0.989360i $$0.546476\pi$$
$$480$$ 0 0
$$481$$ −48.9403 −2.23148
$$482$$ 0 0
$$483$$ 1.37778 0.0626914
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17.3274 0.785180 0.392590 0.919714i $$-0.371579\pi$$
0.392590 + 0.919714i $$0.371579\pi$$
$$488$$ 0 0
$$489$$ −20.8573 −0.943199
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ −3.34614 −0.150703
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.00000 0.0897123
$$498$$ 0 0
$$499$$ −23.3461 −1.04512 −0.522558 0.852603i $$-0.675022\pi$$
−0.522558 + 0.852603i $$0.675022\pi$$
$$500$$ 0 0
$$501$$ −15.3461 −0.685615
$$502$$ 0 0
$$503$$ 0.387152 0.0172623 0.00863113 0.999963i $$-0.497253\pi$$
0.00863113 + 0.999963i $$0.497253\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −28.3274 −1.25806
$$508$$ 0 0
$$509$$ −29.9496 −1.32749 −0.663747 0.747957i $$-0.731035\pi$$
−0.663747 + 0.747957i $$0.731035\pi$$
$$510$$ 0 0
$$511$$ −1.57136 −0.0695129
$$512$$ 0 0
$$513$$ −2.42864 −0.107227
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −5.51114 −0.242380
$$518$$ 0 0
$$519$$ 2.06022 0.0904338
$$520$$ 0 0
$$521$$ −18.5205 −0.811398 −0.405699 0.914007i $$-0.632972\pi$$
−0.405699 + 0.914007i $$0.632972\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 22.9590 1.00011
$$528$$ 0 0
$$529$$ −21.1017 −0.917466
$$530$$ 0 0
$$531$$ −14.1017 −0.611962
$$532$$ 0 0
$$533$$ −52.9403 −2.29310
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10.0000 0.431532
$$538$$ 0 0
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ 14.5906 0.627298 0.313649 0.949539i $$-0.398449\pi$$
0.313649 + 0.949539i $$0.398449\pi$$
$$542$$ 0 0
$$543$$ 12.1017 0.519334
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −18.7556 −0.801930 −0.400965 0.916093i $$-0.631325\pi$$
−0.400965 + 0.916093i $$0.631325\pi$$
$$548$$ 0 0
$$549$$ 6.85728 0.292662
$$550$$ 0 0
$$551$$ 1.83500 0.0781738
$$552$$ 0 0
$$553$$ −4.85728 −0.206553
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −31.8765 −1.35065 −0.675325 0.737520i $$-0.735997\pi$$
−0.675325 + 0.737520i $$0.735997\pi$$
$$558$$ 0 0
$$559$$ 64.9403 2.74668
$$560$$ 0 0
$$561$$ −8.85728 −0.373955
$$562$$ 0 0
$$563$$ −2.01874 −0.0850796 −0.0425398 0.999095i $$-0.513545\pi$$
−0.0425398 + 0.999095i $$0.513545\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −28.9590 −1.21402 −0.607012 0.794693i $$-0.707632\pi$$
−0.607012 + 0.794693i $$0.707632\pi$$
$$570$$ 0 0
$$571$$ −8.97773 −0.375706 −0.187853 0.982197i $$-0.560153\pi$$
−0.187853 + 0.982197i $$0.560153\pi$$
$$572$$ 0 0
$$573$$ 0.488863 0.0204225
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −28.6766 −1.19382 −0.596911 0.802307i $$-0.703606\pi$$
−0.596911 + 0.802307i $$0.703606\pi$$
$$578$$ 0 0
$$579$$ −22.9590 −0.954143
$$580$$ 0 0
$$581$$ −11.6128 −0.481782
$$582$$ 0 0
$$583$$ 18.3684 0.760742
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 45.2070 1.86589 0.932945 0.360018i $$-0.117229\pi$$
0.932945 + 0.360018i $$0.117229\pi$$
$$588$$ 0 0
$$589$$ −12.5906 −0.518786
$$590$$ 0 0
$$591$$ −1.18421 −0.0487118
$$592$$ 0 0
$$593$$ 18.2636 0.749998 0.374999 0.927025i $$-0.377643\pi$$
0.374999 + 0.927025i $$0.377643\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.79706 0.360040
$$598$$ 0 0
$$599$$ 22.7368 0.929002 0.464501 0.885573i $$-0.346234\pi$$
0.464501 + 0.885573i $$0.346234\pi$$
$$600$$ 0 0
$$601$$ 0.488863 0.0199411 0.00997056 0.999950i $$-0.496826\pi$$
0.00997056 + 0.999950i $$0.496826\pi$$
$$602$$ 0 0
$$603$$ 2.75557 0.112215
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.2034 0.820032 0.410016 0.912078i $$-0.365523\pi$$
0.410016 + 0.912078i $$0.365523\pi$$
$$608$$ 0 0
$$609$$ 0.755569 0.0306172
$$610$$ 0 0
$$611$$ 17.7146 0.716654
$$612$$ 0 0
$$613$$ 10.3684 0.418776 0.209388 0.977833i $$-0.432853\pi$$
0.209388 + 0.977833i $$0.432853\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −39.2859 −1.58159 −0.790796 0.612080i $$-0.790333\pi$$
−0.790796 + 0.612080i $$0.790333\pi$$
$$618$$ 0 0
$$619$$ −42.8988 −1.72425 −0.862123 0.506698i $$-0.830866\pi$$
−0.862123 + 0.506698i $$0.830866\pi$$
$$620$$ 0 0
$$621$$ −1.37778 −0.0552886
$$622$$ 0 0
$$623$$ −4.62222 −0.185185
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 4.85728 0.193981
$$628$$ 0 0
$$629$$ 33.7146 1.34429
$$630$$ 0 0
$$631$$ −15.3461 −0.610920 −0.305460 0.952205i $$-0.598810\pi$$
−0.305460 + 0.952205i $$0.598810\pi$$
$$632$$ 0 0
$$633$$ 23.2257 0.923139
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.42864 0.254712
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ 30.6735 1.21153 0.605766 0.795643i $$-0.292867\pi$$
0.605766 + 0.795643i $$0.292867\pi$$
$$642$$ 0 0
$$643$$ 49.0607 1.93477 0.967383 0.253320i $$-0.0815225\pi$$
0.967383 + 0.253320i $$0.0815225\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 15.3461 0.603319 0.301660 0.953416i $$-0.402459\pi$$
0.301660 + 0.953416i $$0.402459\pi$$
$$648$$ 0 0
$$649$$ 28.2034 1.10708
$$650$$ 0 0
$$651$$ −5.18421 −0.203185
$$652$$ 0 0
$$653$$ −19.4697 −0.761906 −0.380953 0.924594i $$-0.624404\pi$$
−0.380953 + 0.924594i $$0.624404\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.57136 0.0613046
$$658$$ 0 0
$$659$$ −30.9403 −1.20526 −0.602631 0.798020i $$-0.705881\pi$$
−0.602631 + 0.798020i $$0.705881\pi$$
$$660$$ 0 0
$$661$$ 47.7975 1.85911 0.929554 0.368685i $$-0.120192\pi$$
0.929554 + 0.368685i $$0.120192\pi$$
$$662$$ 0 0
$$663$$ 28.4701 1.10569
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.04101 0.0403081
$$668$$ 0 0
$$669$$ 15.2257 0.588659
$$670$$ 0 0
$$671$$ −13.7146 −0.529445
$$672$$ 0 0
$$673$$ 27.8163 1.07224 0.536119 0.844142i $$-0.319890\pi$$
0.536119 + 0.844142i $$0.319890\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19.0005 −0.730248 −0.365124 0.930959i $$-0.618973\pi$$
−0.365124 + 0.930959i $$0.618973\pi$$
$$678$$ 0 0
$$679$$ −11.9398 −0.458207
$$680$$ 0 0
$$681$$ 14.3684 0.550599
$$682$$ 0 0
$$683$$ −4.52051 −0.172972 −0.0864862 0.996253i $$-0.527564\pi$$
−0.0864862 + 0.996253i $$0.527564\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 5.61285 0.214143
$$688$$ 0 0
$$689$$ −59.0420 −2.24932
$$690$$ 0 0
$$691$$ 1.18421 0.0450494 0.0225247 0.999746i $$-0.492830\pi$$
0.0225247 + 0.999746i $$0.492830\pi$$
$$692$$ 0 0
$$693$$ 2.00000 0.0759737
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.4701 1.38140
$$698$$ 0 0
$$699$$ 23.2859 0.880754
$$700$$ 0 0
$$701$$ −26.6735 −1.00745 −0.503723 0.863865i $$-0.668037\pi$$
−0.503723 + 0.863865i $$0.668037\pi$$
$$702$$ 0 0
$$703$$ −18.4889 −0.697321
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.47949 −0.0556421
$$708$$ 0 0
$$709$$ −18.2034 −0.683644 −0.341822 0.939765i $$-0.611044\pi$$
−0.341822 + 0.939765i $$0.611044\pi$$
$$710$$ 0 0
$$711$$ 4.85728 0.182162
$$712$$ 0 0
$$713$$ −7.14272 −0.267497
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −8.48886 −0.317022
$$718$$ 0 0
$$719$$ −4.85728 −0.181146 −0.0905730 0.995890i $$-0.528870\pi$$
−0.0905730 + 0.995890i $$0.528870\pi$$
$$720$$ 0 0
$$721$$ 8.85728 0.329862
$$722$$ 0 0
$$723$$ 7.24443 0.269423
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21.0607 −0.781098 −0.390549 0.920582i $$-0.627715\pi$$
−0.390549 + 0.920582i $$0.627715\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −44.7368 −1.65465
$$732$$ 0 0
$$733$$ 9.45091 0.349077 0.174539 0.984650i $$-0.444157\pi$$
0.174539 + 0.984650i $$0.444157\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −5.51114 −0.203005
$$738$$ 0 0
$$739$$ 8.20342 0.301768 0.150884 0.988551i $$-0.451788\pi$$
0.150884 + 0.988551i $$0.451788\pi$$
$$740$$ 0 0
$$741$$ −15.6128 −0.573552
$$742$$ 0 0
$$743$$ −8.33677 −0.305847 −0.152923 0.988238i $$-0.548869\pi$$
−0.152923 + 0.988238i $$0.548869\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 11.6128 0.424892
$$748$$ 0 0
$$749$$ −1.76494 −0.0644894
$$750$$ 0 0
$$751$$ 25.9180 0.945760 0.472880 0.881127i $$-0.343214\pi$$
0.472880 + 0.881127i $$0.343214\pi$$
$$752$$ 0 0
$$753$$ 27.6128 1.00627
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.94025 0.324939 0.162470 0.986714i $$-0.448054\pi$$
0.162470 + 0.986714i $$0.448054\pi$$
$$758$$ 0 0
$$759$$ 2.75557 0.100021
$$760$$ 0 0
$$761$$ −0.825636 −0.0299293 −0.0149646 0.999888i $$-0.504764\pi$$
−0.0149646 + 0.999888i $$0.504764\pi$$
$$762$$ 0 0
$$763$$ −5.61285 −0.203199
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −90.6548 −3.27336
$$768$$ 0 0
$$769$$ 21.2257 0.765418 0.382709 0.923869i $$-0.374991\pi$$
0.382709 + 0.923869i $$0.374991\pi$$
$$770$$ 0 0
$$771$$ −0.428639 −0.0154371
$$772$$ 0 0
$$773$$ 29.4893 1.06066 0.530329 0.847792i $$-0.322068\pi$$
0.530329 + 0.847792i $$0.322068\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −7.61285 −0.273109
$$778$$ 0 0
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 0 0
$$783$$ −0.755569 −0.0270018
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 34.4514 1.22806 0.614030 0.789283i $$-0.289547\pi$$
0.614030 + 0.789283i $$0.289547\pi$$
$$788$$ 0 0
$$789$$ 9.37778 0.333858
$$790$$ 0 0
$$791$$ −11.2859 −0.401281
$$792$$ 0 0
$$793$$ 44.0830 1.56543
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.9175 0.670092 0.335046 0.942202i $$-0.391248\pi$$
0.335046 + 0.942202i $$0.391248\pi$$
$$798$$ 0 0
$$799$$ −12.2034 −0.431726
$$800$$ 0 0
$$801$$ 4.62222 0.163318
$$802$$ 0 0
$$803$$ −3.14272 −0.110904
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.74620 −0.0614692
$$808$$ 0 0
$$809$$ 21.2257 0.746256 0.373128 0.927780i $$-0.378285\pi$$
0.373128 + 0.927780i $$0.378285\pi$$
$$810$$ 0 0
$$811$$ 21.5081 0.755251 0.377625 0.925958i $$-0.376741\pi$$
0.377625 + 0.925958i $$0.376741\pi$$
$$812$$ 0 0
$$813$$ −2.69535 −0.0945299
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.5334 0.858315
$$818$$ 0 0
$$819$$ −6.42864 −0.224635
$$820$$ 0 0
$$821$$ −46.2034 −1.61251 −0.806255 0.591568i $$-0.798509\pi$$
−0.806255 + 0.591568i $$0.798509\pi$$
$$822$$ 0 0
$$823$$ 17.8350 0.621689 0.310845 0.950461i $$-0.399388\pi$$
0.310845 + 0.950461i $$0.399388\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −35.2128 −1.22447 −0.612234 0.790676i $$-0.709729\pi$$
−0.612234 + 0.790676i $$0.709729\pi$$
$$828$$ 0 0
$$829$$ 14.3872 0.499686 0.249843 0.968286i $$-0.419621\pi$$
0.249843 + 0.968286i $$0.419621\pi$$
$$830$$ 0 0
$$831$$ −5.12399 −0.177749
$$832$$ 0 0
$$833$$ −4.42864 −0.153443
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.18421 0.179192
$$838$$ 0 0
$$839$$ −1.51114 −0.0521703 −0.0260851 0.999660i $$-0.508304\pi$$
−0.0260851 + 0.999660i $$0.508304\pi$$
$$840$$ 0 0
$$841$$ −28.4291 −0.980314
$$842$$ 0 0
$$843$$ −23.9813 −0.825959
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 7.00000 0.240523
$$848$$ 0 0
$$849$$ −2.36842 −0.0812838
$$850$$ 0 0
$$851$$ −10.4889 −0.359554
$$852$$ 0 0
$$853$$ −15.4064 −0.527504 −0.263752 0.964591i $$-0.584960\pi$$
−0.263752 + 0.964591i $$0.584960\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 19.8578 0.678328 0.339164 0.940727i $$-0.389856\pi$$
0.339164 + 0.940727i $$0.389856\pi$$
$$858$$ 0 0
$$859$$ −2.42864 −0.0828641 −0.0414321 0.999141i $$-0.513192\pi$$
−0.0414321 + 0.999141i $$0.513192\pi$$
$$860$$ 0 0
$$861$$ −8.23506 −0.280650
$$862$$ 0 0
$$863$$ 39.2958 1.33764 0.668822 0.743423i $$-0.266799\pi$$
0.668822 + 0.743423i $$0.266799\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −2.61285 −0.0887370
$$868$$ 0 0
$$869$$ −9.71456 −0.329544
$$870$$ 0 0
$$871$$ 17.7146 0.600235
$$872$$ 0 0
$$873$$ 11.9398 0.404100
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −56.2864 −1.90066 −0.950328 0.311249i $$-0.899253\pi$$
−0.950328 + 0.311249i $$0.899253\pi$$
$$878$$ 0 0
$$879$$ 8.42864 0.284291
$$880$$ 0 0
$$881$$ −2.33677 −0.0787279 −0.0393640 0.999225i $$-0.512533\pi$$
−0.0393640 + 0.999225i $$0.512533\pi$$
$$882$$ 0 0
$$883$$ −33.7146 −1.13459 −0.567293 0.823516i $$-0.692009\pi$$
−0.567293 + 0.823516i $$0.692009\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −47.8992 −1.60830 −0.804150 0.594427i $$-0.797379\pi$$
−0.804150 + 0.594427i $$0.797379\pi$$
$$888$$ 0 0
$$889$$ 12.8573 0.431219
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 6.69228 0.223949
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −8.85728 −0.295736
$$898$$ 0 0
$$899$$ −3.91703 −0.130640
$$900$$ 0 0
$$901$$ 40.6735 1.35503
$$902$$ 0 0
$$903$$ 10.1017 0.336164
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −23.7591 −0.788908 −0.394454 0.918916i $$-0.629066\pi$$
−0.394454 + 0.918916i $$0.629066\pi$$
$$908$$ 0 0
$$909$$ 1.47949 0.0490717
$$910$$ 0 0
$$911$$ −22.9403 −0.760045 −0.380022 0.924977i $$-0.624084\pi$$
−0.380022 + 0.924977i $$0.624084\pi$$
$$912$$ 0 0
$$913$$ −23.2257 −0.768658
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −2.10171 −0.0694046
$$918$$ 0 0
$$919$$ −16.9777 −0.560043 −0.280022 0.959994i $$-0.590342\pi$$
−0.280022 + 0.959994i $$0.590342\pi$$
$$920$$ 0 0
$$921$$ 22.5718 0.743767
$$922$$ 0 0
$$923$$ −12.8573 −0.423202
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −8.85728 −0.290911
$$928$$ 0 0
$$929$$ 39.3403 1.29071 0.645357 0.763881i $$-0.276709\pi$$
0.645357 + 0.763881i $$0.276709\pi$$
$$930$$ 0 0
$$931$$ 2.42864 0.0795954
$$932$$ 0 0
$$933$$ −24.0830 −0.788441
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −17.7748 −0.580677 −0.290338 0.956924i $$-0.593768\pi$$
−0.290338 + 0.956924i $$0.593768\pi$$
$$938$$ 0 0
$$939$$ 9.65433 0.315057
$$940$$ 0 0
$$941$$ 35.5812 1.15991 0.579957 0.814647i $$-0.303069\pi$$
0.579957 + 0.814647i $$0.303069\pi$$
$$942$$ 0 0
$$943$$ −11.3461 −0.369481
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30.5018 −0.991174 −0.495587 0.868558i $$-0.665047\pi$$
−0.495587 + 0.868558i $$0.665047\pi$$
$$948$$ 0 0
$$949$$ 10.1017 0.327915
$$950$$ 0 0
$$951$$ −6.04149 −0.195909
$$952$$ 0 0
$$953$$ −51.1655 −1.65741 −0.828706 0.559684i $$-0.810923\pi$$
−0.828706 + 0.559684i $$0.810923\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1.51114 0.0488481
$$958$$ 0 0
$$959$$ 15.9398 0.514722
$$960$$ 0 0
$$961$$ −4.12399 −0.133032
$$962$$ 0 0
$$963$$ 1.76494 0.0568743
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −47.8992 −1.54034 −0.770168 0.637841i $$-0.779828\pi$$
−0.770168 + 0.637841i $$0.779828\pi$$
$$968$$ 0 0
$$969$$ 10.7556 0.345519
$$970$$ 0 0
$$971$$ −40.6735 −1.30528 −0.652638 0.757670i $$-0.726338\pi$$
−0.652638 + 0.757670i $$0.726338\pi$$
$$972$$ 0 0
$$973$$ 11.6731 0.374221
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −27.4893 −0.879462 −0.439731 0.898130i $$-0.644926\pi$$
−0.439731 + 0.898130i $$0.644926\pi$$
$$978$$ 0 0
$$979$$ −9.24443 −0.295453
$$980$$ 0 0
$$981$$ 5.61285 0.179204
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 2.75557 0.0877107
$$988$$ 0 0
$$989$$ 13.9180 0.442566
$$990$$ 0 0
$$991$$ 34.6923 1.10204 0.551018 0.834493i $$-0.314239\pi$$
0.551018 + 0.834493i $$0.314239\pi$$
$$992$$ 0 0
$$993$$ 13.5111 0.428763
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −28.6766 −0.908197 −0.454099 0.890951i $$-0.650039\pi$$
−0.454099 + 0.890951i $$0.650039\pi$$
$$998$$ 0 0
$$999$$ 7.61285 0.240860
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 8400.2.a.dg.1.3 3
4.3 odd 2 525.2.a.j.1.3 3
5.2 odd 4 1680.2.t.k.1009.5 6
5.3 odd 4 1680.2.t.k.1009.2 6
5.4 even 2 8400.2.a.dj.1.1 3
12.11 even 2 1575.2.a.x.1.1 3
15.2 even 4 5040.2.t.v.1009.3 6
15.8 even 4 5040.2.t.v.1009.4 6
20.3 even 4 105.2.d.b.64.2 6
20.7 even 4 105.2.d.b.64.5 yes 6
20.19 odd 2 525.2.a.k.1.1 3
28.27 even 2 3675.2.a.bi.1.3 3
60.23 odd 4 315.2.d.e.64.5 6
60.47 odd 4 315.2.d.e.64.2 6
60.59 even 2 1575.2.a.w.1.3 3
140.3 odd 12 735.2.q.f.79.5 12
140.23 even 12 735.2.q.e.214.2 12
140.27 odd 4 735.2.d.b.589.5 6
140.47 odd 12 735.2.q.f.214.5 12
140.67 even 12 735.2.q.e.79.2 12
140.83 odd 4 735.2.d.b.589.2 6
140.87 odd 12 735.2.q.f.79.2 12
140.103 odd 12 735.2.q.f.214.2 12
140.107 even 12 735.2.q.e.214.5 12
140.123 even 12 735.2.q.e.79.5 12
140.139 even 2 3675.2.a.bj.1.1 3
420.83 even 4 2205.2.d.l.1324.5 6
420.167 even 4 2205.2.d.l.1324.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 20.3 even 4
105.2.d.b.64.5 yes 6 20.7 even 4
315.2.d.e.64.2 6 60.47 odd 4
315.2.d.e.64.5 6 60.23 odd 4
525.2.a.j.1.3 3 4.3 odd 2
525.2.a.k.1.1 3 20.19 odd 2
735.2.d.b.589.2 6 140.83 odd 4
735.2.d.b.589.5 6 140.27 odd 4
735.2.q.e.79.2 12 140.67 even 12
735.2.q.e.79.5 12 140.123 even 12
735.2.q.e.214.2 12 140.23 even 12
735.2.q.e.214.5 12 140.107 even 12
735.2.q.f.79.2 12 140.87 odd 12
735.2.q.f.79.5 12 140.3 odd 12
735.2.q.f.214.2 12 140.103 odd 12
735.2.q.f.214.5 12 140.47 odd 12
1575.2.a.w.1.3 3 60.59 even 2
1575.2.a.x.1.1 3 12.11 even 2
1680.2.t.k.1009.2 6 5.3 odd 4
1680.2.t.k.1009.5 6 5.2 odd 4
2205.2.d.l.1324.2 6 420.167 even 4
2205.2.d.l.1324.5 6 420.83 even 4
3675.2.a.bi.1.3 3 28.27 even 2
3675.2.a.bj.1.1 3 140.139 even 2
5040.2.t.v.1009.3 6 15.2 even 4
5040.2.t.v.1009.4 6 15.8 even 4
8400.2.a.dg.1.3 3 1.1 even 1 trivial
8400.2.a.dj.1.1 3 5.4 even 2