# Properties

 Label 3150.2.g.h.2899.2 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1050) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$-1.00000i$$ Character $$\chi$$ = 3150.2899 Dual form 3150.2.g.h.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} -2.00000 q^{11} -1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{22} +1.00000i q^{23} +1.00000 q^{26} +1.00000i q^{28} -5.00000 q^{29} +7.00000 q^{31} +1.00000i q^{32} -3.00000 q^{34} +2.00000i q^{37} -7.00000 q^{41} -11.0000i q^{43} +2.00000 q^{44} -1.00000 q^{46} +8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +1.00000i q^{53} -1.00000 q^{56} -5.00000i q^{58} -5.00000 q^{59} -3.00000 q^{61} +7.00000i q^{62} -1.00000 q^{64} +12.0000i q^{67} -3.00000i q^{68} -12.0000 q^{71} -6.00000i q^{73} -2.00000 q^{74} +2.00000i q^{77} -10.0000 q^{79} -7.00000i q^{82} +11.0000i q^{83} +11.0000 q^{86} +2.00000i q^{88} -10.0000 q^{89} -1.00000 q^{91} -1.00000i q^{92} -8.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{11} + 2q^{14} + 2q^{16} + 2q^{26} - 10q^{29} + 14q^{31} - 6q^{34} - 14q^{41} + 4q^{44} - 2q^{46} - 2q^{49} - 2q^{56} - 10q^{59} - 6q^{61} - 2q^{64} - 24q^{71} - 4q^{74} - 20q^{79} + 22q^{86} - 20q^{89} - 2q^{91} - 16q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 2.00000i − 0.426401i
$$23$$ 1.00000i 0.208514i 0.994550 + 0.104257i $$0.0332465\pi$$
−0.994550 + 0.104257i $$0.966753\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 1.00000i 0.188982i
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.00000 −1.09322 −0.546608 0.837389i $$-0.684081\pi$$
−0.546608 + 0.837389i $$0.684081\pi$$
$$42$$ 0 0
$$43$$ − 11.0000i − 1.67748i −0.544529 0.838742i $$-0.683292\pi$$
0.544529 0.838742i $$-0.316708\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ 1.00000i 0.137361i 0.997639 + 0.0686803i $$0.0218788\pi$$
−0.997639 + 0.0686803i $$0.978121\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ − 5.00000i − 0.656532i
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 7.00000i 0.889001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 3.00000i − 0.363803i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000i 0.227921i
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 7.00000i − 0.773021i
$$83$$ 11.0000i 1.20741i 0.797209 + 0.603703i $$0.206309\pi$$
−0.797209 + 0.603703i $$0.793691\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 11.0000 1.18616
$$87$$ 0 0
$$88$$ 2.00000i 0.213201i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ − 1.00000i − 0.104257i
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ − 11.0000i − 1.08386i −0.840423 0.541931i $$-0.817693\pi$$
0.840423 0.541931i $$-0.182307\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ − 2.00000i − 0.193347i −0.995316 0.0966736i $$-0.969180\pi$$
0.995316 0.0966736i $$-0.0308203\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ − 5.00000i − 0.460287i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 3.00000i − 0.271607i
$$123$$ 0 0
$$124$$ −7.00000 −0.628619
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 22.0000i 1.95218i 0.217357 + 0.976092i $$0.430256\pi$$
−0.217357 + 0.976092i $$0.569744\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 12.0000i − 1.00702i
$$143$$ 2.00000i 0.167248i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ − 1.00000i − 0.0783260i −0.999233 0.0391630i $$-0.987531\pi$$
0.999233 0.0391630i $$-0.0124692\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ −11.0000 −0.853766
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 11.0000i 0.838742i
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ − 1.00000i − 0.0741249i
$$183$$ 0 0
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.00000i − 0.438763i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 17.0000i − 1.21120i −0.795769 0.605600i $$-0.792933\pi$$
0.795769 0.605600i $$-0.207067\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 12.0000i − 0.844317i
$$203$$ 5.00000i 0.350931i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 11.0000 0.766406
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ − 1.00000i − 0.0686803i
$$213$$ 0 0
$$214$$ 2.00000 0.136717
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 7.00000i − 0.475191i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 9.00000i 0.602685i 0.953516 + 0.301342i $$0.0974347\pi$$
−0.953516 + 0.301342i $$0.902565\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 17.0000i − 1.12833i −0.825662 0.564165i $$-0.809198\pi$$
0.825662 0.564165i $$-0.190802\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 5.00000i 0.328266i
$$233$$ − 24.0000i − 1.57229i −0.618041 0.786146i $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5.00000 0.325472
$$237$$ 0 0
$$238$$ 3.00000i 0.194461i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 0 0
$$244$$ 3.00000 0.192055
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ − 7.00000i − 0.444500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −7.00000 −0.441836 −0.220918 0.975292i $$-0.570905\pi$$
−0.220918 + 0.975292i $$0.570905\pi$$
$$252$$ 0 0
$$253$$ − 2.00000i − 0.125739i
$$254$$ −22.0000 −1.38040
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.0000i 1.43470i 0.696713 + 0.717350i $$0.254645\pi$$
−0.696713 + 0.717350i $$0.745355\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 8.00000i 0.494242i
$$263$$ 21.0000i 1.29492i 0.762101 + 0.647458i $$0.224168\pi$$
−0.762101 + 0.647458i $$0.775832\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 3.00000i 0.181902i
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −32.0000 −1.90896 −0.954480 0.298275i $$-0.903589\pi$$
−0.954480 + 0.298275i $$0.903589\pi$$
$$282$$ 0 0
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 7.00000i 0.413197i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6.00000i 0.351123i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 15.0000i 0.868927i
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ −11.0000 −0.634029
$$302$$ − 18.0000i − 1.03578i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ − 2.00000i − 0.113961i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 3.00000i 0.168497i 0.996445 + 0.0842484i $$0.0268489\pi$$
−0.996445 + 0.0842484i $$0.973151\pi$$
$$318$$ 0 0
$$319$$ 10.0000 0.559893
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1.00000i 0.0557278i
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 1.00000 0.0553849
$$327$$ 0 0
$$328$$ 7.00000i 0.386510i
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ − 11.0000i − 0.603703i
$$333$$ 0 0
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 27.0000i 1.47078i 0.677642 + 0.735392i $$0.263002\pi$$
−0.677642 + 0.735392i $$0.736998\pi$$
$$338$$ 12.0000i 0.652714i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −14.0000 −0.758143
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ −11.0000 −0.593080
$$345$$ 0 0
$$346$$ 24.0000 1.29025
$$347$$ − 2.00000i − 0.107366i −0.998558 0.0536828i $$-0.982904\pi$$
0.998558 0.0536828i $$-0.0170960\pi$$
$$348$$ 0 0
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 2.00000i − 0.106600i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ − 10.0000i − 0.528516i
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 18.0000i − 0.946059i
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 23.0000i − 1.20059i −0.799779 0.600295i $$-0.795050\pi$$
0.799779 0.600295i $$-0.204950\pi$$
$$368$$ 1.00000i 0.0521286i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.00000 0.0519174
$$372$$ 0 0
$$373$$ 24.0000i 1.24267i 0.783544 + 0.621336i $$0.213410\pi$$
−0.783544 + 0.621336i $$0.786590\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 5.00000i 0.257513i
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 3.00000i 0.153493i
$$383$$ − 4.00000i − 0.204390i −0.994764 0.102195i $$-0.967413\pi$$
0.994764 0.102195i $$-0.0325866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ − 2.00000i − 0.101535i
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 1.00000i 0.0505076i
$$393$$ 0 0
$$394$$ 17.0000 0.856448
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 13.0000i − 0.652451i −0.945292 0.326226i $$-0.894223\pi$$
0.945292 0.326226i $$-0.105777\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8.00000 0.399501 0.199750 0.979847i $$-0.435987\pi$$
0.199750 + 0.979847i $$0.435987\pi$$
$$402$$ 0 0
$$403$$ − 7.00000i − 0.348695i
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ −5.00000 −0.248146
$$407$$ − 4.00000i − 0.198273i
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 11.0000i 0.541931i
$$413$$ 5.00000i 0.246034i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 25.0000 1.22133 0.610665 0.791889i $$-0.290902\pi$$
0.610665 + 0.791889i $$0.290902\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ − 13.0000i − 0.632830i
$$423$$ 0 0
$$424$$ 1.00000 0.0485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.00000i 0.145180i
$$428$$ 2.00000i 0.0966736i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 7.00000 0.336011
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3.00000i 0.142695i
$$443$$ 16.0000i 0.760183i 0.924949 + 0.380091i $$0.124107\pi$$
−0.924949 + 0.380091i $$0.875893\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −9.00000 −0.426162
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 14.0000 0.659234
$$452$$ − 6.00000i − 0.282216i
$$453$$ 0 0
$$454$$ 17.0000 0.797850
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 33.0000i − 1.54367i −0.635820 0.771837i $$-0.719338\pi$$
0.635820 0.771837i $$-0.280662\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −5.00000 −0.232119
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 33.0000i 1.52706i 0.645774 + 0.763529i $$0.276535\pi$$
−0.645774 + 0.763529i $$0.723465\pi$$
$$468$$ 0 0
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 5.00000i 0.230144i
$$473$$ 22.0000i 1.01156i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 22.0000i 1.00207i
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 3.00000i 0.135804i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ − 15.0000i − 0.675566i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 7.00000 0.314309
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 7.00000i − 0.312425i
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.00000 0.0889108
$$507$$ 0 0
$$508$$ − 22.0000i − 0.976092i
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −23.0000 −1.01449
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16.0000i − 0.703679i
$$518$$ 2.00000i 0.0878750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.00000 0.131432 0.0657162 0.997838i $$-0.479067\pi$$
0.0657162 + 0.997838i $$0.479067\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ −21.0000 −0.915644
$$527$$ 21.0000i 0.914774i
$$528$$ 0 0
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 7.00000i 0.303204i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ − 10.0000i − 0.431131i
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 32.0000 1.37579 0.687894 0.725811i $$-0.258536\pi$$
0.687894 + 0.725811i $$0.258536\pi$$
$$542$$ − 28.0000i − 1.20270i
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 37.0000i 1.58201i 0.611812 + 0.791003i $$0.290441\pi$$
−0.611812 + 0.791003i $$0.709559\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 10.0000i 0.425243i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −11.0000 −0.465250
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 32.0000i − 1.34984i
$$563$$ 31.0000i 1.30649i 0.757145 + 0.653247i $$0.226594\pi$$
−0.757145 + 0.653247i $$0.773406\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 0 0
$$568$$ 12.0000i 0.503509i
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 7.00000 0.292941 0.146470 0.989215i $$-0.453209\pi$$
0.146470 + 0.989215i $$0.453209\pi$$
$$572$$ − 2.00000i − 0.0836242i
$$573$$ 0 0
$$574$$ −7.00000 −0.292174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ 8.00000i 0.332756i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 11.0000 0.456357
$$582$$ 0 0
$$583$$ − 2.00000i − 0.0828315i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ − 27.0000i − 1.11441i −0.830375 0.557205i $$-0.811874\pi$$
0.830375 0.557205i $$-0.188126\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000i 0.0821995i
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −15.0000 −0.614424
$$597$$ 0 0
$$598$$ 1.00000i 0.0408930i
$$599$$ 5.00000 0.204294 0.102147 0.994769i $$-0.467429\pi$$
0.102147 + 0.994769i $$0.467429\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ − 11.0000i − 0.448327i
$$603$$ 0 0
$$604$$ 18.0000 0.732410
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 14.0000i 0.565455i 0.959200 + 0.282727i $$0.0912392\pi$$
−0.959200 + 0.282727i $$0.908761\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 2.00000 0.0805823
$$617$$ 38.0000i 1.52982i 0.644136 + 0.764911i $$0.277217\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000i 0.721734i
$$623$$ 10.0000i 0.400642i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ −38.0000 −1.51276 −0.756378 0.654135i $$-0.773033\pi$$
−0.756378 + 0.654135i $$0.773033\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ 0 0
$$634$$ −3.00000 −0.119145
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.00000i 0.0396214i
$$638$$ 10.0000i 0.395904i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ −1.00000 −0.0394055
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 22.0000i − 0.864909i −0.901656 0.432455i $$-0.857648\pi$$
0.901656 0.432455i $$-0.142352\pi$$
$$648$$ 0 0
$$649$$ 10.0000 0.392534
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 1.00000i 0.0391630i
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −7.00000 −0.273304
$$657$$ 0 0
$$658$$ 8.00000i 0.311872i
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ 42.0000 1.63361 0.816805 0.576913i $$-0.195743\pi$$
0.816805 + 0.576913i $$0.195743\pi$$
$$662$$ 17.0000i 0.660724i
$$663$$ 0 0
$$664$$ 11.0000 0.426883
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 5.00000i − 0.193601i
$$668$$ − 18.0000i − 0.696441i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.00000 0.231627
$$672$$ 0 0
$$673$$ 29.0000i 1.11787i 0.829212 + 0.558934i $$0.188789\pi$$
−0.829212 + 0.558934i $$0.811211\pi$$
$$674$$ −27.0000 −1.04000
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ − 32.0000i − 1.22986i −0.788582 0.614930i $$-0.789184\pi$$
0.788582 0.614930i $$-0.210816\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 14.0000i − 0.536088i
$$683$$ 46.0000i 1.76014i 0.474843 + 0.880071i $$0.342505\pi$$
−0.474843 + 0.880071i $$0.657495\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ − 11.0000i − 0.419371i
$$689$$ 1.00000 0.0380970
$$690$$ 0 0
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ 24.0000i 0.912343i
$$693$$ 0 0
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 21.0000i − 0.795432i
$$698$$ − 15.0000i − 0.567758i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −27.0000 −1.01978 −0.509888 0.860241i $$-0.670313\pi$$
−0.509888 + 0.860241i $$0.670313\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000i 0.374766i
$$713$$ 7.00000i 0.262152i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ 25.0000i 0.932992i
$$719$$ −10.0000 −0.372937 −0.186469 0.982461i $$-0.559704\pi$$
−0.186469 + 0.982461i $$0.559704\pi$$
$$720$$ 0 0
$$721$$ −11.0000 −0.409661
$$722$$ − 19.0000i − 0.707107i
$$723$$ 0 0
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 13.0000i − 0.482143i −0.970507 0.241072i $$-0.922501\pi$$
0.970507 0.241072i $$-0.0774989\pi$$
$$728$$ 1.00000i 0.0370625i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 33.0000 1.22055
$$732$$ 0 0
$$733$$ − 11.0000i − 0.406294i −0.979148 0.203147i $$-0.934883\pi$$
0.979148 0.203147i $$-0.0651170\pi$$
$$734$$ 23.0000 0.848945
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 24.0000i − 0.884051i
$$738$$ 0 0
$$739$$ 25.0000 0.919640 0.459820 0.888012i $$-0.347914\pi$$
0.459820 + 0.888012i $$0.347914\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.00000i 0.0367112i
$$743$$ − 29.0000i − 1.06391i −0.846774 0.531953i $$-0.821458\pi$$
0.846774 0.531953i $$-0.178542\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −24.0000 −0.878702
$$747$$ 0 0
$$748$$ 6.00000i 0.219382i
$$749$$ −2.00000 −0.0730784
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ −5.00000 −0.182089
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 25.0000i 0.908041i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −3.00000 −0.108536
$$765$$ 0 0
$$766$$ 4.00000 0.144526
$$767$$ 5.00000i 0.180540i
$$768$$ 0 0
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ 10.0000i 0.358517i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ − 3.00000i − 0.107280i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.0000i 0.784215i 0.919919 + 0.392108i $$0.128254\pi$$
−0.919919 + 0.392108i $$0.871746\pi$$
$$788$$ 17.0000i 0.605600i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 3.00000i 0.106533i
$$794$$ 13.0000 0.461353
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 8.00000i 0.282490i
$$803$$ 12.0000i 0.423471i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7.00000 0.246564
$$807$$ 0 0
$$808$$ 12.0000i 0.422159i
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −18.0000 −0.632065 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$812$$ − 5.00000i − 0.175466i
$$813$$ 0 0
$$814$$ 4.00000 0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 54.0000i 1.88232i 0.337959 + 0.941161i $$0.390263\pi$$
−0.337959 + 0.941161i $$0.609737\pi$$
$$824$$ −11.0000 −0.383203
$$825$$ 0 0
$$826$$ −5.00000 −0.173972
$$827$$ − 2.00000i − 0.0695468i −0.999395 0.0347734i $$-0.988929\pi$$
0.999395 0.0347734i $$-0.0110710\pi$$
$$828$$ 0 0
$$829$$ 5.00000 0.173657 0.0868286 0.996223i $$-0.472327\pi$$
0.0868286 + 0.996223i $$0.472327\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000i 0.0346688i
$$833$$ − 3.00000i − 0.103944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 25.0000i 0.863611i
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 22.0000i 0.758170i
$$843$$ 0 0
$$844$$ 13.0000 0.447478
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 7.00000i 0.240523i
$$848$$ 1.00000i 0.0343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ 29.0000i 0.992941i 0.868054 + 0.496471i $$0.165371\pi$$
−0.868054 + 0.496471i $$0.834629\pi$$
$$854$$ −3.00000 −0.102658
$$855$$ 0 0
$$856$$ −2.00000 −0.0683586
$$857$$ − 22.0000i − 0.751506i −0.926720 0.375753i $$-0.877384\pi$$
0.926720 0.375753i $$-0.122616\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 3.00000i 0.102180i
$$863$$ 16.0000i 0.544646i 0.962206 + 0.272323i $$0.0877920\pi$$
−0.962206 + 0.272323i $$0.912208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 7.00000i 0.237595i
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ − 35.0000i − 1.18119i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −7.00000 −0.235836 −0.117918 0.993023i $$-0.537622\pi$$
−0.117918 + 0.993023i $$0.537622\pi$$
$$882$$ 0 0
$$883$$ 39.0000i 1.31245i 0.754563 + 0.656227i $$0.227849\pi$$
−0.754563 + 0.656227i $$0.772151\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 0 0
$$886$$ −16.0000 −0.537531
$$887$$ 58.0000i 1.94745i 0.227728 + 0.973725i $$0.426870\pi$$
−0.227728 + 0.973725i $$0.573130\pi$$
$$888$$ 0 0
$$889$$ 22.0000 0.737856
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 9.00000i − 0.301342i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ 20.0000i 0.667409i
$$899$$ −35.0000 −1.16732
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 14.0000i 0.466149i
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 17.0000i 0.564476i 0.959344 + 0.282238i $$0.0910767\pi$$
−0.959344 + 0.282238i $$0.908923\pi$$
$$908$$ 17.0000i 0.564165i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 53.0000 1.75597 0.877984 0.478690i $$-0.158888\pi$$
0.877984 + 0.478690i $$0.158888\pi$$
$$912$$ 0 0
$$913$$ − 22.0000i − 0.728094i
$$914$$ 33.0000 1.09154
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ − 8.00000i − 0.264183i
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 12.0000i − 0.395199i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ − 5.00000i − 0.164133i
$$929$$ −45.0000 −1.47640 −0.738201 0.674581i $$-0.764324\pi$$
−0.738201 + 0.674581i $$0.764324\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 24.0000i 0.786146i
$$933$$ 0 0
$$934$$ −33.0000 −1.07979
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 42.0000i 1.37208i 0.727564 + 0.686040i $$0.240653\pi$$
−0.727564 + 0.686040i $$0.759347\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ − 7.00000i − 0.227951i
$$944$$ −5.00000 −0.162736
$$945$$ 0 0
$$946$$ −22.0000 −0.715282
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 3.00000i − 0.0972306i
$$953$$ 16.0000i 0.518291i 0.965838 + 0.259145i $$0.0834409\pi$$
−0.965838 + 0.259145i $$0.916559\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ − 20.0000i − 0.646171i
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 2.00000i 0.0644826i
$$963$$ 0 0
$$964$$ −22.0000 −0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 20.0000i 0.641171i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −3.00000 −0.0960277
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 0 0
$$979$$ 20.0000 0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 2.00000i − 0.0638226i
$$983$$ 16.0000i 0.510321i 0.966899 + 0.255160i $$0.0821283\pi$$
−0.966899 + 0.255160i $$0.917872\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 15.0000 0.477697
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11.0000 0.349780
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 7.00000i 0.222250i
$$993$$ 0 0
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ − 25.0000i − 0.791361i
$$999$$ 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 3150.2.g.h.2899.2 2
3.2 odd 2 1050.2.g.i.799.1 2
5.2 odd 4 3150.2.a.n.1.1 1
5.3 odd 4 3150.2.a.y.1.1 1
5.4 even 2 inner 3150.2.g.h.2899.1 2
15.2 even 4 1050.2.a.r.1.1 yes 1
15.8 even 4 1050.2.a.b.1.1 1
15.14 odd 2 1050.2.g.i.799.2 2
60.23 odd 4 8400.2.a.ck.1.1 1
60.47 odd 4 8400.2.a.d.1.1 1
105.62 odd 4 7350.2.a.cb.1.1 1
105.83 odd 4 7350.2.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.b.1.1 1 15.8 even 4
1050.2.a.r.1.1 yes 1 15.2 even 4
1050.2.g.i.799.1 2 3.2 odd 2
1050.2.g.i.799.2 2 15.14 odd 2
3150.2.a.n.1.1 1 5.2 odd 4
3150.2.a.y.1.1 1 5.3 odd 4
3150.2.g.h.2899.1 2 5.4 even 2 inner
3150.2.g.h.2899.2 2 1.1 even 1 trivial
7350.2.a.bi.1.1 1 105.83 odd 4
7350.2.a.cb.1.1 1 105.62 odd 4
8400.2.a.d.1.1 1 60.47 odd 4
8400.2.a.ck.1.1 1 60.23 odd 4