Properties

 Label 3150.2.g.q.2899.1 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

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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 210) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} +2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -4.00000 q^{19} -4.00000i q^{22} -8.00000i q^{23} +2.00000 q^{26} +1.00000i q^{28} -2.00000 q^{29} -1.00000i q^{32} -2.00000 q^{34} +6.00000i q^{37} +4.00000i q^{38} +6.00000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -8.00000 q^{46} -1.00000 q^{49} -2.00000i q^{52} -10.0000i q^{53} +1.00000 q^{56} +2.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} -1.00000 q^{64} -12.0000i q^{67} +2.00000i q^{68} +8.00000 q^{71} -10.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -4.00000i q^{77} -16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +4.00000i q^{88} +10.0000 q^{89} +2.00000 q^{91} +8.00000i q^{92} +2.00000i q^{97} +1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 8q^{11} - 2q^{14} + 2q^{16} - 8q^{19} + 4q^{26} - 4q^{29} - 4q^{34} + 12q^{41} - 8q^{44} - 16q^{46} - 2q^{49} + 2q^{56} + 24q^{59} + 28q^{61} - 2q^{64} + 16q^{71} + 12q^{74} + 8q^{76} - 32q^{79} + 8q^{86} + 20q^{89} + 4q^{91} + O(q^{100})$$

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
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 1.00000i 0.188982i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 2.00000i 0.262613i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ − 4.00000i − 0.455842i
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000i 0.426401i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 8.00000i 0.834058i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 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
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ − 12.0000i − 1.10469i
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 14.0000i − 1.26750i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ 8.00000i 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ − 6.00000i − 0.493197i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 0 0
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ − 10.0000i − 0.749532i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ 0 0
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ 2.00000i 0.140372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000i 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 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
$$232$$ − 2.00000i − 0.131306i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 2.00000i 0.129641i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 8.00000i − 0.509028i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000i 0.873296i 0.899632 + 0.436648i $$0.143834\pi$$
−0.899632 + 0.436648i $$0.856166\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 20.0000i 1.23560i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ 12.0000i 0.733017i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 10.0000i 0.585206i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 0 0
$$298$$ 10.0000i 0.579284i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00000i 0.445823i
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 0 0
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 18.0000i 0.980522i 0.871576 + 0.490261i $$0.163099\pi$$
−0.871576 + 0.490261i $$0.836901\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −10.0000 −0.519174
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 16.0000i − 0.818631i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ − 2.00000i − 0.101535i
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 30.0000i 1.50566i 0.658217 + 0.752828i $$0.271311\pi$$
−0.658217 + 0.752828i $$0.728689\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ 24.0000i 1.18964i
$$408$$ 0 0
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 16.0000i 0.782586i
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 14.0000i − 0.677507i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ − 18.0000i − 0.846649i
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000i 0.552345i
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 14.0000i 0.637683i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ 4.00000i 0.180151i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 8.00000i − 0.358849i
$$498$$ 0 0
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ 40.0000i 1.78351i 0.452517 + 0.891756i $$0.350526\pi$$
−0.452517 + 0.891756i $$0.649474\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ − 6.00000i − 0.263625i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 4.00000i − 0.173422i
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 18.0000i 0.776035i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12.0000i − 0.513083i −0.966533 0.256541i $$-0.917417\pi$$
0.966533 0.256541i $$-0.0825830\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 16.0000i 0.680389i
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ 20.0000i 0.842900i 0.906852 + 0.421450i $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 0 0
$$568$$ 8.00000i 0.335673i
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 34.0000i 1.41544i 0.706494 + 0.707719i $$0.250276\pi$$
−0.706494 + 0.707719i $$0.749724\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ − 40.0000i − 1.65663i
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000i 0.246598i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ − 16.0000i − 0.654289i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ − 4.00000i − 0.163028i
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.0000i 0.649420i 0.945814 + 0.324710i $$0.105267\pi$$
−0.945814 + 0.324710i $$0.894733\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 6.00000i − 0.242338i −0.992632 0.121169i $$-0.961336\pi$$
0.992632 0.121169i $$-0.0386643\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 8.00000i − 0.320771i
$$623$$ − 10.0000i − 0.400642i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ 18.0000i 0.718278i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ − 16.0000i − 0.636446i
$$633$$ 0 0
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000i 0.783260i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ − 12.0000i − 0.466393i
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000i 0.619522i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ 30.0000i 1.15642i 0.815890 + 0.578208i $$0.196248\pi$$
−0.815890 + 0.578208i $$0.803752\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 12.0000i − 0.454532i
$$698$$ 14.0000i 0.529908i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ − 24.0000i − 0.905177i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 6.00000i 0.225653i
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 10.0000i 0.374766i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ − 24.0000i − 0.895672i
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ −6.00000 −0.222988
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ − 48.0000i − 1.76810i
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 10.0000i 0.367112i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ 8.00000i 0.292509i
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000i 0.799604i 0.916602 + 0.399802i $$0.130921\pi$$
−0.916602 + 0.399802i $$0.869079\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 14.0000i 0.506834i
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 24.0000i 0.866590i
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000i 0.0719816i
$$773$$ − 26.0000i − 0.935155i −0.883952 0.467578i $$-0.845127\pi$$
0.883952 0.467578i $$-0.154873\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 26.0000i 0.932145i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 16.0000i 0.572159i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 28.0000i 0.994309i
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 14.0000i − 0.494357i
$$803$$ − 40.0000i − 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ − 6.00000i − 0.211079i
$$809$$ −22.0000 −0.773479 −0.386739 0.922189i $$-0.626399\pi$$
−0.386739 + 0.922189i $$0.626399\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 38.0000i − 1.32864i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 0 0
$$823$$ 24.0000i 0.836587i 0.908312 + 0.418294i $$0.137372\pi$$
−0.908312 + 0.418294i $$0.862628\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 2.00000i 0.0692959i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 20.0000i − 0.690889i
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000i 0.896019i
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 5.00000i − 0.171802i
$$848$$ − 10.0000i − 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −14.0000 −0.479070
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 22.0000i 0.751506i 0.926720 + 0.375753i $$0.122616\pi$$
−0.926720 + 0.375753i $$0.877384\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ − 14.0000i − 0.474100i
$$873$$ 0 0
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ 28.0000i 0.942275i 0.882060 + 0.471138i $$0.156156\pi$$
−0.882060 + 0.471138i $$0.843844\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ − 24.0000i − 0.799113i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 0 0
$$913$$ − 48.0000i − 1.58857i
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 20.0000i 0.660458i
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 14.0000i 0.461065i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 2.00000i 0.0656532i
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 22.0000i 0.720634i
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 10.0000i 0.326686i 0.986569 + 0.163343i $$0.0522277\pi$$
−0.986569 + 0.163343i $$0.947772\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ − 48.0000i − 1.56310i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 2.00000i − 0.0648204i
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ − 32.0000i − 1.03387i
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 12.0000i 0.386896i
$$963$$ 0 0
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ − 4.00000i − 0.128234i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 36.0000i − 1.14881i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 22.0000i 0.696747i 0.937356 + 0.348373i $$0.113266\pi$$
−0.937356 + 0.348373i $$0.886734\pi$$
$$998$$ − 12.0000i − 0.379853i
$$999$$ 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.q.2899.1 2
3.2 odd 2 1050.2.g.g.799.2 2
5.2 odd 4 3150.2.a.bp.1.1 1
5.3 odd 4 630.2.a.a.1.1 1
5.4 even 2 inner 3150.2.g.q.2899.2 2
15.2 even 4 1050.2.a.c.1.1 1
15.8 even 4 210.2.a.e.1.1 1
15.14 odd 2 1050.2.g.g.799.1 2
20.3 even 4 5040.2.a.k.1.1 1
35.13 even 4 4410.2.a.t.1.1 1
60.23 odd 4 1680.2.a.j.1.1 1
60.47 odd 4 8400.2.a.ce.1.1 1
105.23 even 12 1470.2.i.a.361.1 2
105.38 odd 12 1470.2.i.j.961.1 2
105.53 even 12 1470.2.i.a.961.1 2
105.62 odd 4 7350.2.a.w.1.1 1
105.68 odd 12 1470.2.i.j.361.1 2
105.83 odd 4 1470.2.a.j.1.1 1
120.53 even 4 6720.2.a.j.1.1 1
120.83 odd 4 6720.2.a.bq.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.e.1.1 1 15.8 even 4
630.2.a.a.1.1 1 5.3 odd 4
1050.2.a.c.1.1 1 15.2 even 4
1050.2.g.g.799.1 2 15.14 odd 2
1050.2.g.g.799.2 2 3.2 odd 2
1470.2.a.j.1.1 1 105.83 odd 4
1470.2.i.a.361.1 2 105.23 even 12
1470.2.i.a.961.1 2 105.53 even 12
1470.2.i.j.361.1 2 105.68 odd 12
1470.2.i.j.961.1 2 105.38 odd 12
1680.2.a.j.1.1 1 60.23 odd 4
3150.2.a.bp.1.1 1 5.2 odd 4
3150.2.g.q.2899.1 2 1.1 even 1 trivial
3150.2.g.q.2899.2 2 5.4 even 2 inner
4410.2.a.t.1.1 1 35.13 even 4
5040.2.a.k.1.1 1 20.3 even 4
6720.2.a.j.1.1 1 120.53 even 4
6720.2.a.bq.1.1 1 120.83 odd 4
7350.2.a.w.1.1 1 105.62 odd 4
8400.2.a.ce.1.1 1 60.47 odd 4