# Properties

 Label 735.2.a.c.1.1 Level $735$ Weight $2$ Character 735.1 Self dual yes Analytic conductor $5.869$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.86900454856$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 735.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} -5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -8.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +7.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{71} +3.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -12.0000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{94} +4.00000 q^{95} -5.00000 q^{96} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ −4.00000 −0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000 0.320256
$$40$$ −3.00000 −0.474342
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ −2.00000 −0.280056
$$52$$ −2.00000 −0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 2.00000 0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −2.00000 −0.248069
$$66$$ 4.00000 0.492366
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000 0.353553
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ −12.0000 −1.27920
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ −5.00000 −0.510310
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ 2.00000 0.184900
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ −3.00000 −0.273861
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 4.00000 0.352180
$$130$$ 2.00000 0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ −1.00000 −0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000 0.671345
$$143$$ −8.00000 −0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ 2.00000 0.166091
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 10.0000 0.821995
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −12.0000 −0.973329
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 5.00000 0.395285
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 4.00000 0.311400
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −2.00000 −0.153393
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 4.00000 0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 7.00000 0.505181
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 12.0000 0.846415
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 10.0000 0.698430
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000 0.686803
$$213$$ −8.00000 −0.548151
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ −10.0000 −0.675737
$$220$$ −4.00000 −0.269680
$$221$$ −4.00000 −0.269069
$$222$$ 10.0000 0.671156
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −2.00000 −0.133038
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 8.00000 0.521862
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 1.00000 0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 2.00000 0.125245
$$256$$ −17.0000 −1.06250
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 2.00000 0.124035
$$261$$ −2.00000 −0.123797
$$262$$ −12.0000 −0.741362
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ −12.0000 −0.733017
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 4.00000 0.236940
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ −5.00000 −0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ −2.00000 −0.117242
$$292$$ 10.0000 0.585206
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ −30.0000 −1.74371
$$297$$ −4.00000 −0.232104
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ −6.00000 −0.344691
$$304$$ 4.00000 0.229416
$$305$$ −2.00000 −0.114520
$$306$$ 2.00000 0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 8.00000 0.447914
$$320$$ −7.00000 −0.391312
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ −1.00000 −0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ 14.0000 0.774202
$$328$$ −30.0000 −1.65647
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 2.00000 0.108625
$$340$$ −2.00000 −0.108465
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 20.0000 1.06600
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 8.00000 0.424596
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −20.0000 −1.05703
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ −10.0000 −0.525588
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ −2.00000 −0.104542
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ −10.0000 −0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ −1.00000 −0.0516398
$$376$$ −24.0000 −1.23771
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ −8.00000 −0.409852
$$382$$ −16.0000 −0.818631
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 4.00000 0.203331
$$388$$ 2.00000 0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 12.0000 0.605320
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ −6.00000 −0.297044
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ −6.00000 −0.295958
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −10.0000 −0.490290
$$417$$ 4.00000 0.195881
$$418$$ −16.0000 −0.782586
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ −8.00000 −0.388973
$$424$$ −30.0000 −1.45693
$$425$$ −2.00000 −0.0970143
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ −8.00000 −0.386244
$$430$$ 4.00000 0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 12.0000 0.572078
$$441$$ 0 0
$$442$$ 4.00000 0.190261
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 10.0000 0.474579
$$445$$ −6.00000 −0.284427
$$446$$ 8.00000 0.378811
$$447$$ 22.0000 1.04056
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 40.0000 1.88353
$$452$$ −2.00000 −0.0940721
$$453$$ −8.00000 −0.375873
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 6.00000 0.280362
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 24.0000 1.11537 0.557687 0.830051i $$-0.311689\pi$$
0.557687 + 0.830051i $$0.311689\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ −14.0000 −0.645086
$$472$$ 12.0000 0.552345
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 5.00000 0.228218
$$481$$ −20.0000 −0.911922
$$482$$ −14.0000 −0.637683
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 2.00000 0.0908153
$$486$$ −1.00000 −0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 4.00000 0.180151
$$494$$ 8.00000 0.359937
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 8.00000 0.354943
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ −4.00000 −0.176604
$$514$$ 18.0000 0.793946
$$515$$ −16.0000 −0.705044
$$516$$ −4.00000 −0.176090
$$517$$ 32.0000 1.40736
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ −6.00000 −0.263117
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ −23.0000 −1.00000
$$530$$ −10.0000 −0.434372
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ −6.00000 −0.259645
$$535$$ 12.0000 0.518805
$$536$$ 36.0000 1.55496
$$537$$ 20.0000 0.863064
$$538$$ 14.0000 0.603583
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 10.0000 0.429141
$$544$$ 10.0000 0.428746
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 4.00000 0.170561
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 10.0000 0.424476
$$556$$ −4.00000 −0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 6.00000 0.253095
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 8.00000 0.336861
$$565$$ −2.00000 −0.0841406
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 16.0000 0.668410
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 2.00000 0.0831172
$$580$$ −2.00000 −0.0830455
$$581$$ 0 0
$$582$$ 2.00000 0.0829027
$$583$$ 40.0000 1.65663
$$584$$ −30.0000 −1.24141
$$585$$ −2.00000 −0.0826898
$$586$$ 6.00000 0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ 6.00000 0.246807
$$592$$ 10.0000 0.410997
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 3.00000 0.122474
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 8.00000 0.325515
$$605$$ −5.00000 −0.203279
$$606$$ 6.00000 0.243733
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 20.0000 0.811107
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ −16.0000 −0.647291
$$612$$ 2.00000 0.0808452
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 10.0000 0.403239
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 16.0000 0.638978
$$628$$ 14.0000 0.558661
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 2.00000 0.0794301
$$635$$ 8.00000 0.317470
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ −8.00000 −0.316723
$$639$$ −8.00000 −0.316475
$$640$$ −3.00000 −0.118585
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ −8.00000 −0.314756
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 3.00000 0.117851
$$649$$ −16.0000 −0.628055
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 46.0000 1.80012 0.900060 0.435767i $$-0.143523\pi$$
0.900060 + 0.435767i $$0.143523\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ −12.0000 −0.468879
$$656$$ 10.0000 0.390434
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ −4.00000 −0.155347
$$664$$ −36.0000 −1.39707
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 12.0000 0.463600
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 9.00000 0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 6.00000 0.229248
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ −4.00000 −0.152499
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ −4.00000 −0.151729
$$696$$ −6.00000 −0.227429
$$697$$ 20.0000 0.757554
$$698$$ −2.00000 −0.0757011
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 40.0000 1.50863
$$704$$ −28.0000 −1.05529
$$705$$ 8.00000 0.301297
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 18.0000 0.674579
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ −20.0000 −0.747435
$$717$$ −16.0000 −0.597531
$$718$$ 24.0000 0.895672
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 14.0000 0.520666
$$724$$ −10.0000 −0.371647
$$725$$ −2.00000 −0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −10.0000 −0.370117
$$731$$ −8.00000 −0.295891
$$732$$ −2.00000 −0.0739221
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.0000 −1.76810
$$738$$ 10.0000 0.368105
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ −10.0000 −0.367607
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −22.0000 −0.806018
$$746$$ 26.0000 0.951928
$$747$$ −12.0000 −0.439057
$$748$$ −8.00000 −0.292509
$$749$$ 0 0
$$750$$ 1.00000 0.0365148
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 8.00000 0.291730
$$753$$ −12.0000 −0.437304
$$754$$ 4.00000 0.145671
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 2.00000 0.0723102
$$766$$ −24.0000 −0.867155
$$767$$ 8.00000 0.288863
$$768$$ −17.0000 −0.613435
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −2.00000 −0.0719816
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ 40.0000 1.43315
$$780$$ 2.00000 0.0716115
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 14.0000 0.499681
$$786$$ −12.0000 −0.428026
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −12.0000 −0.426401
$$793$$ 4.00000 0.142044
$$794$$ −2.00000 −0.0709773
$$795$$ 10.0000 0.354663
$$796$$ −8.00000 −0.283552
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ −5.00000 −0.176777
$$801$$ 6.00000 0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 40.0000 1.41157
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ −18.0000 −0.633238
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ −40.0000 −1.40200
$$815$$ 4.00000 0.140114
$$816$$ 2.00000 0.0700140
$$817$$ −16.0000 −0.559769
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 48.0000 1.67216
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 6.00000 0.208138
$$832$$ 14.0000 0.485363
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000 0.896019
$$843$$ −6.00000 −0.206651
$$844$$ −20.0000 −0.688428
$$845$$ 9.00000 0.309609
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ 12.0000 0.411839
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 8.00000 0.274075
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ −36.0000 −1.23045
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 8.00000 0.273115
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −18.0000 −0.612018
$$866$$ −14.0000 −0.475739
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −2.00000 −0.0678064
$$871$$ 24.0000 0.813209
$$872$$ 42.0000 1.42230
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ 30.0000 1.01303 0.506514 0.862232i $$-0.330934\pi$$
0.506514 + 0.862232i $$0.330934\pi$$
$$878$$ 40.0000 1.34993
$$879$$ −6.00000 −0.202375
$$880$$ −4.00000 −0.134840
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 4.00000 0.134535
$$885$$ −4.00000 −0.134459
$$886$$ 12.0000 0.403148
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −30.0000 −1.00673
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000 0.267860
$$893$$ 32.0000 1.07084
$$894$$ −22.0000 −0.735790
$$895$$ −20.0000 −0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −2.00000 −0.0667409
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ 20.0000 0.666297
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ −10.0000 −0.332411
$$906$$ 8.00000 0.265782
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 48.0000 1.58857
$$914$$ −10.0000 −0.330771
$$915$$ −2.00000 −0.0661180
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 2.00000 0.0660098
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ −18.0000 −0.592798
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −24.0000 −0.788689
$$927$$ 16.0000 0.525509
$$928$$ 10.0000 0.328266
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 24.0000 0.785725
$$934$$ 28.0000 0.916188
$$935$$ −8.00000 −0.261628
$$936$$ 6.00000 0.196116
$$937$$ 54.0000 1.76410 0.882052 0.471153i $$-0.156162\pi$$
0.882052 + 0.471153i $$0.156162\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ −8.00000 −0.260931
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 14.0000 0.456145
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 4.00000 0.129777
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 10.0000 0.323762
$$955$$ −16.0000 −0.517748
$$956$$ 16.0000 0.517477
$$957$$ 8.00000 0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −7.00000 −0.225924
$$961$$ −31.0000 −1.00000
$$962$$ 20.0000 0.644826
$$963$$ −12.0000 −0.386695
$$964$$ −14.0000 −0.450910
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 15.0000 0.482118
$$969$$ 8.00000 0.256997
$$970$$ −2.00000 −0.0642161
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 2.00000 0.0640513
$$976$$ −2.00000 −0.0640184
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ −28.0000 −0.893516
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ −30.0000 −0.956365
$$985$$ −6.00000 −0.191176
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ −4.00000 −0.127128
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 12.0000 0.380808
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 12.0000 0.380235
$$997$$ −54.0000 −1.71020 −0.855099 0.518465i $$-0.826503\pi$$
−0.855099 + 0.518465i $$0.826503\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ −10.0000 −0.316386
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 735.2.a.c.1.1 1
3.2 odd 2 2205.2.a.i.1.1 1
5.4 even 2 3675.2.a.j.1.1 1
7.2 even 3 735.2.i.d.361.1 2
7.3 odd 6 735.2.i.e.226.1 2
7.4 even 3 735.2.i.d.226.1 2
7.5 odd 6 735.2.i.e.361.1 2
7.6 odd 2 15.2.a.a.1.1 1
21.20 even 2 45.2.a.a.1.1 1
28.27 even 2 240.2.a.d.1.1 1
35.13 even 4 75.2.b.b.49.2 2
35.27 even 4 75.2.b.b.49.1 2
35.34 odd 2 75.2.a.b.1.1 1
56.13 odd 2 960.2.a.l.1.1 1
56.27 even 2 960.2.a.a.1.1 1
63.13 odd 6 405.2.e.f.136.1 2
63.20 even 6 405.2.e.c.271.1 2
63.34 odd 6 405.2.e.f.271.1 2
63.41 even 6 405.2.e.c.136.1 2
77.76 even 2 1815.2.a.d.1.1 1
84.83 odd 2 720.2.a.c.1.1 1
91.90 odd 2 2535.2.a.j.1.1 1
105.62 odd 4 225.2.b.b.199.2 2
105.83 odd 4 225.2.b.b.199.1 2
105.104 even 2 225.2.a.b.1.1 1
112.13 odd 4 3840.2.k.m.1921.1 2
112.27 even 4 3840.2.k.r.1921.1 2
112.69 odd 4 3840.2.k.m.1921.2 2
112.83 even 4 3840.2.k.r.1921.2 2
119.118 odd 2 4335.2.a.c.1.1 1
133.132 even 2 5415.2.a.j.1.1 1
140.27 odd 4 1200.2.f.h.49.1 2
140.83 odd 4 1200.2.f.h.49.2 2
140.139 even 2 1200.2.a.e.1.1 1
161.160 even 2 7935.2.a.d.1.1 1
168.83 odd 2 2880.2.a.bc.1.1 1
168.125 even 2 2880.2.a.y.1.1 1
231.230 odd 2 5445.2.a.c.1.1 1
273.272 even 2 7605.2.a.g.1.1 1
280.13 even 4 4800.2.f.bf.3649.2 2
280.27 odd 4 4800.2.f.c.3649.2 2
280.69 odd 2 4800.2.a.t.1.1 1
280.83 odd 4 4800.2.f.c.3649.1 2
280.139 even 2 4800.2.a.bz.1.1 1
280.237 even 4 4800.2.f.bf.3649.1 2
385.384 even 2 9075.2.a.g.1.1 1
420.83 even 4 3600.2.f.e.2449.1 2
420.167 even 4 3600.2.f.e.2449.2 2
420.419 odd 2 3600.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 7.6 odd 2
45.2.a.a.1.1 1 21.20 even 2
75.2.a.b.1.1 1 35.34 odd 2
75.2.b.b.49.1 2 35.27 even 4
75.2.b.b.49.2 2 35.13 even 4
225.2.a.b.1.1 1 105.104 even 2
225.2.b.b.199.1 2 105.83 odd 4
225.2.b.b.199.2 2 105.62 odd 4
240.2.a.d.1.1 1 28.27 even 2
405.2.e.c.136.1 2 63.41 even 6
405.2.e.c.271.1 2 63.20 even 6
405.2.e.f.136.1 2 63.13 odd 6
405.2.e.f.271.1 2 63.34 odd 6
720.2.a.c.1.1 1 84.83 odd 2
735.2.a.c.1.1 1 1.1 even 1 trivial
735.2.i.d.226.1 2 7.4 even 3
735.2.i.d.361.1 2 7.2 even 3
735.2.i.e.226.1 2 7.3 odd 6
735.2.i.e.361.1 2 7.5 odd 6
960.2.a.a.1.1 1 56.27 even 2
960.2.a.l.1.1 1 56.13 odd 2
1200.2.a.e.1.1 1 140.139 even 2
1200.2.f.h.49.1 2 140.27 odd 4
1200.2.f.h.49.2 2 140.83 odd 4
1815.2.a.d.1.1 1 77.76 even 2
2205.2.a.i.1.1 1 3.2 odd 2
2535.2.a.j.1.1 1 91.90 odd 2
2880.2.a.y.1.1 1 168.125 even 2
2880.2.a.bc.1.1 1 168.83 odd 2
3600.2.a.u.1.1 1 420.419 odd 2
3600.2.f.e.2449.1 2 420.83 even 4
3600.2.f.e.2449.2 2 420.167 even 4
3675.2.a.j.1.1 1 5.4 even 2
3840.2.k.m.1921.1 2 112.13 odd 4
3840.2.k.m.1921.2 2 112.69 odd 4
3840.2.k.r.1921.1 2 112.27 even 4
3840.2.k.r.1921.2 2 112.83 even 4
4335.2.a.c.1.1 1 119.118 odd 2
4800.2.a.t.1.1 1 280.69 odd 2
4800.2.a.bz.1.1 1 280.139 even 2
4800.2.f.c.3649.1 2 280.83 odd 4
4800.2.f.c.3649.2 2 280.27 odd 4
4800.2.f.bf.3649.1 2 280.237 even 4
4800.2.f.bf.3649.2 2 280.13 even 4
5415.2.a.j.1.1 1 133.132 even 2
5445.2.a.c.1.1 1 231.230 odd 2
7605.2.a.g.1.1 1 273.272 even 2
7935.2.a.d.1.1 1 161.160 even 2
9075.2.a.g.1.1 1 385.384 even 2