# Properties

 Label 4650.2.d.b.3349.2 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.b.3349.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +12.0000i q^{37} +8.00000i q^{38} +4.00000 q^{39} +10.0000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +4.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000i q^{57} -4.00000i q^{58} -2.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +6.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} -4.00000i q^{73} -12.0000 q^{74} -8.00000 q^{76} -8.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -8.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +8.00000 q^{91} +8.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} +18.0000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 8 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 8 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} + 2 q^{54} + 4 q^{56} - 4 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 12 q^{71} - 24 q^{74} - 16 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 16 q^{86} + 16 q^{91} - 8 q^{94} - 2 q^{96} + 8 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 12.0000i 1.97279i 0.164399 + 0.986394i $$0.447432\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 8.00000i 1.29777i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 4.00000i 0.554700i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 8.00000i 1.05963i
$$58$$ − 4.00000i − 0.525226i
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 6.00000i 0.733017i 0.930415 + 0.366508i $$0.119447\pi$$
−0.930415 + 0.366508i $$0.880553\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ −12.0000 −1.39497
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ − 8.00000i − 0.911685i
$$78$$ 4.00000i 0.452911i
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000i 1.10432i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ − 4.00000i − 0.428845i
$$88$$ 4.00000i 0.426401i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 8.00000i 0.834058i
$$93$$ − 1.00000i − 0.103695i
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 18.0000i 1.82762i 0.406138 + 0.913812i $$0.366875\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ 2.00000i 0.188982i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −8.00000 −0.749269
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ 4.00000i 0.369800i
$$118$$ − 2.00000i − 0.184115i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 10.0000i 0.901670i
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 16.0000i 1.38738i
$$134$$ −6.00000 −0.518321
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 6.00000i 0.503509i
$$143$$ 16.0000i 1.33799i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 3.00000i 0.247436i
$$148$$ − 12.0000i − 0.986394i
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ − 8.00000i − 0.648886i
$$153$$ 2.00000i 0.161690i
$$154$$ 8.00000 0.644658
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 22.0000i 1.75579i 0.478852 + 0.877896i $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 1.00000i 0.0785674i
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ − 8.00000i − 0.609994i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 2.00000i − 0.150329i
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 8.00000i 0.592999i
$$183$$ 10.0000i 0.739221i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 8.00000i 0.585018i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −18.0000 −1.29232
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 10.0000i − 0.712470i −0.934396 0.356235i $$-0.884060\pi$$
0.934396 0.356235i $$-0.115940\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −6.00000 −0.423207
$$202$$ − 18.0000i − 1.26648i
$$203$$ − 8.00000i − 0.561490i
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 8.00000i 0.556038i
$$208$$ − 4.00000i − 0.277350i
$$209$$ −32.0000 −2.21349
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 6.00000i 0.411113i
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 2.00000i − 0.135769i
$$218$$ − 18.0000i − 1.21911i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ − 12.0000i − 0.805387i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ − 8.00000i − 0.529813i
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 4.00000i 0.262613i
$$233$$ − 10.0000i − 0.655122i −0.944830 0.327561i $$-0.893773\pi$$
0.944830 0.327561i $$-0.106227\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 2.00000 0.130189
$$237$$ 8.00000i 0.519656i
$$238$$ 4.00000i 0.259281i
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ − 32.0000i − 2.03611i
$$248$$ 1.00000i 0.0635001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 32.0000i 2.01182i
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ 10.0000i 0.617802i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −16.0000 −0.981023
$$267$$ 0 0
$$268$$ − 6.00000i − 0.366508i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 8.00000i 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ − 8.00000i − 0.480673i −0.970690 0.240337i $$-0.922742\pi$$
0.970690 0.240337i $$-0.0772579\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 2.00000i 0.118888i 0.998232 + 0.0594438i $$0.0189327\pi$$
−0.998232 + 0.0594438i $$0.981067\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ −16.0000 −0.946100
$$287$$ 20.0000i 1.18056i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −18.0000 −1.05518
$$292$$ 4.00000i 0.234082i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 12.0000 0.697486
$$297$$ 4.00000i 0.232104i
$$298$$ 18.0000i 1.04271i
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ − 16.0000i − 0.920697i
$$303$$ − 18.0000i − 1.03407i
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 8.00000i 0.455842i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 20.0000i 1.13047i 0.824931 + 0.565233i $$0.191214\pi$$
−0.824931 + 0.565233i $$0.808786\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 22.0000i 1.23564i 0.786318 + 0.617822i $$0.211985\pi$$
−0.786318 + 0.617822i $$0.788015\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 16.0000 0.895828
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 16.0000i 0.891645i
$$323$$ − 16.0000i − 0.890264i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 6.00000 0.332309
$$327$$ − 18.0000i − 0.995402i
$$328$$ − 10.0000i − 0.552158i
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ − 12.0000i − 0.657596i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 32.0000i 1.74315i 0.490261 + 0.871576i $$0.336901\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ − 8.00000i − 0.432590i
$$343$$ 20.0000i 1.07990i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ − 4.00000i − 0.213201i
$$353$$ 34.0000i 1.80964i 0.425797 + 0.904819i $$0.359994\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$354$$ 2.00000 0.106299
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.00000i 0.211702i
$$358$$ 20.0000i 1.05703i
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 18.0000i 0.946059i
$$363$$ 5.00000i 0.262432i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 1.00000i 0.0518476i
$$373$$ 34.0000i 1.76045i 0.474554 + 0.880227i $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 16.0000i 0.824042i
$$378$$ 2.00000i 0.102869i
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 18.0000i 0.920960i
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ − 8.00000i − 0.406663i
$$388$$ − 18.0000i − 0.913812i
$$389$$ 28.0000 1.41966 0.709828 0.704375i $$-0.248773\pi$$
0.709828 + 0.704375i $$0.248773\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 3.00000i − 0.151523i
$$393$$ 10.0000i 0.504433i
$$394$$ 10.0000 0.503793
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 34.0000i 1.70641i 0.521575 + 0.853206i $$0.325345\pi$$
−0.521575 + 0.853206i $$0.674655\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 4.00000 0.199750 0.0998752 0.995000i $$-0.468156\pi$$
0.0998752 + 0.995000i $$0.468156\pi$$
$$402$$ − 6.00000i − 0.299253i
$$403$$ 4.00000i 0.199254i
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ − 48.0000i − 2.37927i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 14.0000i 0.689730i
$$413$$ − 4.00000i − 0.196827i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ − 4.00000i − 0.195881i
$$418$$ − 32.0000i − 1.56517i
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 24.0000i 1.16830i
$$423$$ − 4.00000i − 0.194487i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 20.0000i 0.967868i
$$428$$ 8.00000i 0.386695i
$$429$$ −16.0000 −0.772487
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 4.00000i 0.192228i 0.995370 + 0.0961139i $$0.0306413\pi$$
−0.995370 + 0.0961139i $$0.969359\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ − 64.0000i − 3.06154i
$$438$$ 4.00000i 0.191127i
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 8.00000i − 0.380521i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 12.0000 0.569495
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 18.0000i 0.851371i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −28.0000 −1.32140 −0.660701 0.750649i $$-0.729741\pi$$
−0.660701 + 0.750649i $$0.729741\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 6.00000i 0.282216i
$$453$$ − 16.0000i − 0.751746i
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 8.00000 0.374634
$$457$$ − 36.0000i − 1.68401i −0.539471 0.842004i $$-0.681376\pi$$
0.539471 0.842004i $$-0.318624\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −40.0000 −1.86299 −0.931493 0.363760i $$-0.881493\pi$$
−0.931493 + 0.363760i $$0.881493\pi$$
$$462$$ 8.00000i 0.372194i
$$463$$ − 36.0000i − 1.67306i −0.547920 0.836531i $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 2.00000i 0.0920575i
$$473$$ − 32.0000i − 1.47136i
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ − 6.00000i − 0.274721i
$$478$$ 4.00000i 0.182956i
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 10.0000i 0.455488i
$$483$$ 16.0000i 0.728025i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 8.00000i 0.360302i
$$494$$ 32.0000 1.43975
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 12.0000i 0.538274i
$$498$$ − 4.00000i − 0.179244i
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 20.0000i 0.892644i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ − 3.00000i − 0.133235i
$$508$$ 4.00000i 0.177471i
$$509$$ −16.0000 −0.709188 −0.354594 0.935020i $$-0.615381\pi$$
−0.354594 + 0.935020i $$0.615381\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ − 16.0000i − 0.703679i
$$518$$ − 24.0000i − 1.05450i
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.00000i 0.0871214i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ − 16.0000i − 0.693688i
$$533$$ − 40.0000i − 1.73259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.00000 0.259161
$$537$$ 20.0000i 0.863064i
$$538$$ 12.0000i 0.517357i
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ 18.0000i 0.772454i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 2.00000i − 0.0855138i −0.999086 0.0427569i $$-0.986386\pi$$
0.999086 0.0427569i $$-0.0136141\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ − 8.00000i − 0.340503i
$$553$$ 16.0000i 0.680389i
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ 32.0000 1.35346
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 6.00000i 0.253095i
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −2.00000 −0.0840663
$$567$$ 2.00000i 0.0839921i
$$568$$ − 6.00000i − 0.251754i
$$569$$ 28.0000 1.17382 0.586911 0.809652i $$-0.300344\pi$$
0.586911 + 0.809652i $$0.300344\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ − 16.0000i − 0.668994i
$$573$$ 18.0000i 0.751961i
$$574$$ −20.0000 −0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 6.00000i 0.249783i 0.992170 + 0.124892i $$0.0398583\pi$$
−0.992170 + 0.124892i $$0.960142\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ − 18.0000i − 0.746124i
$$583$$ − 24.0000i − 0.993978i
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 12.0000i 0.493197i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 32.0000i − 1.30858i
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ − 6.00000i − 0.244339i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ 14.0000i 0.568242i 0.958788 + 0.284121i $$0.0917018\pi$$
−0.958788 + 0.284121i $$0.908298\pi$$
$$608$$ 8.00000i 0.324443i
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 8.00000i 0.323117i 0.986863 + 0.161558i $$0.0516520\pi$$
−0.986863 + 0.161558i $$0.948348\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ − 30.0000i − 1.20289i
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ − 32.0000i − 1.27796i
$$628$$ − 22.0000i − 0.877896i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ 24.0000i 0.953914i
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 12.0000i − 0.475457i
$$638$$ 16.0000i 0.633446i
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −4.00000 −0.157991 −0.0789953 0.996875i $$-0.525171\pi$$
−0.0789953 + 0.996875i $$0.525171\pi$$
$$642$$ 8.00000i 0.315735i
$$643$$ − 8.00000i − 0.315489i −0.987480 0.157745i $$-0.949578\pi$$
0.987480 0.157745i $$-0.0504223\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ 16.0000 0.629512
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 6.00000i 0.234978i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ 18.0000 0.703856
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 4.00000i 0.156055i
$$658$$ − 8.00000i − 0.311872i
$$659$$ 2.00000 0.0779089 0.0389545 0.999241i $$-0.487597\pi$$
0.0389545 + 0.999241i $$0.487597\pi$$
$$660$$ 0 0
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ − 12.0000i − 0.466393i
$$663$$ − 8.00000i − 0.310694i
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 12.0000 0.464991
$$667$$ 32.0000i 1.23904i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ − 2.00000i − 0.0771517i
$$673$$ 8.00000i 0.308377i 0.988041 + 0.154189i $$0.0492764\pi$$
−0.988041 + 0.154189i $$0.950724\pi$$
$$674$$ −32.0000 −1.23259
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −36.0000 −1.38155
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 4.00000i 0.153168i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ − 22.0000i − 0.839352i
$$688$$ 8.00000i 0.304997i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ − 22.0000i − 0.836315i
$$693$$ 8.00000i 0.303895i
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ − 20.0000i − 0.757554i
$$698$$ 30.0000i 1.13552i
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 96.0000i 3.62071i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −34.0000 −1.27961
$$707$$ − 36.0000i − 1.35392i
$$708$$ 2.00000i 0.0751646i
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 4.00000i 0.149383i
$$718$$ − 10.0000i − 0.373197i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 45.0000i 1.67473i
$$723$$ 10.0000i 0.371904i
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ − 8.00000i − 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ − 10.0000i − 0.369611i
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ − 24.0000i − 0.884051i
$$738$$ − 10.0000i − 0.368105i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 32.0000 1.17555
$$742$$ − 12.0000i − 0.440534i
$$743$$ − 16.0000i − 0.586983i −0.955962 0.293492i $$-0.905183\pi$$
0.955962 0.293492i $$-0.0948173\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ − 4.00000i − 0.146352i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 16.0000 0.584627
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 20.0000i 0.728841i
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ − 36.0000i − 1.30844i −0.756303 0.654221i $$-0.772997\pi$$
0.756303 0.654221i $$-0.227003\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ −20.0000 −0.724999 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ − 36.0000i − 1.30329i
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ 8.00000i 0.288863i
$$768$$ 1.00000i 0.0360844i
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 2.00000i 0.0719816i
$$773$$ 14.0000i 0.503545i 0.967786 + 0.251773i $$0.0810135\pi$$
−0.967786 + 0.251773i $$0.918987\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 18.0000 0.646162
$$777$$ − 24.0000i − 0.860995i
$$778$$ 28.0000i 1.00385i
$$779$$ 80.0000 2.86630
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ − 16.0000i − 0.572159i
$$783$$ 4.00000i 0.142948i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −10.0000 −0.356688
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 10.0000i 0.356235i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ − 4.00000i − 0.142134i
$$793$$ − 40.0000i − 1.42044i
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4.00000i 0.141245i
$$803$$ 16.0000i 0.564628i
$$804$$ 6.00000 0.211604
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 12.0000i 0.422420i
$$808$$ 18.0000i 0.633238i
$$809$$ −28.0000 −0.984428 −0.492214 0.870474i $$-0.663812\pi$$
−0.492214 + 0.870474i $$0.663812\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 8.00000i 0.280745i
$$813$$ 8.00000i 0.280572i
$$814$$ 48.0000 1.68240
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ 64.0000i 2.23908i
$$818$$ − 6.00000i − 0.209785i
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ −4.00000 −0.139601 −0.0698005 0.997561i $$-0.522236\pi$$
−0.0698005 + 0.997561i $$0.522236\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 28.0000i 0.976019i 0.872838 + 0.488009i $$0.162277\pi$$
−0.872838 + 0.488009i $$0.837723\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ − 8.00000i − 0.278019i
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 4.00000i 0.138675i
$$833$$ − 6.00000i − 0.207888i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 32.0000 1.10674
$$837$$ 1.00000i 0.0345651i
$$838$$ 10.0000i 0.345444i
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 2.00000i 0.0689246i
$$843$$ 6.00000i 0.206651i
$$844$$ −24.0000 −0.826114
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ 10.0000i 0.343604i
$$848$$ 6.00000i 0.206041i
$$849$$ −2.00000 −0.0686398
$$850$$ 0 0
$$851$$ 96.0000 3.29084
$$852$$ − 6.00000i − 0.205557i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ − 16.0000i − 0.546231i
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −20.0000 −0.681598
$$862$$ − 30.0000i − 1.02180i
$$863$$ 16.0000i 0.544646i 0.962206 + 0.272323i $$0.0877920\pi$$
−0.962206 + 0.272323i $$0.912208\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −4.00000 −0.135926
$$867$$ 13.0000i 0.441503i
$$868$$ 2.00000i 0.0678844i
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 18.0000i 0.609557i
$$873$$ − 18.0000i − 0.609208i
$$874$$ 64.0000 2.16483
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 56.0000i 1.88030i 0.340766 + 0.940148i $$0.389313\pi$$
−0.340766 + 0.940148i $$0.610687\pi$$
$$888$$ 12.0000i 0.402694i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000i 0.267860i
$$893$$ 32.0000i 1.07084i
$$894$$ −18.0000 −0.602010
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ − 32.0000i − 1.06845i
$$898$$ − 28.0000i − 0.934372i
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 40.0000i − 1.33185i
$$903$$ − 16.0000i − 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 10.0000i 0.332045i 0.986122 + 0.166022i $$0.0530924\pi$$
−0.986122 + 0.166022i $$0.946908\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ − 16.0000i − 0.529523i
$$914$$ 36.0000 1.19077
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ 20.0000i 0.660458i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ −2.00000 −0.0659022
$$922$$ − 40.0000i − 1.31733i
$$923$$ − 24.0000i − 0.789970i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 14.0000i 0.459820i
$$928$$ − 4.00000i − 0.131306i
$$929$$ −32.0000 −1.04989 −0.524943 0.851137i $$-0.675913\pi$$
−0.524943 + 0.851137i $$0.675913\pi$$
$$930$$ 0 0
$$931$$ 24.0000 0.786568
$$932$$ 10.0000i 0.327561i
$$933$$ − 30.0000i − 0.982156i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ − 12.0000i − 0.391814i
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ − 22.0000i − 0.716799i
$$943$$ − 80.0000i − 2.60516i
$$944$$ −2.00000 −0.0650945
$$945$$ 0 0
$$946$$ 32.0000 1.04041
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ − 4.00000i − 0.129641i
$$953$$ 22.0000i 0.712650i 0.934362 + 0.356325i $$0.115970\pi$$
−0.934362 + 0.356325i $$0.884030\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −4.00000 −0.129369
$$957$$ 16.0000i 0.517207i
$$958$$ 10.0000i 0.323085i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 48.0000i 1.54758i
$$963$$ 8.00000i 0.257796i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ 16.0000 0.513994
$$970$$ 0 0
$$971$$ −2.00000 −0.0641831 −0.0320915 0.999485i $$-0.510217\pi$$
−0.0320915 + 0.999485i $$0.510217\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 8.00000i − 0.256468i
$$974$$ −12.0000 −0.384505
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 46.0000i 1.47167i 0.677161 + 0.735835i $$0.263210\pi$$
−0.677161 + 0.735835i $$0.736790\pi$$
$$978$$ 6.00000i 0.191859i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ − 8.00000i − 0.255290i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ − 8.00000i − 0.254643i
$$988$$ 32.0000i 1.01806i
$$989$$ 64.0000 2.03508
$$990$$ 0 0
$$991$$ −56.0000 −1.77890 −0.889449 0.457034i $$-0.848912\pi$$
−0.889449 + 0.457034i $$0.848912\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 12.0000i − 0.380808i
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 18.0000i − 0.570066i −0.958518 0.285033i $$-0.907995\pi$$
0.958518 0.285033i $$-0.0920045\pi$$
$$998$$ 36.0000i 1.13956i
$$999$$ 12.0000 0.379663
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 4650.2.d.b.3349.2 2
5.2 odd 4 930.2.a.h.1.1 1
5.3 odd 4 4650.2.a.bg.1.1 1
5.4 even 2 inner 4650.2.d.b.3349.1 2
15.2 even 4 2790.2.a.p.1.1 1
20.7 even 4 7440.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 1 5.2 odd 4
2790.2.a.p.1.1 1 15.2 even 4
4650.2.a.bg.1.1 1 5.3 odd 4
4650.2.d.b.3349.1 2 5.4 even 2 inner
4650.2.d.b.3349.2 2 1.1 even 1 trivial
7440.2.a.m.1.1 1 20.7 even 4