# Properties

 Label 4650.2.d.u.3349.2 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ 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.u.3349.1

## $q$-expansion

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

## 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
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 8.00000i − 1.29777i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 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
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 1.00000i 0.127000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 0 0
$$78$$ 4.00000i 0.452911i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ − 1.00000i − 0.103695i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ 10.0000i 0.985329i 0.870219 + 0.492665i $$0.163977\pi$$
−0.870219 + 0.492665i $$0.836023\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 2.00000i 0.188982i
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ −8.00000 −0.749269
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 6.00000i 0.552345i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 2.00000i 0.181071i
$$123$$ 6.00000i 0.541002i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ − 16.0000i − 1.38738i
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ − 6.00000i − 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ − 3.00000i − 0.247436i
$$148$$ 4.00000i 0.328798i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 8.00000i 0.648886i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 2.00000i − 0.156652i −0.996928 0.0783260i $$-0.975042\pi$$
0.996928 0.0783260i $$-0.0249575\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 8.00000i 0.609994i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 6.00000i − 0.450988i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ − 2.00000i − 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 2.00000 0.141069
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ 0 0
$$208$$ 4.00000i 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 6.00000i 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 2.00000i 0.135769i
$$218$$ − 2.00000i − 0.135457i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ − 4.00000i − 0.268462i
$$223$$ 28.0000i 1.87502i 0.347960 + 0.937509i $$0.386874\pi$$
−0.347960 + 0.937509i $$0.613126\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 8.00000i − 0.529813i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 8.00000i 0.519656i
$$238$$ − 12.0000i − 0.777844i
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ − 32.0000i − 2.03611i
$$248$$ − 1.00000i − 0.0635001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 30.0000i − 1.87135i −0.352865 0.935674i $$-0.614792\pi$$
0.352865 0.935674i $$-0.385208\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 6.00000i − 0.370681i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 16.0000 0.981023
$$267$$ 0 0
$$268$$ − 2.00000i − 0.122169i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 8.00000i 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 12.0000i − 0.708338i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 8.00000i 0.468165i
$$293$$ − 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
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ − 18.0000i − 1.04271i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 8.00000i 0.460348i
$$303$$ 6.00000i 0.344691i
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 34.0000i − 1.94048i −0.242140 0.970241i $$-0.577849\pi$$
0.242140 0.970241i $$-0.422151\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 16.0000i 0.904373i 0.891923 + 0.452187i $$0.149356\pi$$
−0.891923 + 0.452187i $$0.850644\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 48.0000i − 2.67079i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 2.00000 0.110770
$$327$$ 2.00000i 0.110600i
$$328$$ 6.00000i 0.331295i
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 4.00000i 0.219199i
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 20.0000i 1.08947i 0.838608 + 0.544735i $$0.183370\pi$$
−0.838608 + 0.544735i $$0.816630\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 8.00000i 0.432590i
$$343$$ 20.0000i 1.07990i
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ − 36.0000i − 1.93258i −0.257454 0.966291i $$-0.582883\pi$$
0.257454 0.966291i $$-0.417117\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 12.0000i 0.635107i
$$358$$ 0 0
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 2.00000i 0.105118i
$$363$$ 11.0000i 0.577350i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 20.0000i 1.04399i 0.852948 + 0.521996i $$0.174812\pi$$
−0.852948 + 0.521996i $$0.825188\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 1.00000i 0.0518476i
$$373$$ − 38.0000i − 1.96757i −0.179364 0.983783i $$-0.557404\pi$$
0.179364 0.983783i $$-0.442596\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ − 2.00000i − 0.102869i
$$379$$ −32.0000 −1.64373 −0.821865 0.569683i $$-0.807066\pi$$
−0.821865 + 0.569683i $$0.807066\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ − 18.0000i − 0.920960i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 8.00000i 0.406663i
$$388$$ 10.0000i 0.507673i
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ 6.00000i 0.302660i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 2.00000i 0.0997509i
$$403$$ 4.00000i 0.199254i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 10.0000i − 0.492665i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ − 4.00000i − 0.195881i
$$418$$ 0 0
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ − 8.00000i − 0.382255i
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 24.0000i − 1.14156i
$$443$$ 36.0000i 1.71041i 0.518289 + 0.855206i $$0.326569\pi$$
−0.518289 + 0.855206i $$0.673431\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −28.0000 −1.32584
$$447$$ 18.0000i 0.851371i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 18.0000i − 0.846649i
$$453$$ − 8.00000i − 0.375873i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 8.00000 0.374634
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ − 14.0000i − 0.654177i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −36.0000 −1.67669 −0.838344 0.545142i $$-0.816476\pi$$
−0.838344 + 0.545142i $$0.816476\pi$$
$$462$$ 0 0
$$463$$ 16.0000i 0.743583i 0.928316 + 0.371792i $$0.121256\pi$$
−0.928316 + 0.371792i $$0.878744\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 24.0000i 1.11059i 0.831654 + 0.555294i $$0.187394\pi$$
−0.831654 + 0.555294i $$0.812606\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ − 6.00000i − 0.276172i
$$473$$ 0 0
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ − 6.00000i − 0.274721i
$$478$$ − 12.0000i − 0.548867i
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 40.0000i − 1.81257i −0.422664 0.906287i $$-0.638905\pi$$
0.422664 0.906287i $$-0.361095\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ −2.00000 −0.0904431
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 0 0
$$494$$ 32.0000 1.43975
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ − 12.0000i − 0.538274i
$$498$$ − 12.0000i − 0.537733i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.00000i 0.133235i
$$508$$ 16.0000i 0.709885i
$$509$$ −12.0000 −0.531891 −0.265945 0.963988i $$-0.585684\pi$$
−0.265945 + 0.963988i $$0.585684\pi$$
$$510$$ 0 0
$$511$$ 16.0000 0.707798
$$512$$ 1.00000i 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 8.00000i 0.351500i
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 40.0000i 1.74908i 0.484955 + 0.874539i $$0.338836\pi$$
−0.484955 + 0.874539i $$0.661164\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.00000i 0.261364i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 16.0000i 0.693688i
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 32.0000i 1.37452i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 2.00000i 0.0855138i 0.999086 + 0.0427569i $$0.0136141\pi$$
−0.999086 + 0.0427569i $$0.986386\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 16.0000i − 0.680389i
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 42.0000i 1.77960i 0.456354 + 0.889799i $$0.349155\pi$$
−0.456354 + 0.889799i $$0.650845\pi$$
$$558$$ − 1.00000i − 0.0423334i
$$559$$ 32.0000 1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 18.0000i − 0.759284i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ 2.00000i 0.0839921i
$$568$$ 6.00000i 0.251754i
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ 18.0000i 0.751961i
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 22.0000i − 0.915872i −0.888985 0.457936i $$-0.848589\pi$$
0.888985 0.457936i $$-0.151411\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ − 10.0000i − 0.414513i
$$583$$ 0 0
$$584$$ −8.00000 −0.331042
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ − 4.00000i − 0.164399i
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ − 34.0000i − 1.38002i −0.723801 0.690009i $$-0.757607\pi$$
0.723801 0.690009i $$-0.242393\pi$$
$$608$$ − 8.00000i − 0.324443i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 6.00000i 0.242536i
$$613$$ 40.0000i 1.61558i 0.589467 + 0.807792i $$0.299338\pi$$
−0.589467 + 0.807792i $$0.700662\pi$$
$$614$$ 34.0000 1.37213
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 10.0000i 0.402259i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 18.0000i − 0.721734i
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ 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
$$633$$ − 8.00000i − 0.317971i
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 12.0000i 0.475457i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 48.0000 1.88853
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 2.00000i 0.0783260i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 8.00000i 0.312110i
$$658$$ 24.0000i 0.935617i
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ − 4.00000i − 0.155464i
$$663$$ 24.0000i 0.932083i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 0 0
$$668$$ 24.0000i 0.928588i
$$669$$ 28.0000 1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000i 0.0771517i
$$673$$ − 44.0000i − 1.69608i −0.529936 0.848038i $$-0.677784\pi$$
0.529936 0.848038i $$-0.322216\pi$$
$$674$$ −20.0000 −0.770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 14.0000i 0.534133i
$$688$$ − 8.00000i − 0.304997i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 32.0000i 1.20690i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ − 12.0000i − 0.451306i
$$708$$ 6.00000i 0.225494i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ −12.0000 −0.449089
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000i 0.448148i
$$718$$ − 30.0000i − 1.11959i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 45.0000i 1.67473i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ 26.0000i 0.964287i 0.876092 + 0.482143i $$0.160142\pi$$
−0.876092 + 0.482143i $$0.839858\pi$$
$$728$$ 8.00000i 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 2.00000i 0.0739221i
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ −20.0000 −0.738213
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000i 0.220863i
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ 0 0
$$741$$ −32.0000 −1.17555
$$742$$ − 12.0000i − 0.440534i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 38.0000 1.39128
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ − 4.00000i − 0.145382i −0.997354 0.0726912i $$-0.976841\pi$$
0.997354 0.0726912i $$-0.0231588\pi$$
$$758$$ − 32.0000i − 1.16229i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ − 4.00000i − 0.144810i
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000i 0.866590i
$$768$$ − 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
$$771$$ −30.0000 −1.08042
$$772$$ 14.0000i 0.503871i
$$773$$ 54.0000i 1.94225i 0.238581 + 0.971123i $$0.423318\pi$$
−0.238581 + 0.971123i $$0.576682\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ − 8.00000i − 0.286998i
$$778$$ − 24.0000i − 0.860442i
$$779$$ 48.0000 1.71978
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ 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
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ 8.00000i 0.284088i
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ − 16.0000i − 0.566394i
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 36.0000i 1.27120i
$$803$$ 0 0
$$804$$ −2.00000 −0.0705346
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 6.00000i 0.211079i
$$809$$ −36.0000 −1.26569 −0.632846 0.774277i $$-0.718114\pi$$
−0.632846 + 0.774277i $$0.718114\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 64.0000i 2.23908i
$$818$$ − 14.0000i − 0.489499i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ 16.0000i 0.557725i 0.960331 + 0.278862i $$0.0899574\pi$$
−0.960331 + 0.278862i $$0.910043\pi$$
$$824$$ 10.0000 0.348367
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ 22.0000 0.764092 0.382046 0.924143i $$-0.375220\pi$$
0.382046 + 0.924143i $$0.375220\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ − 4.00000i − 0.138675i
$$833$$ 18.0000i 0.623663i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.00000i 0.0345651i
$$838$$ − 30.0000i − 1.03633i
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 26.0000i 0.896019i
$$843$$ 18.0000i 0.619953i
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ − 22.0000i − 0.755929i
$$848$$ 6.00000i 0.206041i
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ 0 0
$$852$$ − 6.00000i − 0.205557i
$$853$$ − 2.00000i − 0.0684787i −0.999414 0.0342393i $$-0.989099\pi$$
0.999414 0.0342393i $$-0.0109009\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 54.0000i 1.84460i 0.386469 + 0.922302i $$0.373695\pi$$
−0.386469 + 0.922302i $$0.626305\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ 6.00000i 0.204361i
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 19.0000i 0.645274i
$$868$$ − 2.00000i − 0.0678844i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 2.00000i 0.0677285i
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ −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
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ 4.00000i 0.134231i
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 28.0000i − 0.937509i
$$893$$ 96.0000i 3.21252i
$$894$$ −18.0000 −0.602010
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ − 12.0000i − 0.400445i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ − 10.0000i − 0.332045i −0.986122 0.166022i $$-0.946908\pi$$
0.986122 0.166022i $$-0.0530924\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 6.00000 0.199007
$$910$$ 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
$$913$$ 0 0
$$914$$ −8.00000 −0.264616
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ − 12.0000i − 0.396275i
$$918$$ − 6.00000i − 0.198030i
$$919$$ 52.0000 1.71532 0.857661 0.514216i $$-0.171917\pi$$
0.857661 + 0.514216i $$0.171917\pi$$
$$920$$ 0 0
$$921$$ −34.0000 −1.12034
$$922$$ − 36.0000i − 1.18560i
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ − 10.0000i − 0.328443i
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 18.0000i 0.589610i
$$933$$ 18.0000i 0.589294i
$$934$$ −24.0000 −0.785304
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ 16.0000 0.522140
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ 14.0000i 0.456145i
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ 6.00000 0.194563
$$952$$ 12.0000i 0.388922i
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 6.00000i 0.193851i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 16.0000i 0.515861i
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 4.00000i − 0.128631i −0.997930 0.0643157i $$-0.979514\pi$$
0.997930 0.0643157i $$-0.0204865\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −48.0000 −1.54198
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 8.00000i 0.256468i
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ − 2.00000i − 0.0639529i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 12.0000i 0.382935i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ 32.0000i 1.01806i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 1.00000i 0.0317500i
$$993$$ 4.00000i 0.126936i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 4.00000 0.126554
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.u.3349.2 2
5.2 odd 4 4650.2.a.d.1.1 1
5.3 odd 4 930.2.a.n.1.1 1
5.4 even 2 inner 4650.2.d.u.3349.1 2
15.8 even 4 2790.2.a.k.1.1 1
20.3 even 4 7440.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.n.1.1 1 5.3 odd 4
2790.2.a.k.1.1 1 15.8 even 4
4650.2.a.d.1.1 1 5.2 odd 4
4650.2.d.u.3349.1 2 5.4 even 2 inner
4650.2.d.u.3349.2 2 1.1 even 1 trivial
7440.2.a.b.1.1 1 20.3 even 4