# Properties

 Label 960.2.f.d.769.1 Level $960$ Weight $2$ Character 960.769 Analytic conductor $7.666$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 769.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 960.769 Dual form 960.2.f.d.769.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +4.00000i q^{13} +(-2.00000 + 1.00000i) q^{15} +8.00000 q^{19} +4.00000 q^{21} +4.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +(8.00000 - 4.00000i) q^{35} +4.00000i q^{37} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} -12.0000i q^{53} -8.00000i q^{57} +6.00000 q^{61} -4.00000i q^{63} +(8.00000 - 4.00000i) q^{65} +12.0000i q^{67} +4.00000 q^{69} +16.0000 q^{71} +(4.00000 + 3.00000i) q^{75} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} +10.0000 q^{89} -16.0000 q^{91} -8.00000i q^{93} +(-8.00000 - 16.0000i) q^{95} +8.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} - 4 q^{15} + 16 q^{19} + 8 q^{21} - 6 q^{25} - 12 q^{29} + 16 q^{31} + 16 q^{35} + 8 q^{39} + 12 q^{41} + 2 q^{45} - 18 q^{49} + 12 q^{61} + 16 q^{65} + 8 q^{69} + 32 q^{71} + 8 q^{75} - 16 q^{79} + 2 q^{81} + 20 q^{89} - 32 q^{91} - 16 q^{95}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 - 4 * q^15 + 16 * q^19 + 8 * q^21 - 6 * q^25 - 12 * q^29 + 16 * q^31 + 16 * q^35 + 8 * q^39 + 12 * q^41 + 2 * q^45 - 18 * q^49 + 12 * q^61 + 16 * q^65 + 8 * q^69 + 32 * q^71 + 8 * q^75 - 16 * q^79 + 2 * q^81 + 20 * q^89 - 32 * q^91 - 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ −1.00000 2.00000i −0.447214 0.894427i
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ −2.00000 + 1.00000i −0.516398 + 0.258199i
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ −3.00000 + 4.00000i −0.600000 + 0.800000i
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 8.00000 4.00000i 1.35225 0.676123i
$$36$$ 0 0
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 1.00000 + 2.00000i 0.149071 + 0.298142i
$$46$$ 0 0
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.0000i 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 8.00000i 1.05963i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 4.00000i 0.503953i
$$64$$ 0 0
$$65$$ 8.00000 4.00000i 0.992278 0.496139i
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 4.00000 + 3.00000i 0.461880 + 0.346410i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −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
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −16.0000 −1.67726
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ −8.00000 16.0000i −0.820783 1.64157i
$$96$$ 0 0
$$97$$ 8.00000i 0.812277i 0.913812 + 0.406138i $$0.133125\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 12.0000i 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 0 0
$$105$$ −4.00000 8.00000i −0.390360 0.780720i
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 8.00000i 0.752577i 0.926503 + 0.376288i $$0.122800\pi$$
−0.926503 + 0.376288i $$0.877200\pi$$
$$114$$ 0 0
$$115$$ 8.00000 4.00000i 0.746004 0.373002i
$$116$$ 0 0
$$117$$ 4.00000i 0.369800i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 11.0000 + 2.00000i 0.983870 + 0.178885i
$$126$$ 0 0
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 32.0000i 2.77475i
$$134$$ 0 0
$$135$$ 2.00000 1.00000i 0.172133 0.0860663i
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 + 12.0000i 0.498273 + 0.996546i
$$146$$ 0 0
$$147$$ 9.00000i 0.742307i
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 16.0000i −0.642575 1.28515i
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ 0 0
$$173$$ 4.00000i 0.304114i 0.988372 + 0.152057i $$0.0485898\pi$$
−0.988372 + 0.152057i $$0.951410\pi$$
$$174$$ 0 0
$$175$$ −16.0000 12.0000i −1.20949 0.907115i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 6.00000i 0.443533i
$$184$$ 0 0
$$185$$ 8.00000 4.00000i 0.588172 0.294086i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 24.0000i 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ 0 0
$$195$$ −4.00000 8.00000i −0.286446 0.572892i
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ 24.0000i 1.68447i
$$204$$ 0 0
$$205$$ −6.00000 12.0000i −0.419058 0.838116i
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ 16.0000i 1.09630i
$$214$$ 0 0
$$215$$ −8.00000 + 4.00000i −0.545595 + 0.272798i
$$216$$ 0 0
$$217$$ 32.0000i 2.17230i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 0 0
$$225$$ 3.00000 4.00000i 0.200000 0.266667i
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.00000i 0.524097i −0.965055 0.262049i $$-0.915602\pi$$
0.965055 0.262049i $$-0.0843981\pi$$
$$234$$ 0 0
$$235$$ 8.00000 4.00000i 0.521862 0.260931i
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 9.00000 + 18.0000i 0.574989 + 1.14998i
$$246$$ 0 0
$$247$$ 32.0000i 2.03611i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 24.0000i 1.49708i −0.663090 0.748539i $$-0.730755\pi$$
0.663090 0.748539i $$-0.269245\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 12.0000i 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ −24.0000 + 12.0000i −1.47431 + 0.737154i
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 16.0000i 0.968364i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4.00000i 0.240337i 0.992754 + 0.120168i $$0.0383434\pi$$
−0.992754 + 0.120168i $$0.961657\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 20.0000i 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 0 0
$$285$$ −16.0000 + 8.00000i −0.947758 + 0.473879i
$$286$$ 0 0
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 0 0
$$293$$ 20.0000i 1.16841i −0.811605 0.584206i $$-0.801406\pi$$
0.811605 0.584206i $$-0.198594\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 0 0
$$303$$ 14.0000i 0.804279i
$$304$$ 0 0
$$305$$ −6.00000 12.0000i −0.343559 0.687118i
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 8.00000i 0.452187i 0.974106 + 0.226093i $$0.0725954\pi$$
−0.974106 + 0.226093i $$0.927405\pi$$
$$314$$ 0 0
$$315$$ −8.00000 + 4.00000i −0.450749 + 0.225374i
$$316$$ 0 0
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −16.0000 12.0000i −0.887520 0.665640i
$$326$$ 0 0
$$327$$ 10.0000i 0.553001i
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 0 0
$$333$$ 4.00000i 0.219199i
$$334$$ 0 0
$$335$$ 24.0000 12.0000i 1.31126 0.655630i
$$336$$ 0 0
$$337$$ 16.0000i 0.871576i −0.900049 0.435788i $$-0.856470\pi$$
0.900049 0.435788i $$-0.143530\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 0 0
$$345$$ −4.00000 8.00000i −0.215353 0.430706i
$$346$$ 0 0
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ 24.0000i 1.27739i 0.769460 + 0.638696i $$0.220526\pi$$
−0.769460 + 0.638696i $$0.779474\pi$$
$$354$$ 0 0
$$355$$ −16.0000 32.0000i −0.849192 1.69838i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 11.0000i 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.0000i 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 48.0000 2.49204
$$372$$ 0 0
$$373$$ 20.0000i 1.03556i 0.855514 + 0.517780i $$0.173242\pi$$
−0.855514 + 0.517780i $$0.826758\pi$$
$$374$$ 0 0
$$375$$ 2.00000 11.0000i 0.103280 0.568038i
$$376$$ 0 0
$$377$$ 24.0000i 1.23606i
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000i 0.203331i
$$388$$ 0 0
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 16.0000i 0.807093i
$$394$$ 0 0
$$395$$ 8.00000 + 16.0000i 0.402524 + 0.805047i
$$396$$ 0 0
$$397$$ 28.0000i 1.40528i −0.711546 0.702640i $$-0.752005\pi$$
0.711546 0.702640i $$-0.247995\pi$$
$$398$$ 0 0
$$399$$ 32.0000 1.60200
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 32.0000i 1.59403i
$$404$$ 0 0
$$405$$ −1.00000 2.00000i −0.0496904 0.0993808i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 24.0000 12.0000i 1.17811 0.589057i
$$416$$ 0 0
$$417$$ 8.00000i 0.391762i
$$418$$ 0 0
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 0 0
$$423$$ 4.00000i 0.194487i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 24.0000i 1.16144i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 40.0000i 1.92228i −0.276066 0.961139i $$-0.589031\pi$$
0.276066 0.961139i $$-0.410969\pi$$
$$434$$ 0 0
$$435$$ 12.0000 6.00000i 0.575356 0.287678i
$$436$$ 0 0
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 20.0000i 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 0 0
$$445$$ −10.0000 20.0000i −0.474045 0.948091i
$$446$$ 0 0
$$447$$ 14.0000i 0.662177i
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 16.0000 + 32.0000i 0.750092 + 1.50018i
$$456$$ 0 0
$$457$$ 8.00000i 0.374224i −0.982339 0.187112i $$-0.940087\pi$$
0.982339 0.187112i $$-0.0599128\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −22.0000 −1.02464 −0.512321 0.858794i $$-0.671214\pi$$
−0.512321 + 0.858794i $$0.671214\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ −16.0000 + 8.00000i −0.741982 + 0.370991i
$$466$$ 0 0
$$467$$ 4.00000i 0.185098i −0.995708 0.0925490i $$-0.970499\pi$$
0.995708 0.0925490i $$-0.0295015\pi$$
$$468$$ 0 0
$$469$$ −48.0000 −2.21643
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −24.0000 + 32.0000i −1.10120 + 1.46826i
$$476$$ 0 0
$$477$$ 12.0000i 0.549442i
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ 16.0000i 0.728025i
$$484$$ 0 0
$$485$$ 16.0000 8.00000i 0.726523 0.363261i
$$486$$ 0 0
$$487$$ 4.00000i 0.181257i 0.995885 + 0.0906287i $$0.0288876\pi$$
−0.995885 + 0.0906287i $$0.971112\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 32.0000 1.44414 0.722070 0.691820i $$-0.243191\pi$$
0.722070 + 0.691820i $$0.243191\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 64.0000i 2.87079i
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 0 0
$$501$$ 4.00000 0.178707
$$502$$ 0 0
$$503$$ 28.0000i 1.24846i −0.781241 0.624229i $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ 0 0
$$505$$ 14.0000 + 28.0000i 0.622992 + 1.24598i
$$506$$ 0 0
$$507$$ 3.00000i 0.133235i
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 8.00000i 0.353209i
$$514$$ 0 0
$$515$$ −24.0000 + 12.0000i −1.05757 + 0.528783i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ 0 0
$$525$$ −12.0000 + 16.0000i −0.523723 + 0.698297i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 24.0000i 1.03956i
$$534$$ 0 0
$$535$$ 24.0000 12.0000i 1.03761 0.518805i
$$536$$ 0 0
$$537$$ 16.0000i 0.690451i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 10.0000 0.429934 0.214967 0.976621i $$-0.431036\pi$$
0.214967 + 0.976621i $$0.431036\pi$$
$$542$$ 0 0
$$543$$ 14.0000i 0.600798i
$$544$$ 0 0
$$545$$ 10.0000 + 20.0000i 0.428353 + 0.856706i
$$546$$ 0 0
$$547$$ 36.0000i 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −48.0000 −2.04487
$$552$$ 0 0
$$553$$ 32.0000i 1.36078i
$$554$$ 0 0
$$555$$ −4.00000 8.00000i −0.169791 0.339581i
$$556$$ 0 0
$$557$$ 4.00000i 0.169485i 0.996403 + 0.0847427i $$0.0270068\pi$$
−0.996403 + 0.0847427i $$0.972993\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 28.0000i 1.18006i 0.807382 + 0.590030i $$0.200884\pi$$
−0.807382 + 0.590030i $$0.799116\pi$$
$$564$$ 0 0
$$565$$ 16.0000 8.00000i 0.673125 0.336563i
$$566$$ 0 0
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ −16.0000 12.0000i −0.667246 0.500435i
$$576$$ 0 0
$$577$$ 16.0000i 0.666089i −0.942911 0.333044i $$-0.891924\pi$$
0.942911 0.333044i $$-0.108076\pi$$
$$578$$ 0 0
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ −48.0000 −1.99138
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −8.00000 + 4.00000i −0.330759 + 0.165380i
$$586$$ 0 0
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ 0 0
$$589$$ 64.0000 2.63707
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ 8.00000i 0.328521i 0.986417 + 0.164260i $$0.0525237\pi$$
−0.986417 + 0.164260i $$0.947476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 11.0000 + 22.0000i 0.447214 + 0.894427i
$$606$$ 0 0
$$607$$ 20.0000i 0.811775i 0.913923 + 0.405887i $$0.133038\pi$$
−0.913923 + 0.405887i $$0.866962\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 0 0
$$613$$ 20.0000i 0.807792i −0.914805 0.403896i $$-0.867656\pi$$
0.914805 0.403896i $$-0.132344\pi$$
$$614$$ 0 0
$$615$$ −12.0000 + 6.00000i −0.483887 + 0.241943i
$$616$$ 0 0
$$617$$ 8.00000i 0.322068i −0.986949 0.161034i $$-0.948517\pi$$
0.986949 0.161034i $$-0.0514829\pi$$
$$618$$ 0 0
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 40.0000i 1.60257i
$$624$$ 0 0
$$625$$ −7.00000 24.0000i −0.280000 0.960000i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 8.00000i 0.317971i
$$634$$ 0 0
$$635$$ 8.00000 4.00000i 0.317470 0.158735i
$$636$$ 0 0
$$637$$ 36.0000i 1.42637i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ −34.0000 −1.34292 −0.671460 0.741041i $$-0.734332\pi$$
−0.671460 + 0.741041i $$0.734332\pi$$
$$642$$ 0 0
$$643$$ 12.0000i 0.473234i 0.971603 + 0.236617i $$0.0760386\pi$$
−0.971603 + 0.236617i $$0.923961\pi$$
$$644$$ 0 0
$$645$$ 4.00000 + 8.00000i 0.157500 + 0.315000i
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 32.0000 1.25418
$$652$$ 0 0
$$653$$ 20.0000i 0.782660i −0.920250 0.391330i $$-0.872015\pi$$
0.920250 0.391330i $$-0.127985\pi$$
$$654$$ 0 0
$$655$$ 16.0000 + 32.0000i 0.625172 + 1.25034i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 64.0000 32.0000i 2.48181 1.24091i
$$666$$ 0 0
$$667$$ 24.0000i 0.929284i
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 8.00000i 0.308377i −0.988041 0.154189i $$-0.950724\pi$$
0.988041 0.154189i $$-0.0492764\pi$$
$$674$$ 0 0
$$675$$ −4.00000 3.00000i −0.153960 0.115470i
$$676$$ 0 0
$$677$$ 12.0000i 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ 0 0
$$679$$ −32.0000 −1.22805
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 2.00000i 0.0763048i
$$688$$ 0 0
$$689$$ 48.0000 1.82865
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.00000 + 16.0000i 0.303457 + 0.606915i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −8.00000 −0.302588
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 32.0000i 1.20690i
$$704$$ 0 0
$$705$$ −4.00000 8.00000i −0.150649 0.301297i
$$706$$ 0 0
$$707$$ 56.0000i 2.10610i
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 32.0000i 1.19841i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 16.0000i 0.597531i
$$718$$ 0 0
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 48.0000 1.78761
$$722$$ 0 0
$$723$$ 14.0000i 0.520666i
$$724$$ 0 0
$$725$$ 18.0000 24.0000i 0.668503 0.891338i
$$726$$ 0 0
$$727$$ 28.0000i 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 4.00000i 0.147743i −0.997268 0.0738717i $$-0.976464\pi$$
0.997268 0.0738717i $$-0.0235355\pi$$
$$734$$ 0 0
$$735$$ 18.0000 9.00000i 0.663940 0.331970i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −40.0000 −1.47142 −0.735712 0.677295i $$-0.763152\pi$$
−0.735712 + 0.677295i $$0.763152\pi$$
$$740$$ 0 0
$$741$$ 32.0000 1.17555
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i −0.975473 0.220119i $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ 0 0
$$745$$ −14.0000 28.0000i −0.512920 1.02584i
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 16.0000i 0.583072i
$$754$$ 0 0
$$755$$ −8.00000 16.0000i −0.291150 0.582300i
$$756$$ 0 0
$$757$$ 12.0000i 0.436147i −0.975932 0.218074i $$-0.930023\pi$$
0.975932 0.218074i $$-0.0699773\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 40.0000i 1.44810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −24.0000 −0.864339
$$772$$ 0 0
$$773$$ 28.0000i 1.00709i −0.863969 0.503545i $$-0.832029\pi$$
0.863969 0.503545i $$-0.167971\pi$$
$$774$$ 0 0
$$775$$ −24.0000 + 32.0000i −0.862105 + 1.14947i
$$776$$ 0 0
$$777$$ 16.0000i 0.573997i
$$778$$ 0 0
$$779$$ 48.0000 1.71978
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 0 0
$$785$$ −8.00000 + 4.00000i −0.285532 + 0.142766i
$$786$$ 0 0
$$787$$ 12.0000i 0.427754i 0.976861 + 0.213877i $$0.0686091\pi$$
−0.976861 + 0.213877i $$0.931391\pi$$
$$788$$ 0 0
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ −32.0000 −1.13779
$$792$$ 0 0
$$793$$ 24.0000i 0.852265i
$$794$$ 0 0
$$795$$ 12.0000 + 24.0000i 0.425596 + 0.851192i
$$796$$ 0 0
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 0 0