# Properties

 Label 784.2.a.l.1.1 Level $784$ Weight $2$ Character 784.1 Self dual yes Analytic conductor $6.260$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} +O(q^{10})$$ $$q-1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} +2.00000 q^{11} +4.00000 q^{15} +1.41421 q^{17} -7.07107 q^{19} +4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +8.48528 q^{31} -2.82843 q^{33} +10.0000 q^{37} +9.89949 q^{41} -2.00000 q^{43} +2.82843 q^{45} +2.82843 q^{47} -2.00000 q^{51} -2.00000 q^{53} -5.65685 q^{55} +10.0000 q^{57} -1.41421 q^{59} -2.82843 q^{61} -12.0000 q^{67} -5.65685 q^{69} +12.0000 q^{71} +1.41421 q^{73} -4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -4.00000 q^{85} -2.82843 q^{87} +7.07107 q^{89} -12.0000 q^{93} +20.0000 q^{95} -9.89949 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} + 4 q^{11} + 8 q^{15} + 8 q^{23} + 6 q^{25} + 4 q^{29} + 20 q^{37} - 4 q^{43} - 4 q^{51} - 4 q^{53} + 20 q^{57} - 24 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} - 8 q^{85} - 24 q^{93} + 40 q^{95} - 4 q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.41421 −0.816497 −0.408248 0.912871i $$-0.633860\pi$$
−0.408248 + 0.912871i $$0.633860\pi$$
$$4$$ 0 0
$$5$$ −2.82843 −1.26491 −0.632456 0.774597i $$-0.717953\pi$$
−0.632456 + 0.774597i $$0.717953\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 0 0
$$17$$ 1.41421 0.342997 0.171499 0.985184i $$-0.445139\pi$$
0.171499 + 0.985184i $$0.445139\pi$$
$$18$$ 0 0
$$19$$ −7.07107 −1.62221 −0.811107 0.584898i $$-0.801135\pi$$
−0.811107 + 0.584898i $$0.801135\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 8.48528 1.52400 0.762001 0.647576i $$-0.224217\pi$$
0.762001 + 0.647576i $$0.224217\pi$$
$$32$$ 0 0
$$33$$ −2.82843 −0.492366
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.89949 1.54604 0.773021 0.634381i $$-0.218745\pi$$
0.773021 + 0.634381i $$0.218745\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −5.65685 −0.762770
$$56$$ 0 0
$$57$$ 10.0000 1.32453
$$58$$ 0 0
$$59$$ −1.41421 −0.184115 −0.0920575 0.995754i $$-0.529344\pi$$
−0.0920575 + 0.995754i $$0.529344\pi$$
$$60$$ 0 0
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ −5.65685 −0.681005
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 1.41421 0.165521 0.0827606 0.996569i $$-0.473626\pi$$
0.0827606 + 0.996569i $$0.473626\pi$$
$$74$$ 0 0
$$75$$ −4.24264 −0.489898
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −5.00000 −0.555556
$$82$$ 0 0
$$83$$ 9.89949 1.08661 0.543305 0.839535i $$-0.317173\pi$$
0.543305 + 0.839535i $$0.317173\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 0 0
$$87$$ −2.82843 −0.303239
$$88$$ 0 0
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −12.0000 −1.24434
$$94$$ 0 0
$$95$$ 20.0000 2.05196
$$96$$ 0 0
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −8.48528 −0.844317 −0.422159 0.906522i $$-0.638727\pi$$
−0.422159 + 0.906522i $$0.638727\pi$$
$$102$$ 0 0
$$103$$ 2.82843 0.278693 0.139347 0.990244i $$-0.455500\pi$$
0.139347 + 0.990244i $$0.455500\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −14.1421 −1.34231
$$112$$ 0 0
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ −11.3137 −1.05501
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −14.0000 −1.26234
$$124$$ 0 0
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 2.82843 0.249029
$$130$$ 0 0
$$131$$ 12.7279 1.11204 0.556022 0.831168i $$-0.312327\pi$$
0.556022 + 0.831168i $$0.312327\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −16.0000 −1.37706
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 9.89949 0.839664 0.419832 0.907602i $$-0.362089\pi$$
0.419832 + 0.907602i $$0.362089\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −5.65685 −0.469776
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ −1.41421 −0.114332
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 11.3137 0.902932 0.451466 0.892288i $$-0.350901\pi$$
0.451466 + 0.892288i $$0.350901\pi$$
$$158$$ 0 0
$$159$$ 2.82843 0.224309
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 7.07107 0.540738
$$172$$ 0 0
$$173$$ 16.9706 1.29025 0.645124 0.764078i $$-0.276806\pi$$
0.645124 + 0.764078i $$0.276806\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2.00000 0.150329
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ 0 0
$$185$$ −28.2843 −2.07950
$$186$$ 0 0
$$187$$ 2.82843 0.206835
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 8.48528 0.601506 0.300753 0.953702i $$-0.402762\pi$$
0.300753 + 0.953702i $$0.402762\pi$$
$$200$$ 0 0
$$201$$ 16.9706 1.19701
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −28.0000 −1.95560
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −14.1421 −0.978232
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ −16.9706 −1.16280
$$214$$ 0 0
$$215$$ 5.65685 0.385794
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ −21.2132 −1.40797 −0.703985 0.710215i $$-0.748598\pi$$
−0.703985 + 0.710215i $$0.748598\pi$$
$$228$$ 0 0
$$229$$ 16.9706 1.12145 0.560723 0.828003i $$-0.310523\pi$$
0.560723 + 0.828003i $$0.310523\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 0 0
$$237$$ −5.65685 −0.367452
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 21.2132 1.36646 0.683231 0.730202i $$-0.260574\pi$$
0.683231 + 0.730202i $$0.260574\pi$$
$$242$$ 0 0
$$243$$ −9.89949 −0.635053
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −14.0000 −0.887214
$$250$$ 0 0
$$251$$ 9.89949 0.624851 0.312425 0.949942i $$-0.398859\pi$$
0.312425 + 0.949942i $$0.398859\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ 0 0
$$255$$ 5.65685 0.354246
$$256$$ 0 0
$$257$$ −12.7279 −0.793946 −0.396973 0.917830i $$-0.629939\pi$$
−0.396973 + 0.917830i $$0.629939\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 5.65685 0.347498
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ 0 0
$$269$$ 11.3137 0.689809 0.344904 0.938638i $$-0.387911\pi$$
0.344904 + 0.938638i $$0.387911\pi$$
$$270$$ 0 0
$$271$$ 22.6274 1.37452 0.687259 0.726413i $$-0.258814\pi$$
0.687259 + 0.726413i $$0.258814\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000 0.361814
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ −8.48528 −0.508001
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ −1.41421 −0.0840663 −0.0420331 0.999116i $$-0.513384\pi$$
−0.0420331 + 0.999116i $$0.513384\pi$$
$$284$$ 0 0
$$285$$ −28.2843 −1.67542
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.0000 −0.882353
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ 19.7990 1.15667 0.578335 0.815800i $$-0.303703\pi$$
0.578335 + 0.815800i $$0.303703\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ 11.3137 0.656488
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 12.0000 0.689382
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ −9.89949 −0.564994 −0.282497 0.959268i $$-0.591163\pi$$
−0.282497 + 0.959268i $$0.591163\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −11.3137 −0.641542 −0.320771 0.947157i $$-0.603942\pi$$
−0.320771 + 0.947157i $$0.603942\pi$$
$$312$$ 0 0
$$313$$ −12.7279 −0.719425 −0.359712 0.933063i $$-0.617125\pi$$
−0.359712 + 0.933063i $$0.617125\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ 0 0
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ −5.65685 −0.315735
$$322$$ 0 0
$$323$$ −10.0000 −0.556415
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.82843 0.156412
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 0 0
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 33.9411 1.85440
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 16.9706 0.921714
$$340$$ 0 0
$$341$$ 16.9706 0.919007
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.41421 0.0752710 0.0376355 0.999292i $$-0.488017\pi$$
0.0376355 + 0.999292i $$0.488017\pi$$
$$354$$ 0 0
$$355$$ −33.9411 −1.80141
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 31.0000 1.63158
$$362$$ 0 0
$$363$$ 9.89949 0.519589
$$364$$ 0 0
$$365$$ −4.00000 −0.209370
$$366$$ 0 0
$$367$$ 28.2843 1.47643 0.738213 0.674567i $$-0.235670\pi$$
0.738213 + 0.674567i $$0.235670\pi$$
$$368$$ 0 0
$$369$$ −9.89949 −0.515347
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 0 0
$$375$$ −8.00000 −0.413118
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 26.0000 1.33553 0.667765 0.744372i $$-0.267251\pi$$
0.667765 + 0.744372i $$0.267251\pi$$
$$380$$ 0 0
$$381$$ 22.6274 1.15924
$$382$$ 0 0
$$383$$ −36.7696 −1.87884 −0.939418 0.342773i $$-0.888634\pi$$
−0.939418 + 0.342773i $$0.888634\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.00000 0.101666
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 5.65685 0.286079
$$392$$ 0 0
$$393$$ −18.0000 −0.907980
$$394$$ 0 0
$$395$$ −11.3137 −0.569254
$$396$$ 0 0
$$397$$ −22.6274 −1.13564 −0.567819 0.823154i $$-0.692213\pi$$
−0.567819 + 0.823154i $$0.692213\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 14.1421 0.702728
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ −38.1838 −1.88807 −0.944033 0.329851i $$-0.893001\pi$$
−0.944033 + 0.329851i $$0.893001\pi$$
$$410$$ 0 0
$$411$$ −16.9706 −0.837096
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −28.0000 −1.37447
$$416$$ 0 0
$$417$$ −14.0000 −0.685583
$$418$$ 0 0
$$419$$ −9.89949 −0.483622 −0.241811 0.970323i $$-0.577741\pi$$
−0.241811 + 0.970323i $$0.577741\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ 4.24264 0.205798
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 29.6985 1.42722 0.713609 0.700544i $$-0.247059\pi$$
0.713609 + 0.700544i $$0.247059\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ 0 0
$$437$$ −28.2843 −1.35302
$$438$$ 0 0
$$439$$ −16.9706 −0.809961 −0.404980 0.914325i $$-0.632722\pi$$
−0.404980 + 0.914325i $$0.632722\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ −20.0000 −0.948091
$$446$$ 0 0
$$447$$ −14.1421 −0.668900
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 19.7990 0.932298
$$452$$ 0 0
$$453$$ −22.6274 −1.06313
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 24.0000 1.12267 0.561336 0.827588i $$-0.310287\pi$$
0.561336 + 0.827588i $$0.310287\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −39.5980 −1.84426 −0.922131 0.386878i $$-0.873553\pi$$
−0.922131 + 0.386878i $$0.873553\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 33.9411 1.57398
$$466$$ 0 0
$$467$$ 32.5269 1.50517 0.752583 0.658497i $$-0.228808\pi$$
0.752583 + 0.658497i $$0.228808\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −16.0000 −0.737241
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ −21.2132 −0.973329
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −31.1127 −1.42158 −0.710788 0.703407i $$-0.751661\pi$$
−0.710788 + 0.703407i $$0.751661\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 28.0000 1.27141
$$486$$ 0 0
$$487$$ −12.0000 −0.543772 −0.271886 0.962329i $$-0.587647\pi$$
−0.271886 + 0.962329i $$0.587647\pi$$
$$488$$ 0 0
$$489$$ 14.1421 0.639529
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 2.82843 0.127386
$$494$$ 0 0
$$495$$ 5.65685 0.254257
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 28.0000 1.25095
$$502$$ 0 0
$$503$$ 39.5980 1.76559 0.882793 0.469762i $$-0.155660\pi$$
0.882793 + 0.469762i $$0.155660\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ 0 0
$$507$$ 18.3848 0.816497
$$508$$ 0 0
$$509$$ −22.6274 −1.00294 −0.501471 0.865174i $$-0.667208\pi$$
−0.501471 + 0.865174i $$0.667208\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −40.0000 −1.76604
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 1.41421 0.0619578 0.0309789 0.999520i $$-0.490138\pi$$
0.0309789 + 0.999520i $$0.490138\pi$$
$$522$$ 0 0
$$523$$ 12.7279 0.556553 0.278277 0.960501i $$-0.410237\pi$$
0.278277 + 0.960501i $$0.410237\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 1.41421 0.0613716
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −11.3137 −0.489134
$$536$$ 0 0
$$537$$ 16.9706 0.732334
$$538$$ 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
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5.65685 0.242313
$$546$$ 0 0
$$547$$ 26.0000 1.11168 0.555840 0.831289i $$-0.312397\pi$$
0.555840 + 0.831289i $$0.312397\pi$$
$$548$$ 0 0
$$549$$ 2.82843 0.120714
$$550$$ 0 0
$$551$$ −14.1421 −0.602475
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 40.0000 1.69791
$$556$$ 0 0
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ −1.41421 −0.0596020 −0.0298010 0.999556i $$-0.509487\pi$$
−0.0298010 + 0.999556i $$0.509487\pi$$
$$564$$ 0 0
$$565$$ 33.9411 1.42791
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ 2.00000 0.0836974 0.0418487 0.999124i $$-0.486675\pi$$
0.0418487 + 0.999124i $$0.486675\pi$$
$$572$$ 0 0
$$573$$ −5.65685 −0.236318
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ 21.2132 0.883117 0.441559 0.897232i $$-0.354426\pi$$
0.441559 + 0.897232i $$0.354426\pi$$
$$578$$ 0 0
$$579$$ 22.6274 0.940363
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −29.6985 −1.22579 −0.612894 0.790165i $$-0.709995\pi$$
−0.612894 + 0.790165i $$0.709995\pi$$
$$588$$ 0 0
$$589$$ −60.0000 −2.47226
$$590$$ 0 0
$$591$$ −2.82843 −0.116346
$$592$$ 0 0
$$593$$ 7.07107 0.290374 0.145187 0.989404i $$-0.453622\pi$$
0.145187 + 0.989404i $$0.453622\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −12.0000 −0.491127
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −29.6985 −1.21143 −0.605713 0.795683i $$-0.707112\pi$$
−0.605713 + 0.795683i $$0.707112\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 0 0
$$605$$ 19.7990 0.804943
$$606$$ 0 0
$$607$$ −16.9706 −0.688814 −0.344407 0.938820i $$-0.611920\pi$$
−0.344407 + 0.938820i $$0.611920\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ 0 0
$$615$$ 39.5980 1.59674
$$616$$ 0 0
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 0 0
$$619$$ 18.3848 0.738947 0.369473 0.929241i $$-0.379538\pi$$
0.369473 + 0.929241i $$0.379538\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 20.0000 0.798723
$$628$$ 0 0
$$629$$ 14.1421 0.563884
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ 0 0
$$633$$ −16.9706 −0.674519
$$634$$ 0 0
$$635$$ 45.2548 1.79588
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 9.89949 0.390398 0.195199 0.980764i $$-0.437465\pi$$
0.195199 + 0.980764i $$0.437465\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ 8.48528 0.333591 0.166795 0.985992i $$-0.446658\pi$$
0.166795 + 0.985992i $$0.446658\pi$$
$$648$$ 0 0
$$649$$ −2.82843 −0.111025
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ −36.0000 −1.40664
$$656$$ 0 0
$$657$$ −1.41421 −0.0551737
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 0 0
$$661$$ −8.48528 −0.330039 −0.165020 0.986290i $$-0.552769\pi$$
−0.165020 + 0.986290i $$0.552769\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000 0.309761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −5.65685 −0.218380
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ 0 0
$$675$$ 16.9706 0.653197
$$676$$ 0 0
$$677$$ 16.9706 0.652232 0.326116 0.945330i $$-0.394260\pi$$
0.326116 + 0.945330i $$0.394260\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 30.0000 1.14960
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −33.9411 −1.29682
$$686$$ 0 0
$$687$$ −24.0000 −0.915657
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 12.7279 0.484193 0.242096 0.970252i $$-0.422165\pi$$
0.242096 + 0.970252i $$0.422165\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −28.0000 −1.06210
$$696$$ 0 0
$$697$$ 14.0000 0.530288
$$698$$ 0 0
$$699$$ −33.9411 −1.28377
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −70.7107 −2.66690
$$704$$ 0 0
$$705$$ 11.3137 0.426099
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 33.9411 1.27111
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −16.9706 −0.633777
$$718$$ 0 0
$$719$$ 2.82843 0.105483 0.0527413 0.998608i $$-0.483204\pi$$
0.0527413 + 0.998608i $$0.483204\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −30.0000 −1.11571
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −19.7990 −0.734304 −0.367152 0.930161i $$-0.619667\pi$$
−0.367152 + 0.930161i $$0.619667\pi$$
$$728$$ 0 0
$$729$$ 29.0000 1.07407
$$730$$ 0 0
$$731$$ −2.82843 −0.104613
$$732$$ 0 0
$$733$$ −42.4264 −1.56706 −0.783528 0.621357i $$-0.786582\pi$$
−0.783528 + 0.621357i $$0.786582\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −24.0000 −0.884051
$$738$$ 0 0
$$739$$ 30.0000 1.10357 0.551784 0.833987i $$-0.313947\pi$$
0.551784 + 0.833987i $$0.313947\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −28.2843 −1.03626
$$746$$ 0 0
$$747$$ −9.89949 −0.362204
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ −14.0000 −0.510188
$$754$$ 0 0
$$755$$ −45.2548 −1.64699
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ −11.3137 −0.410662
$$760$$ 0 0
$$761$$ 7.07107 0.256326 0.128163 0.991753i $$-0.459092\pi$$
0.128163 + 0.991753i $$0.459092\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −29.6985 −1.07095 −0.535477 0.844550i $$-0.679868\pi$$
−0.535477 + 0.844550i $$0.679868\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ −48.0833 −1.72943 −0.864717 0.502259i $$-0.832502\pi$$
−0.864717 + 0.502259i $$0.832502\pi$$
$$774$$ 0 0
$$775$$ 25.4558 0.914401
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −70.0000 −2.50801
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ 11.3137 0.404319
$$784$$ 0 0
$$785$$ −32.0000 −1.14213
$$786$$ 0 0
$$787$$ −1.41421 −0.0504113 −0.0252056 0.999682i $$-0.508024\pi$$
−0.0252056 + 0.999682i $$0.508024\pi$$
$$788$$ 0 0
$$789$$ 16.9706 0.604168
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −8.00000 −0.283731
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −7.07107 −0.249844
$$802$$ 0 0
$$803$$ 2.82843 0.0998130
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −16.0000 −0.563227
$$808$$ 0 0
$$809$$ −16.0000 −0.562530 −0.281265 0.959630i $$-0.590754\pi$$
−0.281265 + 0.959630i $$0.590754\pi$$
$$810$$ 0 0
$$811$$ −29.6985 −1.04285 −0.521427 0.853296i $$-0.674600\pi$$
−0.521427 + 0.853296i $$0.674600\pi$$
$$812$$ 0 0
$$813$$ −32.0000 −1.12229
$$814$$ 0 0
$$815$$ 28.2843 0.990755
$$816$$ 0 0
$$817$$ 14.1421 0.494771
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ −8.48528 −0.295420
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 31.1127 1.08059 0.540294 0.841476i $$-0.318313\pi$$
0.540294 + 0.841476i $$0.318313\pi$$
$$830$$ 0 0
$$831$$ 2.82843 0.0981170
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 56.0000 1.93796
$$836$$ 0 0
$$837$$ 48.0000 1.65912
$$838$$ 0 0
$$839$$ 19.7990 0.683537 0.341769 0.939784i $$-0.388974\pi$$
0.341769 + 0.939784i $$0.388974\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −22.6274 −0.779330
$$844$$ 0 0
$$845$$ 36.7696 1.26491
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2.00000 0.0686398
$$850$$ 0 0
$$851$$ 40.0000 1.37118
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ −20.0000 −0.683986
$$856$$ 0 0
$$857$$ −18.3848 −0.628012 −0.314006 0.949421i $$-0.601671\pi$$
−0.314006 + 0.949421i $$0.601671\pi$$
$$858$$ 0 0
$$859$$ −26.8701 −0.916795 −0.458397 0.888747i $$-0.651576\pi$$
−0.458397 + 0.888747i $$0.651576\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ 0 0
$$867$$ 21.2132 0.720438
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 9.89949 0.335047
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −46.0000 −1.55331 −0.776655 0.629926i $$-0.783085\pi$$
−0.776655 + 0.629926i $$0.783085\pi$$
$$878$$ 0 0
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −29.6985 −1.00057 −0.500284 0.865862i $$-0.666771\pi$$
−0.500284 + 0.865862i $$0.666771\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ −5.65685 −0.190153
$$886$$ 0 0
$$887$$ −36.7696 −1.23460 −0.617300 0.786728i $$-0.711774\pi$$
−0.617300 + 0.786728i $$0.711774\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −10.0000 −0.335013
$$892$$ 0 0
$$893$$ −20.0000 −0.669274
$$894$$ 0 0
$$895$$ 33.9411 1.13453
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 16.9706 0.566000
$$900$$ 0 0
$$901$$ −2.82843 −0.0942286
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 0 0
$$909$$ 8.48528 0.281439
$$910$$ 0 0
$$911$$ 40.0000 1.32526 0.662630 0.748947i $$-0.269440\pi$$
0.662630 + 0.748947i $$0.269440\pi$$
$$912$$ 0 0
$$913$$ 19.7990 0.655251
$$914$$ 0 0
$$915$$ −11.3137 −0.374020
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 30.0000 0.986394
$$926$$ 0 0
$$927$$ −2.82843 −0.0928977
$$928$$ 0 0
$$929$$ −32.5269 −1.06717 −0.533587 0.845745i $$-0.679156\pi$$
−0.533587 + 0.845745i $$0.679156\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 16.0000 0.523816
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ −9.89949 −0.323402 −0.161701 0.986840i $$-0.551698\pi$$
−0.161701 + 0.986840i $$0.551698\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ 31.1127 1.01424 0.507122 0.861874i $$-0.330709\pi$$
0.507122 + 0.861874i $$0.330709\pi$$
$$942$$ 0 0
$$943$$ 39.5980 1.28949
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −14.1421 −0.458590
$$952$$ 0 0
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ −11.3137 −0.366103
$$956$$ 0 0
$$957$$ −5.65685 −0.182860
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ 0 0
$$965$$ 45.2548 1.45680
$$966$$ 0 0
$$967$$ 12.0000 0.385894 0.192947 0.981209i $$-0.438195\pi$$
0.192947 + 0.981209i $$0.438195\pi$$
$$968$$ 0 0
$$969$$ 14.1421 0.454311
$$970$$ 0 0
$$971$$ 32.5269 1.04384 0.521919 0.852995i $$-0.325216\pi$$
0.521919 + 0.852995i $$0.325216\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ 14.1421 0.451985
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 48.0833 1.53362 0.766809 0.641875i $$-0.221843\pi$$
0.766809 + 0.641875i $$0.221843\pi$$
$$984$$ 0 0
$$985$$ −5.65685 −0.180242
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 14.1421 0.448787
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ 0 0
$$997$$ 31.1127 0.985349 0.492675 0.870214i $$-0.336019\pi$$
0.492675 + 0.870214i $$0.336019\pi$$
$$998$$ 0 0
$$999$$ 56.5685 1.78975
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 784.2.a.l.1.1 2
3.2 odd 2 7056.2.a.cl.1.2 2
4.3 odd 2 98.2.a.b.1.2 yes 2
7.2 even 3 784.2.i.m.753.2 4
7.3 odd 6 784.2.i.m.177.1 4
7.4 even 3 784.2.i.m.177.2 4
7.5 odd 6 784.2.i.m.753.1 4
7.6 odd 2 inner 784.2.a.l.1.2 2
8.3 odd 2 3136.2.a.bn.1.1 2
8.5 even 2 3136.2.a.bm.1.2 2
12.11 even 2 882.2.a.n.1.2 2
20.3 even 4 2450.2.c.v.99.2 4
20.7 even 4 2450.2.c.v.99.3 4
20.19 odd 2 2450.2.a.bj.1.1 2
21.20 even 2 7056.2.a.cl.1.1 2
28.3 even 6 98.2.c.c.79.2 4
28.11 odd 6 98.2.c.c.79.1 4
28.19 even 6 98.2.c.c.67.2 4
28.23 odd 6 98.2.c.c.67.1 4
28.27 even 2 98.2.a.b.1.1 2
56.13 odd 2 3136.2.a.bm.1.1 2
56.27 even 2 3136.2.a.bn.1.2 2
84.11 even 6 882.2.g.l.667.1 4
84.23 even 6 882.2.g.l.361.1 4
84.47 odd 6 882.2.g.l.361.2 4
84.59 odd 6 882.2.g.l.667.2 4
84.83 odd 2 882.2.a.n.1.1 2
140.27 odd 4 2450.2.c.v.99.4 4
140.83 odd 4 2450.2.c.v.99.1 4
140.139 even 2 2450.2.a.bj.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 28.27 even 2
98.2.a.b.1.2 yes 2 4.3 odd 2
98.2.c.c.67.1 4 28.23 odd 6
98.2.c.c.67.2 4 28.19 even 6
98.2.c.c.79.1 4 28.11 odd 6
98.2.c.c.79.2 4 28.3 even 6
784.2.a.l.1.1 2 1.1 even 1 trivial
784.2.a.l.1.2 2 7.6 odd 2 inner
784.2.i.m.177.1 4 7.3 odd 6
784.2.i.m.177.2 4 7.4 even 3
784.2.i.m.753.1 4 7.5 odd 6
784.2.i.m.753.2 4 7.2 even 3
882.2.a.n.1.1 2 84.83 odd 2
882.2.a.n.1.2 2 12.11 even 2
882.2.g.l.361.1 4 84.23 even 6
882.2.g.l.361.2 4 84.47 odd 6
882.2.g.l.667.1 4 84.11 even 6
882.2.g.l.667.2 4 84.59 odd 6
2450.2.a.bj.1.1 2 20.19 odd 2
2450.2.a.bj.1.2 2 140.139 even 2
2450.2.c.v.99.1 4 140.83 odd 4
2450.2.c.v.99.2 4 20.3 even 4
2450.2.c.v.99.3 4 20.7 even 4
2450.2.c.v.99.4 4 140.27 odd 4
3136.2.a.bm.1.1 2 56.13 odd 2
3136.2.a.bm.1.2 2 8.5 even 2
3136.2.a.bn.1.1 2 8.3 odd 2
3136.2.a.bn.1.2 2 56.27 even 2
7056.2.a.cl.1.1 2 21.20 even 2
7056.2.a.cl.1.2 2 3.2 odd 2