# Properties

 Label 882.2.a.o.1.1 Level $882$ Weight $2$ Character 882.1 Self dual yes Analytic conductor $7.043$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 882.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} +4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} +7.07107 q^{17} +5.65685 q^{19} -1.41421 q^{20} +4.00000 q^{22} +8.00000 q^{23} -3.00000 q^{25} -4.24264 q^{26} +2.00000 q^{29} +1.00000 q^{32} +7.07107 q^{34} +4.00000 q^{37} +5.65685 q^{38} -1.41421 q^{40} -9.89949 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} -5.65685 q^{47} -3.00000 q^{50} -4.24264 q^{52} +4.00000 q^{53} -5.65685 q^{55} +2.00000 q^{58} +11.3137 q^{59} -1.41421 q^{61} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{67} +7.07107 q^{68} +15.5563 q^{73} +4.00000 q^{74} +5.65685 q^{76} -16.0000 q^{79} -1.41421 q^{80} -9.89949 q^{82} +5.65685 q^{83} -10.0000 q^{85} -4.00000 q^{86} +4.00000 q^{88} -7.07107 q^{89} +8.00000 q^{92} -5.65685 q^{94} -8.00000 q^{95} -7.07107 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{8} + 8q^{11} + 2q^{16} + 8q^{22} + 16q^{23} - 6q^{25} + 4q^{29} + 2q^{32} + 8q^{37} - 8q^{43} + 8q^{44} + 16q^{46} - 6q^{50} + 8q^{53} + 4q^{58} + 2q^{64} + 12q^{65} - 24q^{67} + 8q^{74} - 32q^{79} - 20q^{85} - 8q^{86} + 8q^{88} + 16q^{92} - 16q^{95} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.41421 −0.632456 −0.316228 0.948683i $$-0.602416\pi$$
−0.316228 + 0.948683i $$0.602416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −1.41421 −0.447214
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −4.24264 −1.17670 −0.588348 0.808608i $$-0.700222\pi$$
−0.588348 + 0.808608i $$0.700222\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 7.07107 1.71499 0.857493 0.514496i $$-0.172021\pi$$
0.857493 + 0.514496i $$0.172021\pi$$
$$18$$ 0 0
$$19$$ 5.65685 1.29777 0.648886 0.760886i $$-0.275235\pi$$
0.648886 + 0.760886i $$0.275235\pi$$
$$20$$ −1.41421 −0.316228
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ −4.24264 −0.832050
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 7.07107 1.21268
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 5.65685 0.917663
$$39$$ 0 0
$$40$$ −1.41421 −0.223607
$$41$$ −9.89949 −1.54604 −0.773021 0.634381i $$-0.781255\pi$$
−0.773021 + 0.634381i $$0.781255\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ −5.65685 −0.825137 −0.412568 0.910927i $$-0.635368\pi$$
−0.412568 + 0.910927i $$0.635368\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −3.00000 −0.424264
$$51$$ 0 0
$$52$$ −4.24264 −0.588348
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ −5.65685 −0.762770
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 11.3137 1.47292 0.736460 0.676481i $$-0.236496\pi$$
0.736460 + 0.676481i $$0.236496\pi$$
$$60$$ 0 0
$$61$$ −1.41421 −0.181071 −0.0905357 0.995893i $$-0.528858\pi$$
−0.0905357 + 0.995893i $$0.528858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 7.07107 0.857493
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 15.5563 1.82073 0.910366 0.413803i $$-0.135800\pi$$
0.910366 + 0.413803i $$0.135800\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 5.65685 0.648886
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −1.41421 −0.158114
$$81$$ 0 0
$$82$$ −9.89949 −1.09322
$$83$$ 5.65685 0.620920 0.310460 0.950586i $$-0.399517\pi$$
0.310460 + 0.950586i $$0.399517\pi$$
$$84$$ 0 0
$$85$$ −10.0000 −1.08465
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ −5.65685 −0.583460
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ −7.07107 −0.717958 −0.358979 0.933346i $$-0.616875\pi$$
−0.358979 + 0.933346i $$0.616875\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −3.00000 −0.300000
$$101$$ −12.7279 −1.26648 −0.633238 0.773957i $$-0.718274\pi$$
−0.633238 + 0.773957i $$0.718274\pi$$
$$102$$ 0 0
$$103$$ −5.65685 −0.557386 −0.278693 0.960380i $$-0.589901\pi$$
−0.278693 + 0.960380i $$0.589901\pi$$
$$104$$ −4.24264 −0.416025
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ −5.65685 −0.539360
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −11.3137 −1.05501
$$116$$ 2.00000 0.185695
$$117$$ 0 0
$$118$$ 11.3137 1.04151
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −1.41421 −0.128037
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.3137 1.01193
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 6.00000 0.526235
$$131$$ 16.9706 1.48272 0.741362 0.671105i $$-0.234180\pi$$
0.741362 + 0.671105i $$0.234180\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 7.07107 0.606339
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 5.65685 0.479808 0.239904 0.970797i $$-0.422884\pi$$
0.239904 + 0.970797i $$0.422884\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −16.9706 −1.41915
$$144$$ 0 0
$$145$$ −2.82843 −0.234888
$$146$$ 15.5563 1.28745
$$147$$ 0 0
$$148$$ 4.00000 0.328798
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 5.65685 0.458831
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.3848 1.46726 0.733632 0.679546i $$-0.237823\pi$$
0.733632 + 0.679546i $$0.237823\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 0 0
$$160$$ −1.41421 −0.111803
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −9.89949 −0.773021
$$165$$ 0 0
$$166$$ 5.65685 0.439057
$$167$$ −11.3137 −0.875481 −0.437741 0.899101i $$-0.644221\pi$$
−0.437741 + 0.899101i $$0.644221\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ −10.0000 −0.766965
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ 12.7279 0.967686 0.483843 0.875155i $$-0.339241\pi$$
0.483843 + 0.875155i $$0.339241\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −7.07107 −0.529999
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −12.7279 −0.946059 −0.473029 0.881047i $$-0.656840\pi$$
−0.473029 + 0.881047i $$0.656840\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 8.00000 0.589768
$$185$$ −5.65685 −0.415900
$$186$$ 0 0
$$187$$ 28.2843 2.06835
$$188$$ −5.65685 −0.412568
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −7.07107 −0.507673
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.00000 −0.284988 −0.142494 0.989796i $$-0.545512\pi$$
−0.142494 + 0.989796i $$0.545512\pi$$
$$198$$ 0 0
$$199$$ −16.9706 −1.20301 −0.601506 0.798869i $$-0.705432\pi$$
−0.601506 + 0.798869i $$0.705432\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ −12.7279 −0.895533
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 14.0000 0.977802
$$206$$ −5.65685 −0.394132
$$207$$ 0 0
$$208$$ −4.24264 −0.294174
$$209$$ 22.6274 1.56517
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 4.00000 0.274721
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 5.65685 0.385794
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ −5.65685 −0.381385
$$221$$ −30.0000 −2.01802
$$222$$ 0 0
$$223$$ 16.9706 1.13643 0.568216 0.822879i $$-0.307634\pi$$
0.568216 + 0.822879i $$0.307634\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −16.9706 −1.12638 −0.563188 0.826329i $$-0.690425\pi$$
−0.563188 + 0.826329i $$0.690425\pi$$
$$228$$ 0 0
$$229$$ −12.7279 −0.841085 −0.420542 0.907273i $$-0.638160\pi$$
−0.420542 + 0.907273i $$0.638160\pi$$
$$230$$ −11.3137 −0.746004
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 11.3137 0.736460
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −4.24264 −0.273293 −0.136646 0.990620i $$-0.543632\pi$$
−0.136646 + 0.990620i $$0.543632\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ −1.41421 −0.0905357
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 11.3137 0.715542
$$251$$ 5.65685 0.357057 0.178529 0.983935i $$-0.442866\pi$$
0.178529 + 0.983935i $$0.442866\pi$$
$$252$$ 0 0
$$253$$ 32.0000 2.01182
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.7279 0.793946 0.396973 0.917830i $$-0.370061\pi$$
0.396973 + 0.917830i $$0.370061\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 6.00000 0.372104
$$261$$ 0 0
$$262$$ 16.9706 1.04844
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ −5.65685 −0.347498
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ 1.41421 0.0862261 0.0431131 0.999070i $$-0.486272\pi$$
0.0431131 + 0.999070i $$0.486272\pi$$
$$270$$ 0 0
$$271$$ 5.65685 0.343629 0.171815 0.985129i $$-0.445037\pi$$
0.171815 + 0.985129i $$0.445037\pi$$
$$272$$ 7.07107 0.428746
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 5.65685 0.339276
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −22.6274 −1.34506 −0.672530 0.740070i $$-0.734792\pi$$
−0.672530 + 0.740070i $$0.734792\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −16.9706 −1.00349
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 33.0000 1.94118
$$290$$ −2.82843 −0.166091
$$291$$ 0 0
$$292$$ 15.5563 0.910366
$$293$$ −24.0416 −1.40453 −0.702264 0.711917i $$-0.747827\pi$$
−0.702264 + 0.711917i $$0.747827\pi$$
$$294$$ 0 0
$$295$$ −16.0000 −0.931556
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ −20.0000 −1.15857
$$299$$ −33.9411 −1.96287
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ 5.65685 0.324443
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ −5.65685 −0.322854 −0.161427 0.986885i $$-0.551610\pi$$
−0.161427 + 0.986885i $$0.551610\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.65685 0.320771 0.160385 0.987054i $$-0.448726\pi$$
0.160385 + 0.987054i $$0.448726\pi$$
$$312$$ 0 0
$$313$$ −21.2132 −1.19904 −0.599521 0.800359i $$-0.704642\pi$$
−0.599521 + 0.800359i $$0.704642\pi$$
$$314$$ 18.3848 1.03751
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ 28.0000 1.57264 0.786318 0.617822i $$-0.211985\pi$$
0.786318 + 0.617822i $$0.211985\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ −1.41421 −0.0790569
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 40.0000 2.22566
$$324$$ 0 0
$$325$$ 12.7279 0.706018
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −9.89949 −0.546608
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 5.65685 0.310460
$$333$$ 0 0
$$334$$ −11.3137 −0.619059
$$335$$ 16.9706 0.927201
$$336$$ 0 0
$$337$$ −16.0000 −0.871576 −0.435788 0.900049i $$-0.643530\pi$$
−0.435788 + 0.900049i $$0.643530\pi$$
$$338$$ 5.00000 0.271964
$$339$$ 0 0
$$340$$ −10.0000 −0.542326
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 12.7279 0.684257
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 29.6985 1.58972 0.794862 0.606791i $$-0.207543\pi$$
0.794862 + 0.606791i $$0.207543\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −1.41421 −0.0752710 −0.0376355 0.999292i $$-0.511983\pi$$
−0.0376355 + 0.999292i $$0.511983\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −7.07107 −0.374766
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 13.0000 0.684211
$$362$$ −12.7279 −0.668965
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −22.0000 −1.15153
$$366$$ 0 0
$$367$$ −5.65685 −0.295285 −0.147643 0.989041i $$-0.547169\pi$$
−0.147643 + 0.989041i $$0.547169\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 0 0
$$370$$ −5.65685 −0.294086
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 28.2843 1.46254
$$375$$ 0 0
$$376$$ −5.65685 −0.291730
$$377$$ −8.48528 −0.437014
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ 5.65685 0.289052 0.144526 0.989501i $$-0.453834\pi$$
0.144526 + 0.989501i $$0.453834\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ −7.07107 −0.358979
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 56.5685 2.86079
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4.00000 −0.201517
$$395$$ 22.6274 1.13851
$$396$$ 0 0
$$397$$ −7.07107 −0.354887 −0.177443 0.984131i $$-0.556783\pi$$
−0.177443 + 0.984131i $$0.556783\pi$$
$$398$$ −16.9706 −0.850657
$$399$$ 0 0
$$400$$ −3.00000 −0.150000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −12.7279 −0.633238
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.0000 0.793091
$$408$$ 0 0
$$409$$ 21.2132 1.04893 0.524463 0.851433i $$-0.324266\pi$$
0.524463 + 0.851433i $$0.324266\pi$$
$$410$$ 14.0000 0.691411
$$411$$ 0 0
$$412$$ −5.65685 −0.278693
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ −4.24264 −0.208013
$$417$$ 0 0
$$418$$ 22.6274 1.10674
$$419$$ −22.6274 −1.10542 −0.552711 0.833373i $$-0.686407\pi$$
−0.552711 + 0.833373i $$0.686407\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 4.00000 0.194257
$$425$$ −21.2132 −1.02899
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 5.65685 0.272798
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 4.24264 0.203888 0.101944 0.994790i $$-0.467494\pi$$
0.101944 + 0.994790i $$0.467494\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ 45.2548 2.16483
$$438$$ 0 0
$$439$$ −33.9411 −1.61992 −0.809961 0.586484i $$-0.800512\pi$$
−0.809961 + 0.586484i $$0.800512\pi$$
$$440$$ −5.65685 −0.269680
$$441$$ 0 0
$$442$$ −30.0000 −1.42695
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 16.9706 0.803579
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ −39.5980 −1.86460
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −16.9706 −0.796468
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ −12.7279 −0.594737
$$459$$ 0 0
$$460$$ −11.3137 −0.527504
$$461$$ 1.41421 0.0658665 0.0329332 0.999458i $$-0.489515\pi$$
0.0329332 + 0.999458i $$0.489515\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −5.65685 −0.261768 −0.130884 0.991398i $$-0.541782\pi$$
−0.130884 + 0.991398i $$0.541782\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 11.3137 0.520756
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −16.9706 −0.778663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ 28.2843 1.29234 0.646171 0.763193i $$-0.276369\pi$$
0.646171 + 0.763193i $$0.276369\pi$$
$$480$$ 0 0
$$481$$ −16.9706 −0.773791
$$482$$ −4.24264 −0.193247
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 10.0000 0.454077
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ −1.41421 −0.0640184
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 14.1421 0.636930
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 11.3137 0.505964
$$501$$ 0 0
$$502$$ 5.65685 0.252478
$$503$$ −28.2843 −1.26113 −0.630567 0.776135i $$-0.717177\pi$$
−0.630567 + 0.776135i $$0.717177\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 32.0000 1.42257
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −32.5269 −1.44173 −0.720865 0.693075i $$-0.756255\pi$$
−0.720865 + 0.693075i $$0.756255\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 12.7279 0.561405
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −22.6274 −0.995153
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 6.00000 0.263117
$$521$$ −1.41421 −0.0619578 −0.0309789 0.999520i $$-0.509862\pi$$
−0.0309789 + 0.999520i $$0.509862\pi$$
$$522$$ 0 0
$$523$$ 33.9411 1.48414 0.742071 0.670321i $$-0.233844\pi$$
0.742071 + 0.670321i $$0.233844\pi$$
$$524$$ 16.9706 0.741362
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −5.65685 −0.245718
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 42.0000 1.81922
$$534$$ 0 0
$$535$$ 5.65685 0.244567
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 1.41421 0.0609711
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 5.65685 0.242983
$$543$$ 0 0
$$544$$ 7.07107 0.303170
$$545$$ −5.65685 −0.242313
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ −12.0000 −0.511682
$$551$$ 11.3137 0.481980
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 5.65685 0.239904
$$557$$ 36.0000 1.52537 0.762684 0.646771i $$-0.223881\pi$$
0.762684 + 0.646771i $$0.223881\pi$$
$$558$$ 0 0
$$559$$ 16.9706 0.717778
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 11.3137 0.476816 0.238408 0.971165i $$-0.423374\pi$$
0.238408 + 0.971165i $$0.423374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −22.6274 −0.951101
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ −16.9706 −0.709575
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −24.0000 −1.00087
$$576$$ 0 0
$$577$$ −12.7279 −0.529870 −0.264935 0.964266i $$-0.585351\pi$$
−0.264935 + 0.964266i $$0.585351\pi$$
$$578$$ 33.0000 1.37262
$$579$$ 0 0
$$580$$ −2.82843 −0.117444
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 16.0000 0.662652
$$584$$ 15.5563 0.643726
$$585$$ 0 0
$$586$$ −24.0416 −0.993151
$$587$$ 16.9706 0.700450 0.350225 0.936666i $$-0.386105\pi$$
0.350225 + 0.936666i $$0.386105\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −16.0000 −0.658710
$$591$$ 0 0
$$592$$ 4.00000 0.164399
$$593$$ 35.3553 1.45187 0.725935 0.687763i $$-0.241407\pi$$
0.725935 + 0.687763i $$0.241407\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ −33.9411 −1.38796
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −4.24264 −0.173061 −0.0865305 0.996249i $$-0.527578\pi$$
−0.0865305 + 0.996249i $$0.527578\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −16.0000 −0.651031
$$605$$ −7.07107 −0.287480
$$606$$ 0 0
$$607$$ −16.9706 −0.688814 −0.344407 0.938820i $$-0.611920\pi$$
−0.344407 + 0.938820i $$0.611920\pi$$
$$608$$ 5.65685 0.229416
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ 12.0000 0.484675 0.242338 0.970192i $$-0.422086\pi$$
0.242338 + 0.970192i $$0.422086\pi$$
$$614$$ −5.65685 −0.228292
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ −11.3137 −0.454736 −0.227368 0.973809i $$-0.573012\pi$$
−0.227368 + 0.973809i $$0.573012\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 5.65685 0.226819
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ −21.2132 −0.847850
$$627$$ 0 0
$$628$$ 18.3848 0.733632
$$629$$ 28.2843 1.12777
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 0 0
$$634$$ 28.0000 1.11202
$$635$$ 11.3137 0.448971
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ −1.41421 −0.0559017
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −11.3137 −0.446169 −0.223085 0.974799i $$-0.571613\pi$$
−0.223085 + 0.974799i $$0.571613\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.0000 1.57378
$$647$$ −33.9411 −1.33436 −0.667182 0.744895i $$-0.732500\pi$$
−0.667182 + 0.744895i $$0.732500\pi$$
$$648$$ 0 0
$$649$$ 45.2548 1.77641
$$650$$ 12.7279 0.499230
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ −9.89949 −0.386510
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −4.24264 −0.165020 −0.0825098 0.996590i $$-0.526294\pi$$
−0.0825098 + 0.996590i $$0.526294\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 5.65685 0.219529
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ −11.3137 −0.437741
$$669$$ 0 0
$$670$$ 16.9706 0.655630
$$671$$ −5.65685 −0.218380
$$672$$ 0 0
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ 5.00000 0.192308
$$677$$ −4.24264 −0.163058 −0.0815290 0.996671i $$-0.525980\pi$$
−0.0815290 + 0.996671i $$0.525980\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −10.0000 −0.383482
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −8.48528 −0.324206
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −16.9706 −0.646527
$$690$$ 0 0
$$691$$ 50.9117 1.93677 0.968386 0.249457i $$-0.0802520\pi$$
0.968386 + 0.249457i $$0.0802520\pi$$
$$692$$ 12.7279 0.483843
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ −70.0000 −2.65144
$$698$$ 29.6985 1.12410
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ 22.6274 0.853409
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −1.41421 −0.0532246
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −7.07107 −0.264999
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 16.0000 0.597115
$$719$$ −39.5980 −1.47676 −0.738378 0.674387i $$-0.764408\pi$$
−0.738378 + 0.674387i $$0.764408\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 13.0000 0.483810
$$723$$ 0 0
$$724$$ −12.7279 −0.473029
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −28.2843 −1.04901 −0.524503 0.851409i $$-0.675749\pi$$
−0.524503 + 0.851409i $$0.675749\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −22.0000 −0.814257
$$731$$ −28.2843 −1.04613
$$732$$ 0 0
$$733$$ −12.7279 −0.470117 −0.235058 0.971981i $$-0.575528\pi$$
−0.235058 + 0.971981i $$0.575528\pi$$
$$734$$ −5.65685 −0.208798
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −5.65685 −0.207950
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 28.2843 1.03626
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ 28.2843 1.03418
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ −5.65685 −0.206284
$$753$$ 0 0
$$754$$ −8.48528 −0.309016
$$755$$ 22.6274 0.823496
$$756$$ 0 0
$$757$$ −28.0000 −1.01768 −0.508839 0.860862i $$-0.669925\pi$$
−0.508839 + 0.860862i $$0.669925\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ 9.89949 0.358856 0.179428 0.983771i $$-0.442575\pi$$
0.179428 + 0.983771i $$0.442575\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 5.65685 0.204390
$$767$$ −48.0000 −1.73318
$$768$$ 0 0
$$769$$ 4.24264 0.152994 0.0764968 0.997070i $$-0.475627\pi$$
0.0764968 + 0.997070i $$0.475627\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 14.0000 0.503871
$$773$$ −32.5269 −1.16991 −0.584956 0.811065i $$-0.698888\pi$$
−0.584956 + 0.811065i $$0.698888\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −7.07107 −0.253837
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ −56.0000 −2.00641
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 56.5685 2.02289
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −26.0000 −0.927980
$$786$$ 0 0
$$787$$ −5.65685 −0.201645 −0.100823 0.994904i $$-0.532147\pi$$
−0.100823 + 0.994904i $$0.532147\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ 0 0
$$790$$ 22.6274 0.805047
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 6.00000 0.213066
$$794$$ −7.07107 −0.250943
$$795$$ 0 0
$$796$$ −16.9706 −0.601506
$$797$$ −12.7279 −0.450846 −0.225423 0.974261i $$-0.572376\pi$$
−0.225423 + 0.974261i $$0.572376\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ −3.00000 −0.106066
$$801$$ 0 0
$$802$$ 18.0000 0.635602
$$803$$ 62.2254 2.19589
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −12.7279 −0.447767
$$809$$ −40.0000 −1.40633 −0.703163 0.711029i $$-0.748229\pi$$
−0.703163 + 0.711029i $$0.748229\pi$$
$$810$$ 0 0
$$811$$ −33.9411 −1.19183 −0.595917 0.803046i $$-0.703211\pi$$
−0.595917 + 0.803046i $$0.703211\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 16.0000 0.560800
$$815$$ −5.65685 −0.198151
$$816$$ 0 0
$$817$$ −22.6274 −0.791633
$$818$$ 21.2132 0.741702
$$819$$ 0 0
$$820$$ 14.0000 0.488901
$$821$$ −36.0000 −1.25641 −0.628204 0.778048i $$-0.716210\pi$$
−0.628204 + 0.778048i $$0.716210\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ −5.65685 −0.197066
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ −43.8406 −1.52265 −0.761324 0.648372i $$-0.775450\pi$$
−0.761324 + 0.648372i $$0.775450\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 0 0
$$832$$ −4.24264 −0.147087
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 22.6274 0.782586
$$837$$ 0 0
$$838$$ −22.6274 −0.781651
$$839$$ 45.2548 1.56237 0.781185 0.624299i $$-0.214615\pi$$
0.781185 + 0.624299i $$0.214615\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −6.00000 −0.206774
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ −7.07107 −0.243252
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 4.00000 0.137361
$$849$$ 0 0
$$850$$ −21.2132 −0.727607
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ −21.2132 −0.726326 −0.363163 0.931726i $$-0.618303\pi$$
−0.363163 + 0.931726i $$0.618303\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ −24.0416 −0.821246 −0.410623 0.911805i $$-0.634689\pi$$
−0.410623 + 0.911805i $$0.634689\pi$$
$$858$$ 0 0
$$859$$ −5.65685 −0.193009 −0.0965047 0.995333i $$-0.530766\pi$$
−0.0965047 + 0.995333i $$0.530766\pi$$
$$860$$ 5.65685 0.192897
$$861$$ 0 0
$$862$$ 24.0000 0.817443
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 4.24264 0.144171
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 50.9117 1.72508
$$872$$ 4.00000 0.135457
$$873$$ 0 0
$$874$$ 45.2548 1.53077
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28.0000 −0.945493 −0.472746 0.881199i $$-0.656737\pi$$
−0.472746 + 0.881199i $$0.656737\pi$$
$$878$$ −33.9411 −1.14546
$$879$$ 0 0
$$880$$ −5.65685 −0.190693
$$881$$ −21.2132 −0.714691 −0.357345 0.933972i $$-0.616318\pi$$
−0.357345 + 0.933972i $$0.616318\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −30.0000 −1.00901
$$885$$ 0 0
$$886$$ 20.0000 0.671913
$$887$$ 22.6274 0.759754 0.379877 0.925037i $$-0.375966\pi$$
0.379877 + 0.925037i $$0.375966\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ 0 0
$$892$$ 16.9706 0.568216
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 16.9706 0.567263
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −24.0000 −0.800890
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 28.2843 0.942286
$$902$$ −39.5980 −1.31847
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ −20.0000 −0.664089 −0.332045 0.943264i $$-0.607738\pi$$
−0.332045 + 0.943264i $$0.607738\pi$$
$$908$$ −16.9706 −0.563188
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 0 0
$$913$$ 22.6274 0.748858
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ −12.7279 −0.420542
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ −11.3137 −0.373002
$$921$$ 0 0
$$922$$ 1.41421 0.0465746
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ −32.0000 −1.05159
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ −18.3848 −0.603185 −0.301592 0.953437i $$-0.597518\pi$$
−0.301592 + 0.953437i $$0.597518\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −5.65685 −0.185098
$$935$$ −40.0000 −1.30814
$$936$$ 0 0
$$937$$ −15.5563 −0.508204 −0.254102 0.967177i $$-0.581780\pi$$
−0.254102 + 0.967177i $$0.581780\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ −26.8701 −0.875939 −0.437969 0.898990i $$-0.644302\pi$$
−0.437969 + 0.898990i $$0.644302\pi$$
$$942$$ 0 0
$$943$$ −79.1960 −2.57898
$$944$$ 11.3137 0.368230
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ −66.0000 −2.14245
$$950$$ −16.9706 −0.550598
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.00000 −0.259145 −0.129573 0.991570i $$-0.541361\pi$$
−0.129573 + 0.991570i $$0.541361\pi$$
$$954$$ 0 0
$$955$$ 22.6274 0.732206
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 28.2843 0.913823
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −16.9706 −0.547153
$$963$$ 0 0
$$964$$ −4.24264 −0.136646
$$965$$ −19.7990 −0.637352
$$966$$ 0 0
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ 11.3137 0.363074 0.181537 0.983384i $$-0.441893\pi$$
0.181537 + 0.983384i $$0.441893\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −24.0000 −0.769010
$$975$$ 0 0
$$976$$ −1.41421 −0.0452679
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −28.2843 −0.903969
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ −11.3137 −0.360851 −0.180426 0.983589i $$-0.557748\pi$$
−0.180426 + 0.983589i $$0.557748\pi$$
$$984$$ 0 0
$$985$$ 5.65685 0.180242
$$986$$ 14.1421 0.450377
$$987$$ 0 0
$$988$$ −24.0000 −0.763542
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 24.0000 0.760851
$$996$$ 0 0
$$997$$ 7.07107 0.223943 0.111971 0.993711i $$-0.464283\pi$$
0.111971 + 0.993711i $$0.464283\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 0 0
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 882.2.a.o.1.1 yes 2
3.2 odd 2 882.2.a.m.1.2 yes 2
4.3 odd 2 7056.2.a.ci.1.1 2
7.2 even 3 882.2.g.k.361.2 4
7.3 odd 6 882.2.g.k.667.1 4
7.4 even 3 882.2.g.k.667.2 4
7.5 odd 6 882.2.g.k.361.1 4
7.6 odd 2 inner 882.2.a.o.1.2 yes 2
12.11 even 2 7056.2.a.cs.1.2 2
21.2 odd 6 882.2.g.m.361.1 4
21.5 even 6 882.2.g.m.361.2 4
21.11 odd 6 882.2.g.m.667.1 4
21.17 even 6 882.2.g.m.667.2 4
21.20 even 2 882.2.a.m.1.1 2
28.27 even 2 7056.2.a.ci.1.2 2
84.83 odd 2 7056.2.a.cs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.a.m.1.1 2 21.20 even 2
882.2.a.m.1.2 yes 2 3.2 odd 2
882.2.a.o.1.1 yes 2 1.1 even 1 trivial
882.2.a.o.1.2 yes 2 7.6 odd 2 inner
882.2.g.k.361.1 4 7.5 odd 6
882.2.g.k.361.2 4 7.2 even 3
882.2.g.k.667.1 4 7.3 odd 6
882.2.g.k.667.2 4 7.4 even 3
882.2.g.m.361.1 4 21.2 odd 6
882.2.g.m.361.2 4 21.5 even 6
882.2.g.m.667.1 4 21.11 odd 6
882.2.g.m.667.2 4 21.17 even 6
7056.2.a.ci.1.1 2 4.3 odd 2
7056.2.a.ci.1.2 2 28.27 even 2
7056.2.a.cs.1.1 2 84.83 odd 2
7056.2.a.cs.1.2 2 12.11 even 2