# Properties

 Label 7056.2.a.cy.1.4 Level 7056 Weight 2 Character 7056.1 Self dual yes Analytic conductor 56.342 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1764) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.57794$$ of defining polynomial Character $$\chi$$ $$=$$ 7056.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.74166 q^{5} +O(q^{10})$$ $$q+3.74166 q^{5} +5.29150 q^{11} -4.24264 q^{13} +3.74166 q^{17} -2.82843 q^{19} -5.29150 q^{23} +9.00000 q^{25} +5.29150 q^{29} -8.48528 q^{31} +4.00000 q^{37} +3.74166 q^{41} -8.00000 q^{43} +7.48331 q^{47} +10.5830 q^{53} +19.7990 q^{55} +7.48331 q^{59} +9.89949 q^{61} -15.8745 q^{65} -12.0000 q^{67} +15.8745 q^{71} +1.41421 q^{73} +4.00000 q^{79} +14.9666 q^{83} +14.0000 q^{85} -3.74166 q^{89} -10.5830 q^{95} -9.89949 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 36q^{25} + 16q^{37} - 32q^{43} - 48q^{67} + 16q^{79} + 56q^{85} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.74166 1.67332 0.836660 0.547723i $$-0.184505\pi$$
0.836660 + 0.547723i $$0.184505\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.29150 1.59545 0.797724 0.603023i $$-0.206037\pi$$
0.797724 + 0.603023i $$0.206037\pi$$
$$12$$ 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
$$16$$ 0 0
$$17$$ 3.74166 0.907485 0.453743 0.891133i $$-0.350089\pi$$
0.453743 + 0.891133i $$0.350089\pi$$
$$18$$ 0 0
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −5.29150 −1.10335 −0.551677 0.834058i $$-0.686012\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ 0 0
$$25$$ 9.00000 1.80000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5.29150 0.982607 0.491304 0.870988i $$-0.336521\pi$$
0.491304 + 0.870988i $$0.336521\pi$$
$$30$$ 0 0
$$31$$ −8.48528 −1.52400 −0.762001 0.647576i $$-0.775783\pi$$
−0.762001 + 0.647576i $$0.775783\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.74166 0.584349 0.292174 0.956365i $$-0.405621\pi$$
0.292174 + 0.956365i $$0.405621\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.48331 1.09155 0.545777 0.837931i $$-0.316235\pi$$
0.545777 + 0.837931i $$0.316235\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.5830 1.45369 0.726844 0.686803i $$-0.240986\pi$$
0.726844 + 0.686803i $$0.240986\pi$$
$$54$$ 0 0
$$55$$ 19.7990 2.66970
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 7.48331 0.974245 0.487122 0.873334i $$-0.338047\pi$$
0.487122 + 0.873334i $$0.338047\pi$$
$$60$$ 0 0
$$61$$ 9.89949 1.26750 0.633750 0.773538i $$-0.281515\pi$$
0.633750 + 0.773538i $$0.281515\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −15.8745 −1.96899
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 15.8745 1.88396 0.941979 0.335673i $$-0.108964\pi$$
0.941979 + 0.335673i $$0.108964\pi$$
$$72$$ 0 0
$$73$$ 1.41421 0.165521 0.0827606 0.996569i $$-0.473626\pi$$
0.0827606 + 0.996569i $$0.473626\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 14.9666 1.64280 0.821401 0.570352i $$-0.193193\pi$$
0.821401 + 0.570352i $$0.193193\pi$$
$$84$$ 0 0
$$85$$ 14.0000 1.51851
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.74166 −0.396615 −0.198307 0.980140i $$-0.563544\pi$$
−0.198307 + 0.980140i $$0.563544\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −10.5830 −1.08579
$$96$$ 0 0
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −11.2250 −1.11693 −0.558463 0.829529i $$-0.688609\pi$$
−0.558463 + 0.829529i $$0.688609\pi$$
$$102$$ 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
$$107$$ −5.29150 −0.511549 −0.255774 0.966736i $$-0.582330\pi$$
−0.255774 + 0.966736i $$0.582330\pi$$
$$108$$ 0 0
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −19.7990 −1.84627
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 17.0000 1.54545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 14.9666 1.33866
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.8745 −1.35625 −0.678125 0.734946i $$-0.737207\pi$$
−0.678125 + 0.734946i $$0.737207\pi$$
$$138$$ 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
$$143$$ −22.4499 −1.87736
$$144$$ 0 0
$$145$$ 19.7990 1.64422
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.5830 0.866994 0.433497 0.901155i $$-0.357280\pi$$
0.433497 + 0.901155i $$0.357280\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −31.7490 −2.55014
$$156$$ 0 0
$$157$$ −9.89949 −0.790066 −0.395033 0.918667i $$-0.629267\pi$$
−0.395033 + 0.918667i $$0.629267\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.48331 −0.579076 −0.289538 0.957166i $$-0.593502\pi$$
−0.289538 + 0.957166i $$0.593502\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 11.2250 0.853419 0.426709 0.904389i $$-0.359673\pi$$
0.426709 + 0.904389i $$0.359673\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 15.8745 1.18652 0.593258 0.805012i $$-0.297841\pi$$
0.593258 + 0.805012i $$0.297841\pi$$
$$180$$ 0 0
$$181$$ 4.24264 0.315353 0.157676 0.987491i $$-0.449600\pi$$
0.157676 + 0.987491i $$0.449600\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 14.9666 1.10037
$$186$$ 0 0
$$187$$ 19.7990 1.44785
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.29150 −0.382880 −0.191440 0.981504i $$-0.561316\pi$$
−0.191440 + 0.981504i $$0.561316\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.5830 −0.754008 −0.377004 0.926212i $$-0.623046\pi$$
−0.377004 + 0.926212i $$0.623046\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 14.0000 0.977802
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −14.9666 −1.03526
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −29.9333 −2.04143
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −15.8745 −1.06783
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 22.4499 1.49006 0.745028 0.667034i $$-0.232436\pi$$
0.745028 + 0.667034i $$0.232436\pi$$
$$228$$ 0 0
$$229$$ −12.7279 −0.841085 −0.420542 0.907273i $$-0.638160\pi$$
−0.420542 + 0.907273i $$0.638160\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 15.8745 1.03997 0.519987 0.854174i $$-0.325937\pi$$
0.519987 + 0.854174i $$0.325937\pi$$
$$234$$ 0 0
$$235$$ 28.0000 1.82652
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 15.8745 1.02684 0.513418 0.858138i $$-0.328379\pi$$
0.513418 + 0.858138i $$0.328379\pi$$
$$240$$ 0 0
$$241$$ −29.6985 −1.91305 −0.956524 0.291654i $$-0.905794\pi$$
−0.956524 + 0.291654i $$0.905794\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −7.48331 −0.472343 −0.236171 0.971711i $$-0.575893\pi$$
−0.236171 + 0.971711i $$0.575893\pi$$
$$252$$ 0 0
$$253$$ −28.0000 −1.76034
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −11.2250 −0.700195 −0.350097 0.936713i $$-0.613851\pi$$
−0.350097 + 0.936713i $$0.613851\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −15.8745 −0.978864 −0.489432 0.872041i $$-0.662796\pi$$
−0.489432 + 0.872041i $$0.662796\pi$$
$$264$$ 0 0
$$265$$ 39.5980 2.43248
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.74166 −0.228133 −0.114066 0.993473i $$-0.536388\pi$$
−0.114066 + 0.993473i $$0.536388\pi$$
$$270$$ 0 0
$$271$$ −19.7990 −1.20270 −0.601351 0.798985i $$-0.705371\pi$$
−0.601351 + 0.798985i $$0.705371\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 47.6235 2.87181
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.29150 −0.315665 −0.157832 0.987466i $$-0.550451\pi$$
−0.157832 + 0.987466i $$0.550451\pi$$
$$282$$ 0 0
$$283$$ −31.1127 −1.84946 −0.924729 0.380626i $$-0.875708\pi$$
−0.924729 + 0.380626i $$0.875708\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3.00000 −0.176471
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −26.1916 −1.53013 −0.765065 0.643953i $$-0.777293\pi$$
−0.765065 + 0.643953i $$0.777293\pi$$
$$294$$ 0 0
$$295$$ 28.0000 1.63022
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 22.4499 1.29831
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 37.0405 2.12093
$$306$$ 0 0
$$307$$ −14.1421 −0.807134 −0.403567 0.914950i $$-0.632230\pi$$
−0.403567 + 0.914950i $$0.632230\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −7.48331 −0.424340 −0.212170 0.977233i $$-0.568053\pi$$
−0.212170 + 0.977233i $$0.568053\pi$$
$$312$$ 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
$$317$$ 10.5830 0.594401 0.297200 0.954815i $$-0.403947\pi$$
0.297200 + 0.954815i $$0.403947\pi$$
$$318$$ 0 0
$$319$$ 28.0000 1.56770
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −10.5830 −0.588854
$$324$$ 0 0
$$325$$ −38.1838 −2.11805
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −44.8999 −2.45314
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −44.8999 −2.43147
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −15.8745 −0.852188 −0.426094 0.904679i $$-0.640111\pi$$
−0.426094 + 0.904679i $$0.640111\pi$$
$$348$$ 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
$$352$$ 0 0
$$353$$ −18.7083 −0.995742 −0.497871 0.867251i $$-0.665885\pi$$
−0.497871 + 0.867251i $$0.665885\pi$$
$$354$$ 0 0
$$355$$ 59.3970 3.15246
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.29150 0.279275 0.139637 0.990203i $$-0.455406\pi$$
0.139637 + 0.990203i $$0.455406\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5.29150 0.276970
$$366$$ 0 0
$$367$$ 11.3137 0.590571 0.295285 0.955409i $$-0.404585\pi$$
0.295285 + 0.955409i $$0.404585\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −22.4499 −1.15623
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 14.9666 0.764759 0.382380 0.924005i $$-0.375105\pi$$
0.382380 + 0.924005i $$0.375105\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 5.29150 0.268290 0.134145 0.990962i $$-0.457171\pi$$
0.134145 + 0.990962i $$0.457171\pi$$
$$390$$ 0 0
$$391$$ −19.7990 −1.00128
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 14.9666 0.753053
$$396$$ 0 0
$$397$$ 32.5269 1.63248 0.816239 0.577714i $$-0.196055\pi$$
0.816239 + 0.577714i $$0.196055\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15.8745 0.792735 0.396368 0.918092i $$-0.370271\pi$$
0.396368 + 0.918092i $$0.370271\pi$$
$$402$$ 0 0
$$403$$ 36.0000 1.79329
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 21.1660 1.04916
$$408$$ 0 0
$$409$$ −21.2132 −1.04893 −0.524463 0.851433i $$-0.675734\pi$$
−0.524463 + 0.851433i $$0.675734\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 56.0000 2.74893
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7.48331 0.365584 0.182792 0.983152i $$-0.441487\pi$$
0.182792 + 0.983152i $$0.441487\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 33.6749 1.63347
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.8745 0.764648 0.382324 0.924028i $$-0.375124\pi$$
0.382324 + 0.924028i $$0.375124\pi$$
$$432$$ 0 0
$$433$$ −4.24264 −0.203888 −0.101944 0.994790i $$-0.532506\pi$$
−0.101944 + 0.994790i $$0.532506\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 14.9666 0.715951
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −5.29150 −0.251407 −0.125703 0.992068i $$-0.540119\pi$$
−0.125703 + 0.992068i $$0.540119\pi$$
$$444$$ 0 0
$$445$$ −14.0000 −0.663664
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 19.7990 0.932298
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.74166 −0.174266 −0.0871332 0.996197i $$-0.527771\pi$$
−0.0871332 + 0.996197i $$0.527771\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −37.4166 −1.73143 −0.865716 0.500535i $$-0.833137\pi$$
−0.865716 + 0.500535i $$0.833137\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −42.3320 −1.94643
$$474$$ 0 0
$$475$$ −25.4558 −1.16799
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −37.4166 −1.70961 −0.854803 0.518952i $$-0.826322\pi$$
−0.854803 + 0.518952i $$0.826322\pi$$
$$480$$ 0 0
$$481$$ −16.9706 −0.773791
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −37.0405 −1.68192
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15.8745 −0.716407 −0.358203 0.933644i $$-0.616611\pi$$
−0.358203 + 0.933644i $$0.616611\pi$$
$$492$$ 0 0
$$493$$ 19.7990 0.891702
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14.9666 0.667329 0.333665 0.942692i $$-0.391715\pi$$
0.333665 + 0.942692i $$0.391715\pi$$
$$504$$ 0 0
$$505$$ −42.0000 −1.86898
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 41.1582 1.82431 0.912153 0.409849i $$-0.134419\pi$$
0.912153 + 0.409849i $$0.134419\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 10.5830 0.466343
$$516$$ 0 0
$$517$$ 39.5980 1.74152
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.7083 −0.819625 −0.409812 0.912170i $$-0.634406\pi$$
−0.409812 + 0.912170i $$0.634406\pi$$
$$522$$ 0 0
$$523$$ 16.9706 0.742071 0.371035 0.928619i $$-0.379003\pi$$
0.371035 + 0.928619i $$0.379003\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −31.7490 −1.38301
$$528$$ 0 0
$$529$$ 5.00000 0.217391
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −15.8745 −0.687601
$$534$$ 0 0
$$535$$ −19.7990 −0.855985
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$545$$ 59.8665 2.56440
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −14.9666 −0.637600
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −31.7490 −1.34525 −0.672624 0.739984i $$-0.734833\pi$$
−0.672624 + 0.739984i $$0.734833\pi$$
$$558$$ 0 0
$$559$$ 33.9411 1.43556
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 7.48331 0.315384 0.157692 0.987488i $$-0.449595\pi$$
0.157692 + 0.987488i $$0.449595\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 26.4575 1.10916 0.554578 0.832132i $$-0.312880\pi$$
0.554578 + 0.832132i $$0.312880\pi$$
$$570$$ 0 0
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −47.6235 −1.98604
$$576$$ 0 0
$$577$$ −21.2132 −0.883117 −0.441559 0.897232i $$-0.645574\pi$$
−0.441559 + 0.897232i $$0.645574\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 56.0000 2.31928
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 22.4499 0.926608 0.463304 0.886199i $$-0.346664\pi$$
0.463304 + 0.886199i $$0.346664\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −26.1916 −1.07556 −0.537780 0.843085i $$-0.680737\pi$$
−0.537780 + 0.843085i $$0.680737\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −5.29150 −0.216205 −0.108102 0.994140i $$-0.534477\pi$$
−0.108102 + 0.994140i $$0.534477\pi$$
$$600$$ 0 0
$$601$$ 4.24264 0.173061 0.0865305 0.996249i $$-0.472422\pi$$
0.0865305 + 0.996249i $$0.472422\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 63.6082 2.58604
$$606$$ 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
$$611$$ −31.7490 −1.28443
$$612$$ 0 0
$$613$$ −24.0000 −0.969351 −0.484675 0.874694i $$-0.661062\pi$$
−0.484675 + 0.874694i $$0.661062\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.4575 1.06514 0.532570 0.846386i $$-0.321226\pi$$
0.532570 + 0.846386i $$0.321226\pi$$
$$618$$ 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
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 14.9666 0.596759
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 29.9333 1.18787
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 37.0405 1.46301 0.731506 0.681835i $$-0.238818\pi$$
0.731506 + 0.681835i $$0.238818\pi$$
$$642$$ 0 0
$$643$$ 14.1421 0.557711 0.278856 0.960333i $$-0.410045\pi$$
0.278856 + 0.960333i $$0.410045\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.4499 0.882598 0.441299 0.897360i $$-0.354518\pi$$
0.441299 + 0.897360i $$0.354518\pi$$
$$648$$ 0 0
$$649$$ 39.5980 1.55436
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −15.8745 −0.621217 −0.310609 0.950538i $$-0.600533\pi$$
−0.310609 + 0.950538i $$0.600533\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 47.6235 1.85515 0.927575 0.373638i $$-0.121890\pi$$
0.927575 + 0.373638i $$0.121890\pi$$
$$660$$ 0 0
$$661$$ −21.2132 −0.825098 −0.412549 0.910935i $$-0.635361\pi$$
−0.412549 + 0.910935i $$0.635361\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −28.0000 −1.08416
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 52.3832 2.02223
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −33.6749 −1.29423 −0.647116 0.762391i $$-0.724025\pi$$
−0.647116 + 0.762391i $$0.724025\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 15.8745 0.607421 0.303711 0.952764i $$-0.401774\pi$$
0.303711 + 0.952764i $$0.401774\pi$$
$$684$$ 0 0
$$685$$ −59.3970 −2.26944
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −44.8999 −1.71055
$$690$$ 0 0
$$691$$ 16.9706 0.645591 0.322795 0.946469i $$-0.395377\pi$$
0.322795 + 0.946469i $$0.395377\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 21.1660 0.802873
$$696$$ 0 0
$$697$$ 14.0000 0.530288
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 15.8745 0.599572 0.299786 0.954006i $$-0.403085\pi$$
0.299786 + 0.954006i $$0.403085\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 44.8999 1.68151
$$714$$ 0 0
$$715$$ −84.0000 −3.14142
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 29.9333 1.11632 0.558161 0.829733i $$-0.311507\pi$$
0.558161 + 0.829733i $$0.311507\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 47.6235 1.76869
$$726$$ 0 0
$$727$$ 14.1421 0.524503 0.262251 0.965000i $$-0.415535\pi$$
0.262251 + 0.965000i $$0.415535\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −29.9333 −1.10712
$$732$$ 0 0
$$733$$ −46.6690 −1.72376 −0.861880 0.507112i $$-0.830713\pi$$
−0.861880 + 0.507112i $$0.830713\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −63.4980 −2.33898
$$738$$ 0 0
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 37.0405 1.35888 0.679442 0.733729i $$-0.262222\pi$$
0.679442 + 0.733729i $$0.262222\pi$$
$$744$$ 0 0
$$745$$ 39.5980 1.45076
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 14.9666 0.544691
$$756$$ 0 0
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 41.1582 1.49198 0.745992 0.665955i $$-0.231976\pi$$
0.745992 + 0.665955i $$0.231976\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −31.7490 −1.14639
$$768$$ 0 0
$$769$$ −4.24264 −0.152994 −0.0764968 0.997070i $$-0.524373\pi$$
−0.0764968 + 0.997070i $$0.524373\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −48.6415 −1.74951 −0.874757 0.484561i $$-0.838979\pi$$
−0.874757 + 0.484561i $$0.838979\pi$$
$$774$$ 0 0
$$775$$ −76.3675 −2.74320
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −10.5830 −0.379176
$$780$$ 0 0
$$781$$ 84.0000 3.00576
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −37.0405 −1.32203
$$786$$ 0 0
$$787$$ −39.5980 −1.41152 −0.705758 0.708453i $$-0.749393\pi$$
−0.705758 + 0.708453i $$0.749393\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −42.0000 −1.49146
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −11.2250 −0.397609 −0.198804 0.980039i $$-0.563706\pi$$
−0.198804 + 0.980039i $$0.563706\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 7.48331 0.264080
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 21.1660 0.744157 0.372079 0.928201i $$-0.378645\pi$$
0.372079 + 0.928201i $$0.378645\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 74.8331 2.62129
$$816$$ 0 0
$$817$$ 22.6274 0.791633
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −31.7490 −1.10805 −0.554024 0.832501i $$-0.686908\pi$$
−0.554024 + 0.832501i $$0.686908\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 15.8745 0.552011 0.276005 0.961156i $$-0.410989\pi$$
0.276005 + 0.961156i $$0.410989\pi$$
$$828$$ 0 0
$$829$$ 18.3848 0.638530 0.319265 0.947666i $$-0.396564\pi$$
0.319265 + 0.947666i $$0.396564\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −28.0000 −0.968980
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 7.48331 0.258353 0.129176 0.991622i $$-0.458767\pi$$
0.129176 + 0.991622i $$0.458767\pi$$
$$840$$ 0 0
$$841$$ −1.00000 −0.0344828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 18.7083 0.643585
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −21.1660 −0.725561
$$852$$ 0 0
$$853$$ −55.1543 −1.88845 −0.944224 0.329304i $$-0.893186\pi$$
−0.944224 + 0.329304i $$0.893186\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 41.1582 1.40594 0.702969 0.711220i $$-0.251857\pi$$
0.702969 + 0.711220i $$0.251857\pi$$
$$858$$ 0 0
$$859$$ 19.7990 0.675533 0.337766 0.941230i $$-0.390329\pi$$
0.337766 + 0.941230i $$0.390329\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −5.29150 −0.180125 −0.0900624 0.995936i $$-0.528707\pi$$
−0.0900624 + 0.995936i $$0.528707\pi$$
$$864$$ 0 0
$$865$$ 42.0000 1.42804
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 21.1660 0.718008
$$870$$ 0 0
$$871$$ 50.9117 1.72508
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −56.1249 −1.89089 −0.945447 0.325775i $$-0.894375\pi$$
−0.945447 + 0.325775i $$0.894375\pi$$
$$882$$ 0 0
$$883$$ −8.00000 −0.269221 −0.134611 0.990899i $$-0.542978\pi$$
−0.134611 + 0.990899i $$0.542978\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 37.4166 1.25633 0.628163 0.778082i $$-0.283807\pi$$
0.628163 + 0.778082i $$0.283807\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −21.1660 −0.708294
$$894$$ 0 0
$$895$$ 59.3970 1.98542
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −44.8999 −1.49750
$$900$$ 0 0
$$901$$ 39.5980 1.31920
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 15.8745 0.527686
$$906$$ 0 0
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.29150 −0.175315 −0.0876577 0.996151i $$-0.527938\pi$$
−0.0876577 + 0.996151i $$0.527938\pi$$
$$912$$ 0 0
$$913$$ 79.1960 2.62100
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −67.3498 −2.21685
$$924$$ 0 0
$$925$$ 36.0000 1.18367
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −18.7083 −0.613799 −0.306899 0.951742i $$-0.599292\pi$$
−0.306899 + 0.951742i $$0.599292\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 74.0810 2.42271
$$936$$ 0 0
$$937$$ −35.3553 −1.15501 −0.577504 0.816388i $$-0.695973\pi$$
−0.577504 + 0.816388i $$0.695973\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 26.1916 0.853822 0.426911 0.904294i $$-0.359602\pi$$
0.426911 + 0.904294i $$0.359602\pi$$
$$942$$ 0 0
$$943$$ −19.7990 −0.644744
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −47.6235 −1.54756 −0.773778 0.633457i $$-0.781636\pi$$
−0.773778 + 0.633457i $$0.781636\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 42.3320 1.37127 0.685634 0.727946i $$-0.259525\pi$$
0.685634 + 0.727946i $$0.259525\pi$$
$$954$$ 0 0
$$955$$ −19.7990 −0.640680
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 41.0000 1.32258
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 7.48331 0.240896
$$966$$ 0 0
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.9666 −0.480302 −0.240151 0.970736i $$-0.577197\pi$$
−0.240151 + 0.970736i $$0.577197\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −15.8745 −0.507871 −0.253935 0.967221i $$-0.581725\pi$$
−0.253935 + 0.967221i $$0.581725\pi$$
$$978$$ 0 0
$$979$$ −19.7990 −0.632778
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 14.9666 0.477361 0.238681 0.971098i $$-0.423285\pi$$
0.238681 + 0.971098i $$0.423285\pi$$
$$984$$ 0 0
$$985$$ −39.5980 −1.26170
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 42.3320 1.34608
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.41421 0.0447886 0.0223943 0.999749i $$-0.492871\pi$$
0.0223943 + 0.999749i $$0.492871\pi$$
$$998$$ 0 0
$$999$$ 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 7056.2.a.cy.1.4 4
3.2 odd 2 inner 7056.2.a.cy.1.1 4
4.3 odd 2 1764.2.a.m.1.3 yes 4
7.6 odd 2 inner 7056.2.a.cy.1.2 4
12.11 even 2 1764.2.a.m.1.2 yes 4
21.20 even 2 inner 7056.2.a.cy.1.3 4
28.3 even 6 1764.2.k.m.1549.4 8
28.11 odd 6 1764.2.k.m.1549.2 8
28.19 even 6 1764.2.k.m.361.4 8
28.23 odd 6 1764.2.k.m.361.2 8
28.27 even 2 1764.2.a.m.1.1 4
84.11 even 6 1764.2.k.m.1549.3 8
84.23 even 6 1764.2.k.m.361.3 8
84.47 odd 6 1764.2.k.m.361.1 8
84.59 odd 6 1764.2.k.m.1549.1 8
84.83 odd 2 1764.2.a.m.1.4 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.a.m.1.1 4 28.27 even 2
1764.2.a.m.1.2 yes 4 12.11 even 2
1764.2.a.m.1.3 yes 4 4.3 odd 2
1764.2.a.m.1.4 yes 4 84.83 odd 2
1764.2.k.m.361.1 8 84.47 odd 6
1764.2.k.m.361.2 8 28.23 odd 6
1764.2.k.m.361.3 8 84.23 even 6
1764.2.k.m.361.4 8 28.19 even 6
1764.2.k.m.1549.1 8 84.59 odd 6
1764.2.k.m.1549.2 8 28.11 odd 6
1764.2.k.m.1549.3 8 84.11 even 6
1764.2.k.m.1549.4 8 28.3 even 6
7056.2.a.cy.1.1 4 3.2 odd 2 inner
7056.2.a.cy.1.2 4 7.6 odd 2 inner
7056.2.a.cy.1.3 4 21.20 even 2 inner
7056.2.a.cy.1.4 4 1.1 even 1 trivial