# Properties

 Label 441.6.a.o.1.2 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.2 Root $$2.64575$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} +O(q^{10})$$ $$q+2.64575 q^{2} -25.0000 q^{4} -150.808 q^{8} -799.017 q^{11} +401.000 q^{16} -2114.00 q^{22} -1105.92 q^{23} -3125.00 q^{25} +5386.75 q^{29} +5886.80 q^{32} +8886.00 q^{37} -11748.0 q^{43} +19975.4 q^{44} -2926.00 q^{46} -8267.97 q^{50} -32712.1 q^{53} +14252.0 q^{58} +2743.00 q^{64} +69364.0 q^{67} +84923.3 q^{71} +23510.1 q^{74} +80168.0 q^{79} -31082.3 q^{86} +120498. q^{88} +27648.1 q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 50 q^{4}+O(q^{10})$$ 2 * q - 50 * q^4 $$2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88}+O(q^{100})$$ 2 * q - 50 * q^4 + 802 * q^16 - 4228 * q^22 - 6250 * q^25 + 17772 * q^37 - 23496 * q^43 - 5852 * q^46 + 28504 * q^58 + 5486 * q^64 + 138728 * q^67 + 160336 * q^79 + 240996 * q^88

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.64575 0.467707 0.233854 0.972272i $$-0.424866\pi$$
0.233854 + 0.972272i $$0.424866\pi$$
$$3$$ 0 0
$$4$$ −25.0000 −0.781250
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −150.808 −0.833103
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −799.017 −1.99101 −0.995507 0.0946895i $$-0.969814\pi$$
−0.995507 + 0.0946895i $$0.969814\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 401.000 0.391602
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2114.00 −0.931211
$$23$$ −1105.92 −0.435919 −0.217959 0.975958i $$-0.569940\pi$$
−0.217959 + 0.975958i $$0.569940\pi$$
$$24$$ 0 0
$$25$$ −3125.00 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5386.75 1.18941 0.594705 0.803944i $$-0.297269\pi$$
0.594705 + 0.803944i $$0.297269\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5886.80 1.01626
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8886.00 1.06709 0.533546 0.845771i $$-0.320859\pi$$
0.533546 + 0.845771i $$0.320859\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −11748.0 −0.968931 −0.484465 0.874810i $$-0.660986\pi$$
−0.484465 + 0.874810i $$0.660986\pi$$
$$44$$ 19975.4 1.55548
$$45$$ 0 0
$$46$$ −2926.00 −0.203882
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −8267.97 −0.467707
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −32712.1 −1.59963 −0.799813 0.600250i $$-0.795068\pi$$
−0.799813 + 0.600250i $$0.795068\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 14252.0 0.556296
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 2743.00 0.0837097
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 69364.0 1.88776 0.943881 0.330286i $$-0.107145\pi$$
0.943881 + 0.330286i $$0.107145\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 84923.3 1.99931 0.999657 0.0261794i $$-0.00833410\pi$$
0.999657 + 0.0261794i $$0.00833410\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 23510.1 0.499087
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 80168.0 1.44522 0.722609 0.691257i $$-0.242943\pi$$
0.722609 + 0.691257i $$0.242943\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −31082.3 −0.453176
$$87$$ 0 0
$$88$$ 120498. 1.65872
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 27648.1 0.340562
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 78125.0 0.781250
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −86548.0 −0.748156
$$107$$ 227794. 1.92346 0.961729 0.274003i $$-0.0883478\pi$$
0.961729 + 0.274003i $$0.0883478\pi$$
$$108$$ 0 0
$$109$$ 219582. 1.77023 0.885117 0.465369i $$-0.154078\pi$$
0.885117 + 0.465369i $$0.154078\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −241906. −1.78218 −0.891089 0.453828i $$-0.850058\pi$$
−0.891089 + 0.453828i $$0.850058\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −134669. −0.929227
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 477377. 2.96414
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −262064. −1.44178 −0.720888 0.693051i $$-0.756266\pi$$
−0.720888 + 0.693051i $$0.756266\pi$$
$$128$$ −181120. −0.977107
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 183520. 0.882920
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 260998. 1.18805 0.594027 0.804445i $$-0.297537\pi$$
0.594027 + 0.804445i $$0.297537\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 224686. 0.935094
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −222150. −0.833666
$$149$$ −424474. −1.56634 −0.783168 0.621810i $$-0.786398\pi$$
−0.783168 + 0.621810i $$0.786398\pi$$
$$150$$ 0 0
$$151$$ 261624. 0.933760 0.466880 0.884321i $$-0.345378\pi$$
0.466880 + 0.884321i $$0.345378\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 212105. 0.675939
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 663100. 1.95483 0.977417 0.211318i $$-0.0677757\pi$$
0.977417 + 0.211318i $$0.0677757\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −371293. −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 293700. 0.756977
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −320406. −0.779684
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −627175. −1.46304 −0.731520 0.681820i $$-0.761189\pi$$
−0.731520 + 0.681820i $$0.761189\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 166782. 0.363166
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 103317. 0.204921 0.102461 0.994737i $$-0.467328\pi$$
0.102461 + 0.994737i $$0.467328\pi$$
$$192$$ 0 0
$$193$$ 385902. 0.745734 0.372867 0.927885i $$-0.378375\pi$$
0.372867 + 0.927885i $$0.378375\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.01882e6 −1.87038 −0.935190 0.354146i $$-0.884772\pi$$
−0.935190 + 0.354146i $$0.884772\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 471274. 0.833103
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.09705e6 1.69637 0.848186 0.529699i $$-0.177695\pi$$
0.848186 + 0.529699i $$0.177695\pi$$
$$212$$ 817802. 1.24971
$$213$$ 0 0
$$214$$ 602686. 0.899615
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 580959. 0.827951
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −640024. −0.833538
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −812364. −0.990902
$$233$$ −1.05345e6 −1.27123 −0.635617 0.772004i $$-0.719254\pi$$
−0.635617 + 0.772004i $$0.719254\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1.61113e6 1.82447 0.912233 0.409671i $$-0.134357\pi$$
0.912233 + 0.409671i $$0.134357\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 1.26302e6 1.38635
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 883652. 0.867921
$$254$$ −693356. −0.674329
$$255$$ 0 0
$$256$$ −566975. −0.540709
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −1.63936e6 −1.46145 −0.730725 0.682672i $$-0.760818\pi$$
−0.730725 + 0.682672i $$0.760818\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1.73410e6 −1.47481
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 690536. 0.555661
$$275$$ 2.49693e6 1.99101
$$276$$ 0 0
$$277$$ −2.55145e6 −1.99796 −0.998982 0.0451116i $$-0.985636\pi$$
−0.998982 + 0.0451116i $$0.985636\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −718184. −0.542588 −0.271294 0.962497i $$-0.587452\pi$$
−0.271294 + 0.962497i $$0.587452\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ −2.12308e6 −1.56196
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.41986e6 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.34008e6 −0.888998
$$297$$ 0 0
$$298$$ −1.12305e6 −0.732587
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 692192. 0.436726
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −2.00420e6 −1.12908
$$317$$ 3.57144e6 1.99616 0.998079 0.0619605i $$-0.0197353\pi$$
0.998079 + 0.0619605i $$0.0197353\pi$$
$$318$$ 0 0
$$319$$ −4.30410e6 −2.36813
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 1.75440e6 0.914290
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.97148e6 −1.49074 −0.745371 0.666650i $$-0.767727\pi$$
−0.745371 + 0.666650i $$0.767727\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.15965e6 1.99518 0.997590 0.0693859i $$-0.0221040\pi$$
0.997590 + 0.0693859i $$0.0221040\pi$$
$$338$$ −982349. −0.467707
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 1.77169e6 0.807220
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.85255e6 1.71761 0.858804 0.512304i $$-0.171208\pi$$
0.858804 + 0.512304i $$0.171208\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.70365e6 −2.02338
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −1.65935e6 −0.684275
$$359$$ −2.37241e6 −0.971523 −0.485762 0.874091i $$-0.661458\pi$$
−0.485762 + 0.874091i $$0.661458\pi$$
$$360$$ 0 0
$$361$$ −2.47610e6 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ −443476. −0.170707
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 599302. 0.223035 0.111518 0.993762i $$-0.464429\pi$$
0.111518 + 0.993762i $$0.464429\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 5.59273e6 1.99998 0.999991 0.00429827i $$-0.00136819\pi$$
0.999991 + 0.00429827i $$0.00136819\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 273350. 0.0958431
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 1.02100e6 0.348785
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −5.83451e6 −1.95492 −0.977462 0.211109i $$-0.932292\pi$$
−0.977462 + 0.211109i $$0.932292\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −2.69553e6 −0.874790
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.25312e6 −0.391602
$$401$$ −2.25352e6 −0.699844 −0.349922 0.936779i $$-0.613792\pi$$
−0.349922 + 0.936779i $$0.613792\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7.10006e6 −2.12460
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 2.07477e6 0.570513 0.285257 0.958451i $$-0.407921\pi$$
0.285257 + 0.958451i $$0.407921\pi$$
$$422$$ 2.90253e6 0.793405
$$423$$ 0 0
$$424$$ 4.93324e6 1.33265
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −5.69485e6 −1.50270
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.37311e6 −0.615353 −0.307676 0.951491i $$-0.599551\pi$$
−0.307676 + 0.951491i $$0.599551\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −5.48955e6 −1.38299
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −6.49509e6 −1.57245 −0.786223 0.617942i $$-0.787967\pi$$
−0.786223 + 0.617942i $$0.787967\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 8.30677e6 1.94454 0.972269 0.233866i $$-0.0751378\pi$$
0.972269 + 0.233866i $$0.0751378\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.04766e6 1.39233
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.77969e6 1.29453 0.647267 0.762263i $$-0.275912\pi$$
0.647267 + 0.762263i $$0.275912\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 2.88620e6 0.625711 0.312856 0.949801i $$-0.398714\pi$$
0.312856 + 0.949801i $$0.398714\pi$$
$$464$$ 2.16009e6 0.465775
$$465$$ 0 0
$$466$$ −2.78718e6 −0.594565
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.38685e6 1.92915
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 4.26265e6 0.853316
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −1.19344e7 −2.31573
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.76146e6 0.527615 0.263807 0.964575i $$-0.415022\pi$$
0.263807 + 0.964575i $$0.415022\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6.01881e6 −1.12670 −0.563349 0.826219i $$-0.690487\pi$$
−0.563349 + 0.826219i $$0.690487\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 1.11204e7 1.99925 0.999626 0.0273386i $$-0.00870324\pi$$
0.999626 + 0.0273386i $$0.00870324\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 2.33792e6 0.405933
$$507$$ 0 0
$$508$$ 6.55160e6 1.12639
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 4.29577e6 0.724213
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −4.33733e6 −0.683530
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −5.21328e6 −0.809975
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −1.04606e7 −1.57270
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.11261e7 1.63437 0.817186 0.576374i $$-0.195533\pi$$
0.817186 + 0.576374i $$0.195533\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −2.23604e6 −0.319529 −0.159765 0.987155i $$-0.551074\pi$$
−0.159765 + 0.987155i $$0.551074\pi$$
$$548$$ −6.52495e6 −0.928167
$$549$$ 0 0
$$550$$ 6.60625e6 0.931211
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.75050e6 −0.934462
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1.32485e7 −1.80938 −0.904690 0.426070i $$-0.859898\pi$$
−0.904690 + 0.426070i $$0.859898\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.90014e6 −0.253772
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ −1.28071e7 −1.66564
$$569$$ 1.13522e7 1.46994 0.734971 0.678099i $$-0.237196\pi$$
0.734971 + 0.678099i $$0.237196\pi$$
$$570$$ 0 0
$$571$$ 6.33912e6 0.813653 0.406826 0.913506i $$-0.366635\pi$$
0.406826 + 0.913506i $$0.366635\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.45601e6 0.435919
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −3.75659e6 −0.467707
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 2.61375e7 3.18488
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3.56329e6 0.417875
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.06118e7 1.22370
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.60293e7 1.82536 0.912679 0.408677i $$-0.134010\pi$$
0.912679 + 0.408677i $$0.134010\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −6.54060e6 −0.729500
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 1.75514e7 1.88652 0.943258 0.332060i $$-0.107744\pi$$
0.943258 + 0.332060i $$0.107744\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.75090e7 1.85160 0.925802 0.378008i $$-0.123391\pi$$
0.925802 + 0.378008i $$0.123391\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9.76562e6 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1.99786e7 1.99752 0.998760 0.0497844i $$-0.0158534\pi$$
0.998760 + 0.0497844i $$0.0158534\pi$$
$$632$$ −1.20900e7 −1.20402
$$633$$ 0 0
$$634$$ 9.44913e6 0.933617
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.13876e7 −1.10759
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.73407e7 1.66694 0.833471 0.552563i $$-0.186350\pi$$
0.833471 + 0.552563i $$0.186350\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1.65775e7 −1.52721
$$653$$ 1.88262e7 1.72775 0.863873 0.503710i $$-0.168032\pi$$
0.863873 + 0.503710i $$0.168032\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 9.38990e6 0.842263 0.421131 0.907000i $$-0.361633\pi$$
0.421131 + 0.907000i $$0.361633\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ −7.86179e6 −0.697230
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −5.95734e6 −0.518487
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1.38435e6 −0.117817 −0.0589085 0.998263i $$-0.518762\pi$$
−0.0589085 + 0.998263i $$0.518762\pi$$
$$674$$ 1.10054e7 0.933160
$$675$$ 0 0
$$676$$ 9.28232e6 0.781250
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.18403e7 −0.971206 −0.485603 0.874179i $$-0.661400\pi$$
−0.485603 + 0.874179i $$0.661400\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.71095e6 −0.379435
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 1.01929e7 0.803338
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.55582e7 −1.96443 −0.982213 0.187771i $$-0.939874\pi$$
−0.982213 + 0.187771i $$0.939874\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −2.19170e6 −0.166667
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.43225e7 1.07005 0.535023 0.844837i $$-0.320303\pi$$
0.535023 + 0.844837i $$0.320303\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.56794e7 1.14300
$$717$$ 0 0
$$718$$ −6.27680e6 −0.454388
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −6.55114e6 −0.467707
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1.68336e7 −1.18941
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ −6.51035e6 −0.443006
$$737$$ −5.54230e7 −3.75856
$$738$$ 0 0
$$739$$ −2.64893e6 −0.178427 −0.0892133 0.996013i $$-0.528435\pi$$
−0.0892133 + 0.996013i $$0.528435\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.68294e7 1.11840 0.559199 0.829033i $$-0.311109\pi$$
0.559199 + 0.829033i $$0.311109\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 1.58560e6 0.104315
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.51088e7 −1.62452 −0.812260 0.583295i $$-0.801763\pi$$
−0.812260 + 0.583295i $$0.801763\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 1.78870e7 1.13448 0.567242 0.823551i $$-0.308011\pi$$
0.567242 + 0.823551i $$0.308011\pi$$
$$758$$ 1.47970e7 0.935406
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.58291e6 −0.160095
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −9.64755e6 −0.582604
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −1.54367e7 −0.914332
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −6.78552e7 −3.98066
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 2.54704e7 1.46123
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.83962e7 −1.01626
$$801$$ 0 0
$$802$$ −5.96226e6 −0.327322
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 3.23828e7 1.73957 0.869787 0.493428i $$-0.164256\pi$$
0.869787 + 0.493428i $$0.164256\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −1.87850e7 −0.993689
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.50170e7 −0.777546 −0.388773 0.921334i $$-0.627101\pi$$
−0.388773 + 0.921334i $$0.627101\pi$$
$$822$$ 0 0
$$823$$ 7.08675e6 0.364710 0.182355 0.983233i $$-0.441628\pi$$
0.182355 + 0.983233i $$0.441628\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1.37491e7 0.699054 0.349527 0.936926i $$-0.386342\pi$$
0.349527 + 0.936926i $$0.386342\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 8.50592e6 0.414698
$$842$$ 5.48934e6 0.266833
$$843$$ 0 0
$$844$$ −2.74263e7 −1.32529
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −1.31175e7 −0.626416
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −9.82724e6 −0.465166
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −3.43531e7 −1.60244
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −6.27865e6 −0.287805
$$863$$ 3.39442e7 1.55145 0.775727 0.631068i $$-0.217383\pi$$
0.775727 + 0.631068i $$0.217383\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −6.40556e7 −2.87745
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −3.31147e7 −1.47479
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −4.33428e7 −1.90291 −0.951453 0.307793i $$-0.900410\pi$$
−0.951453 + 0.307793i $$0.900410\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 3.94011e7 1.70062 0.850308 0.526286i $$-0.176416\pi$$
0.850308 + 0.526286i $$0.176416\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −1.71844e7 −0.735445
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.19776e7 0.909474
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 3.64814e7 1.48474
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.48347e7 −1.80966 −0.904828 0.425777i $$-0.860001\pi$$
−0.904828 + 0.425777i $$0.860001\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.13974e7 −0.454997 −0.227498 0.973778i $$-0.573055\pi$$
−0.227498 + 0.973778i $$0.573055\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 1.52916e7 0.605463
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5.06716e7 −1.97914 −0.989569 0.144059i $$-0.953984\pi$$
−0.989569 + 0.144059i $$0.953984\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −2.77688e7 −1.06709
$$926$$ 7.63617e6 0.292650
$$927$$ 0 0
$$928$$ 3.17107e7 1.20875
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2.63363e7 0.993152
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 2.48353e7 0.902279
$$947$$ 4.63272e6 0.167865 0.0839326 0.996471i $$-0.473252\pi$$
0.0839326 + 0.996471i $$0.473252\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 5.51804e7 1.96812 0.984062 0.177824i $$-0.0569057\pi$$
0.984062 + 0.177824i $$0.0569057\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −4.02783e7 −1.42536
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.86292e7 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.00222e7 0.688566 0.344283 0.938866i $$-0.388122\pi$$
0.344283 + 0.938866i $$0.388122\pi$$
$$968$$ −7.19922e7 −2.46943
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 7.30615e6 0.246769
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.25961e7 1.09252 0.546260 0.837616i $$-0.316051\pi$$
0.546260 + 0.837616i $$0.316051\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −1.59243e7 −0.526964
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.29924e7 0.422375
$$990$$ 0 0
$$991$$ 5.73144e7 1.85387 0.926936 0.375219i $$-0.122433\pi$$
0.926936 + 0.375219i $$0.122433\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 2.94217e7 0.935065
$$999$$ 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 441.6.a.o.1.2 yes 2
3.2 odd 2 inner 441.6.a.o.1.1 2
7.6 odd 2 CM 441.6.a.o.1.2 yes 2
21.20 even 2 inner 441.6.a.o.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.o.1.1 2 3.2 odd 2 inner
441.6.a.o.1.1 2 21.20 even 2 inner
441.6.a.o.1.2 yes 2 1.1 even 1 trivial
441.6.a.o.1.2 yes 2 7.6 odd 2 CM