# Properties

 Label 5850.2.a.bp.1.1 Level $5850$ Weight $2$ Character 5850.1 Self dual yes Analytic conductor $46.712$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.7124851824$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1950) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -4.00000 q^{11} -1.00000 q^{13} +1.00000 q^{16} +1.00000 q^{19} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{26} +3.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{37} +1.00000 q^{38} -9.00000 q^{41} +2.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +3.00000 q^{47} -7.00000 q^{49} -1.00000 q^{52} -1.00000 q^{53} +3.00000 q^{58} -10.0000 q^{59} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -9.00000 q^{67} -7.00000 q^{71} +4.00000 q^{73} -5.00000 q^{74} +1.00000 q^{76} +11.0000 q^{79} -9.00000 q^{82} -6.00000 q^{83} +2.00000 q^{86} -4.00000 q^{88} -10.0000 q^{89} -4.00000 q^{92} +3.00000 q^{94} -12.0000 q^{97} -7.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.00000 −1.09952 −0.549762 0.835321i $$-0.685282\pi$$
−0.549762 + 0.835321i $$0.685282\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −7.00000 −0.830747 −0.415374 0.909651i $$-0.636349\pi$$
−0.415374 + 0.909651i $$0.636349\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ −5.00000 −0.581238
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −9.00000 −0.993884
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 3.00000 0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ −10.0000 −0.920575
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 4.00000 0.362143
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −17.0000 −1.50851 −0.754253 0.656584i $$-0.772001\pi$$
−0.754253 + 0.656584i $$0.772001\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −9.00000 −0.777482
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −23.0000 −1.96502 −0.982511 0.186203i $$-0.940382\pi$$
−0.982511 + 0.186203i $$0.940382\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.00000 −0.587427
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −5.00000 −0.410997
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 11.0000 0.875113
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ −7.00000 −0.541676 −0.270838 0.962625i $$-0.587301\pi$$
−0.270838 + 0.962625i $$0.587301\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 3.00000 0.228086 0.114043 0.993476i $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ −10.0000 −0.749532
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 3.00000 0.218797
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 20.0000 1.43963 0.719816 0.694165i $$-0.244226\pi$$
0.719816 + 0.694165i $$0.244226\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ 1.00000 0.0708881 0.0354441 0.999372i $$-0.488715\pi$$
0.0354441 + 0.999372i $$0.488715\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ −1.00000 −0.0686803
$$213$$ 0 0
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −1.00000 −0.0677285
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 10.0000 0.663723 0.331862 0.943328i $$-0.392323\pi$$
0.331862 + 0.943328i $$0.392323\pi$$
$$228$$ 0 0
$$229$$ 5.00000 0.330409 0.165205 0.986259i $$-0.447172\pi$$
0.165205 + 0.986259i $$0.447172\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −10.0000 −0.650945
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −23.0000 −1.45175 −0.725874 0.687828i $$-0.758564\pi$$
−0.725874 + 0.687828i $$0.758564\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −17.0000 −1.06667
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 3.00000 0.185341
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −9.00000 −0.549762
$$269$$ 11.0000 0.670682 0.335341 0.942097i $$-0.391148\pi$$
0.335341 + 0.942097i $$0.391148\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −23.0000 −1.38948
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.00000 0.298275 0.149137 0.988816i $$-0.452350\pi$$
0.149137 + 0.988816i $$0.452350\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −7.00000 −0.415374
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4.00000 0.234082
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −5.00000 −0.290619
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12.0000 0.690522
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3.00000 0.171219 0.0856095 0.996329i $$-0.472716\pi$$
0.0856095 + 0.996329i $$0.472716\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 11.0000 0.618798
$$317$$ −32.0000 −1.79730 −0.898650 0.438667i $$-0.855451\pi$$
−0.898650 + 0.438667i $$0.855451\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ −9.00000 −0.496942
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 0 0
$$334$$ −7.00000 −0.383023
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ 3.00000 0.161281
$$347$$ 17.0000 0.912608 0.456304 0.889824i $$-0.349173\pi$$
0.456304 + 0.889824i $$0.349173\pi$$
$$348$$ 0 0
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ 15.0000 0.798369 0.399185 0.916871i $$-0.369293\pi$$
0.399185 + 0.916871i $$0.369293\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 17.0000 0.897226 0.448613 0.893726i $$-0.351918\pi$$
0.448613 + 0.893726i $$0.351918\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −12.0000 −0.630706
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −21.0000 −1.09619 −0.548096 0.836416i $$-0.684647\pi$$
−0.548096 + 0.836416i $$0.684647\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.0000 0.920960
$$383$$ 5.00000 0.255488 0.127744 0.991807i $$-0.459226\pi$$
0.127744 + 0.991807i $$0.459226\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 20.0000 1.01797
$$387$$ 0 0
$$388$$ −12.0000 −0.609208
$$389$$ 19.0000 0.963338 0.481669 0.876353i $$-0.340031\pi$$
0.481669 + 0.876353i $$0.340031\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ 0 0
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 3.00000 0.150566 0.0752828 0.997162i $$-0.476014\pi$$
0.0752828 + 0.997162i $$0.476014\pi$$
$$398$$ 1.00000 0.0501255
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ −40.0000 −1.97787 −0.988936 0.148340i $$-0.952607\pi$$
−0.988936 + 0.148340i $$0.952607\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −4.00000 −0.195646
$$419$$ 37.0000 1.80757 0.903784 0.427989i $$-0.140778\pi$$
0.903784 + 0.427989i $$0.140778\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 24.0000 1.16830
$$423$$ 0 0
$$424$$ −1.00000 −0.0485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −9.00000 −0.435031
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.00000 −0.0478913
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −19.0000 −0.906821 −0.453410 0.891302i $$-0.649793\pi$$
−0.453410 + 0.891302i $$0.649793\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3.00000 0.142534 0.0712672 0.997457i $$-0.477296\pi$$
0.0712672 + 0.997457i $$0.477296\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14.0000 −0.662919
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 10.0000 0.469323
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.0000 1.49690 0.748448 0.663193i $$-0.230799\pi$$
0.748448 + 0.663193i $$0.230799\pi$$
$$458$$ 5.00000 0.233635
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ −8.00000 −0.370593
$$467$$ −15.0000 −0.694117 −0.347059 0.937843i $$-0.612820\pi$$
−0.347059 + 0.937843i $$0.612820\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −10.0000 −0.460287
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 8.00000 0.365911
$$479$$ −25.0000 −1.14228 −0.571140 0.820853i $$-0.693499\pi$$
−0.571140 + 0.820853i $$0.693499\pi$$
$$480$$ 0 0
$$481$$ 5.00000 0.227980
$$482$$ −12.0000 −0.546585
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −38.0000 −1.72194 −0.860972 0.508652i $$-0.830144\pi$$
−0.860972 + 0.508652i $$0.830144\pi$$
$$488$$ 4.00000 0.181071
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.0000 1.44414 0.722070 0.691820i $$-0.243191\pi$$
0.722070 + 0.691820i $$0.243191\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −1.00000 −0.0449921
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1.00000 −0.0447661 −0.0223831 0.999749i $$-0.507125\pi$$
−0.0223831 + 0.999749i $$0.507125\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −23.0000 −1.02654
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ 0 0
$$508$$ −17.0000 −0.754253
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −24.0000 −1.05859
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −30.0000 −1.31181 −0.655904 0.754844i $$-0.727712\pi$$
−0.655904 + 0.754844i $$0.727712\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 9.00000 0.389833
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ 0 0
$$538$$ 11.0000 0.474244
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ 24.0000 1.03089
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ −23.0000 −0.982511
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.00000 0.127804
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5.00000 0.210912
$$563$$ 21.0000 0.885044 0.442522 0.896758i $$-0.354084\pi$$
0.442522 + 0.896758i $$0.354084\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −7.00000 −0.293713
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12.0000 0.499567 0.249783 0.968302i $$-0.419641\pi$$
0.249783 + 0.968302i $$0.419641\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 16.0000 0.660954
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −5.00000 −0.205499
$$593$$ 1.00000 0.0410651 0.0205325 0.999789i $$-0.493464\pi$$
0.0205325 + 0.999789i $$0.493464\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 45.0000 1.83559 0.917794 0.397057i $$-0.129968\pi$$
0.917794 + 0.397057i $$0.129968\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 12.0000 0.488273
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −7.00000 −0.284121 −0.142061 0.989858i $$-0.545373\pi$$
−0.142061 + 0.989858i $$0.545373\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 3.00000 0.121070
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.0000 1.08698 0.543490 0.839416i $$-0.317103\pi$$
0.543490 + 0.839416i $$0.317103\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ 11.0000 0.437557
$$633$$ 0 0
$$634$$ −32.0000 −1.27088
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.00000 0.277350
$$638$$ −12.0000 −0.475085
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −19.0000 −0.749287 −0.374643 0.927169i $$-0.622235\pi$$
−0.374643 + 0.927169i $$0.622235\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ 40.0000 1.57014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −13.0000 −0.506408 −0.253204 0.967413i $$-0.581484\pi$$
−0.253204 + 0.967413i $$0.581484\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ −7.00000 −0.270838
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ 1.00000 0.0380970
$$690$$ 0 0
$$691$$ −3.00000 −0.114125 −0.0570627 0.998371i $$-0.518173\pi$$
−0.0570627 + 0.998371i $$0.518173\pi$$
$$692$$ 3.00000 0.114043
$$693$$ 0 0
$$694$$ 17.0000 0.645311
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 38.0000 1.43524 0.717620 0.696435i $$-0.245231\pi$$
0.717620 + 0.696435i $$0.245231\pi$$
$$702$$ 0 0
$$703$$ −5.00000 −0.188579
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −10.0000 −0.374766
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 17.0000 0.634434
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −18.0000 −0.669891
$$723$$ 0 0
$$724$$ −12.0000 −0.445976
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −49.0000 −1.80986 −0.904928 0.425564i $$-0.860076\pi$$
−0.904928 + 0.425564i $$0.860076\pi$$
$$734$$ −21.0000 −0.775124
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 36.0000 1.32608
$$738$$ 0 0
$$739$$ 53.0000 1.94964 0.974818 0.223001i $$-0.0715853\pi$$
0.974818 + 0.223001i $$0.0715853\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 27.0000 0.990534 0.495267 0.868741i $$-0.335070\pi$$
0.495267 + 0.868741i $$0.335070\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −43.0000 −1.56909 −0.784546 0.620070i $$-0.787104\pi$$
−0.784546 + 0.620070i $$0.787104\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 0 0
$$754$$ −3.00000 −0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15.0000 0.543750 0.271875 0.962333i $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ 10.0000 0.361079
$$768$$ 0 0
$$769$$ 16.0000 0.576975 0.288487 0.957484i $$-0.406848\pi$$
0.288487 + 0.957484i $$0.406848\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 20.0000 0.719816
$$773$$ 32.0000 1.15096 0.575480 0.817816i $$-0.304815\pi$$
0.575480 + 0.817816i $$0.304815\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ 0 0
$$778$$ 19.0000 0.681183
$$779$$ −9.00000 −0.322458
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 3.00000 0.106466
$$795$$ 0 0
$$796$$ 1.00000 0.0354441
$$797$$ 22.0000 0.779280 0.389640 0.920967i $$-0.372599\pi$$
0.389640 + 0.920967i $$0.372599\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 26.0000 0.918092
$$803$$ −16.0000 −0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 20.0000 0.701000
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ −40.0000 −1.39857
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ 37.0000 1.27814
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 2.00000 0.0689246
$$843$$ 0 0
$$844$$ 24.0000 0.826114
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −1.00000 −0.0343401
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20.0000 0.685591
$$852$$ 0 0
$$853$$ 41.0000 1.40381 0.701907 0.712269i $$-0.252332\pi$$
0.701907 + 0.712269i $$0.252332\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 3.00000 0.102180
$$863$$ −57.0000 −1.94030 −0.970151 0.242500i $$-0.922032\pi$$
−0.970151 + 0.242500i $$0.922032\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −19.0000 −0.645646
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −44.0000 −1.49260
$$870$$ 0 0
$$871$$ 9.00000 0.304953
$$872$$ −1.00000 −0.0338643
$$873$$ 0 0
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −23.0000 −0.776655 −0.388327 0.921521i $$-0.626947\pi$$
−0.388327 + 0.921521i $$0.626947\pi$$
$$878$$ −19.0000 −0.641219
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −56.0000 −1.88669 −0.943344 0.331816i $$-0.892339\pi$$
−0.943344 + 0.331816i $$0.892339\pi$$
$$882$$ 0 0
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 3.00000 0.100787
$$887$$ 44.0000 1.47738 0.738688 0.674048i $$-0.235446\pi$$
0.738688 + 0.674048i $$0.235446\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14.0000 −0.468755
$$893$$ 3.00000 0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 9.00000 0.300334
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 36.0000 1.19867
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −30.0000 −0.996134 −0.498067 0.867139i $$-0.665957\pi$$
−0.498067 + 0.867139i $$0.665957\pi$$
$$908$$ 10.0000 0.331862
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ 24.0000 0.794284
$$914$$ 32.0000 1.05847
$$915$$ 0 0
$$916$$ 5.00000 0.165205
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000 0.790398
$$923$$ 7.00000 0.230408
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ 3.00000 0.0984798
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ −8.00000 −0.262049
$$933$$ 0 0
$$934$$ −15.0000 −0.490815
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34.0000 1.11073 0.555366 0.831606i $$-0.312578\pi$$
0.555366 + 0.831606i $$0.312578\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 36.0000 1.17232
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −56.0000 −1.81402 −0.907009 0.421111i $$-0.861640\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ −25.0000 −0.807713
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 5.00000 0.161206
$$963$$ 0 0
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ 40.0000 1.27841
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 32.0000 1.02116
$$983$$ 32.0000 1.02064 0.510321 0.859984i $$-0.329527\pi$$
0.510321 + 0.859984i $$0.329527\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −1.00000 −0.0318142
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −7.00000 −0.222362 −0.111181 0.993800i $$-0.535463\pi$$
−0.111181 + 0.993800i $$0.535463\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ −1.00000 −0.0316544
$$999$$ 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 5850.2.a.bp.1.1 1
3.2 odd 2 1950.2.a.j.1.1 1
5.2 odd 4 5850.2.e.f.5149.2 2
5.3 odd 4 5850.2.e.f.5149.1 2
5.4 even 2 5850.2.a.l.1.1 1
15.2 even 4 1950.2.e.h.1249.1 2
15.8 even 4 1950.2.e.h.1249.2 2
15.14 odd 2 1950.2.a.s.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.j.1.1 1 3.2 odd 2
1950.2.a.s.1.1 yes 1 15.14 odd 2
1950.2.e.h.1249.1 2 15.2 even 4
1950.2.e.h.1249.2 2 15.8 even 4
5850.2.a.l.1.1 1 5.4 even 2
5850.2.a.bp.1.1 1 1.1 even 1 trivial
5850.2.e.f.5149.1 2 5.3 odd 4
5850.2.e.f.5149.2 2 5.2 odd 4