# Properties

 Label 7440.2.a.bl.1.1 Level $7440$ Weight $2$ Character 7440.1 Self dual yes Analytic conductor $59.409$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4086991038$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7440.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} -2.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} -3.12311 q^{17} +7.68466 q^{19} -2.56155 q^{21} -1.43845 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.12311 q^{29} -1.00000 q^{31} -2.56155 q^{33} -2.56155 q^{35} -3.12311 q^{37} +2.00000 q^{39} +7.12311 q^{41} -12.8078 q^{43} +1.00000 q^{45} -5.12311 q^{47} -0.438447 q^{49} -3.12311 q^{51} +7.43845 q^{53} -2.56155 q^{55} +7.68466 q^{57} +13.1231 q^{59} +6.00000 q^{61} -2.56155 q^{63} +2.00000 q^{65} +15.3693 q^{67} -1.43845 q^{69} +7.68466 q^{71} -10.8078 q^{73} +1.00000 q^{75} +6.56155 q^{77} +4.31534 q^{79} +1.00000 q^{81} -14.2462 q^{83} -3.12311 q^{85} +7.12311 q^{87} -13.6847 q^{89} -5.12311 q^{91} -1.00000 q^{93} +7.68466 q^{95} -6.00000 q^{97} -2.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 + 3 * q^19 - q^21 - 7 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^31 - q^33 - q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 - 5 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 + 2 * q^51 + 19 * q^53 - q^55 + 3 * q^57 + 18 * q^59 + 12 * q^61 - q^63 + 4 * q^65 + 6 * q^67 - 7 * q^69 + 3 * q^71 - q^73 + 2 * q^75 + 9 * q^77 + 21 * q^79 + 2 * q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 15 * q^89 - 2 * q^91 - 2 * q^93 + 3 * q^95 - 12 * q^97 - q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.56155 −0.968176 −0.484088 0.875019i $$-0.660849\pi$$
−0.484088 + 0.875019i $$0.660849\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.56155 −0.772337 −0.386169 0.922428i $$-0.626202\pi$$
−0.386169 + 0.922428i $$0.626202\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −3.12311 −0.757464 −0.378732 0.925506i $$-0.623640\pi$$
−0.378732 + 0.925506i $$0.623640\pi$$
$$18$$ 0 0
$$19$$ 7.68466 1.76298 0.881491 0.472201i $$-0.156540\pi$$
0.881491 + 0.472201i $$0.156540\pi$$
$$20$$ 0 0
$$21$$ −2.56155 −0.558977
$$22$$ 0 0
$$23$$ −1.43845 −0.299937 −0.149968 0.988691i $$-0.547917\pi$$
−0.149968 + 0.988691i $$0.547917\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 7.12311 1.32273 0.661364 0.750065i $$-0.269978\pi$$
0.661364 + 0.750065i $$0.269978\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 0 0
$$33$$ −2.56155 −0.445909
$$34$$ 0 0
$$35$$ −2.56155 −0.432981
$$36$$ 0 0
$$37$$ −3.12311 −0.513435 −0.256718 0.966486i $$-0.582641\pi$$
−0.256718 + 0.966486i $$0.582641\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 7.12311 1.11244 0.556221 0.831034i $$-0.312251\pi$$
0.556221 + 0.831034i $$0.312251\pi$$
$$42$$ 0 0
$$43$$ −12.8078 −1.95317 −0.976583 0.215142i $$-0.930979\pi$$
−0.976583 + 0.215142i $$0.930979\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −5.12311 −0.747282 −0.373641 0.927573i $$-0.621891\pi$$
−0.373641 + 0.927573i $$0.621891\pi$$
$$48$$ 0 0
$$49$$ −0.438447 −0.0626353
$$50$$ 0 0
$$51$$ −3.12311 −0.437322
$$52$$ 0 0
$$53$$ 7.43845 1.02175 0.510875 0.859655i $$-0.329322\pi$$
0.510875 + 0.859655i $$0.329322\pi$$
$$54$$ 0 0
$$55$$ −2.56155 −0.345400
$$56$$ 0 0
$$57$$ 7.68466 1.01786
$$58$$ 0 0
$$59$$ 13.1231 1.70848 0.854241 0.519877i $$-0.174022\pi$$
0.854241 + 0.519877i $$0.174022\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ −2.56155 −0.322725
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 15.3693 1.87766 0.938830 0.344380i $$-0.111911\pi$$
0.938830 + 0.344380i $$0.111911\pi$$
$$68$$ 0 0
$$69$$ −1.43845 −0.173169
$$70$$ 0 0
$$71$$ 7.68466 0.912001 0.456001 0.889979i $$-0.349281\pi$$
0.456001 + 0.889979i $$0.349281\pi$$
$$72$$ 0 0
$$73$$ −10.8078 −1.26495 −0.632477 0.774580i $$-0.717962\pi$$
−0.632477 + 0.774580i $$0.717962\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 6.56155 0.747758
$$78$$ 0 0
$$79$$ 4.31534 0.485514 0.242757 0.970087i $$-0.421948\pi$$
0.242757 + 0.970087i $$0.421948\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −14.2462 −1.56372 −0.781862 0.623451i $$-0.785730\pi$$
−0.781862 + 0.623451i $$0.785730\pi$$
$$84$$ 0 0
$$85$$ −3.12311 −0.338748
$$86$$ 0 0
$$87$$ 7.12311 0.763677
$$88$$ 0 0
$$89$$ −13.6847 −1.45057 −0.725285 0.688448i $$-0.758292\pi$$
−0.725285 + 0.688448i $$0.758292\pi$$
$$90$$ 0 0
$$91$$ −5.12311 −0.537047
$$92$$ 0 0
$$93$$ −1.00000 −0.103695
$$94$$ 0 0
$$95$$ 7.68466 0.788429
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ 0 0
$$99$$ −2.56155 −0.257446
$$100$$ 0 0
$$101$$ −0.561553 −0.0558766 −0.0279383 0.999610i $$-0.508894\pi$$
−0.0279383 + 0.999610i $$0.508894\pi$$
$$102$$ 0 0
$$103$$ 1.75379 0.172806 0.0864030 0.996260i $$-0.472463\pi$$
0.0864030 + 0.996260i $$0.472463\pi$$
$$104$$ 0 0
$$105$$ −2.56155 −0.249982
$$106$$ 0 0
$$107$$ 2.56155 0.247635 0.123817 0.992305i $$-0.460486\pi$$
0.123817 + 0.992305i $$0.460486\pi$$
$$108$$ 0 0
$$109$$ 5.36932 0.514287 0.257144 0.966373i $$-0.417219\pi$$
0.257144 + 0.966373i $$0.417219\pi$$
$$110$$ 0 0
$$111$$ −3.12311 −0.296432
$$112$$ 0 0
$$113$$ −1.68466 −0.158479 −0.0792397 0.996856i $$-0.525249\pi$$
−0.0792397 + 0.996856i $$0.525249\pi$$
$$114$$ 0 0
$$115$$ −1.43845 −0.134136
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ 0 0
$$123$$ 7.12311 0.642269
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ −12.8078 −1.12766
$$130$$ 0 0
$$131$$ 15.3693 1.34282 0.671412 0.741085i $$-0.265688\pi$$
0.671412 + 0.741085i $$0.265688\pi$$
$$132$$ 0 0
$$133$$ −19.6847 −1.70688
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 20.2462 1.72975 0.864875 0.501987i $$-0.167397\pi$$
0.864875 + 0.501987i $$0.167397\pi$$
$$138$$ 0 0
$$139$$ 17.1231 1.45236 0.726181 0.687503i $$-0.241293\pi$$
0.726181 + 0.687503i $$0.241293\pi$$
$$140$$ 0 0
$$141$$ −5.12311 −0.431443
$$142$$ 0 0
$$143$$ −5.12311 −0.428416
$$144$$ 0 0
$$145$$ 7.12311 0.591542
$$146$$ 0 0
$$147$$ −0.438447 −0.0361625
$$148$$ 0 0
$$149$$ 17.0540 1.39712 0.698558 0.715553i $$-0.253825\pi$$
0.698558 + 0.715553i $$0.253825\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −3.12311 −0.252488
$$154$$ 0 0
$$155$$ −1.00000 −0.0803219
$$156$$ 0 0
$$157$$ 17.0540 1.36106 0.680528 0.732722i $$-0.261751\pi$$
0.680528 + 0.732722i $$0.261751\pi$$
$$158$$ 0 0
$$159$$ 7.43845 0.589907
$$160$$ 0 0
$$161$$ 3.68466 0.290392
$$162$$ 0 0
$$163$$ 15.3693 1.20382 0.601909 0.798565i $$-0.294407\pi$$
0.601909 + 0.798565i $$0.294407\pi$$
$$164$$ 0 0
$$165$$ −2.56155 −0.199417
$$166$$ 0 0
$$167$$ 6.56155 0.507748 0.253874 0.967237i $$-0.418295\pi$$
0.253874 + 0.967237i $$0.418295\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 7.68466 0.587661
$$172$$ 0 0
$$173$$ 8.24621 0.626948 0.313474 0.949597i $$-0.398507\pi$$
0.313474 + 0.949597i $$0.398507\pi$$
$$174$$ 0 0
$$175$$ −2.56155 −0.193635
$$176$$ 0 0
$$177$$ 13.1231 0.986393
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −13.0540 −0.970294 −0.485147 0.874433i $$-0.661234\pi$$
−0.485147 + 0.874433i $$0.661234\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ −3.12311 −0.229615
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −2.56155 −0.186326
$$190$$ 0 0
$$191$$ 14.2462 1.03082 0.515410 0.856944i $$-0.327640\pi$$
0.515410 + 0.856944i $$0.327640\pi$$
$$192$$ 0 0
$$193$$ 14.4924 1.04319 0.521594 0.853194i $$-0.325338\pi$$
0.521594 + 0.853194i $$0.325338\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 16.8078 1.19147 0.595735 0.803181i $$-0.296861\pi$$
0.595735 + 0.803181i $$0.296861\pi$$
$$200$$ 0 0
$$201$$ 15.3693 1.08407
$$202$$ 0 0
$$203$$ −18.2462 −1.28063
$$204$$ 0 0
$$205$$ 7.12311 0.497499
$$206$$ 0 0
$$207$$ −1.43845 −0.0999790
$$208$$ 0 0
$$209$$ −19.6847 −1.36162
$$210$$ 0 0
$$211$$ 23.6847 1.63052 0.815260 0.579096i $$-0.196594\pi$$
0.815260 + 0.579096i $$0.196594\pi$$
$$212$$ 0 0
$$213$$ 7.68466 0.526544
$$214$$ 0 0
$$215$$ −12.8078 −0.873482
$$216$$ 0 0
$$217$$ 2.56155 0.173890
$$218$$ 0 0
$$219$$ −10.8078 −0.730321
$$220$$ 0 0
$$221$$ −6.24621 −0.420166
$$222$$ 0 0
$$223$$ 21.1231 1.41451 0.707254 0.706960i $$-0.249934\pi$$
0.707254 + 0.706960i $$0.249934\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −7.68466 −0.510049 −0.255024 0.966935i $$-0.582083\pi$$
−0.255024 + 0.966935i $$0.582083\pi$$
$$228$$ 0 0
$$229$$ −16.5616 −1.09442 −0.547209 0.836996i $$-0.684310\pi$$
−0.547209 + 0.836996i $$0.684310\pi$$
$$230$$ 0 0
$$231$$ 6.56155 0.431718
$$232$$ 0 0
$$233$$ −17.0540 −1.11724 −0.558622 0.829423i $$-0.688670\pi$$
−0.558622 + 0.829423i $$0.688670\pi$$
$$234$$ 0 0
$$235$$ −5.12311 −0.334195
$$236$$ 0 0
$$237$$ 4.31534 0.280312
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 12.2462 0.788848 0.394424 0.918929i $$-0.370944\pi$$
0.394424 + 0.918929i $$0.370944\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.438447 −0.0280114
$$246$$ 0 0
$$247$$ 15.3693 0.977926
$$248$$ 0 0
$$249$$ −14.2462 −0.902817
$$250$$ 0 0
$$251$$ −16.4924 −1.04099 −0.520496 0.853864i $$-0.674253\pi$$
−0.520496 + 0.853864i $$0.674253\pi$$
$$252$$ 0 0
$$253$$ 3.68466 0.231652
$$254$$ 0 0
$$255$$ −3.12311 −0.195576
$$256$$ 0 0
$$257$$ −1.68466 −0.105086 −0.0525431 0.998619i $$-0.516733\pi$$
−0.0525431 + 0.998619i $$0.516733\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 7.12311 0.440909
$$262$$ 0 0
$$263$$ 20.4924 1.26362 0.631808 0.775125i $$-0.282313\pi$$
0.631808 + 0.775125i $$0.282313\pi$$
$$264$$ 0 0
$$265$$ 7.43845 0.456940
$$266$$ 0 0
$$267$$ −13.6847 −0.837487
$$268$$ 0 0
$$269$$ −0.246211 −0.0150118 −0.00750588 0.999972i $$-0.502389\pi$$
−0.00750588 + 0.999972i $$0.502389\pi$$
$$270$$ 0 0
$$271$$ −27.0540 −1.64341 −0.821706 0.569912i $$-0.806977\pi$$
−0.821706 + 0.569912i $$0.806977\pi$$
$$272$$ 0 0
$$273$$ −5.12311 −0.310064
$$274$$ 0 0
$$275$$ −2.56155 −0.154467
$$276$$ 0 0
$$277$$ −5.36932 −0.322611 −0.161305 0.986905i $$-0.551570\pi$$
−0.161305 + 0.986905i $$0.551570\pi$$
$$278$$ 0 0
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 4.87689 0.290931 0.145466 0.989363i $$-0.453532\pi$$
0.145466 + 0.989363i $$0.453532\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 7.68466 0.455200
$$286$$ 0 0
$$287$$ −18.2462 −1.07704
$$288$$ 0 0
$$289$$ −7.24621 −0.426248
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ 0 0
$$293$$ 26.4924 1.54770 0.773852 0.633367i $$-0.218327\pi$$
0.773852 + 0.633367i $$0.218327\pi$$
$$294$$ 0 0
$$295$$ 13.1231 0.764057
$$296$$ 0 0
$$297$$ −2.56155 −0.148636
$$298$$ 0 0
$$299$$ −2.87689 −0.166375
$$300$$ 0 0
$$301$$ 32.8078 1.89101
$$302$$ 0 0
$$303$$ −0.561553 −0.0322604
$$304$$ 0 0
$$305$$ 6.00000 0.343559
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 1.75379 0.0997696
$$310$$ 0 0
$$311$$ 1.75379 0.0994482 0.0497241 0.998763i $$-0.484166\pi$$
0.0497241 + 0.998763i $$0.484166\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ −2.56155 −0.144327
$$316$$ 0 0
$$317$$ 16.8769 0.947901 0.473950 0.880552i $$-0.342828\pi$$
0.473950 + 0.880552i $$0.342828\pi$$
$$318$$ 0 0
$$319$$ −18.2462 −1.02159
$$320$$ 0 0
$$321$$ 2.56155 0.142972
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ 5.36932 0.296924
$$328$$ 0 0
$$329$$ 13.1231 0.723500
$$330$$ 0 0
$$331$$ 6.24621 0.343323 0.171661 0.985156i $$-0.445086\pi$$
0.171661 + 0.985156i $$0.445086\pi$$
$$332$$ 0 0
$$333$$ −3.12311 −0.171145
$$334$$ 0 0
$$335$$ 15.3693 0.839715
$$336$$ 0 0
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ 0 0
$$339$$ −1.68466 −0.0914981
$$340$$ 0 0
$$341$$ 2.56155 0.138716
$$342$$ 0 0
$$343$$ 19.0540 1.02882
$$344$$ 0 0
$$345$$ −1.43845 −0.0774434
$$346$$ 0 0
$$347$$ −14.2462 −0.764777 −0.382388 0.924002i $$-0.624898\pi$$
−0.382388 + 0.924002i $$0.624898\pi$$
$$348$$ 0 0
$$349$$ 5.36932 0.287413 0.143706 0.989620i $$-0.454098\pi$$
0.143706 + 0.989620i $$0.454098\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −0.246211 −0.0131045 −0.00655225 0.999979i $$-0.502086\pi$$
−0.00655225 + 0.999979i $$0.502086\pi$$
$$354$$ 0 0
$$355$$ 7.68466 0.407859
$$356$$ 0 0
$$357$$ 8.00000 0.423405
$$358$$ 0 0
$$359$$ 31.6847 1.67225 0.836126 0.548537i $$-0.184815\pi$$
0.836126 + 0.548537i $$0.184815\pi$$
$$360$$ 0 0
$$361$$ 40.0540 2.10810
$$362$$ 0 0
$$363$$ −4.43845 −0.232958
$$364$$ 0 0
$$365$$ −10.8078 −0.565704
$$366$$ 0 0
$$367$$ −15.3693 −0.802272 −0.401136 0.916019i $$-0.631384\pi$$
−0.401136 + 0.916019i $$0.631384\pi$$
$$368$$ 0 0
$$369$$ 7.12311 0.370814
$$370$$ 0 0
$$371$$ −19.0540 −0.989233
$$372$$ 0 0
$$373$$ −5.68466 −0.294340 −0.147170 0.989111i $$-0.547017\pi$$
−0.147170 + 0.989111i $$0.547017\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 14.2462 0.733717
$$378$$ 0 0
$$379$$ 7.05398 0.362338 0.181169 0.983452i $$-0.442012\pi$$
0.181169 + 0.983452i $$0.442012\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 10.2462 0.523557 0.261778 0.965128i $$-0.415691\pi$$
0.261778 + 0.965128i $$0.415691\pi$$
$$384$$ 0 0
$$385$$ 6.56155 0.334408
$$386$$ 0 0
$$387$$ −12.8078 −0.651055
$$388$$ 0 0
$$389$$ 7.75379 0.393133 0.196566 0.980491i $$-0.437021\pi$$
0.196566 + 0.980491i $$0.437021\pi$$
$$390$$ 0 0
$$391$$ 4.49242 0.227192
$$392$$ 0 0
$$393$$ 15.3693 0.775279
$$394$$ 0 0
$$395$$ 4.31534 0.217128
$$396$$ 0 0
$$397$$ 9.05398 0.454406 0.227203 0.973847i $$-0.427042\pi$$
0.227203 + 0.973847i $$0.427042\pi$$
$$398$$ 0 0
$$399$$ −19.6847 −0.985466
$$400$$ 0 0
$$401$$ −13.6847 −0.683379 −0.341690 0.939813i $$-0.610999\pi$$
−0.341690 + 0.939813i $$0.610999\pi$$
$$402$$ 0 0
$$403$$ −2.00000 −0.0996271
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ −37.3693 −1.84779 −0.923897 0.382642i $$-0.875014\pi$$
−0.923897 + 0.382642i $$0.875014\pi$$
$$410$$ 0 0
$$411$$ 20.2462 0.998672
$$412$$ 0 0
$$413$$ −33.6155 −1.65411
$$414$$ 0 0
$$415$$ −14.2462 −0.699319
$$416$$ 0 0
$$417$$ 17.1231 0.838522
$$418$$ 0 0
$$419$$ −5.75379 −0.281091 −0.140545 0.990074i $$-0.544886\pi$$
−0.140545 + 0.990074i $$0.544886\pi$$
$$420$$ 0 0
$$421$$ −14.4924 −0.706317 −0.353159 0.935563i $$-0.614892\pi$$
−0.353159 + 0.935563i $$0.614892\pi$$
$$422$$ 0 0
$$423$$ −5.12311 −0.249094
$$424$$ 0 0
$$425$$ −3.12311 −0.151493
$$426$$ 0 0
$$427$$ −15.3693 −0.743773
$$428$$ 0 0
$$429$$ −5.12311 −0.247346
$$430$$ 0 0
$$431$$ 30.2462 1.45691 0.728454 0.685094i $$-0.240239\pi$$
0.728454 + 0.685094i $$0.240239\pi$$
$$432$$ 0 0
$$433$$ 35.3002 1.69642 0.848209 0.529661i $$-0.177681\pi$$
0.848209 + 0.529661i $$0.177681\pi$$
$$434$$ 0 0
$$435$$ 7.12311 0.341527
$$436$$ 0 0
$$437$$ −11.0540 −0.528783
$$438$$ 0 0
$$439$$ −9.61553 −0.458924 −0.229462 0.973318i $$-0.573697\pi$$
−0.229462 + 0.973318i $$0.573697\pi$$
$$440$$ 0 0
$$441$$ −0.438447 −0.0208784
$$442$$ 0 0
$$443$$ −23.0540 −1.09533 −0.547664 0.836699i $$-0.684483\pi$$
−0.547664 + 0.836699i $$0.684483\pi$$
$$444$$ 0 0
$$445$$ −13.6847 −0.648715
$$446$$ 0 0
$$447$$ 17.0540 0.806625
$$448$$ 0 0
$$449$$ 10.4924 0.495168 0.247584 0.968866i $$-0.420363\pi$$
0.247584 + 0.968866i $$0.420363\pi$$
$$450$$ 0 0
$$451$$ −18.2462 −0.859181
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −5.12311 −0.240175
$$456$$ 0 0
$$457$$ −24.7386 −1.15722 −0.578612 0.815603i $$-0.696406\pi$$
−0.578612 + 0.815603i $$0.696406\pi$$
$$458$$ 0 0
$$459$$ −3.12311 −0.145774
$$460$$ 0 0
$$461$$ 9.36932 0.436373 0.218186 0.975907i $$-0.429986\pi$$
0.218186 + 0.975907i $$0.429986\pi$$
$$462$$ 0 0
$$463$$ −30.7386 −1.42855 −0.714273 0.699867i $$-0.753242\pi$$
−0.714273 + 0.699867i $$0.753242\pi$$
$$464$$ 0 0
$$465$$ −1.00000 −0.0463739
$$466$$ 0 0
$$467$$ 9.75379 0.451352 0.225676 0.974202i $$-0.427541\pi$$
0.225676 + 0.974202i $$0.427541\pi$$
$$468$$ 0 0
$$469$$ −39.3693 −1.81791
$$470$$ 0 0
$$471$$ 17.0540 0.785806
$$472$$ 0 0
$$473$$ 32.8078 1.50850
$$474$$ 0 0
$$475$$ 7.68466 0.352596
$$476$$ 0 0
$$477$$ 7.43845 0.340583
$$478$$ 0 0
$$479$$ −10.5616 −0.482570 −0.241285 0.970454i $$-0.577569\pi$$
−0.241285 + 0.970454i $$0.577569\pi$$
$$480$$ 0 0
$$481$$ −6.24621 −0.284803
$$482$$ 0 0
$$483$$ 3.68466 0.167658
$$484$$ 0 0
$$485$$ −6.00000 −0.272446
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ 0 0
$$489$$ 15.3693 0.695025
$$490$$ 0 0
$$491$$ 25.9309 1.17024 0.585122 0.810945i $$-0.301047\pi$$
0.585122 + 0.810945i $$0.301047\pi$$
$$492$$ 0 0
$$493$$ −22.2462 −1.00192
$$494$$ 0 0
$$495$$ −2.56155 −0.115133
$$496$$ 0 0
$$497$$ −19.6847 −0.882978
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 6.56155 0.293149
$$502$$ 0 0
$$503$$ −26.2462 −1.17026 −0.585130 0.810939i $$-0.698957\pi$$
−0.585130 + 0.810939i $$0.698957\pi$$
$$504$$ 0 0
$$505$$ −0.561553 −0.0249888
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 19.6155 0.869443 0.434721 0.900565i $$-0.356847\pi$$
0.434721 + 0.900565i $$0.356847\pi$$
$$510$$ 0 0
$$511$$ 27.6847 1.22470
$$512$$ 0 0
$$513$$ 7.68466 0.339286
$$514$$ 0 0
$$515$$ 1.75379 0.0772812
$$516$$ 0 0
$$517$$ 13.1231 0.577154
$$518$$ 0 0
$$519$$ 8.24621 0.361968
$$520$$ 0 0
$$521$$ −32.2462 −1.41273 −0.706366 0.707847i $$-0.749667\pi$$
−0.706366 + 0.707847i $$0.749667\pi$$
$$522$$ 0 0
$$523$$ 22.4233 0.980502 0.490251 0.871581i $$-0.336905\pi$$
0.490251 + 0.871581i $$0.336905\pi$$
$$524$$ 0 0
$$525$$ −2.56155 −0.111795
$$526$$ 0 0
$$527$$ 3.12311 0.136045
$$528$$ 0 0
$$529$$ −20.9309 −0.910038
$$530$$ 0 0
$$531$$ 13.1231 0.569494
$$532$$ 0 0
$$533$$ 14.2462 0.617072
$$534$$ 0 0
$$535$$ 2.56155 0.110746
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 1.12311 0.0483756
$$540$$ 0 0
$$541$$ 19.7538 0.849282 0.424641 0.905362i $$-0.360400\pi$$
0.424641 + 0.905362i $$0.360400\pi$$
$$542$$ 0 0
$$543$$ −13.0540 −0.560200
$$544$$ 0 0
$$545$$ 5.36932 0.229996
$$546$$ 0 0
$$547$$ −26.8769 −1.14917 −0.574587 0.818444i $$-0.694837\pi$$
−0.574587 + 0.818444i $$0.694837\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 54.7386 2.33194
$$552$$ 0 0
$$553$$ −11.0540 −0.470063
$$554$$ 0 0
$$555$$ −3.12311 −0.132568
$$556$$ 0 0
$$557$$ −26.8078 −1.13588 −0.567941 0.823069i $$-0.692260\pi$$
−0.567941 + 0.823069i $$0.692260\pi$$
$$558$$ 0 0
$$559$$ −25.6155 −1.08342
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ −16.4924 −0.695073 −0.347536 0.937667i $$-0.612982\pi$$
−0.347536 + 0.937667i $$0.612982\pi$$
$$564$$ 0 0
$$565$$ −1.68466 −0.0708741
$$566$$ 0 0
$$567$$ −2.56155 −0.107575
$$568$$ 0 0
$$569$$ 30.8078 1.29153 0.645764 0.763537i $$-0.276539\pi$$
0.645764 + 0.763537i $$0.276539\pi$$
$$570$$ 0 0
$$571$$ −41.1231 −1.72095 −0.860474 0.509494i $$-0.829833\pi$$
−0.860474 + 0.509494i $$0.829833\pi$$
$$572$$ 0 0
$$573$$ 14.2462 0.595144
$$574$$ 0 0
$$575$$ −1.43845 −0.0599874
$$576$$ 0 0
$$577$$ 15.1231 0.629583 0.314792 0.949161i $$-0.398065\pi$$
0.314792 + 0.949161i $$0.398065\pi$$
$$578$$ 0 0
$$579$$ 14.4924 0.602285
$$580$$ 0 0
$$581$$ 36.4924 1.51396
$$582$$ 0 0
$$583$$ −19.0540 −0.789135
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ 0.492423 0.0203245 0.0101622 0.999948i $$-0.496765\pi$$
0.0101622 + 0.999948i $$0.496765\pi$$
$$588$$ 0 0
$$589$$ −7.68466 −0.316641
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ 0 0
$$597$$ 16.8078 0.687896
$$598$$ 0 0
$$599$$ −38.4233 −1.56993 −0.784967 0.619538i $$-0.787320\pi$$
−0.784967 + 0.619538i $$0.787320\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 15.3693 0.625887
$$604$$ 0 0
$$605$$ −4.43845 −0.180449
$$606$$ 0 0
$$607$$ −7.05398 −0.286312 −0.143156 0.989700i $$-0.545725\pi$$
−0.143156 + 0.989700i $$0.545725\pi$$
$$608$$ 0 0
$$609$$ −18.2462 −0.739374
$$610$$ 0 0
$$611$$ −10.2462 −0.414517
$$612$$ 0 0
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 7.12311 0.287231
$$616$$ 0 0
$$617$$ −15.4384 −0.621528 −0.310764 0.950487i $$-0.600585\pi$$
−0.310764 + 0.950487i $$0.600585\pi$$
$$618$$ 0 0
$$619$$ −11.3693 −0.456971 −0.228486 0.973547i $$-0.573377\pi$$
−0.228486 + 0.973547i $$0.573377\pi$$
$$620$$ 0 0
$$621$$ −1.43845 −0.0577229
$$622$$ 0 0
$$623$$ 35.0540 1.40441
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −19.6847 −0.786130
$$628$$ 0 0
$$629$$ 9.75379 0.388909
$$630$$ 0 0
$$631$$ −29.3002 −1.16642 −0.583211 0.812321i $$-0.698204\pi$$
−0.583211 + 0.812321i $$0.698204\pi$$
$$632$$ 0 0
$$633$$ 23.6847 0.941381
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −0.876894 −0.0347438
$$638$$ 0 0
$$639$$ 7.68466 0.304000
$$640$$ 0 0
$$641$$ −5.50758 −0.217536 −0.108768 0.994067i $$-0.534691\pi$$
−0.108768 + 0.994067i $$0.534691\pi$$
$$642$$ 0 0
$$643$$ 7.05398 0.278182 0.139091 0.990280i $$-0.455582\pi$$
0.139091 + 0.990280i $$0.455582\pi$$
$$644$$ 0 0
$$645$$ −12.8078 −0.504305
$$646$$ 0 0
$$647$$ 45.9309 1.80573 0.902864 0.429925i $$-0.141460\pi$$
0.902864 + 0.429925i $$0.141460\pi$$
$$648$$ 0 0
$$649$$ −33.6155 −1.31952
$$650$$ 0 0
$$651$$ 2.56155 0.100395
$$652$$ 0 0
$$653$$ 40.2462 1.57496 0.787478 0.616343i $$-0.211386\pi$$
0.787478 + 0.616343i $$0.211386\pi$$
$$654$$ 0 0
$$655$$ 15.3693 0.600529
$$656$$ 0 0
$$657$$ −10.8078 −0.421651
$$658$$ 0 0
$$659$$ 30.7386 1.19741 0.598704 0.800971i $$-0.295683\pi$$
0.598704 + 0.800971i $$0.295683\pi$$
$$660$$ 0 0
$$661$$ 26.4924 1.03044 0.515218 0.857059i $$-0.327711\pi$$
0.515218 + 0.857059i $$0.327711\pi$$
$$662$$ 0 0
$$663$$ −6.24621 −0.242583
$$664$$ 0 0
$$665$$ −19.6847 −0.763338
$$666$$ 0 0
$$667$$ −10.2462 −0.396735
$$668$$ 0 0
$$669$$ 21.1231 0.816666
$$670$$ 0 0
$$671$$ −15.3693 −0.593326
$$672$$ 0 0
$$673$$ 11.7538 0.453075 0.226538 0.974002i $$-0.427259\pi$$
0.226538 + 0.974002i $$0.427259\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 32.4233 1.24613 0.623064 0.782171i $$-0.285888\pi$$
0.623064 + 0.782171i $$0.285888\pi$$
$$678$$ 0 0
$$679$$ 15.3693 0.589820
$$680$$ 0 0
$$681$$ −7.68466 −0.294477
$$682$$ 0 0
$$683$$ −31.6847 −1.21238 −0.606190 0.795320i $$-0.707303\pi$$
−0.606190 + 0.795320i $$0.707303\pi$$
$$684$$ 0 0
$$685$$ 20.2462 0.773568
$$686$$ 0 0
$$687$$ −16.5616 −0.631863
$$688$$ 0 0
$$689$$ 14.8769 0.566765
$$690$$ 0 0
$$691$$ −15.0540 −0.572680 −0.286340 0.958128i $$-0.592439\pi$$
−0.286340 + 0.958128i $$0.592439\pi$$
$$692$$ 0 0
$$693$$ 6.56155 0.249253
$$694$$ 0 0
$$695$$ 17.1231 0.649516
$$696$$ 0 0
$$697$$ −22.2462 −0.842635
$$698$$ 0 0
$$699$$ −17.0540 −0.645041
$$700$$ 0 0
$$701$$ 5.19224 0.196108 0.0980540 0.995181i $$-0.468738\pi$$
0.0980540 + 0.995181i $$0.468738\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ 0 0
$$705$$ −5.12311 −0.192947
$$706$$ 0 0
$$707$$ 1.43845 0.0540984
$$708$$ 0 0
$$709$$ 17.0540 0.640475 0.320238 0.947337i $$-0.396237\pi$$
0.320238 + 0.947337i $$0.396237\pi$$
$$710$$ 0 0
$$711$$ 4.31534 0.161838
$$712$$ 0 0
$$713$$ 1.43845 0.0538703
$$714$$ 0 0
$$715$$ −5.12311 −0.191593
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ 34.8769 1.30069 0.650344 0.759640i $$-0.274625\pi$$
0.650344 + 0.759640i $$0.274625\pi$$
$$720$$ 0 0
$$721$$ −4.49242 −0.167307
$$722$$ 0 0
$$723$$ 12.2462 0.455441
$$724$$ 0 0
$$725$$ 7.12311 0.264546
$$726$$ 0 0
$$727$$ −23.0540 −0.855025 −0.427512 0.904010i $$-0.640610\pi$$
−0.427512 + 0.904010i $$0.640610\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ 0 0
$$733$$ −6.49242 −0.239803 −0.119902 0.992786i $$-0.538258\pi$$
−0.119902 + 0.992786i $$0.538258\pi$$
$$734$$ 0 0
$$735$$ −0.438447 −0.0161724
$$736$$ 0 0
$$737$$ −39.3693 −1.45019
$$738$$ 0 0
$$739$$ −52.9848 −1.94908 −0.974540 0.224216i $$-0.928018\pi$$
−0.974540 + 0.224216i $$0.928018\pi$$
$$740$$ 0 0
$$741$$ 15.3693 0.564606
$$742$$ 0 0
$$743$$ 50.4233 1.84985 0.924926 0.380148i $$-0.124127\pi$$
0.924926 + 0.380148i $$0.124127\pi$$
$$744$$ 0 0
$$745$$ 17.0540 0.624809
$$746$$ 0 0
$$747$$ −14.2462 −0.521242
$$748$$ 0 0
$$749$$ −6.56155 −0.239754
$$750$$ 0 0
$$751$$ −45.1231 −1.64657 −0.823283 0.567631i $$-0.807860\pi$$
−0.823283 + 0.567631i $$0.807860\pi$$
$$752$$ 0 0
$$753$$ −16.4924 −0.601017
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 3.68466 0.133745
$$760$$ 0 0
$$761$$ −20.4233 −0.740344 −0.370172 0.928963i $$-0.620701\pi$$
−0.370172 + 0.928963i $$0.620701\pi$$
$$762$$ 0 0
$$763$$ −13.7538 −0.497921
$$764$$ 0 0
$$765$$ −3.12311 −0.112916
$$766$$ 0 0
$$767$$ 26.2462 0.947696
$$768$$ 0 0
$$769$$ −20.5616 −0.741469 −0.370734 0.928739i $$-0.620894\pi$$
−0.370734 + 0.928739i $$0.620894\pi$$
$$770$$ 0 0
$$771$$ −1.68466 −0.0606715
$$772$$ 0 0
$$773$$ −11.4384 −0.411412 −0.205706 0.978614i $$-0.565949\pi$$
−0.205706 + 0.978614i $$0.565949\pi$$
$$774$$ 0 0
$$775$$ −1.00000 −0.0359211
$$776$$ 0 0
$$777$$ 8.00000 0.286998
$$778$$ 0 0
$$779$$ 54.7386 1.96122
$$780$$ 0 0
$$781$$ −19.6847 −0.704372
$$782$$ 0 0
$$783$$ 7.12311 0.254559
$$784$$ 0 0
$$785$$ 17.0540 0.608682
$$786$$ 0 0
$$787$$ 17.9309 0.639166 0.319583 0.947558i $$-0.396457\pi$$
0.319583 + 0.947558i $$0.396457\pi$$
$$788$$ 0 0
$$789$$ 20.4924 0.729550
$$790$$ 0 0
$$791$$ 4.31534 0.153436
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 0 0
$$795$$ 7.43845 0.263815
$$796$$ 0 0
$$797$$ −26.9848 −0.955852 −0.477926 0.878400i $$-0.658611\pi$$
−0.477926 + 0.878400i $$0.658611\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −13.6847 −0.483524
$$802$$ 0 0
$$803$$ 27.6847 0.976970
$$804$$ 0 0
$$805$$ 3.68466 0.129867
$$806$$ 0 0
$$807$$ −0.246211 −0.00866705
$$808$$ 0 0
$$809$$ −37.6847 −1.32492 −0.662461 0.749096i $$-0.730488\pi$$
−0.662461 + 0.749096i $$0.730488\pi$$
$$810$$ 0 0
$$811$$ 24.6695 0.866263 0.433132 0.901331i $$-0.357408\pi$$
0.433132 + 0.901331i $$0.357408\pi$$
$$812$$ 0 0
$$813$$ −27.0540 −0.948824
$$814$$ 0 0
$$815$$ 15.3693 0.538364
$$816$$ 0 0
$$817$$ −98.4233 −3.44340
$$818$$ 0 0
$$819$$ −5.12311 −0.179016
$$820$$ 0 0
$$821$$ 19.6155 0.684587 0.342293 0.939593i $$-0.388796\pi$$
0.342293 + 0.939593i $$0.388796\pi$$
$$822$$ 0 0
$$823$$ 25.6155 0.892901 0.446451 0.894808i $$-0.352688\pi$$
0.446451 + 0.894808i $$0.352688\pi$$
$$824$$ 0 0
$$825$$ −2.56155 −0.0891818
$$826$$ 0 0
$$827$$ −1.75379 −0.0609852 −0.0304926 0.999535i $$-0.509708\pi$$
−0.0304926 + 0.999535i $$0.509708\pi$$
$$828$$ 0 0
$$829$$ 24.4233 0.848256 0.424128 0.905602i $$-0.360581\pi$$
0.424128 + 0.905602i $$0.360581\pi$$
$$830$$ 0 0
$$831$$ −5.36932 −0.186260
$$832$$ 0 0
$$833$$ 1.36932 0.0474440
$$834$$ 0 0
$$835$$ 6.56155 0.227072
$$836$$ 0 0
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ −6.06913 −0.209530 −0.104765 0.994497i $$-0.533409\pi$$
−0.104765 + 0.994497i $$0.533409\pi$$
$$840$$ 0 0
$$841$$ 21.7386 0.749608
$$842$$ 0 0
$$843$$ 4.87689 0.167969
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 11.3693 0.390654
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4.49242 0.153998
$$852$$ 0 0
$$853$$ 47.7926 1.63639 0.818194 0.574942i $$-0.194976\pi$$
0.818194 + 0.574942i $$0.194976\pi$$
$$854$$ 0 0
$$855$$ 7.68466 0.262810
$$856$$ 0 0
$$857$$ 29.2311 0.998514 0.499257 0.866454i $$-0.333606\pi$$
0.499257 + 0.866454i $$0.333606\pi$$
$$858$$ 0 0
$$859$$ −26.7386 −0.912310 −0.456155 0.889900i $$-0.650774\pi$$
−0.456155 + 0.889900i $$0.650774\pi$$
$$860$$ 0 0
$$861$$ −18.2462 −0.621829
$$862$$ 0 0
$$863$$ −39.5464 −1.34618 −0.673088 0.739563i $$-0.735032\pi$$
−0.673088 + 0.739563i $$0.735032\pi$$
$$864$$ 0 0
$$865$$ 8.24621 0.280380
$$866$$ 0 0
$$867$$ −7.24621 −0.246094
$$868$$ 0 0
$$869$$ −11.0540 −0.374980
$$870$$ 0 0
$$871$$ 30.7386 1.04154
$$872$$ 0 0
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ −2.56155 −0.0865963
$$876$$ 0 0
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ 0 0
$$879$$ 26.4924 0.893567
$$880$$ 0 0
$$881$$ 9.50758 0.320318 0.160159 0.987091i $$-0.448799\pi$$
0.160159 + 0.987091i $$0.448799\pi$$
$$882$$ 0 0
$$883$$ −30.4233 −1.02383 −0.511913 0.859038i $$-0.671063\pi$$
−0.511913 + 0.859038i $$0.671063\pi$$
$$884$$ 0 0
$$885$$ 13.1231 0.441128
$$886$$ 0 0
$$887$$ 32.6307 1.09563 0.547816 0.836599i $$-0.315460\pi$$
0.547816 + 0.836599i $$0.315460\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.56155 −0.0858152
$$892$$ 0 0
$$893$$ −39.3693 −1.31744
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ −2.87689 −0.0960567
$$898$$ 0 0
$$899$$ −7.12311 −0.237569
$$900$$ 0 0
$$901$$ −23.2311 −0.773939
$$902$$ 0 0
$$903$$ 32.8078 1.09177
$$904$$ 0 0
$$905$$ −13.0540 −0.433929
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 0 0
$$909$$ −0.561553 −0.0186255
$$910$$ 0 0
$$911$$ 11.5076 0.381263 0.190632 0.981662i $$-0.438946\pi$$
0.190632 + 0.981662i $$0.438946\pi$$
$$912$$ 0 0
$$913$$ 36.4924 1.20772
$$914$$ 0 0
$$915$$ 6.00000 0.198354
$$916$$ 0 0
$$917$$ −39.3693 −1.30009
$$918$$ 0 0
$$919$$ 1.61553 0.0532914 0.0266457 0.999645i $$-0.491517\pi$$
0.0266457 + 0.999645i $$0.491517\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 15.3693 0.505887
$$924$$ 0 0
$$925$$ −3.12311 −0.102687
$$926$$ 0 0
$$927$$ 1.75379 0.0576020
$$928$$ 0 0
$$929$$ 7.43845 0.244048 0.122024 0.992527i $$-0.461062\pi$$
0.122024 + 0.992527i $$0.461062\pi$$
$$930$$ 0 0
$$931$$ −3.36932 −0.110425
$$932$$ 0 0
$$933$$ 1.75379 0.0574165
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 41.3693 1.35148 0.675738 0.737142i $$-0.263825\pi$$
0.675738 + 0.737142i $$0.263825\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 2.63068 0.0857578 0.0428789 0.999080i $$-0.486347\pi$$
0.0428789 + 0.999080i $$0.486347\pi$$
$$942$$ 0 0
$$943$$ −10.2462 −0.333663
$$944$$ 0 0
$$945$$ −2.56155 −0.0833273
$$946$$ 0 0
$$947$$ 9.12311 0.296461 0.148231 0.988953i $$-0.452642\pi$$
0.148231 + 0.988953i $$0.452642\pi$$
$$948$$ 0 0
$$949$$ −21.6155 −0.701670
$$950$$ 0 0
$$951$$ 16.8769 0.547271
$$952$$ 0 0
$$953$$ 16.1080 0.521788 0.260894 0.965367i $$-0.415983\pi$$
0.260894 + 0.965367i $$0.415983\pi$$
$$954$$ 0 0
$$955$$ 14.2462 0.460997
$$956$$ 0 0
$$957$$ −18.2462 −0.589816
$$958$$ 0 0
$$959$$ −51.8617 −1.67470
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ 2.56155 0.0825449
$$964$$ 0 0
$$965$$ 14.4924 0.466528
$$966$$ 0 0
$$967$$ 42.2462 1.35855 0.679273 0.733885i $$-0.262295\pi$$
0.679273 + 0.733885i $$0.262295\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 37.1231 1.19134 0.595669 0.803230i $$-0.296887\pi$$
0.595669 + 0.803230i $$0.296887\pi$$
$$972$$ 0 0
$$973$$ −43.8617 −1.40614
$$974$$ 0 0
$$975$$ 2.00000 0.0640513
$$976$$ 0 0
$$977$$ 42.9848 1.37521 0.687604 0.726086i $$-0.258663\pi$$
0.687604 + 0.726086i $$0.258663\pi$$
$$978$$ 0 0
$$979$$ 35.0540 1.12033
$$980$$ 0 0
$$981$$ 5.36932 0.171429
$$982$$ 0 0
$$983$$ −40.9848 −1.30721 −0.653607 0.756834i $$-0.726745\pi$$
−0.653607 + 0.756834i $$0.726745\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 13.1231 0.417713
$$988$$ 0 0
$$989$$ 18.4233 0.585827
$$990$$ 0 0
$$991$$ −35.6847 −1.13356 −0.566780 0.823869i $$-0.691811\pi$$
−0.566780 + 0.823869i $$0.691811\pi$$
$$992$$ 0 0
$$993$$ 6.24621 0.198218
$$994$$ 0 0
$$995$$ 16.8078 0.532842
$$996$$ 0 0
$$997$$ −29.5076 −0.934514 −0.467257 0.884121i $$-0.654758\pi$$
−0.467257 + 0.884121i $$0.654758\pi$$
$$998$$ 0 0
$$999$$ −3.12311 −0.0988107
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 7440.2.a.bl.1.1 2
4.3 odd 2 930.2.a.p.1.2 2
12.11 even 2 2790.2.a.be.1.2 2
20.3 even 4 4650.2.d.bd.3349.1 4
20.7 even 4 4650.2.d.bd.3349.4 4
20.19 odd 2 4650.2.a.ce.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 2 4.3 odd 2
2790.2.a.be.1.2 2 12.11 even 2
4650.2.a.ce.1.1 2 20.19 odd 2
4650.2.d.bd.3349.1 4 20.3 even 4
4650.2.d.bd.3349.4 4 20.7 even 4
7440.2.a.bl.1.1 2 1.1 even 1 trivial