# Properties

 Label 2240.2.a.be.1.2 Level $2240$ Weight $2$ Character 2240.1 Self dual yes Analytic conductor $17.886$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2240.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$$ $$q+1.56155 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.561553 q^{9} -1.56155 q^{11} -6.68466 q^{13} -1.56155 q^{15} +7.56155 q^{17} +7.12311 q^{19} +1.56155 q^{21} +3.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} -0.438447 q^{29} +6.24621 q^{31} -2.43845 q^{33} -1.00000 q^{35} +8.24621 q^{37} -10.4384 q^{39} -1.12311 q^{41} +7.12311 q^{43} +0.561553 q^{45} +2.43845 q^{47} +1.00000 q^{49} +11.8078 q^{51} +13.1231 q^{53} +1.56155 q^{55} +11.1231 q^{57} +4.00000 q^{59} +6.87689 q^{61} -0.561553 q^{63} +6.68466 q^{65} -2.24621 q^{67} +4.87689 q^{69} -4.24621 q^{73} +1.56155 q^{75} -1.56155 q^{77} +0.684658 q^{79} -7.00000 q^{81} -12.0000 q^{83} -7.56155 q^{85} -0.684658 q^{87} +5.12311 q^{89} -6.68466 q^{91} +9.75379 q^{93} -7.12311 q^{95} +1.31534 q^{97} +0.876894 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$2q - q^{3} - 2q^{5} + 2q^{7} + 3q^{9} + q^{11} - q^{13} + q^{15} + 11q^{17} + 6q^{19} - q^{21} - 2q^{23} + 2q^{25} - 7q^{27} - 5q^{29} - 4q^{31} - 9q^{33} - 2q^{35} - 25q^{39} + 6q^{41} + 6q^{43} - 3q^{45} + 9q^{47} + 2q^{49} + 3q^{51} + 18q^{53} - q^{55} + 14q^{57} + 8q^{59} + 22q^{61} + 3q^{63} + q^{65} + 12q^{67} + 18q^{69} + 8q^{73} - q^{75} + q^{77} - 11q^{79} - 14q^{81} - 24q^{83} - 11q^{85} + 11q^{87} + 2q^{89} - q^{91} + 36q^{93} - 6q^{95} + 15q^{97} + 10q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.56155 0.901563 0.450781 0.892634i $$-0.351145\pi$$
0.450781 + 0.892634i $$0.351145\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −0.561553 −0.187184
$$10$$ 0 0
$$11$$ −1.56155 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$12$$ 0 0
$$13$$ −6.68466 −1.85399 −0.926995 0.375073i $$-0.877618\pi$$
−0.926995 + 0.375073i $$0.877618\pi$$
$$14$$ 0 0
$$15$$ −1.56155 −0.403191
$$16$$ 0 0
$$17$$ 7.56155 1.83395 0.916973 0.398949i $$-0.130625\pi$$
0.916973 + 0.398949i $$0.130625\pi$$
$$18$$ 0 0
$$19$$ 7.12311 1.63415 0.817076 0.576530i $$-0.195593\pi$$
0.817076 + 0.576530i $$0.195593\pi$$
$$20$$ 0 0
$$21$$ 1.56155 0.340759
$$22$$ 0 0
$$23$$ 3.12311 0.651213 0.325606 0.945505i $$-0.394432\pi$$
0.325606 + 0.945505i $$0.394432\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.56155 −1.07032
$$28$$ 0 0
$$29$$ −0.438447 −0.0814176 −0.0407088 0.999171i $$-0.512962\pi$$
−0.0407088 + 0.999171i $$0.512962\pi$$
$$30$$ 0 0
$$31$$ 6.24621 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$32$$ 0 0
$$33$$ −2.43845 −0.424479
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 8.24621 1.35567 0.677834 0.735215i $$-0.262919\pi$$
0.677834 + 0.735215i $$0.262919\pi$$
$$38$$ 0 0
$$39$$ −10.4384 −1.67149
$$40$$ 0 0
$$41$$ −1.12311 −0.175400 −0.0876998 0.996147i $$-0.527952\pi$$
−0.0876998 + 0.996147i $$0.527952\pi$$
$$42$$ 0 0
$$43$$ 7.12311 1.08626 0.543132 0.839648i $$-0.317238\pi$$
0.543132 + 0.839648i $$0.317238\pi$$
$$44$$ 0 0
$$45$$ 0.561553 0.0837114
$$46$$ 0 0
$$47$$ 2.43845 0.355684 0.177842 0.984059i $$-0.443088\pi$$
0.177842 + 0.984059i $$0.443088\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 11.8078 1.65342
$$52$$ 0 0
$$53$$ 13.1231 1.80260 0.901299 0.433198i $$-0.142615\pi$$
0.901299 + 0.433198i $$0.142615\pi$$
$$54$$ 0 0
$$55$$ 1.56155 0.210560
$$56$$ 0 0
$$57$$ 11.1231 1.47329
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 6.87689 0.880496 0.440248 0.897876i $$-0.354891\pi$$
0.440248 + 0.897876i $$0.354891\pi$$
$$62$$ 0 0
$$63$$ −0.561553 −0.0707490
$$64$$ 0 0
$$65$$ 6.68466 0.829130
$$66$$ 0 0
$$67$$ −2.24621 −0.274418 −0.137209 0.990542i $$-0.543813\pi$$
−0.137209 + 0.990542i $$0.543813\pi$$
$$68$$ 0 0
$$69$$ 4.87689 0.587109
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −4.24621 −0.496981 −0.248491 0.968634i $$-0.579935\pi$$
−0.248491 + 0.968634i $$0.579935\pi$$
$$74$$ 0 0
$$75$$ 1.56155 0.180313
$$76$$ 0 0
$$77$$ −1.56155 −0.177955
$$78$$ 0 0
$$79$$ 0.684658 0.0770301 0.0385150 0.999258i $$-0.487737\pi$$
0.0385150 + 0.999258i $$0.487737\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −7.56155 −0.820166
$$86$$ 0 0
$$87$$ −0.684658 −0.0734031
$$88$$ 0 0
$$89$$ 5.12311 0.543048 0.271524 0.962432i $$-0.412472\pi$$
0.271524 + 0.962432i $$0.412472\pi$$
$$90$$ 0 0
$$91$$ −6.68466 −0.700743
$$92$$ 0 0
$$93$$ 9.75379 1.01142
$$94$$ 0 0
$$95$$ −7.12311 −0.730815
$$96$$ 0 0
$$97$$ 1.31534 0.133553 0.0667764 0.997768i $$-0.478729\pi$$
0.0667764 + 0.997768i $$0.478729\pi$$
$$98$$ 0 0
$$99$$ 0.876894 0.0881312
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −11.8078 −1.16345 −0.581727 0.813384i $$-0.697623\pi$$
−0.581727 + 0.813384i $$0.697623\pi$$
$$104$$ 0 0
$$105$$ −1.56155 −0.152392
$$106$$ 0 0
$$107$$ −15.1231 −1.46201 −0.731003 0.682374i $$-0.760947\pi$$
−0.731003 + 0.682374i $$0.760947\pi$$
$$108$$ 0 0
$$109$$ 4.43845 0.425126 0.212563 0.977147i $$-0.431819\pi$$
0.212563 + 0.977147i $$0.431819\pi$$
$$110$$ 0 0
$$111$$ 12.8769 1.22222
$$112$$ 0 0
$$113$$ 8.24621 0.775738 0.387869 0.921714i $$-0.373211\pi$$
0.387869 + 0.921714i $$0.373211\pi$$
$$114$$ 0 0
$$115$$ −3.12311 −0.291231
$$116$$ 0 0
$$117$$ 3.75379 0.347038
$$118$$ 0 0
$$119$$ 7.56155 0.693166
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ 0 0
$$123$$ −1.75379 −0.158134
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.24621 −0.554262 −0.277131 0.960832i $$-0.589384\pi$$
−0.277131 + 0.960832i $$0.589384\pi$$
$$128$$ 0 0
$$129$$ 11.1231 0.979335
$$130$$ 0 0
$$131$$ −15.1231 −1.32131 −0.660656 0.750689i $$-0.729722\pi$$
−0.660656 + 0.750689i $$0.729722\pi$$
$$132$$ 0 0
$$133$$ 7.12311 0.617652
$$134$$ 0 0
$$135$$ 5.56155 0.478662
$$136$$ 0 0
$$137$$ −7.36932 −0.629603 −0.314802 0.949157i $$-0.601938\pi$$
−0.314802 + 0.949157i $$0.601938\pi$$
$$138$$ 0 0
$$139$$ 21.3693 1.81252 0.906261 0.422719i $$-0.138924\pi$$
0.906261 + 0.422719i $$0.138924\pi$$
$$140$$ 0 0
$$141$$ 3.80776 0.320672
$$142$$ 0 0
$$143$$ 10.4384 0.872907
$$144$$ 0 0
$$145$$ 0.438447 0.0364111
$$146$$ 0 0
$$147$$ 1.56155 0.128795
$$148$$ 0 0
$$149$$ 0.246211 0.0201704 0.0100852 0.999949i $$-0.496790\pi$$
0.0100852 + 0.999949i $$0.496790\pi$$
$$150$$ 0 0
$$151$$ −19.8078 −1.61193 −0.805966 0.591961i $$-0.798354\pi$$
−0.805966 + 0.591961i $$0.798354\pi$$
$$152$$ 0 0
$$153$$ −4.24621 −0.343286
$$154$$ 0 0
$$155$$ −6.24621 −0.501708
$$156$$ 0 0
$$157$$ −4.24621 −0.338885 −0.169442 0.985540i $$-0.554197\pi$$
−0.169442 + 0.985540i $$0.554197\pi$$
$$158$$ 0 0
$$159$$ 20.4924 1.62515
$$160$$ 0 0
$$161$$ 3.12311 0.246135
$$162$$ 0 0
$$163$$ −19.6155 −1.53641 −0.768203 0.640206i $$-0.778849\pi$$
−0.768203 + 0.640206i $$0.778849\pi$$
$$164$$ 0 0
$$165$$ 2.43845 0.189833
$$166$$ 0 0
$$167$$ 4.19224 0.324405 0.162202 0.986757i $$-0.448140\pi$$
0.162202 + 0.986757i $$0.448140\pi$$
$$168$$ 0 0
$$169$$ 31.6847 2.43728
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 23.1771 1.76212 0.881060 0.473004i $$-0.156830\pi$$
0.881060 + 0.473004i $$0.156830\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 6.24621 0.469494
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 5.12311 0.380797 0.190399 0.981707i $$-0.439022\pi$$
0.190399 + 0.981707i $$0.439022\pi$$
$$182$$ 0 0
$$183$$ 10.7386 0.793823
$$184$$ 0 0
$$185$$ −8.24621 −0.606274
$$186$$ 0 0
$$187$$ −11.8078 −0.863469
$$188$$ 0 0
$$189$$ −5.56155 −0.404543
$$190$$ 0 0
$$191$$ −0.684658 −0.0495401 −0.0247701 0.999693i $$-0.507885\pi$$
−0.0247701 + 0.999693i $$0.507885\pi$$
$$192$$ 0 0
$$193$$ 13.1231 0.944622 0.472311 0.881432i $$-0.343420\pi$$
0.472311 + 0.881432i $$0.343420\pi$$
$$194$$ 0 0
$$195$$ 10.4384 0.747513
$$196$$ 0 0
$$197$$ 13.1231 0.934983 0.467491 0.883998i $$-0.345158\pi$$
0.467491 + 0.883998i $$0.345158\pi$$
$$198$$ 0 0
$$199$$ −14.2462 −1.00989 −0.504944 0.863152i $$-0.668487\pi$$
−0.504944 + 0.863152i $$0.668487\pi$$
$$200$$ 0 0
$$201$$ −3.50758 −0.247405
$$202$$ 0 0
$$203$$ −0.438447 −0.0307730
$$204$$ 0 0
$$205$$ 1.12311 0.0784411
$$206$$ 0 0
$$207$$ −1.75379 −0.121897
$$208$$ 0 0
$$209$$ −11.1231 −0.769401
$$210$$ 0 0
$$211$$ 17.5616 1.20899 0.604494 0.796610i $$-0.293376\pi$$
0.604494 + 0.796610i $$0.293376\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −7.12311 −0.485792
$$216$$ 0 0
$$217$$ 6.24621 0.424020
$$218$$ 0 0
$$219$$ −6.63068 −0.448060
$$220$$ 0 0
$$221$$ −50.5464 −3.40012
$$222$$ 0 0
$$223$$ 24.6847 1.65301 0.826503 0.562932i $$-0.190327\pi$$
0.826503 + 0.562932i $$0.190327\pi$$
$$224$$ 0 0
$$225$$ −0.561553 −0.0374369
$$226$$ 0 0
$$227$$ 11.3153 0.751026 0.375513 0.926817i $$-0.377467\pi$$
0.375513 + 0.926817i $$0.377467\pi$$
$$228$$ 0 0
$$229$$ 11.3693 0.751306 0.375653 0.926760i $$-0.377419\pi$$
0.375653 + 0.926760i $$0.377419\pi$$
$$230$$ 0 0
$$231$$ −2.43845 −0.160438
$$232$$ 0 0
$$233$$ −10.8769 −0.712569 −0.356285 0.934378i $$-0.615957\pi$$
−0.356285 + 0.934378i $$0.615957\pi$$
$$234$$ 0 0
$$235$$ −2.43845 −0.159067
$$236$$ 0 0
$$237$$ 1.06913 0.0694475
$$238$$ 0 0
$$239$$ −18.0540 −1.16781 −0.583907 0.811820i $$-0.698477\pi$$
−0.583907 + 0.811820i $$0.698477\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 5.75379 0.369106
$$244$$ 0 0
$$245$$ −1.00000 −0.0638877
$$246$$ 0 0
$$247$$ −47.6155 −3.02970
$$248$$ 0 0
$$249$$ −18.7386 −1.18751
$$250$$ 0 0
$$251$$ −13.3693 −0.843864 −0.421932 0.906628i $$-0.638648\pi$$
−0.421932 + 0.906628i $$0.638648\pi$$
$$252$$ 0 0
$$253$$ −4.87689 −0.306608
$$254$$ 0 0
$$255$$ −11.8078 −0.739431
$$256$$ 0 0
$$257$$ −18.4924 −1.15353 −0.576763 0.816912i $$-0.695684\pi$$
−0.576763 + 0.816912i $$0.695684\pi$$
$$258$$ 0 0
$$259$$ 8.24621 0.512395
$$260$$ 0 0
$$261$$ 0.246211 0.0152401
$$262$$ 0 0
$$263$$ 9.36932 0.577737 0.288868 0.957369i $$-0.406721\pi$$
0.288868 + 0.957369i $$0.406721\pi$$
$$264$$ 0 0
$$265$$ −13.1231 −0.806146
$$266$$ 0 0
$$267$$ 8.00000 0.489592
$$268$$ 0 0
$$269$$ −4.24621 −0.258896 −0.129448 0.991586i $$-0.541321\pi$$
−0.129448 + 0.991586i $$0.541321\pi$$
$$270$$ 0 0
$$271$$ −6.24621 −0.379430 −0.189715 0.981839i $$-0.560756\pi$$
−0.189715 + 0.981839i $$0.560756\pi$$
$$272$$ 0 0
$$273$$ −10.4384 −0.631764
$$274$$ 0 0
$$275$$ −1.56155 −0.0941652
$$276$$ 0 0
$$277$$ 8.24621 0.495467 0.247733 0.968828i $$-0.420314\pi$$
0.247733 + 0.968828i $$0.420314\pi$$
$$278$$ 0 0
$$279$$ −3.50758 −0.209993
$$280$$ 0 0
$$281$$ −19.5616 −1.16694 −0.583472 0.812133i $$-0.698306\pi$$
−0.583472 + 0.812133i $$0.698306\pi$$
$$282$$ 0 0
$$283$$ 4.68466 0.278474 0.139237 0.990259i $$-0.455535\pi$$
0.139237 + 0.990259i $$0.455535\pi$$
$$284$$ 0 0
$$285$$ −11.1231 −0.658876
$$286$$ 0 0
$$287$$ −1.12311 −0.0662948
$$288$$ 0 0
$$289$$ 40.1771 2.36336
$$290$$ 0 0
$$291$$ 2.05398 0.120406
$$292$$ 0 0
$$293$$ −32.0540 −1.87261 −0.936307 0.351184i $$-0.885779\pi$$
−0.936307 + 0.351184i $$0.885779\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 8.68466 0.503935
$$298$$ 0 0
$$299$$ −20.8769 −1.20734
$$300$$ 0 0
$$301$$ 7.12311 0.410569
$$302$$ 0 0
$$303$$ −9.36932 −0.538253
$$304$$ 0 0
$$305$$ −6.87689 −0.393770
$$306$$ 0 0
$$307$$ 28.6847 1.63712 0.818560 0.574421i $$-0.194773\pi$$
0.818560 + 0.574421i $$0.194773\pi$$
$$308$$ 0 0
$$309$$ −18.4384 −1.04893
$$310$$ 0 0
$$311$$ −12.8769 −0.730182 −0.365091 0.930972i $$-0.618962\pi$$
−0.365091 + 0.930972i $$0.618962\pi$$
$$312$$ 0 0
$$313$$ 26.6847 1.50831 0.754153 0.656699i $$-0.228048\pi$$
0.754153 + 0.656699i $$0.228048\pi$$
$$314$$ 0 0
$$315$$ 0.561553 0.0316399
$$316$$ 0 0
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ 0 0
$$319$$ 0.684658 0.0383335
$$320$$ 0 0
$$321$$ −23.6155 −1.31809
$$322$$ 0 0
$$323$$ 53.8617 2.99695
$$324$$ 0 0
$$325$$ −6.68466 −0.370798
$$326$$ 0 0
$$327$$ 6.93087 0.383278
$$328$$ 0 0
$$329$$ 2.43845 0.134436
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 0 0
$$333$$ −4.63068 −0.253760
$$334$$ 0 0
$$335$$ 2.24621 0.122724
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 0 0
$$339$$ 12.8769 0.699377
$$340$$ 0 0
$$341$$ −9.75379 −0.528197
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −4.87689 −0.262563
$$346$$ 0 0
$$347$$ −15.1231 −0.811851 −0.405925 0.913906i $$-0.633051\pi$$
−0.405925 + 0.913906i $$0.633051\pi$$
$$348$$ 0 0
$$349$$ 11.7538 0.629166 0.314583 0.949230i $$-0.398135\pi$$
0.314583 + 0.949230i $$0.398135\pi$$
$$350$$ 0 0
$$351$$ 37.1771 1.98437
$$352$$ 0 0
$$353$$ −2.19224 −0.116681 −0.0583405 0.998297i $$-0.518581\pi$$
−0.0583405 + 0.998297i $$0.518581\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 11.8078 0.624933
$$358$$ 0 0
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 0 0
$$363$$ −13.3693 −0.701707
$$364$$ 0 0
$$365$$ 4.24621 0.222257
$$366$$ 0 0
$$367$$ 14.9309 0.779385 0.389693 0.920945i $$-0.372581\pi$$
0.389693 + 0.920945i $$0.372581\pi$$
$$368$$ 0 0
$$369$$ 0.630683 0.0328321
$$370$$ 0 0
$$371$$ 13.1231 0.681318
$$372$$ 0 0
$$373$$ −15.3693 −0.795793 −0.397897 0.917430i $$-0.630260\pi$$
−0.397897 + 0.917430i $$0.630260\pi$$
$$374$$ 0 0
$$375$$ −1.56155 −0.0806382
$$376$$ 0 0
$$377$$ 2.93087 0.150947
$$378$$ 0 0
$$379$$ −32.4924 −1.66902 −0.834512 0.550990i $$-0.814250\pi$$
−0.834512 + 0.550990i $$0.814250\pi$$
$$380$$ 0 0
$$381$$ −9.75379 −0.499702
$$382$$ 0 0
$$383$$ −9.75379 −0.498395 −0.249198 0.968453i $$-0.580167\pi$$
−0.249198 + 0.968453i $$0.580167\pi$$
$$384$$ 0 0
$$385$$ 1.56155 0.0795841
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ −22.3002 −1.13066 −0.565332 0.824863i $$-0.691252\pi$$
−0.565332 + 0.824863i $$0.691252\pi$$
$$390$$ 0 0
$$391$$ 23.6155 1.19429
$$392$$ 0 0
$$393$$ −23.6155 −1.19125
$$394$$ 0 0
$$395$$ −0.684658 −0.0344489
$$396$$ 0 0
$$397$$ 23.1771 1.16322 0.581612 0.813466i $$-0.302422\pi$$
0.581612 + 0.813466i $$0.302422\pi$$
$$398$$ 0 0
$$399$$ 11.1231 0.556852
$$400$$ 0 0
$$401$$ −12.9309 −0.645737 −0.322868 0.946444i $$-0.604647\pi$$
−0.322868 + 0.946444i $$0.604647\pi$$
$$402$$ 0 0
$$403$$ −41.7538 −2.07990
$$404$$ 0 0
$$405$$ 7.00000 0.347833
$$406$$ 0 0
$$407$$ −12.8769 −0.638284
$$408$$ 0 0
$$409$$ −18.4924 −0.914391 −0.457196 0.889366i $$-0.651146\pi$$
−0.457196 + 0.889366i $$0.651146\pi$$
$$410$$ 0 0
$$411$$ −11.5076 −0.567627
$$412$$ 0 0
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 33.3693 1.63410
$$418$$ 0 0
$$419$$ 18.2462 0.891386 0.445693 0.895186i $$-0.352957\pi$$
0.445693 + 0.895186i $$0.352957\pi$$
$$420$$ 0 0
$$421$$ −3.56155 −0.173579 −0.0867897 0.996227i $$-0.527661\pi$$
−0.0867897 + 0.996227i $$0.527661\pi$$
$$422$$ 0 0
$$423$$ −1.36932 −0.0665785
$$424$$ 0 0
$$425$$ 7.56155 0.366789
$$426$$ 0 0
$$427$$ 6.87689 0.332796
$$428$$ 0 0
$$429$$ 16.3002 0.786980
$$430$$ 0 0
$$431$$ −22.9309 −1.10454 −0.552271 0.833665i $$-0.686238\pi$$
−0.552271 + 0.833665i $$0.686238\pi$$
$$432$$ 0 0
$$433$$ 19.7538 0.949307 0.474653 0.880173i $$-0.342573\pi$$
0.474653 + 0.880173i $$0.342573\pi$$
$$434$$ 0 0
$$435$$ 0.684658 0.0328269
$$436$$ 0 0
$$437$$ 22.2462 1.06418
$$438$$ 0 0
$$439$$ 19.1231 0.912696 0.456348 0.889801i $$-0.349157\pi$$
0.456348 + 0.889801i $$0.349157\pi$$
$$440$$ 0 0
$$441$$ −0.561553 −0.0267406
$$442$$ 0 0
$$443$$ 19.6155 0.931962 0.465981 0.884795i $$-0.345702\pi$$
0.465981 + 0.884795i $$0.345702\pi$$
$$444$$ 0 0
$$445$$ −5.12311 −0.242858
$$446$$ 0 0
$$447$$ 0.384472 0.0181849
$$448$$ 0 0
$$449$$ −21.3153 −1.00593 −0.502967 0.864306i $$-0.667758\pi$$
−0.502967 + 0.864306i $$0.667758\pi$$
$$450$$ 0 0
$$451$$ 1.75379 0.0825827
$$452$$ 0 0
$$453$$ −30.9309 −1.45326
$$454$$ 0 0
$$455$$ 6.68466 0.313382
$$456$$ 0 0
$$457$$ 8.63068 0.403726 0.201863 0.979414i $$-0.435300\pi$$
0.201863 + 0.979414i $$0.435300\pi$$
$$458$$ 0 0
$$459$$ −42.0540 −1.96291
$$460$$ 0 0
$$461$$ −18.8769 −0.879185 −0.439592 0.898197i $$-0.644877\pi$$
−0.439592 + 0.898197i $$0.644877\pi$$
$$462$$ 0 0
$$463$$ −6.24621 −0.290286 −0.145143 0.989411i $$-0.546364\pi$$
−0.145143 + 0.989411i $$0.546364\pi$$
$$464$$ 0 0
$$465$$ −9.75379 −0.452321
$$466$$ 0 0
$$467$$ −25.5616 −1.18285 −0.591424 0.806361i $$-0.701434\pi$$
−0.591424 + 0.806361i $$0.701434\pi$$
$$468$$ 0 0
$$469$$ −2.24621 −0.103720
$$470$$ 0 0
$$471$$ −6.63068 −0.305526
$$472$$ 0 0
$$473$$ −11.1231 −0.511441
$$474$$ 0 0
$$475$$ 7.12311 0.326831
$$476$$ 0 0
$$477$$ −7.36932 −0.337418
$$478$$ 0 0
$$479$$ 17.3693 0.793624 0.396812 0.917900i $$-0.370116\pi$$
0.396812 + 0.917900i $$0.370116\pi$$
$$480$$ 0 0
$$481$$ −55.1231 −2.51340
$$482$$ 0 0
$$483$$ 4.87689 0.221906
$$484$$ 0 0
$$485$$ −1.31534 −0.0597266
$$486$$ 0 0
$$487$$ 3.12311 0.141521 0.0707607 0.997493i $$-0.477457\pi$$
0.0707607 + 0.997493i $$0.477457\pi$$
$$488$$ 0 0
$$489$$ −30.6307 −1.38517
$$490$$ 0 0
$$491$$ 3.31534 0.149619 0.0748096 0.997198i $$-0.476165\pi$$
0.0748096 + 0.997198i $$0.476165\pi$$
$$492$$ 0 0
$$493$$ −3.31534 −0.149315
$$494$$ 0 0
$$495$$ −0.876894 −0.0394135
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.192236 0.00860566 0.00430283 0.999991i $$-0.498630\pi$$
0.00430283 + 0.999991i $$0.498630\pi$$
$$500$$ 0 0
$$501$$ 6.54640 0.292471
$$502$$ 0 0
$$503$$ −29.1771 −1.30094 −0.650471 0.759531i $$-0.725428\pi$$
−0.650471 + 0.759531i $$0.725428\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 49.4773 2.19736
$$508$$ 0 0
$$509$$ −32.7386 −1.45111 −0.725557 0.688162i $$-0.758418\pi$$
−0.725557 + 0.688162i $$0.758418\pi$$
$$510$$ 0 0
$$511$$ −4.24621 −0.187841
$$512$$ 0 0
$$513$$ −39.6155 −1.74907
$$514$$ 0 0
$$515$$ 11.8078 0.520312
$$516$$ 0 0
$$517$$ −3.80776 −0.167465
$$518$$ 0 0
$$519$$ 36.1922 1.58866
$$520$$ 0 0
$$521$$ −12.2462 −0.536516 −0.268258 0.963347i $$-0.586448\pi$$
−0.268258 + 0.963347i $$0.586448\pi$$
$$522$$ 0 0
$$523$$ 12.0000 0.524723 0.262362 0.964970i $$-0.415499\pi$$
0.262362 + 0.964970i $$0.415499\pi$$
$$524$$ 0 0
$$525$$ 1.56155 0.0681518
$$526$$ 0 0
$$527$$ 47.2311 2.05742
$$528$$ 0 0
$$529$$ −13.2462 −0.575922
$$530$$ 0 0
$$531$$ −2.24621 −0.0974773
$$532$$ 0 0
$$533$$ 7.50758 0.325189
$$534$$ 0 0
$$535$$ 15.1231 0.653829
$$536$$ 0 0
$$537$$ 6.24621 0.269544
$$538$$ 0 0
$$539$$ −1.56155 −0.0672608
$$540$$ 0 0
$$541$$ −41.4233 −1.78093 −0.890463 0.455055i $$-0.849620\pi$$
−0.890463 + 0.455055i $$0.849620\pi$$
$$542$$ 0 0
$$543$$ 8.00000 0.343313
$$544$$ 0 0
$$545$$ −4.43845 −0.190122
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 0 0
$$549$$ −3.86174 −0.164815
$$550$$ 0 0
$$551$$ −3.12311 −0.133049
$$552$$ 0 0
$$553$$ 0.684658 0.0291146
$$554$$ 0 0
$$555$$ −12.8769 −0.546594
$$556$$ 0 0
$$557$$ 17.6155 0.746394 0.373197 0.927752i $$-0.378262\pi$$
0.373197 + 0.927752i $$0.378262\pi$$
$$558$$ 0 0
$$559$$ −47.6155 −2.01392
$$560$$ 0 0
$$561$$ −18.4384 −0.778472
$$562$$ 0 0
$$563$$ 7.50758 0.316407 0.158203 0.987407i $$-0.449430\pi$$
0.158203 + 0.987407i $$0.449430\pi$$
$$564$$ 0 0
$$565$$ −8.24621 −0.346921
$$566$$ 0 0
$$567$$ −7.00000 −0.293972
$$568$$ 0 0
$$569$$ 18.9848 0.795886 0.397943 0.917410i $$-0.369724\pi$$
0.397943 + 0.917410i $$0.369724\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ −1.06913 −0.0446636
$$574$$ 0 0
$$575$$ 3.12311 0.130243
$$576$$ 0 0
$$577$$ −3.56155 −0.148269 −0.0741347 0.997248i $$-0.523619\pi$$
−0.0741347 + 0.997248i $$0.523619\pi$$
$$578$$ 0 0
$$579$$ 20.4924 0.851636
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −20.4924 −0.848709
$$584$$ 0 0
$$585$$ −3.75379 −0.155200
$$586$$ 0 0
$$587$$ −10.2462 −0.422906 −0.211453 0.977388i $$-0.567820\pi$$
−0.211453 + 0.977388i $$0.567820\pi$$
$$588$$ 0 0
$$589$$ 44.4924 1.83328
$$590$$ 0 0
$$591$$ 20.4924 0.842946
$$592$$ 0 0
$$593$$ 37.4233 1.53679 0.768395 0.639976i $$-0.221056\pi$$
0.768395 + 0.639976i $$0.221056\pi$$
$$594$$ 0 0
$$595$$ −7.56155 −0.309993
$$596$$ 0 0
$$597$$ −22.2462 −0.910477
$$598$$ 0 0
$$599$$ −46.9309 −1.91754 −0.958772 0.284178i $$-0.908279\pi$$
−0.958772 + 0.284178i $$0.908279\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 1.26137 0.0513668
$$604$$ 0 0
$$605$$ 8.56155 0.348077
$$606$$ 0 0
$$607$$ −8.68466 −0.352499 −0.176250 0.984345i $$-0.556397\pi$$
−0.176250 + 0.984345i $$0.556397\pi$$
$$608$$ 0 0
$$609$$ −0.684658 −0.0277438
$$610$$ 0 0
$$611$$ −16.3002 −0.659435
$$612$$ 0 0
$$613$$ −16.7386 −0.676067 −0.338034 0.941134i $$-0.609762\pi$$
−0.338034 + 0.941134i $$0.609762\pi$$
$$614$$ 0 0
$$615$$ 1.75379 0.0707196
$$616$$ 0 0
$$617$$ −15.7538 −0.634224 −0.317112 0.948388i $$-0.602713\pi$$
−0.317112 + 0.948388i $$0.602713\pi$$
$$618$$ 0 0
$$619$$ −10.6307 −0.427283 −0.213642 0.976912i $$-0.568532\pi$$
−0.213642 + 0.976912i $$0.568532\pi$$
$$620$$ 0 0
$$621$$ −17.3693 −0.697007
$$622$$ 0 0
$$623$$ 5.12311 0.205253
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −17.3693 −0.693664
$$628$$ 0 0
$$629$$ 62.3542 2.48622
$$630$$ 0 0
$$631$$ −27.4233 −1.09170 −0.545852 0.837882i $$-0.683794\pi$$
−0.545852 + 0.837882i $$0.683794\pi$$
$$632$$ 0 0
$$633$$ 27.4233 1.08998
$$634$$ 0 0
$$635$$ 6.24621 0.247873
$$636$$ 0 0
$$637$$ −6.68466 −0.264856
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −22.9848 −0.907847 −0.453923 0.891041i $$-0.649976\pi$$
−0.453923 + 0.891041i $$0.649976\pi$$
$$642$$ 0 0
$$643$$ 30.0540 1.18521 0.592607 0.805492i $$-0.298099\pi$$
0.592607 + 0.805492i $$0.298099\pi$$
$$644$$ 0 0
$$645$$ −11.1231 −0.437972
$$646$$ 0 0
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ 0 0
$$649$$ −6.24621 −0.245185
$$650$$ 0 0
$$651$$ 9.75379 0.382281
$$652$$ 0 0
$$653$$ 38.4924 1.50632 0.753162 0.657835i $$-0.228527\pi$$
0.753162 + 0.657835i $$0.228527\pi$$
$$654$$ 0 0
$$655$$ 15.1231 0.590909
$$656$$ 0 0
$$657$$ 2.38447 0.0930271
$$658$$ 0 0
$$659$$ 0.192236 0.00748845 0.00374422 0.999993i $$-0.498808\pi$$
0.00374422 + 0.999993i $$0.498808\pi$$
$$660$$ 0 0
$$661$$ 17.6155 0.685165 0.342582 0.939488i $$-0.388698\pi$$
0.342582 + 0.939488i $$0.388698\pi$$
$$662$$ 0 0
$$663$$ −78.9309 −3.06542
$$664$$ 0 0
$$665$$ −7.12311 −0.276222
$$666$$ 0 0
$$667$$ −1.36932 −0.0530202
$$668$$ 0 0
$$669$$ 38.5464 1.49029
$$670$$ 0 0
$$671$$ −10.7386 −0.414560
$$672$$ 0 0
$$673$$ 41.6155 1.60416 0.802080 0.597216i $$-0.203727\pi$$
0.802080 + 0.597216i $$0.203727\pi$$
$$674$$ 0 0
$$675$$ −5.56155 −0.214064
$$676$$ 0 0
$$677$$ 1.31534 0.0505527 0.0252763 0.999681i $$-0.491953\pi$$
0.0252763 + 0.999681i $$0.491953\pi$$
$$678$$ 0 0
$$679$$ 1.31534 0.0504782
$$680$$ 0 0
$$681$$ 17.6695 0.677097
$$682$$ 0 0
$$683$$ −44.9848 −1.72130 −0.860649 0.509199i $$-0.829942\pi$$
−0.860649 + 0.509199i $$0.829942\pi$$
$$684$$ 0 0
$$685$$ 7.36932 0.281567
$$686$$ 0 0
$$687$$ 17.7538 0.677349
$$688$$ 0 0
$$689$$ −87.7235 −3.34200
$$690$$ 0 0
$$691$$ −16.4924 −0.627401 −0.313701 0.949522i $$-0.601569\pi$$
−0.313701 + 0.949522i $$0.601569\pi$$
$$692$$ 0 0
$$693$$ 0.876894 0.0333105
$$694$$ 0 0
$$695$$ −21.3693 −0.810584
$$696$$ 0 0
$$697$$ −8.49242 −0.321673
$$698$$ 0 0
$$699$$ −16.9848 −0.642426
$$700$$ 0 0
$$701$$ 9.31534 0.351836 0.175918 0.984405i $$-0.443711\pi$$
0.175918 + 0.984405i $$0.443711\pi$$
$$702$$ 0 0
$$703$$ 58.7386 2.21537
$$704$$ 0 0
$$705$$ −3.80776 −0.143409
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 4.05398 0.152250 0.0761251 0.997098i $$-0.475745\pi$$
0.0761251 + 0.997098i $$0.475745\pi$$
$$710$$ 0 0
$$711$$ −0.384472 −0.0144188
$$712$$ 0 0
$$713$$ 19.5076 0.730565
$$714$$ 0 0
$$715$$ −10.4384 −0.390376
$$716$$ 0 0
$$717$$ −28.1922 −1.05286
$$718$$ 0 0
$$719$$ −23.6155 −0.880711 −0.440355 0.897824i $$-0.645148\pi$$
−0.440355 + 0.897824i $$0.645148\pi$$
$$720$$ 0 0
$$721$$ −11.8078 −0.439744
$$722$$ 0 0
$$723$$ 3.12311 0.116150
$$724$$ 0 0
$$725$$ −0.438447 −0.0162835
$$726$$ 0 0
$$727$$ 48.9848 1.81675 0.908374 0.418159i $$-0.137325\pi$$
0.908374 + 0.418159i $$0.137325\pi$$
$$728$$ 0 0
$$729$$ 29.9848 1.11055
$$730$$ 0 0
$$731$$ 53.8617 1.99215
$$732$$ 0 0
$$733$$ −16.4384 −0.607168 −0.303584 0.952805i $$-0.598183\pi$$
−0.303584 + 0.952805i $$0.598183\pi$$
$$734$$ 0 0
$$735$$ −1.56155 −0.0575987
$$736$$ 0 0
$$737$$ 3.50758 0.129203
$$738$$ 0 0
$$739$$ 6.43845 0.236842 0.118421 0.992963i $$-0.462217\pi$$
0.118421 + 0.992963i $$0.462217\pi$$
$$740$$ 0 0
$$741$$ −74.3542 −2.73147
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ −0.246211 −0.00902048
$$746$$ 0 0
$$747$$ 6.73863 0.246554
$$748$$ 0 0
$$749$$ −15.1231 −0.552586
$$750$$ 0 0
$$751$$ −31.3153 −1.14271 −0.571357 0.820702i $$-0.693583\pi$$
−0.571357 + 0.820702i $$0.693583\pi$$
$$752$$ 0 0
$$753$$ −20.8769 −0.760796
$$754$$ 0 0
$$755$$ 19.8078 0.720878
$$756$$ 0 0
$$757$$ −15.3693 −0.558607 −0.279304 0.960203i $$-0.590104\pi$$
−0.279304 + 0.960203i $$0.590104\pi$$
$$758$$ 0 0
$$759$$ −7.61553 −0.276426
$$760$$ 0 0
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 0 0
$$763$$ 4.43845 0.160683
$$764$$ 0 0
$$765$$ 4.24621 0.153522
$$766$$ 0 0
$$767$$ −26.7386 −0.965476
$$768$$ 0 0
$$769$$ 42.9848 1.55007 0.775037 0.631916i $$-0.217731\pi$$
0.775037 + 0.631916i $$0.217731\pi$$
$$770$$ 0 0
$$771$$ −28.8769 −1.03998
$$772$$ 0 0
$$773$$ 29.8078 1.07211 0.536055 0.844183i $$-0.319914\pi$$
0.536055 + 0.844183i $$0.319914\pi$$
$$774$$ 0 0
$$775$$ 6.24621 0.224371
$$776$$ 0 0
$$777$$ 12.8769 0.461956
$$778$$ 0 0
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 2.43845 0.0871430
$$784$$ 0 0
$$785$$ 4.24621 0.151554
$$786$$ 0 0
$$787$$ −33.1771 −1.18264 −0.591318 0.806439i $$-0.701392\pi$$
−0.591318 + 0.806439i $$0.701392\pi$$
$$788$$ 0 0
$$789$$ 14.6307 0.520866
$$790$$ 0 0
$$791$$ 8.24621 0.293202
$$792$$ 0 0
$$793$$ −45.9697 −1.63243
$$794$$ 0 0
$$795$$ −20.4924 −0.726791
$$796$$ 0 0
$$797$$ −17.8078 −0.630783 −0.315392 0.948962i $$-0.602136\pi$$
−0.315392 + 0.948962i $$0.602136\pi$$
$$798$$ 0 0
$$799$$ 18.4384 0.652305
$$800$$ 0 0
$$801$$ −2.87689 −0.101650
$$802$$ 0 0
$$803$$ 6.63068 0.233992
$$804$$ 0 0
$$805$$ −3.12311 −0.110075
$$806$$ 0 0
$$807$$ −6.63068 −0.233411
$$808$$ 0 0
$$809$$ 4.05398 0.142530 0.0712651 0.997457i $$-0.477296\pi$$
0.0712651 + 0.997457i $$0.477296\pi$$
$$810$$ 0 0
$$811$$ 27.6155 0.969712 0.484856 0.874594i $$-0.338872\pi$$
0.484856 + 0.874594i $$0.338872\pi$$
$$812$$ 0 0
$$813$$ −9.75379 −0.342080
$$814$$ 0 0
$$815$$ 19.6155 0.687102
$$816$$ 0 0
$$817$$ 50.7386 1.77512
$$818$$ 0 0
$$819$$ 3.75379 0.131168
$$820$$ 0 0
$$821$$ −54.3002 −1.89509 −0.947545 0.319623i $$-0.896444\pi$$
−0.947545 + 0.319623i $$0.896444\pi$$
$$822$$ 0 0
$$823$$ −17.7538 −0.618858 −0.309429 0.950923i $$-0.600138\pi$$
−0.309429 + 0.950923i $$0.600138\pi$$
$$824$$ 0 0
$$825$$ −2.43845 −0.0848958
$$826$$ 0 0
$$827$$ −24.8769 −0.865054 −0.432527 0.901621i $$-0.642378\pi$$
−0.432527 + 0.901621i $$0.642378\pi$$
$$828$$ 0 0
$$829$$ −5.61553 −0.195035 −0.0975177 0.995234i $$-0.531090\pi$$
−0.0975177 + 0.995234i $$0.531090\pi$$
$$830$$ 0 0
$$831$$ 12.8769 0.446695
$$832$$ 0 0
$$833$$ 7.56155 0.261992
$$834$$ 0 0
$$835$$ −4.19224 −0.145078
$$836$$ 0 0
$$837$$ −34.7386 −1.20074
$$838$$ 0 0
$$839$$ 19.1231 0.660203 0.330101 0.943945i $$-0.392917\pi$$
0.330101 + 0.943945i $$0.392917\pi$$
$$840$$ 0 0
$$841$$ −28.8078 −0.993371
$$842$$ 0 0
$$843$$ −30.5464 −1.05207
$$844$$ 0 0
$$845$$ −31.6847 −1.08999
$$846$$ 0 0
$$847$$ −8.56155 −0.294178
$$848$$ 0 0
$$849$$ 7.31534 0.251062
$$850$$ 0 0
$$851$$ 25.7538 0.882829
$$852$$ 0 0
$$853$$ −15.7538 −0.539399 −0.269700 0.962944i $$-0.586924\pi$$
−0.269700 + 0.962944i $$0.586924\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ 52.7386 1.80152 0.900759 0.434320i $$-0.143011\pi$$
0.900759 + 0.434320i $$0.143011\pi$$
$$858$$ 0 0
$$859$$ −36.9848 −1.26191 −0.630953 0.775821i $$-0.717336\pi$$
−0.630953 + 0.775821i $$0.717336\pi$$
$$860$$ 0 0
$$861$$ −1.75379 −0.0597690
$$862$$ 0 0
$$863$$ 28.4924 0.969893 0.484947 0.874544i $$-0.338839\pi$$
0.484947 + 0.874544i $$0.338839\pi$$
$$864$$ 0 0
$$865$$ −23.1771 −0.788044
$$866$$ 0 0
$$867$$ 62.7386 2.13072
$$868$$ 0 0
$$869$$ −1.06913 −0.0362678
$$870$$ 0 0
$$871$$ 15.0152 0.508769
$$872$$ 0 0
$$873$$ −0.738634 −0.0249990
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 13.5076 0.456118 0.228059 0.973647i $$-0.426762\pi$$
0.228059 + 0.973647i $$0.426762\pi$$
$$878$$ 0 0
$$879$$ −50.0540 −1.68828
$$880$$ 0 0
$$881$$ 54.1080 1.82294 0.911472 0.411363i $$-0.134947\pi$$
0.911472 + 0.411363i $$0.134947\pi$$
$$882$$ 0 0
$$883$$ −21.7538 −0.732073 −0.366037 0.930600i $$-0.619286\pi$$
−0.366037 + 0.930600i $$0.619286\pi$$
$$884$$ 0 0
$$885$$ −6.24621 −0.209964
$$886$$ 0 0
$$887$$ −36.4924 −1.22530 −0.612648 0.790356i $$-0.709896\pi$$
−0.612648 + 0.790356i $$0.709896\pi$$
$$888$$ 0 0
$$889$$ −6.24621 −0.209491
$$890$$ 0 0
$$891$$ 10.9309 0.366198
$$892$$ 0 0
$$893$$ 17.3693 0.581242
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ −32.6004 −1.08849
$$898$$ 0 0
$$899$$ −2.73863 −0.0913385
$$900$$ 0 0
$$901$$ 99.2311 3.30587
$$902$$ 0 0
$$903$$ 11.1231 0.370154
$$904$$ 0 0
$$905$$ −5.12311 −0.170298
$$906$$ 0 0
$$907$$ 48.1080 1.59740 0.798699 0.601731i $$-0.205522\pi$$
0.798699 + 0.601731i $$0.205522\pi$$
$$908$$ 0 0
$$909$$ 3.36932 0.111753
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ 18.7386 0.620158
$$914$$ 0 0
$$915$$ −10.7386 −0.355008
$$916$$ 0 0
$$917$$ −15.1231 −0.499409
$$918$$ 0 0
$$919$$ 2.43845 0.0804370 0.0402185 0.999191i $$-0.487195\pi$$
0.0402185 + 0.999191i $$0.487195\pi$$
$$920$$ 0 0
$$921$$ 44.7926 1.47597
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 8.24621 0.271134
$$926$$ 0 0
$$927$$ 6.63068 0.217780
$$928$$ 0 0
$$929$$ −2.87689 −0.0943878 −0.0471939 0.998886i $$-0.515028\pi$$
−0.0471939 + 0.998886i $$0.515028\pi$$
$$930$$ 0 0
$$931$$ 7.12311 0.233450
$$932$$ 0 0
$$933$$ −20.1080 −0.658305
$$934$$ 0 0
$$935$$ 11.8078 0.386155
$$936$$ 0 0
$$937$$ 37.8078 1.23513 0.617563 0.786521i $$-0.288120\pi$$
0.617563 + 0.786521i $$0.288120\pi$$
$$938$$ 0 0
$$939$$ 41.6695 1.35983
$$940$$ 0 0
$$941$$ −28.6307 −0.933334 −0.466667 0.884433i $$-0.654545\pi$$
−0.466667 + 0.884433i $$0.654545\pi$$
$$942$$ 0 0
$$943$$ −3.50758 −0.114222
$$944$$ 0 0
$$945$$ 5.56155 0.180917
$$946$$ 0 0
$$947$$ 35.2311 1.14486 0.572428 0.819955i $$-0.306002\pi$$
0.572428 + 0.819955i $$0.306002\pi$$
$$948$$ 0 0
$$949$$ 28.3845 0.921399
$$950$$ 0 0
$$951$$ 15.6155 0.506368
$$952$$ 0 0
$$953$$ −51.8617 −1.67997 −0.839983 0.542612i $$-0.817435\pi$$
−0.839983 + 0.542612i $$0.817435\pi$$
$$954$$ 0 0
$$955$$ 0.684658 0.0221550
$$956$$ 0 0
$$957$$ 1.06913 0.0345601
$$958$$ 0 0
$$959$$ −7.36932 −0.237968
$$960$$ 0 0
$$961$$ 8.01515 0.258553
$$962$$ 0 0
$$963$$ 8.49242 0.273664
$$964$$ 0 0
$$965$$ −13.1231 −0.422448
$$966$$ 0 0
$$967$$ 44.1080 1.41842 0.709208 0.704999i $$-0.249053\pi$$
0.709208 + 0.704999i $$0.249053\pi$$
$$968$$ 0 0
$$969$$ 84.1080 2.70194
$$970$$ 0 0
$$971$$ 22.7386 0.729717 0.364859 0.931063i $$-0.381117\pi$$
0.364859 + 0.931063i $$0.381117\pi$$
$$972$$ 0 0
$$973$$ 21.3693 0.685069
$$974$$ 0 0
$$975$$ −10.4384 −0.334298
$$976$$ 0 0
$$977$$ 10.9848 0.351436 0.175718 0.984441i $$-0.443775\pi$$
0.175718 + 0.984441i $$0.443775\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ −2.49242 −0.0795769
$$982$$ 0 0
$$983$$ −29.1771 −0.930604 −0.465302 0.885152i $$-0.654054\pi$$
−0.465302 + 0.885152i $$0.654054\pi$$
$$984$$ 0 0
$$985$$ −13.1231 −0.418137
$$986$$ 0 0
$$987$$ 3.80776 0.121202
$$988$$ 0 0
$$989$$ 22.2462 0.707388
$$990$$ 0 0
$$991$$ −20.4924 −0.650963 −0.325482 0.945548i $$-0.605526\pi$$
−0.325482 + 0.945548i $$0.605526\pi$$
$$992$$ 0 0
$$993$$ 18.7386 0.594653
$$994$$ 0 0
$$995$$ 14.2462 0.451635
$$996$$ 0 0
$$997$$ 56.9309 1.80302 0.901509 0.432760i $$-0.142460\pi$$
0.901509 + 0.432760i $$0.142460\pi$$
$$998$$ 0 0
$$999$$ −45.8617 −1.45100
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 2240.2.a.be.1.2 2
4.3 odd 2 2240.2.a.bi.1.1 2
8.3 odd 2 560.2.a.g.1.2 2
8.5 even 2 280.2.a.d.1.1 2
24.5 odd 2 2520.2.a.w.1.1 2
24.11 even 2 5040.2.a.bq.1.2 2
40.3 even 4 2800.2.g.u.449.3 4
40.13 odd 4 1400.2.g.k.449.2 4
40.19 odd 2 2800.2.a.bn.1.1 2
40.27 even 4 2800.2.g.u.449.2 4
40.29 even 2 1400.2.a.p.1.2 2
40.37 odd 4 1400.2.g.k.449.3 4
56.5 odd 6 1960.2.q.u.361.1 4
56.13 odd 2 1960.2.a.r.1.2 2
56.27 even 2 3920.2.a.bu.1.1 2
56.37 even 6 1960.2.q.s.361.2 4
56.45 odd 6 1960.2.q.u.961.1 4
56.53 even 6 1960.2.q.s.961.2 4
280.69 odd 2 9800.2.a.by.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.d.1.1 2 8.5 even 2
560.2.a.g.1.2 2 8.3 odd 2
1400.2.a.p.1.2 2 40.29 even 2
1400.2.g.k.449.2 4 40.13 odd 4
1400.2.g.k.449.3 4 40.37 odd 4
1960.2.a.r.1.2 2 56.13 odd 2
1960.2.q.s.361.2 4 56.37 even 6
1960.2.q.s.961.2 4 56.53 even 6
1960.2.q.u.361.1 4 56.5 odd 6
1960.2.q.u.961.1 4 56.45 odd 6
2240.2.a.be.1.2 2 1.1 even 1 trivial
2240.2.a.bi.1.1 2 4.3 odd 2
2520.2.a.w.1.1 2 24.5 odd 2
2800.2.a.bn.1.1 2 40.19 odd 2
2800.2.g.u.449.2 4 40.27 even 4
2800.2.g.u.449.3 4 40.3 even 4
3920.2.a.bu.1.1 2 56.27 even 2
5040.2.a.bq.1.2 2 24.11 even 2
9800.2.a.by.1.1 2 280.69 odd 2