# Properties

 Label 1400.2.a.q.1.2 Level $1400$ Weight $2$ Character 1400.1 Self dual yes Analytic conductor $11.179$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q+2.56155 q^{3} +1.00000 q^{7} +3.56155 q^{9} +2.12311 q^{11} +2.00000 q^{13} -2.56155 q^{17} -0.561553 q^{19} +2.56155 q^{21} +5.56155 q^{23} +1.43845 q^{27} +7.56155 q^{29} -0.876894 q^{31} +5.43845 q^{33} -11.8078 q^{37} +5.12311 q^{39} -6.56155 q^{41} +2.43845 q^{43} +8.24621 q^{47} +1.00000 q^{49} -6.56155 q^{51} +7.12311 q^{53} -1.43845 q^{57} -13.3693 q^{59} +2.87689 q^{61} +3.56155 q^{63} +16.1231 q^{67} +14.2462 q^{69} -10.6847 q^{71} +10.5616 q^{73} +2.12311 q^{77} +6.68466 q^{79} -7.00000 q^{81} -13.6847 q^{83} +19.3693 q^{87} -10.8078 q^{89} +2.00000 q^{91} -2.24621 q^{93} +6.00000 q^{97} +7.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^7 + 3 * q^9 $$2 q + q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{17} + 3 q^{19} + q^{21} + 7 q^{23} + 7 q^{27} + 11 q^{29} - 10 q^{31} + 15 q^{33} - 3 q^{37} + 2 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{49} - 9 q^{51} + 6 q^{53} - 7 q^{57} - 2 q^{59} + 14 q^{61} + 3 q^{63} + 24 q^{67} + 12 q^{69} - 9 q^{71} + 17 q^{73} - 4 q^{77} + q^{79} - 14 q^{81} - 15 q^{83} + 14 q^{87} - q^{89} + 4 q^{91} + 12 q^{93} + 12 q^{97} + 11 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^7 + 3 * q^9 - 4 * q^11 + 4 * q^13 - q^17 + 3 * q^19 + q^21 + 7 * q^23 + 7 * q^27 + 11 * q^29 - 10 * q^31 + 15 * q^33 - 3 * q^37 + 2 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^49 - 9 * q^51 + 6 * q^53 - 7 * q^57 - 2 * q^59 + 14 * q^61 + 3 * q^63 + 24 * q^67 + 12 * q^69 - 9 * q^71 + 17 * q^73 - 4 * q^77 + q^79 - 14 * q^81 - 15 * q^83 + 14 * q^87 - q^89 + 4 * q^91 + 12 * q^93 + 12 * q^97 + 11 * 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
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ 2.12311 0.640140 0.320070 0.947394i $$-0.396293\pi$$
0.320070 + 0.947394i $$0.396293\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.56155 −0.621268 −0.310634 0.950530i $$-0.600541\pi$$
−0.310634 + 0.950530i $$0.600541\pi$$
$$18$$ 0 0
$$19$$ −0.561553 −0.128829 −0.0644145 0.997923i $$-0.520518\pi$$
−0.0644145 + 0.997923i $$0.520518\pi$$
$$20$$ 0 0
$$21$$ 2.56155 0.558977
$$22$$ 0 0
$$23$$ 5.56155 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.43845 0.276829
$$28$$ 0 0
$$29$$ 7.56155 1.40415 0.702073 0.712105i $$-0.252258\pi$$
0.702073 + 0.712105i $$0.252258\pi$$
$$30$$ 0 0
$$31$$ −0.876894 −0.157495 −0.0787474 0.996895i $$-0.525092\pi$$
−0.0787474 + 0.996895i $$0.525092\pi$$
$$32$$ 0 0
$$33$$ 5.43845 0.946712
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −11.8078 −1.94118 −0.970592 0.240730i $$-0.922613\pi$$
−0.970592 + 0.240730i $$0.922613\pi$$
$$38$$ 0 0
$$39$$ 5.12311 0.820353
$$40$$ 0 0
$$41$$ −6.56155 −1.02474 −0.512371 0.858764i $$-0.671233\pi$$
−0.512371 + 0.858764i $$0.671233\pi$$
$$42$$ 0 0
$$43$$ 2.43845 0.371860 0.185930 0.982563i $$-0.440470\pi$$
0.185930 + 0.982563i $$0.440470\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.24621 1.20283 0.601417 0.798935i $$-0.294603\pi$$
0.601417 + 0.798935i $$0.294603\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.56155 −0.918801
$$52$$ 0 0
$$53$$ 7.12311 0.978434 0.489217 0.872162i $$-0.337283\pi$$
0.489217 + 0.872162i $$0.337283\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.43845 −0.190527
$$58$$ 0 0
$$59$$ −13.3693 −1.74054 −0.870268 0.492578i $$-0.836055\pi$$
−0.870268 + 0.492578i $$0.836055\pi$$
$$60$$ 0 0
$$61$$ 2.87689 0.368349 0.184174 0.982894i $$-0.441039\pi$$
0.184174 + 0.982894i $$0.441039\pi$$
$$62$$ 0 0
$$63$$ 3.56155 0.448713
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 16.1231 1.96975 0.984875 0.173264i $$-0.0554314\pi$$
0.984875 + 0.173264i $$0.0554314\pi$$
$$68$$ 0 0
$$69$$ 14.2462 1.71504
$$70$$ 0 0
$$71$$ −10.6847 −1.26804 −0.634018 0.773318i $$-0.718595\pi$$
−0.634018 + 0.773318i $$0.718595\pi$$
$$72$$ 0 0
$$73$$ 10.5616 1.23614 0.618068 0.786125i $$-0.287916\pi$$
0.618068 + 0.786125i $$0.287916\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.12311 0.241950
$$78$$ 0 0
$$79$$ 6.68466 0.752083 0.376041 0.926603i $$-0.377285\pi$$
0.376041 + 0.926603i $$0.377285\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −13.6847 −1.50209 −0.751043 0.660253i $$-0.770449\pi$$
−0.751043 + 0.660253i $$0.770449\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 19.3693 2.07661
$$88$$ 0 0
$$89$$ −10.8078 −1.14562 −0.572810 0.819688i $$-0.694147\pi$$
−0.572810 + 0.819688i $$0.694147\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ −2.24621 −0.232921
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 7.56155 0.759965
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 7.12311 0.701860 0.350930 0.936402i $$-0.385865\pi$$
0.350930 + 0.936402i $$0.385865\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.8078 −1.23817 −0.619087 0.785323i $$-0.712497\pi$$
−0.619087 + 0.785323i $$0.712497\pi$$
$$108$$ 0 0
$$109$$ −8.93087 −0.855422 −0.427711 0.903915i $$-0.640680\pi$$
−0.427711 + 0.903915i $$0.640680\pi$$
$$110$$ 0 0
$$111$$ −30.2462 −2.87084
$$112$$ 0 0
$$113$$ 0.753789 0.0709105 0.0354552 0.999371i $$-0.488712\pi$$
0.0354552 + 0.999371i $$0.488712\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 7.12311 0.658531
$$118$$ 0 0
$$119$$ −2.56155 −0.234817
$$120$$ 0 0
$$121$$ −6.49242 −0.590220
$$122$$ 0 0
$$123$$ −16.8078 −1.51551
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −9.80776 −0.870298 −0.435149 0.900358i $$-0.643304\pi$$
−0.435149 + 0.900358i $$0.643304\pi$$
$$128$$ 0 0
$$129$$ 6.24621 0.549948
$$130$$ 0 0
$$131$$ −1.75379 −0.153229 −0.0766146 0.997061i $$-0.524411\pi$$
−0.0766146 + 0.997061i $$0.524411\pi$$
$$132$$ 0 0
$$133$$ −0.561553 −0.0486928
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.31534 −0.197813 −0.0989065 0.995097i $$-0.531534\pi$$
−0.0989065 + 0.995097i $$0.531534\pi$$
$$138$$ 0 0
$$139$$ −19.6847 −1.66963 −0.834815 0.550530i $$-0.814426\pi$$
−0.834815 + 0.550530i $$0.814426\pi$$
$$140$$ 0 0
$$141$$ 21.1231 1.77889
$$142$$ 0 0
$$143$$ 4.24621 0.355086
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.56155 0.211273
$$148$$ 0 0
$$149$$ 17.5616 1.43870 0.719349 0.694649i $$-0.244440\pi$$
0.719349 + 0.694649i $$0.244440\pi$$
$$150$$ 0 0
$$151$$ −5.56155 −0.452593 −0.226296 0.974058i $$-0.572662\pi$$
−0.226296 + 0.974058i $$0.572662\pi$$
$$152$$ 0 0
$$153$$ −9.12311 −0.737559
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 24.4924 1.95471 0.977354 0.211611i $$-0.0678709\pi$$
0.977354 + 0.211611i $$0.0678709\pi$$
$$158$$ 0 0
$$159$$ 18.2462 1.44702
$$160$$ 0 0
$$161$$ 5.56155 0.438312
$$162$$ 0 0
$$163$$ −13.9309 −1.09115 −0.545575 0.838062i $$-0.683689\pi$$
−0.545575 + 0.838062i $$0.683689\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.6155 −1.51790 −0.758948 0.651152i $$-0.774286\pi$$
−0.758948 + 0.651152i $$0.774286\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 14.4924 1.10184 0.550919 0.834559i $$-0.314277\pi$$
0.550919 + 0.834559i $$0.314277\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −34.2462 −2.57410
$$178$$ 0 0
$$179$$ −8.31534 −0.621518 −0.310759 0.950489i $$-0.600583\pi$$
−0.310759 + 0.950489i $$0.600583\pi$$
$$180$$ 0 0
$$181$$ −2.87689 −0.213838 −0.106919 0.994268i $$-0.534099\pi$$
−0.106919 + 0.994268i $$0.534099\pi$$
$$182$$ 0 0
$$183$$ 7.36932 0.544756
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.43845 −0.397699
$$188$$ 0 0
$$189$$ 1.43845 0.104632
$$190$$ 0 0
$$191$$ 0.630683 0.0456346 0.0228173 0.999740i $$-0.492736\pi$$
0.0228173 + 0.999740i $$0.492736\pi$$
$$192$$ 0 0
$$193$$ 20.6155 1.48394 0.741969 0.670434i $$-0.233892\pi$$
0.741969 + 0.670434i $$0.233892\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.56155 −0.253750 −0.126875 0.991919i $$-0.540495\pi$$
−0.126875 + 0.991919i $$0.540495\pi$$
$$198$$ 0 0
$$199$$ −13.1231 −0.930272 −0.465136 0.885239i $$-0.653995\pi$$
−0.465136 + 0.885239i $$0.653995\pi$$
$$200$$ 0 0
$$201$$ 41.3002 2.91309
$$202$$ 0 0
$$203$$ 7.56155 0.530717
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 19.8078 1.37673
$$208$$ 0 0
$$209$$ −1.19224 −0.0824687
$$210$$ 0 0
$$211$$ 10.5616 0.727087 0.363544 0.931577i $$-0.381567\pi$$
0.363544 + 0.931577i $$0.381567\pi$$
$$212$$ 0 0
$$213$$ −27.3693 −1.87531
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.876894 −0.0595275
$$218$$ 0 0
$$219$$ 27.0540 1.82814
$$220$$ 0 0
$$221$$ −5.12311 −0.344617
$$222$$ 0 0
$$223$$ −0.630683 −0.0422337 −0.0211168 0.999777i $$-0.506722\pi$$
−0.0211168 + 0.999777i $$0.506722\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −13.3693 −0.887353 −0.443676 0.896187i $$-0.646326\pi$$
−0.443676 + 0.896187i $$0.646326\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 5.43845 0.357824
$$232$$ 0 0
$$233$$ 3.56155 0.233325 0.116663 0.993172i $$-0.462780\pi$$
0.116663 + 0.993172i $$0.462780\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 17.1231 1.11227
$$238$$ 0 0
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 0 0
$$241$$ 25.0540 1.61387 0.806934 0.590641i $$-0.201125\pi$$
0.806934 + 0.590641i $$0.201125\pi$$
$$242$$ 0 0
$$243$$ −22.2462 −1.42710
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.12311 −0.0714615
$$248$$ 0 0
$$249$$ −35.0540 −2.22146
$$250$$ 0 0
$$251$$ −12.3153 −0.777337 −0.388669 0.921378i $$-0.627065\pi$$
−0.388669 + 0.921378i $$0.627065\pi$$
$$252$$ 0 0
$$253$$ 11.8078 0.742348
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.24621 −0.264871 −0.132436 0.991192i $$-0.542280\pi$$
−0.132436 + 0.991192i $$0.542280\pi$$
$$258$$ 0 0
$$259$$ −11.8078 −0.733699
$$260$$ 0 0
$$261$$ 26.9309 1.66698
$$262$$ 0 0
$$263$$ −10.6847 −0.658844 −0.329422 0.944183i $$-0.606854\pi$$
−0.329422 + 0.944183i $$0.606854\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −27.6847 −1.69427
$$268$$ 0 0
$$269$$ 11.7538 0.716641 0.358321 0.933599i $$-0.383349\pi$$
0.358321 + 0.933599i $$0.383349\pi$$
$$270$$ 0 0
$$271$$ 16.2462 0.986887 0.493444 0.869778i $$-0.335738\pi$$
0.493444 + 0.869778i $$0.335738\pi$$
$$272$$ 0 0
$$273$$ 5.12311 0.310064
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.4924 −0.630429 −0.315214 0.949021i $$-0.602076\pi$$
−0.315214 + 0.949021i $$0.602076\pi$$
$$278$$ 0 0
$$279$$ −3.12311 −0.186975
$$280$$ 0 0
$$281$$ 11.5616 0.689704 0.344852 0.938657i $$-0.387929\pi$$
0.344852 + 0.938657i $$0.387929\pi$$
$$282$$ 0 0
$$283$$ −13.6847 −0.813469 −0.406734 0.913547i $$-0.633333\pi$$
−0.406734 + 0.913547i $$0.633333\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.56155 −0.387316
$$288$$ 0 0
$$289$$ −10.4384 −0.614026
$$290$$ 0 0
$$291$$ 15.3693 0.900965
$$292$$ 0 0
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.05398 0.177210
$$298$$ 0 0
$$299$$ 11.1231 0.643266
$$300$$ 0 0
$$301$$ 2.43845 0.140550
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.31534 0.246290 0.123145 0.992389i $$-0.460702\pi$$
0.123145 + 0.992389i $$0.460702\pi$$
$$308$$ 0 0
$$309$$ 18.2462 1.03799
$$310$$ 0 0
$$311$$ 2.87689 0.163134 0.0815669 0.996668i $$-0.474008\pi$$
0.0815669 + 0.996668i $$0.474008\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −20.6847 −1.16177 −0.580883 0.813987i $$-0.697293\pi$$
−0.580883 + 0.813987i $$0.697293\pi$$
$$318$$ 0 0
$$319$$ 16.0540 0.898850
$$320$$ 0 0
$$321$$ −32.8078 −1.83115
$$322$$ 0 0
$$323$$ 1.43845 0.0800373
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −22.8769 −1.26510
$$328$$ 0 0
$$329$$ 8.24621 0.454628
$$330$$ 0 0
$$331$$ 27.0000 1.48405 0.742027 0.670370i $$-0.233865\pi$$
0.742027 + 0.670370i $$0.233865\pi$$
$$332$$ 0 0
$$333$$ −42.0540 −2.30454
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −5.68466 −0.309663 −0.154832 0.987941i $$-0.549483\pi$$
−0.154832 + 0.987941i $$0.549483\pi$$
$$338$$ 0 0
$$339$$ 1.93087 0.104870
$$340$$ 0 0
$$341$$ −1.86174 −0.100819
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 22.6155 1.21407 0.607033 0.794677i $$-0.292360\pi$$
0.607033 + 0.794677i $$0.292360\pi$$
$$348$$ 0 0
$$349$$ −12.8769 −0.689284 −0.344642 0.938734i $$-0.612000\pi$$
−0.344642 + 0.938734i $$0.612000\pi$$
$$350$$ 0 0
$$351$$ 2.87689 0.153557
$$352$$ 0 0
$$353$$ −11.7538 −0.625591 −0.312796 0.949820i $$-0.601265\pi$$
−0.312796 + 0.949820i $$0.601265\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.56155 −0.347274
$$358$$ 0 0
$$359$$ −14.6847 −0.775027 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$360$$ 0 0
$$361$$ −18.6847 −0.983403
$$362$$ 0 0
$$363$$ −16.6307 −0.872884
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 17.7538 0.926740 0.463370 0.886165i $$-0.346640\pi$$
0.463370 + 0.886165i $$0.346640\pi$$
$$368$$ 0 0
$$369$$ −23.3693 −1.21656
$$370$$ 0 0
$$371$$ 7.12311 0.369813
$$372$$ 0 0
$$373$$ 14.0540 0.727687 0.363844 0.931460i $$-0.381464\pi$$
0.363844 + 0.931460i $$0.381464\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 15.1231 0.778880
$$378$$ 0 0
$$379$$ −24.8617 −1.27706 −0.638531 0.769596i $$-0.720458\pi$$
−0.638531 + 0.769596i $$0.720458\pi$$
$$380$$ 0 0
$$381$$ −25.1231 −1.28710
$$382$$ 0 0
$$383$$ −20.2462 −1.03453 −0.517267 0.855824i $$-0.673050\pi$$
−0.517267 + 0.855824i $$0.673050\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.68466 0.441466
$$388$$ 0 0
$$389$$ 2.43845 0.123634 0.0618171 0.998087i $$-0.480310\pi$$
0.0618171 + 0.998087i $$0.480310\pi$$
$$390$$ 0 0
$$391$$ −14.2462 −0.720462
$$392$$ 0 0
$$393$$ −4.49242 −0.226613
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 38.4924 1.93188 0.965940 0.258767i $$-0.0833163\pi$$
0.965940 + 0.258767i $$0.0833163\pi$$
$$398$$ 0 0
$$399$$ −1.43845 −0.0720124
$$400$$ 0 0
$$401$$ −31.4924 −1.57266 −0.786328 0.617809i $$-0.788021\pi$$
−0.786328 + 0.617809i $$0.788021\pi$$
$$402$$ 0 0
$$403$$ −1.75379 −0.0873624
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −25.0691 −1.24263
$$408$$ 0 0
$$409$$ −5.19224 −0.256740 −0.128370 0.991726i $$-0.540974\pi$$
−0.128370 + 0.991726i $$0.540974\pi$$
$$410$$ 0 0
$$411$$ −5.93087 −0.292548
$$412$$ 0 0
$$413$$ −13.3693 −0.657861
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −50.4233 −2.46924
$$418$$ 0 0
$$419$$ −18.8078 −0.918819 −0.459410 0.888224i $$-0.651939\pi$$
−0.459410 + 0.888224i $$0.651939\pi$$
$$420$$ 0 0
$$421$$ 5.06913 0.247054 0.123527 0.992341i $$-0.460579\pi$$
0.123527 + 0.992341i $$0.460579\pi$$
$$422$$ 0 0
$$423$$ 29.3693 1.42799
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.87689 0.139223
$$428$$ 0 0
$$429$$ 10.8769 0.525141
$$430$$ 0 0
$$431$$ −13.7538 −0.662497 −0.331248 0.943544i $$-0.607470\pi$$
−0.331248 + 0.943544i $$0.607470\pi$$
$$432$$ 0 0
$$433$$ −35.5464 −1.70825 −0.854125 0.520067i $$-0.825907\pi$$
−0.854125 + 0.520067i $$0.825907\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.12311 −0.149398
$$438$$ 0 0
$$439$$ −7.12311 −0.339967 −0.169984 0.985447i $$-0.554371\pi$$
−0.169984 + 0.985447i $$0.554371\pi$$
$$440$$ 0 0
$$441$$ 3.56155 0.169598
$$442$$ 0 0
$$443$$ 22.5616 1.07193 0.535966 0.844240i $$-0.319948\pi$$
0.535966 + 0.844240i $$0.319948\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 44.9848 2.12771
$$448$$ 0 0
$$449$$ 10.3693 0.489358 0.244679 0.969604i $$-0.421317\pi$$
0.244679 + 0.969604i $$0.421317\pi$$
$$450$$ 0 0
$$451$$ −13.9309 −0.655979
$$452$$ 0 0
$$453$$ −14.2462 −0.669345
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.1080 1.73584 0.867918 0.496707i $$-0.165458\pi$$
0.867918 + 0.496707i $$0.165458\pi$$
$$458$$ 0 0
$$459$$ −3.68466 −0.171985
$$460$$ 0 0
$$461$$ 6.49242 0.302382 0.151191 0.988505i $$-0.451689\pi$$
0.151191 + 0.988505i $$0.451689\pi$$
$$462$$ 0 0
$$463$$ −20.4924 −0.952364 −0.476182 0.879347i $$-0.657980\pi$$
−0.476182 + 0.879347i $$0.657980\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −24.8769 −1.15117 −0.575583 0.817744i $$-0.695225\pi$$
−0.575583 + 0.817744i $$0.695225\pi$$
$$468$$ 0 0
$$469$$ 16.1231 0.744496
$$470$$ 0 0
$$471$$ 62.7386 2.89084
$$472$$ 0 0
$$473$$ 5.17708 0.238042
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 25.3693 1.16158
$$478$$ 0 0
$$479$$ −40.7386 −1.86140 −0.930698 0.365789i $$-0.880799\pi$$
−0.930698 + 0.365789i $$0.880799\pi$$
$$480$$ 0 0
$$481$$ −23.6155 −1.07678
$$482$$ 0 0
$$483$$ 14.2462 0.648225
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 35.5616 1.61145 0.805724 0.592291i $$-0.201777\pi$$
0.805724 + 0.592291i $$0.201777\pi$$
$$488$$ 0 0
$$489$$ −35.6847 −1.61372
$$490$$ 0 0
$$491$$ 20.6847 0.933486 0.466743 0.884393i $$-0.345427\pi$$
0.466743 + 0.884393i $$0.345427\pi$$
$$492$$ 0 0
$$493$$ −19.3693 −0.872350
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.6847 −0.479272
$$498$$ 0 0
$$499$$ −16.4924 −0.738302 −0.369151 0.929369i $$-0.620352\pi$$
−0.369151 + 0.929369i $$0.620352\pi$$
$$500$$ 0 0
$$501$$ −50.2462 −2.24484
$$502$$ 0 0
$$503$$ 19.7538 0.880778 0.440389 0.897807i $$-0.354841\pi$$
0.440389 + 0.897807i $$0.354841\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −23.0540 −1.02386
$$508$$ 0 0
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ 10.5616 0.467216
$$512$$ 0 0
$$513$$ −0.807764 −0.0356637
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 17.5076 0.769982
$$518$$ 0 0
$$519$$ 37.1231 1.62952
$$520$$ 0 0
$$521$$ −27.4384 −1.20210 −0.601050 0.799211i $$-0.705251\pi$$
−0.601050 + 0.799211i $$0.705251\pi$$
$$522$$ 0 0
$$523$$ 28.3153 1.23814 0.619072 0.785334i $$-0.287509\pi$$
0.619072 + 0.785334i $$0.287509\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.24621 0.0978465
$$528$$ 0 0
$$529$$ 7.93087 0.344820
$$530$$ 0 0
$$531$$ −47.6155 −2.06634
$$532$$ 0 0
$$533$$ −13.1231 −0.568425
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −21.3002 −0.919171
$$538$$ 0 0
$$539$$ 2.12311 0.0914486
$$540$$ 0 0
$$541$$ 41.4233 1.78093 0.890463 0.455055i $$-0.150380\pi$$
0.890463 + 0.455055i $$0.150380\pi$$
$$542$$ 0 0
$$543$$ −7.36932 −0.316248
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 33.7386 1.44256 0.721280 0.692644i $$-0.243554\pi$$
0.721280 + 0.692644i $$0.243554\pi$$
$$548$$ 0 0
$$549$$ 10.2462 0.437298
$$550$$ 0 0
$$551$$ −4.24621 −0.180895
$$552$$ 0 0
$$553$$ 6.68466 0.284261
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.56155 0.405136 0.202568 0.979268i $$-0.435071\pi$$
0.202568 + 0.979268i $$0.435071\pi$$
$$558$$ 0 0
$$559$$ 4.87689 0.206271
$$560$$ 0 0
$$561$$ −13.9309 −0.588162
$$562$$ 0 0
$$563$$ −7.12311 −0.300203 −0.150102 0.988671i $$-0.547960\pi$$
−0.150102 + 0.988671i $$0.547960\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −7.00000 −0.293972
$$568$$ 0 0
$$569$$ −19.9848 −0.837808 −0.418904 0.908030i $$-0.637586\pi$$
−0.418904 + 0.908030i $$0.637586\pi$$
$$570$$ 0 0
$$571$$ 27.3153 1.14311 0.571556 0.820563i $$-0.306340\pi$$
0.571556 + 0.820563i $$0.306340\pi$$
$$572$$ 0 0
$$573$$ 1.61553 0.0674897
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −45.5464 −1.89612 −0.948061 0.318090i $$-0.896959\pi$$
−0.948061 + 0.318090i $$0.896959\pi$$
$$578$$ 0 0
$$579$$ 52.8078 2.19462
$$580$$ 0 0
$$581$$ −13.6847 −0.567735
$$582$$ 0 0
$$583$$ 15.1231 0.626335
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.17708 0.337504 0.168752 0.985659i $$-0.446026\pi$$
0.168752 + 0.985659i $$0.446026\pi$$
$$588$$ 0 0
$$589$$ 0.492423 0.0202899
$$590$$ 0 0
$$591$$ −9.12311 −0.375274
$$592$$ 0 0
$$593$$ 12.3153 0.505730 0.252865 0.967502i $$-0.418627\pi$$
0.252865 + 0.967502i $$0.418627\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −33.6155 −1.37579
$$598$$ 0 0
$$599$$ 24.3002 0.992879 0.496439 0.868071i $$-0.334641\pi$$
0.496439 + 0.868071i $$0.334641\pi$$
$$600$$ 0 0
$$601$$ −4.94602 −0.201753 −0.100876 0.994899i $$-0.532165\pi$$
−0.100876 + 0.994899i $$0.532165\pi$$
$$602$$ 0 0
$$603$$ 57.4233 2.33846
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −37.3693 −1.51677 −0.758387 0.651805i $$-0.774012\pi$$
−0.758387 + 0.651805i $$0.774012\pi$$
$$608$$ 0 0
$$609$$ 19.3693 0.784884
$$610$$ 0 0
$$611$$ 16.4924 0.667212
$$612$$ 0 0
$$613$$ −6.68466 −0.269991 −0.134995 0.990846i $$-0.543102\pi$$
−0.134995 + 0.990846i $$0.543102\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 17.8078 0.716914 0.358457 0.933546i $$-0.383303\pi$$
0.358457 + 0.933546i $$0.383303\pi$$
$$618$$ 0 0
$$619$$ 29.3693 1.18045 0.590226 0.807238i $$-0.299039\pi$$
0.590226 + 0.807238i $$0.299039\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ −10.8078 −0.433004
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −3.05398 −0.121964
$$628$$ 0 0
$$629$$ 30.2462 1.20600
$$630$$ 0 0
$$631$$ 34.0540 1.35567 0.677834 0.735215i $$-0.262919\pi$$
0.677834 + 0.735215i $$0.262919\pi$$
$$632$$ 0 0
$$633$$ 27.0540 1.07530
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000 0.0792429
$$638$$ 0 0
$$639$$ −38.0540 −1.50539
$$640$$ 0 0
$$641$$ −16.4384 −0.649280 −0.324640 0.945838i $$-0.605243\pi$$
−0.324640 + 0.945838i $$0.605243\pi$$
$$642$$ 0 0
$$643$$ 2.73863 0.108001 0.0540006 0.998541i $$-0.482803\pi$$
0.0540006 + 0.998541i $$0.482803\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −25.3693 −0.997371 −0.498685 0.866783i $$-0.666184\pi$$
−0.498685 + 0.866783i $$0.666184\pi$$
$$648$$ 0 0
$$649$$ −28.3845 −1.11419
$$650$$ 0 0
$$651$$ −2.24621 −0.0880360
$$652$$ 0 0
$$653$$ 33.8617 1.32511 0.662556 0.749012i $$-0.269472\pi$$
0.662556 + 0.749012i $$0.269472\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 37.6155 1.46752
$$658$$ 0 0
$$659$$ 18.5616 0.723055 0.361528 0.932361i $$-0.382255\pi$$
0.361528 + 0.932361i $$0.382255\pi$$
$$660$$ 0 0
$$661$$ −22.4924 −0.874854 −0.437427 0.899254i $$-0.644110\pi$$
−0.437427 + 0.899254i $$0.644110\pi$$
$$662$$ 0 0
$$663$$ −13.1231 −0.509659
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 42.0540 1.62834
$$668$$ 0 0
$$669$$ −1.61553 −0.0624599
$$670$$ 0 0
$$671$$ 6.10795 0.235795
$$672$$ 0 0
$$673$$ 46.4924 1.79215 0.896076 0.443901i $$-0.146406\pi$$
0.896076 + 0.443901i $$0.146406\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 14.6307 0.562303 0.281151 0.959663i $$-0.409284\pi$$
0.281151 + 0.959663i $$0.409284\pi$$
$$678$$ 0 0
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ −34.2462 −1.31232
$$682$$ 0 0
$$683$$ 34.1231 1.30568 0.652842 0.757494i $$-0.273576\pi$$
0.652842 + 0.757494i $$0.273576\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −35.8617 −1.36821
$$688$$ 0 0
$$689$$ 14.2462 0.542737
$$690$$ 0 0
$$691$$ −17.4384 −0.663390 −0.331695 0.943387i $$-0.607620\pi$$
−0.331695 + 0.943387i $$0.607620\pi$$
$$692$$ 0 0
$$693$$ 7.56155 0.287240
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.8078 0.636639
$$698$$ 0 0
$$699$$ 9.12311 0.345068
$$700$$ 0 0
$$701$$ 44.1080 1.66593 0.832967 0.553322i $$-0.186640\pi$$
0.832967 + 0.553322i $$0.186640\pi$$
$$702$$ 0 0
$$703$$ 6.63068 0.250081
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 37.3693 1.40343 0.701717 0.712456i $$-0.252417\pi$$
0.701717 + 0.712456i $$0.252417\pi$$
$$710$$ 0 0
$$711$$ 23.8078 0.892861
$$712$$ 0 0
$$713$$ −4.87689 −0.182641
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −10.2462 −0.382652
$$718$$ 0 0
$$719$$ 38.2462 1.42634 0.713171 0.700990i $$-0.247258\pi$$
0.713171 + 0.700990i $$0.247258\pi$$
$$720$$ 0 0
$$721$$ 7.12311 0.265278
$$722$$ 0 0
$$723$$ 64.1771 2.38677
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −1.61553 −0.0599166 −0.0299583 0.999551i $$-0.509537\pi$$
−0.0299583 + 0.999551i $$0.509537\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ −6.24621 −0.231024
$$732$$ 0 0
$$733$$ 22.2462 0.821683 0.410841 0.911707i $$-0.365235\pi$$
0.410841 + 0.911707i $$0.365235\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.2311 1.26092
$$738$$ 0 0
$$739$$ 17.1771 0.631869 0.315935 0.948781i $$-0.397682\pi$$
0.315935 + 0.948781i $$0.397682\pi$$
$$740$$ 0 0
$$741$$ −2.87689 −0.105685
$$742$$ 0 0
$$743$$ −18.7386 −0.687454 −0.343727 0.939070i $$-0.611689\pi$$
−0.343727 + 0.939070i $$0.611689\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −48.7386 −1.78325
$$748$$ 0 0
$$749$$ −12.8078 −0.467986
$$750$$ 0 0
$$751$$ 27.3693 0.998721 0.499360 0.866394i $$-0.333568\pi$$
0.499360 + 0.866394i $$0.333568\pi$$
$$752$$ 0 0
$$753$$ −31.5464 −1.14961
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 13.3153 0.483954 0.241977 0.970282i $$-0.422204\pi$$
0.241977 + 0.970282i $$0.422204\pi$$
$$758$$ 0 0
$$759$$ 30.2462 1.09787
$$760$$ 0 0
$$761$$ 34.4233 1.24784 0.623922 0.781487i $$-0.285538\pi$$
0.623922 + 0.781487i $$0.285538\pi$$
$$762$$ 0 0
$$763$$ −8.93087 −0.323319
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −26.7386 −0.965476
$$768$$ 0 0
$$769$$ −33.3002 −1.20084 −0.600418 0.799687i $$-0.704999\pi$$
−0.600418 + 0.799687i $$0.704999\pi$$
$$770$$ 0 0
$$771$$ −10.8769 −0.391722
$$772$$ 0 0
$$773$$ −11.6155 −0.417782 −0.208891 0.977939i $$-0.566985\pi$$
−0.208891 + 0.977939i $$0.566985\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −30.2462 −1.08508
$$778$$ 0 0
$$779$$ 3.68466 0.132017
$$780$$ 0 0
$$781$$ −22.6847 −0.811721
$$782$$ 0 0
$$783$$ 10.8769 0.388708
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 26.2462 0.935576 0.467788 0.883841i $$-0.345051\pi$$
0.467788 + 0.883841i $$0.345051\pi$$
$$788$$ 0 0
$$789$$ −27.3693 −0.974373
$$790$$ 0 0
$$791$$ 0.753789 0.0268016
$$792$$ 0 0
$$793$$ 5.75379 0.204323
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 41.6155 1.47410 0.737049 0.675840i $$-0.236219\pi$$
0.737049 + 0.675840i $$0.236219\pi$$
$$798$$ 0 0
$$799$$ −21.1231 −0.747282
$$800$$ 0 0
$$801$$ −38.4924 −1.36006
$$802$$ 0 0
$$803$$ 22.4233 0.791301
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 30.1080 1.05985
$$808$$ 0 0
$$809$$ 49.4233 1.73763 0.868815 0.495136i $$-0.164882\pi$$
0.868815 + 0.495136i $$0.164882\pi$$
$$810$$ 0 0
$$811$$ 43.6155 1.53155 0.765774 0.643110i $$-0.222356\pi$$
0.765774 + 0.643110i $$0.222356\pi$$
$$812$$ 0 0
$$813$$ 41.6155 1.45952
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.36932 −0.0479063
$$818$$ 0 0
$$819$$ 7.12311 0.248901
$$820$$ 0 0
$$821$$ 45.8617 1.60059 0.800293 0.599609i $$-0.204677\pi$$
0.800293 + 0.599609i $$0.204677\pi$$
$$822$$ 0 0
$$823$$ 12.4384 0.433577 0.216789 0.976219i $$-0.430442\pi$$
0.216789 + 0.976219i $$0.430442\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −9.24621 −0.321522 −0.160761 0.986993i $$-0.551395\pi$$
−0.160761 + 0.986993i $$0.551395\pi$$
$$828$$ 0 0
$$829$$ −2.24621 −0.0780141 −0.0390071 0.999239i $$-0.512419\pi$$
−0.0390071 + 0.999239i $$0.512419\pi$$
$$830$$ 0 0
$$831$$ −26.8769 −0.932349
$$832$$ 0 0
$$833$$ −2.56155 −0.0887525
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.26137 −0.0435992
$$838$$ 0 0
$$839$$ −1.12311 −0.0387739 −0.0193870 0.999812i $$-0.506171\pi$$
−0.0193870 + 0.999812i $$0.506171\pi$$
$$840$$ 0 0
$$841$$ 28.1771 0.971623
$$842$$ 0 0
$$843$$ 29.6155 1.02001
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −6.49242 −0.223082
$$848$$ 0 0
$$849$$ −35.0540 −1.20305
$$850$$ 0 0
$$851$$ −65.6695 −2.25112
$$852$$ 0 0
$$853$$ 10.8769 0.372418 0.186209 0.982510i $$-0.440380\pi$$
0.186209 + 0.982510i $$0.440380\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.94602 0.168953 0.0844765 0.996425i $$-0.473078\pi$$
0.0844765 + 0.996425i $$0.473078\pi$$
$$858$$ 0 0
$$859$$ 34.1771 1.16611 0.583053 0.812434i $$-0.301858\pi$$
0.583053 + 0.812434i $$0.301858\pi$$
$$860$$ 0 0
$$861$$ −16.8078 −0.572807
$$862$$ 0 0
$$863$$ −23.8078 −0.810426 −0.405213 0.914222i $$-0.632803\pi$$
−0.405213 + 0.914222i $$0.632803\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −26.7386 −0.908092
$$868$$ 0 0
$$869$$ 14.1922 0.481439
$$870$$ 0 0
$$871$$ 32.2462 1.09262
$$872$$ 0 0
$$873$$ 21.3693 0.723242
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.6155 0.797440 0.398720 0.917073i $$-0.369455\pi$$
0.398720 + 0.917073i $$0.369455\pi$$
$$878$$ 0 0
$$879$$ −76.8466 −2.59197
$$880$$ 0 0
$$881$$ −40.7386 −1.37252 −0.686260 0.727357i $$-0.740749\pi$$
−0.686260 + 0.727357i $$0.740749\pi$$
$$882$$ 0 0
$$883$$ −1.49242 −0.0502240 −0.0251120 0.999685i $$-0.507994\pi$$
−0.0251120 + 0.999685i $$0.507994\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −49.6155 −1.66593 −0.832963 0.553328i $$-0.813357\pi$$
−0.832963 + 0.553328i $$0.813357\pi$$
$$888$$ 0 0
$$889$$ −9.80776 −0.328942
$$890$$ 0 0
$$891$$ −14.8617 −0.497887
$$892$$ 0 0
$$893$$ −4.63068 −0.154960
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 28.4924 0.951334
$$898$$ 0 0
$$899$$ −6.63068 −0.221146
$$900$$ 0 0
$$901$$ −18.2462 −0.607869
$$902$$ 0 0
$$903$$ 6.24621 0.207861
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −39.4233 −1.30615 −0.653076 0.757292i $$-0.726522\pi$$
−0.653076 + 0.757292i $$0.726522\pi$$
$$912$$ 0 0
$$913$$ −29.0540 −0.961546
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.75379 −0.0579152
$$918$$ 0 0
$$919$$ 18.1922 0.600106 0.300053 0.953922i $$-0.402996\pi$$
0.300053 + 0.953922i $$0.402996\pi$$
$$920$$ 0 0
$$921$$ 11.0540 0.364241
$$922$$ 0 0
$$923$$ −21.3693 −0.703380
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 25.3693 0.833238
$$928$$ 0 0
$$929$$ 56.3542 1.84892 0.924460 0.381279i $$-0.124516\pi$$
0.924460 + 0.381279i $$0.124516\pi$$
$$930$$ 0 0
$$931$$ −0.561553 −0.0184042
$$932$$ 0 0
$$933$$ 7.36932 0.241261
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21.3002 0.695847 0.347923 0.937523i $$-0.386887\pi$$
0.347923 + 0.937523i $$0.386887\pi$$
$$938$$ 0 0
$$939$$ 56.3542 1.83905
$$940$$ 0 0
$$941$$ −1.36932 −0.0446385 −0.0223192 0.999751i $$-0.507105\pi$$
−0.0223192 + 0.999751i $$0.507105\pi$$
$$942$$ 0 0
$$943$$ −36.4924 −1.18836
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 16.4924 0.535932 0.267966 0.963428i $$-0.413649\pi$$
0.267966 + 0.963428i $$0.413649\pi$$
$$948$$ 0 0
$$949$$ 21.1231 0.685685
$$950$$ 0 0
$$951$$ −52.9848 −1.71815
$$952$$ 0 0
$$953$$ −27.8769 −0.903021 −0.451511 0.892266i $$-0.649115\pi$$
−0.451511 + 0.892266i $$0.649115\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 41.1231 1.32932
$$958$$ 0 0
$$959$$ −2.31534 −0.0747663
$$960$$ 0 0
$$961$$ −30.2311 −0.975195
$$962$$ 0 0
$$963$$ −45.6155 −1.46994
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −16.6307 −0.534807 −0.267403 0.963585i $$-0.586166\pi$$
−0.267403 + 0.963585i $$0.586166\pi$$
$$968$$ 0 0
$$969$$ 3.68466 0.118368
$$970$$ 0 0
$$971$$ −2.31534 −0.0743028 −0.0371514 0.999310i $$-0.511828\pi$$
−0.0371514 + 0.999310i $$0.511828\pi$$
$$972$$ 0 0
$$973$$ −19.6847 −0.631061
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.2462 −1.06364 −0.531820 0.846857i $$-0.678492\pi$$
−0.531820 + 0.846857i $$0.678492\pi$$
$$978$$ 0 0
$$979$$ −22.9460 −0.733358
$$980$$ 0 0
$$981$$ −31.8078 −1.01554
$$982$$ 0 0
$$983$$ −62.3542 −1.98879 −0.994394 0.105734i $$-0.966281\pi$$
−0.994394 + 0.105734i $$0.966281\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 21.1231 0.672356
$$988$$ 0 0
$$989$$ 13.5616 0.431232
$$990$$ 0 0
$$991$$ −29.6695 −0.942483 −0.471241 0.882004i $$-0.656194\pi$$
−0.471241 + 0.882004i $$0.656194\pi$$
$$992$$ 0 0
$$993$$ 69.1619 2.19479
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 12.2462 0.387841 0.193921 0.981017i $$-0.437880\pi$$
0.193921 + 0.981017i $$0.437880\pi$$
$$998$$ 0 0
$$999$$ −16.9848 −0.537377
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 1400.2.a.q.1.2 yes 2
4.3 odd 2 2800.2.a.bj.1.1 2
5.2 odd 4 1400.2.g.j.449.1 4
5.3 odd 4 1400.2.g.j.449.4 4
5.4 even 2 1400.2.a.o.1.1 2
7.6 odd 2 9800.2.a.bt.1.1 2
20.3 even 4 2800.2.g.v.449.1 4
20.7 even 4 2800.2.g.v.449.4 4
20.19 odd 2 2800.2.a.bo.1.2 2
35.34 odd 2 9800.2.a.bx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.a.o.1.1 2 5.4 even 2
1400.2.a.q.1.2 yes 2 1.1 even 1 trivial
1400.2.g.j.449.1 4 5.2 odd 4
1400.2.g.j.449.4 4 5.3 odd 4
2800.2.a.bj.1.1 2 4.3 odd 2
2800.2.a.bo.1.2 2 20.19 odd 2
2800.2.g.v.449.1 4 20.3 even 4
2800.2.g.v.449.4 4 20.7 even 4
9800.2.a.bt.1.1 2 7.6 odd 2
9800.2.a.bx.1.2 2 35.34 odd 2