# Properties

 Label 7448.2.a.t.1.1 Level $7448$ Weight $2$ Character 7448.1 Self dual yes Analytic conductor $59.473$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7448 = 2^{3} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4725794254$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7448.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} -7.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +2.00000 q^{29} +6.00000 q^{31} -6.00000 q^{33} -10.0000 q^{37} +8.00000 q^{39} +8.00000 q^{41} +7.00000 q^{43} +1.00000 q^{45} +9.00000 q^{47} +4.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} +2.00000 q^{57} +14.0000 q^{59} +5.00000 q^{61} +4.00000 q^{65} +14.0000 q^{67} -14.0000 q^{69} +8.00000 q^{71} +1.00000 q^{73} -8.00000 q^{75} +10.0000 q^{79} -11.0000 q^{81} +17.0000 q^{83} +2.00000 q^{85} +4.00000 q^{87} +12.0000 q^{93} +1.00000 q^{95} +12.0000 q^{97} -3.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 2.00000 0.516398
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 14.0000 1.82264 0.911322 0.411693i $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 0 0
$$69$$ −14.0000 −1.68540
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ −8.00000 −0.923760
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 17.0000 1.86599 0.932996 0.359886i $$-0.117184\pi$$
0.932996 + 0.359886i $$0.117184\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 12.0000 1.24434
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 11.0000 1.09454 0.547270 0.836956i $$-0.315667\pi$$
0.547270 + 0.836956i $$0.315667\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −20.0000 −1.89832
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 16.0000 1.44267
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 14.0000 1.23263
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −7.00000 −0.598050 −0.299025 0.954245i $$-0.596661\pi$$
−0.299025 + 0.954245i $$0.596661\pi$$
$$138$$ 0 0
$$139$$ 9.00000 0.763370 0.381685 0.924292i $$-0.375344\pi$$
0.381685 + 0.924292i $$0.375344\pi$$
$$140$$ 0 0
$$141$$ 18.0000 1.51587
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ −6.00000 −0.467099
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 28.0000 2.10461
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.0000 0.940647 0.470323 0.882494i $$-0.344137\pi$$
0.470323 + 0.882494i $$0.344137\pi$$
$$192$$ 0 0
$$193$$ −24.0000 −1.72756 −0.863779 0.503871i $$-0.831909\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 0 0
$$195$$ 8.00000 0.572892
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ 1.00000 0.0708881 0.0354441 0.999372i $$-0.488715\pi$$
0.0354441 + 0.999372i $$0.488715\pi$$
$$200$$ 0 0
$$201$$ 28.0000 1.97497
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 8.00000 0.558744
$$206$$ 0 0
$$207$$ −7.00000 −0.486534
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ 0 0
$$213$$ 16.0000 1.09630
$$214$$ 0 0
$$215$$ 7.00000 0.477396
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 0 0
$$237$$ 20.0000 1.29914
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 34.0000 2.15466
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 21.0000 1.32026
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ 10.0000 0.623783 0.311891 0.950118i $$-0.399037\pi$$
0.311891 + 0.950118i $$0.399037\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 15.0000 0.911185 0.455593 0.890188i $$-0.349427\pi$$
0.455593 + 0.890188i $$0.349427\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ −5.00000 −0.300421 −0.150210 0.988654i $$-0.547995\pi$$
−0.150210 + 0.988654i $$0.547995\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ −13.0000 −0.772770 −0.386385 0.922338i $$-0.626276\pi$$
−0.386385 + 0.922338i $$0.626276\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 24.0000 1.40690
$$292$$ 0 0
$$293$$ −24.0000 −1.40209 −0.701047 0.713115i $$-0.747284\pi$$
−0.701047 + 0.713115i $$0.747284\pi$$
$$294$$ 0 0
$$295$$ 14.0000 0.815112
$$296$$ 0 0
$$297$$ 12.0000 0.696311
$$298$$ 0 0
$$299$$ −28.0000 −1.61928
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 22.0000 1.26387
$$304$$ 0 0
$$305$$ 5.00000 0.286299
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 7.00000 0.395663 0.197832 0.980236i $$-0.436610\pi$$
0.197832 + 0.980236i $$0.436610\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4.00000 −0.224662 −0.112331 0.993671i $$-0.535832\pi$$
−0.112331 + 0.993671i $$0.535832\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −36.0000 −2.00932
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ −16.0000 −0.887520
$$326$$ 0 0
$$327$$ −32.0000 −1.76960
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 6.00000 0.329790 0.164895 0.986311i $$-0.447272\pi$$
0.164895 + 0.986311i $$0.447272\pi$$
$$332$$ 0 0
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 14.0000 0.764902
$$336$$ 0 0
$$337$$ 28.0000 1.52526 0.762629 0.646837i $$-0.223908\pi$$
0.762629 + 0.646837i $$0.223908\pi$$
$$338$$ 0 0
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ −18.0000 −0.974755
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −14.0000 −0.753735
$$346$$ 0 0
$$347$$ 13.0000 0.697877 0.348938 0.937146i $$-0.386542\pi$$
0.348938 + 0.937146i $$0.386542\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −25.0000 −1.31945 −0.659725 0.751507i $$-0.729327\pi$$
−0.659725 + 0.751507i $$0.729327\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −4.00000 −0.209946
$$364$$ 0 0
$$365$$ 1.00000 0.0523424
$$366$$ 0 0
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 0 0
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ −18.0000 −0.929516
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ 18.0000 0.924598 0.462299 0.886724i $$-0.347025\pi$$
0.462299 + 0.886724i $$0.347025\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.00000 0.355830
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −14.0000 −0.708010
$$392$$ 0 0
$$393$$ −24.0000 −1.21064
$$394$$ 0 0
$$395$$ 10.0000 0.503155
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 24.0000 1.19553
$$404$$ 0 0
$$405$$ −11.0000 −0.546594
$$406$$ 0 0
$$407$$ 30.0000 1.48704
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 17.0000 0.834497
$$416$$ 0 0
$$417$$ 18.0000 0.881464
$$418$$ 0 0
$$419$$ −23.0000 −1.12362 −0.561812 0.827265i $$-0.689895\pi$$
−0.561812 + 0.827265i $$0.689895\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ −8.00000 −0.388057
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ 0 0
$$437$$ −7.00000 −0.334855
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −32.0000 −1.51017 −0.755087 0.655625i $$-0.772405\pi$$
−0.755087 + 0.655625i $$0.772405\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ −24.0000 −1.12762
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ 7.00000 0.326023 0.163011 0.986624i $$-0.447879\pi$$
0.163011 + 0.986624i $$0.447879\pi$$
$$462$$ 0 0
$$463$$ 33.0000 1.53364 0.766820 0.641862i $$-0.221838\pi$$
0.766820 + 0.641862i $$0.221838\pi$$
$$464$$ 0 0
$$465$$ 12.0000 0.556487
$$466$$ 0 0
$$467$$ −29.0000 −1.34196 −0.670980 0.741475i $$-0.734126\pi$$
−0.670980 + 0.741475i $$0.734126\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ −21.0000 −0.965581
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 7.00000 0.319838 0.159919 0.987130i $$-0.448877\pi$$
0.159919 + 0.987130i $$0.448877\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.0000 0.544892
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ 22.0000 0.994874
$$490$$ 0 0
$$491$$ −27.0000 −1.21849 −0.609246 0.792981i $$-0.708528\pi$$
−0.609246 + 0.792981i $$0.708528\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ −3.00000 −0.134840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13.0000 0.581960 0.290980 0.956729i $$-0.406019\pi$$
0.290980 + 0.956729i $$0.406019\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ 9.00000 0.401290 0.200645 0.979664i $$-0.435696\pi$$
0.200645 + 0.979664i $$0.435696\pi$$
$$504$$ 0 0
$$505$$ 11.0000 0.489494
$$506$$ 0 0
$$507$$ 6.00000 0.266469
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −27.0000 −1.18746
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ −40.0000 −1.74908 −0.874539 0.484955i $$-0.838836\pi$$
−0.874539 + 0.484955i $$0.838836\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 14.0000 0.607548
$$532$$ 0 0
$$533$$ 32.0000 1.38607
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ 0 0
$$537$$ −36.0000 −1.55351
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 33.0000 1.41878 0.709390 0.704816i $$-0.248970\pi$$
0.709390 + 0.704816i $$0.248970\pi$$
$$542$$ 0 0
$$543$$ −32.0000 −1.37325
$$544$$ 0 0
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ −34.0000 −1.45374 −0.726868 0.686778i $$-0.759025\pi$$
−0.726868 + 0.686778i $$0.759025\pi$$
$$548$$ 0 0
$$549$$ 5.00000 0.213395
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −20.0000 −0.848953
$$556$$ 0 0
$$557$$ 35.0000 1.48300 0.741499 0.670954i $$-0.234115\pi$$
0.741499 + 0.670954i $$0.234115\pi$$
$$558$$ 0 0
$$559$$ 28.0000 1.18427
$$560$$ 0 0
$$561$$ −12.0000 −0.506640
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 21.0000 0.878823 0.439411 0.898286i $$-0.355187\pi$$
0.439411 + 0.898286i $$0.355187\pi$$
$$572$$ 0 0
$$573$$ 26.0000 1.08617
$$574$$ 0 0
$$575$$ 28.0000 1.16768
$$576$$ 0 0
$$577$$ −5.00000 −0.208153 −0.104076 0.994569i $$-0.533189\pi$$
−0.104076 + 0.994569i $$0.533189\pi$$
$$578$$ 0 0
$$579$$ −48.0000 −1.99481
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 4.00000 0.165380
$$586$$ 0 0
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 11.0000 0.451716 0.225858 0.974160i $$-0.427481\pi$$
0.225858 + 0.974160i $$0.427481\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.00000 0.0818546
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ 14.0000 0.570124
$$604$$ 0 0
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 36.0000 1.45640
$$612$$ 0 0
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 0 0
$$615$$ 16.0000 0.645182
$$616$$ 0 0
$$617$$ 5.00000 0.201292 0.100646 0.994922i $$-0.467909\pi$$
0.100646 + 0.994922i $$0.467909\pi$$
$$618$$ 0 0
$$619$$ −43.0000 −1.72832 −0.864158 0.503221i $$-0.832148\pi$$
−0.864158 + 0.503221i $$0.832148\pi$$
$$620$$ 0 0
$$621$$ 28.0000 1.12360
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −6.00000 −0.239617
$$628$$ 0 0
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ 25.0000 0.995234 0.497617 0.867397i $$-0.334208\pi$$
0.497617 + 0.867397i $$0.334208\pi$$
$$632$$ 0 0
$$633$$ 44.0000 1.74884
$$634$$ 0 0
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −10.0000 −0.394976 −0.197488 0.980305i $$-0.563278\pi$$
−0.197488 + 0.980305i $$0.563278\pi$$
$$642$$ 0 0
$$643$$ 36.0000 1.41970 0.709851 0.704352i $$-0.248762\pi$$
0.709851 + 0.704352i $$0.248762\pi$$
$$644$$ 0 0
$$645$$ 14.0000 0.551249
$$646$$ 0 0
$$647$$ −7.00000 −0.275198 −0.137599 0.990488i $$-0.543939\pi$$
−0.137599 + 0.990488i $$0.543939\pi$$
$$648$$ 0 0
$$649$$ −42.0000 −1.64864
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 1.00000 0.0390137
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ 16.0000 0.622328 0.311164 0.950356i $$-0.399281\pi$$
0.311164 + 0.950356i $$0.399281\pi$$
$$662$$ 0 0
$$663$$ 16.0000 0.621389
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −14.0000 −0.542082
$$668$$ 0 0
$$669$$ 40.0000 1.54649
$$670$$ 0 0
$$671$$ −15.0000 −0.579069
$$672$$ 0 0
$$673$$ −50.0000 −1.92736 −0.963679 0.267063i $$-0.913947\pi$$
−0.963679 + 0.267063i $$0.913947\pi$$
$$674$$ 0 0
$$675$$ 16.0000 0.615840
$$676$$ 0 0
$$677$$ −32.0000 −1.22986 −0.614930 0.788582i $$-0.710816\pi$$
−0.614930 + 0.788582i $$0.710816\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −16.0000 −0.613121
$$682$$ 0 0
$$683$$ −14.0000 −0.535695 −0.267848 0.963461i $$-0.586312\pi$$
−0.267848 + 0.963461i $$0.586312\pi$$
$$684$$ 0 0
$$685$$ −7.00000 −0.267456
$$686$$ 0 0
$$687$$ 12.0000 0.457829
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9.00000 0.341389
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ 0 0
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ −33.0000 −1.24639 −0.623196 0.782065i $$-0.714166\pi$$
−0.623196 + 0.782065i $$0.714166\pi$$
$$702$$ 0 0
$$703$$ −10.0000 −0.377157
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −13.0000 −0.488225 −0.244113 0.969747i $$-0.578497\pi$$
−0.244113 + 0.969747i $$0.578497\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 0 0
$$713$$ −42.0000 −1.57291
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ −48.0000 −1.79259
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ −49.0000 −1.81731 −0.908655 0.417548i $$-0.862889\pi$$
−0.908655 + 0.417548i $$0.862889\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 14.0000 0.517809
$$732$$ 0 0
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −42.0000 −1.54709
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ 3.00000 0.109911
$$746$$ 0 0
$$747$$ 17.0000 0.621997
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 30.0000 1.09472 0.547358 0.836899i $$-0.315634\pi$$
0.547358 + 0.836899i $$0.315634\pi$$
$$752$$ 0 0
$$753$$ −30.0000 −1.09326
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ −23.0000 −0.835949 −0.417975 0.908459i $$-0.637260\pi$$
−0.417975 + 0.908459i $$0.637260\pi$$
$$758$$ 0 0
$$759$$ 42.0000 1.52450
$$760$$ 0 0
$$761$$ 21.0000 0.761249 0.380625 0.924730i $$-0.375709\pi$$
0.380625 + 0.924730i $$0.375709\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 56.0000 2.02204
$$768$$ 0 0
$$769$$ 15.0000 0.540914 0.270457 0.962732i $$-0.412825\pi$$
0.270457 + 0.962732i $$0.412825\pi$$
$$770$$ 0 0
$$771$$ 20.0000 0.720282
$$772$$ 0 0
$$773$$ 22.0000 0.791285 0.395643 0.918405i $$-0.370522\pi$$
0.395643 + 0.918405i $$0.370522\pi$$
$$774$$ 0 0
$$775$$ −24.0000 −0.862105
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 7.00000 0.249841
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 0 0
$$789$$ 32.0000 1.13923
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 0 0
$$795$$ 12.0000 0.425596
$$796$$ 0 0
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.00000 −0.105868
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 48.0000 1.68968
$$808$$ 0 0
$$809$$ −41.0000 −1.44148 −0.720742 0.693204i $$-0.756199\pi$$
−0.720742 + 0.693204i $$0.756199\pi$$
$$810$$ 0 0
$$811$$ 42.0000 1.47482 0.737410 0.675446i $$-0.236049\pi$$
0.737410 + 0.675446i $$0.236049\pi$$
$$812$$ 0 0
$$813$$ 30.0000 1.05215
$$814$$ 0 0
$$815$$ 11.0000 0.385313
$$816$$ 0 0
$$817$$ 7.00000 0.244899
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.0000 1.08191 0.540954 0.841052i $$-0.318063\pi$$
0.540954 + 0.841052i $$0.318063\pi$$
$$822$$ 0 0
$$823$$ 23.0000 0.801730 0.400865 0.916137i $$-0.368710\pi$$
0.400865 + 0.916137i $$0.368710\pi$$
$$824$$ 0 0
$$825$$ 24.0000 0.835573
$$826$$ 0 0
$$827$$ 8.00000 0.278187 0.139094 0.990279i $$-0.455581\pi$$
0.139094 + 0.990279i $$0.455581\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −12.0000 −0.415277
$$836$$ 0 0
$$837$$ −24.0000 −0.829561
$$838$$ 0 0
$$839$$ 54.0000 1.86429 0.932144 0.362089i $$-0.117936\pi$$
0.932144 + 0.362089i $$0.117936\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −20.0000 −0.688837
$$844$$ 0 0
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −26.0000 −0.892318
$$850$$ 0 0
$$851$$ 70.0000 2.39957
$$852$$ 0 0
$$853$$ 37.0000 1.26686 0.633428 0.773802i $$-0.281647\pi$$
0.633428 + 0.773802i $$0.281647\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −32.0000 −1.09310 −0.546550 0.837427i $$-0.684059\pi$$
−0.546550 + 0.837427i $$0.684059\pi$$
$$858$$ 0 0
$$859$$ −17.0000 −0.580033 −0.290016 0.957022i $$-0.593661\pi$$
−0.290016 + 0.957022i $$0.593661\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −30.0000 −1.02121 −0.510606 0.859815i $$-0.670579\pi$$
−0.510606 + 0.859815i $$0.670579\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ 0 0
$$867$$ −26.0000 −0.883006
$$868$$ 0 0
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 56.0000 1.89749
$$872$$ 0 0
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −8.00000 −0.270141 −0.135070 0.990836i $$-0.543126\pi$$
−0.135070 + 0.990836i $$0.543126\pi$$
$$878$$ 0 0
$$879$$ −48.0000 −1.61900
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 0 0
$$885$$ 28.0000 0.941210
$$886$$ 0 0
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 33.0000 1.10554
$$892$$ 0 0
$$893$$ 9.00000 0.301174
$$894$$ 0 0
$$895$$ −18.0000 −0.601674
$$896$$ 0 0
$$897$$ −56.0000 −1.86979
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.0000 −0.531858
$$906$$ 0 0
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ 0 0
$$909$$ 11.0000 0.364847
$$910$$ 0 0
$$911$$ −58.0000 −1.92163 −0.960813 0.277198i $$-0.910594\pi$$
−0.960813 + 0.277198i $$0.910594\pi$$
$$912$$ 0 0
$$913$$ −51.0000 −1.68785
$$914$$ 0 0
$$915$$ 10.0000 0.330590
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 39.0000 1.28649 0.643246 0.765660i $$-0.277587\pi$$
0.643246 + 0.765660i $$0.277587\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ 0 0
$$923$$ 32.0000 1.05329
$$924$$ 0 0
$$925$$ 40.0000 1.31519
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 51.0000 1.67326 0.836628 0.547772i $$-0.184524\pi$$
0.836628 + 0.547772i $$0.184524\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −48.0000 −1.57145
$$934$$ 0 0
$$935$$ −6.00000 −0.196221
$$936$$ 0 0
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ −56.0000 −1.82361
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −8.00000 −0.259418
$$952$$ 0 0
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ 0 0
$$955$$ 13.0000 0.420670
$$956$$ 0 0
$$957$$ −12.0000 −0.387905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ −24.0000 −0.772587
$$966$$ 0 0
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ 0 0
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −32.0000 −1.02482
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ 8.00000 0.255160 0.127580 0.991828i $$-0.459279\pi$$
0.127580 + 0.991828i $$0.459279\pi$$
$$984$$ 0 0
$$985$$ −3.00000 −0.0955879
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −49.0000 −1.55811
$$990$$ 0 0
$$991$$ −38.0000 −1.20711 −0.603555 0.797321i $$-0.706250\pi$$
−0.603555 + 0.797321i $$0.706250\pi$$
$$992$$ 0 0
$$993$$ 12.0000 0.380808
$$994$$ 0 0
$$995$$ 1.00000 0.0317021
$$996$$ 0 0
$$997$$ −62.0000 −1.96356 −0.981780 0.190022i $$-0.939144\pi$$
−0.981780 + 0.190022i $$0.939144\pi$$
$$998$$ 0 0
$$999$$ 40.0000 1.26554
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 7448.2.a.t.1.1 1
7.2 even 3 1064.2.q.a.305.1 2
7.4 even 3 1064.2.q.a.457.1 yes 2
7.6 odd 2 7448.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.a.305.1 2 7.2 even 3
1064.2.q.a.457.1 yes 2 7.4 even 3
7448.2.a.a.1.1 1 7.6 odd 2
7448.2.a.t.1.1 1 1.1 even 1 trivial