# Properties

 Label 3380.2.a.g.1.1 Level $3380$ Weight $2$ Character 3380.1 Self dual yes Analytic conductor $26.989$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9894358832$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3380.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -1.00000 q^{15} -3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000 q^{31} -3.00000 q^{33} -1.00000 q^{35} +7.00000 q^{37} +9.00000 q^{41} +11.0000 q^{43} +2.00000 q^{45} -6.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000 q^{57} +3.00000 q^{59} +11.0000 q^{61} -2.00000 q^{63} +7.00000 q^{67} -3.00000 q^{69} +3.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +3.00000 q^{85} +3.00000 q^{87} -15.0000 q^{89} +4.00000 q^{93} -7.00000 q^{95} +7.00000 q^{97} +6.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 7.00000 0.927173
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 11.0000 1.40841 0.704203 0.709999i $$-0.251305\pi$$
0.704203 + 0.709999i $$0.251305\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ −7.00000 −0.718185
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 9.00000 0.811503
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −19.0000 −1.68598 −0.842989 0.537931i $$-0.819206\pi$$
−0.842989 + 0.537931i $$0.819206\pi$$
$$128$$ 0 0
$$129$$ 11.0000 0.968496
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 7.00000 0.606977
$$134$$ 0 0
$$135$$ 5.00000 0.430331
$$136$$ 0 0
$$137$$ 15.0000 1.28154 0.640768 0.767734i $$-0.278616\pi$$
0.640768 + 0.767734i $$0.278616\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −14.0000 −1.07061
$$172$$ 0 0
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 3.00000 0.225494
$$178$$ 0 0
$$179$$ −21.0000 −1.56961 −0.784807 0.619740i $$-0.787238\pi$$
−0.784807 + 0.619740i $$0.787238\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 11.0000 0.813143
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −21.0000 −1.49619 −0.748094 0.663593i $$-0.769031\pi$$
−0.748094 + 0.663593i $$0.769031\pi$$
$$198$$ 0 0
$$199$$ 17.0000 1.20510 0.602549 0.798082i $$-0.294152\pi$$
0.602549 + 0.798082i $$0.294152\pi$$
$$200$$ 0 0
$$201$$ 7.00000 0.493742
$$202$$ 0 0
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ −9.00000 −0.628587
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 11.0000 0.757271 0.378636 0.925546i $$-0.376393\pi$$
0.378636 + 0.925546i $$0.376393\pi$$
$$212$$ 0 0
$$213$$ 3.00000 0.205557
$$214$$ 0 0
$$215$$ −11.0000 −0.750194
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ −27.0000 −1.79205 −0.896026 0.444001i $$-0.853559\pi$$
−0.896026 + 0.444001i $$0.853559\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 1.00000 0.0644157 0.0322078 0.999481i $$-0.489746\pi$$
0.0322078 + 0.999481i $$0.489746\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 21.0000 1.32551 0.662754 0.748837i $$-0.269387\pi$$
0.662754 + 0.748837i $$0.269387\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ 9.00000 0.561405 0.280702 0.959795i $$-0.409433\pi$$
0.280702 + 0.959795i $$0.409433\pi$$
$$258$$ 0 0
$$259$$ 7.00000 0.434959
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −3.00000 −0.184988 −0.0924940 0.995713i $$-0.529484\pi$$
−0.0924940 + 0.995713i $$0.529484\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ −15.0000 −0.917985
$$268$$ 0 0
$$269$$ 27.0000 1.64622 0.823110 0.567883i $$-0.192237\pi$$
0.823110 + 0.567883i $$0.192237\pi$$
$$270$$ 0 0
$$271$$ −23.0000 −1.39715 −0.698575 0.715537i $$-0.746182\pi$$
−0.698575 + 0.715537i $$0.746182\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −19.0000 −1.14160 −0.570800 0.821089i $$-0.693367\pi$$
−0.570800 + 0.821089i $$0.693367\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 5.00000 0.297219 0.148610 0.988896i $$-0.452520\pi$$
0.148610 + 0.988896i $$0.452520\pi$$
$$284$$ 0 0
$$285$$ −7.00000 −0.414644
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ 27.0000 1.57736 0.788678 0.614806i $$-0.210766\pi$$
0.788678 + 0.614806i $$0.210766\pi$$
$$294$$ 0 0
$$295$$ −3.00000 −0.174667
$$296$$ 0 0
$$297$$ 15.0000 0.870388
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 11.0000 0.634029
$$302$$ 0 0
$$303$$ −9.00000 −0.517036
$$304$$ 0 0
$$305$$ −11.0000 −0.629858
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 0 0
$$315$$ 2.00000 0.112687
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ −21.0000 −1.16847
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 19.0000 1.04433 0.522167 0.852843i $$-0.325124\pi$$
0.522167 + 0.852843i $$0.325124\pi$$
$$332$$ 0 0
$$333$$ −14.0000 −0.767195
$$334$$ 0 0
$$335$$ −7.00000 −0.382451
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 0 0
$$339$$ 9.00000 0.488813
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ 0 0
$$347$$ −33.0000 −1.77153 −0.885766 0.464131i $$-0.846367\pi$$
−0.885766 + 0.464131i $$0.846367\pi$$
$$348$$ 0 0
$$349$$ 1.00000 0.0535288 0.0267644 0.999642i $$-0.491480\pi$$
0.0267644 + 0.999642i $$0.491480\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −9.00000 −0.479022 −0.239511 0.970894i $$-0.576987\pi$$
−0.239511 + 0.970894i $$0.576987\pi$$
$$354$$ 0 0
$$355$$ −3.00000 −0.159223
$$356$$ 0 0
$$357$$ −3.00000 −0.158777
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 5.00000 0.260998 0.130499 0.991448i $$-0.458342\pi$$
0.130499 + 0.991448i $$0.458342\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ −31.0000 −1.60512 −0.802560 0.596572i $$-0.796529\pi$$
−0.802560 + 0.596572i $$0.796529\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1.00000 0.0513665 0.0256833 0.999670i $$-0.491824\pi$$
0.0256833 + 0.999670i $$0.491824\pi$$
$$380$$ 0 0
$$381$$ −19.0000 −0.973399
$$382$$ 0 0
$$383$$ 9.00000 0.459879 0.229939 0.973205i $$-0.426147\pi$$
0.229939 + 0.973205i $$0.426147\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ 0 0
$$387$$ −22.0000 −1.11832
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −5.00000 −0.250943 −0.125471 0.992097i $$-0.540044\pi$$
−0.125471 + 0.992097i $$0.540044\pi$$
$$398$$ 0 0
$$399$$ 7.00000 0.350438
$$400$$ 0 0
$$401$$ 33.0000 1.64794 0.823971 0.566632i $$-0.191754\pi$$
0.823971 + 0.566632i $$0.191754\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −21.0000 −1.04093
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 15.0000 0.739895
$$412$$ 0 0
$$413$$ 3.00000 0.147620
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ −9.00000 −0.439679 −0.219839 0.975536i $$-0.570553\pi$$
−0.219839 + 0.975536i $$0.570553\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 11.0000 0.532327
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9.00000 0.433515 0.216757 0.976226i $$-0.430452\pi$$
0.216757 + 0.976226i $$0.430452\pi$$
$$432$$ 0 0
$$433$$ 29.0000 1.39365 0.696826 0.717241i $$-0.254595\pi$$
0.696826 + 0.717241i $$0.254595\pi$$
$$434$$ 0 0
$$435$$ −3.00000 −0.143839
$$436$$ 0 0
$$437$$ −21.0000 −1.00457
$$438$$ 0 0
$$439$$ 11.0000 0.525001 0.262501 0.964932i $$-0.415453\pi$$
0.262501 + 0.964932i $$0.415453\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ 21.0000 0.993266
$$448$$ 0 0
$$449$$ −3.00000 −0.141579 −0.0707894 0.997491i $$-0.522552\pi$$
−0.0707894 + 0.997491i $$0.522552\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.00000 −0.233890 −0.116945 0.993138i $$-0.537310\pi$$
−0.116945 + 0.993138i $$0.537310\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −27.0000 −1.25752 −0.628758 0.777601i $$-0.716436\pi$$
−0.628758 + 0.777601i $$0.716436\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ 7.00000 0.323230
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ −33.0000 −1.51734
$$474$$ 0 0
$$475$$ 7.00000 0.321182
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −39.0000 −1.78196 −0.890978 0.454047i $$-0.849980\pi$$
−0.890978 + 0.454047i $$0.849980\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ −7.00000 −0.317854
$$486$$ 0 0
$$487$$ −11.0000 −0.498458 −0.249229 0.968445i $$-0.580177\pi$$
−0.249229 + 0.968445i $$0.580177\pi$$
$$488$$ 0 0
$$489$$ 1.00000 0.0452216
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ −6.00000 −0.269680
$$496$$ 0 0
$$497$$ 3.00000 0.134568
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 3.00000 0.134030
$$502$$ 0 0
$$503$$ 15.0000 0.668817 0.334408 0.942428i $$-0.391463\pi$$
0.334408 + 0.942428i $$0.391463\pi$$
$$504$$ 0 0
$$505$$ 9.00000 0.400495
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9.00000 0.398918 0.199459 0.979906i $$-0.436082\pi$$
0.199459 + 0.979906i $$0.436082\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 0 0
$$513$$ −35.0000 −1.54529
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −3.00000 −0.131685
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ 0 0
$$537$$ −21.0000 −0.906217
$$538$$ 0 0
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 0 0
$$543$$ 2.00000 0.0858282
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −22.0000 −0.938937
$$550$$ 0 0
$$551$$ 21.0000 0.894630
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ −7.00000 −0.297133
$$556$$ 0 0
$$557$$ 39.0000 1.65248 0.826242 0.563316i $$-0.190475\pi$$
0.826242 + 0.563316i $$0.190475\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 9.00000 0.379980
$$562$$ 0 0
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ −9.00000 −0.378633
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 27.0000 1.13190 0.565949 0.824440i $$-0.308510\pi$$
0.565949 + 0.824440i $$0.308510\pi$$
$$570$$ 0 0
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ −3.00000 −0.125327
$$574$$ 0 0
$$575$$ −3.00000 −0.125109
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ −5.00000 −0.207793
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ 0 0
$$589$$ 28.0000 1.15372
$$590$$ 0 0
$$591$$ −21.0000 −0.863825
$$592$$ 0 0
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 0 0
$$597$$ 17.0000 0.695764
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ −14.0000 −0.570124
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ −13.0000 −0.527654 −0.263827 0.964570i $$-0.584985\pi$$
−0.263827 + 0.964570i $$0.584985\pi$$
$$608$$ 0 0
$$609$$ 3.00000 0.121566
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 43.0000 1.73675 0.868377 0.495905i $$-0.165164\pi$$
0.868377 + 0.495905i $$0.165164\pi$$
$$614$$ 0 0
$$615$$ −9.00000 −0.362915
$$616$$ 0 0
$$617$$ −33.0000 −1.32853 −0.664265 0.747497i $$-0.731255\pi$$
−0.664265 + 0.747497i $$0.731255\pi$$
$$618$$ 0 0
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 15.0000 0.601929
$$622$$ 0 0
$$623$$ −15.0000 −0.600962
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −21.0000 −0.838659
$$628$$ 0 0
$$629$$ −21.0000 −0.837325
$$630$$ 0 0
$$631$$ −17.0000 −0.676759 −0.338380 0.941010i $$-0.609879\pi$$
−0.338380 + 0.941010i $$0.609879\pi$$
$$632$$ 0 0
$$633$$ 11.0000 0.437211
$$634$$ 0 0
$$635$$ 19.0000 0.753992
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 27.0000 1.06644 0.533218 0.845978i $$-0.320983\pi$$
0.533218 + 0.845978i $$0.320983\pi$$
$$642$$ 0 0
$$643$$ −11.0000 −0.433798 −0.216899 0.976194i $$-0.569594\pi$$
−0.216899 + 0.976194i $$0.569594\pi$$
$$644$$ 0 0
$$645$$ −11.0000 −0.433125
$$646$$ 0 0
$$647$$ 45.0000 1.76913 0.884566 0.466415i $$-0.154454\pi$$
0.884566 + 0.466415i $$0.154454\pi$$
$$648$$ 0 0
$$649$$ −9.00000 −0.353281
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 0 0
$$653$$ −39.0000 −1.52619 −0.763094 0.646288i $$-0.776321\pi$$
−0.763094 + 0.646288i $$0.776321\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ 4.00000 0.156055
$$658$$ 0 0
$$659$$ −39.0000 −1.51922 −0.759612 0.650376i $$-0.774611\pi$$
−0.759612 + 0.650376i $$0.774611\pi$$
$$660$$ 0 0
$$661$$ 1.00000 0.0388955 0.0194477 0.999811i $$-0.493809\pi$$
0.0194477 + 0.999811i $$0.493809\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −7.00000 −0.271448
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ −33.0000 −1.27395
$$672$$ 0 0
$$673$$ 17.0000 0.655302 0.327651 0.944799i $$-0.393743\pi$$
0.327651 + 0.944799i $$0.393743\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 7.00000 0.268635
$$680$$ 0 0
$$681$$ −27.0000 −1.03464
$$682$$ 0 0
$$683$$ −51.0000 −1.95146 −0.975730 0.218975i $$-0.929729\pi$$
−0.975730 + 0.218975i $$0.929729\pi$$
$$684$$ 0 0
$$685$$ −15.0000 −0.573121
$$686$$ 0 0
$$687$$ 22.0000 0.839352
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −23.0000 −0.874961 −0.437481 0.899228i $$-0.644129\pi$$
−0.437481 + 0.899228i $$0.644129\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ −27.0000 −1.02270
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 49.0000 1.84807
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −9.00000 −0.338480
$$708$$ 0 0
$$709$$ 37.0000 1.38956 0.694782 0.719220i $$-0.255501\pi$$
0.694782 + 0.719220i $$0.255501\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 9.00000 0.335643 0.167822 0.985817i $$-0.446327\pi$$
0.167822 + 0.985817i $$0.446327\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 1.00000 0.0371904
$$724$$ 0 0
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ −52.0000 −1.92857 −0.964287 0.264861i $$-0.914674\pi$$
−0.964287 + 0.264861i $$0.914674\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −33.0000 −1.22055
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ −21.0000 −0.773545
$$738$$ 0 0
$$739$$ −47.0000 −1.72892 −0.864461 0.502699i $$-0.832340\pi$$
−0.864461 + 0.502699i $$0.832340\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ −21.0000 −0.769380
$$746$$ 0 0
$$747$$ −24.0000 −0.878114
$$748$$ 0 0
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ 0 0
$$753$$ 21.0000 0.765283
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ 29.0000 1.05402 0.527011 0.849858i $$-0.323312\pi$$
0.527011 + 0.849858i $$0.323312\pi$$
$$758$$ 0 0
$$759$$ 9.00000 0.326679
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ 9.00000 0.324127
$$772$$ 0 0
$$773$$ 27.0000 0.971123 0.485561 0.874203i $$-0.338615\pi$$
0.485561 + 0.874203i $$0.338615\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 7.00000 0.251124
$$778$$ 0 0
$$779$$ 63.0000 2.25721
$$780$$ 0 0
$$781$$ −9.00000 −0.322045
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 37.0000 1.31891 0.659454 0.751745i $$-0.270788\pi$$
0.659454 + 0.751745i $$0.270788\pi$$
$$788$$ 0 0
$$789$$ −3.00000 −0.106803
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 6.00000 0.212798
$$796$$ 0 0
$$797$$ −51.0000 −1.80651 −0.903256 0.429101i $$-0.858830\pi$$
−0.903256 + 0.429101i $$0.858830\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ 27.0000 0.950445
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −32.0000 −1.12367 −0.561836 0.827249i $$-0.689905\pi$$
−0.561836 + 0.827249i $$0.689905\pi$$
$$812$$ 0 0
$$813$$ −23.0000 −0.806645
$$814$$ 0 0
$$815$$ −1.00000 −0.0350285
$$816$$ 0 0
$$817$$ 77.0000 2.69389
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.0000 −0.523504 −0.261752 0.965135i $$-0.584300\pi$$
−0.261752 + 0.965135i $$0.584300\pi$$
$$822$$ 0 0
$$823$$ −13.0000 −0.453152 −0.226576 0.973994i $$-0.572753\pi$$
−0.226576 + 0.973994i $$0.572753\pi$$
$$824$$ 0 0
$$825$$ −3.00000 −0.104447
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 11.0000 0.382046 0.191023 0.981586i $$-0.438820\pi$$
0.191023 + 0.981586i $$0.438820\pi$$
$$830$$ 0 0
$$831$$ −19.0000 −0.659103
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 0 0
$$839$$ −21.0000 −0.725001 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ −6.00000 −0.206651
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ 5.00000 0.171600
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ 14.0000 0.478790
$$856$$ 0 0
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 9.00000 0.306719
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 3.00000 0.102003
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ −24.0000 −0.814144
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −41.0000 −1.38447 −0.692236 0.721671i $$-0.743374\pi$$
−0.692236 + 0.721671i $$0.743374\pi$$
$$878$$ 0 0
$$879$$ 27.0000 0.910687
$$880$$ 0 0
$$881$$ 27.0000 0.909653 0.454827 0.890580i $$-0.349701\pi$$
0.454827 + 0.890580i $$0.349701\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 0 0
$$885$$ −3.00000 −0.100844
$$886$$ 0 0
$$887$$ 9.00000 0.302190 0.151095 0.988519i $$-0.451720\pi$$
0.151095 + 0.988519i $$0.451720\pi$$
$$888$$ 0 0
$$889$$ −19.0000 −0.637240
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 21.0000 0.701953
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ 0 0
$$903$$ 11.0000 0.366057
$$904$$ 0 0
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ −19.0000 −0.630885 −0.315442 0.948945i $$-0.602153\pi$$
−0.315442 + 0.948945i $$0.602153\pi$$
$$908$$ 0 0
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ −11.0000 −0.363649
$$916$$ 0 0
$$917$$ −12.0000 −0.396275
$$918$$ 0 0
$$919$$ 29.0000 0.956622 0.478311 0.878191i $$-0.341249\pi$$
0.478311 + 0.878191i $$0.341249\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 7.00000 0.230159
$$926$$ 0 0
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ −51.0000 −1.67326 −0.836628 0.547772i $$-0.815476\pi$$
−0.836628 + 0.547772i $$0.815476\pi$$
$$930$$ 0 0
$$931$$ −42.0000 −1.37649
$$932$$ 0 0
$$933$$ 24.0000 0.785725
$$934$$ 0 0
$$935$$ −9.00000 −0.294331
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −22.0000 −0.717943
$$940$$ 0 0
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 0 0
$$943$$ −27.0000 −0.879241
$$944$$ 0 0
$$945$$ 5.00000 0.162650
$$946$$ 0 0
$$947$$ −21.0000 −0.682408 −0.341204 0.939989i $$-0.610835\pi$$
−0.341204 + 0.939989i $$0.610835\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −51.0000 −1.65205 −0.826026 0.563632i $$-0.809404\pi$$
−0.826026 + 0.563632i $$0.809404\pi$$
$$954$$ 0 0
$$955$$ 3.00000 0.0970777
$$956$$ 0 0
$$957$$ −9.00000 −0.290929
$$958$$ 0 0
$$959$$ 15.0000 0.484375
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ 5.00000 0.160956
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ 0 0
$$969$$ −21.0000 −0.674617
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ 5.00000 0.160293
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −9.00000 −0.287936 −0.143968 0.989582i $$-0.545986\pi$$
−0.143968 + 0.989582i $$0.545986\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 36.0000 1.14822 0.574111 0.818778i $$-0.305348\pi$$
0.574111 + 0.818778i $$0.305348\pi$$
$$984$$ 0 0
$$985$$ 21.0000 0.669116
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −33.0000 −1.04934
$$990$$ 0 0
$$991$$ −61.0000 −1.93773 −0.968864 0.247592i $$-0.920361\pi$$
−0.968864 + 0.247592i $$0.920361\pi$$
$$992$$ 0 0
$$993$$ 19.0000 0.602947
$$994$$ 0 0
$$995$$ −17.0000 −0.538936
$$996$$ 0 0
$$997$$ −31.0000 −0.981780 −0.490890 0.871222i $$-0.663328\pi$$
−0.490890 + 0.871222i $$0.663328\pi$$
$$998$$ 0 0
$$999$$ −35.0000 −1.10735
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 3380.2.a.g.1.1 1
13.4 even 6 260.2.i.b.81.1 yes 2
13.5 odd 4 3380.2.f.e.3041.2 2
13.8 odd 4 3380.2.f.e.3041.1 2
13.10 even 6 260.2.i.b.61.1 2
13.12 even 2 3380.2.a.h.1.1 1
39.17 odd 6 2340.2.q.b.2161.1 2
39.23 odd 6 2340.2.q.b.1621.1 2
52.23 odd 6 1040.2.q.j.321.1 2
52.43 odd 6 1040.2.q.j.81.1 2
65.4 even 6 1300.2.i.e.601.1 2
65.17 odd 12 1300.2.bb.a.549.1 4
65.23 odd 12 1300.2.bb.a.1049.1 4
65.43 odd 12 1300.2.bb.a.549.2 4
65.49 even 6 1300.2.i.e.1101.1 2
65.62 odd 12 1300.2.bb.a.1049.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.b.61.1 2 13.10 even 6
260.2.i.b.81.1 yes 2 13.4 even 6
1040.2.q.j.81.1 2 52.43 odd 6
1040.2.q.j.321.1 2 52.23 odd 6
1300.2.i.e.601.1 2 65.4 even 6
1300.2.i.e.1101.1 2 65.49 even 6
1300.2.bb.a.549.1 4 65.17 odd 12
1300.2.bb.a.549.2 4 65.43 odd 12
1300.2.bb.a.1049.1 4 65.23 odd 12
1300.2.bb.a.1049.2 4 65.62 odd 12
2340.2.q.b.1621.1 2 39.23 odd 6
2340.2.q.b.2161.1 2 39.17 odd 6
3380.2.a.g.1.1 1 1.1 even 1 trivial
3380.2.a.h.1.1 1 13.12 even 2
3380.2.f.e.3041.1 2 13.8 odd 4
3380.2.f.e.3041.2 2 13.5 odd 4