# Properties

 Label 3380.2.a.f.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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{15} -3.00000 q^{17} +5.00000 q^{19} -1.00000 q^{21} +9.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -9.00000 q^{29} +8.00000 q^{31} +3.00000 q^{33} +1.00000 q^{35} -7.00000 q^{37} +3.00000 q^{41} -1.00000 q^{43} +2.00000 q^{45} -6.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} +5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{61} +2.00000 q^{63} +5.00000 q^{67} +9.00000 q^{69} +9.00000 q^{71} +2.00000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +3.00000 q^{85} -9.00000 q^{87} +3.00000 q^{89} +8.00000 q^{93} -5.00000 q^{95} +17.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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 $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 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
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 9.00000 1.87663 0.938315 0.345782i $$-0.112386\pi$$
0.938315 + 0.345782i $$0.112386\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 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 $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 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 $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 5.00000 0.662266
$$58$$ 0 0
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ 9.00000 1.08347
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 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
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −5.00000 −0.512989
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 15.0000 1.49256 0.746278 0.665635i $$-0.231839\pi$$
0.746278 + 0.665635i $$0.231839\pi$$
$$102$$ 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
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −7.00000 −0.664411
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 3.00000 0.270501
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 17.0000 1.50851 0.754253 0.656584i $$-0.227999\pi$$
0.754253 + 0.656584i $$0.227999\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −5.00000 −0.433555
$$134$$ 0 0
$$135$$ 5.00000 0.430331
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 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
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ −9.00000 −0.737309 −0.368654 0.929567i $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −9.00000 −0.709299
$$162$$ 0 0
$$163$$ −1.00000 −0.0783260 −0.0391630 0.999233i $$-0.512469\pi$$
−0.0391630 + 0.999233i $$0.512469\pi$$
$$164$$ 0 0
$$165$$ −3.00000 −0.233550
$$166$$ 0 0
$$167$$ −15.0000 −1.16073 −0.580367 0.814355i $$-0.697091\pi$$
−0.580367 + 0.814355i $$0.697091\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −10.0000 −0.764719
$$172$$ 0 0
$$173$$ 21.0000 1.59660 0.798300 0.602260i $$-0.205733\pi$$
0.798300 + 0.602260i $$0.205733\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ 0 0
$$179$$ 15.0000 1.12115 0.560576 0.828103i $$-0.310580\pi$$
0.560576 + 0.828103i $$0.310580\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 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 $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 0 0
$$203$$ 9.00000 0.631676
$$204$$ 0 0
$$205$$ −3.00000 −0.209529
$$206$$ 0 0
$$207$$ −18.0000 −1.25109
$$208$$ 0 0
$$209$$ 15.0000 1.03757
$$210$$ 0 0
$$211$$ −25.0000 −1.72107 −0.860535 0.509390i $$-0.829871\pi$$
−0.860535 + 0.509390i $$0.829871\pi$$
$$212$$ 0 0
$$213$$ 9.00000 0.616670
$$214$$ 0 0
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 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 $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 15.0000 0.995585 0.497792 0.867296i $$-0.334144\pi$$
0.497792 + 0.867296i $$0.334144\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 23.0000 1.48156 0.740780 0.671748i $$-0.234456\pi$$
0.740780 + 0.671748i $$0.234456\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 9.00000 0.568075 0.284037 0.958813i $$-0.408326\pi$$
0.284037 + 0.958813i $$0.408326\pi$$
$$252$$ 0 0
$$253$$ 27.0000 1.69748
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ −27.0000 −1.68421 −0.842107 0.539311i $$-0.818685\pi$$
−0.842107 + 0.539311i $$0.818685\pi$$
$$258$$ 0 0
$$259$$ 7.00000 0.434959
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ 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
$$267$$ 3.00000 0.183597
$$268$$ 0 0
$$269$$ −21.0000 −1.28039 −0.640196 0.768211i $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 0 0
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ 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
$$279$$ −16.0000 −0.957895
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 17.0000 1.01055 0.505273 0.862960i $$-0.331392\pi$$
0.505273 + 0.862960i $$0.331392\pi$$
$$284$$ 0 0
$$285$$ −5.00000 −0.296174
$$286$$ 0 0
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 17.0000 0.996558
$$292$$ 0 0
$$293$$ −3.00000 −0.175262 −0.0876309 0.996153i $$-0.527930\pi$$
−0.0876309 + 0.996153i $$0.527930\pi$$
$$294$$ 0 0
$$295$$ −9.00000 −0.524000
$$296$$ 0 0
$$297$$ −15.0000 −0.870388
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 15.0000 0.861727
$$304$$ 0 0
$$305$$ 1.00000 0.0572598
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −27.0000 −1.51171
$$320$$ 0 0
$$321$$ −3.00000 −0.167444
$$322$$ 0 0
$$323$$ −15.0000 −0.834622
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 14.0000 0.774202
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 0 0
$$333$$ 14.0000 0.767195
$$334$$ 0 0
$$335$$ −5.00000 −0.273179
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ −15.0000 −0.814688
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ −9.00000 −0.484544
$$346$$ 0 0
$$347$$ −9.00000 −0.483145 −0.241573 0.970383i $$-0.577663\pi$$
−0.241573 + 0.970383i $$0.577663\pi$$
$$348$$ 0 0
$$349$$ 35.0000 1.87351 0.936754 0.349990i $$-0.113815\pi$$
0.936754 + 0.349990i $$0.113815\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −27.0000 −1.43706 −0.718532 0.695493i $$-0.755186\pi$$
−0.718532 + 0.695493i $$0.755186\pi$$
$$354$$ 0 0
$$355$$ −9.00000 −0.477670
$$356$$ 0 0
$$357$$ 3.00000 0.158777
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 5.00000 0.258890 0.129445 0.991587i $$-0.458680\pi$$
0.129445 + 0.991587i $$0.458680\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 11.0000 0.565032 0.282516 0.959263i $$-0.408831\pi$$
0.282516 + 0.959263i $$0.408831\pi$$
$$380$$ 0 0
$$381$$ 17.0000 0.870936
$$382$$ 0 0
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ 0 0
$$387$$ 2.00000 0.101666
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −27.0000 −1.36545
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ 29.0000 1.45547 0.727734 0.685859i $$-0.240573\pi$$
0.727734 + 0.685859i $$0.240573\pi$$
$$398$$ 0 0
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 15.0000 0.749064 0.374532 0.927214i $$-0.377803\pi$$
0.374532 + 0.927214i $$0.377803\pi$$
$$402$$ 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 $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ −3.00000 −0.147979
$$412$$ 0 0
$$413$$ −9.00000 −0.442861
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ −21.0000 −1.02592 −0.512959 0.858413i $$-0.671451\pi$$
−0.512959 + 0.858413i $$0.671451\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ 1.00000 0.0483934
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ 5.00000 0.240285 0.120142 0.992757i $$-0.461665\pi$$
0.120142 + 0.992757i $$0.461665\pi$$
$$434$$ 0 0
$$435$$ 9.00000 0.431517
$$436$$ 0 0
$$437$$ 45.0000 2.15264
$$438$$ 0 0
$$439$$ −1.00000 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −3.00000 −0.142214
$$446$$ 0 0
$$447$$ −9.00000 −0.425685
$$448$$ 0 0
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 9.00000 0.423793
$$452$$ 0 0
$$453$$ 20.0000 0.939682
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 41.0000 1.91790 0.958950 0.283577i $$-0.0915211\pi$$
0.958950 + 0.283577i $$0.0915211\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 15.0000 0.698620 0.349310 0.937007i $$-0.386416\pi$$
0.349310 + 0.937007i $$0.386416\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 0 0
$$475$$ 5.00000 0.229416
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −9.00000 −0.409514
$$484$$ 0 0
$$485$$ −17.0000 −0.771930
$$486$$ 0 0
$$487$$ 35.0000 1.58600 0.793001 0.609221i $$-0.208518\pi$$
0.793001 + 0.609221i $$0.208518\pi$$
$$488$$ 0 0
$$489$$ −1.00000 −0.0452216
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ 0 0
$$495$$ 6.00000 0.269680
$$496$$ 0 0
$$497$$ −9.00000 −0.403705
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −15.0000 −0.670151
$$502$$ 0 0
$$503$$ 39.0000 1.73892 0.869462 0.494000i $$-0.164466\pi$$
0.869462 + 0.494000i $$0.164466\pi$$
$$504$$ 0 0
$$505$$ −15.0000 −0.667491
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −33.0000 −1.46270 −0.731350 0.682003i $$-0.761109\pi$$
−0.731350 + 0.682003i $$0.761109\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 0 0
$$513$$ −25.0000 −1.10378
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 21.0000 0.921798
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 5.00000 0.218635 0.109317 0.994007i $$-0.465134\pi$$
0.109317 + 0.994007i $$0.465134\pi$$
$$524$$ 0 0
$$525$$ −1.00000 −0.0436436
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ 58.0000 2.52174
$$530$$ 0 0
$$531$$ −18.0000 −0.781133
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 3.00000 0.129701
$$536$$ 0 0
$$537$$ 15.0000 0.647298
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ −10.0000 −0.429141
$$544$$ 0 0
$$545$$ −14.0000 −0.599694
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −45.0000 −1.91706
$$552$$ 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 $$-0.809525\pi$$
−0.826242 + 0.563316i $$0.809525\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ −33.0000 −1.39078 −0.695392 0.718631i $$-0.744769\pi$$
−0.695392 + 0.718631i $$0.744769\pi$$
$$564$$ 0 0
$$565$$ 15.0000 0.631055
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −33.0000 −1.38343 −0.691716 0.722170i $$-0.743145\pi$$
−0.691716 + 0.722170i $$0.743145\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ −3.00000 −0.125327
$$574$$ 0 0
$$575$$ 9.00000 0.375326
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 0 0
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −27.0000 −1.11441 −0.557205 0.830375i $$-0.688126\pi$$
−0.557205 + 0.830375i $$0.688126\pi$$
$$588$$ 0 0
$$589$$ 40.0000 1.64817
$$590$$ 0 0
$$591$$ −3.00000 −0.123404
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ −3.00000 −0.122988
$$596$$ 0 0
$$597$$ −7.00000 −0.286491
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −0.0407909 −0.0203954 0.999792i $$-0.506493\pi$$
−0.0203954 + 0.999792i $$0.506493\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ 0 0
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 29.0000 1.17130 0.585649 0.810564i $$-0.300840\pi$$
0.585649 + 0.810564i $$0.300840\pi$$
$$614$$ 0 0
$$615$$ −3.00000 −0.120972
$$616$$ 0 0
$$617$$ −27.0000 −1.08698 −0.543490 0.839416i $$-0.682897\pi$$
−0.543490 + 0.839416i $$0.682897\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ −45.0000 −1.80579
$$622$$ 0 0
$$623$$ −3.00000 −0.120192
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 15.0000 0.599042
$$628$$ 0 0
$$629$$ 21.0000 0.837325
$$630$$ 0 0
$$631$$ −7.00000 −0.278666 −0.139333 0.990246i $$-0.544496\pi$$
−0.139333 + 0.990246i $$0.544496\pi$$
$$632$$ 0 0
$$633$$ −25.0000 −0.993661
$$634$$ 0 0
$$635$$ −17.0000 −0.674624
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −18.0000 −0.712069
$$640$$ 0 0
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ −25.0000 −0.985904 −0.492952 0.870057i $$-0.664082\pi$$
−0.492952 + 0.870057i $$0.664082\pi$$
$$644$$ 0 0
$$645$$ 1.00000 0.0393750
$$646$$ 0 0
$$647$$ −27.0000 −1.06148 −0.530740 0.847535i $$-0.678086\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 0 0
$$649$$ 27.0000 1.05984
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 0 0
$$653$$ −27.0000 −1.05659 −0.528296 0.849060i $$-0.677169\pi$$
−0.528296 + 0.849060i $$0.677169\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 45.0000 1.75295 0.876476 0.481446i $$-0.159888\pi$$
0.876476 + 0.481446i $$0.159888\pi$$
$$660$$ 0 0
$$661$$ 35.0000 1.36134 0.680671 0.732589i $$-0.261688\pi$$
0.680671 + 0.732589i $$0.261688\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 5.00000 0.193892
$$666$$ 0 0
$$667$$ −81.0000 −3.13633
$$668$$ 0 0
$$669$$ −19.0000 −0.734582
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −17.0000 −0.652400
$$680$$ 0 0
$$681$$ 15.0000 0.574801
$$682$$ 0 0
$$683$$ −21.0000 −0.803543 −0.401771 0.915740i $$-0.631605\pi$$
−0.401771 + 0.915740i $$0.631605\pi$$
$$684$$ 0 0
$$685$$ 3.00000 0.114624
$$686$$ 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 $$-0.355871\pi$$
0.437481 + 0.899228i $$0.355871\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ −9.00000 −0.340899
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ −35.0000 −1.32005
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15.0000 −0.564133
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ 72.0000 2.69642
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 45.0000 1.67822 0.839108 0.543964i $$-0.183077\pi$$
0.839108 + 0.543964i $$0.183077\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 23.0000 0.855379
$$724$$ 0 0
$$725$$ −9.00000 −0.334252
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ 15.0000 0.552532
$$738$$ 0 0
$$739$$ 11.0000 0.404642 0.202321 0.979319i $$-0.435152\pi$$
0.202321 + 0.979319i $$0.435152\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 33.0000 1.21065 0.605326 0.795977i $$-0.293043\pi$$
0.605326 + 0.795977i $$0.293043\pi$$
$$744$$ 0 0
$$745$$ 9.00000 0.329734
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3.00000 0.109618
$$750$$ 0 0
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ 0 0
$$753$$ 9.00000 0.327978
$$754$$ 0 0
$$755$$ −20.0000 −0.727875
$$756$$ 0 0
$$757$$ 5.00000 0.181728 0.0908640 0.995863i $$-0.471037\pi$$
0.0908640 + 0.995863i $$0.471037\pi$$
$$758$$ 0 0
$$759$$ 27.0000 0.980038
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ −27.0000 −0.972381
$$772$$ 0 0
$$773$$ −15.0000 −0.539513 −0.269756 0.962929i $$-0.586943\pi$$
−0.269756 + 0.962929i $$0.586943\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ 0 0
$$777$$ 7.00000 0.251124
$$778$$ 0 0
$$779$$ 15.0000 0.537431
$$780$$ 0 0
$$781$$ 27.0000 0.966136
$$782$$ 0 0
$$783$$ 45.0000 1.60817
$$784$$ 0 0
$$785$$ −14.0000 −0.499681
$$786$$ 0 0
$$787$$ −49.0000 −1.74666 −0.873331 0.487128i $$-0.838045\pi$$
−0.873331 + 0.487128i $$0.838045\pi$$
$$788$$ 0 0
$$789$$ −3.00000 −0.106803
$$790$$ 0 0
$$791$$ 15.0000 0.533339
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ −27.0000 −0.956389 −0.478195 0.878254i $$-0.658709\pi$$
−0.478195 + 0.878254i $$0.658709\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 9.00000 0.317208
$$806$$ 0 0
$$807$$ −21.0000 −0.739235
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ −13.0000 −0.455930
$$814$$ 0 0
$$815$$ 1.00000 0.0350285
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −21.0000 −0.732905 −0.366453 0.930437i $$-0.619428\pi$$
−0.366453 + 0.930437i $$0.619428\pi$$
$$822$$ 0 0
$$823$$ 47.0000 1.63832 0.819159 0.573567i $$-0.194441\pi$$
0.819159 + 0.573567i $$0.194441\pi$$
$$824$$ 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
$$835$$ 15.0000 0.519096
$$836$$ 0 0
$$837$$ −40.0000 −1.38260
$$838$$ 0 0
$$839$$ −15.0000 −0.517858 −0.258929 0.965896i $$-0.583369\pi$$
−0.258929 + 0.965896i $$0.583369\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ −18.0000 −0.619953
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 17.0000 0.583438
$$850$$ 0 0
$$851$$ −63.0000 −2.15961
$$852$$ 0 0
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 0 0
$$855$$ 10.0000 0.341993
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ 0 0
$$861$$ −3.00000 −0.102240
$$862$$ 0 0
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ −21.0000 −0.714021
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −34.0000 −1.15073
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ 0 0
$$877$$ 17.0000 0.574049 0.287025 0.957923i $$-0.407334\pi$$
0.287025 + 0.957923i $$0.407334\pi$$
$$878$$ 0 0
$$879$$ −3.00000 −0.101187
$$880$$ 0 0
$$881$$ 3.00000 0.101073 0.0505363 0.998722i $$-0.483907\pi$$
0.0505363 + 0.998722i $$0.483907\pi$$
$$882$$ 0 0
$$883$$ 56.0000 1.88455 0.942275 0.334840i $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 0 0
$$885$$ −9.00000 −0.302532
$$886$$ 0 0
$$887$$ 33.0000 1.10803 0.554016 0.832506i $$-0.313095\pi$$
0.554016 + 0.832506i $$0.313095\pi$$
$$888$$ 0 0
$$889$$ −17.0000 −0.570162
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −15.0000 −0.501395
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −72.0000 −2.40133
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 0 0
$$903$$ 1.00000 0.0332779
$$904$$ 0 0
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −55.0000 −1.82625 −0.913123 0.407685i $$-0.866336\pi$$
−0.913123 + 0.407685i $$0.866336\pi$$
$$908$$ 0 0
$$909$$ −30.0000 −0.995037
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 1.00000 0.0330590
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −7.00000 −0.230909 −0.115454 0.993313i $$-0.536832\pi$$
−0.115454 + 0.993313i $$0.536832\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −7.00000 −0.230159
$$926$$ 0 0
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ 3.00000 0.0984268 0.0492134 0.998788i $$-0.484329\pi$$
0.0492134 + 0.998788i $$0.484329\pi$$
$$930$$ 0 0
$$931$$ −30.0000 −0.983210
$$932$$ 0 0
$$933$$ 12.0000 0.392862
$$934$$ 0 0
$$935$$ 9.00000 0.294331
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ 27.0000 0.879241
$$944$$ 0 0
$$945$$ −5.00000 −0.162650
$$946$$ 0 0
$$947$$ −51.0000 −1.65728 −0.828639 0.559784i $$-0.810884\pi$$
−0.828639 + 0.559784i $$0.810884\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ 21.0000 0.680257 0.340128 0.940379i $$-0.389529\pi$$
0.340128 + 0.940379i $$0.389529\pi$$
$$954$$ 0 0
$$955$$ 3.00000 0.0970777
$$956$$ 0 0
$$957$$ −27.0000 −0.872786
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 6.00000 0.193347
$$964$$ 0 0
$$965$$ −5.00000 −0.160956
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 0 0
$$969$$ −15.0000 −0.481869
$$970$$ 0 0
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ −5.00000 −0.160293
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ 0 0
$$979$$ 9.00000 0.287641
$$980$$ 0 0
$$981$$ −28.0000 −0.893971
$$982$$ 0 0
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 0 0
$$985$$ 3.00000 0.0955879
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.00000 −0.286183
$$990$$ 0 0
$$991$$ −49.0000 −1.55654 −0.778268 0.627932i $$-0.783902\pi$$
−0.778268 + 0.627932i $$0.783902\pi$$
$$992$$ 0 0
$$993$$ −19.0000 −0.602947
$$994$$ 0 0
$$995$$ 7.00000 0.221915
$$996$$ 0 0
$$997$$ 5.00000 0.158352 0.0791758 0.996861i $$-0.474771\pi$$
0.0791758 + 0.996861i $$0.474771\pi$$
$$998$$ 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.f.1.1 1
13.3 even 3 260.2.i.a.61.1 2
13.5 odd 4 3380.2.f.d.3041.2 2
13.8 odd 4 3380.2.f.d.3041.1 2
13.9 even 3 260.2.i.a.81.1 yes 2
13.12 even 2 3380.2.a.i.1.1 1
39.29 odd 6 2340.2.q.f.1621.1 2
39.35 odd 6 2340.2.q.f.2161.1 2
52.3 odd 6 1040.2.q.i.321.1 2
52.35 odd 6 1040.2.q.i.81.1 2
65.3 odd 12 1300.2.bb.b.1049.1 4
65.9 even 6 1300.2.i.d.601.1 2
65.22 odd 12 1300.2.bb.b.549.1 4
65.29 even 6 1300.2.i.d.1101.1 2
65.42 odd 12 1300.2.bb.b.1049.2 4
65.48 odd 12 1300.2.bb.b.549.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.a.61.1 2 13.3 even 3
260.2.i.a.81.1 yes 2 13.9 even 3
1040.2.q.i.81.1 2 52.35 odd 6
1040.2.q.i.321.1 2 52.3 odd 6
1300.2.i.d.601.1 2 65.9 even 6
1300.2.i.d.1101.1 2 65.29 even 6
1300.2.bb.b.549.1 4 65.22 odd 12
1300.2.bb.b.549.2 4 65.48 odd 12
1300.2.bb.b.1049.1 4 65.3 odd 12
1300.2.bb.b.1049.2 4 65.42 odd 12
2340.2.q.f.1621.1 2 39.29 odd 6
2340.2.q.f.2161.1 2 39.35 odd 6
3380.2.a.f.1.1 1 1.1 even 1 trivial
3380.2.a.i.1.1 1 13.12 even 2
3380.2.f.d.3041.1 2 13.8 odd 4
3380.2.f.d.3041.2 2 13.5 odd 4