# Properties

 Label 3380.2.a.e.1.1 Level $3380$ Weight $2$ Character 3380.1 Self dual yes Analytic conductor $26.989$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +5.00000 q^{11} -1.00000 q^{15} -1.00000 q^{17} +3.00000 q^{19} +5.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} -5.00000 q^{33} -5.00000 q^{35} -7.00000 q^{37} +5.00000 q^{41} +5.00000 q^{43} -2.00000 q^{45} -12.0000 q^{47} +18.0000 q^{49} +1.00000 q^{51} +2.00000 q^{53} +5.00000 q^{55} -3.00000 q^{57} +11.0000 q^{59} -13.0000 q^{61} +10.0000 q^{63} -3.00000 q^{67} -3.00000 q^{69} -13.0000 q^{71} +2.00000 q^{73} -1.00000 q^{75} -25.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -1.00000 q^{85} +1.00000 q^{87} -7.00000 q^{89} +3.00000 q^{95} -11.0000 q^{97} -10.0000 q^{99} +O(q^{100})$$

## 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 $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −5.00000 −0.870388
$$34$$ 0 0
$$35$$ −5.00000 −0.845154
$$36$$ 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
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ 0 0
$$57$$ −3.00000 −0.397360
$$58$$ 0 0
$$59$$ 11.0000 1.43208 0.716039 0.698060i $$-0.245953\pi$$
0.716039 + 0.698060i $$0.245953\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 0 0
$$63$$ 10.0000 1.25988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ −13.0000 −1.54282 −0.771408 0.636341i $$-0.780447\pi$$
−0.771408 + 0.636341i $$0.780447\pi$$
$$72$$ 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
$$77$$ −25.0000 −2.84901
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 0 0
$$87$$ 1.00000 0.107211
$$88$$ 0 0
$$89$$ −7.00000 −0.741999 −0.370999 0.928633i $$-0.620985\pi$$
−0.370999 + 0.928633i $$0.620985\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 3.00000 0.307794
$$96$$ 0 0
$$97$$ −11.0000 −1.11688 −0.558440 0.829545i $$-0.688600\pi$$
−0.558440 + 0.829545i $$0.688600\pi$$
$$98$$ 0 0
$$99$$ −10.0000 −1.00504
$$100$$ 0 0
$$101$$ −13.0000 −1.29355 −0.646774 0.762682i $$-0.723882\pi$$
−0.646774 + 0.762682i $$0.723882\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 5.00000 0.487950
$$106$$ 0 0
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ 0 0
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −5.00000 −0.450835
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 7.00000 0.621150 0.310575 0.950549i $$-0.399478\pi$$
0.310575 + 0.950549i $$0.399478\pi$$
$$128$$ 0 0
$$129$$ −5.00000 −0.440225
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ −15.0000 −1.30066
$$134$$ 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
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ −11.0000 −0.901155 −0.450578 0.892737i $$-0.648782\pi$$
−0.450578 + 0.892737i $$0.648782\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −15.0000 −1.18217
$$162$$ 0 0
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ 0 0
$$165$$ −5.00000 −0.389249
$$166$$ 0 0
$$167$$ 13.0000 1.00597 0.502985 0.864295i $$-0.332235\pi$$
0.502985 + 0.864295i $$0.332235\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ 0 0
$$173$$ −17.0000 −1.29249 −0.646243 0.763132i $$-0.723661\pi$$
−0.646243 + 0.763132i $$0.723661\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 0 0
$$177$$ −11.0000 −0.826811
$$178$$ 0 0
$$179$$ 11.0000 0.822179 0.411089 0.911595i $$-0.365148\pi$$
0.411089 + 0.911595i $$0.365148\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 13.0000 0.960988
$$184$$ 0 0
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ −5.00000 −0.365636
$$188$$ 0 0
$$189$$ −25.0000 −1.81848
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ −23.0000 −1.65558 −0.827788 0.561041i $$-0.810401\pi$$
−0.827788 + 0.561041i $$0.810401\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −27.0000 −1.92367 −0.961835 0.273629i $$-0.911776\pi$$
−0.961835 + 0.273629i $$0.911776\pi$$
$$198$$ 0 0
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ 5.00000 0.349215
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 15.0000 1.03757
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 0 0
$$213$$ 13.0000 0.890745
$$214$$ 0 0
$$215$$ 5.00000 0.340997
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 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
$$227$$ −17.0000 −1.12833 −0.564165 0.825662i $$-0.690802\pi$$
−0.564165 + 0.825662i $$0.690802\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 25.0000 1.64488
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 0 0
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 18.0000 1.14998
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 15.0000 0.943042
$$254$$ 0 0
$$255$$ 1.00000 0.0626224
$$256$$ 0 0
$$257$$ 15.0000 0.935674 0.467837 0.883815i $$-0.345033\pi$$
0.467837 + 0.883815i $$0.345033\pi$$
$$258$$ 0 0
$$259$$ 35.0000 2.17479
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 11.0000 0.678289 0.339145 0.940734i $$-0.389862\pi$$
0.339145 + 0.940734i $$0.389862\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ 7.00000 0.428393
$$268$$ 0 0
$$269$$ −9.00000 −0.548740 −0.274370 0.961624i $$-0.588469\pi$$
−0.274370 + 0.961624i $$0.588469\pi$$
$$270$$ 0 0
$$271$$ −7.00000 −0.425220 −0.212610 0.977137i $$-0.568196\pi$$
−0.212610 + 0.977137i $$0.568196\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ −13.0000 −0.781094 −0.390547 0.920583i $$-0.627714\pi$$
−0.390547 + 0.920583i $$0.627714\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 23.0000 1.36721 0.683604 0.729853i $$-0.260412\pi$$
0.683604 + 0.729853i $$0.260412\pi$$
$$284$$ 0 0
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ −25.0000 −1.47570
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 11.0000 0.644831
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 11.0000 0.640445
$$296$$ 0 0
$$297$$ 25.0000 1.45065
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −25.0000 −1.44098
$$302$$ 0 0
$$303$$ 13.0000 0.746830
$$304$$ 0 0
$$305$$ −13.0000 −0.744378
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 10.0000 0.563436
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −5.00000 −0.279946
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 18.0000 0.995402
$$328$$ 0 0
$$329$$ 60.0000 3.30791
$$330$$ 0 0
$$331$$ −1.00000 −0.0549650 −0.0274825 0.999622i $$-0.508749\pi$$
−0.0274825 + 0.999622i $$0.508749\pi$$
$$332$$ 0 0
$$333$$ 14.0000 0.767195
$$334$$ 0 0
$$335$$ −3.00000 −0.163908
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 1.00000 0.0543125
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −55.0000 −2.96972
$$344$$ 0 0
$$345$$ −3.00000 −0.161515
$$346$$ 0 0
$$347$$ −27.0000 −1.44944 −0.724718 0.689046i $$-0.758030\pi$$
−0.724718 + 0.689046i $$0.758030\pi$$
$$348$$ 0 0
$$349$$ −35.0000 −1.87351 −0.936754 0.349990i $$-0.886185\pi$$
−0.936754 + 0.349990i $$0.886185\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 5.00000 0.266123 0.133062 0.991108i $$-0.457519\pi$$
0.133062 + 0.991108i $$0.457519\pi$$
$$354$$ 0 0
$$355$$ −13.0000 −0.689968
$$356$$ 0 0
$$357$$ −5.00000 −0.264628
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 0 0
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 3.00000 0.156599 0.0782994 0.996930i $$-0.475051\pi$$
0.0782994 + 0.996930i $$0.475051\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −10.0000 −0.519174
$$372$$ 0 0
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 21.0000 1.07870 0.539349 0.842082i $$-0.318670\pi$$
0.539349 + 0.842082i $$0.318670\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ 0 0
$$383$$ 3.00000 0.153293 0.0766464 0.997058i $$-0.475579\pi$$
0.0766464 + 0.997058i $$0.475579\pi$$
$$384$$ 0 0
$$385$$ −25.0000 −1.27412
$$386$$ 0 0
$$387$$ −10.0000 −0.508329
$$388$$ 0 0
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 13.0000 0.652451 0.326226 0.945292i $$-0.394223\pi$$
0.326226 + 0.945292i $$0.394223\pi$$
$$398$$ 0 0
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −35.0000 −1.73489
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ 0 0
$$413$$ −55.0000 −2.70637
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −13.0000 −0.636613
$$418$$ 0 0
$$419$$ −17.0000 −0.830504 −0.415252 0.909706i $$-0.636307\pi$$
−0.415252 + 0.909706i $$0.636307\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 24.0000 1.16692
$$424$$ 0 0
$$425$$ −1.00000 −0.0485071
$$426$$ 0 0
$$427$$ 65.0000 3.14557
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ 0 0
$$433$$ 7.00000 0.336399 0.168199 0.985753i $$-0.446205\pi$$
0.168199 + 0.985753i $$0.446205\pi$$
$$434$$ 0 0
$$435$$ 1.00000 0.0479463
$$436$$ 0 0
$$437$$ 9.00000 0.430528
$$438$$ 0 0
$$439$$ −29.0000 −1.38409 −0.692047 0.721852i $$-0.743291\pi$$
−0.692047 + 0.721852i $$0.743291\pi$$
$$440$$ 0 0
$$441$$ −36.0000 −1.71429
$$442$$ 0 0
$$443$$ −20.0000 −0.950229 −0.475114 0.879924i $$-0.657593\pi$$
−0.475114 + 0.879924i $$0.657593\pi$$
$$444$$ 0 0
$$445$$ −7.00000 −0.331832
$$446$$ 0 0
$$447$$ 11.0000 0.520282
$$448$$ 0 0
$$449$$ 21.0000 0.991051 0.495526 0.868593i $$-0.334975\pi$$
0.495526 + 0.868593i $$0.334975\pi$$
$$450$$ 0 0
$$451$$ 25.0000 1.17720
$$452$$ 0 0
$$453$$ −24.0000 −1.12762
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ 0 0
$$459$$ −5.00000 −0.233380
$$460$$ 0 0
$$461$$ 33.0000 1.53696 0.768482 0.639872i $$-0.221013\pi$$
0.768482 + 0.639872i $$0.221013\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ 15.0000 0.692636
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ 25.0000 1.14950
$$474$$ 0 0
$$475$$ 3.00000 0.137649
$$476$$ 0 0
$$477$$ −4.00000 −0.183147
$$478$$ 0 0
$$479$$ −11.0000 −0.502603 −0.251301 0.967909i $$-0.580859\pi$$
−0.251301 + 0.967909i $$0.580859\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 15.0000 0.682524
$$484$$ 0 0
$$485$$ −11.0000 −0.499484
$$486$$ 0 0
$$487$$ −17.0000 −0.770344 −0.385172 0.922845i $$-0.625858\pi$$
−0.385172 + 0.922845i $$0.625858\pi$$
$$488$$ 0 0
$$489$$ 5.00000 0.226108
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ 0 0
$$493$$ 1.00000 0.0450377
$$494$$ 0 0
$$495$$ −10.0000 −0.449467
$$496$$ 0 0
$$497$$ 65.0000 2.91565
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −13.0000 −0.580797
$$502$$ 0 0
$$503$$ −11.0000 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$504$$ 0 0
$$505$$ −13.0000 −0.578492
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −3.00000 −0.132973 −0.0664863 0.997787i $$-0.521179\pi$$
−0.0664863 + 0.997787i $$0.521179\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ 0 0
$$513$$ 15.0000 0.662266
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −60.0000 −2.63880
$$518$$ 0 0
$$519$$ 17.0000 0.746217
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ −21.0000 −0.918266 −0.459133 0.888368i $$-0.651840\pi$$
−0.459133 + 0.888368i $$0.651840\pi$$
$$524$$ 0 0
$$525$$ 5.00000 0.218218
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −22.0000 −0.954719
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ 0 0
$$537$$ −11.0000 −0.474685
$$538$$ 0 0
$$539$$ 90.0000 3.87657
$$540$$ 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
$$545$$ −18.0000 −0.771035
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 0 0
$$549$$ 26.0000 1.10965
$$550$$ 0 0
$$551$$ −3.00000 −0.127804
$$552$$ 0 0
$$553$$ 20.0000 0.850487
$$554$$ 0 0
$$555$$ 7.00000 0.297133
$$556$$ 0 0
$$557$$ 13.0000 0.550828 0.275414 0.961326i $$-0.411185\pi$$
0.275414 + 0.961326i $$0.411185\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ 0 0
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ 0 0
$$565$$ −1.00000 −0.0420703
$$566$$ 0 0
$$567$$ −5.00000 −0.209980
$$568$$ 0 0
$$569$$ 39.0000 1.63497 0.817483 0.575953i $$-0.195369\pi$$
0.817483 + 0.575953i $$0.195369\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 15.0000 0.626634
$$574$$ 0 0
$$575$$ 3.00000 0.125109
$$576$$ 0 0
$$577$$ 42.0000 1.74848 0.874241 0.485491i $$-0.161359\pi$$
0.874241 + 0.485491i $$0.161359\pi$$
$$578$$ 0 0
$$579$$ 23.0000 0.955847
$$580$$ 0 0
$$581$$ 60.0000 2.48922
$$582$$ 0 0
$$583$$ 10.0000 0.414158
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.00000 −0.123823 −0.0619116 0.998082i $$-0.519720\pi$$
−0.0619116 + 0.998082i $$0.519720\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 27.0000 1.11063
$$592$$ 0 0
$$593$$ 2.00000 0.0821302 0.0410651 0.999156i $$-0.486925\pi$$
0.0410651 + 0.999156i $$0.486925\pi$$
$$594$$ 0 0
$$595$$ 5.00000 0.204980
$$596$$ 0 0
$$597$$ −21.0000 −0.859473
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 0 0
$$603$$ 6.00000 0.244339
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ −31.0000 −1.25825 −0.629126 0.777304i $$-0.716587\pi$$
−0.629126 + 0.777304i $$0.716587\pi$$
$$608$$ 0 0
$$609$$ −5.00000 −0.202610
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 25.0000 1.00974 0.504870 0.863195i $$-0.331540\pi$$
0.504870 + 0.863195i $$0.331540\pi$$
$$614$$ 0 0
$$615$$ −5.00000 −0.201619
$$616$$ 0 0
$$617$$ −27.0000 −1.08698 −0.543490 0.839416i $$-0.682897\pi$$
−0.543490 + 0.839416i $$0.682897\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 15.0000 0.601929
$$622$$ 0 0
$$623$$ 35.0000 1.40225
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −15.0000 −0.599042
$$628$$ 0 0
$$629$$ 7.00000 0.279108
$$630$$ 0 0
$$631$$ 27.0000 1.07485 0.537427 0.843311i $$-0.319397\pi$$
0.537427 + 0.843311i $$0.319397\pi$$
$$632$$ 0 0
$$633$$ 5.00000 0.198732
$$634$$ 0 0
$$635$$ 7.00000 0.277787
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 26.0000 1.02854
$$640$$ 0 0
$$641$$ 27.0000 1.06644 0.533218 0.845978i $$-0.320983\pi$$
0.533218 + 0.845978i $$0.320983\pi$$
$$642$$ 0 0
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ 0 0
$$645$$ −5.00000 −0.196875
$$646$$ 0 0
$$647$$ −9.00000 −0.353827 −0.176913 0.984226i $$-0.556611\pi$$
−0.176913 + 0.984226i $$0.556611\pi$$
$$648$$ 0 0
$$649$$ 55.0000 2.15894
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 4.00000 0.156293
$$656$$ 0 0
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 17.0000 0.662226 0.331113 0.943591i $$-0.392576\pi$$
0.331113 + 0.943591i $$0.392576\pi$$
$$660$$ 0 0
$$661$$ −3.00000 −0.116686 −0.0583432 0.998297i $$-0.518582\pi$$
−0.0583432 + 0.998297i $$0.518582\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −15.0000 −0.581675
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ −65.0000 −2.50930
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 0 0
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 0 0
$$679$$ 55.0000 2.11071
$$680$$ 0 0
$$681$$ 17.0000 0.651441
$$682$$ 0 0
$$683$$ −49.0000 −1.87493 −0.937466 0.348076i $$-0.886835\pi$$
−0.937466 + 0.348076i $$0.886835\pi$$
$$684$$ 0 0
$$685$$ −3.00000 −0.114624
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 5.00000 0.190209 0.0951045 0.995467i $$-0.469681\pi$$
0.0951045 + 0.995467i $$0.469681\pi$$
$$692$$ 0 0
$$693$$ 50.0000 1.89934
$$694$$ 0 0
$$695$$ 13.0000 0.493118
$$696$$ 0 0
$$697$$ −5.00000 −0.189389
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 26.0000 0.982006 0.491003 0.871158i $$-0.336630\pi$$
0.491003 + 0.871158i $$0.336630\pi$$
$$702$$ 0 0
$$703$$ −21.0000 −0.792030
$$704$$ 0 0
$$705$$ 12.0000 0.451946
$$706$$ 0 0
$$707$$ 65.0000 2.44458
$$708$$ 0 0
$$709$$ −23.0000 −0.863783 −0.431892 0.901926i $$-0.642154\pi$$
−0.431892 + 0.901926i $$0.642154\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −8.00000 −0.298765
$$718$$ 0 0
$$719$$ 33.0000 1.23069 0.615346 0.788257i $$-0.289016\pi$$
0.615346 + 0.788257i $$0.289016\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 11.0000 0.409094
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −5.00000 −0.184932
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 0 0
$$735$$ −18.0000 −0.663940
$$736$$ 0 0
$$737$$ −15.0000 −0.552532
$$738$$ 0 0
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 0 0
$$745$$ −11.0000 −0.403009
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 0 0
$$749$$ 45.0000 1.64426
$$750$$ 0 0
$$751$$ −25.0000 −0.912263 −0.456131 0.889912i $$-0.650765\pi$$
−0.456131 + 0.889912i $$0.650765\pi$$
$$752$$ 0 0
$$753$$ 15.0000 0.546630
$$754$$ 0 0
$$755$$ 24.0000 0.873449
$$756$$ 0 0
$$757$$ −9.00000 −0.327111 −0.163555 0.986534i $$-0.552296\pi$$
−0.163555 + 0.986534i $$0.552296\pi$$
$$758$$ 0 0
$$759$$ −15.0000 −0.544466
$$760$$ 0 0
$$761$$ −19.0000 −0.688749 −0.344375 0.938832i $$-0.611909\pi$$
−0.344375 + 0.938832i $$0.611909\pi$$
$$762$$ 0 0
$$763$$ 90.0000 3.25822
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −19.0000 −0.685158 −0.342579 0.939489i $$-0.611300\pi$$
−0.342579 + 0.939489i $$0.611300\pi$$
$$770$$ 0 0
$$771$$ −15.0000 −0.540212
$$772$$ 0 0
$$773$$ 37.0000 1.33080 0.665399 0.746488i $$-0.268262\pi$$
0.665399 + 0.746488i $$0.268262\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −35.0000 −1.25562
$$778$$ 0 0
$$779$$ 15.0000 0.537431
$$780$$ 0 0
$$781$$ −65.0000 −2.32588
$$782$$ 0 0
$$783$$ −5.00000 −0.178685
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 11.0000 0.392108 0.196054 0.980593i $$-0.437187\pi$$
0.196054 + 0.980593i $$0.437187\pi$$
$$788$$ 0 0
$$789$$ −11.0000 −0.391610
$$790$$ 0 0
$$791$$ 5.00000 0.177780
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −2.00000 −0.0709327
$$796$$ 0 0
$$797$$ 23.0000 0.814702 0.407351 0.913272i $$-0.366453\pi$$
0.407351 + 0.913272i $$0.366453\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ 0 0
$$803$$ 10.0000 0.352892
$$804$$ 0 0
$$805$$ −15.0000 −0.528681
$$806$$ 0 0
$$807$$ 9.00000 0.316815
$$808$$ 0 0
$$809$$ −37.0000 −1.30085 −0.650425 0.759570i $$-0.725409\pi$$
−0.650425 + 0.759570i $$0.725409\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 0 0
$$813$$ 7.00000 0.245501
$$814$$ 0 0
$$815$$ −5.00000 −0.175142
$$816$$ 0 0
$$817$$ 15.0000 0.524784
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.0000 0.872506 0.436253 0.899824i $$-0.356305\pi$$
0.436253 + 0.899824i $$0.356305\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 0 0
$$825$$ −5.00000 −0.174078
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 0 0
$$831$$ 13.0000 0.450965
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 13.0000 0.449884
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 43.0000 1.48452 0.742262 0.670109i $$-0.233753\pi$$
0.742262 + 0.670109i $$0.233753\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 30.0000 1.03325
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −70.0000 −2.40523
$$848$$ 0 0
$$849$$ −23.0000 −0.789358
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 0 0
$$855$$ −6.00000 −0.205196
$$856$$ 0 0
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 25.0000 0.851998
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −17.0000 −0.578017
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ −20.0000 −0.678454
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 22.0000 0.744587
$$874$$ 0 0
$$875$$ −5.00000 −0.169031
$$876$$ 0 0
$$877$$ −31.0000 −1.04680 −0.523398 0.852088i $$-0.675336\pi$$
−0.523398 + 0.852088i $$0.675336\pi$$
$$878$$ 0 0
$$879$$ 7.00000 0.236104
$$880$$ 0 0
$$881$$ −45.0000 −1.51609 −0.758044 0.652203i $$-0.773845\pi$$
−0.758044 + 0.652203i $$0.773845\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ −11.0000 −0.369761
$$886$$ 0 0
$$887$$ 31.0000 1.04088 0.520439 0.853899i $$-0.325768\pi$$
0.520439 + 0.853899i $$0.325768\pi$$
$$888$$ 0 0
$$889$$ −35.0000 −1.17386
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 0 0
$$893$$ −36.0000 −1.20469
$$894$$ 0 0
$$895$$ 11.0000 0.367689
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −2.00000 −0.0666297
$$902$$ 0 0
$$903$$ 25.0000 0.831948
$$904$$ 0 0
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 0 0
$$909$$ 26.0000 0.862366
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ −60.0000 −1.98571
$$914$$ 0 0
$$915$$ 13.0000 0.429767
$$916$$ 0 0
$$917$$ −20.0000 −0.660458
$$918$$ 0 0
$$919$$ 49.0000 1.61636 0.808180 0.588935i $$-0.200453\pi$$
0.808180 + 0.588935i $$0.200453\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −7.00000 −0.230159
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.0000 1.47640 0.738201 0.674581i $$-0.235676\pi$$
0.738201 + 0.674581i $$0.235676\pi$$
$$930$$ 0 0
$$931$$ 54.0000 1.76978
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −5.00000 −0.163517
$$936$$ 0 0
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 0 0
$$943$$ 15.0000 0.488467
$$944$$ 0 0
$$945$$ −25.0000 −0.813250
$$946$$ 0 0
$$947$$ −31.0000 −1.00736 −0.503682 0.863889i $$-0.668022\pi$$
−0.503682 + 0.863889i $$0.668022\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ 3.00000 0.0971795 0.0485898 0.998819i $$-0.484527\pi$$
0.0485898 + 0.998819i $$0.484527\pi$$
$$954$$ 0 0
$$955$$ −15.0000 −0.485389
$$956$$ 0 0
$$957$$ 5.00000 0.161627
$$958$$ 0 0
$$959$$ 15.0000 0.484375
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ 0 0
$$965$$ −23.0000 −0.740396
$$966$$ 0 0
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 0 0
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ −7.00000 −0.224641 −0.112320 0.993672i $$-0.535828\pi$$
−0.112320 + 0.993672i $$0.535828\pi$$
$$972$$ 0 0
$$973$$ −65.0000 −2.08380
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 45.0000 1.43968 0.719839 0.694141i $$-0.244216\pi$$
0.719839 + 0.694141i $$0.244216\pi$$
$$978$$ 0 0
$$979$$ −35.0000 −1.11860
$$980$$ 0 0
$$981$$ 36.0000 1.14939
$$982$$ 0 0
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 0 0
$$985$$ −27.0000 −0.860292
$$986$$ 0 0
$$987$$ −60.0000 −1.90982
$$988$$ 0 0
$$989$$ 15.0000 0.476972
$$990$$ 0 0
$$991$$ 3.00000 0.0952981 0.0476491 0.998864i $$-0.484827\pi$$
0.0476491 + 0.998864i $$0.484827\pi$$
$$992$$ 0 0
$$993$$ 1.00000 0.0317340
$$994$$ 0 0
$$995$$ 21.0000 0.665745
$$996$$ 0 0
$$997$$ 7.00000 0.221692 0.110846 0.993838i $$-0.464644\pi$$
0.110846 + 0.993838i $$0.464644\pi$$
$$998$$ 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.e.1.1 1
13.4 even 6 260.2.i.c.81.1 yes 2
13.5 odd 4 3380.2.f.c.3041.1 2
13.8 odd 4 3380.2.f.c.3041.2 2
13.10 even 6 260.2.i.c.61.1 2
13.12 even 2 3380.2.a.d.1.1 1
39.17 odd 6 2340.2.q.c.2161.1 2
39.23 odd 6 2340.2.q.c.1621.1 2
52.23 odd 6 1040.2.q.f.321.1 2
52.43 odd 6 1040.2.q.f.81.1 2
65.4 even 6 1300.2.i.c.601.1 2
65.17 odd 12 1300.2.bb.c.549.2 4
65.23 odd 12 1300.2.bb.c.1049.2 4
65.43 odd 12 1300.2.bb.c.549.1 4
65.49 even 6 1300.2.i.c.1101.1 2
65.62 odd 12 1300.2.bb.c.1049.1 4

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