# Properties

 Label 4410.2.a.s.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 490) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +8.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{34} -10.0000 q^{37} -6.00000 q^{38} -1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{50} -2.00000 q^{52} -10.0000 q^{53} +4.00000 q^{55} -6.00000 q^{58} -14.0000 q^{59} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +8.00000 q^{68} -12.0000 q^{71} -4.00000 q^{73} +10.0000 q^{74} +6.00000 q^{76} +4.00000 q^{79} +1.00000 q^{80} -4.00000 q^{82} -2.00000 q^{83} +8.00000 q^{85} -4.00000 q^{86} -4.00000 q^{88} +8.00000 q^{89} +4.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −4.00000 −0.441726
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ 14.0000 1.28880
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −8.00000 −0.685994
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.0000 1.00702
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 2.00000 0.155230
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −8.00000 −0.613572
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −8.00000 −0.599625
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ 32.0000 2.34007
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ −16.0000 −1.07628
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ −14.0000 −0.911322
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 22.0000 1.38863 0.694314 0.719672i $$-0.255708\pi$$
0.694314 + 0.719672i $$0.255708\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ −18.0000 −1.11204
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 8.00000 0.485071
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 47.0000 2.76471
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ −14.0000 −0.815112
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 6.00000 0.344124
$$305$$ 10.0000 0.572598
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 48.0000 2.67079
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −4.00000 −0.220863
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −2.00000 −0.109764
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 8.00000 0.433861
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 8.00000 0.423999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 26.0000 1.36653
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.00000 −0.209370
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ 10.0000 0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ −32.0000 −1.65468
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ 12.0000 0.613973
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 4.00000 0.201262
$$396$$ 0 0
$$397$$ −26.0000 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −2.00000 −0.0981761
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ −24.0000 −1.17388
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 8.00000 0.388057
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ 8.00000 0.379236
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ −10.0000 −0.462745 −0.231372 0.972865i $$-0.574322\pi$$
−0.231372 + 0.972865i $$0.574322\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −4.00000 −0.184506
$$471$$ 0 0
$$472$$ 14.0000 0.644402
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8.00000 −0.365911
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 4.00000 0.182195
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −44.0000 −1.99383 −0.996915 0.0784867i $$-0.974991\pi$$
−0.996915 + 0.0784867i $$0.974991\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 48.0000 2.16181
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −22.0000 −0.981908
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ −2.00000 −0.0889988
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −12.0000 −0.532414
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 2.00000 0.0877058
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ 34.0000 1.48672 0.743358 0.668894i $$-0.233232\pi$$
0.743358 + 0.668894i $$0.233232\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −32.0000 −1.39394
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 10.0000 0.434372
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −8.00000 −0.346518
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −26.0000 −1.12094
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ −8.00000 −0.342997
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ −47.0000 −1.95494
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 2.00000 0.0825488 0.0412744 0.999148i $$-0.486858\pi$$
0.0412744 + 0.999148i $$0.486858\pi$$
$$588$$ 0 0
$$589$$ −24.0000 −0.988903
$$590$$ 14.0000 0.576371
$$591$$ 0 0
$$592$$ −10.0000 −0.410997
$$593$$ −20.0000 −0.821302 −0.410651 0.911793i $$-0.634698\pi$$
−0.410651 + 0.911793i $$0.634698\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 28.0000 1.14405 0.572024 0.820237i $$-0.306158\pi$$
0.572024 + 0.820237i $$0.306158\pi$$
$$600$$ 0 0
$$601$$ −40.0000 −1.63163 −0.815817 0.578310i $$-0.803712\pi$$
−0.815817 + 0.578310i $$0.803712\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ 6.00000 0.241160 0.120580 0.992704i $$-0.461525\pi$$
0.120580 + 0.992704i $$0.461525\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −16.0000 −0.641542
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ −80.0000 −3.18981
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ −12.0000 −0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −18.0000 −0.709851 −0.354925 0.934895i $$-0.615494\pi$$
−0.354925 + 0.934895i $$0.615494\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −48.0000 −1.88853
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −56.0000 −2.19819
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 18.0000 0.703318
$$656$$ 4.00000 0.156174
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 2.00000 0.0776151
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −46.0000 −1.76792 −0.883962 0.467559i $$-0.845134\pi$$
−0.883962 + 0.467559i $$0.845134\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −8.00000 −0.306786
$$681$$ 0 0
$$682$$ 16.0000 0.612672
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 2.00000 0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 46.0000 1.74992 0.874961 0.484193i $$-0.160887\pi$$
0.874961 + 0.484193i $$0.160887\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 10.0000 0.379322
$$696$$ 0 0
$$697$$ 32.0000 1.21209
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 38.0000 1.43524 0.717620 0.696435i $$-0.245231\pi$$
0.717620 + 0.696435i $$0.245231\pi$$
$$702$$ 0 0
$$703$$ −60.0000 −2.26294
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ −8.00000 −0.299813
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ −8.00000 −0.298557
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ −26.0000 −0.966282
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −20.0000 −0.741759 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 4.00000 0.148047
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ −10.0000 −0.367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 6.00000 0.219676
$$747$$ 0 0
$$748$$ 32.0000 1.17004
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 28.0000 1.01102
$$768$$ 0 0
$$769$$ −28.0000 −1.00971 −0.504853 0.863205i $$-0.668453\pi$$
−0.504853 + 0.863205i $$0.668453\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −18.0000 −0.647834
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ −48.0000 −1.71758
$$782$$ −32.0000 −1.14432
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −6.00000 −0.213877 −0.106938 0.994266i $$-0.534105\pi$$
−0.106938 + 0.994266i $$0.534105\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ −4.00000 −0.142314
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 32.0000 1.13208
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −14.0000 −0.494357
$$803$$ −16.0000 −0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 40.0000 1.40200
$$815$$ −4.00000 −0.140114
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ −20.0000 −0.699284
$$819$$ 0 0
$$820$$ 4.00000 0.139686
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −44.0000 −1.53003 −0.765015 0.644013i $$-0.777268\pi$$
−0.765015 + 0.644013i $$0.777268\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 2.00000 0.0694210
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 24.0000 0.830057
$$837$$ 0 0
$$838$$ −6.00000 −0.207267
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 0 0
$$850$$ −8.00000 −0.274398
$$851$$ −40.0000 −1.37118
$$852$$ 0 0
$$853$$ −50.0000 −1.71197 −0.855984 0.517003i $$-0.827048\pi$$
−0.855984 + 0.517003i $$0.827048\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −24.0000 −0.819824 −0.409912 0.912125i $$-0.634441\pi$$
−0.409912 + 0.912125i $$0.634441\pi$$
$$858$$ 0 0
$$859$$ −38.0000 −1.29654 −0.648272 0.761409i $$-0.724508\pi$$
−0.648272 + 0.761409i $$0.724508\pi$$
$$860$$ 4.00000 0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 18.0000 0.612018
$$866$$ −40.0000 −1.35926
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −10.0000 −0.338643
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −16.0000 −0.538138
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 4.00000 0.134307 0.0671534 0.997743i $$-0.478608\pi$$
0.0671534 + 0.997743i $$0.478608\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −8.00000 −0.268161
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 24.0000 0.803129
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ −80.0000 −2.66519
$$902$$ −16.0000 −0.532742
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ −26.0000 −0.864269
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ 6.00000 0.197599
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −12.0000 −0.393707 −0.196854 0.980433i $$-0.563072\pi$$
−0.196854 + 0.980433i $$0.563072\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −22.0000 −0.720634
$$933$$ 0 0
$$934$$ 10.0000 0.327210
$$935$$ 32.0000 1.04651
$$936$$ 0 0
$$937$$ −12.0000 −0.392023 −0.196011 0.980602i $$-0.562799\pi$$
−0.196011 + 0.980602i $$0.562799\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 4.00000 0.130466
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ −6.00000 −0.194666
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ −4.00000 −0.129234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −20.0000 −0.644826
$$963$$ 0 0
$$964$$ −4.00000 −0.128831
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ −52.0000 −1.67221 −0.836104 0.548572i $$-0.815172\pi$$
−0.836104 + 0.548572i $$0.815172\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −62.0000 −1.98967 −0.994837 0.101482i $$-0.967641\pi$$
−0.994837 + 0.101482i $$0.967641\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 44.0000 1.40985
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −54.0000 −1.72761 −0.863807 0.503824i $$-0.831926\pi$$
−0.863807 + 0.503824i $$0.831926\pi$$
$$978$$ 0 0
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ 20.0000 0.637901 0.318950 0.947771i $$-0.396670\pi$$
0.318950 + 0.947771i $$0.396670\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ −48.0000 −1.52863
$$987$$ 0 0
$$988$$ −12.0000 −0.381771
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ −36.0000 −1.14358 −0.571789 0.820401i $$-0.693750\pi$$
−0.571789 + 0.820401i $$0.693750\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 0 0
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 4410.2.a.s.1.1 1
3.2 odd 2 490.2.a.f.1.1 1
7.6 odd 2 4410.2.a.i.1.1 1
12.11 even 2 3920.2.a.bg.1.1 1
15.2 even 4 2450.2.c.b.99.2 2
15.8 even 4 2450.2.c.b.99.1 2
15.14 odd 2 2450.2.a.n.1.1 1
21.2 odd 6 490.2.e.e.361.1 2
21.5 even 6 490.2.e.b.361.1 2
21.11 odd 6 490.2.e.e.471.1 2
21.17 even 6 490.2.e.b.471.1 2
21.20 even 2 490.2.a.i.1.1 yes 1
84.83 odd 2 3920.2.a.j.1.1 1
105.62 odd 4 2450.2.c.n.99.2 2
105.83 odd 4 2450.2.c.n.99.1 2
105.104 even 2 2450.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.f.1.1 1 3.2 odd 2
490.2.a.i.1.1 yes 1 21.20 even 2
490.2.e.b.361.1 2 21.5 even 6
490.2.e.b.471.1 2 21.17 even 6
490.2.e.e.361.1 2 21.2 odd 6
490.2.e.e.471.1 2 21.11 odd 6
2450.2.a.d.1.1 1 105.104 even 2
2450.2.a.n.1.1 1 15.14 odd 2
2450.2.c.b.99.1 2 15.8 even 4
2450.2.c.b.99.2 2 15.2 even 4
2450.2.c.n.99.1 2 105.83 odd 4
2450.2.c.n.99.2 2 105.62 odd 4
3920.2.a.j.1.1 1 84.83 odd 2
3920.2.a.bg.1.1 1 12.11 even 2
4410.2.a.i.1.1 1 7.6 odd 2
4410.2.a.s.1.1 1 1.1 even 1 trivial