# Properties

 Label 4410.2.a.o.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +10.0000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} -2.00000 q^{55} +4.00000 q^{59} -8.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} -6.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} -10.0000 q^{82} +4.00000 q^{83} +2.00000 q^{85} +8.00000 q^{86} +2.00000 q^{88} -10.0000 q^{89} +4.00000 q^{92} +8.00000 q^{94} -6.00000 q^{95} -18.0000 q^{97} +O(q^{100})$$

## 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
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 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
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ 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
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 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 $$0.578571\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −10.0000 −1.10432
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 2.00000 0.213201
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 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
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 2.00000 0.190693
$$111$$ 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
$$116$$ 0 0
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 8.00000 0.724286
$$123$$ 0 0
$$124$$ 2.00000 0.179605
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 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
$$155$$ 2.00000 0.160644
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 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
$$170$$ −2.00000 −0.153393
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 10.0000 0.698430
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ −10.0000 −0.607457 −0.303728 0.952759i $$-0.598232\pi$$
−0.303728 + 0.952759i $$0.598232\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −14.0000 −0.845771
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 0 0
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 20.0000 1.15857
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −2.00000 −0.113592
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 2.00000 0.108465
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ −6.00000 −0.318447
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 10.0000 0.528516
$$359$$ −34.0000 −1.79445 −0.897226 0.441572i $$-0.854421\pi$$
−0.897226 + 0.441572i $$0.854421\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 20.0000 1.05118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −2.00000 −0.103975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −18.0000 −0.932005 −0.466002 0.884783i $$-0.654306\pi$$
−0.466002 + 0.884783i $$0.654306\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 0 0
$$382$$ −6.00000 −0.306987
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 0 0
$$388$$ −18.0000 −0.913812
$$389$$ −36.0000 −1.82527 −0.912636 0.408773i $$-0.865957\pi$$
−0.912636 + 0.408773i $$0.865957\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −26.0000 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 32.0000 1.59800 0.799002 0.601329i $$-0.205362\pi$$
0.799002 + 0.601329i $$0.205362\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ 8.00000 0.395575 0.197787 0.980245i $$-0.436624\pi$$
0.197787 + 0.980245i $$0.436624\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 0 0
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 0 0
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ −24.0000 −1.14808
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 0 0
$$442$$ 4.00000 0.190261
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 28.0000 1.32584
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 30.0000 1.40334 0.701670 0.712502i $$-0.252438\pi$$
0.701670 + 0.712502i $$0.252438\pi$$
$$458$$ 16.0000 0.747631
$$459$$ 0 0
$$460$$ 4.00000 0.186501
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −26.0000 −1.18921
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −20.0000 −0.910975
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ −18.0000 −0.817338
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 8.00000 0.362143
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 20.0000 0.892644
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ −2.00000 −0.0889988
$$506$$ 8.00000 0.355643
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 2.00000 0.0877058
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ 22.0000 0.948487
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 10.0000 0.429537
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 14.0000 0.598050
$$549$$ 0 0
$$550$$ 2.00000 0.0852803
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20.0000 0.843649
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ −40.0000 −1.67689 −0.838444 0.544988i $$-0.816534\pi$$
−0.838444 + 0.544988i $$0.816534\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −40.0000 −1.65098 −0.825488 0.564419i $$-0.809100\pi$$
−0.825488 + 0.564419i $$0.809100\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ −4.00000 −0.164677
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ −26.0000 −1.06233 −0.531166 0.847268i $$-0.678246\pi$$
−0.531166 + 0.847268i $$0.678246\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 4.00000 0.162355 0.0811775 0.996700i $$-0.474132\pi$$
0.0811775 + 0.996700i $$0.474132\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 8.00000 0.323911
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 18.0000 0.723481 0.361741 0.932279i $$-0.382183\pi$$
0.361741 + 0.932279i $$0.382183\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ 0 0
$$622$$ 8.00000 0.320771
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ 12.0000 0.473234 0.236617 0.971603i $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 46.0000 1.80012 0.900060 0.435767i $$-0.143523\pi$$
0.900060 + 0.435767i $$0.143523\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 34.0000 1.32445 0.662226 0.749304i $$-0.269612\pi$$
0.662226 + 0.749304i $$0.269612\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ 4.00000 0.154533
$$671$$ 16.0000 0.617673
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2.00000 −0.0766965
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 14.0000 0.534913
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 46.0000 1.74992 0.874961 0.484193i $$-0.160887\pi$$
0.874961 + 0.484193i $$0.160887\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ −14.0000 −0.531050
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 16.0000 0.605609
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 32.0000 1.20862 0.604312 0.796748i $$-0.293448\pi$$
0.604312 + 0.796748i $$0.293448\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −42.0000 −1.57734 −0.788672 0.614815i $$-0.789231\pi$$
−0.788672 + 0.614815i $$0.789231\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ 34.0000 1.26887
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ −20.0000 −0.743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2.00000 −0.0740233
$$731$$ −16.0000 −0.591781
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ −20.0000 −0.732743
$$746$$ 18.0000 0.659027
$$747$$ 0 0
$$748$$ −4.00000 −0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ 44.0000 1.58668 0.793340 0.608778i $$-0.208340\pi$$
0.793340 + 0.608778i $$0.208340\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ 18.0000 0.646162
$$777$$ 0 0
$$778$$ 36.0000 1.29066
$$779$$ −60.0000 −2.14972
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ 50.0000 1.77109 0.885545 0.464553i $$-0.153785\pi$$
0.885545 + 0.464553i $$0.153785\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −32.0000 −1.12996
$$803$$ −4.00000 −0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ −44.0000 −1.54696 −0.773479 0.633822i $$-0.781485\pi$$
−0.773479 + 0.633822i $$0.781485\pi$$
$$810$$ 0 0
$$811$$ 34.0000 1.19390 0.596951 0.802278i $$-0.296379\pi$$
0.596951 + 0.802278i $$0.296379\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.00000 0.140200
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −8.00000 −0.279713
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ −52.0000 −1.81261 −0.906303 0.422628i $$-0.861108\pi$$
−0.906303 + 0.422628i $$0.861108\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −32.0000 −1.11275 −0.556375 0.830932i $$-0.687808\pi$$
−0.556375 + 0.830932i $$0.687808\pi$$
$$828$$ 0 0
$$829$$ −44.0000 −1.52818 −0.764092 0.645108i $$-0.776812\pi$$
−0.764092 + 0.645108i $$0.776812\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −26.0000 −0.896019
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ −22.0000 −0.747590
$$867$$ 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
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ −26.0000 −0.877457
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ 0 0
$$892$$ −28.0000 −0.937509
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 20.0000 0.665927
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ −20.0000 −0.664822
$$906$$ 0 0
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 14.0000 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −30.0000 −0.992312
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ −18.0000 −0.592798
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 20.0000 0.657241
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 26.0000 0.851658
$$933$$ 0 0
$$934$$ 28.0000 0.916188
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ 40.0000 1.30258
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 6.00000 0.194666
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −14.0000 −0.453504 −0.226752 0.973952i $$-0.572811\pi$$
−0.226752 + 0.973952i $$0.572811\pi$$
$$954$$ 0 0
$$955$$ 6.00000 0.194155
$$956$$ 26.0000 0.840900
$$957$$ 0 0
$$958$$ 8.00000 0.258468
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 4.00000 0.128965
$$963$$ 0 0
$$964$$ 20.0000 0.644157
$$965$$ −6.00000 −0.193147
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 7.00000 0.224989
$$969$$ 0 0
$$970$$ 18.0000 0.577945
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 20.0000 0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 18.0000 0.574403
$$983$$ 56.0000 1.78612 0.893061 0.449935i $$-0.148553\pi$$
0.893061 + 0.449935i $$0.148553\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.00000 0.0634043
$$996$$ 0 0
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 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.o.1.1 yes 1
3.2 odd 2 4410.2.a.bb.1.1 yes 1
7.6 odd 2 4410.2.a.d.1.1 1
21.20 even 2 4410.2.a.bk.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.a.d.1.1 1 7.6 odd 2
4410.2.a.o.1.1 yes 1 1.1 even 1 trivial
4410.2.a.bb.1.1 yes 1 3.2 odd 2
4410.2.a.bk.1.1 yes 1 21.20 even 2