# Properties

 Label 3330.2.a.b.1.1 Level $3330$ Weight $2$ Character 3330.1 Self dual yes Analytic conductor $26.590$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1110) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3330.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +5.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -5.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -3.00000 q^{28} +1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +3.00000 q^{35} -1.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +7.00000 q^{41} +3.00000 q^{43} +5.00000 q^{44} -4.00000 q^{46} +2.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -5.00000 q^{53} -5.00000 q^{55} +3.00000 q^{56} -1.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -3.00000 q^{70} +12.0000 q^{71} +1.00000 q^{74} -6.00000 q^{76} -15.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -7.00000 q^{82} -6.00000 q^{83} +3.00000 q^{85} -3.00000 q^{86} -5.00000 q^{88} +18.0000 q^{89} +6.00000 q^{91} +4.00000 q^{92} +6.00000 q^{95} -13.0000 q^{97} -2.00000 q^{98} +O(q^{100})$$

## 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
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 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
$$22$$ −5.00000 −1.06600
$$23$$ 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
$$28$$ −3.00000 −0.566947
$$29$$ 1.00000 0.185695 0.0928477 0.995680i $$-0.470403\pi$$
0.0928477 + 0.995680i $$0.470403\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ 3.00000 0.457496 0.228748 0.973486i $$-0.426537\pi$$
0.228748 + 0.973486i $$0.426537\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −5.00000 −0.686803 −0.343401 0.939189i $$-0.611579\pi$$
−0.343401 + 0.939189i $$0.611579\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 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 $$0.578571\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ −3.00000 −0.358569
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ −15.0000 −1.70941
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −7.00000 −0.773021
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 3.00000 0.325396
$$86$$ −3.00000 −0.323498
$$87$$ 0 0
$$88$$ −5.00000 −0.533002
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 5.00000 0.485643
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 5.00000 0.476731
$$111$$ 0 0
$$112$$ −3.00000 −0.283473
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 1.00000 0.0928477
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ −5.00000 −0.452679
$$123$$ 0 0
$$124$$ −3.00000 −0.269408
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −2.00000 −0.175412
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ 18.0000 1.56080
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 3.00000 0.253546
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ −10.0000 −0.836242
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 0 0
$$154$$ 15.0000 1.20873
$$155$$ 3.00000 0.240966
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −3.00000 −0.230089
$$171$$ 0 0
$$172$$ 3.00000 0.228748
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 5.00000 0.376889
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ −15.0000 −1.09691
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −6.00000 −0.435286
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 0 0
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 13.0000 0.933346
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 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
$$202$$ −12.0000 −0.844317
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ −7.00000 −0.488901
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ −30.0000 −2.07514
$$210$$ 0 0
$$211$$ 19.0000 1.30801 0.654007 0.756489i $$-0.273087\pi$$
0.654007 + 0.756489i $$0.273087\pi$$
$$212$$ −5.00000 −0.343401
$$213$$ 0 0
$$214$$ 2.00000 0.136717
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ 9.00000 0.610960
$$218$$ −5.00000 −0.338643
$$219$$ 0 0
$$220$$ −5.00000 −0.337100
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ 7.00000 0.468755 0.234377 0.972146i $$-0.424695\pi$$
0.234377 + 0.972146i $$0.424695\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ 9.00000 0.598671
$$227$$ 21.0000 1.39382 0.696909 0.717159i $$-0.254558\pi$$
0.696909 + 0.717159i $$0.254558\pi$$
$$228$$ 0 0
$$229$$ 12.0000 0.792982 0.396491 0.918039i $$-0.370228\pi$$
0.396491 + 0.918039i $$0.370228\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ −9.00000 −0.583383
$$239$$ −29.0000 −1.87585 −0.937927 0.346833i $$-0.887257\pi$$
−0.937927 + 0.346833i $$0.887257\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ 0 0
$$244$$ 5.00000 0.320092
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 3.00000 0.190500
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ 20.0000 1.25739
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 2.00000 0.124035
$$261$$ 0 0
$$262$$ 10.0000 0.617802
$$263$$ −5.00000 −0.308313 −0.154157 0.988046i $$-0.549266\pi$$
−0.154157 + 0.988046i $$0.549266\pi$$
$$264$$ 0 0
$$265$$ 5.00000 0.307148
$$266$$ −18.0000 −1.10365
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 16.0000 0.975537 0.487769 0.872973i $$-0.337811\pi$$
0.487769 + 0.872973i $$0.337811\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 5.00000 0.301511
$$276$$ 0 0
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ −5.00000 −0.299880
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 10.0000 0.591312
$$287$$ −21.0000 −1.23959
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 1.00000 0.0587220
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ −9.00000 −0.518751
$$302$$ −10.0000 −0.575435
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ −5.00000 −0.286299
$$306$$ 0 0
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ −15.0000 −0.854704
$$309$$ 0 0
$$310$$ −3.00000 −0.170389
$$311$$ 31.0000 1.75785 0.878924 0.476961i $$-0.158262\pi$$
0.878924 + 0.476961i $$0.158262\pi$$
$$312$$ 0 0
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ −11.0000 −0.620766
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 27.0000 1.51647 0.758236 0.651981i $$-0.226062\pi$$
0.758236 + 0.651981i $$0.226062\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 18.0000 1.00155
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ −11.0000 −0.609234
$$327$$ 0 0
$$328$$ −7.00000 −0.386510
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 3.00000 0.162698
$$341$$ −15.0000 −0.812296
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ −3.00000 −0.161749
$$345$$ 0 0
$$346$$ 13.0000 0.698884
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 3.00000 0.160357
$$351$$ 0 0
$$352$$ −5.00000 −0.266501
$$353$$ 37.0000 1.96931 0.984656 0.174509i $$-0.0558337\pi$$
0.984656 + 0.174509i $$0.0558337\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.00000 −0.156599 −0.0782994 0.996930i $$-0.524949\pi$$
−0.0782994 + 0.996930i $$0.524949\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −1.00000 −0.0519875
$$371$$ 15.0000 0.778761
$$372$$ 0 0
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 15.0000 0.775632
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −2.00000 −0.103005
$$378$$ 0 0
$$379$$ 32.0000 1.64373 0.821865 0.569683i $$-0.192934\pi$$
0.821865 + 0.569683i $$0.192934\pi$$
$$380$$ 6.00000 0.307794
$$381$$ 0 0
$$382$$ −3.00000 −0.153493
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ −18.0000 −0.916176
$$387$$ 0 0
$$388$$ −13.0000 −0.659975
$$389$$ 3.00000 0.152106 0.0760530 0.997104i $$-0.475768\pi$$
0.0760530 + 0.997104i $$0.475768\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ −2.00000 −0.101015
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 4.00000 0.201262
$$396$$ 0 0
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 3.00000 0.148888
$$407$$ −5.00000 −0.247841
$$408$$ 0 0
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 7.00000 0.345705
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ 18.0000 0.885722
$$414$$ 0 0
$$415$$ 6.00000 0.294528
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 30.0000 1.46735
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ −19.0000 −0.924906
$$423$$ 0 0
$$424$$ 5.00000 0.242821
$$425$$ −3.00000 −0.145521
$$426$$ 0 0
$$427$$ −15.0000 −0.725901
$$428$$ −2.00000 −0.0966736
$$429$$ 0 0
$$430$$ 3.00000 0.144673
$$431$$ 7.00000 0.337178 0.168589 0.985686i $$-0.446079\pi$$
0.168589 + 0.985686i $$0.446079\pi$$
$$432$$ 0 0
$$433$$ −8.00000 −0.384455 −0.192228 0.981350i $$-0.561571\pi$$
−0.192228 + 0.981350i $$0.561571\pi$$
$$434$$ −9.00000 −0.432014
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ −24.0000 −1.14808
$$438$$ 0 0
$$439$$ −29.0000 −1.38409 −0.692047 0.721852i $$-0.743291\pi$$
−0.692047 + 0.721852i $$0.743291\pi$$
$$440$$ 5.00000 0.238366
$$441$$ 0 0
$$442$$ −6.00000 −0.285391
$$443$$ −14.0000 −0.665160 −0.332580 0.943075i $$-0.607919\pi$$
−0.332580 + 0.943075i $$0.607919\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ −7.00000 −0.331460
$$447$$ 0 0
$$448$$ −3.00000 −0.141737
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ 35.0000 1.64809
$$452$$ −9.00000 −0.423324
$$453$$ 0 0
$$454$$ −21.0000 −0.985579
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ −5.00000 −0.233890 −0.116945 0.993138i $$-0.537310\pi$$
−0.116945 + 0.993138i $$0.537310\pi$$
$$458$$ −12.0000 −0.560723
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ −21.0000 −0.978068 −0.489034 0.872265i $$-0.662651\pi$$
−0.489034 + 0.872265i $$0.662651\pi$$
$$462$$ 0 0
$$463$$ 34.0000 1.58011 0.790057 0.613033i $$-0.210051\pi$$
0.790057 + 0.613033i $$0.210051\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ 15.0000 0.689701
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 9.00000 0.412514
$$477$$ 0 0
$$478$$ 29.0000 1.32643
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ 13.0000 0.590300
$$486$$ 0 0
$$487$$ −6.00000 −0.271886 −0.135943 0.990717i $$-0.543406\pi$$
−0.135943 + 0.990717i $$0.543406\pi$$
$$488$$ −5.00000 −0.226339
$$489$$ 0 0
$$490$$ 2.00000 0.0903508
$$491$$ −4.00000 −0.180517 −0.0902587 0.995918i $$-0.528769\pi$$
−0.0902587 + 0.995918i $$0.528769\pi$$
$$492$$ 0 0
$$493$$ −3.00000 −0.135113
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ −36.0000 −1.61482
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −24.0000 −1.07117
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ −20.0000 −0.889108
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 26.0000 1.14681
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −3.00000 −0.131812
$$519$$ 0 0
$$520$$ −2.00000 −0.0877058
$$521$$ −3.00000 −0.131432 −0.0657162 0.997838i $$-0.520933\pi$$
−0.0657162 + 0.997838i $$0.520933\pi$$
$$522$$ 0 0
$$523$$ −40.0000 −1.74908 −0.874539 0.484955i $$-0.838836\pi$$
−0.874539 + 0.484955i $$0.838836\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 5.00000 0.218010
$$527$$ 9.00000 0.392046
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −5.00000 −0.217186
$$531$$ 0 0
$$532$$ 18.0000 0.780399
$$533$$ −14.0000 −0.606407
$$534$$ 0 0
$$535$$ 2.00000 0.0864675
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −16.0000 −0.689809
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ −5.00000 −0.214176
$$546$$ 0 0
$$547$$ 19.0000 0.812381 0.406191 0.913788i $$-0.366857\pi$$
0.406191 + 0.913788i $$0.366857\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 0 0
$$550$$ −5.00000 −0.213201
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 12.0000 0.510292
$$554$$ 16.0000 0.679775
$$555$$ 0 0
$$556$$ 5.00000 0.212047
$$557$$ 6.00000 0.254228 0.127114 0.991888i $$-0.459429\pi$$
0.127114 + 0.991888i $$0.459429\pi$$
$$558$$ 0 0
$$559$$ −6.00000 −0.253773
$$560$$ 3.00000 0.126773
$$561$$ 0 0
$$562$$ 2.00000 0.0843649
$$563$$ 17.0000 0.716465 0.358232 0.933632i $$-0.383380\pi$$
0.358232 + 0.933632i $$0.383380\pi$$
$$564$$ 0 0
$$565$$ 9.00000 0.378633
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −46.0000 −1.92842 −0.964210 0.265139i $$-0.914582\pi$$
−0.964210 + 0.265139i $$0.914582\pi$$
$$570$$ 0 0
$$571$$ −21.0000 −0.878823 −0.439411 0.898286i $$-0.644813\pi$$
−0.439411 + 0.898286i $$0.644813\pi$$
$$572$$ −10.0000 −0.418121
$$573$$ 0 0
$$574$$ 21.0000 0.876523
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ −1.00000 −0.0415227
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ −25.0000 −1.03539
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 7.00000 0.289167
$$587$$ −1.00000 −0.0412744 −0.0206372 0.999787i $$-0.506569\pi$$
−0.0206372 + 0.999787i $$0.506569\pi$$
$$588$$ 0 0
$$589$$ 18.0000 0.741677
$$590$$ −6.00000 −0.247016
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ −31.0000 −1.26452 −0.632258 0.774758i $$-0.717872\pi$$
−0.632258 + 0.774758i $$0.717872\pi$$
$$602$$ 9.00000 0.366813
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −34.0000 −1.38002 −0.690009 0.723801i $$-0.742393\pi$$
−0.690009 + 0.723801i $$0.742393\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 5.00000 0.202444
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −43.0000 −1.73675 −0.868377 0.495905i $$-0.834836\pi$$
−0.868377 + 0.495905i $$0.834836\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ 15.0000 0.604367
$$617$$ −34.0000 −1.36879 −0.684394 0.729112i $$-0.739933\pi$$
−0.684394 + 0.729112i $$0.739933\pi$$
$$618$$ 0 0
$$619$$ −27.0000 −1.08522 −0.542611 0.839984i $$-0.682564\pi$$
−0.542611 + 0.839984i $$0.682564\pi$$
$$620$$ 3.00000 0.120483
$$621$$ 0 0
$$622$$ −31.0000 −1.24299
$$623$$ −54.0000 −2.16346
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 30.0000 1.19904
$$627$$ 0 0
$$628$$ 11.0000 0.438948
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ −27.0000 −1.07231
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ −5.00000 −0.197952
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 23.0000 0.908445 0.454223 0.890888i $$-0.349917\pi$$
0.454223 + 0.890888i $$0.349917\pi$$
$$642$$ 0 0
$$643$$ 31.0000 1.22252 0.611260 0.791430i $$-0.290663\pi$$
0.611260 + 0.791430i $$0.290663\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ −18.0000 −0.708201
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ −30.0000 −1.17760
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 11.0000 0.430793
$$653$$ −16.0000 −0.626128 −0.313064 0.949732i $$-0.601356\pi$$
−0.313064 + 0.949732i $$0.601356\pi$$
$$654$$ 0 0
$$655$$ 10.0000 0.390732
$$656$$ 7.00000 0.273304
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −29.0000 −1.12797 −0.563985 0.825785i $$-0.690732\pi$$
−0.563985 + 0.825785i $$0.690732\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ −18.0000 −0.698010
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ 24.0000 0.928588
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ 25.0000 0.965114
$$672$$ 0 0
$$673$$ 48.0000 1.85026 0.925132 0.379646i $$-0.123954\pi$$
0.925132 + 0.379646i $$0.123954\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −10.0000 −0.384331 −0.192166 0.981363i $$-0.561551\pi$$
−0.192166 + 0.981363i $$0.561551\pi$$
$$678$$ 0 0
$$679$$ 39.0000 1.49668
$$680$$ −3.00000 −0.115045
$$681$$ 0 0
$$682$$ 15.0000 0.574380
$$683$$ 29.0000 1.10965 0.554827 0.831966i $$-0.312784\pi$$
0.554827 + 0.831966i $$0.312784\pi$$
$$684$$ 0 0
$$685$$ −10.0000 −0.382080
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ 3.00000 0.114374
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ −33.0000 −1.25538 −0.627690 0.778464i $$-0.715999\pi$$
−0.627690 + 0.778464i $$0.715999\pi$$
$$692$$ −13.0000 −0.494186
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ −5.00000 −0.189661
$$696$$ 0 0
$$697$$ −21.0000 −0.795432
$$698$$ 20.0000 0.757011
$$699$$ 0 0
$$700$$ −3.00000 −0.113389
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ 0 0
$$703$$ 6.00000 0.226294
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −37.0000 −1.39251
$$707$$ −36.0000 −1.35392
$$708$$ 0 0
$$709$$ 49.0000 1.84023 0.920117 0.391644i $$-0.128094\pi$$
0.920117 + 0.391644i $$0.128094\pi$$
$$710$$ 12.0000 0.450352
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ −12.0000 −0.449404
$$714$$ 0 0
$$715$$ 10.0000 0.373979
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −30.0000 −1.11959
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ −22.0000 −0.815935 −0.407967 0.912996i $$-0.633762\pi$$
−0.407967 + 0.912996i $$0.633762\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −9.00000 −0.332877
$$732$$ 0 0
$$733$$ −51.0000 −1.88373 −0.941864 0.335994i $$-0.890928\pi$$
−0.941864 + 0.335994i $$0.890928\pi$$
$$734$$ 3.00000 0.110732
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ −20.0000 −0.736709
$$738$$ 0 0
$$739$$ 1.00000 0.0367856 0.0183928 0.999831i $$-0.494145\pi$$
0.0183928 + 0.999831i $$0.494145\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ 0 0
$$742$$ −15.0000 −0.550667
$$743$$ 3.00000 0.110059 0.0550297 0.998485i $$-0.482475\pi$$
0.0550297 + 0.998485i $$0.482475\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ −15.0000 −0.548454
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 2.00000 0.0728357
$$755$$ −10.0000 −0.363937
$$756$$ 0 0
$$757$$ 30.0000 1.09037 0.545184 0.838316i $$-0.316460\pi$$
0.545184 + 0.838316i $$0.316460\pi$$
$$758$$ −32.0000 −1.16229
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ −25.0000 −0.906249 −0.453125 0.891447i $$-0.649691\pi$$
−0.453125 + 0.891447i $$0.649691\pi$$
$$762$$ 0 0
$$763$$ −15.0000 −0.543036
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 38.0000 1.37032 0.685158 0.728395i $$-0.259733\pi$$
0.685158 + 0.728395i $$0.259733\pi$$
$$770$$ −15.0000 −0.540562
$$771$$ 0 0
$$772$$ 18.0000 0.647834
$$773$$ 17.0000 0.611448 0.305724 0.952120i $$-0.401102\pi$$
0.305724 + 0.952120i $$0.401102\pi$$
$$774$$ 0 0
$$775$$ −3.00000 −0.107763
$$776$$ 13.0000 0.466673
$$777$$ 0 0
$$778$$ −3.00000 −0.107555
$$779$$ −42.0000 −1.50481
$$780$$ 0 0
$$781$$ 60.0000 2.14697
$$782$$ 12.0000 0.429119
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ −11.0000 −0.392607
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ −4.00000 −0.142314
$$791$$ 27.0000 0.960009
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −40.0000 −1.41687 −0.708436 0.705775i $$-0.750599\pi$$
−0.708436 + 0.705775i $$0.750599\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −2.00000 −0.0706225
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ −6.00000 −0.211341
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −3.00000 −0.105279
$$813$$ 0 0
$$814$$ 5.00000 0.175250
$$815$$ −11.0000 −0.385313
$$816$$ 0 0
$$817$$ −18.0000 −0.629740
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ −7.00000 −0.244451
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ 56.0000 1.95204 0.976019 0.217687i $$-0.0698512\pi$$
0.976019 + 0.217687i $$0.0698512\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ −18.0000 −0.626300
$$827$$ −23.0000 −0.799788 −0.399894 0.916561i $$-0.630953\pi$$
−0.399894 + 0.916561i $$0.630953\pi$$
$$828$$ 0 0
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ −30.0000 −1.03757
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ −2.00000 −0.0689246
$$843$$ 0 0
$$844$$ 19.0000 0.654007
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ −42.0000 −1.44314
$$848$$ −5.00000 −0.171701
$$849$$ 0 0
$$850$$ 3.00000 0.102899
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 15.0000 0.513289
$$855$$ 0 0
$$856$$ 2.00000 0.0683586
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −26.0000 −0.887109 −0.443554 0.896248i $$-0.646283\pi$$
−0.443554 + 0.896248i $$0.646283\pi$$
$$860$$ −3.00000 −0.102299
$$861$$ 0 0
$$862$$ −7.00000 −0.238421
$$863$$ −27.0000 −0.919091 −0.459545 0.888154i $$-0.651988\pi$$
−0.459545 + 0.888154i $$0.651988\pi$$
$$864$$ 0 0
$$865$$ 13.0000 0.442013
$$866$$ 8.00000 0.271851
$$867$$ 0 0
$$868$$ 9.00000 0.305480
$$869$$ −20.0000 −0.678454
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −5.00000 −0.169321
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 27.0000 0.911725 0.455863 0.890050i $$-0.349331\pi$$
0.455863 + 0.890050i $$0.349331\pi$$
$$878$$ 29.0000 0.978703
$$879$$ 0 0
$$880$$ −5.00000 −0.168550
$$881$$ 39.0000 1.31394 0.656972 0.753915i $$-0.271837\pi$$
0.656972 + 0.753915i $$0.271837\pi$$
$$882$$ 0 0
$$883$$ −3.00000 −0.100958 −0.0504790 0.998725i $$-0.516075\pi$$
−0.0504790 + 0.998725i $$0.516075\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 14.0000 0.470339
$$887$$ 7.00000 0.235037 0.117518 0.993071i $$-0.462506\pi$$
0.117518 + 0.993071i $$0.462506\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 18.0000 0.603361
$$891$$ 0 0
$$892$$ 7.00000 0.234377
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ 15.0000 0.499722
$$902$$ −35.0000 −1.16537
$$903$$ 0 0
$$904$$ 9.00000 0.299336
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 21.0000 0.696909
$$909$$ 0 0
$$910$$ 6.00000 0.198898
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −30.0000 −0.992855
$$914$$ 5.00000 0.165385
$$915$$ 0 0
$$916$$ 12.0000 0.396491
$$917$$ 30.0000 0.990687
$$918$$ 0 0
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 0 0
$$922$$ 21.0000 0.691598
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ −34.0000 −1.11731
$$927$$ 0 0
$$928$$ −1.00000 −0.0328266
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 0 0
$$933$$ 0 0
$$934$$ −15.0000 −0.490815
$$935$$ 15.0000 0.490552
$$936$$ 0 0
$$937$$ −40.0000 −1.30674 −0.653372 0.757037i $$-0.726646\pi$$
−0.653372 + 0.757037i $$0.726646\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ 0 0
$$943$$ 28.0000 0.911805
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −15.0000 −0.487692
$$947$$ 29.0000 0.942373 0.471187 0.882034i $$-0.343826\pi$$
0.471187 + 0.882034i $$0.343826\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ 0 0
$$952$$ −9.00000 −0.291692
$$953$$ 40.0000 1.29573 0.647864 0.761756i $$-0.275663\pi$$
0.647864 + 0.761756i $$0.275663\pi$$
$$954$$ 0 0
$$955$$ −3.00000 −0.0970777
$$956$$ −29.0000 −0.937927
$$957$$ 0 0
$$958$$ −16.0000 −0.516937
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ 0 0
$$970$$ −13.0000 −0.417405
$$971$$ 35.0000 1.12320 0.561602 0.827408i $$-0.310185\pi$$
0.561602 + 0.827408i $$0.310185\pi$$
$$972$$ 0 0
$$973$$ −15.0000 −0.480878
$$974$$ 6.00000 0.192252
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ 25.0000 0.799821 0.399910 0.916554i $$-0.369041\pi$$
0.399910 + 0.916554i $$0.369041\pi$$
$$978$$ 0 0
$$979$$ 90.0000 2.87641
$$980$$ −2.00000 −0.0638877
$$981$$ 0 0
$$982$$ 4.00000 0.127645
$$983$$ 29.0000 0.924956 0.462478 0.886631i $$-0.346960\pi$$
0.462478 + 0.886631i $$0.346960\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 3.00000 0.0955395
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ 12.0000 0.381578
$$990$$ 0 0
$$991$$ 7.00000 0.222362 0.111181 0.993800i $$-0.464537\pi$$
0.111181 + 0.993800i $$0.464537\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 0 0
$$994$$ 36.0000 1.14185
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ −18.0000 −0.570066 −0.285033 0.958518i $$-0.592005\pi$$
−0.285033 + 0.958518i $$0.592005\pi$$
$$998$$ 0 0
$$999$$ 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 3330.2.a.b.1.1 1
3.2 odd 2 1110.2.a.j.1.1 1
12.11 even 2 8880.2.a.bc.1.1 1
15.14 odd 2 5550.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.j.1.1 1 3.2 odd 2
3330.2.a.b.1.1 1 1.1 even 1 trivial
5550.2.a.t.1.1 1 15.14 odd 2
8880.2.a.bc.1.1 1 12.11 even 2