# Properties

 Label 378.2.a.b.1.1 Level $378$ Weight $2$ Character 378.1 Self dual yes Analytic conductor $3.018$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.01834519640$$ 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 $$\chi$$ $$=$$ 378.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +7.00000 q^{38} +3.00000 q^{40} -9.00000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -4.00000 q^{52} +12.0000 q^{53} -9.00000 q^{55} -1.00000 q^{56} -6.00000 q^{59} +8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +3.00000 q^{70} +9.00000 q^{71} +2.00000 q^{73} +7.00000 q^{74} -7.00000 q^{76} +3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +9.00000 q^{82} +18.0000 q^{85} +10.0000 q^{86} -3.00000 q^{88} +15.0000 q^{89} -4.00000 q^{91} -3.00000 q^{92} -6.00000 q^{94} +21.0000 q^{95} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 3.00000 0.948683
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 7.00000 1.13555
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ −9.00000 −1.21356
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 3.00000 0.358569
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 0 0
$$82$$ 9.00000 0.993884
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 18.0000 1.95237
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −3.00000 −0.312772
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 21.0000 2.15455
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 5.00000 0.492665 0.246332 0.969185i $$-0.420775\pi$$
0.246332 + 0.969185i $$0.420775\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 9.00000 0.858116
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 9.00000 0.839254
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ 5.00000 0.449013
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −12.0000 −1.05247
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ −7.00000 −0.606977
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −3.00000 −0.253546
$$141$$ 0 0
$$142$$ −9.00000 −0.755263
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 7.00000 0.567775
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ −15.0000 −1.20483
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 0 0
$$160$$ 3.00000 0.237171
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ −18.0000 −1.38054
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −15.0000 −1.12430
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ 3.00000 0.221163
$$185$$ 21.0000 1.54395
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ −21.0000 −1.52350
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 27.0000 1.88576
$$206$$ −5.00000 −0.348367
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 30.0000 2.04598
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ −11.0000 −0.745014
$$219$$ 0 0
$$220$$ −9.00000 −0.606780
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 0 0
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ −9.00000 −0.593442
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −18.0000 −1.17419
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ 6.00000 0.388922
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ 8.00000 0.512148
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 28.0000 1.78160
$$248$$ −5.00000 −0.317500
$$249$$ 0 0
$$250$$ −3.00000 −0.189737
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −9.00000 −0.565825
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −21.0000 −1.30994 −0.654972 0.755653i $$-0.727320\pi$$
−0.654972 + 0.755653i $$0.727320\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ 12.0000 0.744208
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ 0 0
$$265$$ −36.0000 −2.21146
$$266$$ 7.00000 0.429198
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −15.0000 −0.914566 −0.457283 0.889321i $$-0.651177\pi$$
−0.457283 + 0.889321i $$0.651177\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ 17.0000 1.02143 0.510716 0.859750i $$-0.329381\pi$$
0.510716 + 0.859750i $$0.329381\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 3.00000 0.179284
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 9.00000 0.534052
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ −9.00000 −0.531253
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 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
$$295$$ 18.0000 1.04800
$$296$$ 7.00000 0.406867
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −10.0000 −0.576390
$$302$$ 10.0000 0.575435
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ −24.0000 −1.37424
$$306$$ 0 0
$$307$$ 29.0000 1.65512 0.827559 0.561379i $$-0.189729\pi$$
0.827559 + 0.561379i $$0.189729\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 15.0000 0.851943
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −3.00000 −0.167705
$$321$$ 0 0
$$322$$ 3.00000 0.167183
$$323$$ 42.0000 2.33694
$$324$$ 0 0
$$325$$ −16.0000 −0.887520
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ 9.00000 0.496942
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 18.0000 0.984916
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 18.0000 0.976187
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 10.0000 0.539164
$$345$$ 0 0
$$346$$ 21.0000 1.12897
$$347$$ 3.00000 0.161048 0.0805242 0.996753i $$-0.474341\pi$$
0.0805242 + 0.996753i $$0.474341\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 0 0
$$355$$ −27.0000 −1.43301
$$356$$ 15.0000 0.794998
$$357$$ 0 0
$$358$$ 24.0000 1.26844
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ −3.00000 −0.156386
$$369$$ 0 0
$$370$$ −21.0000 −1.09174
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ −13.0000 −0.673114 −0.336557 0.941663i $$-0.609263\pi$$
−0.336557 + 0.941663i $$0.609263\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 21.0000 1.07728
$$381$$ 0 0
$$382$$ −15.0000 −0.767467
$$383$$ −30.0000 −1.53293 −0.766464 0.642287i $$-0.777986\pi$$
−0.766464 + 0.642287i $$0.777986\pi$$
$$384$$ 0 0
$$385$$ −9.00000 −0.458682
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ 8.00000 0.406138
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 30.0000 1.50946
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 7.00000 0.350878
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −21.0000 −1.04093
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ −27.0000 −1.33343
$$411$$ 0 0
$$412$$ 5.00000 0.246332
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ 21.0000 1.02714
$$419$$ 18.0000 0.879358 0.439679 0.898155i $$-0.355092\pi$$
0.439679 + 0.898155i $$0.355092\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ −14.0000 −0.681509
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ −24.0000 −1.16417
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −30.0000 −1.44673
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 21.0000 1.00457
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 9.00000 0.429058
$$441$$ 0 0
$$442$$ −24.0000 −1.14156
$$443$$ −3.00000 −0.142534 −0.0712672 0.997457i $$-0.522704\pi$$
−0.0712672 + 0.997457i $$0.522704\pi$$
$$444$$ 0 0
$$445$$ −45.0000 −2.13320
$$446$$ 1.00000 0.0473514
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ −19.0000 −0.888783 −0.444391 0.895833i $$-0.646580\pi$$
−0.444391 + 0.895833i $$0.646580\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 0 0
$$460$$ 9.00000 0.419627
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 0 0
$$463$$ 14.0000 0.650635 0.325318 0.945605i $$-0.394529\pi$$
0.325318 + 0.945605i $$0.394529\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 18.0000 0.830278
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ −30.0000 −1.37940
$$474$$ 0 0
$$475$$ −28.0000 −1.28473
$$476$$ −6.00000 −0.275010
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 28.0000 1.27669
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ −24.0000 −1.08978
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 0 0
$$490$$ 3.00000 0.135526
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −28.0000 −1.25978
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ 9.00000 0.403705
$$498$$ 0 0
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 9.00000 0.400099
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 21.0000 0.926270
$$515$$ −15.0000 −0.660979
$$516$$ 0 0
$$517$$ 18.0000 0.791639
$$518$$ 7.00000 0.307562
$$519$$ 0 0
$$520$$ −12.0000 −0.526235
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 15.0000 0.654031
$$527$$ −30.0000 −1.30682
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 36.0000 1.56374
$$531$$ 0 0
$$532$$ −7.00000 −0.303488
$$533$$ 36.0000 1.55933
$$534$$ 0 0
$$535$$ 36.0000 1.55642
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ 15.0000 0.646696
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ −7.00000 −0.300954 −0.150477 0.988614i $$-0.548081\pi$$
−0.150477 + 0.988614i $$0.548081\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 6.00000 0.257248
$$545$$ −33.0000 −1.41356
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 0 0
$$550$$ −12.0000 −0.511682
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ −17.0000 −0.722261
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ −3.00000 −0.126773
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 54.0000 2.27180
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −9.00000 −0.377632
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 14.0000 0.585882 0.292941 0.956131i $$-0.405366\pi$$
0.292941 + 0.956131i $$0.405366\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ 9.00000 0.375653
$$575$$ −12.0000 −0.500435
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ −35.0000 −1.44215
$$590$$ −18.0000 −0.741048
$$591$$ 0 0
$$592$$ −7.00000 −0.287698
$$593$$ 33.0000 1.35515 0.677574 0.735455i $$-0.263031\pi$$
0.677574 + 0.735455i $$0.263031\pi$$
$$594$$ 0 0
$$595$$ 18.0000 0.737928
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −12.0000 −0.490716
$$599$$ −33.0000 −1.34834 −0.674172 0.738575i $$-0.735499\pi$$
−0.674172 + 0.738575i $$0.735499\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 10.0000 0.407570
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 6.00000 0.243935
$$606$$ 0 0
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 7.00000 0.283887
$$609$$ 0 0
$$610$$ 24.0000 0.971732
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 29.0000 1.17130 0.585649 0.810564i $$-0.300840\pi$$
0.585649 + 0.810564i $$0.300840\pi$$
$$614$$ −29.0000 −1.17034
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ 35.0000 1.40677 0.703384 0.710810i $$-0.251671\pi$$
0.703384 + 0.710810i $$0.251671\pi$$
$$620$$ −15.0000 −0.602414
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 15.0000 0.600962
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 28.0000 1.11911
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ 42.0000 1.67465
$$630$$ 0 0
$$631$$ 38.0000 1.51276 0.756378 0.654135i $$-0.226967\pi$$
0.756378 + 0.654135i $$0.226967\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 0 0
$$634$$ 30.0000 1.19145
$$635$$ −60.0000 −2.38103
$$636$$ 0 0
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 0 0
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ −42.0000 −1.65247
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ 0 0
$$649$$ −18.0000 −0.706562
$$650$$ 16.0000 0.627572
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ −18.0000 −0.703318
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ −6.00000 −0.233904
$$659$$ −9.00000 −0.350590 −0.175295 0.984516i $$-0.556088\pi$$
−0.175295 + 0.984516i $$0.556088\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 21.0000 0.814345
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −18.0000 −0.696441
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ 13.0000 0.500741
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 15.0000 0.576497 0.288248 0.957556i $$-0.406927\pi$$
0.288248 + 0.957556i $$0.406927\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ −18.0000 −0.690268
$$681$$ 0 0
$$682$$ −15.0000 −0.574380
$$683$$ −15.0000 −0.573959 −0.286980 0.957937i $$-0.592651\pi$$
−0.286980 + 0.957937i $$0.592651\pi$$
$$684$$ 0 0
$$685$$ −36.0000 −1.37549
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −10.0000 −0.381246
$$689$$ −48.0000 −1.82865
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −21.0000 −0.798300
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 54.0000 2.04540
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 49.0000 1.84807
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 27.0000 1.01329
$$711$$ 0 0
$$712$$ −15.0000 −0.562149
$$713$$ −15.0000 −0.561754
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ −24.0000 −0.896922
$$717$$ 0 0
$$718$$ 12.0000 0.447836
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 5.00000 0.186210
$$722$$ −30.0000 −1.11648
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 0 0
$$730$$ 6.00000 0.222070
$$731$$ 60.0000 2.21918
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −17.0000 −0.627481
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 21.0000 0.771975
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 13.0000 0.475964
$$747$$ 0 0
$$748$$ −18.0000 −0.658145
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −2.00000 −0.0726433
$$759$$ 0 0
$$760$$ −21.0000 −0.761750
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 11.0000 0.398227
$$764$$ 15.0000 0.542681
$$765$$ 0 0
$$766$$ 30.0000 1.08394
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −4.00000 −0.144244 −0.0721218 0.997396i $$-0.522977\pi$$
−0.0721218 + 0.997396i $$0.522977\pi$$
$$770$$ 9.00000 0.324337
$$771$$ 0 0
$$772$$ 14.0000 0.503871
$$773$$ 39.0000 1.40273 0.701366 0.712801i $$-0.252574\pi$$
0.701366 + 0.712801i $$0.252574\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ −8.00000 −0.287183
$$777$$ 0 0
$$778$$ −12.0000 −0.430221
$$779$$ 63.0000 2.25721
$$780$$ 0 0
$$781$$ 27.0000 0.966136
$$782$$ −18.0000 −0.643679
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 12.0000 0.428298
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ −30.0000 −1.06735
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ −32.0000 −1.13635
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −7.00000 −0.248108
$$797$$ 27.0000 0.956389 0.478195 0.878254i $$-0.341291\pi$$
0.478195 + 0.878254i $$0.341291\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ −4.00000 −0.141421
$$801$$ 0 0
$$802$$ −24.0000 −0.847469
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 9.00000 0.317208
$$806$$ 20.0000 0.704470
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ −24.0000 −0.843795 −0.421898 0.906644i $$-0.638636\pi$$
−0.421898 + 0.906644i $$0.638636\pi$$
$$810$$ 0 0
$$811$$ −25.0000 −0.877869 −0.438934 0.898519i $$-0.644644\pi$$
−0.438934 + 0.898519i $$0.644644\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 21.0000 0.736050
$$815$$ 48.0000 1.68137
$$816$$ 0 0
$$817$$ 70.0000 2.44899
$$818$$ 22.0000 0.769212
$$819$$ 0 0
$$820$$ 27.0000 0.942881
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ −5.00000 −0.174183
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 9.00000 0.312961 0.156480 0.987681i $$-0.449985\pi$$
0.156480 + 0.987681i $$0.449985\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.00000 −0.138675
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 54.0000 1.86875
$$836$$ −21.0000 −0.726300
$$837$$ 0 0
$$838$$ −18.0000 −0.621800
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −17.0000 −0.585859
$$843$$ 0 0
$$844$$ 14.0000 0.481900
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 12.0000 0.412082
$$849$$ 0 0
$$850$$ 24.0000 0.823193
$$851$$ 21.0000 0.719871
$$852$$ 0 0
$$853$$ 8.00000 0.273915 0.136957 0.990577i $$-0.456268\pi$$
0.136957 + 0.990577i $$0.456268\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ −31.0000 −1.05771 −0.528853 0.848713i $$-0.677378\pi$$
−0.528853 + 0.848713i $$0.677378\pi$$
$$860$$ 30.0000 1.02299
$$861$$ 0 0
$$862$$ −3.00000 −0.102180
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 63.0000 2.14206
$$866$$ 16.0000 0.543702
$$867$$ 0 0
$$868$$ 5.00000 0.169711
$$869$$ −30.0000 −1.01768
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −11.0000 −0.372507
$$873$$ 0 0
$$874$$ −21.0000 −0.710336
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ −9.00000 −0.303390
$$881$$ 9.00000 0.303218 0.151609 0.988441i $$-0.451555\pi$$
0.151609 + 0.988441i $$0.451555\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 3.00000 0.100787
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 45.0000 1.50840
$$891$$ 0 0
$$892$$ −1.00000 −0.0334825
$$893$$ −42.0000 −1.40548
$$894$$ 0 0
$$895$$ 72.0000 2.40669
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −72.0000 −2.39867
$$902$$ 27.0000 0.899002
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −46.0000 −1.52740 −0.763702 0.645568i $$-0.776621\pi$$
−0.763702 + 0.645568i $$0.776621\pi$$
$$908$$ −6.00000 −0.199117
$$909$$ 0 0
$$910$$ −12.0000 −0.397796
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 19.0000 0.628464
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ 6.00000 0.198137
$$918$$ 0 0
$$919$$ −52.0000 −1.71532 −0.857661 0.514216i $$-0.828083\pi$$
−0.857661 + 0.514216i $$0.828083\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ 0 0
$$922$$ −27.0000 −0.889198
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ −28.0000 −0.920634
$$926$$ −14.0000 −0.460069
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −6.00000 −0.196326
$$935$$ 54.0000 1.76599
$$936$$ 0 0
$$937$$ 20.0000 0.653372 0.326686 0.945133i $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 0 0
$$940$$ −18.0000 −0.587095
$$941$$ −15.0000 −0.488986 −0.244493 0.969651i $$-0.578622\pi$$
−0.244493 + 0.969651i $$0.578622\pi$$
$$942$$ 0 0
$$943$$ 27.0000 0.879241
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 30.0000 0.975384
$$947$$ −39.0000 −1.26733 −0.633665 0.773608i $$-0.718450\pi$$
−0.633665 + 0.773608i $$0.718450\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 28.0000 0.908440
$$951$$ 0 0
$$952$$ 6.00000 0.194461
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ −45.0000 −1.45617
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −28.0000 −0.902756
$$963$$ 0 0
$$964$$ 26.0000 0.837404
$$965$$ −42.0000 −1.35203
$$966$$ 0 0
$$967$$ −58.0000 −1.86515 −0.932577 0.360971i $$-0.882445\pi$$
−0.932577 + 0.360971i $$0.882445\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ 45.0000 1.43821
$$980$$ −3.00000 −0.0958315
$$981$$ 0 0
$$982$$ 9.00000 0.287202
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ 54.0000 1.72058
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 28.0000 0.890799
$$989$$ 30.0000 0.953945
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ 0 0
$$994$$ −9.00000 −0.285463
$$995$$ 21.0000 0.665745
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 22.0000 0.696398
$$999$$ 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 378.2.a.b.1.1 1
3.2 odd 2 378.2.a.g.1.1 yes 1
4.3 odd 2 3024.2.a.c.1.1 1
5.4 even 2 9450.2.a.cu.1.1 1
7.6 odd 2 2646.2.a.n.1.1 1
9.2 odd 6 1134.2.f.b.757.1 2
9.4 even 3 1134.2.f.o.379.1 2
9.5 odd 6 1134.2.f.b.379.1 2
9.7 even 3 1134.2.f.o.757.1 2
12.11 even 2 3024.2.a.bb.1.1 1
15.14 odd 2 9450.2.a.h.1.1 1
21.20 even 2 2646.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.b.1.1 1 1.1 even 1 trivial
378.2.a.g.1.1 yes 1 3.2 odd 2
1134.2.f.b.379.1 2 9.5 odd 6
1134.2.f.b.757.1 2 9.2 odd 6
1134.2.f.o.379.1 2 9.4 even 3
1134.2.f.o.757.1 2 9.7 even 3
2646.2.a.n.1.1 1 7.6 odd 2
2646.2.a.q.1.1 1 21.20 even 2
3024.2.a.c.1.1 1 4.3 odd 2
3024.2.a.bb.1.1 1 12.11 even 2
9450.2.a.h.1.1 1 15.14 odd 2
9450.2.a.cu.1.1 1 5.4 even 2