# Properties

 Label 3822.2.a.s.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$30.5188236525$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -3.00000 q^{20} +3.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -9.00000 q^{29} +3.00000 q^{30} -5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -3.00000 q^{40} +12.0000 q^{41} -4.00000 q^{43} +3.00000 q^{44} -3.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -9.00000 q^{55} -4.00000 q^{57} -9.00000 q^{58} +9.00000 q^{59} +3.00000 q^{60} -8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -3.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} -4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} -1.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -3.00000 q^{83} +18.0000 q^{85} -4.00000 q^{86} +9.00000 q^{87} +3.00000 q^{88} -3.00000 q^{90} +6.00000 q^{92} +5.00000 q^{93} +12.0000 q^{94} -12.0000 q^{95} -1.00000 q^{96} -5.00000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −3.00000 −0.948683
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ 3.00000 0.639602
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 4.00000 0.800000
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 3.00000 0.547723
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −3.00000 −0.522233
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 1.00000 0.160128
$$40$$ −3.00000 −0.474342
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 3.00000 0.452267
$$45$$ −3.00000 −0.447214
$$46$$ 6.00000 0.884652
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ 6.00000 0.840168
$$52$$ −1.00000 −0.138675
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −9.00000 −1.21356
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ −9.00000 −1.18176
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 3.00000 0.387298
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 3.00000 0.372104
$$66$$ −3.00000 −0.369274
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ −4.00000 −0.461880
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 1.00000 0.111111
$$82$$ 12.0000 1.32518
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 18.0000 1.95237
$$86$$ −4.00000 −0.431331
$$87$$ 9.00000 0.964901
$$88$$ 3.00000 0.319801
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 5.00000 0.518476
$$94$$ 12.0000 1.23771
$$95$$ −12.0000 −1.23117
$$96$$ −1.00000 −0.102062
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 4.00000 0.400000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ −9.00000 −0.858116
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −18.0000 −1.67851
$$116$$ −9.00000 −0.835629
$$117$$ −1.00000 −0.0924500
$$118$$ 9.00000 0.828517
$$119$$ 0 0
$$120$$ 3.00000 0.273861
$$121$$ −2.00000 −0.181818
$$122$$ −8.00000 −0.724286
$$123$$ −12.0000 −1.08200
$$124$$ −5.00000 −0.449013
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 3.00000 0.263117
$$131$$ −21.0000 −1.83478 −0.917389 0.397991i $$-0.869707\pi$$
−0.917389 + 0.397991i $$0.869707\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 3.00000 0.258199
$$136$$ −6.00000 −0.514496
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 6.00000 0.503509
$$143$$ −3.00000 −0.250873
$$144$$ 1.00000 0.0833333
$$145$$ 27.0000 2.24223
$$146$$ −14.0000 −1.15865
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −4.00000 −0.326599
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 15.0000 1.20483
$$156$$ 1.00000 0.0800641
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −1.00000 −0.0795557
$$159$$ 9.00000 0.713746
$$160$$ −3.00000 −0.237171
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 9.00000 0.700649
$$166$$ −3.00000 −0.232845
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 18.0000 1.38054
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 9.00000 0.682288
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ −9.00000 −0.676481
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ −3.00000 −0.223607
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ 8.00000 0.591377
$$184$$ 6.00000 0.442326
$$185$$ 12.0000 0.882258
$$186$$ 5.00000 0.366618
$$187$$ −18.0000 −1.31629
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ −12.0000 −0.870572
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ −5.00000 −0.358979
$$195$$ −3.00000 −0.214834
$$196$$ 0 0
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 4.00000 0.282138
$$202$$ −18.0000 −1.26648
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ −36.0000 −2.51435
$$206$$ −8.00000 −0.557386
$$207$$ 6.00000 0.417029
$$208$$ −1.00000 −0.0693375
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ −6.00000 −0.411113
$$214$$ 3.00000 0.205076
$$215$$ 12.0000 0.818393
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −16.0000 −1.08366
$$219$$ 14.0000 0.946032
$$220$$ −9.00000 −0.606780
$$221$$ 6.00000 0.403604
$$222$$ 4.00000 0.268462
$$223$$ 1.00000 0.0669650 0.0334825 0.999439i $$-0.489340\pi$$
0.0334825 + 0.999439i $$0.489340\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ −12.0000 −0.798228
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ −18.0000 −1.18688
$$231$$ 0 0
$$232$$ −9.00000 −0.590879
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ −36.0000 −2.34838
$$236$$ 9.00000 0.585850
$$237$$ 1.00000 0.0649570
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 3.00000 0.193649
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ −1.00000 −0.0641500
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ −4.00000 −0.254514
$$248$$ −5.00000 −0.317500
$$249$$ 3.00000 0.190117
$$250$$ 3.00000 0.189737
$$251$$ 3.00000 0.189358 0.0946792 0.995508i $$-0.469817\pi$$
0.0946792 + 0.995508i $$0.469817\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ −7.00000 −0.439219
$$255$$ −18.0000 −1.12720
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 3.00000 0.186052
$$261$$ −9.00000 −0.557086
$$262$$ −21.0000 −1.29738
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 27.0000 1.65860
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 3.00000 0.182574
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 12.0000 0.723627
$$276$$ −6.00000 −0.361158
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 12.0000 0.710819
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 27.0000 1.58549
$$291$$ 5.00000 0.293105
$$292$$ −14.0000 −0.819288
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 0 0
$$295$$ −27.0000 −1.57200
$$296$$ −4.00000 −0.232495
$$297$$ −3.00000 −0.174078
$$298$$ 6.00000 0.347571
$$299$$ −6.00000 −0.346989
$$300$$ −4.00000 −0.230940
$$301$$ 0 0
$$302$$ −19.0000 −1.09333
$$303$$ 18.0000 1.03407
$$304$$ 4.00000 0.229416
$$305$$ 24.0000 1.37424
$$306$$ −6.00000 −0.342997
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 15.0000 0.851943
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 19.0000 1.07394 0.536972 0.843600i $$-0.319568\pi$$
0.536972 + 0.843600i $$0.319568\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −1.00000 −0.0562544
$$317$$ −15.0000 −0.842484 −0.421242 0.906948i $$-0.638406\pi$$
−0.421242 + 0.906948i $$0.638406\pi$$
$$318$$ 9.00000 0.504695
$$319$$ −27.0000 −1.51171
$$320$$ −3.00000 −0.167705
$$321$$ −3.00000 −0.167444
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ 8.00000 0.443079
$$327$$ 16.0000 0.884802
$$328$$ 12.0000 0.662589
$$329$$ 0 0
$$330$$ 9.00000 0.495434
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ −3.00000 −0.164646
$$333$$ −4.00000 −0.219199
$$334$$ −12.0000 −0.656611
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −31.0000 −1.68868 −0.844339 0.535810i $$-0.820006\pi$$
−0.844339 + 0.535810i $$0.820006\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 12.0000 0.651751
$$340$$ 18.0000 0.976187
$$341$$ −15.0000 −0.812296
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 18.0000 0.969087
$$346$$ −6.00000 −0.322562
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 3.00000 0.159901
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ −9.00000 −0.478345
$$355$$ −18.0000 −0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 16.0000 0.840941
$$363$$ 2.00000 0.104973
$$364$$ 0 0
$$365$$ 42.0000 2.19838
$$366$$ 8.00000 0.418167
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 12.0000 0.624695
$$370$$ 12.0000 0.623850
$$371$$ 0 0
$$372$$ 5.00000 0.259238
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ −3.00000 −0.154919
$$376$$ 12.0000 0.618853
$$377$$ 9.00000 0.463524
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ −12.0000 −0.615587
$$381$$ 7.00000 0.358621
$$382$$ 6.00000 0.306987
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −1.00000 −0.0508987
$$387$$ −4.00000 −0.203331
$$388$$ −5.00000 −0.253837
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ −3.00000 −0.151911
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 21.0000 1.05931
$$394$$ −18.0000 −0.906827
$$395$$ 3.00000 0.150946
$$396$$ 3.00000 0.150756
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 5.00000 0.249068
$$404$$ −18.0000 −0.895533
$$405$$ −3.00000 −0.149071
$$406$$ 0 0
$$407$$ −12.0000 −0.594818
$$408$$ 6.00000 0.297044
$$409$$ 7.00000 0.346128 0.173064 0.984911i $$-0.444633\pi$$
0.173064 + 0.984911i $$0.444633\pi$$
$$410$$ −36.0000 −1.77791
$$411$$ 12.0000 0.591916
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 9.00000 0.441793
$$416$$ −1.00000 −0.0490290
$$417$$ −4.00000 −0.195881
$$418$$ 12.0000 0.586939
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 12.0000 0.583460
$$424$$ −9.00000 −0.437079
$$425$$ −24.0000 −1.16417
$$426$$ −6.00000 −0.290701
$$427$$ 0 0
$$428$$ 3.00000 0.145010
$$429$$ 3.00000 0.144841
$$430$$ 12.0000 0.578691
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ −27.0000 −1.29455
$$436$$ −16.0000 −0.766261
$$437$$ 24.0000 1.14808
$$438$$ 14.0000 0.668946
$$439$$ 1.00000 0.0477274 0.0238637 0.999715i $$-0.492403\pi$$
0.0238637 + 0.999715i $$0.492403\pi$$
$$440$$ −9.00000 −0.429058
$$441$$ 0 0
$$442$$ 6.00000 0.285391
$$443$$ −3.00000 −0.142534 −0.0712672 0.997457i $$-0.522704\pi$$
−0.0712672 + 0.997457i $$0.522704\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ 1.00000 0.0473514
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 4.00000 0.188562
$$451$$ 36.0000 1.69517
$$452$$ −12.0000 −0.564433
$$453$$ 19.0000 0.892698
$$454$$ 3.00000 0.140797
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −19.0000 −0.888783 −0.444391 0.895833i $$-0.646580\pi$$
−0.444391 + 0.895833i $$0.646580\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ 6.00000 0.280056
$$460$$ −18.0000 −0.839254
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ −15.0000 −0.695608
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 0 0
$$470$$ −36.0000 −1.66056
$$471$$ 2.00000 0.0921551
$$472$$ 9.00000 0.414259
$$473$$ −12.0000 −0.551761
$$474$$ 1.00000 0.0459315
$$475$$ 16.0000 0.734130
$$476$$ 0 0
$$477$$ −9.00000 −0.412082
$$478$$ 6.00000 0.274434
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 3.00000 0.136931
$$481$$ 4.00000 0.182384
$$482$$ −17.0000 −0.774329
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 15.0000 0.681115
$$486$$ −1.00000 −0.0453609
$$487$$ 23.0000 1.04223 0.521115 0.853487i $$-0.325516\pi$$
0.521115 + 0.853487i $$0.325516\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −15.0000 −0.676941 −0.338470 0.940977i $$-0.609909\pi$$
−0.338470 + 0.940977i $$0.609909\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ 54.0000 2.43204
$$494$$ −4.00000 −0.179969
$$495$$ −9.00000 −0.404520
$$496$$ −5.00000 −0.224507
$$497$$ 0 0
$$498$$ 3.00000 0.134433
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 12.0000 0.536120
$$502$$ 3.00000 0.133897
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ 0 0
$$505$$ 54.0000 2.40297
$$506$$ 18.0000 0.800198
$$507$$ −1.00000 −0.0444116
$$508$$ −7.00000 −0.310575
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ −18.0000 −0.797053
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 30.0000 1.32324
$$515$$ 24.0000 1.05757
$$516$$ 4.00000 0.176090
$$517$$ 36.0000 1.58328
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 3.00000 0.131559
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ −9.00000 −0.393919
$$523$$ −8.00000 −0.349816 −0.174908 0.984585i $$-0.555963\pi$$
−0.174908 + 0.984585i $$0.555963\pi$$
$$524$$ −21.0000 −0.917389
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ 30.0000 1.30682
$$528$$ −3.00000 −0.130558
$$529$$ 13.0000 0.565217
$$530$$ 27.0000 1.17281
$$531$$ 9.00000 0.390567
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ −4.00000 −0.172774
$$537$$ −12.0000 −0.517838
$$538$$ −3.00000 −0.129339
$$539$$ 0 0
$$540$$ 3.00000 0.129099
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 7.00000 0.300676
$$543$$ −16.0000 −0.686626
$$544$$ −6.00000 −0.257248
$$545$$ 48.0000 2.05609
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −8.00000 −0.341432
$$550$$ 12.0000 0.511682
$$551$$ −36.0000 −1.53365
$$552$$ −6.00000 −0.255377
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ −12.0000 −0.509372
$$556$$ 4.00000 0.169638
$$557$$ −21.0000 −0.889799 −0.444899 0.895581i $$-0.646761\pi$$
−0.444899 + 0.895581i $$0.646761\pi$$
$$558$$ −5.00000 −0.211667
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ −33.0000 −1.39078 −0.695392 0.718631i $$-0.744769\pi$$
−0.695392 + 0.718631i $$0.744769\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 36.0000 1.51453
$$566$$ −2.00000 −0.0840663
$$567$$ 0 0
$$568$$ 6.00000 0.251754
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ 12.0000 0.502625
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ 24.0000 1.00087
$$576$$ 1.00000 0.0416667
$$577$$ −17.0000 −0.707719 −0.353860 0.935299i $$-0.615131\pi$$
−0.353860 + 0.935299i $$0.615131\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 1.00000 0.0415586
$$580$$ 27.0000 1.12111
$$581$$ 0 0
$$582$$ 5.00000 0.207257
$$583$$ −27.0000 −1.11823
$$584$$ −14.0000 −0.579324
$$585$$ 3.00000 0.124035
$$586$$ −21.0000 −0.867502
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ 0 0
$$589$$ −20.0000 −0.824086
$$590$$ −27.0000 −1.11157
$$591$$ 18.0000 0.740421
$$592$$ −4.00000 −0.164399
$$593$$ 12.0000 0.492781 0.246390 0.969171i $$-0.420755\pi$$
0.246390 + 0.969171i $$0.420755\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 8.00000 0.327418
$$598$$ −6.00000 −0.245358
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ −4.00000 −0.163299
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ −19.0000 −0.773099
$$605$$ 6.00000 0.243935
$$606$$ 18.0000 0.731200
$$607$$ −5.00000 −0.202944 −0.101472 0.994838i $$-0.532355\pi$$
−0.101472 + 0.994838i $$0.532355\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 24.0000 0.971732
$$611$$ −12.0000 −0.485468
$$612$$ −6.00000 −0.242536
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 36.0000 1.45166
$$616$$ 0 0
$$617$$ 48.0000 1.93241 0.966204 0.257780i $$-0.0829910\pi$$
0.966204 + 0.257780i $$0.0829910\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 15.0000 0.602414
$$621$$ −6.00000 −0.240772
$$622$$ 18.0000 0.721734
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ −29.0000 −1.16000
$$626$$ 19.0000 0.759393
$$627$$ −12.0000 −0.479234
$$628$$ −2.00000 −0.0798087
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −13.0000 −0.517522 −0.258761 0.965941i $$-0.583314\pi$$
−0.258761 + 0.965941i $$0.583314\pi$$
$$632$$ −1.00000 −0.0397779
$$633$$ −8.00000 −0.317971
$$634$$ −15.0000 −0.595726
$$635$$ 21.0000 0.833360
$$636$$ 9.00000 0.356873
$$637$$ 0 0
$$638$$ −27.0000 −1.06894
$$639$$ 6.00000 0.237356
$$640$$ −3.00000 −0.118585
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ −3.00000 −0.118401
$$643$$ −8.00000 −0.315489 −0.157745 0.987480i $$-0.550422\pi$$
−0.157745 + 0.987480i $$0.550422\pi$$
$$644$$ 0 0
$$645$$ −12.0000 −0.472500
$$646$$ −24.0000 −0.944267
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 27.0000 1.05984
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 63.0000 2.46161
$$656$$ 12.0000 0.468521
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 9.00000 0.350325
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 32.0000 1.24372
$$663$$ −6.00000 −0.233021
$$664$$ −3.00000 −0.116423
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ −54.0000 −2.09089
$$668$$ −12.0000 −0.464294
$$669$$ −1.00000 −0.0386622
$$670$$ 12.0000 0.463600
$$671$$ −24.0000 −0.926510
$$672$$ 0 0
$$673$$ −31.0000 −1.19496 −0.597481 0.801883i $$-0.703832\pi$$
−0.597481 + 0.801883i $$0.703832\pi$$
$$674$$ −31.0000 −1.19408
$$675$$ −4.00000 −0.153960
$$676$$ 1.00000 0.0384615
$$677$$ −9.00000 −0.345898 −0.172949 0.984931i $$-0.555330\pi$$
−0.172949 + 0.984931i $$0.555330\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 0 0
$$680$$ 18.0000 0.690268
$$681$$ −3.00000 −0.114960
$$682$$ −15.0000 −0.574380
$$683$$ −39.0000 −1.49229 −0.746147 0.665782i $$-0.768098\pi$$
−0.746147 + 0.665782i $$0.768098\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 36.0000 1.37549
$$686$$ 0 0
$$687$$ 26.0000 0.991962
$$688$$ −4.00000 −0.152499
$$689$$ 9.00000 0.342873
$$690$$ 18.0000 0.685248
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ −12.0000 −0.455186
$$696$$ 9.00000 0.341144
$$697$$ −72.0000 −2.72719
$$698$$ 34.0000 1.28692
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 15.0000 0.566542 0.283271 0.959040i $$-0.408580\pi$$
0.283271 + 0.959040i $$0.408580\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −16.0000 −0.603451
$$704$$ 3.00000 0.113067
$$705$$ 36.0000 1.35584
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ −9.00000 −0.338241
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ −18.0000 −0.675528
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ −30.0000 −1.12351
$$714$$ 0 0
$$715$$ 9.00000 0.336581
$$716$$ 12.0000 0.448461
$$717$$ −6.00000 −0.224074
$$718$$ −6.00000 −0.223918
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ −3.00000 −0.111803
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 17.0000 0.632237
$$724$$ 16.0000 0.594635
$$725$$ −36.0000 −1.33701
$$726$$ 2.00000 0.0742270
$$727$$ 37.0000 1.37225 0.686127 0.727482i $$-0.259309\pi$$
0.686127 + 0.727482i $$0.259309\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 42.0000 1.55449
$$731$$ 24.0000 0.887672
$$732$$ 8.00000 0.295689
$$733$$ −50.0000 −1.84679 −0.923396 0.383849i $$-0.874598\pi$$
−0.923396 + 0.383849i $$0.874598\pi$$
$$734$$ 7.00000 0.258375
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −12.0000 −0.442026
$$738$$ 12.0000 0.441726
$$739$$ 38.0000 1.39785 0.698926 0.715194i $$-0.253662\pi$$
0.698926 + 0.715194i $$0.253662\pi$$
$$740$$ 12.0000 0.441129
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 5.00000 0.183309
$$745$$ −18.0000 −0.659469
$$746$$ 14.0000 0.512576
$$747$$ −3.00000 −0.109764
$$748$$ −18.0000 −0.658145
$$749$$ 0 0
$$750$$ −3.00000 −0.109545
$$751$$ −31.0000 −1.13121 −0.565603 0.824678i $$-0.691357\pi$$
−0.565603 + 0.824678i $$0.691357\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −3.00000 −0.109326
$$754$$ 9.00000 0.327761
$$755$$ 57.0000 2.07444
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ −18.0000 −0.653359
$$760$$ −12.0000 −0.435286
$$761$$ −36.0000 −1.30500 −0.652499 0.757789i $$-0.726280\pi$$
−0.652499 + 0.757789i $$0.726280\pi$$
$$762$$ 7.00000 0.253583
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 18.0000 0.650791
$$766$$ 6.00000 0.216789
$$767$$ −9.00000 −0.324971
$$768$$ −1.00000 −0.0360844
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ −1.00000 −0.0359908
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −20.0000 −0.718421
$$776$$ −5.00000 −0.179490
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 48.0000 1.71978
$$780$$ −3.00000 −0.107417
$$781$$ 18.0000 0.644091
$$782$$ −36.0000 −1.28736
$$783$$ 9.00000 0.321634
$$784$$ 0 0
$$785$$ 6.00000 0.214149
$$786$$ 21.0000 0.749045
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −6.00000 −0.213606
$$790$$ 3.00000 0.106735
$$791$$ 0 0
$$792$$ 3.00000 0.106600
$$793$$ 8.00000 0.284088
$$794$$ −2.00000 −0.0709773
$$795$$ −27.0000 −0.957591
$$796$$ −8.00000 −0.283552
$$797$$ −9.00000 −0.318796 −0.159398 0.987214i $$-0.550955\pi$$
−0.159398 + 0.987214i $$0.550955\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 4.00000 0.141421
$$801$$ 0 0
$$802$$ 18.0000 0.635602
$$803$$ −42.0000 −1.48215
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 5.00000 0.176117
$$807$$ 3.00000 0.105605
$$808$$ −18.0000 −0.633238
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ −3.00000 −0.105409
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 0 0
$$813$$ −7.00000 −0.245501
$$814$$ −12.0000 −0.420600
$$815$$ −24.0000 −0.840683
$$816$$ 6.00000 0.210042
$$817$$ −16.0000 −0.559769
$$818$$ 7.00000 0.244749
$$819$$ 0 0
$$820$$ −36.0000 −1.25717
$$821$$ −39.0000 −1.36111 −0.680555 0.732697i $$-0.738261\pi$$
−0.680555 + 0.732697i $$0.738261\pi$$
$$822$$ 12.0000 0.418548
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ 3.00000 0.104320 0.0521601 0.998639i $$-0.483389\pi$$
0.0521601 + 0.998639i $$0.483389\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 9.00000 0.312395
$$831$$ −2.00000 −0.0693792
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ 36.0000 1.24583
$$836$$ 12.0000 0.415029
$$837$$ 5.00000 0.172825
$$838$$ −24.0000 −0.829066
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 8.00000 0.275698
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ −3.00000 −0.103203
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ −9.00000 −0.309061
$$849$$ 2.00000 0.0686398
$$850$$ −24.0000 −0.823193
$$851$$ −24.0000 −0.822709
$$852$$ −6.00000 −0.205557
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ −12.0000 −0.410391
$$856$$ 3.00000 0.102538
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 3.00000 0.102418
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 12.0000 0.409197
$$861$$ 0 0
$$862$$ 30.0000 1.02180
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 18.0000 0.612018
$$866$$ −2.00000 −0.0679628
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ −3.00000 −0.101768
$$870$$ −27.0000 −0.915386
$$871$$ 4.00000 0.135535
$$872$$ −16.0000 −0.541828
$$873$$ −5.00000 −0.169224
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ −58.0000 −1.95852 −0.979260 0.202606i $$-0.935059\pi$$
−0.979260 + 0.202606i $$0.935059\pi$$
$$878$$ 1.00000 0.0337484
$$879$$ 21.0000 0.708312
$$880$$ −9.00000 −0.303390
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ 26.0000 0.874970 0.437485 0.899226i $$-0.355869\pi$$
0.437485 + 0.899226i $$0.355869\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 27.0000 0.907595
$$886$$ −3.00000 −0.100787
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 4.00000 0.134231
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 1.00000 0.0334825
$$893$$ 48.0000 1.60626
$$894$$ −6.00000 −0.200670
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ −12.0000 −0.400445
$$899$$ 45.0000 1.50083
$$900$$ 4.00000 0.133333
$$901$$ 54.0000 1.79900
$$902$$ 36.0000 1.19867
$$903$$ 0 0
$$904$$ −12.0000 −0.399114
$$905$$ −48.0000 −1.59557
$$906$$ 19.0000 0.631233
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 3.00000 0.0995585
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 54.0000 1.78910 0.894550 0.446968i $$-0.147496\pi$$
0.894550 + 0.446968i $$0.147496\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −9.00000 −0.297857
$$914$$ −19.0000 −0.628464
$$915$$ −24.0000 −0.793416
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 6.00000 0.198030
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ −18.0000 −0.593442
$$921$$ 2.00000 0.0659022
$$922$$ −18.0000 −0.592798
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ −16.0000 −0.525793
$$927$$ −8.00000 −0.262754
$$928$$ −9.00000 −0.295439
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ −15.0000 −0.491869
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ −18.0000 −0.589294
$$934$$ 36.0000 1.17796
$$935$$ 54.0000 1.76599
$$936$$ −1.00000 −0.0326860
$$937$$ 13.0000 0.424691 0.212346 0.977195i $$-0.431890\pi$$
0.212346 + 0.977195i $$0.431890\pi$$
$$938$$ 0 0
$$939$$ −19.0000 −0.620042
$$940$$ −36.0000 −1.17419
$$941$$ 45.0000 1.46696 0.733479 0.679712i $$-0.237895\pi$$
0.733479 + 0.679712i $$0.237895\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ 72.0000 2.34464
$$944$$ 9.00000 0.292925
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 60.0000 1.94974 0.974869 0.222779i $$-0.0715128\pi$$
0.974869 + 0.222779i $$0.0715128\pi$$
$$948$$ 1.00000 0.0324785
$$949$$ 14.0000 0.454459
$$950$$ 16.0000 0.519109
$$951$$ 15.0000 0.486408
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ −9.00000 −0.291386
$$955$$ −18.0000 −0.582466
$$956$$ 6.00000 0.194054
$$957$$ 27.0000 0.872786
$$958$$ −6.00000 −0.193851
$$959$$ 0 0
$$960$$ 3.00000 0.0968246
$$961$$ −6.00000 −0.193548
$$962$$ 4.00000 0.128965
$$963$$ 3.00000 0.0966736
$$964$$ −17.0000 −0.547533
$$965$$ 3.00000 0.0965734
$$966$$ 0 0
$$967$$ 29.0000 0.932577 0.466289 0.884633i $$-0.345591\pi$$
0.466289 + 0.884633i $$0.345591\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 24.0000 0.770991
$$970$$ 15.0000 0.481621
$$971$$ 33.0000 1.05902 0.529510 0.848304i $$-0.322376\pi$$
0.529510 + 0.848304i $$0.322376\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 23.0000 0.736968
$$975$$ 4.00000 0.128103
$$976$$ −8.00000 −0.256074
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ −15.0000 −0.478669
$$983$$ −18.0000 −0.574111 −0.287055 0.957914i $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 54.0000 1.72058
$$986$$ 54.0000 1.71971
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ −24.0000 −0.763156
$$990$$ −9.00000 −0.286039
$$991$$ 35.0000 1.11181 0.555906 0.831245i $$-0.312372\pi$$
0.555906 + 0.831245i $$0.312372\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ −32.0000 −1.01549
$$994$$ 0 0
$$995$$ 24.0000 0.760851
$$996$$ 3.00000 0.0950586
$$997$$ 16.0000 0.506725 0.253363 0.967371i $$-0.418463\pi$$
0.253363 + 0.967371i $$0.418463\pi$$
$$998$$ −16.0000 −0.506471
$$999$$ 4.00000 0.126554
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 3822.2.a.s.1.1 1
7.3 odd 6 546.2.i.a.79.1 2
7.5 odd 6 546.2.i.a.235.1 yes 2
7.6 odd 2 3822.2.a.bh.1.1 1
21.5 even 6 1638.2.j.j.235.1 2
21.17 even 6 1638.2.j.j.1171.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.a.79.1 2 7.3 odd 6
546.2.i.a.235.1 yes 2 7.5 odd 6
1638.2.j.j.235.1 2 21.5 even 6
1638.2.j.j.1171.1 2 21.17 even 6
3822.2.a.s.1.1 1 1.1 even 1 trivial
3822.2.a.bh.1.1 1 7.6 odd 2