# Properties

 Label 3648.2.a.z.1.1 Level $3648$ Weight $2$ Character 3648.1 Self dual yes Analytic conductor $29.129$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.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: $$29.1294266574$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3648.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} -1.00000 q^{19} -3.00000 q^{21} +4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -10.0000 q^{31} +5.00000 q^{33} +3.00000 q^{35} +2.00000 q^{39} +11.0000 q^{43} -1.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -10.0000 q^{53} -5.00000 q^{55} -1.00000 q^{57} -4.00000 q^{59} +5.00000 q^{61} -3.00000 q^{63} -2.00000 q^{65} +4.00000 q^{67} +4.00000 q^{69} +8.00000 q^{71} +13.0000 q^{73} -4.00000 q^{75} -15.0000 q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +1.00000 q^{85} +6.00000 q^{87} -6.00000 q^{89} -6.00000 q^{91} -10.0000 q^{93} +1.00000 q^{95} +2.00000 q^{97} +5.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 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 $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 0 0
$$33$$ 5.00000 0.870388
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 13.0000 1.52153 0.760767 0.649025i $$-0.224823\pi$$
0.760767 + 0.649025i $$0.224823\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ −15.0000 −1.70941
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 1.00000 0.108465
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 0 0
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 3.00000 0.292770
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ 8.00000 0.766261 0.383131 0.923694i $$-0.374846\pi$$
0.383131 + 0.923694i $$0.374846\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 0 0
$$129$$ 11.0000 0.968496
$$130$$ 0 0
$$131$$ 19.0000 1.66004 0.830019 0.557735i $$-0.188330\pi$$
0.830019 + 0.557735i $$0.188330\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 0 0
$$139$$ 15.0000 1.27228 0.636142 0.771572i $$-0.280529\pi$$
0.636142 + 0.771572i $$0.280529\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 0 0
$$143$$ 10.0000 0.836242
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 1.00000 0.0819232 0.0409616 0.999161i $$-0.486958\pi$$
0.0409616 + 0.999161i $$0.486958\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 10.0000 0.803219
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ 0 0
$$165$$ −5.00000 −0.389249
$$166$$ 0 0
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 12.0000 0.907115
$$176$$ 0 0
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 5.00000 0.369611
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.00000 −0.365636
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 0 0
$$195$$ −2.00000 −0.143223
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −18.0000 −1.26335
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ 8.00000 0.548151
$$214$$ 0 0
$$215$$ −11.0000 −0.750194
$$216$$ 0 0
$$217$$ 30.0000 2.03653
$$218$$ 0 0
$$219$$ 13.0000 0.878459
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 0 0
$$229$$ 11.0000 0.726900 0.363450 0.931614i $$-0.381599\pi$$
0.363450 + 0.931614i $$0.381599\pi$$
$$230$$ 0 0
$$231$$ −15.0000 −0.986928
$$232$$ 0 0
$$233$$ −23.0000 −1.50678 −0.753390 0.657574i $$-0.771583\pi$$
−0.753390 + 0.657574i $$0.771583\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ 11.0000 0.711531 0.355765 0.934575i $$-0.384220\pi$$
0.355765 + 0.934575i $$0.384220\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 19.0000 1.19927 0.599635 0.800274i $$-0.295313\pi$$
0.599635 + 0.800274i $$0.295313\pi$$
$$252$$ 0 0
$$253$$ 20.0000 1.25739
$$254$$ 0 0
$$255$$ 1.00000 0.0626224
$$256$$ 0 0
$$257$$ −16.0000 −0.998053 −0.499026 0.866587i $$-0.666309\pi$$
−0.499026 + 0.866587i $$0.666309\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −23.0000 −1.41824 −0.709120 0.705087i $$-0.750908\pi$$
−0.709120 + 0.705087i $$0.750908\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ −6.00000 −0.363137
$$274$$ 0 0
$$275$$ −20.0000 −1.20605
$$276$$ 0 0
$$277$$ −25.0000 −1.50210 −0.751052 0.660243i $$-0.770453\pi$$
−0.751052 + 0.660243i $$0.770453\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ 7.00000 0.416107 0.208053 0.978117i $$-0.433287\pi$$
0.208053 + 0.978117i $$0.433287\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 0 0
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ −33.0000 −1.90209
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ −5.00000 −0.286299
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ −27.0000 −1.53103 −0.765515 0.643418i $$-0.777516\pi$$
−0.765515 + 0.643418i $$0.777516\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ 0 0
$$321$$ −18.0000 −1.00466
$$322$$ 0 0
$$323$$ 1.00000 0.0556415
$$324$$ 0 0
$$325$$ −8.00000 −0.443760
$$326$$ 0 0
$$327$$ 8.00000 0.442401
$$328$$ 0 0
$$329$$ −27.0000 −1.48856
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −50.0000 −2.70765
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ −21.0000 −1.12734 −0.563670 0.826000i $$-0.690611\pi$$
−0.563670 + 0.826000i $$0.690611\pi$$
$$348$$ 0 0
$$349$$ 27.0000 1.44528 0.722638 0.691226i $$-0.242929\pi$$
0.722638 + 0.691226i $$0.242929\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 3.00000 0.158777
$$358$$ 0 0
$$359$$ −21.0000 −1.10834 −0.554169 0.832404i $$-0.686964\pi$$
−0.554169 + 0.832404i $$0.686964\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 14.0000 0.734809
$$364$$ 0 0
$$365$$ −13.0000 −0.680451
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 30.0000 1.55752
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 0 0
$$375$$ 9.00000 0.464758
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ 0 0
$$387$$ 11.0000 0.559161
$$388$$ 0 0
$$389$$ −25.0000 −1.26755 −0.633775 0.773517i $$-0.718496\pi$$
−0.633775 + 0.773517i $$0.718496\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 19.0000 0.958423
$$394$$ 0 0
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ 19.0000 0.953583 0.476791 0.879017i $$-0.341800\pi$$
0.476791 + 0.879017i $$0.341800\pi$$
$$398$$ 0 0
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ 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
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ 0 0
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 15.0000 0.734553
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ 9.00000 0.437595
$$424$$ 0 0
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ −15.0000 −0.725901
$$428$$ 0 0
$$429$$ 10.0000 0.482805
$$430$$ 0 0
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 0 0
$$435$$ −6.00000 −0.287678
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −30.0000 −1.43182 −0.715911 0.698192i $$-0.753988\pi$$
−0.715911 + 0.698192i $$0.753988\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ 7.00000 0.332580 0.166290 0.986077i $$-0.446821\pi$$
0.166290 + 0.986077i $$0.446821\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 1.00000 0.0472984
$$448$$ 0 0
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 20.0000 0.939682
$$454$$ 0 0
$$455$$ 6.00000 0.281284
$$456$$ 0 0
$$457$$ 35.0000 1.63723 0.818615 0.574342i $$-0.194742\pi$$
0.818615 + 0.574342i $$0.194742\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −7.00000 −0.326023 −0.163011 0.986624i $$-0.552121\pi$$
−0.163011 + 0.986624i $$0.552121\pi$$
$$462$$ 0 0
$$463$$ −1.00000 −0.0464739 −0.0232370 0.999730i $$-0.507397\pi$$
−0.0232370 + 0.999730i $$0.507397\pi$$
$$464$$ 0 0
$$465$$ 10.0000 0.463739
$$466$$ 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
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 55.0000 2.52890
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −12.0000 −0.546019
$$484$$ 0 0
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ 0 0
$$489$$ 24.0000 1.08532
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 0 0
$$495$$ −5.00000 −0.224733
$$496$$ 0 0
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ −39.0000 −1.74588 −0.872940 0.487828i $$-0.837789\pi$$
−0.872940 + 0.487828i $$0.837789\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −39.0000 −1.72526
$$512$$ 0 0
$$513$$ −1.00000 −0.0441511
$$514$$ 0 0
$$515$$ −10.0000 −0.440653
$$516$$ 0 0
$$517$$ 45.0000 1.97910
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 8.00000 0.350486 0.175243 0.984525i $$-0.443929\pi$$
0.175243 + 0.984525i $$0.443929\pi$$
$$522$$ 0 0
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ 0 0
$$525$$ 12.0000 0.523723
$$526$$ 0 0
$$527$$ 10.0000 0.435607
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 18.0000 0.778208
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 0 0
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ −15.0000 −0.644900 −0.322450 0.946586i $$-0.604506\pi$$
−0.322450 + 0.946586i $$0.604506\pi$$
$$542$$ 0 0
$$543$$ −18.0000 −0.772454
$$544$$ 0 0
$$545$$ −8.00000 −0.342682
$$546$$ 0 0
$$547$$ 10.0000 0.427569 0.213785 0.976881i $$-0.431421\pi$$
0.213785 + 0.976881i $$0.431421\pi$$
$$548$$ 0 0
$$549$$ 5.00000 0.213395
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ −12.0000 −0.510292
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.00000 0.211857 0.105928 0.994374i $$-0.466219\pi$$
0.105928 + 0.994374i $$0.466219\pi$$
$$558$$ 0 0
$$559$$ 22.0000 0.930501
$$560$$ 0 0
$$561$$ −5.00000 −0.211100
$$562$$ 0 0
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 0 0
$$573$$ 15.0000 0.626634
$$574$$ 0 0
$$575$$ −16.0000 −0.667246
$$576$$ 0 0
$$577$$ 35.0000 1.45707 0.728535 0.685009i $$-0.240202\pi$$
0.728535 + 0.685009i $$0.240202\pi$$
$$578$$ 0 0
$$579$$ −16.0000 −0.664937
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −50.0000 −2.07079
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 35.0000 1.44460 0.722302 0.691577i $$-0.243084\pi$$
0.722302 + 0.691577i $$0.243084\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ −3.00000 −0.122988
$$596$$ 0 0
$$597$$ −11.0000 −0.450200
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −44.0000 −1.79480 −0.897399 0.441221i $$-0.854546\pi$$
−0.897399 + 0.441221i $$0.854546\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 0 0
$$609$$ −18.0000 −0.729397
$$610$$ 0 0
$$611$$ 18.0000 0.728202
$$612$$ 0 0
$$613$$ −5.00000 −0.201948 −0.100974 0.994889i $$-0.532196\pi$$
−0.100974 + 0.994889i $$0.532196\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.0000 1.08698 0.543490 0.839416i $$-0.317103\pi$$
0.543490 + 0.839416i $$0.317103\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ 18.0000 0.721155
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −5.00000 −0.199681
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −47.0000 −1.87104 −0.935520 0.353273i $$-0.885069\pi$$
−0.935520 + 0.353273i $$0.885069\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 0 0
$$635$$ 2.00000 0.0793676
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −26.0000 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$642$$ 0 0
$$643$$ −21.0000 −0.828159 −0.414080 0.910241i $$-0.635896\pi$$
−0.414080 + 0.910241i $$0.635896\pi$$
$$644$$ 0 0
$$645$$ −11.0000 −0.433125
$$646$$ 0 0
$$647$$ −25.0000 −0.982851 −0.491426 0.870919i $$-0.663524\pi$$
−0.491426 + 0.870919i $$0.663524\pi$$
$$648$$ 0 0
$$649$$ −20.0000 −0.785069
$$650$$ 0 0
$$651$$ 30.0000 1.17579
$$652$$ 0 0
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 0 0
$$655$$ −19.0000 −0.742391
$$656$$ 0 0
$$657$$ 13.0000 0.507178
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 28.0000 1.08907 0.544537 0.838737i $$-0.316705\pi$$
0.544537 + 0.838737i $$0.316705\pi$$
$$662$$ 0 0
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ −3.00000 −0.116335
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 25.0000 0.965114
$$672$$ 0 0
$$673$$ −44.0000 −1.69608 −0.848038 0.529936i $$-0.822216\pi$$
−0.848038 + 0.529936i $$0.822216\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 6.00000 0.229920
$$682$$ 0 0
$$683$$ −34.0000 −1.30097 −0.650487 0.759517i $$-0.725435\pi$$
−0.650487 + 0.759517i $$0.725435\pi$$
$$684$$ 0 0
$$685$$ 9.00000 0.343872
$$686$$ 0 0
$$687$$ 11.0000 0.419676
$$688$$ 0 0
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −35.0000 −1.33146 −0.665731 0.746191i $$-0.731880\pi$$
−0.665731 + 0.746191i $$0.731880\pi$$
$$692$$ 0 0
$$693$$ −15.0000 −0.569803
$$694$$ 0 0
$$695$$ −15.0000 −0.568982
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −23.0000 −0.869940
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −9.00000 −0.338960
$$706$$ 0 0
$$707$$ −18.0000 −0.676960
$$708$$ 0 0
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 0 0
$$713$$ −40.0000 −1.49801
$$714$$ 0 0
$$715$$ −10.0000 −0.373979
$$716$$ 0 0
$$717$$ 11.0000 0.410803
$$718$$ 0 0
$$719$$ −17.0000 −0.633993 −0.316997 0.948427i $$-0.602674\pi$$
−0.316997 + 0.948427i $$0.602674\pi$$
$$720$$ 0 0
$$721$$ −30.0000 −1.11726
$$722$$ 0 0
$$723$$ −12.0000 −0.446285
$$724$$ 0 0
$$725$$ −24.0000 −0.891338
$$726$$ 0 0
$$727$$ 39.0000 1.44643 0.723215 0.690623i $$-0.242664\pi$$
0.723215 + 0.690623i $$0.242664\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −11.0000 −0.406850
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 0 0
$$735$$ −2.00000 −0.0737711
$$736$$ 0 0
$$737$$ 20.0000 0.736709
$$738$$ 0 0
$$739$$ 7.00000 0.257499 0.128750 0.991677i $$-0.458904\pi$$
0.128750 + 0.991677i $$0.458904\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ −44.0000 −1.61420 −0.807102 0.590412i $$-0.798965\pi$$
−0.807102 + 0.590412i $$0.798965\pi$$
$$744$$ 0 0
$$745$$ −1.00000 −0.0366372
$$746$$ 0 0
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 54.0000 1.97312
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 19.0000 0.692398
$$754$$ 0 0
$$755$$ −20.0000 −0.727875
$$756$$ 0 0
$$757$$ −35.0000 −1.27210 −0.636048 0.771649i $$-0.719432\pi$$
−0.636048 + 0.771649i $$0.719432\pi$$
$$758$$ 0 0
$$759$$ 20.0000 0.725954
$$760$$ 0 0
$$761$$ −1.00000 −0.0362500 −0.0181250 0.999836i $$-0.505770\pi$$
−0.0181250 + 0.999836i $$0.505770\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 0 0
$$765$$ 1.00000 0.0361551
$$766$$ 0 0
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 0 0
$$771$$ −16.0000 −0.576226
$$772$$ 0 0
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ 40.0000 1.43684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 40.0000 1.43131
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ −16.0000 −0.570338 −0.285169 0.958477i $$-0.592050\pi$$
−0.285169 + 0.958477i $$0.592050\pi$$
$$788$$ 0 0
$$789$$ −23.0000 −0.818822
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 0 0
$$795$$ 10.0000 0.354663
$$796$$ 0 0
$$797$$ −40.0000 −1.41687 −0.708436 0.705775i $$-0.750599\pi$$
−0.708436 + 0.705775i $$0.750599\pi$$
$$798$$ 0 0
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 65.0000 2.29380
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 22.0000 0.774437
$$808$$ 0 0
$$809$$ 17.0000 0.597688 0.298844 0.954302i $$-0.403399\pi$$
0.298844 + 0.954302i $$0.403399\pi$$
$$810$$ 0 0
$$811$$ 14.0000 0.491606 0.245803 0.969320i $$-0.420948\pi$$
0.245803 + 0.969320i $$0.420948\pi$$
$$812$$ 0 0
$$813$$ 20.0000 0.701431
$$814$$ 0 0
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ −11.0000 −0.384841
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 9.00000 0.314102 0.157051 0.987590i $$-0.449801\pi$$
0.157051 + 0.987590i $$0.449801\pi$$
$$822$$ 0 0
$$823$$ 5.00000 0.174289 0.0871445 0.996196i $$-0.472226\pi$$
0.0871445 + 0.996196i $$0.472226\pi$$
$$824$$ 0 0
$$825$$ −20.0000 −0.696311
$$826$$ 0 0
$$827$$ −32.0000 −1.11275 −0.556375 0.830932i $$-0.687808\pi$$
−0.556375 + 0.830932i $$0.687808\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −25.0000 −0.867240
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ −18.0000 −0.622916
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 0 0
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −22.0000 −0.757720
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ −42.0000 −1.44314
$$848$$ 0 0
$$849$$ 7.00000 0.240239
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 0 0
$$859$$ 15.0000 0.511793 0.255897 0.966704i $$-0.417629\pi$$
0.255897 + 0.966704i $$0.417629\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 28.0000 0.953131 0.476566 0.879139i $$-0.341881\pi$$
0.476566 + 0.879139i $$0.341881\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ −27.0000 −0.912767
$$876$$ 0 0
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ 0 0
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ 0 0
$$883$$ 31.0000 1.04323 0.521617 0.853180i $$-0.325329\pi$$
0.521617 + 0.853180i $$0.325329\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ 56.0000 1.88030 0.940148 0.340766i $$-0.110687\pi$$
0.940148 + 0.340766i $$0.110687\pi$$
$$888$$ 0 0
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ 0 0
$$893$$ −9.00000 −0.301174
$$894$$ 0 0
$$895$$ 6.00000 0.200558
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 0 0
$$899$$ −60.0000 −2.00111
$$900$$ 0 0
$$901$$ 10.0000 0.333148
$$902$$ 0 0
$$903$$ −33.0000 −1.09817
$$904$$ 0 0
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −50.0000 −1.65657 −0.828287 0.560304i $$-0.810684\pi$$
−0.828287 + 0.560304i $$0.810684\pi$$
$$912$$ 0 0
$$913$$ 20.0000 0.661903
$$914$$ 0 0
$$915$$ −5.00000 −0.165295
$$916$$ 0 0
$$917$$ −57.0000 −1.88231
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 10.0000 0.328443
$$928$$ 0 0
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ −27.0000 −0.883940
$$934$$ 0 0
$$935$$ 5.00000 0.163517
$$936$$ 0 0
$$937$$ −3.00000 −0.0980057 −0.0490029 0.998799i $$-0.515604\pi$$
−0.0490029 + 0.998799i $$0.515604\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 3.00000 0.0975900
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ 26.0000 0.843996
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ 12.0000 0.388718 0.194359 0.980930i $$-0.437737\pi$$
0.194359 + 0.980930i $$0.437737\pi$$
$$954$$ 0 0
$$955$$ −15.0000 −0.485389
$$956$$ 0 0
$$957$$ 30.0000 0.969762
$$958$$ 0 0
$$959$$ 27.0000 0.871875
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ 16.0000 0.515058
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 1.00000 0.0321246
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −45.0000 −1.44263
$$974$$ 0 0
$$975$$ −8.00000 −0.256205
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 8.00000 0.255420
$$982$$ 0 0
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ −27.0000 −0.859419
$$988$$ 0 0
$$989$$ 44.0000 1.39912
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ −32.0000 −1.01549
$$994$$ 0 0
$$995$$ 11.0000 0.348723
$$996$$ 0 0
$$997$$ 35.0000 1.10846 0.554231 0.832363i $$-0.313013\pi$$
0.554231 + 0.832363i $$0.313013\pi$$
$$998$$ 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 3648.2.a.z.1.1 1
4.3 odd 2 3648.2.a.g.1.1 1
8.3 odd 2 912.2.a.i.1.1 1
8.5 even 2 456.2.a.a.1.1 1
24.5 odd 2 1368.2.a.d.1.1 1
24.11 even 2 2736.2.a.i.1.1 1
152.37 odd 2 8664.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.a.1.1 1 8.5 even 2
912.2.a.i.1.1 1 8.3 odd 2
1368.2.a.d.1.1 1 24.5 odd 2
2736.2.a.i.1.1 1 24.11 even 2
3648.2.a.g.1.1 1 4.3 odd 2
3648.2.a.z.1.1 1 1.1 even 1 trivial
8664.2.a.l.1.1 1 152.37 odd 2