# Properties

 Label 19.2.a.a.1.1 Level $19$ Weight $2$ Character 19.1 Self dual yes Analytic conductor $0.152$ Analytic rank $0$ 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] = [19,2,Mod(1,19)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(19, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("19.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 19.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: $$0.151715763840$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 19.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +4.00000 q^{12} -4.00000 q^{13} -6.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} +1.00000 q^{19} -6.00000 q^{20} +2.00000 q^{21} +4.00000 q^{25} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} -6.00000 q^{33} -3.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +8.00000 q^{39} -6.00000 q^{41} -1.00000 q^{43} -6.00000 q^{44} +3.00000 q^{45} -3.00000 q^{47} -8.00000 q^{48} -6.00000 q^{49} +6.00000 q^{51} +8.00000 q^{52} +12.0000 q^{53} +9.00000 q^{55} -2.00000 q^{57} -6.00000 q^{59} +12.0000 q^{60} -1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +6.00000 q^{71} -7.00000 q^{73} -8.00000 q^{75} -2.00000 q^{76} -3.00000 q^{77} +8.00000 q^{79} +12.0000 q^{80} -11.0000 q^{81} +12.0000 q^{83} -4.00000 q^{84} -9.00000 q^{85} -12.0000 q^{87} +12.0000 q^{89} +4.00000 q^{91} +8.00000 q^{93} +3.00000 q^{95} +8.00000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ −2.00000 −1.00000
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 4.00000 1.15470
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ −6.00000 −1.54919
$$16$$ 4.00000 1.00000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ −6.00000 −1.34164
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 2.00000 0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −8.00000 −1.15470
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 8.00000 1.10940
$$53$$ 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
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 12.0000 1.54919
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ −8.00000 −1.00000
$$65$$ −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
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ 0 0
$$75$$ −8.00000 −0.923760
$$76$$ −2.00000 −0.229416
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 12.0000 1.34164
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ −9.00000 −0.976187
$$86$$ 0 0
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 3.00000 0.307794
$$96$$ 0 0
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ −8.00000 −0.800000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ −8.00000 −0.769800
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ −4.00000 −0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −12.0000 −1.11417
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 8.00000 0.718421
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 12.0000 1.04447
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 12.0000 1.03280
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ −13.0000 −1.10265 −0.551323 0.834292i $$-0.685877\pi$$
−0.551323 + 0.834292i $$0.685877\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 4.00000 0.333333
$$145$$ 18.0000 1.49482
$$146$$ 0 0
$$147$$ 12.0000 0.989743
$$148$$ −4.00000 −0.328798
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ −3.00000 −0.242536
$$154$$ 0 0
$$155$$ −12.0000 −0.963863
$$156$$ −16.0000 −1.28103
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ −24.0000 −1.90332
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 12.0000 0.937043
$$165$$ −18.0000 −1.40130
$$166$$ 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
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 2.00000 0.152499
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 12.0000 0.904534
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ −6.00000 −0.447214
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ −9.00000 −0.658145
$$188$$ 6.00000 0.437595
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 16.0000 1.15470
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 24.0000 1.71868
$$196$$ 12.0000 0.857143
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ −12.0000 −0.840168
$$205$$ −18.0000 −1.25717
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −16.0000 −1.10940
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ 14.0000 0.963800 0.481900 0.876226i $$-0.339947\pi$$
0.481900 + 0.876226i $$0.339947\pi$$
$$212$$ −24.0000 −1.64833
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ 14.0000 0.946032
$$220$$ −18.0000 −1.21356
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 5.00000 0.330409 0.165205 0.986259i $$-0.447172\pi$$
0.165205 + 0.986259i $$0.447172\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ −21.0000 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 12.0000 0.781133
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ −24.0000 −1.54919
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 2.00000 0.128037
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 0 0
$$249$$ −24.0000 −1.52094
$$250$$ 0 0
$$251$$ 21.0000 1.32551 0.662754 0.748837i $$-0.269387\pi$$
0.662754 + 0.748837i $$0.269387\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 18.0000 1.12720
$$256$$ 16.0000 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 24.0000 1.48842
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 9.00000 0.554964 0.277482 0.960731i $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ 36.0000 2.21146
$$266$$ 0 0
$$267$$ −24.0000 −1.46878
$$268$$ 8.00000 0.488678
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −12.0000 −0.727607
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ 12.0000 0.723627
$$276$$ 0 0
$$277$$ −19.0000 −1.14160 −0.570800 0.821089i $$-0.693367\pi$$
−0.570800 + 0.821089i $$0.693367\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −13.0000 −0.772770 −0.386385 0.922338i $$-0.626276\pi$$
−0.386385 + 0.922338i $$0.626276\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ 14.0000 0.819288
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ −18.0000 −1.04800
$$296$$ 0 0
$$297$$ 12.0000 0.696311
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 16.0000 0.923760
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 4.00000 0.229416
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 6.00000 0.341882
$$309$$ −28.0000 −1.59286
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ −16.0000 −0.900070
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ −24.0000 −1.34164
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ −3.00000 −0.166924
$$324$$ 22.0000 1.22222
$$325$$ −16.0000 −0.887520
$$326$$ 0 0
$$327$$ 32.0000 1.76960
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −24.0000 −1.31717
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 8.00000 0.436436
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 18.0000 0.976187
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 21.0000 1.12734 0.563670 0.826000i $$-0.309389\pi$$
0.563670 + 0.826000i $$0.309389\pi$$
$$348$$ 24.0000 1.28654
$$349$$ 17.0000 0.909989 0.454995 0.890494i $$-0.349641\pi$$
0.454995 + 0.890494i $$0.349641\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ −24.0000 −1.27200
$$357$$ −6.00000 −0.317554
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 4.00000 0.209946
$$364$$ −8.00000 −0.419314
$$365$$ −21.0000 −1.09919
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ −16.0000 −0.829561
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 6.00000 0.309839
$$376$$ 0 0
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ −9.00000 −0.458682
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ −16.0000 −0.812277
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 30.0000 1.51330
$$394$$ 0 0
$$395$$ 24.0000 1.20757
$$396$$ −6.00000 −0.301511
$$397$$ −7.00000 −0.351320 −0.175660 0.984451i $$-0.556206\pi$$
−0.175660 + 0.984451i $$0.556206\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 16.0000 0.800000
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 16.0000 0.797017
$$404$$ −12.0000 −0.597022
$$405$$ −33.0000 −1.63978
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ −28.0000 −1.37946
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 36.0000 1.76717
$$416$$ 0 0
$$417$$ 26.0000 1.27323
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ −12.0000 −0.585540
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 0 0
$$423$$ −3.00000 −0.145865
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 1.00000 0.0483934
$$428$$ 36.0000 1.74013
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 16.0000 0.769800
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ −36.0000 −1.72607
$$436$$ 32.0000 1.53252
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −3.00000 −0.142534 −0.0712672 0.997457i $$-0.522704\pi$$
−0.0712672 + 0.997457i $$0.522704\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 36.0000 1.70656
$$446$$ 0 0
$$447$$ −42.0000 −1.98653
$$448$$ 8.00000 0.377964
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ −12.0000 −0.564433
$$453$$ 20.0000 0.939682
$$454$$ 0 0
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ −37.0000 −1.73079 −0.865393 0.501093i $$-0.832931\pi$$
−0.865393 + 0.501093i $$0.832931\pi$$
$$458$$ 0 0
$$459$$ −12.0000 −0.560112
$$460$$ 0 0
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 0 0
$$463$$ −31.0000 −1.44069 −0.720346 0.693615i $$-0.756017\pi$$
−0.720346 + 0.693615i $$0.756017\pi$$
$$464$$ 24.0000 1.11417
$$465$$ 24.0000 1.11297
$$466$$ 0 0
$$467$$ −27.0000 −1.24941 −0.624705 0.780860i $$-0.714781\pi$$
−0.624705 + 0.780860i $$0.714781\pi$$
$$468$$ 8.00000 0.369800
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ −6.00000 −0.275010
$$477$$ 12.0000 0.549442
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 4.00000 0.181818
$$485$$ 24.0000 1.08978
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 0 0
$$489$$ −40.0000 −1.80886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −24.0000 −1.08200
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ 9.00000 0.404520
$$496$$ −16.0000 −0.718421
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ 5.00000 0.223831 0.111915 0.993718i $$-0.464301\pi$$
0.111915 + 0.993718i $$0.464301\pi$$
$$500$$ 6.00000 0.268328
$$501$$ 36.0000 1.60836
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ −6.00000 −0.266469
$$508$$ −4.00000 −0.177471
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 42.0000 1.85074
$$516$$ −4.00000 −0.176090
$$517$$ −9.00000 −0.395820
$$518$$ 0 0
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ 30.0000 1.31056
$$525$$ 8.00000 0.349149
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ −24.0000 −1.04447
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 2.00000 0.0867110
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ −54.0000 −2.33462
$$536$$ 0 0
$$537$$ 36.0000 1.55351
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ −24.0000 −1.03280
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ 0 0
$$543$$ −4.00000 −0.171656
$$544$$ 0 0
$$545$$ −48.0000 −2.05609
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −1.00000 −0.0426790
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ −12.0000 −0.509372
$$556$$ 26.0000 1.10265
$$557$$ 21.0000 0.889799 0.444899 0.895581i $$-0.353239\pi$$
0.444899 + 0.895581i $$0.353239\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ −12.0000 −0.507093
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ 6.00000 0.252870 0.126435 0.991975i $$-0.459647\pi$$
0.126435 + 0.991975i $$0.459647\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 18.0000 0.757266
$$566$$ 0 0
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 24.0000 1.00349
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 0 0
$$579$$ 8.00000 0.332469
$$580$$ −36.0000 −1.49482
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 45.0000 1.85735 0.928674 0.370896i $$-0.120949\pi$$
0.928674 + 0.370896i $$0.120949\pi$$
$$588$$ −24.0000 −0.989743
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 8.00000 0.328798
$$593$$ −42.0000 −1.72473 −0.862367 0.506284i $$-0.831019\pi$$
−0.862367 + 0.506284i $$0.831019\pi$$
$$594$$ 0 0
$$595$$ 9.00000 0.368964
$$596$$ −42.0000 −1.72039
$$597$$ −22.0000 −0.900400
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 20.0000 0.813788
$$605$$ −6.00000 −0.243935
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 6.00000 0.242536
$$613$$ 29.0000 1.17130 0.585649 0.810564i $$-0.300840\pi$$
0.585649 + 0.810564i $$0.300840\pi$$
$$614$$ 0 0
$$615$$ 36.0000 1.45166
$$616$$ 0 0
$$617$$ 9.00000 0.362326 0.181163 0.983453i $$-0.442014\pi$$
0.181163 + 0.983453i $$0.442014\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 24.0000 0.963863
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 32.0000 1.28103
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ −6.00000 −0.239617
$$628$$ −28.0000 −1.11732
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 11.0000 0.437903 0.218952 0.975736i $$-0.429736\pi$$
0.218952 + 0.975736i $$0.429736\pi$$
$$632$$ 0 0
$$633$$ −28.0000 −1.11290
$$634$$ 0 0
$$635$$ 6.00000 0.238103
$$636$$ 48.0000 1.90332
$$637$$ 24.0000 0.950915
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −13.0000 −0.512670 −0.256335 0.966588i $$-0.582515\pi$$
−0.256335 + 0.966588i $$0.582515\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ 0 0
$$649$$ −18.0000 −0.706562
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ −40.0000 −1.56652
$$653$$ −39.0000 −1.52619 −0.763094 0.646288i $$-0.776321\pi$$
−0.763094 + 0.646288i $$0.776321\pi$$
$$654$$ 0 0
$$655$$ −45.0000 −1.75830
$$656$$ −24.0000 −0.937043
$$657$$ −7.00000 −0.273096
$$658$$ 0 0
$$659$$ −30.0000 −1.16863 −0.584317 0.811525i $$-0.698638\pi$$
−0.584317 + 0.811525i $$0.698638\pi$$
$$660$$ 36.0000 1.40130
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 0 0
$$663$$ −24.0000 −0.932083
$$664$$ 0 0
$$665$$ −3.00000 −0.116335
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 36.0000 1.39288
$$669$$ 20.0000 0.773245
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 0 0
$$675$$ 16.0000 0.615840
$$676$$ −6.00000 −0.230769
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −9.00000 −0.343872
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ −4.00000 −0.152499
$$689$$ −48.0000 −1.82865
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 36.0000 1.36851
$$693$$ −3.00000 −0.113961
$$694$$ 0 0
$$695$$ −39.0000 −1.47935
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 0 0
$$699$$ 42.0000 1.58859
$$700$$ 8.00000 0.302372
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ −24.0000 −0.904534
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ −24.0000 −0.901975
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −36.0000 −1.34632
$$716$$ 36.0000 1.34538
$$717$$ −30.0000 −1.12037
$$718$$ 0 0
$$719$$ 15.0000 0.559406 0.279703 0.960087i $$-0.409764\pi$$
0.279703 + 0.960087i $$0.409764\pi$$
$$720$$ 12.0000 0.447214
$$721$$ −14.0000 −0.521387
$$722$$ 0 0
$$723$$ 20.0000 0.743808
$$724$$ −4.00000 −0.148659
$$725$$ 24.0000 0.891338
$$726$$ 0 0
$$727$$ −19.0000 −0.704671 −0.352335 0.935874i $$-0.614612\pi$$
−0.352335 + 0.935874i $$0.614612\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ −4.00000 −0.147844
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 0 0
$$735$$ 36.0000 1.32788
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ 11.0000 0.404642 0.202321 0.979319i $$-0.435152\pi$$
0.202321 + 0.979319i $$0.435152\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 63.0000 2.30814
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 18.0000 0.658145
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ −42.0000 −1.53057
$$754$$ 0 0
$$755$$ −30.0000 −1.09181
$$756$$ 8.00000 0.290957
$$757$$ −25.0000 −0.908640 −0.454320 0.890838i $$-0.650118\pi$$
−0.454320 + 0.890838i $$0.650118\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0000 1.19625 0.598125 0.801403i $$-0.295913\pi$$
0.598125 + 0.801403i $$0.295913\pi$$
$$762$$ 0 0
$$763$$ 16.0000 0.579239
$$764$$ −6.00000 −0.217072
$$765$$ −9.00000 −0.325396
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ −32.0000 −1.15470
$$769$$ 23.0000 0.829401 0.414701 0.909958i $$-0.363886\pi$$
0.414701 + 0.909958i $$0.363886\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.00000 0.287926
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ −48.0000 −1.71868
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ −24.0000 −0.857143
$$785$$ 42.0000 1.49904
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ −36.0000 −1.28245
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 0 0
$$795$$ −72.0000 −2.55358
$$796$$ −22.0000 −0.779769
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 0 0
$$803$$ −21.0000 −0.741074
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −48.0000 −1.68968
$$808$$ 0 0
$$809$$ −9.00000 −0.316423 −0.158212 0.987405i $$-0.550573\pi$$
−0.158212 + 0.987405i $$0.550573\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 60.0000 2.10171
$$816$$ 24.0000 0.840168
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 4.00000 0.139771
$$820$$ 36.0000 1.25717
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 0 0
$$823$$ −49.0000 −1.70803 −0.854016 0.520246i $$-0.825840\pi$$
−0.854016 + 0.520246i $$0.825840\pi$$
$$824$$ 0 0
$$825$$ −24.0000 −0.835573
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 0 0
$$831$$ 38.0000 1.31821
$$832$$ 32.0000 1.10940
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ −54.0000 −1.86875
$$836$$ −6.00000 −0.207514
$$837$$ −16.0000 −0.553041
$$838$$ 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
$$843$$ −12.0000 −0.413302
$$844$$ −28.0000 −0.963800
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 48.0000 1.64833
$$849$$ 26.0000 0.892318
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 24.0000 0.822226
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ 3.00000 0.102598
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −49.0000 −1.67186 −0.835929 0.548837i $$-0.815071\pi$$
−0.835929 + 0.548837i $$0.815071\pi$$
$$860$$ 6.00000 0.204598
$$861$$ −12.0000 −0.408959
$$862$$ 0 0
$$863$$ 18.0000 0.612727 0.306364 0.951915i $$-0.400888\pi$$
0.306364 + 0.951915i $$0.400888\pi$$
$$864$$ 0 0
$$865$$ −54.0000 −1.83606
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ −8.00000 −0.271538
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 8.00000 0.270759
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ −28.0000 −0.946032
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 36.0000 1.21356
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 0 0
$$883$$ 47.0000 1.58168 0.790838 0.612026i $$-0.209645\pi$$
0.790838 + 0.612026i $$0.209645\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 36.0000 1.21013
$$886$$ 0 0
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ −33.0000 −1.10554
$$892$$ 20.0000 0.669650
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ −54.0000 −1.80502
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ −8.00000 −0.266667
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ −2.00000 −0.0665558
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ 0 0
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ 36.0000 1.19143
$$914$$ 0 0
$$915$$ 6.00000 0.198354
$$916$$ −10.0000 −0.330409
$$917$$ 15.0000 0.495344
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ −40.0000 −1.31804
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ −12.0000 −0.394771
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 14.0000 0.459820
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 42.0000 1.37576
$$933$$ 6.00000 0.196431
$$934$$ 0 0
$$935$$ −27.0000 −0.882994
$$936$$ 0 0
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 18.0000 0.587095
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −24.0000 −0.781133
$$945$$ −12.0000 −0.390360
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 32.0000 1.03931
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ −48.0000 −1.55487 −0.777436 0.628962i $$-0.783480\pi$$
−0.777436 + 0.628962i $$0.783480\pi$$
$$954$$ 0 0
$$955$$ 9.00000 0.291233
$$956$$ −30.0000 −0.970269
$$957$$ −36.0000 −1.16371
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ 48.0000 1.54919
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 20.0000 0.644157
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ −20.0000 −0.641500
$$973$$ 13.0000 0.416761
$$974$$ 0 0
$$975$$ 32.0000 1.02482
$$976$$ −4.00000 −0.128037
$$977$$ 24.0000 0.767828 0.383914 0.923369i $$-0.374576\pi$$
0.383914 + 0.923369i $$0.374576\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 36.0000 1.14998
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 54.0000 1.72058
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 0 0
$$993$$ 56.0000 1.77711
$$994$$ 0 0
$$995$$ 33.0000 1.04617
$$996$$ 48.0000 1.52094
$$997$$ 17.0000 0.538395 0.269198 0.963085i $$-0.413241\pi$$
0.269198 + 0.963085i $$0.413241\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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 19.2.a.a.1.1 1
3.2 odd 2 171.2.a.b.1.1 1
4.3 odd 2 304.2.a.f.1.1 1
5.2 odd 4 475.2.b.a.324.2 2
5.3 odd 4 475.2.b.a.324.1 2
5.4 even 2 475.2.a.b.1.1 1
7.2 even 3 931.2.f.c.704.1 2
7.3 odd 6 931.2.f.b.324.1 2
7.4 even 3 931.2.f.c.324.1 2
7.5 odd 6 931.2.f.b.704.1 2
7.6 odd 2 931.2.a.a.1.1 1
8.3 odd 2 1216.2.a.b.1.1 1
8.5 even 2 1216.2.a.o.1.1 1
11.10 odd 2 2299.2.a.b.1.1 1
12.11 even 2 2736.2.a.c.1.1 1
13.12 even 2 3211.2.a.a.1.1 1
15.14 odd 2 4275.2.a.i.1.1 1
17.16 even 2 5491.2.a.b.1.1 1
19.2 odd 18 361.2.e.e.99.1 6
19.3 odd 18 361.2.e.e.28.1 6
19.4 even 9 361.2.e.d.54.1 6
19.5 even 9 361.2.e.d.234.1 6
19.6 even 9 361.2.e.d.245.1 6
19.7 even 3 361.2.c.c.68.1 2
19.8 odd 6 361.2.c.a.292.1 2
19.9 even 9 361.2.e.d.62.1 6
19.10 odd 18 361.2.e.e.62.1 6
19.11 even 3 361.2.c.c.292.1 2
19.12 odd 6 361.2.c.a.68.1 2
19.13 odd 18 361.2.e.e.245.1 6
19.14 odd 18 361.2.e.e.234.1 6
19.15 odd 18 361.2.e.e.54.1 6
19.16 even 9 361.2.e.d.28.1 6
19.17 even 9 361.2.e.d.99.1 6
19.18 odd 2 361.2.a.b.1.1 1
20.19 odd 2 7600.2.a.c.1.1 1
21.20 even 2 8379.2.a.j.1.1 1
57.56 even 2 3249.2.a.d.1.1 1
76.75 even 2 5776.2.a.c.1.1 1
95.94 odd 2 9025.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.a.a.1.1 1 1.1 even 1 trivial
171.2.a.b.1.1 1 3.2 odd 2
304.2.a.f.1.1 1 4.3 odd 2
361.2.a.b.1.1 1 19.18 odd 2
361.2.c.a.68.1 2 19.12 odd 6
361.2.c.a.292.1 2 19.8 odd 6
361.2.c.c.68.1 2 19.7 even 3
361.2.c.c.292.1 2 19.11 even 3
361.2.e.d.28.1 6 19.16 even 9
361.2.e.d.54.1 6 19.4 even 9
361.2.e.d.62.1 6 19.9 even 9
361.2.e.d.99.1 6 19.17 even 9
361.2.e.d.234.1 6 19.5 even 9
361.2.e.d.245.1 6 19.6 even 9
361.2.e.e.28.1 6 19.3 odd 18
361.2.e.e.54.1 6 19.15 odd 18
361.2.e.e.62.1 6 19.10 odd 18
361.2.e.e.99.1 6 19.2 odd 18
361.2.e.e.234.1 6 19.14 odd 18
361.2.e.e.245.1 6 19.13 odd 18
475.2.a.b.1.1 1 5.4 even 2
475.2.b.a.324.1 2 5.3 odd 4
475.2.b.a.324.2 2 5.2 odd 4
931.2.a.a.1.1 1 7.6 odd 2
931.2.f.b.324.1 2 7.3 odd 6
931.2.f.b.704.1 2 7.5 odd 6
931.2.f.c.324.1 2 7.4 even 3
931.2.f.c.704.1 2 7.2 even 3
1216.2.a.b.1.1 1 8.3 odd 2
1216.2.a.o.1.1 1 8.5 even 2
2299.2.a.b.1.1 1 11.10 odd 2
2736.2.a.c.1.1 1 12.11 even 2
3211.2.a.a.1.1 1 13.12 even 2
3249.2.a.d.1.1 1 57.56 even 2
4275.2.a.i.1.1 1 15.14 odd 2
5491.2.a.b.1.1 1 17.16 even 2
5776.2.a.c.1.1 1 76.75 even 2
7600.2.a.c.1.1 1 20.19 odd 2
8379.2.a.j.1.1 1 21.20 even 2
9025.2.a.d.1.1 1 95.94 odd 2