# Properties

 Label 9025.2.a.d.1.1 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ 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] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 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
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ −2.00000 −0.377964
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −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 $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −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
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −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
$$65$$ 0 0
$$66$$ 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 $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 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
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 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 $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 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
$$125$$ 0 0
$$126$$ 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
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ −13.0000 −1.10265 −0.551323 0.834292i $$-0.685877\pi$$
−0.551323 + 0.834292i $$0.685877\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 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 $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 3.00000 0.242536
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −16.0000 −1.28103
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 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 $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −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
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 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
$$195$$ 0 0
$$196$$ 12.0000 0.857143
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 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
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −16.0000 −1.10940
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ −24.0000 −1.64833
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ −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
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 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 $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −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 $$0.338771\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 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
$$255$$ 0 0
$$256$$ 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
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 24.0000 1.46878
$$268$$ 8.00000 0.488678
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 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
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
0.570800 + 0.821089i $$0.306633\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 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
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 12.0000 0.696311
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 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 $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 22.0000 1.22222
$$325$$ 0 0
$$326$$ 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 $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 24.0000 1.31717
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −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
$$340$$ 0 0
$$341$$ 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 $$-0.690611\pi$$
−0.563670 + 0.826000i $$0.690611\pi$$
$$348$$ −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 $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$361$$ 0 0
$$362$$ 0 0
$$363$$ 4.00000 0.209946
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 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
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ −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
$$385$$ 0 0
$$386$$ 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
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.00000 0.297409
$$408$$ 0 0
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ −28.0000 −1.37946
$$413$$ 6.00000 0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 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 $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 3.00000 0.145865
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 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 $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 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
$$435$$ 0 0
$$436$$ −32.0000 −1.53252
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ 3.00000 0.142534 0.0712672 0.997457i $$-0.477296\pi$$
0.0712672 + 0.997457i $$0.477296\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 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
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 37.0000 1.73079 0.865393 0.501093i $$-0.167069\pi$$
0.865393 + 0.501093i $$0.167069\pi$$
$$458$$ 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 $$-0.243983\pi$$
0.720346 + 0.693615i $$0.243983\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 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
$$475$$ 0 0
$$476$$ −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
$$485$$ 0 0
$$486$$ 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
$$495$$ 0 0
$$496$$ 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 $$0.464301\pi$$
$$500$$ 0 0
$$501$$ 36.0000 1.60836
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 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
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 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
$$525$$ 0 0
$$526$$ 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
$$532$$ 0 0
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −36.0000 −1.55351
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −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
$$545$$ 0 0
$$546$$ 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
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 26.0000 1.10265
$$557$$ −21.0000 −0.889799 −0.444899 0.895581i $$-0.646761\pi$$
−0.444899 + 0.895581i $$0.646761\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ −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
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 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 $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ 0 0
$$579$$ 8.00000 0.332469
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −45.0000 −1.85735 −0.928674 0.370896i $$-0.879051\pi$$
−0.928674 + 0.370896i $$0.879051\pi$$
$$588$$ −24.0000 −0.989743
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ 8.00000 0.328798
$$593$$ 42.0000 1.72473 0.862367 0.506284i $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −42.0000 −1.72039
$$597$$ −22.0000 −0.900400
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 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 $$-0.699160\pi$$
−0.585649 + 0.810564i $$0.699160\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −9.00000 −0.362326 −0.181163 0.983453i $$-0.557986\pi$$
−0.181163 + 0.983453i $$0.557986\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 32.0000 1.28103
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$635$$ 0 0
$$636$$ 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 $$-0.417485\pi$$
0.256335 + 0.966588i $$0.417485\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −27.0000 −1.06148 −0.530740 0.847535i $$-0.678086\pi$$
−0.530740 + 0.847535i $$0.678086\pi$$
$$648$$ 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 $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 24.0000 0.937043
$$657$$ 7.00000 0.273096
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 0 0
$$661$$ −32.0000 −1.24466 −0.622328 0.782757i $$-0.713813\pi$$
−0.622328 + 0.782757i $$0.713813\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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
$$675$$ 0 0
$$676$$ −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 $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 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
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 0 0
$$699$$ −42.0000 −1.58859
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −24.0000 −0.904534
$$705$$ 0 0
$$706$$ 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
$$715$$ 0 0
$$716$$ −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 $$0.409764\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 0 0
$$723$$ −20.0000 −0.743808
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 19.0000 0.704671 0.352335 0.935874i $$-0.385388\pi$$
0.352335 + 0.935874i $$0.385388\pi$$
$$728$$ 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 $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 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 $$0.435152\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 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 $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −42.0000 −1.53057
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −8.00000 −0.290957
$$757$$ 25.0000 0.908640 0.454320 0.890838i $$-0.349882\pi$$
0.454320 + 0.890838i $$0.349882\pi$$
$$758$$ 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
$$765$$ 0 0
$$766$$ 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
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −4.00000 −0.143499
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ −24.0000 −0.857143
$$785$$ 0 0
$$786$$ 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
$$795$$ 0 0
$$796$$ −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 $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −24.0000 −0.840168
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 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 $$-0.174160\pi$$
0.854016 + 0.520246i $$0.174160\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 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 $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ 0 0
$$831$$ −38.0000 −1.31821
$$832$$ 32.0000 1.10940
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 16.0000 0.553041
$$838$$ 0 0
$$839$$ −18.0000 −0.621429 −0.310715 0.950503i $$-0.600568\pi$$
−0.310715 + 0.950503i $$0.600568\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 12.0000 0.413302
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ 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 $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 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 $$0.815071\pi$$
$$860$$ 0 0
$$861$$ −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
$$865$$ 0 0
$$866$$ 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
$$875$$ 0 0
$$876$$ 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
$$880$$ 0 0
$$881$$ −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 $$-0.790355\pi$$
−0.790838 + 0.612026i $$0.790355\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 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
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ −2.00000 −0.0665558
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 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 $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −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
$$925$$ 0 0
$$926$$ 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
$$931$$ 0 0
$$932$$ −42.0000 −1.37576
$$933$$ 6.00000 0.196431
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 24.0000 0.781133
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ −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
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ 36.0000 1.16371
$$958$$ 0 0
$$959$$ 3.00000 0.0968751
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ −20.0000 −0.641500
$$973$$ −13.0000 −0.416761
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −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
$$980$$ 0 0
$$981$$ 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
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 34.0000 1.08005 0.540023 0.841650i $$-0.318416\pi$$
0.540023 + 0.841650i $$0.318416\pi$$
$$992$$ 0 0
$$993$$ −56.0000 −1.77711
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −48.0000 −1.52094
$$997$$ −17.0000 −0.538395 −0.269198 0.963085i $$-0.586759\pi$$
−0.269198 + 0.963085i $$0.586759\pi$$
$$998$$ 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 9025.2.a.d.1.1 1
5.4 even 2 361.2.a.b.1.1 1
15.14 odd 2 3249.2.a.d.1.1 1
19.18 odd 2 475.2.a.b.1.1 1
20.19 odd 2 5776.2.a.c.1.1 1
57.56 even 2 4275.2.a.i.1.1 1
76.75 even 2 7600.2.a.c.1.1 1
95.4 even 18 361.2.e.e.54.1 6
95.9 even 18 361.2.e.e.62.1 6
95.14 odd 18 361.2.e.d.234.1 6
95.18 even 4 475.2.b.a.324.2 2
95.24 even 18 361.2.e.e.234.1 6
95.29 odd 18 361.2.e.d.62.1 6
95.34 odd 18 361.2.e.d.54.1 6
95.37 even 4 475.2.b.a.324.1 2
95.44 even 18 361.2.e.e.245.1 6
95.49 even 6 361.2.c.a.292.1 2
95.54 even 18 361.2.e.e.28.1 6
95.59 odd 18 361.2.e.d.99.1 6
95.64 even 6 361.2.c.a.68.1 2
95.69 odd 6 361.2.c.c.68.1 2
95.74 even 18 361.2.e.e.99.1 6
95.79 odd 18 361.2.e.d.28.1 6
95.84 odd 6 361.2.c.c.292.1 2
95.89 odd 18 361.2.e.d.245.1 6
95.94 odd 2 19.2.a.a.1.1 1
285.284 even 2 171.2.a.b.1.1 1
380.379 even 2 304.2.a.f.1.1 1
665.94 even 6 931.2.f.b.324.1 2
665.284 odd 6 931.2.f.c.324.1 2
665.474 even 6 931.2.f.b.704.1 2
665.569 odd 6 931.2.f.c.704.1 2
665.664 even 2 931.2.a.a.1.1 1
760.189 odd 2 1216.2.a.o.1.1 1
760.379 even 2 1216.2.a.b.1.1 1
1045.1044 even 2 2299.2.a.b.1.1 1
1140.1139 odd 2 2736.2.a.c.1.1 1
1235.1234 odd 2 3211.2.a.a.1.1 1
1615.1614 odd 2 5491.2.a.b.1.1 1
1995.1994 odd 2 8379.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.a.a.1.1 1 95.94 odd 2
171.2.a.b.1.1 1 285.284 even 2
304.2.a.f.1.1 1 380.379 even 2
361.2.a.b.1.1 1 5.4 even 2
361.2.c.a.68.1 2 95.64 even 6
361.2.c.a.292.1 2 95.49 even 6
361.2.c.c.68.1 2 95.69 odd 6
361.2.c.c.292.1 2 95.84 odd 6
361.2.e.d.28.1 6 95.79 odd 18
361.2.e.d.54.1 6 95.34 odd 18
361.2.e.d.62.1 6 95.29 odd 18
361.2.e.d.99.1 6 95.59 odd 18
361.2.e.d.234.1 6 95.14 odd 18
361.2.e.d.245.1 6 95.89 odd 18
361.2.e.e.28.1 6 95.54 even 18
361.2.e.e.54.1 6 95.4 even 18
361.2.e.e.62.1 6 95.9 even 18
361.2.e.e.99.1 6 95.74 even 18
361.2.e.e.234.1 6 95.24 even 18
361.2.e.e.245.1 6 95.44 even 18
475.2.a.b.1.1 1 19.18 odd 2
475.2.b.a.324.1 2 95.37 even 4
475.2.b.a.324.2 2 95.18 even 4
931.2.a.a.1.1 1 665.664 even 2
931.2.f.b.324.1 2 665.94 even 6
931.2.f.b.704.1 2 665.474 even 6
931.2.f.c.324.1 2 665.284 odd 6
931.2.f.c.704.1 2 665.569 odd 6
1216.2.a.b.1.1 1 760.379 even 2
1216.2.a.o.1.1 1 760.189 odd 2
2299.2.a.b.1.1 1 1045.1044 even 2
2736.2.a.c.1.1 1 1140.1139 odd 2
3211.2.a.a.1.1 1 1235.1234 odd 2
3249.2.a.d.1.1 1 15.14 odd 2
4275.2.a.i.1.1 1 57.56 even 2
5491.2.a.b.1.1 1 1615.1614 odd 2
5776.2.a.c.1.1 1 20.19 odd 2
7600.2.a.c.1.1 1 76.75 even 2
8379.2.a.j.1.1 1 1995.1994 odd 2
9025.2.a.d.1.1 1 1.1 even 1 trivial