# Properties

 Label 1216.2.a.o.1.1 Level $1216$ Weight $2$ Character 1216.1 Self dual yes Analytic conductor $9.710$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$9.70980888579$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −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 $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ −6.00000 −1.54919
$$16$$ 0 0
$$17$$ −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
$$20$$ 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
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −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 $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 3.00000 0.507093
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 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 $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 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 $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 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
$$76$$ 0 0
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 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 $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 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
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 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 $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ 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
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 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
$$124$$ 0 0
$$125$$ 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 $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 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 $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 18.0000 1.49482
$$146$$ 0 0
$$147$$ −12.0000 −0.989743
$$148$$ 0 0
$$149$$ −21.0000 −1.72039 −0.860194 0.509968i $$-0.829657\pi$$
−0.860194 + 0.509968i $$0.829657\pi$$
$$150$$ 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
$$156$$ 0 0
$$157$$ −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 $$0.786445\pi$$
$$164$$ 0 0
$$165$$ 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
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 0 0
$$177$$ 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
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 0 0
$$193$$ −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
$$196$$ 0 0
$$197$$ −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
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 3.00000 0.207514
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ 0 0
$$213$$ 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
$$220$$ 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
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 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
$$236$$ 0 0
$$237$$ 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 $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 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 $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 18.0000 1.12720
$$256$$ 0 0
$$257$$ 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 $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ 36.0000 2.21146
$$266$$ 0 0
$$267$$ 24.0000 1.46878
$$268$$ 0 0
$$269$$ −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 $$0.661532\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
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 $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ 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
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ −18.0000 −1.04800
$$296$$ 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
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 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
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 36.0000 2.00932
$$322$$ 0 0
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ 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 $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ 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 $$0.690611\pi$$
$$348$$ 0 0
$$349$$ −17.0000 −0.909989 −0.454995 0.890494i $$-0.650359\pi$$
−0.454995 + 0.890494i $$0.650359\pi$$
$$350$$ 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
$$356$$ 0 0
$$357$$ 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
$$364$$ 0 0
$$365$$ 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
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 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 $$-0.162020\pi$$
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
$$385$$ −9.00000 −0.458682
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 30.0000 1.51330
$$394$$ 0 0
$$395$$ −24.0000 −1.20757
$$396$$ 0 0
$$397$$ 7.00000 0.351320 0.175660 0.984451i $$-0.443794\pi$$
0.175660 + 0.984451i $$0.443794\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 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
$$404$$ 0 0
$$405$$ 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
$$412$$ 0 0
$$413$$ −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 $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 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
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 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
$$436$$ 0 0
$$437$$ 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 $$-0.477296\pi$$
0.0712672 + 0.997457i $$0.477296\pi$$
$$444$$ 0 0
$$445$$ −36.0000 −1.70656
$$446$$ 0 0
$$447$$ −42.0000 −1.98653
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 0 0
$$453$$ −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 $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ 0 0
$$463$$ −31.0000 −1.44069 −0.720346 0.693615i $$-0.756017\pi$$
−0.720346 + 0.693615i $$0.756017\pi$$
$$464$$ 0 0
$$465$$ 24.0000 1.11297
$$466$$ 0 0
$$467$$ 27.0000 1.24941 0.624705 0.780860i $$-0.285219\pi$$
0.624705 + 0.780860i $$0.285219\pi$$
$$468$$ 0 0
$$469$$ −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
$$476$$ 0 0
$$477$$ −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
$$484$$ 0 0
$$485$$ −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 $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 18.0000 0.810679
$$494$$ 0 0
$$495$$ 9.00000 0.404520
$$496$$ 0 0
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ −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
$$508$$ 0 0
$$509$$ 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
$$516$$ 0 0
$$517$$ 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 $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ 0 0
$$525$$ −8.00000 −0.349149
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ −24.0000 −1.03956
$$534$$ 0 0
$$535$$ −54.0000 −2.33462
$$536$$ 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 $$-0.319400\pi$$
0.537417 + 0.843317i $$0.319400\pi$$
$$542$$ 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 $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 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
$$556$$ 0 0
$$557$$ −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 $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ 0 0
$$565$$ −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 $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 6.00000 0.250654
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 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
$$580$$ 0 0
$$581$$ 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 $$-0.879051\pi$$
−0.928674 + 0.370896i $$0.879051\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 0 0
$$593$$ −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
$$596$$ 0 0
$$597$$ 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
$$604$$ 0 0
$$605$$ 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
$$612$$ 0 0
$$613$$ −29.0000 −1.17130 −0.585649 0.810564i $$-0.699160\pi$$
−0.585649 + 0.810564i $$0.699160\pi$$
$$614$$ 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 $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 6.00000 0.239617
$$628$$ 0 0
$$629$$ 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
$$636$$ 0 0
$$637$$ −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
$$645$$ −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
$$652$$ 0 0
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ −45.0000 −1.75830
$$656$$ 0 0
$$657$$ −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
$$665$$ −3.00000 −0.116335
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −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
$$676$$ 0 0
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 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 $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 9.00000 0.343872
$$686$$ 0 0
$$687$$ −10.0000 −0.381524
$$688$$ 0 0
$$689$$ −48.0000 −1.82865
$$690$$ 0 0
$$691$$ −17.0000 −0.646710 −0.323355 0.946278i $$-0.604811\pi$$
−0.323355 + 0.946278i $$0.604811\pi$$
$$692$$ 0 0
$$693$$ 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
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ 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
$$724$$ 0 0
$$725$$ −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
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 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 $$-0.564848\pi$$
−0.202321 + 0.979319i $$0.564848\pi$$
$$740$$ 0 0
$$741$$ −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
$$748$$ 0 0
$$749$$ −18.0000 −0.657706
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ −42.0000 −1.53057
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ 0 0
$$757$$ 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
$$764$$ 0 0
$$765$$ 9.00000 0.325396
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ 23.0000 0.829401 0.414701 0.909958i $$-0.363886\pi$$
0.414701 + 0.909958i $$0.363886\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ 42.0000 1.49904
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 0 0
$$789$$ 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
$$796$$ 0 0
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 0 0
$$803$$ 21.0000 0.741074
$$804$$ 0 0
$$805$$ 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 $$0.409361\pi$$
$$812$$ 0 0
$$813$$ −32.0000 −1.12229
$$814$$ 0 0
$$815$$ 60.0000 2.10171
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ 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 $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 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
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 54.0000 1.86875
$$836$$ 0 0
$$837$$ 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
$$844$$ 0 0
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 26.0000 0.892318
$$850$$ 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
$$855$$ 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 $$-0.184929\pi$$
0.835929 + 0.548837i $$0.184929\pi$$
$$860$$ 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
$$865$$ −54.0000 −1.83606
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ −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
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 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 $$0.790355\pi$$
$$884$$ 0 0
$$885$$ −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
$$892$$ 0 0
$$893$$ 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
$$900$$ 0 0
$$901$$ 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 $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 0 0
$$913$$ 36.0000 1.19143
$$914$$ 0 0
$$915$$ −6.00000 −0.198354
$$916$$ 0 0
$$917$$ −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
$$924$$ 0 0
$$925$$ −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
$$932$$ 0 0
$$933$$ −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
$$940$$ 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
$$944$$ 0 0
$$945$$ −12.0000 −0.390360
$$946$$ 0 0
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 0 0
$$949$$ −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
$$956$$ 0 0
$$957$$ 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
$$964$$ 0 0
$$965$$ 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 $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 0 0
$$973$$ −13.0000 −0.416761
$$974$$ 0 0
$$975$$ 32.0000 1.02482
$$976$$ 0 0
$$977$$ 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
$$985$$ 54.0000 1.72058
$$986$$ 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 $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 0 0
$$993$$ 56.0000 1.77711
$$994$$ 0 0
$$995$$ −33.0000 −1.04617
$$996$$ 0 0
$$997$$ −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 1216.2.a.o.1.1 1
4.3 odd 2 1216.2.a.b.1.1 1
8.3 odd 2 304.2.a.f.1.1 1
8.5 even 2 19.2.a.a.1.1 1
24.5 odd 2 171.2.a.b.1.1 1
24.11 even 2 2736.2.a.c.1.1 1
40.13 odd 4 475.2.b.a.324.1 2
40.19 odd 2 7600.2.a.c.1.1 1
40.29 even 2 475.2.a.b.1.1 1
40.37 odd 4 475.2.b.a.324.2 2
56.5 odd 6 931.2.f.b.704.1 2
56.13 odd 2 931.2.a.a.1.1 1
56.37 even 6 931.2.f.c.704.1 2
56.45 odd 6 931.2.f.b.324.1 2
56.53 even 6 931.2.f.c.324.1 2
88.21 odd 2 2299.2.a.b.1.1 1
104.77 even 2 3211.2.a.a.1.1 1
120.29 odd 2 4275.2.a.i.1.1 1
136.101 even 2 5491.2.a.b.1.1 1
152.5 even 18 361.2.e.d.234.1 6
152.13 odd 18 361.2.e.e.245.1 6
152.21 odd 18 361.2.e.e.99.1 6
152.29 odd 18 361.2.e.e.62.1 6
152.37 odd 2 361.2.a.b.1.1 1
152.45 even 6 361.2.c.c.68.1 2
152.53 odd 18 361.2.e.e.54.1 6
152.61 even 18 361.2.e.d.54.1 6
152.69 odd 6 361.2.c.a.68.1 2
152.75 even 2 5776.2.a.c.1.1 1
152.85 even 18 361.2.e.d.62.1 6
152.93 even 18 361.2.e.d.99.1 6
152.101 even 18 361.2.e.d.245.1 6
152.109 odd 18 361.2.e.e.234.1 6
152.117 odd 18 361.2.e.e.28.1 6
152.125 even 6 361.2.c.c.292.1 2
152.141 odd 6 361.2.c.a.292.1 2
152.149 even 18 361.2.e.d.28.1 6
168.125 even 2 8379.2.a.j.1.1 1
456.341 even 2 3249.2.a.d.1.1 1
760.189 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 8.5 even 2
171.2.a.b.1.1 1 24.5 odd 2
304.2.a.f.1.1 1 8.3 odd 2
361.2.a.b.1.1 1 152.37 odd 2
361.2.c.a.68.1 2 152.69 odd 6
361.2.c.a.292.1 2 152.141 odd 6
361.2.c.c.68.1 2 152.45 even 6
361.2.c.c.292.1 2 152.125 even 6
361.2.e.d.28.1 6 152.149 even 18
361.2.e.d.54.1 6 152.61 even 18
361.2.e.d.62.1 6 152.85 even 18
361.2.e.d.99.1 6 152.93 even 18
361.2.e.d.234.1 6 152.5 even 18
361.2.e.d.245.1 6 152.101 even 18
361.2.e.e.28.1 6 152.117 odd 18
361.2.e.e.54.1 6 152.53 odd 18
361.2.e.e.62.1 6 152.29 odd 18
361.2.e.e.99.1 6 152.21 odd 18
361.2.e.e.234.1 6 152.109 odd 18
361.2.e.e.245.1 6 152.13 odd 18
475.2.a.b.1.1 1 40.29 even 2
475.2.b.a.324.1 2 40.13 odd 4
475.2.b.a.324.2 2 40.37 odd 4
931.2.a.a.1.1 1 56.13 odd 2
931.2.f.b.324.1 2 56.45 odd 6
931.2.f.b.704.1 2 56.5 odd 6
931.2.f.c.324.1 2 56.53 even 6
931.2.f.c.704.1 2 56.37 even 6
1216.2.a.b.1.1 1 4.3 odd 2
1216.2.a.o.1.1 1 1.1 even 1 trivial
2299.2.a.b.1.1 1 88.21 odd 2
2736.2.a.c.1.1 1 24.11 even 2
3211.2.a.a.1.1 1 104.77 even 2
3249.2.a.d.1.1 1 456.341 even 2
4275.2.a.i.1.1 1 120.29 odd 2
5491.2.a.b.1.1 1 136.101 even 2
5776.2.a.c.1.1 1 152.75 even 2
7600.2.a.c.1.1 1 40.19 odd 2
8379.2.a.j.1.1 1 168.125 even 2
9025.2.a.d.1.1 1 760.189 odd 2