# Properties

 Label 7942.2.a.l.1.1 Level $7942$ Weight $2$ Character 7942.1 Self dual yes Analytic conductor $63.417$ 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] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.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: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7942.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{21} +1.00000 q^{22} -2.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +9.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} +5.00000 q^{43} +1.00000 q^{44} -3.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} +12.0000 q^{51} -1.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} +9.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} +9.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -10.0000 q^{74} +10.0000 q^{75} +2.00000 q^{77} +2.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} -9.00000 q^{83} -4.00000 q^{84} +5.00000 q^{86} -18.0000 q^{87} +1.00000 q^{88} +15.0000 q^{89} -2.00000 q^{91} -16.0000 q^{93} -3.00000 q^{94} -2.00000 q^{96} -1.00000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −2.00000 −0.577350
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 4.00000 0.769800
$$28$$ 2.00000 0.377964
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −2.00000 −0.348155
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ −3.00000 −0.428571
$$50$$ −5.00000 −0.707107
$$51$$ 12.0000 1.68034
$$52$$ −1.00000 −0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 0 0
$$58$$ 9.00000 1.18176
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 10.0000 1.15470
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 2.00000 0.226455
$$79$$ 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
$$82$$ −6.00000 −0.662589
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 5.00000 0.539164
$$87$$ −18.0000 −1.92980
$$88$$ 1.00000 0.106600
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ −16.0000 −1.65912
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 1.00000 0.100504
$$100$$ −5.00000 −0.500000
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 12.0000 1.18818
$$103$$ −19.0000 −1.87213 −0.936063 0.351833i $$-0.885559\pi$$
−0.936063 + 0.351833i $$0.885559\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 20.0000 1.89832
$$112$$ 2.00000 0.188982
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ −1.00000 −0.0924500
$$118$$ 6.00000 0.552345
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −13.0000 −1.17696
$$123$$ 12.0000 1.08200
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −10.0000 −0.880451
$$130$$ 0 0
$$131$$ −3.00000 −0.262111 −0.131056 0.991375i $$-0.541837\pi$$
−0.131056 + 0.991375i $$0.541837\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 0 0
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 9.00000 0.755263
$$143$$ −1.00000 −0.0836242
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 6.00000 0.494872
$$148$$ −10.0000 −0.821995
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 10.0000 0.816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −11.0000 −0.864242
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −9.00000 −0.698535
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 5.00000 0.381246
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −18.0000 −1.36458
$$175$$ −10.0000 −0.755929
$$176$$ 1.00000 0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ 15.0000 1.12430
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 26.0000 1.92198
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −16.0000 −1.17318
$$187$$ −6.00000 −0.438763
$$188$$ −3.00000 −0.218797
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ −2.00000 −0.144338
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ −1.00000 −0.0717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −15.0000 −1.06871 −0.534353 0.845262i $$-0.679445\pi$$
−0.534353 + 0.845262i $$0.679445\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −19.0000 −1.34687 −0.673437 0.739244i $$-0.735183\pi$$
−0.673437 + 0.739244i $$0.735183\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ 20.0000 1.41069
$$202$$ −3.00000 −0.211079
$$203$$ 18.0000 1.26335
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −19.0000 −1.32379
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −18.0000 −1.23334
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ 16.0000 1.08615
$$218$$ 11.0000 0.745014
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 20.0000 1.34231
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −5.00000 −0.333333
$$226$$ 9.00000 0.598671
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 9.00000 0.590879
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ −16.0000 −1.03931
$$238$$ −12.0000 −0.777844
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 10.0000 0.641500
$$244$$ −13.0000 −0.832240
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −10.0000 −0.622573
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ −3.00000 −0.185341
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −30.0000 −1.83597
$$268$$ −10.0000 −0.610847
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 4.00000 0.242091
$$274$$ −9.00000 −0.543710
$$275$$ −5.00000 −0.301511
$$276$$ 0 0
$$277$$ 17.0000 1.02143 0.510716 0.859750i $$-0.329381\pi$$
0.510716 + 0.859750i $$0.329381\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 9.00000 0.534052
$$285$$ 0 0
$$286$$ −1.00000 −0.0591312
$$287$$ −12.0000 −0.708338
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ −4.00000 −0.234082
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 4.00000 0.232104
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 10.0000 0.577350
$$301$$ 10.0000 0.576390
$$302$$ 8.00000 0.460348
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −7.00000 −0.399511 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 38.0000 2.16174
$$310$$ 0 0
$$311$$ −15.0000 −0.850572 −0.425286 0.905059i $$-0.639826\pi$$
−0.425286 + 0.905059i $$0.639826\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 17.0000 0.960897 0.480448 0.877023i $$-0.340474\pi$$
0.480448 + 0.877023i $$0.340474\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −11.0000 −0.611111
$$325$$ 5.00000 0.277350
$$326$$ 2.00000 0.110770
$$327$$ −22.0000 −1.21660
$$328$$ −6.00000 −0.331295
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −9.00000 −0.493939
$$333$$ −10.0000 −0.547997
$$334$$ 6.00000 0.328305
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 5.00000 0.269582
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −33.0000 −1.77153 −0.885766 0.464131i $$-0.846367\pi$$
−0.885766 + 0.464131i $$0.846367\pi$$
$$348$$ −18.0000 −0.964901
$$349$$ 5.00000 0.267644 0.133822 0.991005i $$-0.457275\pi$$
0.133822 + 0.991005i $$0.457275\pi$$
$$350$$ −10.0000 −0.534522
$$351$$ −4.00000 −0.213504
$$352$$ 1.00000 0.0533002
$$353$$ −21.0000 −1.11772 −0.558859 0.829263i $$-0.688761\pi$$
−0.558859 + 0.829263i $$0.688761\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 15.0000 0.794998
$$357$$ 24.0000 1.27021
$$358$$ −12.0000 −0.634220
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ −16.0000 −0.840941
$$363$$ −2.00000 −0.104973
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 26.0000 1.35904
$$367$$ −7.00000 −0.365397 −0.182699 0.983169i $$-0.558483\pi$$
−0.182699 + 0.983169i $$0.558483\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ −16.0000 −0.829561
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ −9.00000 −0.463524
$$378$$ 8.00000 0.411476
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 3.00000 0.153493
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ −16.0000 −0.814379
$$387$$ 5.00000 0.254164
$$388$$ −1.00000 −0.0507673
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ 6.00000 0.302660
$$394$$ −15.0000 −0.755689
$$395$$ 0 0
$$396$$ 1.00000 0.0502519
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −19.0000 −0.952384
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 20.0000 0.997509
$$403$$ −8.00000 −0.398508
$$404$$ −3.00000 −0.149256
$$405$$ 0 0
$$406$$ 18.0000 0.893325
$$407$$ −10.0000 −0.495682
$$408$$ 12.0000 0.594089
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ −19.0000 −0.936063
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −3.00000 −0.145865
$$424$$ −6.00000 −0.291386
$$425$$ 30.0000 1.45521
$$426$$ −18.0000 −0.872103
$$427$$ −26.0000 −1.25823
$$428$$ −9.00000 −0.435031
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 4.00000 0.192450
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 0 0
$$438$$ 8.00000 0.382255
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 6.00000 0.285391
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 20.0000 0.949158
$$445$$ 0 0
$$446$$ −7.00000 −0.331460
$$447$$ −12.0000 −0.567581
$$448$$ 2.00000 0.0944911
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −5.00000 −0.235702
$$451$$ −6.00000 −0.282529
$$452$$ 9.00000 0.423324
$$453$$ −16.0000 −0.751746
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −40.0000 −1.87112 −0.935561 0.353166i $$-0.885105\pi$$
−0.935561 + 0.353166i $$0.885105\pi$$
$$458$$ 20.0000 0.934539
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ −3.00000 −0.139724 −0.0698620 0.997557i $$-0.522256\pi$$
−0.0698620 + 0.997557i $$0.522256\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ 17.0000 0.790057 0.395029 0.918669i $$-0.370735\pi$$
0.395029 + 0.918669i $$0.370735\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ −20.0000 −0.923514
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 6.00000 0.276172
$$473$$ 5.00000 0.229900
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 26.0000 1.18427
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ −7.00000 −0.317200 −0.158600 0.987343i $$-0.550698\pi$$
−0.158600 + 0.987343i $$0.550698\pi$$
$$488$$ −13.0000 −0.588482
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 12.0000 0.541002
$$493$$ −54.0000 −2.43204
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 18.0000 0.807410
$$498$$ 18.0000 0.806599
$$499$$ 26.0000 1.16392 0.581960 0.813217i $$-0.302286\pi$$
0.581960 + 0.813217i $$0.302286\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ −18.0000 −0.803379
$$503$$ −30.0000 −1.33763 −0.668817 0.743427i $$-0.733199\pi$$
−0.668817 + 0.743427i $$0.733199\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 24.0000 1.06588
$$508$$ 2.00000 0.0887357
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −10.0000 −0.440225
$$517$$ −3.00000 −0.131940
$$518$$ −20.0000 −0.878750
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 9.00000 0.393919
$$523$$ 35.0000 1.53044 0.765222 0.643767i $$-0.222629\pi$$
0.765222 + 0.643767i $$0.222629\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 20.0000 0.872872
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ −2.00000 −0.0870388
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ −30.0000 −1.29823
$$535$$ 0 0
$$536$$ −10.0000 −0.431934
$$537$$ 24.0000 1.03568
$$538$$ −18.0000 −0.776035
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 2.00000 0.0859074
$$543$$ 32.0000 1.37325
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 4.00000 0.171184
$$547$$ 29.0000 1.23995 0.619975 0.784621i $$-0.287143\pi$$
0.619975 + 0.784621i $$0.287143\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ −13.0000 −0.554826
$$550$$ −5.00000 −0.213201
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 17.0000 0.722261
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 21.0000 0.889799 0.444899 0.895581i $$-0.353239\pi$$
0.444899 + 0.895581i $$0.353239\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 12.0000 0.506189
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ −22.0000 −0.923913
$$568$$ 9.00000 0.377632
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 23.0000 0.962520 0.481260 0.876578i $$-0.340179\pi$$
0.481260 + 0.876578i $$0.340179\pi$$
$$572$$ −1.00000 −0.0418121
$$573$$ −6.00000 −0.250654
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 32.0000 1.32987
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 2.00000 0.0829027
$$583$$ −6.00000 −0.248495
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ −9.00000 −0.371787
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 6.00000 0.247436
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 30.0000 1.23404
$$592$$ −10.0000 −0.410997
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 38.0000 1.55524
$$598$$ 0 0
$$599$$ −9.00000 −0.367730 −0.183865 0.982952i $$-0.558861\pi$$
−0.183865 + 0.982952i $$0.558861\pi$$
$$600$$ 10.0000 0.408248
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 10.0000 0.407570
$$603$$ −10.0000 −0.407231
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ −36.0000 −1.45879
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ −6.00000 −0.242536
$$613$$ −25.0000 −1.00974 −0.504870 0.863195i $$-0.668460\pi$$
−0.504870 + 0.863195i $$0.668460\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ 2.00000 0.0805823
$$617$$ 21.0000 0.845428 0.422714 0.906263i $$-0.361077\pi$$
0.422714 + 0.906263i $$0.361077\pi$$
$$618$$ 38.0000 1.52858
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −15.0000 −0.601445
$$623$$ 30.0000 1.20192
$$624$$ 2.00000 0.0800641
$$625$$ 25.0000 1.00000
$$626$$ 17.0000 0.679457
$$627$$ 0 0
$$628$$ 14.0000 0.558661
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −13.0000 −0.517522 −0.258761 0.965941i $$-0.583314\pi$$
−0.258761 + 0.965941i $$0.583314\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −40.0000 −1.58986
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 3.00000 0.118864
$$638$$ 9.00000 0.356313
$$639$$ 9.00000 0.356034
$$640$$ 0 0
$$641$$ −3.00000 −0.118493 −0.0592464 0.998243i $$-0.518870\pi$$
−0.0592464 + 0.998243i $$0.518870\pi$$
$$642$$ 18.0000 0.710403
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 21.0000 0.825595 0.412798 0.910823i $$-0.364552\pi$$
0.412798 + 0.910823i $$0.364552\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 6.00000 0.235521
$$650$$ 5.00000 0.196116
$$651$$ −32.0000 −1.25418
$$652$$ 2.00000 0.0783260
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ −22.0000 −0.860268
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −4.00000 −0.156055
$$658$$ −6.00000 −0.233904
$$659$$ −39.0000 −1.51922 −0.759612 0.650376i $$-0.774611\pi$$
−0.759612 + 0.650376i $$0.774611\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ −12.0000 −0.466041
$$664$$ −9.00000 −0.349268
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 0 0
$$668$$ 6.00000 0.232147
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ −13.0000 −0.501859
$$672$$ −4.00000 −0.154303
$$673$$ 38.0000 1.46479 0.732396 0.680879i $$-0.238402\pi$$
0.732396 + 0.680879i $$0.238402\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ −20.0000 −0.769800
$$676$$ −12.0000 −0.461538
$$677$$ 15.0000 0.576497 0.288248 0.957556i $$-0.406927\pi$$
0.288248 + 0.957556i $$0.406927\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ 8.00000 0.306336
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −40.0000 −1.52610
$$688$$ 5.00000 0.190623
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 2.00000 0.0759737
$$694$$ −33.0000 −1.25266
$$695$$ 0 0
$$696$$ −18.0000 −0.682288
$$697$$ 36.0000 1.36360
$$698$$ 5.00000 0.189253
$$699$$ −24.0000 −0.907763
$$700$$ −10.0000 −0.377964
$$701$$ 15.0000 0.566542 0.283271 0.959040i $$-0.408580\pi$$
0.283271 + 0.959040i $$0.408580\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ −6.00000 −0.225653
$$708$$ −12.0000 −0.450988
$$709$$ 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
$$712$$ 15.0000 0.562149
$$713$$ 0 0
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −30.0000 −1.11959
$$719$$ −15.0000 −0.559406 −0.279703 0.960087i $$-0.590236\pi$$
−0.279703 + 0.960087i $$0.590236\pi$$
$$720$$ 0 0
$$721$$ −38.0000 −1.41519
$$722$$ 0 0
$$723$$ −52.0000 −1.93390
$$724$$ −16.0000 −0.594635
$$725$$ −45.0000 −1.67126
$$726$$ −2.00000 −0.0742270
$$727$$ −13.0000 −0.482143 −0.241072 0.970507i $$-0.577499\pi$$
−0.241072 + 0.970507i $$0.577499\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −30.0000 −1.10959
$$732$$ 26.0000 0.960988
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ −7.00000 −0.258375
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ −6.00000 −0.220863
$$739$$ 23.0000 0.846069 0.423034 0.906114i $$-0.360965\pi$$
0.423034 + 0.906114i $$0.360965\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 18.0000 0.660356 0.330178 0.943919i $$-0.392891\pi$$
0.330178 + 0.943919i $$0.392891\pi$$
$$744$$ −16.0000 −0.586588
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ −9.00000 −0.329293
$$748$$ −6.00000 −0.219382
$$749$$ −18.0000 −0.657706
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 36.0000 1.31191
$$754$$ −9.00000 −0.327761
$$755$$ 0 0
$$756$$ 8.00000 0.290957
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 2.00000 0.0726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 22.0000 0.796453
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.00000 −0.216647
$$768$$ −2.00000 −0.0721688
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 36.0000 1.29651
$$772$$ −16.0000 −0.575853
$$773$$ −54.0000 −1.94225 −0.971123 0.238581i $$-0.923318\pi$$
−0.971123 + 0.238581i $$0.923318\pi$$
$$774$$ 5.00000 0.179721
$$775$$ −40.0000 −1.43684
$$776$$ −1.00000 −0.0358979
$$777$$ 40.0000 1.43499
$$778$$ −24.0000 −0.860442
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 9.00000 0.322045
$$782$$ 0 0
$$783$$ 36.0000 1.28654
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 6.00000 0.214013
$$787$$ 35.0000 1.24762 0.623808 0.781578i $$-0.285585\pi$$
0.623808 + 0.781578i $$0.285585\pi$$
$$788$$ −15.0000 −0.534353
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0000 0.640006
$$792$$ 1.00000 0.0355335
$$793$$ 13.0000 0.461644
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −19.0000 −0.673437
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ −5.00000 −0.176777
$$801$$ 15.0000 0.529999
$$802$$ −18.0000 −0.635602
$$803$$ −4.00000 −0.141157
$$804$$ 20.0000 0.705346
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 36.0000 1.26726
$$808$$ −3.00000 −0.105540
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 18.0000 0.631676
$$813$$ −4.00000 −0.140286
$$814$$ −10.0000 −0.350500
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ 0 0
$$818$$ −4.00000 −0.139857
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 18.0000 0.627822
$$823$$ −13.0000 −0.453152 −0.226576 0.973994i $$-0.572753\pi$$
−0.226576 + 0.973994i $$0.572753\pi$$
$$824$$ −19.0000 −0.661896
$$825$$ 10.0000 0.348155
$$826$$ 12.0000 0.417533
$$827$$ 45.0000 1.56480 0.782402 0.622774i $$-0.213994\pi$$
0.782402 + 0.622774i $$0.213994\pi$$
$$828$$ 0 0
$$829$$ −46.0000 −1.59765 −0.798823 0.601566i $$-0.794544\pi$$
−0.798823 + 0.601566i $$0.794544\pi$$
$$830$$ 0 0
$$831$$ −34.0000 −1.17945
$$832$$ −1.00000 −0.0346688
$$833$$ 18.0000 0.623663
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 32.0000 1.10608
$$838$$ 12.0000 0.414533
$$839$$ 9.00000 0.310715 0.155357 0.987858i $$-0.450347\pi$$
0.155357 + 0.987858i $$0.450347\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ −28.0000 −0.964944
$$843$$ −24.0000 −0.826604
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ −3.00000 −0.103142
$$847$$ 2.00000 0.0687208
$$848$$ −6.00000 −0.206041
$$849$$ 8.00000 0.274559
$$850$$ 30.0000 1.02899
$$851$$ 0 0
$$852$$ −18.0000 −0.616670
$$853$$ 2.00000 0.0684787 0.0342393 0.999414i $$-0.489099\pi$$
0.0342393 + 0.999414i $$0.489099\pi$$
$$854$$ −26.0000 −0.889702
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ −34.0000 −1.16007 −0.580033 0.814593i $$-0.696960\pi$$
−0.580033 + 0.814593i $$0.696960\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ −36.0000 −1.22616
$$863$$ −33.0000 −1.12333 −0.561667 0.827364i $$-0.689840\pi$$
−0.561667 + 0.827364i $$0.689840\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 11.0000 0.373795
$$867$$ −38.0000 −1.29055
$$868$$ 16.0000 0.543075
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 11.0000 0.372507
$$873$$ −1.00000 −0.0338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ 38.0000 1.28317 0.641584 0.767052i $$-0.278277\pi$$
0.641584 + 0.767052i $$0.278277\pi$$
$$878$$ 20.0000 0.674967
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 20.0000 0.671156
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ −7.00000 −0.234377
$$893$$ 0 0
$$894$$ −12.0000 −0.401340
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ 72.0000 2.40133
$$900$$ −5.00000 −0.166667
$$901$$ 36.0000 1.19933
$$902$$ −6.00000 −0.199778
$$903$$ −20.0000 −0.665558
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ −3.00000 −0.0995037
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ −9.00000 −0.297857
$$914$$ −40.0000 −1.32308
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ −6.00000 −0.198137
$$918$$ −24.0000 −0.792118
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ −3.00000 −0.0987997
$$923$$ −9.00000 −0.296239
$$924$$ −4.00000 −0.131590
$$925$$ 50.0000 1.64399
$$926$$ 17.0000 0.558655
$$927$$ −19.0000 −0.624042
$$928$$ 9.00000 0.295439
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 12.0000 0.393073
$$933$$ 30.0000 0.982156
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 20.0000 0.653372 0.326686 0.945133i $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$938$$ −20.0000 −0.653023
$$939$$ −34.0000 −1.10955
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −28.0000 −0.912289
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 5.00000 0.162564
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −60.0000 −1.94563
$$952$$ −12.0000 −0.388922
$$953$$ 48.0000 1.55487 0.777436 0.628962i $$-0.216520\pi$$
0.777436 + 0.628962i $$0.216520\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −18.0000 −0.581857
$$958$$ −36.0000 −1.16311
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 10.0000 0.322413
$$963$$ −9.00000 −0.290021
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 10.0000 0.320750
$$973$$ −8.00000 −0.256468
$$974$$ −7.00000 −0.224294
$$975$$ −10.0000 −0.320256
$$976$$ −13.0000 −0.416120
$$977$$ −9.00000 −0.287936 −0.143968 0.989582i $$-0.545986\pi$$
−0.143968 + 0.989582i $$0.545986\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 15.0000 0.479402
$$980$$ 0 0
$$981$$ 11.0000 0.351203
$$982$$ −12.0000 −0.382935
$$983$$ 33.0000 1.05254 0.526268 0.850319i $$-0.323591\pi$$
0.526268 + 0.850319i $$0.323591\pi$$
$$984$$ 12.0000 0.382546
$$985$$ 0 0
$$986$$ −54.0000 −1.71971
$$987$$ 12.0000 0.381964
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 5.00000 0.158830 0.0794151 0.996842i $$-0.474695\pi$$
0.0794151 + 0.996842i $$0.474695\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 56.0000 1.77711
$$994$$ 18.0000 0.570925
$$995$$ 0 0
$$996$$ 18.0000 0.570352
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 26.0000 0.823016
$$999$$ −40.0000 −1.26554
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 7942.2.a.l.1.1 1
19.7 even 3 418.2.e.d.353.1 yes 2
19.11 even 3 418.2.e.d.45.1 2
19.18 odd 2 7942.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.d.45.1 2 19.11 even 3
418.2.e.d.353.1 yes 2 19.7 even 3
7942.2.a.j.1.1 1 19.18 odd 2
7942.2.a.l.1.1 1 1.1 even 1 trivial