# Properties

 Label 2960.2.a.j.1.1 Level $2960$ Weight $2$ Character 2960.1 Self dual yes Analytic conductor $23.636$ 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] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.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: $$23.6357189983$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -4.00000 q^{13} -2.00000 q^{15} +3.00000 q^{17} -2.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} -6.00000 q^{33} -1.00000 q^{35} +1.00000 q^{37} -8.00000 q^{39} +3.00000 q^{41} +1.00000 q^{43} -1.00000 q^{45} -12.0000 q^{47} -6.00000 q^{49} +6.00000 q^{51} +3.00000 q^{53} +3.00000 q^{55} -4.00000 q^{57} -1.00000 q^{61} +1.00000 q^{63} +4.00000 q^{65} +4.00000 q^{67} -12.0000 q^{69} -6.00000 q^{71} -16.0000 q^{73} +2.00000 q^{75} -3.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} +12.0000 q^{83} -3.00000 q^{85} +6.00000 q^{87} -6.00000 q^{89} -4.00000 q^{91} -10.0000 q^{93} +2.00000 q^{95} +17.0000 q^{97} -3.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 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 $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 1.00000 0.164399
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 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
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −16.0000 −1.87266 −0.936329 0.351123i $$-0.885800\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 2.00000 0.230940
$$76$$ 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 $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 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
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 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
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ −24.0000 −2.02116
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ −3.00000 −0.249136
$$146$$ 0 0
$$147$$ −12.0000 −0.989743
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 3.00000 0.242536
$$154$$ 0 0
$$155$$ 5.00000 0.401610
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ 0 0
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ 0 0
$$165$$ 6.00000 0.467099
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 15.0000 1.14043 0.570214 0.821496i $$-0.306860\pi$$
0.570214 + 0.821496i $$0.306860\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −1.00000 −0.0735215
$$186$$ 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 $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 0 0
$$195$$ 8.00000 0.572892
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 3.00000 0.210559
$$204$$ 0 0
$$205$$ −3.00000 −0.209529
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 0 0
$$213$$ −12.0000 −0.822226
$$214$$ 0 0
$$215$$ −1.00000 −0.0681994
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 0 0
$$219$$ −32.0000 −2.16236
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ −17.0000 −1.13840 −0.569202 0.822198i $$-0.692748\pi$$
−0.569202 + 0.822198i $$0.692748\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 27.0000 1.79205 0.896026 0.444001i $$-0.146441\pi$$
0.896026 + 0.444001i $$0.146441\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 12.0000 0.782794
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ −6.00000 −0.375735
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 1.00000 0.0621370
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 0 0
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ −3.00000 −0.184289
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 0 0
$$269$$ −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 $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ −8.00000 −0.484182
$$274$$ 0 0
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 3.00000 0.177084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 34.0000 1.99312
$$292$$ 0 0
$$293$$ 21.0000 1.22683 0.613417 0.789760i $$-0.289795\pi$$
0.613417 + 0.789760i $$0.289795\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 12.0000 0.696311
$$298$$ 0 0
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 0 0
$$305$$ 1.00000 0.0572598
$$306$$ 0 0
$$307$$ 34.0000 1.94048 0.970241 0.242140i $$-0.0778494\pi$$
0.970241 + 0.242140i $$0.0778494\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 9.00000 0.510343 0.255172 0.966896i $$-0.417868\pi$$
0.255172 + 0.966896i $$0.417868\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 0 0
$$315$$ −1.00000 −0.0563436
$$316$$ 0 0
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ 22.0000 1.21660
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −26.0000 −1.42909 −0.714545 0.699590i $$-0.753366\pi$$
−0.714545 + 0.699590i $$0.753366\pi$$
$$332$$ 0 0
$$333$$ 1.00000 0.0547997
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ 0 0
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 16.0000 0.854017
$$352$$ 0 0
$$353$$ −21.0000 −1.11772 −0.558859 0.829263i $$-0.688761\pi$$
−0.558859 + 0.829263i $$0.688761\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ 0 0
$$357$$ 6.00000 0.317554
$$358$$ 0 0
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −4.00000 −0.209946
$$364$$ 0 0
$$365$$ 16.0000 0.837478
$$366$$ 0 0
$$367$$ −35.0000 −1.82699 −0.913493 0.406855i $$-0.866625\pi$$
−0.913493 + 0.406855i $$0.866625\pi$$
$$368$$ 0 0
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ −2.00000 −0.103280
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −40.0000 −2.04926
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 3.00000 0.152894
$$386$$ 0 0
$$387$$ 1.00000 0.0508329
$$388$$ 0 0
$$389$$ 9.00000 0.456318 0.228159 0.973624i $$-0.426729\pi$$
0.228159 + 0.973624i $$0.426729\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ 20.0000 0.996271
$$404$$ 0 0
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ −3.00000 −0.148704
$$408$$ 0 0
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ 24.0000 1.18383
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 26.0000 1.27323
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ 3.00000 0.145521
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ −15.0000 −0.722525 −0.361262 0.932464i $$-0.617654\pi$$
−0.361262 + 0.932464i $$0.617654\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ −6.00000 −0.287678
$$436$$ 0 0
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 0 0
$$453$$ −16.0000 −0.751746
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 17.0000 0.795226 0.397613 0.917553i $$-0.369839\pi$$
0.397613 + 0.917553i $$0.369839\pi$$
$$458$$ 0 0
$$459$$ −12.0000 −0.560112
$$460$$ 0 0
$$461$$ 33.0000 1.53696 0.768482 0.639872i $$-0.221013\pi$$
0.768482 + 0.639872i $$0.221013\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ 10.0000 0.463739
$$466$$ 0 0
$$467$$ −27.0000 −1.24941 −0.624705 0.780860i $$-0.714781\pi$$
−0.624705 + 0.780860i $$0.714781\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −26.0000 −1.19802
$$472$$ 0 0
$$473$$ −3.00000 −0.137940
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 3.00000 0.137361
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 0 0
$$483$$ −12.0000 −0.546019
$$484$$ 0 0
$$485$$ −17.0000 −0.771930
$$486$$ 0 0
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ 0 0
$$489$$ −22.0000 −0.994874
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 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
$$511$$ −16.0000 −0.707798
$$512$$ 0 0
$$513$$ 8.00000 0.353209
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 36.0000 1.58328
$$518$$ 0 0
$$519$$ 30.0000 1.31685
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 2.00000 0.0872872
$$526$$ 0 0
$$527$$ −15.0000 −0.653410
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 4.00000 0.171656
$$544$$ 0 0
$$545$$ −11.0000 −0.471188
$$546$$ 0 0
$$547$$ 1.00000 0.0427569 0.0213785 0.999771i $$-0.493195\pi$$
0.0213785 + 0.999771i $$0.493195\pi$$
$$548$$ 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
$$555$$ −2.00000 −0.0848953
$$556$$ 0 0
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ 3.00000 0.126435 0.0632175 0.998000i $$-0.479864\pi$$
0.0632175 + 0.998000i $$0.479864\pi$$
$$564$$ 0 0
$$565$$ −9.00000 −0.378633
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 31.0000 1.29731 0.648655 0.761083i $$-0.275332\pi$$
0.648655 + 0.761083i $$0.275332\pi$$
$$572$$ 0 0
$$573$$ −6.00000 −0.250654
$$574$$ 0 0
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 0 0
$$579$$ 28.0000 1.16364
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ 4.00000 0.165380
$$586$$ 0 0
$$587$$ −27.0000 −1.11441 −0.557205 0.830375i $$-0.688126\pi$$
−0.557205 + 0.830375i $$0.688126\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ −3.00000 −0.122988
$$596$$ 0 0
$$597$$ 32.0000 1.30967
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 22.0000 0.892952 0.446476 0.894795i $$-0.352679\pi$$
0.446476 + 0.894795i $$0.352679\pi$$
$$608$$ 0 0
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 0 0
$$613$$ 29.0000 1.17130 0.585649 0.810564i $$-0.300840\pi$$
0.585649 + 0.810564i $$0.300840\pi$$
$$614$$ 0 0
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −35.0000 −1.40677 −0.703384 0.710810i $$-0.748329\pi$$
−0.703384 + 0.710810i $$0.748329\pi$$
$$620$$ 0 0
$$621$$ 24.0000 0.963087
$$622$$ 0 0
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ 0 0
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ −10.0000 −0.397464
$$634$$ 0 0
$$635$$ 20.0000 0.793676
$$636$$ 0 0
$$637$$ 24.0000 0.950915
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ 13.0000 0.512670 0.256335 0.966588i $$-0.417485\pi$$
0.256335 + 0.966588i $$0.417485\pi$$
$$644$$ 0 0
$$645$$ −2.00000 −0.0787499
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ 0 0
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −16.0000 −0.624219
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 5.00000 0.194477 0.0972387 0.995261i $$-0.468999\pi$$
0.0972387 + 0.995261i $$0.468999\pi$$
$$662$$ 0 0
$$663$$ −24.0000 −0.932083
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ 0 0
$$669$$ −34.0000 −1.31452
$$670$$ 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
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 17.0000 0.652400
$$680$$ 0 0
$$681$$ 54.0000 2.06928
$$682$$ 0 0
$$683$$ 15.0000 0.573959 0.286980 0.957937i $$-0.407349\pi$$
0.286980 + 0.957937i $$0.407349\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ 28.0000 1.06827
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 19.0000 0.722794 0.361397 0.932412i $$-0.382300\pi$$
0.361397 + 0.932412i $$0.382300\pi$$
$$692$$ 0 0
$$693$$ −3.00000 −0.113961
$$694$$ 0 0
$$695$$ −13.0000 −0.493118
$$696$$ 0 0
$$697$$ 9.00000 0.340899
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 24.0000 0.903892
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 0 0
$$717$$ 18.0000 0.672222
$$718$$ 0 0
$$719$$ 42.0000 1.56634 0.783168 0.621810i $$-0.213603\pi$$
0.783168 + 0.621810i $$0.213603\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 0 0
$$723$$ −56.0000 −2.08266
$$724$$ 0 0
$$725$$ 3.00000 0.111417
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 3.00000 0.110959
$$732$$ 0 0
$$733$$ −31.0000 −1.14501 −0.572506 0.819901i $$-0.694029\pi$$
−0.572506 + 0.819901i $$0.694029\pi$$
$$734$$ 0 0
$$735$$ 12.0000 0.442627
$$736$$ 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
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ 51.0000 1.87101 0.935504 0.353315i $$-0.114946\pi$$
0.935504 + 0.353315i $$0.114946\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ −14.0000 −0.510867 −0.255434 0.966827i $$-0.582218\pi$$
−0.255434 + 0.966827i $$0.582218\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 0 0
$$759$$ 36.0000 1.30672
$$760$$ 0 0
$$761$$ 27.0000 0.978749 0.489375 0.872074i $$-0.337225\pi$$
0.489375 + 0.872074i $$0.337225\pi$$
$$762$$ 0 0
$$763$$ 11.0000 0.398227
$$764$$ 0 0
$$765$$ −3.00000 −0.108465
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 36.0000 1.29651
$$772$$ 0 0
$$773$$ −39.0000 −1.40273 −0.701366 0.712801i $$-0.747426\pi$$
−0.701366 + 0.712801i $$0.747426\pi$$
$$774$$ 0 0
$$775$$ −5.00000 −0.179605
$$776$$ 0 0
$$777$$ 2.00000 0.0717496
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ −12.0000 −0.428845
$$784$$ 0 0
$$785$$ 13.0000 0.463990
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 0 0
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 0 0
$$795$$ −6.00000 −0.212798
$$796$$ 0 0
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 48.0000 1.69388
$$804$$ 0 0
$$805$$ 6.00000 0.211472
$$806$$ 0 0
$$807$$ −36.0000 −1.26726
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 0 0
$$813$$ −4.00000 −0.140286
$$814$$ 0 0
$$815$$ 11.0000 0.385313
$$816$$ 0 0
$$817$$ −2.00000 −0.0699711
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ −6.00000 −0.208893
$$826$$ 0 0
$$827$$ 3.00000 0.104320 0.0521601 0.998639i $$-0.483389\pi$$
0.0521601 + 0.998639i $$0.483389\pi$$
$$828$$ 0 0
$$829$$ 47.0000 1.63238 0.816189 0.577785i $$-0.196083\pi$$
0.816189 + 0.577785i $$0.196083\pi$$
$$830$$ 0 0
$$831$$ −56.0000 −1.94262
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 48.0000 1.65321
$$844$$ 0 0
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ −15.0000 −0.512390 −0.256195 0.966625i $$-0.582469\pi$$
−0.256195 + 0.966625i $$0.582469\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 0 0
$$865$$ −15.0000 −0.510015
$$866$$ 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
$$873$$ 17.0000 0.575363
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ 0 0
$$879$$ 42.0000 1.41662
$$880$$ 0 0
$$881$$ 15.0000 0.505363 0.252681 0.967550i $$-0.418688\pi$$
0.252681 + 0.967550i $$0.418688\pi$$
$$882$$ 0 0
$$883$$ −29.0000 −0.975928 −0.487964 0.872864i $$-0.662260\pi$$
−0.487964 + 0.872864i $$0.662260\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −33.0000 −1.10803 −0.554016 0.832506i $$-0.686905\pi$$
−0.554016 + 0.832506i $$0.686905\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ 33.0000 1.10554
$$892$$ 0 0
$$893$$ 24.0000 0.803129
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 48.0000 1.60267
$$898$$ 0 0
$$899$$ −15.0000 −0.500278
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 2.00000 0.0665558
$$904$$ 0 0
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 2.00000 0.0661180
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 68.0000 2.24068
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 45.0000 1.47640 0.738201 0.674581i $$-0.235676\pi$$
0.738201 + 0.674581i $$0.235676\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 0 0
$$933$$ 18.0000 0.589294
$$934$$ 0 0
$$935$$ 9.00000 0.294331
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ 52.0000 1.69696
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ 0 0
$$947$$ −33.0000 −1.07236 −0.536178 0.844105i $$-0.680132\pi$$
−0.536178 + 0.844105i $$0.680132\pi$$
$$948$$ 0 0
$$949$$ 64.0000 2.07753
$$950$$ 0 0
$$951$$ −42.0000 −1.36194
$$952$$ 0 0
$$953$$ −60.0000 −1.94359 −0.971795 0.235826i $$-0.924220\pi$$
−0.971795 + 0.235826i $$0.924220\pi$$
$$954$$ 0 0
$$955$$ 3.00000 0.0970777
$$956$$ 0 0
$$957$$ −18.0000 −0.581857
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 0 0
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ 0 0
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ −57.0000 −1.82922 −0.914609 0.404341i $$-0.867501\pi$$
−0.914609 + 0.404341i $$0.867501\pi$$
$$972$$ 0 0
$$973$$ 13.0000 0.416761
$$974$$ 0 0
$$975$$ −8.00000 −0.256205
$$976$$ 0 0
$$977$$ −45.0000 −1.43968 −0.719839 0.694141i $$-0.755784\pi$$
−0.719839 + 0.694141i $$0.755784\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ 11.0000 0.351203
$$982$$ 0 0
$$983$$ −51.0000 −1.62665 −0.813324 0.581811i $$-0.802344\pi$$
−0.813324 + 0.581811i $$0.802344\pi$$
$$984$$ 0 0
$$985$$ 6.00000 0.191176
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ −47.0000 −1.49300 −0.746502 0.665383i $$-0.768268\pi$$
−0.746502 + 0.665383i $$0.768268\pi$$
$$992$$ 0 0
$$993$$ −52.0000 −1.65017
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 2960.2.a.j.1.1 1
4.3 odd 2 370.2.a.a.1.1 1
12.11 even 2 3330.2.a.v.1.1 1
20.3 even 4 1850.2.b.g.149.2 2
20.7 even 4 1850.2.b.g.149.1 2
20.19 odd 2 1850.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.a.1.1 1 4.3 odd 2
1850.2.a.o.1.1 1 20.19 odd 2
1850.2.b.g.149.1 2 20.7 even 4
1850.2.b.g.149.2 2 20.3 even 4
2960.2.a.j.1.1 1 1.1 even 1 trivial
3330.2.a.v.1.1 1 12.11 even 2