Properties

 Label 1216.2.a.l.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 152) 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -5.00000 q^{17} +1.00000 q^{19} -3.00000 q^{21} +1.00000 q^{23} -5.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} -4.00000 q^{31} +2.00000 q^{33} -2.00000 q^{37} -1.00000 q^{39} -8.00000 q^{41} -8.00000 q^{43} +8.00000 q^{47} +2.00000 q^{49} -5.00000 q^{51} -9.00000 q^{53} +1.00000 q^{57} +1.00000 q^{59} -14.0000 q^{61} +6.00000 q^{63} +13.0000 q^{67} +1.00000 q^{69} -10.0000 q^{71} +9.00000 q^{73} -5.00000 q^{75} -6.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +10.0000 q^{83} +3.00000 q^{87} -12.0000 q^{89} +3.00000 q^{91} -4.00000 q^{93} +14.0000 q^{97} -4.00000 q^{99} +O(q^{100})$$

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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 1.00000 0.130189 0.0650945 0.997879i $$-0.479265\pi$$
0.0650945 + 0.997879i $$0.479265\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 6.00000 0.755929
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 13.0000 1.58820 0.794101 0.607785i $$-0.207942\pi$$
0.794101 + 0.607785i $$0.207942\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 0 0
$$75$$ −5.00000 −0.577350
$$76$$ 0 0
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.0000 1.45010 0.725052 0.688694i $$-0.241816\pi$$
0.725052 + 0.688694i $$0.241816\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 15.0000 1.37505
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −8.00000 −0.721336
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 7.00000 0.598050 0.299025 0.954245i $$-0.403339\pi$$
0.299025 + 0.954245i $$0.403339\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ −22.0000 −1.79033 −0.895167 0.445730i $$-0.852944\pi$$
−0.895167 + 0.445730i $$0.852944\pi$$
$$152$$ 0 0
$$153$$ 10.0000 0.808452
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4.00000 −0.309529 −0.154765 0.987951i $$-0.549462\pi$$
−0.154765 + 0.987951i $$0.549462\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 15.0000 1.13389
$$176$$ 0 0
$$177$$ 1.00000 0.0751646
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ −14.0000 −1.03491
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −10.0000 −0.731272
$$188$$ 0 0
$$189$$ 15.0000 1.09109
$$190$$ 0 0
$$191$$ −23.0000 −1.66422 −0.832111 0.554609i $$-0.812868\pi$$
−0.832111 + 0.554609i $$0.812868\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ 9.00000 0.637993 0.318997 0.947756i $$-0.396654\pi$$
0.318997 + 0.947756i $$0.396654\pi$$
$$200$$ 0 0
$$201$$ 13.0000 0.916949
$$202$$ 0 0
$$203$$ −9.00000 −0.631676
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.00000 −0.139010
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 21.0000 1.44570 0.722850 0.691005i $$-0.242832\pi$$
0.722850 + 0.691005i $$0.242832\pi$$
$$212$$ 0 0
$$213$$ −10.0000 −0.685189
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 0 0
$$223$$ 14.0000 0.937509 0.468755 0.883328i $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 0 0
$$225$$ 10.0000 0.666667
$$226$$ 0 0
$$227$$ 9.00000 0.597351 0.298675 0.954355i $$-0.403455\pi$$
0.298675 + 0.954355i $$0.403455\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.00000 −0.0636285
$$248$$ 0 0
$$249$$ 10.0000 0.633724
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 2.00000 0.125739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.00000 −0.249513 −0.124757 0.992187i $$-0.539815\pi$$
−0.124757 + 0.992187i $$0.539815\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −32.0000 −1.97320 −0.986602 0.163144i $$-0.947836\pi$$
−0.986602 + 0.163144i $$0.947836\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 25.0000 1.51864 0.759321 0.650716i $$-0.225531\pi$$
0.759321 + 0.650716i $$0.225531\pi$$
$$272$$ 0 0
$$273$$ 3.00000 0.181568
$$274$$ 0 0
$$275$$ −10.0000 −0.603023
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −10.0000 −0.580259
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ 9.00000 0.510343 0.255172 0.966896i $$-0.417868\pi$$
0.255172 + 0.966896i $$0.417868\pi$$
$$312$$ 0 0
$$313$$ 13.0000 0.734803 0.367402 0.930062i $$-0.380247\pi$$
0.367402 + 0.930062i $$0.380247\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3.00000 −0.168497 −0.0842484 0.996445i $$-0.526849\pi$$
−0.0842484 + 0.996445i $$0.526849\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ 0 0
$$323$$ −5.00000 −0.278207
$$324$$ 0 0
$$325$$ 5.00000 0.277350
$$326$$ 0 0
$$327$$ −7.00000 −0.387101
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12.0000 0.653682 0.326841 0.945079i $$-0.394016\pi$$
0.326841 + 0.945079i $$0.394016\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 0 0
$$353$$ −7.00000 −0.372572 −0.186286 0.982496i $$-0.559645\pi$$
−0.186286 + 0.982496i $$0.559645\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 15.0000 0.793884
$$358$$ 0 0
$$359$$ −17.0000 −0.897226 −0.448613 0.893726i $$-0.648082\pi$$
−0.448613 + 0.893726i $$0.648082\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 16.0000 0.832927
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 0 0
$$373$$ 13.0000 0.673114 0.336557 0.941663i $$-0.390737\pi$$
0.336557 + 0.941663i $$0.390737\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −3.00000 −0.154508
$$378$$ 0 0
$$379$$ 9.00000 0.462299 0.231149 0.972918i $$-0.425751\pi$$
0.231149 + 0.972918i $$0.425751\pi$$
$$380$$ 0 0
$$381$$ 6.00000 0.307389
$$382$$ 0 0
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 16.0000 0.813326
$$388$$ 0 0
$$389$$ −34.0000 −1.72387 −0.861934 0.507020i $$-0.830747\pi$$
−0.861934 + 0.507020i $$0.830747\pi$$
$$390$$ 0 0
$$391$$ −5.00000 −0.252861
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.00000 0.200754 0.100377 0.994949i $$-0.467995\pi$$
0.100377 + 0.994949i $$0.467995\pi$$
$$398$$ 0 0
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ −8.00000 −0.395575 −0.197787 0.980245i $$-0.563376\pi$$
−0.197787 + 0.980245i $$0.563376\pi$$
$$410$$ 0 0
$$411$$ 7.00000 0.345285
$$412$$ 0 0
$$413$$ −3.00000 −0.147620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 35.0000 1.70580 0.852898 0.522078i $$-0.174843\pi$$
0.852898 + 0.522078i $$0.174843\pi$$
$$422$$ 0 0
$$423$$ −16.0000 −0.777947
$$424$$ 0 0
$$425$$ 25.0000 1.21268
$$426$$ 0 0
$$427$$ 42.0000 2.03252
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.00000 0.0478365
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 6.00000 0.285069 0.142534 0.989790i $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 8.00000 0.378387
$$448$$ 0 0
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ 0 0
$$453$$ −22.0000 −1.03365
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ 0 0
$$459$$ 25.0000 1.16690
$$460$$ 0 0
$$461$$ −20.0000 −0.931493 −0.465746 0.884918i $$-0.654214\pi$$
−0.465746 + 0.884918i $$0.654214\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ −39.0000 −1.80085
$$470$$ 0 0
$$471$$ 22.0000 1.01371
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −5.00000 −0.229416
$$476$$ 0 0
$$477$$ 18.0000 0.824163
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ −3.00000 −0.136505
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −15.0000 −0.675566
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 30.0000 1.34568
$$498$$ 0 0
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ −4.00000 −0.178707
$$502$$ 0 0
$$503$$ 33.0000 1.47140 0.735699 0.677309i $$-0.236854\pi$$
0.735699 + 0.677309i $$0.236854\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12.0000 −0.532939
$$508$$ 0 0
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ −27.0000 −1.19441
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ −5.00000 −0.218635 −0.109317 0.994007i $$-0.534866\pi$$
−0.109317 + 0.994007i $$0.534866\pi$$
$$524$$ 0 0
$$525$$ 15.0000 0.654654
$$526$$ 0 0
$$527$$ 20.0000 0.871214
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ −2.00000 −0.0867926
$$532$$ 0 0
$$533$$ 8.00000 0.346518
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 28.0000 1.19501
$$550$$ 0 0
$$551$$ 3.00000 0.127804
$$552$$ 0 0
$$553$$ −30.0000 −1.27573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −24.0000 −1.01691 −0.508456 0.861088i $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −10.0000 −0.422200
$$562$$ 0 0
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ −23.0000 −0.960839
$$574$$ 0 0
$$575$$ −5.00000 −0.208514
$$576$$ 0 0
$$577$$ −29.0000 −1.20729 −0.603643 0.797255i $$-0.706285\pi$$
−0.603643 + 0.797255i $$0.706285\pi$$
$$578$$ 0 0
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ −30.0000 −1.24461
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −44.0000 −1.81607 −0.908037 0.418890i $$-0.862419\pi$$
−0.908037 + 0.418890i $$0.862419\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 8.00000 0.329076
$$592$$ 0 0
$$593$$ 2.00000 0.0821302 0.0410651 0.999156i $$-0.486925\pi$$
0.0410651 + 0.999156i $$0.486925\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 9.00000 0.368345
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ −4.00000 −0.163163 −0.0815817 0.996667i $$-0.525997\pi$$
−0.0815817 + 0.996667i $$0.525997\pi$$
$$602$$ 0 0
$$603$$ −26.0000 −1.05880
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −42.0000 −1.70473 −0.852364 0.522949i $$-0.824832\pi$$
−0.852364 + 0.522949i $$0.824832\pi$$
$$608$$ 0 0
$$609$$ −9.00000 −0.364698
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 22.0000 0.884255 0.442127 0.896952i $$-0.354224\pi$$
0.442127 + 0.896952i $$0.354224\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ 36.0000 1.44231
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 2.00000 0.0798723
$$628$$ 0 0
$$629$$ 10.0000 0.398726
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 21.0000 0.834675
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 20.0000 0.791188
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ −46.0000 −1.81406 −0.907031 0.421063i $$-0.861657\pi$$
−0.907031 + 0.421063i $$0.861657\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.00000 0.353827 0.176913 0.984226i $$-0.443389\pi$$
0.176913 + 0.984226i $$0.443389\pi$$
$$648$$ 0 0
$$649$$ 2.00000 0.0785069
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −18.0000 −0.702247
$$658$$ 0 0
$$659$$ 19.0000 0.740135 0.370067 0.929005i $$-0.379335\pi$$
0.370067 + 0.929005i $$0.379335\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ 0 0
$$663$$ 5.00000 0.194184
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.00000 0.116160
$$668$$ 0 0
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ −28.0000 −1.08093
$$672$$ 0 0
$$673$$ −12.0000 −0.462566 −0.231283 0.972887i $$-0.574292\pi$$
−0.231283 + 0.972887i $$0.574292\pi$$
$$674$$ 0 0
$$675$$ 25.0000 0.962250
$$676$$ 0 0
$$677$$ −27.0000 −1.03769 −0.518847 0.854867i $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ 0 0
$$679$$ −42.0000 −1.61181
$$680$$ 0 0
$$681$$ 9.00000 0.344881
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ 0 0
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 0 0
$$693$$ 12.0000 0.455842
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 40.0000 1.51511
$$698$$ 0 0
$$699$$ 26.0000 0.983410
$$700$$ 0 0
$$701$$ −44.0000 −1.66186 −0.830929 0.556379i $$-0.812190\pi$$
−0.830929 + 0.556379i $$0.812190\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −42.0000 −1.57957
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ −20.0000 −0.750059
$$712$$ 0 0
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 9.00000 0.336111
$$718$$ 0 0
$$719$$ −27.0000 −1.00693 −0.503465 0.864016i $$-0.667942\pi$$
−0.503465 + 0.864016i $$0.667942\pi$$
$$720$$ 0 0
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −15.0000 −0.557086
$$726$$ 0 0
$$727$$ −7.00000 −0.259616 −0.129808 0.991539i $$-0.541436\pi$$
−0.129808 + 0.991539i $$0.541436\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 26.0000 0.957722
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ −1.00000 −0.0367359
$$742$$ 0 0
$$743$$ 28.0000 1.02722 0.513610 0.858024i $$-0.328308\pi$$
0.513610 + 0.858024i $$0.328308\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −20.0000 −0.731762
$$748$$ 0 0
$$749$$ −45.0000 −1.64426
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 0 0
$$753$$ −18.0000 −0.655956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 0 0
$$759$$ 2.00000 0.0725954
$$760$$ 0 0
$$761$$ −37.0000 −1.34125 −0.670624 0.741797i $$-0.733974\pi$$
−0.670624 + 0.741797i $$0.733974\pi$$
$$762$$ 0 0
$$763$$ 21.0000 0.760251
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.00000 −0.0361079
$$768$$ 0 0
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ −4.00000 −0.144056
$$772$$ 0 0
$$773$$ 49.0000 1.76241 0.881204 0.472737i $$-0.156734\pi$$
0.881204 + 0.472737i $$0.156734\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 6.00000 0.215249
$$778$$ 0 0
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −20.0000 −0.715656
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ 0 0
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ 54.0000 1.92002
$$792$$ 0 0
$$793$$ 14.0000 0.497155
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.00000 0.106265 0.0531327 0.998587i $$-0.483079\pi$$
0.0531327 + 0.998587i $$0.483079\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ 18.0000 0.635206
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 0 0
$$811$$ 27.0000 0.948098 0.474049 0.880498i $$-0.342792\pi$$
0.474049 + 0.880498i $$0.342792\pi$$
$$812$$ 0 0
$$813$$ 25.0000 0.876788
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ 0 0
$$823$$ 11.0000 0.383436 0.191718 0.981450i $$-0.438594\pi$$
0.191718 + 0.981450i $$0.438594\pi$$
$$824$$ 0 0
$$825$$ −10.0000 −0.348155
$$826$$ 0 0
$$827$$ 33.0000 1.14752 0.573761 0.819023i $$-0.305484\pi$$
0.573761 + 0.819023i $$0.305484\pi$$
$$828$$ 0 0
$$829$$ −39.0000 −1.35453 −0.677263 0.735741i $$-0.736834\pi$$
−0.677263 + 0.735741i $$0.736834\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.0000 −0.346479
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000 0.721569
$$848$$ 0 0
$$849$$ −6.00000 −0.205919
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −44.0000 −1.50301 −0.751506 0.659727i $$-0.770672\pi$$
−0.751506 + 0.659727i $$0.770672\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ −13.0000 −0.440488
$$872$$ 0 0
$$873$$ −28.0000 −0.947656
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.0000 1.24940 0.624701 0.780864i $$-0.285221\pi$$
0.624701 + 0.780864i $$0.285221\pi$$
$$878$$ 0 0
$$879$$ −7.00000 −0.236104
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 46.0000 1.54802 0.774012 0.633171i $$-0.218247\pi$$
0.774012 + 0.633171i $$0.218247\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 0 0
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ 8.00000 0.267710
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.00000 −0.0333890
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ 0 0
$$903$$ 24.0000 0.798670
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −5.00000 −0.166022 −0.0830111 0.996549i $$-0.526454\pi$$
−0.0830111 + 0.996549i $$0.526454\pi$$
$$908$$ 0 0
$$909$$ −28.0000 −0.928701
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ 20.0000 0.661903
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 43.0000 1.41844 0.709220 0.704988i $$-0.249047\pi$$
0.709220 + 0.704988i $$0.249047\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ 9.00000 0.294647
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 49.0000 1.60076 0.800380 0.599493i $$-0.204631\pi$$
0.800380 + 0.599493i $$0.204631\pi$$
$$938$$ 0 0
$$939$$ 13.0000 0.424239
$$940$$ 0 0
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 0 0
$$943$$ −8.00000 −0.260516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ 0 0
$$949$$ −9.00000 −0.292152
$$950$$ 0 0
$$951$$ −3.00000 −0.0972817
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.00000 0.193952
$$958$$ 0 0
$$959$$ −21.0000 −0.678125
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −30.0000 −0.966736
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ −5.00000 −0.160623
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 0 0
$$973$$ 36.0000 1.15411
$$974$$ 0 0
$$975$$ 5.00000 0.160128
$$976$$ 0 0
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 0 0
$$983$$ −10.0000 −0.318950 −0.159475 0.987202i $$-0.550980\pi$$
−0.159475 + 0.987202i $$0.550980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −24.0000 −0.763928
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ 28.0000 0.889449 0.444725 0.895667i $$-0.353302\pi$$
0.444725 + 0.895667i $$0.353302\pi$$
$$992$$ 0 0
$$993$$ −25.0000 −0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16.0000 0.506725 0.253363 0.967371i $$-0.418463\pi$$
0.253363 + 0.967371i $$0.418463\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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.l.1.1 1
4.3 odd 2 1216.2.a.f.1.1 1
8.3 odd 2 152.2.a.b.1.1 1
8.5 even 2 304.2.a.b.1.1 1
24.5 odd 2 2736.2.a.k.1.1 1
24.11 even 2 1368.2.a.g.1.1 1
40.3 even 4 3800.2.d.f.3649.2 2
40.19 odd 2 3800.2.a.d.1.1 1
40.27 even 4 3800.2.d.f.3649.1 2
40.29 even 2 7600.2.a.o.1.1 1
56.27 even 2 7448.2.a.g.1.1 1
152.37 odd 2 5776.2.a.l.1.1 1
152.75 even 2 2888.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.b.1.1 1 8.3 odd 2
304.2.a.b.1.1 1 8.5 even 2
1216.2.a.f.1.1 1 4.3 odd 2
1216.2.a.l.1.1 1 1.1 even 1 trivial
1368.2.a.g.1.1 1 24.11 even 2
2736.2.a.k.1.1 1 24.5 odd 2
2888.2.a.b.1.1 1 152.75 even 2
3800.2.a.d.1.1 1 40.19 odd 2
3800.2.d.f.3649.1 2 40.27 even 4
3800.2.d.f.3649.2 2 40.3 even 4
5776.2.a.l.1.1 1 152.37 odd 2
7448.2.a.g.1.1 1 56.27 even 2
7600.2.a.o.1.1 1 40.29 even 2