# Properties

 Label 2496.4.a.a.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2496.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -10.0000 q^{5} -8.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} -13.0000 q^{13} +30.0000 q^{15} +130.000 q^{17} +20.0000 q^{19} +24.0000 q^{21} -25.0000 q^{25} -27.0000 q^{27} +18.0000 q^{29} -184.000 q^{31} +120.000 q^{33} +80.0000 q^{35} +74.0000 q^{37} +39.0000 q^{39} -362.000 q^{41} -76.0000 q^{43} -90.0000 q^{45} -452.000 q^{47} -279.000 q^{49} -390.000 q^{51} -382.000 q^{53} +400.000 q^{55} -60.0000 q^{57} -464.000 q^{59} -358.000 q^{61} -72.0000 q^{63} +130.000 q^{65} +700.000 q^{67} -748.000 q^{71} +1058.00 q^{73} +75.0000 q^{75} +320.000 q^{77} -976.000 q^{79} +81.0000 q^{81} +1008.00 q^{83} -1300.00 q^{85} -54.0000 q^{87} -386.000 q^{89} +104.000 q^{91} +552.000 q^{93} -200.000 q^{95} -614.000 q^{97} -360.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −10.0000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ −8.00000 −0.431959 −0.215980 0.976398i $$-0.569295\pi$$
−0.215980 + 0.976398i $$0.569295\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −40.0000 −1.09640 −0.548202 0.836346i $$-0.684688\pi$$
−0.548202 + 0.836346i $$0.684688\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 30.0000 0.516398
$$16$$ 0 0
$$17$$ 130.000 1.85468 0.927342 0.374215i $$-0.122088\pi$$
0.927342 + 0.374215i $$0.122088\pi$$
$$18$$ 0 0
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ 24.0000 0.249392
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 18.0000 0.115259 0.0576296 0.998338i $$-0.481646\pi$$
0.0576296 + 0.998338i $$0.481646\pi$$
$$30$$ 0 0
$$31$$ −184.000 −1.06604 −0.533022 0.846101i $$-0.678944\pi$$
−0.533022 + 0.846101i $$0.678944\pi$$
$$32$$ 0 0
$$33$$ 120.000 0.633010
$$34$$ 0 0
$$35$$ 80.0000 0.386356
$$36$$ 0 0
$$37$$ 74.0000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ −362.000 −1.37890 −0.689450 0.724333i $$-0.742148\pi$$
−0.689450 + 0.724333i $$0.742148\pi$$
$$42$$ 0 0
$$43$$ −76.0000 −0.269532 −0.134766 0.990877i $$-0.543028\pi$$
−0.134766 + 0.990877i $$0.543028\pi$$
$$44$$ 0 0
$$45$$ −90.0000 −0.298142
$$46$$ 0 0
$$47$$ −452.000 −1.40279 −0.701393 0.712774i $$-0.747438\pi$$
−0.701393 + 0.712774i $$0.747438\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ −390.000 −1.07080
$$52$$ 0 0
$$53$$ −382.000 −0.990033 −0.495016 0.868884i $$-0.664838\pi$$
−0.495016 + 0.868884i $$0.664838\pi$$
$$54$$ 0 0
$$55$$ 400.000 0.980654
$$56$$ 0 0
$$57$$ −60.0000 −0.139424
$$58$$ 0 0
$$59$$ −464.000 −1.02386 −0.511929 0.859028i $$-0.671069\pi$$
−0.511929 + 0.859028i $$0.671069\pi$$
$$60$$ 0 0
$$61$$ −358.000 −0.751430 −0.375715 0.926735i $$-0.622603\pi$$
−0.375715 + 0.926735i $$0.622603\pi$$
$$62$$ 0 0
$$63$$ −72.0000 −0.143986
$$64$$ 0 0
$$65$$ 130.000 0.248069
$$66$$ 0 0
$$67$$ 700.000 1.27640 0.638199 0.769872i $$-0.279680\pi$$
0.638199 + 0.769872i $$0.279680\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −748.000 −1.25030 −0.625150 0.780505i $$-0.714962\pi$$
−0.625150 + 0.780505i $$0.714962\pi$$
$$72$$ 0 0
$$73$$ 1058.00 1.69629 0.848147 0.529760i $$-0.177718\pi$$
0.848147 + 0.529760i $$0.177718\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ 320.000 0.473602
$$78$$ 0 0
$$79$$ −976.000 −1.38998 −0.694991 0.719018i $$-0.744592\pi$$
−0.694991 + 0.719018i $$0.744592\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1008.00 1.33304 0.666520 0.745487i $$-0.267783\pi$$
0.666520 + 0.745487i $$0.267783\pi$$
$$84$$ 0 0
$$85$$ −1300.00 −1.65888
$$86$$ 0 0
$$87$$ −54.0000 −0.0665449
$$88$$ 0 0
$$89$$ −386.000 −0.459729 −0.229865 0.973223i $$-0.573828\pi$$
−0.229865 + 0.973223i $$0.573828\pi$$
$$90$$ 0 0
$$91$$ 104.000 0.119804
$$92$$ 0 0
$$93$$ 552.000 0.615481
$$94$$ 0 0
$$95$$ −200.000 −0.215995
$$96$$ 0 0
$$97$$ −614.000 −0.642704 −0.321352 0.946960i $$-0.604137\pi$$
−0.321352 + 0.946960i $$0.604137\pi$$
$$98$$ 0 0
$$99$$ −360.000 −0.365468
$$100$$ 0 0
$$101$$ −518.000 −0.510326 −0.255163 0.966898i $$-0.582129\pi$$
−0.255163 + 0.966898i $$0.582129\pi$$
$$102$$ 0 0
$$103$$ 112.000 0.107143 0.0535713 0.998564i $$-0.482940\pi$$
0.0535713 + 0.998564i $$0.482940\pi$$
$$104$$ 0 0
$$105$$ −240.000 −0.223063
$$106$$ 0 0
$$107$$ 372.000 0.336099 0.168050 0.985779i $$-0.446253\pi$$
0.168050 + 0.985779i $$0.446253\pi$$
$$108$$ 0 0
$$109$$ −934.000 −0.820743 −0.410371 0.911918i $$-0.634601\pi$$
−0.410371 + 0.911918i $$0.634601\pi$$
$$110$$ 0 0
$$111$$ −222.000 −0.189832
$$112$$ 0 0
$$113$$ 1914.00 1.59340 0.796699 0.604376i $$-0.206578\pi$$
0.796699 + 0.604376i $$0.206578\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −1040.00 −0.801148
$$120$$ 0 0
$$121$$ 269.000 0.202104
$$122$$ 0 0
$$123$$ 1086.00 0.796108
$$124$$ 0 0
$$125$$ 1500.00 1.07331
$$126$$ 0 0
$$127$$ 1296.00 0.905523 0.452761 0.891632i $$-0.350439\pi$$
0.452761 + 0.891632i $$0.350439\pi$$
$$128$$ 0 0
$$129$$ 228.000 0.155615
$$130$$ 0 0
$$131$$ 892.000 0.594919 0.297460 0.954734i $$-0.403861\pi$$
0.297460 + 0.954734i $$0.403861\pi$$
$$132$$ 0 0
$$133$$ −160.000 −0.104314
$$134$$ 0 0
$$135$$ 270.000 0.172133
$$136$$ 0 0
$$137$$ 2326.00 1.45054 0.725269 0.688466i $$-0.241716\pi$$
0.725269 + 0.688466i $$0.241716\pi$$
$$138$$ 0 0
$$139$$ −1932.00 −1.17892 −0.589461 0.807797i $$-0.700660\pi$$
−0.589461 + 0.807797i $$0.700660\pi$$
$$140$$ 0 0
$$141$$ 1356.00 0.809899
$$142$$ 0 0
$$143$$ 520.000 0.304088
$$144$$ 0 0
$$145$$ −180.000 −0.103091
$$146$$ 0 0
$$147$$ 837.000 0.469623
$$148$$ 0 0
$$149$$ −882.000 −0.484941 −0.242471 0.970159i $$-0.577958\pi$$
−0.242471 + 0.970159i $$0.577958\pi$$
$$150$$ 0 0
$$151$$ −1776.00 −0.957145 −0.478572 0.878048i $$-0.658846\pi$$
−0.478572 + 0.878048i $$0.658846\pi$$
$$152$$ 0 0
$$153$$ 1170.00 0.618228
$$154$$ 0 0
$$155$$ 1840.00 0.953499
$$156$$ 0 0
$$157$$ 2410.00 1.22509 0.612544 0.790436i $$-0.290146\pi$$
0.612544 + 0.790436i $$0.290146\pi$$
$$158$$ 0 0
$$159$$ 1146.00 0.571596
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3212.00 −1.54346 −0.771728 0.635953i $$-0.780607\pi$$
−0.771728 + 0.635953i $$0.780607\pi$$
$$164$$ 0 0
$$165$$ −1200.00 −0.566181
$$166$$ 0 0
$$167$$ 1668.00 0.772896 0.386448 0.922311i $$-0.373702\pi$$
0.386448 + 0.922311i $$0.373702\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 0 0
$$173$$ −3598.00 −1.58122 −0.790609 0.612321i $$-0.790236\pi$$
−0.790609 + 0.612321i $$0.790236\pi$$
$$174$$ 0 0
$$175$$ 200.000 0.0863919
$$176$$ 0 0
$$177$$ 1392.00 0.591125
$$178$$ 0 0
$$179$$ −1068.00 −0.445956 −0.222978 0.974824i $$-0.571578\pi$$
−0.222978 + 0.974824i $$0.571578\pi$$
$$180$$ 0 0
$$181$$ 4786.00 1.96542 0.982709 0.185158i $$-0.0592797\pi$$
0.982709 + 0.185158i $$0.0592797\pi$$
$$182$$ 0 0
$$183$$ 1074.00 0.433838
$$184$$ 0 0
$$185$$ −740.000 −0.294086
$$186$$ 0 0
$$187$$ −5200.00 −2.03348
$$188$$ 0 0
$$189$$ 216.000 0.0831306
$$190$$ 0 0
$$191$$ −1312.00 −0.497031 −0.248516 0.968628i $$-0.579943\pi$$
−0.248516 + 0.968628i $$0.579943\pi$$
$$192$$ 0 0
$$193$$ −350.000 −0.130537 −0.0652683 0.997868i $$-0.520790\pi$$
−0.0652683 + 0.997868i $$0.520790\pi$$
$$194$$ 0 0
$$195$$ −390.000 −0.143223
$$196$$ 0 0
$$197$$ 342.000 0.123688 0.0618439 0.998086i $$-0.480302\pi$$
0.0618439 + 0.998086i $$0.480302\pi$$
$$198$$ 0 0
$$199$$ −3368.00 −1.19975 −0.599877 0.800092i $$-0.704784\pi$$
−0.599877 + 0.800092i $$0.704784\pi$$
$$200$$ 0 0
$$201$$ −2100.00 −0.736928
$$202$$ 0 0
$$203$$ −144.000 −0.0497873
$$204$$ 0 0
$$205$$ 3620.00 1.23333
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −800.000 −0.264771
$$210$$ 0 0
$$211$$ 2004.00 0.653844 0.326922 0.945051i $$-0.393989\pi$$
0.326922 + 0.945051i $$0.393989\pi$$
$$212$$ 0 0
$$213$$ 2244.00 0.721861
$$214$$ 0 0
$$215$$ 760.000 0.241077
$$216$$ 0 0
$$217$$ 1472.00 0.460488
$$218$$ 0 0
$$219$$ −3174.00 −0.979356
$$220$$ 0 0
$$221$$ −1690.00 −0.514397
$$222$$ 0 0
$$223$$ −5608.00 −1.68403 −0.842017 0.539451i $$-0.818632\pi$$
−0.842017 + 0.539451i $$0.818632\pi$$
$$224$$ 0 0
$$225$$ −225.000 −0.0666667
$$226$$ 0 0
$$227$$ 1928.00 0.563726 0.281863 0.959455i $$-0.409048\pi$$
0.281863 + 0.959455i $$0.409048\pi$$
$$228$$ 0 0
$$229$$ 3938.00 1.13638 0.568189 0.822898i $$-0.307644\pi$$
0.568189 + 0.822898i $$0.307644\pi$$
$$230$$ 0 0
$$231$$ −960.000 −0.273434
$$232$$ 0 0
$$233$$ 2562.00 0.720353 0.360176 0.932884i $$-0.382717\pi$$
0.360176 + 0.932884i $$0.382717\pi$$
$$234$$ 0 0
$$235$$ 4520.00 1.25469
$$236$$ 0 0
$$237$$ 2928.00 0.802506
$$238$$ 0 0
$$239$$ 7164.00 1.93891 0.969457 0.245260i $$-0.0788733\pi$$
0.969457 + 0.245260i $$0.0788733\pi$$
$$240$$ 0 0
$$241$$ −6182.00 −1.65236 −0.826178 0.563410i $$-0.809489\pi$$
−0.826178 + 0.563410i $$0.809489\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 2790.00 0.727537
$$246$$ 0 0
$$247$$ −260.000 −0.0669773
$$248$$ 0 0
$$249$$ −3024.00 −0.769631
$$250$$ 0 0
$$251$$ 1396.00 0.351055 0.175527 0.984475i $$-0.443837\pi$$
0.175527 + 0.984475i $$0.443837\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 3900.00 0.957755
$$256$$ 0 0
$$257$$ 6906.00 1.67620 0.838102 0.545514i $$-0.183665\pi$$
0.838102 + 0.545514i $$0.183665\pi$$
$$258$$ 0 0
$$259$$ −592.000 −0.142027
$$260$$ 0 0
$$261$$ 162.000 0.0384197
$$262$$ 0 0
$$263$$ −6848.00 −1.60557 −0.802787 0.596266i $$-0.796650\pi$$
−0.802787 + 0.596266i $$0.796650\pi$$
$$264$$ 0 0
$$265$$ 3820.00 0.885512
$$266$$ 0 0
$$267$$ 1158.00 0.265425
$$268$$ 0 0
$$269$$ 6034.00 1.36766 0.683828 0.729643i $$-0.260314\pi$$
0.683828 + 0.729643i $$0.260314\pi$$
$$270$$ 0 0
$$271$$ 4832.00 1.08311 0.541556 0.840665i $$-0.317836\pi$$
0.541556 + 0.840665i $$0.317836\pi$$
$$272$$ 0 0
$$273$$ −312.000 −0.0691689
$$274$$ 0 0
$$275$$ 1000.00 0.219281
$$276$$ 0 0
$$277$$ 4082.00 0.885428 0.442714 0.896663i $$-0.354016\pi$$
0.442714 + 0.896663i $$0.354016\pi$$
$$278$$ 0 0
$$279$$ −1656.00 −0.355348
$$280$$ 0 0
$$281$$ 3350.00 0.711189 0.355595 0.934640i $$-0.384278\pi$$
0.355595 + 0.934640i $$0.384278\pi$$
$$282$$ 0 0
$$283$$ −7796.00 −1.63754 −0.818770 0.574121i $$-0.805344\pi$$
−0.818770 + 0.574121i $$0.805344\pi$$
$$284$$ 0 0
$$285$$ 600.000 0.124705
$$286$$ 0 0
$$287$$ 2896.00 0.595629
$$288$$ 0 0
$$289$$ 11987.0 2.43985
$$290$$ 0 0
$$291$$ 1842.00 0.371065
$$292$$ 0 0
$$293$$ −3922.00 −0.781999 −0.390999 0.920391i $$-0.627871\pi$$
−0.390999 + 0.920391i $$0.627871\pi$$
$$294$$ 0 0
$$295$$ 4640.00 0.915767
$$296$$ 0 0
$$297$$ 1080.00 0.211003
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 608.000 0.116427
$$302$$ 0 0
$$303$$ 1554.00 0.294637
$$304$$ 0 0
$$305$$ 3580.00 0.672099
$$306$$ 0 0
$$307$$ −5956.00 −1.10725 −0.553627 0.832765i $$-0.686757\pi$$
−0.553627 + 0.832765i $$0.686757\pi$$
$$308$$ 0 0
$$309$$ −336.000 −0.0618588
$$310$$ 0 0
$$311$$ 2352.00 0.428841 0.214421 0.976741i $$-0.431214\pi$$
0.214421 + 0.976741i $$0.431214\pi$$
$$312$$ 0 0
$$313$$ 8442.00 1.52450 0.762252 0.647280i $$-0.224093\pi$$
0.762252 + 0.647280i $$0.224093\pi$$
$$314$$ 0 0
$$315$$ 720.000 0.128785
$$316$$ 0 0
$$317$$ 5550.00 0.983341 0.491670 0.870781i $$-0.336386\pi$$
0.491670 + 0.870781i $$0.336386\pi$$
$$318$$ 0 0
$$319$$ −720.000 −0.126371
$$320$$ 0 0
$$321$$ −1116.00 −0.194047
$$322$$ 0 0
$$323$$ 2600.00 0.447888
$$324$$ 0 0
$$325$$ 325.000 0.0554700
$$326$$ 0 0
$$327$$ 2802.00 0.473856
$$328$$ 0 0
$$329$$ 3616.00 0.605947
$$330$$ 0 0
$$331$$ −140.000 −0.0232480 −0.0116240 0.999932i $$-0.503700\pi$$
−0.0116240 + 0.999932i $$0.503700\pi$$
$$332$$ 0 0
$$333$$ 666.000 0.109599
$$334$$ 0 0
$$335$$ −7000.00 −1.14164
$$336$$ 0 0
$$337$$ −6174.00 −0.997980 −0.498990 0.866608i $$-0.666296\pi$$
−0.498990 + 0.866608i $$0.666296\pi$$
$$338$$ 0 0
$$339$$ −5742.00 −0.919949
$$340$$ 0 0
$$341$$ 7360.00 1.16882
$$342$$ 0 0
$$343$$ 4976.00 0.783320
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2988.00 0.462260 0.231130 0.972923i $$-0.425758\pi$$
0.231130 + 0.972923i $$0.425758\pi$$
$$348$$ 0 0
$$349$$ 162.000 0.0248472 0.0124236 0.999923i $$-0.496045\pi$$
0.0124236 + 0.999923i $$0.496045\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 0 0
$$353$$ −10754.0 −1.62147 −0.810733 0.585416i $$-0.800931\pi$$
−0.810733 + 0.585416i $$0.800931\pi$$
$$354$$ 0 0
$$355$$ 7480.00 1.11830
$$356$$ 0 0
$$357$$ 3120.00 0.462543
$$358$$ 0 0
$$359$$ 3588.00 0.527486 0.263743 0.964593i $$-0.415043\pi$$
0.263743 + 0.964593i $$0.415043\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ −807.000 −0.116685
$$364$$ 0 0
$$365$$ −10580.0 −1.51721
$$366$$ 0 0
$$367$$ 11272.0 1.60325 0.801626 0.597826i $$-0.203968\pi$$
0.801626 + 0.597826i $$0.203968\pi$$
$$368$$ 0 0
$$369$$ −3258.00 −0.459633
$$370$$ 0 0
$$371$$ 3056.00 0.427654
$$372$$ 0 0
$$373$$ 10914.0 1.51503 0.757514 0.652819i $$-0.226414\pi$$
0.757514 + 0.652819i $$0.226414\pi$$
$$374$$ 0 0
$$375$$ −4500.00 −0.619677
$$376$$ 0 0
$$377$$ −234.000 −0.0319671
$$378$$ 0 0
$$379$$ −8100.00 −1.09781 −0.548904 0.835886i $$-0.684955\pi$$
−0.548904 + 0.835886i $$0.684955\pi$$
$$380$$ 0 0
$$381$$ −3888.00 −0.522804
$$382$$ 0 0
$$383$$ 6180.00 0.824499 0.412250 0.911071i $$-0.364743\pi$$
0.412250 + 0.911071i $$0.364743\pi$$
$$384$$ 0 0
$$385$$ −3200.00 −0.423603
$$386$$ 0 0
$$387$$ −684.000 −0.0898441
$$388$$ 0 0
$$389$$ 7522.00 0.980413 0.490206 0.871606i $$-0.336921\pi$$
0.490206 + 0.871606i $$0.336921\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −2676.00 −0.343477
$$394$$ 0 0
$$395$$ 9760.00 1.24324
$$396$$ 0 0
$$397$$ −6078.00 −0.768378 −0.384189 0.923254i $$-0.625519\pi$$
−0.384189 + 0.923254i $$0.625519\pi$$
$$398$$ 0 0
$$399$$ 480.000 0.0602257
$$400$$ 0 0
$$401$$ 1830.00 0.227895 0.113947 0.993487i $$-0.463650\pi$$
0.113947 + 0.993487i $$0.463650\pi$$
$$402$$ 0 0
$$403$$ 2392.00 0.295668
$$404$$ 0 0
$$405$$ −810.000 −0.0993808
$$406$$ 0 0
$$407$$ −2960.00 −0.360496
$$408$$ 0 0
$$409$$ 12434.0 1.50323 0.751616 0.659601i $$-0.229275\pi$$
0.751616 + 0.659601i $$0.229275\pi$$
$$410$$ 0 0
$$411$$ −6978.00 −0.837468
$$412$$ 0 0
$$413$$ 3712.00 0.442265
$$414$$ 0 0
$$415$$ −10080.0 −1.19231
$$416$$ 0 0
$$417$$ 5796.00 0.680651
$$418$$ 0 0
$$419$$ 14188.0 1.65425 0.827123 0.562021i $$-0.189976\pi$$
0.827123 + 0.562021i $$0.189976\pi$$
$$420$$ 0 0
$$421$$ −8638.00 −0.999977 −0.499989 0.866032i $$-0.666662\pi$$
−0.499989 + 0.866032i $$0.666662\pi$$
$$422$$ 0 0
$$423$$ −4068.00 −0.467596
$$424$$ 0 0
$$425$$ −3250.00 −0.370937
$$426$$ 0 0
$$427$$ 2864.00 0.324587
$$428$$ 0 0
$$429$$ −1560.00 −0.175565
$$430$$ 0 0
$$431$$ 4292.00 0.479671 0.239836 0.970813i $$-0.422906\pi$$
0.239836 + 0.970813i $$0.422906\pi$$
$$432$$ 0 0
$$433$$ −5982.00 −0.663918 −0.331959 0.943294i $$-0.607710\pi$$
−0.331959 + 0.943294i $$0.607710\pi$$
$$434$$ 0 0
$$435$$ 540.000 0.0595196
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 256.000 0.0278319 0.0139160 0.999903i $$-0.495570\pi$$
0.0139160 + 0.999903i $$0.495570\pi$$
$$440$$ 0 0
$$441$$ −2511.00 −0.271137
$$442$$ 0 0
$$443$$ −12556.0 −1.34662 −0.673311 0.739359i $$-0.735128\pi$$
−0.673311 + 0.739359i $$0.735128\pi$$
$$444$$ 0 0
$$445$$ 3860.00 0.411194
$$446$$ 0 0
$$447$$ 2646.00 0.279981
$$448$$ 0 0
$$449$$ 5574.00 0.585865 0.292932 0.956133i $$-0.405369\pi$$
0.292932 + 0.956133i $$0.405369\pi$$
$$450$$ 0 0
$$451$$ 14480.0 1.51183
$$452$$ 0 0
$$453$$ 5328.00 0.552608
$$454$$ 0 0
$$455$$ −1040.00 −0.107156
$$456$$ 0 0
$$457$$ 1266.00 0.129586 0.0647932 0.997899i $$-0.479361\pi$$
0.0647932 + 0.997899i $$0.479361\pi$$
$$458$$ 0 0
$$459$$ −3510.00 −0.356934
$$460$$ 0 0
$$461$$ −7554.00 −0.763178 −0.381589 0.924332i $$-0.624623\pi$$
−0.381589 + 0.924332i $$0.624623\pi$$
$$462$$ 0 0
$$463$$ −6752.00 −0.677737 −0.338868 0.940834i $$-0.610044\pi$$
−0.338868 + 0.940834i $$0.610044\pi$$
$$464$$ 0 0
$$465$$ −5520.00 −0.550503
$$466$$ 0 0
$$467$$ −7924.00 −0.785180 −0.392590 0.919714i $$-0.628421\pi$$
−0.392590 + 0.919714i $$0.628421\pi$$
$$468$$ 0 0
$$469$$ −5600.00 −0.551352
$$470$$ 0 0
$$471$$ −7230.00 −0.707305
$$472$$ 0 0
$$473$$ 3040.00 0.295517
$$474$$ 0 0
$$475$$ −500.000 −0.0482980
$$476$$ 0 0
$$477$$ −3438.00 −0.330011
$$478$$ 0 0
$$479$$ −11084.0 −1.05729 −0.528644 0.848844i $$-0.677299\pi$$
−0.528644 + 0.848844i $$0.677299\pi$$
$$480$$ 0 0
$$481$$ −962.000 −0.0911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6140.00 0.574852
$$486$$ 0 0
$$487$$ 4432.00 0.412388 0.206194 0.978511i $$-0.433892\pi$$
0.206194 + 0.978511i $$0.433892\pi$$
$$488$$ 0 0
$$489$$ 9636.00 0.891114
$$490$$ 0 0
$$491$$ 1140.00 0.104781 0.0523905 0.998627i $$-0.483316\pi$$
0.0523905 + 0.998627i $$0.483316\pi$$
$$492$$ 0 0
$$493$$ 2340.00 0.213769
$$494$$ 0 0
$$495$$ 3600.00 0.326885
$$496$$ 0 0
$$497$$ 5984.00 0.540079
$$498$$ 0 0
$$499$$ −1764.00 −0.158251 −0.0791257 0.996865i $$-0.525213\pi$$
−0.0791257 + 0.996865i $$0.525213\pi$$
$$500$$ 0 0
$$501$$ −5004.00 −0.446232
$$502$$ 0 0
$$503$$ 16976.0 1.50482 0.752408 0.658697i $$-0.228892\pi$$
0.752408 + 0.658697i $$0.228892\pi$$
$$504$$ 0 0
$$505$$ 5180.00 0.456449
$$506$$ 0 0
$$507$$ −507.000 −0.0444116
$$508$$ 0 0
$$509$$ −9474.00 −0.825005 −0.412503 0.910956i $$-0.635345\pi$$
−0.412503 + 0.910956i $$0.635345\pi$$
$$510$$ 0 0
$$511$$ −8464.00 −0.732731
$$512$$ 0 0
$$513$$ −540.000 −0.0464748
$$514$$ 0 0
$$515$$ −1120.00 −0.0958313
$$516$$ 0 0
$$517$$ 18080.0 1.53802
$$518$$ 0 0
$$519$$ 10794.0 0.912917
$$520$$ 0 0
$$521$$ 14114.0 1.18684 0.593422 0.804892i $$-0.297777\pi$$
0.593422 + 0.804892i $$0.297777\pi$$
$$522$$ 0 0
$$523$$ −20284.0 −1.69590 −0.847952 0.530074i $$-0.822164\pi$$
−0.847952 + 0.530074i $$0.822164\pi$$
$$524$$ 0 0
$$525$$ −600.000 −0.0498784
$$526$$ 0 0
$$527$$ −23920.0 −1.97718
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ −4176.00 −0.341286
$$532$$ 0 0
$$533$$ 4706.00 0.382438
$$534$$ 0 0
$$535$$ −3720.00 −0.300616
$$536$$ 0 0
$$537$$ 3204.00 0.257473
$$538$$ 0 0
$$539$$ 11160.0 0.891828
$$540$$ 0 0
$$541$$ 14362.0 1.14135 0.570675 0.821176i $$-0.306682\pi$$
0.570675 + 0.821176i $$0.306682\pi$$
$$542$$ 0 0
$$543$$ −14358.0 −1.13473
$$544$$ 0 0
$$545$$ 9340.00 0.734095
$$546$$ 0 0
$$547$$ 20956.0 1.63805 0.819025 0.573757i $$-0.194515\pi$$
0.819025 + 0.573757i $$0.194515\pi$$
$$548$$ 0 0
$$549$$ −3222.00 −0.250477
$$550$$ 0 0
$$551$$ 360.000 0.0278340
$$552$$ 0 0
$$553$$ 7808.00 0.600416
$$554$$ 0 0
$$555$$ 2220.00 0.169791
$$556$$ 0 0
$$557$$ 4134.00 0.314476 0.157238 0.987561i $$-0.449741\pi$$
0.157238 + 0.987561i $$0.449741\pi$$
$$558$$ 0 0
$$559$$ 988.000 0.0747548
$$560$$ 0 0
$$561$$ 15600.0 1.17403
$$562$$ 0 0
$$563$$ 16228.0 1.21479 0.607397 0.794399i $$-0.292214\pi$$
0.607397 + 0.794399i $$0.292214\pi$$
$$564$$ 0 0
$$565$$ −19140.0 −1.42518
$$566$$ 0 0
$$567$$ −648.000 −0.0479955
$$568$$ 0 0
$$569$$ 2514.00 0.185224 0.0926119 0.995702i $$-0.470478\pi$$
0.0926119 + 0.995702i $$0.470478\pi$$
$$570$$ 0 0
$$571$$ 11612.0 0.851046 0.425523 0.904948i $$-0.360090\pi$$
0.425523 + 0.904948i $$0.360090\pi$$
$$572$$ 0 0
$$573$$ 3936.00 0.286961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6354.00 0.458441 0.229221 0.973375i $$-0.426382\pi$$
0.229221 + 0.973375i $$0.426382\pi$$
$$578$$ 0 0
$$579$$ 1050.00 0.0753653
$$580$$ 0 0
$$581$$ −8064.00 −0.575819
$$582$$ 0 0
$$583$$ 15280.0 1.08548
$$584$$ 0 0
$$585$$ 1170.00 0.0826898
$$586$$ 0 0
$$587$$ 13240.0 0.930960 0.465480 0.885059i $$-0.345882\pi$$
0.465480 + 0.885059i $$0.345882\pi$$
$$588$$ 0 0
$$589$$ −3680.00 −0.257439
$$590$$ 0 0
$$591$$ −1026.00 −0.0714112
$$592$$ 0 0
$$593$$ −1146.00 −0.0793602 −0.0396801 0.999212i $$-0.512634\pi$$
−0.0396801 + 0.999212i $$0.512634\pi$$
$$594$$ 0 0
$$595$$ 10400.0 0.716569
$$596$$ 0 0
$$597$$ 10104.0 0.692679
$$598$$ 0 0
$$599$$ 10464.0 0.713769 0.356884 0.934149i $$-0.383839\pi$$
0.356884 + 0.934149i $$0.383839\pi$$
$$600$$ 0 0
$$601$$ 6650.00 0.451346 0.225673 0.974203i $$-0.427542\pi$$
0.225673 + 0.974203i $$0.427542\pi$$
$$602$$ 0 0
$$603$$ 6300.00 0.425466
$$604$$ 0 0
$$605$$ −2690.00 −0.180767
$$606$$ 0 0
$$607$$ −6664.00 −0.445607 −0.222803 0.974863i $$-0.571521\pi$$
−0.222803 + 0.974863i $$0.571521\pi$$
$$608$$ 0 0
$$609$$ 432.000 0.0287447
$$610$$ 0 0
$$611$$ 5876.00 0.389063
$$612$$ 0 0
$$613$$ −2134.00 −0.140606 −0.0703030 0.997526i $$-0.522397\pi$$
−0.0703030 + 0.997526i $$0.522397\pi$$
$$614$$ 0 0
$$615$$ −10860.0 −0.712061
$$616$$ 0 0
$$617$$ −714.000 −0.0465876 −0.0232938 0.999729i $$-0.507415\pi$$
−0.0232938 + 0.999729i $$0.507415\pi$$
$$618$$ 0 0
$$619$$ −29228.0 −1.89786 −0.948928 0.315494i $$-0.897830\pi$$
−0.948928 + 0.315494i $$0.897830\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 3088.00 0.198584
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ 0 0
$$627$$ 2400.00 0.152866
$$628$$ 0 0
$$629$$ 9620.00 0.609816
$$630$$ 0 0
$$631$$ −13536.0 −0.853977 −0.426989 0.904257i $$-0.640426\pi$$
−0.426989 + 0.904257i $$0.640426\pi$$
$$632$$ 0 0
$$633$$ −6012.00 −0.377497
$$634$$ 0 0
$$635$$ −12960.0 −0.809924
$$636$$ 0 0
$$637$$ 3627.00 0.225600
$$638$$ 0 0
$$639$$ −6732.00 −0.416767
$$640$$ 0 0
$$641$$ 17218.0 1.06095 0.530476 0.847700i $$-0.322013\pi$$
0.530476 + 0.847700i $$0.322013\pi$$
$$642$$ 0 0
$$643$$ −15044.0 −0.922671 −0.461335 0.887226i $$-0.652630\pi$$
−0.461335 + 0.887226i $$0.652630\pi$$
$$644$$ 0 0
$$645$$ −2280.00 −0.139186
$$646$$ 0 0
$$647$$ 25176.0 1.52978 0.764892 0.644158i $$-0.222792\pi$$
0.764892 + 0.644158i $$0.222792\pi$$
$$648$$ 0 0
$$649$$ 18560.0 1.12256
$$650$$ 0 0
$$651$$ −4416.00 −0.265863
$$652$$ 0 0
$$653$$ 16034.0 0.960887 0.480443 0.877026i $$-0.340476\pi$$
0.480443 + 0.877026i $$0.340476\pi$$
$$654$$ 0 0
$$655$$ −8920.00 −0.532112
$$656$$ 0 0
$$657$$ 9522.00 0.565432
$$658$$ 0 0
$$659$$ −25356.0 −1.49883 −0.749415 0.662100i $$-0.769665\pi$$
−0.749415 + 0.662100i $$0.769665\pi$$
$$660$$ 0 0
$$661$$ −18310.0 −1.07742 −0.538711 0.842490i $$-0.681089\pi$$
−0.538711 + 0.842490i $$0.681089\pi$$
$$662$$ 0 0
$$663$$ 5070.00 0.296987
$$664$$ 0 0
$$665$$ 1600.00 0.0933013
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16824.0 0.972277
$$670$$ 0 0
$$671$$ 14320.0 0.823871
$$672$$ 0 0
$$673$$ 24802.0 1.42057 0.710287 0.703912i $$-0.248565\pi$$
0.710287 + 0.703912i $$0.248565\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ 22706.0 1.28901 0.644507 0.764598i $$-0.277063\pi$$
0.644507 + 0.764598i $$0.277063\pi$$
$$678$$ 0 0
$$679$$ 4912.00 0.277622
$$680$$ 0 0
$$681$$ −5784.00 −0.325467
$$682$$ 0 0
$$683$$ 14792.0 0.828697 0.414349 0.910118i $$-0.364009\pi$$
0.414349 + 0.910118i $$0.364009\pi$$
$$684$$ 0 0
$$685$$ −23260.0 −1.29740
$$686$$ 0 0
$$687$$ −11814.0 −0.656088
$$688$$ 0 0
$$689$$ 4966.00 0.274586
$$690$$ 0 0
$$691$$ 1148.00 0.0632011 0.0316006 0.999501i $$-0.489940\pi$$
0.0316006 + 0.999501i $$0.489940\pi$$
$$692$$ 0 0
$$693$$ 2880.00 0.157867
$$694$$ 0 0
$$695$$ 19320.0 1.05446
$$696$$ 0 0
$$697$$ −47060.0 −2.55742
$$698$$ 0 0
$$699$$ −7686.00 −0.415896
$$700$$ 0 0
$$701$$ −14870.0 −0.801187 −0.400594 0.916256i $$-0.631196\pi$$
−0.400594 + 0.916256i $$0.631196\pi$$
$$702$$ 0 0
$$703$$ 1480.00 0.0794015
$$704$$ 0 0
$$705$$ −13560.0 −0.724396
$$706$$ 0 0
$$707$$ 4144.00 0.220440
$$708$$ 0 0
$$709$$ 6354.00 0.336572 0.168286 0.985738i $$-0.446177\pi$$
0.168286 + 0.985738i $$0.446177\pi$$
$$710$$ 0 0
$$711$$ −8784.00 −0.463327
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −5200.00 −0.271985
$$716$$ 0 0
$$717$$ −21492.0 −1.11943
$$718$$ 0 0
$$719$$ 9288.00 0.481758 0.240879 0.970555i $$-0.422564\pi$$
0.240879 + 0.970555i $$0.422564\pi$$
$$720$$ 0 0
$$721$$ −896.000 −0.0462813
$$722$$ 0 0
$$723$$ 18546.0 0.953988
$$724$$ 0 0
$$725$$ −450.000 −0.0230518
$$726$$ 0 0
$$727$$ −21544.0 −1.09907 −0.549534 0.835471i $$-0.685195\pi$$
−0.549534 + 0.835471i $$0.685195\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −9880.00 −0.499897
$$732$$ 0 0
$$733$$ −19990.0 −1.00730 −0.503648 0.863909i $$-0.668009\pi$$
−0.503648 + 0.863909i $$0.668009\pi$$
$$734$$ 0 0
$$735$$ −8370.00 −0.420044
$$736$$ 0 0
$$737$$ −28000.0 −1.39945
$$738$$ 0 0
$$739$$ −532.000 −0.0264816 −0.0132408 0.999912i $$-0.504215\pi$$
−0.0132408 + 0.999912i $$0.504215\pi$$
$$740$$ 0 0
$$741$$ 780.000 0.0386694
$$742$$ 0 0
$$743$$ −25452.0 −1.25672 −0.628360 0.777922i $$-0.716274\pi$$
−0.628360 + 0.777922i $$0.716274\pi$$
$$744$$ 0 0
$$745$$ 8820.00 0.433745
$$746$$ 0 0
$$747$$ 9072.00 0.444347
$$748$$ 0 0
$$749$$ −2976.00 −0.145181
$$750$$ 0 0
$$751$$ 6440.00 0.312915 0.156457 0.987685i $$-0.449993\pi$$
0.156457 + 0.987685i $$0.449993\pi$$
$$752$$ 0 0
$$753$$ −4188.00 −0.202682
$$754$$ 0 0
$$755$$ 17760.0 0.856096
$$756$$ 0 0
$$757$$ 786.000 0.0377380 0.0188690 0.999822i $$-0.493993\pi$$
0.0188690 + 0.999822i $$0.493993\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1498.00 −0.0713567 −0.0356784 0.999363i $$-0.511359\pi$$
−0.0356784 + 0.999363i $$0.511359\pi$$
$$762$$ 0 0
$$763$$ 7472.00 0.354528
$$764$$ 0 0
$$765$$ −11700.0 −0.552960
$$766$$ 0 0
$$767$$ 6032.00 0.283967
$$768$$ 0 0
$$769$$ 14738.0 0.691113 0.345556 0.938398i $$-0.387690\pi$$
0.345556 + 0.938398i $$0.387690\pi$$
$$770$$ 0 0
$$771$$ −20718.0 −0.967757
$$772$$ 0 0
$$773$$ 3822.00 0.177837 0.0889184 0.996039i $$-0.471659\pi$$
0.0889184 + 0.996039i $$0.471659\pi$$
$$774$$ 0 0
$$775$$ 4600.00 0.213209
$$776$$ 0 0
$$777$$ 1776.00 0.0819995
$$778$$ 0 0
$$779$$ −7240.00 −0.332991
$$780$$ 0 0
$$781$$ 29920.0 1.37083
$$782$$ 0 0
$$783$$ −486.000 −0.0221816
$$784$$ 0 0
$$785$$ −24100.0 −1.09575
$$786$$ 0 0
$$787$$ 11900.0 0.538995 0.269498 0.963001i $$-0.413142\pi$$
0.269498 + 0.963001i $$0.413142\pi$$
$$788$$ 0 0
$$789$$ 20544.0 0.926978
$$790$$ 0 0
$$791$$ −15312.0 −0.688283
$$792$$ 0 0
$$793$$ 4654.00 0.208409
$$794$$ 0 0
$$795$$ −11460.0 −0.511251
$$796$$ 0 0
$$797$$ 21274.0 0.945500 0.472750 0.881197i $$-0.343261\pi$$
0.472750 + 0.881197i $$0.343261\pi$$
$$798$$ 0 0
$$799$$ −58760.0 −2.60173
$$800$$ 0 0
$$801$$ −3474.00 −0.153243
$$802$$ 0 0
$$803$$ −42320.0 −1.85983
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18102.0 −0.789617
$$808$$ 0 0
$$809$$ −27566.0 −1.19798 −0.598992 0.800755i $$-0.704432\pi$$
−0.598992 + 0.800755i $$0.704432\pi$$
$$810$$ 0 0
$$811$$ 11244.0 0.486844 0.243422 0.969921i $$-0.421730\pi$$
0.243422 + 0.969921i $$0.421730\pi$$
$$812$$ 0 0
$$813$$ −14496.0 −0.625334
$$814$$ 0 0
$$815$$ 32120.0 1.38051
$$816$$ 0 0
$$817$$ −1520.00 −0.0650894
$$818$$ 0 0
$$819$$ 936.000 0.0399347
$$820$$ 0 0
$$821$$ −13554.0 −0.576173 −0.288086 0.957604i $$-0.593019\pi$$
−0.288086 + 0.957604i $$0.593019\pi$$
$$822$$ 0 0
$$823$$ 14384.0 0.609228 0.304614 0.952476i $$-0.401473\pi$$
0.304614 + 0.952476i $$0.401473\pi$$
$$824$$ 0 0
$$825$$ −3000.00 −0.126602
$$826$$ 0 0
$$827$$ 2488.00 0.104615 0.0523073 0.998631i $$-0.483342\pi$$
0.0523073 + 0.998631i $$0.483342\pi$$
$$828$$ 0 0
$$829$$ 20858.0 0.873858 0.436929 0.899496i $$-0.356066\pi$$
0.436929 + 0.899496i $$0.356066\pi$$
$$830$$ 0 0
$$831$$ −12246.0 −0.511202
$$832$$ 0 0
$$833$$ −36270.0 −1.50862
$$834$$ 0 0
$$835$$ −16680.0 −0.691300
$$836$$ 0 0
$$837$$ 4968.00 0.205160
$$838$$ 0 0
$$839$$ 23116.0 0.951195 0.475598 0.879663i $$-0.342232\pi$$
0.475598 + 0.879663i $$0.342232\pi$$
$$840$$ 0 0
$$841$$ −24065.0 −0.986715
$$842$$ 0 0
$$843$$ −10050.0 −0.410605
$$844$$ 0 0
$$845$$ −1690.00 −0.0688021
$$846$$ 0 0
$$847$$ −2152.00 −0.0873006
$$848$$ 0 0
$$849$$ 23388.0 0.945435
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −934.000 −0.0374907 −0.0187453 0.999824i $$-0.505967\pi$$
−0.0187453 + 0.999824i $$0.505967\pi$$
$$854$$ 0 0
$$855$$ −1800.00 −0.0719985
$$856$$ 0 0
$$857$$ 12642.0 0.503900 0.251950 0.967740i $$-0.418928\pi$$
0.251950 + 0.967740i $$0.418928\pi$$
$$858$$ 0 0
$$859$$ 22796.0 0.905459 0.452730 0.891648i $$-0.350450\pi$$
0.452730 + 0.891648i $$0.350450\pi$$
$$860$$ 0 0
$$861$$ −8688.00 −0.343886
$$862$$ 0 0
$$863$$ −76.0000 −0.00299776 −0.00149888 0.999999i $$-0.500477\pi$$
−0.00149888 + 0.999999i $$0.500477\pi$$
$$864$$ 0 0
$$865$$ 35980.0 1.41429
$$866$$ 0 0
$$867$$ −35961.0 −1.40865
$$868$$ 0 0
$$869$$ 39040.0 1.52398
$$870$$ 0 0
$$871$$ −9100.00 −0.354009
$$872$$ 0 0
$$873$$ −5526.00 −0.214235
$$874$$ 0 0
$$875$$ −12000.0 −0.463627
$$876$$ 0 0
$$877$$ 46130.0 1.77617 0.888084 0.459681i $$-0.152036\pi$$
0.888084 + 0.459681i $$0.152036\pi$$
$$878$$ 0 0
$$879$$ 11766.0 0.451487
$$880$$ 0 0
$$881$$ 6682.00 0.255530 0.127765 0.991804i $$-0.459220\pi$$
0.127765 + 0.991804i $$0.459220\pi$$
$$882$$ 0 0
$$883$$ −47404.0 −1.80665 −0.903325 0.428957i $$-0.858881\pi$$
−0.903325 + 0.428957i $$0.858881\pi$$
$$884$$ 0 0
$$885$$ −13920.0 −0.528718
$$886$$ 0 0
$$887$$ 33672.0 1.27463 0.637314 0.770604i $$-0.280045\pi$$
0.637314 + 0.770604i $$0.280045\pi$$
$$888$$ 0 0
$$889$$ −10368.0 −0.391149
$$890$$ 0 0
$$891$$ −3240.00 −0.121823
$$892$$ 0 0
$$893$$ −9040.00 −0.338759
$$894$$ 0 0
$$895$$ 10680.0 0.398875
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −3312.00 −0.122871
$$900$$ 0 0
$$901$$ −49660.0 −1.83620
$$902$$ 0 0
$$903$$ −1824.00 −0.0672192
$$904$$ 0 0
$$905$$ −47860.0 −1.75792
$$906$$ 0 0
$$907$$ 14540.0 0.532296 0.266148 0.963932i $$-0.414249\pi$$
0.266148 + 0.963932i $$0.414249\pi$$
$$908$$ 0 0
$$909$$ −4662.00 −0.170109
$$910$$ 0 0
$$911$$ −7840.00 −0.285127 −0.142564 0.989786i $$-0.545535\pi$$
−0.142564 + 0.989786i $$0.545535\pi$$
$$912$$ 0 0
$$913$$ −40320.0 −1.46155
$$914$$ 0 0
$$915$$ −10740.0 −0.388037
$$916$$ 0 0
$$917$$ −7136.00 −0.256981
$$918$$ 0 0
$$919$$ 47720.0 1.71288 0.856440 0.516246i $$-0.172671\pi$$
0.856440 + 0.516246i $$0.172671\pi$$
$$920$$ 0 0
$$921$$ 17868.0 0.639273
$$922$$ 0 0
$$923$$ 9724.00 0.346771
$$924$$ 0 0
$$925$$ −1850.00 −0.0657596
$$926$$ 0 0
$$927$$ 1008.00 0.0357142
$$928$$ 0 0
$$929$$ 7502.00 0.264944 0.132472 0.991187i $$-0.457709\pi$$
0.132472 + 0.991187i $$0.457709\pi$$
$$930$$ 0 0
$$931$$ −5580.00 −0.196431
$$932$$ 0 0
$$933$$ −7056.00 −0.247592
$$934$$ 0 0
$$935$$ 52000.0 1.81880
$$936$$ 0 0
$$937$$ 22058.0 0.769054 0.384527 0.923114i $$-0.374365\pi$$
0.384527 + 0.923114i $$0.374365\pi$$
$$938$$ 0 0
$$939$$ −25326.0 −0.880173
$$940$$ 0 0
$$941$$ −23338.0 −0.808498 −0.404249 0.914649i $$-0.632467\pi$$
−0.404249 + 0.914649i $$0.632467\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ −2160.00 −0.0743543
$$946$$ 0 0
$$947$$ 30488.0 1.04617 0.523087 0.852279i $$-0.324780\pi$$
0.523087 + 0.852279i $$0.324780\pi$$
$$948$$ 0 0
$$949$$ −13754.0 −0.470468
$$950$$ 0 0
$$951$$ −16650.0 −0.567732
$$952$$ 0 0
$$953$$ 9522.00 0.323660 0.161830 0.986819i $$-0.448260\pi$$
0.161830 + 0.986819i $$0.448260\pi$$
$$954$$ 0 0
$$955$$ 13120.0 0.444558
$$956$$ 0 0
$$957$$ 2160.00 0.0729602
$$958$$ 0 0
$$959$$ −18608.0 −0.626573
$$960$$ 0 0
$$961$$ 4065.00 0.136451
$$962$$ 0 0
$$963$$ 3348.00 0.112033
$$964$$ 0 0
$$965$$ 3500.00 0.116755
$$966$$ 0 0
$$967$$ −7616.00 −0.253272 −0.126636 0.991949i $$-0.540418\pi$$
−0.126636 + 0.991949i $$0.540418\pi$$
$$968$$ 0 0
$$969$$ −7800.00 −0.258588
$$970$$ 0 0
$$971$$ −51316.0 −1.69599 −0.847996 0.530002i $$-0.822191\pi$$
−0.847996 + 0.530002i $$0.822191\pi$$
$$972$$ 0 0
$$973$$ 15456.0 0.509246
$$974$$ 0 0
$$975$$ −975.000 −0.0320256
$$976$$ 0 0
$$977$$ −48666.0 −1.59362 −0.796808 0.604232i $$-0.793480\pi$$
−0.796808 + 0.604232i $$0.793480\pi$$
$$978$$ 0 0
$$979$$ 15440.0 0.504050
$$980$$ 0 0
$$981$$ −8406.00 −0.273581
$$982$$ 0 0
$$983$$ 17388.0 0.564182 0.282091 0.959388i $$-0.408972\pi$$
0.282091 + 0.959388i $$0.408972\pi$$
$$984$$ 0 0
$$985$$ −3420.00 −0.110630
$$986$$ 0 0
$$987$$ −10848.0 −0.349844
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 11496.0 0.368499 0.184249 0.982880i $$-0.441015\pi$$
0.184249 + 0.982880i $$0.441015\pi$$
$$992$$ 0 0
$$993$$ 420.000 0.0134223
$$994$$ 0 0
$$995$$ 33680.0 1.07309
$$996$$ 0 0
$$997$$ −48862.0 −1.55213 −0.776066 0.630652i $$-0.782788\pi$$
−0.776066 + 0.630652i $$0.782788\pi$$
$$998$$ 0 0
$$999$$ −1998.00 −0.0632772
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 2496.4.a.a.1.1 1
4.3 odd 2 2496.4.a.j.1.1 1
8.3 odd 2 624.4.a.d.1.1 1
8.5 even 2 78.4.a.c.1.1 1
24.5 odd 2 234.4.a.h.1.1 1
24.11 even 2 1872.4.a.d.1.1 1
40.29 even 2 1950.4.a.l.1.1 1
104.5 odd 4 1014.4.b.h.337.2 2
104.21 odd 4 1014.4.b.h.337.1 2
104.77 even 2 1014.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 8.5 even 2
234.4.a.h.1.1 1 24.5 odd 2
624.4.a.d.1.1 1 8.3 odd 2
1014.4.a.j.1.1 1 104.77 even 2
1014.4.b.h.337.1 2 104.21 odd 4
1014.4.b.h.337.2 2 104.5 odd 4
1872.4.a.d.1.1 1 24.11 even 2
1950.4.a.l.1.1 1 40.29 even 2
2496.4.a.a.1.1 1 1.1 even 1 trivial
2496.4.a.j.1.1 1 4.3 odd 2