# Properties

 Label 960.4.a.n.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ 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] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +48.0000 q^{11} -2.00000 q^{13} -15.0000 q^{15} -114.000 q^{17} -140.000 q^{19} +12.0000 q^{21} +72.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -210.000 q^{29} +272.000 q^{31} -144.000 q^{33} -20.0000 q^{35} +334.000 q^{37} +6.00000 q^{39} -198.000 q^{41} +268.000 q^{43} +45.0000 q^{45} +216.000 q^{47} -327.000 q^{49} +342.000 q^{51} +78.0000 q^{53} +240.000 q^{55} +420.000 q^{57} -240.000 q^{59} -302.000 q^{61} -36.0000 q^{63} -10.0000 q^{65} -596.000 q^{67} -216.000 q^{69} -768.000 q^{71} -478.000 q^{73} -75.0000 q^{75} -192.000 q^{77} -640.000 q^{79} +81.0000 q^{81} +348.000 q^{83} -570.000 q^{85} +630.000 q^{87} +210.000 q^{89} +8.00000 q^{91} -816.000 q^{93} -700.000 q^{95} -1534.00 q^{97} +432.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −4.00000 −0.215980 −0.107990 0.994152i $$-0.534441\pi$$
−0.107990 + 0.994152i $$0.534441\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 48.0000 1.31569 0.657843 0.753155i $$-0.271469\pi$$
0.657843 + 0.753155i $$0.271469\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.0426692 −0.0213346 0.999772i $$-0.506792\pi$$
−0.0213346 + 0.999772i $$0.506792\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ −114.000 −1.62642 −0.813208 0.581974i $$-0.802281\pi$$
−0.813208 + 0.581974i $$0.802281\pi$$
$$18$$ 0 0
$$19$$ −140.000 −1.69043 −0.845216 0.534425i $$-0.820528\pi$$
−0.845216 + 0.534425i $$0.820528\pi$$
$$20$$ 0 0
$$21$$ 12.0000 0.124696
$$22$$ 0 0
$$23$$ 72.0000 0.652741 0.326370 0.945242i $$-0.394174\pi$$
0.326370 + 0.945242i $$0.394174\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −210.000 −1.34469 −0.672345 0.740238i $$-0.734713\pi$$
−0.672345 + 0.740238i $$0.734713\pi$$
$$30$$ 0 0
$$31$$ 272.000 1.57589 0.787946 0.615745i $$-0.211145\pi$$
0.787946 + 0.615745i $$0.211145\pi$$
$$32$$ 0 0
$$33$$ −144.000 −0.759612
$$34$$ 0 0
$$35$$ −20.0000 −0.0965891
$$36$$ 0 0
$$37$$ 334.000 1.48403 0.742017 0.670381i $$-0.233869\pi$$
0.742017 + 0.670381i $$0.233869\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.0246351
$$40$$ 0 0
$$41$$ −198.000 −0.754205 −0.377102 0.926172i $$-0.623080\pi$$
−0.377102 + 0.926172i $$0.623080\pi$$
$$42$$ 0 0
$$43$$ 268.000 0.950456 0.475228 0.879863i $$-0.342366\pi$$
0.475228 + 0.879863i $$0.342366\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 216.000 0.670358 0.335179 0.942154i $$-0.391203\pi$$
0.335179 + 0.942154i $$0.391203\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ 342.000 0.939011
$$52$$ 0 0
$$53$$ 78.0000 0.202153 0.101077 0.994879i $$-0.467771\pi$$
0.101077 + 0.994879i $$0.467771\pi$$
$$54$$ 0 0
$$55$$ 240.000 0.588393
$$56$$ 0 0
$$57$$ 420.000 0.975971
$$58$$ 0 0
$$59$$ −240.000 −0.529582 −0.264791 0.964306i $$-0.585303\pi$$
−0.264791 + 0.964306i $$0.585303\pi$$
$$60$$ 0 0
$$61$$ −302.000 −0.633888 −0.316944 0.948444i $$-0.602657\pi$$
−0.316944 + 0.948444i $$0.602657\pi$$
$$62$$ 0 0
$$63$$ −36.0000 −0.0719932
$$64$$ 0 0
$$65$$ −10.0000 −0.0190823
$$66$$ 0 0
$$67$$ −596.000 −1.08676 −0.543381 0.839487i $$-0.682856\pi$$
−0.543381 + 0.839487i $$0.682856\pi$$
$$68$$ 0 0
$$69$$ −216.000 −0.376860
$$70$$ 0 0
$$71$$ −768.000 −1.28373 −0.641865 0.766818i $$-0.721839\pi$$
−0.641865 + 0.766818i $$0.721839\pi$$
$$72$$ 0 0
$$73$$ −478.000 −0.766379 −0.383190 0.923670i $$-0.625174\pi$$
−0.383190 + 0.923670i $$0.625174\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −192.000 −0.284161
$$78$$ 0 0
$$79$$ −640.000 −0.911464 −0.455732 0.890117i $$-0.650622\pi$$
−0.455732 + 0.890117i $$0.650622\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 348.000 0.460216 0.230108 0.973165i $$-0.426092\pi$$
0.230108 + 0.973165i $$0.426092\pi$$
$$84$$ 0 0
$$85$$ −570.000 −0.727355
$$86$$ 0 0
$$87$$ 630.000 0.776357
$$88$$ 0 0
$$89$$ 210.000 0.250112 0.125056 0.992150i $$-0.460089\pi$$
0.125056 + 0.992150i $$0.460089\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.00921569
$$92$$ 0 0
$$93$$ −816.000 −0.909841
$$94$$ 0 0
$$95$$ −700.000 −0.755984
$$96$$ 0 0
$$97$$ −1534.00 −1.60571 −0.802856 0.596173i $$-0.796687\pi$$
−0.802856 + 0.596173i $$0.796687\pi$$
$$98$$ 0 0
$$99$$ 432.000 0.438562
$$100$$ 0 0
$$101$$ −1722.00 −1.69649 −0.848245 0.529605i $$-0.822340\pi$$
−0.848245 + 0.529605i $$0.822340\pi$$
$$102$$ 0 0
$$103$$ 1052.00 1.00638 0.503188 0.864177i $$-0.332160\pi$$
0.503188 + 0.864177i $$0.332160\pi$$
$$104$$ 0 0
$$105$$ 60.0000 0.0557657
$$106$$ 0 0
$$107$$ 564.000 0.509570 0.254785 0.966998i $$-0.417995\pi$$
0.254785 + 0.966998i $$0.417995\pi$$
$$108$$ 0 0
$$109$$ 610.000 0.536031 0.268016 0.963415i $$-0.413632\pi$$
0.268016 + 0.963415i $$0.413632\pi$$
$$110$$ 0 0
$$111$$ −1002.00 −0.856807
$$112$$ 0 0
$$113$$ 1302.00 1.08391 0.541955 0.840407i $$-0.317684\pi$$
0.541955 + 0.840407i $$0.317684\pi$$
$$114$$ 0 0
$$115$$ 360.000 0.291915
$$116$$ 0 0
$$117$$ −18.0000 −0.0142231
$$118$$ 0 0
$$119$$ 456.000 0.351273
$$120$$ 0 0
$$121$$ 973.000 0.731029
$$122$$ 0 0
$$123$$ 594.000 0.435440
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −124.000 −0.0866395 −0.0433198 0.999061i $$-0.513793\pi$$
−0.0433198 + 0.999061i $$0.513793\pi$$
$$128$$ 0 0
$$129$$ −804.000 −0.548746
$$130$$ 0 0
$$131$$ −192.000 −0.128054 −0.0640272 0.997948i $$-0.520394\pi$$
−0.0640272 + 0.997948i $$0.520394\pi$$
$$132$$ 0 0
$$133$$ 560.000 0.365099
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ −2514.00 −1.56778 −0.783889 0.620901i $$-0.786767\pi$$
−0.783889 + 0.620901i $$0.786767\pi$$
$$138$$ 0 0
$$139$$ −1340.00 −0.817679 −0.408839 0.912606i $$-0.634066\pi$$
−0.408839 + 0.912606i $$0.634066\pi$$
$$140$$ 0 0
$$141$$ −648.000 −0.387032
$$142$$ 0 0
$$143$$ −96.0000 −0.0561393
$$144$$ 0 0
$$145$$ −1050.00 −0.601364
$$146$$ 0 0
$$147$$ 981.000 0.550418
$$148$$ 0 0
$$149$$ −1410.00 −0.775246 −0.387623 0.921818i $$-0.626704\pi$$
−0.387623 + 0.921818i $$0.626704\pi$$
$$150$$ 0 0
$$151$$ −2128.00 −1.14685 −0.573424 0.819258i $$-0.694385\pi$$
−0.573424 + 0.819258i $$0.694385\pi$$
$$152$$ 0 0
$$153$$ −1026.00 −0.542138
$$154$$ 0 0
$$155$$ 1360.00 0.704760
$$156$$ 0 0
$$157$$ −3026.00 −1.53822 −0.769112 0.639114i $$-0.779301\pi$$
−0.769112 + 0.639114i $$0.779301\pi$$
$$158$$ 0 0
$$159$$ −234.000 −0.116713
$$160$$ 0 0
$$161$$ −288.000 −0.140979
$$162$$ 0 0
$$163$$ −2612.00 −1.25514 −0.627569 0.778561i $$-0.715950\pi$$
−0.627569 + 0.778561i $$0.715950\pi$$
$$164$$ 0 0
$$165$$ −720.000 −0.339709
$$166$$ 0 0
$$167$$ −24.0000 −0.0111208 −0.00556041 0.999985i $$-0.501770\pi$$
−0.00556041 + 0.999985i $$0.501770\pi$$
$$168$$ 0 0
$$169$$ −2193.00 −0.998179
$$170$$ 0 0
$$171$$ −1260.00 −0.563477
$$172$$ 0 0
$$173$$ −1962.00 −0.862243 −0.431122 0.902294i $$-0.641882\pi$$
−0.431122 + 0.902294i $$0.641882\pi$$
$$174$$ 0 0
$$175$$ −100.000 −0.0431959
$$176$$ 0 0
$$177$$ 720.000 0.305754
$$178$$ 0 0
$$179$$ 120.000 0.0501074 0.0250537 0.999686i $$-0.492024\pi$$
0.0250537 + 0.999686i $$0.492024\pi$$
$$180$$ 0 0
$$181$$ −902.000 −0.370415 −0.185208 0.982699i $$-0.559296\pi$$
−0.185208 + 0.982699i $$0.559296\pi$$
$$182$$ 0 0
$$183$$ 906.000 0.365975
$$184$$ 0 0
$$185$$ 1670.00 0.663680
$$186$$ 0 0
$$187$$ −5472.00 −2.13985
$$188$$ 0 0
$$189$$ 108.000 0.0415653
$$190$$ 0 0
$$191$$ −168.000 −0.0636443 −0.0318221 0.999494i $$-0.510131\pi$$
−0.0318221 + 0.999494i $$0.510131\pi$$
$$192$$ 0 0
$$193$$ −1318.00 −0.491563 −0.245782 0.969325i $$-0.579045\pi$$
−0.245782 + 0.969325i $$0.579045\pi$$
$$194$$ 0 0
$$195$$ 30.0000 0.0110172
$$196$$ 0 0
$$197$$ 4014.00 1.45170 0.725852 0.687851i $$-0.241446\pi$$
0.725852 + 0.687851i $$0.241446\pi$$
$$198$$ 0 0
$$199$$ 2000.00 0.712443 0.356222 0.934401i $$-0.384065\pi$$
0.356222 + 0.934401i $$0.384065\pi$$
$$200$$ 0 0
$$201$$ 1788.00 0.627442
$$202$$ 0 0
$$203$$ 840.000 0.290426
$$204$$ 0 0
$$205$$ −990.000 −0.337291
$$206$$ 0 0
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −6720.00 −2.22408
$$210$$ 0 0
$$211$$ 3868.00 1.26201 0.631005 0.775779i $$-0.282643\pi$$
0.631005 + 0.775779i $$0.282643\pi$$
$$212$$ 0 0
$$213$$ 2304.00 0.741162
$$214$$ 0 0
$$215$$ 1340.00 0.425057
$$216$$ 0 0
$$217$$ −1088.00 −0.340361
$$218$$ 0 0
$$219$$ 1434.00 0.442469
$$220$$ 0 0
$$221$$ 228.000 0.0693979
$$222$$ 0 0
$$223$$ −3148.00 −0.945317 −0.472658 0.881246i $$-0.656706\pi$$
−0.472658 + 0.881246i $$0.656706\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −2556.00 −0.747347 −0.373673 0.927560i $$-0.621902\pi$$
−0.373673 + 0.927560i $$0.621902\pi$$
$$228$$ 0 0
$$229$$ 610.000 0.176026 0.0880130 0.996119i $$-0.471948\pi$$
0.0880130 + 0.996119i $$0.471948\pi$$
$$230$$ 0 0
$$231$$ 576.000 0.164061
$$232$$ 0 0
$$233$$ −2058.00 −0.578644 −0.289322 0.957232i $$-0.593430\pi$$
−0.289322 + 0.957232i $$0.593430\pi$$
$$234$$ 0 0
$$235$$ 1080.00 0.299793
$$236$$ 0 0
$$237$$ 1920.00 0.526234
$$238$$ 0 0
$$239$$ 4920.00 1.33158 0.665792 0.746138i $$-0.268094\pi$$
0.665792 + 0.746138i $$0.268094\pi$$
$$240$$ 0 0
$$241$$ −1438.00 −0.384356 −0.192178 0.981360i $$-0.561555\pi$$
−0.192178 + 0.981360i $$0.561555\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −1635.00 −0.426352
$$246$$ 0 0
$$247$$ 280.000 0.0721294
$$248$$ 0 0
$$249$$ −1044.00 −0.265706
$$250$$ 0 0
$$251$$ −792.000 −0.199166 −0.0995829 0.995029i $$-0.531751\pi$$
−0.0995829 + 0.995029i $$0.531751\pi$$
$$252$$ 0 0
$$253$$ 3456.00 0.858802
$$254$$ 0 0
$$255$$ 1710.00 0.419939
$$256$$ 0 0
$$257$$ 2166.00 0.525725 0.262863 0.964833i $$-0.415333\pi$$
0.262863 + 0.964833i $$0.415333\pi$$
$$258$$ 0 0
$$259$$ −1336.00 −0.320521
$$260$$ 0 0
$$261$$ −1890.00 −0.448230
$$262$$ 0 0
$$263$$ 3192.00 0.748392 0.374196 0.927350i $$-0.377919\pi$$
0.374196 + 0.927350i $$0.377919\pi$$
$$264$$ 0 0
$$265$$ 390.000 0.0904057
$$266$$ 0 0
$$267$$ −630.000 −0.144402
$$268$$ 0 0
$$269$$ −5490.00 −1.24435 −0.622177 0.782877i $$-0.713752\pi$$
−0.622177 + 0.782877i $$0.713752\pi$$
$$270$$ 0 0
$$271$$ −6328.00 −1.41845 −0.709223 0.704985i $$-0.750954\pi$$
−0.709223 + 0.704985i $$0.750954\pi$$
$$272$$ 0 0
$$273$$ −24.0000 −0.00532068
$$274$$ 0 0
$$275$$ 1200.00 0.263137
$$276$$ 0 0
$$277$$ 574.000 0.124507 0.0622533 0.998060i $$-0.480171\pi$$
0.0622533 + 0.998060i $$0.480171\pi$$
$$278$$ 0 0
$$279$$ 2448.00 0.525297
$$280$$ 0 0
$$281$$ 4242.00 0.900557 0.450278 0.892888i $$-0.351325\pi$$
0.450278 + 0.892888i $$0.351325\pi$$
$$282$$ 0 0
$$283$$ 628.000 0.131911 0.0659553 0.997823i $$-0.478991\pi$$
0.0659553 + 0.997823i $$0.478991\pi$$
$$284$$ 0 0
$$285$$ 2100.00 0.436468
$$286$$ 0 0
$$287$$ 792.000 0.162893
$$288$$ 0 0
$$289$$ 8083.00 1.64523
$$290$$ 0 0
$$291$$ 4602.00 0.927058
$$292$$ 0 0
$$293$$ 558.000 0.111258 0.0556292 0.998451i $$-0.482284\pi$$
0.0556292 + 0.998451i $$0.482284\pi$$
$$294$$ 0 0
$$295$$ −1200.00 −0.236836
$$296$$ 0 0
$$297$$ −1296.00 −0.253204
$$298$$ 0 0
$$299$$ −144.000 −0.0278520
$$300$$ 0 0
$$301$$ −1072.00 −0.205279
$$302$$ 0 0
$$303$$ 5166.00 0.979468
$$304$$ 0 0
$$305$$ −1510.00 −0.283483
$$306$$ 0 0
$$307$$ 6964.00 1.29465 0.647323 0.762216i $$-0.275888\pi$$
0.647323 + 0.762216i $$0.275888\pi$$
$$308$$ 0 0
$$309$$ −3156.00 −0.581031
$$310$$ 0 0
$$311$$ 2832.00 0.516360 0.258180 0.966097i $$-0.416877\pi$$
0.258180 + 0.966097i $$0.416877\pi$$
$$312$$ 0 0
$$313$$ 8642.00 1.56062 0.780311 0.625392i $$-0.215061\pi$$
0.780311 + 0.625392i $$0.215061\pi$$
$$314$$ 0 0
$$315$$ −180.000 −0.0321964
$$316$$ 0 0
$$317$$ 2214.00 0.392273 0.196137 0.980577i $$-0.437160\pi$$
0.196137 + 0.980577i $$0.437160\pi$$
$$318$$ 0 0
$$319$$ −10080.0 −1.76919
$$320$$ 0 0
$$321$$ −1692.00 −0.294200
$$322$$ 0 0
$$323$$ 15960.0 2.74934
$$324$$ 0 0
$$325$$ −50.0000 −0.00853385
$$326$$ 0 0
$$327$$ −1830.00 −0.309478
$$328$$ 0 0
$$329$$ −864.000 −0.144784
$$330$$ 0 0
$$331$$ −10772.0 −1.78877 −0.894385 0.447299i $$-0.852386\pi$$
−0.894385 + 0.447299i $$0.852386\pi$$
$$332$$ 0 0
$$333$$ 3006.00 0.494678
$$334$$ 0 0
$$335$$ −2980.00 −0.486014
$$336$$ 0 0
$$337$$ −1654.00 −0.267356 −0.133678 0.991025i $$-0.542679\pi$$
−0.133678 + 0.991025i $$0.542679\pi$$
$$338$$ 0 0
$$339$$ −3906.00 −0.625796
$$340$$ 0 0
$$341$$ 13056.0 2.07338
$$342$$ 0 0
$$343$$ 2680.00 0.421885
$$344$$ 0 0
$$345$$ −1080.00 −0.168537
$$346$$ 0 0
$$347$$ −2196.00 −0.339733 −0.169867 0.985467i $$-0.554334\pi$$
−0.169867 + 0.985467i $$0.554334\pi$$
$$348$$ 0 0
$$349$$ −8270.00 −1.26843 −0.634216 0.773156i $$-0.718677\pi$$
−0.634216 + 0.773156i $$0.718677\pi$$
$$350$$ 0 0
$$351$$ 54.0000 0.00821170
$$352$$ 0 0
$$353$$ 10302.0 1.55331 0.776657 0.629923i $$-0.216914\pi$$
0.776657 + 0.629923i $$0.216914\pi$$
$$354$$ 0 0
$$355$$ −3840.00 −0.574102
$$356$$ 0 0
$$357$$ −1368.00 −0.202807
$$358$$ 0 0
$$359$$ −2280.00 −0.335192 −0.167596 0.985856i $$-0.553600\pi$$
−0.167596 + 0.985856i $$0.553600\pi$$
$$360$$ 0 0
$$361$$ 12741.0 1.85756
$$362$$ 0 0
$$363$$ −2919.00 −0.422060
$$364$$ 0 0
$$365$$ −2390.00 −0.342735
$$366$$ 0 0
$$367$$ −8764.00 −1.24653 −0.623266 0.782010i $$-0.714195\pi$$
−0.623266 + 0.782010i $$0.714195\pi$$
$$368$$ 0 0
$$369$$ −1782.00 −0.251402
$$370$$ 0 0
$$371$$ −312.000 −0.0436610
$$372$$ 0 0
$$373$$ 1318.00 0.182958 0.0914792 0.995807i $$-0.470841\pi$$
0.0914792 + 0.995807i $$0.470841\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ 420.000 0.0573769
$$378$$ 0 0
$$379$$ −1100.00 −0.149085 −0.0745425 0.997218i $$-0.523750\pi$$
−0.0745425 + 0.997218i $$0.523750\pi$$
$$380$$ 0 0
$$381$$ 372.000 0.0500214
$$382$$ 0 0
$$383$$ −3528.00 −0.470685 −0.235343 0.971912i $$-0.575621\pi$$
−0.235343 + 0.971912i $$0.575621\pi$$
$$384$$ 0 0
$$385$$ −960.000 −0.127081
$$386$$ 0 0
$$387$$ 2412.00 0.316819
$$388$$ 0 0
$$389$$ 9630.00 1.25517 0.627584 0.778549i $$-0.284044\pi$$
0.627584 + 0.778549i $$0.284044\pi$$
$$390$$ 0 0
$$391$$ −8208.00 −1.06163
$$392$$ 0 0
$$393$$ 576.000 0.0739322
$$394$$ 0 0
$$395$$ −3200.00 −0.407619
$$396$$ 0 0
$$397$$ 3094.00 0.391142 0.195571 0.980690i $$-0.437344\pi$$
0.195571 + 0.980690i $$0.437344\pi$$
$$398$$ 0 0
$$399$$ −1680.00 −0.210790
$$400$$ 0 0
$$401$$ −1638.00 −0.203985 −0.101992 0.994785i $$-0.532522\pi$$
−0.101992 + 0.994785i $$0.532522\pi$$
$$402$$ 0 0
$$403$$ −544.000 −0.0672421
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 16032.0 1.95252
$$408$$ 0 0
$$409$$ −13750.0 −1.66233 −0.831166 0.556024i $$-0.812326\pi$$
−0.831166 + 0.556024i $$0.812326\pi$$
$$410$$ 0 0
$$411$$ 7542.00 0.905157
$$412$$ 0 0
$$413$$ 960.000 0.114379
$$414$$ 0 0
$$415$$ 1740.00 0.205815
$$416$$ 0 0
$$417$$ 4020.00 0.472087
$$418$$ 0 0
$$419$$ 12480.0 1.45510 0.727551 0.686053i $$-0.240658\pi$$
0.727551 + 0.686053i $$0.240658\pi$$
$$420$$ 0 0
$$421$$ −7262.00 −0.840685 −0.420342 0.907366i $$-0.638090\pi$$
−0.420342 + 0.907366i $$0.638090\pi$$
$$422$$ 0 0
$$423$$ 1944.00 0.223453
$$424$$ 0 0
$$425$$ −2850.00 −0.325283
$$426$$ 0 0
$$427$$ 1208.00 0.136907
$$428$$ 0 0
$$429$$ 288.000 0.0324121
$$430$$ 0 0
$$431$$ 9792.00 1.09435 0.547174 0.837019i $$-0.315704\pi$$
0.547174 + 0.837019i $$0.315704\pi$$
$$432$$ 0 0
$$433$$ 1802.00 0.199997 0.0999984 0.994988i $$-0.468116\pi$$
0.0999984 + 0.994988i $$0.468116\pi$$
$$434$$ 0 0
$$435$$ 3150.00 0.347198
$$436$$ 0 0
$$437$$ −10080.0 −1.10341
$$438$$ 0 0
$$439$$ −2320.00 −0.252227 −0.126113 0.992016i $$-0.540250\pi$$
−0.126113 + 0.992016i $$0.540250\pi$$
$$440$$ 0 0
$$441$$ −2943.00 −0.317784
$$442$$ 0 0
$$443$$ −11172.0 −1.19819 −0.599095 0.800678i $$-0.704473\pi$$
−0.599095 + 0.800678i $$0.704473\pi$$
$$444$$ 0 0
$$445$$ 1050.00 0.111853
$$446$$ 0 0
$$447$$ 4230.00 0.447589
$$448$$ 0 0
$$449$$ 6810.00 0.715777 0.357888 0.933764i $$-0.383497\pi$$
0.357888 + 0.933764i $$0.383497\pi$$
$$450$$ 0 0
$$451$$ −9504.00 −0.992297
$$452$$ 0 0
$$453$$ 6384.00 0.662134
$$454$$ 0 0
$$455$$ 40.0000 0.00412138
$$456$$ 0 0
$$457$$ 17066.0 1.74686 0.873429 0.486952i $$-0.161891\pi$$
0.873429 + 0.486952i $$0.161891\pi$$
$$458$$ 0 0
$$459$$ 3078.00 0.313004
$$460$$ 0 0
$$461$$ 18918.0 1.91128 0.955639 0.294541i $$-0.0951667\pi$$
0.955639 + 0.294541i $$0.0951667\pi$$
$$462$$ 0 0
$$463$$ 1052.00 0.105595 0.0527976 0.998605i $$-0.483186\pi$$
0.0527976 + 0.998605i $$0.483186\pi$$
$$464$$ 0 0
$$465$$ −4080.00 −0.406893
$$466$$ 0 0
$$467$$ −11076.0 −1.09751 −0.548754 0.835984i $$-0.684898\pi$$
−0.548754 + 0.835984i $$0.684898\pi$$
$$468$$ 0 0
$$469$$ 2384.00 0.234718
$$470$$ 0 0
$$471$$ 9078.00 0.888094
$$472$$ 0 0
$$473$$ 12864.0 1.25050
$$474$$ 0 0
$$475$$ −3500.00 −0.338086
$$476$$ 0 0
$$477$$ 702.000 0.0673844
$$478$$ 0 0
$$479$$ −9000.00 −0.858498 −0.429249 0.903186i $$-0.641222\pi$$
−0.429249 + 0.903186i $$0.641222\pi$$
$$480$$ 0 0
$$481$$ −668.000 −0.0633226
$$482$$ 0 0
$$483$$ 864.000 0.0813941
$$484$$ 0 0
$$485$$ −7670.00 −0.718096
$$486$$ 0 0
$$487$$ −8764.00 −0.815472 −0.407736 0.913100i $$-0.633682\pi$$
−0.407736 + 0.913100i $$0.633682\pi$$
$$488$$ 0 0
$$489$$ 7836.00 0.724655
$$490$$ 0 0
$$491$$ −5592.00 −0.513978 −0.256989 0.966414i $$-0.582730\pi$$
−0.256989 + 0.966414i $$0.582730\pi$$
$$492$$ 0 0
$$493$$ 23940.0 2.18703
$$494$$ 0 0
$$495$$ 2160.00 0.196131
$$496$$ 0 0
$$497$$ 3072.00 0.277260
$$498$$ 0 0
$$499$$ −4700.00 −0.421645 −0.210823 0.977524i $$-0.567614\pi$$
−0.210823 + 0.977524i $$0.567614\pi$$
$$500$$ 0 0
$$501$$ 72.0000 0.00642060
$$502$$ 0 0
$$503$$ −11808.0 −1.04671 −0.523353 0.852116i $$-0.675319\pi$$
−0.523353 + 0.852116i $$0.675319\pi$$
$$504$$ 0 0
$$505$$ −8610.00 −0.758693
$$506$$ 0 0
$$507$$ 6579.00 0.576299
$$508$$ 0 0
$$509$$ −1170.00 −0.101885 −0.0509424 0.998702i $$-0.516222\pi$$
−0.0509424 + 0.998702i $$0.516222\pi$$
$$510$$ 0 0
$$511$$ 1912.00 0.165522
$$512$$ 0 0
$$513$$ 3780.00 0.325324
$$514$$ 0 0
$$515$$ 5260.00 0.450065
$$516$$ 0 0
$$517$$ 10368.0 0.881981
$$518$$ 0 0
$$519$$ 5886.00 0.497816
$$520$$ 0 0
$$521$$ −16638.0 −1.39909 −0.699543 0.714590i $$-0.746613\pi$$
−0.699543 + 0.714590i $$0.746613\pi$$
$$522$$ 0 0
$$523$$ −15692.0 −1.31198 −0.655988 0.754771i $$-0.727748\pi$$
−0.655988 + 0.754771i $$0.727748\pi$$
$$524$$ 0 0
$$525$$ 300.000 0.0249392
$$526$$ 0 0
$$527$$ −31008.0 −2.56305
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ −2160.00 −0.176527
$$532$$ 0 0
$$533$$ 396.000 0.0321814
$$534$$ 0 0
$$535$$ 2820.00 0.227886
$$536$$ 0 0
$$537$$ −360.000 −0.0289295
$$538$$ 0 0
$$539$$ −15696.0 −1.25431
$$540$$ 0 0
$$541$$ 22018.0 1.74977 0.874887 0.484327i $$-0.160936\pi$$
0.874887 + 0.484327i $$0.160936\pi$$
$$542$$ 0 0
$$543$$ 2706.00 0.213859
$$544$$ 0 0
$$545$$ 3050.00 0.239720
$$546$$ 0 0
$$547$$ 4564.00 0.356751 0.178375 0.983963i $$-0.442916\pi$$
0.178375 + 0.983963i $$0.442916\pi$$
$$548$$ 0 0
$$549$$ −2718.00 −0.211296
$$550$$ 0 0
$$551$$ 29400.0 2.27311
$$552$$ 0 0
$$553$$ 2560.00 0.196858
$$554$$ 0 0
$$555$$ −5010.00 −0.383176
$$556$$ 0 0
$$557$$ 7734.00 0.588331 0.294165 0.955755i $$-0.404958\pi$$
0.294165 + 0.955755i $$0.404958\pi$$
$$558$$ 0 0
$$559$$ −536.000 −0.0405552
$$560$$ 0 0
$$561$$ 16416.0 1.23544
$$562$$ 0 0
$$563$$ 20148.0 1.50824 0.754118 0.656739i $$-0.228065\pi$$
0.754118 + 0.656739i $$0.228065\pi$$
$$564$$ 0 0
$$565$$ 6510.00 0.484739
$$566$$ 0 0
$$567$$ −324.000 −0.0239977
$$568$$ 0 0
$$569$$ −24030.0 −1.77046 −0.885228 0.465156i $$-0.845998\pi$$
−0.885228 + 0.465156i $$0.845998\pi$$
$$570$$ 0 0
$$571$$ −2372.00 −0.173844 −0.0869222 0.996215i $$-0.527703\pi$$
−0.0869222 + 0.996215i $$0.527703\pi$$
$$572$$ 0 0
$$573$$ 504.000 0.0367450
$$574$$ 0 0
$$575$$ 1800.00 0.130548
$$576$$ 0 0
$$577$$ 8546.00 0.616594 0.308297 0.951290i $$-0.400241\pi$$
0.308297 + 0.951290i $$0.400241\pi$$
$$578$$ 0 0
$$579$$ 3954.00 0.283804
$$580$$ 0 0
$$581$$ −1392.00 −0.0993974
$$582$$ 0 0
$$583$$ 3744.00 0.265970
$$584$$ 0 0
$$585$$ −90.0000 −0.00636076
$$586$$ 0 0
$$587$$ 15444.0 1.08593 0.542966 0.839755i $$-0.317301\pi$$
0.542966 + 0.839755i $$0.317301\pi$$
$$588$$ 0 0
$$589$$ −38080.0 −2.66394
$$590$$ 0 0
$$591$$ −12042.0 −0.838142
$$592$$ 0 0
$$593$$ 18342.0 1.27018 0.635089 0.772439i $$-0.280963\pi$$
0.635089 + 0.772439i $$0.280963\pi$$
$$594$$ 0 0
$$595$$ 2280.00 0.157094
$$596$$ 0 0
$$597$$ −6000.00 −0.411329
$$598$$ 0 0
$$599$$ 24600.0 1.67801 0.839006 0.544123i $$-0.183137\pi$$
0.839006 + 0.544123i $$0.183137\pi$$
$$600$$ 0 0
$$601$$ −8998.00 −0.610709 −0.305354 0.952239i $$-0.598775\pi$$
−0.305354 + 0.952239i $$0.598775\pi$$
$$602$$ 0 0
$$603$$ −5364.00 −0.362254
$$604$$ 0 0
$$605$$ 4865.00 0.326926
$$606$$ 0 0
$$607$$ 4076.00 0.272553 0.136277 0.990671i $$-0.456486\pi$$
0.136277 + 0.990671i $$0.456486\pi$$
$$608$$ 0 0
$$609$$ −2520.00 −0.167677
$$610$$ 0 0
$$611$$ −432.000 −0.0286037
$$612$$ 0 0
$$613$$ 4078.00 0.268693 0.134347 0.990934i $$-0.457106\pi$$
0.134347 + 0.990934i $$0.457106\pi$$
$$614$$ 0 0
$$615$$ 2970.00 0.194735
$$616$$ 0 0
$$617$$ 10086.0 0.658099 0.329049 0.944313i $$-0.393272\pi$$
0.329049 + 0.944313i $$0.393272\pi$$
$$618$$ 0 0
$$619$$ −8780.00 −0.570110 −0.285055 0.958511i $$-0.592012\pi$$
−0.285055 + 0.958511i $$0.592012\pi$$
$$620$$ 0 0
$$621$$ −1944.00 −0.125620
$$622$$ 0 0
$$623$$ −840.000 −0.0540191
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 20160.0 1.28407
$$628$$ 0 0
$$629$$ −38076.0 −2.41366
$$630$$ 0 0
$$631$$ 2792.00 0.176145 0.0880727 0.996114i $$-0.471929\pi$$
0.0880727 + 0.996114i $$0.471929\pi$$
$$632$$ 0 0
$$633$$ −11604.0 −0.728622
$$634$$ 0 0
$$635$$ −620.000 −0.0387464
$$636$$ 0 0
$$637$$ 654.000 0.0406788
$$638$$ 0 0
$$639$$ −6912.00 −0.427910
$$640$$ 0 0
$$641$$ 7602.00 0.468426 0.234213 0.972185i $$-0.424749\pi$$
0.234213 + 0.972185i $$0.424749\pi$$
$$642$$ 0 0
$$643$$ −24212.0 −1.48496 −0.742479 0.669869i $$-0.766350\pi$$
−0.742479 + 0.669869i $$0.766350\pi$$
$$644$$ 0 0
$$645$$ −4020.00 −0.245407
$$646$$ 0 0
$$647$$ 9456.00 0.574581 0.287290 0.957844i $$-0.407246\pi$$
0.287290 + 0.957844i $$0.407246\pi$$
$$648$$ 0 0
$$649$$ −11520.0 −0.696764
$$650$$ 0 0
$$651$$ 3264.00 0.196507
$$652$$ 0 0
$$653$$ 9558.00 0.572792 0.286396 0.958111i $$-0.407543\pi$$
0.286396 + 0.958111i $$0.407543\pi$$
$$654$$ 0 0
$$655$$ −960.000 −0.0572676
$$656$$ 0 0
$$657$$ −4302.00 −0.255460
$$658$$ 0 0
$$659$$ 29280.0 1.73078 0.865392 0.501095i $$-0.167069\pi$$
0.865392 + 0.501095i $$0.167069\pi$$
$$660$$ 0 0
$$661$$ 29098.0 1.71223 0.856113 0.516789i $$-0.172873\pi$$
0.856113 + 0.516789i $$0.172873\pi$$
$$662$$ 0 0
$$663$$ −684.000 −0.0400669
$$664$$ 0 0
$$665$$ 2800.00 0.163277
$$666$$ 0 0
$$667$$ −15120.0 −0.877734
$$668$$ 0 0
$$669$$ 9444.00 0.545779
$$670$$ 0 0
$$671$$ −14496.0 −0.833997
$$672$$ 0 0
$$673$$ −11638.0 −0.666585 −0.333293 0.942823i $$-0.608160\pi$$
−0.333293 + 0.942823i $$0.608160\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −3426.00 −0.194493 −0.0972466 0.995260i $$-0.531004\pi$$
−0.0972466 + 0.995260i $$0.531004\pi$$
$$678$$ 0 0
$$679$$ 6136.00 0.346801
$$680$$ 0 0
$$681$$ 7668.00 0.431481
$$682$$ 0 0
$$683$$ 20148.0 1.12876 0.564379 0.825516i $$-0.309116\pi$$
0.564379 + 0.825516i $$0.309116\pi$$
$$684$$ 0 0
$$685$$ −12570.0 −0.701131
$$686$$ 0 0
$$687$$ −1830.00 −0.101629
$$688$$ 0 0
$$689$$ −156.000 −0.00862573
$$690$$ 0 0
$$691$$ 29428.0 1.62011 0.810053 0.586356i $$-0.199438\pi$$
0.810053 + 0.586356i $$0.199438\pi$$
$$692$$ 0 0
$$693$$ −1728.00 −0.0947205
$$694$$ 0 0
$$695$$ −6700.00 −0.365677
$$696$$ 0 0
$$697$$ 22572.0 1.22665
$$698$$ 0 0
$$699$$ 6174.00 0.334080
$$700$$ 0 0
$$701$$ −16242.0 −0.875110 −0.437555 0.899192i $$-0.644155\pi$$
−0.437555 + 0.899192i $$0.644155\pi$$
$$702$$ 0 0
$$703$$ −46760.0 −2.50866
$$704$$ 0 0
$$705$$ −3240.00 −0.173086
$$706$$ 0 0
$$707$$ 6888.00 0.366407
$$708$$ 0 0
$$709$$ −2030.00 −0.107529 −0.0537646 0.998554i $$-0.517122\pi$$
−0.0537646 + 0.998554i $$0.517122\pi$$
$$710$$ 0 0
$$711$$ −5760.00 −0.303821
$$712$$ 0 0
$$713$$ 19584.0 1.02865
$$714$$ 0 0
$$715$$ −480.000 −0.0251063
$$716$$ 0 0
$$717$$ −14760.0 −0.768790
$$718$$ 0 0
$$719$$ 6960.00 0.361007 0.180504 0.983574i $$-0.442227\pi$$
0.180504 + 0.983574i $$0.442227\pi$$
$$720$$ 0 0
$$721$$ −4208.00 −0.217357
$$722$$ 0 0
$$723$$ 4314.00 0.221908
$$724$$ 0 0
$$725$$ −5250.00 −0.268938
$$726$$ 0 0
$$727$$ 18596.0 0.948676 0.474338 0.880343i $$-0.342687\pi$$
0.474338 + 0.880343i $$0.342687\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −30552.0 −1.54584
$$732$$ 0 0
$$733$$ −21242.0 −1.07038 −0.535192 0.844731i $$-0.679761\pi$$
−0.535192 + 0.844731i $$0.679761\pi$$
$$734$$ 0 0
$$735$$ 4905.00 0.246155
$$736$$ 0 0
$$737$$ −28608.0 −1.42984
$$738$$ 0 0
$$739$$ 340.000 0.0169244 0.00846218 0.999964i $$-0.497306\pi$$
0.00846218 + 0.999964i $$0.497306\pi$$
$$740$$ 0 0
$$741$$ −840.000 −0.0416440
$$742$$ 0 0
$$743$$ −21888.0 −1.08074 −0.540372 0.841426i $$-0.681716\pi$$
−0.540372 + 0.841426i $$0.681716\pi$$
$$744$$ 0 0
$$745$$ −7050.00 −0.346701
$$746$$ 0 0
$$747$$ 3132.00 0.153405
$$748$$ 0 0
$$749$$ −2256.00 −0.110057
$$750$$ 0 0
$$751$$ 17792.0 0.864500 0.432250 0.901754i $$-0.357720\pi$$
0.432250 + 0.901754i $$0.357720\pi$$
$$752$$ 0 0
$$753$$ 2376.00 0.114988
$$754$$ 0 0
$$755$$ −10640.0 −0.512886
$$756$$ 0 0
$$757$$ −37346.0 −1.79308 −0.896541 0.442960i $$-0.853928\pi$$
−0.896541 + 0.442960i $$0.853928\pi$$
$$758$$ 0 0
$$759$$ −10368.0 −0.495829
$$760$$ 0 0
$$761$$ −11358.0 −0.541034 −0.270517 0.962715i $$-0.587195\pi$$
−0.270517 + 0.962715i $$0.587195\pi$$
$$762$$ 0 0
$$763$$ −2440.00 −0.115772
$$764$$ 0 0
$$765$$ −5130.00 −0.242452
$$766$$ 0 0
$$767$$ 480.000 0.0225969
$$768$$ 0 0
$$769$$ −34270.0 −1.60703 −0.803516 0.595283i $$-0.797040\pi$$
−0.803516 + 0.595283i $$0.797040\pi$$
$$770$$ 0 0
$$771$$ −6498.00 −0.303528
$$772$$ 0 0
$$773$$ 13278.0 0.617822 0.308911 0.951091i $$-0.400035\pi$$
0.308911 + 0.951091i $$0.400035\pi$$
$$774$$ 0 0
$$775$$ 6800.00 0.315178
$$776$$ 0 0
$$777$$ 4008.00 0.185053
$$778$$ 0 0
$$779$$ 27720.0 1.27493
$$780$$ 0 0
$$781$$ −36864.0 −1.68899
$$782$$ 0 0
$$783$$ 5670.00 0.258786
$$784$$ 0 0
$$785$$ −15130.0 −0.687914
$$786$$ 0 0
$$787$$ 11164.0 0.505659 0.252829 0.967511i $$-0.418639\pi$$
0.252829 + 0.967511i $$0.418639\pi$$
$$788$$ 0 0
$$789$$ −9576.00 −0.432084
$$790$$ 0 0
$$791$$ −5208.00 −0.234103
$$792$$ 0 0
$$793$$ 604.000 0.0270475
$$794$$ 0 0
$$795$$ −1170.00 −0.0521958
$$796$$ 0 0
$$797$$ 5094.00 0.226397 0.113199 0.993572i $$-0.463890\pi$$
0.113199 + 0.993572i $$0.463890\pi$$
$$798$$ 0 0
$$799$$ −24624.0 −1.09028
$$800$$ 0 0
$$801$$ 1890.00 0.0833706
$$802$$ 0 0
$$803$$ −22944.0 −1.00831
$$804$$ 0 0
$$805$$ −1440.00 −0.0630476
$$806$$ 0 0
$$807$$ 16470.0 0.718428
$$808$$ 0 0
$$809$$ −8790.00 −0.382002 −0.191001 0.981590i $$-0.561173\pi$$
−0.191001 + 0.981590i $$0.561173\pi$$
$$810$$ 0 0
$$811$$ −5852.00 −0.253380 −0.126690 0.991942i $$-0.540435\pi$$
−0.126690 + 0.991942i $$0.540435\pi$$
$$812$$ 0 0
$$813$$ 18984.0 0.818940
$$814$$ 0 0
$$815$$ −13060.0 −0.561315
$$816$$ 0 0
$$817$$ −37520.0 −1.60668
$$818$$ 0 0
$$819$$ 72.0000 0.00307190
$$820$$ 0 0
$$821$$ 29478.0 1.25309 0.626546 0.779384i $$-0.284468\pi$$
0.626546 + 0.779384i $$0.284468\pi$$
$$822$$ 0 0
$$823$$ 39332.0 1.66589 0.832945 0.553356i $$-0.186653\pi$$
0.832945 + 0.553356i $$0.186653\pi$$
$$824$$ 0 0
$$825$$ −3600.00 −0.151922
$$826$$ 0 0
$$827$$ −6756.00 −0.284074 −0.142037 0.989861i $$-0.545365\pi$$
−0.142037 + 0.989861i $$0.545365\pi$$
$$828$$ 0 0
$$829$$ −3950.00 −0.165488 −0.0827438 0.996571i $$-0.526368\pi$$
−0.0827438 + 0.996571i $$0.526368\pi$$
$$830$$ 0 0
$$831$$ −1722.00 −0.0718839
$$832$$ 0 0
$$833$$ 37278.0 1.55055
$$834$$ 0 0
$$835$$ −120.000 −0.00497338
$$836$$ 0 0
$$837$$ −7344.00 −0.303280
$$838$$ 0 0
$$839$$ 12360.0 0.508599 0.254300 0.967126i $$-0.418155\pi$$
0.254300 + 0.967126i $$0.418155\pi$$
$$840$$ 0 0
$$841$$ 19711.0 0.808192
$$842$$ 0 0
$$843$$ −12726.0 −0.519937
$$844$$ 0 0
$$845$$ −10965.0 −0.446399
$$846$$ 0 0
$$847$$ −3892.00 −0.157887
$$848$$ 0 0
$$849$$ −1884.00 −0.0761587
$$850$$ 0 0
$$851$$ 24048.0 0.968690
$$852$$ 0 0
$$853$$ 35998.0 1.44496 0.722478 0.691394i $$-0.243003\pi$$
0.722478 + 0.691394i $$0.243003\pi$$
$$854$$ 0 0
$$855$$ −6300.00 −0.251995
$$856$$ 0 0
$$857$$ −21594.0 −0.860720 −0.430360 0.902657i $$-0.641613\pi$$
−0.430360 + 0.902657i $$0.641613\pi$$
$$858$$ 0 0
$$859$$ −9260.00 −0.367808 −0.183904 0.982944i $$-0.558874\pi$$
−0.183904 + 0.982944i $$0.558874\pi$$
$$860$$ 0 0
$$861$$ −2376.00 −0.0940463
$$862$$ 0 0
$$863$$ 31632.0 1.24770 0.623850 0.781544i $$-0.285567\pi$$
0.623850 + 0.781544i $$0.285567\pi$$
$$864$$ 0 0
$$865$$ −9810.00 −0.385607
$$866$$ 0 0
$$867$$ −24249.0 −0.949872
$$868$$ 0 0
$$869$$ −30720.0 −1.19920
$$870$$ 0 0
$$871$$ 1192.00 0.0463713
$$872$$ 0 0
$$873$$ −13806.0 −0.535237
$$874$$ 0 0
$$875$$ −500.000 −0.0193178
$$876$$ 0 0
$$877$$ 39694.0 1.52836 0.764180 0.645003i $$-0.223144\pi$$
0.764180 + 0.645003i $$0.223144\pi$$
$$878$$ 0 0
$$879$$ −1674.00 −0.0642351
$$880$$ 0 0
$$881$$ 1242.00 0.0474961 0.0237480 0.999718i $$-0.492440\pi$$
0.0237480 + 0.999718i $$0.492440\pi$$
$$882$$ 0 0
$$883$$ 2668.00 0.101682 0.0508411 0.998707i $$-0.483810\pi$$
0.0508411 + 0.998707i $$0.483810\pi$$
$$884$$ 0 0
$$885$$ 3600.00 0.136737
$$886$$ 0 0
$$887$$ −4344.00 −0.164439 −0.0822194 0.996614i $$-0.526201\pi$$
−0.0822194 + 0.996614i $$0.526201\pi$$
$$888$$ 0 0
$$889$$ 496.000 0.0187124
$$890$$ 0 0
$$891$$ 3888.00 0.146187
$$892$$ 0 0
$$893$$ −30240.0 −1.13319
$$894$$ 0 0
$$895$$ 600.000 0.0224087
$$896$$ 0 0
$$897$$ 432.000 0.0160803
$$898$$ 0 0
$$899$$ −57120.0 −2.11909
$$900$$ 0 0
$$901$$ −8892.00 −0.328785
$$902$$ 0 0
$$903$$ 3216.00 0.118518
$$904$$ 0 0
$$905$$ −4510.00 −0.165655
$$906$$ 0 0
$$907$$ −4436.00 −0.162398 −0.0811990 0.996698i $$-0.525875\pi$$
−0.0811990 + 0.996698i $$0.525875\pi$$
$$908$$ 0 0
$$909$$ −15498.0 −0.565496
$$910$$ 0 0
$$911$$ 22752.0 0.827450 0.413725 0.910402i $$-0.364227\pi$$
0.413725 + 0.910402i $$0.364227\pi$$
$$912$$ 0 0
$$913$$ 16704.0 0.605500
$$914$$ 0 0
$$915$$ 4530.00 0.163669
$$916$$ 0 0
$$917$$ 768.000 0.0276571
$$918$$ 0 0
$$919$$ −27160.0 −0.974892 −0.487446 0.873153i $$-0.662071\pi$$
−0.487446 + 0.873153i $$0.662071\pi$$
$$920$$ 0 0
$$921$$ −20892.0 −0.747465
$$922$$ 0 0
$$923$$ 1536.00 0.0547758
$$924$$ 0 0
$$925$$ 8350.00 0.296807
$$926$$ 0 0
$$927$$ 9468.00 0.335458
$$928$$ 0 0
$$929$$ −33030.0 −1.16650 −0.583250 0.812292i $$-0.698219\pi$$
−0.583250 + 0.812292i $$0.698219\pi$$
$$930$$ 0 0
$$931$$ 45780.0 1.61158
$$932$$ 0 0
$$933$$ −8496.00 −0.298121
$$934$$ 0 0
$$935$$ −27360.0 −0.956971
$$936$$ 0 0
$$937$$ −29974.0 −1.04505 −0.522523 0.852625i $$-0.675009\pi$$
−0.522523 + 0.852625i $$0.675009\pi$$
$$938$$ 0 0
$$939$$ −25926.0 −0.901026
$$940$$ 0 0
$$941$$ −13962.0 −0.483686 −0.241843 0.970315i $$-0.577752\pi$$
−0.241843 + 0.970315i $$0.577752\pi$$
$$942$$ 0 0
$$943$$ −14256.0 −0.492300
$$944$$ 0 0
$$945$$ 540.000 0.0185886
$$946$$ 0 0
$$947$$ −35196.0 −1.20773 −0.603863 0.797088i $$-0.706373\pi$$
−0.603863 + 0.797088i $$0.706373\pi$$
$$948$$ 0 0
$$949$$ 956.000 0.0327008
$$950$$ 0 0
$$951$$ −6642.00 −0.226479
$$952$$ 0 0
$$953$$ −28338.0 −0.963230 −0.481615 0.876383i $$-0.659950\pi$$
−0.481615 + 0.876383i $$0.659950\pi$$
$$954$$ 0 0
$$955$$ −840.000 −0.0284626
$$956$$ 0 0
$$957$$ 30240.0 1.02144
$$958$$ 0 0
$$959$$ 10056.0 0.338608
$$960$$ 0 0
$$961$$ 44193.0 1.48343
$$962$$ 0 0
$$963$$ 5076.00 0.169857
$$964$$ 0 0
$$965$$ −6590.00 −0.219834
$$966$$ 0 0
$$967$$ −17524.0 −0.582765 −0.291383 0.956607i $$-0.594115\pi$$
−0.291383 + 0.956607i $$0.594115\pi$$
$$968$$ 0 0
$$969$$ −47880.0 −1.58733
$$970$$ 0 0
$$971$$ 26808.0 0.886004 0.443002 0.896521i $$-0.353913\pi$$
0.443002 + 0.896521i $$0.353913\pi$$
$$972$$ 0 0
$$973$$ 5360.00 0.176602
$$974$$ 0 0
$$975$$ 150.000 0.00492702
$$976$$ 0 0
$$977$$ −10914.0 −0.357390 −0.178695 0.983905i $$-0.557187\pi$$
−0.178695 + 0.983905i $$0.557187\pi$$
$$978$$ 0 0
$$979$$ 10080.0 0.329069
$$980$$ 0 0
$$981$$ 5490.00 0.178677
$$982$$ 0 0
$$983$$ 22272.0 0.722652 0.361326 0.932440i $$-0.382324\pi$$
0.361326 + 0.932440i $$0.382324\pi$$
$$984$$ 0 0
$$985$$ 20070.0 0.649222
$$986$$ 0 0
$$987$$ 2592.00 0.0835910
$$988$$ 0 0
$$989$$ 19296.0 0.620402
$$990$$ 0 0
$$991$$ 14072.0 0.451071 0.225536 0.974235i $$-0.427587\pi$$
0.225536 + 0.974235i $$0.427587\pi$$
$$992$$ 0 0
$$993$$ 32316.0 1.03275
$$994$$ 0 0
$$995$$ 10000.0 0.318614
$$996$$ 0 0
$$997$$ −4826.00 −0.153301 −0.0766504 0.997058i $$-0.524423\pi$$
−0.0766504 + 0.997058i $$0.524423\pi$$
$$998$$ 0 0
$$999$$ −9018.00 −0.285602
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 960.4.a.n.1.1 1
4.3 odd 2 960.4.a.bg.1.1 1
8.3 odd 2 240.4.a.b.1.1 1
8.5 even 2 30.4.a.b.1.1 1
24.5 odd 2 90.4.a.c.1.1 1
24.11 even 2 720.4.a.y.1.1 1
40.3 even 4 1200.4.f.r.49.1 2
40.13 odd 4 150.4.c.c.49.1 2
40.19 odd 2 1200.4.a.ba.1.1 1
40.27 even 4 1200.4.f.r.49.2 2
40.29 even 2 150.4.a.b.1.1 1
40.37 odd 4 150.4.c.c.49.2 2
56.13 odd 2 1470.4.a.r.1.1 1
72.5 odd 6 810.4.e.p.541.1 2
72.13 even 6 810.4.e.i.541.1 2
72.29 odd 6 810.4.e.p.271.1 2
72.61 even 6 810.4.e.i.271.1 2
120.29 odd 2 450.4.a.r.1.1 1
120.53 even 4 450.4.c.j.199.2 2
120.77 even 4 450.4.c.j.199.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.b.1.1 1 8.5 even 2
90.4.a.c.1.1 1 24.5 odd 2
150.4.a.b.1.1 1 40.29 even 2
150.4.c.c.49.1 2 40.13 odd 4
150.4.c.c.49.2 2 40.37 odd 4
240.4.a.b.1.1 1 8.3 odd 2
450.4.a.r.1.1 1 120.29 odd 2
450.4.c.j.199.1 2 120.77 even 4
450.4.c.j.199.2 2 120.53 even 4
720.4.a.y.1.1 1 24.11 even 2
810.4.e.i.271.1 2 72.61 even 6
810.4.e.i.541.1 2 72.13 even 6
810.4.e.p.271.1 2 72.29 odd 6
810.4.e.p.541.1 2 72.5 odd 6
960.4.a.n.1.1 1 1.1 even 1 trivial
960.4.a.bg.1.1 1 4.3 odd 2
1200.4.a.ba.1.1 1 40.19 odd 2
1200.4.f.r.49.1 2 40.3 even 4
1200.4.f.r.49.2 2 40.27 even 4
1470.4.a.r.1.1 1 56.13 odd 2