# Properties

 Label 960.4.a.e.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 120) 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} +9.00000 q^{9} +4.00000 q^{11} -54.0000 q^{13} +15.0000 q^{15} +114.000 q^{17} +44.0000 q^{19} -96.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -134.000 q^{29} +272.000 q^{31} -12.0000 q^{33} +98.0000 q^{37} +162.000 q^{39} -6.00000 q^{41} +12.0000 q^{43} -45.0000 q^{45} +200.000 q^{47} -343.000 q^{49} -342.000 q^{51} -654.000 q^{53} -20.0000 q^{55} -132.000 q^{57} +36.0000 q^{59} +442.000 q^{61} +270.000 q^{65} -188.000 q^{67} +288.000 q^{69} +632.000 q^{71} -390.000 q^{73} -75.0000 q^{75} -688.000 q^{79} +81.0000 q^{81} +1188.00 q^{83} -570.000 q^{85} +402.000 q^{87} -694.000 q^{89} -816.000 q^{93} -220.000 q^{95} -1726.00 q^{97} +36.0000 q^{99} +O(q^{100})$$

## 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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 0.109640 0.0548202 0.998496i $$-0.482541\pi$$
0.0548202 + 0.998496i $$0.482541\pi$$
$$12$$ 0 0
$$13$$ −54.0000 −1.15207 −0.576035 0.817425i $$-0.695401\pi$$
−0.576035 + 0.817425i $$0.695401\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 114.000 1.62642 0.813208 0.581974i $$-0.197719\pi$$
0.813208 + 0.581974i $$0.197719\pi$$
$$18$$ 0 0
$$19$$ 44.0000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −96.0000 −0.870321 −0.435161 0.900353i $$-0.643308\pi$$
−0.435161 + 0.900353i $$0.643308\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −134.000 −0.858041 −0.429020 0.903295i $$-0.641141\pi$$
−0.429020 + 0.903295i $$0.641141\pi$$
$$30$$ 0 0
$$31$$ 272.000 1.57589 0.787946 0.615745i $$-0.211145\pi$$
0.787946 + 0.615745i $$0.211145\pi$$
$$32$$ 0 0
$$33$$ −12.0000 −0.0633010
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 98.0000 0.435435 0.217718 0.976012i $$-0.430139\pi$$
0.217718 + 0.976012i $$0.430139\pi$$
$$38$$ 0 0
$$39$$ 162.000 0.665148
$$40$$ 0 0
$$41$$ −6.00000 −0.0228547 −0.0114273 0.999935i $$-0.503638\pi$$
−0.0114273 + 0.999935i $$0.503638\pi$$
$$42$$ 0 0
$$43$$ 12.0000 0.0425577 0.0212789 0.999774i $$-0.493226\pi$$
0.0212789 + 0.999774i $$0.493226\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 200.000 0.620702 0.310351 0.950622i $$-0.399553\pi$$
0.310351 + 0.950622i $$0.399553\pi$$
$$48$$ 0 0
$$49$$ −343.000 −1.00000
$$50$$ 0 0
$$51$$ −342.000 −0.939011
$$52$$ 0 0
$$53$$ −654.000 −1.69498 −0.847489 0.530813i $$-0.821887\pi$$
−0.847489 + 0.530813i $$0.821887\pi$$
$$54$$ 0 0
$$55$$ −20.0000 −0.0490327
$$56$$ 0 0
$$57$$ −132.000 −0.306734
$$58$$ 0 0
$$59$$ 36.0000 0.0794373 0.0397187 0.999211i $$-0.487354\pi$$
0.0397187 + 0.999211i $$0.487354\pi$$
$$60$$ 0 0
$$61$$ 442.000 0.927743 0.463871 0.885903i $$-0.346460\pi$$
0.463871 + 0.885903i $$0.346460\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 270.000 0.515221
$$66$$ 0 0
$$67$$ −188.000 −0.342804 −0.171402 0.985201i $$-0.554830\pi$$
−0.171402 + 0.985201i $$0.554830\pi$$
$$68$$ 0 0
$$69$$ 288.000 0.502480
$$70$$ 0 0
$$71$$ 632.000 1.05640 0.528201 0.849119i $$-0.322867\pi$$
0.528201 + 0.849119i $$0.322867\pi$$
$$72$$ 0 0
$$73$$ −390.000 −0.625288 −0.312644 0.949870i $$-0.601215\pi$$
−0.312644 + 0.949870i $$0.601215\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −688.000 −0.979823 −0.489912 0.871772i $$-0.662971\pi$$
−0.489912 + 0.871772i $$0.662971\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1188.00 1.57108 0.785542 0.618809i $$-0.212384\pi$$
0.785542 + 0.618809i $$0.212384\pi$$
$$84$$ 0 0
$$85$$ −570.000 −0.727355
$$86$$ 0 0
$$87$$ 402.000 0.495390
$$88$$ 0 0
$$89$$ −694.000 −0.826560 −0.413280 0.910604i $$-0.635617\pi$$
−0.413280 + 0.910604i $$0.635617\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −816.000 −0.909841
$$94$$ 0 0
$$95$$ −220.000 −0.237595
$$96$$ 0 0
$$97$$ −1726.00 −1.80669 −0.903344 0.428917i $$-0.858895\pi$$
−0.903344 + 0.428917i $$0.858895\pi$$
$$98$$ 0 0
$$99$$ 36.0000 0.0365468
$$100$$ 0 0
$$101$$ −1182.00 −1.16449 −0.582245 0.813014i $$-0.697825\pi$$
−0.582245 + 0.813014i $$0.697825\pi$$
$$102$$ 0 0
$$103$$ −1968.00 −1.88265 −0.941324 0.337503i $$-0.890418\pi$$
−0.941324 + 0.337503i $$0.890418\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 796.000 0.719180 0.359590 0.933110i $$-0.382917\pi$$
0.359590 + 0.933110i $$0.382917\pi$$
$$108$$ 0 0
$$109$$ −342.000 −0.300529 −0.150264 0.988646i $$-0.548013\pi$$
−0.150264 + 0.988646i $$0.548013\pi$$
$$110$$ 0 0
$$111$$ −294.000 −0.251399
$$112$$ 0 0
$$113$$ 114.000 0.0949046 0.0474523 0.998874i $$-0.484890\pi$$
0.0474523 + 0.998874i $$0.484890\pi$$
$$114$$ 0 0
$$115$$ 480.000 0.389219
$$116$$ 0 0
$$117$$ −486.000 −0.384023
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1315.00 −0.987979
$$122$$ 0 0
$$123$$ 18.0000 0.0131952
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −2344.00 −1.63777 −0.818883 0.573960i $$-0.805406\pi$$
−0.818883 + 0.573960i $$0.805406\pi$$
$$128$$ 0 0
$$129$$ −36.0000 −0.0245707
$$130$$ 0 0
$$131$$ −2164.00 −1.44328 −0.721640 0.692269i $$-0.756611\pi$$
−0.721640 + 0.692269i $$0.756611\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ −2822.00 −1.75985 −0.879926 0.475111i $$-0.842408\pi$$
−0.879926 + 0.475111i $$0.842408\pi$$
$$138$$ 0 0
$$139$$ 1972.00 1.20333 0.601665 0.798749i $$-0.294504\pi$$
0.601665 + 0.798749i $$0.294504\pi$$
$$140$$ 0 0
$$141$$ −600.000 −0.358363
$$142$$ 0 0
$$143$$ −216.000 −0.126313
$$144$$ 0 0
$$145$$ 670.000 0.383727
$$146$$ 0 0
$$147$$ 1029.00 0.577350
$$148$$ 0 0
$$149$$ 1394.00 0.766449 0.383225 0.923655i $$-0.374814\pi$$
0.383225 + 0.923655i $$0.374814\pi$$
$$150$$ 0 0
$$151$$ 2216.00 1.19427 0.597137 0.802139i $$-0.296305\pi$$
0.597137 + 0.802139i $$0.296305\pi$$
$$152$$ 0 0
$$153$$ 1026.00 0.542138
$$154$$ 0 0
$$155$$ −1360.00 −0.704760
$$156$$ 0 0
$$157$$ 954.000 0.484952 0.242476 0.970157i $$-0.422040\pi$$
0.242476 + 0.970157i $$0.422040\pi$$
$$158$$ 0 0
$$159$$ 1962.00 0.978596
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3404.00 −1.63572 −0.817858 0.575419i $$-0.804839\pi$$
−0.817858 + 0.575419i $$0.804839\pi$$
$$164$$ 0 0
$$165$$ 60.0000 0.0283091
$$166$$ 0 0
$$167$$ 832.000 0.385522 0.192761 0.981246i $$-0.438256\pi$$
0.192761 + 0.981246i $$0.438256\pi$$
$$168$$ 0 0
$$169$$ 719.000 0.327264
$$170$$ 0 0
$$171$$ 396.000 0.177093
$$172$$ 0 0
$$173$$ 362.000 0.159089 0.0795444 0.996831i $$-0.474653\pi$$
0.0795444 + 0.996831i $$0.474653\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −108.000 −0.0458631
$$178$$ 0 0
$$179$$ −3252.00 −1.35791 −0.678955 0.734180i $$-0.737567\pi$$
−0.678955 + 0.734180i $$0.737567\pi$$
$$180$$ 0 0
$$181$$ −3086.00 −1.26730 −0.633648 0.773621i $$-0.718443\pi$$
−0.633648 + 0.773621i $$0.718443\pi$$
$$182$$ 0 0
$$183$$ −1326.00 −0.535632
$$184$$ 0 0
$$185$$ −490.000 −0.194733
$$186$$ 0 0
$$187$$ 456.000 0.178321
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4080.00 1.54565 0.772823 0.634621i $$-0.218844\pi$$
0.772823 + 0.634621i $$0.218844\pi$$
$$192$$ 0 0
$$193$$ −2654.00 −0.989840 −0.494920 0.868939i $$-0.664803\pi$$
−0.494920 + 0.868939i $$0.664803\pi$$
$$194$$ 0 0
$$195$$ −810.000 −0.297463
$$196$$ 0 0
$$197$$ −1534.00 −0.554787 −0.277393 0.960756i $$-0.589471\pi$$
−0.277393 + 0.960756i $$0.589471\pi$$
$$198$$ 0 0
$$199$$ −4344.00 −1.54743 −0.773714 0.633536i $$-0.781603\pi$$
−0.773714 + 0.633536i $$0.781603\pi$$
$$200$$ 0 0
$$201$$ 564.000 0.197918
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 30.0000 0.0102209
$$206$$ 0 0
$$207$$ −864.000 −0.290107
$$208$$ 0 0
$$209$$ 176.000 0.0582496
$$210$$ 0 0
$$211$$ −1380.00 −0.450252 −0.225126 0.974330i $$-0.572279\pi$$
−0.225126 + 0.974330i $$0.572279\pi$$
$$212$$ 0 0
$$213$$ −1896.00 −0.609914
$$214$$ 0 0
$$215$$ −60.0000 −0.0190324
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1170.00 0.361010
$$220$$ 0 0
$$221$$ −6156.00 −1.87374
$$222$$ 0 0
$$223$$ 5224.00 1.56872 0.784361 0.620305i $$-0.212991\pi$$
0.784361 + 0.620305i $$0.212991\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 3364.00 0.983597 0.491799 0.870709i $$-0.336340\pi$$
0.491799 + 0.870709i $$0.336340\pi$$
$$228$$ 0 0
$$229$$ −3998.00 −1.15369 −0.576846 0.816853i $$-0.695717\pi$$
−0.576846 + 0.816853i $$0.695717\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3590.00 −1.00939 −0.504697 0.863297i $$-0.668396\pi$$
−0.504697 + 0.863297i $$0.668396\pi$$
$$234$$ 0 0
$$235$$ −1000.00 −0.277586
$$236$$ 0 0
$$237$$ 2064.00 0.565701
$$238$$ 0 0
$$239$$ 1104.00 0.298794 0.149397 0.988777i $$-0.452267\pi$$
0.149397 + 0.988777i $$0.452267\pi$$
$$240$$ 0 0
$$241$$ 1618.00 0.432467 0.216233 0.976342i $$-0.430623\pi$$
0.216233 + 0.976342i $$0.430623\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1715.00 0.447214
$$246$$ 0 0
$$247$$ −2376.00 −0.612070
$$248$$ 0 0
$$249$$ −3564.00 −0.907066
$$250$$ 0 0
$$251$$ 5780.00 1.45351 0.726754 0.686898i $$-0.241028\pi$$
0.726754 + 0.686898i $$0.241028\pi$$
$$252$$ 0 0
$$253$$ −384.000 −0.0954224
$$254$$ 0 0
$$255$$ 1710.00 0.419939
$$256$$ 0 0
$$257$$ 2594.00 0.629608 0.314804 0.949157i $$-0.398061\pi$$
0.314804 + 0.949157i $$0.398061\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1206.00 −0.286014
$$262$$ 0 0
$$263$$ −3696.00 −0.866559 −0.433280 0.901260i $$-0.642644\pi$$
−0.433280 + 0.901260i $$0.642644\pi$$
$$264$$ 0 0
$$265$$ 3270.00 0.758017
$$266$$ 0 0
$$267$$ 2082.00 0.477215
$$268$$ 0 0
$$269$$ 2250.00 0.509981 0.254991 0.966944i $$-0.417928\pi$$
0.254991 + 0.966944i $$0.417928\pi$$
$$270$$ 0 0
$$271$$ −2208.00 −0.494932 −0.247466 0.968897i $$-0.579598\pi$$
−0.247466 + 0.968897i $$0.579598\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 100.000 0.0219281
$$276$$ 0 0
$$277$$ 1682.00 0.364843 0.182422 0.983220i $$-0.441606\pi$$
0.182422 + 0.983220i $$0.441606\pi$$
$$278$$ 0 0
$$279$$ 2448.00 0.525297
$$280$$ 0 0
$$281$$ 7306.00 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −8164.00 −1.71484 −0.857419 0.514618i $$-0.827934\pi$$
−0.857419 + 0.514618i $$0.827934\pi$$
$$284$$ 0 0
$$285$$ 660.000 0.137176
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8083.00 1.64523
$$290$$ 0 0
$$291$$ 5178.00 1.04309
$$292$$ 0 0
$$293$$ 514.000 0.102485 0.0512427 0.998686i $$-0.483682\pi$$
0.0512427 + 0.998686i $$0.483682\pi$$
$$294$$ 0 0
$$295$$ −180.000 −0.0355254
$$296$$ 0 0
$$297$$ −108.000 −0.0211003
$$298$$ 0 0
$$299$$ 5184.00 1.00267
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 3546.00 0.672318
$$304$$ 0 0
$$305$$ −2210.00 −0.414899
$$306$$ 0 0
$$307$$ −2476.00 −0.460302 −0.230151 0.973155i $$-0.573922\pi$$
−0.230151 + 0.973155i $$0.573922\pi$$
$$308$$ 0 0
$$309$$ 5904.00 1.08695
$$310$$ 0 0
$$311$$ −2296.00 −0.418631 −0.209315 0.977848i $$-0.567124\pi$$
−0.209315 + 0.977848i $$0.567124\pi$$
$$312$$ 0 0
$$313$$ −9878.00 −1.78383 −0.891913 0.452207i $$-0.850637\pi$$
−0.891913 + 0.452207i $$0.850637\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2138.00 0.378808 0.189404 0.981899i $$-0.439344\pi$$
0.189404 + 0.981899i $$0.439344\pi$$
$$318$$ 0 0
$$319$$ −536.000 −0.0940760
$$320$$ 0 0
$$321$$ −2388.00 −0.415219
$$322$$ 0 0
$$323$$ 5016.00 0.864080
$$324$$ 0 0
$$325$$ −1350.00 −0.230414
$$326$$ 0 0
$$327$$ 1026.00 0.173510
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6460.00 −1.07273 −0.536365 0.843986i $$-0.680203\pi$$
−0.536365 + 0.843986i $$0.680203\pi$$
$$332$$ 0 0
$$333$$ 882.000 0.145145
$$334$$ 0 0
$$335$$ 940.000 0.153307
$$336$$ 0 0
$$337$$ 626.000 0.101188 0.0505941 0.998719i $$-0.483889\pi$$
0.0505941 + 0.998719i $$0.483889\pi$$
$$338$$ 0 0
$$339$$ −342.000 −0.0547932
$$340$$ 0 0
$$341$$ 1088.00 0.172782
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −1440.00 −0.224716
$$346$$ 0 0
$$347$$ 876.000 0.135522 0.0677610 0.997702i $$-0.478414\pi$$
0.0677610 + 0.997702i $$0.478414\pi$$
$$348$$ 0 0
$$349$$ 9850.00 1.51077 0.755385 0.655282i $$-0.227450\pi$$
0.755385 + 0.655282i $$0.227450\pi$$
$$350$$ 0 0
$$351$$ 1458.00 0.221716
$$352$$ 0 0
$$353$$ −8894.00 −1.34102 −0.670510 0.741901i $$-0.733925\pi$$
−0.670510 + 0.741901i $$0.733925\pi$$
$$354$$ 0 0
$$355$$ −3160.00 −0.472438
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1464.00 −0.215228 −0.107614 0.994193i $$-0.534321\pi$$
−0.107614 + 0.994193i $$0.534321\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ 0 0
$$363$$ 3945.00 0.570410
$$364$$ 0 0
$$365$$ 1950.00 0.279637
$$366$$ 0 0
$$367$$ 7016.00 0.997908 0.498954 0.866628i $$-0.333718\pi$$
0.498954 + 0.866628i $$0.333718\pi$$
$$368$$ 0 0
$$369$$ −54.0000 −0.00761823
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1010.00 0.140203 0.0701016 0.997540i $$-0.477668\pi$$
0.0701016 + 0.997540i $$0.477668\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 7236.00 0.988522
$$378$$ 0 0
$$379$$ 4900.00 0.664106 0.332053 0.943261i $$-0.392259\pi$$
0.332053 + 0.943261i $$0.392259\pi$$
$$380$$ 0 0
$$381$$ 7032.00 0.945565
$$382$$ 0 0
$$383$$ −7800.00 −1.04063 −0.520315 0.853974i $$-0.674186\pi$$
−0.520315 + 0.853974i $$0.674186\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 108.000 0.0141859
$$388$$ 0 0
$$389$$ 12258.0 1.59770 0.798850 0.601530i $$-0.205442\pi$$
0.798850 + 0.601530i $$0.205442\pi$$
$$390$$ 0 0
$$391$$ −10944.0 −1.41550
$$392$$ 0 0
$$393$$ 6492.00 0.833278
$$394$$ 0 0
$$395$$ 3440.00 0.438190
$$396$$ 0 0
$$397$$ −5558.00 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1970.00 0.245329 0.122665 0.992448i $$-0.460856\pi$$
0.122665 + 0.992448i $$0.460856\pi$$
$$402$$ 0 0
$$403$$ −14688.0 −1.81554
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ 392.000 0.0477413
$$408$$ 0 0
$$409$$ 15626.0 1.88913 0.944567 0.328318i $$-0.106482\pi$$
0.944567 + 0.328318i $$0.106482\pi$$
$$410$$ 0 0
$$411$$ 8466.00 1.01605
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −5940.00 −0.702610
$$416$$ 0 0
$$417$$ −5916.00 −0.694743
$$418$$ 0 0
$$419$$ −5412.00 −0.631011 −0.315505 0.948924i $$-0.602174\pi$$
−0.315505 + 0.948924i $$0.602174\pi$$
$$420$$ 0 0
$$421$$ 10690.0 1.23753 0.618763 0.785577i $$-0.287634\pi$$
0.618763 + 0.785577i $$0.287634\pi$$
$$422$$ 0 0
$$423$$ 1800.00 0.206901
$$424$$ 0 0
$$425$$ 2850.00 0.325283
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 648.000 0.0729271
$$430$$ 0 0
$$431$$ 14048.0 1.57000 0.784998 0.619498i $$-0.212664\pi$$
0.784998 + 0.619498i $$0.212664\pi$$
$$432$$ 0 0
$$433$$ 17778.0 1.97311 0.986554 0.163433i $$-0.0522567\pi$$
0.986554 + 0.163433i $$0.0522567\pi$$
$$434$$ 0 0
$$435$$ −2010.00 −0.221545
$$436$$ 0 0
$$437$$ −4224.00 −0.462383
$$438$$ 0 0
$$439$$ −7240.00 −0.787122 −0.393561 0.919299i $$-0.628757\pi$$
−0.393561 + 0.919299i $$0.628757\pi$$
$$440$$ 0 0
$$441$$ −3087.00 −0.333333
$$442$$ 0 0
$$443$$ 11740.0 1.25911 0.629553 0.776957i $$-0.283238\pi$$
0.629553 + 0.776957i $$0.283238\pi$$
$$444$$ 0 0
$$445$$ 3470.00 0.369649
$$446$$ 0 0
$$447$$ −4182.00 −0.442510
$$448$$ 0 0
$$449$$ 15234.0 1.60120 0.800598 0.599202i $$-0.204515\pi$$
0.800598 + 0.599202i $$0.204515\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −0.00250580
$$452$$ 0 0
$$453$$ −6648.00 −0.689515
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3866.00 0.395720 0.197860 0.980230i $$-0.436601\pi$$
0.197860 + 0.980230i $$0.436601\pi$$
$$458$$ 0 0
$$459$$ −3078.00 −0.313004
$$460$$ 0 0
$$461$$ 1706.00 0.172356 0.0861782 0.996280i $$-0.472535\pi$$
0.0861782 + 0.996280i $$0.472535\pi$$
$$462$$ 0 0
$$463$$ −3944.00 −0.395882 −0.197941 0.980214i $$-0.563425\pi$$
−0.197941 + 0.980214i $$0.563425\pi$$
$$464$$ 0 0
$$465$$ 4080.00 0.406893
$$466$$ 0 0
$$467$$ −9452.00 −0.936588 −0.468294 0.883573i $$-0.655131\pi$$
−0.468294 + 0.883573i $$0.655131\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2862.00 −0.279987
$$472$$ 0 0
$$473$$ 48.0000 0.00466605
$$474$$ 0 0
$$475$$ 1100.00 0.106256
$$476$$ 0 0
$$477$$ −5886.00 −0.564993
$$478$$ 0 0
$$479$$ 12544.0 1.19656 0.598278 0.801289i $$-0.295852\pi$$
0.598278 + 0.801289i $$0.295852\pi$$
$$480$$ 0 0
$$481$$ −5292.00 −0.501652
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 8630.00 0.807975
$$486$$ 0 0
$$487$$ −7936.00 −0.738428 −0.369214 0.929344i $$-0.620373\pi$$
−0.369214 + 0.929344i $$0.620373\pi$$
$$488$$ 0 0
$$489$$ 10212.0 0.944382
$$490$$ 0 0
$$491$$ −8412.00 −0.773174 −0.386587 0.922253i $$-0.626346\pi$$
−0.386587 + 0.922253i $$0.626346\pi$$
$$492$$ 0 0
$$493$$ −15276.0 −1.39553
$$494$$ 0 0
$$495$$ −180.000 −0.0163442
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15092.0 −1.35393 −0.676965 0.736016i $$-0.736705\pi$$
−0.676965 + 0.736016i $$0.736705\pi$$
$$500$$ 0 0
$$501$$ −2496.00 −0.222581
$$502$$ 0 0
$$503$$ −6112.00 −0.541790 −0.270895 0.962609i $$-0.587320\pi$$
−0.270895 + 0.962609i $$0.587320\pi$$
$$504$$ 0 0
$$505$$ 5910.00 0.520775
$$506$$ 0 0
$$507$$ −2157.00 −0.188946
$$508$$ 0 0
$$509$$ −2534.00 −0.220663 −0.110332 0.993895i $$-0.535191\pi$$
−0.110332 + 0.993895i $$0.535191\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1188.00 −0.102245
$$514$$ 0 0
$$515$$ 9840.00 0.841946
$$516$$ 0 0
$$517$$ 800.000 0.0680541
$$518$$ 0 0
$$519$$ −1086.00 −0.0918499
$$520$$ 0 0
$$521$$ −9894.00 −0.831985 −0.415992 0.909368i $$-0.636566\pi$$
−0.415992 + 0.909368i $$0.636566\pi$$
$$522$$ 0 0
$$523$$ 16172.0 1.35211 0.676054 0.736852i $$-0.263689\pi$$
0.676054 + 0.736852i $$0.263689\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 31008.0 2.56305
$$528$$ 0 0
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ 324.000 0.0264791
$$532$$ 0 0
$$533$$ 324.000 0.0263302
$$534$$ 0 0
$$535$$ −3980.00 −0.321627
$$536$$ 0 0
$$537$$ 9756.00 0.783990
$$538$$ 0 0
$$539$$ −1372.00 −0.109640
$$540$$ 0 0
$$541$$ 6138.00 0.487788 0.243894 0.969802i $$-0.421575\pi$$
0.243894 + 0.969802i $$0.421575\pi$$
$$542$$ 0 0
$$543$$ 9258.00 0.731674
$$544$$ 0 0
$$545$$ 1710.00 0.134401
$$546$$ 0 0
$$547$$ −21852.0 −1.70809 −0.854044 0.520201i $$-0.825857\pi$$
−0.854044 + 0.520201i $$0.825857\pi$$
$$548$$ 0 0
$$549$$ 3978.00 0.309248
$$550$$ 0 0
$$551$$ −5896.00 −0.455859
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1470.00 0.112429
$$556$$ 0 0
$$557$$ 1962.00 0.149251 0.0746253 0.997212i $$-0.476224\pi$$
0.0746253 + 0.997212i $$0.476224\pi$$
$$558$$ 0 0
$$559$$ −648.000 −0.0490295
$$560$$ 0 0
$$561$$ −1368.00 −0.102954
$$562$$ 0 0
$$563$$ −10876.0 −0.814154 −0.407077 0.913394i $$-0.633452\pi$$
−0.407077 + 0.913394i $$0.633452\pi$$
$$564$$ 0 0
$$565$$ −570.000 −0.0424426
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 5610.00 0.413328 0.206664 0.978412i $$-0.433739\pi$$
0.206664 + 0.978412i $$0.433739\pi$$
$$570$$ 0 0
$$571$$ 5076.00 0.372021 0.186010 0.982548i $$-0.440444\pi$$
0.186010 + 0.982548i $$0.440444\pi$$
$$572$$ 0 0
$$573$$ −12240.0 −0.892379
$$574$$ 0 0
$$575$$ −2400.00 −0.174064
$$576$$ 0 0
$$577$$ −6526.00 −0.470851 −0.235425 0.971892i $$-0.575648\pi$$
−0.235425 + 0.971892i $$0.575648\pi$$
$$578$$ 0 0
$$579$$ 7962.00 0.571484
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2616.00 −0.185838
$$584$$ 0 0
$$585$$ 2430.00 0.171740
$$586$$ 0 0
$$587$$ 2332.00 0.163973 0.0819863 0.996633i $$-0.473874\pi$$
0.0819863 + 0.996633i $$0.473874\pi$$
$$588$$ 0 0
$$589$$ 11968.0 0.837237
$$590$$ 0 0
$$591$$ 4602.00 0.320306
$$592$$ 0 0
$$593$$ −9582.00 −0.663551 −0.331775 0.943358i $$-0.607648\pi$$
−0.331775 + 0.943358i $$0.607648\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13032.0 0.893407
$$598$$ 0 0
$$599$$ 17624.0 1.20217 0.601083 0.799187i $$-0.294736\pi$$
0.601083 + 0.799187i $$0.294736\pi$$
$$600$$ 0 0
$$601$$ −21238.0 −1.44146 −0.720729 0.693217i $$-0.756193\pi$$
−0.720729 + 0.693217i $$0.756193\pi$$
$$602$$ 0 0
$$603$$ −1692.00 −0.114268
$$604$$ 0 0
$$605$$ 6575.00 0.441838
$$606$$ 0 0
$$607$$ −13000.0 −0.869281 −0.434641 0.900604i $$-0.643125\pi$$
−0.434641 + 0.900604i $$0.643125\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10800.0 −0.715092
$$612$$ 0 0
$$613$$ −9214.00 −0.607096 −0.303548 0.952816i $$-0.598171\pi$$
−0.303548 + 0.952816i $$0.598171\pi$$
$$614$$ 0 0
$$615$$ −90.0000 −0.00590106
$$616$$ 0 0
$$617$$ 4474.00 0.291923 0.145961 0.989290i $$-0.453372\pi$$
0.145961 + 0.989290i $$0.453372\pi$$
$$618$$ 0 0
$$619$$ −12556.0 −0.815296 −0.407648 0.913139i $$-0.633651\pi$$
−0.407648 + 0.913139i $$0.633651\pi$$
$$620$$ 0 0
$$621$$ 2592.00 0.167493
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −528.000 −0.0336304
$$628$$ 0 0
$$629$$ 11172.0 0.708198
$$630$$ 0 0
$$631$$ −26936.0 −1.69937 −0.849687 0.527287i $$-0.823209\pi$$
−0.849687 + 0.527287i $$0.823209\pi$$
$$632$$ 0 0
$$633$$ 4140.00 0.259953
$$634$$ 0 0
$$635$$ 11720.0 0.732432
$$636$$ 0 0
$$637$$ 18522.0 1.15207
$$638$$ 0 0
$$639$$ 5688.00 0.352134
$$640$$ 0 0
$$641$$ −19134.0 −1.17901 −0.589507 0.807764i $$-0.700678\pi$$
−0.589507 + 0.807764i $$0.700678\pi$$
$$642$$ 0 0
$$643$$ 12436.0 0.762718 0.381359 0.924427i $$-0.375456\pi$$
0.381359 + 0.924427i $$0.375456\pi$$
$$644$$ 0 0
$$645$$ 180.000 0.0109884
$$646$$ 0 0
$$647$$ 2784.00 0.169166 0.0845829 0.996416i $$-0.473044\pi$$
0.0845829 + 0.996416i $$0.473044\pi$$
$$648$$ 0 0
$$649$$ 144.000 0.00870954
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7318.00 −0.438554 −0.219277 0.975663i $$-0.570370\pi$$
−0.219277 + 0.975663i $$0.570370\pi$$
$$654$$ 0 0
$$655$$ 10820.0 0.645454
$$656$$ 0 0
$$657$$ −3510.00 −0.208429
$$658$$ 0 0
$$659$$ 8108.00 0.479276 0.239638 0.970862i $$-0.422971\pi$$
0.239638 + 0.970862i $$0.422971\pi$$
$$660$$ 0 0
$$661$$ −1230.00 −0.0723774 −0.0361887 0.999345i $$-0.511522\pi$$
−0.0361887 + 0.999345i $$0.511522\pi$$
$$662$$ 0 0
$$663$$ 18468.0 1.08181
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12864.0 0.746771
$$668$$ 0 0
$$669$$ −15672.0 −0.905702
$$670$$ 0 0
$$671$$ 1768.00 0.101718
$$672$$ 0 0
$$673$$ −14078.0 −0.806340 −0.403170 0.915125i $$-0.632092\pi$$
−0.403170 + 0.915125i $$0.632092\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −25246.0 −1.43321 −0.716605 0.697480i $$-0.754305\pi$$
−0.716605 + 0.697480i $$0.754305\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −10092.0 −0.567880
$$682$$ 0 0
$$683$$ 24332.0 1.36316 0.681580 0.731744i $$-0.261293\pi$$
0.681580 + 0.731744i $$0.261293\pi$$
$$684$$ 0 0
$$685$$ 14110.0 0.787030
$$686$$ 0 0
$$687$$ 11994.0 0.666084
$$688$$ 0 0
$$689$$ 35316.0 1.95273
$$690$$ 0 0
$$691$$ 19036.0 1.04799 0.523997 0.851720i $$-0.324440\pi$$
0.523997 + 0.851720i $$0.324440\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −9860.00 −0.538145
$$696$$ 0 0
$$697$$ −684.000 −0.0371712
$$698$$ 0 0
$$699$$ 10770.0 0.582774
$$700$$ 0 0
$$701$$ −28806.0 −1.55205 −0.776025 0.630702i $$-0.782767\pi$$
−0.776025 + 0.630702i $$0.782767\pi$$
$$702$$ 0 0
$$703$$ 4312.00 0.231337
$$704$$ 0 0
$$705$$ 3000.00 0.160265
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 25090.0 1.32902 0.664510 0.747280i $$-0.268640\pi$$
0.664510 + 0.747280i $$0.268640\pi$$
$$710$$ 0 0
$$711$$ −6192.00 −0.326608
$$712$$ 0 0
$$713$$ −26112.0 −1.37153
$$714$$ 0 0
$$715$$ 1080.00 0.0564891
$$716$$ 0 0
$$717$$ −3312.00 −0.172509
$$718$$ 0 0
$$719$$ −36432.0 −1.88969 −0.944843 0.327523i $$-0.893786\pi$$
−0.944843 + 0.327523i $$0.893786\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −4854.00 −0.249685
$$724$$ 0 0
$$725$$ −3350.00 −0.171608
$$726$$ 0 0
$$727$$ −21616.0 −1.10274 −0.551371 0.834260i $$-0.685895\pi$$
−0.551371 + 0.834260i $$0.685895\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1368.00 0.0692166
$$732$$ 0 0
$$733$$ −28102.0 −1.41606 −0.708029 0.706183i $$-0.750416\pi$$
−0.708029 + 0.706183i $$0.750416\pi$$
$$734$$ 0 0
$$735$$ −5145.00 −0.258199
$$736$$ 0 0
$$737$$ −752.000 −0.0375852
$$738$$ 0 0
$$739$$ 764.000 0.0380300 0.0190150 0.999819i $$-0.493947\pi$$
0.0190150 + 0.999819i $$0.493947\pi$$
$$740$$ 0 0
$$741$$ 7128.00 0.353379
$$742$$ 0 0
$$743$$ −6256.00 −0.308897 −0.154448 0.988001i $$-0.549360\pi$$
−0.154448 + 0.988001i $$0.549360\pi$$
$$744$$ 0 0
$$745$$ −6970.00 −0.342766
$$746$$ 0 0
$$747$$ 10692.0 0.523695
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1184.00 −0.0575297 −0.0287648 0.999586i $$-0.509157\pi$$
−0.0287648 + 0.999586i $$0.509157\pi$$
$$752$$ 0 0
$$753$$ −17340.0 −0.839183
$$754$$ 0 0
$$755$$ −11080.0 −0.534096
$$756$$ 0 0
$$757$$ −26446.0 −1.26974 −0.634872 0.772617i $$-0.718947\pi$$
−0.634872 + 0.772617i $$0.718947\pi$$
$$758$$ 0 0
$$759$$ 1152.00 0.0550922
$$760$$ 0 0
$$761$$ 36778.0 1.75191 0.875954 0.482395i $$-0.160233\pi$$
0.875954 + 0.482395i $$0.160233\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −5130.00 −0.242452
$$766$$ 0 0
$$767$$ −1944.00 −0.0915173
$$768$$ 0 0
$$769$$ −10302.0 −0.483094 −0.241547 0.970389i $$-0.577655\pi$$
−0.241547 + 0.970389i $$0.577655\pi$$
$$770$$ 0 0
$$771$$ −7782.00 −0.363504
$$772$$ 0 0
$$773$$ 4674.00 0.217480 0.108740 0.994070i $$-0.465318\pi$$
0.108740 + 0.994070i $$0.465318\pi$$
$$774$$ 0 0
$$775$$ 6800.00 0.315178
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −264.000 −0.0121422
$$780$$ 0 0
$$781$$ 2528.00 0.115825
$$782$$ 0 0
$$783$$ 3618.00 0.165130
$$784$$ 0 0
$$785$$ −4770.00 −0.216877
$$786$$ 0 0
$$787$$ −23084.0 −1.04556 −0.522780 0.852468i $$-0.675105\pi$$
−0.522780 + 0.852468i $$0.675105\pi$$
$$788$$ 0 0
$$789$$ 11088.0 0.500308
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −23868.0 −1.06882
$$794$$ 0 0
$$795$$ −9810.00 −0.437641
$$796$$ 0 0
$$797$$ −10694.0 −0.475283 −0.237642 0.971353i $$-0.576374\pi$$
−0.237642 + 0.971353i $$0.576374\pi$$
$$798$$ 0 0
$$799$$ 22800.0 1.00952
$$800$$ 0 0
$$801$$ −6246.00 −0.275520
$$802$$ 0 0
$$803$$ −1560.00 −0.0685569
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6750.00 −0.294438
$$808$$ 0 0
$$809$$ 9594.00 0.416943 0.208472 0.978028i $$-0.433151\pi$$
0.208472 + 0.978028i $$0.433151\pi$$
$$810$$ 0 0
$$811$$ 10244.0 0.443546 0.221773 0.975098i $$-0.428816\pi$$
0.221773 + 0.975098i $$0.428816\pi$$
$$812$$ 0 0
$$813$$ 6624.00 0.285749
$$814$$ 0 0
$$815$$ 17020.0 0.731515
$$816$$ 0 0
$$817$$ 528.000 0.0226100
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1390.00 −0.0590881 −0.0295441 0.999563i $$-0.509406\pi$$
−0.0295441 + 0.999563i $$0.509406\pi$$
$$822$$ 0 0
$$823$$ 8448.00 0.357811 0.178906 0.983866i $$-0.442744\pi$$
0.178906 + 0.983866i $$0.442744\pi$$
$$824$$ 0 0
$$825$$ −300.000 −0.0126602
$$826$$ 0 0
$$827$$ 41484.0 1.74430 0.872152 0.489234i $$-0.162724\pi$$
0.872152 + 0.489234i $$0.162724\pi$$
$$828$$ 0 0
$$829$$ 31610.0 1.32432 0.662160 0.749363i $$-0.269640\pi$$
0.662160 + 0.749363i $$0.269640\pi$$
$$830$$ 0 0
$$831$$ −5046.00 −0.210642
$$832$$ 0 0
$$833$$ −39102.0 −1.62642
$$834$$ 0 0
$$835$$ −4160.00 −0.172410
$$836$$ 0 0
$$837$$ −7344.00 −0.303280
$$838$$ 0 0
$$839$$ −38264.0 −1.57452 −0.787259 0.616623i $$-0.788500\pi$$
−0.787259 + 0.616623i $$0.788500\pi$$
$$840$$ 0 0
$$841$$ −6433.00 −0.263766
$$842$$ 0 0
$$843$$ −21918.0 −0.895488
$$844$$ 0 0
$$845$$ −3595.00 −0.146357
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 24492.0 0.990063
$$850$$ 0 0
$$851$$ −9408.00 −0.378968
$$852$$ 0 0
$$853$$ −30350.0 −1.21825 −0.609123 0.793076i $$-0.708479\pi$$
−0.609123 + 0.793076i $$0.708479\pi$$
$$854$$ 0 0
$$855$$ −1980.00 −0.0791983
$$856$$ 0 0
$$857$$ −12566.0 −0.500871 −0.250435 0.968133i $$-0.580574\pi$$
−0.250435 + 0.968133i $$0.580574\pi$$
$$858$$ 0 0
$$859$$ 11812.0 0.469174 0.234587 0.972095i $$-0.424626\pi$$
0.234587 + 0.972095i $$0.424626\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 31496.0 1.24234 0.621168 0.783677i $$-0.286658\pi$$
0.621168 + 0.783677i $$0.286658\pi$$
$$864$$ 0 0
$$865$$ −1810.00 −0.0711466
$$866$$ 0 0
$$867$$ −24249.0 −0.949872
$$868$$ 0 0
$$869$$ −2752.00 −0.107428
$$870$$ 0 0
$$871$$ 10152.0 0.394934
$$872$$ 0 0
$$873$$ −15534.0 −0.602229
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7414.00 −0.285465 −0.142733 0.989761i $$-0.545589\pi$$
−0.142733 + 0.989761i $$0.545589\pi$$
$$878$$ 0 0
$$879$$ −1542.00 −0.0591699
$$880$$ 0 0
$$881$$ −22190.0 −0.848581 −0.424291 0.905526i $$-0.639477\pi$$
−0.424291 + 0.905526i $$0.639477\pi$$
$$882$$ 0 0
$$883$$ −10172.0 −0.387673 −0.193836 0.981034i $$-0.562093\pi$$
−0.193836 + 0.981034i $$0.562093\pi$$
$$884$$ 0 0
$$885$$ 540.000 0.0205106
$$886$$ 0 0
$$887$$ 20784.0 0.786763 0.393381 0.919375i $$-0.371305\pi$$
0.393381 + 0.919375i $$0.371305\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 324.000 0.0121823
$$892$$ 0 0
$$893$$ 8800.00 0.329766
$$894$$ 0 0
$$895$$ 16260.0 0.607276
$$896$$ 0 0
$$897$$ −15552.0 −0.578892
$$898$$ 0 0
$$899$$ −36448.0 −1.35218
$$900$$ 0 0
$$901$$ −74556.0 −2.75674
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 15430.0 0.566752
$$906$$ 0 0
$$907$$ −7652.00 −0.280133 −0.140066 0.990142i $$-0.544732\pi$$
−0.140066 + 0.990142i $$0.544732\pi$$
$$908$$ 0 0
$$909$$ −10638.0 −0.388163
$$910$$ 0 0
$$911$$ −19296.0 −0.701762 −0.350881 0.936420i $$-0.614118\pi$$
−0.350881 + 0.936420i $$0.614118\pi$$
$$912$$ 0 0
$$913$$ 4752.00 0.172254
$$914$$ 0 0
$$915$$ 6630.00 0.239542
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 35896.0 1.28847 0.644233 0.764830i $$-0.277177\pi$$
0.644233 + 0.764830i $$0.277177\pi$$
$$920$$ 0 0
$$921$$ 7428.00 0.265756
$$922$$ 0 0
$$923$$ −34128.0 −1.21705
$$924$$ 0 0
$$925$$ 2450.00 0.0870870
$$926$$ 0 0
$$927$$ −17712.0 −0.627550
$$928$$ 0 0
$$929$$ −16350.0 −0.577423 −0.288712 0.957416i $$-0.593227\pi$$
−0.288712 + 0.957416i $$0.593227\pi$$
$$930$$ 0 0
$$931$$ −15092.0 −0.531279
$$932$$ 0 0
$$933$$ 6888.00 0.241697
$$934$$ 0 0
$$935$$ −2280.00 −0.0797476
$$936$$ 0 0
$$937$$ −19686.0 −0.686354 −0.343177 0.939271i $$-0.611503\pi$$
−0.343177 + 0.939271i $$0.611503\pi$$
$$938$$ 0 0
$$939$$ 29634.0 1.02989
$$940$$ 0 0
$$941$$ −56246.0 −1.94853 −0.974265 0.225405i $$-0.927630\pi$$
−0.974265 + 0.225405i $$0.927630\pi$$
$$942$$ 0 0
$$943$$ 576.000 0.0198909
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −11436.0 −0.392418 −0.196209 0.980562i $$-0.562863\pi$$
−0.196209 + 0.980562i $$0.562863\pi$$
$$948$$ 0 0
$$949$$ 21060.0 0.720376
$$950$$ 0 0
$$951$$ −6414.00 −0.218705
$$952$$ 0 0
$$953$$ −22582.0 −0.767579 −0.383789 0.923421i $$-0.625381\pi$$
−0.383789 + 0.923421i $$0.625381\pi$$
$$954$$ 0 0
$$955$$ −20400.0 −0.691234
$$956$$ 0 0
$$957$$ 1608.00 0.0543148
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 44193.0 1.48343
$$962$$ 0 0
$$963$$ 7164.00 0.239727
$$964$$ 0 0
$$965$$ 13270.0 0.442670
$$966$$ 0 0
$$967$$ 2112.00 0.0702351 0.0351175 0.999383i $$-0.488819\pi$$
0.0351175 + 0.999383i $$0.488819\pi$$
$$968$$ 0 0
$$969$$ −15048.0 −0.498877
$$970$$ 0 0
$$971$$ −47964.0 −1.58521 −0.792605 0.609736i $$-0.791275\pi$$
−0.792605 + 0.609736i $$0.791275\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 4050.00 0.133030
$$976$$ 0 0
$$977$$ −10510.0 −0.344160 −0.172080 0.985083i $$-0.555049\pi$$
−0.172080 + 0.985083i $$0.555049\pi$$
$$978$$ 0 0
$$979$$ −2776.00 −0.0906245
$$980$$ 0 0
$$981$$ −3078.00 −0.100176
$$982$$ 0 0
$$983$$ −11488.0 −0.372747 −0.186373 0.982479i $$-0.559673\pi$$
−0.186373 + 0.982479i $$0.559673\pi$$
$$984$$ 0 0
$$985$$ 7670.00 0.248108
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1152.00 −0.0370389
$$990$$ 0 0
$$991$$ 23120.0 0.741101 0.370550 0.928812i $$-0.379169\pi$$
0.370550 + 0.928812i $$0.379169\pi$$
$$992$$ 0 0
$$993$$ 19380.0 0.619341
$$994$$ 0 0
$$995$$ 21720.0 0.692030
$$996$$ 0 0
$$997$$ −30078.0 −0.955446 −0.477723 0.878510i $$-0.658538\pi$$
−0.477723 + 0.878510i $$0.658538\pi$$
$$998$$ 0 0
$$999$$ −2646.00 −0.0837995
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.e.1.1 1
4.3 odd 2 960.4.a.x.1.1 1
8.3 odd 2 120.4.a.d.1.1 1
8.5 even 2 240.4.a.k.1.1 1
24.5 odd 2 720.4.a.i.1.1 1
24.11 even 2 360.4.a.c.1.1 1
40.3 even 4 600.4.f.d.49.1 2
40.13 odd 4 1200.4.f.l.49.2 2
40.19 odd 2 600.4.a.m.1.1 1
40.27 even 4 600.4.f.d.49.2 2
40.29 even 2 1200.4.a.j.1.1 1
40.37 odd 4 1200.4.f.l.49.1 2
120.59 even 2 1800.4.a.s.1.1 1
120.83 odd 4 1800.4.f.m.649.2 2
120.107 odd 4 1800.4.f.m.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.d.1.1 1 8.3 odd 2
240.4.a.k.1.1 1 8.5 even 2
360.4.a.c.1.1 1 24.11 even 2
600.4.a.m.1.1 1 40.19 odd 2
600.4.f.d.49.1 2 40.3 even 4
600.4.f.d.49.2 2 40.27 even 4
720.4.a.i.1.1 1 24.5 odd 2
960.4.a.e.1.1 1 1.1 even 1 trivial
960.4.a.x.1.1 1 4.3 odd 2
1200.4.a.j.1.1 1 40.29 even 2
1200.4.f.l.49.1 2 40.37 odd 4
1200.4.f.l.49.2 2 40.13 odd 4
1800.4.a.s.1.1 1 120.59 even 2
1800.4.f.m.649.1 2 120.107 odd 4
1800.4.f.m.649.2 2 120.83 odd 4