# Properties

 Label 600.4.a.b.1.1 Level $600$ Weight $4$ Character 600.1 Self dual yes Analytic conductor $35.401$ 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] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, 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("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.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: $$35.4011460034$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -10.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -10.0000 q^{7} +9.00000 q^{9} -46.0000 q^{11} +34.0000 q^{13} -66.0000 q^{17} +104.000 q^{19} +30.0000 q^{21} -164.000 q^{23} -27.0000 q^{27} +224.000 q^{29} -72.0000 q^{31} +138.000 q^{33} +22.0000 q^{37} -102.000 q^{39} +194.000 q^{41} -108.000 q^{43} +480.000 q^{47} -243.000 q^{49} +198.000 q^{51} -286.000 q^{53} -312.000 q^{57} +426.000 q^{59} +698.000 q^{61} -90.0000 q^{63} -328.000 q^{67} +492.000 q^{69} +188.000 q^{71} +740.000 q^{73} +460.000 q^{77} +1168.00 q^{79} +81.0000 q^{81} -412.000 q^{83} -672.000 q^{87} +1206.00 q^{89} -340.000 q^{91} +216.000 q^{93} +1384.00 q^{97} -414.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −10.0000 −0.539949 −0.269975 0.962867i $$-0.587015\pi$$
−0.269975 + 0.962867i $$0.587015\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −46.0000 −1.26087 −0.630433 0.776244i $$-0.717123\pi$$
−0.630433 + 0.776244i $$0.717123\pi$$
$$12$$ 0 0
$$13$$ 34.0000 0.725377 0.362689 0.931910i $$-0.381859\pi$$
0.362689 + 0.931910i $$0.381859\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −66.0000 −0.941609 −0.470804 0.882238i $$-0.656036\pi$$
−0.470804 + 0.882238i $$0.656036\pi$$
$$18$$ 0 0
$$19$$ 104.000 1.25575 0.627875 0.778314i $$-0.283925\pi$$
0.627875 + 0.778314i $$0.283925\pi$$
$$20$$ 0 0
$$21$$ 30.0000 0.311740
$$22$$ 0 0
$$23$$ −164.000 −1.48680 −0.743399 0.668848i $$-0.766788\pi$$
−0.743399 + 0.668848i $$0.766788\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 224.000 1.43434 0.717168 0.696900i $$-0.245438\pi$$
0.717168 + 0.696900i $$0.245438\pi$$
$$30$$ 0 0
$$31$$ −72.0000 −0.417148 −0.208574 0.978007i $$-0.566882\pi$$
−0.208574 + 0.978007i $$0.566882\pi$$
$$32$$ 0 0
$$33$$ 138.000 0.727961
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 22.0000 0.0977507 0.0488754 0.998805i $$-0.484436\pi$$
0.0488754 + 0.998805i $$0.484436\pi$$
$$38$$ 0 0
$$39$$ −102.000 −0.418797
$$40$$ 0 0
$$41$$ 194.000 0.738969 0.369484 0.929237i $$-0.379534\pi$$
0.369484 + 0.929237i $$0.379534\pi$$
$$42$$ 0 0
$$43$$ −108.000 −0.383020 −0.191510 0.981491i $$-0.561338\pi$$
−0.191510 + 0.981491i $$0.561338\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 480.000 1.48969 0.744843 0.667240i $$-0.232525\pi$$
0.744843 + 0.667240i $$0.232525\pi$$
$$48$$ 0 0
$$49$$ −243.000 −0.708455
$$50$$ 0 0
$$51$$ 198.000 0.543638
$$52$$ 0 0
$$53$$ −286.000 −0.741229 −0.370614 0.928787i $$-0.620853\pi$$
−0.370614 + 0.928787i $$0.620853\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −312.000 −0.725007
$$58$$ 0 0
$$59$$ 426.000 0.940008 0.470004 0.882664i $$-0.344252\pi$$
0.470004 + 0.882664i $$0.344252\pi$$
$$60$$ 0 0
$$61$$ 698.000 1.46508 0.732539 0.680725i $$-0.238335\pi$$
0.732539 + 0.680725i $$0.238335\pi$$
$$62$$ 0 0
$$63$$ −90.0000 −0.179983
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −328.000 −0.598083 −0.299042 0.954240i $$-0.596667\pi$$
−0.299042 + 0.954240i $$0.596667\pi$$
$$68$$ 0 0
$$69$$ 492.000 0.858403
$$70$$ 0 0
$$71$$ 188.000 0.314246 0.157123 0.987579i $$-0.449778\pi$$
0.157123 + 0.987579i $$0.449778\pi$$
$$72$$ 0 0
$$73$$ 740.000 1.18644 0.593222 0.805039i $$-0.297856\pi$$
0.593222 + 0.805039i $$0.297856\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 460.000 0.680803
$$78$$ 0 0
$$79$$ 1168.00 1.66342 0.831711 0.555209i $$-0.187362\pi$$
0.831711 + 0.555209i $$0.187362\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −412.000 −0.544854 −0.272427 0.962176i $$-0.587826\pi$$
−0.272427 + 0.962176i $$0.587826\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −672.000 −0.828115
$$88$$ 0 0
$$89$$ 1206.00 1.43636 0.718178 0.695859i $$-0.244976\pi$$
0.718178 + 0.695859i $$0.244976\pi$$
$$90$$ 0 0
$$91$$ −340.000 −0.391667
$$92$$ 0 0
$$93$$ 216.000 0.240840
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1384.00 1.44870 0.724350 0.689432i $$-0.242140\pi$$
0.724350 + 0.689432i $$0.242140\pi$$
$$98$$ 0 0
$$99$$ −414.000 −0.420289
$$100$$ 0 0
$$101$$ −1128.00 −1.11129 −0.555645 0.831420i $$-0.687528\pi$$
−0.555645 + 0.831420i $$0.687528\pi$$
$$102$$ 0 0
$$103$$ 758.000 0.725126 0.362563 0.931959i $$-0.381902\pi$$
0.362563 + 0.931959i $$0.381902\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1324.00 −1.19622 −0.598112 0.801413i $$-0.704082\pi$$
−0.598112 + 0.801413i $$0.704082\pi$$
$$108$$ 0 0
$$109$$ 1602.00 1.40774 0.703871 0.710328i $$-0.251454\pi$$
0.703871 + 0.710328i $$0.251454\pi$$
$$110$$ 0 0
$$111$$ −66.0000 −0.0564364
$$112$$ 0 0
$$113$$ 2074.00 1.72660 0.863299 0.504693i $$-0.168394\pi$$
0.863299 + 0.504693i $$0.168394\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 306.000 0.241792
$$118$$ 0 0
$$119$$ 660.000 0.508421
$$120$$ 0 0
$$121$$ 785.000 0.589782
$$122$$ 0 0
$$123$$ −582.000 −0.426644
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 534.000 0.373109 0.186554 0.982445i $$-0.440268\pi$$
0.186554 + 0.982445i $$0.440268\pi$$
$$128$$ 0 0
$$129$$ 324.000 0.221137
$$130$$ 0 0
$$131$$ 1806.00 1.20451 0.602256 0.798303i $$-0.294269\pi$$
0.602256 + 0.798303i $$0.294269\pi$$
$$132$$ 0 0
$$133$$ −1040.00 −0.678041
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1822.00 −1.13623 −0.568117 0.822948i $$-0.692328\pi$$
−0.568117 + 0.822948i $$0.692328\pi$$
$$138$$ 0 0
$$139$$ 532.000 0.324631 0.162315 0.986739i $$-0.448104\pi$$
0.162315 + 0.986739i $$0.448104\pi$$
$$140$$ 0 0
$$141$$ −1440.00 −0.860070
$$142$$ 0 0
$$143$$ −1564.00 −0.914603
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 729.000 0.409027
$$148$$ 0 0
$$149$$ −1284.00 −0.705969 −0.352984 0.935629i $$-0.614833\pi$$
−0.352984 + 0.935629i $$0.614833\pi$$
$$150$$ 0 0
$$151$$ 184.000 0.0991636 0.0495818 0.998770i $$-0.484211\pi$$
0.0495818 + 0.998770i $$0.484211\pi$$
$$152$$ 0 0
$$153$$ −594.000 −0.313870
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3746.00 1.90423 0.952113 0.305748i $$-0.0989064\pi$$
0.952113 + 0.305748i $$0.0989064\pi$$
$$158$$ 0 0
$$159$$ 858.000 0.427949
$$160$$ 0 0
$$161$$ 1640.00 0.802796
$$162$$ 0 0
$$163$$ −1504.00 −0.722714 −0.361357 0.932428i $$-0.617686\pi$$
−0.361357 + 0.932428i $$0.617686\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3012.00 −1.39566 −0.697831 0.716262i $$-0.745851\pi$$
−0.697831 + 0.716262i $$0.745851\pi$$
$$168$$ 0 0
$$169$$ −1041.00 −0.473828
$$170$$ 0 0
$$171$$ 936.000 0.418583
$$172$$ 0 0
$$173$$ 438.000 0.192489 0.0962443 0.995358i $$-0.469317\pi$$
0.0962443 + 0.995358i $$0.469317\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1278.00 −0.542714
$$178$$ 0 0
$$179$$ −1462.00 −0.610475 −0.305237 0.952276i $$-0.598736\pi$$
−0.305237 + 0.952276i $$0.598736\pi$$
$$180$$ 0 0
$$181$$ 586.000 0.240647 0.120323 0.992735i $$-0.461607\pi$$
0.120323 + 0.992735i $$0.461607\pi$$
$$182$$ 0 0
$$183$$ −2094.00 −0.845863
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3036.00 1.18724
$$188$$ 0 0
$$189$$ 270.000 0.103913
$$190$$ 0 0
$$191$$ 60.0000 0.0227301 0.0113650 0.999935i $$-0.496382\pi$$
0.0113650 + 0.999935i $$0.496382\pi$$
$$192$$ 0 0
$$193$$ 4676.00 1.74397 0.871984 0.489534i $$-0.162833\pi$$
0.871984 + 0.489534i $$0.162833\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2286.00 −0.826755 −0.413378 0.910560i $$-0.635651\pi$$
−0.413378 + 0.910560i $$0.635651\pi$$
$$198$$ 0 0
$$199$$ −3536.00 −1.25960 −0.629800 0.776757i $$-0.716863\pi$$
−0.629800 + 0.776757i $$0.716863\pi$$
$$200$$ 0 0
$$201$$ 984.000 0.345304
$$202$$ 0 0
$$203$$ −2240.00 −0.774469
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1476.00 −0.495599
$$208$$ 0 0
$$209$$ −4784.00 −1.58333
$$210$$ 0 0
$$211$$ −3500.00 −1.14194 −0.570971 0.820970i $$-0.693433\pi$$
−0.570971 + 0.820970i $$0.693433\pi$$
$$212$$ 0 0
$$213$$ −564.000 −0.181430
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 720.000 0.225239
$$218$$ 0 0
$$219$$ −2220.00 −0.684994
$$220$$ 0 0
$$221$$ −2244.00 −0.683022
$$222$$ 0 0
$$223$$ −5874.00 −1.76391 −0.881955 0.471333i $$-0.843773\pi$$
−0.881955 + 0.471333i $$0.843773\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 124.000 0.0362563 0.0181281 0.999836i $$-0.494229\pi$$
0.0181281 + 0.999836i $$0.494229\pi$$
$$228$$ 0 0
$$229$$ −1362.00 −0.393028 −0.196514 0.980501i $$-0.562962\pi$$
−0.196514 + 0.980501i $$0.562962\pi$$
$$230$$ 0 0
$$231$$ −1380.00 −0.393062
$$232$$ 0 0
$$233$$ 3870.00 1.08812 0.544060 0.839046i $$-0.316886\pi$$
0.544060 + 0.839046i $$0.316886\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −3504.00 −0.960377
$$238$$ 0 0
$$239$$ 6116.00 1.65528 0.827638 0.561262i $$-0.189684\pi$$
0.827638 + 0.561262i $$0.189684\pi$$
$$240$$ 0 0
$$241$$ −5962.00 −1.59355 −0.796776 0.604274i $$-0.793463\pi$$
−0.796776 + 0.604274i $$0.793463\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3536.00 0.910892
$$248$$ 0 0
$$249$$ 1236.00 0.314572
$$250$$ 0 0
$$251$$ −1490.00 −0.374693 −0.187347 0.982294i $$-0.559989\pi$$
−0.187347 + 0.982294i $$0.559989\pi$$
$$252$$ 0 0
$$253$$ 7544.00 1.87465
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5394.00 1.30922 0.654608 0.755969i $$-0.272834\pi$$
0.654608 + 0.755969i $$0.272834\pi$$
$$258$$ 0 0
$$259$$ −220.000 −0.0527804
$$260$$ 0 0
$$261$$ 2016.00 0.478112
$$262$$ 0 0
$$263$$ 636.000 0.149116 0.0745579 0.997217i $$-0.476245\pi$$
0.0745579 + 0.997217i $$0.476245\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −3618.00 −0.829281
$$268$$ 0 0
$$269$$ −3360.00 −0.761572 −0.380786 0.924663i $$-0.624347\pi$$
−0.380786 + 0.924663i $$0.624347\pi$$
$$270$$ 0 0
$$271$$ 5768.00 1.29292 0.646459 0.762948i $$-0.276249\pi$$
0.646459 + 0.762948i $$0.276249\pi$$
$$272$$ 0 0
$$273$$ 1020.00 0.226129
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1398.00 0.303241 0.151620 0.988439i $$-0.451551\pi$$
0.151620 + 0.988439i $$0.451551\pi$$
$$278$$ 0 0
$$279$$ −648.000 −0.139049
$$280$$ 0 0
$$281$$ −4194.00 −0.890367 −0.445183 0.895439i $$-0.646862\pi$$
−0.445183 + 0.895439i $$0.646862\pi$$
$$282$$ 0 0
$$283$$ 8256.00 1.73416 0.867082 0.498166i $$-0.165993\pi$$
0.867082 + 0.498166i $$0.165993\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1940.00 −0.399006
$$288$$ 0 0
$$289$$ −557.000 −0.113373
$$290$$ 0 0
$$291$$ −4152.00 −0.836407
$$292$$ 0 0
$$293$$ −5534.00 −1.10341 −0.551706 0.834039i $$-0.686023\pi$$
−0.551706 + 0.834039i $$0.686023\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1242.00 0.242654
$$298$$ 0 0
$$299$$ −5576.00 −1.07849
$$300$$ 0 0
$$301$$ 1080.00 0.206811
$$302$$ 0 0
$$303$$ 3384.00 0.641603
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 484.000 0.0899783 0.0449892 0.998987i $$-0.485675\pi$$
0.0449892 + 0.998987i $$0.485675\pi$$
$$308$$ 0 0
$$309$$ −2274.00 −0.418652
$$310$$ 0 0
$$311$$ −2724.00 −0.496668 −0.248334 0.968674i $$-0.579883\pi$$
−0.248334 + 0.968674i $$0.579883\pi$$
$$312$$ 0 0
$$313$$ −5308.00 −0.958549 −0.479275 0.877665i $$-0.659100\pi$$
−0.479275 + 0.877665i $$0.659100\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4218.00 −0.747339 −0.373670 0.927562i $$-0.621901\pi$$
−0.373670 + 0.927562i $$0.621901\pi$$
$$318$$ 0 0
$$319$$ −10304.0 −1.80851
$$320$$ 0 0
$$321$$ 3972.00 0.690640
$$322$$ 0 0
$$323$$ −6864.00 −1.18242
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −4806.00 −0.812760
$$328$$ 0 0
$$329$$ −4800.00 −0.804354
$$330$$ 0 0
$$331$$ −4640.00 −0.770506 −0.385253 0.922811i $$-0.625886\pi$$
−0.385253 + 0.922811i $$0.625886\pi$$
$$332$$ 0 0
$$333$$ 198.000 0.0325836
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8156.00 1.31835 0.659177 0.751987i $$-0.270905\pi$$
0.659177 + 0.751987i $$0.270905\pi$$
$$338$$ 0 0
$$339$$ −6222.00 −0.996851
$$340$$ 0 0
$$341$$ 3312.00 0.525967
$$342$$ 0 0
$$343$$ 5860.00 0.922479
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −4124.00 −0.638006 −0.319003 0.947754i $$-0.603348\pi$$
−0.319003 + 0.947754i $$0.603348\pi$$
$$348$$ 0 0
$$349$$ 3650.00 0.559828 0.279914 0.960025i $$-0.409694\pi$$
0.279914 + 0.960025i $$0.409694\pi$$
$$350$$ 0 0
$$351$$ −918.000 −0.139599
$$352$$ 0 0
$$353$$ −6834.00 −1.03042 −0.515208 0.857065i $$-0.672285\pi$$
−0.515208 + 0.857065i $$0.672285\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −1980.00 −0.293537
$$358$$ 0 0
$$359$$ 3904.00 0.573942 0.286971 0.957939i $$-0.407352\pi$$
0.286971 + 0.957939i $$0.407352\pi$$
$$360$$ 0 0
$$361$$ 3957.00 0.576906
$$362$$ 0 0
$$363$$ −2355.00 −0.340511
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13174.0 1.87378 0.936890 0.349624i $$-0.113691\pi$$
0.936890 + 0.349624i $$0.113691\pi$$
$$368$$ 0 0
$$369$$ 1746.00 0.246323
$$370$$ 0 0
$$371$$ 2860.00 0.400226
$$372$$ 0 0
$$373$$ 1090.00 0.151308 0.0756542 0.997134i $$-0.475895\pi$$
0.0756542 + 0.997134i $$0.475895\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7616.00 1.04043
$$378$$ 0 0
$$379$$ −9220.00 −1.24960 −0.624802 0.780784i $$-0.714820\pi$$
−0.624802 + 0.780784i $$0.714820\pi$$
$$380$$ 0 0
$$381$$ −1602.00 −0.215415
$$382$$ 0 0
$$383$$ 3960.00 0.528320 0.264160 0.964479i $$-0.414905\pi$$
0.264160 + 0.964479i $$0.414905\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −972.000 −0.127673
$$388$$ 0 0
$$389$$ −1788.00 −0.233047 −0.116523 0.993188i $$-0.537175\pi$$
−0.116523 + 0.993188i $$0.537175\pi$$
$$390$$ 0 0
$$391$$ 10824.0 1.39998
$$392$$ 0 0
$$393$$ −5418.00 −0.695425
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9642.00 −1.21894 −0.609469 0.792810i $$-0.708617\pi$$
−0.609469 + 0.792810i $$0.708617\pi$$
$$398$$ 0 0
$$399$$ 3120.00 0.391467
$$400$$ 0 0
$$401$$ −410.000 −0.0510584 −0.0255292 0.999674i $$-0.508127\pi$$
−0.0255292 + 0.999674i $$0.508127\pi$$
$$402$$ 0 0
$$403$$ −2448.00 −0.302589
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1012.00 −0.123251
$$408$$ 0 0
$$409$$ 13766.0 1.66427 0.832133 0.554576i $$-0.187120\pi$$
0.832133 + 0.554576i $$0.187120\pi$$
$$410$$ 0 0
$$411$$ 5466.00 0.656005
$$412$$ 0 0
$$413$$ −4260.00 −0.507557
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1596.00 −0.187426
$$418$$ 0 0
$$419$$ 16998.0 1.98188 0.990939 0.134315i $$-0.0428833\pi$$
0.990939 + 0.134315i $$0.0428833\pi$$
$$420$$ 0 0
$$421$$ −2450.00 −0.283624 −0.141812 0.989894i $$-0.545293\pi$$
−0.141812 + 0.989894i $$0.545293\pi$$
$$422$$ 0 0
$$423$$ 4320.00 0.496562
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6980.00 −0.791068
$$428$$ 0 0
$$429$$ 4692.00 0.528046
$$430$$ 0 0
$$431$$ −9248.00 −1.03355 −0.516776 0.856121i $$-0.672868\pi$$
−0.516776 + 0.856121i $$0.672868\pi$$
$$432$$ 0 0
$$433$$ 5028.00 0.558038 0.279019 0.960286i $$-0.409991\pi$$
0.279019 + 0.960286i $$0.409991\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −17056.0 −1.86705
$$438$$ 0 0
$$439$$ 3120.00 0.339202 0.169601 0.985513i $$-0.445752\pi$$
0.169601 + 0.985513i $$0.445752\pi$$
$$440$$ 0 0
$$441$$ −2187.00 −0.236152
$$442$$ 0 0
$$443$$ 8220.00 0.881589 0.440795 0.897608i $$-0.354697\pi$$
0.440795 + 0.897608i $$0.354697\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3852.00 0.407591
$$448$$ 0 0
$$449$$ −5826.00 −0.612352 −0.306176 0.951975i $$-0.599050\pi$$
−0.306176 + 0.951975i $$0.599050\pi$$
$$450$$ 0 0
$$451$$ −8924.00 −0.931740
$$452$$ 0 0
$$453$$ −552.000 −0.0572521
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16896.0 1.72946 0.864728 0.502240i $$-0.167491\pi$$
0.864728 + 0.502240i $$0.167491\pi$$
$$458$$ 0 0
$$459$$ 1782.00 0.181213
$$460$$ 0 0
$$461$$ −996.000 −0.100625 −0.0503127 0.998734i $$-0.516022\pi$$
−0.0503127 + 0.998734i $$0.516022\pi$$
$$462$$ 0 0
$$463$$ −10046.0 −1.00837 −0.504187 0.863594i $$-0.668208\pi$$
−0.504187 + 0.863594i $$0.668208\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8388.00 0.831157 0.415579 0.909557i $$-0.363579\pi$$
0.415579 + 0.909557i $$0.363579\pi$$
$$468$$ 0 0
$$469$$ 3280.00 0.322935
$$470$$ 0 0
$$471$$ −11238.0 −1.09940
$$472$$ 0 0
$$473$$ 4968.00 0.482936
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −2574.00 −0.247076
$$478$$ 0 0
$$479$$ 11396.0 1.08705 0.543525 0.839393i $$-0.317089\pi$$
0.543525 + 0.839393i $$0.317089\pi$$
$$480$$ 0 0
$$481$$ 748.000 0.0709062
$$482$$ 0 0
$$483$$ −4920.00 −0.463494
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −6454.00 −0.600531 −0.300266 0.953856i $$-0.597075\pi$$
−0.300266 + 0.953856i $$0.597075\pi$$
$$488$$ 0 0
$$489$$ 4512.00 0.417259
$$490$$ 0 0
$$491$$ 18638.0 1.71308 0.856539 0.516083i $$-0.172610\pi$$
0.856539 + 0.516083i $$0.172610\pi$$
$$492$$ 0 0
$$493$$ −14784.0 −1.35058
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1880.00 −0.169677
$$498$$ 0 0
$$499$$ 17768.0 1.59400 0.796999 0.603981i $$-0.206420\pi$$
0.796999 + 0.603981i $$0.206420\pi$$
$$500$$ 0 0
$$501$$ 9036.00 0.805786
$$502$$ 0 0
$$503$$ 7952.00 0.704895 0.352447 0.935832i $$-0.385350\pi$$
0.352447 + 0.935832i $$0.385350\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3123.00 0.273565
$$508$$ 0 0
$$509$$ −12896.0 −1.12300 −0.561498 0.827478i $$-0.689775\pi$$
−0.561498 + 0.827478i $$0.689775\pi$$
$$510$$ 0 0
$$511$$ −7400.00 −0.640620
$$512$$ 0 0
$$513$$ −2808.00 −0.241669
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −22080.0 −1.87829
$$518$$ 0 0
$$519$$ −1314.00 −0.111133
$$520$$ 0 0
$$521$$ −2714.00 −0.228220 −0.114110 0.993468i $$-0.536402\pi$$
−0.114110 + 0.993468i $$0.536402\pi$$
$$522$$ 0 0
$$523$$ 13792.0 1.15312 0.576560 0.817055i $$-0.304395\pi$$
0.576560 + 0.817055i $$0.304395\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4752.00 0.392790
$$528$$ 0 0
$$529$$ 14729.0 1.21057
$$530$$ 0 0
$$531$$ 3834.00 0.313336
$$532$$ 0 0
$$533$$ 6596.00 0.536031
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4386.00 0.352458
$$538$$ 0 0
$$539$$ 11178.0 0.893266
$$540$$ 0 0
$$541$$ 6802.00 0.540556 0.270278 0.962782i $$-0.412884\pi$$
0.270278 + 0.962782i $$0.412884\pi$$
$$542$$ 0 0
$$543$$ −1758.00 −0.138937
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18188.0 1.42169 0.710843 0.703350i $$-0.248313\pi$$
0.710843 + 0.703350i $$0.248313\pi$$
$$548$$ 0 0
$$549$$ 6282.00 0.488359
$$550$$ 0 0
$$551$$ 23296.0 1.80117
$$552$$ 0 0
$$553$$ −11680.0 −0.898163
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −21462.0 −1.63263 −0.816314 0.577608i $$-0.803986\pi$$
−0.816314 + 0.577608i $$0.803986\pi$$
$$558$$ 0 0
$$559$$ −3672.00 −0.277834
$$560$$ 0 0
$$561$$ −9108.00 −0.685455
$$562$$ 0 0
$$563$$ 17244.0 1.29085 0.645424 0.763824i $$-0.276681\pi$$
0.645424 + 0.763824i $$0.276681\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −810.000 −0.0599944
$$568$$ 0 0
$$569$$ 8790.00 0.647620 0.323810 0.946122i $$-0.395036\pi$$
0.323810 + 0.946122i $$0.395036\pi$$
$$570$$ 0 0
$$571$$ −5984.00 −0.438568 −0.219284 0.975661i $$-0.570372\pi$$
−0.219284 + 0.975661i $$0.570372\pi$$
$$572$$ 0 0
$$573$$ −180.000 −0.0131232
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9344.00 0.674170 0.337085 0.941474i $$-0.390559\pi$$
0.337085 + 0.941474i $$0.390559\pi$$
$$578$$ 0 0
$$579$$ −14028.0 −1.00688
$$580$$ 0 0
$$581$$ 4120.00 0.294193
$$582$$ 0 0
$$583$$ 13156.0 0.934590
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6932.00 0.487418 0.243709 0.969848i $$-0.421636\pi$$
0.243709 + 0.969848i $$0.421636\pi$$
$$588$$ 0 0
$$589$$ −7488.00 −0.523833
$$590$$ 0 0
$$591$$ 6858.00 0.477327
$$592$$ 0 0
$$593$$ −9382.00 −0.649701 −0.324850 0.945765i $$-0.605314\pi$$
−0.324850 + 0.945765i $$0.605314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10608.0 0.727230
$$598$$ 0 0
$$599$$ 4096.00 0.279396 0.139698 0.990194i $$-0.455387\pi$$
0.139698 + 0.990194i $$0.455387\pi$$
$$600$$ 0 0
$$601$$ 22962.0 1.55847 0.779234 0.626733i $$-0.215608\pi$$
0.779234 + 0.626733i $$0.215608\pi$$
$$602$$ 0 0
$$603$$ −2952.00 −0.199361
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3490.00 0.233369 0.116684 0.993169i $$-0.462773\pi$$
0.116684 + 0.993169i $$0.462773\pi$$
$$608$$ 0 0
$$609$$ 6720.00 0.447140
$$610$$ 0 0
$$611$$ 16320.0 1.08058
$$612$$ 0 0
$$613$$ −6386.00 −0.420764 −0.210382 0.977619i $$-0.567471\pi$$
−0.210382 + 0.977619i $$0.567471\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 19534.0 1.27457 0.637285 0.770629i $$-0.280058\pi$$
0.637285 + 0.770629i $$0.280058\pi$$
$$618$$ 0 0
$$619$$ 8764.00 0.569071 0.284535 0.958666i $$-0.408161\pi$$
0.284535 + 0.958666i $$0.408161\pi$$
$$620$$ 0 0
$$621$$ 4428.00 0.286134
$$622$$ 0 0
$$623$$ −12060.0 −0.775560
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 14352.0 0.914137
$$628$$ 0 0
$$629$$ −1452.00 −0.0920430
$$630$$ 0 0
$$631$$ 7856.00 0.495630 0.247815 0.968807i $$-0.420288\pi$$
0.247815 + 0.968807i $$0.420288\pi$$
$$632$$ 0 0
$$633$$ 10500.0 0.659301
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −8262.00 −0.513897
$$638$$ 0 0
$$639$$ 1692.00 0.104749
$$640$$ 0 0
$$641$$ −22974.0 −1.41563 −0.707815 0.706398i $$-0.750319\pi$$
−0.707815 + 0.706398i $$0.750319\pi$$
$$642$$ 0 0
$$643$$ 6216.00 0.381237 0.190618 0.981664i $$-0.438951\pi$$
0.190618 + 0.981664i $$0.438951\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −13384.0 −0.813260 −0.406630 0.913593i $$-0.633296\pi$$
−0.406630 + 0.913593i $$0.633296\pi$$
$$648$$ 0 0
$$649$$ −19596.0 −1.18522
$$650$$ 0 0
$$651$$ −2160.00 −0.130042
$$652$$ 0 0
$$653$$ −12882.0 −0.771993 −0.385997 0.922500i $$-0.626142\pi$$
−0.385997 + 0.922500i $$0.626142\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6660.00 0.395482
$$658$$ 0 0
$$659$$ −2082.00 −0.123070 −0.0615351 0.998105i $$-0.519600\pi$$
−0.0615351 + 0.998105i $$0.519600\pi$$
$$660$$ 0 0
$$661$$ −9430.00 −0.554893 −0.277447 0.960741i $$-0.589488\pi$$
−0.277447 + 0.960741i $$0.589488\pi$$
$$662$$ 0 0
$$663$$ 6732.00 0.394343
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −36736.0 −2.13257
$$668$$ 0 0
$$669$$ 17622.0 1.01839
$$670$$ 0 0
$$671$$ −32108.0 −1.84727
$$672$$ 0 0
$$673$$ −3268.00 −0.187180 −0.0935900 0.995611i $$-0.529834\pi$$
−0.0935900 + 0.995611i $$0.529834\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15606.0 0.885949 0.442974 0.896534i $$-0.353923\pi$$
0.442974 + 0.896534i $$0.353923\pi$$
$$678$$ 0 0
$$679$$ −13840.0 −0.782225
$$680$$ 0 0
$$681$$ −372.000 −0.0209326
$$682$$ 0 0
$$683$$ −428.000 −0.0239780 −0.0119890 0.999928i $$-0.503816\pi$$
−0.0119890 + 0.999928i $$0.503816\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4086.00 0.226915
$$688$$ 0 0
$$689$$ −9724.00 −0.537670
$$690$$ 0 0
$$691$$ −6384.00 −0.351460 −0.175730 0.984438i $$-0.556229\pi$$
−0.175730 + 0.984438i $$0.556229\pi$$
$$692$$ 0 0
$$693$$ 4140.00 0.226934
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12804.0 −0.695819
$$698$$ 0 0
$$699$$ −11610.0 −0.628227
$$700$$ 0 0
$$701$$ −12224.0 −0.658622 −0.329311 0.944221i $$-0.606816\pi$$
−0.329311 + 0.944221i $$0.606816\pi$$
$$702$$ 0 0
$$703$$ 2288.00 0.122750
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 11280.0 0.600040
$$708$$ 0 0
$$709$$ 19510.0 1.03345 0.516723 0.856153i $$-0.327152\pi$$
0.516723 + 0.856153i $$0.327152\pi$$
$$710$$ 0 0
$$711$$ 10512.0 0.554474
$$712$$ 0 0
$$713$$ 11808.0 0.620215
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −18348.0 −0.955674
$$718$$ 0 0
$$719$$ −3368.00 −0.174694 −0.0873472 0.996178i $$-0.527839\pi$$
−0.0873472 + 0.996178i $$0.527839\pi$$
$$720$$ 0 0
$$721$$ −7580.00 −0.391531
$$722$$ 0 0
$$723$$ 17886.0 0.920038
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −22134.0 −1.12917 −0.564584 0.825376i $$-0.690963\pi$$
−0.564584 + 0.825376i $$0.690963\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 7128.00 0.360655
$$732$$ 0 0
$$733$$ −32298.0 −1.62750 −0.813748 0.581219i $$-0.802576\pi$$
−0.813748 + 0.581219i $$0.802576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15088.0 0.754103
$$738$$ 0 0
$$739$$ 25104.0 1.24962 0.624808 0.780779i $$-0.285177\pi$$
0.624808 + 0.780779i $$0.285177\pi$$
$$740$$ 0 0
$$741$$ −10608.0 −0.525904
$$742$$ 0 0
$$743$$ 27696.0 1.36752 0.683760 0.729707i $$-0.260343\pi$$
0.683760 + 0.729707i $$0.260343\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −3708.00 −0.181618
$$748$$ 0 0
$$749$$ 13240.0 0.645900
$$750$$ 0 0
$$751$$ −2176.00 −0.105730 −0.0528651 0.998602i $$-0.516835\pi$$
−0.0528651 + 0.998602i $$0.516835\pi$$
$$752$$ 0 0
$$753$$ 4470.00 0.216329
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −25514.0 −1.22500 −0.612498 0.790472i $$-0.709835\pi$$
−0.612498 + 0.790472i $$0.709835\pi$$
$$758$$ 0 0
$$759$$ −22632.0 −1.08233
$$760$$ 0 0
$$761$$ 18238.0 0.868761 0.434380 0.900730i $$-0.356967\pi$$
0.434380 + 0.900730i $$0.356967\pi$$
$$762$$ 0 0
$$763$$ −16020.0 −0.760109
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14484.0 0.681860
$$768$$ 0 0
$$769$$ −14462.0 −0.678170 −0.339085 0.940756i $$-0.610118\pi$$
−0.339085 + 0.940756i $$0.610118\pi$$
$$770$$ 0 0
$$771$$ −16182.0 −0.755876
$$772$$ 0 0
$$773$$ −34034.0 −1.58359 −0.791797 0.610785i $$-0.790854\pi$$
−0.791797 + 0.610785i $$0.790854\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 660.000 0.0304728
$$778$$ 0 0
$$779$$ 20176.0 0.927959
$$780$$ 0 0
$$781$$ −8648.00 −0.396222
$$782$$ 0 0
$$783$$ −6048.00 −0.276038
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −22064.0 −0.999360 −0.499680 0.866210i $$-0.666549\pi$$
−0.499680 + 0.866210i $$0.666549\pi$$
$$788$$ 0 0
$$789$$ −1908.00 −0.0860920
$$790$$ 0 0
$$791$$ −20740.0 −0.932275
$$792$$ 0 0
$$793$$ 23732.0 1.06273
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23334.0 1.03705 0.518527 0.855061i $$-0.326480\pi$$
0.518527 + 0.855061i $$0.326480\pi$$
$$798$$ 0 0
$$799$$ −31680.0 −1.40270
$$800$$ 0 0
$$801$$ 10854.0 0.478786
$$802$$ 0 0
$$803$$ −34040.0 −1.49595
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 10080.0 0.439694
$$808$$ 0 0
$$809$$ −7566.00 −0.328809 −0.164404 0.986393i $$-0.552570\pi$$
−0.164404 + 0.986393i $$0.552570\pi$$
$$810$$ 0 0
$$811$$ 5964.00 0.258230 0.129115 0.991630i $$-0.458786\pi$$
0.129115 + 0.991630i $$0.458786\pi$$
$$812$$ 0 0
$$813$$ −17304.0 −0.746467
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −11232.0 −0.480977
$$818$$ 0 0
$$819$$ −3060.00 −0.130556
$$820$$ 0 0
$$821$$ −11880.0 −0.505012 −0.252506 0.967595i $$-0.581255\pi$$
−0.252506 + 0.967595i $$0.581255\pi$$
$$822$$ 0 0
$$823$$ 1762.00 0.0746287 0.0373144 0.999304i $$-0.488120\pi$$
0.0373144 + 0.999304i $$0.488120\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14124.0 0.593881 0.296941 0.954896i $$-0.404034\pi$$
0.296941 + 0.954896i $$0.404034\pi$$
$$828$$ 0 0
$$829$$ −21350.0 −0.894471 −0.447235 0.894416i $$-0.647591\pi$$
−0.447235 + 0.894416i $$0.647591\pi$$
$$830$$ 0 0
$$831$$ −4194.00 −0.175076
$$832$$ 0 0
$$833$$ 16038.0 0.667087
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1944.00 0.0802801
$$838$$ 0 0
$$839$$ −26136.0 −1.07546 −0.537732 0.843116i $$-0.680719\pi$$
−0.537732 + 0.843116i $$0.680719\pi$$
$$840$$ 0 0
$$841$$ 25787.0 1.05732
$$842$$ 0 0
$$843$$ 12582.0 0.514053
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −7850.00 −0.318452
$$848$$ 0 0
$$849$$ −24768.0 −1.00122
$$850$$ 0 0
$$851$$ −3608.00 −0.145336
$$852$$ 0 0
$$853$$ 7030.00 0.282184 0.141092 0.989997i $$-0.454939\pi$$
0.141092 + 0.989997i $$0.454939\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 9574.00 0.381612 0.190806 0.981628i $$-0.438890\pi$$
0.190806 + 0.981628i $$0.438890\pi$$
$$858$$ 0 0
$$859$$ −43748.0 −1.73767 −0.868837 0.495098i $$-0.835132\pi$$
−0.868837 + 0.495098i $$0.835132\pi$$
$$860$$ 0 0
$$861$$ 5820.00 0.230366
$$862$$ 0 0
$$863$$ −41436.0 −1.63441 −0.817206 0.576345i $$-0.804478\pi$$
−0.817206 + 0.576345i $$0.804478\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 1671.00 0.0654558
$$868$$ 0 0
$$869$$ −53728.0 −2.09735
$$870$$ 0 0
$$871$$ −11152.0 −0.433836
$$872$$ 0 0
$$873$$ 12456.0 0.482900
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −4606.00 −0.177347 −0.0886736 0.996061i $$-0.528263\pi$$
−0.0886736 + 0.996061i $$0.528263\pi$$
$$878$$ 0 0
$$879$$ 16602.0 0.637055
$$880$$ 0 0
$$881$$ −4610.00 −0.176294 −0.0881469 0.996107i $$-0.528094\pi$$
−0.0881469 + 0.996107i $$0.528094\pi$$
$$882$$ 0 0
$$883$$ −23512.0 −0.896084 −0.448042 0.894013i $$-0.647878\pi$$
−0.448042 + 0.894013i $$0.647878\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 30396.0 1.15062 0.575309 0.817936i $$-0.304882\pi$$
0.575309 + 0.817936i $$0.304882\pi$$
$$888$$ 0 0
$$889$$ −5340.00 −0.201460
$$890$$ 0 0
$$891$$ −3726.00 −0.140096
$$892$$ 0 0
$$893$$ 49920.0 1.87067
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 16728.0 0.622666
$$898$$ 0 0
$$899$$ −16128.0 −0.598330
$$900$$ 0 0
$$901$$ 18876.0 0.697948
$$902$$ 0 0
$$903$$ −3240.00 −0.119402
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −3452.00 −0.126375 −0.0631873 0.998002i $$-0.520127\pi$$
−0.0631873 + 0.998002i $$0.520127\pi$$
$$908$$ 0 0
$$909$$ −10152.0 −0.370430
$$910$$ 0 0
$$911$$ 14256.0 0.518466 0.259233 0.965815i $$-0.416530\pi$$
0.259233 + 0.965815i $$0.416530\pi$$
$$912$$ 0 0
$$913$$ 18952.0 0.686988
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −18060.0 −0.650375
$$918$$ 0 0
$$919$$ 8064.00 0.289452 0.144726 0.989472i $$-0.453770\pi$$
0.144726 + 0.989472i $$0.453770\pi$$
$$920$$ 0 0
$$921$$ −1452.00 −0.0519490
$$922$$ 0 0
$$923$$ 6392.00 0.227947
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 6822.00 0.241709
$$928$$ 0 0
$$929$$ −40930.0 −1.44550 −0.722750 0.691109i $$-0.757122\pi$$
−0.722750 + 0.691109i $$0.757122\pi$$
$$930$$ 0 0
$$931$$ −25272.0 −0.889642
$$932$$ 0 0
$$933$$ 8172.00 0.286752
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −9016.00 −0.314344 −0.157172 0.987571i $$-0.550238\pi$$
−0.157172 + 0.987571i $$0.550238\pi$$
$$938$$ 0 0
$$939$$ 15924.0 0.553419
$$940$$ 0 0
$$941$$ −10444.0 −0.361812 −0.180906 0.983500i $$-0.557903\pi$$
−0.180906 + 0.983500i $$0.557903\pi$$
$$942$$ 0 0
$$943$$ −31816.0 −1.09870
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −21516.0 −0.738306 −0.369153 0.929369i $$-0.620352\pi$$
−0.369153 + 0.929369i $$0.620352\pi$$
$$948$$ 0 0
$$949$$ 25160.0 0.860620
$$950$$ 0 0
$$951$$ 12654.0 0.431476
$$952$$ 0 0
$$953$$ 28098.0 0.955072 0.477536 0.878612i $$-0.341530\pi$$
0.477536 + 0.878612i $$0.341530\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 30912.0 1.04414
$$958$$ 0 0
$$959$$ 18220.0 0.613508
$$960$$ 0 0
$$961$$ −24607.0 −0.825988
$$962$$ 0 0
$$963$$ −11916.0 −0.398741
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14558.0 0.484130 0.242065 0.970260i $$-0.422175\pi$$
0.242065 + 0.970260i $$0.422175\pi$$
$$968$$ 0 0
$$969$$ 20592.0 0.682673
$$970$$ 0 0
$$971$$ 24846.0 0.821160 0.410580 0.911825i $$-0.365326\pi$$
0.410580 + 0.911825i $$0.365326\pi$$
$$972$$ 0 0
$$973$$ −5320.00 −0.175284
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12950.0 0.424061 0.212030 0.977263i $$-0.431992\pi$$
0.212030 + 0.977263i $$0.431992\pi$$
$$978$$ 0 0
$$979$$ −55476.0 −1.81105
$$980$$ 0 0
$$981$$ 14418.0 0.469247
$$982$$ 0 0
$$983$$ 26728.0 0.867234 0.433617 0.901097i $$-0.357237\pi$$
0.433617 + 0.901097i $$0.357237\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 14400.0 0.464394
$$988$$ 0 0
$$989$$ 17712.0 0.569473
$$990$$ 0 0
$$991$$ 3880.00 0.124372 0.0621858 0.998065i $$-0.480193\pi$$
0.0621858 + 0.998065i $$0.480193\pi$$
$$992$$ 0 0
$$993$$ 13920.0 0.444852
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18582.0 −0.590269 −0.295134 0.955456i $$-0.595364\pi$$
−0.295134 + 0.955456i $$0.595364\pi$$
$$998$$ 0 0
$$999$$ −594.000 −0.0188121
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 600.4.a.b.1.1 1
3.2 odd 2 1800.4.a.j.1.1 1
4.3 odd 2 1200.4.a.bh.1.1 1
5.2 odd 4 120.4.f.a.49.2 yes 2
5.3 odd 4 120.4.f.a.49.1 2
5.4 even 2 600.4.a.o.1.1 1
15.2 even 4 360.4.f.c.289.1 2
15.8 even 4 360.4.f.c.289.2 2
15.14 odd 2 1800.4.a.y.1.1 1
20.3 even 4 240.4.f.b.49.2 2
20.7 even 4 240.4.f.b.49.1 2
20.19 odd 2 1200.4.a.f.1.1 1
40.3 even 4 960.4.f.g.769.1 2
40.13 odd 4 960.4.f.l.769.2 2
40.27 even 4 960.4.f.g.769.2 2
40.37 odd 4 960.4.f.l.769.1 2
60.23 odd 4 720.4.f.g.289.2 2
60.47 odd 4 720.4.f.g.289.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.a.49.1 2 5.3 odd 4
120.4.f.a.49.2 yes 2 5.2 odd 4
240.4.f.b.49.1 2 20.7 even 4
240.4.f.b.49.2 2 20.3 even 4
360.4.f.c.289.1 2 15.2 even 4
360.4.f.c.289.2 2 15.8 even 4
600.4.a.b.1.1 1 1.1 even 1 trivial
600.4.a.o.1.1 1 5.4 even 2
720.4.f.g.289.1 2 60.47 odd 4
720.4.f.g.289.2 2 60.23 odd 4
960.4.f.g.769.1 2 40.3 even 4
960.4.f.g.769.2 2 40.27 even 4
960.4.f.l.769.1 2 40.37 odd 4
960.4.f.l.769.2 2 40.13 odd 4
1200.4.a.f.1.1 1 20.19 odd 2
1200.4.a.bh.1.1 1 4.3 odd 2
1800.4.a.j.1.1 1 3.2 odd 2
1800.4.a.y.1.1 1 15.14 odd 2