# Properties

 Label 960.4.a.r.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +5.00000 q^{5} +28.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +5.00000 q^{5} +28.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +70.0000 q^{13} -15.0000 q^{15} +102.000 q^{17} +20.0000 q^{19} -84.0000 q^{21} +72.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -306.000 q^{29} +136.000 q^{31} +72.0000 q^{33} +140.000 q^{35} +214.000 q^{37} -210.000 q^{39} -150.000 q^{41} -292.000 q^{43} +45.0000 q^{45} +72.0000 q^{47} +441.000 q^{49} -306.000 q^{51} +414.000 q^{53} -120.000 q^{55} -60.0000 q^{57} -744.000 q^{59} +418.000 q^{61} +252.000 q^{63} +350.000 q^{65} +188.000 q^{67} -216.000 q^{69} -480.000 q^{71} +434.000 q^{73} -75.0000 q^{75} -672.000 q^{77} -1352.00 q^{79} +81.0000 q^{81} -612.000 q^{83} +510.000 q^{85} +918.000 q^{87} -30.0000 q^{89} +1960.00 q^{91} -408.000 q^{93} +100.000 q^{95} -286.000 q^{97} -216.000 q^{99} +O(q^{100})$$

## 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
$$7$$ 28.0000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 0 0
$$13$$ 70.0000 1.49342 0.746712 0.665148i $$-0.231631\pi$$
0.746712 + 0.665148i $$0.231631\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ 102.000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ −84.0000 −0.872872
$$22$$ 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
$$29$$ −306.000 −1.95941 −0.979703 0.200455i $$-0.935758\pi$$
−0.979703 + 0.200455i $$0.935758\pi$$
$$30$$ 0 0
$$31$$ 136.000 0.787946 0.393973 0.919122i $$-0.371100\pi$$
0.393973 + 0.919122i $$0.371100\pi$$
$$32$$ 0 0
$$33$$ 72.0000 0.379806
$$34$$ 0 0
$$35$$ 140.000 0.676123
$$36$$ 0 0
$$37$$ 214.000 0.950848 0.475424 0.879757i $$-0.342295\pi$$
0.475424 + 0.879757i $$0.342295\pi$$
$$38$$ 0 0
$$39$$ −210.000 −0.862229
$$40$$ 0 0
$$41$$ −150.000 −0.571367 −0.285684 0.958324i $$-0.592221\pi$$
−0.285684 + 0.958324i $$0.592221\pi$$
$$42$$ 0 0
$$43$$ −292.000 −1.03557 −0.517786 0.855510i $$-0.673244\pi$$
−0.517786 + 0.855510i $$0.673244\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 72.0000 0.223453 0.111726 0.993739i $$-0.464362\pi$$
0.111726 + 0.993739i $$0.464362\pi$$
$$48$$ 0 0
$$49$$ 441.000 1.28571
$$50$$ 0 0
$$51$$ −306.000 −0.840168
$$52$$ 0 0
$$53$$ 414.000 1.07297 0.536484 0.843911i $$-0.319752\pi$$
0.536484 + 0.843911i $$0.319752\pi$$
$$54$$ 0 0
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ −60.0000 −0.139424
$$58$$ 0 0
$$59$$ −744.000 −1.64170 −0.820852 0.571141i $$-0.806501\pi$$
−0.820852 + 0.571141i $$0.806501\pi$$
$$60$$ 0 0
$$61$$ 418.000 0.877367 0.438684 0.898642i $$-0.355445\pi$$
0.438684 + 0.898642i $$0.355445\pi$$
$$62$$ 0 0
$$63$$ 252.000 0.503953
$$64$$ 0 0
$$65$$ 350.000 0.667879
$$66$$ 0 0
$$67$$ 188.000 0.342804 0.171402 0.985201i $$-0.445170\pi$$
0.171402 + 0.985201i $$0.445170\pi$$
$$68$$ 0 0
$$69$$ −216.000 −0.376860
$$70$$ 0 0
$$71$$ −480.000 −0.802331 −0.401166 0.916006i $$-0.631395\pi$$
−0.401166 + 0.916006i $$0.631395\pi$$
$$72$$ 0 0
$$73$$ 434.000 0.695834 0.347917 0.937525i $$-0.386889\pi$$
0.347917 + 0.937525i $$0.386889\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −672.000 −0.994565
$$78$$ 0 0
$$79$$ −1352.00 −1.92547 −0.962733 0.270452i $$-0.912827\pi$$
−0.962733 + 0.270452i $$0.912827\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −612.000 −0.809346 −0.404673 0.914461i $$-0.632615\pi$$
−0.404673 + 0.914461i $$0.632615\pi$$
$$84$$ 0 0
$$85$$ 510.000 0.650791
$$86$$ 0 0
$$87$$ 918.000 1.13126
$$88$$ 0 0
$$89$$ −30.0000 −0.0357303 −0.0178651 0.999840i $$-0.505687\pi$$
−0.0178651 + 0.999840i $$0.505687\pi$$
$$90$$ 0 0
$$91$$ 1960.00 2.25784
$$92$$ 0 0
$$93$$ −408.000 −0.454921
$$94$$ 0 0
$$95$$ 100.000 0.107998
$$96$$ 0 0
$$97$$ −286.000 −0.299370 −0.149685 0.988734i $$-0.547826\pi$$
−0.149685 + 0.988734i $$0.547826\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ 1542.00 1.51916 0.759578 0.650416i $$-0.225406\pi$$
0.759578 + 0.650416i $$0.225406\pi$$
$$102$$ 0 0
$$103$$ −1172.00 −1.12117 −0.560585 0.828097i $$-0.689424\pi$$
−0.560585 + 0.828097i $$0.689424\pi$$
$$104$$ 0 0
$$105$$ −420.000 −0.390360
$$106$$ 0 0
$$107$$ 1956.00 1.76723 0.883615 0.468214i $$-0.155102\pi$$
0.883615 + 0.468214i $$0.155102\pi$$
$$108$$ 0 0
$$109$$ 1858.00 1.63270 0.816349 0.577559i $$-0.195995\pi$$
0.816349 + 0.577559i $$0.195995\pi$$
$$110$$ 0 0
$$111$$ −642.000 −0.548972
$$112$$ 0 0
$$113$$ 174.000 0.144854 0.0724272 0.997374i $$-0.476926\pi$$
0.0724272 + 0.997374i $$0.476926\pi$$
$$114$$ 0 0
$$115$$ 360.000 0.291915
$$116$$ 0 0
$$117$$ 630.000 0.497808
$$118$$ 0 0
$$119$$ 2856.00 2.20008
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 450.000 0.329879
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 2068.00 1.44492 0.722462 0.691411i $$-0.243010\pi$$
0.722462 + 0.691411i $$0.243010\pi$$
$$128$$ 0 0
$$129$$ 876.000 0.597888
$$130$$ 0 0
$$131$$ 312.000 0.208088 0.104044 0.994573i $$-0.466822\pi$$
0.104044 + 0.994573i $$0.466822\pi$$
$$132$$ 0 0
$$133$$ 560.000 0.365099
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ 2646.00 1.65010 0.825048 0.565063i $$-0.191148\pi$$
0.825048 + 0.565063i $$0.191148\pi$$
$$138$$ 0 0
$$139$$ −1276.00 −0.778625 −0.389313 0.921106i $$-0.627287\pi$$
−0.389313 + 0.921106i $$0.627287\pi$$
$$140$$ 0 0
$$141$$ −216.000 −0.129011
$$142$$ 0 0
$$143$$ −1680.00 −0.982438
$$144$$ 0 0
$$145$$ −1530.00 −0.876273
$$146$$ 0 0
$$147$$ −1323.00 −0.742307
$$148$$ 0 0
$$149$$ 3198.00 1.75832 0.879162 0.476522i $$-0.158103\pi$$
0.879162 + 0.476522i $$0.158103\pi$$
$$150$$ 0 0
$$151$$ 760.000 0.409589 0.204794 0.978805i $$-0.434347\pi$$
0.204794 + 0.978805i $$0.434347\pi$$
$$152$$ 0 0
$$153$$ 918.000 0.485071
$$154$$ 0 0
$$155$$ 680.000 0.352380
$$156$$ 0 0
$$157$$ 166.000 0.0843837 0.0421919 0.999110i $$-0.486566\pi$$
0.0421919 + 0.999110i $$0.486566\pi$$
$$158$$ 0 0
$$159$$ −1242.00 −0.619478
$$160$$ 0 0
$$161$$ 2016.00 0.986851
$$162$$ 0 0
$$163$$ 3020.00 1.45119 0.725597 0.688120i $$-0.241564\pi$$
0.725597 + 0.688120i $$0.241564\pi$$
$$164$$ 0 0
$$165$$ 360.000 0.169854
$$166$$ 0 0
$$167$$ 984.000 0.455953 0.227977 0.973667i $$-0.426789\pi$$
0.227977 + 0.973667i $$0.426789\pi$$
$$168$$ 0 0
$$169$$ 2703.00 1.23031
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 0 0
$$173$$ −1962.00 −0.862243 −0.431122 0.902294i $$-0.641882\pi$$
−0.431122 + 0.902294i $$0.641882\pi$$
$$174$$ 0 0
$$175$$ 700.000 0.302372
$$176$$ 0 0
$$177$$ 2232.00 0.947838
$$178$$ 0 0
$$179$$ 576.000 0.240515 0.120258 0.992743i $$-0.461628\pi$$
0.120258 + 0.992743i $$0.461628\pi$$
$$180$$ 0 0
$$181$$ 1210.00 0.496898 0.248449 0.968645i $$-0.420079\pi$$
0.248449 + 0.968645i $$0.420079\pi$$
$$182$$ 0 0
$$183$$ −1254.00 −0.506548
$$184$$ 0 0
$$185$$ 1070.00 0.425232
$$186$$ 0 0
$$187$$ −2448.00 −0.957302
$$188$$ 0 0
$$189$$ −756.000 −0.290957
$$190$$ 0 0
$$191$$ −3384.00 −1.28198 −0.640989 0.767550i $$-0.721475\pi$$
−0.640989 + 0.767550i $$0.721475\pi$$
$$192$$ 0 0
$$193$$ −2038.00 −0.760096 −0.380048 0.924967i $$-0.624092\pi$$
−0.380048 + 0.924967i $$0.624092\pi$$
$$194$$ 0 0
$$195$$ −1050.00 −0.385600
$$196$$ 0 0
$$197$$ −4098.00 −1.48208 −0.741042 0.671459i $$-0.765668\pi$$
−0.741042 + 0.671459i $$0.765668\pi$$
$$198$$ 0 0
$$199$$ 2248.00 0.800786 0.400393 0.916343i $$-0.368874\pi$$
0.400393 + 0.916343i $$0.368874\pi$$
$$200$$ 0 0
$$201$$ −564.000 −0.197918
$$202$$ 0 0
$$203$$ −8568.00 −2.96234
$$204$$ 0 0
$$205$$ −750.000 −0.255523
$$206$$ 0 0
$$207$$ 648.000 0.217580
$$208$$ 0 0
$$209$$ −480.000 −0.158863
$$210$$ 0 0
$$211$$ 3260.00 1.06364 0.531819 0.846858i $$-0.321509\pi$$
0.531819 + 0.846858i $$0.321509\pi$$
$$212$$ 0 0
$$213$$ 1440.00 0.463226
$$214$$ 0 0
$$215$$ −1460.00 −0.463122
$$216$$ 0 0
$$217$$ 3808.00 1.19126
$$218$$ 0 0
$$219$$ −1302.00 −0.401740
$$220$$ 0 0
$$221$$ 7140.00 2.17325
$$222$$ 0 0
$$223$$ 2980.00 0.894868 0.447434 0.894317i $$-0.352338\pi$$
0.447434 + 0.894317i $$0.352338\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −3180.00 −0.929797 −0.464899 0.885364i $$-0.653909\pi$$
−0.464899 + 0.885364i $$0.653909\pi$$
$$228$$ 0 0
$$229$$ −3374.00 −0.973625 −0.486813 0.873506i $$-0.661841\pi$$
−0.486813 + 0.873506i $$0.661841\pi$$
$$230$$ 0 0
$$231$$ 2016.00 0.574212
$$232$$ 0 0
$$233$$ 1950.00 0.548278 0.274139 0.961690i $$-0.411607\pi$$
0.274139 + 0.961690i $$0.411607\pi$$
$$234$$ 0 0
$$235$$ 360.000 0.0999311
$$236$$ 0 0
$$237$$ 4056.00 1.11167
$$238$$ 0 0
$$239$$ −2232.00 −0.604084 −0.302042 0.953295i $$-0.597668\pi$$
−0.302042 + 0.953295i $$0.597668\pi$$
$$240$$ 0 0
$$241$$ −1822.00 −0.486993 −0.243497 0.969902i $$-0.578294\pi$$
−0.243497 + 0.969902i $$0.578294\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 2205.00 0.574989
$$246$$ 0 0
$$247$$ 1400.00 0.360647
$$248$$ 0 0
$$249$$ 1836.00 0.467276
$$250$$ 0 0
$$251$$ 1488.00 0.374190 0.187095 0.982342i $$-0.440093\pi$$
0.187095 + 0.982342i $$0.440093\pi$$
$$252$$ 0 0
$$253$$ −1728.00 −0.429401
$$254$$ 0 0
$$255$$ −1530.00 −0.375735
$$256$$ 0 0
$$257$$ −2994.00 −0.726695 −0.363347 0.931654i $$-0.618366\pi$$
−0.363347 + 0.931654i $$0.618366\pi$$
$$258$$ 0 0
$$259$$ 5992.00 1.43755
$$260$$ 0 0
$$261$$ −2754.00 −0.653135
$$262$$ 0 0
$$263$$ 2472.00 0.579582 0.289791 0.957090i $$-0.406414\pi$$
0.289791 + 0.957090i $$0.406414\pi$$
$$264$$ 0 0
$$265$$ 2070.00 0.479846
$$266$$ 0 0
$$267$$ 90.0000 0.0206289
$$268$$ 0 0
$$269$$ −3954.00 −0.896207 −0.448103 0.893982i $$-0.647900\pi$$
−0.448103 + 0.893982i $$0.647900\pi$$
$$270$$ 0 0
$$271$$ 2176.00 0.487759 0.243879 0.969806i $$-0.421580\pi$$
0.243879 + 0.969806i $$0.421580\pi$$
$$272$$ 0 0
$$273$$ −5880.00 −1.30357
$$274$$ 0 0
$$275$$ −600.000 −0.131569
$$276$$ 0 0
$$277$$ −1034.00 −0.224285 −0.112143 0.993692i $$-0.535771\pi$$
−0.112143 + 0.993692i $$0.535771\pi$$
$$278$$ 0 0
$$279$$ 1224.00 0.262649
$$280$$ 0 0
$$281$$ −6654.00 −1.41261 −0.706307 0.707906i $$-0.749640\pi$$
−0.706307 + 0.707906i $$0.749640\pi$$
$$282$$ 0 0
$$283$$ −1756.00 −0.368846 −0.184423 0.982847i $$-0.559042\pi$$
−0.184423 + 0.982847i $$0.559042\pi$$
$$284$$ 0 0
$$285$$ −300.000 −0.0623525
$$286$$ 0 0
$$287$$ −4200.00 −0.863826
$$288$$ 0 0
$$289$$ 5491.00 1.11765
$$290$$ 0 0
$$291$$ 858.000 0.172841
$$292$$ 0 0
$$293$$ −3234.00 −0.644820 −0.322410 0.946600i $$-0.604493\pi$$
−0.322410 + 0.946600i $$0.604493\pi$$
$$294$$ 0 0
$$295$$ −3720.00 −0.734192
$$296$$ 0 0
$$297$$ 648.000 0.126602
$$298$$ 0 0
$$299$$ 5040.00 0.974818
$$300$$ 0 0
$$301$$ −8176.00 −1.56564
$$302$$ 0 0
$$303$$ −4626.00 −0.877085
$$304$$ 0 0
$$305$$ 2090.00 0.392371
$$306$$ 0 0
$$307$$ 2036.00 0.378504 0.189252 0.981929i $$-0.439394\pi$$
0.189252 + 0.981929i $$0.439394\pi$$
$$308$$ 0 0
$$309$$ 3516.00 0.647308
$$310$$ 0 0
$$311$$ 96.0000 0.0175037 0.00875187 0.999962i $$-0.497214\pi$$
0.00875187 + 0.999962i $$0.497214\pi$$
$$312$$ 0 0
$$313$$ 1202.00 0.217064 0.108532 0.994093i $$-0.465385\pi$$
0.108532 + 0.994093i $$0.465385\pi$$
$$314$$ 0 0
$$315$$ 1260.00 0.225374
$$316$$ 0 0
$$317$$ 3798.00 0.672924 0.336462 0.941697i $$-0.390770\pi$$
0.336462 + 0.941697i $$0.390770\pi$$
$$318$$ 0 0
$$319$$ 7344.00 1.28898
$$320$$ 0 0
$$321$$ −5868.00 −1.02031
$$322$$ 0 0
$$323$$ 2040.00 0.351420
$$324$$ 0 0
$$325$$ 1750.00 0.298685
$$326$$ 0 0
$$327$$ −5574.00 −0.942639
$$328$$ 0 0
$$329$$ 2016.00 0.337829
$$330$$ 0 0
$$331$$ −5668.00 −0.941213 −0.470606 0.882343i $$-0.655965\pi$$
−0.470606 + 0.882343i $$0.655965\pi$$
$$332$$ 0 0
$$333$$ 1926.00 0.316949
$$334$$ 0 0
$$335$$ 940.000 0.153307
$$336$$ 0 0
$$337$$ −454.000 −0.0733856 −0.0366928 0.999327i $$-0.511682\pi$$
−0.0366928 + 0.999327i $$0.511682\pi$$
$$338$$ 0 0
$$339$$ −522.000 −0.0836317
$$340$$ 0 0
$$341$$ −3264.00 −0.518345
$$342$$ 0 0
$$343$$ 2744.00 0.431959
$$344$$ 0 0
$$345$$ −1080.00 −0.168537
$$346$$ 0 0
$$347$$ −5604.00 −0.866970 −0.433485 0.901161i $$-0.642716\pi$$
−0.433485 + 0.901161i $$0.642716\pi$$
$$348$$ 0 0
$$349$$ 11266.0 1.72795 0.863976 0.503533i $$-0.167967\pi$$
0.863976 + 0.503533i $$0.167967\pi$$
$$350$$ 0 0
$$351$$ −1890.00 −0.287410
$$352$$ 0 0
$$353$$ −6426.00 −0.968899 −0.484450 0.874819i $$-0.660980\pi$$
−0.484450 + 0.874819i $$0.660980\pi$$
$$354$$ 0 0
$$355$$ −2400.00 −0.358813
$$356$$ 0 0
$$357$$ −8568.00 −1.27021
$$358$$ 0 0
$$359$$ −6936.00 −1.01969 −0.509844 0.860267i $$-0.670297\pi$$
−0.509844 + 0.860267i $$0.670297\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ 2265.00 0.327498
$$364$$ 0 0
$$365$$ 2170.00 0.311186
$$366$$ 0 0
$$367$$ 388.000 0.0551865 0.0275932 0.999619i $$-0.491216\pi$$
0.0275932 + 0.999619i $$0.491216\pi$$
$$368$$ 0 0
$$369$$ −1350.00 −0.190456
$$370$$ 0 0
$$371$$ 11592.0 1.62217
$$372$$ 0 0
$$373$$ 8062.00 1.11913 0.559564 0.828787i $$-0.310969\pi$$
0.559564 + 0.828787i $$0.310969\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ −21420.0 −2.92622
$$378$$ 0 0
$$379$$ −3388.00 −0.459182 −0.229591 0.973287i $$-0.573739\pi$$
−0.229591 + 0.973287i $$0.573739\pi$$
$$380$$ 0 0
$$381$$ −6204.00 −0.834227
$$382$$ 0 0
$$383$$ −6984.00 −0.931764 −0.465882 0.884847i $$-0.654263\pi$$
−0.465882 + 0.884847i $$0.654263\pi$$
$$384$$ 0 0
$$385$$ −3360.00 −0.444783
$$386$$ 0 0
$$387$$ −2628.00 −0.345191
$$388$$ 0 0
$$389$$ 2526.00 0.329237 0.164619 0.986357i $$-0.447361\pi$$
0.164619 + 0.986357i $$0.447361\pi$$
$$390$$ 0 0
$$391$$ 7344.00 0.949877
$$392$$ 0 0
$$393$$ −936.000 −0.120140
$$394$$ 0 0
$$395$$ −6760.00 −0.861095
$$396$$ 0 0
$$397$$ −6146.00 −0.776975 −0.388487 0.921454i $$-0.627002\pi$$
−0.388487 + 0.921454i $$0.627002\pi$$
$$398$$ 0 0
$$399$$ −1680.00 −0.210790
$$400$$ 0 0
$$401$$ 9786.00 1.21868 0.609339 0.792910i $$-0.291435\pi$$
0.609339 + 0.792910i $$0.291435\pi$$
$$402$$ 0 0
$$403$$ 9520.00 1.17674
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ −5136.00 −0.625509
$$408$$ 0 0
$$409$$ −886.000 −0.107115 −0.0535573 0.998565i $$-0.517056\pi$$
−0.0535573 + 0.998565i $$0.517056\pi$$
$$410$$ 0 0
$$411$$ −7938.00 −0.952683
$$412$$ 0 0
$$413$$ −20832.0 −2.48202
$$414$$ 0 0
$$415$$ −3060.00 −0.361951
$$416$$ 0 0
$$417$$ 3828.00 0.449539
$$418$$ 0 0
$$419$$ −11352.0 −1.32358 −0.661792 0.749688i $$-0.730204\pi$$
−0.661792 + 0.749688i $$0.730204\pi$$
$$420$$ 0 0
$$421$$ −10190.0 −1.17964 −0.589822 0.807533i $$-0.700802\pi$$
−0.589822 + 0.807533i $$0.700802\pi$$
$$422$$ 0 0
$$423$$ 648.000 0.0744843
$$424$$ 0 0
$$425$$ 2550.00 0.291043
$$426$$ 0 0
$$427$$ 11704.0 1.32645
$$428$$ 0 0
$$429$$ 5040.00 0.567211
$$430$$ 0 0
$$431$$ −2448.00 −0.273587 −0.136794 0.990600i $$-0.543680\pi$$
−0.136794 + 0.990600i $$0.543680\pi$$
$$432$$ 0 0
$$433$$ −7078.00 −0.785559 −0.392779 0.919633i $$-0.628486\pi$$
−0.392779 + 0.919633i $$0.628486\pi$$
$$434$$ 0 0
$$435$$ 4590.00 0.505916
$$436$$ 0 0
$$437$$ 1440.00 0.157631
$$438$$ 0 0
$$439$$ 18088.0 1.96650 0.983250 0.182264i $$-0.0583426\pi$$
0.983250 + 0.182264i $$0.0583426\pi$$
$$440$$ 0 0
$$441$$ 3969.00 0.428571
$$442$$ 0 0
$$443$$ 3852.00 0.413124 0.206562 0.978433i $$-0.433772\pi$$
0.206562 + 0.978433i $$0.433772\pi$$
$$444$$ 0 0
$$445$$ −150.000 −0.0159791
$$446$$ 0 0
$$447$$ −9594.00 −1.01517
$$448$$ 0 0
$$449$$ 6522.00 0.685506 0.342753 0.939426i $$-0.388641\pi$$
0.342753 + 0.939426i $$0.388641\pi$$
$$450$$ 0 0
$$451$$ 3600.00 0.375870
$$452$$ 0 0
$$453$$ −2280.00 −0.236476
$$454$$ 0 0
$$455$$ 9800.00 1.00974
$$456$$ 0 0
$$457$$ 2090.00 0.213930 0.106965 0.994263i $$-0.465887\pi$$
0.106965 + 0.994263i $$0.465887\pi$$
$$458$$ 0 0
$$459$$ −2754.00 −0.280056
$$460$$ 0 0
$$461$$ 9894.00 0.999587 0.499793 0.866145i $$-0.333409\pi$$
0.499793 + 0.866145i $$0.333409\pi$$
$$462$$ 0 0
$$463$$ −3044.00 −0.305544 −0.152772 0.988261i $$-0.548820\pi$$
−0.152772 + 0.988261i $$0.548820\pi$$
$$464$$ 0 0
$$465$$ −2040.00 −0.203447
$$466$$ 0 0
$$467$$ 10236.0 1.01427 0.507137 0.861866i $$-0.330704\pi$$
0.507137 + 0.861866i $$0.330704\pi$$
$$468$$ 0 0
$$469$$ 5264.00 0.518271
$$470$$ 0 0
$$471$$ −498.000 −0.0487190
$$472$$ 0 0
$$473$$ 7008.00 0.681244
$$474$$ 0 0
$$475$$ 500.000 0.0482980
$$476$$ 0 0
$$477$$ 3726.00 0.357656
$$478$$ 0 0
$$479$$ −11496.0 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 14980.0 1.42002
$$482$$ 0 0
$$483$$ −6048.00 −0.569759
$$484$$ 0 0
$$485$$ −1430.00 −0.133882
$$486$$ 0 0
$$487$$ 15316.0 1.42512 0.712561 0.701610i $$-0.247535\pi$$
0.712561 + 0.701610i $$0.247535\pi$$
$$488$$ 0 0
$$489$$ −9060.00 −0.837847
$$490$$ 0 0
$$491$$ −11616.0 −1.06766 −0.533832 0.845591i $$-0.679248\pi$$
−0.533832 + 0.845591i $$0.679248\pi$$
$$492$$ 0 0
$$493$$ −31212.0 −2.85135
$$494$$ 0 0
$$495$$ −1080.00 −0.0980654
$$496$$ 0 0
$$497$$ −13440.0 −1.21301
$$498$$ 0 0
$$499$$ 14996.0 1.34532 0.672658 0.739953i $$-0.265152\pi$$
0.672658 + 0.739953i $$0.265152\pi$$
$$500$$ 0 0
$$501$$ −2952.00 −0.263245
$$502$$ 0 0
$$503$$ 21648.0 1.91896 0.959480 0.281778i $$-0.0909240\pi$$
0.959480 + 0.281778i $$0.0909240\pi$$
$$504$$ 0 0
$$505$$ 7710.00 0.679387
$$506$$ 0 0
$$507$$ −8109.00 −0.710322
$$508$$ 0 0
$$509$$ −3378.00 −0.294160 −0.147080 0.989125i $$-0.546987\pi$$
−0.147080 + 0.989125i $$0.546987\pi$$
$$510$$ 0 0
$$511$$ 12152.0 1.05200
$$512$$ 0 0
$$513$$ −540.000 −0.0464748
$$514$$ 0 0
$$515$$ −5860.00 −0.501403
$$516$$ 0 0
$$517$$ −1728.00 −0.146997
$$518$$ 0 0
$$519$$ 5886.00 0.497816
$$520$$ 0 0
$$521$$ −16158.0 −1.35872 −0.679362 0.733804i $$-0.737743\pi$$
−0.679362 + 0.733804i $$0.737743\pi$$
$$522$$ 0 0
$$523$$ −76.0000 −0.00635420 −0.00317710 0.999995i $$-0.501011\pi$$
−0.00317710 + 0.999995i $$0.501011\pi$$
$$524$$ 0 0
$$525$$ −2100.00 −0.174574
$$526$$ 0 0
$$527$$ 13872.0 1.14663
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ −6696.00 −0.547235
$$532$$ 0 0
$$533$$ −10500.0 −0.853294
$$534$$ 0 0
$$535$$ 9780.00 0.790329
$$536$$ 0 0
$$537$$ −1728.00 −0.138862
$$538$$ 0 0
$$539$$ −10584.0 −0.845798
$$540$$ 0 0
$$541$$ −9278.00 −0.737324 −0.368662 0.929563i $$-0.620184\pi$$
−0.368662 + 0.929563i $$0.620184\pi$$
$$542$$ 0 0
$$543$$ −3630.00 −0.286884
$$544$$ 0 0
$$545$$ 9290.00 0.730165
$$546$$ 0 0
$$547$$ 14564.0 1.13841 0.569206 0.822195i $$-0.307251\pi$$
0.569206 + 0.822195i $$0.307251\pi$$
$$548$$ 0 0
$$549$$ 3762.00 0.292456
$$550$$ 0 0
$$551$$ −6120.00 −0.473177
$$552$$ 0 0
$$553$$ −37856.0 −2.91103
$$554$$ 0 0
$$555$$ −3210.00 −0.245508
$$556$$ 0 0
$$557$$ −2154.00 −0.163856 −0.0819281 0.996638i $$-0.526108\pi$$
−0.0819281 + 0.996638i $$0.526108\pi$$
$$558$$ 0 0
$$559$$ −20440.0 −1.54655
$$560$$ 0 0
$$561$$ 7344.00 0.552699
$$562$$ 0 0
$$563$$ −8700.00 −0.651263 −0.325632 0.945497i $$-0.605577\pi$$
−0.325632 + 0.945497i $$0.605577\pi$$
$$564$$ 0 0
$$565$$ 870.000 0.0647808
$$566$$ 0 0
$$567$$ 2268.00 0.167984
$$568$$ 0 0
$$569$$ 4194.00 0.309001 0.154501 0.987993i $$-0.450623\pi$$
0.154501 + 0.987993i $$0.450623\pi$$
$$570$$ 0 0
$$571$$ −8020.00 −0.587787 −0.293894 0.955838i $$-0.594951\pi$$
−0.293894 + 0.955838i $$0.594951\pi$$
$$572$$ 0 0
$$573$$ 10152.0 0.740150
$$574$$ 0 0
$$575$$ 1800.00 0.130548
$$576$$ 0 0
$$577$$ −2686.00 −0.193795 −0.0968974 0.995294i $$-0.530892\pi$$
−0.0968974 + 0.995294i $$0.530892\pi$$
$$578$$ 0 0
$$579$$ 6114.00 0.438841
$$580$$ 0 0
$$581$$ −17136.0 −1.22362
$$582$$ 0 0
$$583$$ −9936.00 −0.705844
$$584$$ 0 0
$$585$$ 3150.00 0.222626
$$586$$ 0 0
$$587$$ 3012.00 0.211786 0.105893 0.994378i $$-0.466230\pi$$
0.105893 + 0.994378i $$0.466230\pi$$
$$588$$ 0 0
$$589$$ 2720.00 0.190281
$$590$$ 0 0
$$591$$ 12294.0 0.855681
$$592$$ 0 0
$$593$$ −15522.0 −1.07489 −0.537447 0.843298i $$-0.680611\pi$$
−0.537447 + 0.843298i $$0.680611\pi$$
$$594$$ 0 0
$$595$$ 14280.0 0.983904
$$596$$ 0 0
$$597$$ −6744.00 −0.462334
$$598$$ 0 0
$$599$$ −19224.0 −1.31130 −0.655652 0.755063i $$-0.727606\pi$$
−0.655652 + 0.755063i $$0.727606\pi$$
$$600$$ 0 0
$$601$$ −6502.00 −0.441301 −0.220651 0.975353i $$-0.570818\pi$$
−0.220651 + 0.975353i $$0.570818\pi$$
$$602$$ 0 0
$$603$$ 1692.00 0.114268
$$604$$ 0 0
$$605$$ −3775.00 −0.253679
$$606$$ 0 0
$$607$$ −29396.0 −1.96565 −0.982823 0.184552i $$-0.940917\pi$$
−0.982823 + 0.184552i $$0.940917\pi$$
$$608$$ 0 0
$$609$$ 25704.0 1.71031
$$610$$ 0 0
$$611$$ 5040.00 0.333710
$$612$$ 0 0
$$613$$ 10006.0 0.659280 0.329640 0.944107i $$-0.393073\pi$$
0.329640 + 0.944107i $$0.393073\pi$$
$$614$$ 0 0
$$615$$ 2250.00 0.147526
$$616$$ 0 0
$$617$$ 23118.0 1.50842 0.754210 0.656633i $$-0.228020\pi$$
0.754210 + 0.656633i $$0.228020\pi$$
$$618$$ 0 0
$$619$$ 14036.0 0.911397 0.455698 0.890134i $$-0.349390\pi$$
0.455698 + 0.890134i $$0.349390\pi$$
$$620$$ 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
$$627$$ 1440.00 0.0917194
$$628$$ 0 0
$$629$$ 21828.0 1.38369
$$630$$ 0 0
$$631$$ 4288.00 0.270527 0.135264 0.990810i $$-0.456812\pi$$
0.135264 + 0.990810i $$0.456812\pi$$
$$632$$ 0 0
$$633$$ −9780.00 −0.614092
$$634$$ 0 0
$$635$$ 10340.0 0.646190
$$636$$ 0 0
$$637$$ 30870.0 1.92012
$$638$$ 0 0
$$639$$ −4320.00 −0.267444
$$640$$ 0 0
$$641$$ 1314.00 0.0809671 0.0404835 0.999180i $$-0.487110\pi$$
0.0404835 + 0.999180i $$0.487110\pi$$
$$642$$ 0 0
$$643$$ −628.000 −0.0385162 −0.0192581 0.999815i $$-0.506130\pi$$
−0.0192581 + 0.999815i $$0.506130\pi$$
$$644$$ 0 0
$$645$$ 4380.00 0.267383
$$646$$ 0 0
$$647$$ 10944.0 0.664997 0.332498 0.943104i $$-0.392108\pi$$
0.332498 + 0.943104i $$0.392108\pi$$
$$648$$ 0 0
$$649$$ 17856.0 1.07998
$$650$$ 0 0
$$651$$ −11424.0 −0.687776
$$652$$ 0 0
$$653$$ −1098.00 −0.0658010 −0.0329005 0.999459i $$-0.510474\pi$$
−0.0329005 + 0.999459i $$0.510474\pi$$
$$654$$ 0 0
$$655$$ 1560.00 0.0930599
$$656$$ 0 0
$$657$$ 3906.00 0.231945
$$658$$ 0 0
$$659$$ 312.000 0.0184428 0.00922139 0.999957i $$-0.497065\pi$$
0.00922139 + 0.999957i $$0.497065\pi$$
$$660$$ 0 0
$$661$$ −8678.00 −0.510643 −0.255322 0.966856i $$-0.582181\pi$$
−0.255322 + 0.966856i $$0.582181\pi$$
$$662$$ 0 0
$$663$$ −21420.0 −1.25473
$$664$$ 0 0
$$665$$ 2800.00 0.163277
$$666$$ 0 0
$$667$$ −22032.0 −1.27898
$$668$$ 0 0
$$669$$ −8940.00 −0.516652
$$670$$ 0 0
$$671$$ −10032.0 −0.577170
$$672$$ 0 0
$$673$$ −14470.0 −0.828793 −0.414396 0.910097i $$-0.636007\pi$$
−0.414396 + 0.910097i $$0.636007\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ 11838.0 0.672040 0.336020 0.941855i $$-0.390919\pi$$
0.336020 + 0.941855i $$0.390919\pi$$
$$678$$ 0 0
$$679$$ −8008.00 −0.452605
$$680$$ 0 0
$$681$$ 9540.00 0.536819
$$682$$ 0 0
$$683$$ −25548.0 −1.43128 −0.715642 0.698467i $$-0.753866\pi$$
−0.715642 + 0.698467i $$0.753866\pi$$
$$684$$ 0 0
$$685$$ 13230.0 0.737945
$$686$$ 0 0
$$687$$ 10122.0 0.562123
$$688$$ 0 0
$$689$$ 28980.0 1.60239
$$690$$ 0 0
$$691$$ −18412.0 −1.01364 −0.506820 0.862052i $$-0.669179\pi$$
−0.506820 + 0.862052i $$0.669179\pi$$
$$692$$ 0 0
$$693$$ −6048.00 −0.331522
$$694$$ 0 0
$$695$$ −6380.00 −0.348212
$$696$$ 0 0
$$697$$ −15300.0 −0.831462
$$698$$ 0 0
$$699$$ −5850.00 −0.316548
$$700$$ 0 0
$$701$$ 8814.00 0.474893 0.237447 0.971401i $$-0.423690\pi$$
0.237447 + 0.971401i $$0.423690\pi$$
$$702$$ 0 0
$$703$$ 4280.00 0.229621
$$704$$ 0 0
$$705$$ −1080.00 −0.0576953
$$706$$ 0 0
$$707$$ 43176.0 2.29675
$$708$$ 0 0
$$709$$ 17314.0 0.917124 0.458562 0.888662i $$-0.348365\pi$$
0.458562 + 0.888662i $$0.348365\pi$$
$$710$$ 0 0
$$711$$ −12168.0 −0.641822
$$712$$ 0 0
$$713$$ 9792.00 0.514324
$$714$$ 0 0
$$715$$ −8400.00 −0.439360
$$716$$ 0 0
$$717$$ 6696.00 0.348768
$$718$$ 0 0
$$719$$ 768.000 0.0398353 0.0199176 0.999802i $$-0.493660\pi$$
0.0199176 + 0.999802i $$0.493660\pi$$
$$720$$ 0 0
$$721$$ −32816.0 −1.69505
$$722$$ 0 0
$$723$$ 5466.00 0.281166
$$724$$ 0 0
$$725$$ −7650.00 −0.391881
$$726$$ 0 0
$$727$$ 18196.0 0.928270 0.464135 0.885764i $$-0.346365\pi$$
0.464135 + 0.885764i $$0.346365\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −29784.0 −1.50698
$$732$$ 0 0
$$733$$ 18142.0 0.914175 0.457087 0.889422i $$-0.348893\pi$$
0.457087 + 0.889422i $$0.348893\pi$$
$$734$$ 0 0
$$735$$ −6615.00 −0.331970
$$736$$ 0 0
$$737$$ −4512.00 −0.225511
$$738$$ 0 0
$$739$$ −13660.0 −0.679961 −0.339981 0.940432i $$-0.610420\pi$$
−0.339981 + 0.940432i $$0.610420\pi$$
$$740$$ 0 0
$$741$$ −4200.00 −0.208220
$$742$$ 0 0
$$743$$ 12768.0 0.630434 0.315217 0.949020i $$-0.397923\pi$$
0.315217 + 0.949020i $$0.397923\pi$$
$$744$$ 0 0
$$745$$ 15990.0 0.786347
$$746$$ 0 0
$$747$$ −5508.00 −0.269782
$$748$$ 0 0
$$749$$ 54768.0 2.67180
$$750$$ 0 0
$$751$$ −22952.0 −1.11522 −0.557610 0.830103i $$-0.688282\pi$$
−0.557610 + 0.830103i $$0.688282\pi$$
$$752$$ 0 0
$$753$$ −4464.00 −0.216039
$$754$$ 0 0
$$755$$ 3800.00 0.183174
$$756$$ 0 0
$$757$$ −15818.0 −0.759465 −0.379732 0.925096i $$-0.623984\pi$$
−0.379732 + 0.925096i $$0.623984\pi$$
$$758$$ 0 0
$$759$$ 5184.00 0.247915
$$760$$ 0 0
$$761$$ −18558.0 −0.884004 −0.442002 0.897014i $$-0.645732\pi$$
−0.442002 + 0.897014i $$0.645732\pi$$
$$762$$ 0 0
$$763$$ 52024.0 2.46841
$$764$$ 0 0
$$765$$ 4590.00 0.216930
$$766$$ 0 0
$$767$$ −52080.0 −2.45176
$$768$$ 0 0
$$769$$ 14978.0 0.702367 0.351184 0.936307i $$-0.385779\pi$$
0.351184 + 0.936307i $$0.385779\pi$$
$$770$$ 0 0
$$771$$ 8982.00 0.419557
$$772$$ 0 0
$$773$$ −8946.00 −0.416255 −0.208128 0.978102i $$-0.566737\pi$$
−0.208128 + 0.978102i $$0.566737\pi$$
$$774$$ 0 0
$$775$$ 3400.00 0.157589
$$776$$ 0 0
$$777$$ −17976.0 −0.829968
$$778$$ 0 0
$$779$$ −3000.00 −0.137980
$$780$$ 0 0
$$781$$ 11520.0 0.527808
$$782$$ 0 0
$$783$$ 8262.00 0.377088
$$784$$ 0 0
$$785$$ 830.000 0.0377375
$$786$$ 0 0
$$787$$ −18436.0 −0.835035 −0.417517 0.908669i $$-0.637100\pi$$
−0.417517 + 0.908669i $$0.637100\pi$$
$$788$$ 0 0
$$789$$ −7416.00 −0.334622
$$790$$ 0 0
$$791$$ 4872.00 0.218999
$$792$$ 0 0
$$793$$ 29260.0 1.31028
$$794$$ 0 0
$$795$$ −6210.00 −0.277039
$$796$$ 0 0
$$797$$ −16314.0 −0.725058 −0.362529 0.931972i $$-0.618087\pi$$
−0.362529 + 0.931972i $$0.618087\pi$$
$$798$$ 0 0
$$799$$ 7344.00 0.325172
$$800$$ 0 0
$$801$$ −270.000 −0.0119101
$$802$$ 0 0
$$803$$ −10416.0 −0.457749
$$804$$ 0 0
$$805$$ 10080.0 0.441333
$$806$$ 0 0
$$807$$ 11862.0 0.517425
$$808$$ 0 0
$$809$$ −25446.0 −1.10585 −0.552926 0.833231i $$-0.686489\pi$$
−0.552926 + 0.833231i $$0.686489\pi$$
$$810$$ 0 0
$$811$$ 42740.0 1.85056 0.925280 0.379284i $$-0.123830\pi$$
0.925280 + 0.379284i $$0.123830\pi$$
$$812$$ 0 0
$$813$$ −6528.00 −0.281608
$$814$$ 0 0
$$815$$ 15100.0 0.648994
$$816$$ 0 0
$$817$$ −5840.00 −0.250080
$$818$$ 0 0
$$819$$ 17640.0 0.752615
$$820$$ 0 0
$$821$$ −29946.0 −1.27299 −0.636494 0.771282i $$-0.719616\pi$$
−0.636494 + 0.771282i $$0.719616\pi$$
$$822$$ 0 0
$$823$$ 32596.0 1.38059 0.690295 0.723528i $$-0.257481\pi$$
0.690295 + 0.723528i $$0.257481\pi$$
$$824$$ 0 0
$$825$$ 1800.00 0.0759612
$$826$$ 0 0
$$827$$ 3804.00 0.159949 0.0799746 0.996797i $$-0.474516\pi$$
0.0799746 + 0.996797i $$0.474516\pi$$
$$828$$ 0 0
$$829$$ −3278.00 −0.137334 −0.0686669 0.997640i $$-0.521875\pi$$
−0.0686669 + 0.997640i $$0.521875\pi$$
$$830$$ 0 0
$$831$$ 3102.00 0.129491
$$832$$ 0 0
$$833$$ 44982.0 1.87099
$$834$$ 0 0
$$835$$ 4920.00 0.203909
$$836$$ 0 0
$$837$$ −3672.00 −0.151640
$$838$$ 0 0
$$839$$ 5784.00 0.238005 0.119002 0.992894i $$-0.462030\pi$$
0.119002 + 0.992894i $$0.462030\pi$$
$$840$$ 0 0
$$841$$ 69247.0 2.83927
$$842$$ 0 0
$$843$$ 19962.0 0.815573
$$844$$ 0 0
$$845$$ 13515.0 0.550213
$$846$$ 0 0
$$847$$ −21140.0 −0.857590
$$848$$ 0 0
$$849$$ 5268.00 0.212953
$$850$$ 0 0
$$851$$ 15408.0 0.620657
$$852$$ 0 0
$$853$$ −17306.0 −0.694661 −0.347331 0.937743i $$-0.612912\pi$$
−0.347331 + 0.937743i $$0.612912\pi$$
$$854$$ 0 0
$$855$$ 900.000 0.0359992
$$856$$ 0 0
$$857$$ 31134.0 1.24098 0.620488 0.784216i $$-0.286934\pi$$
0.620488 + 0.784216i $$0.286934\pi$$
$$858$$ 0 0
$$859$$ −10780.0 −0.428183 −0.214091 0.976814i $$-0.568679\pi$$
−0.214091 + 0.976814i $$0.568679\pi$$
$$860$$ 0 0
$$861$$ 12600.0 0.498730
$$862$$ 0 0
$$863$$ −3456.00 −0.136319 −0.0681597 0.997674i $$-0.521713\pi$$
−0.0681597 + 0.997674i $$0.521713\pi$$
$$864$$ 0 0
$$865$$ −9810.00 −0.385607
$$866$$ 0 0
$$867$$ −16473.0 −0.645274
$$868$$ 0 0
$$869$$ 32448.0 1.26665
$$870$$ 0 0
$$871$$ 13160.0 0.511951
$$872$$ 0 0
$$873$$ −2574.00 −0.0997900
$$874$$ 0 0
$$875$$ 3500.00 0.135225
$$876$$ 0 0
$$877$$ −2618.00 −0.100802 −0.0504011 0.998729i $$-0.516050\pi$$
−0.0504011 + 0.998729i $$0.516050\pi$$
$$878$$ 0 0
$$879$$ 9702.00 0.372287
$$880$$ 0 0
$$881$$ −26550.0 −1.01531 −0.507657 0.861559i $$-0.669488\pi$$
−0.507657 + 0.861559i $$0.669488\pi$$
$$882$$ 0 0
$$883$$ 27596.0 1.05173 0.525866 0.850567i $$-0.323741\pi$$
0.525866 + 0.850567i $$0.323741\pi$$
$$884$$ 0 0
$$885$$ 11160.0 0.423886
$$886$$ 0 0
$$887$$ 37848.0 1.43271 0.716354 0.697737i $$-0.245810\pi$$
0.716354 + 0.697737i $$0.245810\pi$$
$$888$$ 0 0
$$889$$ 57904.0 2.18452
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ 0 0
$$893$$ 1440.00 0.0539617
$$894$$ 0 0
$$895$$ 2880.00 0.107562
$$896$$ 0 0
$$897$$ −15120.0 −0.562812
$$898$$ 0 0
$$899$$ −41616.0 −1.54391
$$900$$ 0 0
$$901$$ 42228.0 1.56140
$$902$$ 0 0
$$903$$ 24528.0 0.903921
$$904$$ 0 0
$$905$$ 6050.00 0.222220
$$906$$ 0 0
$$907$$ −4804.00 −0.175870 −0.0879351 0.996126i $$-0.528027\pi$$
−0.0879351 + 0.996126i $$0.528027\pi$$
$$908$$ 0 0
$$909$$ 13878.0 0.506385
$$910$$ 0 0
$$911$$ −28608.0 −1.04042 −0.520211 0.854037i $$-0.674147\pi$$
−0.520211 + 0.854037i $$0.674147\pi$$
$$912$$ 0 0
$$913$$ 14688.0 0.532423
$$914$$ 0 0
$$915$$ −6270.00 −0.226535
$$916$$ 0 0
$$917$$ 8736.00 0.314600
$$918$$ 0 0
$$919$$ 40768.0 1.46334 0.731672 0.681657i $$-0.238741\pi$$
0.731672 + 0.681657i $$0.238741\pi$$
$$920$$ 0 0
$$921$$ −6108.00 −0.218529
$$922$$ 0 0
$$923$$ −33600.0 −1.19822
$$924$$ 0 0
$$925$$ 5350.00 0.190170
$$926$$ 0 0
$$927$$ −10548.0 −0.373724
$$928$$ 0 0
$$929$$ 27642.0 0.976216 0.488108 0.872783i $$-0.337687\pi$$
0.488108 + 0.872783i $$0.337687\pi$$
$$930$$ 0 0
$$931$$ 8820.00 0.310487
$$932$$ 0 0
$$933$$ −288.000 −0.0101058
$$934$$ 0 0
$$935$$ −12240.0 −0.428119
$$936$$ 0 0
$$937$$ 28106.0 0.979918 0.489959 0.871746i $$-0.337012\pi$$
0.489959 + 0.871746i $$0.337012\pi$$
$$938$$ 0 0
$$939$$ −3606.00 −0.125322
$$940$$ 0 0
$$941$$ −14730.0 −0.510291 −0.255146 0.966903i $$-0.582123\pi$$
−0.255146 + 0.966903i $$0.582123\pi$$
$$942$$ 0 0
$$943$$ −10800.0 −0.372955
$$944$$ 0 0
$$945$$ −3780.00 −0.130120
$$946$$ 0 0
$$947$$ −9564.00 −0.328182 −0.164091 0.986445i $$-0.552469\pi$$
−0.164091 + 0.986445i $$0.552469\pi$$
$$948$$ 0 0
$$949$$ 30380.0 1.03917
$$950$$ 0 0
$$951$$ −11394.0 −0.388513
$$952$$ 0 0
$$953$$ −53898.0 −1.83203 −0.916017 0.401141i $$-0.868614\pi$$
−0.916017 + 0.401141i $$0.868614\pi$$
$$954$$ 0 0
$$955$$ −16920.0 −0.573318
$$956$$ 0 0
$$957$$ −22032.0 −0.744194
$$958$$ 0 0
$$959$$ 74088.0 2.49471
$$960$$ 0 0
$$961$$ −11295.0 −0.379141
$$962$$ 0 0
$$963$$ 17604.0 0.589077
$$964$$ 0 0
$$965$$ −10190.0 −0.339925
$$966$$ 0 0
$$967$$ −15140.0 −0.503485 −0.251742 0.967794i $$-0.581004\pi$$
−0.251742 + 0.967794i $$0.581004\pi$$
$$968$$ 0 0
$$969$$ −6120.00 −0.202892
$$970$$ 0 0
$$971$$ 23808.0 0.786854 0.393427 0.919356i $$-0.371289\pi$$
0.393427 + 0.919356i $$0.371289\pi$$
$$972$$ 0 0
$$973$$ −35728.0 −1.17717
$$974$$ 0 0
$$975$$ −5250.00 −0.172446
$$976$$ 0 0
$$977$$ 23094.0 0.756236 0.378118 0.925757i $$-0.376571\pi$$
0.378118 + 0.925757i $$0.376571\pi$$
$$978$$ 0 0
$$979$$ 720.000 0.0235049
$$980$$ 0 0
$$981$$ 16722.0 0.544233
$$982$$ 0 0
$$983$$ −7584.00 −0.246075 −0.123038 0.992402i $$-0.539264\pi$$
−0.123038 + 0.992402i $$0.539264\pi$$
$$984$$ 0 0
$$985$$ −20490.0 −0.662808
$$986$$ 0 0
$$987$$ −6048.00 −0.195046
$$988$$ 0 0
$$989$$ −21024.0 −0.675960
$$990$$ 0 0
$$991$$ 26752.0 0.857523 0.428761 0.903418i $$-0.358950\pi$$
0.428761 + 0.903418i $$0.358950\pi$$
$$992$$ 0 0
$$993$$ 17004.0 0.543409
$$994$$ 0 0
$$995$$ 11240.0 0.358123
$$996$$ 0 0
$$997$$ −7778.00 −0.247073 −0.123536 0.992340i $$-0.539424\pi$$
−0.123536 + 0.992340i $$0.539424\pi$$
$$998$$ 0 0
$$999$$ −5778.00 −0.182991
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.r.1.1 1
4.3 odd 2 960.4.a.bc.1.1 1
8.3 odd 2 60.4.a.a.1.1 1
8.5 even 2 240.4.a.i.1.1 1
24.5 odd 2 720.4.a.bb.1.1 1
24.11 even 2 180.4.a.d.1.1 1
40.3 even 4 300.4.d.b.49.1 2
40.13 odd 4 1200.4.f.n.49.2 2
40.19 odd 2 300.4.a.i.1.1 1
40.27 even 4 300.4.d.b.49.2 2
40.29 even 2 1200.4.a.a.1.1 1
40.37 odd 4 1200.4.f.n.49.1 2
72.11 even 6 1620.4.i.f.1081.1 2
72.43 odd 6 1620.4.i.l.1081.1 2
72.59 even 6 1620.4.i.f.541.1 2
72.67 odd 6 1620.4.i.l.541.1 2
120.59 even 2 900.4.a.q.1.1 1
120.83 odd 4 900.4.d.h.649.2 2
120.107 odd 4 900.4.d.h.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.a.1.1 1 8.3 odd 2
180.4.a.d.1.1 1 24.11 even 2
240.4.a.i.1.1 1 8.5 even 2
300.4.a.i.1.1 1 40.19 odd 2
300.4.d.b.49.1 2 40.3 even 4
300.4.d.b.49.2 2 40.27 even 4
720.4.a.bb.1.1 1 24.5 odd 2
900.4.a.q.1.1 1 120.59 even 2
900.4.d.h.649.1 2 120.107 odd 4
900.4.d.h.649.2 2 120.83 odd 4
960.4.a.r.1.1 1 1.1 even 1 trivial
960.4.a.bc.1.1 1 4.3 odd 2
1200.4.a.a.1.1 1 40.29 even 2
1200.4.f.n.49.1 2 40.37 odd 4
1200.4.f.n.49.2 2 40.13 odd 4
1620.4.i.f.541.1 2 72.59 even 6
1620.4.i.f.1081.1 2 72.11 even 6
1620.4.i.l.541.1 2 72.67 odd 6
1620.4.i.l.1081.1 2 72.43 odd 6