# Properties

 Label 1323.4.a.d.1.1 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ 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] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +1.00000 q^{4} +15.0000 q^{5} +21.0000 q^{8} +O(q^{10})$$ $$q-3.00000 q^{2} +1.00000 q^{4} +15.0000 q^{5} +21.0000 q^{8} -45.0000 q^{10} +15.0000 q^{11} -20.0000 q^{13} -71.0000 q^{16} +72.0000 q^{17} -2.00000 q^{19} +15.0000 q^{20} -45.0000 q^{22} -114.000 q^{23} +100.000 q^{25} +60.0000 q^{26} -30.0000 q^{29} -101.000 q^{31} +45.0000 q^{32} -216.000 q^{34} -430.000 q^{37} +6.00000 q^{38} +315.000 q^{40} -30.0000 q^{41} +110.000 q^{43} +15.0000 q^{44} +342.000 q^{46} -330.000 q^{47} -300.000 q^{50} -20.0000 q^{52} -621.000 q^{53} +225.000 q^{55} +90.0000 q^{58} -660.000 q^{59} +376.000 q^{61} +303.000 q^{62} +433.000 q^{64} -300.000 q^{65} -250.000 q^{67} +72.0000 q^{68} +360.000 q^{71} -785.000 q^{73} +1290.00 q^{74} -2.00000 q^{76} +488.000 q^{79} -1065.00 q^{80} +90.0000 q^{82} +489.000 q^{83} +1080.00 q^{85} -330.000 q^{86} +315.000 q^{88} -450.000 q^{89} -114.000 q^{92} +990.000 q^{94} -30.0000 q^{95} +1105.00 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.125000
$$5$$ 15.0000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ 0 0
$$10$$ −45.0000 −1.42302
$$11$$ 15.0000 0.411152 0.205576 0.978641i $$-0.434093\pi$$
0.205576 + 0.978641i $$0.434093\pi$$
$$12$$ 0 0
$$13$$ −20.0000 −0.426692 −0.213346 0.976977i $$-0.568436\pi$$
−0.213346 + 0.976977i $$0.568436\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 72.0000 1.02721 0.513605 0.858027i $$-0.328310\pi$$
0.513605 + 0.858027i $$0.328310\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.0241490 −0.0120745 0.999927i $$-0.503844\pi$$
−0.0120745 + 0.999927i $$0.503844\pi$$
$$20$$ 15.0000 0.167705
$$21$$ 0 0
$$22$$ −45.0000 −0.436092
$$23$$ −114.000 −1.03351 −0.516753 0.856134i $$-0.672859\pi$$
−0.516753 + 0.856134i $$0.672859\pi$$
$$24$$ 0 0
$$25$$ 100.000 0.800000
$$26$$ 60.0000 0.452576
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −30.0000 −0.192099 −0.0960493 0.995377i $$-0.530621\pi$$
−0.0960493 + 0.995377i $$0.530621\pi$$
$$30$$ 0 0
$$31$$ −101.000 −0.585166 −0.292583 0.956240i $$-0.594515\pi$$
−0.292583 + 0.956240i $$0.594515\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 0 0
$$34$$ −216.000 −1.08952
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −430.000 −1.91058 −0.955291 0.295666i $$-0.904458\pi$$
−0.955291 + 0.295666i $$0.904458\pi$$
$$38$$ 6.00000 0.0256139
$$39$$ 0 0
$$40$$ 315.000 1.24515
$$41$$ −30.0000 −0.114273 −0.0571367 0.998366i $$-0.518197\pi$$
−0.0571367 + 0.998366i $$0.518197\pi$$
$$42$$ 0 0
$$43$$ 110.000 0.390113 0.195056 0.980792i $$-0.437511\pi$$
0.195056 + 0.980792i $$0.437511\pi$$
$$44$$ 15.0000 0.0513940
$$45$$ 0 0
$$46$$ 342.000 1.09620
$$47$$ −330.000 −1.02416 −0.512079 0.858938i $$-0.671125\pi$$
−0.512079 + 0.858938i $$0.671125\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −300.000 −0.848528
$$51$$ 0 0
$$52$$ −20.0000 −0.0533366
$$53$$ −621.000 −1.60945 −0.804726 0.593647i $$-0.797688\pi$$
−0.804726 + 0.593647i $$0.797688\pi$$
$$54$$ 0 0
$$55$$ 225.000 0.551618
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 90.0000 0.203751
$$59$$ −660.000 −1.45635 −0.728175 0.685391i $$-0.759631\pi$$
−0.728175 + 0.685391i $$0.759631\pi$$
$$60$$ 0 0
$$61$$ 376.000 0.789211 0.394605 0.918851i $$-0.370881\pi$$
0.394605 + 0.918851i $$0.370881\pi$$
$$62$$ 303.000 0.620662
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ −300.000 −0.572468
$$66$$ 0 0
$$67$$ −250.000 −0.455856 −0.227928 0.973678i $$-0.573195\pi$$
−0.227928 + 0.973678i $$0.573195\pi$$
$$68$$ 72.0000 0.128401
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 360.000 0.601748 0.300874 0.953664i $$-0.402722\pi$$
0.300874 + 0.953664i $$0.402722\pi$$
$$72$$ 0 0
$$73$$ −785.000 −1.25859 −0.629297 0.777165i $$-0.716657\pi$$
−0.629297 + 0.777165i $$0.716657\pi$$
$$74$$ 1290.00 2.02648
$$75$$ 0 0
$$76$$ −2.00000 −0.00301863
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 488.000 0.694991 0.347496 0.937682i $$-0.387032\pi$$
0.347496 + 0.937682i $$0.387032\pi$$
$$80$$ −1065.00 −1.48838
$$81$$ 0 0
$$82$$ 90.0000 0.121205
$$83$$ 489.000 0.646683 0.323342 0.946282i $$-0.395194\pi$$
0.323342 + 0.946282i $$0.395194\pi$$
$$84$$ 0 0
$$85$$ 1080.00 1.37815
$$86$$ −330.000 −0.413777
$$87$$ 0 0
$$88$$ 315.000 0.381581
$$89$$ −450.000 −0.535954 −0.267977 0.963425i $$-0.586355\pi$$
−0.267977 + 0.963425i $$0.586355\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −114.000 −0.129188
$$93$$ 0 0
$$94$$ 990.000 1.08628
$$95$$ −30.0000 −0.0323993
$$96$$ 0 0
$$97$$ 1105.00 1.15666 0.578329 0.815804i $$-0.303705\pi$$
0.578329 + 0.815804i $$0.303705\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 100.000 0.100000
$$101$$ 1425.00 1.40389 0.701945 0.712232i $$-0.252315\pi$$
0.701945 + 0.712232i $$0.252315\pi$$
$$102$$ 0 0
$$103$$ 1060.00 1.01403 0.507014 0.861938i $$-0.330749\pi$$
0.507014 + 0.861938i $$0.330749\pi$$
$$104$$ −420.000 −0.396004
$$105$$ 0 0
$$106$$ 1863.00 1.70708
$$107$$ −1485.00 −1.34169 −0.670843 0.741600i $$-0.734067\pi$$
−0.670843 + 0.741600i $$0.734067\pi$$
$$108$$ 0 0
$$109$$ −862.000 −0.757474 −0.378737 0.925504i $$-0.623641\pi$$
−0.378737 + 0.925504i $$0.623641\pi$$
$$110$$ −675.000 −0.585079
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −690.000 −0.574422 −0.287211 0.957867i $$-0.592728\pi$$
−0.287211 + 0.957867i $$0.592728\pi$$
$$114$$ 0 0
$$115$$ −1710.00 −1.38659
$$116$$ −30.0000 −0.0240123
$$117$$ 0 0
$$118$$ 1980.00 1.54469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1106.00 −0.830954
$$122$$ −1128.00 −0.837085
$$123$$ 0 0
$$124$$ −101.000 −0.0731457
$$125$$ −375.000 −0.268328
$$126$$ 0 0
$$127$$ 1865.00 1.30309 0.651543 0.758611i $$-0.274122\pi$$
0.651543 + 0.758611i $$0.274122\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ 0 0
$$130$$ 900.000 0.607194
$$131$$ −1155.00 −0.770327 −0.385163 0.922848i $$-0.625855\pi$$
−0.385163 + 0.922848i $$0.625855\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 750.000 0.483508
$$135$$ 0 0
$$136$$ 1512.00 0.953330
$$137$$ 2778.00 1.73241 0.866206 0.499686i $$-0.166551\pi$$
0.866206 + 0.499686i $$0.166551\pi$$
$$138$$ 0 0
$$139$$ 1924.00 1.17404 0.587020 0.809572i $$-0.300301\pi$$
0.587020 + 0.809572i $$0.300301\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1080.00 −0.638251
$$143$$ −300.000 −0.175435
$$144$$ 0 0
$$145$$ −450.000 −0.257727
$$146$$ 2355.00 1.33494
$$147$$ 0 0
$$148$$ −430.000 −0.238823
$$149$$ −1455.00 −0.799988 −0.399994 0.916518i $$-0.630988\pi$$
−0.399994 + 0.916518i $$0.630988\pi$$
$$150$$ 0 0
$$151$$ −727.000 −0.391804 −0.195902 0.980623i $$-0.562763\pi$$
−0.195902 + 0.980623i $$0.562763\pi$$
$$152$$ −42.0000 −0.0224122
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1515.00 −0.785082
$$156$$ 0 0
$$157$$ −3260.00 −1.65717 −0.828587 0.559860i $$-0.810855\pi$$
−0.828587 + 0.559860i $$0.810855\pi$$
$$158$$ −1464.00 −0.737149
$$159$$ 0 0
$$160$$ 675.000 0.333521
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2540.00 1.22054 0.610270 0.792193i $$-0.291061\pi$$
0.610270 + 0.792193i $$0.291061\pi$$
$$164$$ −30.0000 −0.0142842
$$165$$ 0 0
$$166$$ −1467.00 −0.685911
$$167$$ 3498.00 1.62086 0.810429 0.585837i $$-0.199234\pi$$
0.810429 + 0.585837i $$0.199234\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ −3240.00 −1.46175
$$171$$ 0 0
$$172$$ 110.000 0.0487641
$$173$$ −1149.00 −0.504953 −0.252476 0.967603i $$-0.581245\pi$$
−0.252476 + 0.967603i $$0.581245\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1065.00 −0.456122
$$177$$ 0 0
$$178$$ 1350.00 0.568465
$$179$$ −315.000 −0.131532 −0.0657659 0.997835i $$-0.520949\pi$$
−0.0657659 + 0.997835i $$0.520949\pi$$
$$180$$ 0 0
$$181$$ −1136.00 −0.466509 −0.233255 0.972416i $$-0.574938\pi$$
−0.233255 + 0.972416i $$0.574938\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −2394.00 −0.959174
$$185$$ −6450.00 −2.56332
$$186$$ 0 0
$$187$$ 1080.00 0.422339
$$188$$ −330.000 −0.128020
$$189$$ 0 0
$$190$$ 90.0000 0.0343647
$$191$$ −2460.00 −0.931934 −0.465967 0.884802i $$-0.654293\pi$$
−0.465967 + 0.884802i $$0.654293\pi$$
$$192$$ 0 0
$$193$$ 965.000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ −3315.00 −1.22682
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2493.00 −0.901619 −0.450809 0.892620i $$-0.648865\pi$$
−0.450809 + 0.892620i $$0.648865\pi$$
$$198$$ 0 0
$$199$$ 511.000 0.182029 0.0910146 0.995850i $$-0.470989\pi$$
0.0910146 + 0.995850i $$0.470989\pi$$
$$200$$ 2100.00 0.742462
$$201$$ 0 0
$$202$$ −4275.00 −1.48905
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −450.000 −0.153314
$$206$$ −3180.00 −1.07554
$$207$$ 0 0
$$208$$ 1420.00 0.473362
$$209$$ −30.0000 −0.00992892
$$210$$ 0 0
$$211$$ −2086.00 −0.680598 −0.340299 0.940317i $$-0.610528\pi$$
−0.340299 + 0.940317i $$0.610528\pi$$
$$212$$ −621.000 −0.201181
$$213$$ 0 0
$$214$$ 4455.00 1.42307
$$215$$ 1650.00 0.523391
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2586.00 0.803422
$$219$$ 0 0
$$220$$ 225.000 0.0689523
$$221$$ −1440.00 −0.438303
$$222$$ 0 0
$$223$$ −5240.00 −1.57353 −0.786763 0.617255i $$-0.788245\pi$$
−0.786763 + 0.617255i $$0.788245\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2070.00 0.609267
$$227$$ 2388.00 0.698225 0.349113 0.937081i $$-0.386483\pi$$
0.349113 + 0.937081i $$0.386483\pi$$
$$228$$ 0 0
$$229$$ −182.000 −0.0525192 −0.0262596 0.999655i $$-0.508360\pi$$
−0.0262596 + 0.999655i $$0.508360\pi$$
$$230$$ 5130.00 1.47071
$$231$$ 0 0
$$232$$ −630.000 −0.178282
$$233$$ −450.000 −0.126526 −0.0632628 0.997997i $$-0.520151\pi$$
−0.0632628 + 0.997997i $$0.520151\pi$$
$$234$$ 0 0
$$235$$ −4950.00 −1.37405
$$236$$ −660.000 −0.182044
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5190.00 −1.40466 −0.702329 0.711853i $$-0.747856\pi$$
−0.702329 + 0.711853i $$0.747856\pi$$
$$240$$ 0 0
$$241$$ 2266.00 0.605668 0.302834 0.953043i $$-0.402067\pi$$
0.302834 + 0.953043i $$0.402067\pi$$
$$242$$ 3318.00 0.881360
$$243$$ 0 0
$$244$$ 376.000 0.0986514
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 40.0000 0.0103042
$$248$$ −2121.00 −0.543079
$$249$$ 0 0
$$250$$ 1125.00 0.284605
$$251$$ −2880.00 −0.724239 −0.362119 0.932132i $$-0.617947\pi$$
−0.362119 + 0.932132i $$0.617947\pi$$
$$252$$ 0 0
$$253$$ −1710.00 −0.424928
$$254$$ −5595.00 −1.38213
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ −4188.00 −1.01650 −0.508250 0.861210i $$-0.669707\pi$$
−0.508250 + 0.861210i $$0.669707\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −300.000 −0.0715585
$$261$$ 0 0
$$262$$ 3465.00 0.817055
$$263$$ 3030.00 0.710410 0.355205 0.934788i $$-0.384411\pi$$
0.355205 + 0.934788i $$0.384411\pi$$
$$264$$ 0 0
$$265$$ −9315.00 −2.15931
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −250.000 −0.0569820
$$269$$ 3510.00 0.795571 0.397785 0.917479i $$-0.369779\pi$$
0.397785 + 0.917479i $$0.369779\pi$$
$$270$$ 0 0
$$271$$ −2999.00 −0.672237 −0.336119 0.941820i $$-0.609114\pi$$
−0.336119 + 0.941820i $$0.609114\pi$$
$$272$$ −5112.00 −1.13956
$$273$$ 0 0
$$274$$ −8334.00 −1.83750
$$275$$ 1500.00 0.328921
$$276$$ 0 0
$$277$$ −7720.00 −1.67455 −0.837274 0.546783i $$-0.815852\pi$$
−0.837274 + 0.546783i $$0.815852\pi$$
$$278$$ −5772.00 −1.24526
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7440.00 1.57948 0.789739 0.613443i $$-0.210216\pi$$
0.789739 + 0.613443i $$0.210216\pi$$
$$282$$ 0 0
$$283$$ −830.000 −0.174341 −0.0871703 0.996193i $$-0.527782\pi$$
−0.0871703 + 0.996193i $$0.527782\pi$$
$$284$$ 360.000 0.0752186
$$285$$ 0 0
$$286$$ 900.000 0.186077
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 271.000 0.0551598
$$290$$ 1350.00 0.273361
$$291$$ 0 0
$$292$$ −785.000 −0.157324
$$293$$ 546.000 0.108866 0.0544329 0.998517i $$-0.482665\pi$$
0.0544329 + 0.998517i $$0.482665\pi$$
$$294$$ 0 0
$$295$$ −9900.00 −1.95390
$$296$$ −9030.00 −1.77317
$$297$$ 0 0
$$298$$ 4365.00 0.848516
$$299$$ 2280.00 0.440989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2181.00 0.415571
$$303$$ 0 0
$$304$$ 142.000 0.0267903
$$305$$ 5640.00 1.05884
$$306$$ 0 0
$$307$$ 5560.00 1.03364 0.516818 0.856096i $$-0.327117\pi$$
0.516818 + 0.856096i $$0.327117\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4545.00 0.832705
$$311$$ −8670.00 −1.58081 −0.790403 0.612587i $$-0.790129\pi$$
−0.790403 + 0.612587i $$0.790129\pi$$
$$312$$ 0 0
$$313$$ −4565.00 −0.824374 −0.412187 0.911099i $$-0.635235\pi$$
−0.412187 + 0.911099i $$0.635235\pi$$
$$314$$ 9780.00 1.75770
$$315$$ 0 0
$$316$$ 488.000 0.0868739
$$317$$ −4233.00 −0.749997 −0.374998 0.927025i $$-0.622357\pi$$
−0.374998 + 0.927025i $$0.622357\pi$$
$$318$$ 0 0
$$319$$ −450.000 −0.0789817
$$320$$ 6495.00 1.13463
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −144.000 −0.0248061
$$324$$ 0 0
$$325$$ −2000.00 −0.341354
$$326$$ −7620.00 −1.29458
$$327$$ 0 0
$$328$$ −630.000 −0.106055
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 542.000 0.0900031 0.0450015 0.998987i $$-0.485671\pi$$
0.0450015 + 0.998987i $$0.485671\pi$$
$$332$$ 489.000 0.0808354
$$333$$ 0 0
$$334$$ −10494.0 −1.71918
$$335$$ −3750.00 −0.611595
$$336$$ 0 0
$$337$$ 5690.00 0.919745 0.459872 0.887985i $$-0.347895\pi$$
0.459872 + 0.887985i $$0.347895\pi$$
$$338$$ 5391.00 0.867550
$$339$$ 0 0
$$340$$ 1080.00 0.172268
$$341$$ −1515.00 −0.240592
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2310.00 0.362055
$$345$$ 0 0
$$346$$ 3447.00 0.535583
$$347$$ −5055.00 −0.782036 −0.391018 0.920383i $$-0.627877\pi$$
−0.391018 + 0.920383i $$0.627877\pi$$
$$348$$ 0 0
$$349$$ −1622.00 −0.248778 −0.124389 0.992234i $$-0.539697\pi$$
−0.124389 + 0.992234i $$0.539697\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 675.000 0.102209
$$353$$ 30.0000 0.00452334 0.00226167 0.999997i $$-0.499280\pi$$
0.00226167 + 0.999997i $$0.499280\pi$$
$$354$$ 0 0
$$355$$ 5400.00 0.807330
$$356$$ −450.000 −0.0669942
$$357$$ 0 0
$$358$$ 945.000 0.139511
$$359$$ 7470.00 1.09819 0.549097 0.835759i $$-0.314972\pi$$
0.549097 + 0.835759i $$0.314972\pi$$
$$360$$ 0 0
$$361$$ −6855.00 −0.999417
$$362$$ 3408.00 0.494808
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11775.0 −1.68858
$$366$$ 0 0
$$367$$ 1375.00 0.195571 0.0977853 0.995208i $$-0.468824\pi$$
0.0977853 + 0.995208i $$0.468824\pi$$
$$368$$ 8094.00 1.14655
$$369$$ 0 0
$$370$$ 19350.0 2.71881
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4840.00 −0.671865 −0.335933 0.941886i $$-0.609051\pi$$
−0.335933 + 0.941886i $$0.609051\pi$$
$$374$$ −3240.00 −0.447958
$$375$$ 0 0
$$376$$ −6930.00 −0.950499
$$377$$ 600.000 0.0819670
$$378$$ 0 0
$$379$$ 1892.00 0.256426 0.128213 0.991747i $$-0.459076\pi$$
0.128213 + 0.991747i $$0.459076\pi$$
$$380$$ −30.0000 −0.00404991
$$381$$ 0 0
$$382$$ 7380.00 0.988465
$$383$$ −10704.0 −1.42806 −0.714032 0.700113i $$-0.753133\pi$$
−0.714032 + 0.700113i $$0.753133\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2895.00 −0.381740
$$387$$ 0 0
$$388$$ 1105.00 0.144582
$$389$$ −7815.00 −1.01860 −0.509301 0.860588i $$-0.670096\pi$$
−0.509301 + 0.860588i $$0.670096\pi$$
$$390$$ 0 0
$$391$$ −8208.00 −1.06163
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 7479.00 0.956311
$$395$$ 7320.00 0.932428
$$396$$ 0 0
$$397$$ −4700.00 −0.594172 −0.297086 0.954851i $$-0.596015\pi$$
−0.297086 + 0.954851i $$0.596015\pi$$
$$398$$ −1533.00 −0.193071
$$399$$ 0 0
$$400$$ −7100.00 −0.887500
$$401$$ 2100.00 0.261519 0.130759 0.991414i $$-0.458258\pi$$
0.130759 + 0.991414i $$0.458258\pi$$
$$402$$ 0 0
$$403$$ 2020.00 0.249686
$$404$$ 1425.00 0.175486
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6450.00 −0.785540
$$408$$ 0 0
$$409$$ 10753.0 1.30000 0.650002 0.759933i $$-0.274768\pi$$
0.650002 + 0.759933i $$0.274768\pi$$
$$410$$ 1350.00 0.162614
$$411$$ 0 0
$$412$$ 1060.00 0.126754
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 7335.00 0.867617
$$416$$ −900.000 −0.106072
$$417$$ 0 0
$$418$$ 90.0000 0.0105312
$$419$$ 2940.00 0.342789 0.171394 0.985203i $$-0.445173\pi$$
0.171394 + 0.985203i $$0.445173\pi$$
$$420$$ 0 0
$$421$$ 8696.00 1.00669 0.503346 0.864085i $$-0.332102\pi$$
0.503346 + 0.864085i $$0.332102\pi$$
$$422$$ 6258.00 0.721883
$$423$$ 0 0
$$424$$ −13041.0 −1.49370
$$425$$ 7200.00 0.821768
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −1485.00 −0.167711
$$429$$ 0 0
$$430$$ −4950.00 −0.555140
$$431$$ −8370.00 −0.935426 −0.467713 0.883880i $$-0.654922\pi$$
−0.467713 + 0.883880i $$0.654922\pi$$
$$432$$ 0 0
$$433$$ 5155.00 0.572133 0.286066 0.958210i $$-0.407652\pi$$
0.286066 + 0.958210i $$0.407652\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −862.000 −0.0946842
$$437$$ 228.000 0.0249582
$$438$$ 0 0
$$439$$ 10987.0 1.19449 0.597245 0.802059i $$-0.296262\pi$$
0.597245 + 0.802059i $$0.296262\pi$$
$$440$$ 4725.00 0.511944
$$441$$ 0 0
$$442$$ 4320.00 0.464890
$$443$$ −1956.00 −0.209780 −0.104890 0.994484i $$-0.533449\pi$$
−0.104890 + 0.994484i $$0.533449\pi$$
$$444$$ 0 0
$$445$$ −6750.00 −0.719058
$$446$$ 15720.0 1.66898
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 8730.00 0.917582 0.458791 0.888544i $$-0.348283\pi$$
0.458791 + 0.888544i $$0.348283\pi$$
$$450$$ 0 0
$$451$$ −450.000 −0.0469838
$$452$$ −690.000 −0.0718028
$$453$$ 0 0
$$454$$ −7164.00 −0.740580
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8665.00 −0.886940 −0.443470 0.896289i $$-0.646253\pi$$
−0.443470 + 0.896289i $$0.646253\pi$$
$$458$$ 546.000 0.0557050
$$459$$ 0 0
$$460$$ −1710.00 −0.173324
$$461$$ −9825.00 −0.992616 −0.496308 0.868147i $$-0.665311\pi$$
−0.496308 + 0.868147i $$0.665311\pi$$
$$462$$ 0 0
$$463$$ −5245.00 −0.526470 −0.263235 0.964732i $$-0.584790\pi$$
−0.263235 + 0.964732i $$0.584790\pi$$
$$464$$ 2130.00 0.213109
$$465$$ 0 0
$$466$$ 1350.00 0.134201
$$467$$ −11007.0 −1.09067 −0.545335 0.838218i $$-0.683598\pi$$
−0.545335 + 0.838218i $$0.683598\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 14850.0 1.45740
$$471$$ 0 0
$$472$$ −13860.0 −1.35161
$$473$$ 1650.00 0.160396
$$474$$ 0 0
$$475$$ −200.000 −0.0193192
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 15570.0 1.48986
$$479$$ 16950.0 1.61684 0.808419 0.588608i $$-0.200324\pi$$
0.808419 + 0.588608i $$0.200324\pi$$
$$480$$ 0 0
$$481$$ 8600.00 0.815231
$$482$$ −6798.00 −0.642408
$$483$$ 0 0
$$484$$ −1106.00 −0.103869
$$485$$ 16575.0 1.55182
$$486$$ 0 0
$$487$$ 10640.0 0.990030 0.495015 0.868885i $$-0.335163\pi$$
0.495015 + 0.868885i $$0.335163\pi$$
$$488$$ 7896.00 0.732449
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1635.00 −0.150278 −0.0751390 0.997173i $$-0.523940\pi$$
−0.0751390 + 0.997173i $$0.523940\pi$$
$$492$$ 0 0
$$493$$ −2160.00 −0.197326
$$494$$ −120.000 −0.0109293
$$495$$ 0 0
$$496$$ 7171.00 0.649168
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15802.0 −1.41762 −0.708812 0.705397i $$-0.750769\pi$$
−0.708812 + 0.705397i $$0.750769\pi$$
$$500$$ −375.000 −0.0335410
$$501$$ 0 0
$$502$$ 8640.00 0.768171
$$503$$ −7866.00 −0.697272 −0.348636 0.937258i $$-0.613355\pi$$
−0.348636 + 0.937258i $$0.613355\pi$$
$$504$$ 0 0
$$505$$ 21375.0 1.88351
$$506$$ 5130.00 0.450704
$$507$$ 0 0
$$508$$ 1865.00 0.162886
$$509$$ −11955.0 −1.04105 −0.520527 0.853845i $$-0.674264\pi$$
−0.520527 + 0.853845i $$0.674264\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ 0 0
$$514$$ 12564.0 1.07816
$$515$$ 15900.0 1.36046
$$516$$ 0 0
$$517$$ −4950.00 −0.421085
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6300.00 −0.531295
$$521$$ 19260.0 1.61957 0.809785 0.586727i $$-0.199584\pi$$
0.809785 + 0.586727i $$0.199584\pi$$
$$522$$ 0 0
$$523$$ 18520.0 1.54842 0.774209 0.632930i $$-0.218148\pi$$
0.774209 + 0.632930i $$0.218148\pi$$
$$524$$ −1155.00 −0.0962909
$$525$$ 0 0
$$526$$ −9090.00 −0.753503
$$527$$ −7272.00 −0.601088
$$528$$ 0 0
$$529$$ 829.000 0.0681351
$$530$$ 27945.0 2.29029
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 600.000 0.0487596
$$534$$ 0 0
$$535$$ −22275.0 −1.80006
$$536$$ −5250.00 −0.423070
$$537$$ 0 0
$$538$$ −10530.0 −0.843830
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8372.00 0.665324 0.332662 0.943046i $$-0.392053\pi$$
0.332662 + 0.943046i $$0.392053\pi$$
$$542$$ 8997.00 0.713015
$$543$$ 0 0
$$544$$ 3240.00 0.255356
$$545$$ −12930.0 −1.01626
$$546$$ 0 0
$$547$$ 17120.0 1.33821 0.669103 0.743170i $$-0.266679\pi$$
0.669103 + 0.743170i $$0.266679\pi$$
$$548$$ 2778.00 0.216552
$$549$$ 0 0
$$550$$ −4500.00 −0.348874
$$551$$ 60.0000 0.00463899
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 23160.0 1.77613
$$555$$ 0 0
$$556$$ 1924.00 0.146755
$$557$$ −10575.0 −0.804447 −0.402224 0.915541i $$-0.631763\pi$$
−0.402224 + 0.915541i $$0.631763\pi$$
$$558$$ 0 0
$$559$$ −2200.00 −0.166458
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −22320.0 −1.67529
$$563$$ 10455.0 0.782639 0.391319 0.920255i $$-0.372019\pi$$
0.391319 + 0.920255i $$0.372019\pi$$
$$564$$ 0 0
$$565$$ −10350.0 −0.770669
$$566$$ 2490.00 0.184916
$$567$$ 0 0
$$568$$ 7560.00 0.558469
$$569$$ 24540.0 1.80803 0.904016 0.427498i $$-0.140605\pi$$
0.904016 + 0.427498i $$0.140605\pi$$
$$570$$ 0 0
$$571$$ 24644.0 1.80616 0.903082 0.429469i $$-0.141299\pi$$
0.903082 + 0.429469i $$0.141299\pi$$
$$572$$ −300.000 −0.0219294
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −11400.0 −0.826805
$$576$$ 0 0
$$577$$ 9610.00 0.693361 0.346681 0.937983i $$-0.387309\pi$$
0.346681 + 0.937983i $$0.387309\pi$$
$$578$$ −813.000 −0.0585058
$$579$$ 0 0
$$580$$ −450.000 −0.0322159
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9315.00 −0.661729
$$584$$ −16485.0 −1.16807
$$585$$ 0 0
$$586$$ −1638.00 −0.115470
$$587$$ 4017.00 0.282452 0.141226 0.989977i $$-0.454896\pi$$
0.141226 + 0.989977i $$0.454896\pi$$
$$588$$ 0 0
$$589$$ 202.000 0.0141312
$$590$$ 29700.0 2.07242
$$591$$ 0 0
$$592$$ 30530.0 2.11955
$$593$$ 594.000 0.0411343 0.0205672 0.999788i $$-0.493453\pi$$
0.0205672 + 0.999788i $$0.493453\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1455.00 −0.0999985
$$597$$ 0 0
$$598$$ −6840.00 −0.467740
$$599$$ −8790.00 −0.599582 −0.299791 0.954005i $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$600$$ 0 0
$$601$$ −9371.00 −0.636025 −0.318013 0.948087i $$-0.603015\pi$$
−0.318013 + 0.948087i $$0.603015\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −727.000 −0.0489755
$$605$$ −16590.0 −1.11484
$$606$$ 0 0
$$607$$ 14560.0 0.973595 0.486798 0.873515i $$-0.338165\pi$$
0.486798 + 0.873515i $$0.338165\pi$$
$$608$$ −90.0000 −0.00600326
$$609$$ 0 0
$$610$$ −16920.0 −1.12307
$$611$$ 6600.00 0.437001
$$612$$ 0 0
$$613$$ −18250.0 −1.20246 −0.601232 0.799074i $$-0.705323\pi$$
−0.601232 + 0.799074i $$0.705323\pi$$
$$614$$ −16680.0 −1.09634
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −19662.0 −1.28292 −0.641461 0.767156i $$-0.721671\pi$$
−0.641461 + 0.767156i $$0.721671\pi$$
$$618$$ 0 0
$$619$$ −12044.0 −0.782050 −0.391025 0.920380i $$-0.627879\pi$$
−0.391025 + 0.920380i $$0.627879\pi$$
$$620$$ −1515.00 −0.0981353
$$621$$ 0 0
$$622$$ 26010.0 1.67670
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −18125.0 −1.16000
$$626$$ 13695.0 0.874381
$$627$$ 0 0
$$628$$ −3260.00 −0.207147
$$629$$ −30960.0 −1.96257
$$630$$ 0 0
$$631$$ 14879.0 0.938706 0.469353 0.883011i $$-0.344487\pi$$
0.469353 + 0.883011i $$0.344487\pi$$
$$632$$ 10248.0 0.645006
$$633$$ 0 0
$$634$$ 12699.0 0.795492
$$635$$ 27975.0 1.74827
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1350.00 0.0837727
$$639$$ 0 0
$$640$$ −24885.0 −1.53698
$$641$$ −8850.00 −0.545326 −0.272663 0.962110i $$-0.587904\pi$$
−0.272663 + 0.962110i $$0.587904\pi$$
$$642$$ 0 0
$$643$$ −18380.0 −1.12727 −0.563636 0.826023i $$-0.690598\pi$$
−0.563636 + 0.826023i $$0.690598\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 432.000 0.0263109
$$647$$ 3888.00 0.236249 0.118124 0.992999i $$-0.462312\pi$$
0.118124 + 0.992999i $$0.462312\pi$$
$$648$$ 0 0
$$649$$ −9900.00 −0.598781
$$650$$ 6000.00 0.362061
$$651$$ 0 0
$$652$$ 2540.00 0.152568
$$653$$ 6789.00 0.406852 0.203426 0.979090i $$-0.434792\pi$$
0.203426 + 0.979090i $$0.434792\pi$$
$$654$$ 0 0
$$655$$ −17325.0 −1.03350
$$656$$ 2130.00 0.126772
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −28335.0 −1.67492 −0.837462 0.546496i $$-0.815962\pi$$
−0.837462 + 0.546496i $$0.815962\pi$$
$$660$$ 0 0
$$661$$ 6082.00 0.357886 0.178943 0.983859i $$-0.442732\pi$$
0.178943 + 0.983859i $$0.442732\pi$$
$$662$$ −1626.00 −0.0954627
$$663$$ 0 0
$$664$$ 10269.0 0.600172
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3420.00 0.198535
$$668$$ 3498.00 0.202607
$$669$$ 0 0
$$670$$ 11250.0 0.648695
$$671$$ 5640.00 0.324486
$$672$$ 0 0
$$673$$ 9965.00 0.570762 0.285381 0.958414i $$-0.407880\pi$$
0.285381 + 0.958414i $$0.407880\pi$$
$$674$$ −17070.0 −0.975537
$$675$$ 0 0
$$676$$ −1797.00 −0.102242
$$677$$ 8130.00 0.461538 0.230769 0.973009i $$-0.425876\pi$$
0.230769 + 0.973009i $$0.425876\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 22680.0 1.27903
$$681$$ 0 0
$$682$$ 4545.00 0.255186
$$683$$ −33516.0 −1.87768 −0.938839 0.344356i $$-0.888097\pi$$
−0.938839 + 0.344356i $$0.888097\pi$$
$$684$$ 0 0
$$685$$ 41670.0 2.32428
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −7810.00 −0.432781
$$689$$ 12420.0 0.686741
$$690$$ 0 0
$$691$$ 22084.0 1.21580 0.607898 0.794015i $$-0.292013\pi$$
0.607898 + 0.794015i $$0.292013\pi$$
$$692$$ −1149.00 −0.0631191
$$693$$ 0 0
$$694$$ 15165.0 0.829475
$$695$$ 28860.0 1.57514
$$696$$ 0 0
$$697$$ −2160.00 −0.117383
$$698$$ 4866.00 0.263869
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 10395.0 0.560077 0.280038 0.959989i $$-0.409653\pi$$
0.280038 + 0.959989i $$0.409653\pi$$
$$702$$ 0 0
$$703$$ 860.000 0.0461387
$$704$$ 6495.00 0.347712
$$705$$ 0 0
$$706$$ −90.0000 −0.00479773
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4804.00 −0.254468 −0.127234 0.991873i $$-0.540610\pi$$
−0.127234 + 0.991873i $$0.540610\pi$$
$$710$$ −16200.0 −0.856303
$$711$$ 0 0
$$712$$ −9450.00 −0.497407
$$713$$ 11514.0 0.604772
$$714$$ 0 0
$$715$$ −4500.00 −0.235371
$$716$$ −315.000 −0.0164415
$$717$$ 0 0
$$718$$ −22410.0 −1.16481
$$719$$ 10980.0 0.569520 0.284760 0.958599i $$-0.408086\pi$$
0.284760 + 0.958599i $$0.408086\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 20565.0 1.06004
$$723$$ 0 0
$$724$$ −1136.00 −0.0583137
$$725$$ −3000.00 −0.153679
$$726$$ 0 0
$$727$$ 25945.0 1.32359 0.661793 0.749687i $$-0.269796\pi$$
0.661793 + 0.749687i $$0.269796\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 35325.0 1.79101
$$731$$ 7920.00 0.400727
$$732$$ 0 0
$$733$$ −18650.0 −0.939773 −0.469886 0.882727i $$-0.655705\pi$$
−0.469886 + 0.882727i $$0.655705\pi$$
$$734$$ −4125.00 −0.207434
$$735$$ 0 0
$$736$$ −5130.00 −0.256922
$$737$$ −3750.00 −0.187426
$$738$$ 0 0
$$739$$ −5128.00 −0.255259 −0.127630 0.991822i $$-0.540737\pi$$
−0.127630 + 0.991822i $$0.540737\pi$$
$$740$$ −6450.00 −0.320414
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 32700.0 1.61460 0.807299 0.590142i $$-0.200928\pi$$
0.807299 + 0.590142i $$0.200928\pi$$
$$744$$ 0 0
$$745$$ −21825.0 −1.07330
$$746$$ 14520.0 0.712621
$$747$$ 0 0
$$748$$ 1080.00 0.0527924
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 21161.0 1.02820 0.514098 0.857731i $$-0.328127\pi$$
0.514098 + 0.857731i $$0.328127\pi$$
$$752$$ 23430.0 1.13618
$$753$$ 0 0
$$754$$ −1800.00 −0.0869392
$$755$$ −10905.0 −0.525660
$$756$$ 0 0
$$757$$ 7130.00 0.342331 0.171165 0.985242i $$-0.445247\pi$$
0.171165 + 0.985242i $$0.445247\pi$$
$$758$$ −5676.00 −0.271981
$$759$$ 0 0
$$760$$ −630.000 −0.0300691
$$761$$ −3360.00 −0.160052 −0.0800262 0.996793i $$-0.525500\pi$$
−0.0800262 + 0.996793i $$0.525500\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2460.00 −0.116492
$$765$$ 0 0
$$766$$ 32112.0 1.51469
$$767$$ 13200.0 0.621414
$$768$$ 0 0
$$769$$ −33473.0 −1.56966 −0.784829 0.619712i $$-0.787249\pi$$
−0.784829 + 0.619712i $$0.787249\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 965.000 0.0449885
$$773$$ −3546.00 −0.164995 −0.0824973 0.996591i $$-0.526290\pi$$
−0.0824973 + 0.996591i $$0.526290\pi$$
$$774$$ 0 0
$$775$$ −10100.0 −0.468133
$$776$$ 23205.0 1.07347
$$777$$ 0 0
$$778$$ 23445.0 1.08039
$$779$$ 60.0000 0.00275959
$$780$$ 0 0
$$781$$ 5400.00 0.247410
$$782$$ 24624.0 1.12603
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −48900.0 −2.22333
$$786$$ 0 0
$$787$$ 31840.0 1.44215 0.721076 0.692856i $$-0.243648\pi$$
0.721076 + 0.692856i $$0.243648\pi$$
$$788$$ −2493.00 −0.112702
$$789$$ 0 0
$$790$$ −21960.0 −0.988990
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −7520.00 −0.336750
$$794$$ 14100.0 0.630214
$$795$$ 0 0
$$796$$ 511.000 0.0227537
$$797$$ −15717.0 −0.698525 −0.349263 0.937025i $$-0.613568\pi$$
−0.349263 + 0.937025i $$0.613568\pi$$
$$798$$ 0 0
$$799$$ −23760.0 −1.05203
$$800$$ 4500.00 0.198874
$$801$$ 0 0
$$802$$ −6300.00 −0.277382
$$803$$ −11775.0 −0.517473
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −6060.00 −0.264832
$$807$$ 0 0
$$808$$ 29925.0 1.30292
$$809$$ −10530.0 −0.457621 −0.228810 0.973471i $$-0.573484\pi$$
−0.228810 + 0.973471i $$0.573484\pi$$
$$810$$ 0 0
$$811$$ 26782.0 1.15961 0.579805 0.814755i $$-0.303129\pi$$
0.579805 + 0.814755i $$0.303129\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 19350.0 0.833191
$$815$$ 38100.0 1.63753
$$816$$ 0 0
$$817$$ −220.000 −0.00942084
$$818$$ −32259.0 −1.37886
$$819$$ 0 0
$$820$$ −450.000 −0.0191642
$$821$$ −10110.0 −0.429770 −0.214885 0.976639i $$-0.568938\pi$$
−0.214885 + 0.976639i $$0.568938\pi$$
$$822$$ 0 0
$$823$$ −12535.0 −0.530914 −0.265457 0.964123i $$-0.585523\pi$$
−0.265457 + 0.964123i $$0.585523\pi$$
$$824$$ 22260.0 0.941097
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −9792.00 −0.411731 −0.205865 0.978580i $$-0.566001\pi$$
−0.205865 + 0.978580i $$0.566001\pi$$
$$828$$ 0 0
$$829$$ 4534.00 0.189955 0.0949773 0.995479i $$-0.469722\pi$$
0.0949773 + 0.995479i $$0.469722\pi$$
$$830$$ −22005.0 −0.920247
$$831$$ 0 0
$$832$$ −8660.00 −0.360855
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 52470.0 2.17461
$$836$$ −30.0000 −0.00124111
$$837$$ 0 0
$$838$$ −8820.00 −0.363582
$$839$$ −8880.00 −0.365401 −0.182701 0.983169i $$-0.558484\pi$$
−0.182701 + 0.983169i $$0.558484\pi$$
$$840$$ 0 0
$$841$$ −23489.0 −0.963098
$$842$$ −26088.0 −1.06776
$$843$$ 0 0
$$844$$ −2086.00 −0.0850747
$$845$$ −26955.0 −1.09737
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 44091.0 1.78548
$$849$$ 0 0
$$850$$ −21600.0 −0.871616
$$851$$ 49020.0 1.97460
$$852$$ 0 0
$$853$$ −2270.00 −0.0911176 −0.0455588 0.998962i $$-0.514507\pi$$
−0.0455588 + 0.998962i $$0.514507\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −31185.0 −1.24519
$$857$$ −19608.0 −0.781560 −0.390780 0.920484i $$-0.627795\pi$$
−0.390780 + 0.920484i $$0.627795\pi$$
$$858$$ 0 0
$$859$$ 952.000 0.0378135 0.0189068 0.999821i $$-0.493981\pi$$
0.0189068 + 0.999821i $$0.493981\pi$$
$$860$$ 1650.00 0.0654239
$$861$$ 0 0
$$862$$ 25110.0 0.992169
$$863$$ 17604.0 0.694377 0.347188 0.937795i $$-0.387136\pi$$
0.347188 + 0.937795i $$0.387136\pi$$
$$864$$ 0 0
$$865$$ −17235.0 −0.677465
$$866$$ −15465.0 −0.606838
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 7320.00 0.285747
$$870$$ 0 0
$$871$$ 5000.00 0.194510
$$872$$ −18102.0 −0.702994
$$873$$ 0 0
$$874$$ −684.000 −0.0264721
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 21890.0 0.842842 0.421421 0.906865i $$-0.361531\pi$$
0.421421 + 0.906865i $$0.361531\pi$$
$$878$$ −32961.0 −1.26695
$$879$$ 0 0
$$880$$ −15975.0 −0.611951
$$881$$ 23940.0 0.915504 0.457752 0.889080i $$-0.348655\pi$$
0.457752 + 0.889080i $$0.348655\pi$$
$$882$$ 0 0
$$883$$ −34990.0 −1.33353 −0.666765 0.745268i $$-0.732322\pi$$
−0.666765 + 0.745268i $$0.732322\pi$$
$$884$$ −1440.00 −0.0547878
$$885$$ 0 0
$$886$$ 5868.00 0.222505
$$887$$ −22188.0 −0.839910 −0.419955 0.907545i $$-0.637954\pi$$
−0.419955 + 0.907545i $$0.637954\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 20250.0 0.762676
$$891$$ 0 0
$$892$$ −5240.00 −0.196691
$$893$$ 660.000 0.0247324
$$894$$ 0 0
$$895$$ −4725.00 −0.176469
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −26190.0 −0.973242
$$899$$ 3030.00 0.112410
$$900$$ 0 0
$$901$$ −44712.0 −1.65324
$$902$$ 1350.00 0.0498338
$$903$$ 0 0
$$904$$ −14490.0 −0.533109
$$905$$ −17040.0 −0.625888
$$906$$ 0 0
$$907$$ 37370.0 1.36808 0.684041 0.729444i $$-0.260221\pi$$
0.684041 + 0.729444i $$0.260221\pi$$
$$908$$ 2388.00 0.0872782
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40710.0 −1.48055 −0.740276 0.672303i $$-0.765305\pi$$
−0.740276 + 0.672303i $$0.765305\pi$$
$$912$$ 0 0
$$913$$ 7335.00 0.265885
$$914$$ 25995.0 0.940742
$$915$$ 0 0
$$916$$ −182.000 −0.00656490
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 20981.0 0.753100 0.376550 0.926396i $$-0.377110\pi$$
0.376550 + 0.926396i $$0.377110\pi$$
$$920$$ −35910.0 −1.28687
$$921$$ 0 0
$$922$$ 29475.0 1.05283
$$923$$ −7200.00 −0.256762
$$924$$ 0 0
$$925$$ −43000.0 −1.52847
$$926$$ 15735.0 0.558406
$$927$$ 0 0
$$928$$ −1350.00 −0.0477542
$$929$$ 20100.0 0.709860 0.354930 0.934893i $$-0.384505\pi$$
0.354930 + 0.934893i $$0.384505\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −450.000 −0.0158157
$$933$$ 0 0
$$934$$ 33021.0 1.15683
$$935$$ 16200.0 0.566627
$$936$$ 0 0
$$937$$ −15635.0 −0.545115 −0.272558 0.962139i $$-0.587870\pi$$
−0.272558 + 0.962139i $$0.587870\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −4950.00 −0.171757
$$941$$ 23955.0 0.829873 0.414937 0.909850i $$-0.363804\pi$$
0.414937 + 0.909850i $$0.363804\pi$$
$$942$$ 0 0
$$943$$ 3420.00 0.118102
$$944$$ 46860.0 1.61564
$$945$$ 0 0
$$946$$ −4950.00 −0.170125
$$947$$ 36393.0 1.24880 0.624400 0.781105i $$-0.285344\pi$$
0.624400 + 0.781105i $$0.285344\pi$$
$$948$$ 0 0
$$949$$ 15700.0 0.537032
$$950$$ 600.000 0.0204911
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −43020.0 −1.46228 −0.731141 0.682227i $$-0.761012\pi$$
−0.731141 + 0.682227i $$0.761012\pi$$
$$954$$ 0 0
$$955$$ −36900.0 −1.25032
$$956$$ −5190.00 −0.175582
$$957$$ 0 0
$$958$$ −50850.0 −1.71492
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −19590.0 −0.657581
$$962$$ −25800.0 −0.864683
$$963$$ 0 0
$$964$$ 2266.00 0.0757084
$$965$$ 14475.0 0.482867
$$966$$ 0 0
$$967$$ −43585.0 −1.44943 −0.724715 0.689049i $$-0.758029\pi$$
−0.724715 + 0.689049i $$0.758029\pi$$
$$968$$ −23226.0 −0.771190
$$969$$ 0 0
$$970$$ −49725.0 −1.64595
$$971$$ −43335.0 −1.43222 −0.716110 0.697987i $$-0.754079\pi$$
−0.716110 + 0.697987i $$0.754079\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −31920.0 −1.05008
$$975$$ 0 0
$$976$$ −26696.0 −0.875531
$$977$$ −30390.0 −0.995151 −0.497575 0.867421i $$-0.665776\pi$$
−0.497575 + 0.867421i $$0.665776\pi$$
$$978$$ 0 0
$$979$$ −6750.00 −0.220358
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 4905.00 0.159394
$$983$$ −59226.0 −1.92168 −0.960842 0.277096i $$-0.910628\pi$$
−0.960842 + 0.277096i $$0.910628\pi$$
$$984$$ 0 0
$$985$$ −37395.0 −1.20965
$$986$$ 6480.00 0.209295
$$987$$ 0 0
$$988$$ 40.0000 0.00128803
$$989$$ −12540.0 −0.403184
$$990$$ 0 0
$$991$$ 8399.00 0.269226 0.134613 0.990898i $$-0.457021\pi$$
0.134613 + 0.990898i $$0.457021\pi$$
$$992$$ −4545.00 −0.145468
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 7665.00 0.244218
$$996$$ 0 0
$$997$$ −13340.0 −0.423753 −0.211877 0.977296i $$-0.567958\pi$$
−0.211877 + 0.977296i $$0.567958\pi$$
$$998$$ 47406.0 1.50362
$$999$$ 0 0
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 1323.4.a.d.1.1 1
3.2 odd 2 1323.4.a.k.1.1 1
7.6 odd 2 27.4.a.a.1.1 1
21.20 even 2 27.4.a.b.1.1 yes 1
28.27 even 2 432.4.a.a.1.1 1
35.13 even 4 675.4.b.b.649.2 2
35.27 even 4 675.4.b.b.649.1 2
35.34 odd 2 675.4.a.j.1.1 1
56.13 odd 2 1728.4.a.bc.1.1 1
56.27 even 2 1728.4.a.bd.1.1 1
63.13 odd 6 81.4.c.c.55.1 2
63.20 even 6 81.4.c.a.28.1 2
63.34 odd 6 81.4.c.c.28.1 2
63.41 even 6 81.4.c.a.55.1 2
84.83 odd 2 432.4.a.n.1.1 1
105.62 odd 4 675.4.b.a.649.2 2
105.83 odd 4 675.4.b.a.649.1 2
105.104 even 2 675.4.a.a.1.1 1
168.83 odd 2 1728.4.a.d.1.1 1
168.125 even 2 1728.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
27.4.a.a.1.1 1 7.6 odd 2
27.4.a.b.1.1 yes 1 21.20 even 2
81.4.c.a.28.1 2 63.20 even 6
81.4.c.a.55.1 2 63.41 even 6
81.4.c.c.28.1 2 63.34 odd 6
81.4.c.c.55.1 2 63.13 odd 6
432.4.a.a.1.1 1 28.27 even 2
432.4.a.n.1.1 1 84.83 odd 2
675.4.a.a.1.1 1 105.104 even 2
675.4.a.j.1.1 1 35.34 odd 2
675.4.b.a.649.1 2 105.83 odd 4
675.4.b.a.649.2 2 105.62 odd 4
675.4.b.b.649.1 2 35.27 even 4
675.4.b.b.649.2 2 35.13 even 4
1323.4.a.d.1.1 1 1.1 even 1 trivial
1323.4.a.k.1.1 1 3.2 odd 2
1728.4.a.c.1.1 1 168.125 even 2
1728.4.a.d.1.1 1 168.83 odd 2
1728.4.a.bc.1.1 1 56.13 odd 2
1728.4.a.bd.1.1 1 56.27 even 2