# Properties

 Label 1521.4.a.c.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ 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] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +1.00000 q^{4} +9.00000 q^{5} +2.00000 q^{7} +21.0000 q^{8} +O(q^{10})$$ $$q-3.00000 q^{2} +1.00000 q^{4} +9.00000 q^{5} +2.00000 q^{7} +21.0000 q^{8} -27.0000 q^{10} -30.0000 q^{11} -6.00000 q^{14} -71.0000 q^{16} +111.000 q^{17} -46.0000 q^{19} +9.00000 q^{20} +90.0000 q^{22} +6.00000 q^{23} -44.0000 q^{25} +2.00000 q^{28} +105.000 q^{29} -100.000 q^{31} +45.0000 q^{32} -333.000 q^{34} +18.0000 q^{35} +17.0000 q^{37} +138.000 q^{38} +189.000 q^{40} +231.000 q^{41} -514.000 q^{43} -30.0000 q^{44} -18.0000 q^{46} +162.000 q^{47} -339.000 q^{49} +132.000 q^{50} -639.000 q^{53} -270.000 q^{55} +42.0000 q^{56} -315.000 q^{58} -600.000 q^{59} +233.000 q^{61} +300.000 q^{62} +433.000 q^{64} +926.000 q^{67} +111.000 q^{68} -54.0000 q^{70} +930.000 q^{71} -253.000 q^{73} -51.0000 q^{74} -46.0000 q^{76} -60.0000 q^{77} -1324.00 q^{79} -639.000 q^{80} -693.000 q^{82} -810.000 q^{83} +999.000 q^{85} +1542.00 q^{86} -630.000 q^{88} -498.000 q^{89} +6.00000 q^{92} -486.000 q^{94} -414.000 q^{95} +1358.00 q^{97} +1017.00 q^{98} +O(q^{100})$$

## 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
$$5$$ 9.00000 0.804984 0.402492 0.915423i $$-0.368144\pi$$
0.402492 + 0.915423i $$0.368144\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.107990 0.0539949 0.998541i $$-0.482805\pi$$
0.0539949 + 0.998541i $$0.482805\pi$$
$$8$$ 21.0000 0.928078
$$9$$ 0 0
$$10$$ −27.0000 −0.853815
$$11$$ −30.0000 −0.822304 −0.411152 0.911567i $$-0.634873\pi$$
−0.411152 + 0.911567i $$0.634873\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −6.00000 −0.114541
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 111.000 1.58361 0.791807 0.610771i $$-0.209140\pi$$
0.791807 + 0.610771i $$0.209140\pi$$
$$18$$ 0 0
$$19$$ −46.0000 −0.555428 −0.277714 0.960664i $$-0.589577\pi$$
−0.277714 + 0.960664i $$0.589577\pi$$
$$20$$ 9.00000 0.100623
$$21$$ 0 0
$$22$$ 90.0000 0.872185
$$23$$ 6.00000 0.0543951 0.0271975 0.999630i $$-0.491342\pi$$
0.0271975 + 0.999630i $$0.491342\pi$$
$$24$$ 0 0
$$25$$ −44.0000 −0.352000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 2.00000 0.0134987
$$29$$ 105.000 0.672345 0.336173 0.941800i $$-0.390867\pi$$
0.336173 + 0.941800i $$0.390867\pi$$
$$30$$ 0 0
$$31$$ −100.000 −0.579372 −0.289686 0.957122i $$-0.593551\pi$$
−0.289686 + 0.957122i $$0.593551\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 0 0
$$34$$ −333.000 −1.67968
$$35$$ 18.0000 0.0869302
$$36$$ 0 0
$$37$$ 17.0000 0.0755347 0.0377673 0.999287i $$-0.487975\pi$$
0.0377673 + 0.999287i $$0.487975\pi$$
$$38$$ 138.000 0.589120
$$39$$ 0 0
$$40$$ 189.000 0.747088
$$41$$ 231.000 0.879906 0.439953 0.898021i $$-0.354995\pi$$
0.439953 + 0.898021i $$0.354995\pi$$
$$42$$ 0 0
$$43$$ −514.000 −1.82289 −0.911445 0.411422i $$-0.865032\pi$$
−0.911445 + 0.411422i $$0.865032\pi$$
$$44$$ −30.0000 −0.102788
$$45$$ 0 0
$$46$$ −18.0000 −0.0576947
$$47$$ 162.000 0.502769 0.251384 0.967887i $$-0.419114\pi$$
0.251384 + 0.967887i $$0.419114\pi$$
$$48$$ 0 0
$$49$$ −339.000 −0.988338
$$50$$ 132.000 0.373352
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −639.000 −1.65610 −0.828051 0.560653i $$-0.810550\pi$$
−0.828051 + 0.560653i $$0.810550\pi$$
$$54$$ 0 0
$$55$$ −270.000 −0.661942
$$56$$ 42.0000 0.100223
$$57$$ 0 0
$$58$$ −315.000 −0.713130
$$59$$ −600.000 −1.32396 −0.661978 0.749524i $$-0.730283\pi$$
−0.661978 + 0.749524i $$0.730283\pi$$
$$60$$ 0 0
$$61$$ 233.000 0.489059 0.244529 0.969642i $$-0.421367\pi$$
0.244529 + 0.969642i $$0.421367\pi$$
$$62$$ 300.000 0.614517
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 926.000 1.68849 0.844246 0.535957i $$-0.180049\pi$$
0.844246 + 0.535957i $$0.180049\pi$$
$$68$$ 111.000 0.197952
$$69$$ 0 0
$$70$$ −54.0000 −0.0922033
$$71$$ 930.000 1.55452 0.777258 0.629182i $$-0.216610\pi$$
0.777258 + 0.629182i $$0.216610\pi$$
$$72$$ 0 0
$$73$$ −253.000 −0.405636 −0.202818 0.979216i $$-0.565010\pi$$
−0.202818 + 0.979216i $$0.565010\pi$$
$$74$$ −51.0000 −0.0801166
$$75$$ 0 0
$$76$$ −46.0000 −0.0694284
$$77$$ −60.0000 −0.0888004
$$78$$ 0 0
$$79$$ −1324.00 −1.88559 −0.942795 0.333373i $$-0.891813\pi$$
−0.942795 + 0.333373i $$0.891813\pi$$
$$80$$ −639.000 −0.893030
$$81$$ 0 0
$$82$$ −693.000 −0.933281
$$83$$ −810.000 −1.07119 −0.535597 0.844474i $$-0.679913\pi$$
−0.535597 + 0.844474i $$0.679913\pi$$
$$84$$ 0 0
$$85$$ 999.000 1.27479
$$86$$ 1542.00 1.93347
$$87$$ 0 0
$$88$$ −630.000 −0.763162
$$89$$ −498.000 −0.593122 −0.296561 0.955014i $$-0.595840\pi$$
−0.296561 + 0.955014i $$0.595840\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 6.00000 0.00679938
$$93$$ 0 0
$$94$$ −486.000 −0.533267
$$95$$ −414.000 −0.447111
$$96$$ 0 0
$$97$$ 1358.00 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 1017.00 1.04829
$$99$$ 0 0
$$100$$ −44.0000 −0.0440000
$$101$$ 357.000 0.351711 0.175856 0.984416i $$-0.443731\pi$$
0.175856 + 0.984416i $$0.443731\pi$$
$$102$$ 0 0
$$103$$ 1118.00 1.06951 0.534756 0.845006i $$-0.320403\pi$$
0.534756 + 0.845006i $$0.320403\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1917.00 1.75656
$$107$$ −714.000 −0.645093 −0.322547 0.946554i $$-0.604539\pi$$
−0.322547 + 0.946554i $$0.604539\pi$$
$$108$$ 0 0
$$109$$ 2006.00 1.76275 0.881376 0.472416i $$-0.156618\pi$$
0.881376 + 0.472416i $$0.156618\pi$$
$$110$$ 810.000 0.702095
$$111$$ 0 0
$$112$$ −142.000 −0.119801
$$113$$ 1119.00 0.931563 0.465782 0.884900i $$-0.345773\pi$$
0.465782 + 0.884900i $$0.345773\pi$$
$$114$$ 0 0
$$115$$ 54.0000 0.0437872
$$116$$ 105.000 0.0840431
$$117$$ 0 0
$$118$$ 1800.00 1.40427
$$119$$ 222.000 0.171014
$$120$$ 0 0
$$121$$ −431.000 −0.323817
$$122$$ −699.000 −0.518725
$$123$$ 0 0
$$124$$ −100.000 −0.0724215
$$125$$ −1521.00 −1.08834
$$126$$ 0 0
$$127$$ −604.000 −0.422018 −0.211009 0.977484i $$-0.567675\pi$$
−0.211009 + 0.977484i $$0.567675\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1584.00 1.05645 0.528224 0.849105i $$-0.322858\pi$$
0.528224 + 0.849105i $$0.322858\pi$$
$$132$$ 0 0
$$133$$ −92.0000 −0.0599805
$$134$$ −2778.00 −1.79092
$$135$$ 0 0
$$136$$ 2331.00 1.46972
$$137$$ −717.000 −0.447135 −0.223567 0.974688i $$-0.571770\pi$$
−0.223567 + 0.974688i $$0.571770\pi$$
$$138$$ 0 0
$$139$$ −820.000 −0.500370 −0.250185 0.968198i $$-0.580492\pi$$
−0.250185 + 0.968198i $$0.580492\pi$$
$$140$$ 18.0000 0.0108663
$$141$$ 0 0
$$142$$ −2790.00 −1.64881
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 945.000 0.541227
$$146$$ 759.000 0.430242
$$147$$ 0 0
$$148$$ 17.0000 0.00944183
$$149$$ 1749.00 0.961635 0.480818 0.876821i $$-0.340340\pi$$
0.480818 + 0.876821i $$0.340340\pi$$
$$150$$ 0 0
$$151$$ −370.000 −0.199405 −0.0997026 0.995017i $$-0.531789\pi$$
−0.0997026 + 0.995017i $$0.531789\pi$$
$$152$$ −966.000 −0.515480
$$153$$ 0 0
$$154$$ 180.000 0.0941871
$$155$$ −900.000 −0.466385
$$156$$ 0 0
$$157$$ −2611.00 −1.32726 −0.663632 0.748059i $$-0.730986\pi$$
−0.663632 + 0.748059i $$0.730986\pi$$
$$158$$ 3972.00 1.99997
$$159$$ 0 0
$$160$$ 405.000 0.200113
$$161$$ 12.0000 0.00587411
$$162$$ 0 0
$$163$$ −1636.00 −0.786144 −0.393072 0.919508i $$-0.628588\pi$$
−0.393072 + 0.919508i $$0.628588\pi$$
$$164$$ 231.000 0.109988
$$165$$ 0 0
$$166$$ 2430.00 1.13617
$$167$$ −264.000 −0.122329 −0.0611645 0.998128i $$-0.519481\pi$$
−0.0611645 + 0.998128i $$0.519481\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −2997.00 −1.35211
$$171$$ 0 0
$$172$$ −514.000 −0.227861
$$173$$ −1410.00 −0.619655 −0.309827 0.950793i $$-0.600271\pi$$
−0.309827 + 0.950793i $$0.600271\pi$$
$$174$$ 0 0
$$175$$ −88.0000 −0.0380124
$$176$$ 2130.00 0.912243
$$177$$ 0 0
$$178$$ 1494.00 0.629101
$$179$$ 474.000 0.197924 0.0989621 0.995091i $$-0.468448\pi$$
0.0989621 + 0.995091i $$0.468448\pi$$
$$180$$ 0 0
$$181$$ 2249.00 0.923574 0.461787 0.886991i $$-0.347208\pi$$
0.461787 + 0.886991i $$0.347208\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 126.000 0.0504828
$$185$$ 153.000 0.0608042
$$186$$ 0 0
$$187$$ −3330.00 −1.30221
$$188$$ 162.000 0.0628461
$$189$$ 0 0
$$190$$ 1242.00 0.474232
$$191$$ −3444.00 −1.30471 −0.652354 0.757915i $$-0.726218\pi$$
−0.652354 + 0.757915i $$0.726218\pi$$
$$192$$ 0 0
$$193$$ −4273.00 −1.59366 −0.796832 0.604201i $$-0.793493\pi$$
−0.796832 + 0.604201i $$0.793493\pi$$
$$194$$ −4074.00 −1.50771
$$195$$ 0 0
$$196$$ −339.000 −0.123542
$$197$$ 1986.00 0.718257 0.359129 0.933288i $$-0.383074\pi$$
0.359129 + 0.933288i $$0.383074\pi$$
$$198$$ 0 0
$$199$$ −2386.00 −0.849945 −0.424973 0.905206i $$-0.639716\pi$$
−0.424973 + 0.905206i $$0.639716\pi$$
$$200$$ −924.000 −0.326683
$$201$$ 0 0
$$202$$ −1071.00 −0.373046
$$203$$ 210.000 0.0726065
$$204$$ 0 0
$$205$$ 2079.00 0.708311
$$206$$ −3354.00 −1.13439
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1380.00 0.456730
$$210$$ 0 0
$$211$$ −1600.00 −0.522031 −0.261016 0.965335i $$-0.584057\pi$$
−0.261016 + 0.965335i $$0.584057\pi$$
$$212$$ −639.000 −0.207013
$$213$$ 0 0
$$214$$ 2142.00 0.684225
$$215$$ −4626.00 −1.46740
$$216$$ 0 0
$$217$$ −200.000 −0.0625663
$$218$$ −6018.00 −1.86968
$$219$$ 0 0
$$220$$ −270.000 −0.0827427
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −3832.00 −1.15072 −0.575358 0.817902i $$-0.695137\pi$$
−0.575358 + 0.817902i $$0.695137\pi$$
$$224$$ 90.0000 0.0268454
$$225$$ 0 0
$$226$$ −3357.00 −0.988072
$$227$$ 1398.00 0.408760 0.204380 0.978892i $$-0.434482\pi$$
0.204380 + 0.978892i $$0.434482\pi$$
$$228$$ 0 0
$$229$$ 4466.00 1.28874 0.644370 0.764714i $$-0.277120\pi$$
0.644370 + 0.764714i $$0.277120\pi$$
$$230$$ −162.000 −0.0464433
$$231$$ 0 0
$$232$$ 2205.00 0.623989
$$233$$ 1638.00 0.460553 0.230277 0.973125i $$-0.426037\pi$$
0.230277 + 0.973125i $$0.426037\pi$$
$$234$$ 0 0
$$235$$ 1458.00 0.404721
$$236$$ −600.000 −0.165494
$$237$$ 0 0
$$238$$ −666.000 −0.181388
$$239$$ 594.000 0.160764 0.0803821 0.996764i $$-0.474386\pi$$
0.0803821 + 0.996764i $$0.474386\pi$$
$$240$$ 0 0
$$241$$ 2303.00 0.615557 0.307779 0.951458i $$-0.400414\pi$$
0.307779 + 0.951458i $$0.400414\pi$$
$$242$$ 1293.00 0.343459
$$243$$ 0 0
$$244$$ 233.000 0.0611324
$$245$$ −3051.00 −0.795597
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2100.00 −0.537702
$$249$$ 0 0
$$250$$ 4563.00 1.15436
$$251$$ −6324.00 −1.59031 −0.795154 0.606407i $$-0.792610\pi$$
−0.795154 + 0.606407i $$0.792610\pi$$
$$252$$ 0 0
$$253$$ −180.000 −0.0447293
$$254$$ 1812.00 0.447618
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ −7833.00 −1.90120 −0.950601 0.310414i $$-0.899532\pi$$
−0.950601 + 0.310414i $$0.899532\pi$$
$$258$$ 0 0
$$259$$ 34.0000 0.00815698
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −4752.00 −1.12053
$$263$$ 3030.00 0.710410 0.355205 0.934788i $$-0.384411\pi$$
0.355205 + 0.934788i $$0.384411\pi$$
$$264$$ 0 0
$$265$$ −5751.00 −1.33314
$$266$$ 276.000 0.0636190
$$267$$ 0 0
$$268$$ 926.000 0.211061
$$269$$ 534.000 0.121036 0.0605178 0.998167i $$-0.480725\pi$$
0.0605178 + 0.998167i $$0.480725\pi$$
$$270$$ 0 0
$$271$$ −3688.00 −0.826679 −0.413340 0.910577i $$-0.635638\pi$$
−0.413340 + 0.910577i $$0.635638\pi$$
$$272$$ −7881.00 −1.75682
$$273$$ 0 0
$$274$$ 2151.00 0.474258
$$275$$ 1320.00 0.289451
$$276$$ 0 0
$$277$$ 1865.00 0.404538 0.202269 0.979330i $$-0.435168\pi$$
0.202269 + 0.979330i $$0.435168\pi$$
$$278$$ 2460.00 0.530723
$$279$$ 0 0
$$280$$ 378.000 0.0806779
$$281$$ −2997.00 −0.636249 −0.318125 0.948049i $$-0.603053\pi$$
−0.318125 + 0.948049i $$0.603053\pi$$
$$282$$ 0 0
$$283$$ −4114.00 −0.864141 −0.432071 0.901840i $$-0.642217\pi$$
−0.432071 + 0.901840i $$0.642217\pi$$
$$284$$ 930.000 0.194315
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 462.000 0.0950209
$$288$$ 0 0
$$289$$ 7408.00 1.50784
$$290$$ −2835.00 −0.574058
$$291$$ 0 0
$$292$$ −253.000 −0.0507045
$$293$$ 4665.00 0.930144 0.465072 0.885273i $$-0.346028\pi$$
0.465072 + 0.885273i $$0.346028\pi$$
$$294$$ 0 0
$$295$$ −5400.00 −1.06576
$$296$$ 357.000 0.0701020
$$297$$ 0 0
$$298$$ −5247.00 −1.01997
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1028.00 −0.196854
$$302$$ 1110.00 0.211501
$$303$$ 0 0
$$304$$ 3266.00 0.616177
$$305$$ 2097.00 0.393685
$$306$$ 0 0
$$307$$ 1502.00 0.279230 0.139615 0.990206i $$-0.455413\pi$$
0.139615 + 0.990206i $$0.455413\pi$$
$$308$$ −60.0000 −0.0111001
$$309$$ 0 0
$$310$$ 2700.00 0.494676
$$311$$ −2106.00 −0.383988 −0.191994 0.981396i $$-0.561495\pi$$
−0.191994 + 0.981396i $$0.561495\pi$$
$$312$$ 0 0
$$313$$ −3898.00 −0.703923 −0.351962 0.936014i $$-0.614485\pi$$
−0.351962 + 0.936014i $$0.614485\pi$$
$$314$$ 7833.00 1.40778
$$315$$ 0 0
$$316$$ −1324.00 −0.235699
$$317$$ −9351.00 −1.65680 −0.828398 0.560140i $$-0.810747\pi$$
−0.828398 + 0.560140i $$0.810747\pi$$
$$318$$ 0 0
$$319$$ −3150.00 −0.552872
$$320$$ 3897.00 0.680778
$$321$$ 0 0
$$322$$ −36.0000 −0.00623044
$$323$$ −5106.00 −0.879583
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4908.00 0.833831
$$327$$ 0 0
$$328$$ 4851.00 0.816621
$$329$$ 324.000 0.0542939
$$330$$ 0 0
$$331$$ −9172.00 −1.52308 −0.761539 0.648119i $$-0.775556\pi$$
−0.761539 + 0.648119i $$0.775556\pi$$
$$332$$ −810.000 −0.133899
$$333$$ 0 0
$$334$$ 792.000 0.129749
$$335$$ 8334.00 1.35921
$$336$$ 0 0
$$337$$ −11089.0 −1.79245 −0.896226 0.443598i $$-0.853702\pi$$
−0.896226 + 0.443598i $$0.853702\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 999.000 0.159348
$$341$$ 3000.00 0.476420
$$342$$ 0 0
$$343$$ −1364.00 −0.214720
$$344$$ −10794.0 −1.69178
$$345$$ 0 0
$$346$$ 4230.00 0.657243
$$347$$ −9762.00 −1.51024 −0.755118 0.655589i $$-0.772420\pi$$
−0.755118 + 0.655589i $$0.772420\pi$$
$$348$$ 0 0
$$349$$ −8290.00 −1.27150 −0.635750 0.771895i $$-0.719309\pi$$
−0.635750 + 0.771895i $$0.719309\pi$$
$$350$$ 264.000 0.0403183
$$351$$ 0 0
$$352$$ −1350.00 −0.204418
$$353$$ −12405.0 −1.87040 −0.935200 0.354119i $$-0.884781\pi$$
−0.935200 + 0.354119i $$0.884781\pi$$
$$354$$ 0 0
$$355$$ 8370.00 1.25136
$$356$$ −498.000 −0.0741403
$$357$$ 0 0
$$358$$ −1422.00 −0.209930
$$359$$ 1098.00 0.161421 0.0807106 0.996738i $$-0.474281\pi$$
0.0807106 + 0.996738i $$0.474281\pi$$
$$360$$ 0 0
$$361$$ −4743.00 −0.691500
$$362$$ −6747.00 −0.979598
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2277.00 −0.326530
$$366$$ 0 0
$$367$$ −5734.00 −0.815565 −0.407783 0.913079i $$-0.633698\pi$$
−0.407783 + 0.913079i $$0.633698\pi$$
$$368$$ −426.000 −0.0603445
$$369$$ 0 0
$$370$$ −459.000 −0.0644926
$$371$$ −1278.00 −0.178842
$$372$$ 0 0
$$373$$ −8971.00 −1.24531 −0.622655 0.782496i $$-0.713946\pi$$
−0.622655 + 0.782496i $$0.713946\pi$$
$$374$$ 9990.00 1.38120
$$375$$ 0 0
$$376$$ 3402.00 0.466608
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 7244.00 0.981792 0.490896 0.871218i $$-0.336669\pi$$
0.490896 + 0.871218i $$0.336669\pi$$
$$380$$ −414.000 −0.0558888
$$381$$ 0 0
$$382$$ 10332.0 1.38385
$$383$$ 6312.00 0.842110 0.421055 0.907035i $$-0.361660\pi$$
0.421055 + 0.907035i $$0.361660\pi$$
$$384$$ 0 0
$$385$$ −540.000 −0.0714830
$$386$$ 12819.0 1.69034
$$387$$ 0 0
$$388$$ 1358.00 0.177686
$$389$$ −3627.00 −0.472741 −0.236370 0.971663i $$-0.575958\pi$$
−0.236370 + 0.971663i $$0.575958\pi$$
$$390$$ 0 0
$$391$$ 666.000 0.0861408
$$392$$ −7119.00 −0.917255
$$393$$ 0 0
$$394$$ −5958.00 −0.761827
$$395$$ −11916.0 −1.51787
$$396$$ 0 0
$$397$$ −3898.00 −0.492783 −0.246392 0.969170i $$-0.579245\pi$$
−0.246392 + 0.969170i $$0.579245\pi$$
$$398$$ 7158.00 0.901503
$$399$$ 0 0
$$400$$ 3124.00 0.390500
$$401$$ 5703.00 0.710210 0.355105 0.934826i $$-0.384445\pi$$
0.355105 + 0.934826i $$0.384445\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 357.000 0.0439639
$$405$$ 0 0
$$406$$ −630.000 −0.0770108
$$407$$ −510.000 −0.0621124
$$408$$ 0 0
$$409$$ 6311.00 0.762980 0.381490 0.924373i $$-0.375411\pi$$
0.381490 + 0.924373i $$0.375411\pi$$
$$410$$ −6237.00 −0.751277
$$411$$ 0 0
$$412$$ 1118.00 0.133689
$$413$$ −1200.00 −0.142974
$$414$$ 0 0
$$415$$ −7290.00 −0.862294
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −4140.00 −0.484435
$$419$$ 2328.00 0.271433 0.135716 0.990748i $$-0.456666\pi$$
0.135716 + 0.990748i $$0.456666\pi$$
$$420$$ 0 0
$$421$$ 2045.00 0.236739 0.118370 0.992970i $$-0.462233\pi$$
0.118370 + 0.992970i $$0.462233\pi$$
$$422$$ 4800.00 0.553697
$$423$$ 0 0
$$424$$ −13419.0 −1.53699
$$425$$ −4884.00 −0.557432
$$426$$ 0 0
$$427$$ 466.000 0.0528134
$$428$$ −714.000 −0.0806367
$$429$$ 0 0
$$430$$ 13878.0 1.55641
$$431$$ −5034.00 −0.562597 −0.281298 0.959620i $$-0.590765\pi$$
−0.281298 + 0.959620i $$0.590765\pi$$
$$432$$ 0 0
$$433$$ 4283.00 0.475353 0.237676 0.971344i $$-0.423614\pi$$
0.237676 + 0.971344i $$0.423614\pi$$
$$434$$ 600.000 0.0663616
$$435$$ 0 0
$$436$$ 2006.00 0.220344
$$437$$ −276.000 −0.0302125
$$438$$ 0 0
$$439$$ −1306.00 −0.141986 −0.0709931 0.997477i $$-0.522617\pi$$
−0.0709931 + 0.997477i $$0.522617\pi$$
$$440$$ −5670.00 −0.614333
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5796.00 0.621617 0.310808 0.950473i $$-0.399400\pi$$
0.310808 + 0.950473i $$0.399400\pi$$
$$444$$ 0 0
$$445$$ −4482.00 −0.477454
$$446$$ 11496.0 1.22052
$$447$$ 0 0
$$448$$ 866.000 0.0913274
$$449$$ −2706.00 −0.284419 −0.142209 0.989837i $$-0.545421\pi$$
−0.142209 + 0.989837i $$0.545421\pi$$
$$450$$ 0 0
$$451$$ −6930.00 −0.723550
$$452$$ 1119.00 0.116445
$$453$$ 0 0
$$454$$ −4194.00 −0.433555
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −829.000 −0.0848555 −0.0424278 0.999100i $$-0.513509\pi$$
−0.0424278 + 0.999100i $$0.513509\pi$$
$$458$$ −13398.0 −1.36692
$$459$$ 0 0
$$460$$ 54.0000 0.00547340
$$461$$ 5493.00 0.554956 0.277478 0.960732i $$-0.410502\pi$$
0.277478 + 0.960732i $$0.410502\pi$$
$$462$$ 0 0
$$463$$ −15346.0 −1.54037 −0.770183 0.637823i $$-0.779835\pi$$
−0.770183 + 0.637823i $$0.779835\pi$$
$$464$$ −7455.00 −0.745883
$$465$$ 0 0
$$466$$ −4914.00 −0.488491
$$467$$ 9594.00 0.950658 0.475329 0.879808i $$-0.342329\pi$$
0.475329 + 0.879808i $$0.342329\pi$$
$$468$$ 0 0
$$469$$ 1852.00 0.182340
$$470$$ −4374.00 −0.429271
$$471$$ 0 0
$$472$$ −12600.0 −1.22873
$$473$$ 15420.0 1.49897
$$474$$ 0 0
$$475$$ 2024.00 0.195511
$$476$$ 222.000 0.0213768
$$477$$ 0 0
$$478$$ −1782.00 −0.170516
$$479$$ 12840.0 1.22479 0.612395 0.790552i $$-0.290206\pi$$
0.612395 + 0.790552i $$0.290206\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −6909.00 −0.652897
$$483$$ 0 0
$$484$$ −431.000 −0.0404771
$$485$$ 12222.0 1.14427
$$486$$ 0 0
$$487$$ −14086.0 −1.31067 −0.655336 0.755337i $$-0.727473\pi$$
−0.655336 + 0.755337i $$0.727473\pi$$
$$488$$ 4893.00 0.453885
$$489$$ 0 0
$$490$$ 9153.00 0.843858
$$491$$ −11694.0 −1.07483 −0.537416 0.843317i $$-0.680600\pi$$
−0.537416 + 0.843317i $$0.680600\pi$$
$$492$$ 0 0
$$493$$ 11655.0 1.06474
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 7100.00 0.642741
$$497$$ 1860.00 0.167872
$$498$$ 0 0
$$499$$ −3688.00 −0.330857 −0.165428 0.986222i $$-0.552901\pi$$
−0.165428 + 0.986222i $$0.552901\pi$$
$$500$$ −1521.00 −0.136042
$$501$$ 0 0
$$502$$ 18972.0 1.68678
$$503$$ 4746.00 0.420703 0.210352 0.977626i $$-0.432539\pi$$
0.210352 + 0.977626i $$0.432539\pi$$
$$504$$ 0 0
$$505$$ 3213.00 0.283122
$$506$$ 540.000 0.0474425
$$507$$ 0 0
$$508$$ −604.000 −0.0527523
$$509$$ 14505.0 1.26311 0.631555 0.775331i $$-0.282417\pi$$
0.631555 + 0.775331i $$0.282417\pi$$
$$510$$ 0 0
$$511$$ −506.000 −0.0438045
$$512$$ 8733.00 0.753804
$$513$$ 0 0
$$514$$ 23499.0 2.01653
$$515$$ 10062.0 0.860941
$$516$$ 0 0
$$517$$ −4860.00 −0.413429
$$518$$ −102.000 −0.00865178
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −5085.00 −0.427597 −0.213798 0.976878i $$-0.568584\pi$$
−0.213798 + 0.976878i $$0.568584\pi$$
$$522$$ 0 0
$$523$$ −10882.0 −0.909821 −0.454911 0.890537i $$-0.650329\pi$$
−0.454911 + 0.890537i $$0.650329\pi$$
$$524$$ 1584.00 0.132056
$$525$$ 0 0
$$526$$ −9090.00 −0.753503
$$527$$ −11100.0 −0.917502
$$528$$ 0 0
$$529$$ −12131.0 −0.997041
$$530$$ 17253.0 1.41400
$$531$$ 0 0
$$532$$ −92.0000 −0.00749757
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −6426.00 −0.519290
$$536$$ 19446.0 1.56705
$$537$$ 0 0
$$538$$ −1602.00 −0.128378
$$539$$ 10170.0 0.812714
$$540$$ 0 0
$$541$$ −4699.00 −0.373430 −0.186715 0.982414i $$-0.559784\pi$$
−0.186715 + 0.982414i $$0.559784\pi$$
$$542$$ 11064.0 0.876826
$$543$$ 0 0
$$544$$ 4995.00 0.393674
$$545$$ 18054.0 1.41899
$$546$$ 0 0
$$547$$ 8270.00 0.646434 0.323217 0.946325i $$-0.395236\pi$$
0.323217 + 0.946325i $$0.395236\pi$$
$$548$$ −717.000 −0.0558918
$$549$$ 0 0
$$550$$ −3960.00 −0.307009
$$551$$ −4830.00 −0.373439
$$552$$ 0 0
$$553$$ −2648.00 −0.203625
$$554$$ −5595.00 −0.429077
$$555$$ 0 0
$$556$$ −820.000 −0.0625463
$$557$$ 22785.0 1.73327 0.866635 0.498943i $$-0.166278\pi$$
0.866635 + 0.498943i $$0.166278\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −1278.00 −0.0964381
$$561$$ 0 0
$$562$$ 8991.00 0.674844
$$563$$ 11928.0 0.892905 0.446452 0.894807i $$-0.352687\pi$$
0.446452 + 0.894807i $$0.352687\pi$$
$$564$$ 0 0
$$565$$ 10071.0 0.749894
$$566$$ 12342.0 0.916560
$$567$$ 0 0
$$568$$ 19530.0 1.44271
$$569$$ 7962.00 0.586616 0.293308 0.956018i $$-0.405244\pi$$
0.293308 + 0.956018i $$0.405244\pi$$
$$570$$ 0 0
$$571$$ 20618.0 1.51110 0.755549 0.655093i $$-0.227370\pi$$
0.755549 + 0.655093i $$0.227370\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1386.00 −0.100785
$$575$$ −264.000 −0.0191471
$$576$$ 0 0
$$577$$ −3493.00 −0.252020 −0.126010 0.992029i $$-0.540217\pi$$
−0.126010 + 0.992029i $$0.540217\pi$$
$$578$$ −22224.0 −1.59930
$$579$$ 0 0
$$580$$ 945.000 0.0676534
$$581$$ −1620.00 −0.115678
$$582$$ 0 0
$$583$$ 19170.0 1.36182
$$584$$ −5313.00 −0.376461
$$585$$ 0 0
$$586$$ −13995.0 −0.986567
$$587$$ −10416.0 −0.732392 −0.366196 0.930538i $$-0.619340\pi$$
−0.366196 + 0.930538i $$0.619340\pi$$
$$588$$ 0 0
$$589$$ 4600.00 0.321799
$$590$$ 16200.0 1.13041
$$591$$ 0 0
$$592$$ −1207.00 −0.0837963
$$593$$ −2061.00 −0.142724 −0.0713618 0.997450i $$-0.522734\pi$$
−0.0713618 + 0.997450i $$0.522734\pi$$
$$594$$ 0 0
$$595$$ 1998.00 0.137664
$$596$$ 1749.00 0.120204
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12456.0 −0.849647 −0.424823 0.905276i $$-0.639664\pi$$
−0.424823 + 0.905276i $$0.639664\pi$$
$$600$$ 0 0
$$601$$ −781.000 −0.0530077 −0.0265039 0.999649i $$-0.508437\pi$$
−0.0265039 + 0.999649i $$0.508437\pi$$
$$602$$ 3084.00 0.208795
$$603$$ 0 0
$$604$$ −370.000 −0.0249256
$$605$$ −3879.00 −0.260667
$$606$$ 0 0
$$607$$ 19304.0 1.29082 0.645408 0.763838i $$-0.276687\pi$$
0.645408 + 0.763838i $$0.276687\pi$$
$$608$$ −2070.00 −0.138075
$$609$$ 0 0
$$610$$ −6291.00 −0.417566
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12041.0 0.793363 0.396681 0.917956i $$-0.370162\pi$$
0.396681 + 0.917956i $$0.370162\pi$$
$$614$$ −4506.00 −0.296168
$$615$$ 0 0
$$616$$ −1260.00 −0.0824137
$$617$$ −9717.00 −0.634022 −0.317011 0.948422i $$-0.602679\pi$$
−0.317011 + 0.948422i $$0.602679\pi$$
$$618$$ 0 0
$$619$$ −21040.0 −1.36619 −0.683093 0.730332i $$-0.739366\pi$$
−0.683093 + 0.730332i $$0.739366\pi$$
$$620$$ −900.000 −0.0582982
$$621$$ 0 0
$$622$$ 6318.00 0.407281
$$623$$ −996.000 −0.0640512
$$624$$ 0 0
$$625$$ −8189.00 −0.524096
$$626$$ 11694.0 0.746623
$$627$$ 0 0
$$628$$ −2611.00 −0.165908
$$629$$ 1887.00 0.119618
$$630$$ 0 0
$$631$$ −5068.00 −0.319737 −0.159868 0.987138i $$-0.551107\pi$$
−0.159868 + 0.987138i $$0.551107\pi$$
$$632$$ −27804.0 −1.74997
$$633$$ 0 0
$$634$$ 28053.0 1.75730
$$635$$ −5436.00 −0.339718
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 9450.00 0.586409
$$639$$ 0 0
$$640$$ −14931.0 −0.922187
$$641$$ −10185.0 −0.627587 −0.313794 0.949491i $$-0.601600\pi$$
−0.313794 + 0.949491i $$0.601600\pi$$
$$642$$ 0 0
$$643$$ 25928.0 1.59020 0.795101 0.606476i $$-0.207418\pi$$
0.795101 + 0.606476i $$0.207418\pi$$
$$644$$ 12.0000 0.000734264 0
$$645$$ 0 0
$$646$$ 15318.0 0.932939
$$647$$ −23160.0 −1.40729 −0.703643 0.710554i $$-0.748444\pi$$
−0.703643 + 0.710554i $$0.748444\pi$$
$$648$$ 0 0
$$649$$ 18000.0 1.08869
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1636.00 −0.0982680
$$653$$ −16626.0 −0.996364 −0.498182 0.867073i $$-0.665999\pi$$
−0.498182 + 0.867073i $$0.665999\pi$$
$$654$$ 0 0
$$655$$ 14256.0 0.850424
$$656$$ −16401.0 −0.976146
$$657$$ 0 0
$$658$$ −972.000 −0.0575874
$$659$$ 14808.0 0.875323 0.437661 0.899140i $$-0.355807\pi$$
0.437661 + 0.899140i $$0.355807\pi$$
$$660$$ 0 0
$$661$$ 4853.00 0.285567 0.142784 0.989754i $$-0.454395\pi$$
0.142784 + 0.989754i $$0.454395\pi$$
$$662$$ 27516.0 1.61547
$$663$$ 0 0
$$664$$ −17010.0 −0.994151
$$665$$ −828.000 −0.0482834
$$666$$ 0 0
$$667$$ 630.000 0.0365723
$$668$$ −264.000 −0.0152911
$$669$$ 0 0
$$670$$ −25002.0 −1.44166
$$671$$ −6990.00 −0.402155
$$672$$ 0 0
$$673$$ −16165.0 −0.925877 −0.462938 0.886391i $$-0.653205\pi$$
−0.462938 + 0.886391i $$0.653205\pi$$
$$674$$ 33267.0 1.90118
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25686.0 1.45819 0.729094 0.684414i $$-0.239942\pi$$
0.729094 + 0.684414i $$0.239942\pi$$
$$678$$ 0 0
$$679$$ 2716.00 0.153506
$$680$$ 20979.0 1.18310
$$681$$ 0 0
$$682$$ −9000.00 −0.505319
$$683$$ 19056.0 1.06758 0.533790 0.845617i $$-0.320767\pi$$
0.533790 + 0.845617i $$0.320767\pi$$
$$684$$ 0 0
$$685$$ −6453.00 −0.359936
$$686$$ 4092.00 0.227745
$$687$$ 0 0
$$688$$ 36494.0 2.02227
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −16390.0 −0.902323 −0.451161 0.892442i $$-0.648990\pi$$
−0.451161 + 0.892442i $$0.648990\pi$$
$$692$$ −1410.00 −0.0774569
$$693$$ 0 0
$$694$$ 29286.0 1.60185
$$695$$ −7380.00 −0.402790
$$696$$ 0 0
$$697$$ 25641.0 1.39343
$$698$$ 24870.0 1.34863
$$699$$ 0 0
$$700$$ −88.0000 −0.00475155
$$701$$ 27846.0 1.50033 0.750163 0.661253i $$-0.229975\pi$$
0.750163 + 0.661253i $$0.229975\pi$$
$$702$$ 0 0
$$703$$ −782.000 −0.0419540
$$704$$ −12990.0 −0.695425
$$705$$ 0 0
$$706$$ 37215.0 1.98386
$$707$$ 714.000 0.0379812
$$708$$ 0 0
$$709$$ −12283.0 −0.650632 −0.325316 0.945605i $$-0.605471\pi$$
−0.325316 + 0.945605i $$0.605471\pi$$
$$710$$ −25110.0 −1.32727
$$711$$ 0 0
$$712$$ −10458.0 −0.550464
$$713$$ −600.000 −0.0315150
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 474.000 0.0247405
$$717$$ 0 0
$$718$$ −3294.00 −0.171213
$$719$$ −25512.0 −1.32328 −0.661639 0.749822i $$-0.730139\pi$$
−0.661639 + 0.749822i $$0.730139\pi$$
$$720$$ 0 0
$$721$$ 2236.00 0.115497
$$722$$ 14229.0 0.733447
$$723$$ 0 0
$$724$$ 2249.00 0.115447
$$725$$ −4620.00 −0.236666
$$726$$ 0 0
$$727$$ 6110.00 0.311702 0.155851 0.987781i $$-0.450188\pi$$
0.155851 + 0.987781i $$0.450188\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 6831.00 0.346338
$$731$$ −57054.0 −2.88676
$$732$$ 0 0
$$733$$ −27127.0 −1.36693 −0.683464 0.729984i $$-0.739527\pi$$
−0.683464 + 0.729984i $$0.739527\pi$$
$$734$$ 17202.0 0.865037
$$735$$ 0 0
$$736$$ 270.000 0.0135222
$$737$$ −27780.0 −1.38845
$$738$$ 0 0
$$739$$ −880.000 −0.0438042 −0.0219021 0.999760i $$-0.506972\pi$$
−0.0219021 + 0.999760i $$0.506972\pi$$
$$740$$ 153.000 0.00760053
$$741$$ 0 0
$$742$$ 3834.00 0.189691
$$743$$ 21876.0 1.08015 0.540076 0.841616i $$-0.318396\pi$$
0.540076 + 0.841616i $$0.318396\pi$$
$$744$$ 0 0
$$745$$ 15741.0 0.774102
$$746$$ 26913.0 1.32085
$$747$$ 0 0
$$748$$ −3330.00 −0.162777
$$749$$ −1428.00 −0.0696635
$$750$$ 0 0
$$751$$ 11798.0 0.573256 0.286628 0.958042i $$-0.407466\pi$$
0.286628 + 0.958042i $$0.407466\pi$$
$$752$$ −11502.0 −0.557759
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −3330.00 −0.160518
$$756$$ 0 0
$$757$$ −8074.00 −0.387655 −0.193827 0.981036i $$-0.562090\pi$$
−0.193827 + 0.981036i $$0.562090\pi$$
$$758$$ −21732.0 −1.04135
$$759$$ 0 0
$$760$$ −8694.00 −0.414953
$$761$$ −19554.0 −0.931448 −0.465724 0.884930i $$-0.654206\pi$$
−0.465724 + 0.884930i $$0.654206\pi$$
$$762$$ 0 0
$$763$$ 4012.00 0.190359
$$764$$ −3444.00 −0.163088
$$765$$ 0 0
$$766$$ −18936.0 −0.893193
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 14030.0 0.657913 0.328956 0.944345i $$-0.393303\pi$$
0.328956 + 0.944345i $$0.393303\pi$$
$$770$$ 1620.00 0.0758192
$$771$$ 0 0
$$772$$ −4273.00 −0.199208
$$773$$ −36042.0 −1.67703 −0.838513 0.544882i $$-0.816574\pi$$
−0.838513 + 0.544882i $$0.816574\pi$$
$$774$$ 0 0
$$775$$ 4400.00 0.203939
$$776$$ 28518.0 1.31925
$$777$$ 0 0
$$778$$ 10881.0 0.501417
$$779$$ −10626.0 −0.488724
$$780$$ 0 0
$$781$$ −27900.0 −1.27828
$$782$$ −1998.00 −0.0913662
$$783$$ 0 0
$$784$$ 24069.0 1.09644
$$785$$ −23499.0 −1.06843
$$786$$ 0 0
$$787$$ 28628.0 1.29667 0.648334 0.761356i $$-0.275466\pi$$
0.648334 + 0.761356i $$0.275466\pi$$
$$788$$ 1986.00 0.0897821
$$789$$ 0 0
$$790$$ 35748.0 1.60995
$$791$$ 2238.00 0.100599
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 11694.0 0.522676
$$795$$ 0 0
$$796$$ −2386.00 −0.106243
$$797$$ 37434.0 1.66371 0.831857 0.554990i $$-0.187278\pi$$
0.831857 + 0.554990i $$0.187278\pi$$
$$798$$ 0 0
$$799$$ 17982.0 0.796192
$$800$$ −1980.00 −0.0875045
$$801$$ 0 0
$$802$$ −17109.0 −0.753292
$$803$$ 7590.00 0.333556
$$804$$ 0 0
$$805$$ 108.000 0.00472857
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 7497.00 0.326415
$$809$$ −37569.0 −1.63270 −0.816351 0.577556i $$-0.804006\pi$$
−0.816351 + 0.577556i $$0.804006\pi$$
$$810$$ 0 0
$$811$$ 5516.00 0.238832 0.119416 0.992844i $$-0.461898\pi$$
0.119416 + 0.992844i $$0.461898\pi$$
$$812$$ 210.000 0.00907581
$$813$$ 0 0
$$814$$ 1530.00 0.0658802
$$815$$ −14724.0 −0.632833
$$816$$ 0 0
$$817$$ 23644.0 1.01248
$$818$$ −18933.0 −0.809263
$$819$$ 0 0
$$820$$ 2079.00 0.0885388
$$821$$ −8778.00 −0.373148 −0.186574 0.982441i $$-0.559738\pi$$
−0.186574 + 0.982441i $$0.559738\pi$$
$$822$$ 0 0
$$823$$ −3088.00 −0.130791 −0.0653955 0.997859i $$-0.520831\pi$$
−0.0653955 + 0.997859i $$0.520831\pi$$
$$824$$ 23478.0 0.992591
$$825$$ 0 0
$$826$$ 3600.00 0.151647
$$827$$ −13176.0 −0.554020 −0.277010 0.960867i $$-0.589343\pi$$
−0.277010 + 0.960867i $$0.589343\pi$$
$$828$$ 0 0
$$829$$ −2359.00 −0.0988317 −0.0494158 0.998778i $$-0.515736\pi$$
−0.0494158 + 0.998778i $$0.515736\pi$$
$$830$$ 21870.0 0.914601
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −37629.0 −1.56515
$$834$$ 0 0
$$835$$ −2376.00 −0.0984729
$$836$$ 1380.00 0.0570913
$$837$$ 0 0
$$838$$ −6984.00 −0.287898
$$839$$ 2676.00 0.110114 0.0550571 0.998483i $$-0.482466\pi$$
0.0550571 + 0.998483i $$0.482466\pi$$
$$840$$ 0 0
$$841$$ −13364.0 −0.547952
$$842$$ −6135.00 −0.251100
$$843$$ 0 0
$$844$$ −1600.00 −0.0652539
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −862.000 −0.0349689
$$848$$ 45369.0 1.83724
$$849$$ 0 0
$$850$$ 14652.0 0.591246
$$851$$ 102.000 0.00410871
$$852$$ 0 0
$$853$$ 2477.00 0.0994266 0.0497133 0.998764i $$-0.484169\pi$$
0.0497133 + 0.998764i $$0.484169\pi$$
$$854$$ −1398.00 −0.0560171
$$855$$ 0 0
$$856$$ −14994.0 −0.598697
$$857$$ 17199.0 0.685539 0.342769 0.939420i $$-0.388635\pi$$
0.342769 + 0.939420i $$0.388635\pi$$
$$858$$ 0 0
$$859$$ 24338.0 0.966708 0.483354 0.875425i $$-0.339418\pi$$
0.483354 + 0.875425i $$0.339418\pi$$
$$860$$ −4626.00 −0.183425
$$861$$ 0 0
$$862$$ 15102.0 0.596724
$$863$$ −25146.0 −0.991865 −0.495933 0.868361i $$-0.665174\pi$$
−0.495933 + 0.868361i $$0.665174\pi$$
$$864$$ 0 0
$$865$$ −12690.0 −0.498813
$$866$$ −12849.0 −0.504188
$$867$$ 0 0
$$868$$ −200.000 −0.00782079
$$869$$ 39720.0 1.55053
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 42126.0 1.63597
$$873$$ 0 0
$$874$$ 828.000 0.0320452
$$875$$ −3042.00 −0.117530
$$876$$ 0 0
$$877$$ 18089.0 0.696490 0.348245 0.937403i $$-0.386778\pi$$
0.348245 + 0.937403i $$0.386778\pi$$
$$878$$ 3918.00 0.150599
$$879$$ 0 0
$$880$$ 19170.0 0.734342
$$881$$ 15099.0 0.577410 0.288705 0.957418i $$-0.406775\pi$$
0.288705 + 0.957418i $$0.406775\pi$$
$$882$$ 0 0
$$883$$ 33488.0 1.27629 0.638143 0.769918i $$-0.279703\pi$$
0.638143 + 0.769918i $$0.279703\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −17388.0 −0.659324
$$887$$ 39768.0 1.50539 0.752694 0.658371i $$-0.228754\pi$$
0.752694 + 0.658371i $$0.228754\pi$$
$$888$$ 0 0
$$889$$ −1208.00 −0.0455737
$$890$$ 13446.0 0.506417
$$891$$ 0 0
$$892$$ −3832.00 −0.143840
$$893$$ −7452.00 −0.279252
$$894$$ 0 0
$$895$$ 4266.00 0.159326
$$896$$ −3318.00 −0.123713
$$897$$ 0 0
$$898$$ 8118.00 0.301672
$$899$$ −10500.0 −0.389538
$$900$$ 0 0
$$901$$ −70929.0 −2.62263
$$902$$ 20790.0 0.767440
$$903$$ 0 0
$$904$$ 23499.0 0.864563
$$905$$ 20241.0 0.743463
$$906$$ 0 0
$$907$$ 32156.0 1.17720 0.588601 0.808424i $$-0.299679\pi$$
0.588601 + 0.808424i $$0.299679\pi$$
$$908$$ 1398.00 0.0510950
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −11520.0 −0.418962 −0.209481 0.977813i $$-0.567177\pi$$
−0.209481 + 0.977813i $$0.567177\pi$$
$$912$$ 0 0
$$913$$ 24300.0 0.880846
$$914$$ 2487.00 0.0900029
$$915$$ 0 0
$$916$$ 4466.00 0.161093
$$917$$ 3168.00 0.114086
$$918$$ 0 0
$$919$$ 4952.00 0.177749 0.0888745 0.996043i $$-0.471673\pi$$
0.0888745 + 0.996043i $$0.471673\pi$$
$$920$$ 1134.00 0.0406379
$$921$$ 0 0
$$922$$ −16479.0 −0.588619
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −748.000 −0.0265882
$$926$$ 46038.0 1.63380
$$927$$ 0 0
$$928$$ 4725.00 0.167140
$$929$$ −8781.00 −0.310113 −0.155057 0.987906i $$-0.549556\pi$$
−0.155057 + 0.987906i $$0.549556\pi$$
$$930$$ 0 0
$$931$$ 15594.0 0.548950
$$932$$ 1638.00 0.0575692
$$933$$ 0 0
$$934$$ −28782.0 −1.00833
$$935$$ −29970.0 −1.04826
$$936$$ 0 0
$$937$$ 50039.0 1.74461 0.872307 0.488959i $$-0.162623\pi$$
0.872307 + 0.488959i $$0.162623\pi$$
$$938$$ −5556.00 −0.193401
$$939$$ 0 0
$$940$$ 1458.00 0.0505901
$$941$$ 50670.0 1.75536 0.877681 0.479246i $$-0.159090\pi$$
0.877681 + 0.479246i $$0.159090\pi$$
$$942$$ 0 0
$$943$$ 1386.00 0.0478625
$$944$$ 42600.0 1.46876
$$945$$ 0 0
$$946$$ −46260.0 −1.58990
$$947$$ −42384.0 −1.45438 −0.727188 0.686438i $$-0.759173\pi$$
−0.727188 + 0.686438i $$0.759173\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −6072.00 −0.207370
$$951$$ 0 0
$$952$$ 4662.00 0.158715
$$953$$ 50538.0 1.71782 0.858912 0.512123i $$-0.171141\pi$$
0.858912 + 0.512123i $$0.171141\pi$$
$$954$$ 0 0
$$955$$ −30996.0 −1.05027
$$956$$ 594.000 0.0200955
$$957$$ 0 0
$$958$$ −38520.0 −1.29909
$$959$$ −1434.00 −0.0482860
$$960$$ 0 0
$$961$$ −19791.0 −0.664328
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 2303.00 0.0769446
$$965$$ −38457.0 −1.28288
$$966$$ 0 0
$$967$$ −6886.00 −0.228996 −0.114498 0.993423i $$-0.536526\pi$$
−0.114498 + 0.993423i $$0.536526\pi$$
$$968$$ −9051.00 −0.300527
$$969$$ 0 0
$$970$$ −36666.0 −1.21368
$$971$$ −9060.00 −0.299433 −0.149716 0.988729i $$-0.547836\pi$$
−0.149716 + 0.988729i $$0.547836\pi$$
$$972$$ 0 0
$$973$$ −1640.00 −0.0540349
$$974$$ 42258.0 1.39018
$$975$$ 0 0
$$976$$ −16543.0 −0.542550
$$977$$ 28311.0 0.927072 0.463536 0.886078i $$-0.346581\pi$$
0.463536 + 0.886078i $$0.346581\pi$$
$$978$$ 0 0
$$979$$ 14940.0 0.487727
$$980$$ −3051.00 −0.0994496
$$981$$ 0 0
$$982$$ 35082.0 1.14003
$$983$$ −4284.00 −0.139001 −0.0695007 0.997582i $$-0.522141\pi$$
−0.0695007 + 0.997582i $$0.522141\pi$$
$$984$$ 0 0
$$985$$ 17874.0 0.578186
$$986$$ −34965.0 −1.12932
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3084.00 −0.0991562
$$990$$ 0 0
$$991$$ −2458.00 −0.0787901 −0.0393950 0.999224i $$-0.512543\pi$$
−0.0393950 + 0.999224i $$0.512543\pi$$
$$992$$ −4500.00 −0.144027
$$993$$ 0 0
$$994$$ −5580.00 −0.178055
$$995$$ −21474.0 −0.684193
$$996$$ 0 0
$$997$$ 24101.0 0.765583 0.382792 0.923835i $$-0.374963\pi$$
0.382792 + 0.923835i $$0.374963\pi$$
$$998$$ 11064.0 0.350927
$$999$$ 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 1521.4.a.c.1.1 1
3.2 odd 2 507.4.a.e.1.1 1
13.3 even 3 117.4.g.b.100.1 2
13.9 even 3 117.4.g.b.55.1 2
13.12 even 2 1521.4.a.j.1.1 1
39.5 even 4 507.4.b.c.337.1 2
39.8 even 4 507.4.b.c.337.2 2
39.29 odd 6 39.4.e.a.22.1 yes 2
39.35 odd 6 39.4.e.a.16.1 2
39.38 odd 2 507.4.a.a.1.1 1
156.35 even 6 624.4.q.b.289.1 2
156.107 even 6 624.4.q.b.529.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.a.16.1 2 39.35 odd 6
39.4.e.a.22.1 yes 2 39.29 odd 6
117.4.g.b.55.1 2 13.9 even 3
117.4.g.b.100.1 2 13.3 even 3
507.4.a.a.1.1 1 39.38 odd 2
507.4.a.e.1.1 1 3.2 odd 2
507.4.b.c.337.1 2 39.5 even 4
507.4.b.c.337.2 2 39.8 even 4
624.4.q.b.289.1 2 156.35 even 6
624.4.q.b.529.1 2 156.107 even 6
1521.4.a.c.1.1 1 1.1 even 1 trivial
1521.4.a.j.1.1 1 13.12 even 2