# Properties

 Label 2160.4.a.bk.1.2 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, 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("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.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: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 8x + 9$$ x^3 - x^2 - 8*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.14974$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{5} +3.85875 q^{7} +O(q^{10})$$ $$q-5.00000 q^{5} +3.85875 q^{7} -42.2783 q^{11} +4.96700 q^{13} -25.8876 q^{17} -28.9958 q^{19} +191.469 q^{23} +25.0000 q^{25} +287.595 q^{29} -52.6915 q^{31} -19.2937 q^{35} -225.550 q^{37} +73.9907 q^{41} +275.370 q^{43} +192.271 q^{47} -328.110 q^{49} +275.204 q^{53} +211.392 q^{55} -497.084 q^{59} +44.0473 q^{61} -24.8350 q^{65} -761.667 q^{67} -264.284 q^{71} +728.781 q^{73} -163.141 q^{77} -664.347 q^{79} -1491.33 q^{83} +129.438 q^{85} +106.783 q^{89} +19.1664 q^{91} +144.979 q^{95} -924.560 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} + 10 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 + 10 * q^7 $$3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53} - 140 q^{55} + 234 q^{59} - 673 q^{61} + 390 q^{65} - 386 q^{67} + 16 q^{71} - 892 q^{73} + 1800 q^{77} - 1263 q^{79} - 1815 q^{83} + 55 q^{85} + 1800 q^{89} - 1284 q^{91} - 355 q^{95} - 840 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.85875 0.208353 0.104176 0.994559i $$-0.466779\pi$$
0.104176 + 0.994559i $$0.466779\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −42.2783 −1.15885 −0.579427 0.815024i $$-0.696724\pi$$
−0.579427 + 0.815024i $$0.696724\pi$$
$$12$$ 0 0
$$13$$ 4.96700 0.105969 0.0529845 0.998595i $$-0.483127\pi$$
0.0529845 + 0.998595i $$0.483127\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −25.8876 −0.369333 −0.184666 0.982801i $$-0.559120\pi$$
−0.184666 + 0.982801i $$0.559120\pi$$
$$18$$ 0 0
$$19$$ −28.9958 −0.350110 −0.175055 0.984559i $$-0.556010\pi$$
−0.175055 + 0.984559i $$0.556010\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 191.469 1.73583 0.867914 0.496715i $$-0.165461\pi$$
0.867914 + 0.496715i $$0.165461\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 287.595 1.84155 0.920776 0.390092i $$-0.127557\pi$$
0.920776 + 0.390092i $$0.127557\pi$$
$$30$$ 0 0
$$31$$ −52.6915 −0.305280 −0.152640 0.988282i $$-0.548777\pi$$
−0.152640 + 0.988282i $$0.548777\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −19.2937 −0.0931782
$$36$$ 0 0
$$37$$ −225.550 −1.00217 −0.501084 0.865398i $$-0.667065\pi$$
−0.501084 + 0.865398i $$0.667065\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 73.9907 0.281839 0.140920 0.990021i $$-0.454994\pi$$
0.140920 + 0.990021i $$0.454994\pi$$
$$42$$ 0 0
$$43$$ 275.370 0.976593 0.488297 0.872678i $$-0.337618\pi$$
0.488297 + 0.872678i $$0.337618\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 192.271 0.596715 0.298357 0.954454i $$-0.403561\pi$$
0.298357 + 0.954454i $$0.403561\pi$$
$$48$$ 0 0
$$49$$ −328.110 −0.956589
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 275.204 0.713249 0.356624 0.934248i $$-0.383928\pi$$
0.356624 + 0.934248i $$0.383928\pi$$
$$54$$ 0 0
$$55$$ 211.392 0.518255
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −497.084 −1.09686 −0.548431 0.836196i $$-0.684775\pi$$
−0.548431 + 0.836196i $$0.684775\pi$$
$$60$$ 0 0
$$61$$ 44.0473 0.0924538 0.0462269 0.998931i $$-0.485280\pi$$
0.0462269 + 0.998931i $$0.485280\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −24.8350 −0.0473908
$$66$$ 0 0
$$67$$ −761.667 −1.38884 −0.694422 0.719568i $$-0.744340\pi$$
−0.694422 + 0.719568i $$0.744340\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −264.284 −0.441757 −0.220879 0.975301i $$-0.570892\pi$$
−0.220879 + 0.975301i $$0.570892\pi$$
$$72$$ 0 0
$$73$$ 728.781 1.16846 0.584229 0.811589i $$-0.301397\pi$$
0.584229 + 0.811589i $$0.301397\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −163.141 −0.241450
$$78$$ 0 0
$$79$$ −664.347 −0.946137 −0.473069 0.881026i $$-0.656854\pi$$
−0.473069 + 0.881026i $$0.656854\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1491.33 −1.97223 −0.986113 0.166075i $$-0.946891\pi$$
−0.986113 + 0.166075i $$0.946891\pi$$
$$84$$ 0 0
$$85$$ 129.438 0.165171
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 106.783 0.127180 0.0635899 0.997976i $$-0.479745\pi$$
0.0635899 + 0.997976i $$0.479745\pi$$
$$90$$ 0 0
$$91$$ 19.1664 0.0220789
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 144.979 0.156574
$$96$$ 0 0
$$97$$ −924.560 −0.967782 −0.483891 0.875128i $$-0.660777\pi$$
−0.483891 + 0.875128i $$0.660777\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 627.973 0.618670 0.309335 0.950953i $$-0.399894\pi$$
0.309335 + 0.950953i $$0.399894\pi$$
$$102$$ 0 0
$$103$$ 224.878 0.215125 0.107562 0.994198i $$-0.465695\pi$$
0.107562 + 0.994198i $$0.465695\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 70.1076 0.0633417 0.0316708 0.999498i $$-0.489917\pi$$
0.0316708 + 0.999498i $$0.489917\pi$$
$$108$$ 0 0
$$109$$ −235.022 −0.206523 −0.103261 0.994654i $$-0.532928\pi$$
−0.103261 + 0.994654i $$0.532928\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1647.70 1.37170 0.685850 0.727743i $$-0.259431\pi$$
0.685850 + 0.727743i $$0.259431\pi$$
$$114$$ 0 0
$$115$$ −957.344 −0.776286
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −99.8936 −0.0769515
$$120$$ 0 0
$$121$$ 456.458 0.342943
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1282.64 0.896189 0.448095 0.893986i $$-0.352103\pi$$
0.448095 + 0.893986i $$0.352103\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1070.64 −0.714066 −0.357033 0.934092i $$-0.616212\pi$$
−0.357033 + 0.934092i $$0.616212\pi$$
$$132$$ 0 0
$$133$$ −111.888 −0.0729465
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2045.70 1.27574 0.637869 0.770145i $$-0.279816\pi$$
0.637869 + 0.770145i $$0.279816\pi$$
$$138$$ 0 0
$$139$$ −784.256 −0.478559 −0.239280 0.970951i $$-0.576911\pi$$
−0.239280 + 0.970951i $$0.576911\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −209.997 −0.122803
$$144$$ 0 0
$$145$$ −1437.97 −0.823567
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1300.68 0.715140 0.357570 0.933886i $$-0.383605\pi$$
0.357570 + 0.933886i $$0.383605\pi$$
$$150$$ 0 0
$$151$$ −1511.72 −0.814718 −0.407359 0.913268i $$-0.633550\pi$$
−0.407359 + 0.913268i $$0.633550\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 263.458 0.136525
$$156$$ 0 0
$$157$$ 167.967 0.0853838 0.0426919 0.999088i $$-0.486407\pi$$
0.0426919 + 0.999088i $$0.486407\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 738.830 0.361664
$$162$$ 0 0
$$163$$ 67.1490 0.0322670 0.0161335 0.999870i $$-0.494864\pi$$
0.0161335 + 0.999870i $$0.494864\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2568.36 −1.19009 −0.595046 0.803692i $$-0.702866\pi$$
−0.595046 + 0.803692i $$0.702866\pi$$
$$168$$ 0 0
$$169$$ −2172.33 −0.988771
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3012.46 −1.32389 −0.661945 0.749553i $$-0.730269\pi$$
−0.661945 + 0.749553i $$0.730269\pi$$
$$174$$ 0 0
$$175$$ 96.4687 0.0416705
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2462.80 −1.02837 −0.514185 0.857679i $$-0.671905\pi$$
−0.514185 + 0.857679i $$0.671905\pi$$
$$180$$ 0 0
$$181$$ −3691.10 −1.51578 −0.757892 0.652380i $$-0.773771\pi$$
−0.757892 + 0.652380i $$0.773771\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1127.75 0.448183
$$186$$ 0 0
$$187$$ 1094.48 0.428003
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1969.02 −0.745932 −0.372966 0.927845i $$-0.621659\pi$$
−0.372966 + 0.927845i $$0.621659\pi$$
$$192$$ 0 0
$$193$$ 519.119 0.193612 0.0968058 0.995303i $$-0.469137\pi$$
0.0968058 + 0.995303i $$0.469137\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3286.33 1.18853 0.594267 0.804268i $$-0.297442\pi$$
0.594267 + 0.804268i $$0.297442\pi$$
$$198$$ 0 0
$$199$$ −3626.47 −1.29183 −0.645914 0.763410i $$-0.723524\pi$$
−0.645914 + 0.763410i $$0.723524\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1109.76 0.383692
$$204$$ 0 0
$$205$$ −369.954 −0.126042
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1225.90 0.405727
$$210$$ 0 0
$$211$$ −1410.39 −0.460166 −0.230083 0.973171i $$-0.573900\pi$$
−0.230083 + 0.973171i $$0.573900\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1376.85 −0.436746
$$216$$ 0 0
$$217$$ −203.323 −0.0636059
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −128.584 −0.0391379
$$222$$ 0 0
$$223$$ −4026.25 −1.20905 −0.604524 0.796587i $$-0.706637\pi$$
−0.604524 + 0.796587i $$0.706637\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2288.33 −0.669083 −0.334542 0.942381i $$-0.608581\pi$$
−0.334542 + 0.942381i $$0.608581\pi$$
$$228$$ 0 0
$$229$$ 2003.69 0.578200 0.289100 0.957299i $$-0.406644\pi$$
0.289100 + 0.957299i $$0.406644\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2837.72 0.797877 0.398939 0.916978i $$-0.369379\pi$$
0.398939 + 0.916978i $$0.369379\pi$$
$$234$$ 0 0
$$235$$ −961.355 −0.266859
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1749.50 0.473496 0.236748 0.971571i $$-0.423918\pi$$
0.236748 + 0.971571i $$0.423918\pi$$
$$240$$ 0 0
$$241$$ 3645.10 0.974279 0.487140 0.873324i $$-0.338040\pi$$
0.487140 + 0.873324i $$0.338040\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1640.55 0.427800
$$246$$ 0 0
$$247$$ −144.022 −0.0371009
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1286.39 0.323491 0.161745 0.986833i $$-0.448288\pi$$
0.161745 + 0.986833i $$0.448288\pi$$
$$252$$ 0 0
$$253$$ −8094.99 −2.01157
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2860.83 −0.694372 −0.347186 0.937796i $$-0.612863\pi$$
−0.347186 + 0.937796i $$0.612863\pi$$
$$258$$ 0 0
$$259$$ −870.341 −0.208805
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −4615.68 −1.08219 −0.541093 0.840963i $$-0.681989\pi$$
−0.541093 + 0.840963i $$0.681989\pi$$
$$264$$ 0 0
$$265$$ −1376.02 −0.318974
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3053.54 −0.692111 −0.346055 0.938214i $$-0.612479\pi$$
−0.346055 + 0.938214i $$0.612479\pi$$
$$270$$ 0 0
$$271$$ 5867.51 1.31523 0.657613 0.753356i $$-0.271566\pi$$
0.657613 + 0.753356i $$0.271566\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1056.96 −0.231771
$$276$$ 0 0
$$277$$ −7256.41 −1.57399 −0.786995 0.616959i $$-0.788364\pi$$
−0.786995 + 0.616959i $$0.788364\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1318.05 −0.279816 −0.139908 0.990165i $$-0.544681\pi$$
−0.139908 + 0.990165i $$0.544681\pi$$
$$282$$ 0 0
$$283$$ −4810.74 −1.01049 −0.505245 0.862976i $$-0.668598\pi$$
−0.505245 + 0.862976i $$0.668598\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 285.511 0.0587220
$$288$$ 0 0
$$289$$ −4242.83 −0.863593
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1512.59 0.301593 0.150796 0.988565i $$-0.451816\pi$$
0.150796 + 0.988565i $$0.451816\pi$$
$$294$$ 0 0
$$295$$ 2485.42 0.490531
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 951.026 0.183944
$$300$$ 0 0
$$301$$ 1062.58 0.203476
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −220.237 −0.0413466
$$306$$ 0 0
$$307$$ −3092.67 −0.574945 −0.287472 0.957789i $$-0.592815\pi$$
−0.287472 + 0.957789i $$0.592815\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −21.7251 −0.00396115 −0.00198058 0.999998i $$-0.500630\pi$$
−0.00198058 + 0.999998i $$0.500630\pi$$
$$312$$ 0 0
$$313$$ 758.733 0.137016 0.0685082 0.997651i $$-0.478176\pi$$
0.0685082 + 0.997651i $$0.478176\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1764.70 −0.312668 −0.156334 0.987704i $$-0.549968\pi$$
−0.156334 + 0.987704i $$0.549968\pi$$
$$318$$ 0 0
$$319$$ −12159.0 −2.13409
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 750.632 0.129307
$$324$$ 0 0
$$325$$ 124.175 0.0211938
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 741.925 0.124327
$$330$$ 0 0
$$331$$ −10098.1 −1.67687 −0.838433 0.545005i $$-0.816528\pi$$
−0.838433 + 0.545005i $$0.816528\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 3808.34 0.621109
$$336$$ 0 0
$$337$$ −4485.28 −0.725012 −0.362506 0.931981i $$-0.618079\pi$$
−0.362506 + 0.931981i $$0.618079\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2227.71 0.353775
$$342$$ 0 0
$$343$$ −2589.64 −0.407661
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 585.137 0.0905239 0.0452620 0.998975i $$-0.485588\pi$$
0.0452620 + 0.998975i $$0.485588\pi$$
$$348$$ 0 0
$$349$$ 3461.97 0.530989 0.265494 0.964112i $$-0.414465\pi$$
0.265494 + 0.964112i $$0.414465\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6797.01 −1.02484 −0.512420 0.858735i $$-0.671251\pi$$
−0.512420 + 0.858735i $$0.671251\pi$$
$$354$$ 0 0
$$355$$ 1321.42 0.197560
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4622.13 0.679518 0.339759 0.940513i $$-0.389655\pi$$
0.339759 + 0.940513i $$0.389655\pi$$
$$360$$ 0 0
$$361$$ −6018.24 −0.877423
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3643.91 −0.522550
$$366$$ 0 0
$$367$$ 147.843 0.0210281 0.0105141 0.999945i $$-0.496653\pi$$
0.0105141 + 0.999945i $$0.496653\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1061.94 0.148607
$$372$$ 0 0
$$373$$ 8840.27 1.22716 0.613582 0.789631i $$-0.289728\pi$$
0.613582 + 0.789631i $$0.289728\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1428.48 0.195148
$$378$$ 0 0
$$379$$ 6063.51 0.821799 0.410899 0.911681i $$-0.365215\pi$$
0.410899 + 0.911681i $$0.365215\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −496.589 −0.0662520 −0.0331260 0.999451i $$-0.510546\pi$$
−0.0331260 + 0.999451i $$0.510546\pi$$
$$384$$ 0 0
$$385$$ 815.707 0.107980
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3432.37 −0.447373 −0.223686 0.974661i $$-0.571809\pi$$
−0.223686 + 0.974661i $$0.571809\pi$$
$$390$$ 0 0
$$391$$ −4956.66 −0.641098
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3321.73 0.423125
$$396$$ 0 0
$$397$$ −15311.8 −1.93571 −0.967856 0.251506i $$-0.919074\pi$$
−0.967856 + 0.251506i $$0.919074\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7088.63 0.882766 0.441383 0.897319i $$-0.354488\pi$$
0.441383 + 0.897319i $$0.354488\pi$$
$$402$$ 0 0
$$403$$ −261.719 −0.0323502
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9535.89 1.16137
$$408$$ 0 0
$$409$$ 9991.23 1.20791 0.603954 0.797019i $$-0.293591\pi$$
0.603954 + 0.797019i $$0.293591\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −1918.12 −0.228534
$$414$$ 0 0
$$415$$ 7456.65 0.882006
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 8172.83 0.952909 0.476455 0.879199i $$-0.341922\pi$$
0.476455 + 0.879199i $$0.341922\pi$$
$$420$$ 0 0
$$421$$ −15475.1 −1.79147 −0.895734 0.444590i $$-0.853350\pi$$
−0.895734 + 0.444590i $$0.853350\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −647.189 −0.0738666
$$426$$ 0 0
$$427$$ 169.968 0.0192630
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3426.00 0.382888 0.191444 0.981504i $$-0.438683\pi$$
0.191444 + 0.981504i $$0.438683\pi$$
$$432$$ 0 0
$$433$$ −1806.95 −0.200546 −0.100273 0.994960i $$-0.531972\pi$$
−0.100273 + 0.994960i $$0.531972\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −5551.80 −0.607731
$$438$$ 0 0
$$439$$ −7974.34 −0.866957 −0.433479 0.901164i $$-0.642714\pi$$
−0.433479 + 0.901164i $$0.642714\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3566.86 0.382544 0.191272 0.981537i $$-0.438739\pi$$
0.191272 + 0.981537i $$0.438739\pi$$
$$444$$ 0 0
$$445$$ −533.916 −0.0568765
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13687.2 1.43862 0.719308 0.694692i $$-0.244459\pi$$
0.719308 + 0.694692i $$0.244459\pi$$
$$450$$ 0 0
$$451$$ −3128.21 −0.326611
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −95.8320 −0.00987401
$$456$$ 0 0
$$457$$ 1691.89 0.173180 0.0865902 0.996244i $$-0.472403\pi$$
0.0865902 + 0.996244i $$0.472403\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17881.6 −1.80657 −0.903285 0.429041i $$-0.858851\pi$$
−0.903285 + 0.429041i $$0.858851\pi$$
$$462$$ 0 0
$$463$$ −11459.6 −1.15027 −0.575135 0.818059i $$-0.695050\pi$$
−0.575135 + 0.818059i $$0.695050\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18329.4 1.81624 0.908119 0.418711i $$-0.137518\pi$$
0.908119 + 0.418711i $$0.137518\pi$$
$$468$$ 0 0
$$469$$ −2939.08 −0.289369
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −11642.2 −1.13173
$$474$$ 0 0
$$475$$ −724.896 −0.0700221
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −15475.4 −1.47618 −0.738090 0.674703i $$-0.764272\pi$$
−0.738090 + 0.674703i $$0.764272\pi$$
$$480$$ 0 0
$$481$$ −1120.31 −0.106199
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4622.80 0.432805
$$486$$ 0 0
$$487$$ 16320.0 1.51854 0.759270 0.650776i $$-0.225556\pi$$
0.759270 + 0.650776i $$0.225556\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6138.33 0.564193 0.282097 0.959386i $$-0.408970\pi$$
0.282097 + 0.959386i $$0.408970\pi$$
$$492$$ 0 0
$$493$$ −7445.13 −0.680146
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −1019.81 −0.0920413
$$498$$ 0 0
$$499$$ −3272.20 −0.293555 −0.146777 0.989170i $$-0.546890\pi$$
−0.146777 + 0.989170i $$0.546890\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 4696.96 0.416356 0.208178 0.978091i $$-0.433247\pi$$
0.208178 + 0.978091i $$0.433247\pi$$
$$504$$ 0 0
$$505$$ −3139.87 −0.276678
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 10011.5 0.871815 0.435907 0.899992i $$-0.356428\pi$$
0.435907 + 0.899992i $$0.356428\pi$$
$$510$$ 0 0
$$511$$ 2812.18 0.243451
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1124.39 −0.0962067
$$516$$ 0 0
$$517$$ −8128.89 −0.691505
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −19483.2 −1.63833 −0.819167 0.573554i $$-0.805564\pi$$
−0.819167 + 0.573554i $$0.805564\pi$$
$$522$$ 0 0
$$523$$ 5181.82 0.433241 0.216621 0.976256i $$-0.430497\pi$$
0.216621 + 0.976256i $$0.430497\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1364.06 0.112750
$$528$$ 0 0
$$529$$ 24493.3 2.01310
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 367.512 0.0298663
$$534$$ 0 0
$$535$$ −350.538 −0.0283273
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 13871.9 1.10855
$$540$$ 0 0
$$541$$ 3178.24 0.252576 0.126288 0.991994i $$-0.459694\pi$$
0.126288 + 0.991994i $$0.459694\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 1175.11 0.0923599
$$546$$ 0 0
$$547$$ 13887.9 1.08556 0.542781 0.839874i $$-0.317371\pi$$
0.542781 + 0.839874i $$0.317371\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8339.05 −0.644747
$$552$$ 0 0
$$553$$ −2563.55 −0.197130
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10574.5 −0.804406 −0.402203 0.915551i $$-0.631755\pi$$
−0.402203 + 0.915551i $$0.631755\pi$$
$$558$$ 0 0
$$559$$ 1367.76 0.103489
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −7815.91 −0.585083 −0.292541 0.956253i $$-0.594501\pi$$
−0.292541 + 0.956253i $$0.594501\pi$$
$$564$$ 0 0
$$565$$ −8238.48 −0.613443
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −20024.0 −1.47531 −0.737655 0.675178i $$-0.764067\pi$$
−0.737655 + 0.675178i $$0.764067\pi$$
$$570$$ 0 0
$$571$$ −23886.5 −1.75065 −0.875323 0.483538i $$-0.839351\pi$$
−0.875323 + 0.483538i $$0.839351\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4786.72 0.347165
$$576$$ 0 0
$$577$$ −18777.2 −1.35478 −0.677389 0.735625i $$-0.736888\pi$$
−0.677389 + 0.735625i $$0.736888\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5754.67 −0.410919
$$582$$ 0 0
$$583$$ −11635.2 −0.826551
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −23209.8 −1.63198 −0.815988 0.578069i $$-0.803806\pi$$
−0.815988 + 0.578069i $$0.803806\pi$$
$$588$$ 0 0
$$589$$ 1527.83 0.106882
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −4204.44 −0.291157 −0.145578 0.989347i $$-0.546504\pi$$
−0.145578 + 0.989347i $$0.546504\pi$$
$$594$$ 0 0
$$595$$ 499.468 0.0344138
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 26962.9 1.83919 0.919593 0.392872i $$-0.128518\pi$$
0.919593 + 0.392872i $$0.128518\pi$$
$$600$$ 0 0
$$601$$ 13595.2 0.922727 0.461363 0.887211i $$-0.347360\pi$$
0.461363 + 0.887211i $$0.347360\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2282.29 −0.153369
$$606$$ 0 0
$$607$$ −14337.5 −0.958720 −0.479360 0.877618i $$-0.659131\pi$$
−0.479360 + 0.877618i $$0.659131\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 955.010 0.0632333
$$612$$ 0 0
$$613$$ 7963.62 0.524710 0.262355 0.964971i $$-0.415501\pi$$
0.262355 + 0.964971i $$0.415501\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −23594.2 −1.53949 −0.769745 0.638352i $$-0.779617\pi$$
−0.769745 + 0.638352i $$0.779617\pi$$
$$618$$ 0 0
$$619$$ −12132.6 −0.787803 −0.393901 0.919153i $$-0.628875\pi$$
−0.393901 + 0.919153i $$0.628875\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 412.049 0.0264982
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 5838.95 0.370134
$$630$$ 0 0
$$631$$ 614.270 0.0387539 0.0193769 0.999812i $$-0.493832\pi$$
0.0193769 + 0.999812i $$0.493832\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6413.21 −0.400788
$$636$$ 0 0
$$637$$ −1629.72 −0.101369
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2057.06 0.126754 0.0633769 0.997990i $$-0.479813\pi$$
0.0633769 + 0.997990i $$0.479813\pi$$
$$642$$ 0 0
$$643$$ −17827.2 −1.09337 −0.546684 0.837339i $$-0.684110\pi$$
−0.546684 + 0.837339i $$0.684110\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −17499.4 −1.06333 −0.531664 0.846955i $$-0.678433\pi$$
−0.531664 + 0.846955i $$0.678433\pi$$
$$648$$ 0 0
$$649$$ 21015.9 1.27110
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6049.52 −0.362536 −0.181268 0.983434i $$-0.558020\pi$$
−0.181268 + 0.983434i $$0.558020\pi$$
$$654$$ 0 0
$$655$$ 5353.22 0.319340
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1436.32 0.0849027 0.0424514 0.999099i $$-0.486483\pi$$
0.0424514 + 0.999099i $$0.486483\pi$$
$$660$$ 0 0
$$661$$ 9287.18 0.546489 0.273245 0.961945i $$-0.411903\pi$$
0.273245 + 0.961945i $$0.411903\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 559.438 0.0326226
$$666$$ 0 0
$$667$$ 55065.4 3.19662
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −1862.25 −0.107141
$$672$$ 0 0
$$673$$ 15014.9 0.860000 0.430000 0.902829i $$-0.358514\pi$$
0.430000 + 0.902829i $$0.358514\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 9147.19 0.519284 0.259642 0.965705i $$-0.416396\pi$$
0.259642 + 0.965705i $$0.416396\pi$$
$$678$$ 0 0
$$679$$ −3567.64 −0.201640
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −10388.4 −0.581992 −0.290996 0.956724i $$-0.593987\pi$$
−0.290996 + 0.956724i $$0.593987\pi$$
$$684$$ 0 0
$$685$$ −10228.5 −0.570527
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 1366.94 0.0755823
$$690$$ 0 0
$$691$$ 25955.9 1.42895 0.714477 0.699659i $$-0.246665\pi$$
0.714477 + 0.699659i $$0.246665\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3921.28 0.214018
$$696$$ 0 0
$$697$$ −1915.44 −0.104093
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 31771.2 1.71181 0.855906 0.517131i $$-0.173000\pi$$
0.855906 + 0.517131i $$0.173000\pi$$
$$702$$ 0 0
$$703$$ 6540.02 0.350870
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 2423.19 0.128902
$$708$$ 0 0
$$709$$ −7002.47 −0.370921 −0.185461 0.982652i $$-0.559378\pi$$
−0.185461 + 0.982652i $$0.559378\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −10088.8 −0.529913
$$714$$ 0 0
$$715$$ 1049.98 0.0549191
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 34634.3 1.79644 0.898221 0.439545i $$-0.144860\pi$$
0.898221 + 0.439545i $$0.144860\pi$$
$$720$$ 0 0
$$721$$ 867.745 0.0448218
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 7189.87 0.368310
$$726$$ 0 0
$$727$$ −10821.6 −0.552064 −0.276032 0.961148i $$-0.589020\pi$$
−0.276032 + 0.961148i $$0.589020\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −7128.66 −0.360688
$$732$$ 0 0
$$733$$ −5576.04 −0.280977 −0.140488 0.990082i $$-0.544867\pi$$
−0.140488 + 0.990082i $$0.544867\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32202.0 1.60947
$$738$$ 0 0
$$739$$ −29317.7 −1.45936 −0.729681 0.683788i $$-0.760331\pi$$
−0.729681 + 0.683788i $$0.760331\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 7521.08 0.371361 0.185681 0.982610i $$-0.440551\pi$$
0.185681 + 0.982610i $$0.440551\pi$$
$$744$$ 0 0
$$745$$ −6503.40 −0.319820
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 270.528 0.0131974
$$750$$ 0 0
$$751$$ 37300.7 1.81241 0.906206 0.422837i $$-0.138966\pi$$
0.906206 + 0.422837i $$0.138966\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 7558.62 0.364353
$$756$$ 0 0
$$757$$ −26432.1 −1.26908 −0.634538 0.772891i $$-0.718810\pi$$
−0.634538 + 0.772891i $$0.718810\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 23440.4 1.11658 0.558288 0.829647i $$-0.311458\pi$$
0.558288 + 0.829647i $$0.311458\pi$$
$$762$$ 0 0
$$763$$ −906.889 −0.0430296
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2469.02 −0.116233
$$768$$ 0 0
$$769$$ 34075.2 1.59790 0.798948 0.601401i $$-0.205390\pi$$
0.798948 + 0.601401i $$0.205390\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −14623.5 −0.680427 −0.340214 0.940348i $$-0.610499\pi$$
−0.340214 + 0.940348i $$0.610499\pi$$
$$774$$ 0 0
$$775$$ −1317.29 −0.0610560
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2145.42 −0.0986749
$$780$$ 0 0
$$781$$ 11173.5 0.511932
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −839.837 −0.0381848
$$786$$ 0 0
$$787$$ 33523.1 1.51839 0.759193 0.650865i $$-0.225594\pi$$
0.759193 + 0.650865i $$0.225594\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6358.04 0.285797
$$792$$ 0 0
$$793$$ 218.783 0.00979725
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 29447.5 1.30876 0.654381 0.756165i $$-0.272929\pi$$
0.654381 + 0.756165i $$0.272929\pi$$
$$798$$ 0 0
$$799$$ −4977.43 −0.220386
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −30811.6 −1.35407
$$804$$ 0 0
$$805$$ −3694.15 −0.161741
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 26891.1 1.16866 0.584328 0.811518i $$-0.301358\pi$$
0.584328 + 0.811518i $$0.301358\pi$$
$$810$$ 0 0
$$811$$ −14912.6 −0.645687 −0.322844 0.946452i $$-0.604639\pi$$
−0.322844 + 0.946452i $$0.604639\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −335.745 −0.0144302
$$816$$ 0 0
$$817$$ −7984.58 −0.341916
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42644.2 −1.81278 −0.906391 0.422439i $$-0.861174\pi$$
−0.906391 + 0.422439i $$0.861174\pi$$
$$822$$ 0 0
$$823$$ 25121.7 1.06402 0.532010 0.846738i $$-0.321437\pi$$
0.532010 + 0.846738i $$0.321437\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 4918.93 0.206830 0.103415 0.994638i $$-0.467023\pi$$
0.103415 + 0.994638i $$0.467023\pi$$
$$828$$ 0 0
$$829$$ −22807.6 −0.955538 −0.477769 0.878486i $$-0.658554\pi$$
−0.477769 + 0.878486i $$0.658554\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 8493.97 0.353300
$$834$$ 0 0
$$835$$ 12841.8 0.532225
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −25772.5 −1.06051 −0.530255 0.847839i $$-0.677904\pi$$
−0.530255 + 0.847839i $$0.677904\pi$$
$$840$$ 0 0
$$841$$ 58321.7 2.39131
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 10861.6 0.442192
$$846$$ 0 0
$$847$$ 1761.35 0.0714532
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −43185.9 −1.73959
$$852$$ 0 0
$$853$$ 41636.0 1.67126 0.835632 0.549289i $$-0.185101\pi$$
0.835632 + 0.549289i $$0.185101\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −17941.6 −0.715137 −0.357569 0.933887i $$-0.616394\pi$$
−0.357569 + 0.933887i $$0.616394\pi$$
$$858$$ 0 0
$$859$$ −23917.0 −0.949985 −0.474993 0.879990i $$-0.657549\pi$$
−0.474993 + 0.879990i $$0.657549\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −9786.53 −0.386022 −0.193011 0.981197i $$-0.561825\pi$$
−0.193011 + 0.981197i $$0.561825\pi$$
$$864$$ 0 0
$$865$$ 15062.3 0.592061
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 28087.5 1.09644
$$870$$ 0 0
$$871$$ −3783.20 −0.147174
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −482.343 −0.0186356
$$876$$ 0 0
$$877$$ −29186.8 −1.12379 −0.561897 0.827207i $$-0.689928\pi$$
−0.561897 + 0.827207i $$0.689928\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −18426.6 −0.704664 −0.352332 0.935875i $$-0.614611\pi$$
−0.352332 + 0.935875i $$0.614611\pi$$
$$882$$ 0 0
$$883$$ −3942.63 −0.150261 −0.0751303 0.997174i $$-0.523937\pi$$
−0.0751303 + 0.997174i $$0.523937\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 436.876 0.0165376 0.00826881 0.999966i $$-0.497368\pi$$
0.00826881 + 0.999966i $$0.497368\pi$$
$$888$$ 0 0
$$889$$ 4949.39 0.186723
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −5575.05 −0.208916
$$894$$ 0 0
$$895$$ 12314.0 0.459901
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −15153.8 −0.562189
$$900$$ 0 0
$$901$$ −7124.36 −0.263426
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18455.5 0.677879
$$906$$ 0 0
$$907$$ 43243.2 1.58309 0.791547 0.611108i $$-0.209276\pi$$
0.791547 + 0.611108i $$0.209276\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 14141.5 0.514301 0.257150 0.966371i $$-0.417216\pi$$
0.257150 + 0.966371i $$0.417216\pi$$
$$912$$ 0 0
$$913$$ 63051.0 2.28552
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4131.34 −0.148778
$$918$$ 0 0
$$919$$ −1506.36 −0.0540697 −0.0270349 0.999634i $$-0.508607\pi$$
−0.0270349 + 0.999634i $$0.508607\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −1312.70 −0.0468126
$$924$$ 0 0
$$925$$ −5638.76 −0.200434
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 32173.3 1.13624 0.568122 0.822944i $$-0.307670\pi$$
0.568122 + 0.822944i $$0.307670\pi$$
$$930$$ 0 0
$$931$$ 9513.82 0.334912
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −5472.42 −0.191409
$$936$$ 0 0
$$937$$ 34833.5 1.21447 0.607236 0.794522i $$-0.292278\pi$$
0.607236 + 0.794522i $$0.292278\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −3709.66 −0.128514 −0.0642569 0.997933i $$-0.520468\pi$$
−0.0642569 + 0.997933i $$0.520468\pi$$
$$942$$ 0 0
$$943$$ 14166.9 0.489224
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −32085.0 −1.10097 −0.550487 0.834844i $$-0.685558\pi$$
−0.550487 + 0.834844i $$0.685558\pi$$
$$948$$ 0 0
$$949$$ 3619.86 0.123820
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 35654.2 1.21191 0.605956 0.795498i $$-0.292791\pi$$
0.605956 + 0.795498i $$0.292791\pi$$
$$954$$ 0 0
$$955$$ 9845.08 0.333591
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 7893.84 0.265803
$$960$$ 0 0
$$961$$ −27014.6 −0.906804
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −2595.60 −0.0865857
$$966$$ 0 0
$$967$$ 32345.4 1.07565 0.537827 0.843055i $$-0.319245\pi$$
0.537827 + 0.843055i $$0.319245\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20137.4 0.665542 0.332771 0.943008i $$-0.392016\pi$$
0.332771 + 0.943008i $$0.392016\pi$$
$$972$$ 0 0
$$973$$ −3026.25 −0.0997091
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −30087.1 −0.985231 −0.492615 0.870247i $$-0.663959\pi$$
−0.492615 + 0.870247i $$0.663959\pi$$
$$978$$ 0 0
$$979$$ −4514.62 −0.147383
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −59417.5 −1.92790 −0.963950 0.266084i $$-0.914270\pi$$
−0.963950 + 0.266084i $$0.914270\pi$$
$$984$$ 0 0
$$985$$ −16431.6 −0.531529
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 52724.8 1.69520
$$990$$ 0 0
$$991$$ 20153.5 0.646011 0.323006 0.946397i $$-0.395307\pi$$
0.323006 + 0.946397i $$0.395307\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 18132.4 0.577723
$$996$$ 0 0
$$997$$ −20772.3 −0.659846 −0.329923 0.944008i $$-0.607023\pi$$
−0.329923 + 0.944008i $$0.607023\pi$$
$$998$$ 0 0
$$999$$ 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 2160.4.a.bk.1.2 3
3.2 odd 2 2160.4.a.bs.1.2 3
4.3 odd 2 1080.4.a.d.1.2 3
12.11 even 2 1080.4.a.j.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.2 3 4.3 odd 2
1080.4.a.j.1.2 yes 3 12.11 even 2
2160.4.a.bk.1.2 3 1.1 even 1 trivial
2160.4.a.bs.1.2 3 3.2 odd 2