# Properties

 Label 1080.4.a.j.1.1 Level $1080$ Weight $4$ Character 1080.1 Self dual yes Analytic conductor $63.722$ 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] = [1080,4,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, 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("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1080.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: $$63.7220628062$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.87370$$ of defining polynomial Character $$\chi$$ $$=$$ 1080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -30.2207 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -30.2207 q^{7} +67.3185 q^{11} -60.7911 q^{13} +61.1347 q^{17} -27.8771 q^{19} -40.4655 q^{23} +25.0000 q^{25} +212.108 q^{29} +167.318 q^{31} -151.103 q^{35} -366.539 q^{37} -363.385 q^{41} +153.839 q^{43} -434.212 q^{47} +570.291 q^{49} +79.6550 q^{53} +336.593 q^{55} +339.414 q^{59} -525.856 q^{61} -303.956 q^{65} +131.916 q^{67} -296.263 q^{71} -1230.34 q^{73} -2034.41 q^{77} -621.772 q^{79} -76.3372 q^{83} +305.673 q^{85} -192.406 q^{89} +1837.15 q^{91} -139.386 q^{95} -874.771 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} - 10 q^{7}+O(q^{10})$$ 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
$$7$$ −30.2207 −1.63176 −0.815882 0.578218i $$-0.803748\pi$$
−0.815882 + 0.578218i $$0.803748\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 67.3185 1.84521 0.922604 0.385748i $$-0.126056\pi$$
0.922604 + 0.385748i $$0.126056\pi$$
$$12$$ 0 0
$$13$$ −60.7911 −1.29696 −0.648478 0.761234i $$-0.724594\pi$$
−0.648478 + 0.761234i $$0.724594\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 61.1347 0.872196 0.436098 0.899899i $$-0.356360\pi$$
0.436098 + 0.899899i $$0.356360\pi$$
$$18$$ 0 0
$$19$$ −27.8771 −0.336603 −0.168301 0.985736i $$-0.553828\pi$$
−0.168301 + 0.985736i $$0.553828\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −40.4655 −0.366854 −0.183427 0.983033i $$-0.558719\pi$$
−0.183427 + 0.983033i $$0.558719\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 212.108 1.35819 0.679095 0.734051i $$-0.262373\pi$$
0.679095 + 0.734051i $$0.262373\pi$$
$$30$$ 0 0
$$31$$ 167.318 0.969394 0.484697 0.874682i $$-0.338930\pi$$
0.484697 + 0.874682i $$0.338930\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −151.103 −0.729747
$$36$$ 0 0
$$37$$ −366.539 −1.62861 −0.814305 0.580437i $$-0.802882\pi$$
−0.814305 + 0.580437i $$0.802882\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −363.385 −1.38418 −0.692088 0.721813i $$-0.743309\pi$$
−0.692088 + 0.721813i $$0.743309\pi$$
$$42$$ 0 0
$$43$$ 153.839 0.545586 0.272793 0.962073i $$-0.412053\pi$$
0.272793 + 0.962073i $$0.412053\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −434.212 −1.34758 −0.673791 0.738922i $$-0.735335\pi$$
−0.673791 + 0.738922i $$0.735335\pi$$
$$48$$ 0 0
$$49$$ 570.291 1.66265
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 79.6550 0.206442 0.103221 0.994658i $$-0.467085\pi$$
0.103221 + 0.994658i $$0.467085\pi$$
$$54$$ 0 0
$$55$$ 336.593 0.825202
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 339.414 0.748948 0.374474 0.927237i $$-0.377823\pi$$
0.374474 + 0.927237i $$0.377823\pi$$
$$60$$ 0 0
$$61$$ −525.856 −1.10375 −0.551877 0.833926i $$-0.686088\pi$$
−0.551877 + 0.833926i $$0.686088\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −303.956 −0.580016
$$66$$ 0 0
$$67$$ 131.916 0.240539 0.120269 0.992741i $$-0.461624\pi$$
0.120269 + 0.992741i $$0.461624\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −296.263 −0.495210 −0.247605 0.968861i $$-0.579644\pi$$
−0.247605 + 0.968861i $$0.579644\pi$$
$$72$$ 0 0
$$73$$ −1230.34 −1.97261 −0.986305 0.164932i $$-0.947259\pi$$
−0.986305 + 0.164932i $$0.947259\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2034.41 −3.01095
$$78$$ 0 0
$$79$$ −621.772 −0.885504 −0.442752 0.896644i $$-0.645998\pi$$
−0.442752 + 0.896644i $$0.645998\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −76.3372 −0.100953 −0.0504765 0.998725i $$-0.516074\pi$$
−0.0504765 + 0.998725i $$0.516074\pi$$
$$84$$ 0 0
$$85$$ 305.673 0.390058
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −192.406 −0.229157 −0.114579 0.993414i $$-0.536552\pi$$
−0.114579 + 0.993414i $$0.536552\pi$$
$$90$$ 0 0
$$91$$ 1837.15 2.11633
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −139.386 −0.150533
$$96$$ 0 0
$$97$$ −874.771 −0.915666 −0.457833 0.889038i $$-0.651374\pi$$
−0.457833 + 0.889038i $$0.651374\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1093.02 −1.07682 −0.538412 0.842682i $$-0.680975\pi$$
−0.538412 + 0.842682i $$0.680975\pi$$
$$102$$ 0 0
$$103$$ 228.932 0.219003 0.109502 0.993987i $$-0.465075\pi$$
0.109502 + 0.993987i $$0.465075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 479.745 0.433446 0.216723 0.976233i $$-0.430463\pi$$
0.216723 + 0.976233i $$0.430463\pi$$
$$108$$ 0 0
$$109$$ −116.246 −0.102150 −0.0510751 0.998695i $$-0.516265\pi$$
−0.0510751 + 0.998695i $$0.516265\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 741.222 0.617065 0.308532 0.951214i $$-0.400162\pi$$
0.308532 + 0.951214i $$0.400162\pi$$
$$114$$ 0 0
$$115$$ −202.327 −0.164062
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1847.53 −1.42322
$$120$$ 0 0
$$121$$ 3200.78 2.40479
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2595.38 −1.81340 −0.906702 0.421771i $$-0.861409\pi$$
−0.906702 + 0.421771i $$0.861409\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1405.62 0.937476 0.468738 0.883337i $$-0.344709\pi$$
0.468738 + 0.883337i $$0.344709\pi$$
$$132$$ 0 0
$$133$$ 842.466 0.549256
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2457.39 −1.53248 −0.766239 0.642556i $$-0.777874\pi$$
−0.766239 + 0.642556i $$0.777874\pi$$
$$138$$ 0 0
$$139$$ −585.391 −0.357210 −0.178605 0.983921i $$-0.557158\pi$$
−0.178605 + 0.983921i $$0.557158\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4092.37 −2.39315
$$144$$ 0 0
$$145$$ 1060.54 0.607401
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 926.505 0.509411 0.254705 0.967019i $$-0.418021\pi$$
0.254705 + 0.967019i $$0.418021\pi$$
$$150$$ 0 0
$$151$$ 411.900 0.221987 0.110993 0.993821i $$-0.464597\pi$$
0.110993 + 0.993821i $$0.464597\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 836.590 0.433526
$$156$$ 0 0
$$157$$ −2959.97 −1.50466 −0.752328 0.658789i $$-0.771069\pi$$
−0.752328 + 0.658789i $$0.771069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1222.89 0.598619
$$162$$ 0 0
$$163$$ 2504.50 1.20348 0.601741 0.798691i $$-0.294474\pi$$
0.601741 + 0.798691i $$0.294474\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2055.42 0.952413 0.476207 0.879333i $$-0.342011\pi$$
0.476207 + 0.879333i $$0.342011\pi$$
$$168$$ 0 0
$$169$$ 1498.56 0.682093
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1724.17 0.757722 0.378861 0.925454i $$-0.376316\pi$$
0.378861 + 0.925454i $$0.376316\pi$$
$$174$$ 0 0
$$175$$ −755.517 −0.326353
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3601.35 −1.50379 −0.751893 0.659285i $$-0.770859\pi$$
−0.751893 + 0.659285i $$0.770859\pi$$
$$180$$ 0 0
$$181$$ −1257.17 −0.516267 −0.258134 0.966109i $$-0.583107\pi$$
−0.258134 + 0.966109i $$0.583107\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1832.69 −0.728337
$$186$$ 0 0
$$187$$ 4115.50 1.60938
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4677.47 −1.77199 −0.885994 0.463697i $$-0.846523\pi$$
−0.885994 + 0.463697i $$0.846523\pi$$
$$192$$ 0 0
$$193$$ −2647.88 −0.987556 −0.493778 0.869588i $$-0.664384\pi$$
−0.493778 + 0.869588i $$0.664384\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1368.91 0.495082 0.247541 0.968877i $$-0.420378\pi$$
0.247541 + 0.968877i $$0.420378\pi$$
$$198$$ 0 0
$$199$$ −5411.43 −1.92767 −0.963835 0.266501i $$-0.914133\pi$$
−0.963835 + 0.266501i $$0.914133\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −6410.06 −2.21625
$$204$$ 0 0
$$205$$ −1816.93 −0.619022
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1876.65 −0.621102
$$210$$ 0 0
$$211$$ 1987.12 0.648337 0.324168 0.945999i $$-0.394916\pi$$
0.324168 + 0.945999i $$0.394916\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 769.194 0.243993
$$216$$ 0 0
$$217$$ −5056.47 −1.58182
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3716.45 −1.13120
$$222$$ 0 0
$$223$$ 3687.95 1.10746 0.553730 0.832697i $$-0.313204\pi$$
0.553730 + 0.832697i $$0.313204\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −859.424 −0.251286 −0.125643 0.992076i $$-0.540099\pi$$
−0.125643 + 0.992076i $$0.540099\pi$$
$$228$$ 0 0
$$229$$ −1862.67 −0.537504 −0.268752 0.963209i $$-0.586611\pi$$
−0.268752 + 0.963209i $$0.586611\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −5011.22 −1.40900 −0.704498 0.709706i $$-0.748828\pi$$
−0.704498 + 0.709706i $$0.748828\pi$$
$$234$$ 0 0
$$235$$ −2171.06 −0.602657
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5057.92 −1.36891 −0.684455 0.729055i $$-0.739960\pi$$
−0.684455 + 0.729055i $$0.739960\pi$$
$$240$$ 0 0
$$241$$ −7276.37 −1.94486 −0.972431 0.233189i $$-0.925084\pi$$
−0.972431 + 0.233189i $$0.925084\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2851.45 0.743562
$$246$$ 0 0
$$247$$ 1694.68 0.436559
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −204.208 −0.0513525 −0.0256762 0.999670i $$-0.508174\pi$$
−0.0256762 + 0.999670i $$0.508174\pi$$
$$252$$ 0 0
$$253$$ −2724.07 −0.676921
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4591.80 −1.11451 −0.557254 0.830342i $$-0.688145\pi$$
−0.557254 + 0.830342i $$0.688145\pi$$
$$258$$ 0 0
$$259$$ 11077.1 2.65751
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 7689.01 1.80276 0.901378 0.433033i $$-0.142557\pi$$
0.901378 + 0.433033i $$0.142557\pi$$
$$264$$ 0 0
$$265$$ 398.275 0.0923239
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −5790.73 −1.31252 −0.656258 0.754536i $$-0.727862\pi$$
−0.656258 + 0.754536i $$0.727862\pi$$
$$270$$ 0 0
$$271$$ 6146.43 1.37775 0.688873 0.724882i $$-0.258106\pi$$
0.688873 + 0.724882i $$0.258106\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1682.96 0.369042
$$276$$ 0 0
$$277$$ 4811.02 1.04356 0.521781 0.853080i $$-0.325268\pi$$
0.521781 + 0.853080i $$0.325268\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2333.19 −0.495325 −0.247663 0.968846i $$-0.579662\pi$$
−0.247663 + 0.968846i $$0.579662\pi$$
$$282$$ 0 0
$$283$$ 8697.84 1.82697 0.913486 0.406870i $$-0.133380\pi$$
0.913486 + 0.406870i $$0.133380\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10981.8 2.25865
$$288$$ 0 0
$$289$$ −1175.55 −0.239273
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1838.13 0.366501 0.183251 0.983066i $$-0.441338\pi$$
0.183251 + 0.983066i $$0.441338\pi$$
$$294$$ 0 0
$$295$$ 1697.07 0.334940
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2459.94 0.475793
$$300$$ 0 0
$$301$$ −4649.12 −0.890268
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −2629.28 −0.493614
$$306$$ 0 0
$$307$$ −4152.70 −0.772010 −0.386005 0.922497i $$-0.626145\pi$$
−0.386005 + 0.922497i $$0.626145\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2861.14 −0.521674 −0.260837 0.965383i $$-0.583999\pi$$
−0.260837 + 0.965383i $$0.583999\pi$$
$$312$$ 0 0
$$313$$ −438.494 −0.0791858 −0.0395929 0.999216i $$-0.512606\pi$$
−0.0395929 + 0.999216i $$0.512606\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6279.80 1.11265 0.556323 0.830966i $$-0.312212\pi$$
0.556323 + 0.830966i $$0.312212\pi$$
$$318$$ 0 0
$$319$$ 14278.8 2.50614
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1704.26 −0.293584
$$324$$ 0 0
$$325$$ −1519.78 −0.259391
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 13122.2 2.19894
$$330$$ 0 0
$$331$$ −7564.20 −1.25609 −0.628046 0.778176i $$-0.716145\pi$$
−0.628046 + 0.778176i $$0.716145\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 659.580 0.107572
$$336$$ 0 0
$$337$$ 3468.51 0.560658 0.280329 0.959904i $$-0.409556\pi$$
0.280329 + 0.959904i $$0.409556\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 11263.6 1.78873
$$342$$ 0 0
$$343$$ −6868.88 −1.08130
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6415.10 0.992452 0.496226 0.868193i $$-0.334719\pi$$
0.496226 + 0.868193i $$0.334719\pi$$
$$348$$ 0 0
$$349$$ −2248.97 −0.344941 −0.172470 0.985015i $$-0.555175\pi$$
−0.172470 + 0.985015i $$0.555175\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7197.94 1.08529 0.542645 0.839962i $$-0.317423\pi$$
0.542645 + 0.839962i $$0.317423\pi$$
$$354$$ 0 0
$$355$$ −1481.31 −0.221465
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8618.50 1.26704 0.633520 0.773726i $$-0.281609\pi$$
0.633520 + 0.773726i $$0.281609\pi$$
$$360$$ 0 0
$$361$$ −6081.87 −0.886699
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6151.70 −0.882178
$$366$$ 0 0
$$367$$ −9025.95 −1.28379 −0.641895 0.766793i $$-0.721851\pi$$
−0.641895 + 0.766793i $$0.721851\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2407.23 −0.336865
$$372$$ 0 0
$$373$$ 7549.69 1.04801 0.524005 0.851715i $$-0.324437\pi$$
0.524005 + 0.851715i $$0.324437\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12894.3 −1.76151
$$378$$ 0 0
$$379$$ 11849.7 1.60601 0.803007 0.595970i $$-0.203232\pi$$
0.803007 + 0.595970i $$0.203232\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 8613.06 1.14910 0.574552 0.818468i $$-0.305176\pi$$
0.574552 + 0.818468i $$0.305176\pi$$
$$384$$ 0 0
$$385$$ −10172.1 −1.34654
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2529.23 0.329658 0.164829 0.986322i $$-0.447293\pi$$
0.164829 + 0.986322i $$0.447293\pi$$
$$390$$ 0 0
$$391$$ −2473.84 −0.319968
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −3108.86 −0.396009
$$396$$ 0 0
$$397$$ −14195.7 −1.79461 −0.897306 0.441410i $$-0.854479\pi$$
−0.897306 + 0.441410i $$0.854479\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 11005.5 1.37054 0.685270 0.728289i $$-0.259684\pi$$
0.685270 + 0.728289i $$0.259684\pi$$
$$402$$ 0 0
$$403$$ −10171.4 −1.25726
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24674.8 −3.00513
$$408$$ 0 0
$$409$$ 14339.1 1.73356 0.866779 0.498692i $$-0.166186\pi$$
0.866779 + 0.498692i $$0.166186\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −10257.3 −1.22211
$$414$$ 0 0
$$415$$ −381.686 −0.0451475
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −5352.51 −0.624075 −0.312037 0.950070i $$-0.601011\pi$$
−0.312037 + 0.950070i $$0.601011\pi$$
$$420$$ 0 0
$$421$$ −2683.89 −0.310701 −0.155350 0.987859i $$-0.549651\pi$$
−0.155350 + 0.987859i $$0.549651\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1528.37 0.174439
$$426$$ 0 0
$$427$$ 15891.7 1.80107
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 10019.0 1.11972 0.559861 0.828587i $$-0.310855\pi$$
0.559861 + 0.828587i $$0.310855\pi$$
$$432$$ 0 0
$$433$$ −5292.20 −0.587361 −0.293680 0.955904i $$-0.594880\pi$$
−0.293680 + 0.955904i $$0.594880\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1128.06 0.123484
$$438$$ 0 0
$$439$$ 9453.33 1.02775 0.513875 0.857865i $$-0.328209\pi$$
0.513875 + 0.857865i $$0.328209\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −2858.23 −0.306543 −0.153272 0.988184i $$-0.548981\pi$$
−0.153272 + 0.988184i $$0.548981\pi$$
$$444$$ 0 0
$$445$$ −962.030 −0.102482
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 9322.81 0.979890 0.489945 0.871753i $$-0.337017\pi$$
0.489945 + 0.871753i $$0.337017\pi$$
$$450$$ 0 0
$$451$$ −24462.5 −2.55409
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 9185.75 0.946449
$$456$$ 0 0
$$457$$ 4960.54 0.507755 0.253878 0.967236i $$-0.418294\pi$$
0.253878 + 0.967236i $$0.418294\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5235.83 0.528974 0.264487 0.964389i $$-0.414797\pi$$
0.264487 + 0.964389i $$0.414797\pi$$
$$462$$ 0 0
$$463$$ 7121.96 0.714871 0.357436 0.933938i $$-0.383651\pi$$
0.357436 + 0.933938i $$0.383651\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1991.23 −0.197308 −0.0986541 0.995122i $$-0.531454\pi$$
−0.0986541 + 0.995122i $$0.531454\pi$$
$$468$$ 0 0
$$469$$ −3986.59 −0.392503
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 10356.2 1.00672
$$474$$ 0 0
$$475$$ −696.928 −0.0673205
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9377.15 0.894474 0.447237 0.894415i $$-0.352408\pi$$
0.447237 + 0.894415i $$0.352408\pi$$
$$480$$ 0 0
$$481$$ 22282.3 2.11224
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4373.86 −0.409498
$$486$$ 0 0
$$487$$ −4855.98 −0.451838 −0.225919 0.974146i $$-0.572539\pi$$
−0.225919 + 0.974146i $$0.572539\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4048.75 0.372133 0.186067 0.982537i $$-0.440426\pi$$
0.186067 + 0.982537i $$0.440426\pi$$
$$492$$ 0 0
$$493$$ 12967.2 1.18461
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8953.27 0.808066
$$498$$ 0 0
$$499$$ 6259.64 0.561563 0.280781 0.959772i $$-0.409406\pi$$
0.280781 + 0.959772i $$0.409406\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 9127.86 0.809128 0.404564 0.914510i $$-0.367423\pi$$
0.404564 + 0.914510i $$0.367423\pi$$
$$504$$ 0 0
$$505$$ −5465.08 −0.481570
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −9495.89 −0.826911 −0.413456 0.910524i $$-0.635678\pi$$
−0.413456 + 0.910524i $$0.635678\pi$$
$$510$$ 0 0
$$511$$ 37181.8 3.21883
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1144.66 0.0979412
$$516$$ 0 0
$$517$$ −29230.5 −2.48657
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12282.6 −1.03284 −0.516421 0.856335i $$-0.672736\pi$$
−0.516421 + 0.856335i $$0.672736\pi$$
$$522$$ 0 0
$$523$$ 8010.57 0.669747 0.334873 0.942263i $$-0.391306\pi$$
0.334873 + 0.942263i $$0.391306\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10228.9 0.845502
$$528$$ 0 0
$$529$$ −10529.5 −0.865418
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 22090.6 1.79521
$$534$$ 0 0
$$535$$ 2398.73 0.193843
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 38391.1 3.06794
$$540$$ 0 0
$$541$$ −15476.8 −1.22995 −0.614974 0.788548i $$-0.710833\pi$$
−0.614974 + 0.788548i $$0.710833\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −581.231 −0.0456829
$$546$$ 0 0
$$547$$ −22837.0 −1.78508 −0.892540 0.450968i $$-0.851079\pi$$
−0.892540 + 0.450968i $$0.851079\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5912.96 −0.457170
$$552$$ 0 0
$$553$$ 18790.4 1.44493
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15800.3 1.20194 0.600971 0.799271i $$-0.294781\pi$$
0.600971 + 0.799271i $$0.294781\pi$$
$$558$$ 0 0
$$559$$ −9352.03 −0.707601
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −13515.7 −1.01176 −0.505879 0.862605i $$-0.668832\pi$$
−0.505879 + 0.862605i $$0.668832\pi$$
$$564$$ 0 0
$$565$$ 3706.11 0.275960
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −19632.0 −1.44642 −0.723211 0.690627i $$-0.757335\pi$$
−0.723211 + 0.690627i $$0.757335\pi$$
$$570$$ 0 0
$$571$$ −12800.7 −0.938166 −0.469083 0.883154i $$-0.655416\pi$$
−0.469083 + 0.883154i $$0.655416\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1011.64 −0.0733707
$$576$$ 0 0
$$577$$ −9214.08 −0.664796 −0.332398 0.943139i $$-0.607858\pi$$
−0.332398 + 0.943139i $$0.607858\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2306.96 0.164731
$$582$$ 0 0
$$583$$ 5362.25 0.380929
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 275.136 0.0193460 0.00967298 0.999953i $$-0.496921\pi$$
0.00967298 + 0.999953i $$0.496921\pi$$
$$588$$ 0 0
$$589$$ −4664.34 −0.326300
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 15753.5 1.09093 0.545464 0.838135i $$-0.316354\pi$$
0.545464 + 0.838135i $$0.316354\pi$$
$$594$$ 0 0
$$595$$ −9237.66 −0.636483
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 13557.3 0.924770 0.462385 0.886679i $$-0.346994\pi$$
0.462385 + 0.886679i $$0.346994\pi$$
$$600$$ 0 0
$$601$$ 23976.7 1.62734 0.813668 0.581330i $$-0.197467\pi$$
0.813668 + 0.581330i $$0.197467\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 16003.9 1.07546
$$606$$ 0 0
$$607$$ −12766.6 −0.853677 −0.426839 0.904328i $$-0.640373\pi$$
−0.426839 + 0.904328i $$0.640373\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 26396.2 1.74775
$$612$$ 0 0
$$613$$ −16776.3 −1.10537 −0.552684 0.833391i $$-0.686396\pi$$
−0.552684 + 0.833391i $$0.686396\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 17941.4 1.17065 0.585327 0.810798i $$-0.300966\pi$$
0.585327 + 0.810798i $$0.300966\pi$$
$$618$$ 0 0
$$619$$ −987.096 −0.0640949 −0.0320475 0.999486i $$-0.510203\pi$$
−0.0320475 + 0.999486i $$0.510203\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 5814.64 0.373930
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −22408.2 −1.42047
$$630$$ 0 0
$$631$$ 18504.9 1.16746 0.583730 0.811948i $$-0.301593\pi$$
0.583730 + 0.811948i $$0.301593\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −12976.9 −0.810979
$$636$$ 0 0
$$637$$ −34668.6 −2.15639
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 10260.2 0.632219 0.316110 0.948723i $$-0.397623\pi$$
0.316110 + 0.948723i $$0.397623\pi$$
$$642$$ 0 0
$$643$$ 6221.29 0.381561 0.190781 0.981633i $$-0.438898\pi$$
0.190781 + 0.981633i $$0.438898\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21446.7 −1.30318 −0.651590 0.758572i $$-0.725898\pi$$
−0.651590 + 0.758572i $$0.725898\pi$$
$$648$$ 0 0
$$649$$ 22848.8 1.38196
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −25051.0 −1.50126 −0.750628 0.660725i $$-0.770249\pi$$
−0.750628 + 0.660725i $$0.770249\pi$$
$$654$$ 0 0
$$655$$ 7028.09 0.419252
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 20.7125 0.00122435 0.000612174 1.00000i $$-0.499805\pi$$
0.000612174 1.00000i $$0.499805\pi$$
$$660$$ 0 0
$$661$$ 23294.7 1.37074 0.685371 0.728194i $$-0.259640\pi$$
0.685371 + 0.728194i $$0.259640\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 4212.33 0.245635
$$666$$ 0 0
$$667$$ −8583.05 −0.498257
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −35399.9 −2.03666
$$672$$ 0 0
$$673$$ 25764.3 1.47569 0.737845 0.674970i $$-0.235843\pi$$
0.737845 + 0.674970i $$0.235843\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 26667.3 1.51390 0.756948 0.653475i $$-0.226689\pi$$
0.756948 + 0.653475i $$0.226689\pi$$
$$678$$ 0 0
$$679$$ 26436.2 1.49415
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 14.8616 0.000832599 0 0.000416299 1.00000i $$-0.499867\pi$$
0.000416299 1.00000i $$0.499867\pi$$
$$684$$ 0 0
$$685$$ −12287.0 −0.685345
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −4842.31 −0.267747
$$690$$ 0 0
$$691$$ 26645.5 1.46692 0.733460 0.679733i $$-0.237904\pi$$
0.733460 + 0.679733i $$0.237904\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −2926.96 −0.159749
$$696$$ 0 0
$$697$$ −22215.4 −1.20727
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −431.952 −0.0232733 −0.0116367 0.999932i $$-0.503704\pi$$
−0.0116367 + 0.999932i $$0.503704\pi$$
$$702$$ 0 0
$$703$$ 10218.0 0.548195
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 33031.7 1.75712
$$708$$ 0 0
$$709$$ 35463.0 1.87848 0.939240 0.343261i $$-0.111532\pi$$
0.939240 + 0.343261i $$0.111532\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −6770.60 −0.355626
$$714$$ 0 0
$$715$$ −20461.8 −1.07025
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −26340.5 −1.36625 −0.683125 0.730302i $$-0.739379\pi$$
−0.683125 + 0.730302i $$0.739379\pi$$
$$720$$ 0 0
$$721$$ −6918.48 −0.357362
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5302.70 0.271638
$$726$$ 0 0
$$727$$ −38279.1 −1.95281 −0.976404 0.215952i $$-0.930715\pi$$
−0.976404 + 0.215952i $$0.930715\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 9404.89 0.475858
$$732$$ 0 0
$$733$$ −16445.7 −0.828700 −0.414350 0.910118i $$-0.635991\pi$$
−0.414350 + 0.910118i $$0.635991\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8880.39 0.443844
$$738$$ 0 0
$$739$$ −5150.65 −0.256387 −0.128193 0.991749i $$-0.540918\pi$$
−0.128193 + 0.991749i $$0.540918\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −8844.91 −0.436727 −0.218364 0.975868i $$-0.570072\pi$$
−0.218364 + 0.975868i $$0.570072\pi$$
$$744$$ 0 0
$$745$$ 4632.52 0.227816
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −14498.2 −0.707282
$$750$$ 0 0
$$751$$ 5774.81 0.280594 0.140297 0.990109i $$-0.455194\pi$$
0.140297 + 0.990109i $$0.455194\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 2059.50 0.0992754
$$756$$ 0 0
$$757$$ 14267.1 0.685003 0.342501 0.939517i $$-0.388726\pi$$
0.342501 + 0.939517i $$0.388726\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −21823.0 −1.03953 −0.519765 0.854309i $$-0.673980\pi$$
−0.519765 + 0.854309i $$0.673980\pi$$
$$762$$ 0 0
$$763$$ 3513.04 0.166685
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −20633.3 −0.971352
$$768$$ 0 0
$$769$$ −3080.07 −0.144435 −0.0722173 0.997389i $$-0.523008\pi$$
−0.0722173 + 0.997389i $$0.523008\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −8834.10 −0.411049 −0.205524 0.978652i $$-0.565890\pi$$
−0.205524 + 0.978652i $$0.565890\pi$$
$$774$$ 0 0
$$775$$ 4182.95 0.193879
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 10130.1 0.465917
$$780$$ 0 0
$$781$$ −19944.0 −0.913766
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −14799.8 −0.672903
$$786$$ 0 0
$$787$$ 11060.1 0.500952 0.250476 0.968123i $$-0.419413\pi$$
0.250476 + 0.968123i $$0.419413\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −22400.3 −1.00690
$$792$$ 0 0
$$793$$ 31967.4 1.43152
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3379.95 0.150218 0.0751091 0.997175i $$-0.476070\pi$$
0.0751091 + 0.997175i $$0.476070\pi$$
$$798$$ 0 0
$$799$$ −26545.4 −1.17536
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −82824.7 −3.63988
$$804$$ 0 0
$$805$$ 6114.47 0.267710
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −38358.8 −1.66703 −0.833513 0.552499i $$-0.813674\pi$$
−0.833513 + 0.552499i $$0.813674\pi$$
$$810$$ 0 0
$$811$$ −9110.42 −0.394464 −0.197232 0.980357i $$-0.563195\pi$$
−0.197232 + 0.980357i $$0.563195\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12522.5 0.538214
$$816$$ 0 0
$$817$$ −4288.58 −0.183646
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8177.12 0.347605 0.173802 0.984781i $$-0.444395\pi$$
0.173802 + 0.984781i $$0.444395\pi$$
$$822$$ 0 0
$$823$$ −4238.40 −0.179515 −0.0897577 0.995964i $$-0.528609\pi$$
−0.0897577 + 0.995964i $$0.528609\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12021.9 −0.505492 −0.252746 0.967533i $$-0.581334\pi$$
−0.252746 + 0.967533i $$0.581334\pi$$
$$828$$ 0 0
$$829$$ 6564.14 0.275009 0.137504 0.990501i $$-0.456092\pi$$
0.137504 + 0.990501i $$0.456092\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 34864.5 1.45016
$$834$$ 0 0
$$835$$ 10277.1 0.425932
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −3645.41 −0.150004 −0.0750021 0.997183i $$-0.523896\pi$$
−0.0750021 + 0.997183i $$0.523896\pi$$
$$840$$ 0 0
$$841$$ 20600.9 0.844679
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 7492.79 0.305041
$$846$$ 0 0
$$847$$ −96729.9 −3.92406
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 14832.2 0.597462
$$852$$ 0 0
$$853$$ −45697.7 −1.83430 −0.917151 0.398540i $$-0.869517\pi$$
−0.917151 + 0.398540i $$0.869517\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −13766.3 −0.548716 −0.274358 0.961628i $$-0.588465\pi$$
−0.274358 + 0.961628i $$0.588465\pi$$
$$858$$ 0 0
$$859$$ 15422.0 0.612563 0.306282 0.951941i $$-0.400915\pi$$
0.306282 + 0.951941i $$0.400915\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 7382.54 0.291199 0.145599 0.989344i $$-0.453489\pi$$
0.145599 + 0.989344i $$0.453489\pi$$
$$864$$ 0 0
$$865$$ 8620.83 0.338863
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −41856.8 −1.63394
$$870$$ 0 0
$$871$$ −8019.32 −0.311968
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −3777.59 −0.145949
$$876$$ 0 0
$$877$$ −12079.7 −0.465112 −0.232556 0.972583i $$-0.574709\pi$$
−0.232556 + 0.972583i $$0.574709\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −25501.7 −0.975227 −0.487613 0.873060i $$-0.662132\pi$$
−0.487613 + 0.873060i $$0.662132\pi$$
$$882$$ 0 0
$$883$$ −29353.2 −1.11870 −0.559351 0.828931i $$-0.688950\pi$$
−0.559351 + 0.828931i $$0.688950\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −17185.8 −0.650556 −0.325278 0.945618i $$-0.605458\pi$$
−0.325278 + 0.945618i $$0.605458\pi$$
$$888$$ 0 0
$$889$$ 78434.1 2.95905
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 12104.6 0.453600
$$894$$ 0 0
$$895$$ −18006.8 −0.672513
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 35489.5 1.31662
$$900$$ 0 0
$$901$$ 4869.68 0.180058
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6285.83 −0.230882
$$906$$ 0 0
$$907$$ −38511.6 −1.40988 −0.704938 0.709269i $$-0.749025\pi$$
−0.704938 + 0.709269i $$0.749025\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 11451.4 0.416467 0.208233 0.978079i $$-0.433229\pi$$
0.208233 + 0.978079i $$0.433229\pi$$
$$912$$ 0 0
$$913$$ −5138.90 −0.186279
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −42478.8 −1.52974
$$918$$ 0 0
$$919$$ 14096.0 0.505968 0.252984 0.967470i $$-0.418588\pi$$
0.252984 + 0.967470i $$0.418588\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 18010.1 0.642265
$$924$$ 0 0
$$925$$ −9163.47 −0.325722
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 15020.3 0.530462 0.265231 0.964185i $$-0.414552\pi$$
0.265231 + 0.964185i $$0.414552\pi$$
$$930$$ 0 0
$$931$$ −15898.1 −0.559654
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 20577.5 0.719739
$$936$$ 0 0
$$937$$ 18430.5 0.642582 0.321291 0.946981i $$-0.395883\pi$$
0.321291 + 0.946981i $$0.395883\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −55141.3 −1.91026 −0.955131 0.296184i $$-0.904286\pi$$
−0.955131 + 0.296184i $$0.904286\pi$$
$$942$$ 0 0
$$943$$ 14704.5 0.507790
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −24231.3 −0.831480 −0.415740 0.909484i $$-0.636477\pi$$
−0.415740 + 0.909484i $$0.636477\pi$$
$$948$$ 0 0
$$949$$ 74793.8 2.55839
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 19107.5 0.649479 0.324739 0.945804i $$-0.394723\pi$$
0.324739 + 0.945804i $$0.394723\pi$$
$$954$$ 0 0
$$955$$ −23387.3 −0.792457
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 74264.2 2.50064
$$960$$ 0 0
$$961$$ −1795.67 −0.0602756
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −13239.4 −0.441649
$$966$$ 0 0
$$967$$ 21993.6 0.731401 0.365701 0.930732i $$-0.380829\pi$$
0.365701 + 0.930732i $$0.380829\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −19409.3 −0.641477 −0.320738 0.947168i $$-0.603931\pi$$
−0.320738 + 0.947168i $$0.603931\pi$$
$$972$$ 0 0
$$973$$ 17690.9 0.582883
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −36384.1 −1.19143 −0.595717 0.803194i $$-0.703132\pi$$
−0.595717 + 0.803194i $$0.703132\pi$$
$$978$$ 0 0
$$979$$ −12952.5 −0.422843
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 4202.43 0.136355 0.0681774 0.997673i $$-0.478282\pi$$
0.0681774 + 0.997673i $$0.478282\pi$$
$$984$$ 0 0
$$985$$ 6844.57 0.221407
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6225.16 −0.200150
$$990$$ 0 0
$$991$$ −17753.0 −0.569064 −0.284532 0.958666i $$-0.591838\pi$$
−0.284532 + 0.958666i $$0.591838\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −27057.2 −0.862080
$$996$$ 0 0
$$997$$ 12519.0 0.397673 0.198837 0.980033i $$-0.436284\pi$$
0.198837 + 0.980033i $$0.436284\pi$$
$$998$$ 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 1080.4.a.j.1.1 yes 3
3.2 odd 2 1080.4.a.d.1.1 3
4.3 odd 2 2160.4.a.bs.1.3 3
12.11 even 2 2160.4.a.bk.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.1 3 3.2 odd 2
1080.4.a.j.1.1 yes 3 1.1 even 1 trivial
2160.4.a.bk.1.3 3 12.11 even 2
2160.4.a.bs.1.3 3 4.3 odd 2