# Properties

 Label 2160.4.a.bs.1.1 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 1.1 Root $$2.72396$$ 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)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -24.0794 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -24.0794 q^{7} -2.95982 q^{11} -22.1759 q^{13} -76.0223 q^{17} +72.1187 q^{19} +176.003 q^{23} +25.0000 q^{25} -42.5134 q^{29} +327.010 q^{31} -120.397 q^{35} +182.089 q^{37} -154.624 q^{41} -173.531 q^{43} +338.059 q^{47} +236.820 q^{49} +26.5490 q^{53} -14.7991 q^{55} -391.670 q^{59} -191.191 q^{61} -110.879 q^{65} +507.583 q^{67} -576.547 q^{71} -390.440 q^{73} +71.2709 q^{77} -1220.43 q^{79} +247.332 q^{83} -380.111 q^{85} -1500.81 q^{89} +533.983 q^{91} +360.594 q^{95} +959.331 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} + 10 q^{7}+O(q^{10})$$ 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
$$7$$ −24.0794 −1.30017 −0.650084 0.759862i $$-0.725266\pi$$
−0.650084 + 0.759862i $$0.725266\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.95982 −0.0811292 −0.0405646 0.999177i $$-0.512916\pi$$
−0.0405646 + 0.999177i $$0.512916\pi$$
$$12$$ 0 0
$$13$$ −22.1759 −0.473114 −0.236557 0.971618i $$-0.576019\pi$$
−0.236557 + 0.971618i $$0.576019\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −76.0223 −1.08459 −0.542297 0.840187i $$-0.682445\pi$$
−0.542297 + 0.840187i $$0.682445\pi$$
$$18$$ 0 0
$$19$$ 72.1187 0.870798 0.435399 0.900238i $$-0.356607\pi$$
0.435399 + 0.900238i $$0.356607\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 176.003 1.59562 0.797810 0.602909i $$-0.205992\pi$$
0.797810 + 0.602909i $$0.205992\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −42.5134 −0.272226 −0.136113 0.990693i $$-0.543461\pi$$
−0.136113 + 0.990693i $$0.543461\pi$$
$$30$$ 0 0
$$31$$ 327.010 1.89460 0.947301 0.320345i $$-0.103799\pi$$
0.947301 + 0.320345i $$0.103799\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −120.397 −0.581453
$$36$$ 0 0
$$37$$ 182.089 0.809061 0.404530 0.914525i $$-0.367435\pi$$
0.404530 + 0.914525i $$0.367435\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −154.624 −0.588981 −0.294491 0.955654i $$-0.595150\pi$$
−0.294491 + 0.955654i $$0.595150\pi$$
$$42$$ 0 0
$$43$$ −173.531 −0.615424 −0.307712 0.951479i $$-0.599563\pi$$
−0.307712 + 0.951479i $$0.599563\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 338.059 1.04917 0.524585 0.851358i $$-0.324221\pi$$
0.524585 + 0.851358i $$0.324221\pi$$
$$48$$ 0 0
$$49$$ 236.820 0.690436
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 26.5490 0.0688074 0.0344037 0.999408i $$-0.489047\pi$$
0.0344037 + 0.999408i $$0.489047\pi$$
$$54$$ 0 0
$$55$$ −14.7991 −0.0362821
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −391.670 −0.864256 −0.432128 0.901812i $$-0.642237\pi$$
−0.432128 + 0.901812i $$0.642237\pi$$
$$60$$ 0 0
$$61$$ −191.191 −0.401303 −0.200652 0.979663i $$-0.564306\pi$$
−0.200652 + 0.979663i $$0.564306\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −110.879 −0.211583
$$66$$ 0 0
$$67$$ 507.583 0.925540 0.462770 0.886478i $$-0.346856\pi$$
0.462770 + 0.886478i $$0.346856\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −576.547 −0.963712 −0.481856 0.876251i $$-0.660037\pi$$
−0.481856 + 0.876251i $$0.660037\pi$$
$$72$$ 0 0
$$73$$ −390.440 −0.625994 −0.312997 0.949754i $$-0.601333\pi$$
−0.312997 + 0.949754i $$0.601333\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 71.2709 0.105482
$$78$$ 0 0
$$79$$ −1220.43 −1.73808 −0.869042 0.494739i $$-0.835264\pi$$
−0.869042 + 0.494739i $$0.835264\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 247.332 0.327088 0.163544 0.986536i $$-0.447708\pi$$
0.163544 + 0.986536i $$0.447708\pi$$
$$84$$ 0 0
$$85$$ −380.111 −0.485045
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1500.81 −1.78748 −0.893740 0.448586i $$-0.851928\pi$$
−0.893740 + 0.448586i $$0.851928\pi$$
$$90$$ 0 0
$$91$$ 533.983 0.615128
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 360.594 0.389433
$$96$$ 0 0
$$97$$ 959.331 1.00418 0.502089 0.864816i $$-0.332565\pi$$
0.502089 + 0.864816i $$0.332565\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −587.011 −0.578314 −0.289157 0.957282i $$-0.593375\pi$$
−0.289157 + 0.957282i $$0.593375\pi$$
$$102$$ 0 0
$$103$$ −1359.95 −1.30097 −0.650483 0.759521i $$-0.725433\pi$$
−0.650483 + 0.759521i $$0.725433\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1041.85 0.941306 0.470653 0.882318i $$-0.344018\pi$$
0.470653 + 0.882318i $$0.344018\pi$$
$$108$$ 0 0
$$109$$ −457.732 −0.402227 −0.201114 0.979568i $$-0.564456\pi$$
−0.201114 + 0.979568i $$0.564456\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −467.527 −0.389214 −0.194607 0.980881i $$-0.562343\pi$$
−0.194607 + 0.980881i $$0.562343\pi$$
$$114$$ 0 0
$$115$$ 880.017 0.713583
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1830.57 1.41015
$$120$$ 0 0
$$121$$ −1322.24 −0.993418
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2368.02 −1.65455 −0.827274 0.561799i $$-0.810109\pi$$
−0.827274 + 0.561799i $$0.810109\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2532.97 1.68937 0.844683 0.535267i $$-0.179789\pi$$
0.844683 + 0.535267i $$0.179789\pi$$
$$132$$ 0 0
$$133$$ −1736.58 −1.13218
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 424.096 0.264474 0.132237 0.991218i $$-0.457784\pi$$
0.132237 + 0.991218i $$0.457784\pi$$
$$138$$ 0 0
$$139$$ −1461.14 −0.891596 −0.445798 0.895134i $$-0.647080\pi$$
−0.445798 + 0.895134i $$0.647080\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 65.6368 0.0383834
$$144$$ 0 0
$$145$$ −212.567 −0.121743
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −497.826 −0.273715 −0.136857 0.990591i $$-0.543700\pi$$
−0.136857 + 0.990591i $$0.543700\pi$$
$$150$$ 0 0
$$151$$ −708.375 −0.381767 −0.190883 0.981613i $$-0.561135\pi$$
−0.190883 + 0.981613i $$0.561135\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1635.05 0.847292
$$156$$ 0 0
$$157$$ 351.998 0.178933 0.0894666 0.995990i $$-0.471484\pi$$
0.0894666 + 0.995990i $$0.471484\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4238.06 −2.07457
$$162$$ 0 0
$$163$$ 791.353 0.380267 0.190133 0.981758i $$-0.439108\pi$$
0.190133 + 0.981758i $$0.439108\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2756.06 1.27707 0.638535 0.769593i $$-0.279541\pi$$
0.638535 + 0.769593i $$0.279541\pi$$
$$168$$ 0 0
$$169$$ −1705.23 −0.776163
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3691.62 −1.62236 −0.811182 0.584794i $$-0.801175\pi$$
−0.811182 + 0.584794i $$0.801175\pi$$
$$174$$ 0 0
$$175$$ −601.986 −0.260034
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2036.15 −0.850217 −0.425109 0.905142i $$-0.639764\pi$$
−0.425109 + 0.905142i $$0.639764\pi$$
$$180$$ 0 0
$$181$$ −3268.74 −1.34234 −0.671170 0.741304i $$-0.734208\pi$$
−0.671170 + 0.741304i $$0.734208\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 910.445 0.361823
$$186$$ 0 0
$$187$$ 225.013 0.0879922
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2192.48 −0.830589 −0.415294 0.909687i $$-0.636321\pi$$
−0.415294 + 0.909687i $$0.636321\pi$$
$$192$$ 0 0
$$193$$ 3326.76 1.24075 0.620376 0.784304i $$-0.286980\pi$$
0.620376 + 0.784304i $$0.286980\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1818.42 0.657648 0.328824 0.944391i $$-0.393348\pi$$
0.328824 + 0.944391i $$0.393348\pi$$
$$198$$ 0 0
$$199$$ 1191.04 0.424275 0.212137 0.977240i $$-0.431958\pi$$
0.212137 + 0.977240i $$0.431958\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1023.70 0.353939
$$204$$ 0 0
$$205$$ −773.121 −0.263400
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −213.459 −0.0706471
$$210$$ 0 0
$$211$$ −2929.49 −0.955803 −0.477902 0.878413i $$-0.658602\pi$$
−0.477902 + 0.878413i $$0.658602\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −867.656 −0.275226
$$216$$ 0 0
$$217$$ −7874.21 −2.46330
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1685.86 0.513137
$$222$$ 0 0
$$223$$ −1479.80 −0.444370 −0.222185 0.975004i $$-0.571319\pi$$
−0.222185 + 0.975004i $$0.571319\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1543.24 0.451228 0.225614 0.974217i $$-0.427561\pi$$
0.225614 + 0.974217i $$0.427561\pi$$
$$228$$ 0 0
$$229$$ 3163.97 0.913018 0.456509 0.889719i $$-0.349100\pi$$
0.456509 + 0.889719i $$0.349100\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1466.95 0.412459 0.206229 0.978504i $$-0.433881\pi$$
0.206229 + 0.978504i $$0.433881\pi$$
$$234$$ 0 0
$$235$$ 1690.29 0.469203
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1346.42 −0.364405 −0.182202 0.983261i $$-0.558323\pi$$
−0.182202 + 0.983261i $$0.558323\pi$$
$$240$$ 0 0
$$241$$ −47.7311 −0.0127578 −0.00637890 0.999980i $$-0.502030\pi$$
−0.00637890 + 0.999980i $$0.502030\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1184.10 0.308772
$$246$$ 0 0
$$247$$ −1599.30 −0.411987
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5074.18 1.27601 0.638007 0.770031i $$-0.279759\pi$$
0.638007 + 0.770031i $$0.279759\pi$$
$$252$$ 0 0
$$253$$ −520.939 −0.129451
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3088.03 −0.749518 −0.374759 0.927122i $$-0.622275\pi$$
−0.374759 + 0.927122i $$0.622275\pi$$
$$258$$ 0 0
$$259$$ −4384.60 −1.05191
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 4777.33 1.12009 0.560043 0.828463i $$-0.310784\pi$$
0.560043 + 0.828463i $$0.310784\pi$$
$$264$$ 0 0
$$265$$ 132.745 0.0307716
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −7972.82 −1.80710 −0.903552 0.428478i $$-0.859050\pi$$
−0.903552 + 0.428478i $$0.859050\pi$$
$$270$$ 0 0
$$271$$ −3478.08 −0.779625 −0.389813 0.920894i $$-0.627460\pi$$
−0.389813 + 0.920894i $$0.627460\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −73.9956 −0.0162258
$$276$$ 0 0
$$277$$ −7166.62 −1.55451 −0.777257 0.629184i $$-0.783389\pi$$
−0.777257 + 0.629184i $$0.783389\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8157.14 1.73172 0.865862 0.500284i $$-0.166771\pi$$
0.865862 + 0.500284i $$0.166771\pi$$
$$282$$ 0 0
$$283$$ −4435.42 −0.931654 −0.465827 0.884876i $$-0.654243\pi$$
−0.465827 + 0.884876i $$0.654243\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3723.26 0.765774
$$288$$ 0 0
$$289$$ 866.384 0.176345
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1811.46 0.361183 0.180592 0.983558i $$-0.442199\pi$$
0.180592 + 0.983558i $$0.442199\pi$$
$$294$$ 0 0
$$295$$ −1958.35 −0.386507
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −3903.03 −0.754911
$$300$$ 0 0
$$301$$ 4178.53 0.800155
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −955.955 −0.179468
$$306$$ 0 0
$$307$$ 9369.97 1.74193 0.870965 0.491345i $$-0.163495\pi$$
0.870965 + 0.491345i $$0.163495\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −824.869 −0.150399 −0.0751994 0.997169i $$-0.523959\pi$$
−0.0751994 + 0.997169i $$0.523959\pi$$
$$312$$ 0 0
$$313$$ 3447.76 0.622616 0.311308 0.950309i $$-0.399233\pi$$
0.311308 + 0.950309i $$0.399233\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3822.49 0.677264 0.338632 0.940919i $$-0.390036\pi$$
0.338632 + 0.940919i $$0.390036\pi$$
$$318$$ 0 0
$$319$$ 125.832 0.0220854
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5482.63 −0.944463
$$324$$ 0 0
$$325$$ −554.397 −0.0946229
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8140.27 −1.36410
$$330$$ 0 0
$$331$$ 297.908 0.0494698 0.0247349 0.999694i $$-0.492126\pi$$
0.0247349 + 0.999694i $$0.492126\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 2537.92 0.413914
$$336$$ 0 0
$$337$$ 738.773 0.119417 0.0597085 0.998216i $$-0.480983\pi$$
0.0597085 + 0.998216i $$0.480983\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −967.891 −0.153707
$$342$$ 0 0
$$343$$ 2556.77 0.402485
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3379.76 −0.522868 −0.261434 0.965221i $$-0.584195\pi$$
−0.261434 + 0.965221i $$0.584195\pi$$
$$348$$ 0 0
$$349$$ −2622.01 −0.402157 −0.201078 0.979575i $$-0.564445\pi$$
−0.201078 + 0.979575i $$0.564445\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3052.95 −0.460318 −0.230159 0.973153i $$-0.573925\pi$$
−0.230159 + 0.973153i $$0.573925\pi$$
$$354$$ 0 0
$$355$$ −2882.73 −0.430985
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −1613.36 −0.237187 −0.118593 0.992943i $$-0.537838\pi$$
−0.118593 + 0.992943i $$0.537838\pi$$
$$360$$ 0 0
$$361$$ −1657.89 −0.241710
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1952.20 −0.279953
$$366$$ 0 0
$$367$$ −5387.79 −0.766323 −0.383161 0.923681i $$-0.625165\pi$$
−0.383161 + 0.923681i $$0.625165\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −639.286 −0.0894611
$$372$$ 0 0
$$373$$ −13742.0 −1.90759 −0.953796 0.300455i $$-0.902861\pi$$
−0.953796 + 0.300455i $$0.902861\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 942.773 0.128794
$$378$$ 0 0
$$379$$ −6648.79 −0.901123 −0.450561 0.892745i $$-0.648776\pi$$
−0.450561 + 0.892745i $$0.648776\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −9086.53 −1.21227 −0.606136 0.795361i $$-0.707281\pi$$
−0.606136 + 0.795361i $$0.707281\pi$$
$$384$$ 0 0
$$385$$ 356.355 0.0471728
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 3098.40 0.403844 0.201922 0.979402i $$-0.435281\pi$$
0.201922 + 0.979402i $$0.435281\pi$$
$$390$$ 0 0
$$391$$ −13380.2 −1.73060
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −6102.13 −0.777294
$$396$$ 0 0
$$397$$ −13686.5 −1.73024 −0.865122 0.501562i $$-0.832759\pi$$
−0.865122 + 0.501562i $$0.832759\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3734.83 −0.465108 −0.232554 0.972583i $$-0.574708\pi$$
−0.232554 + 0.972583i $$0.574708\pi$$
$$402$$ 0 0
$$403$$ −7251.73 −0.896363
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −538.952 −0.0656384
$$408$$ 0 0
$$409$$ 3672.62 0.444009 0.222004 0.975046i $$-0.428740\pi$$
0.222004 + 0.975046i $$0.428740\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 9431.20 1.12368
$$414$$ 0 0
$$415$$ 1236.66 0.146278
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −9111.68 −1.06237 −0.531187 0.847255i $$-0.678254\pi$$
−0.531187 + 0.847255i $$0.678254\pi$$
$$420$$ 0 0
$$421$$ 13640.0 1.57903 0.789514 0.613733i $$-0.210333\pi$$
0.789514 + 0.613733i $$0.210333\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1900.56 −0.216919
$$426$$ 0 0
$$427$$ 4603.77 0.521762
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −9252.97 −1.03411 −0.517053 0.855953i $$-0.672971\pi$$
−0.517053 + 0.855953i $$0.672971\pi$$
$$432$$ 0 0
$$433$$ −8990.85 −0.997858 −0.498929 0.866643i $$-0.666273\pi$$
−0.498929 + 0.866643i $$0.666273\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12693.1 1.38946
$$438$$ 0 0
$$439$$ −10409.3 −1.13169 −0.565843 0.824513i $$-0.691449\pi$$
−0.565843 + 0.824513i $$0.691449\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 3083.63 0.330717 0.165359 0.986233i $$-0.447122\pi$$
0.165359 + 0.986233i $$0.447122\pi$$
$$444$$ 0 0
$$445$$ −7504.05 −0.799385
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 16734.4 1.75890 0.879448 0.475996i $$-0.157912\pi$$
0.879448 + 0.475996i $$0.157912\pi$$
$$450$$ 0 0
$$451$$ 457.660 0.0477836
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2669.92 0.275094
$$456$$ 0 0
$$457$$ −19008.4 −1.94568 −0.972841 0.231473i $$-0.925646\pi$$
−0.972841 + 0.231473i $$0.925646\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 8624.58 0.871338 0.435669 0.900107i $$-0.356512\pi$$
0.435669 + 0.900107i $$0.356512\pi$$
$$462$$ 0 0
$$463$$ −8116.40 −0.814689 −0.407345 0.913275i $$-0.633545\pi$$
−0.407345 + 0.913275i $$0.633545\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3645.17 0.361196 0.180598 0.983557i $$-0.442197\pi$$
0.180598 + 0.983557i $$0.442197\pi$$
$$468$$ 0 0
$$469$$ −12222.3 −1.20336
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 513.622 0.0499289
$$474$$ 0 0
$$475$$ 1802.97 0.174160
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6118.27 −0.583613 −0.291807 0.956477i $$-0.594256\pi$$
−0.291807 + 0.956477i $$0.594256\pi$$
$$480$$ 0 0
$$481$$ −4037.99 −0.382778
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4796.66 0.449082
$$486$$ 0 0
$$487$$ 20400.0 1.89818 0.949091 0.315004i $$-0.102006\pi$$
0.949091 + 0.315004i $$0.102006\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4429.08 0.407091 0.203545 0.979066i $$-0.434754\pi$$
0.203545 + 0.979066i $$0.434754\pi$$
$$492$$ 0 0
$$493$$ 3231.96 0.295254
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 13882.9 1.25299
$$498$$ 0 0
$$499$$ −17665.2 −1.58477 −0.792386 0.610020i $$-0.791161\pi$$
−0.792386 + 0.610020i $$0.791161\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1518.17 −0.134577 −0.0672883 0.997734i $$-0.521435\pi$$
−0.0672883 + 0.997734i $$0.521435\pi$$
$$504$$ 0 0
$$505$$ −2935.05 −0.258630
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 8679.43 0.755813 0.377906 0.925844i $$-0.376644\pi$$
0.377906 + 0.925844i $$0.376644\pi$$
$$510$$ 0 0
$$511$$ 9401.58 0.813897
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −6799.73 −0.581809
$$516$$ 0 0
$$517$$ −1000.59 −0.0851182
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −5080.54 −0.427222 −0.213611 0.976919i $$-0.568522\pi$$
−0.213611 + 0.976919i $$0.568522\pi$$
$$522$$ 0 0
$$523$$ −13889.3 −1.16125 −0.580626 0.814170i $$-0.697192\pi$$
−0.580626 + 0.814170i $$0.697192\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −24860.0 −2.05487
$$528$$ 0 0
$$529$$ 18810.2 1.54600
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 3428.93 0.278656
$$534$$ 0 0
$$535$$ 5209.26 0.420965
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −700.944 −0.0560145
$$540$$ 0 0
$$541$$ −20787.4 −1.65198 −0.825989 0.563686i $$-0.809383\pi$$
−0.825989 + 0.563686i $$0.809383\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2288.66 −0.179882
$$546$$ 0 0
$$547$$ 2015.14 0.157516 0.0787578 0.996894i $$-0.474905\pi$$
0.0787578 + 0.996894i $$0.474905\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3066.01 −0.237054
$$552$$ 0 0
$$553$$ 29387.2 2.25980
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −16460.8 −1.25218 −0.626092 0.779750i $$-0.715346\pi$$
−0.626092 + 0.779750i $$0.715346\pi$$
$$558$$ 0 0
$$559$$ 3848.21 0.291166
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 968.373 0.0724903 0.0362452 0.999343i $$-0.488460\pi$$
0.0362452 + 0.999343i $$0.488460\pi$$
$$564$$ 0 0
$$565$$ −2337.63 −0.174062
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 3305.91 0.243569 0.121785 0.992557i $$-0.461138\pi$$
0.121785 + 0.992557i $$0.461138\pi$$
$$570$$ 0 0
$$571$$ −22291.2 −1.63373 −0.816863 0.576831i $$-0.804289\pi$$
−0.816863 + 0.576831i $$0.804289\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4400.09 0.319124
$$576$$ 0 0
$$577$$ 12887.3 0.929820 0.464910 0.885358i $$-0.346087\pi$$
0.464910 + 0.885358i $$0.346087\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5955.63 −0.425269
$$582$$ 0 0
$$583$$ −78.5805 −0.00558228
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8042.37 0.565493 0.282746 0.959195i $$-0.408755\pi$$
0.282746 + 0.959195i $$0.408755\pi$$
$$588$$ 0 0
$$589$$ 23583.5 1.64982
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4779.03 0.330946 0.165473 0.986214i $$-0.447085\pi$$
0.165473 + 0.986214i $$0.447085\pi$$
$$594$$ 0 0
$$595$$ 9152.87 0.630640
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 4890.20 0.333569 0.166785 0.985993i $$-0.446662\pi$$
0.166785 + 0.985993i $$0.446662\pi$$
$$600$$ 0 0
$$601$$ −2732.86 −0.185484 −0.0927418 0.995690i $$-0.529563\pi$$
−0.0927418 + 0.995690i $$0.529563\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −6611.20 −0.444270
$$606$$ 0 0
$$607$$ 18006.9 1.20408 0.602041 0.798465i $$-0.294355\pi$$
0.602041 + 0.798465i $$0.294355\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7496.76 −0.496377
$$612$$ 0 0
$$613$$ −9779.27 −0.644341 −0.322170 0.946682i $$-0.604412\pi$$
−0.322170 + 0.946682i $$0.604412\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18024.5 −1.17608 −0.588040 0.808832i $$-0.700100\pi$$
−0.588040 + 0.808832i $$0.700100\pi$$
$$618$$ 0 0
$$619$$ −2990.51 −0.194182 −0.0970911 0.995275i $$-0.530954\pi$$
−0.0970911 + 0.995275i $$0.530954\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 36138.7 2.32402
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −13842.8 −0.877503
$$630$$ 0 0
$$631$$ 28203.6 1.77935 0.889673 0.456599i $$-0.150932\pi$$
0.889673 + 0.456599i $$0.150932\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −11840.1 −0.739936
$$636$$ 0 0
$$637$$ −5251.69 −0.326655
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 27172.9 1.67436 0.837180 0.546928i $$-0.184203\pi$$
0.837180 + 0.546928i $$0.184203\pi$$
$$642$$ 0 0
$$643$$ 278.480 0.0170796 0.00853979 0.999964i $$-0.497282\pi$$
0.00853979 + 0.999964i $$0.497282\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −22641.1 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$648$$ 0 0
$$649$$ 1159.28 0.0701164
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 23064.5 1.38221 0.691104 0.722755i $$-0.257125\pi$$
0.691104 + 0.722755i $$0.257125\pi$$
$$654$$ 0 0
$$655$$ 12664.9 0.755507
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 22595.0 1.33563 0.667813 0.744329i $$-0.267231\pi$$
0.667813 + 0.744329i $$0.267231\pi$$
$$660$$ 0 0
$$661$$ −14627.9 −0.860756 −0.430378 0.902649i $$-0.641620\pi$$
−0.430378 + 0.902649i $$0.641620\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8682.89 −0.506328
$$666$$ 0 0
$$667$$ −7482.50 −0.434368
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 565.892 0.0325574
$$672$$ 0 0
$$673$$ 22246.9 1.27423 0.637113 0.770770i $$-0.280128\pi$$
0.637113 + 0.770770i $$0.280128\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −13502.1 −0.766513 −0.383256 0.923642i $$-0.625197\pi$$
−0.383256 + 0.923642i $$0.625197\pi$$
$$678$$ 0 0
$$679$$ −23100.2 −1.30560
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −25582.5 −1.43322 −0.716609 0.697475i $$-0.754307\pi$$
−0.716609 + 0.697475i $$0.754307\pi$$
$$684$$ 0 0
$$685$$ 2120.48 0.118276
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −588.749 −0.0325538
$$690$$ 0 0
$$691$$ 12434.6 0.684565 0.342283 0.939597i $$-0.388800\pi$$
0.342283 + 0.939597i $$0.388800\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −7305.68 −0.398734
$$696$$ 0 0
$$697$$ 11754.9 0.638806
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 13773.1 0.742088 0.371044 0.928615i $$-0.379000\pi$$
0.371044 + 0.928615i $$0.379000\pi$$
$$702$$ 0 0
$$703$$ 13132.0 0.704529
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 14134.9 0.751906
$$708$$ 0 0
$$709$$ −11950.6 −0.633023 −0.316511 0.948589i $$-0.602512\pi$$
−0.316511 + 0.948589i $$0.602512\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 57554.8 3.02306
$$714$$ 0 0
$$715$$ 328.184 0.0171656
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −23596.2 −1.22391 −0.611953 0.790894i $$-0.709616\pi$$
−0.611953 + 0.790894i $$0.709616\pi$$
$$720$$ 0 0
$$721$$ 32746.7 1.69147
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1062.83 −0.0544451
$$726$$ 0 0
$$727$$ 14414.5 0.735358 0.367679 0.929953i $$-0.380152\pi$$
0.367679 + 0.929953i $$0.380152\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 13192.2 0.667486
$$732$$ 0 0
$$733$$ 28989.8 1.46079 0.730397 0.683023i $$-0.239335\pi$$
0.730397 + 0.683023i $$0.239335\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1502.36 −0.0750883
$$738$$ 0 0
$$739$$ −5035.96 −0.250678 −0.125339 0.992114i $$-0.540002\pi$$
−0.125339 + 0.992114i $$0.540002\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36268.2 1.79078 0.895390 0.445282i $$-0.146897\pi$$
0.895390 + 0.445282i $$0.146897\pi$$
$$744$$ 0 0
$$745$$ −2489.13 −0.122409
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −25087.2 −1.22386
$$750$$ 0 0
$$751$$ −6876.86 −0.334141 −0.167071 0.985945i $$-0.553431\pi$$
−0.167071 + 0.985945i $$0.553431\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −3541.88 −0.170731
$$756$$ 0 0
$$757$$ −32381.0 −1.55470 −0.777350 0.629068i $$-0.783437\pi$$
−0.777350 + 0.629068i $$0.783437\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9253.37 0.440781 0.220391 0.975412i $$-0.429267\pi$$
0.220391 + 0.975412i $$0.429267\pi$$
$$762$$ 0 0
$$763$$ 11021.9 0.522963
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8685.64 0.408892
$$768$$ 0 0
$$769$$ 31587.9 1.48126 0.740630 0.671913i $$-0.234527\pi$$
0.740630 + 0.671913i $$0.234527\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −38886.4 −1.80937 −0.904687 0.426077i $$-0.859895\pi$$
−0.904687 + 0.426077i $$0.859895\pi$$
$$774$$ 0 0
$$775$$ 8175.24 0.378920
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −11151.3 −0.512884
$$780$$ 0 0
$$781$$ 1706.48 0.0781851
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1759.99 0.0800214
$$786$$ 0 0
$$787$$ 8792.97 0.398266 0.199133 0.979972i $$-0.436187\pi$$
0.199133 + 0.979972i $$0.436187\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 11257.8 0.506044
$$792$$ 0 0
$$793$$ 4239.83 0.189862
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 21136.5 0.939391 0.469695 0.882829i $$-0.344364\pi$$
0.469695 + 0.882829i $$0.344364\pi$$
$$798$$ 0 0
$$799$$ −25700.0 −1.13792
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1155.63 0.0507864
$$804$$ 0 0
$$805$$ −21190.3 −0.927777
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31974.0 −1.38955 −0.694776 0.719227i $$-0.744496\pi$$
−0.694776 + 0.719227i $$0.744496\pi$$
$$810$$ 0 0
$$811$$ 15694.2 0.679528 0.339764 0.940511i $$-0.389653\pi$$
0.339764 + 0.940511i $$0.389653\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 3956.76 0.170061
$$816$$ 0 0
$$817$$ −12514.8 −0.535910
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 156.635 0.00665848 0.00332924 0.999994i $$-0.498940\pi$$
0.00332924 + 0.999994i $$0.498940\pi$$
$$822$$ 0 0
$$823$$ −13428.1 −0.568743 −0.284372 0.958714i $$-0.591785\pi$$
−0.284372 + 0.958714i $$0.591785\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −33717.9 −1.41776 −0.708880 0.705329i $$-0.750799\pi$$
−0.708880 + 0.705329i $$0.750799\pi$$
$$828$$ 0 0
$$829$$ 38733.5 1.62276 0.811380 0.584518i $$-0.198717\pi$$
0.811380 + 0.584518i $$0.198717\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −18003.6 −0.748843
$$834$$ 0 0
$$835$$ 13780.3 0.571123
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −14106.0 −0.580443 −0.290221 0.956960i $$-0.593729\pi$$
−0.290221 + 0.956960i $$0.593729\pi$$
$$840$$ 0 0
$$841$$ −22581.6 −0.925893
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −8526.15 −0.347111
$$846$$ 0 0
$$847$$ 31838.8 1.29161
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 32048.3 1.29095
$$852$$ 0 0
$$853$$ −24614.3 −0.988015 −0.494007 0.869458i $$-0.664468\pi$$
−0.494007 + 0.869458i $$0.664468\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −21660.2 −0.863360 −0.431680 0.902027i $$-0.642079\pi$$
−0.431680 + 0.902027i $$0.642079\pi$$
$$858$$ 0 0
$$859$$ 26600.0 1.05655 0.528277 0.849072i $$-0.322838\pi$$
0.528277 + 0.849072i $$0.322838\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 47381.0 1.86891 0.934454 0.356083i $$-0.115888\pi$$
0.934454 + 0.356083i $$0.115888\pi$$
$$864$$ 0 0
$$865$$ −18458.1 −0.725543
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 3612.24 0.141009
$$870$$ 0 0
$$871$$ −11256.1 −0.437886
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −3009.93 −0.116291
$$876$$ 0 0
$$877$$ −31597.5 −1.21662 −0.608308 0.793701i $$-0.708151\pi$$
−0.608308 + 0.793701i $$0.708151\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −33206.9 −1.26989 −0.634943 0.772559i $$-0.718976\pi$$
−0.634943 + 0.772559i $$0.718976\pi$$
$$882$$ 0 0
$$883$$ 12479.4 0.475612 0.237806 0.971313i $$-0.423572\pi$$
0.237806 + 0.971313i $$0.423572\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −7331.93 −0.277545 −0.138772 0.990324i $$-0.544316\pi$$
−0.138772 + 0.990324i $$0.544316\pi$$
$$888$$ 0 0
$$889$$ 57020.5 2.15119
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 24380.4 0.913615
$$894$$ 0 0
$$895$$ −10180.7 −0.380229
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −13902.3 −0.515759
$$900$$ 0 0
$$901$$ −2018.32 −0.0746281
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16343.7 −0.600312
$$906$$ 0 0
$$907$$ 4489.18 0.164345 0.0821725 0.996618i $$-0.473814\pi$$
0.0821725 + 0.996618i $$0.473814\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 25710.9 0.935059 0.467529 0.883978i $$-0.345144\pi$$
0.467529 + 0.883978i $$0.345144\pi$$
$$912$$ 0 0
$$913$$ −732.061 −0.0265363
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −60992.6 −2.19646
$$918$$ 0 0
$$919$$ −24637.6 −0.884353 −0.442176 0.896928i $$-0.645793\pi$$
−0.442176 + 0.896928i $$0.645793\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 12785.4 0.455946
$$924$$ 0 0
$$925$$ 4552.23 0.161812
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 6833.00 0.241317 0.120658 0.992694i $$-0.461499\pi$$
0.120658 + 0.992694i $$0.461499\pi$$
$$930$$ 0 0
$$931$$ 17079.1 0.601231
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 1125.06 0.0393513
$$936$$ 0 0
$$937$$ −25120.0 −0.875810 −0.437905 0.899021i $$-0.644279\pi$$
−0.437905 + 0.899021i $$0.644279\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −54122.3 −1.87496 −0.937480 0.348040i $$-0.886847\pi$$
−0.937480 + 0.348040i $$0.886847\pi$$
$$942$$ 0 0
$$943$$ −27214.4 −0.939790
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 47670.7 1.63579 0.817894 0.575370i $$-0.195142\pi$$
0.817894 + 0.575370i $$0.195142\pi$$
$$948$$ 0 0
$$949$$ 8658.36 0.296167
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −11963.3 −0.406642 −0.203321 0.979112i $$-0.565174\pi$$
−0.203321 + 0.979112i $$0.565174\pi$$
$$954$$ 0 0
$$955$$ −10962.4 −0.371451
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −10212.0 −0.343861
$$960$$ 0 0
$$961$$ 77144.3 2.58952
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 16633.8 0.554881
$$966$$ 0 0
$$967$$ 36546.2 1.21535 0.607676 0.794185i $$-0.292102\pi$$
0.607676 + 0.794185i $$0.292102\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −785.854 −0.0259725 −0.0129862 0.999916i $$-0.504134\pi$$
−0.0129862 + 0.999916i $$0.504134\pi$$
$$972$$ 0 0
$$973$$ 35183.3 1.15922
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16159.0 0.529144 0.264572 0.964366i $$-0.414769\pi$$
0.264572 + 0.964366i $$0.414769\pi$$
$$978$$ 0 0
$$979$$ 4442.14 0.145017
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 19453.9 0.631213 0.315607 0.948890i $$-0.397792\pi$$
0.315607 + 0.948890i $$0.397792\pi$$
$$984$$ 0 0
$$985$$ 9092.08 0.294109
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −30542.1 −0.981983
$$990$$ 0 0
$$991$$ −25557.5 −0.819234 −0.409617 0.912258i $$-0.634338\pi$$
−0.409617 + 0.912258i $$0.634338\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 5955.20 0.189741
$$996$$ 0 0
$$997$$ −10058.6 −0.319519 −0.159760 0.987156i $$-0.551072\pi$$
−0.159760 + 0.987156i $$0.551072\pi$$
$$998$$ 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.bs.1.1 3
3.2 odd 2 2160.4.a.bk.1.1 3
4.3 odd 2 1080.4.a.j.1.3 yes 3
12.11 even 2 1080.4.a.d.1.3 3

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