# Properties

 Label 1521.2.b.l.1351.1 Level $1521$ Weight $2$ Character 1521.1351 Analytic conductor $12.145$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.1 Root $$-0.445042i$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1351 Dual form 1521.2.b.l.1351.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.24698i q^{2} -3.04892 q^{4} -1.44504i q^{5} +2.04892i q^{7} +2.35690i q^{8} +O(q^{10})$$ $$q-2.24698i q^{2} -3.04892 q^{4} -1.44504i q^{5} +2.04892i q^{7} +2.35690i q^{8} -3.24698 q^{10} +2.55496i q^{11} +4.60388 q^{14} -0.801938 q^{16} -5.29590 q^{17} +5.85086i q^{19} +4.40581i q^{20} +5.74094 q^{22} -1.89008 q^{23} +2.91185 q^{25} -6.24698i q^{28} -2.26875 q^{29} +4.26875i q^{31} +6.51573i q^{32} +11.8998i q^{34} +2.96077 q^{35} +5.35690i q^{37} +13.1468 q^{38} +3.40581 q^{40} +1.27413i q^{41} -6.13706 q^{43} -7.78986i q^{44} +4.24698i q^{46} +2.95108i q^{47} +2.80194 q^{49} -6.54288i q^{50} -5.52111 q^{53} +3.69202 q^{55} -4.82908 q^{56} +5.09783i q^{58} +12.2078i q^{59} +8.56465 q^{61} +9.59179 q^{62} +13.0368 q^{64} -0.576728i q^{67} +16.1468 q^{68} -6.65279i q^{70} -4.59419i q^{71} -10.5526i q^{73} +12.0368 q^{74} -17.8388i q^{76} -5.23490 q^{77} -15.7778 q^{79} +1.15883i q^{80} +2.86294 q^{82} +7.72348i q^{83} +7.65279i q^{85} +13.7899i q^{86} -6.02177 q^{88} -6.61356i q^{89} +5.76271 q^{92} +6.63102 q^{94} +8.45473 q^{95} -11.9269i q^{97} -6.29590i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62} + 22 q^{64} + 42 q^{68} + 16 q^{74} + 16 q^{77} - 10 q^{79} + 28 q^{82} - 30 q^{88} + 10 q^{94} + 6 q^{95}+O(q^{100})$$ 6 * q - 10 * q^10 + 10 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^22 - 10 * q^23 + 10 * q^25 + 2 * q^29 - 8 * q^35 + 24 * q^38 - 6 * q^40 - 26 * q^43 + 8 * q^49 - 2 * q^53 + 12 * q^55 - 8 * q^56 + 8 * q^61 + 2 * q^62 + 22 * q^64 + 42 * q^68 + 16 * q^74 + 16 * q^77 - 10 * q^79 + 28 * q^82 - 30 * q^88 + 10 * q^94 + 6 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.24698i − 1.58885i −0.607359 0.794427i $$-0.707771\pi$$
0.607359 0.794427i $$-0.292229\pi$$
$$3$$ 0 0
$$4$$ −3.04892 −1.52446
$$5$$ − 1.44504i − 0.646242i −0.946358 0.323121i $$-0.895268\pi$$
0.946358 0.323121i $$-0.104732\pi$$
$$6$$ 0 0
$$7$$ 2.04892i 0.774418i 0.921992 + 0.387209i $$0.126561\pi$$
−0.921992 + 0.387209i $$0.873439\pi$$
$$8$$ 2.35690i 0.833289i
$$9$$ 0 0
$$10$$ −3.24698 −1.02679
$$11$$ 2.55496i 0.770349i 0.922844 + 0.385174i $$0.125859\pi$$
−0.922844 + 0.385174i $$0.874141\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 4.60388 1.23044
$$15$$ 0 0
$$16$$ −0.801938 −0.200484
$$17$$ −5.29590 −1.28444 −0.642222 0.766519i $$-0.721987\pi$$
−0.642222 + 0.766519i $$0.721987\pi$$
$$18$$ 0 0
$$19$$ 5.85086i 1.34228i 0.741331 + 0.671139i $$0.234195\pi$$
−0.741331 + 0.671139i $$0.765805\pi$$
$$20$$ 4.40581i 0.985170i
$$21$$ 0 0
$$22$$ 5.74094 1.22397
$$23$$ −1.89008 −0.394110 −0.197055 0.980392i $$-0.563138\pi$$
−0.197055 + 0.980392i $$0.563138\pi$$
$$24$$ 0 0
$$25$$ 2.91185 0.582371
$$26$$ 0 0
$$27$$ 0 0
$$28$$ − 6.24698i − 1.18057i
$$29$$ −2.26875 −0.421296 −0.210648 0.977562i $$-0.567557\pi$$
−0.210648 + 0.977562i $$0.567557\pi$$
$$30$$ 0 0
$$31$$ 4.26875i 0.766690i 0.923605 + 0.383345i $$0.125228\pi$$
−0.923605 + 0.383345i $$0.874772\pi$$
$$32$$ 6.51573i 1.15183i
$$33$$ 0 0
$$34$$ 11.8998i 2.04079i
$$35$$ 2.96077 0.500462
$$36$$ 0 0
$$37$$ 5.35690i 0.880668i 0.897834 + 0.440334i $$0.145140\pi$$
−0.897834 + 0.440334i $$0.854860\pi$$
$$38$$ 13.1468 2.13268
$$39$$ 0 0
$$40$$ 3.40581 0.538506
$$41$$ 1.27413i 0.198985i 0.995038 + 0.0994926i $$0.0317220\pi$$
−0.995038 + 0.0994926i $$0.968278\pi$$
$$42$$ 0 0
$$43$$ −6.13706 −0.935893 −0.467947 0.883757i $$-0.655006\pi$$
−0.467947 + 0.883757i $$0.655006\pi$$
$$44$$ − 7.78986i − 1.17437i
$$45$$ 0 0
$$46$$ 4.24698i 0.626183i
$$47$$ 2.95108i 0.430460i 0.976563 + 0.215230i $$0.0690501\pi$$
−0.976563 + 0.215230i $$0.930950\pi$$
$$48$$ 0 0
$$49$$ 2.80194 0.400277
$$50$$ − 6.54288i − 0.925302i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −5.52111 −0.758382 −0.379191 0.925318i $$-0.623798\pi$$
−0.379191 + 0.925318i $$0.623798\pi$$
$$54$$ 0 0
$$55$$ 3.69202 0.497832
$$56$$ −4.82908 −0.645314
$$57$$ 0 0
$$58$$ 5.09783i 0.669378i
$$59$$ 12.2078i 1.58931i 0.607059 + 0.794657i $$0.292349\pi$$
−0.607059 + 0.794657i $$0.707651\pi$$
$$60$$ 0 0
$$61$$ 8.56465 1.09659 0.548295 0.836285i $$-0.315277\pi$$
0.548295 + 0.836285i $$0.315277\pi$$
$$62$$ 9.59179 1.21816
$$63$$ 0 0
$$64$$ 13.0368 1.62960
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 0.576728i − 0.0704586i −0.999379 0.0352293i $$-0.988784\pi$$
0.999379 0.0352293i $$-0.0112162\pi$$
$$68$$ 16.1468 1.95808
$$69$$ 0 0
$$70$$ − 6.65279i − 0.795161i
$$71$$ − 4.59419i − 0.545230i −0.962123 0.272615i $$-0.912112\pi$$
0.962123 0.272615i $$-0.0878885\pi$$
$$72$$ 0 0
$$73$$ − 10.5526i − 1.23508i −0.786538 0.617542i $$-0.788128\pi$$
0.786538 0.617542i $$-0.211872\pi$$
$$74$$ 12.0368 1.39925
$$75$$ 0 0
$$76$$ − 17.8388i − 2.04625i
$$77$$ −5.23490 −0.596572
$$78$$ 0 0
$$79$$ −15.7778 −1.77514 −0.887569 0.460674i $$-0.847608\pi$$
−0.887569 + 0.460674i $$0.847608\pi$$
$$80$$ 1.15883i 0.129562i
$$81$$ 0 0
$$82$$ 2.86294 0.316158
$$83$$ 7.72348i 0.847762i 0.905718 + 0.423881i $$0.139333\pi$$
−0.905718 + 0.423881i $$0.860667\pi$$
$$84$$ 0 0
$$85$$ 7.65279i 0.830062i
$$86$$ 13.7899i 1.48700i
$$87$$ 0 0
$$88$$ −6.02177 −0.641923
$$89$$ − 6.61356i − 0.701036i −0.936556 0.350518i $$-0.886005\pi$$
0.936556 0.350518i $$-0.113995\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.76271 0.600804
$$93$$ 0 0
$$94$$ 6.63102 0.683938
$$95$$ 8.45473 0.867437
$$96$$ 0 0
$$97$$ − 11.9269i − 1.21100i −0.795847 0.605498i $$-0.792974\pi$$
0.795847 0.605498i $$-0.207026\pi$$
$$98$$ − 6.29590i − 0.635982i
$$99$$ 0 0
$$100$$ −8.87800 −0.887800
$$101$$ 13.0640 1.29991 0.649957 0.759971i $$-0.274787\pi$$
0.649957 + 0.759971i $$0.274787\pi$$
$$102$$ 0 0
$$103$$ −9.16852 −0.903401 −0.451701 0.892170i $$-0.649182\pi$$
−0.451701 + 0.892170i $$0.649182\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.4058i 1.20496i
$$107$$ 6.89977 0.667026 0.333513 0.942745i $$-0.391766\pi$$
0.333513 + 0.942745i $$0.391766\pi$$
$$108$$ 0 0
$$109$$ − 0.121998i − 0.0116853i −0.999983 0.00584264i $$-0.998140\pi$$
0.999983 0.00584264i $$-0.00185978\pi$$
$$110$$ − 8.29590i − 0.790983i
$$111$$ 0 0
$$112$$ − 1.64310i − 0.155259i
$$113$$ −7.30798 −0.687477 −0.343738 0.939065i $$-0.611693\pi$$
−0.343738 + 0.939065i $$0.611693\pi$$
$$114$$ 0 0
$$115$$ 2.73125i 0.254690i
$$116$$ 6.91723 0.642249
$$117$$ 0 0
$$118$$ 27.4306 2.52519
$$119$$ − 10.8509i − 0.994696i
$$120$$ 0 0
$$121$$ 4.47219 0.406563
$$122$$ − 19.2446i − 1.74232i
$$123$$ 0 0
$$124$$ − 13.0151i − 1.16879i
$$125$$ − 11.4330i − 1.02260i
$$126$$ 0 0
$$127$$ 18.9705 1.68336 0.841678 0.539980i $$-0.181568\pi$$
0.841678 + 0.539980i $$0.181568\pi$$
$$128$$ − 16.2620i − 1.43738i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.25667 −0.284536 −0.142268 0.989828i $$-0.545440\pi$$
−0.142268 + 0.989828i $$0.545440\pi$$
$$132$$ 0 0
$$133$$ −11.9879 −1.03948
$$134$$ −1.29590 −0.111948
$$135$$ 0 0
$$136$$ − 12.4819i − 1.07031i
$$137$$ − 0.792249i − 0.0676864i −0.999427 0.0338432i $$-0.989225\pi$$
0.999427 0.0338432i $$-0.0107747\pi$$
$$138$$ 0 0
$$139$$ −11.3394 −0.961799 −0.480899 0.876776i $$-0.659690\pi$$
−0.480899 + 0.876776i $$0.659690\pi$$
$$140$$ −9.02715 −0.762933
$$141$$ 0 0
$$142$$ −10.3230 −0.866291
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.27844i 0.272260i
$$146$$ −23.7114 −1.96237
$$147$$ 0 0
$$148$$ − 16.3327i − 1.34254i
$$149$$ 8.40581i 0.688631i 0.938854 + 0.344316i $$0.111889\pi$$
−0.938854 + 0.344316i $$0.888111\pi$$
$$150$$ 0 0
$$151$$ 14.1293i 1.14983i 0.818215 + 0.574913i $$0.194964\pi$$
−0.818215 + 0.574913i $$0.805036\pi$$
$$152$$ −13.7899 −1.11851
$$153$$ 0 0
$$154$$ 11.7627i 0.947866i
$$155$$ 6.16852 0.495468
$$156$$ 0 0
$$157$$ −9.43296 −0.752832 −0.376416 0.926451i $$-0.622844\pi$$
−0.376416 + 0.926451i $$0.622844\pi$$
$$158$$ 35.4523i 2.82044i
$$159$$ 0 0
$$160$$ 9.41550 0.744361
$$161$$ − 3.87263i − 0.305206i
$$162$$ 0 0
$$163$$ 8.70410i 0.681758i 0.940107 + 0.340879i $$0.110725\pi$$
−0.940107 + 0.340879i $$0.889275\pi$$
$$164$$ − 3.88471i − 0.303345i
$$165$$ 0 0
$$166$$ 17.3545 1.34697
$$167$$ 23.8538i 1.84587i 0.384961 + 0.922933i $$0.374215\pi$$
−0.384961 + 0.922933i $$0.625785\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 17.1957 1.31885
$$171$$ 0 0
$$172$$ 18.7114 1.42673
$$173$$ −18.8552 −1.43353 −0.716766 0.697314i $$-0.754378\pi$$
−0.716766 + 0.697314i $$0.754378\pi$$
$$174$$ 0 0
$$175$$ 5.96615i 0.450998i
$$176$$ − 2.04892i − 0.154443i
$$177$$ 0 0
$$178$$ −14.8605 −1.11384
$$179$$ 6.02177 0.450088 0.225044 0.974349i $$-0.427747\pi$$
0.225044 + 0.974349i $$0.427747\pi$$
$$180$$ 0 0
$$181$$ 4.77777 0.355129 0.177565 0.984109i $$-0.443178\pi$$
0.177565 + 0.984109i $$0.443178\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ − 4.45473i − 0.328407i
$$185$$ 7.74094 0.569125
$$186$$ 0 0
$$187$$ − 13.5308i − 0.989470i
$$188$$ − 8.99761i − 0.656218i
$$189$$ 0 0
$$190$$ − 18.9976i − 1.37823i
$$191$$ −18.4306 −1.33359 −0.666795 0.745242i $$-0.732334\pi$$
−0.666795 + 0.745242i $$0.732334\pi$$
$$192$$ 0 0
$$193$$ 6.05429i 0.435798i 0.975971 + 0.217899i $$0.0699203\pi$$
−0.975971 + 0.217899i $$0.930080\pi$$
$$194$$ −26.7995 −1.92410
$$195$$ 0 0
$$196$$ −8.54288 −0.610205
$$197$$ − 11.4155i − 0.813321i −0.913579 0.406660i $$-0.866693\pi$$
0.913579 0.406660i $$-0.133307\pi$$
$$198$$ 0 0
$$199$$ 13.9051 0.985710 0.492855 0.870111i $$-0.335953\pi$$
0.492855 + 0.870111i $$0.335953\pi$$
$$200$$ 6.86294i 0.485283i
$$201$$ 0 0
$$202$$ − 29.3545i − 2.06538i
$$203$$ − 4.64848i − 0.326259i
$$204$$ 0 0
$$205$$ 1.84117 0.128593
$$206$$ 20.6015i 1.43537i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −14.9487 −1.03402
$$210$$ 0 0
$$211$$ −13.2446 −0.911795 −0.455897 0.890032i $$-0.650682\pi$$
−0.455897 + 0.890032i $$0.650682\pi$$
$$212$$ 16.8334 1.15612
$$213$$ 0 0
$$214$$ − 15.5036i − 1.05981i
$$215$$ 8.86831i 0.604814i
$$216$$ 0 0
$$217$$ −8.74632 −0.593739
$$218$$ −0.274127 −0.0185662
$$219$$ 0 0
$$220$$ −11.2567 −0.758924
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 7.33513i 0.491196i 0.969372 + 0.245598i $$0.0789844\pi$$
−0.969372 + 0.245598i $$0.921016\pi$$
$$224$$ −13.3502 −0.891997
$$225$$ 0 0
$$226$$ 16.4209i 1.09230i
$$227$$ − 8.67456i − 0.575751i −0.957668 0.287875i $$-0.907051\pi$$
0.957668 0.287875i $$-0.0929489\pi$$
$$228$$ 0 0
$$229$$ − 13.6866i − 0.904439i −0.891907 0.452219i $$-0.850632\pi$$
0.891907 0.452219i $$-0.149368\pi$$
$$230$$ 6.13706 0.404666
$$231$$ 0 0
$$232$$ − 5.34721i − 0.351061i
$$233$$ −5.08815 −0.333336 −0.166668 0.986013i $$-0.553301\pi$$
−0.166668 + 0.986013i $$0.553301\pi$$
$$234$$ 0 0
$$235$$ 4.26444 0.278181
$$236$$ − 37.2204i − 2.42284i
$$237$$ 0 0
$$238$$ −24.3817 −1.58043
$$239$$ − 10.9239i − 0.706611i −0.935508 0.353305i $$-0.885058\pi$$
0.935508 0.353305i $$-0.114942\pi$$
$$240$$ 0 0
$$241$$ 11.9148i 0.767502i 0.923437 + 0.383751i $$0.125368\pi$$
−0.923437 + 0.383751i $$0.874632\pi$$
$$242$$ − 10.0489i − 0.645969i
$$243$$ 0 0
$$244$$ −26.1129 −1.67171
$$245$$ − 4.04892i − 0.258676i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10.0610 −0.638874
$$249$$ 0 0
$$250$$ −25.6896 −1.62475
$$251$$ 22.3478 1.41058 0.705290 0.708919i $$-0.250817\pi$$
0.705290 + 0.708919i $$0.250817\pi$$
$$252$$ 0 0
$$253$$ − 4.82908i − 0.303602i
$$254$$ − 42.6262i − 2.67461i
$$255$$ 0 0
$$256$$ −10.4668 −0.654176
$$257$$ −18.6601 −1.16398 −0.581992 0.813194i $$-0.697727\pi$$
−0.581992 + 0.813194i $$0.697727\pi$$
$$258$$ 0 0
$$259$$ −10.9758 −0.682005
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 7.31767i 0.452087i
$$263$$ −14.3991 −0.887887 −0.443944 0.896055i $$-0.646421\pi$$
−0.443944 + 0.896055i $$0.646421\pi$$
$$264$$ 0 0
$$265$$ 7.97823i 0.490099i
$$266$$ 26.9366i 1.65159i
$$267$$ 0 0
$$268$$ 1.75840i 0.107411i
$$269$$ −0.652793 −0.0398015 −0.0199007 0.999802i $$-0.506335\pi$$
−0.0199007 + 0.999802i $$0.506335\pi$$
$$270$$ 0 0
$$271$$ − 1.99569i − 0.121229i −0.998161 0.0606147i $$-0.980694\pi$$
0.998161 0.0606147i $$-0.0193061\pi$$
$$272$$ 4.24698 0.257511
$$273$$ 0 0
$$274$$ −1.78017 −0.107544
$$275$$ 7.43967i 0.448629i
$$276$$ 0 0
$$277$$ −11.7845 −0.708061 −0.354030 0.935234i $$-0.615189\pi$$
−0.354030 + 0.935234i $$0.615189\pi$$
$$278$$ 25.4795i 1.52816i
$$279$$ 0 0
$$280$$ 6.97823i 0.417029i
$$281$$ 6.47219i 0.386098i 0.981189 + 0.193049i $$0.0618377\pi$$
−0.981189 + 0.193049i $$0.938162\pi$$
$$282$$ 0 0
$$283$$ −6.58104 −0.391202 −0.195601 0.980684i $$-0.562666\pi$$
−0.195601 + 0.980684i $$0.562666\pi$$
$$284$$ 14.0073i 0.831180i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.61058 −0.154098
$$288$$ 0 0
$$289$$ 11.0465 0.649796
$$290$$ 7.36658 0.432581
$$291$$ 0 0
$$292$$ 32.1739i 1.88284i
$$293$$ 24.3381i 1.42185i 0.703269 + 0.710924i $$0.251723\pi$$
−0.703269 + 0.710924i $$0.748277\pi$$
$$294$$ 0 0
$$295$$ 17.6407 1.02708
$$296$$ −12.6256 −0.733851
$$297$$ 0 0
$$298$$ 18.8877 1.09413
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 12.5743i − 0.724773i
$$302$$ 31.7482 1.82691
$$303$$ 0 0
$$304$$ − 4.69202i − 0.269106i
$$305$$ − 12.3763i − 0.708663i
$$306$$ 0 0
$$307$$ − 14.0737i − 0.803227i −0.915809 0.401613i $$-0.868450\pi$$
0.915809 0.401613i $$-0.131550\pi$$
$$308$$ 15.9608 0.909449
$$309$$ 0 0
$$310$$ − 13.8605i − 0.787226i
$$311$$ −29.7700 −1.68810 −0.844051 0.536263i $$-0.819836\pi$$
−0.844051 + 0.536263i $$0.819836\pi$$
$$312$$ 0 0
$$313$$ −7.47889 −0.422732 −0.211366 0.977407i $$-0.567791\pi$$
−0.211366 + 0.977407i $$0.567791\pi$$
$$314$$ 21.1957i 1.19614i
$$315$$ 0 0
$$316$$ 48.1051 2.70613
$$317$$ 30.0301i 1.68666i 0.537396 + 0.843330i $$0.319408\pi$$
−0.537396 + 0.843330i $$0.680592\pi$$
$$318$$ 0 0
$$319$$ − 5.79656i − 0.324545i
$$320$$ − 18.8388i − 1.05312i
$$321$$ 0 0
$$322$$ −8.70171 −0.484927
$$323$$ − 30.9855i − 1.72408i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.5579 1.08321
$$327$$ 0 0
$$328$$ −3.00298 −0.165812
$$329$$ −6.04652 −0.333356
$$330$$ 0 0
$$331$$ 15.7168i 0.863872i 0.901904 + 0.431936i $$0.142169\pi$$
−0.901904 + 0.431936i $$0.857831\pi$$
$$332$$ − 23.5483i − 1.29238i
$$333$$ 0 0
$$334$$ 53.5991 2.93281
$$335$$ −0.833397 −0.0455333
$$336$$ 0 0
$$337$$ −1.95407 −0.106445 −0.0532224 0.998583i $$-0.516949\pi$$
−0.0532224 + 0.998583i $$0.516949\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ − 23.3327i − 1.26540i
$$341$$ −10.9065 −0.590619
$$342$$ 0 0
$$343$$ 20.0834i 1.08440i
$$344$$ − 14.4644i − 0.779869i
$$345$$ 0 0
$$346$$ 42.3672i 2.27767i
$$347$$ 17.1250 0.919317 0.459659 0.888096i $$-0.347972\pi$$
0.459659 + 0.888096i $$0.347972\pi$$
$$348$$ 0 0
$$349$$ − 10.4668i − 0.560276i −0.959960 0.280138i $$-0.909620\pi$$
0.959960 0.280138i $$-0.0903802\pi$$
$$350$$ 13.4058 0.716571
$$351$$ 0 0
$$352$$ −16.6474 −0.887310
$$353$$ 15.5308i 0.826621i 0.910590 + 0.413310i $$0.135628\pi$$
−0.910590 + 0.413310i $$0.864372\pi$$
$$354$$ 0 0
$$355$$ −6.63879 −0.352351
$$356$$ 20.1642i 1.06870i
$$357$$ 0 0
$$358$$ − 13.5308i − 0.715125i
$$359$$ 21.4263i 1.13083i 0.824805 + 0.565417i $$0.191285\pi$$
−0.824805 + 0.565417i $$0.808715\pi$$
$$360$$ 0 0
$$361$$ −15.2325 −0.801711
$$362$$ − 10.7356i − 0.564249i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.2489 −0.798164
$$366$$ 0 0
$$367$$ 34.3032 1.79061 0.895306 0.445452i $$-0.146957\pi$$
0.895306 + 0.445452i $$0.146957\pi$$
$$368$$ 1.51573 0.0790129
$$369$$ 0 0
$$370$$ − 17.3937i − 0.904257i
$$371$$ − 11.3123i − 0.587305i
$$372$$ 0 0
$$373$$ −12.5961 −0.652202 −0.326101 0.945335i $$-0.605735\pi$$
−0.326101 + 0.945335i $$0.605735\pi$$
$$374$$ −30.4034 −1.57212
$$375$$ 0 0
$$376$$ −6.95539 −0.358697
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 16.5386i − 0.849529i −0.905304 0.424765i $$-0.860357\pi$$
0.905304 0.424765i $$-0.139643\pi$$
$$380$$ −25.7778 −1.32237
$$381$$ 0 0
$$382$$ 41.4131i 2.11888i
$$383$$ 7.53617i 0.385080i 0.981289 + 0.192540i $$0.0616726\pi$$
−0.981289 + 0.192540i $$0.938327\pi$$
$$384$$ 0 0
$$385$$ 7.56465i 0.385530i
$$386$$ 13.6039 0.692419
$$387$$ 0 0
$$388$$ 36.3642i 1.84611i
$$389$$ 35.5555 1.80274 0.901369 0.433052i $$-0.142563\pi$$
0.901369 + 0.433052i $$0.142563\pi$$
$$390$$ 0 0
$$391$$ 10.0097 0.506212
$$392$$ 6.60388i 0.333546i
$$393$$ 0 0
$$394$$ −25.6504 −1.29225
$$395$$ 22.7995i 1.14717i
$$396$$ 0 0
$$397$$ 1.35152i 0.0678308i 0.999425 + 0.0339154i $$0.0107977\pi$$
−0.999425 + 0.0339154i $$0.989202\pi$$
$$398$$ − 31.2446i − 1.56615i
$$399$$ 0 0
$$400$$ −2.33513 −0.116756
$$401$$ − 0.579121i − 0.0289199i −0.999895 0.0144600i $$-0.995397\pi$$
0.999895 0.0144600i $$-0.00460291\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −39.8310 −1.98167
$$405$$ 0 0
$$406$$ −10.4450 −0.518379
$$407$$ −13.6866 −0.678422
$$408$$ 0 0
$$409$$ 15.1575i 0.749490i 0.927128 + 0.374745i $$0.122270\pi$$
−0.927128 + 0.374745i $$0.877730\pi$$
$$410$$ − 4.13706i − 0.204315i
$$411$$ 0 0
$$412$$ 27.9541 1.37720
$$413$$ −25.0127 −1.23079
$$414$$ 0 0
$$415$$ 11.1608 0.547860
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 33.5894i 1.64291i
$$419$$ 35.7235 1.74521 0.872603 0.488430i $$-0.162430\pi$$
0.872603 + 0.488430i $$0.162430\pi$$
$$420$$ 0 0
$$421$$ 35.0465i 1.70806i 0.520221 + 0.854032i $$0.325849\pi$$
−0.520221 + 0.854032i $$0.674151\pi$$
$$422$$ 29.7603i 1.44871i
$$423$$ 0 0
$$424$$ − 13.0127i − 0.631951i
$$425$$ −15.4209 −0.748022
$$426$$ 0 0
$$427$$ 17.5483i 0.849220i
$$428$$ −21.0368 −1.01685
$$429$$ 0 0
$$430$$ 19.9269 0.960961
$$431$$ − 34.2814i − 1.65128i −0.564199 0.825639i $$-0.690815\pi$$
0.564199 0.825639i $$-0.309185\pi$$
$$432$$ 0 0
$$433$$ −13.7385 −0.660232 −0.330116 0.943940i $$-0.607088\pi$$
−0.330116 + 0.943940i $$0.607088\pi$$
$$434$$ 19.6528i 0.943364i
$$435$$ 0 0
$$436$$ 0.371961i 0.0178137i
$$437$$ − 11.0586i − 0.529005i
$$438$$ 0 0
$$439$$ −10.2403 −0.488742 −0.244371 0.969682i $$-0.578581\pi$$
−0.244371 + 0.969682i $$0.578581\pi$$
$$440$$ 8.70171i 0.414838i
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.1763 −0.578513 −0.289257 0.957252i $$-0.593408\pi$$
−0.289257 + 0.957252i $$0.593408\pi$$
$$444$$ 0 0
$$445$$ −9.55688 −0.453039
$$446$$ 16.4819 0.780440
$$447$$ 0 0
$$448$$ 26.7114i 1.26199i
$$449$$ 12.9051i 0.609032i 0.952507 + 0.304516i $$0.0984947\pi$$
−0.952507 + 0.304516i $$0.901505\pi$$
$$450$$ 0 0
$$451$$ −3.25534 −0.153288
$$452$$ 22.2814 1.04803
$$453$$ 0 0
$$454$$ −19.4916 −0.914785
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.65710i − 0.217850i −0.994050 0.108925i $$-0.965259\pi$$
0.994050 0.108925i $$-0.0347409\pi$$
$$458$$ −30.7536 −1.43702
$$459$$ 0 0
$$460$$ − 8.32736i − 0.388265i
$$461$$ − 31.5405i − 1.46899i −0.678616 0.734493i $$-0.737420\pi$$
0.678616 0.734493i $$-0.262580\pi$$
$$462$$ 0 0
$$463$$ 17.6504i 0.820284i 0.912022 + 0.410142i $$0.134521\pi$$
−0.912022 + 0.410142i $$0.865479\pi$$
$$464$$ 1.81940 0.0844633
$$465$$ 0 0
$$466$$ 11.4330i 0.529622i
$$467$$ −32.1726 −1.48877 −0.744385 0.667751i $$-0.767257\pi$$
−0.744385 + 0.667751i $$0.767257\pi$$
$$468$$ 0 0
$$469$$ 1.18167 0.0545644
$$470$$ − 9.58211i − 0.441990i
$$471$$ 0 0
$$472$$ −28.7724 −1.32436
$$473$$ − 15.6799i − 0.720964i
$$474$$ 0 0
$$475$$ 17.0368i 0.781704i
$$476$$ 33.0834i 1.51637i
$$477$$ 0 0
$$478$$ −24.5459 −1.12270
$$479$$ 34.8998i 1.59461i 0.603576 + 0.797306i $$0.293742\pi$$
−0.603576 + 0.797306i $$0.706258\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26.7724 1.21945
$$483$$ 0 0
$$484$$ −13.6353 −0.619788
$$485$$ −17.2349 −0.782596
$$486$$ 0 0
$$487$$ − 41.8351i − 1.89573i −0.318676 0.947864i $$-0.603238\pi$$
0.318676 0.947864i $$-0.396762\pi$$
$$488$$ 20.1860i 0.913776i
$$489$$ 0 0
$$490$$ −9.09783 −0.410998
$$491$$ 21.8455 0.985873 0.492936 0.870065i $$-0.335924\pi$$
0.492936 + 0.870065i $$0.335924\pi$$
$$492$$ 0 0
$$493$$ 12.0151 0.541131
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 3.42327i − 0.153709i
$$497$$ 9.41311 0.422236
$$498$$ 0 0
$$499$$ − 23.5472i − 1.05412i −0.849829 0.527058i $$-0.823295\pi$$
0.849829 0.527058i $$-0.176705\pi$$
$$500$$ 34.8582i 1.55890i
$$501$$ 0 0
$$502$$ − 50.2150i − 2.24121i
$$503$$ 7.08682 0.315986 0.157993 0.987440i $$-0.449498\pi$$
0.157993 + 0.987440i $$0.449498\pi$$
$$504$$ 0 0
$$505$$ − 18.8780i − 0.840060i
$$506$$ −10.8509 −0.482379
$$507$$ 0 0
$$508$$ −57.8394 −2.56621
$$509$$ 7.61894i 0.337704i 0.985641 + 0.168852i $$0.0540059\pi$$
−0.985641 + 0.168852i $$0.945994\pi$$
$$510$$ 0 0
$$511$$ 21.6213 0.956471
$$512$$ − 9.00538i − 0.397985i
$$513$$ 0 0
$$514$$ 41.9288i 1.84940i
$$515$$ 13.2489i 0.583816i
$$516$$ 0 0
$$517$$ −7.53989 −0.331604
$$518$$ 24.6625i 1.08361i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 39.5133 1.73111 0.865555 0.500813i $$-0.166966\pi$$
0.865555 + 0.500813i $$0.166966\pi$$
$$522$$ 0 0
$$523$$ −15.8194 −0.691734 −0.345867 0.938284i $$-0.612415\pi$$
−0.345867 + 0.938284i $$0.612415\pi$$
$$524$$ 9.92931 0.433764
$$525$$ 0 0
$$526$$ 32.3545i 1.41072i
$$527$$ − 22.6069i − 0.984770i
$$528$$ 0 0
$$529$$ −19.4276 −0.844678
$$530$$ 17.9269 0.778696
$$531$$ 0 0
$$532$$ 36.5502 1.58465
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 9.97046i − 0.431061i
$$536$$ 1.35929 0.0587123
$$537$$ 0 0
$$538$$ 1.46681i 0.0632388i
$$539$$ 7.15883i 0.308353i
$$540$$ 0 0
$$541$$ 34.4819i 1.48249i 0.671234 + 0.741246i $$0.265765\pi$$
−0.671234 + 0.741246i $$0.734235\pi$$
$$542$$ −4.48427 −0.192616
$$543$$ 0 0
$$544$$ − 34.5066i − 1.47946i
$$545$$ −0.176292 −0.00755152
$$546$$ 0 0
$$547$$ 36.8582 1.57594 0.787970 0.615713i $$-0.211132\pi$$
0.787970 + 0.615713i $$0.211132\pi$$
$$548$$ 2.41550i 0.103185i
$$549$$ 0 0
$$550$$ 16.7168 0.712806
$$551$$ − 13.2741i − 0.565497i
$$552$$ 0 0
$$553$$ − 32.3274i − 1.37470i
$$554$$ 26.4795i 1.12501i
$$555$$ 0 0
$$556$$ 34.5730 1.46622
$$557$$ − 1.27652i − 0.0540879i −0.999634 0.0270439i $$-0.991391\pi$$
0.999634 0.0270439i $$-0.00860940\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.37435 −0.100335
$$561$$ 0 0
$$562$$ 14.5429 0.613454
$$563$$ −9.12737 −0.384673 −0.192336 0.981329i $$-0.561607\pi$$
−0.192336 + 0.981329i $$0.561607\pi$$
$$564$$ 0 0
$$565$$ 10.5603i 0.444277i
$$566$$ 14.7875i 0.621563i
$$567$$ 0 0
$$568$$ 10.8280 0.454334
$$569$$ −5.72156 −0.239860 −0.119930 0.992782i $$-0.538267\pi$$
−0.119930 + 0.992782i $$0.538267\pi$$
$$570$$ 0 0
$$571$$ −7.60148 −0.318112 −0.159056 0.987270i $$-0.550845\pi$$
−0.159056 + 0.987270i $$0.550845\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 5.86592i 0.244839i
$$575$$ −5.50365 −0.229518
$$576$$ 0 0
$$577$$ − 45.1564i − 1.87989i −0.341330 0.939944i $$-0.610877\pi$$
0.341330 0.939944i $$-0.389123\pi$$
$$578$$ − 24.8213i − 1.03243i
$$579$$ 0 0
$$580$$ − 9.99569i − 0.415048i
$$581$$ −15.8248 −0.656522
$$582$$ 0 0
$$583$$ − 14.1062i − 0.584219i
$$584$$ 24.8713 1.02918
$$585$$ 0 0
$$586$$ 54.6872 2.25911
$$587$$ − 32.4040i − 1.33746i −0.743507 0.668728i $$-0.766839\pi$$
0.743507 0.668728i $$-0.233161\pi$$
$$588$$ 0 0
$$589$$ −24.9758 −1.02911
$$590$$ − 39.6383i − 1.63188i
$$591$$ 0 0
$$592$$ − 4.29590i − 0.176560i
$$593$$ − 36.6848i − 1.50647i −0.657754 0.753233i $$-0.728493\pi$$
0.657754 0.753233i $$-0.271507\pi$$
$$594$$ 0 0
$$595$$ −15.6799 −0.642815
$$596$$ − 25.6286i − 1.04979i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 9.99223 0.408271 0.204136 0.978943i $$-0.434562\pi$$
0.204136 + 0.978943i $$0.434562\pi$$
$$600$$ 0 0
$$601$$ −1.81163 −0.0738978 −0.0369489 0.999317i $$-0.511764\pi$$
−0.0369489 + 0.999317i $$0.511764\pi$$
$$602$$ −28.2543 −1.15156
$$603$$ 0 0
$$604$$ − 43.0790i − 1.75286i
$$605$$ − 6.46250i − 0.262738i
$$606$$ 0 0
$$607$$ 11.2161 0.455248 0.227624 0.973749i $$-0.426904\pi$$
0.227624 + 0.973749i $$0.426904\pi$$
$$608$$ −38.1226 −1.54608
$$609$$ 0 0
$$610$$ −27.8092 −1.12596
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20.8944i 0.843917i 0.906615 + 0.421958i $$0.138657\pi$$
−0.906615 + 0.421958i $$0.861343\pi$$
$$614$$ −31.6233 −1.27621
$$615$$ 0 0
$$616$$ − 12.3381i − 0.497117i
$$617$$ 12.0992i 0.487094i 0.969889 + 0.243547i $$0.0783110\pi$$
−0.969889 + 0.243547i $$0.921689\pi$$
$$618$$ 0 0
$$619$$ − 10.5526i − 0.424143i −0.977254 0.212072i $$-0.931979\pi$$
0.977254 0.212072i $$-0.0680210\pi$$
$$620$$ −18.8073 −0.755320
$$621$$ 0 0
$$622$$ 66.8926i 2.68215i
$$623$$ 13.5506 0.542895
$$624$$ 0 0
$$625$$ −1.96184 −0.0784735
$$626$$ 16.8049i 0.671660i
$$627$$ 0 0
$$628$$ 28.7603 1.14766
$$629$$ − 28.3696i − 1.13117i
$$630$$ 0 0
$$631$$ − 13.8514i − 0.551417i −0.961241 0.275709i $$-0.911087\pi$$
0.961241 0.275709i $$-0.0889125\pi$$
$$632$$ − 37.1866i − 1.47920i
$$633$$ 0 0
$$634$$ 67.4771 2.67986
$$635$$ − 27.4131i − 1.08786i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −13.0248 −0.515655
$$639$$ 0 0
$$640$$ −23.4993 −0.928893
$$641$$ 34.9608 1.38087 0.690434 0.723396i $$-0.257420\pi$$
0.690434 + 0.723396i $$0.257420\pi$$
$$642$$ 0 0
$$643$$ − 33.3980i − 1.31709i −0.752541 0.658545i $$-0.771172\pi$$
0.752541 0.658545i $$-0.228828\pi$$
$$644$$ 11.8073i 0.465273i
$$645$$ 0 0
$$646$$ −69.6238 −2.73931
$$647$$ 2.32842 0.0915397 0.0457698 0.998952i $$-0.485426\pi$$
0.0457698 + 0.998952i $$0.485426\pi$$
$$648$$ 0 0
$$649$$ −31.1903 −1.22433
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 26.5381i − 1.03931i
$$653$$ −14.5714 −0.570221 −0.285111 0.958495i $$-0.592030\pi$$
−0.285111 + 0.958495i $$0.592030\pi$$
$$654$$ 0 0
$$655$$ 4.70602i 0.183879i
$$656$$ − 1.02177i − 0.0398934i
$$657$$ 0 0
$$658$$ 13.5864i 0.529654i
$$659$$ −11.1395 −0.433932 −0.216966 0.976179i $$-0.569616\pi$$
−0.216966 + 0.976179i $$0.569616\pi$$
$$660$$ 0 0
$$661$$ − 13.8498i − 0.538694i −0.963043 0.269347i $$-0.913192\pi$$
0.963043 0.269347i $$-0.0868079\pi$$
$$662$$ 35.3153 1.37257
$$663$$ 0 0
$$664$$ −18.2034 −0.706430
$$665$$ 17.3230i 0.671759i
$$666$$ 0 0
$$667$$ 4.28813 0.166037
$$668$$ − 72.7284i − 2.81395i
$$669$$ 0 0
$$670$$ 1.87263i 0.0723458i
$$671$$ 21.8823i 0.844757i
$$672$$ 0 0
$$673$$ 6.52973 0.251703 0.125851 0.992049i $$-0.459834\pi$$
0.125851 + 0.992049i $$0.459834\pi$$
$$674$$ 4.39075i 0.169125i
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 11.3104 0.434693 0.217346 0.976095i $$-0.430260\pi$$
0.217346 + 0.976095i $$0.430260\pi$$
$$678$$ 0 0
$$679$$ 24.4373 0.937816
$$680$$ −18.0368 −0.691681
$$681$$ 0 0
$$682$$ 24.5066i 0.938407i
$$683$$ − 14.1793i − 0.542555i −0.962501 0.271277i $$-0.912554\pi$$
0.962501 0.271277i $$-0.0874461\pi$$
$$684$$ 0 0
$$685$$ −1.14483 −0.0437418
$$686$$ 45.1269 1.72295
$$687$$ 0 0
$$688$$ 4.92154 0.187632
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 30.7952i − 1.17151i −0.810490 0.585753i $$-0.800799\pi$$
0.810490 0.585753i $$-0.199201\pi$$
$$692$$ 57.4878 2.18536
$$693$$ 0 0
$$694$$ − 38.4795i − 1.46066i
$$695$$ 16.3860i 0.621555i
$$696$$ 0 0
$$697$$ − 6.74764i − 0.255585i
$$698$$ −23.5187 −0.890196
$$699$$ 0 0
$$700$$ − 18.1903i − 0.687528i
$$701$$ 6.73184 0.254258 0.127129 0.991886i $$-0.459424\pi$$
0.127129 + 0.991886i $$0.459424\pi$$
$$702$$ 0 0
$$703$$ −31.3424 −1.18210
$$704$$ 33.3086i 1.25536i
$$705$$ 0 0
$$706$$ 34.8974 1.31338
$$707$$ 26.7670i 1.00668i
$$708$$ 0 0
$$709$$ − 47.6252i − 1.78860i −0.447467 0.894300i $$-0.647674\pi$$
0.447467 0.894300i $$-0.352326\pi$$
$$710$$ 14.9172i 0.559834i
$$711$$ 0 0
$$712$$ 15.5875 0.584166
$$713$$ − 8.06829i − 0.302160i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −18.3599 −0.686141
$$717$$ 0 0
$$718$$ 48.1444 1.79673
$$719$$ −5.99330 −0.223512 −0.111756 0.993736i $$-0.535648\pi$$
−0.111756 + 0.993736i $$0.535648\pi$$
$$720$$ 0 0
$$721$$ − 18.7855i − 0.699610i
$$722$$ 34.2271i 1.27380i
$$723$$ 0 0
$$724$$ −14.5670 −0.541380
$$725$$ −6.60627 −0.245351
$$726$$ 0 0
$$727$$ 24.1226 0.894657 0.447329 0.894370i $$-0.352375\pi$$
0.447329 + 0.894370i $$0.352375\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 34.2640i 1.26817i
$$731$$ 32.5013 1.20210
$$732$$ 0 0
$$733$$ − 36.0646i − 1.33208i −0.745918 0.666038i $$-0.767989\pi$$
0.745918 0.666038i $$-0.232011\pi$$
$$734$$ − 77.0786i − 2.84502i
$$735$$ 0 0
$$736$$ − 12.3153i − 0.453947i
$$737$$ 1.47352 0.0542777
$$738$$ 0 0
$$739$$ 27.5254i 1.01254i 0.862375 + 0.506269i $$0.168976\pi$$
−0.862375 + 0.506269i $$0.831024\pi$$
$$740$$ −23.6015 −0.867608
$$741$$ 0 0
$$742$$ −25.4185 −0.933142
$$743$$ − 10.4692i − 0.384078i −0.981387 0.192039i $$-0.938490\pi$$
0.981387 0.192039i $$-0.0615100\pi$$
$$744$$ 0 0
$$745$$ 12.1468 0.445023
$$746$$ 28.3032i 1.03625i
$$747$$ 0 0
$$748$$ 41.2543i 1.50841i
$$749$$ 14.1371i 0.516557i
$$750$$ 0 0
$$751$$ −4.06770 −0.148433 −0.0742163 0.997242i $$-0.523646\pi$$
−0.0742163 + 0.997242i $$0.523646\pi$$
$$752$$ − 2.36658i − 0.0863005i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 20.4174 0.743066
$$756$$ 0 0
$$757$$ 20.4336 0.742670 0.371335 0.928499i $$-0.378900\pi$$
0.371335 + 0.928499i $$0.378900\pi$$
$$758$$ −37.1618 −1.34978
$$759$$ 0 0
$$760$$ 19.9269i 0.722825i
$$761$$ 27.0237i 0.979608i 0.871833 + 0.489804i $$0.162932\pi$$
−0.871833 + 0.489804i $$0.837068\pi$$
$$762$$ 0 0
$$763$$ 0.249964 0.00904929
$$764$$ 56.1933 2.03300
$$765$$ 0 0
$$766$$ 16.9336 0.611837
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 37.9407i 1.36818i 0.729400 + 0.684088i $$0.239799\pi$$
−0.729400 + 0.684088i $$0.760201\pi$$
$$770$$ 16.9976 0.612551
$$771$$ 0 0
$$772$$ − 18.4590i − 0.664355i
$$773$$ − 16.3375i − 0.587620i −0.955864 0.293810i $$-0.905077\pi$$
0.955864 0.293810i $$-0.0949232\pi$$
$$774$$ 0 0
$$775$$ 12.4300i 0.446498i
$$776$$ 28.1105 1.00911
$$777$$ 0 0
$$778$$ − 79.8926i − 2.86429i
$$779$$ −7.45473 −0.267093
$$780$$ 0 0
$$781$$ 11.7380 0.420017
$$782$$ − 22.4916i − 0.804297i
$$783$$ 0 0
$$784$$ −2.24698 −0.0802493
$$785$$ 13.6310i 0.486512i
$$786$$ 0 0
$$787$$ − 18.6907i − 0.666251i −0.942882 0.333126i $$-0.891897\pi$$
0.942882 0.333126i $$-0.108103\pi$$
$$788$$ 34.8049i 1.23987i
$$789$$ 0 0
$$790$$ 51.2301 1.82269
$$791$$ − 14.9734i − 0.532394i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 3.03684 0.107773
$$795$$ 0 0
$$796$$ −42.3957 −1.50267
$$797$$ 29.2519 1.03615 0.518077 0.855334i $$-0.326648\pi$$
0.518077 + 0.855334i $$0.326648\pi$$
$$798$$ 0 0
$$799$$ − 15.6286i − 0.552901i
$$800$$ 18.9729i 0.670792i
$$801$$ 0 0
$$802$$ −1.30127 −0.0459496
$$803$$ 26.9614 0.951446
$$804$$ 0 0
$$805$$ −5.59611 −0.197237
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 30.7904i 1.08320i
$$809$$ 6.65087 0.233832 0.116916 0.993142i $$-0.462699\pi$$
0.116916 + 0.993142i $$0.462699\pi$$
$$810$$ 0 0
$$811$$ − 3.89200i − 0.136667i −0.997663 0.0683333i $$-0.978232\pi$$
0.997663 0.0683333i $$-0.0217681\pi$$
$$812$$ 14.1728i 0.497369i
$$813$$ 0 0
$$814$$ 30.7536i 1.07791i
$$815$$ 12.5778 0.440581
$$816$$ 0 0
$$817$$ − 35.9071i − 1.25623i
$$818$$ 34.0586 1.19083
$$819$$ 0 0
$$820$$ −5.61356 −0.196034
$$821$$ 45.9982i 1.60535i 0.596418 + 0.802674i $$0.296590\pi$$
−0.596418 + 0.802674i $$0.703410\pi$$
$$822$$ 0 0
$$823$$ −7.95300 −0.277224 −0.138612 0.990347i $$-0.544264\pi$$
−0.138612 + 0.990347i $$0.544264\pi$$
$$824$$ − 21.6093i − 0.752794i
$$825$$ 0 0
$$826$$ 56.2030i 1.95555i
$$827$$ − 27.9648i − 0.972432i −0.873839 0.486216i $$-0.838377\pi$$
0.873839 0.486216i $$-0.161623\pi$$
$$828$$ 0 0
$$829$$ −27.6310 −0.959665 −0.479833 0.877360i $$-0.659303\pi$$
−0.479833 + 0.877360i $$0.659303\pi$$
$$830$$ − 25.0780i − 0.870470i
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −14.8388 −0.514133
$$834$$ 0 0
$$835$$ 34.4698 1.19288
$$836$$ 45.5773 1.57632
$$837$$ 0 0
$$838$$ − 80.2699i − 2.77288i
$$839$$ − 28.6848i − 0.990311i −0.868804 0.495155i $$-0.835111\pi$$
0.868804 0.495155i $$-0.164889\pi$$
$$840$$ 0 0
$$841$$ −23.8528 −0.822509
$$842$$ 78.7488 2.71386
$$843$$ 0 0
$$844$$ 40.3817 1.38999
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 9.16315i 0.314849i
$$848$$ 4.42758 0.152044
$$849$$ 0 0
$$850$$ 34.6504i 1.18850i
$$851$$ − 10.1250i − 0.347080i
$$852$$ 0 0
$$853$$ − 43.2078i − 1.47941i −0.672934 0.739703i $$-0.734966\pi$$
0.672934 0.739703i $$-0.265034\pi$$
$$854$$ 39.4306 1.34929
$$855$$ 0 0
$$856$$ 16.2620i 0.555825i
$$857$$ 35.1685 1.20133 0.600667 0.799499i $$-0.294902\pi$$
0.600667 + 0.799499i $$0.294902\pi$$
$$858$$ 0 0
$$859$$ 27.3793 0.934168 0.467084 0.884213i $$-0.345305\pi$$
0.467084 + 0.884213i $$0.345305\pi$$
$$860$$ − 27.0388i − 0.922014i
$$861$$ 0 0
$$862$$ −77.0297 −2.62364
$$863$$ − 41.3913i − 1.40898i −0.709715 0.704489i $$-0.751176\pi$$
0.709715 0.704489i $$-0.248824\pi$$
$$864$$ 0 0
$$865$$ 27.2465i 0.926409i
$$866$$ 30.8702i 1.04901i
$$867$$ 0 0
$$868$$ 26.6668 0.905130
$$869$$ − 40.3116i − 1.36748i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0.287536 0.00973721
$$873$$ 0 0
$$874$$ −24.8485 −0.840512
$$875$$ 23.4252 0.791916
$$876$$ 0 0
$$877$$ 24.7472i 0.835653i 0.908527 + 0.417826i $$0.137208\pi$$
−0.908527 + 0.417826i $$0.862792\pi$$
$$878$$ 23.0097i 0.776539i
$$879$$ 0 0
$$880$$ −2.96077 −0.0998076
$$881$$ −28.5875 −0.963137 −0.481568 0.876409i $$-0.659933\pi$$
−0.481568 + 0.876409i $$0.659933\pi$$
$$882$$ 0 0
$$883$$ −9.61702 −0.323639 −0.161819 0.986820i $$-0.551736\pi$$
−0.161819 + 0.986820i $$0.551736\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 27.3599i 0.919173i
$$887$$ −15.9661 −0.536091 −0.268045 0.963406i $$-0.586378\pi$$
−0.268045 + 0.963406i $$0.586378\pi$$
$$888$$ 0 0
$$889$$ 38.8689i 1.30362i
$$890$$ 21.4741i 0.719814i
$$891$$ 0 0
$$892$$ − 22.3642i − 0.748809i
$$893$$ −17.2664 −0.577797
$$894$$ 0 0
$$895$$ − 8.70171i − 0.290866i
$$896$$ 33.3196 1.11313
$$897$$ 0 0
$$898$$ 28.9976 0.967663
$$899$$ − 9.68473i − 0.323004i
$$900$$ 0 0
$$901$$ 29.2392 0.974099
$$902$$ 7.31468i 0.243552i
$$903$$ 0 0
$$904$$ − 17.2241i − 0.572867i
$$905$$ − 6.90408i − 0.229500i
$$906$$ 0 0
$$907$$ 28.8364 0.957496 0.478748 0.877952i $$-0.341091\pi$$
0.478748 + 0.877952i $$0.341091\pi$$
$$908$$ 26.4480i 0.877709i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.5633 −1.27766 −0.638830 0.769348i $$-0.720581\pi$$
−0.638830 + 0.769348i $$0.720581\pi$$
$$912$$ 0 0
$$913$$ −19.7332 −0.653073
$$914$$ −10.4644 −0.346132
$$915$$ 0 0
$$916$$ 41.7294i 1.37878i
$$917$$ − 6.67264i − 0.220350i
$$918$$ 0 0
$$919$$ 8.87502 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$920$$ −6.43727 −0.212231
$$921$$ 0 0
$$922$$ −70.8708 −2.33401
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 15.5985i 0.512875i
$$926$$ 39.6601 1.30331
$$927$$ 0 0
$$928$$ − 14.7826i − 0.485261i
$$929$$ 24.2295i 0.794945i 0.917614 + 0.397472i $$0.130113\pi$$
−0.917614 + 0.397472i $$0.869887\pi$$
$$930$$ 0 0
$$931$$ 16.3937i 0.537283i
$$932$$ 15.5133 0.508156
$$933$$ 0 0
$$934$$ 72.2911i 2.36544i
$$935$$ −19.5526 −0.639437
$$936$$ 0 0
$$937$$ 17.2644 0.564005 0.282002 0.959414i $$-0.409001\pi$$
0.282002 + 0.959414i $$0.409001\pi$$
$$938$$ − 2.65519i − 0.0866949i
$$939$$ 0 0
$$940$$ −13.0019 −0.424076
$$941$$ 4.34050i 0.141496i 0.997494 + 0.0707482i $$0.0225387\pi$$
−0.997494 + 0.0707482i $$0.977461\pi$$
$$942$$ 0 0
$$943$$ − 2.40821i − 0.0784220i
$$944$$ − 9.78986i − 0.318633i
$$945$$ 0 0
$$946$$ −35.2325 −1.14551
$$947$$ 45.0146i 1.46278i 0.681961 + 0.731389i $$0.261128\pi$$
−0.681961 + 0.731389i $$0.738872\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 38.2814 1.24201
$$951$$ 0 0
$$952$$ 25.5743 0.828869
$$953$$ −46.8859 −1.51878 −0.759391 0.650634i $$-0.774503\pi$$
−0.759391 + 0.650634i $$0.774503\pi$$
$$954$$ 0 0
$$955$$ 26.6329i 0.861822i
$$956$$ 33.3062i 1.07720i
$$957$$ 0 0
$$958$$ 78.4191 2.53361
$$959$$ 1.62325 0.0524176
$$960$$ 0 0
$$961$$ 12.7778 0.412186
$$962$$ 0 0
$$963$$ 0 0
$$964$$ − 36.3274i − 1.17003i
$$965$$ 8.74871 0.281631
$$966$$ 0 0
$$967$$ − 6.29457i − 0.202420i −0.994865 0.101210i $$-0.967729\pi$$
0.994865 0.101210i $$-0.0322714\pi$$
$$968$$ 10.5405i 0.338784i
$$969$$ 0 0
$$970$$ 38.7265i 1.24343i
$$971$$ 41.8068 1.34165 0.670823 0.741618i $$-0.265941\pi$$
0.670823 + 0.741618i $$0.265941\pi$$
$$972$$ 0 0
$$973$$ − 23.2336i − 0.744834i
$$974$$ −94.0025 −3.01203
$$975$$ 0 0
$$976$$ −6.86831 −0.219849
$$977$$ 23.7530i 0.759926i 0.925002 + 0.379963i $$0.124063\pi$$
−0.925002 + 0.379963i $$0.875937\pi$$
$$978$$ 0 0
$$979$$ 16.8974 0.540043
$$980$$ 12.3448i 0.394341i
$$981$$ 0 0
$$982$$ − 49.0863i − 1.56641i
$$983$$ 55.7251i 1.77736i 0.458532 + 0.888678i $$0.348375\pi$$
−0.458532 + 0.888678i $$0.651625\pi$$
$$984$$ 0 0
$$985$$ −16.4959 −0.525602
$$986$$ − 26.9976i − 0.859779i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11.5996 0.368845
$$990$$ 0 0
$$991$$ −35.5512 −1.12932 −0.564661 0.825323i $$-0.690993\pi$$
−0.564661 + 0.825323i $$0.690993\pi$$
$$992$$ −27.8140 −0.883096
$$993$$ 0 0
$$994$$ − 21.1511i − 0.670871i
$$995$$ − 20.0935i − 0.637007i
$$996$$ 0 0
$$997$$ 6.61058 0.209359 0.104680 0.994506i $$-0.466618\pi$$
0.104680 + 0.994506i $$0.466618\pi$$
$$998$$ −52.9101 −1.67484
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.2.b.l.1351.1 6
3.2 odd 2 169.2.b.b.168.6 6
12.11 even 2 2704.2.f.o.337.4 6
13.5 odd 4 1521.2.a.o.1.1 3
13.8 odd 4 1521.2.a.r.1.3 3
13.12 even 2 inner 1521.2.b.l.1351.6 6
39.2 even 12 169.2.c.b.22.1 6
39.5 even 4 169.2.a.c.1.3 yes 3
39.8 even 4 169.2.a.b.1.1 3
39.11 even 12 169.2.c.c.22.3 6
39.17 odd 6 169.2.e.b.23.1 12
39.20 even 12 169.2.c.c.146.3 6
39.23 odd 6 169.2.e.b.147.6 12
39.29 odd 6 169.2.e.b.147.1 12
39.32 even 12 169.2.c.b.146.1 6
39.35 odd 6 169.2.e.b.23.6 12
39.38 odd 2 169.2.b.b.168.1 6
156.47 odd 4 2704.2.a.z.1.2 3
156.83 odd 4 2704.2.a.ba.1.2 3
156.155 even 2 2704.2.f.o.337.3 6
195.44 even 4 4225.2.a.bb.1.1 3
195.164 even 4 4225.2.a.bg.1.3 3
273.83 odd 4 8281.2.a.bj.1.3 3
273.125 odd 4 8281.2.a.bf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 39.8 even 4
169.2.a.c.1.3 yes 3 39.5 even 4
169.2.b.b.168.1 6 39.38 odd 2
169.2.b.b.168.6 6 3.2 odd 2
169.2.c.b.22.1 6 39.2 even 12
169.2.c.b.146.1 6 39.32 even 12
169.2.c.c.22.3 6 39.11 even 12
169.2.c.c.146.3 6 39.20 even 12
169.2.e.b.23.1 12 39.17 odd 6
169.2.e.b.23.6 12 39.35 odd 6
169.2.e.b.147.1 12 39.29 odd 6
169.2.e.b.147.6 12 39.23 odd 6
1521.2.a.o.1.1 3 13.5 odd 4
1521.2.a.r.1.3 3 13.8 odd 4
1521.2.b.l.1351.1 6 1.1 even 1 trivial
1521.2.b.l.1351.6 6 13.12 even 2 inner
2704.2.a.z.1.2 3 156.47 odd 4
2704.2.a.ba.1.2 3 156.83 odd 4
2704.2.f.o.337.3 6 156.155 even 2
2704.2.f.o.337.4 6 12.11 even 2
4225.2.a.bb.1.1 3 195.44 even 4
4225.2.a.bg.1.3 3 195.164 even 4
8281.2.a.bf.1.1 3 273.125 odd 4
8281.2.a.bj.1.3 3 273.83 odd 4