# Properties

 Label 2704.2.f.o.337.3 Level $2704$ Weight $2$ Character 2704.337 Analytic conductor $21.592$ 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] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.f (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: $$21.5915487066$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.3 Root $$-1.80194i$$ of defining polynomial Character $$\chi$$ $$=$$ 2704.337 Dual form 2704.2.f.o.337.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.554958 q^{3} -1.44504i q^{5} +2.04892i q^{7} -2.69202 q^{9} +O(q^{10})$$ $$q+0.554958 q^{3} -1.44504i q^{5} +2.04892i q^{7} -2.69202 q^{9} -2.55496i q^{11} -0.801938i q^{15} +5.29590 q^{17} +5.85086i q^{19} +1.13706i q^{21} -1.89008 q^{23} +2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} +4.26875i q^{31} -1.41789i q^{33} +2.96077 q^{35} -5.35690i q^{37} +1.27413i q^{41} +6.13706 q^{43} +3.89008i q^{45} -2.95108i q^{47} +2.80194 q^{49} +2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} +3.24698i q^{57} -12.2078i q^{59} +8.56465 q^{61} -5.51573i q^{63} -0.576728i q^{67} -1.04892 q^{69} +4.59419i q^{71} +10.5526i q^{73} +1.61596 q^{75} +5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} -7.72348i q^{83} -7.65279i q^{85} +1.25906 q^{87} -6.61356i q^{89} +2.36898i q^{93} +8.45473 q^{95} +11.9269i q^{97} +6.87800i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{3} - 6 q^{9}+O(q^{10})$$ 6 * q + 4 * q^3 - 6 * q^9 $$6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79} - 2 q^{81} + 36 q^{87} + 6 q^{95}+O(q^{100})$$ 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1185$$ $$2367$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0.554958 0.320405 0.160203 0.987084i $$-0.448785\pi$$
0.160203 + 0.987084i $$0.448785\pi$$
$$4$$ 0 0
$$5$$ − 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 $$0.873439\pi$$
$$8$$ 0 0
$$9$$ −2.69202 −0.897340
$$10$$ 0 0
$$11$$ − 2.55496i − 0.770349i −0.922844 0.385174i $$-0.874141\pi$$
0.922844 0.385174i $$-0.125859\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 0.801938i − 0.207059i
$$16$$ 0 0
$$17$$ 5.29590 1.28444 0.642222 0.766519i $$-0.278013\pi$$
0.642222 + 0.766519i $$0.278013\pi$$
$$18$$ 0 0
$$19$$ 5.85086i 1.34228i 0.741331 + 0.671139i $$0.234195\pi$$
−0.741331 + 0.671139i $$0.765805\pi$$
$$20$$ 0 0
$$21$$ 1.13706i 0.248128i
$$22$$ 0 0
$$23$$ −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
$$27$$ −3.15883 −0.607918
$$28$$ 0 0
$$29$$ 2.26875 0.421296 0.210648 0.977562i $$-0.432443\pi$$
0.210648 + 0.977562i $$0.432443\pi$$
$$30$$ 0 0
$$31$$ 4.26875i 0.766690i 0.923605 + 0.383345i $$0.125228\pi$$
−0.923605 + 0.383345i $$0.874772\pi$$
$$32$$ 0 0
$$33$$ − 1.41789i − 0.246824i
$$34$$ 0 0
$$35$$ 2.96077 0.500462
$$36$$ 0 0
$$37$$ − 5.35690i − 0.880668i −0.897834 0.440334i $$-0.854860\pi$$
0.897834 0.440334i $$-0.145140\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 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 $$-0.344994\pi$$
0.467947 + 0.883757i $$0.344994\pi$$
$$44$$ 0 0
$$45$$ 3.89008i 0.579899i
$$46$$ 0 0
$$47$$ − 2.95108i − 0.430460i −0.976563 0.215230i $$-0.930950\pi$$
0.976563 0.215230i $$-0.0690501\pi$$
$$48$$ 0 0
$$49$$ 2.80194 0.400277
$$50$$ 0 0
$$51$$ 2.93900 0.411542
$$52$$ 0 0
$$53$$ 5.52111 0.758382 0.379191 0.925318i $$-0.376202\pi$$
0.379191 + 0.925318i $$0.376202\pi$$
$$54$$ 0 0
$$55$$ −3.69202 −0.497832
$$56$$ 0 0
$$57$$ 3.24698i 0.430073i
$$58$$ 0 0
$$59$$ − 12.2078i − 1.58931i −0.607059 0.794657i $$-0.707651\pi$$
0.607059 0.794657i $$-0.292349\pi$$
$$60$$ 0 0
$$61$$ 8.56465 1.09659 0.548295 0.836285i $$-0.315277\pi$$
0.548295 + 0.836285i $$0.315277\pi$$
$$62$$ 0 0
$$63$$ − 5.51573i − 0.694917i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 0.576728i − 0.0704586i −0.999379 0.0352293i $$-0.988784\pi$$
0.999379 0.0352293i $$-0.0112162\pi$$
$$68$$ 0 0
$$69$$ −1.04892 −0.126275
$$70$$ 0 0
$$71$$ 4.59419i 0.545230i 0.962123 + 0.272615i $$0.0878885\pi$$
−0.962123 + 0.272615i $$0.912112\pi$$
$$72$$ 0 0
$$73$$ 10.5526i 1.23508i 0.786538 + 0.617542i $$0.211872\pi$$
−0.786538 + 0.617542i $$0.788128\pi$$
$$74$$ 0 0
$$75$$ 1.61596 0.186595
$$76$$ 0 0
$$77$$ 5.23490 0.596572
$$78$$ 0 0
$$79$$ 15.7778 1.77514 0.887569 0.460674i $$-0.152392\pi$$
0.887569 + 0.460674i $$0.152392\pi$$
$$80$$ 0 0
$$81$$ 6.32304 0.702560
$$82$$ 0 0
$$83$$ − 7.72348i − 0.847762i −0.905718 0.423881i $$-0.860667\pi$$
0.905718 0.423881i $$-0.139333\pi$$
$$84$$ 0 0
$$85$$ − 7.65279i − 0.830062i
$$86$$ 0 0
$$87$$ 1.25906 0.134986
$$88$$ 0 0
$$89$$ − 6.61356i − 0.701036i −0.936556 0.350518i $$-0.886005\pi$$
0.936556 0.350518i $$-0.113995\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 2.36898i 0.245652i
$$94$$ 0 0
$$95$$ 8.45473 0.867437
$$96$$ 0 0
$$97$$ 11.9269i 1.21100i 0.795847 + 0.605498i $$0.207026\pi$$
−0.795847 + 0.605498i $$0.792974\pi$$
$$98$$ 0 0
$$99$$ 6.87800i 0.691265i
$$100$$ 0 0
$$101$$ −13.0640 −1.29991 −0.649957 0.759971i $$-0.725213\pi$$
−0.649957 + 0.759971i $$0.725213\pi$$
$$102$$ 0 0
$$103$$ 9.16852 0.903401 0.451701 0.892170i $$-0.350818\pi$$
0.451701 + 0.892170i $$0.350818\pi$$
$$104$$ 0 0
$$105$$ 1.64310 0.160351
$$106$$ 0 0
$$107$$ 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 $$0.00185978\pi$$
−0.999983 + 0.00584264i $$0.998140\pi$$
$$110$$ 0 0
$$111$$ − 2.97285i − 0.282171i
$$112$$ 0 0
$$113$$ 7.30798 0.687477 0.343738 0.939065i $$-0.388307\pi$$
0.343738 + 0.939065i $$0.388307\pi$$
$$114$$ 0 0
$$115$$ 2.73125i 0.254690i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 10.8509i 0.994696i
$$120$$ 0 0
$$121$$ 4.47219 0.406563
$$122$$ 0 0
$$123$$ 0.707087i 0.0637559i
$$124$$ 0 0
$$125$$ − 11.4330i − 1.02260i
$$126$$ 0 0
$$127$$ −18.9705 −1.68336 −0.841678 0.539980i $$-0.818432\pi$$
−0.841678 + 0.539980i $$0.818432\pi$$
$$128$$ 0 0
$$129$$ 3.40581 0.299865
$$130$$ 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
$$134$$ 0 0
$$135$$ 4.56465i 0.392862i
$$136$$ 0 0
$$137$$ − 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 $$-0.340310\pi$$
0.480899 + 0.876776i $$0.340310\pi$$
$$140$$ 0 0
$$141$$ − 1.63773i − 0.137922i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 3.27844i − 0.272260i
$$146$$ 0 0
$$147$$ 1.55496 0.128251
$$148$$ 0 0
$$149$$ 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 $$0.805036\pi$$
$$152$$ 0 0
$$153$$ −14.2567 −1.15258
$$154$$ 0 0
$$155$$ 6.16852 0.495468
$$156$$ 0 0
$$157$$ −9.43296 −0.752832 −0.376416 0.926451i $$-0.622844\pi$$
−0.376416 + 0.926451i $$0.622844\pi$$
$$158$$ 0 0
$$159$$ 3.06398 0.242990
$$160$$ 0 0
$$161$$ − 3.87263i − 0.305206i
$$162$$ 0 0
$$163$$ 8.70410i 0.681758i 0.940107 + 0.340879i $$0.110725\pi$$
−0.940107 + 0.340879i $$0.889275\pi$$
$$164$$ 0 0
$$165$$ −2.04892 −0.159508
$$166$$ 0 0
$$167$$ − 23.8538i − 1.84587i −0.384961 0.922933i $$-0.625785\pi$$
0.384961 0.922933i $$-0.374215\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 15.7506i − 1.20448i
$$172$$ 0 0
$$173$$ 18.8552 1.43353 0.716766 0.697314i $$-0.245622\pi$$
0.716766 + 0.697314i $$0.245622\pi$$
$$174$$ 0 0
$$175$$ 5.96615i 0.450998i
$$176$$ 0 0
$$177$$ − 6.77479i − 0.509224i
$$178$$ 0 0
$$179$$ 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
$$183$$ 4.75302 0.351353
$$184$$ 0 0
$$185$$ −7.74094 −0.569125
$$186$$ 0 0
$$187$$ − 13.5308i − 0.989470i
$$188$$ 0 0
$$189$$ − 6.47219i − 0.470782i
$$190$$ 0 0
$$191$$ −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 $$-0.930080\pi$$
0.975971 0.217899i $$-0.0699203\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 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 $$-0.664047\pi$$
−0.492855 + 0.870111i $$0.664047\pi$$
$$200$$ 0 0
$$201$$ − 0.320060i − 0.0225753i
$$202$$ 0 0
$$203$$ 4.64848i 0.326259i
$$204$$ 0 0
$$205$$ 1.84117 0.128593
$$206$$ 0 0
$$207$$ 5.08815 0.353651
$$208$$ 0 0
$$209$$ 14.9487 1.03402
$$210$$ 0 0
$$211$$ 13.2446 0.911795 0.455897 0.890032i $$-0.349318\pi$$
0.455897 + 0.890032i $$0.349318\pi$$
$$212$$ 0 0
$$213$$ 2.54958i 0.174694i
$$214$$ 0 0
$$215$$ − 8.86831i − 0.604814i
$$216$$ 0 0
$$217$$ −8.74632 −0.593739
$$218$$ 0 0
$$219$$ 5.85623i 0.395727i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 7.33513i 0.491196i 0.969372 + 0.245598i $$0.0789844\pi$$
−0.969372 + 0.245598i $$0.921016\pi$$
$$224$$ 0 0
$$225$$ −7.83877 −0.522585
$$226$$ 0 0
$$227$$ 8.67456i 0.575751i 0.957668 + 0.287875i $$0.0929489\pi$$
−0.957668 + 0.287875i $$0.907051\pi$$
$$228$$ 0 0
$$229$$ 13.6866i 0.904439i 0.891907 + 0.452219i $$0.149368\pi$$
−0.891907 + 0.452219i $$0.850632\pi$$
$$230$$ 0 0
$$231$$ 2.90515 0.191145
$$232$$ 0 0
$$233$$ 5.08815 0.333336 0.166668 0.986013i $$-0.446699\pi$$
0.166668 + 0.986013i $$0.446699\pi$$
$$234$$ 0 0
$$235$$ −4.26444 −0.278181
$$236$$ 0 0
$$237$$ 8.75600 0.568764
$$238$$ 0 0
$$239$$ 10.9239i 0.706611i 0.935508 + 0.353305i $$0.114942\pi$$
−0.935508 + 0.353305i $$0.885058\pi$$
$$240$$ 0 0
$$241$$ − 11.9148i − 0.767502i −0.923437 0.383751i $$-0.874632\pi$$
0.923437 0.383751i $$-0.125368\pi$$
$$242$$ 0 0
$$243$$ 12.9855 0.833022
$$244$$ 0 0
$$245$$ − 4.04892i − 0.258676i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 4.28621i − 0.271627i
$$250$$ 0 0
$$251$$ 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
$$254$$ 0 0
$$255$$ − 4.24698i − 0.265956i
$$256$$ 0 0
$$257$$ 18.6601 1.16398 0.581992 0.813194i $$-0.302273\pi$$
0.581992 + 0.813194i $$0.302273\pi$$
$$258$$ 0 0
$$259$$ 10.9758 0.682005
$$260$$ 0 0
$$261$$ −6.10752 −0.378046
$$262$$ 0 0
$$263$$ −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
$$266$$ 0 0
$$267$$ − 3.67025i − 0.224616i
$$268$$ 0 0
$$269$$ 0.652793 0.0398015 0.0199007 0.999802i $$-0.493665\pi$$
0.0199007 + 0.999802i $$0.493665\pi$$
$$270$$ 0 0
$$271$$ − 1.99569i − 0.121229i −0.998161 0.0606147i $$-0.980694\pi$$
0.998161 0.0606147i $$-0.0193061\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 7.43967i − 0.448629i
$$276$$ 0 0
$$277$$ −11.7845 −0.708061 −0.354030 0.935234i $$-0.615189\pi$$
−0.354030 + 0.935234i $$0.615189\pi$$
$$278$$ 0 0
$$279$$ − 11.4916i − 0.687982i
$$280$$ 0 0
$$281$$ 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 $$-0.437334\pi$$
0.195601 + 0.980684i $$0.437334\pi$$
$$284$$ 0 0
$$285$$ 4.69202 0.277931
$$286$$ 0 0
$$287$$ −2.61058 −0.154098
$$288$$ 0 0
$$289$$ 11.0465 0.649796
$$290$$ 0 0
$$291$$ 6.61894i 0.388009i
$$292$$ 0 0
$$293$$ 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
$$296$$ 0 0
$$297$$ 8.07069i 0.468309i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.5743i 0.724773i
$$302$$ 0 0
$$303$$ −7.24996 −0.416500
$$304$$ 0 0
$$305$$ − 12.3763i − 0.708663i
$$306$$ 0 0
$$307$$ − 14.0737i − 0.803227i −0.915809 0.401613i $$-0.868450\pi$$
0.915809 0.401613i $$-0.131550\pi$$
$$308$$ 0 0
$$309$$ 5.08815 0.289455
$$310$$ 0 0
$$311$$ −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 $$0.567791\pi$$
$$314$$ 0 0
$$315$$ −7.97046 −0.449085
$$316$$ 0 0
$$317$$ 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
$$320$$ 0 0
$$321$$ 3.82908 0.213719
$$322$$ 0 0
$$323$$ 30.9855i 1.72408i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0.0677037i 0.00374402i
$$328$$ 0 0
$$329$$ 6.04652 0.333356
$$330$$ 0 0
$$331$$ 15.7168i 0.863872i 0.901904 + 0.431936i $$0.142169\pi$$
−0.901904 + 0.431936i $$0.857831\pi$$
$$332$$ 0 0
$$333$$ 14.4209i 0.790259i
$$334$$ 0 0
$$335$$ −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
$$339$$ 4.05562 0.220271
$$340$$ 0 0
$$341$$ 10.9065 0.590619
$$342$$ 0 0
$$343$$ 20.0834i 1.08440i
$$344$$ 0 0
$$345$$ 1.51573i 0.0816041i
$$346$$ 0 0
$$347$$ 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 $$0.0903802\pi$$
−0.959960 + 0.280138i $$0.909620\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 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
$$356$$ 0 0
$$357$$ 6.02177i 0.318706i
$$358$$ 0 0
$$359$$ − 21.4263i − 1.13083i −0.824805 0.565417i $$-0.808715\pi$$
0.824805 0.565417i $$-0.191285\pi$$
$$360$$ 0 0
$$361$$ −15.2325 −0.801711
$$362$$ 0 0
$$363$$ 2.48188 0.130265
$$364$$ 0 0
$$365$$ 15.2489 0.798164
$$366$$ 0 0
$$367$$ −34.3032 −1.79061 −0.895306 0.445452i $$-0.853043\pi$$
−0.895306 + 0.445452i $$0.853043\pi$$
$$368$$ 0 0
$$369$$ − 3.42998i − 0.178557i
$$370$$ 0 0
$$371$$ 11.3123i 0.587305i
$$372$$ 0 0
$$373$$ −12.5961 −0.652202 −0.326101 0.945335i $$-0.605735\pi$$
−0.326101 + 0.945335i $$0.605735\pi$$
$$374$$ 0 0
$$375$$ − 6.34481i − 0.327645i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 16.5386i − 0.849529i −0.905304 0.424765i $$-0.860357\pi$$
0.905304 0.424765i $$-0.139643\pi$$
$$380$$ 0 0
$$381$$ −10.5278 −0.539356
$$382$$ 0 0
$$383$$ − 7.53617i − 0.385080i −0.981289 0.192540i $$-0.938327\pi$$
0.981289 0.192540i $$-0.0616726\pi$$
$$384$$ 0 0
$$385$$ − 7.56465i − 0.385530i
$$386$$ 0 0
$$387$$ −16.5211 −0.839815
$$388$$ 0 0
$$389$$ −35.5555 −1.80274 −0.901369 0.433052i $$-0.857437\pi$$
−0.901369 + 0.433052i $$0.857437\pi$$
$$390$$ 0 0
$$391$$ −10.0097 −0.506212
$$392$$ 0 0
$$393$$ −1.80731 −0.0911670
$$394$$ 0 0
$$395$$ − 22.7995i − 1.14717i
$$396$$ 0 0
$$397$$ − 1.35152i − 0.0678308i −0.999425 0.0339154i $$-0.989202\pi$$
0.999425 0.0339154i $$-0.0107977\pi$$
$$398$$ 0 0
$$399$$ −6.65279 −0.333056
$$400$$ 0 0
$$401$$ − 0.579121i − 0.0289199i −0.999895 0.0144600i $$-0.995397\pi$$
0.999895 0.0144600i $$-0.00460291\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 9.13706i − 0.454024i
$$406$$ 0 0
$$407$$ −13.6866 −0.678422
$$408$$ 0 0
$$409$$ − 15.1575i − 0.749490i −0.927128 0.374745i $$-0.877730\pi$$
0.927128 0.374745i $$-0.122270\pi$$
$$410$$ 0 0
$$411$$ − 0.439665i − 0.0216871i
$$412$$ 0 0
$$413$$ 25.0127 1.23079
$$414$$ 0 0
$$415$$ −11.1608 −0.547860
$$416$$ 0 0
$$417$$ 6.29291 0.308165
$$418$$ 0 0
$$419$$ 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 $$-0.674151\pi$$
0.520221 0.854032i $$-0.325849\pi$$
$$422$$ 0 0
$$423$$ 7.94438i 0.386269i
$$424$$ 0 0
$$425$$ 15.4209 0.748022
$$426$$ 0 0
$$427$$ 17.5483i 0.849220i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 34.2814i 1.65128i 0.564199 + 0.825639i $$0.309185\pi$$
−0.564199 + 0.825639i $$0.690815\pi$$
$$432$$ 0 0
$$433$$ −13.7385 −0.660232 −0.330116 0.943940i $$-0.607088\pi$$
−0.330116 + 0.943940i $$0.607088\pi$$
$$434$$ 0 0
$$435$$ − 1.81940i − 0.0872334i
$$436$$ 0 0
$$437$$ − 11.0586i − 0.529005i
$$438$$ 0 0
$$439$$ 10.2403 0.488742 0.244371 0.969682i $$-0.421419\pi$$
0.244371 + 0.969682i $$0.421419\pi$$
$$440$$ 0 0
$$441$$ −7.54288 −0.359185
$$442$$ 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
$$446$$ 0 0
$$447$$ 4.66487i 0.220641i
$$448$$ 0 0
$$449$$ 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
$$452$$ 0 0
$$453$$ 7.84117i 0.368410i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.65710i 0.217850i 0.994050 + 0.108925i $$0.0347409\pi$$
−0.994050 + 0.108925i $$0.965259\pi$$
$$458$$ 0 0
$$459$$ −16.7289 −0.780836
$$460$$ 0 0
$$461$$ − 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 $$0.865479\pi$$
$$464$$ 0 0
$$465$$ 3.42327 0.158750
$$466$$ 0 0
$$467$$ −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
$$470$$ 0 0
$$471$$ −5.23490 −0.241211
$$472$$ 0 0
$$473$$ − 15.6799i − 0.720964i
$$474$$ 0 0
$$475$$ 17.0368i 0.781704i
$$476$$ 0 0
$$477$$ −14.8629 −0.680527
$$478$$ 0 0
$$479$$ − 34.8998i − 1.59461i −0.603576 0.797306i $$-0.706258\pi$$
0.603576 0.797306i $$-0.293742\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 2.14914i − 0.0977895i
$$484$$ 0 0
$$485$$ 17.2349 0.782596
$$486$$ 0 0
$$487$$ − 41.8351i − 1.89573i −0.318676 0.947864i $$-0.603238\pi$$
0.318676 0.947864i $$-0.396762\pi$$
$$488$$ 0 0
$$489$$ 4.83041i 0.218439i
$$490$$ 0 0
$$491$$ 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
$$495$$ 9.93900 0.446725
$$496$$ 0 0
$$497$$ −9.41311 −0.422236
$$498$$ 0 0
$$499$$ − 23.5472i − 1.05412i −0.849829 0.527058i $$-0.823295\pi$$
0.849829 0.527058i $$-0.176705\pi$$
$$500$$ 0 0
$$501$$ − 13.2379i − 0.591425i
$$502$$ 0 0
$$503$$ 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
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 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
$$512$$ 0 0
$$513$$ − 18.4819i − 0.815995i
$$514$$ 0 0
$$515$$ − 13.2489i − 0.583816i
$$516$$ 0 0
$$517$$ −7.53989 −0.331604
$$518$$ 0 0
$$519$$ 10.4638 0.459311
$$520$$ 0 0
$$521$$ −39.5133 −1.73111 −0.865555 0.500813i $$-0.833034\pi$$
−0.865555 + 0.500813i $$0.833034\pi$$
$$522$$ 0 0
$$523$$ 15.8194 0.691734 0.345867 0.938284i $$-0.387585\pi$$
0.345867 + 0.938284i $$0.387585\pi$$
$$524$$ 0 0
$$525$$ 3.31096i 0.144502i
$$526$$ 0 0
$$527$$ 22.6069i 0.984770i
$$528$$ 0 0
$$529$$ −19.4276 −0.844678
$$530$$ 0 0
$$531$$ 32.8635i 1.42616i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 9.97046i − 0.431061i
$$536$$ 0 0
$$537$$ 3.34183 0.144211
$$538$$ 0 0
$$539$$ − 7.15883i − 0.308353i
$$540$$ 0 0
$$541$$ − 34.4819i − 1.48249i −0.671234 0.741246i $$-0.734235\pi$$
0.671234 0.741246i $$-0.265765\pi$$
$$542$$ 0 0
$$543$$ 2.65146 0.113785
$$544$$ 0 0
$$545$$ 0.176292 0.00755152
$$546$$ 0 0
$$547$$ −36.8582 −1.57594 −0.787970 0.615713i $$-0.788868\pi$$
−0.787970 + 0.615713i $$0.788868\pi$$
$$548$$ 0 0
$$549$$ −23.0562 −0.984015
$$550$$ 0 0
$$551$$ 13.2741i 0.565497i
$$552$$ 0 0
$$553$$ 32.3274i 1.37470i
$$554$$ 0 0
$$555$$ −4.29590 −0.182351
$$556$$ 0 0
$$557$$ − 1.27652i − 0.0540879i −0.999634 0.0270439i $$-0.991391\pi$$
0.999634 0.0270439i $$-0.00860940\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ − 7.50902i − 0.317031i
$$562$$ 0 0
$$563$$ −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
$$566$$ 0 0
$$567$$ 12.9554i 0.544075i
$$568$$ 0 0
$$569$$ 5.72156 0.239860 0.119930 0.992782i $$-0.461733\pi$$
0.119930 + 0.992782i $$0.461733\pi$$
$$570$$ 0 0
$$571$$ 7.60148 0.318112 0.159056 0.987270i $$-0.449155\pi$$
0.159056 + 0.987270i $$0.449155\pi$$
$$572$$ 0 0
$$573$$ −10.2282 −0.427289
$$574$$ 0 0
$$575$$ −5.50365 −0.229518
$$576$$ 0 0
$$577$$ 45.1564i 1.87989i 0.341330 + 0.939944i $$0.389123\pi$$
−0.341330 + 0.939944i $$0.610877\pi$$
$$578$$ 0 0
$$579$$ − 3.35988i − 0.139632i
$$580$$ 0 0
$$581$$ 15.8248 0.656522
$$582$$ 0 0
$$583$$ − 14.1062i − 0.584219i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 32.4040i 1.33746i 0.743507 + 0.668728i $$0.233161\pi$$
−0.743507 + 0.668728i $$0.766839\pi$$
$$588$$ 0 0
$$589$$ −24.9758 −1.02911
$$590$$ 0 0
$$591$$ − 6.33513i − 0.260592i
$$592$$ 0 0
$$593$$ − 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
$$596$$ 0 0
$$597$$ −7.71678 −0.315827
$$598$$ 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 $$0.511764\pi$$
$$602$$ 0 0
$$603$$ 1.55257i 0.0632253i
$$604$$ 0 0
$$605$$ − 6.46250i − 0.262738i
$$606$$ 0 0
$$607$$ −11.2161 −0.455248 −0.227624 0.973749i $$-0.573096\pi$$
−0.227624 + 0.973749i $$0.573096\pi$$
$$608$$ 0 0
$$609$$ 2.57971i 0.104535i
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 20.8944i − 0.843917i −0.906615 0.421958i $$-0.861343\pi$$
0.906615 0.421958i $$-0.138657\pi$$
$$614$$ 0 0
$$615$$ 1.02177 0.0412018
$$616$$ 0 0
$$617$$ 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 $$-0.0680210\pi$$
$$620$$ 0 0
$$621$$ 5.97046 0.239586
$$622$$ 0 0
$$623$$ 13.5506 0.542895
$$624$$ 0 0
$$625$$ −1.96184 −0.0784735
$$626$$ 0 0
$$627$$ 8.29590 0.331306
$$628$$ 0 0
$$629$$ − 28.3696i − 1.13117i
$$630$$ 0 0
$$631$$ − 13.8514i − 0.551417i −0.961241 0.275709i $$-0.911087\pi$$
0.961241 0.275709i $$-0.0889125\pi$$
$$632$$ 0 0
$$633$$ 7.35019 0.292144
$$634$$ 0 0
$$635$$ 27.4131i 1.08786i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 12.3676i − 0.489257i
$$640$$ 0 0
$$641$$ −34.9608 −1.38087 −0.690434 0.723396i $$-0.742580\pi$$
−0.690434 + 0.723396i $$0.742580\pi$$
$$642$$ 0 0
$$643$$ − 33.3980i − 1.31709i −0.752541 0.658545i $$-0.771172\pi$$
0.752541 0.658545i $$-0.228828\pi$$
$$644$$ 0 0
$$645$$ − 4.92154i − 0.193786i
$$646$$ 0 0
$$647$$ 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
$$651$$ −4.85384 −0.190237
$$652$$ 0 0
$$653$$ 14.5714 0.570221 0.285111 0.958495i $$-0.407970\pi$$
0.285111 + 0.958495i $$0.407970\pi$$
$$654$$ 0 0
$$655$$ 4.70602i 0.183879i
$$656$$ 0 0
$$657$$ − 28.4077i − 1.10829i
$$658$$ 0 0
$$659$$ −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 $$0.0868079\pi$$
−0.963043 + 0.269347i $$0.913192\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 17.3230i 0.671759i
$$666$$ 0 0
$$667$$ −4.28813 −0.166037
$$668$$ 0 0
$$669$$ 4.07069i 0.157382i
$$670$$ 0 0
$$671$$ − 21.8823i − 0.844757i
$$672$$ 0 0
$$673$$ 6.52973 0.251703 0.125851 0.992049i $$-0.459834\pi$$
0.125851 + 0.992049i $$0.459834\pi$$
$$674$$ 0 0
$$675$$ −9.19806 −0.354034
$$676$$ 0 0
$$677$$ −11.3104 −0.434693 −0.217346 0.976095i $$-0.569740\pi$$
−0.217346 + 0.976095i $$0.569740\pi$$
$$678$$ 0 0
$$679$$ −24.4373 −0.937816
$$680$$ 0 0
$$681$$ 4.81402i 0.184474i
$$682$$ 0 0
$$683$$ 14.1793i 0.542555i 0.962501 + 0.271277i $$0.0874461\pi$$
−0.962501 + 0.271277i $$0.912554\pi$$
$$684$$ 0 0
$$685$$ −1.14483 −0.0437418
$$686$$ 0 0
$$687$$ 7.59551i 0.289787i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 30.7952i − 1.17151i −0.810490 0.585753i $$-0.800799\pi$$
0.810490 0.585753i $$-0.199201\pi$$
$$692$$ 0 0
$$693$$ −14.0925 −0.535328
$$694$$ 0 0
$$695$$ − 16.3860i − 0.621555i
$$696$$ 0 0
$$697$$ 6.74764i 0.255585i
$$698$$ 0 0
$$699$$ 2.82371 0.106802
$$700$$ 0 0
$$701$$ −6.73184 −0.254258 −0.127129 0.991886i $$-0.540576\pi$$
−0.127129 + 0.991886i $$0.540576\pi$$
$$702$$ 0 0
$$703$$ 31.3424 1.18210
$$704$$ 0 0
$$705$$ −2.36658 −0.0891307
$$706$$ 0 0
$$707$$ − 26.7670i − 1.00668i
$$708$$ 0 0
$$709$$ 47.6252i 1.78860i 0.447467 + 0.894300i $$0.352326\pi$$
−0.447467 + 0.894300i $$0.647674\pi$$
$$710$$ 0 0
$$711$$ −42.4741 −1.59290
$$712$$ 0 0
$$713$$ − 8.06829i − 0.302160i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.06233i 0.226402i
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ − 6.61224i − 0.245912i
$$724$$ 0 0
$$725$$ 6.60627 0.245351
$$726$$ 0 0
$$727$$ −24.1226 −0.894657 −0.447329 0.894370i $$-0.647625\pi$$
−0.447329 + 0.894370i $$0.647625\pi$$
$$728$$ 0 0
$$729$$ −11.7627 −0.435656
$$730$$ 0 0
$$731$$ 32.5013 1.20210
$$732$$ 0 0
$$733$$ 36.0646i 1.33208i 0.745918 + 0.666038i $$0.232011\pi$$
−0.745918 + 0.666038i $$0.767989\pi$$
$$734$$ 0 0
$$735$$ − 2.24698i − 0.0828811i
$$736$$ 0 0
$$737$$ −1.47352 −0.0542777
$$738$$ 0 0
$$739$$ 27.5254i 1.01254i 0.862375 + 0.506269i $$0.168976\pi$$
−0.862375 + 0.506269i $$0.831024\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10.4692i 0.384078i 0.981387 + 0.192039i $$0.0615100\pi$$
−0.981387 + 0.192039i $$0.938490\pi$$
$$744$$ 0 0
$$745$$ 12.1468 0.445023
$$746$$ 0 0
$$747$$ 20.7918i 0.760731i
$$748$$ 0 0
$$749$$ 14.1371i 0.516557i
$$750$$ 0 0
$$751$$ 4.06770 0.148433 0.0742163 0.997242i $$-0.476354\pi$$
0.0742163 + 0.997242i $$0.476354\pi$$
$$752$$ 0 0
$$753$$ 12.4021 0.451957
$$754$$ 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 $$0.378900\pi$$
$$758$$ 0 0
$$759$$ 2.67994i 0.0972757i
$$760$$ 0 0
$$761$$ 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
$$764$$ 0 0
$$765$$ 20.6015i 0.744848i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 37.9407i − 1.36818i −0.729400 0.684088i $$-0.760201\pi$$
0.729400 0.684088i $$-0.239799\pi$$
$$770$$ 0 0
$$771$$ 10.3556 0.372947
$$772$$ 0 0
$$773$$ − 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
$$776$$ 0 0
$$777$$ 6.09113 0.218518
$$778$$ 0 0
$$779$$ −7.45473 −0.267093
$$780$$ 0 0
$$781$$ 11.7380 0.420017
$$782$$ 0 0
$$783$$ −7.16660 −0.256114
$$784$$ 0 0
$$785$$ 13.6310i 0.486512i
$$786$$ 0 0
$$787$$ − 18.6907i − 0.666251i −0.942882 0.333126i $$-0.891897\pi$$
0.942882 0.333126i $$-0.108103\pi$$
$$788$$ 0 0
$$789$$ −7.99090 −0.284484
$$790$$ 0 0
$$791$$ 14.9734i 0.532394i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ − 4.42758i − 0.157030i
$$796$$ 0 0
$$797$$ −29.2519 −1.03615 −0.518077 0.855334i $$-0.673352\pi$$
−0.518077 + 0.855334i $$0.673352\pi$$
$$798$$ 0 0
$$799$$ − 15.6286i − 0.552901i
$$800$$ 0 0
$$801$$ 17.8039i 0.629068i
$$802$$ 0 0
$$803$$ 26.9614 0.951446
$$804$$ 0 0
$$805$$ −5.59611 −0.197237
$$806$$ 0 0
$$807$$ 0.362273 0.0127526
$$808$$ 0 0
$$809$$ −6.65087 −0.233832 −0.116916 0.993142i $$-0.537301\pi$$
−0.116916 + 0.993142i $$0.537301\pi$$
$$810$$ 0 0
$$811$$ − 3.89200i − 0.136667i −0.997663 0.0683333i $$-0.978232\pi$$
0.997663 0.0683333i $$-0.0217681\pi$$
$$812$$ 0 0
$$813$$ − 1.10752i − 0.0388425i
$$814$$ 0 0
$$815$$ 12.5778 0.440581
$$816$$ 0 0
$$817$$ 35.9071i 1.25623i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 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 $$-0.455736\pi$$
0.138612 + 0.990347i $$0.455736\pi$$
$$824$$ 0 0
$$825$$ − 4.12870i − 0.143743i
$$826$$ 0 0
$$827$$ 27.9648i 0.972432i 0.873839 + 0.486216i $$0.161623\pi$$
−0.873839 + 0.486216i $$0.838377\pi$$
$$828$$ 0 0
$$829$$ −27.6310 −0.959665 −0.479833 0.877360i $$-0.659303\pi$$
−0.479833 + 0.877360i $$0.659303\pi$$
$$830$$ 0 0
$$831$$ −6.53989 −0.226866
$$832$$ 0 0
$$833$$ 14.8388 0.514133
$$834$$ 0 0
$$835$$ −34.4698 −1.19288
$$836$$ 0 0
$$837$$ − 13.4843i − 0.466085i
$$838$$ 0 0
$$839$$ 28.6848i 0.990311i 0.868804 + 0.495155i $$0.164889\pi$$
−0.868804 + 0.495155i $$0.835111\pi$$
$$840$$ 0 0
$$841$$ −23.8528 −0.822509
$$842$$ 0 0
$$843$$ 3.59179i 0.123708i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 9.16315i 0.314849i
$$848$$ 0 0
$$849$$ 3.65220 0.125343
$$850$$ 0 0
$$851$$ 10.1250i 0.347080i
$$852$$ 0 0
$$853$$ 43.2078i 1.47941i 0.672934 + 0.739703i $$0.265034\pi$$
−0.672934 + 0.739703i $$0.734966\pi$$
$$854$$ 0 0
$$855$$ −22.7603 −0.778386
$$856$$ 0 0
$$857$$ −35.1685 −1.20133 −0.600667 0.799499i $$-0.705098\pi$$
−0.600667 + 0.799499i $$0.705098\pi$$
$$858$$ 0 0
$$859$$ −27.3793 −0.934168 −0.467084 0.884213i $$-0.654695\pi$$
−0.467084 + 0.884213i $$0.654695\pi$$
$$860$$ 0 0
$$861$$ −1.44876 −0.0493737
$$862$$ 0 0
$$863$$ 41.3913i 1.40898i 0.709715 + 0.704489i $$0.248824\pi$$
−0.709715 + 0.704489i $$0.751176\pi$$
$$864$$ 0 0
$$865$$ − 27.2465i − 0.926409i
$$866$$ 0 0
$$867$$ 6.13036 0.208198
$$868$$ 0 0
$$869$$ − 40.3116i − 1.36748i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 32.1075i − 1.08668i
$$874$$ 0 0
$$875$$ 23.4252 0.791916
$$876$$ 0 0
$$877$$ − 24.7472i − 0.835653i −0.908527 0.417826i $$-0.862792\pi$$
0.908527 0.417826i $$-0.137208\pi$$
$$878$$ 0 0
$$879$$ 13.5066i 0.455567i
$$880$$ 0 0
$$881$$ 28.5875 0.963137 0.481568 0.876409i $$-0.340067\pi$$
0.481568 + 0.876409i $$0.340067\pi$$
$$882$$ 0 0
$$883$$ 9.61702 0.323639 0.161819 0.986820i $$-0.448264\pi$$
0.161819 + 0.986820i $$0.448264\pi$$
$$884$$ 0 0
$$885$$ −9.78986 −0.329082
$$886$$ 0 0
$$887$$ −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
$$890$$ 0 0
$$891$$ − 16.1551i − 0.541217i
$$892$$ 0 0
$$893$$ 17.2664 0.577797
$$894$$ 0 0
$$895$$ − 8.70171i − 0.290866i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9.68473i 0.323004i
$$900$$ 0 0
$$901$$ 29.2392 0.974099
$$902$$ 0 0
$$903$$ 6.97823i 0.232221i
$$904$$ 0 0
$$905$$ − 6.90408i − 0.229500i
$$906$$ 0 0
$$907$$ −28.8364 −0.957496 −0.478748 0.877952i $$-0.658909\pi$$
−0.478748 + 0.877952i $$0.658909\pi$$
$$908$$ 0 0
$$909$$ 35.1685 1.16647
$$910$$ 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
$$914$$ 0 0
$$915$$ − 6.86831i − 0.227059i
$$916$$ 0 0
$$917$$ − 6.67264i − 0.220350i
$$918$$ 0 0
$$919$$ −8.87502 −0.292760 −0.146380 0.989228i $$-0.546762\pi$$
−0.146380 + 0.989228i $$0.546762\pi$$
$$920$$ 0 0
$$921$$ − 7.81030i − 0.257358i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 15.5985i − 0.512875i
$$926$$ 0 0
$$927$$ −24.6819 −0.810659
$$928$$ 0 0
$$929$$ 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
$$932$$ 0 0
$$933$$ −16.5211 −0.540877
$$934$$ 0 0
$$935$$ −19.5526 −0.639437
$$936$$ 0 0
$$937$$ 17.2644 0.564005 0.282002 0.959414i $$-0.409001\pi$$
0.282002 + 0.959414i $$0.409001\pi$$
$$938$$ 0 0
$$939$$ −4.15047 −0.135446
$$940$$ 0 0
$$941$$ 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
$$944$$ 0 0
$$945$$ −9.35258 −0.304240
$$946$$ 0 0
$$947$$ − 45.0146i − 1.46278i −0.681961 0.731389i $$-0.738872\pi$$
0.681961 0.731389i $$-0.261128\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 16.6655i 0.540415i
$$952$$ 0 0
$$953$$ 46.8859 1.51878 0.759391 0.650634i $$-0.225497\pi$$
0.759391 + 0.650634i $$0.225497\pi$$
$$954$$ 0 0
$$955$$ 26.6329i 0.861822i
$$956$$ 0 0
$$957$$ − 3.21685i − 0.103986i
$$958$$ 0 0
$$959$$ 1.62325 0.0524176
$$960$$ 0 0
$$961$$ 12.7778 0.412186
$$962$$ 0 0
$$963$$ −18.5743 −0.598550
$$964$$ 0 0
$$965$$ −8.74871 −0.281631
$$966$$ 0 0
$$967$$ − 6.29457i − 0.202420i −0.994865 0.101210i $$-0.967729\pi$$
0.994865 0.101210i $$-0.0322714\pi$$
$$968$$ 0 0
$$969$$ 17.1957i 0.552404i
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 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
$$980$$ 0 0
$$981$$ − 0.328421i − 0.0104857i
$$982$$ 0 0
$$983$$ − 55.7251i − 1.77736i −0.458532 0.888678i $$-0.651625\pi$$
0.458532 0.888678i $$-0.348375\pi$$
$$984$$ 0 0
$$985$$ −16.4959 −0.525602
$$986$$ 0 0
$$987$$ 3.35557 0.106809
$$988$$ 0 0
$$989$$ −11.5996 −0.368845
$$990$$ 0 0
$$991$$ 35.5512 1.12932 0.564661 0.825323i $$-0.309007\pi$$
0.564661 + 0.825323i $$0.309007\pi$$
$$992$$ 0 0
$$993$$ 8.72215i 0.276789i
$$994$$ 0 0
$$995$$ 20.0935i 0.637007i
$$996$$ 0 0
$$997$$ 6.61058 0.209359 0.104680 0.994506i $$-0.466618\pi$$
0.104680 + 0.994506i $$0.466618\pi$$
$$998$$ 0 0
$$999$$ 16.9215i 0.535374i
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 2704.2.f.o.337.3 6
4.3 odd 2 169.2.b.b.168.1 6
12.11 even 2 1521.2.b.l.1351.6 6
13.5 odd 4 2704.2.a.z.1.2 3
13.8 odd 4 2704.2.a.ba.1.2 3
13.12 even 2 inner 2704.2.f.o.337.4 6
52.3 odd 6 169.2.e.b.147.6 12
52.7 even 12 169.2.c.b.146.1 6
52.11 even 12 169.2.c.b.22.1 6
52.15 even 12 169.2.c.c.22.3 6
52.19 even 12 169.2.c.c.146.3 6
52.23 odd 6 169.2.e.b.147.1 12
52.31 even 4 169.2.a.b.1.1 3
52.35 odd 6 169.2.e.b.23.1 12
52.43 odd 6 169.2.e.b.23.6 12
52.47 even 4 169.2.a.c.1.3 yes 3
52.51 odd 2 169.2.b.b.168.6 6
156.47 odd 4 1521.2.a.o.1.1 3
156.83 odd 4 1521.2.a.r.1.3 3
156.155 even 2 1521.2.b.l.1351.1 6
260.99 even 4 4225.2.a.bb.1.1 3
260.239 even 4 4225.2.a.bg.1.3 3
364.83 odd 4 8281.2.a.bf.1.1 3
364.307 odd 4 8281.2.a.bj.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 52.31 even 4
169.2.a.c.1.3 yes 3 52.47 even 4
169.2.b.b.168.1 6 4.3 odd 2
169.2.b.b.168.6 6 52.51 odd 2
169.2.c.b.22.1 6 52.11 even 12
169.2.c.b.146.1 6 52.7 even 12
169.2.c.c.22.3 6 52.15 even 12
169.2.c.c.146.3 6 52.19 even 12
169.2.e.b.23.1 12 52.35 odd 6
169.2.e.b.23.6 12 52.43 odd 6
169.2.e.b.147.1 12 52.23 odd 6
169.2.e.b.147.6 12 52.3 odd 6
1521.2.a.o.1.1 3 156.47 odd 4
1521.2.a.r.1.3 3 156.83 odd 4
1521.2.b.l.1351.1 6 156.155 even 2
1521.2.b.l.1351.6 6 12.11 even 2
2704.2.a.z.1.2 3 13.5 odd 4
2704.2.a.ba.1.2 3 13.8 odd 4
2704.2.f.o.337.3 6 1.1 even 1 trivial
2704.2.f.o.337.4 6 13.12 even 2 inner
4225.2.a.bb.1.1 3 260.99 even 4
4225.2.a.bg.1.3 3 260.239 even 4
8281.2.a.bf.1.1 3 364.83 odd 4
8281.2.a.bj.1.3 3 364.307 odd 4