# Properties

 Label 2704.2.a.ba.1.2 Level $2704$ Weight $2$ Character 2704.1 Self dual yes Analytic conductor $21.592$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(1,2704)]

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, 0]))

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

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

chi := DirichletCharacter("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.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: $$21.5915487066$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 2704.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.554958 q^{3} +1.44504 q^{5} +2.04892 q^{7} -2.69202 q^{9} +O(q^{10})$$ $$q+0.554958 q^{3} +1.44504 q^{5} +2.04892 q^{7} -2.69202 q^{9} -2.55496 q^{11} +0.801938 q^{15} -5.29590 q^{17} -5.85086 q^{19} +1.13706 q^{21} +1.89008 q^{23} -2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} -4.26875 q^{31} -1.41789 q^{33} +2.96077 q^{35} -5.35690 q^{37} -1.27413 q^{41} -6.13706 q^{43} -3.89008 q^{45} -2.95108 q^{47} -2.80194 q^{49} -2.93900 q^{51} +5.52111 q^{53} -3.69202 q^{55} -3.24698 q^{57} -12.2078 q^{59} +8.56465 q^{61} -5.51573 q^{63} +0.576728 q^{67} +1.04892 q^{69} -4.59419 q^{71} +10.5526 q^{73} -1.61596 q^{75} -5.23490 q^{77} +15.7778 q^{79} +6.32304 q^{81} +7.72348 q^{83} -7.65279 q^{85} +1.25906 q^{87} -6.61356 q^{89} -2.36898 q^{93} -8.45473 q^{95} -11.9269 q^{97} +6.87800 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 4 q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 4 * q^5 - 3 * q^7 - 3 * q^9 $$3 q + 2 q^{3} + 4 q^{5} - 3 q^{7} - 3 q^{9} - 8 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} - 5 q^{31} - 10 q^{33} - 4 q^{35} - 12 q^{37} + 7 q^{41} - 13 q^{43} - 11 q^{45} - 18 q^{47} - 4 q^{49} + q^{51} + q^{53} - 6 q^{55} - 5 q^{57} - 19 q^{59} + 4 q^{61} - 4 q^{63} - q^{67} - 6 q^{69} - 27 q^{71} - 9 q^{73} - 15 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} - 7 q^{83} - 5 q^{85} + 18 q^{87} + 11 q^{89} - 22 q^{93} - 3 q^{95} - 7 q^{97} + q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 4 * q^5 - 3 * q^7 - 3 * q^9 - 8 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 - 2 * q^21 + 5 * q^23 - 5 * q^25 - q^27 - q^29 - 5 * q^31 - 10 * q^33 - 4 * q^35 - 12 * q^37 + 7 * q^41 - 13 * q^43 - 11 * q^45 - 18 * q^47 - 4 * q^49 + q^51 + q^53 - 6 * q^55 - 5 * q^57 - 19 * q^59 + 4 * q^61 - 4 * q^63 - q^67 - 6 * q^69 - 27 * q^71 - 9 * q^73 - 15 * q^75 + 8 * q^77 + 5 * q^79 - q^81 - 7 * q^83 - 5 * q^85 + 18 * q^87 + 11 * q^89 - 22 * q^93 - 3 * q^95 - 7 * q^97 + q^99

## 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
$$5$$ 1.44504 0.646242 0.323121 0.946358i $$-0.395268\pi$$
0.323121 + 0.946358i $$0.395268\pi$$
$$6$$ 0 0
$$7$$ 2.04892 0.774418 0.387209 0.921992i $$-0.373439\pi$$
0.387209 + 0.921992i $$0.373439\pi$$
$$8$$ 0 0
$$9$$ −2.69202 −0.897340
$$10$$ 0 0
$$11$$ −2.55496 −0.770349 −0.385174 0.922844i $$-0.625859\pi$$
−0.385174 + 0.922844i $$0.625859\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0.801938 0.207059
$$16$$ 0 0
$$17$$ −5.29590 −1.28444 −0.642222 0.766519i $$-0.721987\pi$$
−0.642222 + 0.766519i $$0.721987\pi$$
$$18$$ 0 0
$$19$$ −5.85086 −1.34228 −0.671139 0.741331i $$-0.734195\pi$$
−0.671139 + 0.741331i $$0.734195\pi$$
$$20$$ 0 0
$$21$$ 1.13706 0.248128
$$22$$ 0 0
$$23$$ 1.89008 0.394110 0.197055 0.980392i $$-0.436862\pi$$
0.197055 + 0.980392i $$0.436862\pi$$
$$24$$ 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
$$31$$ −4.26875 −0.766690 −0.383345 0.923605i $$-0.625228\pi$$
−0.383345 + 0.923605i $$0.625228\pi$$
$$32$$ 0 0
$$33$$ −1.41789 −0.246824
$$34$$ 0 0
$$35$$ 2.96077 0.500462
$$36$$ 0 0
$$37$$ −5.35690 −0.880668 −0.440334 0.897834i $$-0.645140\pi$$
−0.440334 + 0.897834i $$0.645140\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.27413 −0.198985 −0.0994926 0.995038i $$-0.531722\pi$$
−0.0994926 + 0.995038i $$0.531722\pi$$
$$42$$ 0 0
$$43$$ −6.13706 −0.935893 −0.467947 0.883757i $$-0.655006\pi$$
−0.467947 + 0.883757i $$0.655006\pi$$
$$44$$ 0 0
$$45$$ −3.89008 −0.579899
$$46$$ 0 0
$$47$$ −2.95108 −0.430460 −0.215230 0.976563i $$-0.569050\pi$$
−0.215230 + 0.976563i $$0.569050\pi$$
$$48$$ 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
$$57$$ −3.24698 −0.430073
$$58$$ 0 0
$$59$$ −12.2078 −1.58931 −0.794657 0.607059i $$-0.792349\pi$$
−0.794657 + 0.607059i $$0.792349\pi$$
$$60$$ 0 0
$$61$$ 8.56465 1.09659 0.548295 0.836285i $$-0.315277\pi$$
0.548295 + 0.836285i $$0.315277\pi$$
$$62$$ 0 0
$$63$$ −5.51573 −0.694917
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.576728 0.0704586 0.0352293 0.999379i $$-0.488784\pi$$
0.0352293 + 0.999379i $$0.488784\pi$$
$$68$$ 0 0
$$69$$ 1.04892 0.126275
$$70$$ 0 0
$$71$$ −4.59419 −0.545230 −0.272615 0.962123i $$-0.587888\pi$$
−0.272615 + 0.962123i $$0.587888\pi$$
$$72$$ 0 0
$$73$$ 10.5526 1.23508 0.617542 0.786538i $$-0.288128\pi$$
0.617542 + 0.786538i $$0.288128\pi$$
$$74$$ 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
$$83$$ 7.72348 0.847762 0.423881 0.905718i $$-0.360667\pi$$
0.423881 + 0.905718i $$0.360667\pi$$
$$84$$ 0 0
$$85$$ −7.65279 −0.830062
$$86$$ 0 0
$$87$$ 1.25906 0.134986
$$88$$ 0 0
$$89$$ −6.61356 −0.701036 −0.350518 0.936556i $$-0.613995\pi$$
−0.350518 + 0.936556i $$0.613995\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.36898 −0.245652
$$94$$ 0 0
$$95$$ −8.45473 −0.867437
$$96$$ 0 0
$$97$$ −11.9269 −1.21100 −0.605498 0.795847i $$-0.707026\pi$$
−0.605498 + 0.795847i $$0.707026\pi$$
$$98$$ 0 0
$$99$$ 6.87800 0.691265
$$100$$ 0 0
$$101$$ 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
$$105$$ 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
$$109$$ −0.121998 −0.0116853 −0.00584264 0.999983i $$-0.501860\pi$$
−0.00584264 + 0.999983i $$0.501860\pi$$
$$110$$ 0 0
$$111$$ −2.97285 −0.282171
$$112$$ 0 0
$$113$$ 7.30798 0.687477 0.343738 0.939065i $$-0.388307\pi$$
0.343738 + 0.939065i $$0.388307\pi$$
$$114$$ 0 0
$$115$$ 2.73125 0.254690
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −10.8509 −0.994696
$$120$$ 0 0
$$121$$ −4.47219 −0.406563
$$122$$ 0 0
$$123$$ −0.707087 −0.0637559
$$124$$ 0 0
$$125$$ −11.4330 −1.02260
$$126$$ 0 0
$$127$$ 18.9705 1.68336 0.841678 0.539980i $$-0.181568\pi$$
0.841678 + 0.539980i $$0.181568\pi$$
$$128$$ 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
$$135$$ −4.56465 −0.392862
$$136$$ 0 0
$$137$$ −0.792249 −0.0676864 −0.0338432 0.999427i $$-0.510775\pi$$
−0.0338432 + 0.999427i $$0.510775\pi$$
$$138$$ 0 0
$$139$$ 11.3394 0.961799 0.480899 0.876776i $$-0.340310\pi$$
0.480899 + 0.876776i $$0.340310\pi$$
$$140$$ 0 0
$$141$$ −1.63773 −0.137922
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.27844 0.272260
$$146$$ 0 0
$$147$$ −1.55496 −0.128251
$$148$$ 0 0
$$149$$ −8.40581 −0.688631 −0.344316 0.938854i $$-0.611889\pi$$
−0.344316 + 0.938854i $$0.611889\pi$$
$$150$$ 0 0
$$151$$ 14.1293 1.14983 0.574913 0.818215i $$-0.305036\pi$$
0.574913 + 0.818215i $$0.305036\pi$$
$$152$$ 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
$$161$$ 3.87263 0.305206
$$162$$ 0 0
$$163$$ 8.70410 0.681758 0.340879 0.940107i $$-0.389275\pi$$
0.340879 + 0.940107i $$0.389275\pi$$
$$164$$ 0 0
$$165$$ −2.04892 −0.159508
$$166$$ 0 0
$$167$$ −23.8538 −1.84587 −0.922933 0.384961i $$-0.874215\pi$$
−0.922933 + 0.384961i $$0.874215\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 15.7506 1.20448
$$172$$ 0 0
$$173$$ −18.8552 −1.43353 −0.716766 0.697314i $$-0.754378\pi$$
−0.716766 + 0.697314i $$0.754378\pi$$
$$174$$ 0 0
$$175$$ −5.96615 −0.450998
$$176$$ 0 0
$$177$$ −6.77479 −0.509224
$$178$$ 0 0
$$179$$ −6.02177 −0.450088 −0.225044 0.974349i $$-0.572253\pi$$
−0.225044 + 0.974349i $$0.572253\pi$$
$$180$$ 0 0
$$181$$ −4.77777 −0.355129 −0.177565 0.984109i $$-0.556822\pi$$
−0.177565 + 0.984109i $$0.556822\pi$$
$$182$$ 0 0
$$183$$ 4.75302 0.351353
$$184$$ 0 0
$$185$$ −7.74094 −0.569125
$$186$$ 0 0
$$187$$ 13.5308 0.989470
$$188$$ 0 0
$$189$$ −6.47219 −0.470782
$$190$$ 0 0
$$191$$ −18.4306 −1.33359 −0.666795 0.745242i $$-0.732334\pi$$
−0.666795 + 0.745242i $$0.732334\pi$$
$$192$$ 0 0
$$193$$ −6.05429 −0.435798 −0.217899 0.975971i $$-0.569920\pi$$
−0.217899 + 0.975971i $$0.569920\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.4155 0.813321 0.406660 0.913579i $$-0.366693\pi$$
0.406660 + 0.913579i $$0.366693\pi$$
$$198$$ 0 0
$$199$$ 13.9051 0.985710 0.492855 0.870111i $$-0.335953\pi$$
0.492855 + 0.870111i $$0.335953\pi$$
$$200$$ 0 0
$$201$$ 0.320060 0.0225753
$$202$$ 0 0
$$203$$ 4.64848 0.326259
$$204$$ 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
$$213$$ −2.54958 −0.174694
$$214$$ 0 0
$$215$$ −8.86831 −0.604814
$$216$$ 0 0
$$217$$ −8.74632 −0.593739
$$218$$ 0 0
$$219$$ 5.85623 0.395727
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −7.33513 −0.491196 −0.245598 0.969372i $$-0.578984\pi$$
−0.245598 + 0.969372i $$0.578984\pi$$
$$224$$ 0 0
$$225$$ 7.83877 0.522585
$$226$$ 0 0
$$227$$ −8.67456 −0.575751 −0.287875 0.957668i $$-0.592949\pi$$
−0.287875 + 0.957668i $$0.592949\pi$$
$$228$$ 0 0
$$229$$ 13.6866 0.904439 0.452219 0.891907i $$-0.350632\pi$$
0.452219 + 0.891907i $$0.350632\pi$$
$$230$$ 0 0
$$231$$ −2.90515 −0.191145
$$232$$ 0 0
$$233$$ −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
$$236$$ 0 0
$$237$$ 8.75600 0.568764
$$238$$ 0 0
$$239$$ −10.9239 −0.706611 −0.353305 0.935508i $$-0.614942\pi$$
−0.353305 + 0.935508i $$0.614942\pi$$
$$240$$ 0 0
$$241$$ −11.9148 −0.767502 −0.383751 0.923437i $$-0.625368\pi$$
−0.383751 + 0.923437i $$0.625368\pi$$
$$242$$ 0 0
$$243$$ 12.9855 0.833022
$$244$$ 0 0
$$245$$ −4.04892 −0.258676
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.28621 0.271627
$$250$$ 0 0
$$251$$ −22.3478 −1.41058 −0.705290 0.708919i $$-0.749183\pi$$
−0.705290 + 0.708919i $$0.749183\pi$$
$$252$$ 0 0
$$253$$ −4.82908 −0.303602
$$254$$ 0 0
$$255$$ −4.24698 −0.265956
$$256$$ 0 0
$$257$$ −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
$$261$$ −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
$$265$$ 7.97823 0.490099
$$266$$ 0 0
$$267$$ −3.67025 −0.224616
$$268$$ 0 0
$$269$$ 0.652793 0.0398015 0.0199007 0.999802i $$-0.493665\pi$$
0.0199007 + 0.999802i $$0.493665\pi$$
$$270$$ 0 0
$$271$$ −1.99569 −0.121229 −0.0606147 0.998161i $$-0.519306\pi$$
−0.0606147 + 0.998161i $$0.519306\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 7.43967 0.448629
$$276$$ 0 0
$$277$$ 11.7845 0.708061 0.354030 0.935234i $$-0.384811\pi$$
0.354030 + 0.935234i $$0.384811\pi$$
$$278$$ 0 0
$$279$$ 11.4916 0.687982
$$280$$ 0 0
$$281$$ 6.47219 0.386098 0.193049 0.981189i $$-0.438162\pi$$
0.193049 + 0.981189i $$0.438162\pi$$
$$282$$ 0 0
$$283$$ −6.58104 −0.391202 −0.195601 0.980684i $$-0.562666\pi$$
−0.195601 + 0.980684i $$0.562666\pi$$
$$284$$ 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
$$291$$ −6.61894 −0.388009
$$292$$ 0 0
$$293$$ 24.3381 1.42185 0.710924 0.703269i $$-0.248277\pi$$
0.710924 + 0.703269i $$0.248277\pi$$
$$294$$ 0 0
$$295$$ −17.6407 −1.02708
$$296$$ 0 0
$$297$$ 8.07069 0.468309
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.5743 −0.724773
$$302$$ 0 0
$$303$$ 7.24996 0.416500
$$304$$ 0 0
$$305$$ 12.3763 0.708663
$$306$$ 0 0
$$307$$ −14.0737 −0.803227 −0.401613 0.915809i $$-0.631550\pi$$
−0.401613 + 0.915809i $$0.631550\pi$$
$$308$$ 0 0
$$309$$ −5.08815 −0.289455
$$310$$ 0 0
$$311$$ 29.7700 1.68810 0.844051 0.536263i $$-0.180164\pi$$
0.844051 + 0.536263i $$0.180164\pi$$
$$312$$ 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
$$317$$ −30.0301 −1.68666 −0.843330 0.537396i $$-0.819408\pi$$
−0.843330 + 0.537396i $$0.819408\pi$$
$$318$$ 0 0
$$319$$ −5.79656 −0.324545
$$320$$ 0 0
$$321$$ 3.82908 0.213719
$$322$$ 0 0
$$323$$ 30.9855 1.72408
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −0.0677037 −0.00374402
$$328$$ 0 0
$$329$$ −6.04652 −0.333356
$$330$$ 0 0
$$331$$ −15.7168 −0.863872 −0.431936 0.901904i $$-0.642169\pi$$
−0.431936 + 0.901904i $$0.642169\pi$$
$$332$$ 0 0
$$333$$ 14.4209 0.790259
$$334$$ 0 0
$$335$$ 0.833397 0.0455333
$$336$$ 0 0
$$337$$ 1.95407 0.106445 0.0532224 0.998583i $$-0.483051\pi$$
0.0532224 + 0.998583i $$0.483051\pi$$
$$338$$ 0 0
$$339$$ 4.05562 0.220271
$$340$$ 0 0
$$341$$ 10.9065 0.590619
$$342$$ 0 0
$$343$$ −20.0834 −1.08440
$$344$$ 0 0
$$345$$ 1.51573 0.0816041
$$346$$ 0 0
$$347$$ 17.1250 0.919317 0.459659 0.888096i $$-0.347972\pi$$
0.459659 + 0.888096i $$0.347972\pi$$
$$348$$ 0 0
$$349$$ 10.4668 0.560276 0.280138 0.959960i $$-0.409620\pi$$
0.280138 + 0.959960i $$0.409620\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −15.5308 −0.826621 −0.413310 0.910590i $$-0.635628\pi$$
−0.413310 + 0.910590i $$0.635628\pi$$
$$354$$ 0 0
$$355$$ −6.63879 −0.352351
$$356$$ 0 0
$$357$$ −6.02177 −0.318706
$$358$$ 0 0
$$359$$ −21.4263 −1.13083 −0.565417 0.824805i $$-0.691285\pi$$
−0.565417 + 0.824805i $$0.691285\pi$$
$$360$$ 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
$$369$$ 3.42998 0.178557
$$370$$ 0 0
$$371$$ 11.3123 0.587305
$$372$$ 0 0
$$373$$ −12.5961 −0.652202 −0.326101 0.945335i $$-0.605735\pi$$
−0.326101 + 0.945335i $$0.605735\pi$$
$$374$$ 0 0
$$375$$ −6.34481 −0.327645
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.5386 0.849529 0.424765 0.905304i $$-0.360357\pi$$
0.424765 + 0.905304i $$0.360357\pi$$
$$380$$ 0 0
$$381$$ 10.5278 0.539356
$$382$$ 0 0
$$383$$ 7.53617 0.385080 0.192540 0.981289i $$-0.438327\pi$$
0.192540 + 0.981289i $$0.438327\pi$$
$$384$$ 0 0
$$385$$ −7.56465 −0.385530
$$386$$ 0 0
$$387$$ 16.5211 0.839815
$$388$$ 0 0
$$389$$ 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
$$392$$ 0 0
$$393$$ −1.80731 −0.0911670
$$394$$ 0 0
$$395$$ 22.7995 1.14717
$$396$$ 0 0
$$397$$ −1.35152 −0.0678308 −0.0339154 0.999425i $$-0.510798\pi$$
−0.0339154 + 0.999425i $$0.510798\pi$$
$$398$$ 0 0
$$399$$ −6.65279 −0.333056
$$400$$ 0 0
$$401$$ −0.579121 −0.0289199 −0.0144600 0.999895i $$-0.504603\pi$$
−0.0144600 + 0.999895i $$0.504603\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 9.13706 0.454024
$$406$$ 0 0
$$407$$ 13.6866 0.678422
$$408$$ 0 0
$$409$$ 15.1575 0.749490 0.374745 0.927128i $$-0.377730\pi$$
0.374745 + 0.927128i $$0.377730\pi$$
$$410$$ 0 0
$$411$$ −0.439665 −0.0216871
$$412$$ 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
$$421$$ 35.0465 1.70806 0.854032 0.520221i $$-0.174151\pi$$
0.854032 + 0.520221i $$0.174151\pi$$
$$422$$ 0 0
$$423$$ 7.94438 0.386269
$$424$$ 0 0
$$425$$ 15.4209 0.748022
$$426$$ 0 0
$$427$$ 17.5483 0.849220
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.2814 −1.65128 −0.825639 0.564199i $$-0.809185\pi$$
−0.825639 + 0.564199i $$0.809185\pi$$
$$432$$ 0 0
$$433$$ 13.7385 0.660232 0.330116 0.943940i $$-0.392912\pi$$
0.330116 + 0.943940i $$0.392912\pi$$
$$434$$ 0 0
$$435$$ 1.81940 0.0872334
$$436$$ 0 0
$$437$$ −11.0586 −0.529005
$$438$$ 0 0
$$439$$ −10.2403 −0.488742 −0.244371 0.969682i $$-0.578581\pi$$
−0.244371 + 0.969682i $$0.578581\pi$$
$$440$$ 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
$$447$$ −4.66487 −0.220641
$$448$$ 0 0
$$449$$ 12.9051 0.609032 0.304516 0.952507i $$-0.401505\pi$$
0.304516 + 0.952507i $$0.401505\pi$$
$$450$$ 0 0
$$451$$ 3.25534 0.153288
$$452$$ 0 0
$$453$$ 7.84117 0.368410
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4.65710 −0.217850 −0.108925 0.994050i $$-0.534741\pi$$
−0.108925 + 0.994050i $$0.534741\pi$$
$$458$$ 0 0
$$459$$ 16.7289 0.780836
$$460$$ 0 0
$$461$$ 31.5405 1.46899 0.734493 0.678616i $$-0.237420\pi$$
0.734493 + 0.678616i $$0.237420\pi$$
$$462$$ 0 0
$$463$$ 17.6504 0.820284 0.410142 0.912022i $$-0.365479\pi$$
0.410142 + 0.912022i $$0.365479\pi$$
$$464$$ 0 0
$$465$$ −3.42327 −0.158750
$$466$$ 0 0
$$467$$ 32.1726 1.48877 0.744385 0.667751i $$-0.232743\pi$$
0.744385 + 0.667751i $$0.232743\pi$$
$$468$$ 0 0
$$469$$ 1.18167 0.0545644
$$470$$ 0 0
$$471$$ −5.23490 −0.241211
$$472$$ 0 0
$$473$$ 15.6799 0.720964
$$474$$ 0 0
$$475$$ 17.0368 0.781704
$$476$$ 0 0
$$477$$ −14.8629 −0.680527
$$478$$ 0 0
$$479$$ −34.8998 −1.59461 −0.797306 0.603576i $$-0.793742\pi$$
−0.797306 + 0.603576i $$0.793742\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 2.14914 0.0977895
$$484$$ 0 0
$$485$$ −17.2349 −0.782596
$$486$$ 0 0
$$487$$ 41.8351 1.89573 0.947864 0.318676i $$-0.103238\pi$$
0.947864 + 0.318676i $$0.103238\pi$$
$$488$$ 0 0
$$489$$ 4.83041 0.218439
$$490$$ 0 0
$$491$$ −21.8455 −0.985873 −0.492936 0.870065i $$-0.664076\pi$$
−0.492936 + 0.870065i $$0.664076\pi$$
$$492$$ 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
$$499$$ 23.5472 1.05412 0.527058 0.849829i $$-0.323295\pi$$
0.527058 + 0.849829i $$0.323295\pi$$
$$500$$ 0 0
$$501$$ −13.2379 −0.591425
$$502$$ 0 0
$$503$$ 7.08682 0.315986 0.157993 0.987440i $$-0.449498\pi$$
0.157993 + 0.987440i $$0.449498\pi$$
$$504$$ 0 0
$$505$$ 18.8780 0.840060
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −7.61894 −0.337704 −0.168852 0.985641i $$-0.554006\pi$$
−0.168852 + 0.985641i $$0.554006\pi$$
$$510$$ 0 0
$$511$$ 21.6213 0.956471
$$512$$ 0 0
$$513$$ 18.4819 0.815995
$$514$$ 0 0
$$515$$ −13.2489 −0.583816
$$516$$ 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
$$525$$ −3.31096 −0.144502
$$526$$ 0 0
$$527$$ 22.6069 0.984770
$$528$$ 0 0
$$529$$ −19.4276 −0.844678
$$530$$ 0 0
$$531$$ 32.8635 1.42616
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 9.97046 0.431061
$$536$$ 0 0
$$537$$ −3.34183 −0.144211
$$538$$ 0 0
$$539$$ 7.15883 0.308353
$$540$$ 0 0
$$541$$ −34.4819 −1.48249 −0.741246 0.671234i $$-0.765765\pi$$
−0.741246 + 0.671234i $$0.765765\pi$$
$$542$$ 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
$$551$$ −13.2741 −0.565497
$$552$$ 0 0
$$553$$ 32.3274 1.37470
$$554$$ 0 0
$$555$$ −4.29590 −0.182351
$$556$$ 0 0
$$557$$ −1.27652 −0.0540879 −0.0270439 0.999634i $$-0.508609\pi$$
−0.0270439 + 0.999634i $$0.508609\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 7.50902 0.317031
$$562$$ 0 0
$$563$$ 9.12737 0.384673 0.192336 0.981329i $$-0.438393\pi$$
0.192336 + 0.981329i $$0.438393\pi$$
$$564$$ 0 0
$$565$$ 10.5603 0.444277
$$566$$ 0 0
$$567$$ 12.9554 0.544075
$$568$$ 0 0
$$569$$ −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
$$573$$ −10.2282 −0.427289
$$574$$ 0 0
$$575$$ −5.50365 −0.229518
$$576$$ 0 0
$$577$$ −45.1564 −1.87989 −0.939944 0.341330i $$-0.889123\pi$$
−0.939944 + 0.341330i $$0.889123\pi$$
$$578$$ 0 0
$$579$$ −3.35988 −0.139632
$$580$$ 0 0
$$581$$ 15.8248 0.656522
$$582$$ 0 0
$$583$$ −14.1062 −0.584219
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −32.4040 −1.33746 −0.668728 0.743507i $$-0.733161\pi$$
−0.668728 + 0.743507i $$0.733161\pi$$
$$588$$ 0 0
$$589$$ 24.9758 1.02911
$$590$$ 0 0
$$591$$ 6.33513 0.260592
$$592$$ 0 0
$$593$$ −36.6848 −1.50647 −0.753233 0.657754i $$-0.771507\pi$$
−0.753233 + 0.657754i $$0.771507\pi$$
$$594$$ 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
$$603$$ −1.55257 −0.0632253
$$604$$ 0 0
$$605$$ −6.46250 −0.262738
$$606$$ 0 0
$$607$$ −11.2161 −0.455248 −0.227624 0.973749i $$-0.573096\pi$$
−0.227624 + 0.973749i $$0.573096\pi$$
$$608$$ 0 0
$$609$$ 2.57971 0.104535
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20.8944 0.843917 0.421958 0.906615i $$-0.361343\pi$$
0.421958 + 0.906615i $$0.361343\pi$$
$$614$$ 0 0
$$615$$ −1.02177 −0.0412018
$$616$$ 0 0
$$617$$ −12.0992 −0.487094 −0.243547 0.969889i $$-0.578311\pi$$
−0.243547 + 0.969889i $$0.578311\pi$$
$$618$$ 0 0
$$619$$ −10.5526 −0.424143 −0.212072 0.977254i $$-0.568021\pi$$
−0.212072 + 0.977254i $$0.568021\pi$$
$$620$$ 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
$$629$$ 28.3696 1.13117
$$630$$ 0 0
$$631$$ −13.8514 −0.551417 −0.275709 0.961241i $$-0.588913\pi$$
−0.275709 + 0.961241i $$0.588913\pi$$
$$632$$ 0 0
$$633$$ 7.35019 0.292144
$$634$$ 0 0
$$635$$ 27.4131 1.08786
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 12.3676 0.489257
$$640$$ 0 0
$$641$$ 34.9608 1.38087 0.690434 0.723396i $$-0.257420\pi$$
0.690434 + 0.723396i $$0.257420\pi$$
$$642$$ 0 0
$$643$$ 33.3980 1.31709 0.658545 0.752541i $$-0.271172\pi$$
0.658545 + 0.752541i $$0.271172\pi$$
$$644$$ 0 0
$$645$$ −4.92154 −0.193786
$$646$$ 0 0
$$647$$ −2.32842 −0.0915397 −0.0457698 0.998952i $$-0.514574\pi$$
−0.0457698 + 0.998952i $$0.514574\pi$$
$$648$$ 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
$$655$$ −4.70602 −0.183879
$$656$$ 0 0
$$657$$ −28.4077 −1.10829
$$658$$ 0 0
$$659$$ −11.1395 −0.433932 −0.216966 0.976179i $$-0.569616\pi$$
−0.216966 + 0.976179i $$0.569616\pi$$
$$660$$ 0 0
$$661$$ 13.8498 0.538694 0.269347 0.963043i $$-0.413192\pi$$
0.269347 + 0.963043i $$0.413192\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −17.3230 −0.671759
$$666$$ 0 0
$$667$$ 4.28813 0.166037
$$668$$ 0 0
$$669$$ −4.07069 −0.157382
$$670$$ 0 0
$$671$$ −21.8823 −0.844757
$$672$$ 0 0
$$673$$ −6.52973 −0.251703 −0.125851 0.992049i $$-0.540166\pi$$
−0.125851 + 0.992049i $$0.540166\pi$$
$$674$$ 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
$$681$$ −4.81402 −0.184474
$$682$$ 0 0
$$683$$ 14.1793 0.542555 0.271277 0.962501i $$-0.412554\pi$$
0.271277 + 0.962501i $$0.412554\pi$$
$$684$$ 0 0
$$685$$ −1.14483 −0.0437418
$$686$$ 0 0
$$687$$ 7.59551 0.289787
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 30.7952 1.17151 0.585753 0.810490i $$-0.300799\pi$$
0.585753 + 0.810490i $$0.300799\pi$$
$$692$$ 0 0
$$693$$ 14.0925 0.535328
$$694$$ 0 0
$$695$$ 16.3860 0.621555
$$696$$ 0 0
$$697$$ 6.74764 0.255585
$$698$$ 0 0
$$699$$ −2.82371 −0.106802
$$700$$ 0 0
$$701$$ 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
$$704$$ 0 0
$$705$$ −2.36658 −0.0891307
$$706$$ 0 0
$$707$$ 26.7670 1.00668
$$708$$ 0 0
$$709$$ 47.6252 1.78860 0.894300 0.447467i $$-0.147674\pi$$
0.894300 + 0.447467i $$0.147674\pi$$
$$710$$ 0 0
$$711$$ −42.4741 −1.59290
$$712$$ 0 0
$$713$$ −8.06829 −0.302160
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.06233 −0.226402
$$718$$ 0 0
$$719$$ 5.99330 0.223512 0.111756 0.993736i $$-0.464352\pi$$
0.111756 + 0.993736i $$0.464352\pi$$
$$720$$ 0 0
$$721$$ −18.7855 −0.699610
$$722$$ 0 0
$$723$$ −6.61224 −0.245912
$$724$$ 0 0
$$725$$ −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
$$729$$ −11.7627 −0.435656
$$730$$ 0 0
$$731$$ 32.5013 1.20210
$$732$$ 0 0
$$733$$ −36.0646 −1.33208 −0.666038 0.745918i $$-0.732011\pi$$
−0.666038 + 0.745918i $$0.732011\pi$$
$$734$$ 0 0
$$735$$ −2.24698 −0.0828811
$$736$$ 0 0
$$737$$ −1.47352 −0.0542777
$$738$$ 0 0
$$739$$ 27.5254 1.01254 0.506269 0.862375i $$-0.331024\pi$$
0.506269 + 0.862375i $$0.331024\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −10.4692 −0.384078 −0.192039 0.981387i $$-0.561510\pi$$
−0.192039 + 0.981387i $$0.561510\pi$$
$$744$$ 0 0
$$745$$ −12.1468 −0.445023
$$746$$ 0 0
$$747$$ −20.7918 −0.760731
$$748$$ 0 0
$$749$$ 14.1371 0.516557
$$750$$ 0 0
$$751$$ −4.06770 −0.148433 −0.0742163 0.997242i $$-0.523646\pi$$
−0.0742163 + 0.997242i $$0.523646\pi$$
$$752$$ 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
$$759$$ −2.67994 −0.0972757
$$760$$ 0 0
$$761$$ 27.0237 0.979608 0.489804 0.871833i $$-0.337068\pi$$
0.489804 + 0.871833i $$0.337068\pi$$
$$762$$ 0 0
$$763$$ −0.249964 −0.00904929
$$764$$ 0 0
$$765$$ 20.6015 0.744848
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 37.9407 1.36818 0.684088 0.729400i $$-0.260201\pi$$
0.684088 + 0.729400i $$0.260201\pi$$
$$770$$ 0 0
$$771$$ −10.3556 −0.372947
$$772$$ 0 0
$$773$$ 16.3375 0.587620 0.293810 0.955864i $$-0.405077\pi$$
0.293810 + 0.955864i $$0.405077\pi$$
$$774$$ 0 0
$$775$$ 12.4300 0.446498
$$776$$ 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
$$785$$ −13.6310 −0.486512
$$786$$ 0 0
$$787$$ −18.6907 −0.666251 −0.333126 0.942882i $$-0.608103\pi$$
−0.333126 + 0.942882i $$0.608103\pi$$
$$788$$ 0 0
$$789$$ −7.99090 −0.284484
$$790$$ 0 0
$$791$$ 14.9734 0.532394
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 4.42758 0.157030
$$796$$ 0 0
$$797$$ 29.2519 1.03615 0.518077 0.855334i $$-0.326648\pi$$
0.518077 + 0.855334i $$0.326648\pi$$
$$798$$ 0 0
$$799$$ 15.6286 0.552901
$$800$$ 0 0
$$801$$ 17.8039 0.629068
$$802$$ 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
$$811$$ 3.89200 0.136667 0.0683333 0.997663i $$-0.478232\pi$$
0.0683333 + 0.997663i $$0.478232\pi$$
$$812$$ 0 0
$$813$$ −1.10752 −0.0388425
$$814$$ 0 0
$$815$$ 12.5778 0.440581
$$816$$ 0 0
$$817$$ 35.9071 1.25623
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −45.9982 −1.60535 −0.802674 0.596418i $$-0.796590\pi$$
−0.802674 + 0.596418i $$0.796590\pi$$
$$822$$ 0 0
$$823$$ −7.95300 −0.277224 −0.138612 0.990347i $$-0.544264\pi$$
−0.138612 + 0.990347i $$0.544264\pi$$
$$824$$ 0 0
$$825$$ 4.12870 0.143743
$$826$$ 0 0
$$827$$ 27.9648 0.972432 0.486216 0.873839i $$-0.338377\pi$$
0.486216 + 0.873839i $$0.338377\pi$$
$$828$$ 0 0
$$829$$ 27.6310 0.959665 0.479833 0.877360i $$-0.340697\pi$$
0.479833 + 0.877360i $$0.340697\pi$$
$$830$$ 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
$$837$$ 13.4843 0.466085
$$838$$ 0 0
$$839$$ 28.6848 0.990311 0.495155 0.868804i $$-0.335111\pi$$
0.495155 + 0.868804i $$0.335111\pi$$
$$840$$ 0 0
$$841$$ −23.8528 −0.822509
$$842$$ 0 0
$$843$$ 3.59179 0.123708
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −9.16315 −0.314849
$$848$$ 0 0
$$849$$ −3.65220 −0.125343
$$850$$ 0 0
$$851$$ −10.1250 −0.347080
$$852$$ 0 0
$$853$$ 43.2078 1.47941 0.739703 0.672934i $$-0.234966\pi$$
0.739703 + 0.672934i $$0.234966\pi$$
$$854$$ 0 0
$$855$$ 22.7603 0.778386
$$856$$ 0 0
$$857$$ 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 $$-0.654695\pi$$
−0.467084 + 0.884213i $$0.654695\pi$$
$$860$$ 0 0
$$861$$ −1.44876 −0.0493737
$$862$$ 0 0
$$863$$ −41.3913 −1.40898 −0.704489 0.709715i $$-0.748824\pi$$
−0.704489 + 0.709715i $$0.748824\pi$$
$$864$$ 0 0
$$865$$ −27.2465 −0.926409
$$866$$ 0 0
$$867$$ 6.13036 0.208198
$$868$$ 0 0
$$869$$ −40.3116 −1.36748
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 32.1075 1.08668
$$874$$ 0 0
$$875$$ −23.4252 −0.791916
$$876$$ 0 0
$$877$$ 24.7472 0.835653 0.417826 0.908527i $$-0.362792\pi$$
0.417826 + 0.908527i $$0.362792\pi$$
$$878$$ 0 0
$$879$$ 13.5066 0.455567
$$880$$ 0 0
$$881$$ −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
$$885$$ −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
$$889$$ 38.8689 1.30362
$$890$$ 0 0
$$891$$ −16.1551 −0.541217
$$892$$ 0 0
$$893$$ 17.2664 0.577797
$$894$$ 0 0
$$895$$ −8.70171 −0.290866
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −9.68473 −0.323004
$$900$$ 0 0
$$901$$ −29.2392 −0.974099
$$902$$ 0 0
$$903$$ −6.97823 −0.232221
$$904$$ 0 0
$$905$$ −6.90408 −0.229500
$$906$$ 0 0
$$907$$ 28.8364 0.957496 0.478748 0.877952i $$-0.341091\pi$$
0.478748 + 0.877952i $$0.341091\pi$$
$$908$$ 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
$$915$$ 6.86831 0.227059
$$916$$ 0 0
$$917$$ −6.67264 −0.220350
$$918$$ 0 0
$$919$$ −8.87502 −0.292760 −0.146380 0.989228i $$-0.546762\pi$$
−0.146380 + 0.989228i $$0.546762\pi$$
$$920$$ 0 0
$$921$$ −7.81030 −0.257358
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 15.5985 0.512875
$$926$$ 0 0
$$927$$ 24.6819 0.810659
$$928$$ 0 0
$$929$$ −24.2295 −0.794945 −0.397472 0.917614i $$-0.630113\pi$$
−0.397472 + 0.917614i $$0.630113\pi$$
$$930$$ 0 0
$$931$$ 16.3937 0.537283
$$932$$ 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
$$941$$ −4.34050 −0.141496 −0.0707482 0.997494i $$-0.522539\pi$$
−0.0707482 + 0.997494i $$0.522539\pi$$
$$942$$ 0 0
$$943$$ −2.40821 −0.0784220
$$944$$ 0 0
$$945$$ −9.35258 −0.304240
$$946$$ 0 0
$$947$$ −45.0146 −1.46278 −0.731389 0.681961i $$-0.761128\pi$$
−0.731389 + 0.681961i $$0.761128\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −16.6655 −0.540415
$$952$$ 0 0
$$953$$ −46.8859 −1.51878 −0.759391 0.650634i $$-0.774503\pi$$
−0.759391 + 0.650634i $$0.774503\pi$$
$$954$$ 0 0
$$955$$ −26.6329 −0.861822
$$956$$ 0 0
$$957$$ −3.21685 −0.103986
$$958$$ 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
$$967$$ 6.29457 0.202420 0.101210 0.994865i $$-0.467729\pi$$
0.101210 + 0.994865i $$0.467729\pi$$
$$968$$ 0 0
$$969$$ 17.1957 0.552404
$$970$$ 0 0
$$971$$ 41.8068 1.34165 0.670823 0.741618i $$-0.265941\pi$$
0.670823 + 0.741618i $$0.265941\pi$$
$$972$$ 0 0
$$973$$ 23.2336 0.744834
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −23.7530 −0.759926 −0.379963 0.925002i $$-0.624063\pi$$
−0.379963 + 0.925002i $$0.624063\pi$$
$$978$$ 0 0
$$979$$ 16.8974 0.540043
$$980$$ 0 0
$$981$$ 0.328421 0.0104857
$$982$$ 0 0
$$983$$ −55.7251 −1.77736 −0.888678 0.458532i $$-0.848375\pi$$
−0.888678 + 0.458532i $$0.848375\pi$$
$$984$$ 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
$$993$$ −8.72215 −0.276789
$$994$$ 0 0
$$995$$ 20.0935 0.637007
$$996$$ 0 0
$$997$$ 6.61058 0.209359 0.104680 0.994506i $$-0.466618\pi$$
0.104680 + 0.994506i $$0.466618\pi$$
$$998$$ 0 0
$$999$$ 16.9215 0.535374
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.a.ba.1.2 3
4.3 odd 2 169.2.a.c.1.3 yes 3
12.11 even 2 1521.2.a.o.1.1 3
13.5 odd 4 2704.2.f.o.337.3 6
13.8 odd 4 2704.2.f.o.337.4 6
13.12 even 2 2704.2.a.z.1.2 3
20.19 odd 2 4225.2.a.bb.1.1 3
28.27 even 2 8281.2.a.bj.1.3 3
52.3 odd 6 169.2.c.b.22.1 6
52.7 even 12 169.2.e.b.23.6 12
52.11 even 12 169.2.e.b.147.1 12
52.15 even 12 169.2.e.b.147.6 12
52.19 even 12 169.2.e.b.23.1 12
52.23 odd 6 169.2.c.c.22.3 6
52.31 even 4 169.2.b.b.168.1 6
52.35 odd 6 169.2.c.b.146.1 6
52.43 odd 6 169.2.c.c.146.3 6
52.47 even 4 169.2.b.b.168.6 6
52.51 odd 2 169.2.a.b.1.1 3
156.47 odd 4 1521.2.b.l.1351.1 6
156.83 odd 4 1521.2.b.l.1351.6 6
156.155 even 2 1521.2.a.r.1.3 3
260.259 odd 2 4225.2.a.bg.1.3 3
364.363 even 2 8281.2.a.bf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 52.51 odd 2
169.2.a.c.1.3 yes 3 4.3 odd 2
169.2.b.b.168.1 6 52.31 even 4
169.2.b.b.168.6 6 52.47 even 4
169.2.c.b.22.1 6 52.3 odd 6
169.2.c.b.146.1 6 52.35 odd 6
169.2.c.c.22.3 6 52.23 odd 6
169.2.c.c.146.3 6 52.43 odd 6
169.2.e.b.23.1 12 52.19 even 12
169.2.e.b.23.6 12 52.7 even 12
169.2.e.b.147.1 12 52.11 even 12
169.2.e.b.147.6 12 52.15 even 12
1521.2.a.o.1.1 3 12.11 even 2
1521.2.a.r.1.3 3 156.155 even 2
1521.2.b.l.1351.1 6 156.47 odd 4
1521.2.b.l.1351.6 6 156.83 odd 4
2704.2.a.z.1.2 3 13.12 even 2
2704.2.a.ba.1.2 3 1.1 even 1 trivial
2704.2.f.o.337.3 6 13.5 odd 4
2704.2.f.o.337.4 6 13.8 odd 4
4225.2.a.bb.1.1 3 20.19 odd 2
4225.2.a.bg.1.3 3 260.259 odd 2
8281.2.a.bf.1.1 3 364.363 even 2
8281.2.a.bj.1.3 3 28.27 even 2