# Properties

 Label 8281.2.a.bf.1.1 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.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: $$66.1241179138$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ 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 1.1 Root $$1.80194$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.44504 q^{5} -1.24698 q^{6} -2.35690 q^{8} -2.69202 q^{9} +O(q^{10})$$ $$q-2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.44504 q^{5} -1.24698 q^{6} -2.35690 q^{8} -2.69202 q^{9} -3.24698 q^{10} -2.55496 q^{11} +1.69202 q^{12} +0.801938 q^{15} -0.801938 q^{16} +5.29590 q^{17} +6.04892 q^{18} +5.85086 q^{19} +4.40581 q^{20} +5.74094 q^{22} -1.89008 q^{23} -1.30798 q^{24} -2.91185 q^{25} -3.15883 q^{27} +2.26875 q^{29} -1.80194 q^{30} +4.26875 q^{31} +6.51573 q^{32} -1.41789 q^{33} -11.8998 q^{34} -8.20775 q^{36} +5.35690 q^{37} -13.1468 q^{38} -3.40581 q^{40} -1.27413 q^{41} +6.13706 q^{43} -7.78986 q^{44} -3.89008 q^{45} +4.24698 q^{46} +2.95108 q^{47} -0.445042 q^{48} +6.54288 q^{50} +2.93900 q^{51} +5.52111 q^{53} +7.09783 q^{54} -3.69202 q^{55} +3.24698 q^{57} -5.09783 q^{58} +12.2078 q^{59} +2.44504 q^{60} -8.56465 q^{61} -9.59179 q^{62} -13.0368 q^{64} +3.18598 q^{66} +0.576728 q^{67} +16.1468 q^{68} -1.04892 q^{69} -4.59419 q^{71} +6.34481 q^{72} +10.5526 q^{73} -12.0368 q^{74} -1.61596 q^{75} +17.8388 q^{76} -15.7778 q^{79} -1.15883 q^{80} +6.32304 q^{81} +2.86294 q^{82} -7.72348 q^{83} +7.65279 q^{85} -13.7899 q^{86} +1.25906 q^{87} +6.02177 q^{88} -6.61356 q^{89} +8.74094 q^{90} -5.76271 q^{92} +2.36898 q^{93} -6.63102 q^{94} +8.45473 q^{95} +3.61596 q^{96} -11.9269 q^{97} +6.87800 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 2 q^{3} + 4 q^{5} + q^{6} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 2 * q^3 + 4 * q^5 + q^6 - 3 * q^8 - 3 * q^9 $$3 q - 2 q^{2} + 2 q^{3} + 4 q^{5} + q^{6} - 3 q^{8} - 3 q^{9} - 5 q^{10} - 8 q^{11} - 2 q^{15} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} + 3 q^{22} - 5 q^{23} - 9 q^{24} - 5 q^{25} - q^{27} - q^{29} - q^{30} + 5 q^{31} + 7 q^{32} - 10 q^{33} - 13 q^{34} - 7 q^{36} + 12 q^{37} - 12 q^{38} + 3 q^{40} + 7 q^{41} + 13 q^{43} - 11 q^{45} + 8 q^{46} + 18 q^{47} - q^{48} + q^{50} - q^{51} + q^{53} + 3 q^{54} - 6 q^{55} + 5 q^{57} + 3 q^{58} + 19 q^{59} + 7 q^{60} - 4 q^{61} - q^{62} - 11 q^{64} - 5 q^{66} - q^{67} + 21 q^{68} + 6 q^{69} - 27 q^{71} - 4 q^{72} - 9 q^{73} - 8 q^{74} - 15 q^{75} + 21 q^{76} - 5 q^{79} + 5 q^{80} - q^{81} + 14 q^{82} + 7 q^{83} + 5 q^{85} - 18 q^{86} + 18 q^{87} + 15 q^{88} + 11 q^{89} + 12 q^{90} + 22 q^{93} - 5 q^{94} + 3 q^{95} + 21 q^{96} - 7 q^{97} + q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 2 * q^3 + 4 * q^5 + q^6 - 3 * q^8 - 3 * q^9 - 5 * q^10 - 8 * q^11 - 2 * q^15 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 + 3 * q^22 - 5 * q^23 - 9 * q^24 - 5 * q^25 - q^27 - q^29 - q^30 + 5 * q^31 + 7 * q^32 - 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 3 * q^40 + 7 * q^41 + 13 * q^43 - 11 * q^45 + 8 * q^46 + 18 * q^47 - q^48 + q^50 - q^51 + q^53 + 3 * q^54 - 6 * q^55 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 7 * q^60 - 4 * q^61 - q^62 - 11 * q^64 - 5 * q^66 - q^67 + 21 * q^68 + 6 * q^69 - 27 * q^71 - 4 * q^72 - 9 * q^73 - 8 * q^74 - 15 * q^75 + 21 * q^76 - 5 * q^79 + 5 * q^80 - q^81 + 14 * q^82 + 7 * q^83 + 5 * q^85 - 18 * q^86 + 18 * q^87 + 15 * q^88 + 11 * q^89 + 12 * q^90 + 22 * q^93 - 5 * q^94 + 3 * q^95 + 21 * q^96 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.24698 −1.58885 −0.794427 0.607359i $$-0.792229\pi$$
−0.794427 + 0.607359i $$0.792229\pi$$
$$3$$ 0.554958 0.320405 0.160203 0.987084i $$-0.448785\pi$$
0.160203 + 0.987084i $$0.448785\pi$$
$$4$$ 3.04892 1.52446
$$5$$ 1.44504 0.646242 0.323121 0.946358i $$-0.395268\pi$$
0.323121 + 0.946358i $$0.395268\pi$$
$$6$$ −1.24698 −0.509077
$$7$$ 0 0
$$8$$ −2.35690 −0.833289
$$9$$ −2.69202 −0.897340
$$10$$ −3.24698 −1.02679
$$11$$ −2.55496 −0.770349 −0.385174 0.922844i $$-0.625859\pi$$
−0.385174 + 0.922844i $$0.625859\pi$$
$$12$$ 1.69202 0.488445
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 0.801938 0.207059
$$16$$ −0.801938 −0.200484
$$17$$ 5.29590 1.28444 0.642222 0.766519i $$-0.278013\pi$$
0.642222 + 0.766519i $$0.278013\pi$$
$$18$$ 6.04892 1.42574
$$19$$ 5.85086 1.34228 0.671139 0.741331i $$-0.265805\pi$$
0.671139 + 0.741331i $$0.265805\pi$$
$$20$$ 4.40581 0.985170
$$21$$ 0 0
$$22$$ 5.74094 1.22397
$$23$$ −1.89008 −0.394110 −0.197055 0.980392i $$-0.563138\pi$$
−0.197055 + 0.980392i $$0.563138\pi$$
$$24$$ −1.30798 −0.266990
$$25$$ −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 $$0.432443\pi$$
$$30$$ −1.80194 −0.328987
$$31$$ 4.26875 0.766690 0.383345 0.923605i $$-0.374772\pi$$
0.383345 + 0.923605i $$0.374772\pi$$
$$32$$ 6.51573 1.15183
$$33$$ −1.41789 −0.246824
$$34$$ −11.8998 −2.04079
$$35$$ 0 0
$$36$$ −8.20775 −1.36796
$$37$$ 5.35690 0.880668 0.440334 0.897834i $$-0.354860\pi$$
0.440334 + 0.897834i $$0.354860\pi$$
$$38$$ −13.1468 −2.13268
$$39$$ 0 0
$$40$$ −3.40581 −0.538506
$$41$$ −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 $$-0.344994\pi$$
0.467947 + 0.883757i $$0.344994\pi$$
$$44$$ −7.78986 −1.17437
$$45$$ −3.89008 −0.579899
$$46$$ 4.24698 0.626183
$$47$$ 2.95108 0.430460 0.215230 0.976563i $$-0.430950\pi$$
0.215230 + 0.976563i $$0.430950\pi$$
$$48$$ −0.445042 −0.0642363
$$49$$ 0 0
$$50$$ 6.54288 0.925302
$$51$$ 2.93900 0.411542
$$52$$ 0 0
$$53$$ 5.52111 0.758382 0.379191 0.925318i $$-0.376202\pi$$
0.379191 + 0.925318i $$0.376202\pi$$
$$54$$ 7.09783 0.965893
$$55$$ −3.69202 −0.497832
$$56$$ 0 0
$$57$$ 3.24698 0.430073
$$58$$ −5.09783 −0.669378
$$59$$ 12.2078 1.58931 0.794657 0.607059i $$-0.207651\pi$$
0.794657 + 0.607059i $$0.207651\pi$$
$$60$$ 2.44504 0.315654
$$61$$ −8.56465 −1.09659 −0.548295 0.836285i $$-0.684723\pi$$
−0.548295 + 0.836285i $$0.684723\pi$$
$$62$$ −9.59179 −1.21816
$$63$$ 0 0
$$64$$ −13.0368 −1.62960
$$65$$ 0 0
$$66$$ 3.18598 0.392167
$$67$$ 0.576728 0.0704586 0.0352293 0.999379i $$-0.488784\pi$$
0.0352293 + 0.999379i $$0.488784\pi$$
$$68$$ 16.1468 1.95808
$$69$$ −1.04892 −0.126275
$$70$$ 0 0
$$71$$ −4.59419 −0.545230 −0.272615 0.962123i $$-0.587888\pi$$
−0.272615 + 0.962123i $$0.587888\pi$$
$$72$$ 6.34481 0.747744
$$73$$ 10.5526 1.23508 0.617542 0.786538i $$-0.288128\pi$$
0.617542 + 0.786538i $$0.288128\pi$$
$$74$$ −12.0368 −1.39925
$$75$$ −1.61596 −0.186595
$$76$$ 17.8388 2.04625
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −15.7778 −1.77514 −0.887569 0.460674i $$-0.847608\pi$$
−0.887569 + 0.460674i $$0.847608\pi$$
$$80$$ −1.15883 −0.129562
$$81$$ 6.32304 0.702560
$$82$$ 2.86294 0.316158
$$83$$ −7.72348 −0.847762 −0.423881 0.905718i $$-0.639333\pi$$
−0.423881 + 0.905718i $$0.639333\pi$$
$$84$$ 0 0
$$85$$ 7.65279 0.830062
$$86$$ −13.7899 −1.48700
$$87$$ 1.25906 0.134986
$$88$$ 6.02177 0.641923
$$89$$ −6.61356 −0.701036 −0.350518 0.936556i $$-0.613995\pi$$
−0.350518 + 0.936556i $$0.613995\pi$$
$$90$$ 8.74094 0.921376
$$91$$ 0 0
$$92$$ −5.76271 −0.600804
$$93$$ 2.36898 0.245652
$$94$$ −6.63102 −0.683938
$$95$$ 8.45473 0.867437
$$96$$ 3.61596 0.369052
$$97$$ −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
$$100$$ −8.87800 −0.887800
$$101$$ −13.0640 −1.29991 −0.649957 0.759971i $$-0.725213\pi$$
−0.649957 + 0.759971i $$0.725213\pi$$
$$102$$ −6.60388 −0.653881
$$103$$ −9.16852 −0.903401 −0.451701 0.892170i $$-0.649182\pi$$
−0.451701 + 0.892170i $$0.649182\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −12.4058 −1.20496
$$107$$ −6.89977 −0.667026 −0.333513 0.942745i $$-0.608234\pi$$
−0.333513 + 0.942745i $$0.608234\pi$$
$$108$$ −9.63102 −0.926746
$$109$$ 0.121998 0.0116853 0.00584264 0.999983i $$-0.498140\pi$$
0.00584264 + 0.999983i $$0.498140\pi$$
$$110$$ 8.29590 0.790983
$$111$$ 2.97285 0.282171
$$112$$ 0 0
$$113$$ 7.30798 0.687477 0.343738 0.939065i $$-0.388307\pi$$
0.343738 + 0.939065i $$0.388307\pi$$
$$114$$ −7.29590 −0.683323
$$115$$ −2.73125 −0.254690
$$116$$ 6.91723 0.642249
$$117$$ 0 0
$$118$$ −27.4306 −2.52519
$$119$$ 0 0
$$120$$ −1.89008 −0.172540
$$121$$ −4.47219 −0.406563
$$122$$ 19.2446 1.74232
$$123$$ −0.707087 −0.0637559
$$124$$ 13.0151 1.16879
$$125$$ −11.4330 −1.02260
$$126$$ 0 0
$$127$$ −18.9705 −1.68336 −0.841678 0.539980i $$-0.818432\pi$$
−0.841678 + 0.539980i $$0.818432\pi$$
$$128$$ 16.2620 1.43738
$$129$$ 3.40581 0.299865
$$130$$ 0 0
$$131$$ −3.25667 −0.284536 −0.142268 0.989828i $$-0.545440\pi$$
−0.142268 + 0.989828i $$0.545440\pi$$
$$132$$ −4.32304 −0.376273
$$133$$ 0 0
$$134$$ −1.29590 −0.111948
$$135$$ −4.56465 −0.392862
$$136$$ −12.4819 −1.07031
$$137$$ 0.792249 0.0676864 0.0338432 0.999427i $$-0.489225\pi$$
0.0338432 + 0.999427i $$0.489225\pi$$
$$138$$ 2.35690 0.200632
$$139$$ 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
$$142$$ 10.3230 0.866291
$$143$$ 0 0
$$144$$ 2.15883 0.179903
$$145$$ 3.27844 0.272260
$$146$$ −23.7114 −1.96237
$$147$$ 0 0
$$148$$ 16.3327 1.34254
$$149$$ 8.40581 0.688631 0.344316 0.938854i $$-0.388111\pi$$
0.344316 + 0.938854i $$0.388111\pi$$
$$150$$ 3.63102 0.296472
$$151$$ 14.1293 1.14983 0.574913 0.818215i $$-0.305036\pi$$
0.574913 + 0.818215i $$0.305036\pi$$
$$152$$ −13.7899 −1.11851
$$153$$ −14.2567 −1.15258
$$154$$ 0 0
$$155$$ 6.16852 0.495468
$$156$$ 0 0
$$157$$ 9.43296 0.752832 0.376416 0.926451i $$-0.377156\pi$$
0.376416 + 0.926451i $$0.377156\pi$$
$$158$$ 35.4523 2.82044
$$159$$ 3.06398 0.242990
$$160$$ 9.41550 0.744361
$$161$$ 0 0
$$162$$ −14.2078 −1.11627
$$163$$ 8.70410 0.681758 0.340879 0.940107i $$-0.389275\pi$$
0.340879 + 0.940107i $$0.389275\pi$$
$$164$$ −3.88471 −0.303345
$$165$$ −2.04892 −0.159508
$$166$$ 17.3545 1.34697
$$167$$ 23.8538 1.84587 0.922933 0.384961i $$-0.125785\pi$$
0.922933 + 0.384961i $$0.125785\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −17.1957 −1.31885
$$171$$ −15.7506 −1.20448
$$172$$ 18.7114 1.42673
$$173$$ 18.8552 1.43353 0.716766 0.697314i $$-0.245622\pi$$
0.716766 + 0.697314i $$0.245622\pi$$
$$174$$ −2.82908 −0.214472
$$175$$ 0 0
$$176$$ 2.04892 0.154443
$$177$$ 6.77479 0.509224
$$178$$ 14.8605 1.11384
$$179$$ 6.02177 0.450088 0.225044 0.974349i $$-0.427747\pi$$
0.225044 + 0.974349i $$0.427747\pi$$
$$180$$ −11.8605 −0.884033
$$181$$ 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
$$184$$ 4.45473 0.328407
$$185$$ 7.74094 0.569125
$$186$$ −5.32304 −0.390305
$$187$$ −13.5308 −0.989470
$$188$$ 8.99761 0.656218
$$189$$ 0 0
$$190$$ −18.9976 −1.37823
$$191$$ 18.4306 1.33359 0.666795 0.745242i $$-0.267666\pi$$
0.666795 + 0.745242i $$0.267666\pi$$
$$192$$ −7.23490 −0.522134
$$193$$ 6.05429 0.435798 0.217899 0.975971i $$-0.430080\pi$$
0.217899 + 0.975971i $$0.430080\pi$$
$$194$$ 26.7995 1.92410
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −11.4155 −0.813321 −0.406660 0.913579i $$-0.633307\pi$$
−0.406660 + 0.913579i $$0.633307\pi$$
$$198$$ −15.4547 −1.09832
$$199$$ 13.9051 0.985710 0.492855 0.870111i $$-0.335953\pi$$
0.492855 + 0.870111i $$0.335953\pi$$
$$200$$ 6.86294 0.485283
$$201$$ 0.320060 0.0225753
$$202$$ 29.3545 2.06538
$$203$$ 0 0
$$204$$ 8.96077 0.627379
$$205$$ −1.84117 −0.128593
$$206$$ 20.6015 1.43537
$$207$$ 5.08815 0.353651
$$208$$ 0 0
$$209$$ −14.9487 −1.03402
$$210$$ 0 0
$$211$$ −13.2446 −0.911795 −0.455897 0.890032i $$-0.650682\pi$$
−0.455897 + 0.890032i $$0.650682\pi$$
$$212$$ 16.8334 1.15612
$$213$$ −2.54958 −0.174694
$$214$$ 15.5036 1.05981
$$215$$ 8.86831 0.604814
$$216$$ 7.44504 0.506571
$$217$$ 0 0
$$218$$ −0.274127 −0.0185662
$$219$$ 5.85623 0.395727
$$220$$ −11.2567 −0.758924
$$221$$ 0 0
$$222$$ −6.67994 −0.448328
$$223$$ 7.33513 0.491196 0.245598 0.969372i $$-0.421016\pi$$
0.245598 + 0.969372i $$0.421016\pi$$
$$224$$ 0 0
$$225$$ 7.83877 0.522585
$$226$$ −16.4209 −1.09230
$$227$$ 8.67456 0.575751 0.287875 0.957668i $$-0.407051\pi$$
0.287875 + 0.957668i $$0.407051\pi$$
$$228$$ 9.89977 0.655628
$$229$$ 13.6866 0.904439 0.452219 0.891907i $$-0.350632\pi$$
0.452219 + 0.891907i $$0.350632\pi$$
$$230$$ 6.13706 0.404666
$$231$$ 0 0
$$232$$ −5.34721 −0.351061
$$233$$ −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
$$236$$ 37.2204 2.42284
$$237$$ −8.75600 −0.568764
$$238$$ 0 0
$$239$$ −10.9239 −0.706611 −0.353305 0.935508i $$-0.614942\pi$$
−0.353305 + 0.935508i $$0.614942\pi$$
$$240$$ −0.643104 −0.0415122
$$241$$ −11.9148 −0.767502 −0.383751 0.923437i $$-0.625368\pi$$
−0.383751 + 0.923437i $$0.625368\pi$$
$$242$$ 10.0489 0.645969
$$243$$ 12.9855 0.833022
$$244$$ −26.1129 −1.67171
$$245$$ 0 0
$$246$$ 1.58881 0.101299
$$247$$ 0 0
$$248$$ −10.0610 −0.638874
$$249$$ −4.28621 −0.271627
$$250$$ 25.6896 1.62475
$$251$$ −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
$$254$$ 42.6262 2.67461
$$255$$ 4.24698 0.265956
$$256$$ −10.4668 −0.654176
$$257$$ 18.6601 1.16398 0.581992 0.813194i $$-0.302273\pi$$
0.581992 + 0.813194i $$0.302273\pi$$
$$258$$ −7.65279 −0.476442
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.10752 −0.378046
$$262$$ 7.31767 0.452087
$$263$$ 14.3991 0.887887 0.443944 0.896055i $$-0.353579\pi$$
0.443944 + 0.896055i $$0.353579\pi$$
$$264$$ 3.34183 0.205675
$$265$$ 7.97823 0.490099
$$266$$ 0 0
$$267$$ −3.67025 −0.224616
$$268$$ 1.75840 0.107411
$$269$$ −0.652793 −0.0398015 −0.0199007 0.999802i $$-0.506335\pi$$
−0.0199007 + 0.999802i $$0.506335\pi$$
$$270$$ 10.2567 0.624201
$$271$$ 1.99569 0.121229 0.0606147 0.998161i $$-0.480694\pi$$
0.0606147 + 0.998161i $$0.480694\pi$$
$$272$$ −4.24698 −0.257511
$$273$$ 0 0
$$274$$ −1.78017 −0.107544
$$275$$ 7.43967 0.448629
$$276$$ −3.19806 −0.192501
$$277$$ 11.7845 0.708061 0.354030 0.935234i $$-0.384811\pi$$
0.354030 + 0.935234i $$0.384811\pi$$
$$278$$ −25.4795 −1.52816
$$279$$ −11.4916 −0.687982
$$280$$ 0 0
$$281$$ −6.47219 −0.386098 −0.193049 0.981189i $$-0.561838\pi$$
−0.193049 + 0.981189i $$0.561838\pi$$
$$282$$ −3.67994 −0.219137
$$283$$ −6.58104 −0.391202 −0.195601 0.980684i $$-0.562666\pi$$
−0.195601 + 0.980684i $$0.562666\pi$$
$$284$$ −14.0073 −0.831180
$$285$$ 4.69202 0.277931
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −17.5405 −1.03358
$$289$$ 11.0465 0.649796
$$290$$ −7.36658 −0.432581
$$291$$ −6.61894 −0.388009
$$292$$ 32.1739 1.88284
$$293$$ 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
$$296$$ −12.6256 −0.733851
$$297$$ 8.07069 0.468309
$$298$$ −18.8877 −1.09413
$$299$$ 0 0
$$300$$ −4.92692 −0.284456
$$301$$ 0 0
$$302$$ −31.7482 −1.82691
$$303$$ −7.24996 −0.416500
$$304$$ −4.69202 −0.269106
$$305$$ −12.3763 −0.708663
$$306$$ 32.0344 1.83129
$$307$$ 14.0737 0.803227 0.401613 0.915809i $$-0.368450\pi$$
0.401613 + 0.915809i $$0.368450\pi$$
$$308$$ 0 0
$$309$$ −5.08815 −0.289455
$$310$$ −13.8605 −0.787226
$$311$$ 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 $$-0.432209\pi$$
0.211366 + 0.977407i $$0.432209\pi$$
$$314$$ −21.1957 −1.19614
$$315$$ 0 0
$$316$$ −48.1051 −2.70613
$$317$$ 30.0301 1.68666 0.843330 0.537396i $$-0.180592\pi$$
0.843330 + 0.537396i $$0.180592\pi$$
$$318$$ −6.88471 −0.386075
$$319$$ −5.79656 −0.324545
$$320$$ −18.8388 −1.05312
$$321$$ −3.82908 −0.213719
$$322$$ 0 0
$$323$$ 30.9855 1.72408
$$324$$ 19.2784 1.07102
$$325$$ 0 0
$$326$$ −19.5579 −1.08321
$$327$$ 0.0677037 0.00374402
$$328$$ 3.00298 0.165812
$$329$$ 0 0
$$330$$ 4.60388 0.253435
$$331$$ −15.7168 −0.863872 −0.431936 0.901904i $$-0.642169\pi$$
−0.431936 + 0.901904i $$0.642169\pi$$
$$332$$ −23.5483 −1.29238
$$333$$ −14.4209 −0.790259
$$334$$ −53.5991 −2.93281
$$335$$ 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
$$340$$ 23.3327 1.26540
$$341$$ −10.9065 −0.590619
$$342$$ 35.3913 1.91374
$$343$$ 0 0
$$344$$ −14.4644 −0.779869
$$345$$ −1.51573 −0.0816041
$$346$$ −42.3672 −2.27767
$$347$$ −17.1250 −0.919317 −0.459659 0.888096i $$-0.652028\pi$$
−0.459659 + 0.888096i $$0.652028\pi$$
$$348$$ 3.83877 0.205780
$$349$$ 10.4668 0.560276 0.280138 0.959960i $$-0.409620\pi$$
0.280138 + 0.959960i $$0.409620\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −16.6474 −0.887310
$$353$$ −15.5308 −0.826621 −0.413310 0.910590i $$-0.635628\pi$$
−0.413310 + 0.910590i $$0.635628\pi$$
$$354$$ −15.2228 −0.809084
$$355$$ −6.63879 −0.352351
$$356$$ −20.1642 −1.06870
$$357$$ 0 0
$$358$$ −13.5308 −0.715125
$$359$$ −21.4263 −1.13083 −0.565417 0.824805i $$-0.691285\pi$$
−0.565417 + 0.824805i $$0.691285\pi$$
$$360$$ 9.16852 0.483224
$$361$$ 15.2325 0.801711
$$362$$ −10.7356 −0.564249
$$363$$ −2.48188 −0.130265
$$364$$ 0 0
$$365$$ 15.2489 0.798164
$$366$$ 10.6799 0.558249
$$367$$ −34.3032 −1.79061 −0.895306 0.445452i $$-0.853043\pi$$
−0.895306 + 0.445452i $$0.853043\pi$$
$$368$$ 1.51573 0.0790129
$$369$$ 3.42998 0.178557
$$370$$ −17.3937 −0.904257
$$371$$ 0 0
$$372$$ 7.22282 0.374486
$$373$$ −12.5961 −0.652202 −0.326101 0.945335i $$-0.605735\pi$$
−0.326101 + 0.945335i $$0.605735\pi$$
$$374$$ 30.4034 1.57212
$$375$$ −6.34481 −0.327645
$$376$$ −6.95539 −0.358697
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.5386 0.849529 0.424765 0.905304i $$-0.360357\pi$$
0.424765 + 0.905304i $$0.360357\pi$$
$$380$$ 25.7778 1.32237
$$381$$ −10.5278 −0.539356
$$382$$ −41.4131 −2.11888
$$383$$ −7.53617 −0.385080 −0.192540 0.981289i $$-0.561673\pi$$
−0.192540 + 0.981289i $$0.561673\pi$$
$$384$$ 9.02475 0.460543
$$385$$ 0 0
$$386$$ −13.6039 −0.692419
$$387$$ −16.5211 −0.839815
$$388$$ −36.3642 −1.84611
$$389$$ 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
$$394$$ 25.6504 1.29225
$$395$$ −22.7995 −1.14717
$$396$$ 20.9705 1.05381
$$397$$ −1.35152 −0.0678308 −0.0339154 0.999425i $$-0.510798\pi$$
−0.0339154 + 0.999425i $$0.510798\pi$$
$$398$$ −31.2446 −1.56615
$$399$$ 0 0
$$400$$ 2.33513 0.116756
$$401$$ 0.579121 0.0289199 0.0144600 0.999895i $$-0.495397\pi$$
0.0144600 + 0.999895i $$0.495397\pi$$
$$402$$ −0.719169 −0.0358689
$$403$$ 0 0
$$404$$ −39.8310 −1.98167
$$405$$ 9.13706 0.454024
$$406$$ 0 0
$$407$$ −13.6866 −0.678422
$$408$$ −6.92692 −0.342934
$$409$$ 15.1575 0.749490 0.374745 0.927128i $$-0.377730\pi$$
0.374745 + 0.927128i $$0.377730\pi$$
$$410$$ 4.13706 0.204315
$$411$$ 0.439665 0.0216871
$$412$$ −27.9541 −1.37720
$$413$$ 0 0
$$414$$ −11.4330 −0.561899
$$415$$ −11.1608 −0.547860
$$416$$ 0 0
$$417$$ 6.29291 0.308165
$$418$$ 33.5894 1.64291
$$419$$ 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 $$-0.825849\pi$$
−0.854032 + 0.520221i $$0.825849\pi$$
$$422$$ 29.7603 1.44871
$$423$$ −7.94438 −0.386269
$$424$$ −13.0127 −0.631951
$$425$$ −15.4209 −0.748022
$$426$$ 5.72886 0.277564
$$427$$ 0 0
$$428$$ −21.0368 −1.01685
$$429$$ 0 0
$$430$$ −19.9269 −0.960961
$$431$$ −34.2814 −1.65128 −0.825639 0.564199i $$-0.809185\pi$$
−0.825639 + 0.564199i $$0.809185\pi$$
$$432$$ 2.53319 0.121878
$$433$$ −13.7385 −0.660232 −0.330116 0.943940i $$-0.607088\pi$$
−0.330116 + 0.943940i $$0.607088\pi$$
$$434$$ 0 0
$$435$$ 1.81940 0.0872334
$$436$$ 0.371961 0.0178137
$$437$$ −11.0586 −0.529005
$$438$$ −13.1588 −0.628753
$$439$$ −10.2403 −0.488742 −0.244371 0.969682i $$-0.578581\pi$$
−0.244371 + 0.969682i $$0.578581\pi$$
$$440$$ 8.70171 0.414838
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.1763 0.578513 0.289257 0.957252i $$-0.406592\pi$$
0.289257 + 0.957252i $$0.406592\pi$$
$$444$$ 9.06398 0.430158
$$445$$ −9.55688 −0.453039
$$446$$ −16.4819 −0.780440
$$447$$ 4.66487 0.220641
$$448$$ 0 0
$$449$$ −12.9051 −0.609032 −0.304516 0.952507i $$-0.598495\pi$$
−0.304516 + 0.952507i $$0.598495\pi$$
$$450$$ −17.6136 −0.830311
$$451$$ 3.25534 0.153288
$$452$$ 22.2814 1.04803
$$453$$ 7.84117 0.368410
$$454$$ −19.4916 −0.914785
$$455$$ 0 0
$$456$$ −7.65279 −0.358375
$$457$$ 4.65710 0.217850 0.108925 0.994050i $$-0.465259\pi$$
0.108925 + 0.994050i $$0.465259\pi$$
$$458$$ −30.7536 −1.43702
$$459$$ −16.7289 −0.780836
$$460$$ −8.32736 −0.388265
$$461$$ 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 $$0.365479\pi$$
$$464$$ −1.81940 −0.0844633
$$465$$ 3.42327 0.158750
$$466$$ 11.4330 0.529622
$$467$$ 32.1726 1.48877 0.744385 0.667751i $$-0.232743\pi$$
0.744385 + 0.667751i $$0.232743\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −9.58211 −0.441990
$$471$$ 5.23490 0.241211
$$472$$ −28.7724 −1.32436
$$473$$ −15.6799 −0.720964
$$474$$ 19.6746 0.903683
$$475$$ −17.0368 −0.781704
$$476$$ 0 0
$$477$$ −14.8629 −0.680527
$$478$$ 24.5459 1.12270
$$479$$ 34.8998 1.59461 0.797306 0.603576i $$-0.206258\pi$$
0.797306 + 0.603576i $$0.206258\pi$$
$$480$$ 5.22521 0.238497
$$481$$ 0 0
$$482$$ 26.7724 1.21945
$$483$$ 0 0
$$484$$ −13.6353 −0.619788
$$485$$ −17.2349 −0.782596
$$486$$ −29.1782 −1.32355
$$487$$ 41.8351 1.89573 0.947864 0.318676i $$-0.103238\pi$$
0.947864 + 0.318676i $$0.103238\pi$$
$$488$$ 20.1860 0.913776
$$489$$ 4.83041 0.218439
$$490$$ 0 0
$$491$$ 21.8455 0.985873 0.492936 0.870065i $$-0.335924\pi$$
0.492936 + 0.870065i $$0.335924\pi$$
$$492$$ −2.15585 −0.0971932
$$493$$ 12.0151 0.541131
$$494$$ 0 0
$$495$$ 9.93900 0.446725
$$496$$ −3.42327 −0.153709
$$497$$ 0 0
$$498$$ 9.63102 0.431576
$$499$$ 23.5472 1.05412 0.527058 0.849829i $$-0.323295\pi$$
0.527058 + 0.849829i $$0.323295\pi$$
$$500$$ −34.8582 −1.55890
$$501$$ 13.2379 0.591425
$$502$$ 50.2150 2.24121
$$503$$ 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
$$506$$ −10.8509 −0.482379
$$507$$ 0 0
$$508$$ −57.8394 −2.56621
$$509$$ −7.61894 −0.337704 −0.168852 0.985641i $$-0.554006\pi$$
−0.168852 + 0.985641i $$0.554006\pi$$
$$510$$ −9.54288 −0.422566
$$511$$ 0 0
$$512$$ −9.00538 −0.397985
$$513$$ −18.4819 −0.815995
$$514$$ −41.9288 −1.84940
$$515$$ −13.2489 −0.583816
$$516$$ 10.3840 0.457132
$$517$$ −7.53989 −0.331604
$$518$$ 0 0
$$519$$ 10.4638 0.459311
$$520$$ 0 0
$$521$$ 39.5133 1.73111 0.865555 0.500813i $$-0.166966\pi$$
0.865555 + 0.500813i $$0.166966\pi$$
$$522$$ 13.7235 0.600660
$$523$$ 15.8194 0.691734 0.345867 0.938284i $$-0.387585\pi$$
0.345867 + 0.938284i $$0.387585\pi$$
$$524$$ −9.92931 −0.433764
$$525$$ 0 0
$$526$$ −32.3545 −1.41072
$$527$$ 22.6069 0.984770
$$528$$ 1.13706 0.0494843
$$529$$ −19.4276 −0.844678
$$530$$ −17.9269 −0.778696
$$531$$ −32.8635 −1.42616
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 8.24698 0.356882
$$535$$ −9.97046 −0.431061
$$536$$ −1.35929 −0.0587123
$$537$$ 3.34183 0.144211
$$538$$ 1.46681 0.0632388
$$539$$ 0 0
$$540$$ −13.9172 −0.598902
$$541$$ 34.4819 1.48249 0.741246 0.671234i $$-0.234235\pi$$
0.741246 + 0.671234i $$0.234235\pi$$
$$542$$ −4.48427 −0.192616
$$543$$ 2.65146 0.113785
$$544$$ 34.5066 1.47946
$$545$$ 0.176292 0.00755152
$$546$$ 0 0
$$547$$ 36.8582 1.57594 0.787970 0.615713i $$-0.211132\pi$$
0.787970 + 0.615713i $$0.211132\pi$$
$$548$$ 2.41550 0.103185
$$549$$ 23.0562 0.984015
$$550$$ −16.7168 −0.712806
$$551$$ 13.2741 0.565497
$$552$$ 2.47219 0.105223
$$553$$ 0 0
$$554$$ −26.4795 −1.12501
$$555$$ 4.29590 0.182351
$$556$$ 34.5730 1.46622
$$557$$ 1.27652 0.0540879 0.0270439 0.999634i $$-0.491391\pi$$
0.0270439 + 0.999634i $$0.491391\pi$$
$$558$$ 25.8213 1.09310
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −7.50902 −0.317031
$$562$$ 14.5429 0.613454
$$563$$ 9.12737 0.384673 0.192336 0.981329i $$-0.438393\pi$$
0.192336 + 0.981329i $$0.438393\pi$$
$$564$$ 4.99330 0.210256
$$565$$ 10.5603 0.444277
$$566$$ 14.7875 0.621563
$$567$$ 0 0
$$568$$ 10.8280 0.454334
$$569$$ −5.72156 −0.239860 −0.119930 0.992782i $$-0.538267\pi$$
−0.119930 + 0.992782i $$0.538267\pi$$
$$570$$ −10.5429 −0.441593
$$571$$ 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
$$576$$ 35.0954 1.46231
$$577$$ −45.1564 −1.87989 −0.939944 0.341330i $$-0.889123\pi$$
−0.939944 + 0.341330i $$0.889123\pi$$
$$578$$ −24.8213 −1.03243
$$579$$ 3.35988 0.139632
$$580$$ 9.99569 0.415048
$$581$$ 0 0
$$582$$ 14.8726 0.616490
$$583$$ −14.1062 −0.584219
$$584$$ −24.8713 −1.02918
$$585$$ 0 0
$$586$$ −54.6872 −2.25911
$$587$$ 32.4040 1.33746 0.668728 0.743507i $$-0.266839\pi$$
0.668728 + 0.743507i $$0.266839\pi$$
$$588$$ 0 0
$$589$$ 24.9758 1.02911
$$590$$ −39.6383 −1.63188
$$591$$ −6.33513 −0.260592
$$592$$ −4.29590 −0.176560
$$593$$ −36.6848 −1.50647 −0.753233 0.657754i $$-0.771507\pi$$
−0.753233 + 0.657754i $$0.771507\pi$$
$$594$$ −18.1347 −0.744075
$$595$$ 0 0
$$596$$ 25.6286 1.04979
$$597$$ 7.71678 0.315827
$$598$$ 0 0
$$599$$ −9.99223 −0.408271 −0.204136 0.978943i $$-0.565438\pi$$
−0.204136 + 0.978943i $$0.565438\pi$$
$$600$$ 3.80864 0.155487
$$601$$ 1.81163 0.0738978 0.0369489 0.999317i $$-0.488236\pi$$
0.0369489 + 0.999317i $$0.488236\pi$$
$$602$$ 0 0
$$603$$ −1.55257 −0.0632253
$$604$$ 43.0790 1.75286
$$605$$ −6.46250 −0.262738
$$606$$ 16.2905 0.661757
$$607$$ −11.2161 −0.455248 −0.227624 0.973749i $$-0.573096\pi$$
−0.227624 + 0.973749i $$0.573096\pi$$
$$608$$ 38.1226 1.54608
$$609$$ 0 0
$$610$$ 27.8092 1.12596
$$611$$ 0 0
$$612$$ −43.4674 −1.75707
$$613$$ −20.8944 −0.843917 −0.421958 0.906615i $$-0.638657\pi$$
−0.421958 + 0.906615i $$0.638657\pi$$
$$614$$ −31.6233 −1.27621
$$615$$ −1.02177 −0.0412018
$$616$$ 0 0
$$617$$ 12.0992 0.487094 0.243547 0.969889i $$-0.421689\pi$$
0.243547 + 0.969889i $$0.421689\pi$$
$$618$$ 11.4330 0.459901
$$619$$ 10.5526 0.424143 0.212072 0.977254i $$-0.431979\pi$$
0.212072 + 0.977254i $$0.431979\pi$$
$$620$$ 18.8073 0.755320
$$621$$ 5.97046 0.239586
$$622$$ −66.8926 −2.68215
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.96184 −0.0784735
$$626$$ −16.8049 −0.671660
$$627$$ −8.29590 −0.331306
$$628$$ 28.7603 1.14766
$$629$$ 28.3696 1.13117
$$630$$ 0 0
$$631$$ −13.8514 −0.551417 −0.275709 0.961241i $$-0.588913\pi$$
−0.275709 + 0.961241i $$0.588913\pi$$
$$632$$ 37.1866 1.47920
$$633$$ −7.35019 −0.292144
$$634$$ −67.4771 −2.67986
$$635$$ −27.4131 −1.08786
$$636$$ 9.34183 0.370428
$$637$$ 0 0
$$638$$ 13.0248 0.515655
$$639$$ 12.3676 0.489257
$$640$$ 23.4993 0.928893
$$641$$ 34.9608 1.38087 0.690434 0.723396i $$-0.257420\pi$$
0.690434 + 0.723396i $$0.257420\pi$$
$$642$$ 8.60388 0.339568
$$643$$ −33.3980 −1.31709 −0.658545 0.752541i $$-0.728828\pi$$
−0.658545 + 0.752541i $$0.728828\pi$$
$$644$$ 0 0
$$645$$ 4.92154 0.193786
$$646$$ −69.6238 −2.73931
$$647$$ −2.32842 −0.0915397 −0.0457698 0.998952i $$-0.514574\pi$$
−0.0457698 + 0.998952i $$0.514574\pi$$
$$648$$ −14.9028 −0.585436
$$649$$ −31.1903 −1.22433
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 26.5381 1.03931
$$653$$ 14.5714 0.570221 0.285111 0.958495i $$-0.407970\pi$$
0.285111 + 0.958495i $$0.407970\pi$$
$$654$$ −0.152129 −0.00594871
$$655$$ −4.70602 −0.183879
$$656$$ 1.02177 0.0398934
$$657$$ −28.4077 −1.10829
$$658$$ 0 0
$$659$$ 11.1395 0.433932 0.216966 0.976179i $$-0.430384\pi$$
0.216966 + 0.976179i $$0.430384\pi$$
$$660$$ −6.24698 −0.243163
$$661$$ 13.8498 0.538694 0.269347 0.963043i $$-0.413192\pi$$
0.269347 + 0.963043i $$0.413192\pi$$
$$662$$ 35.3153 1.37257
$$663$$ 0 0
$$664$$ 18.2034 0.706430
$$665$$ 0 0
$$666$$ 32.4034 1.25561
$$667$$ −4.28813 −0.166037
$$668$$ 72.7284 2.81395
$$669$$ 4.07069 0.157382
$$670$$ −1.87263 −0.0723458
$$671$$ 21.8823 0.844757
$$672$$ 0 0
$$673$$ −6.52973 −0.251703 −0.125851 0.992049i $$-0.540166\pi$$
−0.125851 + 0.992049i $$0.540166\pi$$
$$674$$ −4.39075 −0.169125
$$675$$ 9.19806 0.354034
$$676$$ 0 0
$$677$$ 11.3104 0.434693 0.217346 0.976095i $$-0.430260\pi$$
0.217346 + 0.976095i $$0.430260\pi$$
$$678$$ −9.11290 −0.349979
$$679$$ 0 0
$$680$$ −18.0368 −0.691681
$$681$$ 4.81402 0.184474
$$682$$ 24.5066 0.938407
$$683$$ 14.1793 0.542555 0.271277 0.962501i $$-0.412554\pi$$
0.271277 + 0.962501i $$0.412554\pi$$
$$684$$ −48.0224 −1.83618
$$685$$ 1.14483 0.0437418
$$686$$ 0 0
$$687$$ 7.59551 0.289787
$$688$$ −4.92154 −0.187632
$$689$$ 0 0
$$690$$ 3.40581 0.129657
$$691$$ −30.7952 −1.17151 −0.585753 0.810490i $$-0.699201\pi$$
−0.585753 + 0.810490i $$0.699201\pi$$
$$692$$ 57.4878 2.18536
$$693$$ 0 0
$$694$$ 38.4795 1.46066
$$695$$ 16.3860 0.621555
$$696$$ −2.96748 −0.112482
$$697$$ −6.74764 −0.255585
$$698$$ −23.5187 −0.890196
$$699$$ −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
$$704$$ 33.3086 1.25536
$$705$$ 2.36658 0.0891307
$$706$$ 34.8974 1.31338
$$707$$ 0 0
$$708$$ 20.6558 0.776292
$$709$$ −47.6252 −1.78860 −0.894300 0.447467i $$-0.852326\pi$$
−0.894300 + 0.447467i $$0.852326\pi$$
$$710$$ 14.9172 0.559834
$$711$$ 42.4741 1.59290
$$712$$ 15.5875 0.584166
$$713$$ −8.06829 −0.302160
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18.3599 0.686141
$$717$$ −6.06233 −0.226402
$$718$$ 48.1444 1.79673
$$719$$ 5.99330 0.223512 0.111756 0.993736i $$-0.464352\pi$$
0.111756 + 0.993736i $$0.464352\pi$$
$$720$$ 3.11960 0.116261
$$721$$ 0 0
$$722$$ −34.2271 −1.27380
$$723$$ −6.61224 −0.245912
$$724$$ 14.5670 0.541380
$$725$$ −6.60627 −0.245351
$$726$$ 5.57673 0.206972
$$727$$ 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
$$730$$ −34.2640 −1.26817
$$731$$ 32.5013 1.20210
$$732$$ −14.4916 −0.535624
$$733$$ −36.0646 −1.33208 −0.666038 0.745918i $$-0.732011\pi$$
−0.666038 + 0.745918i $$0.732011\pi$$
$$734$$ 77.0786 2.84502
$$735$$ 0 0
$$736$$ −12.3153 −0.453947
$$737$$ −1.47352 −0.0542777
$$738$$ −7.70709 −0.283702
$$739$$ 27.5254 1.01254 0.506269 0.862375i $$-0.331024\pi$$
0.506269 + 0.862375i $$0.331024\pi$$
$$740$$ 23.6015 0.867608
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −10.4692 −0.384078 −0.192039 0.981387i $$-0.561510\pi$$
−0.192039 + 0.981387i $$0.561510\pi$$
$$744$$ −5.58343 −0.204699
$$745$$ 12.1468 0.445023
$$746$$ 28.3032 1.03625
$$747$$ 20.7918 0.760731
$$748$$ −41.2543 −1.50841
$$749$$ 0 0
$$750$$ 14.2567 0.520580
$$751$$ 4.06770 0.148433 0.0742163 0.997242i $$-0.476354\pi$$
0.0742163 + 0.997242i $$0.476354\pi$$
$$752$$ −2.36658 −0.0863005
$$753$$ −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 $$0.378900\pi$$
$$758$$ −37.1618 −1.34978
$$759$$ 2.67994 0.0972757
$$760$$ −19.9269 −0.722825
$$761$$ 27.0237 0.979608 0.489804 0.871833i $$-0.337068\pi$$
0.489804 + 0.871833i $$0.337068\pi$$
$$762$$ 23.6558 0.856958
$$763$$ 0 0
$$764$$ 56.1933 2.03300
$$765$$ −20.6015 −0.744848
$$766$$ 16.9336 0.611837
$$767$$ 0 0
$$768$$ −5.80864 −0.209601
$$769$$ 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
$$772$$ 18.4590 0.664355
$$773$$ 16.3375 0.587620 0.293810 0.955864i $$-0.405077\pi$$
0.293810 + 0.955864i $$0.405077\pi$$
$$774$$ 37.1226 1.33434
$$775$$ −12.4300 −0.446498
$$776$$ 28.1105 1.00911
$$777$$ 0 0
$$778$$ −79.8926 −2.86429
$$779$$ −7.45473 −0.267093
$$780$$ 0 0
$$781$$ 11.7380 0.420017
$$782$$ 22.4916 0.804297
$$783$$ −7.16660 −0.256114
$$784$$ 0 0
$$785$$ 13.6310 0.486512
$$786$$ 4.06100 0.144851
$$787$$ 18.6907 0.666251 0.333126 0.942882i $$-0.391897\pi$$
0.333126 + 0.942882i $$0.391897\pi$$
$$788$$ −34.8049 −1.23987
$$789$$ 7.99090 0.284484
$$790$$ 51.2301 1.82269
$$791$$ 0 0
$$792$$ −16.2107 −0.576023
$$793$$ 0 0
$$794$$ 3.03684 0.107773
$$795$$ 4.42758 0.157030
$$796$$ 42.3957 1.50267
$$797$$ −29.2519 −1.03615 −0.518077 0.855334i $$-0.673352\pi$$
−0.518077 + 0.855334i $$0.673352\pi$$
$$798$$ 0 0
$$799$$ 15.6286 0.552901
$$800$$ −18.9729 −0.670792
$$801$$ 17.8039 0.629068
$$802$$ −1.30127 −0.0459496
$$803$$ −26.9614 −0.951446
$$804$$ 0.975837 0.0344151
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −0.362273 −0.0127526
$$808$$ 30.7904 1.08320
$$809$$ −6.65087 −0.233832 −0.116916 0.993142i $$-0.537301\pi$$
−0.116916 + 0.993142i $$0.537301\pi$$
$$810$$ −20.5308 −0.721379
$$811$$ −3.89200 −0.136667 −0.0683333 0.997663i $$-0.521768\pi$$
−0.0683333 + 0.997663i $$0.521768\pi$$
$$812$$ 0 0
$$813$$ 1.10752 0.0388425
$$814$$ 30.7536 1.07791
$$815$$ 12.5778 0.440581
$$816$$ −2.35690 −0.0825079
$$817$$ 35.9071 1.25623
$$818$$ −34.0586 −1.19083
$$819$$ 0 0
$$820$$ −5.61356 −0.196034
$$821$$ 45.9982 1.60535 0.802674 0.596418i $$-0.203410\pi$$
0.802674 + 0.596418i $$0.203410\pi$$
$$822$$ −0.987918 −0.0344576
$$823$$ 7.95300 0.277224 0.138612 0.990347i $$-0.455736\pi$$
0.138612 + 0.990347i $$0.455736\pi$$
$$824$$ 21.6093 0.752794
$$825$$ 4.12870 0.143743
$$826$$ 0 0
$$827$$ 27.9648 0.972432 0.486216 0.873839i $$-0.338377\pi$$
0.486216 + 0.873839i $$0.338377\pi$$
$$828$$ 15.5133 0.539126
$$829$$ −27.6310 −0.959665 −0.479833 0.877360i $$-0.659303\pi$$
−0.479833 + 0.877360i $$0.659303\pi$$
$$830$$ 25.0780 0.870470
$$831$$ 6.53989 0.226866
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −14.1400 −0.489630
$$835$$ 34.4698 1.19288
$$836$$ −45.5773 −1.57632
$$837$$ −13.4843 −0.466085
$$838$$ −80.2699 −2.77288
$$839$$ −28.6848 −0.990311 −0.495155 0.868804i $$-0.664889\pi$$
−0.495155 + 0.868804i $$0.664889\pi$$
$$840$$ 0 0
$$841$$ −23.8528 −0.822509
$$842$$ 78.7488 2.71386
$$843$$ −3.59179 −0.123708
$$844$$ −40.3817 −1.38999
$$845$$ 0 0
$$846$$ 17.8509 0.613725
$$847$$ 0 0
$$848$$ −4.42758 −0.152044
$$849$$ −3.65220 −0.125343
$$850$$ 34.6504 1.18850
$$851$$ −10.1250 −0.347080
$$852$$ −7.77346 −0.266314
$$853$$ 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
$$856$$ 16.2620 0.555825
$$857$$ −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 $$0.654695\pi$$
$$860$$ 27.0388 0.922014
$$861$$ 0 0
$$862$$ 77.0297 2.62364
$$863$$ −41.3913 −1.40898 −0.704489 0.709715i $$-0.748824\pi$$
−0.704489 + 0.709715i $$0.748824\pi$$
$$864$$ −20.5821 −0.700217
$$865$$ 27.2465 0.926409
$$866$$ 30.8702 1.04901
$$867$$ 6.13036 0.208198
$$868$$ 0 0
$$869$$ 40.3116 1.36748
$$870$$ −4.08815 −0.138601
$$871$$ 0 0
$$872$$ −0.287536 −0.00973721
$$873$$ 32.1075 1.08668
$$874$$ 24.8485 0.840512
$$875$$ 0 0
$$876$$ 17.8552 0.603270
$$877$$ −24.7472 −0.835653 −0.417826 0.908527i $$-0.637208\pi$$
−0.417826 + 0.908527i $$0.637208\pi$$
$$878$$ 23.0097 0.776539
$$879$$ 13.5066 0.455567
$$880$$ 2.96077 0.0998076
$$881$$ 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
$$886$$ −27.3599 −0.919173
$$887$$ −15.9661 −0.536091 −0.268045 0.963406i $$-0.586378\pi$$
−0.268045 + 0.963406i $$0.586378\pi$$
$$888$$ −7.00670 −0.235130
$$889$$ 0 0
$$890$$ 21.4741 0.719814
$$891$$ −16.1551 −0.541217
$$892$$ 22.3642 0.748809
$$893$$ 17.2664 0.577797
$$894$$ −10.4819 −0.350566
$$895$$ 8.70171 0.290866
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 28.9976 0.967663
$$899$$ 9.68473 0.323004
$$900$$ 23.8998 0.796659
$$901$$ 29.2392 0.974099
$$902$$ −7.31468 −0.243552
$$903$$ 0 0
$$904$$ −17.2241 −0.572867
$$905$$ 6.90408 0.229500
$$906$$ −17.6189 −0.585350
$$907$$ −28.8364 −0.957496 −0.478748 0.877952i $$-0.658909\pi$$
−0.478748 + 0.877952i $$0.658909\pi$$
$$908$$ 26.4480 0.877709
$$909$$ 35.1685 1.16647
$$910$$ 0 0
$$911$$ 38.5633 1.27766 0.638830 0.769348i $$-0.279419\pi$$
0.638830 + 0.769348i $$0.279419\pi$$
$$912$$ −2.60388 −0.0862229
$$913$$ 19.7332 0.653073
$$914$$ −10.4644 −0.346132
$$915$$ −6.86831 −0.227059
$$916$$ 41.7294 1.37878
$$917$$ 0 0
$$918$$ 37.5894 1.24064
$$919$$ 8.87502 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$920$$ 6.43727 0.212231
$$921$$ 7.81030 0.257358
$$922$$ −70.8708 −2.33401
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −15.5985 −0.512875
$$926$$ −39.6601 −1.30331
$$927$$ 24.6819 0.810659
$$928$$ 14.7826 0.485261
$$929$$ −24.2295 −0.794945 −0.397472 0.917614i $$-0.630113\pi$$
−0.397472 + 0.917614i $$0.630113\pi$$
$$930$$ −7.69202 −0.252231
$$931$$ 0 0
$$932$$ −15.5133 −0.508156
$$933$$ 16.5211 0.540877
$$934$$ −72.2911 −2.36544
$$935$$ −19.5526 −0.639437
$$936$$ 0 0
$$937$$ −17.2644 −0.564005 −0.282002 0.959414i $$-0.590999\pi$$
−0.282002 + 0.959414i $$0.590999\pi$$
$$938$$ 0 0
$$939$$ 4.15047 0.135446
$$940$$ 13.0019 0.424076
$$941$$ −4.34050 −0.141496 −0.0707482 0.997494i $$-0.522539\pi$$
−0.0707482 + 0.997494i $$0.522539\pi$$
$$942$$ −11.7627 −0.383250
$$943$$ 2.40821 0.0784220
$$944$$ −9.78986 −0.318633
$$945$$ 0 0
$$946$$ 35.2325 1.14551
$$947$$ −45.0146 −1.46278 −0.731389 0.681961i $$-0.761128\pi$$
−0.731389 + 0.681961i $$0.761128\pi$$
$$948$$ −26.6963 −0.867057
$$949$$ 0 0
$$950$$ 38.2814 1.24201
$$951$$ 16.6655 0.540415
$$952$$ 0 0
$$953$$ −46.8859 −1.51878 −0.759391 0.650634i $$-0.774503\pi$$
−0.759391 + 0.650634i $$0.774503\pi$$
$$954$$ 33.3967 1.08126
$$955$$ 26.6329 0.861822
$$956$$ −33.3062 −1.07720
$$957$$ −3.21685 −0.103986
$$958$$ −78.4191 −2.53361
$$959$$ 0 0
$$960$$ −10.4547 −0.337425
$$961$$ −12.7778 −0.412186
$$962$$ 0 0
$$963$$ 18.5743 0.598550
$$964$$ −36.3274 −1.17003
$$965$$ 8.74871 0.281631
$$966$$ 0 0
$$967$$ 6.29457 0.202420 0.101210 0.994865i $$-0.467729\pi$$
0.101210 + 0.994865i $$0.467729\pi$$
$$968$$ 10.5405 0.338784
$$969$$ 17.1957 0.552404
$$970$$ 38.7265 1.24343
$$971$$ 41.8068 1.34165 0.670823 0.741618i $$-0.265941\pi$$
0.670823 + 0.741618i $$0.265941\pi$$
$$972$$ 39.5918 1.26991
$$973$$ 0 0
$$974$$ −94.0025 −3.01203
$$975$$ 0 0
$$976$$ 6.86831 0.219849
$$977$$ 23.7530 0.759926 0.379963 0.925002i $$-0.375937\pi$$
0.379963 + 0.925002i $$0.375937\pi$$
$$978$$ −10.8538 −0.347067
$$979$$ 16.8974 0.540043
$$980$$ 0 0
$$981$$ −0.328421 −0.0104857
$$982$$ −49.0863 −1.56641
$$983$$ 55.7251 1.77736 0.888678 0.458532i $$-0.151625\pi$$
0.888678 + 0.458532i $$0.151625\pi$$
$$984$$ 1.66653 0.0531270
$$985$$ −16.4959 −0.525602
$$986$$ −26.9976 −0.859779
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −11.5996 −0.368845
$$990$$ −22.3327 −0.709781
$$991$$ −35.5512 −1.12932 −0.564661 0.825323i $$-0.690993\pi$$
−0.564661 + 0.825323i $$0.690993\pi$$
$$992$$ 27.8140 0.883096
$$993$$ −8.72215 −0.276789
$$994$$ 0 0
$$995$$ 20.0935 0.637007
$$996$$ −13.0683 −0.414085
$$997$$ −6.61058 −0.209359 −0.104680 0.994506i $$-0.533382\pi$$
−0.104680 + 0.994506i $$0.533382\pi$$
$$998$$ −52.9101 −1.67484
$$999$$ −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 8281.2.a.bf.1.1 3
7.6 odd 2 169.2.a.b.1.1 3
13.12 even 2 8281.2.a.bj.1.3 3
21.20 even 2 1521.2.a.r.1.3 3
28.27 even 2 2704.2.a.z.1.2 3
35.34 odd 2 4225.2.a.bg.1.3 3
91.6 even 12 169.2.e.b.23.6 12
91.20 even 12 169.2.e.b.23.1 12
91.34 even 4 169.2.b.b.168.1 6
91.41 even 12 169.2.e.b.147.1 12
91.48 odd 6 169.2.c.c.146.3 6
91.55 odd 6 169.2.c.c.22.3 6
91.62 odd 6 169.2.c.b.22.1 6
91.69 odd 6 169.2.c.b.146.1 6
91.76 even 12 169.2.e.b.147.6 12
91.83 even 4 169.2.b.b.168.6 6
91.90 odd 2 169.2.a.c.1.3 yes 3
273.83 odd 4 1521.2.b.l.1351.1 6
273.125 odd 4 1521.2.b.l.1351.6 6
273.272 even 2 1521.2.a.o.1.1 3
364.83 odd 4 2704.2.f.o.337.4 6
364.307 odd 4 2704.2.f.o.337.3 6
364.363 even 2 2704.2.a.ba.1.2 3
455.454 odd 2 4225.2.a.bb.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 7.6 odd 2
169.2.a.c.1.3 yes 3 91.90 odd 2
169.2.b.b.168.1 6 91.34 even 4
169.2.b.b.168.6 6 91.83 even 4
169.2.c.b.22.1 6 91.62 odd 6
169.2.c.b.146.1 6 91.69 odd 6
169.2.c.c.22.3 6 91.55 odd 6
169.2.c.c.146.3 6 91.48 odd 6
169.2.e.b.23.1 12 91.20 even 12
169.2.e.b.23.6 12 91.6 even 12
169.2.e.b.147.1 12 91.41 even 12
169.2.e.b.147.6 12 91.76 even 12
1521.2.a.o.1.1 3 273.272 even 2
1521.2.a.r.1.3 3 21.20 even 2
1521.2.b.l.1351.1 6 273.83 odd 4
1521.2.b.l.1351.6 6 273.125 odd 4
2704.2.a.z.1.2 3 28.27 even 2
2704.2.a.ba.1.2 3 364.363 even 2
2704.2.f.o.337.3 6 364.307 odd 4
2704.2.f.o.337.4 6 364.83 odd 4
4225.2.a.bb.1.1 3 455.454 odd 2
4225.2.a.bg.1.3 3 35.34 odd 2
8281.2.a.bf.1.1 3 1.1 even 1 trivial
8281.2.a.bj.1.3 3 13.12 even 2