# Properties

 Label 8281.2.a.bj.1.3 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$1$$ 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: $$\Z[a_1, a_2]$$ 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 1.3 Root $$-1.24698$$ 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 $$-0.207771\pi$$
0.794427 + 0.607359i $$0.207771\pi$$
$$3$$ 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 $$-0.604732\pi$$
−0.323121 + 0.946358i $$0.604732\pi$$
$$6$$ 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 $$-0.374141\pi$$
0.385174 + 0.922844i $$0.374141\pi$$
$$12$$ 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 $$-0.734195\pi$$
−0.671139 + 0.741331i $$0.734195\pi$$
$$20$$ −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 $$-0.625228\pi$$
−0.383345 + 0.923605i $$0.625228\pi$$
$$32$$ −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 $$-0.645140\pi$$
−0.440334 + 0.897834i $$0.645140\pi$$
$$38$$ −13.1468 −2.13268
$$39$$ 0 0
$$40$$ −3.40581 −0.538506
$$41$$ 1.27413 0.198985 0.0994926 0.995038i $$-0.468278\pi$$
0.0994926 + 0.995038i $$0.468278\pi$$
$$42$$ 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 $$-0.569050\pi$$
−0.215230 + 0.976563i $$0.569050\pi$$
$$48$$ −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 $$-0.792349\pi$$
−0.794657 + 0.607059i $$0.792349\pi$$
$$60$$ −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 $$-0.511216\pi$$
−0.0352293 + 0.999379i $$0.511216\pi$$
$$68$$ 16.1468 1.95808
$$69$$ −1.04892 −0.126275
$$70$$ 0 0
$$71$$ 4.59419 0.545230 0.272615 0.962123i $$-0.412112\pi$$
0.272615 + 0.962123i $$0.412112\pi$$
$$72$$ −6.34481 −0.747744
$$73$$ −10.5526 −1.23508 −0.617542 0.786538i $$-0.711872\pi$$
−0.617542 + 0.786538i $$0.711872\pi$$
$$74$$ −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 $$-0.360667\pi$$
0.423881 + 0.905718i $$0.360667\pi$$
$$84$$ 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 $$-0.386005\pi$$
0.350518 + 0.936556i $$0.386005\pi$$
$$90$$ 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 $$-0.292974\pi$$
0.605498 + 0.795847i $$0.292974\pi$$
$$98$$ 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 $$-0.501860\pi$$
−0.00584264 + 0.999983i $$0.501860\pi$$
$$110$$ −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 $$-0.510775\pi$$
−0.0338432 + 0.999427i $$0.510775\pi$$
$$138$$ −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 $$-0.611889\pi$$
−0.344316 + 0.938854i $$0.611889\pi$$
$$150$$ −3.63102 −0.296472
$$151$$ −14.1293 −1.14983 −0.574913 0.818215i $$-0.694964\pi$$
−0.574913 + 0.818215i $$0.694964\pi$$
$$152$$ −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 $$-0.610725\pi$$
−0.340879 + 0.940107i $$0.610725\pi$$
$$164$$ 3.88471 0.303345
$$165$$ −2.04892 −0.159508
$$166$$ 17.3545 1.34697
$$167$$ −23.8538 −1.84587 −0.922933 0.384961i $$-0.874215\pi$$
−0.922933 + 0.384961i $$0.874215\pi$$
$$168$$ 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 $$-0.569920\pi$$
−0.217899 + 0.975971i $$0.569920\pi$$
$$194$$ 26.7995 1.92410
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.4155 0.813321 0.406660 0.913579i $$-0.366693\pi$$
0.406660 + 0.913579i $$0.366693\pi$$
$$198$$ −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 $$-0.578984\pi$$
−0.245598 + 0.969372i $$0.578984\pi$$
$$224$$ 0 0
$$225$$ 7.83877 0.522585
$$226$$ 16.4209 1.09230
$$227$$ −8.67456 −0.575751 −0.287875 0.957668i $$-0.592949\pi$$
−0.287875 + 0.957668i $$0.592949\pi$$
$$228$$ −9.89977 −0.655628
$$229$$ −13.6866 −0.904439 −0.452219 0.891907i $$-0.649368\pi$$
−0.452219 + 0.891907i $$0.649368\pi$$
$$230$$ 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 $$-0.385058\pi$$
0.353305 + 0.935508i $$0.385058\pi$$
$$240$$ 0.643104 0.0415122
$$241$$ 11.9148 0.767502 0.383751 0.923437i $$-0.374632\pi$$
0.383751 + 0.923437i $$0.374632\pi$$
$$242$$ −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 $$-0.519306\pi$$
−0.0606147 + 0.998161i $$0.519306\pi$$
$$272$$ −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 $$-0.438162\pi$$
0.193049 + 0.981189i $$0.438162\pi$$
$$282$$ −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 $$-0.751723\pi$$
−0.710924 + 0.703269i $$0.751723\pi$$
$$294$$ 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 $$-0.631550\pi$$
−0.401613 + 0.915809i $$0.631550\pi$$
$$308$$ 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 $$-0.819408\pi$$
−0.843330 + 0.537396i $$0.819408\pi$$
$$318$$ 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 $$-0.357831\pi$$
0.431936 + 0.901904i $$0.357831\pi$$
$$332$$ 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 $$-0.590380\pi$$
−0.280138 + 0.959960i $$0.590380\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −16.6474 −0.887310
$$353$$ 15.5308 0.826621 0.413310 0.910590i $$-0.364372\pi$$
0.413310 + 0.910590i $$0.364372\pi$$
$$354$$ −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 $$-0.308715\pi$$
0.565417 + 0.824805i $$0.308715\pi$$
$$360$$ 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 $$-0.639643\pi$$
−0.424765 + 0.905304i $$0.639643\pi$$
$$380$$ 25.7778 1.32237
$$381$$ −10.5278 −0.539356
$$382$$ 41.4131 2.11888
$$383$$ 7.53617 0.385080 0.192540 0.981289i $$-0.438327\pi$$
0.192540 + 0.981289i $$0.438327\pi$$
$$384$$ −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 $$-0.489202\pi$$
0.0339154 + 0.999425i $$0.489202\pi$$
$$398$$ 31.2446 1.56615
$$399$$ 0 0
$$400$$ 2.33513 0.116756
$$401$$ −0.579121 −0.0289199 −0.0144600 0.999895i $$-0.504603\pi$$
−0.0144600 + 0.999895i $$0.504603\pi$$
$$402$$ −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 $$-0.622270\pi$$
−0.374745 + 0.927128i $$0.622270\pi$$
$$410$$ −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 $$-0.174151\pi$$
0.854032 + 0.520221i $$0.174151\pi$$
$$422$$ −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 $$-0.190815\pi$$
0.825639 + 0.564199i $$0.190815\pi$$
$$432$$ 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 $$-0.401505\pi$$
0.304516 + 0.952507i $$0.401505\pi$$
$$450$$ 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 $$-0.534741\pi$$
−0.108925 + 0.994050i $$0.534741\pi$$
$$458$$ −30.7536 −1.43702
$$459$$ −16.7289 −0.780836
$$460$$ 8.32736 0.388265
$$461$$ −31.5405 −1.46899 −0.734493 0.678616i $$-0.762580\pi$$
−0.734493 + 0.678616i $$0.762580\pi$$
$$462$$ 0 0
$$463$$ −17.6504 −0.820284 −0.410142 0.912022i $$-0.634521\pi$$
−0.410142 + 0.912022i $$0.634521\pi$$
$$464$$ −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 $$-0.793742\pi$$
−0.797306 + 0.603576i $$0.793742\pi$$
$$480$$ 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 $$-0.896762\pi$$
−0.947864 + 0.318676i $$0.896762\pi$$
$$488$$ −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 $$-0.676705\pi$$
−0.527058 + 0.849829i $$0.676705\pi$$
$$500$$ 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 $$-0.445994\pi$$
0.168852 + 0.985641i $$0.445994\pi$$
$$510$$ −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 $$-0.765765\pi$$
−0.741246 + 0.671234i $$0.765765\pi$$
$$542$$ −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 $$-0.508609\pi$$
−0.0270439 + 0.999634i $$0.508609\pi$$
$$558$$ 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 $$-0.110877\pi$$
0.939944 + 0.341330i $$0.110877\pi$$
$$578$$ 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 $$-0.733161\pi$$
−0.668728 + 0.743507i $$0.733161\pi$$
$$588$$ 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 $$-0.228493\pi$$
0.753233 + 0.657754i $$0.228493\pi$$
$$594$$ −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 $$-0.361343\pi$$
0.421958 + 0.906615i $$0.361343\pi$$
$$614$$ −31.6233 −1.27621
$$615$$ −1.02177 −0.0412018
$$616$$ 0 0
$$617$$ −12.0992 −0.487094 −0.243547 0.969889i $$-0.578311\pi$$
−0.243547 + 0.969889i $$0.578311\pi$$
$$618$$ −11.4330 −0.459901
$$619$$ −10.5526 −0.424143 −0.212072 0.977254i $$-0.568021\pi$$
−0.212072 + 0.977254i $$0.568021\pi$$
$$620$$ 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 $$-0.411087\pi$$
0.275709 + 0.961241i $$0.411087\pi$$
$$632$$ −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 $$-0.271172\pi$$
0.658545 + 0.752541i $$0.271172\pi$$
$$644$$ 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 $$-0.586808\pi$$
−0.269347 + 0.963043i $$0.586808\pi$$
$$662$$ 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 $$-0.587446\pi$$
−0.271277 + 0.962501i $$0.587446\pi$$
$$684$$ 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 $$-0.300799\pi$$
0.585753 + 0.810490i $$0.300799\pi$$
$$692$$ 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 $$-0.147674\pi$$
0.894300 + 0.447467i $$0.147674\pi$$
$$710$$ −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 $$-0.267989\pi$$
0.666038 + 0.745918i $$0.267989\pi$$
$$734$$ −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 $$-0.668976\pi$$
−0.506269 + 0.862375i $$0.668976\pi$$
$$740$$ 23.6015 0.867608
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10.4692 0.384078 0.192039 0.981387i $$-0.438490\pi$$
0.192039 + 0.981387i $$0.438490\pi$$
$$744$$ −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 $$-0.662932\pi$$
−0.489804 + 0.871833i $$0.662932\pi$$
$$762$$ −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 $$-0.739799\pi$$
−0.684088 + 0.729400i $$0.739799\pi$$
$$770$$ 0 0
$$771$$ 10.3556 0.372947
$$772$$ −18.4590 −0.664355
$$773$$ −16.3375 −0.587620 −0.293810 0.955864i $$-0.594923\pi$$
−0.293810 + 0.955864i $$0.594923\pi$$
$$774$$ −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 $$-0.608103\pi$$
−0.333126 + 0.942882i $$0.608103\pi$$
$$788$$ 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 $$-0.478232\pi$$
0.0683333 + 0.997663i $$0.478232\pi$$
$$812$$ 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 $$-0.796590\pi$$
−0.802674 + 0.596418i $$0.796590\pi$$
$$822$$ −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 $$-0.661623\pi$$
−0.486216 + 0.873839i $$0.661623\pi$$
$$828$$ 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 $$-0.335111\pi$$
0.495155 + 0.868804i $$0.335111\pi$$
$$840$$ 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 $$-0.765034\pi$$
−0.739703 + 0.672934i $$0.765034\pi$$
$$854$$ 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 $$-0.251176\pi$$
0.704489 + 0.709715i $$0.251176\pi$$
$$864$$ 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 $$-0.362792\pi$$
0.417826 + 0.908527i $$0.362792\pi$$
$$878$$ −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 $$-0.369887\pi$$
0.397472 + 0.917614i $$0.369887\pi$$
$$930$$ 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 $$-0.477461\pi$$
0.0707482 + 0.997494i $$0.477461\pi$$
$$942$$ 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 $$-0.238872\pi$$
0.731389 + 0.681961i $$0.238872\pi$$
$$948$$ −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 $$-0.532271\pi$$
−0.101210 + 0.994865i $$0.532271\pi$$
$$968$$ −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 $$-0.624063\pi$$
−0.379963 + 0.925002i $$0.624063\pi$$
$$978$$ −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 $$-0.848375\pi$$
−0.888678 + 0.458532i $$0.848375\pi$$
$$984$$ 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.bj.1.3 3
7.6 odd 2 169.2.a.c.1.3 yes 3
13.12 even 2 8281.2.a.bf.1.1 3
21.20 even 2 1521.2.a.o.1.1 3
28.27 even 2 2704.2.a.ba.1.2 3
35.34 odd 2 4225.2.a.bb.1.1 3
91.6 even 12 169.2.e.b.23.1 12
91.20 even 12 169.2.e.b.23.6 12
91.34 even 4 169.2.b.b.168.6 6
91.41 even 12 169.2.e.b.147.6 12
91.48 odd 6 169.2.c.b.146.1 6
91.55 odd 6 169.2.c.b.22.1 6
91.62 odd 6 169.2.c.c.22.3 6
91.69 odd 6 169.2.c.c.146.3 6
91.76 even 12 169.2.e.b.147.1 12
91.83 even 4 169.2.b.b.168.1 6
91.90 odd 2 169.2.a.b.1.1 3
273.83 odd 4 1521.2.b.l.1351.6 6
273.125 odd 4 1521.2.b.l.1351.1 6
273.272 even 2 1521.2.a.r.1.3 3
364.83 odd 4 2704.2.f.o.337.3 6
364.307 odd 4 2704.2.f.o.337.4 6
364.363 even 2 2704.2.a.z.1.2 3
455.454 odd 2 4225.2.a.bg.1.3 3

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