# Properties

 Label 1521.2.a.o.1.1 Level $1521$ Weight $2$ Character 1521.1 Self dual yes Analytic conductor $12.145$ 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] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$12.1452461474$$ 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 1.1 Root $$1.80194$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.24698 q^{2} +3.04892 q^{4} -1.44504 q^{5} -2.04892 q^{7} -2.35690 q^{8} +O(q^{10})$$ $$q-2.24698 q^{2} +3.04892 q^{4} -1.44504 q^{5} -2.04892 q^{7} -2.35690 q^{8} +3.24698 q^{10} -2.55496 q^{11} +4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} +5.85086 q^{19} -4.40581 q^{20} +5.74094 q^{22} +1.89008 q^{23} -2.91185 q^{25} -6.24698 q^{28} -2.26875 q^{29} +4.26875 q^{31} +6.51573 q^{32} -11.8998 q^{34} +2.96077 q^{35} -5.35690 q^{37} -13.1468 q^{38} +3.40581 q^{40} +1.27413 q^{41} +6.13706 q^{43} -7.78986 q^{44} -4.24698 q^{46} -2.95108 q^{47} -2.80194 q^{49} +6.54288 q^{50} -5.52111 q^{53} +3.69202 q^{55} +4.82908 q^{56} +5.09783 q^{58} -12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -13.0368 q^{64} -0.576728 q^{67} +16.1468 q^{68} -6.65279 q^{70} -4.59419 q^{71} +10.5526 q^{73} +12.0368 q^{74} +17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +1.15883 q^{80} -2.86294 q^{82} +7.72348 q^{83} -7.65279 q^{85} -13.7899 q^{86} +6.02177 q^{88} +6.61356 q^{89} +5.76271 q^{92} +6.63102 q^{94} -8.45473 q^{95} -11.9269 q^{97} +6.29590 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 - 4 * q^5 + 3 * q^7 - 3 * q^8 $$3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35} - 12 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 13 q^{43} - 8 q^{46} - 18 q^{47} - 4 q^{49} + q^{50} - q^{53} + 6 q^{55} + 4 q^{56} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} + q^{67} + 21 q^{68} - 2 q^{70} - 27 q^{71} - 9 q^{73} + 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - 5 q^{80} - 14 q^{82} - 7 q^{83} - 5 q^{85} - 18 q^{86} + 15 q^{88} - 11 q^{89} + 5 q^{94} - 3 q^{95} - 7 q^{97} + 5 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 - 4 * q^5 + 3 * q^7 - 3 * q^8 + 5 * q^10 - 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 + 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 - 14 * q^28 + q^29 + 5 * q^31 + 7 * q^32 - 13 * q^34 - 4 * q^35 - 12 * q^37 - 12 * q^38 - 3 * q^40 - 7 * q^41 + 13 * q^43 - 8 * q^46 - 18 * q^47 - 4 * q^49 + q^50 - q^53 + 6 * q^55 + 4 * q^56 - 3 * q^58 - 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 + q^67 + 21 * q^68 - 2 * q^70 - 27 * q^71 - 9 * q^73 + 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - 5 * q^80 - 14 * q^82 - 7 * q^83 - 5 * q^85 - 18 * q^86 + 15 * q^88 - 11 * q^89 + 5 * q^94 - 3 * q^95 - 7 * q^97 + 5 * q^98

## 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 0
$$4$$ 3.04892 1.52446
$$5$$ −1.44504 −0.646242 −0.323121 0.946358i $$-0.604732\pi$$
−0.323121 + 0.946358i $$0.604732\pi$$
$$6$$ 0 0
$$7$$ −2.04892 −0.774418 −0.387209 0.921992i $$-0.626561\pi$$
−0.387209 + 0.921992i $$0.626561\pi$$
$$8$$ −2.35690 −0.833289
$$9$$ 0 0
$$10$$ 3.24698 1.02679
$$11$$ −2.55496 −0.770349 −0.385174 0.922844i $$-0.625859\pi$$
−0.385174 + 0.922844i $$0.625859\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 4.60388 1.23044
$$15$$ 0 0
$$16$$ −0.801938 −0.200484
$$17$$ 5.29590 1.28444 0.642222 0.766519i $$-0.278013\pi$$
0.642222 + 0.766519i $$0.278013\pi$$
$$18$$ 0 0
$$19$$ 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 $$-0.436862\pi$$
0.197055 + 0.980392i $$0.436862\pi$$
$$24$$ 0 0
$$25$$ −2.91185 −0.582371
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −6.24698 −1.18057
$$29$$ −2.26875 −0.421296 −0.210648 0.977562i $$-0.567557\pi$$
−0.210648 + 0.977562i $$0.567557\pi$$
$$30$$ 0 0
$$31$$ 4.26875 0.766690 0.383345 0.923605i $$-0.374772\pi$$
0.383345 + 0.923605i $$0.374772\pi$$
$$32$$ 6.51573 1.15183
$$33$$ 0 0
$$34$$ −11.8998 −2.04079
$$35$$ 2.96077 0.500462
$$36$$ 0 0
$$37$$ −5.35690 −0.880668 −0.440334 0.897834i $$-0.645140\pi$$
−0.440334 + 0.897834i $$0.645140\pi$$
$$38$$ −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
$$45$$ 0 0
$$46$$ −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 0
$$49$$ −2.80194 −0.400277
$$50$$ 6.54288 0.925302
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −5.52111 −0.758382 −0.379191 0.925318i $$-0.623798\pi$$
−0.379191 + 0.925318i $$0.623798\pi$$
$$54$$ 0 0
$$55$$ 3.69202 0.497832
$$56$$ 4.82908 0.645314
$$57$$ 0 0
$$58$$ 5.09783 0.669378
$$59$$ −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 $$0.315277\pi$$
$$62$$ −9.59179 −1.21816
$$63$$ 0 0
$$64$$ −13.0368 −1.62960
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.576728 −0.0704586 −0.0352293 0.999379i $$-0.511216\pi$$
−0.0352293 + 0.999379i $$0.511216\pi$$
$$68$$ 16.1468 1.95808
$$69$$ 0 0
$$70$$ −6.65279 −0.795161
$$71$$ −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 $$0.288128\pi$$
$$74$$ 12.0368 1.39925
$$75$$ 0 0
$$76$$ 17.8388 2.04625
$$77$$ 5.23490 0.596572
$$78$$ 0 0
$$79$$ −15.7778 −1.77514 −0.887569 0.460674i $$-0.847608\pi$$
−0.887569 + 0.460674i $$0.847608\pi$$
$$80$$ 1.15883 0.129562
$$81$$ 0 0
$$82$$ −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
$$87$$ 0 0
$$88$$ 6.02177 0.641923
$$89$$ 6.61356 0.701036 0.350518 0.936556i $$-0.386005\pi$$
0.350518 + 0.936556i $$0.386005\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 5.76271 0.600804
$$93$$ 0 0
$$94$$ 6.63102 0.683938
$$95$$ −8.45473 −0.867437
$$96$$ 0 0
$$97$$ −11.9269 −1.21100 −0.605498 0.795847i $$-0.707026\pi$$
−0.605498 + 0.795847i $$0.707026\pi$$
$$98$$ 6.29590 0.635982
$$99$$ 0 0
$$100$$ −8.87800 −0.887800
$$101$$ −13.0640 −1.29991 −0.649957 0.759971i $$-0.725213\pi$$
−0.649957 + 0.759971i $$0.725213\pi$$
$$102$$ 0 0
$$103$$ 9.16852 0.903401 0.451701 0.892170i $$-0.350818\pi$$
0.451701 + 0.892170i $$0.350818\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.4058 1.20496
$$107$$ 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 $$0.501860\pi$$
$$110$$ −8.29590 −0.790983
$$111$$ 0 0
$$112$$ 1.64310 0.155259
$$113$$ −7.30798 −0.687477 −0.343738 0.939065i $$-0.611693\pi$$
−0.343738 + 0.939065i $$0.611693\pi$$
$$114$$ 0 0
$$115$$ −2.73125 −0.254690
$$116$$ −6.91723 −0.642249
$$117$$ 0 0
$$118$$ 27.4306 2.52519
$$119$$ −10.8509 −0.994696
$$120$$ 0 0
$$121$$ −4.47219 −0.406563
$$122$$ −19.2446 −1.74232
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.25667 −0.284536 −0.142268 0.989828i $$-0.545440\pi$$
−0.142268 + 0.989828i $$0.545440\pi$$
$$132$$ 0 0
$$133$$ −11.9879 −1.03948
$$134$$ 1.29590 0.111948
$$135$$ 0 0
$$136$$ −12.4819 −1.07031
$$137$$ 0.792249 0.0676864 0.0338432 0.999427i $$-0.489225\pi$$
0.0338432 + 0.999427i $$0.489225\pi$$
$$138$$ 0 0
$$139$$ −11.3394 −0.961799 −0.480899 0.876776i $$-0.659690\pi$$
−0.480899 + 0.876776i $$0.659690\pi$$
$$140$$ 9.02715 0.762933
$$141$$ 0 0
$$142$$ 10.3230 0.866291
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.27844 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 $$0.388111\pi$$
$$150$$ 0 0
$$151$$ −14.1293 −1.14983 −0.574913 0.818215i $$-0.694964\pi$$
−0.574913 + 0.818215i $$0.694964\pi$$
$$152$$ −13.7899 −1.11851
$$153$$ 0 0
$$154$$ −11.7627 −0.947866
$$155$$ −6.16852 −0.495468
$$156$$ 0 0
$$157$$ −9.43296 −0.752832 −0.376416 0.926451i $$-0.622844\pi$$
−0.376416 + 0.926451i $$0.622844\pi$$
$$158$$ 35.4523 2.82044
$$159$$ 0 0
$$160$$ −9.41550 −0.744361
$$161$$ −3.87263 −0.305206
$$162$$ 0 0
$$163$$ −8.70410 −0.681758 −0.340879 0.940107i $$-0.610725\pi$$
−0.340879 + 0.940107i $$0.610725\pi$$
$$164$$ 3.88471 0.303345
$$165$$ 0 0
$$166$$ −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
$$171$$ 0 0
$$172$$ 18.7114 1.42673
$$173$$ 18.8552 1.43353 0.716766 0.697314i $$-0.245622\pi$$
0.716766 + 0.697314i $$0.245622\pi$$
$$174$$ 0 0
$$175$$ 5.96615 0.450998
$$176$$ 2.04892 0.154443
$$177$$ 0 0
$$178$$ −14.8605 −1.11384
$$179$$ −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
$$183$$ 0 0
$$184$$ −4.45473 −0.328407
$$185$$ 7.74094 0.569125
$$186$$ 0 0
$$187$$ −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 $$-0.732334\pi$$
−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 $$0.569920\pi$$
$$194$$ 26.7995 1.92410
$$195$$ 0 0
$$196$$ −8.54288 −0.610205
$$197$$ −11.4155 −0.813321 −0.406660 0.913579i $$-0.633307\pi$$
−0.406660 + 0.913579i $$0.633307\pi$$
$$198$$ 0 0
$$199$$ −13.9051 −0.985710 −0.492855 0.870111i $$-0.664047\pi$$
−0.492855 + 0.870111i $$0.664047\pi$$
$$200$$ 6.86294 0.485283
$$201$$ 0 0
$$202$$ 29.3545 2.06538
$$203$$ 4.64848 0.326259
$$204$$ 0 0
$$205$$ −1.84117 −0.128593
$$206$$ −20.6015 −1.43537
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −14.9487 −1.03402
$$210$$ 0 0
$$211$$ −13.2446 −0.911795 −0.455897 0.890032i $$-0.650682\pi$$
−0.455897 + 0.890032i $$0.650682\pi$$
$$212$$ −16.8334 −1.15612
$$213$$ 0 0
$$214$$ −15.5036 −1.05981
$$215$$ −8.86831 −0.604814
$$216$$ 0 0
$$217$$ −8.74632 −0.593739
$$218$$ 0.274127 0.0185662
$$219$$ 0 0
$$220$$ 11.2567 0.758924
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 7.33513 0.491196 0.245598 0.969372i $$-0.421016\pi$$
0.245598 + 0.969372i $$0.421016\pi$$
$$224$$ −13.3502 −0.891997
$$225$$ 0 0
$$226$$ 16.4209 1.09230
$$227$$ −8.67456 −0.575751 −0.287875 0.957668i $$-0.592949\pi$$
−0.287875 + 0.957668i $$0.592949\pi$$
$$228$$ 0 0
$$229$$ 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 $$-0.446699\pi$$
0.166668 + 0.986013i $$0.446699\pi$$
$$234$$ 0 0
$$235$$ 4.26444 0.278181
$$236$$ −37.2204 −2.42284
$$237$$ 0 0
$$238$$ 24.3817 1.58043
$$239$$ −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 $$0.625368\pi$$
$$242$$ 10.0489 0.645969
$$243$$ 0 0
$$244$$ 26.1129 1.67171
$$245$$ 4.04892 0.258676
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10.0610 −0.638874
$$249$$ 0 0
$$250$$ −25.6896 −1.62475
$$251$$ −22.3478 −1.41058 −0.705290 0.708919i $$-0.749183\pi$$
−0.705290 + 0.708919i $$0.749183\pi$$
$$252$$ 0 0
$$253$$ −4.82908 −0.303602
$$254$$ 42.6262 2.67461
$$255$$ 0 0
$$256$$ −10.4668 −0.654176
$$257$$ 18.6601 1.16398 0.581992 0.813194i $$-0.302273\pi$$
0.581992 + 0.813194i $$0.302273\pi$$
$$258$$ 0 0
$$259$$ 10.9758 0.682005
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 7.31767 0.452087
$$263$$ −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
$$266$$ 26.9366 1.65159
$$267$$ 0 0
$$268$$ −1.75840 −0.107411
$$269$$ −0.652793 −0.0398015 −0.0199007 0.999802i $$-0.506335\pi$$
−0.0199007 + 0.999802i $$0.506335\pi$$
$$270$$ 0 0
$$271$$ 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
$$276$$ 0 0
$$277$$ 11.7845 0.708061 0.354030 0.935234i $$-0.384811\pi$$
0.354030 + 0.935234i $$0.384811\pi$$
$$278$$ 25.4795 1.52816
$$279$$ 0 0
$$280$$ −6.97823 −0.417029
$$281$$ −6.47219 −0.386098 −0.193049 0.981189i $$-0.561838\pi$$
−0.193049 + 0.981189i $$0.561838\pi$$
$$282$$ 0 0
$$283$$ 6.58104 0.391202 0.195601 0.980684i $$-0.437334\pi$$
0.195601 + 0.980684i $$0.437334\pi$$
$$284$$ −14.0073 −0.831180
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.61058 −0.154098
$$288$$ 0 0
$$289$$ 11.0465 0.649796
$$290$$ −7.36658 −0.432581
$$291$$ 0 0
$$292$$ 32.1739 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
$$297$$ 0 0
$$298$$ −18.8877 −1.09413
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.5743 −0.724773
$$302$$ 31.7482 1.82691
$$303$$ 0 0
$$304$$ −4.69202 −0.269106
$$305$$ −12.3763 −0.708663
$$306$$ 0 0
$$307$$ 14.0737 0.803227 0.401613 0.915809i $$-0.368450\pi$$
0.401613 + 0.915809i $$0.368450\pi$$
$$308$$ 15.9608 0.909449
$$309$$ 0 0
$$310$$ 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 $$-0.567791\pi$$
−0.211366 + 0.977407i $$0.567791\pi$$
$$314$$ 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 $$0.180592\pi$$
$$318$$ 0 0
$$319$$ 5.79656 0.324545
$$320$$ 18.8388 1.05312
$$321$$ 0 0
$$322$$ 8.70171 0.484927
$$323$$ 30.9855 1.72408
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.5579 1.08321
$$327$$ 0 0
$$328$$ −3.00298 −0.165812
$$329$$ 6.04652 0.333356
$$330$$ 0 0
$$331$$ 15.7168 0.863872 0.431936 0.901904i $$-0.357831\pi$$
0.431936 + 0.901904i $$0.357831\pi$$
$$332$$ 23.5483 1.29238
$$333$$ 0 0
$$334$$ 53.5991 2.93281
$$335$$ 0.833397 0.0455333
$$336$$ 0 0
$$337$$ 1.95407 0.106445 0.0532224 0.998583i $$-0.483051\pi$$
0.0532224 + 0.998583i $$0.483051\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −23.3327 −1.26540
$$341$$ −10.9065 −0.590619
$$342$$ 0 0
$$343$$ 20.0834 1.08440
$$344$$ −14.4644 −0.779869
$$345$$ 0 0
$$346$$ −42.3672 −2.27767
$$347$$ 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 $$0.409620\pi$$
$$350$$ −13.4058 −0.716571
$$351$$ 0 0
$$352$$ −16.6474 −0.887310
$$353$$ 15.5308 0.826621 0.413310 0.910590i $$-0.364372\pi$$
0.413310 + 0.910590i $$0.364372\pi$$
$$354$$ 0 0
$$355$$ 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 $$0.691285\pi$$
$$360$$ 0 0
$$361$$ 15.2325 0.801711
$$362$$ 10.7356 0.564249
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.2489 −0.798164
$$366$$ 0 0
$$367$$ 34.3032 1.79061 0.895306 0.445452i $$-0.146957\pi$$
0.895306 + 0.445452i $$0.146957\pi$$
$$368$$ −1.51573 −0.0790129
$$369$$ 0 0
$$370$$ −17.3937 −0.904257
$$371$$ 11.3123 0.587305
$$372$$ 0 0
$$373$$ −12.5961 −0.652202 −0.326101 0.945335i $$-0.605735\pi$$
−0.326101 + 0.945335i $$0.605735\pi$$
$$374$$ 30.4034 1.57212
$$375$$ 0 0
$$376$$ 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
$$381$$ 0 0
$$382$$ 41.4131 2.11888
$$383$$ 7.53617 0.385080 0.192540 0.981289i $$-0.438327\pi$$
0.192540 + 0.981289i $$0.438327\pi$$
$$384$$ 0 0
$$385$$ −7.56465 −0.385530
$$386$$ 13.6039 0.692419
$$387$$ 0 0
$$388$$ −36.3642 −1.84611
$$389$$ −35.5555 −1.80274 −0.901369 0.433052i $$-0.857437\pi$$
−0.901369 + 0.433052i $$0.857437\pi$$
$$390$$ 0 0
$$391$$ 10.0097 0.506212
$$392$$ 6.60388 0.333546
$$393$$ 0 0
$$394$$ 25.6504 1.29225
$$395$$ 22.7995 1.14717
$$396$$ 0 0
$$397$$ −1.35152 −0.0678308 −0.0339154 0.999425i $$-0.510798\pi$$
−0.0339154 + 0.999425i $$0.510798\pi$$
$$398$$ 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 0
$$403$$ 0 0
$$404$$ −39.8310 −1.98167
$$405$$ 0 0
$$406$$ −10.4450 −0.518379
$$407$$ 13.6866 0.678422
$$408$$ 0 0
$$409$$ 15.1575 0.749490 0.374745 0.927128i $$-0.377730\pi$$
0.374745 + 0.927128i $$0.377730\pi$$
$$410$$ 4.13706 0.204315
$$411$$ 0 0
$$412$$ 27.9541 1.37720
$$413$$ 25.0127 1.23079
$$414$$ 0 0
$$415$$ −11.1608 −0.547860
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 33.5894 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
$$423$$ 0 0
$$424$$ 13.0127 0.631951
$$425$$ −15.4209 −0.748022
$$426$$ 0 0
$$427$$ −17.5483 −0.849220
$$428$$ 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 $$0.809185\pi$$
$$432$$ 0 0
$$433$$ 13.7385 0.660232 0.330116 0.943940i $$-0.392912\pi$$
0.330116 + 0.943940i $$0.392912\pi$$
$$434$$ 19.6528 0.943364
$$435$$ 0 0
$$436$$ −0.371961 −0.0178137
$$437$$ 11.0586 0.529005
$$438$$ 0 0
$$439$$ 10.2403 0.488742 0.244371 0.969682i $$-0.421419\pi$$
0.244371 + 0.969682i $$0.421419\pi$$
$$440$$ −8.70171 −0.414838
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.1763 −0.578513 −0.289257 0.957252i $$-0.593408\pi$$
−0.289257 + 0.957252i $$0.593408\pi$$
$$444$$ 0 0
$$445$$ −9.55688 −0.453039
$$446$$ −16.4819 −0.780440
$$447$$ 0 0
$$448$$ 26.7114 1.26199
$$449$$ −12.9051 −0.609032 −0.304516 0.952507i $$-0.598495\pi$$
−0.304516 + 0.952507i $$0.598495\pi$$
$$450$$ 0 0
$$451$$ −3.25534 −0.153288
$$452$$ −22.2814 −1.04803
$$453$$ 0 0
$$454$$ 19.4916 0.914785
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4.65710 −0.217850 −0.108925 0.994050i $$-0.534741\pi$$
−0.108925 + 0.994050i $$0.534741\pi$$
$$458$$ −30.7536 −1.43702
$$459$$ 0 0
$$460$$ −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
$$465$$ 0 0
$$466$$ −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
$$469$$ 1.18167 0.0545644
$$470$$ −9.58211 −0.441990
$$471$$ 0 0
$$472$$ 28.7724 1.32436
$$473$$ −15.6799 −0.720964
$$474$$ 0 0
$$475$$ −17.0368 −0.781704
$$476$$ −33.0834 −1.51637
$$477$$ 0 0
$$478$$ 24.5459 1.12270
$$479$$ −34.8998 −1.59461 −0.797306 0.603576i $$-0.793742\pi$$
−0.797306 + 0.603576i $$0.793742\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 26.7724 1.21945
$$483$$ 0 0
$$484$$ −13.6353 −0.619788
$$485$$ 17.2349 0.782596
$$486$$ 0 0
$$487$$ −41.8351 −1.89573 −0.947864 0.318676i $$-0.896762\pi$$
−0.947864 + 0.318676i $$0.896762\pi$$
$$488$$ −20.1860 −0.913776
$$489$$ 0 0
$$490$$ −9.09783 −0.410998
$$491$$ −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
$$495$$ 0 0
$$496$$ −3.42327 −0.153709
$$497$$ 9.41311 0.422236
$$498$$ 0 0
$$499$$ −23.5472 −1.05412 −0.527058 0.849829i $$-0.676705\pi$$
−0.527058 + 0.849829i $$0.676705\pi$$
$$500$$ 34.8582 1.55890
$$501$$ 0 0
$$502$$ 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 $$0.445994\pi$$
$$510$$ 0 0
$$511$$ −21.6213 −0.956471
$$512$$ −9.00538 −0.397985
$$513$$ 0 0
$$514$$ −41.9288 −1.84940
$$515$$ −13.2489 −0.583816
$$516$$ 0 0
$$517$$ 7.53989 0.331604
$$518$$ −24.6625 −1.08361
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 39.5133 1.73111 0.865555 0.500813i $$-0.166966\pi$$
0.865555 + 0.500813i $$0.166966\pi$$
$$522$$ 0 0
$$523$$ −15.8194 −0.691734 −0.345867 0.938284i $$-0.612415\pi$$
−0.345867 + 0.938284i $$0.612415\pi$$
$$524$$ −9.92931 −0.433764
$$525$$ 0 0
$$526$$ 32.3545 1.41072
$$527$$ 22.6069 0.984770
$$528$$ 0 0
$$529$$ −19.4276 −0.844678
$$530$$ −17.9269 −0.778696
$$531$$ 0 0
$$532$$ −36.5502 −1.58465
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.97046 −0.431061
$$536$$ 1.35929 0.0587123
$$537$$ 0 0
$$538$$ 1.46681 0.0632388
$$539$$ 7.15883 0.308353
$$540$$ 0 0
$$541$$ −34.4819 −1.48249 −0.741246 0.671234i $$-0.765765\pi$$
−0.741246 + 0.671234i $$0.765765\pi$$
$$542$$ −4.48427 −0.192616
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ −16.7168 −0.712806
$$551$$ −13.2741 −0.565497
$$552$$ 0 0
$$553$$ 32.3274 1.37470
$$554$$ −26.4795 −1.12501
$$555$$ 0 0
$$556$$ −34.5730 −1.46622
$$557$$ 1.27652 0.0540879 0.0270439 0.999634i $$-0.491391\pi$$
0.0270439 + 0.999634i $$0.491391\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.37435 −0.100335
$$561$$ 0 0
$$562$$ 14.5429 0.613454
$$563$$ 9.12737 0.384673 0.192336 0.981329i $$-0.438393\pi$$
0.192336 + 0.981329i $$0.438393\pi$$
$$564$$ 0 0
$$565$$ 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 $$-0.461733\pi$$
0.119930 + 0.992782i $$0.461733\pi$$
$$570$$ 0 0
$$571$$ 7.60148 0.318112 0.159056 0.987270i $$-0.449155\pi$$
0.159056 + 0.987270i $$0.449155\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 5.86592 0.244839
$$575$$ −5.50365 −0.229518
$$576$$ 0 0
$$577$$ −45.1564 −1.87989 −0.939944 0.341330i $$-0.889123\pi$$
−0.939944 + 0.341330i $$0.889123\pi$$
$$578$$ −24.8213 −1.03243
$$579$$ 0 0
$$580$$ 9.99569 0.415048
$$581$$ −15.8248 −0.656522
$$582$$ 0 0
$$583$$ 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
$$591$$ 0 0
$$592$$ 4.29590 0.176560
$$593$$ 36.6848 1.50647 0.753233 0.657754i $$-0.228493\pi$$
0.753233 + 0.657754i $$0.228493\pi$$
$$594$$ 0 0
$$595$$ 15.6799 0.642815
$$596$$ 25.6286 1.04979
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 9.99223 0.408271 0.204136 0.978943i $$-0.434562\pi$$
0.204136 + 0.978943i $$0.434562\pi$$
$$600$$ 0 0
$$601$$ −1.81163 −0.0738978 −0.0369489 0.999317i $$-0.511764\pi$$
−0.0369489 + 0.999317i $$0.511764\pi$$
$$602$$ 28.2543 1.15156
$$603$$ 0 0
$$604$$ −43.0790 −1.75286
$$605$$ 6.46250 0.262738
$$606$$ 0 0
$$607$$ 11.2161 0.455248 0.227624 0.973749i $$-0.426904\pi$$
0.227624 + 0.973749i $$0.426904\pi$$
$$608$$ 38.1226 1.54608
$$609$$ 0 0
$$610$$ 27.8092 1.12596
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20.8944 0.843917 0.421958 0.906615i $$-0.361343\pi$$
0.421958 + 0.906615i $$0.361343\pi$$
$$614$$ −31.6233 −1.27621
$$615$$ 0 0
$$616$$ −12.3381 −0.497117
$$617$$ 12.0992 0.487094 0.243547 0.969889i $$-0.421689\pi$$
0.243547 + 0.969889i $$0.421689\pi$$
$$618$$ 0 0
$$619$$ 10.5526 0.424143 0.212072 0.977254i $$-0.431979\pi$$
0.212072 + 0.977254i $$0.431979\pi$$
$$620$$ −18.8073 −0.755320
$$621$$ 0 0
$$622$$ −66.8926 −2.68215
$$623$$ −13.5506 −0.542895
$$624$$ 0 0
$$625$$ −1.96184 −0.0784735
$$626$$ 16.8049 0.671660
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ −67.4771 −2.67986
$$635$$ 27.4131 1.08786
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −13.0248 −0.515655
$$639$$ 0 0
$$640$$ −23.4993 −0.928893
$$641$$ −34.9608 −1.38087 −0.690434 0.723396i $$-0.742580\pi$$
−0.690434 + 0.723396i $$0.742580\pi$$
$$642$$ 0 0
$$643$$ −33.3980 −1.31709 −0.658545 0.752541i $$-0.728828\pi$$
−0.658545 + 0.752541i $$0.728828\pi$$
$$644$$ −11.8073 −0.465273
$$645$$ 0 0
$$646$$ −69.6238 −2.73931
$$647$$ −2.32842 −0.0915397 −0.0457698 0.998952i $$-0.514574\pi$$
−0.0457698 + 0.998952i $$0.514574\pi$$
$$648$$ 0 0
$$649$$ 31.1903 1.22433
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −26.5381 −1.03931
$$653$$ −14.5714 −0.570221 −0.285111 0.958495i $$-0.592030\pi$$
−0.285111 + 0.958495i $$0.592030\pi$$
$$654$$ 0 0
$$655$$ 4.70602 0.183879
$$656$$ −1.02177 −0.0398934
$$657$$ 0 0
$$658$$ −13.5864 −0.529654
$$659$$ −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 $$0.413192\pi$$
$$662$$ −35.3153 −1.37257
$$663$$ 0 0
$$664$$ −18.2034 −0.706430
$$665$$ 17.3230 0.671759
$$666$$ 0 0
$$667$$ −4.28813 −0.166037
$$668$$ −72.7284 −2.81395
$$669$$ 0 0
$$670$$ −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
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 11.3104 0.434693 0.217346 0.976095i $$-0.430260\pi$$
0.217346 + 0.976095i $$0.430260\pi$$
$$678$$ 0 0
$$679$$ 24.4373 0.937816
$$680$$ 18.0368 0.691681
$$681$$ 0 0
$$682$$ 24.5066 0.938407
$$683$$ 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
$$686$$ −45.1269 −1.72295
$$687$$ 0 0
$$688$$ −4.92154 −0.187632
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −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
$$696$$ 0 0
$$697$$ 6.74764 0.255585
$$698$$ −23.5187 −0.890196
$$699$$ 0 0
$$700$$ 18.1903 0.687528
$$701$$ −6.73184 −0.254258 −0.127129 0.991886i $$-0.540576\pi$$
−0.127129 + 0.991886i $$0.540576\pi$$
$$702$$ 0 0
$$703$$ −31.3424 −1.18210
$$704$$ 33.3086 1.25536
$$705$$ 0 0
$$706$$ −34.8974 −1.31338
$$707$$ 26.7670 1.00668
$$708$$ 0 0
$$709$$ 47.6252 1.78860 0.894300 0.447467i $$-0.147674\pi$$
0.894300 + 0.447467i $$0.147674\pi$$
$$710$$ −14.9172 −0.559834
$$711$$ 0 0
$$712$$ −15.5875 −0.584166
$$713$$ 8.06829 0.302160
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −18.3599 −0.686141
$$717$$ 0 0
$$718$$ 48.1444 1.79673
$$719$$ 5.99330 0.223512 0.111756 0.993736i $$-0.464352\pi$$
0.111756 + 0.993736i $$0.464352\pi$$
$$720$$ 0 0
$$721$$ −18.7855 −0.699610
$$722$$ −34.2271 −1.27380
$$723$$ 0 0
$$724$$ −14.5670 −0.541380
$$725$$ 6.60627 0.245351
$$726$$ 0 0
$$727$$ −24.1226 −0.894657 −0.447329 0.894370i $$-0.647625\pi$$
−0.447329 + 0.894370i $$0.647625\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 34.2640 1.26817
$$731$$ 32.5013 1.20210
$$732$$ 0 0
$$733$$ −36.0646 −1.33208 −0.666038 0.745918i $$-0.732011\pi$$
−0.666038 + 0.745918i $$0.732011\pi$$
$$734$$ −77.0786 −2.84502
$$735$$ 0 0
$$736$$ 12.3153 0.453947
$$737$$ 1.47352 0.0542777
$$738$$ 0 0
$$739$$ −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
$$742$$ −25.4185 −0.933142
$$743$$ −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
$$746$$ 28.3032 1.03625
$$747$$ 0 0
$$748$$ −41.2543 −1.50841
$$749$$ −14.1371 −0.516557
$$750$$ 0 0
$$751$$ 4.06770 0.148433 0.0742163 0.997242i $$-0.476354\pi$$
0.0742163 + 0.997242i $$0.476354\pi$$
$$752$$ 2.36658 0.0863005
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 20.4174 0.743066
$$756$$ 0 0
$$757$$ 20.4336 0.742670 0.371335 0.928499i $$-0.378900\pi$$
0.371335 + 0.928499i $$0.378900\pi$$
$$758$$ 37.1618 1.34978
$$759$$ 0 0
$$760$$ 19.9269 0.722825
$$761$$ −27.0237 −0.979608 −0.489804 0.871833i $$-0.662932\pi$$
−0.489804 + 0.871833i $$0.662932\pi$$
$$762$$ 0 0
$$763$$ 0.249964 0.00904929
$$764$$ −56.1933 −2.03300
$$765$$ 0 0
$$766$$ −16.9336 −0.611837
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 37.9407 1.36818 0.684088 0.729400i $$-0.260201\pi$$
0.684088 + 0.729400i $$0.260201\pi$$
$$770$$ 16.9976 0.612551
$$771$$ 0 0
$$772$$ −18.4590 −0.664355
$$773$$ −16.3375 −0.587620 −0.293810 0.955864i $$-0.594923\pi$$
−0.293810 + 0.955864i $$0.594923\pi$$
$$774$$ 0 0
$$775$$ −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
$$783$$ 0 0
$$784$$ 2.24698 0.0802493
$$785$$ 13.6310 0.486512
$$786$$ 0 0
$$787$$ 18.6907 0.666251 0.333126 0.942882i $$-0.391897\pi$$
0.333126 + 0.942882i $$0.391897\pi$$
$$788$$ −34.8049 −1.23987
$$789$$ 0 0
$$790$$ −51.2301 −1.82269
$$791$$ 14.9734 0.532394
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 3.03684 0.107773
$$795$$ 0 0
$$796$$ −42.3957 −1.50267
$$797$$ −29.2519 −1.03615 −0.518077 0.855334i $$-0.673352\pi$$
−0.518077 + 0.855334i $$0.673352\pi$$
$$798$$ 0 0
$$799$$ −15.6286 −0.552901
$$800$$ −18.9729 −0.670792
$$801$$ 0 0
$$802$$ −1.30127 −0.0459496
$$803$$ −26.9614 −0.951446
$$804$$ 0 0
$$805$$ 5.59611 0.197237
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 30.7904 1.08320
$$809$$ 6.65087 0.233832 0.116916 0.993142i $$-0.462699\pi$$
0.116916 + 0.993142i $$0.462699\pi$$
$$810$$ 0 0
$$811$$ −3.89200 −0.136667 −0.0683333 0.997663i $$-0.521768\pi$$
−0.0683333 + 0.997663i $$0.521768\pi$$
$$812$$ 14.1728 0.497369
$$813$$ 0 0
$$814$$ −30.7536 −1.07791
$$815$$ 12.5778 0.440581
$$816$$ 0 0
$$817$$ 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 0
$$823$$ 7.95300 0.277224 0.138612 0.990347i $$-0.455736\pi$$
0.138612 + 0.990347i $$0.455736\pi$$
$$824$$ −21.6093 −0.752794
$$825$$ 0 0
$$826$$ −56.2030 −1.95555
$$827$$ 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 $$0.340697\pi$$
$$830$$ 25.0780 0.870470
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −14.8388 −0.514133
$$834$$ 0 0
$$835$$ 34.4698 1.19288
$$836$$ −45.5773 −1.57632
$$837$$ 0 0
$$838$$ −80.2699 −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
$$843$$ 0 0
$$844$$ −40.3817 −1.38999
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 9.16315 0.314849
$$848$$ 4.42758 0.152044
$$849$$ 0 0
$$850$$ 34.6504 1.18850
$$851$$ −10.1250 −0.347080
$$852$$ 0 0
$$853$$ 43.2078 1.47941 0.739703 0.672934i $$-0.234966\pi$$
0.739703 + 0.672934i $$0.234966\pi$$
$$854$$ 39.4306 1.34929
$$855$$ 0 0
$$856$$ −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 $$-0.345305\pi$$
0.467084 + 0.884213i $$0.345305\pi$$
$$860$$ −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 $$0.748824\pi$$
$$864$$ 0 0
$$865$$ −27.2465 −0.926409
$$866$$ −30.8702 −1.04901
$$867$$ 0 0
$$868$$ −26.6668 −0.905130
$$869$$ 40.3116 1.36748
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0.287536 0.00973721
$$873$$ 0 0
$$874$$ −24.8485 −0.840512
$$875$$ −23.4252 −0.791916
$$876$$ 0 0
$$877$$ 24.7472 0.835653 0.417826 0.908527i $$-0.362792\pi$$
0.417826 + 0.908527i $$0.362792\pi$$
$$878$$ −23.0097 −0.776539
$$879$$ 0 0
$$880$$ −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
$$885$$ 0 0
$$886$$ 27.3599 0.919173
$$887$$ −15.9661 −0.536091 −0.268045 0.963406i $$-0.586378\pi$$
−0.268045 + 0.963406i $$0.586378\pi$$
$$888$$ 0 0
$$889$$ 38.8689 1.30362
$$890$$ 21.4741 0.719814
$$891$$ 0 0
$$892$$ 22.3642 0.748809
$$893$$ −17.2664 −0.577797
$$894$$ 0 0
$$895$$ 8.70171 0.290866
$$896$$ −33.3196 −1.11313
$$897$$ 0 0
$$898$$ 28.9976 0.967663
$$899$$ −9.68473 −0.323004
$$900$$ 0 0
$$901$$ −29.2392 −0.974099
$$902$$ 7.31468 0.243552
$$903$$ 0 0
$$904$$ 17.2241 0.572867
$$905$$ 6.90408 0.229500
$$906$$ 0 0
$$907$$ −28.8364 −0.957496 −0.478748 0.877952i $$-0.658909\pi$$
−0.478748 + 0.877952i $$0.658909\pi$$
$$908$$ −26.4480 −0.877709
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.5633 −1.27766 −0.638830 0.769348i $$-0.720581\pi$$
−0.638830 + 0.769348i $$0.720581\pi$$
$$912$$ 0 0
$$913$$ −19.7332 −0.653073
$$914$$ 10.4644 0.346132
$$915$$ 0 0
$$916$$ 41.7294 1.37878
$$917$$ 6.67264 0.220350
$$918$$ 0 0
$$919$$ 8.87502 0.292760 0.146380 0.989228i $$-0.453238\pi$$
0.146380 + 0.989228i $$0.453238\pi$$
$$920$$ 6.43727 0.212231
$$921$$ 0 0
$$922$$ 70.8708 2.33401
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 15.5985 0.512875
$$926$$ 39.6601 1.30331
$$927$$ 0 0
$$928$$ −14.7826 −0.485261
$$929$$ 24.2295 0.794945 0.397472 0.917614i $$-0.369887\pi$$
0.397472 + 0.917614i $$0.369887\pi$$
$$930$$ 0 0
$$931$$ −16.3937 −0.537283
$$932$$ 15.5133 0.508156
$$933$$ 0 0
$$934$$ −72.2911 −2.36544
$$935$$ 19.5526 0.639437
$$936$$ 0 0
$$937$$ 17.2644 0.564005 0.282002 0.959414i $$-0.409001\pi$$
0.282002 + 0.959414i $$0.409001\pi$$
$$938$$ −2.65519 −0.0866949
$$939$$ 0 0
$$940$$ 13.0019 0.424076
$$941$$ 4.34050 0.141496 0.0707482 0.997494i $$-0.477461\pi$$
0.0707482 + 0.997494i $$0.477461\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.761128\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 38.2814 1.24201
$$951$$ 0 0
$$952$$ 25.5743 0.828869
$$953$$ 46.8859 1.51878 0.759391 0.650634i $$-0.225497\pi$$
0.759391 + 0.650634i $$0.225497\pi$$
$$954$$ 0 0
$$955$$ 26.6329 0.861822
$$956$$ −33.3062 −1.07720
$$957$$ 0 0
$$958$$ 78.4191 2.53361
$$959$$ −1.62325 −0.0524176
$$960$$ 0 0
$$961$$ −12.7778 −0.412186
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −36.3274 −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
$$969$$ 0 0
$$970$$ −38.7265 −1.24343
$$971$$ 41.8068 1.34165 0.670823 0.741618i $$-0.265941\pi$$
0.670823 + 0.741618i $$0.265941\pi$$
$$972$$ 0 0
$$973$$ 23.2336 0.744834
$$974$$ 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 $$0.375937\pi$$
$$978$$ 0 0
$$979$$ −16.8974 −0.540043
$$980$$ 12.3448 0.394341
$$981$$ 0 0
$$982$$ 49.0863 1.56641
$$983$$ −55.7251 −1.77736 −0.888678 0.458532i $$-0.848375\pi$$
−0.888678 + 0.458532i $$0.848375\pi$$
$$984$$ 0 0
$$985$$ 16.4959 0.525602
$$986$$ 26.9976 0.859779
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11.5996 0.368845
$$990$$ 0 0
$$991$$ −35.5512 −1.12932 −0.564661 0.825323i $$-0.690993\pi$$
−0.564661 + 0.825323i $$0.690993\pi$$
$$992$$ 27.8140 0.883096
$$993$$ 0 0
$$994$$ −21.1511 −0.670871
$$995$$ 20.0935 0.637007
$$996$$ 0 0
$$997$$ 6.61058 0.209359 0.104680 0.994506i $$-0.466618\pi$$
0.104680 + 0.994506i $$0.466618\pi$$
$$998$$ 52.9101 1.67484
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.2.a.o.1.1 3
3.2 odd 2 169.2.a.c.1.3 yes 3
12.11 even 2 2704.2.a.ba.1.2 3
13.5 odd 4 1521.2.b.l.1351.6 6
13.8 odd 4 1521.2.b.l.1351.1 6
13.12 even 2 1521.2.a.r.1.3 3
15.14 odd 2 4225.2.a.bb.1.1 3
21.20 even 2 8281.2.a.bj.1.3 3
39.2 even 12 169.2.e.b.147.6 12
39.5 even 4 169.2.b.b.168.1 6
39.8 even 4 169.2.b.b.168.6 6
39.11 even 12 169.2.e.b.147.1 12
39.17 odd 6 169.2.c.c.146.3 6
39.20 even 12 169.2.e.b.23.6 12
39.23 odd 6 169.2.c.c.22.3 6
39.29 odd 6 169.2.c.b.22.1 6
39.32 even 12 169.2.e.b.23.1 12
39.35 odd 6 169.2.c.b.146.1 6
39.38 odd 2 169.2.a.b.1.1 3
156.47 odd 4 2704.2.f.o.337.4 6
156.83 odd 4 2704.2.f.o.337.3 6
156.155 even 2 2704.2.a.z.1.2 3
195.194 odd 2 4225.2.a.bg.1.3 3
273.272 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 39.38 odd 2
169.2.a.c.1.3 yes 3 3.2 odd 2
169.2.b.b.168.1 6 39.5 even 4
169.2.b.b.168.6 6 39.8 even 4
169.2.c.b.22.1 6 39.29 odd 6
169.2.c.b.146.1 6 39.35 odd 6
169.2.c.c.22.3 6 39.23 odd 6
169.2.c.c.146.3 6 39.17 odd 6
169.2.e.b.23.1 12 39.32 even 12
169.2.e.b.23.6 12 39.20 even 12
169.2.e.b.147.1 12 39.11 even 12
169.2.e.b.147.6 12 39.2 even 12
1521.2.a.o.1.1 3 1.1 even 1 trivial
1521.2.a.r.1.3 3 13.12 even 2
1521.2.b.l.1351.1 6 13.8 odd 4
1521.2.b.l.1351.6 6 13.5 odd 4
2704.2.a.z.1.2 3 156.155 even 2
2704.2.a.ba.1.2 3 12.11 even 2
2704.2.f.o.337.3 6 156.83 odd 4
2704.2.f.o.337.4 6 156.47 odd 4
4225.2.a.bb.1.1 3 15.14 odd 2
4225.2.a.bg.1.3 3 195.194 odd 2
8281.2.a.bf.1.1 3 273.272 even 2
8281.2.a.bj.1.3 3 21.20 even 2