# Properties

 Label 2704.2.a.ba.1.3 Level $2704$ Weight $2$ Character 2704.1 Self dual yes Analytic conductor $21.592$ 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] = [2704,2,Mod(1,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$21.5915487066$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.24698$$ of defining polynomial Character $$\chi$$ $$=$$ 2704.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.24698 q^{3} -0.246980 q^{5} -2.35690 q^{7} +2.04892 q^{9} +O(q^{10})$$ $$q+2.24698 q^{3} -0.246980 q^{5} -2.35690 q^{7} +2.04892 q^{9} -4.24698 q^{11} -0.554958 q^{15} +2.15883 q^{17} -0.0881460 q^{19} -5.29590 q^{21} -1.49396 q^{23} -4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} -6.63102 q^{31} -9.54288 q^{33} +0.582105 q^{35} -5.69202 q^{37} +11.5918 q^{41} +0.295897 q^{43} -0.506041 q^{45} -7.35690 q^{47} -1.44504 q^{49} +4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} -0.198062 q^{57} -6.78017 q^{59} +3.47219 q^{61} -4.82908 q^{63} +7.67994 q^{67} -3.35690 q^{69} -8.66487 q^{71} -6.73556 q^{73} -11.0978 q^{75} +10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} +1.60925 q^{83} -0.533188 q^{85} +10.4058 q^{87} +2.88471 q^{89} -14.8998 q^{93} +0.0217703 q^{95} +8.05861 q^{97} -8.70171 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 4 q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 4 * q^5 - 3 * q^7 - 3 * q^9 $$3 q + 2 q^{3} + 4 q^{5} - 3 q^{7} - 3 q^{9} - 8 q^{11} - 2 q^{15} - 2 q^{17} - 4 q^{19} - 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} - 5 q^{31} - 10 q^{33} - 4 q^{35} - 12 q^{37} + 7 q^{41} - 13 q^{43} - 11 q^{45} - 18 q^{47} - 4 q^{49} + q^{51} + q^{53} - 6 q^{55} - 5 q^{57} - 19 q^{59} + 4 q^{61} - 4 q^{63} - q^{67} - 6 q^{69} - 27 q^{71} - 9 q^{73} - 15 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} - 7 q^{83} - 5 q^{85} + 18 q^{87} + 11 q^{89} - 22 q^{93} - 3 q^{95} - 7 q^{97} + q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 4 * q^5 - 3 * q^7 - 3 * q^9 - 8 * q^11 - 2 * q^15 - 2 * q^17 - 4 * q^19 - 2 * q^21 + 5 * q^23 - 5 * q^25 - q^27 - q^29 - 5 * q^31 - 10 * q^33 - 4 * q^35 - 12 * q^37 + 7 * q^41 - 13 * q^43 - 11 * q^45 - 18 * q^47 - 4 * q^49 + q^51 + q^53 - 6 * q^55 - 5 * q^57 - 19 * q^59 + 4 * q^61 - 4 * q^63 - q^67 - 6 * q^69 - 27 * q^71 - 9 * q^73 - 15 * q^75 + 8 * q^77 + 5 * q^79 - q^81 - 7 * q^83 - 5 * q^85 + 18 * q^87 + 11 * q^89 - 22 * q^93 - 3 * q^95 - 7 * q^97 + q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.24698 1.29729 0.648647 0.761089i $$-0.275335\pi$$
0.648647 + 0.761089i $$0.275335\pi$$
$$4$$ 0 0
$$5$$ −0.246980 −0.110453 −0.0552263 0.998474i $$-0.517588\pi$$
−0.0552263 + 0.998474i $$0.517588\pi$$
$$6$$ 0 0
$$7$$ −2.35690 −0.890823 −0.445411 0.895326i $$-0.646943\pi$$
−0.445411 + 0.895326i $$0.646943\pi$$
$$8$$ 0 0
$$9$$ 2.04892 0.682972
$$10$$ 0 0
$$11$$ −4.24698 −1.28051 −0.640256 0.768161i $$-0.721172\pi$$
−0.640256 + 0.768161i $$0.721172\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −0.554958 −0.143290
$$16$$ 0 0
$$17$$ 2.15883 0.523594 0.261797 0.965123i $$-0.415685\pi$$
0.261797 + 0.965123i $$0.415685\pi$$
$$18$$ 0 0
$$19$$ −0.0881460 −0.0202221 −0.0101110 0.999949i $$-0.503218\pi$$
−0.0101110 + 0.999949i $$0.503218\pi$$
$$20$$ 0 0
$$21$$ −5.29590 −1.15566
$$22$$ 0 0
$$23$$ −1.49396 −0.311512 −0.155756 0.987796i $$-0.549781\pi$$
−0.155756 + 0.987796i $$0.549781\pi$$
$$24$$ 0 0
$$25$$ −4.93900 −0.987800
$$26$$ 0 0
$$27$$ −2.13706 −0.411278
$$28$$ 0 0
$$29$$ 4.63102 0.859959 0.429980 0.902839i $$-0.358521\pi$$
0.429980 + 0.902839i $$0.358521\pi$$
$$30$$ 0 0
$$31$$ −6.63102 −1.19097 −0.595483 0.803368i $$-0.703039\pi$$
−0.595483 + 0.803368i $$0.703039\pi$$
$$32$$ 0 0
$$33$$ −9.54288 −1.66120
$$34$$ 0 0
$$35$$ 0.582105 0.0983937
$$36$$ 0 0
$$37$$ −5.69202 −0.935763 −0.467881 0.883791i $$-0.654983\pi$$
−0.467881 + 0.883791i $$0.654983\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.5918 1.81033 0.905167 0.425056i $$-0.139746\pi$$
0.905167 + 0.425056i $$0.139746\pi$$
$$42$$ 0 0
$$43$$ 0.295897 0.0451239 0.0225619 0.999745i $$-0.492818\pi$$
0.0225619 + 0.999745i $$0.492818\pi$$
$$44$$ 0 0
$$45$$ −0.506041 −0.0754361
$$46$$ 0 0
$$47$$ −7.35690 −1.07311 −0.536557 0.843864i $$-0.680275\pi$$
−0.536557 + 0.843864i $$0.680275\pi$$
$$48$$ 0 0
$$49$$ −1.44504 −0.206435
$$50$$ 0 0
$$51$$ 4.85086 0.679256
$$52$$ 0 0
$$53$$ −10.3937 −1.42769 −0.713844 0.700304i $$-0.753048\pi$$
−0.713844 + 0.700304i $$0.753048\pi$$
$$54$$ 0 0
$$55$$ 1.04892 0.141436
$$56$$ 0 0
$$57$$ −0.198062 −0.0262340
$$58$$ 0 0
$$59$$ −6.78017 −0.882703 −0.441351 0.897334i $$-0.645501\pi$$
−0.441351 + 0.897334i $$0.645501\pi$$
$$60$$ 0 0
$$61$$ 3.47219 0.444568 0.222284 0.974982i $$-0.428649\pi$$
0.222284 + 0.974982i $$0.428649\pi$$
$$62$$ 0 0
$$63$$ −4.82908 −0.608407
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.67994 0.938254 0.469127 0.883131i $$-0.344569\pi$$
0.469127 + 0.883131i $$0.344569\pi$$
$$68$$ 0 0
$$69$$ −3.35690 −0.404123
$$70$$ 0 0
$$71$$ −8.66487 −1.02833 −0.514166 0.857691i $$-0.671898\pi$$
−0.514166 + 0.857691i $$0.671898\pi$$
$$72$$ 0 0
$$73$$ −6.73556 −0.788338 −0.394169 0.919038i $$-0.628968\pi$$
−0.394169 + 0.919038i $$0.628968\pi$$
$$74$$ 0 0
$$75$$ −11.0978 −1.28147
$$76$$ 0 0
$$77$$ 10.0097 1.14071
$$78$$ 0 0
$$79$$ −9.97046 −1.12176 −0.560882 0.827896i $$-0.689538\pi$$
−0.560882 + 0.827896i $$0.689538\pi$$
$$80$$ 0 0
$$81$$ −10.9487 −1.21652
$$82$$ 0 0
$$83$$ 1.60925 0.176638 0.0883192 0.996092i $$-0.471850\pi$$
0.0883192 + 0.996092i $$0.471850\pi$$
$$84$$ 0 0
$$85$$ −0.533188 −0.0578323
$$86$$ 0 0
$$87$$ 10.4058 1.11562
$$88$$ 0 0
$$89$$ 2.88471 0.305778 0.152889 0.988243i $$-0.451142\pi$$
0.152889 + 0.988243i $$0.451142\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −14.8998 −1.54503
$$94$$ 0 0
$$95$$ 0.0217703 0.00223358
$$96$$ 0 0
$$97$$ 8.05861 0.818227 0.409114 0.912483i $$-0.365838\pi$$
0.409114 + 0.912483i $$0.365838\pi$$
$$98$$ 0 0
$$99$$ −8.70171 −0.874555
$$100$$ 0 0
$$101$$ −13.3545 −1.32882 −0.664411 0.747367i $$-0.731318\pi$$
−0.664411 + 0.747367i $$0.731318\pi$$
$$102$$ 0 0
$$103$$ −1.36227 −0.134229 −0.0671144 0.997745i $$-0.521379\pi$$
−0.0671144 + 0.997745i $$0.521379\pi$$
$$104$$ 0 0
$$105$$ 1.30798 0.127646
$$106$$ 0 0
$$107$$ −3.26875 −0.316002 −0.158001 0.987439i $$-0.550505\pi$$
−0.158001 + 0.987439i $$0.550505\pi$$
$$108$$ 0 0
$$109$$ −15.7017 −1.50395 −0.751976 0.659191i $$-0.770899\pi$$
−0.751976 + 0.659191i $$0.770899\pi$$
$$110$$ 0 0
$$111$$ −12.7899 −1.21396
$$112$$ 0 0
$$113$$ 12.0489 1.13347 0.566733 0.823901i $$-0.308207\pi$$
0.566733 + 0.823901i $$0.308207\pi$$
$$114$$ 0 0
$$115$$ 0.368977 0.0344073
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −5.08815 −0.466430
$$120$$ 0 0
$$121$$ 7.03684 0.639712
$$122$$ 0 0
$$123$$ 26.0465 2.34854
$$124$$ 0 0
$$125$$ 2.45473 0.219558
$$126$$ 0 0
$$127$$ 9.80731 0.870258 0.435129 0.900368i $$-0.356703\pi$$
0.435129 + 0.900368i $$0.356703\pi$$
$$128$$ 0 0
$$129$$ 0.664874 0.0585389
$$130$$ 0 0
$$131$$ 6.57673 0.574611 0.287306 0.957839i $$-0.407240\pi$$
0.287306 + 0.957839i $$0.407240\pi$$
$$132$$ 0 0
$$133$$ 0.207751 0.0180143
$$134$$ 0 0
$$135$$ 0.527811 0.0454267
$$136$$ 0 0
$$137$$ −6.21983 −0.531396 −0.265698 0.964056i $$-0.585602\pi$$
−0.265698 + 0.964056i $$0.585602\pi$$
$$138$$ 0 0
$$139$$ 14.7071 1.24744 0.623719 0.781648i $$-0.285621\pi$$
0.623719 + 0.781648i $$0.285621\pi$$
$$140$$ 0 0
$$141$$ −16.5308 −1.39214
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −1.14377 −0.0949848
$$146$$ 0 0
$$147$$ −3.24698 −0.267806
$$148$$ 0 0
$$149$$ −4.33513 −0.355147 −0.177574 0.984108i $$-0.556825\pi$$
−0.177574 + 0.984108i $$0.556825\pi$$
$$150$$ 0 0
$$151$$ 3.94438 0.320989 0.160494 0.987037i $$-0.448691\pi$$
0.160494 + 0.987037i $$0.448691\pi$$
$$152$$ 0 0
$$153$$ 4.42327 0.357600
$$154$$ 0 0
$$155$$ 1.63773 0.131545
$$156$$ 0 0
$$157$$ 4.45473 0.355526 0.177763 0.984073i $$-0.443114\pi$$
0.177763 + 0.984073i $$0.443114\pi$$
$$158$$ 0 0
$$159$$ −23.3545 −1.85213
$$160$$ 0 0
$$161$$ 3.52111 0.277502
$$162$$ 0 0
$$163$$ 16.1588 1.26566 0.632829 0.774292i $$-0.281894\pi$$
0.632829 + 0.774292i $$0.281894\pi$$
$$164$$ 0 0
$$165$$ 2.35690 0.183484
$$166$$ 0 0
$$167$$ 16.1172 1.24719 0.623594 0.781749i $$-0.285672\pi$$
0.623594 + 0.781749i $$0.285672\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −0.180604 −0.0138111
$$172$$ 0 0
$$173$$ −21.5362 −1.63736 −0.818682 0.574247i $$-0.805295\pi$$
−0.818682 + 0.574247i $$0.805295\pi$$
$$174$$ 0 0
$$175$$ 11.6407 0.879955
$$176$$ 0 0
$$177$$ −15.2349 −1.14513
$$178$$ 0 0
$$179$$ −11.4330 −0.854540 −0.427270 0.904124i $$-0.640525\pi$$
−0.427270 + 0.904124i $$0.640525\pi$$
$$180$$ 0 0
$$181$$ 20.9705 1.55872 0.779361 0.626575i $$-0.215544\pi$$
0.779361 + 0.626575i $$0.215544\pi$$
$$182$$ 0 0
$$183$$ 7.80194 0.576736
$$184$$ 0 0
$$185$$ 1.40581 0.103357
$$186$$ 0 0
$$187$$ −9.16852 −0.670469
$$188$$ 0 0
$$189$$ 5.03684 0.366376
$$190$$ 0 0
$$191$$ 14.4373 1.04464 0.522322 0.852748i $$-0.325066\pi$$
0.522322 + 0.852748i $$0.325066\pi$$
$$192$$ 0 0
$$193$$ 13.5797 0.977489 0.488745 0.872427i $$-0.337455\pi$$
0.488745 + 0.872427i $$0.337455\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.560335 0.0399222 0.0199611 0.999801i $$-0.493646\pi$$
0.0199611 + 0.999801i $$0.493646\pi$$
$$198$$ 0 0
$$199$$ −11.4916 −0.814616 −0.407308 0.913291i $$-0.633532\pi$$
−0.407308 + 0.913291i $$0.633532\pi$$
$$200$$ 0 0
$$201$$ 17.2567 1.21719
$$202$$ 0 0
$$203$$ −10.9148 −0.766071
$$204$$ 0 0
$$205$$ −2.86294 −0.199956
$$206$$ 0 0
$$207$$ −3.06100 −0.212754
$$208$$ 0 0
$$209$$ 0.374354 0.0258946
$$210$$ 0 0
$$211$$ −8.78448 −0.604748 −0.302374 0.953189i $$-0.597779\pi$$
−0.302374 + 0.953189i $$0.597779\pi$$
$$212$$ 0 0
$$213$$ −19.4698 −1.33405
$$214$$ 0 0
$$215$$ −0.0730805 −0.00498405
$$216$$ 0 0
$$217$$ 15.6286 1.06094
$$218$$ 0 0
$$219$$ −15.1347 −1.02271
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −2.25906 −0.151278 −0.0756390 0.997135i $$-0.524100\pi$$
−0.0756390 + 0.997135i $$0.524100\pi$$
$$224$$ 0 0
$$225$$ −10.1196 −0.674640
$$226$$ 0 0
$$227$$ −6.96615 −0.462359 −0.231180 0.972911i $$-0.574259\pi$$
−0.231180 + 0.972911i $$0.574259\pi$$
$$228$$ 0 0
$$229$$ 24.1739 1.59746 0.798728 0.601692i $$-0.205507\pi$$
0.798728 + 0.601692i $$0.205507\pi$$
$$230$$ 0 0
$$231$$ 22.4916 1.47984
$$232$$ 0 0
$$233$$ −3.06100 −0.200533 −0.100266 0.994961i $$-0.531969\pi$$
−0.100266 + 0.994961i $$0.531969\pi$$
$$234$$ 0 0
$$235$$ 1.81700 0.118528
$$236$$ 0 0
$$237$$ −22.4034 −1.45526
$$238$$ 0 0
$$239$$ −25.1468 −1.62661 −0.813304 0.581839i $$-0.802333\pi$$
−0.813304 + 0.581839i $$0.802333\pi$$
$$240$$ 0 0
$$241$$ 20.2664 1.30547 0.652735 0.757586i $$-0.273621\pi$$
0.652735 + 0.757586i $$0.273621\pi$$
$$242$$ 0 0
$$243$$ −18.1903 −1.16691
$$244$$ 0 0
$$245$$ 0.356896 0.0228012
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 3.61596 0.229152
$$250$$ 0 0
$$251$$ 23.7211 1.49726 0.748631 0.662987i $$-0.230712\pi$$
0.748631 + 0.662987i $$0.230712\pi$$
$$252$$ 0 0
$$253$$ 6.34481 0.398895
$$254$$ 0 0
$$255$$ −1.19806 −0.0750256
$$256$$ 0 0
$$257$$ 14.2241 0.887278 0.443639 0.896206i $$-0.353687\pi$$
0.443639 + 0.896206i $$0.353687\pi$$
$$258$$ 0 0
$$259$$ 13.4155 0.833599
$$260$$ 0 0
$$261$$ 9.48858 0.587329
$$262$$ 0 0
$$263$$ 17.0954 1.05415 0.527075 0.849819i $$-0.323289\pi$$
0.527075 + 0.849819i $$0.323289\pi$$
$$264$$ 0 0
$$265$$ 2.56704 0.157692
$$266$$ 0 0
$$267$$ 6.48188 0.396684
$$268$$ 0 0
$$269$$ −6.46681 −0.394288 −0.197144 0.980374i $$-0.563167\pi$$
−0.197144 + 0.980374i $$0.563167\pi$$
$$270$$ 0 0
$$271$$ 6.44803 0.391690 0.195845 0.980635i $$-0.437255\pi$$
0.195845 + 0.980635i $$0.437255\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 20.9758 1.26489
$$276$$ 0 0
$$277$$ 13.4601 0.808739 0.404370 0.914596i $$-0.367491\pi$$
0.404370 + 0.914596i $$0.367491\pi$$
$$278$$ 0 0
$$279$$ −13.5864 −0.813398
$$280$$ 0 0
$$281$$ −5.03684 −0.300472 −0.150236 0.988650i $$-0.548003\pi$$
−0.150236 + 0.988650i $$0.548003\pi$$
$$282$$ 0 0
$$283$$ −22.1280 −1.31537 −0.657686 0.753293i $$-0.728464\pi$$
−0.657686 + 0.753293i $$0.728464\pi$$
$$284$$ 0 0
$$285$$ 0.0489173 0.00289761
$$286$$ 0 0
$$287$$ −27.3207 −1.61269
$$288$$ 0 0
$$289$$ −12.3394 −0.725849
$$290$$ 0 0
$$291$$ 18.1075 1.06148
$$292$$ 0 0
$$293$$ −14.9463 −0.873172 −0.436586 0.899663i $$-0.643813\pi$$
−0.436586 + 0.899663i $$0.643813\pi$$
$$294$$ 0 0
$$295$$ 1.67456 0.0974968
$$296$$ 0 0
$$297$$ 9.07606 0.526647
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −0.697398 −0.0401974
$$302$$ 0 0
$$303$$ −30.0073 −1.72387
$$304$$ 0 0
$$305$$ −0.857560 −0.0491037
$$306$$ 0 0
$$307$$ 19.1293 1.09177 0.545883 0.837861i $$-0.316194\pi$$
0.545883 + 0.837861i $$0.316194\pi$$
$$308$$ 0 0
$$309$$ −3.06100 −0.174134
$$310$$ 0 0
$$311$$ 0.269815 0.0152998 0.00764990 0.999971i $$-0.497565\pi$$
0.00764990 + 0.999971i $$0.497565\pi$$
$$312$$ 0 0
$$313$$ −23.3937 −1.32229 −0.661146 0.750257i $$-0.729930\pi$$
−0.661146 + 0.750257i $$0.729930\pi$$
$$314$$ 0 0
$$315$$ 1.19269 0.0672002
$$316$$ 0 0
$$317$$ 13.9952 0.786050 0.393025 0.919528i $$-0.371429\pi$$
0.393025 + 0.919528i $$0.371429\pi$$
$$318$$ 0 0
$$319$$ −19.6679 −1.10119
$$320$$ 0 0
$$321$$ −7.34481 −0.409948
$$322$$ 0 0
$$323$$ −0.190293 −0.0105882
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −35.2814 −1.95107
$$328$$ 0 0
$$329$$ 17.3394 0.955954
$$330$$ 0 0
$$331$$ 17.8213 0.979548 0.489774 0.871849i $$-0.337079\pi$$
0.489774 + 0.871849i $$0.337079\pi$$
$$332$$ 0 0
$$333$$ −11.6625 −0.639100
$$334$$ 0 0
$$335$$ −1.89679 −0.103633
$$336$$ 0 0
$$337$$ −27.8485 −1.51700 −0.758501 0.651672i $$-0.774068\pi$$
−0.758501 + 0.651672i $$0.774068\pi$$
$$338$$ 0 0
$$339$$ 27.0737 1.47044
$$340$$ 0 0
$$341$$ 28.1618 1.52505
$$342$$ 0 0
$$343$$ 19.9041 1.07472
$$344$$ 0 0
$$345$$ 0.829085 0.0446364
$$346$$ 0 0
$$347$$ −1.50365 −0.0807200 −0.0403600 0.999185i $$-0.512850\pi$$
−0.0403600 + 0.999185i $$0.512850\pi$$
$$348$$ 0 0
$$349$$ 14.1860 0.759358 0.379679 0.925118i $$-0.376034\pi$$
0.379679 + 0.925118i $$0.376034\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7.16852 0.381542 0.190771 0.981635i $$-0.438901\pi$$
0.190771 + 0.981635i $$0.438901\pi$$
$$354$$ 0 0
$$355$$ 2.14005 0.113582
$$356$$ 0 0
$$357$$ −11.4330 −0.605096
$$358$$ 0 0
$$359$$ 19.8853 1.04951 0.524753 0.851255i $$-0.324158\pi$$
0.524753 + 0.851255i $$0.324158\pi$$
$$360$$ 0 0
$$361$$ −18.9922 −0.999591
$$362$$ 0 0
$$363$$ 15.8116 0.829895
$$364$$ 0 0
$$365$$ 1.66355 0.0870740
$$366$$ 0 0
$$367$$ −1.08383 −0.0565757 −0.0282878 0.999600i $$-0.509006\pi$$
−0.0282878 + 0.999600i $$0.509006\pi$$
$$368$$ 0 0
$$369$$ 23.7506 1.23641
$$370$$ 0 0
$$371$$ 24.4969 1.27182
$$372$$ 0 0
$$373$$ −6.13036 −0.317418 −0.158709 0.987325i $$-0.550733\pi$$
−0.158709 + 0.987325i $$0.550733\pi$$
$$374$$ 0 0
$$375$$ 5.51573 0.284831
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −2.40880 −0.123732 −0.0618658 0.998084i $$-0.519705\pi$$
−0.0618658 + 0.998084i $$0.519705\pi$$
$$380$$ 0 0
$$381$$ 22.0368 1.12898
$$382$$ 0 0
$$383$$ −30.3913 −1.55292 −0.776462 0.630164i $$-0.782988\pi$$
−0.776462 + 0.630164i $$0.782988\pi$$
$$384$$ 0 0
$$385$$ −2.47219 −0.125994
$$386$$ 0 0
$$387$$ 0.606268 0.0308184
$$388$$ 0 0
$$389$$ −15.9409 −0.808237 −0.404118 0.914707i $$-0.632422\pi$$
−0.404118 + 0.914707i $$0.632422\pi$$
$$390$$ 0 0
$$391$$ −3.22521 −0.163106
$$392$$ 0 0
$$393$$ 14.7778 0.745440
$$394$$ 0 0
$$395$$ 2.46250 0.123902
$$396$$ 0 0
$$397$$ −16.9148 −0.848931 −0.424466 0.905444i $$-0.639538\pi$$
−0.424466 + 0.905444i $$0.639538\pi$$
$$398$$ 0 0
$$399$$ 0.466812 0.0233698
$$400$$ 0 0
$$401$$ −26.6625 −1.33146 −0.665730 0.746192i $$-0.731880\pi$$
−0.665730 + 0.746192i $$0.731880\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 2.70410 0.134368
$$406$$ 0 0
$$407$$ 24.1739 1.19826
$$408$$ 0 0
$$409$$ −28.5163 −1.41004 −0.705021 0.709187i $$-0.749062\pi$$
−0.705021 + 0.709187i $$0.749062\pi$$
$$410$$ 0 0
$$411$$ −13.9758 −0.689377
$$412$$ 0 0
$$413$$ 15.9801 0.786332
$$414$$ 0 0
$$415$$ −0.397452 −0.0195102
$$416$$ 0 0
$$417$$ 33.0465 1.61830
$$418$$ 0 0
$$419$$ 29.6093 1.44651 0.723253 0.690583i $$-0.242646\pi$$
0.723253 + 0.690583i $$0.242646\pi$$
$$420$$ 0 0
$$421$$ 11.6606 0.568301 0.284151 0.958780i $$-0.408288\pi$$
0.284151 + 0.958780i $$0.408288\pi$$
$$422$$ 0 0
$$423$$ −15.0737 −0.732907
$$424$$ 0 0
$$425$$ −10.6625 −0.517206
$$426$$ 0 0
$$427$$ −8.18359 −0.396032
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4.34913 0.209490 0.104745 0.994499i $$-0.466597\pi$$
0.104745 + 0.994499i $$0.466597\pi$$
$$432$$ 0 0
$$433$$ −14.3884 −0.691460 −0.345730 0.938334i $$-0.612369\pi$$
−0.345730 + 0.938334i $$0.612369\pi$$
$$434$$ 0 0
$$435$$ −2.57002 −0.123223
$$436$$ 0 0
$$437$$ 0.131687 0.00629942
$$438$$ 0 0
$$439$$ 20.2325 0.965645 0.482822 0.875718i $$-0.339612\pi$$
0.482822 + 0.875718i $$0.339612\pi$$
$$440$$ 0 0
$$441$$ −2.96077 −0.140989
$$442$$ 0 0
$$443$$ −8.12200 −0.385888 −0.192944 0.981210i $$-0.561804\pi$$
−0.192944 + 0.981210i $$0.561804\pi$$
$$444$$ 0 0
$$445$$ −0.712464 −0.0337740
$$446$$ 0 0
$$447$$ −9.74094 −0.460731
$$448$$ 0 0
$$449$$ −12.4916 −0.589513 −0.294757 0.955572i $$-0.595239\pi$$
−0.294757 + 0.955572i $$0.595239\pi$$
$$450$$ 0 0
$$451$$ −49.2301 −2.31816
$$452$$ 0 0
$$453$$ 8.86294 0.416417
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.98121 −0.279789 −0.139895 0.990166i $$-0.544676\pi$$
−0.139895 + 0.990166i $$0.544676\pi$$
$$458$$ 0 0
$$459$$ −4.61356 −0.215343
$$460$$ 0 0
$$461$$ 2.05669 0.0957895 0.0478947 0.998852i $$-0.484749\pi$$
0.0478947 + 0.998852i $$0.484749\pi$$
$$462$$ 0 0
$$463$$ −8.44935 −0.392675 −0.196337 0.980536i $$-0.562905\pi$$
−0.196337 + 0.980536i $$0.562905\pi$$
$$464$$ 0 0
$$465$$ 3.67994 0.170653
$$466$$ 0 0
$$467$$ −33.5139 −1.55084 −0.775420 0.631446i $$-0.782462\pi$$
−0.775420 + 0.631446i $$0.782462\pi$$
$$468$$ 0 0
$$469$$ −18.1008 −0.835818
$$470$$ 0 0
$$471$$ 10.0097 0.461222
$$472$$ 0 0
$$473$$ −1.25667 −0.0577817
$$474$$ 0 0
$$475$$ 0.435353 0.0199754
$$476$$ 0 0
$$477$$ −21.2959 −0.975072
$$478$$ 0 0
$$479$$ −24.7313 −1.13000 −0.565000 0.825091i $$-0.691124\pi$$
−0.565000 + 0.825091i $$0.691124\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 7.91185 0.360002
$$484$$ 0 0
$$485$$ −1.99031 −0.0903754
$$486$$ 0 0
$$487$$ −37.7555 −1.71087 −0.855433 0.517913i $$-0.826709\pi$$
−0.855433 + 0.517913i $$0.826709\pi$$
$$488$$ 0 0
$$489$$ 36.3086 1.64193
$$490$$ 0 0
$$491$$ −31.3110 −1.41304 −0.706522 0.707691i $$-0.749737\pi$$
−0.706522 + 0.707691i $$0.749737\pi$$
$$492$$ 0 0
$$493$$ 9.99761 0.450270
$$494$$ 0 0
$$495$$ 2.14914 0.0965969
$$496$$ 0 0
$$497$$ 20.4222 0.916061
$$498$$ 0 0
$$499$$ 21.4873 0.961902 0.480951 0.876748i $$-0.340292\pi$$
0.480951 + 0.876748i $$0.340292\pi$$
$$500$$ 0 0
$$501$$ 36.2150 1.61797
$$502$$ 0 0
$$503$$ −37.5924 −1.67616 −0.838081 0.545546i $$-0.816322\pi$$
−0.838081 + 0.545546i $$0.816322\pi$$
$$504$$ 0 0
$$505$$ 3.29829 0.146772
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 17.1075 0.758278 0.379139 0.925340i $$-0.376220\pi$$
0.379139 + 0.925340i $$0.376220\pi$$
$$510$$ 0 0
$$511$$ 15.8750 0.702269
$$512$$ 0 0
$$513$$ 0.188374 0.00831690
$$514$$ 0 0
$$515$$ 0.336454 0.0148259
$$516$$ 0 0
$$517$$ 31.2446 1.37414
$$518$$ 0 0
$$519$$ −48.3913 −2.12414
$$520$$ 0 0
$$521$$ −19.8465 −0.869493 −0.434746 0.900553i $$-0.643162\pi$$
−0.434746 + 0.900553i $$0.643162\pi$$
$$522$$ 0 0
$$523$$ 11.4300 0.499798 0.249899 0.968272i $$-0.419603\pi$$
0.249899 + 0.968272i $$0.419603\pi$$
$$524$$ 0 0
$$525$$ 26.1564 1.14156
$$526$$ 0 0
$$527$$ −14.3153 −0.623583
$$528$$ 0 0
$$529$$ −20.7681 −0.902960
$$530$$ 0 0
$$531$$ −13.8920 −0.602862
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0.807315 0.0349033
$$536$$ 0 0
$$537$$ −25.6896 −1.10859
$$538$$ 0 0
$$539$$ 6.13706 0.264342
$$540$$ 0 0
$$541$$ −16.1884 −0.695993 −0.347996 0.937496i $$-0.613138\pi$$
−0.347996 + 0.937496i $$0.613138\pi$$
$$542$$ 0 0
$$543$$ 47.1202 2.02212
$$544$$ 0 0
$$545$$ 3.87800 0.166115
$$546$$ 0 0
$$547$$ −5.33081 −0.227929 −0.113965 0.993485i $$-0.536355\pi$$
−0.113965 + 0.993485i $$0.536355\pi$$
$$548$$ 0 0
$$549$$ 7.11423 0.303628
$$550$$ 0 0
$$551$$ −0.408206 −0.0173902
$$552$$ 0 0
$$553$$ 23.4993 0.999293
$$554$$ 0 0
$$555$$ 3.15883 0.134085
$$556$$ 0 0
$$557$$ −7.39075 −0.313156 −0.156578 0.987666i $$-0.550046\pi$$
−0.156578 + 0.987666i $$0.550046\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −20.6015 −0.869795
$$562$$ 0 0
$$563$$ 9.47889 0.399488 0.199744 0.979848i $$-0.435989\pi$$
0.199744 + 0.979848i $$0.435989\pi$$
$$564$$ 0 0
$$565$$ −2.97584 −0.125194
$$566$$ 0 0
$$567$$ 25.8049 1.08370
$$568$$ 0 0
$$569$$ −10.1438 −0.425249 −0.212624 0.977134i $$-0.568201\pi$$
−0.212624 + 0.977134i $$0.568201\pi$$
$$570$$ 0 0
$$571$$ 14.0925 0.589751 0.294876 0.955536i $$-0.404722\pi$$
0.294876 + 0.955536i $$0.404722\pi$$
$$572$$ 0 0
$$573$$ 32.4403 1.35521
$$574$$ 0 0
$$575$$ 7.37867 0.307712
$$576$$ 0 0
$$577$$ −25.1545 −1.04720 −0.523598 0.851965i $$-0.675411\pi$$
−0.523598 + 0.851965i $$0.675411\pi$$
$$578$$ 0 0
$$579$$ 30.5133 1.26809
$$580$$ 0 0
$$581$$ −3.79284 −0.157354
$$582$$ 0 0
$$583$$ 44.1420 1.82817
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 43.8353 1.80928 0.904639 0.426180i $$-0.140141\pi$$
0.904639 + 0.426180i $$0.140141\pi$$
$$588$$ 0 0
$$589$$ 0.584498 0.0240838
$$590$$ 0 0
$$591$$ 1.25906 0.0517909
$$592$$ 0 0
$$593$$ 24.9965 1.02648 0.513242 0.858244i $$-0.328444\pi$$
0.513242 + 0.858244i $$0.328444\pi$$
$$594$$ 0 0
$$595$$ 1.25667 0.0515184
$$596$$ 0 0
$$597$$ −25.8213 −1.05680
$$598$$ 0 0
$$599$$ 6.24027 0.254971 0.127485 0.991840i $$-0.459309\pi$$
0.127485 + 0.991840i $$0.459309\pi$$
$$600$$ 0 0
$$601$$ 6.32975 0.258196 0.129098 0.991632i $$-0.458792\pi$$
0.129098 + 0.991632i $$0.458792\pi$$
$$602$$ 0 0
$$603$$ 15.7356 0.640802
$$604$$ 0 0
$$605$$ −1.73795 −0.0706579
$$606$$ 0 0
$$607$$ 43.6480 1.77162 0.885809 0.464050i $$-0.153604\pi$$
0.885809 + 0.464050i $$0.153604\pi$$
$$608$$ 0 0
$$609$$ −24.5254 −0.993820
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 25.9541 1.04827 0.524137 0.851634i $$-0.324388\pi$$
0.524137 + 0.851634i $$0.324388\pi$$
$$614$$ 0 0
$$615$$ −6.43296 −0.259402
$$616$$ 0 0
$$617$$ −45.9396 −1.84946 −0.924729 0.380626i $$-0.875709\pi$$
−0.924729 + 0.380626i $$0.875709\pi$$
$$618$$ 0 0
$$619$$ 6.73556 0.270725 0.135363 0.990796i $$-0.456780\pi$$
0.135363 + 0.990796i $$0.456780\pi$$
$$620$$ 0 0
$$621$$ 3.19269 0.128118
$$622$$ 0 0
$$623$$ −6.79895 −0.272394
$$624$$ 0 0
$$625$$ 24.0887 0.963549
$$626$$ 0 0
$$627$$ 0.841166 0.0335930
$$628$$ 0 0
$$629$$ −12.2881 −0.489960
$$630$$ 0 0
$$631$$ 45.0998 1.79539 0.897696 0.440614i $$-0.145239\pi$$
0.897696 + 0.440614i $$0.145239\pi$$
$$632$$ 0 0
$$633$$ −19.7385 −0.784537
$$634$$ 0 0
$$635$$ −2.42221 −0.0961223
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −17.7536 −0.702322
$$640$$ 0 0
$$641$$ 32.5821 1.28692 0.643458 0.765482i $$-0.277499\pi$$
0.643458 + 0.765482i $$0.277499\pi$$
$$642$$ 0 0
$$643$$ 25.5754 1.00860 0.504298 0.863530i $$-0.331751\pi$$
0.504298 + 0.863530i $$0.331751\pi$$
$$644$$ 0 0
$$645$$ −0.164210 −0.00646578
$$646$$ 0 0
$$647$$ 30.1715 1.18616 0.593082 0.805142i $$-0.297911\pi$$
0.593082 + 0.805142i $$0.297911\pi$$
$$648$$ 0 0
$$649$$ 28.7952 1.13031
$$650$$ 0 0
$$651$$ 35.1172 1.37635
$$652$$ 0 0
$$653$$ 36.9028 1.44412 0.722058 0.691832i $$-0.243196\pi$$
0.722058 + 0.691832i $$0.243196\pi$$
$$654$$ 0 0
$$655$$ −1.62432 −0.0634673
$$656$$ 0 0
$$657$$ −13.8006 −0.538413
$$658$$ 0 0
$$659$$ −23.6866 −0.922701 −0.461350 0.887218i $$-0.652635\pi$$
−0.461350 + 0.887218i $$0.652635\pi$$
$$660$$ 0 0
$$661$$ 31.7590 1.23528 0.617641 0.786460i $$-0.288089\pi$$
0.617641 + 0.786460i $$0.288089\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −0.0513102 −0.00198973
$$666$$ 0 0
$$667$$ −6.91856 −0.267888
$$668$$ 0 0
$$669$$ −5.07606 −0.196252
$$670$$ 0 0
$$671$$ −14.7463 −0.569275
$$672$$ 0 0
$$673$$ −7.50232 −0.289193 −0.144597 0.989491i $$-0.546188\pi$$
−0.144597 + 0.989491i $$0.546188\pi$$
$$674$$ 0 0
$$675$$ 10.5550 0.406261
$$676$$ 0 0
$$677$$ −35.0315 −1.34637 −0.673184 0.739475i $$-0.735074\pi$$
−0.673184 + 0.739475i $$0.735074\pi$$
$$678$$ 0 0
$$679$$ −18.9933 −0.728896
$$680$$ 0 0
$$681$$ −15.6528 −0.599816
$$682$$ 0 0
$$683$$ −24.0834 −0.921524 −0.460762 0.887524i $$-0.652424\pi$$
−0.460762 + 0.887524i $$0.652424\pi$$
$$684$$ 0 0
$$685$$ 1.53617 0.0586941
$$686$$ 0 0
$$687$$ 54.3183 2.07237
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.01447 0.0766342 0.0383171 0.999266i $$-0.487800\pi$$
0.0383171 + 0.999266i $$0.487800\pi$$
$$692$$ 0 0
$$693$$ 20.5090 0.779073
$$694$$ 0 0
$$695$$ −3.63235 −0.137783
$$696$$ 0 0
$$697$$ 25.0248 0.947880
$$698$$ 0 0
$$699$$ −6.87800 −0.260150
$$700$$ 0 0
$$701$$ −48.8189 −1.84387 −0.921933 0.387350i $$-0.873390\pi$$
−0.921933 + 0.387350i $$0.873390\pi$$
$$702$$ 0 0
$$703$$ 0.501729 0.0189231
$$704$$ 0 0
$$705$$ 4.08277 0.153766
$$706$$ 0 0
$$707$$ 31.4752 1.18375
$$708$$ 0 0
$$709$$ 20.8060 0.781385 0.390693 0.920521i $$-0.372236\pi$$
0.390693 + 0.920521i $$0.372236\pi$$
$$710$$ 0 0
$$711$$ −20.4286 −0.766134
$$712$$ 0 0
$$713$$ 9.90648 0.371000
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −56.5042 −2.11019
$$718$$ 0 0
$$719$$ −21.4306 −0.799225 −0.399613 0.916684i $$-0.630855\pi$$
−0.399613 + 0.916684i $$0.630855\pi$$
$$720$$ 0 0
$$721$$ 3.21073 0.119574
$$722$$ 0 0
$$723$$ 45.5381 1.69358
$$724$$ 0 0
$$725$$ −22.8726 −0.849468
$$726$$ 0 0
$$727$$ −13.4862 −0.500175 −0.250088 0.968223i $$-0.580459\pi$$
−0.250088 + 0.968223i $$0.580459\pi$$
$$728$$ 0 0
$$729$$ −8.02715 −0.297302
$$730$$ 0 0
$$731$$ 0.638792 0.0236266
$$732$$ 0 0
$$733$$ 43.5424 1.60828 0.804138 0.594443i $$-0.202627\pi$$
0.804138 + 0.594443i $$0.202627\pi$$
$$734$$ 0 0
$$735$$ 0.801938 0.0295799
$$736$$ 0 0
$$737$$ −32.6165 −1.20145
$$738$$ 0 0
$$739$$ 20.0543 0.737709 0.368855 0.929487i $$-0.379750\pi$$
0.368855 + 0.929487i $$0.379750\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −33.1685 −1.21684 −0.608418 0.793617i $$-0.708195\pi$$
−0.608418 + 0.793617i $$0.708195\pi$$
$$744$$ 0 0
$$745$$ 1.07069 0.0392270
$$746$$ 0 0
$$747$$ 3.29722 0.120639
$$748$$ 0 0
$$749$$ 7.70410 0.281502
$$750$$ 0 0
$$751$$ −39.2814 −1.43340 −0.716700 0.697382i $$-0.754348\pi$$
−0.716700 + 0.697382i $$0.754348\pi$$
$$752$$ 0 0
$$753$$ 53.3008 1.94239
$$754$$ 0 0
$$755$$ −0.974181 −0.0354541
$$756$$ 0 0
$$757$$ −46.6426 −1.69526 −0.847628 0.530592i $$-0.821970\pi$$
−0.847628 + 0.530592i $$0.821970\pi$$
$$758$$ 0 0
$$759$$ 14.2567 0.517484
$$760$$ 0 0
$$761$$ 21.8984 0.793818 0.396909 0.917858i $$-0.370083\pi$$
0.396909 + 0.917858i $$0.370083\pi$$
$$762$$ 0 0
$$763$$ 37.0073 1.33975
$$764$$ 0 0
$$765$$ −1.09246 −0.0394979
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −46.7096 −1.68439 −0.842196 0.539172i $$-0.818737\pi$$
−0.842196 + 0.539172i $$0.818737\pi$$
$$770$$ 0 0
$$771$$ 31.9614 1.15106
$$772$$ 0 0
$$773$$ 30.2416 1.08771 0.543857 0.839178i $$-0.316963\pi$$
0.543857 + 0.839178i $$0.316963\pi$$
$$774$$ 0 0
$$775$$ 32.7506 1.17644
$$776$$ 0 0
$$777$$ 30.1444 1.08142
$$778$$ 0 0
$$779$$ −1.02177 −0.0366087
$$780$$ 0 0
$$781$$ 36.7995 1.31679
$$782$$ 0 0
$$783$$ −9.89679 −0.353682
$$784$$ 0 0
$$785$$ −1.10023 −0.0392688
$$786$$ 0 0
$$787$$ 28.7023 1.02313 0.511563 0.859246i $$-0.329067\pi$$
0.511563 + 0.859246i $$0.329067\pi$$
$$788$$ 0 0
$$789$$ 38.4131 1.36754
$$790$$ 0 0
$$791$$ −28.3980 −1.00972
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 5.76809 0.204573
$$796$$ 0 0
$$797$$ −18.5418 −0.656785 −0.328392 0.944541i $$-0.606507\pi$$
−0.328392 + 0.944541i $$0.606507\pi$$
$$798$$ 0 0
$$799$$ −15.8823 −0.561876
$$800$$ 0 0
$$801$$ 5.91053 0.208838
$$802$$ 0 0
$$803$$ 28.6058 1.00948
$$804$$ 0 0
$$805$$ −0.869641 −0.0306508
$$806$$ 0 0
$$807$$ −14.5308 −0.511508
$$808$$ 0 0
$$809$$ −10.0677 −0.353962 −0.176981 0.984214i $$-0.556633\pi$$
−0.176981 + 0.984214i $$0.556633\pi$$
$$810$$ 0 0
$$811$$ −10.0285 −0.352147 −0.176074 0.984377i $$-0.556340\pi$$
−0.176074 + 0.984377i $$0.556340\pi$$
$$812$$ 0 0
$$813$$ 14.4886 0.508137
$$814$$ 0 0
$$815$$ −3.99090 −0.139795
$$816$$ 0 0
$$817$$ −0.0260821 −0.000912498 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26.1704 0.913355 0.456677 0.889632i $$-0.349039\pi$$
0.456677 + 0.889632i $$0.349039\pi$$
$$822$$ 0 0
$$823$$ −1.82238 −0.0635242 −0.0317621 0.999495i $$-0.510112\pi$$
−0.0317621 + 0.999495i $$0.510112\pi$$
$$824$$ 0 0
$$825$$ 47.1323 1.64094
$$826$$ 0 0
$$827$$ −32.2941 −1.12298 −0.561488 0.827485i $$-0.689771\pi$$
−0.561488 + 0.827485i $$0.689771\pi$$
$$828$$ 0 0
$$829$$ 15.1002 0.524453 0.262226 0.965006i $$-0.415543\pi$$
0.262226 + 0.965006i $$0.415543\pi$$
$$830$$ 0 0
$$831$$ 30.2446 1.04917
$$832$$ 0 0
$$833$$ −3.11960 −0.108088
$$834$$ 0 0
$$835$$ −3.98062 −0.137755
$$836$$ 0 0
$$837$$ 14.1709 0.489818
$$838$$ 0 0
$$839$$ −32.9965 −1.13917 −0.569584 0.821933i $$-0.692896\pi$$
−0.569584 + 0.821933i $$0.692896\pi$$
$$840$$ 0 0
$$841$$ −7.55363 −0.260470
$$842$$ 0 0
$$843$$ −11.3177 −0.389801
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −16.5851 −0.569870
$$848$$ 0 0
$$849$$ −49.7211 −1.70642
$$850$$ 0 0
$$851$$ 8.50365 0.291501
$$852$$ 0 0
$$853$$ 37.7802 1.29357 0.646784 0.762673i $$-0.276113\pi$$
0.646784 + 0.762673i $$0.276113\pi$$
$$854$$ 0 0
$$855$$ 0.0446055 0.00152547
$$856$$ 0 0
$$857$$ 27.3623 0.934677 0.467339 0.884078i $$-0.345213\pi$$
0.467339 + 0.884078i $$0.345213\pi$$
$$858$$ 0 0
$$859$$ 20.0629 0.684538 0.342269 0.939602i $$-0.388805\pi$$
0.342269 + 0.939602i $$0.388805\pi$$
$$860$$ 0 0
$$861$$ −61.3889 −2.09213
$$862$$ 0 0
$$863$$ −6.14483 −0.209173 −0.104586 0.994516i $$-0.533352\pi$$
−0.104586 + 0.994516i $$0.533352\pi$$
$$864$$ 0 0
$$865$$ 5.31900 0.180851
$$866$$ 0 0
$$867$$ −27.7265 −0.941640
$$868$$ 0 0
$$869$$ 42.3443 1.43643
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 16.5114 0.558827
$$874$$ 0 0
$$875$$ −5.78554 −0.195587
$$876$$ 0 0
$$877$$ 13.5077 0.456123 0.228061 0.973647i $$-0.426761\pi$$
0.228061 + 0.973647i $$0.426761\pi$$
$$878$$ 0 0
$$879$$ −33.5840 −1.13276
$$880$$ 0 0
$$881$$ −5.23431 −0.176348 −0.0881741 0.996105i $$-0.528103\pi$$
−0.0881741 + 0.996105i $$0.528103\pi$$
$$882$$ 0 0
$$883$$ 4.57301 0.153894 0.0769470 0.997035i $$-0.475483\pi$$
0.0769470 + 0.997035i $$0.475483\pi$$
$$884$$ 0 0
$$885$$ 3.76271 0.126482
$$886$$ 0 0
$$887$$ 1.64071 0.0550897 0.0275448 0.999621i $$-0.491231\pi$$
0.0275448 + 0.999621i $$0.491231\pi$$
$$888$$ 0 0
$$889$$ −23.1148 −0.775246
$$890$$ 0 0
$$891$$ 46.4989 1.55777
$$892$$ 0 0
$$893$$ 0.648481 0.0217006
$$894$$ 0 0
$$895$$ 2.82371 0.0943861
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −30.7084 −1.02418
$$900$$ 0 0
$$901$$ −22.4383 −0.747529
$$902$$ 0 0
$$903$$ −1.56704 −0.0521478
$$904$$ 0 0
$$905$$ −5.17928 −0.172165
$$906$$ 0 0
$$907$$ −8.10215 −0.269027 −0.134514 0.990912i $$-0.542947\pi$$
−0.134514 + 0.990912i $$0.542947\pi$$
$$908$$ 0 0
$$909$$ −27.3623 −0.907549
$$910$$ 0 0
$$911$$ 9.18119 0.304187 0.152093 0.988366i $$-0.451399\pi$$
0.152093 + 0.988366i $$0.451399\pi$$
$$912$$ 0 0
$$913$$ −6.83446 −0.226188
$$914$$ 0 0
$$915$$ −1.92692 −0.0637020
$$916$$ 0 0
$$917$$ −15.5007 −0.511877
$$918$$ 0 0
$$919$$ −27.5036 −0.907262 −0.453631 0.891190i $$-0.649872\pi$$
−0.453631 + 0.891190i $$0.649872\pi$$
$$920$$ 0 0
$$921$$ 42.9831 1.41634
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 28.1129 0.924346
$$926$$ 0 0
$$927$$ −2.79118 −0.0916745
$$928$$ 0 0
$$929$$ −24.2131 −0.794407 −0.397203 0.917731i $$-0.630019\pi$$
−0.397203 + 0.917731i $$0.630019\pi$$
$$930$$ 0 0
$$931$$ 0.127375 0.00417454
$$932$$ 0 0
$$933$$ 0.606268 0.0198483
$$934$$ 0 0
$$935$$ 2.26444 0.0740550
$$936$$ 0 0
$$937$$ 11.1830 0.365333 0.182666 0.983175i $$-0.441527\pi$$
0.182666 + 0.983175i $$0.441527\pi$$
$$938$$ 0 0
$$939$$ −52.5652 −1.71540
$$940$$ 0 0
$$941$$ 15.9638 0.520404 0.260202 0.965554i $$-0.416211\pi$$
0.260202 + 0.965554i $$0.416211\pi$$
$$942$$ 0 0
$$943$$ −17.3177 −0.563941
$$944$$ 0 0
$$945$$ −1.24400 −0.0404672
$$946$$ 0 0
$$947$$ 6.51466 0.211698 0.105849 0.994382i $$-0.466244\pi$$
0.105849 + 0.994382i $$0.466244\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 31.4470 1.01974
$$952$$ 0 0
$$953$$ 47.6469 1.54344 0.771718 0.635965i $$-0.219398\pi$$
0.771718 + 0.635965i $$0.219398\pi$$
$$954$$ 0 0
$$955$$ −3.56571 −0.115384
$$956$$ 0 0
$$957$$ −44.1933 −1.42857
$$958$$ 0 0
$$959$$ 14.6595 0.473380
$$960$$ 0 0
$$961$$ 12.9705 0.418402
$$962$$ 0 0
$$963$$ −6.69740 −0.215821
$$964$$ 0 0
$$965$$ −3.35391 −0.107966
$$966$$ 0 0
$$967$$ −43.8122 −1.40891 −0.704453 0.709751i $$-0.748808\pi$$
−0.704453 + 0.709751i $$0.748808\pi$$
$$968$$ 0 0
$$969$$ −0.427583 −0.0137360
$$970$$ 0 0
$$971$$ −4.29483 −0.137828 −0.0689139 0.997623i $$-0.521953\pi$$
−0.0689139 + 0.997623i $$0.521953\pi$$
$$972$$ 0 0
$$973$$ −34.6631 −1.11125
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −26.8019 −0.857470 −0.428735 0.903430i $$-0.641041\pi$$
−0.428735 + 0.903430i $$0.641041\pi$$
$$978$$ 0 0
$$979$$ −12.2513 −0.391553
$$980$$ 0 0
$$981$$ −32.1715 −1.02716
$$982$$ 0 0
$$983$$ 27.2495 0.869124 0.434562 0.900642i $$-0.356903\pi$$
0.434562 + 0.900642i $$0.356903\pi$$
$$984$$ 0 0
$$985$$ −0.138391 −0.00440951
$$986$$ 0 0
$$987$$ 38.9614 1.24015
$$988$$ 0 0
$$989$$ −0.442058 −0.0140566
$$990$$ 0 0
$$991$$ −24.3889 −0.774740 −0.387370 0.921924i $$-0.626616\pi$$
−0.387370 + 0.921924i $$0.626616\pi$$
$$992$$ 0 0
$$993$$ 40.0441 1.27076
$$994$$ 0 0
$$995$$ 2.83818 0.0899764
$$996$$ 0 0
$$997$$ 31.3207 0.991935 0.495967 0.868341i $$-0.334814\pi$$
0.495967 + 0.868341i $$0.334814\pi$$
$$998$$ 0 0
$$999$$ 12.1642 0.384859
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2704.2.a.ba.1.3 3
4.3 odd 2 169.2.a.c.1.1 yes 3
12.11 even 2 1521.2.a.o.1.3 3
13.5 odd 4 2704.2.f.o.337.6 6
13.8 odd 4 2704.2.f.o.337.5 6
13.12 even 2 2704.2.a.z.1.3 3
20.19 odd 2 4225.2.a.bb.1.3 3
28.27 even 2 8281.2.a.bj.1.1 3
52.3 odd 6 169.2.c.b.22.3 6
52.7 even 12 169.2.e.b.23.2 12
52.11 even 12 169.2.e.b.147.5 12
52.15 even 12 169.2.e.b.147.2 12
52.19 even 12 169.2.e.b.23.5 12
52.23 odd 6 169.2.c.c.22.1 6
52.31 even 4 169.2.b.b.168.5 6
52.35 odd 6 169.2.c.b.146.3 6
52.43 odd 6 169.2.c.c.146.1 6
52.47 even 4 169.2.b.b.168.2 6
52.51 odd 2 169.2.a.b.1.3 3
156.47 odd 4 1521.2.b.l.1351.5 6
156.83 odd 4 1521.2.b.l.1351.2 6
156.155 even 2 1521.2.a.r.1.1 3
260.259 odd 2 4225.2.a.bg.1.1 3
364.363 even 2 8281.2.a.bf.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 52.51 odd 2
169.2.a.c.1.1 yes 3 4.3 odd 2
169.2.b.b.168.2 6 52.47 even 4
169.2.b.b.168.5 6 52.31 even 4
169.2.c.b.22.3 6 52.3 odd 6
169.2.c.b.146.3 6 52.35 odd 6
169.2.c.c.22.1 6 52.23 odd 6
169.2.c.c.146.1 6 52.43 odd 6
169.2.e.b.23.2 12 52.7 even 12
169.2.e.b.23.5 12 52.19 even 12
169.2.e.b.147.2 12 52.15 even 12
169.2.e.b.147.5 12 52.11 even 12
1521.2.a.o.1.3 3 12.11 even 2
1521.2.a.r.1.1 3 156.155 even 2
1521.2.b.l.1351.2 6 156.83 odd 4
1521.2.b.l.1351.5 6 156.47 odd 4
2704.2.a.z.1.3 3 13.12 even 2
2704.2.a.ba.1.3 3 1.1 even 1 trivial
2704.2.f.o.337.5 6 13.8 odd 4
2704.2.f.o.337.6 6 13.5 odd 4
4225.2.a.bb.1.3 3 20.19 odd 2
4225.2.a.bg.1.1 3 260.259 odd 2
8281.2.a.bf.1.3 3 364.363 even 2
8281.2.a.bj.1.1 3 28.27 even 2