# Properties

 Label 3040.2.a.u.1.1 Level $3040$ Weight $2$ Character 3040.1 Self dual yes Analytic conductor $24.275$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.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: $$24.2745222145$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.387268.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2$$ x^5 - x^4 - 7*x^3 + 4*x^2 + 12*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.88930$$ of defining polynomial Character $$\chi$$ $$=$$ 3040.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.45876 q^{3} -1.00000 q^{5} +4.18444 q^{7} +8.96304 q^{9} +O(q^{10})$$ $$q-3.45876 q^{3} -1.00000 q^{5} +4.18444 q^{7} +8.96304 q^{9} +1.91967 q^{11} -5.31770 q^{13} +3.45876 q^{15} -6.96519 q^{17} +1.00000 q^{19} -14.4730 q^{21} -2.55761 q^{23} +1.00000 q^{25} -20.6248 q^{27} +2.90445 q^{29} +4.26651 q^{31} -6.63968 q^{33} -4.18444 q^{35} +6.07710 q^{37} +18.3927 q^{39} -6.91753 q^{41} +3.64887 q^{43} -8.96304 q^{45} +6.29283 q^{47} +10.5095 q^{49} +24.0909 q^{51} -12.7150 q^{53} -1.91967 q^{55} -3.45876 q^{57} +5.24303 q^{59} -3.37102 q^{61} +37.5053 q^{63} +5.31770 q^{65} +8.20081 q^{67} +8.84616 q^{69} -16.3533 q^{71} -3.30133 q^{73} -3.45876 q^{75} +8.03274 q^{77} -1.02418 q^{79} +44.4470 q^{81} +4.34256 q^{83} +6.96519 q^{85} -10.0458 q^{87} -1.45564 q^{89} -22.2516 q^{91} -14.7569 q^{93} -1.00000 q^{95} -8.51561 q^{97} +17.2061 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10})$$ 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 $$5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37} + 18 q^{39} - 8 q^{41} - 12 q^{43} - 7 q^{45} + 16 q^{47} + 7 q^{49} + 14 q^{51} - 18 q^{53} + 2 q^{55} - 4 q^{57} - 8 q^{59} + 2 q^{61} + 36 q^{63} + 4 q^{65} - 10 q^{67} - 22 q^{69} - 18 q^{71} - 28 q^{73} - 4 q^{75} - 28 q^{77} + 14 q^{79} + 25 q^{81} - 8 q^{83} + 12 q^{85} + 24 q^{87} - 30 q^{89} - 28 q^{91} - 24 q^{93} - 5 q^{95} - 18 q^{97} + 14 q^{99}+O(q^{100})$$ 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^11 - 4 * q^13 + 4 * q^15 - 12 * q^17 + 5 * q^19 - 10 * q^21 + 8 * q^23 + 5 * q^25 - 16 * q^27 - 6 * q^29 + 10 * q^31 - 18 * q^33 - 4 * q^35 - 6 * q^37 + 18 * q^39 - 8 * q^41 - 12 * q^43 - 7 * q^45 + 16 * q^47 + 7 * q^49 + 14 * q^51 - 18 * q^53 + 2 * q^55 - 4 * q^57 - 8 * q^59 + 2 * q^61 + 36 * q^63 + 4 * q^65 - 10 * q^67 - 22 * q^69 - 18 * q^71 - 28 * q^73 - 4 * q^75 - 28 * q^77 + 14 * q^79 + 25 * q^81 - 8 * q^83 + 12 * q^85 + 24 * q^87 - 30 * q^89 - 28 * q^91 - 24 * q^93 - 5 * q^95 - 18 * q^97 + 14 * 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
$$3$$ −3.45876 −1.99692 −0.998459 0.0554941i $$-0.982327\pi$$
−0.998459 + 0.0554941i $$0.982327\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.18444 1.58157 0.790785 0.612094i $$-0.209673\pi$$
0.790785 + 0.612094i $$0.209673\pi$$
$$8$$ 0 0
$$9$$ 8.96304 2.98768
$$10$$ 0 0
$$11$$ 1.91967 0.578802 0.289401 0.957208i $$-0.406544\pi$$
0.289401 + 0.957208i $$0.406544\pi$$
$$12$$ 0 0
$$13$$ −5.31770 −1.47486 −0.737432 0.675421i $$-0.763962\pi$$
−0.737432 + 0.675421i $$0.763962\pi$$
$$14$$ 0 0
$$15$$ 3.45876 0.893049
$$16$$ 0 0
$$17$$ −6.96519 −1.68931 −0.844653 0.535314i $$-0.820193\pi$$
−0.844653 + 0.535314i $$0.820193\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −14.4730 −3.15827
$$22$$ 0 0
$$23$$ −2.55761 −0.533298 −0.266649 0.963794i $$-0.585916\pi$$
−0.266649 + 0.963794i $$0.585916\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −20.6248 −3.96924
$$28$$ 0 0
$$29$$ 2.90445 0.539343 0.269672 0.962952i $$-0.413085\pi$$
0.269672 + 0.962952i $$0.413085\pi$$
$$30$$ 0 0
$$31$$ 4.26651 0.766289 0.383144 0.923688i $$-0.374841\pi$$
0.383144 + 0.923688i $$0.374841\pi$$
$$32$$ 0 0
$$33$$ −6.63968 −1.15582
$$34$$ 0 0
$$35$$ −4.18444 −0.707300
$$36$$ 0 0
$$37$$ 6.07710 0.999070 0.499535 0.866294i $$-0.333504\pi$$
0.499535 + 0.866294i $$0.333504\pi$$
$$38$$ 0 0
$$39$$ 18.3927 2.94518
$$40$$ 0 0
$$41$$ −6.91753 −1.08034 −0.540168 0.841557i $$-0.681639\pi$$
−0.540168 + 0.841557i $$0.681639\pi$$
$$42$$ 0 0
$$43$$ 3.64887 0.556448 0.278224 0.960516i $$-0.410254\pi$$
0.278224 + 0.960516i $$0.410254\pi$$
$$44$$ 0 0
$$45$$ −8.96304 −1.33613
$$46$$ 0 0
$$47$$ 6.29283 0.917904 0.458952 0.888461i $$-0.348225\pi$$
0.458952 + 0.888461i $$0.348225\pi$$
$$48$$ 0 0
$$49$$ 10.5095 1.50136
$$50$$ 0 0
$$51$$ 24.0909 3.37341
$$52$$ 0 0
$$53$$ −12.7150 −1.74655 −0.873273 0.487232i $$-0.838007\pi$$
−0.873273 + 0.487232i $$0.838007\pi$$
$$54$$ 0 0
$$55$$ −1.91967 −0.258848
$$56$$ 0 0
$$57$$ −3.45876 −0.458124
$$58$$ 0 0
$$59$$ 5.24303 0.682585 0.341292 0.939957i $$-0.389135\pi$$
0.341292 + 0.939957i $$0.389135\pi$$
$$60$$ 0 0
$$61$$ −3.37102 −0.431615 −0.215808 0.976436i $$-0.569238\pi$$
−0.215808 + 0.976436i $$0.569238\pi$$
$$62$$ 0 0
$$63$$ 37.5053 4.72523
$$64$$ 0 0
$$65$$ 5.31770 0.659579
$$66$$ 0 0
$$67$$ 8.20081 1.00189 0.500944 0.865480i $$-0.332986\pi$$
0.500944 + 0.865480i $$0.332986\pi$$
$$68$$ 0 0
$$69$$ 8.84616 1.06495
$$70$$ 0 0
$$71$$ −16.3533 −1.94078 −0.970388 0.241552i $$-0.922344\pi$$
−0.970388 + 0.241552i $$0.922344\pi$$
$$72$$ 0 0
$$73$$ −3.30133 −0.386391 −0.193196 0.981160i $$-0.561885\pi$$
−0.193196 + 0.981160i $$0.561885\pi$$
$$74$$ 0 0
$$75$$ −3.45876 −0.399384
$$76$$ 0 0
$$77$$ 8.03274 0.915416
$$78$$ 0 0
$$79$$ −1.02418 −0.115229 −0.0576145 0.998339i $$-0.518349\pi$$
−0.0576145 + 0.998339i $$0.518349\pi$$
$$80$$ 0 0
$$81$$ 44.4470 4.93856
$$82$$ 0 0
$$83$$ 4.34256 0.476658 0.238329 0.971184i $$-0.423400\pi$$
0.238329 + 0.971184i $$0.423400\pi$$
$$84$$ 0 0
$$85$$ 6.96519 0.755481
$$86$$ 0 0
$$87$$ −10.0458 −1.07702
$$88$$ 0 0
$$89$$ −1.45564 −0.154297 −0.0771487 0.997020i $$-0.524582\pi$$
−0.0771487 + 0.997020i $$0.524582\pi$$
$$90$$ 0 0
$$91$$ −22.2516 −2.33260
$$92$$ 0 0
$$93$$ −14.7569 −1.53022
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −8.51561 −0.864630 −0.432315 0.901723i $$-0.642303\pi$$
−0.432315 + 0.901723i $$0.642303\pi$$
$$98$$ 0 0
$$99$$ 17.2061 1.72928
$$100$$ 0 0
$$101$$ −2.08033 −0.207001 −0.103500 0.994629i $$-0.533004\pi$$
−0.103500 + 0.994629i $$0.533004\pi$$
$$102$$ 0 0
$$103$$ 14.2598 1.40506 0.702530 0.711654i $$-0.252054\pi$$
0.702530 + 0.711654i $$0.252054\pi$$
$$104$$ 0 0
$$105$$ 14.4730 1.41242
$$106$$ 0 0
$$107$$ −8.01874 −0.775201 −0.387600 0.921827i $$-0.626696\pi$$
−0.387600 + 0.921827i $$0.626696\pi$$
$$108$$ 0 0
$$109$$ −8.22558 −0.787867 −0.393934 0.919139i $$-0.628886\pi$$
−0.393934 + 0.919139i $$0.628886\pi$$
$$110$$ 0 0
$$111$$ −21.0193 −1.99506
$$112$$ 0 0
$$113$$ 16.3280 1.53601 0.768005 0.640444i $$-0.221250\pi$$
0.768005 + 0.640444i $$0.221250\pi$$
$$114$$ 0 0
$$115$$ 2.55761 0.238498
$$116$$ 0 0
$$117$$ −47.6628 −4.40642
$$118$$ 0 0
$$119$$ −29.1454 −2.67176
$$120$$ 0 0
$$121$$ −7.31487 −0.664988
$$122$$ 0 0
$$123$$ 23.9261 2.15734
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −9.89012 −0.877606 −0.438803 0.898583i $$-0.644597\pi$$
−0.438803 + 0.898583i $$0.644597\pi$$
$$128$$ 0 0
$$129$$ −12.6206 −1.11118
$$130$$ 0 0
$$131$$ 5.98438 0.522858 0.261429 0.965223i $$-0.415806\pi$$
0.261429 + 0.965223i $$0.415806\pi$$
$$132$$ 0 0
$$133$$ 4.18444 0.362837
$$134$$ 0 0
$$135$$ 20.6248 1.77510
$$136$$ 0 0
$$137$$ −11.1258 −0.950545 −0.475273 0.879839i $$-0.657651\pi$$
−0.475273 + 0.879839i $$0.657651\pi$$
$$138$$ 0 0
$$139$$ −16.5011 −1.39960 −0.699801 0.714338i $$-0.746728\pi$$
−0.699801 + 0.714338i $$0.746728\pi$$
$$140$$ 0 0
$$141$$ −21.7654 −1.83298
$$142$$ 0 0
$$143$$ −10.2082 −0.853654
$$144$$ 0 0
$$145$$ −2.90445 −0.241202
$$146$$ 0 0
$$147$$ −36.3500 −2.99810
$$148$$ 0 0
$$149$$ 17.7917 1.45756 0.728778 0.684750i $$-0.240088\pi$$
0.728778 + 0.684750i $$0.240088\pi$$
$$150$$ 0 0
$$151$$ −0.913243 −0.0743187 −0.0371593 0.999309i $$-0.511831\pi$$
−0.0371593 + 0.999309i $$0.511831\pi$$
$$152$$ 0 0
$$153$$ −62.4293 −5.04711
$$154$$ 0 0
$$155$$ −4.26651 −0.342695
$$156$$ 0 0
$$157$$ −2.28213 −0.182134 −0.0910669 0.995845i $$-0.529028\pi$$
−0.0910669 + 0.995845i $$0.529028\pi$$
$$158$$ 0 0
$$159$$ 43.9783 3.48771
$$160$$ 0 0
$$161$$ −10.7022 −0.843448
$$162$$ 0 0
$$163$$ −2.34685 −0.183819 −0.0919096 0.995767i $$-0.529297\pi$$
−0.0919096 + 0.995767i $$0.529297\pi$$
$$164$$ 0 0
$$165$$ 6.63968 0.516898
$$166$$ 0 0
$$167$$ −14.3130 −1.10757 −0.553787 0.832658i $$-0.686818\pi$$
−0.553787 + 0.832658i $$0.686818\pi$$
$$168$$ 0 0
$$169$$ 15.2779 1.17522
$$170$$ 0 0
$$171$$ 8.96304 0.685421
$$172$$ 0 0
$$173$$ 11.4702 0.872061 0.436030 0.899932i $$-0.356384\pi$$
0.436030 + 0.899932i $$0.356384\pi$$
$$174$$ 0 0
$$175$$ 4.18444 0.316314
$$176$$ 0 0
$$177$$ −18.1344 −1.36307
$$178$$ 0 0
$$179$$ 0.494634 0.0369707 0.0184853 0.999829i $$-0.494116\pi$$
0.0184853 + 0.999829i $$0.494116\pi$$
$$180$$ 0 0
$$181$$ 5.02418 0.373444 0.186722 0.982413i $$-0.440214\pi$$
0.186722 + 0.982413i $$0.440214\pi$$
$$182$$ 0 0
$$183$$ 11.6596 0.861901
$$184$$ 0 0
$$185$$ −6.07710 −0.446798
$$186$$ 0 0
$$187$$ −13.3709 −0.977774
$$188$$ 0 0
$$189$$ −86.3031 −6.27763
$$190$$ 0 0
$$191$$ −4.01736 −0.290686 −0.145343 0.989381i $$-0.546429\pi$$
−0.145343 + 0.989381i $$0.546429\pi$$
$$192$$ 0 0
$$193$$ −15.3265 −1.10322 −0.551612 0.834101i $$-0.685987\pi$$
−0.551612 + 0.834101i $$0.685987\pi$$
$$194$$ 0 0
$$195$$ −18.3927 −1.31713
$$196$$ 0 0
$$197$$ −2.20197 −0.156884 −0.0784420 0.996919i $$-0.524995\pi$$
−0.0784420 + 0.996919i $$0.524995\pi$$
$$198$$ 0 0
$$199$$ −11.4972 −0.815013 −0.407506 0.913202i $$-0.633602\pi$$
−0.407506 + 0.913202i $$0.633602\pi$$
$$200$$ 0 0
$$201$$ −28.3647 −2.00069
$$202$$ 0 0
$$203$$ 12.1535 0.853009
$$204$$ 0 0
$$205$$ 6.91753 0.483141
$$206$$ 0 0
$$207$$ −22.9239 −1.59332
$$208$$ 0 0
$$209$$ 1.91967 0.132786
$$210$$ 0 0
$$211$$ −2.98080 −0.205207 −0.102603 0.994722i $$-0.532717\pi$$
−0.102603 + 0.994722i $$0.532717\pi$$
$$212$$ 0 0
$$213$$ 56.5621 3.87557
$$214$$ 0 0
$$215$$ −3.64887 −0.248851
$$216$$ 0 0
$$217$$ 17.8530 1.21194
$$218$$ 0 0
$$219$$ 11.4185 0.771592
$$220$$ 0 0
$$221$$ 37.0388 2.49150
$$222$$ 0 0
$$223$$ −16.1105 −1.07884 −0.539419 0.842038i $$-0.681356\pi$$
−0.539419 + 0.842038i $$0.681356\pi$$
$$224$$ 0 0
$$225$$ 8.96304 0.597536
$$226$$ 0 0
$$227$$ −16.7992 −1.11500 −0.557500 0.830177i $$-0.688240\pi$$
−0.557500 + 0.830177i $$0.688240\pi$$
$$228$$ 0 0
$$229$$ −17.1521 −1.13344 −0.566720 0.823910i $$-0.691788\pi$$
−0.566720 + 0.823910i $$0.691788\pi$$
$$230$$ 0 0
$$231$$ −27.7834 −1.82801
$$232$$ 0 0
$$233$$ −12.4713 −0.817019 −0.408509 0.912754i $$-0.633951\pi$$
−0.408509 + 0.912754i $$0.633951\pi$$
$$234$$ 0 0
$$235$$ −6.29283 −0.410499
$$236$$ 0 0
$$237$$ 3.54239 0.230103
$$238$$ 0 0
$$239$$ −0.600913 −0.0388698 −0.0194349 0.999811i $$-0.506187\pi$$
−0.0194349 + 0.999811i $$0.506187\pi$$
$$240$$ 0 0
$$241$$ 21.5643 1.38908 0.694538 0.719456i $$-0.255609\pi$$
0.694538 + 0.719456i $$0.255609\pi$$
$$242$$ 0 0
$$243$$ −91.8575 −5.89266
$$244$$ 0 0
$$245$$ −10.5095 −0.671430
$$246$$ 0 0
$$247$$ −5.31770 −0.338357
$$248$$ 0 0
$$249$$ −15.0199 −0.951847
$$250$$ 0 0
$$251$$ −6.34667 −0.400598 −0.200299 0.979735i $$-0.564191\pi$$
−0.200299 + 0.979735i $$0.564191\pi$$
$$252$$ 0 0
$$253$$ −4.90976 −0.308674
$$254$$ 0 0
$$255$$ −24.0909 −1.50863
$$256$$ 0 0
$$257$$ −24.1830 −1.50849 −0.754246 0.656592i $$-0.771997\pi$$
−0.754246 + 0.656592i $$0.771997\pi$$
$$258$$ 0 0
$$259$$ 25.4293 1.58010
$$260$$ 0 0
$$261$$ 26.0327 1.61139
$$262$$ 0 0
$$263$$ 4.80017 0.295991 0.147996 0.988988i $$-0.452718\pi$$
0.147996 + 0.988988i $$0.452718\pi$$
$$264$$ 0 0
$$265$$ 12.7150 0.781079
$$266$$ 0 0
$$267$$ 5.03471 0.308119
$$268$$ 0 0
$$269$$ −4.88050 −0.297569 −0.148785 0.988870i $$-0.547536\pi$$
−0.148785 + 0.988870i $$0.547536\pi$$
$$270$$ 0 0
$$271$$ −1.60226 −0.0973301 −0.0486651 0.998815i $$-0.515497\pi$$
−0.0486651 + 0.998815i $$0.515497\pi$$
$$272$$ 0 0
$$273$$ 76.9630 4.65801
$$274$$ 0 0
$$275$$ 1.91967 0.115760
$$276$$ 0 0
$$277$$ 17.3012 1.03953 0.519765 0.854309i $$-0.326020\pi$$
0.519765 + 0.854309i $$0.326020\pi$$
$$278$$ 0 0
$$279$$ 38.2410 2.28943
$$280$$ 0 0
$$281$$ −24.7440 −1.47610 −0.738052 0.674744i $$-0.764254\pi$$
−0.738052 + 0.674744i $$0.764254\pi$$
$$282$$ 0 0
$$283$$ −3.78687 −0.225106 −0.112553 0.993646i $$-0.535903\pi$$
−0.112553 + 0.993646i $$0.535903\pi$$
$$284$$ 0 0
$$285$$ 3.45876 0.204879
$$286$$ 0 0
$$287$$ −28.9460 −1.70863
$$288$$ 0 0
$$289$$ 31.5138 1.85375
$$290$$ 0 0
$$291$$ 29.4535 1.72659
$$292$$ 0 0
$$293$$ −14.1812 −0.828475 −0.414238 0.910169i $$-0.635952\pi$$
−0.414238 + 0.910169i $$0.635952\pi$$
$$294$$ 0 0
$$295$$ −5.24303 −0.305261
$$296$$ 0 0
$$297$$ −39.5927 −2.29740
$$298$$ 0 0
$$299$$ 13.6006 0.786542
$$300$$ 0 0
$$301$$ 15.2685 0.880061
$$302$$ 0 0
$$303$$ 7.19537 0.413363
$$304$$ 0 0
$$305$$ 3.37102 0.193024
$$306$$ 0 0
$$307$$ 12.9379 0.738404 0.369202 0.929349i $$-0.379631\pi$$
0.369202 + 0.929349i $$0.379631\pi$$
$$308$$ 0 0
$$309$$ −49.3213 −2.80579
$$310$$ 0 0
$$311$$ 9.62875 0.545996 0.272998 0.962015i $$-0.411985\pi$$
0.272998 + 0.962015i $$0.411985\pi$$
$$312$$ 0 0
$$313$$ −17.5614 −0.992628 −0.496314 0.868143i $$-0.665314\pi$$
−0.496314 + 0.868143i $$0.665314\pi$$
$$314$$ 0 0
$$315$$ −37.5053 −2.11319
$$316$$ 0 0
$$317$$ −7.85153 −0.440986 −0.220493 0.975389i $$-0.570767\pi$$
−0.220493 + 0.975389i $$0.570767\pi$$
$$318$$ 0 0
$$319$$ 5.57559 0.312173
$$320$$ 0 0
$$321$$ 27.7349 1.54801
$$322$$ 0 0
$$323$$ −6.96519 −0.387553
$$324$$ 0 0
$$325$$ −5.31770 −0.294973
$$326$$ 0 0
$$327$$ 28.4503 1.57331
$$328$$ 0 0
$$329$$ 26.3320 1.45173
$$330$$ 0 0
$$331$$ −22.3426 −1.22806 −0.614031 0.789282i $$-0.710453\pi$$
−0.614031 + 0.789282i $$0.710453\pi$$
$$332$$ 0 0
$$333$$ 54.4694 2.98490
$$334$$ 0 0
$$335$$ −8.20081 −0.448058
$$336$$ 0 0
$$337$$ 32.9016 1.79227 0.896133 0.443786i $$-0.146365\pi$$
0.896133 + 0.443786i $$0.146365\pi$$
$$338$$ 0 0
$$339$$ −56.4747 −3.06728
$$340$$ 0 0
$$341$$ 8.19030 0.443529
$$342$$ 0 0
$$343$$ 14.6855 0.792942
$$344$$ 0 0
$$345$$ −8.84616 −0.476261
$$346$$ 0 0
$$347$$ 8.38387 0.450070 0.225035 0.974351i $$-0.427750\pi$$
0.225035 + 0.974351i $$0.427750\pi$$
$$348$$ 0 0
$$349$$ 4.05969 0.217310 0.108655 0.994080i $$-0.465346\pi$$
0.108655 + 0.994080i $$0.465346\pi$$
$$350$$ 0 0
$$351$$ 109.676 5.85409
$$352$$ 0 0
$$353$$ −9.56784 −0.509245 −0.254622 0.967041i $$-0.581951\pi$$
−0.254622 + 0.967041i $$0.581951\pi$$
$$354$$ 0 0
$$355$$ 16.3533 0.867941
$$356$$ 0 0
$$357$$ 100.807 5.33528
$$358$$ 0 0
$$359$$ 25.1347 1.32656 0.663280 0.748372i $$-0.269164\pi$$
0.663280 + 0.748372i $$0.269164\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 25.3004 1.32793
$$364$$ 0 0
$$365$$ 3.30133 0.172799
$$366$$ 0 0
$$367$$ 13.1236 0.685047 0.342523 0.939509i $$-0.388718\pi$$
0.342523 + 0.939509i $$0.388718\pi$$
$$368$$ 0 0
$$369$$ −62.0021 −3.22770
$$370$$ 0 0
$$371$$ −53.2053 −2.76228
$$372$$ 0 0
$$373$$ −19.6739 −1.01867 −0.509337 0.860567i $$-0.670109\pi$$
−0.509337 + 0.860567i $$0.670109\pi$$
$$374$$ 0 0
$$375$$ 3.45876 0.178610
$$376$$ 0 0
$$377$$ −15.4450 −0.795458
$$378$$ 0 0
$$379$$ −33.9979 −1.74636 −0.873178 0.487401i $$-0.837945\pi$$
−0.873178 + 0.487401i $$0.837945\pi$$
$$380$$ 0 0
$$381$$ 34.2076 1.75251
$$382$$ 0 0
$$383$$ 0.534885 0.0273314 0.0136657 0.999907i $$-0.495650\pi$$
0.0136657 + 0.999907i $$0.495650\pi$$
$$384$$ 0 0
$$385$$ −8.03274 −0.409386
$$386$$ 0 0
$$387$$ 32.7050 1.66249
$$388$$ 0 0
$$389$$ −13.2296 −0.670769 −0.335385 0.942081i $$-0.608866\pi$$
−0.335385 + 0.942081i $$0.608866\pi$$
$$390$$ 0 0
$$391$$ 17.8142 0.900903
$$392$$ 0 0
$$393$$ −20.6986 −1.04410
$$394$$ 0 0
$$395$$ 1.02418 0.0515320
$$396$$ 0 0
$$397$$ 11.5381 0.579081 0.289541 0.957166i $$-0.406498\pi$$
0.289541 + 0.957166i $$0.406498\pi$$
$$398$$ 0 0
$$399$$ −14.4730 −0.724556
$$400$$ 0 0
$$401$$ −23.6658 −1.18181 −0.590907 0.806739i $$-0.701230\pi$$
−0.590907 + 0.806739i $$0.701230\pi$$
$$402$$ 0 0
$$403$$ −22.6880 −1.13017
$$404$$ 0 0
$$405$$ −44.4470 −2.20859
$$406$$ 0 0
$$407$$ 11.6660 0.578264
$$408$$ 0 0
$$409$$ −18.0206 −0.891062 −0.445531 0.895267i $$-0.646985\pi$$
−0.445531 + 0.895267i $$0.646985\pi$$
$$410$$ 0 0
$$411$$ 38.4817 1.89816
$$412$$ 0 0
$$413$$ 21.9392 1.07956
$$414$$ 0 0
$$415$$ −4.34256 −0.213168
$$416$$ 0 0
$$417$$ 57.0732 2.79489
$$418$$ 0 0
$$419$$ −18.6880 −0.912970 −0.456485 0.889731i $$-0.650892\pi$$
−0.456485 + 0.889731i $$0.650892\pi$$
$$420$$ 0 0
$$421$$ −2.27530 −0.110892 −0.0554458 0.998462i $$-0.517658\pi$$
−0.0554458 + 0.998462i $$0.517658\pi$$
$$422$$ 0 0
$$423$$ 56.4030 2.74241
$$424$$ 0 0
$$425$$ −6.96519 −0.337861
$$426$$ 0 0
$$427$$ −14.1059 −0.682630
$$428$$ 0 0
$$429$$ 35.3078 1.70468
$$430$$ 0 0
$$431$$ −3.25578 −0.156826 −0.0784128 0.996921i $$-0.524985\pi$$
−0.0784128 + 0.996921i $$0.524985\pi$$
$$432$$ 0 0
$$433$$ −0.690637 −0.0331899 −0.0165949 0.999862i $$-0.505283\pi$$
−0.0165949 + 0.999862i $$0.505283\pi$$
$$434$$ 0 0
$$435$$ 10.0458 0.481660
$$436$$ 0 0
$$437$$ −2.55761 −0.122347
$$438$$ 0 0
$$439$$ 11.5529 0.551391 0.275695 0.961245i $$-0.411092\pi$$
0.275695 + 0.961245i $$0.411092\pi$$
$$440$$ 0 0
$$441$$ 94.1975 4.48560
$$442$$ 0 0
$$443$$ −35.0042 −1.66310 −0.831551 0.555448i $$-0.812547\pi$$
−0.831551 + 0.555448i $$0.812547\pi$$
$$444$$ 0 0
$$445$$ 1.45564 0.0690039
$$446$$ 0 0
$$447$$ −61.5374 −2.91062
$$448$$ 0 0
$$449$$ −7.59989 −0.358661 −0.179330 0.983789i $$-0.557393\pi$$
−0.179330 + 0.983789i $$0.557393\pi$$
$$450$$ 0 0
$$451$$ −13.2794 −0.625301
$$452$$ 0 0
$$453$$ 3.15869 0.148408
$$454$$ 0 0
$$455$$ 22.2516 1.04317
$$456$$ 0 0
$$457$$ −4.49322 −0.210184 −0.105092 0.994463i $$-0.533514\pi$$
−0.105092 + 0.994463i $$0.533514\pi$$
$$458$$ 0 0
$$459$$ 143.655 6.70526
$$460$$ 0 0
$$461$$ −28.3207 −1.31902 −0.659512 0.751694i $$-0.729237\pi$$
−0.659512 + 0.751694i $$0.729237\pi$$
$$462$$ 0 0
$$463$$ −11.3519 −0.527569 −0.263784 0.964582i $$-0.584971\pi$$
−0.263784 + 0.964582i $$0.584971\pi$$
$$464$$ 0 0
$$465$$ 14.7569 0.684333
$$466$$ 0 0
$$467$$ 4.27998 0.198054 0.0990270 0.995085i $$-0.468427\pi$$
0.0990270 + 0.995085i $$0.468427\pi$$
$$468$$ 0 0
$$469$$ 34.3158 1.58456
$$470$$ 0 0
$$471$$ 7.89335 0.363706
$$472$$ 0 0
$$473$$ 7.00462 0.322073
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −113.966 −5.21812
$$478$$ 0 0
$$479$$ 18.9310 0.864979 0.432490 0.901639i $$-0.357635\pi$$
0.432490 + 0.901639i $$0.357635\pi$$
$$480$$ 0 0
$$481$$ −32.3162 −1.47349
$$482$$ 0 0
$$483$$ 37.0162 1.68430
$$484$$ 0 0
$$485$$ 8.51561 0.386674
$$486$$ 0 0
$$487$$ 29.3181 1.32853 0.664265 0.747497i $$-0.268745\pi$$
0.664265 + 0.747497i $$0.268745\pi$$
$$488$$ 0 0
$$489$$ 8.11718 0.367072
$$490$$ 0 0
$$491$$ 3.21099 0.144910 0.0724550 0.997372i $$-0.476917\pi$$
0.0724550 + 0.997372i $$0.476917\pi$$
$$492$$ 0 0
$$493$$ −20.2301 −0.911116
$$494$$ 0 0
$$495$$ −17.2061 −0.773356
$$496$$ 0 0
$$497$$ −68.4293 −3.06947
$$498$$ 0 0
$$499$$ 3.85861 0.172735 0.0863675 0.996263i $$-0.472474\pi$$
0.0863675 + 0.996263i $$0.472474\pi$$
$$500$$ 0 0
$$501$$ 49.5053 2.21174
$$502$$ 0 0
$$503$$ −21.4587 −0.956797 −0.478399 0.878143i $$-0.658783\pi$$
−0.478399 + 0.878143i $$0.658783\pi$$
$$504$$ 0 0
$$505$$ 2.08033 0.0925735
$$506$$ 0 0
$$507$$ −52.8427 −2.34683
$$508$$ 0 0
$$509$$ −13.6077 −0.603152 −0.301576 0.953442i $$-0.597513\pi$$
−0.301576 + 0.953442i $$0.597513\pi$$
$$510$$ 0 0
$$511$$ −13.8142 −0.611105
$$512$$ 0 0
$$513$$ −20.6248 −0.910605
$$514$$ 0 0
$$515$$ −14.2598 −0.628362
$$516$$ 0 0
$$517$$ 12.0802 0.531285
$$518$$ 0 0
$$519$$ −39.6726 −1.74143
$$520$$ 0 0
$$521$$ −36.3168 −1.59107 −0.795535 0.605908i $$-0.792810\pi$$
−0.795535 + 0.605908i $$0.792810\pi$$
$$522$$ 0 0
$$523$$ 38.9982 1.70527 0.852636 0.522506i $$-0.175003\pi$$
0.852636 + 0.522506i $$0.175003\pi$$
$$524$$ 0 0
$$525$$ −14.4730 −0.631653
$$526$$ 0 0
$$527$$ −29.7171 −1.29450
$$528$$ 0 0
$$529$$ −16.4586 −0.715593
$$530$$ 0 0
$$531$$ 46.9935 2.03935
$$532$$ 0 0
$$533$$ 36.7853 1.59335
$$534$$ 0 0
$$535$$ 8.01874 0.346680
$$536$$ 0 0
$$537$$ −1.71082 −0.0738274
$$538$$ 0 0
$$539$$ 20.1749 0.868992
$$540$$ 0 0
$$541$$ 18.8267 0.809421 0.404711 0.914445i $$-0.367372\pi$$
0.404711 + 0.914445i $$0.367372\pi$$
$$542$$ 0 0
$$543$$ −17.3774 −0.745738
$$544$$ 0 0
$$545$$ 8.22558 0.352345
$$546$$ 0 0
$$547$$ −19.3493 −0.827317 −0.413659 0.910432i $$-0.635749\pi$$
−0.413659 + 0.910432i $$0.635749\pi$$
$$548$$ 0 0
$$549$$ −30.2146 −1.28953
$$550$$ 0 0
$$551$$ 2.90445 0.123734
$$552$$ 0 0
$$553$$ −4.28561 −0.182243
$$554$$ 0 0
$$555$$ 21.0193 0.892218
$$556$$ 0 0
$$557$$ −29.3012 −1.24153 −0.620766 0.783996i $$-0.713178\pi$$
−0.620766 + 0.783996i $$0.713178\pi$$
$$558$$ 0 0
$$559$$ −19.4036 −0.820685
$$560$$ 0 0
$$561$$ 46.2466 1.95253
$$562$$ 0 0
$$563$$ 20.2988 0.855492 0.427746 0.903899i $$-0.359308\pi$$
0.427746 + 0.903899i $$0.359308\pi$$
$$564$$ 0 0
$$565$$ −16.3280 −0.686924
$$566$$ 0 0
$$567$$ 185.986 7.81068
$$568$$ 0 0
$$569$$ −1.49266 −0.0625757 −0.0312879 0.999510i $$-0.509961\pi$$
−0.0312879 + 0.999510i $$0.509961\pi$$
$$570$$ 0 0
$$571$$ −23.4085 −0.979615 −0.489808 0.871830i $$-0.662933\pi$$
−0.489808 + 0.871830i $$0.662933\pi$$
$$572$$ 0 0
$$573$$ 13.8951 0.580476
$$574$$ 0 0
$$575$$ −2.55761 −0.106660
$$576$$ 0 0
$$577$$ −29.4279 −1.22510 −0.612549 0.790433i $$-0.709856\pi$$
−0.612549 + 0.790433i $$0.709856\pi$$
$$578$$ 0 0
$$579$$ 53.0107 2.20305
$$580$$ 0 0
$$581$$ 18.1712 0.753868
$$582$$ 0 0
$$583$$ −24.4087 −1.01090
$$584$$ 0 0
$$585$$ 47.6628 1.97061
$$586$$ 0 0
$$587$$ −38.3731 −1.58383 −0.791914 0.610632i $$-0.790915\pi$$
−0.791914 + 0.610632i $$0.790915\pi$$
$$588$$ 0 0
$$589$$ 4.26651 0.175799
$$590$$ 0 0
$$591$$ 7.61610 0.313284
$$592$$ 0 0
$$593$$ −12.1564 −0.499203 −0.249601 0.968349i $$-0.580300\pi$$
−0.249601 + 0.968349i $$0.580300\pi$$
$$594$$ 0 0
$$595$$ 29.1454 1.19485
$$596$$ 0 0
$$597$$ 39.7660 1.62751
$$598$$ 0 0
$$599$$ 4.87208 0.199068 0.0995340 0.995034i $$-0.468265\pi$$
0.0995340 + 0.995034i $$0.468265\pi$$
$$600$$ 0 0
$$601$$ −4.82141 −0.196669 −0.0983347 0.995153i $$-0.531352\pi$$
−0.0983347 + 0.995153i $$0.531352\pi$$
$$602$$ 0 0
$$603$$ 73.5042 2.99332
$$604$$ 0 0
$$605$$ 7.31487 0.297392
$$606$$ 0 0
$$607$$ −14.9897 −0.608414 −0.304207 0.952606i $$-0.598391\pi$$
−0.304207 + 0.952606i $$0.598391\pi$$
$$608$$ 0 0
$$609$$ −42.0361 −1.70339
$$610$$ 0 0
$$611$$ −33.4634 −1.35378
$$612$$ 0 0
$$613$$ 11.0133 0.444823 0.222412 0.974953i $$-0.428607\pi$$
0.222412 + 0.974953i $$0.428607\pi$$
$$614$$ 0 0
$$615$$ −23.9261 −0.964793
$$616$$ 0 0
$$617$$ 11.3041 0.455088 0.227544 0.973768i $$-0.426931\pi$$
0.227544 + 0.973768i $$0.426931\pi$$
$$618$$ 0 0
$$619$$ −15.8466 −0.636927 −0.318463 0.947935i $$-0.603167\pi$$
−0.318463 + 0.947935i $$0.603167\pi$$
$$620$$ 0 0
$$621$$ 52.7500 2.11679
$$622$$ 0 0
$$623$$ −6.09104 −0.244032
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −6.63968 −0.265163
$$628$$ 0 0
$$629$$ −42.3282 −1.68773
$$630$$ 0 0
$$631$$ 22.3835 0.891074 0.445537 0.895263i $$-0.353013\pi$$
0.445537 + 0.895263i $$0.353013\pi$$
$$632$$ 0 0
$$633$$ 10.3099 0.409781
$$634$$ 0 0
$$635$$ 9.89012 0.392478
$$636$$ 0 0
$$637$$ −55.8866 −2.21431
$$638$$ 0 0
$$639$$ −146.575 −5.79842
$$640$$ 0 0
$$641$$ 31.2552 1.23451 0.617253 0.786765i $$-0.288246\pi$$
0.617253 + 0.786765i $$0.288246\pi$$
$$642$$ 0 0
$$643$$ 28.6100 1.12827 0.564134 0.825683i $$-0.309210\pi$$
0.564134 + 0.825683i $$0.309210\pi$$
$$644$$ 0 0
$$645$$ 12.6206 0.496935
$$646$$ 0 0
$$647$$ 17.0323 0.669610 0.334805 0.942287i $$-0.391329\pi$$
0.334805 + 0.942287i $$0.391329\pi$$
$$648$$ 0 0
$$649$$ 10.0649 0.395081
$$650$$ 0 0
$$651$$ −61.7492 −2.42014
$$652$$ 0 0
$$653$$ −27.2694 −1.06714 −0.533568 0.845757i $$-0.679149\pi$$
−0.533568 + 0.845757i $$0.679149\pi$$
$$654$$ 0 0
$$655$$ −5.98438 −0.233829
$$656$$ 0 0
$$657$$ −29.5899 −1.15441
$$658$$ 0 0
$$659$$ 37.0311 1.44253 0.721264 0.692660i $$-0.243562\pi$$
0.721264 + 0.692660i $$0.243562\pi$$
$$660$$ 0 0
$$661$$ 6.03237 0.234632 0.117316 0.993095i $$-0.462571\pi$$
0.117316 + 0.993095i $$0.462571\pi$$
$$662$$ 0 0
$$663$$ −128.108 −4.97531
$$664$$ 0 0
$$665$$ −4.18444 −0.162266
$$666$$ 0 0
$$667$$ −7.42845 −0.287631
$$668$$ 0 0
$$669$$ 55.7223 2.15435
$$670$$ 0 0
$$671$$ −6.47125 −0.249820
$$672$$ 0 0
$$673$$ −37.8118 −1.45754 −0.728768 0.684761i $$-0.759907\pi$$
−0.728768 + 0.684761i $$0.759907\pi$$
$$674$$ 0 0
$$675$$ −20.6248 −0.793847
$$676$$ 0 0
$$677$$ −31.0356 −1.19279 −0.596397 0.802690i $$-0.703402\pi$$
−0.596397 + 0.802690i $$0.703402\pi$$
$$678$$ 0 0
$$679$$ −35.6331 −1.36747
$$680$$ 0 0
$$681$$ 58.1044 2.22657
$$682$$ 0 0
$$683$$ 6.68835 0.255923 0.127961 0.991779i $$-0.459157\pi$$
0.127961 + 0.991779i $$0.459157\pi$$
$$684$$ 0 0
$$685$$ 11.1258 0.425097
$$686$$ 0 0
$$687$$ 59.3249 2.26339
$$688$$ 0 0
$$689$$ 67.6148 2.57592
$$690$$ 0 0
$$691$$ −27.6759 −1.05284 −0.526421 0.850224i $$-0.676466\pi$$
−0.526421 + 0.850224i $$0.676466\pi$$
$$692$$ 0 0
$$693$$ 71.9978 2.73497
$$694$$ 0 0
$$695$$ 16.5011 0.625921
$$696$$ 0 0
$$697$$ 48.1819 1.82502
$$698$$ 0 0
$$699$$ 43.1351 1.63152
$$700$$ 0 0
$$701$$ −10.6234 −0.401242 −0.200621 0.979669i $$-0.564296\pi$$
−0.200621 + 0.979669i $$0.564296\pi$$
$$702$$ 0 0
$$703$$ 6.07710 0.229202
$$704$$ 0 0
$$705$$ 21.7654 0.819733
$$706$$ 0 0
$$707$$ −8.70502 −0.327386
$$708$$ 0 0
$$709$$ 34.0768 1.27978 0.639891 0.768466i $$-0.278980\pi$$
0.639891 + 0.768466i $$0.278980\pi$$
$$710$$ 0 0
$$711$$ −9.17976 −0.344268
$$712$$ 0 0
$$713$$ −10.9121 −0.408660
$$714$$ 0 0
$$715$$ 10.2082 0.381766
$$716$$ 0 0
$$717$$ 2.07842 0.0776199
$$718$$ 0 0
$$719$$ 25.8319 0.963367 0.481683 0.876345i $$-0.340026\pi$$
0.481683 + 0.876345i $$0.340026\pi$$
$$720$$ 0 0
$$721$$ 59.6693 2.22220
$$722$$ 0 0
$$723$$ −74.5857 −2.77387
$$724$$ 0 0
$$725$$ 2.90445 0.107869
$$726$$ 0 0
$$727$$ 41.7046 1.54674 0.773369 0.633956i $$-0.218570\pi$$
0.773369 + 0.633956i $$0.218570\pi$$
$$728$$ 0 0
$$729$$ 184.372 6.82860
$$730$$ 0 0
$$731$$ −25.4151 −0.940010
$$732$$ 0 0
$$733$$ 13.3033 0.491370 0.245685 0.969350i $$-0.420987\pi$$
0.245685 + 0.969350i $$0.420987\pi$$
$$734$$ 0 0
$$735$$ 36.3500 1.34079
$$736$$ 0 0
$$737$$ 15.7428 0.579895
$$738$$ 0 0
$$739$$ −37.7826 −1.38985 −0.694927 0.719080i $$-0.744563\pi$$
−0.694927 + 0.719080i $$0.744563\pi$$
$$740$$ 0 0
$$741$$ 18.3927 0.675671
$$742$$ 0 0
$$743$$ −9.99052 −0.366517 −0.183258 0.983065i $$-0.558665\pi$$
−0.183258 + 0.983065i $$0.558665\pi$$
$$744$$ 0 0
$$745$$ −17.7917 −0.651839
$$746$$ 0 0
$$747$$ 38.9226 1.42410
$$748$$ 0 0
$$749$$ −33.5539 −1.22603
$$750$$ 0 0
$$751$$ 44.6739 1.63017 0.815087 0.579338i $$-0.196689\pi$$
0.815087 + 0.579338i $$0.196689\pi$$
$$752$$ 0 0
$$753$$ 21.9516 0.799962
$$754$$ 0 0
$$755$$ 0.913243 0.0332363
$$756$$ 0 0
$$757$$ 42.9608 1.56144 0.780718 0.624883i $$-0.214853\pi$$
0.780718 + 0.624883i $$0.214853\pi$$
$$758$$ 0 0
$$759$$ 16.9817 0.616396
$$760$$ 0 0
$$761$$ −45.0314 −1.63239 −0.816194 0.577778i $$-0.803920\pi$$
−0.816194 + 0.577778i $$0.803920\pi$$
$$762$$ 0 0
$$763$$ −34.4194 −1.24607
$$764$$ 0 0
$$765$$ 62.4293 2.25714
$$766$$ 0 0
$$767$$ −27.8809 −1.00672
$$768$$ 0 0
$$769$$ 41.6483 1.50188 0.750938 0.660372i $$-0.229602\pi$$
0.750938 + 0.660372i $$0.229602\pi$$
$$770$$ 0 0
$$771$$ 83.6431 3.01233
$$772$$ 0 0
$$773$$ 42.6831 1.53520 0.767602 0.640927i $$-0.221450\pi$$
0.767602 + 0.640927i $$0.221450\pi$$
$$774$$ 0 0
$$775$$ 4.26651 0.153258
$$776$$ 0 0
$$777$$ −87.9539 −3.15533
$$778$$ 0 0
$$779$$ −6.91753 −0.247846
$$780$$ 0 0
$$781$$ −31.3929 −1.12332
$$782$$ 0 0
$$783$$ −59.9036 −2.14078
$$784$$ 0 0
$$785$$ 2.28213 0.0814527
$$786$$ 0 0
$$787$$ 16.4161 0.585170 0.292585 0.956240i $$-0.405485\pi$$
0.292585 + 0.956240i $$0.405485\pi$$
$$788$$ 0 0
$$789$$ −16.6027 −0.591070
$$790$$ 0 0
$$791$$ 68.3236 2.42931
$$792$$ 0 0
$$793$$ 17.9261 0.636574
$$794$$ 0 0
$$795$$ −43.9783 −1.55975
$$796$$ 0 0
$$797$$ 6.04841 0.214246 0.107123 0.994246i $$-0.465836\pi$$
0.107123 + 0.994246i $$0.465836\pi$$
$$798$$ 0 0
$$799$$ −43.8308 −1.55062
$$800$$ 0 0
$$801$$ −13.0470 −0.460992
$$802$$ 0 0
$$803$$ −6.33746 −0.223644
$$804$$ 0 0
$$805$$ 10.7022 0.377201
$$806$$ 0 0
$$807$$ 16.8805 0.594222
$$808$$ 0 0
$$809$$ −40.1166 −1.41043 −0.705213 0.708996i $$-0.749149\pi$$
−0.705213 + 0.708996i $$0.749149\pi$$
$$810$$ 0 0
$$811$$ 50.6027 1.77690 0.888450 0.458973i $$-0.151782\pi$$
0.888450 + 0.458973i $$0.151782\pi$$
$$812$$ 0 0
$$813$$ 5.54182 0.194360
$$814$$ 0 0
$$815$$ 2.34685 0.0822064
$$816$$ 0 0
$$817$$ 3.64887 0.127658
$$818$$ 0 0
$$819$$ −199.442 −6.96907
$$820$$ 0 0
$$821$$ −7.91047 −0.276077 −0.138039 0.990427i $$-0.544080\pi$$
−0.138039 + 0.990427i $$0.544080\pi$$
$$822$$ 0 0
$$823$$ −18.5164 −0.645442 −0.322721 0.946494i $$-0.604598\pi$$
−0.322721 + 0.946494i $$0.604598\pi$$
$$824$$ 0 0
$$825$$ −6.63968 −0.231164
$$826$$ 0 0
$$827$$ 16.9938 0.590931 0.295466 0.955353i $$-0.404525\pi$$
0.295466 + 0.955353i $$0.404525\pi$$
$$828$$ 0 0
$$829$$ 17.7915 0.617926 0.308963 0.951074i $$-0.400018\pi$$
0.308963 + 0.951074i $$0.400018\pi$$
$$830$$ 0 0
$$831$$ −59.8408 −2.07586
$$832$$ 0 0
$$833$$ −73.2010 −2.53626
$$834$$ 0 0
$$835$$ 14.3130 0.495322
$$836$$ 0 0
$$837$$ −87.9958 −3.04158
$$838$$ 0 0
$$839$$ 42.6250 1.47158 0.735789 0.677211i $$-0.236811\pi$$
0.735789 + 0.677211i $$0.236811\pi$$
$$840$$ 0 0
$$841$$ −20.5642 −0.709109
$$842$$ 0 0
$$843$$ 85.5837 2.94766
$$844$$ 0 0
$$845$$ −15.2779 −0.525576
$$846$$ 0 0
$$847$$ −30.6086 −1.05173
$$848$$ 0 0
$$849$$ 13.0979 0.449518
$$850$$ 0 0
$$851$$ −15.5428 −0.532802
$$852$$ 0 0
$$853$$ −9.19246 −0.314744 −0.157372 0.987539i $$-0.550302\pi$$
−0.157372 + 0.987539i $$0.550302\pi$$
$$854$$ 0 0
$$855$$ −8.96304 −0.306530
$$856$$ 0 0
$$857$$ 4.17610 0.142653 0.0713265 0.997453i $$-0.477277\pi$$
0.0713265 + 0.997453i $$0.477277\pi$$
$$858$$ 0 0
$$859$$ −26.0354 −0.888317 −0.444159 0.895948i $$-0.646497\pi$$
−0.444159 + 0.895948i $$0.646497\pi$$
$$860$$ 0 0
$$861$$ 100.117 3.41199
$$862$$ 0 0
$$863$$ 34.0880 1.16037 0.580185 0.814485i $$-0.302980\pi$$
0.580185 + 0.814485i $$0.302980\pi$$
$$864$$ 0 0
$$865$$ −11.4702 −0.389997
$$866$$ 0 0
$$867$$ −108.999 −3.70180
$$868$$ 0 0
$$869$$ −1.96608 −0.0666948
$$870$$ 0 0
$$871$$ −43.6094 −1.47765
$$872$$ 0 0
$$873$$ −76.3258 −2.58324
$$874$$ 0 0
$$875$$ −4.18444 −0.141460
$$876$$ 0 0
$$877$$ −17.2442 −0.582297 −0.291148 0.956678i $$-0.594037\pi$$
−0.291148 + 0.956678i $$0.594037\pi$$
$$878$$ 0 0
$$879$$ 49.0495 1.65440
$$880$$ 0 0
$$881$$ 1.50151 0.0505873 0.0252937 0.999680i $$-0.491948\pi$$
0.0252937 + 0.999680i $$0.491948\pi$$
$$882$$ 0 0
$$883$$ −9.28230 −0.312374 −0.156187 0.987727i $$-0.549920\pi$$
−0.156187 + 0.987727i $$0.549920\pi$$
$$884$$ 0 0
$$885$$ 18.1344 0.609582
$$886$$ 0 0
$$887$$ −26.3714 −0.885467 −0.442733 0.896653i $$-0.645991\pi$$
−0.442733 + 0.896653i $$0.645991\pi$$
$$888$$ 0 0
$$889$$ −41.3846 −1.38800
$$890$$ 0 0
$$891$$ 85.3236 2.85845
$$892$$ 0 0
$$893$$ 6.29283 0.210582
$$894$$ 0 0
$$895$$ −0.494634 −0.0165338
$$896$$ 0 0
$$897$$ −47.0412 −1.57066
$$898$$ 0 0
$$899$$ 12.3919 0.413293
$$900$$ 0 0
$$901$$ 88.5626 2.95045
$$902$$ 0 0
$$903$$ −52.8101 −1.75741
$$904$$ 0 0
$$905$$ −5.02418 −0.167009
$$906$$ 0 0
$$907$$ −15.6537 −0.519772 −0.259886 0.965639i $$-0.583685\pi$$
−0.259886 + 0.965639i $$0.583685\pi$$
$$908$$ 0 0
$$909$$ −18.6461 −0.618452
$$910$$ 0 0
$$911$$ 55.5024 1.83888 0.919438 0.393235i $$-0.128644\pi$$
0.919438 + 0.393235i $$0.128644\pi$$
$$912$$ 0 0
$$913$$ 8.33628 0.275891
$$914$$ 0 0
$$915$$ −11.6596 −0.385454
$$916$$ 0 0
$$917$$ 25.0413 0.826937
$$918$$ 0 0
$$919$$ −49.9719 −1.64842 −0.824211 0.566284i $$-0.808381\pi$$
−0.824211 + 0.566284i $$0.808381\pi$$
$$920$$ 0 0
$$921$$ −44.7491 −1.47453
$$922$$ 0 0
$$923$$ 86.9617 2.86238
$$924$$ 0 0
$$925$$ 6.07710 0.199814
$$926$$ 0 0
$$927$$ 127.811 4.19787
$$928$$ 0 0
$$929$$ 4.21840 0.138401 0.0692006 0.997603i $$-0.477955\pi$$
0.0692006 + 0.997603i $$0.477955\pi$$
$$930$$ 0 0
$$931$$ 10.5095 0.344437
$$932$$ 0 0
$$933$$ −33.3036 −1.09031
$$934$$ 0 0
$$935$$ 13.3709 0.437274
$$936$$ 0 0
$$937$$ 7.31403 0.238939 0.119469 0.992838i $$-0.461881\pi$$
0.119469 + 0.992838i $$0.461881\pi$$
$$938$$ 0 0
$$939$$ 60.7407 1.98220
$$940$$ 0 0
$$941$$ −47.3976 −1.54512 −0.772558 0.634944i $$-0.781023\pi$$
−0.772558 + 0.634944i $$0.781023\pi$$
$$942$$ 0 0
$$943$$ 17.6923 0.576141
$$944$$ 0 0
$$945$$ 86.3031 2.80744
$$946$$ 0 0
$$947$$ 34.0009 1.10488 0.552440 0.833553i $$-0.313697\pi$$
0.552440 + 0.833553i $$0.313697\pi$$
$$948$$ 0 0
$$949$$ 17.5555 0.569875
$$950$$ 0 0
$$951$$ 27.1566 0.880612
$$952$$ 0 0
$$953$$ −56.6039 −1.83358 −0.916790 0.399370i $$-0.869229\pi$$
−0.916790 + 0.399370i $$0.869229\pi$$
$$954$$ 0 0
$$955$$ 4.01736 0.129999
$$956$$ 0 0
$$957$$ −19.2846 −0.623384
$$958$$ 0 0
$$959$$ −46.5555 −1.50335
$$960$$ 0 0
$$961$$ −12.7969 −0.412802
$$962$$ 0 0
$$963$$ −71.8723 −2.31605
$$964$$ 0 0
$$965$$ 15.3265 0.493377
$$966$$ 0 0
$$967$$ 33.7007 1.08374 0.541871 0.840462i $$-0.317716\pi$$
0.541871 + 0.840462i $$0.317716\pi$$
$$968$$ 0 0
$$969$$ 24.0909 0.773912
$$970$$ 0 0
$$971$$ −14.4952 −0.465173 −0.232587 0.972576i $$-0.574719\pi$$
−0.232587 + 0.972576i $$0.574719\pi$$
$$972$$ 0 0
$$973$$ −69.0477 −2.21357
$$974$$ 0 0
$$975$$ 18.3927 0.589037
$$976$$ 0 0
$$977$$ 8.67759 0.277621 0.138810 0.990319i $$-0.455672\pi$$
0.138810 + 0.990319i $$0.455672\pi$$
$$978$$ 0 0
$$979$$ −2.79434 −0.0893077
$$980$$ 0 0
$$981$$ −73.7262 −2.35390
$$982$$ 0 0
$$983$$ 15.5754 0.496778 0.248389 0.968660i $$-0.420099\pi$$
0.248389 + 0.968660i $$0.420099\pi$$
$$984$$ 0 0
$$985$$ 2.20197 0.0701606
$$986$$ 0 0
$$987$$ −91.0762 −2.89899
$$988$$ 0 0
$$989$$ −9.33238 −0.296752
$$990$$ 0 0
$$991$$ −35.5941 −1.13068 −0.565342 0.824857i $$-0.691256\pi$$
−0.565342 + 0.824857i $$0.691256\pi$$
$$992$$ 0 0
$$993$$ 77.2779 2.45234
$$994$$ 0 0
$$995$$ 11.4972 0.364485
$$996$$ 0 0
$$997$$ 32.2203 1.02043 0.510213 0.860048i $$-0.329567\pi$$
0.510213 + 0.860048i $$0.329567\pi$$
$$998$$ 0 0
$$999$$ −125.339 −3.96555
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 3040.2.a.u.1.1 5
4.3 odd 2 3040.2.a.x.1.5 yes 5
8.3 odd 2 6080.2.a.ci.1.1 5
8.5 even 2 6080.2.a.cl.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.1 5 1.1 even 1 trivial
3040.2.a.x.1.5 yes 5 4.3 odd 2
6080.2.a.ci.1.1 5 8.3 odd 2
6080.2.a.cl.1.5 5 8.5 even 2