# Properties

 Label 4840.2.a.z.1.2 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.09529$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.29496 q^{3} +1.00000 q^{5} +0.227777 q^{7} -1.32307 q^{9} +O(q^{10})$$ $$q-1.29496 q^{3} +1.00000 q^{5} +0.227777 q^{7} -1.32307 q^{9} +3.34478 q^{13} -1.29496 q^{15} -6.04981 q^{17} +2.22137 q^{19} -0.294963 q^{21} -3.47726 q^{23} +1.00000 q^{25} +5.59822 q^{27} -0.0307867 q^{29} +2.21625 q^{31} +0.227777 q^{35} +8.74728 q^{37} -4.33136 q^{39} -10.2404 q^{41} +3.74411 q^{43} -1.32307 q^{45} -5.31399 q^{47} -6.94812 q^{49} +7.83428 q^{51} -10.2028 q^{53} -2.87660 q^{57} +8.34399 q^{59} +2.24553 q^{61} -0.301365 q^{63} +3.34478 q^{65} +3.47330 q^{67} +4.50292 q^{69} -15.0637 q^{71} +3.66389 q^{73} -1.29496 q^{75} +11.6212 q^{79} -3.28027 q^{81} -1.52274 q^{83} -6.04981 q^{85} +0.0398677 q^{87} -10.0058 q^{89} +0.761864 q^{91} -2.86996 q^{93} +2.22137 q^{95} -8.33570 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 4 * q^5 + 7 * q^7 - 4 * q^9 $$4 q - 2 q^{3} + 4 q^{5} + 7 q^{7} - 4 q^{9} - 3 q^{13} - 2 q^{15} - 11 q^{17} - 2 q^{19} + 2 q^{21} - 11 q^{23} + 4 q^{25} + q^{27} - 4 q^{29} - 17 q^{31} + 7 q^{35} - 3 q^{37} - q^{39} - 13 q^{41} + 7 q^{43} - 4 q^{45} - q^{47} - 5 q^{49} + q^{51} - 15 q^{53} - 17 q^{57} - 17 q^{59} - 4 q^{61} - 15 q^{63} - 3 q^{65} + 7 q^{67} + 4 q^{69} - 15 q^{71} - 7 q^{73} - 2 q^{75} + 12 q^{79} - 8 q^{81} - 9 q^{83} - 11 q^{85} + 23 q^{87} - 12 q^{89} - 24 q^{91} - 11 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 4 * q^5 + 7 * q^7 - 4 * q^9 - 3 * q^13 - 2 * q^15 - 11 * q^17 - 2 * q^19 + 2 * q^21 - 11 * q^23 + 4 * q^25 + q^27 - 4 * q^29 - 17 * q^31 + 7 * q^35 - 3 * q^37 - q^39 - 13 * q^41 + 7 * q^43 - 4 * q^45 - q^47 - 5 * q^49 + q^51 - 15 * q^53 - 17 * q^57 - 17 * q^59 - 4 * q^61 - 15 * q^63 - 3 * q^65 + 7 * q^67 + 4 * q^69 - 15 * q^71 - 7 * q^73 - 2 * q^75 + 12 * q^79 - 8 * q^81 - 9 * q^83 - 11 * q^85 + 23 * q^87 - 12 * q^89 - 24 * q^91 - 11 * q^93 - 2 * q^95 + 2 * q^97

## 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
$$3$$ −1.29496 −0.747647 −0.373824 0.927500i $$-0.621953\pi$$
−0.373824 + 0.927500i $$0.621953\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0.227777 0.0860917 0.0430458 0.999073i $$-0.486294\pi$$
0.0430458 + 0.999073i $$0.486294\pi$$
$$8$$ 0 0
$$9$$ −1.32307 −0.441024
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 3.34478 0.927674 0.463837 0.885921i $$-0.346472\pi$$
0.463837 + 0.885921i $$0.346472\pi$$
$$14$$ 0 0
$$15$$ −1.29496 −0.334358
$$16$$ 0 0
$$17$$ −6.04981 −1.46730 −0.733648 0.679530i $$-0.762184\pi$$
−0.733648 + 0.679530i $$0.762184\pi$$
$$18$$ 0 0
$$19$$ 2.22137 0.509618 0.254809 0.966991i $$-0.417987\pi$$
0.254809 + 0.966991i $$0.417987\pi$$
$$20$$ 0 0
$$21$$ −0.294963 −0.0643662
$$22$$ 0 0
$$23$$ −3.47726 −0.725059 −0.362529 0.931972i $$-0.618087\pi$$
−0.362529 + 0.931972i $$0.618087\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.59822 1.07738
$$28$$ 0 0
$$29$$ −0.0307867 −0.00571695 −0.00285848 0.999996i $$-0.500910\pi$$
−0.00285848 + 0.999996i $$0.500910\pi$$
$$30$$ 0 0
$$31$$ 2.21625 0.398050 0.199025 0.979994i $$-0.436222\pi$$
0.199025 + 0.979994i $$0.436222\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.227777 0.0385014
$$36$$ 0 0
$$37$$ 8.74728 1.43804 0.719022 0.694987i $$-0.244590\pi$$
0.719022 + 0.694987i $$0.244590\pi$$
$$38$$ 0 0
$$39$$ −4.33136 −0.693573
$$40$$ 0 0
$$41$$ −10.2404 −1.59928 −0.799641 0.600478i $$-0.794977\pi$$
−0.799641 + 0.600478i $$0.794977\pi$$
$$42$$ 0 0
$$43$$ 3.74411 0.570972 0.285486 0.958383i $$-0.407845\pi$$
0.285486 + 0.958383i $$0.407845\pi$$
$$44$$ 0 0
$$45$$ −1.32307 −0.197232
$$46$$ 0 0
$$47$$ −5.31399 −0.775125 −0.387563 0.921843i $$-0.626683\pi$$
−0.387563 + 0.921843i $$0.626683\pi$$
$$48$$ 0 0
$$49$$ −6.94812 −0.992588
$$50$$ 0 0
$$51$$ 7.83428 1.09702
$$52$$ 0 0
$$53$$ −10.2028 −1.40147 −0.700734 0.713423i $$-0.747144\pi$$
−0.700734 + 0.713423i $$0.747144\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.87660 −0.381015
$$58$$ 0 0
$$59$$ 8.34399 1.08629 0.543147 0.839637i $$-0.317233\pi$$
0.543147 + 0.839637i $$0.317233\pi$$
$$60$$ 0 0
$$61$$ 2.24553 0.287510 0.143755 0.989613i $$-0.454082\pi$$
0.143755 + 0.989613i $$0.454082\pi$$
$$62$$ 0 0
$$63$$ −0.301365 −0.0379685
$$64$$ 0 0
$$65$$ 3.34478 0.414869
$$66$$ 0 0
$$67$$ 3.47330 0.424332 0.212166 0.977234i $$-0.431948\pi$$
0.212166 + 0.977234i $$0.431948\pi$$
$$68$$ 0 0
$$69$$ 4.50292 0.542088
$$70$$ 0 0
$$71$$ −15.0637 −1.78773 −0.893867 0.448332i $$-0.852018\pi$$
−0.893867 + 0.448332i $$0.852018\pi$$
$$72$$ 0 0
$$73$$ 3.66389 0.428826 0.214413 0.976743i $$-0.431216\pi$$
0.214413 + 0.976743i $$0.431216\pi$$
$$74$$ 0 0
$$75$$ −1.29496 −0.149529
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.6212 1.30749 0.653744 0.756716i $$-0.273197\pi$$
0.653744 + 0.756716i $$0.273197\pi$$
$$80$$ 0 0
$$81$$ −3.28027 −0.364474
$$82$$ 0 0
$$83$$ −1.52274 −0.167142 −0.0835712 0.996502i $$-0.526633\pi$$
−0.0835712 + 0.996502i $$0.526633\pi$$
$$84$$ 0 0
$$85$$ −6.04981 −0.656194
$$86$$ 0 0
$$87$$ 0.0398677 0.00427426
$$88$$ 0 0
$$89$$ −10.0058 −1.06062 −0.530309 0.847805i $$-0.677924\pi$$
−0.530309 + 0.847805i $$0.677924\pi$$
$$90$$ 0 0
$$91$$ 0.761864 0.0798650
$$92$$ 0 0
$$93$$ −2.86996 −0.297601
$$94$$ 0 0
$$95$$ 2.22137 0.227908
$$96$$ 0 0
$$97$$ −8.33570 −0.846362 −0.423181 0.906045i $$-0.639087\pi$$
−0.423181 + 0.906045i $$0.639087\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.72279 −0.171424 −0.0857118 0.996320i $$-0.527316\pi$$
−0.0857118 + 0.996320i $$0.527316\pi$$
$$102$$ 0 0
$$103$$ −1.95452 −0.192585 −0.0962923 0.995353i $$-0.530698\pi$$
−0.0962923 + 0.995353i $$0.530698\pi$$
$$104$$ 0 0
$$105$$ −0.294963 −0.0287854
$$106$$ 0 0
$$107$$ −13.4004 −1.29547 −0.647735 0.761866i $$-0.724283\pi$$
−0.647735 + 0.761866i $$0.724283\pi$$
$$108$$ 0 0
$$109$$ −7.46306 −0.714831 −0.357416 0.933945i $$-0.616342\pi$$
−0.357416 + 0.933945i $$0.616342\pi$$
$$110$$ 0 0
$$111$$ −11.3274 −1.07515
$$112$$ 0 0
$$113$$ 5.29252 0.497878 0.248939 0.968519i $$-0.419918\pi$$
0.248939 + 0.968519i $$0.419918\pi$$
$$114$$ 0 0
$$115$$ −3.47726 −0.324256
$$116$$ 0 0
$$117$$ −4.42538 −0.409126
$$118$$ 0 0
$$119$$ −1.37801 −0.126322
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 13.2609 1.19570
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 12.1933 1.08198 0.540989 0.841030i $$-0.318050\pi$$
0.540989 + 0.841030i $$0.318050\pi$$
$$128$$ 0 0
$$129$$ −4.84849 −0.426886
$$130$$ 0 0
$$131$$ 2.06530 0.180446 0.0902229 0.995922i $$-0.471242\pi$$
0.0902229 + 0.995922i $$0.471242\pi$$
$$132$$ 0 0
$$133$$ 0.505978 0.0438739
$$134$$ 0 0
$$135$$ 5.59822 0.481818
$$136$$ 0 0
$$137$$ 14.4436 1.23400 0.617001 0.786963i $$-0.288348\pi$$
0.617001 + 0.786963i $$0.288348\pi$$
$$138$$ 0 0
$$139$$ 16.3421 1.38612 0.693059 0.720881i $$-0.256262\pi$$
0.693059 + 0.720881i $$0.256262\pi$$
$$140$$ 0 0
$$141$$ 6.88142 0.579520
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −0.0307867 −0.00255670
$$146$$ 0 0
$$147$$ 8.99755 0.742106
$$148$$ 0 0
$$149$$ −14.9817 −1.22735 −0.613674 0.789559i $$-0.710309\pi$$
−0.613674 + 0.789559i $$0.710309\pi$$
$$150$$ 0 0
$$151$$ −12.3281 −1.00325 −0.501624 0.865086i $$-0.667264\pi$$
−0.501624 + 0.865086i $$0.667264\pi$$
$$152$$ 0 0
$$153$$ 8.00433 0.647112
$$154$$ 0 0
$$155$$ 2.21625 0.178014
$$156$$ 0 0
$$157$$ −20.8442 −1.66355 −0.831775 0.555112i $$-0.812675\pi$$
−0.831775 + 0.555112i $$0.812675\pi$$
$$158$$ 0 0
$$159$$ 13.2123 1.04780
$$160$$ 0 0
$$161$$ −0.792040 −0.0624215
$$162$$ 0 0
$$163$$ 0.564984 0.0442530 0.0221265 0.999755i $$-0.492956\pi$$
0.0221265 + 0.999755i $$0.492956\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.4354 1.73611 0.868053 0.496471i $$-0.165371\pi$$
0.868053 + 0.496471i $$0.165371\pi$$
$$168$$ 0 0
$$169$$ −1.81247 −0.139421
$$170$$ 0 0
$$171$$ −2.93904 −0.224754
$$172$$ 0 0
$$173$$ −2.16572 −0.164656 −0.0823281 0.996605i $$-0.526236\pi$$
−0.0823281 + 0.996605i $$0.526236\pi$$
$$174$$ 0 0
$$175$$ 0.227777 0.0172183
$$176$$ 0 0
$$177$$ −10.8052 −0.812165
$$178$$ 0 0
$$179$$ −22.1568 −1.65608 −0.828038 0.560671i $$-0.810543\pi$$
−0.828038 + 0.560671i $$0.810543\pi$$
$$180$$ 0 0
$$181$$ −8.93677 −0.664265 −0.332132 0.943233i $$-0.607768\pi$$
−0.332132 + 0.943233i $$0.607768\pi$$
$$182$$ 0 0
$$183$$ −2.90787 −0.214956
$$184$$ 0 0
$$185$$ 8.74728 0.643113
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.27515 0.0927532
$$190$$ 0 0
$$191$$ −11.5080 −0.832693 −0.416347 0.909206i $$-0.636690\pi$$
−0.416347 + 0.909206i $$0.636690\pi$$
$$192$$ 0 0
$$193$$ −8.31192 −0.598305 −0.299153 0.954205i $$-0.596704\pi$$
−0.299153 + 0.954205i $$0.596704\pi$$
$$194$$ 0 0
$$195$$ −4.33136 −0.310175
$$196$$ 0 0
$$197$$ −3.64765 −0.259885 −0.129942 0.991522i $$-0.541479\pi$$
−0.129942 + 0.991522i $$0.541479\pi$$
$$198$$ 0 0
$$199$$ −6.34757 −0.449967 −0.224984 0.974363i $$-0.572233\pi$$
−0.224984 + 0.974363i $$0.572233\pi$$
$$200$$ 0 0
$$201$$ −4.49780 −0.317250
$$202$$ 0 0
$$203$$ −0.00701251 −0.000492182 0
$$204$$ 0 0
$$205$$ −10.2404 −0.715221
$$206$$ 0 0
$$207$$ 4.60066 0.319768
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 19.2543 1.32552 0.662761 0.748831i $$-0.269385\pi$$
0.662761 + 0.748831i $$0.269385\pi$$
$$212$$ 0 0
$$213$$ 19.5070 1.33659
$$214$$ 0 0
$$215$$ 3.74411 0.255347
$$216$$ 0 0
$$217$$ 0.504811 0.0342688
$$218$$ 0 0
$$219$$ −4.74460 −0.320611
$$220$$ 0 0
$$221$$ −20.2353 −1.36117
$$222$$ 0 0
$$223$$ −21.7301 −1.45516 −0.727579 0.686024i $$-0.759355\pi$$
−0.727579 + 0.686024i $$0.759355\pi$$
$$224$$ 0 0
$$225$$ −1.32307 −0.0882047
$$226$$ 0 0
$$227$$ 6.00475 0.398549 0.199275 0.979944i $$-0.436141\pi$$
0.199275 + 0.979944i $$0.436141\pi$$
$$228$$ 0 0
$$229$$ −23.2079 −1.53362 −0.766810 0.641874i $$-0.778157\pi$$
−0.766810 + 0.641874i $$0.778157\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.2183 −1.58659 −0.793296 0.608836i $$-0.791637\pi$$
−0.793296 + 0.608836i $$0.791637\pi$$
$$234$$ 0 0
$$235$$ −5.31399 −0.346646
$$236$$ 0 0
$$237$$ −15.0490 −0.977539
$$238$$ 0 0
$$239$$ 18.3873 1.18937 0.594686 0.803958i $$-0.297276\pi$$
0.594686 + 0.803958i $$0.297276\pi$$
$$240$$ 0 0
$$241$$ 12.3990 0.798692 0.399346 0.916800i $$-0.369237\pi$$
0.399346 + 0.916800i $$0.369237\pi$$
$$242$$ 0 0
$$243$$ −12.5468 −0.804879
$$244$$ 0 0
$$245$$ −6.94812 −0.443899
$$246$$ 0 0
$$247$$ 7.43000 0.472760
$$248$$ 0 0
$$249$$ 1.97189 0.124964
$$250$$ 0 0
$$251$$ −21.7939 −1.37562 −0.687810 0.725890i $$-0.741428\pi$$
−0.687810 + 0.725890i $$0.741428\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 7.83428 0.490602
$$256$$ 0 0
$$257$$ 14.7089 0.917518 0.458759 0.888561i $$-0.348294\pi$$
0.458759 + 0.888561i $$0.348294\pi$$
$$258$$ 0 0
$$259$$ 1.99243 0.123804
$$260$$ 0 0
$$261$$ 0.0407330 0.00252131
$$262$$ 0 0
$$263$$ −27.4694 −1.69384 −0.846918 0.531724i $$-0.821544\pi$$
−0.846918 + 0.531724i $$0.821544\pi$$
$$264$$ 0 0
$$265$$ −10.2028 −0.626755
$$266$$ 0 0
$$267$$ 12.9572 0.792968
$$268$$ 0 0
$$269$$ −1.41916 −0.0865274 −0.0432637 0.999064i $$-0.513776\pi$$
−0.0432637 + 0.999064i $$0.513776\pi$$
$$270$$ 0 0
$$271$$ −10.3498 −0.628708 −0.314354 0.949306i $$-0.601788\pi$$
−0.314354 + 0.949306i $$0.601788\pi$$
$$272$$ 0 0
$$273$$ −0.986585 −0.0597108
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −17.8058 −1.06985 −0.534923 0.844901i $$-0.679660\pi$$
−0.534923 + 0.844901i $$0.679660\pi$$
$$278$$ 0 0
$$279$$ −2.93226 −0.175550
$$280$$ 0 0
$$281$$ −17.1393 −1.02245 −0.511223 0.859448i $$-0.670807\pi$$
−0.511223 + 0.859448i $$0.670807\pi$$
$$282$$ 0 0
$$283$$ −9.92524 −0.589995 −0.294997 0.955498i $$-0.595319\pi$$
−0.294997 + 0.955498i $$0.595319\pi$$
$$284$$ 0 0
$$285$$ −2.87660 −0.170395
$$286$$ 0 0
$$287$$ −2.33253 −0.137685
$$288$$ 0 0
$$289$$ 19.6002 1.15296
$$290$$ 0 0
$$291$$ 10.7944 0.632780
$$292$$ 0 0
$$293$$ 18.1035 1.05762 0.528809 0.848741i $$-0.322639\pi$$
0.528809 + 0.848741i $$0.322639\pi$$
$$294$$ 0 0
$$295$$ 8.34399 0.485806
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −11.6307 −0.672618
$$300$$ 0 0
$$301$$ 0.852824 0.0491559
$$302$$ 0 0
$$303$$ 2.23094 0.128164
$$304$$ 0 0
$$305$$ 2.24553 0.128578
$$306$$ 0 0
$$307$$ 6.00433 0.342685 0.171343 0.985211i $$-0.445189\pi$$
0.171343 + 0.985211i $$0.445189\pi$$
$$308$$ 0 0
$$309$$ 2.53103 0.143985
$$310$$ 0 0
$$311$$ −14.8766 −0.843574 −0.421787 0.906695i $$-0.638597\pi$$
−0.421787 + 0.906695i $$0.638597\pi$$
$$312$$ 0 0
$$313$$ −2.41939 −0.136752 −0.0683760 0.997660i $$-0.521782\pi$$
−0.0683760 + 0.997660i $$0.521782\pi$$
$$314$$ 0 0
$$315$$ −0.301365 −0.0169800
$$316$$ 0 0
$$317$$ 19.2375 1.08049 0.540244 0.841508i $$-0.318332\pi$$
0.540244 + 0.841508i $$0.318332\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 17.3531 0.968554
$$322$$ 0 0
$$323$$ −13.4389 −0.747761
$$324$$ 0 0
$$325$$ 3.34478 0.185535
$$326$$ 0 0
$$327$$ 9.66438 0.534441
$$328$$ 0 0
$$329$$ −1.21041 −0.0667318
$$330$$ 0 0
$$331$$ −13.8876 −0.763330 −0.381665 0.924301i $$-0.624649\pi$$
−0.381665 + 0.924301i $$0.624649\pi$$
$$332$$ 0 0
$$333$$ −11.5733 −0.634212
$$334$$ 0 0
$$335$$ 3.47330 0.189767
$$336$$ 0 0
$$337$$ 3.08479 0.168039 0.0840196 0.996464i $$-0.473224\pi$$
0.0840196 + 0.996464i $$0.473224\pi$$
$$338$$ 0 0
$$339$$ −6.85361 −0.372237
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −3.17706 −0.171545
$$344$$ 0 0
$$345$$ 4.50292 0.242429
$$346$$ 0 0
$$347$$ 28.6219 1.53650 0.768252 0.640148i $$-0.221127\pi$$
0.768252 + 0.640148i $$0.221127\pi$$
$$348$$ 0 0
$$349$$ −16.8202 −0.900365 −0.450182 0.892937i $$-0.648641\pi$$
−0.450182 + 0.892937i $$0.648641\pi$$
$$350$$ 0 0
$$351$$ 18.7248 0.999455
$$352$$ 0 0
$$353$$ −14.6656 −0.780573 −0.390287 0.920693i $$-0.627624\pi$$
−0.390287 + 0.920693i $$0.627624\pi$$
$$354$$ 0 0
$$355$$ −15.0637 −0.799499
$$356$$ 0 0
$$357$$ 1.78447 0.0944442
$$358$$ 0 0
$$359$$ 30.8757 1.62956 0.814780 0.579771i $$-0.196858\pi$$
0.814780 + 0.579771i $$0.196858\pi$$
$$360$$ 0 0
$$361$$ −14.0655 −0.740289
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.66389 0.191777
$$366$$ 0 0
$$367$$ −3.63145 −0.189560 −0.0947800 0.995498i $$-0.530215\pi$$
−0.0947800 + 0.995498i $$0.530215\pi$$
$$368$$ 0 0
$$369$$ 13.5488 0.705321
$$370$$ 0 0
$$371$$ −2.32397 −0.120655
$$372$$ 0 0
$$373$$ −29.2244 −1.51318 −0.756590 0.653890i $$-0.773136\pi$$
−0.756590 + 0.653890i $$0.773136\pi$$
$$374$$ 0 0
$$375$$ −1.29496 −0.0668716
$$376$$ 0 0
$$377$$ −0.102975 −0.00530347
$$378$$ 0 0
$$379$$ 3.65327 0.187656 0.0938278 0.995588i $$-0.470090\pi$$
0.0938278 + 0.995588i $$0.470090\pi$$
$$380$$ 0 0
$$381$$ −15.7898 −0.808937
$$382$$ 0 0
$$383$$ −14.2462 −0.727949 −0.363975 0.931409i $$-0.618581\pi$$
−0.363975 + 0.931409i $$0.618581\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.95373 −0.251812
$$388$$ 0 0
$$389$$ −8.24632 −0.418105 −0.209052 0.977904i $$-0.567038\pi$$
−0.209052 + 0.977904i $$0.567038\pi$$
$$390$$ 0 0
$$391$$ 21.0368 1.06388
$$392$$ 0 0
$$393$$ −2.67448 −0.134910
$$394$$ 0 0
$$395$$ 11.6212 0.584726
$$396$$ 0 0
$$397$$ −2.57097 −0.129033 −0.0645167 0.997917i $$-0.520551\pi$$
−0.0645167 + 0.997917i $$0.520551\pi$$
$$398$$ 0 0
$$399$$ −0.655223 −0.0328022
$$400$$ 0 0
$$401$$ −38.9710 −1.94612 −0.973060 0.230551i $$-0.925947\pi$$
−0.973060 + 0.230551i $$0.925947\pi$$
$$402$$ 0 0
$$403$$ 7.41286 0.369261
$$404$$ 0 0
$$405$$ −3.28027 −0.162998
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 3.42256 0.169234 0.0846172 0.996414i $$-0.473033\pi$$
0.0846172 + 0.996414i $$0.473033\pi$$
$$410$$ 0 0
$$411$$ −18.7039 −0.922598
$$412$$ 0 0
$$413$$ 1.90057 0.0935209
$$414$$ 0 0
$$415$$ −1.52274 −0.0747484
$$416$$ 0 0
$$417$$ −21.1624 −1.03633
$$418$$ 0 0
$$419$$ 10.5914 0.517426 0.258713 0.965954i $$-0.416702\pi$$
0.258713 + 0.965954i $$0.416702\pi$$
$$420$$ 0 0
$$421$$ 2.37055 0.115534 0.0577668 0.998330i $$-0.481602\pi$$
0.0577668 + 0.998330i $$0.481602\pi$$
$$422$$ 0 0
$$423$$ 7.03079 0.341849
$$424$$ 0 0
$$425$$ −6.04981 −0.293459
$$426$$ 0 0
$$427$$ 0.511479 0.0247522
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 26.2360 1.26374 0.631872 0.775073i $$-0.282287\pi$$
0.631872 + 0.775073i $$0.282287\pi$$
$$432$$ 0 0
$$433$$ −20.0338 −0.962762 −0.481381 0.876512i $$-0.659865\pi$$
−0.481381 + 0.876512i $$0.659865\pi$$
$$434$$ 0 0
$$435$$ 0.0398677 0.00191151
$$436$$ 0 0
$$437$$ −7.72430 −0.369503
$$438$$ 0 0
$$439$$ −21.9546 −1.04783 −0.523917 0.851769i $$-0.675530\pi$$
−0.523917 + 0.851769i $$0.675530\pi$$
$$440$$ 0 0
$$441$$ 9.19285 0.437755
$$442$$ 0 0
$$443$$ 16.3419 0.776428 0.388214 0.921569i $$-0.373092\pi$$
0.388214 + 0.921569i $$0.373092\pi$$
$$444$$ 0 0
$$445$$ −10.0058 −0.474323
$$446$$ 0 0
$$447$$ 19.4007 0.917623
$$448$$ 0 0
$$449$$ −38.0476 −1.79558 −0.897788 0.440429i $$-0.854826\pi$$
−0.897788 + 0.440429i $$0.854826\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 15.9645 0.750076
$$454$$ 0 0
$$455$$ 0.761864 0.0357167
$$456$$ 0 0
$$457$$ 29.3016 1.37067 0.685336 0.728227i $$-0.259656\pi$$
0.685336 + 0.728227i $$0.259656\pi$$
$$458$$ 0 0
$$459$$ −33.8682 −1.58083
$$460$$ 0 0
$$461$$ 12.1136 0.564188 0.282094 0.959387i $$-0.408971\pi$$
0.282094 + 0.959387i $$0.408971\pi$$
$$462$$ 0 0
$$463$$ −3.92452 −0.182388 −0.0911940 0.995833i $$-0.529068\pi$$
−0.0911940 + 0.995833i $$0.529068\pi$$
$$464$$ 0 0
$$465$$ −2.86996 −0.133091
$$466$$ 0 0
$$467$$ 35.1795 1.62792 0.813958 0.580924i $$-0.197309\pi$$
0.813958 + 0.580924i $$0.197309\pi$$
$$468$$ 0 0
$$469$$ 0.791139 0.0365314
$$470$$ 0 0
$$471$$ 26.9925 1.24375
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 2.22137 0.101924
$$476$$ 0 0
$$477$$ 13.4991 0.618080
$$478$$ 0 0
$$479$$ −34.9726 −1.59794 −0.798969 0.601372i $$-0.794621\pi$$
−0.798969 + 0.601372i $$0.794621\pi$$
$$480$$ 0 0
$$481$$ 29.2577 1.33404
$$482$$ 0 0
$$483$$ 1.02566 0.0466693
$$484$$ 0 0
$$485$$ −8.33570 −0.378504
$$486$$ 0 0
$$487$$ 10.6381 0.482058 0.241029 0.970518i $$-0.422515\pi$$
0.241029 + 0.970518i $$0.422515\pi$$
$$488$$ 0 0
$$489$$ −0.731634 −0.0330856
$$490$$ 0 0
$$491$$ −29.0985 −1.31320 −0.656598 0.754241i $$-0.728005\pi$$
−0.656598 + 0.754241i $$0.728005\pi$$
$$492$$ 0 0
$$493$$ 0.186254 0.00838846
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3.43117 −0.153909
$$498$$ 0 0
$$499$$ −23.1487 −1.03628 −0.518140 0.855296i $$-0.673375\pi$$
−0.518140 + 0.855296i $$0.673375\pi$$
$$500$$ 0 0
$$501$$ −29.0531 −1.29799
$$502$$ 0 0
$$503$$ −12.6609 −0.564521 −0.282261 0.959338i $$-0.591084\pi$$
−0.282261 + 0.959338i $$0.591084\pi$$
$$504$$ 0 0
$$505$$ −1.72279 −0.0766630
$$506$$ 0 0
$$507$$ 2.34708 0.104237
$$508$$ 0 0
$$509$$ 9.86065 0.437066 0.218533 0.975830i $$-0.429873\pi$$
0.218533 + 0.975830i $$0.429873\pi$$
$$510$$ 0 0
$$511$$ 0.834550 0.0369183
$$512$$ 0 0
$$513$$ 12.4357 0.549051
$$514$$ 0 0
$$515$$ −1.95452 −0.0861264
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.80452 0.123105
$$520$$ 0 0
$$521$$ −2.52164 −0.110475 −0.0552376 0.998473i $$-0.517592\pi$$
−0.0552376 + 0.998473i $$0.517592\pi$$
$$522$$ 0 0
$$523$$ −20.7213 −0.906079 −0.453039 0.891490i $$-0.649660\pi$$
−0.453039 + 0.891490i $$0.649660\pi$$
$$524$$ 0 0
$$525$$ −0.294963 −0.0128732
$$526$$ 0 0
$$527$$ −13.4079 −0.584057
$$528$$ 0 0
$$529$$ −10.9087 −0.474290
$$530$$ 0 0
$$531$$ −11.0397 −0.479082
$$532$$ 0 0
$$533$$ −34.2519 −1.48361
$$534$$ 0 0
$$535$$ −13.4004 −0.579351
$$536$$ 0 0
$$537$$ 28.6922 1.23816
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −26.7777 −1.15126 −0.575631 0.817710i $$-0.695243\pi$$
−0.575631 + 0.817710i $$0.695243\pi$$
$$542$$ 0 0
$$543$$ 11.5728 0.496636
$$544$$ 0 0
$$545$$ −7.46306 −0.319682
$$546$$ 0 0
$$547$$ 4.57034 0.195414 0.0977068 0.995215i $$-0.468849\pi$$
0.0977068 + 0.995215i $$0.468849\pi$$
$$548$$ 0 0
$$549$$ −2.97099 −0.126799
$$550$$ 0 0
$$551$$ −0.0683889 −0.00291346
$$552$$ 0 0
$$553$$ 2.64704 0.112564
$$554$$ 0 0
$$555$$ −11.3274 −0.480822
$$556$$ 0 0
$$557$$ 31.7834 1.34670 0.673352 0.739322i $$-0.264854\pi$$
0.673352 + 0.739322i $$0.264854\pi$$
$$558$$ 0 0
$$559$$ 12.5232 0.529676
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 9.29858 0.391888 0.195944 0.980615i $$-0.437223\pi$$
0.195944 + 0.980615i $$0.437223\pi$$
$$564$$ 0 0
$$565$$ 5.29252 0.222658
$$566$$ 0 0
$$567$$ −0.747170 −0.0313782
$$568$$ 0 0
$$569$$ 47.1667 1.97733 0.988665 0.150139i $$-0.0479720\pi$$
0.988665 + 0.150139i $$0.0479720\pi$$
$$570$$ 0 0
$$571$$ 4.32635 0.181052 0.0905260 0.995894i $$-0.471145\pi$$
0.0905260 + 0.995894i $$0.471145\pi$$
$$572$$ 0 0
$$573$$ 14.9025 0.622561
$$574$$ 0 0
$$575$$ −3.47726 −0.145012
$$576$$ 0 0
$$577$$ 8.77746 0.365410 0.182705 0.983168i $$-0.441515\pi$$
0.182705 + 0.983168i $$0.441515\pi$$
$$578$$ 0 0
$$579$$ 10.7636 0.447321
$$580$$ 0 0
$$581$$ −0.346845 −0.0143896
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −4.42538 −0.182967
$$586$$ 0 0
$$587$$ −20.6919 −0.854047 −0.427024 0.904240i $$-0.640438\pi$$
−0.427024 + 0.904240i $$0.640438\pi$$
$$588$$ 0 0
$$589$$ 4.92312 0.202854
$$590$$ 0 0
$$591$$ 4.72358 0.194302
$$592$$ 0 0
$$593$$ 46.8273 1.92296 0.961482 0.274866i $$-0.0886336\pi$$
0.961482 + 0.274866i $$0.0886336\pi$$
$$594$$ 0 0
$$595$$ −1.37801 −0.0564929
$$596$$ 0 0
$$597$$ 8.21986 0.336417
$$598$$ 0 0
$$599$$ −31.0055 −1.26685 −0.633425 0.773804i $$-0.718351\pi$$
−0.633425 + 0.773804i $$0.718351\pi$$
$$600$$ 0 0
$$601$$ −42.0290 −1.71440 −0.857199 0.514986i $$-0.827797\pi$$
−0.857199 + 0.514986i $$0.827797\pi$$
$$602$$ 0 0
$$603$$ −4.59543 −0.187140
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −10.8594 −0.440769 −0.220385 0.975413i $$-0.570731\pi$$
−0.220385 + 0.975413i $$0.570731\pi$$
$$608$$ 0 0
$$609$$ 0.00908094 0.000367978 0
$$610$$ 0 0
$$611$$ −17.7741 −0.719064
$$612$$ 0 0
$$613$$ 37.0203 1.49524 0.747618 0.664129i $$-0.231197\pi$$
0.747618 + 0.664129i $$0.231197\pi$$
$$614$$ 0 0
$$615$$ 13.2609 0.534733
$$616$$ 0 0
$$617$$ 37.3620 1.50414 0.752068 0.659085i $$-0.229056\pi$$
0.752068 + 0.659085i $$0.229056\pi$$
$$618$$ 0 0
$$619$$ −39.7254 −1.59670 −0.798350 0.602194i $$-0.794293\pi$$
−0.798350 + 0.602194i $$0.794293\pi$$
$$620$$ 0 0
$$621$$ −19.4665 −0.781162
$$622$$ 0 0
$$623$$ −2.27910 −0.0913103
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −52.9194 −2.11004
$$630$$ 0 0
$$631$$ 7.44019 0.296189 0.148095 0.988973i $$-0.452686\pi$$
0.148095 + 0.988973i $$0.452686\pi$$
$$632$$ 0 0
$$633$$ −24.9336 −0.991022
$$634$$ 0 0
$$635$$ 12.1933 0.483875
$$636$$ 0 0
$$637$$ −23.2399 −0.920798
$$638$$ 0 0
$$639$$ 19.9304 0.788433
$$640$$ 0 0
$$641$$ −1.86589 −0.0736984 −0.0368492 0.999321i $$-0.511732\pi$$
−0.0368492 + 0.999321i $$0.511732\pi$$
$$642$$ 0 0
$$643$$ 45.7333 1.80354 0.901772 0.432212i $$-0.142267\pi$$
0.901772 + 0.432212i $$0.142267\pi$$
$$644$$ 0 0
$$645$$ −4.84849 −0.190909
$$646$$ 0 0
$$647$$ −34.0395 −1.33823 −0.669115 0.743159i $$-0.733327\pi$$
−0.669115 + 0.743159i $$0.733327\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −0.653712 −0.0256210
$$652$$ 0 0
$$653$$ 3.10780 0.121617 0.0608087 0.998149i $$-0.480632\pi$$
0.0608087 + 0.998149i $$0.480632\pi$$
$$654$$ 0 0
$$655$$ 2.06530 0.0806978
$$656$$ 0 0
$$657$$ −4.84759 −0.189122
$$658$$ 0 0
$$659$$ 10.1518 0.395459 0.197729 0.980257i $$-0.436643\pi$$
0.197729 + 0.980257i $$0.436643\pi$$
$$660$$ 0 0
$$661$$ −44.7642 −1.74113 −0.870564 0.492056i $$-0.836246\pi$$
−0.870564 + 0.492056i $$0.836246\pi$$
$$662$$ 0 0
$$663$$ 26.2039 1.01768
$$664$$ 0 0
$$665$$ 0.505978 0.0196210
$$666$$ 0 0
$$667$$ 0.107053 0.00414513
$$668$$ 0 0
$$669$$ 28.1397 1.08794
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 5.36472 0.206795 0.103397 0.994640i $$-0.467029\pi$$
0.103397 + 0.994640i $$0.467029\pi$$
$$674$$ 0 0
$$675$$ 5.59822 0.215475
$$676$$ 0 0
$$677$$ 11.3147 0.434860 0.217430 0.976076i $$-0.430233\pi$$
0.217430 + 0.976076i $$0.430233\pi$$
$$678$$ 0 0
$$679$$ −1.89868 −0.0728647
$$680$$ 0 0
$$681$$ −7.77592 −0.297974
$$682$$ 0 0
$$683$$ 31.0817 1.18931 0.594655 0.803981i $$-0.297289\pi$$
0.594655 + 0.803981i $$0.297289\pi$$
$$684$$ 0 0
$$685$$ 14.4436 0.551862
$$686$$ 0 0
$$687$$ 30.0534 1.14661
$$688$$ 0 0
$$689$$ −34.1262 −1.30011
$$690$$ 0 0
$$691$$ 3.95742 0.150547 0.0752737 0.997163i $$-0.476017\pi$$
0.0752737 + 0.997163i $$0.476017\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.3421 0.619891
$$696$$ 0 0
$$697$$ 61.9525 2.34662
$$698$$ 0 0
$$699$$ 31.3618 1.18621
$$700$$ 0 0
$$701$$ −22.9309 −0.866089 −0.433045 0.901372i $$-0.642561\pi$$
−0.433045 + 0.901372i $$0.642561\pi$$
$$702$$ 0 0
$$703$$ 19.4310 0.732854
$$704$$ 0 0
$$705$$ 6.88142 0.259169
$$706$$ 0 0
$$707$$ −0.392411 −0.0147581
$$708$$ 0 0
$$709$$ 32.5067 1.22081 0.610407 0.792088i $$-0.291006\pi$$
0.610407 + 0.792088i $$0.291006\pi$$
$$710$$ 0 0
$$711$$ −15.3757 −0.576633
$$712$$ 0 0
$$713$$ −7.70648 −0.288610
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −23.8108 −0.889231
$$718$$ 0 0
$$719$$ 30.4360 1.13507 0.567536 0.823348i $$-0.307897\pi$$
0.567536 + 0.823348i $$0.307897\pi$$
$$720$$ 0 0
$$721$$ −0.445195 −0.0165799
$$722$$ 0 0
$$723$$ −16.0563 −0.597140
$$724$$ 0 0
$$725$$ −0.0307867 −0.00114339
$$726$$ 0 0
$$727$$ 36.6338 1.35867 0.679337 0.733827i $$-0.262268\pi$$
0.679337 + 0.733827i $$0.262268\pi$$
$$728$$ 0 0
$$729$$ 26.0885 0.966240
$$730$$ 0 0
$$731$$ −22.6512 −0.837785
$$732$$ 0 0
$$733$$ −32.1710 −1.18826 −0.594132 0.804368i $$-0.702504\pi$$
−0.594132 + 0.804368i $$0.702504\pi$$
$$734$$ 0 0
$$735$$ 8.99755 0.331880
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −20.7794 −0.764382 −0.382191 0.924083i $$-0.624830\pi$$
−0.382191 + 0.924083i $$0.624830\pi$$
$$740$$ 0 0
$$741$$ −9.62158 −0.353457
$$742$$ 0 0
$$743$$ 37.8417 1.38828 0.694140 0.719840i $$-0.255785\pi$$
0.694140 + 0.719840i $$0.255785\pi$$
$$744$$ 0 0
$$745$$ −14.9817 −0.548887
$$746$$ 0 0
$$747$$ 2.01469 0.0737138
$$748$$ 0 0
$$749$$ −3.05231 −0.111529
$$750$$ 0 0
$$751$$ −13.0447 −0.476007 −0.238004 0.971264i $$-0.576493\pi$$
−0.238004 + 0.971264i $$0.576493\pi$$
$$752$$ 0 0
$$753$$ 28.2223 1.02848
$$754$$ 0 0
$$755$$ −12.3281 −0.448666
$$756$$ 0 0
$$757$$ 41.1704 1.49636 0.748181 0.663495i $$-0.230927\pi$$
0.748181 + 0.663495i $$0.230927\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 43.7344 1.58537 0.792686 0.609630i $$-0.208682\pi$$
0.792686 + 0.609630i $$0.208682\pi$$
$$762$$ 0 0
$$763$$ −1.69991 −0.0615410
$$764$$ 0 0
$$765$$ 8.00433 0.289397
$$766$$ 0 0
$$767$$ 27.9088 1.00773
$$768$$ 0 0
$$769$$ 17.3914 0.627151 0.313575 0.949563i $$-0.398473\pi$$
0.313575 + 0.949563i $$0.398473\pi$$
$$770$$ 0 0
$$771$$ −19.0475 −0.685979
$$772$$ 0 0
$$773$$ 24.8805 0.894890 0.447445 0.894311i $$-0.352334\pi$$
0.447445 + 0.894311i $$0.352334\pi$$
$$774$$ 0 0
$$775$$ 2.21625 0.0796101
$$776$$ 0 0
$$777$$ −2.58012 −0.0925614
$$778$$ 0 0
$$779$$ −22.7478 −0.815023
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −0.172351 −0.00615931
$$784$$ 0 0
$$785$$ −20.8442 −0.743963
$$786$$ 0 0
$$787$$ −27.0641 −0.964731 −0.482365 0.875970i $$-0.660222\pi$$
−0.482365 + 0.875970i $$0.660222\pi$$
$$788$$ 0 0
$$789$$ 35.5718 1.26639
$$790$$ 0 0
$$791$$ 1.20551 0.0428632
$$792$$ 0 0
$$793$$ 7.51078 0.266716
$$794$$ 0 0
$$795$$ 13.2123 0.468592
$$796$$ 0 0
$$797$$ −12.0392 −0.426449 −0.213224 0.977003i $$-0.568397\pi$$
−0.213224 + 0.977003i $$0.568397\pi$$
$$798$$ 0 0
$$799$$ 32.1487 1.13734
$$800$$ 0 0
$$801$$ 13.2384 0.467757
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −0.792040 −0.0279157
$$806$$ 0 0
$$807$$ 1.83775 0.0646920
$$808$$ 0 0
$$809$$ −27.3914 −0.963031 −0.481516 0.876438i $$-0.659914\pi$$
−0.481516 + 0.876438i $$0.659914\pi$$
$$810$$ 0 0
$$811$$ −54.3420 −1.90820 −0.954102 0.299481i $$-0.903186\pi$$
−0.954102 + 0.299481i $$0.903186\pi$$
$$812$$ 0 0
$$813$$ 13.4026 0.470051
$$814$$ 0 0
$$815$$ 0.564984 0.0197905
$$816$$ 0 0
$$817$$ 8.31708 0.290978
$$818$$ 0 0
$$819$$ −1.00800 −0.0352224
$$820$$ 0 0
$$821$$ 39.7997 1.38902 0.694509 0.719484i $$-0.255622\pi$$
0.694509 + 0.719484i $$0.255622\pi$$
$$822$$ 0 0
$$823$$ 23.8835 0.832526 0.416263 0.909244i $$-0.363340\pi$$
0.416263 + 0.909244i $$0.363340\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.83591 −0.272481 −0.136241 0.990676i $$-0.543502\pi$$
−0.136241 + 0.990676i $$0.543502\pi$$
$$828$$ 0 0
$$829$$ 36.1073 1.25406 0.627028 0.778996i $$-0.284271\pi$$
0.627028 + 0.778996i $$0.284271\pi$$
$$830$$ 0 0
$$831$$ 23.0578 0.799868
$$832$$ 0 0
$$833$$ 42.0348 1.45642
$$834$$ 0 0
$$835$$ 22.4354 0.776410
$$836$$ 0 0
$$837$$ 12.4071 0.428850
$$838$$ 0 0
$$839$$ 25.3952 0.876741 0.438371 0.898794i $$-0.355556\pi$$
0.438371 + 0.898794i $$0.355556\pi$$
$$840$$ 0 0
$$841$$ −28.9991 −0.999967
$$842$$ 0 0
$$843$$ 22.1948 0.764428
$$844$$ 0 0
$$845$$ −1.81247 −0.0623508
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 12.8528 0.441108
$$850$$ 0 0
$$851$$ −30.4166 −1.04267
$$852$$ 0 0
$$853$$ 17.0660 0.584329 0.292165 0.956368i $$-0.405624\pi$$
0.292165 + 0.956368i $$0.405624\pi$$
$$854$$ 0 0
$$855$$ −2.93904 −0.100513
$$856$$ 0 0
$$857$$ −31.2827 −1.06860 −0.534298 0.845296i $$-0.679424\pi$$
−0.534298 + 0.845296i $$0.679424\pi$$
$$858$$ 0 0
$$859$$ 50.6101 1.72679 0.863396 0.504526i $$-0.168333\pi$$
0.863396 + 0.504526i $$0.168333\pi$$
$$860$$ 0 0
$$861$$ 3.02054 0.102940
$$862$$ 0 0
$$863$$ −29.9414 −1.01922 −0.509609 0.860406i $$-0.670210\pi$$
−0.509609 + 0.860406i $$0.670210\pi$$
$$864$$ 0 0
$$865$$ −2.16572 −0.0736365
$$866$$ 0 0
$$867$$ −25.3816 −0.862004
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 11.6174 0.393641
$$872$$ 0 0
$$873$$ 11.0287 0.373266
$$874$$ 0 0
$$875$$ 0.227777 0.00770027
$$876$$ 0 0
$$877$$ −23.7971 −0.803571 −0.401785 0.915734i $$-0.631610\pi$$
−0.401785 + 0.915734i $$0.631610\pi$$
$$878$$ 0 0
$$879$$ −23.4434 −0.790726
$$880$$ 0 0
$$881$$ 34.7141 1.16955 0.584773 0.811197i $$-0.301184\pi$$
0.584773 + 0.811197i $$0.301184\pi$$
$$882$$ 0 0
$$883$$ −7.25118 −0.244022 −0.122011 0.992529i $$-0.538934\pi$$
−0.122011 + 0.992529i $$0.538934\pi$$
$$884$$ 0 0
$$885$$ −10.8052 −0.363211
$$886$$ 0 0
$$887$$ 14.3106 0.480503 0.240251 0.970711i $$-0.422770\pi$$
0.240251 + 0.970711i $$0.422770\pi$$
$$888$$ 0 0
$$889$$ 2.77735 0.0931492
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11.8044 −0.395018
$$894$$ 0 0
$$895$$ −22.1568 −0.740620
$$896$$ 0 0
$$897$$ 15.0613 0.502881
$$898$$ 0 0
$$899$$ −0.0682311 −0.00227564
$$900$$ 0 0
$$901$$ 61.7253 2.05637
$$902$$ 0 0
$$903$$ −1.10437 −0.0367513
$$904$$ 0 0
$$905$$ −8.93677 −0.297068
$$906$$ 0 0
$$907$$ 55.7899 1.85247 0.926237 0.376941i $$-0.123024\pi$$
0.926237 + 0.376941i $$0.123024\pi$$
$$908$$ 0 0
$$909$$ 2.27937 0.0756019
$$910$$ 0 0
$$911$$ 38.8903 1.28849 0.644246 0.764819i $$-0.277171\pi$$
0.644246 + 0.764819i $$0.277171\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −2.90787 −0.0961313
$$916$$ 0 0
$$917$$ 0.470427 0.0155349
$$918$$ 0 0
$$919$$ 37.3365 1.23162 0.615808 0.787896i $$-0.288830\pi$$
0.615808 + 0.787896i $$0.288830\pi$$
$$920$$ 0 0
$$921$$ −7.77539 −0.256208
$$922$$ 0 0
$$923$$ −50.3848 −1.65844
$$924$$ 0 0
$$925$$ 8.74728 0.287609
$$926$$ 0 0
$$927$$ 2.58597 0.0849344
$$928$$ 0 0
$$929$$ 46.8843 1.53822 0.769112 0.639114i $$-0.220699\pi$$
0.769112 + 0.639114i $$0.220699\pi$$
$$930$$ 0 0
$$931$$ −15.4344 −0.505841
$$932$$ 0 0
$$933$$ 19.2646 0.630696
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −53.2268 −1.73884 −0.869422 0.494070i $$-0.835509\pi$$
−0.869422 + 0.494070i $$0.835509\pi$$
$$938$$ 0 0
$$939$$ 3.13302 0.102242
$$940$$ 0 0
$$941$$ 26.1871 0.853676 0.426838 0.904328i $$-0.359628\pi$$
0.426838 + 0.904328i $$0.359628\pi$$
$$942$$ 0 0
$$943$$ 35.6085 1.15957
$$944$$ 0 0
$$945$$ 1.27515 0.0414805
$$946$$ 0 0
$$947$$ 27.9451 0.908095 0.454047 0.890977i $$-0.349980\pi$$
0.454047 + 0.890977i $$0.349980\pi$$
$$948$$ 0 0
$$949$$ 12.2549 0.397811
$$950$$ 0 0
$$951$$ −24.9119 −0.807824
$$952$$ 0 0
$$953$$ 7.31788 0.237049 0.118525 0.992951i $$-0.462184\pi$$
0.118525 + 0.992951i $$0.462184\pi$$
$$954$$ 0 0
$$955$$ −11.5080 −0.372392
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 3.28992 0.106237
$$960$$ 0 0
$$961$$ −26.0882 −0.841556
$$962$$ 0 0
$$963$$ 17.7297 0.571333
$$964$$ 0 0
$$965$$ −8.31192 −0.267570
$$966$$ 0 0
$$967$$ 39.7018 1.27673 0.638363 0.769736i $$-0.279612\pi$$
0.638363 + 0.769736i $$0.279612\pi$$
$$968$$ 0 0
$$969$$ 17.4029 0.559061
$$970$$ 0 0
$$971$$ 59.0922 1.89636 0.948179 0.317737i $$-0.102923\pi$$
0.948179 + 0.317737i $$0.102923\pi$$
$$972$$ 0 0
$$973$$ 3.72236 0.119333
$$974$$ 0 0
$$975$$ −4.33136 −0.138715
$$976$$ 0 0
$$977$$ −51.0338 −1.63272 −0.816359 0.577545i $$-0.804011\pi$$
−0.816359 + 0.577545i $$0.804011\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 9.87415 0.315257
$$982$$ 0 0
$$983$$ −21.5904 −0.688626 −0.344313 0.938855i $$-0.611888\pi$$
−0.344313 + 0.938855i $$0.611888\pi$$
$$984$$ 0 0
$$985$$ −3.64765 −0.116224
$$986$$ 0 0
$$987$$ 1.56743 0.0498918
$$988$$ 0 0
$$989$$ −13.0193 −0.413988
$$990$$ 0 0
$$991$$ 33.7197 1.07114 0.535570 0.844491i $$-0.320097\pi$$
0.535570 + 0.844491i $$0.320097\pi$$
$$992$$ 0 0
$$993$$ 17.9839 0.570701
$$994$$ 0 0
$$995$$ −6.34757 −0.201231
$$996$$ 0 0
$$997$$ −14.5044 −0.459360 −0.229680 0.973266i $$-0.573768\pi$$
−0.229680 + 0.973266i $$0.573768\pi$$
$$998$$ 0 0
$$999$$ 48.9692 1.54932
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 4840.2.a.z.1.2 4
4.3 odd 2 9680.2.a.ct.1.3 4
11.2 odd 10 440.2.y.a.81.1 8
11.6 odd 10 440.2.y.a.201.1 yes 8
11.10 odd 2 4840.2.a.y.1.2 4
44.35 even 10 880.2.bo.d.81.2 8
44.39 even 10 880.2.bo.d.641.2 8
44.43 even 2 9680.2.a.cu.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.1 8 11.2 odd 10
440.2.y.a.201.1 yes 8 11.6 odd 10
880.2.bo.d.81.2 8 44.35 even 10
880.2.bo.d.641.2 8 44.39 even 10
4840.2.a.y.1.2 4 11.10 odd 2
4840.2.a.z.1.2 4 1.1 even 1 trivial
9680.2.a.ct.1.3 4 4.3 odd 2
9680.2.a.cu.1.3 4 44.43 even 2