# Properties

 Label 2352.2.a.bf.1.2 Level $2352$ Weight $2$ Character 2352.1 Self dual yes Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.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: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} -3.27492 q^{11} +6.27492 q^{13} +3.27492 q^{15} -4.00000 q^{17} +6.27492 q^{19} -4.00000 q^{23} +5.72508 q^{25} +1.00000 q^{27} +5.27492 q^{29} +1.00000 q^{31} -3.27492 q^{33} -2.27492 q^{37} +6.27492 q^{39} -4.54983 q^{41} -0.274917 q^{43} +3.27492 q^{45} +6.00000 q^{47} -4.00000 q^{51} +9.27492 q^{53} -10.7251 q^{55} +6.27492 q^{57} +1.27492 q^{59} +10.0000 q^{61} +20.5498 q^{65} +0.274917 q^{67} -4.00000 q^{69} -2.00000 q^{71} -4.27492 q^{73} +5.72508 q^{75} -11.5498 q^{79} +1.00000 q^{81} -7.27492 q^{83} -13.0997 q^{85} +5.27492 q^{87} -10.5498 q^{89} +1.00000 q^{93} +20.5498 q^{95} +8.72508 q^{97} -3.27492 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^5 + 2 * q^9 $$2 q + 2 q^{3} - q^{5} + 2 q^{9} + q^{11} + 5 q^{13} - q^{15} - 8 q^{17} + 5 q^{19} - 8 q^{23} + 19 q^{25} + 2 q^{27} + 3 q^{29} + 2 q^{31} + q^{33} + 3 q^{37} + 5 q^{39} + 6 q^{41} + 7 q^{43} - q^{45} + 12 q^{47} - 8 q^{51} + 11 q^{53} - 29 q^{55} + 5 q^{57} - 5 q^{59} + 20 q^{61} + 26 q^{65} - 7 q^{67} - 8 q^{69} - 4 q^{71} - q^{73} + 19 q^{75} - 8 q^{79} + 2 q^{81} - 7 q^{83} + 4 q^{85} + 3 q^{87} - 6 q^{89} + 2 q^{93} + 26 q^{95} + 25 q^{97} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^5 + 2 * q^9 + q^11 + 5 * q^13 - q^15 - 8 * q^17 + 5 * q^19 - 8 * q^23 + 19 * q^25 + 2 * q^27 + 3 * q^29 + 2 * q^31 + q^33 + 3 * q^37 + 5 * q^39 + 6 * q^41 + 7 * q^43 - q^45 + 12 * q^47 - 8 * q^51 + 11 * q^53 - 29 * q^55 + 5 * q^57 - 5 * q^59 + 20 * q^61 + 26 * q^65 - 7 * q^67 - 8 * q^69 - 4 * q^71 - q^73 + 19 * q^75 - 8 * q^79 + 2 * q^81 - 7 * q^83 + 4 * q^85 + 3 * q^87 - 6 * q^89 + 2 * q^93 + 26 * q^95 + 25 * 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 3.27492 1.46459 0.732294 0.680989i $$-0.238450\pi$$
0.732294 + 0.680989i $$0.238450\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.27492 −0.987425 −0.493712 0.869625i $$-0.664360\pi$$
−0.493712 + 0.869625i $$0.664360\pi$$
$$12$$ 0 0
$$13$$ 6.27492 1.74035 0.870174 0.492744i $$-0.164006\pi$$
0.870174 + 0.492744i $$0.164006\pi$$
$$14$$ 0 0
$$15$$ 3.27492 0.845580
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 6.27492 1.43956 0.719782 0.694200i $$-0.244242\pi$$
0.719782 + 0.694200i $$0.244242\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 5.72508 1.14502
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 5.27492 0.979528 0.489764 0.871855i $$-0.337083\pi$$
0.489764 + 0.871855i $$0.337083\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ −3.27492 −0.570090
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.27492 −0.373994 −0.186997 0.982360i $$-0.559875\pi$$
−0.186997 + 0.982360i $$0.559875\pi$$
$$38$$ 0 0
$$39$$ 6.27492 1.00479
$$40$$ 0 0
$$41$$ −4.54983 −0.710565 −0.355282 0.934759i $$-0.615615\pi$$
−0.355282 + 0.934759i $$0.615615\pi$$
$$42$$ 0 0
$$43$$ −0.274917 −0.0419245 −0.0209622 0.999780i $$-0.506673\pi$$
−0.0209622 + 0.999780i $$0.506673\pi$$
$$44$$ 0 0
$$45$$ 3.27492 0.488196
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ 9.27492 1.27401 0.637004 0.770861i $$-0.280173\pi$$
0.637004 + 0.770861i $$0.280173\pi$$
$$54$$ 0 0
$$55$$ −10.7251 −1.44617
$$56$$ 0 0
$$57$$ 6.27492 0.831133
$$58$$ 0 0
$$59$$ 1.27492 0.165980 0.0829900 0.996550i $$-0.473553\pi$$
0.0829900 + 0.996550i $$0.473553\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 20.5498 2.54889
$$66$$ 0 0
$$67$$ 0.274917 0.0335865 0.0167932 0.999859i $$-0.494654\pi$$
0.0167932 + 0.999859i $$0.494654\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ −4.27492 −0.500341 −0.250171 0.968202i $$-0.580487\pi$$
−0.250171 + 0.968202i $$0.580487\pi$$
$$74$$ 0 0
$$75$$ 5.72508 0.661076
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.5498 −1.29946 −0.649729 0.760166i $$-0.725118\pi$$
−0.649729 + 0.760166i $$0.725118\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.27492 −0.798526 −0.399263 0.916836i $$-0.630734\pi$$
−0.399263 + 0.916836i $$0.630734\pi$$
$$84$$ 0 0
$$85$$ −13.0997 −1.42086
$$86$$ 0 0
$$87$$ 5.27492 0.565530
$$88$$ 0 0
$$89$$ −10.5498 −1.11828 −0.559140 0.829073i $$-0.688869\pi$$
−0.559140 + 0.829073i $$0.688869\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ 20.5498 2.10837
$$96$$ 0 0
$$97$$ 8.72508 0.885898 0.442949 0.896547i $$-0.353932\pi$$
0.442949 + 0.896547i $$0.353932\pi$$
$$98$$ 0 0
$$99$$ −3.27492 −0.329142
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −10.8248 −1.06659 −0.533297 0.845928i $$-0.679047\pi$$
−0.533297 + 0.845928i $$0.679047\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.8248 1.52984 0.764918 0.644127i $$-0.222779\pi$$
0.764918 + 0.644127i $$0.222779\pi$$
$$108$$ 0 0
$$109$$ 16.8248 1.61152 0.805759 0.592243i $$-0.201757\pi$$
0.805759 + 0.592243i $$0.201757\pi$$
$$110$$ 0 0
$$111$$ −2.27492 −0.215926
$$112$$ 0 0
$$113$$ −4.54983 −0.428012 −0.214006 0.976832i $$-0.568651\pi$$
−0.214006 + 0.976832i $$0.568651\pi$$
$$114$$ 0 0
$$115$$ −13.0997 −1.22155
$$116$$ 0 0
$$117$$ 6.27492 0.580116
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −0.274917 −0.0249925
$$122$$ 0 0
$$123$$ −4.54983 −0.410245
$$124$$ 0 0
$$125$$ 2.37459 0.212389
$$126$$ 0 0
$$127$$ 6.45017 0.572360 0.286180 0.958176i $$-0.407615\pi$$
0.286180 + 0.958176i $$0.407615\pi$$
$$128$$ 0 0
$$129$$ −0.274917 −0.0242051
$$130$$ 0 0
$$131$$ −7.27492 −0.635612 −0.317806 0.948156i $$-0.602946\pi$$
−0.317806 + 0.948156i $$0.602946\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.27492 0.281860
$$136$$ 0 0
$$137$$ 1.45017 0.123896 0.0619480 0.998079i $$-0.480269\pi$$
0.0619480 + 0.998079i $$0.480269\pi$$
$$138$$ 0 0
$$139$$ −8.27492 −0.701869 −0.350935 0.936400i $$-0.614136\pi$$
−0.350935 + 0.936400i $$0.614136\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −20.5498 −1.71846
$$144$$ 0 0
$$145$$ 17.2749 1.43460
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.5498 −1.19197 −0.595984 0.802996i $$-0.703238\pi$$
−0.595984 + 0.802996i $$0.703238\pi$$
$$150$$ 0 0
$$151$$ −22.3746 −1.82082 −0.910409 0.413709i $$-0.864233\pi$$
−0.910409 + 0.413709i $$0.864233\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 3.27492 0.263048
$$156$$ 0 0
$$157$$ −0.549834 −0.0438816 −0.0219408 0.999759i $$-0.506985\pi$$
−0.0219408 + 0.999759i $$0.506985\pi$$
$$158$$ 0 0
$$159$$ 9.27492 0.735549
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 0 0
$$165$$ −10.7251 −0.834947
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 26.3746 2.02881
$$170$$ 0 0
$$171$$ 6.27492 0.479855
$$172$$ 0 0
$$173$$ −22.5498 −1.71443 −0.857216 0.514957i $$-0.827808\pi$$
−0.857216 + 0.514957i $$0.827808\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.27492 0.0958286
$$178$$ 0 0
$$179$$ 16.5498 1.23699 0.618496 0.785788i $$-0.287742\pi$$
0.618496 + 0.785788i $$0.287742\pi$$
$$180$$ 0 0
$$181$$ −18.8248 −1.39923 −0.699616 0.714519i $$-0.746646\pi$$
−0.699616 + 0.714519i $$0.746646\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ −7.45017 −0.547747
$$186$$ 0 0
$$187$$ 13.0997 0.957943
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.54983 −0.329214 −0.164607 0.986359i $$-0.552636\pi$$
−0.164607 + 0.986359i $$0.552636\pi$$
$$192$$ 0 0
$$193$$ 0.450166 0.0324036 0.0162018 0.999869i $$-0.494843\pi$$
0.0162018 + 0.999869i $$0.494843\pi$$
$$194$$ 0 0
$$195$$ 20.5498 1.47160
$$196$$ 0 0
$$197$$ −1.45017 −0.103320 −0.0516600 0.998665i $$-0.516451\pi$$
−0.0516600 + 0.998665i $$0.516451\pi$$
$$198$$ 0 0
$$199$$ 5.09967 0.361506 0.180753 0.983529i $$-0.442147\pi$$
0.180753 + 0.983529i $$0.442147\pi$$
$$200$$ 0 0
$$201$$ 0.274917 0.0193912
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −14.9003 −1.04068
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −20.5498 −1.42146
$$210$$ 0 0
$$211$$ 27.6495 1.90347 0.951735 0.306921i $$-0.0992986\pi$$
0.951735 + 0.306921i $$0.0992986\pi$$
$$212$$ 0 0
$$213$$ −2.00000 −0.137038
$$214$$ 0 0
$$215$$ −0.900331 −0.0614021
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −4.27492 −0.288872
$$220$$ 0 0
$$221$$ −25.0997 −1.68839
$$222$$ 0 0
$$223$$ 1.27492 0.0853748 0.0426874 0.999088i $$-0.486408\pi$$
0.0426874 + 0.999088i $$0.486408\pi$$
$$224$$ 0 0
$$225$$ 5.72508 0.381672
$$226$$ 0 0
$$227$$ 11.2749 0.748343 0.374171 0.927360i $$-0.377927\pi$$
0.374171 + 0.927360i $$0.377927\pi$$
$$228$$ 0 0
$$229$$ 18.2749 1.20764 0.603820 0.797121i $$-0.293644\pi$$
0.603820 + 0.797121i $$0.293644\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −0.549834 −0.0360209 −0.0180104 0.999838i $$-0.505733\pi$$
−0.0180104 + 0.999838i $$0.505733\pi$$
$$234$$ 0 0
$$235$$ 19.6495 1.28179
$$236$$ 0 0
$$237$$ −11.5498 −0.750242
$$238$$ 0 0
$$239$$ −15.4502 −0.999388 −0.499694 0.866202i $$-0.666554\pi$$
−0.499694 + 0.866202i $$0.666554\pi$$
$$240$$ 0 0
$$241$$ 9.82475 0.632868 0.316434 0.948615i $$-0.397514\pi$$
0.316434 + 0.948615i $$0.397514\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 39.3746 2.50534
$$248$$ 0 0
$$249$$ −7.27492 −0.461029
$$250$$ 0 0
$$251$$ 18.3746 1.15979 0.579897 0.814690i $$-0.303093\pi$$
0.579897 + 0.814690i $$0.303093\pi$$
$$252$$ 0 0
$$253$$ 13.0997 0.823569
$$254$$ 0 0
$$255$$ −13.0997 −0.820333
$$256$$ 0 0
$$257$$ 11.0997 0.692378 0.346189 0.938165i $$-0.387476\pi$$
0.346189 + 0.938165i $$0.387476\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 5.27492 0.326509
$$262$$ 0 0
$$263$$ 9.45017 0.582722 0.291361 0.956613i $$-0.405892\pi$$
0.291361 + 0.956613i $$0.405892\pi$$
$$264$$ 0 0
$$265$$ 30.3746 1.86590
$$266$$ 0 0
$$267$$ −10.5498 −0.645639
$$268$$ 0 0
$$269$$ 20.7251 1.26363 0.631815 0.775119i $$-0.282310\pi$$
0.631815 + 0.775119i $$0.282310\pi$$
$$270$$ 0 0
$$271$$ −1.27492 −0.0774457 −0.0387229 0.999250i $$-0.512329\pi$$
−0.0387229 + 0.999250i $$0.512329\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −18.7492 −1.13062
$$276$$ 0 0
$$277$$ −26.8248 −1.61174 −0.805872 0.592090i $$-0.798303\pi$$
−0.805872 + 0.592090i $$0.798303\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ −26.5498 −1.58383 −0.791915 0.610631i $$-0.790916\pi$$
−0.791915 + 0.610631i $$0.790916\pi$$
$$282$$ 0 0
$$283$$ 25.9244 1.54105 0.770523 0.637412i $$-0.219995\pi$$
0.770523 + 0.637412i $$0.219995\pi$$
$$284$$ 0 0
$$285$$ 20.5498 1.21727
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 8.72508 0.511473
$$292$$ 0 0
$$293$$ −27.8248 −1.62554 −0.812770 0.582585i $$-0.802041\pi$$
−0.812770 + 0.582585i $$0.802041\pi$$
$$294$$ 0 0
$$295$$ 4.17525 0.243092
$$296$$ 0 0
$$297$$ −3.27492 −0.190030
$$298$$ 0 0
$$299$$ −25.0997 −1.45155
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ 32.7492 1.87521
$$306$$ 0 0
$$307$$ −11.3746 −0.649182 −0.324591 0.945854i $$-0.605227\pi$$
−0.324591 + 0.945854i $$0.605227\pi$$
$$308$$ 0 0
$$309$$ −10.8248 −0.615799
$$310$$ 0 0
$$311$$ 4.54983 0.257997 0.128999 0.991645i $$-0.458824\pi$$
0.128999 + 0.991645i $$0.458824\pi$$
$$312$$ 0 0
$$313$$ −19.5498 −1.10502 −0.552511 0.833506i $$-0.686330\pi$$
−0.552511 + 0.833506i $$0.686330\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −27.8248 −1.56279 −0.781397 0.624034i $$-0.785493\pi$$
−0.781397 + 0.624034i $$0.785493\pi$$
$$318$$ 0 0
$$319$$ −17.2749 −0.967210
$$320$$ 0 0
$$321$$ 15.8248 0.883252
$$322$$ 0 0
$$323$$ −25.0997 −1.39658
$$324$$ 0 0
$$325$$ 35.9244 1.99273
$$326$$ 0 0
$$327$$ 16.8248 0.930411
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.17525 −0.0645975 −0.0322987 0.999478i $$-0.510283\pi$$
−0.0322987 + 0.999478i $$0.510283\pi$$
$$332$$ 0 0
$$333$$ −2.27492 −0.124665
$$334$$ 0 0
$$335$$ 0.900331 0.0491903
$$336$$ 0 0
$$337$$ −24.0997 −1.31279 −0.656396 0.754416i $$-0.727920\pi$$
−0.656396 + 0.754416i $$0.727920\pi$$
$$338$$ 0 0
$$339$$ −4.54983 −0.247113
$$340$$ 0 0
$$341$$ −3.27492 −0.177347
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −13.0997 −0.705262
$$346$$ 0 0
$$347$$ −30.1993 −1.62119 −0.810593 0.585610i $$-0.800855\pi$$
−0.810593 + 0.585610i $$0.800855\pi$$
$$348$$ 0 0
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 6.27492 0.334930
$$352$$ 0 0
$$353$$ −20.5498 −1.09376 −0.546879 0.837212i $$-0.684184\pi$$
−0.546879 + 0.837212i $$0.684184\pi$$
$$354$$ 0 0
$$355$$ −6.54983 −0.347629
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 19.6495 1.03706 0.518531 0.855059i $$-0.326479\pi$$
0.518531 + 0.855059i $$0.326479\pi$$
$$360$$ 0 0
$$361$$ 20.3746 1.07235
$$362$$ 0 0
$$363$$ −0.274917 −0.0144294
$$364$$ 0 0
$$365$$ −14.0000 −0.732793
$$366$$ 0 0
$$367$$ −22.0997 −1.15359 −0.576797 0.816888i $$-0.695698\pi$$
−0.576797 + 0.816888i $$0.695698\pi$$
$$368$$ 0 0
$$369$$ −4.54983 −0.236855
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.27492 −0.324903 −0.162451 0.986717i $$-0.551940\pi$$
−0.162451 + 0.986717i $$0.551940\pi$$
$$374$$ 0 0
$$375$$ 2.37459 0.122623
$$376$$ 0 0
$$377$$ 33.0997 1.70472
$$378$$ 0 0
$$379$$ −13.1752 −0.676767 −0.338384 0.941008i $$-0.609880\pi$$
−0.338384 + 0.941008i $$0.609880\pi$$
$$380$$ 0 0
$$381$$ 6.45017 0.330452
$$382$$ 0 0
$$383$$ −10.5498 −0.539071 −0.269536 0.962990i $$-0.586870\pi$$
−0.269536 + 0.962990i $$0.586870\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.274917 −0.0139748
$$388$$ 0 0
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ −7.27492 −0.366971
$$394$$ 0 0
$$395$$ −37.8248 −1.90317
$$396$$ 0 0
$$397$$ −3.37459 −0.169366 −0.0846828 0.996408i $$-0.526988\pi$$
−0.0846828 + 0.996408i $$0.526988\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 6.27492 0.312576
$$404$$ 0 0
$$405$$ 3.27492 0.162732
$$406$$ 0 0
$$407$$ 7.45017 0.369291
$$408$$ 0 0
$$409$$ 10.4502 0.516727 0.258364 0.966048i $$-0.416817\pi$$
0.258364 + 0.966048i $$0.416817\pi$$
$$410$$ 0 0
$$411$$ 1.45017 0.0715314
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −23.8248 −1.16951
$$416$$ 0 0
$$417$$ −8.27492 −0.405224
$$418$$ 0 0
$$419$$ −28.5498 −1.39475 −0.697375 0.716706i $$-0.745649\pi$$
−0.697375 + 0.716706i $$0.745649\pi$$
$$420$$ 0 0
$$421$$ 8.82475 0.430092 0.215046 0.976604i $$-0.431010\pi$$
0.215046 + 0.976604i $$0.431010\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ −22.9003 −1.11083
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −20.5498 −0.992155
$$430$$ 0 0
$$431$$ 17.6495 0.850147 0.425073 0.905159i $$-0.360248\pi$$
0.425073 + 0.905159i $$0.360248\pi$$
$$432$$ 0 0
$$433$$ 3.17525 0.152593 0.0762963 0.997085i $$-0.475690\pi$$
0.0762963 + 0.997085i $$0.475690\pi$$
$$434$$ 0 0
$$435$$ 17.2749 0.828269
$$436$$ 0 0
$$437$$ −25.0997 −1.20068
$$438$$ 0 0
$$439$$ −17.2749 −0.824487 −0.412243 0.911074i $$-0.635255\pi$$
−0.412243 + 0.911074i $$0.635255\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6.37459 0.302866 0.151433 0.988468i $$-0.451611\pi$$
0.151433 + 0.988468i $$0.451611\pi$$
$$444$$ 0 0
$$445$$ −34.5498 −1.63782
$$446$$ 0 0
$$447$$ −14.5498 −0.688184
$$448$$ 0 0
$$449$$ 20.5498 0.969807 0.484903 0.874568i $$-0.338855\pi$$
0.484903 + 0.874568i $$0.338855\pi$$
$$450$$ 0 0
$$451$$ 14.9003 0.701629
$$452$$ 0 0
$$453$$ −22.3746 −1.05125
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 36.6495 1.71439 0.857196 0.514991i $$-0.172205\pi$$
0.857196 + 0.514991i $$0.172205\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −3.64950 −0.169974 −0.0849872 0.996382i $$-0.527085\pi$$
−0.0849872 + 0.996382i $$0.527085\pi$$
$$462$$ 0 0
$$463$$ −13.1752 −0.612306 −0.306153 0.951982i $$-0.599042\pi$$
−0.306153 + 0.951982i $$0.599042\pi$$
$$464$$ 0 0
$$465$$ 3.27492 0.151871
$$466$$ 0 0
$$467$$ −40.5498 −1.87642 −0.938211 0.346063i $$-0.887518\pi$$
−0.938211 + 0.346063i $$0.887518\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −0.549834 −0.0253350
$$472$$ 0 0
$$473$$ 0.900331 0.0413973
$$474$$ 0 0
$$475$$ 35.9244 1.64833
$$476$$ 0 0
$$477$$ 9.27492 0.424669
$$478$$ 0 0
$$479$$ 5.45017 0.249024 0.124512 0.992218i $$-0.460263\pi$$
0.124512 + 0.992218i $$0.460263\pi$$
$$480$$ 0 0
$$481$$ −14.2749 −0.650880
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 28.5739 1.29748
$$486$$ 0 0
$$487$$ −1.00000 −0.0453143 −0.0226572 0.999743i $$-0.507213\pi$$
−0.0226572 + 0.999743i $$0.507213\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −36.9244 −1.66638 −0.833188 0.552990i $$-0.813487\pi$$
−0.833188 + 0.552990i $$0.813487\pi$$
$$492$$ 0 0
$$493$$ −21.0997 −0.950281
$$494$$ 0 0
$$495$$ −10.7251 −0.482057
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −32.2749 −1.44482 −0.722412 0.691463i $$-0.756966\pi$$
−0.722412 + 0.691463i $$0.756966\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ 0 0
$$503$$ 37.6495 1.67871 0.839354 0.543585i $$-0.182933\pi$$
0.839354 + 0.543585i $$0.182933\pi$$
$$504$$ 0 0
$$505$$ −19.6495 −0.874391
$$506$$ 0 0
$$507$$ 26.3746 1.17134
$$508$$ 0 0
$$509$$ −11.2749 −0.499752 −0.249876 0.968278i $$-0.580390\pi$$
−0.249876 + 0.968278i $$0.580390\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 6.27492 0.277044
$$514$$ 0 0
$$515$$ −35.4502 −1.56212
$$516$$ 0 0
$$517$$ −19.6495 −0.864184
$$518$$ 0 0
$$519$$ −22.5498 −0.989828
$$520$$ 0 0
$$521$$ −14.5498 −0.637440 −0.318720 0.947849i $$-0.603253\pi$$
−0.318720 + 0.947849i $$0.603253\pi$$
$$522$$ 0 0
$$523$$ 17.7251 0.775064 0.387532 0.921856i $$-0.373328\pi$$
0.387532 + 0.921856i $$0.373328\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 1.27492 0.0553267
$$532$$ 0 0
$$533$$ −28.5498 −1.23663
$$534$$ 0 0
$$535$$ 51.8248 2.24058
$$536$$ 0 0
$$537$$ 16.5498 0.714178
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.27492 −0.355766 −0.177883 0.984052i $$-0.556925\pi$$
−0.177883 + 0.984052i $$0.556925\pi$$
$$542$$ 0 0
$$543$$ −18.8248 −0.807847
$$544$$ 0 0
$$545$$ 55.0997 2.36021
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 33.0997 1.41009
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −7.45017 −0.316242
$$556$$ 0 0
$$557$$ −29.8248 −1.26372 −0.631858 0.775084i $$-0.717707\pi$$
−0.631858 + 0.775084i $$0.717707\pi$$
$$558$$ 0 0
$$559$$ −1.72508 −0.0729632
$$560$$ 0 0
$$561$$ 13.0997 0.553068
$$562$$ 0 0
$$563$$ −15.2749 −0.643761 −0.321881 0.946780i $$-0.604315\pi$$
−0.321881 + 0.946780i $$0.604315\pi$$
$$564$$ 0 0
$$565$$ −14.9003 −0.626862
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 11.4502 0.480016 0.240008 0.970771i $$-0.422850\pi$$
0.240008 + 0.970771i $$0.422850\pi$$
$$570$$ 0 0
$$571$$ −8.27492 −0.346295 −0.173147 0.984896i $$-0.555394\pi$$
−0.173147 + 0.984896i $$0.555394\pi$$
$$572$$ 0 0
$$573$$ −4.54983 −0.190072
$$574$$ 0 0
$$575$$ −22.9003 −0.955010
$$576$$ 0 0
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ 0 0
$$579$$ 0.450166 0.0187082
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −30.3746 −1.25799
$$584$$ 0 0
$$585$$ 20.5498 0.849631
$$586$$ 0 0
$$587$$ −9.27492 −0.382817 −0.191408 0.981510i $$-0.561305\pi$$
−0.191408 + 0.981510i $$0.561305\pi$$
$$588$$ 0 0
$$589$$ 6.27492 0.258553
$$590$$ 0 0
$$591$$ −1.45017 −0.0596518
$$592$$ 0 0
$$593$$ −0.549834 −0.0225790 −0.0112895 0.999936i $$-0.503594\pi$$
−0.0112895 + 0.999936i $$0.503594\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.09967 0.208716
$$598$$ 0 0
$$599$$ −22.5498 −0.921361 −0.460681 0.887566i $$-0.652395\pi$$
−0.460681 + 0.887566i $$0.652395\pi$$
$$600$$ 0 0
$$601$$ 4.09967 0.167229 0.0836145 0.996498i $$-0.473354\pi$$
0.0836145 + 0.996498i $$0.473354\pi$$
$$602$$ 0 0
$$603$$ 0.274917 0.0111955
$$604$$ 0 0
$$605$$ −0.900331 −0.0366037
$$606$$ 0 0
$$607$$ 7.00000 0.284121 0.142061 0.989858i $$-0.454627\pi$$
0.142061 + 0.989858i $$0.454627\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 37.6495 1.52314
$$612$$ 0 0
$$613$$ −8.54983 −0.345325 −0.172662 0.984981i $$-0.555237\pi$$
−0.172662 + 0.984981i $$0.555237\pi$$
$$614$$ 0 0
$$615$$ −14.9003 −0.600839
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −28.8248 −1.15856 −0.579282 0.815127i $$-0.696667\pi$$
−0.579282 + 0.815127i $$0.696667\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −20.8488 −0.833954
$$626$$ 0 0
$$627$$ −20.5498 −0.820681
$$628$$ 0 0
$$629$$ 9.09967 0.362828
$$630$$ 0 0
$$631$$ −19.8248 −0.789211 −0.394605 0.918851i $$-0.629119\pi$$
−0.394605 + 0.918851i $$0.629119\pi$$
$$632$$ 0 0
$$633$$ 27.6495 1.09897
$$634$$ 0 0
$$635$$ 21.1238 0.838271
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ −3.64950 −0.144147 −0.0720734 0.997399i $$-0.522962\pi$$
−0.0720734 + 0.997399i $$0.522962\pi$$
$$642$$ 0 0
$$643$$ −5.37459 −0.211953 −0.105976 0.994369i $$-0.533797\pi$$
−0.105976 + 0.994369i $$0.533797\pi$$
$$644$$ 0 0
$$645$$ −0.900331 −0.0354505
$$646$$ 0 0
$$647$$ 34.0000 1.33668 0.668339 0.743857i $$-0.267006\pi$$
0.668339 + 0.743857i $$0.267006\pi$$
$$648$$ 0 0
$$649$$ −4.17525 −0.163893
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6.92442 −0.270974 −0.135487 0.990779i $$-0.543260\pi$$
−0.135487 + 0.990779i $$0.543260\pi$$
$$654$$ 0 0
$$655$$ −23.8248 −0.930910
$$656$$ 0 0
$$657$$ −4.27492 −0.166780
$$658$$ 0 0
$$659$$ 42.1993 1.64385 0.821926 0.569594i $$-0.192899\pi$$
0.821926 + 0.569594i $$0.192899\pi$$
$$660$$ 0 0
$$661$$ 7.17525 0.279085 0.139542 0.990216i $$-0.455437\pi$$
0.139542 + 0.990216i $$0.455437\pi$$
$$662$$ 0 0
$$663$$ −25.0997 −0.974790
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −21.0997 −0.816982
$$668$$ 0 0
$$669$$ 1.27492 0.0492911
$$670$$ 0 0
$$671$$ −32.7492 −1.26427
$$672$$ 0 0
$$673$$ 41.5498 1.60163 0.800814 0.598913i $$-0.204400\pi$$
0.800814 + 0.598913i $$0.204400\pi$$
$$674$$ 0 0
$$675$$ 5.72508 0.220359
$$676$$ 0 0
$$677$$ −13.2749 −0.510197 −0.255098 0.966915i $$-0.582108\pi$$
−0.255098 + 0.966915i $$0.582108\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 11.2749 0.432056
$$682$$ 0 0
$$683$$ −12.1752 −0.465873 −0.232936 0.972492i $$-0.574833\pi$$
−0.232936 + 0.972492i $$0.574833\pi$$
$$684$$ 0 0
$$685$$ 4.74917 0.181457
$$686$$ 0 0
$$687$$ 18.2749 0.697232
$$688$$ 0 0
$$689$$ 58.1993 2.21722
$$690$$ 0 0
$$691$$ 10.8248 0.411793 0.205896 0.978574i $$-0.433989\pi$$
0.205896 + 0.978574i $$0.433989\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −27.0997 −1.02795
$$696$$ 0 0
$$697$$ 18.1993 0.689349
$$698$$ 0 0
$$699$$ −0.549834 −0.0207966
$$700$$ 0 0
$$701$$ −12.9244 −0.488149 −0.244074 0.969757i $$-0.578484\pi$$
−0.244074 + 0.969757i $$0.578484\pi$$
$$702$$ 0 0
$$703$$ −14.2749 −0.538389
$$704$$ 0 0
$$705$$ 19.6495 0.740043
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −28.1993 −1.05905 −0.529524 0.848295i $$-0.677630\pi$$
−0.529524 + 0.848295i $$0.677630\pi$$
$$710$$ 0 0
$$711$$ −11.5498 −0.433153
$$712$$ 0 0
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ −67.2990 −2.51684
$$716$$ 0 0
$$717$$ −15.4502 −0.576997
$$718$$ 0 0
$$719$$ −28.1993 −1.05166 −0.525829 0.850590i $$-0.676245\pi$$
−0.525829 + 0.850590i $$0.676245\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 9.82475 0.365386
$$724$$ 0 0
$$725$$ 30.1993 1.12158
$$726$$ 0 0
$$727$$ −31.5498 −1.17012 −0.585059 0.810991i $$-0.698929\pi$$
−0.585059 + 0.810991i $$0.698929\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.09967 0.0406727
$$732$$ 0 0
$$733$$ 5.92442 0.218823 0.109412 0.993997i $$-0.465103\pi$$
0.109412 + 0.993997i $$0.465103\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −0.900331 −0.0331641
$$738$$ 0 0
$$739$$ −1.37459 −0.0505650 −0.0252825 0.999680i $$-0.508049\pi$$
−0.0252825 + 0.999680i $$0.508049\pi$$
$$740$$ 0 0
$$741$$ 39.3746 1.44646
$$742$$ 0 0
$$743$$ 44.1993 1.62152 0.810758 0.585381i $$-0.199055\pi$$
0.810758 + 0.585381i $$0.199055\pi$$
$$744$$ 0 0
$$745$$ −47.6495 −1.74574
$$746$$ 0 0
$$747$$ −7.27492 −0.266175
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 10.4502 0.381332 0.190666 0.981655i $$-0.438935\pi$$
0.190666 + 0.981655i $$0.438935\pi$$
$$752$$ 0 0
$$753$$ 18.3746 0.669607
$$754$$ 0 0
$$755$$ −73.2749 −2.66675
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 0 0
$$759$$ 13.0997 0.475488
$$760$$ 0 0
$$761$$ −25.0997 −0.909862 −0.454931 0.890527i $$-0.650336\pi$$
−0.454931 + 0.890527i $$0.650336\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −13.0997 −0.473620
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −32.6495 −1.17737 −0.588686 0.808362i $$-0.700354\pi$$
−0.588686 + 0.808362i $$0.700354\pi$$
$$770$$ 0 0
$$771$$ 11.0997 0.399745
$$772$$ 0 0
$$773$$ 3.09967 0.111487 0.0557437 0.998445i $$-0.482247\pi$$
0.0557437 + 0.998445i $$0.482247\pi$$
$$774$$ 0 0
$$775$$ 5.72508 0.205651
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −28.5498 −1.02290
$$780$$ 0 0
$$781$$ 6.54983 0.234372
$$782$$ 0 0
$$783$$ 5.27492 0.188510
$$784$$ 0 0
$$785$$ −1.80066 −0.0642684
$$786$$ 0 0
$$787$$ −2.54983 −0.0908918 −0.0454459 0.998967i $$-0.514471\pi$$
−0.0454459 + 0.998967i $$0.514471\pi$$
$$788$$ 0 0
$$789$$ 9.45017 0.336435
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 62.7492 2.22829
$$794$$ 0 0
$$795$$ 30.3746 1.07728
$$796$$ 0 0
$$797$$ −31.4743 −1.11488 −0.557438 0.830219i $$-0.688215\pi$$
−0.557438 + 0.830219i $$0.688215\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ −10.5498 −0.372760
$$802$$ 0 0
$$803$$ 14.0000 0.494049
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.7251 0.729557
$$808$$ 0 0
$$809$$ −29.6495 −1.04242 −0.521211 0.853428i $$-0.674519\pi$$
−0.521211 + 0.853428i $$0.674519\pi$$
$$810$$ 0 0
$$811$$ 42.5498 1.49413 0.747063 0.664753i $$-0.231463\pi$$
0.747063 + 0.664753i $$0.231463\pi$$
$$812$$ 0 0
$$813$$ −1.27492 −0.0447133
$$814$$ 0 0
$$815$$ 39.2990 1.37658
$$816$$ 0 0
$$817$$ −1.72508 −0.0603530
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.7251 0.444108 0.222054 0.975034i $$-0.428724\pi$$
0.222054 + 0.975034i $$0.428724\pi$$
$$822$$ 0 0
$$823$$ 34.1993 1.19211 0.596057 0.802942i $$-0.296733\pi$$
0.596057 + 0.802942i $$0.296733\pi$$
$$824$$ 0 0
$$825$$ −18.7492 −0.652762
$$826$$ 0 0
$$827$$ 44.0241 1.53087 0.765434 0.643515i $$-0.222524\pi$$
0.765434 + 0.643515i $$0.222524\pi$$
$$828$$ 0 0
$$829$$ 30.2749 1.05149 0.525746 0.850642i $$-0.323786\pi$$
0.525746 + 0.850642i $$0.323786\pi$$
$$830$$ 0 0
$$831$$ −26.8248 −0.930540
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 19.6495 0.679999
$$836$$ 0 0
$$837$$ 1.00000 0.0345651
$$838$$ 0 0
$$839$$ 30.1993 1.04260 0.521298 0.853374i $$-0.325448\pi$$
0.521298 + 0.853374i $$0.325448\pi$$
$$840$$ 0 0
$$841$$ −1.17525 −0.0405258
$$842$$ 0 0
$$843$$ −26.5498 −0.914425
$$844$$ 0 0
$$845$$ 86.3746 2.97138
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 25.9244 0.889724
$$850$$ 0 0
$$851$$ 9.09967 0.311933
$$852$$ 0 0
$$853$$ 13.3746 0.457937 0.228969 0.973434i $$-0.426465\pi$$
0.228969 + 0.973434i $$0.426465\pi$$
$$854$$ 0 0
$$855$$ 20.5498 0.702790
$$856$$ 0 0
$$857$$ −13.4502 −0.459449 −0.229724 0.973256i $$-0.573782\pi$$
−0.229724 + 0.973256i $$0.573782\pi$$
$$858$$ 0 0
$$859$$ 55.6495 1.89874 0.949368 0.314165i $$-0.101725\pi$$
0.949368 + 0.314165i $$0.101725\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −19.0997 −0.650160 −0.325080 0.945686i $$-0.605391\pi$$
−0.325080 + 0.945686i $$0.605391\pi$$
$$864$$ 0 0
$$865$$ −73.8488 −2.51094
$$866$$ 0 0
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 37.8248 1.28312
$$870$$ 0 0
$$871$$ 1.72508 0.0584522
$$872$$ 0 0
$$873$$ 8.72508 0.295299
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46.7492 1.57861 0.789304 0.614003i $$-0.210442\pi$$
0.789304 + 0.614003i $$0.210442\pi$$
$$878$$ 0 0
$$879$$ −27.8248 −0.938506
$$880$$ 0 0
$$881$$ 9.45017 0.318384 0.159192 0.987248i $$-0.449111\pi$$
0.159192 + 0.987248i $$0.449111\pi$$
$$882$$ 0 0
$$883$$ −7.37459 −0.248175 −0.124087 0.992271i $$-0.539600\pi$$
−0.124087 + 0.992271i $$0.539600\pi$$
$$884$$ 0 0
$$885$$ 4.17525 0.140349
$$886$$ 0 0
$$887$$ −33.6495 −1.12984 −0.564920 0.825146i $$-0.691093\pi$$
−0.564920 + 0.825146i $$0.691093\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.27492 −0.109714
$$892$$ 0 0
$$893$$ 37.6495 1.25989
$$894$$ 0 0
$$895$$ 54.1993 1.81168
$$896$$ 0 0
$$897$$ −25.0997 −0.838054
$$898$$ 0 0
$$899$$ 5.27492 0.175928
$$900$$ 0 0
$$901$$ −37.0997 −1.23597
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −61.6495 −2.04930
$$906$$ 0 0
$$907$$ 29.7251 0.987005 0.493503 0.869744i $$-0.335716\pi$$
0.493503 + 0.869744i $$0.335716\pi$$
$$908$$ 0 0
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −53.8488 −1.78409 −0.892046 0.451945i $$-0.850730\pi$$
−0.892046 + 0.451945i $$0.850730\pi$$
$$912$$ 0 0
$$913$$ 23.8248 0.788484
$$914$$ 0 0
$$915$$ 32.7492 1.08265
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 2.82475 0.0931800 0.0465900 0.998914i $$-0.485165\pi$$
0.0465900 + 0.998914i $$0.485165\pi$$
$$920$$ 0 0
$$921$$ −11.3746 −0.374805
$$922$$ 0 0
$$923$$ −12.5498 −0.413083
$$924$$ 0 0
$$925$$ −13.0241 −0.428229
$$926$$ 0 0
$$927$$ −10.8248 −0.355531
$$928$$ 0 0
$$929$$ −40.1993 −1.31890 −0.659449 0.751750i $$-0.729210\pi$$
−0.659449 + 0.751750i $$0.729210\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 4.54983 0.148955
$$934$$ 0 0
$$935$$ 42.9003 1.40299
$$936$$ 0 0
$$937$$ −6.09967 −0.199267 −0.0996337 0.995024i $$-0.531767\pi$$
−0.0996337 + 0.995024i $$0.531767\pi$$
$$938$$ 0 0
$$939$$ −19.5498 −0.637985
$$940$$ 0 0
$$941$$ 51.8248 1.68944 0.844719 0.535210i $$-0.179767\pi$$
0.844719 + 0.535210i $$0.179767\pi$$
$$942$$ 0 0
$$943$$ 18.1993 0.592652
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 19.4502 0.632045 0.316023 0.948752i $$-0.397652\pi$$
0.316023 + 0.948752i $$0.397652\pi$$
$$948$$ 0 0
$$949$$ −26.8248 −0.870768
$$950$$ 0 0
$$951$$ −27.8248 −0.902279
$$952$$ 0 0
$$953$$ 55.6495 1.80266 0.901332 0.433129i $$-0.142590\pi$$
0.901332 + 0.433129i $$0.142590\pi$$
$$954$$ 0 0
$$955$$ −14.9003 −0.482163
$$956$$ 0 0
$$957$$ −17.2749 −0.558419
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 15.8248 0.509945
$$964$$ 0 0
$$965$$ 1.47425 0.0474579
$$966$$ 0 0
$$967$$ 53.5498 1.72205 0.861023 0.508566i $$-0.169824\pi$$
0.861023 + 0.508566i $$0.169824\pi$$
$$968$$ 0 0
$$969$$ −25.0997 −0.806318
$$970$$ 0 0
$$971$$ −56.5739 −1.81554 −0.907772 0.419464i $$-0.862218\pi$$
−0.907772 + 0.419464i $$0.862218\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 35.9244 1.15050
$$976$$ 0 0
$$977$$ 8.90033 0.284747 0.142373 0.989813i $$-0.454527\pi$$
0.142373 + 0.989813i $$0.454527\pi$$
$$978$$ 0 0
$$979$$ 34.5498 1.10422
$$980$$ 0 0
$$981$$ 16.8248 0.537173
$$982$$ 0 0
$$983$$ −39.2990 −1.25344 −0.626722 0.779243i $$-0.715604\pi$$
−0.626722 + 0.779243i $$0.715604\pi$$
$$984$$ 0 0
$$985$$ −4.74917 −0.151321
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.09967 0.0349674
$$990$$ 0 0
$$991$$ 14.0997 0.447891 0.223945 0.974602i $$-0.428106\pi$$
0.223945 + 0.974602i $$0.428106\pi$$
$$992$$ 0 0
$$993$$ −1.17525 −0.0372954
$$994$$ 0 0
$$995$$ 16.7010 0.529457
$$996$$ 0 0
$$997$$ −44.8248 −1.41961 −0.709807 0.704396i $$-0.751218\pi$$
−0.709807 + 0.704396i $$0.751218\pi$$
$$998$$ 0 0
$$999$$ −2.27492 −0.0719752
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 2352.2.a.bf.1.2 2
3.2 odd 2 7056.2.a.cu.1.1 2
4.3 odd 2 1176.2.a.k.1.2 2
7.2 even 3 336.2.q.g.193.1 4
7.3 odd 6 2352.2.q.bf.961.2 4
7.4 even 3 336.2.q.g.289.1 4
7.5 odd 6 2352.2.q.bf.1537.2 4
7.6 odd 2 2352.2.a.ba.1.1 2
8.3 odd 2 9408.2.a.ec.1.1 2
8.5 even 2 9408.2.a.dp.1.1 2
12.11 even 2 3528.2.a.bk.1.1 2
21.2 odd 6 1008.2.s.r.865.2 4
21.11 odd 6 1008.2.s.r.289.2 4
21.20 even 2 7056.2.a.ch.1.2 2
28.3 even 6 1176.2.q.l.961.2 4
28.11 odd 6 168.2.q.c.121.1 yes 4
28.19 even 6 1176.2.q.l.361.2 4
28.23 odd 6 168.2.q.c.25.1 4
28.27 even 2 1176.2.a.n.1.1 2
56.11 odd 6 1344.2.q.w.961.2 4
56.13 odd 2 9408.2.a.dw.1.2 2
56.27 even 2 9408.2.a.dj.1.2 2
56.37 even 6 1344.2.q.x.193.2 4
56.51 odd 6 1344.2.q.w.193.2 4
56.53 even 6 1344.2.q.x.961.2 4
84.11 even 6 504.2.s.i.289.2 4
84.23 even 6 504.2.s.i.361.2 4
84.47 odd 6 3528.2.s.bk.361.1 4
84.59 odd 6 3528.2.s.bk.3313.1 4
84.83 odd 2 3528.2.a.bd.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.1 4 28.23 odd 6
168.2.q.c.121.1 yes 4 28.11 odd 6
336.2.q.g.193.1 4 7.2 even 3
336.2.q.g.289.1 4 7.4 even 3
504.2.s.i.289.2 4 84.11 even 6
504.2.s.i.361.2 4 84.23 even 6
1008.2.s.r.289.2 4 21.11 odd 6
1008.2.s.r.865.2 4 21.2 odd 6
1176.2.a.k.1.2 2 4.3 odd 2
1176.2.a.n.1.1 2 28.27 even 2
1176.2.q.l.361.2 4 28.19 even 6
1176.2.q.l.961.2 4 28.3 even 6
1344.2.q.w.193.2 4 56.51 odd 6
1344.2.q.w.961.2 4 56.11 odd 6
1344.2.q.x.193.2 4 56.37 even 6
1344.2.q.x.961.2 4 56.53 even 6
2352.2.a.ba.1.1 2 7.6 odd 2
2352.2.a.bf.1.2 2 1.1 even 1 trivial
2352.2.q.bf.961.2 4 7.3 odd 6
2352.2.q.bf.1537.2 4 7.5 odd 6
3528.2.a.bd.1.2 2 84.83 odd 2
3528.2.a.bk.1.1 2 12.11 even 2
3528.2.s.bk.361.1 4 84.47 odd 6
3528.2.s.bk.3313.1 4 84.59 odd 6
7056.2.a.ch.1.2 2 21.20 even 2
7056.2.a.cu.1.1 2 3.2 odd 2
9408.2.a.dj.1.2 2 56.27 even 2
9408.2.a.dp.1.1 2 8.5 even 2
9408.2.a.dw.1.2 2 56.13 odd 2
9408.2.a.ec.1.1 2 8.3 odd 2