# Properties

 Label 2576.2.a.bb.1.4 Level $2576$ Weight $2$ Character 2576.1 Self dual yes Analytic conductor $20.569$ Analytic rank $0$ 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] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2576 = 2^{4} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2576.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: $$20.5694635607$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.6963152.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10$$ x^5 - 2*x^4 - 10*x^3 + 10*x^2 + 29*x + 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 644) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.76321$$ of defining polynomial Character $$\chi$$ $$=$$ 2576.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.76321 q^{3} -3.11657 q^{5} +1.00000 q^{7} +0.108911 q^{9} +O(q^{10})$$ $$q+1.76321 q^{3} -3.11657 q^{5} +1.00000 q^{7} +0.108911 q^{9} +5.23940 q^{11} +6.34832 q^{13} -5.49516 q^{15} -0.585105 q^{17} -8.48889 q^{19} +1.76321 q^{21} +1.00000 q^{23} +4.71298 q^{25} -5.09760 q^{27} -1.59780 q^{29} +5.97640 q^{31} +9.23817 q^{33} -3.11657 q^{35} +10.0153 q^{37} +11.1934 q^{39} +1.17811 q^{41} +2.96247 q^{43} -0.339428 q^{45} +2.55043 q^{47} +1.00000 q^{49} -1.03166 q^{51} +8.27107 q^{53} -16.3289 q^{55} -14.9677 q^{57} -14.6004 q^{59} +6.15410 q^{61} +0.108911 q^{63} -19.7849 q^{65} +8.52015 q^{67} +1.76321 q^{69} -8.38625 q^{71} -0.358396 q^{73} +8.30998 q^{75} +5.23940 q^{77} +11.0216 q^{79} -9.31487 q^{81} +10.7446 q^{83} +1.82352 q^{85} -2.81726 q^{87} +18.1369 q^{89} +6.34832 q^{91} +10.5376 q^{93} +26.4562 q^{95} -8.38723 q^{97} +0.570629 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 3 q^{3} + 2 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10})$$ 5 * q - 3 * q^3 + 2 * q^5 + 5 * q^7 + 10 * q^9 $$5 q - 3 q^{3} + 2 q^{5} + 5 q^{7} + 10 q^{9} - 2 q^{11} + 13 q^{13} - 4 q^{15} + 4 q^{17} - 12 q^{19} - 3 q^{21} + 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} + 24 q^{33} + 2 q^{35} - 4 q^{37} - 3 q^{39} + q^{41} + 8 q^{43} - 16 q^{45} - 5 q^{47} + 5 q^{49} + 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} + 20 q^{61} + 10 q^{63} - 12 q^{65} + 12 q^{67} - 3 q^{69} - 9 q^{71} - 9 q^{73} - 35 q^{75} - 2 q^{77} + 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} + 32 q^{89} + 13 q^{91} - 15 q^{93} + 36 q^{95} + 4 q^{97} - 28 q^{99}+O(q^{100})$$ 5 * q - 3 * q^3 + 2 * q^5 + 5 * q^7 + 10 * q^9 - 2 * q^11 + 13 * q^13 - 4 * q^15 + 4 * q^17 - 12 * q^19 - 3 * q^21 + 5 * q^23 + 19 * q^25 - 15 * q^27 + 13 * q^29 + 3 * q^31 + 24 * q^33 + 2 * q^35 - 4 * q^37 - 3 * q^39 + q^41 + 8 * q^43 - 16 * q^45 - 5 * q^47 + 5 * q^49 + 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 12 * q^59 + 20 * q^61 + 10 * q^63 - 12 * q^65 + 12 * q^67 - 3 * q^69 - 9 * q^71 - 9 * q^73 - 35 * q^75 - 2 * q^77 + 8 * q^79 - 11 * q^81 + 28 * q^83 + 16 * q^85 + 15 * q^87 + 32 * q^89 + 13 * q^91 - 15 * q^93 + 36 * q^95 + 4 * q^97 - 28 * 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
$$3$$ 1.76321 1.01799 0.508995 0.860769i $$-0.330017\pi$$
0.508995 + 0.860769i $$0.330017\pi$$
$$4$$ 0 0
$$5$$ −3.11657 −1.39377 −0.696885 0.717183i $$-0.745431\pi$$
−0.696885 + 0.717183i $$0.745431\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0.108911 0.0363037
$$10$$ 0 0
$$11$$ 5.23940 1.57974 0.789870 0.613274i $$-0.210148\pi$$
0.789870 + 0.613274i $$0.210148\pi$$
$$12$$ 0 0
$$13$$ 6.34832 1.76071 0.880353 0.474319i $$-0.157306\pi$$
0.880353 + 0.474319i $$0.157306\pi$$
$$14$$ 0 0
$$15$$ −5.49516 −1.41884
$$16$$ 0 0
$$17$$ −0.585105 −0.141909 −0.0709544 0.997480i $$-0.522604\pi$$
−0.0709544 + 0.997480i $$0.522604\pi$$
$$18$$ 0 0
$$19$$ −8.48889 −1.94748 −0.973742 0.227653i $$-0.926895\pi$$
−0.973742 + 0.227653i $$0.926895\pi$$
$$20$$ 0 0
$$21$$ 1.76321 0.384764
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 4.71298 0.942597
$$26$$ 0 0
$$27$$ −5.09760 −0.981033
$$28$$ 0 0
$$29$$ −1.59780 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$30$$ 0 0
$$31$$ 5.97640 1.07339 0.536696 0.843776i $$-0.319672\pi$$
0.536696 + 0.843776i $$0.319672\pi$$
$$32$$ 0 0
$$33$$ 9.23817 1.60816
$$34$$ 0 0
$$35$$ −3.11657 −0.526796
$$36$$ 0 0
$$37$$ 10.0153 1.64651 0.823253 0.567674i $$-0.192157\pi$$
0.823253 + 0.567674i $$0.192157\pi$$
$$38$$ 0 0
$$39$$ 11.1934 1.79238
$$40$$ 0 0
$$41$$ 1.17811 0.183989 0.0919946 0.995760i $$-0.470676\pi$$
0.0919946 + 0.995760i $$0.470676\pi$$
$$42$$ 0 0
$$43$$ 2.96247 0.451772 0.225886 0.974154i $$-0.427472\pi$$
0.225886 + 0.974154i $$0.427472\pi$$
$$44$$ 0 0
$$45$$ −0.339428 −0.0505990
$$46$$ 0 0
$$47$$ 2.55043 0.372018 0.186009 0.982548i $$-0.440445\pi$$
0.186009 + 0.982548i $$0.440445\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.03166 −0.144462
$$52$$ 0 0
$$53$$ 8.27107 1.13612 0.568059 0.822988i $$-0.307694\pi$$
0.568059 + 0.822988i $$0.307694\pi$$
$$54$$ 0 0
$$55$$ −16.3289 −2.20180
$$56$$ 0 0
$$57$$ −14.9677 −1.98252
$$58$$ 0 0
$$59$$ −14.6004 −1.90081 −0.950406 0.311012i $$-0.899332\pi$$
−0.950406 + 0.311012i $$0.899332\pi$$
$$60$$ 0 0
$$61$$ 6.15410 0.787951 0.393976 0.919121i $$-0.371099\pi$$
0.393976 + 0.919121i $$0.371099\pi$$
$$62$$ 0 0
$$63$$ 0.108911 0.0137215
$$64$$ 0 0
$$65$$ −19.7849 −2.45402
$$66$$ 0 0
$$67$$ 8.52015 1.04090 0.520451 0.853892i $$-0.325764\pi$$
0.520451 + 0.853892i $$0.325764\pi$$
$$68$$ 0 0
$$69$$ 1.76321 0.212266
$$70$$ 0 0
$$71$$ −8.38625 −0.995265 −0.497632 0.867388i $$-0.665797\pi$$
−0.497632 + 0.867388i $$0.665797\pi$$
$$72$$ 0 0
$$73$$ −0.358396 −0.0419471 −0.0209735 0.999780i $$-0.506677\pi$$
−0.0209735 + 0.999780i $$0.506677\pi$$
$$74$$ 0 0
$$75$$ 8.30998 0.959554
$$76$$ 0 0
$$77$$ 5.23940 0.597086
$$78$$ 0 0
$$79$$ 11.0216 1.24002 0.620012 0.784592i $$-0.287128\pi$$
0.620012 + 0.784592i $$0.287128\pi$$
$$80$$ 0 0
$$81$$ −9.31487 −1.03499
$$82$$ 0 0
$$83$$ 10.7446 1.17938 0.589689 0.807630i $$-0.299250\pi$$
0.589689 + 0.807630i $$0.299250\pi$$
$$84$$ 0 0
$$85$$ 1.82352 0.197788
$$86$$ 0 0
$$87$$ −2.81726 −0.302042
$$88$$ 0 0
$$89$$ 18.1369 1.92251 0.961255 0.275662i $$-0.0888970\pi$$
0.961255 + 0.275662i $$0.0888970\pi$$
$$90$$ 0 0
$$91$$ 6.34832 0.665484
$$92$$ 0 0
$$93$$ 10.5376 1.09270
$$94$$ 0 0
$$95$$ 26.4562 2.71435
$$96$$ 0 0
$$97$$ −8.38723 −0.851594 −0.425797 0.904819i $$-0.640006\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$98$$ 0 0
$$99$$ 0.570629 0.0573503
$$100$$ 0 0
$$101$$ 6.79708 0.676335 0.338168 0.941086i $$-0.390193\pi$$
0.338168 + 0.941086i $$0.390193\pi$$
$$102$$ 0 0
$$103$$ 9.75955 0.961637 0.480819 0.876820i $$-0.340340\pi$$
0.480819 + 0.876820i $$0.340340\pi$$
$$104$$ 0 0
$$105$$ −5.49516 −0.536273
$$106$$ 0 0
$$107$$ 6.96247 0.673087 0.336544 0.941668i $$-0.390742\pi$$
0.336544 + 0.941668i $$0.390742\pi$$
$$108$$ 0 0
$$109$$ −2.03794 −0.195199 −0.0975994 0.995226i $$-0.531116\pi$$
−0.0975994 + 0.995226i $$0.531116\pi$$
$$110$$ 0 0
$$111$$ 17.6591 1.67613
$$112$$ 0 0
$$113$$ 3.20815 0.301797 0.150898 0.988549i $$-0.451783\pi$$
0.150898 + 0.988549i $$0.451783\pi$$
$$114$$ 0 0
$$115$$ −3.11657 −0.290621
$$116$$ 0 0
$$117$$ 0.691401 0.0639201
$$118$$ 0 0
$$119$$ −0.585105 −0.0536365
$$120$$ 0 0
$$121$$ 16.4514 1.49558
$$122$$ 0 0
$$123$$ 2.07725 0.187299
$$124$$ 0 0
$$125$$ 0.894506 0.0800070
$$126$$ 0 0
$$127$$ −4.83720 −0.429232 −0.214616 0.976698i $$-0.568850\pi$$
−0.214616 + 0.976698i $$0.568850\pi$$
$$128$$ 0 0
$$129$$ 5.22346 0.459900
$$130$$ 0 0
$$131$$ −13.9917 −1.22246 −0.611231 0.791453i $$-0.709325\pi$$
−0.611231 + 0.791453i $$0.709325\pi$$
$$132$$ 0 0
$$133$$ −8.48889 −0.736080
$$134$$ 0 0
$$135$$ 15.8870 1.36734
$$136$$ 0 0
$$137$$ 3.65910 0.312618 0.156309 0.987708i $$-0.450040\pi$$
0.156309 + 0.987708i $$0.450040\pi$$
$$138$$ 0 0
$$139$$ −16.4852 −1.39826 −0.699130 0.714995i $$-0.746429\pi$$
−0.699130 + 0.714995i $$0.746429\pi$$
$$140$$ 0 0
$$141$$ 4.49694 0.378711
$$142$$ 0 0
$$143$$ 33.2614 2.78146
$$144$$ 0 0
$$145$$ 4.97965 0.413537
$$146$$ 0 0
$$147$$ 1.76321 0.145427
$$148$$ 0 0
$$149$$ −15.7596 −1.29107 −0.645536 0.763729i $$-0.723366\pi$$
−0.645536 + 0.763729i $$0.723366\pi$$
$$150$$ 0 0
$$151$$ −6.13049 −0.498892 −0.249446 0.968389i $$-0.580249\pi$$
−0.249446 + 0.968389i $$0.580249\pi$$
$$152$$ 0 0
$$153$$ −0.0637243 −0.00515181
$$154$$ 0 0
$$155$$ −18.6258 −1.49606
$$156$$ 0 0
$$157$$ 3.65811 0.291949 0.145974 0.989288i $$-0.453368\pi$$
0.145974 + 0.989288i $$0.453368\pi$$
$$158$$ 0 0
$$159$$ 14.5836 1.15656
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ 17.9381 1.40502 0.702509 0.711675i $$-0.252063\pi$$
0.702509 + 0.711675i $$0.252063\pi$$
$$164$$ 0 0
$$165$$ −28.7914 −2.24141
$$166$$ 0 0
$$167$$ −0.419935 −0.0324956 −0.0162478 0.999868i $$-0.505172\pi$$
−0.0162478 + 0.999868i $$0.505172\pi$$
$$168$$ 0 0
$$169$$ 27.3011 2.10009
$$170$$ 0 0
$$171$$ −0.924533 −0.0707008
$$172$$ 0 0
$$173$$ 2.48889 0.189227 0.0946134 0.995514i $$-0.469839\pi$$
0.0946134 + 0.995514i $$0.469839\pi$$
$$174$$ 0 0
$$175$$ 4.71298 0.356268
$$176$$ 0 0
$$177$$ −25.7436 −1.93501
$$178$$ 0 0
$$179$$ 2.64201 0.197473 0.0987365 0.995114i $$-0.468520\pi$$
0.0987365 + 0.995114i $$0.468520\pi$$
$$180$$ 0 0
$$181$$ 1.38950 0.103281 0.0516405 0.998666i $$-0.483555\pi$$
0.0516405 + 0.998666i $$0.483555\pi$$
$$182$$ 0 0
$$183$$ 10.8510 0.802127
$$184$$ 0 0
$$185$$ −31.2134 −2.29485
$$186$$ 0 0
$$187$$ −3.06560 −0.224179
$$188$$ 0 0
$$189$$ −5.09760 −0.370796
$$190$$ 0 0
$$191$$ −13.0682 −0.945578 −0.472789 0.881176i $$-0.656753\pi$$
−0.472789 + 0.881176i $$0.656753\pi$$
$$192$$ 0 0
$$193$$ 21.5964 1.55454 0.777270 0.629167i $$-0.216604\pi$$
0.777270 + 0.629167i $$0.216604\pi$$
$$194$$ 0 0
$$195$$ −34.8850 −2.49817
$$196$$ 0 0
$$197$$ 10.2410 0.729643 0.364822 0.931077i $$-0.381130\pi$$
0.364822 + 0.931077i $$0.381130\pi$$
$$198$$ 0 0
$$199$$ 21.6744 1.53646 0.768229 0.640175i $$-0.221138\pi$$
0.768229 + 0.640175i $$0.221138\pi$$
$$200$$ 0 0
$$201$$ 15.0228 1.05963
$$202$$ 0 0
$$203$$ −1.59780 −0.112144
$$204$$ 0 0
$$205$$ −3.67164 −0.256439
$$206$$ 0 0
$$207$$ 0.108911 0.00756983
$$208$$ 0 0
$$209$$ −44.4767 −3.07652
$$210$$ 0 0
$$211$$ −15.8499 −1.09115 −0.545577 0.838061i $$-0.683689\pi$$
−0.545577 + 0.838061i $$0.683689\pi$$
$$212$$ 0 0
$$213$$ −14.7867 −1.01317
$$214$$ 0 0
$$215$$ −9.23273 −0.629667
$$216$$ 0 0
$$217$$ 5.97640 0.405704
$$218$$ 0 0
$$219$$ −0.631927 −0.0427017
$$220$$ 0 0
$$221$$ −3.71443 −0.249860
$$222$$ 0 0
$$223$$ −1.04860 −0.0702197 −0.0351098 0.999383i $$-0.511178\pi$$
−0.0351098 + 0.999383i $$0.511178\pi$$
$$224$$ 0 0
$$225$$ 0.513296 0.0342197
$$226$$ 0 0
$$227$$ −27.9455 −1.85481 −0.927403 0.374063i $$-0.877964\pi$$
−0.927403 + 0.374063i $$0.877964\pi$$
$$228$$ 0 0
$$229$$ −17.8239 −1.17783 −0.588917 0.808193i $$-0.700446\pi$$
−0.588917 + 0.808193i $$0.700446\pi$$
$$230$$ 0 0
$$231$$ 9.23817 0.607827
$$232$$ 0 0
$$233$$ 9.53528 0.624677 0.312339 0.949971i $$-0.398888\pi$$
0.312339 + 0.949971i $$0.398888\pi$$
$$234$$ 0 0
$$235$$ −7.94858 −0.518508
$$236$$ 0 0
$$237$$ 19.4334 1.26233
$$238$$ 0 0
$$239$$ −13.6190 −0.880939 −0.440469 0.897768i $$-0.645188\pi$$
−0.440469 + 0.897768i $$0.645188\pi$$
$$240$$ 0 0
$$241$$ 16.1115 1.03783 0.518917 0.854824i $$-0.326335\pi$$
0.518917 + 0.854824i $$0.326335\pi$$
$$242$$ 0 0
$$243$$ −1.13128 −0.0725718
$$244$$ 0 0
$$245$$ −3.11657 −0.199110
$$246$$ 0 0
$$247$$ −53.8901 −3.42895
$$248$$ 0 0
$$249$$ 18.9451 1.20060
$$250$$ 0 0
$$251$$ −17.7822 −1.12240 −0.561201 0.827680i $$-0.689660\pi$$
−0.561201 + 0.827680i $$0.689660\pi$$
$$252$$ 0 0
$$253$$ 5.23940 0.329399
$$254$$ 0 0
$$255$$ 3.21525 0.201347
$$256$$ 0 0
$$257$$ −11.3733 −0.709447 −0.354724 0.934971i $$-0.615425\pi$$
−0.354724 + 0.934971i $$0.615425\pi$$
$$258$$ 0 0
$$259$$ 10.0153 0.622321
$$260$$ 0 0
$$261$$ −0.174018 −0.0107714
$$262$$ 0 0
$$263$$ −8.22726 −0.507315 −0.253657 0.967294i $$-0.581634\pi$$
−0.253657 + 0.967294i $$0.581634\pi$$
$$264$$ 0 0
$$265$$ −25.7773 −1.58349
$$266$$ 0 0
$$267$$ 31.9792 1.95710
$$268$$ 0 0
$$269$$ 3.31078 0.201862 0.100931 0.994893i $$-0.467818\pi$$
0.100931 + 0.994893i $$0.467818\pi$$
$$270$$ 0 0
$$271$$ −24.3447 −1.47883 −0.739416 0.673248i $$-0.764898\pi$$
−0.739416 + 0.673248i $$0.764898\pi$$
$$272$$ 0 0
$$273$$ 11.1934 0.677456
$$274$$ 0 0
$$275$$ 24.6932 1.48906
$$276$$ 0 0
$$277$$ −14.1492 −0.850144 −0.425072 0.905160i $$-0.639751\pi$$
−0.425072 + 0.905160i $$0.639751\pi$$
$$278$$ 0 0
$$279$$ 0.650895 0.0389681
$$280$$ 0 0
$$281$$ 10.2231 0.609856 0.304928 0.952375i $$-0.401368\pi$$
0.304928 + 0.952375i $$0.401368\pi$$
$$282$$ 0 0
$$283$$ −5.80480 −0.345060 −0.172530 0.985004i $$-0.555194\pi$$
−0.172530 + 0.985004i $$0.555194\pi$$
$$284$$ 0 0
$$285$$ 46.6478 2.76318
$$286$$ 0 0
$$287$$ 1.17811 0.0695414
$$288$$ 0 0
$$289$$ −16.6577 −0.979862
$$290$$ 0 0
$$291$$ −14.7885 −0.866914
$$292$$ 0 0
$$293$$ −31.1117 −1.81757 −0.908783 0.417269i $$-0.862987\pi$$
−0.908783 + 0.417269i $$0.862987\pi$$
$$294$$ 0 0
$$295$$ 45.5032 2.64930
$$296$$ 0 0
$$297$$ −26.7084 −1.54978
$$298$$ 0 0
$$299$$ 6.34832 0.367133
$$300$$ 0 0
$$301$$ 2.96247 0.170754
$$302$$ 0 0
$$303$$ 11.9847 0.688502
$$304$$ 0 0
$$305$$ −19.1797 −1.09822
$$306$$ 0 0
$$307$$ 4.55784 0.260130 0.130065 0.991505i $$-0.458481\pi$$
0.130065 + 0.991505i $$0.458481\pi$$
$$308$$ 0 0
$$309$$ 17.2081 0.978937
$$310$$ 0 0
$$311$$ −1.08872 −0.0617358 −0.0308679 0.999523i $$-0.509827\pi$$
−0.0308679 + 0.999523i $$0.509827\pi$$
$$312$$ 0 0
$$313$$ 16.9086 0.955731 0.477866 0.878433i $$-0.341411\pi$$
0.477866 + 0.878433i $$0.341411\pi$$
$$314$$ 0 0
$$315$$ −0.339428 −0.0191246
$$316$$ 0 0
$$317$$ −4.26439 −0.239512 −0.119756 0.992803i $$-0.538211\pi$$
−0.119756 + 0.992803i $$0.538211\pi$$
$$318$$ 0 0
$$319$$ −8.37152 −0.468715
$$320$$ 0 0
$$321$$ 12.2763 0.685196
$$322$$ 0 0
$$323$$ 4.96689 0.276365
$$324$$ 0 0
$$325$$ 29.9195 1.65964
$$326$$ 0 0
$$327$$ −3.59331 −0.198710
$$328$$ 0 0
$$329$$ 2.55043 0.140610
$$330$$ 0 0
$$331$$ −16.9450 −0.931380 −0.465690 0.884948i $$-0.654194\pi$$
−0.465690 + 0.884948i $$0.654194\pi$$
$$332$$ 0 0
$$333$$ 1.09078 0.0597742
$$334$$ 0 0
$$335$$ −26.5536 −1.45078
$$336$$ 0 0
$$337$$ 10.7547 0.585847 0.292924 0.956136i $$-0.405372\pi$$
0.292924 + 0.956136i $$0.405372\pi$$
$$338$$ 0 0
$$339$$ 5.65663 0.307226
$$340$$ 0 0
$$341$$ 31.3128 1.69568
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −5.49516 −0.295850
$$346$$ 0 0
$$347$$ −30.2105 −1.62178 −0.810892 0.585195i $$-0.801018\pi$$
−0.810892 + 0.585195i $$0.801018\pi$$
$$348$$ 0 0
$$349$$ −20.2309 −1.08294 −0.541469 0.840721i $$-0.682132\pi$$
−0.541469 + 0.840721i $$0.682132\pi$$
$$350$$ 0 0
$$351$$ −32.3612 −1.72731
$$352$$ 0 0
$$353$$ 14.0953 0.750218 0.375109 0.926981i $$-0.377605\pi$$
0.375109 + 0.926981i $$0.377605\pi$$
$$354$$ 0 0
$$355$$ 26.1363 1.38717
$$356$$ 0 0
$$357$$ −1.03166 −0.0546014
$$358$$ 0 0
$$359$$ 16.0086 0.844903 0.422452 0.906385i $$-0.361170\pi$$
0.422452 + 0.906385i $$0.361170\pi$$
$$360$$ 0 0
$$361$$ 53.0612 2.79270
$$362$$ 0 0
$$363$$ 29.0072 1.52248
$$364$$ 0 0
$$365$$ 1.11696 0.0584646
$$366$$ 0 0
$$367$$ 28.8277 1.50479 0.752397 0.658710i $$-0.228898\pi$$
0.752397 + 0.658710i $$0.228898\pi$$
$$368$$ 0 0
$$369$$ 0.128309 0.00667948
$$370$$ 0 0
$$371$$ 8.27107 0.429412
$$372$$ 0 0
$$373$$ −18.6273 −0.964484 −0.482242 0.876038i $$-0.660177\pi$$
−0.482242 + 0.876038i $$0.660177\pi$$
$$374$$ 0 0
$$375$$ 1.57720 0.0814464
$$376$$ 0 0
$$377$$ −10.1433 −0.522409
$$378$$ 0 0
$$379$$ −14.8138 −0.760936 −0.380468 0.924794i $$-0.624237\pi$$
−0.380468 + 0.924794i $$0.624237\pi$$
$$380$$ 0 0
$$381$$ −8.52901 −0.436954
$$382$$ 0 0
$$383$$ −0.611968 −0.0312701 −0.0156351 0.999878i $$-0.504977\pi$$
−0.0156351 + 0.999878i $$0.504977\pi$$
$$384$$ 0 0
$$385$$ −16.3289 −0.832200
$$386$$ 0 0
$$387$$ 0.322645 0.0164010
$$388$$ 0 0
$$389$$ 11.4514 0.580607 0.290303 0.956935i $$-0.406244\pi$$
0.290303 + 0.956935i $$0.406244\pi$$
$$390$$ 0 0
$$391$$ −0.585105 −0.0295900
$$392$$ 0 0
$$393$$ −24.6703 −1.24445
$$394$$ 0 0
$$395$$ −34.3495 −1.72831
$$396$$ 0 0
$$397$$ 15.9228 0.799140 0.399570 0.916703i $$-0.369159\pi$$
0.399570 + 0.916703i $$0.369159\pi$$
$$398$$ 0 0
$$399$$ −14.9677 −0.749322
$$400$$ 0 0
$$401$$ −35.1157 −1.75359 −0.876797 0.480861i $$-0.840324\pi$$
−0.876797 + 0.480861i $$0.840324\pi$$
$$402$$ 0 0
$$403$$ 37.9400 1.88993
$$404$$ 0 0
$$405$$ 29.0304 1.44253
$$406$$ 0 0
$$407$$ 52.4743 2.60105
$$408$$ 0 0
$$409$$ 24.6726 1.21998 0.609991 0.792408i $$-0.291173\pi$$
0.609991 + 0.792408i $$0.291173\pi$$
$$410$$ 0 0
$$411$$ 6.45176 0.318242
$$412$$ 0 0
$$413$$ −14.6004 −0.718439
$$414$$ 0 0
$$415$$ −33.4864 −1.64378
$$416$$ 0 0
$$417$$ −29.0669 −1.42341
$$418$$ 0 0
$$419$$ −23.0411 −1.12563 −0.562816 0.826582i $$-0.690282\pi$$
−0.562816 + 0.826582i $$0.690282\pi$$
$$420$$ 0 0
$$421$$ 11.8048 0.575331 0.287665 0.957731i $$-0.407121\pi$$
0.287665 + 0.957731i $$0.407121\pi$$
$$422$$ 0 0
$$423$$ 0.277770 0.0135056
$$424$$ 0 0
$$425$$ −2.75759 −0.133763
$$426$$ 0 0
$$427$$ 6.15410 0.297818
$$428$$ 0 0
$$429$$ 58.6468 2.83150
$$430$$ 0 0
$$431$$ 3.56873 0.171899 0.0859497 0.996299i $$-0.472608\pi$$
0.0859497 + 0.996299i $$0.472608\pi$$
$$432$$ 0 0
$$433$$ −41.0992 −1.97510 −0.987550 0.157305i $$-0.949719\pi$$
−0.987550 + 0.157305i $$0.949719\pi$$
$$434$$ 0 0
$$435$$ 8.78017 0.420977
$$436$$ 0 0
$$437$$ −8.48889 −0.406079
$$438$$ 0 0
$$439$$ −31.6361 −1.50991 −0.754954 0.655778i $$-0.772341\pi$$
−0.754954 + 0.655778i $$0.772341\pi$$
$$440$$ 0 0
$$441$$ 0.108911 0.00518624
$$442$$ 0 0
$$443$$ −3.30070 −0.156821 −0.0784106 0.996921i $$-0.524985\pi$$
−0.0784106 + 0.996921i $$0.524985\pi$$
$$444$$ 0 0
$$445$$ −56.5249 −2.67954
$$446$$ 0 0
$$447$$ −27.7874 −1.31430
$$448$$ 0 0
$$449$$ −19.8820 −0.938289 −0.469145 0.883121i $$-0.655438\pi$$
−0.469145 + 0.883121i $$0.655438\pi$$
$$450$$ 0 0
$$451$$ 6.17257 0.290655
$$452$$ 0 0
$$453$$ −10.8093 −0.507868
$$454$$ 0 0
$$455$$ −19.7849 −0.927532
$$456$$ 0 0
$$457$$ 15.2707 0.714332 0.357166 0.934041i $$-0.383743\pi$$
0.357166 + 0.934041i $$0.383743\pi$$
$$458$$ 0 0
$$459$$ 2.98263 0.139217
$$460$$ 0 0
$$461$$ −8.68922 −0.404697 −0.202349 0.979314i $$-0.564857\pi$$
−0.202349 + 0.979314i $$0.564857\pi$$
$$462$$ 0 0
$$463$$ −8.57355 −0.398447 −0.199223 0.979954i $$-0.563842\pi$$
−0.199223 + 0.979954i $$0.563842\pi$$
$$464$$ 0 0
$$465$$ −32.8413 −1.52298
$$466$$ 0 0
$$467$$ −30.1327 −1.39437 −0.697187 0.716889i $$-0.745565\pi$$
−0.697187 + 0.716889i $$0.745565\pi$$
$$468$$ 0 0
$$469$$ 8.52015 0.393424
$$470$$ 0 0
$$471$$ 6.45001 0.297201
$$472$$ 0 0
$$473$$ 15.5216 0.713683
$$474$$ 0 0
$$475$$ −40.0080 −1.83569
$$476$$ 0 0
$$477$$ 0.900810 0.0412452
$$478$$ 0 0
$$479$$ −4.74988 −0.217027 −0.108514 0.994095i $$-0.534609\pi$$
−0.108514 + 0.994095i $$0.534609\pi$$
$$480$$ 0 0
$$481$$ 63.5803 2.89901
$$482$$ 0 0
$$483$$ 1.76321 0.0802289
$$484$$ 0 0
$$485$$ 26.1394 1.18693
$$486$$ 0 0
$$487$$ 18.9728 0.859741 0.429870 0.902891i $$-0.358559\pi$$
0.429870 + 0.902891i $$0.358559\pi$$
$$488$$ 0 0
$$489$$ 31.6286 1.43029
$$490$$ 0 0
$$491$$ −6.18017 −0.278907 −0.139453 0.990229i $$-0.544535\pi$$
−0.139453 + 0.990229i $$0.544535\pi$$
$$492$$ 0 0
$$493$$ 0.934881 0.0421049
$$494$$ 0 0
$$495$$ −1.77840 −0.0799332
$$496$$ 0 0
$$497$$ −8.38625 −0.376175
$$498$$ 0 0
$$499$$ 20.9756 0.938997 0.469498 0.882933i $$-0.344435\pi$$
0.469498 + 0.882933i $$0.344435\pi$$
$$500$$ 0 0
$$501$$ −0.740435 −0.0330802
$$502$$ 0 0
$$503$$ 3.42597 0.152756 0.0763782 0.997079i $$-0.475664\pi$$
0.0763782 + 0.997079i $$0.475664\pi$$
$$504$$ 0 0
$$505$$ −21.1836 −0.942656
$$506$$ 0 0
$$507$$ 48.1376 2.13787
$$508$$ 0 0
$$509$$ −35.6250 −1.57905 −0.789526 0.613718i $$-0.789673\pi$$
−0.789526 + 0.613718i $$0.789673\pi$$
$$510$$ 0 0
$$511$$ −0.358396 −0.0158545
$$512$$ 0 0
$$513$$ 43.2729 1.91055
$$514$$ 0 0
$$515$$ −30.4163 −1.34030
$$516$$ 0 0
$$517$$ 13.3627 0.587692
$$518$$ 0 0
$$519$$ 4.38844 0.192631
$$520$$ 0 0
$$521$$ −28.6482 −1.25510 −0.627551 0.778576i $$-0.715942\pi$$
−0.627551 + 0.778576i $$0.715942\pi$$
$$522$$ 0 0
$$523$$ 14.7717 0.645921 0.322961 0.946412i $$-0.395322\pi$$
0.322961 + 0.946412i $$0.395322\pi$$
$$524$$ 0 0
$$525$$ 8.30998 0.362677
$$526$$ 0 0
$$527$$ −3.49682 −0.152324
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −1.59015 −0.0690064
$$532$$ 0 0
$$533$$ 7.47899 0.323951
$$534$$ 0 0
$$535$$ −21.6990 −0.938129
$$536$$ 0 0
$$537$$ 4.65842 0.201025
$$538$$ 0 0
$$539$$ 5.23940 0.225677
$$540$$ 0 0
$$541$$ −12.3001 −0.528824 −0.264412 0.964410i $$-0.585178\pi$$
−0.264412 + 0.964410i $$0.585178\pi$$
$$542$$ 0 0
$$543$$ 2.44999 0.105139
$$544$$ 0 0
$$545$$ 6.35136 0.272062
$$546$$ 0 0
$$547$$ 20.4059 0.872495 0.436247 0.899827i $$-0.356307\pi$$
0.436247 + 0.899827i $$0.356307\pi$$
$$548$$ 0 0
$$549$$ 0.670249 0.0286055
$$550$$ 0 0
$$551$$ 13.5635 0.577827
$$552$$ 0 0
$$553$$ 11.0216 0.468685
$$554$$ 0 0
$$555$$ −55.0357 −2.33614
$$556$$ 0 0
$$557$$ 13.9455 0.590889 0.295444 0.955360i $$-0.404532\pi$$
0.295444 + 0.955360i $$0.404532\pi$$
$$558$$ 0 0
$$559$$ 18.8067 0.795438
$$560$$ 0 0
$$561$$ −5.40530 −0.228212
$$562$$ 0 0
$$563$$ 22.3868 0.943492 0.471746 0.881734i $$-0.343624\pi$$
0.471746 + 0.881734i $$0.343624\pi$$
$$564$$ 0 0
$$565$$ −9.99840 −0.420636
$$566$$ 0 0
$$567$$ −9.31487 −0.391188
$$568$$ 0 0
$$569$$ −23.0581 −0.966645 −0.483322 0.875442i $$-0.660570\pi$$
−0.483322 + 0.875442i $$0.660570\pi$$
$$570$$ 0 0
$$571$$ −0.606257 −0.0253711 −0.0126855 0.999920i $$-0.504038\pi$$
−0.0126855 + 0.999920i $$0.504038\pi$$
$$572$$ 0 0
$$573$$ −23.0419 −0.962589
$$574$$ 0 0
$$575$$ 4.71298 0.196545
$$576$$ 0 0
$$577$$ 5.57828 0.232227 0.116113 0.993236i $$-0.462956\pi$$
0.116113 + 0.993236i $$0.462956\pi$$
$$578$$ 0 0
$$579$$ 38.0789 1.58251
$$580$$ 0 0
$$581$$ 10.7446 0.445763
$$582$$ 0 0
$$583$$ 43.3355 1.79477
$$584$$ 0 0
$$585$$ −2.15480 −0.0890899
$$586$$ 0 0
$$587$$ 15.5804 0.643071 0.321536 0.946898i $$-0.395801\pi$$
0.321536 + 0.946898i $$0.395801\pi$$
$$588$$ 0 0
$$589$$ −50.7330 −2.09042
$$590$$ 0 0
$$591$$ 18.0571 0.742769
$$592$$ 0 0
$$593$$ −12.4788 −0.512443 −0.256222 0.966618i $$-0.582478\pi$$
−0.256222 + 0.966618i $$0.582478\pi$$
$$594$$ 0 0
$$595$$ 1.82352 0.0747569
$$596$$ 0 0
$$597$$ 38.2165 1.56410
$$598$$ 0 0
$$599$$ 25.6897 1.04965 0.524827 0.851209i $$-0.324130\pi$$
0.524827 + 0.851209i $$0.324130\pi$$
$$600$$ 0 0
$$601$$ 14.1712 0.578055 0.289028 0.957321i $$-0.406668\pi$$
0.289028 + 0.957321i $$0.406668\pi$$
$$602$$ 0 0
$$603$$ 0.927938 0.0377885
$$604$$ 0 0
$$605$$ −51.2717 −2.08449
$$606$$ 0 0
$$607$$ 38.6010 1.56677 0.783383 0.621539i $$-0.213492\pi$$
0.783383 + 0.621539i $$0.213492\pi$$
$$608$$ 0 0
$$609$$ −2.81726 −0.114161
$$610$$ 0 0
$$611$$ 16.1909 0.655015
$$612$$ 0 0
$$613$$ −2.40423 −0.0971058 −0.0485529 0.998821i $$-0.515461\pi$$
−0.0485529 + 0.998821i $$0.515461\pi$$
$$614$$ 0 0
$$615$$ −6.47388 −0.261052
$$616$$ 0 0
$$617$$ 24.5675 0.989051 0.494526 0.869163i $$-0.335342\pi$$
0.494526 + 0.869163i $$0.335342\pi$$
$$618$$ 0 0
$$619$$ −31.0657 −1.24864 −0.624318 0.781171i $$-0.714623\pi$$
−0.624318 + 0.781171i $$0.714623\pi$$
$$620$$ 0 0
$$621$$ −5.09760 −0.204560
$$622$$ 0 0
$$623$$ 18.1369 0.726640
$$624$$ 0 0
$$625$$ −26.3527 −1.05411
$$626$$ 0 0
$$627$$ −78.4218 −3.13187
$$628$$ 0 0
$$629$$ −5.86001 −0.233654
$$630$$ 0 0
$$631$$ 15.9461 0.634805 0.317402 0.948291i $$-0.397189\pi$$
0.317402 + 0.948291i $$0.397189\pi$$
$$632$$ 0 0
$$633$$ −27.9468 −1.11078
$$634$$ 0 0
$$635$$ 15.0755 0.598252
$$636$$ 0 0
$$637$$ 6.34832 0.251529
$$638$$ 0 0
$$639$$ −0.913355 −0.0361317
$$640$$ 0 0
$$641$$ −34.1109 −1.34730 −0.673650 0.739051i $$-0.735274\pi$$
−0.673650 + 0.739051i $$0.735274\pi$$
$$642$$ 0 0
$$643$$ −22.7215 −0.896050 −0.448025 0.894021i $$-0.647872\pi$$
−0.448025 + 0.894021i $$0.647872\pi$$
$$644$$ 0 0
$$645$$ −16.2792 −0.640995
$$646$$ 0 0
$$647$$ 4.53740 0.178384 0.0891918 0.996014i $$-0.471572\pi$$
0.0891918 + 0.996014i $$0.471572\pi$$
$$648$$ 0 0
$$649$$ −76.4975 −3.00279
$$650$$ 0 0
$$651$$ 10.5376 0.413003
$$652$$ 0 0
$$653$$ −38.9525 −1.52433 −0.762164 0.647384i $$-0.775863\pi$$
−0.762164 + 0.647384i $$0.775863\pi$$
$$654$$ 0 0
$$655$$ 43.6061 1.70383
$$656$$ 0 0
$$657$$ −0.0390332 −0.00152283
$$658$$ 0 0
$$659$$ 14.8983 0.580357 0.290179 0.956973i $$-0.406285\pi$$
0.290179 + 0.956973i $$0.406285\pi$$
$$660$$ 0 0
$$661$$ 17.2118 0.669461 0.334731 0.942314i $$-0.391355\pi$$
0.334731 + 0.942314i $$0.391355\pi$$
$$662$$ 0 0
$$663$$ −6.54932 −0.254355
$$664$$ 0 0
$$665$$ 26.4562 1.02593
$$666$$ 0 0
$$667$$ −1.59780 −0.0618671
$$668$$ 0 0
$$669$$ −1.84891 −0.0714829
$$670$$ 0 0
$$671$$ 32.2438 1.24476
$$672$$ 0 0
$$673$$ 19.7685 0.762018 0.381009 0.924571i $$-0.375577\pi$$
0.381009 + 0.924571i $$0.375577\pi$$
$$674$$ 0 0
$$675$$ −24.0249 −0.924719
$$676$$ 0 0
$$677$$ −36.0117 −1.38404 −0.692020 0.721878i $$-0.743279\pi$$
−0.692020 + 0.721878i $$0.743279\pi$$
$$678$$ 0 0
$$679$$ −8.38723 −0.321872
$$680$$ 0 0
$$681$$ −49.2738 −1.88817
$$682$$ 0 0
$$683$$ −11.0796 −0.423949 −0.211975 0.977275i $$-0.567989\pi$$
−0.211975 + 0.977275i $$0.567989\pi$$
$$684$$ 0 0
$$685$$ −11.4038 −0.435718
$$686$$ 0 0
$$687$$ −31.4272 −1.19902
$$688$$ 0 0
$$689$$ 52.5073 2.00037
$$690$$ 0 0
$$691$$ 16.5328 0.628936 0.314468 0.949268i $$-0.398174\pi$$
0.314468 + 0.949268i $$0.398174\pi$$
$$692$$ 0 0
$$693$$ 0.570629 0.0216764
$$694$$ 0 0
$$695$$ 51.3773 1.94885
$$696$$ 0 0
$$697$$ −0.689315 −0.0261097
$$698$$ 0 0
$$699$$ 16.8127 0.635915
$$700$$ 0 0
$$701$$ 17.2642 0.652058 0.326029 0.945360i $$-0.394289\pi$$
0.326029 + 0.945360i $$0.394289\pi$$
$$702$$ 0 0
$$703$$ −85.0189 −3.20655
$$704$$ 0 0
$$705$$ −14.0150 −0.527836
$$706$$ 0 0
$$707$$ 6.79708 0.255631
$$708$$ 0 0
$$709$$ −3.98260 −0.149570 −0.0747849 0.997200i $$-0.523827\pi$$
−0.0747849 + 0.997200i $$0.523827\pi$$
$$710$$ 0 0
$$711$$ 1.20037 0.0450174
$$712$$ 0 0
$$713$$ 5.97640 0.223818
$$714$$ 0 0
$$715$$ −103.661 −3.87671
$$716$$ 0 0
$$717$$ −24.0131 −0.896787
$$718$$ 0 0
$$719$$ 7.33439 0.273527 0.136763 0.990604i $$-0.456330\pi$$
0.136763 + 0.990604i $$0.456330\pi$$
$$720$$ 0 0
$$721$$ 9.75955 0.363465
$$722$$ 0 0
$$723$$ 28.4080 1.05651
$$724$$ 0 0
$$725$$ −7.53041 −0.279672
$$726$$ 0 0
$$727$$ −4.06498 −0.150762 −0.0753809 0.997155i $$-0.524017\pi$$
−0.0753809 + 0.997155i $$0.524017\pi$$
$$728$$ 0 0
$$729$$ 25.9499 0.961108
$$730$$ 0 0
$$731$$ −1.73335 −0.0641104
$$732$$ 0 0
$$733$$ 17.2247 0.636207 0.318104 0.948056i $$-0.396954\pi$$
0.318104 + 0.948056i $$0.396954\pi$$
$$734$$ 0 0
$$735$$ −5.49516 −0.202692
$$736$$ 0 0
$$737$$ 44.6405 1.64435
$$738$$ 0 0
$$739$$ −13.0829 −0.481262 −0.240631 0.970617i $$-0.577354\pi$$
−0.240631 + 0.970617i $$0.577354\pi$$
$$740$$ 0 0
$$741$$ −95.0197 −3.49063
$$742$$ 0 0
$$743$$ 9.58553 0.351659 0.175830 0.984421i $$-0.443739\pi$$
0.175830 + 0.984421i $$0.443739\pi$$
$$744$$ 0 0
$$745$$ 49.1157 1.79946
$$746$$ 0 0
$$747$$ 1.17021 0.0428157
$$748$$ 0 0
$$749$$ 6.96247 0.254403
$$750$$ 0 0
$$751$$ −39.3151 −1.43463 −0.717314 0.696750i $$-0.754629\pi$$
−0.717314 + 0.696750i $$0.754629\pi$$
$$752$$ 0 0
$$753$$ −31.3537 −1.14259
$$754$$ 0 0
$$755$$ 19.1061 0.695342
$$756$$ 0 0
$$757$$ −30.7971 −1.11934 −0.559670 0.828716i $$-0.689072\pi$$
−0.559670 + 0.828716i $$0.689072\pi$$
$$758$$ 0 0
$$759$$ 9.23817 0.335324
$$760$$ 0 0
$$761$$ 19.0651 0.691110 0.345555 0.938399i $$-0.387691\pi$$
0.345555 + 0.938399i $$0.387691\pi$$
$$762$$ 0 0
$$763$$ −2.03794 −0.0737782
$$764$$ 0 0
$$765$$ 0.198601 0.00718044
$$766$$ 0 0
$$767$$ −92.6880 −3.34677
$$768$$ 0 0
$$769$$ 14.0861 0.507959 0.253980 0.967210i $$-0.418260\pi$$
0.253980 + 0.967210i $$0.418260\pi$$
$$770$$ 0 0
$$771$$ −20.0535 −0.722210
$$772$$ 0 0
$$773$$ −5.46538 −0.196576 −0.0982879 0.995158i $$-0.531337\pi$$
−0.0982879 + 0.995158i $$0.531337\pi$$
$$774$$ 0 0
$$775$$ 28.1667 1.01178
$$776$$ 0 0
$$777$$ 17.6591 0.633517
$$778$$ 0 0
$$779$$ −10.0008 −0.358316
$$780$$ 0 0
$$781$$ −43.9390 −1.57226
$$782$$ 0 0
$$783$$ 8.14494 0.291077
$$784$$ 0 0
$$785$$ −11.4007 −0.406910
$$786$$ 0 0
$$787$$ −36.3921 −1.29724 −0.648618 0.761114i $$-0.724653\pi$$
−0.648618 + 0.761114i $$0.724653\pi$$
$$788$$ 0 0
$$789$$ −14.5064 −0.516441
$$790$$ 0 0
$$791$$ 3.20815 0.114069
$$792$$ 0 0
$$793$$ 39.0682 1.38735
$$794$$ 0 0
$$795$$ −45.4509 −1.61198
$$796$$ 0 0
$$797$$ 43.5810 1.54372 0.771858 0.635794i $$-0.219327\pi$$
0.771858 + 0.635794i $$0.219327\pi$$
$$798$$ 0 0
$$799$$ −1.49227 −0.0527927
$$800$$ 0 0
$$801$$ 1.97531 0.0697941
$$802$$ 0 0
$$803$$ −1.87778 −0.0662654
$$804$$ 0 0
$$805$$ −3.11657 −0.109845
$$806$$ 0 0
$$807$$ 5.83761 0.205494
$$808$$ 0 0
$$809$$ 5.11013 0.179663 0.0898313 0.995957i $$-0.471367\pi$$
0.0898313 + 0.995957i $$0.471367\pi$$
$$810$$ 0 0
$$811$$ 2.06953 0.0726712 0.0363356 0.999340i $$-0.488431\pi$$
0.0363356 + 0.999340i $$0.488431\pi$$
$$812$$ 0 0
$$813$$ −42.9248 −1.50544
$$814$$ 0 0
$$815$$ −55.9052 −1.95827
$$816$$ 0 0
$$817$$ −25.1481 −0.879819
$$818$$ 0 0
$$819$$ 0.691401 0.0241595
$$820$$ 0 0
$$821$$ 2.87359 0.100289 0.0501446 0.998742i $$-0.484032\pi$$
0.0501446 + 0.998742i $$0.484032\pi$$
$$822$$ 0 0
$$823$$ −3.81222 −0.132886 −0.0664428 0.997790i $$-0.521165\pi$$
−0.0664428 + 0.997790i $$0.521165\pi$$
$$824$$ 0 0
$$825$$ 43.5394 1.51585
$$826$$ 0 0
$$827$$ 6.87210 0.238966 0.119483 0.992836i $$-0.461876\pi$$
0.119483 + 0.992836i $$0.461876\pi$$
$$828$$ 0 0
$$829$$ 5.53374 0.192195 0.0960973 0.995372i $$-0.469364\pi$$
0.0960973 + 0.995372i $$0.469364\pi$$
$$830$$ 0 0
$$831$$ −24.9480 −0.865438
$$832$$ 0 0
$$833$$ −0.585105 −0.0202727
$$834$$ 0 0
$$835$$ 1.30876 0.0452914
$$836$$ 0 0
$$837$$ −30.4653 −1.05303
$$838$$ 0 0
$$839$$ −1.16338 −0.0401642 −0.0200821 0.999798i $$-0.506393\pi$$
−0.0200821 + 0.999798i $$0.506393\pi$$
$$840$$ 0 0
$$841$$ −26.4470 −0.911967
$$842$$ 0 0
$$843$$ 18.0254 0.620827
$$844$$ 0 0
$$845$$ −85.0857 −2.92704
$$846$$ 0 0
$$847$$ 16.4514 0.565275
$$848$$ 0 0
$$849$$ −10.2351 −0.351267
$$850$$ 0 0
$$851$$ 10.0153 0.343320
$$852$$ 0 0
$$853$$ 5.30694 0.181706 0.0908531 0.995864i $$-0.471041\pi$$
0.0908531 + 0.995864i $$0.471041\pi$$
$$854$$ 0 0
$$855$$ 2.88137 0.0985407
$$856$$ 0 0
$$857$$ −1.84412 −0.0629938 −0.0314969 0.999504i $$-0.510027\pi$$
−0.0314969 + 0.999504i $$0.510027\pi$$
$$858$$ 0 0
$$859$$ −45.1077 −1.53905 −0.769527 0.638615i $$-0.779508\pi$$
−0.769527 + 0.638615i $$0.779508\pi$$
$$860$$ 0 0
$$861$$ 2.07725 0.0707924
$$862$$ 0 0
$$863$$ −2.75173 −0.0936701 −0.0468351 0.998903i $$-0.514914\pi$$
−0.0468351 + 0.998903i $$0.514914\pi$$
$$864$$ 0 0
$$865$$ −7.75679 −0.263739
$$866$$ 0 0
$$867$$ −29.3709 −0.997490
$$868$$ 0 0
$$869$$ 57.7465 1.95892
$$870$$ 0 0
$$871$$ 54.0886 1.83272
$$872$$ 0 0
$$873$$ −0.913461 −0.0309160
$$874$$ 0 0
$$875$$ 0.894506 0.0302398
$$876$$ 0 0
$$877$$ 32.9340 1.11210 0.556052 0.831148i $$-0.312316\pi$$
0.556052 + 0.831148i $$0.312316\pi$$
$$878$$ 0 0
$$879$$ −54.8565 −1.85026
$$880$$ 0 0
$$881$$ 40.6201 1.36853 0.684264 0.729235i $$-0.260124\pi$$
0.684264 + 0.729235i $$0.260124\pi$$
$$882$$ 0 0
$$883$$ −0.817246 −0.0275025 −0.0137513 0.999905i $$-0.504377\pi$$
−0.0137513 + 0.999905i $$0.504377\pi$$
$$884$$ 0 0
$$885$$ 80.2316 2.69696
$$886$$ 0 0
$$887$$ −4.29164 −0.144099 −0.0720496 0.997401i $$-0.522954\pi$$
−0.0720496 + 0.997401i $$0.522954\pi$$
$$888$$ 0 0
$$889$$ −4.83720 −0.162235
$$890$$ 0 0
$$891$$ −48.8044 −1.63501
$$892$$ 0 0
$$893$$ −21.6503 −0.724500
$$894$$ 0 0
$$895$$ −8.23399 −0.275232
$$896$$ 0 0
$$897$$ 11.1934 0.373737
$$898$$ 0 0
$$899$$ −9.54909 −0.318480
$$900$$ 0 0
$$901$$ −4.83944 −0.161225
$$902$$ 0 0
$$903$$ 5.22346 0.173826
$$904$$ 0 0
$$905$$ −4.33048 −0.143950
$$906$$ 0 0
$$907$$ 43.7845 1.45384 0.726920 0.686723i $$-0.240951\pi$$
0.726920 + 0.686723i $$0.240951\pi$$
$$908$$ 0 0
$$909$$ 0.740277 0.0245534
$$910$$ 0 0
$$911$$ 48.8009 1.61684 0.808422 0.588603i $$-0.200322\pi$$
0.808422 + 0.588603i $$0.200322\pi$$
$$912$$ 0 0
$$913$$ 56.2955 1.86311
$$914$$ 0 0
$$915$$ −33.8178 −1.11798
$$916$$ 0 0
$$917$$ −13.9917 −0.462047
$$918$$ 0 0
$$919$$ −10.5677 −0.348596 −0.174298 0.984693i $$-0.555766\pi$$
−0.174298 + 0.984693i $$0.555766\pi$$
$$920$$ 0 0
$$921$$ 8.03644 0.264810
$$922$$ 0 0
$$923$$ −53.2386 −1.75237
$$924$$ 0 0
$$925$$ 47.2020 1.55199
$$926$$ 0 0
$$927$$ 1.06292 0.0349109
$$928$$ 0 0
$$929$$ 5.51933 0.181083 0.0905417 0.995893i $$-0.471140\pi$$
0.0905417 + 0.995893i $$0.471140\pi$$
$$930$$ 0 0
$$931$$ −8.48889 −0.278212
$$932$$ 0 0
$$933$$ −1.91965 −0.0628465
$$934$$ 0 0
$$935$$ 9.55415 0.312454
$$936$$ 0 0
$$937$$ −51.4761 −1.68165 −0.840826 0.541305i $$-0.817930\pi$$
−0.840826 + 0.541305i $$0.817930\pi$$
$$938$$ 0 0
$$939$$ 29.8134 0.972925
$$940$$ 0 0
$$941$$ 23.0595 0.751718 0.375859 0.926677i $$-0.377348\pi$$
0.375859 + 0.926677i $$0.377348\pi$$
$$942$$ 0 0
$$943$$ 1.17811 0.0383644
$$944$$ 0 0
$$945$$ 15.8870 0.516804
$$946$$ 0 0
$$947$$ −27.0207 −0.878054 −0.439027 0.898474i $$-0.644677\pi$$
−0.439027 + 0.898474i $$0.644677\pi$$
$$948$$ 0 0
$$949$$ −2.27521 −0.0738564
$$950$$ 0 0
$$951$$ −7.51902 −0.243821
$$952$$ 0 0
$$953$$ 29.6466 0.960348 0.480174 0.877173i $$-0.340574\pi$$
0.480174 + 0.877173i $$0.340574\pi$$
$$954$$ 0 0
$$955$$ 40.7278 1.31792
$$956$$ 0 0
$$957$$ −14.7608 −0.477147
$$958$$ 0 0
$$959$$ 3.65910 0.118158
$$960$$ 0 0
$$961$$ 4.71731 0.152171
$$962$$ 0 0
$$963$$ 0.758289 0.0244355
$$964$$ 0 0
$$965$$ −67.3065 −2.16667
$$966$$ 0 0
$$967$$ 41.6275 1.33865 0.669324 0.742970i $$-0.266584\pi$$
0.669324 + 0.742970i $$0.266584\pi$$
$$968$$ 0 0
$$969$$ 8.75767 0.281337
$$970$$ 0 0
$$971$$ −37.4436 −1.20162 −0.600811 0.799391i $$-0.705155\pi$$
−0.600811 + 0.799391i $$0.705155\pi$$
$$972$$ 0 0
$$973$$ −16.4852 −0.528492
$$974$$ 0 0
$$975$$ 52.7544 1.68949
$$976$$ 0 0
$$977$$ 25.8568 0.827231 0.413616 0.910452i $$-0.364266\pi$$
0.413616 + 0.910452i $$0.364266\pi$$
$$978$$ 0 0
$$979$$ 95.0266 3.03706
$$980$$ 0 0
$$981$$ −0.221953 −0.00708643
$$982$$ 0 0
$$983$$ −33.8201 −1.07869 −0.539347 0.842084i $$-0.681329\pi$$
−0.539347 + 0.842084i $$0.681329\pi$$
$$984$$ 0 0
$$985$$ −31.9168 −1.01696
$$986$$ 0 0
$$987$$ 4.49694 0.143139
$$988$$ 0 0
$$989$$ 2.96247 0.0942010
$$990$$ 0 0
$$991$$ 26.9278 0.855390 0.427695 0.903923i $$-0.359326\pi$$
0.427695 + 0.903923i $$0.359326\pi$$
$$992$$ 0 0
$$993$$ −29.8776 −0.948136
$$994$$ 0 0
$$995$$ −67.5497 −2.14147
$$996$$ 0 0
$$997$$ 4.87456 0.154379 0.0771894 0.997016i $$-0.475405\pi$$
0.0771894 + 0.997016i $$0.475405\pi$$
$$998$$ 0 0
$$999$$ −51.0540 −1.61528
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 2576.2.a.bb.1.4 5
4.3 odd 2 644.2.a.d.1.2 5
12.11 even 2 5796.2.a.t.1.5 5
28.27 even 2 4508.2.a.f.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.2 5 4.3 odd 2
2576.2.a.bb.1.4 5 1.1 even 1 trivial
4508.2.a.f.1.4 5 28.27 even 2
5796.2.a.t.1.5 5 12.11 even 2