# Properties

 Label 520.2.a.g.1.2 Level $520$ Weight $2$ Character 520.1 Self dual yes Analytic conductor $4.152$ 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] = [520,2,Mod(1,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 520.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: $$4.15222090511$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} +O(q^{10})$$ $$q+3.23607 q^{3} -1.00000 q^{5} +7.47214 q^{9} +0.763932 q^{11} -1.00000 q^{13} -3.23607 q^{15} +2.00000 q^{17} +0.763932 q^{19} -3.23607 q^{23} +1.00000 q^{25} +14.4721 q^{27} +8.47214 q^{29} -5.70820 q^{31} +2.47214 q^{33} -8.47214 q^{37} -3.23607 q^{39} -10.9443 q^{41} -3.23607 q^{43} -7.47214 q^{45} +12.9443 q^{47} -7.00000 q^{49} +6.47214 q^{51} -10.9443 q^{53} -0.763932 q^{55} +2.47214 q^{57} -5.70820 q^{59} -4.47214 q^{61} +1.00000 q^{65} +10.4721 q^{67} -10.4721 q^{69} -0.763932 q^{71} -7.52786 q^{73} +3.23607 q^{75} +6.47214 q^{79} +24.4164 q^{81} +4.00000 q^{83} -2.00000 q^{85} +27.4164 q^{87} +10.0000 q^{89} -18.4721 q^{93} -0.763932 q^{95} -10.0000 q^{97} +5.70820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 6 q^{9} + 6 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} + 20 q^{27} + 8 q^{29} + 2 q^{31} - 4 q^{33} - 8 q^{37} - 2 q^{39} - 4 q^{41} - 2 q^{43} - 6 q^{45} + 8 q^{47} - 14 q^{49} + 4 q^{51} - 4 q^{53} - 6 q^{55} - 4 q^{57} + 2 q^{59} + 2 q^{65} + 12 q^{67} - 12 q^{69} - 6 q^{71} - 24 q^{73} + 2 q^{75} + 4 q^{79} + 22 q^{81} + 8 q^{83} - 4 q^{85} + 28 q^{87} + 20 q^{89} - 28 q^{93} - 6 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 6 * q^9 + 6 * q^11 - 2 * q^13 - 2 * q^15 + 4 * q^17 + 6 * q^19 - 2 * q^23 + 2 * q^25 + 20 * q^27 + 8 * q^29 + 2 * q^31 - 4 * q^33 - 8 * q^37 - 2 * q^39 - 4 * q^41 - 2 * q^43 - 6 * q^45 + 8 * q^47 - 14 * q^49 + 4 * q^51 - 4 * q^53 - 6 * q^55 - 4 * q^57 + 2 * q^59 + 2 * q^65 + 12 * q^67 - 12 * q^69 - 6 * q^71 - 24 * q^73 + 2 * q^75 + 4 * q^79 + 22 * q^81 + 8 * q^83 - 4 * q^85 + 28 * q^87 + 20 * q^89 - 28 * q^93 - 6 * q^95 - 20 * q^97 - 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.23607 1.86834 0.934172 0.356822i $$-0.116140\pi$$
0.934172 + 0.356822i $$0.116140\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 7.47214 2.49071
$$10$$ 0 0
$$11$$ 0.763932 0.230334 0.115167 0.993346i $$-0.463260\pi$$
0.115167 + 0.993346i $$0.463260\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ −3.23607 −0.835549
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 0.763932 0.175258 0.0876290 0.996153i $$-0.472071\pi$$
0.0876290 + 0.996153i $$0.472071\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.23607 −0.674767 −0.337383 0.941367i $$-0.609542\pi$$
−0.337383 + 0.941367i $$0.609542\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 14.4721 2.78516
$$28$$ 0 0
$$29$$ 8.47214 1.57324 0.786618 0.617440i $$-0.211830\pi$$
0.786618 + 0.617440i $$0.211830\pi$$
$$30$$ 0 0
$$31$$ −5.70820 −1.02522 −0.512612 0.858620i $$-0.671322\pi$$
−0.512612 + 0.858620i $$0.671322\pi$$
$$32$$ 0 0
$$33$$ 2.47214 0.430344
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.47214 −1.39281 −0.696405 0.717649i $$-0.745218\pi$$
−0.696405 + 0.717649i $$0.745218\pi$$
$$38$$ 0 0
$$39$$ −3.23607 −0.518186
$$40$$ 0 0
$$41$$ −10.9443 −1.70921 −0.854604 0.519280i $$-0.826200\pi$$
−0.854604 + 0.519280i $$0.826200\pi$$
$$42$$ 0 0
$$43$$ −3.23607 −0.493496 −0.246748 0.969080i $$-0.579362\pi$$
−0.246748 + 0.969080i $$0.579362\pi$$
$$44$$ 0 0
$$45$$ −7.47214 −1.11388
$$46$$ 0 0
$$47$$ 12.9443 1.88812 0.944058 0.329779i $$-0.106974\pi$$
0.944058 + 0.329779i $$0.106974\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 6.47214 0.906280
$$52$$ 0 0
$$53$$ −10.9443 −1.50331 −0.751656 0.659556i $$-0.770744\pi$$
−0.751656 + 0.659556i $$0.770744\pi$$
$$54$$ 0 0
$$55$$ −0.763932 −0.103009
$$56$$ 0 0
$$57$$ 2.47214 0.327442
$$58$$ 0 0
$$59$$ −5.70820 −0.743145 −0.371572 0.928404i $$-0.621181\pi$$
−0.371572 + 0.928404i $$0.621181\pi$$
$$60$$ 0 0
$$61$$ −4.47214 −0.572598 −0.286299 0.958140i $$-0.592425\pi$$
−0.286299 + 0.958140i $$0.592425\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 10.4721 1.27938 0.639688 0.768635i $$-0.279064\pi$$
0.639688 + 0.768635i $$0.279064\pi$$
$$68$$ 0 0
$$69$$ −10.4721 −1.26070
$$70$$ 0 0
$$71$$ −0.763932 −0.0906621 −0.0453310 0.998972i $$-0.514434\pi$$
−0.0453310 + 0.998972i $$0.514434\pi$$
$$72$$ 0 0
$$73$$ −7.52786 −0.881070 −0.440535 0.897735i $$-0.645211\pi$$
−0.440535 + 0.897735i $$0.645211\pi$$
$$74$$ 0 0
$$75$$ 3.23607 0.373669
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.47214 0.728172 0.364086 0.931365i $$-0.381381\pi$$
0.364086 + 0.931365i $$0.381381\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ 0 0
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 27.4164 2.93935
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −18.4721 −1.91547
$$94$$ 0 0
$$95$$ −0.763932 −0.0783778
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 5.70820 0.573696
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −6.29180 −0.619949 −0.309975 0.950745i $$-0.600321\pi$$
−0.309975 + 0.950745i $$0.600321\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.1803 −1.56421 −0.782106 0.623145i $$-0.785855\pi$$
−0.782106 + 0.623145i $$0.785855\pi$$
$$108$$ 0 0
$$109$$ −1.05573 −0.101120 −0.0505602 0.998721i $$-0.516101\pi$$
−0.0505602 + 0.998721i $$0.516101\pi$$
$$110$$ 0 0
$$111$$ −27.4164 −2.60225
$$112$$ 0 0
$$113$$ 19.8885 1.87096 0.935478 0.353384i $$-0.114969\pi$$
0.935478 + 0.353384i $$0.114969\pi$$
$$114$$ 0 0
$$115$$ 3.23607 0.301765
$$116$$ 0 0
$$117$$ −7.47214 −0.690799
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 0 0
$$123$$ −35.4164 −3.19339
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −9.70820 −0.861464 −0.430732 0.902480i $$-0.641745\pi$$
−0.430732 + 0.902480i $$0.641745\pi$$
$$128$$ 0 0
$$129$$ −10.4721 −0.922020
$$130$$ 0 0
$$131$$ 16.9443 1.48043 0.740214 0.672371i $$-0.234724\pi$$
0.740214 + 0.672371i $$0.234724\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −14.4721 −1.24556
$$136$$ 0 0
$$137$$ −19.8885 −1.69919 −0.849596 0.527433i $$-0.823154\pi$$
−0.849596 + 0.527433i $$0.823154\pi$$
$$138$$ 0 0
$$139$$ 13.5279 1.14742 0.573709 0.819059i $$-0.305504\pi$$
0.573709 + 0.819059i $$0.305504\pi$$
$$140$$ 0 0
$$141$$ 41.8885 3.52765
$$142$$ 0 0
$$143$$ −0.763932 −0.0638832
$$144$$ 0 0
$$145$$ −8.47214 −0.703573
$$146$$ 0 0
$$147$$ −22.6525 −1.86834
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 10.6525 0.866886 0.433443 0.901181i $$-0.357299\pi$$
0.433443 + 0.901181i $$0.357299\pi$$
$$152$$ 0 0
$$153$$ 14.9443 1.20817
$$154$$ 0 0
$$155$$ 5.70820 0.458494
$$156$$ 0 0
$$157$$ −10.9443 −0.873448 −0.436724 0.899596i $$-0.643861\pi$$
−0.436724 + 0.899596i $$0.643861\pi$$
$$158$$ 0 0
$$159$$ −35.4164 −2.80870
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.4721 −0.820241 −0.410120 0.912031i $$-0.634513\pi$$
−0.410120 + 0.912031i $$0.634513\pi$$
$$164$$ 0 0
$$165$$ −2.47214 −0.192456
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 5.70820 0.436517
$$172$$ 0 0
$$173$$ −10.9443 −0.832078 −0.416039 0.909347i $$-0.636582\pi$$
−0.416039 + 0.909347i $$0.636582\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −18.4721 −1.38845
$$178$$ 0 0
$$179$$ 16.9443 1.26647 0.633237 0.773958i $$-0.281726\pi$$
0.633237 + 0.773958i $$0.281726\pi$$
$$180$$ 0 0
$$181$$ 0.472136 0.0350936 0.0175468 0.999846i $$-0.494414\pi$$
0.0175468 + 0.999846i $$0.494414\pi$$
$$182$$ 0 0
$$183$$ −14.4721 −1.06981
$$184$$ 0 0
$$185$$ 8.47214 0.622884
$$186$$ 0 0
$$187$$ 1.52786 0.111728
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.8885 −1.29437 −0.647185 0.762333i $$-0.724054\pi$$
−0.647185 + 0.762333i $$0.724054\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 3.23607 0.231740
$$196$$ 0 0
$$197$$ 19.8885 1.41700 0.708500 0.705711i $$-0.249372\pi$$
0.708500 + 0.705711i $$0.249372\pi$$
$$198$$ 0 0
$$199$$ 20.9443 1.48470 0.742350 0.670012i $$-0.233711\pi$$
0.742350 + 0.670012i $$0.233711\pi$$
$$200$$ 0 0
$$201$$ 33.8885 2.39031
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 10.9443 0.764381
$$206$$ 0 0
$$207$$ −24.1803 −1.68065
$$208$$ 0 0
$$209$$ 0.583592 0.0403679
$$210$$ 0 0
$$211$$ 8.94427 0.615749 0.307875 0.951427i $$-0.400382\pi$$
0.307875 + 0.951427i $$0.400382\pi$$
$$212$$ 0 0
$$213$$ −2.47214 −0.169388
$$214$$ 0 0
$$215$$ 3.23607 0.220698
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −24.3607 −1.64614
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ −9.88854 −0.662186 −0.331093 0.943598i $$-0.607417\pi$$
−0.331093 + 0.943598i $$0.607417\pi$$
$$224$$ 0 0
$$225$$ 7.47214 0.498142
$$226$$ 0 0
$$227$$ 18.4721 1.22604 0.613019 0.790068i $$-0.289955\pi$$
0.613019 + 0.790068i $$0.289955\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.9443 −0.716983 −0.358492 0.933533i $$-0.616709\pi$$
−0.358492 + 0.933533i $$0.616709\pi$$
$$234$$ 0 0
$$235$$ −12.9443 −0.844391
$$236$$ 0 0
$$237$$ 20.9443 1.36048
$$238$$ 0 0
$$239$$ 2.65248 0.171574 0.0857872 0.996313i $$-0.472659\pi$$
0.0857872 + 0.996313i $$0.472659\pi$$
$$240$$ 0 0
$$241$$ −15.8885 −1.02347 −0.511736 0.859143i $$-0.670997\pi$$
−0.511736 + 0.859143i $$0.670997\pi$$
$$242$$ 0 0
$$243$$ 35.5967 2.28353
$$244$$ 0 0
$$245$$ 7.00000 0.447214
$$246$$ 0 0
$$247$$ −0.763932 −0.0486078
$$248$$ 0 0
$$249$$ 12.9443 0.820310
$$250$$ 0 0
$$251$$ 2.47214 0.156040 0.0780199 0.996952i $$-0.475140\pi$$
0.0780199 + 0.996952i $$0.475140\pi$$
$$252$$ 0 0
$$253$$ −2.47214 −0.155422
$$254$$ 0 0
$$255$$ −6.47214 −0.405301
$$256$$ 0 0
$$257$$ 5.05573 0.315368 0.157684 0.987490i $$-0.449597\pi$$
0.157684 + 0.987490i $$0.449597\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 63.3050 3.91848
$$262$$ 0 0
$$263$$ −1.70820 −0.105332 −0.0526662 0.998612i $$-0.516772\pi$$
−0.0526662 + 0.998612i $$0.516772\pi$$
$$264$$ 0 0
$$265$$ 10.9443 0.672301
$$266$$ 0 0
$$267$$ 32.3607 1.98044
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 18.6525 1.13306 0.566529 0.824042i $$-0.308286\pi$$
0.566529 + 0.824042i $$0.308286\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.763932 0.0460668
$$276$$ 0 0
$$277$$ 6.94427 0.417241 0.208620 0.977997i $$-0.433103\pi$$
0.208620 + 0.977997i $$0.433103\pi$$
$$278$$ 0 0
$$279$$ −42.6525 −2.55354
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 30.6525 1.82210 0.911050 0.412295i $$-0.135273\pi$$
0.911050 + 0.412295i $$0.135273\pi$$
$$284$$ 0 0
$$285$$ −2.47214 −0.146437
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −32.3607 −1.89702
$$292$$ 0 0
$$293$$ 22.3607 1.30632 0.653162 0.757218i $$-0.273442\pi$$
0.653162 + 0.757218i $$0.273442\pi$$
$$294$$ 0 0
$$295$$ 5.70820 0.332344
$$296$$ 0 0
$$297$$ 11.0557 0.641518
$$298$$ 0 0
$$299$$ 3.23607 0.187147
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 45.3050 2.60270
$$304$$ 0 0
$$305$$ 4.47214 0.256074
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −20.3607 −1.15828
$$310$$ 0 0
$$311$$ 11.4164 0.647365 0.323683 0.946166i $$-0.395079\pi$$
0.323683 + 0.946166i $$0.395079\pi$$
$$312$$ 0 0
$$313$$ −20.8328 −1.17754 −0.588770 0.808300i $$-0.700388\pi$$
−0.588770 + 0.808300i $$0.700388\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 30.3607 1.70523 0.852613 0.522543i $$-0.175017\pi$$
0.852613 + 0.522543i $$0.175017\pi$$
$$318$$ 0 0
$$319$$ 6.47214 0.362370
$$320$$ 0 0
$$321$$ −52.3607 −2.92249
$$322$$ 0 0
$$323$$ 1.52786 0.0850126
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −3.41641 −0.188928
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −0.763932 −0.0419895 −0.0209948 0.999780i $$-0.506683\pi$$
−0.0209948 + 0.999780i $$0.506683\pi$$
$$332$$ 0 0
$$333$$ −63.3050 −3.46909
$$334$$ 0 0
$$335$$ −10.4721 −0.572154
$$336$$ 0 0
$$337$$ −30.0000 −1.63420 −0.817102 0.576493i $$-0.804421\pi$$
−0.817102 + 0.576493i $$0.804421\pi$$
$$338$$ 0 0
$$339$$ 64.3607 3.49559
$$340$$ 0 0
$$341$$ −4.36068 −0.236144
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 10.4721 0.563801
$$346$$ 0 0
$$347$$ 20.7639 1.11467 0.557333 0.830289i $$-0.311825\pi$$
0.557333 + 0.830289i $$0.311825\pi$$
$$348$$ 0 0
$$349$$ 14.9443 0.799949 0.399974 0.916526i $$-0.369019\pi$$
0.399974 + 0.916526i $$0.369019\pi$$
$$350$$ 0 0
$$351$$ −14.4721 −0.772465
$$352$$ 0 0
$$353$$ −12.4721 −0.663825 −0.331912 0.943310i $$-0.607694\pi$$
−0.331912 + 0.943310i $$0.607694\pi$$
$$354$$ 0 0
$$355$$ 0.763932 0.0405453
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 7.23607 0.381905 0.190953 0.981599i $$-0.438842\pi$$
0.190953 + 0.981599i $$0.438842\pi$$
$$360$$ 0 0
$$361$$ −18.4164 −0.969285
$$362$$ 0 0
$$363$$ −33.7082 −1.76922
$$364$$ 0 0
$$365$$ 7.52786 0.394026
$$366$$ 0 0
$$367$$ −24.1803 −1.26220 −0.631102 0.775700i $$-0.717397\pi$$
−0.631102 + 0.775700i $$0.717397\pi$$
$$368$$ 0 0
$$369$$ −81.7771 −4.25715
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −2.94427 −0.152449 −0.0762243 0.997091i $$-0.524287\pi$$
−0.0762243 + 0.997091i $$0.524287\pi$$
$$374$$ 0 0
$$375$$ −3.23607 −0.167110
$$376$$ 0 0
$$377$$ −8.47214 −0.436337
$$378$$ 0 0
$$379$$ 16.7639 0.861105 0.430553 0.902565i $$-0.358319\pi$$
0.430553 + 0.902565i $$0.358319\pi$$
$$380$$ 0 0
$$381$$ −31.4164 −1.60951
$$382$$ 0 0
$$383$$ −17.8885 −0.914062 −0.457031 0.889451i $$-0.651087\pi$$
−0.457031 + 0.889451i $$0.651087\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −24.1803 −1.22916
$$388$$ 0 0
$$389$$ 31.8885 1.61681 0.808407 0.588624i $$-0.200330\pi$$
0.808407 + 0.588624i $$0.200330\pi$$
$$390$$ 0 0
$$391$$ −6.47214 −0.327310
$$392$$ 0 0
$$393$$ 54.8328 2.76595
$$394$$ 0 0
$$395$$ −6.47214 −0.325649
$$396$$ 0 0
$$397$$ −6.58359 −0.330421 −0.165211 0.986258i $$-0.552830\pi$$
−0.165211 + 0.986258i $$0.552830\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7.88854 −0.393935 −0.196968 0.980410i $$-0.563109\pi$$
−0.196968 + 0.980410i $$0.563109\pi$$
$$402$$ 0 0
$$403$$ 5.70820 0.284346
$$404$$ 0 0
$$405$$ −24.4164 −1.21326
$$406$$ 0 0
$$407$$ −6.47214 −0.320812
$$408$$ 0 0
$$409$$ 32.8328 1.62348 0.811739 0.584020i $$-0.198521\pi$$
0.811739 + 0.584020i $$0.198521\pi$$
$$410$$ 0 0
$$411$$ −64.3607 −3.17468
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −4.00000 −0.196352
$$416$$ 0 0
$$417$$ 43.7771 2.14377
$$418$$ 0 0
$$419$$ −0.583592 −0.0285103 −0.0142552 0.999898i $$-0.504538\pi$$
−0.0142552 + 0.999898i $$0.504538\pi$$
$$420$$ 0 0
$$421$$ 3.88854 0.189516 0.0947580 0.995500i $$-0.469792\pi$$
0.0947580 + 0.995500i $$0.469792\pi$$
$$422$$ 0 0
$$423$$ 96.7214 4.70275
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −2.47214 −0.119356
$$430$$ 0 0
$$431$$ 38.0689 1.83371 0.916857 0.399216i $$-0.130718\pi$$
0.916857 + 0.399216i $$0.130718\pi$$
$$432$$ 0 0
$$433$$ 24.8328 1.19339 0.596694 0.802469i $$-0.296480\pi$$
0.596694 + 0.802469i $$0.296480\pi$$
$$434$$ 0 0
$$435$$ −27.4164 −1.31452
$$436$$ 0 0
$$437$$ −2.47214 −0.118258
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ −52.3050 −2.49071
$$442$$ 0 0
$$443$$ −22.6525 −1.07625 −0.538126 0.842865i $$-0.680867\pi$$
−0.538126 + 0.842865i $$0.680867\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 0 0
$$447$$ −19.4164 −0.918365
$$448$$ 0 0
$$449$$ −9.05573 −0.427366 −0.213683 0.976903i $$-0.568546\pi$$
−0.213683 + 0.976903i $$0.568546\pi$$
$$450$$ 0 0
$$451$$ −8.36068 −0.393689
$$452$$ 0 0
$$453$$ 34.4721 1.61964
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7.88854 0.369011 0.184505 0.982832i $$-0.440932\pi$$
0.184505 + 0.982832i $$0.440932\pi$$
$$458$$ 0 0
$$459$$ 28.9443 1.35100
$$460$$ 0 0
$$461$$ −4.11146 −0.191490 −0.0957448 0.995406i $$-0.530523\pi$$
−0.0957448 + 0.995406i $$0.530523\pi$$
$$462$$ 0 0
$$463$$ −19.4164 −0.902357 −0.451178 0.892434i $$-0.648996\pi$$
−0.451178 + 0.892434i $$0.648996\pi$$
$$464$$ 0 0
$$465$$ 18.4721 0.856625
$$466$$ 0 0
$$467$$ −1.70820 −0.0790463 −0.0395231 0.999219i $$-0.512584\pi$$
−0.0395231 + 0.999219i $$0.512584\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −35.4164 −1.63190
$$472$$ 0 0
$$473$$ −2.47214 −0.113669
$$474$$ 0 0
$$475$$ 0.763932 0.0350516
$$476$$ 0 0
$$477$$ −81.7771 −3.74432
$$478$$ 0 0
$$479$$ 9.12461 0.416914 0.208457 0.978032i $$-0.433156\pi$$
0.208457 + 0.978032i $$0.433156\pi$$
$$480$$ 0 0
$$481$$ 8.47214 0.386296
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 10.0000 0.454077
$$486$$ 0 0
$$487$$ 29.3050 1.32793 0.663967 0.747762i $$-0.268872\pi$$
0.663967 + 0.747762i $$0.268872\pi$$
$$488$$ 0 0
$$489$$ −33.8885 −1.53249
$$490$$ 0 0
$$491$$ −34.4721 −1.55571 −0.777853 0.628446i $$-0.783691\pi$$
−0.777853 + 0.628446i $$0.783691\pi$$
$$492$$ 0 0
$$493$$ 16.9443 0.763132
$$494$$ 0 0
$$495$$ −5.70820 −0.256565
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 37.7082 1.68805 0.844026 0.536303i $$-0.180180\pi$$
0.844026 + 0.536303i $$0.180180\pi$$
$$500$$ 0 0
$$501$$ 25.8885 1.15661
$$502$$ 0 0
$$503$$ 14.2918 0.637240 0.318620 0.947883i $$-0.396781\pi$$
0.318620 + 0.947883i $$0.396781\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 0 0
$$507$$ 3.23607 0.143719
$$508$$ 0 0
$$509$$ −9.05573 −0.401388 −0.200694 0.979654i $$-0.564320\pi$$
−0.200694 + 0.979654i $$0.564320\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 11.0557 0.488122
$$514$$ 0 0
$$515$$ 6.29180 0.277250
$$516$$ 0 0
$$517$$ 9.88854 0.434898
$$518$$ 0 0
$$519$$ −35.4164 −1.55461
$$520$$ 0 0
$$521$$ −6.58359 −0.288432 −0.144216 0.989546i $$-0.546066\pi$$
−0.144216 + 0.989546i $$0.546066\pi$$
$$522$$ 0 0
$$523$$ −19.5967 −0.856906 −0.428453 0.903564i $$-0.640941\pi$$
−0.428453 + 0.903564i $$0.640941\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.4164 −0.497307
$$528$$ 0 0
$$529$$ −12.5279 −0.544690
$$530$$ 0 0
$$531$$ −42.6525 −1.85096
$$532$$ 0 0
$$533$$ 10.9443 0.474049
$$534$$ 0 0
$$535$$ 16.1803 0.699537
$$536$$ 0 0
$$537$$ 54.8328 2.36621
$$538$$ 0 0
$$539$$ −5.34752 −0.230334
$$540$$ 0 0
$$541$$ −17.0557 −0.733283 −0.366642 0.930362i $$-0.619492\pi$$
−0.366642 + 0.930362i $$0.619492\pi$$
$$542$$ 0 0
$$543$$ 1.52786 0.0655669
$$544$$ 0 0
$$545$$ 1.05573 0.0452224
$$546$$ 0 0
$$547$$ 35.5967 1.52201 0.761004 0.648748i $$-0.224707\pi$$
0.761004 + 0.648748i $$0.224707\pi$$
$$548$$ 0 0
$$549$$ −33.4164 −1.42618
$$550$$ 0 0
$$551$$ 6.47214 0.275722
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 27.4164 1.16376
$$556$$ 0 0
$$557$$ 25.4164 1.07693 0.538464 0.842649i $$-0.319005\pi$$
0.538464 + 0.842649i $$0.319005\pi$$
$$558$$ 0 0
$$559$$ 3.23607 0.136871
$$560$$ 0 0
$$561$$ 4.94427 0.208747
$$562$$ 0 0
$$563$$ −14.2918 −0.602327 −0.301164 0.953572i $$-0.597375\pi$$
−0.301164 + 0.953572i $$0.597375\pi$$
$$564$$ 0 0
$$565$$ −19.8885 −0.836717
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 41.4164 1.73627 0.868133 0.496332i $$-0.165320\pi$$
0.868133 + 0.496332i $$0.165320\pi$$
$$570$$ 0 0
$$571$$ −13.5279 −0.566123 −0.283062 0.959102i $$-0.591350\pi$$
−0.283062 + 0.959102i $$0.591350\pi$$
$$572$$ 0 0
$$573$$ −57.8885 −2.41833
$$574$$ 0 0
$$575$$ −3.23607 −0.134953
$$576$$ 0 0
$$577$$ −41.4164 −1.72419 −0.862094 0.506749i $$-0.830847\pi$$
−0.862094 + 0.506749i $$0.830847\pi$$
$$578$$ 0 0
$$579$$ 19.4164 0.806918
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −8.36068 −0.346264
$$584$$ 0 0
$$585$$ 7.47214 0.308935
$$586$$ 0 0
$$587$$ −13.5279 −0.558355 −0.279177 0.960240i $$-0.590062\pi$$
−0.279177 + 0.960240i $$0.590062\pi$$
$$588$$ 0 0
$$589$$ −4.36068 −0.179679
$$590$$ 0 0
$$591$$ 64.3607 2.64744
$$592$$ 0 0
$$593$$ −0.111456 −0.00457696 −0.00228848 0.999997i $$-0.500728\pi$$
−0.00228848 + 0.999997i $$0.500728\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 67.7771 2.77393
$$598$$ 0 0
$$599$$ −6.47214 −0.264444 −0.132222 0.991220i $$-0.542211\pi$$
−0.132222 + 0.991220i $$0.542211\pi$$
$$600$$ 0 0
$$601$$ −15.8885 −0.648107 −0.324054 0.946039i $$-0.605046\pi$$
−0.324054 + 0.946039i $$0.605046\pi$$
$$602$$ 0 0
$$603$$ 78.2492 3.18655
$$604$$ 0 0
$$605$$ 10.4164 0.423487
$$606$$ 0 0
$$607$$ −34.0689 −1.38281 −0.691407 0.722466i $$-0.743009\pi$$
−0.691407 + 0.722466i $$0.743009\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.9443 −0.523669
$$612$$ 0 0
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 0 0
$$615$$ 35.4164 1.42813
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −2.65248 −0.106612 −0.0533060 0.998578i $$-0.516976\pi$$
−0.0533060 + 0.998578i $$0.516976\pi$$
$$620$$ 0 0
$$621$$ −46.8328 −1.87934
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 1.88854 0.0754212
$$628$$ 0 0
$$629$$ −16.9443 −0.675612
$$630$$ 0 0
$$631$$ −43.0132 −1.71233 −0.856163 0.516705i $$-0.827158\pi$$
−0.856163 + 0.516705i $$0.827158\pi$$
$$632$$ 0 0
$$633$$ 28.9443 1.15043
$$634$$ 0 0
$$635$$ 9.70820 0.385258
$$636$$ 0 0
$$637$$ 7.00000 0.277350
$$638$$ 0 0
$$639$$ −5.70820 −0.225813
$$640$$ 0 0
$$641$$ −14.0000 −0.552967 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$642$$ 0 0
$$643$$ 15.0557 0.593740 0.296870 0.954918i $$-0.404057\pi$$
0.296870 + 0.954918i $$0.404057\pi$$
$$644$$ 0 0
$$645$$ 10.4721 0.412340
$$646$$ 0 0
$$647$$ 19.2361 0.756248 0.378124 0.925755i $$-0.376569\pi$$
0.378124 + 0.925755i $$0.376569\pi$$
$$648$$ 0 0
$$649$$ −4.36068 −0.171172
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −12.8328 −0.502187 −0.251093 0.967963i $$-0.580790\pi$$
−0.251093 + 0.967963i $$0.580790\pi$$
$$654$$ 0 0
$$655$$ −16.9443 −0.662067
$$656$$ 0 0
$$657$$ −56.2492 −2.19449
$$658$$ 0 0
$$659$$ −33.3050 −1.29738 −0.648688 0.761054i $$-0.724682\pi$$
−0.648688 + 0.761054i $$0.724682\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 0 0
$$663$$ −6.47214 −0.251357
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −27.4164 −1.06157
$$668$$ 0 0
$$669$$ −32.0000 −1.23719
$$670$$ 0 0
$$671$$ −3.41641 −0.131889
$$672$$ 0 0
$$673$$ 43.8885 1.69178 0.845890 0.533358i $$-0.179070\pi$$
0.845890 + 0.533358i $$0.179070\pi$$
$$674$$ 0 0
$$675$$ 14.4721 0.557033
$$676$$ 0 0
$$677$$ −12.8328 −0.493205 −0.246603 0.969117i $$-0.579314\pi$$
−0.246603 + 0.969117i $$0.579314\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 59.7771 2.29066
$$682$$ 0 0
$$683$$ −44.3607 −1.69741 −0.848707 0.528863i $$-0.822618\pi$$
−0.848707 + 0.528863i $$0.822618\pi$$
$$684$$ 0 0
$$685$$ 19.8885 0.759902
$$686$$ 0 0
$$687$$ 32.3607 1.23464
$$688$$ 0 0
$$689$$ 10.9443 0.416944
$$690$$ 0 0
$$691$$ 43.0132 1.63630 0.818149 0.575007i $$-0.195001\pi$$
0.818149 + 0.575007i $$0.195001\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −13.5279 −0.513141
$$696$$ 0 0
$$697$$ −21.8885 −0.829088
$$698$$ 0 0
$$699$$ −35.4164 −1.33957
$$700$$ 0 0
$$701$$ −51.8885 −1.95980 −0.979902 0.199481i $$-0.936074\pi$$
−0.979902 + 0.199481i $$0.936074\pi$$
$$702$$ 0 0
$$703$$ −6.47214 −0.244101
$$704$$ 0 0
$$705$$ −41.8885 −1.57761
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −15.8885 −0.596707 −0.298353 0.954455i $$-0.596437\pi$$
−0.298353 + 0.954455i $$0.596437\pi$$
$$710$$ 0 0
$$711$$ 48.3607 1.81367
$$712$$ 0 0
$$713$$ 18.4721 0.691787
$$714$$ 0 0
$$715$$ 0.763932 0.0285694
$$716$$ 0 0
$$717$$ 8.58359 0.320560
$$718$$ 0 0
$$719$$ −46.8328 −1.74657 −0.873285 0.487210i $$-0.838015\pi$$
−0.873285 + 0.487210i $$0.838015\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −51.4164 −1.91220
$$724$$ 0 0
$$725$$ 8.47214 0.314647
$$726$$ 0 0
$$727$$ −25.7082 −0.953465 −0.476732 0.879049i $$-0.658179\pi$$
−0.476732 + 0.879049i $$0.658179\pi$$
$$728$$ 0 0
$$729$$ 41.9443 1.55349
$$730$$ 0 0
$$731$$ −6.47214 −0.239381
$$732$$ 0 0
$$733$$ 2.00000 0.0738717 0.0369358 0.999318i $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ 0 0
$$735$$ 22.6525 0.835549
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ 35.0132 1.28798 0.643990 0.765034i $$-0.277278\pi$$
0.643990 + 0.765034i $$0.277278\pi$$
$$740$$ 0 0
$$741$$ −2.47214 −0.0908162
$$742$$ 0 0
$$743$$ 6.11146 0.224208 0.112104 0.993697i $$-0.464241\pi$$
0.112104 + 0.993697i $$0.464241\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ 29.8885 1.09356
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 46.4721 1.69579 0.847896 0.530162i $$-0.177869\pi$$
0.847896 + 0.530162i $$0.177869\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ −10.6525 −0.387683
$$756$$ 0 0
$$757$$ −28.8328 −1.04795 −0.523973 0.851735i $$-0.675551\pi$$
−0.523973 + 0.851735i $$0.675551\pi$$
$$758$$ 0 0
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −14.9443 −0.540311
$$766$$ 0 0
$$767$$ 5.70820 0.206111
$$768$$ 0 0
$$769$$ 14.9443 0.538904 0.269452 0.963014i $$-0.413157\pi$$
0.269452 + 0.963014i $$0.413157\pi$$
$$770$$ 0 0
$$771$$ 16.3607 0.589215
$$772$$ 0 0
$$773$$ −28.2492 −1.01605 −0.508027 0.861341i $$-0.669625\pi$$
−0.508027 + 0.861341i $$0.669625\pi$$
$$774$$ 0 0
$$775$$ −5.70820 −0.205045
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8.36068 −0.299552
$$780$$ 0 0
$$781$$ −0.583592 −0.0208826
$$782$$ 0 0
$$783$$ 122.610 4.38172
$$784$$ 0 0
$$785$$ 10.9443 0.390618
$$786$$ 0 0
$$787$$ −21.8885 −0.780242 −0.390121 0.920764i $$-0.627567\pi$$
−0.390121 + 0.920764i $$0.627567\pi$$
$$788$$ 0 0
$$789$$ −5.52786 −0.196797
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4.47214 0.158810
$$794$$ 0 0
$$795$$ 35.4164 1.25609
$$796$$ 0 0
$$797$$ −22.0000 −0.779280 −0.389640 0.920967i $$-0.627401\pi$$
−0.389640 + 0.920967i $$0.627401\pi$$
$$798$$ 0 0
$$799$$ 25.8885 0.915871
$$800$$ 0 0
$$801$$ 74.7214 2.64015
$$802$$ 0 0
$$803$$ −5.75078 −0.202940
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 19.4164 0.683490
$$808$$ 0 0
$$809$$ 4.47214 0.157232 0.0786160 0.996905i $$-0.474950\pi$$
0.0786160 + 0.996905i $$0.474950\pi$$
$$810$$ 0 0
$$811$$ 39.2361 1.37776 0.688882 0.724873i $$-0.258102\pi$$
0.688882 + 0.724873i $$0.258102\pi$$
$$812$$ 0 0
$$813$$ 60.3607 2.11694
$$814$$ 0 0
$$815$$ 10.4721 0.366823
$$816$$ 0 0
$$817$$ −2.47214 −0.0864891
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 5.05573 0.176446 0.0882231 0.996101i $$-0.471881\pi$$
0.0882231 + 0.996101i $$0.471881\pi$$
$$822$$ 0 0
$$823$$ −21.1246 −0.736358 −0.368179 0.929755i $$-0.620019\pi$$
−0.368179 + 0.929755i $$0.620019\pi$$
$$824$$ 0 0
$$825$$ 2.47214 0.0860687
$$826$$ 0 0
$$827$$ 26.8328 0.933068 0.466534 0.884503i $$-0.345502\pi$$
0.466534 + 0.884503i $$0.345502\pi$$
$$828$$ 0 0
$$829$$ 37.4164 1.29953 0.649763 0.760137i $$-0.274868\pi$$
0.649763 + 0.760137i $$0.274868\pi$$
$$830$$ 0 0
$$831$$ 22.4721 0.779550
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ −82.6099 −2.85542
$$838$$ 0 0
$$839$$ −50.6525 −1.74872 −0.874359 0.485280i $$-0.838718\pi$$
−0.874359 + 0.485280i $$0.838718\pi$$
$$840$$ 0 0
$$841$$ 42.7771 1.47507
$$842$$ 0 0
$$843$$ 32.3607 1.11456
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 99.1935 3.40431
$$850$$ 0 0
$$851$$ 27.4164 0.939822
$$852$$ 0 0
$$853$$ 12.4721 0.427038 0.213519 0.976939i $$-0.431508\pi$$
0.213519 + 0.976939i $$0.431508\pi$$
$$854$$ 0 0
$$855$$ −5.70820 −0.195216
$$856$$ 0 0
$$857$$ 21.0557 0.719250 0.359625 0.933097i $$-0.382905\pi$$
0.359625 + 0.933097i $$0.382905\pi$$
$$858$$ 0 0
$$859$$ 11.6393 0.397128 0.198564 0.980088i $$-0.436372\pi$$
0.198564 + 0.980088i $$0.436372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 19.0557 0.648665 0.324332 0.945943i $$-0.394860\pi$$
0.324332 + 0.945943i $$0.394860\pi$$
$$864$$ 0 0
$$865$$ 10.9443 0.372116
$$866$$ 0 0
$$867$$ −42.0689 −1.42873
$$868$$ 0 0
$$869$$ 4.94427 0.167723
$$870$$ 0 0
$$871$$ −10.4721 −0.354835
$$872$$ 0 0
$$873$$ −74.7214 −2.52893
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −17.7771 −0.600290 −0.300145 0.953894i $$-0.597035\pi$$
−0.300145 + 0.953894i $$0.597035\pi$$
$$878$$ 0 0
$$879$$ 72.3607 2.44067
$$880$$ 0 0
$$881$$ −9.63932 −0.324757 −0.162378 0.986729i $$-0.551917\pi$$
−0.162378 + 0.986729i $$0.551917\pi$$
$$882$$ 0 0
$$883$$ −14.2918 −0.480957 −0.240479 0.970654i $$-0.577304\pi$$
−0.240479 + 0.970654i $$0.577304\pi$$
$$884$$ 0 0
$$885$$ 18.4721 0.620934
$$886$$ 0 0
$$887$$ 4.76393 0.159957 0.0799786 0.996797i $$-0.474515\pi$$
0.0799786 + 0.996797i $$0.474515\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 18.6525 0.624881
$$892$$ 0 0
$$893$$ 9.88854 0.330908
$$894$$ 0 0
$$895$$ −16.9443 −0.566385
$$896$$ 0 0
$$897$$ 10.4721 0.349654
$$898$$ 0 0
$$899$$ −48.3607 −1.61292
$$900$$ 0 0
$$901$$ −21.8885 −0.729213
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −0.472136 −0.0156943
$$906$$ 0 0
$$907$$ −41.7082 −1.38490 −0.692449 0.721467i $$-0.743468\pi$$
−0.692449 + 0.721467i $$0.743468\pi$$
$$908$$ 0 0
$$909$$ 104.610 3.46969
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 3.05573 0.101130
$$914$$ 0 0
$$915$$ 14.4721 0.478434
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −38.4721 −1.26908 −0.634539 0.772891i $$-0.718810\pi$$
−0.634539 + 0.772891i $$0.718810\pi$$
$$920$$ 0 0
$$921$$ −38.8328 −1.27958
$$922$$ 0 0
$$923$$ 0.763932 0.0251451
$$924$$ 0 0
$$925$$ −8.47214 −0.278562
$$926$$ 0 0
$$927$$ −47.0132 −1.54411
$$928$$ 0 0
$$929$$ −18.9443 −0.621541 −0.310771 0.950485i $$-0.600587\pi$$
−0.310771 + 0.950485i $$0.600587\pi$$
$$930$$ 0 0
$$931$$ −5.34752 −0.175258
$$932$$ 0 0
$$933$$ 36.9443 1.20950
$$934$$ 0 0
$$935$$ −1.52786 −0.0499665
$$936$$ 0 0
$$937$$ −44.8328 −1.46462 −0.732312 0.680969i $$-0.761559\pi$$
−0.732312 + 0.680969i $$0.761559\pi$$
$$938$$ 0 0
$$939$$ −67.4164 −2.20005
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ 35.4164 1.15332
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −39.0557 −1.26914 −0.634570 0.772865i $$-0.718823\pi$$
−0.634570 + 0.772865i $$0.718823\pi$$
$$948$$ 0 0
$$949$$ 7.52786 0.244365
$$950$$ 0 0
$$951$$ 98.2492 3.18595
$$952$$ 0 0
$$953$$ −7.16718 −0.232168 −0.116084 0.993239i $$-0.537034\pi$$
−0.116084 + 0.993239i $$0.537034\pi$$
$$954$$ 0 0
$$955$$ 17.8885 0.578860
$$956$$ 0 0
$$957$$ 20.9443 0.677032
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.58359 0.0510836
$$962$$ 0 0
$$963$$ −120.902 −3.89600
$$964$$ 0 0
$$965$$ −6.00000 −0.193147
$$966$$ 0 0
$$967$$ 60.1378 1.93390 0.966950 0.254966i $$-0.0820642\pi$$
0.966950 + 0.254966i $$0.0820642\pi$$
$$968$$ 0 0
$$969$$ 4.94427 0.158833
$$970$$ 0 0
$$971$$ 26.8328 0.861106 0.430553 0.902565i $$-0.358319\pi$$
0.430553 + 0.902565i $$0.358319\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −3.23607 −0.103637
$$976$$ 0 0
$$977$$ −15.5279 −0.496780 −0.248390 0.968660i $$-0.579902\pi$$
−0.248390 + 0.968660i $$0.579902\pi$$
$$978$$ 0 0
$$979$$ 7.63932 0.244154
$$980$$ 0 0
$$981$$ −7.88854 −0.251862
$$982$$ 0 0
$$983$$ 7.63932 0.243656 0.121828 0.992551i $$-0.461124\pi$$
0.121828 + 0.992551i $$0.461124\pi$$
$$984$$ 0 0
$$985$$ −19.8885 −0.633702
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 10.4721 0.332995
$$990$$ 0 0
$$991$$ −3.05573 −0.0970684 −0.0485342 0.998822i $$-0.515455\pi$$
−0.0485342 + 0.998822i $$0.515455\pi$$
$$992$$ 0 0
$$993$$ −2.47214 −0.0784509
$$994$$ 0 0
$$995$$ −20.9443 −0.663978
$$996$$ 0 0
$$997$$ 0.832816 0.0263755 0.0131878 0.999913i $$-0.495802\pi$$
0.0131878 + 0.999913i $$0.495802\pi$$
$$998$$ 0 0
$$999$$ −122.610 −3.87921
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 520.2.a.g.1.2 2
3.2 odd 2 4680.2.a.bc.1.2 2
4.3 odd 2 1040.2.a.i.1.1 2
5.2 odd 4 2600.2.d.j.1249.1 4
5.3 odd 4 2600.2.d.j.1249.4 4
5.4 even 2 2600.2.a.o.1.1 2
8.3 odd 2 4160.2.a.bm.1.2 2
8.5 even 2 4160.2.a.x.1.1 2
12.11 even 2 9360.2.a.cs.1.1 2
13.12 even 2 6760.2.a.t.1.2 2
20.19 odd 2 5200.2.a.bz.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
520.2.a.g.1.2 2 1.1 even 1 trivial
1040.2.a.i.1.1 2 4.3 odd 2
2600.2.a.o.1.1 2 5.4 even 2
2600.2.d.j.1249.1 4 5.2 odd 4
2600.2.d.j.1249.4 4 5.3 odd 4
4160.2.a.x.1.1 2 8.5 even 2
4160.2.a.bm.1.2 2 8.3 odd 2
4680.2.a.bc.1.2 2 3.2 odd 2
5200.2.a.bz.1.2 2 20.19 odd 2
6760.2.a.t.1.2 2 13.12 even 2
9360.2.a.cs.1.1 2 12.11 even 2