# Properties

 Label 1305.2.a.r.1.1 Level $1305$ Weight $2$ Character 1305.1 Self dual yes Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.43828$$ of defining polynomial Character $$\chi$$ $$=$$ 1305.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.43828 q^{2} +0.0686587 q^{4} +1.00000 q^{5} -2.74301 q^{7} +2.77782 q^{8} +O(q^{10})$$ $$q-1.43828 q^{2} +0.0686587 q^{4} +1.00000 q^{5} -2.74301 q^{7} +2.77782 q^{8} -1.43828 q^{10} -2.74301 q^{11} -5.14744 q^{13} +3.94523 q^{14} -4.13260 q^{16} +3.72913 q^{17} -0.404431 q^{19} +0.0686587 q^{20} +3.94523 q^{22} +5.45825 q^{23} +1.00000 q^{25} +7.40348 q^{26} -0.188331 q^{28} +1.00000 q^{29} +1.45825 q^{31} +0.388222 q^{32} -5.36354 q^{34} -2.74301 q^{35} +6.76702 q^{37} +0.581686 q^{38} +2.77782 q^{40} -9.78090 q^{41} +4.43220 q^{43} -0.188331 q^{44} -7.85051 q^{46} -2.60569 q^{47} +0.524103 q^{49} -1.43828 q^{50} -0.353416 q^{52} +6.43220 q^{53} -2.74301 q^{55} -7.61958 q^{56} -1.43828 q^{58} +9.91822 q^{59} -13.0816 q^{61} -2.09738 q^{62} +7.70683 q^{64} -5.14744 q^{65} +12.4961 q^{67} +0.256037 q^{68} +3.94523 q^{70} +11.3487 q^{71} -10.7670 q^{73} -9.73289 q^{74} -0.0277677 q^{76} +7.52410 q^{77} +14.1576 q^{79} -4.13260 q^{80} +14.0677 q^{82} -1.62334 q^{83} +3.72913 q^{85} -6.37476 q^{86} -7.61958 q^{88} +8.87281 q^{89} +14.1195 q^{91} +0.374756 q^{92} +3.74772 q^{94} -0.404431 q^{95} +7.82084 q^{97} -0.753809 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q + 3 * q^2 + 5 * q^4 + 4 * q^5 + 2 * q^7 + 12 * q^8 $$4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29} - 4 q^{31} + 17 q^{32} - q^{34} + 2 q^{35} - 16 q^{37} + 10 q^{38} + 12 q^{40} + 12 q^{41} + 2 q^{43} - 9 q^{44} - 8 q^{46} + 12 q^{47} + 6 q^{49} + 3 q^{50} - 3 q^{52} + 10 q^{53} + 2 q^{55} + 3 q^{58} - 2 q^{59} - 26 q^{61} - 20 q^{62} + 34 q^{64} - 8 q^{65} + 2 q^{67} - 9 q^{68} + 3 q^{70} + 10 q^{71} - 48 q^{74} + 16 q^{76} + 34 q^{77} + 22 q^{79} + 11 q^{80} + 38 q^{82} + 10 q^{83} + 10 q^{85} + 4 q^{86} + 4 q^{89} - 8 q^{91} - 28 q^{92} + 39 q^{94} - 2 q^{95} - 22 q^{97} - 34 q^{98}+O(q^{100})$$ 4 * q + 3 * q^2 + 5 * q^4 + 4 * q^5 + 2 * q^7 + 12 * q^8 + 3 * q^10 + 2 * q^11 - 8 * q^13 + 3 * q^14 + 11 * q^16 + 10 * q^17 - 2 * q^19 + 5 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 7 * q^26 - 9 * q^28 + 4 * q^29 - 4 * q^31 + 17 * q^32 - q^34 + 2 * q^35 - 16 * q^37 + 10 * q^38 + 12 * q^40 + 12 * q^41 + 2 * q^43 - 9 * q^44 - 8 * q^46 + 12 * q^47 + 6 * q^49 + 3 * q^50 - 3 * q^52 + 10 * q^53 + 2 * q^55 + 3 * q^58 - 2 * q^59 - 26 * q^61 - 20 * q^62 + 34 * q^64 - 8 * q^65 + 2 * q^67 - 9 * q^68 + 3 * q^70 + 10 * q^71 - 48 * q^74 + 16 * q^76 + 34 * q^77 + 22 * q^79 + 11 * q^80 + 38 * q^82 + 10 * q^83 + 10 * q^85 + 4 * q^86 + 4 * q^89 - 8 * q^91 - 28 * q^92 + 39 * q^94 - 2 * q^95 - 22 * q^97 - 34 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.43828 −1.01702 −0.508510 0.861056i $$-0.669803\pi$$
−0.508510 + 0.861056i $$0.669803\pi$$
$$3$$ 0 0
$$4$$ 0.0686587 0.0343293
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.74301 −1.03676 −0.518380 0.855150i $$-0.673465\pi$$
−0.518380 + 0.855150i $$0.673465\pi$$
$$8$$ 2.77782 0.982106
$$9$$ 0 0
$$10$$ −1.43828 −0.454825
$$11$$ −2.74301 −0.827049 −0.413524 0.910493i $$-0.635702\pi$$
−0.413524 + 0.910493i $$0.635702\pi$$
$$12$$ 0 0
$$13$$ −5.14744 −1.42764 −0.713822 0.700328i $$-0.753037\pi$$
−0.713822 + 0.700328i $$0.753037\pi$$
$$14$$ 3.94523 1.05441
$$15$$ 0 0
$$16$$ −4.13260 −1.03315
$$17$$ 3.72913 0.904446 0.452223 0.891905i $$-0.350631\pi$$
0.452223 + 0.891905i $$0.350631\pi$$
$$18$$ 0 0
$$19$$ −0.404431 −0.0927827 −0.0463914 0.998923i $$-0.514772\pi$$
−0.0463914 + 0.998923i $$0.514772\pi$$
$$20$$ 0.0686587 0.0153525
$$21$$ 0 0
$$22$$ 3.94523 0.841125
$$23$$ 5.45825 1.13812 0.569062 0.822295i $$-0.307306\pi$$
0.569062 + 0.822295i $$0.307306\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 7.40348 1.45194
$$27$$ 0 0
$$28$$ −0.188331 −0.0355913
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 1.45825 0.261910 0.130955 0.991388i $$-0.458196\pi$$
0.130955 + 0.991388i $$0.458196\pi$$
$$32$$ 0.388222 0.0686287
$$33$$ 0 0
$$34$$ −5.36354 −0.919839
$$35$$ −2.74301 −0.463653
$$36$$ 0 0
$$37$$ 6.76702 1.11249 0.556245 0.831018i $$-0.312241\pi$$
0.556245 + 0.831018i $$0.312241\pi$$
$$38$$ 0.581686 0.0943619
$$39$$ 0 0
$$40$$ 2.77782 0.439211
$$41$$ −9.78090 −1.52752 −0.763760 0.645500i $$-0.776649\pi$$
−0.763760 + 0.645500i $$0.776649\pi$$
$$42$$ 0 0
$$43$$ 4.43220 0.675904 0.337952 0.941163i $$-0.390266\pi$$
0.337952 + 0.941163i $$0.390266\pi$$
$$44$$ −0.188331 −0.0283920
$$45$$ 0 0
$$46$$ −7.85051 −1.15749
$$47$$ −2.60569 −0.380079 −0.190040 0.981776i $$-0.560862\pi$$
−0.190040 + 0.981776i $$0.560862\pi$$
$$48$$ 0 0
$$49$$ 0.524103 0.0748719
$$50$$ −1.43828 −0.203404
$$51$$ 0 0
$$52$$ −0.353416 −0.0490100
$$53$$ 6.43220 0.883530 0.441765 0.897131i $$-0.354352\pi$$
0.441765 + 0.897131i $$0.354352\pi$$
$$54$$ 0 0
$$55$$ −2.74301 −0.369867
$$56$$ −7.61958 −1.01821
$$57$$ 0 0
$$58$$ −1.43828 −0.188856
$$59$$ 9.91822 1.29124 0.645621 0.763658i $$-0.276599\pi$$
0.645621 + 0.763658i $$0.276599\pi$$
$$60$$ 0 0
$$61$$ −13.0816 −1.67493 −0.837463 0.546494i $$-0.815962\pi$$
−0.837463 + 0.546494i $$0.815962\pi$$
$$62$$ −2.09738 −0.266367
$$63$$ 0 0
$$64$$ 7.70683 0.963354
$$65$$ −5.14744 −0.638461
$$66$$ 0 0
$$67$$ 12.4961 1.52665 0.763323 0.646017i $$-0.223566\pi$$
0.763323 + 0.646017i $$0.223566\pi$$
$$68$$ 0.256037 0.0310490
$$69$$ 0 0
$$70$$ 3.94523 0.471545
$$71$$ 11.3487 1.34684 0.673422 0.739259i $$-0.264824\pi$$
0.673422 + 0.739259i $$0.264824\pi$$
$$72$$ 0 0
$$73$$ −10.7670 −1.26018 −0.630092 0.776520i $$-0.716983\pi$$
−0.630092 + 0.776520i $$0.716983\pi$$
$$74$$ −9.73289 −1.13143
$$75$$ 0 0
$$76$$ −0.0277677 −0.00318517
$$77$$ 7.52410 0.857451
$$78$$ 0 0
$$79$$ 14.1576 1.59285 0.796425 0.604737i $$-0.206722\pi$$
0.796425 + 0.604737i $$0.206722\pi$$
$$80$$ −4.13260 −0.462039
$$81$$ 0 0
$$82$$ 14.0677 1.55352
$$83$$ −1.62334 −0.178184 −0.0890922 0.996023i $$-0.528397\pi$$
−0.0890922 + 0.996023i $$0.528397\pi$$
$$84$$ 0 0
$$85$$ 3.72913 0.404481
$$86$$ −6.37476 −0.687408
$$87$$ 0 0
$$88$$ −7.61958 −0.812250
$$89$$ 8.87281 0.940516 0.470258 0.882529i $$-0.344161\pi$$
0.470258 + 0.882529i $$0.344161\pi$$
$$90$$ 0 0
$$91$$ 14.1195 1.48012
$$92$$ 0.374756 0.0390711
$$93$$ 0 0
$$94$$ 3.74772 0.386548
$$95$$ −0.404431 −0.0414937
$$96$$ 0 0
$$97$$ 7.82084 0.794086 0.397043 0.917800i $$-0.370036\pi$$
0.397043 + 0.917800i $$0.370036\pi$$
$$98$$ −0.753809 −0.0761462
$$99$$ 0 0
$$100$$ 0.0686587 0.00686587
$$101$$ 4.88033 0.485611 0.242805 0.970075i $$-0.421932\pi$$
0.242805 + 0.970075i $$0.421932\pi$$
$$102$$ 0 0
$$103$$ −0.294881 −0.0290555 −0.0145277 0.999894i $$-0.504624\pi$$
−0.0145277 + 0.999894i $$0.504624\pi$$
$$104$$ −14.2986 −1.40210
$$105$$ 0 0
$$106$$ −9.25132 −0.898568
$$107$$ 13.7809 1.33225 0.666125 0.745840i $$-0.267952\pi$$
0.666125 + 0.745840i $$0.267952\pi$$
$$108$$ 0 0
$$109$$ −6.20126 −0.593973 −0.296987 0.954882i $$-0.595982\pi$$
−0.296987 + 0.954882i $$0.595982\pi$$
$$110$$ 3.94523 0.376162
$$111$$ 0 0
$$112$$ 11.3358 1.07113
$$113$$ 10.5658 0.993943 0.496971 0.867767i $$-0.334445\pi$$
0.496971 + 0.867767i $$0.334445\pi$$
$$114$$ 0 0
$$115$$ 5.45825 0.508985
$$116$$ 0.0686587 0.00637480
$$117$$ 0 0
$$118$$ −14.2652 −1.31322
$$119$$ −10.2290 −0.937694
$$120$$ 0 0
$$121$$ −3.47590 −0.315991
$$122$$ 18.8150 1.70343
$$123$$ 0 0
$$124$$ 0.100122 0.00899119
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 8.54175 0.757958 0.378979 0.925405i $$-0.376275\pi$$
0.378979 + 0.925405i $$0.376275\pi$$
$$128$$ −11.8611 −1.04838
$$129$$ 0 0
$$130$$ 7.40348 0.649328
$$131$$ −13.3050 −1.16246 −0.581232 0.813738i $$-0.697429\pi$$
−0.581232 + 0.813738i $$0.697429\pi$$
$$132$$ 0 0
$$133$$ 1.10936 0.0961934
$$134$$ −17.9730 −1.55263
$$135$$ 0 0
$$136$$ 10.3588 0.888262
$$137$$ 18.2253 1.55709 0.778545 0.627589i $$-0.215958\pi$$
0.778545 + 0.627589i $$0.215958\pi$$
$$138$$ 0 0
$$139$$ −5.66142 −0.480195 −0.240098 0.970749i $$-0.577179\pi$$
−0.240098 + 0.970749i $$0.577179\pi$$
$$140$$ −0.188331 −0.0159169
$$141$$ 0 0
$$142$$ −16.3226 −1.36977
$$143$$ 14.1195 1.18073
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 15.4860 1.28163
$$147$$ 0 0
$$148$$ 0.464614 0.0381911
$$149$$ 11.9182 0.976378 0.488189 0.872738i $$-0.337658\pi$$
0.488189 + 0.872738i $$0.337658\pi$$
$$150$$ 0 0
$$151$$ 7.19114 0.585207 0.292603 0.956234i $$-0.405478\pi$$
0.292603 + 0.956234i $$0.405478\pi$$
$$152$$ −1.12343 −0.0911225
$$153$$ 0 0
$$154$$ −10.8218 −0.872045
$$155$$ 1.45825 0.117130
$$156$$ 0 0
$$157$$ −4.31457 −0.344340 −0.172170 0.985067i $$-0.555078\pi$$
−0.172170 + 0.985067i $$0.555078\pi$$
$$158$$ −20.3626 −1.61996
$$159$$ 0 0
$$160$$ 0.388222 0.0306917
$$161$$ −14.9720 −1.17996
$$162$$ 0 0
$$163$$ 21.6436 1.69526 0.847628 0.530591i $$-0.178030\pi$$
0.847628 + 0.530591i $$0.178030\pi$$
$$164$$ −0.671544 −0.0524388
$$165$$ 0 0
$$166$$ 2.33482 0.181217
$$167$$ 16.4303 1.27141 0.635707 0.771930i $$-0.280709\pi$$
0.635707 + 0.771930i $$0.280709\pi$$
$$168$$ 0 0
$$169$$ 13.4961 1.03816
$$170$$ −5.36354 −0.411365
$$171$$ 0 0
$$172$$ 0.304309 0.0232033
$$173$$ 4.64939 0.353487 0.176743 0.984257i $$-0.443444\pi$$
0.176743 + 0.984257i $$0.443444\pi$$
$$174$$ 0 0
$$175$$ −2.74301 −0.207352
$$176$$ 11.3358 0.854466
$$177$$ 0 0
$$178$$ −12.7616 −0.956523
$$179$$ −7.62334 −0.569795 −0.284897 0.958558i $$-0.591960\pi$$
−0.284897 + 0.958558i $$0.591960\pi$$
$$180$$ 0 0
$$181$$ −21.3050 −1.58359 −0.791794 0.610788i $$-0.790853\pi$$
−0.791794 + 0.610788i $$0.790853\pi$$
$$182$$ −20.3078 −1.50532
$$183$$ 0 0
$$184$$ 15.1620 1.11776
$$185$$ 6.76702 0.497521
$$186$$ 0 0
$$187$$ −10.2290 −0.748021
$$188$$ −0.178903 −0.0130479
$$189$$ 0 0
$$190$$ 0.581686 0.0421999
$$191$$ −3.07597 −0.222570 −0.111285 0.993789i $$-0.535497\pi$$
−0.111285 + 0.993789i $$0.535497\pi$$
$$192$$ 0 0
$$193$$ 6.77454 0.487642 0.243821 0.969820i $$-0.421599\pi$$
0.243821 + 0.969820i $$0.421599\pi$$
$$194$$ −11.2486 −0.807601
$$195$$ 0 0
$$196$$ 0.0359842 0.00257030
$$197$$ 13.6455 0.972201 0.486100 0.873903i $$-0.338419\pi$$
0.486100 + 0.873903i $$0.338419\pi$$
$$198$$ 0 0
$$199$$ −6.33858 −0.449330 −0.224665 0.974436i $$-0.572129\pi$$
−0.224665 + 0.974436i $$0.572129\pi$$
$$200$$ 2.77782 0.196421
$$201$$ 0 0
$$202$$ −7.01929 −0.493876
$$203$$ −2.74301 −0.192522
$$204$$ 0 0
$$205$$ −9.78090 −0.683128
$$206$$ 0.424122 0.0295500
$$207$$ 0 0
$$208$$ 21.2723 1.47497
$$209$$ 1.10936 0.0767358
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ 0.441626 0.0303310
$$213$$ 0 0
$$214$$ −19.8208 −1.35492
$$215$$ 4.43220 0.302273
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 8.91917 0.604082
$$219$$ 0 0
$$220$$ −0.188331 −0.0126973
$$221$$ −19.1955 −1.29123
$$222$$ 0 0
$$223$$ −2.20126 −0.147407 −0.0737037 0.997280i $$-0.523482\pi$$
−0.0737037 + 0.997280i $$0.523482\pi$$
$$224$$ −1.06490 −0.0711515
$$225$$ 0 0
$$226$$ −15.1965 −1.01086
$$227$$ −12.1853 −0.808769 −0.404384 0.914589i $$-0.632514\pi$$
−0.404384 + 0.914589i $$0.632514\pi$$
$$228$$ 0 0
$$229$$ −3.16337 −0.209041 −0.104521 0.994523i $$-0.533331\pi$$
−0.104521 + 0.994523i $$0.533331\pi$$
$$230$$ −7.85051 −0.517647
$$231$$ 0 0
$$232$$ 2.77782 0.182373
$$233$$ −23.8087 −1.55976 −0.779879 0.625930i $$-0.784719\pi$$
−0.779879 + 0.625930i $$0.784719\pi$$
$$234$$ 0 0
$$235$$ −2.60569 −0.169977
$$236$$ 0.680972 0.0443275
$$237$$ 0 0
$$238$$ 14.7122 0.953653
$$239$$ 6.02025 0.389417 0.194709 0.980861i $$-0.437624\pi$$
0.194709 + 0.980861i $$0.437624\pi$$
$$240$$ 0 0
$$241$$ 24.5517 1.58151 0.790756 0.612131i $$-0.209688\pi$$
0.790756 + 0.612131i $$0.209688\pi$$
$$242$$ 4.99932 0.321369
$$243$$ 0 0
$$244$$ −0.898165 −0.0574991
$$245$$ 0.524103 0.0334837
$$246$$ 0 0
$$247$$ 2.08178 0.132461
$$248$$ 4.05076 0.257223
$$249$$ 0 0
$$250$$ −1.43828 −0.0909650
$$251$$ 27.5162 1.73681 0.868403 0.495858i $$-0.165146\pi$$
0.868403 + 0.495858i $$0.165146\pi$$
$$252$$ 0 0
$$253$$ −14.9720 −0.941284
$$254$$ −12.2855 −0.770858
$$255$$ 0 0
$$256$$ 1.64589 0.102868
$$257$$ −15.6436 −0.975820 −0.487910 0.872894i $$-0.662241\pi$$
−0.487910 + 0.872894i $$0.662241\pi$$
$$258$$ 0 0
$$259$$ −18.5620 −1.15339
$$260$$ −0.353416 −0.0219180
$$261$$ 0 0
$$262$$ 19.1364 1.18225
$$263$$ 18.6175 1.14801 0.574003 0.818853i $$-0.305390\pi$$
0.574003 + 0.818853i $$0.305390\pi$$
$$264$$ 0 0
$$265$$ 6.43220 0.395127
$$266$$ −1.59557 −0.0978306
$$267$$ 0 0
$$268$$ 0.857969 0.0524088
$$269$$ −10.9006 −0.664620 −0.332310 0.943170i $$-0.607828\pi$$
−0.332310 + 0.943170i $$0.607828\pi$$
$$270$$ 0 0
$$271$$ 17.7809 1.08011 0.540056 0.841629i $$-0.318403\pi$$
0.540056 + 0.841629i $$0.318403\pi$$
$$272$$ −15.4110 −0.934429
$$273$$ 0 0
$$274$$ −26.2131 −1.58359
$$275$$ −2.74301 −0.165410
$$276$$ 0 0
$$277$$ −20.6612 −1.24141 −0.620706 0.784043i $$-0.713154\pi$$
−0.620706 + 0.784043i $$0.713154\pi$$
$$278$$ 8.14273 0.488368
$$279$$ 0 0
$$280$$ −7.61958 −0.455357
$$281$$ −28.2652 −1.68616 −0.843080 0.537787i $$-0.819260\pi$$
−0.843080 + 0.537787i $$0.819260\pi$$
$$282$$ 0 0
$$283$$ −3.21139 −0.190897 −0.0954485 0.995434i $$-0.530429\pi$$
−0.0954485 + 0.995434i $$0.530429\pi$$
$$284$$ 0.779187 0.0462362
$$285$$ 0 0
$$286$$ −20.3078 −1.20083
$$287$$ 26.8291 1.58367
$$288$$ 0 0
$$289$$ −3.09362 −0.181978
$$290$$ −1.43828 −0.0844589
$$291$$ 0 0
$$292$$ −0.739249 −0.0432613
$$293$$ 3.99624 0.233463 0.116731 0.993164i $$-0.462758\pi$$
0.116731 + 0.993164i $$0.462758\pi$$
$$294$$ 0 0
$$295$$ 9.91822 0.577461
$$296$$ 18.7975 1.09258
$$297$$ 0 0
$$298$$ −17.1418 −0.992996
$$299$$ −28.0960 −1.62484
$$300$$ 0 0
$$301$$ −12.1576 −0.700750
$$302$$ −10.3429 −0.595167
$$303$$ 0 0
$$304$$ 1.67135 0.0958586
$$305$$ −13.0816 −0.749050
$$306$$ 0 0
$$307$$ −12.0555 −0.688046 −0.344023 0.938961i $$-0.611790\pi$$
−0.344023 + 0.938961i $$0.611790\pi$$
$$308$$ 0.516595 0.0294357
$$309$$ 0 0
$$310$$ −2.09738 −0.119123
$$311$$ −15.6873 −0.889544 −0.444772 0.895644i $$-0.646715\pi$$
−0.444772 + 0.895644i $$0.646715\pi$$
$$312$$ 0 0
$$313$$ −33.8726 −1.91459 −0.957297 0.289107i $$-0.906641\pi$$
−0.957297 + 0.289107i $$0.906641\pi$$
$$314$$ 6.20558 0.350201
$$315$$ 0 0
$$316$$ 0.972040 0.0546815
$$317$$ −1.75689 −0.0986770 −0.0493385 0.998782i $$-0.515711\pi$$
−0.0493385 + 0.998782i $$0.515711\pi$$
$$318$$ 0 0
$$319$$ −2.74301 −0.153579
$$320$$ 7.70683 0.430825
$$321$$ 0 0
$$322$$ 21.5340 1.20004
$$323$$ −1.50817 −0.0839170
$$324$$ 0 0
$$325$$ −5.14744 −0.285529
$$326$$ −31.1296 −1.72411
$$327$$ 0 0
$$328$$ −27.1695 −1.50019
$$329$$ 7.14744 0.394051
$$330$$ 0 0
$$331$$ −22.4505 −1.23399 −0.616997 0.786966i $$-0.711651\pi$$
−0.616997 + 0.786966i $$0.711651\pi$$
$$332$$ −0.111456 −0.00611695
$$333$$ 0 0
$$334$$ −23.6314 −1.29305
$$335$$ 12.4961 0.682737
$$336$$ 0 0
$$337$$ 15.3886 0.838273 0.419136 0.907923i $$-0.362333\pi$$
0.419136 + 0.907923i $$0.362333\pi$$
$$338$$ −19.4113 −1.05583
$$339$$ 0 0
$$340$$ 0.256037 0.0138855
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 17.7634 0.959136
$$344$$ 12.3118 0.663809
$$345$$ 0 0
$$346$$ −6.68714 −0.359503
$$347$$ 12.3504 0.663005 0.331503 0.943454i $$-0.392444\pi$$
0.331503 + 0.943454i $$0.392444\pi$$
$$348$$ 0 0
$$349$$ −1.94446 −0.104085 −0.0520424 0.998645i $$-0.516573\pi$$
−0.0520424 + 0.998645i $$0.516573\pi$$
$$350$$ 3.94523 0.210881
$$351$$ 0 0
$$352$$ −1.06490 −0.0567592
$$353$$ −6.72517 −0.357945 −0.178972 0.983854i $$-0.557277\pi$$
−0.178972 + 0.983854i $$0.557277\pi$$
$$354$$ 0 0
$$355$$ 11.3487 0.602327
$$356$$ 0.609195 0.0322873
$$357$$ 0 0
$$358$$ 10.9645 0.579493
$$359$$ −15.2114 −0.802826 −0.401413 0.915897i $$-0.631481\pi$$
−0.401413 + 0.915897i $$0.631481\pi$$
$$360$$ 0 0
$$361$$ −18.8364 −0.991391
$$362$$ 30.6426 1.61054
$$363$$ 0 0
$$364$$ 0.969425 0.0508117
$$365$$ −10.7670 −0.563571
$$366$$ 0 0
$$367$$ 13.2189 0.690021 0.345011 0.938599i $$-0.387875\pi$$
0.345011 + 0.938599i $$0.387875\pi$$
$$368$$ −22.5568 −1.17585
$$369$$ 0 0
$$370$$ −9.73289 −0.505989
$$371$$ −17.6436 −0.916009
$$372$$ 0 0
$$373$$ 26.8050 1.38791 0.693956 0.720017i $$-0.255866\pi$$
0.693956 + 0.720017i $$0.255866\pi$$
$$374$$ 14.7122 0.760752
$$375$$ 0 0
$$376$$ −7.23813 −0.373278
$$377$$ −5.14744 −0.265107
$$378$$ 0 0
$$379$$ −24.4228 −1.25451 −0.627257 0.778813i $$-0.715822\pi$$
−0.627257 + 0.778813i $$0.715822\pi$$
$$380$$ −0.0277677 −0.00142445
$$381$$ 0 0
$$382$$ 4.42412 0.226358
$$383$$ −27.2372 −1.39176 −0.695879 0.718159i $$-0.744985\pi$$
−0.695879 + 0.718159i $$0.744985\pi$$
$$384$$ 0 0
$$385$$ 7.52410 0.383464
$$386$$ −9.74370 −0.495942
$$387$$ 0 0
$$388$$ 0.536968 0.0272604
$$389$$ 12.1994 0.618532 0.309266 0.950976i $$-0.399917\pi$$
0.309266 + 0.950976i $$0.399917\pi$$
$$390$$ 0 0
$$391$$ 20.3545 1.02937
$$392$$ 1.45586 0.0735322
$$393$$ 0 0
$$394$$ −19.6261 −0.988748
$$395$$ 14.1576 0.712344
$$396$$ 0 0
$$397$$ −17.7254 −0.889611 −0.444805 0.895627i $$-0.646727\pi$$
−0.444805 + 0.895627i $$0.646727\pi$$
$$398$$ 9.11667 0.456977
$$399$$ 0 0
$$400$$ −4.13260 −0.206630
$$401$$ 2.48773 0.124231 0.0621157 0.998069i $$-0.480215\pi$$
0.0621157 + 0.998069i $$0.480215\pi$$
$$402$$ 0 0
$$403$$ −7.50627 −0.373914
$$404$$ 0.335077 0.0166707
$$405$$ 0 0
$$406$$ 3.94523 0.195798
$$407$$ −18.5620 −0.920084
$$408$$ 0 0
$$409$$ 37.7194 1.86510 0.932551 0.361038i $$-0.117577\pi$$
0.932551 + 0.361038i $$0.117577\pi$$
$$410$$ 14.0677 0.694754
$$411$$ 0 0
$$412$$ −0.0202461 −0.000997455 0
$$413$$ −27.2058 −1.33871
$$414$$ 0 0
$$415$$ −1.62334 −0.0796865
$$416$$ −1.99835 −0.0979773
$$417$$ 0 0
$$418$$ −1.59557 −0.0780419
$$419$$ 2.29488 0.112112 0.0560561 0.998428i $$-0.482147\pi$$
0.0560561 + 0.998428i $$0.482147\pi$$
$$420$$ 0 0
$$421$$ 35.9662 1.75289 0.876443 0.481505i $$-0.159910\pi$$
0.876443 + 0.481505i $$0.159910\pi$$
$$422$$ −2.87657 −0.140029
$$423$$ 0 0
$$424$$ 17.8675 0.867721
$$425$$ 3.72913 0.180889
$$426$$ 0 0
$$427$$ 35.8829 1.73650
$$428$$ 0.946178 0.0457353
$$429$$ 0 0
$$430$$ −6.37476 −0.307418
$$431$$ −22.6974 −1.09330 −0.546648 0.837363i $$-0.684096\pi$$
−0.546648 + 0.837363i $$0.684096\pi$$
$$432$$ 0 0
$$433$$ −11.4108 −0.548368 −0.274184 0.961677i $$-0.588408\pi$$
−0.274184 + 0.961677i $$0.588408\pi$$
$$434$$ 5.75313 0.276159
$$435$$ 0 0
$$436$$ −0.425770 −0.0203907
$$437$$ −2.20748 −0.105598
$$438$$ 0 0
$$439$$ −7.36654 −0.351586 −0.175793 0.984427i $$-0.556249\pi$$
−0.175793 + 0.984427i $$0.556249\pi$$
$$440$$ −7.61958 −0.363249
$$441$$ 0 0
$$442$$ 27.6085 1.31320
$$443$$ −36.5423 −1.73617 −0.868087 0.496411i $$-0.834651\pi$$
−0.868087 + 0.496411i $$0.834651\pi$$
$$444$$ 0 0
$$445$$ 8.87281 0.420611
$$446$$ 3.16604 0.149916
$$447$$ 0 0
$$448$$ −21.1399 −0.998767
$$449$$ 39.6182 1.86970 0.934850 0.355043i $$-0.115534\pi$$
0.934850 + 0.355043i $$0.115534\pi$$
$$450$$ 0 0
$$451$$ 26.8291 1.26333
$$452$$ 0.725431 0.0341214
$$453$$ 0 0
$$454$$ 17.5260 0.822534
$$455$$ 14.1195 0.661931
$$456$$ 0 0
$$457$$ 28.5220 1.33420 0.667102 0.744967i $$-0.267535\pi$$
0.667102 + 0.744967i $$0.267535\pi$$
$$458$$ 4.54982 0.212599
$$459$$ 0 0
$$460$$ 0.374756 0.0174731
$$461$$ 22.6696 1.05583 0.527915 0.849297i $$-0.322974\pi$$
0.527915 + 0.849297i $$0.322974\pi$$
$$462$$ 0 0
$$463$$ 28.5517 1.32691 0.663455 0.748217i $$-0.269090\pi$$
0.663455 + 0.748217i $$0.269090\pi$$
$$464$$ −4.13260 −0.191851
$$465$$ 0 0
$$466$$ 34.2436 1.58630
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −34.2770 −1.58277
$$470$$ 3.74772 0.172870
$$471$$ 0 0
$$472$$ 27.5510 1.26814
$$473$$ −12.1576 −0.559005
$$474$$ 0 0
$$475$$ −0.404431 −0.0185565
$$476$$ −0.702312 −0.0321904
$$477$$ 0 0
$$478$$ −8.65882 −0.396045
$$479$$ 4.43410 0.202599 0.101300 0.994856i $$-0.467700\pi$$
0.101300 + 0.994856i $$0.467700\pi$$
$$480$$ 0 0
$$481$$ −34.8328 −1.58824
$$482$$ −35.3123 −1.60843
$$483$$ 0 0
$$484$$ −0.238650 −0.0108477
$$485$$ 7.82084 0.355126
$$486$$ 0 0
$$487$$ 14.1035 0.639093 0.319546 0.947571i $$-0.396469\pi$$
0.319546 + 0.947571i $$0.396469\pi$$
$$488$$ −36.3382 −1.64496
$$489$$ 0 0
$$490$$ −0.753809 −0.0340536
$$491$$ −39.6376 −1.78882 −0.894410 0.447249i $$-0.852404\pi$$
−0.894410 + 0.447249i $$0.852404\pi$$
$$492$$ 0 0
$$493$$ 3.72913 0.167951
$$494$$ −2.99419 −0.134715
$$495$$ 0 0
$$496$$ −6.02638 −0.270592
$$497$$ −31.1296 −1.39635
$$498$$ 0 0
$$499$$ 6.33858 0.283754 0.141877 0.989884i $$-0.454686\pi$$
0.141877 + 0.989884i $$0.454686\pi$$
$$500$$ 0.0686587 0.00307051
$$501$$ 0 0
$$502$$ −39.5761 −1.76637
$$503$$ 19.0638 0.850011 0.425005 0.905191i $$-0.360272\pi$$
0.425005 + 0.905191i $$0.360272\pi$$
$$504$$ 0 0
$$505$$ 4.88033 0.217172
$$506$$ 21.5340 0.957305
$$507$$ 0 0
$$508$$ 0.586465 0.0260202
$$509$$ 12.8145 0.567992 0.283996 0.958826i $$-0.408340\pi$$
0.283996 + 0.958826i $$0.408340\pi$$
$$510$$ 0 0
$$511$$ 29.5340 1.30651
$$512$$ 21.3549 0.943760
$$513$$ 0 0
$$514$$ 22.4999 0.992428
$$515$$ −0.294881 −0.0129940
$$516$$ 0 0
$$517$$ 7.14744 0.314344
$$518$$ 26.6974 1.17302
$$519$$ 0 0
$$520$$ −14.2986 −0.627037
$$521$$ 35.4800 1.55441 0.777204 0.629249i $$-0.216637\pi$$
0.777204 + 0.629249i $$0.216637\pi$$
$$522$$ 0 0
$$523$$ −14.9227 −0.652525 −0.326263 0.945279i $$-0.605789\pi$$
−0.326263 + 0.945279i $$0.605789\pi$$
$$524$$ −0.913504 −0.0399066
$$525$$ 0 0
$$526$$ −26.7773 −1.16754
$$527$$ 5.43801 0.236883
$$528$$ 0 0
$$529$$ 6.79252 0.295327
$$530$$ −9.25132 −0.401852
$$531$$ 0 0
$$532$$ 0.0761670 0.00330226
$$533$$ 50.3466 2.18075
$$534$$ 0 0
$$535$$ 13.7809 0.595800
$$536$$ 34.7120 1.49933
$$537$$ 0 0
$$538$$ 15.6781 0.675931
$$539$$ −1.43762 −0.0619227
$$540$$ 0 0
$$541$$ −22.8644 −0.983017 −0.491509 0.870873i $$-0.663554\pi$$
−0.491509 + 0.870873i $$0.663554\pi$$
$$542$$ −25.5740 −1.09850
$$543$$ 0 0
$$544$$ 1.44773 0.0620709
$$545$$ −6.20126 −0.265633
$$546$$ 0 0
$$547$$ −35.3290 −1.51056 −0.755279 0.655404i $$-0.772498\pi$$
−0.755279 + 0.655404i $$0.772498\pi$$
$$548$$ 1.25132 0.0534539
$$549$$ 0 0
$$550$$ 3.94523 0.168225
$$551$$ −0.404431 −0.0172293
$$552$$ 0 0
$$553$$ −38.8343 −1.65140
$$554$$ 29.7167 1.26254
$$555$$ 0 0
$$556$$ −0.388706 −0.0164848
$$557$$ −32.7994 −1.38976 −0.694878 0.719127i $$-0.744542\pi$$
−0.694878 + 0.719127i $$0.744542\pi$$
$$558$$ 0 0
$$559$$ −22.8145 −0.964950
$$560$$ 11.3358 0.479024
$$561$$ 0 0
$$562$$ 40.6534 1.71486
$$563$$ −16.8169 −0.708747 −0.354374 0.935104i $$-0.615306\pi$$
−0.354374 + 0.935104i $$0.615306\pi$$
$$564$$ 0 0
$$565$$ 10.5658 0.444505
$$566$$ 4.61888 0.194146
$$567$$ 0 0
$$568$$ 31.5246 1.32274
$$569$$ −8.88033 −0.372283 −0.186141 0.982523i $$-0.559598\pi$$
−0.186141 + 0.982523i $$0.559598\pi$$
$$570$$ 0 0
$$571$$ 25.1240 1.05141 0.525703 0.850668i $$-0.323802\pi$$
0.525703 + 0.850668i $$0.323802\pi$$
$$572$$ 0.969425 0.0405337
$$573$$ 0 0
$$574$$ −38.5879 −1.61063
$$575$$ 5.45825 0.227625
$$576$$ 0 0
$$577$$ −13.9026 −0.578774 −0.289387 0.957212i $$-0.593451\pi$$
−0.289387 + 0.957212i $$0.593451\pi$$
$$578$$ 4.44950 0.185075
$$579$$ 0 0
$$580$$ 0.0686587 0.00285090
$$581$$ 4.45283 0.184735
$$582$$ 0 0
$$583$$ −17.6436 −0.730723
$$584$$ −29.9088 −1.23763
$$585$$ 0 0
$$586$$ −5.74772 −0.237436
$$587$$ 20.9942 0.866523 0.433262 0.901268i $$-0.357363\pi$$
0.433262 + 0.901268i $$0.357363\pi$$
$$588$$ 0 0
$$589$$ −0.589762 −0.0243007
$$590$$ −14.2652 −0.587289
$$591$$ 0 0
$$592$$ −27.9654 −1.14937
$$593$$ −3.54003 −0.145372 −0.0726859 0.997355i $$-0.523157\pi$$
−0.0726859 + 0.997355i $$0.523157\pi$$
$$594$$ 0 0
$$595$$ −10.2290 −0.419349
$$596$$ 0.818289 0.0335184
$$597$$ 0 0
$$598$$ 40.4100 1.65249
$$599$$ 32.4886 1.32745 0.663725 0.747977i $$-0.268975\pi$$
0.663725 + 0.747977i $$0.268975\pi$$
$$600$$ 0 0
$$601$$ −12.5620 −0.512414 −0.256207 0.966622i $$-0.582473\pi$$
−0.256207 + 0.966622i $$0.582473\pi$$
$$602$$ 17.4860 0.712677
$$603$$ 0 0
$$604$$ 0.493734 0.0200898
$$605$$ −3.47590 −0.141315
$$606$$ 0 0
$$607$$ −0.369141 −0.0149830 −0.00749149 0.999972i $$-0.502385\pi$$
−0.00749149 + 0.999972i $$0.502385\pi$$
$$608$$ −0.157009 −0.00636756
$$609$$ 0 0
$$610$$ 18.8150 0.761798
$$611$$ 13.4126 0.542618
$$612$$ 0 0
$$613$$ 44.7017 1.80549 0.902743 0.430181i $$-0.141550\pi$$
0.902743 + 0.430181i $$0.141550\pi$$
$$614$$ 17.3393 0.699756
$$615$$ 0 0
$$616$$ 20.9006 0.842108
$$617$$ 5.91014 0.237933 0.118967 0.992898i $$-0.462042\pi$$
0.118967 + 0.992898i $$0.462042\pi$$
$$618$$ 0 0
$$619$$ −44.2187 −1.77730 −0.888650 0.458586i $$-0.848356\pi$$
−0.888650 + 0.458586i $$0.848356\pi$$
$$620$$ 0.100122 0.00402098
$$621$$ 0 0
$$622$$ 22.5628 0.904684
$$623$$ −24.3382 −0.975089
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 48.7184 1.94718
$$627$$ 0 0
$$628$$ −0.296233 −0.0118210
$$629$$ 25.2351 1.00619
$$630$$ 0 0
$$631$$ −29.5702 −1.17717 −0.588586 0.808435i $$-0.700315\pi$$
−0.588586 + 0.808435i $$0.700315\pi$$
$$632$$ 39.3271 1.56435
$$633$$ 0 0
$$634$$ 2.52691 0.100356
$$635$$ 8.54175 0.338969
$$636$$ 0 0
$$637$$ −2.69779 −0.106890
$$638$$ 3.94523 0.156193
$$639$$ 0 0
$$640$$ −11.8611 −0.468849
$$641$$ −26.6335 −1.05196 −0.525979 0.850497i $$-0.676301\pi$$
−0.525979 + 0.850497i $$0.676301\pi$$
$$642$$ 0 0
$$643$$ 8.16597 0.322035 0.161017 0.986952i $$-0.448523\pi$$
0.161017 + 0.986952i $$0.448523\pi$$
$$644$$ −1.02796 −0.0405073
$$645$$ 0 0
$$646$$ 2.16918 0.0853452
$$647$$ −20.8381 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$648$$ 0 0
$$649$$ −27.2058 −1.06792
$$650$$ 7.40348 0.290388
$$651$$ 0 0
$$652$$ 1.48602 0.0581970
$$653$$ −13.4267 −0.525428 −0.262714 0.964874i $$-0.584618\pi$$
−0.262714 + 0.964874i $$0.584618\pi$$
$$654$$ 0 0
$$655$$ −13.3050 −0.519870
$$656$$ 40.4206 1.57816
$$657$$ 0 0
$$658$$ −10.2800 −0.400758
$$659$$ 42.9744 1.67405 0.837023 0.547167i $$-0.184294\pi$$
0.837023 + 0.547167i $$0.184294\pi$$
$$660$$ 0 0
$$661$$ −10.0178 −0.389649 −0.194824 0.980838i $$-0.562414\pi$$
−0.194824 + 0.980838i $$0.562414\pi$$
$$662$$ 32.2902 1.25500
$$663$$ 0 0
$$664$$ −4.50933 −0.174996
$$665$$ 1.10936 0.0430190
$$666$$ 0 0
$$667$$ 5.45825 0.211344
$$668$$ 1.12808 0.0436468
$$669$$ 0 0
$$670$$ −17.9730 −0.694357
$$671$$ 35.8829 1.38525
$$672$$ 0 0
$$673$$ 23.0878 0.889970 0.444985 0.895538i $$-0.353209\pi$$
0.444985 + 0.895538i $$0.353209\pi$$
$$674$$ −22.1332 −0.852540
$$675$$ 0 0
$$676$$ 0.926627 0.0356395
$$677$$ −4.95555 −0.190457 −0.0952287 0.995455i $$-0.530358\pi$$
−0.0952287 + 0.995455i $$0.530358\pi$$
$$678$$ 0 0
$$679$$ −21.4526 −0.823277
$$680$$ 10.3588 0.397243
$$681$$ 0 0
$$682$$ 5.75313 0.220299
$$683$$ 36.8010 1.40815 0.704075 0.710126i $$-0.251362\pi$$
0.704075 + 0.710126i $$0.251362\pi$$
$$684$$ 0 0
$$685$$ 18.2253 0.696352
$$686$$ −25.5489 −0.975460
$$687$$ 0 0
$$688$$ −18.3165 −0.698311
$$689$$ −33.1094 −1.26137
$$690$$ 0 0
$$691$$ 15.8246 0.601996 0.300998 0.953625i $$-0.402680\pi$$
0.300998 + 0.953625i $$0.402680\pi$$
$$692$$ 0.319221 0.0121350
$$693$$ 0 0
$$694$$ −17.7634 −0.674289
$$695$$ −5.66142 −0.214750
$$696$$ 0 0
$$697$$ −36.4742 −1.38156
$$698$$ 2.79669 0.105856
$$699$$ 0 0
$$700$$ −0.188331 −0.00711826
$$701$$ 42.6156 1.60957 0.804785 0.593567i $$-0.202281\pi$$
0.804785 + 0.593567i $$0.202281\pi$$
$$702$$ 0 0
$$703$$ −2.73679 −0.103220
$$704$$ −21.1399 −0.796741
$$705$$ 0 0
$$706$$ 9.67270 0.364037
$$707$$ −13.3868 −0.503462
$$708$$ 0 0
$$709$$ 35.1795 1.32119 0.660597 0.750740i $$-0.270303\pi$$
0.660597 + 0.750740i $$0.270303\pi$$
$$710$$ −16.3226 −0.612578
$$711$$ 0 0
$$712$$ 24.6470 0.923686
$$713$$ 7.95951 0.298086
$$714$$ 0 0
$$715$$ 14.1195 0.528039
$$716$$ −0.523408 −0.0195607
$$717$$ 0 0
$$718$$ 21.8783 0.816490
$$719$$ 29.4563 1.09854 0.549268 0.835646i $$-0.314907\pi$$
0.549268 + 0.835646i $$0.314907\pi$$
$$720$$ 0 0
$$721$$ 0.808861 0.0301236
$$722$$ 27.0921 1.00826
$$723$$ 0 0
$$724$$ −1.46277 −0.0543635
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 46.3391 1.71862 0.859311 0.511454i $$-0.170893\pi$$
0.859311 + 0.511454i $$0.170893\pi$$
$$728$$ 39.2213 1.45364
$$729$$ 0 0
$$730$$ 15.4860 0.573163
$$731$$ 16.5282 0.611319
$$732$$ 0 0
$$733$$ −3.73344 −0.137898 −0.0689489 0.997620i $$-0.521965\pi$$
−0.0689489 + 0.997620i $$0.521965\pi$$
$$734$$ −19.0125 −0.701765
$$735$$ 0 0
$$736$$ 2.11902 0.0781080
$$737$$ −34.2770 −1.26261
$$738$$ 0 0
$$739$$ −6.48021 −0.238378 −0.119189 0.992872i $$-0.538030\pi$$
−0.119189 + 0.992872i $$0.538030\pi$$
$$740$$ 0.464614 0.0170796
$$741$$ 0 0
$$742$$ 25.3765 0.931600
$$743$$ 51.0638 1.87335 0.936674 0.350203i $$-0.113888\pi$$
0.936674 + 0.350203i $$0.113888\pi$$
$$744$$ 0 0
$$745$$ 11.9182 0.436650
$$746$$ −38.5533 −1.41153
$$747$$ 0 0
$$748$$ −0.702312 −0.0256791
$$749$$ −37.8011 −1.38122
$$750$$ 0 0
$$751$$ 24.6494 0.899469 0.449735 0.893162i $$-0.351519\pi$$
0.449735 + 0.893162i $$0.351519\pi$$
$$752$$ 10.7683 0.392679
$$753$$ 0 0
$$754$$ 7.40348 0.269619
$$755$$ 7.19114 0.261712
$$756$$ 0 0
$$757$$ −38.6833 −1.40597 −0.702985 0.711205i $$-0.748150\pi$$
−0.702985 + 0.711205i $$0.748150\pi$$
$$758$$ 35.1269 1.27587
$$759$$ 0 0
$$760$$ −1.12343 −0.0407512
$$761$$ 3.23725 0.117350 0.0586750 0.998277i $$-0.481312\pi$$
0.0586750 + 0.998277i $$0.481312\pi$$
$$762$$ 0 0
$$763$$ 17.0101 0.615808
$$764$$ −0.211192 −0.00764067
$$765$$ 0 0
$$766$$ 39.1749 1.41545
$$767$$ −51.0534 −1.84343
$$768$$ 0 0
$$769$$ 46.3166 1.67022 0.835111 0.550082i $$-0.185404\pi$$
0.835111 + 0.550082i $$0.185404\pi$$
$$770$$ −10.8218 −0.389990
$$771$$ 0 0
$$772$$ 0.465131 0.0167404
$$773$$ 27.0341 0.972350 0.486175 0.873861i $$-0.338392\pi$$
0.486175 + 0.873861i $$0.338392\pi$$
$$774$$ 0 0
$$775$$ 1.45825 0.0523820
$$776$$ 21.7248 0.779877
$$777$$ 0 0
$$778$$ −17.5461 −0.629059
$$779$$ 3.95569 0.141727
$$780$$ 0 0
$$781$$ −31.1296 −1.11390
$$782$$ −29.2756 −1.04689
$$783$$ 0 0
$$784$$ −2.16591 −0.0773540
$$785$$ −4.31457 −0.153994
$$786$$ 0 0
$$787$$ −39.1870 −1.39687 −0.698434 0.715675i $$-0.746119\pi$$
−0.698434 + 0.715675i $$0.746119\pi$$
$$788$$ 0.936881 0.0333750
$$789$$ 0 0
$$790$$ −20.3626 −0.724468
$$791$$ −28.9820 −1.03048
$$792$$ 0 0
$$793$$ 67.3367 2.39120
$$794$$ 25.4941 0.904752
$$795$$ 0 0
$$796$$ −0.435198 −0.0154252
$$797$$ 53.2933 1.88775 0.943873 0.330307i $$-0.107152\pi$$
0.943873 + 0.330307i $$0.107152\pi$$
$$798$$ 0 0
$$799$$ −9.71696 −0.343761
$$800$$ 0.388222 0.0137257
$$801$$ 0 0
$$802$$ −3.57807 −0.126346
$$803$$ 29.5340 1.04223
$$804$$ 0 0
$$805$$ −14.9720 −0.527695
$$806$$ 10.7961 0.380278
$$807$$ 0 0
$$808$$ 13.5567 0.476921
$$809$$ −0.613214 −0.0215595 −0.0107797 0.999942i $$-0.503431\pi$$
−0.0107797 + 0.999942i $$0.503431\pi$$
$$810$$ 0 0
$$811$$ −0.230936 −0.00810927 −0.00405463 0.999992i $$-0.501291\pi$$
−0.00405463 + 0.999992i $$0.501291\pi$$
$$812$$ −0.188331 −0.00660914
$$813$$ 0 0
$$814$$ 26.6974 0.935744
$$815$$ 21.6436 0.758142
$$816$$ 0 0
$$817$$ −1.79252 −0.0627122
$$818$$ −54.2511 −1.89685
$$819$$ 0 0
$$820$$ −0.671544 −0.0234513
$$821$$ −25.7254 −0.897821 −0.448911 0.893577i $$-0.648188\pi$$
−0.448911 + 0.893577i $$0.648188\pi$$
$$822$$ 0 0
$$823$$ −38.2258 −1.33247 −0.666234 0.745743i $$-0.732095\pi$$
−0.666234 + 0.745743i $$0.732095\pi$$
$$824$$ −0.819125 −0.0285356
$$825$$ 0 0
$$826$$ 39.1296 1.36149
$$827$$ −24.9199 −0.866551 −0.433275 0.901262i $$-0.642642\pi$$
−0.433275 + 0.901262i $$0.642642\pi$$
$$828$$ 0 0
$$829$$ −1.65301 −0.0574115 −0.0287057 0.999588i $$-0.509139\pi$$
−0.0287057 + 0.999588i $$0.509139\pi$$
$$830$$ 2.33482 0.0810427
$$831$$ 0 0
$$832$$ −39.6705 −1.37533
$$833$$ 1.95445 0.0677176
$$834$$ 0 0
$$835$$ 16.4303 0.568594
$$836$$ 0.0761670 0.00263429
$$837$$ 0 0
$$838$$ −3.30069 −0.114020
$$839$$ 31.4757 1.08666 0.543331 0.839519i $$-0.317163\pi$$
0.543331 + 0.839519i $$0.317163\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −51.7296 −1.78272
$$843$$ 0 0
$$844$$ 0.137317 0.00472666
$$845$$ 13.4961 0.464281
$$846$$ 0 0
$$847$$ 9.53442 0.327607
$$848$$ −26.5817 −0.912820
$$849$$ 0 0
$$850$$ −5.36354 −0.183968
$$851$$ 36.9361 1.26615
$$852$$ 0 0
$$853$$ −30.3753 −1.04003 −0.520015 0.854157i $$-0.674074\pi$$
−0.520015 + 0.854157i $$0.674074\pi$$
$$854$$ −51.6098 −1.76605
$$855$$ 0 0
$$856$$ 38.2808 1.30841
$$857$$ 6.53423 0.223205 0.111602 0.993753i $$-0.464402\pi$$
0.111602 + 0.993753i $$0.464402\pi$$
$$858$$ 0 0
$$859$$ 27.0220 0.921977 0.460989 0.887406i $$-0.347495\pi$$
0.460989 + 0.887406i $$0.347495\pi$$
$$860$$ 0.304309 0.0103768
$$861$$ 0 0
$$862$$ 32.6453 1.11190
$$863$$ −31.9587 −1.08789 −0.543944 0.839122i $$-0.683069\pi$$
−0.543944 + 0.839122i $$0.683069\pi$$
$$864$$ 0 0
$$865$$ 4.64939 0.158084
$$866$$ 16.4120 0.557701
$$867$$ 0 0
$$868$$ −0.274635 −0.00932171
$$869$$ −38.8343 −1.31736
$$870$$ 0 0
$$871$$ −64.3232 −2.17951
$$872$$ −17.2260 −0.583345
$$873$$ 0 0
$$874$$ 3.17499 0.107396
$$875$$ −2.74301 −0.0927307
$$876$$ 0 0
$$877$$ −50.0679 −1.69067 −0.845336 0.534235i $$-0.820600\pi$$
−0.845336 + 0.534235i $$0.820600\pi$$
$$878$$ 10.5952 0.357570
$$879$$ 0 0
$$880$$ 11.3358 0.382129
$$881$$ 13.6817 0.460947 0.230474 0.973079i $$-0.425972\pi$$
0.230474 + 0.973079i $$0.425972\pi$$
$$882$$ 0 0
$$883$$ 29.5693 0.995087 0.497543 0.867439i $$-0.334236\pi$$
0.497543 + 0.867439i $$0.334236\pi$$
$$884$$ −1.31793 −0.0443269
$$885$$ 0 0
$$886$$ 52.5581 1.76572
$$887$$ −4.35130 −0.146102 −0.0730512 0.997328i $$-0.523274\pi$$
−0.0730512 + 0.997328i $$0.523274\pi$$
$$888$$ 0 0
$$889$$ −23.4301 −0.785820
$$890$$ −12.7616 −0.427770
$$891$$ 0 0
$$892$$ −0.151136 −0.00506040
$$893$$ 1.05382 0.0352648
$$894$$ 0 0
$$895$$ −7.62334 −0.254820
$$896$$ 32.5350 1.08692
$$897$$ 0 0
$$898$$ −56.9822 −1.90152
$$899$$ 1.45825 0.0486354
$$900$$ 0 0
$$901$$ 23.9865 0.799105
$$902$$ −38.5879 −1.28484
$$903$$ 0 0
$$904$$ 29.3497 0.976157
$$905$$ −21.3050 −0.708202
$$906$$ 0 0
$$907$$ −52.3965 −1.73980 −0.869899 0.493230i $$-0.835816\pi$$
−0.869899 + 0.493230i $$0.835816\pi$$
$$908$$ −0.836629 −0.0277645
$$909$$ 0 0
$$910$$ −20.3078 −0.673197
$$911$$ −7.85085 −0.260110 −0.130055 0.991507i $$-0.541515\pi$$
−0.130055 + 0.991507i $$0.541515\pi$$
$$912$$ 0 0
$$913$$ 4.45283 0.147367
$$914$$ −41.0227 −1.35691
$$915$$ 0 0
$$916$$ −0.217193 −0.00717625
$$917$$ 36.4958 1.20520
$$918$$ 0 0
$$919$$ 13.4979 0.445253 0.222627 0.974904i $$-0.428537\pi$$
0.222627 + 0.974904i $$0.428537\pi$$
$$920$$ 15.1620 0.499877
$$921$$ 0 0
$$922$$ −32.6054 −1.07380
$$923$$ −58.4168 −1.92281
$$924$$ 0 0
$$925$$ 6.76702 0.222498
$$926$$ −41.0654 −1.34949
$$927$$ 0 0
$$928$$ 0.388222 0.0127440
$$929$$ −11.6177 −0.381165 −0.190583 0.981671i $$-0.561038\pi$$
−0.190583 + 0.981671i $$0.561038\pi$$
$$930$$ 0 0
$$931$$ −0.211963 −0.00694682
$$932$$ −1.63467 −0.0535455
$$933$$ 0 0
$$934$$ −11.5063 −0.376497
$$935$$ −10.2290 −0.334525
$$936$$ 0 0
$$937$$ −42.6740 −1.39410 −0.697049 0.717024i $$-0.745504\pi$$
−0.697049 + 0.717024i $$0.745504\pi$$
$$938$$ 49.3001 1.60971
$$939$$ 0 0
$$940$$ −0.178903 −0.00583519
$$941$$ 5.91479 0.192817 0.0964083 0.995342i $$-0.469265\pi$$
0.0964083 + 0.995342i $$0.469265\pi$$
$$942$$ 0 0
$$943$$ −53.3866 −1.73851
$$944$$ −40.9881 −1.33405
$$945$$ 0 0
$$946$$ 17.4860 0.568520
$$947$$ 45.0915 1.46528 0.732639 0.680618i $$-0.238289\pi$$
0.732639 + 0.680618i $$0.238289\pi$$
$$948$$ 0 0
$$949$$ 55.4226 1.79909
$$950$$ 0.581686 0.0188724
$$951$$ 0 0
$$952$$ −28.4144 −0.920915
$$953$$ −30.3166 −0.982053 −0.491026 0.871145i $$-0.663378\pi$$
−0.491026 + 0.871145i $$0.663378\pi$$
$$954$$ 0 0
$$955$$ −3.07597 −0.0995362
$$956$$ 0.413342 0.0133684
$$957$$ 0 0
$$958$$ −6.37750 −0.206048
$$959$$ −49.9921 −1.61433
$$960$$ 0 0
$$961$$ −28.8735 −0.931403
$$962$$ 50.0995 1.61527
$$963$$ 0 0
$$964$$ 1.68569 0.0542923
$$965$$ 6.77454 0.218080
$$966$$ 0 0
$$967$$ 5.99809 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$968$$ −9.65540 −0.310336
$$969$$ 0 0
$$970$$ −11.2486 −0.361170
$$971$$ −28.5140 −0.915057 −0.457529 0.889195i $$-0.651265\pi$$
−0.457529 + 0.889195i $$0.651265\pi$$
$$972$$ 0 0
$$973$$ 15.5293 0.497848
$$974$$ −20.2849 −0.649970
$$975$$ 0 0
$$976$$ 54.0610 1.73045
$$977$$ 5.53813 0.177180 0.0885902 0.996068i $$-0.471764\pi$$
0.0885902 + 0.996068i $$0.471764\pi$$
$$978$$ 0 0
$$979$$ −24.3382 −0.777852
$$980$$ 0.0359842 0.00114947
$$981$$ 0 0
$$982$$ 57.0101 1.81926
$$983$$ −7.37086 −0.235094 −0.117547 0.993067i $$-0.537503\pi$$
−0.117547 + 0.993067i $$0.537503\pi$$
$$984$$ 0 0
$$985$$ 13.6455 0.434781
$$986$$ −5.36354 −0.170810
$$987$$ 0 0
$$988$$ 0.142932 0.00454729
$$989$$ 24.1921 0.769263
$$990$$ 0 0
$$991$$ 3.68167 0.116952 0.0584760 0.998289i $$-0.481376\pi$$
0.0584760 + 0.998289i $$0.481376\pi$$
$$992$$ 0.566126 0.0179745
$$993$$ 0 0
$$994$$ 44.7732 1.42012
$$995$$ −6.33858 −0.200946
$$996$$ 0 0
$$997$$ 36.8747 1.16783 0.583916 0.811814i $$-0.301520\pi$$
0.583916 + 0.811814i $$0.301520\pi$$
$$998$$ −9.11667 −0.288583
$$999$$ 0 0
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 1305.2.a.r.1.1 4
3.2 odd 2 435.2.a.j.1.4 4
5.4 even 2 6525.2.a.bi.1.4 4
12.11 even 2 6960.2.a.co.1.4 4
15.2 even 4 2175.2.c.n.349.6 8
15.8 even 4 2175.2.c.n.349.3 8
15.14 odd 2 2175.2.a.v.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 3.2 odd 2
1305.2.a.r.1.1 4 1.1 even 1 trivial
2175.2.a.v.1.1 4 15.14 odd 2
2175.2.c.n.349.3 8 15.8 even 4
2175.2.c.n.349.6 8 15.2 even 4
6525.2.a.bi.1.4 4 5.4 even 2
6960.2.a.co.1.4 4 12.11 even 2