# Properties

 Label 9405.2.a.z.1.2 Level $9405$ Weight $2$ Character 9405.1 Self dual yes Analytic conductor $75.099$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9405.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: $$75.0993031010$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*x^2 - 2*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.59744$$ of defining polynomial Character $$\chi$$ $$=$$ 9405.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} +O(q^{10})$$ $$q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} -1.21244 q^{10} +1.00000 q^{11} -0.439786 q^{13} +4.08605 q^{14} -2.65916 q^{16} +3.08872 q^{17} -1.00000 q^{19} -0.529980 q^{20} -1.21244 q^{22} -1.45571 q^{23} +1.00000 q^{25} +0.533216 q^{26} +1.78608 q^{28} -5.85137 q^{29} -6.80721 q^{31} -2.91083 q^{32} -3.74490 q^{34} -3.37010 q^{35} -2.73402 q^{37} +1.21244 q^{38} +3.06746 q^{40} +3.67980 q^{41} +4.59330 q^{43} -0.529980 q^{44} +1.76497 q^{46} -0.210971 q^{47} +4.35755 q^{49} -1.21244 q^{50} +0.233078 q^{52} -5.17998 q^{53} +1.00000 q^{55} -10.3376 q^{56} +7.09446 q^{58} +10.8732 q^{59} -5.66286 q^{61} +8.25336 q^{62} +8.84754 q^{64} -0.439786 q^{65} +6.14764 q^{67} -1.63696 q^{68} +4.08605 q^{70} +7.15175 q^{71} +4.15351 q^{73} +3.31484 q^{74} +0.529980 q^{76} -3.37010 q^{77} +15.4881 q^{79} -2.65916 q^{80} -4.46155 q^{82} -3.35537 q^{83} +3.08872 q^{85} -5.56911 q^{86} +3.06746 q^{88} +6.26557 q^{89} +1.48212 q^{91} +0.771498 q^{92} +0.255791 q^{94} -1.00000 q^{95} -6.70633 q^{97} -5.28329 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.21244 −0.857327 −0.428664 0.903464i $$-0.641015\pi$$
−0.428664 + 0.903464i $$0.641015\pi$$
$$3$$ 0 0
$$4$$ −0.529980 −0.264990
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.37010 −1.27378 −0.636888 0.770956i $$-0.719779\pi$$
−0.636888 + 0.770956i $$0.719779\pi$$
$$8$$ 3.06746 1.08451
$$9$$ 0 0
$$10$$ −1.21244 −0.383408
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −0.439786 −0.121975 −0.0609873 0.998139i $$-0.519425\pi$$
−0.0609873 + 0.998139i $$0.519425\pi$$
$$14$$ 4.08605 1.09204
$$15$$ 0 0
$$16$$ −2.65916 −0.664790
$$17$$ 3.08872 0.749125 0.374563 0.927202i $$-0.377793\pi$$
0.374563 + 0.927202i $$0.377793\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ −0.529980 −0.118507
$$21$$ 0 0
$$22$$ −1.21244 −0.258494
$$23$$ −1.45571 −0.303537 −0.151768 0.988416i $$-0.548497\pi$$
−0.151768 + 0.988416i $$0.548497\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.533216 0.104572
$$27$$ 0 0
$$28$$ 1.78608 0.337538
$$29$$ −5.85137 −1.08657 −0.543286 0.839548i $$-0.682820\pi$$
−0.543286 + 0.839548i $$0.682820\pi$$
$$30$$ 0 0
$$31$$ −6.80721 −1.22261 −0.611306 0.791394i $$-0.709355\pi$$
−0.611306 + 0.791394i $$0.709355\pi$$
$$32$$ −2.91083 −0.514567
$$33$$ 0 0
$$34$$ −3.74490 −0.642245
$$35$$ −3.37010 −0.569650
$$36$$ 0 0
$$37$$ −2.73402 −0.449470 −0.224735 0.974420i $$-0.572152\pi$$
−0.224735 + 0.974420i $$0.572152\pi$$
$$38$$ 1.21244 0.196684
$$39$$ 0 0
$$40$$ 3.06746 0.485008
$$41$$ 3.67980 0.574687 0.287344 0.957828i $$-0.407228\pi$$
0.287344 + 0.957828i $$0.407228\pi$$
$$42$$ 0 0
$$43$$ 4.59330 0.700471 0.350236 0.936662i $$-0.386102\pi$$
0.350236 + 0.936662i $$0.386102\pi$$
$$44$$ −0.529980 −0.0798975
$$45$$ 0 0
$$46$$ 1.76497 0.260230
$$47$$ −0.210971 −0.0307733 −0.0153867 0.999882i $$-0.504898\pi$$
−0.0153867 + 0.999882i $$0.504898\pi$$
$$48$$ 0 0
$$49$$ 4.35755 0.622507
$$50$$ −1.21244 −0.171465
$$51$$ 0 0
$$52$$ 0.233078 0.0323221
$$53$$ −5.17998 −0.711525 −0.355762 0.934576i $$-0.615779\pi$$
−0.355762 + 0.934576i $$0.615779\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ −10.3376 −1.38142
$$57$$ 0 0
$$58$$ 7.09446 0.931548
$$59$$ 10.8732 1.41557 0.707785 0.706428i $$-0.249695\pi$$
0.707785 + 0.706428i $$0.249695\pi$$
$$60$$ 0 0
$$61$$ −5.66286 −0.725055 −0.362527 0.931973i $$-0.618086\pi$$
−0.362527 + 0.931973i $$0.618086\pi$$
$$62$$ 8.25336 1.04818
$$63$$ 0 0
$$64$$ 8.84754 1.10594
$$65$$ −0.439786 −0.0545487
$$66$$ 0 0
$$67$$ 6.14764 0.751054 0.375527 0.926811i $$-0.377462\pi$$
0.375527 + 0.926811i $$0.377462\pi$$
$$68$$ −1.63696 −0.198511
$$69$$ 0 0
$$70$$ 4.08605 0.488377
$$71$$ 7.15175 0.848756 0.424378 0.905485i $$-0.360493\pi$$
0.424378 + 0.905485i $$0.360493\pi$$
$$72$$ 0 0
$$73$$ 4.15351 0.486132 0.243066 0.970010i $$-0.421847\pi$$
0.243066 + 0.970010i $$0.421847\pi$$
$$74$$ 3.31484 0.385343
$$75$$ 0 0
$$76$$ 0.529980 0.0607929
$$77$$ −3.37010 −0.384058
$$78$$ 0 0
$$79$$ 15.4881 1.74254 0.871272 0.490800i $$-0.163295\pi$$
0.871272 + 0.490800i $$0.163295\pi$$
$$80$$ −2.65916 −0.297303
$$81$$ 0 0
$$82$$ −4.46155 −0.492695
$$83$$ −3.35537 −0.368299 −0.184150 0.982898i $$-0.558953\pi$$
−0.184150 + 0.982898i $$0.558953\pi$$
$$84$$ 0 0
$$85$$ 3.08872 0.335019
$$86$$ −5.56911 −0.600533
$$87$$ 0 0
$$88$$ 3.06746 0.326992
$$89$$ 6.26557 0.664149 0.332075 0.943253i $$-0.392251\pi$$
0.332075 + 0.943253i $$0.392251\pi$$
$$90$$ 0 0
$$91$$ 1.48212 0.155368
$$92$$ 0.771498 0.0804342
$$93$$ 0 0
$$94$$ 0.255791 0.0263828
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −6.70633 −0.680924 −0.340462 0.940258i $$-0.610584\pi$$
−0.340462 + 0.940258i $$0.610584\pi$$
$$98$$ −5.28329 −0.533692
$$99$$ 0 0
$$100$$ −0.529980 −0.0529980
$$101$$ −19.2123 −1.91169 −0.955846 0.293868i $$-0.905057\pi$$
−0.955846 + 0.293868i $$0.905057\pi$$
$$102$$ 0 0
$$103$$ −15.0086 −1.47884 −0.739418 0.673246i $$-0.764899\pi$$
−0.739418 + 0.673246i $$0.764899\pi$$
$$104$$ −1.34902 −0.132283
$$105$$ 0 0
$$106$$ 6.28043 0.610010
$$107$$ 7.63909 0.738499 0.369250 0.929330i $$-0.379615\pi$$
0.369250 + 0.929330i $$0.379615\pi$$
$$108$$ 0 0
$$109$$ 0.0458556 0.00439217 0.00219609 0.999998i $$-0.499301\pi$$
0.00219609 + 0.999998i $$0.499301\pi$$
$$110$$ −1.21244 −0.115602
$$111$$ 0 0
$$112$$ 8.96163 0.846795
$$113$$ −8.43432 −0.793434 −0.396717 0.917941i $$-0.629851\pi$$
−0.396717 + 0.917941i $$0.629851\pi$$
$$114$$ 0 0
$$115$$ −1.45571 −0.135746
$$116$$ 3.10111 0.287931
$$117$$ 0 0
$$118$$ −13.1831 −1.21361
$$119$$ −10.4093 −0.954218
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 6.86590 0.621609
$$123$$ 0 0
$$124$$ 3.60769 0.323980
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 1.50239 0.133316 0.0666579 0.997776i $$-0.478766\pi$$
0.0666579 + 0.997776i $$0.478766\pi$$
$$128$$ −4.90548 −0.433588
$$129$$ 0 0
$$130$$ 0.533216 0.0467661
$$131$$ 0.608483 0.0531634 0.0265817 0.999647i $$-0.491538\pi$$
0.0265817 + 0.999647i $$0.491538\pi$$
$$132$$ 0 0
$$133$$ 3.37010 0.292224
$$134$$ −7.45367 −0.643899
$$135$$ 0 0
$$136$$ 9.47453 0.812434
$$137$$ 21.4534 1.83289 0.916443 0.400166i $$-0.131047\pi$$
0.916443 + 0.400166i $$0.131047\pi$$
$$138$$ 0 0
$$139$$ −8.85639 −0.751189 −0.375595 0.926784i $$-0.622561\pi$$
−0.375595 + 0.926784i $$0.622561\pi$$
$$140$$ 1.78608 0.150952
$$141$$ 0 0
$$142$$ −8.67109 −0.727662
$$143$$ −0.439786 −0.0367767
$$144$$ 0 0
$$145$$ −5.85137 −0.485930
$$146$$ −5.03590 −0.416774
$$147$$ 0 0
$$148$$ 1.44898 0.119105
$$149$$ 2.25215 0.184503 0.0922515 0.995736i $$-0.470594\pi$$
0.0922515 + 0.995736i $$0.470594\pi$$
$$150$$ 0 0
$$151$$ −17.3616 −1.41286 −0.706432 0.707781i $$-0.749696\pi$$
−0.706432 + 0.707781i $$0.749696\pi$$
$$152$$ −3.06746 −0.248804
$$153$$ 0 0
$$154$$ 4.08605 0.329264
$$155$$ −6.80721 −0.546769
$$156$$ 0 0
$$157$$ −3.75886 −0.299990 −0.149995 0.988687i $$-0.547926\pi$$
−0.149995 + 0.988687i $$0.547926\pi$$
$$158$$ −18.7784 −1.49393
$$159$$ 0 0
$$160$$ −2.91083 −0.230122
$$161$$ 4.90589 0.386638
$$162$$ 0 0
$$163$$ −11.8435 −0.927655 −0.463828 0.885925i $$-0.653524\pi$$
−0.463828 + 0.885925i $$0.653524\pi$$
$$164$$ −1.95022 −0.152286
$$165$$ 0 0
$$166$$ 4.06819 0.315753
$$167$$ −6.33806 −0.490454 −0.245227 0.969466i $$-0.578863\pi$$
−0.245227 + 0.969466i $$0.578863\pi$$
$$168$$ 0 0
$$169$$ −12.8066 −0.985122
$$170$$ −3.74490 −0.287221
$$171$$ 0 0
$$172$$ −2.43436 −0.185618
$$173$$ 8.21796 0.624800 0.312400 0.949951i $$-0.398867\pi$$
0.312400 + 0.949951i $$0.398867\pi$$
$$174$$ 0 0
$$175$$ −3.37010 −0.254755
$$176$$ −2.65916 −0.200442
$$177$$ 0 0
$$178$$ −7.59665 −0.569393
$$179$$ −11.2448 −0.840476 −0.420238 0.907414i $$-0.638053\pi$$
−0.420238 + 0.907414i $$0.638053\pi$$
$$180$$ 0 0
$$181$$ 8.27670 0.615202 0.307601 0.951515i $$-0.400474\pi$$
0.307601 + 0.951515i $$0.400474\pi$$
$$182$$ −1.79699 −0.133202
$$183$$ 0 0
$$184$$ −4.46533 −0.329189
$$185$$ −2.73402 −0.201009
$$186$$ 0 0
$$187$$ 3.08872 0.225870
$$188$$ 0.111811 0.00815462
$$189$$ 0 0
$$190$$ 1.21244 0.0879599
$$191$$ −3.21777 −0.232830 −0.116415 0.993201i $$-0.537140\pi$$
−0.116415 + 0.993201i $$0.537140\pi$$
$$192$$ 0 0
$$193$$ 19.0447 1.37087 0.685433 0.728136i $$-0.259613\pi$$
0.685433 + 0.728136i $$0.259613\pi$$
$$194$$ 8.13104 0.583775
$$195$$ 0 0
$$196$$ −2.30941 −0.164958
$$197$$ −2.04266 −0.145534 −0.0727668 0.997349i $$-0.523183\pi$$
−0.0727668 + 0.997349i $$0.523183\pi$$
$$198$$ 0 0
$$199$$ 4.61202 0.326937 0.163469 0.986549i $$-0.447732\pi$$
0.163469 + 0.986549i $$0.447732\pi$$
$$200$$ 3.06746 0.216902
$$201$$ 0 0
$$202$$ 23.2938 1.63895
$$203$$ 19.7197 1.38405
$$204$$ 0 0
$$205$$ 3.67980 0.257008
$$206$$ 18.1970 1.26785
$$207$$ 0 0
$$208$$ 1.16946 0.0810876
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −16.5102 −1.13661 −0.568303 0.822819i $$-0.692400\pi$$
−0.568303 + 0.822819i $$0.692400\pi$$
$$212$$ 2.74528 0.188547
$$213$$ 0 0
$$214$$ −9.26197 −0.633135
$$215$$ 4.59330 0.313260
$$216$$ 0 0
$$217$$ 22.9410 1.55733
$$218$$ −0.0555973 −0.00376553
$$219$$ 0 0
$$220$$ −0.529980 −0.0357312
$$221$$ −1.35838 −0.0913742
$$222$$ 0 0
$$223$$ 3.43650 0.230125 0.115062 0.993358i $$-0.463293\pi$$
0.115062 + 0.993358i $$0.463293\pi$$
$$224$$ 9.80979 0.655444
$$225$$ 0 0
$$226$$ 10.2261 0.680233
$$227$$ 9.66762 0.641663 0.320831 0.947136i $$-0.396038\pi$$
0.320831 + 0.947136i $$0.396038\pi$$
$$228$$ 0 0
$$229$$ 11.7897 0.779084 0.389542 0.921009i $$-0.372633\pi$$
0.389542 + 0.921009i $$0.372633\pi$$
$$230$$ 1.76497 0.116379
$$231$$ 0 0
$$232$$ −17.9488 −1.17840
$$233$$ −2.00908 −0.131619 −0.0658097 0.997832i $$-0.520963\pi$$
−0.0658097 + 0.997832i $$0.520963\pi$$
$$234$$ 0 0
$$235$$ −0.210971 −0.0137622
$$236$$ −5.76258 −0.375112
$$237$$ 0 0
$$238$$ 12.6207 0.818077
$$239$$ 6.36104 0.411461 0.205731 0.978609i $$-0.434043\pi$$
0.205731 + 0.978609i $$0.434043\pi$$
$$240$$ 0 0
$$241$$ −28.3775 −1.82796 −0.913978 0.405763i $$-0.867006\pi$$
−0.913978 + 0.405763i $$0.867006\pi$$
$$242$$ −1.21244 −0.0779388
$$243$$ 0 0
$$244$$ 3.00120 0.192132
$$245$$ 4.35755 0.278394
$$246$$ 0 0
$$247$$ 0.439786 0.0279829
$$248$$ −20.8808 −1.32594
$$249$$ 0 0
$$250$$ −1.21244 −0.0766817
$$251$$ 17.7932 1.12310 0.561550 0.827443i $$-0.310205\pi$$
0.561550 + 0.827443i $$0.310205\pi$$
$$252$$ 0 0
$$253$$ −1.45571 −0.0915198
$$254$$ −1.82157 −0.114295
$$255$$ 0 0
$$256$$ −11.7475 −0.734217
$$257$$ 22.5168 1.40456 0.702279 0.711902i $$-0.252166\pi$$
0.702279 + 0.711902i $$0.252166\pi$$
$$258$$ 0 0
$$259$$ 9.21391 0.572524
$$260$$ 0.233078 0.0144549
$$261$$ 0 0
$$262$$ −0.737751 −0.0455784
$$263$$ 5.74826 0.354453 0.177226 0.984170i $$-0.443288\pi$$
0.177226 + 0.984170i $$0.443288\pi$$
$$264$$ 0 0
$$265$$ −5.17998 −0.318204
$$266$$ −4.08605 −0.250532
$$267$$ 0 0
$$268$$ −3.25813 −0.199022
$$269$$ 5.69265 0.347087 0.173544 0.984826i $$-0.444478\pi$$
0.173544 + 0.984826i $$0.444478\pi$$
$$270$$ 0 0
$$271$$ −16.0881 −0.977282 −0.488641 0.872485i $$-0.662507\pi$$
−0.488641 + 0.872485i $$0.662507\pi$$
$$272$$ −8.21341 −0.498011
$$273$$ 0 0
$$274$$ −26.0110 −1.57138
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −3.44717 −0.207120 −0.103560 0.994623i $$-0.533023\pi$$
−0.103560 + 0.994623i $$0.533023\pi$$
$$278$$ 10.7379 0.644015
$$279$$ 0 0
$$280$$ −10.3376 −0.617792
$$281$$ 8.74423 0.521637 0.260819 0.965388i $$-0.416008\pi$$
0.260819 + 0.965388i $$0.416008\pi$$
$$282$$ 0 0
$$283$$ 30.1987 1.79513 0.897564 0.440885i $$-0.145335\pi$$
0.897564 + 0.440885i $$0.145335\pi$$
$$284$$ −3.79028 −0.224912
$$285$$ 0 0
$$286$$ 0.533216 0.0315297
$$287$$ −12.4013 −0.732024
$$288$$ 0 0
$$289$$ −7.45980 −0.438812
$$290$$ 7.09446 0.416601
$$291$$ 0 0
$$292$$ −2.20128 −0.128820
$$293$$ 11.1213 0.649714 0.324857 0.945763i $$-0.394684\pi$$
0.324857 + 0.945763i $$0.394684\pi$$
$$294$$ 0 0
$$295$$ 10.8732 0.633062
$$296$$ −8.38649 −0.487455
$$297$$ 0 0
$$298$$ −2.73060 −0.158179
$$299$$ 0.640201 0.0370238
$$300$$ 0 0
$$301$$ −15.4799 −0.892244
$$302$$ 21.0499 1.21129
$$303$$ 0 0
$$304$$ 2.65916 0.152513
$$305$$ −5.66286 −0.324254
$$306$$ 0 0
$$307$$ 24.8969 1.42094 0.710469 0.703728i $$-0.248483\pi$$
0.710469 + 0.703728i $$0.248483\pi$$
$$308$$ 1.78608 0.101772
$$309$$ 0 0
$$310$$ 8.25336 0.468760
$$311$$ −9.24020 −0.523963 −0.261982 0.965073i $$-0.584376\pi$$
−0.261982 + 0.965073i $$0.584376\pi$$
$$312$$ 0 0
$$313$$ −14.8438 −0.839021 −0.419511 0.907750i $$-0.637798\pi$$
−0.419511 + 0.907750i $$0.637798\pi$$
$$314$$ 4.55740 0.257189
$$315$$ 0 0
$$316$$ −8.20837 −0.461757
$$317$$ −33.8110 −1.89902 −0.949509 0.313740i $$-0.898418\pi$$
−0.949509 + 0.313740i $$0.898418\pi$$
$$318$$ 0 0
$$319$$ −5.85137 −0.327614
$$320$$ 8.84754 0.494593
$$321$$ 0 0
$$322$$ −5.94811 −0.331475
$$323$$ −3.08872 −0.171861
$$324$$ 0 0
$$325$$ −0.439786 −0.0243949
$$326$$ 14.3596 0.795304
$$327$$ 0 0
$$328$$ 11.2876 0.623255
$$329$$ 0.710993 0.0391983
$$330$$ 0 0
$$331$$ 11.5116 0.632737 0.316369 0.948636i $$-0.397536\pi$$
0.316369 + 0.948636i $$0.397536\pi$$
$$332$$ 1.77828 0.0975956
$$333$$ 0 0
$$334$$ 7.68455 0.420480
$$335$$ 6.14764 0.335882
$$336$$ 0 0
$$337$$ 25.3591 1.38140 0.690698 0.723143i $$-0.257303\pi$$
0.690698 + 0.723143i $$0.257303\pi$$
$$338$$ 15.5273 0.844572
$$339$$ 0 0
$$340$$ −1.63696 −0.0887766
$$341$$ −6.80721 −0.368631
$$342$$ 0 0
$$343$$ 8.90531 0.480841
$$344$$ 14.0897 0.759668
$$345$$ 0 0
$$346$$ −9.96382 −0.535658
$$347$$ 9.66755 0.518982 0.259491 0.965746i $$-0.416445\pi$$
0.259491 + 0.965746i $$0.416445\pi$$
$$348$$ 0 0
$$349$$ 30.9349 1.65591 0.827953 0.560797i $$-0.189505\pi$$
0.827953 + 0.560797i $$0.189505\pi$$
$$350$$ 4.08605 0.218409
$$351$$ 0 0
$$352$$ −2.91083 −0.155148
$$353$$ −10.9128 −0.580830 −0.290415 0.956901i $$-0.593793\pi$$
−0.290415 + 0.956901i $$0.593793\pi$$
$$354$$ 0 0
$$355$$ 7.15175 0.379575
$$356$$ −3.32063 −0.175993
$$357$$ 0 0
$$358$$ 13.6337 0.720563
$$359$$ 29.0146 1.53133 0.765667 0.643237i $$-0.222409\pi$$
0.765667 + 0.643237i $$0.222409\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −10.0350 −0.527429
$$363$$ 0 0
$$364$$ −0.785494 −0.0411711
$$365$$ 4.15351 0.217405
$$366$$ 0 0
$$367$$ 21.1172 1.10231 0.551154 0.834404i $$-0.314188\pi$$
0.551154 + 0.834404i $$0.314188\pi$$
$$368$$ 3.87097 0.201788
$$369$$ 0 0
$$370$$ 3.31484 0.172331
$$371$$ 17.4570 0.906324
$$372$$ 0 0
$$373$$ 24.3430 1.26043 0.630215 0.776421i $$-0.282967\pi$$
0.630215 + 0.776421i $$0.282967\pi$$
$$374$$ −3.74490 −0.193644
$$375$$ 0 0
$$376$$ −0.647146 −0.0333740
$$377$$ 2.57335 0.132534
$$378$$ 0 0
$$379$$ 12.5458 0.644433 0.322216 0.946666i $$-0.395572\pi$$
0.322216 + 0.946666i $$0.395572\pi$$
$$380$$ 0.529980 0.0271874
$$381$$ 0 0
$$382$$ 3.90137 0.199611
$$383$$ −8.27908 −0.423041 −0.211521 0.977374i $$-0.567842\pi$$
−0.211521 + 0.977374i $$0.567842\pi$$
$$384$$ 0 0
$$385$$ −3.37010 −0.171756
$$386$$ −23.0906 −1.17528
$$387$$ 0 0
$$388$$ 3.55422 0.180438
$$389$$ 7.91740 0.401428 0.200714 0.979650i $$-0.435674\pi$$
0.200714 + 0.979650i $$0.435674\pi$$
$$390$$ 0 0
$$391$$ −4.49629 −0.227387
$$392$$ 13.3666 0.675116
$$393$$ 0 0
$$394$$ 2.47661 0.124770
$$395$$ 15.4881 0.779290
$$396$$ 0 0
$$397$$ −24.3775 −1.22347 −0.611737 0.791062i $$-0.709529\pi$$
−0.611737 + 0.791062i $$0.709529\pi$$
$$398$$ −5.59181 −0.280292
$$399$$ 0 0
$$400$$ −2.65916 −0.132958
$$401$$ −32.6120 −1.62857 −0.814283 0.580469i $$-0.802869\pi$$
−0.814283 + 0.580469i $$0.802869\pi$$
$$402$$ 0 0
$$403$$ 2.99372 0.149128
$$404$$ 10.1821 0.506579
$$405$$ 0 0
$$406$$ −23.9090 −1.18658
$$407$$ −2.73402 −0.135520
$$408$$ 0 0
$$409$$ −36.7429 −1.81682 −0.908409 0.418083i $$-0.862702\pi$$
−0.908409 + 0.418083i $$0.862702\pi$$
$$410$$ −4.46155 −0.220340
$$411$$ 0 0
$$412$$ 7.95423 0.391877
$$413$$ −36.6437 −1.80312
$$414$$ 0 0
$$415$$ −3.35537 −0.164708
$$416$$ 1.28014 0.0627642
$$417$$ 0 0
$$418$$ 1.21244 0.0593026
$$419$$ 20.7229 1.01238 0.506191 0.862422i $$-0.331053\pi$$
0.506191 + 0.862422i $$0.331053\pi$$
$$420$$ 0 0
$$421$$ −23.3391 −1.13748 −0.568738 0.822518i $$-0.692568\pi$$
−0.568738 + 0.822518i $$0.692568\pi$$
$$422$$ 20.0176 0.974443
$$423$$ 0 0
$$424$$ −15.8894 −0.771656
$$425$$ 3.08872 0.149825
$$426$$ 0 0
$$427$$ 19.0844 0.923558
$$428$$ −4.04857 −0.195695
$$429$$ 0 0
$$430$$ −5.56911 −0.268567
$$431$$ −2.35724 −0.113544 −0.0567721 0.998387i $$-0.518081\pi$$
−0.0567721 + 0.998387i $$0.518081\pi$$
$$432$$ 0 0
$$433$$ 37.3199 1.79348 0.896741 0.442555i $$-0.145928\pi$$
0.896741 + 0.442555i $$0.145928\pi$$
$$434$$ −27.8146 −1.33515
$$435$$ 0 0
$$436$$ −0.0243026 −0.00116388
$$437$$ 1.45571 0.0696361
$$438$$ 0 0
$$439$$ 26.3625 1.25821 0.629106 0.777320i $$-0.283421\pi$$
0.629106 + 0.777320i $$0.283421\pi$$
$$440$$ 3.06746 0.146235
$$441$$ 0 0
$$442$$ 1.64695 0.0783376
$$443$$ 7.31633 0.347609 0.173805 0.984780i $$-0.444394\pi$$
0.173805 + 0.984780i $$0.444394\pi$$
$$444$$ 0 0
$$445$$ 6.26557 0.297017
$$446$$ −4.16656 −0.197292
$$447$$ 0 0
$$448$$ −29.8171 −1.40872
$$449$$ 0.872582 0.0411797 0.0205898 0.999788i $$-0.493446\pi$$
0.0205898 + 0.999788i $$0.493446\pi$$
$$450$$ 0 0
$$451$$ 3.67980 0.173275
$$452$$ 4.47002 0.210252
$$453$$ 0 0
$$454$$ −11.7215 −0.550115
$$455$$ 1.48212 0.0694829
$$456$$ 0 0
$$457$$ −2.04749 −0.0957774 −0.0478887 0.998853i $$-0.515249\pi$$
−0.0478887 + 0.998853i $$0.515249\pi$$
$$458$$ −14.2943 −0.667930
$$459$$ 0 0
$$460$$ 0.771498 0.0359713
$$461$$ −26.6538 −1.24139 −0.620695 0.784052i $$-0.713149\pi$$
−0.620695 + 0.784052i $$0.713149\pi$$
$$462$$ 0 0
$$463$$ −19.5335 −0.907798 −0.453899 0.891053i $$-0.649967\pi$$
−0.453899 + 0.891053i $$0.649967\pi$$
$$464$$ 15.5597 0.722343
$$465$$ 0 0
$$466$$ 2.43590 0.112841
$$467$$ 29.3029 1.35598 0.677988 0.735073i $$-0.262852\pi$$
0.677988 + 0.735073i $$0.262852\pi$$
$$468$$ 0 0
$$469$$ −20.7182 −0.956675
$$470$$ 0.255791 0.0117987
$$471$$ 0 0
$$472$$ 33.3531 1.53520
$$473$$ 4.59330 0.211200
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 5.51671 0.252858
$$477$$ 0 0
$$478$$ −7.71240 −0.352757
$$479$$ 0.965639 0.0441212 0.0220606 0.999757i $$-0.492977\pi$$
0.0220606 + 0.999757i $$0.492977\pi$$
$$480$$ 0 0
$$481$$ 1.20238 0.0548239
$$482$$ 34.4061 1.56716
$$483$$ 0 0
$$484$$ −0.529980 −0.0240900
$$485$$ −6.70633 −0.304519
$$486$$ 0 0
$$487$$ −3.36125 −0.152313 −0.0761564 0.997096i $$-0.524265\pi$$
−0.0761564 + 0.997096i $$0.524265\pi$$
$$488$$ −17.3706 −0.786329
$$489$$ 0 0
$$490$$ −5.28329 −0.238675
$$491$$ 32.5832 1.47046 0.735230 0.677818i $$-0.237074\pi$$
0.735230 + 0.677818i $$0.237074\pi$$
$$492$$ 0 0
$$493$$ −18.0733 −0.813978
$$494$$ −0.533216 −0.0239905
$$495$$ 0 0
$$496$$ 18.1015 0.812780
$$497$$ −24.1021 −1.08113
$$498$$ 0 0
$$499$$ 7.53868 0.337477 0.168739 0.985661i $$-0.446031\pi$$
0.168739 + 0.985661i $$0.446031\pi$$
$$500$$ −0.529980 −0.0237014
$$501$$ 0 0
$$502$$ −21.5733 −0.962864
$$503$$ 9.02729 0.402507 0.201254 0.979539i $$-0.435498\pi$$
0.201254 + 0.979539i $$0.435498\pi$$
$$504$$ 0 0
$$505$$ −19.2123 −0.854935
$$506$$ 1.76497 0.0784624
$$507$$ 0 0
$$508$$ −0.796238 −0.0353273
$$509$$ 20.8922 0.926031 0.463015 0.886350i $$-0.346767\pi$$
0.463015 + 0.886350i $$0.346767\pi$$
$$510$$ 0 0
$$511$$ −13.9977 −0.619223
$$512$$ 24.0541 1.06305
$$513$$ 0 0
$$514$$ −27.3003 −1.20417
$$515$$ −15.0086 −0.661356
$$516$$ 0 0
$$517$$ −0.210971 −0.00927850
$$518$$ −11.1713 −0.490841
$$519$$ 0 0
$$520$$ −1.34902 −0.0591586
$$521$$ 43.6640 1.91296 0.956478 0.291803i $$-0.0942553\pi$$
0.956478 + 0.291803i $$0.0942553\pi$$
$$522$$ 0 0
$$523$$ 26.4677 1.15735 0.578675 0.815558i $$-0.303570\pi$$
0.578675 + 0.815558i $$0.303570\pi$$
$$524$$ −0.322484 −0.0140878
$$525$$ 0 0
$$526$$ −6.96944 −0.303882
$$527$$ −21.0256 −0.915889
$$528$$ 0 0
$$529$$ −20.8809 −0.907865
$$530$$ 6.28043 0.272805
$$531$$ 0 0
$$532$$ −1.78608 −0.0774365
$$533$$ −1.61832 −0.0700973
$$534$$ 0 0
$$535$$ 7.63909 0.330267
$$536$$ 18.8576 0.814526
$$537$$ 0 0
$$538$$ −6.90202 −0.297567
$$539$$ 4.35755 0.187693
$$540$$ 0 0
$$541$$ −1.04162 −0.0447829 −0.0223915 0.999749i $$-0.507128\pi$$
−0.0223915 + 0.999749i $$0.507128\pi$$
$$542$$ 19.5059 0.837851
$$543$$ 0 0
$$544$$ −8.99075 −0.385475
$$545$$ 0.0458556 0.00196424
$$546$$ 0 0
$$547$$ −5.27609 −0.225589 −0.112795 0.993618i $$-0.535980\pi$$
−0.112795 + 0.993618i $$0.535980\pi$$
$$548$$ −11.3699 −0.485696
$$549$$ 0 0
$$550$$ −1.21244 −0.0516988
$$551$$ 5.85137 0.249277
$$552$$ 0 0
$$553$$ −52.1963 −2.21961
$$554$$ 4.17949 0.177570
$$555$$ 0 0
$$556$$ 4.69371 0.199058
$$557$$ 12.9219 0.547517 0.273759 0.961798i $$-0.411733\pi$$
0.273759 + 0.961798i $$0.411733\pi$$
$$558$$ 0 0
$$559$$ −2.02007 −0.0854397
$$560$$ 8.96163 0.378698
$$561$$ 0 0
$$562$$ −10.6019 −0.447214
$$563$$ 8.76657 0.369467 0.184733 0.982789i $$-0.440858\pi$$
0.184733 + 0.982789i $$0.440858\pi$$
$$564$$ 0 0
$$565$$ −8.43432 −0.354834
$$566$$ −36.6142 −1.53901
$$567$$ 0 0
$$568$$ 21.9377 0.920485
$$569$$ 14.7828 0.619726 0.309863 0.950781i $$-0.399717\pi$$
0.309863 + 0.950781i $$0.399717\pi$$
$$570$$ 0 0
$$571$$ 22.6278 0.946943 0.473472 0.880809i $$-0.343001\pi$$
0.473472 + 0.880809i $$0.343001\pi$$
$$572$$ 0.233078 0.00974547
$$573$$ 0 0
$$574$$ 15.0358 0.627584
$$575$$ −1.45571 −0.0607074
$$576$$ 0 0
$$577$$ 3.73371 0.155436 0.0777182 0.996975i $$-0.475237\pi$$
0.0777182 + 0.996975i $$0.475237\pi$$
$$578$$ 9.04459 0.376205
$$579$$ 0 0
$$580$$ 3.10111 0.128767
$$581$$ 11.3079 0.469131
$$582$$ 0 0
$$583$$ −5.17998 −0.214533
$$584$$ 12.7407 0.527215
$$585$$ 0 0
$$586$$ −13.4840 −0.557018
$$587$$ 19.6389 0.810584 0.405292 0.914187i $$-0.367170\pi$$
0.405292 + 0.914187i $$0.367170\pi$$
$$588$$ 0 0
$$589$$ 6.80721 0.280486
$$590$$ −13.1831 −0.542741
$$591$$ 0 0
$$592$$ 7.27020 0.298803
$$593$$ 32.8110 1.34739 0.673693 0.739011i $$-0.264707\pi$$
0.673693 + 0.739011i $$0.264707\pi$$
$$594$$ 0 0
$$595$$ −10.4093 −0.426739
$$596$$ −1.19359 −0.0488914
$$597$$ 0 0
$$598$$ −0.776208 −0.0317415
$$599$$ 41.5943 1.69950 0.849750 0.527187i $$-0.176753\pi$$
0.849750 + 0.527187i $$0.176753\pi$$
$$600$$ 0 0
$$601$$ −16.1727 −0.659698 −0.329849 0.944034i $$-0.606998\pi$$
−0.329849 + 0.944034i $$0.606998\pi$$
$$602$$ 18.7685 0.764945
$$603$$ 0 0
$$604$$ 9.20127 0.374395
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −14.9877 −0.608330 −0.304165 0.952619i $$-0.598377\pi$$
−0.304165 + 0.952619i $$0.598377\pi$$
$$608$$ 2.91083 0.118050
$$609$$ 0 0
$$610$$ 6.86590 0.277992
$$611$$ 0.0927821 0.00375356
$$612$$ 0 0
$$613$$ 19.1145 0.772027 0.386014 0.922493i $$-0.373852\pi$$
0.386014 + 0.922493i $$0.373852\pi$$
$$614$$ −30.1860 −1.21821
$$615$$ 0 0
$$616$$ −10.3376 −0.416515
$$617$$ 1.20030 0.0483223 0.0241612 0.999708i $$-0.492309\pi$$
0.0241612 + 0.999708i $$0.492309\pi$$
$$618$$ 0 0
$$619$$ −44.2783 −1.77969 −0.889847 0.456258i $$-0.849189\pi$$
−0.889847 + 0.456258i $$0.849189\pi$$
$$620$$ 3.60769 0.144888
$$621$$ 0 0
$$622$$ 11.2032 0.449208
$$623$$ −21.1156 −0.845978
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 17.9973 0.719316
$$627$$ 0 0
$$628$$ 1.99212 0.0794942
$$629$$ −8.44462 −0.336709
$$630$$ 0 0
$$631$$ 7.76811 0.309244 0.154622 0.987974i $$-0.450584\pi$$
0.154622 + 0.987974i $$0.450584\pi$$
$$632$$ 47.5090 1.88981
$$633$$ 0 0
$$634$$ 40.9940 1.62808
$$635$$ 1.50239 0.0596206
$$636$$ 0 0
$$637$$ −1.91639 −0.0759301
$$638$$ 7.09446 0.280872
$$639$$ 0 0
$$640$$ −4.90548 −0.193906
$$641$$ −27.0513 −1.06846 −0.534231 0.845339i $$-0.679399\pi$$
−0.534231 + 0.845339i $$0.679399\pi$$
$$642$$ 0 0
$$643$$ −15.1507 −0.597485 −0.298742 0.954334i $$-0.596567\pi$$
−0.298742 + 0.954334i $$0.596567\pi$$
$$644$$ −2.60002 −0.102455
$$645$$ 0 0
$$646$$ 3.74490 0.147341
$$647$$ 22.7874 0.895866 0.447933 0.894067i $$-0.352160\pi$$
0.447933 + 0.894067i $$0.352160\pi$$
$$648$$ 0 0
$$649$$ 10.8732 0.426810
$$650$$ 0.533216 0.0209144
$$651$$ 0 0
$$652$$ 6.27682 0.245819
$$653$$ −36.1144 −1.41327 −0.706633 0.707580i $$-0.749787\pi$$
−0.706633 + 0.707580i $$0.749787\pi$$
$$654$$ 0 0
$$655$$ 0.608483 0.0237754
$$656$$ −9.78517 −0.382047
$$657$$ 0 0
$$658$$ −0.862040 −0.0336058
$$659$$ 5.96009 0.232172 0.116086 0.993239i $$-0.462965\pi$$
0.116086 + 0.993239i $$0.462965\pi$$
$$660$$ 0 0
$$661$$ 47.5667 1.85013 0.925065 0.379808i $$-0.124010\pi$$
0.925065 + 0.379808i $$0.124010\pi$$
$$662$$ −13.9572 −0.542463
$$663$$ 0 0
$$664$$ −10.2924 −0.399425
$$665$$ 3.37010 0.130687
$$666$$ 0 0
$$667$$ 8.51791 0.329815
$$668$$ 3.35905 0.129965
$$669$$ 0 0
$$670$$ −7.45367 −0.287960
$$671$$ −5.66286 −0.218612
$$672$$ 0 0
$$673$$ 23.3646 0.900639 0.450319 0.892868i $$-0.351310\pi$$
0.450319 + 0.892868i $$0.351310\pi$$
$$674$$ −30.7465 −1.18431
$$675$$ 0 0
$$676$$ 6.78723 0.261047
$$677$$ 22.7258 0.873423 0.436712 0.899602i $$-0.356143\pi$$
0.436712 + 0.899602i $$0.356143\pi$$
$$678$$ 0 0
$$679$$ 22.6010 0.867346
$$680$$ 9.47453 0.363331
$$681$$ 0 0
$$682$$ 8.25336 0.316038
$$683$$ 6.08405 0.232800 0.116400 0.993202i $$-0.462865\pi$$
0.116400 + 0.993202i $$0.462865\pi$$
$$684$$ 0 0
$$685$$ 21.4534 0.819691
$$686$$ −10.7972 −0.412239
$$687$$ 0 0
$$688$$ −12.2143 −0.465667
$$689$$ 2.27808 0.0867880
$$690$$ 0 0
$$691$$ 10.2313 0.389215 0.194608 0.980881i $$-0.437657\pi$$
0.194608 + 0.980881i $$0.437657\pi$$
$$692$$ −4.35535 −0.165566
$$693$$ 0 0
$$694$$ −11.7214 −0.444937
$$695$$ −8.85639 −0.335942
$$696$$ 0 0
$$697$$ 11.3659 0.430513
$$698$$ −37.5068 −1.41965
$$699$$ 0 0
$$700$$ 1.78608 0.0675076
$$701$$ −20.1154 −0.759749 −0.379875 0.925038i $$-0.624033\pi$$
−0.379875 + 0.925038i $$0.624033\pi$$
$$702$$ 0 0
$$703$$ 2.73402 0.103115
$$704$$ 8.84754 0.333454
$$705$$ 0 0
$$706$$ 13.2312 0.497962
$$707$$ 64.7472 2.43507
$$708$$ 0 0
$$709$$ −18.6469 −0.700301 −0.350150 0.936694i $$-0.613870\pi$$
−0.350150 + 0.936694i $$0.613870\pi$$
$$710$$ −8.67109 −0.325420
$$711$$ 0 0
$$712$$ 19.2194 0.720277
$$713$$ 9.90934 0.371108
$$714$$ 0 0
$$715$$ −0.439786 −0.0164471
$$716$$ 5.95952 0.222718
$$717$$ 0 0
$$718$$ −35.1786 −1.31285
$$719$$ 25.8776 0.965073 0.482537 0.875876i $$-0.339716\pi$$
0.482537 + 0.875876i $$0.339716\pi$$
$$720$$ 0 0
$$721$$ 50.5803 1.88371
$$722$$ −1.21244 −0.0451225
$$723$$ 0 0
$$724$$ −4.38648 −0.163022
$$725$$ −5.85137 −0.217314
$$726$$ 0 0
$$727$$ 48.1800 1.78690 0.893450 0.449164i $$-0.148278\pi$$
0.893450 + 0.449164i $$0.148278\pi$$
$$728$$ 4.54634 0.168499
$$729$$ 0 0
$$730$$ −5.03590 −0.186387
$$731$$ 14.1874 0.524741
$$732$$ 0 0
$$733$$ 5.47157 0.202097 0.101049 0.994881i $$-0.467780\pi$$
0.101049 + 0.994881i $$0.467780\pi$$
$$734$$ −25.6034 −0.945038
$$735$$ 0 0
$$736$$ 4.23733 0.156190
$$737$$ 6.14764 0.226451
$$738$$ 0 0
$$739$$ 11.0454 0.406311 0.203156 0.979146i $$-0.434880\pi$$
0.203156 + 0.979146i $$0.434880\pi$$
$$740$$ 1.44898 0.0532654
$$741$$ 0 0
$$742$$ −21.1657 −0.777016
$$743$$ 16.8303 0.617443 0.308721 0.951153i $$-0.400099\pi$$
0.308721 + 0.951153i $$0.400099\pi$$
$$744$$ 0 0
$$745$$ 2.25215 0.0825122
$$746$$ −29.5145 −1.08060
$$747$$ 0 0
$$748$$ −1.63696 −0.0598532
$$749$$ −25.7445 −0.940683
$$750$$ 0 0
$$751$$ −8.34710 −0.304590 −0.152295 0.988335i $$-0.548666\pi$$
−0.152295 + 0.988335i $$0.548666\pi$$
$$752$$ 0.561007 0.0204578
$$753$$ 0 0
$$754$$ −3.12004 −0.113625
$$755$$ −17.3616 −0.631852
$$756$$ 0 0
$$757$$ 2.18852 0.0795431 0.0397716 0.999209i $$-0.487337\pi$$
0.0397716 + 0.999209i $$0.487337\pi$$
$$758$$ −15.2110 −0.552490
$$759$$ 0 0
$$760$$ −3.06746 −0.111268
$$761$$ −33.4322 −1.21192 −0.605958 0.795496i $$-0.707210\pi$$
−0.605958 + 0.795496i $$0.707210\pi$$
$$762$$ 0 0
$$763$$ −0.154538 −0.00559464
$$764$$ 1.70535 0.0616975
$$765$$ 0 0
$$766$$ 10.0379 0.362685
$$767$$ −4.78188 −0.172664
$$768$$ 0 0
$$769$$ 35.7102 1.28774 0.643871 0.765134i $$-0.277327\pi$$
0.643871 + 0.765134i $$0.277327\pi$$
$$770$$ 4.08605 0.147251
$$771$$ 0 0
$$772$$ −10.0933 −0.363266
$$773$$ −12.1707 −0.437749 −0.218875 0.975753i $$-0.570239\pi$$
−0.218875 + 0.975753i $$0.570239\pi$$
$$774$$ 0 0
$$775$$ −6.80721 −0.244522
$$776$$ −20.5714 −0.738469
$$777$$ 0 0
$$778$$ −9.59940 −0.344155
$$779$$ −3.67980 −0.131842
$$780$$ 0 0
$$781$$ 7.15175 0.255910
$$782$$ 5.45150 0.194945
$$783$$ 0 0
$$784$$ −11.5874 −0.413837
$$785$$ −3.75886 −0.134159
$$786$$ 0 0
$$787$$ 47.9347 1.70869 0.854344 0.519707i $$-0.173959\pi$$
0.854344 + 0.519707i $$0.173959\pi$$
$$788$$ 1.08257 0.0385649
$$789$$ 0 0
$$790$$ −18.7784 −0.668106
$$791$$ 28.4245 1.01066
$$792$$ 0 0
$$793$$ 2.49044 0.0884383
$$794$$ 29.5564 1.04892
$$795$$ 0 0
$$796$$ −2.44428 −0.0866351
$$797$$ −50.1321 −1.77577 −0.887885 0.460066i $$-0.847826\pi$$
−0.887885 + 0.460066i $$0.847826\pi$$
$$798$$ 0 0
$$799$$ −0.651631 −0.0230531
$$800$$ −2.91083 −0.102913
$$801$$ 0 0
$$802$$ 39.5402 1.39621
$$803$$ 4.15351 0.146574
$$804$$ 0 0
$$805$$ 4.90589 0.172910
$$806$$ −3.62971 −0.127851
$$807$$ 0 0
$$808$$ −58.9328 −2.07325
$$809$$ 39.8704 1.40177 0.700884 0.713275i $$-0.252789\pi$$
0.700884 + 0.713275i $$0.252789\pi$$
$$810$$ 0 0
$$811$$ −54.0597 −1.89829 −0.949147 0.314835i $$-0.898051\pi$$
−0.949147 + 0.314835i $$0.898051\pi$$
$$812$$ −10.4510 −0.366759
$$813$$ 0 0
$$814$$ 3.31484 0.116185
$$815$$ −11.8435 −0.414860
$$816$$ 0 0
$$817$$ −4.59330 −0.160699
$$818$$ 44.5487 1.55761
$$819$$ 0 0
$$820$$ −1.95022 −0.0681046
$$821$$ −28.2714 −0.986677 −0.493339 0.869837i $$-0.664224\pi$$
−0.493339 + 0.869837i $$0.664224\pi$$
$$822$$ 0 0
$$823$$ −28.9439 −1.00892 −0.504461 0.863434i $$-0.668309\pi$$
−0.504461 + 0.863434i $$0.668309\pi$$
$$824$$ −46.0381 −1.60381
$$825$$ 0 0
$$826$$ 44.4285 1.54586
$$827$$ 6.79361 0.236237 0.118118 0.993000i $$-0.462314\pi$$
0.118118 + 0.993000i $$0.462314\pi$$
$$828$$ 0 0
$$829$$ 4.79089 0.166394 0.0831972 0.996533i $$-0.473487\pi$$
0.0831972 + 0.996533i $$0.473487\pi$$
$$830$$ 4.06819 0.141209
$$831$$ 0 0
$$832$$ −3.89102 −0.134897
$$833$$ 13.4593 0.466336
$$834$$ 0 0
$$835$$ −6.33806 −0.219338
$$836$$ 0.529980 0.0183297
$$837$$ 0 0
$$838$$ −25.1254 −0.867942
$$839$$ 46.8222 1.61648 0.808241 0.588852i $$-0.200420\pi$$
0.808241 + 0.588852i $$0.200420\pi$$
$$840$$ 0 0
$$841$$ 5.23853 0.180639
$$842$$ 28.2973 0.975190
$$843$$ 0 0
$$844$$ 8.75005 0.301189
$$845$$ −12.8066 −0.440560
$$846$$ 0 0
$$847$$ −3.37010 −0.115798
$$848$$ 13.7744 0.473015
$$849$$ 0 0
$$850$$ −3.74490 −0.128449
$$851$$ 3.97994 0.136431
$$852$$ 0 0
$$853$$ 4.99555 0.171044 0.0855222 0.996336i $$-0.472744\pi$$
0.0855222 + 0.996336i $$0.472744\pi$$
$$854$$ −23.1387 −0.791791
$$855$$ 0 0
$$856$$ 23.4326 0.800910
$$857$$ −3.32386 −0.113541 −0.0567705 0.998387i $$-0.518080\pi$$
−0.0567705 + 0.998387i $$0.518080\pi$$
$$858$$ 0 0
$$859$$ −32.5496 −1.11058 −0.555288 0.831658i $$-0.687392\pi$$
−0.555288 + 0.831658i $$0.687392\pi$$
$$860$$ −2.43436 −0.0830108
$$861$$ 0 0
$$862$$ 2.85802 0.0973446
$$863$$ −7.39803 −0.251832 −0.125916 0.992041i $$-0.540187\pi$$
−0.125916 + 0.992041i $$0.540187\pi$$
$$864$$ 0 0
$$865$$ 8.21796 0.279419
$$866$$ −45.2483 −1.53760
$$867$$ 0 0
$$868$$ −12.1583 −0.412678
$$869$$ 15.4881 0.525397
$$870$$ 0 0
$$871$$ −2.70365 −0.0916096
$$872$$ 0.140660 0.00476335
$$873$$ 0 0
$$874$$ −1.76497 −0.0597009
$$875$$ −3.37010 −0.113930
$$876$$ 0 0
$$877$$ 40.5912 1.37067 0.685333 0.728230i $$-0.259657\pi$$
0.685333 + 0.728230i $$0.259657\pi$$
$$878$$ −31.9630 −1.07870
$$879$$ 0 0
$$880$$ −2.65916 −0.0896403
$$881$$ −16.4133 −0.552978 −0.276489 0.961017i $$-0.589171\pi$$
−0.276489 + 0.961017i $$0.589171\pi$$
$$882$$ 0 0
$$883$$ 41.3719 1.39227 0.696137 0.717909i $$-0.254900\pi$$
0.696137 + 0.717909i $$0.254900\pi$$
$$884$$ 0.719912 0.0242133
$$885$$ 0 0
$$886$$ −8.87064 −0.298015
$$887$$ −11.9068 −0.399791 −0.199896 0.979817i $$-0.564060\pi$$
−0.199896 + 0.979817i $$0.564060\pi$$
$$888$$ 0 0
$$889$$ −5.06321 −0.169814
$$890$$ −7.59665 −0.254640
$$891$$ 0 0
$$892$$ −1.82128 −0.0609808
$$893$$ 0.210971 0.00705988
$$894$$ 0 0
$$895$$ −11.2448 −0.375872
$$896$$ 16.5320 0.552294
$$897$$ 0 0
$$898$$ −1.05796 −0.0353045
$$899$$ 39.8315 1.32846
$$900$$ 0 0
$$901$$ −15.9995 −0.533021
$$902$$ −4.46155 −0.148553
$$903$$ 0 0
$$904$$ −25.8719 −0.860487
$$905$$ 8.27670 0.275127
$$906$$ 0 0
$$907$$ −18.1787 −0.603614 −0.301807 0.953369i $$-0.597590\pi$$
−0.301807 + 0.953369i $$0.597590\pi$$
$$908$$ −5.12365 −0.170034
$$909$$ 0 0
$$910$$ −1.79699 −0.0595696
$$911$$ −32.5463 −1.07831 −0.539154 0.842207i $$-0.681256\pi$$
−0.539154 + 0.842207i $$0.681256\pi$$
$$912$$ 0 0
$$913$$ −3.35537 −0.111046
$$914$$ 2.48246 0.0821125
$$915$$ 0 0
$$916$$ −6.24830 −0.206450
$$917$$ −2.05065 −0.0677183
$$918$$ 0 0
$$919$$ −22.8963 −0.755280 −0.377640 0.925952i $$-0.623264\pi$$
−0.377640 + 0.925952i $$0.623264\pi$$
$$920$$ −4.46533 −0.147218
$$921$$ 0 0
$$922$$ 32.3162 1.06428
$$923$$ −3.14524 −0.103527
$$924$$ 0 0
$$925$$ −2.73402 −0.0898940
$$926$$ 23.6832 0.778280
$$927$$ 0 0
$$928$$ 17.0324 0.559115
$$929$$ 39.9363 1.31027 0.655135 0.755512i $$-0.272612\pi$$
0.655135 + 0.755512i $$0.272612\pi$$
$$930$$ 0 0
$$931$$ −4.35755 −0.142813
$$932$$ 1.06477 0.0348778
$$933$$ 0 0
$$934$$ −35.5281 −1.16252
$$935$$ 3.08872 0.101012
$$936$$ 0 0
$$937$$ 51.4680 1.68139 0.840693 0.541511i $$-0.182148\pi$$
0.840693 + 0.541511i $$0.182148\pi$$
$$938$$ 25.1196 0.820184
$$939$$ 0 0
$$940$$ 0.111811 0.00364686
$$941$$ −56.3979 −1.83852 −0.919259 0.393652i $$-0.871211\pi$$
−0.919259 + 0.393652i $$0.871211\pi$$
$$942$$ 0 0
$$943$$ −5.35672 −0.174439
$$944$$ −28.9136 −0.941057
$$945$$ 0 0
$$946$$ −5.56911 −0.181068
$$947$$ −26.6837 −0.867103 −0.433552 0.901129i $$-0.642740\pi$$
−0.433552 + 0.901129i $$0.642740\pi$$
$$948$$ 0 0
$$949$$ −1.82666 −0.0592957
$$950$$ 1.21244 0.0393369
$$951$$ 0 0
$$952$$ −31.9301 −1.03486
$$953$$ 14.9629 0.484695 0.242348 0.970190i $$-0.422083\pi$$
0.242348 + 0.970190i $$0.422083\pi$$
$$954$$ 0 0
$$955$$ −3.21777 −0.104125
$$956$$ −3.37122 −0.109033
$$957$$ 0 0
$$958$$ −1.17078 −0.0378263
$$959$$ −72.2999 −2.33469
$$960$$ 0 0
$$961$$ 15.3382 0.494779
$$962$$ −1.45782 −0.0470021
$$963$$ 0 0
$$964$$ 15.0395 0.484390
$$965$$ 19.0447 0.613070
$$966$$ 0 0
$$967$$ 19.7493 0.635094 0.317547 0.948242i $$-0.397141\pi$$
0.317547 + 0.948242i $$0.397141\pi$$
$$968$$ 3.06746 0.0985919
$$969$$ 0 0
$$970$$ 8.13104 0.261072
$$971$$ 53.2159 1.70778 0.853889 0.520455i $$-0.174238\pi$$
0.853889 + 0.520455i $$0.174238\pi$$
$$972$$ 0 0
$$973$$ 29.8469 0.956847
$$974$$ 4.07533 0.130582
$$975$$ 0 0
$$976$$ 15.0585 0.482009
$$977$$ 49.9728 1.59877 0.799385 0.600819i $$-0.205159\pi$$
0.799385 + 0.600819i $$0.205159\pi$$
$$978$$ 0 0
$$979$$ 6.26557 0.200249
$$980$$ −2.30941 −0.0737715
$$981$$ 0 0
$$982$$ −39.5053 −1.26067
$$983$$ −29.2283 −0.932239 −0.466119 0.884722i $$-0.654348\pi$$
−0.466119 + 0.884722i $$0.654348\pi$$
$$984$$ 0 0
$$985$$ −2.04266 −0.0650846
$$986$$ 21.9128 0.697846
$$987$$ 0 0
$$988$$ −0.233078 −0.00741519
$$989$$ −6.68651 −0.212619
$$990$$ 0 0
$$991$$ −7.92503 −0.251747 −0.125873 0.992046i $$-0.540173\pi$$
−0.125873 + 0.992046i $$0.540173\pi$$
$$992$$ 19.8147 0.629116
$$993$$ 0 0
$$994$$ 29.2224 0.926879
$$995$$ 4.61202 0.146211
$$996$$ 0 0
$$997$$ 11.9229 0.377601 0.188800 0.982015i $$-0.439540\pi$$
0.188800 + 0.982015i $$0.439540\pi$$
$$998$$ −9.14022 −0.289329
$$999$$ 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 9405.2.a.z.1.2 6
3.2 odd 2 1045.2.a.f.1.5 6
15.14 odd 2 5225.2.a.l.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.f.1.5 6 3.2 odd 2
5225.2.a.l.1.2 6 15.14 odd 2
9405.2.a.z.1.2 6 1.1 even 1 trivial