# Properties

 Label 5225.2.a.s.1.10 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $15$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13$$ x^15 - 5*x^14 - 11*x^13 + 85*x^12 + 6*x^11 - 537*x^10 + 327*x^9 + 1556*x^8 - 1451*x^7 - 2033*x^6 + 2316*x^5 + 927*x^4 - 1295*x^3 + 17*x^2 + 103*x - 13 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.10 Root $$-0.324814$$ of defining polynomial Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +O(q^{10})$$ $$q+0.324814 q^{2} -3.24319 q^{3} -1.89450 q^{4} -1.05343 q^{6} -0.166007 q^{7} -1.26499 q^{8} +7.51826 q^{9} +1.00000 q^{11} +6.14420 q^{12} -6.05331 q^{13} -0.0539213 q^{14} +3.37811 q^{16} -0.875666 q^{17} +2.44204 q^{18} -1.00000 q^{19} +0.538390 q^{21} +0.324814 q^{22} -4.61502 q^{23} +4.10259 q^{24} -1.96620 q^{26} -14.6536 q^{27} +0.314499 q^{28} -1.00709 q^{29} +8.35612 q^{31} +3.62723 q^{32} -3.24319 q^{33} -0.284429 q^{34} -14.2433 q^{36} +7.03003 q^{37} -0.324814 q^{38} +19.6320 q^{39} -8.32057 q^{41} +0.174877 q^{42} -6.62275 q^{43} -1.89450 q^{44} -1.49902 q^{46} +8.43004 q^{47} -10.9558 q^{48} -6.97244 q^{49} +2.83995 q^{51} +11.4680 q^{52} +10.0757 q^{53} -4.75968 q^{54} +0.209996 q^{56} +3.24319 q^{57} -0.327117 q^{58} +4.98050 q^{59} +7.32714 q^{61} +2.71418 q^{62} -1.24808 q^{63} -5.57804 q^{64} -1.05343 q^{66} +0.360331 q^{67} +1.65895 q^{68} +14.9674 q^{69} -7.74042 q^{71} -9.51050 q^{72} +6.39400 q^{73} +2.28345 q^{74} +1.89450 q^{76} -0.166007 q^{77} +6.37676 q^{78} -0.309265 q^{79} +24.9695 q^{81} -2.70264 q^{82} +3.15788 q^{83} -1.01998 q^{84} -2.15116 q^{86} +3.26618 q^{87} -1.26499 q^{88} +13.0206 q^{89} +1.00489 q^{91} +8.74314 q^{92} -27.1005 q^{93} +2.73819 q^{94} -11.7638 q^{96} -10.6444 q^{97} -2.26475 q^{98} +7.51826 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10})$$ 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 15 * q^7 - 15 * q^8 + 19 * q^9 $$15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24} - 17 q^{26} - 13 q^{27} - 30 q^{28} + 5 q^{29} + 6 q^{31} - 40 q^{32} - 4 q^{33} + 17 q^{34} + 28 q^{36} - 13 q^{37} + 5 q^{38} - 22 q^{39} - 30 q^{42} - 36 q^{43} + 17 q^{44} + 13 q^{46} + 6 q^{47} - 14 q^{48} + 16 q^{49} + 4 q^{51} - 50 q^{52} - 9 q^{53} + 9 q^{54} - 18 q^{56} + 4 q^{57} - 2 q^{58} - 7 q^{59} - 2 q^{61} - 11 q^{62} - 39 q^{63} + 17 q^{64} - q^{66} - 35 q^{67} + 18 q^{68} - 9 q^{69} + 13 q^{71} - 68 q^{72} - 2 q^{73} + 13 q^{74} - 17 q^{76} - 15 q^{77} + 10 q^{78} + 6 q^{79} + 11 q^{81} - 14 q^{82} - 30 q^{83} - 6 q^{84} - 25 q^{86} - 19 q^{87} - 15 q^{88} + 55 q^{89} + 26 q^{91} + 18 q^{92} - 14 q^{93} - 22 q^{94} - 17 q^{96} - 28 q^{97} + 22 q^{98} + 19 q^{99}+O(q^{100})$$ 15 * q - 5 * q^2 - 4 * q^3 + 17 * q^4 - q^6 - 15 * q^7 - 15 * q^8 + 19 * q^9 + 15 * q^11 - 9 * q^12 - 13 * q^13 + 9 * q^14 + 21 * q^16 - 11 * q^17 - 16 * q^18 - 15 * q^19 - 5 * q^22 - 10 * q^23 + 17 * q^24 - 17 * q^26 - 13 * q^27 - 30 * q^28 + 5 * q^29 + 6 * q^31 - 40 * q^32 - 4 * q^33 + 17 * q^34 + 28 * q^36 - 13 * q^37 + 5 * q^38 - 22 * q^39 - 30 * q^42 - 36 * q^43 + 17 * q^44 + 13 * q^46 + 6 * q^47 - 14 * q^48 + 16 * q^49 + 4 * q^51 - 50 * q^52 - 9 * q^53 + 9 * q^54 - 18 * q^56 + 4 * q^57 - 2 * q^58 - 7 * q^59 - 2 * q^61 - 11 * q^62 - 39 * q^63 + 17 * q^64 - q^66 - 35 * q^67 + 18 * q^68 - 9 * q^69 + 13 * q^71 - 68 * q^72 - 2 * q^73 + 13 * q^74 - 17 * q^76 - 15 * q^77 + 10 * q^78 + 6 * q^79 + 11 * q^81 - 14 * q^82 - 30 * q^83 - 6 * q^84 - 25 * q^86 - 19 * q^87 - 15 * q^88 + 55 * q^89 + 26 * q^91 + 18 * q^92 - 14 * q^93 - 22 * q^94 - 17 * q^96 - 28 * q^97 + 22 * q^98 + 19 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.324814 0.229678 0.114839 0.993384i $$-0.463365\pi$$
0.114839 + 0.993384i $$0.463365\pi$$
$$3$$ −3.24319 −1.87245 −0.936227 0.351395i $$-0.885708\pi$$
−0.936227 + 0.351395i $$0.885708\pi$$
$$4$$ −1.89450 −0.947248
$$5$$ 0 0
$$6$$ −1.05343 −0.430062
$$7$$ −0.166007 −0.0627446 −0.0313723 0.999508i $$-0.509988\pi$$
−0.0313723 + 0.999508i $$0.509988\pi$$
$$8$$ −1.26499 −0.447240
$$9$$ 7.51826 2.50609
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 6.14420 1.77368
$$13$$ −6.05331 −1.67889 −0.839443 0.543447i $$-0.817119\pi$$
−0.839443 + 0.543447i $$0.817119\pi$$
$$14$$ −0.0539213 −0.0144111
$$15$$ 0 0
$$16$$ 3.37811 0.844527
$$17$$ −0.875666 −0.212380 −0.106190 0.994346i $$-0.533865\pi$$
−0.106190 + 0.994346i $$0.533865\pi$$
$$18$$ 2.44204 0.575593
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.538390 0.117486
$$22$$ 0.324814 0.0692506
$$23$$ −4.61502 −0.962299 −0.481149 0.876639i $$-0.659781\pi$$
−0.481149 + 0.876639i $$0.659781\pi$$
$$24$$ 4.10259 0.837437
$$25$$ 0 0
$$26$$ −1.96620 −0.385604
$$27$$ −14.6536 −2.82008
$$28$$ 0.314499 0.0594347
$$29$$ −1.00709 −0.187012 −0.0935059 0.995619i $$-0.529807\pi$$
−0.0935059 + 0.995619i $$0.529807\pi$$
$$30$$ 0 0
$$31$$ 8.35612 1.50080 0.750402 0.660982i $$-0.229860\pi$$
0.750402 + 0.660982i $$0.229860\pi$$
$$32$$ 3.62723 0.641210
$$33$$ −3.24319 −0.564566
$$34$$ −0.284429 −0.0487791
$$35$$ 0 0
$$36$$ −14.2433 −2.37389
$$37$$ 7.03003 1.15573 0.577865 0.816133i $$-0.303886\pi$$
0.577865 + 0.816133i $$0.303886\pi$$
$$38$$ −0.324814 −0.0526918
$$39$$ 19.6320 3.14364
$$40$$ 0 0
$$41$$ −8.32057 −1.29945 −0.649727 0.760167i $$-0.725117\pi$$
−0.649727 + 0.760167i $$0.725117\pi$$
$$42$$ 0.174877 0.0269841
$$43$$ −6.62275 −1.00996 −0.504980 0.863131i $$-0.668500\pi$$
−0.504980 + 0.863131i $$0.668500\pi$$
$$44$$ −1.89450 −0.285606
$$45$$ 0 0
$$46$$ −1.49902 −0.221019
$$47$$ 8.43004 1.22965 0.614824 0.788664i $$-0.289227\pi$$
0.614824 + 0.788664i $$0.289227\pi$$
$$48$$ −10.9558 −1.58134
$$49$$ −6.97244 −0.996063
$$50$$ 0 0
$$51$$ 2.83995 0.397673
$$52$$ 11.4680 1.59032
$$53$$ 10.0757 1.38401 0.692005 0.721893i $$-0.256728\pi$$
0.692005 + 0.721893i $$0.256728\pi$$
$$54$$ −4.75968 −0.647711
$$55$$ 0 0
$$56$$ 0.209996 0.0280619
$$57$$ 3.24319 0.429571
$$58$$ −0.327117 −0.0429525
$$59$$ 4.98050 0.648406 0.324203 0.945987i $$-0.394904\pi$$
0.324203 + 0.945987i $$0.394904\pi$$
$$60$$ 0 0
$$61$$ 7.32714 0.938144 0.469072 0.883160i $$-0.344588\pi$$
0.469072 + 0.883160i $$0.344588\pi$$
$$62$$ 2.71418 0.344702
$$63$$ −1.24808 −0.157243
$$64$$ −5.57804 −0.697255
$$65$$ 0 0
$$66$$ −1.05343 −0.129669
$$67$$ 0.360331 0.0440214 0.0220107 0.999758i $$-0.492993\pi$$
0.0220107 + 0.999758i $$0.492993\pi$$
$$68$$ 1.65895 0.201177
$$69$$ 14.9674 1.80186
$$70$$ 0 0
$$71$$ −7.74042 −0.918619 −0.459310 0.888276i $$-0.651903\pi$$
−0.459310 + 0.888276i $$0.651903\pi$$
$$72$$ −9.51050 −1.12082
$$73$$ 6.39400 0.748361 0.374180 0.927356i $$-0.377924\pi$$
0.374180 + 0.927356i $$0.377924\pi$$
$$74$$ 2.28345 0.265446
$$75$$ 0 0
$$76$$ 1.89450 0.217314
$$77$$ −0.166007 −0.0189182
$$78$$ 6.37676 0.722025
$$79$$ −0.309265 −0.0347950 −0.0173975 0.999849i $$-0.505538\pi$$
−0.0173975 + 0.999849i $$0.505538\pi$$
$$80$$ 0 0
$$81$$ 24.9695 2.77438
$$82$$ −2.70264 −0.298456
$$83$$ 3.15788 0.346623 0.173311 0.984867i $$-0.444553\pi$$
0.173311 + 0.984867i $$0.444553\pi$$
$$84$$ −1.01998 −0.111289
$$85$$ 0 0
$$86$$ −2.15116 −0.231966
$$87$$ 3.26618 0.350171
$$88$$ −1.26499 −0.134848
$$89$$ 13.0206 1.38018 0.690088 0.723726i $$-0.257572\pi$$
0.690088 + 0.723726i $$0.257572\pi$$
$$90$$ 0 0
$$91$$ 1.00489 0.105341
$$92$$ 8.74314 0.911535
$$93$$ −27.1005 −2.81019
$$94$$ 2.73819 0.282423
$$95$$ 0 0
$$96$$ −11.7638 −1.20064
$$97$$ −10.6444 −1.08077 −0.540385 0.841418i $$-0.681721\pi$$
−0.540385 + 0.841418i $$0.681721\pi$$
$$98$$ −2.26475 −0.228774
$$99$$ 7.51826 0.755614
$$100$$ 0 0
$$101$$ 10.8769 1.08229 0.541145 0.840929i $$-0.317991\pi$$
0.541145 + 0.840929i $$0.317991\pi$$
$$102$$ 0.922455 0.0913367
$$103$$ 5.78563 0.570075 0.285037 0.958516i $$-0.407994\pi$$
0.285037 + 0.958516i $$0.407994\pi$$
$$104$$ 7.65736 0.750866
$$105$$ 0 0
$$106$$ 3.27274 0.317877
$$107$$ 4.75657 0.459835 0.229918 0.973210i $$-0.426154\pi$$
0.229918 + 0.973210i $$0.426154\pi$$
$$108$$ 27.7611 2.67131
$$109$$ 14.0827 1.34888 0.674438 0.738332i $$-0.264386\pi$$
0.674438 + 0.738332i $$0.264386\pi$$
$$110$$ 0 0
$$111$$ −22.7997 −2.16405
$$112$$ −0.560788 −0.0529895
$$113$$ 16.5897 1.56063 0.780314 0.625387i $$-0.215059\pi$$
0.780314 + 0.625387i $$0.215059\pi$$
$$114$$ 1.05343 0.0986630
$$115$$ 0 0
$$116$$ 1.90793 0.177147
$$117$$ −45.5104 −4.20744
$$118$$ 1.61774 0.148925
$$119$$ 0.145366 0.0133257
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.37996 0.215471
$$123$$ 26.9852 2.43317
$$124$$ −15.8306 −1.42163
$$125$$ 0 0
$$126$$ −0.405394 −0.0361154
$$127$$ −16.0352 −1.42290 −0.711448 0.702738i $$-0.751961\pi$$
−0.711448 + 0.702738i $$0.751961\pi$$
$$128$$ −9.06628 −0.801354
$$129$$ 21.4788 1.89110
$$130$$ 0 0
$$131$$ 9.50050 0.830063 0.415031 0.909807i $$-0.363771\pi$$
0.415031 + 0.909807i $$0.363771\pi$$
$$132$$ 6.14420 0.534784
$$133$$ 0.166007 0.0143946
$$134$$ 0.117040 0.0101108
$$135$$ 0 0
$$136$$ 1.10771 0.0949850
$$137$$ −14.1245 −1.20674 −0.603368 0.797463i $$-0.706175\pi$$
−0.603368 + 0.797463i $$0.706175\pi$$
$$138$$ 4.86161 0.413848
$$139$$ 8.78232 0.744907 0.372453 0.928051i $$-0.378517\pi$$
0.372453 + 0.928051i $$0.378517\pi$$
$$140$$ 0 0
$$141$$ −27.3402 −2.30246
$$142$$ −2.51420 −0.210987
$$143$$ −6.05331 −0.506203
$$144$$ 25.3975 2.11646
$$145$$ 0 0
$$146$$ 2.07686 0.171882
$$147$$ 22.6129 1.86508
$$148$$ −13.3184 −1.09476
$$149$$ −6.56136 −0.537528 −0.268764 0.963206i $$-0.586615\pi$$
−0.268764 + 0.963206i $$0.586615\pi$$
$$150$$ 0 0
$$151$$ −15.9594 −1.29876 −0.649380 0.760464i $$-0.724971\pi$$
−0.649380 + 0.760464i $$0.724971\pi$$
$$152$$ 1.26499 0.102604
$$153$$ −6.58349 −0.532244
$$154$$ −0.0539213 −0.00434510
$$155$$ 0 0
$$156$$ −37.1928 −2.97781
$$157$$ 13.8417 1.10469 0.552343 0.833617i $$-0.313734\pi$$
0.552343 + 0.833617i $$0.313734\pi$$
$$158$$ −0.100454 −0.00799166
$$159$$ −32.6775 −2.59150
$$160$$ 0 0
$$161$$ 0.766124 0.0603790
$$162$$ 8.11043 0.637216
$$163$$ 0.673322 0.0527386 0.0263693 0.999652i $$-0.491605\pi$$
0.0263693 + 0.999652i $$0.491605\pi$$
$$164$$ 15.7633 1.23091
$$165$$ 0 0
$$166$$ 1.02572 0.0796117
$$167$$ 9.05874 0.700986 0.350493 0.936565i $$-0.386014\pi$$
0.350493 + 0.936565i $$0.386014\pi$$
$$168$$ −0.681057 −0.0525447
$$169$$ 23.6426 1.81866
$$170$$ 0 0
$$171$$ −7.51826 −0.574936
$$172$$ 12.5468 0.956682
$$173$$ −21.7359 −1.65255 −0.826275 0.563267i $$-0.809544\pi$$
−0.826275 + 0.563267i $$0.809544\pi$$
$$174$$ 1.06090 0.0804267
$$175$$ 0 0
$$176$$ 3.37811 0.254634
$$177$$ −16.1527 −1.21411
$$178$$ 4.22926 0.316996
$$179$$ −1.39240 −0.104073 −0.0520363 0.998645i $$-0.516571\pi$$
−0.0520363 + 0.998645i $$0.516571\pi$$
$$180$$ 0 0
$$181$$ 12.5467 0.932587 0.466294 0.884630i $$-0.345589\pi$$
0.466294 + 0.884630i $$0.345589\pi$$
$$182$$ 0.326402 0.0241945
$$183$$ −23.7633 −1.75663
$$184$$ 5.83794 0.430379
$$185$$ 0 0
$$186$$ −8.80261 −0.645438
$$187$$ −0.875666 −0.0640351
$$188$$ −15.9707 −1.16478
$$189$$ 2.43259 0.176945
$$190$$ 0 0
$$191$$ −14.7271 −1.06561 −0.532807 0.846237i $$-0.678863\pi$$
−0.532807 + 0.846237i $$0.678863\pi$$
$$192$$ 18.0906 1.30558
$$193$$ −11.7487 −0.845689 −0.422845 0.906202i $$-0.638968\pi$$
−0.422845 + 0.906202i $$0.638968\pi$$
$$194$$ −3.45744 −0.248229
$$195$$ 0 0
$$196$$ 13.2093 0.943519
$$197$$ −13.0704 −0.931224 −0.465612 0.884989i $$-0.654166\pi$$
−0.465612 + 0.884989i $$0.654166\pi$$
$$198$$ 2.44204 0.173548
$$199$$ −16.7629 −1.18829 −0.594145 0.804358i $$-0.702509\pi$$
−0.594145 + 0.804358i $$0.702509\pi$$
$$200$$ 0 0
$$201$$ −1.16862 −0.0824281
$$202$$ 3.53297 0.248579
$$203$$ 0.167183 0.0117340
$$204$$ −5.38027 −0.376694
$$205$$ 0 0
$$206$$ 1.87925 0.130934
$$207$$ −34.6969 −2.41160
$$208$$ −20.4487 −1.41786
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −19.4607 −1.33973 −0.669866 0.742482i $$-0.733648\pi$$
−0.669866 + 0.742482i $$0.733648\pi$$
$$212$$ −19.0885 −1.31100
$$213$$ 25.1036 1.72007
$$214$$ 1.54500 0.105614
$$215$$ 0 0
$$216$$ 18.5366 1.26125
$$217$$ −1.38717 −0.0941673
$$218$$ 4.57425 0.309807
$$219$$ −20.7369 −1.40127
$$220$$ 0 0
$$221$$ 5.30068 0.356563
$$222$$ −7.40566 −0.497035
$$223$$ −11.4123 −0.764224 −0.382112 0.924116i $$-0.624803\pi$$
−0.382112 + 0.924116i $$0.624803\pi$$
$$224$$ −0.602144 −0.0402324
$$225$$ 0 0
$$226$$ 5.38857 0.358442
$$227$$ −4.01475 −0.266468 −0.133234 0.991085i $$-0.542536\pi$$
−0.133234 + 0.991085i $$0.542536\pi$$
$$228$$ −6.14420 −0.406910
$$229$$ 26.3140 1.73888 0.869440 0.494038i $$-0.164480\pi$$
0.869440 + 0.494038i $$0.164480\pi$$
$$230$$ 0 0
$$231$$ 0.538390 0.0354235
$$232$$ 1.27395 0.0836392
$$233$$ 14.6801 0.961724 0.480862 0.876796i $$-0.340324\pi$$
0.480862 + 0.876796i $$0.340324\pi$$
$$234$$ −14.7824 −0.966356
$$235$$ 0 0
$$236$$ −9.43554 −0.614202
$$237$$ 1.00300 0.0651521
$$238$$ 0.0472170 0.00306063
$$239$$ −28.5560 −1.84713 −0.923567 0.383437i $$-0.874740\pi$$
−0.923567 + 0.383437i $$0.874740\pi$$
$$240$$ 0 0
$$241$$ −22.3426 −1.43921 −0.719607 0.694381i $$-0.755678\pi$$
−0.719607 + 0.694381i $$0.755678\pi$$
$$242$$ 0.324814 0.0208798
$$243$$ −37.0199 −2.37483
$$244$$ −13.8812 −0.888655
$$245$$ 0 0
$$246$$ 8.76516 0.558846
$$247$$ 6.05331 0.385163
$$248$$ −10.5704 −0.671220
$$249$$ −10.2416 −0.649035
$$250$$ 0 0
$$251$$ 11.5146 0.726795 0.363398 0.931634i $$-0.381617\pi$$
0.363398 + 0.931634i $$0.381617\pi$$
$$252$$ 2.36448 0.148948
$$253$$ −4.61502 −0.290144
$$254$$ −5.20847 −0.326808
$$255$$ 0 0
$$256$$ 8.21122 0.513201
$$257$$ −10.5478 −0.657955 −0.328978 0.944338i $$-0.606704\pi$$
−0.328978 + 0.944338i $$0.606704\pi$$
$$258$$ 6.97662 0.434345
$$259$$ −1.16703 −0.0725158
$$260$$ 0 0
$$261$$ −7.57156 −0.468668
$$262$$ 3.08590 0.190647
$$263$$ −20.3165 −1.25277 −0.626386 0.779513i $$-0.715466\pi$$
−0.626386 + 0.779513i $$0.715466\pi$$
$$264$$ 4.10259 0.252497
$$265$$ 0 0
$$266$$ 0.0539213 0.00330612
$$267$$ −42.2281 −2.58432
$$268$$ −0.682645 −0.0416992
$$269$$ −16.4623 −1.00373 −0.501863 0.864947i $$-0.667352\pi$$
−0.501863 + 0.864947i $$0.667352\pi$$
$$270$$ 0 0
$$271$$ 23.2292 1.41107 0.705537 0.708673i $$-0.250706\pi$$
0.705537 + 0.708673i $$0.250706\pi$$
$$272$$ −2.95809 −0.179361
$$273$$ −3.25904 −0.197246
$$274$$ −4.58783 −0.277161
$$275$$ 0 0
$$276$$ −28.3556 −1.70681
$$277$$ 17.7363 1.06567 0.532837 0.846218i $$-0.321126\pi$$
0.532837 + 0.846218i $$0.321126\pi$$
$$278$$ 2.85262 0.171089
$$279$$ 62.8235 3.76114
$$280$$ 0 0
$$281$$ −25.6914 −1.53262 −0.766310 0.642471i $$-0.777909\pi$$
−0.766310 + 0.642471i $$0.777909\pi$$
$$282$$ −8.88048 −0.528825
$$283$$ −5.17198 −0.307442 −0.153721 0.988114i $$-0.549126\pi$$
−0.153721 + 0.988114i $$0.549126\pi$$
$$284$$ 14.6642 0.870160
$$285$$ 0 0
$$286$$ −1.96620 −0.116264
$$287$$ 1.38127 0.0815337
$$288$$ 27.2705 1.60693
$$289$$ −16.2332 −0.954895
$$290$$ 0 0
$$291$$ 34.5216 2.02369
$$292$$ −12.1134 −0.708883
$$293$$ −24.8835 −1.45371 −0.726854 0.686792i $$-0.759018\pi$$
−0.726854 + 0.686792i $$0.759018\pi$$
$$294$$ 7.34500 0.428369
$$295$$ 0 0
$$296$$ −8.89289 −0.516889
$$297$$ −14.6536 −0.850286
$$298$$ −2.13122 −0.123458
$$299$$ 27.9362 1.61559
$$300$$ 0 0
$$301$$ 1.09942 0.0633695
$$302$$ −5.18384 −0.298297
$$303$$ −35.2758 −2.02654
$$304$$ −3.37811 −0.193748
$$305$$ 0 0
$$306$$ −2.13841 −0.122245
$$307$$ 1.37200 0.0783042 0.0391521 0.999233i $$-0.487534\pi$$
0.0391521 + 0.999233i $$0.487534\pi$$
$$308$$ 0.314499 0.0179202
$$309$$ −18.7639 −1.06744
$$310$$ 0 0
$$311$$ 1.75534 0.0995360 0.0497680 0.998761i $$-0.484152\pi$$
0.0497680 + 0.998761i $$0.484152\pi$$
$$312$$ −24.8343 −1.40596
$$313$$ 18.5787 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$314$$ 4.49597 0.253722
$$315$$ 0 0
$$316$$ 0.585901 0.0329595
$$317$$ −4.00886 −0.225160 −0.112580 0.993643i $$-0.535911\pi$$
−0.112580 + 0.993643i $$0.535911\pi$$
$$318$$ −10.6141 −0.595210
$$319$$ −1.00709 −0.0563862
$$320$$ 0 0
$$321$$ −15.4265 −0.861021
$$322$$ 0.248848 0.0138677
$$323$$ 0.875666 0.0487234
$$324$$ −47.3045 −2.62803
$$325$$ 0 0
$$326$$ 0.218704 0.0121129
$$327$$ −45.6728 −2.52571
$$328$$ 10.5254 0.581169
$$329$$ −1.39944 −0.0771537
$$330$$ 0 0
$$331$$ 6.40426 0.352010 0.176005 0.984389i $$-0.443682\pi$$
0.176005 + 0.984389i $$0.443682\pi$$
$$332$$ −5.98260 −0.328338
$$333$$ 52.8536 2.89636
$$334$$ 2.94240 0.161001
$$335$$ 0 0
$$336$$ 1.81874 0.0992204
$$337$$ −26.4195 −1.43916 −0.719582 0.694407i $$-0.755667\pi$$
−0.719582 + 0.694407i $$0.755667\pi$$
$$338$$ 7.67945 0.417707
$$339$$ −53.8035 −2.92221
$$340$$ 0 0
$$341$$ 8.35612 0.452509
$$342$$ −2.44204 −0.132050
$$343$$ 2.31952 0.125242
$$344$$ 8.37769 0.451695
$$345$$ 0 0
$$346$$ −7.06013 −0.379555
$$347$$ 6.14023 0.329625 0.164812 0.986325i $$-0.447298\pi$$
0.164812 + 0.986325i $$0.447298\pi$$
$$348$$ −6.18776 −0.331699
$$349$$ 17.0701 0.913742 0.456871 0.889533i $$-0.348970\pi$$
0.456871 + 0.889533i $$0.348970\pi$$
$$350$$ 0 0
$$351$$ 88.7026 4.73459
$$352$$ 3.62723 0.193332
$$353$$ 24.1803 1.28699 0.643495 0.765451i $$-0.277484\pi$$
0.643495 + 0.765451i $$0.277484\pi$$
$$354$$ −5.24662 −0.278855
$$355$$ 0 0
$$356$$ −24.6674 −1.30737
$$357$$ −0.471450 −0.0249518
$$358$$ −0.452270 −0.0239032
$$359$$ −27.5984 −1.45659 −0.728294 0.685265i $$-0.759687\pi$$
−0.728294 + 0.685265i $$0.759687\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.07534 0.214195
$$363$$ −3.24319 −0.170223
$$364$$ −1.90376 −0.0997841
$$365$$ 0 0
$$366$$ −7.71865 −0.403460
$$367$$ −0.102872 −0.00536987 −0.00268494 0.999996i $$-0.500855\pi$$
−0.00268494 + 0.999996i $$0.500855\pi$$
$$368$$ −15.5900 −0.812687
$$369$$ −62.5562 −3.25655
$$370$$ 0 0
$$371$$ −1.67264 −0.0868392
$$372$$ 51.3417 2.66194
$$373$$ −11.1255 −0.576055 −0.288027 0.957622i $$-0.592999\pi$$
−0.288027 + 0.957622i $$0.592999\pi$$
$$374$$ −0.284429 −0.0147075
$$375$$ 0 0
$$376$$ −10.6639 −0.549948
$$377$$ 6.09623 0.313972
$$378$$ 0.790138 0.0406403
$$379$$ −19.2468 −0.988643 −0.494321 0.869279i $$-0.664583\pi$$
−0.494321 + 0.869279i $$0.664583\pi$$
$$380$$ 0 0
$$381$$ 52.0053 2.66431
$$382$$ −4.78356 −0.244748
$$383$$ −8.55556 −0.437169 −0.218584 0.975818i $$-0.570144\pi$$
−0.218584 + 0.975818i $$0.570144\pi$$
$$384$$ 29.4037 1.50050
$$385$$ 0 0
$$386$$ −3.81614 −0.194236
$$387$$ −49.7916 −2.53105
$$388$$ 20.1657 1.02376
$$389$$ 16.7276 0.848125 0.424063 0.905633i $$-0.360604\pi$$
0.424063 + 0.905633i $$0.360604\pi$$
$$390$$ 0 0
$$391$$ 4.04122 0.204373
$$392$$ 8.82005 0.445480
$$393$$ −30.8119 −1.55425
$$394$$ −4.24543 −0.213882
$$395$$ 0 0
$$396$$ −14.2433 −0.715753
$$397$$ 37.0868 1.86133 0.930667 0.365867i $$-0.119228\pi$$
0.930667 + 0.365867i $$0.119228\pi$$
$$398$$ −5.44482 −0.272924
$$399$$ −0.538390 −0.0269532
$$400$$ 0 0
$$401$$ −4.15586 −0.207534 −0.103767 0.994602i $$-0.533090\pi$$
−0.103767 + 0.994602i $$0.533090\pi$$
$$402$$ −0.379584 −0.0189319
$$403$$ −50.5822 −2.51968
$$404$$ −20.6062 −1.02520
$$405$$ 0 0
$$406$$ 0.0543035 0.00269504
$$407$$ 7.03003 0.348466
$$408$$ −3.59250 −0.177855
$$409$$ −19.6949 −0.973851 −0.486926 0.873443i $$-0.661882\pi$$
−0.486926 + 0.873443i $$0.661882\pi$$
$$410$$ 0 0
$$411$$ 45.8083 2.25956
$$412$$ −10.9608 −0.540002
$$413$$ −0.826796 −0.0406840
$$414$$ −11.2701 −0.553893
$$415$$ 0 0
$$416$$ −21.9568 −1.07652
$$417$$ −28.4827 −1.39480
$$418$$ −0.324814 −0.0158872
$$419$$ −4.19043 −0.204716 −0.102358 0.994748i $$-0.532639\pi$$
−0.102358 + 0.994748i $$0.532639\pi$$
$$420$$ 0 0
$$421$$ 22.5663 1.09981 0.549907 0.835226i $$-0.314663\pi$$
0.549907 + 0.835226i $$0.314663\pi$$
$$422$$ −6.32111 −0.307707
$$423$$ 63.3792 3.08160
$$424$$ −12.7457 −0.618985
$$425$$ 0 0
$$426$$ 8.15401 0.395063
$$427$$ −1.21635 −0.0588635
$$428$$ −9.01131 −0.435578
$$429$$ 19.6320 0.947843
$$430$$ 0 0
$$431$$ −18.3524 −0.884002 −0.442001 0.897014i $$-0.645731\pi$$
−0.442001 + 0.897014i $$0.645731\pi$$
$$432$$ −49.5013 −2.38163
$$433$$ 1.78082 0.0855810 0.0427905 0.999084i $$-0.486375\pi$$
0.0427905 + 0.999084i $$0.486375\pi$$
$$434$$ −0.450572 −0.0216282
$$435$$ 0 0
$$436$$ −26.6796 −1.27772
$$437$$ 4.61502 0.220766
$$438$$ −6.73564 −0.321841
$$439$$ −26.8306 −1.28055 −0.640277 0.768144i $$-0.721180\pi$$
−0.640277 + 0.768144i $$0.721180\pi$$
$$440$$ 0 0
$$441$$ −52.4206 −2.49622
$$442$$ 1.72174 0.0818946
$$443$$ 15.4460 0.733862 0.366931 0.930248i $$-0.380409\pi$$
0.366931 + 0.930248i $$0.380409\pi$$
$$444$$ 43.1939 2.04989
$$445$$ 0 0
$$446$$ −3.70687 −0.175525
$$447$$ 21.2797 1.00650
$$448$$ 0.925991 0.0437490
$$449$$ −14.7640 −0.696758 −0.348379 0.937354i $$-0.613268\pi$$
−0.348379 + 0.937354i $$0.613268\pi$$
$$450$$ 0 0
$$451$$ −8.32057 −0.391800
$$452$$ −31.4291 −1.47830
$$453$$ 51.7594 2.43187
$$454$$ −1.30405 −0.0612020
$$455$$ 0 0
$$456$$ −4.10259 −0.192121
$$457$$ 16.2215 0.758811 0.379406 0.925230i $$-0.376128\pi$$
0.379406 + 0.925230i $$0.376128\pi$$
$$458$$ 8.54717 0.399383
$$459$$ 12.8316 0.598929
$$460$$ 0 0
$$461$$ 13.4764 0.627659 0.313830 0.949479i $$-0.398388\pi$$
0.313830 + 0.949479i $$0.398388\pi$$
$$462$$ 0.174877 0.00813600
$$463$$ 28.6461 1.33130 0.665648 0.746266i $$-0.268155\pi$$
0.665648 + 0.746266i $$0.268155\pi$$
$$464$$ −3.40205 −0.157936
$$465$$ 0 0
$$466$$ 4.76829 0.220887
$$467$$ −17.0725 −0.790023 −0.395011 0.918676i $$-0.629259\pi$$
−0.395011 + 0.918676i $$0.629259\pi$$
$$468$$ 86.2192 3.98549
$$469$$ −0.0598173 −0.00276211
$$470$$ 0 0
$$471$$ −44.8911 −2.06847
$$472$$ −6.30027 −0.289993
$$473$$ −6.62275 −0.304514
$$474$$ 0.325790 0.0149640
$$475$$ 0 0
$$476$$ −0.275396 −0.0126228
$$477$$ 75.7521 3.46845
$$478$$ −9.27539 −0.424246
$$479$$ 28.3507 1.29538 0.647689 0.761905i $$-0.275736\pi$$
0.647689 + 0.761905i $$0.275736\pi$$
$$480$$ 0 0
$$481$$ −42.5550 −1.94034
$$482$$ −7.25720 −0.330556
$$483$$ −2.48468 −0.113057
$$484$$ −1.89450 −0.0861134
$$485$$ 0 0
$$486$$ −12.0246 −0.545447
$$487$$ 11.4047 0.516798 0.258399 0.966038i $$-0.416805\pi$$
0.258399 + 0.966038i $$0.416805\pi$$
$$488$$ −9.26874 −0.419576
$$489$$ −2.18371 −0.0987507
$$490$$ 0 0
$$491$$ −29.9154 −1.35006 −0.675032 0.737788i $$-0.735870\pi$$
−0.675032 + 0.737788i $$0.735870\pi$$
$$492$$ −51.1233 −2.30482
$$493$$ 0.881874 0.0397176
$$494$$ 1.96620 0.0884636
$$495$$ 0 0
$$496$$ 28.2279 1.26747
$$497$$ 1.28496 0.0576384
$$498$$ −3.32662 −0.149069
$$499$$ −37.9248 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$500$$ 0 0
$$501$$ −29.3792 −1.31256
$$502$$ 3.74010 0.166929
$$503$$ −2.74270 −0.122291 −0.0611456 0.998129i $$-0.519475\pi$$
−0.0611456 + 0.998129i $$0.519475\pi$$
$$504$$ 1.57881 0.0703256
$$505$$ 0 0
$$506$$ −1.49902 −0.0666397
$$507$$ −76.6774 −3.40536
$$508$$ 30.3787 1.34784
$$509$$ 0.635877 0.0281847 0.0140924 0.999901i $$-0.495514\pi$$
0.0140924 + 0.999901i $$0.495514\pi$$
$$510$$ 0 0
$$511$$ −1.06145 −0.0469556
$$512$$ 20.7997 0.919225
$$513$$ 14.6536 0.646971
$$514$$ −3.42608 −0.151118
$$515$$ 0 0
$$516$$ −40.6915 −1.79134
$$517$$ 8.43004 0.370753
$$518$$ −0.379068 −0.0166553
$$519$$ 70.4936 3.09433
$$520$$ 0 0
$$521$$ 28.8718 1.26490 0.632449 0.774602i $$-0.282050\pi$$
0.632449 + 0.774602i $$0.282050\pi$$
$$522$$ −2.45935 −0.107643
$$523$$ 20.2611 0.885956 0.442978 0.896532i $$-0.353922\pi$$
0.442978 + 0.896532i $$0.353922\pi$$
$$524$$ −17.9987 −0.786275
$$525$$ 0 0
$$526$$ −6.59910 −0.287734
$$527$$ −7.31717 −0.318741
$$528$$ −10.9558 −0.476791
$$529$$ −1.70157 −0.0739814
$$530$$ 0 0
$$531$$ 37.4447 1.62496
$$532$$ −0.314499 −0.0136352
$$533$$ 50.3670 2.18164
$$534$$ −13.7163 −0.593561
$$535$$ 0 0
$$536$$ −0.455814 −0.0196882
$$537$$ 4.51580 0.194871
$$538$$ −5.34720 −0.230534
$$539$$ −6.97244 −0.300324
$$540$$ 0 0
$$541$$ 25.4105 1.09248 0.546242 0.837628i $$-0.316058\pi$$
0.546242 + 0.837628i $$0.316058\pi$$
$$542$$ 7.54518 0.324093
$$543$$ −40.6912 −1.74623
$$544$$ −3.17624 −0.136180
$$545$$ 0 0
$$546$$ −1.05858 −0.0453032
$$547$$ 0.458489 0.0196036 0.00980178 0.999952i $$-0.496880\pi$$
0.00980178 + 0.999952i $$0.496880\pi$$
$$548$$ 26.7588 1.14308
$$549$$ 55.0874 2.35107
$$550$$ 0 0
$$551$$ 1.00709 0.0429034
$$552$$ −18.9335 −0.805865
$$553$$ 0.0513400 0.00218320
$$554$$ 5.76101 0.244762
$$555$$ 0 0
$$556$$ −16.6381 −0.705611
$$557$$ 43.5515 1.84534 0.922669 0.385593i $$-0.126003\pi$$
0.922669 + 0.385593i $$0.126003\pi$$
$$558$$ 20.4059 0.863852
$$559$$ 40.0896 1.69561
$$560$$ 0 0
$$561$$ 2.83995 0.119903
$$562$$ −8.34492 −0.352009
$$563$$ −46.9523 −1.97880 −0.989401 0.145208i $$-0.953615\pi$$
−0.989401 + 0.145208i $$0.953615\pi$$
$$564$$ 51.7959 2.18100
$$565$$ 0 0
$$566$$ −1.67993 −0.0706128
$$567$$ −4.14509 −0.174078
$$568$$ 9.79153 0.410844
$$569$$ 30.2122 1.26656 0.633282 0.773922i $$-0.281708\pi$$
0.633282 + 0.773922i $$0.281708\pi$$
$$570$$ 0 0
$$571$$ 23.2284 0.972077 0.486038 0.873938i $$-0.338442\pi$$
0.486038 + 0.873938i $$0.338442\pi$$
$$572$$ 11.4680 0.479500
$$573$$ 47.7627 1.99531
$$574$$ 0.448656 0.0187265
$$575$$ 0 0
$$576$$ −41.9371 −1.74738
$$577$$ −9.07629 −0.377851 −0.188926 0.981991i $$-0.560500\pi$$
−0.188926 + 0.981991i $$0.560500\pi$$
$$578$$ −5.27277 −0.219318
$$579$$ 38.1032 1.58351
$$580$$ 0 0
$$581$$ −0.524229 −0.0217487
$$582$$ 11.2131 0.464798
$$583$$ 10.0757 0.417295
$$584$$ −8.08832 −0.334697
$$585$$ 0 0
$$586$$ −8.08250 −0.333885
$$587$$ 36.7331 1.51614 0.758069 0.652174i $$-0.226143\pi$$
0.758069 + 0.652174i $$0.226143\pi$$
$$588$$ −42.8401 −1.76670
$$589$$ −8.35612 −0.344308
$$590$$ 0 0
$$591$$ 42.3896 1.74368
$$592$$ 23.7482 0.976044
$$593$$ 28.5431 1.17213 0.586063 0.810266i $$-0.300677\pi$$
0.586063 + 0.810266i $$0.300677\pi$$
$$594$$ −4.75968 −0.195292
$$595$$ 0 0
$$596$$ 12.4305 0.509172
$$597$$ 54.3652 2.22502
$$598$$ 9.07406 0.371066
$$599$$ −45.9055 −1.87565 −0.937825 0.347109i $$-0.887164\pi$$
−0.937825 + 0.347109i $$0.887164\pi$$
$$600$$ 0 0
$$601$$ 34.0331 1.38824 0.694120 0.719860i $$-0.255794\pi$$
0.694120 + 0.719860i $$0.255794\pi$$
$$602$$ 0.357107 0.0145546
$$603$$ 2.70906 0.110321
$$604$$ 30.2351 1.23025
$$605$$ 0 0
$$606$$ −11.4581 −0.465452
$$607$$ −3.84641 −0.156121 −0.0780606 0.996949i $$-0.524873\pi$$
−0.0780606 + 0.996949i $$0.524873\pi$$
$$608$$ −3.62723 −0.147104
$$609$$ −0.542207 −0.0219713
$$610$$ 0 0
$$611$$ −51.0297 −2.06444
$$612$$ 12.4724 0.504167
$$613$$ −17.5232 −0.707754 −0.353877 0.935292i $$-0.615137\pi$$
−0.353877 + 0.935292i $$0.615137\pi$$
$$614$$ 0.445645 0.0179848
$$615$$ 0 0
$$616$$ 0.209996 0.00846098
$$617$$ 14.0815 0.566900 0.283450 0.958987i $$-0.408521\pi$$
0.283450 + 0.958987i $$0.408521\pi$$
$$618$$ −6.09477 −0.245167
$$619$$ −23.1027 −0.928577 −0.464288 0.885684i $$-0.653690\pi$$
−0.464288 + 0.885684i $$0.653690\pi$$
$$620$$ 0 0
$$621$$ 67.6265 2.71376
$$622$$ 0.570158 0.0228613
$$623$$ −2.16150 −0.0865985
$$624$$ 66.3191 2.65489
$$625$$ 0 0
$$626$$ 6.03462 0.241192
$$627$$ 3.24319 0.129520
$$628$$ −26.2230 −1.04641
$$629$$ −6.15596 −0.245454
$$630$$ 0 0
$$631$$ 3.60839 0.143648 0.0718239 0.997417i $$-0.477118\pi$$
0.0718239 + 0.997417i $$0.477118\pi$$
$$632$$ 0.391216 0.0155617
$$633$$ 63.1148 2.50859
$$634$$ −1.30213 −0.0517144
$$635$$ 0 0
$$636$$ 61.9074 2.45479
$$637$$ 42.2064 1.67228
$$638$$ −0.327117 −0.0129507
$$639$$ −58.1945 −2.30214
$$640$$ 0 0
$$641$$ −17.5350 −0.692589 −0.346294 0.938126i $$-0.612560\pi$$
−0.346294 + 0.938126i $$0.612560\pi$$
$$642$$ −5.01073 −0.197758
$$643$$ −19.6051 −0.773151 −0.386575 0.922258i $$-0.626342\pi$$
−0.386575 + 0.922258i $$0.626342\pi$$
$$644$$ −1.45142 −0.0571939
$$645$$ 0 0
$$646$$ 0.284429 0.0111907
$$647$$ −41.2895 −1.62326 −0.811629 0.584173i $$-0.801419\pi$$
−0.811629 + 0.584173i $$0.801419\pi$$
$$648$$ −31.5860 −1.24082
$$649$$ 4.98050 0.195502
$$650$$ 0 0
$$651$$ 4.49885 0.176324
$$652$$ −1.27561 −0.0499566
$$653$$ 8.64148 0.338167 0.169084 0.985602i $$-0.445919\pi$$
0.169084 + 0.985602i $$0.445919\pi$$
$$654$$ −14.8351 −0.580100
$$655$$ 0 0
$$656$$ −28.1078 −1.09742
$$657$$ 48.0717 1.87546
$$658$$ −0.454558 −0.0177205
$$659$$ −2.34694 −0.0914237 −0.0457118 0.998955i $$-0.514556\pi$$
−0.0457118 + 0.998955i $$0.514556\pi$$
$$660$$ 0 0
$$661$$ −37.8564 −1.47244 −0.736222 0.676740i $$-0.763392\pi$$
−0.736222 + 0.676740i $$0.763392\pi$$
$$662$$ 2.08019 0.0808490
$$663$$ −17.1911 −0.667647
$$664$$ −3.99468 −0.155024
$$665$$ 0 0
$$666$$ 17.1676 0.665230
$$667$$ 4.64774 0.179961
$$668$$ −17.1617 −0.664008
$$669$$ 37.0122 1.43097
$$670$$ 0 0
$$671$$ 7.32714 0.282861
$$672$$ 1.95287 0.0753334
$$673$$ −32.1381 −1.23883 −0.619417 0.785062i $$-0.712631\pi$$
−0.619417 + 0.785062i $$0.712631\pi$$
$$674$$ −8.58144 −0.330545
$$675$$ 0 0
$$676$$ −44.7908 −1.72272
$$677$$ −46.2653 −1.77812 −0.889060 0.457791i $$-0.848641\pi$$
−0.889060 + 0.457791i $$0.848641\pi$$
$$678$$ −17.4761 −0.671167
$$679$$ 1.76703 0.0678125
$$680$$ 0 0
$$681$$ 13.0206 0.498950
$$682$$ 2.71418 0.103931
$$683$$ 3.51505 0.134500 0.0672498 0.997736i $$-0.478578\pi$$
0.0672498 + 0.997736i $$0.478578\pi$$
$$684$$ 14.2433 0.544607
$$685$$ 0 0
$$686$$ 0.753412 0.0287654
$$687$$ −85.3414 −3.25598
$$688$$ −22.3724 −0.852938
$$689$$ −60.9917 −2.32360
$$690$$ 0 0
$$691$$ 6.47549 0.246339 0.123170 0.992386i $$-0.460694\pi$$
0.123170 + 0.992386i $$0.460694\pi$$
$$692$$ 41.1786 1.56537
$$693$$ −1.24808 −0.0474107
$$694$$ 1.99443 0.0757077
$$695$$ 0 0
$$696$$ −4.13167 −0.156611
$$697$$ 7.28604 0.275979
$$698$$ 5.54461 0.209867
$$699$$ −47.6102 −1.80078
$$700$$ 0 0
$$701$$ −14.8438 −0.560643 −0.280321 0.959906i $$-0.590441\pi$$
−0.280321 + 0.959906i $$0.590441\pi$$
$$702$$ 28.8118 1.08743
$$703$$ −7.03003 −0.265142
$$704$$ −5.57804 −0.210230
$$705$$ 0 0
$$706$$ 7.85411 0.295593
$$707$$ −1.80564 −0.0679079
$$708$$ 30.6012 1.15006
$$709$$ −25.8169 −0.969572 −0.484786 0.874633i $$-0.661103\pi$$
−0.484786 + 0.874633i $$0.661103\pi$$
$$710$$ 0 0
$$711$$ −2.32514 −0.0871994
$$712$$ −16.4708 −0.617270
$$713$$ −38.5637 −1.44422
$$714$$ −0.153134 −0.00573088
$$715$$ 0 0
$$716$$ 2.63789 0.0985826
$$717$$ 92.6124 3.45868
$$718$$ −8.96435 −0.334547
$$719$$ 25.3406 0.945044 0.472522 0.881319i $$-0.343344\pi$$
0.472522 + 0.881319i $$0.343344\pi$$
$$720$$ 0 0
$$721$$ −0.960452 −0.0357691
$$722$$ 0.324814 0.0120883
$$723$$ 72.4613 2.69486
$$724$$ −23.7696 −0.883391
$$725$$ 0 0
$$726$$ −1.05343 −0.0390965
$$727$$ −3.83993 −0.142415 −0.0712075 0.997462i $$-0.522685\pi$$
−0.0712075 + 0.997462i $$0.522685\pi$$
$$728$$ −1.27117 −0.0471128
$$729$$ 45.1542 1.67238
$$730$$ 0 0
$$731$$ 5.79932 0.214496
$$732$$ 45.0195 1.66397
$$733$$ −19.8037 −0.731466 −0.365733 0.930720i $$-0.619182\pi$$
−0.365733 + 0.930720i $$0.619182\pi$$
$$734$$ −0.0334142 −0.00123334
$$735$$ 0 0
$$736$$ −16.7397 −0.617035
$$737$$ 0.360331 0.0132730
$$738$$ −20.3191 −0.747958
$$739$$ −43.0878 −1.58501 −0.792506 0.609864i $$-0.791224\pi$$
−0.792506 + 0.609864i $$0.791224\pi$$
$$740$$ 0 0
$$741$$ −19.6320 −0.721200
$$742$$ −0.543297 −0.0199451
$$743$$ 16.0472 0.588714 0.294357 0.955696i $$-0.404895\pi$$
0.294357 + 0.955696i $$0.404895\pi$$
$$744$$ 34.2817 1.25683
$$745$$ 0 0
$$746$$ −3.61371 −0.132307
$$747$$ 23.7418 0.868667
$$748$$ 1.65895 0.0606571
$$749$$ −0.789622 −0.0288522
$$750$$ 0 0
$$751$$ −34.8124 −1.27032 −0.635161 0.772380i $$-0.719066\pi$$
−0.635161 + 0.772380i $$0.719066\pi$$
$$752$$ 28.4776 1.03847
$$753$$ −37.3440 −1.36089
$$754$$ 1.98014 0.0721124
$$755$$ 0 0
$$756$$ −4.60853 −0.167611
$$757$$ 31.5794 1.14777 0.573886 0.818935i $$-0.305435\pi$$
0.573886 + 0.818935i $$0.305435\pi$$
$$758$$ −6.25164 −0.227070
$$759$$ 14.9674 0.543281
$$760$$ 0 0
$$761$$ −26.0445 −0.944113 −0.472056 0.881568i $$-0.656488\pi$$
−0.472056 + 0.881568i $$0.656488\pi$$
$$762$$ 16.8920 0.611934
$$763$$ −2.33782 −0.0846346
$$764$$ 27.9004 1.00940
$$765$$ 0 0
$$766$$ −2.77897 −0.100408
$$767$$ −30.1485 −1.08860
$$768$$ −26.6305 −0.960946
$$769$$ −1.27892 −0.0461189 −0.0230595 0.999734i $$-0.507341\pi$$
−0.0230595 + 0.999734i $$0.507341\pi$$
$$770$$ 0 0
$$771$$ 34.2086 1.23199
$$772$$ 22.2578 0.801077
$$773$$ 43.3371 1.55873 0.779364 0.626571i $$-0.215542\pi$$
0.779364 + 0.626571i $$0.215542\pi$$
$$774$$ −16.1730 −0.581326
$$775$$ 0 0
$$776$$ 13.4650 0.483364
$$777$$ 3.78490 0.135782
$$778$$ 5.43337 0.194796
$$779$$ 8.32057 0.298115
$$780$$ 0 0
$$781$$ −7.74042 −0.276974
$$782$$ 1.31264 0.0469401
$$783$$ 14.7574 0.527388
$$784$$ −23.5537 −0.841202
$$785$$ 0 0
$$786$$ −10.0081 −0.356978
$$787$$ −15.9911 −0.570022 −0.285011 0.958524i $$-0.591997\pi$$
−0.285011 + 0.958524i $$0.591997\pi$$
$$788$$ 24.7617 0.882100
$$789$$ 65.8903 2.34576
$$790$$ 0 0
$$791$$ −2.75400 −0.0979210
$$792$$ −9.51050 −0.337941
$$793$$ −44.3535 −1.57504
$$794$$ 12.0463 0.427508
$$795$$ 0 0
$$796$$ 31.7572 1.12561
$$797$$ −28.8110 −1.02054 −0.510269 0.860015i $$-0.670454\pi$$
−0.510269 + 0.860015i $$0.670454\pi$$
$$798$$ −0.174877 −0.00619057
$$799$$ −7.38190 −0.261153
$$800$$ 0 0
$$801$$ 97.8919 3.45884
$$802$$ −1.34988 −0.0476659
$$803$$ 6.39400 0.225639
$$804$$ 2.21395 0.0780799
$$805$$ 0 0
$$806$$ −16.4298 −0.578715
$$807$$ 53.3904 1.87943
$$808$$ −13.7591 −0.484044
$$809$$ 47.1962 1.65933 0.829666 0.558261i $$-0.188531\pi$$
0.829666 + 0.558261i $$0.188531\pi$$
$$810$$ 0 0
$$811$$ 21.4242 0.752304 0.376152 0.926558i $$-0.377247\pi$$
0.376152 + 0.926558i $$0.377247\pi$$
$$812$$ −0.316728 −0.0111150
$$813$$ −75.3367 −2.64217
$$814$$ 2.28345 0.0800349
$$815$$ 0 0
$$816$$ 9.59365 0.335845
$$817$$ 6.62275 0.231701
$$818$$ −6.39718 −0.223672
$$819$$ 7.55502 0.263994
$$820$$ 0 0
$$821$$ 9.70806 0.338814 0.169407 0.985546i $$-0.445815\pi$$
0.169407 + 0.985546i $$0.445815\pi$$
$$822$$ 14.8792 0.518971
$$823$$ −41.7049 −1.45374 −0.726871 0.686774i $$-0.759026\pi$$
−0.726871 + 0.686774i $$0.759026\pi$$
$$824$$ −7.31874 −0.254960
$$825$$ 0 0
$$826$$ −0.268555 −0.00934422
$$827$$ 19.1359 0.665420 0.332710 0.943029i $$-0.392037\pi$$
0.332710 + 0.943029i $$0.392037\pi$$
$$828$$ 65.7332 2.28439
$$829$$ −53.7809 −1.86789 −0.933943 0.357421i $$-0.883656\pi$$
−0.933943 + 0.357421i $$0.883656\pi$$
$$830$$ 0 0
$$831$$ −57.5222 −1.99542
$$832$$ 33.7656 1.17061
$$833$$ 6.10553 0.211544
$$834$$ −9.25158 −0.320356
$$835$$ 0 0
$$836$$ 1.89450 0.0655225
$$837$$ −122.447 −4.23238
$$838$$ −1.36111 −0.0470188
$$839$$ −11.9762 −0.413464 −0.206732 0.978398i $$-0.566283\pi$$
−0.206732 + 0.978398i $$0.566283\pi$$
$$840$$ 0 0
$$841$$ −27.9858 −0.965027
$$842$$ 7.32985 0.252603
$$843$$ 83.3220 2.86976
$$844$$ 36.8683 1.26906
$$845$$ 0 0
$$846$$ 20.5865 0.707777
$$847$$ −0.166007 −0.00570405
$$848$$ 34.0369 1.16883
$$849$$ 16.7737 0.575672
$$850$$ 0 0
$$851$$ −32.4437 −1.11216
$$852$$ −47.5587 −1.62934
$$853$$ −36.1717 −1.23850 −0.619248 0.785196i $$-0.712562\pi$$
−0.619248 + 0.785196i $$0.712562\pi$$
$$854$$ −0.395089 −0.0135197
$$855$$ 0 0
$$856$$ −6.01700 −0.205657
$$857$$ 13.2431 0.452375 0.226187 0.974084i $$-0.427374\pi$$
0.226187 + 0.974084i $$0.427374\pi$$
$$858$$ 6.37676 0.217699
$$859$$ −18.5582 −0.633198 −0.316599 0.948560i $$-0.602541\pi$$
−0.316599 + 0.948560i $$0.602541\pi$$
$$860$$ 0 0
$$861$$ −4.47971 −0.152668
$$862$$ −5.96111 −0.203036
$$863$$ 5.72788 0.194979 0.0974896 0.995237i $$-0.468919\pi$$
0.0974896 + 0.995237i $$0.468919\pi$$
$$864$$ −53.1518 −1.80826
$$865$$ 0 0
$$866$$ 0.578437 0.0196561
$$867$$ 52.6473 1.78800
$$868$$ 2.62799 0.0891997
$$869$$ −0.309265 −0.0104911
$$870$$ 0 0
$$871$$ −2.18120 −0.0739070
$$872$$ −17.8144 −0.603272
$$873$$ −80.0270 −2.70850
$$874$$ 1.49902 0.0507052
$$875$$ 0 0
$$876$$ 39.2860 1.32735
$$877$$ −1.40932 −0.0475893 −0.0237947 0.999717i $$-0.507575\pi$$
−0.0237947 + 0.999717i $$0.507575\pi$$
$$878$$ −8.71495 −0.294115
$$879$$ 80.7017 2.72200
$$880$$ 0 0
$$881$$ 21.6242 0.728538 0.364269 0.931294i $$-0.381319\pi$$
0.364269 + 0.931294i $$0.381319\pi$$
$$882$$ −17.0270 −0.573327
$$883$$ −45.4323 −1.52892 −0.764460 0.644671i $$-0.776994\pi$$
−0.764460 + 0.644671i $$0.776994\pi$$
$$884$$ −10.0421 −0.337753
$$885$$ 0 0
$$886$$ 5.01708 0.168552
$$887$$ −35.7575 −1.20062 −0.600310 0.799767i $$-0.704956\pi$$
−0.600310 + 0.799767i $$0.704956\pi$$
$$888$$ 28.8413 0.967851
$$889$$ 2.66195 0.0892791
$$890$$ 0 0
$$891$$ 24.9695 0.836508
$$892$$ 21.6205 0.723909
$$893$$ −8.43004 −0.282101
$$894$$ 6.91195 0.231170
$$895$$ 0 0
$$896$$ 1.50506 0.0502806
$$897$$ −90.6022 −3.02512
$$898$$ −4.79557 −0.160030
$$899$$ −8.41536 −0.280668
$$900$$ 0 0
$$901$$ −8.82299 −0.293937
$$902$$ −2.70264 −0.0899880
$$903$$ −3.56562 −0.118657
$$904$$ −20.9858 −0.697976
$$905$$ 0 0
$$906$$ 16.8122 0.558547
$$907$$ −11.5725 −0.384257 −0.192129 0.981370i $$-0.561539\pi$$
−0.192129 + 0.981370i $$0.561539\pi$$
$$908$$ 7.60593 0.252412
$$909$$ 81.7753 2.71232
$$910$$ 0 0
$$911$$ 19.3672 0.641665 0.320832 0.947136i $$-0.396037\pi$$
0.320832 + 0.947136i $$0.396037\pi$$
$$912$$ 10.9558 0.362784
$$913$$ 3.15788 0.104511
$$914$$ 5.26898 0.174282
$$915$$ 0 0
$$916$$ −49.8518 −1.64715
$$917$$ −1.57715 −0.0520819
$$918$$ 4.16789 0.137561
$$919$$ 35.9029 1.18433 0.592163 0.805818i $$-0.298274\pi$$
0.592163 + 0.805818i $$0.298274\pi$$
$$920$$ 0 0
$$921$$ −4.44966 −0.146621
$$922$$ 4.37733 0.144160
$$923$$ 46.8552 1.54226
$$924$$ −1.01998 −0.0335548
$$925$$ 0 0
$$926$$ 9.30464 0.305769
$$927$$ 43.4979 1.42866
$$928$$ −3.65294 −0.119914
$$929$$ −26.0456 −0.854529 −0.427265 0.904127i $$-0.640523\pi$$
−0.427265 + 0.904127i $$0.640523\pi$$
$$930$$ 0 0
$$931$$ 6.97244 0.228513
$$932$$ −27.8113 −0.910991
$$933$$ −5.69289 −0.186377
$$934$$ −5.54540 −0.181451
$$935$$ 0 0
$$936$$ 57.5700 1.88174
$$937$$ −59.8785 −1.95615 −0.978073 0.208262i $$-0.933219\pi$$
−0.978073 + 0.208262i $$0.933219\pi$$
$$938$$ −0.0194295 −0.000634395 0
$$939$$ −60.2542 −1.96632
$$940$$ 0 0
$$941$$ 17.0743 0.556606 0.278303 0.960493i $$-0.410228\pi$$
0.278303 + 0.960493i $$0.410228\pi$$
$$942$$ −14.5813 −0.475083
$$943$$ 38.3996 1.25046
$$944$$ 16.8247 0.547596
$$945$$ 0 0
$$946$$ −2.15116 −0.0699403
$$947$$ −9.94311 −0.323108 −0.161554 0.986864i $$-0.551651\pi$$
−0.161554 + 0.986864i $$0.551651\pi$$
$$948$$ −1.90019 −0.0617152
$$949$$ −38.7049 −1.25641
$$950$$ 0 0
$$951$$ 13.0015 0.421602
$$952$$ −0.183887 −0.00595980
$$953$$ −29.7554 −0.963871 −0.481935 0.876207i $$-0.660066\pi$$
−0.481935 + 0.876207i $$0.660066\pi$$
$$954$$ 24.6053 0.796627
$$955$$ 0 0
$$956$$ 54.0992 1.74969
$$957$$ 3.26618 0.105581
$$958$$ 9.20872 0.297520
$$959$$ 2.34476 0.0757161
$$960$$ 0 0
$$961$$ 38.8247 1.25241
$$962$$ −13.8224 −0.445654
$$963$$ 35.7612 1.15239
$$964$$ 42.3280 1.36329
$$965$$ 0 0
$$966$$ −0.807060 −0.0259667
$$967$$ −21.9469 −0.705765 −0.352883 0.935668i $$-0.614798\pi$$
−0.352883 + 0.935668i $$0.614798\pi$$
$$968$$ −1.26499 −0.0406582
$$969$$ −2.83995 −0.0912323
$$970$$ 0 0
$$971$$ 14.5115 0.465695 0.232847 0.972513i $$-0.425196\pi$$
0.232847 + 0.972513i $$0.425196\pi$$
$$972$$ 70.1341 2.24955
$$973$$ −1.45792 −0.0467389
$$974$$ 3.70442 0.118697
$$975$$ 0 0
$$976$$ 24.7519 0.792288
$$977$$ 50.3307 1.61022 0.805111 0.593124i $$-0.202106\pi$$
0.805111 + 0.593124i $$0.202106\pi$$
$$978$$ −0.709299 −0.0226809
$$979$$ 13.0206 0.416139
$$980$$ 0 0
$$981$$ 105.877 3.38040
$$982$$ −9.71695 −0.310080
$$983$$ −9.71499 −0.309860 −0.154930 0.987925i $$-0.549515\pi$$
−0.154930 + 0.987925i $$0.549515\pi$$
$$984$$ −34.1359 −1.08821
$$985$$ 0 0
$$986$$ 0.286445 0.00912227
$$987$$ 4.53865 0.144467
$$988$$ −11.4680 −0.364845
$$989$$ 30.5641 0.971883
$$990$$ 0 0
$$991$$ 24.1420 0.766897 0.383449 0.923562i $$-0.374736\pi$$
0.383449 + 0.923562i $$0.374736\pi$$
$$992$$ 30.3096 0.962330
$$993$$ −20.7702 −0.659123
$$994$$ 0.417373 0.0132383
$$995$$ 0 0
$$996$$ 19.4027 0.614797
$$997$$ 5.23428 0.165771 0.0828856 0.996559i $$-0.473586\pi$$
0.0828856 + 0.996559i $$0.473586\pi$$
$$998$$ −12.3185 −0.389936
$$999$$ −103.015 −3.25925
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 5225.2.a.s.1.10 15
5.4 even 2 5225.2.a.x.1.6 yes 15

By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.s.1.10 15 1.1 even 1 trivial
5225.2.a.x.1.6 yes 15 5.4 even 2