# Properties

 Label 5225.2.a.g.1.1 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 1.1 Root $$-1.41421$$ 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)

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -0.828427 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.414214 q^{2} +2.00000 q^{3} -1.82843 q^{4} -0.828427 q^{6} -0.828427 q^{7} +1.58579 q^{8} +1.00000 q^{9} -1.00000 q^{11} -3.65685 q^{12} +6.82843 q^{13} +0.343146 q^{14} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.00000 q^{19} -1.65685 q^{21} +0.414214 q^{22} +7.65685 q^{23} +3.17157 q^{24} -2.82843 q^{26} -4.00000 q^{27} +1.51472 q^{28} -4.82843 q^{29} -6.82843 q^{31} -4.41421 q^{32} -2.00000 q^{33} -2.82843 q^{34} -1.82843 q^{36} -8.48528 q^{37} +0.414214 q^{38} +13.6569 q^{39} +6.48528 q^{41} +0.686292 q^{42} -0.828427 q^{43} +1.82843 q^{44} -3.17157 q^{46} +11.6569 q^{47} +6.00000 q^{48} -6.31371 q^{49} +13.6569 q^{51} -12.4853 q^{52} +10.8284 q^{53} +1.65685 q^{54} -1.31371 q^{56} -2.00000 q^{57} +2.00000 q^{58} -2.82843 q^{59} -2.00000 q^{61} +2.82843 q^{62} -0.828427 q^{63} -4.17157 q^{64} +0.828427 q^{66} +6.00000 q^{67} -12.4853 q^{68} +15.3137 q^{69} -14.8284 q^{71} +1.58579 q^{72} +1.17157 q^{73} +3.51472 q^{74} +1.82843 q^{76} +0.828427 q^{77} -5.65685 q^{78} +5.65685 q^{79} -11.0000 q^{81} -2.68629 q^{82} -6.48528 q^{83} +3.02944 q^{84} +0.343146 q^{86} -9.65685 q^{87} -1.58579 q^{88} -4.34315 q^{89} -5.65685 q^{91} -14.0000 q^{92} -13.6569 q^{93} -4.82843 q^{94} -8.82843 q^{96} +1.17157 q^{97} +2.61522 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{11} + 4 q^{12} + 8 q^{13} + 12 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 2 q^{19} + 8 q^{21} - 2 q^{22} + 4 q^{23} + 12 q^{24} - 8 q^{27} + 20 q^{28} - 4 q^{29} - 8 q^{31} - 6 q^{32} - 4 q^{33} + 2 q^{36} - 2 q^{38} + 16 q^{39} - 4 q^{41} + 24 q^{42} + 4 q^{43} - 2 q^{44} - 12 q^{46} + 12 q^{47} + 12 q^{48} + 10 q^{49} + 16 q^{51} - 8 q^{52} + 16 q^{53} - 8 q^{54} + 20 q^{56} - 4 q^{57} + 4 q^{58} - 4 q^{61} + 4 q^{63} - 14 q^{64} - 4 q^{66} + 12 q^{67} - 8 q^{68} + 8 q^{69} - 24 q^{71} + 6 q^{72} + 8 q^{73} + 24 q^{74} - 2 q^{76} - 4 q^{77} - 22 q^{81} - 28 q^{82} + 4 q^{83} + 40 q^{84} + 12 q^{86} - 8 q^{87} - 6 q^{88} - 20 q^{89} - 28 q^{92} - 16 q^{93} - 4 q^{94} - 12 q^{96} + 8 q^{97} + 42 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 4 * q^6 + 4 * q^7 + 6 * q^8 + 2 * q^9 - 2 * q^11 + 4 * q^12 + 8 * q^13 + 12 * q^14 + 6 * q^16 + 8 * q^17 + 2 * q^18 - 2 * q^19 + 8 * q^21 - 2 * q^22 + 4 * q^23 + 12 * q^24 - 8 * q^27 + 20 * q^28 - 4 * q^29 - 8 * q^31 - 6 * q^32 - 4 * q^33 + 2 * q^36 - 2 * q^38 + 16 * q^39 - 4 * q^41 + 24 * q^42 + 4 * q^43 - 2 * q^44 - 12 * q^46 + 12 * q^47 + 12 * q^48 + 10 * q^49 + 16 * q^51 - 8 * q^52 + 16 * q^53 - 8 * q^54 + 20 * q^56 - 4 * q^57 + 4 * q^58 - 4 * q^61 + 4 * q^63 - 14 * q^64 - 4 * q^66 + 12 * q^67 - 8 * q^68 + 8 * q^69 - 24 * q^71 + 6 * q^72 + 8 * q^73 + 24 * q^74 - 2 * q^76 - 4 * q^77 - 22 * q^81 - 28 * q^82 + 4 * q^83 + 40 * q^84 + 12 * q^86 - 8 * q^87 - 6 * q^88 - 20 * q^89 - 28 * q^92 - 16 * q^93 - 4 * q^94 - 12 * q^96 + 8 * q^97 + 42 * q^98 - 2 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ −1.82843 −0.914214
$$5$$ 0 0
$$6$$ −0.828427 −0.338204
$$7$$ −0.828427 −0.313116 −0.156558 0.987669i $$-0.550040\pi$$
−0.156558 + 0.987669i $$0.550040\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −3.65685 −1.05564
$$13$$ 6.82843 1.89386 0.946932 0.321433i $$-0.104164\pi$$
0.946932 + 0.321433i $$0.104164\pi$$
$$14$$ 0.343146 0.0917096
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 6.82843 1.65614 0.828068 0.560627i $$-0.189440\pi$$
0.828068 + 0.560627i $$0.189440\pi$$
$$18$$ −0.414214 −0.0976311
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −1.65685 −0.361555
$$22$$ 0.414214 0.0883106
$$23$$ 7.65685 1.59656 0.798282 0.602284i $$-0.205742\pi$$
0.798282 + 0.602284i $$0.205742\pi$$
$$24$$ 3.17157 0.647395
$$25$$ 0 0
$$26$$ −2.82843 −0.554700
$$27$$ −4.00000 −0.769800
$$28$$ 1.51472 0.286255
$$29$$ −4.82843 −0.896616 −0.448308 0.893879i $$-0.647973\pi$$
−0.448308 + 0.893879i $$0.647973\pi$$
$$30$$ 0 0
$$31$$ −6.82843 −1.22642 −0.613211 0.789919i $$-0.710122\pi$$
−0.613211 + 0.789919i $$0.710122\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ −2.00000 −0.348155
$$34$$ −2.82843 −0.485071
$$35$$ 0 0
$$36$$ −1.82843 −0.304738
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ 0.414214 0.0671943
$$39$$ 13.6569 2.18685
$$40$$ 0 0
$$41$$ 6.48528 1.01283 0.506415 0.862290i $$-0.330970\pi$$
0.506415 + 0.862290i $$0.330970\pi$$
$$42$$ 0.686292 0.105897
$$43$$ −0.828427 −0.126334 −0.0631670 0.998003i $$-0.520120\pi$$
−0.0631670 + 0.998003i $$0.520120\pi$$
$$44$$ 1.82843 0.275646
$$45$$ 0 0
$$46$$ −3.17157 −0.467623
$$47$$ 11.6569 1.70033 0.850163 0.526519i $$-0.176503\pi$$
0.850163 + 0.526519i $$0.176503\pi$$
$$48$$ 6.00000 0.866025
$$49$$ −6.31371 −0.901958
$$50$$ 0 0
$$51$$ 13.6569 1.91234
$$52$$ −12.4853 −1.73140
$$53$$ 10.8284 1.48740 0.743699 0.668514i $$-0.233069\pi$$
0.743699 + 0.668514i $$0.233069\pi$$
$$54$$ 1.65685 0.225469
$$55$$ 0 0
$$56$$ −1.31371 −0.175552
$$57$$ −2.00000 −0.264906
$$58$$ 2.00000 0.262613
$$59$$ −2.82843 −0.368230 −0.184115 0.982905i $$-0.558942\pi$$
−0.184115 + 0.982905i $$0.558942\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 2.82843 0.359211
$$63$$ −0.828427 −0.104372
$$64$$ −4.17157 −0.521447
$$65$$ 0 0
$$66$$ 0.828427 0.101972
$$67$$ 6.00000 0.733017 0.366508 0.930415i $$-0.380553\pi$$
0.366508 + 0.930415i $$0.380553\pi$$
$$68$$ −12.4853 −1.51406
$$69$$ 15.3137 1.84355
$$70$$ 0 0
$$71$$ −14.8284 −1.75981 −0.879905 0.475149i $$-0.842394\pi$$
−0.879905 + 0.475149i $$0.842394\pi$$
$$72$$ 1.58579 0.186887
$$73$$ 1.17157 0.137122 0.0685611 0.997647i $$-0.478159\pi$$
0.0685611 + 0.997647i $$0.478159\pi$$
$$74$$ 3.51472 0.408578
$$75$$ 0 0
$$76$$ 1.82843 0.209735
$$77$$ 0.828427 0.0944080
$$78$$ −5.65685 −0.640513
$$79$$ 5.65685 0.636446 0.318223 0.948016i $$-0.396914\pi$$
0.318223 + 0.948016i $$0.396914\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ −2.68629 −0.296651
$$83$$ −6.48528 −0.711852 −0.355926 0.934514i $$-0.615835\pi$$
−0.355926 + 0.934514i $$0.615835\pi$$
$$84$$ 3.02944 0.330539
$$85$$ 0 0
$$86$$ 0.343146 0.0370024
$$87$$ −9.65685 −1.03532
$$88$$ −1.58579 −0.169045
$$89$$ −4.34315 −0.460373 −0.230186 0.973147i $$-0.573934\pi$$
−0.230186 + 0.973147i $$0.573934\pi$$
$$90$$ 0 0
$$91$$ −5.65685 −0.592999
$$92$$ −14.0000 −1.45960
$$93$$ −13.6569 −1.41615
$$94$$ −4.82843 −0.498014
$$95$$ 0 0
$$96$$ −8.82843 −0.901048
$$97$$ 1.17157 0.118955 0.0594776 0.998230i $$-0.481057\pi$$
0.0594776 + 0.998230i $$0.481057\pi$$
$$98$$ 2.61522 0.264177
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 9.31371 0.926749 0.463374 0.886163i $$-0.346639\pi$$
0.463374 + 0.886163i $$0.346639\pi$$
$$102$$ −5.65685 −0.560112
$$103$$ −4.34315 −0.427943 −0.213971 0.976840i $$-0.568640\pi$$
−0.213971 + 0.976840i $$0.568640\pi$$
$$104$$ 10.8284 1.06181
$$105$$ 0 0
$$106$$ −4.48528 −0.435649
$$107$$ 18.9706 1.83395 0.916977 0.398941i $$-0.130622\pi$$
0.916977 + 0.398941i $$0.130622\pi$$
$$108$$ 7.31371 0.703762
$$109$$ 0.828427 0.0793489 0.0396745 0.999213i $$-0.487368\pi$$
0.0396745 + 0.999213i $$0.487368\pi$$
$$110$$ 0 0
$$111$$ −16.9706 −1.61077
$$112$$ −2.48528 −0.234837
$$113$$ 10.1421 0.954092 0.477046 0.878878i $$-0.341708\pi$$
0.477046 + 0.878878i $$0.341708\pi$$
$$114$$ 0.828427 0.0775893
$$115$$ 0 0
$$116$$ 8.82843 0.819699
$$117$$ 6.82843 0.631288
$$118$$ 1.17157 0.107852
$$119$$ −5.65685 −0.518563
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0.828427 0.0750023
$$123$$ 12.9706 1.16952
$$124$$ 12.4853 1.12121
$$125$$ 0 0
$$126$$ 0.343146 0.0305699
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ 10.5563 0.933058
$$129$$ −1.65685 −0.145878
$$130$$ 0 0
$$131$$ 7.31371 0.639002 0.319501 0.947586i $$-0.396485\pi$$
0.319501 + 0.947586i $$0.396485\pi$$
$$132$$ 3.65685 0.318288
$$133$$ 0.828427 0.0718337
$$134$$ −2.48528 −0.214696
$$135$$ 0 0
$$136$$ 10.8284 0.928530
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ −6.34315 −0.539964
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 23.3137 1.96337
$$142$$ 6.14214 0.515437
$$143$$ −6.82843 −0.571022
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −0.485281 −0.0401622
$$147$$ −12.6274 −1.04149
$$148$$ 15.5147 1.27530
$$149$$ −8.34315 −0.683497 −0.341749 0.939791i $$-0.611019\pi$$
−0.341749 + 0.939791i $$0.611019\pi$$
$$150$$ 0 0
$$151$$ 10.3431 0.841713 0.420857 0.907127i $$-0.361730\pi$$
0.420857 + 0.907127i $$0.361730\pi$$
$$152$$ −1.58579 −0.128624
$$153$$ 6.82843 0.552046
$$154$$ −0.343146 −0.0276515
$$155$$ 0 0
$$156$$ −24.9706 −1.99925
$$157$$ 12.0000 0.957704 0.478852 0.877896i $$-0.341053\pi$$
0.478852 + 0.877896i $$0.341053\pi$$
$$158$$ −2.34315 −0.186411
$$159$$ 21.6569 1.71750
$$160$$ 0 0
$$161$$ −6.34315 −0.499910
$$162$$ 4.55635 0.357981
$$163$$ −2.00000 −0.156652 −0.0783260 0.996928i $$-0.524958\pi$$
−0.0783260 + 0.996928i $$0.524958\pi$$
$$164$$ −11.8579 −0.925944
$$165$$ 0 0
$$166$$ 2.68629 0.208497
$$167$$ 8.34315 0.645612 0.322806 0.946465i $$-0.395374\pi$$
0.322806 + 0.946465i $$0.395374\pi$$
$$168$$ −2.62742 −0.202710
$$169$$ 33.6274 2.58672
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 1.51472 0.115496
$$173$$ 12.4853 0.949238 0.474619 0.880191i $$-0.342586\pi$$
0.474619 + 0.880191i $$0.342586\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ −5.65685 −0.425195
$$178$$ 1.79899 0.134840
$$179$$ 16.4853 1.23217 0.616084 0.787681i $$-0.288718\pi$$
0.616084 + 0.787681i $$0.288718\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 2.34315 0.173686
$$183$$ −4.00000 −0.295689
$$184$$ 12.1421 0.895130
$$185$$ 0 0
$$186$$ 5.65685 0.414781
$$187$$ −6.82843 −0.499344
$$188$$ −21.3137 −1.55446
$$189$$ 3.31371 0.241037
$$190$$ 0 0
$$191$$ 16.9706 1.22795 0.613973 0.789327i $$-0.289570\pi$$
0.613973 + 0.789327i $$0.289570\pi$$
$$192$$ −8.34315 −0.602115
$$193$$ −14.1421 −1.01797 −0.508987 0.860774i $$-0.669980\pi$$
−0.508987 + 0.860774i $$0.669980\pi$$
$$194$$ −0.485281 −0.0348412
$$195$$ 0 0
$$196$$ 11.5442 0.824583
$$197$$ −6.14214 −0.437609 −0.218805 0.975769i $$-0.570216\pi$$
−0.218805 + 0.975769i $$0.570216\pi$$
$$198$$ 0.414214 0.0294369
$$199$$ −5.65685 −0.401004 −0.200502 0.979693i $$-0.564257\pi$$
−0.200502 + 0.979693i $$0.564257\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ −3.85786 −0.271438
$$203$$ 4.00000 0.280745
$$204$$ −24.9706 −1.74829
$$205$$ 0 0
$$206$$ 1.79899 0.125342
$$207$$ 7.65685 0.532188
$$208$$ 20.4853 1.42040
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 15.3137 1.05424 0.527120 0.849791i $$-0.323272\pi$$
0.527120 + 0.849791i $$0.323272\pi$$
$$212$$ −19.7990 −1.35980
$$213$$ −29.6569 −2.03205
$$214$$ −7.85786 −0.537153
$$215$$ 0 0
$$216$$ −6.34315 −0.431596
$$217$$ 5.65685 0.384012
$$218$$ −0.343146 −0.0232408
$$219$$ 2.34315 0.158335
$$220$$ 0 0
$$221$$ 46.6274 3.13650
$$222$$ 7.02944 0.471785
$$223$$ 11.6569 0.780601 0.390300 0.920688i $$-0.372371\pi$$
0.390300 + 0.920688i $$0.372371\pi$$
$$224$$ 3.65685 0.244334
$$225$$ 0 0
$$226$$ −4.20101 −0.279447
$$227$$ 12.3431 0.819243 0.409622 0.912255i $$-0.365661\pi$$
0.409622 + 0.912255i $$0.365661\pi$$
$$228$$ 3.65685 0.242181
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 1.65685 0.109013
$$232$$ −7.65685 −0.502697
$$233$$ −4.48528 −0.293841 −0.146920 0.989148i $$-0.546936\pi$$
−0.146920 + 0.989148i $$0.546936\pi$$
$$234$$ −2.82843 −0.184900
$$235$$ 0 0
$$236$$ 5.17157 0.336641
$$237$$ 11.3137 0.734904
$$238$$ 2.34315 0.151884
$$239$$ 4.68629 0.303131 0.151565 0.988447i $$-0.451569\pi$$
0.151565 + 0.988447i $$0.451569\pi$$
$$240$$ 0 0
$$241$$ 6.48528 0.417754 0.208877 0.977942i $$-0.433019\pi$$
0.208877 + 0.977942i $$0.433019\pi$$
$$242$$ −0.414214 −0.0266267
$$243$$ −10.0000 −0.641500
$$244$$ 3.65685 0.234106
$$245$$ 0 0
$$246$$ −5.37258 −0.342543
$$247$$ −6.82843 −0.434482
$$248$$ −10.8284 −0.687606
$$249$$ −12.9706 −0.821976
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 1.51472 0.0954183
$$253$$ −7.65685 −0.481382
$$254$$ −5.79899 −0.363861
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ −18.1421 −1.13168 −0.565838 0.824517i $$-0.691447\pi$$
−0.565838 + 0.824517i $$0.691447\pi$$
$$258$$ 0.686292 0.0427266
$$259$$ 7.02944 0.436788
$$260$$ 0 0
$$261$$ −4.82843 −0.298872
$$262$$ −3.02944 −0.187159
$$263$$ −18.4853 −1.13985 −0.569926 0.821696i $$-0.693028\pi$$
−0.569926 + 0.821696i $$0.693028\pi$$
$$264$$ −3.17157 −0.195197
$$265$$ 0 0
$$266$$ −0.343146 −0.0210396
$$267$$ −8.68629 −0.531592
$$268$$ −10.9706 −0.670134
$$269$$ −22.9706 −1.40054 −0.700270 0.713878i $$-0.746937\pi$$
−0.700270 + 0.713878i $$0.746937\pi$$
$$270$$ 0 0
$$271$$ 11.3137 0.687259 0.343629 0.939105i $$-0.388344\pi$$
0.343629 + 0.939105i $$0.388344\pi$$
$$272$$ 20.4853 1.24210
$$273$$ −11.3137 −0.684737
$$274$$ −3.31371 −0.200188
$$275$$ 0 0
$$276$$ −28.0000 −1.68540
$$277$$ −18.8284 −1.13129 −0.565645 0.824649i $$-0.691373\pi$$
−0.565645 + 0.824649i $$0.691373\pi$$
$$278$$ −1.65685 −0.0993715
$$279$$ −6.82843 −0.408807
$$280$$ 0 0
$$281$$ −19.4558 −1.16064 −0.580319 0.814389i $$-0.697072\pi$$
−0.580319 + 0.814389i $$0.697072\pi$$
$$282$$ −9.65685 −0.575057
$$283$$ −6.48528 −0.385510 −0.192755 0.981247i $$-0.561742\pi$$
−0.192755 + 0.981247i $$0.561742\pi$$
$$284$$ 27.1127 1.60884
$$285$$ 0 0
$$286$$ 2.82843 0.167248
$$287$$ −5.37258 −0.317134
$$288$$ −4.41421 −0.260110
$$289$$ 29.6274 1.74279
$$290$$ 0 0
$$291$$ 2.34315 0.137358
$$292$$ −2.14214 −0.125359
$$293$$ −26.1421 −1.52724 −0.763620 0.645666i $$-0.776580\pi$$
−0.763620 + 0.645666i $$0.776580\pi$$
$$294$$ 5.23045 0.305046
$$295$$ 0 0
$$296$$ −13.4558 −0.782105
$$297$$ 4.00000 0.232104
$$298$$ 3.45584 0.200192
$$299$$ 52.2843 3.02368
$$300$$ 0 0
$$301$$ 0.686292 0.0395572
$$302$$ −4.28427 −0.246532
$$303$$ 18.6274 1.07012
$$304$$ −3.00000 −0.172062
$$305$$ 0 0
$$306$$ −2.82843 −0.161690
$$307$$ 26.9706 1.53929 0.769646 0.638471i $$-0.220433\pi$$
0.769646 + 0.638471i $$0.220433\pi$$
$$308$$ −1.51472 −0.0863091
$$309$$ −8.68629 −0.494146
$$310$$ 0 0
$$311$$ 0.970563 0.0550356 0.0275178 0.999621i $$-0.491240\pi$$
0.0275178 + 0.999621i $$0.491240\pi$$
$$312$$ 21.6569 1.22608
$$313$$ 16.9706 0.959233 0.479616 0.877478i $$-0.340776\pi$$
0.479616 + 0.877478i $$0.340776\pi$$
$$314$$ −4.97056 −0.280505
$$315$$ 0 0
$$316$$ −10.3431 −0.581847
$$317$$ −6.14214 −0.344977 −0.172488 0.985012i $$-0.555181\pi$$
−0.172488 + 0.985012i $$0.555181\pi$$
$$318$$ −8.97056 −0.503044
$$319$$ 4.82843 0.270340
$$320$$ 0 0
$$321$$ 37.9411 2.11767
$$322$$ 2.62742 0.146420
$$323$$ −6.82843 −0.379944
$$324$$ 20.1127 1.11737
$$325$$ 0 0
$$326$$ 0.828427 0.0458823
$$327$$ 1.65685 0.0916242
$$328$$ 10.2843 0.567854
$$329$$ −9.65685 −0.532400
$$330$$ 0 0
$$331$$ −6.14214 −0.337602 −0.168801 0.985650i $$-0.553990\pi$$
−0.168801 + 0.985650i $$0.553990\pi$$
$$332$$ 11.8579 0.650785
$$333$$ −8.48528 −0.464991
$$334$$ −3.45584 −0.189095
$$335$$ 0 0
$$336$$ −4.97056 −0.271166
$$337$$ 2.82843 0.154074 0.0770371 0.997028i $$-0.475454\pi$$
0.0770371 + 0.997028i $$0.475454\pi$$
$$338$$ −13.9289 −0.757634
$$339$$ 20.2843 1.10169
$$340$$ 0 0
$$341$$ 6.82843 0.369780
$$342$$ 0.414214 0.0223981
$$343$$ 11.0294 0.595534
$$344$$ −1.31371 −0.0708304
$$345$$ 0 0
$$346$$ −5.17157 −0.278025
$$347$$ 24.8284 1.33286 0.666430 0.745568i $$-0.267822\pi$$
0.666430 + 0.745568i $$0.267822\pi$$
$$348$$ 17.6569 0.946507
$$349$$ −27.6569 −1.48044 −0.740219 0.672366i $$-0.765278\pi$$
−0.740219 + 0.672366i $$0.765278\pi$$
$$350$$ 0 0
$$351$$ −27.3137 −1.45790
$$352$$ 4.41421 0.235278
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 2.34315 0.124537
$$355$$ 0 0
$$356$$ 7.94113 0.420879
$$357$$ −11.3137 −0.598785
$$358$$ −6.82843 −0.360894
$$359$$ 16.9706 0.895672 0.447836 0.894116i $$-0.352195\pi$$
0.447836 + 0.894116i $$0.352195\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −5.79899 −0.304788
$$363$$ 2.00000 0.104973
$$364$$ 10.3431 0.542128
$$365$$ 0 0
$$366$$ 1.65685 0.0866052
$$367$$ −20.3431 −1.06190 −0.530952 0.847402i $$-0.678165\pi$$
−0.530952 + 0.847402i $$0.678165\pi$$
$$368$$ 22.9706 1.19742
$$369$$ 6.48528 0.337610
$$370$$ 0 0
$$371$$ −8.97056 −0.465728
$$372$$ 24.9706 1.29466
$$373$$ −17.1716 −0.889110 −0.444555 0.895751i $$-0.646638\pi$$
−0.444555 + 0.895751i $$0.646638\pi$$
$$374$$ 2.82843 0.146254
$$375$$ 0 0
$$376$$ 18.4853 0.953306
$$377$$ −32.9706 −1.69807
$$378$$ −1.37258 −0.0705981
$$379$$ 10.8284 0.556219 0.278109 0.960549i $$-0.410292\pi$$
0.278109 + 0.960549i $$0.410292\pi$$
$$380$$ 0 0
$$381$$ 28.0000 1.43448
$$382$$ −7.02944 −0.359657
$$383$$ −26.9706 −1.37813 −0.689066 0.724699i $$-0.741979\pi$$
−0.689066 + 0.724699i $$0.741979\pi$$
$$384$$ 21.1127 1.07740
$$385$$ 0 0
$$386$$ 5.85786 0.298157
$$387$$ −0.828427 −0.0421113
$$388$$ −2.14214 −0.108750
$$389$$ 13.3137 0.675032 0.337516 0.941320i $$-0.390413\pi$$
0.337516 + 0.941320i $$0.390413\pi$$
$$390$$ 0 0
$$391$$ 52.2843 2.64413
$$392$$ −10.0122 −0.505692
$$393$$ 14.6274 0.737856
$$394$$ 2.54416 0.128173
$$395$$ 0 0
$$396$$ 1.82843 0.0918819
$$397$$ 7.31371 0.367065 0.183532 0.983014i $$-0.441247\pi$$
0.183532 + 0.983014i $$0.441247\pi$$
$$398$$ 2.34315 0.117451
$$399$$ 1.65685 0.0829465
$$400$$ 0 0
$$401$$ 5.31371 0.265354 0.132677 0.991159i $$-0.457643\pi$$
0.132677 + 0.991159i $$0.457643\pi$$
$$402$$ −4.97056 −0.247909
$$403$$ −46.6274 −2.32268
$$404$$ −17.0294 −0.847246
$$405$$ 0 0
$$406$$ −1.65685 −0.0822283
$$407$$ 8.48528 0.420600
$$408$$ 21.6569 1.07217
$$409$$ −24.8284 −1.22769 −0.613843 0.789428i $$-0.710377\pi$$
−0.613843 + 0.789428i $$0.710377\pi$$
$$410$$ 0 0
$$411$$ 16.0000 0.789222
$$412$$ 7.94113 0.391231
$$413$$ 2.34315 0.115299
$$414$$ −3.17157 −0.155874
$$415$$ 0 0
$$416$$ −30.1421 −1.47784
$$417$$ 8.00000 0.391762
$$418$$ −0.414214 −0.0202598
$$419$$ −20.9706 −1.02448 −0.512240 0.858843i $$-0.671184\pi$$
−0.512240 + 0.858843i $$0.671184\pi$$
$$420$$ 0 0
$$421$$ −26.9706 −1.31446 −0.657232 0.753688i $$-0.728273\pi$$
−0.657232 + 0.753688i $$0.728273\pi$$
$$422$$ −6.34315 −0.308780
$$423$$ 11.6569 0.566776
$$424$$ 17.1716 0.833925
$$425$$ 0 0
$$426$$ 12.2843 0.595175
$$427$$ 1.65685 0.0801808
$$428$$ −34.6863 −1.67663
$$429$$ −13.6569 −0.659359
$$430$$ 0 0
$$431$$ 6.62742 0.319231 0.159616 0.987179i $$-0.448974\pi$$
0.159616 + 0.987179i $$0.448974\pi$$
$$432$$ −12.0000 −0.577350
$$433$$ 1.17157 0.0563022 0.0281511 0.999604i $$-0.491038\pi$$
0.0281511 + 0.999604i $$0.491038\pi$$
$$434$$ −2.34315 −0.112475
$$435$$ 0 0
$$436$$ −1.51472 −0.0725419
$$437$$ −7.65685 −0.366277
$$438$$ −0.970563 −0.0463753
$$439$$ 18.3431 0.875471 0.437735 0.899104i $$-0.355781\pi$$
0.437735 + 0.899104i $$0.355781\pi$$
$$440$$ 0 0
$$441$$ −6.31371 −0.300653
$$442$$ −19.3137 −0.918659
$$443$$ 9.31371 0.442508 0.221254 0.975216i $$-0.428985\pi$$
0.221254 + 0.975216i $$0.428985\pi$$
$$444$$ 31.0294 1.47259
$$445$$ 0 0
$$446$$ −4.82843 −0.228633
$$447$$ −16.6863 −0.789235
$$448$$ 3.45584 0.163273
$$449$$ −2.97056 −0.140190 −0.0700948 0.997540i $$-0.522330\pi$$
−0.0700948 + 0.997540i $$0.522330\pi$$
$$450$$ 0 0
$$451$$ −6.48528 −0.305380
$$452$$ −18.5442 −0.872244
$$453$$ 20.6863 0.971927
$$454$$ −5.11270 −0.239951
$$455$$ 0 0
$$456$$ −3.17157 −0.148523
$$457$$ −15.7990 −0.739046 −0.369523 0.929222i $$-0.620479\pi$$
−0.369523 + 0.929222i $$0.620479\pi$$
$$458$$ 9.11270 0.425808
$$459$$ −27.3137 −1.27489
$$460$$ 0 0
$$461$$ −22.2843 −1.03788 −0.518941 0.854810i $$-0.673674\pi$$
−0.518941 + 0.854810i $$0.673674\pi$$
$$462$$ −0.686292 −0.0319292
$$463$$ 30.2843 1.40743 0.703715 0.710483i $$-0.251523\pi$$
0.703715 + 0.710483i $$0.251523\pi$$
$$464$$ −14.4853 −0.672462
$$465$$ 0 0
$$466$$ 1.85786 0.0860639
$$467$$ 4.34315 0.200977 0.100488 0.994938i $$-0.467959\pi$$
0.100488 + 0.994938i $$0.467959\pi$$
$$468$$ −12.4853 −0.577132
$$469$$ −4.97056 −0.229519
$$470$$ 0 0
$$471$$ 24.0000 1.10586
$$472$$ −4.48528 −0.206452
$$473$$ 0.828427 0.0380911
$$474$$ −4.68629 −0.215248
$$475$$ 0 0
$$476$$ 10.3431 0.474077
$$477$$ 10.8284 0.495800
$$478$$ −1.94113 −0.0887850
$$479$$ −3.31371 −0.151407 −0.0757036 0.997130i $$-0.524120\pi$$
−0.0757036 + 0.997130i $$0.524120\pi$$
$$480$$ 0 0
$$481$$ −57.9411 −2.64189
$$482$$ −2.68629 −0.122357
$$483$$ −12.6863 −0.577246
$$484$$ −1.82843 −0.0831103
$$485$$ 0 0
$$486$$ 4.14214 0.187891
$$487$$ 15.6569 0.709480 0.354740 0.934965i $$-0.384569\pi$$
0.354740 + 0.934965i $$0.384569\pi$$
$$488$$ −3.17157 −0.143570
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −36.9706 −1.66846 −0.834229 0.551418i $$-0.814087\pi$$
−0.834229 + 0.551418i $$0.814087\pi$$
$$492$$ −23.7157 −1.06919
$$493$$ −32.9706 −1.48492
$$494$$ 2.82843 0.127257
$$495$$ 0 0
$$496$$ −20.4853 −0.919816
$$497$$ 12.2843 0.551025
$$498$$ 5.37258 0.240751
$$499$$ 0.686292 0.0307226 0.0153613 0.999882i $$-0.495110\pi$$
0.0153613 + 0.999882i $$0.495110\pi$$
$$500$$ 0 0
$$501$$ 16.6863 0.745489
$$502$$ 1.65685 0.0739490
$$503$$ 41.7990 1.86372 0.931862 0.362812i $$-0.118183\pi$$
0.931862 + 0.362812i $$0.118183\pi$$
$$504$$ −1.31371 −0.0585172
$$505$$ 0 0
$$506$$ 3.17157 0.140994
$$507$$ 67.2548 2.98689
$$508$$ −25.5980 −1.13573
$$509$$ 31.6569 1.40317 0.701583 0.712588i $$-0.252477\pi$$
0.701583 + 0.712588i $$0.252477\pi$$
$$510$$ 0 0
$$511$$ −0.970563 −0.0429352
$$512$$ −22.7574 −1.00574
$$513$$ 4.00000 0.176604
$$514$$ 7.51472 0.331460
$$515$$ 0 0
$$516$$ 3.02944 0.133364
$$517$$ −11.6569 −0.512668
$$518$$ −2.91169 −0.127932
$$519$$ 24.9706 1.09609
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 6.97056 0.304801 0.152401 0.988319i $$-0.451300\pi$$
0.152401 + 0.988319i $$0.451300\pi$$
$$524$$ −13.3726 −0.584184
$$525$$ 0 0
$$526$$ 7.65685 0.333855
$$527$$ −46.6274 −2.03112
$$528$$ −6.00000 −0.261116
$$529$$ 35.6274 1.54902
$$530$$ 0 0
$$531$$ −2.82843 −0.122743
$$532$$ −1.51472 −0.0656714
$$533$$ 44.2843 1.91816
$$534$$ 3.59798 0.155700
$$535$$ 0 0
$$536$$ 9.51472 0.410973
$$537$$ 32.9706 1.42278
$$538$$ 9.51472 0.410209
$$539$$ 6.31371 0.271951
$$540$$ 0 0
$$541$$ −12.3431 −0.530673 −0.265337 0.964156i $$-0.585483\pi$$
−0.265337 + 0.964156i $$0.585483\pi$$
$$542$$ −4.68629 −0.201293
$$543$$ 28.0000 1.20160
$$544$$ −30.1421 −1.29233
$$545$$ 0 0
$$546$$ 4.68629 0.200555
$$547$$ −34.2843 −1.46589 −0.732945 0.680288i $$-0.761855\pi$$
−0.732945 + 0.680288i $$0.761855\pi$$
$$548$$ −14.6274 −0.624852
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 4.82843 0.205698
$$552$$ 24.2843 1.03361
$$553$$ −4.68629 −0.199281
$$554$$ 7.79899 0.331347
$$555$$ 0 0
$$556$$ −7.31371 −0.310170
$$557$$ 23.1127 0.979316 0.489658 0.871914i $$-0.337122\pi$$
0.489658 + 0.871914i $$0.337122\pi$$
$$558$$ 2.82843 0.119737
$$559$$ −5.65685 −0.239259
$$560$$ 0 0
$$561$$ −13.6569 −0.576593
$$562$$ 8.05887 0.339943
$$563$$ −18.9706 −0.799514 −0.399757 0.916621i $$-0.630905\pi$$
−0.399757 + 0.916621i $$0.630905\pi$$
$$564$$ −42.6274 −1.79494
$$565$$ 0 0
$$566$$ 2.68629 0.112913
$$567$$ 9.11270 0.382697
$$568$$ −23.5147 −0.986656
$$569$$ −32.8284 −1.37624 −0.688120 0.725597i $$-0.741564\pi$$
−0.688120 + 0.725597i $$0.741564\pi$$
$$570$$ 0 0
$$571$$ −24.2843 −1.01627 −0.508133 0.861279i $$-0.669664\pi$$
−0.508133 + 0.861279i $$0.669664\pi$$
$$572$$ 12.4853 0.522036
$$573$$ 33.9411 1.41791
$$574$$ 2.22540 0.0928863
$$575$$ 0 0
$$576$$ −4.17157 −0.173816
$$577$$ 32.0000 1.33218 0.666089 0.745873i $$-0.267967\pi$$
0.666089 + 0.745873i $$0.267967\pi$$
$$578$$ −12.2721 −0.510451
$$579$$ −28.2843 −1.17545
$$580$$ 0 0
$$581$$ 5.37258 0.222892
$$582$$ −0.970563 −0.0402311
$$583$$ −10.8284 −0.448468
$$584$$ 1.85786 0.0768790
$$585$$ 0 0
$$586$$ 10.8284 0.447318
$$587$$ 22.2843 0.919770 0.459885 0.887978i $$-0.347891\pi$$
0.459885 + 0.887978i $$0.347891\pi$$
$$588$$ 23.0883 0.952146
$$589$$ 6.82843 0.281360
$$590$$ 0 0
$$591$$ −12.2843 −0.505307
$$592$$ −25.4558 −1.04623
$$593$$ −5.45584 −0.224045 −0.112022 0.993706i $$-0.535733\pi$$
−0.112022 + 0.993706i $$0.535733\pi$$
$$594$$ −1.65685 −0.0679816
$$595$$ 0 0
$$596$$ 15.2548 0.624862
$$597$$ −11.3137 −0.463039
$$598$$ −21.6569 −0.885615
$$599$$ 30.8284 1.25962 0.629808 0.776751i $$-0.283134\pi$$
0.629808 + 0.776751i $$0.283134\pi$$
$$600$$ 0 0
$$601$$ −25.1127 −1.02437 −0.512184 0.858876i $$-0.671163\pi$$
−0.512184 + 0.858876i $$0.671163\pi$$
$$602$$ −0.284271 −0.0115860
$$603$$ 6.00000 0.244339
$$604$$ −18.9117 −0.769506
$$605$$ 0 0
$$606$$ −7.71573 −0.313430
$$607$$ −28.3431 −1.15041 −0.575206 0.818008i $$-0.695078\pi$$
−0.575206 + 0.818008i $$0.695078\pi$$
$$608$$ 4.41421 0.179020
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 79.5980 3.22019
$$612$$ −12.4853 −0.504688
$$613$$ −13.1716 −0.531995 −0.265997 0.963974i $$-0.585701\pi$$
−0.265997 + 0.963974i $$0.585701\pi$$
$$614$$ −11.1716 −0.450848
$$615$$ 0 0
$$616$$ 1.31371 0.0529308
$$617$$ 21.6569 0.871872 0.435936 0.899978i $$-0.356417\pi$$
0.435936 + 0.899978i $$0.356417\pi$$
$$618$$ 3.59798 0.144732
$$619$$ −26.6274 −1.07025 −0.535123 0.844774i $$-0.679735\pi$$
−0.535123 + 0.844774i $$0.679735\pi$$
$$620$$ 0 0
$$621$$ −30.6274 −1.22904
$$622$$ −0.402020 −0.0161195
$$623$$ 3.59798 0.144150
$$624$$ 40.9706 1.64014
$$625$$ 0 0
$$626$$ −7.02944 −0.280953
$$627$$ 2.00000 0.0798723
$$628$$ −21.9411 −0.875546
$$629$$ −57.9411 −2.31026
$$630$$ 0 0
$$631$$ 32.9706 1.31254 0.656269 0.754527i $$-0.272134\pi$$
0.656269 + 0.754527i $$0.272134\pi$$
$$632$$ 8.97056 0.356830
$$633$$ 30.6274 1.21733
$$634$$ 2.54416 0.101041
$$635$$ 0 0
$$636$$ −39.5980 −1.57016
$$637$$ −43.1127 −1.70819
$$638$$ −2.00000 −0.0791808
$$639$$ −14.8284 −0.586604
$$640$$ 0 0
$$641$$ 7.65685 0.302428 0.151214 0.988501i $$-0.451682\pi$$
0.151214 + 0.988501i $$0.451682\pi$$
$$642$$ −15.7157 −0.620250
$$643$$ 47.9411 1.89061 0.945307 0.326183i $$-0.105762\pi$$
0.945307 + 0.326183i $$0.105762\pi$$
$$644$$ 11.5980 0.457024
$$645$$ 0 0
$$646$$ 2.82843 0.111283
$$647$$ 2.68629 0.105609 0.0528045 0.998605i $$-0.483184\pi$$
0.0528045 + 0.998605i $$0.483184\pi$$
$$648$$ −17.4437 −0.685251
$$649$$ 2.82843 0.111025
$$650$$ 0 0
$$651$$ 11.3137 0.443419
$$652$$ 3.65685 0.143213
$$653$$ −34.6274 −1.35508 −0.677538 0.735488i $$-0.736953\pi$$
−0.677538 + 0.735488i $$0.736953\pi$$
$$654$$ −0.686292 −0.0268361
$$655$$ 0 0
$$656$$ 19.4558 0.759623
$$657$$ 1.17157 0.0457074
$$658$$ 4.00000 0.155936
$$659$$ 26.6274 1.03726 0.518628 0.855000i $$-0.326443\pi$$
0.518628 + 0.855000i $$0.326443\pi$$
$$660$$ 0 0
$$661$$ −5.31371 −0.206679 −0.103340 0.994646i $$-0.532953\pi$$
−0.103340 + 0.994646i $$0.532953\pi$$
$$662$$ 2.54416 0.0988814
$$663$$ 93.2548 3.62172
$$664$$ −10.2843 −0.399107
$$665$$ 0 0
$$666$$ 3.51472 0.136193
$$667$$ −36.9706 −1.43151
$$668$$ −15.2548 −0.590227
$$669$$ 23.3137 0.901360
$$670$$ 0 0
$$671$$ 2.00000 0.0772091
$$672$$ 7.31371 0.282132
$$673$$ 22.1421 0.853517 0.426758 0.904366i $$-0.359656\pi$$
0.426758 + 0.904366i $$0.359656\pi$$
$$674$$ −1.17157 −0.0451273
$$675$$ 0 0
$$676$$ −61.4853 −2.36482
$$677$$ 19.5147 0.750012 0.375006 0.927022i $$-0.377641\pi$$
0.375006 + 0.927022i $$0.377641\pi$$
$$678$$ −8.40202 −0.322678
$$679$$ −0.970563 −0.0372468
$$680$$ 0 0
$$681$$ 24.6863 0.945981
$$682$$ −2.82843 −0.108306
$$683$$ 16.6274 0.636230 0.318115 0.948052i $$-0.396950\pi$$
0.318115 + 0.948052i $$0.396950\pi$$
$$684$$ 1.82843 0.0699117
$$685$$ 0 0
$$686$$ −4.56854 −0.174428
$$687$$ −44.0000 −1.67870
$$688$$ −2.48528 −0.0947505
$$689$$ 73.9411 2.81693
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −22.8284 −0.867807
$$693$$ 0.828427 0.0314693
$$694$$ −10.2843 −0.390386
$$695$$ 0 0
$$696$$ −15.3137 −0.580465
$$697$$ 44.2843 1.67739
$$698$$ 11.4558 0.433610
$$699$$ −8.97056 −0.339298
$$700$$ 0 0
$$701$$ 34.2843 1.29490 0.647450 0.762108i $$-0.275836\pi$$
0.647450 + 0.762108i $$0.275836\pi$$
$$702$$ 11.3137 0.427008
$$703$$ 8.48528 0.320028
$$704$$ 4.17157 0.157222
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −7.71573 −0.290180
$$708$$ 10.3431 0.388719
$$709$$ 6.68629 0.251109 0.125554 0.992087i $$-0.459929\pi$$
0.125554 + 0.992087i $$0.459929\pi$$
$$710$$ 0 0
$$711$$ 5.65685 0.212149
$$712$$ −6.88730 −0.258113
$$713$$ −52.2843 −1.95806
$$714$$ 4.68629 0.175380
$$715$$ 0 0
$$716$$ −30.1421 −1.12646
$$717$$ 9.37258 0.350026
$$718$$ −7.02944 −0.262336
$$719$$ 14.6274 0.545511 0.272755 0.962083i $$-0.412065\pi$$
0.272755 + 0.962083i $$0.412065\pi$$
$$720$$ 0 0
$$721$$ 3.59798 0.133996
$$722$$ −0.414214 −0.0154154
$$723$$ 12.9706 0.482380
$$724$$ −25.5980 −0.951341
$$725$$ 0 0
$$726$$ −0.828427 −0.0307458
$$727$$ −13.3137 −0.493778 −0.246889 0.969044i $$-0.579408\pi$$
−0.246889 + 0.969044i $$0.579408\pi$$
$$728$$ −8.97056 −0.332471
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −5.65685 −0.209226
$$732$$ 7.31371 0.270322
$$733$$ −8.48528 −0.313411 −0.156706 0.987645i $$-0.550087\pi$$
−0.156706 + 0.987645i $$0.550087\pi$$
$$734$$ 8.42641 0.311024
$$735$$ 0 0
$$736$$ −33.7990 −1.24585
$$737$$ −6.00000 −0.221013
$$738$$ −2.68629 −0.0988838
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ −13.6569 −0.501697
$$742$$ 3.71573 0.136409
$$743$$ 10.9706 0.402471 0.201235 0.979543i $$-0.435504\pi$$
0.201235 + 0.979543i $$0.435504\pi$$
$$744$$ −21.6569 −0.793979
$$745$$ 0 0
$$746$$ 7.11270 0.260414
$$747$$ −6.48528 −0.237284
$$748$$ 12.4853 0.456507
$$749$$ −15.7157 −0.574240
$$750$$ 0 0
$$751$$ 36.4853 1.33137 0.665683 0.746234i $$-0.268140\pi$$
0.665683 + 0.746234i $$0.268140\pi$$
$$752$$ 34.9706 1.27525
$$753$$ −8.00000 −0.291536
$$754$$ 13.6569 0.497353
$$755$$ 0 0
$$756$$ −6.05887 −0.220359
$$757$$ 20.0000 0.726912 0.363456 0.931611i $$-0.381597\pi$$
0.363456 + 0.931611i $$0.381597\pi$$
$$758$$ −4.48528 −0.162913
$$759$$ −15.3137 −0.555852
$$760$$ 0 0
$$761$$ −47.9411 −1.73786 −0.868932 0.494931i $$-0.835193\pi$$
−0.868932 + 0.494931i $$0.835193\pi$$
$$762$$ −11.5980 −0.420150
$$763$$ −0.686292 −0.0248454
$$764$$ −31.0294 −1.12261
$$765$$ 0 0
$$766$$ 11.1716 0.403645
$$767$$ −19.3137 −0.697378
$$768$$ 7.94113 0.286551
$$769$$ 8.34315 0.300862 0.150431 0.988621i $$-0.451934\pi$$
0.150431 + 0.988621i $$0.451934\pi$$
$$770$$ 0 0
$$771$$ −36.2843 −1.30675
$$772$$ 25.8579 0.930645
$$773$$ −5.17157 −0.186009 −0.0930043 0.995666i $$-0.529647\pi$$
−0.0930043 + 0.995666i $$0.529647\pi$$
$$774$$ 0.343146 0.0123341
$$775$$ 0 0
$$776$$ 1.85786 0.0666934
$$777$$ 14.0589 0.504359
$$778$$ −5.51472 −0.197712
$$779$$ −6.48528 −0.232359
$$780$$ 0 0
$$781$$ 14.8284 0.530603
$$782$$ −21.6569 −0.774448
$$783$$ 19.3137 0.690216
$$784$$ −18.9411 −0.676469
$$785$$ 0 0
$$786$$ −6.05887 −0.216113
$$787$$ 2.97056 0.105889 0.0529446 0.998597i $$-0.483139\pi$$
0.0529446 + 0.998597i $$0.483139\pi$$
$$788$$ 11.2304 0.400068
$$789$$ −36.9706 −1.31619
$$790$$ 0 0
$$791$$ −8.40202 −0.298741
$$792$$ −1.58579 −0.0563485
$$793$$ −13.6569 −0.484969
$$794$$ −3.02944 −0.107511
$$795$$ 0 0
$$796$$ 10.3431 0.366603
$$797$$ −10.8284 −0.383563 −0.191781 0.981438i $$-0.561426\pi$$
−0.191781 + 0.981438i $$0.561426\pi$$
$$798$$ −0.686292 −0.0242945
$$799$$ 79.5980 2.81597
$$800$$ 0 0
$$801$$ −4.34315 −0.153458
$$802$$ −2.20101 −0.0777204
$$803$$ −1.17157 −0.0413439
$$804$$ −21.9411 −0.773804
$$805$$ 0 0
$$806$$ 19.3137 0.680296
$$807$$ −45.9411 −1.61720
$$808$$ 14.7696 0.519591
$$809$$ −53.3137 −1.87441 −0.937205 0.348779i $$-0.886596\pi$$
−0.937205 + 0.348779i $$0.886596\pi$$
$$810$$ 0 0
$$811$$ −16.2843 −0.571818 −0.285909 0.958257i $$-0.592296\pi$$
−0.285909 + 0.958257i $$0.592296\pi$$
$$812$$ −7.31371 −0.256661
$$813$$ 22.6274 0.793578
$$814$$ −3.51472 −0.123191
$$815$$ 0 0
$$816$$ 40.9706 1.43426
$$817$$ 0.828427 0.0289830
$$818$$ 10.2843 0.359581
$$819$$ −5.65685 −0.197666
$$820$$ 0 0
$$821$$ −51.9411 −1.81276 −0.906379 0.422466i $$-0.861165\pi$$
−0.906379 + 0.422466i $$0.861165\pi$$
$$822$$ −6.62742 −0.231158
$$823$$ −9.31371 −0.324655 −0.162328 0.986737i $$-0.551900\pi$$
−0.162328 + 0.986737i $$0.551900\pi$$
$$824$$ −6.88730 −0.239931
$$825$$ 0 0
$$826$$ −0.970563 −0.0337702
$$827$$ 14.6863 0.510692 0.255346 0.966850i $$-0.417811\pi$$
0.255346 + 0.966850i $$0.417811\pi$$
$$828$$ −14.0000 −0.486534
$$829$$ 52.9117 1.83770 0.918849 0.394608i $$-0.129120\pi$$
0.918849 + 0.394608i $$0.129120\pi$$
$$830$$ 0 0
$$831$$ −37.6569 −1.30630
$$832$$ −28.4853 −0.987549
$$833$$ −43.1127 −1.49377
$$834$$ −3.31371 −0.114744
$$835$$ 0 0
$$836$$ −1.82843 −0.0632375
$$837$$ 27.3137 0.944100
$$838$$ 8.68629 0.300063
$$839$$ 7.79899 0.269251 0.134626 0.990897i $$-0.457017\pi$$
0.134626 + 0.990897i $$0.457017\pi$$
$$840$$ 0 0
$$841$$ −5.68629 −0.196079
$$842$$ 11.1716 0.384998
$$843$$ −38.9117 −1.34019
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ −4.82843 −0.166005
$$847$$ −0.828427 −0.0284651
$$848$$ 32.4853 1.11555
$$849$$ −12.9706 −0.445149
$$850$$ 0 0
$$851$$ −64.9706 −2.22716
$$852$$ 54.2254 1.85773
$$853$$ 56.4853 1.93402 0.967010 0.254740i $$-0.0819899\pi$$
0.967010 + 0.254740i $$0.0819899\pi$$
$$854$$ −0.686292 −0.0234844
$$855$$ 0 0
$$856$$ 30.0833 1.02822
$$857$$ 48.0833 1.64249 0.821246 0.570574i $$-0.193279\pi$$
0.821246 + 0.570574i $$0.193279\pi$$
$$858$$ 5.65685 0.193122
$$859$$ −47.3137 −1.61432 −0.807161 0.590331i $$-0.798997\pi$$
−0.807161 + 0.590331i $$0.798997\pi$$
$$860$$ 0 0
$$861$$ −10.7452 −0.366194
$$862$$ −2.74517 −0.0935007
$$863$$ −42.9706 −1.46273 −0.731367 0.681984i $$-0.761118\pi$$
−0.731367 + 0.681984i $$0.761118\pi$$
$$864$$ 17.6569 0.600698
$$865$$ 0 0
$$866$$ −0.485281 −0.0164905
$$867$$ 59.2548 2.01240
$$868$$ −10.3431 −0.351069
$$869$$ −5.65685 −0.191896
$$870$$ 0 0
$$871$$ 40.9706 1.38823
$$872$$ 1.31371 0.0444878
$$873$$ 1.17157 0.0396517
$$874$$ 3.17157 0.107280
$$875$$ 0 0
$$876$$ −4.28427 −0.144752
$$877$$ −2.14214 −0.0723348 −0.0361674 0.999346i $$-0.511515\pi$$
−0.0361674 + 0.999346i $$0.511515\pi$$
$$878$$ −7.59798 −0.256419
$$879$$ −52.2843 −1.76350
$$880$$ 0 0
$$881$$ −31.9411 −1.07612 −0.538062 0.842905i $$-0.680843\pi$$
−0.538062 + 0.842905i $$0.680843\pi$$
$$882$$ 2.61522 0.0880592
$$883$$ 33.3137 1.12110 0.560548 0.828122i $$-0.310591\pi$$
0.560548 + 0.828122i $$0.310591\pi$$
$$884$$ −85.2548 −2.86743
$$885$$ 0 0
$$886$$ −3.85786 −0.129607
$$887$$ 15.9411 0.535251 0.267625 0.963523i $$-0.413761\pi$$
0.267625 + 0.963523i $$0.413761\pi$$
$$888$$ −26.9117 −0.903097
$$889$$ −11.5980 −0.388984
$$890$$ 0 0
$$891$$ 11.0000 0.368514
$$892$$ −21.3137 −0.713636
$$893$$ −11.6569 −0.390082
$$894$$ 6.91169 0.231161
$$895$$ 0 0
$$896$$ −8.74517 −0.292155
$$897$$ 104.569 3.49144
$$898$$ 1.23045 0.0410606
$$899$$ 32.9706 1.09963
$$900$$ 0 0
$$901$$ 73.9411 2.46334
$$902$$ 2.68629 0.0894437
$$903$$ 1.37258 0.0456767
$$904$$ 16.0833 0.534921
$$905$$ 0 0
$$906$$ −8.56854 −0.284671
$$907$$ 28.6274 0.950558 0.475279 0.879835i $$-0.342347\pi$$
0.475279 + 0.879835i $$0.342347\pi$$
$$908$$ −22.5685 −0.748963
$$909$$ 9.31371 0.308916
$$910$$ 0 0
$$911$$ 25.1716 0.833971 0.416986 0.908913i $$-0.363087\pi$$
0.416986 + 0.908913i $$0.363087\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ 6.48528 0.214631
$$914$$ 6.54416 0.216461
$$915$$ 0 0
$$916$$ 40.2254 1.32908
$$917$$ −6.05887 −0.200082
$$918$$ 11.3137 0.373408
$$919$$ −48.9706 −1.61539 −0.807695 0.589601i $$-0.799285\pi$$
−0.807695 + 0.589601i $$0.799285\pi$$
$$920$$ 0 0
$$921$$ 53.9411 1.77742
$$922$$ 9.23045 0.303989
$$923$$ −101.255 −3.33284
$$924$$ −3.02944 −0.0996612
$$925$$ 0 0
$$926$$ −12.5442 −0.412227
$$927$$ −4.34315 −0.142648
$$928$$ 21.3137 0.699657
$$929$$ 55.9411 1.83537 0.917684 0.397310i $$-0.130056\pi$$
0.917684 + 0.397310i $$0.130056\pi$$
$$930$$ 0 0
$$931$$ 6.31371 0.206923
$$932$$ 8.20101 0.268633
$$933$$ 1.94113 0.0635496
$$934$$ −1.79899 −0.0588647
$$935$$ 0 0
$$936$$ 10.8284 0.353938
$$937$$ −19.1127 −0.624385 −0.312192 0.950019i $$-0.601063\pi$$
−0.312192 + 0.950019i $$0.601063\pi$$
$$938$$ 2.05887 0.0672246
$$939$$ 33.9411 1.10763
$$940$$ 0 0
$$941$$ −23.8579 −0.777744 −0.388872 0.921292i $$-0.627135\pi$$
−0.388872 + 0.921292i $$0.627135\pi$$
$$942$$ −9.94113 −0.323899
$$943$$ 49.6569 1.61705
$$944$$ −8.48528 −0.276172
$$945$$ 0 0
$$946$$ −0.343146 −0.0111566
$$947$$ −58.2843 −1.89398 −0.946992 0.321257i $$-0.895895\pi$$
−0.946992 + 0.321257i $$0.895895\pi$$
$$948$$ −20.6863 −0.671860
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ −12.2843 −0.398345
$$952$$ −8.97056 −0.290738
$$953$$ 39.1127 1.26698 0.633492 0.773749i $$-0.281621\pi$$
0.633492 + 0.773749i $$0.281621\pi$$
$$954$$ −4.48528 −0.145216
$$955$$ 0 0
$$956$$ −8.56854 −0.277126
$$957$$ 9.65685 0.312162
$$958$$ 1.37258 0.0443461
$$959$$ −6.62742 −0.214010
$$960$$ 0 0
$$961$$ 15.6274 0.504110
$$962$$ 24.0000 0.773791
$$963$$ 18.9706 0.611318
$$964$$ −11.8579 −0.381916
$$965$$ 0 0
$$966$$ 5.25483 0.169072
$$967$$ −12.1421 −0.390465 −0.195232 0.980757i $$-0.562546\pi$$
−0.195232 + 0.980757i $$0.562546\pi$$
$$968$$ 1.58579 0.0509691
$$969$$ −13.6569 −0.438721
$$970$$ 0 0
$$971$$ −15.1127 −0.484990 −0.242495 0.970153i $$-0.577966\pi$$
−0.242495 + 0.970153i $$0.577966\pi$$
$$972$$ 18.2843 0.586468
$$973$$ −3.31371 −0.106233
$$974$$ −6.48528 −0.207802
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ −7.79899 −0.249512 −0.124756 0.992187i $$-0.539815\pi$$
−0.124756 + 0.992187i $$0.539815\pi$$
$$978$$ 1.65685 0.0529804
$$979$$ 4.34315 0.138808
$$980$$ 0 0
$$981$$ 0.828427 0.0264496
$$982$$ 15.3137 0.488680
$$983$$ 8.34315 0.266105 0.133053 0.991109i $$-0.457522\pi$$
0.133053 + 0.991109i $$0.457522\pi$$
$$984$$ 20.5685 0.655701
$$985$$ 0 0
$$986$$ 13.6569 0.434923
$$987$$ −19.3137 −0.614762
$$988$$ 12.4853 0.397210
$$989$$ −6.34315 −0.201700
$$990$$ 0 0
$$991$$ 22.8284 0.725169 0.362584 0.931951i $$-0.381894\pi$$
0.362584 + 0.931951i $$0.381894\pi$$
$$992$$ 30.1421 0.957014
$$993$$ −12.2843 −0.389830
$$994$$ −5.08831 −0.161391
$$995$$ 0 0
$$996$$ 23.7157 0.751462
$$997$$ −16.4853 −0.522094 −0.261047 0.965326i $$-0.584068\pi$$
−0.261047 + 0.965326i $$0.584068\pi$$
$$998$$ −0.284271 −0.00899845
$$999$$ 33.9411 1.07385
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.g.1.1 2
5.2 odd 4 1045.2.b.a.419.2 4
5.3 odd 4 1045.2.b.a.419.3 yes 4
5.4 even 2 5225.2.a.d.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.a.419.2 4 5.2 odd 4
1045.2.b.a.419.3 yes 4 5.3 odd 4
5225.2.a.d.1.2 2 5.4 even 2
5225.2.a.g.1.1 2 1.1 even 1 trivial