# Properties

 Label 5225.2.a.h.1.4 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.71250$$ 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)

 $$f(q)$$ $$=$$ $$q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} +2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +O(q^{10})$$ $$q+0.779856 q^{2} +2.98063 q^{3} -1.39182 q^{4} +2.32446 q^{6} -1.06736 q^{7} -2.64513 q^{8} +5.88418 q^{9} +1.00000 q^{11} -4.14852 q^{12} +0.0563258 q^{13} -0.832387 q^{14} +0.720827 q^{16} +4.53628 q^{17} +4.58881 q^{18} -1.00000 q^{19} -3.18141 q^{21} +0.779856 q^{22} +1.07949 q^{23} -7.88418 q^{24} +0.0439260 q^{26} +8.59667 q^{27} +1.48558 q^{28} +0.299905 q^{29} +9.18548 q^{31} +5.85241 q^{32} +2.98063 q^{33} +3.53764 q^{34} -8.18974 q^{36} -4.50448 q^{37} -0.779856 q^{38} +0.167887 q^{39} +12.0009 q^{41} -2.48104 q^{42} -10.7260 q^{43} -1.39182 q^{44} +0.841844 q^{46} -2.89630 q^{47} +2.14852 q^{48} -5.86074 q^{49} +13.5210 q^{51} -0.0783957 q^{52} +12.3213 q^{53} +6.70416 q^{54} +2.82331 q^{56} -2.98063 q^{57} +0.233882 q^{58} -1.14582 q^{59} -8.09599 q^{61} +7.16335 q^{62} -6.28054 q^{63} +3.12238 q^{64} +2.32446 q^{66} -11.2733 q^{67} -6.31370 q^{68} +3.21755 q^{69} +13.4948 q^{71} -15.5644 q^{72} +11.1470 q^{73} -3.51284 q^{74} +1.39182 q^{76} -1.06736 q^{77} +0.130927 q^{78} +11.4250 q^{79} +7.97100 q^{81} +9.35899 q^{82} +13.9802 q^{83} +4.42797 q^{84} -8.36475 q^{86} +0.893906 q^{87} -2.64513 q^{88} -0.183185 q^{89} -0.0601200 q^{91} -1.50246 q^{92} +27.3785 q^{93} -2.25870 q^{94} +17.4439 q^{96} -5.66263 q^{97} -4.57053 q^{98} +5.88418 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10})$$ 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 $$5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24} - 6 q^{26} + 11 q^{27} + 10 q^{28} + 10 q^{29} + 11 q^{31} - 14 q^{32} - q^{33} - 4 q^{34} - 26 q^{36} - q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{41} + 16 q^{42} - 20 q^{43} + 6 q^{44} - 4 q^{46} + 20 q^{47} - 4 q^{48} + 3 q^{49} + 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 38 q^{56} + q^{57} + 6 q^{58} + 3 q^{59} - 10 q^{61} + 6 q^{62} - 24 q^{63} - 2 q^{66} - 9 q^{67} - 24 q^{68} - 5 q^{69} + 23 q^{71} + 12 q^{72} + 8 q^{74} - 6 q^{76} - 6 q^{77} + 22 q^{78} + 44 q^{79} + q^{81} + 30 q^{82} + 14 q^{83} + 14 q^{84} + 52 q^{86} - 28 q^{87} - 6 q^{88} - 27 q^{89} + 24 q^{91} - 58 q^{92} + 27 q^{93} - 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100})$$ 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.779856 0.551441 0.275721 0.961238i $$-0.411084\pi$$
0.275721 + 0.961238i $$0.411084\pi$$
$$3$$ 2.98063 1.72087 0.860435 0.509561i $$-0.170192\pi$$
0.860435 + 0.509561i $$0.170192\pi$$
$$4$$ −1.39182 −0.695912
$$5$$ 0 0
$$6$$ 2.32446 0.948959
$$7$$ −1.06736 −0.403424 −0.201712 0.979445i $$-0.564651\pi$$
−0.201712 + 0.979445i $$0.564651\pi$$
$$8$$ −2.64513 −0.935196
$$9$$ 5.88418 1.96139
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −4.14852 −1.19757
$$13$$ 0.0563258 0.0156220 0.00781098 0.999969i $$-0.497514\pi$$
0.00781098 + 0.999969i $$0.497514\pi$$
$$14$$ −0.832387 −0.222465
$$15$$ 0 0
$$16$$ 0.720827 0.180207
$$17$$ 4.53628 1.10021 0.550104 0.835096i $$-0.314588\pi$$
0.550104 + 0.835096i $$0.314588\pi$$
$$18$$ 4.58881 1.08159
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −3.18141 −0.694241
$$22$$ 0.779856 0.166266
$$23$$ 1.07949 0.225088 0.112544 0.993647i $$-0.464100\pi$$
0.112544 + 0.993647i $$0.464100\pi$$
$$24$$ −7.88418 −1.60935
$$25$$ 0 0
$$26$$ 0.0439260 0.00861460
$$27$$ 8.59667 1.65443
$$28$$ 1.48558 0.280748
$$29$$ 0.299905 0.0556909 0.0278455 0.999612i $$-0.491135\pi$$
0.0278455 + 0.999612i $$0.491135\pi$$
$$30$$ 0 0
$$31$$ 9.18548 1.64976 0.824880 0.565307i $$-0.191242\pi$$
0.824880 + 0.565307i $$0.191242\pi$$
$$32$$ 5.85241 1.03457
$$33$$ 2.98063 0.518862
$$34$$ 3.53764 0.606701
$$35$$ 0 0
$$36$$ −8.18974 −1.36496
$$37$$ −4.50448 −0.740531 −0.370266 0.928926i $$-0.620733\pi$$
−0.370266 + 0.928926i $$0.620733\pi$$
$$38$$ −0.779856 −0.126509
$$39$$ 0.167887 0.0268834
$$40$$ 0 0
$$41$$ 12.0009 1.87423 0.937115 0.349020i $$-0.113486\pi$$
0.937115 + 0.349020i $$0.113486\pi$$
$$42$$ −2.48104 −0.382833
$$43$$ −10.7260 −1.63570 −0.817851 0.575430i $$-0.804835\pi$$
−0.817851 + 0.575430i $$0.804835\pi$$
$$44$$ −1.39182 −0.209826
$$45$$ 0 0
$$46$$ 0.841844 0.124123
$$47$$ −2.89630 −0.422469 −0.211235 0.977435i $$-0.567748\pi$$
−0.211235 + 0.977435i $$0.567748\pi$$
$$48$$ 2.14852 0.310112
$$49$$ −5.86074 −0.837249
$$50$$ 0 0
$$51$$ 13.5210 1.89332
$$52$$ −0.0783957 −0.0108715
$$53$$ 12.3213 1.69246 0.846230 0.532818i $$-0.178867\pi$$
0.846230 + 0.532818i $$0.178867\pi$$
$$54$$ 6.70416 0.912321
$$55$$ 0 0
$$56$$ 2.82331 0.377281
$$57$$ −2.98063 −0.394795
$$58$$ 0.233882 0.0307103
$$59$$ −1.14582 −0.149173 −0.0745863 0.997215i $$-0.523764\pi$$
−0.0745863 + 0.997215i $$0.523764\pi$$
$$60$$ 0 0
$$61$$ −8.09599 −1.03659 −0.518293 0.855203i $$-0.673432\pi$$
−0.518293 + 0.855203i $$0.673432\pi$$
$$62$$ 7.16335 0.909746
$$63$$ −6.28054 −0.791273
$$64$$ 3.12238 0.390298
$$65$$ 0 0
$$66$$ 2.32446 0.286122
$$67$$ −11.2733 −1.37726 −0.688628 0.725115i $$-0.741787\pi$$
−0.688628 + 0.725115i $$0.741787\pi$$
$$68$$ −6.31370 −0.765649
$$69$$ 3.21755 0.387348
$$70$$ 0 0
$$71$$ 13.4948 1.60154 0.800771 0.598970i $$-0.204423\pi$$
0.800771 + 0.598970i $$0.204423\pi$$
$$72$$ −15.5644 −1.83429
$$73$$ 11.1470 1.30466 0.652330 0.757935i $$-0.273792\pi$$
0.652330 + 0.757935i $$0.273792\pi$$
$$74$$ −3.51284 −0.408360
$$75$$ 0 0
$$76$$ 1.39182 0.159653
$$77$$ −1.06736 −0.121637
$$78$$ 0.130927 0.0148246
$$79$$ 11.4250 1.28541 0.642706 0.766113i $$-0.277812\pi$$
0.642706 + 0.766113i $$0.277812\pi$$
$$80$$ 0 0
$$81$$ 7.97100 0.885666
$$82$$ 9.35899 1.03353
$$83$$ 13.9802 1.53452 0.767261 0.641335i $$-0.221619\pi$$
0.767261 + 0.641335i $$0.221619\pi$$
$$84$$ 4.42797 0.483131
$$85$$ 0 0
$$86$$ −8.36475 −0.901994
$$87$$ 0.893906 0.0958368
$$88$$ −2.64513 −0.281972
$$89$$ −0.183185 −0.0194176 −0.00970878 0.999953i $$-0.503090\pi$$
−0.00970878 + 0.999953i $$0.503090\pi$$
$$90$$ 0 0
$$91$$ −0.0601200 −0.00630228
$$92$$ −1.50246 −0.156642
$$93$$ 27.3785 2.83902
$$94$$ −2.25870 −0.232967
$$95$$ 0 0
$$96$$ 17.4439 1.78036
$$97$$ −5.66263 −0.574953 −0.287477 0.957788i $$-0.592816\pi$$
−0.287477 + 0.957788i $$0.592816\pi$$
$$98$$ −4.57053 −0.461694
$$99$$ 5.88418 0.591382
$$100$$ 0 0
$$101$$ 8.00759 0.796785 0.398392 0.917215i $$-0.369568\pi$$
0.398392 + 0.917215i $$0.369568\pi$$
$$102$$ 10.5444 1.04405
$$103$$ −6.18725 −0.609648 −0.304824 0.952409i $$-0.598598\pi$$
−0.304824 + 0.952409i $$0.598598\pi$$
$$104$$ −0.148989 −0.0146096
$$105$$ 0 0
$$106$$ 9.60883 0.933292
$$107$$ 7.29027 0.704777 0.352388 0.935854i $$-0.385370\pi$$
0.352388 + 0.935854i $$0.385370\pi$$
$$108$$ −11.9651 −1.15134
$$109$$ 7.79895 0.747004 0.373502 0.927629i $$-0.378157\pi$$
0.373502 + 0.927629i $$0.378157\pi$$
$$110$$ 0 0
$$111$$ −13.4262 −1.27436
$$112$$ −0.769382 −0.0726998
$$113$$ 0.430558 0.0405035 0.0202517 0.999795i $$-0.493553\pi$$
0.0202517 + 0.999795i $$0.493553\pi$$
$$114$$ −2.32446 −0.217706
$$115$$ 0 0
$$116$$ −0.417415 −0.0387560
$$117$$ 0.331431 0.0306408
$$118$$ −0.893572 −0.0822600
$$119$$ −4.84184 −0.443851
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −6.31370 −0.571616
$$123$$ 35.7704 3.22531
$$124$$ −12.7846 −1.14809
$$125$$ 0 0
$$126$$ −4.89791 −0.436341
$$127$$ 3.13888 0.278531 0.139265 0.990255i $$-0.455526\pi$$
0.139265 + 0.990255i $$0.455526\pi$$
$$128$$ −9.26981 −0.819343
$$129$$ −31.9703 −2.81483
$$130$$ 0 0
$$131$$ 12.4315 1.08615 0.543075 0.839684i $$-0.317260\pi$$
0.543075 + 0.839684i $$0.317260\pi$$
$$132$$ −4.14852 −0.361082
$$133$$ 1.06736 0.0925519
$$134$$ −8.79157 −0.759476
$$135$$ 0 0
$$136$$ −11.9991 −1.02891
$$137$$ −13.8301 −1.18159 −0.590794 0.806822i $$-0.701186\pi$$
−0.590794 + 0.806822i $$0.701186\pi$$
$$138$$ 2.50923 0.213600
$$139$$ 15.9968 1.35683 0.678417 0.734677i $$-0.262666\pi$$
0.678417 + 0.734677i $$0.262666\pi$$
$$140$$ 0 0
$$141$$ −8.63281 −0.727014
$$142$$ 10.5240 0.883157
$$143$$ 0.0563258 0.00471020
$$144$$ 4.24147 0.353456
$$145$$ 0 0
$$146$$ 8.69307 0.719443
$$147$$ −17.4687 −1.44080
$$148$$ 6.26944 0.515345
$$149$$ 11.3620 0.930812 0.465406 0.885097i $$-0.345908\pi$$
0.465406 + 0.885097i $$0.345908\pi$$
$$150$$ 0 0
$$151$$ 4.66341 0.379503 0.189751 0.981832i $$-0.439232\pi$$
0.189751 + 0.981832i $$0.439232\pi$$
$$152$$ 2.64513 0.214549
$$153$$ 26.6923 2.15794
$$154$$ −0.832387 −0.0670757
$$155$$ 0 0
$$156$$ −0.233669 −0.0187085
$$157$$ 9.05206 0.722433 0.361217 0.932482i $$-0.382361\pi$$
0.361217 + 0.932482i $$0.382361\pi$$
$$158$$ 8.90984 0.708829
$$159$$ 36.7253 2.91250
$$160$$ 0 0
$$161$$ −1.15220 −0.0908062
$$162$$ 6.21623 0.488393
$$163$$ −2.36761 −0.185446 −0.0927229 0.995692i $$-0.529557\pi$$
−0.0927229 + 0.995692i $$0.529557\pi$$
$$164$$ −16.7032 −1.30430
$$165$$ 0 0
$$166$$ 10.9025 0.846199
$$167$$ 9.27361 0.717613 0.358807 0.933412i $$-0.383184\pi$$
0.358807 + 0.933412i $$0.383184\pi$$
$$168$$ 8.41526 0.649251
$$169$$ −12.9968 −0.999756
$$170$$ 0 0
$$171$$ −5.88418 −0.449974
$$172$$ 14.9287 1.13831
$$173$$ −10.7172 −0.814812 −0.407406 0.913247i $$-0.633567\pi$$
−0.407406 + 0.913247i $$0.633567\pi$$
$$174$$ 0.697118 0.0528484
$$175$$ 0 0
$$176$$ 0.720827 0.0543344
$$177$$ −3.41526 −0.256707
$$178$$ −0.142858 −0.0107076
$$179$$ −22.9070 −1.71215 −0.856073 0.516854i $$-0.827103\pi$$
−0.856073 + 0.516854i $$0.827103\pi$$
$$180$$ 0 0
$$181$$ 5.90522 0.438931 0.219466 0.975620i $$-0.429569\pi$$
0.219466 + 0.975620i $$0.429569\pi$$
$$182$$ −0.0468849 −0.00347534
$$183$$ −24.1312 −1.78383
$$184$$ −2.85539 −0.210502
$$185$$ 0 0
$$186$$ 21.3513 1.56555
$$187$$ 4.53628 0.331725
$$188$$ 4.03114 0.294001
$$189$$ −9.17575 −0.667438
$$190$$ 0 0
$$191$$ 6.44628 0.466437 0.233218 0.972424i $$-0.425074\pi$$
0.233218 + 0.972424i $$0.425074\pi$$
$$192$$ 9.30668 0.671652
$$193$$ −1.43606 −0.103370 −0.0516849 0.998663i $$-0.516459\pi$$
−0.0516849 + 0.998663i $$0.516459\pi$$
$$194$$ −4.41604 −0.317053
$$195$$ 0 0
$$196$$ 8.15713 0.582652
$$197$$ 4.75399 0.338708 0.169354 0.985555i $$-0.445832\pi$$
0.169354 + 0.985555i $$0.445832\pi$$
$$198$$ 4.58881 0.326112
$$199$$ −2.36002 −0.167298 −0.0836489 0.996495i $$-0.526657\pi$$
−0.0836489 + 0.996495i $$0.526657\pi$$
$$200$$ 0 0
$$201$$ −33.6017 −2.37008
$$202$$ 6.24476 0.439380
$$203$$ −0.320107 −0.0224671
$$204$$ −18.8188 −1.31758
$$205$$ 0 0
$$206$$ −4.82517 −0.336185
$$207$$ 6.35189 0.441487
$$208$$ 0.0406011 0.00281518
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −24.5133 −1.68756 −0.843781 0.536687i $$-0.819676\pi$$
−0.843781 + 0.536687i $$0.819676\pi$$
$$212$$ −17.1491 −1.17780
$$213$$ 40.2232 2.75605
$$214$$ 5.68536 0.388643
$$215$$ 0 0
$$216$$ −22.7393 −1.54722
$$217$$ −9.80422 −0.665554
$$218$$ 6.08205 0.411929
$$219$$ 33.2252 2.24515
$$220$$ 0 0
$$221$$ 0.255509 0.0171874
$$222$$ −10.4705 −0.702734
$$223$$ −8.47427 −0.567479 −0.283740 0.958901i $$-0.591575\pi$$
−0.283740 + 0.958901i $$0.591575\pi$$
$$224$$ −6.24663 −0.417371
$$225$$ 0 0
$$226$$ 0.335773 0.0223353
$$227$$ −13.7491 −0.912558 −0.456279 0.889837i $$-0.650818\pi$$
−0.456279 + 0.889837i $$0.650818\pi$$
$$228$$ 4.14852 0.274742
$$229$$ −1.99640 −0.131926 −0.0659629 0.997822i $$-0.521012\pi$$
−0.0659629 + 0.997822i $$0.521012\pi$$
$$230$$ 0 0
$$231$$ −3.18141 −0.209321
$$232$$ −0.793288 −0.0520819
$$233$$ −11.3481 −0.743437 −0.371718 0.928346i $$-0.621231\pi$$
−0.371718 + 0.928346i $$0.621231\pi$$
$$234$$ 0.258468 0.0168966
$$235$$ 0 0
$$236$$ 1.59478 0.103811
$$237$$ 34.0537 2.21203
$$238$$ −3.77594 −0.244758
$$239$$ 5.21045 0.337036 0.168518 0.985699i $$-0.446102\pi$$
0.168518 + 0.985699i $$0.446102\pi$$
$$240$$ 0 0
$$241$$ 6.60827 0.425676 0.212838 0.977087i $$-0.431729\pi$$
0.212838 + 0.977087i $$0.431729\pi$$
$$242$$ 0.779856 0.0501310
$$243$$ −2.03139 −0.130314
$$244$$ 11.2682 0.721373
$$245$$ 0 0
$$246$$ 27.8957 1.77857
$$247$$ −0.0563258 −0.00358393
$$248$$ −24.2968 −1.54285
$$249$$ 41.6697 2.64071
$$250$$ 0 0
$$251$$ 24.0024 1.51502 0.757510 0.652823i $$-0.226415\pi$$
0.757510 + 0.652823i $$0.226415\pi$$
$$252$$ 8.74141 0.550657
$$253$$ 1.07949 0.0678667
$$254$$ 2.44788 0.153593
$$255$$ 0 0
$$256$$ −13.4739 −0.842117
$$257$$ 5.68903 0.354872 0.177436 0.984132i $$-0.443220\pi$$
0.177436 + 0.984132i $$0.443220\pi$$
$$258$$ −24.9322 −1.55221
$$259$$ 4.80790 0.298748
$$260$$ 0 0
$$261$$ 1.76469 0.109232
$$262$$ 9.69481 0.598948
$$263$$ −13.9857 −0.862395 −0.431197 0.902258i $$-0.641909\pi$$
−0.431197 + 0.902258i $$0.641909\pi$$
$$264$$ −7.88418 −0.485237
$$265$$ 0 0
$$266$$ 0.832387 0.0510369
$$267$$ −0.546007 −0.0334151
$$268$$ 15.6905 0.958450
$$269$$ 15.4020 0.939075 0.469538 0.882912i $$-0.344421\pi$$
0.469538 + 0.882912i $$0.344421\pi$$
$$270$$ 0 0
$$271$$ −25.3911 −1.54240 −0.771199 0.636595i $$-0.780342\pi$$
−0.771199 + 0.636595i $$0.780342\pi$$
$$272$$ 3.26987 0.198265
$$273$$ −0.179196 −0.0108454
$$274$$ −10.7855 −0.651577
$$275$$ 0 0
$$276$$ −4.47827 −0.269560
$$277$$ −19.9798 −1.20047 −0.600235 0.799824i $$-0.704926\pi$$
−0.600235 + 0.799824i $$0.704926\pi$$
$$278$$ 12.4752 0.748214
$$279$$ 54.0490 3.23583
$$280$$ 0 0
$$281$$ 6.18130 0.368745 0.184373 0.982856i $$-0.440975\pi$$
0.184373 + 0.982856i $$0.440975\pi$$
$$282$$ −6.73235 −0.400906
$$283$$ 7.12127 0.423316 0.211658 0.977344i $$-0.432114\pi$$
0.211658 + 0.977344i $$0.432114\pi$$
$$284$$ −18.7825 −1.11453
$$285$$ 0 0
$$286$$ 0.0439260 0.00259740
$$287$$ −12.8093 −0.756110
$$288$$ 34.4366 2.02920
$$289$$ 3.57781 0.210459
$$290$$ 0 0
$$291$$ −16.8782 −0.989420
$$292$$ −15.5147 −0.907929
$$293$$ −19.1342 −1.11783 −0.558916 0.829224i $$-0.688783\pi$$
−0.558916 + 0.829224i $$0.688783\pi$$
$$294$$ −13.6231 −0.794514
$$295$$ 0 0
$$296$$ 11.9149 0.692542
$$297$$ 8.59667 0.498829
$$298$$ 8.86073 0.513288
$$299$$ 0.0608030 0.00351633
$$300$$ 0 0
$$301$$ 11.4485 0.659882
$$302$$ 3.63679 0.209274
$$303$$ 23.8677 1.37116
$$304$$ −0.720827 −0.0413422
$$305$$ 0 0
$$306$$ 20.8161 1.18998
$$307$$ −6.82573 −0.389565 −0.194782 0.980846i $$-0.562400\pi$$
−0.194782 + 0.980846i $$0.562400\pi$$
$$308$$ 1.48558 0.0846487
$$309$$ −18.4419 −1.04912
$$310$$ 0 0
$$311$$ −12.8609 −0.729276 −0.364638 0.931149i $$-0.618807\pi$$
−0.364638 + 0.931149i $$0.618807\pi$$
$$312$$ −0.444083 −0.0251412
$$313$$ 3.01707 0.170535 0.0852673 0.996358i $$-0.472826\pi$$
0.0852673 + 0.996358i $$0.472826\pi$$
$$314$$ 7.05930 0.398380
$$315$$ 0 0
$$316$$ −15.9016 −0.894534
$$317$$ −17.7857 −0.998943 −0.499471 0.866330i $$-0.666472\pi$$
−0.499471 + 0.866330i $$0.666472\pi$$
$$318$$ 28.6404 1.60607
$$319$$ 0.299905 0.0167914
$$320$$ 0 0
$$321$$ 21.7296 1.21283
$$322$$ −0.898551 −0.0500743
$$323$$ −4.53628 −0.252405
$$324$$ −11.0942 −0.616346
$$325$$ 0 0
$$326$$ −1.84640 −0.102262
$$327$$ 23.2458 1.28550
$$328$$ −31.7441 −1.75277
$$329$$ 3.09140 0.170434
$$330$$ 0 0
$$331$$ −26.3860 −1.45030 −0.725152 0.688589i $$-0.758230\pi$$
−0.725152 + 0.688589i $$0.758230\pi$$
$$332$$ −19.4579 −1.06789
$$333$$ −26.5051 −1.45247
$$334$$ 7.23207 0.395722
$$335$$ 0 0
$$336$$ −2.29325 −0.125107
$$337$$ 13.2024 0.719181 0.359590 0.933110i $$-0.382916\pi$$
0.359590 + 0.933110i $$0.382916\pi$$
$$338$$ −10.1357 −0.551307
$$339$$ 1.28334 0.0697012
$$340$$ 0 0
$$341$$ 9.18548 0.497422
$$342$$ −4.58881 −0.248134
$$343$$ 13.7271 0.741191
$$344$$ 28.3718 1.52970
$$345$$ 0 0
$$346$$ −8.35786 −0.449321
$$347$$ −16.4809 −0.884740 −0.442370 0.896833i $$-0.645862\pi$$
−0.442370 + 0.896833i $$0.645862\pi$$
$$348$$ −1.24416 −0.0666940
$$349$$ −19.0165 −1.01793 −0.508966 0.860787i $$-0.669972\pi$$
−0.508966 + 0.860787i $$0.669972\pi$$
$$350$$ 0 0
$$351$$ 0.484214 0.0258455
$$352$$ 5.85241 0.311934
$$353$$ −13.1955 −0.702325 −0.351162 0.936315i $$-0.614213\pi$$
−0.351162 + 0.936315i $$0.614213\pi$$
$$354$$ −2.66341 −0.141559
$$355$$ 0 0
$$356$$ 0.254961 0.0135129
$$357$$ −14.4318 −0.763810
$$358$$ −17.8641 −0.944148
$$359$$ −2.88756 −0.152400 −0.0761998 0.997093i $$-0.524279\pi$$
−0.0761998 + 0.997093i $$0.524279\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.60522 0.242045
$$363$$ 2.98063 0.156443
$$364$$ 0.0836765 0.00438584
$$365$$ 0 0
$$366$$ −18.8188 −0.983676
$$367$$ 2.10833 0.110054 0.0550269 0.998485i $$-0.482476\pi$$
0.0550269 + 0.998485i $$0.482476\pi$$
$$368$$ 0.778123 0.0405624
$$369$$ 70.6156 3.67610
$$370$$ 0 0
$$371$$ −13.1513 −0.682780
$$372$$ −38.1061 −1.97571
$$373$$ −8.07305 −0.418007 −0.209003 0.977915i $$-0.567022\pi$$
−0.209003 + 0.977915i $$0.567022\pi$$
$$374$$ 3.53764 0.182927
$$375$$ 0 0
$$376$$ 7.66111 0.395091
$$377$$ 0.0168924 0.000870002 0
$$378$$ −7.15576 −0.368053
$$379$$ 14.3355 0.736365 0.368183 0.929754i $$-0.379980\pi$$
0.368183 + 0.929754i $$0.379980\pi$$
$$380$$ 0 0
$$381$$ 9.35586 0.479315
$$382$$ 5.02717 0.257212
$$383$$ 9.79867 0.500689 0.250344 0.968157i $$-0.419456\pi$$
0.250344 + 0.968157i $$0.419456\pi$$
$$384$$ −27.6299 −1.40998
$$385$$ 0 0
$$386$$ −1.11992 −0.0570024
$$387$$ −63.1138 −3.20825
$$388$$ 7.88139 0.400117
$$389$$ −19.9236 −1.01016 −0.505082 0.863071i $$-0.668538\pi$$
−0.505082 + 0.863071i $$0.668538\pi$$
$$390$$ 0 0
$$391$$ 4.89685 0.247644
$$392$$ 15.5024 0.782992
$$393$$ 37.0539 1.86912
$$394$$ 3.70743 0.186778
$$395$$ 0 0
$$396$$ −8.18974 −0.411550
$$397$$ −13.9941 −0.702342 −0.351171 0.936311i $$-0.614216\pi$$
−0.351171 + 0.936311i $$0.614216\pi$$
$$398$$ −1.84048 −0.0922549
$$399$$ 3.18141 0.159270
$$400$$ 0 0
$$401$$ −18.6293 −0.930301 −0.465151 0.885232i $$-0.654000\pi$$
−0.465151 + 0.885232i $$0.654000\pi$$
$$402$$ −26.2045 −1.30696
$$403$$ 0.517380 0.0257725
$$404$$ −11.1452 −0.554492
$$405$$ 0 0
$$406$$ −0.249637 −0.0123893
$$407$$ −4.50448 −0.223279
$$408$$ −35.7648 −1.77062
$$409$$ −30.8428 −1.52508 −0.762540 0.646941i $$-0.776048\pi$$
−0.762540 + 0.646941i $$0.776048\pi$$
$$410$$ 0 0
$$411$$ −41.2226 −2.03336
$$412$$ 8.61157 0.424262
$$413$$ 1.22300 0.0601799
$$414$$ 4.95356 0.243454
$$415$$ 0 0
$$416$$ 0.329642 0.0161620
$$417$$ 47.6807 2.33493
$$418$$ −0.779856 −0.0381440
$$419$$ 37.9347 1.85323 0.926616 0.376009i $$-0.122704\pi$$
0.926616 + 0.376009i $$0.122704\pi$$
$$420$$ 0 0
$$421$$ −18.4642 −0.899891 −0.449945 0.893056i $$-0.648557\pi$$
−0.449945 + 0.893056i $$0.648557\pi$$
$$422$$ −19.1168 −0.930592
$$423$$ −17.0423 −0.828627
$$424$$ −32.5915 −1.58278
$$425$$ 0 0
$$426$$ 31.3683 1.51980
$$427$$ 8.64134 0.418184
$$428$$ −10.1468 −0.490463
$$429$$ 0.167887 0.00810564
$$430$$ 0 0
$$431$$ 33.6742 1.62203 0.811013 0.585027i $$-0.198916\pi$$
0.811013 + 0.585027i $$0.198916\pi$$
$$432$$ 6.19671 0.298139
$$433$$ −11.4041 −0.548047 −0.274024 0.961723i $$-0.588355\pi$$
−0.274024 + 0.961723i $$0.588355\pi$$
$$434$$ −7.64588 −0.367014
$$435$$ 0 0
$$436$$ −10.8548 −0.519849
$$437$$ −1.07949 −0.0516388
$$438$$ 25.9108 1.23807
$$439$$ −3.76890 −0.179880 −0.0899399 0.995947i $$-0.528667\pi$$
−0.0899399 + 0.995947i $$0.528667\pi$$
$$440$$ 0 0
$$441$$ −34.4856 −1.64217
$$442$$ 0.199261 0.00947786
$$443$$ −4.51841 −0.214676 −0.107338 0.994223i $$-0.534233\pi$$
−0.107338 + 0.994223i $$0.534233\pi$$
$$444$$ 18.6869 0.886842
$$445$$ 0 0
$$446$$ −6.60871 −0.312931
$$447$$ 33.8660 1.60181
$$448$$ −3.33271 −0.157456
$$449$$ −37.3611 −1.76318 −0.881590 0.472016i $$-0.843526\pi$$
−0.881590 + 0.472016i $$0.843526\pi$$
$$450$$ 0 0
$$451$$ 12.0009 0.565102
$$452$$ −0.599261 −0.0281869
$$453$$ 13.8999 0.653075
$$454$$ −10.7223 −0.503222
$$455$$ 0 0
$$456$$ 7.88418 0.369210
$$457$$ −9.37773 −0.438672 −0.219336 0.975649i $$-0.570389\pi$$
−0.219336 + 0.975649i $$0.570389\pi$$
$$458$$ −1.55690 −0.0727494
$$459$$ 38.9969 1.82022
$$460$$ 0 0
$$461$$ 31.6785 1.47542 0.737708 0.675120i $$-0.235908\pi$$
0.737708 + 0.675120i $$0.235908\pi$$
$$462$$ −2.48104 −0.115429
$$463$$ 37.1284 1.72550 0.862752 0.505628i $$-0.168739\pi$$
0.862752 + 0.505628i $$0.168739\pi$$
$$464$$ 0.216179 0.0100359
$$465$$ 0 0
$$466$$ −8.84986 −0.409962
$$467$$ 15.9678 0.738902 0.369451 0.929250i $$-0.379546\pi$$
0.369451 + 0.929250i $$0.379546\pi$$
$$468$$ −0.461294 −0.0213233
$$469$$ 12.0327 0.555619
$$470$$ 0 0
$$471$$ 26.9809 1.24321
$$472$$ 3.03084 0.139506
$$473$$ −10.7260 −0.493183
$$474$$ 26.5570 1.21980
$$475$$ 0 0
$$476$$ 6.73900 0.308882
$$477$$ 72.5006 3.31958
$$478$$ 4.06340 0.185855
$$479$$ 28.1729 1.28725 0.643627 0.765340i $$-0.277429\pi$$
0.643627 + 0.765340i $$0.277429\pi$$
$$480$$ 0 0
$$481$$ −0.253718 −0.0115686
$$482$$ 5.15350 0.234735
$$483$$ −3.43429 −0.156266
$$484$$ −1.39182 −0.0632648
$$485$$ 0 0
$$486$$ −1.58419 −0.0718604
$$487$$ −24.0613 −1.09032 −0.545161 0.838331i $$-0.683532\pi$$
−0.545161 + 0.838331i $$0.683532\pi$$
$$488$$ 21.4150 0.969410
$$489$$ −7.05699 −0.319128
$$490$$ 0 0
$$491$$ −29.0197 −1.30964 −0.654821 0.755784i $$-0.727256\pi$$
−0.654821 + 0.755784i $$0.727256\pi$$
$$492$$ −49.7861 −2.24453
$$493$$ 1.36045 0.0612716
$$494$$ −0.0439260 −0.00197632
$$495$$ 0 0
$$496$$ 6.62114 0.297298
$$497$$ −14.4039 −0.646102
$$498$$ 32.4964 1.45620
$$499$$ −15.0257 −0.672642 −0.336321 0.941747i $$-0.609183\pi$$
−0.336321 + 0.941747i $$0.609183\pi$$
$$500$$ 0 0
$$501$$ 27.6412 1.23492
$$502$$ 18.7184 0.835445
$$503$$ 28.7785 1.28317 0.641585 0.767052i $$-0.278277\pi$$
0.641585 + 0.767052i $$0.278277\pi$$
$$504$$ 16.6129 0.739996
$$505$$ 0 0
$$506$$ 0.841844 0.0374245
$$507$$ −38.7388 −1.72045
$$508$$ −4.36878 −0.193833
$$509$$ −10.4772 −0.464395 −0.232197 0.972669i $$-0.574592\pi$$
−0.232197 + 0.972669i $$0.574592\pi$$
$$510$$ 0 0
$$511$$ −11.8979 −0.526332
$$512$$ 8.03194 0.354965
$$513$$ −8.59667 −0.379552
$$514$$ 4.43662 0.195691
$$515$$ 0 0
$$516$$ 44.4971 1.95888
$$517$$ −2.89630 −0.127379
$$518$$ 3.74947 0.164742
$$519$$ −31.9440 −1.40219
$$520$$ 0 0
$$521$$ −24.7590 −1.08471 −0.542357 0.840148i $$-0.682468\pi$$
−0.542357 + 0.840148i $$0.682468\pi$$
$$522$$ 1.37621 0.0602349
$$523$$ −14.8566 −0.649635 −0.324818 0.945777i $$-0.605303\pi$$
−0.324818 + 0.945777i $$0.605303\pi$$
$$524$$ −17.3025 −0.755865
$$525$$ 0 0
$$526$$ −10.9068 −0.475560
$$527$$ 41.6679 1.81508
$$528$$ 2.14852 0.0935023
$$529$$ −21.8347 −0.949335
$$530$$ 0 0
$$531$$ −6.74219 −0.292586
$$532$$ −1.48558 −0.0644080
$$533$$ 0.675962 0.0292792
$$534$$ −0.425807 −0.0184265
$$535$$ 0 0
$$536$$ 29.8195 1.28801
$$537$$ −68.2773 −2.94638
$$538$$ 12.0113 0.517845
$$539$$ −5.86074 −0.252440
$$540$$ 0 0
$$541$$ 3.88960 0.167227 0.0836134 0.996498i $$-0.473354\pi$$
0.0836134 + 0.996498i $$0.473354\pi$$
$$542$$ −19.8014 −0.850541
$$543$$ 17.6013 0.755343
$$544$$ 26.5481 1.13824
$$545$$ 0 0
$$546$$ −0.139747 −0.00598061
$$547$$ −17.5180 −0.749015 −0.374507 0.927224i $$-0.622188\pi$$
−0.374507 + 0.927224i $$0.622188\pi$$
$$548$$ 19.2491 0.822283
$$549$$ −47.6382 −2.03315
$$550$$ 0 0
$$551$$ −0.299905 −0.0127764
$$552$$ −8.51086 −0.362246
$$553$$ −12.1946 −0.518567
$$554$$ −15.5814 −0.661988
$$555$$ 0 0
$$556$$ −22.2648 −0.944237
$$557$$ −28.9860 −1.22818 −0.614088 0.789237i $$-0.710476\pi$$
−0.614088 + 0.789237i $$0.710476\pi$$
$$558$$ 42.1504 1.78437
$$559$$ −0.604152 −0.0255529
$$560$$ 0 0
$$561$$ 13.5210 0.570856
$$562$$ 4.82052 0.203341
$$563$$ −29.6112 −1.24796 −0.623982 0.781438i $$-0.714486\pi$$
−0.623982 + 0.781438i $$0.714486\pi$$
$$564$$ 12.0154 0.505938
$$565$$ 0 0
$$566$$ 5.55356 0.233434
$$567$$ −8.50793 −0.357299
$$568$$ −35.6957 −1.49776
$$569$$ −4.05818 −0.170128 −0.0850640 0.996375i $$-0.527109\pi$$
−0.0850640 + 0.996375i $$0.527109\pi$$
$$570$$ 0 0
$$571$$ 20.3380 0.851117 0.425559 0.904931i $$-0.360078\pi$$
0.425559 + 0.904931i $$0.360078\pi$$
$$572$$ −0.0783957 −0.00327789
$$573$$ 19.2140 0.802677
$$574$$ −9.98942 −0.416950
$$575$$ 0 0
$$576$$ 18.3726 0.765527
$$577$$ −24.4768 −1.01898 −0.509492 0.860476i $$-0.670167\pi$$
−0.509492 + 0.860476i $$0.670167\pi$$
$$578$$ 2.79017 0.116056
$$579$$ −4.28037 −0.177886
$$580$$ 0 0
$$581$$ −14.9219 −0.619064
$$582$$ −13.1626 −0.545607
$$583$$ 12.3213 0.510296
$$584$$ −29.4854 −1.22011
$$585$$ 0 0
$$586$$ −14.9219 −0.616419
$$587$$ 3.34628 0.138116 0.0690579 0.997613i $$-0.478001\pi$$
0.0690579 + 0.997613i $$0.478001\pi$$
$$588$$ 24.3134 1.00267
$$589$$ −9.18548 −0.378481
$$590$$ 0 0
$$591$$ 14.1699 0.582872
$$592$$ −3.24695 −0.133449
$$593$$ 39.2063 1.61001 0.805006 0.593267i $$-0.202162\pi$$
0.805006 + 0.593267i $$0.202162\pi$$
$$594$$ 6.70416 0.275075
$$595$$ 0 0
$$596$$ −15.8139 −0.647764
$$597$$ −7.03437 −0.287898
$$598$$ 0.0474175 0.00193905
$$599$$ 5.72987 0.234116 0.117058 0.993125i $$-0.462654\pi$$
0.117058 + 0.993125i $$0.462654\pi$$
$$600$$ 0 0
$$601$$ 13.9163 0.567656 0.283828 0.958875i $$-0.408395\pi$$
0.283828 + 0.958875i $$0.408395\pi$$
$$602$$ 8.92820 0.363886
$$603$$ −66.3343 −2.70134
$$604$$ −6.49065 −0.264101
$$605$$ 0 0
$$606$$ 18.6134 0.756116
$$607$$ −0.156175 −0.00633897 −0.00316948 0.999995i $$-0.501009\pi$$
−0.00316948 + 0.999995i $$0.501009\pi$$
$$608$$ −5.85241 −0.237347
$$609$$ −0.954120 −0.0386629
$$610$$ 0 0
$$611$$ −0.163137 −0.00659980
$$612$$ −37.1509 −1.50174
$$613$$ −40.1902 −1.62327 −0.811634 0.584166i $$-0.801422\pi$$
−0.811634 + 0.584166i $$0.801422\pi$$
$$614$$ −5.32308 −0.214822
$$615$$ 0 0
$$616$$ 2.82331 0.113755
$$617$$ −12.8606 −0.517747 −0.258874 0.965911i $$-0.583351\pi$$
−0.258874 + 0.965911i $$0.583351\pi$$
$$618$$ −14.3820 −0.578531
$$619$$ −2.03398 −0.0817526 −0.0408763 0.999164i $$-0.513015\pi$$
−0.0408763 + 0.999164i $$0.513015\pi$$
$$620$$ 0 0
$$621$$ 9.27999 0.372393
$$622$$ −10.0297 −0.402153
$$623$$ 0.195524 0.00783352
$$624$$ 0.121017 0.00484456
$$625$$ 0 0
$$626$$ 2.35288 0.0940398
$$627$$ −2.98063 −0.119035
$$628$$ −12.5989 −0.502750
$$629$$ −20.4336 −0.814739
$$630$$ 0 0
$$631$$ 37.6984 1.50075 0.750375 0.661012i $$-0.229873\pi$$
0.750375 + 0.661012i $$0.229873\pi$$
$$632$$ −30.2206 −1.20211
$$633$$ −73.0651 −2.90408
$$634$$ −13.8703 −0.550858
$$635$$ 0 0
$$636$$ −51.1151 −2.02685
$$637$$ −0.330111 −0.0130795
$$638$$ 0.233882 0.00925950
$$639$$ 79.4060 3.14125
$$640$$ 0 0
$$641$$ 2.84360 0.112315 0.0561576 0.998422i $$-0.482115\pi$$
0.0561576 + 0.998422i $$0.482115\pi$$
$$642$$ 16.9460 0.668804
$$643$$ −30.6389 −1.20828 −0.604140 0.796878i $$-0.706483\pi$$
−0.604140 + 0.796878i $$0.706483\pi$$
$$644$$ 1.60366 0.0631932
$$645$$ 0 0
$$646$$ −3.53764 −0.139187
$$647$$ 6.76617 0.266006 0.133003 0.991116i $$-0.457538\pi$$
0.133003 + 0.991116i $$0.457538\pi$$
$$648$$ −21.0844 −0.828272
$$649$$ −1.14582 −0.0449772
$$650$$ 0 0
$$651$$ −29.2228 −1.14533
$$652$$ 3.29530 0.129054
$$653$$ −11.9015 −0.465743 −0.232872 0.972508i $$-0.574812\pi$$
−0.232872 + 0.972508i $$0.574812\pi$$
$$654$$ 18.1284 0.708876
$$655$$ 0 0
$$656$$ 8.65059 0.337749
$$657$$ 65.5910 2.55895
$$658$$ 2.41085 0.0939845
$$659$$ −11.2353 −0.437664 −0.218832 0.975763i $$-0.570225\pi$$
−0.218832 + 0.975763i $$0.570225\pi$$
$$660$$ 0 0
$$661$$ −22.9273 −0.891768 −0.445884 0.895091i $$-0.647111\pi$$
−0.445884 + 0.895091i $$0.647111\pi$$
$$662$$ −20.5772 −0.799757
$$663$$ 0.761580 0.0295773
$$664$$ −36.9794 −1.43508
$$665$$ 0 0
$$666$$ −20.6702 −0.800953
$$667$$ 0.323743 0.0125354
$$668$$ −12.9072 −0.499396
$$669$$ −25.2587 −0.976558
$$670$$ 0 0
$$671$$ −8.09599 −0.312542
$$672$$ −18.6189 −0.718240
$$673$$ 3.09828 0.119430 0.0597150 0.998215i $$-0.480981\pi$$
0.0597150 + 0.998215i $$0.480981\pi$$
$$674$$ 10.2960 0.396586
$$675$$ 0 0
$$676$$ 18.0893 0.695743
$$677$$ −33.3985 −1.28361 −0.641804 0.766868i $$-0.721814\pi$$
−0.641804 + 0.766868i $$0.721814\pi$$
$$678$$ 1.00082 0.0384361
$$679$$ 6.04407 0.231950
$$680$$ 0 0
$$681$$ −40.9810 −1.57039
$$682$$ 7.16335 0.274299
$$683$$ −6.01634 −0.230209 −0.115104 0.993353i $$-0.536720\pi$$
−0.115104 + 0.993353i $$0.536720\pi$$
$$684$$ 8.18974 0.313143
$$685$$ 0 0
$$686$$ 10.7051 0.408723
$$687$$ −5.95054 −0.227027
$$688$$ −7.73160 −0.294765
$$689$$ 0.694007 0.0264396
$$690$$ 0 0
$$691$$ −15.6730 −0.596227 −0.298114 0.954530i $$-0.596357\pi$$
−0.298114 + 0.954530i $$0.596357\pi$$
$$692$$ 14.9164 0.567038
$$693$$ −6.28054 −0.238578
$$694$$ −12.8527 −0.487882
$$695$$ 0 0
$$696$$ −2.36450 −0.0896262
$$697$$ 54.4395 2.06204
$$698$$ −14.8302 −0.561330
$$699$$ −33.8244 −1.27936
$$700$$ 0 0
$$701$$ 18.0567 0.681992 0.340996 0.940065i $$-0.389236\pi$$
0.340996 + 0.940065i $$0.389236\pi$$
$$702$$ 0.377617 0.0142523
$$703$$ 4.50448 0.169890
$$704$$ 3.12238 0.117679
$$705$$ 0 0
$$706$$ −10.2906 −0.387291
$$707$$ −8.54699 −0.321442
$$708$$ 4.75344 0.178645
$$709$$ 16.2048 0.608586 0.304293 0.952579i $$-0.401580\pi$$
0.304293 + 0.952579i $$0.401580\pi$$
$$710$$ 0 0
$$711$$ 67.2266 2.52120
$$712$$ 0.484549 0.0181592
$$713$$ 9.91560 0.371342
$$714$$ −11.2547 −0.421196
$$715$$ 0 0
$$716$$ 31.8825 1.19150
$$717$$ 15.5304 0.579995
$$718$$ −2.25188 −0.0840395
$$719$$ −27.9403 −1.04200 −0.520998 0.853558i $$-0.674440\pi$$
−0.520998 + 0.853558i $$0.674440\pi$$
$$720$$ 0 0
$$721$$ 6.60403 0.245947
$$722$$ 0.779856 0.0290232
$$723$$ 19.6968 0.732533
$$724$$ −8.21903 −0.305458
$$725$$ 0 0
$$726$$ 2.32446 0.0862690
$$727$$ 30.7020 1.13867 0.569337 0.822104i $$-0.307200\pi$$
0.569337 + 0.822104i $$0.307200\pi$$
$$728$$ 0.159025 0.00589387
$$729$$ −29.9678 −1.10992
$$730$$ 0 0
$$731$$ −48.6562 −1.79961
$$732$$ 33.5864 1.24139
$$733$$ −44.9333 −1.65965 −0.829826 0.558023i $$-0.811560\pi$$
−0.829826 + 0.558023i $$0.811560\pi$$
$$734$$ 1.64419 0.0606882
$$735$$ 0 0
$$736$$ 6.31760 0.232870
$$737$$ −11.2733 −0.415258
$$738$$ 55.0700 2.02715
$$739$$ −3.19172 −0.117409 −0.0587047 0.998275i $$-0.518697\pi$$
−0.0587047 + 0.998275i $$0.518697\pi$$
$$740$$ 0 0
$$741$$ −0.167887 −0.00616747
$$742$$ −10.2561 −0.376513
$$743$$ −17.4348 −0.639619 −0.319810 0.947482i $$-0.603619\pi$$
−0.319810 + 0.947482i $$0.603619\pi$$
$$744$$ −72.4199 −2.65504
$$745$$ 0 0
$$746$$ −6.29581 −0.230506
$$747$$ 82.2617 3.00980
$$748$$ −6.31370 −0.230852
$$749$$ −7.78135 −0.284324
$$750$$ 0 0
$$751$$ 9.55633 0.348715 0.174358 0.984682i $$-0.444215\pi$$
0.174358 + 0.984682i $$0.444215\pi$$
$$752$$ −2.08773 −0.0761317
$$753$$ 71.5425 2.60715
$$754$$ 0.0131736 0.000479755 0
$$755$$ 0 0
$$756$$ 12.7710 0.464478
$$757$$ 48.5259 1.76370 0.881852 0.471527i $$-0.156297\pi$$
0.881852 + 0.471527i $$0.156297\pi$$
$$758$$ 11.1796 0.406062
$$759$$ 3.21755 0.116790
$$760$$ 0 0
$$761$$ 38.6035 1.39937 0.699687 0.714449i $$-0.253323\pi$$
0.699687 + 0.714449i $$0.253323\pi$$
$$762$$ 7.29622 0.264314
$$763$$ −8.32429 −0.301360
$$764$$ −8.97210 −0.324599
$$765$$ 0 0
$$766$$ 7.64155 0.276101
$$767$$ −0.0645391 −0.00233037
$$768$$ −40.1607 −1.44917
$$769$$ 53.2658 1.92081 0.960406 0.278603i $$-0.0898712\pi$$
0.960406 + 0.278603i $$0.0898712\pi$$
$$770$$ 0 0
$$771$$ 16.9569 0.610688
$$772$$ 1.99874 0.0719363
$$773$$ −9.31748 −0.335126 −0.167563 0.985861i $$-0.553590\pi$$
−0.167563 + 0.985861i $$0.553590\pi$$
$$774$$ −49.2196 −1.76916
$$775$$ 0 0
$$776$$ 14.9784 0.537694
$$777$$ 14.3306 0.514107
$$778$$ −15.5375 −0.557047
$$779$$ −12.0009 −0.429978
$$780$$ 0 0
$$781$$ 13.4948 0.482883
$$782$$ 3.81884 0.136561
$$783$$ 2.57818 0.0921367
$$784$$ −4.22458 −0.150878
$$785$$ 0 0
$$786$$ 28.8967 1.03071
$$787$$ 15.0616 0.536889 0.268444 0.963295i $$-0.413490\pi$$
0.268444 + 0.963295i $$0.413490\pi$$
$$788$$ −6.61672 −0.235711
$$789$$ −41.6862 −1.48407
$$790$$ 0 0
$$791$$ −0.459561 −0.0163401
$$792$$ −15.5644 −0.553058
$$793$$ −0.456013 −0.0161935
$$794$$ −10.9133 −0.387300
$$795$$ 0 0
$$796$$ 3.28474 0.116425
$$797$$ 19.0593 0.675114 0.337557 0.941305i $$-0.390399\pi$$
0.337557 + 0.941305i $$0.390399\pi$$
$$798$$ 2.48104 0.0878279
$$799$$ −13.1384 −0.464804
$$800$$ 0 0
$$801$$ −1.07789 −0.0380855
$$802$$ −14.5281 −0.513006
$$803$$ 11.1470 0.393370
$$804$$ 46.7676 1.64937
$$805$$ 0 0
$$806$$ 0.403481 0.0142120
$$807$$ 45.9077 1.61603
$$808$$ −21.1811 −0.745150
$$809$$ −15.2273 −0.535363 −0.267681 0.963507i $$-0.586257\pi$$
−0.267681 + 0.963507i $$0.586257\pi$$
$$810$$ 0 0
$$811$$ −43.3354 −1.52171 −0.760857 0.648920i $$-0.775221\pi$$
−0.760857 + 0.648920i $$0.775221\pi$$
$$812$$ 0.445532 0.0156351
$$813$$ −75.6814 −2.65426
$$814$$ −3.51284 −0.123125
$$815$$ 0 0
$$816$$ 9.74628 0.341188
$$817$$ 10.7260 0.375256
$$818$$ −24.0530 −0.840992
$$819$$ −0.353756 −0.0123612
$$820$$ 0 0
$$821$$ 19.4279 0.678040 0.339020 0.940779i $$-0.389905\pi$$
0.339020 + 0.940779i $$0.389905\pi$$
$$822$$ −32.1477 −1.12128
$$823$$ −2.35683 −0.0821539 −0.0410769 0.999156i $$-0.513079\pi$$
−0.0410769 + 0.999156i $$0.513079\pi$$
$$824$$ 16.3661 0.570141
$$825$$ 0 0
$$826$$ 0.953763 0.0331857
$$827$$ 29.8127 1.03669 0.518344 0.855172i $$-0.326549\pi$$
0.518344 + 0.855172i $$0.326549\pi$$
$$828$$ −8.84072 −0.307236
$$829$$ 52.6510 1.82864 0.914322 0.404988i $$-0.132724\pi$$
0.914322 + 0.404988i $$0.132724\pi$$
$$830$$ 0 0
$$831$$ −59.5524 −2.06585
$$832$$ 0.175871 0.00609722
$$833$$ −26.5859 −0.921148
$$834$$ 37.1841 1.28758
$$835$$ 0 0
$$836$$ 1.39182 0.0481373
$$837$$ 78.9645 2.72941
$$838$$ 29.5836 1.02195
$$839$$ 29.7892 1.02844 0.514218 0.857660i $$-0.328082\pi$$
0.514218 + 0.857660i $$0.328082\pi$$
$$840$$ 0 0
$$841$$ −28.9101 −0.996899
$$842$$ −14.3994 −0.496237
$$843$$ 18.4242 0.634563
$$844$$ 34.1182 1.17440
$$845$$ 0 0
$$846$$ −13.2906 −0.456939
$$847$$ −1.06736 −0.0366749
$$848$$ 8.88152 0.304992
$$849$$ 21.2259 0.728471
$$850$$ 0 0
$$851$$ −4.86252 −0.166685
$$852$$ −55.9836 −1.91797
$$853$$ 35.2393 1.20657 0.603285 0.797526i $$-0.293858\pi$$
0.603285 + 0.797526i $$0.293858\pi$$
$$854$$ 6.73900 0.230604
$$855$$ 0 0
$$856$$ −19.2837 −0.659105
$$857$$ 21.5877 0.737421 0.368711 0.929544i $$-0.379799\pi$$
0.368711 + 0.929544i $$0.379799\pi$$
$$858$$ 0.130927 0.00446979
$$859$$ 27.7669 0.947396 0.473698 0.880687i $$-0.342919\pi$$
0.473698 + 0.880687i $$0.342919\pi$$
$$860$$ 0 0
$$861$$ −38.1799 −1.30117
$$862$$ 26.2610 0.894453
$$863$$ −7.11774 −0.242291 −0.121145 0.992635i $$-0.538657\pi$$
−0.121145 + 0.992635i $$0.538657\pi$$
$$864$$ 50.3112 1.71162
$$865$$ 0 0
$$866$$ −8.89357 −0.302216
$$867$$ 10.6641 0.362173
$$868$$ 13.6458 0.463167
$$869$$ 11.4250 0.387566
$$870$$ 0 0
$$871$$ −0.634980 −0.0215155
$$872$$ −20.6293 −0.698595
$$873$$ −33.3199 −1.12771
$$874$$ −0.841844 −0.0284758
$$875$$ 0 0
$$876$$ −46.2436 −1.56243
$$877$$ 44.1031 1.48925 0.744627 0.667481i $$-0.232627\pi$$
0.744627 + 0.667481i $$0.232627\pi$$
$$878$$ −2.93920 −0.0991931
$$879$$ −57.0321 −1.92364
$$880$$ 0 0
$$881$$ −37.9204 −1.27757 −0.638785 0.769385i $$-0.720563\pi$$
−0.638785 + 0.769385i $$0.720563\pi$$
$$882$$ −26.8938 −0.905562
$$883$$ −4.85252 −0.163300 −0.0816502 0.996661i $$-0.526019\pi$$
−0.0816502 + 0.996661i $$0.526019\pi$$
$$884$$ −0.355624 −0.0119609
$$885$$ 0 0
$$886$$ −3.52371 −0.118381
$$887$$ −11.2754 −0.378591 −0.189295 0.981920i $$-0.560620\pi$$
−0.189295 + 0.981920i $$0.560620\pi$$
$$888$$ 35.5141 1.19177
$$889$$ −3.35032 −0.112366
$$890$$ 0 0
$$891$$ 7.97100 0.267038
$$892$$ 11.7947 0.394916
$$893$$ 2.89630 0.0969210
$$894$$ 26.4106 0.883302
$$895$$ 0 0
$$896$$ 9.89423 0.330543
$$897$$ 0.181231 0.00605114
$$898$$ −29.1363 −0.972290
$$899$$ 2.75477 0.0918767
$$900$$ 0 0
$$901$$ 55.8928 1.86206
$$902$$ 9.35899 0.311620
$$903$$ 34.1239 1.13557
$$904$$ −1.13888 −0.0378787
$$905$$ 0 0
$$906$$ 10.8399 0.360133
$$907$$ −24.0191 −0.797540 −0.398770 0.917051i $$-0.630563\pi$$
−0.398770 + 0.917051i $$0.630563\pi$$
$$908$$ 19.1363 0.635061
$$909$$ 47.1181 1.56281
$$910$$ 0 0
$$911$$ 7.48540 0.248002 0.124001 0.992282i $$-0.460427\pi$$
0.124001 + 0.992282i $$0.460427\pi$$
$$912$$ −2.14852 −0.0711446
$$913$$ 13.9802 0.462676
$$914$$ −7.31328 −0.241902
$$915$$ 0 0
$$916$$ 2.77864 0.0918089
$$917$$ −13.2689 −0.438179
$$918$$ 30.4119 1.00374
$$919$$ 40.0187 1.32009 0.660047 0.751224i $$-0.270536\pi$$
0.660047 + 0.751224i $$0.270536\pi$$
$$920$$ 0 0
$$921$$ −20.3450 −0.670390
$$922$$ 24.7047 0.813605
$$923$$ 0.760108 0.0250193
$$924$$ 4.42797 0.145669
$$925$$ 0 0
$$926$$ 28.9548 0.951514
$$927$$ −36.4069 −1.19576
$$928$$ 1.75517 0.0576161
$$929$$ −32.3099 −1.06005 −0.530027 0.847981i $$-0.677818\pi$$
−0.530027 + 0.847981i $$0.677818\pi$$
$$930$$ 0 0
$$931$$ 5.86074 0.192078
$$932$$ 15.7945 0.517367
$$933$$ −38.3337 −1.25499
$$934$$ 12.4526 0.407461
$$935$$ 0 0
$$936$$ −0.876679 −0.0286552
$$937$$ 54.1000 1.76737 0.883684 0.468083i $$-0.155055\pi$$
0.883684 + 0.468083i $$0.155055\pi$$
$$938$$ 9.38378 0.306391
$$939$$ 8.99277 0.293468
$$940$$ 0 0
$$941$$ 26.6955 0.870248 0.435124 0.900371i $$-0.356705\pi$$
0.435124 + 0.900371i $$0.356705\pi$$
$$942$$ 21.0412 0.685559
$$943$$ 12.9548 0.421868
$$944$$ −0.825935 −0.0268819
$$945$$ 0 0
$$946$$ −8.36475 −0.271961
$$947$$ 39.4463 1.28183 0.640916 0.767611i $$-0.278555\pi$$
0.640916 + 0.767611i $$0.278555\pi$$
$$948$$ −47.3968 −1.53938
$$949$$ 0.627865 0.0203814
$$950$$ 0 0
$$951$$ −53.0126 −1.71905
$$952$$ 12.8073 0.415088
$$953$$ 48.7523 1.57924 0.789621 0.613595i $$-0.210277\pi$$
0.789621 + 0.613595i $$0.210277\pi$$
$$954$$ 56.5400 1.83055
$$955$$ 0 0
$$956$$ −7.25203 −0.234547
$$957$$ 0.893906 0.0288959
$$958$$ 21.9708 0.709845
$$959$$ 14.7618 0.476682
$$960$$ 0 0
$$961$$ 53.3730 1.72171
$$962$$ −0.197864 −0.00637938
$$963$$ 42.8972 1.38234
$$964$$ −9.19755 −0.296233
$$965$$ 0 0
$$966$$ −2.67825 −0.0861713
$$967$$ 1.39478 0.0448532 0.0224266 0.999748i $$-0.492861\pi$$
0.0224266 + 0.999748i $$0.492861\pi$$
$$968$$ −2.64513 −0.0850178
$$969$$ −13.5210 −0.434356
$$970$$ 0 0
$$971$$ −3.11733 −0.100040 −0.0500200 0.998748i $$-0.515929\pi$$
−0.0500200 + 0.998748i $$0.515929\pi$$
$$972$$ 2.82734 0.0906870
$$973$$ −17.0744 −0.547380
$$974$$ −18.7644 −0.601249
$$975$$ 0 0
$$976$$ −5.83580 −0.186800
$$977$$ 22.3690 0.715647 0.357823 0.933789i $$-0.383519\pi$$
0.357823 + 0.933789i $$0.383519\pi$$
$$978$$ −5.50343 −0.175980
$$979$$ −0.183185 −0.00585462
$$980$$ 0 0
$$981$$ 45.8904 1.46517
$$982$$ −22.6312 −0.722191
$$983$$ −42.2421 −1.34731 −0.673657 0.739044i $$-0.735277\pi$$
−0.673657 + 0.739044i $$0.735277\pi$$
$$984$$ −94.6174 −3.01629
$$985$$ 0 0
$$986$$ 1.06096 0.0337877
$$987$$ 9.21433 0.293295
$$988$$ 0.0783957 0.00249410
$$989$$ −11.5786 −0.368178
$$990$$ 0 0
$$991$$ −1.69828 −0.0539477 −0.0269738 0.999636i $$-0.508587\pi$$
−0.0269738 + 0.999636i $$0.508587\pi$$
$$992$$ 53.7572 1.70679
$$993$$ −78.6469 −2.49578
$$994$$ −11.2329 −0.356287
$$995$$ 0 0
$$996$$ −57.9970 −1.83770
$$997$$ −18.9376 −0.599759 −0.299879 0.953977i $$-0.596946\pi$$
−0.299879 + 0.953977i $$0.596946\pi$$
$$998$$ −11.7179 −0.370923
$$999$$ −38.7235 −1.22516
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.h.1.4 5
5.4 even 2 209.2.a.c.1.2 5
15.14 odd 2 1881.2.a.k.1.4 5
20.19 odd 2 3344.2.a.t.1.5 5
55.54 odd 2 2299.2.a.n.1.4 5
95.94 odd 2 3971.2.a.h.1.4 5

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.2 5 5.4 even 2
1881.2.a.k.1.4 5 15.14 odd 2
2299.2.a.n.1.4 5 55.54 odd 2
3344.2.a.t.1.5 5 20.19 odd 2
3971.2.a.h.1.4 5 95.94 odd 2
5225.2.a.h.1.4 5 1.1 even 1 trivial