# Properties

 Label 5225.2.a.n.1.4 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Learn more

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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 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 $$0.456669$$ 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)

 $$f(q)$$ $$=$$ $$q+0.456669 q^{2} +0.835165 q^{3} -1.79145 q^{4} +0.381394 q^{6} -4.69915 q^{7} -1.73144 q^{8} -2.30250 q^{9} +O(q^{10})$$ $$q+0.456669 q^{2} +0.835165 q^{3} -1.79145 q^{4} +0.381394 q^{6} -4.69915 q^{7} -1.73144 q^{8} -2.30250 q^{9} -1.00000 q^{11} -1.49616 q^{12} -5.89016 q^{13} -2.14596 q^{14} +2.79221 q^{16} -7.06513 q^{17} -1.05148 q^{18} +1.00000 q^{19} -3.92457 q^{21} -0.456669 q^{22} -1.06348 q^{23} -1.44604 q^{24} -2.68985 q^{26} -4.42846 q^{27} +8.41832 q^{28} -7.62662 q^{29} +0.901295 q^{31} +4.73799 q^{32} -0.835165 q^{33} -3.22642 q^{34} +4.12482 q^{36} +2.71758 q^{37} +0.456669 q^{38} -4.91925 q^{39} +0.788714 q^{41} -1.79223 q^{42} -0.714571 q^{43} +1.79145 q^{44} -0.485656 q^{46} -3.96368 q^{47} +2.33196 q^{48} +15.0820 q^{49} -5.90055 q^{51} +10.5519 q^{52} +9.69714 q^{53} -2.02234 q^{54} +8.13629 q^{56} +0.835165 q^{57} -3.48284 q^{58} -7.33476 q^{59} +8.15179 q^{61} +0.411593 q^{62} +10.8198 q^{63} -3.42073 q^{64} -0.381394 q^{66} -7.86697 q^{67} +12.6569 q^{68} -0.888178 q^{69} -3.13400 q^{71} +3.98664 q^{72} +6.49076 q^{73} +1.24103 q^{74} -1.79145 q^{76} +4.69915 q^{77} -2.24647 q^{78} +12.0148 q^{79} +3.20900 q^{81} +0.360181 q^{82} -16.3902 q^{83} +7.03068 q^{84} -0.326322 q^{86} -6.36948 q^{87} +1.73144 q^{88} -12.9487 q^{89} +27.6788 q^{91} +1.90517 q^{92} +0.752730 q^{93} -1.81009 q^{94} +3.95701 q^{96} -8.14414 q^{97} +6.88750 q^{98} +2.30250 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9}+O(q^{10})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 - 2 * q^6 - 10 * q^7 + 9 * q^8 + 11 * q^9 $$7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} - 7 q^{11} + 16 q^{12} + 4 q^{13} + 6 q^{14} + 27 q^{16} - 2 q^{17} - 9 q^{18} + 7 q^{19} - 14 q^{21} - q^{22} - 10 q^{23} - 2 q^{24} - 8 q^{26} + 4 q^{27} - 26 q^{28} - 18 q^{29} + 24 q^{31} + 49 q^{32} + 2 q^{33} - 6 q^{34} + 29 q^{36} + q^{38} + 24 q^{39} - 12 q^{41} + 44 q^{42} - 2 q^{43} - 15 q^{44} - 4 q^{46} - 8 q^{47} + 72 q^{48} + 17 q^{49} - 24 q^{51} + 60 q^{52} - 2 q^{53} - 52 q^{54} + 26 q^{56} - 2 q^{57} + 8 q^{58} - 10 q^{59} + 14 q^{61} - 14 q^{62} + 55 q^{64} + 2 q^{66} - 8 q^{67} + 18 q^{68} - 6 q^{69} + 10 q^{71} - 53 q^{72} + 6 q^{73} + 26 q^{74} + 15 q^{76} + 10 q^{77} - 22 q^{78} + 52 q^{79} - q^{81} - 24 q^{82} + 10 q^{83} - 6 q^{84} + 8 q^{86} - 6 q^{87} - 9 q^{88} + 12 q^{91} - 2 q^{93} + 24 q^{94} + 6 q^{96} + 24 q^{97} - 19 q^{98} - 11 q^{99}+O(q^{100})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 - 2 * q^6 - 10 * q^7 + 9 * q^8 + 11 * q^9 - 7 * q^11 + 16 * q^12 + 4 * q^13 + 6 * q^14 + 27 * q^16 - 2 * q^17 - 9 * q^18 + 7 * q^19 - 14 * q^21 - q^22 - 10 * q^23 - 2 * q^24 - 8 * q^26 + 4 * q^27 - 26 * q^28 - 18 * q^29 + 24 * q^31 + 49 * q^32 + 2 * q^33 - 6 * q^34 + 29 * q^36 + q^38 + 24 * q^39 - 12 * q^41 + 44 * q^42 - 2 * q^43 - 15 * q^44 - 4 * q^46 - 8 * q^47 + 72 * q^48 + 17 * q^49 - 24 * q^51 + 60 * q^52 - 2 * q^53 - 52 * q^54 + 26 * q^56 - 2 * q^57 + 8 * q^58 - 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 2 * q^66 - 8 * q^67 + 18 * q^68 - 6 * q^69 + 10 * q^71 - 53 * q^72 + 6 * q^73 + 26 * q^74 + 15 * q^76 + 10 * q^77 - 22 * q^78 + 52 * q^79 - q^81 - 24 * q^82 + 10 * q^83 - 6 * q^84 + 8 * q^86 - 6 * q^87 - 9 * q^88 + 12 * q^91 - 2 * q^93 + 24 * q^94 + 6 * q^96 + 24 * q^97 - 19 * q^98 - 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.456669 0.322914 0.161457 0.986880i $$-0.448381\pi$$
0.161457 + 0.986880i $$0.448381\pi$$
$$3$$ 0.835165 0.482183 0.241091 0.970502i $$-0.422495\pi$$
0.241091 + 0.970502i $$0.422495\pi$$
$$4$$ −1.79145 −0.895727
$$5$$ 0 0
$$6$$ 0.381394 0.155703
$$7$$ −4.69915 −1.77611 −0.888057 0.459734i $$-0.847945\pi$$
−0.888057 + 0.459734i $$0.847945\pi$$
$$8$$ −1.73144 −0.612156
$$9$$ −2.30250 −0.767500
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −1.49616 −0.431904
$$13$$ −5.89016 −1.63364 −0.816818 0.576895i $$-0.804264\pi$$
−0.816818 + 0.576895i $$0.804264\pi$$
$$14$$ −2.14596 −0.573531
$$15$$ 0 0
$$16$$ 2.79221 0.698053
$$17$$ −7.06513 −1.71355 −0.856773 0.515694i $$-0.827534\pi$$
−0.856773 + 0.515694i $$0.827534\pi$$
$$18$$ −1.05148 −0.247836
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −3.92457 −0.856411
$$22$$ −0.456669 −0.0973621
$$23$$ −1.06348 −0.221750 −0.110875 0.993834i $$-0.535365\pi$$
−0.110875 + 0.993834i $$0.535365\pi$$
$$24$$ −1.44604 −0.295171
$$25$$ 0 0
$$26$$ −2.68985 −0.527523
$$27$$ −4.42846 −0.852258
$$28$$ 8.41832 1.59091
$$29$$ −7.62662 −1.41623 −0.708114 0.706098i $$-0.750454\pi$$
−0.708114 + 0.706098i $$0.750454\pi$$
$$30$$ 0 0
$$31$$ 0.901295 0.161877 0.0809387 0.996719i $$-0.474208\pi$$
0.0809387 + 0.996719i $$0.474208\pi$$
$$32$$ 4.73799 0.837567
$$33$$ −0.835165 −0.145384
$$34$$ −3.22642 −0.553327
$$35$$ 0 0
$$36$$ 4.12482 0.687470
$$37$$ 2.71758 0.446768 0.223384 0.974731i $$-0.428290\pi$$
0.223384 + 0.974731i $$0.428290\pi$$
$$38$$ 0.456669 0.0740815
$$39$$ −4.91925 −0.787711
$$40$$ 0 0
$$41$$ 0.788714 0.123176 0.0615882 0.998102i $$-0.480383\pi$$
0.0615882 + 0.998102i $$0.480383\pi$$
$$42$$ −1.79223 −0.276547
$$43$$ −0.714571 −0.108971 −0.0544855 0.998515i $$-0.517352\pi$$
−0.0544855 + 0.998515i $$0.517352\pi$$
$$44$$ 1.79145 0.270072
$$45$$ 0 0
$$46$$ −0.485656 −0.0716061
$$47$$ −3.96368 −0.578163 −0.289081 0.957305i $$-0.593350\pi$$
−0.289081 + 0.957305i $$0.593350\pi$$
$$48$$ 2.33196 0.336589
$$49$$ 15.0820 2.15458
$$50$$ 0 0
$$51$$ −5.90055 −0.826242
$$52$$ 10.5519 1.46329
$$53$$ 9.69714 1.33200 0.666002 0.745950i $$-0.268004\pi$$
0.666002 + 0.745950i $$0.268004\pi$$
$$54$$ −2.02234 −0.275206
$$55$$ 0 0
$$56$$ 8.13629 1.08726
$$57$$ 0.835165 0.110620
$$58$$ −3.48284 −0.457319
$$59$$ −7.33476 −0.954904 −0.477452 0.878658i $$-0.658440\pi$$
−0.477452 + 0.878658i $$0.658440\pi$$
$$60$$ 0 0
$$61$$ 8.15179 1.04373 0.521865 0.853028i $$-0.325236\pi$$
0.521865 + 0.853028i $$0.325236\pi$$
$$62$$ 0.411593 0.0522724
$$63$$ 10.8198 1.36317
$$64$$ −3.42073 −0.427592
$$65$$ 0 0
$$66$$ −0.381394 −0.0469463
$$67$$ −7.86697 −0.961104 −0.480552 0.876966i $$-0.659564\pi$$
−0.480552 + 0.876966i $$0.659564\pi$$
$$68$$ 12.6569 1.53487
$$69$$ −0.888178 −0.106924
$$70$$ 0 0
$$71$$ −3.13400 −0.371937 −0.185969 0.982556i $$-0.559542\pi$$
−0.185969 + 0.982556i $$0.559542\pi$$
$$72$$ 3.98664 0.469830
$$73$$ 6.49076 0.759687 0.379843 0.925051i $$-0.375978\pi$$
0.379843 + 0.925051i $$0.375978\pi$$
$$74$$ 1.24103 0.144267
$$75$$ 0 0
$$76$$ −1.79145 −0.205494
$$77$$ 4.69915 0.535518
$$78$$ −2.24647 −0.254363
$$79$$ 12.0148 1.35177 0.675884 0.737008i $$-0.263762\pi$$
0.675884 + 0.737008i $$0.263762\pi$$
$$80$$ 0 0
$$81$$ 3.20900 0.356556
$$82$$ 0.360181 0.0397753
$$83$$ −16.3902 −1.79906 −0.899528 0.436863i $$-0.856089\pi$$
−0.899528 + 0.436863i $$0.856089\pi$$
$$84$$ 7.03068 0.767110
$$85$$ 0 0
$$86$$ −0.326322 −0.0351882
$$87$$ −6.36948 −0.682880
$$88$$ 1.73144 0.184572
$$89$$ −12.9487 −1.37256 −0.686281 0.727336i $$-0.740758\pi$$
−0.686281 + 0.727336i $$0.740758\pi$$
$$90$$ 0 0
$$91$$ 27.6788 2.90152
$$92$$ 1.90517 0.198628
$$93$$ 0.752730 0.0780545
$$94$$ −1.81009 −0.186697
$$95$$ 0 0
$$96$$ 3.95701 0.403860
$$97$$ −8.14414 −0.826912 −0.413456 0.910524i $$-0.635679\pi$$
−0.413456 + 0.910524i $$0.635679\pi$$
$$98$$ 6.88750 0.695742
$$99$$ 2.30250 0.231410
$$100$$ 0 0
$$101$$ −15.3713 −1.52950 −0.764750 0.644327i $$-0.777138\pi$$
−0.764750 + 0.644327i $$0.777138\pi$$
$$102$$ −2.69460 −0.266805
$$103$$ 8.83682 0.870718 0.435359 0.900257i $$-0.356622\pi$$
0.435359 + 0.900257i $$0.356622\pi$$
$$104$$ 10.1984 1.00004
$$105$$ 0 0
$$106$$ 4.42838 0.430122
$$107$$ 9.04360 0.874277 0.437139 0.899394i $$-0.355992\pi$$
0.437139 + 0.899394i $$0.355992\pi$$
$$108$$ 7.93338 0.763390
$$109$$ −18.5950 −1.78107 −0.890537 0.454911i $$-0.849671\pi$$
−0.890537 + 0.454911i $$0.849671\pi$$
$$110$$ 0 0
$$111$$ 2.26963 0.215424
$$112$$ −13.1210 −1.23982
$$113$$ 4.42423 0.416196 0.208098 0.978108i $$-0.433273\pi$$
0.208098 + 0.978108i $$0.433273\pi$$
$$114$$ 0.381394 0.0357208
$$115$$ 0 0
$$116$$ 13.6627 1.26855
$$117$$ 13.5621 1.25382
$$118$$ −3.34955 −0.308352
$$119$$ 33.2001 3.04345
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 3.72267 0.337035
$$123$$ 0.658706 0.0593935
$$124$$ −1.61463 −0.144998
$$125$$ 0 0
$$126$$ 4.94106 0.440185
$$127$$ 0.338657 0.0300510 0.0150255 0.999887i $$-0.495217\pi$$
0.0150255 + 0.999887i $$0.495217\pi$$
$$128$$ −11.0381 −0.975642
$$129$$ −0.596784 −0.0525439
$$130$$ 0 0
$$131$$ −4.55211 −0.397720 −0.198860 0.980028i $$-0.563724\pi$$
−0.198860 + 0.980028i $$0.563724\pi$$
$$132$$ 1.49616 0.130224
$$133$$ −4.69915 −0.407468
$$134$$ −3.59260 −0.310353
$$135$$ 0 0
$$136$$ 12.2328 1.04896
$$137$$ 0.654263 0.0558974 0.0279487 0.999609i $$-0.491102\pi$$
0.0279487 + 0.999609i $$0.491102\pi$$
$$138$$ −0.405603 −0.0345272
$$139$$ 11.0479 0.937072 0.468536 0.883444i $$-0.344782\pi$$
0.468536 + 0.883444i $$0.344782\pi$$
$$140$$ 0 0
$$141$$ −3.31033 −0.278780
$$142$$ −1.43120 −0.120104
$$143$$ 5.89016 0.492560
$$144$$ −6.42907 −0.535756
$$145$$ 0 0
$$146$$ 2.96413 0.245313
$$147$$ 12.5960 1.03890
$$148$$ −4.86842 −0.400182
$$149$$ −7.57743 −0.620767 −0.310384 0.950611i $$-0.600457\pi$$
−0.310384 + 0.950611i $$0.600457\pi$$
$$150$$ 0 0
$$151$$ −6.95296 −0.565824 −0.282912 0.959146i $$-0.591300\pi$$
−0.282912 + 0.959146i $$0.591300\pi$$
$$152$$ −1.73144 −0.140438
$$153$$ 16.2675 1.31515
$$154$$ 2.14596 0.172926
$$155$$ 0 0
$$156$$ 8.81261 0.705574
$$157$$ 14.7427 1.17659 0.588297 0.808645i $$-0.299799\pi$$
0.588297 + 0.808645i $$0.299799\pi$$
$$158$$ 5.48678 0.436504
$$159$$ 8.09871 0.642270
$$160$$ 0 0
$$161$$ 4.99744 0.393853
$$162$$ 1.46545 0.115137
$$163$$ 4.94015 0.386942 0.193471 0.981106i $$-0.438025\pi$$
0.193471 + 0.981106i $$0.438025\pi$$
$$164$$ −1.41294 −0.110332
$$165$$ 0 0
$$166$$ −7.48488 −0.580940
$$167$$ −22.6032 −1.74909 −0.874544 0.484946i $$-0.838839\pi$$
−0.874544 + 0.484946i $$0.838839\pi$$
$$168$$ 6.79515 0.524257
$$169$$ 21.6940 1.66877
$$170$$ 0 0
$$171$$ −2.30250 −0.176077
$$172$$ 1.28012 0.0976083
$$173$$ 8.18620 0.622385 0.311193 0.950347i $$-0.399272\pi$$
0.311193 + 0.950347i $$0.399272\pi$$
$$174$$ −2.90874 −0.220511
$$175$$ 0 0
$$176$$ −2.79221 −0.210471
$$177$$ −6.12573 −0.460438
$$178$$ −5.91328 −0.443219
$$179$$ −14.0830 −1.05261 −0.526305 0.850296i $$-0.676423\pi$$
−0.526305 + 0.850296i $$0.676423\pi$$
$$180$$ 0 0
$$181$$ −14.3260 −1.06484 −0.532422 0.846479i $$-0.678718\pi$$
−0.532422 + 0.846479i $$0.678718\pi$$
$$182$$ 12.6400 0.936941
$$183$$ 6.80809 0.503268
$$184$$ 1.84134 0.135746
$$185$$ 0 0
$$186$$ 0.343748 0.0252049
$$187$$ 7.06513 0.516653
$$188$$ 7.10076 0.517876
$$189$$ 20.8100 1.51371
$$190$$ 0 0
$$191$$ 0.394967 0.0285788 0.0142894 0.999898i $$-0.495451\pi$$
0.0142894 + 0.999898i $$0.495451\pi$$
$$192$$ −2.85688 −0.206177
$$193$$ −9.03423 −0.650298 −0.325149 0.945663i $$-0.605414\pi$$
−0.325149 + 0.945663i $$0.605414\pi$$
$$194$$ −3.71918 −0.267021
$$195$$ 0 0
$$196$$ −27.0188 −1.92991
$$197$$ −7.85313 −0.559512 −0.279756 0.960071i $$-0.590254\pi$$
−0.279756 + 0.960071i $$0.590254\pi$$
$$198$$ 1.05148 0.0747254
$$199$$ 7.81540 0.554019 0.277009 0.960867i $$-0.410657\pi$$
0.277009 + 0.960867i $$0.410657\pi$$
$$200$$ 0 0
$$201$$ −6.57022 −0.463427
$$202$$ −7.01959 −0.493897
$$203$$ 35.8386 2.51538
$$204$$ 10.5706 0.740087
$$205$$ 0 0
$$206$$ 4.03550 0.281167
$$207$$ 2.44865 0.170193
$$208$$ −16.4466 −1.14037
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −21.5955 −1.48670 −0.743348 0.668905i $$-0.766763\pi$$
−0.743348 + 0.668905i $$0.766763\pi$$
$$212$$ −17.3720 −1.19311
$$213$$ −2.61741 −0.179342
$$214$$ 4.12993 0.282316
$$215$$ 0 0
$$216$$ 7.66761 0.521715
$$217$$ −4.23533 −0.287513
$$218$$ −8.49174 −0.575133
$$219$$ 5.42086 0.366308
$$220$$ 0 0
$$221$$ 41.6147 2.79931
$$222$$ 1.03647 0.0695632
$$223$$ 6.95854 0.465979 0.232989 0.972479i $$-0.425149\pi$$
0.232989 + 0.972479i $$0.425149\pi$$
$$224$$ −22.2646 −1.48761
$$225$$ 0 0
$$226$$ 2.02041 0.134395
$$227$$ −7.67819 −0.509619 −0.254810 0.966991i $$-0.582013\pi$$
−0.254810 + 0.966991i $$0.582013\pi$$
$$228$$ −1.49616 −0.0990855
$$229$$ 5.83925 0.385869 0.192934 0.981212i $$-0.438200\pi$$
0.192934 + 0.981212i $$0.438200\pi$$
$$230$$ 0 0
$$231$$ 3.92457 0.258218
$$232$$ 13.2050 0.866952
$$233$$ 21.8431 1.43099 0.715494 0.698619i $$-0.246202\pi$$
0.715494 + 0.698619i $$0.246202\pi$$
$$234$$ 6.19338 0.404874
$$235$$ 0 0
$$236$$ 13.1399 0.855333
$$237$$ 10.0343 0.651799
$$238$$ 15.1615 0.982772
$$239$$ 4.21004 0.272325 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$240$$ 0 0
$$241$$ −17.4276 −1.12261 −0.561304 0.827610i $$-0.689700\pi$$
−0.561304 + 0.827610i $$0.689700\pi$$
$$242$$ 0.456669 0.0293558
$$243$$ 15.9654 1.02418
$$244$$ −14.6036 −0.934897
$$245$$ 0 0
$$246$$ 0.300810 0.0191790
$$247$$ −5.89016 −0.374782
$$248$$ −1.56054 −0.0990942
$$249$$ −13.6885 −0.867474
$$250$$ 0 0
$$251$$ 2.14594 0.135451 0.0677254 0.997704i $$-0.478426\pi$$
0.0677254 + 0.997704i $$0.478426\pi$$
$$252$$ −19.3832 −1.22102
$$253$$ 1.06348 0.0668602
$$254$$ 0.154654 0.00970387
$$255$$ 0 0
$$256$$ 1.80070 0.112544
$$257$$ 5.23899 0.326799 0.163399 0.986560i $$-0.447754\pi$$
0.163399 + 0.986560i $$0.447754\pi$$
$$258$$ −0.272533 −0.0169672
$$259$$ −12.7703 −0.793510
$$260$$ 0 0
$$261$$ 17.5603 1.08695
$$262$$ −2.07881 −0.128429
$$263$$ −12.2597 −0.755967 −0.377983 0.925812i $$-0.623382\pi$$
−0.377983 + 0.925812i $$0.623382\pi$$
$$264$$ 1.44604 0.0889974
$$265$$ 0 0
$$266$$ −2.14596 −0.131577
$$267$$ −10.8143 −0.661826
$$268$$ 14.0933 0.860886
$$269$$ −9.37273 −0.571465 −0.285733 0.958309i $$-0.592237\pi$$
−0.285733 + 0.958309i $$0.592237\pi$$
$$270$$ 0 0
$$271$$ −13.3131 −0.808716 −0.404358 0.914601i $$-0.632505\pi$$
−0.404358 + 0.914601i $$0.632505\pi$$
$$272$$ −19.7273 −1.19615
$$273$$ 23.1163 1.39906
$$274$$ 0.298781 0.0180500
$$275$$ 0 0
$$276$$ 1.59113 0.0957748
$$277$$ 14.5641 0.875073 0.437537 0.899201i $$-0.355851\pi$$
0.437537 + 0.899201i $$0.355851\pi$$
$$278$$ 5.04524 0.302593
$$279$$ −2.07523 −0.124241
$$280$$ 0 0
$$281$$ 20.2411 1.20748 0.603741 0.797180i $$-0.293676\pi$$
0.603741 + 0.797180i $$0.293676\pi$$
$$282$$ −1.51172 −0.0900219
$$283$$ −14.6230 −0.869248 −0.434624 0.900612i $$-0.643119\pi$$
−0.434624 + 0.900612i $$0.643119\pi$$
$$284$$ 5.61442 0.333154
$$285$$ 0 0
$$286$$ 2.68985 0.159054
$$287$$ −3.70629 −0.218775
$$288$$ −10.9092 −0.642833
$$289$$ 32.9161 1.93624
$$290$$ 0 0
$$291$$ −6.80170 −0.398723
$$292$$ −11.6279 −0.680472
$$293$$ −27.6524 −1.61547 −0.807737 0.589543i $$-0.799308\pi$$
−0.807737 + 0.589543i $$0.799308\pi$$
$$294$$ 5.75220 0.335475
$$295$$ 0 0
$$296$$ −4.70533 −0.273491
$$297$$ 4.42846 0.256965
$$298$$ −3.46037 −0.200454
$$299$$ 6.26404 0.362259
$$300$$ 0 0
$$301$$ 3.35788 0.193545
$$302$$ −3.17520 −0.182712
$$303$$ −12.8376 −0.737499
$$304$$ 2.79221 0.160144
$$305$$ 0 0
$$306$$ 7.42884 0.424679
$$307$$ 0.756065 0.0431509 0.0215755 0.999767i $$-0.493132\pi$$
0.0215755 + 0.999767i $$0.493132\pi$$
$$308$$ −8.41832 −0.479678
$$309$$ 7.38020 0.419845
$$310$$ 0 0
$$311$$ 1.13352 0.0642761 0.0321381 0.999483i $$-0.489768\pi$$
0.0321381 + 0.999483i $$0.489768\pi$$
$$312$$ 8.51738 0.482202
$$313$$ 10.9012 0.616172 0.308086 0.951359i $$-0.400312\pi$$
0.308086 + 0.951359i $$0.400312\pi$$
$$314$$ 6.73252 0.379938
$$315$$ 0 0
$$316$$ −21.5239 −1.21082
$$317$$ −8.32326 −0.467481 −0.233741 0.972299i $$-0.575097\pi$$
−0.233741 + 0.972299i $$0.575097\pi$$
$$318$$ 3.69843 0.207398
$$319$$ 7.62662 0.427009
$$320$$ 0 0
$$321$$ 7.55289 0.421561
$$322$$ 2.28217 0.127181
$$323$$ −7.06513 −0.393114
$$324$$ −5.74878 −0.319377
$$325$$ 0 0
$$326$$ 2.25601 0.124949
$$327$$ −15.5299 −0.858803
$$328$$ −1.36561 −0.0754031
$$329$$ 18.6260 1.02688
$$330$$ 0 0
$$331$$ −2.47466 −0.136020 −0.0680098 0.997685i $$-0.521665\pi$$
−0.0680098 + 0.997685i $$0.521665\pi$$
$$332$$ 29.3622 1.61146
$$333$$ −6.25723 −0.342894
$$334$$ −10.3222 −0.564804
$$335$$ 0 0
$$336$$ −10.9582 −0.597820
$$337$$ 30.3242 1.65187 0.825933 0.563768i $$-0.190649\pi$$
0.825933 + 0.563768i $$0.190649\pi$$
$$338$$ 9.90696 0.538867
$$339$$ 3.69496 0.200683
$$340$$ 0 0
$$341$$ −0.901295 −0.0488079
$$342$$ −1.05148 −0.0568575
$$343$$ −37.9788 −2.05066
$$344$$ 1.23724 0.0667073
$$345$$ 0 0
$$346$$ 3.73838 0.200977
$$347$$ 10.4332 0.560081 0.280041 0.959988i $$-0.409652\pi$$
0.280041 + 0.959988i $$0.409652\pi$$
$$348$$ 11.4106 0.611674
$$349$$ 4.89049 0.261782 0.130891 0.991397i $$-0.458216\pi$$
0.130891 + 0.991397i $$0.458216\pi$$
$$350$$ 0 0
$$351$$ 26.0843 1.39228
$$352$$ −4.73799 −0.252536
$$353$$ −34.1254 −1.81631 −0.908155 0.418634i $$-0.862509\pi$$
−0.908155 + 0.418634i $$0.862509\pi$$
$$354$$ −2.79743 −0.148682
$$355$$ 0 0
$$356$$ 23.1970 1.22944
$$357$$ 27.7276 1.46750
$$358$$ −6.43125 −0.339902
$$359$$ 10.5366 0.556103 0.278051 0.960566i $$-0.410311\pi$$
0.278051 + 0.960566i $$0.410311\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −6.54224 −0.343853
$$363$$ 0.835165 0.0438348
$$364$$ −49.5852 −2.59897
$$365$$ 0 0
$$366$$ 3.10904 0.162512
$$367$$ −31.1277 −1.62485 −0.812427 0.583062i $$-0.801854\pi$$
−0.812427 + 0.583062i $$0.801854\pi$$
$$368$$ −2.96945 −0.154793
$$369$$ −1.81601 −0.0945378
$$370$$ 0 0
$$371$$ −45.5684 −2.36579
$$372$$ −1.34848 −0.0699155
$$373$$ −28.9881 −1.50095 −0.750473 0.660901i $$-0.770174\pi$$
−0.750473 + 0.660901i $$0.770174\pi$$
$$374$$ 3.22642 0.166834
$$375$$ 0 0
$$376$$ 6.86288 0.353926
$$377$$ 44.9220 2.31360
$$378$$ 9.50329 0.488796
$$379$$ −26.4204 −1.35712 −0.678561 0.734544i $$-0.737396\pi$$
−0.678561 + 0.734544i $$0.737396\pi$$
$$380$$ 0 0
$$381$$ 0.282835 0.0144901
$$382$$ 0.180369 0.00922849
$$383$$ 13.6280 0.696360 0.348180 0.937428i $$-0.386800\pi$$
0.348180 + 0.937428i $$0.386800\pi$$
$$384$$ −9.21866 −0.470438
$$385$$ 0 0
$$386$$ −4.12565 −0.209990
$$387$$ 1.64530 0.0836353
$$388$$ 14.5899 0.740688
$$389$$ −15.7277 −0.797428 −0.398714 0.917075i $$-0.630543\pi$$
−0.398714 + 0.917075i $$0.630543\pi$$
$$390$$ 0 0
$$391$$ 7.51360 0.379979
$$392$$ −26.1136 −1.31894
$$393$$ −3.80176 −0.191774
$$394$$ −3.58628 −0.180674
$$395$$ 0 0
$$396$$ −4.12482 −0.207280
$$397$$ −16.8523 −0.845790 −0.422895 0.906179i $$-0.638986\pi$$
−0.422895 + 0.906179i $$0.638986\pi$$
$$398$$ 3.56905 0.178900
$$399$$ −3.92457 −0.196474
$$400$$ 0 0
$$401$$ 30.3744 1.51682 0.758412 0.651775i $$-0.225975\pi$$
0.758412 + 0.651775i $$0.225975\pi$$
$$402$$ −3.00041 −0.149647
$$403$$ −5.30877 −0.264449
$$404$$ 27.5370 1.37001
$$405$$ 0 0
$$406$$ 16.3664 0.812250
$$407$$ −2.71758 −0.134706
$$408$$ 10.2164 0.505789
$$409$$ −1.31844 −0.0651925 −0.0325962 0.999469i $$-0.510378\pi$$
−0.0325962 + 0.999469i $$0.510378\pi$$
$$410$$ 0 0
$$411$$ 0.546417 0.0269528
$$412$$ −15.8307 −0.779925
$$413$$ 34.4672 1.69602
$$414$$ 1.11822 0.0549577
$$415$$ 0 0
$$416$$ −27.9075 −1.36828
$$417$$ 9.22683 0.451840
$$418$$ −0.456669 −0.0223364
$$419$$ −30.3257 −1.48151 −0.740753 0.671778i $$-0.765531\pi$$
−0.740753 + 0.671778i $$0.765531\pi$$
$$420$$ 0 0
$$421$$ 8.27204 0.403154 0.201577 0.979473i $$-0.435393\pi$$
0.201577 + 0.979473i $$0.435393\pi$$
$$422$$ −9.86200 −0.480075
$$423$$ 9.12638 0.443740
$$424$$ −16.7900 −0.815395
$$425$$ 0 0
$$426$$ −1.19529 −0.0579119
$$427$$ −38.3065 −1.85378
$$428$$ −16.2012 −0.783114
$$429$$ 4.91925 0.237504
$$430$$ 0 0
$$431$$ 22.9574 1.10582 0.552909 0.833241i $$-0.313518\pi$$
0.552909 + 0.833241i $$0.313518\pi$$
$$432$$ −12.3652 −0.594921
$$433$$ −11.4207 −0.548846 −0.274423 0.961609i $$-0.588487\pi$$
−0.274423 + 0.961609i $$0.588487\pi$$
$$434$$ −1.93414 −0.0928417
$$435$$ 0 0
$$436$$ 33.3120 1.59536
$$437$$ −1.06348 −0.0508730
$$438$$ 2.47554 0.118286
$$439$$ 5.43642 0.259466 0.129733 0.991549i $$-0.458588\pi$$
0.129733 + 0.991549i $$0.458588\pi$$
$$440$$ 0 0
$$441$$ −34.7264 −1.65364
$$442$$ 19.0041 0.903935
$$443$$ 34.7494 1.65100 0.825498 0.564405i $$-0.190894\pi$$
0.825498 + 0.564405i $$0.190894\pi$$
$$444$$ −4.06593 −0.192961
$$445$$ 0 0
$$446$$ 3.17775 0.150471
$$447$$ −6.32840 −0.299323
$$448$$ 16.0745 0.759451
$$449$$ −11.3505 −0.535662 −0.267831 0.963466i $$-0.586307\pi$$
−0.267831 + 0.963466i $$0.586307\pi$$
$$450$$ 0 0
$$451$$ −0.788714 −0.0371391
$$452$$ −7.92580 −0.372798
$$453$$ −5.80687 −0.272830
$$454$$ −3.50639 −0.164563
$$455$$ 0 0
$$456$$ −1.44604 −0.0677169
$$457$$ −6.89453 −0.322513 −0.161256 0.986913i $$-0.551555\pi$$
−0.161256 + 0.986913i $$0.551555\pi$$
$$458$$ 2.66660 0.124602
$$459$$ 31.2877 1.46038
$$460$$ 0 0
$$461$$ 13.5620 0.631643 0.315822 0.948819i $$-0.397720\pi$$
0.315822 + 0.948819i $$0.397720\pi$$
$$462$$ 1.79223 0.0833820
$$463$$ 18.6923 0.868704 0.434352 0.900743i $$-0.356977\pi$$
0.434352 + 0.900743i $$0.356977\pi$$
$$464$$ −21.2951 −0.988602
$$465$$ 0 0
$$466$$ 9.97505 0.462085
$$467$$ 14.7156 0.680957 0.340479 0.940252i $$-0.389411\pi$$
0.340479 + 0.940252i $$0.389411\pi$$
$$468$$ −24.2959 −1.12308
$$469$$ 36.9681 1.70703
$$470$$ 0 0
$$471$$ 12.3126 0.567333
$$472$$ 12.6997 0.584550
$$473$$ 0.714571 0.0328560
$$474$$ 4.58236 0.210475
$$475$$ 0 0
$$476$$ −59.4765 −2.72610
$$477$$ −22.3277 −1.02231
$$478$$ 1.92259 0.0879374
$$479$$ −16.3467 −0.746901 −0.373450 0.927650i $$-0.621825\pi$$
−0.373450 + 0.927650i $$0.621825\pi$$
$$480$$ 0 0
$$481$$ −16.0070 −0.729856
$$482$$ −7.95863 −0.362505
$$483$$ 4.17368 0.189909
$$484$$ −1.79145 −0.0814297
$$485$$ 0 0
$$486$$ 7.29091 0.330723
$$487$$ 24.8625 1.12663 0.563314 0.826243i $$-0.309526\pi$$
0.563314 + 0.826243i $$0.309526\pi$$
$$488$$ −14.1143 −0.638926
$$489$$ 4.12584 0.186577
$$490$$ 0 0
$$491$$ 17.5025 0.789878 0.394939 0.918707i $$-0.370766\pi$$
0.394939 + 0.918707i $$0.370766\pi$$
$$492$$ −1.18004 −0.0532003
$$493$$ 53.8830 2.42677
$$494$$ −2.68985 −0.121022
$$495$$ 0 0
$$496$$ 2.51661 0.112999
$$497$$ 14.7271 0.660603
$$498$$ −6.25111 −0.280119
$$499$$ −3.13006 −0.140121 −0.0700603 0.997543i $$-0.522319\pi$$
−0.0700603 + 0.997543i $$0.522319\pi$$
$$500$$ 0 0
$$501$$ −18.8774 −0.843380
$$502$$ 0.979986 0.0437389
$$503$$ 8.98965 0.400829 0.200414 0.979711i $$-0.435771\pi$$
0.200414 + 0.979711i $$0.435771\pi$$
$$504$$ −18.7338 −0.834471
$$505$$ 0 0
$$506$$ 0.485656 0.0215901
$$507$$ 18.1180 0.804650
$$508$$ −0.606689 −0.0269175
$$509$$ 23.6416 1.04790 0.523949 0.851750i $$-0.324458\pi$$
0.523949 + 0.851750i $$0.324458\pi$$
$$510$$ 0 0
$$511$$ −30.5011 −1.34929
$$512$$ 22.8986 1.01198
$$513$$ −4.42846 −0.195521
$$514$$ 2.39248 0.105528
$$515$$ 0 0
$$516$$ 1.06911 0.0470650
$$517$$ 3.96368 0.174323
$$518$$ −5.83181 −0.256235
$$519$$ 6.83683 0.300103
$$520$$ 0 0
$$521$$ 36.0036 1.57735 0.788673 0.614813i $$-0.210768\pi$$
0.788673 + 0.614813i $$0.210768\pi$$
$$522$$ 8.01923 0.350992
$$523$$ −7.08081 −0.309622 −0.154811 0.987944i $$-0.549477\pi$$
−0.154811 + 0.987944i $$0.549477\pi$$
$$524$$ 8.15489 0.356248
$$525$$ 0 0
$$526$$ −5.59863 −0.244112
$$527$$ −6.36777 −0.277384
$$528$$ −2.33196 −0.101485
$$529$$ −21.8690 −0.950827
$$530$$ 0 0
$$531$$ 16.8883 0.732889
$$532$$ 8.41832 0.364980
$$533$$ −4.64565 −0.201225
$$534$$ −4.93856 −0.213712
$$535$$ 0 0
$$536$$ 13.6212 0.588345
$$537$$ −11.7616 −0.507550
$$538$$ −4.28023 −0.184534
$$539$$ −15.0820 −0.649630
$$540$$ 0 0
$$541$$ −4.23609 −0.182124 −0.0910619 0.995845i $$-0.529026\pi$$
−0.0910619 + 0.995845i $$0.529026\pi$$
$$542$$ −6.07970 −0.261145
$$543$$ −11.9646 −0.513449
$$544$$ −33.4745 −1.43521
$$545$$ 0 0
$$546$$ 10.5565 0.451777
$$547$$ −9.53317 −0.407609 −0.203805 0.979012i $$-0.565331\pi$$
−0.203805 + 0.979012i $$0.565331\pi$$
$$548$$ −1.17208 −0.0500688
$$549$$ −18.7695 −0.801063
$$550$$ 0 0
$$551$$ −7.62662 −0.324905
$$552$$ 1.53783 0.0654542
$$553$$ −56.4593 −2.40089
$$554$$ 6.65098 0.282573
$$555$$ 0 0
$$556$$ −19.7918 −0.839361
$$557$$ −2.85948 −0.121160 −0.0605800 0.998163i $$-0.519295\pi$$
−0.0605800 + 0.998163i $$0.519295\pi$$
$$558$$ −0.947694 −0.0401191
$$559$$ 4.20894 0.178019
$$560$$ 0 0
$$561$$ 5.90055 0.249121
$$562$$ 9.24347 0.389912
$$563$$ −12.8091 −0.539841 −0.269920 0.962883i $$-0.586997\pi$$
−0.269920 + 0.962883i $$0.586997\pi$$
$$564$$ 5.93030 0.249711
$$565$$ 0 0
$$566$$ −6.67787 −0.280692
$$567$$ −15.0796 −0.633284
$$568$$ 5.42633 0.227684
$$569$$ −10.5273 −0.441327 −0.220664 0.975350i $$-0.570822\pi$$
−0.220664 + 0.975350i $$0.570822\pi$$
$$570$$ 0 0
$$571$$ −24.9055 −1.04226 −0.521132 0.853476i $$-0.674490\pi$$
−0.521132 + 0.853476i $$0.674490\pi$$
$$572$$ −10.5519 −0.441199
$$573$$ 0.329863 0.0137802
$$574$$ −1.69255 −0.0706455
$$575$$ 0 0
$$576$$ 7.87624 0.328177
$$577$$ 26.7799 1.11486 0.557430 0.830224i $$-0.311787\pi$$
0.557430 + 0.830224i $$0.311787\pi$$
$$578$$ 15.0317 0.625238
$$579$$ −7.54507 −0.313562
$$580$$ 0 0
$$581$$ 77.0200 3.19533
$$582$$ −3.10612 −0.128753
$$583$$ −9.69714 −0.401615
$$584$$ −11.2384 −0.465047
$$585$$ 0 0
$$586$$ −12.6280 −0.521658
$$587$$ −18.7991 −0.775920 −0.387960 0.921676i $$-0.626820\pi$$
−0.387960 + 0.921676i $$0.626820\pi$$
$$588$$ −22.5651 −0.930570
$$589$$ 0.901295 0.0371372
$$590$$ 0 0
$$591$$ −6.55866 −0.269787
$$592$$ 7.58807 0.311868
$$593$$ 3.68216 0.151208 0.0756041 0.997138i $$-0.475911\pi$$
0.0756041 + 0.997138i $$0.475911\pi$$
$$594$$ 2.02234 0.0829776
$$595$$ 0 0
$$596$$ 13.5746 0.556038
$$597$$ 6.52715 0.267138
$$598$$ 2.86059 0.116978
$$599$$ 43.2659 1.76780 0.883899 0.467677i $$-0.154909\pi$$
0.883899 + 0.467677i $$0.154909\pi$$
$$600$$ 0 0
$$601$$ −14.4072 −0.587683 −0.293842 0.955854i $$-0.594934\pi$$
−0.293842 + 0.955854i $$0.594934\pi$$
$$602$$ 1.53344 0.0624983
$$603$$ 18.1137 0.737647
$$604$$ 12.4559 0.506824
$$605$$ 0 0
$$606$$ −5.86251 −0.238148
$$607$$ 6.06496 0.246169 0.123084 0.992396i $$-0.460721\pi$$
0.123084 + 0.992396i $$0.460721\pi$$
$$608$$ 4.73799 0.192151
$$609$$ 29.9312 1.21287
$$610$$ 0 0
$$611$$ 23.3467 0.944508
$$612$$ −29.1424 −1.17801
$$613$$ 3.51855 0.142113 0.0710565 0.997472i $$-0.477363\pi$$
0.0710565 + 0.997472i $$0.477363\pi$$
$$614$$ 0.345271 0.0139340
$$615$$ 0 0
$$616$$ −8.13629 −0.327821
$$617$$ −7.17642 −0.288912 −0.144456 0.989511i $$-0.546143\pi$$
−0.144456 + 0.989511i $$0.546143\pi$$
$$618$$ 3.37031 0.135574
$$619$$ −8.27421 −0.332568 −0.166284 0.986078i $$-0.553177\pi$$
−0.166284 + 0.986078i $$0.553177\pi$$
$$620$$ 0 0
$$621$$ 4.70956 0.188988
$$622$$ 0.517644 0.0207556
$$623$$ 60.8480 2.43783
$$624$$ −13.7356 −0.549864
$$625$$ 0 0
$$626$$ 4.97824 0.198970
$$627$$ −0.835165 −0.0333533
$$628$$ −26.4108 −1.05391
$$629$$ −19.2001 −0.765557
$$630$$ 0 0
$$631$$ −9.09560 −0.362090 −0.181045 0.983475i $$-0.557948\pi$$
−0.181045 + 0.983475i $$0.557948\pi$$
$$632$$ −20.8029 −0.827493
$$633$$ −18.0358 −0.716859
$$634$$ −3.80097 −0.150956
$$635$$ 0 0
$$636$$ −14.5085 −0.575298
$$637$$ −88.8356 −3.51980
$$638$$ 3.48284 0.137887
$$639$$ 7.21604 0.285462
$$640$$ 0 0
$$641$$ 16.4386 0.649285 0.324642 0.945837i $$-0.394756\pi$$
0.324642 + 0.945837i $$0.394756\pi$$
$$642$$ 3.44917 0.136128
$$643$$ −7.04594 −0.277865 −0.138932 0.990302i $$-0.544367\pi$$
−0.138932 + 0.990302i $$0.544367\pi$$
$$644$$ −8.95268 −0.352785
$$645$$ 0 0
$$646$$ −3.22642 −0.126942
$$647$$ −18.1024 −0.711679 −0.355840 0.934547i $$-0.615805\pi$$
−0.355840 + 0.934547i $$0.615805\pi$$
$$648$$ −5.55619 −0.218268
$$649$$ 7.33476 0.287914
$$650$$ 0 0
$$651$$ −3.53719 −0.138634
$$652$$ −8.85005 −0.346595
$$653$$ −37.2985 −1.45960 −0.729802 0.683659i $$-0.760388\pi$$
−0.729802 + 0.683659i $$0.760388\pi$$
$$654$$ −7.09200 −0.277319
$$655$$ 0 0
$$656$$ 2.20226 0.0859837
$$657$$ −14.9450 −0.583059
$$658$$ 8.50590 0.331594
$$659$$ 13.2223 0.515068 0.257534 0.966269i $$-0.417090\pi$$
0.257534 + 0.966269i $$0.417090\pi$$
$$660$$ 0 0
$$661$$ −17.6212 −0.685385 −0.342692 0.939448i $$-0.611339\pi$$
−0.342692 + 0.939448i $$0.611339\pi$$
$$662$$ −1.13010 −0.0439226
$$663$$ 34.7552 1.34978
$$664$$ 28.3786 1.10130
$$665$$ 0 0
$$666$$ −2.85748 −0.110725
$$667$$ 8.11073 0.314049
$$668$$ 40.4926 1.56671
$$669$$ 5.81153 0.224687
$$670$$ 0 0
$$671$$ −8.15179 −0.314696
$$672$$ −18.5946 −0.717301
$$673$$ −32.2335 −1.24251 −0.621255 0.783609i $$-0.713377\pi$$
−0.621255 + 0.783609i $$0.713377\pi$$
$$674$$ 13.8481 0.533410
$$675$$ 0 0
$$676$$ −38.8637 −1.49476
$$677$$ 10.2788 0.395045 0.197523 0.980298i $$-0.436710\pi$$
0.197523 + 0.980298i $$0.436710\pi$$
$$678$$ 1.68737 0.0648031
$$679$$ 38.2706 1.46869
$$680$$ 0 0
$$681$$ −6.41255 −0.245730
$$682$$ −0.411593 −0.0157607
$$683$$ −25.4018 −0.971973 −0.485986 0.873966i $$-0.661540\pi$$
−0.485986 + 0.873966i $$0.661540\pi$$
$$684$$ 4.12482 0.157716
$$685$$ 0 0
$$686$$ −17.3437 −0.662186
$$687$$ 4.87674 0.186059
$$688$$ −1.99523 −0.0760676
$$689$$ −57.1177 −2.17601
$$690$$ 0 0
$$691$$ −2.58092 −0.0981828 −0.0490914 0.998794i $$-0.515633\pi$$
−0.0490914 + 0.998794i $$0.515633\pi$$
$$692$$ −14.6652 −0.557487
$$693$$ −10.8198 −0.411010
$$694$$ 4.76450 0.180858
$$695$$ 0 0
$$696$$ 11.0284 0.418029
$$697$$ −5.57236 −0.211068
$$698$$ 2.23334 0.0845330
$$699$$ 18.2426 0.689998
$$700$$ 0 0
$$701$$ −30.0612 −1.13540 −0.567698 0.823237i $$-0.692166\pi$$
−0.567698 + 0.823237i $$0.692166\pi$$
$$702$$ 11.9119 0.449586
$$703$$ 2.71758 0.102496
$$704$$ 3.42073 0.128924
$$705$$ 0 0
$$706$$ −15.5840 −0.586511
$$707$$ 72.2321 2.71657
$$708$$ 10.9740 0.412427
$$709$$ −6.90683 −0.259391 −0.129696 0.991554i $$-0.541400\pi$$
−0.129696 + 0.991554i $$0.541400\pi$$
$$710$$ 0 0
$$711$$ −27.6640 −1.03748
$$712$$ 22.4199 0.840222
$$713$$ −0.958506 −0.0358963
$$714$$ 12.6623 0.473875
$$715$$ 0 0
$$716$$ 25.2290 0.942851
$$717$$ 3.51608 0.131310
$$718$$ 4.81175 0.179573
$$719$$ 13.1525 0.490506 0.245253 0.969459i $$-0.421129\pi$$
0.245253 + 0.969459i $$0.421129\pi$$
$$720$$ 0 0
$$721$$ −41.5256 −1.54649
$$722$$ 0.456669 0.0169955
$$723$$ −14.5549 −0.541302
$$724$$ 25.6644 0.953810
$$725$$ 0 0
$$726$$ 0.381394 0.0141548
$$727$$ 46.8434 1.73733 0.868663 0.495404i $$-0.164980\pi$$
0.868663 + 0.495404i $$0.164980\pi$$
$$728$$ −47.9241 −1.77618
$$729$$ 3.70675 0.137287
$$730$$ 0 0
$$731$$ 5.04854 0.186727
$$732$$ −12.1964 −0.450791
$$733$$ 31.9162 1.17885 0.589426 0.807822i $$-0.299354\pi$$
0.589426 + 0.807822i $$0.299354\pi$$
$$734$$ −14.2151 −0.524688
$$735$$ 0 0
$$736$$ −5.03874 −0.185731
$$737$$ 7.86697 0.289784
$$738$$ −0.829316 −0.0305276
$$739$$ 30.0302 1.10468 0.552339 0.833620i $$-0.313736\pi$$
0.552339 + 0.833620i $$0.313736\pi$$
$$740$$ 0 0
$$741$$ −4.91925 −0.180713
$$742$$ −20.8096 −0.763946
$$743$$ 42.6871 1.56604 0.783019 0.621998i $$-0.213679\pi$$
0.783019 + 0.621998i $$0.213679\pi$$
$$744$$ −1.30331 −0.0477815
$$745$$ 0 0
$$746$$ −13.2380 −0.484676
$$747$$ 37.7384 1.38078
$$748$$ −12.6569 −0.462780
$$749$$ −42.4972 −1.55282
$$750$$ 0 0
$$751$$ 30.7211 1.12103 0.560514 0.828145i $$-0.310604\pi$$
0.560514 + 0.828145i $$0.310604\pi$$
$$752$$ −11.0675 −0.403589
$$753$$ 1.79222 0.0653120
$$754$$ 20.5145 0.747093
$$755$$ 0 0
$$756$$ −37.2802 −1.35587
$$757$$ 49.0714 1.78353 0.891765 0.452498i $$-0.149467\pi$$
0.891765 + 0.452498i $$0.149467\pi$$
$$758$$ −12.0654 −0.438233
$$759$$ 0.888178 0.0322388
$$760$$ 0 0
$$761$$ −24.8178 −0.899645 −0.449822 0.893118i $$-0.648513\pi$$
−0.449822 + 0.893118i $$0.648513\pi$$
$$762$$ 0.129162 0.00467904
$$763$$ 87.3806 3.16339
$$764$$ −0.707565 −0.0255988
$$765$$ 0 0
$$766$$ 6.22350 0.224864
$$767$$ 43.2029 1.55997
$$768$$ 1.50388 0.0542666
$$769$$ −38.9862 −1.40588 −0.702940 0.711250i $$-0.748130\pi$$
−0.702940 + 0.711250i $$0.748130\pi$$
$$770$$ 0 0
$$771$$ 4.37542 0.157577
$$772$$ 16.1844 0.582489
$$773$$ −0.998099 −0.0358991 −0.0179496 0.999839i $$-0.505714\pi$$
−0.0179496 + 0.999839i $$0.505714\pi$$
$$774$$ 0.751357 0.0270070
$$775$$ 0 0
$$776$$ 14.1011 0.506199
$$777$$ −10.6653 −0.382617
$$778$$ −7.18237 −0.257500
$$779$$ 0.788714 0.0282586
$$780$$ 0 0
$$781$$ 3.13400 0.112143
$$782$$ 3.43123 0.122700
$$783$$ 33.7742 1.20699
$$784$$ 42.1123 1.50401
$$785$$ 0 0
$$786$$ −1.73615 −0.0619263
$$787$$ −18.7420 −0.668082 −0.334041 0.942559i $$-0.608412\pi$$
−0.334041 + 0.942559i $$0.608412\pi$$
$$788$$ 14.0685 0.501170
$$789$$ −10.2389 −0.364514
$$790$$ 0 0
$$791$$ −20.7901 −0.739211
$$792$$ −3.98664 −0.141659
$$793$$ −48.0153 −1.70507
$$794$$ −7.69590 −0.273117
$$795$$ 0 0
$$796$$ −14.0009 −0.496250
$$797$$ −5.05128 −0.178925 −0.0894627 0.995990i $$-0.528515\pi$$
−0.0894627 + 0.995990i $$0.528515\pi$$
$$798$$ −1.79223 −0.0634442
$$799$$ 28.0039 0.990708
$$800$$ 0 0
$$801$$ 29.8144 1.05344
$$802$$ 13.8710 0.489803
$$803$$ −6.49076 −0.229054
$$804$$ 11.7702 0.415104
$$805$$ 0 0
$$806$$ −2.42435 −0.0853941
$$807$$ −7.82777 −0.275551
$$808$$ 26.6144 0.936293
$$809$$ −4.98214 −0.175163 −0.0875813 0.996157i $$-0.527914\pi$$
−0.0875813 + 0.996157i $$0.527914\pi$$
$$810$$ 0 0
$$811$$ 25.3145 0.888913 0.444456 0.895800i $$-0.353397\pi$$
0.444456 + 0.895800i $$0.353397\pi$$
$$812$$ −64.2033 −2.25309
$$813$$ −11.1187 −0.389949
$$814$$ −1.24103 −0.0434982
$$815$$ 0 0
$$816$$ −16.4756 −0.576761
$$817$$ −0.714571 −0.0249997
$$818$$ −0.602089 −0.0210515
$$819$$ −63.7303 −2.22692
$$820$$ 0 0
$$821$$ 5.81543 0.202960 0.101480 0.994838i $$-0.467642\pi$$
0.101480 + 0.994838i $$0.467642\pi$$
$$822$$ 0.249532 0.00870342
$$823$$ −47.7881 −1.66579 −0.832895 0.553431i $$-0.813318\pi$$
−0.832895 + 0.553431i $$0.813318\pi$$
$$824$$ −15.3004 −0.533015
$$825$$ 0 0
$$826$$ 15.7401 0.547667
$$827$$ 39.0692 1.35857 0.679284 0.733875i $$-0.262290\pi$$
0.679284 + 0.733875i $$0.262290\pi$$
$$828$$ −4.38665 −0.152447
$$829$$ −7.57440 −0.263070 −0.131535 0.991312i $$-0.541991\pi$$
−0.131535 + 0.991312i $$0.541991\pi$$
$$830$$ 0 0
$$831$$ 12.1634 0.421945
$$832$$ 20.1487 0.698529
$$833$$ −106.557 −3.69197
$$834$$ 4.21361 0.145905
$$835$$ 0 0
$$836$$ 1.79145 0.0619587
$$837$$ −3.99135 −0.137961
$$838$$ −13.8488 −0.478398
$$839$$ −28.0343 −0.967850 −0.483925 0.875110i $$-0.660789\pi$$
−0.483925 + 0.875110i $$0.660789\pi$$
$$840$$ 0 0
$$841$$ 29.1653 1.00570
$$842$$ 3.77758 0.130184
$$843$$ 16.9046 0.582227
$$844$$ 38.6874 1.33167
$$845$$ 0 0
$$846$$ 4.16773 0.143290
$$847$$ −4.69915 −0.161465
$$848$$ 27.0765 0.929811
$$849$$ −12.2126 −0.419136
$$850$$ 0 0
$$851$$ −2.89008 −0.0990708
$$852$$ 4.68896 0.160641
$$853$$ 27.9605 0.957349 0.478674 0.877992i $$-0.341117\pi$$
0.478674 + 0.877992i $$0.341117\pi$$
$$854$$ −17.4934 −0.598612
$$855$$ 0 0
$$856$$ −15.6584 −0.535194
$$857$$ −17.8805 −0.610787 −0.305393 0.952226i $$-0.598788\pi$$
−0.305393 + 0.952226i $$0.598788\pi$$
$$858$$ 2.24647 0.0766932
$$859$$ 7.76017 0.264774 0.132387 0.991198i $$-0.457736\pi$$
0.132387 + 0.991198i $$0.457736\pi$$
$$860$$ 0 0
$$861$$ −3.09536 −0.105490
$$862$$ 10.4839 0.357084
$$863$$ −31.9699 −1.08827 −0.544135 0.838998i $$-0.683142\pi$$
−0.544135 + 0.838998i $$0.683142\pi$$
$$864$$ −20.9820 −0.713823
$$865$$ 0 0
$$866$$ −5.21549 −0.177230
$$867$$ 27.4903 0.933621
$$868$$ 7.58739 0.257533
$$869$$ −12.0148 −0.407574
$$870$$ 0 0
$$871$$ 46.3377 1.57009
$$872$$ 32.1960 1.09029
$$873$$ 18.7519 0.634655
$$874$$ −0.485656 −0.0164276
$$875$$ 0 0
$$876$$ −9.71122 −0.328112
$$877$$ 2.12224 0.0716630 0.0358315 0.999358i $$-0.488592\pi$$
0.0358315 + 0.999358i $$0.488592\pi$$
$$878$$ 2.48264 0.0837852
$$879$$ −23.0944 −0.778953
$$880$$ 0 0
$$881$$ −15.1859 −0.511627 −0.255814 0.966726i $$-0.582343\pi$$
−0.255814 + 0.966726i $$0.582343\pi$$
$$882$$ −15.8585 −0.533982
$$883$$ −29.5964 −0.995999 −0.498000 0.867177i $$-0.665932\pi$$
−0.498000 + 0.867177i $$0.665932\pi$$
$$884$$ −74.5509 −2.50742
$$885$$ 0 0
$$886$$ 15.8690 0.533129
$$887$$ −23.2340 −0.780121 −0.390061 0.920789i $$-0.627546\pi$$
−0.390061 + 0.920789i $$0.627546\pi$$
$$888$$ −3.92972 −0.131873
$$889$$ −1.59140 −0.0533740
$$890$$ 0 0
$$891$$ −3.20900 −0.107506
$$892$$ −12.4659 −0.417389
$$893$$ −3.96368 −0.132640
$$894$$ −2.88998 −0.0966555
$$895$$ 0 0
$$896$$ 51.8699 1.73285
$$897$$ 5.23151 0.174675
$$898$$ −5.18340 −0.172972
$$899$$ −6.87384 −0.229255
$$900$$ 0 0
$$901$$ −68.5116 −2.28245
$$902$$ −0.360181 −0.0119927
$$903$$ 2.80438 0.0933240
$$904$$ −7.66028 −0.254777
$$905$$ 0 0
$$906$$ −2.65181 −0.0881006
$$907$$ 35.4985 1.17871 0.589354 0.807875i $$-0.299382\pi$$
0.589354 + 0.807875i $$0.299382\pi$$
$$908$$ 13.7551 0.456480
$$909$$ 35.3924 1.17389
$$910$$ 0 0
$$911$$ −38.5075 −1.27581 −0.637905 0.770115i $$-0.720199\pi$$
−0.637905 + 0.770115i $$0.720199\pi$$
$$912$$ 2.33196 0.0772189
$$913$$ 16.3902 0.542436
$$914$$ −3.14852 −0.104144
$$915$$ 0 0
$$916$$ −10.4608 −0.345633
$$917$$ 21.3911 0.706395
$$918$$ 14.2881 0.471577
$$919$$ 20.9347 0.690572 0.345286 0.938497i $$-0.387782\pi$$
0.345286 + 0.938497i $$0.387782\pi$$
$$920$$ 0 0
$$921$$ 0.631439 0.0208066
$$922$$ 6.19332 0.203966
$$923$$ 18.4598 0.607610
$$924$$ −7.03068 −0.231292
$$925$$ 0 0
$$926$$ 8.53618 0.280516
$$927$$ −20.3468 −0.668276
$$928$$ −36.1349 −1.18619
$$929$$ −0.536958 −0.0176170 −0.00880851 0.999961i $$-0.502804\pi$$
−0.00880851 + 0.999961i $$0.502804\pi$$
$$930$$ 0 0
$$931$$ 15.0820 0.494294
$$932$$ −39.1309 −1.28177
$$933$$ 0.946677 0.0309928
$$934$$ 6.72016 0.219890
$$935$$ 0 0
$$936$$ −23.4819 −0.767531
$$937$$ −34.4984 −1.12701 −0.563507 0.826112i $$-0.690548\pi$$
−0.563507 + 0.826112i $$0.690548\pi$$
$$938$$ 16.8822 0.551223
$$939$$ 9.10430 0.297107
$$940$$ 0 0
$$941$$ 17.8959 0.583391 0.291695 0.956511i $$-0.405781\pi$$
0.291695 + 0.956511i $$0.405781\pi$$
$$942$$ 5.62276 0.183200
$$943$$ −0.838778 −0.0273144
$$944$$ −20.4802 −0.666574
$$945$$ 0 0
$$946$$ 0.326322 0.0106097
$$947$$ −16.0979 −0.523111 −0.261556 0.965188i $$-0.584235\pi$$
−0.261556 + 0.965188i $$0.584235\pi$$
$$948$$ −17.9760 −0.583834
$$949$$ −38.2316 −1.24105
$$950$$ 0 0
$$951$$ −6.95130 −0.225411
$$952$$ −57.4840 −1.86307
$$953$$ −29.2828 −0.948562 −0.474281 0.880374i $$-0.657292\pi$$
−0.474281 + 0.880374i $$0.657292\pi$$
$$954$$ −10.1963 −0.330119
$$955$$ 0 0
$$956$$ −7.54210 −0.243929
$$957$$ 6.36948 0.205896
$$958$$ −7.46504 −0.241184
$$959$$ −3.07448 −0.0992802
$$960$$ 0 0
$$961$$ −30.1877 −0.973796
$$962$$ −7.30989 −0.235680
$$963$$ −20.8229 −0.671008
$$964$$ 31.2207 1.00555
$$965$$ 0 0
$$966$$ 1.90599 0.0613243
$$967$$ 39.4644 1.26909 0.634544 0.772887i $$-0.281188\pi$$
0.634544 + 0.772887i $$0.281188\pi$$
$$968$$ −1.73144 −0.0556505
$$969$$ −5.90055 −0.189553
$$970$$ 0 0
$$971$$ 27.0245 0.867257 0.433628 0.901092i $$-0.357233\pi$$
0.433628 + 0.901092i $$0.357233\pi$$
$$972$$ −28.6013 −0.917388
$$973$$ −51.9159 −1.66435
$$974$$ 11.3539 0.363804
$$975$$ 0 0
$$976$$ 22.7615 0.728579
$$977$$ 55.2922 1.76895 0.884477 0.466584i $$-0.154516\pi$$
0.884477 + 0.466584i $$0.154516\pi$$
$$978$$ 1.88414 0.0602482
$$979$$ 12.9487 0.413843
$$980$$ 0 0
$$981$$ 42.8149 1.36697
$$982$$ 7.99286 0.255062
$$983$$ −36.0221 −1.14893 −0.574463 0.818531i $$-0.694789\pi$$
−0.574463 + 0.818531i $$0.694789\pi$$
$$984$$ −1.14051 −0.0363581
$$985$$ 0 0
$$986$$ 24.6067 0.783637
$$987$$ 15.5557 0.495145
$$988$$ 10.5519 0.335702
$$989$$ 0.759929 0.0241643
$$990$$ 0 0
$$991$$ 50.7944 1.61354 0.806768 0.590868i $$-0.201215\pi$$
0.806768 + 0.590868i $$0.201215\pi$$
$$992$$ 4.27033 0.135583
$$993$$ −2.06675 −0.0655863
$$994$$ 6.72543 0.213318
$$995$$ 0 0
$$996$$ 24.5223 0.777019
$$997$$ 41.8525 1.32548 0.662741 0.748849i $$-0.269393\pi$$
0.662741 + 0.748849i $$0.269393\pi$$
$$998$$ −1.42940 −0.0452469
$$999$$ −12.0347 −0.380761
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.n.1.4 7
5.4 even 2 209.2.a.d.1.4 7
15.14 odd 2 1881.2.a.p.1.4 7
20.19 odd 2 3344.2.a.ba.1.5 7
55.54 odd 2 2299.2.a.q.1.4 7
95.94 odd 2 3971.2.a.i.1.4 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.4 7 5.4 even 2
1881.2.a.p.1.4 7 15.14 odd 2
2299.2.a.q.1.4 7 55.54 odd 2
3344.2.a.ba.1.5 7 20.19 odd 2
3971.2.a.i.1.4 7 95.94 odd 2
5225.2.a.n.1.4 7 1.1 even 1 trivial