# Properties

 Label 5225.2.a.n.1.6 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $7$ 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: $$7$$ 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 1.6 Root $$2.61330$$ 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)

 $$f(q)$$ $$=$$ $$q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} -3.12549 q^{6} -3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} +O(q^{10})$$ $$q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} -3.12549 q^{6} -3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} -1.00000 q^{11} -5.77587 q^{12} +1.47857 q^{13} -9.45570 q^{14} +9.66398 q^{16} +3.27003 q^{17} -4.10185 q^{18} +1.00000 q^{19} +4.32745 q^{21} -2.61330 q^{22} +7.45793 q^{23} -8.84313 q^{24} +3.86395 q^{26} +5.46521 q^{27} -17.4740 q^{28} +1.02535 q^{29} +1.64921 q^{31} +10.4670 q^{32} +1.19599 q^{33} +8.54558 q^{34} -7.58018 q^{36} +6.71293 q^{37} +2.61330 q^{38} -1.76836 q^{39} -3.92451 q^{41} +11.3089 q^{42} -5.38113 q^{43} -4.82936 q^{44} +19.4898 q^{46} +3.71597 q^{47} -11.5580 q^{48} +6.09205 q^{49} -3.91093 q^{51} +7.14054 q^{52} +0.102902 q^{53} +14.2823 q^{54} -26.7536 q^{56} -1.19599 q^{57} +2.67955 q^{58} +13.2986 q^{59} -6.49664 q^{61} +4.30989 q^{62} +5.67929 q^{63} +8.02543 q^{64} +3.12549 q^{66} +3.70989 q^{67} +15.7921 q^{68} -8.91962 q^{69} +6.32968 q^{71} -11.6056 q^{72} +1.37759 q^{73} +17.5429 q^{74} +4.82936 q^{76} +3.61829 q^{77} -4.62125 q^{78} +13.6725 q^{79} -1.82753 q^{81} -10.2559 q^{82} -5.44061 q^{83} +20.8988 q^{84} -14.0625 q^{86} -1.22631 q^{87} -7.39397 q^{88} +12.1357 q^{89} -5.34990 q^{91} +36.0170 q^{92} -1.97244 q^{93} +9.71096 q^{94} -12.5184 q^{96} +13.7910 q^{97} +15.9204 q^{98} +1.56960 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61330 1.84789 0.923943 0.382531i $$-0.124948\pi$$
0.923943 + 0.382531i $$0.124948\pi$$
$$3$$ −1.19599 −0.690506 −0.345253 0.938510i $$-0.612207\pi$$
−0.345253 + 0.938510i $$0.612207\pi$$
$$4$$ 4.82936 2.41468
$$5$$ 0 0
$$6$$ −3.12549 −1.27598
$$7$$ −3.61829 −1.36759 −0.683793 0.729676i $$-0.739671\pi$$
−0.683793 + 0.729676i $$0.739671\pi$$
$$8$$ 7.39397 2.61416
$$9$$ −1.56960 −0.523201
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −5.77587 −1.66735
$$13$$ 1.47857 0.410081 0.205041 0.978753i $$-0.434267\pi$$
0.205041 + 0.978753i $$0.434267\pi$$
$$14$$ −9.45570 −2.52714
$$15$$ 0 0
$$16$$ 9.66398 2.41600
$$17$$ 3.27003 0.793099 0.396549 0.918013i $$-0.370208\pi$$
0.396549 + 0.918013i $$0.370208\pi$$
$$18$$ −4.10185 −0.966816
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 4.32745 0.944327
$$22$$ −2.61330 −0.557158
$$23$$ 7.45793 1.55509 0.777543 0.628830i $$-0.216466\pi$$
0.777543 + 0.628830i $$0.216466\pi$$
$$24$$ −8.84313 −1.80510
$$25$$ 0 0
$$26$$ 3.86395 0.757783
$$27$$ 5.46521 1.05178
$$28$$ −17.4740 −3.30228
$$29$$ 1.02535 0.190403 0.0952013 0.995458i $$-0.469651\pi$$
0.0952013 + 0.995458i $$0.469651\pi$$
$$30$$ 0 0
$$31$$ 1.64921 0.296207 0.148104 0.988972i $$-0.452683\pi$$
0.148104 + 0.988972i $$0.452683\pi$$
$$32$$ 10.4670 1.85032
$$33$$ 1.19599 0.208195
$$34$$ 8.54558 1.46556
$$35$$ 0 0
$$36$$ −7.58018 −1.26336
$$37$$ 6.71293 1.10360 0.551799 0.833977i $$-0.313941\pi$$
0.551799 + 0.833977i $$0.313941\pi$$
$$38$$ 2.61330 0.423934
$$39$$ −1.76836 −0.283164
$$40$$ 0 0
$$41$$ −3.92451 −0.612905 −0.306453 0.951886i $$-0.599142\pi$$
−0.306453 + 0.951886i $$0.599142\pi$$
$$42$$ 11.3089 1.74501
$$43$$ −5.38113 −0.820614 −0.410307 0.911947i $$-0.634578\pi$$
−0.410307 + 0.911947i $$0.634578\pi$$
$$44$$ −4.82936 −0.728053
$$45$$ 0 0
$$46$$ 19.4898 2.87362
$$47$$ 3.71597 0.542030 0.271015 0.962575i $$-0.412641\pi$$
0.271015 + 0.962575i $$0.412641\pi$$
$$48$$ −11.5580 −1.66826
$$49$$ 6.09205 0.870292
$$50$$ 0 0
$$51$$ −3.91093 −0.547640
$$52$$ 7.14054 0.990215
$$53$$ 0.102902 0.0141347 0.00706733 0.999975i $$-0.497750\pi$$
0.00706733 + 0.999975i $$0.497750\pi$$
$$54$$ 14.2823 1.94357
$$55$$ 0 0
$$56$$ −26.7536 −3.57509
$$57$$ −1.19599 −0.158413
$$58$$ 2.67955 0.351842
$$59$$ 13.2986 1.73134 0.865668 0.500619i $$-0.166894\pi$$
0.865668 + 0.500619i $$0.166894\pi$$
$$60$$ 0 0
$$61$$ −6.49664 −0.831809 −0.415905 0.909408i $$-0.636535\pi$$
−0.415905 + 0.909408i $$0.636535\pi$$
$$62$$ 4.30989 0.547357
$$63$$ 5.67929 0.715523
$$64$$ 8.02543 1.00318
$$65$$ 0 0
$$66$$ 3.12549 0.384721
$$67$$ 3.70989 0.453235 0.226618 0.973984i $$-0.427233\pi$$
0.226618 + 0.973984i $$0.427233\pi$$
$$68$$ 15.7921 1.91508
$$69$$ −8.91962 −1.07380
$$70$$ 0 0
$$71$$ 6.32968 0.751194 0.375597 0.926783i $$-0.377438\pi$$
0.375597 + 0.926783i $$0.377438\pi$$
$$72$$ −11.6056 −1.36773
$$73$$ 1.37759 0.161235 0.0806173 0.996745i $$-0.474311\pi$$
0.0806173 + 0.996745i $$0.474311\pi$$
$$74$$ 17.5429 2.03932
$$75$$ 0 0
$$76$$ 4.82936 0.553965
$$77$$ 3.61829 0.412343
$$78$$ −4.62125 −0.523254
$$79$$ 13.6725 1.53828 0.769141 0.639079i $$-0.220684\pi$$
0.769141 + 0.639079i $$0.220684\pi$$
$$80$$ 0 0
$$81$$ −1.82753 −0.203059
$$82$$ −10.2559 −1.13258
$$83$$ −5.44061 −0.597184 −0.298592 0.954381i $$-0.596517\pi$$
−0.298592 + 0.954381i $$0.596517\pi$$
$$84$$ 20.8988 2.28025
$$85$$ 0 0
$$86$$ −14.0625 −1.51640
$$87$$ −1.22631 −0.131474
$$88$$ −7.39397 −0.788200
$$89$$ 12.1357 1.28638 0.643191 0.765706i $$-0.277610\pi$$
0.643191 + 0.765706i $$0.277610\pi$$
$$90$$ 0 0
$$91$$ −5.34990 −0.560822
$$92$$ 36.0170 3.75503
$$93$$ −1.97244 −0.204533
$$94$$ 9.71096 1.00161
$$95$$ 0 0
$$96$$ −12.5184 −1.27766
$$97$$ 13.7910 1.40026 0.700131 0.714014i $$-0.253125\pi$$
0.700131 + 0.714014i $$0.253125\pi$$
$$98$$ 15.9204 1.60820
$$99$$ 1.56960 0.157751
$$100$$ 0 0
$$101$$ −11.0029 −1.09483 −0.547413 0.836863i $$-0.684387\pi$$
−0.547413 + 0.836863i $$0.684387\pi$$
$$102$$ −10.2204 −1.01198
$$103$$ 4.99191 0.491867 0.245934 0.969287i $$-0.420906\pi$$
0.245934 + 0.969287i $$0.420906\pi$$
$$104$$ 10.9325 1.07202
$$105$$ 0 0
$$106$$ 0.268914 0.0261192
$$107$$ −7.31345 −0.707018 −0.353509 0.935431i $$-0.615012\pi$$
−0.353509 + 0.935431i $$0.615012\pi$$
$$108$$ 26.3934 2.53971
$$109$$ −1.44482 −0.138389 −0.0691944 0.997603i $$-0.522043\pi$$
−0.0691944 + 0.997603i $$0.522043\pi$$
$$110$$ 0 0
$$111$$ −8.02861 −0.762042
$$112$$ −34.9671 −3.30408
$$113$$ 12.0369 1.13234 0.566169 0.824289i $$-0.308425\pi$$
0.566169 + 0.824289i $$0.308425\pi$$
$$114$$ −3.12549 −0.292729
$$115$$ 0 0
$$116$$ 4.95178 0.459761
$$117$$ −2.32077 −0.214555
$$118$$ 34.7534 3.19931
$$119$$ −11.8319 −1.08463
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −16.9777 −1.53709
$$123$$ 4.69368 0.423215
$$124$$ 7.96463 0.715245
$$125$$ 0 0
$$126$$ 14.8417 1.32220
$$127$$ 4.69692 0.416784 0.208392 0.978045i $$-0.433177\pi$$
0.208392 + 0.978045i $$0.433177\pi$$
$$128$$ 0.0389415 0.00344197
$$129$$ 6.43578 0.566639
$$130$$ 0 0
$$131$$ 3.74466 0.327173 0.163586 0.986529i $$-0.447694\pi$$
0.163586 + 0.986529i $$0.447694\pi$$
$$132$$ 5.77587 0.502725
$$133$$ −3.61829 −0.313746
$$134$$ 9.69507 0.837526
$$135$$ 0 0
$$136$$ 24.1785 2.07329
$$137$$ 15.7595 1.34643 0.673213 0.739449i $$-0.264914\pi$$
0.673213 + 0.739449i $$0.264914\pi$$
$$138$$ −23.3097 −1.98425
$$139$$ −2.52822 −0.214440 −0.107220 0.994235i $$-0.534195\pi$$
−0.107220 + 0.994235i $$0.534195\pi$$
$$140$$ 0 0
$$141$$ −4.44427 −0.374275
$$142$$ 16.5414 1.38812
$$143$$ −1.47857 −0.123644
$$144$$ −15.1686 −1.26405
$$145$$ 0 0
$$146$$ 3.60006 0.297943
$$147$$ −7.28604 −0.600942
$$148$$ 32.4191 2.66484
$$149$$ 1.84902 0.151477 0.0757387 0.997128i $$-0.475869\pi$$
0.0757387 + 0.997128i $$0.475869\pi$$
$$150$$ 0 0
$$151$$ −15.3184 −1.24659 −0.623296 0.781986i $$-0.714207\pi$$
−0.623296 + 0.781986i $$0.714207\pi$$
$$152$$ 7.39397 0.599730
$$153$$ −5.13265 −0.414950
$$154$$ 9.45570 0.761962
$$155$$ 0 0
$$156$$ −8.54003 −0.683750
$$157$$ −24.6631 −1.96833 −0.984165 0.177254i $$-0.943278\pi$$
−0.984165 + 0.177254i $$0.943278\pi$$
$$158$$ 35.7305 2.84257
$$159$$ −0.123070 −0.00976006
$$160$$ 0 0
$$161$$ −26.9850 −2.12671
$$162$$ −4.77590 −0.375230
$$163$$ 3.72149 0.291490 0.145745 0.989322i $$-0.453442\pi$$
0.145745 + 0.989322i $$0.453442\pi$$
$$164$$ −18.9529 −1.47997
$$165$$ 0 0
$$166$$ −14.2180 −1.10353
$$167$$ −2.64758 −0.204876 −0.102438 0.994739i $$-0.532664\pi$$
−0.102438 + 0.994739i $$0.532664\pi$$
$$168$$ 31.9970 2.46863
$$169$$ −10.8138 −0.831833
$$170$$ 0 0
$$171$$ −1.56960 −0.120031
$$172$$ −25.9874 −1.98152
$$173$$ −7.34552 −0.558469 −0.279235 0.960223i $$-0.590081\pi$$
−0.279235 + 0.960223i $$0.590081\pi$$
$$174$$ −3.20472 −0.242949
$$175$$ 0 0
$$176$$ −9.66398 −0.728450
$$177$$ −15.9051 −1.19550
$$178$$ 31.7143 2.37708
$$179$$ 9.55394 0.714095 0.357047 0.934086i $$-0.383783\pi$$
0.357047 + 0.934086i $$0.383783\pi$$
$$180$$ 0 0
$$181$$ 6.02638 0.447937 0.223969 0.974596i $$-0.428099\pi$$
0.223969 + 0.974596i $$0.428099\pi$$
$$182$$ −13.9809 −1.03633
$$183$$ 7.76993 0.574369
$$184$$ 55.1437 4.06525
$$185$$ 0 0
$$186$$ −5.15459 −0.377953
$$187$$ −3.27003 −0.239128
$$188$$ 17.9458 1.30883
$$189$$ −19.7747 −1.43840
$$190$$ 0 0
$$191$$ 17.7069 1.28123 0.640613 0.767864i $$-0.278680\pi$$
0.640613 + 0.767864i $$0.278680\pi$$
$$192$$ −9.59835 −0.692701
$$193$$ −3.69348 −0.265863 −0.132931 0.991125i $$-0.542439\pi$$
−0.132931 + 0.991125i $$0.542439\pi$$
$$194$$ 36.0400 2.58752
$$195$$ 0 0
$$196$$ 29.4207 2.10148
$$197$$ 25.7789 1.83667 0.918336 0.395802i $$-0.129533\pi$$
0.918336 + 0.395802i $$0.129533\pi$$
$$198$$ 4.10185 0.291506
$$199$$ 18.8953 1.33945 0.669726 0.742608i $$-0.266411\pi$$
0.669726 + 0.742608i $$0.266411\pi$$
$$200$$ 0 0
$$201$$ −4.43700 −0.312962
$$202$$ −28.7538 −2.02311
$$203$$ −3.71002 −0.260392
$$204$$ −18.8873 −1.32237
$$205$$ 0 0
$$206$$ 13.0454 0.908914
$$207$$ −11.7060 −0.813623
$$208$$ 14.2889 0.990755
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −10.1993 −0.702150 −0.351075 0.936347i $$-0.614184\pi$$
−0.351075 + 0.936347i $$0.614184\pi$$
$$212$$ 0.496950 0.0341306
$$213$$ −7.57024 −0.518704
$$214$$ −19.1123 −1.30649
$$215$$ 0 0
$$216$$ 40.4096 2.74953
$$217$$ −5.96733 −0.405089
$$218$$ −3.77576 −0.255727
$$219$$ −1.64759 −0.111334
$$220$$ 0 0
$$221$$ 4.83497 0.325235
$$222$$ −20.9812 −1.40817
$$223$$ 0.262700 0.0175917 0.00879584 0.999961i $$-0.497200\pi$$
0.00879584 + 0.999961i $$0.497200\pi$$
$$224$$ −37.8726 −2.53047
$$225$$ 0 0
$$226$$ 31.4561 2.09243
$$227$$ −27.3256 −1.81367 −0.906834 0.421489i $$-0.861508\pi$$
−0.906834 + 0.421489i $$0.861508\pi$$
$$228$$ −5.77587 −0.382516
$$229$$ −5.53171 −0.365546 −0.182773 0.983155i $$-0.558507\pi$$
−0.182773 + 0.983155i $$0.558507\pi$$
$$230$$ 0 0
$$231$$ −4.32745 −0.284725
$$232$$ 7.58141 0.497744
$$233$$ −27.4733 −1.79984 −0.899918 0.436059i $$-0.856374\pi$$
−0.899918 + 0.436059i $$0.856374\pi$$
$$234$$ −6.06487 −0.396473
$$235$$ 0 0
$$236$$ 64.2239 4.18062
$$237$$ −16.3523 −1.06219
$$238$$ −30.9204 −2.00427
$$239$$ 1.40339 0.0907781 0.0453890 0.998969i $$-0.485547\pi$$
0.0453890 + 0.998969i $$0.485547\pi$$
$$240$$ 0 0
$$241$$ −20.7696 −1.33789 −0.668944 0.743312i $$-0.733254\pi$$
−0.668944 + 0.743312i $$0.733254\pi$$
$$242$$ 2.61330 0.167990
$$243$$ −14.2099 −0.911566
$$244$$ −31.3746 −2.00855
$$245$$ 0 0
$$246$$ 12.2660 0.782052
$$247$$ 1.47857 0.0940791
$$248$$ 12.1942 0.774334
$$249$$ 6.50692 0.412359
$$250$$ 0 0
$$251$$ −1.29936 −0.0820148 −0.0410074 0.999159i $$-0.513057\pi$$
−0.0410074 + 0.999159i $$0.513057\pi$$
$$252$$ 27.4273 1.72776
$$253$$ −7.45793 −0.468876
$$254$$ 12.2745 0.770169
$$255$$ 0 0
$$256$$ −15.9491 −0.996819
$$257$$ −3.41219 −0.212847 −0.106423 0.994321i $$-0.533940\pi$$
−0.106423 + 0.994321i $$0.533940\pi$$
$$258$$ 16.8187 1.04708
$$259$$ −24.2893 −1.50927
$$260$$ 0 0
$$261$$ −1.60939 −0.0996189
$$262$$ 9.78594 0.604577
$$263$$ −14.2418 −0.878189 −0.439094 0.898441i $$-0.644701\pi$$
−0.439094 + 0.898441i $$0.644701\pi$$
$$264$$ 8.84313 0.544257
$$265$$ 0 0
$$266$$ −9.45570 −0.579766
$$267$$ −14.5142 −0.888254
$$268$$ 17.9164 1.09442
$$269$$ 5.39477 0.328925 0.164462 0.986383i $$-0.447411\pi$$
0.164462 + 0.986383i $$0.447411\pi$$
$$270$$ 0 0
$$271$$ −11.0624 −0.671995 −0.335998 0.941863i $$-0.609073\pi$$
−0.335998 + 0.941863i $$0.609073\pi$$
$$272$$ 31.6015 1.91612
$$273$$ 6.39843 0.387251
$$274$$ 41.1844 2.48804
$$275$$ 0 0
$$276$$ −43.0760 −2.59287
$$277$$ −14.0808 −0.846036 −0.423018 0.906121i $$-0.639029\pi$$
−0.423018 + 0.906121i $$0.639029\pi$$
$$278$$ −6.60700 −0.396261
$$279$$ −2.58861 −0.154976
$$280$$ 0 0
$$281$$ −10.8199 −0.645459 −0.322729 0.946491i $$-0.604600\pi$$
−0.322729 + 0.946491i $$0.604600\pi$$
$$282$$ −11.6142 −0.691617
$$283$$ −1.90947 −0.113506 −0.0567532 0.998388i $$-0.518075\pi$$
−0.0567532 + 0.998388i $$0.518075\pi$$
$$284$$ 30.5683 1.81389
$$285$$ 0 0
$$286$$ −3.86395 −0.228480
$$287$$ 14.2000 0.838201
$$288$$ −16.4290 −0.968089
$$289$$ −6.30690 −0.370994
$$290$$ 0 0
$$291$$ −16.4939 −0.966890
$$292$$ 6.65287 0.389330
$$293$$ −3.63550 −0.212388 −0.106194 0.994345i $$-0.533867\pi$$
−0.106194 + 0.994345i $$0.533867\pi$$
$$294$$ −19.0406 −1.11047
$$295$$ 0 0
$$296$$ 49.6352 2.88499
$$297$$ −5.46521 −0.317124
$$298$$ 4.83205 0.279913
$$299$$ 11.0271 0.637712
$$300$$ 0 0
$$301$$ 19.4705 1.12226
$$302$$ −40.0316 −2.30356
$$303$$ 13.1593 0.755984
$$304$$ 9.66398 0.554267
$$305$$ 0 0
$$306$$ −13.4132 −0.766781
$$307$$ 23.5329 1.34310 0.671548 0.740961i $$-0.265630\pi$$
0.671548 + 0.740961i $$0.265630\pi$$
$$308$$ 17.4740 0.995675
$$309$$ −5.97028 −0.339637
$$310$$ 0 0
$$311$$ −16.0026 −0.907425 −0.453713 0.891148i $$-0.649901\pi$$
−0.453713 + 0.891148i $$0.649901\pi$$
$$312$$ −13.0752 −0.740237
$$313$$ −16.0034 −0.904566 −0.452283 0.891875i $$-0.649390\pi$$
−0.452283 + 0.891875i $$0.649390\pi$$
$$314$$ −64.4522 −3.63725
$$315$$ 0 0
$$316$$ 66.0296 3.71446
$$317$$ 20.8766 1.17255 0.586273 0.810113i $$-0.300595\pi$$
0.586273 + 0.810113i $$0.300595\pi$$
$$318$$ −0.321619 −0.0180355
$$319$$ −1.02535 −0.0574086
$$320$$ 0 0
$$321$$ 8.74683 0.488200
$$322$$ −70.5199 −3.92992
$$323$$ 3.27003 0.181949
$$324$$ −8.82581 −0.490323
$$325$$ 0 0
$$326$$ 9.72539 0.538640
$$327$$ 1.72800 0.0955584
$$328$$ −29.0177 −1.60224
$$329$$ −13.4455 −0.741273
$$330$$ 0 0
$$331$$ 15.1136 0.830721 0.415360 0.909657i $$-0.363655\pi$$
0.415360 + 0.909657i $$0.363655\pi$$
$$332$$ −26.2746 −1.44201
$$333$$ −10.5366 −0.577404
$$334$$ −6.91893 −0.378587
$$335$$ 0 0
$$336$$ 41.8204 2.28149
$$337$$ 12.2766 0.668751 0.334376 0.942440i $$-0.391475\pi$$
0.334376 + 0.942440i $$0.391475\pi$$
$$338$$ −28.2598 −1.53713
$$339$$ −14.3961 −0.781886
$$340$$ 0 0
$$341$$ −1.64921 −0.0893098
$$342$$ −4.10185 −0.221803
$$343$$ 3.28525 0.177387
$$344$$ −39.7879 −2.14522
$$345$$ 0 0
$$346$$ −19.1961 −1.03199
$$347$$ 30.9067 1.65916 0.829580 0.558387i $$-0.188580\pi$$
0.829580 + 0.558387i $$0.188580\pi$$
$$348$$ −5.92229 −0.317468
$$349$$ −23.9024 −1.27947 −0.639733 0.768597i $$-0.720955\pi$$
−0.639733 + 0.768597i $$0.720955\pi$$
$$350$$ 0 0
$$351$$ 8.08069 0.431315
$$352$$ −10.4670 −0.557892
$$353$$ −15.0158 −0.799210 −0.399605 0.916687i $$-0.630853\pi$$
−0.399605 + 0.916687i $$0.630853\pi$$
$$354$$ −41.5648 −2.20914
$$355$$ 0 0
$$356$$ 58.6076 3.10620
$$357$$ 14.1509 0.748945
$$358$$ 24.9673 1.31957
$$359$$ 14.0826 0.743251 0.371626 0.928383i $$-0.378800\pi$$
0.371626 + 0.928383i $$0.378800\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 15.7488 0.827737
$$363$$ −1.19599 −0.0627733
$$364$$ −25.8366 −1.35420
$$365$$ 0 0
$$366$$ 20.3052 1.06137
$$367$$ −21.3142 −1.11259 −0.556296 0.830984i $$-0.687778\pi$$
−0.556296 + 0.830984i $$0.687778\pi$$
$$368$$ 72.0733 3.75708
$$369$$ 6.15992 0.320673
$$370$$ 0 0
$$371$$ −0.372329 −0.0193304
$$372$$ −9.52564 −0.493881
$$373$$ 2.42088 0.125349 0.0626743 0.998034i $$-0.480037\pi$$
0.0626743 + 0.998034i $$0.480037\pi$$
$$374$$ −8.54558 −0.441882
$$375$$ 0 0
$$376$$ 27.4758 1.41696
$$377$$ 1.51605 0.0780806
$$378$$ −51.6774 −2.65800
$$379$$ 8.87535 0.455896 0.227948 0.973673i $$-0.426798\pi$$
0.227948 + 0.973673i $$0.426798\pi$$
$$380$$ 0 0
$$381$$ −5.61748 −0.287792
$$382$$ 46.2735 2.36756
$$383$$ 3.54065 0.180919 0.0904595 0.995900i $$-0.471166\pi$$
0.0904595 + 0.995900i $$0.471166\pi$$
$$384$$ −0.0465737 −0.00237670
$$385$$ 0 0
$$386$$ −9.65219 −0.491284
$$387$$ 8.44624 0.429346
$$388$$ 66.6016 3.38118
$$389$$ −16.7041 −0.846933 −0.423466 0.905912i $$-0.639187\pi$$
−0.423466 + 0.905912i $$0.639187\pi$$
$$390$$ 0 0
$$391$$ 24.3877 1.23334
$$392$$ 45.0444 2.27509
$$393$$ −4.47858 −0.225915
$$394$$ 67.3681 3.39396
$$395$$ 0 0
$$396$$ 7.58018 0.380918
$$397$$ −28.2073 −1.41568 −0.707841 0.706372i $$-0.750331\pi$$
−0.707841 + 0.706372i $$0.750331\pi$$
$$398$$ 49.3792 2.47515
$$399$$ 4.32745 0.216643
$$400$$ 0 0
$$401$$ −36.2078 −1.80813 −0.904065 0.427395i $$-0.859431\pi$$
−0.904065 + 0.427395i $$0.859431\pi$$
$$402$$ −11.5952 −0.578317
$$403$$ 2.43847 0.121469
$$404$$ −53.1368 −2.64365
$$405$$ 0 0
$$406$$ −9.69540 −0.481175
$$407$$ −6.71293 −0.332748
$$408$$ −28.9173 −1.43162
$$409$$ 36.9236 1.82576 0.912878 0.408233i $$-0.133855\pi$$
0.912878 + 0.408233i $$0.133855\pi$$
$$410$$ 0 0
$$411$$ −18.8482 −0.929715
$$412$$ 24.1077 1.18770
$$413$$ −48.1184 −2.36775
$$414$$ −30.5913 −1.50348
$$415$$ 0 0
$$416$$ 15.4762 0.758781
$$417$$ 3.02373 0.148072
$$418$$ −2.61330 −0.127821
$$419$$ 18.0690 0.882726 0.441363 0.897329i $$-0.354495\pi$$
0.441363 + 0.897329i $$0.354495\pi$$
$$420$$ 0 0
$$421$$ −8.57629 −0.417983 −0.208991 0.977918i $$-0.567018\pi$$
−0.208991 + 0.977918i $$0.567018\pi$$
$$422$$ −26.6539 −1.29749
$$423$$ −5.83260 −0.283591
$$424$$ 0.760853 0.0369503
$$425$$ 0 0
$$426$$ −19.7833 −0.958506
$$427$$ 23.5067 1.13757
$$428$$ −35.3193 −1.70722
$$429$$ 1.76836 0.0853771
$$430$$ 0 0
$$431$$ 4.28147 0.206231 0.103116 0.994669i $$-0.467119\pi$$
0.103116 + 0.994669i $$0.467119\pi$$
$$432$$ 52.8157 2.54110
$$433$$ −18.2035 −0.874804 −0.437402 0.899266i $$-0.644101\pi$$
−0.437402 + 0.899266i $$0.644101\pi$$
$$434$$ −15.5945 −0.748558
$$435$$ 0 0
$$436$$ −6.97756 −0.334165
$$437$$ 7.45793 0.356761
$$438$$ −4.30564 −0.205732
$$439$$ 29.4442 1.40529 0.702647 0.711538i $$-0.252001\pi$$
0.702647 + 0.711538i $$0.252001\pi$$
$$440$$ 0 0
$$441$$ −9.56210 −0.455338
$$442$$ 12.6352 0.600997
$$443$$ 24.3633 1.15753 0.578767 0.815493i $$-0.303534\pi$$
0.578767 + 0.815493i $$0.303534\pi$$
$$444$$ −38.7730 −1.84009
$$445$$ 0 0
$$446$$ 0.686514 0.0325074
$$447$$ −2.21141 −0.104596
$$448$$ −29.0384 −1.37193
$$449$$ 38.7776 1.83003 0.915015 0.403421i $$-0.132179\pi$$
0.915015 + 0.403421i $$0.132179\pi$$
$$450$$ 0 0
$$451$$ 3.92451 0.184798
$$452$$ 58.1306 2.73423
$$453$$ 18.3206 0.860779
$$454$$ −71.4102 −3.35145
$$455$$ 0 0
$$456$$ −8.84313 −0.414118
$$457$$ 7.47672 0.349746 0.174873 0.984591i $$-0.444048\pi$$
0.174873 + 0.984591i $$0.444048\pi$$
$$458$$ −14.4560 −0.675487
$$459$$ 17.8714 0.834165
$$460$$ 0 0
$$461$$ 15.9782 0.744181 0.372091 0.928196i $$-0.378641\pi$$
0.372091 + 0.928196i $$0.378641\pi$$
$$462$$ −11.3089 −0.526139
$$463$$ 6.10221 0.283594 0.141797 0.989896i $$-0.454712\pi$$
0.141797 + 0.989896i $$0.454712\pi$$
$$464$$ 9.90896 0.460012
$$465$$ 0 0
$$466$$ −71.7961 −3.32589
$$467$$ 26.6373 1.23263 0.616313 0.787502i $$-0.288626\pi$$
0.616313 + 0.787502i $$0.288626\pi$$
$$468$$ −11.2078 −0.518082
$$469$$ −13.4235 −0.619838
$$470$$ 0 0
$$471$$ 29.4969 1.35914
$$472$$ 98.3298 4.52599
$$473$$ 5.38113 0.247424
$$474$$ −42.7334 −1.96281
$$475$$ 0 0
$$476$$ −57.1406 −2.61904
$$477$$ −0.161515 −0.00739527
$$478$$ 3.66750 0.167747
$$479$$ −16.2094 −0.740626 −0.370313 0.928907i $$-0.620750\pi$$
−0.370313 + 0.928907i $$0.620750\pi$$
$$480$$ 0 0
$$481$$ 9.92553 0.452565
$$482$$ −54.2773 −2.47226
$$483$$ 32.2738 1.46851
$$484$$ 4.82936 0.219516
$$485$$ 0 0
$$486$$ −37.1348 −1.68447
$$487$$ 4.82448 0.218618 0.109309 0.994008i $$-0.465136\pi$$
0.109309 + 0.994008i $$0.465136\pi$$
$$488$$ −48.0360 −2.17449
$$489$$ −4.45088 −0.201276
$$490$$ 0 0
$$491$$ −8.53579 −0.385215 −0.192607 0.981276i $$-0.561694\pi$$
−0.192607 + 0.981276i $$0.561694\pi$$
$$492$$ 22.6675 1.02193
$$493$$ 3.35293 0.151008
$$494$$ 3.86395 0.173847
$$495$$ 0 0
$$496$$ 15.9380 0.715635
$$497$$ −22.9026 −1.02732
$$498$$ 17.0046 0.761993
$$499$$ −13.4325 −0.601319 −0.300660 0.953732i $$-0.597207\pi$$
−0.300660 + 0.953732i $$0.597207\pi$$
$$500$$ 0 0
$$501$$ 3.16648 0.141468
$$502$$ −3.39562 −0.151554
$$503$$ 1.66487 0.0742327 0.0371163 0.999311i $$-0.488183\pi$$
0.0371163 + 0.999311i $$0.488183\pi$$
$$504$$ 41.9925 1.87049
$$505$$ 0 0
$$506$$ −19.4898 −0.866429
$$507$$ 12.9333 0.574386
$$508$$ 22.6831 1.00640
$$509$$ −14.9684 −0.663463 −0.331731 0.943374i $$-0.607633\pi$$
−0.331731 + 0.943374i $$0.607633\pi$$
$$510$$ 0 0
$$511$$ −4.98452 −0.220502
$$512$$ −41.7577 −1.84545
$$513$$ 5.46521 0.241295
$$514$$ −8.91709 −0.393316
$$515$$ 0 0
$$516$$ 31.0807 1.36825
$$517$$ −3.71597 −0.163428
$$518$$ −63.4754 −2.78895
$$519$$ 8.78518 0.385627
$$520$$ 0 0
$$521$$ −7.89123 −0.345721 −0.172861 0.984946i $$-0.555301\pi$$
−0.172861 + 0.984946i $$0.555301\pi$$
$$522$$ −4.20583 −0.184084
$$523$$ 22.3062 0.975381 0.487690 0.873017i $$-0.337840\pi$$
0.487690 + 0.873017i $$0.337840\pi$$
$$524$$ 18.0843 0.790017
$$525$$ 0 0
$$526$$ −37.2182 −1.62279
$$527$$ 5.39297 0.234922
$$528$$ 11.5580 0.502999
$$529$$ 32.6207 1.41829
$$530$$ 0 0
$$531$$ −20.8736 −0.905837
$$532$$ −17.4740 −0.757595
$$533$$ −5.80266 −0.251341
$$534$$ −37.9300 −1.64139
$$535$$ 0 0
$$536$$ 27.4308 1.18483
$$537$$ −11.4264 −0.493087
$$538$$ 14.0982 0.607815
$$539$$ −6.09205 −0.262403
$$540$$ 0 0
$$541$$ 38.9694 1.67542 0.837712 0.546112i $$-0.183893\pi$$
0.837712 + 0.546112i $$0.183893\pi$$
$$542$$ −28.9095 −1.24177
$$543$$ −7.20750 −0.309304
$$544$$ 34.2273 1.46749
$$545$$ 0 0
$$546$$ 16.7211 0.715595
$$547$$ −37.7503 −1.61409 −0.807043 0.590492i $$-0.798934\pi$$
−0.807043 + 0.590492i $$0.798934\pi$$
$$548$$ 76.1083 3.25119
$$549$$ 10.1971 0.435204
$$550$$ 0 0
$$551$$ 1.02535 0.0436814
$$552$$ −65.9514 −2.80708
$$553$$ −49.4713 −2.10373
$$554$$ −36.7975 −1.56338
$$555$$ 0 0
$$556$$ −12.2097 −0.517805
$$557$$ 3.17436 0.134502 0.0672511 0.997736i $$-0.478577\pi$$
0.0672511 + 0.997736i $$0.478577\pi$$
$$558$$ −6.76482 −0.286378
$$559$$ −7.95637 −0.336519
$$560$$ 0 0
$$561$$ 3.91093 0.165120
$$562$$ −28.2756 −1.19273
$$563$$ 19.9431 0.840503 0.420252 0.907408i $$-0.361942\pi$$
0.420252 + 0.907408i $$0.361942\pi$$
$$564$$ −21.4630 −0.903754
$$565$$ 0 0
$$566$$ −4.99004 −0.209747
$$567$$ 6.61255 0.277701
$$568$$ 46.8015 1.96375
$$569$$ 36.6424 1.53613 0.768064 0.640374i $$-0.221220\pi$$
0.768064 + 0.640374i $$0.221220\pi$$
$$570$$ 0 0
$$571$$ 11.5300 0.482515 0.241258 0.970461i $$-0.422440\pi$$
0.241258 + 0.970461i $$0.422440\pi$$
$$572$$ −7.14054 −0.298561
$$573$$ −21.1773 −0.884694
$$574$$ 37.1090 1.54890
$$575$$ 0 0
$$576$$ −12.5968 −0.524865
$$577$$ −28.5590 −1.18893 −0.594463 0.804123i $$-0.702635\pi$$
−0.594463 + 0.804123i $$0.702635\pi$$
$$578$$ −16.4818 −0.685554
$$579$$ 4.41737 0.183580
$$580$$ 0 0
$$581$$ 19.6857 0.816701
$$582$$ −43.1036 −1.78670
$$583$$ −0.102902 −0.00426176
$$584$$ 10.1859 0.421494
$$585$$ 0 0
$$586$$ −9.50067 −0.392469
$$587$$ −18.1461 −0.748969 −0.374484 0.927233i $$-0.622180\pi$$
−0.374484 + 0.927233i $$0.622180\pi$$
$$588$$ −35.1869 −1.45108
$$589$$ 1.64921 0.0679546
$$590$$ 0 0
$$591$$ −30.8314 −1.26823
$$592$$ 64.8736 2.66629
$$593$$ −15.3085 −0.628644 −0.314322 0.949316i $$-0.601777\pi$$
−0.314322 + 0.949316i $$0.601777\pi$$
$$594$$ −14.2823 −0.586008
$$595$$ 0 0
$$596$$ 8.92957 0.365769
$$597$$ −22.5986 −0.924900
$$598$$ 28.8171 1.17842
$$599$$ 17.8806 0.730580 0.365290 0.930894i $$-0.380970\pi$$
0.365290 + 0.930894i $$0.380970\pi$$
$$600$$ 0 0
$$601$$ −32.5803 −1.32898 −0.664489 0.747298i $$-0.731351\pi$$
−0.664489 + 0.747298i $$0.731351\pi$$
$$602$$ 50.8823 2.07381
$$603$$ −5.82306 −0.237133
$$604$$ −73.9779 −3.01012
$$605$$ 0 0
$$606$$ 34.3893 1.39697
$$607$$ −43.6494 −1.77167 −0.885837 0.463997i $$-0.846415\pi$$
−0.885837 + 0.463997i $$0.846415\pi$$
$$608$$ 10.4670 0.424492
$$609$$ 4.43715 0.179802
$$610$$ 0 0
$$611$$ 5.49432 0.222276
$$612$$ −24.7874 −1.00197
$$613$$ −0.843061 −0.0340509 −0.0170254 0.999855i $$-0.505420\pi$$
−0.0170254 + 0.999855i $$0.505420\pi$$
$$614$$ 61.4987 2.48189
$$615$$ 0 0
$$616$$ 26.7536 1.07793
$$617$$ −4.89882 −0.197219 −0.0986094 0.995126i $$-0.531439\pi$$
−0.0986094 + 0.995126i $$0.531439\pi$$
$$618$$ −15.6021 −0.627611
$$619$$ −14.8704 −0.597691 −0.298846 0.954301i $$-0.596602\pi$$
−0.298846 + 0.954301i $$0.596602\pi$$
$$620$$ 0 0
$$621$$ 40.7591 1.63561
$$622$$ −41.8197 −1.67682
$$623$$ −43.9105 −1.75924
$$624$$ −17.0894 −0.684122
$$625$$ 0 0
$$626$$ −41.8217 −1.67153
$$627$$ 1.19599 0.0477633
$$628$$ −119.107 −4.75289
$$629$$ 21.9515 0.875263
$$630$$ 0 0
$$631$$ 2.83922 0.113027 0.0565137 0.998402i $$-0.482002\pi$$
0.0565137 + 0.998402i $$0.482002\pi$$
$$632$$ 101.094 4.02132
$$633$$ 12.1983 0.484839
$$634$$ 54.5569 2.16673
$$635$$ 0 0
$$636$$ −0.594348 −0.0235674
$$637$$ 9.00751 0.356891
$$638$$ −2.67955 −0.106084
$$639$$ −9.93508 −0.393026
$$640$$ 0 0
$$641$$ 20.6746 0.816599 0.408299 0.912848i $$-0.366122\pi$$
0.408299 + 0.912848i $$0.366122\pi$$
$$642$$ 22.8581 0.902138
$$643$$ 24.6254 0.971130 0.485565 0.874201i $$-0.338614\pi$$
0.485565 + 0.874201i $$0.338614\pi$$
$$644$$ −130.320 −5.13533
$$645$$ 0 0
$$646$$ 8.54558 0.336222
$$647$$ −45.3626 −1.78339 −0.891693 0.452640i $$-0.850482\pi$$
−0.891693 + 0.452640i $$0.850482\pi$$
$$648$$ −13.5127 −0.530830
$$649$$ −13.2986 −0.522017
$$650$$ 0 0
$$651$$ 7.13688 0.279716
$$652$$ 17.9724 0.703854
$$653$$ −26.1012 −1.02142 −0.510710 0.859753i $$-0.670617\pi$$
−0.510710 + 0.859753i $$0.670617\pi$$
$$654$$ 4.51578 0.176581
$$655$$ 0 0
$$656$$ −37.9264 −1.48078
$$657$$ −2.16227 −0.0843582
$$658$$ −35.1371 −1.36979
$$659$$ −45.8507 −1.78609 −0.893045 0.449967i $$-0.851436\pi$$
−0.893045 + 0.449967i $$0.851436\pi$$
$$660$$ 0 0
$$661$$ −15.4225 −0.599864 −0.299932 0.953961i $$-0.596964\pi$$
−0.299932 + 0.953961i $$0.596964\pi$$
$$662$$ 39.4965 1.53508
$$663$$ −5.78258 −0.224577
$$664$$ −40.2277 −1.56114
$$665$$ 0 0
$$666$$ −27.5354 −1.06698
$$667$$ 7.64698 0.296092
$$668$$ −12.7861 −0.494709
$$669$$ −0.314187 −0.0121472
$$670$$ 0 0
$$671$$ 6.49664 0.250800
$$672$$ 45.2953 1.74730
$$673$$ 37.8633 1.45952 0.729762 0.683702i $$-0.239631\pi$$
0.729762 + 0.683702i $$0.239631\pi$$
$$674$$ 32.0826 1.23578
$$675$$ 0 0
$$676$$ −52.2239 −2.00861
$$677$$ 25.3510 0.974320 0.487160 0.873313i $$-0.338033\pi$$
0.487160 + 0.873313i $$0.338033\pi$$
$$678$$ −37.6213 −1.44484
$$679$$ −49.8998 −1.91498
$$680$$ 0 0
$$681$$ 32.6812 1.25235
$$682$$ −4.30989 −0.165034
$$683$$ 21.7513 0.832290 0.416145 0.909298i $$-0.363381\pi$$
0.416145 + 0.909298i $$0.363381\pi$$
$$684$$ −7.58018 −0.289835
$$685$$ 0 0
$$686$$ 8.58534 0.327790
$$687$$ 6.61588 0.252412
$$688$$ −52.0031 −1.98260
$$689$$ 0.152147 0.00579636
$$690$$ 0 0
$$691$$ 17.3051 0.658318 0.329159 0.944275i $$-0.393235\pi$$
0.329159 + 0.944275i $$0.393235\pi$$
$$692$$ −35.4741 −1.34852
$$693$$ −5.67929 −0.215738
$$694$$ 80.7687 3.06594
$$695$$ 0 0
$$696$$ −9.06730 −0.343695
$$697$$ −12.8333 −0.486095
$$698$$ −62.4643 −2.36431
$$699$$ 32.8578 1.24280
$$700$$ 0 0
$$701$$ −29.6923 −1.12146 −0.560732 0.827997i $$-0.689480\pi$$
−0.560732 + 0.827997i $$0.689480\pi$$
$$702$$ 21.1173 0.797021
$$703$$ 6.71293 0.253183
$$704$$ −8.02543 −0.302470
$$705$$ 0 0
$$706$$ −39.2409 −1.47685
$$707$$ 39.8116 1.49727
$$708$$ −76.8112 −2.88674
$$709$$ −21.4898 −0.807067 −0.403534 0.914965i $$-0.632218\pi$$
−0.403534 + 0.914965i $$0.632218\pi$$
$$710$$ 0 0
$$711$$ −21.4605 −0.804831
$$712$$ 89.7310 3.36281
$$713$$ 12.2997 0.460627
$$714$$ 36.9806 1.38396
$$715$$ 0 0
$$716$$ 46.1394 1.72431
$$717$$ −1.67845 −0.0626828
$$718$$ 36.8021 1.37344
$$719$$ 9.61388 0.358537 0.179269 0.983800i $$-0.442627\pi$$
0.179269 + 0.983800i $$0.442627\pi$$
$$720$$ 0 0
$$721$$ −18.0622 −0.672671
$$722$$ 2.61330 0.0972571
$$723$$ 24.8403 0.923821
$$724$$ 29.1036 1.08163
$$725$$ 0 0
$$726$$ −3.12549 −0.115998
$$727$$ −6.84046 −0.253699 −0.126849 0.991922i $$-0.540486\pi$$
−0.126849 + 0.991922i $$0.540486\pi$$
$$728$$ −39.5570 −1.46608
$$729$$ 22.4775 0.832501
$$730$$ 0 0
$$731$$ −17.5965 −0.650828
$$732$$ 37.5238 1.38692
$$733$$ −38.2277 −1.41197 −0.705987 0.708225i $$-0.749496\pi$$
−0.705987 + 0.708225i $$0.749496\pi$$
$$734$$ −55.7005 −2.05594
$$735$$ 0 0
$$736$$ 78.0620 2.87740
$$737$$ −3.70989 −0.136656
$$738$$ 16.0978 0.592567
$$739$$ −17.7157 −0.651682 −0.325841 0.945425i $$-0.605647\pi$$
−0.325841 + 0.945425i $$0.605647\pi$$
$$740$$ 0 0
$$741$$ −1.76836 −0.0649622
$$742$$ −0.973009 −0.0357203
$$743$$ −21.3028 −0.781525 −0.390763 0.920491i $$-0.627789\pi$$
−0.390763 + 0.920491i $$0.627789\pi$$
$$744$$ −14.5842 −0.534682
$$745$$ 0 0
$$746$$ 6.32650 0.231630
$$747$$ 8.53960 0.312447
$$748$$ −15.7921 −0.577418
$$749$$ 26.4622 0.966908
$$750$$ 0 0
$$751$$ 25.1552 0.917926 0.458963 0.888455i $$-0.348221\pi$$
0.458963 + 0.888455i $$0.348221\pi$$
$$752$$ 35.9111 1.30954
$$753$$ 1.55402 0.0566317
$$754$$ 3.96190 0.144284
$$755$$ 0 0
$$756$$ −95.4992 −3.47327
$$757$$ −15.5116 −0.563780 −0.281890 0.959447i $$-0.590961\pi$$
−0.281890 + 0.959447i $$0.590961\pi$$
$$758$$ 23.1940 0.842443
$$759$$ 8.91962 0.323762
$$760$$ 0 0
$$761$$ 35.2085 1.27631 0.638154 0.769908i $$-0.279698\pi$$
0.638154 + 0.769908i $$0.279698\pi$$
$$762$$ −14.6802 −0.531807
$$763$$ 5.22779 0.189259
$$764$$ 85.5129 3.09375
$$765$$ 0 0
$$766$$ 9.25280 0.334317
$$767$$ 19.6630 0.709988
$$768$$ 19.0750 0.688310
$$769$$ −5.57667 −0.201100 −0.100550 0.994932i $$-0.532060\pi$$
−0.100550 + 0.994932i $$0.532060\pi$$
$$770$$ 0 0
$$771$$ 4.08095 0.146972
$$772$$ −17.8371 −0.641973
$$773$$ −35.9175 −1.29186 −0.645931 0.763396i $$-0.723531\pi$$
−0.645931 + 0.763396i $$0.723531\pi$$
$$774$$ 22.0726 0.793383
$$775$$ 0 0
$$776$$ 101.970 3.66052
$$777$$ 29.0499 1.04216
$$778$$ −43.6530 −1.56503
$$779$$ −3.92451 −0.140610
$$780$$ 0 0
$$781$$ −6.32968 −0.226494
$$782$$ 63.7324 2.27906
$$783$$ 5.60375 0.200262
$$784$$ 58.8734 2.10262
$$785$$ 0 0
$$786$$ −11.7039 −0.417464
$$787$$ −7.53242 −0.268502 −0.134251 0.990947i $$-0.542863\pi$$
−0.134251 + 0.990947i $$0.542863\pi$$
$$788$$ 124.496 4.43497
$$789$$ 17.0331 0.606395
$$790$$ 0 0
$$791$$ −43.5531 −1.54857
$$792$$ 11.6056 0.412387
$$793$$ −9.60573 −0.341110
$$794$$ −73.7141 −2.61602
$$795$$ 0 0
$$796$$ 91.2522 3.23435
$$797$$ 49.3837 1.74926 0.874629 0.484792i $$-0.161105\pi$$
0.874629 + 0.484792i $$0.161105\pi$$
$$798$$ 11.3089 0.400332
$$799$$ 12.1513 0.429883
$$800$$ 0 0
$$801$$ −19.0482 −0.673036
$$802$$ −94.6219 −3.34122
$$803$$ −1.37759 −0.0486141
$$804$$ −21.4278 −0.755702
$$805$$ 0 0
$$806$$ 6.37247 0.224461
$$807$$ −6.45210 −0.227125
$$808$$ −81.3549 −2.86205
$$809$$ 34.4637 1.21168 0.605840 0.795587i $$-0.292837\pi$$
0.605840 + 0.795587i $$0.292837\pi$$
$$810$$ 0 0
$$811$$ 20.0278 0.703272 0.351636 0.936137i $$-0.385625\pi$$
0.351636 + 0.936137i $$0.385625\pi$$
$$812$$ −17.9170 −0.628763
$$813$$ 13.2306 0.464017
$$814$$ −17.5429 −0.614879
$$815$$ 0 0
$$816$$ −37.7952 −1.32310
$$817$$ −5.38113 −0.188262
$$818$$ 96.4926 3.37379
$$819$$ 8.39722 0.293423
$$820$$ 0 0
$$821$$ 6.45245 0.225192 0.112596 0.993641i $$-0.464083\pi$$
0.112596 + 0.993641i $$0.464083\pi$$
$$822$$ −49.2562 −1.71801
$$823$$ 43.0190 1.49955 0.749774 0.661694i $$-0.230162\pi$$
0.749774 + 0.661694i $$0.230162\pi$$
$$824$$ 36.9100 1.28582
$$825$$ 0 0
$$826$$ −125.748 −4.37533
$$827$$ 43.1557 1.50067 0.750335 0.661058i $$-0.229892\pi$$
0.750335 + 0.661058i $$0.229892\pi$$
$$828$$ −56.5324 −1.96464
$$829$$ 13.4937 0.468656 0.234328 0.972158i $$-0.424711\pi$$
0.234328 + 0.972158i $$0.424711\pi$$
$$830$$ 0 0
$$831$$ 16.8406 0.584193
$$832$$ 11.8662 0.411385
$$833$$ 19.9212 0.690228
$$834$$ 7.90191 0.273621
$$835$$ 0 0
$$836$$ −4.82936 −0.167027
$$837$$ 9.01329 0.311545
$$838$$ 47.2197 1.63118
$$839$$ −14.6851 −0.506988 −0.253494 0.967337i $$-0.581580\pi$$
−0.253494 + 0.967337i $$0.581580\pi$$
$$840$$ 0 0
$$841$$ −27.9487 −0.963747
$$842$$ −22.4124 −0.772384
$$843$$ 12.9405 0.445693
$$844$$ −49.2562 −1.69547
$$845$$ 0 0
$$846$$ −15.2424 −0.524043
$$847$$ −3.61829 −0.124326
$$848$$ 0.994441 0.0341493
$$849$$ 2.28372 0.0783769
$$850$$ 0 0
$$851$$ 50.0645 1.71619
$$852$$ −36.5594 −1.25250
$$853$$ 51.5775 1.76598 0.882990 0.469392i $$-0.155527\pi$$
0.882990 + 0.469392i $$0.155527\pi$$
$$854$$ 61.4303 2.10210
$$855$$ 0 0
$$856$$ −54.0755 −1.84826
$$857$$ 46.6355 1.59304 0.796520 0.604612i $$-0.206672\pi$$
0.796520 + 0.604612i $$0.206672\pi$$
$$858$$ 4.62125 0.157767
$$859$$ −18.6711 −0.637048 −0.318524 0.947915i $$-0.603187\pi$$
−0.318524 + 0.947915i $$0.603187\pi$$
$$860$$ 0 0
$$861$$ −16.9831 −0.578783
$$862$$ 11.1888 0.381091
$$863$$ −43.0160 −1.46428 −0.732141 0.681153i $$-0.761479\pi$$
−0.732141 + 0.681153i $$0.761479\pi$$
$$864$$ 57.2042 1.94613
$$865$$ 0 0
$$866$$ −47.5712 −1.61654
$$867$$ 7.54300 0.256174
$$868$$ −28.8184 −0.978160
$$869$$ −13.6725 −0.463809
$$870$$ 0 0
$$871$$ 5.48533 0.185863
$$872$$ −10.6830 −0.361771
$$873$$ −21.6464 −0.732619
$$874$$ 19.4898 0.659253
$$875$$ 0 0
$$876$$ −7.95678 −0.268835
$$877$$ −56.0429 −1.89243 −0.946217 0.323534i $$-0.895129\pi$$
−0.946217 + 0.323534i $$0.895129\pi$$
$$878$$ 76.9466 2.59682
$$879$$ 4.34803 0.146655
$$880$$ 0 0
$$881$$ 17.9947 0.606257 0.303128 0.952950i $$-0.401969\pi$$
0.303128 + 0.952950i $$0.401969\pi$$
$$882$$ −24.9887 −0.841412
$$883$$ −1.99550 −0.0671540 −0.0335770 0.999436i $$-0.510690\pi$$
−0.0335770 + 0.999436i $$0.510690\pi$$
$$884$$ 23.3498 0.785339
$$885$$ 0 0
$$886$$ 63.6687 2.13899
$$887$$ −7.20927 −0.242063 −0.121032 0.992649i $$-0.538620\pi$$
−0.121032 + 0.992649i $$0.538620\pi$$
$$888$$ −59.3633 −1.99210
$$889$$ −16.9948 −0.569988
$$890$$ 0 0
$$891$$ 1.82753 0.0612246
$$892$$ 1.26867 0.0424782
$$893$$ 3.71597 0.124350
$$894$$ −5.77909 −0.193282
$$895$$ 0 0
$$896$$ −0.140902 −0.00470720
$$897$$ −13.1883 −0.440344
$$898$$ 101.338 3.38168
$$899$$ 1.69102 0.0563986
$$900$$ 0 0
$$901$$ 0.336492 0.0112102
$$902$$ 10.2559 0.341485
$$903$$ −23.2866 −0.774928
$$904$$ 89.0006 2.96012
$$905$$ 0 0
$$906$$ 47.8774 1.59062
$$907$$ 25.7832 0.856116 0.428058 0.903751i $$-0.359198\pi$$
0.428058 + 0.903751i $$0.359198\pi$$
$$908$$ −131.965 −4.37942
$$909$$ 17.2701 0.572814
$$910$$ 0 0
$$911$$ 31.1334 1.03149 0.515747 0.856741i $$-0.327514\pi$$
0.515747 + 0.856741i $$0.327514\pi$$
$$912$$ −11.5580 −0.382725
$$913$$ 5.44061 0.180058
$$914$$ 19.5389 0.646291
$$915$$ 0 0
$$916$$ −26.7146 −0.882676
$$917$$ −13.5493 −0.447437
$$918$$ 46.7034 1.54144
$$919$$ 5.42725 0.179029 0.0895143 0.995986i $$-0.471469\pi$$
0.0895143 + 0.995986i $$0.471469\pi$$
$$920$$ 0 0
$$921$$ −28.1452 −0.927416
$$922$$ 41.7560 1.37516
$$923$$ 9.35887 0.308051
$$924$$ −20.8988 −0.687520
$$925$$ 0 0
$$926$$ 15.9469 0.524049
$$927$$ −7.83531 −0.257345
$$928$$ 10.7323 0.352305
$$929$$ −21.6025 −0.708756 −0.354378 0.935102i $$-0.615307\pi$$
−0.354378 + 0.935102i $$0.615307\pi$$
$$930$$ 0 0
$$931$$ 6.09205 0.199659
$$932$$ −132.678 −4.34603
$$933$$ 19.1390 0.626583
$$934$$ 69.6113 2.27775
$$935$$ 0 0
$$936$$ −17.1597 −0.560882
$$937$$ −31.6840 −1.03507 −0.517536 0.855661i $$-0.673151\pi$$
−0.517536 + 0.855661i $$0.673151\pi$$
$$938$$ −35.0796 −1.14539
$$939$$ 19.1399 0.624608
$$940$$ 0 0
$$941$$ −5.63987 −0.183854 −0.0919272 0.995766i $$-0.529303\pi$$
−0.0919272 + 0.995766i $$0.529303\pi$$
$$942$$ 77.0843 2.51154
$$943$$ −29.2687 −0.953120
$$944$$ 128.518 4.18290
$$945$$ 0 0
$$946$$ 14.0625 0.457212
$$947$$ 51.7369 1.68122 0.840611 0.541639i $$-0.182196\pi$$
0.840611 + 0.541639i $$0.182196\pi$$
$$948$$ −78.9709 −2.56486
$$949$$ 2.03686 0.0661194
$$950$$ 0 0
$$951$$ −24.9682 −0.809650
$$952$$ −87.4850 −2.83540
$$953$$ 16.2553 0.526561 0.263281 0.964719i $$-0.415196\pi$$
0.263281 + 0.964719i $$0.415196\pi$$
$$954$$ −0.422088 −0.0136656
$$955$$ 0 0
$$956$$ 6.77750 0.219200
$$957$$ 1.22631 0.0396410
$$958$$ −42.3601 −1.36859
$$959$$ −57.0225 −1.84135
$$960$$ 0 0
$$961$$ −28.2801 −0.912261
$$962$$ 25.9384 0.836289
$$963$$ 11.4792 0.369913
$$964$$ −100.304 −3.23057
$$965$$ 0 0
$$966$$ 84.3413 2.71364
$$967$$ 54.5004 1.75261 0.876307 0.481754i $$-0.160000\pi$$
0.876307 + 0.481754i $$0.160000\pi$$
$$968$$ 7.39397 0.237651
$$969$$ −3.91093 −0.125637
$$970$$ 0 0
$$971$$ −34.2679 −1.09971 −0.549855 0.835260i $$-0.685317\pi$$
−0.549855 + 0.835260i $$0.685317\pi$$
$$972$$ −68.6247 −2.20114
$$973$$ 9.14783 0.293266
$$974$$ 12.6078 0.403981
$$975$$ 0 0
$$976$$ −62.7834 −2.00965
$$977$$ −55.5644 −1.77766 −0.888831 0.458234i $$-0.848482\pi$$
−0.888831 + 0.458234i $$0.848482\pi$$
$$978$$ −11.6315 −0.371934
$$979$$ −12.1357 −0.387858
$$980$$ 0 0
$$981$$ 2.26780 0.0724052
$$982$$ −22.3066 −0.711832
$$983$$ 41.3971 1.32036 0.660181 0.751106i $$-0.270479\pi$$
0.660181 + 0.751106i $$0.270479\pi$$
$$984$$ 34.7049 1.10635
$$985$$ 0 0
$$986$$ 8.76221 0.279046
$$987$$ 16.0807 0.511853
$$988$$ 7.14054 0.227171
$$989$$ −40.1321 −1.27613
$$990$$ 0 0
$$991$$ 60.8219 1.93207 0.966036 0.258406i $$-0.0831973\pi$$
0.966036 + 0.258406i $$0.0831973\pi$$
$$992$$ 17.2623 0.548077
$$993$$ −18.0758 −0.573618
$$994$$ −59.8515 −1.89837
$$995$$ 0 0
$$996$$ 31.4242 0.995715
$$997$$ −26.0627 −0.825414 −0.412707 0.910864i $$-0.635417\pi$$
−0.412707 + 0.910864i $$0.635417\pi$$
$$998$$ −35.1031 −1.11117
$$999$$ 36.6876 1.16074
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.1.6 7
5.4 even 2 209.2.a.d.1.2 7
15.14 odd 2 1881.2.a.p.1.6 7
20.19 odd 2 3344.2.a.ba.1.4 7
55.54 odd 2 2299.2.a.q.1.6 7
95.94 odd 2 3971.2.a.i.1.6 7

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 7 5.4 even 2
1881.2.a.p.1.6 7 15.14 odd 2
2299.2.a.q.1.6 7 55.54 odd 2
3344.2.a.ba.1.4 7 20.19 odd 2
3971.2.a.i.1.6 7 95.94 odd 2
5225.2.a.n.1.6 7 1.1 even 1 trivial