# Properties

 Label 4400.2.a.cb.1.2 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.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: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 9x^{2} + 3x + 2$$ x^4 - x^3 - 9*x^2 + 3*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.655762$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} +O(q^{10})$$ $$q-0.655762 q^{3} +0.415806 q^{7} -2.56998 q^{9} -1.00000 q^{11} +4.00000 q^{13} -6.51558 q^{17} -5.20406 q^{19} -0.272670 q^{21} -8.54830 q^{23} +3.65258 q^{27} +0.895717 q^{29} +6.73836 q^{31} +0.655762 q^{33} +8.96410 q^{37} -2.62305 q^{39} +10.0998 q^{41} +4.78825 q^{43} -5.61986 q^{47} -6.82711 q^{49} +4.27267 q^{51} +10.0357 q^{53} +3.41262 q^{57} +1.63408 q^{59} +7.10428 q^{61} -1.06861 q^{63} -10.6914 q^{67} +5.60565 q^{69} +6.19302 q^{71} +3.16839 q^{73} -0.415806 q^{77} +11.2682 q^{79} +5.31471 q^{81} -16.2429 q^{83} -0.587377 q^{87} +9.56998 q^{89} +1.66323 q^{91} -4.41876 q^{93} +0.591657 q^{97} +2.56998 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} - q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q - q^3 - q^7 + 7 * q^9 $$4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49} + 23 q^{51} + 5 q^{53} - 15 q^{57} - 6 q^{59} + 29 q^{61} - 40 q^{63} - 6 q^{67} - 30 q^{69} + q^{71} + 18 q^{73} + q^{77} + 20 q^{79} + 44 q^{81} - 26 q^{83} - 31 q^{87} + 21 q^{89} - 4 q^{91} + 25 q^{93} - 4 q^{97} - 7 q^{99}+O(q^{100})$$ 4 * q - q^3 - q^7 + 7 * q^9 - 4 * q^11 + 16 * q^13 + 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 + 15 * q^31 + q^33 + 5 * q^37 - 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 + 23 * q^51 + 5 * q^53 - 15 * q^57 - 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 + q^71 + 18 * q^73 + q^77 + 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 - 4 * q^91 + 25 * q^93 - 4 * q^97 - 7 * 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 0
$$3$$ −0.655762 −0.378604 −0.189302 0.981919i $$-0.560623\pi$$
−0.189302 + 0.981919i $$0.560623\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.415806 0.157160 0.0785800 0.996908i $$-0.474961\pi$$
0.0785800 + 0.996908i $$0.474961\pi$$
$$8$$ 0 0
$$9$$ −2.56998 −0.856659
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.51558 −1.58026 −0.790130 0.612939i $$-0.789987\pi$$
−0.790130 + 0.612939i $$0.789987\pi$$
$$18$$ 0 0
$$19$$ −5.20406 −1.19389 −0.596946 0.802281i $$-0.703619\pi$$
−0.596946 + 0.802281i $$0.703619\pi$$
$$20$$ 0 0
$$21$$ −0.272670 −0.0595015
$$22$$ 0 0
$$23$$ −8.54830 −1.78244 −0.891221 0.453568i $$-0.850151\pi$$
−0.891221 + 0.453568i $$0.850151\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.65258 0.702939
$$28$$ 0 0
$$29$$ 0.895717 0.166331 0.0831653 0.996536i $$-0.473497\pi$$
0.0831653 + 0.996536i $$0.473497\pi$$
$$30$$ 0 0
$$31$$ 6.73836 1.21025 0.605123 0.796132i $$-0.293124\pi$$
0.605123 + 0.796132i $$0.293124\pi$$
$$32$$ 0 0
$$33$$ 0.655762 0.114153
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.96410 1.47369 0.736845 0.676062i $$-0.236315\pi$$
0.736845 + 0.676062i $$0.236315\pi$$
$$38$$ 0 0
$$39$$ −2.62305 −0.420024
$$40$$ 0 0
$$41$$ 10.0998 1.57732 0.788660 0.614830i $$-0.210775\pi$$
0.788660 + 0.614830i $$0.210775\pi$$
$$42$$ 0 0
$$43$$ 4.78825 0.730201 0.365101 0.930968i $$-0.381035\pi$$
0.365101 + 0.930968i $$0.381035\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −5.61986 −0.819741 −0.409871 0.912144i $$-0.634426\pi$$
−0.409871 + 0.912144i $$0.634426\pi$$
$$48$$ 0 0
$$49$$ −6.82711 −0.975301
$$50$$ 0 0
$$51$$ 4.27267 0.598293
$$52$$ 0 0
$$53$$ 10.0357 1.37851 0.689253 0.724521i $$-0.257939\pi$$
0.689253 + 0.724521i $$0.257939\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.41262 0.452013
$$58$$ 0 0
$$59$$ 1.63408 0.212739 0.106370 0.994327i $$-0.466077\pi$$
0.106370 + 0.994327i $$0.466077\pi$$
$$60$$ 0 0
$$61$$ 7.10428 0.909610 0.454805 0.890591i $$-0.349709\pi$$
0.454805 + 0.890591i $$0.349709\pi$$
$$62$$ 0 0
$$63$$ −1.06861 −0.134633
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −10.6914 −1.30617 −0.653083 0.757286i $$-0.726525\pi$$
−0.653083 + 0.757286i $$0.726525\pi$$
$$68$$ 0 0
$$69$$ 5.60565 0.674841
$$70$$ 0 0
$$71$$ 6.19302 0.734977 0.367488 0.930028i $$-0.380218\pi$$
0.367488 + 0.930028i $$0.380218\pi$$
$$72$$ 0 0
$$73$$ 3.16839 0.370832 0.185416 0.982660i $$-0.440637\pi$$
0.185416 + 0.982660i $$0.440637\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.415806 −0.0473855
$$78$$ 0 0
$$79$$ 11.2682 1.26777 0.633884 0.773428i $$-0.281460\pi$$
0.633884 + 0.773428i $$0.281460\pi$$
$$80$$ 0 0
$$81$$ 5.31471 0.590523
$$82$$ 0 0
$$83$$ −16.2429 −1.78289 −0.891446 0.453128i $$-0.850308\pi$$
−0.891446 + 0.453128i $$0.850308\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.587377 −0.0629735
$$88$$ 0 0
$$89$$ 9.56998 1.01442 0.507208 0.861824i $$-0.330678\pi$$
0.507208 + 0.861824i $$0.330678\pi$$
$$90$$ 0 0
$$91$$ 1.66323 0.174353
$$92$$ 0 0
$$93$$ −4.41876 −0.458204
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.591657 0.0600737 0.0300368 0.999549i $$-0.490438\pi$$
0.0300368 + 0.999549i $$0.490438\pi$$
$$98$$ 0 0
$$99$$ 2.56998 0.258292
$$100$$ 0 0
$$101$$ −4.62305 −0.460010 −0.230005 0.973189i $$-0.573874\pi$$
−0.230005 + 0.973189i $$0.573874\pi$$
$$102$$ 0 0
$$103$$ 8.24291 0.812198 0.406099 0.913829i $$-0.366889\pi$$
0.406099 + 0.913829i $$0.366889\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −17.6199 −1.70338 −0.851688 0.524049i $$-0.824421\pi$$
−0.851688 + 0.524049i $$0.824421\pi$$
$$108$$ 0 0
$$109$$ 3.66323 0.350873 0.175437 0.984491i $$-0.443866\pi$$
0.175437 + 0.984491i $$0.443866\pi$$
$$110$$ 0 0
$$111$$ −5.87832 −0.557945
$$112$$ 0 0
$$113$$ −11.4797 −1.07992 −0.539959 0.841691i $$-0.681560\pi$$
−0.539959 + 0.841691i $$0.681560\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −10.2799 −0.950378
$$118$$ 0 0
$$119$$ −2.70922 −0.248354
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −6.62305 −0.597180
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 9.13995 0.811040 0.405520 0.914086i $$-0.367091\pi$$
0.405520 + 0.914086i $$0.367091\pi$$
$$128$$ 0 0
$$129$$ −3.13995 −0.276457
$$130$$ 0 0
$$131$$ 2.48123 0.216787 0.108393 0.994108i $$-0.465429\pi$$
0.108393 + 0.994108i $$0.465429\pi$$
$$132$$ 0 0
$$133$$ −2.16388 −0.187632
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 7.40834 0.632937 0.316469 0.948603i $$-0.397503\pi$$
0.316469 + 0.948603i $$0.397503\pi$$
$$138$$ 0 0
$$139$$ 1.69166 0.143485 0.0717424 0.997423i $$-0.477144\pi$$
0.0717424 + 0.997423i $$0.477144\pi$$
$$140$$ 0 0
$$141$$ 3.68529 0.310358
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.47696 0.369253
$$148$$ 0 0
$$149$$ 3.74940 0.307163 0.153581 0.988136i $$-0.450919\pi$$
0.153581 + 0.988136i $$0.450919\pi$$
$$150$$ 0 0
$$151$$ 15.1400 1.23207 0.616036 0.787718i $$-0.288738\pi$$
0.616036 + 0.787718i $$0.288738\pi$$
$$152$$ 0 0
$$153$$ 16.7449 1.35374
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.5871 0.924755 0.462378 0.886683i $$-0.346996\pi$$
0.462378 + 0.886683i $$0.346996\pi$$
$$158$$ 0 0
$$159$$ −6.58101 −0.521908
$$160$$ 0 0
$$161$$ −3.55444 −0.280129
$$162$$ 0 0
$$163$$ −6.82392 −0.534491 −0.267245 0.963629i $$-0.586113\pi$$
−0.267245 + 0.963629i $$0.586113\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.10428 −0.549746 −0.274873 0.961481i $$-0.588636\pi$$
−0.274873 + 0.961481i $$0.588636\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 13.3743 1.02276
$$172$$ 0 0
$$173$$ 9.31152 0.707942 0.353971 0.935256i $$-0.384831\pi$$
0.353971 + 0.935256i $$0.384831\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.07157 −0.0805440
$$178$$ 0 0
$$179$$ 26.6368 1.99093 0.995464 0.0951359i $$-0.0303286\pi$$
0.995464 + 0.0951359i $$0.0303286\pi$$
$$180$$ 0 0
$$181$$ 1.01103 0.0751495 0.0375748 0.999294i $$-0.488037\pi$$
0.0375748 + 0.999294i $$0.488037\pi$$
$$182$$ 0 0
$$183$$ −4.65872 −0.344382
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.51558 0.476466
$$188$$ 0 0
$$189$$ 1.51877 0.110474
$$190$$ 0 0
$$191$$ −10.5655 −0.764490 −0.382245 0.924061i $$-0.624849\pi$$
−0.382245 + 0.924061i $$0.624849\pi$$
$$192$$ 0 0
$$193$$ 11.8271 0.851334 0.425667 0.904880i $$-0.360040\pi$$
0.425667 + 0.904880i $$0.360040\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.31152 −0.0934422 −0.0467211 0.998908i $$-0.514877\pi$$
−0.0467211 + 0.998908i $$0.514877\pi$$
$$198$$ 0 0
$$199$$ 11.8271 0.838401 0.419201 0.907894i $$-0.362310\pi$$
0.419201 + 0.907894i $$0.362310\pi$$
$$200$$ 0 0
$$201$$ 7.01103 0.494520
$$202$$ 0 0
$$203$$ 0.372445 0.0261405
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 21.9689 1.52695
$$208$$ 0 0
$$209$$ 5.20406 0.359972
$$210$$ 0 0
$$211$$ 8.37244 0.576383 0.288191 0.957573i $$-0.406946\pi$$
0.288191 + 0.957573i $$0.406946\pi$$
$$212$$ 0 0
$$213$$ −4.06115 −0.278265
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.80185 0.190202
$$218$$ 0 0
$$219$$ −2.07771 −0.140398
$$220$$ 0 0
$$221$$ −26.0623 −1.75314
$$222$$ 0 0
$$223$$ 22.1461 1.48301 0.741506 0.670946i $$-0.234112\pi$$
0.741506 + 0.670946i $$0.234112\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 21.8194 1.44821 0.724103 0.689692i $$-0.242254\pi$$
0.724103 + 0.689692i $$0.242254\pi$$
$$228$$ 0 0
$$229$$ 2.80247 0.185192 0.0925962 0.995704i $$-0.470483\pi$$
0.0925962 + 0.995704i $$0.470483\pi$$
$$230$$ 0 0
$$231$$ 0.272670 0.0179404
$$232$$ 0 0
$$233$$ 8.24424 0.540098 0.270049 0.962847i $$-0.412960\pi$$
0.270049 + 0.962847i $$0.412960\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −7.38923 −0.479982
$$238$$ 0 0
$$239$$ 11.2682 0.728877 0.364438 0.931227i $$-0.381261\pi$$
0.364438 + 0.931227i $$0.381261\pi$$
$$240$$ 0 0
$$241$$ 22.3860 1.44201 0.721006 0.692929i $$-0.243680\pi$$
0.721006 + 0.692929i $$0.243680\pi$$
$$242$$ 0 0
$$243$$ −14.4429 −0.926514
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.8162 −1.32451
$$248$$ 0 0
$$249$$ 10.6515 0.675010
$$250$$ 0 0
$$251$$ −3.84265 −0.242546 −0.121273 0.992619i $$-0.538698\pi$$
−0.121273 + 0.992619i $$0.538698\pi$$
$$252$$ 0 0
$$253$$ 8.54830 0.537427
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 10.6230 0.662648 0.331324 0.943517i $$-0.392505\pi$$
0.331324 + 0.943517i $$0.392505\pi$$
$$258$$ 0 0
$$259$$ 3.72733 0.231605
$$260$$ 0 0
$$261$$ −2.30197 −0.142489
$$262$$ 0 0
$$263$$ −5.79276 −0.357197 −0.178598 0.983922i $$-0.557156\pi$$
−0.178598 + 0.983922i $$0.557156\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.27563 −0.384062
$$268$$ 0 0
$$269$$ −8.20857 −0.500485 −0.250243 0.968183i $$-0.580510\pi$$
−0.250243 + 0.968183i $$0.580510\pi$$
$$270$$ 0 0
$$271$$ −27.3111 −1.65903 −0.829515 0.558485i $$-0.811383\pi$$
−0.829515 + 0.558485i $$0.811383\pi$$
$$272$$ 0 0
$$273$$ −1.09068 −0.0660109
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.4735 −0.869631 −0.434815 0.900520i $$-0.643186\pi$$
−0.434815 + 0.900520i $$0.643186\pi$$
$$278$$ 0 0
$$279$$ −17.3174 −1.03677
$$280$$ 0 0
$$281$$ 16.2799 0.971178 0.485589 0.874187i $$-0.338605\pi$$
0.485589 + 0.874187i $$0.338605\pi$$
$$282$$ 0 0
$$283$$ −10.5169 −0.625165 −0.312583 0.949891i $$-0.601194\pi$$
−0.312583 + 0.949891i $$0.601194\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.19955 0.247892
$$288$$ 0 0
$$289$$ 25.4528 1.49722
$$290$$ 0 0
$$291$$ −0.387986 −0.0227441
$$292$$ 0 0
$$293$$ −9.37695 −0.547807 −0.273904 0.961757i $$-0.588315\pi$$
−0.273904 + 0.961757i $$0.588315\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.65258 −0.211944
$$298$$ 0 0
$$299$$ −34.1932 −1.97744
$$300$$ 0 0
$$301$$ 1.99099 0.114758
$$302$$ 0 0
$$303$$ 3.03162 0.174162
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.09978 −0.119840 −0.0599202 0.998203i $$-0.519085\pi$$
−0.0599202 + 0.998203i $$0.519085\pi$$
$$308$$ 0 0
$$309$$ −5.40539 −0.307502
$$310$$ 0 0
$$311$$ 20.3724 1.15522 0.577608 0.816314i $$-0.303986\pi$$
0.577608 + 0.816314i $$0.303986\pi$$
$$312$$ 0 0
$$313$$ −23.8968 −1.35073 −0.675364 0.737485i $$-0.736013\pi$$
−0.675364 + 0.737485i $$0.736013\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.86114 −0.329195 −0.164597 0.986361i $$-0.552632\pi$$
−0.164597 + 0.986361i $$0.552632\pi$$
$$318$$ 0 0
$$319$$ −0.895717 −0.0501506
$$320$$ 0 0
$$321$$ 11.5544 0.644906
$$322$$ 0 0
$$323$$ 33.9075 1.88666
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.40220 −0.132842
$$328$$ 0 0
$$329$$ −2.33677 −0.128831
$$330$$ 0 0
$$331$$ 0.574484 0.0315765 0.0157882 0.999875i $$-0.494974\pi$$
0.0157882 + 0.999875i $$0.494974\pi$$
$$332$$ 0 0
$$333$$ −23.0375 −1.26245
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4.10747 −0.223748 −0.111874 0.993722i $$-0.535685\pi$$
−0.111874 + 0.993722i $$0.535685\pi$$
$$338$$ 0 0
$$339$$ 7.52794 0.408862
$$340$$ 0 0
$$341$$ −6.73836 −0.364903
$$342$$ 0 0
$$343$$ −5.74940 −0.310438
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8.30834 0.446015 0.223008 0.974817i $$-0.428413\pi$$
0.223008 + 0.974817i $$0.428413\pi$$
$$348$$ 0 0
$$349$$ −18.4858 −0.989523 −0.494762 0.869029i $$-0.664745\pi$$
−0.494762 + 0.869029i $$0.664745\pi$$
$$350$$ 0 0
$$351$$ 14.6103 0.779841
$$352$$ 0 0
$$353$$ 26.5199 1.41151 0.705755 0.708456i $$-0.250608\pi$$
0.705755 + 0.708456i $$0.250608\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.77660 0.0940278
$$358$$ 0 0
$$359$$ 18.8226 0.993419 0.496709 0.867917i $$-0.334541\pi$$
0.496709 + 0.867917i $$0.334541\pi$$
$$360$$ 0 0
$$361$$ 8.08222 0.425380
$$362$$ 0 0
$$363$$ −0.655762 −0.0344186
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 10.2115 0.533037 0.266519 0.963830i $$-0.414127\pi$$
0.266519 + 0.963830i $$0.414127\pi$$
$$368$$ 0 0
$$369$$ −25.9562 −1.35122
$$370$$ 0 0
$$371$$ 4.17289 0.216646
$$372$$ 0 0
$$373$$ −32.3363 −1.67431 −0.837156 0.546965i $$-0.815783\pi$$
−0.837156 + 0.546965i $$0.815783\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.58287 0.184527
$$378$$ 0 0
$$379$$ −15.3686 −0.789434 −0.394717 0.918803i $$-0.629157\pi$$
−0.394717 + 0.918803i $$0.629157\pi$$
$$380$$ 0 0
$$381$$ −5.99363 −0.307063
$$382$$ 0 0
$$383$$ 13.6662 0.698309 0.349155 0.937065i $$-0.386469\pi$$
0.349155 + 0.937065i $$0.386469\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −12.3057 −0.625534
$$388$$ 0 0
$$389$$ −9.01103 −0.456878 −0.228439 0.973558i $$-0.573362\pi$$
−0.228439 + 0.973558i $$0.573362\pi$$
$$390$$ 0 0
$$391$$ 55.6971 2.81672
$$392$$ 0 0
$$393$$ −1.62710 −0.0820763
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.45466 −0.173384 −0.0866922 0.996235i $$-0.527630\pi$$
−0.0866922 + 0.996235i $$0.527630\pi$$
$$398$$ 0 0
$$399$$ 1.41899 0.0710384
$$400$$ 0 0
$$401$$ −1.62756 −0.0812762 −0.0406381 0.999174i $$-0.512939\pi$$
−0.0406381 + 0.999174i $$0.512939\pi$$
$$402$$ 0 0
$$403$$ 26.9535 1.34265
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.96410 −0.444334
$$408$$ 0 0
$$409$$ 6.54534 0.323646 0.161823 0.986820i $$-0.448263\pi$$
0.161823 + 0.986820i $$0.448263\pi$$
$$410$$ 0 0
$$411$$ −4.85811 −0.239633
$$412$$ 0 0
$$413$$ 0.679461 0.0334341
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1.10933 −0.0543239
$$418$$ 0 0
$$419$$ 34.4795 1.68443 0.842216 0.539141i $$-0.181251\pi$$
0.842216 + 0.539141i $$0.181251\pi$$
$$420$$ 0 0
$$421$$ 6.54534 0.319000 0.159500 0.987198i $$-0.449012\pi$$
0.159500 + 0.987198i $$0.449012\pi$$
$$422$$ 0 0
$$423$$ 14.4429 0.702239
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.95401 0.142954
$$428$$ 0 0
$$429$$ 2.62305 0.126642
$$430$$ 0 0
$$431$$ −36.1711 −1.74230 −0.871151 0.491016i $$-0.836626\pi$$
−0.871151 + 0.491016i $$0.836626\pi$$
$$432$$ 0 0
$$433$$ −38.1027 −1.83110 −0.915550 0.402204i $$-0.868244\pi$$
−0.915550 + 0.402204i $$0.868244\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 44.4858 2.12805
$$438$$ 0 0
$$439$$ −12.1282 −0.578848 −0.289424 0.957201i $$-0.593464\pi$$
−0.289424 + 0.957201i $$0.593464\pi$$
$$440$$ 0 0
$$441$$ 17.5455 0.835500
$$442$$ 0 0
$$443$$ 16.5483 0.786233 0.393117 0.919489i $$-0.371397\pi$$
0.393117 + 0.919489i $$0.371397\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −2.45871 −0.116293
$$448$$ 0 0
$$449$$ 15.6056 0.736476 0.368238 0.929732i $$-0.379961\pi$$
0.368238 + 0.929732i $$0.379961\pi$$
$$450$$ 0 0
$$451$$ −10.0998 −0.475580
$$452$$ 0 0
$$453$$ −9.92820 −0.466468
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.87761 −0.274943 −0.137471 0.990506i $$-0.543898\pi$$
−0.137471 + 0.990506i $$0.543898\pi$$
$$458$$ 0 0
$$459$$ −23.7987 −1.11083
$$460$$ 0 0
$$461$$ 36.7495 1.71159 0.855797 0.517312i $$-0.173067\pi$$
0.855797 + 0.517312i $$0.173067\pi$$
$$462$$ 0 0
$$463$$ 19.3081 0.897324 0.448662 0.893702i $$-0.351901\pi$$
0.448662 + 0.893702i $$0.351901\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 11.4129 0.528127 0.264064 0.964505i $$-0.414937\pi$$
0.264064 + 0.964505i $$0.414937\pi$$
$$468$$ 0 0
$$469$$ −4.44556 −0.205277
$$470$$ 0 0
$$471$$ −7.59841 −0.350116
$$472$$ 0 0
$$473$$ −4.78825 −0.220164
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −25.7914 −1.18091
$$478$$ 0 0
$$479$$ −29.2307 −1.33559 −0.667793 0.744347i $$-0.732761\pi$$
−0.667793 + 0.744347i $$0.732761\pi$$
$$480$$ 0 0
$$481$$ 35.8564 1.63491
$$482$$ 0 0
$$483$$ 2.33086 0.106058
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 33.2932 1.50866 0.754329 0.656496i $$-0.227962\pi$$
0.754329 + 0.656496i $$0.227962\pi$$
$$488$$ 0 0
$$489$$ 4.47487 0.202361
$$490$$ 0 0
$$491$$ 24.5151 1.10635 0.553176 0.833064i $$-0.313416\pi$$
0.553176 + 0.833064i $$0.313416\pi$$
$$492$$ 0 0
$$493$$ −5.83612 −0.262846
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.57510 0.115509
$$498$$ 0 0
$$499$$ 18.9093 0.846497 0.423249 0.906014i $$-0.360890\pi$$
0.423249 + 0.906014i $$0.360890\pi$$
$$500$$ 0 0
$$501$$ 4.65872 0.208136
$$502$$ 0 0
$$503$$ −9.42031 −0.420031 −0.210016 0.977698i $$-0.567351\pi$$
−0.210016 + 0.977698i $$0.567351\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.96729 −0.0873702
$$508$$ 0 0
$$509$$ −16.7804 −0.743778 −0.371889 0.928277i $$-0.621290\pi$$
−0.371889 + 0.928277i $$0.621290\pi$$
$$510$$ 0 0
$$511$$ 1.31744 0.0582799
$$512$$ 0 0
$$513$$ −19.0082 −0.839234
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5.61986 0.247161
$$518$$ 0 0
$$519$$ −6.10614 −0.268030
$$520$$ 0 0
$$521$$ −37.8273 −1.65724 −0.828621 0.559810i $$-0.810874\pi$$
−0.828621 + 0.559810i $$0.810874\pi$$
$$522$$ 0 0
$$523$$ 10.0998 0.441632 0.220816 0.975315i $$-0.429128\pi$$
0.220816 + 0.975315i $$0.429128\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −43.9044 −1.91250
$$528$$ 0 0
$$529$$ 50.0734 2.17710
$$530$$ 0 0
$$531$$ −4.19955 −0.182245
$$532$$ 0 0
$$533$$ 40.3991 1.74988
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −17.4674 −0.753774
$$538$$ 0 0
$$539$$ 6.82711 0.294064
$$540$$ 0 0
$$541$$ −29.9269 −1.28666 −0.643329 0.765590i $$-0.722447\pi$$
−0.643329 + 0.765590i $$0.722447\pi$$
$$542$$ 0 0
$$543$$ −0.662997 −0.0284519
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 23.7630 1.01603 0.508016 0.861347i $$-0.330379\pi$$
0.508016 + 0.861347i $$0.330379\pi$$
$$548$$ 0 0
$$549$$ −18.2578 −0.779226
$$550$$ 0 0
$$551$$ −4.66137 −0.198581
$$552$$ 0 0
$$553$$ 4.68537 0.199242
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24.3363 1.03116 0.515581 0.856841i $$-0.327576\pi$$
0.515581 + 0.856841i $$0.327576\pi$$
$$558$$ 0 0
$$559$$ 19.1530 0.810086
$$560$$ 0 0
$$561$$ −4.27267 −0.180392
$$562$$ 0 0
$$563$$ −8.85368 −0.373138 −0.186569 0.982442i $$-0.559737\pi$$
−0.186569 + 0.982442i $$0.559737\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.20989 0.0928066
$$568$$ 0 0
$$569$$ 26.3860 1.10616 0.553080 0.833128i $$-0.313452\pi$$
0.553080 + 0.833128i $$0.313452\pi$$
$$570$$ 0 0
$$571$$ −6.45280 −0.270041 −0.135021 0.990843i $$-0.543110\pi$$
−0.135021 + 0.990843i $$0.543110\pi$$
$$572$$ 0 0
$$573$$ 6.92843 0.289439
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −12.1745 −0.506832 −0.253416 0.967357i $$-0.581554\pi$$
−0.253416 + 0.967357i $$0.581554\pi$$
$$578$$ 0 0
$$579$$ −7.75576 −0.322319
$$580$$ 0 0
$$581$$ −6.75390 −0.280199
$$582$$ 0 0
$$583$$ −10.0357 −0.415635
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −22.9521 −0.947336 −0.473668 0.880704i $$-0.657070\pi$$
−0.473668 + 0.880704i $$0.657070\pi$$
$$588$$ 0 0
$$589$$ −35.0668 −1.44490
$$590$$ 0 0
$$591$$ 0.860047 0.0353776
$$592$$ 0 0
$$593$$ −10.4081 −0.427410 −0.213705 0.976898i $$-0.568553\pi$$
−0.213705 + 0.976898i $$0.568553\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −7.75576 −0.317422
$$598$$ 0 0
$$599$$ 8.98913 0.367286 0.183643 0.982993i $$-0.441211\pi$$
0.183643 + 0.982993i $$0.441211\pi$$
$$600$$ 0 0
$$601$$ −3.57650 −0.145889 −0.0729443 0.997336i $$-0.523240\pi$$
−0.0729443 + 0.997336i $$0.523240\pi$$
$$602$$ 0 0
$$603$$ 27.4767 1.11894
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 27.0235 1.09685 0.548424 0.836200i $$-0.315228\pi$$
0.548424 + 0.836200i $$0.315228\pi$$
$$608$$ 0 0
$$609$$ −0.244235 −0.00989691
$$610$$ 0 0
$$611$$ −22.4795 −0.909421
$$612$$ 0 0
$$613$$ 44.5572 1.79965 0.899823 0.436254i $$-0.143695\pi$$
0.899823 + 0.436254i $$0.143695\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 15.2966 0.615818 0.307909 0.951416i $$-0.400371\pi$$
0.307909 + 0.951416i $$0.400371\pi$$
$$618$$ 0 0
$$619$$ −14.5655 −0.585436 −0.292718 0.956199i $$-0.594560\pi$$
−0.292718 + 0.956199i $$0.594560\pi$$
$$620$$ 0 0
$$621$$ −31.2233 −1.25295
$$622$$ 0 0
$$623$$ 3.97926 0.159426
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −3.41262 −0.136287
$$628$$ 0 0
$$629$$ −58.4063 −2.32881
$$630$$ 0 0
$$631$$ 22.0779 0.878906 0.439453 0.898266i $$-0.355172\pi$$
0.439453 + 0.898266i $$0.355172\pi$$
$$632$$ 0 0
$$633$$ −5.49033 −0.218221
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −27.3084 −1.08200
$$638$$ 0 0
$$639$$ −15.9159 −0.629624
$$640$$ 0 0
$$641$$ −31.2152 −1.23293 −0.616463 0.787384i $$-0.711435\pi$$
−0.616463 + 0.787384i $$0.711435\pi$$
$$642$$ 0 0
$$643$$ −10.4498 −0.412102 −0.206051 0.978541i $$-0.566061\pi$$
−0.206051 + 0.978541i $$0.566061\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26.1307 1.02730 0.513652 0.857999i $$-0.328292\pi$$
0.513652 + 0.857999i $$0.328292\pi$$
$$648$$ 0 0
$$649$$ −1.63408 −0.0641433
$$650$$ 0 0
$$651$$ −1.83735 −0.0720114
$$652$$ 0 0
$$653$$ −44.2097 −1.73006 −0.865030 0.501719i $$-0.832701\pi$$
−0.865030 + 0.501719i $$0.832701\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −8.14268 −0.317676
$$658$$ 0 0
$$659$$ −6.96706 −0.271398 −0.135699 0.990750i $$-0.543328\pi$$
−0.135699 + 0.990750i $$0.543328\pi$$
$$660$$ 0 0
$$661$$ −9.19753 −0.357743 −0.178871 0.983872i $$-0.557245\pi$$
−0.178871 + 0.983872i $$0.557245\pi$$
$$662$$ 0 0
$$663$$ 17.0907 0.663747
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −7.65686 −0.296475
$$668$$ 0 0
$$669$$ −14.5226 −0.561475
$$670$$ 0 0
$$671$$ −7.10428 −0.274258
$$672$$ 0 0
$$673$$ 8.72415 0.336291 0.168146 0.985762i $$-0.446222\pi$$
0.168146 + 0.985762i $$0.446222\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −8.76618 −0.336912 −0.168456 0.985709i $$-0.553878\pi$$
−0.168456 + 0.985709i $$0.553878\pi$$
$$678$$ 0 0
$$679$$ 0.246015 0.00944118
$$680$$ 0 0
$$681$$ −14.3083 −0.548297
$$682$$ 0 0
$$683$$ −49.2320 −1.88381 −0.941906 0.335877i $$-0.890967\pi$$
−0.941906 + 0.335877i $$0.890967\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −1.83775 −0.0701146
$$688$$ 0 0
$$689$$ 40.1427 1.52931
$$690$$ 0 0
$$691$$ 18.1483 0.690395 0.345198 0.938530i $$-0.387812\pi$$
0.345198 + 0.938530i $$0.387812\pi$$
$$692$$ 0 0
$$693$$ 1.06861 0.0405932
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −65.8059 −2.49258
$$698$$ 0 0
$$699$$ −5.40626 −0.204483
$$700$$ 0 0
$$701$$ 22.6587 0.855808 0.427904 0.903824i $$-0.359252\pi$$
0.427904 + 0.903824i $$0.359252\pi$$
$$702$$ 0 0
$$703$$ −46.6497 −1.75943
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1.92229 −0.0722952
$$708$$ 0 0
$$709$$ −36.7366 −1.37967 −0.689836 0.723966i $$-0.742317\pi$$
−0.689836 + 0.723966i $$0.742317\pi$$
$$710$$ 0 0
$$711$$ −28.9589 −1.08604
$$712$$ 0 0
$$713$$ −57.6015 −2.15719
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −7.38923 −0.275956
$$718$$ 0 0
$$719$$ −8.74738 −0.326222 −0.163111 0.986608i $$-0.552153\pi$$
−0.163111 + 0.986608i $$0.552153\pi$$
$$720$$ 0 0
$$721$$ 3.42745 0.127645
$$722$$ 0 0
$$723$$ −14.6799 −0.545952
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2.20887 −0.0819226 −0.0409613 0.999161i $$-0.513042\pi$$
−0.0409613 + 0.999161i $$0.513042\pi$$
$$728$$ 0 0
$$729$$ −6.47301 −0.239741
$$730$$ 0 0
$$731$$ −31.1982 −1.15391
$$732$$ 0 0
$$733$$ −3.58552 −0.132434 −0.0662171 0.997805i $$-0.521093\pi$$
−0.0662171 + 0.997805i $$0.521093\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.6914 0.393824
$$738$$ 0 0
$$739$$ 24.9314 0.917116 0.458558 0.888665i $$-0.348366\pi$$
0.458558 + 0.888665i $$0.348366\pi$$
$$740$$ 0 0
$$741$$ 13.6505 0.501463
$$742$$ 0 0
$$743$$ 21.4257 0.786032 0.393016 0.919532i $$-0.371432\pi$$
0.393016 + 0.919532i $$0.371432\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 41.7439 1.52733
$$748$$ 0 0
$$749$$ −7.32645 −0.267703
$$750$$ 0 0
$$751$$ −28.1582 −1.02751 −0.513754 0.857938i $$-0.671746\pi$$
−0.513754 + 0.857938i $$0.671746\pi$$
$$752$$ 0 0
$$753$$ 2.51986 0.0918288
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.7513 0.390761 0.195381 0.980728i $$-0.437406\pi$$
0.195381 + 0.980728i $$0.437406\pi$$
$$758$$ 0 0
$$759$$ −5.60565 −0.203472
$$760$$ 0 0
$$761$$ −38.0429 −1.37905 −0.689527 0.724260i $$-0.742182\pi$$
−0.689527 + 0.724260i $$0.742182\pi$$
$$762$$ 0 0
$$763$$ 1.52319 0.0551433
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.53632 0.236013
$$768$$ 0 0
$$769$$ 26.2280 0.945805 0.472903 0.881115i $$-0.343206\pi$$
0.472903 + 0.881115i $$0.343206\pi$$
$$770$$ 0 0
$$771$$ −6.96619 −0.250881
$$772$$ 0 0
$$773$$ 8.94499 0.321729 0.160864 0.986977i $$-0.448572\pi$$
0.160864 + 0.986977i $$0.448572\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −2.44424 −0.0876867
$$778$$ 0 0
$$779$$ −52.5598 −1.88315
$$780$$ 0 0
$$781$$ −6.19302 −0.221604
$$782$$ 0 0
$$783$$ 3.27168 0.116920
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −29.9716 −1.06837 −0.534185 0.845367i $$-0.679382\pi$$
−0.534185 + 0.845367i $$0.679382\pi$$
$$788$$ 0 0
$$789$$ 3.79867 0.135236
$$790$$ 0 0
$$791$$ −4.77332 −0.169720
$$792$$ 0 0
$$793$$ 28.4171 1.00912
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.80022 0.240876 0.120438 0.992721i $$-0.461570\pi$$
0.120438 + 0.992721i $$0.461570\pi$$
$$798$$ 0 0
$$799$$ 36.6167 1.29541
$$800$$ 0 0
$$801$$ −24.5946 −0.869008
$$802$$ 0 0
$$803$$ −3.16839 −0.111810
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 5.38286 0.189486
$$808$$ 0 0
$$809$$ 53.1246 1.86776 0.933880 0.357586i $$-0.116400\pi$$
0.933880 + 0.357586i $$0.116400\pi$$
$$810$$ 0 0
$$811$$ 30.0487 1.05515 0.527577 0.849507i $$-0.323101\pi$$
0.527577 + 0.849507i $$0.323101\pi$$
$$812$$ 0 0
$$813$$ 17.9096 0.628116
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.9183 −0.871782
$$818$$ 0 0
$$819$$ −4.27445 −0.149361
$$820$$ 0 0
$$821$$ 19.9069 0.694756 0.347378 0.937725i $$-0.387072\pi$$
0.347378 + 0.937725i $$0.387072\pi$$
$$822$$ 0 0
$$823$$ 33.2491 1.15899 0.579495 0.814976i $$-0.303250\pi$$
0.579495 + 0.814976i $$0.303250\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.3770 −0.986766 −0.493383 0.869812i $$-0.664240\pi$$
−0.493383 + 0.869812i $$0.664240\pi$$
$$828$$ 0 0
$$829$$ 3.63143 0.126125 0.0630625 0.998010i $$-0.479913\pi$$
0.0630625 + 0.998010i $$0.479913\pi$$
$$830$$ 0 0
$$831$$ 9.49120 0.329246
$$832$$ 0 0
$$833$$ 44.4826 1.54123
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 24.6124 0.850729
$$838$$ 0 0
$$839$$ 3.45660 0.119335 0.0596675 0.998218i $$-0.480996\pi$$
0.0596675 + 0.998218i $$0.480996\pi$$
$$840$$ 0 0
$$841$$ −28.1977 −0.972334
$$842$$ 0 0
$$843$$ −10.6757 −0.367692
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.415806 0.0142873
$$848$$ 0 0
$$849$$ 6.89659 0.236690
$$850$$ 0 0
$$851$$ −76.6278 −2.62677
$$852$$ 0 0
$$853$$ −47.4452 −1.62449 −0.812246 0.583315i $$-0.801755\pi$$
−0.812246 + 0.583315i $$0.801755\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 35.8271 1.22383 0.611915 0.790923i $$-0.290399\pi$$
0.611915 + 0.790923i $$0.290399\pi$$
$$858$$ 0 0
$$859$$ 39.0539 1.33250 0.666252 0.745727i $$-0.267898\pi$$
0.666252 + 0.745727i $$0.267898\pi$$
$$860$$ 0 0
$$861$$ −2.75390 −0.0938528
$$862$$ 0 0
$$863$$ 5.75072 0.195757 0.0978784 0.995198i $$-0.468794\pi$$
0.0978784 + 0.995198i $$0.468794\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −16.6910 −0.566855
$$868$$ 0 0
$$869$$ −11.2682 −0.382246
$$870$$ 0 0
$$871$$ −42.7657 −1.44906
$$872$$ 0 0
$$873$$ −1.52054 −0.0514626
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −8.32776 −0.281208 −0.140604 0.990066i $$-0.544905\pi$$
−0.140604 + 0.990066i $$0.544905\pi$$
$$878$$ 0 0
$$879$$ 6.14905 0.207402
$$880$$ 0 0
$$881$$ −49.0954 −1.65407 −0.827033 0.562153i $$-0.809973\pi$$
−0.827033 + 0.562153i $$0.809973\pi$$
$$882$$ 0 0
$$883$$ −10.7526 −0.361853 −0.180927 0.983497i $$-0.557910\pi$$
−0.180927 + 0.983497i $$0.557910\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −53.0591 −1.78155 −0.890776 0.454443i $$-0.849838\pi$$
−0.890776 + 0.454443i $$0.849838\pi$$
$$888$$ 0 0
$$889$$ 3.80045 0.127463
$$890$$ 0 0
$$891$$ −5.31471 −0.178049
$$892$$ 0 0
$$893$$ 29.2461 0.978683
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 22.4226 0.748668
$$898$$ 0 0
$$899$$ 6.03567 0.201301
$$900$$ 0 0
$$901$$ −65.3882 −2.17840
$$902$$ 0 0
$$903$$ −1.30561 −0.0434481
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 7.03885 0.233721 0.116861 0.993148i $$-0.462717\pi$$
0.116861 + 0.993148i $$0.462717\pi$$
$$908$$ 0 0
$$909$$ 11.8811 0.394072
$$910$$ 0 0
$$911$$ −32.0980 −1.06345 −0.531727 0.846916i $$-0.678457\pi$$
−0.531727 + 0.846916i $$0.678457\pi$$
$$912$$ 0 0
$$913$$ 16.2429 0.537562
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.03171 0.0340702
$$918$$ 0 0
$$919$$ 1.05960 0.0349529 0.0174764 0.999847i $$-0.494437\pi$$
0.0174764 + 0.999847i $$0.494437\pi$$
$$920$$ 0 0
$$921$$ 1.37695 0.0453721
$$922$$ 0 0
$$923$$ 24.7721 0.815383
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −21.1841 −0.695777
$$928$$ 0 0
$$929$$ −12.9981 −0.426455 −0.213228 0.977003i $$-0.568398\pi$$
−0.213228 + 0.977003i $$0.568398\pi$$
$$930$$ 0 0
$$931$$ 35.5286 1.16440
$$932$$ 0 0
$$933$$ −13.3595 −0.437370
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39.9846 1.30624 0.653120 0.757254i $$-0.273460\pi$$
0.653120 + 0.757254i $$0.273460\pi$$
$$938$$ 0 0
$$939$$ 15.6706 0.511391
$$940$$ 0 0
$$941$$ 34.0267 1.10924 0.554619 0.832105i $$-0.312864\pi$$
0.554619 + 0.832105i $$0.312864\pi$$
$$942$$ 0 0
$$943$$ −86.3359 −2.81148
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −27.1506 −0.882276 −0.441138 0.897439i $$-0.645425\pi$$
−0.441138 + 0.897439i $$0.645425\pi$$
$$948$$ 0 0
$$949$$ 12.6735 0.411401
$$950$$ 0 0
$$951$$ 3.84351 0.124634
$$952$$ 0 0
$$953$$ −43.1169 −1.39669 −0.698346 0.715760i $$-0.746080\pi$$
−0.698346 + 0.715760i $$0.746080\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0.587377 0.0189872
$$958$$ 0 0
$$959$$ 3.08044 0.0994725
$$960$$ 0 0
$$961$$ 14.4055 0.464695
$$962$$ 0 0
$$963$$ 45.2826 1.45921
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.0641 0.387955 0.193978 0.981006i $$-0.437861\pi$$
0.193978 + 0.981006i $$0.437861\pi$$
$$968$$ 0 0
$$969$$ −22.2352 −0.714298
$$970$$ 0 0
$$971$$ 3.81156 0.122319 0.0611595 0.998128i $$-0.480520\pi$$
0.0611595 + 0.998128i $$0.480520\pi$$
$$972$$ 0 0
$$973$$ 0.703403 0.0225501
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39.4202 1.26116 0.630581 0.776123i $$-0.282816\pi$$
0.630581 + 0.776123i $$0.282816\pi$$
$$978$$ 0 0
$$979$$ −9.56998 −0.305858
$$980$$ 0 0
$$981$$ −9.41440 −0.300579
$$982$$ 0 0
$$983$$ 8.67651 0.276738 0.138369 0.990381i $$-0.455814\pi$$
0.138369 + 0.990381i $$0.455814\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 1.53237 0.0487758
$$988$$ 0 0
$$989$$ −40.9314 −1.30154
$$990$$ 0 0
$$991$$ −35.7256 −1.13486 −0.567430 0.823422i $$-0.692062\pi$$
−0.567430 + 0.823422i $$0.692062\pi$$
$$992$$ 0 0
$$993$$ −0.376725 −0.0119550
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26.3453 −0.834365 −0.417183 0.908823i $$-0.636982\pi$$
−0.417183 + 0.908823i $$0.636982\pi$$
$$998$$ 0 0
$$999$$ 32.7421 1.03591
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 4400.2.a.cb.1.2 4
4.3 odd 2 2200.2.a.y.1.3 4
5.2 odd 4 880.2.b.j.529.6 8
5.3 odd 4 880.2.b.j.529.3 8
5.4 even 2 4400.2.a.ce.1.3 4
20.3 even 4 440.2.b.d.89.6 yes 8
20.7 even 4 440.2.b.d.89.3 8
20.19 odd 2 2200.2.a.x.1.2 4
60.23 odd 4 3960.2.d.f.3169.7 8
60.47 odd 4 3960.2.d.f.3169.8 8

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.3 8 20.7 even 4
440.2.b.d.89.6 yes 8 20.3 even 4
880.2.b.j.529.3 8 5.3 odd 4
880.2.b.j.529.6 8 5.2 odd 4
2200.2.a.x.1.2 4 20.19 odd 2
2200.2.a.y.1.3 4 4.3 odd 2
3960.2.d.f.3169.7 8 60.23 odd 4
3960.2.d.f.3169.8 8 60.47 odd 4
4400.2.a.cb.1.2 4 1.1 even 1 trivial
4400.2.a.ce.1.3 4 5.4 even 2