# Properties

 Label 1296.3.e.d.161.4 Level $1296$ Weight $3$ Character 1296.161 Analytic conductor $35.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: no (minimal twist has level 162) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.4 Root $$1.93185i$$ of defining polynomial Character $$\chi$$ $$=$$ 1296.161 Dual form 1296.3.e.d.161.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.79555i q^{5} +8.39230 q^{7} +O(q^{10})$$ $$q+5.79555i q^{5} +8.39230 q^{7} -14.6969i q^{11} -21.1962 q^{13} +7.76457i q^{17} -24.3923 q^{19} +14.6969i q^{23} -8.58846 q^{25} +35.4940i q^{29} -8.00000 q^{31} +48.6381i q^{35} -60.5692 q^{37} -33.6365i q^{41} -9.17691 q^{43} +16.9706i q^{47} +21.4308 q^{49} +25.7605i q^{53} +85.1769 q^{55} -61.6706i q^{59} -13.0000 q^{61} -122.843i q^{65} -21.1769 q^{67} +101.214i q^{71} +40.4115 q^{73} -123.341i q^{77} -98.7461 q^{79} -103.488i q^{83} -45.0000 q^{85} +134.130i q^{89} -177.885 q^{91} -141.367i q^{95} -75.1384 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^7 $$4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^7 - 64 * q^13 - 56 * q^19 + 28 * q^25 - 32 * q^31 - 76 * q^37 + 88 * q^43 + 252 * q^49 + 216 * q^55 - 52 * q^61 + 40 * q^67 + 224 * q^73 - 104 * q^79 - 180 * q^85 - 88 * q^91 + 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 0
$$4$$ 0 0
$$5$$ 5.79555i 1.15911i 0.814933 + 0.579555i $$0.196774\pi$$
−0.814933 + 0.579555i $$0.803226\pi$$
$$6$$ 0 0
$$7$$ 8.39230 1.19890 0.599450 0.800412i $$-0.295386\pi$$
0.599450 + 0.800412i $$0.295386\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 14.6969i − 1.33609i −0.744123 0.668043i $$-0.767132\pi$$
0.744123 0.668043i $$-0.232868\pi$$
$$12$$ 0 0
$$13$$ −21.1962 −1.63047 −0.815237 0.579128i $$-0.803393\pi$$
−0.815237 + 0.579128i $$0.803393\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.76457i 0.456739i 0.973574 + 0.228370i $$0.0733395\pi$$
−0.973574 + 0.228370i $$0.926660\pi$$
$$18$$ 0 0
$$19$$ −24.3923 −1.28381 −0.641903 0.766786i $$-0.721855\pi$$
−0.641903 + 0.766786i $$0.721855\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 14.6969i 0.638997i 0.947587 + 0.319499i $$0.103514\pi$$
−0.947587 + 0.319499i $$0.896486\pi$$
$$24$$ 0 0
$$25$$ −8.58846 −0.343538
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 35.4940i 1.22393i 0.790884 + 0.611966i $$0.209621\pi$$
−0.790884 + 0.611966i $$0.790379\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −0.258065 −0.129032 0.991640i $$-0.541187\pi$$
−0.129032 + 0.991640i $$0.541187\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 48.6381i 1.38966i
$$36$$ 0 0
$$37$$ −60.5692 −1.63701 −0.818503 0.574502i $$-0.805196\pi$$
−0.818503 + 0.574502i $$0.805196\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 33.6365i − 0.820403i −0.911995 0.410201i $$-0.865458\pi$$
0.911995 0.410201i $$-0.134542\pi$$
$$42$$ 0 0
$$43$$ −9.17691 −0.213417 −0.106708 0.994290i $$-0.534031\pi$$
−0.106708 + 0.994290i $$0.534031\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 16.9706i 0.361076i 0.983568 + 0.180538i $$0.0577838\pi$$
−0.983568 + 0.180538i $$0.942216\pi$$
$$48$$ 0 0
$$49$$ 21.4308 0.437363
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 25.7605i 0.486046i 0.970020 + 0.243023i $$0.0781391\pi$$
−0.970020 + 0.243023i $$0.921861\pi$$
$$54$$ 0 0
$$55$$ 85.1769 1.54867
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 61.6706i − 1.04526i −0.852558 0.522632i $$-0.824950\pi$$
0.852558 0.522632i $$-0.175050\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −0.213115 −0.106557 0.994307i $$-0.533983\pi$$
−0.106557 + 0.994307i $$0.533983\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 122.843i − 1.88990i
$$66$$ 0 0
$$67$$ −21.1769 −0.316073 −0.158037 0.987433i $$-0.550516\pi$$
−0.158037 + 0.987433i $$0.550516\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 101.214i 1.42555i 0.701392 + 0.712776i $$0.252562\pi$$
−0.701392 + 0.712776i $$0.747438\pi$$
$$72$$ 0 0
$$73$$ 40.4115 0.553583 0.276791 0.960930i $$-0.410729\pi$$
0.276791 + 0.960930i $$0.410729\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 123.341i − 1.60183i
$$78$$ 0 0
$$79$$ −98.7461 −1.24995 −0.624976 0.780644i $$-0.714891\pi$$
−0.624976 + 0.780644i $$0.714891\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 103.488i − 1.24684i −0.781887 0.623420i $$-0.785743\pi$$
0.781887 0.623420i $$-0.214257\pi$$
$$84$$ 0 0
$$85$$ −45.0000 −0.529412
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 134.130i 1.50708i 0.657403 + 0.753539i $$0.271655\pi$$
−0.657403 + 0.753539i $$0.728345\pi$$
$$90$$ 0 0
$$91$$ −177.885 −1.95478
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 141.367i − 1.48807i
$$96$$ 0 0
$$97$$ −75.1384 −0.774623 −0.387312 0.921949i $$-0.626596\pi$$
−0.387312 + 0.921949i $$0.626596\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 29.0893i − 0.288013i −0.989577 0.144006i $$-0.954001\pi$$
0.989577 0.144006i $$-0.0459986\pi$$
$$102$$ 0 0
$$103$$ −96.7077 −0.938909 −0.469455 0.882957i $$-0.655549\pi$$
−0.469455 + 0.882957i $$0.655549\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 177.582i − 1.65964i −0.558030 0.829821i $$-0.688442\pi$$
0.558030 0.829821i $$-0.311558\pi$$
$$108$$ 0 0
$$109$$ 61.9423 0.568278 0.284139 0.958783i $$-0.408292\pi$$
0.284139 + 0.958783i $$0.408292\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 109.811i 0.971778i 0.874020 + 0.485889i $$0.161504\pi$$
−0.874020 + 0.485889i $$0.838496\pi$$
$$114$$ 0 0
$$115$$ −85.1769 −0.740669
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 65.1626i 0.547585i
$$120$$ 0 0
$$121$$ −95.0000 −0.785124
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 95.1140i 0.760912i
$$126$$ 0 0
$$127$$ −141.177 −1.11163 −0.555815 0.831306i $$-0.687594\pi$$
−0.555815 + 0.831306i $$0.687594\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 53.1853i 0.405995i 0.979179 + 0.202997i $$0.0650683\pi$$
−0.979179 + 0.202997i $$0.934932\pi$$
$$132$$ 0 0
$$133$$ −204.708 −1.53916
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.63453i 0.0119309i 0.999982 + 0.00596545i $$0.00189887\pi$$
−0.999982 + 0.00596545i $$0.998101\pi$$
$$138$$ 0 0
$$139$$ 19.2154 0.138240 0.0691201 0.997608i $$-0.477981\pi$$
0.0691201 + 0.997608i $$0.477981\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 311.519i 2.17845i
$$144$$ 0 0
$$145$$ −205.708 −1.41867
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 94.2818i 0.632764i 0.948632 + 0.316382i $$0.102468\pi$$
−0.948632 + 0.316382i $$0.897532\pi$$
$$150$$ 0 0
$$151$$ −32.0000 −0.211921 −0.105960 0.994370i $$-0.533792\pi$$
−0.105960 + 0.994370i $$0.533792\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 46.3644i − 0.299125i
$$156$$ 0 0
$$157$$ 0.292342 0.00186205 0.000931025 1.00000i $$-0.499704\pi$$
0.000931025 1.00000i $$0.499704\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 123.341i 0.766094i
$$162$$ 0 0
$$163$$ −28.7846 −0.176593 −0.0882963 0.996094i $$-0.528142\pi$$
−0.0882963 + 0.996094i $$0.528142\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 56.0682i − 0.335737i −0.985809 0.167869i $$-0.946312\pi$$
0.985809 0.167869i $$-0.0536885\pi$$
$$168$$ 0 0
$$169$$ 280.277 1.65844
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 220.952i − 1.27718i −0.769548 0.638589i $$-0.779518\pi$$
0.769548 0.638589i $$-0.220482\pi$$
$$174$$ 0 0
$$175$$ −72.0770 −0.411868
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 248.347i 1.38741i 0.720258 + 0.693706i $$0.244023\pi$$
−0.720258 + 0.693706i $$0.755977\pi$$
$$180$$ 0 0
$$181$$ −158.277 −0.874458 −0.437229 0.899350i $$-0.644040\pi$$
−0.437229 + 0.899350i $$0.644040\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 351.032i − 1.89747i
$$186$$ 0 0
$$187$$ 114.115 0.610243
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 34.5503i − 0.180892i −0.995901 0.0904459i $$-0.971171\pi$$
0.995901 0.0904459i $$-0.0288292\pi$$
$$192$$ 0 0
$$193$$ −55.0000 −0.284974 −0.142487 0.989797i $$-0.545510\pi$$
−0.142487 + 0.989797i $$0.545510\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 35.7170i 0.181305i 0.995883 + 0.0906524i $$0.0288952\pi$$
−0.995883 + 0.0906524i $$0.971105\pi$$
$$198$$ 0 0
$$199$$ 375.138 1.88512 0.942559 0.334040i $$-0.108412\pi$$
0.942559 + 0.334040i $$0.108412\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 297.877i 1.46737i
$$204$$ 0 0
$$205$$ 194.942 0.950938
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 358.492i 1.71527i
$$210$$ 0 0
$$211$$ −307.454 −1.45713 −0.728563 0.684978i $$-0.759812\pi$$
−0.728563 + 0.684978i $$0.759812\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 53.1853i − 0.247374i
$$216$$ 0 0
$$217$$ −67.1384 −0.309394
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 164.579i − 0.744702i
$$222$$ 0 0
$$223$$ 338.592 1.51835 0.759175 0.650886i $$-0.225602\pi$$
0.759175 + 0.650886i $$0.225602\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 123.950i − 0.546037i −0.962009 0.273019i $$-0.911978\pi$$
0.962009 0.273019i $$-0.0880220\pi$$
$$228$$ 0 0
$$229$$ 74.0577 0.323396 0.161698 0.986840i $$-0.448303\pi$$
0.161698 + 0.986840i $$0.448303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 273.223i − 1.17263i −0.810082 0.586316i $$-0.800578\pi$$
0.810082 0.586316i $$-0.199422\pi$$
$$234$$ 0 0
$$235$$ −98.3538 −0.418527
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 452.885i − 1.89492i −0.319877 0.947459i $$-0.603641\pi$$
0.319877 0.947459i $$-0.396359\pi$$
$$240$$ 0 0
$$241$$ −382.688 −1.58792 −0.793959 0.607971i $$-0.791984\pi$$
−0.793959 + 0.607971i $$0.791984\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 124.203i 0.506952i
$$246$$ 0 0
$$247$$ 517.023 2.09321
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 73.9307i − 0.294544i −0.989096 0.147272i $$-0.952951\pi$$
0.989096 0.147272i $$-0.0470493\pi$$
$$252$$ 0 0
$$253$$ 216.000 0.853755
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 159.809i 0.621824i 0.950439 + 0.310912i $$0.100634\pi$$
−0.950439 + 0.310912i $$0.899366\pi$$
$$258$$ 0 0
$$259$$ −508.315 −1.96261
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 259.106i 0.985193i 0.870258 + 0.492596i $$0.163952\pi$$
−0.870258 + 0.492596i $$0.836048\pi$$
$$264$$ 0 0
$$265$$ −149.296 −0.563382
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 24.8168i − 0.0922556i −0.998936 0.0461278i $$-0.985312\pi$$
0.998936 0.0461278i $$-0.0146881\pi$$
$$270$$ 0 0
$$271$$ −98.1154 −0.362050 −0.181025 0.983479i $$-0.557941\pi$$
−0.181025 + 0.983479i $$0.557941\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 126.224i 0.458996i
$$276$$ 0 0
$$277$$ −1.41532 −0.00510945 −0.00255472 0.999997i $$-0.500813\pi$$
−0.00255472 + 0.999997i $$0.500813\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 100.716i 0.358421i 0.983811 + 0.179211i $$0.0573544\pi$$
−0.983811 + 0.179211i $$0.942646\pi$$
$$282$$ 0 0
$$283$$ 47.2923 0.167111 0.0835554 0.996503i $$-0.473372\pi$$
0.0835554 + 0.996503i $$0.473372\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 282.288i − 0.983582i
$$288$$ 0 0
$$289$$ 228.711 0.791389
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 333.616i 1.13862i 0.822123 + 0.569310i $$0.192790\pi$$
−0.822123 + 0.569310i $$0.807210\pi$$
$$294$$ 0 0
$$295$$ 357.415 1.21158
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 311.519i − 1.04187i
$$300$$ 0 0
$$301$$ −77.0155 −0.255865
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 75.3422i − 0.247024i
$$306$$ 0 0
$$307$$ −10.3538 −0.0337258 −0.0168629 0.999858i $$-0.505368\pi$$
−0.0168629 + 0.999858i $$0.505368\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 9.54047i − 0.0306768i −0.999882 0.0153384i $$-0.995117\pi$$
0.999882 0.0153384i $$-0.00488255\pi$$
$$312$$ 0 0
$$313$$ −479.277 −1.53124 −0.765618 0.643295i $$-0.777567\pi$$
−0.765618 + 0.643295i $$0.777567\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 179.662i − 0.566758i −0.959008 0.283379i $$-0.908545\pi$$
0.959008 0.283379i $$-0.0914554\pi$$
$$318$$ 0 0
$$319$$ 521.654 1.63528
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 189.396i − 0.586365i
$$324$$ 0 0
$$325$$ 182.042 0.560130
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 142.422i 0.432894i
$$330$$ 0 0
$$331$$ −295.454 −0.892610 −0.446305 0.894881i $$-0.647260\pi$$
−0.446305 + 0.894881i $$0.647260\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 122.732i − 0.366364i
$$336$$ 0 0
$$337$$ 489.261 1.45181 0.725907 0.687793i $$-0.241420\pi$$
0.725907 + 0.687793i $$0.241420\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 117.576i 0.344796i
$$342$$ 0 0
$$343$$ −231.369 −0.674546
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 666.682i 1.92127i 0.277807 + 0.960637i $$0.410392\pi$$
−0.277807 + 0.960637i $$0.589608\pi$$
$$348$$ 0 0
$$349$$ 511.969 1.46696 0.733480 0.679711i $$-0.237895\pi$$
0.733480 + 0.679711i $$0.237895\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 586.681i − 1.66199i −0.556283 0.830993i $$-0.687773\pi$$
0.556283 0.830993i $$-0.312227\pi$$
$$354$$ 0 0
$$355$$ −586.592 −1.65237
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 534.573i − 1.48906i −0.667589 0.744530i $$-0.732674\pi$$
0.667589 0.744530i $$-0.267326\pi$$
$$360$$ 0 0
$$361$$ 233.985 0.648157
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 234.207i 0.641664i
$$366$$ 0 0
$$367$$ −15.2923 −0.0416685 −0.0208343 0.999783i $$-0.506632\pi$$
−0.0208343 + 0.999783i $$0.506632\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 216.190i 0.582721i
$$372$$ 0 0
$$373$$ 652.985 1.75063 0.875314 0.483554i $$-0.160654\pi$$
0.875314 + 0.483554i $$0.160654\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 752.337i − 1.99559i
$$378$$ 0 0
$$379$$ 655.215 1.72880 0.864400 0.502804i $$-0.167698\pi$$
0.864400 + 0.502804i $$0.167698\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 299.541i − 0.782092i −0.920371 0.391046i $$-0.872113\pi$$
0.920371 0.391046i $$-0.127887\pi$$
$$384$$ 0 0
$$385$$ 714.831 1.85670
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 464.558i 1.19424i 0.802153 + 0.597119i $$0.203688\pi$$
−0.802153 + 0.597119i $$0.796312\pi$$
$$390$$ 0 0
$$391$$ −114.115 −0.291855
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 572.289i − 1.44883i
$$396$$ 0 0
$$397$$ −185.708 −0.467777 −0.233889 0.972263i $$-0.575145\pi$$
−0.233889 + 0.972263i $$0.575145\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 62.8592i 0.156756i 0.996924 + 0.0783780i $$0.0249741\pi$$
−0.996924 + 0.0783780i $$0.975026\pi$$
$$402$$ 0 0
$$403$$ 169.569 0.420767
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 890.182i 2.18718i
$$408$$ 0 0
$$409$$ −327.281 −0.800197 −0.400099 0.916472i $$-0.631024\pi$$
−0.400099 + 0.916472i $$0.631024\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 517.558i − 1.25317i
$$414$$ 0 0
$$415$$ 599.769 1.44523
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 649.875i 1.55101i 0.631339 + 0.775507i $$0.282506\pi$$
−0.631339 + 0.775507i $$0.717494\pi$$
$$420$$ 0 0
$$421$$ 3.31913 0.00788391 0.00394196 0.999992i $$-0.498745\pi$$
0.00394196 + 0.999992i $$0.498745\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 66.6857i − 0.156908i
$$426$$ 0 0
$$427$$ −109.100 −0.255503
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 803.502i 1.86427i 0.362107 + 0.932136i $$0.382057\pi$$
−0.362107 + 0.932136i $$0.617943\pi$$
$$432$$ 0 0
$$433$$ −93.1230 −0.215065 −0.107532 0.994202i $$-0.534295\pi$$
−0.107532 + 0.994202i $$0.534295\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 358.492i − 0.820348i
$$438$$ 0 0
$$439$$ 472.000 1.07517 0.537585 0.843209i $$-0.319337\pi$$
0.537585 + 0.843209i $$0.319337\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 569.689i 1.28598i 0.765875 + 0.642989i $$0.222306\pi$$
−0.765875 + 0.642989i $$0.777694\pi$$
$$444$$ 0 0
$$445$$ −777.358 −1.74687
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 559.115i 1.24524i 0.782523 + 0.622622i $$0.213933\pi$$
−0.782523 + 0.622622i $$0.786067\pi$$
$$450$$ 0 0
$$451$$ −494.354 −1.09613
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 1030.94i − 2.26580i
$$456$$ 0 0
$$457$$ 604.296 1.32231 0.661155 0.750249i $$-0.270066\pi$$
0.661155 + 0.750249i $$0.270066\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5.13459i 0.0111379i 0.999984 + 0.00556897i $$0.00177267\pi$$
−0.999984 + 0.00556897i $$0.998227\pi$$
$$462$$ 0 0
$$463$$ 161.492 0.348795 0.174398 0.984675i $$-0.444202\pi$$
0.174398 + 0.984675i $$0.444202\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 503.025i − 1.07714i −0.842581 0.538570i $$-0.818965\pi$$
0.842581 0.538570i $$-0.181035\pi$$
$$468$$ 0 0
$$469$$ −177.723 −0.378941
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 134.873i 0.285143i
$$474$$ 0 0
$$475$$ 209.492 0.441036
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 213.796i 0.446339i 0.974780 + 0.223170i $$0.0716404\pi$$
−0.974780 + 0.223170i $$0.928360\pi$$
$$480$$ 0 0
$$481$$ 1283.83 2.66909
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 435.469i − 0.897874i
$$486$$ 0 0
$$487$$ −8.63071 −0.0177222 −0.00886110 0.999961i $$-0.502821\pi$$
−0.00886110 + 0.999961i $$0.502821\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 922.459i 1.87873i 0.342912 + 0.939367i $$0.388587\pi$$
−0.342912 + 0.939367i $$0.611413\pi$$
$$492$$ 0 0
$$493$$ −275.596 −0.559018
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 849.420i 1.70909i
$$498$$ 0 0
$$499$$ 434.592 0.870926 0.435463 0.900207i $$-0.356585\pi$$
0.435463 + 0.900207i $$0.356585\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 144.087i 0.286454i 0.989690 + 0.143227i $$0.0457480\pi$$
−0.989690 + 0.143227i $$0.954252\pi$$
$$504$$ 0 0
$$505$$ 168.588 0.333839
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 697.272i 1.36989i 0.728596 + 0.684943i $$0.240173\pi$$
−0.728596 + 0.684943i $$0.759827\pi$$
$$510$$ 0 0
$$511$$ 339.146 0.663691
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 560.475i − 1.08830i
$$516$$ 0 0
$$517$$ 249.415 0.482428
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 426.962i 0.819504i 0.912197 + 0.409752i $$0.134385\pi$$
−0.912197 + 0.409752i $$0.865615\pi$$
$$522$$ 0 0
$$523$$ −179.762 −0.343712 −0.171856 0.985122i $$-0.554976\pi$$
−0.171856 + 0.985122i $$0.554976\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 62.1166i − 0.117868i
$$528$$ 0 0
$$529$$ 313.000 0.591682
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 712.965i 1.33764i
$$534$$ 0 0
$$535$$ 1029.18 1.92371
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 314.967i − 0.584354i
$$540$$ 0 0
$$541$$ 708.734 1.31005 0.655023 0.755609i $$-0.272659\pi$$
0.655023 + 0.755609i $$0.272659\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 358.990i 0.658697i
$$546$$ 0 0
$$547$$ 196.231 0.358740 0.179370 0.983782i $$-0.442594\pi$$
0.179370 + 0.983782i $$0.442594\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 865.781i − 1.57129i
$$552$$ 0 0
$$553$$ −828.708 −1.49857
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 353.610i − 0.634848i −0.948284 0.317424i $$-0.897182\pi$$
0.948284 0.317424i $$-0.102818\pi$$
$$558$$ 0 0
$$559$$ 194.515 0.347970
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 373.026i 0.662568i 0.943531 + 0.331284i $$0.107482\pi$$
−0.943531 + 0.331284i $$0.892518\pi$$
$$564$$ 0 0
$$565$$ −636.415 −1.12640
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 60.1177i − 0.105655i −0.998604 0.0528275i $$-0.983177\pi$$
0.998604 0.0528275i $$-0.0168233\pi$$
$$570$$ 0 0
$$571$$ −86.1999 −0.150963 −0.0754815 0.997147i $$-0.524049\pi$$
−0.0754815 + 0.997147i $$0.524049\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 126.224i − 0.219520i
$$576$$ 0 0
$$577$$ 709.123 1.22898 0.614491 0.788924i $$-0.289361\pi$$
0.614491 + 0.788924i $$0.289361\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 868.501i − 1.49484i
$$582$$ 0 0
$$583$$ 378.600 0.649399
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 962.285i − 1.63933i −0.572845 0.819664i $$-0.694160\pi$$
0.572845 0.819664i $$-0.305840\pi$$
$$588$$ 0 0
$$589$$ 195.138 0.331305
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 104.350i 0.175969i 0.996122 + 0.0879847i $$0.0280427\pi$$
−0.996122 + 0.0879847i $$0.971957\pi$$
$$594$$ 0 0
$$595$$ −377.654 −0.634712
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 286.063i 0.477567i 0.971073 + 0.238784i $$0.0767486\pi$$
−0.971073 + 0.238784i $$0.923251\pi$$
$$600$$ 0 0
$$601$$ −280.415 −0.466581 −0.233291 0.972407i $$-0.574949\pi$$
−0.233291 + 0.972407i $$0.574949\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 550.578i − 0.910046i
$$606$$ 0 0
$$607$$ −737.731 −1.21537 −0.607686 0.794177i $$-0.707902\pi$$
−0.607686 + 0.794177i $$0.707902\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 359.711i − 0.588724i
$$612$$ 0 0
$$613$$ −679.415 −1.10834 −0.554172 0.832402i $$-0.686965\pi$$
−0.554172 + 0.832402i $$0.686965\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 472.649i 0.766044i 0.923739 + 0.383022i $$0.125117\pi$$
−0.923739 + 0.383022i $$0.874883\pi$$
$$618$$ 0 0
$$619$$ −886.354 −1.43191 −0.715956 0.698145i $$-0.754009\pi$$
−0.715956 + 0.698145i $$0.754009\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1125.66i 1.80684i
$$624$$ 0 0
$$625$$ −765.950 −1.22552
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 470.294i − 0.747685i
$$630$$ 0 0
$$631$$ 729.108 1.15548 0.577740 0.816221i $$-0.303935\pi$$
0.577740 + 0.816221i $$0.303935\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 818.199i − 1.28850i
$$636$$ 0 0
$$637$$ −454.250 −0.713108
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 870.195i − 1.35756i −0.734342 0.678780i $$-0.762509\pi$$
0.734342 0.678780i $$-0.237491\pi$$
$$642$$ 0 0
$$643$$ 618.123 0.961311 0.480656 0.876910i $$-0.340399\pi$$
0.480656 + 0.876910i $$0.340399\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 425.439i 0.657556i 0.944407 + 0.328778i $$0.106637\pi$$
−0.944407 + 0.328778i $$0.893363\pi$$
$$648$$ 0 0
$$649$$ −906.369 −1.39656
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 188.808i − 0.289140i −0.989495 0.144570i $$-0.953820\pi$$
0.989495 0.144570i $$-0.0461799\pi$$
$$654$$ 0 0
$$655$$ −308.238 −0.470593
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 660.470i − 1.00223i −0.865380 0.501116i $$-0.832923\pi$$
0.865380 0.501116i $$-0.167077\pi$$
$$660$$ 0 0
$$661$$ −589.831 −0.892331 −0.446165 0.894951i $$-0.647211\pi$$
−0.446165 + 0.894951i $$0.647211\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 1186.39i − 1.78405i
$$666$$ 0 0
$$667$$ −521.654 −0.782090
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 191.060i 0.284739i
$$672$$ 0 0
$$673$$ −391.431 −0.581621 −0.290810 0.956781i $$-0.593925\pi$$
−0.290810 + 0.956781i $$0.593925\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 693.661i 1.02461i 0.858804 + 0.512305i $$0.171208\pi$$
−0.858804 + 0.512305i $$0.828792\pi$$
$$678$$ 0 0
$$679$$ −630.585 −0.928696
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 678.170i 0.992928i 0.868057 + 0.496464i $$0.165368\pi$$
−0.868057 + 0.496464i $$0.834632\pi$$
$$684$$ 0 0
$$685$$ −9.47303 −0.0138292
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 546.022i − 0.792485i
$$690$$ 0 0
$$691$$ −675.777 −0.977969 −0.488985 0.872292i $$-0.662633\pi$$
−0.488985 + 0.872292i $$0.662633\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 111.364i 0.160236i
$$696$$ 0 0
$$697$$ 261.173 0.374710
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 797.260i 1.13732i 0.822573 + 0.568659i $$0.192538\pi$$
−0.822573 + 0.568659i $$0.807462\pi$$
$$702$$ 0 0
$$703$$ 1477.42 2.10160
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 244.126i − 0.345298i
$$708$$ 0 0
$$709$$ 173.504 0.244716 0.122358 0.992486i $$-0.460954\pi$$
0.122358 + 0.992486i $$0.460954\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 117.576i − 0.164903i
$$714$$ 0 0
$$715$$ −1805.42 −2.52507
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 188.177i − 0.261721i −0.991401 0.130860i $$-0.958226\pi$$
0.991401 0.130860i $$-0.0417740\pi$$
$$720$$ 0 0
$$721$$ −811.600 −1.12566
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 304.839i − 0.420468i
$$726$$ 0 0
$$727$$ −711.777 −0.979060 −0.489530 0.871986i $$-0.662832\pi$$
−0.489530 + 0.871986i $$0.662832\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 71.2548i − 0.0974758i
$$732$$ 0 0
$$733$$ 1226.83 1.67371 0.836856 0.547423i $$-0.184391\pi$$
0.836856 + 0.547423i $$0.184391\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 311.236i 0.422301i
$$738$$ 0 0
$$739$$ −741.892 −1.00391 −0.501957 0.864893i $$-0.667386\pi$$
−0.501957 + 0.864893i $$0.667386\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 781.984i − 1.05247i −0.850340 0.526234i $$-0.823604\pi$$
0.850340 0.526234i $$-0.176396\pi$$
$$744$$ 0 0
$$745$$ −546.415 −0.733443
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 1490.32i − 1.98975i
$$750$$ 0 0
$$751$$ 1033.33 1.37594 0.687970 0.725739i $$-0.258502\pi$$
0.687970 + 0.725739i $$0.258502\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 185.458i − 0.245639i
$$756$$ 0 0
$$757$$ 473.877 0.625993 0.312997 0.949754i $$-0.398667\pi$$
0.312997 + 0.949754i $$0.398667\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 940.455i − 1.23581i −0.786251 0.617907i $$-0.787981\pi$$
0.786251 0.617907i $$-0.212019\pi$$
$$762$$ 0 0
$$763$$ 519.839 0.681309
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1307.18i 1.70428i
$$768$$ 0 0
$$769$$ 980.815 1.27544 0.637721 0.770267i $$-0.279877\pi$$
0.637721 + 0.770267i $$0.279877\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1042.20i 1.34825i 0.738618 + 0.674124i $$0.235479\pi$$
−0.738618 + 0.674124i $$0.764521\pi$$
$$774$$ 0 0
$$775$$ 68.7077 0.0886550
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 820.472i 1.05324i
$$780$$ 0 0
$$781$$ 1487.54 1.90466
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.69428i 0.00215832i
$$786$$ 0 0
$$787$$ 144.700 0.183863 0.0919315 0.995765i $$-0.470696\pi$$
0.0919315 + 0.995765i $$0.470696\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 921.567i 1.16507i
$$792$$ 0 0
$$793$$ 275.550 0.347478
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 155.729i 0.195394i 0.995216 + 0.0976972i $$0.0311477\pi$$
−0.995216 + 0.0976972i $$0.968852\pi$$
$$798$$ 0 0
$$799$$ −131.769 −0.164918
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 593.926i − 0.739634i
$$804$$ 0 0
$$805$$ −714.831 −0.887988
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1216.92i 1.50422i 0.659035 + 0.752112i $$0.270965\pi$$
−0.659035 + 0.752112i $$0.729035\pi$$
$$810$$ 0 0
$$811$$ −1243.31 −1.53305 −0.766527 0.642212i $$-0.778017\pi$$
−0.766527 + 0.642212i $$0.778017\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 166.823i − 0.204691i
$$816$$ 0 0
$$817$$ 223.846 0.273985
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 13.2854i − 0.0161820i −0.999967 0.00809098i $$-0.997425\pi$$
0.999967 0.00809098i $$-0.00257547\pi$$
$$822$$ 0 0
$$823$$ 214.200 0.260267 0.130134 0.991496i $$-0.458459\pi$$
0.130134 + 0.991496i $$0.458459\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1221.27i 1.47675i 0.674391 + 0.738374i $$0.264406\pi$$
−0.674391 + 0.738374i $$0.735594\pi$$
$$828$$ 0 0
$$829$$ −358.431 −0.432365 −0.216183 0.976353i $$-0.569361\pi$$
−0.216183 + 0.976353i $$0.569361\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 166.401i 0.199761i
$$834$$ 0 0
$$835$$ 324.946 0.389157
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 327.597i − 0.390461i −0.980757 0.195231i $$-0.937454\pi$$
0.980757 0.195231i $$-0.0625456\pi$$
$$840$$ 0 0
$$841$$ −418.827 −0.498011
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1624.36i 1.92232i
$$846$$ 0 0
$$847$$ −797.269 −0.941286
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 890.182i − 1.04604i
$$852$$ 0 0
$$853$$ −52.8306 −0.0619351 −0.0309675 0.999520i $$-0.509859\pi$$
−0.0309675 + 0.999520i $$0.509859\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1248.42i 1.45673i 0.685187 + 0.728367i $$0.259720\pi$$
−0.685187 + 0.728367i $$0.740280\pi$$
$$858$$ 0 0
$$859$$ 446.739 0.520068 0.260034 0.965599i $$-0.416266\pi$$
0.260034 + 0.965599i $$0.416266\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 635.297i 0.736150i 0.929796 + 0.368075i $$0.119983\pi$$
−0.929796 + 0.368075i $$0.880017\pi$$
$$864$$ 0 0
$$865$$ 1280.54 1.48039
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1451.27i 1.67004i
$$870$$ 0 0
$$871$$ 448.869 0.515349
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 798.226i 0.912258i
$$876$$ 0 0
$$877$$ −788.692 −0.899307 −0.449653 0.893203i $$-0.648453\pi$$
−0.449653 + 0.893203i $$0.648453\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 558.506i 0.633945i 0.948435 + 0.316972i $$0.102666\pi$$
−0.948435 + 0.316972i $$0.897334\pi$$
$$882$$ 0 0
$$883$$ −313.338 −0.354857 −0.177428 0.984134i $$-0.556778\pi$$
−0.177428 + 0.984134i $$0.556778\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 597.701i − 0.673845i −0.941532 0.336923i $$-0.890614\pi$$
0.941532 0.336923i $$-0.109386\pi$$
$$888$$ 0 0
$$889$$ −1184.80 −1.33273
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 413.951i − 0.463551i
$$894$$ 0 0
$$895$$ −1439.31 −1.60816
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 283.952i − 0.315854i
$$900$$ 0 0
$$901$$ −200.019 −0.221997
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 917.302i − 1.01359i
$$906$$ 0 0
$$907$$ −1086.51 −1.19791 −0.598957 0.800781i $$-0.704418\pi$$
−0.598957 + 0.800781i $$0.704418\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 710.283i − 0.779674i −0.920884 0.389837i $$-0.872531\pi$$
0.920884 0.389837i $$-0.127469\pi$$
$$912$$ 0 0
$$913$$ −1520.95 −1.66589
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 446.347i 0.486747i
$$918$$ 0 0
$$919$$ −1540.01 −1.67574 −0.837871 0.545868i $$-0.816200\pi$$
−0.837871 + 0.545868i $$0.816200\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 2145.35i − 2.32432i
$$924$$ 0 0
$$925$$ 520.196 0.562374
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 1582.59i − 1.70355i −0.523911 0.851773i $$-0.675527\pi$$
0.523911 0.851773i $$-0.324473\pi$$
$$930$$ 0 0
$$931$$ −522.746 −0.561489
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 661.362i 0.707339i
$$936$$ 0 0
$$937$$ −1747.40 −1.86489 −0.932444 0.361315i $$-0.882328\pi$$
−0.932444 + 0.361315i $$0.882328\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 367.579i − 0.390626i −0.980741 0.195313i $$-0.937428\pi$$
0.980741 0.195313i $$-0.0625722\pi$$
$$942$$ 0 0
$$943$$ 494.354 0.524235
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 190.168i 0.200811i 0.994947 + 0.100406i $$0.0320140\pi$$
−0.994947 + 0.100406i $$0.967986\pi$$
$$948$$ 0 0
$$949$$ −856.569 −0.902602
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 861.754i − 0.904254i −0.891954 0.452127i $$-0.850665\pi$$
0.891954 0.452127i $$-0.149335\pi$$
$$954$$ 0 0
$$955$$ 200.238 0.209674
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 13.7175i 0.0143040i
$$960$$ 0 0
$$961$$ −897.000 −0.933403
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 318.756i − 0.330317i
$$966$$ 0 0
$$967$$ −1275.31 −1.31884 −0.659418 0.751776i $$-0.729197\pi$$
−0.659418 + 0.751776i $$0.729197\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 1238.69i − 1.27568i −0.770168 0.637841i $$-0.779828\pi$$
0.770168 0.637841i $$-0.220172\pi$$
$$972$$ 0 0
$$973$$ 161.261 0.165736
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18.4936i 0.0189290i 0.999955 + 0.00946448i $$0.00301268\pi$$
−0.999955 + 0.00946448i $$0.996987\pi$$
$$978$$ 0 0
$$979$$ 1971.30 2.01359
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 658.643i − 0.670033i −0.942212 0.335017i $$-0.891258\pi$$
0.942212 0.335017i $$-0.108742\pi$$
$$984$$ 0 0
$$985$$ −207.000 −0.210152
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 134.873i − 0.136373i
$$990$$ 0 0
$$991$$ −608.484 −0.614010 −0.307005 0.951708i $$-0.599327\pi$$
−0.307005 + 0.951708i $$0.599327\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2174.14i 2.18506i
$$996$$ 0 0
$$997$$ −617.246 −0.619103 −0.309552 0.950883i $$-0.600179\pi$$
−0.309552 + 0.950883i $$0.600179\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 1296.3.e.d.161.4 4
3.2 odd 2 inner 1296.3.e.d.161.1 4
4.3 odd 2 162.3.b.b.161.2 4
9.2 odd 6 1296.3.q.o.1025.4 8
9.4 even 3 1296.3.q.o.593.4 8
9.5 odd 6 1296.3.q.o.593.1 8
9.7 even 3 1296.3.q.o.1025.1 8
12.11 even 2 162.3.b.b.161.3 yes 4
36.7 odd 6 162.3.d.c.53.1 8
36.11 even 6 162.3.d.c.53.4 8
36.23 even 6 162.3.d.c.107.1 8
36.31 odd 6 162.3.d.c.107.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
162.3.b.b.161.2 4 4.3 odd 2
162.3.b.b.161.3 yes 4 12.11 even 2
162.3.d.c.53.1 8 36.7 odd 6
162.3.d.c.53.4 8 36.11 even 6
162.3.d.c.107.1 8 36.23 even 6
162.3.d.c.107.4 8 36.31 odd 6
1296.3.e.d.161.1 4 3.2 odd 2 inner
1296.3.e.d.161.4 4 1.1 even 1 trivial
1296.3.q.o.593.1 8 9.5 odd 6
1296.3.q.o.593.4 8 9.4 even 3
1296.3.q.o.1025.1 8 9.7 even 3
1296.3.q.o.1025.4 8 9.2 odd 6