# Properties

 Label 4600.2.e.h.4049.1 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.h.4049.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +2.00000 q^{9} +2.00000 q^{11} -5.00000i q^{13} +4.00000i q^{17} +2.00000 q^{19} -1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +7.00000 q^{31} -2.00000i q^{33} +2.00000i q^{37} -5.00000 q^{39} -9.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +7.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} -2.00000i q^{57} +2.00000 q^{61} +2.00000i q^{67} -1.00000 q^{69} -1.00000 q^{71} +1.00000i q^{73} +14.0000 q^{79} +1.00000 q^{81} -3.00000i q^{87} -16.0000 q^{89} -7.00000i q^{93} +4.00000i q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 4 q^{11} + 4 q^{19} + 6 q^{29} + 14 q^{31} - 10 q^{39} - 18 q^{41} + 14 q^{49} + 8 q^{51} + 4 q^{61} - 2 q^{69} - 2 q^{71} + 28 q^{79} + 2 q^{81} - 32 q^{89} + 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 4 * q^11 + 4 * q^19 + 6 * q^29 + 14 * q^31 - 10 * q^39 - 18 * q^41 + 14 * q^49 + 8 * q^51 + 4 * q^61 - 2 * q^69 - 2 * q^71 + 28 * q^79 + 2 * q^81 - 32 * q^89 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.00000i − 0.577350i −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 5.00000i − 1.38675i −0.720577 0.693375i $$-0.756123\pi$$
0.720577 0.693375i $$-0.243877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.00000i − 0.962250i
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ − 2.00000i − 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 9.00000i 1.31278i 0.754420 + 0.656392i $$0.227918\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −1.00000 −0.118678 −0.0593391 0.998238i $$-0.518899\pi$$
−0.0593391 + 0.998238i $$0.518899\pi$$
$$72$$ 0 0
$$73$$ 1.00000i 0.117041i 0.998286 + 0.0585206i $$0.0186383\pi$$
−0.998286 + 0.0585206i $$0.981362\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 3.00000i − 0.321634i
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 7.00000i − 0.725866i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.00000i 0.406138i 0.979164 + 0.203069i $$0.0650917\pi$$
−0.979164 + 0.203069i $$0.934908\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 10.0000i − 0.966736i −0.875417 0.483368i $$-0.839413\pi$$
0.875417 0.483368i $$-0.160587\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ − 16.0000i − 1.50515i −0.658505 0.752577i $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 10.0000i − 0.924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 9.00000i 0.811503i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 7.00000i 0.621150i 0.950549 + 0.310575i $$0.100522\pi$$
−0.950549 + 0.310575i $$0.899478\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 0 0
$$143$$ − 10.0000i − 0.836242i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 7.00000i − 0.577350i
$$148$$ 0 0
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ 15.0000 1.22068 0.610341 0.792139i $$-0.291032\pi$$
0.610341 + 0.792139i $$0.291032\pi$$
$$152$$ 0 0
$$153$$ 8.00000i 0.646762i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 23.0000i − 1.80150i −0.434339 0.900750i $$-0.643018\pi$$
0.434339 0.900750i $$-0.356982\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −9.00000 −0.672692 −0.336346 0.941739i $$-0.609191\pi$$
−0.336346 + 0.941739i $$0.609191\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ − 2.00000i − 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 14.0000 1.01300 0.506502 0.862239i $$-0.330938\pi$$
0.506502 + 0.862239i $$0.330938\pi$$
$$192$$ 0 0
$$193$$ 5.00000i 0.359908i 0.983675 + 0.179954i $$0.0575949\pi$$
−0.983675 + 0.179954i $$0.942405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 25.0000i − 1.78118i −0.454811 0.890588i $$-0.650293\pi$$
0.454811 0.890588i $$-0.349707\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 2.00000 0.141069
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ 0 0
$$213$$ 1.00000i 0.0685189i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1.00000 0.0675737
$$220$$ 0 0
$$221$$ 20.0000 1.34535
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 25.0000i − 1.63780i −0.573933 0.818902i $$-0.694583\pi$$
0.573933 0.818902i $$-0.305417\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 14.0000i − 0.909398i
$$238$$ 0 0
$$239$$ 9.00000 0.582162 0.291081 0.956698i $$-0.405985\pi$$
0.291081 + 0.956698i $$0.405985\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ − 16.0000i − 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 10.0000i − 0.636285i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ − 2.00000i − 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 19.0000i − 1.18519i −0.805502 0.592594i $$-0.798104\pi$$
0.805502 0.592594i $$-0.201896\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 14.0000i 0.863277i 0.902047 + 0.431638i $$0.142064\pi$$
−0.902047 + 0.431638i $$0.857936\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000i 0.979184i
$$268$$ 0 0
$$269$$ 7.00000 0.426798 0.213399 0.976965i $$-0.431547\pi$$
0.213399 + 0.976965i $$0.431547\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 13.0000i − 0.781094i −0.920583 0.390547i $$-0.872286\pi$$
0.920583 0.390547i $$-0.127714\pi$$
$$278$$ 0 0
$$279$$ 14.0000 0.838158
$$280$$ 0 0
$$281$$ −32.0000 −1.90896 −0.954480 0.298275i $$-0.903589\pi$$
−0.954480 + 0.298275i $$0.903589\pi$$
$$282$$ 0 0
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ 0 0
$$293$$ 16.0000i 0.934730i 0.884064 + 0.467365i $$0.154797\pi$$
−0.884064 + 0.467365i $$0.845203\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 10.0000i − 0.580259i
$$298$$ 0 0
$$299$$ −5.00000 −0.289157
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 10.0000i − 0.574485i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ 8.00000i 0.452187i 0.974106 + 0.226093i $$0.0725954\pi$$
−0.974106 + 0.226093i $$0.927405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ −10.0000 −0.558146
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4.00000i − 0.221201i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 0 0
$$333$$ 4.00000i 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 8.00000i − 0.435788i −0.975972 0.217894i $$-0.930081\pi$$
0.975972 0.217894i $$-0.0699187\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 14.0000 0.758143
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 28.0000i 1.50312i 0.659665 + 0.751559i $$0.270698\pi$$
−0.659665 + 0.751559i $$0.729302\pi$$
$$348$$ 0 0
$$349$$ −17.0000 −0.909989 −0.454995 0.890494i $$-0.650359\pi$$
−0.454995 + 0.890494i $$0.650359\pi$$
$$350$$ 0 0
$$351$$ −25.0000 −1.33440
$$352$$ 0 0
$$353$$ − 27.0000i − 1.43706i −0.695493 0.718532i $$-0.744814\pi$$
0.695493 0.718532i $$-0.255186\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −30.0000 −1.58334 −0.791670 0.610949i $$-0.790788\pi$$
−0.791670 + 0.610949i $$0.790788\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 7.00000i 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 15.0000i − 0.772539i
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 7.00000 0.358621
$$382$$ 0 0
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ − 15.0000i − 0.756650i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.00000i − 0.150566i −0.997162 0.0752828i $$-0.976014\pi$$
0.997162 0.0752828i $$-0.0239860\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 35.0000i − 1.74347i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000i 0.198273i
$$408$$ 0 0
$$409$$ 35.0000 1.73064 0.865319 0.501221i $$-0.167116\pi$$
0.865319 + 0.501221i $$0.167116\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 9.00000i 0.440732i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 0 0
$$423$$ 18.0000i 0.875190i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −10.0000 −0.482805
$$430$$ 0 0
$$431$$ 28.0000 1.34871 0.674356 0.738406i $$-0.264421\pi$$
0.674356 + 0.738406i $$0.264421\pi$$
$$432$$ 0 0
$$433$$ − 6.00000i − 0.288342i −0.989553 0.144171i $$-0.953949\pi$$
0.989553 0.144171i $$-0.0460515\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2.00000i − 0.0956730i
$$438$$ 0 0
$$439$$ 15.0000 0.715911 0.357955 0.933739i $$-0.383474\pi$$
0.357955 + 0.933739i $$0.383474\pi$$
$$440$$ 0 0
$$441$$ 14.0000 0.666667
$$442$$ 0 0
$$443$$ − 25.0000i − 1.18779i −0.804544 0.593893i $$-0.797590\pi$$
0.804544 0.593893i $$-0.202410\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.0000i 0.662177i
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ 0 0
$$453$$ − 15.0000i − 0.704761i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 32.0000i − 1.49690i −0.663193 0.748448i $$-0.730799\pi$$
0.663193 0.748448i $$-0.269201\pi$$
$$458$$ 0 0
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ −7.00000 −0.326023 −0.163011 0.986624i $$-0.552121\pi$$
−0.163011 + 0.986624i $$0.552121\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ 0 0
$$473$$ − 8.00000i − 0.367840i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 12.0000i − 0.549442i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 23.0000i 1.04223i 0.853487 + 0.521115i $$0.174484\pi$$
−0.853487 + 0.521115i $$0.825516\pi$$
$$488$$ 0 0
$$489$$ −23.0000 −1.04010
$$490$$ 0 0
$$491$$ 1.00000 0.0451294 0.0225647 0.999745i $$-0.492817\pi$$
0.0225647 + 0.999745i $$0.492817\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3.00000 −0.134298 −0.0671492 0.997743i $$-0.521390\pi$$
−0.0671492 + 0.997743i $$0.521390\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ 22.0000i 0.980932i 0.871460 + 0.490466i $$0.163173\pi$$
−0.871460 + 0.490466i $$0.836827\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 12.0000i 0.532939i
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ − 10.0000i − 0.441511i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 28.0000i 1.21970i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 45.0000i 1.94917i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 9.00000i 0.388379i
$$538$$ 0 0
$$539$$ 14.0000 0.603023
$$540$$ 0 0
$$541$$ −35.0000 −1.50477 −0.752384 0.658725i $$-0.771096\pi$$
−0.752384 + 0.658725i $$0.771096\pi$$
$$542$$ 0 0
$$543$$ 2.00000i 0.0858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 7.00000i 0.299298i 0.988739 + 0.149649i $$0.0478144\pi$$
−0.988739 + 0.149649i $$0.952186\pi$$
$$548$$ 0 0
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24.0000i 1.01691i 0.861088 + 0.508456i $$0.169784\pi$$
−0.861088 + 0.508456i $$0.830216\pi$$
$$558$$ 0 0
$$559$$ −20.0000 −0.845910
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −2.00000 −0.0838444 −0.0419222 0.999121i $$-0.513348\pi$$
−0.0419222 + 0.999121i $$0.513348\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ − 14.0000i − 0.584858i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 11.0000i 0.457936i 0.973434 + 0.228968i $$0.0735351\pi$$
−0.973434 + 0.228968i $$0.926465\pi$$
$$578$$ 0 0
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 12.0000i − 0.496989i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9.00000i 0.371470i 0.982600 + 0.185735i $$0.0594666\pi$$
−0.982600 + 0.185735i $$0.940533\pi$$
$$588$$ 0 0
$$589$$ 14.0000 0.576860
$$590$$ 0 0
$$591$$ −25.0000 −1.02836
$$592$$ 0 0
$$593$$ − 2.00000i − 0.0821302i −0.999156 0.0410651i $$-0.986925\pi$$
0.999156 0.0410651i $$-0.0130751\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 10.0000i − 0.409273i
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 5.00000 0.203954 0.101977 0.994787i $$-0.467483\pi$$
0.101977 + 0.994787i $$0.467483\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 20.0000i 0.811775i 0.913923 + 0.405887i $$0.133038\pi$$
−0.913923 + 0.405887i $$0.866962\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 45.0000 1.82051
$$612$$ 0 0
$$613$$ 12.0000i 0.484675i 0.970192 + 0.242338i $$0.0779142\pi$$
−0.970192 + 0.242338i $$0.922086\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.0000i 1.04672i 0.852111 + 0.523360i $$0.175322\pi$$
−0.852111 + 0.523360i $$0.824678\pi$$
$$618$$ 0 0
$$619$$ 48.0000 1.92928 0.964641 0.263566i $$-0.0848986\pi$$
0.964641 + 0.263566i $$0.0848986\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 4.00000i − 0.159745i
$$628$$ 0 0
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ − 28.0000i − 1.11290i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 35.0000i − 1.38675i
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ − 34.0000i − 1.34083i −0.741987 0.670415i $$-0.766116\pi$$
0.741987 0.670415i $$-0.233884\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 7.00000i − 0.275198i −0.990488 0.137599i $$-0.956061\pi$$
0.990488 0.137599i $$-0.0439386\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19.0000i 0.743527i 0.928327 + 0.371764i $$0.121247\pi$$
−0.928327 + 0.371764i $$0.878753\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ − 20.0000i − 0.776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 3.00000i − 0.116160i
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ − 9.00000i − 0.346925i −0.984841 0.173462i $$-0.944505\pi$$
0.984841 0.173462i $$-0.0554955\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30.0000i 1.15299i 0.817099 + 0.576497i $$0.195581\pi$$
−0.817099 + 0.576497i $$0.804419\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 5.00000i 0.191320i 0.995414 + 0.0956598i $$0.0304961\pi$$
−0.995414 + 0.0956598i $$0.969504\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −30.0000 −1.14291
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ 0 0
$$699$$ −25.0000 −0.945587
$$700$$ 0 0
$$701$$ −16.0000 −0.604312 −0.302156 0.953259i $$-0.597706\pi$$
−0.302156 + 0.953259i $$0.597706\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 12.0000 0.450669 0.225335 0.974281i $$-0.427652\pi$$
0.225335 + 0.974281i $$0.427652\pi$$
$$710$$ 0 0
$$711$$ 28.0000 1.05008
$$712$$ 0 0
$$713$$ − 7.00000i − 0.262152i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 9.00000i − 0.336111i
$$718$$ 0 0
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 10.0000i 0.371904i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 46.0000i − 1.70605i −0.521874 0.853023i $$-0.674767\pi$$
0.521874 0.853023i $$-0.325233\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 0 0
$$733$$ 52.0000i 1.92066i 0.278859 + 0.960332i $$0.410044\pi$$
−0.278859 + 0.960332i $$0.589956\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000i 0.147342i
$$738$$ 0 0
$$739$$ 23.0000 0.846069 0.423034 0.906114i $$-0.360965\pi$$
0.423034 + 0.906114i $$0.360965\pi$$
$$740$$ 0 0
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ − 8.00000i − 0.291536i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 26.0000i − 0.944986i −0.881334 0.472493i $$-0.843354\pi$$
0.881334 0.472493i $$-0.156646\pi$$
$$758$$ 0 0
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ 13.0000 0.471250 0.235625 0.971844i $$-0.424286\pi$$
0.235625 + 0.971844i $$0.424286\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ −19.0000 −0.684268
$$772$$ 0 0
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ −2.00000 −0.0715656
$$782$$ 0 0
$$783$$ − 15.0000i − 0.536056i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ 0 0
$$789$$ 14.0000 0.498413
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 10.0000i − 0.355110i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ −32.0000 −1.13066
$$802$$ 0 0
$$803$$ 2.00000i 0.0705785i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 7.00000i − 0.246412i
$$808$$ 0 0
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ −29.0000 −1.01833 −0.509164 0.860670i $$-0.670045\pi$$
−0.509164 + 0.860670i $$0.670045\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 8.00000i − 0.279885i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ 3.00000i 0.104573i 0.998632 + 0.0522867i $$0.0166510\pi$$
−0.998632 + 0.0522867i $$0.983349\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000i 0.278187i 0.990279 + 0.139094i $$0.0444189\pi$$
−0.990279 + 0.139094i $$0.955581\pi$$
$$828$$ 0 0
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ −13.0000 −0.450965
$$832$$ 0 0
$$833$$ 28.0000i 0.970143i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 35.0000i − 1.20978i
$$838$$ 0 0
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 32.0000i 1.10214i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 41.0000i 1.40053i 0.713881 + 0.700267i $$0.246936\pi$$
−0.713881 + 0.700267i $$0.753064\pi$$
$$858$$ 0 0
$$859$$ −43.0000 −1.46714 −0.733571 0.679613i $$-0.762148\pi$$
−0.733571 + 0.679613i $$0.762148\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 57.0000i 1.94030i 0.242500 + 0.970151i $$0.422032\pi$$
−0.242500 + 0.970151i $$0.577968\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 0 0
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 0 0
$$873$$ 8.00000i 0.270759i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ 0 0
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ −28.0000 −0.943344 −0.471672 0.881774i $$-0.656349\pi$$
−0.471672 + 0.881774i $$0.656349\pi$$
$$882$$ 0 0
$$883$$ 52.0000i 1.74994i 0.484178 + 0.874970i $$0.339119\pi$$
−0.484178 + 0.874970i $$0.660881\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 33.0000i 1.10803i 0.832506 + 0.554016i $$0.186905\pi$$
−0.832506 + 0.554016i $$0.813095\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ 18.0000i 0.602347i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 5.00000i 0.166945i
$$898$$ 0 0
$$899$$ 21.0000 0.700389
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 30.0000i 0.996134i 0.867139 + 0.498067i $$0.165957\pi$$
−0.867139 + 0.498067i $$0.834043\pi$$
$$908$$ 0 0
$$909$$ 20.0000 0.663358
$$910$$ 0 0
$$911$$ 50.0000 1.65657 0.828287 0.560304i $$-0.189316\pi$$
0.828287 + 0.560304i $$0.189316\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −60.0000 −1.97922 −0.989609 0.143787i $$-0.954072\pi$$
−0.989609 + 0.143787i $$0.954072\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ 0 0
$$923$$ 5.00000i 0.164577i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −23.0000 −0.754606 −0.377303 0.926090i $$-0.623148\pi$$
−0.377303 + 0.926090i $$0.623148\pi$$
$$930$$ 0 0
$$931$$ 14.0000 0.458831
$$932$$ 0 0
$$933$$ 9.00000i 0.294647i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ 0 0
$$939$$ 8.00000 0.261070
$$940$$ 0 0
$$941$$ −48.0000 −1.56476 −0.782378 0.622804i $$-0.785993\pi$$
−0.782378 + 0.622804i $$0.785993\pi$$
$$942$$ 0 0
$$943$$ 9.00000i 0.293080i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 15.0000i − 0.487435i −0.969846 0.243717i $$-0.921633\pi$$
0.969846 0.243717i $$-0.0783669\pi$$
$$948$$ 0 0
$$949$$ 5.00000 0.162307
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ − 26.0000i − 0.842223i −0.907009 0.421111i $$-0.861640\pi$$
0.907009 0.421111i $$-0.138360\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 6.00000i − 0.193952i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ − 20.0000i − 0.644491i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 3.00000i 0.0964735i 0.998836 + 0.0482367i $$0.0153602\pi$$
−0.998836 + 0.0482367i $$0.984640\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 22.0000 0.706014 0.353007 0.935621i $$-0.385159\pi$$
0.353007 + 0.935621i $$0.385159\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 26.0000i 0.831814i 0.909407 + 0.415907i $$0.136536\pi$$
−0.909407 + 0.415907i $$0.863464\pi$$
$$978$$ 0 0
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ 8.00000 0.255420
$$982$$ 0 0
$$983$$ 50.0000i 1.59475i 0.603483 + 0.797376i $$0.293779\pi$$
−0.603483 + 0.797376i $$0.706221\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 19.0000i 0.602947i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 14.0000i − 0.443384i −0.975117 0.221692i $$-0.928842\pi$$
0.975117 0.221692i $$-0.0711580\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 4600.2.e.h.4049.1 2
5.2 odd 4 920.2.a.b.1.1 1
5.3 odd 4 4600.2.a.j.1.1 1
5.4 even 2 inner 4600.2.e.h.4049.2 2
15.2 even 4 8280.2.a.e.1.1 1
20.3 even 4 9200.2.a.n.1.1 1
20.7 even 4 1840.2.a.g.1.1 1
40.27 even 4 7360.2.a.g.1.1 1
40.37 odd 4 7360.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.b.1.1 1 5.2 odd 4
1840.2.a.g.1.1 1 20.7 even 4
4600.2.a.j.1.1 1 5.3 odd 4
4600.2.e.h.4049.1 2 1.1 even 1 trivial
4600.2.e.h.4049.2 2 5.4 even 2 inner
7360.2.a.g.1.1 1 40.27 even 4
7360.2.a.t.1.1 1 40.37 odd 4
8280.2.a.e.1.1 1 15.2 even 4
9200.2.a.n.1.1 1 20.3 even 4