# Properties

 Label 4600.2.e.g.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.g.4049.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -1.00000i q^{13} -4.00000i q^{17} +4.00000 q^{19} -2.00000 q^{21} -1.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} -1.00000 q^{31} -8.00000i q^{37} -1.00000 q^{39} -5.00000 q^{41} +6.00000i q^{43} +9.00000i q^{47} +3.00000 q^{49} -4.00000 q^{51} -2.00000i q^{53} -4.00000i q^{57} -4.00000i q^{63} +4.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} -7.00000i q^{73} -4.00000 q^{79} +1.00000 q^{81} -8.00000i q^{83} -3.00000i q^{87} +14.0000 q^{89} -2.00000 q^{91} +1.00000i q^{93} -14.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 8 q^{19} - 4 q^{21} + 6 q^{29} - 2 q^{31} - 2 q^{39} - 10 q^{41} + 6 q^{49} - 8 q^{51} - 2 q^{69} + 6 q^{71} - 8 q^{79} + 2 q^{81} + 28 q^{89} - 4 q^{91}+O(q^{100})$$ 2 * q + 4 * q^9 + 8 * q^19 - 4 * q^21 + 6 * q^29 - 2 * q^31 - 2 * q^39 - 10 * q^41 + 6 * q^49 - 8 * q^51 - 2 * q^69 + 6 * q^71 - 8 * q^79 + 2 * q^81 + 28 * q^89 - 4 * q^91

## 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
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 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
$$31$$ −1.00000 −0.179605 −0.0898027 0.995960i $$-0.528624\pi$$
−0.0898027 + 0.995960i $$0.528624\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 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
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ − 4.00000i − 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ − 7.00000i − 0.819288i −0.912245 0.409644i $$-0.865653\pi$$
0.912245 0.409644i $$-0.134347\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 3.00000i − 0.321634i
$$88$$ 0 0
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 1.00000i 0.103695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 5.00000i 0.450835i
$$124$$ 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
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −1.00000 −0.0848189 −0.0424094 0.999100i $$-0.513503\pi$$
−0.0424094 + 0.999100i $$0.513503\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 3.00000i − 0.247436i
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −13.0000 −1.05792 −0.528962 0.848645i $$-0.677419\pi$$
−0.528962 + 0.848645i $$0.677419\pi$$
$$152$$ 0 0
$$153$$ − 8.00000i − 0.646762i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 24.0000i 1.85718i 0.371113 + 0.928588i $$0.378976\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 0 0
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 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
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −10.0000 −0.727393
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ 9.00000i 0.647834i 0.946085 + 0.323917i $$0.105000\pi$$
−0.946085 + 0.323917i $$0.895000\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 13.0000i − 0.926212i −0.886303 0.463106i $$-0.846735\pi$$
0.886303 0.463106i $$-0.153265\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ − 6.00000i − 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ − 3.00000i − 0.205557i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.00000i 0.135769i
$$218$$ 0 0
$$219$$ −7.00000 −0.473016
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.0000i 0.663723i 0.943328 + 0.331862i $$0.107677\pi$$
−0.943328 + 0.331862i $$0.892323\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 21.0000i − 1.37576i −0.725826 0.687878i $$-0.758542\pi$$
0.725826 0.687878i $$-0.241458\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4.00000i 0.259828i
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ − 16.0000i − 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ 0 0
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 3.00000i − 0.187135i −0.995613 0.0935674i $$-0.970173\pi$$
0.995613 0.0935674i $$-0.0298271\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 14.0000i − 0.856786i
$$268$$ 0 0
$$269$$ −25.0000 −1.52428 −0.762138 0.647414i $$-0.775850\pi$$
−0.762138 + 0.647414i $$0.775850\pi$$
$$270$$ 0 0
$$271$$ −28.0000 −1.70088 −0.850439 0.526073i $$-0.823664\pi$$
−0.850439 + 0.526073i $$0.823664\pi$$
$$272$$ 0 0
$$273$$ 2.00000i 0.121046i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.00000i 0.180253i 0.995930 + 0.0901263i $$0.0287271\pi$$
−0.995930 + 0.0901263i $$0.971273\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 10.0000i 0.594438i 0.954809 + 0.297219i $$0.0960592\pi$$
−0.954809 + 0.297219i $$0.903941\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.0000i 0.590281i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 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
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ − 2.00000i − 0.114897i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ 4.00000i 0.226093i 0.993590 + 0.113047i $$0.0360610\pi$$
−0.993590 + 0.113047i $$0.963939\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 26.0000i 1.46031i 0.683284 + 0.730153i $$0.260551\pi$$
−0.683284 + 0.730153i $$0.739449\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 16.0000i − 0.890264i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.00000i 0.110600i
$$328$$ 0 0
$$329$$ 18.0000 0.992372
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 0 0
$$333$$ − 16.0000i − 0.876795i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 10.0000i − 0.544735i −0.962193 0.272367i $$-0.912193\pi$$
0.962193 0.272367i $$-0.0878066\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ − 3.00000i − 0.159674i −0.996808 0.0798369i $$-0.974560\pi$$
0.996808 0.0798369i $$-0.0254400\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000i 0.423405i
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 11.0000i 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 36.0000i − 1.87918i −0.342296 0.939592i $$-0.611204\pi$$
0.342296 0.939592i $$-0.388796\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ − 6.00000i − 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 3.00000i − 0.154508i
$$378$$ 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
$$383$$ − 2.00000i − 0.102195i −0.998694 0.0510976i $$-0.983728\pi$$
0.998694 0.0510976i $$-0.0162720\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.0000i 0.609994i
$$388$$ 0 0
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ − 3.00000i − 0.151330i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7.00000i − 0.351320i −0.984451 0.175660i $$-0.943794\pi$$
0.984451 0.175660i $$-0.0562059\pi$$
$$398$$ 0 0
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ 1.00000i 0.0498135i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.0000 0.939490 0.469745 0.882802i $$-0.344346\pi$$
0.469745 + 0.882802i $$0.344346\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1.00000i 0.0489702i
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 18.0000i 0.875190i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 4.00000i − 0.191346i
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 6.00000 0.285714
$$442$$ 0 0
$$443$$ − 5.00000i − 0.237557i −0.992921 0.118779i $$-0.962102\pi$$
0.992921 0.118779i $$-0.0378979\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 18.0000i 0.851371i
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 13.0000i 0.610793i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 40.0000i − 1.87112i −0.353166 0.935561i $$-0.614895\pi$$
0.353166 0.935561i $$-0.385105\pi$$
$$458$$ 0 0
$$459$$ −20.0000 −0.933520
$$460$$ 0 0
$$461$$ −3.00000 −0.139724 −0.0698620 0.997557i $$-0.522256\pi$$
−0.0698620 + 0.997557i $$0.522256\pi$$
$$462$$ 0 0
$$463$$ 20.0000i 0.929479i 0.885448 + 0.464739i $$0.153852\pi$$
−0.885448 + 0.464739i $$0.846148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.0000i 0.832941i 0.909149 + 0.416470i $$0.136733\pi$$
−0.909149 + 0.416470i $$0.863267\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 4.00000i − 0.183147i
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ 2.00000i 0.0910032i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 5.00000i − 0.226572i −0.993562 0.113286i $$-0.963862\pi$$
0.993562 0.113286i $$-0.0361376\pi$$
$$488$$ 0 0
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ −27.0000 −1.21849 −0.609246 0.792981i $$-0.708528\pi$$
−0.609246 + 0.792981i $$0.708528\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ −19.0000 −0.850557 −0.425278 0.905063i $$-0.639824\pi$$
−0.425278 + 0.905063i $$0.639824\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ 0 0
$$503$$ 30.0000i 1.33763i 0.743427 + 0.668817i $$0.233199\pi$$
−0.743427 + 0.668817i $$0.766801\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 12.0000i − 0.532939i
$$508$$ 0 0
$$509$$ 19.0000 0.842160 0.421080 0.907023i $$-0.361651\pi$$
0.421080 + 0.907023i $$0.361651\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ 0 0
$$513$$ − 20.0000i − 0.883022i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ 38.0000i 1.66162i 0.556553 + 0.830812i $$0.312124\pi$$
−0.556553 + 0.830812i $$0.687876\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 4.00000i 0.174243i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.00000i 0.216574i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 9.00000i 0.388379i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ 0 0
$$543$$ 12.0000i 0.514969i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 1.00000i − 0.0427569i −0.999771 0.0213785i $$-0.993195\pi$$
0.999771 0.0213785i $$-0.00680549\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 34.0000i 1.44063i 0.693649 + 0.720313i $$0.256002\pi$$
−0.693649 + 0.720313i $$0.743998\pi$$
$$558$$ 0 0
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ 24.0000 1.00437 0.502184 0.864761i $$-0.332530\pi$$
0.502184 + 0.864761i $$0.332530\pi$$
$$572$$ 0 0
$$573$$ 10.0000i 0.417756i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.0000i 0.957503i 0.877951 + 0.478751i $$0.158910\pi$$
−0.877951 + 0.478751i $$0.841090\pi$$
$$578$$ 0 0
$$579$$ 9.00000 0.374027
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 29.0000i 1.19696i 0.801138 + 0.598479i $$0.204228\pi$$
−0.801138 + 0.598479i $$0.795772\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −13.0000 −0.534749
$$592$$ 0 0
$$593$$ − 26.0000i − 1.06769i −0.845582 0.533846i $$-0.820746\pi$$
0.845582 0.533846i $$-0.179254\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 20.0000i − 0.818546i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 45.0000 1.83559 0.917794 0.397057i $$-0.129968\pi$$
0.917794 + 0.397057i $$0.129968\pi$$
$$602$$ 0 0
$$603$$ 8.00000i 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 0 0
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ 4.00000i 0.161558i 0.996732 + 0.0807792i $$0.0257409\pi$$
−0.996732 + 0.0807792i $$0.974259\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 28.0000i − 1.12724i −0.826035 0.563619i $$-0.809409\pi$$
0.826035 0.563619i $$-0.190591\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ 0 0
$$623$$ − 28.0000i − 1.12180i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −32.0000 −1.27592
$$630$$ 0 0
$$631$$ 26.0000 1.03504 0.517522 0.855670i $$-0.326855\pi$$
0.517522 + 0.855670i $$0.326855\pi$$
$$632$$ 0 0
$$633$$ 8.00000i 0.317971i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 3.00000i − 0.118864i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 21.0000i 0.825595i 0.910823 + 0.412798i $$0.135448\pi$$
−0.910823 + 0.412798i $$0.864552\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 2.00000 0.0783862
$$652$$ 0 0
$$653$$ 35.0000i 1.36966i 0.728705 + 0.684828i $$0.240123\pi$$
−0.728705 + 0.684828i $$0.759877\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 14.0000i − 0.546192i
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 0 0
$$663$$ 4.00000i 0.155347i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 3.00000i − 0.116160i
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 43.0000i 1.65753i 0.559598 + 0.828764i $$0.310955\pi$$
−0.559598 + 0.828764i $$0.689045\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ 0 0
$$679$$ −28.0000 −1.07454
$$680$$ 0 0
$$681$$ 10.0000 0.383201
$$682$$ 0 0
$$683$$ 9.00000i 0.344375i 0.985064 + 0.172188i $$0.0550836\pi$$
−0.985064 + 0.172188i $$0.944916\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ 0 0
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 20.0000i 0.757554i
$$698$$ 0 0
$$699$$ −21.0000 −0.794293
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ − 32.0000i − 1.20690i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 4.00000i − 0.150435i
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 1.00000i 0.0374503i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 21.0000i − 0.784259i
$$718$$ 0 0
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ − 22.0000i − 0.818189i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 26.0000i − 0.964287i −0.876092 0.482143i $$-0.839858\pi$$
0.876092 0.482143i $$-0.160142\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 28.0000i 1.03420i 0.855924 + 0.517102i $$0.172989\pi$$
−0.855924 + 0.517102i $$0.827011\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 0 0
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 16.0000i − 0.585409i
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ 0 0
$$753$$ 12.0000i 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 40.0000i − 1.45382i −0.686730 0.726912i $$-0.740955\pi$$
0.686730 0.726912i $$-0.259045\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 5.00000 0.181250 0.0906249 0.995885i $$-0.471114\pi$$
0.0906249 + 0.995885i $$0.471114\pi$$
$$762$$ 0 0
$$763$$ 4.00000i 0.144810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ −3.00000 −0.108042
$$772$$ 0 0
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 16.0000i 0.573997i
$$778$$ 0 0
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 15.0000i − 0.536056i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 24.0000i − 0.850124i −0.905164 0.425062i $$-0.860252\pi$$
0.905164 0.425062i $$-0.139748\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ 28.0000 0.989331
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 25.0000i 0.880042i
$$808$$ 0 0
$$809$$ 46.0000 1.61727 0.808637 0.588308i $$-0.200206\pi$$
0.808637 + 0.588308i $$0.200206\pi$$
$$810$$ 0 0
$$811$$ 39.0000 1.36948 0.684738 0.728790i $$-0.259917\pi$$
0.684738 + 0.728790i $$0.259917\pi$$
$$812$$ 0 0
$$813$$ 28.0000i 0.982003i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24.0000i 0.839654i
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 34.0000 1.18661 0.593304 0.804978i $$-0.297823\pi$$
0.593304 + 0.804978i $$0.297823\pi$$
$$822$$ 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
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 3.00000 0.104069
$$832$$ 0 0
$$833$$ − 12.0000i − 0.415775i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.00000i 0.172825i
$$838$$ 0 0
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ − 6.00000i − 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000i 0.755929i
$$848$$ 0 0
$$849$$ 10.0000 0.343199
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 7.00000i − 0.239115i −0.992827 0.119558i $$-0.961852\pi$$
0.992827 0.119558i $$-0.0381477\pi$$
$$858$$ 0 0
$$859$$ 45.0000 1.53538 0.767690 0.640821i $$-0.221406\pi$$
0.767690 + 0.640821i $$0.221406\pi$$
$$860$$ 0 0
$$861$$ 10.0000 0.340799
$$862$$ 0 0
$$863$$ − 11.0000i − 0.374444i −0.982318 0.187222i $$-0.940052\pi$$
0.982318 0.187222i $$-0.0599484\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 1.00000i − 0.0339618i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ − 28.0000i − 0.947656i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000i 0.0675352i 0.999430 + 0.0337676i $$0.0107506\pi$$
−0.999430 + 0.0337676i $$0.989249\pi$$
$$878$$ 0 0
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ 56.0000 1.88669 0.943344 0.331816i $$-0.107661\pi$$
0.943344 + 0.331816i $$0.107661\pi$$
$$882$$ 0 0
$$883$$ − 4.00000i − 0.134611i −0.997732 0.0673054i $$-0.978560\pi$$
0.997732 0.0673054i $$-0.0214402\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 3.00000i − 0.100730i −0.998731 0.0503651i $$-0.983962\pi$$
0.998731 0.0503651i $$-0.0160385\pi$$
$$888$$ 0 0
$$889$$ 14.0000 0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 36.0000i 1.20469i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.00000i 0.0333890i
$$898$$ 0 0
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ 0 0
$$903$$ − 12.0000i − 0.399335i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 0 0
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 6.00000i − 0.198137i
$$918$$ 0 0
$$919$$ −46.0000 −1.51740 −0.758700 0.651440i $$-0.774165\pi$$
−0.758700 + 0.651440i $$0.774165\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ − 3.00000i − 0.0987462i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 16.0000i − 0.525509i
$$928$$ 0 0
$$929$$ 45.0000 1.47640 0.738201 0.674581i $$-0.235676\pi$$
0.738201 + 0.674581i $$0.235676\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 0 0
$$933$$ 9.00000i 0.294647i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 36.0000i 1.17607i 0.808836 + 0.588034i $$0.200098\pi$$
−0.808836 + 0.588034i $$0.799902\pi$$
$$938$$ 0 0
$$939$$ 4.00000 0.130535
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ 5.00000i 0.162822i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 45.0000i 1.46230i 0.682215 + 0.731152i $$0.261017\pi$$
−0.682215 + 0.731152i $$0.738983\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ 0 0
$$951$$ 26.0000 0.843108
$$952$$ 0 0
$$953$$ 6.00000i 0.194359i 0.995267 + 0.0971795i $$0.0309821\pi$$
−0.995267 + 0.0971795i $$0.969018\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 8.00000i 0.257796i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 23.0000i 0.739630i 0.929105 + 0.369815i $$0.120579\pi$$
−0.929105 + 0.369815i $$0.879421\pi$$
$$968$$ 0 0
$$969$$ −16.0000 −0.513994
$$970$$ 0 0
$$971$$ 50.0000 1.60458 0.802288 0.596937i $$-0.203616\pi$$
0.802288 + 0.596937i $$0.203616\pi$$
$$972$$ 0 0
$$973$$ 2.00000i 0.0641171i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 30.0000i 0.956851i 0.878128 + 0.478426i $$0.158792\pi$$
−0.878128 + 0.478426i $$0.841208\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 18.0000i − 0.572946i
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ 19.0000i 0.602947i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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.g.4049.1 2
5.2 odd 4 4600.2.a.f.1.1 1
5.3 odd 4 920.2.a.d.1.1 1
5.4 even 2 inner 4600.2.e.g.4049.2 2
15.8 even 4 8280.2.a.o.1.1 1
20.3 even 4 1840.2.a.b.1.1 1
20.7 even 4 9200.2.a.z.1.1 1
40.3 even 4 7360.2.a.u.1.1 1
40.13 odd 4 7360.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.d.1.1 1 5.3 odd 4
1840.2.a.b.1.1 1 20.3 even 4
4600.2.a.f.1.1 1 5.2 odd 4
4600.2.e.g.4049.1 2 1.1 even 1 trivial
4600.2.e.g.4049.2 2 5.4 even 2 inner
7360.2.a.j.1.1 1 40.13 odd 4
7360.2.a.u.1.1 1 40.3 even 4
8280.2.a.o.1.1 1 15.8 even 4
9200.2.a.z.1.1 1 20.7 even 4