# Properties

 Label 4600.2.e.r.4049.4 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.24681024.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 9$$ x^6 + 12*x^4 + 36*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.4 Root $$0.523976i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.r.4049.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} +O(q^{10})$$ $$q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} -1.67750 q^{11} -2.67750i q^{13} +1.67750i q^{17} -7.92692 q^{19} -0.249425 q^{21} -1.00000i q^{23} +3.00000i q^{27} +2.20147 q^{29} +6.77340 q^{31} -0.878968i q^{33} +4.00000i q^{37} +1.40294 q^{39} +5.97487 q^{41} +0.402945i q^{43} +1.79853i q^{47} +6.77340 q^{49} -0.878968 q^{51} +10.8059i q^{53} -4.15352i q^{57} -9.45090 q^{59} +6.32250 q^{61} +1.29738i q^{63} +14.7100i q^{67} +0.523976 q^{69} +9.97487 q^{71} +6.10557i q^{73} -0.798528i q^{77} +6.60442 q^{81} -4.49885i q^{83} +1.15352i q^{87} +3.59706 q^{89} +1.27455 q^{91} +3.54910i q^{93} -6.08044i q^{97} -4.57193 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100})$$ 6 * q - 6 * q^9 + 6 * q^11 - 6 * q^19 + 24 * q^21 - 6 * q^29 + 12 * q^31 - 30 * q^39 - 12 * q^41 + 12 * q^49 + 30 * q^51 - 12 * q^59 + 54 * q^61 + 12 * q^71 - 18 * q^81 + 60 * q^89 + 30 * q^91 - 18 * 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
$$3$$ 0.523976i 0.302518i 0.988494 + 0.151259i $$0.0483327\pi$$
−0.988494 + 0.151259i $$0.951667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.476024i 0.179920i 0.995945 + 0.0899600i $$0.0286739\pi$$
−0.995945 + 0.0899600i $$0.971326\pi$$
$$8$$ 0 0
$$9$$ 2.72545 0.908483
$$10$$ 0 0
$$11$$ −1.67750 −0.505784 −0.252892 0.967495i $$-0.581382\pi$$
−0.252892 + 0.967495i $$0.581382\pi$$
$$12$$ 0 0
$$13$$ − 2.67750i − 0.742604i −0.928512 0.371302i $$-0.878911\pi$$
0.928512 0.371302i $$-0.121089\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.67750i 0.406853i 0.979090 + 0.203426i $$0.0652077\pi$$
−0.979090 + 0.203426i $$0.934792\pi$$
$$18$$ 0 0
$$19$$ −7.92692 −1.81856 −0.909280 0.416184i $$-0.863367\pi$$
−0.909280 + 0.416184i $$0.863367\pi$$
$$20$$ 0 0
$$21$$ −0.249425 −0.0544290
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.00000i 0.577350i
$$28$$ 0 0
$$29$$ 2.20147 0.408803 0.204402 0.978887i $$-0.434475\pi$$
0.204402 + 0.978887i $$0.434475\pi$$
$$30$$ 0 0
$$31$$ 6.77340 1.21654 0.608269 0.793731i $$-0.291864\pi$$
0.608269 + 0.793731i $$0.291864\pi$$
$$32$$ 0 0
$$33$$ − 0.878968i − 0.153009i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 0 0
$$39$$ 1.40294 0.224651
$$40$$ 0 0
$$41$$ 5.97487 0.933119 0.466559 0.884490i $$-0.345493\pi$$
0.466559 + 0.884490i $$0.345493\pi$$
$$42$$ 0 0
$$43$$ 0.402945i 0.0614485i 0.999528 + 0.0307242i $$0.00978137\pi$$
−0.999528 + 0.0307242i $$0.990219\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.79853i 0.262342i 0.991360 + 0.131171i $$0.0418737\pi$$
−0.991360 + 0.131171i $$0.958126\pi$$
$$48$$ 0 0
$$49$$ 6.77340 0.967629
$$50$$ 0 0
$$51$$ −0.878968 −0.123080
$$52$$ 0 0
$$53$$ 10.8059i 1.48430i 0.670232 + 0.742152i $$0.266195\pi$$
−0.670232 + 0.742152i $$0.733805\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.15352i − 0.550147i
$$58$$ 0 0
$$59$$ −9.45090 −1.23040 −0.615201 0.788370i $$-0.710925\pi$$
−0.615201 + 0.788370i $$0.710925\pi$$
$$60$$ 0 0
$$61$$ 6.32250 0.809514 0.404757 0.914424i $$-0.367356\pi$$
0.404757 + 0.914424i $$0.367356\pi$$
$$62$$ 0 0
$$63$$ 1.29738i 0.163454i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.7100i 1.79711i 0.438860 + 0.898555i $$0.355382\pi$$
−0.438860 + 0.898555i $$0.644618\pi$$
$$68$$ 0 0
$$69$$ 0.523976 0.0630793
$$70$$ 0 0
$$71$$ 9.97487 1.18380 0.591900 0.806012i $$-0.298378\pi$$
0.591900 + 0.806012i $$0.298378\pi$$
$$72$$ 0 0
$$73$$ 6.10557i 0.714603i 0.933989 + 0.357301i $$0.116303\pi$$
−0.933989 + 0.357301i $$0.883697\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 0.798528i − 0.0910007i
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 6.60442 0.733824
$$82$$ 0 0
$$83$$ − 4.49885i − 0.493813i −0.969039 0.246906i $$-0.920586\pi$$
0.969039 0.246906i $$-0.0794141\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.15352i 0.123670i
$$88$$ 0 0
$$89$$ 3.59706 0.381287 0.190644 0.981659i $$-0.438943\pi$$
0.190644 + 0.981659i $$0.438943\pi$$
$$90$$ 0 0
$$91$$ 1.27455 0.133609
$$92$$ 0 0
$$93$$ 3.54910i 0.368025i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 6.08044i − 0.617375i −0.951163 0.308688i $$-0.900110\pi$$
0.951163 0.308688i $$-0.0998898\pi$$
$$98$$ 0 0
$$99$$ −4.57193 −0.459496
$$100$$ 0 0
$$101$$ 11.4509 1.13941 0.569703 0.821850i $$-0.307058\pi$$
0.569703 + 0.821850i $$0.307058\pi$$
$$102$$ 0 0
$$103$$ 2.32250i 0.228843i 0.993432 + 0.114422i $$0.0365015\pi$$
−0.993432 + 0.114422i $$0.963499\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 7.85384i − 0.759260i −0.925138 0.379630i $$-0.876051\pi$$
0.925138 0.379630i $$-0.123949\pi$$
$$108$$ 0 0
$$109$$ −13.9269 −1.33396 −0.666979 0.745077i $$-0.732413\pi$$
−0.666979 + 0.745077i $$0.732413\pi$$
$$110$$ 0 0
$$111$$ −2.09591 −0.198935
$$112$$ 0 0
$$113$$ 10.0000i 0.940721i 0.882474 + 0.470360i $$0.155876\pi$$
−0.882474 + 0.470360i $$0.844124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 7.29738i − 0.674643i
$$118$$ 0 0
$$119$$ −0.798528 −0.0732009
$$120$$ 0 0
$$121$$ −8.18601 −0.744182
$$122$$ 0 0
$$123$$ 3.13069i 0.282285i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 11.1033i − 0.985256i −0.870240 0.492628i $$-0.836036\pi$$
0.870240 0.492628i $$-0.163964\pi$$
$$128$$ 0 0
$$129$$ −0.211133 −0.0185893
$$130$$ 0 0
$$131$$ 7.65237 0.668591 0.334295 0.942468i $$-0.391502\pi$$
0.334295 + 0.942468i $$0.391502\pi$$
$$132$$ 0 0
$$133$$ − 3.77340i − 0.327195i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 21.3778i − 1.82643i −0.407478 0.913215i $$-0.633592\pi$$
0.407478 0.913215i $$-0.366408\pi$$
$$138$$ 0 0
$$139$$ 8.60442 0.729817 0.364909 0.931043i $$-0.381100\pi$$
0.364909 + 0.931043i $$0.381100\pi$$
$$140$$ 0 0
$$141$$ −0.942386 −0.0793632
$$142$$ 0 0
$$143$$ 4.49149i 0.375597i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 3.54910i 0.292725i
$$148$$ 0 0
$$149$$ −4.97487 −0.407558 −0.203779 0.979017i $$-0.565322\pi$$
−0.203779 + 0.979017i $$0.565322\pi$$
$$150$$ 0 0
$$151$$ −0.926921 −0.0754318 −0.0377159 0.999289i $$-0.512008\pi$$
−0.0377159 + 0.999289i $$0.512008\pi$$
$$152$$ 0 0
$$153$$ 4.57193i 0.369619i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.3527i 1.78394i 0.452096 + 0.891970i $$0.350677\pi$$
−0.452096 + 0.891970i $$0.649323\pi$$
$$158$$ 0 0
$$159$$ −5.66203 −0.449028
$$160$$ 0 0
$$161$$ 0.476024 0.0375159
$$162$$ 0 0
$$163$$ 7.93658i 0.621641i 0.950469 + 0.310821i $$0.100604\pi$$
−0.950469 + 0.310821i $$0.899396\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.8059i 0.990949i 0.868622 + 0.495475i $$0.165006\pi$$
−0.868622 + 0.495475i $$0.834994\pi$$
$$168$$ 0 0
$$169$$ 5.83102 0.448540
$$170$$ 0 0
$$171$$ −21.6044 −1.65213
$$172$$ 0 0
$$173$$ 24.6752i 1.87602i 0.346608 + 0.938010i $$0.387334\pi$$
−0.346608 + 0.938010i $$0.612666\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 4.95205i − 0.372219i
$$178$$ 0 0
$$179$$ 13.1033 0.979384 0.489692 0.871895i $$-0.337109\pi$$
0.489692 + 0.871895i $$0.337109\pi$$
$$180$$ 0 0
$$181$$ 9.52398 0.707912 0.353956 0.935262i $$-0.384836\pi$$
0.353956 + 0.935262i $$0.384836\pi$$
$$182$$ 0 0
$$183$$ 3.31284i 0.244892i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 2.81399i − 0.205780i
$$188$$ 0 0
$$189$$ −1.42807 −0.103877
$$190$$ 0 0
$$191$$ 5.85384 0.423569 0.211785 0.977316i $$-0.432072\pi$$
0.211785 + 0.977316i $$0.432072\pi$$
$$192$$ 0 0
$$193$$ − 4.41261i − 0.317626i −0.987309 0.158813i $$-0.949233\pi$$
0.987309 0.158813i $$-0.0507667\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 11.4258i 0.814053i 0.913416 + 0.407026i $$0.133434\pi$$
−0.913416 + 0.407026i $$0.866566\pi$$
$$198$$ 0 0
$$199$$ 0.307039 0.0217654 0.0108827 0.999941i $$-0.496536\pi$$
0.0108827 + 0.999941i $$0.496536\pi$$
$$200$$ 0 0
$$201$$ −7.70768 −0.543658
$$202$$ 0 0
$$203$$ 1.04795i 0.0735519i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.72545i − 0.189432i
$$208$$ 0 0
$$209$$ 13.2974 0.919799
$$210$$ 0 0
$$211$$ 17.3047 1.19131 0.595654 0.803241i $$-0.296893\pi$$
0.595654 + 0.803241i $$0.296893\pi$$
$$212$$ 0 0
$$213$$ 5.22660i 0.358121i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.22430i 0.218880i
$$218$$ 0 0
$$219$$ −3.19917 −0.216180
$$220$$ 0 0
$$221$$ 4.49149 0.302130
$$222$$ 0 0
$$223$$ 1.90409i 0.127508i 0.997966 + 0.0637538i $$0.0203072\pi$$
−0.997966 + 0.0637538i $$0.979693\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.3550i 1.15189i 0.817488 + 0.575946i $$0.195366\pi$$
−0.817488 + 0.575946i $$0.804634\pi$$
$$228$$ 0 0
$$229$$ 6.19181 0.409166 0.204583 0.978849i $$-0.434416\pi$$
0.204583 + 0.978849i $$0.434416\pi$$
$$230$$ 0 0
$$231$$ 0.418410 0.0275293
$$232$$ 0 0
$$233$$ 15.7985i 1.03500i 0.855684 + 0.517498i $$0.173137\pi$$
−0.855684 + 0.517498i $$0.826863\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.34763 −0.410594 −0.205297 0.978700i $$-0.565816\pi$$
−0.205297 + 0.978700i $$0.565816\pi$$
$$240$$ 0 0
$$241$$ 5.69296 0.366716 0.183358 0.983046i $$-0.441303\pi$$
0.183358 + 0.983046i $$0.441303\pi$$
$$242$$ 0 0
$$243$$ 12.4606i 0.799345i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 21.2243i 1.35047i
$$248$$ 0 0
$$249$$ 2.35729 0.149387
$$250$$ 0 0
$$251$$ −28.9246 −1.82571 −0.912853 0.408288i $$-0.866126\pi$$
−0.912853 + 0.408288i $$0.866126\pi$$
$$252$$ 0 0
$$253$$ 1.67750i 0.105463i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 14.5085i − 0.905016i −0.891760 0.452508i $$-0.850529\pi$$
0.891760 0.452508i $$-0.149471\pi$$
$$258$$ 0 0
$$259$$ −1.90409 −0.118315
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 15.7734i 0.972630i 0.873784 + 0.486315i $$0.161659\pi$$
−0.873784 + 0.486315i $$0.838341\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 1.88477i 0.115346i
$$268$$ 0 0
$$269$$ 6.20147 0.378110 0.189055 0.981966i $$-0.439457\pi$$
0.189055 + 0.981966i $$0.439457\pi$$
$$270$$ 0 0
$$271$$ −25.5696 −1.55324 −0.776622 0.629967i $$-0.783069\pi$$
−0.776622 + 0.629967i $$0.783069\pi$$
$$272$$ 0 0
$$273$$ 0.667835i 0.0404192i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.03829i 0.422890i 0.977390 + 0.211445i $$0.0678169\pi$$
−0.977390 + 0.211445i $$0.932183\pi$$
$$278$$ 0 0
$$279$$ 18.4606 1.10520
$$280$$ 0 0
$$281$$ 29.0627 1.73373 0.866867 0.498540i $$-0.166130\pi$$
0.866867 + 0.498540i $$0.166130\pi$$
$$282$$ 0 0
$$283$$ 18.6141i 1.10649i 0.833018 + 0.553246i $$0.186611\pi$$
−0.833018 + 0.553246i $$0.813389\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.84418i 0.167887i
$$288$$ 0 0
$$289$$ 14.1860 0.834471
$$290$$ 0 0
$$291$$ 3.18601 0.186767
$$292$$ 0 0
$$293$$ − 23.4006i − 1.36708i −0.729913 0.683540i $$-0.760439\pi$$
0.729913 0.683540i $$-0.239561\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 5.03249i − 0.292015i
$$298$$ 0 0
$$299$$ −2.67750 −0.154844
$$300$$ 0 0
$$301$$ −0.191811 −0.0110558
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1.53134i − 0.0873981i −0.999045 0.0436990i $$-0.986086\pi$$
0.999045 0.0436990i $$-0.0139143\pi$$
$$308$$ 0 0
$$309$$ −1.21694 −0.0692291
$$310$$ 0 0
$$311$$ −7.55646 −0.428488 −0.214244 0.976780i $$-0.568729\pi$$
−0.214244 + 0.976780i $$0.568729\pi$$
$$312$$ 0 0
$$313$$ 2.32987i 0.131692i 0.997830 + 0.0658459i $$0.0209746\pi$$
−0.997830 + 0.0658459i $$0.979025\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 15.3778i − 0.863704i −0.901944 0.431852i $$-0.857860\pi$$
0.901944 0.431852i $$-0.142140\pi$$
$$318$$ 0 0
$$319$$ −3.69296 −0.206766
$$320$$ 0 0
$$321$$ 4.11523 0.229690
$$322$$ 0 0
$$323$$ − 13.2974i − 0.739886i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 7.29738i − 0.403546i
$$328$$ 0 0
$$329$$ −0.856142 −0.0472006
$$330$$ 0 0
$$331$$ 22.9115 1.25933 0.629664 0.776868i $$-0.283193\pi$$
0.629664 + 0.776868i $$0.283193\pi$$
$$332$$ 0 0
$$333$$ 10.9018i 0.597415i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.5410i 1.44578i 0.690963 + 0.722890i $$0.257187\pi$$
−0.690963 + 0.722890i $$0.742813\pi$$
$$338$$ 0 0
$$339$$ −5.23976 −0.284585
$$340$$ 0 0
$$341$$ −11.3624 −0.615306
$$342$$ 0 0
$$343$$ 6.55646i 0.354016i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 3.27455i − 0.175787i −0.996130 0.0878936i $$-0.971986\pi$$
0.996130 0.0878936i $$-0.0280135\pi$$
$$348$$ 0 0
$$349$$ −32.0553 −1.71588 −0.857941 0.513749i $$-0.828256\pi$$
−0.857941 + 0.513749i $$0.828256\pi$$
$$350$$ 0 0
$$351$$ 8.03249 0.428742
$$352$$ 0 0
$$353$$ − 11.1535i − 0.593642i −0.954933 0.296821i $$-0.904074\pi$$
0.954933 0.296821i $$-0.0959265\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 0.418410i − 0.0221446i
$$358$$ 0 0
$$359$$ −22.4029 −1.18238 −0.591191 0.806532i $$-0.701342\pi$$
−0.591191 + 0.806532i $$0.701342\pi$$
$$360$$ 0 0
$$361$$ 43.8361 2.30716
$$362$$ 0 0
$$363$$ − 4.28927i − 0.225129i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 14.2877i 0.745813i 0.927869 + 0.372906i $$0.121639\pi$$
−0.927869 + 0.372906i $$0.878361\pi$$
$$368$$ 0 0
$$369$$ 16.2842 0.847722
$$370$$ 0 0
$$371$$ −5.14386 −0.267056
$$372$$ 0 0
$$373$$ 1.23976i 0.0641925i 0.999485 + 0.0320963i $$0.0102183\pi$$
−0.999485 + 0.0320963i $$0.989782\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 5.89443i − 0.303579i
$$378$$ 0 0
$$379$$ −4.38748 −0.225370 −0.112685 0.993631i $$-0.535945\pi$$
−0.112685 + 0.993631i $$0.535945\pi$$
$$380$$ 0 0
$$381$$ 5.81785 0.298057
$$382$$ 0 0
$$383$$ − 18.4989i − 0.945247i −0.881264 0.472624i $$-0.843307\pi$$
0.881264 0.472624i $$-0.156693\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.09821i 0.0558249i
$$388$$ 0 0
$$389$$ −21.6775 −1.09909 −0.549546 0.835463i $$-0.685199\pi$$
−0.549546 + 0.835463i $$0.685199\pi$$
$$390$$ 0 0
$$391$$ 1.67750 0.0848346
$$392$$ 0 0
$$393$$ 4.00966i 0.202261i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 20.6849i − 1.03814i −0.854731 0.519072i $$-0.826278\pi$$
0.854731 0.519072i $$-0.173722\pi$$
$$398$$ 0 0
$$399$$ 1.97717 0.0989825
$$400$$ 0 0
$$401$$ −28.5638 −1.42641 −0.713205 0.700956i $$-0.752757\pi$$
−0.713205 + 0.700956i $$0.752757\pi$$
$$402$$ 0 0
$$403$$ − 18.1358i − 0.903406i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 6.70998i − 0.332602i
$$408$$ 0 0
$$409$$ −16.7734 −0.829391 −0.414696 0.909960i $$-0.636112\pi$$
−0.414696 + 0.909960i $$0.636112\pi$$
$$410$$ 0 0
$$411$$ 11.2015 0.552528
$$412$$ 0 0
$$413$$ − 4.49885i − 0.221374i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.50851i 0.220783i
$$418$$ 0 0
$$419$$ 4.59476 0.224469 0.112234 0.993682i $$-0.464199\pi$$
0.112234 + 0.993682i $$0.464199\pi$$
$$420$$ 0 0
$$421$$ 31.8310 1.55135 0.775674 0.631133i $$-0.217410\pi$$
0.775674 + 0.631133i $$0.217410\pi$$
$$422$$ 0 0
$$423$$ 4.90179i 0.238333i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.00966i 0.145648i
$$428$$ 0 0
$$429$$ −2.35343 −0.113625
$$430$$ 0 0
$$431$$ −16.0650 −0.773823 −0.386911 0.922117i $$-0.626458\pi$$
−0.386911 + 0.922117i $$0.626458\pi$$
$$432$$ 0 0
$$433$$ 7.68486i 0.369311i 0.982803 + 0.184655i $$0.0591169\pi$$
−0.982803 + 0.184655i $$0.940883\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.92692i 0.379196i
$$438$$ 0 0
$$439$$ −15.9822 −0.762790 −0.381395 0.924412i $$-0.624556\pi$$
−0.381395 + 0.924412i $$0.624556\pi$$
$$440$$ 0 0
$$441$$ 18.4606 0.879074
$$442$$ 0 0
$$443$$ 3.57929i 0.170057i 0.996379 + 0.0850286i $$0.0270982\pi$$
−0.996379 + 0.0850286i $$0.972902\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 2.60672i − 0.123293i
$$448$$ 0 0
$$449$$ 28.5217 1.34602 0.673011 0.739633i $$-0.265001\pi$$
0.673011 + 0.739633i $$0.265001\pi$$
$$450$$ 0 0
$$451$$ −10.0228 −0.471956
$$452$$ 0 0
$$453$$ − 0.485685i − 0.0228195i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.49885i − 0.210447i −0.994449 0.105224i $$-0.966444\pi$$
0.994449 0.105224i $$-0.0335559\pi$$
$$458$$ 0 0
$$459$$ −5.03249 −0.234896
$$460$$ 0 0
$$461$$ 2.29738 0.107000 0.0534998 0.998568i $$-0.482962\pi$$
0.0534998 + 0.998568i $$0.482962\pi$$
$$462$$ 0 0
$$463$$ 33.4966i 1.55672i 0.627820 + 0.778358i $$0.283947\pi$$
−0.627820 + 0.778358i $$0.716053\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 5.23976i − 0.242467i −0.992624 0.121234i $$-0.961315\pi$$
0.992624 0.121234i $$-0.0386850\pi$$
$$468$$ 0 0
$$469$$ −7.00230 −0.323336
$$470$$ 0 0
$$471$$ −11.7123 −0.539674
$$472$$ 0 0
$$473$$ − 0.675938i − 0.0310797i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 29.4509i 1.34846i
$$478$$ 0 0
$$479$$ 2.40294 0.109793 0.0548967 0.998492i $$-0.482517\pi$$
0.0548967 + 0.998492i $$0.482517\pi$$
$$480$$ 0 0
$$481$$ 10.7100 0.488333
$$482$$ 0 0
$$483$$ 0.249425i 0.0113492i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 25.8944i 1.17339i 0.809808 + 0.586694i $$0.199571\pi$$
−0.809808 + 0.586694i $$0.800429\pi$$
$$488$$ 0 0
$$489$$ −4.15858 −0.188058
$$490$$ 0 0
$$491$$ −0.489189 −0.0220768 −0.0110384 0.999939i $$-0.503514\pi$$
−0.0110384 + 0.999939i $$0.503514\pi$$
$$492$$ 0 0
$$493$$ 3.69296i 0.166323i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.74828i 0.212989i
$$498$$ 0 0
$$499$$ 38.4583 1.72163 0.860814 0.508920i $$-0.169955\pi$$
0.860814 + 0.508920i $$0.169955\pi$$
$$500$$ 0 0
$$501$$ −6.70998 −0.299780
$$502$$ 0 0
$$503$$ − 37.8960i − 1.68970i −0.535004 0.844849i $$-0.679690\pi$$
0.535004 0.844849i $$-0.320310\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.05531i 0.135691i
$$508$$ 0 0
$$509$$ −8.70032 −0.385635 −0.192818 0.981235i $$-0.561763\pi$$
−0.192818 + 0.981235i $$0.561763\pi$$
$$510$$ 0 0
$$511$$ −2.90639 −0.128571
$$512$$ 0 0
$$513$$ − 23.7808i − 1.04995i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 3.01702i − 0.132689i
$$518$$ 0 0
$$519$$ −12.9292 −0.567530
$$520$$ 0 0
$$521$$ 25.1129 1.10022 0.550109 0.835093i $$-0.314586\pi$$
0.550109 + 0.835093i $$0.314586\pi$$
$$522$$ 0 0
$$523$$ − 12.7100i − 0.555769i −0.960615 0.277884i $$-0.910367\pi$$
0.960615 0.277884i $$-0.0896332\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11.3624i 0.494952i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −25.7579 −1.11780
$$532$$ 0 0
$$533$$ − 15.9977i − 0.692937i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.86580i 0.296281i
$$538$$ 0 0
$$539$$ −11.3624 −0.489411
$$540$$ 0 0
$$541$$ −0.251725 −0.0108225 −0.00541124 0.999985i $$-0.501722\pi$$
−0.00541124 + 0.999985i $$0.501722\pi$$
$$542$$ 0 0
$$543$$ 4.99034i 0.214156i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 24.3395i 1.04068i 0.853958 + 0.520342i $$0.174195\pi$$
−0.853958 + 0.520342i $$0.825805\pi$$
$$548$$ 0 0
$$549$$ 17.2317 0.735429
$$550$$ 0 0
$$551$$ −17.4509 −0.743433
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 23.4966i − 0.995581i −0.867297 0.497790i $$-0.834145\pi$$
0.867297 0.497790i $$-0.165855\pi$$
$$558$$ 0 0
$$559$$ 1.07888 0.0456319
$$560$$ 0 0
$$561$$ 1.47447 0.0622520
$$562$$ 0 0
$$563$$ 19.5159i 0.822496i 0.911523 + 0.411248i $$0.134907\pi$$
−0.911523 + 0.411248i $$0.865093\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3.14386i 0.132030i
$$568$$ 0 0
$$569$$ 8.51817 0.357100 0.178550 0.983931i $$-0.442859\pi$$
0.178550 + 0.983931i $$0.442859\pi$$
$$570$$ 0 0
$$571$$ −27.4811 −1.15005 −0.575024 0.818137i $$-0.695007\pi$$
−0.575024 + 0.818137i $$0.695007\pi$$
$$572$$ 0 0
$$573$$ 3.06728i 0.128137i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 26.1512i − 1.08869i −0.838862 0.544345i $$-0.816778\pi$$
0.838862 0.544345i $$-0.183222\pi$$
$$578$$ 0 0
$$579$$ 2.31210 0.0960877
$$580$$ 0 0
$$581$$ 2.14156 0.0888468
$$582$$ 0 0
$$583$$ − 18.1268i − 0.750737i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 34.3276i − 1.41685i −0.705786 0.708425i $$-0.749406\pi$$
0.705786 0.708425i $$-0.250594\pi$$
$$588$$ 0 0
$$589$$ −53.6922 −2.21235
$$590$$ 0 0
$$591$$ −5.98683 −0.246265
$$592$$ 0 0
$$593$$ − 15.9497i − 0.654978i −0.944855 0.327489i $$-0.893798\pi$$
0.944855 0.327489i $$-0.106202\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0.160881i 0.00658443i
$$598$$ 0 0
$$599$$ −40.5940 −1.65863 −0.829313 0.558784i $$-0.811268\pi$$
−0.829313 + 0.558784i $$0.811268\pi$$
$$600$$ 0 0
$$601$$ −29.0302 −1.18417 −0.592083 0.805877i $$-0.701694\pi$$
−0.592083 + 0.805877i $$0.701694\pi$$
$$602$$ 0 0
$$603$$ 40.0913i 1.63264i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.9977i 0.689915i 0.938618 + 0.344958i $$0.112107\pi$$
−0.938618 + 0.344958i $$0.887893\pi$$
$$608$$ 0 0
$$609$$ −0.549103 −0.0222508
$$610$$ 0 0
$$611$$ 4.81555 0.194816
$$612$$ 0 0
$$613$$ − 30.3024i − 1.22390i −0.790895 0.611952i $$-0.790385\pi$$
0.790895 0.611952i $$-0.209615\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.13069i − 0.246812i −0.992356 0.123406i $$-0.960618\pi$$
0.992356 0.123406i $$-0.0393818\pi$$
$$618$$ 0 0
$$619$$ 33.7305 1.35574 0.677872 0.735180i $$-0.262902\pi$$
0.677872 + 0.735180i $$0.262902\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 1.71228i 0.0686012i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 6.96751i 0.278256i
$$628$$ 0 0
$$629$$ −6.70998 −0.267545
$$630$$ 0 0
$$631$$ 4.85614 0.193320 0.0966600 0.995317i $$-0.469184\pi$$
0.0966600 + 0.995317i $$0.469184\pi$$
$$632$$ 0 0
$$633$$ 9.06728i 0.360392i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 18.1358i − 0.718565i
$$638$$ 0 0
$$639$$ 27.1860 1.07546
$$640$$ 0 0
$$641$$ 7.78887 0.307642 0.153821 0.988099i $$-0.450842\pi$$
0.153821 + 0.988099i $$0.450842\pi$$
$$642$$ 0 0
$$643$$ − 43.6427i − 1.72110i −0.509366 0.860550i $$-0.670120\pi$$
0.509366 0.860550i $$-0.329880\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 15.4606i − 0.607817i −0.952701 0.303909i $$-0.901708\pi$$
0.952701 0.303909i $$-0.0982918\pi$$
$$648$$ 0 0
$$649$$ 15.8538 0.622318
$$650$$ 0 0
$$651$$ −1.68946 −0.0662150
$$652$$ 0 0
$$653$$ 9.70613i 0.379830i 0.981801 + 0.189915i $$0.0608213\pi$$
−0.981801 + 0.189915i $$0.939179\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 16.6404i 0.649204i
$$658$$ 0 0
$$659$$ −50.7100 −1.97538 −0.987690 0.156422i $$-0.950004\pi$$
−0.987690 + 0.156422i $$0.950004\pi$$
$$660$$ 0 0
$$661$$ 33.0132 1.28406 0.642032 0.766678i $$-0.278092\pi$$
0.642032 + 0.766678i $$0.278092\pi$$
$$662$$ 0 0
$$663$$ 2.35343i 0.0913998i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 2.20147i − 0.0852413i
$$668$$ 0 0
$$669$$ −0.997701 −0.0385733
$$670$$ 0 0
$$671$$ −10.6060 −0.409439
$$672$$ 0 0
$$673$$ − 19.2494i − 0.742011i −0.928631 0.371005i $$-0.879013\pi$$
0.928631 0.371005i $$-0.120987\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 25.1941i − 0.968288i −0.874988 0.484144i $$-0.839131\pi$$
0.874988 0.484144i $$-0.160869\pi$$
$$678$$ 0 0
$$679$$ 2.89443 0.111078
$$680$$ 0 0
$$681$$ −9.09361 −0.348468
$$682$$ 0 0
$$683$$ − 9.48339i − 0.362872i −0.983403 0.181436i $$-0.941926\pi$$
0.983403 0.181436i $$-0.0580745\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3.24436i 0.123780i
$$688$$ 0 0
$$689$$ 28.9327 1.10225
$$690$$ 0 0
$$691$$ −23.8538 −0.907443 −0.453721 0.891144i $$-0.649904\pi$$
−0.453721 + 0.891144i $$0.649904\pi$$
$$692$$ 0 0
$$693$$ − 2.17635i − 0.0826726i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10.0228i 0.379642i
$$698$$ 0 0
$$699$$ −8.27806 −0.313105
$$700$$ 0 0
$$701$$ 19.8767 0.750731 0.375366 0.926877i $$-0.377517\pi$$
0.375366 + 0.926877i $$0.377517\pi$$
$$702$$ 0 0
$$703$$ − 31.7077i − 1.19588i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 5.45090i 0.205002i
$$708$$ 0 0
$$709$$ 6.72545 0.252580 0.126290 0.991993i $$-0.459693\pi$$
0.126290 + 0.991993i $$0.459693\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 6.77340i − 0.253666i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 3.32601i − 0.124212i
$$718$$ 0 0
$$719$$ 38.2294 1.42571 0.712857 0.701309i $$-0.247401\pi$$
0.712857 + 0.701309i $$0.247401\pi$$
$$720$$ 0 0
$$721$$ −1.10557 −0.0411735
$$722$$ 0 0
$$723$$ 2.98298i 0.110938i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 3.72315i − 0.138084i −0.997614 0.0690420i $$-0.978006\pi$$
0.997614 0.0690420i $$-0.0219942\pi$$
$$728$$ 0 0
$$729$$ 13.2842 0.492008
$$730$$ 0 0
$$731$$ −0.675938 −0.0250005
$$732$$ 0 0
$$733$$ 8.49885i 0.313912i 0.987606 + 0.156956i $$0.0501681\pi$$
−0.987606 + 0.156956i $$0.949832\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 24.6759i − 0.908950i
$$738$$ 0 0
$$739$$ 0.251725 0.00925984 0.00462992 0.999989i $$-0.498526\pi$$
0.00462992 + 0.999989i $$0.498526\pi$$
$$740$$ 0 0
$$741$$ −11.1210 −0.408541
$$742$$ 0 0
$$743$$ 29.7305i 1.09071i 0.838206 + 0.545353i $$0.183605\pi$$
−0.838206 + 0.545353i $$0.816395\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 12.2614i − 0.448621i
$$748$$ 0 0
$$749$$ 3.73861 0.136606
$$750$$ 0 0
$$751$$ 10.6597 0.388979 0.194490 0.980905i $$-0.437695\pi$$
0.194490 + 0.980905i $$0.437695\pi$$
$$752$$ 0 0
$$753$$ − 15.1558i − 0.552309i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 16.1918i − 0.588501i −0.955728 0.294251i $$-0.904930\pi$$
0.955728 0.294251i $$-0.0950701\pi$$
$$758$$ 0 0
$$759$$ −0.878968 −0.0319045
$$760$$ 0 0
$$761$$ −46.9343 −1.70137 −0.850683 0.525679i $$-0.823811\pi$$
−0.850683 + 0.525679i $$0.823811\pi$$
$$762$$ 0 0
$$763$$ − 6.62954i − 0.240006i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 25.3047i 0.913701i
$$768$$ 0 0
$$769$$ 35.9954 1.29803 0.649014 0.760777i $$-0.275182\pi$$
0.649014 + 0.760777i $$0.275182\pi$$
$$770$$ 0 0
$$771$$ 7.60212 0.273784
$$772$$ 0 0
$$773$$ 21.8229i 0.784916i 0.919770 + 0.392458i $$0.128375\pi$$
−0.919770 + 0.392458i $$0.871625\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 0.997701i − 0.0357923i
$$778$$ 0 0
$$779$$ −47.3624 −1.69693
$$780$$ 0 0
$$781$$ −16.7328 −0.598747
$$782$$ 0 0
$$783$$ 6.60442i 0.236023i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 7.16318i − 0.255340i −0.991817 0.127670i $$-0.959250\pi$$
0.991817 0.127670i $$-0.0407498\pi$$
$$788$$ 0 0
$$789$$ −8.26489 −0.294238
$$790$$ 0 0
$$791$$ −4.76024 −0.169255
$$792$$ 0 0
$$793$$ − 16.9285i − 0.601148i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 48.0604i − 1.70239i −0.524853 0.851193i $$-0.675880\pi$$
0.524853 0.851193i $$-0.324120\pi$$
$$798$$ 0 0
$$799$$ −3.01702 −0.106735
$$800$$ 0 0
$$801$$ 9.80359 0.346393
$$802$$ 0 0
$$803$$ − 10.2421i − 0.361435i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.24943i 0.114385i
$$808$$ 0 0
$$809$$ −0.0611183 −0.00214880 −0.00107440 0.999999i $$-0.500342\pi$$
−0.00107440 + 0.999999i $$0.500342\pi$$
$$810$$ 0 0
$$811$$ 25.9092 0.909794 0.454897 0.890544i $$-0.349676\pi$$
0.454897 + 0.890544i $$0.349676\pi$$
$$812$$ 0 0
$$813$$ − 13.3979i − 0.469884i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 3.19411i − 0.111748i
$$818$$ 0 0
$$819$$ 3.47372 0.121382
$$820$$ 0 0
$$821$$ −21.0936 −0.736172 −0.368086 0.929792i $$-0.619987\pi$$
−0.368086 + 0.929792i $$0.619987\pi$$
$$822$$ 0 0
$$823$$ 37.2641i 1.29895i 0.760384 + 0.649473i $$0.225011\pi$$
−0.760384 + 0.649473i $$0.774989\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 4.45320i − 0.154853i −0.996998 0.0774264i $$-0.975330\pi$$
0.996998 0.0774264i $$-0.0246703\pi$$
$$828$$ 0 0
$$829$$ 38.1300 1.32431 0.662154 0.749368i $$-0.269642\pi$$
0.662154 + 0.749368i $$0.269642\pi$$
$$830$$ 0 0
$$831$$ −3.68790 −0.127932
$$832$$ 0 0
$$833$$ 11.3624i 0.393682i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 20.3202i 0.702369i
$$838$$ 0 0
$$839$$ 7.15858 0.247142 0.123571 0.992336i $$-0.460565\pi$$
0.123571 + 0.992336i $$0.460565\pi$$
$$840$$ 0 0
$$841$$ −24.1535 −0.832880
$$842$$ 0 0
$$843$$ 15.2282i 0.524486i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3.89673i − 0.133893i
$$848$$ 0 0
$$849$$ −9.75334 −0.334734
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ 49.6199i 1.69895i 0.527627 + 0.849476i $$0.323082\pi$$
−0.527627 + 0.849476i $$0.676918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 44.0360i − 1.50424i −0.659026 0.752120i $$-0.729031\pi$$
0.659026 0.752120i $$-0.270969\pi$$
$$858$$ 0 0
$$859$$ 49.5062 1.68913 0.844565 0.535453i $$-0.179859\pi$$
0.844565 + 0.535453i $$0.179859\pi$$
$$860$$ 0 0
$$861$$ −1.49028 −0.0507887
$$862$$ 0 0
$$863$$ − 20.7962i − 0.707912i −0.935262 0.353956i $$-0.884836\pi$$
0.935262 0.353956i $$-0.115164\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 7.43313i 0.252442i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 39.3859 1.33454
$$872$$ 0 0
$$873$$ − 16.5719i − 0.560875i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 5.42071i − 0.183044i −0.995803 0.0915222i $$-0.970827\pi$$
0.995803 0.0915222i $$-0.0291732\pi$$
$$878$$ 0 0
$$879$$ 12.2614 0.413566
$$880$$ 0 0
$$881$$ −45.8036 −1.54316 −0.771581 0.636131i $$-0.780534\pi$$
−0.771581 + 0.636131i $$0.780534\pi$$
$$882$$ 0 0
$$883$$ − 57.1860i − 1.92446i −0.272234 0.962231i $$-0.587762\pi$$
0.272234 0.962231i $$-0.412238\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 15.5371i − 0.521686i −0.965381 0.260843i $$-0.915999\pi$$
0.965381 0.260843i $$-0.0840005\pi$$
$$888$$ 0 0
$$889$$ 5.28542 0.177267
$$890$$ 0 0
$$891$$ −11.0789 −0.371157
$$892$$ 0 0
$$893$$ − 14.2568i − 0.477085i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 1.40294i − 0.0468430i
$$898$$ 0 0
$$899$$ 14.9115 0.497325
$$900$$ 0 0
$$901$$ −18.1268 −0.603892
$$902$$ 0 0
$$903$$ − 0.100505i − 0.00334458i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 32.8515i − 1.09082i −0.838170 0.545409i $$-0.816374\pi$$
0.838170 0.545409i $$-0.183626\pi$$
$$908$$ 0 0
$$909$$ 31.2088 1.03513
$$910$$ 0 0
$$911$$ 8.95205 0.296595 0.148297 0.988943i $$-0.452621\pi$$
0.148297 + 0.988943i $$0.452621\pi$$
$$912$$ 0 0
$$913$$ 7.54680i 0.249763i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 3.64271i 0.120293i
$$918$$ 0 0
$$919$$ 49.8229 1.64351 0.821753 0.569844i $$-0.192996\pi$$
0.821753 + 0.569844i $$0.192996\pi$$
$$920$$ 0 0
$$921$$ 0.802385 0.0264395
$$922$$ 0 0
$$923$$ − 26.7077i − 0.879094i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 6.32987i 0.207900i
$$928$$ 0 0
$$929$$ −25.0723 −0.822597 −0.411298 0.911501i $$-0.634925\pi$$
−0.411298 + 0.911501i $$0.634925\pi$$
$$930$$ 0 0
$$931$$ −53.6922 −1.75969
$$932$$ 0 0
$$933$$ − 3.95941i − 0.129625i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21.3322i 0.696891i 0.937329 + 0.348446i $$0.113290\pi$$
−0.937329 + 0.348446i $$0.886710\pi$$
$$938$$ 0 0
$$939$$ −1.22079 −0.0398391
$$940$$ 0 0
$$941$$ 3.93428 0.128254 0.0641270 0.997942i $$-0.479574\pi$$
0.0641270 + 0.997942i $$0.479574\pi$$
$$942$$ 0 0
$$943$$ − 5.97487i − 0.194569i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 48.6420i 1.58065i 0.612687 + 0.790326i $$0.290089\pi$$
−0.612687 + 0.790326i $$0.709911\pi$$
$$948$$ 0 0
$$949$$ 16.3476 0.530667
$$950$$ 0 0
$$951$$ 8.05761 0.261286
$$952$$ 0 0
$$953$$ 15.5623i 0.504111i 0.967713 + 0.252056i $$0.0811066\pi$$
−0.967713 + 0.252056i $$0.918893\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 1.93502i − 0.0625505i
$$958$$ 0 0
$$959$$ 10.1763 0.328611
$$960$$ 0 0
$$961$$ 14.8790 0.479967
$$962$$ 0 0
$$963$$ − 21.4052i − 0.689774i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 6.80083i 0.218700i 0.994003 + 0.109350i $$0.0348769\pi$$
−0.994003 + 0.109350i $$0.965123\pi$$
$$968$$ 0 0
$$969$$ 6.96751 0.223829
$$970$$ 0 0
$$971$$ 6.46406 0.207442 0.103721 0.994606i $$-0.466925\pi$$
0.103721 + 0.994606i $$0.466925\pi$$
$$972$$ 0 0
$$973$$ 4.09591i 0.131309i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 9.18601i 0.293886i 0.989145 + 0.146943i $$0.0469435\pi$$
−0.989145 + 0.146943i $$0.953057\pi$$
$$978$$ 0 0
$$979$$ −6.03405 −0.192849
$$980$$ 0 0
$$981$$ −37.9571 −1.21188
$$982$$ 0 0
$$983$$ − 15.3705i − 0.490241i −0.969493 0.245121i $$-0.921172\pi$$
0.969493 0.245121i $$-0.0788276\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 0.448598i − 0.0142790i
$$988$$ 0 0
$$989$$ 0.402945 0.0128129
$$990$$ 0 0
$$991$$ −52.0109 −1.65218 −0.826090 0.563538i $$-0.809440\pi$$
−0.826090 + 0.563538i $$0.809440\pi$$
$$992$$ 0 0
$$993$$ 12.0051i 0.380969i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 15.7123i 0.497613i 0.968553 + 0.248807i $$0.0800383\pi$$
−0.968553 + 0.248807i $$0.919962\pi$$
$$998$$ 0 0
$$999$$ −12.0000 −0.379663
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.r.4049.4 6
5.2 odd 4 4600.2.a.y.1.2 3
5.3 odd 4 920.2.a.g.1.2 3
5.4 even 2 inner 4600.2.e.r.4049.3 6
15.8 even 4 8280.2.a.bo.1.2 3
20.3 even 4 1840.2.a.t.1.2 3
20.7 even 4 9200.2.a.cd.1.2 3
40.3 even 4 7360.2.a.ca.1.2 3
40.13 odd 4 7360.2.a.cb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.2 3 5.3 odd 4
1840.2.a.t.1.2 3 20.3 even 4
4600.2.a.y.1.2 3 5.2 odd 4
4600.2.e.r.4049.3 6 5.4 even 2 inner
4600.2.e.r.4049.4 6 1.1 even 1 trivial
7360.2.a.ca.1.2 3 40.3 even 4
7360.2.a.cb.1.2 3 40.13 odd 4
8280.2.a.bo.1.2 3 15.8 even 4
9200.2.a.cd.1.2 3 20.7 even 4