# Properties

 Label 4800.2.f.a.3649.2 Level $4800$ Weight $2$ Character 4800.3649 Analytic conductor $38.328$ 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] = [4800,2,Mod(3649,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (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: $$38.3281929702$$ 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 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4800.3649 Dual form 4800.2.f.a.3649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -5.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} +3.00000i q^{13} +2.00000i q^{17} -1.00000 q^{19} +5.00000 q^{21} +2.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} -3.00000 q^{31} -6.00000i q^{33} -6.00000i q^{37} -3.00000 q^{39} +4.00000 q^{41} +11.0000i q^{43} -10.0000i q^{47} -18.0000 q^{49} -2.00000 q^{51} +8.00000i q^{53} -1.00000i q^{57} +6.00000 q^{59} -3.00000 q^{61} +5.00000i q^{63} +1.00000i q^{67} -2.00000 q^{69} +12.0000 q^{71} +10.0000i q^{73} +30.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +6.00000i q^{87} +16.0000 q^{89} +15.0000 q^{91} -3.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 12 q^{11} - 2 q^{19} + 10 q^{21} + 12 q^{29} - 6 q^{31} - 6 q^{39} + 8 q^{41} - 36 q^{49} - 4 q^{51} + 12 q^{59} - 6 q^{61} - 4 q^{69} + 24 q^{71} - 16 q^{79} + 2 q^{81} + 32 q^{89} + 30 q^{91} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 12 * q^11 - 2 * q^19 + 10 * q^21 + 12 * q^29 - 6 * q^31 - 6 * q^39 + 8 * q^41 - 36 * q^49 - 4 * q^51 + 12 * q^59 - 6 * q^61 - 4 * q^69 + 24 * q^71 - 16 * q^79 + 2 * q^81 + 32 * q^89 + 30 * q^91 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 5.00000i − 1.88982i −0.327327 0.944911i $$-0.606148\pi$$
0.327327 0.944911i $$-0.393852\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ 3.00000i 0.832050i 0.909353 + 0.416025i $$0.136577\pi$$
−0.909353 + 0.416025i $$0.863423\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ 0 0
$$23$$ 2.00000i 0.417029i 0.978019 + 0.208514i $$0.0668628\pi$$
−0.978019 + 0.208514i $$0.933137\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ − 6.00000i − 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 11.0000i 1.67748i 0.544529 + 0.838742i $$0.316708\pi$$
−0.544529 + 0.838742i $$0.683292\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ 0 0
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.00000i − 0.132453i
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 0 0
$$63$$ 5.00000i 0.629941i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.00000i 0.122169i 0.998133 + 0.0610847i $$0.0194560\pi$$
−0.998133 + 0.0610847i $$0.980544\pi$$
$$68$$ 0 0
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 30.0000i 3.41882i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 15.0000 1.57243
$$92$$ 0 0
$$93$$ − 3.00000i − 0.311086i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.00000i 0.710742i 0.934725 + 0.355371i $$0.115646\pi$$
−0.934725 + 0.355371i $$0.884354\pi$$
$$98$$ 0 0
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 3.00000i − 0.277350i
$$118$$ 0 0
$$119$$ 10.0000 0.916698
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 0 0
$$123$$ 4.00000i 0.360668i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ −11.0000 −0.968496
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ 5.00000i 0.433555i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 0 0
$$143$$ − 18.0000i − 1.50524i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 18.0000i − 1.48461i
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −9.00000 −0.732410 −0.366205 0.930534i $$-0.619343\pi$$
−0.366205 + 0.930534i $$0.619343\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.00000i 0.558661i 0.960195 + 0.279330i $$0.0901125\pi$$
−0.960195 + 0.279330i $$0.909888\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 10.0000 0.788110
$$162$$ 0 0
$$163$$ − 7.00000i − 0.548282i −0.961689 0.274141i $$-0.911606\pi$$
0.961689 0.274141i $$-0.0883936\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ 4.00000 0.307692
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000i 0.450988i
$$178$$ 0 0
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ 19.0000 1.41226 0.706129 0.708083i $$-0.250440\pi$$
0.706129 + 0.708083i $$0.250440\pi$$
$$182$$ 0 0
$$183$$ − 3.00000i − 0.221766i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 12.0000i − 0.877527i
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 0 0
$$193$$ 3.00000i 0.215945i 0.994154 + 0.107972i $$0.0344358\pi$$
−0.994154 + 0.107972i $$0.965564\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 0 0
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ 0 0
$$201$$ −1.00000 −0.0705346
$$202$$ 0 0
$$203$$ − 30.0000i − 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 2.00000i − 0.139010i
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 0 0
$$213$$ 12.0000i 0.822226i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 15.0000i 1.01827i
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ − 3.00000i − 0.200895i −0.994942 0.100447i $$-0.967973\pi$$
0.994942 0.100447i $$-0.0320274\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −3.00000 −0.198246 −0.0991228 0.995075i $$-0.531604\pi$$
−0.0991228 + 0.995075i $$0.531604\pi$$
$$230$$ 0 0
$$231$$ −30.0000 −1.97386
$$232$$ 0 0
$$233$$ 20.0000i 1.31024i 0.755523 + 0.655122i $$0.227383\pi$$
−0.755523 + 0.655122i $$0.772617\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3.00000i − 0.190885i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ − 12.0000i − 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 12.0000i 0.739952i 0.929041 + 0.369976i $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000i 0.979184i
$$268$$ 0 0
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ 15.0000i 0.907841i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.00000i 0.0600842i 0.999549 + 0.0300421i $$0.00956413\pi$$
−0.999549 + 0.0300421i $$0.990436\pi$$
$$278$$ 0 0
$$279$$ 3.00000 0.179605
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 13.0000i 0.772770i 0.922338 + 0.386385i $$0.126276\pi$$
−0.922338 + 0.386385i $$0.873724\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 20.0000i − 1.18056i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −7.00000 −0.410347
$$292$$ 0 0
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 6.00000i 0.348155i
$$298$$ 0 0
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 55.0000 3.17015
$$302$$ 0 0
$$303$$ 8.00000i 0.459588i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 13.0000i − 0.741949i −0.928643 0.370975i $$-0.879024\pi$$
0.928643 0.370975i $$-0.120976\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ − 29.0000i − 1.63918i −0.572953 0.819588i $$-0.694202\pi$$
0.572953 0.819588i $$-0.305798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 16.0000i − 0.898650i −0.893368 0.449325i $$-0.851665\pi$$
0.893368 0.449325i $$-0.148335\pi$$
$$318$$ 0 0
$$319$$ −36.0000 −2.01561
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ − 2.00000i − 0.111283i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 7.00000i − 0.387101i
$$328$$ 0 0
$$329$$ −50.0000 −2.75659
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.0000i 1.25289i 0.779466 + 0.626445i $$0.215491\pi$$
−0.779466 + 0.626445i $$0.784509\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ 0 0
$$343$$ 55.0000i 2.96972i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 10.0000i 0.529256i
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 25.0000i 1.31216i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000i 0.678594i 0.940679 + 0.339297i $$0.110189\pi$$
−0.940679 + 0.339297i $$0.889811\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 0 0
$$373$$ 25.0000i 1.29445i 0.762299 + 0.647225i $$0.224071\pi$$
−0.762299 + 0.647225i $$0.775929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 18.0000i 0.927047i
$$378$$ 0 0
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ − 20.0000i − 1.02195i −0.859595 0.510976i $$-0.829284\pi$$
0.859595 0.510976i $$-0.170716\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 11.0000i − 0.559161i
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 16.0000i 0.807093i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.0000i 0.552074i 0.961147 + 0.276037i $$0.0890213\pi$$
−0.961147 + 0.276037i $$0.910979\pi$$
$$398$$ 0 0
$$399$$ −5.00000 −0.250313
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ − 9.00000i − 0.448322i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 36.0000i 1.78445i
$$408$$ 0 0
$$409$$ 11.0000 0.543915 0.271957 0.962309i $$-0.412329\pi$$
0.271957 + 0.962309i $$0.412329\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 0 0
$$413$$ − 30.0000i − 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000i 0.587643i
$$418$$ 0 0
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 0 0
$$423$$ 10.0000i 0.486217i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.0000i 0.725901i
$$428$$ 0 0
$$429$$ 18.0000 0.869048
$$430$$ 0 0
$$431$$ −22.0000 −1.05970 −0.529851 0.848091i $$-0.677752\pi$$
−0.529851 + 0.848091i $$0.677752\pi$$
$$432$$ 0 0
$$433$$ 19.0000i 0.913082i 0.889702 + 0.456541i $$0.150912\pi$$
−0.889702 + 0.456541i $$0.849088\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2.00000i − 0.0956730i
$$438$$ 0 0
$$439$$ −5.00000 −0.238637 −0.119318 0.992856i $$-0.538071\pi$$
−0.119318 + 0.992856i $$0.538071\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ − 9.00000i − 0.422857i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 16.0000 0.745194 0.372597 0.927993i $$-0.378467\pi$$
0.372597 + 0.927993i $$0.378467\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.0000i 0.647843i 0.946084 + 0.323921i $$0.105001\pi$$
−0.946084 + 0.323921i $$0.894999\pi$$
$$468$$ 0 0
$$469$$ 5.00000 0.230879
$$470$$ 0 0
$$471$$ −7.00000 −0.322543
$$472$$ 0 0
$$473$$ − 66.0000i − 3.03468i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 8.00000i − 0.366295i
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 18.0000 0.820729
$$482$$ 0 0
$$483$$ 10.0000i 0.455016i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 3.00000i − 0.135943i −0.997687 0.0679715i $$-0.978347\pi$$
0.997687 0.0679715i $$-0.0216527\pi$$
$$488$$ 0 0
$$489$$ 7.00000 0.316551
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 60.0000i − 2.69137i
$$498$$ 0 0
$$499$$ −17.0000 −0.761025 −0.380512 0.924776i $$-0.624252\pi$$
−0.380512 + 0.924776i $$0.624252\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 4.00000i 0.177646i
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 50.0000 2.21187
$$512$$ 0 0
$$513$$ 1.00000i 0.0441511i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 60.0000i 2.63880i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ 21.0000i 0.918266i 0.888368 + 0.459133i $$0.151840\pi$$
−0.888368 + 0.459133i $$0.848160\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 6.00000i − 0.261364i
$$528$$ 0 0
$$529$$ 19.0000 0.826087
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 2.00000i − 0.0863064i
$$538$$ 0 0
$$539$$ 108.000 4.65189
$$540$$ 0 0
$$541$$ −17.0000 −0.730887 −0.365444 0.930834i $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ 19.0000i 0.815368i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 16.0000i − 0.684111i −0.939680 0.342055i $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ 0 0
$$549$$ 3.00000 0.128037
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 40.0000i 1.70097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 14.0000i − 0.593199i −0.955002 0.296600i $$-0.904147\pi$$
0.955002 0.296600i $$-0.0958526\pi$$
$$558$$ 0 0
$$559$$ −33.0000 −1.39575
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 5.00000i − 0.209980i
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 1.00000 0.0418487 0.0209243 0.999781i $$-0.493339\pi$$
0.0209243 + 0.999781i $$0.493339\pi$$
$$572$$ 0 0
$$573$$ 10.0000i 0.417756i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 43.0000i − 1.79011i −0.445952 0.895057i $$-0.647135\pi$$
0.445952 0.895057i $$-0.352865\pi$$
$$578$$ 0 0
$$579$$ −3.00000 −0.124676
$$580$$ 0 0
$$581$$ −30.0000 −1.24461
$$582$$ 0 0
$$583$$ − 48.0000i − 1.98796i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 24.0000i − 0.990586i −0.868726 0.495293i $$-0.835061\pi$$
0.868726 0.495293i $$-0.164939\pi$$
$$588$$ 0 0
$$589$$ 3.00000 0.123613
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 4.00000i 0.164260i 0.996622 + 0.0821302i $$0.0261723\pi$$
−0.996622 + 0.0821302i $$0.973828\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 21.0000i 0.859473i
$$598$$ 0 0
$$599$$ 44.0000 1.79779 0.898896 0.438163i $$-0.144371\pi$$
0.898896 + 0.438163i $$0.144371\pi$$
$$600$$ 0 0
$$601$$ −13.0000 −0.530281 −0.265141 0.964210i $$-0.585418\pi$$
−0.265141 + 0.964210i $$0.585418\pi$$
$$602$$ 0 0
$$603$$ − 1.00000i − 0.0407231i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ 30.0000 1.21367
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 32.0000i 1.28827i 0.764911 + 0.644136i $$0.222783\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 0 0
$$619$$ 19.0000 0.763674 0.381837 0.924230i $$-0.375291\pi$$
0.381837 + 0.924230i $$0.375291\pi$$
$$620$$ 0 0
$$621$$ 2.00000 0.0802572
$$622$$ 0 0
$$623$$ − 80.0000i − 3.20513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 6.00000i 0.239617i
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −31.0000 −1.23409 −0.617045 0.786928i $$-0.711670\pi$$
−0.617045 + 0.786928i $$0.711670\pi$$
$$632$$ 0 0
$$633$$ − 15.0000i − 0.596196i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 54.0000i − 2.13956i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 0 0
$$643$$ − 12.0000i − 0.473234i −0.971603 0.236617i $$-0.923961\pi$$
0.971603 0.236617i $$-0.0760386\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ −15.0000 −0.587896
$$652$$ 0 0
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 10.0000i − 0.390137i
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ − 6.00000i − 0.233021i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000i 0.464642i
$$668$$ 0 0
$$669$$ 3.00000 0.115987
$$670$$ 0 0
$$671$$ 18.0000 0.694882
$$672$$ 0 0
$$673$$ − 34.0000i − 1.31060i −0.755367 0.655302i $$-0.772541\pi$$
0.755367 0.655302i $$-0.227459\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ 35.0000 1.34318
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 3.00000i − 0.114457i
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 0 0
$$693$$ − 30.0000i − 1.13961i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 8.00000i 0.303022i
$$698$$ 0 0
$$699$$ −20.0000 −0.756469
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 0 0
$$703$$ 6.00000i 0.226294i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 40.0000i − 1.50435i
$$708$$ 0 0
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ − 6.00000i − 0.224702i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000i 0.896296i
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 0 0
$$723$$ − 7.00000i − 0.260333i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 13.0000i − 0.482143i −0.970507 0.241072i $$-0.922501\pi$$
0.970507 0.241072i $$-0.0774989\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −22.0000 −0.813699
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 6.00000i − 0.221013i
$$738$$ 0 0
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 3.00000 0.110208
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ 40.0000 1.46157
$$750$$ 0 0
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 0 0
$$753$$ − 20.0000i − 0.728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 3.00000i − 0.109037i −0.998513 0.0545184i $$-0.982638\pi$$
0.998513 0.0545184i $$-0.0173624\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −52.0000 −1.88500 −0.942499 0.334208i $$-0.891531\pi$$
−0.942499 + 0.334208i $$0.891531\pi$$
$$762$$ 0 0
$$763$$ 35.0000i 1.26709i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.0000i 0.649942i
$$768$$ 0 0
$$769$$ −27.0000 −0.973645 −0.486822 0.873501i $$-0.661844\pi$$
−0.486822 + 0.873501i $$0.661844\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 0 0
$$773$$ − 28.0000i − 1.00709i −0.863969 0.503545i $$-0.832029\pi$$
0.863969 0.503545i $$-0.167971\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 30.0000i − 1.07624i
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ −72.0000 −2.57636
$$782$$ 0 0
$$783$$ − 6.00000i − 0.214423i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.00000i 0.106938i 0.998569 + 0.0534692i $$0.0170279\pi$$
−0.998569 + 0.0534692i $$0.982972\pi$$
$$788$$ 0 0
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 60.0000 2.13335
$$792$$ 0 0
$$793$$ − 9.00000i − 0.319599i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 28.0000i 0.991811i 0.868377 + 0.495905i $$0.165164\pi$$
−0.868377 + 0.495905i $$0.834836\pi$$
$$798$$ 0 0
$$799$$ 20.0000 0.707549
$$800$$ 0 0
$$801$$ −16.0000 −0.565332
$$802$$ 0 0
$$803$$ − 60.0000i − 2.11735i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 30.0000i − 1.05605i
$$808$$ 0 0
$$809$$ 24.0000 0.843795 0.421898 0.906644i $$-0.361364\pi$$
0.421898 + 0.906644i $$0.361364\pi$$
$$810$$ 0 0
$$811$$ 9.00000 0.316033 0.158016 0.987436i $$-0.449490\pi$$
0.158016 + 0.987436i $$0.449490\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 11.0000i − 0.384841i
$$818$$ 0 0
$$819$$ −15.0000 −0.524142
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 49.0000i 1.70803i 0.520246 + 0.854016i $$0.325840\pi$$
−0.520246 + 0.854016i $$0.674160\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.0000i 1.11275i 0.830932 + 0.556375i $$0.187808\pi$$
−0.830932 + 0.556375i $$0.812192\pi$$
$$828$$ 0 0
$$829$$ 50.0000 1.73657 0.868286 0.496064i $$-0.165222\pi$$
0.868286 + 0.496064i $$0.165222\pi$$
$$830$$ 0 0
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ − 36.0000i − 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.00000i 0.103695i
$$838$$ 0 0
$$839$$ 26.0000 0.897620 0.448810 0.893627i $$-0.351848\pi$$
0.448810 + 0.893627i $$0.351848\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 26.0000i 0.895488i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 125.000i − 4.29505i
$$848$$ 0 0
$$849$$ −13.0000 −0.446159
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ − 55.0000i − 1.88316i −0.336784 0.941582i $$-0.609339\pi$$
0.336784 0.941582i $$-0.390661\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 52.0000i 1.77629i 0.459567 + 0.888143i $$0.348005\pi$$
−0.459567 + 0.888143i $$0.651995\pi$$
$$858$$ 0 0
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 20.0000 0.681598
$$862$$ 0 0
$$863$$ 8.00000i 0.272323i 0.990687 + 0.136162i $$0.0434766\pi$$
−0.990687 + 0.136162i $$0.956523\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 0 0
$$873$$ − 7.00000i − 0.236914i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 17.0000i − 0.574049i −0.957923 0.287025i $$-0.907334\pi$$
0.957923 0.287025i $$-0.0926662\pi$$
$$878$$ 0 0
$$879$$ −2.00000 −0.0674583
$$880$$ 0 0
$$881$$ −40.0000 −1.34763 −0.673817 0.738898i $$-0.735346\pi$$
−0.673817 + 0.738898i $$0.735346\pi$$
$$882$$ 0 0
$$883$$ 19.0000i 0.639401i 0.947519 + 0.319700i $$0.103582\pi$$
−0.947519 + 0.319700i $$0.896418\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 42.0000i 1.41022i 0.709097 + 0.705111i $$0.249103\pi$$
−0.709097 + 0.705111i $$0.750897\pi$$
$$888$$ 0 0
$$889$$ 40.0000 1.34156
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 0 0
$$893$$ 10.0000i 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 6.00000i − 0.200334i
$$898$$ 0 0
$$899$$ −18.0000 −0.600334
$$900$$ 0 0
$$901$$ −16.0000 −0.533037
$$902$$ 0 0
$$903$$ 55.0000i 1.83029i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ 0 0
$$909$$ −8.00000 −0.265343
$$910$$ 0 0
$$911$$ 14.0000 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$912$$ 0 0
$$913$$ 36.0000i 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 80.0000i − 2.64183i
$$918$$ 0 0
$$919$$ 1.00000 0.0329870 0.0164935 0.999864i $$-0.494750\pi$$
0.0164935 + 0.999864i $$0.494750\pi$$
$$920$$ 0 0
$$921$$ 13.0000 0.428365
$$922$$ 0 0
$$923$$ 36.0000i 1.18495i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.00000i 0.131377i
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 0 0
$$933$$ 14.0000i 0.458339i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 23.0000i − 0.751377i −0.926746 0.375689i $$-0.877406\pi$$
0.926746 0.375689i $$-0.122594\pi$$
$$938$$ 0 0
$$939$$ 29.0000 0.946379
$$940$$ 0 0
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ 0 0
$$943$$ 8.00000i 0.260516i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.00000i 0.0649913i 0.999472 + 0.0324956i $$0.0103455\pi$$
−0.999472 + 0.0324956i $$0.989654\pi$$
$$948$$ 0 0
$$949$$ −30.0000 −0.973841
$$950$$ 0 0
$$951$$ 16.0000 0.518836
$$952$$ 0 0
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 36.0000i − 1.16371i
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 8.00000i − 0.257796i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ 0 0
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 46.0000 1.47621 0.738105 0.674686i $$-0.235721\pi$$
0.738105 + 0.674686i $$0.235721\pi$$
$$972$$ 0 0
$$973$$ − 60.0000i − 1.92351i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 10.0000i − 0.319928i −0.987123 0.159964i $$-0.948862\pi$$
0.987123 0.159964i $$-0.0511379\pi$$
$$978$$ 0 0
$$979$$ −96.0000 −3.06817
$$980$$ 0 0
$$981$$ 7.00000 0.223493
$$982$$ 0 0
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 50.0000i − 1.59152i
$$988$$ 0 0
$$989$$ −22.0000 −0.699559
$$990$$ 0 0
$$991$$ 25.0000 0.794151 0.397076 0.917786i $$-0.370025\pi$$
0.397076 + 0.917786i $$0.370025\pi$$
$$992$$ 0 0
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 54.0000i − 1.71020i −0.518465 0.855099i $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 4800.2.f.a.3649.2 2
4.3 odd 2 4800.2.f.bj.3649.1 2
5.2 odd 4 4800.2.a.cs.1.1 1
5.3 odd 4 4800.2.a.a.1.1 1
5.4 even 2 inner 4800.2.f.a.3649.1 2
8.3 odd 2 600.2.f.a.49.2 2
8.5 even 2 1200.2.f.i.49.1 2
20.3 even 4 4800.2.a.ct.1.1 1
20.7 even 4 4800.2.a.b.1.1 1
20.19 odd 2 4800.2.f.bj.3649.2 2
24.5 odd 2 3600.2.f.b.2449.1 2
24.11 even 2 1800.2.f.k.649.2 2
40.3 even 4 600.2.a.e.1.1 1
40.13 odd 4 1200.2.a.j.1.1 1
40.19 odd 2 600.2.f.a.49.1 2
40.27 even 4 600.2.a.f.1.1 yes 1
40.29 even 2 1200.2.f.i.49.2 2
40.37 odd 4 1200.2.a.i.1.1 1
120.29 odd 2 3600.2.f.b.2449.2 2
120.53 even 4 3600.2.a.a.1.1 1
120.59 even 2 1800.2.f.k.649.1 2
120.77 even 4 3600.2.a.bq.1.1 1
120.83 odd 4 1800.2.a.x.1.1 1
120.107 odd 4 1800.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.e.1.1 1 40.3 even 4
600.2.a.f.1.1 yes 1 40.27 even 4
600.2.f.a.49.1 2 40.19 odd 2
600.2.f.a.49.2 2 8.3 odd 2
1200.2.a.i.1.1 1 40.37 odd 4
1200.2.a.j.1.1 1 40.13 odd 4
1200.2.f.i.49.1 2 8.5 even 2
1200.2.f.i.49.2 2 40.29 even 2
1800.2.a.a.1.1 1 120.107 odd 4
1800.2.a.x.1.1 1 120.83 odd 4
1800.2.f.k.649.1 2 120.59 even 2
1800.2.f.k.649.2 2 24.11 even 2
3600.2.a.a.1.1 1 120.53 even 4
3600.2.a.bq.1.1 1 120.77 even 4
3600.2.f.b.2449.1 2 24.5 odd 2
3600.2.f.b.2449.2 2 120.29 odd 2
4800.2.a.a.1.1 1 5.3 odd 4
4800.2.a.b.1.1 1 20.7 even 4
4800.2.a.cs.1.1 1 5.2 odd 4
4800.2.a.ct.1.1 1 20.3 even 4
4800.2.f.a.3649.1 2 5.4 even 2 inner
4800.2.f.a.3649.2 2 1.1 even 1 trivial
4800.2.f.bj.3649.1 2 4.3 odd 2
4800.2.f.bj.3649.2 2 20.19 odd 2