# Properties

 Label 6900.2.f.d.6349.2 Level $6900$ Weight $2$ Character 6900.6349 Analytic conductor $55.097$ 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] = [6900,2,Mod(6349,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, 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("6900.6349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.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: $$55.0967773947$$ 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 1380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 6349.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 6900.6349 Dual form 6900.2.f.d.6349.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +2.00000i q^{13} -6.00000i q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000i q^{23} -1.00000i q^{27} -6.00000 q^{29} -4.00000 q^{31} -8.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} -12.0000i q^{47} -9.00000 q^{49} +6.00000 q^{51} -6.00000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -10.0000 q^{61} -4.00000i q^{63} -8.00000i q^{67} +1.00000 q^{69} -6.00000 q^{71} +2.00000i q^{73} +10.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -6.00000i q^{87} -6.00000 q^{89} -8.00000 q^{91} -4.00000i q^{93} -8.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{19} - 8 q^{21} - 12 q^{29} - 8 q^{31} - 4 q^{39} + 12 q^{41} - 18 q^{49} + 12 q^{51} + 12 q^{59} - 20 q^{61} + 2 q^{69} - 12 q^{71} + 20 q^{79} + 2 q^{81} - 12 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^19 - 8 * q^21 - 12 * q^29 - 8 * q^31 - 4 * q^39 + 12 * q^41 - 18 * q^49 + 12 * q^51 + 12 * q^59 - 20 * q^61 + 2 * q^69 - 12 * q^71 + 20 * q^79 + 2 * q^81 - 12 * q^89 - 16 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1201$$ $$3451$$ $$4601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 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 $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 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
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ − 4.00000i − 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 6.00000i − 0.643268i
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 8.00000i − 0.812277i −0.913812 0.406138i $$-0.866875\pi$$
0.913812 0.406138i $$-0.133125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 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 $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 9.00000i − 0.742307i
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ − 24.0000i − 1.82469i −0.409426 0.912343i $$-0.634271\pi$$
0.409426 0.912343i $$-0.365729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000i 0.450988i
$$178$$ 0 0
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ − 10.0000i − 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ − 24.0000i − 1.68447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000i 0.0695048i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ − 6.00000i − 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 16.0000i − 1.08615i
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 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
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.0000i 0.649570i
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4.00000i − 0.254514i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 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
$$257$$ − 12.0000i − 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ 0 0
$$259$$ 32.0000 1.98838
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 0 0
$$273$$ − 8.00000i − 0.484182i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 14.0000i − 0.841178i −0.907251 0.420589i $$-0.861823\pi$$
0.907251 0.420589i $$-0.138177\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ − 28.0000i − 1.58265i −0.611393 0.791327i $$-0.709391\pi$$
0.611393 0.791327i $$-0.290609\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.00000i − 0.110600i
$$328$$ 0 0
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 0 0
$$333$$ 8.00000i 0.438397i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 32.0000i − 1.74315i −0.490261 0.871576i $$-0.663099\pi$$
0.490261 0.871576i $$-0.336901\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ 36.0000i 1.91609i 0.286623 + 0.958043i $$0.407467\pi$$
−0.286623 + 0.958043i $$0.592533\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 24.0000i 1.27021i
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 11.0000i − 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ 20.0000i 1.03556i 0.855514 + 0.517780i $$0.173242\pi$$
−0.855514 + 0.517780i $$0.826758\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 0 0
$$393$$ 6.00000i 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 0 0
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ 24.0000i 1.18096i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 20.0000i − 0.979404i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ 12.0000i 0.583460i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 40.0000i − 1.93574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ − 16.0000i − 0.768911i −0.923144 0.384455i $$-0.874389\pi$$
0.923144 0.384455i $$-0.125611\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.00000i 0.0956730i
$$438$$ 0 0
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.00000i 0.283790i
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 32.0000i − 1.49690i −0.663193 0.748448i $$-0.730799\pi$$
0.663193 0.748448i $$-0.269201\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ − 40.0000i − 1.85896i −0.368875 0.929479i $$-0.620257\pi$$
0.368875 0.929479i $$-0.379743\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 0 0
$$483$$ 4.00000i 0.182006i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 0 0
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ 36.0000i 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 24.0000i − 1.07655i
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 24.0000i − 1.07011i −0.844818 0.535054i $$-0.820291\pi$$
0.844818 0.535054i $$-0.179709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000i 0.399704i
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 0 0
$$513$$ 2.00000i 0.0883022i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000i 1.04546i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 2.00000i 0.0858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 44.0000i − 1.88130i −0.339372 0.940652i $$-0.610215\pi$$
0.339372 0.940652i $$-0.389785\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 40.0000i 1.70097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 34.0000i 1.41544i 0.706494 + 0.707719i $$0.250276\pi$$
−0.706494 + 0.707719i $$0.749724\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 12.0000i − 0.492781i −0.969171 0.246390i $$-0.920755\pi$$
0.969171 0.246390i $$-0.0792446\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 10.0000i 0.409273i
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 8.00000i 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ 0 0
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ − 4.00000i − 0.161558i −0.996732 0.0807792i $$-0.974259\pi$$
0.996732 0.0807792i $$-0.0257409\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 0 0
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ − 24.0000i − 0.961540i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −48.0000 −1.91389
$$630$$ 0 0
$$631$$ −34.0000 −1.35352 −0.676759 0.736204i $$-0.736616\pi$$
−0.676759 + 0.736204i $$0.736616\pi$$
$$632$$ 0 0
$$633$$ 8.00000i 0.317971i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 18.0000i − 0.713186i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 32.0000i 1.26196i 0.775800 + 0.630978i $$0.217346\pi$$
−0.775800 + 0.630978i $$0.782654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 0 0
$$653$$ 24.0000i 0.939193i 0.882881 + 0.469596i $$0.155601\pi$$
−0.882881 + 0.469596i $$0.844399\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 0 0
$$663$$ 12.0000i 0.466041i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.00000i 0.232321i
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 46.0000i − 1.77317i −0.462566 0.886585i $$-0.653071\pi$$
0.462566 0.886585i $$-0.346929\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 0 0
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ 0 0
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ 16.0000i 0.603451i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 24.0000i − 0.902613i
$$708$$ 0 0
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 0 0
$$713$$ 4.00000i 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000i 0.224074i
$$718$$ 0 0
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ 0 0
$$723$$ − 10.0000i − 0.371904i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000i 1.03846i 0.854634 + 0.519231i $$0.173782\pi$$
−0.854634 + 0.519231i $$0.826218\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 20.0000i 0.738717i 0.929287 + 0.369358i $$0.120423\pi$$
−0.929287 + 0.369358i $$0.879577\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 16.0000i 0.581530i 0.956795 + 0.290765i $$0.0939098\pi$$
−0.956795 + 0.290765i $$0.906090\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ − 8.00000i − 0.289619i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000i 0.433295i
$$768$$ 0 0
$$769$$ −38.0000 −1.37032 −0.685158 0.728395i $$-0.740267\pi$$
−0.685158 + 0.728395i $$0.740267\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 32.0000i 1.14799i
$$778$$ 0 0
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000i 0.211210i
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 32.0000i 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ 0 0
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ 54.0000i 1.87099i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ − 6.00000i − 0.206651i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 44.0000i − 1.51186i
$$848$$ 0 0
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 34.0000i − 1.16414i −0.813139 0.582069i $$-0.802243\pi$$
0.813139 0.582069i $$-0.197757\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.0000i 1.63965i 0.572615 + 0.819824i $$0.305929\pi$$
−0.572615 + 0.819824i $$0.694071\pi$$
$$858$$ 0 0
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 8.00000i 0.270759i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 0 0
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 0 0
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 24.0000i 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.00000i 0.0667781i
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 32.0000i − 1.06489i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 24.0000i 0.792550i
$$918$$ 0 0
$$919$$ −26.0000 −0.857661 −0.428830 0.903385i $$-0.641074\pi$$
−0.428830 + 0.903385i $$0.641074\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ − 12.0000i − 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 8.00000i − 0.262754i
$$928$$ 0 0
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 0 0
$$933$$ − 18.0000i − 0.589294i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.00000i 0.130674i 0.997863 + 0.0653372i $$0.0208123\pi$$
−0.997863 + 0.0653372i $$0.979188\pi$$
$$938$$ 0 0
$$939$$ 28.0000 0.913745
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 0 0
$$943$$ − 6.00000i − 0.195387i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 54.0000i 1.74923i 0.484817 + 0.874616i $$0.338886\pi$$
−0.484817 + 0.874616i $$0.661114\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 0 0
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ − 80.0000i − 2.56468i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 48.0000i 1.52786i
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.0000i 1.45683i 0.685134 + 0.728417i $$0.259744\pi$$
−0.685134 + 0.728417i $$0.740256\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 6900.2.f.d.6349.2 2
5.2 odd 4 1380.2.a.d.1.1 1
5.3 odd 4 6900.2.a.e.1.1 1
5.4 even 2 inner 6900.2.f.d.6349.1 2
15.2 even 4 4140.2.a.a.1.1 1
20.7 even 4 5520.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.d.1.1 1 5.2 odd 4
4140.2.a.a.1.1 1 15.2 even 4
5520.2.a.o.1.1 1 20.7 even 4
6900.2.a.e.1.1 1 5.3 odd 4
6900.2.f.d.6349.1 2 5.4 even 2 inner
6900.2.f.d.6349.2 2 1.1 even 1 trivial