# Properties

 Label 5400.2.f.j.649.2 Level $5400$ Weight $2$ Character 5400.649 Analytic conductor $43.119$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(649,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, 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("5400.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5400 = 2^{3} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5400.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: $$43.1192170915$$ 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: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5400.649 Dual form 5400.2.f.j.649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ $$q-2.00000 q^{11} +3.00000i q^{17} +1.00000 q^{19} -3.00000i q^{23} +4.00000 q^{29} -5.00000 q^{31} -10.0000i q^{37} -6.00000 q^{41} -6.00000i q^{43} +8.00000i q^{47} +7.00000 q^{49} -3.00000i q^{53} +5.00000 q^{61} +2.00000i q^{67} -2.00000 q^{71} +6.00000i q^{73} +11.0000 q^{79} -9.00000i q^{83} +10.0000 q^{89} -8.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 4 q^{11} + 2 q^{19} + 8 q^{29} - 10 q^{31} - 12 q^{41} + 14 q^{49} + 10 q^{61} - 4 q^{71} + 22 q^{79} + 20 q^{89}+O(q^{100})$$ 2 * q - 4 * q^11 + 2 * q^19 + 8 * q^29 - 10 * q^31 - 12 * q^41 + 14 * q^49 + 10 * q^61 - 4 * q^71 + 22 * q^79 + 20 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$2701$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.00000i 0.727607i 0.931476 + 0.363803i $$0.118522\pi$$
−0.931476 + 0.363803i $$0.881478\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 3.00000i − 0.412082i −0.978543 0.206041i $$-0.933942\pi$$
0.978543 0.206041i $$-0.0660580\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 22.0000i − 1.95218i −0.217357 0.976092i $$-0.569744\pi$$
0.217357 0.976092i $$-0.430256\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 7.00000i − 0.598050i −0.954245 0.299025i $$-0.903339\pi$$
0.954245 0.299025i $$-0.0966615\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 2.00000i − 0.156652i −0.996928 0.0783260i $$-0.975042\pi$$
0.996928 0.0783260i $$-0.0249575\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 17.0000i − 1.31550i −0.753237 0.657750i $$-0.771508\pi$$
0.753237 0.657750i $$-0.228492\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 19.0000i 1.44454i 0.691609 + 0.722272i $$0.256902\pi$$
−0.691609 + 0.722272i $$0.743098\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 17.0000 1.26360 0.631800 0.775131i $$-0.282316\pi$$
0.631800 + 0.775131i $$0.282316\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.00000i − 0.438763i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ − 20.0000i − 1.43963i −0.694165 0.719816i $$-0.744226\pi$$
0.694165 0.719816i $$-0.255774\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.00000i 0.356235i 0.984009 + 0.178118i $$0.0570008\pi$$
−0.984009 + 0.178118i $$0.942999\pi$$
$$198$$ 0 0
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 2.00000i 0.133930i 0.997755 + 0.0669650i $$0.0213316\pi$$
−0.997755 + 0.0669650i $$0.978668\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.0000i 1.12833i 0.825662 + 0.564165i $$0.190802\pi$$
−0.825662 + 0.564165i $$0.809198\pi$$
$$228$$ 0 0
$$229$$ 13.0000 0.859064 0.429532 0.903052i $$-0.358679\pi$$
0.429532 + 0.903052i $$0.358679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 10.0000i 0.655122i 0.944830 + 0.327561i $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −30.0000 −1.94054 −0.970269 0.242028i $$-0.922188\pi$$
−0.970269 + 0.242028i $$0.922188\pi$$
$$240$$ 0 0
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 6.00000i 0.377217i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 23.0000i − 1.43470i −0.696713 0.717350i $$-0.745355\pi$$
0.696713 0.717350i $$-0.254645\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ 3.00000 0.182237 0.0911185 0.995840i $$-0.470956\pi$$
0.0911185 + 0.995840i $$0.470956\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 28.0000 1.67034 0.835170 0.549992i $$-0.185369\pi$$
0.835170 + 0.549992i $$0.185369\pi$$
$$282$$ 0 0
$$283$$ 2.00000i 0.118888i 0.998232 + 0.0594438i $$0.0189327\pi$$
−0.998232 + 0.0594438i $$0.981067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 11.0000i 0.642627i 0.946973 + 0.321313i $$0.104124\pi$$
−0.946973 + 0.321313i $$0.895876\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 26.0000i − 1.48390i −0.670456 0.741949i $$-0.733902\pi$$
0.670456 0.741949i $$-0.266098\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ − 20.0000i − 1.13047i −0.824931 0.565233i $$-0.808786\pi$$
0.824931 0.565233i $$-0.191214\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 11.0000i 0.617822i 0.951091 + 0.308911i $$0.0999645\pi$$
−0.951091 + 0.308911i $$0.900036\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.00000i 0.166924i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 8.00000i − 0.435788i −0.975972 0.217894i $$-0.930081\pi$$
0.975972 0.217894i $$-0.0699187\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 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
$$349$$ 7.00000 0.374701 0.187351 0.982293i $$-0.440010\pi$$
0.187351 + 0.982293i $$0.440010\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 10.0000i 0.532246i 0.963939 + 0.266123i $$0.0857428\pi$$
−0.963939 + 0.266123i $$0.914257\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 3.00000i − 0.153293i −0.997058 0.0766464i $$-0.975579\pi$$
0.997058 0.0766464i $$-0.0244213\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.00000i 0.200754i 0.994949 + 0.100377i $$0.0320049\pi$$
−0.994949 + 0.100377i $$0.967995\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000i 0.991363i
$$408$$ 0 0
$$409$$ 31.0000 1.53285 0.766426 0.642333i $$-0.222033\pi$$
0.766426 + 0.642333i $$0.222033\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 0 0
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ − 24.0000i − 1.15337i −0.816968 0.576683i $$-0.804347\pi$$
0.816968 0.576683i $$-0.195653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3.00000i − 0.143509i
$$438$$ 0 0
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 29.0000i − 1.37783i −0.724841 0.688916i $$-0.758087\pi$$
0.724841 0.688916i $$-0.241913\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.0000i 0.748448i 0.927338 + 0.374224i $$0.122091\pi$$
−0.927338 + 0.374224i $$0.877909\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 34.0000 1.58354 0.791769 0.610821i $$-0.209160\pi$$
0.791769 + 0.610821i $$0.209160\pi$$
$$462$$ 0 0
$$463$$ 14.0000i 0.650635i 0.945605 + 0.325318i $$0.105471\pi$$
−0.945605 + 0.325318i $$0.894529\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 7.00000i − 0.323921i −0.986797 0.161961i $$-0.948218\pi$$
0.986797 0.161961i $$-0.0517818\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −43.0000 −1.92494 −0.962472 0.271380i $$-0.912520\pi$$
−0.962472 + 0.271380i $$0.912520\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 35.0000i − 1.56057i −0.625422 0.780286i $$-0.715073\pi$$
0.625422 0.780286i $$-0.284927\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 28.0000 1.24108 0.620539 0.784176i $$-0.286914\pi$$
0.620539 + 0.784176i $$0.286914\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16.0000i − 0.703679i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ − 28.0000i − 1.22435i −0.790721 0.612177i $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 15.0000i − 0.653410i
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −14.0000 −0.603023
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 18.0000i − 0.769624i −0.922995 0.384812i $$-0.874266\pi$$
0.922995 0.384812i $$-0.125734\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 28.0000i − 1.18006i −0.807382 0.590030i $$-0.799116\pi$$
0.807382 0.590030i $$-0.200884\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ −9.00000 −0.376638 −0.188319 0.982108i $$-0.560304\pi$$
−0.188319 + 0.982108i $$0.560304\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 46.0000i 1.91501i 0.288425 + 0.957503i $$0.406868\pi$$
−0.288425 + 0.957503i $$0.593132\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 6.00000i 0.248495i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 23.0000i 0.949312i 0.880172 + 0.474656i $$0.157427\pi$$
−0.880172 + 0.474656i $$0.842573\pi$$
$$588$$ 0 0
$$589$$ −5.00000 −0.206021
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.00000i 0.123195i 0.998101 + 0.0615976i $$0.0196196\pi$$
−0.998101 + 0.0615976i $$0.980380\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 17.0000 0.693444 0.346722 0.937968i $$-0.387295\pi$$
0.346722 + 0.937968i $$0.387295\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 10.0000i − 0.405887i −0.979190 0.202944i $$-0.934949\pi$$
0.979190 0.202944i $$-0.0650509\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 25.0000i − 1.00646i −0.864152 0.503231i $$-0.832144\pi$$
0.864152 0.503231i $$-0.167856\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 30.0000 1.19618
$$630$$ 0 0
$$631$$ −19.0000 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 39.0000i 1.53325i 0.642096 + 0.766624i $$0.278065\pi$$
−0.642096 + 0.766624i $$0.721935\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 11.0000i − 0.430463i −0.976563 0.215232i $$-0.930949\pi$$
0.976563 0.215232i $$-0.0690506\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 12.0000i − 0.464642i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ − 8.00000i − 0.308377i −0.988041 0.154189i $$-0.950724\pi$$
0.988041 0.154189i $$-0.0492764\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 50.0000i 1.92166i 0.277145 + 0.960828i $$0.410612\pi$$
−0.277145 + 0.960828i $$0.589388\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 23.0000i 0.880071i 0.897980 + 0.440035i $$0.145034\pi$$
−0.897980 + 0.440035i $$0.854966\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 13.0000 0.494543 0.247272 0.968946i $$-0.420466\pi$$
0.247272 + 0.968946i $$0.420466\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 18.0000i − 0.681799i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 15.0000i 0.561754i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 44.0000i 1.63187i 0.578144 + 0.815935i $$0.303777\pi$$
−0.578144 + 0.815935i $$0.696223\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 18.0000 0.665754
$$732$$ 0 0
$$733$$ − 8.00000i − 0.295487i −0.989026 0.147743i $$-0.952799\pi$$
0.989026 0.147743i $$-0.0472010\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 4.00000i − 0.147342i
$$738$$ 0 0
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 53.0000 1.93400 0.966999 0.254781i $$-0.0820034\pi$$
0.966999 + 0.254781i $$0.0820034\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 50.0000i 1.81728i 0.417579 + 0.908640i $$0.362879\pi$$
−0.417579 + 0.908640i $$0.637121\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −44.0000 −1.59500 −0.797499 0.603320i $$-0.793844\pi$$
−0.797499 + 0.603320i $$0.793844\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −21.0000 −0.757279 −0.378640 0.925544i $$-0.623608\pi$$
−0.378640 + 0.925544i $$0.623608\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 3.00000i − 0.107903i −0.998544 0.0539513i $$-0.982818\pi$$
0.998544 0.0539513i $$-0.0171816\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 38.0000i 1.35455i 0.735728 + 0.677277i $$0.236840\pi$$
−0.735728 + 0.677277i $$0.763160\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 23.0000i − 0.814702i −0.913272 0.407351i $$-0.866453\pi$$
0.913272 0.407351i $$-0.133547\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 12.0000i − 0.423471i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 6.00000i − 0.209913i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 34.0000 1.18661 0.593304 0.804978i $$-0.297823\pi$$
0.593304 + 0.804978i $$0.297823\pi$$
$$822$$ 0 0
$$823$$ − 6.00000i − 0.209147i −0.994517 0.104573i $$-0.966652\pi$$
0.994517 0.104573i $$-0.0333477\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 21.0000i − 0.730242i −0.930960 0.365121i $$-0.881028\pi$$
0.930960 0.365121i $$-0.118972\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 21.0000i 0.727607i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −30.0000 −1.02839
$$852$$ 0 0
$$853$$ 8.00000i 0.273915i 0.990577 + 0.136957i $$0.0437323\pi$$
−0.990577 + 0.136957i $$0.956268\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 1.00000i − 0.0341593i −0.999854 0.0170797i $$-0.994563\pi$$
0.999854 0.0170797i $$-0.00543689\pi$$
$$858$$ 0 0
$$859$$ −37.0000 −1.26242 −0.631212 0.775610i $$-0.717442\pi$$
−0.631212 + 0.775610i $$0.717442\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 23.0000i − 0.782929i −0.920193 0.391465i $$-0.871969\pi$$
0.920193 0.391465i $$-0.128031\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −22.0000 −0.746299
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 4.00000 0.134763 0.0673817 0.997727i $$-0.478535\pi$$
0.0673817 + 0.997727i $$0.478535\pi$$
$$882$$ 0 0
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 33.0000i − 1.10803i −0.832506 0.554016i $$-0.813095\pi$$
0.832506 0.554016i $$-0.186905\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 8.00000i 0.267710i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −20.0000 −0.667037
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ 18.0000i 0.595713i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.0000i 0.392023i 0.980602 + 0.196011i $$0.0627990\pi$$
−0.980602 + 0.196011i $$0.937201\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ 0 0
$$943$$ 18.0000i 0.586161i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 3.00000i − 0.0974869i −0.998811 0.0487435i $$-0.984478\pi$$
0.998811 0.0487435i $$-0.0155217\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 42.0000i 1.35063i 0.737530 + 0.675314i $$0.235992\pi$$
−0.737530 + 0.675314i $$0.764008\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 50.0000i − 1.59964i −0.600239 0.799821i $$-0.704928\pi$$
0.600239 0.799821i $$-0.295072\pi$$
$$978$$ 0 0
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 11.0000i − 0.350846i −0.984493 0.175423i $$-0.943871\pi$$
0.984493 0.175423i $$-0.0561292\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −18.0000 −0.572367
$$990$$ 0 0
$$991$$ 1.00000 0.0317660 0.0158830 0.999874i $$-0.494944\pi$$
0.0158830 + 0.999874i $$0.494944\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 5400.2.f.j.649.2 2
3.2 odd 2 5400.2.f.r.649.1 2
5.2 odd 4 1080.2.a.d.1.1 1
5.3 odd 4 5400.2.a.w.1.1 1
5.4 even 2 inner 5400.2.f.j.649.1 2
15.2 even 4 1080.2.a.j.1.1 yes 1
15.8 even 4 5400.2.a.x.1.1 1
15.14 odd 2 5400.2.f.r.649.2 2
20.7 even 4 2160.2.a.f.1.1 1
40.27 even 4 8640.2.a.bs.1.1 1
40.37 odd 4 8640.2.a.bt.1.1 1
45.2 even 12 3240.2.q.e.1081.1 2
45.7 odd 12 3240.2.q.s.1081.1 2
45.22 odd 12 3240.2.q.s.2161.1 2
45.32 even 12 3240.2.q.e.2161.1 2
60.47 odd 4 2160.2.a.r.1.1 1
120.77 even 4 8640.2.a.o.1.1 1
120.107 odd 4 8640.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.d.1.1 1 5.2 odd 4
1080.2.a.j.1.1 yes 1 15.2 even 4
2160.2.a.f.1.1 1 20.7 even 4
2160.2.a.r.1.1 1 60.47 odd 4
3240.2.q.e.1081.1 2 45.2 even 12
3240.2.q.e.2161.1 2 45.32 even 12
3240.2.q.s.1081.1 2 45.7 odd 12
3240.2.q.s.2161.1 2 45.22 odd 12
5400.2.a.w.1.1 1 5.3 odd 4
5400.2.a.x.1.1 1 15.8 even 4
5400.2.f.j.649.1 2 5.4 even 2 inner
5400.2.f.j.649.2 2 1.1 even 1 trivial
5400.2.f.r.649.1 2 3.2 odd 2
5400.2.f.r.649.2 2 15.14 odd 2
8640.2.a.o.1.1 1 120.77 even 4
8640.2.a.p.1.1 1 120.107 odd 4
8640.2.a.bs.1.1 1 40.27 even 4
8640.2.a.bt.1.1 1 40.37 odd 4