# Properties

 Label 5184.2.a.s.1.1 Level $5184$ Weight $2$ Character 5184.1 Self dual yes Analytic conductor $41.394$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(1,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.a (trivial)

## 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: yes Analytic conductor: $$41.3944484078$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5184.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -3.00000 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -3.00000 q^{7} -5.00000 q^{11} +5.00000 q^{13} -2.00000 q^{17} +4.00000 q^{19} -1.00000 q^{23} -4.00000 q^{25} +9.00000 q^{29} -1.00000 q^{31} -3.00000 q^{35} +6.00000 q^{37} +3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{47} +2.00000 q^{49} -2.00000 q^{53} -5.00000 q^{55} -11.0000 q^{59} -7.00000 q^{61} +5.00000 q^{65} +1.00000 q^{67} +4.00000 q^{71} -2.00000 q^{73} +15.0000 q^{77} +1.00000 q^{79} -1.00000 q^{83} -2.00000 q^{85} -18.0000 q^{89} -15.0000 q^{91} +4.00000 q^{95} -13.0000 q^{97} +O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605 −0.0898027 0.995960i $$-0.528624\pi$$
−0.0898027 + 0.995960i $$0.528624\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5.00000 0.620174
$$66$$ 0 0
$$67$$ 1.00000 0.122169 0.0610847 0.998133i $$-0.480544\pi$$
0.0610847 + 0.998133i $$0.480544\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 15.0000 1.70941
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ −15.0000 −1.57243
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ −5.00000 −0.492665 −0.246332 0.969185i $$-0.579225\pi$$
−0.246332 + 0.969185i $$0.579225\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.00000 0.436852 0.218426 0.975854i $$-0.429908\pi$$
0.218426 + 0.975854i $$0.429908\pi$$
$$132$$ 0 0
$$133$$ −12.0000 −1.04053
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ −3.00000 −0.254457 −0.127228 0.991873i $$-0.540608\pi$$
−0.127228 + 0.991873i $$0.540608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −25.0000 −2.09061
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −19.0000 −1.55654 −0.778270 0.627929i $$-0.783903\pi$$
−0.778270 + 0.627929i $$0.783903\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.00000 −0.0803219
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −21.0000 −1.62503 −0.812514 0.582941i $$-0.801902\pi$$
−0.812514 + 0.582941i $$0.801902\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ 12.0000 0.907115
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 10.0000 0.731272
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −27.0000 −1.95365 −0.976826 0.214036i $$-0.931339\pi$$
−0.976826 + 0.214036i $$0.931339\pi$$
$$192$$ 0 0
$$193$$ −13.0000 −0.935760 −0.467880 0.883792i $$-0.654982\pi$$
−0.467880 + 0.883792i $$0.654982\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −27.0000 −1.89503
$$204$$ 0 0
$$205$$ 3.00000 0.209529
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −20.0000 −1.38343
$$210$$ 0 0
$$211$$ −7.00000 −0.481900 −0.240950 0.970538i $$-0.577459\pi$$
−0.240950 + 0.970538i $$0.577459\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1.00000 −0.0681994
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −10.0000 −0.672673
$$222$$ 0 0
$$223$$ −11.0000 −0.736614 −0.368307 0.929704i $$-0.620063\pi$$
−0.368307 + 0.929704i $$0.620063\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −21.0000 −1.39382 −0.696909 0.717159i $$-0.745442\pi$$
−0.696909 + 0.717159i $$0.745442\pi$$
$$228$$ 0 0
$$229$$ 17.0000 1.12339 0.561696 0.827344i $$-0.310149\pi$$
0.561696 + 0.827344i $$0.310149\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −17.0000 −1.09964 −0.549819 0.835284i $$-0.685303\pi$$
−0.549819 + 0.835284i $$0.685303\pi$$
$$240$$ 0 0
$$241$$ 15.0000 0.966235 0.483117 0.875556i $$-0.339504\pi$$
0.483117 + 0.875556i $$0.339504\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ 20.0000 1.27257
$$248$$ 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
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7.00000 0.436648 0.218324 0.975876i $$-0.429941\pi$$
0.218324 + 0.975876i $$0.429941\pi$$
$$258$$ 0 0
$$259$$ −18.0000 −1.11847
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7.00000 −0.431638 −0.215819 0.976433i $$-0.569242\pi$$
−0.215819 + 0.976433i $$0.569242\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 20.0000 1.20605
$$276$$ 0 0
$$277$$ 29.0000 1.74244 0.871221 0.490892i $$-0.163329\pi$$
0.871221 + 0.490892i $$0.163329\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 19.0000 1.13344 0.566722 0.823909i $$-0.308211\pi$$
0.566722 + 0.823909i $$0.308211\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.00000 −0.531253
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ −11.0000 −0.640445
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.00000 −0.289157
$$300$$ 0 0
$$301$$ 3.00000 0.172917
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.00000 −0.400819
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.00000 −0.510343 −0.255172 0.966896i $$-0.582132\pi$$
−0.255172 + 0.966896i $$0.582132\pi$$
$$312$$ 0 0
$$313$$ −9.00000 −0.508710 −0.254355 0.967111i $$-0.581863\pi$$
−0.254355 + 0.967111i $$0.581863\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −15.0000 −0.842484 −0.421242 0.906948i $$-0.638406\pi$$
−0.421242 + 0.906948i $$0.638406\pi$$
$$318$$ 0 0
$$319$$ −45.0000 −2.51952
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −20.0000 −1.10940
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ 19.0000 1.04433 0.522167 0.852843i $$-0.325124\pi$$
0.522167 + 0.852843i $$0.325124\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 1.00000 0.0546358
$$336$$ 0 0
$$337$$ −9.00000 −0.490261 −0.245131 0.969490i $$-0.578831\pi$$
−0.245131 + 0.969490i $$0.578831\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 5.00000 0.270765
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 9.00000 0.483145 0.241573 0.970383i $$-0.422337\pi$$
0.241573 + 0.970383i $$0.422337\pi$$
$$348$$ 0 0
$$349$$ 21.0000 1.12410 0.562052 0.827102i $$-0.310012\pi$$
0.562052 + 0.827102i $$0.310012\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −5.00000 −0.266123 −0.133062 0.991108i $$-0.542481\pi$$
−0.133062 + 0.991108i $$0.542481\pi$$
$$354$$ 0 0
$$355$$ 4.00000 0.212298
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ −23.0000 −1.20059 −0.600295 0.799779i $$-0.704950\pi$$
−0.600295 + 0.799779i $$0.704950\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −35.0000 −1.81223 −0.906116 0.423030i $$-0.860966\pi$$
−0.906116 + 0.423030i $$0.860966\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 45.0000 2.31762
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 27.0000 1.37964 0.689818 0.723983i $$-0.257691\pi$$
0.689818 + 0.723983i $$0.257691\pi$$
$$384$$ 0 0
$$385$$ 15.0000 0.764471
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −35.0000 −1.77457 −0.887285 0.461221i $$-0.847411\pi$$
−0.887285 + 0.461221i $$0.847411\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1.00000 0.0503155
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −21.0000 −1.04869 −0.524345 0.851506i $$-0.675690\pi$$
−0.524345 + 0.851506i $$0.675690\pi$$
$$402$$ 0 0
$$403$$ −5.00000 −0.249068
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −30.0000 −1.48704
$$408$$ 0 0
$$409$$ −33.0000 −1.63174 −0.815872 0.578232i $$-0.803743\pi$$
−0.815872 + 0.578232i $$0.803743\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 33.0000 1.62382
$$414$$ 0 0
$$415$$ −1.00000 −0.0490881
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 0 0
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8.00000 0.388057
$$426$$ 0 0
$$427$$ 21.0000 1.01626
$$428$$ 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
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ 33.0000 1.57500 0.787502 0.616312i $$-0.211374\pi$$
0.787502 + 0.616312i $$0.211374\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 23.0000 1.09276 0.546381 0.837536i $$-0.316005\pi$$
0.546381 + 0.837536i $$0.316005\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −15.0000 −0.706322
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −15.0000 −0.703211
$$456$$ 0 0
$$457$$ 11.0000 0.514558 0.257279 0.966337i $$-0.417174\pi$$
0.257279 + 0.966337i $$0.417174\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 25.0000 1.16437 0.582183 0.813058i $$-0.302199\pi$$
0.582183 + 0.813058i $$0.302199\pi$$
$$462$$ 0 0
$$463$$ 23.0000 1.06890 0.534450 0.845200i $$-0.320519\pi$$
0.534450 + 0.845200i $$0.320519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −3.00000 −0.138527
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.00000 0.229900
$$474$$ 0 0
$$475$$ −16.0000 −0.734130
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1.00000 0.0456912 0.0228456 0.999739i $$-0.492727\pi$$
0.0228456 + 0.999739i $$0.492727\pi$$
$$480$$ 0 0
$$481$$ 30.0000 1.36788
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −13.0000 −0.590300
$$486$$ 0 0
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −27.0000 −1.21849 −0.609246 0.792981i $$-0.708528\pi$$
−0.609246 + 0.792981i $$0.708528\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ −27.0000 −1.20869 −0.604343 0.796724i $$-0.706564\pi$$
−0.604343 + 0.796724i $$0.706564\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ −3.00000 −0.133498
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −3.00000 −0.132973 −0.0664863 0.997787i $$-0.521179\pi$$
−0.0664863 + 0.997787i $$0.521179\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −5.00000 −0.220326
$$516$$ 0 0
$$517$$ 15.0000 0.659699
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ −40.0000 −1.74908 −0.874539 0.484955i $$-0.838836\pi$$
−0.874539 + 0.484955i $$0.838836\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.00000 0.0871214
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 15.0000 0.649722
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −10.0000 −0.430730
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ 19.0000 0.812381 0.406191 0.913788i $$-0.366857\pi$$
0.406191 + 0.913788i $$0.366857\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ −3.00000 −0.127573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ 0 0
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.00000 0.0421450 0.0210725 0.999778i $$-0.493292\pi$$
0.0210725 + 0.999778i $$0.493292\pi$$
$$564$$ 0 0
$$565$$ −9.00000 −0.378633
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −25.0000 −1.04805 −0.524027 0.851701i $$-0.675571\pi$$
−0.524027 + 0.851701i $$0.675571\pi$$
$$570$$ 0 0
$$571$$ −15.0000 −0.627730 −0.313865 0.949468i $$-0.601624\pi$$
−0.313865 + 0.949468i $$0.601624\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ 10.0000 0.414158
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −17.0000 −0.701665 −0.350833 0.936438i $$-0.614101\pi$$
−0.350833 + 0.936438i $$0.614101\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 6.00000 0.245976
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −21.0000 −0.858037 −0.429018 0.903296i $$-0.641140\pi$$
−0.429018 + 0.903296i $$0.641140\pi$$
$$600$$ 0 0
$$601$$ 19.0000 0.775026 0.387513 0.921864i $$-0.373334\pi$$
0.387513 + 0.921864i $$0.373334\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ −13.0000 −0.527654 −0.263827 0.964570i $$-0.584985\pi$$
−0.263827 + 0.964570i $$0.584985\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −15.0000 −0.606835
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.00000 0.120775 0.0603877 0.998175i $$-0.480766\pi$$
0.0603877 + 0.998175i $$0.480766\pi$$
$$618$$ 0 0
$$619$$ 11.0000 0.442127 0.221064 0.975259i $$-0.429047\pi$$
0.221064 + 0.975259i $$0.429047\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 54.0000 2.16346
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ 10.0000 0.396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.00000 −0.0394976 −0.0197488 0.999805i $$-0.506287\pi$$
−0.0197488 + 0.999805i $$0.506287\pi$$
$$642$$ 0 0
$$643$$ −15.0000 −0.591542 −0.295771 0.955259i $$-0.595577\pi$$
−0.295771 + 0.955259i $$0.595577\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ 55.0000 2.15894
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 33.0000 1.29139 0.645695 0.763596i $$-0.276568\pi$$
0.645695 + 0.763596i $$0.276568\pi$$
$$654$$ 0 0
$$655$$ 5.00000 0.195366
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −21.0000 −0.818044 −0.409022 0.912525i $$-0.634130\pi$$
−0.409022 + 0.912525i $$0.634130\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 35.0000 1.35116
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −35.0000 −1.34516 −0.672580 0.740025i $$-0.734814\pi$$
−0.672580 + 0.740025i $$0.734814\pi$$
$$678$$ 0 0
$$679$$ 39.0000 1.49668
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ 3.00000 0.114624
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −10.0000 −0.380970
$$690$$ 0 0
$$691$$ −41.0000 −1.55971 −0.779857 0.625958i $$-0.784708\pi$$
−0.779857 + 0.625958i $$0.784708\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.00000 −0.113796
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 9.00000 0.338480
$$708$$ 0 0
$$709$$ −11.0000 −0.413114 −0.206557 0.978435i $$-0.566226\pi$$
−0.206557 + 0.978435i $$0.566226\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.00000 0.0374503
$$714$$ 0 0
$$715$$ −25.0000 −0.934947
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 32.0000 1.19340 0.596699 0.802465i $$-0.296479\pi$$
0.596699 + 0.802465i $$0.296479\pi$$
$$720$$ 0 0
$$721$$ 15.0000 0.558629
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −36.0000 −1.33701
$$726$$ 0 0
$$727$$ −39.0000 −1.44643 −0.723215 0.690623i $$-0.757336\pi$$
−0.723215 + 0.690623i $$0.757336\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ 0 0
$$733$$ 13.0000 0.480166 0.240083 0.970752i $$-0.422825\pi$$
0.240083 + 0.970752i $$0.422825\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −5.00000 −0.184177
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 11.0000 0.403551 0.201775 0.979432i $$-0.435329\pi$$
0.201775 + 0.979432i $$0.435329\pi$$
$$744$$ 0 0
$$745$$ −19.0000 −0.696106
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 36.0000 1.31541
$$750$$ 0 0
$$751$$ 27.0000 0.985244 0.492622 0.870243i $$-0.336039\pi$$
0.492622 + 0.870243i $$0.336039\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 17.0000 0.618693
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −5.00000 −0.181250 −0.0906249 0.995885i $$-0.528886\pi$$
−0.0906249 + 0.995885i $$0.528886\pi$$
$$762$$ 0 0
$$763$$ 30.0000 1.08607
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −55.0000 −1.98593
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −42.0000 −1.51064 −0.755318 0.655359i $$-0.772517\pi$$
−0.755318 + 0.655359i $$0.772517\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ −20.0000 −0.715656
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −7.00000 −0.249841
$$786$$ 0 0
$$787$$ 53.0000 1.88925 0.944623 0.328158i $$-0.106428\pi$$
0.944623 + 0.328158i $$0.106428\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 27.0000 0.960009
$$792$$ 0 0
$$793$$ −35.0000 −1.24289
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 13.0000 0.460484 0.230242 0.973133i $$-0.426048\pi$$
0.230242 + 0.973133i $$0.426048\pi$$
$$798$$ 0 0
$$799$$ 6.00000 0.212265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 10.0000 0.352892
$$804$$ 0 0
$$805$$ 3.00000 0.105736
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −50.0000 −1.75791 −0.878953 0.476908i $$-0.841757\pi$$
−0.878953 + 0.476908i $$0.841757\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12.0000 0.420342
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 33.0000 1.15171 0.575854 0.817553i $$-0.304670\pi$$
0.575854 + 0.817553i $$0.304670\pi$$
$$822$$ 0 0
$$823$$ −49.0000 −1.70803 −0.854016 0.520246i $$-0.825840\pi$$
−0.854016 + 0.520246i $$0.825840\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ −21.0000 −0.726735
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 21.0000 0.725001 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ −42.0000 −1.44314
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ 1.00000 0.0342393 0.0171197 0.999853i $$-0.494550\pi$$
0.0171197 + 0.999853i $$0.494550\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.0000 0.922302 0.461151 0.887322i $$-0.347437\pi$$
0.461151 + 0.887322i $$0.347437\pi$$
$$858$$ 0 0
$$859$$ 9.00000 0.307076 0.153538 0.988143i $$-0.450933\pi$$
0.153538 + 0.988143i $$0.450933\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 56.0000 1.90626 0.953131 0.302558i $$-0.0978405\pi$$
0.953131 + 0.302558i $$0.0978405\pi$$
$$864$$ 0 0
$$865$$ −3.00000 −0.102003
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −5.00000 −0.169613
$$870$$ 0 0
$$871$$ 5.00000 0.169419
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 27.0000 0.912767
$$876$$ 0 0
$$877$$ −3.00000 −0.101303 −0.0506514 0.998716i $$-0.516130\pi$$
−0.0506514 + 0.998716i $$0.516130\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26.0000 −0.875962 −0.437981 0.898984i $$-0.644306\pi$$
−0.437981 + 0.898984i $$0.644306\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 55.0000 1.84672 0.923360 0.383936i $$-0.125432\pi$$
0.923360 + 0.383936i $$0.125432\pi$$
$$888$$ 0 0
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −9.00000 −0.300167
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.00000 0.199447
$$906$$ 0 0
$$907$$ −49.0000 −1.62702 −0.813509 0.581552i $$-0.802446\pi$$
−0.813509 + 0.581552i $$0.802446\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 41.0000 1.35839 0.679195 0.733958i $$-0.262329\pi$$
0.679195 + 0.733958i $$0.262329\pi$$
$$912$$ 0 0
$$913$$ 5.00000 0.165476
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −15.0000 −0.495344
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 20.0000 0.658308
$$924$$ 0 0
$$925$$ −24.0000 −0.789115
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −29.0000 −0.951459 −0.475730 0.879592i $$-0.657816\pi$$
−0.475730 + 0.879592i $$0.657816\pi$$
$$930$$ 0 0
$$931$$ 8.00000 0.262189
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 10.0000 0.327035
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 33.0000 1.07577 0.537885 0.843018i $$-0.319224\pi$$
0.537885 + 0.843018i $$0.319224\pi$$
$$942$$ 0 0
$$943$$ −3.00000 −0.0976934
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ −27.0000 −0.873699
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −13.0000 −0.418485
$$966$$ 0 0
$$967$$ −25.0000 −0.803946 −0.401973 0.915652i $$-0.631675\pi$$
−0.401973 + 0.915652i $$0.631675\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 9.00000 0.288527
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39.0000 1.24772 0.623860 0.781536i $$-0.285563\pi$$
0.623860 + 0.781536i $$0.285563\pi$$
$$978$$ 0 0
$$979$$ 90.0000 2.87641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −15.0000 −0.478426 −0.239213 0.970967i $$-0.576889\pi$$
−0.239213 + 0.970967i $$0.576889\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.00000 0.0317982
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −20.0000 −0.634043
$$996$$ 0 0
$$997$$ −19.0000 −0.601736 −0.300868 0.953666i $$-0.597276\pi$$
−0.300868 + 0.953666i $$0.597276\pi$$
$$998$$ 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 5184.2.a.s.1.1 1
3.2 odd 2 5184.2.a.i.1.1 1
4.3 odd 2 5184.2.a.x.1.1 1
8.3 odd 2 1296.2.a.e.1.1 1
8.5 even 2 648.2.a.a.1.1 1
9.2 odd 6 1728.2.i.h.577.1 2
9.4 even 3 576.2.i.d.385.1 2
9.5 odd 6 1728.2.i.h.1153.1 2
9.7 even 3 576.2.i.d.193.1 2
12.11 even 2 5184.2.a.n.1.1 1
24.5 odd 2 648.2.a.c.1.1 1
24.11 even 2 1296.2.a.i.1.1 1
36.7 odd 6 576.2.i.c.193.1 2
36.11 even 6 1728.2.i.g.577.1 2
36.23 even 6 1728.2.i.g.1153.1 2
36.31 odd 6 576.2.i.c.385.1 2
72.5 odd 6 216.2.i.a.73.1 2
72.11 even 6 432.2.i.a.145.1 2
72.13 even 6 72.2.i.a.25.1 2
72.29 odd 6 216.2.i.a.145.1 2
72.43 odd 6 144.2.i.b.49.1 2
72.59 even 6 432.2.i.a.289.1 2
72.61 even 6 72.2.i.a.49.1 yes 2
72.67 odd 6 144.2.i.b.97.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.a.25.1 2 72.13 even 6
72.2.i.a.49.1 yes 2 72.61 even 6
144.2.i.b.49.1 2 72.43 odd 6
144.2.i.b.97.1 2 72.67 odd 6
216.2.i.a.73.1 2 72.5 odd 6
216.2.i.a.145.1 2 72.29 odd 6
432.2.i.a.145.1 2 72.11 even 6
432.2.i.a.289.1 2 72.59 even 6
576.2.i.c.193.1 2 36.7 odd 6
576.2.i.c.385.1 2 36.31 odd 6
576.2.i.d.193.1 2 9.7 even 3
576.2.i.d.385.1 2 9.4 even 3
648.2.a.a.1.1 1 8.5 even 2
648.2.a.c.1.1 1 24.5 odd 2
1296.2.a.e.1.1 1 8.3 odd 2
1296.2.a.i.1.1 1 24.11 even 2
1728.2.i.g.577.1 2 36.11 even 6
1728.2.i.g.1153.1 2 36.23 even 6
1728.2.i.h.577.1 2 9.2 odd 6
1728.2.i.h.1153.1 2 9.5 odd 6
5184.2.a.i.1.1 1 3.2 odd 2
5184.2.a.n.1.1 1 12.11 even 2
5184.2.a.s.1.1 1 1.1 even 1 trivial
5184.2.a.x.1.1 1 4.3 odd 2