# Properties

 Label 9408.2.a.dc.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} -4.00000 q^{13} +3.00000 q^{15} -4.00000 q^{17} +8.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} +7.00000 q^{29} -11.0000 q^{31} +1.00000 q^{33} -4.00000 q^{37} -4.00000 q^{39} +4.00000 q^{41} +2.00000 q^{43} +3.00000 q^{45} +2.00000 q^{47} -4.00000 q^{51} +11.0000 q^{53} +3.00000 q^{55} +7.00000 q^{59} +10.0000 q^{61} -12.0000 q^{65} -10.0000 q^{67} +8.00000 q^{69} +6.00000 q^{71} +6.00000 q^{73} +4.00000 q^{75} +11.0000 q^{79} +1.00000 q^{81} +11.0000 q^{83} -12.0000 q^{85} +7.00000 q^{87} -6.00000 q^{89} -11.0000 q^{93} -7.00000 q^{97} +1.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 7.00000 1.29987 0.649934 0.759991i $$-0.274797\pi$$
0.649934 + 0.759991i $$0.274797\pi$$
$$30$$ 0 0
$$31$$ −11.0000 −1.97566 −0.987829 0.155543i $$-0.950287\pi$$
−0.987829 + 0.155543i $$0.950287\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 7.00000 0.911322 0.455661 0.890153i $$-0.349403\pi$$
0.455661 + 0.890153i $$0.349403\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 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
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 11.0000 1.20741 0.603703 0.797209i $$-0.293691\pi$$
0.603703 + 0.797209i $$0.293691\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ 0 0
$$87$$ 7.00000 0.750479
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −11.0000 −1.14065
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.00000 0.676716 0.338358 0.941018i $$-0.390129\pi$$
0.338358 + 0.941018i $$0.390129\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ 24.0000 2.23801
$$116$$ 0 0
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 4.00000 0.360668
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 17.0000 1.50851 0.754253 0.656584i $$-0.227999\pi$$
0.754253 + 0.656584i $$0.227999\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ −22.0000 −1.86602 −0.933008 0.359856i $$-0.882826\pi$$
−0.933008 + 0.359856i $$0.882826\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 21.0000 1.74396
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 11.0000 0.895167 0.447584 0.894242i $$-0.352285\pi$$
0.447584 + 0.894242i $$0.352285\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ −33.0000 −2.65062
$$156$$ 0 0
$$157$$ −12.0000 −0.957704 −0.478852 0.877896i $$-0.658947\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 0 0
$$159$$ 11.0000 0.872357
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ 22.0000 1.70241 0.851206 0.524832i $$-0.175872\pi$$
0.851206 + 0.524832i $$0.175872\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 7.00000 0.526152
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ −12.0000 −0.859338
$$196$$ 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 $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −10.0000 −0.705346
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ 0 0
$$223$$ −9.00000 −0.602685 −0.301342 0.953516i $$-0.597435\pi$$
−0.301342 + 0.953516i $$0.597435\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ −7.00000 −0.464606 −0.232303 0.972643i $$-0.574626\pi$$
−0.232303 + 0.972643i $$0.574626\pi$$
$$228$$ 0 0
$$229$$ −8.00000 −0.528655 −0.264327 0.964433i $$-0.585150\pi$$
−0.264327 + 0.964433i $$0.585150\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ 11.0000 0.714527
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 11.0000 0.697097
$$250$$ 0 0
$$251$$ −25.0000 −1.57799 −0.788993 0.614402i $$-0.789397\pi$$
−0.788993 + 0.614402i $$0.789397\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ 0 0
$$255$$ −12.0000 −0.751469
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 7.00000 0.433289
$$262$$ 0 0
$$263$$ 26.0000 1.60323 0.801614 0.597841i $$-0.203975\pi$$
0.801614 + 0.597841i $$0.203975\pi$$
$$264$$ 0 0
$$265$$ 33.0000 2.02717
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ 1.00000 0.0607457 0.0303728 0.999539i $$-0.490331\pi$$
0.0303728 + 0.999539i $$0.490331\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −32.0000 −1.92269 −0.961347 0.275340i $$-0.911209\pi$$
−0.961347 + 0.275340i $$0.911209\pi$$
$$278$$ 0 0
$$279$$ −11.0000 −0.658553
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ −2.00000 −0.118888 −0.0594438 0.998232i $$-0.518933\pi$$
−0.0594438 + 0.998232i $$0.518933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −7.00000 −0.410347
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 21.0000 1.22267
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 30.0000 1.71780
$$306$$ 0 0
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −17.0000 −0.954815 −0.477408 0.878682i $$-0.658423\pi$$
−0.477408 + 0.878682i $$0.658423\pi$$
$$318$$ 0 0
$$319$$ 7.00000 0.391925
$$320$$ 0 0
$$321$$ 7.00000 0.390702
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −16.0000 −0.887520
$$326$$ 0 0
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −30.0000 −1.63908
$$336$$ 0 0
$$337$$ −15.0000 −0.817102 −0.408551 0.912735i $$-0.633966\pi$$
−0.408551 + 0.912735i $$0.633966\pi$$
$$338$$ 0 0
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ −11.0000 −0.595683
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 24.0000 1.29212
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 2.00000 0.105556 0.0527780 0.998606i $$-0.483192\pi$$
0.0527780 + 0.998606i $$0.483192\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 18.0000 0.942163
$$366$$ 0 0
$$367$$ −17.0000 −0.887393 −0.443696 0.896177i $$-0.646333\pi$$
−0.443696 + 0.896177i $$0.646333\pi$$
$$368$$ 0 0
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 12.0000 0.621336 0.310668 0.950518i $$-0.399447\pi$$
0.310668 + 0.950518i $$0.399447\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ −28.0000 −1.44207
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 17.0000 0.870936
$$382$$ 0 0
$$383$$ −2.00000 −0.102195 −0.0510976 0.998694i $$-0.516272\pi$$
−0.0510976 + 0.998694i $$0.516272\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.00000 0.101666
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −32.0000 −1.61831
$$392$$ 0 0
$$393$$ 3.00000 0.151330
$$394$$ 0 0
$$395$$ 33.0000 1.66041
$$396$$ 0 0
$$397$$ −4.00000 −0.200754 −0.100377 0.994949i $$-0.532005\pi$$
−0.100377 + 0.994949i $$0.532005\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 16.0000 0.799002 0.399501 0.916733i $$-0.369183\pi$$
0.399501 + 0.916733i $$0.369183\pi$$
$$402$$ 0 0
$$403$$ 44.0000 2.19180
$$404$$ 0 0
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ −7.00000 −0.346128 −0.173064 0.984911i $$-0.555367\pi$$
−0.173064 + 0.984911i $$0.555367\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 33.0000 1.61991
$$416$$ 0 0
$$417$$ −22.0000 −1.07734
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 0 0
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ −16.0000 −0.776114
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 21.0000 1.00687
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −15.0000 −0.715911 −0.357955 0.933739i $$-0.616526\pi$$
−0.357955 + 0.933739i $$0.616526\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 21.0000 0.997740 0.498870 0.866677i $$-0.333748\pi$$
0.498870 + 0.866677i $$0.333748\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 0 0
$$453$$ 11.0000 0.516825
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ −33.0000 −1.53034
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 0 0
$$473$$ 2.00000 0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 11.0000 0.503655
$$478$$ 0 0
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −21.0000 −0.953561
$$486$$ 0 0
$$487$$ −3.00000 −0.135943 −0.0679715 0.997687i $$-0.521653\pi$$
−0.0679715 + 0.997687i $$0.521653\pi$$
$$488$$ 0 0
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 37.0000 1.66979 0.834893 0.550412i $$-0.185529\pi$$
0.834893 + 0.550412i $$0.185529\pi$$
$$492$$ 0 0
$$493$$ −28.0000 −1.26106
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ 22.0000 0.982888
$$502$$ 0 0
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ 3.00000 0.133235
$$508$$ 0 0
$$509$$ −43.0000 −1.90594 −0.952971 0.303062i $$-0.901991\pi$$
−0.952971 + 0.303062i $$0.901991\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 48.0000 2.11513
$$516$$ 0 0
$$517$$ 2.00000 0.0879599
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 44.0000 1.91667
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 7.00000 0.303774
$$532$$ 0 0
$$533$$ −16.0000 −0.693037
$$534$$ 0 0
$$535$$ 21.0000 0.907909
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 0 0
$$543$$ 12.0000 0.514969
$$544$$ 0 0
$$545$$ −30.0000 −1.28506
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −12.0000 −0.509372
$$556$$ 0 0
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ 19.0000 0.800755 0.400377 0.916350i $$-0.368879\pi$$
0.400377 + 0.916350i $$0.368879\pi$$
$$564$$ 0 0
$$565$$ 36.0000 1.51453
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 22.0000 0.920671 0.460336 0.887745i $$-0.347729\pi$$
0.460336 + 0.887745i $$0.347729\pi$$
$$572$$ 0 0
$$573$$ 24.0000 1.00261
$$574$$ 0 0
$$575$$ 32.0000 1.33449
$$576$$ 0 0
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ 0 0
$$579$$ −19.0000 −0.789613
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 11.0000 0.455573
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ −15.0000 −0.619116 −0.309558 0.950881i $$-0.600181\pi$$
−0.309558 + 0.950881i $$0.600181\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ 18.0000 0.735460 0.367730 0.929933i $$-0.380135\pi$$
0.367730 + 0.929933i $$0.380135\pi$$
$$600$$ 0 0
$$601$$ 37.0000 1.50926 0.754631 0.656150i $$-0.227816\pi$$
0.754631 + 0.656150i $$0.227816\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ 0 0
$$605$$ −30.0000 −1.21967
$$606$$ 0 0
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ −43.0000 −1.71180 −0.855901 0.517139i $$-0.826997\pi$$
−0.855901 + 0.517139i $$0.826997\pi$$
$$632$$ 0 0
$$633$$ −2.00000 −0.0794929
$$634$$ 0 0
$$635$$ 51.0000 2.02387
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −10.0000 −0.394976 −0.197488 0.980305i $$-0.563278\pi$$
−0.197488 + 0.980305i $$0.563278\pi$$
$$642$$ 0 0
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ 30.0000 1.17942 0.589711 0.807614i $$-0.299242\pi$$
0.589711 + 0.807614i $$0.299242\pi$$
$$648$$ 0 0
$$649$$ 7.00000 0.274774
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −43.0000 −1.68272 −0.841360 0.540475i $$-0.818245\pi$$
−0.841360 + 0.540475i $$0.818245\pi$$
$$654$$ 0 0
$$655$$ 9.00000 0.351659
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ 16.0000 0.621389
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 56.0000 2.16833
$$668$$ 0 0
$$669$$ −9.00000 −0.347960
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ 45.0000 1.73462 0.867311 0.497766i $$-0.165846\pi$$
0.867311 + 0.497766i $$0.165846\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 15.0000 0.576497 0.288248 0.957556i $$-0.406927\pi$$
0.288248 + 0.957556i $$0.406927\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −7.00000 −0.268241
$$682$$ 0 0
$$683$$ −21.0000 −0.803543 −0.401771 0.915740i $$-0.631605\pi$$
−0.401771 + 0.915740i $$0.631605\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ −8.00000 −0.305219
$$688$$ 0 0
$$689$$ −44.0000 −1.67627
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −66.0000 −2.50352
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ 0 0
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −33.0000 −1.24639 −0.623196 0.782065i $$-0.714166\pi$$
−0.623196 + 0.782065i $$0.714166\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 6.00000 0.225973
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ 11.0000 0.412532
$$712$$ 0 0
$$713$$ −88.0000 −3.29563
$$714$$ 0 0
$$715$$ −12.0000 −0.448775
$$716$$ 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
$$723$$ 25.0000 0.929760
$$724$$ 0 0
$$725$$ 28.0000 1.03989
$$726$$ 0 0
$$727$$ −7.00000 −0.259616 −0.129808 0.991539i $$-0.541436\pi$$
−0.129808 + 0.991539i $$0.541436\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ 0 0
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −50.0000 −1.83432 −0.917161 0.398517i $$-0.869525\pi$$
−0.917161 + 0.398517i $$0.869525\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 0 0
$$747$$ 11.0000 0.402469
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −3.00000 −0.109472 −0.0547358 0.998501i $$-0.517432\pi$$
−0.0547358 + 0.998501i $$0.517432\pi$$
$$752$$ 0 0
$$753$$ −25.0000 −0.911051
$$754$$ 0 0
$$755$$ 33.0000 1.20099
$$756$$ 0 0
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 0 0
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −12.0000 −0.433861
$$766$$ 0 0
$$767$$ −28.0000 −1.01102
$$768$$ 0 0
$$769$$ 3.00000 0.108183 0.0540914 0.998536i $$-0.482774\pi$$
0.0540914 + 0.998536i $$0.482774\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ 0 0
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ 0 0
$$775$$ −44.0000 −1.58053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ 0 0
$$783$$ 7.00000 0.250160
$$784$$ 0 0
$$785$$ −36.0000 −1.28490
$$786$$ 0 0
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 0 0
$$789$$ 26.0000 0.925625
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −40.0000 −1.42044
$$794$$ 0 0
$$795$$ 33.0000 1.17039
$$796$$ 0 0
$$797$$ −25.0000 −0.885545 −0.442773 0.896634i $$-0.646005\pi$$
−0.442773 + 0.896634i $$0.646005\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 21.0000 0.739235
$$808$$ 0 0
$$809$$ 40.0000 1.40633 0.703163 0.711029i $$-0.251771\pi$$
0.703163 + 0.711029i $$0.251771\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ 1.00000 0.0350715
$$814$$ 0 0
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −37.0000 −1.29131 −0.645654 0.763630i $$-0.723415\pi$$
−0.645654 + 0.763630i $$0.723415\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ 21.0000 0.730242 0.365121 0.930960i $$-0.381028\pi$$
0.365121 + 0.930960i $$0.381028\pi$$
$$828$$ 0 0
$$829$$ 24.0000 0.833554 0.416777 0.909009i $$-0.363160\pi$$
0.416777 + 0.909009i $$0.363160\pi$$
$$830$$ 0 0
$$831$$ −32.0000 −1.11007
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 66.0000 2.28402
$$836$$ 0 0
$$837$$ −11.0000 −0.380216
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ 0 0
$$843$$ −30.0000 −1.03325
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −2.00000 −0.0686398
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 34.0000 1.16007 0.580033 0.814593i $$-0.303040\pi$$
0.580033 + 0.814593i $$0.303040\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −18.0000 −0.612727 −0.306364 0.951915i $$-0.599112\pi$$
−0.306364 + 0.951915i $$0.599112\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 11.0000 0.373149
$$870$$ 0 0
$$871$$ 40.0000 1.35535
$$872$$ 0 0
$$873$$ −7.00000 −0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ 0 0
$$879$$ −7.00000 −0.236104
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 0 0
$$885$$ 21.0000 0.705907
$$886$$ 0 0
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 36.0000 1.20335
$$896$$ 0 0
$$897$$ −32.0000 −1.06845
$$898$$ 0 0
$$899$$ −77.0000 −2.56809
$$900$$ 0 0
$$901$$ −44.0000 −1.46585
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 36.0000 1.19668
$$906$$ 0 0
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 58.0000 1.92163 0.960813 0.277198i $$-0.0894057\pi$$
0.960813 + 0.277198i $$0.0894057\pi$$
$$912$$ 0 0
$$913$$ 11.0000 0.364047
$$914$$ 0 0
$$915$$ 30.0000 0.991769
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −12.0000 −0.392862
$$934$$ 0 0
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ 13.0000 0.424691 0.212346 0.977195i $$-0.431890\pi$$
0.212346 + 0.977195i $$0.431890\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 0 0
$$941$$ −49.0000 −1.59735 −0.798677 0.601760i $$-0.794466\pi$$
−0.798677 + 0.601760i $$0.794466\pi$$
$$942$$ 0 0
$$943$$ 32.0000 1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 40.0000 1.29983 0.649913 0.760009i $$-0.274805\pi$$
0.649913 + 0.760009i $$0.274805\pi$$
$$948$$ 0 0
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ −17.0000 −0.551263
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ 72.0000 2.32987
$$956$$ 0 0
$$957$$ 7.00000 0.226278
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 90.0000 2.90323
$$962$$ 0 0
$$963$$ 7.00000 0.225572
$$964$$ 0 0
$$965$$ −57.0000 −1.83489
$$966$$ 0 0
$$967$$ −5.00000 −0.160789 −0.0803946 0.996763i $$-0.525618\pi$$
−0.0803946 + 0.996763i $$0.525618\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.0000 −0.866471 −0.433236 0.901281i $$-0.642628\pi$$
−0.433236 + 0.901281i $$0.642628\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −16.0000 −0.512410
$$976$$ 0 0
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 0 0
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 23.0000 0.730619 0.365310 0.930886i $$-0.380963\pi$$
0.365310 + 0.930886i $$0.380963\pi$$
$$992$$ 0 0
$$993$$ −12.0000 −0.380808
$$994$$ 0 0
$$995$$ 60.0000 1.90213
$$996$$ 0 0
$$997$$ 2.00000 0.0633406 0.0316703 0.999498i $$-0.489917\pi$$
0.0316703 + 0.999498i $$0.489917\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 9408.2.a.dc.1.1 1
4.3 odd 2 9408.2.a.bl.1.1 1
7.3 odd 6 1344.2.q.u.961.1 2
7.5 odd 6 1344.2.q.u.193.1 2
7.6 odd 2 9408.2.a.e.1.1 1
8.3 odd 2 4704.2.a.s.1.1 1
8.5 even 2 4704.2.a.b.1.1 1
28.3 even 6 1344.2.q.k.961.1 2
28.19 even 6 1344.2.q.k.193.1 2
28.27 even 2 9408.2.a.bt.1.1 1
56.3 even 6 672.2.q.f.289.1 yes 2
56.5 odd 6 672.2.q.a.193.1 2
56.13 odd 2 4704.2.a.bf.1.1 1
56.19 even 6 672.2.q.f.193.1 yes 2
56.27 even 2 4704.2.a.o.1.1 1
56.45 odd 6 672.2.q.a.289.1 yes 2
168.5 even 6 2016.2.s.n.865.1 2
168.59 odd 6 2016.2.s.k.289.1 2
168.101 even 6 2016.2.s.n.289.1 2
168.131 odd 6 2016.2.s.k.865.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.a.193.1 2 56.5 odd 6
672.2.q.a.289.1 yes 2 56.45 odd 6
672.2.q.f.193.1 yes 2 56.19 even 6
672.2.q.f.289.1 yes 2 56.3 even 6
1344.2.q.k.193.1 2 28.19 even 6
1344.2.q.k.961.1 2 28.3 even 6
1344.2.q.u.193.1 2 7.5 odd 6
1344.2.q.u.961.1 2 7.3 odd 6
2016.2.s.k.289.1 2 168.59 odd 6
2016.2.s.k.865.1 2 168.131 odd 6
2016.2.s.n.289.1 2 168.101 even 6
2016.2.s.n.865.1 2 168.5 even 6
4704.2.a.b.1.1 1 8.5 even 2
4704.2.a.o.1.1 1 56.27 even 2
4704.2.a.s.1.1 1 8.3 odd 2
4704.2.a.bf.1.1 1 56.13 odd 2
9408.2.a.e.1.1 1 7.6 odd 2
9408.2.a.bl.1.1 1 4.3 odd 2
9408.2.a.bt.1.1 1 28.27 even 2
9408.2.a.dc.1.1 1 1.1 even 1 trivial