# Properties

 Label 882.2.a.c.1.1 Level $882$ Weight $2$ Character 882.1 Self dual yes Analytic conductor $7.043$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 882.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{16} +4.00000 q^{17} +8.00000 q^{19} -1.00000 q^{20} +5.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} +5.00000 q^{29} +3.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.00000 q^{37} -8.00000 q^{38} +1.00000 q^{40} +2.00000 q^{43} -5.00000 q^{44} -4.00000 q^{46} +6.00000 q^{47} +4.00000 q^{50} +9.00000 q^{53} +5.00000 q^{55} -5.00000 q^{58} +11.0000 q^{59} -6.00000 q^{61} -3.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} +4.00000 q^{68} -2.00000 q^{71} +10.0000 q^{73} +4.00000 q^{74} +8.00000 q^{76} +3.00000 q^{79} -1.00000 q^{80} +7.00000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +5.00000 q^{88} +6.00000 q^{89} +4.00000 q^{92} -6.00000 q^{94} -8.00000 q^{95} +7.00000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 5.00000 1.06600
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ −8.00000 −1.29777
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 4.00000 0.565685
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 5.00000 0.674200
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −5.00000 −0.656532
$$59$$ 11.0000 1.43208 0.716039 0.698060i $$-0.245953\pi$$
0.716039 + 0.698060i $$0.245953\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −3.00000 −0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 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
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.00000 0.768350 0.384175 0.923260i $$-0.374486\pi$$
0.384175 + 0.923260i $$0.374486\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 5.00000 0.533002
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −4.00000 −0.400000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −5.00000 −0.476731
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ −11.0000 −1.01263
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ 3.00000 0.269408
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ 9.00000 0.798621 0.399310 0.916816i $$-0.369250\pi$$
0.399310 + 0.916816i $$0.369250\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.00000 −0.0873704 −0.0436852 0.999045i $$-0.513910\pi$$
−0.0436852 + 0.999045i $$0.513910\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2.00000 0.167836
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −5.00000 −0.415227
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ −8.00000 −0.648886
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −3.00000 −0.238667
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −7.00000 −0.543305
$$167$$ 14.0000 1.08335 0.541676 0.840587i $$-0.317790\pi$$
0.541676 + 0.840587i $$0.317790\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −20.0000 −1.46254
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 8.00000 0.580381
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 5.00000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ −7.00000 −0.502571
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −40.0000 −2.76686
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ 9.00000 0.618123
$$213$$ 0 0
$$214$$ 3.00000 0.205076
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 5.00000 0.337100
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 16.0000 1.06430
$$227$$ −3.00000 −0.199117 −0.0995585 0.995032i $$-0.531743\pi$$
−0.0995585 + 0.995032i $$0.531743\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −5.00000 −0.328266
$$233$$ 4.00000 0.262049 0.131024 0.991379i $$-0.458173\pi$$
0.131024 + 0.991379i $$0.458173\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 11.0000 0.716039
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ 0 0
$$250$$ −9.00000 −0.569210
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ −20.0000 −1.25739
$$254$$ −9.00000 −0.564710
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1.00000 0.0617802
$$263$$ 30.0000 1.84988 0.924940 0.380114i $$-0.124115\pi$$
0.924940 + 0.380114i $$0.124115\pi$$
$$264$$ 0 0
$$265$$ −9.00000 −0.552866
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ −31.0000 −1.89010 −0.945052 0.326921i $$-0.893989\pi$$
−0.945052 + 0.326921i $$0.893989\pi$$
$$270$$ 0 0
$$271$$ 15.0000 0.911185 0.455593 0.890188i $$-0.349427\pi$$
0.455593 + 0.890188i $$0.349427\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 20.0000 1.20605
$$276$$ 0 0
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ 10.0000 0.594438 0.297219 0.954809i $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 5.00000 0.293610
$$291$$ 0 0
$$292$$ 10.0000 0.585206
$$293$$ 21.0000 1.22683 0.613417 0.789760i $$-0.289795\pi$$
0.613417 + 0.789760i $$0.289795\pi$$
$$294$$ 0 0
$$295$$ −11.0000 −0.640445
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −19.0000 −1.09333
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 6.00000 0.343559
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 3.00000 0.170389
$$311$$ 32.0000 1.81455 0.907277 0.420534i $$-0.138157\pi$$
0.907277 + 0.420534i $$0.138157\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 3.00000 0.168763
$$317$$ −3.00000 −0.168497 −0.0842484 0.996445i $$-0.526849\pi$$
−0.0842484 + 0.996445i $$0.526849\pi$$
$$318$$ 0 0
$$319$$ −25.0000 −1.39973
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 32.0000 1.78053
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 7.00000 0.384175
$$333$$ 0 0
$$334$$ −14.0000 −0.766046
$$335$$ 2.00000 0.109272
$$336$$ 0 0
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ −4.00000 −0.216930
$$341$$ −15.0000 −0.812296
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 2.00000 0.106149
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ 17.0000 0.887393 0.443696 0.896177i $$-0.353667\pi$$
0.443696 + 0.896177i $$0.353667\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 0 0
$$370$$ −4.00000 −0.207950
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ 20.0000 1.03418
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.00000 −0.254493
$$387$$ 0 0
$$388$$ 7.00000 0.355371
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ −3.00000 −0.150946
$$396$$ 0 0
$$397$$ 36.0000 1.80679 0.903394 0.428811i $$-0.141067\pi$$
0.903394 + 0.428811i $$0.141067\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ −4.00000 −0.200000
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0000 0.991363
$$408$$ 0 0
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −7.00000 −0.343616
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 40.0000 1.95646
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ −2.00000 −0.0973585
$$423$$ 0 0
$$424$$ −9.00000 −0.437079
$$425$$ −16.0000 −0.776114
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.00000 −0.145010
$$429$$ 0 0
$$430$$ 2.00000 0.0964486
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 32.0000 1.53077
$$438$$ 0 0
$$439$$ 15.0000 0.715911 0.357955 0.933739i $$-0.383474\pi$$
0.357955 + 0.933739i $$0.383474\pi$$
$$440$$ −5.00000 −0.238366
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −17.0000 −0.807694 −0.403847 0.914826i $$-0.632327\pi$$
−0.403847 + 0.914826i $$0.632327\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 7.00000 0.331460
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −16.0000 −0.755087 −0.377543 0.925992i $$-0.623231\pi$$
−0.377543 + 0.925992i $$0.623231\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −16.0000 −0.752577
$$453$$ 0 0
$$454$$ 3.00000 0.140797
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 31.0000 1.45012 0.725059 0.688686i $$-0.241812\pi$$
0.725059 + 0.688686i $$0.241812\pi$$
$$458$$ 20.0000 0.934539
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ −4.00000 −0.185296
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 6.00000 0.276759
$$471$$ 0 0
$$472$$ −11.0000 −0.506316
$$473$$ −10.0000 −0.459800
$$474$$ 0 0
$$475$$ −32.0000 −1.46826
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 25.0000 1.13872
$$483$$ 0 0
$$484$$ 14.0000 0.636364
$$485$$ −7.00000 −0.317854
$$486$$ 0 0
$$487$$ 5.00000 0.226572 0.113286 0.993562i $$-0.463862\pi$$
0.113286 + 0.993562i $$0.463862\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ 9.00000 0.402492
$$501$$ 0 0
$$502$$ 21.0000 0.937276
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 20.0000 0.889108
$$507$$ 0 0
$$508$$ 9.00000 0.399310
$$509$$ −15.0000 −0.664863 −0.332432 0.943127i $$-0.607869\pi$$
−0.332432 + 0.943127i $$0.607869\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −30.0000 −1.31940
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ −1.00000 −0.0436852
$$525$$ 0 0
$$526$$ −30.0000 −1.30806
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 9.00000 0.390935
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 3.00000 0.129701
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ 31.0000 1.33650
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −15.0000 −0.644305
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 0 0
$$550$$ −20.0000 −0.852803
$$551$$ 40.0000 1.70406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 16.0000 0.679775
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 23.0000 0.974541 0.487271 0.873251i $$-0.337993\pi$$
0.487271 + 0.873251i $$0.337993\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000 0.0843649
$$563$$ −17.0000 −0.716465 −0.358232 0.933632i $$-0.616620\pi$$
−0.358232 + 0.933632i $$0.616620\pi$$
$$564$$ 0 0
$$565$$ 16.0000 0.673125
$$566$$ −10.0000 −0.420331
$$567$$ 0 0
$$568$$ 2.00000 0.0839181
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −30.0000 −1.25546 −0.627730 0.778431i $$-0.716016\pi$$
−0.627730 + 0.778431i $$0.716016\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −16.0000 −0.667246
$$576$$ 0 0
$$577$$ 31.0000 1.29055 0.645273 0.763952i $$-0.276743\pi$$
0.645273 + 0.763952i $$0.276743\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ −5.00000 −0.207614
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −45.0000 −1.86371
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −21.0000 −0.867502
$$587$$ −35.0000 −1.44460 −0.722302 0.691577i $$-0.756916\pi$$
−0.722302 + 0.691577i $$0.756916\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 11.0000 0.452863
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 19.0000 0.773099
$$605$$ −14.0000 −0.569181
$$606$$ 0 0
$$607$$ −27.0000 −1.09590 −0.547948 0.836512i $$-0.684591\pi$$
−0.547948 + 0.836512i $$0.684591\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ 0 0
$$610$$ −6.00000 −0.242933
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12.0000 0.484675 0.242338 0.970192i $$-0.422086\pi$$
0.242338 + 0.970192i $$0.422086\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ −3.00000 −0.120483
$$621$$ 0 0
$$622$$ −32.0000 −1.28308
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ −1.00000 −0.0399680
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −19.0000 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$632$$ −3.00000 −0.119334
$$633$$ 0 0
$$634$$ 3.00000 0.119145
$$635$$ −9.00000 −0.357154
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 25.0000 0.989759
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −26.0000 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −32.0000 −1.25902
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ −55.0000 −2.15894
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 1.00000 0.0390732
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ −7.00000 −0.271653
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.0000 0.774403
$$668$$ 14.0000 0.541676
$$669$$ 0 0
$$670$$ −2.00000 −0.0772667
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ −9.00000 −0.346667
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 27.0000 1.03769 0.518847 0.854867i $$-0.326361\pi$$
0.518847 + 0.854867i $$0.326361\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ 15.0000 0.574380
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 14.0000 0.531050
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 14.0000 0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5.00000 0.188847 0.0944237 0.995532i $$-0.469899\pi$$
0.0944237 + 0.995532i $$0.469899\pi$$
$$702$$ 0 0
$$703$$ −32.0000 −1.20690
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ −2.00000 −0.0750587
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 10.0000 0.373197
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −45.0000 −1.67473
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −20.0000 −0.742781
$$726$$ 0 0
$$727$$ 7.00000 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 10.0000 0.370117
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ −17.0000 −0.627481
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 10.0000 0.368355
$$738$$ 0 0
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −30.0000 −1.10059 −0.550297 0.834969i $$-0.685485\pi$$
−0.550297 + 0.834969i $$0.685485\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 32.0000 1.17160
$$747$$ 0 0
$$748$$ −20.0000 −0.731272
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 45.0000 1.64207 0.821037 0.570875i $$-0.193396\pi$$
0.821037 + 0.570875i $$0.193396\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −19.0000 −0.691481
$$756$$ 0 0
$$757$$ −54.0000 −1.96266 −0.981332 0.192323i $$-0.938398\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 8.00000 0.290191
$$761$$ −8.00000 −0.290000 −0.145000 0.989432i $$-0.546318\pi$$
−0.145000 + 0.989432i $$0.546318\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −34.0000 −1.22847
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.00000 0.179954
$$773$$ −10.0000 −0.359675 −0.179838 0.983696i $$-0.557557\pi$$
−0.179838 + 0.983696i $$0.557557\pi$$
$$774$$ 0 0
$$775$$ −12.0000 −0.431053
$$776$$ −7.00000 −0.251285
$$777$$ 0 0
$$778$$ −2.00000 −0.0717035
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ −16.0000 −0.572159
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 4.00000 0.142766
$$786$$ 0 0
$$787$$ −18.0000 −0.641631 −0.320815 0.947142i $$-0.603957\pi$$
−0.320815 + 0.947142i $$0.603957\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 3.00000 0.106735
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −36.0000 −1.27759
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −21.0000 −0.743858 −0.371929 0.928261i $$-0.621304\pi$$
−0.371929 + 0.928261i $$0.621304\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 4.00000 0.141421
$$801$$ 0 0
$$802$$ 24.0000 0.847469
$$803$$ −50.0000 −1.76446
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 10.0000 0.351799
$$809$$ −40.0000 −1.40633 −0.703163 0.711029i $$-0.748229\pi$$
−0.703163 + 0.711029i $$0.748229\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −20.0000 −0.701000
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 25.0000 0.874105
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.0000 0.872506 0.436253 0.899824i $$-0.356305\pi$$
0.436253 + 0.899824i $$0.356305\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −9.00000 −0.312961 −0.156480 0.987681i $$-0.550015\pi$$
−0.156480 + 0.987681i $$0.550015\pi$$
$$828$$ 0 0
$$829$$ −32.0000 −1.11141 −0.555703 0.831381i $$-0.687551\pi$$
−0.555703 + 0.831381i $$0.687551\pi$$
$$830$$ 7.00000 0.242974
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −14.0000 −0.484490
$$836$$ −40.0000 −1.38343
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 28.0000 0.966667 0.483334 0.875436i $$-0.339426\pi$$
0.483334 + 0.875436i $$0.339426\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −30.0000 −1.03387
$$843$$ 0 0
$$844$$ 2.00000 0.0688428
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 9.00000 0.309061
$$849$$ 0 0
$$850$$ 16.0000 0.548795
$$851$$ −16.0000 −0.548473
$$852$$ 0 0
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −34.0000 −1.16007 −0.580033 0.814593i $$-0.696960\pi$$
−0.580033 + 0.814593i $$0.696960\pi$$
$$860$$ −2.00000 −0.0681994
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −10.0000 −0.340404 −0.170202 0.985409i $$-0.554442\pi$$
−0.170202 + 0.985409i $$0.554442\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ −15.0000 −0.506225
$$879$$ 0 0
$$880$$ 5.00000 0.168550
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ −40.0000 −1.34611 −0.673054 0.739594i $$-0.735018\pi$$
−0.673054 + 0.739594i $$0.735018\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 17.0000 0.571126
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ 0 0
$$892$$ −7.00000 −0.234377
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 16.0000 0.533927
$$899$$ 15.0000 0.500278
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 16.0000 0.532152
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −3.00000 −0.0995585
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ −35.0000 −1.15833
$$914$$ −31.0000 −1.02539
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 0 0
$$922$$ −14.0000 −0.461065
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 16.0000 0.526077
$$926$$ −16.0000 −0.525793
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 4.00000 0.131024
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ 20.0000 0.654070
$$936$$ 0 0
$$937$$ 35.0000 1.14340 0.571700 0.820463i $$-0.306284\pi$$
0.571700 + 0.820463i $$0.306284\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −6.00000 −0.195698
$$941$$ 11.0000 0.358590 0.179295 0.983795i $$-0.442618\pi$$
0.179295 + 0.983795i $$0.442618\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 11.0000 0.358020
$$945$$ 0 0
$$946$$ 10.0000 0.325128
$$947$$ 32.0000 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 32.0000 1.03822
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2.00000 −0.0647864 −0.0323932 0.999475i $$-0.510313\pi$$
−0.0323932 + 0.999475i $$0.510313\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 38.0000 1.22772
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −25.0000 −0.805196
$$965$$ −5.00000 −0.160956
$$966$$ 0 0
$$967$$ −61.0000 −1.96163 −0.980814 0.194946i $$-0.937547\pi$$
−0.980814 + 0.194946i $$0.937547\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ 0 0
$$970$$ 7.00000 0.224756
$$971$$ −15.0000 −0.481373 −0.240686 0.970603i $$-0.577373\pi$$
−0.240686 + 0.970603i $$0.577373\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −5.00000 −0.160210
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 9.00000 0.287202
$$983$$ 60.0000 1.91370 0.956851 0.290578i $$-0.0938475\pi$$
0.956851 + 0.290578i $$0.0938475\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ −20.0000 −0.636930
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 47.0000 1.49300 0.746502 0.665383i $$-0.231732\pi$$
0.746502 + 0.665383i $$0.231732\pi$$
$$992$$ −3.00000 −0.0952501
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ −10.0000 −0.316544
$$999$$ 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 882.2.a.c.1.1 1
3.2 odd 2 294.2.a.e.1.1 1
4.3 odd 2 7056.2.a.w.1.1 1
7.2 even 3 126.2.g.c.109.1 2
7.3 odd 6 882.2.g.i.667.1 2
7.4 even 3 126.2.g.c.37.1 2
7.5 odd 6 882.2.g.i.361.1 2
7.6 odd 2 882.2.a.d.1.1 1
12.11 even 2 2352.2.a.t.1.1 1
15.14 odd 2 7350.2.a.bl.1.1 1
21.2 odd 6 42.2.e.a.25.1 2
21.5 even 6 294.2.e.b.67.1 2
21.11 odd 6 42.2.e.a.37.1 yes 2
21.17 even 6 294.2.e.b.79.1 2
21.20 even 2 294.2.a.f.1.1 1
24.5 odd 2 9408.2.a.ce.1.1 1
24.11 even 2 9408.2.a.q.1.1 1
28.11 odd 6 1008.2.s.k.289.1 2
28.23 odd 6 1008.2.s.k.865.1 2
28.27 even 2 7056.2.a.bl.1.1 1
63.2 odd 6 1134.2.h.e.109.1 2
63.4 even 3 1134.2.h.l.541.1 2
63.11 odd 6 1134.2.e.l.919.1 2
63.16 even 3 1134.2.h.l.109.1 2
63.23 odd 6 1134.2.e.l.865.1 2
63.25 even 3 1134.2.e.e.919.1 2
63.32 odd 6 1134.2.h.e.541.1 2
63.58 even 3 1134.2.e.e.865.1 2
84.11 even 6 336.2.q.b.289.1 2
84.23 even 6 336.2.q.b.193.1 2
84.47 odd 6 2352.2.q.u.1537.1 2
84.59 odd 6 2352.2.q.u.961.1 2
84.83 odd 2 2352.2.a.f.1.1 1
105.2 even 12 1050.2.o.a.949.2 4
105.23 even 12 1050.2.o.a.949.1 4
105.32 even 12 1050.2.o.a.499.1 4
105.44 odd 6 1050.2.i.l.151.1 2
105.53 even 12 1050.2.o.a.499.2 4
105.74 odd 6 1050.2.i.l.751.1 2
105.104 even 2 7350.2.a.q.1.1 1
168.11 even 6 1344.2.q.s.961.1 2
168.53 odd 6 1344.2.q.g.961.1 2
168.83 odd 2 9408.2.a.cr.1.1 1
168.107 even 6 1344.2.q.s.193.1 2
168.125 even 2 9408.2.a.z.1.1 1
168.149 odd 6 1344.2.q.g.193.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.a.25.1 2 21.2 odd 6
42.2.e.a.37.1 yes 2 21.11 odd 6
126.2.g.c.37.1 2 7.4 even 3
126.2.g.c.109.1 2 7.2 even 3
294.2.a.e.1.1 1 3.2 odd 2
294.2.a.f.1.1 1 21.20 even 2
294.2.e.b.67.1 2 21.5 even 6
294.2.e.b.79.1 2 21.17 even 6
336.2.q.b.193.1 2 84.23 even 6
336.2.q.b.289.1 2 84.11 even 6
882.2.a.c.1.1 1 1.1 even 1 trivial
882.2.a.d.1.1 1 7.6 odd 2
882.2.g.i.361.1 2 7.5 odd 6
882.2.g.i.667.1 2 7.3 odd 6
1008.2.s.k.289.1 2 28.11 odd 6
1008.2.s.k.865.1 2 28.23 odd 6
1050.2.i.l.151.1 2 105.44 odd 6
1050.2.i.l.751.1 2 105.74 odd 6
1050.2.o.a.499.1 4 105.32 even 12
1050.2.o.a.499.2 4 105.53 even 12
1050.2.o.a.949.1 4 105.23 even 12
1050.2.o.a.949.2 4 105.2 even 12
1134.2.e.e.865.1 2 63.58 even 3
1134.2.e.e.919.1 2 63.25 even 3
1134.2.e.l.865.1 2 63.23 odd 6
1134.2.e.l.919.1 2 63.11 odd 6
1134.2.h.e.109.1 2 63.2 odd 6
1134.2.h.e.541.1 2 63.32 odd 6
1134.2.h.l.109.1 2 63.16 even 3
1134.2.h.l.541.1 2 63.4 even 3
1344.2.q.g.193.1 2 168.149 odd 6
1344.2.q.g.961.1 2 168.53 odd 6
1344.2.q.s.193.1 2 168.107 even 6
1344.2.q.s.961.1 2 168.11 even 6
2352.2.a.f.1.1 1 84.83 odd 2
2352.2.a.t.1.1 1 12.11 even 2
2352.2.q.u.961.1 2 84.59 odd 6
2352.2.q.u.1537.1 2 84.47 odd 6
7056.2.a.w.1.1 1 4.3 odd 2
7056.2.a.bl.1.1 1 28.27 even 2
7350.2.a.q.1.1 1 105.104 even 2
7350.2.a.bl.1.1 1 15.14 odd 2
9408.2.a.q.1.1 1 24.11 even 2
9408.2.a.z.1.1 1 168.125 even 2
9408.2.a.ce.1.1 1 24.5 odd 2
9408.2.a.cr.1.1 1 168.83 odd 2