Properties

 Label 8400.2.a.br.1.1 Level $8400$ Weight $2$ Character 8400.1 Self dual yes Analytic conductor $67.074$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8400.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$67.0743376979$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8400.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.00000 q^{13} -2.00000 q^{19} -1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{27} +2.00000 q^{29} +6.00000 q^{31} -2.00000 q^{33} -8.00000 q^{37} -2.00000 q^{39} -10.0000 q^{41} -12.0000 q^{47} +1.00000 q^{49} +2.00000 q^{53} -2.00000 q^{57} +2.00000 q^{61} -1.00000 q^{63} +4.00000 q^{67} +8.00000 q^{69} -14.0000 q^{71} +2.00000 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +16.0000 q^{83} +2.00000 q^{87} -6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{93} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 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
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ −14.0000 −0.959264
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ −30.0000 −1.98246 −0.991228 0.132164i $$-0.957808\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4.00000 −0.259828
$$238$$ 0 0
$$239$$ 10.0000 0.646846 0.323423 0.946254i $$-0.395166\pi$$
0.323423 + 0.946254i $$0.395166\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 0 0
$$273$$ 2.00000 0.121046
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −24.0000 −1.44202 −0.721010 0.692925i $$-0.756322\pi$$
−0.721010 + 0.692925i $$0.756322\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.0000 0.590281
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ −8.00000 −0.467365 −0.233682 0.972313i $$-0.575078\pi$$
−0.233682 + 0.972313i $$0.575078\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −6.00000 −0.331801
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ 0 0
$$333$$ −8.00000 −0.438397
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 0 0
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −34.0000 −1.79445 −0.897226 0.441572i $$-0.854421\pi$$
−0.897226 + 0.441572i $$0.854421\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ −20.0000 −1.03556 −0.517780 0.855514i $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 30.0000 1.50566 0.752828 0.658217i $$-0.228689\pi$$
0.752828 + 0.658217i $$0.228689\pi$$
$$398$$ 0 0
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ −12.0000 −0.597763
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.0000 0.793091
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −22.0000 −1.05970 −0.529851 0.848091i $$-0.677752\pi$$
−0.529851 + 0.848091i $$0.677752\pi$$
$$432$$ 0 0
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ 6.00000 0.286364 0.143182 0.989696i $$-0.454267\pi$$
0.143182 + 0.989696i $$0.454267\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 20.0000 0.941763
$$452$$ 0 0
$$453$$ −16.0000 −0.751746
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.0000 0.561336 0.280668 0.959805i $$-0.409444\pi$$
0.280668 + 0.959805i $$0.409444\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 0 0
$$483$$ −8.00000 −0.364013
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −26.0000 −1.17336 −0.586682 0.809818i $$-0.699566\pi$$
−0.586682 + 0.809818i $$0.699566\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 14.0000 0.627986
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 10.0000 0.438108 0.219054 0.975713i $$-0.429703\pi$$
0.219054 + 0.975713i $$0.429703\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ 0 0
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ 46.0000 1.97769 0.988847 0.148933i $$-0.0475840\pi$$
0.988847 + 0.148933i $$0.0475840\pi$$
$$542$$ 0 0
$$543$$ −2.00000 −0.0858282
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ −16.0000 −0.664937
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ 8.00000 0.328521 0.164260 0.986417i $$-0.447476\pi$$
0.164260 + 0.986417i $$0.447476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10.0000 −0.409273
$$598$$ 0 0
$$599$$ 14.0000 0.572024 0.286012 0.958226i $$-0.407670\pi$$
0.286012 + 0.958226i $$0.407670\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 0 0
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 4.00000 0.159745
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −12.0000 −0.473234 −0.236617 0.971603i $$-0.576039\pi$$
−0.236617 + 0.971603i $$0.576039\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −6.00000 −0.235159
$$652$$ 0 0
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 42.0000 1.63609 0.818044 0.575156i $$-0.195059\pi$$
0.818044 + 0.575156i $$0.195059\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 20.0000 0.770943 0.385472 0.922720i $$-0.374039\pi$$
0.385472 + 0.922720i $$0.374039\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 24.0000 0.922395 0.461197 0.887298i $$-0.347420\pi$$
0.461197 + 0.887298i $$0.347420\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −30.0000 −1.14457
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −30.0000 −1.14125 −0.570627 0.821209i $$-0.693300\pi$$
−0.570627 + 0.821209i $$0.693300\pi$$
$$692$$ 0 0
$$693$$ 2.00000 0.0759737
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ 0 0
$$703$$ 16.0000 0.603451
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 48.0000 1.79761
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 10.0000 0.373457
$$718$$ 0 0
$$719$$ −52.0000 −1.93927 −0.969636 0.244551i $$-0.921359\pi$$
−0.969636 + 0.244551i $$0.921359\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ −10.0000 −0.371904
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 48.0000 1.78022 0.890111 0.455744i $$-0.150627\pi$$
0.890111 + 0.455744i $$0.150627\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 16.0000 0.585409
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −44.0000 −1.59921 −0.799604 0.600528i $$-0.794957\pi$$
−0.799604 + 0.600528i $$0.794957\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ 6.00000 0.217215
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 0 0
$$773$$ 40.0000 1.43870 0.719350 0.694648i $$-0.244440\pi$$
0.719350 + 0.694648i $$0.244440\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 8.00000 0.286998
$$778$$ 0 0
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −52.0000 −1.85360 −0.926800 0.375555i $$-0.877452\pi$$
−0.926800 + 0.375555i $$0.877452\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ −4.00000 −0.142044
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ −4.00000 −0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18.0000 −0.633630
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −18.0000 −0.632065 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$812$$ 0 0
$$813$$ −14.0000 −0.491001
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ 0 0
$$831$$ −24.0000 −0.832551
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 7.00000 0.240523
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ −64.0000 −2.19389
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.00000 0.136637 0.0683187 0.997664i $$-0.478237\pi$$
0.0683187 + 0.997664i $$0.478237\pi$$
$$858$$ 0 0
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ 0 0
$$861$$ 10.0000 0.340799
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −17.0000 −0.577350
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 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
$$879$$ −8.00000 −0.269833
$$880$$ 0 0
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ 0 0
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 24.0000 0.803129
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −42.0000 −1.39152 −0.695761 0.718273i $$-0.744933\pi$$
−0.695761 + 0.718273i $$0.744933\pi$$
$$912$$ 0 0
$$913$$ −32.0000 −1.05905
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 12.0000 0.396275
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ 0 0
$$923$$ 28.0000 0.921631
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 24.0000 0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ 0 0
$$943$$ −80.0000 −2.60516
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −4.00000 −0.129302
$$958$$ 0 0
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 4.00000 0.128898
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ −14.0000 −0.448819
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 0 0
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 12.0000 0.381964
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 0 0
$$993$$ 32.0000 1.01549
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8400.2.a.br.1.1 1
4.3 odd 2 4200.2.a.k.1.1 1
5.2 odd 4 1680.2.t.c.1009.1 2
5.3 odd 4 1680.2.t.c.1009.2 2
5.4 even 2 8400.2.a.u.1.1 1
15.2 even 4 5040.2.t.n.1009.1 2
15.8 even 4 5040.2.t.n.1009.2 2
20.3 even 4 840.2.t.a.169.1 2
20.7 even 4 840.2.t.a.169.2 yes 2
20.19 odd 2 4200.2.a.x.1.1 1
60.23 odd 4 2520.2.t.c.1009.2 2
60.47 odd 4 2520.2.t.c.1009.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.a.169.1 2 20.3 even 4
840.2.t.a.169.2 yes 2 20.7 even 4
1680.2.t.c.1009.1 2 5.2 odd 4
1680.2.t.c.1009.2 2 5.3 odd 4
2520.2.t.c.1009.1 2 60.47 odd 4
2520.2.t.c.1009.2 2 60.23 odd 4
4200.2.a.k.1.1 1 4.3 odd 2
4200.2.a.x.1.1 1 20.19 odd 2
5040.2.t.n.1009.1 2 15.2 even 4
5040.2.t.n.1009.2 2 15.8 even 4
8400.2.a.u.1.1 1 5.4 even 2
8400.2.a.br.1.1 1 1.1 even 1 trivial