# Properties

 Label 1176.4.a.ba.1.2 Level $1176$ Weight $4$ Character 1176.1 Self dual yes Analytic conductor $69.386$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.3862461668$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-6.46638$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -0.726342 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -0.726342 q^{5} +9.00000 q^{9} -64.4895 q^{11} -71.8475 q^{13} +2.17903 q^{15} -48.9034 q^{17} -34.3968 q^{19} -0.903350 q^{23} -124.472 q^{25} -27.0000 q^{27} +226.686 q^{29} +275.795 q^{31} +193.468 q^{33} +295.209 q^{37} +215.543 q^{39} -186.604 q^{41} -455.317 q^{43} -6.53708 q^{45} -282.334 q^{47} +146.710 q^{51} +356.214 q^{53} +46.8414 q^{55} +103.191 q^{57} -729.370 q^{59} -274.353 q^{61} +52.1859 q^{65} +193.272 q^{67} +2.71005 q^{69} +40.5277 q^{71} +206.472 q^{73} +373.417 q^{75} +937.741 q^{79} +81.0000 q^{81} -911.607 q^{83} +35.5206 q^{85} -680.059 q^{87} +949.975 q^{89} -827.385 q^{93} +24.9839 q^{95} -39.4687 q^{97} -580.405 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} - 4q^{5} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} - 4q^{5} + 36q^{9} + 14q^{11} - 22q^{13} + 12q^{15} - 96q^{17} + 26q^{19} + 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} - 42q^{33} + 562q^{37} + 66q^{39} - 428q^{41} - 258q^{43} - 36q^{45} + 80q^{47} + 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} + 276q^{61} + 2196q^{65} + 150q^{67} - 288q^{69} - 848q^{71} + 218q^{73} - 330q^{75} + 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} + 344q^{89} + 714q^{93} + 2004q^{95} + 622q^{97} + 126q^{99} + O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −0.726342 −0.0649660 −0.0324830 0.999472i $$-0.510341\pi$$
−0.0324830 + 0.999472i $$0.510341\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −64.4895 −1.76766 −0.883832 0.467804i $$-0.845045\pi$$
−0.883832 + 0.467804i $$0.845045\pi$$
$$12$$ 0 0
$$13$$ −71.8475 −1.53284 −0.766420 0.642340i $$-0.777964\pi$$
−0.766420 + 0.642340i $$0.777964\pi$$
$$14$$ 0 0
$$15$$ 2.17903 0.0375081
$$16$$ 0 0
$$17$$ −48.9034 −0.697694 −0.348847 0.937180i $$-0.613427\pi$$
−0.348847 + 0.937180i $$0.613427\pi$$
$$18$$ 0 0
$$19$$ −34.3968 −0.415325 −0.207663 0.978201i $$-0.566586\pi$$
−0.207663 + 0.978201i $$0.566586\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.903350 −0.00818963 −0.00409482 0.999992i $$-0.501303\pi$$
−0.00409482 + 0.999992i $$0.501303\pi$$
$$24$$ 0 0
$$25$$ −124.472 −0.995779
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 226.686 1.45154 0.725769 0.687939i $$-0.241484\pi$$
0.725769 + 0.687939i $$0.241484\pi$$
$$30$$ 0 0
$$31$$ 275.795 1.59788 0.798940 0.601411i $$-0.205395\pi$$
0.798940 + 0.601411i $$0.205395\pi$$
$$32$$ 0 0
$$33$$ 193.468 1.02056
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 295.209 1.31168 0.655839 0.754901i $$-0.272315\pi$$
0.655839 + 0.754901i $$0.272315\pi$$
$$38$$ 0 0
$$39$$ 215.543 0.884985
$$40$$ 0 0
$$41$$ −186.604 −0.710798 −0.355399 0.934715i $$-0.615655\pi$$
−0.355399 + 0.934715i $$0.615655\pi$$
$$42$$ 0 0
$$43$$ −455.317 −1.61477 −0.807386 0.590023i $$-0.799119\pi$$
−0.807386 + 0.590023i $$0.799119\pi$$
$$44$$ 0 0
$$45$$ −6.53708 −0.0216553
$$46$$ 0 0
$$47$$ −282.334 −0.876228 −0.438114 0.898920i $$-0.644353\pi$$
−0.438114 + 0.898920i $$0.644353\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 146.710 0.402814
$$52$$ 0 0
$$53$$ 356.214 0.923204 0.461602 0.887087i $$-0.347275\pi$$
0.461602 + 0.887087i $$0.347275\pi$$
$$54$$ 0 0
$$55$$ 46.8414 0.114838
$$56$$ 0 0
$$57$$ 103.191 0.239788
$$58$$ 0 0
$$59$$ −729.370 −1.60942 −0.804711 0.593667i $$-0.797680\pi$$
−0.804711 + 0.593667i $$0.797680\pi$$
$$60$$ 0 0
$$61$$ −274.353 −0.575856 −0.287928 0.957652i $$-0.592966\pi$$
−0.287928 + 0.957652i $$0.592966\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 52.1859 0.0995825
$$66$$ 0 0
$$67$$ 193.272 0.352417 0.176209 0.984353i $$-0.443617\pi$$
0.176209 + 0.984353i $$0.443617\pi$$
$$68$$ 0 0
$$69$$ 2.71005 0.00472829
$$70$$ 0 0
$$71$$ 40.5277 0.0677429 0.0338715 0.999426i $$-0.489216\pi$$
0.0338715 + 0.999426i $$0.489216\pi$$
$$72$$ 0 0
$$73$$ 206.472 0.331038 0.165519 0.986207i $$-0.447070\pi$$
0.165519 + 0.986207i $$0.447070\pi$$
$$74$$ 0 0
$$75$$ 373.417 0.574914
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 937.741 1.33549 0.667747 0.744388i $$-0.267259\pi$$
0.667747 + 0.744388i $$0.267259\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −911.607 −1.20556 −0.602782 0.797906i $$-0.705941\pi$$
−0.602782 + 0.797906i $$0.705941\pi$$
$$84$$ 0 0
$$85$$ 35.5206 0.0453264
$$86$$ 0 0
$$87$$ −680.059 −0.838045
$$88$$ 0 0
$$89$$ 949.975 1.13143 0.565715 0.824601i $$-0.308600\pi$$
0.565715 + 0.824601i $$0.308600\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −827.385 −0.922536
$$94$$ 0 0
$$95$$ 24.9839 0.0269820
$$96$$ 0 0
$$97$$ −39.4687 −0.0413138 −0.0206569 0.999787i $$-0.506576\pi$$
−0.0206569 + 0.999787i $$0.506576\pi$$
$$98$$ 0 0
$$99$$ −580.405 −0.589221
$$100$$ 0 0
$$101$$ 316.874 0.312180 0.156090 0.987743i $$-0.450111\pi$$
0.156090 + 0.987743i $$0.450111\pi$$
$$102$$ 0 0
$$103$$ 322.634 0.308641 0.154321 0.988021i $$-0.450681\pi$$
0.154321 + 0.988021i $$0.450681\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 681.451 0.615686 0.307843 0.951437i $$-0.400393\pi$$
0.307843 + 0.951437i $$0.400393\pi$$
$$108$$ 0 0
$$109$$ −455.967 −0.400677 −0.200338 0.979727i $$-0.564204\pi$$
−0.200338 + 0.979727i $$0.564204\pi$$
$$110$$ 0 0
$$111$$ −885.627 −0.757297
$$112$$ 0 0
$$113$$ −796.025 −0.662688 −0.331344 0.943510i $$-0.607502\pi$$
−0.331344 + 0.943510i $$0.607502\pi$$
$$114$$ 0 0
$$115$$ 0.656141 0.000532048 0
$$116$$ 0 0
$$117$$ −646.628 −0.510947
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2827.89 2.12464
$$122$$ 0 0
$$123$$ 559.813 0.410379
$$124$$ 0 0
$$125$$ 181.202 0.129658
$$126$$ 0 0
$$127$$ 2333.92 1.63072 0.815362 0.578952i $$-0.196538\pi$$
0.815362 + 0.578952i $$0.196538\pi$$
$$128$$ 0 0
$$129$$ 1365.95 0.932289
$$130$$ 0 0
$$131$$ 1886.98 1.25852 0.629259 0.777195i $$-0.283358\pi$$
0.629259 + 0.777195i $$0.283358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 19.6112 0.0125027
$$136$$ 0 0
$$137$$ 2947.26 1.83797 0.918983 0.394297i $$-0.129012\pi$$
0.918983 + 0.394297i $$0.129012\pi$$
$$138$$ 0 0
$$139$$ −955.433 −0.583013 −0.291506 0.956569i $$-0.594156\pi$$
−0.291506 + 0.956569i $$0.594156\pi$$
$$140$$ 0 0
$$141$$ 847.003 0.505890
$$142$$ 0 0
$$143$$ 4633.41 2.70955
$$144$$ 0 0
$$145$$ −164.652 −0.0943006
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2183.74 1.20066 0.600332 0.799751i $$-0.295035\pi$$
0.600332 + 0.799751i $$0.295035\pi$$
$$150$$ 0 0
$$151$$ 404.719 0.218116 0.109058 0.994035i $$-0.465217\pi$$
0.109058 + 0.994035i $$0.465217\pi$$
$$152$$ 0 0
$$153$$ −440.130 −0.232565
$$154$$ 0 0
$$155$$ −200.322 −0.103808
$$156$$ 0 0
$$157$$ 929.582 0.472540 0.236270 0.971687i $$-0.424075\pi$$
0.236270 + 0.971687i $$0.424075\pi$$
$$158$$ 0 0
$$159$$ −1068.64 −0.533012
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1425.68 −0.685079 −0.342540 0.939503i $$-0.611287\pi$$
−0.342540 + 0.939503i $$0.611287\pi$$
$$164$$ 0 0
$$165$$ −140.524 −0.0663018
$$166$$ 0 0
$$167$$ −4185.15 −1.93926 −0.969631 0.244572i $$-0.921353\pi$$
−0.969631 + 0.244572i $$0.921353\pi$$
$$168$$ 0 0
$$169$$ 2965.07 1.34960
$$170$$ 0 0
$$171$$ −309.572 −0.138442
$$172$$ 0 0
$$173$$ −2235.55 −0.982462 −0.491231 0.871029i $$-0.663453\pi$$
−0.491231 + 0.871029i $$0.663453\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 2188.11 0.929200
$$178$$ 0 0
$$179$$ −941.479 −0.393126 −0.196563 0.980491i $$-0.562978\pi$$
−0.196563 + 0.980491i $$0.562978\pi$$
$$180$$ 0 0
$$181$$ 467.540 0.192000 0.0960000 0.995381i $$-0.469395\pi$$
0.0960000 + 0.995381i $$0.469395\pi$$
$$182$$ 0 0
$$183$$ 823.058 0.332471
$$184$$ 0 0
$$185$$ −214.423 −0.0852144
$$186$$ 0 0
$$187$$ 3153.75 1.23329
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 275.403 0.104332 0.0521660 0.998638i $$-0.483387\pi$$
0.0521660 + 0.998638i $$0.483387\pi$$
$$192$$ 0 0
$$193$$ 1640.30 0.611767 0.305884 0.952069i $$-0.401048\pi$$
0.305884 + 0.952069i $$0.401048\pi$$
$$194$$ 0 0
$$195$$ −156.558 −0.0574940
$$196$$ 0 0
$$197$$ 1303.88 0.471560 0.235780 0.971806i $$-0.424236\pi$$
0.235780 + 0.971806i $$0.424236\pi$$
$$198$$ 0 0
$$199$$ 1327.36 0.472833 0.236417 0.971652i $$-0.424027\pi$$
0.236417 + 0.971652i $$0.424027\pi$$
$$200$$ 0 0
$$201$$ −579.817 −0.203468
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 135.539 0.0461777
$$206$$ 0 0
$$207$$ −8.13015 −0.00272988
$$208$$ 0 0
$$209$$ 2218.23 0.734155
$$210$$ 0 0
$$211$$ −4753.28 −1.55085 −0.775426 0.631439i $$-0.782465\pi$$
−0.775426 + 0.631439i $$0.782465\pi$$
$$212$$ 0 0
$$213$$ −121.583 −0.0391114
$$214$$ 0 0
$$215$$ 330.716 0.104905
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −619.417 −0.191125
$$220$$ 0 0
$$221$$ 3513.58 1.06945
$$222$$ 0 0
$$223$$ −513.149 −0.154094 −0.0770470 0.997027i $$-0.524549\pi$$
−0.0770470 + 0.997027i $$0.524549\pi$$
$$224$$ 0 0
$$225$$ −1120.25 −0.331926
$$226$$ 0 0
$$227$$ −3309.34 −0.967616 −0.483808 0.875174i $$-0.660747\pi$$
−0.483808 + 0.875174i $$0.660747\pi$$
$$228$$ 0 0
$$229$$ 5909.47 1.70528 0.852639 0.522501i $$-0.175001\pi$$
0.852639 + 0.522501i $$0.175001\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −353.572 −0.0994131 −0.0497065 0.998764i $$-0.515829\pi$$
−0.0497065 + 0.998764i $$0.515829\pi$$
$$234$$ 0 0
$$235$$ 205.071 0.0569250
$$236$$ 0 0
$$237$$ −2813.22 −0.771048
$$238$$ 0 0
$$239$$ −1652.55 −0.447259 −0.223629 0.974674i $$-0.571791\pi$$
−0.223629 + 0.974674i $$0.571791\pi$$
$$240$$ 0 0
$$241$$ 3106.95 0.830440 0.415220 0.909721i $$-0.363705\pi$$
0.415220 + 0.909721i $$0.363705\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2471.33 0.636627
$$248$$ 0 0
$$249$$ 2734.82 0.696033
$$250$$ 0 0
$$251$$ −1771.77 −0.445551 −0.222776 0.974870i $$-0.571512\pi$$
−0.222776 + 0.974870i $$0.571512\pi$$
$$252$$ 0 0
$$253$$ 58.2566 0.0144765
$$254$$ 0 0
$$255$$ −106.562 −0.0261692
$$256$$ 0 0
$$257$$ −1478.13 −0.358766 −0.179383 0.983779i $$-0.557410\pi$$
−0.179383 + 0.983779i $$0.557410\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2040.18 0.483846
$$262$$ 0 0
$$263$$ −1431.97 −0.335739 −0.167869 0.985809i $$-0.553689\pi$$
−0.167869 + 0.985809i $$0.553689\pi$$
$$264$$ 0 0
$$265$$ −258.734 −0.0599769
$$266$$ 0 0
$$267$$ −2849.93 −0.653231
$$268$$ 0 0
$$269$$ −285.398 −0.0646879 −0.0323439 0.999477i $$-0.510297\pi$$
−0.0323439 + 0.999477i $$0.510297\pi$$
$$270$$ 0 0
$$271$$ 5249.33 1.17666 0.588328 0.808622i $$-0.299786\pi$$
0.588328 + 0.808622i $$0.299786\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 8027.16 1.76020
$$276$$ 0 0
$$277$$ 7147.87 1.55045 0.775224 0.631687i $$-0.217637\pi$$
0.775224 + 0.631687i $$0.217637\pi$$
$$278$$ 0 0
$$279$$ 2482.16 0.532627
$$280$$ 0 0
$$281$$ 4228.36 0.897661 0.448831 0.893617i $$-0.351841\pi$$
0.448831 + 0.893617i $$0.351841\pi$$
$$282$$ 0 0
$$283$$ −8342.38 −1.75231 −0.876154 0.482031i $$-0.839899\pi$$
−0.876154 + 0.482031i $$0.839899\pi$$
$$284$$ 0 0
$$285$$ −74.9516 −0.0155781
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2521.46 −0.513223
$$290$$ 0 0
$$291$$ 118.406 0.0238525
$$292$$ 0 0
$$293$$ −5038.84 −1.00468 −0.502342 0.864669i $$-0.667528\pi$$
−0.502342 + 0.864669i $$0.667528\pi$$
$$294$$ 0 0
$$295$$ 529.772 0.104558
$$296$$ 0 0
$$297$$ 1741.22 0.340187
$$298$$ 0 0
$$299$$ 64.9035 0.0125534
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −950.623 −0.180237
$$304$$ 0 0
$$305$$ 199.274 0.0374111
$$306$$ 0 0
$$307$$ 4869.67 0.905300 0.452650 0.891688i $$-0.350479\pi$$
0.452650 + 0.891688i $$0.350479\pi$$
$$308$$ 0 0
$$309$$ −967.901 −0.178194
$$310$$ 0 0
$$311$$ 1452.84 0.264897 0.132449 0.991190i $$-0.457716\pi$$
0.132449 + 0.991190i $$0.457716\pi$$
$$312$$ 0 0
$$313$$ −5696.27 −1.02867 −0.514333 0.857591i $$-0.671960\pi$$
−0.514333 + 0.857591i $$0.671960\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3471.47 −0.615070 −0.307535 0.951537i $$-0.599504\pi$$
−0.307535 + 0.951537i $$0.599504\pi$$
$$318$$ 0 0
$$319$$ −14618.9 −2.56583
$$320$$ 0 0
$$321$$ −2044.35 −0.355466
$$322$$ 0 0
$$323$$ 1682.12 0.289770
$$324$$ 0 0
$$325$$ 8943.03 1.52637
$$326$$ 0 0
$$327$$ 1367.90 0.231331
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 6254.38 1.03859 0.519293 0.854597i $$-0.326195\pi$$
0.519293 + 0.854597i $$0.326195\pi$$
$$332$$ 0 0
$$333$$ 2656.88 0.437226
$$334$$ 0 0
$$335$$ −140.382 −0.0228951
$$336$$ 0 0
$$337$$ 8006.96 1.29426 0.647132 0.762378i $$-0.275968\pi$$
0.647132 + 0.762378i $$0.275968\pi$$
$$338$$ 0 0
$$339$$ 2388.08 0.382603
$$340$$ 0 0
$$341$$ −17785.9 −2.82451
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −1.96842 −0.000307178 0
$$346$$ 0 0
$$347$$ 7635.59 1.18127 0.590634 0.806940i $$-0.298878\pi$$
0.590634 + 0.806940i $$0.298878\pi$$
$$348$$ 0 0
$$349$$ 10358.0 1.58869 0.794345 0.607468i $$-0.207815\pi$$
0.794345 + 0.607468i $$0.207815\pi$$
$$350$$ 0 0
$$351$$ 1939.88 0.294995
$$352$$ 0 0
$$353$$ −3384.23 −0.510268 −0.255134 0.966906i $$-0.582120\pi$$
−0.255134 + 0.966906i $$0.582120\pi$$
$$354$$ 0 0
$$355$$ −29.4369 −0.00440099
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8194.61 −1.20472 −0.602361 0.798224i $$-0.705773\pi$$
−0.602361 + 0.798224i $$0.705773\pi$$
$$360$$ 0 0
$$361$$ −5675.86 −0.827505
$$362$$ 0 0
$$363$$ −8483.67 −1.22666
$$364$$ 0 0
$$365$$ −149.970 −0.0215062
$$366$$ 0 0
$$367$$ −805.510 −0.114570 −0.0572851 0.998358i $$-0.518244\pi$$
−0.0572851 + 0.998358i $$0.518244\pi$$
$$368$$ 0 0
$$369$$ −1679.44 −0.236933
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7905.71 1.09743 0.548716 0.836009i $$-0.315117\pi$$
0.548716 + 0.836009i $$0.315117\pi$$
$$374$$ 0 0
$$375$$ −543.607 −0.0748580
$$376$$ 0 0
$$377$$ −16286.8 −2.22497
$$378$$ 0 0
$$379$$ 3324.24 0.450540 0.225270 0.974296i $$-0.427674\pi$$
0.225270 + 0.974296i $$0.427674\pi$$
$$380$$ 0 0
$$381$$ −7001.76 −0.941499
$$382$$ 0 0
$$383$$ 8470.76 1.13012 0.565060 0.825050i $$-0.308853\pi$$
0.565060 + 0.825050i $$0.308853\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4097.85 −0.538258
$$388$$ 0 0
$$389$$ −7908.40 −1.03078 −0.515388 0.856957i $$-0.672352\pi$$
−0.515388 + 0.856957i $$0.672352\pi$$
$$390$$ 0 0
$$391$$ 44.1769 0.00571386
$$392$$ 0 0
$$393$$ −5660.93 −0.726606
$$394$$ 0 0
$$395$$ −681.120 −0.0867617
$$396$$ 0 0
$$397$$ 9781.58 1.23658 0.618292 0.785949i $$-0.287825\pi$$
0.618292 + 0.785949i $$0.287825\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −795.653 −0.0990849 −0.0495424 0.998772i $$-0.515776\pi$$
−0.0495424 + 0.998772i $$0.515776\pi$$
$$402$$ 0 0
$$403$$ −19815.2 −2.44929
$$404$$ 0 0
$$405$$ −58.8337 −0.00721844
$$406$$ 0 0
$$407$$ −19037.9 −2.31860
$$408$$ 0 0
$$409$$ −8584.72 −1.03787 −0.518933 0.854815i $$-0.673671\pi$$
−0.518933 + 0.854815i $$0.673671\pi$$
$$410$$ 0 0
$$411$$ −8841.78 −1.06115
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 662.138 0.0783207
$$416$$ 0 0
$$417$$ 2866.30 0.336603
$$418$$ 0 0
$$419$$ 7447.09 0.868292 0.434146 0.900843i $$-0.357050\pi$$
0.434146 + 0.900843i $$0.357050\pi$$
$$420$$ 0 0
$$421$$ −4446.76 −0.514779 −0.257390 0.966308i $$-0.582862\pi$$
−0.257390 + 0.966308i $$0.582862\pi$$
$$422$$ 0 0
$$423$$ −2541.01 −0.292076
$$424$$ 0 0
$$425$$ 6087.12 0.694750
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −13900.2 −1.56436
$$430$$ 0 0
$$431$$ −8963.70 −1.00178 −0.500889 0.865512i $$-0.666994\pi$$
−0.500889 + 0.865512i $$0.666994\pi$$
$$432$$ 0 0
$$433$$ 16173.6 1.79504 0.897520 0.440973i $$-0.145367\pi$$
0.897520 + 0.440973i $$0.145367\pi$$
$$434$$ 0 0
$$435$$ 493.955 0.0544445
$$436$$ 0 0
$$437$$ 31.0724 0.00340136
$$438$$ 0 0
$$439$$ 6298.43 0.684755 0.342378 0.939562i $$-0.388768\pi$$
0.342378 + 0.939562i $$0.388768\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16784.7 −1.80015 −0.900074 0.435738i $$-0.856488\pi$$
−0.900074 + 0.435738i $$0.856488\pi$$
$$444$$ 0 0
$$445$$ −690.007 −0.0735044
$$446$$ 0 0
$$447$$ −6551.22 −0.693203
$$448$$ 0 0
$$449$$ 2733.88 0.287349 0.143674 0.989625i $$-0.454108\pi$$
0.143674 + 0.989625i $$0.454108\pi$$
$$450$$ 0 0
$$451$$ 12034.0 1.25645
$$452$$ 0 0
$$453$$ −1214.16 −0.125930
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9229.20 0.944691 0.472345 0.881414i $$-0.343408\pi$$
0.472345 + 0.881414i $$0.343408\pi$$
$$458$$ 0 0
$$459$$ 1320.39 0.134271
$$460$$ 0 0
$$461$$ 19726.7 1.99298 0.996491 0.0837048i $$-0.0266753\pi$$
0.996491 + 0.0837048i $$0.0266753\pi$$
$$462$$ 0 0
$$463$$ 368.924 0.0370310 0.0185155 0.999829i $$-0.494106\pi$$
0.0185155 + 0.999829i $$0.494106\pi$$
$$464$$ 0 0
$$465$$ 600.965 0.0599335
$$466$$ 0 0
$$467$$ 13309.6 1.31883 0.659414 0.751780i $$-0.270804\pi$$
0.659414 + 0.751780i $$0.270804\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2788.75 −0.272821
$$472$$ 0 0
$$473$$ 29363.2 2.85438
$$474$$ 0 0
$$475$$ 4281.46 0.413572
$$476$$ 0 0
$$477$$ 3205.93 0.307735
$$478$$ 0 0
$$479$$ −11562.8 −1.10296 −0.551479 0.834189i $$-0.685936\pi$$
−0.551479 + 0.834189i $$0.685936\pi$$
$$480$$ 0 0
$$481$$ −21210.0 −2.01059
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 28.6677 0.00268399
$$486$$ 0 0
$$487$$ 2628.13 0.244542 0.122271 0.992497i $$-0.460982\pi$$
0.122271 + 0.992497i $$0.460982\pi$$
$$488$$ 0 0
$$489$$ 4277.04 0.395531
$$490$$ 0 0
$$491$$ −12319.9 −1.13236 −0.566181 0.824281i $$-0.691580\pi$$
−0.566181 + 0.824281i $$0.691580\pi$$
$$492$$ 0 0
$$493$$ −11085.7 −1.01273
$$494$$ 0 0
$$495$$ 421.573 0.0382794
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −3305.81 −0.296570 −0.148285 0.988945i $$-0.547375\pi$$
−0.148285 + 0.988945i $$0.547375\pi$$
$$500$$ 0 0
$$501$$ 12555.5 1.11963
$$502$$ 0 0
$$503$$ 3072.72 0.272377 0.136189 0.990683i $$-0.456515\pi$$
0.136189 + 0.990683i $$0.456515\pi$$
$$504$$ 0 0
$$505$$ −230.159 −0.0202811
$$506$$ 0 0
$$507$$ −8895.20 −0.779190
$$508$$ 0 0
$$509$$ −13569.8 −1.18167 −0.590836 0.806791i $$-0.701202\pi$$
−0.590836 + 0.806791i $$0.701202\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 928.715 0.0799294
$$514$$ 0 0
$$515$$ −234.342 −0.0200512
$$516$$ 0 0
$$517$$ 18207.6 1.54888
$$518$$ 0 0
$$519$$ 6706.66 0.567225
$$520$$ 0 0
$$521$$ 10733.4 0.902566 0.451283 0.892381i $$-0.350967\pi$$
0.451283 + 0.892381i $$0.350967\pi$$
$$522$$ 0 0
$$523$$ −8348.15 −0.697971 −0.348986 0.937128i $$-0.613474\pi$$
−0.348986 + 0.937128i $$0.613474\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −13487.3 −1.11483
$$528$$ 0 0
$$529$$ −12166.2 −0.999933
$$530$$ 0 0
$$531$$ −6564.33 −0.536474
$$532$$ 0 0
$$533$$ 13407.1 1.08954
$$534$$ 0 0
$$535$$ −494.967 −0.0399987
$$536$$ 0 0
$$537$$ 2824.44 0.226971
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −23107.3 −1.83634 −0.918170 0.396186i $$-0.870334\pi$$
−0.918170 + 0.396186i $$0.870334\pi$$
$$542$$ 0 0
$$543$$ −1402.62 −0.110851
$$544$$ 0 0
$$545$$ 331.188 0.0260304
$$546$$ 0 0
$$547$$ 13935.2 1.08926 0.544630 0.838676i $$-0.316670\pi$$
0.544630 + 0.838676i $$0.316670\pi$$
$$548$$ 0 0
$$549$$ −2469.17 −0.191952
$$550$$ 0 0
$$551$$ −7797.29 −0.602860
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 643.268 0.0491986
$$556$$ 0 0
$$557$$ −15047.5 −1.14467 −0.572337 0.820018i $$-0.693963\pi$$
−0.572337 + 0.820018i $$0.693963\pi$$
$$558$$ 0 0
$$559$$ 32713.4 2.47519
$$560$$ 0 0
$$561$$ −9461.25 −0.712040
$$562$$ 0 0
$$563$$ −15442.2 −1.15597 −0.577983 0.816049i $$-0.696160\pi$$
−0.577983 + 0.816049i $$0.696160\pi$$
$$564$$ 0 0
$$565$$ 578.187 0.0430522
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13362.1 −0.984481 −0.492240 0.870459i $$-0.663822\pi$$
−0.492240 + 0.870459i $$0.663822\pi$$
$$570$$ 0 0
$$571$$ −22185.3 −1.62596 −0.812981 0.582290i $$-0.802157\pi$$
−0.812981 + 0.582290i $$0.802157\pi$$
$$572$$ 0 0
$$573$$ −826.208 −0.0602362
$$574$$ 0 0
$$575$$ 112.442 0.00815507
$$576$$ 0 0
$$577$$ 18951.4 1.36734 0.683671 0.729791i $$-0.260383\pi$$
0.683671 + 0.729791i $$0.260383\pi$$
$$578$$ 0 0
$$579$$ −4920.89 −0.353204
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −22972.1 −1.63191
$$584$$ 0 0
$$585$$ 469.673 0.0331942
$$586$$ 0 0
$$587$$ −19579.5 −1.37672 −0.688359 0.725371i $$-0.741668\pi$$
−0.688359 + 0.725371i $$0.741668\pi$$
$$588$$ 0 0
$$589$$ −9486.48 −0.663640
$$590$$ 0 0
$$591$$ −3911.63 −0.272255
$$592$$ 0 0
$$593$$ −5613.01 −0.388699 −0.194350 0.980932i $$-0.562260\pi$$
−0.194350 + 0.980932i $$0.562260\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −3982.07 −0.272990
$$598$$ 0 0
$$599$$ 21925.5 1.49558 0.747790 0.663936i $$-0.231115\pi$$
0.747790 + 0.663936i $$0.231115\pi$$
$$600$$ 0 0
$$601$$ 2067.51 0.140326 0.0701628 0.997536i $$-0.477648\pi$$
0.0701628 + 0.997536i $$0.477648\pi$$
$$602$$ 0 0
$$603$$ 1739.45 0.117472
$$604$$ 0 0
$$605$$ −2054.02 −0.138029
$$606$$ 0 0
$$607$$ 10167.2 0.679859 0.339930 0.940451i $$-0.389597\pi$$
0.339930 + 0.940451i $$0.389597\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 20285.0 1.34312
$$612$$ 0 0
$$613$$ −18186.1 −1.19825 −0.599127 0.800654i $$-0.704485\pi$$
−0.599127 + 0.800654i $$0.704485\pi$$
$$614$$ 0 0
$$615$$ −406.616 −0.0266607
$$616$$ 0 0
$$617$$ 6584.41 0.429625 0.214812 0.976655i $$-0.431086\pi$$
0.214812 + 0.976655i $$0.431086\pi$$
$$618$$ 0 0
$$619$$ 7779.72 0.505159 0.252579 0.967576i $$-0.418721\pi$$
0.252579 + 0.967576i $$0.418721\pi$$
$$620$$ 0 0
$$621$$ 24.3905 0.00157610
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15427.4 0.987356
$$626$$ 0 0
$$627$$ −6654.70 −0.423865
$$628$$ 0 0
$$629$$ −14436.7 −0.915150
$$630$$ 0 0
$$631$$ 3787.78 0.238969 0.119484 0.992836i $$-0.461876\pi$$
0.119484 + 0.992836i $$0.461876\pi$$
$$632$$ 0 0
$$633$$ 14259.9 0.895384
$$634$$ 0 0
$$635$$ −1695.22 −0.105942
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 364.749 0.0225810
$$640$$ 0 0
$$641$$ −19930.9 −1.22812 −0.614058 0.789261i $$-0.710464\pi$$
−0.614058 + 0.789261i $$0.710464\pi$$
$$642$$ 0 0
$$643$$ −3185.18 −0.195352 −0.0976759 0.995218i $$-0.531141\pi$$
−0.0976759 + 0.995218i $$0.531141\pi$$
$$644$$ 0 0
$$645$$ −992.148 −0.0605671
$$646$$ 0 0
$$647$$ 16943.0 1.02952 0.514759 0.857335i $$-0.327881\pi$$
0.514759 + 0.857335i $$0.327881\pi$$
$$648$$ 0 0
$$649$$ 47036.7 2.84492
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −20078.8 −1.20328 −0.601640 0.798767i $$-0.705486\pi$$
−0.601640 + 0.798767i $$0.705486\pi$$
$$654$$ 0 0
$$655$$ −1370.59 −0.0817609
$$656$$ 0 0
$$657$$ 1858.25 0.110346
$$658$$ 0 0
$$659$$ −9442.46 −0.558158 −0.279079 0.960268i $$-0.590029\pi$$
−0.279079 + 0.960268i $$0.590029\pi$$
$$660$$ 0 0
$$661$$ −761.990 −0.0448381 −0.0224190 0.999749i $$-0.507137\pi$$
−0.0224190 + 0.999749i $$0.507137\pi$$
$$662$$ 0 0
$$663$$ −10540.8 −0.617449
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −204.777 −0.0118876
$$668$$ 0 0
$$669$$ 1539.45 0.0889663
$$670$$ 0 0
$$671$$ 17692.8 1.01792
$$672$$ 0 0
$$673$$ −16111.6 −0.922818 −0.461409 0.887188i $$-0.652656\pi$$
−0.461409 + 0.887188i $$0.652656\pi$$
$$674$$ 0 0
$$675$$ 3360.76 0.191638
$$676$$ 0 0
$$677$$ −31241.5 −1.77357 −0.886786 0.462180i $$-0.847067\pi$$
−0.886786 + 0.462180i $$0.847067\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 9928.03 0.558653
$$682$$ 0 0
$$683$$ −30113.5 −1.68706 −0.843530 0.537082i $$-0.819527\pi$$
−0.843530 + 0.537082i $$0.819527\pi$$
$$684$$ 0 0
$$685$$ −2140.72 −0.119405
$$686$$ 0 0
$$687$$ −17728.4 −0.984542
$$688$$ 0 0
$$689$$ −25593.1 −1.41512
$$690$$ 0 0
$$691$$ 9750.85 0.536816 0.268408 0.963305i $$-0.413502\pi$$
0.268408 + 0.963305i $$0.413502\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 693.971 0.0378760
$$696$$ 0 0
$$697$$ 9125.58 0.495920
$$698$$ 0 0
$$699$$ 1060.71 0.0573962
$$700$$ 0 0
$$701$$ 11851.6 0.638559 0.319280 0.947661i $$-0.396559\pi$$
0.319280 + 0.947661i $$0.396559\pi$$
$$702$$ 0 0
$$703$$ −10154.3 −0.544773
$$704$$ 0 0
$$705$$ −615.214 −0.0328657
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1322.39 0.0700472 0.0350236 0.999386i $$-0.488849\pi$$
0.0350236 + 0.999386i $$0.488849\pi$$
$$710$$ 0 0
$$711$$ 8439.67 0.445165
$$712$$ 0 0
$$713$$ −249.140 −0.0130861
$$714$$ 0 0
$$715$$ −3365.44 −0.176028
$$716$$ 0 0
$$717$$ 4957.66 0.258225
$$718$$ 0 0
$$719$$ −7301.56 −0.378723 −0.189362 0.981907i $$-0.560642\pi$$
−0.189362 + 0.981907i $$0.560642\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −9320.84 −0.479455
$$724$$ 0 0
$$725$$ −28216.2 −1.44541
$$726$$ 0 0
$$727$$ 6088.72 0.310616 0.155308 0.987866i $$-0.450363\pi$$
0.155308 + 0.987866i $$0.450363\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 22266.5 1.12662
$$732$$ 0 0
$$733$$ 26578.5 1.33929 0.669644 0.742683i $$-0.266447\pi$$
0.669644 + 0.742683i $$0.266447\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12464.0 −0.622955
$$738$$ 0 0
$$739$$ 25219.8 1.25538 0.627689 0.778464i $$-0.284001\pi$$
0.627689 + 0.778464i $$0.284001\pi$$
$$740$$ 0 0
$$741$$ −7413.98 −0.367557
$$742$$ 0 0
$$743$$ 2634.28 0.130071 0.0650353 0.997883i $$-0.479284\pi$$
0.0650353 + 0.997883i $$0.479284\pi$$
$$744$$ 0 0
$$745$$ −1586.14 −0.0780023
$$746$$ 0 0
$$747$$ −8204.46 −0.401855
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 5767.29 0.280228 0.140114 0.990135i $$-0.455253\pi$$
0.140114 + 0.990135i $$0.455253\pi$$
$$752$$ 0 0
$$753$$ 5315.32 0.257239
$$754$$ 0 0
$$755$$ −293.965 −0.0141702
$$756$$ 0 0
$$757$$ 33378.7 1.60260 0.801302 0.598260i $$-0.204141\pi$$
0.801302 + 0.598260i $$0.204141\pi$$
$$758$$ 0 0
$$759$$ −174.770 −0.00835802
$$760$$ 0 0
$$761$$ −3626.09 −0.172727 −0.0863637 0.996264i $$-0.527525\pi$$
−0.0863637 + 0.996264i $$0.527525\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 319.685 0.0151088
$$766$$ 0 0
$$767$$ 52403.4 2.46698
$$768$$ 0 0
$$769$$ −3426.15 −0.160663 −0.0803316 0.996768i $$-0.525598\pi$$
−0.0803316 + 0.996768i $$0.525598\pi$$
$$770$$ 0 0
$$771$$ 4434.38 0.207134
$$772$$ 0 0
$$773$$ −13108.2 −0.609920 −0.304960 0.952365i $$-0.598643\pi$$
−0.304960 + 0.952365i $$0.598643\pi$$
$$774$$ 0 0
$$775$$ −34328.9 −1.59114
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6418.60 0.295212
$$780$$ 0 0
$$781$$ −2613.61 −0.119747
$$782$$ 0 0
$$783$$ −6120.53 −0.279348
$$784$$ 0 0
$$785$$ −675.194 −0.0306990
$$786$$ 0 0
$$787$$ −39319.7 −1.78094 −0.890468 0.455045i $$-0.849623\pi$$
−0.890468 + 0.455045i $$0.849623\pi$$
$$788$$ 0 0
$$789$$ 4295.92 0.193839
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 19711.5 0.882696
$$794$$ 0 0
$$795$$ 776.201 0.0346277
$$796$$ 0 0
$$797$$ 14597.8 0.648785 0.324393 0.945923i $$-0.394840\pi$$
0.324393 + 0.945923i $$0.394840\pi$$
$$798$$ 0 0
$$799$$ 13807.1 0.611339
$$800$$ 0 0
$$801$$ 8549.78 0.377143
$$802$$ 0 0
$$803$$ −13315.3 −0.585164
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 856.194 0.0373475
$$808$$ 0 0
$$809$$ 26827.4 1.16589 0.582943 0.812513i $$-0.301901\pi$$
0.582943 + 0.812513i $$0.301901\pi$$
$$810$$ 0 0
$$811$$ 5177.09 0.224158 0.112079 0.993699i $$-0.464249\pi$$
0.112079 + 0.993699i $$0.464249\pi$$
$$812$$ 0 0
$$813$$ −15748.0 −0.679343
$$814$$ 0 0
$$815$$ 1035.53 0.0445068
$$816$$ 0 0
$$817$$ 15661.5 0.670656
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 20356.8 0.865355 0.432678 0.901549i $$-0.357569\pi$$
0.432678 + 0.901549i $$0.357569\pi$$
$$822$$ 0 0
$$823$$ 57.5738 0.00243851 0.00121926 0.999999i $$-0.499612\pi$$
0.00121926 + 0.999999i $$0.499612\pi$$
$$824$$ 0 0
$$825$$ −24081.5 −1.01625
$$826$$ 0 0
$$827$$ −12296.6 −0.517044 −0.258522 0.966005i $$-0.583235\pi$$
−0.258522 + 0.966005i $$0.583235\pi$$
$$828$$ 0 0
$$829$$ −22871.7 −0.958224 −0.479112 0.877754i $$-0.659041\pi$$
−0.479112 + 0.877754i $$0.659041\pi$$
$$830$$ 0 0
$$831$$ −21443.6 −0.895151
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3039.85 0.125986
$$836$$ 0 0
$$837$$ −7446.47 −0.307512
$$838$$ 0 0
$$839$$ −37571.3 −1.54601 −0.773007 0.634398i $$-0.781248\pi$$
−0.773007 + 0.634398i $$0.781248\pi$$
$$840$$ 0 0
$$841$$ 26997.6 1.10696
$$842$$ 0 0
$$843$$ −12685.1 −0.518265
$$844$$ 0 0
$$845$$ −2153.65 −0.0876780
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 25027.1 1.01170
$$850$$ 0 0
$$851$$ −266.677 −0.0107422
$$852$$ 0 0
$$853$$ 38354.5 1.53955 0.769774 0.638317i $$-0.220369\pi$$
0.769774 + 0.638317i $$0.220369\pi$$
$$854$$ 0 0
$$855$$ 224.855 0.00899401
$$856$$ 0 0
$$857$$ −36545.7 −1.45668 −0.728341 0.685215i $$-0.759708\pi$$
−0.728341 + 0.685215i $$0.759708\pi$$
$$858$$ 0 0
$$859$$ 18564.3 0.737374 0.368687 0.929554i $$-0.379807\pi$$
0.368687 + 0.929554i $$0.379807\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 37457.7 1.47749 0.738746 0.673984i $$-0.235418\pi$$
0.738746 + 0.673984i $$0.235418\pi$$
$$864$$ 0 0
$$865$$ 1623.78 0.0638266
$$866$$ 0 0
$$867$$ 7564.39 0.296309
$$868$$ 0 0
$$869$$ −60474.4 −2.36071
$$870$$ 0 0
$$871$$ −13886.1 −0.540199
$$872$$ 0 0
$$873$$ −355.218 −0.0137713
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27128.8 −1.04455 −0.522277 0.852776i $$-0.674917\pi$$
−0.522277 + 0.852776i $$0.674917\pi$$
$$878$$ 0 0
$$879$$ 15116.5 0.580055
$$880$$ 0 0
$$881$$ −23647.8 −0.904329 −0.452165 0.891935i $$-0.649348\pi$$
−0.452165 + 0.891935i $$0.649348\pi$$
$$882$$ 0 0
$$883$$ −4488.47 −0.171063 −0.0855316 0.996335i $$-0.527259\pi$$
−0.0855316 + 0.996335i $$0.527259\pi$$
$$884$$ 0 0
$$885$$ −1589.32 −0.0603664
$$886$$ 0 0
$$887$$ 41026.8 1.55304 0.776520 0.630093i $$-0.216983\pi$$
0.776520 + 0.630093i $$0.216983\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −5223.65 −0.196407
$$892$$ 0 0
$$893$$ 9711.41 0.363919
$$894$$ 0 0
$$895$$ 683.836 0.0255398
$$896$$ 0 0
$$897$$ −194.710 −0.00724771
$$898$$ 0 0
$$899$$ 62519.0 2.31938
$$900$$ 0 0
$$901$$ −17420.1 −0.644114
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −339.594 −0.0124735
$$906$$ 0 0
$$907$$ −736.305 −0.0269555 −0.0134777 0.999909i $$-0.504290\pi$$
−0.0134777 + 0.999909i $$0.504290\pi$$
$$908$$ 0 0
$$909$$ 2851.87 0.104060
$$910$$ 0 0
$$911$$ 1287.54 0.0468256 0.0234128 0.999726i $$-0.492547\pi$$
0.0234128 + 0.999726i $$0.492547\pi$$
$$912$$ 0 0
$$913$$ 58789.0 2.13103
$$914$$ 0 0
$$915$$ −597.821 −0.0215993
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 34280.0 1.23046 0.615230 0.788348i $$-0.289063\pi$$
0.615230 + 0.788348i $$0.289063\pi$$
$$920$$ 0 0
$$921$$ −14609.0 −0.522675
$$922$$ 0 0
$$923$$ −2911.81 −0.103839
$$924$$ 0 0
$$925$$ −36745.4 −1.30614
$$926$$ 0 0
$$927$$ 2903.70 0.102880
$$928$$ 0 0
$$929$$ 31475.8 1.11161 0.555806 0.831312i $$-0.312410\pi$$
0.555806 + 0.831312i $$0.312410\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −4358.52 −0.152939
$$934$$ 0 0
$$935$$ −2290.70 −0.0801219
$$936$$ 0 0
$$937$$ −52560.6 −1.83253 −0.916264 0.400575i $$-0.868810\pi$$
−0.916264 + 0.400575i $$0.868810\pi$$
$$938$$ 0 0
$$939$$ 17088.8 0.593900
$$940$$ 0 0
$$941$$ −15976.7 −0.553482 −0.276741 0.960945i $$-0.589254\pi$$
−0.276741 + 0.960945i $$0.589254\pi$$
$$942$$ 0 0
$$943$$ 168.569 0.00582118
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8504.43 −0.291824 −0.145912 0.989298i $$-0.546612\pi$$
−0.145912 + 0.989298i $$0.546612\pi$$
$$948$$ 0 0
$$949$$ −14834.5 −0.507428
$$950$$ 0 0
$$951$$ 10414.4 0.355111
$$952$$ 0 0
$$953$$ 29417.5 0.999923 0.499961 0.866048i $$-0.333348\pi$$
0.499961 + 0.866048i $$0.333348\pi$$
$$954$$ 0 0
$$955$$ −200.036 −0.00677804
$$956$$ 0 0
$$957$$ 43856.6 1.48138
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 46272.0 1.55322
$$962$$ 0 0
$$963$$ 6133.06 0.205229
$$964$$ 0 0
$$965$$ −1191.42 −0.0397441
$$966$$ 0 0
$$967$$ 3461.97 0.115129 0.0575643 0.998342i $$-0.481667\pi$$
0.0575643 + 0.998342i $$0.481667\pi$$
$$968$$ 0 0
$$969$$ −5046.36 −0.167299
$$970$$ 0 0
$$971$$ 43558.8 1.43962 0.719809 0.694172i $$-0.244229\pi$$
0.719809 + 0.694172i $$0.244229\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −26829.1 −0.881250
$$976$$ 0 0
$$977$$ −5504.95 −0.180265 −0.0901325 0.995930i $$-0.528729\pi$$
−0.0901325 + 0.995930i $$0.528729\pi$$
$$978$$ 0 0
$$979$$ −61263.4 −1.99999
$$980$$ 0 0
$$981$$ −4103.71 −0.133559
$$982$$ 0 0
$$983$$ 30789.7 0.999023 0.499511 0.866307i $$-0.333513\pi$$
0.499511 + 0.866307i $$0.333513\pi$$
$$984$$ 0 0
$$985$$ −947.060 −0.0306354
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 411.311 0.0132244
$$990$$ 0 0
$$991$$ −21629.1 −0.693312 −0.346656 0.937992i $$-0.612683\pi$$
−0.346656 + 0.937992i $$0.612683\pi$$
$$992$$ 0 0
$$993$$ −18763.1 −0.599627
$$994$$ 0 0
$$995$$ −964.114 −0.0307181
$$996$$ 0 0
$$997$$ 38195.8 1.21331 0.606656 0.794964i $$-0.292510\pi$$
0.606656 + 0.794964i $$0.292510\pi$$
$$998$$ 0 0
$$999$$ −7970.64 −0.252432
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 1176.4.a.ba.1.2 4
4.3 odd 2 2352.4.a.cp.1.2 4
7.3 odd 6 168.4.q.f.121.2 yes 8
7.5 odd 6 168.4.q.f.25.2 8
7.6 odd 2 1176.4.a.bd.1.3 4
21.5 even 6 504.4.s.j.361.3 8
21.17 even 6 504.4.s.j.289.3 8
28.3 even 6 336.4.q.m.289.2 8
28.19 even 6 336.4.q.m.193.2 8
28.27 even 2 2352.4.a.cm.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.2 8 7.5 odd 6
168.4.q.f.121.2 yes 8 7.3 odd 6
336.4.q.m.193.2 8 28.19 even 6
336.4.q.m.289.2 8 28.3 even 6
504.4.s.j.289.3 8 21.17 even 6
504.4.s.j.361.3 8 21.5 even 6
1176.4.a.ba.1.2 4 1.1 even 1 trivial
1176.4.a.bd.1.3 4 7.6 odd 2
2352.4.a.cm.1.3 4 28.27 even 2
2352.4.a.cp.1.2 4 4.3 odd 2