# Properties

 Label 76.3.c.a.37.1 Level $76$ Weight $3$ Character 76.37 Analytic conductor $2.071$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-29})$$ Defining polynomial: $$x^{2} + 29$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 37.1 Root $$-5.38516i$$ of defining polynomial Character $$\chi$$ $$=$$ 76.37 Dual form 76.3.c.a.37.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.38516i q^{3} -4.00000 q^{5} -1.00000 q^{7} -20.0000 q^{9} +O(q^{10})$$ $$q-5.38516i q^{3} -4.00000 q^{5} -1.00000 q^{7} -20.0000 q^{9} +14.0000 q^{11} -16.1555i q^{13} +21.5407i q^{15} +23.0000 q^{17} +(10.0000 - 16.1555i) q^{19} +5.38516i q^{21} -1.00000 q^{23} -9.00000 q^{25} +59.2368i q^{27} +48.4665i q^{29} +32.3110i q^{31} -75.3923i q^{33} +4.00000 q^{35} -32.3110i q^{37} -87.0000 q^{39} -32.3110i q^{41} +68.0000 q^{43} +80.0000 q^{45} +26.0000 q^{47} -48.0000 q^{49} -123.859i q^{51} +80.7775i q^{53} -56.0000 q^{55} +(-87.0000 - 53.8516i) q^{57} -16.1555i q^{59} -40.0000 q^{61} +20.0000 q^{63} +64.6220i q^{65} +16.1555i q^{67} +5.38516i q^{69} +32.3110i q^{71} -7.00000 q^{73} +48.4665i q^{75} -14.0000 q^{77} -96.9330i q^{79} +139.000 q^{81} +32.0000 q^{83} -92.0000 q^{85} +261.000 q^{87} +129.244i q^{89} +16.1555i q^{91} +174.000 q^{93} +(-40.0000 + 64.6220i) q^{95} +96.9330i q^{97} -280.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} - 2q^{7} - 40q^{9} + O(q^{10})$$ $$2q - 8q^{5} - 2q^{7} - 40q^{9} + 28q^{11} + 46q^{17} + 20q^{19} - 2q^{23} - 18q^{25} + 8q^{35} - 174q^{39} + 136q^{43} + 160q^{45} + 52q^{47} - 96q^{49} - 112q^{55} - 174q^{57} - 80q^{61} + 40q^{63} - 14q^{73} - 28q^{77} + 278q^{81} + 64q^{83} - 184q^{85} + 522q^{87} + 348q^{93} - 80q^{95} - 560q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$3$$ 5.38516i 1.79505i −0.440959 0.897527i $$-0.645361\pi$$
0.440959 0.897527i $$-0.354639\pi$$
$$4$$ 0 0
$$5$$ −4.00000 −0.800000 −0.400000 0.916515i $$-0.630990\pi$$
−0.400000 + 0.916515i $$0.630990\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.142857 −0.0714286 0.997446i $$-0.522756\pi$$
−0.0714286 + 0.997446i $$0.522756\pi$$
$$8$$ 0 0
$$9$$ −20.0000 −2.22222
$$10$$ 0 0
$$11$$ 14.0000 1.27273 0.636364 0.771389i $$-0.280438\pi$$
0.636364 + 0.771389i $$0.280438\pi$$
$$12$$ 0 0
$$13$$ 16.1555i 1.24273i −0.783521 0.621365i $$-0.786578\pi$$
0.783521 0.621365i $$-0.213422\pi$$
$$14$$ 0 0
$$15$$ 21.5407i 1.43604i
$$16$$ 0 0
$$17$$ 23.0000 1.35294 0.676471 0.736470i $$-0.263509\pi$$
0.676471 + 0.736470i $$0.263509\pi$$
$$18$$ 0 0
$$19$$ 10.0000 16.1555i 0.526316 0.850289i
$$20$$ 0 0
$$21$$ 5.38516i 0.256436i
$$22$$ 0 0
$$23$$ −1.00000 −0.0434783 −0.0217391 0.999764i $$-0.506920\pi$$
−0.0217391 + 0.999764i $$0.506920\pi$$
$$24$$ 0 0
$$25$$ −9.00000 −0.360000
$$26$$ 0 0
$$27$$ 59.2368i 2.19396i
$$28$$ 0 0
$$29$$ 48.4665i 1.67126i 0.549294 + 0.835629i $$0.314897\pi$$
−0.549294 + 0.835629i $$0.685103\pi$$
$$30$$ 0 0
$$31$$ 32.3110i 1.04229i 0.853468 + 0.521145i $$0.174495\pi$$
−0.853468 + 0.521145i $$0.825505\pi$$
$$32$$ 0 0
$$33$$ 75.3923i 2.28462i
$$34$$ 0 0
$$35$$ 4.00000 0.114286
$$36$$ 0 0
$$37$$ 32.3110i 0.873270i −0.899639 0.436635i $$-0.856170\pi$$
0.899639 0.436635i $$-0.143830\pi$$
$$38$$ 0 0
$$39$$ −87.0000 −2.23077
$$40$$ 0 0
$$41$$ 32.3110i 0.788073i −0.919095 0.394036i $$-0.871078\pi$$
0.919095 0.394036i $$-0.128922\pi$$
$$42$$ 0 0
$$43$$ 68.0000 1.58140 0.790698 0.612207i $$-0.209718\pi$$
0.790698 + 0.612207i $$0.209718\pi$$
$$44$$ 0 0
$$45$$ 80.0000 1.77778
$$46$$ 0 0
$$47$$ 26.0000 0.553191 0.276596 0.960986i $$-0.410794\pi$$
0.276596 + 0.960986i $$0.410794\pi$$
$$48$$ 0 0
$$49$$ −48.0000 −0.979592
$$50$$ 0 0
$$51$$ 123.859i 2.42860i
$$52$$ 0 0
$$53$$ 80.7775i 1.52410i 0.647516 + 0.762052i $$0.275808\pi$$
−0.647516 + 0.762052i $$0.724192\pi$$
$$54$$ 0 0
$$55$$ −56.0000 −1.01818
$$56$$ 0 0
$$57$$ −87.0000 53.8516i −1.52632 0.944766i
$$58$$ 0 0
$$59$$ 16.1555i 0.273822i −0.990583 0.136911i $$-0.956283\pi$$
0.990583 0.136911i $$-0.0437174\pi$$
$$60$$ 0 0
$$61$$ −40.0000 −0.655738 −0.327869 0.944723i $$-0.606330\pi$$
−0.327869 + 0.944723i $$0.606330\pi$$
$$62$$ 0 0
$$63$$ 20.0000 0.317460
$$64$$ 0 0
$$65$$ 64.6220i 0.994184i
$$66$$ 0 0
$$67$$ 16.1555i 0.241127i 0.992706 + 0.120563i $$0.0384701\pi$$
−0.992706 + 0.120563i $$0.961530\pi$$
$$68$$ 0 0
$$69$$ 5.38516i 0.0780459i
$$70$$ 0 0
$$71$$ 32.3110i 0.455084i 0.973768 + 0.227542i $$0.0730690\pi$$
−0.973768 + 0.227542i $$0.926931\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.0958904 −0.0479452 0.998850i $$-0.515267\pi$$
−0.0479452 + 0.998850i $$0.515267\pi$$
$$74$$ 0 0
$$75$$ 48.4665i 0.646220i
$$76$$ 0 0
$$77$$ −14.0000 −0.181818
$$78$$ 0 0
$$79$$ 96.9330i 1.22700i −0.789695 0.613500i $$-0.789761\pi$$
0.789695 0.613500i $$-0.210239\pi$$
$$80$$ 0 0
$$81$$ 139.000 1.71605
$$82$$ 0 0
$$83$$ 32.0000 0.385542 0.192771 0.981244i $$-0.438253\pi$$
0.192771 + 0.981244i $$0.438253\pi$$
$$84$$ 0 0
$$85$$ −92.0000 −1.08235
$$86$$ 0 0
$$87$$ 261.000 3.00000
$$88$$ 0 0
$$89$$ 129.244i 1.45218i 0.687600 + 0.726090i $$0.258664\pi$$
−0.687600 + 0.726090i $$0.741336\pi$$
$$90$$ 0 0
$$91$$ 16.1555i 0.177533i
$$92$$ 0 0
$$93$$ 174.000 1.87097
$$94$$ 0 0
$$95$$ −40.0000 + 64.6220i −0.421053 + 0.680231i
$$96$$ 0 0
$$97$$ 96.9330i 0.999309i 0.866225 + 0.499654i $$0.166540\pi$$
−0.866225 + 0.499654i $$0.833460\pi$$
$$98$$ 0 0
$$99$$ −280.000 −2.82828
$$100$$ 0 0
$$101$$ 14.0000 0.138614 0.0693069 0.997595i $$-0.477921\pi$$
0.0693069 + 0.997595i $$0.477921\pi$$
$$102$$ 0 0
$$103$$ 129.244i 1.25480i −0.778699 0.627398i $$-0.784120\pi$$
0.778699 0.627398i $$-0.215880\pi$$
$$104$$ 0 0
$$105$$ 21.5407i 0.205149i
$$106$$ 0 0
$$107$$ 16.1555i 0.150986i −0.997146 0.0754930i $$-0.975947\pi$$
0.997146 0.0754930i $$-0.0240530\pi$$
$$108$$ 0 0
$$109$$ 16.1555i 0.148216i −0.997250 0.0741078i $$-0.976389\pi$$
0.997250 0.0741078i $$-0.0236109\pi$$
$$110$$ 0 0
$$111$$ −174.000 −1.56757
$$112$$ 0 0
$$113$$ 96.9330i 0.857814i −0.903349 0.428907i $$-0.858899\pi$$
0.903349 0.428907i $$-0.141101\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.0347826
$$116$$ 0 0
$$117$$ 323.110i 2.76162i
$$118$$ 0 0
$$119$$ −23.0000 −0.193277
$$120$$ 0 0
$$121$$ 75.0000 0.619835
$$122$$ 0 0
$$123$$ −174.000 −1.41463
$$124$$ 0 0
$$125$$ 136.000 1.08800
$$126$$ 0 0
$$127$$ 64.6220i 0.508834i 0.967095 + 0.254417i $$0.0818836\pi$$
−0.967095 + 0.254417i $$0.918116\pi$$
$$128$$ 0 0
$$129$$ 366.191i 2.83869i
$$130$$ 0 0
$$131$$ 56.0000 0.427481 0.213740 0.976890i $$-0.431435\pi$$
0.213740 + 0.976890i $$0.431435\pi$$
$$132$$ 0 0
$$133$$ −10.0000 + 16.1555i −0.0751880 + 0.121470i
$$134$$ 0 0
$$135$$ 236.947i 1.75516i
$$136$$ 0 0
$$137$$ −19.0000 −0.138686 −0.0693431 0.997593i $$-0.522090\pi$$
−0.0693431 + 0.997593i $$0.522090\pi$$
$$138$$ 0 0
$$139$$ 122.000 0.877698 0.438849 0.898561i $$-0.355386\pi$$
0.438849 + 0.898561i $$0.355386\pi$$
$$140$$ 0 0
$$141$$ 140.014i 0.993009i
$$142$$ 0 0
$$143$$ 226.177i 1.58166i
$$144$$ 0 0
$$145$$ 193.866i 1.33701i
$$146$$ 0 0
$$147$$ 258.488i 1.75842i
$$148$$ 0 0
$$149$$ −82.0000 −0.550336 −0.275168 0.961396i $$-0.588733\pi$$
−0.275168 + 0.961396i $$0.588733\pi$$
$$150$$ 0 0
$$151$$ 226.177i 1.49786i 0.662649 + 0.748930i $$0.269432\pi$$
−0.662649 + 0.748930i $$0.730568\pi$$
$$152$$ 0 0
$$153$$ −460.000 −3.00654
$$154$$ 0 0
$$155$$ 129.244i 0.833832i
$$156$$ 0 0
$$157$$ 242.000 1.54140 0.770701 0.637197i $$-0.219906\pi$$
0.770701 + 0.637197i $$0.219906\pi$$
$$158$$ 0 0
$$159$$ 435.000 2.73585
$$160$$ 0 0
$$161$$ 1.00000 0.00621118
$$162$$ 0 0
$$163$$ −214.000 −1.31288 −0.656442 0.754377i $$-0.727939\pi$$
−0.656442 + 0.754377i $$0.727939\pi$$
$$164$$ 0 0
$$165$$ 301.569i 1.82769i
$$166$$ 0 0
$$167$$ 32.3110i 0.193479i −0.995310 0.0967395i $$-0.969159\pi$$
0.995310 0.0967395i $$-0.0308414\pi$$
$$168$$ 0 0
$$169$$ −92.0000 −0.544379
$$170$$ 0 0
$$171$$ −200.000 + 323.110i −1.16959 + 1.88953i
$$172$$ 0 0
$$173$$ 161.555i 0.933844i 0.884299 + 0.466922i $$0.154637\pi$$
−0.884299 + 0.466922i $$0.845363\pi$$
$$174$$ 0 0
$$175$$ 9.00000 0.0514286
$$176$$ 0 0
$$177$$ −87.0000 −0.491525
$$178$$ 0 0
$$179$$ 290.799i 1.62457i −0.583257 0.812287i $$-0.698222\pi$$
0.583257 0.812287i $$-0.301778\pi$$
$$180$$ 0 0
$$181$$ 96.9330i 0.535541i −0.963483 0.267771i $$-0.913713\pi$$
0.963483 0.267771i $$-0.0862869\pi$$
$$182$$ 0 0
$$183$$ 215.407i 1.17709i
$$184$$ 0 0
$$185$$ 129.244i 0.698616i
$$186$$ 0 0
$$187$$ 322.000 1.72193
$$188$$ 0 0
$$189$$ 59.2368i 0.313422i
$$190$$ 0 0
$$191$$ −67.0000 −0.350785 −0.175393 0.984499i $$-0.556119\pi$$
−0.175393 + 0.984499i $$0.556119\pi$$
$$192$$ 0 0
$$193$$ 96.9330i 0.502243i 0.967956 + 0.251122i $$0.0807994\pi$$
−0.967956 + 0.251122i $$0.919201\pi$$
$$194$$ 0 0
$$195$$ 348.000 1.78462
$$196$$ 0 0
$$197$$ −142.000 −0.720812 −0.360406 0.932796i $$-0.617362\pi$$
−0.360406 + 0.932796i $$0.617362\pi$$
$$198$$ 0 0
$$199$$ 263.000 1.32161 0.660804 0.750558i $$-0.270215\pi$$
0.660804 + 0.750558i $$0.270215\pi$$
$$200$$ 0 0
$$201$$ 87.0000 0.432836
$$202$$ 0 0
$$203$$ 48.4665i 0.238751i
$$204$$ 0 0
$$205$$ 129.244i 0.630458i
$$206$$ 0 0
$$207$$ 20.0000 0.0966184
$$208$$ 0 0
$$209$$ 140.000 226.177i 0.669856 1.08219i
$$210$$ 0 0
$$211$$ 80.7775i 0.382832i 0.981509 + 0.191416i $$0.0613079\pi$$
−0.981509 + 0.191416i $$0.938692\pi$$
$$212$$ 0 0
$$213$$ 174.000 0.816901
$$214$$ 0 0
$$215$$ −272.000 −1.26512
$$216$$ 0 0
$$217$$ 32.3110i 0.148899i
$$218$$ 0 0
$$219$$ 37.6962i 0.172129i
$$220$$ 0 0
$$221$$ 371.576i 1.68134i
$$222$$ 0 0
$$223$$ 258.488i 1.15914i 0.814923 + 0.579569i $$0.196779\pi$$
−0.814923 + 0.579569i $$0.803221\pi$$
$$224$$ 0 0
$$225$$ 180.000 0.800000
$$226$$ 0 0
$$227$$ 80.7775i 0.355848i 0.984044 + 0.177924i $$0.0569381\pi$$
−0.984044 + 0.177924i $$0.943062\pi$$
$$228$$ 0 0
$$229$$ −304.000 −1.32751 −0.663755 0.747950i $$-0.731038\pi$$
−0.663755 + 0.747950i $$0.731038\pi$$
$$230$$ 0 0
$$231$$ 75.3923i 0.326374i
$$232$$ 0 0
$$233$$ 2.00000 0.00858369 0.00429185 0.999991i $$-0.498634\pi$$
0.00429185 + 0.999991i $$0.498634\pi$$
$$234$$ 0 0
$$235$$ −104.000 −0.442553
$$236$$ 0 0
$$237$$ −522.000 −2.20253
$$238$$ 0 0
$$239$$ 89.0000 0.372385 0.186192 0.982513i $$-0.440385\pi$$
0.186192 + 0.982513i $$0.440385\pi$$
$$240$$ 0 0
$$241$$ 161.555i 0.670352i 0.942155 + 0.335176i $$0.108796\pi$$
−0.942155 + 0.335176i $$0.891204\pi$$
$$242$$ 0 0
$$243$$ 215.407i 0.886447i
$$244$$ 0 0
$$245$$ 192.000 0.783673
$$246$$ 0 0
$$247$$ −261.000 161.555i −1.05668 0.654069i
$$248$$ 0 0
$$249$$ 172.325i 0.692069i
$$250$$ 0 0
$$251$$ −370.000 −1.47410 −0.737052 0.675836i $$-0.763783\pi$$
−0.737052 + 0.675836i $$0.763783\pi$$
$$252$$ 0 0
$$253$$ −14.0000 −0.0553360
$$254$$ 0 0
$$255$$ 495.435i 1.94288i
$$256$$ 0 0
$$257$$ 193.866i 0.754342i 0.926144 + 0.377171i $$0.123103\pi$$
−0.926144 + 0.377171i $$0.876897\pi$$
$$258$$ 0 0
$$259$$ 32.3110i 0.124753i
$$260$$ 0 0
$$261$$ 969.330i 3.71391i
$$262$$ 0 0
$$263$$ −394.000 −1.49810 −0.749049 0.662514i $$-0.769489\pi$$
−0.749049 + 0.662514i $$0.769489\pi$$
$$264$$ 0 0
$$265$$ 323.110i 1.21928i
$$266$$ 0 0
$$267$$ 696.000 2.60674
$$268$$ 0 0
$$269$$ 96.9330i 0.360346i −0.983635 0.180173i $$-0.942334\pi$$
0.983635 0.180173i $$-0.0576657\pi$$
$$270$$ 0 0
$$271$$ −403.000 −1.48708 −0.743542 0.668689i $$-0.766856\pi$$
−0.743542 + 0.668689i $$0.766856\pi$$
$$272$$ 0 0
$$273$$ 87.0000 0.318681
$$274$$ 0 0
$$275$$ −126.000 −0.458182
$$276$$ 0 0
$$277$$ 206.000 0.743682 0.371841 0.928296i $$-0.378727\pi$$
0.371841 + 0.928296i $$0.378727\pi$$
$$278$$ 0 0
$$279$$ 646.220i 2.31620i
$$280$$ 0 0
$$281$$ 258.488i 0.919886i 0.887948 + 0.459943i $$0.152130\pi$$
−0.887948 + 0.459943i $$0.847870\pi$$
$$282$$ 0 0
$$283$$ 56.0000 0.197880 0.0989399 0.995093i $$-0.468455\pi$$
0.0989399 + 0.995093i $$0.468455\pi$$
$$284$$ 0 0
$$285$$ 348.000 + 215.407i 1.22105 + 0.755813i
$$286$$ 0 0
$$287$$ 32.3110i 0.112582i
$$288$$ 0 0
$$289$$ 240.000 0.830450
$$290$$ 0 0
$$291$$ 522.000 1.79381
$$292$$ 0 0
$$293$$ 80.7775i 0.275691i 0.990454 + 0.137846i $$0.0440177\pi$$
−0.990454 + 0.137846i $$0.955982\pi$$
$$294$$ 0 0
$$295$$ 64.6220i 0.219058i
$$296$$ 0 0
$$297$$ 829.315i 2.79231i
$$298$$ 0 0
$$299$$ 16.1555i 0.0540318i
$$300$$ 0 0
$$301$$ −68.0000 −0.225914
$$302$$ 0 0
$$303$$ 75.3923i 0.248819i
$$304$$ 0 0
$$305$$ 160.000 0.524590
$$306$$ 0 0
$$307$$ 420.043i 1.36822i −0.729380 0.684109i $$-0.760191\pi$$
0.729380 0.684109i $$-0.239809\pi$$
$$308$$ 0 0
$$309$$ −696.000 −2.25243
$$310$$ 0 0
$$311$$ −265.000 −0.852090 −0.426045 0.904702i $$-0.640093\pi$$
−0.426045 + 0.904702i $$0.640093\pi$$
$$312$$ 0 0
$$313$$ 77.0000 0.246006 0.123003 0.992406i $$-0.460747\pi$$
0.123003 + 0.992406i $$0.460747\pi$$
$$314$$ 0 0
$$315$$ −80.0000 −0.253968
$$316$$ 0 0
$$317$$ 403.887i 1.27409i −0.770826 0.637046i $$-0.780156\pi$$
0.770826 0.637046i $$-0.219844\pi$$
$$318$$ 0 0
$$319$$ 678.531i 2.12706i
$$320$$ 0 0
$$321$$ −87.0000 −0.271028
$$322$$ 0 0
$$323$$ 230.000 371.576i 0.712074 1.15039i
$$324$$ 0 0
$$325$$ 145.399i 0.447383i
$$326$$ 0 0
$$327$$ −87.0000 −0.266055
$$328$$ 0 0
$$329$$ −26.0000 −0.0790274
$$330$$ 0 0
$$331$$ 436.198i 1.31782i 0.752222 + 0.658910i $$0.228982\pi$$
−0.752222 + 0.658910i $$0.771018\pi$$
$$332$$ 0 0
$$333$$ 646.220i 1.94060i
$$334$$ 0 0
$$335$$ 64.6220i 0.192901i
$$336$$ 0 0
$$337$$ 387.732i 1.15054i 0.817964 + 0.575270i $$0.195103\pi$$
−0.817964 + 0.575270i $$0.804897\pi$$
$$338$$ 0 0
$$339$$ −522.000 −1.53982
$$340$$ 0 0
$$341$$ 452.354i 1.32655i
$$342$$ 0 0
$$343$$ 97.0000 0.282799
$$344$$ 0 0
$$345$$ 21.5407i 0.0624367i
$$346$$ 0 0
$$347$$ 392.000 1.12968 0.564841 0.825199i $$-0.308937\pi$$
0.564841 + 0.825199i $$0.308937\pi$$
$$348$$ 0 0
$$349$$ 410.000 1.17479 0.587393 0.809302i $$-0.300154\pi$$
0.587393 + 0.809302i $$0.300154\pi$$
$$350$$ 0 0
$$351$$ 957.000 2.72650
$$352$$ 0 0
$$353$$ −481.000 −1.36261 −0.681303 0.732001i $$-0.738586\pi$$
−0.681303 + 0.732001i $$0.738586\pi$$
$$354$$ 0 0
$$355$$ 129.244i 0.364067i
$$356$$ 0 0
$$357$$ 123.859i 0.346943i
$$358$$ 0 0
$$359$$ −109.000 −0.303621 −0.151811 0.988410i $$-0.548510\pi$$
−0.151811 + 0.988410i $$0.548510\pi$$
$$360$$ 0 0
$$361$$ −161.000 323.110i −0.445983 0.895041i
$$362$$ 0 0
$$363$$ 403.887i 1.11264i
$$364$$ 0 0
$$365$$ 28.0000 0.0767123
$$366$$ 0 0
$$367$$ 50.0000 0.136240 0.0681199 0.997677i $$-0.478300\pi$$
0.0681199 + 0.997677i $$0.478300\pi$$
$$368$$ 0 0
$$369$$ 646.220i 1.75127i
$$370$$ 0 0
$$371$$ 80.7775i 0.217729i
$$372$$ 0 0
$$373$$ 500.820i 1.34268i −0.741149 0.671341i $$-0.765719\pi$$
0.741149 0.671341i $$-0.234281\pi$$
$$374$$ 0 0
$$375$$ 732.382i 1.95302i
$$376$$ 0 0
$$377$$ 783.000 2.07692
$$378$$ 0 0
$$379$$ 210.021i 0.554146i −0.960849 0.277073i $$-0.910636\pi$$
0.960849 0.277073i $$-0.0893644\pi$$
$$380$$ 0 0
$$381$$ 348.000 0.913386
$$382$$ 0 0
$$383$$ 258.488i 0.674903i −0.941343 0.337452i $$-0.890435\pi$$
0.941343 0.337452i $$-0.109565\pi$$
$$384$$ 0 0
$$385$$ 56.0000 0.145455
$$386$$ 0 0
$$387$$ −1360.00 −3.51421
$$388$$ 0 0
$$389$$ 578.000 1.48586 0.742931 0.669368i $$-0.233435\pi$$
0.742931 + 0.669368i $$0.233435\pi$$
$$390$$ 0 0
$$391$$ −23.0000 −0.0588235
$$392$$ 0 0
$$393$$ 301.569i 0.767352i
$$394$$ 0 0
$$395$$ 387.732i 0.981600i
$$396$$ 0 0
$$397$$ −658.000 −1.65743 −0.828715 0.559670i $$-0.810928\pi$$
−0.828715 + 0.559670i $$0.810928\pi$$
$$398$$ 0 0
$$399$$ 87.0000 + 53.8516i 0.218045 + 0.134967i
$$400$$ 0 0
$$401$$ 775.464i 1.93382i 0.255109 + 0.966912i $$0.417889\pi$$
−0.255109 + 0.966912i $$0.582111\pi$$
$$402$$ 0 0
$$403$$ 522.000 1.29529
$$404$$ 0 0
$$405$$ −556.000 −1.37284
$$406$$ 0 0
$$407$$ 452.354i 1.11143i
$$408$$ 0 0
$$409$$ 355.421i 0.869000i −0.900672 0.434500i $$-0.856925\pi$$
0.900672 0.434500i $$-0.143075\pi$$
$$410$$ 0 0
$$411$$ 102.318i 0.248949i
$$412$$ 0 0
$$413$$ 16.1555i 0.0391174i
$$414$$ 0 0
$$415$$ −128.000 −0.308434
$$416$$ 0 0
$$417$$ 656.990i 1.57552i
$$418$$ 0 0
$$419$$ 8.00000 0.0190931 0.00954654 0.999954i $$-0.496961\pi$$
0.00954654 + 0.999954i $$0.496961\pi$$
$$420$$ 0 0
$$421$$ 500.820i 1.18960i −0.803875 0.594798i $$-0.797232\pi$$
0.803875 0.594798i $$-0.202768\pi$$
$$422$$ 0 0
$$423$$ −520.000 −1.22931
$$424$$ 0 0
$$425$$ −207.000 −0.487059
$$426$$ 0 0
$$427$$ 40.0000 0.0936768
$$428$$ 0 0
$$429$$ −1218.00 −2.83916
$$430$$ 0 0
$$431$$ 646.220i 1.49935i 0.661806 + 0.749675i $$0.269790\pi$$
−0.661806 + 0.749675i $$0.730210\pi$$
$$432$$ 0 0
$$433$$ 193.866i 0.447727i −0.974620 0.223864i $$-0.928133\pi$$
0.974620 0.223864i $$-0.0718670\pi$$
$$434$$ 0 0
$$435$$ −1044.00 −2.40000
$$436$$ 0 0
$$437$$ −10.0000 + 16.1555i −0.0228833 + 0.0369691i
$$438$$ 0 0
$$439$$ 323.110i 0.736013i 0.929823 + 0.368007i $$0.119960\pi$$
−0.929823 + 0.368007i $$0.880040\pi$$
$$440$$ 0 0
$$441$$ 960.000 2.17687
$$442$$ 0 0
$$443$$ 182.000 0.410835 0.205418 0.978674i $$-0.434145\pi$$
0.205418 + 0.978674i $$0.434145\pi$$
$$444$$ 0 0
$$445$$ 516.976i 1.16174i
$$446$$ 0 0
$$447$$ 441.584i 0.987883i
$$448$$ 0 0
$$449$$ 420.043i 0.935507i 0.883859 + 0.467754i $$0.154937\pi$$
−0.883859 + 0.467754i $$0.845063\pi$$
$$450$$ 0 0
$$451$$ 452.354i 1.00300i
$$452$$ 0 0
$$453$$ 1218.00 2.68874
$$454$$ 0 0
$$455$$ 64.6220i 0.142026i
$$456$$ 0 0
$$457$$ 317.000 0.693654 0.346827 0.937929i $$-0.387259\pi$$
0.346827 + 0.937929i $$0.387259\pi$$
$$458$$ 0 0
$$459$$ 1362.45i 2.96829i
$$460$$ 0 0
$$461$$ −172.000 −0.373102 −0.186551 0.982445i $$-0.559731\pi$$
−0.186551 + 0.982445i $$0.559731\pi$$
$$462$$ 0 0
$$463$$ 146.000 0.315335 0.157667 0.987492i $$-0.449603\pi$$
0.157667 + 0.987492i $$0.449603\pi$$
$$464$$ 0 0
$$465$$ −696.000 −1.49677
$$466$$ 0 0
$$467$$ 662.000 1.41756 0.708779 0.705430i $$-0.249246\pi$$
0.708779 + 0.705430i $$0.249246\pi$$
$$468$$ 0 0
$$469$$ 16.1555i 0.0344467i
$$470$$ 0 0
$$471$$ 1303.21i 2.76690i
$$472$$ 0 0
$$473$$ 952.000 2.01268
$$474$$ 0 0
$$475$$ −90.0000 + 145.399i −0.189474 + 0.306104i
$$476$$ 0 0
$$477$$ 1615.55i 3.38690i
$$478$$ 0 0
$$479$$ −130.000 −0.271399 −0.135699 0.990750i $$-0.543328\pi$$
−0.135699 + 0.990750i $$0.543328\pi$$
$$480$$ 0 0
$$481$$ −522.000 −1.08524
$$482$$ 0 0
$$483$$ 5.38516i 0.0111494i
$$484$$ 0 0
$$485$$ 387.732i 0.799447i
$$486$$ 0 0
$$487$$ 387.732i 0.796164i −0.917350 0.398082i $$-0.869676\pi$$
0.917350 0.398082i $$-0.130324\pi$$
$$488$$ 0 0
$$489$$ 1152.43i 2.35670i
$$490$$ 0 0
$$491$$ −280.000 −0.570265 −0.285132 0.958488i $$-0.592038\pi$$
−0.285132 + 0.958488i $$0.592038\pi$$
$$492$$ 0 0
$$493$$ 1114.73i 2.26111i
$$494$$ 0 0
$$495$$ 1120.00 2.26263
$$496$$ 0 0
$$497$$ 32.3110i 0.0650120i
$$498$$ 0 0
$$499$$ −460.000 −0.921844 −0.460922 0.887441i $$-0.652481\pi$$
−0.460922 + 0.887441i $$0.652481\pi$$
$$500$$ 0 0
$$501$$ −174.000 −0.347305
$$502$$ 0 0
$$503$$ 263.000 0.522863 0.261431 0.965222i $$-0.415805\pi$$
0.261431 + 0.965222i $$0.415805\pi$$
$$504$$ 0 0
$$505$$ −56.0000 −0.110891
$$506$$ 0 0
$$507$$ 495.435i 0.977190i
$$508$$ 0 0
$$509$$ 290.799i 0.571314i 0.958332 + 0.285657i $$0.0922118\pi$$
−0.958332 + 0.285657i $$0.907788\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.0136986
$$512$$ 0 0
$$513$$ 957.000 + 592.368i 1.86550 + 1.15471i
$$514$$ 0 0
$$515$$ 516.976i 1.00384i
$$516$$ 0 0
$$517$$ 364.000 0.704062
$$518$$ 0 0
$$519$$ 870.000 1.67630
$$520$$ 0 0
$$521$$ 226.177i 0.434121i 0.976158 + 0.217060i $$0.0696469\pi$$
−0.976158 + 0.217060i $$0.930353\pi$$
$$522$$ 0 0
$$523$$ 80.7775i 0.154450i −0.997014 0.0772251i $$-0.975394\pi$$
0.997014 0.0772251i $$-0.0246060\pi$$
$$524$$ 0 0
$$525$$ 48.4665i 0.0923171i
$$526$$ 0 0
$$527$$ 743.153i 1.41016i
$$528$$ 0 0
$$529$$ −528.000 −0.998110
$$530$$ 0 0
$$531$$ 323.110i 0.608493i
$$532$$ 0 0
$$533$$ −522.000 −0.979362
$$534$$ 0 0
$$535$$ 64.6220i 0.120789i
$$536$$ 0 0
$$537$$ −1566.00 −2.91620
$$538$$ 0 0
$$539$$ −672.000 −1.24675
$$540$$ 0 0
$$541$$ −640.000 −1.18299 −0.591497 0.806307i $$-0.701463\pi$$
−0.591497 + 0.806307i $$0.701463\pi$$
$$542$$ 0 0
$$543$$ −522.000 −0.961326
$$544$$ 0 0
$$545$$ 64.6220i 0.118572i
$$546$$ 0 0
$$547$$ 32.3110i 0.0590694i −0.999564 0.0295347i $$-0.990597\pi$$
0.999564 0.0295347i $$-0.00940256\pi$$
$$548$$ 0 0
$$549$$ 800.000 1.45719
$$550$$ 0 0
$$551$$ 783.000 + 484.665i 1.42105 + 0.879609i
$$552$$ 0 0
$$553$$ 96.9330i 0.175286i
$$554$$ 0 0
$$555$$ 696.000 1.25405
$$556$$ 0 0
$$557$$ −844.000 −1.51526 −0.757630 0.652684i $$-0.773643\pi$$
−0.757630 + 0.652684i $$0.773643\pi$$
$$558$$ 0 0
$$559$$ 1098.57i 1.96525i
$$560$$ 0 0
$$561$$ 1734.02i 3.09095i
$$562$$ 0 0
$$563$$ 420.043i 0.746080i −0.927815 0.373040i $$-0.878315\pi$$
0.927815 0.373040i $$-0.121685\pi$$
$$564$$ 0 0
$$565$$ 387.732i 0.686251i
$$566$$ 0 0
$$567$$ −139.000 −0.245150
$$568$$ 0 0
$$569$$ 290.799i 0.511070i −0.966800 0.255535i $$-0.917748\pi$$
0.966800 0.255535i $$-0.0822516\pi$$
$$570$$ 0 0
$$571$$ 458.000 0.802102 0.401051 0.916056i $$-0.368645\pi$$
0.401051 + 0.916056i $$0.368645\pi$$
$$572$$ 0 0
$$573$$ 360.806i 0.629679i
$$574$$ 0 0
$$575$$ 9.00000 0.0156522
$$576$$ 0 0
$$577$$ −85.0000 −0.147314 −0.0736568 0.997284i $$-0.523467\pi$$
−0.0736568 + 0.997284i $$0.523467\pi$$
$$578$$ 0 0
$$579$$ 522.000 0.901554
$$580$$ 0 0
$$581$$ −32.0000 −0.0550775
$$582$$ 0 0
$$583$$ 1130.88i 1.93977i
$$584$$ 0 0
$$585$$ 1292.44i 2.20930i
$$586$$ 0 0
$$587$$ 488.000 0.831346 0.415673 0.909514i $$-0.363546\pi$$
0.415673 + 0.909514i $$0.363546\pi$$
$$588$$ 0 0
$$589$$ 522.000 + 323.110i 0.886248 + 0.548574i
$$590$$ 0 0
$$591$$ 764.693i 1.29390i
$$592$$ 0 0
$$593$$ −610.000 −1.02867 −0.514334 0.857590i $$-0.671961\pi$$
−0.514334 + 0.857590i $$0.671961\pi$$
$$594$$ 0 0
$$595$$ 92.0000 0.154622
$$596$$ 0 0
$$597$$ 1416.30i 2.37236i
$$598$$ 0 0
$$599$$ 323.110i 0.539416i 0.962942 + 0.269708i $$0.0869271\pi$$
−0.962942 + 0.269708i $$0.913073\pi$$
$$600$$ 0 0
$$601$$ 96.9330i 0.161286i 0.996743 + 0.0806431i $$0.0256974\pi$$
−0.996743 + 0.0806431i $$0.974303\pi$$
$$602$$ 0 0
$$603$$ 323.110i 0.535837i
$$604$$ 0 0
$$605$$ −300.000 −0.495868
$$606$$ 0 0
$$607$$ 1001.64i 1.65015i −0.565024 0.825075i $$-0.691133\pi$$
0.565024 0.825075i $$-0.308867\pi$$
$$608$$ 0 0
$$609$$ −261.000 −0.428571
$$610$$ 0 0
$$611$$ 420.043i 0.687468i
$$612$$ 0 0
$$613$$ 200.000 0.326264 0.163132 0.986604i $$-0.447840\pi$$
0.163132 + 0.986604i $$0.447840\pi$$
$$614$$ 0 0
$$615$$ 696.000 1.13171
$$616$$ 0 0
$$617$$ 530.000 0.858995 0.429498 0.903068i $$-0.358691\pi$$
0.429498 + 0.903068i $$0.358691\pi$$
$$618$$ 0 0
$$619$$ −256.000 −0.413570 −0.206785 0.978386i $$-0.566300\pi$$
−0.206785 + 0.978386i $$0.566300\pi$$
$$620$$ 0 0
$$621$$ 59.2368i 0.0953894i
$$622$$ 0 0
$$623$$ 129.244i 0.207454i
$$624$$ 0 0
$$625$$ −319.000 −0.510400
$$626$$ 0 0
$$627$$ −1218.00 753.923i −1.94258 1.20243i
$$628$$ 0 0
$$629$$ 743.153i 1.18148i
$$630$$ 0 0
$$631$$ −826.000 −1.30903 −0.654517 0.756048i $$-0.727128\pi$$
−0.654517 + 0.756048i $$0.727128\pi$$
$$632$$ 0 0
$$633$$ 435.000 0.687204
$$634$$ 0 0
$$635$$ 258.488i 0.407068i
$$636$$ 0 0
$$637$$ 775.464i 1.21737i
$$638$$ 0 0
$$639$$ 646.220i 1.01130i
$$640$$ 0 0
$$641$$ 581.598i 0.907329i 0.891173 + 0.453664i $$0.149884\pi$$
−0.891173 + 0.453664i $$0.850116\pi$$
$$642$$ 0 0
$$643$$ 170.000 0.264386 0.132193 0.991224i $$-0.457798\pi$$
0.132193 + 0.991224i $$0.457798\pi$$
$$644$$ 0 0
$$645$$ 1464.76i 2.27095i
$$646$$ 0 0
$$647$$ −25.0000 −0.0386399 −0.0193199 0.999813i $$-0.506150\pi$$
−0.0193199 + 0.999813i $$0.506150\pi$$
$$648$$ 0 0
$$649$$ 226.177i 0.348501i
$$650$$ 0 0
$$651$$ −174.000 −0.267281
$$652$$ 0 0
$$653$$ −1054.00 −1.61409 −0.807044 0.590491i $$-0.798934\pi$$
−0.807044 + 0.590491i $$0.798934\pi$$
$$654$$ 0 0
$$655$$ −224.000 −0.341985
$$656$$ 0 0
$$657$$ 140.000 0.213090
$$658$$ 0 0
$$659$$ 1243.97i 1.88767i 0.330420 + 0.943834i $$0.392810\pi$$
−0.330420 + 0.943834i $$0.607190\pi$$
$$660$$ 0 0
$$661$$ 533.131i 0.806553i 0.915078 + 0.403276i $$0.132129\pi$$
−0.915078 + 0.403276i $$0.867871\pi$$
$$662$$ 0 0
$$663$$ −2001.00 −3.01810
$$664$$ 0 0
$$665$$ 40.0000 64.6220i 0.0601504 0.0971759i
$$666$$ 0 0
$$667$$ 48.4665i 0.0726634i
$$668$$ 0 0
$$669$$ 1392.00 2.08072
$$670$$ 0 0
$$671$$ −560.000 −0.834575
$$672$$ 0 0
$$673$$ 290.799i 0.432093i 0.976383 + 0.216047i $$0.0693164\pi$$
−0.976383 + 0.216047i $$0.930684\pi$$
$$674$$ 0 0
$$675$$ 533.131i 0.789824i
$$676$$ 0 0
$$677$$ 953.174i 1.40794i −0.710231 0.703969i $$-0.751409\pi$$
0.710231 0.703969i $$-0.248591\pi$$
$$678$$ 0 0
$$679$$ 96.9330i 0.142758i
$$680$$ 0 0
$$681$$ 435.000 0.638767
$$682$$ 0 0
$$683$$ 872.397i 1.27730i −0.769497 0.638651i $$-0.779493\pi$$
0.769497 0.638651i $$-0.220507\pi$$
$$684$$ 0 0
$$685$$ 76.0000 0.110949
$$686$$ 0 0
$$687$$ 1637.09i 2.38296i
$$688$$ 0 0
$$689$$ 1305.00 1.89405
$$690$$ 0 0
$$691$$ 668.000 0.966715 0.483357 0.875423i $$-0.339417\pi$$
0.483357 + 0.875423i $$0.339417\pi$$
$$692$$ 0 0
$$693$$ 280.000 0.404040
$$694$$ 0 0
$$695$$ −488.000 −0.702158
$$696$$ 0 0
$$697$$ 743.153i 1.06622i
$$698$$ 0 0
$$699$$ 10.7703i 0.0154082i
$$700$$ 0 0
$$701$$ −700.000 −0.998573 −0.499287 0.866437i $$-0.666405\pi$$
−0.499287 + 0.866437i $$0.666405\pi$$
$$702$$ 0 0
$$703$$ −522.000 323.110i −0.742532 0.459616i
$$704$$ 0 0
$$705$$ 560.057i 0.794407i
$$706$$ 0 0
$$707$$ −14.0000 −0.0198020
$$708$$ 0 0
$$709$$ 1292.00 1.82228 0.911142 0.412092i $$-0.135202\pi$$
0.911142 + 0.412092i $$0.135202\pi$$
$$710$$ 0 0
$$711$$ 1938.66i 2.72667i
$$712$$ 0 0
$$713$$ 32.3110i 0.0453170i
$$714$$ 0 0
$$715$$ 904.708i 1.26533i
$$716$$ 0 0
$$717$$ 479.280i 0.668451i
$$718$$ 0 0
$$719$$ 1061.00 1.47566 0.737830 0.674986i $$-0.235850\pi$$
0.737830 + 0.674986i $$0.235850\pi$$
$$720$$ 0 0
$$721$$ 129.244i 0.179257i
$$722$$ 0 0
$$723$$ 870.000 1.20332
$$724$$ 0 0
$$725$$ 436.198i 0.601653i
$$726$$ 0 0
$$727$$ 83.0000 0.114168 0.0570839 0.998369i $$-0.481820\pi$$
0.0570839 + 0.998369i $$0.481820\pi$$
$$728$$ 0 0
$$729$$ 91.0000 0.124829
$$730$$ 0 0
$$731$$ 1564.00 2.13953
$$732$$ 0 0
$$733$$ 296.000 0.403820 0.201910 0.979404i $$-0.435285\pi$$
0.201910 + 0.979404i $$0.435285\pi$$
$$734$$ 0 0
$$735$$ 1033.95i 1.40674i
$$736$$ 0 0
$$737$$ 226.177i 0.306889i
$$738$$ 0 0
$$739$$ 548.000 0.741543 0.370771 0.928724i $$-0.379093\pi$$
0.370771 + 0.928724i $$0.379093\pi$$
$$740$$ 0 0
$$741$$ −870.000 + 1405.53i −1.17409 + 1.89680i
$$742$$ 0 0
$$743$$ 613.909i 0.826257i −0.910673 0.413128i $$-0.864436\pi$$
0.910673 0.413128i $$-0.135564\pi$$
$$744$$ 0 0
$$745$$ 328.000 0.440268
$$746$$ 0 0
$$747$$ −640.000 −0.856760
$$748$$ 0 0
$$749$$ 16.1555i 0.0215694i
$$750$$ 0 0
$$751$$ 420.043i 0.559311i −0.960100 0.279656i $$-0.909780\pi$$
0.960100 0.279656i $$-0.0902203\pi$$
$$752$$ 0 0
$$753$$ 1992.51i 2.64610i
$$754$$ 0 0
$$755$$ 904.708i 1.19829i
$$756$$ 0 0
$$757$$ 188.000 0.248349 0.124174 0.992260i $$-0.460372\pi$$
0.124174 + 0.992260i $$0.460372\pi$$
$$758$$ 0 0
$$759$$ 75.3923i 0.0993311i
$$760$$ 0 0
$$761$$ −1183.00 −1.55453 −0.777267 0.629171i $$-0.783394\pi$$
−0.777267 + 0.629171i $$0.783394\pi$$
$$762$$ 0 0
$$763$$ 16.1555i 0.0211736i
$$764$$ 0 0
$$765$$ 1840.00 2.40523
$$766$$ 0 0
$$767$$ −261.000 −0.340287
$$768$$ 0 0
$$769$$ −1177.00 −1.53056 −0.765280 0.643698i $$-0.777399\pi$$
−0.765280 + 0.643698i $$0.777399\pi$$
$$770$$ 0 0
$$771$$ 1044.00 1.35409
$$772$$ 0 0
$$773$$ 1017.80i 1.31668i −0.752719 0.658342i $$-0.771258\pi$$
0.752719 0.658342i $$-0.228742\pi$$
$$774$$ 0 0
$$775$$ 290.799i 0.375224i
$$776$$ 0 0
$$777$$ 174.000 0.223938
$$778$$ 0 0
$$779$$ −522.000 323.110i −0.670090 0.414775i
$$780$$ 0 0
$$781$$ 452.354i 0.579198i
$$782$$ 0 0
$$783$$ −2871.00 −3.66667
$$784$$ 0 0
$$785$$ −968.000 −1.23312
$$786$$ 0 0
$$787$$ 113.088i 0.143696i −0.997416 0.0718478i $$-0.977110\pi$$
0.997416 0.0718478i $$-0.0228896\pi$$
$$788$$ 0 0
$$789$$ 2121.75i 2.68917i
$$790$$ 0 0
$$791$$ 96.9330i 0.122545i
$$792$$ 0 0
$$793$$ 646.220i 0.814905i
$$794$$ 0 0
$$795$$ −1740.00 −2.18868
$$796$$ 0 0
$$797$$ 177.710i 0.222974i 0.993766 + 0.111487i $$0.0355614\pi$$
−0.993766 + 0.111487i $$0.964439\pi$$
$$798$$ 0 0
$$799$$ 598.000 0.748436
$$800$$ 0 0