# Properties

 Label 4032.3.d.j.449.3 Level $4032$ Weight $3$ Character 4032.449 Analytic conductor $109.864$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 4032.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$109.864042590$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.3 Root $$-1.16372i$$ of defining polynomial Character $$\chi$$ $$=$$ 4032.449 Dual form 4032.3.d.j.449.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.06910i q^{5} +2.64575 q^{7} +O(q^{10})$$ $$q+6.06910i q^{5} +2.64575 q^{7} -12.1382i q^{11} +18.5830 q^{13} -10.9015i q^{17} +20.0000 q^{19} +12.1382i q^{23} -11.8340 q^{25} -41.8367i q^{29} -25.1660 q^{31} +16.0573i q^{35} -38.0000 q^{37} -60.6337i q^{41} +83.4980 q^{43} +16.9706i q^{47} +7.00000 q^{49} -94.0424i q^{53} +73.6680 q^{55} -58.2175i q^{59} -15.6680 q^{61} +112.782i q^{65} -132.664 q^{67} +12.1382i q^{71} -76.9150 q^{73} -32.1147i q^{77} -33.6680 q^{79} -60.5764i q^{83} +66.1621 q^{85} -4.77506i q^{89} +49.1660 q^{91} +121.382i q^{95} -188.413 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 32 q^{13} + 80 q^{19} - 132 q^{25} - 16 q^{31} - 152 q^{37} + 80 q^{43} + 28 q^{49} + 464 q^{55} - 232 q^{61} - 192 q^{67} - 96 q^{73} - 304 q^{79} - 328 q^{85} + 112 q^{91} - 288 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 6.06910i 1.21382i 0.794770 + 0.606910i $$0.207591\pi$$
−0.794770 + 0.606910i $$0.792409\pi$$
$$6$$ 0 0
$$7$$ 2.64575 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 12.1382i − 1.10347i −0.834019 0.551736i $$-0.813965\pi$$
0.834019 0.551736i $$-0.186035\pi$$
$$12$$ 0 0
$$13$$ 18.5830 1.42946 0.714731 0.699399i $$-0.246549\pi$$
0.714731 + 0.699399i $$0.246549\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 10.9015i − 0.641262i −0.947204 0.320631i $$-0.896105\pi$$
0.947204 0.320631i $$-0.103895\pi$$
$$18$$ 0 0
$$19$$ 20.0000 1.05263 0.526316 0.850289i $$-0.323573\pi$$
0.526316 + 0.850289i $$0.323573\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 12.1382i 0.527748i 0.964557 + 0.263874i $$0.0850003\pi$$
−0.964557 + 0.263874i $$0.915000\pi$$
$$24$$ 0 0
$$25$$ −11.8340 −0.473360
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 41.8367i − 1.44264i −0.692600 0.721322i $$-0.743535\pi$$
0.692600 0.721322i $$-0.256465\pi$$
$$30$$ 0 0
$$31$$ −25.1660 −0.811807 −0.405903 0.913916i $$-0.633043\pi$$
−0.405903 + 0.913916i $$0.633043\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 16.0573i 0.458781i
$$36$$ 0 0
$$37$$ −38.0000 −1.02703 −0.513514 0.858082i $$-0.671656\pi$$
−0.513514 + 0.858082i $$0.671656\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 60.6337i − 1.47887i −0.673227 0.739435i $$-0.735092\pi$$
0.673227 0.739435i $$-0.264908\pi$$
$$42$$ 0 0
$$43$$ 83.4980 1.94181 0.970907 0.239455i $$-0.0769689\pi$$
0.970907 + 0.239455i $$0.0769689\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 16.9706i 0.361076i 0.983568 + 0.180538i $$0.0577838\pi$$
−0.983568 + 0.180538i $$0.942216\pi$$
$$48$$ 0 0
$$49$$ 7.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 94.0424i − 1.77439i −0.461399 0.887193i $$-0.652652\pi$$
0.461399 0.887193i $$-0.347348\pi$$
$$54$$ 0 0
$$55$$ 73.6680 1.33942
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 58.2175i − 0.986738i −0.869820 0.493369i $$-0.835765\pi$$
0.869820 0.493369i $$-0.164235\pi$$
$$60$$ 0 0
$$61$$ −15.6680 −0.256852 −0.128426 0.991719i $$-0.540992\pi$$
−0.128426 + 0.991719i $$0.540992\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 112.782i 1.73511i
$$66$$ 0 0
$$67$$ −132.664 −1.98006 −0.990030 0.140856i $$-0.955015\pi$$
−0.990030 + 0.140856i $$0.955015\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.1382i 0.170961i 0.996340 + 0.0854803i $$0.0272425\pi$$
−0.996340 + 0.0854803i $$0.972758\pi$$
$$72$$ 0 0
$$73$$ −76.9150 −1.05363 −0.526815 0.849980i $$-0.676614\pi$$
−0.526815 + 0.849980i $$0.676614\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 32.1147i − 0.417074i
$$78$$ 0 0
$$79$$ −33.6680 −0.426177 −0.213088 0.977033i $$-0.568352\pi$$
−0.213088 + 0.977033i $$0.568352\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 60.5764i − 0.729836i −0.931040 0.364918i $$-0.881097\pi$$
0.931040 0.364918i $$-0.118903\pi$$
$$84$$ 0 0
$$85$$ 66.1621 0.778377
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 4.77506i − 0.0536523i −0.999640 0.0268262i $$-0.991460\pi$$
0.999640 0.0268262i $$-0.00854006\pi$$
$$90$$ 0 0
$$91$$ 49.1660 0.540286
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 121.382i 1.27771i
$$96$$ 0 0
$$97$$ −188.413 −1.94240 −0.971201 0.238260i $$-0.923423\pi$$
−0.971201 + 0.238260i $$0.923423\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 106.713i − 1.05656i −0.849069 0.528282i $$-0.822836\pi$$
0.849069 0.528282i $$-0.177164\pi$$
$$102$$ 0 0
$$103$$ −131.498 −1.27668 −0.638340 0.769755i $$-0.720379\pi$$
−0.638340 + 0.769755i $$0.720379\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 82.3793i 0.769900i 0.922937 + 0.384950i $$0.125781\pi$$
−0.922937 + 0.384950i $$0.874219\pi$$
$$108$$ 0 0
$$109$$ −33.8301 −0.310367 −0.155184 0.987886i $$-0.549597\pi$$
−0.155184 + 0.987886i $$0.549597\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 28.5190i − 0.252381i −0.992006 0.126190i $$-0.959725\pi$$
0.992006 0.126190i $$-0.0402750\pi$$
$$114$$ 0 0
$$115$$ −73.6680 −0.640591
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 28.8426i − 0.242374i
$$120$$ 0 0
$$121$$ −26.3360 −0.217653
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 79.9059i 0.639247i
$$126$$ 0 0
$$127$$ −129.668 −1.02101 −0.510504 0.859875i $$-0.670541\pi$$
−0.510504 + 0.859875i $$0.670541\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 148.017i − 1.12990i −0.825124 0.564952i $$-0.808895\pi$$
0.825124 0.564952i $$-0.191105\pi$$
$$132$$ 0 0
$$133$$ 52.9150 0.397857
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 76.9573i − 0.561732i −0.959747 0.280866i $$-0.909378\pi$$
0.959747 0.280866i $$-0.0906216\pi$$
$$138$$ 0 0
$$139$$ 217.328 1.56351 0.781756 0.623585i $$-0.214324\pi$$
0.781756 + 0.623585i $$0.214324\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 225.564i − 1.57737i
$$144$$ 0 0
$$145$$ 253.911 1.75111
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 161.925i − 1.08674i −0.839492 0.543371i $$-0.817148\pi$$
0.839492 0.543371i $$-0.182852\pi$$
$$150$$ 0 0
$$151$$ 93.1660 0.616993 0.308497 0.951225i $$-0.400174\pi$$
0.308497 + 0.951225i $$0.400174\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 152.735i − 0.985388i
$$156$$ 0 0
$$157$$ 184.996 1.17832 0.589159 0.808017i $$-0.299459\pi$$
0.589159 + 0.808017i $$0.299459\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 32.1147i 0.199470i
$$162$$ 0 0
$$163$$ 86.9961 0.533718 0.266859 0.963736i $$-0.414014\pi$$
0.266859 + 0.963736i $$0.414014\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 60.5764i − 0.362733i −0.983416 0.181366i $$-0.941948\pi$$
0.983416 0.181366i $$-0.0580520\pi$$
$$168$$ 0 0
$$169$$ 176.328 1.04336
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 162.572i − 0.939721i −0.882741 0.469860i $$-0.844304\pi$$
0.882741 0.469860i $$-0.155696\pi$$
$$174$$ 0 0
$$175$$ −31.3098 −0.178913
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 223.091i 1.24632i 0.782095 + 0.623159i $$0.214151\pi$$
−0.782095 + 0.623159i $$0.785849\pi$$
$$180$$ 0 0
$$181$$ −188.915 −1.04373 −0.521865 0.853028i $$-0.674763\pi$$
−0.521865 + 0.853028i $$0.674763\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 230.626i − 1.24663i
$$186$$ 0 0
$$187$$ −132.324 −0.707616
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 228.038i 1.19391i 0.802273 + 0.596957i $$0.203624\pi$$
−0.802273 + 0.596957i $$0.796376\pi$$
$$192$$ 0 0
$$193$$ 134.000 0.694301 0.347150 0.937810i $$-0.387149\pi$$
0.347150 + 0.937810i $$0.387149\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 188.560i 0.957157i 0.878045 + 0.478579i $$0.158848\pi$$
−0.878045 + 0.478579i $$0.841152\pi$$
$$198$$ 0 0
$$199$$ −102.494 −0.515046 −0.257523 0.966272i $$-0.582906\pi$$
−0.257523 + 0.966272i $$0.582906\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 110.689i − 0.545268i
$$204$$ 0 0
$$205$$ 367.992 1.79508
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 242.764i − 1.16155i
$$210$$ 0 0
$$211$$ −84.5020 −0.400483 −0.200242 0.979747i $$-0.564173\pi$$
−0.200242 + 0.979747i $$0.564173\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 506.758i 2.35701i
$$216$$ 0 0
$$217$$ −66.5830 −0.306834
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 202.582i − 0.916660i
$$222$$ 0 0
$$223$$ 158.494 0.710736 0.355368 0.934727i $$-0.384356\pi$$
0.355368 + 0.934727i $$0.384356\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 101.823i 0.448561i 0.974525 + 0.224281i $$0.0720032\pi$$
−0.974525 + 0.224281i $$0.927997\pi$$
$$228$$ 0 0
$$229$$ 268.915 1.17430 0.587151 0.809478i $$-0.300250\pi$$
0.587151 + 0.809478i $$0.300250\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 26.2748i − 0.112767i −0.998409 0.0563836i $$-0.982043\pi$$
0.998409 0.0563836i $$-0.0179570\pi$$
$$234$$ 0 0
$$235$$ −102.996 −0.438281
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 92.2733i − 0.386081i −0.981191 0.193040i $$-0.938165\pi$$
0.981191 0.193040i $$-0.0618348\pi$$
$$240$$ 0 0
$$241$$ 343.247 1.42426 0.712131 0.702047i $$-0.247730\pi$$
0.712131 + 0.702047i $$0.247730\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 42.4837i 0.173403i
$$246$$ 0 0
$$247$$ 371.660 1.50470
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 356.382i 1.41985i 0.704278 + 0.709924i $$0.251271\pi$$
−0.704278 + 0.709924i $$0.748729\pi$$
$$252$$ 0 0
$$253$$ 147.336 0.582356
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 254.730i − 0.991169i −0.868560 0.495584i $$-0.834954\pi$$
0.868560 0.495584i $$-0.165046\pi$$
$$258$$ 0 0
$$259$$ −100.539 −0.388180
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 261.979i − 0.996117i −0.867143 0.498059i $$-0.834046\pi$$
0.867143 0.498059i $$-0.165954\pi$$
$$264$$ 0 0
$$265$$ 570.753 2.15378
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 93.6246i − 0.348047i −0.984742 0.174023i $$-0.944323\pi$$
0.984742 0.174023i $$-0.0556768\pi$$
$$270$$ 0 0
$$271$$ −1.16601 −0.00430262 −0.00215131 0.999998i $$-0.500685\pi$$
−0.00215131 + 0.999998i $$0.500685\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 143.643i 0.522339i
$$276$$ 0 0
$$277$$ −32.0000 −0.115523 −0.0577617 0.998330i $$-0.518396\pi$$
−0.0577617 + 0.998330i $$0.518396\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 166.757i 0.593441i 0.954964 + 0.296721i $$0.0958930\pi$$
−0.954964 + 0.296721i $$0.904107\pi$$
$$282$$ 0 0
$$283$$ 16.3399 0.0577381 0.0288691 0.999583i $$-0.490809\pi$$
0.0288691 + 0.999583i $$0.490809\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 160.422i − 0.558961i
$$288$$ 0 0
$$289$$ 170.158 0.588782
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 368.921i − 1.25912i −0.776953 0.629558i $$-0.783236\pi$$
0.776953 0.629558i $$-0.216764\pi$$
$$294$$ 0 0
$$295$$ 353.328 1.19772
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 225.564i 0.754396i
$$300$$ 0 0
$$301$$ 220.915 0.733937
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 95.0906i − 0.311772i
$$306$$ 0 0
$$307$$ −192.664 −0.627570 −0.313785 0.949494i $$-0.601597\pi$$
−0.313785 + 0.949494i $$0.601597\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 131.276i − 0.422109i −0.977474 0.211055i $$-0.932310\pi$$
0.977474 0.211055i $$-0.0676898\pi$$
$$312$$ 0 0
$$313$$ 43.3281 0.138428 0.0692142 0.997602i $$-0.477951\pi$$
0.0692142 + 0.997602i $$0.477951\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 251.724i 0.794083i 0.917801 + 0.397042i $$0.129963\pi$$
−0.917801 + 0.397042i $$0.870037\pi$$
$$318$$ 0 0
$$319$$ −507.822 −1.59192
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 218.029i − 0.675013i
$$324$$ 0 0
$$325$$ −219.911 −0.676650
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 44.8999i 0.136474i
$$330$$ 0 0
$$331$$ 361.490 1.09212 0.546058 0.837748i $$-0.316128\pi$$
0.546058 + 0.837748i $$0.316128\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 805.151i − 2.40344i
$$336$$ 0 0
$$337$$ −298.834 −0.886748 −0.443374 0.896337i $$-0.646219\pi$$
−0.443374 + 0.896337i $$0.646219\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 305.470i 0.895807i
$$342$$ 0 0
$$343$$ 18.5203 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 206.120i 0.594006i 0.954876 + 0.297003i $$0.0959872\pi$$
−0.954876 + 0.297003i $$0.904013\pi$$
$$348$$ 0 0
$$349$$ −434.324 −1.24448 −0.622241 0.782826i $$-0.713778\pi$$
−0.622241 + 0.782826i $$0.713778\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 185.439i − 0.525324i −0.964888 0.262662i $$-0.915400\pi$$
0.964888 0.262662i $$-0.0846005\pi$$
$$354$$ 0 0
$$355$$ −73.6680 −0.207515
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 516.767i 1.43946i 0.694254 + 0.719731i $$0.255735\pi$$
−0.694254 + 0.719731i $$0.744265\pi$$
$$360$$ 0 0
$$361$$ 39.0000 0.108033
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 466.805i − 1.27892i
$$366$$ 0 0
$$367$$ 117.490 0.320137 0.160068 0.987106i $$-0.448829\pi$$
0.160068 + 0.987106i $$0.448829\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 248.813i − 0.670655i
$$372$$ 0 0
$$373$$ 402.664 1.07953 0.539764 0.841816i $$-0.318513\pi$$
0.539764 + 0.841816i $$0.318513\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 777.451i − 2.06221i
$$378$$ 0 0
$$379$$ 398.834 1.05233 0.526166 0.850382i $$-0.323629\pi$$
0.526166 + 0.850382i $$0.323629\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 744.804i − 1.94466i −0.233614 0.972329i $$-0.575055\pi$$
0.233614 0.972329i $$-0.424945\pi$$
$$384$$ 0 0
$$385$$ 194.907 0.506252
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 535.162i 1.37574i 0.725834 + 0.687869i $$0.241454\pi$$
−0.725834 + 0.687869i $$0.758546\pi$$
$$390$$ 0 0
$$391$$ 132.324 0.338425
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 204.334i − 0.517302i
$$396$$ 0 0
$$397$$ 94.3241 0.237592 0.118796 0.992919i $$-0.462096\pi$$
0.118796 + 0.992919i $$0.462096\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 103.593i − 0.258335i −0.991623 0.129168i $$-0.958769\pi$$
0.991623 0.129168i $$-0.0412306\pi$$
$$402$$ 0 0
$$403$$ −467.660 −1.16045
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 461.252i 1.13330i
$$408$$ 0 0
$$409$$ −9.75689 −0.0238555 −0.0119277 0.999929i $$-0.503797\pi$$
−0.0119277 + 0.999929i $$0.503797\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 154.029i − 0.372952i
$$414$$ 0 0
$$415$$ 367.644 0.885890
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 339.411i 0.810051i 0.914305 + 0.405025i $$0.132737\pi$$
−0.914305 + 0.405025i $$0.867263\pi$$
$$420$$ 0 0
$$421$$ 599.320 1.42356 0.711782 0.702401i $$-0.247889\pi$$
0.711782 + 0.702401i $$0.247889\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 129.008i 0.303548i
$$426$$ 0 0
$$427$$ −41.4536 −0.0970810
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 710.978i − 1.64960i −0.565424 0.824800i $$-0.691288\pi$$
0.565424 0.824800i $$-0.308712\pi$$
$$432$$ 0 0
$$433$$ 377.984 0.872943 0.436471 0.899718i $$-0.356228\pi$$
0.436471 + 0.899718i $$0.356228\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 242.764i 0.555524i
$$438$$ 0 0
$$439$$ −528.146 −1.20307 −0.601533 0.798848i $$-0.705443\pi$$
−0.601533 + 0.798848i $$0.705443\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 36.6438i 0.0827174i 0.999144 + 0.0413587i $$0.0131686\pi$$
−0.999144 + 0.0413587i $$0.986831\pi$$
$$444$$ 0 0
$$445$$ 28.9803 0.0651243
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 397.612i − 0.885550i −0.896633 0.442775i $$-0.853994\pi$$
0.896633 0.442775i $$-0.146006\pi$$
$$450$$ 0 0
$$451$$ −735.984 −1.63189
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 298.393i 0.655810i
$$456$$ 0 0
$$457$$ 344.324 0.753445 0.376722 0.926326i $$-0.377051\pi$$
0.376722 + 0.926326i $$0.377051\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 370.936i 0.804634i 0.915500 + 0.402317i $$0.131795\pi$$
−0.915500 + 0.402317i $$0.868205\pi$$
$$462$$ 0 0
$$463$$ −78.3320 −0.169184 −0.0845918 0.996416i $$-0.526959\pi$$
−0.0845918 + 0.996416i $$0.526959\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 399.758i − 0.856014i −0.903775 0.428007i $$-0.859216\pi$$
0.903775 0.428007i $$-0.140784\pi$$
$$468$$ 0 0
$$469$$ −350.996 −0.748392
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 1013.52i − 2.14274i
$$474$$ 0 0
$$475$$ −236.680 −0.498273
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 703.328i 1.46833i 0.678973 + 0.734163i $$0.262425\pi$$
−0.678973 + 0.734163i $$0.737575\pi$$
$$480$$ 0 0
$$481$$ −706.154 −1.46810
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 1143.50i − 2.35773i
$$486$$ 0 0
$$487$$ 82.5098 0.169425 0.0847124 0.996405i $$-0.473003\pi$$
0.0847124 + 0.996405i $$0.473003\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 184.203i − 0.375158i −0.982249 0.187579i $$-0.939936\pi$$
0.982249 0.187579i $$-0.0600641\pi$$
$$492$$ 0 0
$$493$$ −456.081 −0.925114
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.1147i 0.0646170i
$$498$$ 0 0
$$499$$ 752.810 1.50864 0.754319 0.656508i $$-0.227967\pi$$
0.754319 + 0.656508i $$0.227967\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 662.540i 1.31718i 0.752504 + 0.658588i $$0.228846\pi$$
−0.752504 + 0.658588i $$0.771154\pi$$
$$504$$ 0 0
$$505$$ 647.652 1.28248
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 949.115i − 1.86467i −0.361601 0.932333i $$-0.617770\pi$$
0.361601 0.932333i $$-0.382230\pi$$
$$510$$ 0 0
$$511$$ −203.498 −0.398235
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 798.075i − 1.54966i
$$516$$ 0 0
$$517$$ 205.992 0.398437
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 714.344i 1.37110i 0.728025 + 0.685551i $$0.240439\pi$$
−0.728025 + 0.685551i $$0.759561\pi$$
$$522$$ 0 0
$$523$$ −232.000 −0.443595 −0.221797 0.975093i $$-0.571192\pi$$
−0.221797 + 0.975093i $$0.571192\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 274.346i 0.520581i
$$528$$ 0 0
$$529$$ 381.664 0.721482
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 1126.76i − 2.11399i
$$534$$ 0 0
$$535$$ −499.969 −0.934521
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 84.9674i − 0.157639i
$$540$$ 0 0
$$541$$ −165.668 −0.306225 −0.153113 0.988209i $$-0.548930\pi$$
−0.153113 + 0.988209i $$0.548930\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 205.318i − 0.376730i
$$546$$ 0 0
$$547$$ 295.676 0.540541 0.270270 0.962784i $$-0.412887\pi$$
0.270270 + 0.962784i $$0.412887\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 836.734i − 1.51857i
$$552$$ 0 0
$$553$$ −89.0771 −0.161080
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 76.8426i − 0.137958i −0.997618 0.0689790i $$-0.978026\pi$$
0.997618 0.0689790i $$-0.0219742\pi$$
$$558$$ 0 0
$$559$$ 1551.64 2.77575
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1016.33i 1.80521i 0.430470 + 0.902605i $$0.358348\pi$$
−0.430470 + 0.902605i $$0.641652\pi$$
$$564$$ 0 0
$$565$$ 173.085 0.306345
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 586.533i − 1.03081i −0.856946 0.515406i $$-0.827641\pi$$
0.856946 0.515406i $$-0.172359\pi$$
$$570$$ 0 0
$$571$$ 951.644 1.66663 0.833314 0.552800i $$-0.186441\pi$$
0.833314 + 0.552800i $$0.186441\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 143.643i − 0.249815i
$$576$$ 0 0
$$577$$ −148.672 −0.257664 −0.128832 0.991666i $$-0.541123\pi$$
−0.128832 + 0.991666i $$0.541123\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 160.270i − 0.275852i
$$582$$ 0 0
$$583$$ −1141.51 −1.95799
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 332.564i 0.566548i 0.959039 + 0.283274i $$0.0914206\pi$$
−0.959039 + 0.283274i $$0.908579\pi$$
$$588$$ 0 0
$$589$$ −503.320 −0.854533
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 217.251i 0.366359i 0.983079 + 0.183180i $$0.0586390\pi$$
−0.983079 + 0.183180i $$0.941361\pi$$
$$594$$ 0 0
$$595$$ 175.048 0.294199
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 172.179i 0.287444i 0.989618 + 0.143722i $$0.0459072\pi$$
−0.989618 + 0.143722i $$0.954093\pi$$
$$600$$ 0 0
$$601$$ −418.000 −0.695507 −0.347754 0.937586i $$-0.613055\pi$$
−0.347754 + 0.937586i $$0.613055\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 159.836i − 0.264191i
$$606$$ 0 0
$$607$$ −627.158 −1.03321 −0.516605 0.856224i $$-0.672804\pi$$
−0.516605 + 0.856224i $$0.672804\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 315.364i 0.516144i
$$612$$ 0 0
$$613$$ 279.328 0.455674 0.227837 0.973699i $$-0.426835\pi$$
0.227837 + 0.973699i $$0.426835\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 358.380i 0.580843i 0.956899 + 0.290422i $$0.0937955\pi$$
−0.956899 + 0.290422i $$0.906204\pi$$
$$618$$ 0 0
$$619$$ −983.644 −1.58909 −0.794543 0.607208i $$-0.792290\pi$$
−0.794543 + 0.607208i $$0.792290\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 12.6336i − 0.0202787i
$$624$$ 0 0
$$625$$ −780.806 −1.24929
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 414.256i 0.658594i
$$630$$ 0 0
$$631$$ 298.996 0.473845 0.236922 0.971529i $$-0.423861\pi$$
0.236922 + 0.971529i $$0.423861\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 786.968i − 1.23932i
$$636$$ 0 0
$$637$$ 130.081 0.204209
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 311.957i − 0.486672i −0.969942 0.243336i $$-0.921758\pi$$
0.969942 0.243336i $$-0.0782418\pi$$
$$642$$ 0 0
$$643$$ −604.000 −0.939347 −0.469673 0.882840i $$-0.655628\pi$$
−0.469673 + 0.882840i $$0.655628\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 179.600i − 0.277588i −0.990321 0.138794i $$-0.955677\pi$$
0.990321 0.138794i $$-0.0443226\pi$$
$$648$$ 0 0
$$649$$ −706.656 −1.08884
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 481.892i − 0.737966i −0.929436 0.368983i $$-0.879706\pi$$
0.929436 0.368983i $$-0.120294\pi$$
$$654$$ 0 0
$$655$$ 898.332 1.37150
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 877.408i 1.33142i 0.746209 + 0.665711i $$0.231872\pi$$
−0.746209 + 0.665711i $$0.768128\pi$$
$$660$$ 0 0
$$661$$ 521.644 0.789175 0.394587 0.918858i $$-0.370888\pi$$
0.394587 + 0.918858i $$0.370888\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 321.147i 0.482927i
$$666$$ 0 0
$$667$$ 507.822 0.761353
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 190.181i 0.283429i
$$672$$ 0 0
$$673$$ −659.992 −0.980672 −0.490336 0.871534i $$-0.663126\pi$$
−0.490336 + 0.871534i $$0.663126\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 1016.28i − 1.50115i −0.660787 0.750573i $$-0.729778\pi$$
0.660787 0.750573i $$-0.270222\pi$$
$$678$$ 0 0
$$679$$ −498.494 −0.734159
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 235.114i − 0.344238i −0.985076 0.172119i $$-0.944939\pi$$
0.985076 0.172119i $$-0.0550613\pi$$
$$684$$ 0 0
$$685$$ 467.061 0.681841
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 1747.59i − 2.53642i
$$690$$ 0 0
$$691$$ −50.9803 −0.0737776 −0.0368888 0.999319i $$-0.511745\pi$$
−0.0368888 + 0.999319i $$0.511745\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1318.99i 1.89782i
$$696$$ 0 0
$$697$$ −660.996 −0.948344
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 141.530i − 0.201898i −0.994892 0.100949i $$-0.967812\pi$$
0.994892 0.100949i $$-0.0321879\pi$$
$$702$$ 0 0
$$703$$ −760.000 −1.08108
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 282.336i − 0.399344i
$$708$$ 0 0
$$709$$ 55.4980 0.0782765 0.0391382 0.999234i $$-0.487539\pi$$
0.0391382 + 0.999234i $$0.487539\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 305.470i − 0.428429i
$$714$$ 0 0
$$715$$ 1368.97 1.91465
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1009.03i 1.40338i 0.712484 + 0.701688i $$0.247570\pi$$
−0.712484 + 0.701688i $$0.752430\pi$$
$$720$$ 0 0
$$721$$ −347.911 −0.482540
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 495.095i 0.682890i
$$726$$ 0 0
$$727$$ 365.182 0.502313 0.251157 0.967946i $$-0.419189\pi$$
0.251157 + 0.967946i $$0.419189\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 910.251i − 1.24521i
$$732$$ 0 0
$$733$$ 353.077 0.481688 0.240844 0.970564i $$-0.422576\pi$$
0.240844 + 0.970564i $$0.422576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1610.30i 2.18494i
$$738$$ 0 0
$$739$$ 329.684 0.446121 0.223061 0.974805i $$-0.428395\pi$$
0.223061 + 0.974805i $$0.428395\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 112.061i 0.150822i 0.997153 + 0.0754112i $$0.0240270\pi$$
−0.997153 + 0.0754112i $$0.975973\pi$$
$$744$$ 0 0
$$745$$ 982.737 1.31911
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 217.955i 0.290995i
$$750$$ 0 0
$$751$$ 144.826 0.192844 0.0964222 0.995341i $$-0.469260\pi$$
0.0964222 + 0.995341i $$0.469260\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 565.434i 0.748919i
$$756$$ 0 0
$$757$$ −78.1699 −0.103263 −0.0516314 0.998666i $$-0.516442\pi$$
−0.0516314 + 0.998666i $$0.516442\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1465.50i 1.92576i 0.269928 + 0.962880i $$0.413000\pi$$
−0.269928 + 0.962880i $$0.587000\pi$$
$$762$$ 0 0
$$763$$ −89.5059 −0.117308
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1081.86i − 1.41050i
$$768$$ 0 0
$$769$$ 729.320 0.948401 0.474200 0.880417i $$-0.342737\pi$$
0.474200 + 0.880417i $$0.342737\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 434.559i 0.562172i 0.959683 + 0.281086i $$0.0906947\pi$$
−0.959683 + 0.281086i $$0.909305\pi$$
$$774$$ 0 0
$$775$$ 297.814 0.384277
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 1212.67i − 1.55671i
$$780$$ 0 0
$$781$$ 147.336 0.188650
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1122.76i 1.43027i
$$786$$ 0 0
$$787$$ −15.3517 −0.0195066 −0.00975331 0.999952i $$-0.503105\pi$$
−0.00975331 + 0.999952i $$0.503105\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 75.4543i − 0.0953910i
$$792$$ 0 0
$$793$$ −291.158 −0.367160
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1043.48i 1.30927i 0.755947 + 0.654633i $$0.227177\pi$$
−0.755947 + 0.654633i $$0.772823\pi$$
$$798$$ 0 0
$$799$$ 185.004 0.231544
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 933.610i 1.16265i
$$804$$ 0 0
$$805$$ −194.907 −0.242121
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1041.31i 1.28716i 0.765378 + 0.643581i $$0.222552\pi$$
−0.765378 + 0.643581i $$0.777448\pi$$
$$810$$ 0 0
$$811$$ 502.316 0.619379 0.309689 0.950838i $$-0.399775\pi$$
0.309689 + 0.950838i $$0.399775\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 527.988i 0.647838i
$$816$$ 0 0
$$817$$ 1669.96 2.04402
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 23.1137i 0.0281531i 0.999901 + 0.0140765i $$0.00448085\pi$$
−0.999901 + 0.0140765i $$0.995519\pi$$
$$822$$ 0 0
$$823$$ 600.664 0.729847 0.364923 0.931038i $$-0.381095\pi$$
0.364923 + 0.931038i $$0.381095\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 1309.21i − 1.58308i −0.611118 0.791540i $$-0.709280\pi$$
0.611118 0.791540i $$-0.290720\pi$$
$$828$$ 0 0
$$829$$ 621.919 0.750204 0.375102 0.926984i $$-0.377608\pi$$
0.375102 + 0.926984i $$0.377608\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 76.3102i − 0.0916089i
$$834$$ 0 0
$$835$$ 367.644 0.440293
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1190.30i − 1.41871i −0.704851 0.709355i $$-0.748986\pi$$
0.704851 0.709355i $$-0.251014\pi$$
$$840$$ 0 0
$$841$$ −909.308 −1.08122
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1070.15i 1.26645i
$$846$$ 0 0
$$847$$ −69.6784 −0.0822649
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 461.252i − 0.542011i
$$852$$ 0 0
$$853$$ −137.012 −0.160623 −0.0803117 0.996770i $$-0.525592\pi$$
−0.0803117 + 0.996770i $$0.525592\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 466.141i 0.543922i 0.962308 + 0.271961i $$0.0876722\pi$$
−0.962308 + 0.271961i $$0.912328\pi$$
$$858$$ 0 0
$$859$$ 23.9843 0.0279211 0.0139606 0.999903i $$-0.495556\pi$$
0.0139606 + 0.999903i $$0.495556\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 0.114603i 0 0.000132796i −1.00000 6.63982e-5i $$-0.999979\pi$$
1.00000 6.63982e-5i $$-2.11352e-5\pi$$
$$864$$ 0 0
$$865$$ 986.664 1.14065
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 408.669i 0.470275i
$$870$$ 0 0
$$871$$ −2465.30 −2.83042
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 211.411i 0.241613i
$$876$$ 0 0
$$877$$ −997.304 −1.13718 −0.568589 0.822622i $$-0.692510\pi$$
−0.568589 + 0.822622i $$0.692510\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 935.649i 1.06203i 0.847362 + 0.531015i $$0.178189\pi$$
−0.847362 + 0.531015i $$0.821811\pi$$
$$882$$ 0 0
$$883$$ −1549.47 −1.75478 −0.877392 0.479774i $$-0.840719\pi$$
−0.877392 + 0.479774i $$0.840719\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 894.493i 1.00845i 0.863573 + 0.504224i $$0.168221\pi$$
−0.863573 + 0.504224i $$0.831779\pi$$
$$888$$ 0 0
$$889$$ −343.069 −0.385905
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 339.411i 0.380080i
$$894$$ 0 0
$$895$$ −1353.96 −1.51281
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1052.86i 1.17115i
$$900$$ 0 0
$$901$$ −1025.20 −1.13785
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 1146.54i − 1.26690i
$$906$$ 0 0
$$907$$ −135.838 −0.149766 −0.0748831 0.997192i $$-0.523858\pi$$
−0.0748831 + 0.997192i $$0.523858\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1242.01i 1.36335i 0.731655 + 0.681675i $$0.238748\pi$$
−0.731655 + 0.681675i $$0.761252\pi$$
$$912$$ 0 0
$$913$$ −735.289 −0.805355
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 391.617i − 0.427063i
$$918$$ 0 0
$$919$$ 388.162 0.422374 0.211187 0.977446i $$-0.432267\pi$$
0.211187 + 0.977446i $$0.432267\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 225.564i 0.244382i
$$924$$ 0 0
$$925$$ 449.692 0.486153
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 621.694i − 0.669207i −0.942359 0.334604i $$-0.891398\pi$$
0.942359 0.334604i $$-0.108602\pi$$
$$930$$ 0 0
$$931$$ 140.000 0.150376
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 803.089i − 0.858918i
$$936$$ 0 0
$$937$$ 1262.00 1.34685 0.673426 0.739255i $$-0.264822\pi$$
0.673426 + 0.739255i $$0.264822\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 672.410i − 0.714569i −0.933996 0.357285i $$-0.883703\pi$$
0.933996 0.357285i $$-0.116297\pi$$
$$942$$ 0 0
$$943$$ 735.984 0.780471
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1159.75i 1.22465i 0.790605 + 0.612327i $$0.209766\pi$$
−0.790605 + 0.612327i $$0.790234\pi$$
$$948$$ 0 0
$$949$$ −1429.31 −1.50612
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 163.104i − 0.171148i −0.996332 0.0855740i $$-0.972728\pi$$
0.996332 0.0855740i $$-0.0272724\pi$$
$$954$$ 0 0
$$955$$ −1383.98 −1.44920
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 203.610i − 0.212315i
$$960$$ 0 0
$$961$$ −327.672 −0.340970
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 813.260i 0.842756i
$$966$$ 0 0
$$967$$ −887.012 −0.917282 −0.458641 0.888622i $$-0.651664\pi$$
−0.458641 + 0.888622i $$0.651664\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 1416.32i − 1.45862i −0.684183 0.729310i $$-0.739841\pi$$
0.684183 0.729310i $$-0.260159\pi$$
$$972$$ 0 0
$$973$$ 574.996 0.590952
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 339.051i 0.347032i 0.984831 + 0.173516i $$0.0555129\pi$$
−0.984831 + 0.173516i $$0.944487\pi$$
$$978$$ 0 0
$$979$$ −57.9606 −0.0592039
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 487.887i 0.496324i 0.968718 + 0.248162i $$0.0798266\pi$$
−0.968718 + 0.248162i $$0.920173\pi$$
$$984$$ 0 0
$$985$$ −1144.39 −1.16182
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1013.52i 1.02479i
$$990$$ 0 0
$$991$$ 937.474 0.945988 0.472994 0.881066i $$-0.343173\pi$$
0.472994 + 0.881066i $$0.343173\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 622.047i − 0.625173i
$$996$$ 0 0
$$997$$ −461.012 −0.462399 −0.231200 0.972906i $$-0.574265\pi$$
−0.231200 + 0.972906i $$0.574265\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4032.3.d.j.449.3 4
3.2 odd 2 inner 4032.3.d.j.449.2 4
4.3 odd 2 4032.3.d.i.449.3 4
8.3 odd 2 126.3.b.a.71.1 4
8.5 even 2 1008.3.d.a.449.2 4
12.11 even 2 4032.3.d.i.449.2 4
24.5 odd 2 1008.3.d.a.449.3 4
24.11 even 2 126.3.b.a.71.4 yes 4
40.3 even 4 3150.3.c.b.449.4 8
40.19 odd 2 3150.3.e.e.701.4 4
40.27 even 4 3150.3.c.b.449.6 8
56.3 even 6 882.3.s.i.863.3 8
56.11 odd 6 882.3.s.e.863.4 8
56.19 even 6 882.3.s.i.557.2 8
56.27 even 2 882.3.b.f.197.2 4
56.51 odd 6 882.3.s.e.557.1 8
72.11 even 6 1134.3.q.c.701.3 8
72.43 odd 6 1134.3.q.c.701.2 8
72.59 even 6 1134.3.q.c.1079.2 8
72.67 odd 6 1134.3.q.c.1079.3 8
120.59 even 2 3150.3.e.e.701.2 4
120.83 odd 4 3150.3.c.b.449.7 8
120.107 odd 4 3150.3.c.b.449.1 8
168.11 even 6 882.3.s.e.863.1 8
168.59 odd 6 882.3.s.i.863.2 8
168.83 odd 2 882.3.b.f.197.3 4
168.107 even 6 882.3.s.e.557.4 8
168.131 odd 6 882.3.s.i.557.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 8.3 odd 2
126.3.b.a.71.4 yes 4 24.11 even 2
882.3.b.f.197.2 4 56.27 even 2
882.3.b.f.197.3 4 168.83 odd 2
882.3.s.e.557.1 8 56.51 odd 6
882.3.s.e.557.4 8 168.107 even 6
882.3.s.e.863.1 8 168.11 even 6
882.3.s.e.863.4 8 56.11 odd 6
882.3.s.i.557.2 8 56.19 even 6
882.3.s.i.557.3 8 168.131 odd 6
882.3.s.i.863.2 8 168.59 odd 6
882.3.s.i.863.3 8 56.3 even 6
1008.3.d.a.449.2 4 8.5 even 2
1008.3.d.a.449.3 4 24.5 odd 2
1134.3.q.c.701.2 8 72.43 odd 6
1134.3.q.c.701.3 8 72.11 even 6
1134.3.q.c.1079.2 8 72.59 even 6
1134.3.q.c.1079.3 8 72.67 odd 6
3150.3.c.b.449.1 8 120.107 odd 4
3150.3.c.b.449.4 8 40.3 even 4
3150.3.c.b.449.6 8 40.27 even 4
3150.3.c.b.449.7 8 120.83 odd 4
3150.3.e.e.701.2 4 120.59 even 2
3150.3.e.e.701.4 4 40.19 odd 2
4032.3.d.i.449.2 4 12.11 even 2
4032.3.d.i.449.3 4 4.3 odd 2
4032.3.d.j.449.2 4 3.2 odd 2 inner
4032.3.d.j.449.3 4 1.1 even 1 trivial