# Properties

 Label 3150.3.e.e.701.2 Level $3150$ Weight $3$ Character 3150.701 Analytic conductor $85.831$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3150.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$85.8312832735$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 701.2 Root $$-1.16372i$$ of defining polynomial Character $$\chi$$ $$=$$ 3150.701 Dual form 3150.3.e.e.701.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$$ $$q-1.41421i q^{2} -2.00000 q^{4} +2.64575 q^{7} +2.82843i q^{8} +12.1382i q^{11} +18.5830 q^{13} -3.74166i q^{14} +4.00000 q^{16} -10.9015i q^{17} +20.0000 q^{19} +17.1660 q^{22} -12.1382i q^{23} -26.2803i q^{26} -5.29150 q^{28} -41.8367i q^{29} +25.1660 q^{31} -5.65685i q^{32} -15.4170 q^{34} -38.0000 q^{37} -28.2843i q^{38} +60.6337i q^{41} -83.4980 q^{43} -24.2764i q^{44} -17.1660 q^{46} -16.9706i q^{47} +7.00000 q^{49} -37.1660 q^{52} +94.0424i q^{53} +7.48331i q^{56} -59.1660 q^{58} +58.2175i q^{59} +15.6680 q^{61} -35.5901i q^{62} -8.00000 q^{64} +132.664 q^{67} +21.8029i q^{68} +12.1382i q^{71} +76.9150 q^{73} +53.7401i q^{74} -40.0000 q^{76} +32.1147i q^{77} +33.6680 q^{79} +85.7490 q^{82} -60.5764i q^{83} +118.084i q^{86} -34.3320 q^{88} +4.77506i q^{89} +49.1660 q^{91} +24.2764i q^{92} -24.0000 q^{94} +188.413 q^{97} -9.89949i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + O(q^{10})$$ $$4 q - 8 q^{4} + 32 q^{13} + 16 q^{16} + 80 q^{19} - 16 q^{22} + 16 q^{31} - 104 q^{34} - 152 q^{37} - 80 q^{43} + 16 q^{46} + 28 q^{49} - 64 q^{52} - 152 q^{58} + 232 q^{61} - 32 q^{64} + 192 q^{67} + 96 q^{73} - 160 q^{76} + 304 q^{79} + 216 q^{82} + 32 q^{88} + 112 q^{91} - 96 q^{94} + 288 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.41421i − 0.707107i
$$3$$ 0 0
$$4$$ −2.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.64575 0.377964
$$8$$ 2.82843i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 12.1382i 1.10347i 0.834019 + 0.551736i $$0.186035\pi$$
−0.834019 + 0.551736i $$0.813965\pi$$
$$12$$ 0 0
$$13$$ 18.5830 1.42946 0.714731 0.699399i $$-0.246549\pi$$
0.714731 + 0.699399i $$0.246549\pi$$
$$14$$ − 3.74166i − 0.267261i
$$15$$ 0 0
$$16$$ 4.00000 0.250000
$$17$$ − 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
$$22$$ 17.1660 0.780273
$$23$$ − 12.1382i − 0.527748i −0.964557 0.263874i $$-0.915000\pi$$
0.964557 0.263874i $$-0.0850003\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ − 26.2803i − 1.01078i
$$27$$ 0 0
$$28$$ −5.29150 −0.188982
$$29$$ − 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 $$-0.366957\pi$$
0.405903 + 0.913916i $$0.366957\pi$$
$$32$$ − 5.65685i − 0.176777i
$$33$$ 0 0
$$34$$ −15.4170 −0.453441
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −38.0000 −1.02703 −0.513514 0.858082i $$-0.671656\pi$$
−0.513514 + 0.858082i $$0.671656\pi$$
$$38$$ − 28.2843i − 0.744323i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 60.6337i 1.47887i 0.673227 + 0.739435i $$0.264908\pi$$
−0.673227 + 0.739435i $$0.735092\pi$$
$$42$$ 0 0
$$43$$ −83.4980 −1.94181 −0.970907 0.239455i $$-0.923031\pi$$
−0.970907 + 0.239455i $$0.923031\pi$$
$$44$$ − 24.2764i − 0.551736i
$$45$$ 0 0
$$46$$ −17.1660 −0.373174
$$47$$ − 16.9706i − 0.361076i −0.983568 0.180538i $$-0.942216\pi$$
0.983568 0.180538i $$-0.0577838\pi$$
$$48$$ 0 0
$$49$$ 7.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −37.1660 −0.714731
$$53$$ 94.0424i 1.77439i 0.461399 + 0.887193i $$0.347348\pi$$
−0.461399 + 0.887193i $$0.652652\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 7.48331i 0.133631i
$$57$$ 0 0
$$58$$ −59.1660 −1.02010
$$59$$ 58.2175i 0.986738i 0.869820 + 0.493369i $$0.164235\pi$$
−0.869820 + 0.493369i $$0.835765\pi$$
$$60$$ 0 0
$$61$$ 15.6680 0.256852 0.128426 0.991719i $$-0.459008\pi$$
0.128426 + 0.991719i $$0.459008\pi$$
$$62$$ − 35.5901i − 0.574034i
$$63$$ 0 0
$$64$$ −8.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 132.664 1.98006 0.990030 0.140856i $$-0.0449853\pi$$
0.990030 + 0.140856i $$0.0449853\pi$$
$$68$$ 21.8029i 0.320631i
$$69$$ 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 $$-0.323386\pi$$
0.526815 + 0.849980i $$0.323386\pi$$
$$74$$ 53.7401i 0.726218i
$$75$$ 0 0
$$76$$ −40.0000 −0.526316
$$77$$ 32.1147i 0.417074i
$$78$$ 0 0
$$79$$ 33.6680 0.426177 0.213088 0.977033i $$-0.431648\pi$$
0.213088 + 0.977033i $$0.431648\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 85.7490 1.04572
$$83$$ − 60.5764i − 0.729836i −0.931040 0.364918i $$-0.881097\pi$$
0.931040 0.364918i $$-0.118903\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 118.084i 1.37307i
$$87$$ 0 0
$$88$$ −34.3320 −0.390137
$$89$$ 4.77506i 0.0536523i 0.999640 + 0.0268262i $$0.00854006\pi$$
−0.999640 + 0.0268262i $$0.991460\pi$$
$$90$$ 0 0
$$91$$ 49.1660 0.540286
$$92$$ 24.2764i 0.263874i
$$93$$ 0 0
$$94$$ −24.0000 −0.255319
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 188.413 1.94240 0.971201 0.238260i $$-0.0765771\pi$$
0.971201 + 0.238260i $$0.0765771\pi$$
$$98$$ − 9.89949i − 0.101015i
$$99$$ 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 $$0.720379\pi$$
$$104$$ 52.5607i 0.505391i
$$105$$ 0 0
$$106$$ 132.996 1.25468
$$107$$ 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 $$-0.450403\pi$$
0.155184 + 0.987886i $$0.450403\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 10.5830 0.0944911
$$113$$ − 28.5190i − 0.252381i −0.992006 0.126190i $$-0.959725\pi$$
0.992006 0.126190i $$-0.0402750\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 83.6734i 0.721322i
$$117$$ 0 0
$$118$$ 82.3320 0.697729
$$119$$ − 28.8426i − 0.242374i
$$120$$ 0 0
$$121$$ −26.3360 −0.217653
$$122$$ − 22.1579i − 0.181622i
$$123$$ 0 0
$$124$$ −50.3320 −0.405903
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −129.668 −1.02101 −0.510504 0.859875i $$-0.670541\pi$$
−0.510504 + 0.859875i $$0.670541\pi$$
$$128$$ 11.3137i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 148.017i 1.12990i 0.825124 + 0.564952i $$0.191105\pi$$
−0.825124 + 0.564952i $$0.808895\pi$$
$$132$$ 0 0
$$133$$ 52.9150 0.397857
$$134$$ − 187.615i − 1.40011i
$$135$$ 0 0
$$136$$ 30.8340 0.226721
$$137$$ − 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
$$142$$ 17.1660 0.120887
$$143$$ 225.564i 1.57737i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ − 108.774i − 0.745029i
$$147$$ 0 0
$$148$$ 76.0000 0.513514
$$149$$ − 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 $$-0.599826\pi$$
−0.308497 + 0.951225i $$0.599826\pi$$
$$152$$ 56.5685i 0.372161i
$$153$$ 0 0
$$154$$ 45.4170 0.294916
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 184.996 1.17832 0.589159 0.808017i $$-0.299459\pi$$
0.589159 + 0.808017i $$0.299459\pi$$
$$158$$ − 47.6137i − 0.301353i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 32.1147i − 0.199470i
$$162$$ 0 0
$$163$$ −86.9961 −0.533718 −0.266859 0.963736i $$-0.585986\pi$$
−0.266859 + 0.963736i $$0.585986\pi$$
$$164$$ − 121.267i − 0.739435i
$$165$$ 0 0
$$166$$ −85.6680 −0.516072
$$167$$ 60.5764i 0.362733i 0.983416 + 0.181366i $$0.0580520\pi$$
−0.983416 + 0.181366i $$0.941948\pi$$
$$168$$ 0 0
$$169$$ 176.328 1.04336
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 166.996 0.970907
$$173$$ 162.572i 0.939721i 0.882741 + 0.469860i $$0.155696\pi$$
−0.882741 + 0.469860i $$0.844304\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 48.5528i 0.275868i
$$177$$ 0 0
$$178$$ 6.75295 0.0379379
$$179$$ − 223.091i − 1.24632i −0.782095 0.623159i $$-0.785849\pi$$
0.782095 0.623159i $$-0.214151\pi$$
$$180$$ 0 0
$$181$$ 188.915 1.04373 0.521865 0.853028i $$-0.325237\pi$$
0.521865 + 0.853028i $$0.325237\pi$$
$$182$$ − 69.5312i − 0.382040i
$$183$$ 0 0
$$184$$ 34.3320 0.186587
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 132.324 0.707616
$$188$$ 33.9411i 0.180538i
$$189$$ 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 $$-0.612851\pi$$
−0.347150 + 0.937810i $$0.612851\pi$$
$$194$$ − 266.456i − 1.37349i
$$195$$ 0 0
$$196$$ −14.0000 −0.0714286
$$197$$ − 188.560i − 0.957157i −0.878045 0.478579i $$-0.841152\pi$$
0.878045 0.478579i $$-0.158848\pi$$
$$198$$ 0 0
$$199$$ 102.494 0.515046 0.257523 0.966272i $$-0.417094\pi$$
0.257523 + 0.966272i $$0.417094\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −150.915 −0.747104
$$203$$ − 110.689i − 0.545268i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 185.966i 0.902749i
$$207$$ 0 0
$$208$$ 74.3320 0.357365
$$209$$ 242.764i 1.16155i
$$210$$ 0 0
$$211$$ −84.5020 −0.400483 −0.200242 0.979747i $$-0.564173\pi$$
−0.200242 + 0.979747i $$0.564173\pi$$
$$212$$ − 188.085i − 0.887193i
$$213$$ 0 0
$$214$$ 116.502 0.544402
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 66.5830 0.306834
$$218$$ − 47.8429i − 0.219463i
$$219$$ 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 $$0.384356\pi$$
$$224$$ − 14.9666i − 0.0668153i
$$225$$ 0 0
$$226$$ −40.3320 −0.178460
$$227$$ 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 $$-0.699750\pi$$
−0.587151 + 0.809478i $$0.699750\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 118.332 0.510052
$$233$$ − 26.2748i − 0.112767i −0.998409 0.0563836i $$-0.982043\pi$$
0.998409 0.0563836i $$-0.0179570\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ − 116.435i − 0.493369i
$$237$$ 0 0
$$238$$ −40.7895 −0.171385
$$239$$ − 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 $$0.247730\pi$$
$$242$$ 37.2447i 0.153904i
$$243$$ 0 0
$$244$$ −31.3360 −0.128426
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 371.660 1.50470
$$248$$ 71.1802i 0.287017i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 356.382i − 1.41985i −0.704278 0.709924i $$-0.748729\pi$$
0.704278 0.709924i $$-0.251271\pi$$
$$252$$ 0 0
$$253$$ 147.336 0.582356
$$254$$ 183.378i 0.721961i
$$255$$ 0 0
$$256$$ 16.0000 0.0625000
$$257$$ − 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
$$262$$ 209.328 0.798962
$$263$$ 261.979i 0.996117i 0.867143 + 0.498059i $$0.165954\pi$$
−0.867143 + 0.498059i $$0.834046\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ − 74.8331i − 0.281328i
$$267$$ 0 0
$$268$$ −265.328 −0.990030
$$269$$ − 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 $$-0.499315\pi$$
0.00215131 + 0.999998i $$0.499315\pi$$
$$272$$ − 43.6058i − 0.160316i
$$273$$ 0 0
$$274$$ −108.834 −0.397204
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −32.0000 −0.115523 −0.0577617 0.998330i $$-0.518396\pi$$
−0.0577617 + 0.998330i $$0.518396\pi$$
$$278$$ − 307.348i − 1.10557i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 166.757i − 0.593441i −0.954964 0.296721i $$-0.904107\pi$$
0.954964 0.296721i $$-0.0958930\pi$$
$$282$$ 0 0
$$283$$ −16.3399 −0.0577381 −0.0288691 0.999583i $$-0.509191\pi$$
−0.0288691 + 0.999583i $$0.509191\pi$$
$$284$$ − 24.2764i − 0.0854803i
$$285$$ 0 0
$$286$$ 318.996 1.11537
$$287$$ 160.422i 0.558961i
$$288$$ 0 0
$$289$$ 170.158 0.588782
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −153.830 −0.526815
$$293$$ 368.921i 1.25912i 0.776953 + 0.629558i $$0.216764\pi$$
−0.776953 + 0.629558i $$0.783236\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ − 107.480i − 0.363109i
$$297$$ 0 0
$$298$$ −228.996 −0.768443
$$299$$ − 225.564i − 0.754396i
$$300$$ 0 0
$$301$$ −220.915 −0.733937
$$302$$ 131.757i 0.436280i
$$303$$ 0 0
$$304$$ 80.0000 0.263158
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 192.664 0.627570 0.313785 0.949494i $$-0.398403\pi$$
0.313785 + 0.949494i $$0.398403\pi$$
$$308$$ − 64.2293i − 0.208537i
$$309$$ 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 $$-0.522049\pi$$
−0.0692142 + 0.997602i $$0.522049\pi$$
$$314$$ − 261.624i − 0.833197i
$$315$$ 0 0
$$316$$ −67.3360 −0.213088
$$317$$ − 251.724i − 0.794083i −0.917801 0.397042i $$-0.870037\pi$$
0.917801 0.397042i $$-0.129963\pi$$
$$318$$ 0 0
$$319$$ 507.822 1.59192
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −45.4170 −0.141047
$$323$$ − 218.029i − 0.675013i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 123.031i 0.377396i
$$327$$ 0 0
$$328$$ −171.498 −0.522860
$$329$$ − 44.8999i − 0.136474i
$$330$$ 0 0
$$331$$ 361.490 1.09212 0.546058 0.837748i $$-0.316128\pi$$
0.546058 + 0.837748i $$0.316128\pi$$
$$332$$ 121.153i 0.364918i
$$333$$ 0 0
$$334$$ 85.6680 0.256491
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 298.834 0.886748 0.443374 0.896337i $$-0.353781\pi$$
0.443374 + 0.896337i $$0.353781\pi$$
$$338$$ − 249.366i − 0.737768i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 305.470i 0.895807i
$$342$$ 0 0
$$343$$ 18.5203 0.0539949
$$344$$ − 236.168i − 0.686535i
$$345$$ 0 0
$$346$$ 229.911 0.664483
$$347$$ 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 $$-0.286222\pi$$
0.622241 + 0.782826i $$0.286222\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 68.6640 0.195068
$$353$$ − 185.439i − 0.525324i −0.964888 0.262662i $$-0.915400\pi$$
0.964888 0.262662i $$-0.0846005\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ − 9.55012i − 0.0268262i
$$357$$ 0 0
$$358$$ −315.498 −0.881279
$$359$$ 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
$$362$$ − 267.166i − 0.738028i
$$363$$ 0 0
$$364$$ −98.3320 −0.270143
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 117.490 0.320137 0.160068 0.987106i $$-0.448829\pi$$
0.160068 + 0.987106i $$0.448829\pi$$
$$368$$ − 48.5528i − 0.131937i
$$369$$ 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 $$0.318513\pi$$
$$374$$ − 187.135i − 0.500360i
$$375$$ 0 0
$$376$$ 48.0000 0.127660
$$377$$ − 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
$$382$$ 322.494 0.844225
$$383$$ 744.804i 1.94466i 0.233614 + 0.972329i $$0.424945\pi$$
−0.233614 + 0.972329i $$0.575055\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 189.505i 0.490945i
$$387$$ 0 0
$$388$$ −376.826 −0.971201
$$389$$ 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
$$392$$ 19.7990i 0.0505076i
$$393$$ 0 0
$$394$$ −266.664 −0.676812
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 94.3241 0.237592 0.118796 0.992919i $$-0.462096\pi$$
0.118796 + 0.992919i $$0.462096\pi$$
$$398$$ − 144.949i − 0.364192i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 103.593i 0.258335i 0.991623 + 0.129168i $$0.0412306\pi$$
−0.991623 + 0.129168i $$0.958769\pi$$
$$402$$ 0 0
$$403$$ 467.660 1.16045
$$404$$ 213.426i 0.528282i
$$405$$ 0 0
$$406$$ −156.539 −0.385563
$$407$$ − 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
$$412$$ 262.996 0.638340
$$413$$ 154.029i 0.372952i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ − 105.121i − 0.252696i
$$417$$ 0 0
$$418$$ 343.320 0.821340
$$419$$ − 339.411i − 0.810051i −0.914305 0.405025i $$-0.867263\pi$$
0.914305 0.405025i $$-0.132737\pi$$
$$420$$ 0 0
$$421$$ −599.320 −1.42356 −0.711782 0.702401i $$-0.752111\pi$$
−0.711782 + 0.702401i $$0.752111\pi$$
$$422$$ 119.504i 0.283184i
$$423$$ 0 0
$$424$$ −265.992 −0.627340
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 41.4536 0.0970810
$$428$$ − 164.759i − 0.384950i
$$429$$ 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 $$-0.643772\pi$$
−0.436471 + 0.899718i $$0.643772\pi$$
$$434$$ − 94.1626i − 0.216964i
$$435$$ 0 0
$$436$$ −67.6601 −0.155184
$$437$$ − 242.764i − 0.555524i
$$438$$ 0 0
$$439$$ 528.146 1.20307 0.601533 0.798848i $$-0.294557\pi$$
0.601533 + 0.798848i $$0.294557\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −286.494 −0.648177
$$443$$ 36.6438i 0.0827174i 0.999144 + 0.0413587i $$0.0131686\pi$$
−0.999144 + 0.0413587i $$0.986831\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ − 224.144i − 0.502566i
$$447$$ 0 0
$$448$$ −21.1660 −0.0472456
$$449$$ 397.612i 0.885550i 0.896633 + 0.442775i $$0.146006\pi$$
−0.896633 + 0.442775i $$0.853994\pi$$
$$450$$ 0 0
$$451$$ −735.984 −1.63189
$$452$$ 57.0381i 0.126190i
$$453$$ 0 0
$$454$$ 144.000 0.317181
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −344.324 −0.753445 −0.376722 0.926326i $$-0.622949\pi$$
−0.376722 + 0.926326i $$0.622949\pi$$
$$458$$ 380.303i 0.830357i
$$459$$ 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 $$0.526959\pi$$
$$464$$ − 167.347i − 0.360661i
$$465$$ 0 0
$$466$$ −37.1581 −0.0797385
$$467$$ − 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
$$472$$ −164.664 −0.348864
$$473$$ − 1013.52i − 2.14274i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 57.6851i 0.121187i
$$477$$ 0 0
$$478$$ −130.494 −0.273000
$$479$$ 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
$$482$$ − 485.425i − 1.00711i
$$483$$ 0 0
$$484$$ 52.6719 0.108826
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 82.5098 0.169425 0.0847124 0.996405i $$-0.473003\pi$$
0.0847124 + 0.996405i $$0.473003\pi$$
$$488$$ 44.3157i 0.0908109i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 184.203i 0.375158i 0.982249 + 0.187579i $$0.0600641\pi$$
−0.982249 + 0.187579i $$0.939936\pi$$
$$492$$ 0 0
$$493$$ −456.081 −0.925114
$$494$$ − 525.607i − 1.06398i
$$495$$ 0 0
$$496$$ 100.664 0.202952
$$497$$ 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
$$502$$ −504.000 −1.00398
$$503$$ − 662.540i − 1.31718i −0.752504 0.658588i $$-0.771154\pi$$
0.752504 0.658588i $$-0.228846\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ − 208.365i − 0.411788i
$$507$$ 0 0
$$508$$ 259.336 0.510504
$$509$$ − 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
$$512$$ − 22.6274i − 0.0441942i
$$513$$ 0 0
$$514$$ −360.243 −0.700862
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 205.992 0.398437
$$518$$ 142.183i 0.274485i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 714.344i − 1.37110i −0.728025 0.685551i $$-0.759561\pi$$
0.728025 0.685551i $$-0.240439\pi$$
$$522$$ 0 0
$$523$$ 232.000 0.443595 0.221797 0.975093i $$-0.428808\pi$$
0.221797 + 0.975093i $$0.428808\pi$$
$$524$$ − 296.035i − 0.564952i
$$525$$ 0 0
$$526$$ 370.494 0.704361
$$527$$ − 274.346i − 0.520581i
$$528$$ 0 0
$$529$$ 381.664 0.721482
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −105.830 −0.198929
$$533$$ 1126.76i 2.11399i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 375.231i 0.700057i
$$537$$ 0 0
$$538$$ −132.405 −0.246106
$$539$$ 84.9674i 0.157639i
$$540$$ 0 0
$$541$$ 165.668 0.306225 0.153113 0.988209i $$-0.451070\pi$$
0.153113 + 0.988209i $$0.451070\pi$$
$$542$$ − 1.64899i − 0.00304241i
$$543$$ 0 0
$$544$$ −61.6680 −0.113360
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −295.676 −0.540541 −0.270270 0.962784i $$-0.587113\pi$$
−0.270270 + 0.962784i $$0.587113\pi$$
$$548$$ 153.915i 0.280866i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 836.734i − 1.51857i
$$552$$ 0 0
$$553$$ 89.0771 0.161080
$$554$$ 45.2548i 0.0816874i
$$555$$ 0 0
$$556$$ −434.656 −0.781756
$$557$$ 76.8426i 0.137958i 0.997618 + 0.0689790i $$0.0219742\pi$$
−0.997618 + 0.0689790i $$0.978026\pi$$
$$558$$ 0 0
$$559$$ −1551.64 −2.77575
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −235.830 −0.419626
$$563$$ 1016.33i 1.80521i 0.430470 + 0.902605i $$0.358348\pi$$
−0.430470 + 0.902605i $$0.641652\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 23.1081i 0.0408270i
$$567$$ 0 0
$$568$$ −34.3320 −0.0604437
$$569$$ 586.533i 1.03081i 0.856946 + 0.515406i $$0.172359\pi$$
−0.856946 + 0.515406i $$0.827641\pi$$
$$570$$ 0 0
$$571$$ 951.644 1.66663 0.833314 0.552800i $$-0.186441\pi$$
0.833314 + 0.552800i $$0.186441\pi$$
$$572$$ − 451.129i − 0.788686i
$$573$$ 0 0
$$574$$ 226.871 0.395245
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 148.672 0.257664 0.128832 0.991666i $$-0.458877\pi$$
0.128832 + 0.991666i $$0.458877\pi$$
$$578$$ − 240.640i − 0.416332i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 160.270i − 0.275852i
$$582$$ 0 0
$$583$$ −1141.51 −1.95799
$$584$$ 217.549i 0.372515i
$$585$$ 0 0
$$586$$ 521.733 0.890330
$$587$$ 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
$$592$$ −152.000 −0.256757
$$593$$ 217.251i 0.366359i 0.983079 + 0.183180i $$0.0586390\pi$$
−0.983079 + 0.183180i $$0.941361\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 323.849i 0.543371i
$$597$$ 0 0
$$598$$ −318.996 −0.533438
$$599$$ 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 $$0.613055\pi$$
$$602$$ 312.421i 0.518972i
$$603$$ 0 0
$$604$$ 186.332 0.308497
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −627.158 −1.03321 −0.516605 0.856224i $$-0.672804\pi$$
−0.516605 + 0.856224i $$0.672804\pi$$
$$608$$ − 113.137i − 0.186081i
$$609$$ 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 $$0.426835\pi$$
$$614$$ − 272.468i − 0.443759i
$$615$$ 0 0
$$616$$ −90.8340 −0.147458
$$617$$ 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
$$622$$ −185.652 −0.298476
$$623$$ 12.6336i 0.0202787i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 61.2752i 0.0978836i
$$627$$ 0 0
$$628$$ −369.992 −0.589159
$$629$$ 414.256i 0.658594i
$$630$$ 0 0
$$631$$ −298.996 −0.473845 −0.236922 0.971529i $$-0.576139\pi$$
−0.236922 + 0.971529i $$0.576139\pi$$
$$632$$ 95.2274i 0.150676i
$$633$$ 0 0
$$634$$ −355.992 −0.561502
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 130.081 0.204209
$$638$$ − 718.169i − 1.12566i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 311.957i 0.486672i 0.969942 + 0.243336i $$0.0782418\pi$$
−0.969942 + 0.243336i $$0.921758\pi$$
$$642$$ 0 0
$$643$$ 604.000 0.939347 0.469673 0.882840i $$-0.344372\pi$$
0.469673 + 0.882840i $$0.344372\pi$$
$$644$$ 64.2293i 0.0997350i
$$645$$ 0 0
$$646$$ −308.340 −0.477306
$$647$$ 179.600i 0.277588i 0.990321 + 0.138794i $$0.0443226\pi$$
−0.990321 + 0.138794i $$0.955677\pi$$
$$648$$ 0 0
$$649$$ −706.656 −1.08884
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 173.992 0.266859
$$653$$ 481.892i 0.737966i 0.929436 + 0.368983i $$0.120294\pi$$
−0.929436 + 0.368983i $$0.879706\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 242.535i 0.369718i
$$657$$ 0 0
$$658$$ −63.4980 −0.0965016
$$659$$ − 877.408i − 1.33142i −0.746209 0.665711i $$-0.768128\pi$$
0.746209 0.665711i $$-0.231872\pi$$
$$660$$ 0 0
$$661$$ −521.644 −0.789175 −0.394587 0.918858i $$-0.629112\pi$$
−0.394587 + 0.918858i $$0.629112\pi$$
$$662$$ − 511.224i − 0.772242i
$$663$$ 0 0
$$664$$ 171.336 0.258036
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −507.822 −0.761353
$$668$$ − 121.153i − 0.181366i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 190.181i 0.283429i
$$672$$ 0 0
$$673$$ 659.992 0.980672 0.490336 0.871534i $$-0.336874\pi$$
0.490336 + 0.871534i $$0.336874\pi$$
$$674$$ − 422.615i − 0.627025i
$$675$$ 0 0
$$676$$ −352.656 −0.521681
$$677$$ 1016.28i 1.50115i 0.660787 + 0.750573i $$0.270222\pi$$
−0.660787 + 0.750573i $$0.729778\pi$$
$$678$$ 0 0
$$679$$ 498.494 0.734159
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 432.000 0.633431
$$683$$ − 235.114i − 0.344238i −0.985076 0.172119i $$-0.944939\pi$$
0.985076 0.172119i $$-0.0550613\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ − 26.1916i − 0.0381802i
$$687$$ 0 0
$$688$$ −333.992 −0.485454
$$689$$ 1747.59i 2.53642i
$$690$$ 0 0
$$691$$ −50.9803 −0.0737776 −0.0368888 0.999319i $$-0.511745\pi$$
−0.0368888 + 0.999319i $$0.511745\pi$$
$$692$$ − 325.143i − 0.469860i
$$693$$ 0 0
$$694$$ 291.498 0.420026
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 660.996 0.948344
$$698$$ − 614.227i − 0.879982i
$$699$$ 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
$$704$$ − 97.1056i − 0.137934i
$$705$$ 0 0
$$706$$ −262.251 −0.371460
$$707$$ − 282.336i − 0.399344i
$$708$$ 0 0
$$709$$ −55.4980 −0.0782765 −0.0391382 0.999234i $$-0.512461\pi$$
−0.0391382 + 0.999234i $$0.512461\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −13.5059 −0.0189690
$$713$$ − 305.470i − 0.428429i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 446.182i 0.623159i
$$717$$ 0 0
$$718$$ 730.818 1.01785
$$719$$ 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
$$722$$ − 55.1543i − 0.0763910i
$$723$$ 0 0
$$724$$ −377.830 −0.521865
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 365.182 0.502313 0.251157 0.967946i $$-0.419189\pi$$
0.251157 + 0.967946i $$0.419189\pi$$
$$728$$ 139.062i 0.191020i
$$729$$ 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 $$0.422576\pi$$
$$734$$ − 166.156i − 0.226371i
$$735$$ 0 0
$$736$$ −68.6640 −0.0932935
$$737$$ 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
$$742$$ 351.875 0.474224
$$743$$ − 112.061i − 0.150822i −0.997153 0.0754112i $$-0.975973\pi$$
0.997153 0.0754112i $$-0.0240270\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ − 569.453i − 0.763342i
$$747$$ 0 0
$$748$$ −264.648 −0.353808
$$749$$ 217.955i 0.290995i
$$750$$ 0 0
$$751$$ −144.826 −0.192844 −0.0964222 0.995341i $$-0.530740\pi$$
−0.0964222 + 0.995341i $$0.530740\pi$$
$$752$$ − 67.8823i − 0.0902690i
$$753$$ 0 0
$$754$$ −1099.48 −1.45820
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −78.1699 −0.103263 −0.0516314 0.998666i $$-0.516442\pi$$
−0.0516314 + 0.998666i $$0.516442\pi$$
$$758$$ − 564.036i − 0.744111i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 1465.50i − 1.92576i −0.269928 0.962880i $$-0.587000\pi$$
0.269928 0.962880i $$-0.413000\pi$$
$$762$$ 0 0
$$763$$ 89.5059 0.117308
$$764$$ − 456.076i − 0.596957i
$$765$$ 0 0
$$766$$ 1053.31 1.37508
$$767$$ 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
$$772$$ 268.000 0.347150
$$773$$ − 434.559i − 0.562172i −0.959683 0.281086i $$-0.909305\pi$$
0.959683 0.281086i $$-0.0906947\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 532.913i 0.686743i
$$777$$ 0 0
$$778$$ 756.834 0.972794
$$779$$ 1212.67i 1.55671i
$$780$$ 0 0
$$781$$ −147.336 −0.188650
$$782$$ 187.135i 0.239303i
$$783$$ 0 0
$$784$$ 28.0000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 15.3517 0.0195066 0.00975331 0.999952i $$-0.496895\pi$$
0.00975331 + 0.999952i $$0.496895\pi$$
$$788$$ 377.120i 0.478579i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 75.4543i − 0.0953910i
$$792$$ 0 0
$$793$$ 291.158 0.367160
$$794$$ − 133.394i − 0.168003i
$$795$$ 0 0
$$796$$ −204.988 −0.257523
$$797$$ − 1043.48i − 1.30927i −0.755947 0.654633i $$-0.772823\pi$$
0.755947 0.654633i $$-0.227177\pi$$
$$798$$ 0 0
$$799$$ −185.004 −0.231544
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 146.502 0.182671
$$803$$ 933.610i 1.16265i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ − 661.371i − 0.820560i
$$807$$ 0 0
$$808$$ 301.830 0.373552
$$809$$ − 1041.31i − 1.28716i −0.765378 0.643581i $$-0.777448\pi$$
0.765378 0.643581i $$-0.222552\pi$$
$$810$$ 0 0
$$811$$ 502.316 0.619379 0.309689 0.950838i $$-0.399775\pi$$
0.309689 + 0.950838i $$0.399775\pi$$
$$812$$ 221.379i 0.272634i
$$813$$ 0 0
$$814$$ −652.308 −0.801362
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1669.96 −2.04402
$$818$$ 13.7983i 0.0168684i
$$819$$ 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 $$0.381095\pi$$
$$824$$ − 371.933i − 0.451375i
$$825$$ 0 0
$$826$$ 217.830 0.263717
$$827$$ − 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 $$-0.622392\pi$$
−0.375102 + 0.926984i $$0.622392\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −148.664 −0.178683
$$833$$ − 76.3102i − 0.0916089i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ − 485.528i − 0.580775i
$$837$$ 0 0
$$838$$ −480.000 −0.572792
$$839$$ − 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
$$842$$ 847.567i 1.00661i
$$843$$ 0 0
$$844$$ 169.004 0.200242
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −69.6784 −0.0822649
$$848$$ 376.170i 0.443596i
$$849$$ 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 $$0.525592\pi$$
$$854$$ − 58.6242i − 0.0686466i
$$855$$ 0 0
$$856$$ −233.004 −0.272201
$$857$$ 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
$$862$$ −1005.47 −1.16644
$$863$$ 0.114603i 0 0.000132796i 1.00000 6.63982e-5i $$2.11352e-5\pi$$
−1.00000 6.63982e-5i $$0.999979\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 534.550i 0.617264i
$$867$$ 0 0
$$868$$ −133.166 −0.153417
$$869$$ 408.669i 0.470275i
$$870$$ 0 0
$$871$$ 2465.30 2.83042
$$872$$ 95.6858i 0.109731i
$$873$$ 0 0
$$874$$ −343.320 −0.392815
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −997.304 −1.13718 −0.568589 0.822622i $$-0.692510\pi$$
−0.568589 + 0.822622i $$0.692510\pi$$
$$878$$ − 746.912i − 0.850697i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 935.649i − 1.06203i −0.847362 0.531015i $$-0.821811\pi$$
0.847362 0.531015i $$-0.178189\pi$$
$$882$$ 0 0
$$883$$ 1549.47 1.75478 0.877392 0.479774i $$-0.159281\pi$$
0.877392 + 0.479774i $$0.159281\pi$$
$$884$$ 405.164i 0.458330i
$$885$$ 0 0
$$886$$ 51.8222 0.0584900
$$887$$ − 894.493i − 1.00845i −0.863573 0.504224i $$-0.831779\pi$$
0.863573 0.504224i $$-0.168221\pi$$
$$888$$ 0 0
$$889$$ −343.069 −0.385905
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −316.988 −0.355368
$$893$$ − 339.411i − 0.380080i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 29.9333i 0.0334077i
$$897$$ 0 0
$$898$$ 562.308 0.626179
$$899$$ − 1052.86i − 1.17115i
$$900$$ 0 0
$$901$$ 1025.20 1.13785
$$902$$ 1040.84i 1.15392i
$$903$$ 0 0
$$904$$ 80.6640 0.0892301
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 135.838 0.149766 0.0748831 0.997192i $$-0.476142\pi$$
0.0748831 + 0.997192i $$0.476142\pi$$
$$908$$ − 203.647i − 0.224281i
$$909$$ 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
$$914$$ 486.948i 0.532766i
$$915$$ 0 0
$$916$$ 537.830 0.587151
$$917$$ 391.617i 0.427063i
$$918$$ 0 0
$$919$$ −388.162 −0.422374 −0.211187 0.977446i $$-0.567733\pi$$
−0.211187 + 0.977446i $$0.567733\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 524.583 0.568962
$$923$$ 225.564i 0.244382i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 110.778i 0.119631i
$$927$$ 0 0
$$928$$ −236.664 −0.255026
$$929$$ 621.694i 0.669207i 0.942359 + 0.334604i $$0.108602\pi$$
−0.942359 + 0.334604i $$0.891398\pi$$
$$930$$ 0 0
$$931$$ 140.000 0.150376
$$932$$ 52.5495i 0.0563836i
$$933$$ 0 0
$$934$$ −565.344 −0.605293
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1262.00 −1.34685 −0.673426 0.739255i $$-0.735178\pi$$
−0.673426 + 0.739255i $$0.735178\pi$$
$$938$$ − 496.383i − 0.529193i
$$939$$ 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
$$944$$ 232.870i 0.246684i
$$945$$ 0 0
$$946$$ −1433.33 −1.51515
$$947$$ 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
$$952$$ 81.5791 0.0856923
$$953$$ − 163.104i − 0.171148i −0.996332 0.0855740i $$-0.972728\pi$$
0.996332 0.0855740i $$-0.0272724\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 184.547i 0.193040i
$$957$$ 0 0
$$958$$ 994.656 1.03826
$$959$$ − 203.610i − 0.212315i
$$960$$ 0 0
$$961$$ −327.672 −0.340970
$$962$$ 998.653i 1.03810i
$$963$$ 0 0
$$964$$ −686.494 −0.712131
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −887.012 −0.917282 −0.458641 0.888622i $$-0.651664\pi$$
−0.458641 + 0.888622i $$0.651664\pi$$
$$968$$ − 74.4893i − 0.0769518i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1416.32i 1.45862i 0.684183 + 0.729310i $$0.260159\pi$$
−0.684183 + 0.729310i $$0.739841\pi$$
$$972$$ 0 0
$$973$$ 574.996 0.590952
$$974$$ − 116.687i − 0.119801i
$$975$$ 0 0
$$976$$ 62.6719 0.0642130
$$977$$ 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
$$982$$ 260.502 0.265277
$$983$$ − 487.887i − 0.496324i −0.968718 0.248162i $$-0.920173\pi$$
0.968718 0.248162i $$-0.0798266\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 644.996i 0.654154i
$$987$$ 0 0
$$988$$ −743.320 −0.752348
$$989$$ 1013.52i 1.02479i
$$990$$ 0 0
$$991$$ −937.474 −0.945988 −0.472994 0.881066i $$-0.656827\pi$$
−0.472994 + 0.881066i $$0.656827\pi$$
$$992$$ − 142.360i − 0.143509i
$$993$$ 0 0
$$994$$ 45.4170 0.0456911
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −461.012 −0.462399 −0.231200 0.972906i $$-0.574265\pi$$
−0.231200 + 0.972906i $$0.574265\pi$$
$$998$$ − 1064.63i − 1.06677i
$$999$$ 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 3150.3.e.e.701.2 4
3.2 odd 2 inner 3150.3.e.e.701.4 4
5.2 odd 4 3150.3.c.b.449.7 8
5.3 odd 4 3150.3.c.b.449.1 8
5.4 even 2 126.3.b.a.71.4 yes 4
15.2 even 4 3150.3.c.b.449.4 8
15.8 even 4 3150.3.c.b.449.6 8
15.14 odd 2 126.3.b.a.71.1 4
20.19 odd 2 1008.3.d.a.449.3 4
35.4 even 6 882.3.s.e.863.1 8
35.9 even 6 882.3.s.e.557.4 8
35.19 odd 6 882.3.s.i.557.3 8
35.24 odd 6 882.3.s.i.863.2 8
35.34 odd 2 882.3.b.f.197.3 4
40.19 odd 2 4032.3.d.j.449.2 4
40.29 even 2 4032.3.d.i.449.2 4
45.4 even 6 1134.3.q.c.1079.2 8
45.14 odd 6 1134.3.q.c.1079.3 8
45.29 odd 6 1134.3.q.c.701.2 8
45.34 even 6 1134.3.q.c.701.3 8
60.59 even 2 1008.3.d.a.449.2 4
105.44 odd 6 882.3.s.e.557.1 8
105.59 even 6 882.3.s.i.863.3 8
105.74 odd 6 882.3.s.e.863.4 8
105.89 even 6 882.3.s.i.557.2 8
105.104 even 2 882.3.b.f.197.2 4
120.29 odd 2 4032.3.d.i.449.3 4
120.59 even 2 4032.3.d.j.449.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 15.14 odd 2
126.3.b.a.71.4 yes 4 5.4 even 2
882.3.b.f.197.2 4 105.104 even 2
882.3.b.f.197.3 4 35.34 odd 2
882.3.s.e.557.1 8 105.44 odd 6
882.3.s.e.557.4 8 35.9 even 6
882.3.s.e.863.1 8 35.4 even 6
882.3.s.e.863.4 8 105.74 odd 6
882.3.s.i.557.2 8 105.89 even 6
882.3.s.i.557.3 8 35.19 odd 6
882.3.s.i.863.2 8 35.24 odd 6
882.3.s.i.863.3 8 105.59 even 6
1008.3.d.a.449.2 4 60.59 even 2
1008.3.d.a.449.3 4 20.19 odd 2
1134.3.q.c.701.2 8 45.29 odd 6
1134.3.q.c.701.3 8 45.34 even 6
1134.3.q.c.1079.2 8 45.4 even 6
1134.3.q.c.1079.3 8 45.14 odd 6
3150.3.c.b.449.1 8 5.3 odd 4
3150.3.c.b.449.4 8 15.2 even 4
3150.3.c.b.449.6 8 15.8 even 4
3150.3.c.b.449.7 8 5.2 odd 4
3150.3.e.e.701.2 4 1.1 even 1 trivial
3150.3.e.e.701.4 4 3.2 odd 2 inner
4032.3.d.i.449.2 4 40.29 even 2
4032.3.d.i.449.3 4 120.29 odd 2
4032.3.d.j.449.2 4 40.19 odd 2
4032.3.d.j.449.3 4 120.59 even 2