# Properties

 Label 825.4.c.l.199.2 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.2230106176.1 Defining polynomial: $$x^{6} + 41 x^{4} + 452 x^{2} + 676$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$-4.59056i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.l.199.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.59056i q^{2} -3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} +16.1465i q^{7} -11.1590i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-3.59056i q^{2} -3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} +16.1465i q^{7} -11.1590i q^{8} -9.00000 q^{9} +11.0000 q^{11} +14.6764i q^{12} -54.1214i q^{13} +57.9749 q^{14} -79.2041 q^{16} +107.010i q^{17} +32.3150i q^{18} -48.7496 q^{19} +48.4394 q^{21} -39.4962i q^{22} +11.9498i q^{23} -33.4771 q^{24} -194.326 q^{26} +27.0000i q^{27} -78.9905i q^{28} -239.733 q^{29} -82.0851 q^{31} +195.115i q^{32} -33.0000i q^{33} +384.224 q^{34} +44.0291 q^{36} +21.7573i q^{37} +175.038i q^{38} -162.364 q^{39} -124.835 q^{41} -173.925i q^{42} +224.459i q^{43} -53.8133 q^{44} +42.9064 q^{46} +186.832i q^{47} +237.612i q^{48} +82.2913 q^{49} +321.029 q^{51} +264.768i q^{52} +233.997i q^{53} +96.9451 q^{54} +180.179 q^{56} +146.249i q^{57} +860.774i q^{58} -232.936 q^{59} +163.849 q^{61} +294.731i q^{62} -145.318i q^{63} +66.9386 q^{64} -118.488 q^{66} +876.918i q^{67} -523.503i q^{68} +35.8493 q^{69} -733.141 q^{71} +100.431i q^{72} +1161.97i q^{73} +78.1208 q^{74} +238.489 q^{76} +177.611i q^{77} +582.978i q^{78} +588.831 q^{79} +81.0000 q^{81} +448.226i q^{82} -1161.06i q^{83} -236.971 q^{84} +805.933 q^{86} +719.198i q^{87} -122.749i q^{88} +1042.16 q^{89} +873.869 q^{91} -58.4597i q^{92} +246.255i q^{93} +670.831 q^{94} +585.345 q^{96} -1546.63i q^{97} -295.472i q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + O(q^{10})$$ $$6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44} + 2320 q^{46} + 314 q^{49} + 1308 q^{51} - 216 q^{54} + 2736 q^{56} + 2088 q^{59} + 1284 q^{61} - 2332 q^{64} + 264 q^{66} - 1200 q^{69} - 1088 q^{71} - 3072 q^{74} + 3992 q^{76} + 3172 q^{79} + 486 q^{81} - 2040 q^{84} + 7136 q^{86} + 4244 q^{89} - 16 q^{91} + 4304 q^{94} + 4128 q^{96} - 594 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.59056i − 1.26945i −0.772736 0.634727i $$-0.781112\pi$$
0.772736 0.634727i $$-0.218888\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −4.89212 −0.611515
$$5$$ 0 0
$$6$$ −10.7717 −0.732920
$$7$$ 16.1465i 0.871828i 0.899988 + 0.435914i $$0.143575\pi$$
−0.899988 + 0.435914i $$0.856425\pi$$
$$8$$ − 11.1590i − 0.493164i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 14.6764i 0.353058i
$$13$$ − 54.1214i − 1.15466i −0.816511 0.577329i $$-0.804095\pi$$
0.816511 0.577329i $$-0.195905\pi$$
$$14$$ 57.9749 1.10675
$$15$$ 0 0
$$16$$ −79.2041 −1.23756
$$17$$ 107.010i 1.52668i 0.645995 + 0.763342i $$0.276443\pi$$
−0.645995 + 0.763342i $$0.723557\pi$$
$$18$$ 32.3150i 0.423152i
$$19$$ −48.7496 −0.588628 −0.294314 0.955709i $$-0.595091\pi$$
−0.294314 + 0.955709i $$0.595091\pi$$
$$20$$ 0 0
$$21$$ 48.4394 0.503350
$$22$$ − 39.4962i − 0.382755i
$$23$$ 11.9498i 0.108335i 0.998532 + 0.0541674i $$0.0172505\pi$$
−0.998532 + 0.0541674i $$0.982750\pi$$
$$24$$ −33.4771 −0.284729
$$25$$ 0 0
$$26$$ −194.326 −1.46579
$$27$$ 27.0000i 0.192450i
$$28$$ − 78.9905i − 0.533136i
$$29$$ −239.733 −1.53508 −0.767538 0.641003i $$-0.778519\pi$$
−0.767538 + 0.641003i $$0.778519\pi$$
$$30$$ 0 0
$$31$$ −82.0851 −0.475578 −0.237789 0.971317i $$-0.576423\pi$$
−0.237789 + 0.971317i $$0.576423\pi$$
$$32$$ 195.115i 1.07787i
$$33$$ − 33.0000i − 0.174078i
$$34$$ 384.224 1.93806
$$35$$ 0 0
$$36$$ 44.0291 0.203838
$$37$$ 21.7573i 0.0966723i 0.998831 + 0.0483361i $$0.0153919\pi$$
−0.998831 + 0.0483361i $$0.984608\pi$$
$$38$$ 175.038i 0.747236i
$$39$$ −162.364 −0.666643
$$40$$ 0 0
$$41$$ −124.835 −0.475510 −0.237755 0.971325i $$-0.576412\pi$$
−0.237755 + 0.971325i $$0.576412\pi$$
$$42$$ − 173.925i − 0.638980i
$$43$$ 224.459i 0.796039i 0.917377 + 0.398019i $$0.130302\pi$$
−0.917377 + 0.398019i $$0.869698\pi$$
$$44$$ −53.8133 −0.184379
$$45$$ 0 0
$$46$$ 42.9064 0.137526
$$47$$ 186.832i 0.579835i 0.957052 + 0.289917i $$0.0936278\pi$$
−0.957052 + 0.289917i $$0.906372\pi$$
$$48$$ 237.612i 0.714508i
$$49$$ 82.2913 0.239916
$$50$$ 0 0
$$51$$ 321.029 0.881431
$$52$$ 264.768i 0.706091i
$$53$$ 233.997i 0.606453i 0.952919 + 0.303226i $$0.0980638\pi$$
−0.952919 + 0.303226i $$0.901936\pi$$
$$54$$ 96.9451 0.244307
$$55$$ 0 0
$$56$$ 180.179 0.429954
$$57$$ 146.249i 0.339844i
$$58$$ 860.774i 1.94871i
$$59$$ −232.936 −0.513996 −0.256998 0.966412i $$-0.582733\pi$$
−0.256998 + 0.966412i $$0.582733\pi$$
$$60$$ 0 0
$$61$$ 163.849 0.343913 0.171957 0.985105i $$-0.444991\pi$$
0.171957 + 0.985105i $$0.444991\pi$$
$$62$$ 294.731i 0.603725i
$$63$$ − 145.318i − 0.290609i
$$64$$ 66.9386 0.130739
$$65$$ 0 0
$$66$$ −118.488 −0.220984
$$67$$ 876.918i 1.59899i 0.600670 + 0.799497i $$0.294900\pi$$
−0.600670 + 0.799497i $$0.705100\pi$$
$$68$$ − 523.503i − 0.933590i
$$69$$ 35.8493 0.0625471
$$70$$ 0 0
$$71$$ −733.141 −1.22546 −0.612731 0.790291i $$-0.709929\pi$$
−0.612731 + 0.790291i $$0.709929\pi$$
$$72$$ 100.431i 0.164388i
$$73$$ 1161.97i 1.86299i 0.363750 + 0.931496i $$0.381496\pi$$
−0.363750 + 0.931496i $$0.618504\pi$$
$$74$$ 78.1208 0.122721
$$75$$ 0 0
$$76$$ 238.489 0.359954
$$77$$ 177.611i 0.262866i
$$78$$ 582.978i 0.846272i
$$79$$ 588.831 0.838591 0.419296 0.907850i $$-0.362277\pi$$
0.419296 + 0.907850i $$0.362277\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 448.226i 0.603638i
$$83$$ − 1161.06i − 1.53546i −0.640772 0.767731i $$-0.721386\pi$$
0.640772 0.767731i $$-0.278614\pi$$
$$84$$ −236.971 −0.307806
$$85$$ 0 0
$$86$$ 805.933 1.01053
$$87$$ 719.198i 0.886277i
$$88$$ − 122.749i − 0.148695i
$$89$$ 1042.16 1.24122 0.620610 0.784120i $$-0.286885\pi$$
0.620610 + 0.784120i $$0.286885\pi$$
$$90$$ 0 0
$$91$$ 873.869 1.00666
$$92$$ − 58.4597i − 0.0662483i
$$93$$ 246.255i 0.274575i
$$94$$ 670.831 0.736074
$$95$$ 0 0
$$96$$ 585.345 0.622307
$$97$$ − 1546.63i − 1.61893i −0.587169 0.809464i $$-0.699758\pi$$
0.587169 0.809464i $$-0.300242\pi$$
$$98$$ − 295.472i − 0.304563i
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −662.282 −0.652470 −0.326235 0.945289i $$-0.605780\pi$$
−0.326235 + 0.945289i $$0.605780\pi$$
$$102$$ − 1152.67i − 1.11894i
$$103$$ 399.592i 0.382262i 0.981565 + 0.191131i $$0.0612156\pi$$
−0.981565 + 0.191131i $$0.938784\pi$$
$$104$$ −603.942 −0.569436
$$105$$ 0 0
$$106$$ 840.181 0.769864
$$107$$ 1591.22i 1.43765i 0.695189 + 0.718827i $$0.255321\pi$$
−0.695189 + 0.718827i $$0.744679\pi$$
$$108$$ − 132.087i − 0.117686i
$$109$$ −755.128 −0.663561 −0.331780 0.943357i $$-0.607649\pi$$
−0.331780 + 0.943357i $$0.607649\pi$$
$$110$$ 0 0
$$111$$ 65.2718 0.0558138
$$112$$ − 1278.87i − 1.07894i
$$113$$ − 1145.65i − 0.953753i −0.878970 0.476876i $$-0.841769\pi$$
0.878970 0.476876i $$-0.158231\pi$$
$$114$$ 525.115 0.431417
$$115$$ 0 0
$$116$$ 1172.80 0.938722
$$117$$ 487.092i 0.384886i
$$118$$ 836.372i 0.652494i
$$119$$ −1727.83 −1.33101
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 588.309i − 0.436582i
$$123$$ 374.504i 0.274536i
$$124$$ 401.570 0.290823
$$125$$ 0 0
$$126$$ −521.774 −0.368915
$$127$$ − 1461.29i − 1.02101i −0.859875 0.510505i $$-0.829458\pi$$
0.859875 0.510505i $$-0.170542\pi$$
$$128$$ 1320.57i 0.911900i
$$129$$ 673.377 0.459593
$$130$$ 0 0
$$131$$ −1524.94 −1.01706 −0.508528 0.861045i $$-0.669810\pi$$
−0.508528 + 0.861045i $$0.669810\pi$$
$$132$$ 161.440i 0.106451i
$$133$$ − 787.134i − 0.513182i
$$134$$ 3148.63 2.02985
$$135$$ 0 0
$$136$$ 1194.12 0.752906
$$137$$ 2125.68i 1.32561i 0.748792 + 0.662805i $$0.230634\pi$$
−0.748792 + 0.662805i $$0.769366\pi$$
$$138$$ − 128.719i − 0.0794007i
$$139$$ 1774.28 1.08268 0.541339 0.840805i $$-0.317918\pi$$
0.541339 + 0.840805i $$0.317918\pi$$
$$140$$ 0 0
$$141$$ 560.496 0.334768
$$142$$ 2632.39i 1.55567i
$$143$$ − 595.335i − 0.348143i
$$144$$ 712.837 0.412521
$$145$$ 0 0
$$146$$ 4172.13 2.36498
$$147$$ − 246.874i − 0.138516i
$$148$$ − 106.439i − 0.0591165i
$$149$$ −1575.78 −0.866393 −0.433197 0.901299i $$-0.642614\pi$$
−0.433197 + 0.901299i $$0.642614\pi$$
$$150$$ 0 0
$$151$$ 420.978 0.226879 0.113439 0.993545i $$-0.463813\pi$$
0.113439 + 0.993545i $$0.463813\pi$$
$$152$$ 543.998i 0.290290i
$$153$$ − 963.086i − 0.508895i
$$154$$ 637.724 0.333696
$$155$$ 0 0
$$156$$ 794.304 0.407662
$$157$$ 2224.30i 1.13069i 0.824854 + 0.565345i $$0.191257\pi$$
−0.824854 + 0.565345i $$0.808743\pi$$
$$158$$ − 2114.23i − 1.06455i
$$159$$ 701.992 0.350136
$$160$$ 0 0
$$161$$ −192.947 −0.0944492
$$162$$ − 290.835i − 0.141051i
$$163$$ − 3093.37i − 1.48645i −0.669040 0.743226i $$-0.733295\pi$$
0.669040 0.743226i $$-0.266705\pi$$
$$164$$ 610.706 0.290781
$$165$$ 0 0
$$166$$ −4168.87 −1.94920
$$167$$ 2416.43i 1.11970i 0.828595 + 0.559848i $$0.189140\pi$$
−0.828595 + 0.559848i $$0.810860\pi$$
$$168$$ − 540.537i − 0.248234i
$$169$$ −732.122 −0.333237
$$170$$ 0 0
$$171$$ 438.746 0.196209
$$172$$ − 1098.08i − 0.486789i
$$173$$ − 3758.02i − 1.65154i −0.564005 0.825771i $$-0.690740\pi$$
0.564005 0.825771i $$-0.309260\pi$$
$$174$$ 2582.32 1.12509
$$175$$ 0 0
$$176$$ −871.245 −0.373140
$$177$$ 698.809i 0.296756i
$$178$$ − 3741.93i − 1.57567i
$$179$$ −2533.99 −1.05810 −0.529049 0.848591i $$-0.677451\pi$$
−0.529049 + 0.848591i $$0.677451\pi$$
$$180$$ 0 0
$$181$$ −13.8995 −0.00570798 −0.00285399 0.999996i $$-0.500908\pi$$
−0.00285399 + 0.999996i $$0.500908\pi$$
$$182$$ − 3137.68i − 1.27791i
$$183$$ − 491.547i − 0.198558i
$$184$$ 133.348 0.0534268
$$185$$ 0 0
$$186$$ 884.194 0.348561
$$187$$ 1177.10i 0.460312i
$$188$$ − 914.004i − 0.354577i
$$189$$ −435.955 −0.167783
$$190$$ 0 0
$$191$$ 3495.39 1.32417 0.662087 0.749427i $$-0.269671\pi$$
0.662087 + 0.749427i $$0.269671\pi$$
$$192$$ − 200.816i − 0.0754824i
$$193$$ 3469.33i 1.29393i 0.762522 + 0.646963i $$0.223961\pi$$
−0.762522 + 0.646963i $$0.776039\pi$$
$$194$$ −5553.25 −2.05516
$$195$$ 0 0
$$196$$ −402.579 −0.146712
$$197$$ − 3638.39i − 1.31586i −0.753079 0.657930i $$-0.771432\pi$$
0.753079 0.657930i $$-0.228568\pi$$
$$198$$ 355.465i 0.127585i
$$199$$ −51.6049 −0.0183828 −0.00919140 0.999958i $$-0.502926\pi$$
−0.00919140 + 0.999958i $$0.502926\pi$$
$$200$$ 0 0
$$201$$ 2630.75 0.923179
$$202$$ 2377.96i 0.828281i
$$203$$ − 3870.84i − 1.33832i
$$204$$ −1570.51 −0.539008
$$205$$ 0 0
$$206$$ 1434.76 0.485265
$$207$$ − 107.548i − 0.0361116i
$$208$$ 4286.63i 1.42896i
$$209$$ −536.246 −0.177478
$$210$$ 0 0
$$211$$ 2084.57 0.680131 0.340065 0.940402i $$-0.389551\pi$$
0.340065 + 0.940402i $$0.389551\pi$$
$$212$$ − 1144.74i − 0.370855i
$$213$$ 2199.42i 0.707521i
$$214$$ 5713.37 1.82504
$$215$$ 0 0
$$216$$ 301.294 0.0949095
$$217$$ − 1325.38i − 0.414622i
$$218$$ 2711.33i 0.842361i
$$219$$ 3485.91 1.07560
$$220$$ 0 0
$$221$$ 5791.50 1.76280
$$222$$ − 234.362i − 0.0708531i
$$223$$ 1887.36i 0.566757i 0.959008 + 0.283378i $$0.0914552\pi$$
−0.959008 + 0.283378i $$0.908545\pi$$
$$224$$ −3150.42 −0.939715
$$225$$ 0 0
$$226$$ −4113.54 −1.21075
$$227$$ 1150.73i 0.336462i 0.985748 + 0.168231i $$0.0538055\pi$$
−0.985748 + 0.168231i $$0.946194\pi$$
$$228$$ − 715.466i − 0.207820i
$$229$$ −4106.79 −1.18508 −0.592542 0.805540i $$-0.701875\pi$$
−0.592542 + 0.805540i $$0.701875\pi$$
$$230$$ 0 0
$$231$$ 532.834 0.151766
$$232$$ 2675.18i 0.757045i
$$233$$ − 5733.58i − 1.61210i −0.591848 0.806050i $$-0.701601\pi$$
0.591848 0.806050i $$-0.298399\pi$$
$$234$$ 1748.93 0.488596
$$235$$ 0 0
$$236$$ 1139.55 0.314316
$$237$$ − 1766.49i − 0.484161i
$$238$$ 6203.86i 1.68965i
$$239$$ −6036.18 −1.63367 −0.816837 0.576868i $$-0.804275\pi$$
−0.816837 + 0.576868i $$0.804275\pi$$
$$240$$ 0 0
$$241$$ 3720.90 0.994540 0.497270 0.867596i $$-0.334336\pi$$
0.497270 + 0.867596i $$0.334336\pi$$
$$242$$ − 434.458i − 0.115405i
$$243$$ − 243.000i − 0.0641500i
$$244$$ −801.568 −0.210308
$$245$$ 0 0
$$246$$ 1344.68 0.348511
$$247$$ 2638.39i 0.679664i
$$248$$ 915.990i 0.234538i
$$249$$ −3483.19 −0.886500
$$250$$ 0 0
$$251$$ −3809.88 −0.958077 −0.479038 0.877794i $$-0.659015\pi$$
−0.479038 + 0.877794i $$0.659015\pi$$
$$252$$ 710.914i 0.177712i
$$253$$ 131.447i 0.0326641i
$$254$$ −5246.84 −1.29613
$$255$$ 0 0
$$256$$ 5277.10 1.28835
$$257$$ − 1225.95i − 0.297559i −0.988870 0.148780i $$-0.952465\pi$$
0.988870 0.148780i $$-0.0475345\pi$$
$$258$$ − 2417.80i − 0.583433i
$$259$$ −351.303 −0.0842816
$$260$$ 0 0
$$261$$ 2157.59 0.511692
$$262$$ 5475.38i 1.29111i
$$263$$ 7397.68i 1.73445i 0.497916 + 0.867225i $$0.334099\pi$$
−0.497916 + 0.867225i $$0.665901\pi$$
$$264$$ −368.248 −0.0858489
$$265$$ 0 0
$$266$$ −2826.25 −0.651461
$$267$$ − 3126.47i − 0.716618i
$$268$$ − 4289.99i − 0.977808i
$$269$$ 3214.48 0.728589 0.364295 0.931284i $$-0.381310\pi$$
0.364295 + 0.931284i $$0.381310\pi$$
$$270$$ 0 0
$$271$$ −7377.37 −1.65367 −0.826833 0.562448i $$-0.809860\pi$$
−0.826833 + 0.562448i $$0.809860\pi$$
$$272$$ − 8475.60i − 1.88937i
$$273$$ − 2621.61i − 0.581197i
$$274$$ 7632.36 1.68280
$$275$$ 0 0
$$276$$ −175.379 −0.0382485
$$277$$ 810.606i 0.175829i 0.996128 + 0.0879144i $$0.0280202\pi$$
−0.996128 + 0.0879144i $$0.971980\pi$$
$$278$$ − 6370.65i − 1.37441i
$$279$$ 738.766 0.158526
$$280$$ 0 0
$$281$$ 1114.72 0.236651 0.118325 0.992975i $$-0.462247\pi$$
0.118325 + 0.992975i $$0.462247\pi$$
$$282$$ − 2012.49i − 0.424972i
$$283$$ 2265.40i 0.475844i 0.971284 + 0.237922i $$0.0764663\pi$$
−0.971284 + 0.237922i $$0.923534\pi$$
$$284$$ 3586.61 0.749388
$$285$$ 0 0
$$286$$ −2137.59 −0.441951
$$287$$ − 2015.64i − 0.414563i
$$288$$ − 1756.03i − 0.359289i
$$289$$ −6538.04 −1.33076
$$290$$ 0 0
$$291$$ −4639.88 −0.934689
$$292$$ − 5684.50i − 1.13925i
$$293$$ 3802.06i 0.758084i 0.925380 + 0.379042i $$0.123746\pi$$
−0.925380 + 0.379042i $$0.876254\pi$$
$$294$$ −886.416 −0.175840
$$295$$ 0 0
$$296$$ 242.790 0.0476753
$$297$$ 297.000i 0.0580259i
$$298$$ 5657.92i 1.09985i
$$299$$ 646.738 0.125090
$$300$$ 0 0
$$301$$ −3624.22 −0.694009
$$302$$ − 1511.55i − 0.288013i
$$303$$ 1986.85i 0.376704i
$$304$$ 3861.17 0.728465
$$305$$ 0 0
$$306$$ −3458.02 −0.646019
$$307$$ 1356.35i 0.252153i 0.992021 + 0.126077i $$0.0402385\pi$$
−0.992021 + 0.126077i $$0.959761\pi$$
$$308$$ − 868.895i − 0.160746i
$$309$$ 1198.78 0.220699
$$310$$ 0 0
$$311$$ −8078.07 −1.47288 −0.736440 0.676503i $$-0.763495\pi$$
−0.736440 + 0.676503i $$0.763495\pi$$
$$312$$ 1811.83i 0.328764i
$$313$$ 5761.54i 1.04045i 0.854029 + 0.520226i $$0.174152\pi$$
−0.854029 + 0.520226i $$0.825848\pi$$
$$314$$ 7986.48 1.43536
$$315$$ 0 0
$$316$$ −2880.63 −0.512811
$$317$$ − 5107.00i − 0.904851i −0.891802 0.452426i $$-0.850559\pi$$
0.891802 0.452426i $$-0.149441\pi$$
$$318$$ − 2520.54i − 0.444481i
$$319$$ −2637.06 −0.462843
$$320$$ 0 0
$$321$$ 4773.66 0.830030
$$322$$ 692.786i 0.119899i
$$323$$ − 5216.67i − 0.898648i
$$324$$ −396.262 −0.0679461
$$325$$ 0 0
$$326$$ −11106.9 −1.88698
$$327$$ 2265.38i 0.383107i
$$328$$ 1393.03i 0.234504i
$$329$$ −3016.68 −0.505516
$$330$$ 0 0
$$331$$ −2780.94 −0.461796 −0.230898 0.972978i $$-0.574166\pi$$
−0.230898 + 0.972978i $$0.574166\pi$$
$$332$$ 5680.06i 0.938958i
$$333$$ − 195.816i − 0.0322241i
$$334$$ 8676.35 1.42140
$$335$$ 0 0
$$336$$ −3836.60 −0.622928
$$337$$ − 4939.28i − 0.798397i −0.916865 0.399198i $$-0.869288\pi$$
0.916865 0.399198i $$-0.130712\pi$$
$$338$$ 2628.73i 0.423029i
$$339$$ −3436.96 −0.550649
$$340$$ 0 0
$$341$$ −902.936 −0.143392
$$342$$ − 1575.34i − 0.249079i
$$343$$ 6866.96i 1.08099i
$$344$$ 2504.74 0.392578
$$345$$ 0 0
$$346$$ −13493.4 −2.09656
$$347$$ 2711.58i 0.419496i 0.977755 + 0.209748i $$0.0672644\pi$$
−0.977755 + 0.209748i $$0.932736\pi$$
$$348$$ − 3518.40i − 0.541971i
$$349$$ −5496.03 −0.842967 −0.421484 0.906836i $$-0.638490\pi$$
−0.421484 + 0.906836i $$0.638490\pi$$
$$350$$ 0 0
$$351$$ 1461.28 0.222214
$$352$$ 2146.26i 0.324989i
$$353$$ 6372.23i 0.960792i 0.877052 + 0.480396i $$0.159507\pi$$
−0.877052 + 0.480396i $$0.840493\pi$$
$$354$$ 2509.12 0.376718
$$355$$ 0 0
$$356$$ −5098.36 −0.759024
$$357$$ 5183.48i 0.768456i
$$358$$ 9098.46i 1.34321i
$$359$$ −630.622 −0.0927101 −0.0463551 0.998925i $$-0.514761\pi$$
−0.0463551 + 0.998925i $$0.514761\pi$$
$$360$$ 0 0
$$361$$ −4482.48 −0.653518
$$362$$ 49.9071i 0.00724602i
$$363$$ − 363.000i − 0.0524864i
$$364$$ −4275.07 −0.615590
$$365$$ 0 0
$$366$$ −1764.93 −0.252061
$$367$$ 5374.49i 0.764431i 0.924073 + 0.382216i $$0.124839\pi$$
−0.924073 + 0.382216i $$0.875161\pi$$
$$368$$ − 946.471i − 0.134071i
$$369$$ 1123.51 0.158503
$$370$$ 0 0
$$371$$ −3778.23 −0.528722
$$372$$ − 1204.71i − 0.167907i
$$373$$ 7520.12i 1.04391i 0.852974 + 0.521953i $$0.174796\pi$$
−0.852974 + 0.521953i $$0.825204\pi$$
$$374$$ 4226.46 0.584346
$$375$$ 0 0
$$376$$ 2084.86 0.285954
$$377$$ 12974.7i 1.77249i
$$378$$ 1565.32i 0.212993i
$$379$$ 12509.1 1.69538 0.847690 0.530492i $$-0.177993\pi$$
0.847690 + 0.530492i $$0.177993\pi$$
$$380$$ 0 0
$$381$$ −4383.86 −0.589481
$$382$$ − 12550.4i − 1.68098i
$$383$$ − 11149.9i − 1.48755i −0.668430 0.743775i $$-0.733033\pi$$
0.668430 0.743775i $$-0.266967\pi$$
$$384$$ 3961.72 0.526486
$$385$$ 0 0
$$386$$ 12456.8 1.64258
$$387$$ − 2020.13i − 0.265346i
$$388$$ 7566.28i 0.989999i
$$389$$ −3194.22 −0.416333 −0.208166 0.978093i $$-0.566750\pi$$
−0.208166 + 0.978093i $$0.566750\pi$$
$$390$$ 0 0
$$391$$ −1278.74 −0.165393
$$392$$ − 918.292i − 0.118318i
$$393$$ 4574.81i 0.587198i
$$394$$ −13063.8 −1.67042
$$395$$ 0 0
$$396$$ 484.320 0.0614596
$$397$$ 584.410i 0.0738809i 0.999317 + 0.0369404i $$0.0117612\pi$$
−0.999317 + 0.0369404i $$0.988239\pi$$
$$398$$ 185.291i 0.0233361i
$$399$$ −2361.40 −0.296286
$$400$$ 0 0
$$401$$ −6951.24 −0.865657 −0.432829 0.901476i $$-0.642484\pi$$
−0.432829 + 0.901476i $$0.642484\pi$$
$$402$$ − 9445.88i − 1.17193i
$$403$$ 4442.56i 0.549130i
$$404$$ 3239.96 0.398995
$$405$$ 0 0
$$406$$ −13898.5 −1.69894
$$407$$ 239.330i 0.0291478i
$$408$$ − 3582.37i − 0.434690i
$$409$$ −11754.1 −1.42104 −0.710519 0.703678i $$-0.751540\pi$$
−0.710519 + 0.703678i $$0.751540\pi$$
$$410$$ 0 0
$$411$$ 6377.03 0.765342
$$412$$ − 1954.85i − 0.233759i
$$413$$ − 3761.10i − 0.448116i
$$414$$ −386.157 −0.0458420
$$415$$ 0 0
$$416$$ 10559.9 1.24457
$$417$$ − 5322.83i − 0.625084i
$$418$$ 1925.42i 0.225300i
$$419$$ −11829.8 −1.37929 −0.689646 0.724146i $$-0.742234\pi$$
−0.689646 + 0.724146i $$0.742234\pi$$
$$420$$ 0 0
$$421$$ −2000.07 −0.231538 −0.115769 0.993276i $$-0.536933\pi$$
−0.115769 + 0.993276i $$0.536933\pi$$
$$422$$ − 7484.76i − 0.863395i
$$423$$ − 1681.49i − 0.193278i
$$424$$ 2611.18 0.299081
$$425$$ 0 0
$$426$$ 7897.16 0.898166
$$427$$ 2645.58i 0.299833i
$$428$$ − 7784.43i − 0.879147i
$$429$$ −1786.00 −0.201000
$$430$$ 0 0
$$431$$ −8064.11 −0.901240 −0.450620 0.892716i $$-0.648797\pi$$
−0.450620 + 0.892716i $$0.648797\pi$$
$$432$$ − 2138.51i − 0.238169i
$$433$$ − 10710.3i − 1.18869i −0.804210 0.594345i $$-0.797411\pi$$
0.804210 0.594345i $$-0.202589\pi$$
$$434$$ −4758.87 −0.526344
$$435$$ 0 0
$$436$$ 3694.18 0.405777
$$437$$ − 582.546i − 0.0637688i
$$438$$ − 12516.4i − 1.36542i
$$439$$ −4658.08 −0.506419 −0.253210 0.967411i $$-0.581486\pi$$
−0.253210 + 0.967411i $$0.581486\pi$$
$$440$$ 0 0
$$441$$ −740.622 −0.0799721
$$442$$ − 20794.7i − 2.23779i
$$443$$ 2094.73i 0.224658i 0.993671 + 0.112329i $$0.0358310\pi$$
−0.993671 + 0.112329i $$0.964169\pi$$
$$444$$ −319.318 −0.0341310
$$445$$ 0 0
$$446$$ 6776.67 0.719472
$$447$$ 4727.33i 0.500212i
$$448$$ 1080.82i 0.113982i
$$449$$ 1402.21 0.147382 0.0736909 0.997281i $$-0.476522\pi$$
0.0736909 + 0.997281i $$0.476522\pi$$
$$450$$ 0 0
$$451$$ −1373.18 −0.143372
$$452$$ 5604.67i 0.583234i
$$453$$ − 1262.93i − 0.130989i
$$454$$ 4131.78 0.427124
$$455$$ 0 0
$$456$$ 1632.00 0.167599
$$457$$ 4914.19i 0.503011i 0.967856 + 0.251506i $$0.0809257\pi$$
−0.967856 + 0.251506i $$0.919074\pi$$
$$458$$ 14745.7i 1.50441i
$$459$$ −2889.26 −0.293810
$$460$$ 0 0
$$461$$ −2214.08 −0.223688 −0.111844 0.993726i $$-0.535676\pi$$
−0.111844 + 0.993726i $$0.535676\pi$$
$$462$$ − 1913.17i − 0.192660i
$$463$$ − 5567.02i − 0.558793i −0.960176 0.279396i $$-0.909866\pi$$
0.960176 0.279396i $$-0.0901344\pi$$
$$464$$ 18987.8 1.89976
$$465$$ 0 0
$$466$$ −20586.8 −2.04649
$$467$$ − 497.054i − 0.0492525i −0.999697 0.0246263i $$-0.992160\pi$$
0.999697 0.0246263i $$-0.00783957\pi$$
$$468$$ − 2382.91i − 0.235364i
$$469$$ −14159.1 −1.39405
$$470$$ 0 0
$$471$$ 6672.90 0.652804
$$472$$ 2599.35i 0.253484i
$$473$$ 2469.05i 0.240015i
$$474$$ −6342.70 −0.614620
$$475$$ 0 0
$$476$$ 8452.73 0.813929
$$477$$ − 2105.97i − 0.202151i
$$478$$ 21673.3i 2.07388i
$$479$$ 9349.28 0.891815 0.445908 0.895079i $$-0.352881\pi$$
0.445908 + 0.895079i $$0.352881\pi$$
$$480$$ 0 0
$$481$$ 1177.53 0.111624
$$482$$ − 13360.1i − 1.26252i
$$483$$ 578.840i 0.0545303i
$$484$$ −591.946 −0.0555923
$$485$$ 0 0
$$486$$ −872.506 −0.0814355
$$487$$ − 197.750i − 0.0184003i −0.999958 0.00920013i $$-0.997071\pi$$
0.999958 0.00920013i $$-0.00292853\pi$$
$$488$$ − 1828.40i − 0.169606i
$$489$$ −9280.12 −0.858204
$$490$$ 0 0
$$491$$ −9997.05 −0.918861 −0.459430 0.888214i $$-0.651946\pi$$
−0.459430 + 0.888214i $$0.651946\pi$$
$$492$$ − 1832.12i − 0.167883i
$$493$$ − 25653.7i − 2.34358i
$$494$$ 9473.31 0.862803
$$495$$ 0 0
$$496$$ 6501.48 0.588558
$$497$$ − 11837.6i − 1.06839i
$$498$$ 12506.6i 1.12537i
$$499$$ 8714.73 0.781813 0.390907 0.920430i $$-0.372162\pi$$
0.390907 + 0.920430i $$0.372162\pi$$
$$500$$ 0 0
$$501$$ 7249.30 0.646457
$$502$$ 13679.6i 1.21623i
$$503$$ 5978.53i 0.529959i 0.964254 + 0.264979i $$0.0853652\pi$$
−0.964254 + 0.264979i $$0.914635\pi$$
$$504$$ −1621.61 −0.143318
$$505$$ 0 0
$$506$$ 471.970 0.0414657
$$507$$ 2196.36i 0.192394i
$$508$$ 7148.79i 0.624363i
$$509$$ 8205.79 0.714569 0.357284 0.933996i $$-0.383703\pi$$
0.357284 + 0.933996i $$0.383703\pi$$
$$510$$ 0 0
$$511$$ −18761.7 −1.62421
$$512$$ − 8383.17i − 0.723608i
$$513$$ − 1316.24i − 0.113281i
$$514$$ −4401.85 −0.377738
$$515$$ 0 0
$$516$$ −3294.24 −0.281048
$$517$$ 2055.15i 0.174827i
$$518$$ 1261.38i 0.106992i
$$519$$ −11274.1 −0.953519
$$520$$ 0 0
$$521$$ −5266.06 −0.442822 −0.221411 0.975181i $$-0.571066\pi$$
−0.221411 + 0.975181i $$0.571066\pi$$
$$522$$ − 7746.97i − 0.649570i
$$523$$ − 22398.6i − 1.87270i −0.351068 0.936350i $$-0.614181\pi$$
0.351068 0.936350i $$-0.385819\pi$$
$$524$$ 7460.18 0.621945
$$525$$ 0 0
$$526$$ 26561.8 2.20181
$$527$$ − 8783.89i − 0.726057i
$$528$$ 2613.74i 0.215432i
$$529$$ 12024.2 0.988264
$$530$$ 0 0
$$531$$ 2096.43 0.171332
$$532$$ 3850.75i 0.313818i
$$533$$ 6756.22i 0.549052i
$$534$$ −11225.8 −0.909714
$$535$$ 0 0
$$536$$ 9785.56 0.788566
$$537$$ 7601.98i 0.610893i
$$538$$ − 11541.8i − 0.924911i
$$539$$ 905.205 0.0723375
$$540$$ 0 0
$$541$$ −13030.2 −1.03551 −0.517757 0.855528i $$-0.673233\pi$$
−0.517757 + 0.855528i $$0.673233\pi$$
$$542$$ 26488.9i 2.09925i
$$543$$ 41.6986i 0.00329550i
$$544$$ −20879.1 −1.64556
$$545$$ 0 0
$$546$$ −9413.04 −0.737804
$$547$$ − 10448.9i − 0.816747i −0.912815 0.408374i $$-0.866096\pi$$
0.912815 0.408374i $$-0.133904\pi$$
$$548$$ − 10399.1i − 0.810631i
$$549$$ −1474.64 −0.114638
$$550$$ 0 0
$$551$$ 11686.9 0.903588
$$552$$ − 400.044i − 0.0308460i
$$553$$ 9507.55i 0.731107i
$$554$$ 2910.53 0.223207
$$555$$ 0 0
$$556$$ −8679.97 −0.662073
$$557$$ 11448.7i 0.870911i 0.900210 + 0.435456i $$0.143413\pi$$
−0.900210 + 0.435456i $$0.856587\pi$$
$$558$$ − 2652.58i − 0.201242i
$$559$$ 12148.0 0.919153
$$560$$ 0 0
$$561$$ 3531.31 0.265762
$$562$$ − 4002.49i − 0.300418i
$$563$$ − 26035.8i − 1.94898i −0.224422 0.974492i $$-0.572049\pi$$
0.224422 0.974492i $$-0.427951\pi$$
$$564$$ −2742.01 −0.204715
$$565$$ 0 0
$$566$$ 8134.05 0.604063
$$567$$ 1307.86i 0.0968697i
$$568$$ 8181.15i 0.604354i
$$569$$ −25075.0 −1.84745 −0.923725 0.383057i $$-0.874871\pi$$
−0.923725 + 0.383057i $$0.874871\pi$$
$$570$$ 0 0
$$571$$ 19056.6 1.39666 0.698330 0.715776i $$-0.253927\pi$$
0.698330 + 0.715776i $$0.253927\pi$$
$$572$$ 2912.45i 0.212894i
$$573$$ − 10486.2i − 0.764512i
$$574$$ −7237.28 −0.526268
$$575$$ 0 0
$$576$$ −602.447 −0.0435798
$$577$$ 6482.55i 0.467716i 0.972271 + 0.233858i $$0.0751351\pi$$
−0.972271 + 0.233858i $$0.924865\pi$$
$$578$$ 23475.2i 1.68934i
$$579$$ 10408.0 0.747048
$$580$$ 0 0
$$581$$ 18747.1 1.33866
$$582$$ 16659.8i 1.18654i
$$583$$ 2573.97i 0.182852i
$$584$$ 12966.5 0.918762
$$585$$ 0 0
$$586$$ 13651.5 0.962353
$$587$$ − 24650.3i − 1.73327i −0.498946 0.866633i $$-0.666280\pi$$
0.498946 0.866633i $$-0.333720\pi$$
$$588$$ 1207.74i 0.0847045i
$$589$$ 4001.61 0.279938
$$590$$ 0 0
$$591$$ −10915.2 −0.759712
$$592$$ − 1723.27i − 0.119638i
$$593$$ 23163.2i 1.60405i 0.597292 + 0.802024i $$0.296243\pi$$
−0.597292 + 0.802024i $$0.703757\pi$$
$$594$$ 1066.40 0.0736612
$$595$$ 0 0
$$596$$ 7708.88 0.529812
$$597$$ 154.815i 0.0106133i
$$598$$ − 2322.15i − 0.158796i
$$599$$ −5355.44 −0.365304 −0.182652 0.983178i $$-0.558468\pi$$
−0.182652 + 0.983178i $$0.558468\pi$$
$$600$$ 0 0
$$601$$ −20417.6 −1.38578 −0.692889 0.721045i $$-0.743662\pi$$
−0.692889 + 0.721045i $$0.743662\pi$$
$$602$$ 13013.0i 0.881012i
$$603$$ − 7892.26i − 0.532998i
$$604$$ −2059.48 −0.138740
$$605$$ 0 0
$$606$$ 7133.89 0.478208
$$607$$ 6749.81i 0.451345i 0.974203 + 0.225673i $$0.0724580\pi$$
−0.974203 + 0.225673i $$0.927542\pi$$
$$608$$ − 9511.77i − 0.634463i
$$609$$ −11612.5 −0.772681
$$610$$ 0 0
$$611$$ 10111.6 0.669511
$$612$$ 4711.53i 0.311197i
$$613$$ − 30321.0i − 1.99780i −0.0468613 0.998901i $$-0.514922\pi$$
0.0468613 0.998901i $$-0.485078\pi$$
$$614$$ 4870.06 0.320097
$$615$$ 0 0
$$616$$ 1981.97 0.129636
$$617$$ − 15236.8i − 0.994182i −0.867699 0.497091i $$-0.834402\pi$$
0.867699 0.497091i $$-0.165598\pi$$
$$618$$ − 4304.28i − 0.280168i
$$619$$ 20875.4 1.35550 0.677749 0.735293i $$-0.262956\pi$$
0.677749 + 0.735293i $$0.262956\pi$$
$$620$$ 0 0
$$621$$ −322.644 −0.0208490
$$622$$ 29004.8i 1.86975i
$$623$$ 16827.2i 1.08213i
$$624$$ 12859.9 0.825013
$$625$$ 0 0
$$626$$ 20687.2 1.32081
$$627$$ 1608.74i 0.102467i
$$628$$ − 10881.5i − 0.691434i
$$629$$ −2328.24 −0.147588
$$630$$ 0 0
$$631$$ 27966.1 1.76437 0.882183 0.470907i $$-0.156073\pi$$
0.882183 + 0.470907i $$0.156073\pi$$
$$632$$ − 6570.79i − 0.413563i
$$633$$ − 6253.70i − 0.392674i
$$634$$ −18337.0 −1.14867
$$635$$ 0 0
$$636$$ −3434.23 −0.214113
$$637$$ − 4453.72i − 0.277022i
$$638$$ 9468.52i 0.587558i
$$639$$ 6598.27 0.408487
$$640$$ 0 0
$$641$$ −17992.0 −1.10865 −0.554323 0.832301i $$-0.687023\pi$$
−0.554323 + 0.832301i $$0.687023\pi$$
$$642$$ − 17140.1i − 1.05369i
$$643$$ − 9448.64i − 0.579499i −0.957102 0.289750i $$-0.906428\pi$$
0.957102 0.289750i $$-0.0935721\pi$$
$$644$$ 943.918 0.0577571
$$645$$ 0 0
$$646$$ −18730.8 −1.14079
$$647$$ 7429.22i 0.451426i 0.974194 + 0.225713i $$0.0724712\pi$$
−0.974194 + 0.225713i $$0.927529\pi$$
$$648$$ − 903.882i − 0.0547960i
$$649$$ −2562.30 −0.154976
$$650$$ 0 0
$$651$$ −3976.15 −0.239382
$$652$$ 15133.2i 0.908988i
$$653$$ − 4488.63i − 0.268995i −0.990914 0.134497i $$-0.957058\pi$$
0.990914 0.134497i $$-0.0429420\pi$$
$$654$$ 8134.00 0.486337
$$655$$ 0 0
$$656$$ 9887.42 0.588474
$$657$$ − 10457.7i − 0.620998i
$$658$$ 10831.6i 0.641730i
$$659$$ 25326.7 1.49710 0.748550 0.663078i $$-0.230750\pi$$
0.748550 + 0.663078i $$0.230750\pi$$
$$660$$ 0 0
$$661$$ 15192.3 0.893969 0.446984 0.894542i $$-0.352498\pi$$
0.446984 + 0.894542i $$0.352498\pi$$
$$662$$ 9985.15i 0.586229i
$$663$$ − 17374.5i − 1.01775i
$$664$$ −12956.4 −0.757235
$$665$$ 0 0
$$666$$ −703.087 −0.0409070
$$667$$ − 2864.75i − 0.166302i
$$668$$ − 11821.5i − 0.684711i
$$669$$ 5662.07 0.327217
$$670$$ 0 0
$$671$$ 1802.34 0.103694
$$672$$ 9451.25i 0.542545i
$$673$$ 11718.2i 0.671180i 0.942008 + 0.335590i $$0.108936\pi$$
−0.942008 + 0.335590i $$0.891064\pi$$
$$674$$ −17734.8 −1.01353
$$675$$ 0 0
$$676$$ 3581.63 0.203779
$$677$$ 15649.1i 0.888397i 0.895928 + 0.444198i $$0.146511\pi$$
−0.895928 + 0.444198i $$0.853489\pi$$
$$678$$ 12340.6i 0.699024i
$$679$$ 24972.6 1.41143
$$680$$ 0 0
$$681$$ 3452.20 0.194257
$$682$$ 3242.05i 0.182030i
$$683$$ − 18162.8i − 1.01754i −0.860903 0.508770i $$-0.830101\pi$$
0.860903 0.508770i $$-0.169899\pi$$
$$684$$ −2146.40 −0.119985
$$685$$ 0 0
$$686$$ 24656.2 1.37227
$$687$$ 12320.4i 0.684208i
$$688$$ − 17778.1i − 0.985149i
$$689$$ 12664.2 0.700246
$$690$$ 0 0
$$691$$ 29606.7 1.62995 0.814973 0.579500i $$-0.196752\pi$$
0.814973 + 0.579500i $$0.196752\pi$$
$$692$$ 18384.7i 1.00994i
$$693$$ − 1598.50i − 0.0876220i
$$694$$ 9736.08 0.532531
$$695$$ 0 0
$$696$$ 8025.55 0.437080
$$697$$ − 13358.5i − 0.725953i
$$698$$ 19733.8i 1.07011i
$$699$$ −17200.7 −0.930746
$$700$$ 0 0
$$701$$ −26164.7 −1.40974 −0.704870 0.709336i $$-0.748995\pi$$
−0.704870 + 0.709336i $$0.748995\pi$$
$$702$$ − 5246.80i − 0.282091i
$$703$$ − 1060.66i − 0.0569040i
$$704$$ 736.324 0.0394194
$$705$$ 0 0
$$706$$ 22879.9 1.21968
$$707$$ − 10693.5i − 0.568842i
$$708$$ − 3418.66i − 0.181470i
$$709$$ 14508.9 0.768535 0.384268 0.923222i $$-0.374454\pi$$
0.384268 + 0.923222i $$0.374454\pi$$
$$710$$ 0 0
$$711$$ −5299.48 −0.279530
$$712$$ − 11629.5i − 0.612125i
$$713$$ − 980.898i − 0.0515216i
$$714$$ 18611.6 0.975520
$$715$$ 0 0
$$716$$ 12396.6 0.647043
$$717$$ 18108.5i 0.943202i
$$718$$ 2264.28i 0.117691i
$$719$$ 2545.80 0.132048 0.0660239 0.997818i $$-0.478969\pi$$
0.0660239 + 0.997818i $$0.478969\pi$$
$$720$$ 0 0
$$721$$ −6452.01 −0.333267
$$722$$ 16094.6i 0.829611i
$$723$$ − 11162.7i − 0.574198i
$$724$$ 67.9982 0.00349051
$$725$$ 0 0
$$726$$ −1303.37 −0.0666291
$$727$$ 18984.8i 0.968511i 0.874927 + 0.484255i $$0.160909\pi$$
−0.874927 + 0.484255i $$0.839091\pi$$
$$728$$ − 9751.54i − 0.496451i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −24019.2 −1.21530
$$732$$ 2404.70i 0.121421i
$$733$$ 36740.4i 1.85134i 0.378326 + 0.925672i $$0.376500\pi$$
−0.378326 + 0.925672i $$0.623500\pi$$
$$734$$ 19297.4 0.970411
$$735$$ 0 0
$$736$$ −2331.58 −0.116770
$$737$$ 9646.10i 0.482115i
$$738$$ − 4034.04i − 0.201213i
$$739$$ −1755.17 −0.0873682 −0.0436841 0.999045i $$-0.513910\pi$$
−0.0436841 + 0.999045i $$0.513910\pi$$
$$740$$ 0 0
$$741$$ 7915.18 0.392404
$$742$$ 13566.0i 0.671189i
$$743$$ 17972.6i 0.887419i 0.896171 + 0.443709i $$0.146338\pi$$
−0.896171 + 0.443709i $$0.853662\pi$$
$$744$$ 2747.97 0.135411
$$745$$ 0 0
$$746$$ 27001.4 1.32519
$$747$$ 10449.6i 0.511821i
$$748$$ − 5758.54i − 0.281488i
$$749$$ −25692.6 −1.25339
$$750$$ 0 0
$$751$$ 22704.9 1.10321 0.551606 0.834105i $$-0.314015\pi$$
0.551606 + 0.834105i $$0.314015\pi$$
$$752$$ − 14797.9i − 0.717583i
$$753$$ 11429.6i 0.553146i
$$754$$ 46586.3 2.25010
$$755$$ 0 0
$$756$$ 2132.74 0.102602
$$757$$ − 39860.3i − 1.91380i −0.290414 0.956901i $$-0.593793\pi$$
0.290414 0.956901i $$-0.406207\pi$$
$$758$$ − 44914.6i − 2.15221i
$$759$$ 394.342 0.0188587
$$760$$ 0 0
$$761$$ 19631.4 0.935135 0.467568 0.883957i $$-0.345130\pi$$
0.467568 + 0.883957i $$0.345130\pi$$
$$762$$ 15740.5i 0.748319i
$$763$$ − 12192.7i − 0.578511i
$$764$$ −17099.8 −0.809752
$$765$$ 0 0
$$766$$ −40034.3 −1.88838
$$767$$ 12606.8i 0.593490i
$$768$$ − 15831.3i − 0.743832i
$$769$$ −35029.9 −1.64267 −0.821333 0.570449i $$-0.806769\pi$$
−0.821333 + 0.570449i $$0.806769\pi$$
$$770$$ 0 0
$$771$$ −3677.86 −0.171796
$$772$$ − 16972.4i − 0.791255i
$$773$$ − 26736.9i − 1.24406i −0.782992 0.622032i $$-0.786307\pi$$
0.782992 0.622032i $$-0.213693\pi$$
$$774$$ −7253.40 −0.336845
$$775$$ 0 0
$$776$$ −17258.8 −0.798398
$$777$$ 1053.91i 0.0486600i
$$778$$ 11469.0i 0.528515i
$$779$$ 6085.64 0.279898
$$780$$ 0 0
$$781$$ −8064.55 −0.369491
$$782$$ 4591.39i 0.209959i
$$783$$ − 6472.78i − 0.295426i
$$784$$ −6517.81 −0.296912
$$785$$ 0 0
$$786$$ 16426.1 0.745421
$$787$$ − 4799.45i − 0.217385i −0.994075 0.108693i $$-0.965334\pi$$
0.994075 0.108693i $$-0.0346664\pi$$
$$788$$ 17799.4i 0.804668i
$$789$$ 22193.0 1.00139
$$790$$ 0 0
$$791$$ 18498.3 0.831508
$$792$$ 1104.74i 0.0495649i
$$793$$ − 8867.72i − 0.397102i
$$794$$ 2098.36 0.0937884
$$795$$ 0 0
$$796$$ 252.457 0.0112413
$$797$$ − 38438.9i − 1.70838i −0.519963 0.854189i $$-0.674054\pi$$
0.519963 0.854189i $$-0.325946\pi$$
$$798$$ 8478.76i 0.376121i
$$799$$ −19992.8 −0.885224
$$800$$ 0 0
$$801$$ −9379.42 −0.413740
$$802$$ 24958.9i 1.09891i
$$803$$ 12781.7i 0.561713i
$$804$$ −12870.0 −0.564538
$$805$$ 0 0
$$806$$ 15951.3 0.697096
$$807$$ − 9643.45i − 0.420651i
$$808$$ 7390.42i 0.321775i
$$809$$ −9960.91 −0.432889 −0.216444 0.976295i $$-0.569446\pi$$
−0.216444 + 0.976295i $$0.569446\pi$$
$$810$$ 0 0
$$811$$ −34199.9 −1.48079 −0.740396 0.672171i $$-0.765362\pi$$
−0.740396 + 0.672171i $$0.765362\pi$$
$$812$$ 18936.6i 0.818404i
$$813$$ 22132.1i 0.954744i
$$814$$ 859.329 0.0370018
$$815$$ 0 0
$$816$$ −25426.8 −1.09083
$$817$$ − 10942.3i − 0.468570i
$$818$$ 42203.9i 1.80394i
$$819$$ −7864.82 −0.335555
$$820$$ 0 0
$$821$$ −31796.4 −1.35165 −0.675823 0.737064i $$-0.736212\pi$$
−0.675823 + 0.737064i $$0.736212\pi$$
$$822$$ − 22897.1i − 0.971567i
$$823$$ − 10783.6i − 0.456733i −0.973575 0.228366i $$-0.926662\pi$$
0.973575 0.228366i $$-0.0733384\pi$$
$$824$$ 4459.07 0.188518
$$825$$ 0 0
$$826$$ −13504.5 −0.568862
$$827$$ − 44197.5i − 1.85840i −0.369576 0.929201i $$-0.620497\pi$$
0.369576 0.929201i $$-0.379503\pi$$
$$828$$ 526.137i 0.0220828i
$$829$$ −22487.7 −0.942137 −0.471069 0.882097i $$-0.656132\pi$$
−0.471069 + 0.882097i $$0.656132\pi$$
$$830$$ 0 0
$$831$$ 2431.82 0.101515
$$832$$ − 3622.81i − 0.150959i
$$833$$ 8805.96i 0.366276i
$$834$$ −19111.9 −0.793516
$$835$$ 0 0
$$836$$ 2623.38 0.108530
$$837$$ − 2216.30i − 0.0915250i
$$838$$ 42475.6i 1.75095i
$$839$$ −22491.0 −0.925477 −0.462738 0.886495i $$-0.653133\pi$$
−0.462738 + 0.886495i $$0.653133\pi$$
$$840$$ 0 0
$$841$$ 33082.7 1.35646
$$842$$ 7181.37i 0.293927i
$$843$$ − 3344.17i − 0.136630i
$$844$$ −10198.0 −0.415910
$$845$$ 0 0
$$846$$ −6037.48 −0.245358
$$847$$ 1953.72i 0.0792571i
$$848$$ − 18533.5i − 0.750524i
$$849$$ 6796.19 0.274729
$$850$$ 0 0
$$851$$ −259.994 −0.0104730
$$852$$ − 10759.8i − 0.432660i
$$853$$ 22327.9i 0.896241i 0.893973 + 0.448120i $$0.147906\pi$$
−0.893973 + 0.448120i $$0.852094\pi$$
$$854$$ 9499.12 0.380624
$$855$$ 0 0
$$856$$ 17756.5 0.708999
$$857$$ 14505.9i 0.578193i 0.957300 + 0.289096i $$0.0933548\pi$$
−0.957300 + 0.289096i $$0.906645\pi$$
$$858$$ 6412.76i 0.255161i
$$859$$ 8411.45 0.334104 0.167052 0.985948i $$-0.446575\pi$$
0.167052 + 0.985948i $$0.446575\pi$$
$$860$$ 0 0
$$861$$ −6046.92 −0.239348
$$862$$ 28954.7i 1.14408i
$$863$$ 32499.6i 1.28192i 0.767573 + 0.640961i $$0.221464\pi$$
−0.767573 + 0.640961i $$0.778536\pi$$
$$864$$ −5268.10 −0.207436
$$865$$ 0 0
$$866$$ −38455.9 −1.50899
$$867$$ 19614.1i 0.768316i
$$868$$ 6483.94i 0.253548i
$$869$$ 6477.15 0.252845
$$870$$ 0 0
$$871$$ 47460.0 1.84629
$$872$$ 8426.50i 0.327245i
$$873$$ 13919.6i 0.539643i
$$874$$ −2091.67 −0.0809516
$$875$$ 0 0
$$876$$ −17053.5 −0.657745
$$877$$ 32183.1i 1.23916i 0.784933 + 0.619581i $$0.212697\pi$$
−0.784933 + 0.619581i $$0.787303\pi$$
$$878$$ 16725.1i 0.642876i
$$879$$ 11406.2 0.437680
$$880$$ 0 0
$$881$$ −6246.34 −0.238870 −0.119435 0.992842i $$-0.538108\pi$$
−0.119435 + 0.992842i $$0.538108\pi$$
$$882$$ 2659.25i 0.101521i
$$883$$ − 11801.1i − 0.449762i −0.974386 0.224881i $$-0.927801\pi$$
0.974386 0.224881i $$-0.0721993\pi$$
$$884$$ −28332.7 −1.07798
$$885$$ 0 0
$$886$$ 7521.24 0.285193
$$887$$ − 32375.1i − 1.22553i −0.790264 0.612767i $$-0.790056\pi$$
0.790264 0.612767i $$-0.209944\pi$$
$$888$$ − 728.371i − 0.0275254i
$$889$$ 23594.6 0.890145
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ − 9233.17i − 0.346580i
$$893$$ − 9107.98i − 0.341307i
$$894$$ 16973.8 0.634997
$$895$$ 0 0
$$896$$ −21322.6 −0.795020
$$897$$ − 1940.21i − 0.0722205i
$$898$$ − 5034.72i − 0.187095i
$$899$$ 19678.5 0.730049
$$900$$ 0 0
$$901$$ −25039.9 −0.925861
$$902$$ 4930.49i 0.182004i
$$903$$ 10872.7i 0.400686i
$$904$$ −12784.4 −0.470357
$$905$$ 0 0
$$906$$ −4534.64 −0.166284
$$907$$ 19592.8i 0.717277i 0.933477 + 0.358638i $$0.116759\pi$$
−0.933477 + 0.358638i $$0.883241\pi$$
$$908$$ − 5629.53i − 0.205752i
$$909$$ 5960.54 0.217490
$$910$$ 0 0
$$911$$ 18673.8 0.679133 0.339567 0.940582i $$-0.389720\pi$$
0.339567 + 0.940582i $$0.389720\pi$$
$$912$$ − 11583.5i − 0.420579i
$$913$$ − 12771.7i − 0.462959i
$$914$$ 17644.7 0.638550
$$915$$ 0 0
$$916$$ 20090.9 0.724696
$$917$$ − 24622.4i − 0.886698i
$$918$$ 10374.1i 0.372979i
$$919$$ 4572.90 0.164142 0.0820708 0.996627i $$-0.473847\pi$$
0.0820708 + 0.996627i $$0.473847\pi$$
$$920$$ 0 0
$$921$$ 4069.05 0.145581
$$922$$ 7949.78i 0.283961i
$$923$$ 39678.6i 1.41499i
$$924$$ −2606.69 −0.0928070
$$925$$ 0 0
$$926$$ −19988.7 −0.709362
$$927$$ − 3596.33i − 0.127421i
$$928$$ − 46775.4i − 1.65461i
$$929$$ 44222.1 1.56176 0.780882 0.624679i $$-0.214770\pi$$
0.780882 + 0.624679i $$0.214770\pi$$
$$930$$ 0 0
$$931$$ −4011.67 −0.141221
$$932$$ 28049.3i 0.985823i
$$933$$ 24234.2i 0.850367i
$$934$$ −1784.70 −0.0625238
$$935$$ 0 0
$$936$$ 5435.48 0.189812
$$937$$ − 9218.62i − 0.321408i −0.987003 0.160704i $$-0.948624\pi$$
0.987003 0.160704i $$-0.0513764\pi$$
$$938$$ 50839.2i 1.76968i
$$939$$ 17284.6 0.600705
$$940$$ 0 0
$$941$$ −26484.9 −0.917516 −0.458758 0.888561i $$-0.651706\pi$$
−0.458758 + 0.888561i $$0.651706\pi$$
$$942$$ − 23959.4i − 0.828705i
$$943$$ − 1491.75i − 0.0515142i
$$944$$ 18449.5 0.636103
$$945$$ 0 0
$$946$$ 8865.26 0.304688
$$947$$ 44972.4i 1.54320i 0.636110 + 0.771599i $$0.280543\pi$$
−0.636110 + 0.771599i $$0.719457\pi$$
$$948$$ 8641.90i 0.296072i
$$949$$ 62887.5 2.15112
$$950$$ 0 0
$$951$$ −15321.0 −0.522416
$$952$$ 19280.9i 0.656404i
$$953$$ − 2052.50i − 0.0697659i −0.999391 0.0348829i $$-0.988894\pi$$
0.999391 0.0348829i $$-0.0111058\pi$$
$$954$$ −7561.63 −0.256621
$$955$$ 0 0
$$956$$ 29529.7 0.999016
$$957$$ 7911.18i 0.267223i
$$958$$ − 33569.2i − 1.13212i
$$959$$ −34322.2 −1.15570
$$960$$ 0 0
$$961$$ −23053.0 −0.773826
$$962$$ − 4228.00i − 0.141701i
$$963$$ − 14321.0i − 0.479218i
$$964$$ −18203.1 −0.608176
$$965$$ 0 0
$$966$$ 2078.36 0.0692237
$$967$$ − 14950.9i − 0.497195i −0.968607 0.248598i $$-0.920030\pi$$
0.968607 0.248598i $$-0.0799697\pi$$
$$968$$ − 1350.24i − 0.0448331i
$$969$$ −15650.0 −0.518835
$$970$$ 0 0
$$971$$ 26751.9 0.884151 0.442076 0.896978i $$-0.354242\pi$$
0.442076 + 0.896978i $$0.354242\pi$$
$$972$$ 1188.78i 0.0392287i
$$973$$ 28648.3i 0.943908i
$$974$$ −710.034 −0.0233583
$$975$$ 0 0
$$976$$ −12977.5 −0.425615
$$977$$ 56893.7i 1.86304i 0.363690 + 0.931520i $$0.381517\pi$$
−0.363690 + 0.931520i $$0.618483\pi$$
$$978$$ 33320.8i 1.08945i
$$979$$ 11463.7 0.374242
$$980$$ 0 0
$$981$$ 6796.15 0.221187
$$982$$ 35895.0i 1.16645i
$$983$$ − 34633.1i − 1.12373i −0.827230 0.561863i $$-0.810085\pi$$
0.827230 0.561863i $$-0.189915\pi$$
$$984$$ 4179.10 0.135391
$$985$$ 0 0
$$986$$ −92111.0 −2.97506
$$987$$ 9050.03i 0.291860i
$$988$$ − 12907.3i − 0.415625i
$$989$$ −2682.23 −0.0862386
$$990$$ 0 0
$$991$$ 1961.79 0.0628843 0.0314422 0.999506i $$-0.489990\pi$$
0.0314422 + 0.999506i $$0.489990\pi$$
$$992$$ − 16016.0i − 0.512610i
$$993$$ 8342.83i 0.266618i
$$994$$ −42503.8 −1.35628
$$995$$ 0 0
$$996$$ 17040.2 0.542108
$$997$$ − 13096.5i − 0.416017i −0.978127 0.208009i $$-0.933302\pi$$
0.978127 0.208009i $$-0.0666982\pi$$
$$998$$ − 31290.8i − 0.992476i
$$999$$ −587.447 −0.0186046
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 825.4.c.l.199.2 6
5.2 odd 4 165.4.a.d.1.3 3
5.3 odd 4 825.4.a.s.1.1 3
5.4 even 2 inner 825.4.c.l.199.5 6
15.2 even 4 495.4.a.l.1.1 3
15.8 even 4 2475.4.a.s.1.3 3
55.32 even 4 1815.4.a.s.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 5.2 odd 4
495.4.a.l.1.1 3 15.2 even 4
825.4.a.s.1.1 3 5.3 odd 4
825.4.c.l.199.2 6 1.1 even 1 trivial
825.4.c.l.199.5 6 5.4 even 2 inner
1815.4.a.s.1.1 3 55.32 even 4
2475.4.a.s.1.3 3 15.8 even 4