# Properties

 Label 1815.4.a.s.1.1 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.23612.1 Defining polynomial: $$x^{3} - x^{2} - 20 x + 26$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.59056$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} +10.7717 q^{6} +16.1465 q^{7} +11.1590 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} +10.7717 q^{6} +16.1465 q^{7} +11.1590 q^{8} +9.00000 q^{9} +17.9528 q^{10} -14.6764 q^{12} +54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} +107.010 q^{17} -32.3150 q^{18} -48.7496 q^{19} -24.4606 q^{20} -48.4394 q^{21} +11.9498 q^{23} -33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} +78.9905 q^{28} -239.733 q^{29} -53.8584 q^{30} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} -162.364 q^{39} -55.7952 q^{40} +124.835 q^{41} +173.925 q^{42} -224.459 q^{43} -45.0000 q^{45} -42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} -89.7640 q^{50} -321.029 q^{51} +264.768 q^{52} +233.997 q^{53} +96.9451 q^{54} +180.179 q^{56} +146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} -163.849 q^{61} +294.731 q^{62} +145.318 q^{63} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} +100.431 q^{72} -1161.97 q^{73} +78.1208 q^{74} -75.0000 q^{75} -238.489 q^{76} +582.978 q^{78} +588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} +1161.06 q^{83} -236.971 q^{84} -535.048 q^{85} +805.933 q^{86} +719.198 q^{87} -1042.16 q^{89} +161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} +670.831 q^{94} +243.748 q^{95} -585.345 q^{96} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} + O(q^{10})$$ $$3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} - 20 q^{10} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} + 218 q^{17} + 36 q^{18} - 146 q^{19} - 110 q^{20} - 12 q^{21} - 200 q^{23} - 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} + 340 q^{28} - 68 q^{29} + 60 q^{30} - 68 q^{31} + 688 q^{32} - 176 q^{34} - 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} - 240 q^{40} + 196 q^{41} + 168 q^{42} + 524 q^{43} - 135 q^{45} - 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} + 100 q^{50} - 654 q^{51} - 1020 q^{52} - 158 q^{53} - 108 q^{54} + 1368 q^{56} + 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} - 642 q^{61} - 88 q^{62} + 36 q^{63} + 1166 q^{64} - 236 q^{67} - 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} + 432 q^{72} - 900 q^{73} - 1536 q^{74} - 225 q^{75} - 1996 q^{76} + 1524 q^{78} + 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} + 1582 q^{83} - 1020 q^{84} - 1090 q^{85} + 3568 q^{86} + 204 q^{87} - 2122 q^{89} - 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} + 2152 q^{94} + 730 q^{95} - 2064 q^{96} + 618 q^{97} - 572 q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.59056 −1.26945 −0.634727 0.772736i $$-0.718888\pi$$
−0.634727 + 0.772736i $$0.718888\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 4.89212 0.611515
$$5$$ −5.00000 −0.447214
$$6$$ 10.7717 0.732920
$$7$$ 16.1465 0.871828 0.435914 0.899988i $$-0.356425\pi$$
0.435914 + 0.899988i $$0.356425\pi$$
$$8$$ 11.1590 0.493164
$$9$$ 9.00000 0.333333
$$10$$ 17.9528 0.567717
$$11$$ 0 0
$$12$$ −14.6764 −0.353058
$$13$$ 54.1214 1.15466 0.577329 0.816511i $$-0.304095\pi$$
0.577329 + 0.816511i $$0.304095\pi$$
$$14$$ −57.9749 −1.10675
$$15$$ 15.0000 0.258199
$$16$$ −79.2041 −1.23756
$$17$$ 107.010 1.52668 0.763342 0.645995i $$-0.223557\pi$$
0.763342 + 0.645995i $$0.223557\pi$$
$$18$$ −32.3150 −0.423152
$$19$$ −48.7496 −0.588628 −0.294314 0.955709i $$-0.595091\pi$$
−0.294314 + 0.955709i $$0.595091\pi$$
$$20$$ −24.4606 −0.273478
$$21$$ −48.4394 −0.503350
$$22$$ 0 0
$$23$$ 11.9498 0.108335 0.0541674 0.998532i $$-0.482750\pi$$
0.0541674 + 0.998532i $$0.482750\pi$$
$$24$$ −33.4771 −0.284729
$$25$$ 25.0000 0.200000
$$26$$ −194.326 −1.46579
$$27$$ −27.0000 −0.192450
$$28$$ 78.9905 0.533136
$$29$$ −239.733 −1.53508 −0.767538 0.641003i $$-0.778519\pi$$
−0.767538 + 0.641003i $$0.778519\pi$$
$$30$$ −53.8584 −0.327772
$$31$$ −82.0851 −0.475578 −0.237789 0.971317i $$-0.576423\pi$$
−0.237789 + 0.971317i $$0.576423\pi$$
$$32$$ 195.115 1.07787
$$33$$ 0 0
$$34$$ −384.224 −1.93806
$$35$$ −80.7324 −0.389893
$$36$$ 44.0291 0.203838
$$37$$ −21.7573 −0.0966723 −0.0483361 0.998831i $$-0.515392\pi$$
−0.0483361 + 0.998831i $$0.515392\pi$$
$$38$$ 175.038 0.747236
$$39$$ −162.364 −0.666643
$$40$$ −55.7952 −0.220550
$$41$$ 124.835 0.475510 0.237755 0.971325i $$-0.423588\pi$$
0.237755 + 0.971325i $$0.423588\pi$$
$$42$$ 173.925 0.638980
$$43$$ −224.459 −0.796039 −0.398019 0.917377i $$-0.630302\pi$$
−0.398019 + 0.917377i $$0.630302\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ −42.9064 −0.137526
$$47$$ −186.832 −0.579835 −0.289917 0.957052i $$-0.593628\pi$$
−0.289917 + 0.957052i $$0.593628\pi$$
$$48$$ 237.612 0.714508
$$49$$ −82.2913 −0.239916
$$50$$ −89.7640 −0.253891
$$51$$ −321.029 −0.881431
$$52$$ 264.768 0.706091
$$53$$ 233.997 0.606453 0.303226 0.952919i $$-0.401936\pi$$
0.303226 + 0.952919i $$0.401936\pi$$
$$54$$ 96.9451 0.244307
$$55$$ 0 0
$$56$$ 180.179 0.429954
$$57$$ 146.249 0.339844
$$58$$ 860.774 1.94871
$$59$$ 232.936 0.513996 0.256998 0.966412i $$-0.417267\pi$$
0.256998 + 0.966412i $$0.417267\pi$$
$$60$$ 73.3818 0.157892
$$61$$ −163.849 −0.343913 −0.171957 0.985105i $$-0.555009\pi$$
−0.171957 + 0.985105i $$0.555009\pi$$
$$62$$ 294.731 0.603725
$$63$$ 145.318 0.290609
$$64$$ −66.9386 −0.130739
$$65$$ −270.607 −0.516379
$$66$$ 0 0
$$67$$ −876.918 −1.59899 −0.799497 0.600670i $$-0.794900\pi$$
−0.799497 + 0.600670i $$0.794900\pi$$
$$68$$ 523.503 0.933590
$$69$$ −35.8493 −0.0625471
$$70$$ 289.874 0.494952
$$71$$ −733.141 −1.22546 −0.612731 0.790291i $$-0.709929\pi$$
−0.612731 + 0.790291i $$0.709929\pi$$
$$72$$ 100.431 0.164388
$$73$$ −1161.97 −1.86299 −0.931496 0.363750i $$-0.881496\pi$$
−0.931496 + 0.363750i $$0.881496\pi$$
$$74$$ 78.1208 0.122721
$$75$$ −75.0000 −0.115470
$$76$$ −238.489 −0.359954
$$77$$ 0 0
$$78$$ 582.978 0.846272
$$79$$ 588.831 0.838591 0.419296 0.907850i $$-0.362277\pi$$
0.419296 + 0.907850i $$0.362277\pi$$
$$80$$ 396.021 0.553456
$$81$$ 81.0000 0.111111
$$82$$ −448.226 −0.603638
$$83$$ 1161.06 1.53546 0.767731 0.640772i $$-0.221386\pi$$
0.767731 + 0.640772i $$0.221386\pi$$
$$84$$ −236.971 −0.307806
$$85$$ −535.048 −0.682754
$$86$$ 805.933 1.01053
$$87$$ 719.198 0.886277
$$88$$ 0 0
$$89$$ −1042.16 −1.24122 −0.620610 0.784120i $$-0.713115\pi$$
−0.620610 + 0.784120i $$0.713115\pi$$
$$90$$ 161.575 0.189239
$$91$$ 873.869 1.00666
$$92$$ 58.4597 0.0662483
$$93$$ 246.255 0.274575
$$94$$ 670.831 0.736074
$$95$$ 243.748 0.263242
$$96$$ −585.345 −0.622307
$$97$$ 1546.63 1.61893 0.809464 0.587169i $$-0.199758\pi$$
0.809464 + 0.587169i $$0.199758\pi$$
$$98$$ 295.472 0.304563
$$99$$ 0 0
$$100$$ 122.303 0.122303
$$101$$ 662.282 0.652470 0.326235 0.945289i $$-0.394220\pi$$
0.326235 + 0.945289i $$0.394220\pi$$
$$102$$ 1152.67 1.11894
$$103$$ 399.592 0.382262 0.191131 0.981565i $$-0.438784\pi$$
0.191131 + 0.981565i $$0.438784\pi$$
$$104$$ 603.942 0.569436
$$105$$ 242.197 0.225105
$$106$$ −840.181 −0.769864
$$107$$ 1591.22 1.43765 0.718827 0.695189i $$-0.244679\pi$$
0.718827 + 0.695189i $$0.244679\pi$$
$$108$$ −132.087 −0.117686
$$109$$ −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
$$112$$ −1278.87 −1.07894
$$113$$ −1145.65 −0.953753 −0.476876 0.878970i $$-0.658231\pi$$
−0.476876 + 0.878970i $$0.658231\pi$$
$$114$$ −525.115 −0.431417
$$115$$ −59.7488 −0.0484488
$$116$$ −1172.80 −0.938722
$$117$$ 487.092 0.384886
$$118$$ −836.372 −0.652494
$$119$$ 1727.83 1.33101
$$120$$ 167.386 0.127334
$$121$$ 0 0
$$122$$ 588.309 0.436582
$$123$$ −374.504 −0.274536
$$124$$ −401.570 −0.290823
$$125$$ −125.000 −0.0894427
$$126$$ −521.774 −0.368915
$$127$$ −1461.29 −1.02101 −0.510505 0.859875i $$-0.670542\pi$$
−0.510505 + 0.859875i $$0.670542\pi$$
$$128$$ −1320.57 −0.911900
$$129$$ 673.377 0.459593
$$130$$ 971.630 0.655520
$$131$$ 1524.94 1.01706 0.508528 0.861045i $$-0.330190\pi$$
0.508528 + 0.861045i $$0.330190\pi$$
$$132$$ 0 0
$$133$$ −787.134 −0.513182
$$134$$ 3148.63 2.02985
$$135$$ 135.000 0.0860663
$$136$$ 1194.12 0.752906
$$137$$ −2125.68 −1.32561 −0.662805 0.748792i $$-0.730634\pi$$
−0.662805 + 0.748792i $$0.730634\pi$$
$$138$$ 128.719 0.0794007
$$139$$ 1774.28 1.08268 0.541339 0.840805i $$-0.317918\pi$$
0.541339 + 0.840805i $$0.317918\pi$$
$$140$$ −394.952 −0.238425
$$141$$ 560.496 0.334768
$$142$$ 2632.39 1.55567
$$143$$ 0 0
$$144$$ −712.837 −0.412521
$$145$$ 1198.66 0.686507
$$146$$ 4172.13 2.36498
$$147$$ 246.874 0.138516
$$148$$ −106.439 −0.0591165
$$149$$ −1575.78 −0.866393 −0.433197 0.901299i $$-0.642614\pi$$
−0.433197 + 0.901299i $$0.642614\pi$$
$$150$$ 269.292 0.146584
$$151$$ −420.978 −0.226879 −0.113439 0.993545i $$-0.536187\pi$$
−0.113439 + 0.993545i $$0.536187\pi$$
$$152$$ −543.998 −0.290290
$$153$$ 963.086 0.508895
$$154$$ 0 0
$$155$$ 410.425 0.212685
$$156$$ −794.304 −0.407662
$$157$$ −2224.30 −1.13069 −0.565345 0.824854i $$-0.691257\pi$$
−0.565345 + 0.824854i $$0.691257\pi$$
$$158$$ −2114.23 −1.06455
$$159$$ −701.992 −0.350136
$$160$$ −975.574 −0.482037
$$161$$ 192.947 0.0944492
$$162$$ −290.835 −0.141051
$$163$$ −3093.37 −1.48645 −0.743226 0.669040i $$-0.766705\pi$$
−0.743226 + 0.669040i $$0.766705\pi$$
$$164$$ 610.706 0.290781
$$165$$ 0 0
$$166$$ −4168.87 −1.94920
$$167$$ 2416.43 1.11970 0.559848 0.828595i $$-0.310860\pi$$
0.559848 + 0.828595i $$0.310860\pi$$
$$168$$ −540.537 −0.248234
$$169$$ 732.122 0.333237
$$170$$ 1921.12 0.866725
$$171$$ −438.746 −0.196209
$$172$$ −1098.08 −0.486789
$$173$$ 3758.02 1.65154 0.825771 0.564005i $$-0.190740\pi$$
0.825771 + 0.564005i $$0.190740\pi$$
$$174$$ −2582.32 −1.12509
$$175$$ 403.662 0.174366
$$176$$ 0 0
$$177$$ −698.809 −0.296756
$$178$$ 3741.93 1.57567
$$179$$ 2533.99 1.05810 0.529049 0.848591i $$-0.322549\pi$$
0.529049 + 0.848591i $$0.322549\pi$$
$$180$$ −220.145 −0.0911592
$$181$$ −13.8995 −0.00570798 −0.00285399 0.999996i $$-0.500908\pi$$
−0.00285399 + 0.999996i $$0.500908\pi$$
$$182$$ −3137.68 −1.27791
$$183$$ 491.547 0.198558
$$184$$ 133.348 0.0534268
$$185$$ 108.786 0.0432332
$$186$$ −884.194 −0.348561
$$187$$ 0 0
$$188$$ −914.004 −0.354577
$$189$$ −435.955 −0.167783
$$190$$ −875.192 −0.334174
$$191$$ 3495.39 1.32417 0.662087 0.749427i $$-0.269671\pi$$
0.662087 + 0.749427i $$0.269671\pi$$
$$192$$ 200.816 0.0754824
$$193$$ −3469.33 −1.29393 −0.646963 0.762522i $$-0.723961\pi$$
−0.646963 + 0.762522i $$0.723961\pi$$
$$194$$ −5553.25 −2.05516
$$195$$ 811.820 0.298132
$$196$$ −402.579 −0.146712
$$197$$ −3638.39 −1.31586 −0.657930 0.753079i $$-0.728568\pi$$
−0.657930 + 0.753079i $$0.728568\pi$$
$$198$$ 0 0
$$199$$ 51.6049 0.0183828 0.00919140 0.999958i $$-0.497074\pi$$
0.00919140 + 0.999958i $$0.497074\pi$$
$$200$$ 278.976 0.0986329
$$201$$ 2630.75 0.923179
$$202$$ −2377.96 −0.828281
$$203$$ −3870.84 −1.33832
$$204$$ −1570.51 −0.539008
$$205$$ −624.173 −0.212654
$$206$$ −1434.76 −0.485265
$$207$$ 107.548 0.0361116
$$208$$ −4286.63 −1.42896
$$209$$ 0 0
$$210$$ −869.623 −0.285761
$$211$$ −2084.57 −0.680131 −0.340065 0.940402i $$-0.610449\pi$$
−0.340065 + 0.940402i $$0.610449\pi$$
$$212$$ 1144.74 0.370855
$$213$$ 2199.42 0.707521
$$214$$ −5713.37 −1.82504
$$215$$ 1122.29 0.355999
$$216$$ −301.294 −0.0949095
$$217$$ −1325.38 −0.414622
$$218$$ 2711.33 0.842361
$$219$$ 3485.91 1.07560
$$220$$ 0 0
$$221$$ 5791.50 1.76280
$$222$$ −234.362 −0.0708531
$$223$$ 1887.36 0.566757 0.283378 0.959008i $$-0.408545\pi$$
0.283378 + 0.959008i $$0.408545\pi$$
$$224$$ 3150.42 0.939715
$$225$$ 225.000 0.0666667
$$226$$ 4113.54 1.21075
$$227$$ 1150.73 0.336462 0.168231 0.985748i $$-0.446194\pi$$
0.168231 + 0.985748i $$0.446194\pi$$
$$228$$ 715.466 0.207820
$$229$$ 4106.79 1.18508 0.592542 0.805540i $$-0.298125\pi$$
0.592542 + 0.805540i $$0.298125\pi$$
$$230$$ 214.532 0.0615035
$$231$$ 0 0
$$232$$ −2675.18 −0.757045
$$233$$ 5733.58 1.61210 0.806050 0.591848i $$-0.201601\pi$$
0.806050 + 0.591848i $$0.201601\pi$$
$$234$$ −1748.93 −0.488596
$$235$$ 934.159 0.259310
$$236$$ 1139.55 0.314316
$$237$$ −1766.49 −0.484161
$$238$$ −6203.86 −1.68965
$$239$$ −6036.18 −1.63367 −0.816837 0.576868i $$-0.804275\pi$$
−0.816837 + 0.576868i $$0.804275\pi$$
$$240$$ −1188.06 −0.319538
$$241$$ −3720.90 −0.994540 −0.497270 0.867596i $$-0.665664\pi$$
−0.497270 + 0.867596i $$0.665664\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ −801.568 −0.210308
$$245$$ 411.457 0.107294
$$246$$ 1344.68 0.348511
$$247$$ −2638.39 −0.679664
$$248$$ −915.990 −0.234538
$$249$$ −3483.19 −0.886500
$$250$$ 448.820 0.113543
$$251$$ −3809.88 −0.958077 −0.479038 0.877794i $$-0.659015\pi$$
−0.479038 + 0.877794i $$0.659015\pi$$
$$252$$ 710.914 0.177712
$$253$$ 0 0
$$254$$ 5246.84 1.29613
$$255$$ 1605.14 0.394188
$$256$$ 5277.10 1.28835
$$257$$ 1225.95 0.297559 0.148780 0.988870i $$-0.452465\pi$$
0.148780 + 0.988870i $$0.452465\pi$$
$$258$$ −2417.80 −0.583433
$$259$$ −351.303 −0.0842816
$$260$$ −1323.84 −0.315773
$$261$$ −2157.59 −0.511692
$$262$$ −5475.38 −1.29111
$$263$$ −7397.68 −1.73445 −0.867225 0.497916i $$-0.834099\pi$$
−0.867225 + 0.497916i $$0.834099\pi$$
$$264$$ 0 0
$$265$$ −1169.99 −0.271214
$$266$$ 2826.25 0.651461
$$267$$ 3126.47 0.716618
$$268$$ −4289.99 −0.977808
$$269$$ −3214.48 −0.728589 −0.364295 0.931284i $$-0.618690\pi$$
−0.364295 + 0.931284i $$0.618690\pi$$
$$270$$ −484.726 −0.109257
$$271$$ 7377.37 1.65367 0.826833 0.562448i $$-0.190140\pi$$
0.826833 + 0.562448i $$0.190140\pi$$
$$272$$ −8475.60 −1.88937
$$273$$ −2621.61 −0.581197
$$274$$ 7632.36 1.68280
$$275$$ 0 0
$$276$$ −175.379 −0.0382485
$$277$$ 810.606 0.175829 0.0879144 0.996128i $$-0.471980\pi$$
0.0879144 + 0.996128i $$0.471980\pi$$
$$278$$ −6370.65 −1.37441
$$279$$ −738.766 −0.158526
$$280$$ −900.895 −0.192281
$$281$$ −1114.72 −0.236651 −0.118325 0.992975i $$-0.537753\pi$$
−0.118325 + 0.992975i $$0.537753\pi$$
$$282$$ −2012.49 −0.424972
$$283$$ −2265.40 −0.475844 −0.237922 0.971284i $$-0.576466\pi$$
−0.237922 + 0.971284i $$0.576466\pi$$
$$284$$ −3586.61 −0.749388
$$285$$ −731.244 −0.151983
$$286$$ 0 0
$$287$$ 2015.64 0.414563
$$288$$ 1756.03 0.359289
$$289$$ 6538.04 1.33076
$$290$$ −4303.87 −0.871490
$$291$$ −4639.88 −0.934689
$$292$$ −5684.50 −1.13925
$$293$$ −3802.06 −0.758084 −0.379042 0.925380i $$-0.623746\pi$$
−0.379042 + 0.925380i $$0.623746\pi$$
$$294$$ −886.416 −0.175840
$$295$$ −1164.68 −0.229866
$$296$$ −242.790 −0.0476753
$$297$$ 0 0
$$298$$ 5657.92 1.09985
$$299$$ 646.738 0.125090
$$300$$ −366.909 −0.0706116
$$301$$ −3624.22 −0.694009
$$302$$ 1511.55 0.288013
$$303$$ −1986.85 −0.376704
$$304$$ 3861.17 0.728465
$$305$$ 819.244 0.153803
$$306$$ −3458.02 −0.646019
$$307$$ 1356.35 0.252153 0.126077 0.992021i $$-0.459761\pi$$
0.126077 + 0.992021i $$0.459761\pi$$
$$308$$ 0 0
$$309$$ −1198.78 −0.220699
$$310$$ −1473.66 −0.269994
$$311$$ −8078.07 −1.47288 −0.736440 0.676503i $$-0.763495\pi$$
−0.736440 + 0.676503i $$0.763495\pi$$
$$312$$ −1811.83 −0.328764
$$313$$ 5761.54 1.04045 0.520226 0.854029i $$-0.325848\pi$$
0.520226 + 0.854029i $$0.325848\pi$$
$$314$$ 7986.48 1.43536
$$315$$ −726.591 −0.129964
$$316$$ 2880.63 0.512811
$$317$$ 5107.00 0.904851 0.452426 0.891802i $$-0.350559\pi$$
0.452426 + 0.891802i $$0.350559\pi$$
$$318$$ 2520.54 0.444481
$$319$$ 0 0
$$320$$ 334.693 0.0584684
$$321$$ −4773.66 −0.830030
$$322$$ −692.786 −0.119899
$$323$$ −5216.67 −0.898648
$$324$$ 396.262 0.0679461
$$325$$ 1353.03 0.230932
$$326$$ 11106.9 1.88698
$$327$$ 2265.38 0.383107
$$328$$ 1393.03 0.234504
$$329$$ −3016.68 −0.505516
$$330$$ 0 0
$$331$$ −2780.94 −0.461796 −0.230898 0.972978i $$-0.574166\pi$$
−0.230898 + 0.972978i $$0.574166\pi$$
$$332$$ 5680.06 0.938958
$$333$$ −195.816 −0.0322241
$$334$$ −8676.35 −1.42140
$$335$$ 4384.59 0.715092
$$336$$ 3836.60 0.622928
$$337$$ −4939.28 −0.798397 −0.399198 0.916865i $$-0.630712\pi$$
−0.399198 + 0.916865i $$0.630712\pi$$
$$338$$ −2628.73 −0.423029
$$339$$ 3436.96 0.550649
$$340$$ −2617.52 −0.417514
$$341$$ 0 0
$$342$$ 1575.34 0.249079
$$343$$ −6866.96 −1.08099
$$344$$ −2504.74 −0.392578
$$345$$ 179.247 0.0279719
$$346$$ −13493.4 −2.09656
$$347$$ 2711.58 0.419496 0.209748 0.977755i $$-0.432736\pi$$
0.209748 + 0.977755i $$0.432736\pi$$
$$348$$ 3518.40 0.541971
$$349$$ −5496.03 −0.842967 −0.421484 0.906836i $$-0.638490\pi$$
−0.421484 + 0.906836i $$0.638490\pi$$
$$350$$ −1449.37 −0.221349
$$351$$ −1461.28 −0.222214
$$352$$ 0 0
$$353$$ 6372.23 0.960792 0.480396 0.877052i $$-0.340493\pi$$
0.480396 + 0.877052i $$0.340493\pi$$
$$354$$ 2509.12 0.376718
$$355$$ 3665.71 0.548043
$$356$$ −5098.36 −0.759024
$$357$$ −5183.48 −0.768456
$$358$$ −9098.46 −1.34321
$$359$$ −630.622 −0.0927101 −0.0463551 0.998925i $$-0.514761\pi$$
−0.0463551 + 0.998925i $$0.514761\pi$$
$$360$$ −502.157 −0.0735166
$$361$$ −4482.48 −0.653518
$$362$$ 49.9071 0.00724602
$$363$$ 0 0
$$364$$ 4275.07 0.615590
$$365$$ 5809.86 0.833156
$$366$$ −1764.93 −0.252061
$$367$$ −5374.49 −0.764431 −0.382216 0.924073i $$-0.624839\pi$$
−0.382216 + 0.924073i $$0.624839\pi$$
$$368$$ −946.471 −0.134071
$$369$$ 1123.51 0.158503
$$370$$ −390.604 −0.0548825
$$371$$ 3778.23 0.528722
$$372$$ 1204.71 0.167907
$$373$$ −7520.12 −1.04391 −0.521953 0.852974i $$-0.674796\pi$$
−0.521953 + 0.852974i $$0.674796\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ −2084.86 −0.285954
$$377$$ −12974.7 −1.77249
$$378$$ 1565.32 0.212993
$$379$$ −12509.1 −1.69538 −0.847690 0.530492i $$-0.822007\pi$$
−0.847690 + 0.530492i $$0.822007\pi$$
$$380$$ 1192.44 0.160977
$$381$$ 4383.86 0.589481
$$382$$ −12550.4 −1.68098
$$383$$ −11149.9 −1.48755 −0.743775 0.668430i $$-0.766967\pi$$
−0.743775 + 0.668430i $$0.766967\pi$$
$$384$$ 3961.72 0.526486
$$385$$ 0 0
$$386$$ 12456.8 1.64258
$$387$$ −2020.13 −0.265346
$$388$$ 7566.28 0.989999
$$389$$ 3194.22 0.416333 0.208166 0.978093i $$-0.433250\pi$$
0.208166 + 0.978093i $$0.433250\pi$$
$$390$$ −2914.89 −0.378465
$$391$$ 1278.74 0.165393
$$392$$ −918.292 −0.118318
$$393$$ −4574.81 −0.587198
$$394$$ 13063.8 1.67042
$$395$$ −2944.16 −0.375029
$$396$$ 0 0
$$397$$ −584.410 −0.0738809 −0.0369404 0.999317i $$-0.511761\pi$$
−0.0369404 + 0.999317i $$0.511761\pi$$
$$398$$ −185.291 −0.0233361
$$399$$ 2361.40 0.296286
$$400$$ −1980.10 −0.247513
$$401$$ −6951.24 −0.865657 −0.432829 0.901476i $$-0.642484\pi$$
−0.432829 + 0.901476i $$0.642484\pi$$
$$402$$ −9445.88 −1.17193
$$403$$ −4442.56 −0.549130
$$404$$ 3239.96 0.398995
$$405$$ −405.000 −0.0496904
$$406$$ 13898.5 1.69894
$$407$$ 0 0
$$408$$ −3582.37 −0.434690
$$409$$ −11754.1 −1.42104 −0.710519 0.703678i $$-0.751540\pi$$
−0.710519 + 0.703678i $$0.751540\pi$$
$$410$$ 2241.13 0.269955
$$411$$ 6377.03 0.765342
$$412$$ 1954.85 0.233759
$$413$$ 3761.10 0.448116
$$414$$ −386.157 −0.0458420
$$415$$ −5805.32 −0.686680
$$416$$ 10559.9 1.24457
$$417$$ −5322.83 −0.625084
$$418$$ 0 0
$$419$$ 11829.8 1.37929 0.689646 0.724146i $$-0.257766\pi$$
0.689646 + 0.724146i $$0.257766\pi$$
$$420$$ 1184.86 0.137655
$$421$$ −2000.07 −0.231538 −0.115769 0.993276i $$-0.536933\pi$$
−0.115769 + 0.993276i $$0.536933\pi$$
$$422$$ 7484.76 0.863395
$$423$$ −1681.49 −0.193278
$$424$$ 2611.18 0.299081
$$425$$ 2675.24 0.305337
$$426$$ −7897.16 −0.898166
$$427$$ −2645.58 −0.299833
$$428$$ 7784.43 0.879147
$$429$$ 0 0
$$430$$ −4029.67 −0.451925
$$431$$ 8064.11 0.901240 0.450620 0.892716i $$-0.351203\pi$$
0.450620 + 0.892716i $$0.351203\pi$$
$$432$$ 2138.51 0.238169
$$433$$ −10710.3 −1.18869 −0.594345 0.804210i $$-0.702589\pi$$
−0.594345 + 0.804210i $$0.702589\pi$$
$$434$$ 4758.87 0.526344
$$435$$ −3595.99 −0.396355
$$436$$ −3694.18 −0.405777
$$437$$ −582.546 −0.0637688
$$438$$ −12516.4 −1.36542
$$439$$ −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
$$442$$ −20794.7 −2.23779
$$443$$ 2094.73 0.224658 0.112329 0.993671i $$-0.464169\pi$$
0.112329 + 0.993671i $$0.464169\pi$$
$$444$$ 319.318 0.0341310
$$445$$ 5210.79 0.555090
$$446$$ −6776.67 −0.719472
$$447$$ 4727.33 0.500212
$$448$$ −1080.82 −0.113982
$$449$$ −1402.21 −0.147382 −0.0736909 0.997281i $$-0.523478\pi$$
−0.0736909 + 0.997281i $$0.523478\pi$$
$$450$$ −807.876 −0.0846303
$$451$$ 0 0
$$452$$ −5604.67 −0.583234
$$453$$ 1262.93 0.130989
$$454$$ −4131.78 −0.427124
$$455$$ −4369.35 −0.450194
$$456$$ 1632.00 0.167599
$$457$$ 4914.19 0.503011 0.251506 0.967856i $$-0.419074\pi$$
0.251506 + 0.967856i $$0.419074\pi$$
$$458$$ −14745.7 −1.50441
$$459$$ −2889.26 −0.293810
$$460$$ −292.298 −0.0296271
$$461$$ 2214.08 0.223688 0.111844 0.993726i $$-0.464324\pi$$
0.111844 + 0.993726i $$0.464324\pi$$
$$462$$ 0 0
$$463$$ −5567.02 −0.558793 −0.279396 0.960176i $$-0.590134\pi$$
−0.279396 + 0.960176i $$0.590134\pi$$
$$464$$ 18987.8 1.89976
$$465$$ −1231.28 −0.122794
$$466$$ −20586.8 −2.04649
$$467$$ 497.054 0.0492525 0.0246263 0.999697i $$-0.492160\pi$$
0.0246263 + 0.999697i $$0.492160\pi$$
$$468$$ 2382.91 0.235364
$$469$$ −14159.1 −1.39405
$$470$$ −3354.15 −0.329182
$$471$$ 6672.90 0.652804
$$472$$ 2599.35 0.253484
$$473$$ 0 0
$$474$$ 6342.70 0.614620
$$475$$ −1218.74 −0.117726
$$476$$ 8452.73 0.813929
$$477$$ 2105.97 0.202151
$$478$$ 21673.3 2.07388
$$479$$ 9349.28 0.891815 0.445908 0.895079i $$-0.352881\pi$$
0.445908 + 0.895079i $$0.352881\pi$$
$$480$$ 2926.72 0.278304
$$481$$ −1177.53 −0.111624
$$482$$ 13360.1 1.26252
$$483$$ −578.840 −0.0545303
$$484$$ 0 0
$$485$$ −7733.13 −0.724007
$$486$$ 872.506 0.0814355
$$487$$ 197.750 0.0184003 0.00920013 0.999958i $$-0.497071\pi$$
0.00920013 + 0.999958i $$0.497071\pi$$
$$488$$ −1828.40 −0.169606
$$489$$ 9280.12 0.858204
$$490$$ −1477.36 −0.136205
$$491$$ 9997.05 0.918861 0.459430 0.888214i $$-0.348054\pi$$
0.459430 + 0.888214i $$0.348054\pi$$
$$492$$ −1832.12 −0.167883
$$493$$ −25653.7 −2.34358
$$494$$ 9473.31 0.862803
$$495$$ 0 0
$$496$$ 6501.48 0.588558
$$497$$ −11837.6 −1.06839
$$498$$ 12506.6 1.12537
$$499$$ −8714.73 −0.781813 −0.390907 0.920430i $$-0.627838\pi$$
−0.390907 + 0.920430i $$0.627838\pi$$
$$500$$ −611.515 −0.0546955
$$501$$ −7249.30 −0.646457
$$502$$ 13679.6 1.21623
$$503$$ −5978.53 −0.529959 −0.264979 0.964254i $$-0.585365\pi$$
−0.264979 + 0.964254i $$0.585365\pi$$
$$504$$ 1621.61 0.143318
$$505$$ −3311.41 −0.291794
$$506$$ 0 0
$$507$$ −2196.36 −0.192394
$$508$$ −7148.79 −0.624363
$$509$$ −8205.79 −0.714569 −0.357284 0.933996i $$-0.616297\pi$$
−0.357284 + 0.933996i $$0.616297\pi$$
$$510$$ −5763.36 −0.500404
$$511$$ −18761.7 −1.62421
$$512$$ −8383.17 −0.723608
$$513$$ 1316.24 0.113281
$$514$$ −4401.85 −0.377738
$$515$$ −1997.96 −0.170953
$$516$$ 3294.24 0.281048
$$517$$ 0 0
$$518$$ 1261.38 0.106992
$$519$$ −11274.1 −0.953519
$$520$$ −3019.71 −0.254660
$$521$$ −5266.06 −0.442822 −0.221411 0.975181i $$-0.571066\pi$$
−0.221411 + 0.975181i $$0.571066\pi$$
$$522$$ 7746.97 0.649570
$$523$$ 22398.6 1.87270 0.936350 0.351068i $$-0.114181\pi$$
0.936350 + 0.351068i $$0.114181\pi$$
$$524$$ 7460.18 0.621945
$$525$$ −1210.99 −0.100670
$$526$$ 26561.8 2.20181
$$527$$ −8783.89 −0.726057
$$528$$ 0 0
$$529$$ −12024.2 −0.988264
$$530$$ 4200.90 0.344294
$$531$$ 2096.43 0.171332
$$532$$ −3850.75 −0.313818
$$533$$ 6756.22 0.549052
$$534$$ −11225.8 −0.909714
$$535$$ −7956.10 −0.642938
$$536$$ −9785.56 −0.788566
$$537$$ −7601.98 −0.610893
$$538$$ 11541.8 0.924911
$$539$$ 0 0
$$540$$ 660.436 0.0526308
$$541$$ 13030.2 1.03551 0.517757 0.855528i $$-0.326767\pi$$
0.517757 + 0.855528i $$0.326767\pi$$
$$542$$ −26488.9 −2.09925
$$543$$ 41.6986 0.00329550
$$544$$ 20879.1 1.64556
$$545$$ 3775.64 0.296753
$$546$$ 9413.04 0.737804
$$547$$ −10448.9 −0.816747 −0.408374 0.912815i $$-0.633904\pi$$
−0.408374 + 0.912815i $$0.633904\pi$$
$$548$$ −10399.1 −0.810631
$$549$$ −1474.64 −0.114638
$$550$$ 0 0
$$551$$ 11686.9 0.903588
$$552$$ −400.044 −0.0308460
$$553$$ 9507.55 0.731107
$$554$$ −2910.53 −0.223207
$$555$$ −326.359 −0.0249607
$$556$$ 8679.97 0.662073
$$557$$ 11448.7 0.870911 0.435456 0.900210i $$-0.356587\pi$$
0.435456 + 0.900210i $$0.356587\pi$$
$$558$$ 2652.58 0.201242
$$559$$ −12148.0 −0.919153
$$560$$ 6394.34 0.482518
$$561$$ 0 0
$$562$$ 4002.49 0.300418
$$563$$ 26035.8 1.94898 0.974492 0.224422i $$-0.0720494\pi$$
0.974492 + 0.224422i $$0.0720494\pi$$
$$564$$ 2742.01 0.204715
$$565$$ 5728.27 0.426531
$$566$$ 8134.05 0.604063
$$567$$ 1307.86 0.0968697
$$568$$ −8181.15 −0.604354
$$569$$ −25075.0 −1.84745 −0.923725 0.383057i $$-0.874871\pi$$
−0.923725 + 0.383057i $$0.874871\pi$$
$$570$$ 2625.57 0.192935
$$571$$ −19056.6 −1.39666 −0.698330 0.715776i $$-0.746073\pi$$
−0.698330 + 0.715776i $$0.746073\pi$$
$$572$$ 0 0
$$573$$ −10486.2 −0.764512
$$574$$ −7237.28 −0.526268
$$575$$ 298.744 0.0216669
$$576$$ −602.447 −0.0435798
$$577$$ −6482.55 −0.467716 −0.233858 0.972271i $$-0.575135\pi$$
−0.233858 + 0.972271i $$0.575135\pi$$
$$578$$ −23475.2 −1.68934
$$579$$ 10408.0 0.747048
$$580$$ 5864.00 0.419809
$$581$$ 18747.1 1.33866
$$582$$ 16659.8 1.18654
$$583$$ 0 0
$$584$$ −12966.5 −0.918762
$$585$$ −2435.46 −0.172126
$$586$$ 13651.5 0.962353
$$587$$ 24650.3 1.73327 0.866633 0.498946i $$-0.166280\pi$$
0.866633 + 0.498946i $$0.166280\pi$$
$$588$$ 1207.74 0.0847045
$$589$$ 4001.61 0.279938
$$590$$ 4181.86 0.291804
$$591$$ 10915.2 0.759712
$$592$$ 1723.27 0.119638
$$593$$ −23163.2 −1.60405 −0.802024 0.597292i $$-0.796243\pi$$
−0.802024 + 0.597292i $$0.796243\pi$$
$$594$$ 0 0
$$595$$ −8639.13 −0.595244
$$596$$ −7708.88 −0.529812
$$597$$ −154.815 −0.0106133
$$598$$ −2322.15 −0.158796
$$599$$ 5355.44 0.365304 0.182652 0.983178i $$-0.441532\pi$$
0.182652 + 0.983178i $$0.441532\pi$$
$$600$$ −836.928 −0.0569457
$$601$$ 20417.6 1.38578 0.692889 0.721045i $$-0.256338\pi$$
0.692889 + 0.721045i $$0.256338\pi$$
$$602$$ 13013.0 0.881012
$$603$$ −7892.26 −0.532998
$$604$$ −2059.48 −0.138740
$$605$$ 0 0
$$606$$ 7133.89 0.478208
$$607$$ 6749.81 0.451345 0.225673 0.974203i $$-0.427542\pi$$
0.225673 + 0.974203i $$0.427542\pi$$
$$608$$ −9511.77 −0.634463
$$609$$ 11612.5 0.772681
$$610$$ −2941.55 −0.195245
$$611$$ −10111.6 −0.669511
$$612$$ 4711.53 0.311197
$$613$$ 30321.0 1.99780 0.998901 0.0468613i $$-0.0149219\pi$$
0.998901 + 0.0468613i $$0.0149219\pi$$
$$614$$ −4870.06 −0.320097
$$615$$ 1872.52 0.122776
$$616$$ 0 0
$$617$$ 15236.8 0.994182 0.497091 0.867699i $$-0.334402\pi$$
0.497091 + 0.867699i $$0.334402\pi$$
$$618$$ 4304.28 0.280168
$$619$$ −20875.4 −1.35550 −0.677749 0.735293i $$-0.737044\pi$$
−0.677749 + 0.735293i $$0.737044\pi$$
$$620$$ 2007.85 0.130060
$$621$$ −322.644 −0.0208490
$$622$$ 29004.8 1.86975
$$623$$ −16827.2 −1.08213
$$624$$ 12859.9 0.825013
$$625$$ 625.000 0.0400000
$$626$$ −20687.2 −1.32081
$$627$$ 0 0
$$628$$ −10881.5 −0.691434
$$629$$ −2328.24 −0.147588
$$630$$ 2608.87 0.164984
$$631$$ 27966.1 1.76437 0.882183 0.470907i $$-0.156073\pi$$
0.882183 + 0.470907i $$0.156073\pi$$
$$632$$ 6570.79 0.413563
$$633$$ 6253.70 0.392674
$$634$$ −18337.0 −1.14867
$$635$$ 7306.44 0.456610
$$636$$ −3434.23 −0.214113
$$637$$ −4453.72 −0.277022
$$638$$ 0 0
$$639$$ −6598.27 −0.408487
$$640$$ 6602.86 0.407814
$$641$$ −17992.0 −1.10865 −0.554323 0.832301i $$-0.687023\pi$$
−0.554323 + 0.832301i $$0.687023\pi$$
$$642$$ 17140.1 1.05369
$$643$$ −9448.64 −0.579499 −0.289750 0.957102i $$-0.593572\pi$$
−0.289750 + 0.957102i $$0.593572\pi$$
$$644$$ 943.918 0.0577571
$$645$$ −3366.88 −0.205536
$$646$$ 18730.8 1.14079
$$647$$ −7429.22 −0.451426 −0.225713 0.974194i $$-0.572471\pi$$
−0.225713 + 0.974194i $$0.572471\pi$$
$$648$$ 903.882 0.0547960
$$649$$ 0 0
$$650$$ −4858.15 −0.293157
$$651$$ 3976.15 0.239382
$$652$$ −15133.2 −0.908988
$$653$$ −4488.63 −0.268995 −0.134497 0.990914i $$-0.542942\pi$$
−0.134497 + 0.990914i $$0.542942\pi$$
$$654$$ −8134.00 −0.486337
$$655$$ −7624.69 −0.454842
$$656$$ −9887.42 −0.588474
$$657$$ −10457.7 −0.620998
$$658$$ 10831.6 0.641730
$$659$$ 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 $$0.352498\pi$$
$$662$$ 9985.15 0.586229
$$663$$ −17374.5 −1.01775
$$664$$ 12956.4 0.757235
$$665$$ 3935.67 0.229502
$$666$$ 703.087 0.0409070
$$667$$ −2864.75 −0.166302
$$668$$ 11821.5 0.684711
$$669$$ −5662.07 −0.327217
$$670$$ −15743.1 −0.907776
$$671$$ 0 0
$$672$$ −9451.25 −0.542545
$$673$$ −11718.2 −0.671180 −0.335590 0.942008i $$-0.608936\pi$$
−0.335590 + 0.942008i $$0.608936\pi$$
$$674$$ 17734.8 1.01353
$$675$$ −675.000 −0.0384900
$$676$$ 3581.63 0.203779
$$677$$ 15649.1 0.888397 0.444198 0.895928i $$-0.353489\pi$$
0.444198 + 0.895928i $$0.353489\pi$$
$$678$$ −12340.6 −0.699024
$$679$$ 24972.6 1.41143
$$680$$ −5970.61 −0.336710
$$681$$ −3452.20 −0.194257
$$682$$ 0 0
$$683$$ −18162.8 −1.01754 −0.508770 0.860903i $$-0.669899\pi$$
−0.508770 + 0.860903i $$0.669899\pi$$
$$684$$ −2146.40 −0.119985
$$685$$ 10628.4 0.592831
$$686$$ 24656.2 1.37227
$$687$$ −12320.4 −0.684208
$$688$$ 17778.1 0.985149
$$689$$ 12664.2 0.700246
$$690$$ −643.595 −0.0355091
$$691$$ 29606.7 1.62995 0.814973 0.579500i $$-0.196752\pi$$
0.814973 + 0.579500i $$0.196752\pi$$
$$692$$ 18384.7 1.00994
$$693$$ 0 0
$$694$$ −9736.08 −0.532531
$$695$$ −8871.38 −0.484188
$$696$$ 8025.55 0.437080
$$697$$ 13358.5 0.725953
$$698$$ 19733.8 1.07011
$$699$$ −17200.7 −0.930746
$$700$$ 1974.76 0.106627
$$701$$ 26164.7 1.40974 0.704870 0.709336i $$-0.251005\pi$$
0.704870 + 0.709336i $$0.251005\pi$$
$$702$$ 5246.80 0.282091
$$703$$ 1060.66 0.0569040
$$704$$ 0 0
$$705$$ −2802.48 −0.149713
$$706$$ −22879.9 −1.21968
$$707$$ 10693.5 0.568842
$$708$$ −3418.66 −0.181470
$$709$$ −14508.9 −0.768535 −0.384268 0.923222i $$-0.625546\pi$$
−0.384268 + 0.923222i $$0.625546\pi$$
$$710$$ −13161.9 −0.695716
$$711$$ 5299.48 0.279530
$$712$$ −11629.5 −0.612125
$$713$$ −980.898 −0.0515216
$$714$$ 18611.6 0.975520
$$715$$ 0 0
$$716$$ 12396.6 0.647043
$$717$$ 18108.5 0.943202
$$718$$ 2264.28 0.117691
$$719$$ −2545.80 −0.132048 −0.0660239 0.997818i $$-0.521031\pi$$
−0.0660239 + 0.997818i $$0.521031\pi$$
$$720$$ 3564.19 0.184485
$$721$$ 6452.01 0.333267
$$722$$ 16094.6 0.829611
$$723$$ 11162.7 0.574198
$$724$$ −67.9982 −0.00349051
$$725$$ −5993.31 −0.307015
$$726$$ 0 0
$$727$$ −18984.8 −0.968511 −0.484255 0.874927i $$-0.660909\pi$$
−0.484255 + 0.874927i $$0.660909\pi$$
$$728$$ 9751.54 0.496451
$$729$$ 729.000 0.0370370
$$730$$ −20860.6 −1.05765
$$731$$ −24019.2 −1.21530
$$732$$ 2404.70 0.121421
$$733$$ −36740.4 −1.85134 −0.925672 0.378326i $$-0.876500\pi$$
−0.925672 + 0.378326i $$0.876500\pi$$
$$734$$ 19297.4 0.970411
$$735$$ −1234.37 −0.0619461
$$736$$ 2331.58 0.116770
$$737$$ 0 0
$$738$$ −4034.04 −0.201213
$$739$$ −1755.17 −0.0873682 −0.0436841 0.999045i $$-0.513910\pi$$
−0.0436841 + 0.999045i $$0.513910\pi$$
$$740$$ 532.196 0.0264377
$$741$$ 7915.18 0.392404
$$742$$ −13566.0 −0.671189
$$743$$ −17972.6 −0.887419 −0.443709 0.896171i $$-0.646338\pi$$
−0.443709 + 0.896171i $$0.646338\pi$$
$$744$$ 2747.97 0.135411
$$745$$ 7878.88 0.387463
$$746$$ 27001.4 1.32519
$$747$$ 10449.6 0.511821
$$748$$ 0 0
$$749$$ 25692.6 1.25339
$$750$$ −1346.46 −0.0655544
$$751$$ 22704.9 1.10321 0.551606 0.834105i $$-0.314015\pi$$
0.551606 + 0.834105i $$0.314015\pi$$
$$752$$ 14797.9 0.717583
$$753$$ 11429.6 0.553146
$$754$$ 46586.3 2.25010
$$755$$ 2104.89 0.101463
$$756$$ −2132.74 −0.102602
$$757$$ 39860.3 1.91380 0.956901 0.290414i $$-0.0937929\pi$$
0.956901 + 0.290414i $$0.0937929\pi$$
$$758$$ 44914.6 2.15221
$$759$$ 0 0
$$760$$ 2719.99 0.129822
$$761$$ −19631.4 −0.935135 −0.467568 0.883957i $$-0.654870\pi$$
−0.467568 + 0.883957i $$0.654870\pi$$
$$762$$ −15740.5 −0.748319
$$763$$ −12192.7 −0.578511
$$764$$ 17099.8 0.809752
$$765$$ −4815.43 −0.227585
$$766$$ 40034.3 1.88838
$$767$$ 12606.8 0.593490
$$768$$ −15831.3 −0.743832
$$769$$ −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
$$772$$ −16972.4 −0.791255
$$773$$ −26736.9 −1.24406 −0.622032 0.782992i $$-0.713693\pi$$
−0.622032 + 0.782992i $$0.713693\pi$$
$$774$$ 7253.40 0.336845
$$775$$ −2052.13 −0.0951156
$$776$$ 17258.8 0.798398
$$777$$ 1053.91 0.0486600
$$778$$ −11469.0 −0.528515
$$779$$ −6085.64 −0.279898
$$780$$ 3971.52 0.182312
$$781$$ 0 0
$$782$$ −4591.39 −0.209959
$$783$$ 6472.78 0.295426
$$784$$ 6517.81 0.296912
$$785$$ 11121.5 0.505660
$$786$$ 16426.1 0.745421
$$787$$ −4799.45 −0.217385 −0.108693 0.994075i $$-0.534666\pi$$
−0.108693 + 0.994075i $$0.534666\pi$$
$$788$$ −17799.4 −0.804668
$$789$$ 22193.0 1.00139
$$790$$ 10571.2 0.476083
$$791$$ −18498.3 −0.831508
$$792$$ 0 0
$$793$$ −8867.72 −0.397102
$$794$$ 2098.36 0.0937884
$$795$$ 3509.96 0.156585
$$796$$ 252.457 0.0112413
$$797$$ 38438.9 1.70838 0.854189 0.519963i $$-0.174054\pi$$
0.854189 + 0.519963i $$0.174054\pi$$
$$798$$ −8478.76 −0.376121
$$799$$ −19992.8 −0.885224
$$800$$ 4877.87 0.215574
$$801$$ −9379.42 −0.413740
$$802$$ 24958.9 1.09891
$$803$$ 0 0
$$804$$ 12870.0 0.564538
$$805$$ −964.733 −0.0422390
$$806$$ 15951.3 0.697096
$$807$$ 9643.45 0.420651
$$808$$ 7390.42 0.321775
$$809$$ −9960.91 −0.432889 −0.216444 0.976295i $$-0.569446\pi$$
−0.216444 + 0.976295i $$0.569446\pi$$
$$810$$ 1454.18 0.0630797
$$811$$ 34199.9 1.48079 0.740396 0.672171i $$-0.234638\pi$$
0.740396 + 0.672171i $$0.234638\pi$$
$$812$$ −18936.6 −0.818404
$$813$$ −22132.1 −0.954744
$$814$$ 0 0
$$815$$ 15466.9 0.664762
$$816$$ 25426.8 1.09083
$$817$$ 10942.3 0.468570
$$818$$ 42203.9 1.80394
$$819$$ 7864.82 0.335555
$$820$$ −3053.53 −0.130041
$$821$$ 31796.4 1.35165 0.675823 0.737064i $$-0.263788\pi$$
0.675823 + 0.737064i $$0.263788\pi$$
$$822$$ −22897.1 −0.971567
$$823$$ −10783.6 −0.456733 −0.228366 0.973575i $$-0.573338\pi$$
−0.228366 + 0.973575i $$0.573338\pi$$
$$824$$ 4459.07 0.188518
$$825$$ 0 0
$$826$$ −13504.5 −0.568862
$$827$$ −44197.5 −1.85840 −0.929201 0.369576i $$-0.879503\pi$$
−0.929201 + 0.369576i $$0.879503\pi$$
$$828$$ 526.137 0.0220828
$$829$$ 22487.7 0.942137 0.471069 0.882097i $$-0.343868\pi$$
0.471069 + 0.882097i $$0.343868\pi$$
$$830$$ 20844.4 0.871709
$$831$$ −2431.82 −0.101515
$$832$$ −3622.81 −0.150959
$$833$$ −8805.96 −0.366276
$$834$$ 19111.9 0.793516
$$835$$ −12082.2 −0.500743
$$836$$ 0 0
$$837$$ 2216.30 0.0915250
$$838$$ −42475.6 −1.75095
$$839$$ 22491.0 0.925477 0.462738 0.886495i $$-0.346867\pi$$
0.462738 + 0.886495i $$0.346867\pi$$
$$840$$ 2702.69 0.111014
$$841$$ 33082.7 1.35646
$$842$$ 7181.37 0.293927
$$843$$ 3344.17 0.136630
$$844$$ −10198.0 −0.415910
$$845$$ −3660.61 −0.149028
$$846$$ 6037.48 0.245358
$$847$$ 0 0
$$848$$ −18533.5 −0.750524
$$849$$ 6796.19 0.274729
$$850$$ −9605.60 −0.387611
$$851$$ −259.994 −0.0104730
$$852$$ 10759.8 0.432660
$$853$$ −22327.9 −0.896241 −0.448120 0.893973i $$-0.647906\pi$$
−0.448120 + 0.893973i $$0.647906\pi$$
$$854$$ 9499.12 0.380624
$$855$$ 2193.73 0.0877474
$$856$$ 17756.5 0.708999
$$857$$ 14505.9 0.578193 0.289096 0.957300i $$-0.406645\pi$$
0.289096 + 0.957300i $$0.406645\pi$$
$$858$$ 0 0
$$859$$ −8411.45 −0.334104 −0.167052 0.985948i $$-0.553425\pi$$
−0.167052 + 0.985948i $$0.553425\pi$$
$$860$$ 5490.40 0.217699
$$861$$ −6046.92 −0.239348
$$862$$ −28954.7 −1.14408
$$863$$ 32499.6 1.28192 0.640961 0.767573i $$-0.278536\pi$$
0.640961 + 0.767573i $$0.278536\pi$$
$$864$$ −5268.10 −0.207436
$$865$$ −18790.1 −0.738592
$$866$$ 38455.9 1.50899
$$867$$ −19614.1 −0.768316
$$868$$ −6483.94 −0.253548
$$869$$ 0 0
$$870$$ 12911.6 0.503155
$$871$$ −47460.0 −1.84629
$$872$$ −8426.50 −0.327245
$$873$$ 13919.6 0.539643
$$874$$ 2091.67 0.0809516
$$875$$ −2018.31 −0.0779786
$$876$$ 17053.5 0.657745
$$877$$ 32183.1 1.23916 0.619581 0.784933i $$-0.287303\pi$$
0.619581 + 0.784933i $$0.287303\pi$$
$$878$$ 16725.1 0.642876
$$879$$ 11406.2 0.437680
$$880$$ 0 0
$$881$$ −6246.34 −0.238870 −0.119435 0.992842i $$-0.538108\pi$$
−0.119435 + 0.992842i $$0.538108\pi$$
$$882$$ 2659.25 0.101521
$$883$$ −11801.1 −0.449762 −0.224881 0.974386i $$-0.572199\pi$$
−0.224881 + 0.974386i $$0.572199\pi$$
$$884$$ 28332.7 1.07798
$$885$$ 3494.05 0.132713
$$886$$ −7521.24 −0.285193
$$887$$ −32375.1 −1.22553 −0.612767 0.790264i $$-0.709944\pi$$
−0.612767 + 0.790264i $$0.709944\pi$$
$$888$$ 728.371 0.0275254
$$889$$ −23594.6 −0.890145
$$890$$ −18709.7 −0.704662
$$891$$ 0 0
$$892$$ 9233.17 0.346580
$$893$$ 9107.98 0.341307
$$894$$ −16973.8 −0.634997
$$895$$ −12670.0 −0.473196
$$896$$ −21322.6 −0.795020
$$897$$ −1940.21 −0.0722205
$$898$$ 5034.72 0.187095
$$899$$ 19678.5 0.730049
$$900$$ 1100.73 0.0407677
$$901$$ 25039.9 0.925861
$$902$$ 0 0
$$903$$ 10872.7 0.400686
$$904$$ −12784.4 −0.470357
$$905$$ 69.4977 0.00255269
$$906$$ −4534.64 −0.166284
$$907$$ −19592.8 −0.717277 −0.358638 0.933477i $$-0.616759\pi$$
−0.358638 + 0.933477i $$0.616759\pi$$
$$908$$ 5629.53 0.205752
$$909$$ 5960.54 0.217490
$$910$$ 15688.4 0.571500
$$911$$ 18673.8 0.679133 0.339567 0.940582i $$-0.389720\pi$$
0.339567 + 0.940582i $$0.389720\pi$$
$$912$$ −11583.5 −0.420579
$$913$$ 0 0
$$914$$ −17644.7 −0.638550
$$915$$ −2457.73 −0.0887980
$$916$$ 20090.9 0.724696
$$917$$ 24622.4 0.886698
$$918$$ 10374.1 0.372979
$$919$$ 4572.90 0.164142 0.0820708 0.996627i $$-0.473847\pi$$
0.0820708 + 0.996627i $$0.473847\pi$$
$$920$$ −666.739 −0.0238932
$$921$$ −4069.05 −0.145581
$$922$$ −7949.78 −0.283961
$$923$$ −39678.6 −1.41499
$$924$$ 0 0
$$925$$ −543.932 −0.0193345
$$926$$ 19988.7 0.709362
$$927$$ 3596.33 0.127421
$$928$$ −46775.4 −1.65461
$$929$$ −44222.1 −1.56176 −0.780882 0.624679i $$-0.785230\pi$$
−0.780882 + 0.624679i $$0.785230\pi$$
$$930$$ 4420.97 0.155881
$$931$$ 4011.67 0.141221
$$932$$ 28049.3 0.985823
$$933$$ 24234.2 0.850367
$$934$$ −1784.70 −0.0625238
$$935$$ 0 0
$$936$$ 5435.48 0.189812
$$937$$ −9218.62 −0.321408 −0.160704 0.987003i $$-0.551376\pi$$
−0.160704 + 0.987003i $$0.551376\pi$$
$$938$$ 50839.2 1.76968
$$939$$ −17284.6 −0.600705
$$940$$ 4570.02 0.158572
$$941$$ 26484.9 0.917516 0.458758 0.888561i $$-0.348294\pi$$
0.458758 + 0.888561i $$0.348294\pi$$
$$942$$ −23959.4 −0.828705
$$943$$ 1491.75 0.0515142
$$944$$ −18449.5 −0.636103
$$945$$ 2179.77 0.0750350
$$946$$ 0 0
$$947$$ −44972.4 −1.54320 −0.771599 0.636110i $$-0.780543\pi$$
−0.771599 + 0.636110i $$0.780543\pi$$
$$948$$ −8641.90 −0.296072
$$949$$ −62887.5 −2.15112
$$950$$ 4375.96 0.149447
$$951$$ −15321.0 −0.522416
$$952$$ 19280.9 0.656404
$$953$$ 2052.50 0.0697659 0.0348829 0.999391i $$-0.488894\pi$$
0.0348829 + 0.999391i $$0.488894\pi$$
$$954$$ −7561.63 −0.256621
$$955$$ −17476.9 −0.592189
$$956$$ −29529.7 −0.999016
$$957$$ 0 0
$$958$$ −33569.2 −1.13212
$$959$$ −34322.2 −1.15570
$$960$$ −1004.08 −0.0337568
$$961$$ −23053.0 −0.773826
$$962$$ 4228.00 0.141701
$$963$$ 14321.0 0.479218
$$964$$ −18203.1 −0.608176
$$965$$ 17346.6 0.578661
$$966$$ 2078.36 0.0692237
$$967$$ −14950.9 −0.497195 −0.248598 0.968607i $$-0.579970\pi$$
−0.248598 + 0.968607i $$0.579970\pi$$
$$968$$ 0 0
$$969$$ 15650.0 0.518835
$$970$$ 27766.3 0.919094
$$971$$ 26751.9 0.884151 0.442076 0.896978i $$-0.354242\pi$$
0.442076 + 0.896978i $$0.354242\pi$$
$$972$$ −1188.78 −0.0392287
$$973$$ 28648.3 0.943908
$$974$$ −710.034 −0.0233583
$$975$$ −4059.10 −0.133329
$$976$$ 12977.5 0.425615
$$977$$ −56893.7 −1.86304 −0.931520 0.363690i $$-0.881517\pi$$
−0.931520 + 0.363690i $$0.881517\pi$$
$$978$$ −33320.8 −1.08945
$$979$$ 0 0
$$980$$ 2012.89 0.0656118
$$981$$ −6796.15 −0.221187
$$982$$ −35895.0 −1.16645
$$983$$ −34633.1 −1.12373 −0.561863 0.827230i $$-0.689915\pi$$
−0.561863 + 0.827230i $$0.689915\pi$$
$$984$$ −4179.10 −0.135391
$$985$$ 18191.9 0.588470
$$986$$ 92111.0 2.97506
$$987$$ 9050.03 0.291860
$$988$$ −12907.3 −0.415625
$$989$$ −2682.23 −0.0862386
$$990$$ 0 0
$$991$$ 1961.79 0.0628843 0.0314422 0.999506i $$-0.489990\pi$$
0.0314422 + 0.999506i $$0.489990\pi$$
$$992$$ −16016.0 −0.512610
$$993$$ 8342.83 0.266618
$$994$$ 42503.8 1.35628
$$995$$ −258.025 −0.00822103
$$996$$ −17040.2 −0.542108
$$997$$ −13096.5 −0.416017 −0.208009 0.978127i $$-0.566698\pi$$
−0.208009 + 0.978127i $$0.566698\pi$$
$$998$$ 31290.8 0.992476
$$999$$ 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 1815.4.a.s.1.1 3
11.10 odd 2 165.4.a.d.1.3 3
33.32 even 2 495.4.a.l.1.1 3
55.32 even 4 825.4.c.l.199.5 6
55.43 even 4 825.4.c.l.199.2 6
55.54 odd 2 825.4.a.s.1.1 3
165.164 even 2 2475.4.a.s.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 11.10 odd 2
495.4.a.l.1.1 3 33.32 even 2
825.4.a.s.1.1 3 55.54 odd 2
825.4.c.l.199.2 6 55.43 even 4
825.4.c.l.199.5 6 55.32 even 4
1815.4.a.s.1.1 3 1.1 even 1 trivial
2475.4.a.s.1.3 3 165.164 even 2