# Properties

 Label 825.6.a.s.1.2 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400$$ x^9 - x^8 - 229*x^7 + 267*x^6 + 16434*x^5 - 16568*x^4 - 405504*x^3 + 202288*x^2 + 2184608*x + 1190400 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-8.18565$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} -121.000 q^{11} -315.043 q^{12} -728.536 q^{13} -1339.68 q^{14} -918.818 q^{16} -1475.58 q^{17} -663.037 q^{18} -341.835 q^{19} -1472.95 q^{21} +990.463 q^{22} -2943.46 q^{23} +221.366 q^{24} +5963.54 q^{26} -729.000 q^{27} +5728.94 q^{28} -2823.82 q^{29} +5163.10 q^{31} +8308.19 q^{32} +1089.00 q^{33} +12078.5 q^{34} +2835.39 q^{36} -4254.93 q^{37} +2798.14 q^{38} +6556.83 q^{39} +5874.23 q^{41} +12057.1 q^{42} -1662.76 q^{43} -4235.58 q^{44} +24094.1 q^{46} +5981.04 q^{47} +8269.36 q^{48} +9978.09 q^{49} +13280.2 q^{51} -25502.3 q^{52} -27557.3 q^{53} +5967.34 q^{54} -4025.45 q^{56} +3076.51 q^{57} +23114.8 q^{58} +8248.03 q^{59} -2342.27 q^{61} -42263.3 q^{62} +13256.6 q^{63} -38605.8 q^{64} -8914.17 q^{66} -49456.4 q^{67} -51652.2 q^{68} +26491.1 q^{69} -24643.4 q^{71} -1992.29 q^{72} +23000.9 q^{73} +34829.4 q^{74} -11965.9 q^{76} -19803.0 q^{77} -53671.9 q^{78} +87214.6 q^{79} +6561.00 q^{81} -48084.4 q^{82} -90598.7 q^{83} -51560.4 q^{84} +13610.8 q^{86} +25414.4 q^{87} +2976.14 q^{88} +86239.2 q^{89} -119233. q^{91} -103035. q^{92} -46467.9 q^{93} -48958.7 q^{94} -74773.8 q^{96} -13088.6 q^{97} -81677.1 q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10})$$ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 $$9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100})$$ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −8.18565 −1.44703 −0.723516 0.690308i $$-0.757475\pi$$
−0.723516 + 0.690308i $$0.757475\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 35.0048 1.09390
$$5$$ 0 0
$$6$$ 73.6708 0.835444
$$7$$ 163.661 1.26241 0.631206 0.775615i $$-0.282560\pi$$
0.631206 + 0.775615i $$0.282560\pi$$
$$8$$ −24.5962 −0.135876
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ −315.043 −0.631563
$$13$$ −728.536 −1.19562 −0.597809 0.801638i $$-0.703962\pi$$
−0.597809 + 0.801638i $$0.703962\pi$$
$$14$$ −1339.68 −1.82675
$$15$$ 0 0
$$16$$ −918.818 −0.897283
$$17$$ −1475.58 −1.23834 −0.619169 0.785258i $$-0.712531\pi$$
−0.619169 + 0.785258i $$0.712531\pi$$
$$18$$ −663.037 −0.482344
$$19$$ −341.835 −0.217236 −0.108618 0.994084i $$-0.534643\pi$$
−0.108618 + 0.994084i $$0.534643\pi$$
$$20$$ 0 0
$$21$$ −1472.95 −0.728854
$$22$$ 990.463 0.436296
$$23$$ −2943.46 −1.16021 −0.580107 0.814541i $$-0.696989\pi$$
−0.580107 + 0.814541i $$0.696989\pi$$
$$24$$ 221.366 0.0784480
$$25$$ 0 0
$$26$$ 5963.54 1.73010
$$27$$ −729.000 −0.192450
$$28$$ 5728.94 1.38095
$$29$$ −2823.82 −0.623508 −0.311754 0.950163i $$-0.600916\pi$$
−0.311754 + 0.950163i $$0.600916\pi$$
$$30$$ 0 0
$$31$$ 5163.10 0.964953 0.482476 0.875909i $$-0.339737\pi$$
0.482476 + 0.875909i $$0.339737\pi$$
$$32$$ 8308.19 1.43427
$$33$$ 1089.00 0.174078
$$34$$ 12078.5 1.79191
$$35$$ 0 0
$$36$$ 2835.39 0.364633
$$37$$ −4254.93 −0.510962 −0.255481 0.966814i $$-0.582234\pi$$
−0.255481 + 0.966814i $$0.582234\pi$$
$$38$$ 2798.14 0.314348
$$39$$ 6556.83 0.690291
$$40$$ 0 0
$$41$$ 5874.23 0.545747 0.272873 0.962050i $$-0.412026\pi$$
0.272873 + 0.962050i $$0.412026\pi$$
$$42$$ 12057.1 1.05468
$$43$$ −1662.76 −0.137138 −0.0685691 0.997646i $$-0.521843\pi$$
−0.0685691 + 0.997646i $$0.521843\pi$$
$$44$$ −4235.58 −0.329823
$$45$$ 0 0
$$46$$ 24094.1 1.67886
$$47$$ 5981.04 0.394941 0.197471 0.980309i $$-0.436727\pi$$
0.197471 + 0.980309i $$0.436727\pi$$
$$48$$ 8269.36 0.518047
$$49$$ 9978.09 0.593686
$$50$$ 0 0
$$51$$ 13280.2 0.714955
$$52$$ −25502.3 −1.30789
$$53$$ −27557.3 −1.34755 −0.673777 0.738935i $$-0.735329\pi$$
−0.673777 + 0.738935i $$0.735329\pi$$
$$54$$ 5967.34 0.278481
$$55$$ 0 0
$$56$$ −4025.45 −0.171532
$$57$$ 3076.51 0.125421
$$58$$ 23114.8 0.902236
$$59$$ 8248.03 0.308475 0.154238 0.988034i $$-0.450708\pi$$
0.154238 + 0.988034i $$0.450708\pi$$
$$60$$ 0 0
$$61$$ −2342.27 −0.0805959 −0.0402980 0.999188i $$-0.512831\pi$$
−0.0402980 + 0.999188i $$0.512831\pi$$
$$62$$ −42263.3 −1.39632
$$63$$ 13256.6 0.420804
$$64$$ −38605.8 −1.17815
$$65$$ 0 0
$$66$$ −8914.17 −0.251896
$$67$$ −49456.4 −1.34597 −0.672985 0.739656i $$-0.734988\pi$$
−0.672985 + 0.739656i $$0.734988\pi$$
$$68$$ −51652.2 −1.35462
$$69$$ 26491.1 0.669849
$$70$$ 0 0
$$71$$ −24643.4 −0.580169 −0.290085 0.957001i $$-0.593683\pi$$
−0.290085 + 0.957001i $$0.593683\pi$$
$$72$$ −1992.29 −0.0452920
$$73$$ 23000.9 0.505170 0.252585 0.967575i $$-0.418719\pi$$
0.252585 + 0.967575i $$0.418719\pi$$
$$74$$ 34829.4 0.739378
$$75$$ 0 0
$$76$$ −11965.9 −0.237635
$$77$$ −19803.0 −0.380632
$$78$$ −53671.9 −0.998873
$$79$$ 87214.6 1.57225 0.786124 0.618068i $$-0.212085\pi$$
0.786124 + 0.618068i $$0.212085\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −48084.4 −0.789713
$$83$$ −90598.7 −1.44353 −0.721766 0.692137i $$-0.756669\pi$$
−0.721766 + 0.692137i $$0.756669\pi$$
$$84$$ −51560.4 −0.797294
$$85$$ 0 0
$$86$$ 13610.8 0.198443
$$87$$ 25414.4 0.359983
$$88$$ 2976.14 0.0409681
$$89$$ 86239.2 1.15406 0.577032 0.816722i $$-0.304211\pi$$
0.577032 + 0.816722i $$0.304211\pi$$
$$90$$ 0 0
$$91$$ −119233. −1.50936
$$92$$ −103035. −1.26916
$$93$$ −46467.9 −0.557116
$$94$$ −48958.7 −0.571492
$$95$$ 0 0
$$96$$ −74773.8 −0.828078
$$97$$ −13088.6 −0.141242 −0.0706211 0.997503i $$-0.522498\pi$$
−0.0706211 + 0.997503i $$0.522498\pi$$
$$98$$ −81677.1 −0.859083
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ 34292.9 0.334503 0.167252 0.985914i $$-0.446511\pi$$
0.167252 + 0.985914i $$0.446511\pi$$
$$102$$ −108707. −1.03456
$$103$$ 32773.7 0.304391 0.152196 0.988350i $$-0.451366\pi$$
0.152196 + 0.988350i $$0.451366\pi$$
$$104$$ 17919.2 0.162456
$$105$$ 0 0
$$106$$ 225574. 1.94995
$$107$$ −203291. −1.71656 −0.858278 0.513186i $$-0.828465\pi$$
−0.858278 + 0.513186i $$0.828465\pi$$
$$108$$ −25518.5 −0.210521
$$109$$ 203531. 1.64084 0.820418 0.571764i $$-0.193741\pi$$
0.820418 + 0.571764i $$0.193741\pi$$
$$110$$ 0 0
$$111$$ 38294.4 0.295004
$$112$$ −150375. −1.13274
$$113$$ 140927. 1.03824 0.519120 0.854702i $$-0.326260\pi$$
0.519120 + 0.854702i $$0.326260\pi$$
$$114$$ −25183.2 −0.181489
$$115$$ 0 0
$$116$$ −98847.3 −0.682055
$$117$$ −59011.4 −0.398540
$$118$$ −67515.4 −0.446373
$$119$$ −241495. −1.56329
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 19173.0 0.116625
$$123$$ −52868.1 −0.315087
$$124$$ 180733. 1.05556
$$125$$ 0 0
$$126$$ −108514. −0.608917
$$127$$ 2686.76 0.0147815 0.00739076 0.999973i $$-0.497647\pi$$
0.00739076 + 0.999973i $$0.497647\pi$$
$$128$$ 50150.9 0.270554
$$129$$ 14964.8 0.0791767
$$130$$ 0 0
$$131$$ −280946. −1.43036 −0.715179 0.698941i $$-0.753655\pi$$
−0.715179 + 0.698941i $$0.753655\pi$$
$$132$$ 38120.2 0.190424
$$133$$ −55945.2 −0.274242
$$134$$ 404832. 1.94766
$$135$$ 0 0
$$136$$ 36293.5 0.168260
$$137$$ 272454. 1.24020 0.620100 0.784523i $$-0.287092\pi$$
0.620100 + 0.784523i $$0.287092\pi$$
$$138$$ −216847. −0.969293
$$139$$ 324210. 1.42328 0.711638 0.702547i $$-0.247954\pi$$
0.711638 + 0.702547i $$0.247954\pi$$
$$140$$ 0 0
$$141$$ −53829.4 −0.228019
$$142$$ 201722. 0.839523
$$143$$ 88152.9 0.360493
$$144$$ −74424.2 −0.299094
$$145$$ 0 0
$$146$$ −188277. −0.730997
$$147$$ −89802.8 −0.342765
$$148$$ −148943. −0.558941
$$149$$ −325211. −1.20005 −0.600024 0.799982i $$-0.704843\pi$$
−0.600024 + 0.799982i $$0.704843\pi$$
$$150$$ 0 0
$$151$$ −399144. −1.42458 −0.712290 0.701885i $$-0.752342\pi$$
−0.712290 + 0.701885i $$0.752342\pi$$
$$152$$ 8407.83 0.0295172
$$153$$ −119522. −0.412779
$$154$$ 162101. 0.550786
$$155$$ 0 0
$$156$$ 229520. 0.755109
$$157$$ 82871.8 0.268323 0.134161 0.990959i $$-0.457166\pi$$
0.134161 + 0.990959i $$0.457166\pi$$
$$158$$ −713907. −2.27509
$$159$$ 248015. 0.778011
$$160$$ 0 0
$$161$$ −481730. −1.46467
$$162$$ −53706.0 −0.160781
$$163$$ 311672. 0.918818 0.459409 0.888225i $$-0.348061\pi$$
0.459409 + 0.888225i $$0.348061\pi$$
$$164$$ 205626. 0.596992
$$165$$ 0 0
$$166$$ 741609. 2.08884
$$167$$ 129998. 0.360700 0.180350 0.983602i $$-0.442277\pi$$
0.180350 + 0.983602i $$0.442277\pi$$
$$168$$ 36229.0 0.0990338
$$169$$ 159472. 0.429505
$$170$$ 0 0
$$171$$ −27688.6 −0.0724121
$$172$$ −58204.5 −0.150015
$$173$$ −744520. −1.89130 −0.945651 0.325183i $$-0.894574\pi$$
−0.945651 + 0.325183i $$0.894574\pi$$
$$174$$ −208033. −0.520906
$$175$$ 0 0
$$176$$ 111177. 0.270541
$$177$$ −74232.3 −0.178098
$$178$$ −705924. −1.66997
$$179$$ 309968. 0.723077 0.361539 0.932357i $$-0.382252\pi$$
0.361539 + 0.932357i $$0.382252\pi$$
$$180$$ 0 0
$$181$$ −656347. −1.48915 −0.744573 0.667541i $$-0.767347\pi$$
−0.744573 + 0.667541i $$0.767347\pi$$
$$182$$ 976002. 2.18410
$$183$$ 21080.5 0.0465321
$$184$$ 72397.7 0.157645
$$185$$ 0 0
$$186$$ 380370. 0.806164
$$187$$ 178545. 0.373373
$$188$$ 209365. 0.432026
$$189$$ −119309. −0.242951
$$190$$ 0 0
$$191$$ −336716. −0.667853 −0.333926 0.942599i $$-0.608374\pi$$
−0.333926 + 0.942599i $$0.608374\pi$$
$$192$$ 347452. 0.680208
$$193$$ 599122. 1.15777 0.578884 0.815410i $$-0.303488\pi$$
0.578884 + 0.815410i $$0.303488\pi$$
$$194$$ 107139. 0.204382
$$195$$ 0 0
$$196$$ 349281. 0.649433
$$197$$ 365021. 0.670120 0.335060 0.942197i $$-0.391243\pi$$
0.335060 + 0.942197i $$0.391243\pi$$
$$198$$ 80227.5 0.145432
$$199$$ 333867. 0.597641 0.298820 0.954309i $$-0.403407\pi$$
0.298820 + 0.954309i $$0.403407\pi$$
$$200$$ 0 0
$$201$$ 445108. 0.777096
$$202$$ −280709. −0.484037
$$203$$ −462151. −0.787125
$$204$$ 464870. 0.782089
$$205$$ 0 0
$$206$$ −268274. −0.440463
$$207$$ −238420. −0.386738
$$208$$ 669392. 1.07281
$$209$$ 41362.0 0.0654992
$$210$$ 0 0
$$211$$ −980083. −1.51550 −0.757751 0.652543i $$-0.773702\pi$$
−0.757751 + 0.652543i $$0.773702\pi$$
$$212$$ −964636. −1.47409
$$213$$ 221791. 0.334961
$$214$$ 1.66406e6 2.48391
$$215$$ 0 0
$$216$$ 17930.6 0.0261493
$$217$$ 845000. 1.21817
$$218$$ −1.66604e6 −2.37434
$$219$$ −207008. −0.291660
$$220$$ 0 0
$$221$$ 1.07501e6 1.48058
$$222$$ −313464. −0.426880
$$223$$ 80967.8 0.109031 0.0545155 0.998513i $$-0.482639\pi$$
0.0545155 + 0.998513i $$0.482639\pi$$
$$224$$ 1.35973e6 1.81064
$$225$$ 0 0
$$226$$ −1.15358e6 −1.50236
$$227$$ −524287. −0.675312 −0.337656 0.941270i $$-0.609634\pi$$
−0.337656 + 0.941270i $$0.609634\pi$$
$$228$$ 107693. 0.137198
$$229$$ 816517. 1.02891 0.514454 0.857518i $$-0.327995\pi$$
0.514454 + 0.857518i $$0.327995\pi$$
$$230$$ 0 0
$$231$$ 178227. 0.219758
$$232$$ 69455.2 0.0847198
$$233$$ 338979. 0.409056 0.204528 0.978861i $$-0.434434\pi$$
0.204528 + 0.978861i $$0.434434\pi$$
$$234$$ 483047. 0.576699
$$235$$ 0 0
$$236$$ 288721. 0.337441
$$237$$ −784931. −0.907738
$$238$$ 1.97679e6 2.26214
$$239$$ 98246.5 0.111256 0.0556278 0.998452i $$-0.482284\pi$$
0.0556278 + 0.998452i $$0.482284\pi$$
$$240$$ 0 0
$$241$$ 1.00307e6 1.11247 0.556233 0.831026i $$-0.312246\pi$$
0.556233 + 0.831026i $$0.312246\pi$$
$$242$$ −119846. −0.131548
$$243$$ −59049.0 −0.0641500
$$244$$ −81990.8 −0.0881639
$$245$$ 0 0
$$246$$ 432759. 0.455941
$$247$$ 249039. 0.259732
$$248$$ −126992. −0.131114
$$249$$ 815388. 0.833424
$$250$$ 0 0
$$251$$ 989640. 0.991501 0.495750 0.868465i $$-0.334893\pi$$
0.495750 + 0.868465i $$0.334893\pi$$
$$252$$ 464044. 0.460318
$$253$$ 356158. 0.349817
$$254$$ −21992.8 −0.0213893
$$255$$ 0 0
$$256$$ 824867. 0.786655
$$257$$ 1.15250e6 1.08845 0.544225 0.838939i $$-0.316824\pi$$
0.544225 + 0.838939i $$0.316824\pi$$
$$258$$ −122497. −0.114571
$$259$$ −696369. −0.645045
$$260$$ 0 0
$$261$$ −228729. −0.207836
$$262$$ 2.29973e6 2.06977
$$263$$ 1.03041e6 0.918589 0.459294 0.888284i $$-0.348102\pi$$
0.459294 + 0.888284i $$0.348102\pi$$
$$264$$ −26785.2 −0.0236530
$$265$$ 0 0
$$266$$ 457948. 0.396836
$$267$$ −776153. −0.666299
$$268$$ −1.73121e6 −1.47236
$$269$$ −2.24263e6 −1.88963 −0.944814 0.327607i $$-0.893758\pi$$
−0.944814 + 0.327607i $$0.893758\pi$$
$$270$$ 0 0
$$271$$ 2.17227e6 1.79676 0.898382 0.439216i $$-0.144744\pi$$
0.898382 + 0.439216i $$0.144744\pi$$
$$272$$ 1.35579e6 1.11114
$$273$$ 1.07310e6 0.871432
$$274$$ −2.23021e6 −1.79461
$$275$$ 0 0
$$276$$ 927315. 0.732748
$$277$$ −1.04100e6 −0.815178 −0.407589 0.913165i $$-0.633630\pi$$
−0.407589 + 0.913165i $$0.633630\pi$$
$$278$$ −2.65387e6 −2.05952
$$279$$ 418211. 0.321651
$$280$$ 0 0
$$281$$ −1.34424e6 −1.01557 −0.507786 0.861483i $$-0.669536\pi$$
−0.507786 + 0.861483i $$0.669536\pi$$
$$282$$ 440628. 0.329951
$$283$$ −828093. −0.614629 −0.307314 0.951608i $$-0.599430\pi$$
−0.307314 + 0.951608i $$0.599430\pi$$
$$284$$ −862637. −0.634647
$$285$$ 0 0
$$286$$ −721588. −0.521644
$$287$$ 961385. 0.688958
$$288$$ 672964. 0.478091
$$289$$ 757468. 0.533482
$$290$$ 0 0
$$291$$ 117798. 0.0815462
$$292$$ 805142. 0.552606
$$293$$ 1.31305e6 0.893538 0.446769 0.894649i $$-0.352575\pi$$
0.446769 + 0.894649i $$0.352575\pi$$
$$294$$ 735094. 0.495992
$$295$$ 0 0
$$296$$ 104655. 0.0694274
$$297$$ 88209.0 0.0580259
$$298$$ 2.66206e6 1.73651
$$299$$ 2.14441e6 1.38717
$$300$$ 0 0
$$301$$ −272130. −0.173125
$$302$$ 3.26725e6 2.06141
$$303$$ −308636. −0.193126
$$304$$ 314084. 0.194922
$$305$$ 0 0
$$306$$ 978362. 0.597305
$$307$$ −138234. −0.0837082 −0.0418541 0.999124i $$-0.513326\pi$$
−0.0418541 + 0.999124i $$0.513326\pi$$
$$308$$ −693201. −0.416373
$$309$$ −294963. −0.175740
$$310$$ 0 0
$$311$$ 1.17643e6 0.689710 0.344855 0.938656i $$-0.387928\pi$$
0.344855 + 0.938656i $$0.387928\pi$$
$$312$$ −161273. −0.0937939
$$313$$ −919942. −0.530762 −0.265381 0.964144i $$-0.585498\pi$$
−0.265381 + 0.964144i $$0.585498\pi$$
$$314$$ −678359. −0.388271
$$315$$ 0 0
$$316$$ 3.05293e6 1.71988
$$317$$ 1.96510e6 1.09834 0.549171 0.835710i $$-0.314944\pi$$
0.549171 + 0.835710i $$0.314944\pi$$
$$318$$ −2.03016e6 −1.12581
$$319$$ 341682. 0.187995
$$320$$ 0 0
$$321$$ 1.82962e6 0.991054
$$322$$ 3.94327e6 2.11942
$$323$$ 504403. 0.269012
$$324$$ 229666. 0.121544
$$325$$ 0 0
$$326$$ −2.55124e6 −1.32956
$$327$$ −1.83178e6 −0.947337
$$328$$ −144484. −0.0741539
$$329$$ 978866. 0.498579
$$330$$ 0 0
$$331$$ 1.98383e6 0.995255 0.497628 0.867391i $$-0.334205\pi$$
0.497628 + 0.867391i $$0.334205\pi$$
$$332$$ −3.17139e6 −1.57908
$$333$$ −344650. −0.170321
$$334$$ −1.06412e6 −0.521945
$$335$$ 0 0
$$336$$ 1.35338e6 0.653989
$$337$$ 3.28452e6 1.57543 0.787713 0.616043i $$-0.211265\pi$$
0.787713 + 0.616043i $$0.211265\pi$$
$$338$$ −1.30538e6 −0.621507
$$339$$ −1.26834e6 −0.599428
$$340$$ 0 0
$$341$$ −624735. −0.290944
$$342$$ 226649. 0.104783
$$343$$ −1.11763e6 −0.512936
$$344$$ 40897.5 0.0186338
$$345$$ 0 0
$$346$$ 6.09437e6 2.73677
$$347$$ −2.23106e6 −0.994691 −0.497345 0.867553i $$-0.665692\pi$$
−0.497345 + 0.867553i $$0.665692\pi$$
$$348$$ 889625. 0.393785
$$349$$ 1.14634e6 0.503790 0.251895 0.967755i $$-0.418946\pi$$
0.251895 + 0.967755i $$0.418946\pi$$
$$350$$ 0 0
$$351$$ 531103. 0.230097
$$352$$ −1.00529e6 −0.432449
$$353$$ −1.17280e6 −0.500941 −0.250470 0.968124i $$-0.580585\pi$$
−0.250470 + 0.968124i $$0.580585\pi$$
$$354$$ 607639. 0.257714
$$355$$ 0 0
$$356$$ 3.01879e6 1.26243
$$357$$ 2.17346e6 0.902568
$$358$$ −2.53729e6 −1.04632
$$359$$ 718129. 0.294081 0.147040 0.989130i $$-0.453025\pi$$
0.147040 + 0.989130i $$0.453025\pi$$
$$360$$ 0 0
$$361$$ −2.35925e6 −0.952808
$$362$$ 5.37263e6 2.15484
$$363$$ −131769. −0.0524864
$$364$$ −4.17374e6 −1.65109
$$365$$ 0 0
$$366$$ −172557. −0.0673334
$$367$$ −933545. −0.361801 −0.180901 0.983501i $$-0.557901\pi$$
−0.180901 + 0.983501i $$0.557901\pi$$
$$368$$ 2.70450e6 1.04104
$$369$$ 475812. 0.181916
$$370$$ 0 0
$$371$$ −4.51006e6 −1.70117
$$372$$ −1.62660e6 −0.609429
$$373$$ 4.48359e6 1.66860 0.834302 0.551307i $$-0.185871\pi$$
0.834302 + 0.551307i $$0.185871\pi$$
$$374$$ −1.46150e6 −0.540283
$$375$$ 0 0
$$376$$ −147111. −0.0536630
$$377$$ 2.05726e6 0.745478
$$378$$ 976623. 0.351558
$$379$$ 2.61145e6 0.933863 0.466931 0.884294i $$-0.345360\pi$$
0.466931 + 0.884294i $$0.345360\pi$$
$$380$$ 0 0
$$381$$ −24180.8 −0.00853411
$$382$$ 2.75624e6 0.966404
$$383$$ −2.87502e6 −1.00148 −0.500741 0.865597i $$-0.666939\pi$$
−0.500741 + 0.865597i $$0.666939\pi$$
$$384$$ −451358. −0.156204
$$385$$ 0 0
$$386$$ −4.90420e6 −1.67533
$$387$$ −134683. −0.0457127
$$388$$ −458164. −0.154505
$$389$$ 231232. 0.0774771 0.0387386 0.999249i $$-0.487666\pi$$
0.0387386 + 0.999249i $$0.487666\pi$$
$$390$$ 0 0
$$391$$ 4.34329e6 1.43674
$$392$$ −245423. −0.0806677
$$393$$ 2.52852e6 0.825818
$$394$$ −2.98794e6 −0.969685
$$395$$ 0 0
$$396$$ −343082. −0.109941
$$397$$ 2.01862e6 0.642804 0.321402 0.946943i $$-0.395846\pi$$
0.321402 + 0.946943i $$0.395846\pi$$
$$398$$ −2.73291e6 −0.864805
$$399$$ 503507. 0.158334
$$400$$ 0 0
$$401$$ −4.53630e6 −1.40877 −0.704386 0.709817i $$-0.748777\pi$$
−0.704386 + 0.709817i $$0.748777\pi$$
$$402$$ −3.64349e6 −1.12448
$$403$$ −3.76150e6 −1.15372
$$404$$ 1.20041e6 0.365913
$$405$$ 0 0
$$406$$ 3.78300e6 1.13899
$$407$$ 514847. 0.154061
$$408$$ −326642. −0.0971452
$$409$$ 629053. 0.185943 0.0929714 0.995669i $$-0.470363\pi$$
0.0929714 + 0.995669i $$0.470363\pi$$
$$410$$ 0 0
$$411$$ −2.45209e6 −0.716030
$$412$$ 1.14724e6 0.332973
$$413$$ 1.34988e6 0.389423
$$414$$ 1.95162e6 0.559622
$$415$$ 0 0
$$416$$ −6.05282e6 −1.71484
$$417$$ −2.91789e6 −0.821729
$$418$$ −338575. −0.0947794
$$419$$ 1.00216e6 0.278870 0.139435 0.990231i $$-0.455471\pi$$
0.139435 + 0.990231i $$0.455471\pi$$
$$420$$ 0 0
$$421$$ −1.43349e6 −0.394175 −0.197087 0.980386i $$-0.563148\pi$$
−0.197087 + 0.980386i $$0.563148\pi$$
$$422$$ 8.02261e6 2.19298
$$423$$ 484464. 0.131647
$$424$$ 677803. 0.183100
$$425$$ 0 0
$$426$$ −1.81550e6 −0.484699
$$427$$ −383340. −0.101745
$$428$$ −7.11614e6 −1.87774
$$429$$ −793376. −0.208131
$$430$$ 0 0
$$431$$ −4.80577e6 −1.24615 −0.623074 0.782163i $$-0.714116\pi$$
−0.623074 + 0.782163i $$0.714116\pi$$
$$432$$ 669818. 0.172682
$$433$$ 1.53507e6 0.393467 0.196733 0.980457i $$-0.436967\pi$$
0.196733 + 0.980457i $$0.436967\pi$$
$$434$$ −6.91687e6 −1.76273
$$435$$ 0 0
$$436$$ 7.12458e6 1.79491
$$437$$ 1.00618e6 0.252040
$$438$$ 1.69450e6 0.422042
$$439$$ 7.14894e6 1.77044 0.885218 0.465176i $$-0.154009\pi$$
0.885218 + 0.465176i $$0.154009\pi$$
$$440$$ 0 0
$$441$$ 808225. 0.197895
$$442$$ −8.79966e6 −2.14245
$$443$$ −712048. −0.172385 −0.0861926 0.996278i $$-0.527470\pi$$
−0.0861926 + 0.996278i $$0.527470\pi$$
$$444$$ 1.34049e6 0.322705
$$445$$ 0 0
$$446$$ −662773. −0.157771
$$447$$ 2.92689e6 0.692849
$$448$$ −6.31828e6 −1.48732
$$449$$ −3.18417e6 −0.745385 −0.372693 0.927955i $$-0.621565\pi$$
−0.372693 + 0.927955i $$0.621565\pi$$
$$450$$ 0 0
$$451$$ −710782. −0.164549
$$452$$ 4.93311e6 1.13573
$$453$$ 3.59229e6 0.822481
$$454$$ 4.29163e6 0.977197
$$455$$ 0 0
$$456$$ −75670.4 −0.0170418
$$457$$ −8.26972e6 −1.85225 −0.926127 0.377213i $$-0.876883\pi$$
−0.926127 + 0.377213i $$0.876883\pi$$
$$458$$ −6.68372e6 −1.48886
$$459$$ 1.07570e6 0.238318
$$460$$ 0 0
$$461$$ 7.66057e6 1.67884 0.839419 0.543485i $$-0.182896\pi$$
0.839419 + 0.543485i $$0.182896\pi$$
$$462$$ −1.45891e6 −0.317997
$$463$$ −203047. −0.0440194 −0.0220097 0.999758i $$-0.507006\pi$$
−0.0220097 + 0.999758i $$0.507006\pi$$
$$464$$ 2.59458e6 0.559463
$$465$$ 0 0
$$466$$ −2.77476e6 −0.591917
$$467$$ −3.28726e6 −0.697497 −0.348748 0.937216i $$-0.613393\pi$$
−0.348748 + 0.937216i $$0.613393\pi$$
$$468$$ −2.06568e6 −0.435962
$$469$$ −8.09411e6 −1.69917
$$470$$ 0 0
$$471$$ −745846. −0.154916
$$472$$ −202870. −0.0419144
$$473$$ 201194. 0.0413487
$$474$$ 6.42517e6 1.31353
$$475$$ 0 0
$$476$$ −8.45348e6 −1.71009
$$477$$ −2.23214e6 −0.449185
$$478$$ −804211. −0.160990
$$479$$ −3.08986e6 −0.615319 −0.307659 0.951497i $$-0.599546\pi$$
−0.307659 + 0.951497i $$0.599546\pi$$
$$480$$ 0 0
$$481$$ 3.09987e6 0.610916
$$482$$ −8.21074e6 −1.60977
$$483$$ 4.33557e6 0.845627
$$484$$ 512505. 0.0994454
$$485$$ 0 0
$$486$$ 483354. 0.0928271
$$487$$ −2.37051e6 −0.452918 −0.226459 0.974021i $$-0.572715\pi$$
−0.226459 + 0.974021i $$0.572715\pi$$
$$488$$ 57611.0 0.0109510
$$489$$ −2.80505e6 −0.530480
$$490$$ 0 0
$$491$$ −7.56399e6 −1.41595 −0.707974 0.706239i $$-0.750391\pi$$
−0.707974 + 0.706239i $$0.750391\pi$$
$$492$$ −1.85064e6 −0.344674
$$493$$ 4.16676e6 0.772114
$$494$$ −2.03855e6 −0.375840
$$495$$ 0 0
$$496$$ −4.74395e6 −0.865836
$$497$$ −4.03318e6 −0.732413
$$498$$ −6.67448e6 −1.20599
$$499$$ −1.56759e6 −0.281826 −0.140913 0.990022i $$-0.545004\pi$$
−0.140913 + 0.990022i $$0.545004\pi$$
$$500$$ 0 0
$$501$$ −1.16999e6 −0.208251
$$502$$ −8.10084e6 −1.43473
$$503$$ 2.34418e6 0.413115 0.206557 0.978435i $$-0.433774\pi$$
0.206557 + 0.978435i $$0.433774\pi$$
$$504$$ −326061. −0.0571772
$$505$$ 0 0
$$506$$ −2.91538e6 −0.506197
$$507$$ −1.43525e6 −0.247975
$$508$$ 94049.3 0.0161695
$$509$$ 8.79759e6 1.50511 0.752557 0.658527i $$-0.228820\pi$$
0.752557 + 0.658527i $$0.228820\pi$$
$$510$$ 0 0
$$511$$ 3.76436e6 0.637734
$$512$$ −8.35690e6 −1.40887
$$513$$ 249198. 0.0418071
$$514$$ −9.43397e6 −1.57502
$$515$$ 0 0
$$516$$ 523841. 0.0866114
$$517$$ −723706. −0.119079
$$518$$ 5.70023e6 0.933400
$$519$$ 6.70068e6 1.09194
$$520$$ 0 0
$$521$$ −3.43077e6 −0.553729 −0.276865 0.960909i $$-0.589295\pi$$
−0.276865 + 0.960909i $$0.589295\pi$$
$$522$$ 1.87230e6 0.300745
$$523$$ 1.24453e7 1.98953 0.994763 0.102212i $$-0.0325920\pi$$
0.994763 + 0.102212i $$0.0325920\pi$$
$$524$$ −9.83446e6 −1.56467
$$525$$ 0 0
$$526$$ −8.43458e6 −1.32923
$$527$$ −7.61854e6 −1.19494
$$528$$ −1.00059e6 −0.156197
$$529$$ 2.22758e6 0.346095
$$530$$ 0 0
$$531$$ 668090. 0.102825
$$532$$ −1.95835e6 −0.299993
$$533$$ −4.27959e6 −0.652505
$$534$$ 6.35331e6 0.964155
$$535$$ 0 0
$$536$$ 1.21644e6 0.182885
$$537$$ −2.78972e6 −0.417469
$$538$$ 1.83574e7 2.73435
$$539$$ −1.20735e6 −0.179003
$$540$$ 0 0
$$541$$ 4.87769e6 0.716508 0.358254 0.933624i $$-0.383372\pi$$
0.358254 + 0.933624i $$0.383372\pi$$
$$542$$ −1.77814e7 −2.59997
$$543$$ 5.90713e6 0.859759
$$544$$ −1.22594e7 −1.77611
$$545$$ 0 0
$$546$$ −8.78402e6 −1.26099
$$547$$ 3.49347e6 0.499217 0.249608 0.968347i $$-0.419698\pi$$
0.249608 + 0.968347i $$0.419698\pi$$
$$548$$ 9.53719e6 1.35665
$$549$$ −189724. −0.0268653
$$550$$ 0 0
$$551$$ 965280. 0.135449
$$552$$ −651580. −0.0910164
$$553$$ 1.42737e7 1.98483
$$554$$ 8.52128e6 1.17959
$$555$$ 0 0
$$556$$ 1.13489e7 1.55692
$$557$$ 1.34838e7 1.84151 0.920757 0.390137i $$-0.127572\pi$$
0.920757 + 0.390137i $$0.127572\pi$$
$$558$$ −3.42333e6 −0.465439
$$559$$ 1.21138e6 0.163965
$$560$$ 0 0
$$561$$ −1.60690e6 −0.215567
$$562$$ 1.10035e7 1.46956
$$563$$ 2.07215e6 0.275519 0.137759 0.990466i $$-0.456010\pi$$
0.137759 + 0.990466i $$0.456010\pi$$
$$564$$ −1.88429e6 −0.249430
$$565$$ 0 0
$$566$$ 6.77847e6 0.889387
$$567$$ 1.07378e6 0.140268
$$568$$ 606133. 0.0788310
$$569$$ 1.11686e7 1.44617 0.723085 0.690759i $$-0.242723\pi$$
0.723085 + 0.690759i $$0.242723\pi$$
$$570$$ 0 0
$$571$$ 1.04065e7 1.33572 0.667861 0.744286i $$-0.267210\pi$$
0.667861 + 0.744286i $$0.267210\pi$$
$$572$$ 3.08577e6 0.394343
$$573$$ 3.03045e6 0.385585
$$574$$ −7.86956e6 −0.996944
$$575$$ 0 0
$$576$$ −3.12707e6 −0.392718
$$577$$ 4.15724e6 0.519835 0.259918 0.965631i $$-0.416305\pi$$
0.259918 + 0.965631i $$0.416305\pi$$
$$578$$ −6.20037e6 −0.771965
$$579$$ −5.39209e6 −0.668438
$$580$$ 0 0
$$581$$ −1.48275e7 −1.82233
$$582$$ −964249. −0.118000
$$583$$ 3.33443e6 0.406303
$$584$$ −565734. −0.0686405
$$585$$ 0 0
$$586$$ −1.07482e7 −1.29298
$$587$$ −8.38623e6 −1.00455 −0.502275 0.864708i $$-0.667504\pi$$
−0.502275 + 0.864708i $$0.667504\pi$$
$$588$$ −3.14353e6 −0.374951
$$589$$ −1.76493e6 −0.209623
$$590$$ 0 0
$$591$$ −3.28519e6 −0.386894
$$592$$ 3.90951e6 0.458477
$$593$$ −2.10957e6 −0.246353 −0.123177 0.992385i $$-0.539308\pi$$
−0.123177 + 0.992385i $$0.539308\pi$$
$$594$$ −722048. −0.0839653
$$595$$ 0 0
$$596$$ −1.13839e7 −1.31273
$$597$$ −3.00480e6 −0.345048
$$598$$ −1.75534e7 −2.00728
$$599$$ 3.77702e6 0.430113 0.215057 0.976602i $$-0.431006\pi$$
0.215057 + 0.976602i $$0.431006\pi$$
$$600$$ 0 0
$$601$$ 7.88038e6 0.889940 0.444970 0.895545i $$-0.353214\pi$$
0.444970 + 0.895545i $$0.353214\pi$$
$$602$$ 2.22756e6 0.250517
$$603$$ −4.00597e6 −0.448657
$$604$$ −1.39719e7 −1.55835
$$605$$ 0 0
$$606$$ 2.52638e6 0.279459
$$607$$ 5.43393e6 0.598608 0.299304 0.954158i $$-0.403246\pi$$
0.299304 + 0.954158i $$0.403246\pi$$
$$608$$ −2.84003e6 −0.311576
$$609$$ 4.15936e6 0.454447
$$610$$ 0 0
$$611$$ −4.35741e6 −0.472199
$$612$$ −4.18383e6 −0.451539
$$613$$ −1.36825e7 −1.47067 −0.735333 0.677706i $$-0.762974\pi$$
−0.735333 + 0.677706i $$0.762974\pi$$
$$614$$ 1.13153e6 0.121128
$$615$$ 0 0
$$616$$ 487079. 0.0517187
$$617$$ −1.24288e7 −1.31437 −0.657184 0.753730i $$-0.728253\pi$$
−0.657184 + 0.753730i $$0.728253\pi$$
$$618$$ 2.41446e6 0.254302
$$619$$ 1.29365e7 1.35704 0.678518 0.734583i $$-0.262622\pi$$
0.678518 + 0.734583i $$0.262622\pi$$
$$620$$ 0 0
$$621$$ 2.14578e6 0.223283
$$622$$ −9.62988e6 −0.998033
$$623$$ 1.41140e7 1.45690
$$624$$ −6.02453e6 −0.619386
$$625$$ 0 0
$$626$$ 7.53032e6 0.768029
$$627$$ −372258. −0.0378160
$$628$$ 2.90091e6 0.293518
$$629$$ 6.27848e6 0.632744
$$630$$ 0 0
$$631$$ 572856. 0.0572759 0.0286379 0.999590i $$-0.490883\pi$$
0.0286379 + 0.999590i $$0.490883\pi$$
$$632$$ −2.14514e6 −0.213631
$$633$$ 8.82075e6 0.874976
$$634$$ −1.60856e7 −1.58933
$$635$$ 0 0
$$636$$ 8.68172e6 0.851066
$$637$$ −7.26940e6 −0.709823
$$638$$ −2.79689e6 −0.272034
$$639$$ −1.99612e6 −0.193390
$$640$$ 0 0
$$641$$ 2.13431e6 0.205170 0.102585 0.994724i $$-0.467289\pi$$
0.102585 + 0.994724i $$0.467289\pi$$
$$642$$ −1.49766e7 −1.43409
$$643$$ −4.15866e6 −0.396667 −0.198333 0.980135i $$-0.563553\pi$$
−0.198333 + 0.980135i $$0.563553\pi$$
$$644$$ −1.68629e7 −1.60220
$$645$$ 0 0
$$646$$ −4.12887e6 −0.389269
$$647$$ 3.44527e6 0.323565 0.161783 0.986826i $$-0.448276\pi$$
0.161783 + 0.986826i $$0.448276\pi$$
$$648$$ −161375. −0.0150973
$$649$$ −998012. −0.0930088
$$650$$ 0 0
$$651$$ −7.60500e6 −0.703310
$$652$$ 1.09100e7 1.00509
$$653$$ 2.88090e6 0.264390 0.132195 0.991224i $$-0.457798\pi$$
0.132195 + 0.991224i $$0.457798\pi$$
$$654$$ 1.49943e7 1.37083
$$655$$ 0 0
$$656$$ −5.39735e6 −0.489689
$$657$$ 1.86307e6 0.168390
$$658$$ −8.01265e6 −0.721459
$$659$$ 1.31524e7 1.17975 0.589876 0.807494i $$-0.299177\pi$$
0.589876 + 0.807494i $$0.299177\pi$$
$$660$$ 0 0
$$661$$ −4.14045e6 −0.368591 −0.184295 0.982871i $$-0.559000\pi$$
−0.184295 + 0.982871i $$0.559000\pi$$
$$662$$ −1.62389e7 −1.44017
$$663$$ −9.67510e6 −0.854814
$$664$$ 2.22838e6 0.196141
$$665$$ 0 0
$$666$$ 2.82118e6 0.246459
$$667$$ 8.31179e6 0.723402
$$668$$ 4.55057e6 0.394570
$$669$$ −728710. −0.0629491
$$670$$ 0 0
$$671$$ 283415. 0.0243006
$$672$$ −1.22376e7 −1.04538
$$673$$ 8.46081e6 0.720070 0.360035 0.932939i $$-0.382765\pi$$
0.360035 + 0.932939i $$0.382765\pi$$
$$674$$ −2.68860e7 −2.27969
$$675$$ 0 0
$$676$$ 5.58229e6 0.469835
$$677$$ 7.85095e6 0.658340 0.329170 0.944271i $$-0.393231\pi$$
0.329170 + 0.944271i $$0.393231\pi$$
$$678$$ 1.03822e7 0.867391
$$679$$ −2.14210e6 −0.178306
$$680$$ 0 0
$$681$$ 4.71858e6 0.389891
$$682$$ 5.11386e6 0.421005
$$683$$ 1.84419e7 1.51271 0.756354 0.654163i $$-0.226979\pi$$
0.756354 + 0.654163i $$0.226979\pi$$
$$684$$ −969234. −0.0792116
$$685$$ 0 0
$$686$$ 9.14853e6 0.742234
$$687$$ −7.34866e6 −0.594040
$$688$$ 1.52777e6 0.123052
$$689$$ 2.00765e7 1.61116
$$690$$ 0 0
$$691$$ −1.74029e7 −1.38652 −0.693259 0.720688i $$-0.743826\pi$$
−0.693259 + 0.720688i $$0.743826\pi$$
$$692$$ −2.60618e7 −2.06890
$$693$$ −1.60405e6 −0.126877
$$694$$ 1.82627e7 1.43935
$$695$$ 0 0
$$696$$ −625097. −0.0489130
$$697$$ −8.66787e6 −0.675819
$$698$$ −9.38353e6 −0.729000
$$699$$ −3.05081e6 −0.236169
$$700$$ 0 0
$$701$$ −4.85133e6 −0.372877 −0.186438 0.982467i $$-0.559695\pi$$
−0.186438 + 0.982467i $$0.559695\pi$$
$$702$$ −4.34742e6 −0.332958
$$703$$ 1.45448e6 0.110999
$$704$$ 4.67130e6 0.355227
$$705$$ 0 0
$$706$$ 9.60010e6 0.724877
$$707$$ 5.61242e6 0.422281
$$708$$ −2.59848e6 −0.194822
$$709$$ −7.10939e6 −0.531150 −0.265575 0.964090i $$-0.585562\pi$$
−0.265575 + 0.964090i $$0.585562\pi$$
$$710$$ 0 0
$$711$$ 7.06438e6 0.524083
$$712$$ −2.12115e6 −0.156809
$$713$$ −1.51973e7 −1.11955
$$714$$ −1.77911e7 −1.30604
$$715$$ 0 0
$$716$$ 1.08504e7 0.790974
$$717$$ −884218. −0.0642335
$$718$$ −5.87835e6 −0.425544
$$719$$ 1.99693e6 0.144059 0.0720297 0.997402i $$-0.477052\pi$$
0.0720297 + 0.997402i $$0.477052\pi$$
$$720$$ 0 0
$$721$$ 5.36379e6 0.384267
$$722$$ 1.93120e7 1.37874
$$723$$ −9.02760e6 −0.642283
$$724$$ −2.29753e7 −1.62898
$$725$$ 0 0
$$726$$ 1.07861e6 0.0759495
$$727$$ −1.47012e7 −1.03161 −0.515806 0.856705i $$-0.672507\pi$$
−0.515806 + 0.856705i $$0.672507\pi$$
$$728$$ 2.93268e6 0.205086
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 2.45353e6 0.169823
$$732$$ 737917. 0.0509014
$$733$$ 1.03899e7 0.714249 0.357125 0.934057i $$-0.383757\pi$$
0.357125 + 0.934057i $$0.383757\pi$$
$$734$$ 7.64166e6 0.523538
$$735$$ 0 0
$$736$$ −2.44548e7 −1.66406
$$737$$ 5.98422e6 0.405825
$$738$$ −3.89483e6 −0.263238
$$739$$ 1.40217e7 0.944470 0.472235 0.881473i $$-0.343447\pi$$
0.472235 + 0.881473i $$0.343447\pi$$
$$740$$ 0 0
$$741$$ −2.24135e6 −0.149956
$$742$$ 3.69178e7 2.46165
$$743$$ 1.62282e7 1.07845 0.539224 0.842162i $$-0.318717\pi$$
0.539224 + 0.842162i $$0.318717\pi$$
$$744$$ 1.14293e6 0.0756986
$$745$$ 0 0
$$746$$ −3.67010e7 −2.41452
$$747$$ −7.33849e6 −0.481178
$$748$$ 6.24992e6 0.408433
$$749$$ −3.32708e7 −2.16700
$$750$$ 0 0
$$751$$ −2.21572e7 −1.43356 −0.716779 0.697300i $$-0.754385\pi$$
−0.716779 + 0.697300i $$0.754385\pi$$
$$752$$ −5.49549e6 −0.354374
$$753$$ −8.90676e6 −0.572443
$$754$$ −1.68400e7 −1.07873
$$755$$ 0 0
$$756$$ −4.17640e6 −0.265765
$$757$$ 2.95113e7 1.87176 0.935878 0.352325i $$-0.114609\pi$$
0.935878 + 0.352325i $$0.114609\pi$$
$$758$$ −2.13764e7 −1.35133
$$759$$ −3.20542e6 −0.201967
$$760$$ 0 0
$$761$$ 1.82482e7 1.14224 0.571122 0.820865i $$-0.306508\pi$$
0.571122 + 0.820865i $$0.306508\pi$$
$$762$$ 197936. 0.0123491
$$763$$ 3.33103e7 2.07141
$$764$$ −1.17867e7 −0.730564
$$765$$ 0 0
$$766$$ 2.35339e7 1.44918
$$767$$ −6.00899e6 −0.368819
$$768$$ −7.42380e6 −0.454175
$$769$$ −1.89729e7 −1.15696 −0.578480 0.815697i $$-0.696354\pi$$
−0.578480 + 0.815697i $$0.696354\pi$$
$$770$$ 0 0
$$771$$ −1.03725e7 −0.628417
$$772$$ 2.09721e7 1.26648
$$773$$ 9.39706e6 0.565644 0.282822 0.959172i $$-0.408729\pi$$
0.282822 + 0.959172i $$0.408729\pi$$
$$774$$ 1.10247e6 0.0661477
$$775$$ 0 0
$$776$$ 321930. 0.0191914
$$777$$ 6.26732e6 0.372417
$$778$$ −1.89278e6 −0.112112
$$779$$ −2.00802e6 −0.118556
$$780$$ 0 0
$$781$$ 2.98185e6 0.174928
$$782$$ −3.55527e7 −2.07900
$$783$$ 2.05857e6 0.119994
$$784$$ −9.16804e6 −0.532705
$$785$$ 0 0
$$786$$ −2.06975e7 −1.19498
$$787$$ 2.29536e7 1.32103 0.660517 0.750811i $$-0.270337\pi$$
0.660517 + 0.750811i $$0.270337\pi$$
$$788$$ 1.27775e7 0.733044
$$789$$ −9.27370e6 −0.530347
$$790$$ 0 0
$$791$$ 2.30643e7 1.31069
$$792$$ 241067. 0.0136560
$$793$$ 1.70643e6 0.0963620
$$794$$ −1.65237e7 −0.930157
$$795$$ 0 0
$$796$$ 1.16869e7 0.653759
$$797$$ 1.05979e7 0.590984 0.295492 0.955345i $$-0.404516\pi$$
0.295492 + 0.955345i $$0.404516\pi$$
$$798$$ −4.12153e6 −0.229114
$$799$$ −8.82549e6 −0.489071
$$800$$ 0 0
$$801$$ 6.98538e6 0.384688
$$802$$ 3.71325e7 2.03854
$$803$$ −2.78311e6 −0.152315
$$804$$ 1.55809e7 0.850065
$$805$$ 0 0
$$806$$ 3.07903e7 1.66946
$$807$$ 2.01836e7 1.09098
$$808$$ −843473. −0.0454509
$$809$$ −1.95635e7 −1.05093 −0.525467 0.850814i $$-0.676109\pi$$
−0.525467 + 0.850814i $$0.676109\pi$$
$$810$$ 0 0
$$811$$ −3.23306e7 −1.72609 −0.863043 0.505131i $$-0.831444\pi$$
−0.863043 + 0.505131i $$0.831444\pi$$
$$812$$ −1.61775e7 −0.861036
$$813$$ −1.95504e7 −1.03736
$$814$$ −4.21436e6 −0.222931
$$815$$ 0 0
$$816$$ −1.22021e7 −0.641517
$$817$$ 568389. 0.0297914
$$818$$ −5.14921e6 −0.269065
$$819$$ −9.65790e6 −0.503122
$$820$$ 0 0
$$821$$ 3.15924e7 1.63578 0.817890 0.575375i $$-0.195144\pi$$
0.817890 + 0.575375i $$0.195144\pi$$
$$822$$ 2.00719e7 1.03612
$$823$$ −1.52405e7 −0.784333 −0.392167 0.919894i $$-0.628274\pi$$
−0.392167 + 0.919894i $$0.628274\pi$$
$$824$$ −806107. −0.0413594
$$825$$ 0 0
$$826$$ −1.10497e7 −0.563507
$$827$$ −2.87105e7 −1.45975 −0.729873 0.683583i $$-0.760421\pi$$
−0.729873 + 0.683583i $$0.760421\pi$$
$$828$$ −8.34584e6 −0.423052
$$829$$ −2.45816e7 −1.24229 −0.621146 0.783695i $$-0.713333\pi$$
−0.621146 + 0.783695i $$0.713333\pi$$
$$830$$ 0 0
$$831$$ 9.36903e6 0.470643
$$832$$ 2.81257e7 1.40862
$$833$$ −1.47234e7 −0.735185
$$834$$ 2.38848e7 1.18907
$$835$$ 0 0
$$836$$ 1.44787e6 0.0716495
$$837$$ −3.76390e6 −0.185705
$$838$$ −8.20334e6 −0.403534
$$839$$ −1.59309e6 −0.0781333 −0.0390667 0.999237i $$-0.512438\pi$$
−0.0390667 + 0.999237i $$0.512438\pi$$
$$840$$ 0 0
$$841$$ −1.25372e7 −0.611238
$$842$$ 1.17340e7 0.570383
$$843$$ 1.20981e7 0.586341
$$844$$ −3.43076e7 −1.65781
$$845$$ 0 0
$$846$$ −3.96565e6 −0.190497
$$847$$ 2.39617e6 0.114765
$$848$$ 2.53201e7 1.20914
$$849$$ 7.45283e6 0.354856
$$850$$ 0 0
$$851$$ 1.25242e7 0.592825
$$852$$ 7.76373e6 0.366414
$$853$$ −9.71355e6 −0.457094 −0.228547 0.973533i $$-0.573397\pi$$
−0.228547 + 0.973533i $$0.573397\pi$$
$$854$$ 3.13789e6 0.147229
$$855$$ 0 0
$$856$$ 5.00017e6 0.233239
$$857$$ 2.81098e7 1.30739 0.653696 0.756758i $$-0.273218\pi$$
0.653696 + 0.756758i $$0.273218\pi$$
$$858$$ 6.49429e6 0.301171
$$859$$ 1.96952e7 0.910703 0.455352 0.890312i $$-0.349514\pi$$
0.455352 + 0.890312i $$0.349514\pi$$
$$860$$ 0 0
$$861$$ −8.65247e6 −0.397770
$$862$$ 3.93383e7 1.80322
$$863$$ 9.07936e6 0.414981 0.207490 0.978237i $$-0.433470\pi$$
0.207490 + 0.978237i $$0.433470\pi$$
$$864$$ −6.05667e6 −0.276026
$$865$$ 0 0
$$866$$ −1.25655e7 −0.569359
$$867$$ −6.81722e6 −0.308006
$$868$$ 2.95791e7 1.33255
$$869$$ −1.05530e7 −0.474051
$$870$$ 0 0
$$871$$ 3.60308e7 1.60927
$$872$$ −5.00609e6 −0.222950
$$873$$ −1.06018e6 −0.0470807
$$874$$ −8.23619e6 −0.364710
$$875$$ 0 0
$$876$$ −7.24628e6 −0.319047
$$877$$ −1.35513e7 −0.594954 −0.297477 0.954729i $$-0.596145\pi$$
−0.297477 + 0.954729i $$0.596145\pi$$
$$878$$ −5.85187e7 −2.56188
$$879$$ −1.18175e7 −0.515884
$$880$$ 0 0
$$881$$ 3.41684e7 1.48315 0.741574 0.670871i $$-0.234080\pi$$
0.741574 + 0.670871i $$0.234080\pi$$
$$882$$ −6.61584e6 −0.286361
$$883$$ 4.27723e7 1.84612 0.923061 0.384654i $$-0.125679\pi$$
0.923061 + 0.384654i $$0.125679\pi$$
$$884$$ 3.76305e7 1.61961
$$885$$ 0 0
$$886$$ 5.82857e6 0.249447
$$887$$ 1.84459e7 0.787210 0.393605 0.919280i $$-0.371228\pi$$
0.393605 + 0.919280i $$0.371228\pi$$
$$888$$ −941896. −0.0400839
$$889$$ 439719. 0.0186604
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ 2.83426e6 0.119269
$$893$$ −2.04453e6 −0.0857955
$$894$$ −2.39585e7 −1.00257
$$895$$ 0 0
$$896$$ 8.20777e6 0.341551
$$897$$ −1.92997e7 −0.800885
$$898$$ 2.60645e7 1.07860
$$899$$ −1.45797e7 −0.601656
$$900$$ 0 0
$$901$$ 4.06628e7 1.66873
$$902$$ 5.81821e6 0.238107
$$903$$ 2.44917e6 0.0999537
$$904$$ −3.46626e6 −0.141072
$$905$$ 0 0
$$906$$ −2.94052e7 −1.19016
$$907$$ 4.12614e7 1.66543 0.832713 0.553704i $$-0.186786\pi$$
0.832713 + 0.553704i $$0.186786\pi$$
$$908$$ −1.83525e7 −0.738723
$$909$$ 2.77772e6 0.111501
$$910$$ 0 0
$$911$$ 2.05996e7 0.822360 0.411180 0.911554i $$-0.365117\pi$$
0.411180 + 0.911554i $$0.365117\pi$$
$$912$$ −2.82675e6 −0.112538
$$913$$ 1.09624e7 0.435241
$$914$$ 6.76930e7 2.68027
$$915$$ 0 0
$$916$$ 2.85820e7 1.12552
$$917$$ −4.59801e7 −1.80570
$$918$$ −8.80526e6 −0.344854
$$919$$ 5.10704e6 0.199471 0.0997356 0.995014i $$-0.468200\pi$$
0.0997356 + 0.995014i $$0.468200\pi$$
$$920$$ 0 0
$$921$$ 1.24410e6 0.0483290
$$922$$ −6.27067e7 −2.42933
$$923$$ 1.79536e7 0.693661
$$924$$ 6.23881e6 0.240393
$$925$$ 0 0
$$926$$ 1.66207e6 0.0636975
$$927$$ 2.65467e6 0.101464
$$928$$ −2.34609e7 −0.894281
$$929$$ −213931. −0.00813271 −0.00406636 0.999992i $$-0.501294\pi$$
−0.00406636 + 0.999992i $$0.501294\pi$$
$$930$$ 0 0
$$931$$ −3.41086e6 −0.128970
$$932$$ 1.18659e7 0.447466
$$933$$ −1.05879e7 −0.398205
$$934$$ 2.69084e7 1.00930
$$935$$ 0 0
$$936$$ 1.45146e6 0.0541520
$$937$$ 1.71430e7 0.637878 0.318939 0.947775i $$-0.396674\pi$$
0.318939 + 0.947775i $$0.396674\pi$$
$$938$$ 6.62555e7 2.45875
$$939$$ 8.27948e6 0.306436
$$940$$ 0 0
$$941$$ 3.27424e7 1.20542 0.602708 0.797962i $$-0.294088\pi$$
0.602708 + 0.797962i $$0.294088\pi$$
$$942$$ 6.10523e6 0.224169
$$943$$ −1.72905e7 −0.633183
$$944$$ −7.57844e6 −0.276789
$$945$$ 0 0
$$946$$ −1.64690e6 −0.0598329
$$947$$ −2.30829e7 −0.836403 −0.418201 0.908354i $$-0.637339\pi$$
−0.418201 + 0.908354i $$0.637339\pi$$
$$948$$ −2.74763e7 −0.992975
$$949$$ −1.67570e7 −0.603991
$$950$$ 0 0
$$951$$ −1.76859e7 −0.634128
$$952$$ 5.93985e6 0.212414
$$953$$ 2.57216e7 0.917414 0.458707 0.888587i $$-0.348313\pi$$
0.458707 + 0.888587i $$0.348313\pi$$
$$954$$ 1.82715e7 0.649984
$$955$$ 0 0
$$956$$ 3.43910e6 0.121703
$$957$$ −3.07514e6 −0.108539
$$958$$ 2.52925e7 0.890385
$$959$$ 4.45902e7 1.56564
$$960$$ 0 0
$$961$$ −1.97158e6 −0.0688663
$$962$$ −2.53745e7 −0.884014
$$963$$ −1.64665e7 −0.572185
$$964$$ 3.51121e7 1.21693
$$965$$ 0 0
$$966$$ −3.54895e7 −1.22365
$$967$$ 1.86464e6 0.0641253 0.0320627 0.999486i $$-0.489792\pi$$
0.0320627 + 0.999486i $$0.489792\pi$$
$$968$$ −360113. −0.0123524
$$969$$ −4.53963e6 −0.155314
$$970$$ 0 0
$$971$$ 2.90344e7 0.988245 0.494122 0.869392i $$-0.335489\pi$$
0.494122 + 0.869392i $$0.335489\pi$$
$$972$$ −2.06700e6 −0.0701737
$$973$$ 5.30606e7 1.79676
$$974$$ 1.94042e7 0.655387
$$975$$ 0 0
$$976$$ 2.15212e6 0.0723174
$$977$$ −5.86267e6 −0.196498 −0.0982492 0.995162i $$-0.531324\pi$$
−0.0982492 + 0.995162i $$0.531324\pi$$
$$978$$ 2.29612e7 0.767621
$$979$$ −1.04349e7 −0.347963
$$980$$ 0 0
$$981$$ 1.64860e7 0.546945
$$982$$ 6.19161e7 2.04892
$$983$$ −2.93292e7 −0.968093 −0.484047 0.875042i $$-0.660833\pi$$
−0.484047 + 0.875042i $$0.660833\pi$$
$$984$$ 1.30035e6 0.0428128
$$985$$ 0 0
$$986$$ −3.41076e7 −1.11727
$$987$$ −8.80980e6 −0.287855
$$988$$ 8.71756e6 0.284120
$$989$$ 4.89426e6 0.159109
$$990$$ 0 0
$$991$$ 2.75751e7 0.891935 0.445968 0.895049i $$-0.352860\pi$$
0.445968 + 0.895049i $$0.352860\pi$$
$$992$$ 4.28960e7 1.38401
$$993$$ −1.78545e7 −0.574611
$$994$$ 3.30141e7 1.05982
$$995$$ 0 0
$$996$$ 2.85425e7 0.911682
$$997$$ 1.69502e7 0.540052 0.270026 0.962853i $$-0.412968\pi$$
0.270026 + 0.962853i $$0.412968\pi$$
$$998$$ 1.28317e7 0.407811
$$999$$ 3.10185e6 0.0983347
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 825.6.a.s.1.2 yes 9
5.4 even 2 825.6.a.r.1.8 9

By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.8 9 5.4 even 2
825.6.a.s.1.2 yes 9 1.1 even 1 trivial