# Properties

 Label 825.6.a.s.1.1 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Learn more

## 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 1.1 Root $$-11.1064$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-11.1064 q^{2} -9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} -76.1461 q^{7} -659.202 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-11.1064 q^{2} -9.00000 q^{3} +91.3531 q^{4} +99.9580 q^{6} -76.1461 q^{7} -659.202 q^{8} +81.0000 q^{9} -121.000 q^{11} -822.178 q^{12} +470.783 q^{13} +845.713 q^{14} +4398.09 q^{16} +494.756 q^{17} -899.622 q^{18} -44.8076 q^{19} +685.315 q^{21} +1343.88 q^{22} -4242.60 q^{23} +5932.82 q^{24} -5228.73 q^{26} -729.000 q^{27} -6956.19 q^{28} -786.920 q^{29} +5594.34 q^{31} -27752.7 q^{32} +1089.00 q^{33} -5494.98 q^{34} +7399.60 q^{36} -6453.89 q^{37} +497.653 q^{38} -4237.05 q^{39} +7703.22 q^{41} -7611.42 q^{42} +2425.31 q^{43} -11053.7 q^{44} +47120.2 q^{46} -18816.2 q^{47} -39582.8 q^{48} -11008.8 q^{49} -4452.80 q^{51} +43007.5 q^{52} +34459.8 q^{53} +8096.60 q^{54} +50195.7 q^{56} +403.268 q^{57} +8739.88 q^{58} +27937.1 q^{59} +14661.0 q^{61} -62133.2 q^{62} -6167.84 q^{63} +167495. q^{64} -12094.9 q^{66} +47205.0 q^{67} +45197.5 q^{68} +38183.4 q^{69} -39501.5 q^{71} -53395.4 q^{72} -26993.8 q^{73} +71679.8 q^{74} -4093.31 q^{76} +9213.68 q^{77} +47058.6 q^{78} -12293.2 q^{79} +6561.00 q^{81} -85555.4 q^{82} +70308.6 q^{83} +62605.7 q^{84} -26936.6 q^{86} +7082.28 q^{87} +79763.4 q^{88} -143142. q^{89} -35848.3 q^{91} -387574. q^{92} -50349.1 q^{93} +208981. q^{94} +249774. q^{96} -91322.4 q^{97} +122268. q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −11.1064 −1.96336 −0.981680 0.190536i $$-0.938978\pi$$
−0.981680 + 0.190536i $$0.938978\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 91.3531 2.85478
$$5$$ 0 0
$$6$$ 99.9580 1.13355
$$7$$ −76.1461 −0.587358 −0.293679 0.955904i $$-0.594880\pi$$
−0.293679 + 0.955904i $$0.594880\pi$$
$$8$$ −659.202 −3.64161
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ −822.178 −1.64821
$$13$$ 470.783 0.772614 0.386307 0.922370i $$-0.373750\pi$$
0.386307 + 0.922370i $$0.373750\pi$$
$$14$$ 845.713 1.15320
$$15$$ 0 0
$$16$$ 4398.09 4.29501
$$17$$ 494.756 0.415211 0.207605 0.978213i $$-0.433433\pi$$
0.207605 + 0.978213i $$0.433433\pi$$
$$18$$ −899.622 −0.654454
$$19$$ −44.8076 −0.0284752 −0.0142376 0.999899i $$-0.504532\pi$$
−0.0142376 + 0.999899i $$0.504532\pi$$
$$20$$ 0 0
$$21$$ 685.315 0.339111
$$22$$ 1343.88 0.591975
$$23$$ −4242.60 −1.67229 −0.836146 0.548507i $$-0.815196\pi$$
−0.836146 + 0.548507i $$0.815196\pi$$
$$24$$ 5932.82 2.10249
$$25$$ 0 0
$$26$$ −5228.73 −1.51692
$$27$$ −729.000 −0.192450
$$28$$ −6956.19 −1.67678
$$29$$ −786.920 −0.173754 −0.0868772 0.996219i $$-0.527689\pi$$
−0.0868772 + 0.996219i $$0.527689\pi$$
$$30$$ 0 0
$$31$$ 5594.34 1.04555 0.522775 0.852471i $$-0.324897\pi$$
0.522775 + 0.852471i $$0.324897\pi$$
$$32$$ −27752.7 −4.79104
$$33$$ 1089.00 0.174078
$$34$$ −5494.98 −0.815209
$$35$$ 0 0
$$36$$ 7399.60 0.951595
$$37$$ −6453.89 −0.775028 −0.387514 0.921864i $$-0.626666\pi$$
−0.387514 + 0.921864i $$0.626666\pi$$
$$38$$ 497.653 0.0559072
$$39$$ −4237.05 −0.446069
$$40$$ 0 0
$$41$$ 7703.22 0.715670 0.357835 0.933785i $$-0.383515\pi$$
0.357835 + 0.933785i $$0.383515\pi$$
$$42$$ −7611.42 −0.665798
$$43$$ 2425.31 0.200031 0.100015 0.994986i $$-0.468111\pi$$
0.100015 + 0.994986i $$0.468111\pi$$
$$44$$ −11053.7 −0.860750
$$45$$ 0 0
$$46$$ 47120.2 3.28331
$$47$$ −18816.2 −1.24247 −0.621237 0.783622i $$-0.713370\pi$$
−0.621237 + 0.783622i $$0.713370\pi$$
$$48$$ −39582.8 −2.47973
$$49$$ −11008.8 −0.655011
$$50$$ 0 0
$$51$$ −4452.80 −0.239722
$$52$$ 43007.5 2.20565
$$53$$ 34459.8 1.68509 0.842545 0.538626i $$-0.181056\pi$$
0.842545 + 0.538626i $$0.181056\pi$$
$$54$$ 8096.60 0.377849
$$55$$ 0 0
$$56$$ 50195.7 2.13893
$$57$$ 403.268 0.0164402
$$58$$ 8739.88 0.341142
$$59$$ 27937.1 1.04485 0.522423 0.852687i $$-0.325028\pi$$
0.522423 + 0.852687i $$0.325028\pi$$
$$60$$ 0 0
$$61$$ 14661.0 0.504475 0.252237 0.967665i $$-0.418834\pi$$
0.252237 + 0.967665i $$0.418834\pi$$
$$62$$ −62133.2 −2.05279
$$63$$ −6167.84 −0.195786
$$64$$ 167495. 5.11154
$$65$$ 0 0
$$66$$ −12094.9 −0.341777
$$67$$ 47205.0 1.28470 0.642349 0.766412i $$-0.277960\pi$$
0.642349 + 0.766412i $$0.277960\pi$$
$$68$$ 45197.5 1.18534
$$69$$ 38183.4 0.965498
$$70$$ 0 0
$$71$$ −39501.5 −0.929967 −0.464983 0.885319i $$-0.653940\pi$$
−0.464983 + 0.885319i $$0.653940\pi$$
$$72$$ −53395.4 −1.21387
$$73$$ −26993.8 −0.592867 −0.296433 0.955054i $$-0.595797\pi$$
−0.296433 + 0.955054i $$0.595797\pi$$
$$74$$ 71679.8 1.52166
$$75$$ 0 0
$$76$$ −4093.31 −0.0812907
$$77$$ 9213.68 0.177095
$$78$$ 47058.6 0.875795
$$79$$ −12293.2 −0.221614 −0.110807 0.993842i $$-0.535344\pi$$
−0.110807 + 0.993842i $$0.535344\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −85555.4 −1.40512
$$83$$ 70308.6 1.12025 0.560123 0.828409i $$-0.310754\pi$$
0.560123 + 0.828409i $$0.310754\pi$$
$$84$$ 62605.7 0.968090
$$85$$ 0 0
$$86$$ −26936.6 −0.392733
$$87$$ 7082.28 0.100317
$$88$$ 79763.4 1.09799
$$89$$ −143142. −1.91554 −0.957769 0.287539i $$-0.907163\pi$$
−0.957769 + 0.287539i $$0.907163\pi$$
$$90$$ 0 0
$$91$$ −35848.3 −0.453801
$$92$$ −387574. −4.77403
$$93$$ −50349.1 −0.603648
$$94$$ 208981. 2.43943
$$95$$ 0 0
$$96$$ 249774. 2.76611
$$97$$ −91322.4 −0.985480 −0.492740 0.870177i $$-0.664005\pi$$
−0.492740 + 0.870177i $$0.664005\pi$$
$$98$$ 122268. 1.28602
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ −141045. −1.37580 −0.687899 0.725806i $$-0.741467\pi$$
−0.687899 + 0.725806i $$0.741467\pi$$
$$102$$ 49454.8 0.470661
$$103$$ 139369. 1.29442 0.647208 0.762314i $$-0.275937\pi$$
0.647208 + 0.762314i $$0.275937\pi$$
$$104$$ −310341. −2.81356
$$105$$ 0 0
$$106$$ −382726. −3.30844
$$107$$ 189660. 1.60146 0.800729 0.599027i $$-0.204446\pi$$
0.800729 + 0.599027i $$0.204446\pi$$
$$108$$ −66596.4 −0.549404
$$109$$ −61287.5 −0.494089 −0.247045 0.969004i $$-0.579459\pi$$
−0.247045 + 0.969004i $$0.579459\pi$$
$$110$$ 0 0
$$111$$ 58085.0 0.447462
$$112$$ −334898. −2.52271
$$113$$ −243824. −1.79631 −0.898153 0.439683i $$-0.855091\pi$$
−0.898153 + 0.439683i $$0.855091\pi$$
$$114$$ −4478.88 −0.0322780
$$115$$ 0 0
$$116$$ −71887.6 −0.496031
$$117$$ 38133.5 0.257538
$$118$$ −310282. −2.05141
$$119$$ −37673.8 −0.243877
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −162832. −0.990466
$$123$$ −69329.0 −0.413192
$$124$$ 511060. 2.98482
$$125$$ 0 0
$$126$$ 68502.7 0.384398
$$127$$ −101881. −0.560512 −0.280256 0.959925i $$-0.590419\pi$$
−0.280256 + 0.959925i $$0.590419\pi$$
$$128$$ −972186. −5.24474
$$129$$ −21827.8 −0.115488
$$130$$ 0 0
$$131$$ −235642. −1.19971 −0.599853 0.800110i $$-0.704774\pi$$
−0.599853 + 0.800110i $$0.704774\pi$$
$$132$$ 99483.5 0.496954
$$133$$ 3411.92 0.0167252
$$134$$ −524280. −2.52233
$$135$$ 0 0
$$136$$ −326144. −1.51204
$$137$$ −180612. −0.822139 −0.411070 0.911604i $$-0.634845\pi$$
−0.411070 + 0.911604i $$0.634845\pi$$
$$138$$ −424081. −1.89562
$$139$$ 307919. 1.35176 0.675879 0.737013i $$-0.263764\pi$$
0.675879 + 0.737013i $$0.263764\pi$$
$$140$$ 0 0
$$141$$ 169346. 0.717343
$$142$$ 438721. 1.82586
$$143$$ −56964.8 −0.232952
$$144$$ 356245. 1.43167
$$145$$ 0 0
$$146$$ 299805. 1.16401
$$147$$ 99078.9 0.378171
$$148$$ −589583. −2.21254
$$149$$ 325175. 1.19992 0.599959 0.800030i $$-0.295183\pi$$
0.599959 + 0.800030i $$0.295183\pi$$
$$150$$ 0 0
$$151$$ 44535.7 0.158952 0.0794761 0.996837i $$-0.474675\pi$$
0.0794761 + 0.996837i $$0.474675\pi$$
$$152$$ 29537.2 0.103696
$$153$$ 40075.2 0.138404
$$154$$ −102331. −0.347701
$$155$$ 0 0
$$156$$ −387068. −1.27343
$$157$$ 354519. 1.14786 0.573932 0.818903i $$-0.305417\pi$$
0.573932 + 0.818903i $$0.305417\pi$$
$$158$$ 136534. 0.435108
$$159$$ −310138. −0.972887
$$160$$ 0 0
$$161$$ 323057. 0.982234
$$162$$ −72869.4 −0.218151
$$163$$ 97011.8 0.285993 0.142996 0.989723i $$-0.454326\pi$$
0.142996 + 0.989723i $$0.454326\pi$$
$$164$$ 703713. 2.04308
$$165$$ 0 0
$$166$$ −780879. −2.19945
$$167$$ −385119. −1.06857 −0.534287 0.845303i $$-0.679420\pi$$
−0.534287 + 0.845303i $$0.679420\pi$$
$$168$$ −451761. −1.23491
$$169$$ −149656. −0.403067
$$170$$ 0 0
$$171$$ −3629.41 −0.00949175
$$172$$ 221560. 0.571045
$$173$$ −544249. −1.38255 −0.691277 0.722590i $$-0.742952\pi$$
−0.691277 + 0.722590i $$0.742952\pi$$
$$174$$ −78659.0 −0.196959
$$175$$ 0 0
$$176$$ −532169. −1.29499
$$177$$ −251434. −0.603242
$$178$$ 1.58979e6 3.76089
$$179$$ −265639. −0.619668 −0.309834 0.950791i $$-0.600273\pi$$
−0.309834 + 0.950791i $$0.600273\pi$$
$$180$$ 0 0
$$181$$ −748969. −1.69929 −0.849645 0.527355i $$-0.823184\pi$$
−0.849645 + 0.527355i $$0.823184\pi$$
$$182$$ 398148. 0.890975
$$183$$ −131949. −0.291259
$$184$$ 2.79673e6 6.08984
$$185$$ 0 0
$$186$$ 559199. 1.18518
$$187$$ −59865.5 −0.125191
$$188$$ −1.71892e6 −3.54700
$$189$$ 55510.5 0.113037
$$190$$ 0 0
$$191$$ −964220. −1.91246 −0.956231 0.292613i $$-0.905475\pi$$
−0.956231 + 0.292613i $$0.905475\pi$$
$$192$$ −1.50745e6 −2.95115
$$193$$ 167314. 0.323325 0.161662 0.986846i $$-0.448314\pi$$
0.161662 + 0.986846i $$0.448314\pi$$
$$194$$ 1.01427e6 1.93485
$$195$$ 0 0
$$196$$ −1.00568e6 −1.86991
$$197$$ 99359.5 0.182408 0.0912040 0.995832i $$-0.470928\pi$$
0.0912040 + 0.995832i $$0.470928\pi$$
$$198$$ 108854. 0.197325
$$199$$ −56809.9 −0.101693 −0.0508466 0.998706i $$-0.516192\pi$$
−0.0508466 + 0.998706i $$0.516192\pi$$
$$200$$ 0 0
$$201$$ −424845. −0.741721
$$202$$ 1.56651e6 2.70119
$$203$$ 59920.9 0.102056
$$204$$ −406778. −0.684355
$$205$$ 0 0
$$206$$ −1.54790e6 −2.54140
$$207$$ −343650. −0.557431
$$208$$ 2.07055e6 3.31839
$$209$$ 5421.72 0.00858561
$$210$$ 0 0
$$211$$ 529807. 0.819241 0.409620 0.912256i $$-0.365661\pi$$
0.409620 + 0.912256i $$0.365661\pi$$
$$212$$ 3.14801e6 4.81057
$$213$$ 355513. 0.536916
$$214$$ −2.10645e6 −3.14424
$$215$$ 0 0
$$216$$ 480558. 0.700828
$$217$$ −425987. −0.614112
$$218$$ 680686. 0.970076
$$219$$ 242944. 0.342292
$$220$$ 0 0
$$221$$ 232923. 0.320798
$$222$$ −645118. −0.878530
$$223$$ 757914. 1.02061 0.510303 0.859995i $$-0.329533\pi$$
0.510303 + 0.859995i $$0.329533\pi$$
$$224$$ 2.11326e6 2.81406
$$225$$ 0 0
$$226$$ 2.70802e6 3.52680
$$227$$ 794647. 1.02355 0.511776 0.859119i $$-0.328988\pi$$
0.511776 + 0.859119i $$0.328988\pi$$
$$228$$ 36839.8 0.0469332
$$229$$ −191327. −0.241095 −0.120548 0.992708i $$-0.538465\pi$$
−0.120548 + 0.992708i $$0.538465\pi$$
$$230$$ 0 0
$$231$$ −82923.2 −0.102246
$$232$$ 518739. 0.632746
$$233$$ −845860. −1.02072 −0.510362 0.859960i $$-0.670489\pi$$
−0.510362 + 0.859960i $$0.670489\pi$$
$$234$$ −423527. −0.505640
$$235$$ 0 0
$$236$$ 2.55214e6 2.98281
$$237$$ 110639. 0.127949
$$238$$ 418422. 0.478819
$$239$$ 1.41892e6 1.60681 0.803403 0.595435i $$-0.203020\pi$$
0.803403 + 0.595435i $$0.203020\pi$$
$$240$$ 0 0
$$241$$ −902862. −1.00133 −0.500667 0.865640i $$-0.666912\pi$$
−0.500667 + 0.865640i $$0.666912\pi$$
$$242$$ −162609. −0.178487
$$243$$ −59049.0 −0.0641500
$$244$$ 1.33933e6 1.44017
$$245$$ 0 0
$$246$$ 769999. 0.811245
$$247$$ −21094.7 −0.0220004
$$248$$ −3.68780e6 −3.80748
$$249$$ −632778. −0.646774
$$250$$ 0 0
$$251$$ −58009.8 −0.0581188 −0.0290594 0.999578i $$-0.509251\pi$$
−0.0290594 + 0.999578i $$0.509251\pi$$
$$252$$ −563451. −0.558927
$$253$$ 513354. 0.504215
$$254$$ 1.13154e6 1.10049
$$255$$ 0 0
$$256$$ 5.43769e6 5.18579
$$257$$ −661274. −0.624523 −0.312262 0.949996i $$-0.601087\pi$$
−0.312262 + 0.949996i $$0.601087\pi$$
$$258$$ 242430. 0.226744
$$259$$ 491439. 0.455219
$$260$$ 0 0
$$261$$ −63740.5 −0.0579181
$$262$$ 2.61715e6 2.35546
$$263$$ 1.09488e6 0.976058 0.488029 0.872827i $$-0.337716\pi$$
0.488029 + 0.872827i $$0.337716\pi$$
$$264$$ −717871. −0.633923
$$265$$ 0 0
$$266$$ −37894.3 −0.0328375
$$267$$ 1.28827e6 1.10594
$$268$$ 4.31232e6 3.66754
$$269$$ −206823. −0.174268 −0.0871341 0.996197i $$-0.527771\pi$$
−0.0871341 + 0.996197i $$0.527771\pi$$
$$270$$ 0 0
$$271$$ −298237. −0.246683 −0.123341 0.992364i $$-0.539361\pi$$
−0.123341 + 0.992364i $$0.539361\pi$$
$$272$$ 2.17598e6 1.78334
$$273$$ 322635. 0.262002
$$274$$ 2.00596e6 1.61416
$$275$$ 0 0
$$276$$ 3.48817e6 2.75629
$$277$$ −1.71987e6 −1.34678 −0.673390 0.739287i $$-0.735163\pi$$
−0.673390 + 0.739287i $$0.735163\pi$$
$$278$$ −3.41988e6 −2.65399
$$279$$ 453141. 0.348516
$$280$$ 0 0
$$281$$ 2.04137e6 1.54225 0.771126 0.636683i $$-0.219694\pi$$
0.771126 + 0.636683i $$0.219694\pi$$
$$282$$ −1.88083e6 −1.40840
$$283$$ 1.62392e6 1.20531 0.602656 0.798001i $$-0.294109\pi$$
0.602656 + 0.798001i $$0.294109\pi$$
$$284$$ −3.60858e6 −2.65485
$$285$$ 0 0
$$286$$ 632676. 0.457369
$$287$$ −586571. −0.420354
$$288$$ −2.24797e6 −1.59701
$$289$$ −1.17507e6 −0.827600
$$290$$ 0 0
$$291$$ 821901. 0.568967
$$292$$ −2.46597e6 −1.69251
$$293$$ 2.37278e6 1.61469 0.807344 0.590082i $$-0.200904\pi$$
0.807344 + 0.590082i $$0.200904\pi$$
$$294$$ −1.10041e6 −0.742485
$$295$$ 0 0
$$296$$ 4.25442e6 2.82235
$$297$$ 88209.0 0.0580259
$$298$$ −3.61154e6 −2.35587
$$299$$ −1.99734e6 −1.29204
$$300$$ 0 0
$$301$$ −184678. −0.117490
$$302$$ −494634. −0.312080
$$303$$ 1.26941e6 0.794318
$$304$$ −197068. −0.122301
$$305$$ 0 0
$$306$$ −445093. −0.271736
$$307$$ 1.29238e6 0.782608 0.391304 0.920261i $$-0.372024\pi$$
0.391304 + 0.920261i $$0.372024\pi$$
$$308$$ 841699. 0.505568
$$309$$ −1.25432e6 −0.747331
$$310$$ 0 0
$$311$$ −1.55279e6 −0.910357 −0.455179 0.890400i $$-0.650425\pi$$
−0.455179 + 0.890400i $$0.650425\pi$$
$$312$$ 2.79307e6 1.62441
$$313$$ −2.14996e6 −1.24042 −0.620210 0.784436i $$-0.712953\pi$$
−0.620210 + 0.784436i $$0.712953\pi$$
$$314$$ −3.93745e6 −2.25367
$$315$$ 0 0
$$316$$ −1.12302e6 −0.632660
$$317$$ 1.34306e6 0.750669 0.375335 0.926889i $$-0.377528\pi$$
0.375335 + 0.926889i $$0.377528\pi$$
$$318$$ 3.44453e6 1.91013
$$319$$ 95217.3 0.0523889
$$320$$ 0 0
$$321$$ −1.70694e6 −0.924602
$$322$$ −3.58802e6 −1.92848
$$323$$ −22168.8 −0.0118232
$$324$$ 599368. 0.317198
$$325$$ 0 0
$$326$$ −1.07746e6 −0.561507
$$327$$ 551587. 0.285263
$$328$$ −5.07798e6 −2.60619
$$329$$ 1.43278e6 0.729777
$$330$$ 0 0
$$331$$ 1.24921e6 0.626706 0.313353 0.949637i $$-0.398548\pi$$
0.313353 + 0.949637i $$0.398548\pi$$
$$332$$ 6.42291e6 3.19806
$$333$$ −522765. −0.258343
$$334$$ 4.27731e6 2.09799
$$335$$ 0 0
$$336$$ 3.01408e6 1.45649
$$337$$ −3.36237e6 −1.61277 −0.806383 0.591394i $$-0.798578\pi$$
−0.806383 + 0.591394i $$0.798578\pi$$
$$338$$ 1.66215e6 0.791366
$$339$$ 2.19442e6 1.03710
$$340$$ 0 0
$$341$$ −676915. −0.315245
$$342$$ 40309.9 0.0186357
$$343$$ 2.11806e6 0.972084
$$344$$ −1.59877e6 −0.728435
$$345$$ 0 0
$$346$$ 6.04467e6 2.71445
$$347$$ −3.03255e6 −1.35202 −0.676012 0.736890i $$-0.736293\pi$$
−0.676012 + 0.736890i $$0.736293\pi$$
$$348$$ 646988. 0.286384
$$349$$ 690808. 0.303595 0.151797 0.988412i $$-0.451494\pi$$
0.151797 + 0.988412i $$0.451494\pi$$
$$350$$ 0 0
$$351$$ −343201. −0.148690
$$352$$ 3.35808e6 1.44455
$$353$$ 3.16374e6 1.35134 0.675669 0.737205i $$-0.263855\pi$$
0.675669 + 0.737205i $$0.263855\pi$$
$$354$$ 2.79254e6 1.18438
$$355$$ 0 0
$$356$$ −1.30764e7 −5.46845
$$357$$ 339064. 0.140803
$$358$$ 2.95030e6 1.21663
$$359$$ −392579. −0.160765 −0.0803825 0.996764i $$-0.525614\pi$$
−0.0803825 + 0.996764i $$0.525614\pi$$
$$360$$ 0 0
$$361$$ −2.47409e6 −0.999189
$$362$$ 8.31839e6 3.33632
$$363$$ −131769. −0.0524864
$$364$$ −3.27486e6 −1.29550
$$365$$ 0 0
$$366$$ 1.46549e6 0.571846
$$367$$ 2.92576e6 1.13390 0.566949 0.823753i $$-0.308124\pi$$
0.566949 + 0.823753i $$0.308124\pi$$
$$368$$ −1.86593e7 −7.18251
$$369$$ 623961. 0.238557
$$370$$ 0 0
$$371$$ −2.62398e6 −0.989751
$$372$$ −4.59954e6 −1.72329
$$373$$ 3.29066e6 1.22465 0.612323 0.790607i $$-0.290235\pi$$
0.612323 + 0.790607i $$0.290235\pi$$
$$374$$ 664893. 0.245795
$$375$$ 0 0
$$376$$ 1.24037e7 4.52461
$$377$$ −370469. −0.134245
$$378$$ −616525. −0.221933
$$379$$ 2.02091e6 0.722683 0.361342 0.932434i $$-0.382319\pi$$
0.361342 + 0.932434i $$0.382319\pi$$
$$380$$ 0 0
$$381$$ 916932. 0.323612
$$382$$ 1.07091e7 3.75485
$$383$$ 1.49093e6 0.519349 0.259674 0.965696i $$-0.416385\pi$$
0.259674 + 0.965696i $$0.416385\pi$$
$$384$$ 8.74967e6 3.02805
$$385$$ 0 0
$$386$$ −1.85826e6 −0.634803
$$387$$ 196450. 0.0666769
$$388$$ −8.34258e6 −2.81333
$$389$$ 4.98939e6 1.67176 0.835878 0.548915i $$-0.184959\pi$$
0.835878 + 0.548915i $$0.184959\pi$$
$$390$$ 0 0
$$391$$ −2.09905e6 −0.694354
$$392$$ 7.25700e6 2.38529
$$393$$ 2.12078e6 0.692651
$$394$$ −1.10353e6 −0.358133
$$395$$ 0 0
$$396$$ −895352. −0.286917
$$397$$ 2.73054e6 0.869504 0.434752 0.900550i $$-0.356836\pi$$
0.434752 + 0.900550i $$0.356836\pi$$
$$398$$ 630956. 0.199660
$$399$$ −30707.3 −0.00965628
$$400$$ 0 0
$$401$$ −2.44533e6 −0.759409 −0.379705 0.925108i $$-0.623974\pi$$
−0.379705 + 0.925108i $$0.623974\pi$$
$$402$$ 4.71852e6 1.45627
$$403$$ 2.63372e6 0.807807
$$404$$ −1.28849e7 −3.92761
$$405$$ 0 0
$$406$$ −665509. −0.200373
$$407$$ 780921. 0.233680
$$408$$ 2.93530e6 0.872975
$$409$$ 5.04690e6 1.49182 0.745909 0.666047i $$-0.232015\pi$$
0.745909 + 0.666047i $$0.232015\pi$$
$$410$$ 0 0
$$411$$ 1.62551e6 0.474662
$$412$$ 1.27318e7 3.69528
$$413$$ −2.12731e6 −0.613698
$$414$$ 3.81673e6 1.09444
$$415$$ 0 0
$$416$$ −1.30655e7 −3.70163
$$417$$ −2.77127e6 −0.780438
$$418$$ −60216.0 −0.0168566
$$419$$ −30652.8 −0.00852971 −0.00426486 0.999991i $$-0.501358\pi$$
−0.00426486 + 0.999991i $$0.501358\pi$$
$$420$$ 0 0
$$421$$ 2.31067e6 0.635379 0.317689 0.948195i $$-0.397093\pi$$
0.317689 + 0.948195i $$0.397093\pi$$
$$422$$ −5.88427e6 −1.60847
$$423$$ −1.52411e6 −0.414158
$$424$$ −2.27160e7 −6.13644
$$425$$ 0 0
$$426$$ −3.94849e6 −1.05416
$$427$$ −1.11638e6 −0.296307
$$428$$ 1.73260e7 4.57182
$$429$$ 512683. 0.134495
$$430$$ 0 0
$$431$$ 7.38790e6 1.91570 0.957851 0.287266i $$-0.0927464\pi$$
0.957851 + 0.287266i $$0.0927464\pi$$
$$432$$ −3.20621e6 −0.826575
$$433$$ 3.90981e6 1.00216 0.501079 0.865402i $$-0.332937\pi$$
0.501079 + 0.865402i $$0.332937\pi$$
$$434$$ 4.73120e6 1.20572
$$435$$ 0 0
$$436$$ −5.59880e6 −1.41052
$$437$$ 190100. 0.0476189
$$438$$ −2.69825e6 −0.672042
$$439$$ 2.57832e6 0.638522 0.319261 0.947667i $$-0.396565\pi$$
0.319261 + 0.947667i $$0.396565\pi$$
$$440$$ 0 0
$$441$$ −891710. −0.218337
$$442$$ −2.58695e6 −0.629842
$$443$$ 300253. 0.0726905 0.0363452 0.999339i $$-0.488428\pi$$
0.0363452 + 0.999339i $$0.488428\pi$$
$$444$$ 5.30625e6 1.27741
$$445$$ 0 0
$$446$$ −8.41774e6 −2.00382
$$447$$ −2.92658e6 −0.692773
$$448$$ −1.27541e7 −3.00230
$$449$$ −3.40803e6 −0.797788 −0.398894 0.916997i $$-0.630606\pi$$
−0.398894 + 0.916997i $$0.630606\pi$$
$$450$$ 0 0
$$451$$ −932090. −0.215783
$$452$$ −2.22741e7 −5.12807
$$453$$ −400822. −0.0917710
$$454$$ −8.82571e6 −2.00960
$$455$$ 0 0
$$456$$ −265835. −0.0598688
$$457$$ −3.76619e6 −0.843552 −0.421776 0.906700i $$-0.638593\pi$$
−0.421776 + 0.906700i $$0.638593\pi$$
$$458$$ 2.12497e6 0.473356
$$459$$ −360677. −0.0799074
$$460$$ 0 0
$$461$$ −8.18562e6 −1.79390 −0.896952 0.442128i $$-0.854224\pi$$
−0.896952 + 0.442128i $$0.854224\pi$$
$$462$$ 920981. 0.200746
$$463$$ −950861. −0.206141 −0.103071 0.994674i $$-0.532867\pi$$
−0.103071 + 0.994674i $$0.532867\pi$$
$$464$$ −3.46095e6 −0.746277
$$465$$ 0 0
$$466$$ 9.39449e6 2.00405
$$467$$ 3.61259e6 0.766526 0.383263 0.923639i $$-0.374800\pi$$
0.383263 + 0.923639i $$0.374800\pi$$
$$468$$ 3.48361e6 0.735216
$$469$$ −3.59448e6 −0.754577
$$470$$ 0 0
$$471$$ −3.19067e6 −0.662720
$$472$$ −1.84162e7 −3.80492
$$473$$ −293463. −0.0603116
$$474$$ −1.22880e6 −0.251210
$$475$$ 0 0
$$476$$ −3.44162e6 −0.696218
$$477$$ 2.79124e6 0.561697
$$478$$ −1.57592e7 −3.15474
$$479$$ 8.09916e6 1.61288 0.806439 0.591318i $$-0.201392\pi$$
0.806439 + 0.591318i $$0.201392\pi$$
$$480$$ 0 0
$$481$$ −3.03838e6 −0.598797
$$482$$ 1.00276e7 1.96598
$$483$$ −2.90752e6 −0.567093
$$484$$ 1.33750e6 0.259526
$$485$$ 0 0
$$486$$ 655824. 0.125950
$$487$$ −2.22319e6 −0.424771 −0.212386 0.977186i $$-0.568123\pi$$
−0.212386 + 0.977186i $$0.568123\pi$$
$$488$$ −9.66457e6 −1.83710
$$489$$ −873106. −0.165118
$$490$$ 0 0
$$491$$ 1.80556e6 0.337993 0.168996 0.985617i $$-0.445947\pi$$
0.168996 + 0.985617i $$0.445947\pi$$
$$492$$ −6.33342e6 −1.17958
$$493$$ −389333. −0.0721447
$$494$$ 234287. 0.0431947
$$495$$ 0 0
$$496$$ 2.46044e7 4.49065
$$497$$ 3.00788e6 0.546223
$$498$$ 7.02791e6 1.26985
$$499$$ −3.16604e6 −0.569201 −0.284600 0.958646i $$-0.591861\pi$$
−0.284600 + 0.958646i $$0.591861\pi$$
$$500$$ 0 0
$$501$$ 3.46607e6 0.616941
$$502$$ 644282. 0.114108
$$503$$ 4.86714e6 0.857737 0.428869 0.903367i $$-0.358912\pi$$
0.428869 + 0.903367i $$0.358912\pi$$
$$504$$ 4.06585e6 0.712976
$$505$$ 0 0
$$506$$ −5.70154e6 −0.989956
$$507$$ 1.34690e6 0.232711
$$508$$ −9.30717e6 −1.60014
$$509$$ 4.22678e6 0.723128 0.361564 0.932347i $$-0.382243\pi$$
0.361564 + 0.932347i $$0.382243\pi$$
$$510$$ 0 0
$$511$$ 2.05547e6 0.348225
$$512$$ −2.92835e7 −4.93683
$$513$$ 32664.7 0.00548006
$$514$$ 7.34441e6 1.22616
$$515$$ 0 0
$$516$$ −1.99404e6 −0.329693
$$517$$ 2.27676e6 0.374620
$$518$$ −5.45814e6 −0.893758
$$519$$ 4.89824e6 0.798218
$$520$$ 0 0
$$521$$ 1.11140e7 1.79380 0.896901 0.442232i $$-0.145813\pi$$
0.896901 + 0.442232i $$0.145813\pi$$
$$522$$ 707931. 0.113714
$$523$$ −532067. −0.0850574 −0.0425287 0.999095i $$-0.513541\pi$$
−0.0425287 + 0.999095i $$0.513541\pi$$
$$524$$ −2.15267e7 −3.42490
$$525$$ 0 0
$$526$$ −1.21602e7 −1.91635
$$527$$ 2.76783e6 0.434124
$$528$$ 4.78952e6 0.747665
$$529$$ 1.15633e7 1.79656
$$530$$ 0 0
$$531$$ 2.26291e6 0.348282
$$532$$ 311690. 0.0477467
$$533$$ 3.62655e6 0.552937
$$534$$ −1.43081e7 −2.17135
$$535$$ 0 0
$$536$$ −3.11176e7 −4.67837
$$537$$ 2.39075e6 0.357765
$$538$$ 2.29707e6 0.342151
$$539$$ 1.33206e6 0.197493
$$540$$ 0 0
$$541$$ −9.77951e6 −1.43656 −0.718280 0.695754i $$-0.755070\pi$$
−0.718280 + 0.695754i $$0.755070\pi$$
$$542$$ 3.31235e6 0.484327
$$543$$ 6.74072e6 0.981086
$$544$$ −1.37308e7 −1.98929
$$545$$ 0 0
$$546$$ −3.58333e6 −0.514405
$$547$$ −8.61940e6 −1.23171 −0.615855 0.787859i $$-0.711189\pi$$
−0.615855 + 0.787859i $$0.711189\pi$$
$$548$$ −1.64995e7 −2.34703
$$549$$ 1.18754e6 0.168158
$$550$$ 0 0
$$551$$ 35260.0 0.00494770
$$552$$ −2.51706e7 −3.51597
$$553$$ 936079. 0.130167
$$554$$ 1.91017e7 2.64422
$$555$$ 0 0
$$556$$ 2.81293e7 3.85898
$$557$$ 1.37932e7 1.88376 0.941881 0.335946i $$-0.109056\pi$$
0.941881 + 0.335946i $$0.109056\pi$$
$$558$$ −5.03279e6 −0.684263
$$559$$ 1.14180e6 0.154547
$$560$$ 0 0
$$561$$ 538789. 0.0722789
$$562$$ −2.26723e7 −3.02800
$$563$$ 8.21998e6 1.09295 0.546474 0.837476i $$-0.315970\pi$$
0.546474 + 0.837476i $$0.315970\pi$$
$$564$$ 1.54703e7 2.04786
$$565$$ 0 0
$$566$$ −1.80360e7 −2.36646
$$567$$ −499595. −0.0652620
$$568$$ 2.60394e7 3.38658
$$569$$ 8.25501e6 1.06890 0.534450 0.845200i $$-0.320519\pi$$
0.534450 + 0.845200i $$0.320519\pi$$
$$570$$ 0 0
$$571$$ 9.47816e6 1.21656 0.608280 0.793722i $$-0.291860\pi$$
0.608280 + 0.793722i $$0.291860\pi$$
$$572$$ −5.20391e6 −0.665028
$$573$$ 8.67798e6 1.10416
$$574$$ 6.51472e6 0.825307
$$575$$ 0 0
$$576$$ 1.35671e7 1.70385
$$577$$ −5.03255e6 −0.629287 −0.314644 0.949210i $$-0.601885\pi$$
−0.314644 + 0.949210i $$0.601885\pi$$
$$578$$ 1.30509e7 1.62488
$$579$$ −1.50583e6 −0.186672
$$580$$ 0 0
$$581$$ −5.35373e6 −0.657985
$$582$$ −9.12840e6 −1.11709
$$583$$ −4.16964e6 −0.508074
$$584$$ 1.77944e7 2.15899
$$585$$ 0 0
$$586$$ −2.63531e7 −3.17021
$$587$$ −2.93358e6 −0.351401 −0.175700 0.984444i $$-0.556219\pi$$
−0.175700 + 0.984444i $$0.556219\pi$$
$$588$$ 9.05116e6 1.07960
$$589$$ −250669. −0.0297723
$$590$$ 0 0
$$591$$ −894236. −0.105313
$$592$$ −2.83848e7 −3.32875
$$593$$ 1.08060e7 1.26191 0.630953 0.775821i $$-0.282664\pi$$
0.630953 + 0.775821i $$0.282664\pi$$
$$594$$ −979688. −0.113926
$$595$$ 0 0
$$596$$ 2.97058e7 3.42551
$$597$$ 511289. 0.0587125
$$598$$ 2.21834e7 2.53673
$$599$$ 3.28617e6 0.374217 0.187108 0.982339i $$-0.440088\pi$$
0.187108 + 0.982339i $$0.440088\pi$$
$$600$$ 0 0
$$601$$ 6.11530e6 0.690608 0.345304 0.938491i $$-0.387776\pi$$
0.345304 + 0.938491i $$0.387776\pi$$
$$602$$ 2.05112e6 0.230675
$$603$$ 3.82360e6 0.428233
$$604$$ 4.06848e6 0.453774
$$605$$ 0 0
$$606$$ −1.40986e7 −1.55953
$$607$$ −7.27336e6 −0.801241 −0.400621 0.916244i $$-0.631205\pi$$
−0.400621 + 0.916244i $$0.631205\pi$$
$$608$$ 1.24353e6 0.136426
$$609$$ −539288. −0.0589220
$$610$$ 0 0
$$611$$ −8.85836e6 −0.959954
$$612$$ 3.66100e6 0.395113
$$613$$ −7.10635e6 −0.763827 −0.381914 0.924198i $$-0.624735\pi$$
−0.381914 + 0.924198i $$0.624735\pi$$
$$614$$ −1.43538e7 −1.53654
$$615$$ 0 0
$$616$$ −6.07368e6 −0.644911
$$617$$ −5.11598e6 −0.541023 −0.270512 0.962717i $$-0.587193\pi$$
−0.270512 + 0.962717i $$0.587193\pi$$
$$618$$ 1.39311e7 1.46728
$$619$$ −2.88443e6 −0.302576 −0.151288 0.988490i $$-0.548342\pi$$
−0.151288 + 0.988490i $$0.548342\pi$$
$$620$$ 0 0
$$621$$ 3.09285e6 0.321833
$$622$$ 1.72460e7 1.78736
$$623$$ 1.08997e7 1.12511
$$624$$ −1.86349e7 −1.91587
$$625$$ 0 0
$$626$$ 2.38784e7 2.43539
$$627$$ −48795.5 −0.00495690
$$628$$ 3.23864e7 3.27690
$$629$$ −3.19310e6 −0.321800
$$630$$ 0 0
$$631$$ −8.37374e6 −0.837233 −0.418616 0.908163i $$-0.637485\pi$$
−0.418616 + 0.908163i $$0.637485\pi$$
$$632$$ 8.10370e6 0.807031
$$633$$ −4.76826e6 −0.472989
$$634$$ −1.49167e7 −1.47383
$$635$$ 0 0
$$636$$ −2.83321e7 −2.77738
$$637$$ −5.18274e6 −0.506071
$$638$$ −1.05753e6 −0.102858
$$639$$ −3.19962e6 −0.309989
$$640$$ 0 0
$$641$$ 1.14615e7 1.10179 0.550893 0.834576i $$-0.314287\pi$$
0.550893 + 0.834576i $$0.314287\pi$$
$$642$$ 1.89580e7 1.81533
$$643$$ 1.69107e7 1.61300 0.806498 0.591237i $$-0.201360\pi$$
0.806498 + 0.591237i $$0.201360\pi$$
$$644$$ 2.95123e7 2.80407
$$645$$ 0 0
$$646$$ 246217. 0.0232133
$$647$$ 1.21273e7 1.13895 0.569473 0.822010i $$-0.307147\pi$$
0.569473 + 0.822010i $$0.307147\pi$$
$$648$$ −4.32502e6 −0.404623
$$649$$ −3.38039e6 −0.315033
$$650$$ 0 0
$$651$$ 3.83389e6 0.354558
$$652$$ 8.86233e6 0.816448
$$653$$ −1.09620e7 −1.00602 −0.503010 0.864281i $$-0.667774\pi$$
−0.503010 + 0.864281i $$0.667774\pi$$
$$654$$ −6.12617e6 −0.560073
$$655$$ 0 0
$$656$$ 3.38795e7 3.07381
$$657$$ −2.18650e6 −0.197622
$$658$$ −1.59131e7 −1.43282
$$659$$ 2.65796e6 0.238416 0.119208 0.992869i $$-0.461964\pi$$
0.119208 + 0.992869i $$0.461964\pi$$
$$660$$ 0 0
$$661$$ 8.86632e6 0.789295 0.394648 0.918833i $$-0.370867\pi$$
0.394648 + 0.918833i $$0.370867\pi$$
$$662$$ −1.38742e7 −1.23045
$$663$$ −2.09631e6 −0.185213
$$664$$ −4.63476e7 −4.07950
$$665$$ 0 0
$$666$$ 5.80606e6 0.507220
$$667$$ 3.33858e6 0.290568
$$668$$ −3.51819e7 −3.05055
$$669$$ −6.82123e6 −0.589247
$$670$$ 0 0
$$671$$ −1.77398e6 −0.152105
$$672$$ −1.90193e7 −1.62470
$$673$$ 1.56559e7 1.33242 0.666209 0.745765i $$-0.267916\pi$$
0.666209 + 0.745765i $$0.267916\pi$$
$$674$$ 3.73440e7 3.16644
$$675$$ 0 0
$$676$$ −1.36715e7 −1.15067
$$677$$ −2.51293e6 −0.210721 −0.105361 0.994434i $$-0.533600\pi$$
−0.105361 + 0.994434i $$0.533600\pi$$
$$678$$ −2.43722e7 −2.03620
$$679$$ 6.95385e6 0.578830
$$680$$ 0 0
$$681$$ −7.15183e6 −0.590948
$$682$$ 7.51812e6 0.618940
$$683$$ −1.40231e7 −1.15025 −0.575125 0.818066i $$-0.695047\pi$$
−0.575125 + 0.818066i $$0.695047\pi$$
$$684$$ −331558. −0.0270969
$$685$$ 0 0
$$686$$ −2.35242e7 −1.90855
$$687$$ 1.72195e6 0.139196
$$688$$ 1.06668e7 0.859135
$$689$$ 1.62231e7 1.30192
$$690$$ 0 0
$$691$$ −2.03284e7 −1.61960 −0.809799 0.586707i $$-0.800424\pi$$
−0.809799 + 0.586707i $$0.800424\pi$$
$$692$$ −4.97188e7 −3.94690
$$693$$ 746308. 0.0590317
$$694$$ 3.36809e7 2.65451
$$695$$ 0 0
$$696$$ −4.66865e6 −0.365316
$$697$$ 3.81122e6 0.297154
$$698$$ −7.67243e6 −0.596066
$$699$$ 7.61274e6 0.589315
$$700$$ 0 0
$$701$$ 8.73851e6 0.671649 0.335825 0.941925i $$-0.390985\pi$$
0.335825 + 0.941925i $$0.390985\pi$$
$$702$$ 3.81174e6 0.291932
$$703$$ 289183. 0.0220691
$$704$$ −2.02669e7 −1.54119
$$705$$ 0 0
$$706$$ −3.51379e7 −2.65316
$$707$$ 1.07401e7 0.808086
$$708$$ −2.29693e7 −1.72212
$$709$$ 6.30404e6 0.470981 0.235491 0.971877i $$-0.424330\pi$$
0.235491 + 0.971877i $$0.424330\pi$$
$$710$$ 0 0
$$711$$ −995748. −0.0738713
$$712$$ 9.43592e7 6.97564
$$713$$ −2.37345e7 −1.74846
$$714$$ −3.76579e6 −0.276446
$$715$$ 0 0
$$716$$ −2.42669e7 −1.76902
$$717$$ −1.27703e7 −0.927690
$$718$$ 4.36016e6 0.315640
$$719$$ 6.08432e6 0.438924 0.219462 0.975621i $$-0.429570\pi$$
0.219462 + 0.975621i $$0.429570\pi$$
$$720$$ 0 0
$$721$$ −1.06124e7 −0.760285
$$722$$ 2.74784e7 1.96177
$$723$$ 8.12575e6 0.578120
$$724$$ −6.84207e7 −4.85111
$$725$$ 0 0
$$726$$ 1.46349e6 0.103050
$$727$$ 2.10256e7 1.47541 0.737704 0.675125i $$-0.235910\pi$$
0.737704 + 0.675125i $$0.235910\pi$$
$$728$$ 2.36313e7 1.65257
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 1.19994e6 0.0830550
$$732$$ −1.20540e7 −0.831480
$$733$$ −9.82965e6 −0.675738 −0.337869 0.941193i $$-0.609706\pi$$
−0.337869 + 0.941193i $$0.609706\pi$$
$$734$$ −3.24948e7 −2.22625
$$735$$ 0 0
$$736$$ 1.17743e8 8.01202
$$737$$ −5.71180e6 −0.387351
$$738$$ −6.92999e6 −0.468373
$$739$$ −1.51937e7 −1.02342 −0.511709 0.859159i $$-0.670987\pi$$
−0.511709 + 0.859159i $$0.670987\pi$$
$$740$$ 0 0
$$741$$ 189852. 0.0127019
$$742$$ 2.91431e7 1.94324
$$743$$ −4.92706e6 −0.327428 −0.163714 0.986508i $$-0.552347\pi$$
−0.163714 + 0.986508i $$0.552347\pi$$
$$744$$ 3.31902e7 2.19825
$$745$$ 0 0
$$746$$ −3.65475e7 −2.40442
$$747$$ 5.69500e6 0.373415
$$748$$ −5.46890e6 −0.357393
$$749$$ −1.44419e7 −0.940629
$$750$$ 0 0
$$751$$ −1.07515e6 −0.0695617 −0.0347809 0.999395i $$-0.511073\pi$$
−0.0347809 + 0.999395i $$0.511073\pi$$
$$752$$ −8.27554e7 −5.33644
$$753$$ 522088. 0.0335549
$$754$$ 4.11459e6 0.263572
$$755$$ 0 0
$$756$$ 5.07106e6 0.322697
$$757$$ −1.80869e7 −1.14716 −0.573580 0.819150i $$-0.694446\pi$$
−0.573580 + 0.819150i $$0.694446\pi$$
$$758$$ −2.24451e7 −1.41889
$$759$$ −4.62019e6 −0.291109
$$760$$ 0 0
$$761$$ 540730. 0.0338469 0.0169234 0.999857i $$-0.494613\pi$$
0.0169234 + 0.999857i $$0.494613\pi$$
$$762$$ −1.01838e7 −0.635367
$$763$$ 4.66681e6 0.290207
$$764$$ −8.80845e7 −5.45967
$$765$$ 0 0
$$766$$ −1.65589e7 −1.01967
$$767$$ 1.31523e7 0.807262
$$768$$ −4.89392e7 −2.99402
$$769$$ 8.60213e6 0.524554 0.262277 0.964993i $$-0.415527\pi$$
0.262277 + 0.964993i $$0.415527\pi$$
$$770$$ 0 0
$$771$$ 5.95147e6 0.360569
$$772$$ 1.52847e7 0.923023
$$773$$ 1.45721e7 0.877150 0.438575 0.898695i $$-0.355483\pi$$
0.438575 + 0.898695i $$0.355483\pi$$
$$774$$ −2.18187e6 −0.130911
$$775$$ 0 0
$$776$$ 6.01999e7 3.58874
$$777$$ −4.42295e6 −0.262821
$$778$$ −5.54143e7 −3.28226
$$779$$ −345163. −0.0203789
$$780$$ 0 0
$$781$$ 4.77968e6 0.280395
$$782$$ 2.33130e7 1.36327
$$783$$ 573665. 0.0334390
$$784$$ −4.84176e7 −2.81328
$$785$$ 0 0
$$786$$ −2.35543e7 −1.35992
$$787$$ 9.86427e6 0.567712 0.283856 0.958867i $$-0.408386\pi$$
0.283856 + 0.958867i $$0.408386\pi$$
$$788$$ 9.07680e6 0.520736
$$789$$ −9.85388e6 −0.563527
$$790$$ 0 0
$$791$$ 1.85663e7 1.05507
$$792$$ 6.46084e6 0.365996
$$793$$ 6.90216e6 0.389764
$$794$$ −3.03266e7 −1.70715
$$795$$ 0 0
$$796$$ −5.18976e6 −0.290312
$$797$$ 1.18967e7 0.663410 0.331705 0.943383i $$-0.392376\pi$$
0.331705 + 0.943383i $$0.392376\pi$$
$$798$$ 341049. 0.0189588
$$799$$ −9.30943e6 −0.515889
$$800$$ 0 0
$$801$$ −1.15945e7 −0.638513
$$802$$ 2.71589e7 1.49099
$$803$$ 3.26625e6 0.178756
$$804$$ −3.88109e7 −2.11745
$$805$$ 0 0
$$806$$ −2.92513e7 −1.58602
$$807$$ 1.86141e6 0.100614
$$808$$ 9.29773e7 5.01012
$$809$$ 1.09831e7 0.590005 0.295002 0.955497i $$-0.404680\pi$$
0.295002 + 0.955497i $$0.404680\pi$$
$$810$$ 0 0
$$811$$ −3.01879e6 −0.161169 −0.0805845 0.996748i $$-0.525679\pi$$
−0.0805845 + 0.996748i $$0.525679\pi$$
$$812$$ 5.47396e6 0.291348
$$813$$ 2.68413e6 0.142422
$$814$$ −8.67325e6 −0.458797
$$815$$ 0 0
$$816$$ −1.95838e7 −1.02961
$$817$$ −108672. −0.00569593
$$818$$ −5.60531e7 −2.92898
$$819$$ −2.90372e6 −0.151267
$$820$$ 0 0
$$821$$ −5.39706e6 −0.279447 −0.139723 0.990191i $$-0.544621\pi$$
−0.139723 + 0.990191i $$0.544621\pi$$
$$822$$ −1.80536e7 −0.931933
$$823$$ 2.28710e7 1.17702 0.588512 0.808488i $$-0.299714\pi$$
0.588512 + 0.808488i $$0.299714\pi$$
$$824$$ −9.18725e7 −4.71376
$$825$$ 0 0
$$826$$ 2.36268e7 1.20491
$$827$$ −7.63569e6 −0.388226 −0.194113 0.980979i $$-0.562183\pi$$
−0.194113 + 0.980979i $$0.562183\pi$$
$$828$$ −3.13935e7 −1.59134
$$829$$ −7.03185e6 −0.355372 −0.177686 0.984087i $$-0.556861\pi$$
−0.177686 + 0.984087i $$0.556861\pi$$
$$830$$ 0 0
$$831$$ 1.54789e7 0.777564
$$832$$ 7.88538e7 3.94925
$$833$$ −5.44665e6 −0.271968
$$834$$ 3.07789e7 1.53228
$$835$$ 0 0
$$836$$ 495291. 0.0245101
$$837$$ −4.07827e6 −0.201216
$$838$$ 340443. 0.0167469
$$839$$ 6.66791e6 0.327028 0.163514 0.986541i $$-0.447717\pi$$
0.163514 + 0.986541i $$0.447717\pi$$
$$840$$ 0 0
$$841$$ −1.98919e7 −0.969809
$$842$$ −2.56633e7 −1.24748
$$843$$ −1.83723e7 −0.890420
$$844$$ 4.83995e7 2.33876
$$845$$ 0 0
$$846$$ 1.69275e7 0.813142
$$847$$ −1.11486e6 −0.0533962
$$848$$ 1.51557e8 7.23748
$$849$$ −1.46153e7 −0.695888
$$850$$ 0 0
$$851$$ 2.73812e7 1.29607
$$852$$ 3.24772e7 1.53278
$$853$$ 3.51845e7 1.65569 0.827845 0.560956i $$-0.189566\pi$$
0.827845 + 0.560956i $$0.189566\pi$$
$$854$$ 1.23990e7 0.581758
$$855$$ 0 0
$$856$$ −1.25024e8 −5.83189
$$857$$ −1.17588e7 −0.546905 −0.273452 0.961886i $$-0.588166\pi$$
−0.273452 + 0.961886i $$0.588166\pi$$
$$858$$ −5.69409e6 −0.264062
$$859$$ 3.68112e7 1.70215 0.851074 0.525045i $$-0.175952\pi$$
0.851074 + 0.525045i $$0.175952\pi$$
$$860$$ 0 0
$$861$$ 5.27914e6 0.242692
$$862$$ −8.20533e7 −3.76121
$$863$$ 3.57608e7 1.63448 0.817242 0.576295i $$-0.195502\pi$$
0.817242 + 0.576295i $$0.195502\pi$$
$$864$$ 2.02317e7 0.922037
$$865$$ 0 0
$$866$$ −4.34241e7 −1.96760
$$867$$ 1.05757e7 0.477815
$$868$$ −3.89153e7 −1.75316
$$869$$ 1.48748e6 0.0668191
$$870$$ 0 0
$$871$$ 2.22233e7 0.992576
$$872$$ 4.04008e7 1.79928
$$873$$ −7.39711e6 −0.328493
$$874$$ −2.11134e6 −0.0934931
$$875$$ 0 0
$$876$$ 2.21937e7 0.977169
$$877$$ 4.23836e7 1.86080 0.930398 0.366550i $$-0.119461\pi$$
0.930398 + 0.366550i $$0.119461\pi$$
$$878$$ −2.86360e7 −1.25365
$$879$$ −2.13550e7 −0.932240
$$880$$ 0 0
$$881$$ 1.28486e7 0.557721 0.278861 0.960332i $$-0.410043\pi$$
0.278861 + 0.960332i $$0.410043\pi$$
$$882$$ 9.90373e6 0.428674
$$883$$ 1.50474e7 0.649469 0.324735 0.945805i $$-0.394725\pi$$
0.324735 + 0.945805i $$0.394725\pi$$
$$884$$ 2.12782e7 0.915809
$$885$$ 0 0
$$886$$ −3.33474e6 −0.142718
$$887$$ −1.40467e7 −0.599466 −0.299733 0.954023i $$-0.596898\pi$$
−0.299733 + 0.954023i $$0.596898\pi$$
$$888$$ −3.82897e7 −1.62948
$$889$$ 7.75787e6 0.329221
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ 6.92378e7 2.91361
$$893$$ 843109. 0.0353798
$$894$$ 3.25039e7 1.36016
$$895$$ 0 0
$$896$$ 7.40282e7 3.08054
$$897$$ 1.79761e7 0.745958
$$898$$ 3.78511e7 1.56634
$$899$$ −4.40230e6 −0.181669
$$900$$ 0 0
$$901$$ 1.70492e7 0.699668
$$902$$ 1.03522e7 0.423659
$$903$$ 1.66211e6 0.0678327
$$904$$ 1.60729e8 6.54145
$$905$$ 0 0
$$906$$ 4.45170e6 0.180180
$$907$$ −8.25845e6 −0.333335 −0.166667 0.986013i $$-0.553301\pi$$
−0.166667 + 0.986013i $$0.553301\pi$$
$$908$$ 7.25935e7 2.92202
$$909$$ −1.14247e7 −0.458600
$$910$$ 0 0
$$911$$ 2.01230e7 0.803335 0.401667 0.915786i $$-0.368431\pi$$
0.401667 + 0.915786i $$0.368431\pi$$
$$912$$ 1.77361e6 0.0706108
$$913$$ −8.50735e6 −0.337767
$$914$$ 4.18290e7 1.65620
$$915$$ 0 0
$$916$$ −1.74783e7 −0.688274
$$917$$ 1.79433e7 0.704657
$$918$$ 4.00584e6 0.156887
$$919$$ 1.70445e7 0.665725 0.332862 0.942975i $$-0.391986\pi$$
0.332862 + 0.942975i $$0.391986\pi$$
$$920$$ 0 0
$$921$$ −1.16314e7 −0.451839
$$922$$ 9.09131e7 3.52208
$$923$$ −1.85966e7 −0.718506
$$924$$ −7.57529e6 −0.291890
$$925$$ 0 0
$$926$$ 1.05607e7 0.404729
$$927$$ 1.12889e7 0.431472
$$928$$ 2.18392e7 0.832465
$$929$$ 1.53002e7 0.581644 0.290822 0.956777i $$-0.406071\pi$$
0.290822 + 0.956777i $$0.406071\pi$$
$$930$$ 0 0
$$931$$ 493276. 0.0186516
$$932$$ −7.72719e7 −2.91395
$$933$$ 1.39751e7 0.525595
$$934$$ −4.01231e7 −1.50497
$$935$$ 0 0
$$936$$ −2.51377e7 −0.937854
$$937$$ −2.49555e7 −0.928574 −0.464287 0.885685i $$-0.653689\pi$$
−0.464287 + 0.885685i $$0.653689\pi$$
$$938$$ 3.99219e7 1.48151
$$939$$ 1.93496e7 0.716157
$$940$$ 0 0
$$941$$ 2.67710e7 0.985576 0.492788 0.870149i $$-0.335978\pi$$
0.492788 + 0.870149i $$0.335978\pi$$
$$942$$ 3.54370e7 1.30116
$$943$$ −3.26817e7 −1.19681
$$944$$ 1.22870e8 4.48762
$$945$$ 0 0
$$946$$ 3.25933e6 0.118413
$$947$$ −3.32255e7 −1.20392 −0.601958 0.798528i $$-0.705612\pi$$
−0.601958 + 0.798528i $$0.705612\pi$$
$$948$$ 1.01072e7 0.365266
$$949$$ −1.27082e7 −0.458057
$$950$$ 0 0
$$951$$ −1.20876e7 −0.433399
$$952$$ 2.48346e7 0.888107
$$953$$ 2.27164e7 0.810229 0.405114 0.914266i $$-0.367232\pi$$
0.405114 + 0.914266i $$0.367232\pi$$
$$954$$ −3.10008e7 −1.10281
$$955$$ 0 0
$$956$$ 1.29623e8 4.58709
$$957$$ −856956. −0.0302467
$$958$$ −8.99529e7 −3.16666
$$959$$ 1.37529e7 0.482890
$$960$$ 0 0
$$961$$ 2.66748e6 0.0931736
$$962$$ 3.37456e7 1.17566
$$963$$ 1.53624e7 0.533819
$$964$$ −8.24792e7 −2.85859
$$965$$ 0 0
$$966$$ 3.22922e7 1.11341
$$967$$ −4.69627e7 −1.61505 −0.807526 0.589832i $$-0.799194\pi$$
−0.807526 + 0.589832i $$0.799194\pi$$
$$968$$ −9.65138e6 −0.331056
$$969$$ 199519. 0.00682615
$$970$$ 0 0
$$971$$ −261687. −0.00890706 −0.00445353 0.999990i $$-0.501418\pi$$
−0.00445353 + 0.999990i $$0.501418\pi$$
$$972$$ −5.39431e6 −0.183135
$$973$$ −2.34468e7 −0.793966
$$974$$ 2.46918e7 0.833979
$$975$$ 0 0
$$976$$ 6.44805e7 2.16672
$$977$$ −3.02629e7 −1.01432 −0.507159 0.861853i $$-0.669304\pi$$
−0.507159 + 0.861853i $$0.669304\pi$$
$$978$$ 9.69710e6 0.324186
$$979$$ 1.73201e7 0.577556
$$980$$ 0 0
$$981$$ −4.96429e6 −0.164696
$$982$$ −2.00533e7 −0.663601
$$983$$ −2.18315e7 −0.720610 −0.360305 0.932835i $$-0.617327\pi$$
−0.360305 + 0.932835i $$0.617327\pi$$
$$984$$ 4.57018e7 1.50469
$$985$$ 0 0
$$986$$ 4.32411e6 0.141646
$$987$$ −1.28950e7 −0.421337
$$988$$ −1.92706e6 −0.0628064
$$989$$ −1.02896e7 −0.334510
$$990$$ 0 0
$$991$$ −5.46573e7 −1.76792 −0.883962 0.467558i $$-0.845134\pi$$
−0.883962 + 0.467558i $$0.845134\pi$$
$$992$$ −1.55258e8 −5.00927
$$993$$ −1.12428e7 −0.361829
$$994$$ −3.34069e7 −1.07243
$$995$$ 0 0
$$996$$ −5.78062e7 −1.84640
$$997$$ −1.96397e7 −0.625745 −0.312872 0.949795i $$-0.601291\pi$$
−0.312872 + 0.949795i $$0.601291\pi$$
$$998$$ 3.51635e7 1.11755
$$999$$ 4.70488e6 0.149154
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.1.1 yes 9
5.4 even 2 825.6.a.r.1.9 9

By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.9 9 5.4 even 2
825.6.a.s.1.1 yes 9 1.1 even 1 trivial