# Properties

 Label 165.6.a.a.1.1 Level $165$ Weight $6$ Character 165.1 Self dual yes Analytic conductor $26.463$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.4633302691$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.34253.1 Defining polynomial: $$x^{3} - x^{2} - 52x + 48$$ x^3 - x^2 - 52*x + 48 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$7.25531$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-9.25531 q^{2} +9.00000 q^{3} +53.6607 q^{4} +25.0000 q^{5} -83.2977 q^{6} +36.4478 q^{7} -200.476 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-9.25531 q^{2} +9.00000 q^{3} +53.6607 q^{4} +25.0000 q^{5} -83.2977 q^{6} +36.4478 q^{7} -200.476 q^{8} +81.0000 q^{9} -231.383 q^{10} -121.000 q^{11} +482.946 q^{12} -878.032 q^{13} -337.336 q^{14} +225.000 q^{15} +138.327 q^{16} +155.385 q^{17} -749.680 q^{18} -1932.65 q^{19} +1341.52 q^{20} +328.030 q^{21} +1119.89 q^{22} +1927.38 q^{23} -1804.29 q^{24} +625.000 q^{25} +8126.46 q^{26} +729.000 q^{27} +1955.81 q^{28} -480.444 q^{29} -2082.44 q^{30} +1759.49 q^{31} +5134.98 q^{32} -1089.00 q^{33} -1438.14 q^{34} +911.195 q^{35} +4346.51 q^{36} -1898.87 q^{37} +17887.3 q^{38} -7902.29 q^{39} -5011.91 q^{40} -4500.03 q^{41} -3036.02 q^{42} -4475.49 q^{43} -6492.94 q^{44} +2025.00 q^{45} -17838.5 q^{46} -12371.2 q^{47} +1244.94 q^{48} -15478.6 q^{49} -5784.57 q^{50} +1398.47 q^{51} -47115.8 q^{52} +2145.12 q^{53} -6747.12 q^{54} -3025.00 q^{55} -7306.92 q^{56} -17393.8 q^{57} +4446.66 q^{58} -15857.9 q^{59} +12073.7 q^{60} -36447.7 q^{61} -16284.6 q^{62} +2952.27 q^{63} -51952.3 q^{64} -21950.8 q^{65} +10079.0 q^{66} -15668.5 q^{67} +8338.07 q^{68} +17346.4 q^{69} -8433.39 q^{70} +10689.5 q^{71} -16238.6 q^{72} +12172.6 q^{73} +17574.6 q^{74} +5625.00 q^{75} -103707. q^{76} -4410.19 q^{77} +73138.1 q^{78} -87205.6 q^{79} +3458.17 q^{80} +6561.00 q^{81} +41649.2 q^{82} +97230.6 q^{83} +17602.3 q^{84} +3884.63 q^{85} +41422.0 q^{86} -4324.00 q^{87} +24257.6 q^{88} +38639.2 q^{89} -18742.0 q^{90} -32002.4 q^{91} +103424. q^{92} +15835.4 q^{93} +114499. q^{94} -48316.2 q^{95} +46214.8 q^{96} -36754.5 q^{97} +143259. q^{98} -9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q - 7 * q^2 + 27 * q^3 + 25 * q^4 + 75 * q^5 - 63 * q^6 - 172 * q^7 - 231 * q^8 + 243 * q^9 $$3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100})$$ 3 * q - 7 * q^2 + 27 * q^3 + 25 * q^4 + 75 * q^5 - 63 * q^6 - 172 * q^7 - 231 * q^8 + 243 * q^9 - 175 * q^10 - 363 * q^11 + 225 * q^12 - 654 * q^13 - 728 * q^14 + 675 * q^15 - 415 * q^16 - 2366 * q^17 - 567 * q^18 - 2872 * q^19 + 625 * q^20 - 1548 * q^21 + 847 * q^22 + 2272 * q^23 - 2079 * q^24 + 1875 * q^25 + 3422 * q^26 + 2187 * q^27 + 4592 * q^28 - 7738 * q^29 - 1575 * q^30 + 568 * q^31 + 1001 * q^32 - 3267 * q^33 + 2506 * q^34 - 4300 * q^35 + 2025 * q^36 - 9126 * q^37 + 13076 * q^38 - 5886 * q^39 - 5775 * q^40 - 8758 * q^41 - 6552 * q^42 - 14672 * q^43 - 3025 * q^44 + 6075 * q^45 - 28768 * q^46 - 19392 * q^47 - 3735 * q^48 - 26629 * q^49 - 4375 * q^50 - 21294 * q^51 - 61506 * q^52 - 4598 * q^53 - 5103 * q^54 - 9075 * q^55 + 2688 * q^56 - 25848 * q^57 + 8550 * q^58 - 9348 * q^59 + 5625 * q^60 - 60078 * q^61 - 14096 * q^62 - 13932 * q^63 - 7087 * q^64 - 16350 * q^65 + 7623 * q^66 - 38468 * q^67 + 59778 * q^68 + 20448 * q^69 - 18200 * q^70 - 74032 * q^71 - 18711 * q^72 - 44442 * q^73 + 82542 * q^74 + 16875 * q^75 - 98708 * q^76 + 20812 * q^77 + 30798 * q^78 - 108116 * q^79 - 10375 * q^80 + 19683 * q^81 - 92230 * q^82 - 81892 * q^83 + 41328 * q^84 - 59150 * q^85 + 126412 * q^86 - 69642 * q^87 + 27951 * q^88 + 167342 * q^89 - 14175 * q^90 - 31832 * q^91 + 72960 * q^92 + 5112 * q^93 + 12728 * q^94 - 71800 * q^95 + 9009 * q^96 + 159702 * q^97 + 163121 * q^98 - 29403 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −9.25531 −1.63612 −0.818061 0.575131i $$-0.804951\pi$$
−0.818061 + 0.575131i $$0.804951\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 53.6607 1.67690
$$5$$ 25.0000 0.447214
$$6$$ −83.2977 −0.944616
$$7$$ 36.4478 0.281142 0.140571 0.990071i $$-0.455106\pi$$
0.140571 + 0.990071i $$0.455106\pi$$
$$8$$ −200.476 −1.10749
$$9$$ 81.0000 0.333333
$$10$$ −231.383 −0.731696
$$11$$ −121.000 −0.301511
$$12$$ 482.946 0.968156
$$13$$ −878.032 −1.44096 −0.720480 0.693476i $$-0.756079\pi$$
−0.720480 + 0.693476i $$0.756079\pi$$
$$14$$ −337.336 −0.459983
$$15$$ 225.000 0.258199
$$16$$ 138.327 0.135085
$$17$$ 155.385 0.130403 0.0652014 0.997872i $$-0.479231\pi$$
0.0652014 + 0.997872i $$0.479231\pi$$
$$18$$ −749.680 −0.545374
$$19$$ −1932.65 −1.22820 −0.614100 0.789228i $$-0.710481\pi$$
−0.614100 + 0.789228i $$0.710481\pi$$
$$20$$ 1341.52 0.749931
$$21$$ 328.030 0.162318
$$22$$ 1119.89 0.493309
$$23$$ 1927.38 0.759709 0.379855 0.925046i $$-0.375974\pi$$
0.379855 + 0.925046i $$0.375974\pi$$
$$24$$ −1804.29 −0.639407
$$25$$ 625.000 0.200000
$$26$$ 8126.46 2.35759
$$27$$ 729.000 0.192450
$$28$$ 1955.81 0.471447
$$29$$ −480.444 −0.106084 −0.0530418 0.998592i $$-0.516892\pi$$
−0.0530418 + 0.998592i $$0.516892\pi$$
$$30$$ −2082.44 −0.422445
$$31$$ 1759.49 0.328838 0.164419 0.986391i $$-0.447425\pi$$
0.164419 + 0.986391i $$0.447425\pi$$
$$32$$ 5134.98 0.886470
$$33$$ −1089.00 −0.174078
$$34$$ −1438.14 −0.213355
$$35$$ 911.195 0.125731
$$36$$ 4346.51 0.558965
$$37$$ −1898.87 −0.228029 −0.114015 0.993479i $$-0.536371\pi$$
−0.114015 + 0.993479i $$0.536371\pi$$
$$38$$ 17887.3 2.00949
$$39$$ −7902.29 −0.831939
$$40$$ −5011.91 −0.495282
$$41$$ −4500.03 −0.418076 −0.209038 0.977907i $$-0.567033\pi$$
−0.209038 + 0.977907i $$0.567033\pi$$
$$42$$ −3036.02 −0.265572
$$43$$ −4475.49 −0.369121 −0.184561 0.982821i $$-0.559086\pi$$
−0.184561 + 0.982821i $$0.559086\pi$$
$$44$$ −6492.94 −0.505603
$$45$$ 2025.00 0.149071
$$46$$ −17838.5 −1.24298
$$47$$ −12371.2 −0.816895 −0.408448 0.912782i $$-0.633930\pi$$
−0.408448 + 0.912782i $$0.633930\pi$$
$$48$$ 1244.94 0.0779912
$$49$$ −15478.6 −0.920959
$$50$$ −5784.57 −0.327224
$$51$$ 1398.47 0.0752881
$$52$$ −47115.8 −2.41634
$$53$$ 2145.12 0.104897 0.0524483 0.998624i $$-0.483298\pi$$
0.0524483 + 0.998624i $$0.483298\pi$$
$$54$$ −6747.12 −0.314872
$$55$$ −3025.00 −0.134840
$$56$$ −7306.92 −0.311361
$$57$$ −17393.8 −0.709102
$$58$$ 4446.66 0.173566
$$59$$ −15857.9 −0.593084 −0.296542 0.955020i $$-0.595834\pi$$
−0.296542 + 0.955020i $$0.595834\pi$$
$$60$$ 12073.7 0.432973
$$61$$ −36447.7 −1.25414 −0.627070 0.778963i $$-0.715746\pi$$
−0.627070 + 0.778963i $$0.715746\pi$$
$$62$$ −16284.6 −0.538020
$$63$$ 2952.27 0.0937141
$$64$$ −51952.3 −1.58546
$$65$$ −21950.8 −0.644417
$$66$$ 10079.0 0.284812
$$67$$ −15668.5 −0.426424 −0.213212 0.977006i $$-0.568393\pi$$
−0.213212 + 0.977006i $$0.568393\pi$$
$$68$$ 8338.07 0.218672
$$69$$ 17346.4 0.438618
$$70$$ −8433.39 −0.205711
$$71$$ 10689.5 0.251659 0.125830 0.992052i $$-0.459841\pi$$
0.125830 + 0.992052i $$0.459841\pi$$
$$72$$ −16238.6 −0.369162
$$73$$ 12172.6 0.267347 0.133674 0.991025i $$-0.457323\pi$$
0.133674 + 0.991025i $$0.457323\pi$$
$$74$$ 17574.6 0.373084
$$75$$ 5625.00 0.115470
$$76$$ −103707. −2.05956
$$77$$ −4410.19 −0.0847676
$$78$$ 73138.1 1.36115
$$79$$ −87205.6 −1.57209 −0.786043 0.618171i $$-0.787874\pi$$
−0.786043 + 0.618171i $$0.787874\pi$$
$$80$$ 3458.17 0.0604117
$$81$$ 6561.00 0.111111
$$82$$ 41649.2 0.684024
$$83$$ 97230.6 1.54920 0.774600 0.632451i $$-0.217951\pi$$
0.774600 + 0.632451i $$0.217951\pi$$
$$84$$ 17602.3 0.272190
$$85$$ 3884.63 0.0583179
$$86$$ 41422.0 0.603928
$$87$$ −4324.00 −0.0612473
$$88$$ 24257.6 0.333919
$$89$$ 38639.2 0.517075 0.258537 0.966001i $$-0.416759\pi$$
0.258537 + 0.966001i $$0.416759\pi$$
$$90$$ −18742.0 −0.243899
$$91$$ −32002.4 −0.405115
$$92$$ 103424. 1.27395
$$93$$ 15835.4 0.189855
$$94$$ 114499. 1.33654
$$95$$ −48316.2 −0.549268
$$96$$ 46214.8 0.511804
$$97$$ −36754.5 −0.396626 −0.198313 0.980139i $$-0.563546\pi$$
−0.198313 + 0.980139i $$0.563546\pi$$
$$98$$ 143259. 1.50680
$$99$$ −9801.00 −0.100504
$$100$$ 33537.9 0.335379
$$101$$ −185487. −1.80929 −0.904647 0.426161i $$-0.859866\pi$$
−0.904647 + 0.426161i $$0.859866\pi$$
$$102$$ −12943.2 −0.123181
$$103$$ 36890.7 0.342629 0.171315 0.985216i $$-0.445199\pi$$
0.171315 + 0.985216i $$0.445199\pi$$
$$104$$ 176025. 1.59584
$$105$$ 8200.76 0.0725907
$$106$$ −19853.7 −0.171624
$$107$$ −124996. −1.05545 −0.527725 0.849415i $$-0.676955\pi$$
−0.527725 + 0.849415i $$0.676955\pi$$
$$108$$ 39118.6 0.322719
$$109$$ −150975. −1.21714 −0.608568 0.793502i $$-0.708256\pi$$
−0.608568 + 0.793502i $$0.708256\pi$$
$$110$$ 27997.3 0.220615
$$111$$ −17089.8 −0.131653
$$112$$ 5041.71 0.0379780
$$113$$ 157970. 1.16380 0.581899 0.813261i $$-0.302310\pi$$
0.581899 + 0.813261i $$0.302310\pi$$
$$114$$ 160985. 1.16018
$$115$$ 48184.5 0.339752
$$116$$ −25781.0 −0.177891
$$117$$ −71120.6 −0.480320
$$118$$ 146770. 0.970359
$$119$$ 5663.45 0.0366618
$$120$$ −45107.1 −0.285951
$$121$$ 14641.0 0.0909091
$$122$$ 337335. 2.05193
$$123$$ −40500.3 −0.241377
$$124$$ 94415.4 0.551428
$$125$$ 15625.0 0.0894427
$$126$$ −27324.2 −0.153328
$$127$$ 268814. 1.47891 0.739457 0.673204i $$-0.235082\pi$$
0.739457 + 0.673204i $$0.235082\pi$$
$$128$$ 316515. 1.70753
$$129$$ −40279.4 −0.213112
$$130$$ 203161. 1.05435
$$131$$ −366914. −1.86804 −0.934020 0.357220i $$-0.883725\pi$$
−0.934020 + 0.357220i $$0.883725\pi$$
$$132$$ −58436.5 −0.291910
$$133$$ −70440.9 −0.345299
$$134$$ 145017. 0.697682
$$135$$ 18225.0 0.0860663
$$136$$ −31151.0 −0.144419
$$137$$ −182927. −0.832678 −0.416339 0.909209i $$-0.636687\pi$$
−0.416339 + 0.909209i $$0.636687\pi$$
$$138$$ −160546. −0.717633
$$139$$ 8429.85 0.0370069 0.0185035 0.999829i $$-0.494110\pi$$
0.0185035 + 0.999829i $$0.494110\pi$$
$$140$$ 48895.4 0.210837
$$141$$ −111341. −0.471635
$$142$$ −98934.8 −0.411745
$$143$$ 106242. 0.434466
$$144$$ 11204.5 0.0450282
$$145$$ −12011.1 −0.0474420
$$146$$ −112661. −0.437413
$$147$$ −139307. −0.531716
$$148$$ −101895. −0.382381
$$149$$ 271810. 1.00300 0.501499 0.865158i $$-0.332782\pi$$
0.501499 + 0.865158i $$0.332782\pi$$
$$150$$ −52061.1 −0.188923
$$151$$ −236565. −0.844322 −0.422161 0.906521i $$-0.638728\pi$$
−0.422161 + 0.906521i $$0.638728\pi$$
$$152$$ 387450. 1.36021
$$153$$ 12586.2 0.0434676
$$154$$ 40817.6 0.138690
$$155$$ 43987.2 0.147061
$$156$$ −424042. −1.39508
$$157$$ 211824. 0.685846 0.342923 0.939364i $$-0.388583\pi$$
0.342923 + 0.939364i $$0.388583\pi$$
$$158$$ 807114. 2.57213
$$159$$ 19306.1 0.0605621
$$160$$ 128375. 0.396441
$$161$$ 70248.7 0.213586
$$162$$ −60724.1 −0.181791
$$163$$ −341315. −1.00620 −0.503102 0.864227i $$-0.667808\pi$$
−0.503102 + 0.864227i $$0.667808\pi$$
$$164$$ −241475. −0.701071
$$165$$ −27225.0 −0.0778499
$$166$$ −899899. −2.53468
$$167$$ −180548. −0.500958 −0.250479 0.968122i $$-0.580588\pi$$
−0.250479 + 0.968122i $$0.580588\pi$$
$$168$$ −65762.3 −0.179764
$$169$$ 399647. 1.07637
$$170$$ −35953.4 −0.0954153
$$171$$ −156545. −0.409400
$$172$$ −240158. −0.618978
$$173$$ −643322. −1.63423 −0.817114 0.576476i $$-0.804428\pi$$
−0.817114 + 0.576476i $$0.804428\pi$$
$$174$$ 40019.9 0.100208
$$175$$ 22779.9 0.0562285
$$176$$ −16737.5 −0.0407296
$$177$$ −142721. −0.342417
$$178$$ −357618. −0.845998
$$179$$ −266922. −0.622662 −0.311331 0.950302i $$-0.600775\pi$$
−0.311331 + 0.950302i $$0.600775\pi$$
$$180$$ 108663. 0.249977
$$181$$ 281529. 0.638745 0.319372 0.947629i $$-0.396528\pi$$
0.319372 + 0.947629i $$0.396528\pi$$
$$182$$ 296192. 0.662818
$$183$$ −328030. −0.724078
$$184$$ −386393. −0.841366
$$185$$ −47471.7 −0.101978
$$186$$ −146561. −0.310626
$$187$$ −18801.6 −0.0393179
$$188$$ −663846. −1.36985
$$189$$ 26570.5 0.0541059
$$190$$ 447182. 0.898669
$$191$$ 933723. 1.85197 0.925987 0.377556i $$-0.123235\pi$$
0.925987 + 0.377556i $$0.123235\pi$$
$$192$$ −467571. −0.915365
$$193$$ 300124. 0.579973 0.289986 0.957031i $$-0.406349\pi$$
0.289986 + 0.957031i $$0.406349\pi$$
$$194$$ 340174. 0.648929
$$195$$ −197557. −0.372054
$$196$$ −830590. −1.54435
$$197$$ −944940. −1.73476 −0.867378 0.497649i $$-0.834197\pi$$
−0.867378 + 0.497649i $$0.834197\pi$$
$$198$$ 90711.2 0.164436
$$199$$ −1.00821e6 −1.80475 −0.902374 0.430954i $$-0.858177\pi$$
−0.902374 + 0.430954i $$0.858177\pi$$
$$200$$ −125298. −0.221497
$$201$$ −141017. −0.246196
$$202$$ 1.71674e6 2.96023
$$203$$ −17511.1 −0.0298246
$$204$$ 75042.6 0.126250
$$205$$ −112501. −0.186969
$$206$$ −341435. −0.560583
$$207$$ 156118. 0.253236
$$208$$ −121455. −0.194652
$$209$$ 233851. 0.370316
$$210$$ −75900.5 −0.118767
$$211$$ 497479. 0.769253 0.384626 0.923072i $$-0.374330\pi$$
0.384626 + 0.923072i $$0.374330\pi$$
$$212$$ 115108. 0.175901
$$213$$ 96205.7 0.145295
$$214$$ 1.15688e6 1.72684
$$215$$ −111887. −0.165076
$$216$$ −146147. −0.213136
$$217$$ 64129.5 0.0924504
$$218$$ 1.39732e6 1.99138
$$219$$ 109553. 0.154353
$$220$$ −162324. −0.226113
$$221$$ −136433. −0.187905
$$222$$ 158172. 0.215400
$$223$$ 1.14136e6 1.53695 0.768477 0.639878i $$-0.221015\pi$$
0.768477 + 0.639878i $$0.221015\pi$$
$$224$$ 187159. 0.249224
$$225$$ 50625.0 0.0666667
$$226$$ −1.46206e6 −1.90411
$$227$$ 669451. 0.862292 0.431146 0.902282i $$-0.358109\pi$$
0.431146 + 0.902282i $$0.358109\pi$$
$$228$$ −933366. −1.18909
$$229$$ 588061. 0.741026 0.370513 0.928827i $$-0.379182\pi$$
0.370513 + 0.928827i $$0.379182\pi$$
$$230$$ −445962. −0.555876
$$231$$ −39691.7 −0.0489406
$$232$$ 96317.6 0.117486
$$233$$ 199417. 0.240642 0.120321 0.992735i $$-0.461608\pi$$
0.120321 + 0.992735i $$0.461608\pi$$
$$234$$ 658243. 0.785862
$$235$$ −309279. −0.365327
$$236$$ −850947. −0.994541
$$237$$ −784850. −0.907645
$$238$$ −52416.9 −0.0599832
$$239$$ −408055. −0.462088 −0.231044 0.972943i $$-0.574214\pi$$
−0.231044 + 0.972943i $$0.574214\pi$$
$$240$$ 31123.5 0.0348787
$$241$$ 1.24022e6 1.37548 0.687742 0.725956i $$-0.258602\pi$$
0.687742 + 0.725956i $$0.258602\pi$$
$$242$$ −135507. −0.148738
$$243$$ 59049.0 0.0641500
$$244$$ −1.95581e6 −2.10306
$$245$$ −386964. −0.411865
$$246$$ 374842. 0.394922
$$247$$ 1.69693e6 1.76979
$$248$$ −352736. −0.364183
$$249$$ 875075. 0.894431
$$250$$ −144614. −0.146339
$$251$$ 30660.6 0.0307183 0.0153591 0.999882i $$-0.495111\pi$$
0.0153591 + 0.999882i $$0.495111\pi$$
$$252$$ 158421. 0.157149
$$253$$ −233213. −0.229061
$$254$$ −2.48796e6 −2.41968
$$255$$ 34961.7 0.0336699
$$256$$ −1.26697e6 −1.20828
$$257$$ −687971. −0.649737 −0.324868 0.945759i $$-0.605320\pi$$
−0.324868 + 0.945759i $$0.605320\pi$$
$$258$$ 372798. 0.348678
$$259$$ −69209.6 −0.0641087
$$260$$ −1.17790e6 −1.08062
$$261$$ −38916.0 −0.0353612
$$262$$ 3.39590e6 3.05634
$$263$$ 1.70103e6 1.51643 0.758216 0.652004i $$-0.226071\pi$$
0.758216 + 0.652004i $$0.226071\pi$$
$$264$$ 218319. 0.192788
$$265$$ 53628.0 0.0469112
$$266$$ 651952. 0.564952
$$267$$ 347753. 0.298533
$$268$$ −840785. −0.715069
$$269$$ −982644. −0.827972 −0.413986 0.910283i $$-0.635864\pi$$
−0.413986 + 0.910283i $$0.635864\pi$$
$$270$$ −168678. −0.140815
$$271$$ −276206. −0.228459 −0.114230 0.993454i $$-0.536440\pi$$
−0.114230 + 0.993454i $$0.536440\pi$$
$$272$$ 21493.9 0.0176154
$$273$$ −288021. −0.233893
$$274$$ 1.69305e6 1.36236
$$275$$ −75625.0 −0.0603023
$$276$$ 930820. 0.735517
$$277$$ 148776. 0.116502 0.0582509 0.998302i $$-0.481448\pi$$
0.0582509 + 0.998302i $$0.481448\pi$$
$$278$$ −78020.8 −0.0605478
$$279$$ 142519. 0.109613
$$280$$ −182673. −0.139245
$$281$$ 357772. 0.270297 0.135148 0.990825i $$-0.456849\pi$$
0.135148 + 0.990825i $$0.456849\pi$$
$$282$$ 1.03049e6 0.771652
$$283$$ −492090. −0.365240 −0.182620 0.983184i $$-0.558458\pi$$
−0.182620 + 0.983184i $$0.558458\pi$$
$$284$$ 573607. 0.422006
$$285$$ −434846. −0.317120
$$286$$ −983301. −0.710839
$$287$$ −164016. −0.117539
$$288$$ 415934. 0.295490
$$289$$ −1.39571e6 −0.982995
$$290$$ 111166. 0.0776209
$$291$$ −330791. −0.228992
$$292$$ 653189. 0.448314
$$293$$ 498120. 0.338973 0.169487 0.985532i $$-0.445789\pi$$
0.169487 + 0.985532i $$0.445789\pi$$
$$294$$ 1.28933e6 0.869952
$$295$$ −396448. −0.265235
$$296$$ 380678. 0.252539
$$297$$ −88209.0 −0.0580259
$$298$$ −2.51568e6 −1.64103
$$299$$ −1.69230e6 −1.09471
$$300$$ 301841. 0.193631
$$301$$ −163122. −0.103776
$$302$$ 2.18948e6 1.38141
$$303$$ −1.66938e6 −1.04460
$$304$$ −267337. −0.165911
$$305$$ −911194. −0.560869
$$306$$ −116489. −0.0711183
$$307$$ −998760. −0.604805 −0.302402 0.953180i $$-0.597789\pi$$
−0.302402 + 0.953180i $$0.597789\pi$$
$$308$$ −236654. −0.142147
$$309$$ 332017. 0.197817
$$310$$ −407115. −0.240610
$$311$$ 1.88783e6 1.10678 0.553389 0.832923i $$-0.313334\pi$$
0.553389 + 0.832923i $$0.313334\pi$$
$$312$$ 1.58422e6 0.921360
$$313$$ −2.34787e6 −1.35461 −0.677303 0.735704i $$-0.736851\pi$$
−0.677303 + 0.735704i $$0.736851\pi$$
$$314$$ −1.96050e6 −1.12213
$$315$$ 73806.8 0.0419102
$$316$$ −4.67951e6 −2.63623
$$317$$ 562721. 0.314518 0.157259 0.987557i $$-0.449734\pi$$
0.157259 + 0.987557i $$0.449734\pi$$
$$318$$ −178684. −0.0990870
$$319$$ 58133.7 0.0319854
$$320$$ −1.29881e6 −0.709038
$$321$$ −1.12497e6 −0.609364
$$322$$ −650173. −0.349454
$$323$$ −300305. −0.160161
$$324$$ 352068. 0.186322
$$325$$ −548770. −0.288192
$$326$$ 3.15897e6 1.64627
$$327$$ −1.35878e6 −0.702713
$$328$$ 902149. 0.463013
$$329$$ −450902. −0.229664
$$330$$ 251976. 0.127372
$$331$$ −1.00593e6 −0.504657 −0.252328 0.967642i $$-0.581196\pi$$
−0.252328 + 0.967642i $$0.581196\pi$$
$$332$$ 5.21746e6 2.59785
$$333$$ −153808. −0.0760098
$$334$$ 1.67103e6 0.819628
$$335$$ −391714. −0.190703
$$336$$ 45375.4 0.0219266
$$337$$ −280192. −0.134394 −0.0671971 0.997740i $$-0.521406\pi$$
−0.0671971 + 0.997740i $$0.521406\pi$$
$$338$$ −3.69886e6 −1.76107
$$339$$ 1.42173e6 0.671919
$$340$$ 208452. 0.0977931
$$341$$ −212898. −0.0991485
$$342$$ 1.44887e6 0.669829
$$343$$ −1.17674e6 −0.540063
$$344$$ 897229. 0.408796
$$345$$ 433660. 0.196156
$$346$$ 5.95414e6 2.67380
$$347$$ 2.10913e6 0.940328 0.470164 0.882579i $$-0.344195\pi$$
0.470164 + 0.882579i $$0.344195\pi$$
$$348$$ −232029. −0.102705
$$349$$ 3.88469e6 1.70723 0.853617 0.520901i $$-0.174404\pi$$
0.853617 + 0.520901i $$0.174404\pi$$
$$350$$ −210835. −0.0919967
$$351$$ −640085. −0.277313
$$352$$ −621333. −0.267281
$$353$$ 1.35663e6 0.579463 0.289732 0.957108i $$-0.406434\pi$$
0.289732 + 0.957108i $$0.406434\pi$$
$$354$$ 1.32093e6 0.560237
$$355$$ 267238. 0.112545
$$356$$ 2.07341e6 0.867081
$$357$$ 50971.0 0.0211667
$$358$$ 2.47045e6 1.01875
$$359$$ 3.93436e6 1.61116 0.805579 0.592488i $$-0.201854\pi$$
0.805579 + 0.592488i $$0.201854\pi$$
$$360$$ −405964. −0.165094
$$361$$ 1.25904e6 0.508476
$$362$$ −2.60564e6 −1.04506
$$363$$ 131769. 0.0524864
$$364$$ −1.71727e6 −0.679336
$$365$$ 304315. 0.119561
$$366$$ 3.03602e6 1.18468
$$367$$ −2.82588e6 −1.09519 −0.547594 0.836744i $$-0.684456\pi$$
−0.547594 + 0.836744i $$0.684456\pi$$
$$368$$ 266608. 0.102625
$$369$$ −364502. −0.139359
$$370$$ 439365. 0.166848
$$371$$ 78184.9 0.0294909
$$372$$ 849738. 0.318367
$$373$$ 4.58790e6 1.70743 0.853713 0.520744i $$-0.174345\pi$$
0.853713 + 0.520744i $$0.174345\pi$$
$$374$$ 174015. 0.0643290
$$375$$ 140625. 0.0516398
$$376$$ 2.48013e6 0.904699
$$377$$ 421845. 0.152862
$$378$$ −245918. −0.0885239
$$379$$ 2.84827e6 1.01855 0.509277 0.860603i $$-0.329913\pi$$
0.509277 + 0.860603i $$0.329913\pi$$
$$380$$ −2.59268e6 −0.921065
$$381$$ 2.41933e6 0.853852
$$382$$ −8.64190e6 −3.03006
$$383$$ −2.78467e6 −0.970013 −0.485006 0.874511i $$-0.661183\pi$$
−0.485006 + 0.874511i $$0.661183\pi$$
$$384$$ 2.84863e6 0.985845
$$385$$ −110255. −0.0379092
$$386$$ −2.77774e6 −0.948906
$$387$$ −362514. −0.123040
$$388$$ −1.97227e6 −0.665101
$$389$$ 3.95277e6 1.32442 0.662212 0.749316i $$-0.269618\pi$$
0.662212 + 0.749316i $$0.269618\pi$$
$$390$$ 1.82845e6 0.608726
$$391$$ 299486. 0.0990682
$$392$$ 3.10308e6 1.01995
$$393$$ −3.30223e6 −1.07851
$$394$$ 8.74571e6 2.83827
$$395$$ −2.18014e6 −0.703059
$$396$$ −525928. −0.168534
$$397$$ −5.55351e6 −1.76844 −0.884221 0.467068i $$-0.845310\pi$$
−0.884221 + 0.467068i $$0.845310\pi$$
$$398$$ 9.33125e6 2.95279
$$399$$ −633968. −0.199359
$$400$$ 86454.2 0.0270169
$$401$$ −279266. −0.0867277 −0.0433639 0.999059i $$-0.513807\pi$$
−0.0433639 + 0.999059i $$0.513807\pi$$
$$402$$ 1.30515e6 0.402807
$$403$$ −1.54489e6 −0.473843
$$404$$ −9.95334e6 −3.03400
$$405$$ 164025. 0.0496904
$$406$$ 162071. 0.0487967
$$407$$ 229763. 0.0687534
$$408$$ −280359. −0.0833805
$$409$$ 5.17128e6 1.52859 0.764293 0.644869i $$-0.223088\pi$$
0.764293 + 0.644869i $$0.223088\pi$$
$$410$$ 1.04123e6 0.305905
$$411$$ −1.64635e6 −0.480747
$$412$$ 1.97958e6 0.574554
$$413$$ −577987. −0.166741
$$414$$ −1.44492e6 −0.414326
$$415$$ 2.43076e6 0.692824
$$416$$ −4.50868e6 −1.27737
$$417$$ 75868.7 0.0213660
$$418$$ −2.16436e6 −0.605883
$$419$$ −6.04152e6 −1.68117 −0.840585 0.541680i $$-0.817788\pi$$
−0.840585 + 0.541680i $$0.817788\pi$$
$$420$$ 440058. 0.121727
$$421$$ −895386. −0.246210 −0.123105 0.992394i $$-0.539285\pi$$
−0.123105 + 0.992394i $$0.539285\pi$$
$$422$$ −4.60432e6 −1.25859
$$423$$ −1.00207e6 −0.272298
$$424$$ −430045. −0.116171
$$425$$ 97115.7 0.0260806
$$426$$ −890413. −0.237721
$$427$$ −1.32844e6 −0.352592
$$428$$ −6.70738e6 −1.76988
$$429$$ 956177. 0.250839
$$430$$ 1.03555e6 0.270085
$$431$$ −1.60867e6 −0.417132 −0.208566 0.978008i $$-0.566880\pi$$
−0.208566 + 0.978008i $$0.566880\pi$$
$$432$$ 100840. 0.0259971
$$433$$ 1.86039e6 0.476853 0.238427 0.971161i $$-0.423368\pi$$
0.238427 + 0.971161i $$0.423368\pi$$
$$434$$ −593538. −0.151260
$$435$$ −108100. −0.0273906
$$436$$ −8.10142e6 −2.04101
$$437$$ −3.72495e6 −0.933075
$$438$$ −1.01395e6 −0.252540
$$439$$ 2.75051e6 0.681164 0.340582 0.940215i $$-0.389376\pi$$
0.340582 + 0.940215i $$0.389376\pi$$
$$440$$ 606441. 0.149333
$$441$$ −1.25376e6 −0.306986
$$442$$ 1.26273e6 0.307436
$$443$$ 3.15804e6 0.764555 0.382278 0.924048i $$-0.375140\pi$$
0.382278 + 0.924048i $$0.375140\pi$$
$$444$$ −917051. −0.220768
$$445$$ 965981. 0.231243
$$446$$ −1.05636e7 −2.51464
$$447$$ 2.44629e6 0.579081
$$448$$ −1.89355e6 −0.445740
$$449$$ −127508. −0.0298484 −0.0149242 0.999889i $$-0.504751\pi$$
−0.0149242 + 0.999889i $$0.504751\pi$$
$$450$$ −468550. −0.109075
$$451$$ 544504. 0.126055
$$452$$ 8.47675e6 1.95157
$$453$$ −2.12908e6 −0.487469
$$454$$ −6.19598e6 −1.41082
$$455$$ −800059. −0.181173
$$456$$ 3.48705e6 0.785319
$$457$$ 2.19209e6 0.490984 0.245492 0.969399i $$-0.421050\pi$$
0.245492 + 0.969399i $$0.421050\pi$$
$$458$$ −5.44268e6 −1.21241
$$459$$ 113276. 0.0250960
$$460$$ 2.58561e6 0.569729
$$461$$ 5.20229e6 1.14010 0.570050 0.821610i $$-0.306924\pi$$
0.570050 + 0.821610i $$0.306924\pi$$
$$462$$ 367359. 0.0800728
$$463$$ 2.66624e6 0.578025 0.289013 0.957325i $$-0.406673\pi$$
0.289013 + 0.957325i $$0.406673\pi$$
$$464$$ −66458.3 −0.0143303
$$465$$ 395885. 0.0849057
$$466$$ −1.84566e6 −0.393720
$$467$$ −2.35578e6 −0.499853 −0.249926 0.968265i $$-0.580406\pi$$
−0.249926 + 0.968265i $$0.580406\pi$$
$$468$$ −3.81638e6 −0.805447
$$469$$ −571084. −0.119886
$$470$$ 2.86247e6 0.597719
$$471$$ 1.90642e6 0.395973
$$472$$ 3.17914e6 0.656832
$$473$$ 541534. 0.111294
$$474$$ 7.26403e6 1.48502
$$475$$ −1.20791e6 −0.245640
$$476$$ 303905. 0.0614780
$$477$$ 173755. 0.0349655
$$478$$ 3.77668e6 0.756032
$$479$$ −4.78329e6 −0.952551 −0.476276 0.879296i $$-0.658014\pi$$
−0.476276 + 0.879296i $$0.658014\pi$$
$$480$$ 1.15537e6 0.228886
$$481$$ 1.66727e6 0.328581
$$482$$ −1.14786e7 −2.25046
$$483$$ 632239. 0.123314
$$484$$ 785646. 0.152445
$$485$$ −918863. −0.177377
$$486$$ −546517. −0.104957
$$487$$ −3.31515e6 −0.633405 −0.316702 0.948525i $$-0.602576\pi$$
−0.316702 + 0.948525i $$0.602576\pi$$
$$488$$ 7.30691e6 1.38894
$$489$$ −3.07183e6 −0.580932
$$490$$ 3.58147e6 0.673862
$$491$$ −3.02276e6 −0.565847 −0.282924 0.959142i $$-0.591304\pi$$
−0.282924 + 0.959142i $$0.591304\pi$$
$$492$$ −2.17327e6 −0.404763
$$493$$ −74653.9 −0.0138336
$$494$$ −1.57056e7 −2.89559
$$495$$ −245025. −0.0449467
$$496$$ 243384. 0.0444210
$$497$$ 389610. 0.0707520
$$498$$ −8.09909e6 −1.46340
$$499$$ 4.46530e6 0.802785 0.401392 0.915906i $$-0.368526\pi$$
0.401392 + 0.915906i $$0.368526\pi$$
$$500$$ 838448. 0.149986
$$501$$ −1.62493e6 −0.289228
$$502$$ −283774. −0.0502589
$$503$$ −1.77528e6 −0.312858 −0.156429 0.987689i $$-0.549998\pi$$
−0.156429 + 0.987689i $$0.549998\pi$$
$$504$$ −591861. −0.103787
$$505$$ −4.63717e6 −0.809141
$$506$$ 2.15846e6 0.374772
$$507$$ 3.59683e6 0.621441
$$508$$ 1.44248e7 2.47999
$$509$$ −485180. −0.0830058 −0.0415029 0.999138i $$-0.513215\pi$$
−0.0415029 + 0.999138i $$0.513215\pi$$
$$510$$ −323581. −0.0550880
$$511$$ 443664. 0.0751627
$$512$$ 1.59770e6 0.269353
$$513$$ −1.40890e6 −0.236367
$$514$$ 6.36739e6 1.06305
$$515$$ 922269. 0.153228
$$516$$ −2.16142e6 −0.357367
$$517$$ 1.49691e6 0.246303
$$518$$ 640556. 0.104890
$$519$$ −5.78989e6 −0.943522
$$520$$ 4.40061e6 0.713682
$$521$$ 9.92932e6 1.60260 0.801300 0.598262i $$-0.204142\pi$$
0.801300 + 0.598262i $$0.204142\pi$$
$$522$$ 360179. 0.0578552
$$523$$ 5.04767e6 0.806931 0.403466 0.914995i $$-0.367805\pi$$
0.403466 + 0.914995i $$0.367805\pi$$
$$524$$ −1.96889e7 −3.13251
$$525$$ 205019. 0.0324635
$$526$$ −1.57436e7 −2.48107
$$527$$ 273398. 0.0428815
$$528$$ −150638. −0.0235152
$$529$$ −2.72156e6 −0.422842
$$530$$ −496343. −0.0767525
$$531$$ −1.28449e6 −0.197695
$$532$$ −3.77990e6 −0.579031
$$533$$ 3.95117e6 0.602432
$$534$$ −3.21856e6 −0.488437
$$535$$ −3.12491e6 −0.472011
$$536$$ 3.14117e6 0.472258
$$537$$ −2.40230e6 −0.359494
$$538$$ 9.09467e6 1.35466
$$539$$ 1.87291e6 0.277680
$$540$$ 977966. 0.144324
$$541$$ −4.46393e6 −0.655729 −0.327864 0.944725i $$-0.606329\pi$$
−0.327864 + 0.944725i $$0.606329\pi$$
$$542$$ 2.55637e6 0.373788
$$543$$ 2.53376e6 0.368779
$$544$$ 797900. 0.115598
$$545$$ −3.77438e6 −0.544319
$$546$$ 2.66572e6 0.382678
$$547$$ −6.62530e6 −0.946755 −0.473377 0.880860i $$-0.656965\pi$$
−0.473377 + 0.880860i $$0.656965\pi$$
$$548$$ −9.81601e6 −1.39632
$$549$$ −2.95227e6 −0.418047
$$550$$ 699932. 0.0986619
$$551$$ 928530. 0.130292
$$552$$ −3.47754e6 −0.485763
$$553$$ −3.17845e6 −0.441980
$$554$$ −1.37696e6 −0.190611
$$555$$ −427246. −0.0588769
$$556$$ 452351. 0.0620568
$$557$$ −717580. −0.0980014 −0.0490007 0.998799i $$-0.515604\pi$$
−0.0490007 + 0.998799i $$0.515604\pi$$
$$558$$ −1.31905e6 −0.179340
$$559$$ 3.92962e6 0.531889
$$560$$ 126043. 0.0169843
$$561$$ −169214. −0.0227002
$$562$$ −3.31129e6 −0.442238
$$563$$ 1.35097e7 1.79629 0.898143 0.439703i $$-0.144916\pi$$
0.898143 + 0.439703i $$0.144916\pi$$
$$564$$ −5.97461e6 −0.790882
$$565$$ 3.94924e6 0.520466
$$566$$ 4.55444e6 0.597577
$$567$$ 239134. 0.0312380
$$568$$ −2.14300e6 −0.278709
$$569$$ 1.35535e7 1.75498 0.877488 0.479598i $$-0.159218\pi$$
0.877488 + 0.479598i $$0.159218\pi$$
$$570$$ 4.02463e6 0.518847
$$571$$ 7.82130e6 1.00390 0.501948 0.864898i $$-0.332617\pi$$
0.501948 + 0.864898i $$0.332617\pi$$
$$572$$ 5.70101e6 0.728554
$$573$$ 8.40351e6 1.06924
$$574$$ 1.51802e6 0.192308
$$575$$ 1.20461e6 0.151942
$$576$$ −4.20814e6 −0.528486
$$577$$ 218443. 0.0273149 0.0136574 0.999907i $$-0.495653\pi$$
0.0136574 + 0.999907i $$0.495653\pi$$
$$578$$ 1.29177e7 1.60830
$$579$$ 2.70112e6 0.334847
$$580$$ −644524. −0.0795553
$$581$$ 3.54384e6 0.435546
$$582$$ 3.06157e6 0.374659
$$583$$ −259559. −0.0316275
$$584$$ −2.44031e6 −0.296083
$$585$$ −1.77802e6 −0.214806
$$586$$ −4.61026e6 −0.554602
$$587$$ −5.46003e6 −0.654033 −0.327017 0.945019i $$-0.606043\pi$$
−0.327017 + 0.945019i $$0.606043\pi$$
$$588$$ −7.47531e6 −0.891632
$$589$$ −3.40048e6 −0.403879
$$590$$ 3.66925e6 0.433958
$$591$$ −8.50446e6 −1.00156
$$592$$ −262664. −0.0308033
$$593$$ 1.41186e7 1.64875 0.824375 0.566044i $$-0.191527\pi$$
0.824375 + 0.566044i $$0.191527\pi$$
$$594$$ 816401. 0.0949374
$$595$$ 141586. 0.0163956
$$596$$ 1.45855e7 1.68192
$$597$$ −9.07385e6 −1.04197
$$598$$ 1.56628e7 1.79108
$$599$$ 8.02044e6 0.913338 0.456669 0.889637i $$-0.349042\pi$$
0.456669 + 0.889637i $$0.349042\pi$$
$$600$$ −1.12768e6 −0.127881
$$601$$ −1.20301e7 −1.35857 −0.679286 0.733874i $$-0.737710\pi$$
−0.679286 + 0.733874i $$0.737710\pi$$
$$602$$ 1.50974e6 0.169790
$$603$$ −1.26915e6 −0.142141
$$604$$ −1.26942e7 −1.41584
$$605$$ 366025. 0.0406558
$$606$$ 1.54506e7 1.70909
$$607$$ −1.58863e7 −1.75005 −0.875025 0.484078i $$-0.839155\pi$$
−0.875025 + 0.484078i $$0.839155\pi$$
$$608$$ −9.92412e6 −1.08876
$$609$$ −157600. −0.0172192
$$610$$ 8.43338e6 0.917650
$$611$$ 1.08623e7 1.17711
$$612$$ 675384. 0.0728907
$$613$$ 1.27701e7 1.37260 0.686301 0.727318i $$-0.259233\pi$$
0.686301 + 0.727318i $$0.259233\pi$$
$$614$$ 9.24383e6 0.989535
$$615$$ −1.01251e6 −0.107947
$$616$$ 884137. 0.0938789
$$617$$ 5.32363e6 0.562982 0.281491 0.959564i $$-0.409171\pi$$
0.281491 + 0.959564i $$0.409171\pi$$
$$618$$ −3.07292e6 −0.323653
$$619$$ −5.79882e6 −0.608293 −0.304147 0.952625i $$-0.598371\pi$$
−0.304147 + 0.952625i $$0.598371\pi$$
$$620$$ 2.36038e6 0.246606
$$621$$ 1.40506e6 0.146206
$$622$$ −1.74724e7 −1.81083
$$623$$ 1.40832e6 0.145372
$$624$$ −1.09310e6 −0.112382
$$625$$ 390625. 0.0400000
$$626$$ 2.17302e7 2.21630
$$627$$ 2.10466e6 0.213802
$$628$$ 1.13666e7 1.15009
$$629$$ −295056. −0.0297357
$$630$$ −683105. −0.0685703
$$631$$ −8.25264e6 −0.825124 −0.412562 0.910929i $$-0.635366\pi$$
−0.412562 + 0.910929i $$0.635366\pi$$
$$632$$ 1.74826e7 1.74106
$$633$$ 4.47731e6 0.444128
$$634$$ −5.20816e6 −0.514590
$$635$$ 6.72036e6 0.661391
$$636$$ 1.03598e6 0.101556
$$637$$ 1.35907e7 1.32707
$$638$$ −538045. −0.0523320
$$639$$ 865852. 0.0838864
$$640$$ 7.91287e6 0.763632
$$641$$ 1.14111e7 1.09694 0.548471 0.836169i $$-0.315210\pi$$
0.548471 + 0.836169i $$0.315210\pi$$
$$642$$ 1.04119e7 0.996994
$$643$$ −3.26961e6 −0.311866 −0.155933 0.987768i $$-0.549838\pi$$
−0.155933 + 0.987768i $$0.549838\pi$$
$$644$$ 3.76959e6 0.358162
$$645$$ −1.00698e6 −0.0953067
$$646$$ 2.77941e6 0.262043
$$647$$ −9.95068e6 −0.934527 −0.467264 0.884118i $$-0.654760\pi$$
−0.467264 + 0.884118i $$0.654760\pi$$
$$648$$ −1.31532e6 −0.123054
$$649$$ 1.91881e6 0.178822
$$650$$ 5.07903e6 0.471517
$$651$$ 577166. 0.0533763
$$652$$ −1.83152e7 −1.68730
$$653$$ 1.52022e7 1.39515 0.697577 0.716509i $$-0.254261\pi$$
0.697577 + 0.716509i $$0.254261\pi$$
$$654$$ 1.25759e7 1.14973
$$655$$ −9.17285e6 −0.835413
$$656$$ −622474. −0.0564757
$$657$$ 985979. 0.0891157
$$658$$ 4.17324e6 0.375758
$$659$$ −9.63232e6 −0.864007 −0.432004 0.901872i $$-0.642193\pi$$
−0.432004 + 0.901872i $$0.642193\pi$$
$$660$$ −1.46091e6 −0.130546
$$661$$ 2.04631e7 1.82166 0.910832 0.412778i $$-0.135442\pi$$
0.910832 + 0.412778i $$0.135442\pi$$
$$662$$ 9.31016e6 0.825681
$$663$$ −1.22790e6 −0.108487
$$664$$ −1.94924e7 −1.71572
$$665$$ −1.76102e6 −0.154422
$$666$$ 1.42354e6 0.124361
$$667$$ −925997. −0.0805926
$$668$$ −9.68832e6 −0.840054
$$669$$ 1.02722e7 0.887361
$$670$$ 3.62543e6 0.312013
$$671$$ 4.41018e6 0.378137
$$672$$ 1.68443e6 0.143890
$$673$$ 1.57773e6 0.134275 0.0671376 0.997744i $$-0.478613\pi$$
0.0671376 + 0.997744i $$0.478613\pi$$
$$674$$ 2.59326e6 0.219885
$$675$$ 455625. 0.0384900
$$676$$ 2.14454e7 1.80496
$$677$$ 6.92750e6 0.580904 0.290452 0.956890i $$-0.406194\pi$$
0.290452 + 0.956890i $$0.406194\pi$$
$$678$$ −1.31585e7 −1.09934
$$679$$ −1.33962e6 −0.111509
$$680$$ −778776. −0.0645862
$$681$$ 6.02506e6 0.497845
$$682$$ 1.97044e6 0.162219
$$683$$ −277554. −0.0227665 −0.0113833 0.999935i $$-0.503623\pi$$
−0.0113833 + 0.999935i $$0.503623\pi$$
$$684$$ −8.40029e6 −0.686521
$$685$$ −4.57318e6 −0.372385
$$686$$ 1.08911e7 0.883609
$$687$$ 5.29255e6 0.427832
$$688$$ −619080. −0.0498627
$$689$$ −1.88348e6 −0.151152
$$690$$ −4.01366e6 −0.320935
$$691$$ −2.12446e7 −1.69259 −0.846297 0.532712i $$-0.821173\pi$$
−0.846297 + 0.532712i $$0.821173\pi$$
$$692$$ −3.45211e7 −2.74043
$$693$$ −357225. −0.0282559
$$694$$ −1.95206e7 −1.53849
$$695$$ 210746. 0.0165500
$$696$$ 866858. 0.0678305
$$697$$ −699238. −0.0545184
$$698$$ −3.59540e7 −2.79324
$$699$$ 1.79475e6 0.138935
$$700$$ 1.22238e6 0.0942893
$$701$$ −1.06971e7 −0.822184 −0.411092 0.911594i $$-0.634853\pi$$
−0.411092 + 0.911594i $$0.634853\pi$$
$$702$$ 5.92419e6 0.453718
$$703$$ 3.66985e6 0.280066
$$704$$ 6.28623e6 0.478034
$$705$$ −2.78351e6 −0.210921
$$706$$ −1.25561e7 −0.948072
$$707$$ −6.76058e6 −0.508669
$$708$$ −7.65853e6 −0.574199
$$709$$ 6.52521e6 0.487505 0.243752 0.969838i $$-0.421622\pi$$
0.243752 + 0.969838i $$0.421622\pi$$
$$710$$ −2.47337e6 −0.184138
$$711$$ −7.06365e6 −0.524029
$$712$$ −7.74625e6 −0.572653
$$713$$ 3.39120e6 0.249821
$$714$$ −471753. −0.0346313
$$715$$ 2.65605e6 0.194299
$$716$$ −1.43232e7 −1.04414
$$717$$ −3.67250e6 −0.266786
$$718$$ −3.64137e7 −2.63605
$$719$$ −2.12413e7 −1.53235 −0.766176 0.642631i $$-0.777843\pi$$
−0.766176 + 0.642631i $$0.777843\pi$$
$$720$$ 280112. 0.0201372
$$721$$ 1.34459e6 0.0963276
$$722$$ −1.16528e7 −0.831928
$$723$$ 1.11620e7 0.794135
$$724$$ 1.51071e7 1.07111
$$725$$ −300278. −0.0212167
$$726$$ −1.21956e6 −0.0858742
$$727$$ −1.06687e7 −0.748646 −0.374323 0.927298i $$-0.622125\pi$$
−0.374323 + 0.927298i $$0.622125\pi$$
$$728$$ 6.41571e6 0.448659
$$729$$ 531441. 0.0370370
$$730$$ −2.81652e6 −0.195617
$$731$$ −695424. −0.0481345
$$732$$ −1.76023e7 −1.21420
$$733$$ −2.30788e7 −1.58655 −0.793274 0.608864i $$-0.791625\pi$$
−0.793274 + 0.608864i $$0.791625\pi$$
$$734$$ 2.61544e7 1.79186
$$735$$ −3.48268e6 −0.237791
$$736$$ 9.89705e6 0.673459
$$737$$ 1.89589e6 0.128572
$$738$$ 3.37358e6 0.228008
$$739$$ −2.27313e7 −1.53113 −0.765566 0.643358i $$-0.777541\pi$$
−0.765566 + 0.643358i $$0.777541\pi$$
$$740$$ −2.54736e6 −0.171006
$$741$$ 1.52724e7 1.02179
$$742$$ −723625. −0.0482507
$$743$$ −1.49328e7 −0.992360 −0.496180 0.868220i $$-0.665264\pi$$
−0.496180 + 0.868220i $$0.665264\pi$$
$$744$$ −3.17462e6 −0.210261
$$745$$ 6.79525e6 0.448554
$$746$$ −4.24624e7 −2.79356
$$747$$ 7.87568e6 0.516400
$$748$$ −1.00891e6 −0.0659321
$$749$$ −4.55584e6 −0.296732
$$750$$ −1.30153e6 −0.0844890
$$751$$ −1.92964e7 −1.24846 −0.624232 0.781239i $$-0.714588\pi$$
−0.624232 + 0.781239i $$0.714588\pi$$
$$752$$ −1.71126e6 −0.110350
$$753$$ 275946. 0.0177352
$$754$$ −3.90431e6 −0.250101
$$755$$ −5.91412e6 −0.377592
$$756$$ 1.42579e6 0.0907300
$$757$$ 1.35397e7 0.858756 0.429378 0.903125i $$-0.358733\pi$$
0.429378 + 0.903125i $$0.358733\pi$$
$$758$$ −2.63616e7 −1.66648
$$759$$ −2.09891e6 −0.132248
$$760$$ 9.68626e6 0.608306
$$761$$ 2.60669e7 1.63165 0.815825 0.578299i $$-0.196283\pi$$
0.815825 + 0.578299i $$0.196283\pi$$
$$762$$ −2.23916e7 −1.39701
$$763$$ −5.50271e6 −0.342188
$$764$$ 5.01042e7 3.10557
$$765$$ 314655. 0.0194393
$$766$$ 2.57730e7 1.58706
$$767$$ 1.39238e7 0.854611
$$768$$ −1.14027e7 −0.697598
$$769$$ −1.65354e7 −1.00832 −0.504162 0.863609i $$-0.668199\pi$$
−0.504162 + 0.863609i $$0.668199\pi$$
$$770$$ 1.02044e6 0.0620242
$$771$$ −6.19174e6 −0.375126
$$772$$ 1.61049e7 0.972554
$$773$$ 8.95978e6 0.539323 0.269661 0.962955i $$-0.413088\pi$$
0.269661 + 0.962955i $$0.413088\pi$$
$$774$$ 3.35518e6 0.201309
$$775$$ 1.09968e6 0.0657677
$$776$$ 7.36841e6 0.439258
$$777$$ −622887. −0.0370132
$$778$$ −3.65841e7 −2.16692
$$779$$ 8.69698e6 0.513482
$$780$$ −1.06011e7 −0.623897
$$781$$ −1.29343e6 −0.0758781
$$782$$ −2.77183e6 −0.162088
$$783$$ −350244. −0.0204158
$$784$$ −2.14110e6 −0.124407
$$785$$ 5.29561e6 0.306720
$$786$$ 3.05631e7 1.76458
$$787$$ −2.31723e7 −1.33362 −0.666812 0.745226i $$-0.732341\pi$$
−0.666812 + 0.745226i $$0.732341\pi$$
$$788$$ −5.07061e7 −2.90901
$$789$$ 1.53093e7 0.875512
$$790$$ 2.01779e7 1.15029
$$791$$ 5.75765e6 0.327193
$$792$$ 1.96487e6 0.111306
$$793$$ 3.20023e7 1.80717
$$794$$ 5.13994e7 2.89339
$$795$$ 482652. 0.0270842
$$796$$ −5.41010e7 −3.02638
$$797$$ −3.40811e7 −1.90050 −0.950250 0.311489i $$-0.899172\pi$$
−0.950250 + 0.311489i $$0.899172\pi$$
$$798$$ 5.86757e6 0.326175
$$799$$ −1.92230e6 −0.106525
$$800$$ 3.20936e6 0.177294
$$801$$ 3.12978e6 0.172358
$$802$$ 2.58470e6 0.141897
$$803$$ −1.47288e6 −0.0806082
$$804$$ −7.56706e6 −0.412845
$$805$$ 1.75622e6 0.0955188
$$806$$ 1.42984e7 0.775265
$$807$$ −8.84380e6 −0.478030
$$808$$ 3.71857e7 2.00377
$$809$$ 1.77957e6 0.0955969 0.0477985 0.998857i $$-0.484779\pi$$
0.0477985 + 0.998857i $$0.484779\pi$$
$$810$$ −1.51810e6 −0.0812996
$$811$$ 1.28099e7 0.683900 0.341950 0.939718i $$-0.388913\pi$$
0.341950 + 0.939718i $$0.388913\pi$$
$$812$$ −939660. −0.0500127
$$813$$ −2.48585e6 −0.131901
$$814$$ −2.12653e6 −0.112489
$$815$$ −8.53287e6 −0.449988
$$816$$ 193445. 0.0101703
$$817$$ 8.64955e6 0.453355
$$818$$ −4.78618e7 −2.50095
$$819$$ −2.59219e6 −0.135038
$$820$$ −6.03687e6 −0.313528
$$821$$ 1.47980e7 0.766205 0.383102 0.923706i $$-0.374856\pi$$
0.383102 + 0.923706i $$0.374856\pi$$
$$822$$ 1.52374e7 0.786561
$$823$$ −1.17405e7 −0.604207 −0.302103 0.953275i $$-0.597689\pi$$
−0.302103 + 0.953275i $$0.597689\pi$$
$$824$$ −7.39572e6 −0.379457
$$825$$ −680625. −0.0348155
$$826$$ 5.34945e6 0.272809
$$827$$ 1.61763e7 0.822461 0.411231 0.911531i $$-0.365099\pi$$
0.411231 + 0.911531i $$0.365099\pi$$
$$828$$ 8.37738e6 0.424651
$$829$$ −1.56514e7 −0.790984 −0.395492 0.918470i $$-0.629426\pi$$
−0.395492 + 0.918470i $$0.629426\pi$$
$$830$$ −2.24975e7 −1.13354
$$831$$ 1.33898e6 0.0672623
$$832$$ 4.56158e7 2.28458
$$833$$ −2.40514e6 −0.120096
$$834$$ −702188. −0.0349573
$$835$$ −4.51370e6 −0.224035
$$836$$ 1.25486e7 0.620982
$$837$$ 1.28267e6 0.0632850
$$838$$ 5.59161e7 2.75060
$$839$$ 2.70692e7 1.32761 0.663806 0.747905i $$-0.268940\pi$$
0.663806 + 0.747905i $$0.268940\pi$$
$$840$$ −1.64406e6 −0.0803931
$$841$$ −2.02803e7 −0.988746
$$842$$ 8.28707e6 0.402829
$$843$$ 3.21995e6 0.156056
$$844$$ 2.66951e7 1.28996
$$845$$ 9.99119e6 0.481366
$$846$$ 9.27442e6 0.445513
$$847$$ 533632. 0.0255584
$$848$$ 296727. 0.0141699
$$849$$ −4.42881e6 −0.210871
$$850$$ −898836. −0.0426710
$$851$$ −3.65984e6 −0.173236
$$852$$ 5.16246e6 0.243645
$$853$$ −2.43979e7 −1.14810 −0.574050 0.818820i $$-0.694629\pi$$
−0.574050 + 0.818820i $$0.694629\pi$$
$$854$$ 1.22951e7 0.576884
$$855$$ −3.91362e6 −0.183089
$$856$$ 2.50588e7 1.16889
$$857$$ 1.33841e7 0.622496 0.311248 0.950329i $$-0.399253\pi$$
0.311248 + 0.950329i $$0.399253\pi$$
$$858$$ −8.84971e6 −0.410403
$$859$$ −2.60324e7 −1.20374 −0.601869 0.798595i $$-0.705577\pi$$
−0.601869 + 0.798595i $$0.705577\pi$$
$$860$$ −6.00394e6 −0.276815
$$861$$ −1.47615e6 −0.0678612
$$862$$ 1.48887e7 0.682479
$$863$$ −2.45378e7 −1.12152 −0.560762 0.827977i $$-0.689492\pi$$
−0.560762 + 0.827977i $$0.689492\pi$$
$$864$$ 3.74340e6 0.170601
$$865$$ −1.60830e7 −0.730849
$$866$$ −1.72185e7 −0.780190
$$867$$ −1.25614e7 −0.567532
$$868$$ 3.44123e6 0.155030
$$869$$ 1.05519e7 0.474002
$$870$$ 1.00050e6 0.0448144
$$871$$ 1.37575e7 0.614460
$$872$$ 3.02669e7 1.34796
$$873$$ −2.97712e6 −0.132209
$$874$$ 3.44755e7 1.52662
$$875$$ 569497. 0.0251461
$$876$$ 5.87870e6 0.258834
$$877$$ 1.99034e7 0.873833 0.436917 0.899502i $$-0.356070\pi$$
0.436917 + 0.899502i $$0.356070\pi$$
$$878$$ −2.54568e7 −1.11447
$$879$$ 4.48308e6 0.195706
$$880$$ −418438. −0.0182148
$$881$$ −3.90880e7 −1.69669 −0.848346 0.529442i $$-0.822401\pi$$
−0.848346 + 0.529442i $$0.822401\pi$$
$$882$$ 1.16040e7 0.502267
$$883$$ 2.14490e7 0.925776 0.462888 0.886417i $$-0.346813\pi$$
0.462888 + 0.886417i $$0.346813\pi$$
$$884$$ −7.32109e6 −0.315098
$$885$$ −3.56803e6 −0.153134
$$886$$ −2.92287e7 −1.25091
$$887$$ 6.08624e6 0.259741 0.129870 0.991531i $$-0.458544\pi$$
0.129870 + 0.991531i $$0.458544\pi$$
$$888$$ 3.42610e6 0.145803
$$889$$ 9.79769e6 0.415786
$$890$$ −8.94045e6 −0.378342
$$891$$ −793881. −0.0335013
$$892$$ 6.12462e7 2.57731
$$893$$ 2.39091e7 1.00331
$$894$$ −2.26412e7 −0.947447
$$895$$ −6.67306e6 −0.278463
$$896$$ 1.15363e7 0.480060
$$897$$ −1.52307e7 −0.632031
$$898$$ 1.18012e6 0.0488356
$$899$$ −845336. −0.0348843
$$900$$ 2.71657e6 0.111793
$$901$$ 333319. 0.0136788
$$902$$ −5.03955e6 −0.206241
$$903$$ −1.46810e6 −0.0599149
$$904$$ −3.16691e7 −1.28889
$$905$$ 7.03823e6 0.285655
$$906$$ 1.97053e7 0.797560
$$907$$ −2.14459e7 −0.865618 −0.432809 0.901486i $$-0.642477\pi$$
−0.432809 + 0.901486i $$0.642477\pi$$
$$908$$ 3.59232e7 1.44597
$$909$$ −1.50244e7 −0.603098
$$910$$ 7.40479e6 0.296421
$$911$$ 1.03374e7 0.412682 0.206341 0.978480i $$-0.433844\pi$$
0.206341 + 0.978480i $$0.433844\pi$$
$$912$$ −2.40603e6 −0.0957888
$$913$$ −1.17649e7 −0.467102
$$914$$ −2.02884e7 −0.803310
$$915$$ −8.20074e6 −0.323818
$$916$$ 3.15557e7 1.24262
$$917$$ −1.33732e7 −0.525185
$$918$$ −1.04840e6 −0.0410602
$$919$$ −4.75142e7 −1.85581 −0.927907 0.372811i $$-0.878394\pi$$
−0.927907 + 0.372811i $$0.878394\pi$$
$$920$$ −9.65984e6 −0.376271
$$921$$ −8.98884e6 −0.349184
$$922$$ −4.81488e7 −1.86534
$$923$$ −9.38575e6 −0.362631
$$924$$ −2.12988e6 −0.0820683
$$925$$ −1.18679e6 −0.0456059
$$926$$ −2.46769e7 −0.945720
$$927$$ 2.98815e6 0.114210
$$928$$ −2.46707e6 −0.0940398
$$929$$ 6.08962e6 0.231500 0.115750 0.993278i $$-0.463073\pi$$
0.115750 + 0.993278i $$0.463073\pi$$
$$930$$ −3.66404e6 −0.138916
$$931$$ 2.99146e7 1.13112
$$932$$ 1.07008e7 0.403532
$$933$$ 1.69904e7 0.638999
$$934$$ 2.18034e7 0.817820
$$935$$ −470040. −0.0175835
$$936$$ 1.42580e7 0.531947
$$937$$ −2.73781e6 −0.101872 −0.0509360 0.998702i $$-0.516220\pi$$
−0.0509360 + 0.998702i $$0.516220\pi$$
$$938$$ 5.28556e6 0.196148
$$939$$ −2.11308e7 −0.782082
$$940$$ −1.65961e7 −0.612615
$$941$$ 3.73849e7 1.37633 0.688165 0.725554i $$-0.258417\pi$$
0.688165 + 0.725554i $$0.258417\pi$$
$$942$$ −1.76445e7 −0.647861
$$943$$ −8.67326e6 −0.317617
$$944$$ −2.19358e6 −0.0801166
$$945$$ 664261. 0.0241969
$$946$$ −5.01206e6 −0.182091
$$947$$ −4.08261e7 −1.47932 −0.739661 0.672979i $$-0.765014\pi$$
−0.739661 + 0.672979i $$0.765014\pi$$
$$948$$ −4.21156e7 −1.52203
$$949$$ −1.06879e7 −0.385237
$$950$$ 1.11795e7 0.401897
$$951$$ 5.06449e6 0.181587
$$952$$ −1.13539e6 −0.0406024
$$953$$ −4.60553e7 −1.64266 −0.821330 0.570453i $$-0.806768\pi$$
−0.821330 + 0.570453i $$0.806768\pi$$
$$954$$ −1.60815e6 −0.0572079
$$955$$ 2.33431e7 0.828228
$$956$$ −2.18965e7 −0.774873
$$957$$ 523204. 0.0184668
$$958$$ 4.42709e7 1.55849
$$959$$ −6.66730e6 −0.234101
$$960$$ −1.16893e7 −0.409363
$$961$$ −2.55333e7 −0.891865
$$962$$ −1.54311e7 −0.537599
$$963$$ −1.01247e7 −0.351817
$$964$$ 6.65509e7 2.30654
$$965$$ 7.50310e6 0.259372
$$966$$ −5.85156e6 −0.201757
$$967$$ 1.48162e7 0.509532 0.254766 0.967003i $$-0.418002\pi$$
0.254766 + 0.967003i $$0.418002\pi$$
$$968$$ −2.93517e6 −0.100680
$$969$$ −2.70275e6 −0.0924689
$$970$$ 8.50436e6 0.290210
$$971$$ −3.27641e7 −1.11519 −0.557597 0.830112i $$-0.688277\pi$$
−0.557597 + 0.830112i $$0.688277\pi$$
$$972$$ 3.16861e6 0.107573
$$973$$ 307250. 0.0104042
$$974$$ 3.06828e7 1.03633
$$975$$ −4.93893e6 −0.166388
$$976$$ −5.04170e6 −0.169415
$$977$$ −4.06772e6 −0.136337 −0.0681686 0.997674i $$-0.521716\pi$$
−0.0681686 + 0.997674i $$0.521716\pi$$
$$978$$ 2.84307e7 0.950476
$$979$$ −4.67535e6 −0.155904
$$980$$ −2.07647e7 −0.690655
$$981$$ −1.22290e7 −0.405712
$$982$$ 2.79765e7 0.925795
$$983$$ 3.76095e7 1.24141 0.620704 0.784045i $$-0.286847\pi$$
0.620704 + 0.784045i $$0.286847\pi$$
$$984$$ 8.11934e6 0.267321
$$985$$ −2.36235e7 −0.775807
$$986$$ 690944. 0.0226335
$$987$$ −4.05812e6 −0.132596
$$988$$ 9.10583e7 2.96775
$$989$$ −8.62596e6 −0.280425
$$990$$ 2.26778e6 0.0735382
$$991$$ 1.02888e7 0.332797 0.166399 0.986059i $$-0.446786\pi$$
0.166399 + 0.986059i $$0.446786\pi$$
$$992$$ 9.03495e6 0.291505
$$993$$ −9.05334e6 −0.291364
$$994$$ −3.60596e6 −0.115759
$$995$$ −2.52052e7 −0.807108
$$996$$ 4.69571e7 1.49987
$$997$$ 4.37776e7 1.39481 0.697403 0.716680i $$-0.254339\pi$$
0.697403 + 0.716680i $$0.254339\pi$$
$$998$$ −4.13277e7 −1.31345
$$999$$ −1.38428e6 −0.0438843
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 165.6.a.a.1.1 3
3.2 odd 2 495.6.a.e.1.3 3
5.4 even 2 825.6.a.j.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.1 3 1.1 even 1 trivial
495.6.a.e.1.3 3 3.2 odd 2
825.6.a.j.1.3 3 5.4 even 2