# Properties

 Label 150.6.a.a.1.1 Level $150$ Weight $6$ Character 150.1 Self dual yes Analytic conductor $24.058$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.0575729719$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -47.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} -47.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +222.000 q^{11} -144.000 q^{12} -101.000 q^{13} +188.000 q^{14} +256.000 q^{16} -162.000 q^{17} -324.000 q^{18} +1685.00 q^{19} +423.000 q^{21} -888.000 q^{22} -306.000 q^{23} +576.000 q^{24} +404.000 q^{26} -729.000 q^{27} -752.000 q^{28} +7890.00 q^{29} -8593.00 q^{31} -1024.00 q^{32} -1998.00 q^{33} +648.000 q^{34} +1296.00 q^{36} -8642.00 q^{37} -6740.00 q^{38} +909.000 q^{39} -18168.0 q^{41} -1692.00 q^{42} -14351.0 q^{43} +3552.00 q^{44} +1224.00 q^{46} +1098.00 q^{47} -2304.00 q^{48} -14598.0 q^{49} +1458.00 q^{51} -1616.00 q^{52} -17916.0 q^{53} +2916.00 q^{54} +3008.00 q^{56} -15165.0 q^{57} -31560.0 q^{58} +17610.0 q^{59} -21853.0 q^{61} +34372.0 q^{62} -3807.00 q^{63} +4096.00 q^{64} +7992.00 q^{66} -107.000 q^{67} -2592.00 q^{68} +2754.00 q^{69} -40728.0 q^{71} -5184.00 q^{72} -34706.0 q^{73} +34568.0 q^{74} +26960.0 q^{76} -10434.0 q^{77} -3636.00 q^{78} -69160.0 q^{79} +6561.00 q^{81} +72672.0 q^{82} +108534. q^{83} +6768.00 q^{84} +57404.0 q^{86} -71010.0 q^{87} -14208.0 q^{88} +35040.0 q^{89} +4747.00 q^{91} -4896.00 q^{92} +77337.0 q^{93} -4392.00 q^{94} +9216.00 q^{96} +823.000 q^{97} +58392.0 q^{98} +17982.0 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −0.707107
$$3$$ −9.00000 −0.577350
$$4$$ 16.0000 0.500000
$$5$$ 0 0
$$6$$ 36.0000 0.408248
$$7$$ −47.0000 −0.362537 −0.181269 0.983434i $$-0.558020\pi$$
−0.181269 + 0.983434i $$0.558020\pi$$
$$8$$ −64.0000 −0.353553
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 222.000 0.553186 0.276593 0.960987i $$-0.410795\pi$$
0.276593 + 0.960987i $$0.410795\pi$$
$$12$$ −144.000 −0.288675
$$13$$ −101.000 −0.165754 −0.0828768 0.996560i $$-0.526411\pi$$
−0.0828768 + 0.996560i $$0.526411\pi$$
$$14$$ 188.000 0.256353
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −162.000 −0.135954 −0.0679771 0.997687i $$-0.521654\pi$$
−0.0679771 + 0.997687i $$0.521654\pi$$
$$18$$ −324.000 −0.235702
$$19$$ 1685.00 1.07082 0.535409 0.844593i $$-0.320157\pi$$
0.535409 + 0.844593i $$0.320157\pi$$
$$20$$ 0 0
$$21$$ 423.000 0.209311
$$22$$ −888.000 −0.391162
$$23$$ −306.000 −0.120615 −0.0603076 0.998180i $$-0.519208\pi$$
−0.0603076 + 0.998180i $$0.519208\pi$$
$$24$$ 576.000 0.204124
$$25$$ 0 0
$$26$$ 404.000 0.117206
$$27$$ −729.000 −0.192450
$$28$$ −752.000 −0.181269
$$29$$ 7890.00 1.74214 0.871068 0.491163i $$-0.163428\pi$$
0.871068 + 0.491163i $$0.163428\pi$$
$$30$$ 0 0
$$31$$ −8593.00 −1.60598 −0.802991 0.595991i $$-0.796759\pi$$
−0.802991 + 0.595991i $$0.796759\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ −1998.00 −0.319382
$$34$$ 648.000 0.0961342
$$35$$ 0 0
$$36$$ 1296.00 0.166667
$$37$$ −8642.00 −1.03779 −0.518896 0.854838i $$-0.673657\pi$$
−0.518896 + 0.854838i $$0.673657\pi$$
$$38$$ −6740.00 −0.757183
$$39$$ 909.000 0.0956979
$$40$$ 0 0
$$41$$ −18168.0 −1.68790 −0.843951 0.536420i $$-0.819777\pi$$
−0.843951 + 0.536420i $$0.819777\pi$$
$$42$$ −1692.00 −0.148005
$$43$$ −14351.0 −1.18362 −0.591808 0.806079i $$-0.701586\pi$$
−0.591808 + 0.806079i $$0.701586\pi$$
$$44$$ 3552.00 0.276593
$$45$$ 0 0
$$46$$ 1224.00 0.0852878
$$47$$ 1098.00 0.0725033 0.0362516 0.999343i $$-0.488458\pi$$
0.0362516 + 0.999343i $$0.488458\pi$$
$$48$$ −2304.00 −0.144338
$$49$$ −14598.0 −0.868567
$$50$$ 0 0
$$51$$ 1458.00 0.0784932
$$52$$ −1616.00 −0.0828768
$$53$$ −17916.0 −0.876095 −0.438048 0.898952i $$-0.644330\pi$$
−0.438048 + 0.898952i $$0.644330\pi$$
$$54$$ 2916.00 0.136083
$$55$$ 0 0
$$56$$ 3008.00 0.128176
$$57$$ −15165.0 −0.618237
$$58$$ −31560.0 −1.23188
$$59$$ 17610.0 0.658612 0.329306 0.944223i $$-0.393185\pi$$
0.329306 + 0.944223i $$0.393185\pi$$
$$60$$ 0 0
$$61$$ −21853.0 −0.751946 −0.375973 0.926631i $$-0.622691\pi$$
−0.375973 + 0.926631i $$0.622691\pi$$
$$62$$ 34372.0 1.13560
$$63$$ −3807.00 −0.120846
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ 7992.00 0.225837
$$67$$ −107.000 −0.00291204 −0.00145602 0.999999i $$-0.500463\pi$$
−0.00145602 + 0.999999i $$0.500463\pi$$
$$68$$ −2592.00 −0.0679771
$$69$$ 2754.00 0.0696372
$$70$$ 0 0
$$71$$ −40728.0 −0.958842 −0.479421 0.877585i $$-0.659153\pi$$
−0.479421 + 0.877585i $$0.659153\pi$$
$$72$$ −5184.00 −0.117851
$$73$$ −34706.0 −0.762250 −0.381125 0.924524i $$-0.624463\pi$$
−0.381125 + 0.924524i $$0.624463\pi$$
$$74$$ 34568.0 0.733829
$$75$$ 0 0
$$76$$ 26960.0 0.535409
$$77$$ −10434.0 −0.200551
$$78$$ −3636.00 −0.0676686
$$79$$ −69160.0 −1.24677 −0.623386 0.781914i $$-0.714244\pi$$
−0.623386 + 0.781914i $$0.714244\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 72672.0 1.19353
$$83$$ 108534. 1.72930 0.864650 0.502374i $$-0.167540\pi$$
0.864650 + 0.502374i $$0.167540\pi$$
$$84$$ 6768.00 0.104656
$$85$$ 0 0
$$86$$ 57404.0 0.836943
$$87$$ −71010.0 −1.00582
$$88$$ −14208.0 −0.195581
$$89$$ 35040.0 0.468910 0.234455 0.972127i $$-0.424670\pi$$
0.234455 + 0.972127i $$0.424670\pi$$
$$90$$ 0 0
$$91$$ 4747.00 0.0600919
$$92$$ −4896.00 −0.0603076
$$93$$ 77337.0 0.927214
$$94$$ −4392.00 −0.0512676
$$95$$ 0 0
$$96$$ 9216.00 0.102062
$$97$$ 823.000 0.00888118 0.00444059 0.999990i $$-0.498587\pi$$
0.00444059 + 0.999990i $$0.498587\pi$$
$$98$$ 58392.0 0.614169
$$99$$ 17982.0 0.184395
$$100$$ 0 0
$$101$$ −33828.0 −0.329969 −0.164984 0.986296i $$-0.552757\pi$$
−0.164984 + 0.986296i $$0.552757\pi$$
$$102$$ −5832.00 −0.0555031
$$103$$ 133444. 1.23938 0.619692 0.784845i $$-0.287257\pi$$
0.619692 + 0.784845i $$0.287257\pi$$
$$104$$ 6464.00 0.0586028
$$105$$ 0 0
$$106$$ 71664.0 0.619493
$$107$$ −81252.0 −0.686080 −0.343040 0.939321i $$-0.611457\pi$$
−0.343040 + 0.939321i $$0.611457\pi$$
$$108$$ −11664.0 −0.0962250
$$109$$ −217015. −1.74954 −0.874769 0.484540i $$-0.838987\pi$$
−0.874769 + 0.484540i $$0.838987\pi$$
$$110$$ 0 0
$$111$$ 77778.0 0.599169
$$112$$ −12032.0 −0.0906343
$$113$$ 138324. 1.01906 0.509532 0.860452i $$-0.329819\pi$$
0.509532 + 0.860452i $$0.329819\pi$$
$$114$$ 60660.0 0.437160
$$115$$ 0 0
$$116$$ 126240. 0.871068
$$117$$ −8181.00 −0.0552512
$$118$$ −70440.0 −0.465709
$$119$$ 7614.00 0.0492885
$$120$$ 0 0
$$121$$ −111767. −0.693985
$$122$$ 87412.0 0.531706
$$123$$ 163512. 0.974511
$$124$$ −137488. −0.802991
$$125$$ 0 0
$$126$$ 15228.0 0.0854509
$$127$$ 256048. 1.40868 0.704340 0.709863i $$-0.251243\pi$$
0.704340 + 0.709863i $$0.251243\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ 129159. 0.683361
$$130$$ 0 0
$$131$$ 118452. 0.603065 0.301533 0.953456i $$-0.402502\pi$$
0.301533 + 0.953456i $$0.402502\pi$$
$$132$$ −31968.0 −0.159691
$$133$$ −79195.0 −0.388212
$$134$$ 428.000 0.00205912
$$135$$ 0 0
$$136$$ 10368.0 0.0480671
$$137$$ 13218.0 0.0601678 0.0300839 0.999547i $$-0.490423\pi$$
0.0300839 + 0.999547i $$0.490423\pi$$
$$138$$ −11016.0 −0.0492409
$$139$$ −350740. −1.53974 −0.769872 0.638199i $$-0.779680\pi$$
−0.769872 + 0.638199i $$0.779680\pi$$
$$140$$ 0 0
$$141$$ −9882.00 −0.0418598
$$142$$ 162912. 0.678004
$$143$$ −22422.0 −0.0916926
$$144$$ 20736.0 0.0833333
$$145$$ 0 0
$$146$$ 138824. 0.538992
$$147$$ 131382. 0.501467
$$148$$ −138272. −0.518896
$$149$$ 109890. 0.405502 0.202751 0.979230i $$-0.435012\pi$$
0.202751 + 0.979230i $$0.435012\pi$$
$$150$$ 0 0
$$151$$ −172603. −0.616036 −0.308018 0.951381i $$-0.599666\pi$$
−0.308018 + 0.951381i $$0.599666\pi$$
$$152$$ −107840. −0.378592
$$153$$ −13122.0 −0.0453181
$$154$$ 41736.0 0.141811
$$155$$ 0 0
$$156$$ 14544.0 0.0478489
$$157$$ 349993. 1.13321 0.566605 0.823990i $$-0.308257\pi$$
0.566605 + 0.823990i $$0.308257\pi$$
$$158$$ 276640. 0.881601
$$159$$ 161244. 0.505814
$$160$$ 0 0
$$161$$ 14382.0 0.0437275
$$162$$ −26244.0 −0.0785674
$$163$$ −192581. −0.567733 −0.283867 0.958864i $$-0.591617\pi$$
−0.283867 + 0.958864i $$0.591617\pi$$
$$164$$ −290688. −0.843951
$$165$$ 0 0
$$166$$ −434136. −1.22280
$$167$$ −580692. −1.61122 −0.805610 0.592447i $$-0.798162\pi$$
−0.805610 + 0.592447i $$0.798162\pi$$
$$168$$ −27072.0 −0.0740026
$$169$$ −361092. −0.972526
$$170$$ 0 0
$$171$$ 136485. 0.356940
$$172$$ −229616. −0.591808
$$173$$ −738126. −1.87506 −0.937530 0.347904i $$-0.886894\pi$$
−0.937530 + 0.347904i $$0.886894\pi$$
$$174$$ 284040. 0.711224
$$175$$ 0 0
$$176$$ 56832.0 0.138297
$$177$$ −158490. −0.380250
$$178$$ −140160. −0.331569
$$179$$ 497370. 1.16024 0.580119 0.814532i $$-0.303006\pi$$
0.580119 + 0.814532i $$0.303006\pi$$
$$180$$ 0 0
$$181$$ −333163. −0.755893 −0.377947 0.925827i $$-0.623370\pi$$
−0.377947 + 0.925827i $$0.623370\pi$$
$$182$$ −18988.0 −0.0424914
$$183$$ 196677. 0.434136
$$184$$ 19584.0 0.0426439
$$185$$ 0 0
$$186$$ −309348. −0.655639
$$187$$ −35964.0 −0.0752080
$$188$$ 17568.0 0.0362516
$$189$$ 34263.0 0.0697703
$$190$$ 0 0
$$191$$ −40638.0 −0.0806026 −0.0403013 0.999188i $$-0.512832\pi$$
−0.0403013 + 0.999188i $$0.512832\pi$$
$$192$$ −36864.0 −0.0721688
$$193$$ −494651. −0.955885 −0.477942 0.878391i $$-0.658617\pi$$
−0.477942 + 0.878391i $$0.658617\pi$$
$$194$$ −3292.00 −0.00627994
$$195$$ 0 0
$$196$$ −233568. −0.434283
$$197$$ −552342. −1.01401 −0.507005 0.861943i $$-0.669248\pi$$
−0.507005 + 0.861943i $$0.669248\pi$$
$$198$$ −71928.0 −0.130387
$$199$$ 685625. 1.22731 0.613655 0.789575i $$-0.289699\pi$$
0.613655 + 0.789575i $$0.289699\pi$$
$$200$$ 0 0
$$201$$ 963.000 0.00168126
$$202$$ 135312. 0.233323
$$203$$ −370830. −0.631589
$$204$$ 23328.0 0.0392466
$$205$$ 0 0
$$206$$ −533776. −0.876377
$$207$$ −24786.0 −0.0402050
$$208$$ −25856.0 −0.0414384
$$209$$ 374070. 0.592362
$$210$$ 0 0
$$211$$ 749477. 1.15892 0.579458 0.815002i $$-0.303264\pi$$
0.579458 + 0.815002i $$0.303264\pi$$
$$212$$ −286656. −0.438048
$$213$$ 366552. 0.553588
$$214$$ 325008. 0.485132
$$215$$ 0 0
$$216$$ 46656.0 0.0680414
$$217$$ 403871. 0.582228
$$218$$ 868060. 1.23711
$$219$$ 312354. 0.440085
$$220$$ 0 0
$$221$$ 16362.0 0.0225349
$$222$$ −311112. −0.423676
$$223$$ −169271. −0.227940 −0.113970 0.993484i $$-0.536357\pi$$
−0.113970 + 0.993484i $$0.536357\pi$$
$$224$$ 48128.0 0.0640882
$$225$$ 0 0
$$226$$ −553296. −0.720587
$$227$$ 46488.0 0.0598792 0.0299396 0.999552i $$-0.490468\pi$$
0.0299396 + 0.999552i $$0.490468\pi$$
$$228$$ −242640. −0.309119
$$229$$ −90115.0 −0.113556 −0.0567778 0.998387i $$-0.518083\pi$$
−0.0567778 + 0.998387i $$0.518083\pi$$
$$230$$ 0 0
$$231$$ 93906.0 0.115788
$$232$$ −504960. −0.615938
$$233$$ −1.06414e6 −1.28413 −0.642063 0.766652i $$-0.721921\pi$$
−0.642063 + 0.766652i $$0.721921\pi$$
$$234$$ 32724.0 0.0390685
$$235$$ 0 0
$$236$$ 281760. 0.329306
$$237$$ 622440. 0.719825
$$238$$ −30456.0 −0.0348522
$$239$$ 1.15158e6 1.30407 0.652033 0.758191i $$-0.273916\pi$$
0.652033 + 0.758191i $$0.273916\pi$$
$$240$$ 0 0
$$241$$ 856217. 0.949601 0.474801 0.880093i $$-0.342520\pi$$
0.474801 + 0.880093i $$0.342520\pi$$
$$242$$ 447068. 0.490722
$$243$$ −59049.0 −0.0641500
$$244$$ −349648. −0.375973
$$245$$ 0 0
$$246$$ −654048. −0.689084
$$247$$ −170185. −0.177492
$$248$$ 549952. 0.567800
$$249$$ −976806. −0.998412
$$250$$ 0 0
$$251$$ −207708. −0.208098 −0.104049 0.994572i $$-0.533180\pi$$
−0.104049 + 0.994572i $$0.533180\pi$$
$$252$$ −60912.0 −0.0604229
$$253$$ −67932.0 −0.0667226
$$254$$ −1.02419e6 −0.996087
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 1.45319e6 1.37243 0.686213 0.727401i $$-0.259272\pi$$
0.686213 + 0.727401i $$0.259272\pi$$
$$258$$ −516636. −0.483209
$$259$$ 406174. 0.376238
$$260$$ 0 0
$$261$$ 639090. 0.580712
$$262$$ −473808. −0.426431
$$263$$ −169296. −0.150924 −0.0754618 0.997149i $$-0.524043\pi$$
−0.0754618 + 0.997149i $$0.524043\pi$$
$$264$$ 127872. 0.112919
$$265$$ 0 0
$$266$$ 316780. 0.274507
$$267$$ −315360. −0.270725
$$268$$ −1712.00 −0.00145602
$$269$$ −1.58109e6 −1.33222 −0.666110 0.745854i $$-0.732042\pi$$
−0.666110 + 0.745854i $$0.732042\pi$$
$$270$$ 0 0
$$271$$ 822512. 0.680329 0.340165 0.940366i $$-0.389517\pi$$
0.340165 + 0.940366i $$0.389517\pi$$
$$272$$ −41472.0 −0.0339886
$$273$$ −42723.0 −0.0346941
$$274$$ −52872.0 −0.0425451
$$275$$ 0 0
$$276$$ 44064.0 0.0348186
$$277$$ 546823. 0.428201 0.214100 0.976812i $$-0.431318\pi$$
0.214100 + 0.976812i $$0.431318\pi$$
$$278$$ 1.40296e6 1.08876
$$279$$ −696033. −0.535327
$$280$$ 0 0
$$281$$ −1.09250e6 −0.825382 −0.412691 0.910871i $$-0.635411\pi$$
−0.412691 + 0.910871i $$0.635411\pi$$
$$282$$ 39528.0 0.0295993
$$283$$ 2.48480e6 1.84427 0.922136 0.386865i $$-0.126442\pi$$
0.922136 + 0.386865i $$0.126442\pi$$
$$284$$ −651648. −0.479421
$$285$$ 0 0
$$286$$ 89688.0 0.0648365
$$287$$ 853896. 0.611928
$$288$$ −82944.0 −0.0589256
$$289$$ −1.39361e6 −0.981516
$$290$$ 0 0
$$291$$ −7407.00 −0.00512755
$$292$$ −555296. −0.381125
$$293$$ 341394. 0.232320 0.116160 0.993231i $$-0.462941\pi$$
0.116160 + 0.993231i $$0.462941\pi$$
$$294$$ −525528. −0.354591
$$295$$ 0 0
$$296$$ 553088. 0.366915
$$297$$ −161838. −0.106461
$$298$$ −439560. −0.286733
$$299$$ 30906.0 0.0199924
$$300$$ 0 0
$$301$$ 674497. 0.429105
$$302$$ 690412. 0.435603
$$303$$ 304452. 0.190508
$$304$$ 431360. 0.267705
$$305$$ 0 0
$$306$$ 52488.0 0.0320447
$$307$$ −2.02898e6 −1.22866 −0.614329 0.789050i $$-0.710573\pi$$
−0.614329 + 0.789050i $$0.710573\pi$$
$$308$$ −166944. −0.100275
$$309$$ −1.20100e6 −0.715559
$$310$$ 0 0
$$311$$ −206598. −0.121123 −0.0605613 0.998164i $$-0.519289\pi$$
−0.0605613 + 0.998164i $$0.519289\pi$$
$$312$$ −58176.0 −0.0338343
$$313$$ 3.34223e6 1.92830 0.964152 0.265352i $$-0.0854881\pi$$
0.964152 + 0.265352i $$0.0854881\pi$$
$$314$$ −1.39997e6 −0.801300
$$315$$ 0 0
$$316$$ −1.10656e6 −0.623386
$$317$$ 2.53289e6 1.41569 0.707844 0.706368i $$-0.249668\pi$$
0.707844 + 0.706368i $$0.249668\pi$$
$$318$$ −644976. −0.357664
$$319$$ 1.75158e6 0.963725
$$320$$ 0 0
$$321$$ 731268. 0.396108
$$322$$ −57528.0 −0.0309200
$$323$$ −272970. −0.145582
$$324$$ 104976. 0.0555556
$$325$$ 0 0
$$326$$ 770324. 0.401448
$$327$$ 1.95314e6 1.01010
$$328$$ 1.16275e6 0.596764
$$329$$ −51606.0 −0.0262851
$$330$$ 0 0
$$331$$ 602132. 0.302080 0.151040 0.988528i $$-0.451738\pi$$
0.151040 + 0.988528i $$0.451738\pi$$
$$332$$ 1.73654e6 0.864650
$$333$$ −700002. −0.345930
$$334$$ 2.32277e6 1.13930
$$335$$ 0 0
$$336$$ 108288. 0.0523278
$$337$$ −209777. −0.100620 −0.0503099 0.998734i $$-0.516021\pi$$
−0.0503099 + 0.998734i $$0.516021\pi$$
$$338$$ 1.44437e6 0.687680
$$339$$ −1.24492e6 −0.588357
$$340$$ 0 0
$$341$$ −1.90765e6 −0.888407
$$342$$ −545940. −0.252394
$$343$$ 1.47603e6 0.677425
$$344$$ 918464. 0.418472
$$345$$ 0 0
$$346$$ 2.95250e6 1.32587
$$347$$ −4.02166e6 −1.79301 −0.896503 0.443037i $$-0.853901\pi$$
−0.896503 + 0.443037i $$0.853901\pi$$
$$348$$ −1.13616e6 −0.502911
$$349$$ 8330.00 0.00366085 0.00183042 0.999998i $$-0.499417\pi$$
0.00183042 + 0.999998i $$0.499417\pi$$
$$350$$ 0 0
$$351$$ 73629.0 0.0318993
$$352$$ −227328. −0.0977904
$$353$$ −1.95001e6 −0.832912 −0.416456 0.909156i $$-0.636728\pi$$
−0.416456 + 0.909156i $$0.636728\pi$$
$$354$$ 633960. 0.268877
$$355$$ 0 0
$$356$$ 560640. 0.234455
$$357$$ −68526.0 −0.0284567
$$358$$ −1.98948e6 −0.820412
$$359$$ 2.27088e6 0.929947 0.464973 0.885325i $$-0.346064\pi$$
0.464973 + 0.885325i $$0.346064\pi$$
$$360$$ 0 0
$$361$$ 363126. 0.146652
$$362$$ 1.33265e6 0.534497
$$363$$ 1.00590e6 0.400673
$$364$$ 75952.0 0.0300459
$$365$$ 0 0
$$366$$ −786708. −0.306981
$$367$$ −2.86154e6 −1.10901 −0.554503 0.832181i $$-0.687092\pi$$
−0.554503 + 0.832181i $$0.687092\pi$$
$$368$$ −78336.0 −0.0301538
$$369$$ −1.47161e6 −0.562634
$$370$$ 0 0
$$371$$ 842052. 0.317617
$$372$$ 1.23739e6 0.463607
$$373$$ −615311. −0.228993 −0.114497 0.993424i $$-0.536526\pi$$
−0.114497 + 0.993424i $$0.536526\pi$$
$$374$$ 143856. 0.0531801
$$375$$ 0 0
$$376$$ −70272.0 −0.0256338
$$377$$ −796890. −0.288765
$$378$$ −137052. −0.0493351
$$379$$ 5.39878e6 1.93062 0.965311 0.261103i $$-0.0840863\pi$$
0.965311 + 0.261103i $$0.0840863\pi$$
$$380$$ 0 0
$$381$$ −2.30443e6 −0.813301
$$382$$ 162552. 0.0569946
$$383$$ −1.08688e6 −0.378602 −0.189301 0.981919i $$-0.560622\pi$$
−0.189301 + 0.981919i $$0.560622\pi$$
$$384$$ 147456. 0.0510310
$$385$$ 0 0
$$386$$ 1.97860e6 0.675913
$$387$$ −1.16243e6 −0.394539
$$388$$ 13168.0 0.00444059
$$389$$ −3.48432e6 −1.16747 −0.583733 0.811946i $$-0.698408\pi$$
−0.583733 + 0.811946i $$0.698408\pi$$
$$390$$ 0 0
$$391$$ 49572.0 0.0163981
$$392$$ 934272. 0.307085
$$393$$ −1.06607e6 −0.348180
$$394$$ 2.20937e6 0.717014
$$395$$ 0 0
$$396$$ 287712. 0.0921977
$$397$$ −3.26591e6 −1.03999 −0.519993 0.854170i $$-0.674065\pi$$
−0.519993 + 0.854170i $$0.674065\pi$$
$$398$$ −2.74250e6 −0.867839
$$399$$ 712755. 0.224134
$$400$$ 0 0
$$401$$ −4.27319e6 −1.32706 −0.663531 0.748149i $$-0.730943\pi$$
−0.663531 + 0.748149i $$0.730943\pi$$
$$402$$ −3852.00 −0.00118883
$$403$$ 867893. 0.266197
$$404$$ −541248. −0.164984
$$405$$ 0 0
$$406$$ 1.48332e6 0.446601
$$407$$ −1.91852e6 −0.574092
$$408$$ −93312.0 −0.0277515
$$409$$ −1.45188e6 −0.429162 −0.214581 0.976706i $$-0.568839\pi$$
−0.214581 + 0.976706i $$0.568839\pi$$
$$410$$ 0 0
$$411$$ −118962. −0.0347379
$$412$$ 2.13510e6 0.619692
$$413$$ −827670. −0.238771
$$414$$ 99144.0 0.0284293
$$415$$ 0 0
$$416$$ 103424. 0.0293014
$$417$$ 3.15666e6 0.888971
$$418$$ −1.49628e6 −0.418863
$$419$$ 559380. 0.155658 0.0778291 0.996967i $$-0.475201\pi$$
0.0778291 + 0.996967i $$0.475201\pi$$
$$420$$ 0 0
$$421$$ −3.91470e6 −1.07645 −0.538224 0.842802i $$-0.680905\pi$$
−0.538224 + 0.842802i $$0.680905\pi$$
$$422$$ −2.99791e6 −0.819478
$$423$$ 88938.0 0.0241678
$$424$$ 1.14662e6 0.309746
$$425$$ 0 0
$$426$$ −1.46621e6 −0.391446
$$427$$ 1.02709e6 0.272608
$$428$$ −1.30003e6 −0.343040
$$429$$ 201798. 0.0529387
$$430$$ 0 0
$$431$$ −3.57500e6 −0.927006 −0.463503 0.886095i $$-0.653408\pi$$
−0.463503 + 0.886095i $$0.653408\pi$$
$$432$$ −186624. −0.0481125
$$433$$ −7.15969e6 −1.83516 −0.917581 0.397548i $$-0.869861\pi$$
−0.917581 + 0.397548i $$0.869861\pi$$
$$434$$ −1.61548e6 −0.411698
$$435$$ 0 0
$$436$$ −3.47224e6 −0.874769
$$437$$ −515610. −0.129157
$$438$$ −1.24942e6 −0.311187
$$439$$ 1.71790e6 0.425437 0.212719 0.977114i $$-0.431768\pi$$
0.212719 + 0.977114i $$0.431768\pi$$
$$440$$ 0 0
$$441$$ −1.18244e6 −0.289522
$$442$$ −65448.0 −0.0159346
$$443$$ −3.39670e6 −0.822332 −0.411166 0.911560i $$-0.634878\pi$$
−0.411166 + 0.911560i $$0.634878\pi$$
$$444$$ 1.24445e6 0.299584
$$445$$ 0 0
$$446$$ 677084. 0.161178
$$447$$ −989010. −0.234116
$$448$$ −192512. −0.0453172
$$449$$ −3.39606e6 −0.794986 −0.397493 0.917605i $$-0.630120\pi$$
−0.397493 + 0.917605i $$0.630120\pi$$
$$450$$ 0 0
$$451$$ −4.03330e6 −0.933724
$$452$$ 2.21318e6 0.509532
$$453$$ 1.55343e6 0.355668
$$454$$ −185952. −0.0423410
$$455$$ 0 0
$$456$$ 970560. 0.218580
$$457$$ 4.52814e6 1.01421 0.507106 0.861883i $$-0.330715\pi$$
0.507106 + 0.861883i $$0.330715\pi$$
$$458$$ 360460. 0.0802959
$$459$$ 118098. 0.0261644
$$460$$ 0 0
$$461$$ −1.27895e6 −0.280285 −0.140143 0.990131i $$-0.544756\pi$$
−0.140143 + 0.990131i $$0.544756\pi$$
$$462$$ −375624. −0.0818744
$$463$$ 7.19862e6 1.56062 0.780310 0.625393i $$-0.215061\pi$$
0.780310 + 0.625393i $$0.215061\pi$$
$$464$$ 2.01984e6 0.435534
$$465$$ 0 0
$$466$$ 4.25654e6 0.908014
$$467$$ −4.83034e6 −1.02491 −0.512455 0.858714i $$-0.671264\pi$$
−0.512455 + 0.858714i $$0.671264\pi$$
$$468$$ −130896. −0.0276256
$$469$$ 5029.00 0.00105572
$$470$$ 0 0
$$471$$ −3.14994e6 −0.654259
$$472$$ −1.12704e6 −0.232854
$$473$$ −3.18592e6 −0.654760
$$474$$ −2.48976e6 −0.508993
$$475$$ 0 0
$$476$$ 121824. 0.0246442
$$477$$ −1.45120e6 −0.292032
$$478$$ −4.60632e6 −0.922113
$$479$$ 748650. 0.149087 0.0745435 0.997218i $$-0.476250\pi$$
0.0745435 + 0.997218i $$0.476250\pi$$
$$480$$ 0 0
$$481$$ 872842. 0.172018
$$482$$ −3.42487e6 −0.671469
$$483$$ −129438. −0.0252461
$$484$$ −1.78827e6 −0.346993
$$485$$ 0 0
$$486$$ 236196. 0.0453609
$$487$$ 5.16394e6 0.986641 0.493320 0.869848i $$-0.335783\pi$$
0.493320 + 0.869848i $$0.335783\pi$$
$$488$$ 1.39859e6 0.265853
$$489$$ 1.73323e6 0.327781
$$490$$ 0 0
$$491$$ 8.54287e6 1.59919 0.799595 0.600539i $$-0.205047\pi$$
0.799595 + 0.600539i $$0.205047\pi$$
$$492$$ 2.61619e6 0.487256
$$493$$ −1.27818e6 −0.236851
$$494$$ 680740. 0.125506
$$495$$ 0 0
$$496$$ −2.19981e6 −0.401495
$$497$$ 1.91422e6 0.347616
$$498$$ 3.90722e6 0.705984
$$499$$ −4.20588e6 −0.756145 −0.378072 0.925776i $$-0.623413\pi$$
−0.378072 + 0.925776i $$0.623413\pi$$
$$500$$ 0 0
$$501$$ 5.22623e6 0.930238
$$502$$ 830832. 0.147148
$$503$$ 8.18342e6 1.44217 0.721083 0.692849i $$-0.243645\pi$$
0.721083 + 0.692849i $$0.243645\pi$$
$$504$$ 243648. 0.0427254
$$505$$ 0 0
$$506$$ 271728. 0.0471800
$$507$$ 3.24983e6 0.561488
$$508$$ 4.09677e6 0.704340
$$509$$ 3.85923e6 0.660247 0.330123 0.943938i $$-0.392910\pi$$
0.330123 + 0.943938i $$0.392910\pi$$
$$510$$ 0 0
$$511$$ 1.63118e6 0.276344
$$512$$ −262144. −0.0441942
$$513$$ −1.22837e6 −0.206079
$$514$$ −5.81275e6 −0.970452
$$515$$ 0 0
$$516$$ 2.06654e6 0.341681
$$517$$ 243756. 0.0401078
$$518$$ −1.62470e6 −0.266040
$$519$$ 6.64313e6 1.08257
$$520$$ 0 0
$$521$$ 4.55410e6 0.735036 0.367518 0.930016i $$-0.380208\pi$$
0.367518 + 0.930016i $$0.380208\pi$$
$$522$$ −2.55636e6 −0.410625
$$523$$ −4.82224e6 −0.770894 −0.385447 0.922730i $$-0.625953\pi$$
−0.385447 + 0.922730i $$0.625953\pi$$
$$524$$ 1.89523e6 0.301533
$$525$$ 0 0
$$526$$ 677184. 0.106719
$$527$$ 1.39207e6 0.218340
$$528$$ −511488. −0.0798455
$$529$$ −6.34271e6 −0.985452
$$530$$ 0 0
$$531$$ 1.42641e6 0.219537
$$532$$ −1.26712e6 −0.194106
$$533$$ 1.83497e6 0.279776
$$534$$ 1.26144e6 0.191432
$$535$$ 0 0
$$536$$ 6848.00 0.00102956
$$537$$ −4.47633e6 −0.669864
$$538$$ 6.32436e6 0.942022
$$539$$ −3.24076e6 −0.480479
$$540$$ 0 0
$$541$$ 362537. 0.0532549 0.0266274 0.999645i $$-0.491523\pi$$
0.0266274 + 0.999645i $$0.491523\pi$$
$$542$$ −3.29005e6 −0.481065
$$543$$ 2.99847e6 0.436415
$$544$$ 165888. 0.0240335
$$545$$ 0 0
$$546$$ 170892. 0.0245324
$$547$$ −3.11439e6 −0.445046 −0.222523 0.974927i $$-0.571429\pi$$
−0.222523 + 0.974927i $$0.571429\pi$$
$$548$$ 211488. 0.0300839
$$549$$ −1.77009e6 −0.250649
$$550$$ 0 0
$$551$$ 1.32947e7 1.86551
$$552$$ −176256. −0.0246205
$$553$$ 3.25052e6 0.452002
$$554$$ −2.18729e6 −0.302784
$$555$$ 0 0
$$556$$ −5.61184e6 −0.769872
$$557$$ 7.99304e6 1.09163 0.545813 0.837907i $$-0.316221\pi$$
0.545813 + 0.837907i $$0.316221\pi$$
$$558$$ 2.78413e6 0.378534
$$559$$ 1.44945e6 0.196189
$$560$$ 0 0
$$561$$ 323676. 0.0434214
$$562$$ 4.36999e6 0.583633
$$563$$ 1.23236e7 1.63857 0.819286 0.573385i $$-0.194370\pi$$
0.819286 + 0.573385i $$0.194370\pi$$
$$564$$ −158112. −0.0209299
$$565$$ 0 0
$$566$$ −9.93920e6 −1.30410
$$567$$ −308367. −0.0402819
$$568$$ 2.60659e6 0.339002
$$569$$ 1.01364e7 1.31252 0.656258 0.754537i $$-0.272138\pi$$
0.656258 + 0.754537i $$0.272138\pi$$
$$570$$ 0 0
$$571$$ 6.53084e6 0.838260 0.419130 0.907926i $$-0.362335\pi$$
0.419130 + 0.907926i $$0.362335\pi$$
$$572$$ −358752. −0.0458463
$$573$$ 365742. 0.0465359
$$574$$ −3.41558e6 −0.432698
$$575$$ 0 0
$$576$$ 331776. 0.0416667
$$577$$ 1.24453e6 0.155621 0.0778103 0.996968i $$-0.475207\pi$$
0.0778103 + 0.996968i $$0.475207\pi$$
$$578$$ 5.57445e6 0.694037
$$579$$ 4.45186e6 0.551880
$$580$$ 0 0
$$581$$ −5.10110e6 −0.626936
$$582$$ 29628.0 0.00362573
$$583$$ −3.97735e6 −0.484644
$$584$$ 2.22118e6 0.269496
$$585$$ 0 0
$$586$$ −1.36558e6 −0.164275
$$587$$ 1.33403e6 0.159797 0.0798987 0.996803i $$-0.474540\pi$$
0.0798987 + 0.996803i $$0.474540\pi$$
$$588$$ 2.10211e6 0.250734
$$589$$ −1.44792e7 −1.71972
$$590$$ 0 0
$$591$$ 4.97108e6 0.585439
$$592$$ −2.21235e6 −0.259448
$$593$$ 1.19401e7 1.39435 0.697177 0.716899i $$-0.254439\pi$$
0.697177 + 0.716899i $$0.254439\pi$$
$$594$$ 647352. 0.0752791
$$595$$ 0 0
$$596$$ 1.75824e6 0.202751
$$597$$ −6.17062e6 −0.708587
$$598$$ −123624. −0.0141368
$$599$$ −7.16430e6 −0.815843 −0.407922 0.913017i $$-0.633746\pi$$
−0.407922 + 0.913017i $$0.633746\pi$$
$$600$$ 0 0
$$601$$ 1.15163e6 0.130055 0.0650273 0.997883i $$-0.479287\pi$$
0.0650273 + 0.997883i $$0.479287\pi$$
$$602$$ −2.69799e6 −0.303423
$$603$$ −8667.00 −0.000970679 0
$$604$$ −2.76165e6 −0.308018
$$605$$ 0 0
$$606$$ −1.21781e6 −0.134709
$$607$$ 1.34268e7 1.47911 0.739554 0.673097i $$-0.235037\pi$$
0.739554 + 0.673097i $$0.235037\pi$$
$$608$$ −1.72544e6 −0.189296
$$609$$ 3.33747e6 0.364648
$$610$$ 0 0
$$611$$ −110898. −0.0120177
$$612$$ −209952. −0.0226590
$$613$$ −1.20184e7 −1.29180 −0.645900 0.763422i $$-0.723518\pi$$
−0.645900 + 0.763422i $$0.723518\pi$$
$$614$$ 8.11591e6 0.868793
$$615$$ 0 0
$$616$$ 667776. 0.0709054
$$617$$ 6.98519e6 0.738695 0.369348 0.929291i $$-0.379581\pi$$
0.369348 + 0.929291i $$0.379581\pi$$
$$618$$ 4.80398e6 0.505977
$$619$$ −8.20625e6 −0.860832 −0.430416 0.902631i $$-0.641633\pi$$
−0.430416 + 0.902631i $$0.641633\pi$$
$$620$$ 0 0
$$621$$ 223074. 0.0232124
$$622$$ 826392. 0.0856466
$$623$$ −1.64688e6 −0.169997
$$624$$ 232704. 0.0239245
$$625$$ 0 0
$$626$$ −1.33689e7 −1.36352
$$627$$ −3.36663e6 −0.342000
$$628$$ 5.59989e6 0.566605
$$629$$ 1.40000e6 0.141092
$$630$$ 0 0
$$631$$ −1.07686e7 −1.07668 −0.538338 0.842729i $$-0.680947\pi$$
−0.538338 + 0.842729i $$0.680947\pi$$
$$632$$ 4.42624e6 0.440801
$$633$$ −6.74529e6 −0.669101
$$634$$ −1.01316e7 −1.00104
$$635$$ 0 0
$$636$$ 2.57990e6 0.252907
$$637$$ 1.47440e6 0.143968
$$638$$ −7.00632e6 −0.681457
$$639$$ −3.29897e6 −0.319614
$$640$$ 0 0
$$641$$ 1.92571e7 1.85117 0.925585 0.378539i $$-0.123573\pi$$
0.925585 + 0.378539i $$0.123573\pi$$
$$642$$ −2.92507e6 −0.280091
$$643$$ −1.00999e7 −0.963364 −0.481682 0.876346i $$-0.659974\pi$$
−0.481682 + 0.876346i $$0.659974\pi$$
$$644$$ 230112. 0.0218637
$$645$$ 0 0
$$646$$ 1.09188e6 0.102942
$$647$$ −7.52113e6 −0.706354 −0.353177 0.935556i $$-0.614899\pi$$
−0.353177 + 0.935556i $$0.614899\pi$$
$$648$$ −419904. −0.0392837
$$649$$ 3.90942e6 0.364335
$$650$$ 0 0
$$651$$ −3.63484e6 −0.336150
$$652$$ −3.08130e6 −0.283867
$$653$$ 2.67197e6 0.245216 0.122608 0.992455i $$-0.460874\pi$$
0.122608 + 0.992455i $$0.460874\pi$$
$$654$$ −7.81254e6 −0.714246
$$655$$ 0 0
$$656$$ −4.65101e6 −0.421976
$$657$$ −2.81119e6 −0.254083
$$658$$ 206424. 0.0185864
$$659$$ 6.99948e6 0.627845 0.313922 0.949449i $$-0.398357\pi$$
0.313922 + 0.949449i $$0.398357\pi$$
$$660$$ 0 0
$$661$$ 408122. 0.0363318 0.0181659 0.999835i $$-0.494217\pi$$
0.0181659 + 0.999835i $$0.494217\pi$$
$$662$$ −2.40853e6 −0.213603
$$663$$ −147258. −0.0130105
$$664$$ −6.94618e6 −0.611400
$$665$$ 0 0
$$666$$ 2.80001e6 0.244610
$$667$$ −2.41434e6 −0.210128
$$668$$ −9.29107e6 −0.805610
$$669$$ 1.52344e6 0.131601
$$670$$ 0 0
$$671$$ −4.85137e6 −0.415966
$$672$$ −433152. −0.0370013
$$673$$ −1.74939e7 −1.48885 −0.744423 0.667709i $$-0.767275\pi$$
−0.744423 + 0.667709i $$0.767275\pi$$
$$674$$ 839108. 0.0711489
$$675$$ 0 0
$$676$$ −5.77747e6 −0.486263
$$677$$ 8.67440e6 0.727391 0.363695 0.931518i $$-0.381515\pi$$
0.363695 + 0.931518i $$0.381515\pi$$
$$678$$ 4.97966e6 0.416031
$$679$$ −38681.0 −0.00321976
$$680$$ 0 0
$$681$$ −418392. −0.0345713
$$682$$ 7.63058e6 0.628198
$$683$$ −1.18478e7 −0.971822 −0.485911 0.874008i $$-0.661512\pi$$
−0.485911 + 0.874008i $$0.661512\pi$$
$$684$$ 2.18376e6 0.178470
$$685$$ 0 0
$$686$$ −5.90414e6 −0.479012
$$687$$ 811035. 0.0655613
$$688$$ −3.67386e6 −0.295904
$$689$$ 1.80952e6 0.145216
$$690$$ 0 0
$$691$$ 9.47775e6 0.755110 0.377555 0.925987i $$-0.376765\pi$$
0.377555 + 0.925987i $$0.376765\pi$$
$$692$$ −1.18100e7 −0.937530
$$693$$ −845154. −0.0668502
$$694$$ 1.60866e7 1.26785
$$695$$ 0 0
$$696$$ 4.54464e6 0.355612
$$697$$ 2.94322e6 0.229478
$$698$$ −33320.0 −0.00258861
$$699$$ 9.57722e6 0.741390
$$700$$ 0 0
$$701$$ −2.28147e7 −1.75355 −0.876777 0.480898i $$-0.840311\pi$$
−0.876777 + 0.480898i $$0.840311\pi$$
$$702$$ −294516. −0.0225562
$$703$$ −1.45618e7 −1.11129
$$704$$ 909312. 0.0691483
$$705$$ 0 0
$$706$$ 7.80002e6 0.588958
$$707$$ 1.58992e6 0.119626
$$708$$ −2.53584e6 −0.190125
$$709$$ 1.27436e7 0.952090 0.476045 0.879421i $$-0.342070\pi$$
0.476045 + 0.879421i $$0.342070\pi$$
$$710$$ 0 0
$$711$$ −5.60196e6 −0.415591
$$712$$ −2.24256e6 −0.165785
$$713$$ 2.62946e6 0.193706
$$714$$ 274104. 0.0201219
$$715$$ 0 0
$$716$$ 7.95792e6 0.580119
$$717$$ −1.03642e7 −0.752902
$$718$$ −9.08352e6 −0.657572
$$719$$ 2.44929e6 0.176692 0.0883462 0.996090i $$-0.471842\pi$$
0.0883462 + 0.996090i $$0.471842\pi$$
$$720$$ 0 0
$$721$$ −6.27187e6 −0.449323
$$722$$ −1.45250e6 −0.103699
$$723$$ −7.70595e6 −0.548252
$$724$$ −5.33061e6 −0.377947
$$725$$ 0 0
$$726$$ −4.02361e6 −0.283318
$$727$$ 415033. 0.0291237 0.0145619 0.999894i $$-0.495365\pi$$
0.0145619 + 0.999894i $$0.495365\pi$$
$$728$$ −303808. −0.0212457
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 2.32486e6 0.160918
$$732$$ 3.14683e6 0.217068
$$733$$ 1.72877e7 1.18844 0.594221 0.804302i $$-0.297461\pi$$
0.594221 + 0.804302i $$0.297461\pi$$
$$734$$ 1.14461e7 0.784186
$$735$$ 0 0
$$736$$ 313344. 0.0213219
$$737$$ −23754.0 −0.00161090
$$738$$ 5.88643e6 0.397843
$$739$$ 5.18834e6 0.349476 0.174738 0.984615i $$-0.444092\pi$$
0.174738 + 0.984615i $$0.444092\pi$$
$$740$$ 0 0
$$741$$ 1.53166e6 0.102475
$$742$$ −3.36821e6 −0.224589
$$743$$ −4.79572e6 −0.318700 −0.159350 0.987222i $$-0.550940\pi$$
−0.159350 + 0.987222i $$0.550940\pi$$
$$744$$ −4.94957e6 −0.327820
$$745$$ 0 0
$$746$$ 2.46124e6 0.161923
$$747$$ 8.79125e6 0.576434
$$748$$ −575424. −0.0376040
$$749$$ 3.81884e6 0.248730
$$750$$ 0 0
$$751$$ −1.85654e7 −1.20117 −0.600585 0.799561i $$-0.705066\pi$$
−0.600585 + 0.799561i $$0.705066\pi$$
$$752$$ 281088. 0.0181258
$$753$$ 1.86937e6 0.120146
$$754$$ 3.18756e6 0.204188
$$755$$ 0 0
$$756$$ 548208. 0.0348852
$$757$$ −2.82068e7 −1.78902 −0.894508 0.447053i $$-0.852474\pi$$
−0.894508 + 0.447053i $$0.852474\pi$$
$$758$$ −2.15951e7 −1.36516
$$759$$ 611388. 0.0385223
$$760$$ 0 0
$$761$$ 6.56161e6 0.410723 0.205361 0.978686i $$-0.434163\pi$$
0.205361 + 0.978686i $$0.434163\pi$$
$$762$$ 9.21773e6 0.575091
$$763$$ 1.01997e7 0.634273
$$764$$ −650208. −0.0403013
$$765$$ 0 0
$$766$$ 4.34750e6 0.267712
$$767$$ −1.77861e6 −0.109167
$$768$$ −589824. −0.0360844
$$769$$ 2.20930e7 1.34722 0.673610 0.739087i $$-0.264743\pi$$
0.673610 + 0.739087i $$0.264743\pi$$
$$770$$ 0 0
$$771$$ −1.30787e7 −0.792371
$$772$$ −7.91442e6 −0.477942
$$773$$ 3.00787e7 1.81055 0.905276 0.424824i $$-0.139664\pi$$
0.905276 + 0.424824i $$0.139664\pi$$
$$774$$ 4.64972e6 0.278981
$$775$$ 0 0
$$776$$ −52672.0 −0.00313997
$$777$$ −3.65557e6 −0.217221
$$778$$ 1.39373e7 0.825523
$$779$$ −3.06131e7 −1.80744
$$780$$ 0 0
$$781$$ −9.04162e6 −0.530418
$$782$$ −198288. −0.0115952
$$783$$ −5.75181e6 −0.335274
$$784$$ −3.73709e6 −0.217142
$$785$$ 0 0
$$786$$ 4.26427e6 0.246200
$$787$$ 3.28954e6 0.189321 0.0946605 0.995510i $$-0.469823\pi$$
0.0946605 + 0.995510i $$0.469823\pi$$
$$788$$ −8.83747e6 −0.507005
$$789$$ 1.52366e6 0.0871358
$$790$$ 0 0
$$791$$ −6.50123e6 −0.369449
$$792$$ −1.15085e6 −0.0651936
$$793$$ 2.20715e6 0.124638
$$794$$ 1.30636e7 0.735381
$$795$$ 0 0
$$796$$ 1.09700e7 0.613655
$$797$$ −6.71053e6 −0.374206 −0.187103 0.982340i $$-0.559910\pi$$
−0.187103 + 0.982340i $$0.559910\pi$$
$$798$$ −2.85102e6 −0.158487
$$799$$ −177876. −0.00985713
$$800$$ 0 0
$$801$$ 2.83824e6 0.156303
$$802$$ 1.70928e7 0.938374
$$803$$ −7.70473e6 −0.421666
$$804$$ 15408.0 0.000840632 0
$$805$$ 0 0
$$806$$ −3.47157e6 −0.188230
$$807$$ 1.42298e7 0.769157
$$808$$ 2.16499e6 0.116662
$$809$$ 8.74254e6 0.469641 0.234821 0.972039i $$-0.424550\pi$$
0.234821 + 0.972039i $$0.424550\pi$$
$$810$$ 0 0
$$811$$ −2.48410e7 −1.32622 −0.663112 0.748520i $$-0.730765\pi$$
−0.663112 + 0.748520i $$0.730765\pi$$
$$812$$ −5.93328e6 −0.315795
$$813$$ −7.40261e6 −0.392788
$$814$$ 7.67410e6 0.405944
$$815$$ 0 0
$$816$$ 373248. 0.0196233
$$817$$ −2.41814e7 −1.26744
$$818$$ 5.80750e6 0.303463
$$819$$ 384507. 0.0200306
$$820$$ 0 0
$$821$$ 2.12219e7 1.09882 0.549409 0.835554i $$-0.314853\pi$$
0.549409 + 0.835554i $$0.314853\pi$$
$$822$$ 475848. 0.0245634
$$823$$ 8.70659e6 0.448073 0.224036 0.974581i $$-0.428077\pi$$
0.224036 + 0.974581i $$0.428077\pi$$
$$824$$ −8.54042e6 −0.438189
$$825$$ 0 0
$$826$$ 3.31068e6 0.168837
$$827$$ −3.71184e7 −1.88723 −0.943617 0.331040i $$-0.892600\pi$$
−0.943617 + 0.331040i $$0.892600\pi$$
$$828$$ −396576. −0.0201025
$$829$$ 1.01765e6 0.0514295 0.0257147 0.999669i $$-0.491814\pi$$
0.0257147 + 0.999669i $$0.491814\pi$$
$$830$$ 0 0
$$831$$ −4.92141e6 −0.247222
$$832$$ −413696. −0.0207192
$$833$$ 2.36488e6 0.118085
$$834$$ −1.26266e7 −0.628598
$$835$$ 0 0
$$836$$ 5.98512e6 0.296181
$$837$$ 6.26430e6 0.309071
$$838$$ −2.23752e6 −0.110067
$$839$$ −3.36194e7 −1.64887 −0.824433 0.565960i $$-0.808506\pi$$
−0.824433 + 0.565960i $$0.808506\pi$$
$$840$$ 0 0
$$841$$ 4.17410e7 2.03504
$$842$$ 1.56588e7 0.761164
$$843$$ 9.83248e6 0.476534
$$844$$ 1.19916e7 0.579458
$$845$$ 0 0
$$846$$ −355752. −0.0170892
$$847$$ 5.25305e6 0.251596
$$848$$ −4.58650e6 −0.219024
$$849$$ −2.23632e7 −1.06479
$$850$$ 0 0
$$851$$ 2.64445e6 0.125173
$$852$$ 5.86483e6 0.276794
$$853$$ 3.52574e7 1.65912 0.829559 0.558419i $$-0.188592\pi$$
0.829559 + 0.558419i $$0.188592\pi$$
$$854$$ −4.10836e6 −0.192763
$$855$$ 0 0
$$856$$ 5.20013e6 0.242566
$$857$$ 3.14941e7 1.46480 0.732398 0.680877i $$-0.238401\pi$$
0.732398 + 0.680877i $$0.238401\pi$$
$$858$$ −807192. −0.0374333
$$859$$ 1.19344e7 0.551848 0.275924 0.961180i $$-0.411016\pi$$
0.275924 + 0.961180i $$0.411016\pi$$
$$860$$ 0 0
$$861$$ −7.68506e6 −0.353297
$$862$$ 1.43000e7 0.655492
$$863$$ −8.70442e6 −0.397844 −0.198922 0.980015i $$-0.563744\pi$$
−0.198922 + 0.980015i $$0.563744\pi$$
$$864$$ 746496. 0.0340207
$$865$$ 0 0
$$866$$ 2.86388e7 1.29766
$$867$$ 1.25425e7 0.566679
$$868$$ 6.46194e6 0.291114
$$869$$ −1.53535e7 −0.689697
$$870$$ 0 0
$$871$$ 10807.0 0.000482681 0
$$872$$ 1.38890e7 0.618555
$$873$$ 66663.0 0.00296039
$$874$$ 2.06244e6 0.0913277
$$875$$ 0 0
$$876$$ 4.99766e6 0.220043
$$877$$ −1.17999e7 −0.518059 −0.259029 0.965869i $$-0.583403\pi$$
−0.259029 + 0.965869i $$0.583403\pi$$
$$878$$ −6.87158e6 −0.300829
$$879$$ −3.07255e6 −0.134130
$$880$$ 0 0
$$881$$ −2.73840e7 −1.18866 −0.594330 0.804221i $$-0.702583\pi$$
−0.594330 + 0.804221i $$0.702583\pi$$
$$882$$ 4.72975e6 0.204723
$$883$$ 8.80577e6 0.380072 0.190036 0.981777i $$-0.439140\pi$$
0.190036 + 0.981777i $$0.439140\pi$$
$$884$$ 261792. 0.0112675
$$885$$ 0 0
$$886$$ 1.35868e7 0.581477
$$887$$ −250122. −0.0106744 −0.00533719 0.999986i $$-0.501699\pi$$
−0.00533719 + 0.999986i $$0.501699\pi$$
$$888$$ −4.97779e6 −0.211838
$$889$$ −1.20343e7 −0.510699
$$890$$ 0 0
$$891$$ 1.45654e6 0.0614651
$$892$$ −2.70834e6 −0.113970
$$893$$ 1.85013e6 0.0776379
$$894$$ 3.95604e6 0.165545
$$895$$ 0 0
$$896$$ 770048. 0.0320441
$$897$$ −278154. −0.0115426
$$898$$ 1.35842e7 0.562140
$$899$$ −6.77988e7 −2.79784
$$900$$ 0 0
$$901$$ 2.90239e6 0.119109
$$902$$ 1.61332e7 0.660243
$$903$$ −6.07047e6 −0.247744
$$904$$ −8.85274e6 −0.360294
$$905$$ 0 0
$$906$$ −6.21371e6 −0.251496
$$907$$ 3.24955e7 1.31161 0.655806 0.754929i $$-0.272329\pi$$
0.655806 + 0.754929i $$0.272329\pi$$
$$908$$ 743808. 0.0299396
$$909$$ −2.74007e6 −0.109990
$$910$$ 0 0
$$911$$ 4.24595e7 1.69504 0.847518 0.530766i $$-0.178096\pi$$
0.847518 + 0.530766i $$0.178096\pi$$
$$912$$ −3.88224e6 −0.154559
$$913$$ 2.40945e7 0.956625
$$914$$ −1.81126e7 −0.717157
$$915$$ 0 0
$$916$$ −1.44184e6 −0.0567778
$$917$$ −5.56724e6 −0.218634
$$918$$ −472392. −0.0185010
$$919$$ 1.41629e7 0.553176 0.276588 0.960989i $$-0.410796\pi$$
0.276588 + 0.960989i $$0.410796\pi$$
$$920$$ 0 0
$$921$$ 1.82608e7 0.709366
$$922$$ 5.11579e6 0.198192
$$923$$ 4.11353e6 0.158932
$$924$$ 1.50250e6 0.0578940
$$925$$ 0 0
$$926$$ −2.87945e7 −1.10352
$$927$$ 1.08090e7 0.413128
$$928$$ −8.07936e6 −0.307969
$$929$$ 4.37292e7 1.66239 0.831194 0.555982i $$-0.187658\pi$$
0.831194 + 0.555982i $$0.187658\pi$$
$$930$$ 0 0
$$931$$ −2.45976e7 −0.930077
$$932$$ −1.70262e7 −0.642063
$$933$$ 1.85938e6 0.0699302
$$934$$ 1.93214e7 0.724721
$$935$$ 0 0
$$936$$ 523584. 0.0195343
$$937$$ −5.73509e6 −0.213398 −0.106699 0.994291i $$-0.534028\pi$$
−0.106699 + 0.994291i $$0.534028\pi$$
$$938$$ −20116.0 −0.000746508 0
$$939$$ −3.00801e7 −1.11331
$$940$$ 0 0
$$941$$ −3.37395e7 −1.24212 −0.621061 0.783762i $$-0.713298\pi$$
−0.621061 + 0.783762i $$0.713298\pi$$
$$942$$ 1.25997e7 0.462631
$$943$$ 5.55941e6 0.203587
$$944$$ 4.50816e6 0.164653
$$945$$ 0 0
$$946$$ 1.27437e7 0.462985
$$947$$ 3.07342e7 1.11365 0.556823 0.830631i $$-0.312020\pi$$
0.556823 + 0.830631i $$0.312020\pi$$
$$948$$ 9.95904e6 0.359912
$$949$$ 3.50531e6 0.126346
$$950$$ 0 0
$$951$$ −2.27960e7 −0.817348
$$952$$ −487296. −0.0174261
$$953$$ 2.51847e7 0.898264 0.449132 0.893465i $$-0.351733\pi$$
0.449132 + 0.893465i $$0.351733\pi$$
$$954$$ 5.80478e6 0.206498
$$955$$ 0 0
$$956$$ 1.84253e7 0.652033
$$957$$ −1.57642e7 −0.556407
$$958$$ −2.99460e6 −0.105420
$$959$$ −621246. −0.0218131
$$960$$ 0 0
$$961$$ 4.52105e7 1.57918
$$962$$ −3.49137e6 −0.121635
$$963$$ −6.58141e6 −0.228693
$$964$$ 1.36995e7 0.474801
$$965$$ 0 0
$$966$$ 517752. 0.0178517
$$967$$ −1.44556e7 −0.497130 −0.248565 0.968615i $$-0.579959\pi$$
−0.248565 + 0.968615i $$0.579959\pi$$
$$968$$ 7.15309e6 0.245361
$$969$$ 2.45673e6 0.0840520
$$970$$ 0 0
$$971$$ −1.06974e7 −0.364109 −0.182054 0.983288i $$-0.558275\pi$$
−0.182054 + 0.983288i $$0.558275\pi$$
$$972$$ −944784. −0.0320750
$$973$$ 1.64848e7 0.558214
$$974$$ −2.06558e7 −0.697660
$$975$$ 0 0
$$976$$ −5.59437e6 −0.187986
$$977$$ 8.41568e6 0.282067 0.141034 0.990005i $$-0.454957\pi$$
0.141034 + 0.990005i $$0.454957\pi$$
$$978$$ −6.93292e6 −0.231776
$$979$$ 7.77888e6 0.259394
$$980$$ 0 0
$$981$$ −1.75782e7 −0.583180
$$982$$ −3.41715e7 −1.13080
$$983$$ −3.89409e7 −1.28535 −0.642676 0.766138i $$-0.722176\pi$$
−0.642676 + 0.766138i $$0.722176\pi$$
$$984$$ −1.04648e7 −0.344542
$$985$$ 0 0
$$986$$ 5.11272e6 0.167479
$$987$$ 464454. 0.0151757
$$988$$ −2.72296e6 −0.0887460
$$989$$ 4.39141e6 0.142762
$$990$$ 0 0
$$991$$ 4.84592e7 1.56745 0.783723 0.621111i $$-0.213318\pi$$
0.783723 + 0.621111i $$0.213318\pi$$
$$992$$ 8.79923e6 0.283900
$$993$$ −5.41919e6 −0.174406
$$994$$ −7.65686e6 −0.245802
$$995$$ 0 0
$$996$$ −1.56289e7 −0.499206
$$997$$ −3.84733e7 −1.22581 −0.612903 0.790158i $$-0.709998\pi$$
−0.612903 + 0.790158i $$0.709998\pi$$
$$998$$ 1.68235e7 0.534675
$$999$$ 6.30002e6 0.199723
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 150.6.a.a.1.1 1
3.2 odd 2 450.6.a.p.1.1 1
5.2 odd 4 150.6.c.g.49.1 2
5.3 odd 4 150.6.c.g.49.2 2
5.4 even 2 150.6.a.m.1.1 yes 1
15.2 even 4 450.6.c.e.199.2 2
15.8 even 4 450.6.c.e.199.1 2
15.14 odd 2 450.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 1.1 even 1 trivial
150.6.a.m.1.1 yes 1 5.4 even 2
150.6.c.g.49.1 2 5.2 odd 4
150.6.c.g.49.2 2 5.3 odd 4
450.6.a.i.1.1 1 15.14 odd 2
450.6.a.p.1.1 1 3.2 odd 2
450.6.c.e.199.1 2 15.8 even 4
450.6.c.e.199.2 2 15.2 even 4