# Properties

 Label 450.6.a.bf.1.2 Level $450$ Weight $6$ Character 450.1 Self dual yes Analytic conductor $72.173$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{4081})$$ Defining polynomial: $$x^{2} - x - 1020$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-31.4414$$ of defining polynomial Character $$\chi$$ $$=$$ 450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +16.0000 q^{4} +241.648 q^{7} +64.0000 q^{8} +O(q^{10})$$ $$q+4.00000 q^{2} +16.0000 q^{4} +241.648 q^{7} +64.0000 q^{8} -653.296 q^{11} -828.296 q^{13} +966.592 q^{14} +256.000 q^{16} -2162.48 q^{17} +1254.24 q^{19} -2613.18 q^{22} -3746.48 q^{23} -3313.18 q^{26} +3866.37 q^{28} -2466.70 q^{29} -1895.76 q^{31} +1024.00 q^{32} -8649.92 q^{34} -1050.45 q^{37} +5016.96 q^{38} -1960.56 q^{41} -11017.6 q^{43} -10452.7 q^{44} -14985.9 q^{46} +23064.8 q^{47} +41586.8 q^{49} -13252.7 q^{52} +27329.4 q^{53} +15465.5 q^{56} -9866.82 q^{58} -35225.7 q^{59} +2685.33 q^{61} -7583.04 q^{62} +4096.00 q^{64} -48125.1 q^{67} -34599.7 q^{68} -27237.7 q^{71} +6547.77 q^{73} -4201.78 q^{74} +20067.8 q^{76} -157868. q^{77} -65037.3 q^{79} -7842.23 q^{82} +62567.2 q^{83} -44070.3 q^{86} -41811.0 q^{88} -17926.2 q^{89} -200156. q^{91} -59943.7 q^{92} +92259.2 q^{94} -95974.9 q^{97} +166347. q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} + O(q^{10})$$ $$2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} - 540q^{11} - 890q^{13} + 400q^{14} + 512q^{16} - 492q^{17} + 592q^{19} - 2160q^{22} - 3660q^{23} - 3560q^{26} + 1600q^{28} - 5700q^{29} - 5708q^{31} + 2048q^{32} - 1968q^{34} - 11300q^{37} + 2368q^{38} - 15420q^{41} - 6320q^{43} - 8640q^{44} - 14640q^{46} + 7800q^{47} + 44844q^{49} - 14240q^{52} + 27828q^{53} + 6400q^{56} - 22800q^{58} - 50520q^{59} - 29126q^{61} - 22832q^{62} + 8192q^{64} - 97400q^{67} - 7872q^{68} - 6180q^{71} - 32900q^{73} - 45200q^{74} + 9472q^{76} - 173916q^{77} + 7912q^{79} - 61680q^{82} + 163464q^{83} - 25280q^{86} - 34560q^{88} - 164640q^{89} - 191416q^{91} - 58560q^{92} + 31200q^{94} - 52430q^{97} + 179376q^{98} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 241.648 1.86397 0.931984 0.362500i $$-0.118077\pi$$
0.931984 + 0.362500i $$0.118077\pi$$
$$8$$ 64.0000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −653.296 −1.62790 −0.813951 0.580933i $$-0.802688\pi$$
−0.813951 + 0.580933i $$0.802688\pi$$
$$12$$ 0 0
$$13$$ −828.296 −1.35934 −0.679669 0.733519i $$-0.737876\pi$$
−0.679669 + 0.733519i $$0.737876\pi$$
$$14$$ 966.592 1.31802
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ −2162.48 −1.81481 −0.907403 0.420262i $$-0.861938\pi$$
−0.907403 + 0.420262i $$0.861938\pi$$
$$18$$ 0 0
$$19$$ 1254.24 0.797071 0.398535 0.917153i $$-0.369519\pi$$
0.398535 + 0.917153i $$0.369519\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2613.18 −1.15110
$$23$$ −3746.48 −1.47674 −0.738370 0.674396i $$-0.764404\pi$$
−0.738370 + 0.674396i $$0.764404\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3313.18 −0.961197
$$27$$ 0 0
$$28$$ 3866.37 0.931984
$$29$$ −2466.70 −0.544656 −0.272328 0.962205i $$-0.587794\pi$$
−0.272328 + 0.962205i $$0.587794\pi$$
$$30$$ 0 0
$$31$$ −1895.76 −0.354306 −0.177153 0.984183i $$-0.556689\pi$$
−0.177153 + 0.984183i $$0.556689\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ −8649.92 −1.28326
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1050.45 −0.126145 −0.0630724 0.998009i $$-0.520090\pi$$
−0.0630724 + 0.998009i $$0.520090\pi$$
$$38$$ 5016.96 0.563614
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1960.56 −0.182146 −0.0910730 0.995844i $$-0.529030\pi$$
−0.0910730 + 0.995844i $$0.529030\pi$$
$$42$$ 0 0
$$43$$ −11017.6 −0.908688 −0.454344 0.890826i $$-0.650126\pi$$
−0.454344 + 0.890826i $$0.650126\pi$$
$$44$$ −10452.7 −0.813951
$$45$$ 0 0
$$46$$ −14985.9 −1.04421
$$47$$ 23064.8 1.52302 0.761509 0.648154i $$-0.224459\pi$$
0.761509 + 0.648154i $$0.224459\pi$$
$$48$$ 0 0
$$49$$ 41586.8 2.47437
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −13252.7 −0.679669
$$53$$ 27329.4 1.33641 0.668205 0.743977i $$-0.267063\pi$$
0.668205 + 0.743977i $$0.267063\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 15465.5 0.659012
$$57$$ 0 0
$$58$$ −9866.82 −0.385130
$$59$$ −35225.7 −1.31744 −0.658718 0.752390i $$-0.728901\pi$$
−0.658718 + 0.752390i $$0.728901\pi$$
$$60$$ 0 0
$$61$$ 2685.33 0.0924002 0.0462001 0.998932i $$-0.485289\pi$$
0.0462001 + 0.998932i $$0.485289\pi$$
$$62$$ −7583.04 −0.250532
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −48125.1 −1.30974 −0.654869 0.755743i $$-0.727276\pi$$
−0.654869 + 0.755743i $$0.727276\pi$$
$$68$$ −34599.7 −0.907403
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −27237.7 −0.641245 −0.320622 0.947207i $$-0.603892\pi$$
−0.320622 + 0.947207i $$0.603892\pi$$
$$72$$ 0 0
$$73$$ 6547.77 0.143809 0.0719046 0.997412i $$-0.477092\pi$$
0.0719046 + 0.997412i $$0.477092\pi$$
$$74$$ −4201.78 −0.0891978
$$75$$ 0 0
$$76$$ 20067.8 0.398535
$$77$$ −157868. −3.03436
$$78$$ 0 0
$$79$$ −65037.3 −1.17245 −0.586226 0.810148i $$-0.699387\pi$$
−0.586226 + 0.810148i $$0.699387\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −7842.23 −0.128797
$$83$$ 62567.2 0.996900 0.498450 0.866919i $$-0.333903\pi$$
0.498450 + 0.866919i $$0.333903\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −44070.3 −0.642539
$$87$$ 0 0
$$88$$ −41811.0 −0.575551
$$89$$ −17926.2 −0.239891 −0.119946 0.992780i $$-0.538272\pi$$
−0.119946 + 0.992780i $$0.538272\pi$$
$$90$$ 0 0
$$91$$ −200156. −2.53376
$$92$$ −59943.7 −0.738370
$$93$$ 0 0
$$94$$ 92259.2 1.07694
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −95974.9 −1.03569 −0.517843 0.855475i $$-0.673265\pi$$
−0.517843 + 0.855475i $$0.673265\pi$$
$$98$$ 166347. 1.74965
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −41533.7 −0.405133 −0.202567 0.979269i $$-0.564928\pi$$
−0.202567 + 0.979269i $$0.564928\pi$$
$$102$$ 0 0
$$103$$ 41380.4 0.384328 0.192164 0.981363i $$-0.438449\pi$$
0.192164 + 0.981363i $$0.438449\pi$$
$$104$$ −53011.0 −0.480598
$$105$$ 0 0
$$106$$ 109317. 0.944985
$$107$$ −11403.4 −0.0962885 −0.0481442 0.998840i $$-0.515331\pi$$
−0.0481442 + 0.998840i $$0.515331\pi$$
$$108$$ 0 0
$$109$$ 98362.4 0.792981 0.396490 0.918039i $$-0.370228\pi$$
0.396490 + 0.918039i $$0.370228\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 61861.9 0.465992
$$113$$ 10076.8 0.0742377 0.0371189 0.999311i $$-0.488182\pi$$
0.0371189 + 0.999311i $$0.488182\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −39467.3 −0.272328
$$117$$ 0 0
$$118$$ −140903. −0.931568
$$119$$ −522559. −3.38274
$$120$$ 0 0
$$121$$ 265745. 1.65007
$$122$$ 10741.3 0.0653368
$$123$$ 0 0
$$124$$ −30332.2 −0.177153
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 255129. 1.40362 0.701812 0.712362i $$-0.252374\pi$$
0.701812 + 0.712362i $$0.252374\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 65433.3 0.333135 0.166568 0.986030i $$-0.446732\pi$$
0.166568 + 0.986030i $$0.446732\pi$$
$$132$$ 0 0
$$133$$ 303085. 1.48571
$$134$$ −192500. −0.926124
$$135$$ 0 0
$$136$$ −138399. −0.641631
$$137$$ −67354.0 −0.306593 −0.153296 0.988180i $$-0.548989\pi$$
−0.153296 + 0.988180i $$0.548989\pi$$
$$138$$ 0 0
$$139$$ −93837.8 −0.411946 −0.205973 0.978558i $$-0.566036\pi$$
−0.205973 + 0.978558i $$0.566036\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −108951. −0.453429
$$143$$ 541123. 2.21287
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 26191.1 0.101688
$$147$$ 0 0
$$148$$ −16807.1 −0.0630724
$$149$$ −432235. −1.59498 −0.797489 0.603334i $$-0.793839\pi$$
−0.797489 + 0.603334i $$0.793839\pi$$
$$150$$ 0 0
$$151$$ −255822. −0.913053 −0.456527 0.889710i $$-0.650907\pi$$
−0.456527 + 0.889710i $$0.650907\pi$$
$$152$$ 80271.4 0.281807
$$153$$ 0 0
$$154$$ −631471. −2.14561
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 58837.3 0.190504 0.0952519 0.995453i $$-0.469634\pi$$
0.0952519 + 0.995453i $$0.469634\pi$$
$$158$$ −260149. −0.829048
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −905330. −2.75259
$$162$$ 0 0
$$163$$ −155006. −0.456962 −0.228481 0.973548i $$-0.573376\pi$$
−0.228481 + 0.973548i $$0.573376\pi$$
$$164$$ −31368.9 −0.0910730
$$165$$ 0 0
$$166$$ 250269. 0.704914
$$167$$ −588656. −1.63332 −0.816658 0.577121i $$-0.804176\pi$$
−0.816658 + 0.577121i $$0.804176\pi$$
$$168$$ 0 0
$$169$$ 314782. 0.847798
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −176281. −0.454344
$$173$$ 192049. 0.487862 0.243931 0.969793i $$-0.421563\pi$$
0.243931 + 0.969793i $$0.421563\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −167244. −0.406976
$$177$$ 0 0
$$178$$ −71704.9 −0.169629
$$179$$ 350715. 0.818130 0.409065 0.912505i $$-0.365855\pi$$
0.409065 + 0.912505i $$0.365855\pi$$
$$180$$ 0 0
$$181$$ 524637. 1.19032 0.595158 0.803608i $$-0.297089\pi$$
0.595158 + 0.803608i $$0.297089\pi$$
$$182$$ −800625. −1.79164
$$183$$ 0 0
$$184$$ −239775. −0.522106
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.41274e6 2.95433
$$188$$ 369037. 0.761509
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −63662.7 −0.126270 −0.0631352 0.998005i $$-0.520110\pi$$
−0.0631352 + 0.998005i $$0.520110\pi$$
$$192$$ 0 0
$$193$$ 717477. 1.38648 0.693242 0.720705i $$-0.256182\pi$$
0.693242 + 0.720705i $$0.256182\pi$$
$$194$$ −383900. −0.732341
$$195$$ 0 0
$$196$$ 665389. 1.23719
$$197$$ −287465. −0.527740 −0.263870 0.964558i $$-0.584999\pi$$
−0.263870 + 0.964558i $$0.584999\pi$$
$$198$$ 0 0
$$199$$ 643619. 1.15212 0.576058 0.817409i $$-0.304590\pi$$
0.576058 + 0.817409i $$0.304590\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −166135. −0.286472
$$203$$ −596074. −1.01522
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 165522. 0.271761
$$207$$ 0 0
$$208$$ −212044. −0.339834
$$209$$ −819391. −1.29755
$$210$$ 0 0
$$211$$ −873827. −1.35120 −0.675600 0.737269i $$-0.736115\pi$$
−0.675600 + 0.737269i $$0.736115\pi$$
$$212$$ 437270. 0.668205
$$213$$ 0 0
$$214$$ −45613.5 −0.0680862
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −458107. −0.660416
$$218$$ 393449. 0.560722
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.79117e6 2.46693
$$222$$ 0 0
$$223$$ 288496. 0.388488 0.194244 0.980953i $$-0.437775\pi$$
0.194244 + 0.980953i $$0.437775\pi$$
$$224$$ 247448. 0.329506
$$225$$ 0 0
$$226$$ 40307.0 0.0524940
$$227$$ 125358. 0.161469 0.0807343 0.996736i $$-0.474273\pi$$
0.0807343 + 0.996736i $$0.474273\pi$$
$$228$$ 0 0
$$229$$ −698183. −0.879793 −0.439897 0.898048i $$-0.644985\pi$$
−0.439897 + 0.898048i $$0.644985\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −157869. −0.192565
$$233$$ 984588. 1.18813 0.594066 0.804416i $$-0.297522\pi$$
0.594066 + 0.804416i $$0.297522\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −563611. −0.658718
$$237$$ 0 0
$$238$$ −2.09024e6 −2.39196
$$239$$ −131049. −0.148402 −0.0742011 0.997243i $$-0.523641\pi$$
−0.0742011 + 0.997243i $$0.523641\pi$$
$$240$$ 0 0
$$241$$ −1.23632e6 −1.37116 −0.685579 0.727998i $$-0.740451\pi$$
−0.685579 + 0.727998i $$0.740451\pi$$
$$242$$ 1.06298e6 1.16677
$$243$$ 0 0
$$244$$ 42965.3 0.0462001
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.03888e6 −1.08349
$$248$$ −121329. −0.125266
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −611918. −0.613068 −0.306534 0.951860i $$-0.599169\pi$$
−0.306534 + 0.951860i $$0.599169\pi$$
$$252$$ 0 0
$$253$$ 2.44756e6 2.40399
$$254$$ 1.02052e6 0.992513
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 1.64515e6 1.55372 0.776858 0.629675i $$-0.216812\pi$$
0.776858 + 0.629675i $$0.216812\pi$$
$$258$$ 0 0
$$259$$ −253838. −0.235130
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 261733. 0.235562
$$263$$ −268273. −0.239159 −0.119580 0.992825i $$-0.538155\pi$$
−0.119580 + 0.992825i $$0.538155\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1.21234e6 1.05056
$$267$$ 0 0
$$268$$ −770001. −0.654869
$$269$$ 947177. 0.798087 0.399044 0.916932i $$-0.369342\pi$$
0.399044 + 0.916932i $$0.369342\pi$$
$$270$$ 0 0
$$271$$ 609159. 0.503857 0.251929 0.967746i $$-0.418935\pi$$
0.251929 + 0.967746i $$0.418935\pi$$
$$272$$ −553595. −0.453701
$$273$$ 0 0
$$274$$ −269416. −0.216794
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −741976. −0.581019 −0.290510 0.956872i $$-0.593825\pi$$
−0.290510 + 0.956872i $$0.593825\pi$$
$$278$$ −375351. −0.291290
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 423084. 0.319639 0.159820 0.987146i $$-0.448909\pi$$
0.159820 + 0.987146i $$0.448909\pi$$
$$282$$ 0 0
$$283$$ 445130. 0.330385 0.165192 0.986261i $$-0.447175\pi$$
0.165192 + 0.986261i $$0.447175\pi$$
$$284$$ −435803. −0.320622
$$285$$ 0 0
$$286$$ 2.16449e6 1.56473
$$287$$ −473765. −0.339514
$$288$$ 0 0
$$289$$ 3.25647e6 2.29352
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 104764. 0.0719046
$$293$$ −2.45992e6 −1.67399 −0.836994 0.547211i $$-0.815689\pi$$
−0.836994 + 0.547211i $$0.815689\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −67228.5 −0.0445989
$$297$$ 0 0
$$298$$ −1.72894e6 −1.12782
$$299$$ 3.10320e6 2.00739
$$300$$ 0 0
$$301$$ −2.66238e6 −1.69376
$$302$$ −1.02329e6 −0.645626
$$303$$ 0 0
$$304$$ 321086. 0.199268
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −374253. −0.226631 −0.113315 0.993559i $$-0.536147\pi$$
−0.113315 + 0.993559i $$0.536147\pi$$
$$308$$ −2.52588e6 −1.51718
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.22568e6 −1.30485 −0.652427 0.757851i $$-0.726249\pi$$
−0.652427 + 0.757851i $$0.726249\pi$$
$$312$$ 0 0
$$313$$ −3.05869e6 −1.76471 −0.882357 0.470580i $$-0.844045\pi$$
−0.882357 + 0.470580i $$0.844045\pi$$
$$314$$ 235349. 0.134706
$$315$$ 0 0
$$316$$ −1.04060e6 −0.586226
$$317$$ 520610. 0.290981 0.145490 0.989360i $$-0.453524\pi$$
0.145490 + 0.989360i $$0.453524\pi$$
$$318$$ 0 0
$$319$$ 1.61149e6 0.886646
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.62132e6 −1.94638
$$323$$ −2.71227e6 −1.44653
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −620025. −0.323121
$$327$$ 0 0
$$328$$ −125476. −0.0643983
$$329$$ 5.57357e6 2.83886
$$330$$ 0 0
$$331$$ −2.12755e6 −1.06736 −0.533679 0.845687i $$-0.679191\pi$$
−0.533679 + 0.845687i $$0.679191\pi$$
$$332$$ 1.00108e6 0.498450
$$333$$ 0 0
$$334$$ −2.35462e6 −1.15493
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3.08383e6 1.47916 0.739581 0.673067i $$-0.235024\pi$$
0.739581 + 0.673067i $$0.235024\pi$$
$$338$$ 1.25913e6 0.599484
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.23849e6 0.576776
$$342$$ 0 0
$$343$$ 5.98799e6 2.74819
$$344$$ −705125. −0.321270
$$345$$ 0 0
$$346$$ 768196. 0.344970
$$347$$ 4.04955e6 1.80544 0.902719 0.430231i $$-0.141568\pi$$
0.902719 + 0.430231i $$0.141568\pi$$
$$348$$ 0 0
$$349$$ −493135. −0.216722 −0.108361 0.994112i $$-0.534560\pi$$
−0.108361 + 0.994112i $$0.534560\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −668975. −0.287775
$$353$$ 1.34815e6 0.575841 0.287920 0.957654i $$-0.407036\pi$$
0.287920 + 0.957654i $$0.407036\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −286820. −0.119946
$$357$$ 0 0
$$358$$ 1.40286e6 0.578505
$$359$$ −1.64749e6 −0.674664 −0.337332 0.941386i $$-0.609524\pi$$
−0.337332 + 0.941386i $$0.609524\pi$$
$$360$$ 0 0
$$361$$ −902980. −0.364678
$$362$$ 2.09855e6 0.841681
$$363$$ 0 0
$$364$$ −3.20250e6 −1.26688
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 27918.4 0.0108199 0.00540997 0.999985i $$-0.498278\pi$$
0.00540997 + 0.999985i $$0.498278\pi$$
$$368$$ −959099. −0.369185
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.60409e6 2.49103
$$372$$ 0 0
$$373$$ −2.80536e6 −1.04404 −0.522019 0.852934i $$-0.674821\pi$$
−0.522019 + 0.852934i $$0.674821\pi$$
$$374$$ 5.65096e6 2.08902
$$375$$ 0 0
$$376$$ 1.47615e6 0.538468
$$377$$ 2.04316e6 0.740371
$$378$$ 0 0
$$379$$ 1.70014e6 0.607976 0.303988 0.952676i $$-0.401682\pi$$
0.303988 + 0.952676i $$0.401682\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −254651. −0.0892867
$$383$$ 627913. 0.218727 0.109363 0.994002i $$-0.465119\pi$$
0.109363 + 0.994002i $$0.465119\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.86991e6 0.980392
$$387$$ 0 0
$$388$$ −1.53560e6 −0.517843
$$389$$ −3.14339e6 −1.05323 −0.526616 0.850103i $$-0.676540\pi$$
−0.526616 + 0.850103i $$0.676540\pi$$
$$390$$ 0 0
$$391$$ 8.10169e6 2.68000
$$392$$ 2.66156e6 0.874823
$$393$$ 0 0
$$394$$ −1.14986e6 −0.373169
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1.10773e6 0.352743 0.176371 0.984324i $$-0.443564\pi$$
0.176371 + 0.984324i $$0.443564\pi$$
$$398$$ 2.57448e6 0.814669
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.41563e6 −0.750187 −0.375093 0.926987i $$-0.622389\pi$$
−0.375093 + 0.926987i $$0.622389\pi$$
$$402$$ 0 0
$$403$$ 1.57025e6 0.481622
$$404$$ −664540. −0.202567
$$405$$ 0 0
$$406$$ −2.38430e6 −0.717869
$$407$$ 686252. 0.205351
$$408$$ 0 0
$$409$$ 3.69830e6 1.09319 0.546593 0.837398i $$-0.315925\pi$$
0.546593 + 0.837398i $$0.315925\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 662087. 0.192164
$$413$$ −8.51222e6 −2.45566
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −848175. −0.240299
$$417$$ 0 0
$$418$$ −3.27756e6 −0.917509
$$419$$ 3.45478e6 0.961357 0.480679 0.876897i $$-0.340390\pi$$
0.480679 + 0.876897i $$0.340390\pi$$
$$420$$ 0 0
$$421$$ −4.83980e6 −1.33083 −0.665415 0.746474i $$-0.731745\pi$$
−0.665415 + 0.746474i $$0.731745\pi$$
$$422$$ −3.49531e6 −0.955442
$$423$$ 0 0
$$424$$ 1.74908e6 0.472493
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 648905. 0.172231
$$428$$ −182454. −0.0481442
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.54923e6 1.43893 0.719464 0.694529i $$-0.244387\pi$$
0.719464 + 0.694529i $$0.244387\pi$$
$$432$$ 0 0
$$433$$ 814302. 0.208721 0.104360 0.994540i $$-0.466720\pi$$
0.104360 + 0.994540i $$0.466720\pi$$
$$434$$ −1.83243e6 −0.466984
$$435$$ 0 0
$$436$$ 1.57380e6 0.396490
$$437$$ −4.69899e6 −1.17707
$$438$$ 0 0
$$439$$ −4.02462e6 −0.996697 −0.498349 0.866977i $$-0.666060\pi$$
−0.498349 + 0.866977i $$0.666060\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 7.16470e6 1.74438
$$443$$ 4.46382e6 1.08068 0.540341 0.841446i $$-0.318295\pi$$
0.540341 + 0.841446i $$0.318295\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 1.15399e6 0.274703
$$447$$ 0 0
$$448$$ 989791. 0.232996
$$449$$ 4.66889e6 1.09294 0.546472 0.837477i $$-0.315970\pi$$
0.546472 + 0.837477i $$0.315970\pi$$
$$450$$ 0 0
$$451$$ 1.28082e6 0.296516
$$452$$ 161228. 0.0371189
$$453$$ 0 0
$$454$$ 501433. 0.114176
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.71340e6 1.05571 0.527854 0.849335i $$-0.322997\pi$$
0.527854 + 0.849335i $$0.322997\pi$$
$$458$$ −2.79273e6 −0.622108
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.23975e6 −1.36746 −0.683731 0.729734i $$-0.739644\pi$$
−0.683731 + 0.729734i $$0.739644\pi$$
$$462$$ 0 0
$$463$$ 3.33992e6 0.724076 0.362038 0.932163i $$-0.382081\pi$$
0.362038 + 0.932163i $$0.382081\pi$$
$$464$$ −631476. −0.136164
$$465$$ 0 0
$$466$$ 3.93835e6 0.840136
$$467$$ −722497. −0.153301 −0.0766503 0.997058i $$-0.524422\pi$$
−0.0766503 + 0.997058i $$0.524422\pi$$
$$468$$ 0 0
$$469$$ −1.16293e7 −2.44131
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −2.25444e6 −0.465784
$$473$$ 7.19774e6 1.47926
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −8.36095e6 −1.69137
$$477$$ 0 0
$$478$$ −524197. −0.104936
$$479$$ 5.37714e6 1.07081 0.535405 0.844595i $$-0.320159\pi$$
0.535405 + 0.844595i $$0.320159\pi$$
$$480$$ 0 0
$$481$$ 870080. 0.171473
$$482$$ −4.94527e6 −0.969556
$$483$$ 0 0
$$484$$ 4.25192e6 0.825034
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 1.72084e6 0.328790 0.164395 0.986395i $$-0.447433\pi$$
0.164395 + 0.986395i $$0.447433\pi$$
$$488$$ 171861. 0.0326684
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7.31449e6 −1.36924 −0.684621 0.728899i $$-0.740032\pi$$
−0.684621 + 0.728899i $$0.740032\pi$$
$$492$$ 0 0
$$493$$ 5.33420e6 0.988444
$$494$$ −4.15553e6 −0.766142
$$495$$ 0 0
$$496$$ −485314. −0.0885766
$$497$$ −6.58193e6 −1.19526
$$498$$ 0 0
$$499$$ −7.29878e6 −1.31220 −0.656098 0.754676i $$-0.727794\pi$$
−0.656098 + 0.754676i $$0.727794\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −2.44767e6 −0.433505
$$503$$ −923591. −0.162764 −0.0813822 0.996683i $$-0.525933\pi$$
−0.0813822 + 0.996683i $$0.525933\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 9.79025e6 1.69988
$$507$$ 0 0
$$508$$ 4.08207e6 0.701812
$$509$$ −8.81318e6 −1.50778 −0.753891 0.657000i $$-0.771825\pi$$
−0.753891 + 0.657000i $$0.771825\pi$$
$$510$$ 0 0
$$511$$ 1.58226e6 0.268056
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ 6.58059e6 1.09864
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.50682e7 −2.47933
$$518$$ −1.01535e6 −0.166262
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1.66995e6 −0.269531 −0.134765 0.990878i $$-0.543028\pi$$
−0.134765 + 0.990878i $$0.543028\pi$$
$$522$$ 0 0
$$523$$ −2.85438e6 −0.456308 −0.228154 0.973625i $$-0.573269\pi$$
−0.228154 + 0.973625i $$0.573269\pi$$
$$524$$ 1.04693e6 0.166568
$$525$$ 0 0
$$526$$ −1.07309e6 −0.169111
$$527$$ 4.09954e6 0.642997
$$528$$ 0 0
$$529$$ 7.59978e6 1.18076
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.84936e6 0.742857
$$533$$ 1.62392e6 0.247598
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −3.08000e6 −0.463062
$$537$$ 0 0
$$538$$ 3.78871e6 0.564333
$$539$$ −2.71685e7 −4.02804
$$540$$ 0 0
$$541$$ −7.85655e6 −1.15409 −0.577044 0.816713i $$-0.695794\pi$$
−0.577044 + 0.816713i $$0.695794\pi$$
$$542$$ 2.43664e6 0.356281
$$543$$ 0 0
$$544$$ −2.21438e6 −0.320815
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.07748e7 1.53972 0.769860 0.638213i $$-0.220326\pi$$
0.769860 + 0.638213i $$0.220326\pi$$
$$548$$ −1.07766e6 −0.153296
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.09384e6 −0.434129
$$552$$ 0 0
$$553$$ −1.57161e7 −2.18541
$$554$$ −2.96791e6 −0.410843
$$555$$ 0 0
$$556$$ −1.50140e6 −0.205973
$$557$$ 7.62158e6 1.04090 0.520448 0.853893i $$-0.325765\pi$$
0.520448 + 0.853893i $$0.325765\pi$$
$$558$$ 0 0
$$559$$ 9.12581e6 1.23521
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.69233e6 0.226019
$$563$$ 4.25551e6 0.565823 0.282911 0.959146i $$-0.408700\pi$$
0.282911 + 0.959146i $$0.408700\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.78052e6 0.233617
$$567$$ 0 0
$$568$$ −1.74321e6 −0.226714
$$569$$ −5.99899e6 −0.776779 −0.388389 0.921495i $$-0.626968\pi$$
−0.388389 + 0.921495i $$0.626968\pi$$
$$570$$ 0 0
$$571$$ 9.20283e6 1.18122 0.590611 0.806957i $$-0.298887\pi$$
0.590611 + 0.806957i $$0.298887\pi$$
$$572$$ 8.65796e6 1.10643
$$573$$ 0 0
$$574$$ −1.89506e6 −0.240073
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −4.03879e6 −0.505024 −0.252512 0.967594i $$-0.581257\pi$$
−0.252512 + 0.967594i $$0.581257\pi$$
$$578$$ 1.30259e7 1.62176
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1.51192e7 1.85819
$$582$$ 0 0
$$583$$ −1.78542e7 −2.17555
$$584$$ 419058. 0.0508442
$$585$$ 0 0
$$586$$ −9.83969e6 −1.18369
$$587$$ −1.65997e7 −1.98840 −0.994200 0.107544i $$-0.965701\pi$$
−0.994200 + 0.107544i $$0.965701\pi$$
$$588$$ 0 0
$$589$$ −2.37774e6 −0.282407
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −268914. −0.0315362
$$593$$ 1.09900e7 1.28340 0.641698 0.766957i $$-0.278230\pi$$
0.641698 + 0.766957i $$0.278230\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.91577e6 −0.797489
$$597$$ 0 0
$$598$$ 1.24128e7 1.41944
$$599$$ −4.47167e6 −0.509217 −0.254608 0.967044i $$-0.581947\pi$$
−0.254608 + 0.967044i $$0.581947\pi$$
$$600$$ 0 0
$$601$$ 1.60311e7 1.81041 0.905206 0.424974i $$-0.139717\pi$$
0.905206 + 0.424974i $$0.139717\pi$$
$$602$$ −1.06495e7 −1.19767
$$603$$ 0 0
$$604$$ −4.09316e6 −0.456527
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 720250. 0.0793435 0.0396718 0.999213i $$-0.487369\pi$$
0.0396718 + 0.999213i $$0.487369\pi$$
$$608$$ 1.28434e6 0.140904
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.91045e7 −2.07030
$$612$$ 0 0
$$613$$ 1.46218e7 1.57163 0.785813 0.618464i $$-0.212245\pi$$
0.785813 + 0.618464i $$0.212245\pi$$
$$614$$ −1.49701e6 −0.160252
$$615$$ 0 0
$$616$$ −1.01035e7 −1.07281
$$617$$ −4.57269e6 −0.483569 −0.241784 0.970330i $$-0.577733\pi$$
−0.241784 + 0.970330i $$0.577733\pi$$
$$618$$ 0 0
$$619$$ −4.62486e6 −0.485145 −0.242573 0.970133i $$-0.577991\pi$$
−0.242573 + 0.970133i $$0.577991\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −8.90273e6 −0.922671
$$623$$ −4.33184e6 −0.447149
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −1.22348e7 −1.24784
$$627$$ 0 0
$$628$$ 941397. 0.0952519
$$629$$ 2.27157e6 0.228928
$$630$$ 0 0
$$631$$ −1.03670e7 −1.03653 −0.518263 0.855221i $$-0.673421\pi$$
−0.518263 + 0.855221i $$0.673421\pi$$
$$632$$ −4.16239e6 −0.414524
$$633$$ 0 0
$$634$$ 2.08244e6 0.205755
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.44462e7 −3.36351
$$638$$ 6.44595e6 0.626954
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4.62432e6 0.444532 0.222266 0.974986i $$-0.428655\pi$$
0.222266 + 0.974986i $$0.428655\pi$$
$$642$$ 0 0
$$643$$ −1.13460e7 −1.08222 −0.541111 0.840951i $$-0.681996\pi$$
−0.541111 + 0.840951i $$0.681996\pi$$
$$644$$ −1.44853e7 −1.37630
$$645$$ 0 0
$$646$$ −1.08491e7 −1.02285
$$647$$ −1.10482e7 −1.03760 −0.518800 0.854895i $$-0.673621\pi$$
−0.518800 + 0.854895i $$0.673621\pi$$
$$648$$ 0 0
$$649$$ 2.30128e7 2.14466
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −2.48010e6 −0.228481
$$653$$ 1.37432e6 0.126126 0.0630631 0.998010i $$-0.479913\pi$$
0.0630631 + 0.998010i $$0.479913\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −501902. −0.0455365
$$657$$ 0 0
$$658$$ 2.22943e7 2.00738
$$659$$ 1.28061e7 1.14869 0.574344 0.818614i $$-0.305257\pi$$
0.574344 + 0.818614i $$0.305257\pi$$
$$660$$ 0 0
$$661$$ −104794. −0.00932894 −0.00466447 0.999989i $$-0.501485\pi$$
−0.00466447 + 0.999989i $$0.501485\pi$$
$$662$$ −8.51021e6 −0.754737
$$663$$ 0 0
$$664$$ 4.00430e6 0.352457
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.24146e6 0.804315
$$668$$ −9.41850e6 −0.816658
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −1.75432e6 −0.150419
$$672$$ 0 0
$$673$$ −1.80490e7 −1.53608 −0.768042 0.640400i $$-0.778769\pi$$
−0.768042 + 0.640400i $$0.778769\pi$$
$$674$$ 1.23353e7 1.04593
$$675$$ 0 0
$$676$$ 5.03651e6 0.423899
$$677$$ −1.29914e7 −1.08939 −0.544697 0.838633i $$-0.683355\pi$$
−0.544697 + 0.838633i $$0.683355\pi$$
$$678$$ 0 0
$$679$$ −2.31922e7 −1.93049
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 4.95397e6 0.407842
$$683$$ 3.91843e6 0.321411 0.160705 0.987002i $$-0.448623\pi$$
0.160705 + 0.987002i $$0.448623\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 2.39520e7 1.94326
$$687$$ 0 0
$$688$$ −2.82050e6 −0.227172
$$689$$ −2.26368e7 −1.81663
$$690$$ 0 0
$$691$$ −3.11077e6 −0.247840 −0.123920 0.992292i $$-0.539547\pi$$
−0.123920 + 0.992292i $$0.539547\pi$$
$$692$$ 3.07278e6 0.243931
$$693$$ 0 0
$$694$$ 1.61982e7 1.27664
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.23967e6 0.330560
$$698$$ −1.97254e6 −0.153245
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.94916e7 −1.49814 −0.749070 0.662490i $$-0.769500\pi$$
−0.749070 + 0.662490i $$0.769500\pi$$
$$702$$ 0 0
$$703$$ −1.31751e6 −0.100546
$$704$$ −2.67590e6 −0.203488
$$705$$ 0 0
$$706$$ 5.39261e6 0.407181
$$707$$ −1.00366e7 −0.755155
$$708$$ 0 0
$$709$$ −8.13693e6 −0.607918 −0.303959 0.952685i $$-0.598309\pi$$
−0.303959 + 0.952685i $$0.598309\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −1.14728e6 −0.0848143
$$713$$ 7.10243e6 0.523218
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 5.61144e6 0.409065
$$717$$ 0 0
$$718$$ −6.58997e6 −0.477059
$$719$$ 1.14993e7 0.829566 0.414783 0.909920i $$-0.363857\pi$$
0.414783 + 0.909920i $$0.363857\pi$$
$$720$$ 0 0
$$721$$ 9.99951e6 0.716375
$$722$$ −3.61192e6 −0.257867
$$723$$ 0 0
$$724$$ 8.39419e6 0.595158
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.10897e7 1.47991 0.739953 0.672658i $$-0.234848\pi$$
0.739953 + 0.672658i $$0.234848\pi$$
$$728$$ −1.28100e7 −0.895820
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2.38253e7 1.64909
$$732$$ 0 0
$$733$$ −2.20274e7 −1.51427 −0.757134 0.653259i $$-0.773401\pi$$
−0.757134 + 0.653259i $$0.773401\pi$$
$$734$$ 111673. 0.00765085
$$735$$ 0 0
$$736$$ −3.83640e6 −0.261053
$$737$$ 3.14399e7 2.13213
$$738$$ 0 0
$$739$$ −1.26088e7 −0.849305 −0.424653 0.905356i $$-0.639604\pi$$
−0.424653 + 0.905356i $$0.639604\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.64164e7 1.76142
$$743$$ 5.90968e6 0.392728 0.196364 0.980531i $$-0.437087\pi$$
0.196364 + 0.980531i $$0.437087\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.12214e7 −0.738247
$$747$$ 0 0
$$748$$ 2.26039e7 1.47716
$$749$$ −2.75561e6 −0.179479
$$750$$ 0 0
$$751$$ −1.15585e7 −0.747830 −0.373915 0.927463i $$-0.621985\pi$$
−0.373915 + 0.927463i $$0.621985\pi$$
$$752$$ 5.90459e6 0.380755
$$753$$ 0 0
$$754$$ 8.17265e6 0.523521
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.64988e7 −1.68068 −0.840342 0.542057i $$-0.817646\pi$$
−0.840342 + 0.542057i $$0.817646\pi$$
$$758$$ 6.80056e6 0.429904
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.54716e6 −0.0968442 −0.0484221 0.998827i $$-0.515419\pi$$
−0.0484221 + 0.998827i $$0.515419\pi$$
$$762$$ 0 0
$$763$$ 2.37691e7 1.47809
$$764$$ −1.01860e6 −0.0631352
$$765$$ 0 0
$$766$$ 2.51165e6 0.154663
$$767$$ 2.91773e7 1.79084
$$768$$ 0 0
$$769$$ 1.16191e7 0.708529 0.354265 0.935145i $$-0.384731\pi$$
0.354265 + 0.935145i $$0.384731\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.14796e7 0.693242
$$773$$ 1.16233e7 0.699649 0.349824 0.936815i $$-0.386241\pi$$
0.349824 + 0.936815i $$0.386241\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.14239e6 −0.366171
$$777$$ 0 0
$$778$$ −1.25736e7 −0.744748
$$779$$ −2.45901e6 −0.145183
$$780$$ 0 0
$$781$$ 1.77943e7 1.04388
$$782$$ 3.24068e7 1.89504
$$783$$ 0 0
$$784$$ 1.06462e7 0.618594
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −2.03314e7 −1.17012 −0.585059 0.810991i $$-0.698929\pi$$
−0.585059 + 0.810991i $$0.698929\pi$$
$$788$$ −4.59945e6 −0.263870
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.43503e6 0.138377
$$792$$ 0 0
$$793$$ −2.22425e6 −0.125603
$$794$$ 4.43092e6 0.249427
$$795$$ 0 0
$$796$$ 1.02979e7 0.576058
$$797$$ 6.32170e6 0.352523 0.176262 0.984343i $$-0.443599\pi$$
0.176262 + 0.984343i $$0.443599\pi$$
$$798$$ 0 0
$$799$$ −4.98772e7 −2.76398
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −9.66252e6 −0.530462
$$803$$ −4.27764e6 −0.234107
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6.28100e6 0.340558
$$807$$ 0 0
$$808$$ −2.65816e6 −0.143236
$$809$$ 1.84549e7 0.991380 0.495690 0.868499i $$-0.334915\pi$$
0.495690 + 0.868499i $$0.334915\pi$$
$$810$$ 0 0
$$811$$ 3.00263e7 1.60306 0.801530 0.597954i $$-0.204020\pi$$
0.801530 + 0.597954i $$0.204020\pi$$
$$812$$ −9.53719e6 −0.507610
$$813$$ 0 0
$$814$$ 2.74501e6 0.145205
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.38187e7 −0.724289
$$818$$ 1.47932e7 0.772999
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 2.51609e7 1.30277 0.651387 0.758746i $$-0.274188\pi$$
0.651387 + 0.758746i $$0.274188\pi$$
$$822$$ 0 0
$$823$$ 7.97388e6 0.410365 0.205182 0.978724i $$-0.434221\pi$$
0.205182 + 0.978724i $$0.434221\pi$$
$$824$$ 2.64835e6 0.135880
$$825$$ 0 0
$$826$$ −3.40489e7 −1.73641
$$827$$ 1.37135e7 0.697243 0.348622 0.937264i $$-0.386650\pi$$
0.348622 + 0.937264i $$0.386650\pi$$
$$828$$ 0 0
$$829$$ −1.62566e7 −0.821569 −0.410785 0.911732i $$-0.634745\pi$$
−0.410785 + 0.911732i $$0.634745\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −3.39270e6 −0.169917
$$833$$ −8.99307e7 −4.49051
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1.31103e7 −0.648777
$$837$$ 0 0
$$838$$ 1.38191e7 0.679782
$$839$$ 2.36734e7 1.16106 0.580532 0.814237i $$-0.302844\pi$$
0.580532 + 0.814237i $$0.302844\pi$$
$$840$$ 0 0
$$841$$ −1.44265e7 −0.703350
$$842$$ −1.93592e7 −0.941039
$$843$$ 0 0
$$844$$ −1.39812e7 −0.675600
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.42168e7 3.07567
$$848$$ 6.99632e6 0.334103
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 3.93547e6 0.186283
$$852$$ 0 0
$$853$$ 2.53714e7 1.19391 0.596956 0.802274i $$-0.296377\pi$$
0.596956 + 0.802274i $$0.296377\pi$$
$$854$$ 2.59562e6 0.121786
$$855$$ 0 0
$$856$$ −729817. −0.0340431
$$857$$ −2.37850e7 −1.10624 −0.553122 0.833100i $$-0.686564\pi$$
−0.553122 + 0.833100i $$0.686564\pi$$
$$858$$ 0 0
$$859$$ −8.55700e6 −0.395675 −0.197838 0.980235i $$-0.563392\pi$$
−0.197838 + 0.980235i $$0.563392\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 2.21969e7 1.01748
$$863$$ −3.91623e7 −1.78995 −0.894977 0.446113i $$-0.852808\pi$$
−0.894977 + 0.446113i $$0.852808\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 3.25721e6 0.147588
$$867$$ 0 0
$$868$$ −7.32971e6 −0.330208
$$869$$ 4.24886e7 1.90864
$$870$$ 0 0
$$871$$ 3.98618e7 1.78038
$$872$$ 6.29519e6 0.280361
$$873$$ 0 0
$$874$$ −1.87960e7 −0.832311
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.35047e7 1.03194 0.515972 0.856605i $$-0.327431\pi$$
0.515972 + 0.856605i $$0.327431\pi$$
$$878$$ −1.60985e7 −0.704771
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −4.50902e7 −1.95723 −0.978617 0.205691i $$-0.934056\pi$$
−0.978617 + 0.205691i $$0.934056\pi$$
$$882$$ 0 0
$$883$$ −1.95876e7 −0.845434 −0.422717 0.906262i $$-0.638924\pi$$
−0.422717 + 0.906262i $$0.638924\pi$$
$$884$$ 2.86588e7 1.23347
$$885$$ 0 0
$$886$$ 1.78553e7 0.764157
$$887$$ −9.54399e6 −0.407306 −0.203653 0.979043i $$-0.565281\pi$$
−0.203653 + 0.979043i $$0.565281\pi$$
$$888$$ 0 0
$$889$$ 6.16515e7 2.61631
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.61594e6 0.194244
$$893$$ 2.89288e7 1.21395
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 3.95916e6 0.164753
$$897$$ 0 0
$$898$$ 1.86756e7 0.772828
$$899$$ 4.67628e6 0.192975
$$900$$ 0 0
$$901$$ −5.90992e7 −2.42532
$$902$$ 5.12330e6 0.209668
$$903$$ 0 0
$$904$$ 644912. 0.0262470
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.08034e7 0.436054 0.218027 0.975943i $$-0.430038\pi$$
0.218027 + 0.975943i $$0.430038\pi$$
$$908$$ 2.00573e6 0.0807343
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 3.88914e7 1.55259 0.776296 0.630368i $$-0.217096\pi$$
0.776296 + 0.630368i $$0.217096\pi$$
$$912$$ 0 0
$$913$$ −4.08749e7 −1.62286
$$914$$ 1.88536e7 0.746498
$$915$$ 0 0
$$916$$ −1.11709e7 −0.439897
$$917$$ 1.58118e7 0.620953
$$918$$ 0 0
$$919$$ 2.89714e7 1.13157 0.565785 0.824553i $$-0.308573\pi$$
0.565785 + 0.824553i $$0.308573\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −2.49590e7 −0.966941
$$923$$ 2.25609e7 0.871668
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 1.33597e7 0.511999
$$927$$ 0 0
$$928$$ −2.52590e6 −0.0962824
$$929$$ 3.61023e7 1.37245 0.686224 0.727391i $$-0.259267\pi$$
0.686224 + 0.727391i $$0.259267\pi$$
$$930$$ 0 0
$$931$$ 5.21599e7 1.97225
$$932$$ 1.57534e7 0.594066
$$933$$ 0 0
$$934$$ −2.88999e6 −0.108400
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.02966e7 −0.383129 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$938$$ −4.65173e7 −1.72627
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.55696e7 −0.573195 −0.286597 0.958051i $$-0.592524\pi$$
−0.286597 + 0.958051i $$0.592524\pi$$
$$942$$ 0 0
$$943$$ 7.34519e6 0.268982
$$944$$ −9.01778e6 −0.329359
$$945$$ 0 0
$$946$$ 2.87910e7 1.04599
$$947$$ 3.54260e7 1.28365 0.641826 0.766850i $$-0.278177\pi$$
0.641826 + 0.766850i $$0.278177\pi$$
$$948$$ 0 0
$$949$$ −5.42350e6 −0.195485
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −3.34438e7 −1.19598
$$953$$ −5.47490e6 −0.195274 −0.0976369 0.995222i $$-0.531128\pi$$
−0.0976369 + 0.995222i $$0.531128\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −2.09679e6 −0.0742011
$$957$$ 0 0
$$958$$ 2.15086e7 0.757177
$$959$$ −1.62760e7 −0.571479
$$960$$ 0 0
$$961$$ −2.50352e7 −0.874467
$$962$$ 3.48032e6 0.121250
$$963$$ 0 0
$$964$$ −1.97811e7 −0.685579
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.65533e7 −0.569269 −0.284635 0.958636i $$-0.591872\pi$$
−0.284635 + 0.958636i $$0.591872\pi$$
$$968$$ 1.70077e7 0.583387
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −1.83751e6 −0.0625435 −0.0312718 0.999511i $$-0.509956\pi$$
−0.0312718 + 0.999511i $$0.509956\pi$$
$$972$$ 0 0
$$973$$ −2.26757e7 −0.767855
$$974$$ 6.88338e6 0.232490
$$975$$ 0 0
$$976$$ 687445. 0.0231001
$$977$$ −516874. −0.0173240 −0.00866200 0.999962i $$-0.502757\pi$$
−0.00866200 + 0.999962i $$0.502757\pi$$
$$978$$ 0 0
$$979$$ 1.17111e7 0.390519
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −2.92580e7 −0.968201
$$983$$ 4.15798e7 1.37246 0.686229 0.727386i $$-0.259265\pi$$
0.686229 + 0.727386i $$0.259265\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 2.13368e7 0.698935
$$987$$ 0 0
$$988$$ −1.66221e7 −0.541744
$$989$$ 4.12771e7 1.34190
$$990$$ 0 0
$$991$$ −1.61859e7 −0.523544 −0.261772 0.965130i $$-0.584307\pi$$
−0.261772 + 0.965130i $$0.584307\pi$$
$$992$$ −1.94126e6 −0.0626331
$$993$$ 0 0
$$994$$ −2.63277e7 −0.845176
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −2.74426e7 −0.874355 −0.437178 0.899375i $$-0.644022\pi$$
−0.437178 + 0.899375i $$0.644022\pi$$
$$998$$ −2.91951e7 −0.927863
$$999$$ 0 0
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 450.6.a.bf.1.2 yes 2
3.2 odd 2 450.6.a.ba.1.2 yes 2
5.2 odd 4 450.6.c.p.199.4 4
5.3 odd 4 450.6.c.p.199.1 4
5.4 even 2 450.6.a.y.1.1 2
15.2 even 4 450.6.c.q.199.2 4
15.8 even 4 450.6.c.q.199.3 4
15.14 odd 2 450.6.a.bd.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
450.6.a.y.1.1 2 5.4 even 2
450.6.a.ba.1.2 yes 2 3.2 odd 2
450.6.a.bd.1.1 yes 2 15.14 odd 2
450.6.a.bf.1.2 yes 2 1.1 even 1 trivial
450.6.c.p.199.1 4 5.3 odd 4
450.6.c.p.199.4 4 5.2 odd 4
450.6.c.q.199.2 4 15.2 even 4
450.6.c.q.199.3 4 15.8 even 4