# Properties

 Label 1200.4.a.bt.1.1 Level $1200$ Weight $4$ Character 1200.1 Self dual yes Analytic conductor $70.802$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -16.2094 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -16.2094 q^{7} +9.00000 q^{9} +40.2094 q^{11} -19.7906 q^{13} -83.0156 q^{17} +48.8375 q^{19} -48.6281 q^{21} +1.61250 q^{23} +27.0000 q^{27} -24.5344 q^{29} +12.4187 q^{31} +120.628 q^{33} -325.884 q^{37} -59.3719 q^{39} -242.419 q^{41} +367.350 q^{43} -204.544 q^{47} -80.2562 q^{49} -249.047 q^{51} +61.5281 q^{53} +146.512 q^{57} +112.209 q^{59} +477.350 q^{61} -145.884 q^{63} +558.094 q^{67} +4.83749 q^{69} -558.281 q^{71} -1011.77 q^{73} -651.769 q^{77} -1150.47 q^{79} +81.0000 q^{81} -1157.92 q^{83} -73.6032 q^{87} +96.9751 q^{89} +320.794 q^{91} +37.2562 q^{93} +1152.37 q^{97} +361.884 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} + 42 q^{11} - 78 q^{13} - 102 q^{17} - 56 q^{19} + 18 q^{21} - 48 q^{23} + 54 q^{27} - 318 q^{29} - 52 q^{31} + 126 q^{33} - 306 q^{37} - 234 q^{39} - 408 q^{41} + 120 q^{43} + 180 q^{47} + 70 q^{49} - 306 q^{51} - 402 q^{53} - 168 q^{57} + 186 q^{59} + 340 q^{61} + 54 q^{63} + 732 q^{67} - 144 q^{69} + 36 q^{71} - 1332 q^{73} - 612 q^{77} - 380 q^{79} + 162 q^{81} - 984 q^{83} - 954 q^{87} + 1116 q^{89} - 972 q^{91} - 156 q^{93} + 768 q^{97} + 378 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 6 * q^7 + 18 * q^9 + 42 * q^11 - 78 * q^13 - 102 * q^17 - 56 * q^19 + 18 * q^21 - 48 * q^23 + 54 * q^27 - 318 * q^29 - 52 * q^31 + 126 * q^33 - 306 * q^37 - 234 * q^39 - 408 * q^41 + 120 * q^43 + 180 * q^47 + 70 * q^49 - 306 * q^51 - 402 * q^53 - 168 * q^57 + 186 * q^59 + 340 * q^61 + 54 * q^63 + 732 * q^67 - 144 * q^69 + 36 * q^71 - 1332 * q^73 - 612 * q^77 - 380 * q^79 + 162 * q^81 - 984 * q^83 - 954 * q^87 + 1116 * q^89 - 972 * q^91 - 156 * q^93 + 768 * q^97 + 378 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −16.2094 −0.875224 −0.437612 0.899164i $$-0.644176\pi$$
−0.437612 + 0.899164i $$0.644176\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 40.2094 1.10214 0.551072 0.834458i $$-0.314219\pi$$
0.551072 + 0.834458i $$0.314219\pi$$
$$12$$ 0 0
$$13$$ −19.7906 −0.422226 −0.211113 0.977462i $$-0.567709\pi$$
−0.211113 + 0.977462i $$0.567709\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −83.0156 −1.18437 −0.592184 0.805803i $$-0.701734\pi$$
−0.592184 + 0.805803i $$0.701734\pi$$
$$18$$ 0 0
$$19$$ 48.8375 0.589689 0.294844 0.955545i $$-0.404732\pi$$
0.294844 + 0.955545i $$0.404732\pi$$
$$20$$ 0 0
$$21$$ −48.6281 −0.505311
$$22$$ 0 0
$$23$$ 1.61250 0.0146186 0.00730932 0.999973i $$-0.497673\pi$$
0.00730932 + 0.999973i $$0.497673\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −24.5344 −0.157101 −0.0785504 0.996910i $$-0.525029\pi$$
−0.0785504 + 0.996910i $$0.525029\pi$$
$$30$$ 0 0
$$31$$ 12.4187 0.0719507 0.0359754 0.999353i $$-0.488546\pi$$
0.0359754 + 0.999353i $$0.488546\pi$$
$$32$$ 0 0
$$33$$ 120.628 0.636323
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −325.884 −1.44797 −0.723987 0.689813i $$-0.757693\pi$$
−0.723987 + 0.689813i $$0.757693\pi$$
$$38$$ 0 0
$$39$$ −59.3719 −0.243772
$$40$$ 0 0
$$41$$ −242.419 −0.923401 −0.461701 0.887036i $$-0.652761\pi$$
−0.461701 + 0.887036i $$0.652761\pi$$
$$42$$ 0 0
$$43$$ 367.350 1.30280 0.651399 0.758735i $$-0.274182\pi$$
0.651399 + 0.758735i $$0.274182\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −204.544 −0.634804 −0.317402 0.948291i $$-0.602810\pi$$
−0.317402 + 0.948291i $$0.602810\pi$$
$$48$$ 0 0
$$49$$ −80.2562 −0.233983
$$50$$ 0 0
$$51$$ −249.047 −0.683795
$$52$$ 0 0
$$53$$ 61.5281 0.159463 0.0797314 0.996816i $$-0.474594\pi$$
0.0797314 + 0.996816i $$0.474594\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 146.512 0.340457
$$58$$ 0 0
$$59$$ 112.209 0.247600 0.123800 0.992307i $$-0.460492\pi$$
0.123800 + 0.992307i $$0.460492\pi$$
$$60$$ 0 0
$$61$$ 477.350 1.00194 0.500970 0.865464i $$-0.332977\pi$$
0.500970 + 0.865464i $$0.332977\pi$$
$$62$$ 0 0
$$63$$ −145.884 −0.291741
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 558.094 1.01764 0.508821 0.860872i $$-0.330082\pi$$
0.508821 + 0.860872i $$0.330082\pi$$
$$68$$ 0 0
$$69$$ 4.83749 0.00844008
$$70$$ 0 0
$$71$$ −558.281 −0.933180 −0.466590 0.884474i $$-0.654518\pi$$
−0.466590 + 0.884474i $$0.654518\pi$$
$$72$$ 0 0
$$73$$ −1011.77 −1.62217 −0.811086 0.584927i $$-0.801123\pi$$
−0.811086 + 0.584927i $$0.801123\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −651.769 −0.964623
$$78$$ 0 0
$$79$$ −1150.47 −1.63845 −0.819227 0.573470i $$-0.805597\pi$$
−0.819227 + 0.573470i $$0.805597\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1157.92 −1.53131 −0.765655 0.643251i $$-0.777585\pi$$
−0.765655 + 0.643251i $$0.777585\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −73.6032 −0.0907022
$$88$$ 0 0
$$89$$ 96.9751 0.115498 0.0577491 0.998331i $$-0.481608\pi$$
0.0577491 + 0.998331i $$0.481608\pi$$
$$90$$ 0 0
$$91$$ 320.794 0.369542
$$92$$ 0 0
$$93$$ 37.2562 0.0415408
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1152.37 1.20625 0.603123 0.797648i $$-0.293923\pi$$
0.603123 + 0.797648i $$0.293923\pi$$
$$98$$ 0 0
$$99$$ 361.884 0.367381
$$100$$ 0 0
$$101$$ −1156.49 −1.13936 −0.569679 0.821867i $$-0.692932\pi$$
−0.569679 + 0.821867i $$0.692932\pi$$
$$102$$ 0 0
$$103$$ −1333.70 −1.27585 −0.637927 0.770096i $$-0.720208\pi$$
−0.637927 + 0.770096i $$0.720208\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −798.263 −0.721224 −0.360612 0.932716i $$-0.617432\pi$$
−0.360612 + 0.932716i $$0.617432\pi$$
$$108$$ 0 0
$$109$$ −985.119 −0.865663 −0.432831 0.901475i $$-0.642485\pi$$
−0.432831 + 0.901475i $$0.642485\pi$$
$$110$$ 0 0
$$111$$ −977.653 −0.835988
$$112$$ 0 0
$$113$$ −1888.25 −1.57196 −0.785979 0.618253i $$-0.787841\pi$$
−0.785979 + 0.618253i $$0.787841\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −178.116 −0.140742
$$118$$ 0 0
$$119$$ 1345.63 1.03659
$$120$$ 0 0
$$121$$ 285.794 0.214721
$$122$$ 0 0
$$123$$ −727.256 −0.533126
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −620.859 −0.433798 −0.216899 0.976194i $$-0.569594\pi$$
−0.216899 + 0.976194i $$0.569594\pi$$
$$128$$ 0 0
$$129$$ 1102.05 0.752171
$$130$$ 0 0
$$131$$ 2588.35 1.72630 0.863151 0.504947i $$-0.168488\pi$$
0.863151 + 0.504947i $$0.168488\pi$$
$$132$$ 0 0
$$133$$ −791.625 −0.516110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1656.29 1.03289 0.516447 0.856319i $$-0.327254\pi$$
0.516447 + 0.856319i $$0.327254\pi$$
$$138$$ 0 0
$$139$$ 153.256 0.0935182 0.0467591 0.998906i $$-0.485111\pi$$
0.0467591 + 0.998906i $$0.485111\pi$$
$$140$$ 0 0
$$141$$ −613.631 −0.366504
$$142$$ 0 0
$$143$$ −795.769 −0.465353
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −240.769 −0.135090
$$148$$ 0 0
$$149$$ 1483.38 0.815591 0.407795 0.913073i $$-0.366298\pi$$
0.407795 + 0.913073i $$0.366298\pi$$
$$150$$ 0 0
$$151$$ 394.281 0.212491 0.106246 0.994340i $$-0.466117\pi$$
0.106246 + 0.994340i $$0.466117\pi$$
$$152$$ 0 0
$$153$$ −747.141 −0.394789
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1727.05 −0.877922 −0.438961 0.898506i $$-0.644653\pi$$
−0.438961 + 0.898506i $$0.644653\pi$$
$$158$$ 0 0
$$159$$ 184.584 0.0920659
$$160$$ 0 0
$$161$$ −26.1376 −0.0127946
$$162$$ 0 0
$$163$$ −2034.28 −0.977529 −0.488764 0.872416i $$-0.662552\pi$$
−0.488764 + 0.872416i $$0.662552\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 192.900 0.0893835 0.0446918 0.999001i $$-0.485769\pi$$
0.0446918 + 0.999001i $$0.485769\pi$$
$$168$$ 0 0
$$169$$ −1805.33 −0.821726
$$170$$ 0 0
$$171$$ 439.537 0.196563
$$172$$ 0 0
$$173$$ 1239.91 0.544905 0.272452 0.962169i $$-0.412165\pi$$
0.272452 + 0.962169i $$0.412165\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 336.628 0.142952
$$178$$ 0 0
$$179$$ 2636.86 1.10105 0.550525 0.834818i $$-0.314427\pi$$
0.550525 + 0.834818i $$0.314427\pi$$
$$180$$ 0 0
$$181$$ 3317.58 1.36240 0.681199 0.732099i $$-0.261459\pi$$
0.681199 + 0.732099i $$0.261459\pi$$
$$182$$ 0 0
$$183$$ 1432.05 0.578471
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3338.01 −1.30534
$$188$$ 0 0
$$189$$ −437.653 −0.168437
$$190$$ 0 0
$$191$$ 624.506 0.236585 0.118292 0.992979i $$-0.462258\pi$$
0.118292 + 0.992979i $$0.462258\pi$$
$$192$$ 0 0
$$193$$ 436.144 0.162665 0.0813324 0.996687i $$-0.474082\pi$$
0.0813324 + 0.996687i $$0.474082\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3355.81 −1.21366 −0.606831 0.794831i $$-0.707560\pi$$
−0.606831 + 0.794831i $$0.707560\pi$$
$$198$$ 0 0
$$199$$ −3799.77 −1.35356 −0.676780 0.736185i $$-0.736625\pi$$
−0.676780 + 0.736185i $$0.736625\pi$$
$$200$$ 0 0
$$201$$ 1674.28 0.587536
$$202$$ 0 0
$$203$$ 397.687 0.137498
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 14.5125 0.00487288
$$208$$ 0 0
$$209$$ 1963.72 0.649922
$$210$$ 0 0
$$211$$ −2365.27 −0.771715 −0.385857 0.922558i $$-0.626094\pi$$
−0.385857 + 0.922558i $$0.626094\pi$$
$$212$$ 0 0
$$213$$ −1674.84 −0.538772
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −201.300 −0.0629730
$$218$$ 0 0
$$219$$ −3035.31 −0.936562
$$220$$ 0 0
$$221$$ 1642.93 0.500070
$$222$$ 0 0
$$223$$ −3328.58 −0.999545 −0.499772 0.866157i $$-0.666583\pi$$
−0.499772 + 0.866157i $$0.666583\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −527.100 −0.154118 −0.0770592 0.997027i $$-0.524553\pi$$
−0.0770592 + 0.997027i $$0.524553\pi$$
$$228$$ 0 0
$$229$$ 2566.06 0.740479 0.370240 0.928936i $$-0.379276\pi$$
0.370240 + 0.928936i $$0.379276\pi$$
$$230$$ 0 0
$$231$$ −1955.31 −0.556925
$$232$$ 0 0
$$233$$ −5534.99 −1.55626 −0.778132 0.628101i $$-0.783832\pi$$
−0.778132 + 0.628101i $$0.783832\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −3451.41 −0.945962
$$238$$ 0 0
$$239$$ −1010.01 −0.273355 −0.136678 0.990616i $$-0.543642\pi$$
−0.136678 + 0.990616i $$0.543642\pi$$
$$240$$ 0 0
$$241$$ −4074.29 −1.08900 −0.544498 0.838762i $$-0.683280\pi$$
−0.544498 + 0.838762i $$0.683280\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −966.525 −0.248982
$$248$$ 0 0
$$249$$ −3473.77 −0.884103
$$250$$ 0 0
$$251$$ −1773.98 −0.446107 −0.223054 0.974806i $$-0.571602\pi$$
−0.223054 + 0.974806i $$0.571602\pi$$
$$252$$ 0 0
$$253$$ 64.8375 0.0161119
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −662.784 −0.160869 −0.0804345 0.996760i $$-0.525631\pi$$
−0.0804345 + 0.996760i $$0.525631\pi$$
$$258$$ 0 0
$$259$$ 5282.38 1.26730
$$260$$ 0 0
$$261$$ −220.810 −0.0523669
$$262$$ 0 0
$$263$$ 712.312 0.167008 0.0835039 0.996507i $$-0.473389\pi$$
0.0835039 + 0.996507i $$0.473389\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 290.925 0.0666829
$$268$$ 0 0
$$269$$ −3136.41 −0.710894 −0.355447 0.934696i $$-0.615671\pi$$
−0.355447 + 0.934696i $$0.615671\pi$$
$$270$$ 0 0
$$271$$ 2275.69 0.510105 0.255053 0.966927i $$-0.417907\pi$$
0.255053 + 0.966927i $$0.417907\pi$$
$$272$$ 0 0
$$273$$ 962.381 0.213355
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5171.00 −1.12164 −0.560821 0.827937i $$-0.689515\pi$$
−0.560821 + 0.827937i $$0.689515\pi$$
$$278$$ 0 0
$$279$$ 111.769 0.0239836
$$280$$ 0 0
$$281$$ 2240.14 0.475571 0.237785 0.971318i $$-0.423578\pi$$
0.237785 + 0.971318i $$0.423578\pi$$
$$282$$ 0 0
$$283$$ 225.244 0.0473123 0.0236561 0.999720i $$-0.492469\pi$$
0.0236561 + 0.999720i $$0.492469\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3929.46 0.808183
$$288$$ 0 0
$$289$$ 1978.59 0.402726
$$290$$ 0 0
$$291$$ 3457.12 0.696427
$$292$$ 0 0
$$293$$ −1139.86 −0.227274 −0.113637 0.993522i $$-0.536250\pi$$
−0.113637 + 0.993522i $$0.536250\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1085.65 0.212108
$$298$$ 0 0
$$299$$ −31.9123 −0.00617237
$$300$$ 0 0
$$301$$ −5954.51 −1.14024
$$302$$ 0 0
$$303$$ −3469.47 −0.657808
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5244.86 0.975049 0.487525 0.873109i $$-0.337900\pi$$
0.487525 + 0.873109i $$0.337900\pi$$
$$308$$ 0 0
$$309$$ −4001.09 −0.736615
$$310$$ 0 0
$$311$$ 5188.26 0.945977 0.472989 0.881068i $$-0.343175\pi$$
0.472989 + 0.881068i $$0.343175\pi$$
$$312$$ 0 0
$$313$$ −486.656 −0.0878832 −0.0439416 0.999034i $$-0.513992\pi$$
−0.0439416 + 0.999034i $$0.513992\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4218.87 0.747493 0.373747 0.927531i $$-0.378073\pi$$
0.373747 + 0.927531i $$0.378073\pi$$
$$318$$ 0 0
$$319$$ −986.512 −0.173148
$$320$$ 0 0
$$321$$ −2394.79 −0.416399
$$322$$ 0 0
$$323$$ −4054.27 −0.698408
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2955.36 −0.499791
$$328$$ 0 0
$$329$$ 3315.53 0.555595
$$330$$ 0 0
$$331$$ −7439.94 −1.23546 −0.617728 0.786392i $$-0.711947\pi$$
−0.617728 + 0.786392i $$0.711947\pi$$
$$332$$ 0 0
$$333$$ −2932.96 −0.482658
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6555.39 1.05963 0.529815 0.848113i $$-0.322261\pi$$
0.529815 + 0.848113i $$0.322261\pi$$
$$338$$ 0 0
$$339$$ −5664.74 −0.907571
$$340$$ 0 0
$$341$$ 499.350 0.0793000
$$342$$ 0 0
$$343$$ 6860.72 1.08001
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1950.56 0.301763 0.150881 0.988552i $$-0.451789\pi$$
0.150881 + 0.988552i $$0.451789\pi$$
$$348$$ 0 0
$$349$$ −1426.74 −0.218830 −0.109415 0.993996i $$-0.534898\pi$$
−0.109415 + 0.993996i $$0.534898\pi$$
$$350$$ 0 0
$$351$$ −534.347 −0.0812573
$$352$$ 0 0
$$353$$ 7078.96 1.06735 0.533676 0.845689i $$-0.320810\pi$$
0.533676 + 0.845689i $$0.320810\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4036.89 0.598474
$$358$$ 0 0
$$359$$ 5409.79 0.795314 0.397657 0.917534i $$-0.369823\pi$$
0.397657 + 0.917534i $$0.369823\pi$$
$$360$$ 0 0
$$361$$ −4473.90 −0.652267
$$362$$ 0 0
$$363$$ 857.381 0.123969
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4940.09 0.702645 0.351322 0.936255i $$-0.385732\pi$$
0.351322 + 0.936255i $$0.385732\pi$$
$$368$$ 0 0
$$369$$ −2181.77 −0.307800
$$370$$ 0 0
$$371$$ −997.332 −0.139566
$$372$$ 0 0
$$373$$ −12891.9 −1.78959 −0.894797 0.446473i $$-0.852680\pi$$
−0.894797 + 0.446473i $$0.852680\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 485.551 0.0663320
$$378$$ 0 0
$$379$$ 9475.15 1.28418 0.642092 0.766627i $$-0.278067\pi$$
0.642092 + 0.766627i $$0.278067\pi$$
$$380$$ 0 0
$$381$$ −1862.58 −0.250453
$$382$$ 0 0
$$383$$ −5800.97 −0.773931 −0.386966 0.922094i $$-0.626477\pi$$
−0.386966 + 0.922094i $$0.626477\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3306.15 0.434266
$$388$$ 0 0
$$389$$ 13779.7 1.79603 0.898016 0.439962i $$-0.145008\pi$$
0.898016 + 0.439962i $$0.145008\pi$$
$$390$$ 0 0
$$391$$ −133.862 −0.0173138
$$392$$ 0 0
$$393$$ 7765.06 0.996680
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2816.46 0.356056 0.178028 0.984025i $$-0.443028\pi$$
0.178028 + 0.984025i $$0.443028\pi$$
$$398$$ 0 0
$$399$$ −2374.88 −0.297976
$$400$$ 0 0
$$401$$ 11986.4 1.49270 0.746352 0.665551i $$-0.231804\pi$$
0.746352 + 0.665551i $$0.231804\pi$$
$$402$$ 0 0
$$403$$ −245.775 −0.0303794
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −13103.6 −1.59588
$$408$$ 0 0
$$409$$ −3339.07 −0.403683 −0.201841 0.979418i $$-0.564693\pi$$
−0.201841 + 0.979418i $$0.564693\pi$$
$$410$$ 0 0
$$411$$ 4968.87 0.596342
$$412$$ 0 0
$$413$$ −1818.84 −0.216706
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 459.769 0.0539927
$$418$$ 0 0
$$419$$ −1688.52 −0.196873 −0.0984363 0.995143i $$-0.531384\pi$$
−0.0984363 + 0.995143i $$0.531384\pi$$
$$420$$ 0 0
$$421$$ −2664.27 −0.308429 −0.154214 0.988037i $$-0.549285\pi$$
−0.154214 + 0.988037i $$0.549285\pi$$
$$422$$ 0 0
$$423$$ −1840.89 −0.211601
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7737.54 −0.876923
$$428$$ 0 0
$$429$$ −2387.31 −0.268672
$$430$$ 0 0
$$431$$ −12266.0 −1.37084 −0.685420 0.728148i $$-0.740381\pi$$
−0.685420 + 0.728148i $$0.740381\pi$$
$$432$$ 0 0
$$433$$ 15647.3 1.73664 0.868318 0.496008i $$-0.165201\pi$$
0.868318 + 0.496008i $$0.165201\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 78.7503 0.00862045
$$438$$ 0 0
$$439$$ −16131.0 −1.75373 −0.876867 0.480733i $$-0.840371\pi$$
−0.876867 + 0.480733i $$0.840371\pi$$
$$440$$ 0 0
$$441$$ −722.306 −0.0779944
$$442$$ 0 0
$$443$$ 10053.7 1.07825 0.539127 0.842225i $$-0.318754\pi$$
0.539127 + 0.842225i $$0.318754\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4450.13 0.470882
$$448$$ 0 0
$$449$$ 7477.71 0.785957 0.392979 0.919548i $$-0.371445\pi$$
0.392979 + 0.919548i $$0.371445\pi$$
$$450$$ 0 0
$$451$$ −9747.51 −1.01772
$$452$$ 0 0
$$453$$ 1182.84 0.122682
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1363.46 −0.139562 −0.0697812 0.997562i $$-0.522230\pi$$
−0.0697812 + 0.997562i $$0.522230\pi$$
$$458$$ 0 0
$$459$$ −2241.42 −0.227932
$$460$$ 0 0
$$461$$ 5276.77 0.533109 0.266555 0.963820i $$-0.414115\pi$$
0.266555 + 0.963820i $$0.414115\pi$$
$$462$$ 0 0
$$463$$ 5740.02 0.576159 0.288079 0.957607i $$-0.406983\pi$$
0.288079 + 0.957607i $$0.406983\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6233.36 0.617657 0.308828 0.951118i $$-0.400063\pi$$
0.308828 + 0.951118i $$0.400063\pi$$
$$468$$ 0 0
$$469$$ −9046.35 −0.890664
$$470$$ 0 0
$$471$$ −5181.16 −0.506869
$$472$$ 0 0
$$473$$ 14770.9 1.43587
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 553.753 0.0531543
$$478$$ 0 0
$$479$$ 19688.2 1.87803 0.939013 0.343881i $$-0.111742\pi$$
0.939013 + 0.343881i $$0.111742\pi$$
$$480$$ 0 0
$$481$$ 6449.46 0.611372
$$482$$ 0 0
$$483$$ −78.4127 −0.00738696
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3955.08 −0.368012 −0.184006 0.982925i $$-0.558907\pi$$
−0.184006 + 0.982925i $$0.558907\pi$$
$$488$$ 0 0
$$489$$ −6102.84 −0.564377
$$490$$ 0 0
$$491$$ 13893.5 1.27699 0.638497 0.769624i $$-0.279557\pi$$
0.638497 + 0.769624i $$0.279557\pi$$
$$492$$ 0 0
$$493$$ 2036.74 0.186065
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 9049.39 0.816741
$$498$$ 0 0
$$499$$ −13523.7 −1.21324 −0.606618 0.794993i $$-0.707474\pi$$
−0.606618 + 0.794993i $$0.707474\pi$$
$$500$$ 0 0
$$501$$ 578.700 0.0516056
$$502$$ 0 0
$$503$$ 13135.4 1.16437 0.582184 0.813057i $$-0.302198\pi$$
0.582184 + 0.813057i $$0.302198\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −5415.99 −0.474423
$$508$$ 0 0
$$509$$ −2222.71 −0.193556 −0.0967778 0.995306i $$-0.530854\pi$$
−0.0967778 + 0.995306i $$0.530854\pi$$
$$510$$ 0 0
$$511$$ 16400.1 1.41976
$$512$$ 0 0
$$513$$ 1318.61 0.113486
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8224.57 −0.699645
$$518$$ 0 0
$$519$$ 3719.73 0.314601
$$520$$ 0 0
$$521$$ −4916.42 −0.413421 −0.206710 0.978402i $$-0.566276\pi$$
−0.206710 + 0.978402i $$0.566276\pi$$
$$522$$ 0 0
$$523$$ 17743.4 1.48349 0.741746 0.670681i $$-0.233998\pi$$
0.741746 + 0.670681i $$0.233998\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1030.95 −0.0852161
$$528$$ 0 0
$$529$$ −12164.4 −0.999786
$$530$$ 0 0
$$531$$ 1009.88 0.0825334
$$532$$ 0 0
$$533$$ 4797.62 0.389884
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 7910.58 0.635692
$$538$$ 0 0
$$539$$ −3227.05 −0.257883
$$540$$ 0 0
$$541$$ −12671.3 −1.00699 −0.503495 0.863998i $$-0.667953\pi$$
−0.503495 + 0.863998i $$0.667953\pi$$
$$542$$ 0 0
$$543$$ 9952.74 0.786580
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 5250.90 0.410443 0.205221 0.978716i $$-0.434209\pi$$
0.205221 + 0.978716i $$0.434209\pi$$
$$548$$ 0 0
$$549$$ 4296.15 0.333980
$$550$$ 0 0
$$551$$ −1198.20 −0.0926406
$$552$$ 0 0
$$553$$ 18648.4 1.43401
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −25830.2 −1.96492 −0.982462 0.186465i $$-0.940297\pi$$
−0.982462 + 0.186465i $$0.940297\pi$$
$$558$$ 0 0
$$559$$ −7270.09 −0.550075
$$560$$ 0 0
$$561$$ −10014.0 −0.753640
$$562$$ 0 0
$$563$$ 2021.14 0.151298 0.0756490 0.997135i $$-0.475897\pi$$
0.0756490 + 0.997135i $$0.475897\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1312.96 −0.0972471
$$568$$ 0 0
$$569$$ 8706.51 0.641469 0.320734 0.947169i $$-0.396070\pi$$
0.320734 + 0.947169i $$0.396070\pi$$
$$570$$ 0 0
$$571$$ 12194.5 0.893740 0.446870 0.894599i $$-0.352539\pi$$
0.446870 + 0.894599i $$0.352539\pi$$
$$572$$ 0 0
$$573$$ 1873.52 0.136592
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15264.0 1.10130 0.550649 0.834737i $$-0.314380\pi$$
0.550649 + 0.834737i $$0.314380\pi$$
$$578$$ 0 0
$$579$$ 1308.43 0.0939146
$$580$$ 0 0
$$581$$ 18769.2 1.34024
$$582$$ 0 0
$$583$$ 2474.01 0.175751
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −8456.89 −0.594639 −0.297319 0.954778i $$-0.596093\pi$$
−0.297319 + 0.954778i $$0.596093\pi$$
$$588$$ 0 0
$$589$$ 606.500 0.0424285
$$590$$ 0 0
$$591$$ −10067.4 −0.700708
$$592$$ 0 0
$$593$$ 1225.23 0.0848467 0.0424234 0.999100i $$-0.486492\pi$$
0.0424234 + 0.999100i $$0.486492\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −11399.3 −0.781478
$$598$$ 0 0
$$599$$ 16060.0 1.09548 0.547741 0.836648i $$-0.315488\pi$$
0.547741 + 0.836648i $$0.315488\pi$$
$$600$$ 0 0
$$601$$ 9699.93 0.658350 0.329175 0.944269i $$-0.393229\pi$$
0.329175 + 0.944269i $$0.393229\pi$$
$$602$$ 0 0
$$603$$ 5022.84 0.339214
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23661.2 1.58217 0.791087 0.611703i $$-0.209515\pi$$
0.791087 + 0.611703i $$0.209515\pi$$
$$608$$ 0 0
$$609$$ 1193.06 0.0793847
$$610$$ 0 0
$$611$$ 4048.05 0.268030
$$612$$ 0 0
$$613$$ 8085.63 0.532749 0.266375 0.963870i $$-0.414174\pi$$
0.266375 + 0.963870i $$0.414174\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11035.1 0.720029 0.360014 0.932947i $$-0.382772\pi$$
0.360014 + 0.932947i $$0.382772\pi$$
$$618$$ 0 0
$$619$$ 16826.3 1.09258 0.546290 0.837596i $$-0.316040\pi$$
0.546290 + 0.837596i $$0.316040\pi$$
$$620$$ 0 0
$$621$$ 43.5374 0.00281336
$$622$$ 0 0
$$623$$ −1571.90 −0.101087
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 5891.17 0.375233
$$628$$ 0 0
$$629$$ 27053.5 1.71493
$$630$$ 0 0
$$631$$ 3705.91 0.233803 0.116902 0.993144i $$-0.462704\pi$$
0.116902 + 0.993144i $$0.462704\pi$$
$$632$$ 0 0
$$633$$ −7095.81 −0.445550
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1588.32 0.0987937
$$638$$ 0 0
$$639$$ −5024.53 −0.311060
$$640$$ 0 0
$$641$$ −24597.4 −1.51566 −0.757829 0.652453i $$-0.773740\pi$$
−0.757829 + 0.652453i $$0.773740\pi$$
$$642$$ 0 0
$$643$$ −21479.5 −1.31737 −0.658685 0.752419i $$-0.728887\pi$$
−0.658685 + 0.752419i $$0.728887\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27119.7 1.64789 0.823946 0.566668i $$-0.191768\pi$$
0.823946 + 0.566668i $$0.191768\pi$$
$$648$$ 0 0
$$649$$ 4511.87 0.272891
$$650$$ 0 0
$$651$$ −603.900 −0.0363575
$$652$$ 0 0
$$653$$ −18476.4 −1.10725 −0.553627 0.832765i $$-0.686757\pi$$
−0.553627 + 0.832765i $$0.686757\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −9105.92 −0.540724
$$658$$ 0 0
$$659$$ 19273.5 1.13928 0.569641 0.821894i $$-0.307082\pi$$
0.569641 + 0.821894i $$0.307082\pi$$
$$660$$ 0 0
$$661$$ 25605.3 1.50670 0.753352 0.657618i $$-0.228436\pi$$
0.753352 + 0.657618i $$0.228436\pi$$
$$662$$ 0 0
$$663$$ 4928.79 0.288716
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −39.5616 −0.00229660
$$668$$ 0 0
$$669$$ −9985.75 −0.577087
$$670$$ 0 0
$$671$$ 19193.9 1.10428
$$672$$ 0 0
$$673$$ −7855.52 −0.449938 −0.224969 0.974366i $$-0.572228\pi$$
−0.224969 + 0.974366i $$0.572228\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6763.09 0.383939 0.191970 0.981401i $$-0.438512\pi$$
0.191970 + 0.981401i $$0.438512\pi$$
$$678$$ 0 0
$$679$$ −18679.3 −1.05574
$$680$$ 0 0
$$681$$ −1581.30 −0.0889803
$$682$$ 0 0
$$683$$ 15608.6 0.874447 0.437224 0.899353i $$-0.355962\pi$$
0.437224 + 0.899353i $$0.355962\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 7698.17 0.427516
$$688$$ 0 0
$$689$$ −1217.68 −0.0673293
$$690$$ 0 0
$$691$$ −6203.15 −0.341504 −0.170752 0.985314i $$-0.554620\pi$$
−0.170752 + 0.985314i $$0.554620\pi$$
$$692$$ 0 0
$$693$$ −5865.92 −0.321541
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 20124.5 1.09365
$$698$$ 0 0
$$699$$ −16605.0 −0.898509
$$700$$ 0 0
$$701$$ −16507.9 −0.889435 −0.444718 0.895671i $$-0.646696\pi$$
−0.444718 + 0.895671i $$0.646696\pi$$
$$702$$ 0 0
$$703$$ −15915.4 −0.853854
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18746.0 0.997193
$$708$$ 0 0
$$709$$ −25539.6 −1.35283 −0.676416 0.736520i $$-0.736468\pi$$
−0.676416 + 0.736520i $$0.736468\pi$$
$$710$$ 0 0
$$711$$ −10354.2 −0.546151
$$712$$ 0 0
$$713$$ 20.0252 0.00105182
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −3030.02 −0.157822
$$718$$ 0 0
$$719$$ 7353.45 0.381415 0.190708 0.981647i $$-0.438922\pi$$
0.190708 + 0.981647i $$0.438922\pi$$
$$720$$ 0 0
$$721$$ 21618.4 1.11666
$$722$$ 0 0
$$723$$ −12222.9 −0.628732
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21696.5 −1.10685 −0.553424 0.832900i $$-0.686679\pi$$
−0.553424 + 0.832900i $$0.686679\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −30495.8 −1.54299
$$732$$ 0 0
$$733$$ −90.2714 −0.00454877 −0.00227439 0.999997i $$-0.500724\pi$$
−0.00227439 + 0.999997i $$0.500724\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 22440.6 1.12159
$$738$$ 0 0
$$739$$ −14273.1 −0.710479 −0.355239 0.934775i $$-0.615601\pi$$
−0.355239 + 0.934775i $$0.615601\pi$$
$$740$$ 0 0
$$741$$ −2899.57 −0.143750
$$742$$ 0 0
$$743$$ 15866.6 0.783429 0.391715 0.920087i $$-0.371882\pi$$
0.391715 + 0.920087i $$0.371882\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −10421.3 −0.510437
$$748$$ 0 0
$$749$$ 12939.3 0.631232
$$750$$ 0 0
$$751$$ −26776.9 −1.30107 −0.650534 0.759477i $$-0.725455\pi$$
−0.650534 + 0.759477i $$0.725455\pi$$
$$752$$ 0 0
$$753$$ −5321.95 −0.257560
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −30478.0 −1.46333 −0.731666 0.681663i $$-0.761257\pi$$
−0.731666 + 0.681663i $$0.761257\pi$$
$$758$$ 0 0
$$759$$ 194.512 0.00930218
$$760$$ 0 0
$$761$$ 29104.7 1.38639 0.693195 0.720750i $$-0.256202\pi$$
0.693195 + 0.720750i $$0.256202\pi$$
$$762$$ 0 0
$$763$$ 15968.2 0.757649
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2220.69 −0.104543
$$768$$ 0 0
$$769$$ 4170.65 0.195575 0.0977876 0.995207i $$-0.468823\pi$$
0.0977876 + 0.995207i $$0.468823\pi$$
$$770$$ 0 0
$$771$$ −1988.35 −0.0928778
$$772$$ 0 0
$$773$$ 17738.5 0.825367 0.412684 0.910874i $$-0.364591\pi$$
0.412684 + 0.910874i $$0.364591\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 15847.1 0.731677
$$778$$ 0 0
$$779$$ −11839.1 −0.544519
$$780$$ 0 0
$$781$$ −22448.1 −1.02850
$$782$$ 0 0
$$783$$ −662.429 −0.0302341
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −3807.92 −0.172475 −0.0862374 0.996275i $$-0.527484\pi$$
−0.0862374 + 0.996275i $$0.527484\pi$$
$$788$$ 0 0
$$789$$ 2136.94 0.0964220
$$790$$ 0 0
$$791$$ 30607.3 1.37582
$$792$$ 0 0
$$793$$ −9447.06 −0.423045
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −23840.3 −1.05956 −0.529779 0.848136i $$-0.677725\pi$$
−0.529779 + 0.848136i $$0.677725\pi$$
$$798$$ 0 0
$$799$$ 16980.3 0.751841
$$800$$ 0 0
$$801$$ 872.775 0.0384994
$$802$$ 0 0
$$803$$ −40682.6 −1.78787
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −9409.24 −0.410435
$$808$$ 0 0
$$809$$ −1984.22 −0.0862316 −0.0431158 0.999070i $$-0.513728\pi$$
−0.0431158 + 0.999070i $$0.513728\pi$$
$$810$$ 0 0
$$811$$ 9713.78 0.420588 0.210294 0.977638i $$-0.432558\pi$$
0.210294 + 0.977638i $$0.432558\pi$$
$$812$$ 0 0
$$813$$ 6827.08 0.294509
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 17940.5 0.768246
$$818$$ 0 0
$$819$$ 2887.14 0.123181
$$820$$ 0 0
$$821$$ −19235.4 −0.817686 −0.408843 0.912605i $$-0.634068\pi$$
−0.408843 + 0.912605i $$0.634068\pi$$
$$822$$ 0 0
$$823$$ −12717.6 −0.538650 −0.269325 0.963049i $$-0.586801\pi$$
−0.269325 + 0.963049i $$0.586801\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6744.75 −0.283601 −0.141800 0.989895i $$-0.545289\pi$$
−0.141800 + 0.989895i $$0.545289\pi$$
$$828$$ 0 0
$$829$$ 3404.22 0.142622 0.0713108 0.997454i $$-0.477282\pi$$
0.0713108 + 0.997454i $$0.477282\pi$$
$$830$$ 0 0
$$831$$ −15513.0 −0.647581
$$832$$ 0 0
$$833$$ 6662.52 0.277122
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 335.306 0.0138469
$$838$$ 0 0
$$839$$ 21361.9 0.879015 0.439508 0.898239i $$-0.355153\pi$$
0.439508 + 0.898239i $$0.355153\pi$$
$$840$$ 0 0
$$841$$ −23787.1 −0.975319
$$842$$ 0 0
$$843$$ 6720.41 0.274571
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4632.54 −0.187929
$$848$$ 0 0
$$849$$ 675.732 0.0273158
$$850$$ 0 0
$$851$$ −525.488 −0.0211674
$$852$$ 0 0
$$853$$ 10728.9 0.430657 0.215328 0.976542i $$-0.430918\pi$$
0.215328 + 0.976542i $$0.430918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42895.2 1.70977 0.854885 0.518817i $$-0.173627\pi$$
0.854885 + 0.518817i $$0.173627\pi$$
$$858$$ 0 0
$$859$$ −35530.5 −1.41127 −0.705637 0.708574i $$-0.749339\pi$$
−0.705637 + 0.708574i $$0.749339\pi$$
$$860$$ 0 0
$$861$$ 11788.4 0.466605
$$862$$ 0 0
$$863$$ 5704.35 0.225004 0.112502 0.993652i $$-0.464114\pi$$
0.112502 + 0.993652i $$0.464114\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5935.78 0.232514
$$868$$ 0 0
$$869$$ −46259.6 −1.80581
$$870$$ 0 0
$$871$$ −11045.0 −0.429674
$$872$$ 0 0
$$873$$ 10371.4 0.402082
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −50249.0 −1.93476 −0.967382 0.253324i $$-0.918476\pi$$
−0.967382 + 0.253324i $$0.918476\pi$$
$$878$$ 0 0
$$879$$ −3419.58 −0.131217
$$880$$ 0 0
$$881$$ −26864.5 −1.02734 −0.513672 0.857987i $$-0.671715\pi$$
−0.513672 + 0.857987i $$0.671715\pi$$
$$882$$ 0 0
$$883$$ 18942.1 0.721918 0.360959 0.932582i $$-0.382449\pi$$
0.360959 + 0.932582i $$0.382449\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −25344.8 −0.959409 −0.479705 0.877430i $$-0.659256\pi$$
−0.479705 + 0.877430i $$0.659256\pi$$
$$888$$ 0 0
$$889$$ 10063.7 0.379670
$$890$$ 0 0
$$891$$ 3256.96 0.122460
$$892$$ 0 0
$$893$$ −9989.40 −0.374337
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −95.7370 −0.00356362
$$898$$ 0 0
$$899$$ −304.686 −0.0113035
$$900$$ 0 0
$$901$$ −5107.79 −0.188863
$$902$$ 0 0
$$903$$ −17863.5 −0.658318
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4800.11 0.175728 0.0878639 0.996132i $$-0.471996\pi$$
0.0878639 + 0.996132i $$0.471996\pi$$
$$908$$ 0 0
$$909$$ −10408.4 −0.379786
$$910$$ 0 0
$$911$$ 25731.7 0.935819 0.467909 0.883776i $$-0.345007\pi$$
0.467909 + 0.883776i $$0.345007\pi$$
$$912$$ 0 0
$$913$$ −46559.4 −1.68772
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −41955.6 −1.51090
$$918$$ 0 0
$$919$$ 12751.9 0.457722 0.228861 0.973459i $$-0.426500\pi$$
0.228861 + 0.973459i $$0.426500\pi$$
$$920$$ 0 0
$$921$$ 15734.6 0.562945
$$922$$ 0 0
$$923$$ 11048.7 0.394012
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −12003.3 −0.425285
$$928$$ 0 0
$$929$$ 15557.8 0.549444 0.274722 0.961524i $$-0.411414\pi$$
0.274722 + 0.961524i $$0.411414\pi$$
$$930$$ 0 0
$$931$$ −3919.51 −0.137977
$$932$$ 0 0
$$933$$ 15564.8 0.546160
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23858.0 0.831811 0.415905 0.909408i $$-0.363465\pi$$
0.415905 + 0.909408i $$0.363465\pi$$
$$938$$ 0 0
$$939$$ −1459.97 −0.0507394
$$940$$ 0 0
$$941$$ 9748.00 0.337700 0.168850 0.985642i $$-0.445995\pi$$
0.168850 + 0.985642i $$0.445995\pi$$
$$942$$ 0 0
$$943$$ −390.899 −0.0134989
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −51537.0 −1.76845 −0.884227 0.467057i $$-0.845314\pi$$
−0.884227 + 0.467057i $$0.845314\pi$$
$$948$$ 0 0
$$949$$ 20023.5 0.684923
$$950$$ 0 0
$$951$$ 12656.6 0.431566
$$952$$ 0 0
$$953$$ 5631.36 0.191414 0.0957071 0.995410i $$-0.469489\pi$$
0.0957071 + 0.995410i $$0.469489\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −2959.54 −0.0999668
$$958$$ 0 0
$$959$$ −26847.4 −0.904013
$$960$$ 0 0
$$961$$ −29636.8 −0.994823
$$962$$ 0 0
$$963$$ −7184.36 −0.240408
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −43360.9 −1.44198 −0.720989 0.692946i $$-0.756312\pi$$
−0.720989 + 0.692946i $$0.756312\pi$$
$$968$$ 0 0
$$969$$ −12162.8 −0.403226
$$970$$ 0 0
$$971$$ −12920.0 −0.427007 −0.213503 0.976942i $$-0.568487\pi$$
−0.213503 + 0.976942i $$0.568487\pi$$
$$972$$ 0 0
$$973$$ −2484.19 −0.0818493
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 10650.4 0.348759 0.174379 0.984679i $$-0.444208\pi$$
0.174379 + 0.984679i $$0.444208\pi$$
$$978$$ 0 0
$$979$$ 3899.31 0.127296
$$980$$ 0 0
$$981$$ −8866.07 −0.288554
$$982$$ 0 0
$$983$$ −49450.3 −1.60450 −0.802248 0.596991i $$-0.796363\pi$$
−0.802248 + 0.596991i $$0.796363\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 9946.58 0.320773
$$988$$ 0 0
$$989$$ 592.351 0.0190452
$$990$$ 0 0
$$991$$ −9410.47 −0.301648 −0.150824 0.988561i $$-0.548193\pi$$
−0.150824 + 0.988561i $$0.548193\pi$$
$$992$$ 0 0
$$993$$ −22319.8 −0.713291
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −532.117 −0.0169030 −0.00845151 0.999964i $$-0.502690\pi$$
−0.00845151 + 0.999964i $$0.502690\pi$$
$$998$$ 0 0
$$999$$ −8798.88 −0.278663
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 1200.4.a.bt.1.1 2
4.3 odd 2 75.4.a.c.1.1 2
5.2 odd 4 240.4.f.f.49.2 4
5.3 odd 4 240.4.f.f.49.4 4
5.4 even 2 1200.4.a.bn.1.2 2
12.11 even 2 225.4.a.o.1.2 2
15.2 even 4 720.4.f.j.289.2 4
15.8 even 4 720.4.f.j.289.1 4
20.3 even 4 15.4.b.a.4.4 yes 4
20.7 even 4 15.4.b.a.4.1 4
20.19 odd 2 75.4.a.f.1.2 2
40.3 even 4 960.4.f.q.769.3 4
40.13 odd 4 960.4.f.p.769.1 4
40.27 even 4 960.4.f.q.769.1 4
40.37 odd 4 960.4.f.p.769.3 4
60.23 odd 4 45.4.b.b.19.1 4
60.47 odd 4 45.4.b.b.19.4 4
60.59 even 2 225.4.a.i.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.1 4 20.7 even 4
15.4.b.a.4.4 yes 4 20.3 even 4
45.4.b.b.19.1 4 60.23 odd 4
45.4.b.b.19.4 4 60.47 odd 4
75.4.a.c.1.1 2 4.3 odd 2
75.4.a.f.1.2 2 20.19 odd 2
225.4.a.i.1.1 2 60.59 even 2
225.4.a.o.1.2 2 12.11 even 2
240.4.f.f.49.2 4 5.2 odd 4
240.4.f.f.49.4 4 5.3 odd 4
720.4.f.j.289.1 4 15.8 even 4
720.4.f.j.289.2 4 15.2 even 4
960.4.f.p.769.1 4 40.13 odd 4
960.4.f.p.769.3 4 40.37 odd 4
960.4.f.q.769.1 4 40.27 even 4
960.4.f.q.769.3 4 40.3 even 4
1200.4.a.bn.1.2 2 5.4 even 2
1200.4.a.bt.1.1 2 1.1 even 1 trivial