# Properties

 Label 3920.2.a.bt.1.2 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} +O(q^{10})$$ $$q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} -6.37228 q^{11} -4.37228 q^{13} +2.37228 q^{15} +0.372281 q^{17} -4.74456 q^{19} +4.74456 q^{23} +1.00000 q^{25} -0.883156 q^{27} -4.37228 q^{29} -8.00000 q^{31} -15.1168 q^{33} -2.00000 q^{37} -10.3723 q^{39} -6.74456 q^{41} +8.74456 q^{43} +2.62772 q^{45} -7.11684 q^{47} +0.883156 q^{51} +10.7446 q^{53} -6.37228 q^{55} -11.2554 q^{57} +8.00000 q^{59} +2.74456 q^{61} -4.37228 q^{65} +4.00000 q^{67} +11.2554 q^{69} -8.00000 q^{71} +6.00000 q^{73} +2.37228 q^{75} -15.1168 q^{79} -9.97825 q^{81} -9.48913 q^{83} +0.372281 q^{85} -10.3723 q^{87} -14.7446 q^{89} -18.9783 q^{93} -4.74456 q^{95} +9.86141 q^{97} -16.7446 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 2q^{5} + 11q^{9} + O(q^{10})$$ $$2q - q^{3} + 2q^{5} + 11q^{9} - 7q^{11} - 3q^{13} - q^{15} - 5q^{17} + 2q^{19} - 2q^{23} + 2q^{25} - 19q^{27} - 3q^{29} - 16q^{31} - 13q^{33} - 4q^{37} - 15q^{39} - 2q^{41} + 6q^{43} + 11q^{45} + 3q^{47} + 19q^{51} + 10q^{53} - 7q^{55} - 34q^{57} + 16q^{59} - 6q^{61} - 3q^{65} + 8q^{67} + 34q^{69} - 16q^{71} + 12q^{73} - q^{75} - 13q^{79} + 26q^{81} + 4q^{83} - 5q^{85} - 15q^{87} - 18q^{89} + 8q^{93} + 2q^{95} - 9q^{97} - 22q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.37228 1.36964 0.684819 0.728714i $$-0.259881\pi$$
0.684819 + 0.728714i $$0.259881\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.62772 0.875906
$$10$$ 0 0
$$11$$ −6.37228 −1.92132 −0.960658 0.277736i $$-0.910416\pi$$
−0.960658 + 0.277736i $$0.910416\pi$$
$$12$$ 0 0
$$13$$ −4.37228 −1.21265 −0.606326 0.795216i $$-0.707357\pi$$
−0.606326 + 0.795216i $$0.707357\pi$$
$$14$$ 0 0
$$15$$ 2.37228 0.612520
$$16$$ 0 0
$$17$$ 0.372281 0.0902915 0.0451457 0.998980i $$-0.485625\pi$$
0.0451457 + 0.998980i $$0.485625\pi$$
$$18$$ 0 0
$$19$$ −4.74456 −1.08848 −0.544239 0.838930i $$-0.683181\pi$$
−0.544239 + 0.838930i $$0.683181\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.74456 0.989310 0.494655 0.869090i $$-0.335294\pi$$
0.494655 + 0.869090i $$0.335294\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −0.883156 −0.169963
$$28$$ 0 0
$$29$$ −4.37228 −0.811912 −0.405956 0.913893i $$-0.633061\pi$$
−0.405956 + 0.913893i $$0.633061\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ −15.1168 −2.63150
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −10.3723 −1.66089
$$40$$ 0 0
$$41$$ −6.74456 −1.05332 −0.526662 0.850075i $$-0.676557\pi$$
−0.526662 + 0.850075i $$0.676557\pi$$
$$42$$ 0 0
$$43$$ 8.74456 1.33353 0.666767 0.745267i $$-0.267678\pi$$
0.666767 + 0.745267i $$0.267678\pi$$
$$44$$ 0 0
$$45$$ 2.62772 0.391717
$$46$$ 0 0
$$47$$ −7.11684 −1.03810 −0.519049 0.854744i $$-0.673714\pi$$
−0.519049 + 0.854744i $$0.673714\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0.883156 0.123667
$$52$$ 0 0
$$53$$ 10.7446 1.47588 0.737940 0.674867i $$-0.235799\pi$$
0.737940 + 0.674867i $$0.235799\pi$$
$$54$$ 0 0
$$55$$ −6.37228 −0.859238
$$56$$ 0 0
$$57$$ −11.2554 −1.49082
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 2.74456 0.351405 0.175703 0.984443i $$-0.443780\pi$$
0.175703 + 0.984443i $$0.443780\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.37228 −0.542315
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 11.2554 1.35500
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 2.37228 0.273927
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −15.1168 −1.70078 −0.850389 0.526155i $$-0.823633\pi$$
−0.850389 + 0.526155i $$0.823633\pi$$
$$80$$ 0 0
$$81$$ −9.97825 −1.10869
$$82$$ 0 0
$$83$$ −9.48913 −1.04157 −0.520783 0.853689i $$-0.674360\pi$$
−0.520783 + 0.853689i $$0.674360\pi$$
$$84$$ 0 0
$$85$$ 0.372281 0.0403796
$$86$$ 0 0
$$87$$ −10.3723 −1.11203
$$88$$ 0 0
$$89$$ −14.7446 −1.56292 −0.781460 0.623955i $$-0.785525\pi$$
−0.781460 + 0.623955i $$0.785525\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −18.9783 −1.96795
$$94$$ 0 0
$$95$$ −4.74456 −0.486782
$$96$$ 0 0
$$97$$ 9.86141 1.00127 0.500637 0.865657i $$-0.333099\pi$$
0.500637 + 0.865657i $$0.333099\pi$$
$$98$$ 0 0
$$99$$ −16.7446 −1.68289
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −5.62772 −0.554516 −0.277258 0.960796i $$-0.589426\pi$$
−0.277258 + 0.960796i $$0.589426\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.74456 −0.845369 −0.422684 0.906277i $$-0.638912\pi$$
−0.422684 + 0.906277i $$0.638912\pi$$
$$108$$ 0 0
$$109$$ 0.372281 0.0356581 0.0178290 0.999841i $$-0.494325\pi$$
0.0178290 + 0.999841i $$0.494325\pi$$
$$110$$ 0 0
$$111$$ −4.74456 −0.450334
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 4.74456 0.442433
$$116$$ 0 0
$$117$$ −11.4891 −1.06217
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 29.6060 2.69145
$$122$$ 0 0
$$123$$ −16.0000 −1.44267
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 20.7446 1.82646
$$130$$ 0 0
$$131$$ −4.74456 −0.414534 −0.207267 0.978284i $$-0.566457\pi$$
−0.207267 + 0.978284i $$0.566457\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −0.883156 −0.0760100
$$136$$ 0 0
$$137$$ 14.7446 1.25971 0.629857 0.776712i $$-0.283114\pi$$
0.629857 + 0.776712i $$0.283114\pi$$
$$138$$ 0 0
$$139$$ −4.74456 −0.402429 −0.201214 0.979547i $$-0.564489\pi$$
−0.201214 + 0.979547i $$0.564489\pi$$
$$140$$ 0 0
$$141$$ −16.8832 −1.42182
$$142$$ 0 0
$$143$$ 27.8614 2.32989
$$144$$ 0 0
$$145$$ −4.37228 −0.363098
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.4891 1.26892 0.634459 0.772956i $$-0.281223\pi$$
0.634459 + 0.772956i $$0.281223\pi$$
$$150$$ 0 0
$$151$$ −15.1168 −1.23019 −0.615096 0.788452i $$-0.710883\pi$$
−0.615096 + 0.788452i $$0.710883\pi$$
$$152$$ 0 0
$$153$$ 0.978251 0.0790869
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 15.4891 1.23617 0.618083 0.786113i $$-0.287909\pi$$
0.618083 + 0.786113i $$0.287909\pi$$
$$158$$ 0 0
$$159$$ 25.4891 2.02142
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 16.7446 1.31154 0.655768 0.754963i $$-0.272345\pi$$
0.655768 + 0.754963i $$0.272345\pi$$
$$164$$ 0 0
$$165$$ −15.1168 −1.17684
$$166$$ 0 0
$$167$$ −5.62772 −0.435486 −0.217743 0.976006i $$-0.569869\pi$$
−0.217743 + 0.976006i $$0.569869\pi$$
$$168$$ 0 0
$$169$$ 6.11684 0.470526
$$170$$ 0 0
$$171$$ −12.4674 −0.953404
$$172$$ 0 0
$$173$$ 0.372281 0.0283040 0.0141520 0.999900i $$-0.495495\pi$$
0.0141520 + 0.999900i $$0.495495\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 18.9783 1.42649
$$178$$ 0 0
$$179$$ 22.9783 1.71748 0.858738 0.512416i $$-0.171249\pi$$
0.858738 + 0.512416i $$0.171249\pi$$
$$180$$ 0 0
$$181$$ −16.2337 −1.20664 −0.603320 0.797499i $$-0.706156\pi$$
−0.603320 + 0.797499i $$0.706156\pi$$
$$182$$ 0 0
$$183$$ 6.51087 0.481298
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ −2.37228 −0.173478
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.86141 −0.279402 −0.139701 0.990194i $$-0.544614\pi$$
−0.139701 + 0.990194i $$0.544614\pi$$
$$192$$ 0 0
$$193$$ 6.74456 0.485484 0.242742 0.970091i $$-0.421953\pi$$
0.242742 + 0.970091i $$0.421953\pi$$
$$194$$ 0 0
$$195$$ −10.3723 −0.742774
$$196$$ 0 0
$$197$$ −8.23369 −0.586626 −0.293313 0.956016i $$-0.594758\pi$$
−0.293313 + 0.956016i $$0.594758\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 9.48913 0.669311
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.74456 −0.471061
$$206$$ 0 0
$$207$$ 12.4674 0.866543
$$208$$ 0 0
$$209$$ 30.2337 2.09131
$$210$$ 0 0
$$211$$ 14.3723 0.989429 0.494714 0.869056i $$-0.335273\pi$$
0.494714 + 0.869056i $$0.335273\pi$$
$$212$$ 0 0
$$213$$ −18.9783 −1.30037
$$214$$ 0 0
$$215$$ 8.74456 0.596374
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 14.2337 0.961823
$$220$$ 0 0
$$221$$ −1.62772 −0.109492
$$222$$ 0 0
$$223$$ −5.62772 −0.376860 −0.188430 0.982087i $$-0.560340\pi$$
−0.188430 + 0.982087i $$0.560340\pi$$
$$224$$ 0 0
$$225$$ 2.62772 0.175181
$$226$$ 0 0
$$227$$ 19.8614 1.31825 0.659124 0.752034i $$-0.270927\pi$$
0.659124 + 0.752034i $$0.270927\pi$$
$$228$$ 0 0
$$229$$ 12.2337 0.808425 0.404212 0.914665i $$-0.367546\pi$$
0.404212 + 0.914665i $$0.367546\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.25544 −0.0822464 −0.0411232 0.999154i $$-0.513094\pi$$
−0.0411232 + 0.999154i $$0.513094\pi$$
$$234$$ 0 0
$$235$$ −7.11684 −0.464252
$$236$$ 0 0
$$237$$ −35.8614 −2.32945
$$238$$ 0 0
$$239$$ −13.6277 −0.881504 −0.440752 0.897629i $$-0.645288\pi$$
−0.440752 + 0.897629i $$0.645288\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 0 0
$$243$$ −21.0217 −1.34855
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 20.7446 1.31994
$$248$$ 0 0
$$249$$ −22.5109 −1.42657
$$250$$ 0 0
$$251$$ 4.74456 0.299474 0.149737 0.988726i $$-0.452157\pi$$
0.149737 + 0.988726i $$0.452157\pi$$
$$252$$ 0 0
$$253$$ −30.2337 −1.90078
$$254$$ 0 0
$$255$$ 0.883156 0.0553054
$$256$$ 0 0
$$257$$ 23.4891 1.46521 0.732606 0.680653i $$-0.238304\pi$$
0.732606 + 0.680653i $$0.238304\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −11.4891 −0.711159
$$262$$ 0 0
$$263$$ −22.2337 −1.37099 −0.685494 0.728078i $$-0.740414\pi$$
−0.685494 + 0.728078i $$0.740414\pi$$
$$264$$ 0 0
$$265$$ 10.7446 0.660033
$$266$$ 0 0
$$267$$ −34.9783 −2.14063
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 9.48913 0.576423 0.288212 0.957567i $$-0.406939\pi$$
0.288212 + 0.957567i $$0.406939\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6.37228 −0.384263
$$276$$ 0 0
$$277$$ 24.9783 1.50080 0.750399 0.660985i $$-0.229861\pi$$
0.750399 + 0.660985i $$0.229861\pi$$
$$278$$ 0 0
$$279$$ −21.0217 −1.25854
$$280$$ 0 0
$$281$$ 18.6060 1.10994 0.554970 0.831871i $$-0.312730\pi$$
0.554970 + 0.831871i $$0.312730\pi$$
$$282$$ 0 0
$$283$$ −8.88316 −0.528049 −0.264024 0.964516i $$-0.585050\pi$$
−0.264024 + 0.964516i $$0.585050\pi$$
$$284$$ 0 0
$$285$$ −11.2554 −0.666715
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.8614 −0.991847
$$290$$ 0 0
$$291$$ 23.3940 1.37138
$$292$$ 0 0
$$293$$ −25.1168 −1.46734 −0.733671 0.679505i $$-0.762195\pi$$
−0.733671 + 0.679505i $$0.762195\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 0 0
$$297$$ 5.62772 0.326553
$$298$$ 0 0
$$299$$ −20.7446 −1.19969
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 14.2337 0.817704
$$304$$ 0 0
$$305$$ 2.74456 0.157153
$$306$$ 0 0
$$307$$ 31.1168 1.77593 0.887966 0.459909i $$-0.152118\pi$$
0.887966 + 0.459909i $$0.152118\pi$$
$$308$$ 0 0
$$309$$ −13.3505 −0.759485
$$310$$ 0 0
$$311$$ −12.7446 −0.722678 −0.361339 0.932435i $$-0.617680\pi$$
−0.361339 + 0.932435i $$0.617680\pi$$
$$312$$ 0 0
$$313$$ −2.88316 −0.162966 −0.0814828 0.996675i $$-0.525966\pi$$
−0.0814828 + 0.996675i $$0.525966\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 0 0
$$319$$ 27.8614 1.55994
$$320$$ 0 0
$$321$$ −20.7446 −1.15785
$$322$$ 0 0
$$323$$ −1.76631 −0.0982802
$$324$$ 0 0
$$325$$ −4.37228 −0.242531
$$326$$ 0 0
$$327$$ 0.883156 0.0488386
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 0 0
$$333$$ −5.25544 −0.287996
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −7.48913 −0.407959 −0.203979 0.978975i $$-0.565388\pi$$
−0.203979 + 0.978975i $$0.565388\pi$$
$$338$$ 0 0
$$339$$ 4.74456 0.257689
$$340$$ 0 0
$$341$$ 50.9783 2.76063
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 11.2554 0.605972
$$346$$ 0 0
$$347$$ 24.7446 1.32836 0.664179 0.747574i $$-0.268781\pi$$
0.664179 + 0.747574i $$0.268781\pi$$
$$348$$ 0 0
$$349$$ −19.4891 −1.04323 −0.521614 0.853181i $$-0.674670\pi$$
−0.521614 + 0.853181i $$0.674670\pi$$
$$350$$ 0 0
$$351$$ 3.86141 0.206107
$$352$$ 0 0
$$353$$ 1.86141 0.0990727 0.0495363 0.998772i $$-0.484226\pi$$
0.0495363 + 0.998772i $$0.484226\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 3.51087 0.184783
$$362$$ 0 0
$$363$$ 70.2337 3.68631
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 19.8614 1.03676 0.518378 0.855151i $$-0.326536\pi$$
0.518378 + 0.855151i $$0.326536\pi$$
$$368$$ 0 0
$$369$$ −17.7228 −0.922613
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −30.7446 −1.59189 −0.795947 0.605367i $$-0.793026\pi$$
−0.795947 + 0.605367i $$0.793026\pi$$
$$374$$ 0 0
$$375$$ 2.37228 0.122504
$$376$$ 0 0
$$377$$ 19.1168 0.984568
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ −18.9783 −0.972285
$$382$$ 0 0
$$383$$ −17.4891 −0.893653 −0.446826 0.894621i $$-0.647446\pi$$
−0.446826 + 0.894621i $$0.647446\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 22.9783 1.16805
$$388$$ 0 0
$$389$$ 17.8614 0.905609 0.452805 0.891610i $$-0.350424\pi$$
0.452805 + 0.891610i $$0.350424\pi$$
$$390$$ 0 0
$$391$$ 1.76631 0.0893262
$$392$$ 0 0
$$393$$ −11.2554 −0.567762
$$394$$ 0 0
$$395$$ −15.1168 −0.760611
$$396$$ 0 0
$$397$$ −31.6277 −1.58735 −0.793675 0.608342i $$-0.791835\pi$$
−0.793675 + 0.608342i $$0.791835\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −38.6060 −1.92789 −0.963945 0.266101i $$-0.914264\pi$$
−0.963945 + 0.266101i $$0.914264\pi$$
$$402$$ 0 0
$$403$$ 34.9783 1.74239
$$404$$ 0 0
$$405$$ −9.97825 −0.495823
$$406$$ 0 0
$$407$$ 12.7446 0.631725
$$408$$ 0 0
$$409$$ −11.4891 −0.568101 −0.284050 0.958809i $$-0.591678\pi$$
−0.284050 + 0.958809i $$0.591678\pi$$
$$410$$ 0 0
$$411$$ 34.9783 1.72535
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −9.48913 −0.465803
$$416$$ 0 0
$$417$$ −11.2554 −0.551181
$$418$$ 0 0
$$419$$ −14.5109 −0.708903 −0.354451 0.935074i $$-0.615332\pi$$
−0.354451 + 0.935074i $$0.615332\pi$$
$$420$$ 0 0
$$421$$ −18.6060 −0.906799 −0.453400 0.891307i $$-0.649789\pi$$
−0.453400 + 0.891307i $$0.649789\pi$$
$$422$$ 0 0
$$423$$ −18.7011 −0.909277
$$424$$ 0 0
$$425$$ 0.372281 0.0180583
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 66.0951 3.19110
$$430$$ 0 0
$$431$$ −18.3723 −0.884962 −0.442481 0.896778i $$-0.645902\pi$$
−0.442481 + 0.896778i $$0.645902\pi$$
$$432$$ 0 0
$$433$$ −28.9783 −1.39261 −0.696303 0.717748i $$-0.745173\pi$$
−0.696303 + 0.717748i $$0.745173\pi$$
$$434$$ 0 0
$$435$$ −10.3723 −0.497313
$$436$$ 0 0
$$437$$ −22.5109 −1.07684
$$438$$ 0 0
$$439$$ −6.23369 −0.297518 −0.148759 0.988874i $$-0.547528\pi$$
−0.148759 + 0.988874i $$0.547528\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 16.7446 0.795558 0.397779 0.917481i $$-0.369781\pi$$
0.397779 + 0.917481i $$0.369781\pi$$
$$444$$ 0 0
$$445$$ −14.7446 −0.698959
$$446$$ 0 0
$$447$$ 36.7446 1.73796
$$448$$ 0 0
$$449$$ 17.1168 0.807794 0.403897 0.914805i $$-0.367655\pi$$
0.403897 + 0.914805i $$0.367655\pi$$
$$450$$ 0 0
$$451$$ 42.9783 2.02377
$$452$$ 0 0
$$453$$ −35.8614 −1.68492
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.25544 0.245839 0.122919 0.992417i $$-0.460774\pi$$
0.122919 + 0.992417i $$0.460774\pi$$
$$458$$ 0 0
$$459$$ −0.328782 −0.0153463
$$460$$ 0 0
$$461$$ 1.25544 0.0584715 0.0292358 0.999573i $$-0.490693\pi$$
0.0292358 + 0.999573i $$0.490693\pi$$
$$462$$ 0 0
$$463$$ 6.51087 0.302586 0.151293 0.988489i $$-0.451656\pi$$
0.151293 + 0.988489i $$0.451656\pi$$
$$464$$ 0 0
$$465$$ −18.9783 −0.880095
$$466$$ 0 0
$$467$$ 8.60597 0.398237 0.199118 0.979975i $$-0.436192\pi$$
0.199118 + 0.979975i $$0.436192\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 36.7446 1.69310
$$472$$ 0 0
$$473$$ −55.7228 −2.56214
$$474$$ 0 0
$$475$$ −4.74456 −0.217695
$$476$$ 0 0
$$477$$ 28.2337 1.29273
$$478$$ 0 0
$$479$$ 22.2337 1.01588 0.507942 0.861392i $$-0.330407\pi$$
0.507942 + 0.861392i $$0.330407\pi$$
$$480$$ 0 0
$$481$$ 8.74456 0.398718
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 9.86141 0.447783
$$486$$ 0 0
$$487$$ −30.2337 −1.37002 −0.685010 0.728534i $$-0.740202\pi$$
−0.685010 + 0.728534i $$0.740202\pi$$
$$488$$ 0 0
$$489$$ 39.7228 1.79633
$$490$$ 0 0
$$491$$ −35.1168 −1.58480 −0.792400 0.610001i $$-0.791169\pi$$
−0.792400 + 0.610001i $$0.791169\pi$$
$$492$$ 0 0
$$493$$ −1.62772 −0.0733088
$$494$$ 0 0
$$495$$ −16.7446 −0.752612
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −31.8614 −1.42631 −0.713156 0.701005i $$-0.752735\pi$$
−0.713156 + 0.701005i $$0.752735\pi$$
$$500$$ 0 0
$$501$$ −13.3505 −0.596458
$$502$$ 0 0
$$503$$ −18.3723 −0.819180 −0.409590 0.912270i $$-0.634328\pi$$
−0.409590 + 0.912270i $$0.634328\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 14.5109 0.644451
$$508$$ 0 0
$$509$$ 40.9783 1.81633 0.908165 0.418613i $$-0.137484\pi$$
0.908165 + 0.418613i $$0.137484\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 4.19019 0.185001
$$514$$ 0 0
$$515$$ −5.62772 −0.247987
$$516$$ 0 0
$$517$$ 45.3505 1.99451
$$518$$ 0 0
$$519$$ 0.883156 0.0387662
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −36.4674 −1.59461 −0.797304 0.603579i $$-0.793741\pi$$
−0.797304 + 0.603579i $$0.793741\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.97825 −0.129735
$$528$$ 0 0
$$529$$ −0.489125 −0.0212663
$$530$$ 0 0
$$531$$ 21.0217 0.912266
$$532$$ 0 0
$$533$$ 29.4891 1.27732
$$534$$ 0 0
$$535$$ −8.74456 −0.378060
$$536$$ 0 0
$$537$$ 54.5109 2.35232
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −31.3505 −1.34786 −0.673932 0.738793i $$-0.735396\pi$$
−0.673932 + 0.738793i $$0.735396\pi$$
$$542$$ 0 0
$$543$$ −38.5109 −1.65266
$$544$$ 0 0
$$545$$ 0.372281 0.0159468
$$546$$ 0 0
$$547$$ 30.9783 1.32453 0.662267 0.749268i $$-0.269594\pi$$
0.662267 + 0.749268i $$0.269594\pi$$
$$548$$ 0 0
$$549$$ 7.21194 0.307798
$$550$$ 0 0
$$551$$ 20.7446 0.883748
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −4.74456 −0.201395
$$556$$ 0 0
$$557$$ −3.76631 −0.159584 −0.0797919 0.996812i $$-0.525426\pi$$
−0.0797919 + 0.996812i $$0.525426\pi$$
$$558$$ 0 0
$$559$$ −38.2337 −1.61711
$$560$$ 0 0
$$561$$ −5.62772 −0.237602
$$562$$ 0 0
$$563$$ 17.4891 0.737079 0.368539 0.929612i $$-0.379858\pi$$
0.368539 + 0.929612i $$0.379858\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −24.9783 −1.04714 −0.523571 0.851982i $$-0.675401\pi$$
−0.523571 + 0.851982i $$0.675401\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ −9.16034 −0.382679
$$574$$ 0 0
$$575$$ 4.74456 0.197862
$$576$$ 0 0
$$577$$ 22.6060 0.941099 0.470549 0.882374i $$-0.344056\pi$$
0.470549 + 0.882374i $$0.344056\pi$$
$$578$$ 0 0
$$579$$ 16.0000 0.664937
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −68.4674 −2.83563
$$584$$ 0 0
$$585$$ −11.4891 −0.475017
$$586$$ 0 0
$$587$$ −34.9783 −1.44371 −0.721853 0.692046i $$-0.756710\pi$$
−0.721853 + 0.692046i $$0.756710\pi$$
$$588$$ 0 0
$$589$$ 37.9565 1.56397
$$590$$ 0 0
$$591$$ −19.5326 −0.803465
$$592$$ 0 0
$$593$$ 19.6277 0.806014 0.403007 0.915197i $$-0.367965\pi$$
0.403007 + 0.915197i $$0.367965\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −37.9565 −1.55346
$$598$$ 0 0
$$599$$ 32.6060 1.33224 0.666122 0.745843i $$-0.267953\pi$$
0.666122 + 0.745843i $$0.267953\pi$$
$$600$$ 0 0
$$601$$ −16.5109 −0.673493 −0.336746 0.941595i $$-0.609326\pi$$
−0.336746 + 0.941595i $$0.609326\pi$$
$$602$$ 0 0
$$603$$ 10.5109 0.428036
$$604$$ 0 0
$$605$$ 29.6060 1.20365
$$606$$ 0 0
$$607$$ 24.6060 0.998725 0.499363 0.866393i $$-0.333568\pi$$
0.499363 + 0.866393i $$0.333568\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 31.1168 1.25885
$$612$$ 0 0
$$613$$ −8.51087 −0.343751 −0.171875 0.985119i $$-0.554983\pi$$
−0.171875 + 0.985119i $$0.554983\pi$$
$$614$$ 0 0
$$615$$ −16.0000 −0.645182
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −12.7446 −0.512247 −0.256124 0.966644i $$-0.582445\pi$$
−0.256124 + 0.966644i $$0.582445\pi$$
$$620$$ 0 0
$$621$$ −4.19019 −0.168146
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 71.7228 2.86433
$$628$$ 0 0
$$629$$ −0.744563 −0.0296877
$$630$$ 0 0
$$631$$ −2.37228 −0.0944390 −0.0472195 0.998885i $$-0.515036\pi$$
−0.0472195 + 0.998885i $$0.515036\pi$$
$$632$$ 0 0
$$633$$ 34.0951 1.35516
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −21.0217 −0.831608
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 18.3723 0.724532 0.362266 0.932075i $$-0.382003\pi$$
0.362266 + 0.932075i $$0.382003\pi$$
$$644$$ 0 0
$$645$$ 20.7446 0.816816
$$646$$ 0 0
$$647$$ −16.0000 −0.629025 −0.314512 0.949253i $$-0.601841\pi$$
−0.314512 + 0.949253i $$0.601841\pi$$
$$648$$ 0 0
$$649$$ −50.9783 −2.00107
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ −4.74456 −0.185385
$$656$$ 0 0
$$657$$ 15.7663 0.615102
$$658$$ 0 0
$$659$$ 11.1168 0.433051 0.216525 0.976277i $$-0.430528\pi$$
0.216525 + 0.976277i $$0.430528\pi$$
$$660$$ 0 0
$$661$$ −14.7446 −0.573497 −0.286749 0.958006i $$-0.592574\pi$$
−0.286749 + 0.958006i $$0.592574\pi$$
$$662$$ 0 0
$$663$$ −3.86141 −0.149965
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −20.7446 −0.803233
$$668$$ 0 0
$$669$$ −13.3505 −0.516161
$$670$$ 0 0
$$671$$ −17.4891 −0.675160
$$672$$ 0 0
$$673$$ 25.7228 0.991542 0.495771 0.868453i $$-0.334886\pi$$
0.495771 + 0.868453i $$0.334886\pi$$
$$674$$ 0 0
$$675$$ −0.883156 −0.0339927
$$676$$ 0 0
$$677$$ −15.3505 −0.589969 −0.294984 0.955502i $$-0.595314\pi$$
−0.294984 + 0.955502i $$0.595314\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 47.1168 1.80552
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 14.7446 0.563361
$$686$$ 0 0
$$687$$ 29.0217 1.10725
$$688$$ 0 0
$$689$$ −46.9783 −1.78973
$$690$$ 0 0
$$691$$ −9.48913 −0.360983 −0.180492 0.983577i $$-0.557769\pi$$
−0.180492 + 0.983577i $$0.557769\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.74456 −0.179972
$$696$$ 0 0
$$697$$ −2.51087 −0.0951062
$$698$$ 0 0
$$699$$ −2.97825 −0.112648
$$700$$ 0 0
$$701$$ 2.13859 0.0807736 0.0403868 0.999184i $$-0.487141\pi$$
0.0403868 + 0.999184i $$0.487141\pi$$
$$702$$ 0 0
$$703$$ 9.48913 0.357889
$$704$$ 0 0
$$705$$ −16.8832 −0.635856
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.60597 0.248092 0.124046 0.992276i $$-0.460413\pi$$
0.124046 + 0.992276i $$0.460413\pi$$
$$710$$ 0 0
$$711$$ −39.7228 −1.48972
$$712$$ 0 0
$$713$$ −37.9565 −1.42148
$$714$$ 0 0
$$715$$ 27.8614 1.04196
$$716$$ 0 0
$$717$$ −32.3288 −1.20734
$$718$$ 0 0
$$719$$ 3.25544 0.121407 0.0607037 0.998156i $$-0.480666\pi$$
0.0607037 + 0.998156i $$0.480666\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −61.6793 −2.29388
$$724$$ 0 0
$$725$$ −4.37228 −0.162382
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ −19.9348 −0.738324
$$730$$ 0 0
$$731$$ 3.25544 0.120407
$$732$$ 0 0
$$733$$ 10.1386 0.374477 0.187239 0.982314i $$-0.440046\pi$$
0.187239 + 0.982314i $$0.440046\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −25.4891 −0.938904
$$738$$ 0 0
$$739$$ 20.6060 0.758003 0.379001 0.925396i $$-0.376268\pi$$
0.379001 + 0.925396i $$0.376268\pi$$
$$740$$ 0 0
$$741$$ 49.2119 1.80785
$$742$$ 0 0
$$743$$ 6.51087 0.238861 0.119430 0.992843i $$-0.461893\pi$$
0.119430 + 0.992843i $$0.461893\pi$$
$$744$$ 0 0
$$745$$ 15.4891 0.567478
$$746$$ 0 0
$$747$$ −24.9348 −0.912315
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −20.1386 −0.734868 −0.367434 0.930050i $$-0.619764\pi$$
−0.367434 + 0.930050i $$0.619764\pi$$
$$752$$ 0 0
$$753$$ 11.2554 0.410171
$$754$$ 0 0
$$755$$ −15.1168 −0.550158
$$756$$ 0 0
$$757$$ −3.76631 −0.136889 −0.0684445 0.997655i $$-0.521804\pi$$
−0.0684445 + 0.997655i $$0.521804\pi$$
$$758$$ 0 0
$$759$$ −71.7228 −2.60337
$$760$$ 0 0
$$761$$ −4.97825 −0.180461 −0.0902307 0.995921i $$-0.528760\pi$$
−0.0902307 + 0.995921i $$0.528760\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0.978251 0.0353687
$$766$$ 0 0
$$767$$ −34.9783 −1.26299
$$768$$ 0 0
$$769$$ −3.48913 −0.125821 −0.0629105 0.998019i $$-0.520038\pi$$
−0.0629105 + 0.998019i $$0.520038\pi$$
$$770$$ 0 0
$$771$$ 55.7228 2.00681
$$772$$ 0 0
$$773$$ −4.37228 −0.157260 −0.0786300 0.996904i $$-0.525055\pi$$
−0.0786300 + 0.996904i $$0.525055\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 32.0000 1.14652
$$780$$ 0 0
$$781$$ 50.9783 1.82415
$$782$$ 0 0
$$783$$ 3.86141 0.137995
$$784$$ 0 0
$$785$$ 15.4891 0.552831
$$786$$ 0 0
$$787$$ −31.1168 −1.10920 −0.554598 0.832119i $$-0.687128\pi$$
−0.554598 + 0.832119i $$0.687128\pi$$
$$788$$ 0 0
$$789$$ −52.7446 −1.87776
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 0 0
$$795$$ 25.4891 0.904006
$$796$$ 0 0
$$797$$ −15.6277 −0.553562 −0.276781 0.960933i $$-0.589268\pi$$
−0.276781 + 0.960933i $$0.589268\pi$$
$$798$$ 0 0
$$799$$ −2.64947 −0.0937314
$$800$$ 0 0
$$801$$ −38.7446 −1.36897
$$802$$ 0 0
$$803$$ −38.2337 −1.34924
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.2337 0.501050
$$808$$ 0 0
$$809$$ 20.3723 0.716251 0.358126 0.933673i $$-0.383416\pi$$
0.358126 + 0.933673i $$0.383416\pi$$
$$810$$ 0 0
$$811$$ 12.7446 0.447522 0.223761 0.974644i $$-0.428166\pi$$
0.223761 + 0.974644i $$0.428166\pi$$
$$812$$ 0 0
$$813$$ 22.5109 0.789491
$$814$$ 0 0
$$815$$ 16.7446 0.586536
$$816$$ 0 0
$$817$$ −41.4891 −1.45152
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25.1168 −0.876584 −0.438292 0.898833i $$-0.644416\pi$$
−0.438292 + 0.898833i $$0.644416\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 0 0
$$825$$ −15.1168 −0.526301
$$826$$ 0 0
$$827$$ −24.7446 −0.860453 −0.430226 0.902721i $$-0.641566\pi$$
−0.430226 + 0.902721i $$0.641566\pi$$
$$828$$ 0 0
$$829$$ 12.2337 0.424894 0.212447 0.977173i $$-0.431857\pi$$
0.212447 + 0.977173i $$0.431857\pi$$
$$830$$ 0 0
$$831$$ 59.2554 2.05555
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5.62772 −0.194755
$$836$$ 0 0
$$837$$ 7.06525 0.244211
$$838$$ 0 0
$$839$$ −11.2554 −0.388581 −0.194290 0.980944i $$-0.562240\pi$$
−0.194290 + 0.980944i $$0.562240\pi$$
$$840$$ 0 0
$$841$$ −9.88316 −0.340798
$$842$$ 0 0
$$843$$ 44.1386 1.52021
$$844$$ 0 0
$$845$$ 6.11684 0.210426
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −21.0733 −0.723235
$$850$$ 0 0
$$851$$ −9.48913 −0.325283
$$852$$ 0 0
$$853$$ 39.4891 1.35208 0.676041 0.736864i $$-0.263694\pi$$
0.676041 + 0.736864i $$0.263694\pi$$
$$854$$ 0 0
$$855$$ −12.4674 −0.426375
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 44.4674 1.51721 0.758604 0.651552i $$-0.225882\pi$$
0.758604 + 0.651552i $$0.225882\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −25.4891 −0.867660 −0.433830 0.900995i $$-0.642838\pi$$
−0.433830 + 0.900995i $$0.642838\pi$$
$$864$$ 0 0
$$865$$ 0.372281 0.0126579
$$866$$ 0 0
$$867$$ −40.0000 −1.35847
$$868$$ 0 0
$$869$$ 96.3288 3.26773
$$870$$ 0 0
$$871$$ −17.4891 −0.592596
$$872$$ 0 0
$$873$$ 25.9130 0.877022
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.02175 −0.237108 −0.118554 0.992948i $$-0.537826\pi$$
−0.118554 + 0.992948i $$0.537826\pi$$
$$878$$ 0 0
$$879$$ −59.5842 −2.00973
$$880$$ 0 0
$$881$$ 25.2554 0.850877 0.425439 0.904987i $$-0.360120\pi$$
0.425439 + 0.904987i $$0.360120\pi$$
$$882$$ 0 0
$$883$$ 10.5109 0.353719 0.176860 0.984236i $$-0.443406\pi$$
0.176860 + 0.984236i $$0.443406\pi$$
$$884$$ 0 0
$$885$$ 18.9783 0.637947
$$886$$ 0 0
$$887$$ −13.0217 −0.437228 −0.218614 0.975811i $$-0.570153\pi$$
−0.218614 + 0.975811i $$0.570153\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 63.5842 2.13015
$$892$$ 0 0
$$893$$ 33.7663 1.12995
$$894$$ 0 0
$$895$$ 22.9783 0.768078
$$896$$ 0 0
$$897$$ −49.2119 −1.64314
$$898$$ 0 0
$$899$$ 34.9783 1.16659
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.2337 −0.539626
$$906$$ 0 0
$$907$$ −11.7228 −0.389250 −0.194625 0.980878i $$-0.562349\pi$$
−0.194625 + 0.980878i $$0.562349\pi$$
$$908$$ 0 0
$$909$$ 15.7663 0.522936
$$910$$ 0 0
$$911$$ 45.9565 1.52261 0.761303 0.648396i $$-0.224560\pi$$
0.761303 + 0.648396i $$0.224560\pi$$
$$912$$ 0 0
$$913$$ 60.4674 2.00118
$$914$$ 0 0
$$915$$ 6.51087 0.215243
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −13.6277 −0.449537 −0.224768 0.974412i $$-0.572163\pi$$
−0.224768 + 0.974412i $$0.572163\pi$$
$$920$$ 0 0
$$921$$ 73.8179 2.43238
$$922$$ 0 0
$$923$$ 34.9783 1.15132
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ −14.7881 −0.485704
$$928$$ 0 0
$$929$$ 7.76631 0.254804 0.127402 0.991851i $$-0.459336\pi$$
0.127402 + 0.991851i $$0.459336\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −30.2337 −0.989807
$$934$$ 0 0
$$935$$ −2.37228 −0.0775819
$$936$$ 0 0
$$937$$ −28.0951 −0.917827 −0.458913 0.888481i $$-0.651761\pi$$
−0.458913 + 0.888481i $$0.651761\pi$$
$$938$$ 0 0
$$939$$ −6.83966 −0.223204
$$940$$ 0 0
$$941$$ −32.2337 −1.05079 −0.525394 0.850859i $$-0.676082\pi$$
−0.525394 + 0.850859i $$0.676082\pi$$
$$942$$ 0 0
$$943$$ −32.0000 −1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ −26.2337 −0.851582
$$950$$ 0 0
$$951$$ 33.2119 1.07697
$$952$$ 0 0
$$953$$ 37.2554 1.20682 0.603411 0.797430i $$-0.293808\pi$$
0.603411 + 0.797430i $$0.293808\pi$$
$$954$$ 0 0
$$955$$ −3.86141 −0.124952
$$956$$ 0 0
$$957$$ 66.0951 2.13655
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −22.9783 −0.740464
$$964$$ 0 0
$$965$$ 6.74456 0.217115
$$966$$ 0 0
$$967$$ −1.76631 −0.0568008 −0.0284004 0.999597i $$-0.509041\pi$$
−0.0284004 + 0.999597i $$0.509041\pi$$
$$968$$ 0 0
$$969$$ −4.19019 −0.134608
$$970$$ 0 0
$$971$$ 33.4891 1.07472 0.537359 0.843354i $$-0.319422\pi$$
0.537359 + 0.843354i $$0.319422\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −10.3723 −0.332179
$$976$$ 0 0
$$977$$ 52.9783 1.69492 0.847462 0.530856i $$-0.178129\pi$$
0.847462 + 0.530856i $$0.178129\pi$$
$$978$$ 0 0
$$979$$ 93.9565 3.00286
$$980$$ 0 0
$$981$$ 0.978251 0.0312331
$$982$$ 0 0
$$983$$ −10.3723 −0.330824 −0.165412 0.986225i $$-0.552895\pi$$
−0.165412 + 0.986225i $$0.552895\pi$$
$$984$$ 0 0
$$985$$ −8.23369 −0.262347
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 41.4891 1.31928
$$990$$ 0 0
$$991$$ 37.9565 1.20573 0.602864 0.797844i $$-0.294026\pi$$
0.602864 + 0.797844i $$0.294026\pi$$
$$992$$ 0 0
$$993$$ −28.4674 −0.903385
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 6.88316 0.217992 0.108996 0.994042i $$-0.465236\pi$$
0.108996 + 0.994042i $$0.465236\pi$$
$$998$$ 0 0
$$999$$ 1.76631 0.0558836
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 3920.2.a.bt.1.2 2
4.3 odd 2 1960.2.a.s.1.1 2
7.6 odd 2 560.2.a.h.1.1 2
20.19 odd 2 9800.2.a.bu.1.2 2
21.20 even 2 5040.2.a.by.1.2 2
28.3 even 6 1960.2.q.t.961.1 4
28.11 odd 6 1960.2.q.r.961.2 4
28.19 even 6 1960.2.q.t.361.1 4
28.23 odd 6 1960.2.q.r.361.2 4
28.27 even 2 280.2.a.c.1.2 2
35.13 even 4 2800.2.g.r.449.2 4
35.27 even 4 2800.2.g.r.449.3 4
35.34 odd 2 2800.2.a.bk.1.2 2
56.13 odd 2 2240.2.a.bg.1.2 2
56.27 even 2 2240.2.a.bk.1.1 2
84.83 odd 2 2520.2.a.x.1.1 2
140.27 odd 4 1400.2.g.i.449.2 4
140.83 odd 4 1400.2.g.i.449.3 4
140.139 even 2 1400.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.2 2 28.27 even 2
560.2.a.h.1.1 2 7.6 odd 2
1400.2.a.r.1.1 2 140.139 even 2
1400.2.g.i.449.2 4 140.27 odd 4
1400.2.g.i.449.3 4 140.83 odd 4
1960.2.a.s.1.1 2 4.3 odd 2
1960.2.q.r.361.2 4 28.23 odd 6
1960.2.q.r.961.2 4 28.11 odd 6
1960.2.q.t.361.1 4 28.19 even 6
1960.2.q.t.961.1 4 28.3 even 6
2240.2.a.bg.1.2 2 56.13 odd 2
2240.2.a.bk.1.1 2 56.27 even 2
2520.2.a.x.1.1 2 84.83 odd 2
2800.2.a.bk.1.2 2 35.34 odd 2
2800.2.g.r.449.2 4 35.13 even 4
2800.2.g.r.449.3 4 35.27 even 4
3920.2.a.bt.1.2 2 1.1 even 1 trivial
5040.2.a.by.1.2 2 21.20 even 2
9800.2.a.bu.1.2 2 20.19 odd 2