# Properties

 Label 35.3.d.a.6.2 Level $35$ Weight $3$ Character 35.6 Analytic conductor $0.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 6.2 Root $$2.23607i$$ of defining polynomial Character $$\chi$$ $$=$$ 35.6 Dual form 35.3.d.a.6.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +4.47214i q^{3} -3.00000 q^{4} +2.23607i q^{5} -4.47214i q^{6} +7.00000 q^{7} +7.00000 q^{8} -11.0000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +4.47214i q^{3} -3.00000 q^{4} +2.23607i q^{5} -4.47214i q^{6} +7.00000 q^{7} +7.00000 q^{8} -11.0000 q^{9} -2.23607i q^{10} +2.00000 q^{11} -13.4164i q^{12} -13.4164i q^{13} -7.00000 q^{14} -10.0000 q^{15} +5.00000 q^{16} +26.8328i q^{17} +11.0000 q^{18} -13.4164i q^{19} -6.70820i q^{20} +31.3050i q^{21} -2.00000 q^{22} +26.0000 q^{23} +31.3050i q^{24} -5.00000 q^{25} +13.4164i q^{26} -8.94427i q^{27} -21.0000 q^{28} -22.0000 q^{29} +10.0000 q^{30} -53.6656i q^{31} -33.0000 q^{32} +8.94427i q^{33} -26.8328i q^{34} +15.6525i q^{35} +33.0000 q^{36} +14.0000 q^{37} +13.4164i q^{38} +60.0000 q^{39} +15.6525i q^{40} -26.8328i q^{41} -31.3050i q^{42} -34.0000 q^{43} -6.00000 q^{44} -24.5967i q^{45} -26.0000 q^{46} -26.8328i q^{47} +22.3607i q^{48} +49.0000 q^{49} +5.00000 q^{50} -120.000 q^{51} +40.2492i q^{52} -34.0000 q^{53} +8.94427i q^{54} +4.47214i q^{55} +49.0000 q^{56} +60.0000 q^{57} +22.0000 q^{58} -40.2492i q^{59} +30.0000 q^{60} +93.9149i q^{61} +53.6656i q^{62} -77.0000 q^{63} +13.0000 q^{64} +30.0000 q^{65} -8.94427i q^{66} +14.0000 q^{67} -80.4984i q^{68} +116.276i q^{69} -15.6525i q^{70} +62.0000 q^{71} -77.0000 q^{72} -53.6656i q^{73} -14.0000 q^{74} -22.3607i q^{75} +40.2492i q^{76} +14.0000 q^{77} -60.0000 q^{78} +38.0000 q^{79} +11.1803i q^{80} -59.0000 q^{81} +26.8328i q^{82} +40.2492i q^{83} -93.9149i q^{84} -60.0000 q^{85} +34.0000 q^{86} -98.3870i q^{87} +14.0000 q^{88} -26.8328i q^{89} +24.5967i q^{90} -93.9149i q^{91} -78.0000 q^{92} +240.000 q^{93} +26.8328i q^{94} +30.0000 q^{95} -147.580i q^{96} -26.8328i q^{97} -49.0000 q^{98} -22.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^7 + 14 * q^8 - 22 * q^9 $$2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9} + 4 q^{11} - 14 q^{14} - 20 q^{15} + 10 q^{16} + 22 q^{18} - 4 q^{22} + 52 q^{23} - 10 q^{25} - 42 q^{28} - 44 q^{29} + 20 q^{30} - 66 q^{32} + 66 q^{36} + 28 q^{37} + 120 q^{39} - 68 q^{43} - 12 q^{44} - 52 q^{46} + 98 q^{49} + 10 q^{50} - 240 q^{51} - 68 q^{53} + 98 q^{56} + 120 q^{57} + 44 q^{58} + 60 q^{60} - 154 q^{63} + 26 q^{64} + 60 q^{65} + 28 q^{67} + 124 q^{71} - 154 q^{72} - 28 q^{74} + 28 q^{77} - 120 q^{78} + 76 q^{79} - 118 q^{81} - 120 q^{85} + 68 q^{86} + 28 q^{88} - 156 q^{92} + 480 q^{93} + 60 q^{95} - 98 q^{98} - 44 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 6 * q^4 + 14 * q^7 + 14 * q^8 - 22 * q^9 + 4 * q^11 - 14 * q^14 - 20 * q^15 + 10 * q^16 + 22 * q^18 - 4 * q^22 + 52 * q^23 - 10 * q^25 - 42 * q^28 - 44 * q^29 + 20 * q^30 - 66 * q^32 + 66 * q^36 + 28 * q^37 + 120 * q^39 - 68 * q^43 - 12 * q^44 - 52 * q^46 + 98 * q^49 + 10 * q^50 - 240 * q^51 - 68 * q^53 + 98 * q^56 + 120 * q^57 + 44 * q^58 + 60 * q^60 - 154 * q^63 + 26 * q^64 + 60 * q^65 + 28 * q^67 + 124 * q^71 - 154 * q^72 - 28 * q^74 + 28 * q^77 - 120 * q^78 + 76 * q^79 - 118 * q^81 - 120 * q^85 + 68 * q^86 + 28 * q^88 - 156 * q^92 + 480 * q^93 + 60 * q^95 - 98 * q^98 - 44 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.500000 −0.250000 0.968246i $$-0.580431\pi$$
−0.250000 + 0.968246i $$0.580431\pi$$
$$3$$ 4.47214i 1.49071i 0.666667 + 0.745356i $$0.267720\pi$$
−0.666667 + 0.745356i $$0.732280\pi$$
$$4$$ −3.00000 −0.750000
$$5$$ 2.23607i 0.447214i
$$6$$ − 4.47214i − 0.745356i
$$7$$ 7.00000 1.00000
$$8$$ 7.00000 0.875000
$$9$$ −11.0000 −1.22222
$$10$$ − 2.23607i − 0.223607i
$$11$$ 2.00000 0.181818 0.0909091 0.995859i $$-0.471023\pi$$
0.0909091 + 0.995859i $$0.471023\pi$$
$$12$$ − 13.4164i − 1.11803i
$$13$$ − 13.4164i − 1.03203i −0.856579 0.516016i $$-0.827415\pi$$
0.856579 0.516016i $$-0.172585\pi$$
$$14$$ −7.00000 −0.500000
$$15$$ −10.0000 −0.666667
$$16$$ 5.00000 0.312500
$$17$$ 26.8328i 1.57840i 0.614136 + 0.789200i $$0.289505\pi$$
−0.614136 + 0.789200i $$0.710495\pi$$
$$18$$ 11.0000 0.611111
$$19$$ − 13.4164i − 0.706127i −0.935599 0.353063i $$-0.885140\pi$$
0.935599 0.353063i $$-0.114860\pi$$
$$20$$ − 6.70820i − 0.335410i
$$21$$ 31.3050i 1.49071i
$$22$$ −2.00000 −0.0909091
$$23$$ 26.0000 1.13043 0.565217 0.824942i $$-0.308792\pi$$
0.565217 + 0.824942i $$0.308792\pi$$
$$24$$ 31.3050i 1.30437i
$$25$$ −5.00000 −0.200000
$$26$$ 13.4164i 0.516016i
$$27$$ − 8.94427i − 0.331269i
$$28$$ −21.0000 −0.750000
$$29$$ −22.0000 −0.758621 −0.379310 0.925270i $$-0.623839\pi$$
−0.379310 + 0.925270i $$0.623839\pi$$
$$30$$ 10.0000 0.333333
$$31$$ − 53.6656i − 1.73115i −0.500780 0.865575i $$-0.666953\pi$$
0.500780 0.865575i $$-0.333047\pi$$
$$32$$ −33.0000 −1.03125
$$33$$ 8.94427i 0.271039i
$$34$$ − 26.8328i − 0.789200i
$$35$$ 15.6525i 0.447214i
$$36$$ 33.0000 0.916667
$$37$$ 14.0000 0.378378 0.189189 0.981941i $$-0.439414\pi$$
0.189189 + 0.981941i $$0.439414\pi$$
$$38$$ 13.4164i 0.353063i
$$39$$ 60.0000 1.53846
$$40$$ 15.6525i 0.391312i
$$41$$ − 26.8328i − 0.654459i −0.944945 0.327229i $$-0.893885\pi$$
0.944945 0.327229i $$-0.106115\pi$$
$$42$$ − 31.3050i − 0.745356i
$$43$$ −34.0000 −0.790698 −0.395349 0.918531i $$-0.629376\pi$$
−0.395349 + 0.918531i $$0.629376\pi$$
$$44$$ −6.00000 −0.136364
$$45$$ − 24.5967i − 0.546594i
$$46$$ −26.0000 −0.565217
$$47$$ − 26.8328i − 0.570911i −0.958392 0.285455i $$-0.907855\pi$$
0.958392 0.285455i $$-0.0921449\pi$$
$$48$$ 22.3607i 0.465847i
$$49$$ 49.0000 1.00000
$$50$$ 5.00000 0.100000
$$51$$ −120.000 −2.35294
$$52$$ 40.2492i 0.774024i
$$53$$ −34.0000 −0.641509 −0.320755 0.947162i $$-0.603937\pi$$
−0.320755 + 0.947162i $$0.603937\pi$$
$$54$$ 8.94427i 0.165635i
$$55$$ 4.47214i 0.0813116i
$$56$$ 49.0000 0.875000
$$57$$ 60.0000 1.05263
$$58$$ 22.0000 0.379310
$$59$$ − 40.2492i − 0.682190i −0.940029 0.341095i $$-0.889202\pi$$
0.940029 0.341095i $$-0.110798\pi$$
$$60$$ 30.0000 0.500000
$$61$$ 93.9149i 1.53959i 0.638293 + 0.769794i $$0.279641\pi$$
−0.638293 + 0.769794i $$0.720359\pi$$
$$62$$ 53.6656i 0.865575i
$$63$$ −77.0000 −1.22222
$$64$$ 13.0000 0.203125
$$65$$ 30.0000 0.461538
$$66$$ − 8.94427i − 0.135519i
$$67$$ 14.0000 0.208955 0.104478 0.994527i $$-0.466683\pi$$
0.104478 + 0.994527i $$0.466683\pi$$
$$68$$ − 80.4984i − 1.18380i
$$69$$ 116.276i 1.68515i
$$70$$ − 15.6525i − 0.223607i
$$71$$ 62.0000 0.873239 0.436620 0.899646i $$-0.356176\pi$$
0.436620 + 0.899646i $$0.356176\pi$$
$$72$$ −77.0000 −1.06944
$$73$$ − 53.6656i − 0.735146i −0.929995 0.367573i $$-0.880189\pi$$
0.929995 0.367573i $$-0.119811\pi$$
$$74$$ −14.0000 −0.189189
$$75$$ − 22.3607i − 0.298142i
$$76$$ 40.2492i 0.529595i
$$77$$ 14.0000 0.181818
$$78$$ −60.0000 −0.769231
$$79$$ 38.0000 0.481013 0.240506 0.970648i $$-0.422687\pi$$
0.240506 + 0.970648i $$0.422687\pi$$
$$80$$ 11.1803i 0.139754i
$$81$$ −59.0000 −0.728395
$$82$$ 26.8328i 0.327229i
$$83$$ 40.2492i 0.484930i 0.970160 + 0.242465i $$0.0779560\pi$$
−0.970160 + 0.242465i $$0.922044\pi$$
$$84$$ − 93.9149i − 1.11803i
$$85$$ −60.0000 −0.705882
$$86$$ 34.0000 0.395349
$$87$$ − 98.3870i − 1.13088i
$$88$$ 14.0000 0.159091
$$89$$ − 26.8328i − 0.301492i −0.988573 0.150746i $$-0.951832\pi$$
0.988573 0.150746i $$-0.0481676\pi$$
$$90$$ 24.5967i 0.273297i
$$91$$ − 93.9149i − 1.03203i
$$92$$ −78.0000 −0.847826
$$93$$ 240.000 2.58065
$$94$$ 26.8328i 0.285455i
$$95$$ 30.0000 0.315789
$$96$$ − 147.580i − 1.53730i
$$97$$ − 26.8328i − 0.276627i −0.990388 0.138313i $$-0.955832\pi$$
0.990388 0.138313i $$-0.0441681\pi$$
$$98$$ −49.0000 −0.500000
$$99$$ −22.0000 −0.222222
$$100$$ 15.0000 0.150000
$$101$$ 67.0820i 0.664179i 0.943248 + 0.332089i $$0.107754\pi$$
−0.943248 + 0.332089i $$0.892246\pi$$
$$102$$ 120.000 1.17647
$$103$$ 160.997i 1.56308i 0.623857 + 0.781538i $$0.285565\pi$$
−0.623857 + 0.781538i $$0.714435\pi$$
$$104$$ − 93.9149i − 0.903027i
$$105$$ −70.0000 −0.666667
$$106$$ 34.0000 0.320755
$$107$$ −106.000 −0.990654 −0.495327 0.868707i $$-0.664952\pi$$
−0.495327 + 0.868707i $$0.664952\pi$$
$$108$$ 26.8328i 0.248452i
$$109$$ −142.000 −1.30275 −0.651376 0.758755i $$-0.725808\pi$$
−0.651376 + 0.758755i $$0.725808\pi$$
$$110$$ − 4.47214i − 0.0406558i
$$111$$ 62.6099i 0.564053i
$$112$$ 35.0000 0.312500
$$113$$ −34.0000 −0.300885 −0.150442 0.988619i $$-0.548070\pi$$
−0.150442 + 0.988619i $$0.548070\pi$$
$$114$$ −60.0000 −0.526316
$$115$$ 58.1378i 0.505546i
$$116$$ 66.0000 0.568966
$$117$$ 147.580i 1.26137i
$$118$$ 40.2492i 0.341095i
$$119$$ 187.830i 1.57840i
$$120$$ −70.0000 −0.583333
$$121$$ −117.000 −0.966942
$$122$$ − 93.9149i − 0.769794i
$$123$$ 120.000 0.975610
$$124$$ 160.997i 1.29836i
$$125$$ − 11.1803i − 0.0894427i
$$126$$ 77.0000 0.611111
$$127$$ 194.000 1.52756 0.763780 0.645477i $$-0.223341\pi$$
0.763780 + 0.645477i $$0.223341\pi$$
$$128$$ 119.000 0.929688
$$129$$ − 152.053i − 1.17870i
$$130$$ −30.0000 −0.230769
$$131$$ − 120.748i − 0.921738i −0.887468 0.460869i $$-0.847538\pi$$
0.887468 0.460869i $$-0.152462\pi$$
$$132$$ − 26.8328i − 0.203279i
$$133$$ − 93.9149i − 0.706127i
$$134$$ −14.0000 −0.104478
$$135$$ 20.0000 0.148148
$$136$$ 187.830i 1.38110i
$$137$$ −166.000 −1.21168 −0.605839 0.795587i $$-0.707163\pi$$
−0.605839 + 0.795587i $$0.707163\pi$$
$$138$$ − 116.276i − 0.842576i
$$139$$ − 93.9149i − 0.675646i −0.941210 0.337823i $$-0.890309\pi$$
0.941210 0.337823i $$-0.109691\pi$$
$$140$$ − 46.9574i − 0.335410i
$$141$$ 120.000 0.851064
$$142$$ −62.0000 −0.436620
$$143$$ − 26.8328i − 0.187642i
$$144$$ −55.0000 −0.381944
$$145$$ − 49.1935i − 0.339265i
$$146$$ 53.6656i 0.367573i
$$147$$ 219.135i 1.49071i
$$148$$ −42.0000 −0.283784
$$149$$ −142.000 −0.953020 −0.476510 0.879169i $$-0.658098\pi$$
−0.476510 + 0.879169i $$0.658098\pi$$
$$150$$ 22.3607i 0.149071i
$$151$$ 2.00000 0.0132450 0.00662252 0.999978i $$-0.497892\pi$$
0.00662252 + 0.999978i $$0.497892\pi$$
$$152$$ − 93.9149i − 0.617861i
$$153$$ − 295.161i − 1.92916i
$$154$$ −14.0000 −0.0909091
$$155$$ 120.000 0.774194
$$156$$ −180.000 −1.15385
$$157$$ − 67.0820i − 0.427274i −0.976913 0.213637i $$-0.931469\pi$$
0.976913 0.213637i $$-0.0685310\pi$$
$$158$$ −38.0000 −0.240506
$$159$$ − 152.053i − 0.956306i
$$160$$ − 73.7902i − 0.461189i
$$161$$ 182.000 1.13043
$$162$$ 59.0000 0.364198
$$163$$ −34.0000 −0.208589 −0.104294 0.994546i $$-0.533258\pi$$
−0.104294 + 0.994546i $$0.533258\pi$$
$$164$$ 80.4984i 0.490844i
$$165$$ −20.0000 −0.121212
$$166$$ − 40.2492i − 0.242465i
$$167$$ − 107.331i − 0.642702i −0.946960 0.321351i $$-0.895863\pi$$
0.946960 0.321351i $$-0.104137\pi$$
$$168$$ 219.135i 1.30437i
$$169$$ −11.0000 −0.0650888
$$170$$ 60.0000 0.352941
$$171$$ 147.580i 0.863044i
$$172$$ 102.000 0.593023
$$173$$ 147.580i 0.853066i 0.904472 + 0.426533i $$0.140265\pi$$
−0.904472 + 0.426533i $$0.859735\pi$$
$$174$$ 98.3870i 0.565442i
$$175$$ −35.0000 −0.200000
$$176$$ 10.0000 0.0568182
$$177$$ 180.000 1.01695
$$178$$ 26.8328i 0.150746i
$$179$$ 218.000 1.21788 0.608939 0.793217i $$-0.291596\pi$$
0.608939 + 0.793217i $$0.291596\pi$$
$$180$$ 73.7902i 0.409946i
$$181$$ − 254.912i − 1.40835i −0.710025 0.704176i $$-0.751317\pi$$
0.710025 0.704176i $$-0.248683\pi$$
$$182$$ 93.9149i 0.516016i
$$183$$ −420.000 −2.29508
$$184$$ 182.000 0.989130
$$185$$ 31.3050i 0.169216i
$$186$$ −240.000 −1.29032
$$187$$ 53.6656i 0.286982i
$$188$$ 80.4984i 0.428183i
$$189$$ − 62.6099i − 0.331269i
$$190$$ −30.0000 −0.157895
$$191$$ −58.0000 −0.303665 −0.151832 0.988406i $$-0.548517\pi$$
−0.151832 + 0.988406i $$0.548517\pi$$
$$192$$ 58.1378i 0.302801i
$$193$$ 206.000 1.06736 0.533679 0.845687i $$-0.320809\pi$$
0.533679 + 0.845687i $$0.320809\pi$$
$$194$$ 26.8328i 0.138313i
$$195$$ 134.164i 0.688021i
$$196$$ −147.000 −0.750000
$$197$$ −226.000 −1.14721 −0.573604 0.819133i $$-0.694455\pi$$
−0.573604 + 0.819133i $$0.694455\pi$$
$$198$$ 22.0000 0.111111
$$199$$ 134.164i 0.674191i 0.941470 + 0.337096i $$0.109445\pi$$
−0.941470 + 0.337096i $$0.890555\pi$$
$$200$$ −35.0000 −0.175000
$$201$$ 62.6099i 0.311492i
$$202$$ − 67.0820i − 0.332089i
$$203$$ −154.000 −0.758621
$$204$$ 360.000 1.76471
$$205$$ 60.0000 0.292683
$$206$$ − 160.997i − 0.781538i
$$207$$ −286.000 −1.38164
$$208$$ − 67.0820i − 0.322510i
$$209$$ − 26.8328i − 0.128387i
$$210$$ 70.0000 0.333333
$$211$$ −118.000 −0.559242 −0.279621 0.960111i $$-0.590209\pi$$
−0.279621 + 0.960111i $$0.590209\pi$$
$$212$$ 102.000 0.481132
$$213$$ 277.272i 1.30175i
$$214$$ 106.000 0.495327
$$215$$ − 76.0263i − 0.353611i
$$216$$ − 62.6099i − 0.289861i
$$217$$ − 375.659i − 1.73115i
$$218$$ 142.000 0.651376
$$219$$ 240.000 1.09589
$$220$$ − 13.4164i − 0.0609837i
$$221$$ 360.000 1.62896
$$222$$ − 62.6099i − 0.282027i
$$223$$ 80.4984i 0.360980i 0.983577 + 0.180490i $$0.0577683\pi$$
−0.983577 + 0.180490i $$0.942232\pi$$
$$224$$ −231.000 −1.03125
$$225$$ 55.0000 0.244444
$$226$$ 34.0000 0.150442
$$227$$ 254.912i 1.12296i 0.827491 + 0.561480i $$0.189768\pi$$
−0.827491 + 0.561480i $$0.810232\pi$$
$$228$$ −180.000 −0.789474
$$229$$ 13.4164i 0.0585869i 0.999571 + 0.0292935i $$0.00932573\pi$$
−0.999571 + 0.0292935i $$0.990674\pi$$
$$230$$ − 58.1378i − 0.252773i
$$231$$ 62.6099i 0.271039i
$$232$$ −154.000 −0.663793
$$233$$ −214.000 −0.918455 −0.459227 0.888319i $$-0.651874\pi$$
−0.459227 + 0.888319i $$0.651874\pi$$
$$234$$ − 147.580i − 0.630686i
$$235$$ 60.0000 0.255319
$$236$$ 120.748i 0.511643i
$$237$$ 169.941i 0.717051i
$$238$$ − 187.830i − 0.789200i
$$239$$ 98.0000 0.410042 0.205021 0.978758i $$-0.434274\pi$$
0.205021 + 0.978758i $$0.434274\pi$$
$$240$$ −50.0000 −0.208333
$$241$$ − 160.997i − 0.668037i −0.942567 0.334018i $$-0.891595\pi$$
0.942567 0.334018i $$-0.108405\pi$$
$$242$$ 117.000 0.483471
$$243$$ − 344.354i − 1.41710i
$$244$$ − 281.745i − 1.15469i
$$245$$ 109.567i 0.447214i
$$246$$ −120.000 −0.487805
$$247$$ −180.000 −0.728745
$$248$$ − 375.659i − 1.51476i
$$249$$ −180.000 −0.722892
$$250$$ 11.1803i 0.0447214i
$$251$$ 335.410i 1.33630i 0.744029 + 0.668148i $$0.232913\pi$$
−0.744029 + 0.668148i $$0.767087\pi$$
$$252$$ 231.000 0.916667
$$253$$ 52.0000 0.205534
$$254$$ −194.000 −0.763780
$$255$$ − 268.328i − 1.05227i
$$256$$ −171.000 −0.667969
$$257$$ − 134.164i − 0.522039i −0.965333 0.261020i $$-0.915941\pi$$
0.965333 0.261020i $$-0.0840587\pi$$
$$258$$ 152.053i 0.589351i
$$259$$ 98.0000 0.378378
$$260$$ −90.0000 −0.346154
$$261$$ 242.000 0.927203
$$262$$ 120.748i 0.460869i
$$263$$ −34.0000 −0.129278 −0.0646388 0.997909i $$-0.520590\pi$$
−0.0646388 + 0.997909i $$0.520590\pi$$
$$264$$ 62.6099i 0.237159i
$$265$$ − 76.0263i − 0.286892i
$$266$$ 93.9149i 0.353063i
$$267$$ 120.000 0.449438
$$268$$ −42.0000 −0.156716
$$269$$ 254.912i 0.947627i 0.880625 + 0.473814i $$0.157123\pi$$
−0.880625 + 0.473814i $$0.842877\pi$$
$$270$$ −20.0000 −0.0740741
$$271$$ 321.994i 1.18817i 0.804403 + 0.594084i $$0.202486\pi$$
−0.804403 + 0.594084i $$0.797514\pi$$
$$272$$ 134.164i 0.493250i
$$273$$ 420.000 1.53846
$$274$$ 166.000 0.605839
$$275$$ −10.0000 −0.0363636
$$276$$ − 348.827i − 1.26386i
$$277$$ 14.0000 0.0505415 0.0252708 0.999681i $$-0.491955\pi$$
0.0252708 + 0.999681i $$0.491955\pi$$
$$278$$ 93.9149i 0.337823i
$$279$$ 590.322i 2.11585i
$$280$$ 109.567i 0.391312i
$$281$$ 2.00000 0.00711744 0.00355872 0.999994i $$-0.498867\pi$$
0.00355872 + 0.999994i $$0.498867\pi$$
$$282$$ −120.000 −0.425532
$$283$$ − 93.9149i − 0.331855i −0.986138 0.165927i $$-0.946938\pi$$
0.986138 0.165927i $$-0.0530617\pi$$
$$284$$ −186.000 −0.654930
$$285$$ 134.164i 0.470751i
$$286$$ 26.8328i 0.0938210i
$$287$$ − 187.830i − 0.654459i
$$288$$ 363.000 1.26042
$$289$$ −431.000 −1.49135
$$290$$ 49.1935i 0.169633i
$$291$$ 120.000 0.412371
$$292$$ 160.997i 0.551359i
$$293$$ 335.410i 1.14474i 0.819994 + 0.572372i $$0.193977\pi$$
−0.819994 + 0.572372i $$0.806023\pi$$
$$294$$ − 219.135i − 0.745356i
$$295$$ 90.0000 0.305085
$$296$$ 98.0000 0.331081
$$297$$ − 17.8885i − 0.0602308i
$$298$$ 142.000 0.476510
$$299$$ − 348.827i − 1.16664i
$$300$$ 67.0820i 0.223607i
$$301$$ −238.000 −0.790698
$$302$$ −2.00000 −0.00662252
$$303$$ −300.000 −0.990099
$$304$$ − 67.0820i − 0.220665i
$$305$$ −210.000 −0.688525
$$306$$ 295.161i 0.964578i
$$307$$ 201.246i 0.655525i 0.944760 + 0.327762i $$0.106295\pi$$
−0.944760 + 0.327762i $$0.893705\pi$$
$$308$$ −42.0000 −0.136364
$$309$$ −720.000 −2.33010
$$310$$ −120.000 −0.387097
$$311$$ − 509.823i − 1.63930i −0.572862 0.819652i $$-0.694167\pi$$
0.572862 0.819652i $$-0.305833\pi$$
$$312$$ 420.000 1.34615
$$313$$ 321.994i 1.02873i 0.857570 + 0.514367i $$0.171973\pi$$
−0.857570 + 0.514367i $$0.828027\pi$$
$$314$$ 67.0820i 0.213637i
$$315$$ − 172.177i − 0.546594i
$$316$$ −114.000 −0.360759
$$317$$ 374.000 1.17981 0.589905 0.807472i $$-0.299165\pi$$
0.589905 + 0.807472i $$0.299165\pi$$
$$318$$ 152.053i 0.478153i
$$319$$ −44.0000 −0.137931
$$320$$ 29.0689i 0.0908403i
$$321$$ − 474.046i − 1.47678i
$$322$$ −182.000 −0.565217
$$323$$ 360.000 1.11455
$$324$$ 177.000 0.546296
$$325$$ 67.0820i 0.206406i
$$326$$ 34.0000 0.104294
$$327$$ − 635.043i − 1.94203i
$$328$$ − 187.830i − 0.572652i
$$329$$ − 187.830i − 0.570911i
$$330$$ 20.0000 0.0606061
$$331$$ 482.000 1.45619 0.728097 0.685474i $$-0.240405\pi$$
0.728097 + 0.685474i $$0.240405\pi$$
$$332$$ − 120.748i − 0.363698i
$$333$$ −154.000 −0.462462
$$334$$ 107.331i 0.321351i
$$335$$ 31.3050i 0.0934476i
$$336$$ 156.525i 0.465847i
$$337$$ 494.000 1.46588 0.732938 0.680296i $$-0.238149\pi$$
0.732938 + 0.680296i $$0.238149\pi$$
$$338$$ 11.0000 0.0325444
$$339$$ − 152.053i − 0.448533i
$$340$$ 180.000 0.529412
$$341$$ − 107.331i − 0.314754i
$$342$$ − 147.580i − 0.431522i
$$343$$ 343.000 1.00000
$$344$$ −238.000 −0.691860
$$345$$ −260.000 −0.753623
$$346$$ − 147.580i − 0.426533i
$$347$$ −346.000 −0.997118 −0.498559 0.866856i $$-0.666137\pi$$
−0.498559 + 0.866856i $$0.666137\pi$$
$$348$$ 295.161i 0.848164i
$$349$$ − 335.410i − 0.961061i −0.876978 0.480530i $$-0.840444\pi$$
0.876978 0.480530i $$-0.159556\pi$$
$$350$$ 35.0000 0.100000
$$351$$ −120.000 −0.341880
$$352$$ −66.0000 −0.187500
$$353$$ 26.8328i 0.0760136i 0.999277 + 0.0380068i $$0.0121009\pi$$
−0.999277 + 0.0380068i $$0.987899\pi$$
$$354$$ −180.000 −0.508475
$$355$$ 138.636i 0.390525i
$$356$$ 80.4984i 0.226119i
$$357$$ −840.000 −2.35294
$$358$$ −218.000 −0.608939
$$359$$ 338.000 0.941504 0.470752 0.882266i $$-0.343983\pi$$
0.470752 + 0.882266i $$0.343983\pi$$
$$360$$ − 172.177i − 0.478270i
$$361$$ 181.000 0.501385
$$362$$ 254.912i 0.704176i
$$363$$ − 523.240i − 1.44143i
$$364$$ 281.745i 0.774024i
$$365$$ 120.000 0.328767
$$366$$ 420.000 1.14754
$$367$$ 295.161i 0.804253i 0.915584 + 0.402127i $$0.131729\pi$$
−0.915584 + 0.402127i $$0.868271\pi$$
$$368$$ 130.000 0.353261
$$369$$ 295.161i 0.799894i
$$370$$ − 31.3050i − 0.0846080i
$$371$$ −238.000 −0.641509
$$372$$ −720.000 −1.93548
$$373$$ 86.0000 0.230563 0.115282 0.993333i $$-0.463223\pi$$
0.115282 + 0.993333i $$0.463223\pi$$
$$374$$ − 53.6656i − 0.143491i
$$375$$ 50.0000 0.133333
$$376$$ − 187.830i − 0.499547i
$$377$$ 295.161i 0.782920i
$$378$$ 62.6099i 0.165635i
$$379$$ −262.000 −0.691293 −0.345646 0.938365i $$-0.612340\pi$$
−0.345646 + 0.938365i $$0.612340\pi$$
$$380$$ −90.0000 −0.236842
$$381$$ 867.594i 2.27715i
$$382$$ 58.0000 0.151832
$$383$$ − 563.489i − 1.47125i −0.677388 0.735625i $$-0.736888\pi$$
0.677388 0.735625i $$-0.263112\pi$$
$$384$$ 532.184i 1.38590i
$$385$$ 31.3050i 0.0813116i
$$386$$ −206.000 −0.533679
$$387$$ 374.000 0.966408
$$388$$ 80.4984i 0.207470i
$$389$$ 698.000 1.79434 0.897172 0.441681i $$-0.145618\pi$$
0.897172 + 0.441681i $$0.145618\pi$$
$$390$$ − 134.164i − 0.344010i
$$391$$ 697.653i 1.78428i
$$392$$ 343.000 0.875000
$$393$$ 540.000 1.37405
$$394$$ 226.000 0.573604
$$395$$ 84.9706i 0.215115i
$$396$$ 66.0000 0.166667
$$397$$ 308.577i 0.777273i 0.921391 + 0.388636i $$0.127054\pi$$
−0.921391 + 0.388636i $$0.872946\pi$$
$$398$$ − 134.164i − 0.337096i
$$399$$ 420.000 1.05263
$$400$$ −25.0000 −0.0625000
$$401$$ −538.000 −1.34165 −0.670823 0.741618i $$-0.734059\pi$$
−0.670823 + 0.741618i $$0.734059\pi$$
$$402$$ − 62.6099i − 0.155746i
$$403$$ −720.000 −1.78660
$$404$$ − 201.246i − 0.498134i
$$405$$ − 131.928i − 0.325748i
$$406$$ 154.000 0.379310
$$407$$ 28.0000 0.0687961
$$408$$ −840.000 −2.05882
$$409$$ 295.161i 0.721665i 0.932631 + 0.360832i $$0.117507\pi$$
−0.932631 + 0.360832i $$0.882493\pi$$
$$410$$ −60.0000 −0.146341
$$411$$ − 742.375i − 1.80626i
$$412$$ − 482.991i − 1.17231i
$$413$$ − 281.745i − 0.682190i
$$414$$ 286.000 0.690821
$$415$$ −90.0000 −0.216867
$$416$$ 442.741i 1.06428i
$$417$$ 420.000 1.00719
$$418$$ 26.8328i 0.0641933i
$$419$$ − 818.401i − 1.95322i −0.215009 0.976612i $$-0.568978\pi$$
0.215009 0.976612i $$-0.431022\pi$$
$$420$$ 210.000 0.500000
$$421$$ −118.000 −0.280285 −0.140143 0.990131i $$-0.544756\pi$$
−0.140143 + 0.990131i $$0.544756\pi$$
$$422$$ 118.000 0.279621
$$423$$ 295.161i 0.697780i
$$424$$ −238.000 −0.561321
$$425$$ − 134.164i − 0.315680i
$$426$$ − 277.272i − 0.650874i
$$427$$ 657.404i 1.53959i
$$428$$ 318.000 0.742991
$$429$$ 120.000 0.279720
$$430$$ 76.0263i 0.176805i
$$431$$ −718.000 −1.66589 −0.832947 0.553353i $$-0.813348\pi$$
−0.832947 + 0.553353i $$0.813348\pi$$
$$432$$ − 44.7214i − 0.103522i
$$433$$ 509.823i 1.17742i 0.808344 + 0.588711i $$0.200364\pi$$
−0.808344 + 0.588711i $$0.799636\pi$$
$$434$$ 375.659i 0.865575i
$$435$$ 220.000 0.505747
$$436$$ 426.000 0.977064
$$437$$ − 348.827i − 0.798230i
$$438$$ −240.000 −0.547945
$$439$$ − 26.8328i − 0.0611226i −0.999533 0.0305613i $$-0.990271\pi$$
0.999533 0.0305613i $$-0.00972948\pi$$
$$440$$ 31.3050i 0.0711476i
$$441$$ −539.000 −1.22222
$$442$$ −360.000 −0.814480
$$443$$ −634.000 −1.43115 −0.715576 0.698535i $$-0.753836\pi$$
−0.715576 + 0.698535i $$0.753836\pi$$
$$444$$ − 187.830i − 0.423040i
$$445$$ 60.0000 0.134831
$$446$$ − 80.4984i − 0.180490i
$$447$$ − 635.043i − 1.42068i
$$448$$ 91.0000 0.203125
$$449$$ 338.000 0.752784 0.376392 0.926461i $$-0.377165\pi$$
0.376392 + 0.926461i $$0.377165\pi$$
$$450$$ −55.0000 −0.122222
$$451$$ − 53.6656i − 0.118993i
$$452$$ 102.000 0.225664
$$453$$ 8.94427i 0.0197445i
$$454$$ − 254.912i − 0.561480i
$$455$$ 210.000 0.461538
$$456$$ 420.000 0.921053
$$457$$ −466.000 −1.01969 −0.509847 0.860265i $$-0.670298\pi$$
−0.509847 + 0.860265i $$0.670298\pi$$
$$458$$ − 13.4164i − 0.0292935i
$$459$$ 240.000 0.522876
$$460$$ − 174.413i − 0.379159i
$$461$$ − 442.741i − 0.960394i −0.877161 0.480197i $$-0.840565\pi$$
0.877161 0.480197i $$-0.159435\pi$$
$$462$$ − 62.6099i − 0.135519i
$$463$$ 206.000 0.444924 0.222462 0.974941i $$-0.428591\pi$$
0.222462 + 0.974941i $$0.428591\pi$$
$$464$$ −110.000 −0.237069
$$465$$ 536.656i 1.15410i
$$466$$ 214.000 0.459227
$$467$$ 362.243i 0.775681i 0.921727 + 0.387840i $$0.126779\pi$$
−0.921727 + 0.387840i $$0.873221\pi$$
$$468$$ − 442.741i − 0.946029i
$$469$$ 98.0000 0.208955
$$470$$ −60.0000 −0.127660
$$471$$ 300.000 0.636943
$$472$$ − 281.745i − 0.596916i
$$473$$ −68.0000 −0.143763
$$474$$ − 169.941i − 0.358526i
$$475$$ 67.0820i 0.141225i
$$476$$ − 563.489i − 1.18380i
$$477$$ 374.000 0.784067
$$478$$ −98.0000 −0.205021
$$479$$ 214.663i 0.448147i 0.974572 + 0.224074i $$0.0719356\pi$$
−0.974572 + 0.224074i $$0.928064\pi$$
$$480$$ 330.000 0.687500
$$481$$ − 187.830i − 0.390498i
$$482$$ 160.997i 0.334018i
$$483$$ 813.929i 1.68515i
$$484$$ 351.000 0.725207
$$485$$ 60.0000 0.123711
$$486$$ 344.354i 0.708548i
$$487$$ −166.000 −0.340862 −0.170431 0.985370i $$-0.554516\pi$$
−0.170431 + 0.985370i $$0.554516\pi$$
$$488$$ 657.404i 1.34714i
$$489$$ − 152.053i − 0.310946i
$$490$$ − 109.567i − 0.223607i
$$491$$ −838.000 −1.70672 −0.853360 0.521321i $$-0.825439\pi$$
−0.853360 + 0.521321i $$0.825439\pi$$
$$492$$ −360.000 −0.731707
$$493$$ − 590.322i − 1.19741i
$$494$$ 180.000 0.364372
$$495$$ − 49.1935i − 0.0993808i
$$496$$ − 268.328i − 0.540984i
$$497$$ 434.000 0.873239
$$498$$ 180.000 0.361446
$$499$$ −262.000 −0.525050 −0.262525 0.964925i $$-0.584555\pi$$
−0.262525 + 0.964925i $$0.584555\pi$$
$$500$$ 33.5410i 0.0670820i
$$501$$ 480.000 0.958084
$$502$$ − 335.410i − 0.668148i
$$503$$ 429.325i 0.853529i 0.904363 + 0.426764i $$0.140347\pi$$
−0.904363 + 0.426764i $$0.859653\pi$$
$$504$$ −539.000 −1.06944
$$505$$ −150.000 −0.297030
$$506$$ −52.0000 −0.102767
$$507$$ − 49.1935i − 0.0970286i
$$508$$ −582.000 −1.14567
$$509$$ 898.899i 1.76601i 0.469363 + 0.883005i $$0.344484\pi$$
−0.469363 + 0.883005i $$0.655516\pi$$
$$510$$ 268.328i 0.526134i
$$511$$ − 375.659i − 0.735146i
$$512$$ −305.000 −0.595703
$$513$$ −120.000 −0.233918
$$514$$ 134.164i 0.261020i
$$515$$ −360.000 −0.699029
$$516$$ 456.158i 0.884027i
$$517$$ − 53.6656i − 0.103802i
$$518$$ −98.0000 −0.189189
$$519$$ −660.000 −1.27168
$$520$$ 210.000 0.403846
$$521$$ 724.486i 1.39057i 0.718735 + 0.695284i $$0.244721\pi$$
−0.718735 + 0.695284i $$0.755279\pi$$
$$522$$ −242.000 −0.463602
$$523$$ − 523.240i − 1.00046i −0.865893 0.500229i $$-0.833249\pi$$
0.865893 0.500229i $$-0.166751\pi$$
$$524$$ 362.243i 0.691303i
$$525$$ − 156.525i − 0.298142i
$$526$$ 34.0000 0.0646388
$$527$$ 1440.00 2.73245
$$528$$ 44.7214i 0.0846995i
$$529$$ 147.000 0.277883
$$530$$ 76.0263i 0.143446i
$$531$$ 442.741i 0.833788i
$$532$$ 281.745i 0.529595i
$$533$$ −360.000 −0.675422
$$534$$ −120.000 −0.224719
$$535$$ − 237.023i − 0.443034i
$$536$$ 98.0000 0.182836
$$537$$ 974.926i 1.81550i
$$538$$ − 254.912i − 0.473814i
$$539$$ 98.0000 0.181818
$$540$$ −60.0000 −0.111111
$$541$$ 842.000 1.55638 0.778189 0.628031i $$-0.216139\pi$$
0.778189 + 0.628031i $$0.216139\pi$$
$$542$$ − 321.994i − 0.594084i
$$543$$ 1140.00 2.09945
$$544$$ − 885.483i − 1.62773i
$$545$$ − 317.522i − 0.582609i
$$546$$ −420.000 −0.769231
$$547$$ 134.000 0.244973 0.122486 0.992470i $$-0.460913\pi$$
0.122486 + 0.992470i $$0.460913\pi$$
$$548$$ 498.000 0.908759
$$549$$ − 1033.06i − 1.88172i
$$550$$ 10.0000 0.0181818
$$551$$ 295.161i 0.535682i
$$552$$ 813.929i 1.47451i
$$553$$ 266.000 0.481013
$$554$$ −14.0000 −0.0252708
$$555$$ −140.000 −0.252252
$$556$$ 281.745i 0.506735i
$$557$$ −706.000 −1.26750 −0.633752 0.773536i $$-0.718486\pi$$
−0.633752 + 0.773536i $$0.718486\pi$$
$$558$$ − 590.322i − 1.05792i
$$559$$ 456.158i 0.816025i
$$560$$ 78.2624i 0.139754i
$$561$$ −240.000 −0.427807
$$562$$ −2.00000 −0.00355872
$$563$$ − 13.4164i − 0.0238302i −0.999929 0.0119151i $$-0.996207\pi$$
0.999929 0.0119151i $$-0.00379279\pi$$
$$564$$ −360.000 −0.638298
$$565$$ − 76.0263i − 0.134560i
$$566$$ 93.9149i 0.165927i
$$567$$ −413.000 −0.728395
$$568$$ 434.000 0.764085
$$569$$ −82.0000 −0.144112 −0.0720562 0.997401i $$-0.522956\pi$$
−0.0720562 + 0.997401i $$0.522956\pi$$
$$570$$ − 134.164i − 0.235376i
$$571$$ −118.000 −0.206655 −0.103327 0.994647i $$-0.532949\pi$$
−0.103327 + 0.994647i $$0.532949\pi$$
$$572$$ 80.4984i 0.140732i
$$573$$ − 259.384i − 0.452677i
$$574$$ 187.830i 0.327229i
$$575$$ −130.000 −0.226087
$$576$$ −143.000 −0.248264
$$577$$ − 885.483i − 1.53463i −0.641269 0.767316i $$-0.721592\pi$$
0.641269 0.767316i $$-0.278408\pi$$
$$578$$ 431.000 0.745675
$$579$$ 921.260i 1.59112i
$$580$$ 147.580i 0.254449i
$$581$$ 281.745i 0.484930i
$$582$$ −120.000 −0.206186
$$583$$ −68.0000 −0.116638
$$584$$ − 375.659i − 0.643252i
$$585$$ −330.000 −0.564103
$$586$$ − 335.410i − 0.572372i
$$587$$ − 791.568i − 1.34850i −0.738504 0.674249i $$-0.764468\pi$$
0.738504 0.674249i $$-0.235532\pi$$
$$588$$ − 657.404i − 1.11803i
$$589$$ −720.000 −1.22241
$$590$$ −90.0000 −0.152542
$$591$$ − 1010.70i − 1.71016i
$$592$$ 70.0000 0.118243
$$593$$ − 134.164i − 0.226246i −0.993581 0.113123i $$-0.963915\pi$$
0.993581 0.113123i $$-0.0360855\pi$$
$$594$$ 17.8885i 0.0301154i
$$595$$ −420.000 −0.705882
$$596$$ 426.000 0.714765
$$597$$ −600.000 −1.00503
$$598$$ 348.827i 0.583322i
$$599$$ 398.000 0.664441 0.332220 0.943202i $$-0.392202\pi$$
0.332220 + 0.943202i $$0.392202\pi$$
$$600$$ − 156.525i − 0.260875i
$$601$$ 134.164i 0.223235i 0.993751 + 0.111617i $$0.0356031\pi$$
−0.993751 + 0.111617i $$0.964397\pi$$
$$602$$ 238.000 0.395349
$$603$$ −154.000 −0.255390
$$604$$ −6.00000 −0.00993377
$$605$$ − 261.620i − 0.432430i
$$606$$ 300.000 0.495050
$$607$$ − 939.149i − 1.54720i −0.633676 0.773598i $$-0.718455\pi$$
0.633676 0.773598i $$-0.281545\pi$$
$$608$$ 442.741i 0.728193i
$$609$$ − 688.709i − 1.13088i
$$610$$ 210.000 0.344262
$$611$$ −360.000 −0.589198
$$612$$ 885.483i 1.44687i
$$613$$ 206.000 0.336052 0.168026 0.985783i $$-0.446261\pi$$
0.168026 + 0.985783i $$0.446261\pi$$
$$614$$ − 201.246i − 0.327762i
$$615$$ 268.328i 0.436306i
$$616$$ 98.0000 0.159091
$$617$$ 494.000 0.800648 0.400324 0.916374i $$-0.368898\pi$$
0.400324 + 0.916374i $$0.368898\pi$$
$$618$$ 720.000 1.16505
$$619$$ 120.748i 0.195069i 0.995232 + 0.0975345i $$0.0310956\pi$$
−0.995232 + 0.0975345i $$0.968904\pi$$
$$620$$ −360.000 −0.580645
$$621$$ − 232.551i − 0.374478i
$$622$$ 509.823i 0.819652i
$$623$$ − 187.830i − 0.301492i
$$624$$ 300.000 0.480769
$$625$$ 25.0000 0.0400000
$$626$$ − 321.994i − 0.514367i
$$627$$ 120.000 0.191388
$$628$$ 201.246i 0.320456i
$$629$$ 375.659i 0.597233i
$$630$$ 172.177i 0.273297i
$$631$$ 542.000 0.858954 0.429477 0.903078i $$-0.358698\pi$$
0.429477 + 0.903078i $$0.358698\pi$$
$$632$$ 266.000 0.420886
$$633$$ − 527.712i − 0.833668i
$$634$$ −374.000 −0.589905
$$635$$ 433.797i 0.683145i
$$636$$ 456.158i 0.717229i
$$637$$ − 657.404i − 1.03203i
$$638$$ 44.0000 0.0689655
$$639$$ −682.000 −1.06729
$$640$$ 266.092i 0.415769i
$$641$$ −298.000 −0.464899 −0.232449 0.972609i $$-0.574674\pi$$
−0.232449 + 0.972609i $$0.574674\pi$$
$$642$$ 474.046i 0.738390i
$$643$$ 1006.23i 1.56490i 0.622714 + 0.782450i $$0.286030\pi$$
−0.622714 + 0.782450i $$0.713970\pi$$
$$644$$ −546.000 −0.847826
$$645$$ 340.000 0.527132
$$646$$ −360.000 −0.557276
$$647$$ 643.988i 0.995344i 0.867365 + 0.497672i $$0.165812\pi$$
−0.867365 + 0.497672i $$0.834188\pi$$
$$648$$ −413.000 −0.637346
$$649$$ − 80.4984i − 0.124035i
$$650$$ − 67.0820i − 0.103203i
$$651$$ 1680.00 2.58065
$$652$$ 102.000 0.156442
$$653$$ −154.000 −0.235835 −0.117917 0.993023i $$-0.537622\pi$$
−0.117917 + 0.993023i $$0.537622\pi$$
$$654$$ 635.043i 0.971014i
$$655$$ 270.000 0.412214
$$656$$ − 134.164i − 0.204518i
$$657$$ 590.322i 0.898511i
$$658$$ 187.830i 0.285455i
$$659$$ 338.000 0.512898 0.256449 0.966558i $$-0.417447\pi$$
0.256449 + 0.966558i $$0.417447\pi$$
$$660$$ 60.0000 0.0909091
$$661$$ − 576.906i − 0.872777i −0.899758 0.436388i $$-0.856257\pi$$
0.899758 0.436388i $$-0.143743\pi$$
$$662$$ −482.000 −0.728097
$$663$$ 1609.97i 2.42831i
$$664$$ 281.745i 0.424314i
$$665$$ 210.000 0.315789
$$666$$ 154.000 0.231231
$$667$$ −572.000 −0.857571
$$668$$ 321.994i 0.482027i
$$669$$ −360.000 −0.538117
$$670$$ − 31.3050i − 0.0467238i
$$671$$ 187.830i 0.279925i
$$672$$ − 1033.06i − 1.53730i
$$673$$ −814.000 −1.20951 −0.604755 0.796412i $$-0.706729\pi$$
−0.604755 + 0.796412i $$0.706729\pi$$
$$674$$ −494.000 −0.732938
$$675$$ 44.7214i 0.0662539i
$$676$$ 33.0000 0.0488166
$$677$$ − 684.237i − 1.01069i −0.862918 0.505345i $$-0.831365\pi$$
0.862918 0.505345i $$-0.168635\pi$$
$$678$$ 152.053i 0.224266i
$$679$$ − 187.830i − 0.276627i
$$680$$ −420.000 −0.617647
$$681$$ −1140.00 −1.67401
$$682$$ 107.331i 0.157377i
$$683$$ 926.000 1.35578 0.677892 0.735162i $$-0.262894\pi$$
0.677892 + 0.735162i $$0.262894\pi$$
$$684$$ − 442.741i − 0.647283i
$$685$$ − 371.187i − 0.541879i
$$686$$ −343.000 −0.500000
$$687$$ −60.0000 −0.0873362
$$688$$ −170.000 −0.247093
$$689$$ 456.158i 0.662058i
$$690$$ 260.000 0.376812
$$691$$ 576.906i 0.834885i 0.908703 + 0.417443i $$0.137073\pi$$
−0.908703 + 0.417443i $$0.862927\pi$$
$$692$$ − 442.741i − 0.639800i
$$693$$ −154.000 −0.222222
$$694$$ 346.000 0.498559
$$695$$ 210.000 0.302158
$$696$$ − 688.709i − 0.989524i
$$697$$ 720.000 1.03300
$$698$$ 335.410i 0.480530i
$$699$$ − 957.037i − 1.36915i
$$700$$ 105.000 0.150000
$$701$$ 362.000 0.516405 0.258203 0.966091i $$-0.416870\pi$$
0.258203 + 0.966091i $$0.416870\pi$$
$$702$$ 120.000 0.170940
$$703$$ − 187.830i − 0.267183i
$$704$$ 26.0000 0.0369318
$$705$$ 268.328i 0.380607i
$$706$$ − 26.8328i − 0.0380068i
$$707$$ 469.574i 0.664179i
$$708$$ −540.000 −0.762712
$$709$$ 1058.00 1.49224 0.746121 0.665810i $$-0.231914\pi$$
0.746121 + 0.665810i $$0.231914\pi$$
$$710$$ − 138.636i − 0.195262i
$$711$$ −418.000 −0.587904
$$712$$ − 187.830i − 0.263806i
$$713$$ − 1395.31i − 1.95695i
$$714$$ 840.000 1.17647
$$715$$ 60.0000 0.0839161
$$716$$ −654.000 −0.913408
$$717$$ 438.269i 0.611254i
$$718$$ −338.000 −0.470752
$$719$$ − 482.991i − 0.671753i −0.941906 0.335877i $$-0.890967\pi$$
0.941906 0.335877i $$-0.109033\pi$$
$$720$$ − 122.984i − 0.170811i
$$721$$ 1126.98i 1.56308i
$$722$$ −181.000 −0.250693
$$723$$ 720.000 0.995851
$$724$$ 764.735i 1.05626i
$$725$$ 110.000 0.151724
$$726$$ 523.240i 0.720716i
$$727$$ 1126.98i 1.55018i 0.631853 + 0.775088i $$0.282295\pi$$
−0.631853 + 0.775088i $$0.717705\pi$$
$$728$$ − 657.404i − 0.903027i
$$729$$ 1009.00 1.38409
$$730$$ −120.000 −0.164384
$$731$$ − 912.316i − 1.24804i
$$732$$ 1260.00 1.72131
$$733$$ − 1301.39i − 1.77543i −0.460392 0.887716i $$-0.652291\pi$$
0.460392 0.887716i $$-0.347709\pi$$
$$734$$ − 295.161i − 0.402127i
$$735$$ −490.000 −0.666667
$$736$$ −858.000 −1.16576
$$737$$ 28.0000 0.0379919
$$738$$ − 295.161i − 0.399947i
$$739$$ −982.000 −1.32882 −0.664411 0.747367i $$-0.731318\pi$$
−0.664411 + 0.747367i $$0.731318\pi$$
$$740$$ − 93.9149i − 0.126912i
$$741$$ − 804.984i − 1.08635i
$$742$$ 238.000 0.320755
$$743$$ −694.000 −0.934051 −0.467026 0.884244i $$-0.654674\pi$$
−0.467026 + 0.884244i $$0.654674\pi$$
$$744$$ 1680.00 2.25806
$$745$$ − 317.522i − 0.426204i
$$746$$ −86.0000 −0.115282
$$747$$ − 442.741i − 0.592693i
$$748$$ − 160.997i − 0.215236i
$$749$$ −742.000 −0.990654
$$750$$ −50.0000 −0.0666667
$$751$$ 242.000 0.322237 0.161119 0.986935i $$-0.448490\pi$$
0.161119 + 0.986935i $$0.448490\pi$$
$$752$$ − 134.164i − 0.178410i
$$753$$ −1500.00 −1.99203
$$754$$ − 295.161i − 0.391460i
$$755$$ 4.47214i 0.00592336i
$$756$$ 187.830i 0.248452i
$$757$$ −106.000 −0.140026 −0.0700132 0.997546i $$-0.522304\pi$$
−0.0700132 + 0.997546i $$0.522304\pi$$
$$758$$ 262.000 0.345646
$$759$$ 232.551i 0.306391i
$$760$$ 210.000 0.276316
$$761$$ 1100.15i 1.44566i 0.691027 + 0.722829i $$0.257158\pi$$
−0.691027 + 0.722829i $$0.742842\pi$$
$$762$$ − 867.594i − 1.13858i
$$763$$ −994.000 −1.30275
$$764$$ 174.000 0.227749
$$765$$ 660.000 0.862745
$$766$$ 563.489i 0.735625i
$$767$$ −540.000 −0.704042
$$768$$ − 764.735i − 0.995749i
$$769$$ 1126.98i 1.46551i 0.680492 + 0.732756i $$0.261766\pi$$
−0.680492 + 0.732756i $$0.738234\pi$$
$$770$$ − 31.3050i − 0.0406558i
$$771$$ 600.000 0.778210
$$772$$ −618.000 −0.800518
$$773$$ 818.401i 1.05873i 0.848393 + 0.529367i $$0.177570\pi$$
−0.848393 + 0.529367i $$0.822430\pi$$
$$774$$ −374.000 −0.483204
$$775$$ 268.328i 0.346230i
$$776$$ − 187.830i − 0.242049i
$$777$$ 438.269i 0.564053i
$$778$$ −698.000 −0.897172
$$779$$ −360.000 −0.462131
$$780$$ − 402.492i − 0.516016i
$$781$$ 124.000 0.158771
$$782$$ − 697.653i − 0.892140i
$$783$$ 196.774i 0.251308i
$$784$$ 245.000 0.312500
$$785$$ 150.000 0.191083
$$786$$ −540.000 −0.687023
$$787$$ 684.237i 0.869424i 0.900569 + 0.434712i $$0.143150\pi$$
−0.900569 + 0.434712i $$0.856850\pi$$
$$788$$ 678.000 0.860406
$$789$$ − 152.053i − 0.192716i
$$790$$ − 84.9706i − 0.107558i
$$791$$ −238.000 −0.300885
$$792$$ −154.000 −0.194444
$$793$$ 1260.00 1.58890
$$794$$ − 308.577i − 0.388636i
$$795$$ 340.000 0.427673
$$796$$ − 402.492i − 0.505644i
$$797$$ 308.577i 0.387174i 0.981083 + 0.193587i $$0.0620121\pi$$
−0.981083 + 0.193587i $$0.937988\pi$$
$$798$$ −420.000 −0.526316
$$799$$ 720.000 0.901126
$$800$$ 165.000 0.206250
$$801$$ 295.161i 0.368491i
$$802$$ 538.000 0.670823
$$803$$ − 107.331i − 0.133663i
$$804$$ − 187.830i − 0.233619i
$$805$$ 406.964i 0.505546i
$$806$$ 720.000 0.893300
$$807$$ −1140.00 −1.41264
$$808$$ 469.574i 0.581156i
$$809$$ 1358.00 1.67862 0.839308 0.543657i $$-0.182961\pi$$
0.839308 + 0.543657i $$0.182961\pi$$
$$810$$ 131.928i 0.162874i
$$811$$ − 308.577i − 0.380490i −0.981737 0.190245i $$-0.939072\pi$$
0.981737 0.190245i $$-0.0609282\pi$$
$$812$$ 462.000 0.568966
$$813$$ −1440.00 −1.77122
$$814$$ −28.0000 −0.0343980
$$815$$ − 76.0263i − 0.0932838i
$$816$$ −600.000 −0.735294
$$817$$ 456.158i 0.558333i
$$818$$ − 295.161i − 0.360832i
$$819$$ 1033.06i 1.26137i
$$820$$ −180.000 −0.219512
$$821$$ 482.000 0.587089 0.293544 0.955945i $$-0.405165\pi$$
0.293544 + 0.955945i $$0.405165\pi$$
$$822$$ 742.375i 0.903132i
$$823$$ 926.000 1.12515 0.562576 0.826746i $$-0.309810\pi$$
0.562576 + 0.826746i $$0.309810\pi$$
$$824$$ 1126.98i 1.36769i
$$825$$ − 44.7214i − 0.0542077i
$$826$$ 281.745i 0.341095i
$$827$$ −226.000 −0.273277 −0.136638 0.990621i $$-0.543630\pi$$
−0.136638 + 0.990621i $$0.543630\pi$$
$$828$$ 858.000 1.03623
$$829$$ − 1462.39i − 1.76404i −0.471213 0.882020i $$-0.656184\pi$$
0.471213 0.882020i $$-0.343816\pi$$
$$830$$ 90.0000 0.108434
$$831$$ 62.6099i 0.0753428i
$$832$$ − 174.413i − 0.209631i
$$833$$ 1314.81i 1.57840i
$$834$$ −420.000 −0.503597
$$835$$ 240.000 0.287425
$$836$$ 80.4984i 0.0962900i
$$837$$ −480.000 −0.573477
$$838$$ 818.401i 0.976612i
$$839$$ − 831.817i − 0.991439i −0.868483 0.495719i $$-0.834904\pi$$
0.868483 0.495719i $$-0.165096\pi$$
$$840$$ −490.000 −0.583333
$$841$$ −357.000 −0.424495
$$842$$ 118.000 0.140143
$$843$$ 8.94427i 0.0106100i
$$844$$ 354.000 0.419431
$$845$$ − 24.5967i − 0.0291086i
$$846$$ − 295.161i − 0.348890i
$$847$$ −819.000 −0.966942
$$848$$ −170.000 −0.200472
$$849$$ 420.000 0.494700
$$850$$ 134.164i 0.157840i
$$851$$ 364.000 0.427732
$$852$$ − 831.817i − 0.976311i
$$853$$ − 40.2492i − 0.0471855i −0.999722 0.0235927i $$-0.992489\pi$$
0.999722 0.0235927i $$-0.00751050\pi$$
$$854$$ − 657.404i − 0.769794i
$$855$$ −330.000 −0.385965
$$856$$ −742.000 −0.866822
$$857$$ 268.328i 0.313102i 0.987670 + 0.156551i $$0.0500375\pi$$
−0.987670 + 0.156551i $$0.949962\pi$$
$$858$$ −120.000 −0.139860
$$859$$ − 308.577i − 0.359229i −0.983737 0.179614i $$-0.942515\pi$$
0.983737 0.179614i $$-0.0574850\pi$$
$$860$$ 228.079i 0.265208i
$$861$$ 840.000 0.975610
$$862$$ 718.000 0.832947
$$863$$ −514.000 −0.595597 −0.297798 0.954629i $$-0.596252\pi$$
−0.297798 + 0.954629i $$0.596252\pi$$
$$864$$ 295.161i 0.341621i
$$865$$ −330.000 −0.381503
$$866$$ − 509.823i − 0.588711i
$$867$$ − 1927.49i − 2.22317i
$$868$$ 1126.98i 1.29836i
$$869$$ 76.0000 0.0874568
$$870$$ −220.000 −0.252874
$$871$$ − 187.830i − 0.215648i
$$872$$ −994.000 −1.13991
$$873$$ 295.161i 0.338100i
$$874$$ 348.827i 0.399115i
$$875$$ − 78.2624i − 0.0894427i
$$876$$ −720.000 −0.821918
$$877$$ −1306.00 −1.48917 −0.744584 0.667529i $$-0.767352\pi$$
−0.744584 + 0.667529i $$0.767352\pi$$
$$878$$ 26.8328i 0.0305613i
$$879$$ −1500.00 −1.70648
$$880$$ 22.3607i 0.0254099i
$$881$$ − 1126.98i − 1.27920i −0.768706 0.639602i $$-0.779099\pi$$
0.768706 0.639602i $$-0.220901\pi$$
$$882$$ 539.000 0.611111
$$883$$ 1526.00 1.72820 0.864100 0.503321i $$-0.167889\pi$$
0.864100 + 0.503321i $$0.167889\pi$$
$$884$$ −1080.00 −1.22172
$$885$$ 402.492i 0.454793i
$$886$$ 634.000 0.715576
$$887$$ 1556.30i 1.75457i 0.479970 + 0.877285i $$0.340648\pi$$
−0.479970 + 0.877285i $$0.659352\pi$$
$$888$$ 438.269i 0.493547i
$$889$$ 1358.00 1.52756
$$890$$ −60.0000 −0.0674157
$$891$$ −118.000 −0.132435
$$892$$ − 241.495i − 0.270735i
$$893$$ −360.000 −0.403135
$$894$$ 635.043i 0.710339i
$$895$$ 487.463i 0.544651i
$$896$$ 833.000 0.929688
$$897$$ 1560.00 1.73913
$$898$$ −338.000 −0.376392
$$899$$ 1180.64i 1.31329i
$$900$$ −165.000 −0.183333
$$901$$ − 912.316i − 1.01256i
$$902$$ 53.6656i 0.0594963i
$$903$$ − 1064.37i − 1.17870i
$$904$$ −238.000 −0.263274
$$905$$ 570.000 0.629834
$$906$$ − 8.94427i − 0.00987226i
$$907$$ 734.000 0.809261 0.404631 0.914480i $$-0.367400\pi$$
0.404631 + 0.914480i $$0.367400\pi$$
$$908$$ − 764.735i − 0.842219i
$$909$$ − 737.902i − 0.811774i
$$910$$ −210.000 −0.230769
$$911$$ 1202.00 1.31943 0.659715 0.751516i $$-0.270677\pi$$
0.659715 + 0.751516i $$0.270677\pi$$
$$912$$ 300.000 0.328947
$$913$$ 80.4984i 0.0881692i
$$914$$ 466.000 0.509847
$$915$$ − 939.149i − 1.02639i
$$916$$ − 40.2492i − 0.0439402i
$$917$$ − 845.234i − 0.921738i
$$918$$ −240.000 −0.261438
$$919$$ −1282.00 −1.39499 −0.697497 0.716587i $$-0.745703\pi$$
−0.697497 + 0.716587i $$0.745703\pi$$
$$920$$ 406.964i 0.442353i
$$921$$ −900.000 −0.977199
$$922$$ 442.741i 0.480197i
$$923$$ − 831.817i − 0.901210i
$$924$$ − 187.830i − 0.203279i
$$925$$ −70.0000 −0.0756757
$$926$$ −206.000 −0.222462
$$927$$ − 1770.97i − 1.91043i
$$928$$ 726.000 0.782328
$$929$$ − 1126.98i − 1.21311i −0.795042 0.606554i $$-0.792551\pi$$
0.795042 0.606554i $$-0.207449\pi$$
$$930$$ − 536.656i − 0.577050i
$$931$$ − 657.404i − 0.706127i
$$932$$ 642.000 0.688841
$$933$$ 2280.00 2.44373
$$934$$ − 362.243i − 0.387840i
$$935$$ −120.000 −0.128342
$$936$$ 1033.06i 1.10370i
$$937$$ 214.663i 0.229096i 0.993418 + 0.114548i $$0.0365419\pi$$
−0.993418 + 0.114548i $$0.963458\pi$$
$$938$$ −98.0000 −0.104478
$$939$$ −1440.00 −1.53355
$$940$$ −180.000 −0.191489
$$941$$ 845.234i 0.898229i 0.893474 + 0.449115i $$0.148261\pi$$
−0.893474 + 0.449115i $$0.851739\pi$$
$$942$$ −300.000 −0.318471
$$943$$ − 697.653i − 0.739823i
$$944$$ − 201.246i − 0.213184i
$$945$$ 140.000 0.148148
$$946$$ 68.0000 0.0718816
$$947$$ 734.000 0.775079 0.387540 0.921853i $$-0.373325\pi$$
0.387540 + 0.921853i $$0.373325\pi$$
$$948$$ − 509.823i − 0.537789i
$$949$$ −720.000 −0.758693
$$950$$ − 67.0820i − 0.0706127i
$$951$$ 1672.58i 1.75876i
$$952$$ 1314.81i 1.38110i
$$953$$ −934.000 −0.980063 −0.490031 0.871705i $$-0.663015\pi$$
−0.490031 + 0.871705i $$0.663015\pi$$
$$954$$ −374.000 −0.392034
$$955$$ − 129.692i − 0.135803i
$$956$$ −294.000 −0.307531
$$957$$ − 196.774i − 0.205615i
$$958$$ − 214.663i − 0.224074i
$$959$$ −1162.00 −1.21168
$$960$$ −130.000 −0.135417
$$961$$ −1919.00 −1.99688
$$962$$ 187.830i 0.195249i
$$963$$ 1166.00 1.21080
$$964$$ 482.991i 0.501028i
$$965$$ 460.630i 0.477337i
$$966$$ − 813.929i − 0.842576i
$$967$$ 314.000 0.324716 0.162358 0.986732i $$-0.448090\pi$$
0.162358 + 0.986732i $$0.448090\pi$$
$$968$$ −819.000 −0.846074
$$969$$ 1609.97i 1.66147i
$$970$$ −60.0000 −0.0618557
$$971$$ − 147.580i − 0.151988i −0.997108 0.0759941i $$-0.975787\pi$$
0.997108 0.0759941i $$-0.0242130\pi$$
$$972$$ 1033.06i 1.06282i
$$973$$ − 657.404i − 0.675646i
$$974$$ 166.000 0.170431
$$975$$ −300.000 −0.307692
$$976$$ 469.574i 0.481121i
$$977$$ −1486.00 −1.52098 −0.760491 0.649348i $$-0.775042\pi$$
−0.760491 + 0.649348i $$0.775042\pi$$
$$978$$ 152.053i 0.155473i
$$979$$ − 53.6656i − 0.0548168i
$$980$$ − 328.702i − 0.335410i
$$981$$ 1562.00 1.59225
$$982$$ 838.000 0.853360
$$983$$ − 965.981i − 0.982687i −0.870966 0.491344i $$-0.836506\pi$$
0.870966 0.491344i $$-0.163494\pi$$
$$984$$ 840.000 0.853659
$$985$$ − 505.351i − 0.513047i
$$986$$ 590.322i 0.598704i
$$987$$ 840.000 0.851064
$$988$$ 540.000 0.546559
$$989$$ −884.000 −0.893832
$$990$$ 49.1935i 0.0496904i
$$991$$ −58.0000 −0.0585267 −0.0292634 0.999572i $$-0.509316\pi$$
−0.0292634 + 0.999572i $$0.509316\pi$$
$$992$$ 1770.97i 1.78525i
$$993$$ 2155.57i 2.17076i
$$994$$ −434.000 −0.436620
$$995$$ −300.000 −0.301508
$$996$$ 540.000 0.542169
$$997$$ − 630.571i − 0.632469i −0.948681 0.316234i $$-0.897581\pi$$
0.948681 0.316234i $$-0.102419\pi$$
$$998$$ 262.000 0.262525
$$999$$ − 125.220i − 0.125345i
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 35.3.d.a.6.2 yes 2
3.2 odd 2 315.3.h.b.181.1 2
4.3 odd 2 560.3.f.a.321.1 2
5.2 odd 4 175.3.c.d.174.2 4
5.3 odd 4 175.3.c.d.174.3 4
5.4 even 2 175.3.d.f.76.1 2
7.2 even 3 245.3.h.b.31.1 4
7.3 odd 6 245.3.h.b.166.1 4
7.4 even 3 245.3.h.b.166.2 4
7.5 odd 6 245.3.h.b.31.2 4
7.6 odd 2 inner 35.3.d.a.6.1 2
21.20 even 2 315.3.h.b.181.2 2
28.27 even 2 560.3.f.a.321.2 2
35.13 even 4 175.3.c.d.174.4 4
35.27 even 4 175.3.c.d.174.1 4
35.34 odd 2 175.3.d.f.76.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.d.a.6.1 2 7.6 odd 2 inner
35.3.d.a.6.2 yes 2 1.1 even 1 trivial
175.3.c.d.174.1 4 35.27 even 4
175.3.c.d.174.2 4 5.2 odd 4
175.3.c.d.174.3 4 5.3 odd 4
175.3.c.d.174.4 4 35.13 even 4
175.3.d.f.76.1 2 5.4 even 2
175.3.d.f.76.2 2 35.34 odd 2
245.3.h.b.31.1 4 7.2 even 3
245.3.h.b.31.2 4 7.5 odd 6
245.3.h.b.166.1 4 7.3 odd 6
245.3.h.b.166.2 4 7.4 even 3
315.3.h.b.181.1 2 3.2 odd 2
315.3.h.b.181.2 2 21.20 even 2
560.3.f.a.321.1 2 4.3 odd 2
560.3.f.a.321.2 2 28.27 even 2