# Properties

 Label 825.2.a.n.1.1 Level $825$ Weight $2$ Character 825.1 Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} +1.21432 q^{6} +4.90321 q^{7} +3.06668 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} +1.21432 q^{6} +4.90321 q^{7} +3.06668 q^{8} +1.00000 q^{9} -1.00000 q^{11} +0.525428 q^{12} +4.14764 q^{13} -5.95407 q^{14} -2.67307 q^{16} -5.33185 q^{17} -1.21432 q^{18} -5.18421 q^{19} -4.90321 q^{21} +1.21432 q^{22} +4.00000 q^{23} -3.06668 q^{24} -5.03657 q^{26} -1.00000 q^{27} -2.57628 q^{28} +1.80642 q^{29} +2.62222 q^{31} -2.88739 q^{32} +1.00000 q^{33} +6.47457 q^{34} -0.525428 q^{36} +5.80642 q^{37} +6.29529 q^{38} -4.14764 q^{39} +1.80642 q^{41} +5.95407 q^{42} +4.90321 q^{43} +0.525428 q^{44} -4.85728 q^{46} +7.05086 q^{47} +2.67307 q^{48} +17.0415 q^{49} +5.33185 q^{51} -2.17929 q^{52} -7.18421 q^{53} +1.21432 q^{54} +15.0366 q^{56} +5.18421 q^{57} -2.19358 q^{58} +1.67307 q^{59} +0.755569 q^{61} -3.18421 q^{62} +4.90321 q^{63} +8.85236 q^{64} -1.21432 q^{66} -4.85728 q^{67} +2.80150 q^{68} -4.00000 q^{69} +0.428639 q^{71} +3.06668 q^{72} +12.7096 q^{73} -7.05086 q^{74} +2.72393 q^{76} -4.90321 q^{77} +5.03657 q^{78} -6.42864 q^{79} +1.00000 q^{81} -2.19358 q^{82} -2.90321 q^{83} +2.57628 q^{84} -5.95407 q^{86} -1.80642 q^{87} -3.06668 q^{88} +0.622216 q^{89} +20.3368 q^{91} -2.10171 q^{92} -2.62222 q^{93} -8.56199 q^{94} +2.88739 q^{96} -2.75557 q^{97} -20.6938 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{6} + 8 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{11} - 5 q^{12} + 6 q^{13} + 2 q^{14} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 2 q^{19} - 8 q^{21} - 3 q^{22} + 12 q^{23} - 9 q^{24} - 8 q^{26} - 3 q^{27} + 12 q^{28} - 8 q^{29} + 8 q^{31} + 11 q^{32} + 3 q^{33} + 26 q^{34} + 5 q^{36} + 4 q^{37} + 6 q^{38} - 6 q^{39} - 8 q^{41} - 2 q^{42} + 8 q^{43} - 5 q^{44} + 12 q^{46} + 8 q^{47} - 5 q^{48} + 11 q^{49} - 4 q^{51} - 26 q^{52} - 8 q^{53} - 3 q^{54} + 38 q^{56} + 2 q^{57} - 20 q^{58} - 8 q^{59} + 2 q^{61} + 4 q^{62} + 8 q^{63} + 33 q^{64} + 3 q^{66} + 12 q^{67} + 28 q^{68} - 12 q^{69} - 12 q^{71} + 9 q^{72} + 18 q^{73} - 8 q^{74} - 18 q^{76} - 8 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 20 q^{82} - 2 q^{83} - 12 q^{84} + 2 q^{86} + 8 q^{87} - 9 q^{88} + 2 q^{89} + 8 q^{91} + 20 q^{92} - 8 q^{93} - 12 q^{94} - 11 q^{96} - 8 q^{97} - 29 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 5 * q^4 - 3 * q^6 + 8 * q^7 + 9 * q^8 + 3 * q^9 - 3 * q^11 - 5 * q^12 + 6 * q^13 + 2 * q^14 + 5 * q^16 + 4 * q^17 + 3 * q^18 - 2 * q^19 - 8 * q^21 - 3 * q^22 + 12 * q^23 - 9 * q^24 - 8 * q^26 - 3 * q^27 + 12 * q^28 - 8 * q^29 + 8 * q^31 + 11 * q^32 + 3 * q^33 + 26 * q^34 + 5 * q^36 + 4 * q^37 + 6 * q^38 - 6 * q^39 - 8 * q^41 - 2 * q^42 + 8 * q^43 - 5 * q^44 + 12 * q^46 + 8 * q^47 - 5 * q^48 + 11 * q^49 - 4 * q^51 - 26 * q^52 - 8 * q^53 - 3 * q^54 + 38 * q^56 + 2 * q^57 - 20 * q^58 - 8 * q^59 + 2 * q^61 + 4 * q^62 + 8 * q^63 + 33 * q^64 + 3 * q^66 + 12 * q^67 + 28 * q^68 - 12 * q^69 - 12 * q^71 + 9 * q^72 + 18 * q^73 - 8 * q^74 - 18 * q^76 - 8 * q^77 + 8 * q^78 - 6 * q^79 + 3 * q^81 - 20 * q^82 - 2 * q^83 - 12 * q^84 + 2 * q^86 + 8 * q^87 - 9 * q^88 + 2 * q^89 + 8 * q^91 + 20 * q^92 - 8 * q^93 - 12 * q^94 - 11 * q^96 - 8 * q^97 - 29 * q^98 - 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.21432 −0.858654 −0.429327 0.903149i $$-0.641249\pi$$
−0.429327 + 0.903149i $$0.641249\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −0.525428 −0.262714
$$5$$ 0 0
$$6$$ 1.21432 0.495744
$$7$$ 4.90321 1.85324 0.926620 0.375999i $$-0.122700\pi$$
0.926620 + 0.375999i $$0.122700\pi$$
$$8$$ 3.06668 1.08423
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0.525428 0.151678
$$13$$ 4.14764 1.15035 0.575175 0.818031i $$-0.304934\pi$$
0.575175 + 0.818031i $$0.304934\pi$$
$$14$$ −5.95407 −1.59129
$$15$$ 0 0
$$16$$ −2.67307 −0.668268
$$17$$ −5.33185 −1.29316 −0.646582 0.762845i $$-0.723802\pi$$
−0.646582 + 0.762845i $$0.723802\pi$$
$$18$$ −1.21432 −0.286218
$$19$$ −5.18421 −1.18934 −0.594669 0.803970i $$-0.702717\pi$$
−0.594669 + 0.803970i $$0.702717\pi$$
$$20$$ 0 0
$$21$$ −4.90321 −1.06997
$$22$$ 1.21432 0.258894
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −3.06668 −0.625983
$$25$$ 0 0
$$26$$ −5.03657 −0.987752
$$27$$ −1.00000 −0.192450
$$28$$ −2.57628 −0.486872
$$29$$ 1.80642 0.335444 0.167722 0.985834i $$-0.446359\pi$$
0.167722 + 0.985834i $$0.446359\pi$$
$$30$$ 0 0
$$31$$ 2.62222 0.470964 0.235482 0.971879i $$-0.424333\pi$$
0.235482 + 0.971879i $$0.424333\pi$$
$$32$$ −2.88739 −0.510423
$$33$$ 1.00000 0.174078
$$34$$ 6.47457 1.11038
$$35$$ 0 0
$$36$$ −0.525428 −0.0875713
$$37$$ 5.80642 0.954570 0.477285 0.878749i $$-0.341621\pi$$
0.477285 + 0.878749i $$0.341621\pi$$
$$38$$ 6.29529 1.02123
$$39$$ −4.14764 −0.664154
$$40$$ 0 0
$$41$$ 1.80642 0.282116 0.141058 0.990001i $$-0.454950\pi$$
0.141058 + 0.990001i $$0.454950\pi$$
$$42$$ 5.95407 0.918732
$$43$$ 4.90321 0.747733 0.373866 0.927483i $$-0.378032\pi$$
0.373866 + 0.927483i $$0.378032\pi$$
$$44$$ 0.525428 0.0792112
$$45$$ 0 0
$$46$$ −4.85728 −0.716167
$$47$$ 7.05086 1.02847 0.514236 0.857648i $$-0.328075\pi$$
0.514236 + 0.857648i $$0.328075\pi$$
$$48$$ 2.67307 0.385825
$$49$$ 17.0415 2.43450
$$50$$ 0 0
$$51$$ 5.33185 0.746609
$$52$$ −2.17929 −0.302213
$$53$$ −7.18421 −0.986827 −0.493413 0.869795i $$-0.664251\pi$$
−0.493413 + 0.869795i $$0.664251\pi$$
$$54$$ 1.21432 0.165248
$$55$$ 0 0
$$56$$ 15.0366 2.00935
$$57$$ 5.18421 0.686665
$$58$$ −2.19358 −0.288031
$$59$$ 1.67307 0.217815 0.108908 0.994052i $$-0.465265\pi$$
0.108908 + 0.994052i $$0.465265\pi$$
$$60$$ 0 0
$$61$$ 0.755569 0.0967407 0.0483703 0.998829i $$-0.484597\pi$$
0.0483703 + 0.998829i $$0.484597\pi$$
$$62$$ −3.18421 −0.404395
$$63$$ 4.90321 0.617747
$$64$$ 8.85236 1.10654
$$65$$ 0 0
$$66$$ −1.21432 −0.149472
$$67$$ −4.85728 −0.593411 −0.296706 0.954969i $$-0.595888\pi$$
−0.296706 + 0.954969i $$0.595888\pi$$
$$68$$ 2.80150 0.339732
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 0.428639 0.0508701 0.0254351 0.999676i $$-0.491903\pi$$
0.0254351 + 0.999676i $$0.491903\pi$$
$$72$$ 3.06668 0.361411
$$73$$ 12.7096 1.48755 0.743775 0.668430i $$-0.233033\pi$$
0.743775 + 0.668430i $$0.233033\pi$$
$$74$$ −7.05086 −0.819645
$$75$$ 0 0
$$76$$ 2.72393 0.312456
$$77$$ −4.90321 −0.558773
$$78$$ 5.03657 0.570279
$$79$$ −6.42864 −0.723278 −0.361639 0.932318i $$-0.617783\pi$$
−0.361639 + 0.932318i $$0.617783\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.19358 −0.242240
$$83$$ −2.90321 −0.318669 −0.159334 0.987225i $$-0.550935\pi$$
−0.159334 + 0.987225i $$0.550935\pi$$
$$84$$ 2.57628 0.281095
$$85$$ 0 0
$$86$$ −5.95407 −0.642044
$$87$$ −1.80642 −0.193669
$$88$$ −3.06668 −0.326909
$$89$$ 0.622216 0.0659547 0.0329774 0.999456i $$-0.489501\pi$$
0.0329774 + 0.999456i $$0.489501\pi$$
$$90$$ 0 0
$$91$$ 20.3368 2.13187
$$92$$ −2.10171 −0.219118
$$93$$ −2.62222 −0.271911
$$94$$ −8.56199 −0.883102
$$95$$ 0 0
$$96$$ 2.88739 0.294693
$$97$$ −2.75557 −0.279786 −0.139893 0.990167i $$-0.544676\pi$$
−0.139893 + 0.990167i $$0.544676\pi$$
$$98$$ −20.6938 −2.09039
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −17.8064 −1.77181 −0.885903 0.463871i $$-0.846460\pi$$
−0.885903 + 0.463871i $$0.846460\pi$$
$$102$$ −6.47457 −0.641078
$$103$$ 4.94914 0.487654 0.243827 0.969819i $$-0.421597\pi$$
0.243827 + 0.969819i $$0.421597\pi$$
$$104$$ 12.7195 1.24725
$$105$$ 0 0
$$106$$ 8.72393 0.847343
$$107$$ 11.1985 1.08260 0.541300 0.840830i $$-0.317932\pi$$
0.541300 + 0.840830i $$0.317932\pi$$
$$108$$ 0.525428 0.0505593
$$109$$ 15.7146 1.50518 0.752591 0.658488i $$-0.228804\pi$$
0.752591 + 0.658488i $$0.228804\pi$$
$$110$$ 0 0
$$111$$ −5.80642 −0.551121
$$112$$ −13.1066 −1.23846
$$113$$ −1.76494 −0.166031 −0.0830156 0.996548i $$-0.526455\pi$$
−0.0830156 + 0.996548i $$0.526455\pi$$
$$114$$ −6.29529 −0.589608
$$115$$ 0 0
$$116$$ −0.949145 −0.0881259
$$117$$ 4.14764 0.383450
$$118$$ −2.03164 −0.187028
$$119$$ −26.1432 −2.39654
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −0.917502 −0.0830667
$$123$$ −1.80642 −0.162880
$$124$$ −1.37778 −0.123729
$$125$$ 0 0
$$126$$ −5.95407 −0.530430
$$127$$ 18.7096 1.66021 0.830106 0.557606i $$-0.188280\pi$$
0.830106 + 0.557606i $$0.188280\pi$$
$$128$$ −4.97481 −0.439715
$$129$$ −4.90321 −0.431704
$$130$$ 0 0
$$131$$ −1.24443 −0.108726 −0.0543632 0.998521i $$-0.517313\pi$$
−0.0543632 + 0.998521i $$0.517313\pi$$
$$132$$ −0.525428 −0.0457326
$$133$$ −25.4193 −2.20413
$$134$$ 5.89829 0.509535
$$135$$ 0 0
$$136$$ −16.3511 −1.40209
$$137$$ 18.7971 1.60594 0.802970 0.596019i $$-0.203252\pi$$
0.802970 + 0.596019i $$0.203252\pi$$
$$138$$ 4.85728 0.413479
$$139$$ 14.0415 1.19098 0.595492 0.803361i $$-0.296957\pi$$
0.595492 + 0.803361i $$0.296957\pi$$
$$140$$ 0 0
$$141$$ −7.05086 −0.593789
$$142$$ −0.520505 −0.0436798
$$143$$ −4.14764 −0.346843
$$144$$ −2.67307 −0.222756
$$145$$ 0 0
$$146$$ −15.4336 −1.27729
$$147$$ −17.0415 −1.40556
$$148$$ −3.05086 −0.250779
$$149$$ −3.05086 −0.249936 −0.124968 0.992161i $$-0.539883\pi$$
−0.124968 + 0.992161i $$0.539883\pi$$
$$150$$ 0 0
$$151$$ −0.326929 −0.0266051 −0.0133026 0.999912i $$-0.504234\pi$$
−0.0133026 + 0.999912i $$0.504234\pi$$
$$152$$ −15.8983 −1.28952
$$153$$ −5.33185 −0.431055
$$154$$ 5.95407 0.479792
$$155$$ 0 0
$$156$$ 2.17929 0.174483
$$157$$ −19.9081 −1.58884 −0.794421 0.607367i $$-0.792226\pi$$
−0.794421 + 0.607367i $$0.792226\pi$$
$$158$$ 7.80642 0.621046
$$159$$ 7.18421 0.569745
$$160$$ 0 0
$$161$$ 19.6128 1.54571
$$162$$ −1.21432 −0.0954060
$$163$$ 12.1748 0.953607 0.476804 0.879010i $$-0.341795\pi$$
0.476804 + 0.879010i $$0.341795\pi$$
$$164$$ −0.949145 −0.0741158
$$165$$ 0 0
$$166$$ 3.52543 0.273626
$$167$$ 13.0049 1.00635 0.503176 0.864184i $$-0.332165\pi$$
0.503176 + 0.864184i $$0.332165\pi$$
$$168$$ −15.0366 −1.16010
$$169$$ 4.20294 0.323303
$$170$$ 0 0
$$171$$ −5.18421 −0.396446
$$172$$ −2.57628 −0.196440
$$173$$ 13.8938 1.05633 0.528165 0.849142i $$-0.322880\pi$$
0.528165 + 0.849142i $$0.322880\pi$$
$$174$$ 2.19358 0.166295
$$175$$ 0 0
$$176$$ 2.67307 0.201490
$$177$$ −1.67307 −0.125756
$$178$$ −0.755569 −0.0566323
$$179$$ 12.8573 0.960998 0.480499 0.876995i $$-0.340456\pi$$
0.480499 + 0.876995i $$0.340456\pi$$
$$180$$ 0 0
$$181$$ 0.917502 0.0681974 0.0340987 0.999418i $$-0.489144\pi$$
0.0340987 + 0.999418i $$0.489144\pi$$
$$182$$ −24.6953 −1.83054
$$183$$ −0.755569 −0.0558532
$$184$$ 12.2667 0.904314
$$185$$ 0 0
$$186$$ 3.18421 0.233477
$$187$$ 5.33185 0.389904
$$188$$ −3.70471 −0.270194
$$189$$ −4.90321 −0.356656
$$190$$ 0 0
$$191$$ 14.3684 1.03966 0.519831 0.854269i $$-0.325995\pi$$
0.519831 + 0.854269i $$0.325995\pi$$
$$192$$ −8.85236 −0.638864
$$193$$ −11.7605 −0.846539 −0.423269 0.906004i $$-0.639118\pi$$
−0.423269 + 0.906004i $$0.639118\pi$$
$$194$$ 3.34614 0.240239
$$195$$ 0 0
$$196$$ −8.95407 −0.639576
$$197$$ −3.82071 −0.272215 −0.136107 0.990694i $$-0.543459\pi$$
−0.136107 + 0.990694i $$0.543459\pi$$
$$198$$ 1.21432 0.0862979
$$199$$ −13.7146 −0.972199 −0.486100 0.873903i $$-0.661581\pi$$
−0.486100 + 0.873903i $$0.661581\pi$$
$$200$$ 0 0
$$201$$ 4.85728 0.342606
$$202$$ 21.6227 1.52137
$$203$$ 8.85728 0.621659
$$204$$ −2.80150 −0.196144
$$205$$ 0 0
$$206$$ −6.00984 −0.418726
$$207$$ 4.00000 0.278019
$$208$$ −11.0869 −0.768741
$$209$$ 5.18421 0.358599
$$210$$ 0 0
$$211$$ 1.95851 0.134830 0.0674148 0.997725i $$-0.478525\pi$$
0.0674148 + 0.997725i $$0.478525\pi$$
$$212$$ 3.77478 0.259253
$$213$$ −0.428639 −0.0293699
$$214$$ −13.5986 −0.929578
$$215$$ 0 0
$$216$$ −3.06668 −0.208661
$$217$$ 12.8573 0.872809
$$218$$ −19.0825 −1.29243
$$219$$ −12.7096 −0.858838
$$220$$ 0 0
$$221$$ −22.1146 −1.48759
$$222$$ 7.05086 0.473222
$$223$$ −26.0098 −1.74175 −0.870874 0.491506i $$-0.836446\pi$$
−0.870874 + 0.491506i $$0.836446\pi$$
$$224$$ −14.1575 −0.945937
$$225$$ 0 0
$$226$$ 2.14320 0.142563
$$227$$ −6.34122 −0.420882 −0.210441 0.977607i $$-0.567490\pi$$
−0.210441 + 0.977607i $$0.567490\pi$$
$$228$$ −2.72393 −0.180396
$$229$$ −23.3274 −1.54152 −0.770759 0.637127i $$-0.780123\pi$$
−0.770759 + 0.637127i $$0.780123\pi$$
$$230$$ 0 0
$$231$$ 4.90321 0.322608
$$232$$ 5.53972 0.363700
$$233$$ −1.42372 −0.0932708 −0.0466354 0.998912i $$-0.514850\pi$$
−0.0466354 + 0.998912i $$0.514850\pi$$
$$234$$ −5.03657 −0.329251
$$235$$ 0 0
$$236$$ −0.879077 −0.0572231
$$237$$ 6.42864 0.417585
$$238$$ 31.7462 2.05780
$$239$$ −18.9590 −1.22636 −0.613178 0.789945i $$-0.710109\pi$$
−0.613178 + 0.789945i $$0.710109\pi$$
$$240$$ 0 0
$$241$$ −1.34614 −0.0867126 −0.0433563 0.999060i $$-0.513805\pi$$
−0.0433563 + 0.999060i $$0.513805\pi$$
$$242$$ −1.21432 −0.0780594
$$243$$ −1.00000 −0.0641500
$$244$$ −0.396997 −0.0254151
$$245$$ 0 0
$$246$$ 2.19358 0.139857
$$247$$ −21.5022 −1.36816
$$248$$ 8.04149 0.510635
$$249$$ 2.90321 0.183984
$$250$$ 0 0
$$251$$ 1.08250 0.0683267 0.0341633 0.999416i $$-0.489123\pi$$
0.0341633 + 0.999416i $$0.489123\pi$$
$$252$$ −2.57628 −0.162291
$$253$$ −4.00000 −0.251478
$$254$$ −22.7195 −1.42555
$$255$$ 0 0
$$256$$ −11.6637 −0.728981
$$257$$ −0.133353 −0.00831834 −0.00415917 0.999991i $$-0.501324\pi$$
−0.00415917 + 0.999991i $$0.501324\pi$$
$$258$$ 5.95407 0.370684
$$259$$ 28.4701 1.76905
$$260$$ 0 0
$$261$$ 1.80642 0.111815
$$262$$ 1.51114 0.0933584
$$263$$ −0.147643 −0.00910407 −0.00455203 0.999990i $$-0.501449\pi$$
−0.00455203 + 0.999990i $$0.501449\pi$$
$$264$$ 3.06668 0.188741
$$265$$ 0 0
$$266$$ 30.8671 1.89258
$$267$$ −0.622216 −0.0380790
$$268$$ 2.55215 0.155897
$$269$$ −26.8573 −1.63752 −0.818759 0.574138i $$-0.805337\pi$$
−0.818759 + 0.574138i $$0.805337\pi$$
$$270$$ 0 0
$$271$$ 3.08250 0.187248 0.0936242 0.995608i $$-0.470155\pi$$
0.0936242 + 0.995608i $$0.470155\pi$$
$$272$$ 14.2524 0.864180
$$273$$ −20.3368 −1.23084
$$274$$ −22.8256 −1.37895
$$275$$ 0 0
$$276$$ 2.10171 0.126508
$$277$$ −8.70964 −0.523311 −0.261656 0.965161i $$-0.584268\pi$$
−0.261656 + 0.965161i $$0.584268\pi$$
$$278$$ −17.0509 −1.02264
$$279$$ 2.62222 0.156988
$$280$$ 0 0
$$281$$ −20.3783 −1.21567 −0.607833 0.794065i $$-0.707961\pi$$
−0.607833 + 0.794065i $$0.707961\pi$$
$$282$$ 8.56199 0.509859
$$283$$ −6.32248 −0.375833 −0.187916 0.982185i $$-0.560173\pi$$
−0.187916 + 0.982185i $$0.560173\pi$$
$$284$$ −0.225219 −0.0133643
$$285$$ 0 0
$$286$$ 5.03657 0.297818
$$287$$ 8.85728 0.522829
$$288$$ −2.88739 −0.170141
$$289$$ 11.4286 0.672273
$$290$$ 0 0
$$291$$ 2.75557 0.161534
$$292$$ −6.67799 −0.390800
$$293$$ 16.6780 0.974339 0.487169 0.873308i $$-0.338029\pi$$
0.487169 + 0.873308i $$0.338029\pi$$
$$294$$ 20.6938 1.20689
$$295$$ 0 0
$$296$$ 17.8064 1.03498
$$297$$ 1.00000 0.0580259
$$298$$ 3.70471 0.214608
$$299$$ 16.5906 0.959458
$$300$$ 0 0
$$301$$ 24.0415 1.38573
$$302$$ 0.396997 0.0228446
$$303$$ 17.8064 1.02295
$$304$$ 13.8578 0.794797
$$305$$ 0 0
$$306$$ 6.47457 0.370127
$$307$$ −9.58565 −0.547082 −0.273541 0.961860i $$-0.588195\pi$$
−0.273541 + 0.961860i $$0.588195\pi$$
$$308$$ 2.57628 0.146797
$$309$$ −4.94914 −0.281547
$$310$$ 0 0
$$311$$ 14.5303 0.823941 0.411970 0.911197i $$-0.364841\pi$$
0.411970 + 0.911197i $$0.364841\pi$$
$$312$$ −12.7195 −0.720099
$$313$$ −21.0321 −1.18881 −0.594403 0.804167i $$-0.702612\pi$$
−0.594403 + 0.804167i $$0.702612\pi$$
$$314$$ 24.1748 1.36427
$$315$$ 0 0
$$316$$ 3.37778 0.190015
$$317$$ −0.990632 −0.0556394 −0.0278197 0.999613i $$-0.508856\pi$$
−0.0278197 + 0.999613i $$0.508856\pi$$
$$318$$ −8.72393 −0.489213
$$319$$ −1.80642 −0.101140
$$320$$ 0 0
$$321$$ −11.1985 −0.625039
$$322$$ −23.8163 −1.32723
$$323$$ 27.6414 1.53801
$$324$$ −0.525428 −0.0291904
$$325$$ 0 0
$$326$$ −14.7841 −0.818818
$$327$$ −15.7146 −0.869017
$$328$$ 5.53972 0.305880
$$329$$ 34.5718 1.90601
$$330$$ 0 0
$$331$$ −17.5812 −0.966350 −0.483175 0.875524i $$-0.660517\pi$$
−0.483175 + 0.875524i $$0.660517\pi$$
$$332$$ 1.52543 0.0837187
$$333$$ 5.80642 0.318190
$$334$$ −15.7921 −0.864107
$$335$$ 0 0
$$336$$ 13.1066 0.715025
$$337$$ −3.16992 −0.172676 −0.0863382 0.996266i $$-0.527517\pi$$
−0.0863382 + 0.996266i $$0.527517\pi$$
$$338$$ −5.10372 −0.277606
$$339$$ 1.76494 0.0958582
$$340$$ 0 0
$$341$$ −2.62222 −0.142001
$$342$$ 6.29529 0.340410
$$343$$ 49.2355 2.65847
$$344$$ 15.0366 0.810717
$$345$$ 0 0
$$346$$ −16.8716 −0.907021
$$347$$ 4.97634 0.267144 0.133572 0.991039i $$-0.457355\pi$$
0.133572 + 0.991039i $$0.457355\pi$$
$$348$$ 0.949145 0.0508795
$$349$$ 18.2034 0.974407 0.487203 0.873289i $$-0.338017\pi$$
0.487203 + 0.873289i $$0.338017\pi$$
$$350$$ 0 0
$$351$$ −4.14764 −0.221385
$$352$$ 2.88739 0.153898
$$353$$ 22.4099 1.19276 0.596379 0.802703i $$-0.296605\pi$$
0.596379 + 0.802703i $$0.296605\pi$$
$$354$$ 2.03164 0.107981
$$355$$ 0 0
$$356$$ −0.326929 −0.0173272
$$357$$ 26.1432 1.38364
$$358$$ −15.6128 −0.825165
$$359$$ 21.3274 1.12562 0.562809 0.826587i $$-0.309721\pi$$
0.562809 + 0.826587i $$0.309721\pi$$
$$360$$ 0 0
$$361$$ 7.87601 0.414527
$$362$$ −1.11414 −0.0585579
$$363$$ −1.00000 −0.0524864
$$364$$ −10.6855 −0.560072
$$365$$ 0 0
$$366$$ 0.917502 0.0479586
$$367$$ 35.1338 1.83397 0.916985 0.398921i $$-0.130615\pi$$
0.916985 + 0.398921i $$0.130615\pi$$
$$368$$ −10.6923 −0.557374
$$369$$ 1.80642 0.0940387
$$370$$ 0 0
$$371$$ −35.2257 −1.82883
$$372$$ 1.37778 0.0714348
$$373$$ −17.0049 −0.880481 −0.440241 0.897880i $$-0.645107\pi$$
−0.440241 + 0.897880i $$0.645107\pi$$
$$374$$ −6.47457 −0.334792
$$375$$ 0 0
$$376$$ 21.6227 1.11511
$$377$$ 7.49240 0.385878
$$378$$ 5.95407 0.306244
$$379$$ 2.36842 0.121657 0.0608287 0.998148i $$-0.480626\pi$$
0.0608287 + 0.998148i $$0.480626\pi$$
$$380$$ 0 0
$$381$$ −18.7096 −0.958524
$$382$$ −17.4479 −0.892710
$$383$$ 1.21585 0.0621271 0.0310635 0.999517i $$-0.490111\pi$$
0.0310635 + 0.999517i $$0.490111\pi$$
$$384$$ 4.97481 0.253870
$$385$$ 0 0
$$386$$ 14.2810 0.726884
$$387$$ 4.90321 0.249244
$$388$$ 1.44785 0.0735035
$$389$$ 2.26671 0.114927 0.0574633 0.998348i $$-0.481699\pi$$
0.0574633 + 0.998348i $$0.481699\pi$$
$$390$$ 0 0
$$391$$ −21.3274 −1.07857
$$392$$ 52.2607 2.63957
$$393$$ 1.24443 0.0627733
$$394$$ 4.63957 0.233738
$$395$$ 0 0
$$396$$ 0.525428 0.0264037
$$397$$ −18.4889 −0.927929 −0.463965 0.885854i $$-0.653574\pi$$
−0.463965 + 0.885854i $$0.653574\pi$$
$$398$$ 16.6539 0.834782
$$399$$ 25.4193 1.27256
$$400$$ 0 0
$$401$$ 17.5625 0.877028 0.438514 0.898724i $$-0.355505\pi$$
0.438514 + 0.898724i $$0.355505\pi$$
$$402$$ −5.89829 −0.294180
$$403$$ 10.8760 0.541773
$$404$$ 9.35599 0.465478
$$405$$ 0 0
$$406$$ −10.7556 −0.533790
$$407$$ −5.80642 −0.287814
$$408$$ 16.3511 0.809498
$$409$$ −21.3461 −1.05550 −0.527749 0.849400i $$-0.676964\pi$$
−0.527749 + 0.849400i $$0.676964\pi$$
$$410$$ 0 0
$$411$$ −18.7971 −0.927190
$$412$$ −2.60042 −0.128113
$$413$$ 8.20342 0.403664
$$414$$ −4.85728 −0.238722
$$415$$ 0 0
$$416$$ −11.9759 −0.587165
$$417$$ −14.0415 −0.687615
$$418$$ −6.29529 −0.307913
$$419$$ 28.8573 1.40977 0.704885 0.709321i $$-0.250999\pi$$
0.704885 + 0.709321i $$0.250999\pi$$
$$420$$ 0 0
$$421$$ −35.4893 −1.72964 −0.864822 0.502078i $$-0.832569\pi$$
−0.864822 + 0.502078i $$0.832569\pi$$
$$422$$ −2.37826 −0.115772
$$423$$ 7.05086 0.342824
$$424$$ −22.0316 −1.06995
$$425$$ 0 0
$$426$$ 0.520505 0.0252186
$$427$$ 3.70471 0.179284
$$428$$ −5.88400 −0.284414
$$429$$ 4.14764 0.200250
$$430$$ 0 0
$$431$$ 9.24443 0.445289 0.222644 0.974900i $$-0.428531\pi$$
0.222644 + 0.974900i $$0.428531\pi$$
$$432$$ 2.67307 0.128608
$$433$$ −6.28544 −0.302059 −0.151030 0.988529i $$-0.548259\pi$$
−0.151030 + 0.988529i $$0.548259\pi$$
$$434$$ −15.6128 −0.749441
$$435$$ 0 0
$$436$$ −8.25686 −0.395432
$$437$$ −20.7368 −0.991977
$$438$$ 15.4336 0.737444
$$439$$ −36.5303 −1.74350 −0.871749 0.489952i $$-0.837014\pi$$
−0.871749 + 0.489952i $$0.837014\pi$$
$$440$$ 0 0
$$441$$ 17.0415 0.811499
$$442$$ 26.8542 1.27732
$$443$$ −38.2766 −1.81857 −0.909287 0.416170i $$-0.863372\pi$$
−0.909287 + 0.416170i $$0.863372\pi$$
$$444$$ 3.05086 0.144787
$$445$$ 0 0
$$446$$ 31.5843 1.49556
$$447$$ 3.05086 0.144300
$$448$$ 43.4050 2.05069
$$449$$ −31.8479 −1.50300 −0.751498 0.659735i $$-0.770668\pi$$
−0.751498 + 0.659735i $$0.770668\pi$$
$$450$$ 0 0
$$451$$ −1.80642 −0.0850612
$$452$$ 0.927346 0.0436187
$$453$$ 0.326929 0.0153605
$$454$$ 7.70027 0.361391
$$455$$ 0 0
$$456$$ 15.8983 0.744506
$$457$$ 1.39207 0.0651185 0.0325592 0.999470i $$-0.489634\pi$$
0.0325592 + 0.999470i $$0.489634\pi$$
$$458$$ 28.3269 1.32363
$$459$$ 5.33185 0.248870
$$460$$ 0 0
$$461$$ −7.70471 −0.358844 −0.179422 0.983772i $$-0.557423\pi$$
−0.179422 + 0.983772i $$0.557423\pi$$
$$462$$ −5.95407 −0.277008
$$463$$ 4.68244 0.217611 0.108806 0.994063i $$-0.465297\pi$$
0.108806 + 0.994063i $$0.465297\pi$$
$$464$$ −4.82870 −0.224167
$$465$$ 0 0
$$466$$ 1.72885 0.0800873
$$467$$ 12.8573 0.594964 0.297482 0.954727i $$-0.403853\pi$$
0.297482 + 0.954727i $$0.403853\pi$$
$$468$$ −2.17929 −0.100738
$$469$$ −23.8163 −1.09973
$$470$$ 0 0
$$471$$ 19.9081 0.917318
$$472$$ 5.13077 0.236163
$$473$$ −4.90321 −0.225450
$$474$$ −7.80642 −0.358561
$$475$$ 0 0
$$476$$ 13.7364 0.629605
$$477$$ −7.18421 −0.328942
$$478$$ 23.0223 1.05301
$$479$$ 8.38715 0.383219 0.191609 0.981471i $$-0.438629\pi$$
0.191609 + 0.981471i $$0.438629\pi$$
$$480$$ 0 0
$$481$$ 24.0830 1.09809
$$482$$ 1.63465 0.0744561
$$483$$ −19.6128 −0.892415
$$484$$ −0.525428 −0.0238831
$$485$$ 0 0
$$486$$ 1.21432 0.0550827
$$487$$ 9.83500 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$488$$ 2.31708 0.104890
$$489$$ −12.1748 −0.550565
$$490$$ 0 0
$$491$$ −32.9403 −1.48657 −0.743286 0.668973i $$-0.766734\pi$$
−0.743286 + 0.668973i $$0.766734\pi$$
$$492$$ 0.949145 0.0427908
$$493$$ −9.63158 −0.433785
$$494$$ 26.1106 1.17477
$$495$$ 0 0
$$496$$ −7.00937 −0.314730
$$497$$ 2.10171 0.0942746
$$498$$ −3.52543 −0.157978
$$499$$ −1.63158 −0.0730397 −0.0365199 0.999333i $$-0.511627\pi$$
−0.0365199 + 0.999333i $$0.511627\pi$$
$$500$$ 0 0
$$501$$ −13.0049 −0.581017
$$502$$ −1.31450 −0.0586689
$$503$$ 41.8622 1.86654 0.933272 0.359171i $$-0.116941\pi$$
0.933272 + 0.359171i $$0.116941\pi$$
$$504$$ 15.0366 0.669782
$$505$$ 0 0
$$506$$ 4.85728 0.215932
$$507$$ −4.20294 −0.186659
$$508$$ −9.83056 −0.436160
$$509$$ −38.8573 −1.72232 −0.861159 0.508335i $$-0.830261\pi$$
−0.861159 + 0.508335i $$0.830261\pi$$
$$510$$ 0 0
$$511$$ 62.3180 2.75679
$$512$$ 24.1131 1.06566
$$513$$ 5.18421 0.228888
$$514$$ 0.161933 0.00714257
$$515$$ 0 0
$$516$$ 2.57628 0.113415
$$517$$ −7.05086 −0.310096
$$518$$ −34.5718 −1.51900
$$519$$ −13.8938 −0.609872
$$520$$ 0 0
$$521$$ 11.1111 0.486785 0.243393 0.969928i $$-0.421740\pi$$
0.243393 + 0.969928i $$0.421740\pi$$
$$522$$ −2.19358 −0.0960102
$$523$$ 27.3002 1.19375 0.596877 0.802332i $$-0.296408\pi$$
0.596877 + 0.802332i $$0.296408\pi$$
$$524$$ 0.653858 0.0285639
$$525$$ 0 0
$$526$$ 0.179286 0.00781724
$$527$$ −13.9813 −0.609033
$$528$$ −2.67307 −0.116330
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 1.67307 0.0726051
$$532$$ 13.3560 0.579055
$$533$$ 7.49240 0.324532
$$534$$ 0.755569 0.0326967
$$535$$ 0 0
$$536$$ −14.8957 −0.643396
$$537$$ −12.8573 −0.554833
$$538$$ 32.6133 1.40606
$$539$$ −17.0415 −0.734029
$$540$$ 0 0
$$541$$ −16.1017 −0.692267 −0.346133 0.938185i $$-0.612506\pi$$
−0.346133 + 0.938185i $$0.612506\pi$$
$$542$$ −3.74314 −0.160782
$$543$$ −0.917502 −0.0393738
$$544$$ 15.3951 0.660061
$$545$$ 0 0
$$546$$ 24.6953 1.05686
$$547$$ 40.0370 1.71186 0.855930 0.517091i $$-0.172985\pi$$
0.855930 + 0.517091i $$0.172985\pi$$
$$548$$ −9.87649 −0.421903
$$549$$ 0.755569 0.0322469
$$550$$ 0 0
$$551$$ −9.36488 −0.398957
$$552$$ −12.2667 −0.522106
$$553$$ −31.5210 −1.34041
$$554$$ 10.5763 0.449343
$$555$$ 0 0
$$556$$ −7.37778 −0.312888
$$557$$ −28.2908 −1.19872 −0.599361 0.800479i $$-0.704578\pi$$
−0.599361 + 0.800479i $$0.704578\pi$$
$$558$$ −3.18421 −0.134798
$$559$$ 20.3368 0.860154
$$560$$ 0 0
$$561$$ −5.33185 −0.225111
$$562$$ 24.7457 1.04384
$$563$$ −32.7926 −1.38204 −0.691022 0.722834i $$-0.742839\pi$$
−0.691022 + 0.722834i $$0.742839\pi$$
$$564$$ 3.70471 0.155997
$$565$$ 0 0
$$566$$ 7.67752 0.322710
$$567$$ 4.90321 0.205916
$$568$$ 1.31450 0.0551551
$$569$$ −8.88586 −0.372515 −0.186257 0.982501i $$-0.559636\pi$$
−0.186257 + 0.982501i $$0.559636\pi$$
$$570$$ 0 0
$$571$$ −10.6953 −0.447586 −0.223793 0.974637i $$-0.571844\pi$$
−0.223793 + 0.974637i $$0.571844\pi$$
$$572$$ 2.17929 0.0911205
$$573$$ −14.3684 −0.600249
$$574$$ −10.7556 −0.448929
$$575$$ 0 0
$$576$$ 8.85236 0.368848
$$577$$ −27.1338 −1.12960 −0.564798 0.825229i $$-0.691046\pi$$
−0.564798 + 0.825229i $$0.691046\pi$$
$$578$$ −13.8780 −0.577250
$$579$$ 11.7605 0.488749
$$580$$ 0 0
$$581$$ −14.2351 −0.590570
$$582$$ −3.34614 −0.138702
$$583$$ 7.18421 0.297540
$$584$$ 38.9763 1.61285
$$585$$ 0 0
$$586$$ −20.2524 −0.836620
$$587$$ −10.9590 −0.452326 −0.226163 0.974089i $$-0.572618\pi$$
−0.226163 + 0.974089i $$0.572618\pi$$
$$588$$ 8.95407 0.369260
$$589$$ −13.5941 −0.560136
$$590$$ 0 0
$$591$$ 3.82071 0.157163
$$592$$ −15.5210 −0.637908
$$593$$ −23.7003 −0.973253 −0.486627 0.873610i $$-0.661773\pi$$
−0.486627 + 0.873610i $$0.661773\pi$$
$$594$$ −1.21432 −0.0498241
$$595$$ 0 0
$$596$$ 1.60300 0.0656616
$$597$$ 13.7146 0.561299
$$598$$ −20.1463 −0.823842
$$599$$ 41.7146 1.70441 0.852205 0.523208i $$-0.175265\pi$$
0.852205 + 0.523208i $$0.175265\pi$$
$$600$$ 0 0
$$601$$ 14.5906 0.595162 0.297581 0.954697i $$-0.403820\pi$$
0.297581 + 0.954697i $$0.403820\pi$$
$$602$$ −29.1941 −1.18986
$$603$$ −4.85728 −0.197804
$$604$$ 0.171778 0.00698953
$$605$$ 0 0
$$606$$ −21.6227 −0.878362
$$607$$ −19.9826 −0.811071 −0.405535 0.914079i $$-0.632915\pi$$
−0.405535 + 0.914079i $$0.632915\pi$$
$$608$$ 14.9688 0.607066
$$609$$ −8.85728 −0.358915
$$610$$ 0 0
$$611$$ 29.2444 1.18310
$$612$$ 2.80150 0.113244
$$613$$ 19.0781 0.770555 0.385278 0.922801i $$-0.374106\pi$$
0.385278 + 0.922801i $$0.374106\pi$$
$$614$$ 11.6400 0.469754
$$615$$ 0 0
$$616$$ −15.0366 −0.605840
$$617$$ −39.3590 −1.58454 −0.792268 0.610174i $$-0.791100\pi$$
−0.792268 + 0.610174i $$0.791100\pi$$
$$618$$ 6.00984 0.241751
$$619$$ −23.0923 −0.928160 −0.464080 0.885793i $$-0.653615\pi$$
−0.464080 + 0.885793i $$0.653615\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ −17.6445 −0.707480
$$623$$ 3.05086 0.122230
$$624$$ 11.0869 0.443833
$$625$$ 0 0
$$626$$ 25.5397 1.02077
$$627$$ −5.18421 −0.207037
$$628$$ 10.4603 0.417411
$$629$$ −30.9590 −1.23442
$$630$$ 0 0
$$631$$ −25.5111 −1.01558 −0.507791 0.861480i $$-0.669538\pi$$
−0.507791 + 0.861480i $$0.669538\pi$$
$$632$$ −19.7146 −0.784203
$$633$$ −1.95851 −0.0778439
$$634$$ 1.20294 0.0477750
$$635$$ 0 0
$$636$$ −3.77478 −0.149680
$$637$$ 70.6820 2.80052
$$638$$ 2.19358 0.0868445
$$639$$ 0.428639 0.0169567
$$640$$ 0 0
$$641$$ −6.25380 −0.247010 −0.123505 0.992344i $$-0.539414\pi$$
−0.123505 + 0.992344i $$0.539414\pi$$
$$642$$ 13.5986 0.536692
$$643$$ 6.84743 0.270036 0.135018 0.990843i $$-0.456891\pi$$
0.135018 + 0.990843i $$0.456891\pi$$
$$644$$ −10.3051 −0.406079
$$645$$ 0 0
$$646$$ −33.5655 −1.32062
$$647$$ 20.2953 0.797890 0.398945 0.916975i $$-0.369376\pi$$
0.398945 + 0.916975i $$0.369376\pi$$
$$648$$ 3.06668 0.120470
$$649$$ −1.67307 −0.0656738
$$650$$ 0 0
$$651$$ −12.8573 −0.503916
$$652$$ −6.39700 −0.250526
$$653$$ −10.6222 −0.415679 −0.207840 0.978163i $$-0.566643\pi$$
−0.207840 + 0.978163i $$0.566643\pi$$
$$654$$ 19.0825 0.746185
$$655$$ 0 0
$$656$$ −4.82870 −0.188529
$$657$$ 12.7096 0.495850
$$658$$ −41.9813 −1.63660
$$659$$ 10.1017 0.393507 0.196753 0.980453i $$-0.436960\pi$$
0.196753 + 0.980453i $$0.436960\pi$$
$$660$$ 0 0
$$661$$ 21.6128 0.840642 0.420321 0.907375i $$-0.361917\pi$$
0.420321 + 0.907375i $$0.361917\pi$$
$$662$$ 21.3492 0.829760
$$663$$ 22.1146 0.858861
$$664$$ −8.90321 −0.345512
$$665$$ 0 0
$$666$$ −7.05086 −0.273215
$$667$$ 7.22570 0.279780
$$668$$ −6.83314 −0.264382
$$669$$ 26.0098 1.00560
$$670$$ 0 0
$$671$$ −0.755569 −0.0291684
$$672$$ 14.1575 0.546137
$$673$$ 10.2208 0.393982 0.196991 0.980405i $$-0.436883\pi$$
0.196991 + 0.980405i $$0.436883\pi$$
$$674$$ 3.84929 0.148269
$$675$$ 0 0
$$676$$ −2.20834 −0.0849363
$$677$$ 13.9224 0.535082 0.267541 0.963546i $$-0.413789\pi$$
0.267541 + 0.963546i $$0.413789\pi$$
$$678$$ −2.14320 −0.0823090
$$679$$ −13.5111 −0.518510
$$680$$ 0 0
$$681$$ 6.34122 0.242996
$$682$$ 3.18421 0.121930
$$683$$ −10.3970 −0.397830 −0.198915 0.980017i $$-0.563742\pi$$
−0.198915 + 0.980017i $$0.563742\pi$$
$$684$$ 2.72393 0.104152
$$685$$ 0 0
$$686$$ −59.7877 −2.28270
$$687$$ 23.3274 0.889996
$$688$$ −13.1066 −0.499686
$$689$$ −29.7975 −1.13520
$$690$$ 0 0
$$691$$ −0.977725 −0.0371944 −0.0185972 0.999827i $$-0.505920\pi$$
−0.0185972 + 0.999827i $$0.505920\pi$$
$$692$$ −7.30021 −0.277512
$$693$$ −4.90321 −0.186258
$$694$$ −6.04287 −0.229384
$$695$$ 0 0
$$696$$ −5.53972 −0.209982
$$697$$ −9.63158 −0.364822
$$698$$ −22.1048 −0.836678
$$699$$ 1.42372 0.0538499
$$700$$ 0 0
$$701$$ 48.9688 1.84953 0.924764 0.380542i $$-0.124263\pi$$
0.924764 + 0.380542i $$0.124263\pi$$
$$702$$ 5.03657 0.190093
$$703$$ −30.1017 −1.13531
$$704$$ −8.85236 −0.333636
$$705$$ 0 0
$$706$$ −27.2128 −1.02417
$$707$$ −87.3087 −3.28358
$$708$$ 0.879077 0.0330378
$$709$$ −37.2672 −1.39960 −0.699799 0.714340i $$-0.746727\pi$$
−0.699799 + 0.714340i $$0.746727\pi$$
$$710$$ 0 0
$$711$$ −6.42864 −0.241093
$$712$$ 1.90813 0.0715103
$$713$$ 10.4889 0.392811
$$714$$ −31.7462 −1.18807
$$715$$ 0 0
$$716$$ −6.75557 −0.252467
$$717$$ 18.9590 0.708036
$$718$$ −25.8983 −0.966516
$$719$$ 5.83500 0.217609 0.108804 0.994063i $$-0.465298\pi$$
0.108804 + 0.994063i $$0.465298\pi$$
$$720$$ 0 0
$$721$$ 24.2667 0.903739
$$722$$ −9.56400 −0.355935
$$723$$ 1.34614 0.0500635
$$724$$ −0.482081 −0.0179164
$$725$$ 0 0
$$726$$ 1.21432 0.0450676
$$727$$ −46.8385 −1.73715 −0.868573 0.495562i $$-0.834962\pi$$
−0.868573 + 0.495562i $$0.834962\pi$$
$$728$$ 62.3663 2.31145
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −26.1432 −0.966941
$$732$$ 0.396997 0.0146734
$$733$$ −45.2083 −1.66981 −0.834904 0.550395i $$-0.814477\pi$$
−0.834904 + 0.550395i $$0.814477\pi$$
$$734$$ −42.6637 −1.57475
$$735$$ 0 0
$$736$$ −11.5496 −0.425723
$$737$$ 4.85728 0.178920
$$738$$ −2.19358 −0.0807467
$$739$$ 5.65433 0.207998 0.103999 0.994577i $$-0.466836\pi$$
0.103999 + 0.994577i $$0.466836\pi$$
$$740$$ 0 0
$$741$$ 21.5022 0.789905
$$742$$ 42.7753 1.57033
$$743$$ 4.50622 0.165317 0.0826585 0.996578i $$-0.473659\pi$$
0.0826585 + 0.996578i $$0.473659\pi$$
$$744$$ −8.04149 −0.294815
$$745$$ 0 0
$$746$$ 20.6494 0.756029
$$747$$ −2.90321 −0.106223
$$748$$ −2.80150 −0.102433
$$749$$ 54.9086 2.00632
$$750$$ 0 0
$$751$$ 47.5121 1.73374 0.866870 0.498534i $$-0.166128\pi$$
0.866870 + 0.498534i $$0.166128\pi$$
$$752$$ −18.8474 −0.687295
$$753$$ −1.08250 −0.0394484
$$754$$ −9.09817 −0.331336
$$755$$ 0 0
$$756$$ 2.57628 0.0936985
$$757$$ −46.6637 −1.69602 −0.848011 0.529979i $$-0.822200\pi$$
−0.848011 + 0.529979i $$0.822200\pi$$
$$758$$ −2.87601 −0.104462
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 14.9304 0.541227 0.270613 0.962688i $$-0.412773\pi$$
0.270613 + 0.962688i $$0.412773\pi$$
$$762$$ 22.7195 0.823040
$$763$$ 77.0518 2.78946
$$764$$ −7.54956 −0.273134
$$765$$ 0 0
$$766$$ −1.47643 −0.0533457
$$767$$ 6.93930 0.250564
$$768$$ 11.6637 0.420878
$$769$$ 38.8573 1.40123 0.700615 0.713540i $$-0.252909\pi$$
0.700615 + 0.713540i $$0.252909\pi$$
$$770$$ 0 0
$$771$$ 0.133353 0.00480259
$$772$$ 6.17929 0.222397
$$773$$ −36.3368 −1.30694 −0.653471 0.756951i $$-0.726688\pi$$
−0.653471 + 0.756951i $$0.726688\pi$$
$$774$$ −5.95407 −0.214015
$$775$$ 0 0
$$776$$ −8.45044 −0.303353
$$777$$ −28.4701 −1.02136
$$778$$ −2.75251 −0.0986821
$$779$$ −9.36488 −0.335532
$$780$$ 0 0
$$781$$ −0.428639 −0.0153379
$$782$$ 25.8983 0.926121
$$783$$ −1.80642 −0.0645563
$$784$$ −45.5531 −1.62690
$$785$$ 0 0
$$786$$ −1.51114 −0.0539005
$$787$$ −33.5482 −1.19586 −0.597932 0.801547i $$-0.704011\pi$$
−0.597932 + 0.801547i $$0.704011\pi$$
$$788$$ 2.00751 0.0715145
$$789$$ 0.147643 0.00525624
$$790$$ 0 0
$$791$$ −8.65386 −0.307696
$$792$$ −3.06668 −0.108970
$$793$$ 3.13383 0.111286
$$794$$ 22.4514 0.796770
$$795$$ 0 0
$$796$$ 7.20601 0.255410
$$797$$ −16.1334 −0.571473 −0.285736 0.958308i $$-0.592238\pi$$
−0.285736 + 0.958308i $$0.592238\pi$$
$$798$$ −30.8671 −1.09268
$$799$$ −37.5941 −1.32998
$$800$$ 0 0
$$801$$ 0.622216 0.0219849
$$802$$ −21.3265 −0.753063
$$803$$ −12.7096 −0.448513
$$804$$ −2.55215 −0.0900073
$$805$$ 0 0
$$806$$ −13.2070 −0.465195
$$807$$ 26.8573 0.945421
$$808$$ −54.6065 −1.92105
$$809$$ 25.7431 0.905081 0.452540 0.891744i $$-0.350518\pi$$
0.452540 + 0.891744i $$0.350518\pi$$
$$810$$ 0 0
$$811$$ 13.4509 0.472325 0.236163 0.971714i $$-0.424110\pi$$
0.236163 + 0.971714i $$0.424110\pi$$
$$812$$ −4.65386 −0.163318
$$813$$ −3.08250 −0.108108
$$814$$ 7.05086 0.247132
$$815$$ 0 0
$$816$$ −14.2524 −0.498934
$$817$$ −25.4193 −0.889308
$$818$$ 25.9210 0.906308
$$819$$ 20.3368 0.710624
$$820$$ 0 0
$$821$$ 24.1748 0.843708 0.421854 0.906664i $$-0.361380\pi$$
0.421854 + 0.906664i $$0.361380\pi$$
$$822$$ 22.8256 0.796135
$$823$$ −40.9117 −1.42609 −0.713046 0.701118i $$-0.752685\pi$$
−0.713046 + 0.701118i $$0.752685\pi$$
$$824$$ 15.1774 0.528731
$$825$$ 0 0
$$826$$ −9.96158 −0.346608
$$827$$ −20.1476 −0.700602 −0.350301 0.936637i $$-0.613921\pi$$
−0.350301 + 0.936637i $$0.613921\pi$$
$$828$$ −2.10171 −0.0730395
$$829$$ 31.4322 1.09168 0.545842 0.837888i $$-0.316210\pi$$
0.545842 + 0.837888i $$0.316210\pi$$
$$830$$ 0 0
$$831$$ 8.70964 0.302134
$$832$$ 36.7164 1.27291
$$833$$ −90.8627 −3.14821
$$834$$ 17.0509 0.590423
$$835$$ 0 0
$$836$$ −2.72393 −0.0942089
$$837$$ −2.62222 −0.0906370
$$838$$ −35.0420 −1.21050
$$839$$ −52.8988 −1.82627 −0.913134 0.407659i $$-0.866345\pi$$
−0.913134 + 0.407659i $$0.866345\pi$$
$$840$$ 0 0
$$841$$ −25.7368 −0.887477
$$842$$ 43.0954 1.48517
$$843$$ 20.3783 0.701865
$$844$$ −1.02906 −0.0354216
$$845$$ 0 0
$$846$$ −8.56199 −0.294367
$$847$$ 4.90321 0.168476
$$848$$ 19.2039 0.659465
$$849$$ 6.32248 0.216987
$$850$$ 0 0
$$851$$ 23.2257 0.796167
$$852$$ 0.225219 0.00771588
$$853$$ −46.9229 −1.60661 −0.803305 0.595568i $$-0.796927\pi$$
−0.803305 + 0.595568i $$0.796927\pi$$
$$854$$ −4.49871 −0.153943
$$855$$ 0 0
$$856$$ 34.3422 1.17379
$$857$$ −25.1481 −0.859043 −0.429522 0.903057i $$-0.641318\pi$$
−0.429522 + 0.903057i $$0.641318\pi$$
$$858$$ −5.03657 −0.171945
$$859$$ 1.84791 0.0630499 0.0315250 0.999503i $$-0.489964\pi$$
0.0315250 + 0.999503i $$0.489964\pi$$
$$860$$ 0 0
$$861$$ −8.85728 −0.301855
$$862$$ −11.2257 −0.382349
$$863$$ 32.6824 1.11252 0.556262 0.831007i $$-0.312235\pi$$
0.556262 + 0.831007i $$0.312235\pi$$
$$864$$ 2.88739 0.0982310
$$865$$ 0 0
$$866$$ 7.63254 0.259364
$$867$$ −11.4286 −0.388137
$$868$$ −6.75557 −0.229299
$$869$$ 6.42864 0.218077
$$870$$ 0 0
$$871$$ −20.1463 −0.682630
$$872$$ 48.1915 1.63197
$$873$$ −2.75557 −0.0932619
$$874$$ 25.1811 0.851765
$$875$$ 0 0
$$876$$ 6.67799 0.225628
$$877$$ 49.1798 1.66068 0.830341 0.557255i $$-0.188145\pi$$
0.830341 + 0.557255i $$0.188145\pi$$
$$878$$ 44.3595 1.49706
$$879$$ −16.6780 −0.562535
$$880$$ 0 0
$$881$$ 33.8163 1.13930 0.569650 0.821888i $$-0.307079\pi$$
0.569650 + 0.821888i $$0.307079\pi$$
$$882$$ −20.6938 −0.696797
$$883$$ 24.7368 0.832461 0.416230 0.909259i $$-0.363351\pi$$
0.416230 + 0.909259i $$0.363351\pi$$
$$884$$ 11.6196 0.390810
$$885$$ 0 0
$$886$$ 46.4800 1.56153
$$887$$ 7.64004 0.256528 0.128264 0.991740i $$-0.459060\pi$$
0.128264 + 0.991740i $$0.459060\pi$$
$$888$$ −17.8064 −0.597544
$$889$$ 91.7373 3.07677
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 13.6663 0.457581
$$893$$ −36.5531 −1.22320
$$894$$ −3.70471 −0.123904
$$895$$ 0 0
$$896$$ −24.3926 −0.814898
$$897$$ −16.5906 −0.553943
$$898$$ 38.6735 1.29055
$$899$$ 4.73683 0.157982
$$900$$ 0 0
$$901$$ 38.3051 1.27613
$$902$$ 2.19358 0.0730381
$$903$$ −24.0415 −0.800051
$$904$$ −5.41249 −0.180017
$$905$$ 0 0
$$906$$ −0.396997 −0.0131893
$$907$$ −30.3970 −1.00932 −0.504658 0.863319i $$-0.668381\pi$$
−0.504658 + 0.863319i $$0.668381\pi$$
$$908$$ 3.33185 0.110571
$$909$$ −17.8064 −0.590602
$$910$$ 0 0
$$911$$ −45.3274 −1.50176 −0.750882 0.660436i $$-0.770371\pi$$
−0.750882 + 0.660436i $$0.770371\pi$$
$$912$$ −13.8578 −0.458876
$$913$$ 2.90321 0.0960823
$$914$$ −1.69042 −0.0559142
$$915$$ 0 0
$$916$$ 12.2569 0.404978
$$917$$ −6.10171 −0.201496
$$918$$ −6.47457 −0.213693
$$919$$ −16.3269 −0.538576 −0.269288 0.963060i $$-0.586788\pi$$
−0.269288 + 0.963060i $$0.586788\pi$$
$$920$$ 0 0
$$921$$ 9.58565 0.315858
$$922$$ 9.35599 0.308123
$$923$$ 1.77784 0.0585184
$$924$$ −2.57628 −0.0847535
$$925$$ 0 0
$$926$$ −5.68598 −0.186853
$$927$$ 4.94914 0.162551
$$928$$ −5.21585 −0.171219
$$929$$ −29.6128 −0.971566 −0.485783 0.874079i $$-0.661465\pi$$
−0.485783 + 0.874079i $$0.661465\pi$$
$$930$$ 0 0
$$931$$ −88.3466 −2.89544
$$932$$ 0.748060 0.0245035
$$933$$ −14.5303 −0.475702
$$934$$ −15.6128 −0.510868
$$935$$ 0 0
$$936$$ 12.7195 0.415749
$$937$$ 8.92195 0.291467 0.145734 0.989324i $$-0.453446\pi$$
0.145734 + 0.989324i $$0.453446\pi$$
$$938$$ 28.9206 0.944290
$$939$$ 21.0321 0.686357
$$940$$ 0 0
$$941$$ −22.2766 −0.726195 −0.363097 0.931751i $$-0.618281\pi$$
−0.363097 + 0.931751i $$0.618281\pi$$
$$942$$ −24.1748 −0.787659
$$943$$ 7.22570 0.235301
$$944$$ −4.47224 −0.145559
$$945$$ 0 0
$$946$$ 5.95407 0.193583
$$947$$ 44.4612 1.44480 0.722398 0.691477i $$-0.243040\pi$$
0.722398 + 0.691477i $$0.243040\pi$$
$$948$$ −3.37778 −0.109705
$$949$$ 52.7150 1.71120
$$950$$ 0 0
$$951$$ 0.990632 0.0321234
$$952$$ −80.1727 −2.59841
$$953$$ 20.5575 0.665924 0.332962 0.942940i $$-0.391952\pi$$
0.332962 + 0.942940i $$0.391952\pi$$
$$954$$ 8.72393 0.282448
$$955$$ 0 0
$$956$$ 9.96158 0.322180
$$957$$ 1.80642 0.0583934
$$958$$ −10.1847 −0.329052
$$959$$ 92.1659 2.97619
$$960$$ 0 0
$$961$$ −24.1240 −0.778193
$$962$$ −29.2444 −0.942878
$$963$$ 11.1985 0.360867
$$964$$ 0.707300 0.0227806
$$965$$ 0 0
$$966$$ 23.8163 0.766276
$$967$$ 27.4839 0.883824 0.441912 0.897058i $$-0.354300\pi$$
0.441912 + 0.897058i $$0.354300\pi$$
$$968$$ 3.06668 0.0985667
$$969$$ −27.6414 −0.887971
$$970$$ 0 0
$$971$$ −5.81532 −0.186622 −0.0933112 0.995637i $$-0.529745\pi$$
−0.0933112 + 0.995637i $$0.529745\pi$$
$$972$$ 0.525428 0.0168531
$$973$$ 68.8484 2.20718
$$974$$ −11.9428 −0.382673
$$975$$ 0 0
$$976$$ −2.01969 −0.0646487
$$977$$ 48.3912 1.54817 0.774085 0.633081i $$-0.218210\pi$$
0.774085 + 0.633081i $$0.218210\pi$$
$$978$$ 14.7841 0.472745
$$979$$ −0.622216 −0.0198861
$$980$$ 0 0
$$981$$ 15.7146 0.501727
$$982$$ 40.0000 1.27645
$$983$$ −49.9724 −1.59387 −0.796936 0.604064i $$-0.793547\pi$$
−0.796936 + 0.604064i $$0.793547\pi$$
$$984$$ −5.53972 −0.176600
$$985$$ 0 0
$$986$$ 11.6958 0.372471
$$987$$ −34.5718 −1.10043
$$988$$ 11.2979 0.359433
$$989$$ 19.6128 0.623652
$$990$$ 0 0
$$991$$ −7.35905 −0.233768 −0.116884 0.993146i $$-0.537291\pi$$
−0.116884 + 0.993146i $$0.537291\pi$$
$$992$$ −7.57136 −0.240391
$$993$$ 17.5812 0.557923
$$994$$ −2.55215 −0.0809492
$$995$$ 0 0
$$996$$ −1.52543 −0.0483350
$$997$$ −4.33138 −0.137176 −0.0685880 0.997645i $$-0.521849\pi$$
−0.0685880 + 0.997645i $$0.521849\pi$$
$$998$$ 1.98126 0.0627158
$$999$$ −5.80642 −0.183707
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 825.2.a.n.1.1 3
3.2 odd 2 2475.2.a.y.1.3 3
5.2 odd 4 165.2.c.a.34.3 6
5.3 odd 4 165.2.c.a.34.4 yes 6
5.4 even 2 825.2.a.h.1.3 3
11.10 odd 2 9075.2.a.cc.1.3 3
15.2 even 4 495.2.c.d.199.4 6
15.8 even 4 495.2.c.d.199.3 6
15.14 odd 2 2475.2.a.be.1.1 3
20.3 even 4 2640.2.d.i.529.5 6
20.7 even 4 2640.2.d.i.529.2 6
55.32 even 4 1815.2.c.d.364.4 6
55.43 even 4 1815.2.c.d.364.3 6
55.54 odd 2 9075.2.a.ck.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.3 6 5.2 odd 4
165.2.c.a.34.4 yes 6 5.3 odd 4
495.2.c.d.199.3 6 15.8 even 4
495.2.c.d.199.4 6 15.2 even 4
825.2.a.h.1.3 3 5.4 even 2
825.2.a.n.1.1 3 1.1 even 1 trivial
1815.2.c.d.364.3 6 55.43 even 4
1815.2.c.d.364.4 6 55.32 even 4
2475.2.a.y.1.3 3 3.2 odd 2
2475.2.a.be.1.1 3 15.14 odd 2
2640.2.d.i.529.2 6 20.7 even 4
2640.2.d.i.529.5 6 20.3 even 4
9075.2.a.cc.1.3 3 11.10 odd 2
9075.2.a.ck.1.1 3 55.54 odd 2