# Properties

 Label 2475.2.a.y.1.3 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ 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 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.21432 q^{2} -0.525428 q^{4} +4.90321 q^{7} -3.06668 q^{8} +O(q^{10})$$ $$q+1.21432 q^{2} -0.525428 q^{4} +4.90321 q^{7} -3.06668 q^{8} +1.00000 q^{11} +4.14764 q^{13} +5.95407 q^{14} -2.67307 q^{16} +5.33185 q^{17} -5.18421 q^{19} +1.21432 q^{22} -4.00000 q^{23} +5.03657 q^{26} -2.57628 q^{28} -1.80642 q^{29} +2.62222 q^{31} +2.88739 q^{32} +6.47457 q^{34} +5.80642 q^{37} -6.29529 q^{38} -1.80642 q^{41} +4.90321 q^{43} -0.525428 q^{44} -4.85728 q^{46} -7.05086 q^{47} +17.0415 q^{49} -2.17929 q^{52} +7.18421 q^{53} -15.0366 q^{56} -2.19358 q^{58} -1.67307 q^{59} +0.755569 q^{61} +3.18421 q^{62} +8.85236 q^{64} -4.85728 q^{67} -2.80150 q^{68} -0.428639 q^{71} +12.7096 q^{73} +7.05086 q^{74} +2.72393 q^{76} +4.90321 q^{77} -6.42864 q^{79} -2.19358 q^{82} +2.90321 q^{83} +5.95407 q^{86} -3.06668 q^{88} -0.622216 q^{89} +20.3368 q^{91} +2.10171 q^{92} -8.56199 q^{94} -2.75557 q^{97} +20.6938 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 $$3 q - 3 q^{2} + 5 q^{4} + 8 q^{7} - 9 q^{8} + 3 q^{11} + 6 q^{13} - 2 q^{14} + 5 q^{16} - 4 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 8 q^{26} + 12 q^{28} + 8 q^{29} + 8 q^{31} - 11 q^{32} + 26 q^{34} + 4 q^{37} - 6 q^{38} + 8 q^{41} + 8 q^{43} + 5 q^{44} + 12 q^{46} - 8 q^{47} + 11 q^{49} - 26 q^{52} + 8 q^{53} - 38 q^{56} - 20 q^{58} + 8 q^{59} + 2 q^{61} - 4 q^{62} + 33 q^{64} + 12 q^{67} - 28 q^{68} + 12 q^{71} + 18 q^{73} + 8 q^{74} - 18 q^{76} + 8 q^{77} - 6 q^{79} - 20 q^{82} + 2 q^{83} - 2 q^{86} - 9 q^{88} - 2 q^{89} + 8 q^{91} - 20 q^{92} - 12 q^{94} - 8 q^{97} + 29 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 5 * q^4 + 8 * q^7 - 9 * q^8 + 3 * q^11 + 6 * q^13 - 2 * q^14 + 5 * q^16 - 4 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 8 * q^26 + 12 * q^28 + 8 * q^29 + 8 * q^31 - 11 * q^32 + 26 * q^34 + 4 * q^37 - 6 * q^38 + 8 * q^41 + 8 * q^43 + 5 * q^44 + 12 * q^46 - 8 * q^47 + 11 * q^49 - 26 * q^52 + 8 * q^53 - 38 * q^56 - 20 * q^58 + 8 * q^59 + 2 * q^61 - 4 * q^62 + 33 * q^64 + 12 * q^67 - 28 * q^68 + 12 * q^71 + 18 * q^73 + 8 * q^74 - 18 * q^76 + 8 * q^77 - 6 * q^79 - 20 * q^82 + 2 * q^83 - 2 * q^86 - 9 * q^88 - 2 * q^89 + 8 * q^91 - 20 * q^92 - 12 * q^94 - 8 * q^97 + 29 * q^98

## 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 $$-0.358751\pi$$
0.429327 + 0.903149i $$0.358751\pi$$
$$3$$ 0 0
$$4$$ −0.525428 −0.262714
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.90321 1.85324 0.926620 0.375999i $$-0.122700\pi$$
0.926620 + 0.375999i $$0.122700\pi$$
$$8$$ −3.06668 −1.08423
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 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 $$-0.276198\pi$$
0.646582 + 0.762845i $$0.276198\pi$$
$$18$$ 0 0
$$19$$ −5.18421 −1.18934 −0.594669 0.803970i $$-0.702717\pi$$
−0.594669 + 0.803970i $$0.702717\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.21432 0.258894
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.03657 0.987752
$$27$$ 0 0
$$28$$ −2.57628 −0.486872
$$29$$ −1.80642 −0.335444 −0.167722 0.985834i $$-0.553641\pi$$
−0.167722 + 0.985834i $$0.553641\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ 6.47457 1.11038
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.80642 0.954570 0.477285 0.878749i $$-0.341621\pi$$
0.477285 + 0.878749i $$0.341621\pi$$
$$38$$ −6.29529 −1.02123
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.80642 −0.282116 −0.141058 0.990001i $$-0.545050\pi$$
−0.141058 + 0.990001i $$0.545050\pi$$
$$42$$ 0 0
$$43$$ 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 $$-0.671925\pi$$
−0.514236 + 0.857648i $$0.671925\pi$$
$$48$$ 0 0
$$49$$ 17.0415 2.43450
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.17929 −0.302213
$$53$$ 7.18421 0.986827 0.493413 0.869795i $$-0.335749\pi$$
0.493413 + 0.869795i $$0.335749\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −15.0366 −2.00935
$$57$$ 0 0
$$58$$ −2.19358 −0.288031
$$59$$ −1.67307 −0.217815 −0.108908 0.994052i $$-0.534735\pi$$
−0.108908 + 0.994052i $$0.534735\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ 8.85236 1.10654
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.85728 −0.593411 −0.296706 0.954969i $$-0.595888\pi$$
−0.296706 + 0.954969i $$0.595888\pi$$
$$68$$ −2.80150 −0.339732
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −0.428639 −0.0508701 −0.0254351 0.999676i $$-0.508097\pi$$
−0.0254351 + 0.999676i $$0.508097\pi$$
$$72$$ 0 0
$$73$$ 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
$$78$$ 0 0
$$79$$ −6.42864 −0.723278 −0.361639 0.932318i $$-0.617783\pi$$
−0.361639 + 0.932318i $$0.617783\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −2.19358 −0.242240
$$83$$ 2.90321 0.318669 0.159334 0.987225i $$-0.449065\pi$$
0.159334 + 0.987225i $$0.449065\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 5.95407 0.642044
$$87$$ 0 0
$$88$$ −3.06668 −0.326909
$$89$$ −0.622216 −0.0659547 −0.0329774 0.999456i $$-0.510499\pi$$
−0.0329774 + 0.999456i $$0.510499\pi$$
$$90$$ 0 0
$$91$$ 20.3368 2.13187
$$92$$ 2.10171 0.219118
$$93$$ 0 0
$$94$$ −8.56199 −0.883102
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.75557 −0.279786 −0.139893 0.990167i $$-0.544676\pi$$
−0.139893 + 0.990167i $$0.544676\pi$$
$$98$$ 20.6938 2.09039
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 17.8064 1.77181 0.885903 0.463871i $$-0.153540\pi$$
0.885903 + 0.463871i $$0.153540\pi$$
$$102$$ 0 0
$$103$$ 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 $$-0.682068\pi$$
−0.541300 + 0.840830i $$0.682068\pi$$
$$108$$ 0 0
$$109$$ 15.7146 1.50518 0.752591 0.658488i $$-0.228804\pi$$
0.752591 + 0.658488i $$0.228804\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −13.1066 −1.23846
$$113$$ 1.76494 0.166031 0.0830156 0.996548i $$-0.473545\pi$$
0.0830156 + 0.996548i $$0.473545\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.949145 0.0881259
$$117$$ 0 0
$$118$$ −2.03164 −0.187028
$$119$$ 26.1432 2.39654
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0.917502 0.0830667
$$123$$ 0 0
$$124$$ −1.37778 −0.123729
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 18.7096 1.66021 0.830106 0.557606i $$-0.188280\pi$$
0.830106 + 0.557606i $$0.188280\pi$$
$$128$$ 4.97481 0.439715
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.24443 0.108726 0.0543632 0.998521i $$-0.482687\pi$$
0.0543632 + 0.998521i $$0.482687\pi$$
$$132$$ 0 0
$$133$$ −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 $$-0.796748\pi$$
−0.802970 + 0.596019i $$0.796748\pi$$
$$138$$ 0 0
$$139$$ 14.0415 1.19098 0.595492 0.803361i $$-0.296957\pi$$
0.595492 + 0.803361i $$0.296957\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −0.520505 −0.0436798
$$143$$ 4.14764 0.346843
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 15.4336 1.27729
$$147$$ 0 0
$$148$$ −3.05086 −0.250779
$$149$$ 3.05086 0.249936 0.124968 0.992161i $$-0.460117\pi$$
0.124968 + 0.992161i $$0.460117\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 5.95407 0.479792
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −19.9081 −1.58884 −0.794421 0.607367i $$-0.792226\pi$$
−0.794421 + 0.607367i $$0.792226\pi$$
$$158$$ −7.80642 −0.621046
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −19.6128 −1.54571
$$162$$ 0 0
$$163$$ 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 $$-0.667835\pi$$
−0.503176 + 0.864184i $$0.667835\pi$$
$$168$$ 0 0
$$169$$ 4.20294 0.323303
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2.57628 −0.196440
$$173$$ −13.8938 −1.05633 −0.528165 0.849142i $$-0.677120\pi$$
−0.528165 + 0.849142i $$0.677120\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.67307 −0.201490
$$177$$ 0 0
$$178$$ −0.755569 −0.0566323
$$179$$ −12.8573 −0.960998 −0.480499 0.876995i $$-0.659544\pi$$
−0.480499 + 0.876995i $$0.659544\pi$$
$$180$$ 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 0
$$184$$ 12.2667 0.904314
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.33185 0.389904
$$188$$ 3.70471 0.270194
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −14.3684 −1.03966 −0.519831 0.854269i $$-0.674005\pi$$
−0.519831 + 0.854269i $$0.674005\pi$$
$$192$$ 0 0
$$193$$ −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 $$-0.456541\pi$$
0.136107 + 0.990694i $$0.456541\pi$$
$$198$$ 0 0
$$199$$ −13.7146 −0.972199 −0.486100 0.873903i $$-0.661581\pi$$
−0.486100 + 0.873903i $$0.661581\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 21.6227 1.52137
$$203$$ −8.85728 −0.621659
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6.00984 0.418726
$$207$$ 0 0
$$208$$ −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 0
$$214$$ −13.5986 −0.929578
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.8573 0.872809
$$218$$ 19.0825 1.29243
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 22.1146 1.48759
$$222$$ 0 0
$$223$$ −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 $$-0.432510\pi$$
0.210441 + 0.977607i $$0.432510\pi$$
$$228$$ 0 0
$$229$$ −23.3274 −1.54152 −0.770759 0.637127i $$-0.780123\pi$$
−0.770759 + 0.637127i $$0.780123\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 5.53972 0.363700
$$233$$ 1.42372 0.0932708 0.0466354 0.998912i $$-0.485150\pi$$
0.0466354 + 0.998912i $$0.485150\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0.879077 0.0572231
$$237$$ 0 0
$$238$$ 31.7462 2.05780
$$239$$ 18.9590 1.22636 0.613178 0.789945i $$-0.289891\pi$$
0.613178 + 0.789945i $$0.289891\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ −0.396997 −0.0254151
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −21.5022 −1.36816
$$248$$ −8.04149 −0.510635
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.08250 −0.0683267 −0.0341633 0.999416i $$-0.510877\pi$$
−0.0341633 + 0.999416i $$0.510877\pi$$
$$252$$ 0 0
$$253$$ −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 $$-0.498676\pi$$
0.00415917 + 0.999991i $$0.498676\pi$$
$$258$$ 0 0
$$259$$ 28.4701 1.76905
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1.51114 0.0933584
$$263$$ 0.147643 0.00910407 0.00455203 0.999990i $$-0.498551\pi$$
0.00455203 + 0.999990i $$0.498551\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −30.8671 −1.89258
$$267$$ 0 0
$$268$$ 2.55215 0.155897
$$269$$ 26.8573 1.63752 0.818759 0.574138i $$-0.194663\pi$$
0.818759 + 0.574138i $$0.194663\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ −22.8256 −1.37895
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.70964 −0.523311 −0.261656 0.965161i $$-0.584268\pi$$
−0.261656 + 0.965161i $$0.584268\pi$$
$$278$$ 17.0509 1.02264
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 20.3783 1.21567 0.607833 0.794065i $$-0.292039\pi$$
0.607833 + 0.794065i $$0.292039\pi$$
$$282$$ 0 0
$$283$$ −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
$$288$$ 0 0
$$289$$ 11.4286 0.672273
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −6.67799 −0.390800
$$293$$ −16.6780 −0.974339 −0.487169 0.873308i $$-0.661971\pi$$
−0.487169 + 0.873308i $$0.661971\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −17.8064 −1.03498
$$297$$ 0 0
$$298$$ 3.70471 0.214608
$$299$$ −16.5906 −0.959458
$$300$$ 0 0
$$301$$ 24.0415 1.38573
$$302$$ −0.396997 −0.0228446
$$303$$ 0 0
$$304$$ 13.8578 0.794797
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −9.58565 −0.547082 −0.273541 0.961860i $$-0.588195\pi$$
−0.273541 + 0.961860i $$0.588195\pi$$
$$308$$ −2.57628 −0.146797
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −14.5303 −0.823941 −0.411970 0.911197i $$-0.635159\pi$$
−0.411970 + 0.911197i $$0.635159\pi$$
$$312$$ 0 0
$$313$$ −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 $$-0.491144\pi$$
0.0278197 + 0.999613i $$0.491144\pi$$
$$318$$ 0 0
$$319$$ −1.80642 −0.101140
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −23.8163 −1.32723
$$323$$ −27.6414 −1.53801
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 14.7841 0.818818
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ −15.7921 −0.864107
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3.16992 −0.172676 −0.0863382 0.996266i $$-0.527517\pi$$
−0.0863382 + 0.996266i $$0.527517\pi$$
$$338$$ 5.10372 0.277606
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2.62222 0.142001
$$342$$ 0 0
$$343$$ 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 $$-0.542645\pi$$
−0.133572 + 0.991039i $$0.542645\pi$$
$$348$$ 0 0
$$349$$ 18.2034 0.974407 0.487203 0.873289i $$-0.338017\pi$$
0.487203 + 0.873289i $$0.338017\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.88739 0.153898
$$353$$ −22.4099 −1.19276 −0.596379 0.802703i $$-0.703395\pi$$
−0.596379 + 0.802703i $$0.703395\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0.326929 0.0173272
$$357$$ 0 0
$$358$$ −15.6128 −0.825165
$$359$$ −21.3274 −1.12562 −0.562809 0.826587i $$-0.690279\pi$$
−0.562809 + 0.826587i $$0.690279\pi$$
$$360$$ 0 0
$$361$$ 7.87601 0.414527
$$362$$ 1.11414 0.0585579
$$363$$ 0 0
$$364$$ −10.6855 −0.560072
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 35.1338 1.83397 0.916985 0.398921i $$-0.130615\pi$$
0.916985 + 0.398921i $$0.130615\pi$$
$$368$$ 10.6923 0.557374
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 35.2257 1.82883
$$372$$ 0 0
$$373$$ −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
$$378$$ 0 0
$$379$$ 2.36842 0.121657 0.0608287 0.998148i $$-0.480626\pi$$
0.0608287 + 0.998148i $$0.480626\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −17.4479 −0.892710
$$383$$ −1.21585 −0.0621271 −0.0310635 0.999517i $$-0.509889\pi$$
−0.0310635 + 0.999517i $$0.509889\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.2810 −0.726884
$$387$$ 0 0
$$388$$ 1.44785 0.0735035
$$389$$ −2.26671 −0.114927 −0.0574633 0.998348i $$-0.518301\pi$$
−0.0574633 + 0.998348i $$0.518301\pi$$
$$390$$ 0 0
$$391$$ −21.3274 −1.07857
$$392$$ −52.2607 −2.63957
$$393$$ 0 0
$$394$$ 4.63957 0.233738
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −18.4889 −0.927929 −0.463965 0.885854i $$-0.653574\pi$$
−0.463965 + 0.885854i $$0.653574\pi$$
$$398$$ −16.6539 −0.834782
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −17.5625 −0.877028 −0.438514 0.898724i $$-0.644495\pi$$
−0.438514 + 0.898724i $$0.644495\pi$$
$$402$$ 0 0
$$403$$ 10.8760 0.541773
$$404$$ −9.35599 −0.465478
$$405$$ 0 0
$$406$$ −10.7556 −0.533790
$$407$$ 5.80642 0.287814
$$408$$ 0 0
$$409$$ −21.3461 −1.05550 −0.527749 0.849400i $$-0.676964\pi$$
−0.527749 + 0.849400i $$0.676964\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.60042 −0.128113
$$413$$ −8.20342 −0.403664
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 11.9759 0.587165
$$417$$ 0 0
$$418$$ −6.29529 −0.307913
$$419$$ −28.8573 −1.40977 −0.704885 0.709321i $$-0.749001\pi$$
−0.704885 + 0.709321i $$0.749001\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ −22.0316 −1.06995
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.70471 0.179284
$$428$$ 5.88400 0.284414
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −9.24443 −0.445289 −0.222644 0.974900i $$-0.571469\pi$$
−0.222644 + 0.974900i $$0.571469\pi$$
$$432$$ 0 0
$$433$$ −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
$$438$$ 0 0
$$439$$ −36.5303 −1.74350 −0.871749 0.489952i $$-0.837014\pi$$
−0.871749 + 0.489952i $$0.837014\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 26.8542 1.27732
$$443$$ 38.2766 1.81857 0.909287 0.416170i $$-0.136628\pi$$
0.909287 + 0.416170i $$0.136628\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −31.5843 −1.49556
$$447$$ 0 0
$$448$$ 43.4050 2.05069
$$449$$ 31.8479 1.50300 0.751498 0.659735i $$-0.229332\pi$$
0.751498 + 0.659735i $$0.229332\pi$$
$$450$$ 0 0
$$451$$ −1.80642 −0.0850612
$$452$$ −0.927346 −0.0436187
$$453$$ 0 0
$$454$$ 7.70027 0.361391
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.39207 0.0651185 0.0325592 0.999470i $$-0.489634\pi$$
0.0325592 + 0.999470i $$0.489634\pi$$
$$458$$ −28.3269 −1.32363
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 7.70471 0.358844 0.179422 0.983772i $$-0.442577\pi$$
0.179422 + 0.983772i $$0.442577\pi$$
$$462$$ 0 0
$$463$$ 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 $$-0.596147\pi$$
−0.297482 + 0.954727i $$0.596147\pi$$
$$468$$ 0 0
$$469$$ −23.8163 −1.09973
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 5.13077 0.236163
$$473$$ 4.90321 0.225450
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −13.7364 −0.629605
$$477$$ 0 0
$$478$$ 23.0223 1.05301
$$479$$ −8.38715 −0.383219 −0.191609 0.981471i $$-0.561371\pi$$
−0.191609 + 0.981471i $$0.561371\pi$$
$$480$$ 0 0
$$481$$ 24.0830 1.09809
$$482$$ −1.63465 −0.0744561
$$483$$ 0 0
$$484$$ −0.525428 −0.0238831
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9.83500 0.445667 0.222833 0.974857i $$-0.428469\pi$$
0.222833 + 0.974857i $$0.428469\pi$$
$$488$$ −2.31708 −0.104890
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 32.9403 1.48657 0.743286 0.668973i $$-0.233266\pi$$
0.743286 + 0.668973i $$0.233266\pi$$
$$492$$ 0 0
$$493$$ −9.63158 −0.433785
$$494$$ −26.1106 −1.17477
$$495$$ 0 0
$$496$$ −7.00937 −0.314730
$$497$$ −2.10171 −0.0942746
$$498$$ 0 0
$$499$$ −1.63158 −0.0730397 −0.0365199 0.999333i $$-0.511627\pi$$
−0.0365199 + 0.999333i $$0.511627\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −1.31450 −0.0586689
$$503$$ −41.8622 −1.86654 −0.933272 0.359171i $$-0.883059\pi$$
−0.933272 + 0.359171i $$0.883059\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.85728 −0.215932
$$507$$ 0 0
$$508$$ −9.83056 −0.436160
$$509$$ 38.8573 1.72232 0.861159 0.508335i $$-0.169739\pi$$
0.861159 + 0.508335i $$0.169739\pi$$
$$510$$ 0 0
$$511$$ 62.3180 2.75679
$$512$$ −24.1131 −1.06566
$$513$$ 0 0
$$514$$ 0.161933 0.00714257
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −7.05086 −0.310096
$$518$$ 34.5718 1.51900
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11.1111 −0.486785 −0.243393 0.969928i $$-0.578260\pi$$
−0.243393 + 0.969928i $$0.578260\pi$$
$$522$$ 0 0
$$523$$ 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
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 13.3560 0.579055
$$533$$ −7.49240 −0.324532
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 14.8957 0.643396
$$537$$ 0 0
$$538$$ 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 0
$$544$$ 15.3951 0.660061
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 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 0
$$550$$ 0 0
$$551$$ 9.36488 0.398957
$$552$$ 0 0
$$553$$ −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 $$-0.295422\pi$$
0.599361 + 0.800479i $$0.295422\pi$$
$$558$$ 0 0
$$559$$ 20.3368 0.860154
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 24.7457 1.04384
$$563$$ 32.7926 1.38204 0.691022 0.722834i $$-0.257161\pi$$
0.691022 + 0.722834i $$0.257161\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −7.67752 −0.322710
$$567$$ 0 0
$$568$$ 1.31450 0.0551551
$$569$$ 8.88586 0.372515 0.186257 0.982501i $$-0.440364\pi$$
0.186257 + 0.982501i $$0.440364\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −10.7556 −0.448929
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −27.1338 −1.12960 −0.564798 0.825229i $$-0.691046\pi$$
−0.564798 + 0.825229i $$0.691046\pi$$
$$578$$ 13.8780 0.577250
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 14.2351 0.590570
$$582$$ 0 0
$$583$$ 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 $$-0.427382\pi$$
0.226163 + 0.974089i $$0.427382\pi$$
$$588$$ 0 0
$$589$$ −13.5941 −0.560136
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −15.5210 −0.637908
$$593$$ 23.7003 0.973253 0.486627 0.873610i $$-0.338227\pi$$
0.486627 + 0.873610i $$0.338227\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.60300 −0.0656616
$$597$$ 0 0
$$598$$ −20.1463 −0.823842
$$599$$ −41.7146 −1.70441 −0.852205 0.523208i $$-0.824735\pi$$
−0.852205 + 0.523208i $$0.824735\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 0.171778 0.00698953
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −19.9826 −0.811071 −0.405535 0.914079i $$-0.632915\pi$$
−0.405535 + 0.914079i $$0.632915\pi$$
$$608$$ −14.9688 −0.607066
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −29.2444 −1.18310
$$612$$ 0 0
$$613$$ 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 $$-0.208900\pi$$
0.792268 + 0.610174i $$0.208900\pi$$
$$618$$ 0 0
$$619$$ −23.0923 −0.928160 −0.464080 0.885793i $$-0.653615\pi$$
−0.464080 + 0.885793i $$0.653615\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −17.6445 −0.707480
$$623$$ −3.05086 −0.122230
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −25.5397 −1.02077
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 1.20294 0.0477750
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 70.6820 2.80052
$$638$$ −2.19358 −0.0868445
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 6.25380 0.247010 0.123505 0.992344i $$-0.460586\pi$$
0.123505 + 0.992344i $$0.460586\pi$$
$$642$$ 0 0
$$643$$ 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 $$-0.630624\pi$$
−0.398945 + 0.916975i $$0.630624\pi$$
$$648$$ 0 0
$$649$$ −1.67307 −0.0656738
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.39700 −0.250526
$$653$$ 10.6222 0.415679 0.207840 0.978163i $$-0.433357\pi$$
0.207840 + 0.978163i $$0.433357\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.82870 0.188529
$$657$$ 0 0
$$658$$ −41.9813 −1.63660
$$659$$ −10.1017 −0.393507 −0.196753 0.980453i $$-0.563040\pi$$
−0.196753 + 0.980453i $$0.563040\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ −8.90321 −0.345512
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.22570 0.279780
$$668$$ 6.83314 0.264382
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.755569 0.0291684
$$672$$ 0 0
$$673$$ 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 $$-0.586211\pi$$
−0.267541 + 0.963546i $$0.586211\pi$$
$$678$$ 0 0
$$679$$ −13.5111 −0.518510
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3.18421 0.121930
$$683$$ 10.3970 0.397830 0.198915 0.980017i $$-0.436258\pi$$
0.198915 + 0.980017i $$0.436258\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 59.7877 2.28270
$$687$$ 0 0
$$688$$ −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
$$693$$ 0 0
$$694$$ −6.04287 −0.229384
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.63158 −0.364822
$$698$$ 22.1048 0.836678
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −48.9688 −1.84953 −0.924764 0.380542i $$-0.875737\pi$$
−0.924764 + 0.380542i $$0.875737\pi$$
$$702$$ 0 0
$$703$$ −30.1017 −1.13531
$$704$$ 8.85236 0.333636
$$705$$ 0 0
$$706$$ −27.2128 −1.02417
$$707$$ 87.3087 3.28358
$$708$$ 0 0
$$709$$ −37.2672 −1.39960 −0.699799 0.714340i $$-0.746727\pi$$
−0.699799 + 0.714340i $$0.746727\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1.90813 0.0715103
$$713$$ −10.4889 −0.392811
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.75557 0.252467
$$717$$ 0 0
$$718$$ −25.8983 −0.966516
$$719$$ −5.83500 −0.217609 −0.108804 0.994063i $$-0.534702\pi$$
−0.108804 + 0.994063i $$0.534702\pi$$
$$720$$ 0 0
$$721$$ 24.2667 0.903739
$$722$$ 9.56400 0.355935
$$723$$ 0 0
$$724$$ −0.482081 −0.0179164
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −46.8385 −1.73715 −0.868573 0.495562i $$-0.834962\pi$$
−0.868573 + 0.495562i $$0.834962\pi$$
$$728$$ −62.3663 −2.31145
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 26.1432 0.966941
$$732$$ 0 0
$$733$$ −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
$$738$$ 0 0
$$739$$ 5.65433 0.207998 0.103999 0.994577i $$-0.466836\pi$$
0.103999 + 0.994577i $$0.466836\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 42.7753 1.57033
$$743$$ −4.50622 −0.165317 −0.0826585 0.996578i $$-0.526341\pi$$
−0.0826585 + 0.996578i $$0.526341\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −20.6494 −0.756029
$$747$$ 0 0
$$748$$ −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
$$753$$ 0 0
$$754$$ −9.09817 −0.331336
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −46.6637 −1.69602 −0.848011 0.529979i $$-0.822200\pi$$
−0.848011 + 0.529979i $$0.822200\pi$$
$$758$$ 2.87601 0.104462
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14.9304 −0.541227 −0.270613 0.962688i $$-0.587227\pi$$
−0.270613 + 0.962688i $$0.587227\pi$$
$$762$$ 0 0
$$763$$ 77.0518 2.78946
$$764$$ 7.54956 0.273134
$$765$$ 0 0
$$766$$ −1.47643 −0.0533457
$$767$$ −6.93930 −0.250564
$$768$$ 0 0
$$769$$ 38.8573 1.40123 0.700615 0.713540i $$-0.252909\pi$$
0.700615 + 0.713540i $$0.252909\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 6.17929 0.222397
$$773$$ 36.3368 1.30694 0.653471 0.756951i $$-0.273312\pi$$
0.653471 + 0.756951i $$0.273312\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 8.45044 0.303353
$$777$$ 0 0
$$778$$ −2.75251 −0.0986821
$$779$$ 9.36488 0.335532
$$780$$ 0 0
$$781$$ −0.428639 −0.0153379
$$782$$ −25.8983 −0.926121
$$783$$ 0 0
$$784$$ −45.5531 −1.62690
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −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 0
$$790$$ 0 0
$$791$$ 8.65386 0.307696
$$792$$ 0 0
$$793$$ 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 $$-0.407762\pi$$
0.285736 + 0.958308i $$0.407762\pi$$
$$798$$ 0 0
$$799$$ −37.5941 −1.32998
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −21.3265 −0.753063
$$803$$ 12.7096 0.448513
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.2070 0.465195
$$807$$ 0 0
$$808$$ −54.6065 −1.92105
$$809$$ −25.7431 −0.905081 −0.452540 0.891744i $$-0.649482\pi$$
−0.452540 + 0.891744i $$0.649482\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ 7.05086 0.247132
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −25.4193 −0.889308
$$818$$ −25.9210 −0.906308
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −24.1748 −0.843708 −0.421854 0.906664i $$-0.638620\pi$$
−0.421854 + 0.906664i $$0.638620\pi$$
$$822$$ 0 0
$$823$$ −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 $$-0.386079\pi$$
0.350301 + 0.936637i $$0.386079\pi$$
$$828$$ 0 0
$$829$$ 31.4322 1.09168 0.545842 0.837888i $$-0.316210\pi$$
0.545842 + 0.837888i $$0.316210\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 36.7164 1.27291
$$833$$ 90.8627 3.14821
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.72393 0.0942089
$$837$$ 0 0
$$838$$ −35.0420 −1.21050
$$839$$ 52.8988 1.82627 0.913134 0.407659i $$-0.133655\pi$$
0.913134 + 0.407659i $$0.133655\pi$$
$$840$$ 0 0
$$841$$ −25.7368 −0.887477
$$842$$ −43.0954 −1.48517
$$843$$ 0 0
$$844$$ −1.02906 −0.0354216
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.90321 0.168476
$$848$$ −19.2039 −0.659465
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −23.2257 −0.796167
$$852$$ 0 0
$$853$$ −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 $$-0.358682\pi$$
0.429522 + 0.903057i $$0.358682\pi$$
$$858$$ 0 0
$$859$$ 1.84791 0.0630499 0.0315250 0.999503i $$-0.489964\pi$$
0.0315250 + 0.999503i $$0.489964\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −11.2257 −0.382349
$$863$$ −32.6824 −1.11252 −0.556262 0.831007i $$-0.687765\pi$$
−0.556262 + 0.831007i $$0.687765\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −7.63254 −0.259364
$$867$$ 0 0
$$868$$ −6.75557 −0.229299
$$869$$ −6.42864 −0.218077
$$870$$ 0 0
$$871$$ −20.1463 −0.682630
$$872$$ −48.1915 −1.63197
$$873$$ 0 0
$$874$$ 25.1811 0.851765
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 49.1798 1.66068 0.830341 0.557255i $$-0.188145\pi$$
0.830341 + 0.557255i $$0.188145\pi$$
$$878$$ −44.3595 −1.49706
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −33.8163 −1.13930 −0.569650 0.821888i $$-0.692921\pi$$
−0.569650 + 0.821888i $$0.692921\pi$$
$$882$$ 0 0
$$883$$ 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 $$-0.540940\pi$$
−0.128264 + 0.991740i $$0.540940\pi$$
$$888$$ 0 0
$$889$$ 91.7373 3.07677
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.6663 0.457581
$$893$$ 36.5531 1.22320
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 24.3926 0.814898
$$897$$ 0 0
$$898$$ 38.6735 1.29055
$$899$$ −4.73683 −0.157982
$$900$$ 0 0
$$901$$ 38.3051 1.27613
$$902$$ −2.19358 −0.0730381
$$903$$ 0 0
$$904$$ −5.41249 −0.180017
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −30.3970 −1.00932 −0.504658 0.863319i $$-0.668381\pi$$
−0.504658 + 0.863319i $$0.668381\pi$$
$$908$$ −3.33185 −0.110571
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45.3274 1.50176 0.750882 0.660436i $$-0.229629\pi$$
0.750882 + 0.660436i $$0.229629\pi$$
$$912$$ 0 0
$$913$$ 2.90321 0.0960823
$$914$$ 1.69042 0.0559142
$$915$$ 0 0
$$916$$ 12.2569 0.404978
$$917$$ 6.10171 0.201496
$$918$$ 0 0
$$919$$ −16.3269 −0.538576 −0.269288 0.963060i $$-0.586788\pi$$
−0.269288 + 0.963060i $$0.586788\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 9.35599 0.308123
$$923$$ −1.77784 −0.0585184
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 5.68598 0.186853
$$927$$ 0 0
$$928$$ −5.21585 −0.171219
$$929$$ 29.6128 0.971566 0.485783 0.874079i $$-0.338535\pi$$
0.485783 + 0.874079i $$0.338535\pi$$
$$930$$ 0 0
$$931$$ −88.3466 −2.89544
$$932$$ −0.748060 −0.0245035
$$933$$ 0 0
$$934$$ −15.6128 −0.510868
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 8.92195 0.291467 0.145734 0.989324i $$-0.453446\pi$$
0.145734 + 0.989324i $$0.453446\pi$$
$$938$$ −28.9206 −0.944290
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 22.2766 0.726195 0.363097 0.931751i $$-0.381719\pi$$
0.363097 + 0.931751i $$0.381719\pi$$
$$942$$ 0 0
$$943$$ 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 $$-0.756960\pi$$
−0.722398 + 0.691477i $$0.756960\pi$$
$$948$$ 0 0
$$949$$ 52.7150 1.71120
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −80.1727 −2.59841
$$953$$ −20.5575 −0.665924 −0.332962 0.942940i $$-0.608048\pi$$
−0.332962 + 0.942940i $$0.608048\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −9.96158 −0.322180
$$957$$ 0 0
$$958$$ −10.1847 −0.329052
$$959$$ −92.1659 −2.97619
$$960$$ 0 0
$$961$$ −24.1240 −0.778193
$$962$$ 29.2444 0.942878
$$963$$ 0 0
$$964$$ 0.707300 0.0227806
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 27.4839 0.883824 0.441912 0.897058i $$-0.354300\pi$$
0.441912 + 0.897058i $$0.354300\pi$$
$$968$$ −3.06668 −0.0985667
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5.81532 0.186622 0.0933112 0.995637i $$-0.470255\pi$$
0.0933112 + 0.995637i $$0.470255\pi$$
$$972$$ 0 0
$$973$$ 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 $$-0.781790\pi$$
−0.774085 + 0.633081i $$0.781790\pi$$
$$978$$ 0 0
$$979$$ −0.622216 −0.0198861
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 40.0000 1.27645
$$983$$ 49.9724 1.59387 0.796936 0.604064i $$-0.206453\pi$$
0.796936 + 0.604064i $$0.206453\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −11.6958 −0.372471
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ −2.55215 −0.0809492
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −4.33138 −0.137176 −0.0685880 0.997645i $$-0.521849\pi$$
−0.0685880 + 0.997645i $$0.521849\pi$$
$$998$$ −1.98126 −0.0627158
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2475.2.a.y.1.3 3
3.2 odd 2 825.2.a.n.1.1 3
5.2 odd 4 495.2.c.d.199.4 6
5.3 odd 4 495.2.c.d.199.3 6
5.4 even 2 2475.2.a.be.1.1 3
15.2 even 4 165.2.c.a.34.3 6
15.8 even 4 165.2.c.a.34.4 yes 6
15.14 odd 2 825.2.a.h.1.3 3
33.32 even 2 9075.2.a.cc.1.3 3
60.23 odd 4 2640.2.d.i.529.5 6
60.47 odd 4 2640.2.d.i.529.2 6
165.32 odd 4 1815.2.c.d.364.4 6
165.98 odd 4 1815.2.c.d.364.3 6
165.164 even 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 15.2 even 4
165.2.c.a.34.4 yes 6 15.8 even 4
495.2.c.d.199.3 6 5.3 odd 4
495.2.c.d.199.4 6 5.2 odd 4
825.2.a.h.1.3 3 15.14 odd 2
825.2.a.n.1.1 3 3.2 odd 2
1815.2.c.d.364.3 6 165.98 odd 4
1815.2.c.d.364.4 6 165.32 odd 4
2475.2.a.y.1.3 3 1.1 even 1 trivial
2475.2.a.be.1.1 3 5.4 even 2
2640.2.d.i.529.2 6 60.47 odd 4
2640.2.d.i.529.5 6 60.23 odd 4
9075.2.a.cc.1.3 3 33.32 even 2
9075.2.a.ck.1.1 3 165.164 even 2