# Properties

 Label 825.2.a.l.1.2 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 1.2 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.903212 q^{7} -1.21432 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.311108 q^{2} +1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.903212 q^{7} -1.21432 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.90321 q^{12} +2.90321 q^{13} -0.280996 q^{14} +3.42864 q^{16} +2.28100 q^{17} +0.311108 q^{18} +2.42864 q^{19} -0.903212 q^{21} +0.311108 q^{22} +4.00000 q^{23} -1.21432 q^{24} +0.903212 q^{26} +1.00000 q^{27} +1.71900 q^{28} +7.05086 q^{29} -2.62222 q^{31} +3.49532 q^{32} +1.00000 q^{33} +0.709636 q^{34} -1.90321 q^{36} +5.80642 q^{37} +0.755569 q^{38} +2.90321 q^{39} -10.6637 q^{41} -0.280996 q^{42} +10.7096 q^{43} -1.90321 q^{44} +1.24443 q^{46} +0.949145 q^{47} +3.42864 q^{48} -6.18421 q^{49} +2.28100 q^{51} -5.52543 q^{52} +0.815792 q^{53} +0.311108 q^{54} +1.09679 q^{56} +2.42864 q^{57} +2.19358 q^{58} -1.67307 q^{59} -7.24443 q^{61} -0.815792 q^{62} -0.903212 q^{63} -5.76986 q^{64} +0.311108 q^{66} +12.8573 q^{67} -4.34122 q^{68} +4.00000 q^{69} +9.28592 q^{71} -1.21432 q^{72} -5.65878 q^{73} +1.80642 q^{74} -4.62222 q^{76} -0.903212 q^{77} +0.903212 q^{78} -16.5303 q^{79} +1.00000 q^{81} -3.31756 q^{82} +7.76049 q^{83} +1.71900 q^{84} +3.33185 q^{86} +7.05086 q^{87} -1.21432 q^{88} +6.13335 q^{89} -2.62222 q^{91} -7.61285 q^{92} -2.62222 q^{93} +0.295286 q^{94} +3.49532 q^{96} -12.4701 q^{97} -1.92396 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 + 3 * q^3 + q^4 + q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 $$3 q + q^{2} + 3 q^{3} + q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{11} + q^{12} + 2 q^{13} + 6 q^{14} - 3 q^{16} + q^{18} - 6 q^{19} + 4 q^{21} + q^{22} + 12 q^{23} + 3 q^{24} - 4 q^{26} + 3 q^{27} + 12 q^{28} + 8 q^{29} - 8 q^{31} - 3 q^{32} + 3 q^{33} - 18 q^{34} + q^{36} + 4 q^{37} + 2 q^{38} + 2 q^{39} + 8 q^{41} + 6 q^{42} + 12 q^{43} + q^{44} + 4 q^{46} + 16 q^{47} - 3 q^{48} - 5 q^{49} - 10 q^{52} + 16 q^{53} + q^{54} + 10 q^{56} - 6 q^{57} + 20 q^{58} + 8 q^{59} - 22 q^{61} - 16 q^{62} + 4 q^{63} - 11 q^{64} + q^{66} + 12 q^{67} - 20 q^{68} + 12 q^{69} - 12 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 14 q^{76} + 4 q^{77} - 4 q^{78} - 10 q^{79} + 3 q^{81} + 4 q^{82} - 10 q^{83} + 12 q^{84} - 10 q^{86} + 8 q^{87} + 3 q^{88} + 18 q^{89} - 8 q^{91} + 4 q^{92} - 8 q^{93} - 12 q^{94} - 3 q^{96} + 16 q^{97} + 21 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + q^2 + 3 * q^3 + q^4 + q^6 + 4 * q^7 + 3 * q^8 + 3 * q^9 + 3 * q^11 + q^12 + 2 * q^13 + 6 * q^14 - 3 * q^16 + q^18 - 6 * q^19 + 4 * q^21 + q^22 + 12 * q^23 + 3 * q^24 - 4 * q^26 + 3 * q^27 + 12 * q^28 + 8 * q^29 - 8 * q^31 - 3 * q^32 + 3 * q^33 - 18 * q^34 + q^36 + 4 * q^37 + 2 * q^38 + 2 * q^39 + 8 * q^41 + 6 * q^42 + 12 * q^43 + q^44 + 4 * q^46 + 16 * q^47 - 3 * q^48 - 5 * q^49 - 10 * q^52 + 16 * q^53 + q^54 + 10 * q^56 - 6 * q^57 + 20 * q^58 + 8 * q^59 - 22 * q^61 - 16 * q^62 + 4 * q^63 - 11 * q^64 + q^66 + 12 * q^67 - 20 * q^68 + 12 * q^69 - 12 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 14 * q^76 + 4 * q^77 - 4 * q^78 - 10 * q^79 + 3 * q^81 + 4 * q^82 - 10 * q^83 + 12 * q^84 - 10 * q^86 + 8 * q^87 + 3 * q^88 + 18 * q^89 - 8 * q^91 + 4 * q^92 - 8 * q^93 - 12 * q^94 - 3 * q^96 + 16 * q^97 + 21 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.311108 0.219986 0.109993 0.993932i $$-0.464917\pi$$
0.109993 + 0.993932i $$0.464917\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.90321 −0.951606
$$5$$ 0 0
$$6$$ 0.311108 0.127009
$$7$$ −0.903212 −0.341382 −0.170691 0.985325i $$-0.554600\pi$$
−0.170691 + 0.985325i $$0.554600\pi$$
$$8$$ −1.21432 −0.429327
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −1.90321 −0.549410
$$13$$ 2.90321 0.805206 0.402603 0.915375i $$-0.368106\pi$$
0.402603 + 0.915375i $$0.368106\pi$$
$$14$$ −0.280996 −0.0750994
$$15$$ 0 0
$$16$$ 3.42864 0.857160
$$17$$ 2.28100 0.553223 0.276611 0.960982i $$-0.410789\pi$$
0.276611 + 0.960982i $$0.410789\pi$$
$$18$$ 0.311108 0.0733288
$$19$$ 2.42864 0.557168 0.278584 0.960412i $$-0.410135\pi$$
0.278584 + 0.960412i $$0.410135\pi$$
$$20$$ 0 0
$$21$$ −0.903212 −0.197097
$$22$$ 0.311108 0.0663284
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.21432 −0.247872
$$25$$ 0 0
$$26$$ 0.903212 0.177134
$$27$$ 1.00000 0.192450
$$28$$ 1.71900 0.324861
$$29$$ 7.05086 1.30931 0.654655 0.755927i $$-0.272814\pi$$
0.654655 + 0.755927i $$0.272814\pi$$
$$30$$ 0 0
$$31$$ −2.62222 −0.470964 −0.235482 0.971879i $$-0.575667\pi$$
−0.235482 + 0.971879i $$0.575667\pi$$
$$32$$ 3.49532 0.617890
$$33$$ 1.00000 0.174078
$$34$$ 0.709636 0.121702
$$35$$ 0 0
$$36$$ −1.90321 −0.317202
$$37$$ 5.80642 0.954570 0.477285 0.878749i $$-0.341621\pi$$
0.477285 + 0.878749i $$0.341621\pi$$
$$38$$ 0.755569 0.122569
$$39$$ 2.90321 0.464886
$$40$$ 0 0
$$41$$ −10.6637 −1.66539 −0.832695 0.553731i $$-0.813203\pi$$
−0.832695 + 0.553731i $$0.813203\pi$$
$$42$$ −0.280996 −0.0433587
$$43$$ 10.7096 1.63320 0.816602 0.577201i $$-0.195855\pi$$
0.816602 + 0.577201i $$0.195855\pi$$
$$44$$ −1.90321 −0.286920
$$45$$ 0 0
$$46$$ 1.24443 0.183481
$$47$$ 0.949145 0.138447 0.0692235 0.997601i $$-0.477948\pi$$
0.0692235 + 0.997601i $$0.477948\pi$$
$$48$$ 3.42864 0.494881
$$49$$ −6.18421 −0.883458
$$50$$ 0 0
$$51$$ 2.28100 0.319403
$$52$$ −5.52543 −0.766239
$$53$$ 0.815792 0.112058 0.0560288 0.998429i $$-0.482156\pi$$
0.0560288 + 0.998429i $$0.482156\pi$$
$$54$$ 0.311108 0.0423364
$$55$$ 0 0
$$56$$ 1.09679 0.146564
$$57$$ 2.42864 0.321681
$$58$$ 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
$$61$$ −7.24443 −0.927554 −0.463777 0.885952i $$-0.653506\pi$$
−0.463777 + 0.885952i $$0.653506\pi$$
$$62$$ −0.815792 −0.103606
$$63$$ −0.903212 −0.113794
$$64$$ −5.76986 −0.721232
$$65$$ 0 0
$$66$$ 0.311108 0.0382947
$$67$$ 12.8573 1.57077 0.785383 0.619010i $$-0.212466\pi$$
0.785383 + 0.619010i $$0.212466\pi$$
$$68$$ −4.34122 −0.526450
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 9.28592 1.10204 0.551018 0.834493i $$-0.314240\pi$$
0.551018 + 0.834493i $$0.314240\pi$$
$$72$$ −1.21432 −0.143109
$$73$$ −5.65878 −0.662310 −0.331155 0.943576i $$-0.607438\pi$$
−0.331155 + 0.943576i $$0.607438\pi$$
$$74$$ 1.80642 0.209993
$$75$$ 0 0
$$76$$ −4.62222 −0.530204
$$77$$ −0.903212 −0.102931
$$78$$ 0.903212 0.102269
$$79$$ −16.5303 −1.85981 −0.929905 0.367800i $$-0.880111\pi$$
−0.929905 + 0.367800i $$0.880111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −3.31756 −0.366363
$$83$$ 7.76049 0.851825 0.425912 0.904764i $$-0.359953\pi$$
0.425912 + 0.904764i $$0.359953\pi$$
$$84$$ 1.71900 0.187559
$$85$$ 0 0
$$86$$ 3.33185 0.359283
$$87$$ 7.05086 0.755931
$$88$$ −1.21432 −0.129447
$$89$$ 6.13335 0.650134 0.325067 0.945691i $$-0.394613\pi$$
0.325067 + 0.945691i $$0.394613\pi$$
$$90$$ 0 0
$$91$$ −2.62222 −0.274883
$$92$$ −7.61285 −0.793694
$$93$$ −2.62222 −0.271911
$$94$$ 0.295286 0.0304565
$$95$$ 0 0
$$96$$ 3.49532 0.356739
$$97$$ −12.4701 −1.26615 −0.633075 0.774091i $$-0.718207\pi$$
−0.633075 + 0.774091i $$0.718207\pi$$
$$98$$ −1.92396 −0.194349
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 16.1748 1.60946 0.804728 0.593643i $$-0.202311\pi$$
0.804728 + 0.593643i $$0.202311\pi$$
$$102$$ 0.709636 0.0702644
$$103$$ −17.1526 −1.69009 −0.845046 0.534693i $$-0.820427\pi$$
−0.845046 + 0.534693i $$0.820427\pi$$
$$104$$ −3.52543 −0.345697
$$105$$ 0 0
$$106$$ 0.253799 0.0246512
$$107$$ −13.5669 −1.31156 −0.655782 0.754951i $$-0.727661\pi$$
−0.655782 + 0.754951i $$0.727661\pi$$
$$108$$ −1.90321 −0.183137
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 5.80642 0.551121
$$112$$ −3.09679 −0.292619
$$113$$ 14.2351 1.33912 0.669561 0.742757i $$-0.266482\pi$$
0.669561 + 0.742757i $$0.266482\pi$$
$$114$$ 0.755569 0.0707655
$$115$$ 0 0
$$116$$ −13.4193 −1.24595
$$117$$ 2.90321 0.268402
$$118$$ −0.520505 −0.0479164
$$119$$ −2.06022 −0.188860
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −2.25380 −0.204049
$$123$$ −10.6637 −0.961514
$$124$$ 4.99063 0.448172
$$125$$ 0 0
$$126$$ −0.280996 −0.0250331
$$127$$ 11.0049 0.976529 0.488264 0.872696i $$-0.337630\pi$$
0.488264 + 0.872696i $$0.337630\pi$$
$$128$$ −8.78568 −0.776552
$$129$$ 10.7096 0.942931
$$130$$ 0 0
$$131$$ 1.24443 0.108726 0.0543632 0.998521i $$-0.482687\pi$$
0.0543632 + 0.998521i $$0.482687\pi$$
$$132$$ −1.90321 −0.165653
$$133$$ −2.19358 −0.190207
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −2.76986 −0.237513
$$137$$ −4.42864 −0.378364 −0.189182 0.981942i $$-0.560584\pi$$
−0.189182 + 0.981942i $$0.560584\pi$$
$$138$$ 1.24443 0.105933
$$139$$ 0.917502 0.0778215 0.0389108 0.999243i $$-0.487611\pi$$
0.0389108 + 0.999243i $$0.487611\pi$$
$$140$$ 0 0
$$141$$ 0.949145 0.0799324
$$142$$ 2.88892 0.242433
$$143$$ 2.90321 0.242779
$$144$$ 3.42864 0.285720
$$145$$ 0 0
$$146$$ −1.76049 −0.145699
$$147$$ −6.18421 −0.510065
$$148$$ −11.0509 −0.908375
$$149$$ 2.19358 0.179705 0.0898524 0.995955i $$-0.471360\pi$$
0.0898524 + 0.995955i $$0.471360\pi$$
$$150$$ 0 0
$$151$$ −10.4286 −0.848671 −0.424335 0.905505i $$-0.639492\pi$$
−0.424335 + 0.905505i $$0.639492\pi$$
$$152$$ −2.94914 −0.239207
$$153$$ 2.28100 0.184408
$$154$$ −0.280996 −0.0226433
$$155$$ 0 0
$$156$$ −5.52543 −0.442388
$$157$$ 5.80642 0.463403 0.231702 0.972787i $$-0.425571\pi$$
0.231702 + 0.972787i $$0.425571\pi$$
$$158$$ −5.14272 −0.409133
$$159$$ 0.815792 0.0646965
$$160$$ 0 0
$$161$$ −3.61285 −0.284732
$$162$$ 0.311108 0.0244429
$$163$$ 11.0509 0.865570 0.432785 0.901497i $$-0.357531\pi$$
0.432785 + 0.901497i $$0.357531\pi$$
$$164$$ 20.2953 1.58480
$$165$$ 0 0
$$166$$ 2.41435 0.187390
$$167$$ 13.9541 1.07980 0.539899 0.841730i $$-0.318462\pi$$
0.539899 + 0.841730i $$0.318462\pi$$
$$168$$ 1.09679 0.0846190
$$169$$ −4.57136 −0.351643
$$170$$ 0 0
$$171$$ 2.42864 0.185723
$$172$$ −20.3827 −1.55417
$$173$$ −18.8430 −1.43261 −0.716303 0.697789i $$-0.754167\pi$$
−0.716303 + 0.697789i $$0.754167\pi$$
$$174$$ 2.19358 0.166295
$$175$$ 0 0
$$176$$ 3.42864 0.258443
$$177$$ −1.67307 −0.125756
$$178$$ 1.90813 0.143021
$$179$$ 4.85728 0.363050 0.181525 0.983386i $$-0.441897\pi$$
0.181525 + 0.983386i $$0.441897\pi$$
$$180$$ 0 0
$$181$$ −16.7971 −1.24852 −0.624258 0.781219i $$-0.714598\pi$$
−0.624258 + 0.781219i $$0.714598\pi$$
$$182$$ −0.815792 −0.0604705
$$183$$ −7.24443 −0.535524
$$184$$ −4.85728 −0.358083
$$185$$ 0 0
$$186$$ −0.815792 −0.0598168
$$187$$ 2.28100 0.166803
$$188$$ −1.80642 −0.131747
$$189$$ −0.903212 −0.0656990
$$190$$ 0 0
$$191$$ 16.8573 1.21975 0.609875 0.792498i $$-0.291220\pi$$
0.609875 + 0.792498i $$0.291220\pi$$
$$192$$ −5.76986 −0.416404
$$193$$ −24.6178 −1.77203 −0.886013 0.463661i $$-0.846536\pi$$
−0.886013 + 0.463661i $$0.846536\pi$$
$$194$$ −3.87955 −0.278536
$$195$$ 0 0
$$196$$ 11.7699 0.840704
$$197$$ 14.8716 1.05956 0.529778 0.848137i $$-0.322275\pi$$
0.529778 + 0.848137i $$0.322275\pi$$
$$198$$ 0.311108 0.0221095
$$199$$ −11.2257 −0.795768 −0.397884 0.917436i $$-0.630255\pi$$
−0.397884 + 0.917436i $$0.630255\pi$$
$$200$$ 0 0
$$201$$ 12.8573 0.906883
$$202$$ 5.03212 0.354059
$$203$$ −6.36842 −0.446975
$$204$$ −4.34122 −0.303946
$$205$$ 0 0
$$206$$ −5.33630 −0.371797
$$207$$ 4.00000 0.278019
$$208$$ 9.95407 0.690190
$$209$$ 2.42864 0.167993
$$210$$ 0 0
$$211$$ −11.9398 −0.821968 −0.410984 0.911643i $$-0.634815\pi$$
−0.410984 + 0.911643i $$0.634815\pi$$
$$212$$ −1.55262 −0.106635
$$213$$ 9.28592 0.636261
$$214$$ −4.22077 −0.288526
$$215$$ 0 0
$$216$$ −1.21432 −0.0826240
$$217$$ 2.36842 0.160779
$$218$$ −3.11108 −0.210709
$$219$$ −5.65878 −0.382385
$$220$$ 0 0
$$221$$ 6.62222 0.445458
$$222$$ 1.80642 0.121239
$$223$$ 21.8064 1.46027 0.730133 0.683305i $$-0.239458\pi$$
0.730133 + 0.683305i $$0.239458\pi$$
$$224$$ −3.15701 −0.210937
$$225$$ 0 0
$$226$$ 4.42864 0.294589
$$227$$ 3.19850 0.212292 0.106146 0.994351i $$-0.466149\pi$$
0.106146 + 0.994351i $$0.466149\pi$$
$$228$$ −4.62222 −0.306114
$$229$$ 7.12399 0.470766 0.235383 0.971903i $$-0.424366\pi$$
0.235383 + 0.971903i $$0.424366\pi$$
$$230$$ 0 0
$$231$$ −0.903212 −0.0594270
$$232$$ −8.56199 −0.562122
$$233$$ −19.5254 −1.27915 −0.639577 0.768727i $$-0.720890\pi$$
−0.639577 + 0.768727i $$0.720890\pi$$
$$234$$ 0.903212 0.0590448
$$235$$ 0 0
$$236$$ 3.18421 0.207274
$$237$$ −16.5303 −1.07376
$$238$$ −0.640951 −0.0415467
$$239$$ −21.9813 −1.42185 −0.710925 0.703268i $$-0.751723\pi$$
−0.710925 + 0.703268i $$0.751723\pi$$
$$240$$ 0 0
$$241$$ 5.34614 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$242$$ 0.311108 0.0199988
$$243$$ 1.00000 0.0641500
$$244$$ 13.7877 0.882666
$$245$$ 0 0
$$246$$ −3.31756 −0.211520
$$247$$ 7.05086 0.448635
$$248$$ 3.18421 0.202197
$$249$$ 7.76049 0.491801
$$250$$ 0 0
$$251$$ −23.7748 −1.50065 −0.750325 0.661069i $$-0.770103\pi$$
−0.750325 + 0.661069i $$0.770103\pi$$
$$252$$ 1.71900 0.108287
$$253$$ 4.00000 0.251478
$$254$$ 3.42372 0.214823
$$255$$ 0 0
$$256$$ 8.80642 0.550401
$$257$$ −8.13335 −0.507345 −0.253672 0.967290i $$-0.581638\pi$$
−0.253672 + 0.967290i $$0.581638\pi$$
$$258$$ 3.33185 0.207432
$$259$$ −5.24443 −0.325873
$$260$$ 0 0
$$261$$ 7.05086 0.436437
$$262$$ 0.387152 0.0239183
$$263$$ 22.9032 1.41227 0.706136 0.708076i $$-0.250437\pi$$
0.706136 + 0.708076i $$0.250437\pi$$
$$264$$ −1.21432 −0.0747362
$$265$$ 0 0
$$266$$ −0.682439 −0.0418430
$$267$$ 6.13335 0.375355
$$268$$ −24.4701 −1.49475
$$269$$ 11.8350 0.721593 0.360796 0.932645i $$-0.382505\pi$$
0.360796 + 0.932645i $$0.382505\pi$$
$$270$$ 0 0
$$271$$ 14.8988 0.905036 0.452518 0.891755i $$-0.350526\pi$$
0.452518 + 0.891755i $$0.350526\pi$$
$$272$$ 7.82071 0.474200
$$273$$ −2.62222 −0.158704
$$274$$ −1.37778 −0.0832350
$$275$$ 0 0
$$276$$ −7.61285 −0.458240
$$277$$ −27.6686 −1.66245 −0.831223 0.555939i $$-0.812359\pi$$
−0.831223 + 0.555939i $$0.812359\pi$$
$$278$$ 0.285442 0.0171197
$$279$$ −2.62222 −0.156988
$$280$$ 0 0
$$281$$ 9.80642 0.585002 0.292501 0.956265i $$-0.405512\pi$$
0.292501 + 0.956265i $$0.405512\pi$$
$$282$$ 0.295286 0.0175840
$$283$$ −19.0049 −1.12973 −0.564863 0.825185i $$-0.691071\pi$$
−0.564863 + 0.825185i $$0.691071\pi$$
$$284$$ −17.6731 −1.04870
$$285$$ 0 0
$$286$$ 0.903212 0.0534080
$$287$$ 9.63158 0.568534
$$288$$ 3.49532 0.205963
$$289$$ −11.7971 −0.693944
$$290$$ 0 0
$$291$$ −12.4701 −0.731012
$$292$$ 10.7699 0.630258
$$293$$ −30.7511 −1.79650 −0.898250 0.439485i $$-0.855161\pi$$
−0.898250 + 0.439485i $$0.855161\pi$$
$$294$$ −1.92396 −0.112207
$$295$$ 0 0
$$296$$ −7.05086 −0.409823
$$297$$ 1.00000 0.0580259
$$298$$ 0.682439 0.0395326
$$299$$ 11.6128 0.671588
$$300$$ 0 0
$$301$$ −9.67307 −0.557547
$$302$$ −3.24443 −0.186696
$$303$$ 16.1748 0.929220
$$304$$ 8.32693 0.477582
$$305$$ 0 0
$$306$$ 0.709636 0.0405672
$$307$$ −13.4938 −0.770131 −0.385065 0.922889i $$-0.625821\pi$$
−0.385065 + 0.922889i $$0.625821\pi$$
$$308$$ 1.71900 0.0979493
$$309$$ −17.1526 −0.975775
$$310$$ 0 0
$$311$$ −17.5526 −0.995318 −0.497659 0.867373i $$-0.665807\pi$$
−0.497659 + 0.867373i $$0.665807\pi$$
$$312$$ −3.52543 −0.199588
$$313$$ 14.3970 0.813766 0.406883 0.913480i $$-0.366616\pi$$
0.406883 + 0.913480i $$0.366616\pi$$
$$314$$ 1.80642 0.101942
$$315$$ 0 0
$$316$$ 31.4608 1.76981
$$317$$ 29.4608 1.65468 0.827341 0.561701i $$-0.189853\pi$$
0.827341 + 0.561701i $$0.189853\pi$$
$$318$$ 0.253799 0.0142324
$$319$$ 7.05086 0.394772
$$320$$ 0 0
$$321$$ −13.5669 −0.757231
$$322$$ −1.12399 −0.0626372
$$323$$ 5.53972 0.308238
$$324$$ −1.90321 −0.105734
$$325$$ 0 0
$$326$$ 3.43801 0.190414
$$327$$ −10.0000 −0.553001
$$328$$ 12.9491 0.714997
$$329$$ −0.857279 −0.0472633
$$330$$ 0 0
$$331$$ −2.62222 −0.144130 −0.0720650 0.997400i $$-0.522959\pi$$
−0.0720650 + 0.997400i $$0.522959\pi$$
$$332$$ −14.7699 −0.810601
$$333$$ 5.80642 0.318190
$$334$$ 4.34122 0.237541
$$335$$ 0 0
$$336$$ −3.09679 −0.168944
$$337$$ −5.00492 −0.272635 −0.136318 0.990665i $$-0.543527\pi$$
−0.136318 + 0.990665i $$0.543527\pi$$
$$338$$ −1.42219 −0.0773567
$$339$$ 14.2351 0.773143
$$340$$ 0 0
$$341$$ −2.62222 −0.142001
$$342$$ 0.755569 0.0408565
$$343$$ 11.9081 0.642979
$$344$$ −13.0049 −0.701178
$$345$$ 0 0
$$346$$ −5.86220 −0.315154
$$347$$ −22.8113 −1.22458 −0.612289 0.790634i $$-0.709751\pi$$
−0.612289 + 0.790634i $$0.709751\pi$$
$$348$$ −13.4193 −0.719348
$$349$$ 21.2257 1.13619 0.568093 0.822965i $$-0.307681\pi$$
0.568093 + 0.822965i $$0.307681\pi$$
$$350$$ 0 0
$$351$$ 2.90321 0.154962
$$352$$ 3.49532 0.186301
$$353$$ 7.18421 0.382377 0.191188 0.981553i $$-0.438766\pi$$
0.191188 + 0.981553i $$0.438766\pi$$
$$354$$ −0.520505 −0.0276645
$$355$$ 0 0
$$356$$ −11.6731 −0.618672
$$357$$ −2.06022 −0.109039
$$358$$ 1.51114 0.0798661
$$359$$ −14.1017 −0.744260 −0.372130 0.928181i $$-0.621372\pi$$
−0.372130 + 0.928181i $$0.621372\pi$$
$$360$$ 0 0
$$361$$ −13.1017 −0.689564
$$362$$ −5.22570 −0.274656
$$363$$ 1.00000 0.0524864
$$364$$ 4.99063 0.261580
$$365$$ 0 0
$$366$$ −2.25380 −0.117808
$$367$$ −3.90813 −0.204003 −0.102001 0.994784i $$-0.532525\pi$$
−0.102001 + 0.994784i $$0.532525\pi$$
$$368$$ 13.7146 0.714921
$$369$$ −10.6637 −0.555130
$$370$$ 0 0
$$371$$ −0.736833 −0.0382545
$$372$$ 4.99063 0.258752
$$373$$ 12.9763 0.671890 0.335945 0.941882i $$-0.390944\pi$$
0.335945 + 0.941882i $$0.390944\pi$$
$$374$$ 0.709636 0.0366944
$$375$$ 0 0
$$376$$ −1.15257 −0.0594390
$$377$$ 20.4701 1.05427
$$378$$ −0.280996 −0.0144529
$$379$$ 36.0830 1.85346 0.926729 0.375731i $$-0.122608\pi$$
0.926729 + 0.375731i $$0.122608\pi$$
$$380$$ 0 0
$$381$$ 11.0049 0.563799
$$382$$ 5.24443 0.268328
$$383$$ 20.2953 1.03704 0.518520 0.855065i $$-0.326483\pi$$
0.518520 + 0.855065i $$0.326483\pi$$
$$384$$ −8.78568 −0.448342
$$385$$ 0 0
$$386$$ −7.65878 −0.389822
$$387$$ 10.7096 0.544401
$$388$$ 23.7333 1.20488
$$389$$ 30.4701 1.54490 0.772448 0.635078i $$-0.219032\pi$$
0.772448 + 0.635078i $$0.219032\pi$$
$$390$$ 0 0
$$391$$ 9.12399 0.461420
$$392$$ 7.50961 0.379292
$$393$$ 1.24443 0.0627733
$$394$$ 4.62666 0.233088
$$395$$ 0 0
$$396$$ −1.90321 −0.0956400
$$397$$ 4.97773 0.249825 0.124912 0.992168i $$-0.460135\pi$$
0.124912 + 0.992168i $$0.460135\pi$$
$$398$$ −3.49240 −0.175058
$$399$$ −2.19358 −0.109816
$$400$$ 0 0
$$401$$ −1.86665 −0.0932159 −0.0466079 0.998913i $$-0.514841\pi$$
−0.0466079 + 0.998913i $$0.514841\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −7.61285 −0.379223
$$404$$ −30.7841 −1.53157
$$405$$ 0 0
$$406$$ −1.98126 −0.0983285
$$407$$ 5.80642 0.287814
$$408$$ −2.76986 −0.137128
$$409$$ −3.63158 −0.179570 −0.0897851 0.995961i $$-0.528618\pi$$
−0.0897851 + 0.995961i $$0.528618\pi$$
$$410$$ 0 0
$$411$$ −4.42864 −0.218449
$$412$$ 32.6450 1.60830
$$413$$ 1.51114 0.0743582
$$414$$ 1.24443 0.0611605
$$415$$ 0 0
$$416$$ 10.1476 0.497529
$$417$$ 0.917502 0.0449303
$$418$$ 0.755569 0.0369561
$$419$$ 4.85728 0.237294 0.118647 0.992937i $$-0.462144\pi$$
0.118647 + 0.992937i $$0.462144\pi$$
$$420$$ 0 0
$$421$$ 22.6321 1.10302 0.551510 0.834169i $$-0.314052\pi$$
0.551510 + 0.834169i $$0.314052\pi$$
$$422$$ −3.71456 −0.180822
$$423$$ 0.949145 0.0461490
$$424$$ −0.990632 −0.0481093
$$425$$ 0 0
$$426$$ 2.88892 0.139969
$$427$$ 6.54326 0.316650
$$428$$ 25.8207 1.24809
$$429$$ 2.90321 0.140168
$$430$$ 0 0
$$431$$ 1.24443 0.0599421 0.0299711 0.999551i $$-0.490458\pi$$
0.0299711 + 0.999551i $$0.490458\pi$$
$$432$$ 3.42864 0.164960
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0.736833 0.0353691
$$435$$ 0 0
$$436$$ 19.0321 0.911473
$$437$$ 9.71456 0.464710
$$438$$ −1.76049 −0.0841195
$$439$$ −2.42864 −0.115913 −0.0579563 0.998319i $$-0.518458\pi$$
−0.0579563 + 0.998319i $$0.518458\pi$$
$$440$$ 0 0
$$441$$ −6.18421 −0.294486
$$442$$ 2.06022 0.0979948
$$443$$ 31.0509 1.47527 0.737635 0.675199i $$-0.235942\pi$$
0.737635 + 0.675199i $$0.235942\pi$$
$$444$$ −11.0509 −0.524450
$$445$$ 0 0
$$446$$ 6.78415 0.321239
$$447$$ 2.19358 0.103753
$$448$$ 5.21141 0.246216
$$449$$ −37.3590 −1.76308 −0.881541 0.472107i $$-0.843494\pi$$
−0.881541 + 0.472107i $$0.843494\pi$$
$$450$$ 0 0
$$451$$ −10.6637 −0.502134
$$452$$ −27.0923 −1.27432
$$453$$ −10.4286 −0.489980
$$454$$ 0.995078 0.0467013
$$455$$ 0 0
$$456$$ −2.94914 −0.138106
$$457$$ 8.73822 0.408757 0.204378 0.978892i $$-0.434483\pi$$
0.204378 + 0.978892i $$0.434483\pi$$
$$458$$ 2.21633 0.103562
$$459$$ 2.28100 0.106468
$$460$$ 0 0
$$461$$ 31.7877 1.48050 0.740250 0.672332i $$-0.234707\pi$$
0.740250 + 0.672332i $$0.234707\pi$$
$$462$$ −0.280996 −0.0130731
$$463$$ 12.0919 0.561957 0.280978 0.959714i $$-0.409341\pi$$
0.280978 + 0.959714i $$0.409341\pi$$
$$464$$ 24.1748 1.12229
$$465$$ 0 0
$$466$$ −6.07451 −0.281396
$$467$$ −15.3461 −0.710135 −0.355067 0.934841i $$-0.615542\pi$$
−0.355067 + 0.934841i $$0.615542\pi$$
$$468$$ −5.52543 −0.255413
$$469$$ −11.6128 −0.536231
$$470$$ 0 0
$$471$$ 5.80642 0.267546
$$472$$ 2.03164 0.0935139
$$473$$ 10.7096 0.492430
$$474$$ −5.14272 −0.236213
$$475$$ 0 0
$$476$$ 3.92104 0.179721
$$477$$ 0.815792 0.0373525
$$478$$ −6.83854 −0.312788
$$479$$ 5.89829 0.269500 0.134750 0.990880i $$-0.456977\pi$$
0.134750 + 0.990880i $$0.456977\pi$$
$$480$$ 0 0
$$481$$ 16.8573 0.768626
$$482$$ 1.66323 0.0757579
$$483$$ −3.61285 −0.164390
$$484$$ −1.90321 −0.0865096
$$485$$ 0 0
$$486$$ 0.311108 0.0141121
$$487$$ 31.3461 1.42043 0.710215 0.703985i $$-0.248598\pi$$
0.710215 + 0.703985i $$0.248598\pi$$
$$488$$ 8.79706 0.398224
$$489$$ 11.0509 0.499737
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 20.2953 0.914982
$$493$$ 16.0830 0.724341
$$494$$ 2.19358 0.0986937
$$495$$ 0 0
$$496$$ −8.99063 −0.403691
$$497$$ −8.38715 −0.376215
$$498$$ 2.41435 0.108190
$$499$$ 15.1427 0.677881 0.338941 0.940808i $$-0.389931\pi$$
0.338941 + 0.940808i $$0.389931\pi$$
$$500$$ 0 0
$$501$$ 13.9541 0.623422
$$502$$ −7.39652 −0.330123
$$503$$ 26.0370 1.16093 0.580467 0.814284i $$-0.302870\pi$$
0.580467 + 0.814284i $$0.302870\pi$$
$$504$$ 1.09679 0.0488548
$$505$$ 0 0
$$506$$ 1.24443 0.0553217
$$507$$ −4.57136 −0.203021
$$508$$ −20.9447 −0.929271
$$509$$ −24.5718 −1.08913 −0.544564 0.838719i $$-0.683305\pi$$
−0.544564 + 0.838719i $$0.683305\pi$$
$$510$$ 0 0
$$511$$ 5.11108 0.226101
$$512$$ 20.3111 0.897633
$$513$$ 2.42864 0.107227
$$514$$ −2.53035 −0.111609
$$515$$ 0 0
$$516$$ −20.3827 −0.897299
$$517$$ 0.949145 0.0417433
$$518$$ −1.63158 −0.0716877
$$519$$ −18.8430 −0.827115
$$520$$ 0 0
$$521$$ −4.88892 −0.214188 −0.107094 0.994249i $$-0.534155\pi$$
−0.107094 + 0.994249i $$0.534155\pi$$
$$522$$ 2.19358 0.0960102
$$523$$ −4.22077 −0.184562 −0.0922808 0.995733i $$-0.529416\pi$$
−0.0922808 + 0.995733i $$0.529416\pi$$
$$524$$ −2.36842 −0.103465
$$525$$ 0 0
$$526$$ 7.12537 0.310681
$$527$$ −5.98126 −0.260548
$$528$$ 3.42864 0.149212
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −1.67307 −0.0726051
$$532$$ 4.17484 0.181002
$$533$$ −30.9590 −1.34098
$$534$$ 1.90813 0.0825730
$$535$$ 0 0
$$536$$ −15.6128 −0.674372
$$537$$ 4.85728 0.209607
$$538$$ 3.68196 0.158741
$$539$$ −6.18421 −0.266373
$$540$$ 0 0
$$541$$ −13.6128 −0.585262 −0.292631 0.956225i $$-0.594531\pi$$
−0.292631 + 0.956225i $$0.594531\pi$$
$$542$$ 4.63512 0.199096
$$543$$ −16.7971 −0.720831
$$544$$ 7.97280 0.341831
$$545$$ 0 0
$$546$$ −0.815792 −0.0349127
$$547$$ −7.48394 −0.319990 −0.159995 0.987118i $$-0.551148\pi$$
−0.159995 + 0.987118i $$0.551148\pi$$
$$548$$ 8.42864 0.360054
$$549$$ −7.24443 −0.309185
$$550$$ 0 0
$$551$$ 17.1240 0.729506
$$552$$ −4.85728 −0.206740
$$553$$ 14.9304 0.634906
$$554$$ −8.60793 −0.365716
$$555$$ 0 0
$$556$$ −1.74620 −0.0740554
$$557$$ −32.2908 −1.36821 −0.684103 0.729385i $$-0.739806\pi$$
−0.684103 + 0.729385i $$0.739806\pi$$
$$558$$ −0.815792 −0.0345352
$$559$$ 31.0923 1.31507
$$560$$ 0 0
$$561$$ 2.28100 0.0963037
$$562$$ 3.05086 0.128693
$$563$$ −7.49378 −0.315825 −0.157913 0.987453i $$-0.550476\pi$$
−0.157913 + 0.987453i $$0.550476\pi$$
$$564$$ −1.80642 −0.0760642
$$565$$ 0 0
$$566$$ −5.91258 −0.248524
$$567$$ −0.903212 −0.0379313
$$568$$ −11.2761 −0.473134
$$569$$ −12.9491 −0.542856 −0.271428 0.962459i $$-0.587496\pi$$
−0.271428 + 0.962459i $$0.587496\pi$$
$$570$$ 0 0
$$571$$ −15.2859 −0.639696 −0.319848 0.947469i $$-0.603632\pi$$
−0.319848 + 0.947469i $$0.603632\pi$$
$$572$$ −5.52543 −0.231030
$$573$$ 16.8573 0.704223
$$574$$ 2.99646 0.125070
$$575$$ 0 0
$$576$$ −5.76986 −0.240411
$$577$$ −28.4415 −1.18404 −0.592019 0.805924i $$-0.701669\pi$$
−0.592019 + 0.805924i $$0.701669\pi$$
$$578$$ −3.67016 −0.152658
$$579$$ −24.6178 −1.02308
$$580$$ 0 0
$$581$$ −7.00937 −0.290798
$$582$$ −3.87955 −0.160813
$$583$$ 0.815792 0.0337866
$$584$$ 6.87157 0.284348
$$585$$ 0 0
$$586$$ −9.56691 −0.395206
$$587$$ 8.47013 0.349600 0.174800 0.984604i $$-0.444072\pi$$
0.174800 + 0.984604i $$0.444072\pi$$
$$588$$ 11.7699 0.485381
$$589$$ −6.36842 −0.262406
$$590$$ 0 0
$$591$$ 14.8716 0.611735
$$592$$ 19.9081 0.818219
$$593$$ −26.5763 −1.09136 −0.545679 0.837995i $$-0.683728\pi$$
−0.545679 + 0.837995i $$0.683728\pi$$
$$594$$ 0.311108 0.0127649
$$595$$ 0 0
$$596$$ −4.17484 −0.171008
$$597$$ −11.2257 −0.459437
$$598$$ 3.61285 0.147740
$$599$$ −8.77430 −0.358508 −0.179254 0.983803i $$-0.557368\pi$$
−0.179254 + 0.983803i $$0.557368\pi$$
$$600$$ 0 0
$$601$$ −41.8163 −1.70572 −0.852861 0.522139i $$-0.825134\pi$$
−0.852861 + 0.522139i $$0.825134\pi$$
$$602$$ −3.00937 −0.122653
$$603$$ 12.8573 0.523589
$$604$$ 19.8479 0.807600
$$605$$ 0 0
$$606$$ 5.03212 0.204416
$$607$$ −29.9353 −1.21504 −0.607519 0.794305i $$-0.707835\pi$$
−0.607519 + 0.794305i $$0.707835\pi$$
$$608$$ 8.48886 0.344269
$$609$$ −6.36842 −0.258061
$$610$$ 0 0
$$611$$ 2.75557 0.111478
$$612$$ −4.34122 −0.175483
$$613$$ −26.1289 −1.05534 −0.527668 0.849450i $$-0.676934\pi$$
−0.527668 + 0.849450i $$0.676934\pi$$
$$614$$ −4.19802 −0.169418
$$615$$ 0 0
$$616$$ 1.09679 0.0441909
$$617$$ 3.66323 0.147476 0.0737380 0.997278i $$-0.476507\pi$$
0.0737380 + 0.997278i $$0.476507\pi$$
$$618$$ −5.33630 −0.214657
$$619$$ 43.2958 1.74020 0.870102 0.492872i $$-0.164053\pi$$
0.870102 + 0.492872i $$0.164053\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −5.46076 −0.218956
$$623$$ −5.53972 −0.221944
$$624$$ 9.95407 0.398482
$$625$$ 0 0
$$626$$ 4.47902 0.179018
$$627$$ 2.42864 0.0969905
$$628$$ −11.0509 −0.440977
$$629$$ 13.2444 0.528090
$$630$$ 0 0
$$631$$ 8.97773 0.357398 0.178699 0.983904i $$-0.442811\pi$$
0.178699 + 0.983904i $$0.442811\pi$$
$$632$$ 20.0731 0.798466
$$633$$ −11.9398 −0.474564
$$634$$ 9.16547 0.364007
$$635$$ 0 0
$$636$$ −1.55262 −0.0615656
$$637$$ −17.9541 −0.711366
$$638$$ 2.19358 0.0868445
$$639$$ 9.28592 0.367345
$$640$$ 0 0
$$641$$ 9.21279 0.363883 0.181942 0.983309i $$-0.441762\pi$$
0.181942 + 0.983309i $$0.441762\pi$$
$$642$$ −4.22077 −0.166581
$$643$$ 16.3783 0.645896 0.322948 0.946417i $$-0.395326\pi$$
0.322948 + 0.946417i $$0.395326\pi$$
$$644$$ 6.87601 0.270953
$$645$$ 0 0
$$646$$ 1.72345 0.0678082
$$647$$ −9.80642 −0.385530 −0.192765 0.981245i $$-0.561746\pi$$
−0.192765 + 0.981245i $$0.561746\pi$$
$$648$$ −1.21432 −0.0477030
$$649$$ −1.67307 −0.0656738
$$650$$ 0 0
$$651$$ 2.36842 0.0928256
$$652$$ −21.0321 −0.823681
$$653$$ −33.0736 −1.29427 −0.647135 0.762375i $$-0.724033\pi$$
−0.647135 + 0.762375i $$0.724033\pi$$
$$654$$ −3.11108 −0.121653
$$655$$ 0 0
$$656$$ −36.5620 −1.42751
$$657$$ −5.65878 −0.220770
$$658$$ −0.266706 −0.0103973
$$659$$ 34.1017 1.32841 0.664207 0.747549i $$-0.268769\pi$$
0.664207 + 0.747549i $$0.268769\pi$$
$$660$$ 0 0
$$661$$ −5.40943 −0.210402 −0.105201 0.994451i $$-0.533549\pi$$
−0.105201 + 0.994451i $$0.533549\pi$$
$$662$$ −0.815792 −0.0317066
$$663$$ 6.62222 0.257186
$$664$$ −9.42372 −0.365711
$$665$$ 0 0
$$666$$ 1.80642 0.0699975
$$667$$ 28.2034 1.09204
$$668$$ −26.5575 −1.02754
$$669$$ 21.8064 0.843085
$$670$$ 0 0
$$671$$ −7.24443 −0.279668
$$672$$ −3.15701 −0.121784
$$673$$ −24.1476 −0.930823 −0.465412 0.885094i $$-0.654094\pi$$
−0.465412 + 0.885094i $$0.654094\pi$$
$$674$$ −1.55707 −0.0599761
$$675$$ 0 0
$$676$$ 8.70027 0.334626
$$677$$ −26.2810 −1.01006 −0.505030 0.863102i $$-0.668519\pi$$
−0.505030 + 0.863102i $$0.668519\pi$$
$$678$$ 4.42864 0.170081
$$679$$ 11.2632 0.432241
$$680$$ 0 0
$$681$$ 3.19850 0.122567
$$682$$ −0.815792 −0.0312383
$$683$$ −15.3176 −0.586110 −0.293055 0.956096i $$-0.594672\pi$$
−0.293055 + 0.956096i $$0.594672\pi$$
$$684$$ −4.62222 −0.176735
$$685$$ 0 0
$$686$$ 3.70471 0.141447
$$687$$ 7.12399 0.271797
$$688$$ 36.7195 1.39992
$$689$$ 2.36842 0.0902295
$$690$$ 0 0
$$691$$ 15.0223 0.571474 0.285737 0.958308i $$-0.407762\pi$$
0.285737 + 0.958308i $$0.407762\pi$$
$$692$$ 35.8622 1.36328
$$693$$ −0.903212 −0.0343102
$$694$$ −7.09679 −0.269390
$$695$$ 0 0
$$696$$ −8.56199 −0.324541
$$697$$ −24.3239 −0.921332
$$698$$ 6.60348 0.249945
$$699$$ −19.5254 −0.738519
$$700$$ 0 0
$$701$$ −19.9081 −0.751920 −0.375960 0.926636i $$-0.622687\pi$$
−0.375960 + 0.926636i $$0.622687\pi$$
$$702$$ 0.903212 0.0340895
$$703$$ 14.1017 0.531856
$$704$$ −5.76986 −0.217460
$$705$$ 0 0
$$706$$ 2.23506 0.0841177
$$707$$ −14.6093 −0.549440
$$708$$ 3.18421 0.119670
$$709$$ −13.5081 −0.507306 −0.253653 0.967295i $$-0.581632\pi$$
−0.253653 + 0.967295i $$0.581632\pi$$
$$710$$ 0 0
$$711$$ −16.5303 −0.619937
$$712$$ −7.44785 −0.279120
$$713$$ −10.4889 −0.392811
$$714$$ −0.640951 −0.0239870
$$715$$ 0 0
$$716$$ −9.24443 −0.345481
$$717$$ −21.9813 −0.820905
$$718$$ −4.38715 −0.163727
$$719$$ −16.0830 −0.599794 −0.299897 0.953972i $$-0.596952\pi$$
−0.299897 + 0.953972i $$0.596952\pi$$
$$720$$ 0 0
$$721$$ 15.4924 0.576967
$$722$$ −4.07604 −0.151695
$$723$$ 5.34614 0.198825
$$724$$ 31.9684 1.18809
$$725$$ 0 0
$$726$$ 0.311108 0.0115463
$$727$$ 23.6128 0.875752 0.437876 0.899035i $$-0.355731\pi$$
0.437876 + 0.899035i $$0.355731\pi$$
$$728$$ 3.18421 0.118015
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 24.4286 0.903526
$$732$$ 13.7877 0.509608
$$733$$ −30.0459 −1.10977 −0.554886 0.831926i $$-0.687238\pi$$
−0.554886 + 0.831926i $$0.687238\pi$$
$$734$$ −1.21585 −0.0448779
$$735$$ 0 0
$$736$$ 13.9813 0.515356
$$737$$ 12.8573 0.473604
$$738$$ −3.31756 −0.122121
$$739$$ −24.4099 −0.897933 −0.448966 0.893549i $$-0.648208\pi$$
−0.448966 + 0.893549i $$0.648208\pi$$
$$740$$ 0 0
$$741$$ 7.05086 0.259020
$$742$$ −0.229234 −0.00841546
$$743$$ −33.1798 −1.21725 −0.608624 0.793459i $$-0.708278\pi$$
−0.608624 + 0.793459i $$0.708278\pi$$
$$744$$ 3.18421 0.116739
$$745$$ 0 0
$$746$$ 4.03704 0.147807
$$747$$ 7.76049 0.283942
$$748$$ −4.34122 −0.158731
$$749$$ 12.2538 0.447744
$$750$$ 0 0
$$751$$ −22.5718 −0.823658 −0.411829 0.911261i $$-0.635110\pi$$
−0.411829 + 0.911261i $$0.635110\pi$$
$$752$$ 3.25428 0.118671
$$753$$ −23.7748 −0.866401
$$754$$ 6.36842 0.231924
$$755$$ 0 0
$$756$$ 1.71900 0.0625196
$$757$$ −4.94914 −0.179880 −0.0899399 0.995947i $$-0.528667\pi$$
−0.0899399 + 0.995947i $$0.528667\pi$$
$$758$$ 11.2257 0.407736
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 14.6637 0.531559 0.265779 0.964034i $$-0.414371\pi$$
0.265779 + 0.964034i $$0.414371\pi$$
$$762$$ 3.42372 0.124028
$$763$$ 9.03212 0.326985
$$764$$ −32.0830 −1.16072
$$765$$ 0 0
$$766$$ 6.31402 0.228135
$$767$$ −4.85728 −0.175386
$$768$$ 8.80642 0.317774
$$769$$ −44.5718 −1.60730 −0.803651 0.595101i $$-0.797112\pi$$
−0.803651 + 0.595101i $$0.797112\pi$$
$$770$$ 0 0
$$771$$ −8.13335 −0.292916
$$772$$ 46.8528 1.68627
$$773$$ −17.3145 −0.622759 −0.311380 0.950286i $$-0.600791\pi$$
−0.311380 + 0.950286i $$0.600791\pi$$
$$774$$ 3.33185 0.119761
$$775$$ 0 0
$$776$$ 15.1427 0.543592
$$777$$ −5.24443 −0.188143
$$778$$ 9.47949 0.339856
$$779$$ −25.8983 −0.927903
$$780$$ 0 0
$$781$$ 9.28592 0.332276
$$782$$ 2.83854 0.101506
$$783$$ 7.05086 0.251977
$$784$$ −21.2034 −0.757265
$$785$$ 0 0
$$786$$ 0.387152 0.0138093
$$787$$ −36.5161 −1.30166 −0.650828 0.759225i $$-0.725578\pi$$
−0.650828 + 0.759225i $$0.725578\pi$$
$$788$$ −28.3037 −1.00828
$$789$$ 22.9032 0.815376
$$790$$ 0 0
$$791$$ −12.8573 −0.457152
$$792$$ −1.21432 −0.0431490
$$793$$ −21.0321 −0.746872
$$794$$ 1.54861 0.0549581
$$795$$ 0 0
$$796$$ 21.3649 0.757258
$$797$$ 14.3180 0.507171 0.253585 0.967313i $$-0.418390\pi$$
0.253585 + 0.967313i $$0.418390\pi$$
$$798$$ −0.682439 −0.0241581
$$799$$ 2.16500 0.0765921
$$800$$ 0 0
$$801$$ 6.13335 0.216711
$$802$$ −0.580728 −0.0205062
$$803$$ −5.65878 −0.199694
$$804$$ −24.4701 −0.862995
$$805$$ 0 0
$$806$$ −2.36842 −0.0834239
$$807$$ 11.8350 0.416612
$$808$$ −19.6414 −0.690983
$$809$$ 32.0544 1.12697 0.563486 0.826125i $$-0.309460\pi$$
0.563486 + 0.826125i $$0.309460\pi$$
$$810$$ 0 0
$$811$$ −8.44738 −0.296627 −0.148314 0.988940i $$-0.547385\pi$$
−0.148314 + 0.988940i $$0.547385\pi$$
$$812$$ 12.1204 0.425344
$$813$$ 14.8988 0.522523
$$814$$ 1.80642 0.0633151
$$815$$ 0 0
$$816$$ 7.82071 0.273780
$$817$$ 26.0098 0.909969
$$818$$ −1.12981 −0.0395030
$$819$$ −2.62222 −0.0916276
$$820$$ 0 0
$$821$$ 17.2159 0.600837 0.300419 0.953807i $$-0.402874\pi$$
0.300419 + 0.953807i $$0.402874\pi$$
$$822$$ −1.37778 −0.0480557
$$823$$ −12.7654 −0.444974 −0.222487 0.974936i $$-0.571418\pi$$
−0.222487 + 0.974936i $$0.571418\pi$$
$$824$$ 20.8287 0.725602
$$825$$ 0 0
$$826$$ 0.470127 0.0163578
$$827$$ −8.70964 −0.302864 −0.151432 0.988468i $$-0.548388\pi$$
−0.151432 + 0.988468i $$0.548388\pi$$
$$828$$ −7.61285 −0.264565
$$829$$ −8.32693 −0.289206 −0.144603 0.989490i $$-0.546191\pi$$
−0.144603 + 0.989490i $$0.546191\pi$$
$$830$$ 0 0
$$831$$ −27.6686 −0.959814
$$832$$ −16.7511 −0.580741
$$833$$ −14.1062 −0.488749
$$834$$ 0.285442 0.00988405
$$835$$ 0 0
$$836$$ −4.62222 −0.159863
$$837$$ −2.62222 −0.0906370
$$838$$ 1.51114 0.0522014
$$839$$ 12.8988 0.445315 0.222657 0.974897i $$-0.428527\pi$$
0.222657 + 0.974897i $$0.428527\pi$$
$$840$$ 0 0
$$841$$ 20.7146 0.714295
$$842$$ 7.04101 0.242649
$$843$$ 9.80642 0.337751
$$844$$ 22.7239 0.782190
$$845$$ 0 0
$$846$$ 0.295286 0.0101522
$$847$$ −0.903212 −0.0310347
$$848$$ 2.79706 0.0960513
$$849$$ −19.0049 −0.652247
$$850$$ 0 0
$$851$$ 23.2257 0.796167
$$852$$ −17.6731 −0.605469
$$853$$ 19.6686 0.673441 0.336720 0.941605i $$-0.390682\pi$$
0.336720 + 0.941605i $$0.390682\pi$$
$$854$$ 2.03566 0.0696588
$$855$$ 0 0
$$856$$ 16.4746 0.563089
$$857$$ −31.8207 −1.08697 −0.543487 0.839417i $$-0.682896\pi$$
−0.543487 + 0.839417i $$0.682896\pi$$
$$858$$ 0.903212 0.0308351
$$859$$ 27.8292 0.949519 0.474760 0.880116i $$-0.342535\pi$$
0.474760 + 0.880116i $$0.342535\pi$$
$$860$$ 0 0
$$861$$ 9.63158 0.328243
$$862$$ 0.387152 0.0131865
$$863$$ 4.82870 0.164371 0.0821854 0.996617i $$-0.473810\pi$$
0.0821854 + 0.996617i $$0.473810\pi$$
$$864$$ 3.49532 0.118913
$$865$$ 0 0
$$866$$ −4.97773 −0.169150
$$867$$ −11.7971 −0.400649
$$868$$ −4.50760 −0.152998
$$869$$ −16.5303 −0.560754
$$870$$ 0 0
$$871$$ 37.3274 1.26479
$$872$$ 12.1432 0.411221
$$873$$ −12.4701 −0.422050
$$874$$ 3.02227 0.102230
$$875$$ 0 0
$$876$$ 10.7699 0.363880
$$877$$ −21.9826 −0.742301 −0.371151 0.928573i $$-0.621037\pi$$
−0.371151 + 0.928573i $$0.621037\pi$$
$$878$$ −0.755569 −0.0254992
$$879$$ −30.7511 −1.03721
$$880$$ 0 0
$$881$$ −12.1017 −0.407717 −0.203858 0.979000i $$-0.565348\pi$$
−0.203858 + 0.979000i $$0.565348\pi$$
$$882$$ −1.92396 −0.0647830
$$883$$ −8.73683 −0.294018 −0.147009 0.989135i $$-0.546965\pi$$
−0.147009 + 0.989135i $$0.546965\pi$$
$$884$$ −12.6035 −0.423901
$$885$$ 0 0
$$886$$ 9.66016 0.324540
$$887$$ 19.8524 0.666577 0.333288 0.942825i $$-0.391842\pi$$
0.333288 + 0.942825i $$0.391842\pi$$
$$888$$ −7.05086 −0.236611
$$889$$ −9.93978 −0.333369
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −41.5022 −1.38960
$$893$$ 2.30513 0.0771383
$$894$$ 0.682439 0.0228242
$$895$$ 0 0
$$896$$ 7.93533 0.265101
$$897$$ 11.6128 0.387742
$$898$$ −11.6227 −0.387854
$$899$$ −18.4889 −0.616638
$$900$$ 0 0
$$901$$ 1.86082 0.0619928
$$902$$ −3.31756 −0.110463
$$903$$ −9.67307 −0.321900
$$904$$ −17.2859 −0.574921
$$905$$ 0 0
$$906$$ −3.24443 −0.107789
$$907$$ −32.8287 −1.09006 −0.545030 0.838417i $$-0.683482\pi$$
−0.545030 + 0.838417i $$0.683482\pi$$
$$908$$ −6.08742 −0.202018
$$909$$ 16.1748 0.536486
$$910$$ 0 0
$$911$$ −16.3497 −0.541689 −0.270845 0.962623i $$-0.587303\pi$$
−0.270845 + 0.962623i $$0.587303\pi$$
$$912$$ 8.32693 0.275732
$$913$$ 7.76049 0.256835
$$914$$ 2.71853 0.0899209
$$915$$ 0 0
$$916$$ −13.5585 −0.447984
$$917$$ −1.12399 −0.0371173
$$918$$ 0.709636 0.0234215
$$919$$ 20.0228 0.660490 0.330245 0.943895i $$-0.392869\pi$$
0.330245 + 0.943895i $$0.392869\pi$$
$$920$$ 0 0
$$921$$ −13.4938 −0.444635
$$922$$ 9.88940 0.325690
$$923$$ 26.9590 0.887366
$$924$$ 1.71900 0.0565511
$$925$$ 0 0
$$926$$ 3.76187 0.123623
$$927$$ −17.1526 −0.563364
$$928$$ 24.6450 0.809011
$$929$$ −43.5308 −1.42820 −0.714100 0.700044i $$-0.753164\pi$$
−0.714100 + 0.700044i $$0.753164\pi$$
$$930$$ 0 0
$$931$$ −15.0192 −0.492235
$$932$$ 37.1610 1.21725
$$933$$ −17.5526 −0.574647
$$934$$ −4.77430 −0.156220
$$935$$ 0 0
$$936$$ −3.52543 −0.115232
$$937$$ 43.4563 1.41966 0.709828 0.704375i $$-0.248773\pi$$
0.709828 + 0.704375i $$0.248773\pi$$
$$938$$ −3.61285 −0.117964
$$939$$ 14.3970 0.469828
$$940$$ 0 0
$$941$$ −23.7244 −0.773393 −0.386697 0.922207i $$-0.626384\pi$$
−0.386697 + 0.922207i $$0.626384\pi$$
$$942$$ 1.80642 0.0588565
$$943$$ −42.6548 −1.38903
$$944$$ −5.73636 −0.186703
$$945$$ 0 0
$$946$$ 3.33185 0.108328
$$947$$ 11.7047 0.380352 0.190176 0.981750i $$-0.439094\pi$$
0.190176 + 0.981750i $$0.439094\pi$$
$$948$$ 31.4608 1.02180
$$949$$ −16.4286 −0.533296
$$950$$ 0 0
$$951$$ 29.4608 0.955331
$$952$$ 2.50177 0.0810828
$$953$$ 46.1258 1.49416 0.747081 0.664733i $$-0.231455\pi$$
0.747081 + 0.664733i $$0.231455\pi$$
$$954$$ 0.253799 0.00821705
$$955$$ 0 0
$$956$$ 41.8350 1.35304
$$957$$ 7.05086 0.227922
$$958$$ 1.83500 0.0592863
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −24.1240 −0.778193
$$962$$ 5.24443 0.169087
$$963$$ −13.5669 −0.437188
$$964$$ −10.1748 −0.327710
$$965$$ 0 0
$$966$$ −1.12399 −0.0361636
$$967$$ 17.0495 0.548274 0.274137 0.961691i $$-0.411608\pi$$
0.274137 + 0.961691i $$0.411608\pi$$
$$968$$ −1.21432 −0.0390297
$$969$$ 5.53972 0.177961
$$970$$ 0 0
$$971$$ −58.1847 −1.86724 −0.933618 0.358271i $$-0.883366\pi$$
−0.933618 + 0.358271i $$0.883366\pi$$
$$972$$ −1.90321 −0.0610456
$$973$$ −0.828699 −0.0265669
$$974$$ 9.75203 0.312475
$$975$$ 0 0
$$976$$ −24.8385 −0.795062
$$977$$ −51.7373 −1.65522 −0.827612 0.561301i $$-0.810301\pi$$
−0.827612 + 0.561301i $$0.810301\pi$$
$$978$$ 3.43801 0.109935
$$979$$ 6.13335 0.196023
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ −2.48886 −0.0794228
$$983$$ −26.3970 −0.841933 −0.420967 0.907076i $$-0.638309\pi$$
−0.420967 + 0.907076i $$0.638309\pi$$
$$984$$ 12.9491 0.412804
$$985$$ 0 0
$$986$$ 5.00354 0.159345
$$987$$ −0.857279 −0.0272875
$$988$$ −13.4193 −0.426924
$$989$$ 42.8385 1.36219
$$990$$ 0 0
$$991$$ −23.0923 −0.733552 −0.366776 0.930309i $$-0.619539\pi$$
−0.366776 + 0.930309i $$0.619539\pi$$
$$992$$ −9.16547 −0.291004
$$993$$ −2.62222 −0.0832135
$$994$$ −2.60931 −0.0827622
$$995$$ 0 0
$$996$$ −14.7699 −0.468001
$$997$$ 12.9131 0.408961 0.204480 0.978871i $$-0.434450\pi$$
0.204480 + 0.978871i $$0.434450\pi$$
$$998$$ 4.71102 0.149125
$$999$$ 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.l.1.2 3
3.2 odd 2 2475.2.a.ba.1.2 3
5.2 odd 4 165.2.c.b.34.4 yes 6
5.3 odd 4 165.2.c.b.34.3 6
5.4 even 2 825.2.a.j.1.2 3
11.10 odd 2 9075.2.a.cg.1.2 3
15.2 even 4 495.2.c.e.199.3 6
15.8 even 4 495.2.c.e.199.4 6
15.14 odd 2 2475.2.a.bc.1.2 3
20.3 even 4 2640.2.d.h.529.2 6
20.7 even 4 2640.2.d.h.529.5 6
55.32 even 4 1815.2.c.e.364.3 6
55.43 even 4 1815.2.c.e.364.4 6
55.54 odd 2 9075.2.a.ch.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.b.34.3 6 5.3 odd 4
165.2.c.b.34.4 yes 6 5.2 odd 4
495.2.c.e.199.3 6 15.2 even 4
495.2.c.e.199.4 6 15.8 even 4
825.2.a.j.1.2 3 5.4 even 2
825.2.a.l.1.2 3 1.1 even 1 trivial
1815.2.c.e.364.3 6 55.32 even 4
1815.2.c.e.364.4 6 55.43 even 4
2475.2.a.ba.1.2 3 3.2 odd 2
2475.2.a.bc.1.2 3 15.14 odd 2
2640.2.d.h.529.2 6 20.3 even 4
2640.2.d.h.529.5 6 20.7 even 4
9075.2.a.cg.1.2 3 11.10 odd 2
9075.2.a.ch.1.2 3 55.54 odd 2