# Properties

 Label 2640.2.a.be.1.2 Level $2640$ Weight $2$ Character 2640.1 Self dual yes Analytic conductor $21.081$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2640.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$21.0805061336$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 2640.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} -1.07838 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} -1.07838 q^{7} +1.00000 q^{9} -1.00000 q^{11} -4.34017 q^{13} -1.00000 q^{15} +7.75872 q^{17} -5.26180 q^{19} +1.07838 q^{21} +2.15676 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.41855 q^{29} +4.68035 q^{31} +1.00000 q^{33} -1.07838 q^{35} -2.00000 q^{37} +4.34017 q^{39} -9.41855 q^{41} -7.60197 q^{43} +1.00000 q^{45} -4.68035 q^{47} -5.83710 q^{49} -7.75872 q^{51} +0.156755 q^{53} -1.00000 q^{55} +5.26180 q^{57} -6.15676 q^{59} -4.15676 q^{61} -1.07838 q^{63} -4.34017 q^{65} +8.68035 q^{67} -2.15676 q^{69} +4.68035 q^{71} -10.4969 q^{73} -1.00000 q^{75} +1.07838 q^{77} +8.09890 q^{79} +1.00000 q^{81} +11.0205 q^{83} +7.75872 q^{85} -1.41855 q^{87} -12.8371 q^{89} +4.68035 q^{91} -4.68035 q^{93} -5.26180 q^{95} +14.6803 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^15 - 2 * q^17 - 8 * q^19 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 8 * q^31 + 3 * q^33 - 6 * q^37 + 2 * q^39 - 14 * q^41 - 4 * q^43 + 3 * q^45 + 8 * q^47 + 11 * q^49 + 2 * q^51 - 6 * q^53 - 3 * q^55 + 8 * q^57 - 12 * q^59 - 6 * q^61 - 2 * q^65 + 4 * q^67 - 8 * q^71 - 14 * q^73 - 3 * q^75 - 12 * q^79 + 3 * q^81 - 2 * q^85 + 10 * q^87 - 10 * q^89 - 8 * q^91 + 8 * q^93 - 8 * q^95 + 22 * q^97 - 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 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.07838 −0.407588 −0.203794 0.979014i $$-0.565327\pi$$
−0.203794 + 0.979014i $$0.565327\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ −4.34017 −1.20375 −0.601874 0.798591i $$-0.705579\pi$$
−0.601874 + 0.798591i $$0.705579\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 7.75872 1.88177 0.940883 0.338730i $$-0.109997\pi$$
0.940883 + 0.338730i $$0.109997\pi$$
$$18$$ 0 0
$$19$$ −5.26180 −1.20714 −0.603569 0.797311i $$-0.706255\pi$$
−0.603569 + 0.797311i $$0.706255\pi$$
$$20$$ 0 0
$$21$$ 1.07838 0.235321
$$22$$ 0 0
$$23$$ 2.15676 0.449715 0.224857 0.974392i $$-0.427808\pi$$
0.224857 + 0.974392i $$0.427808\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 1.41855 0.263418 0.131709 0.991288i $$-0.457954\pi$$
0.131709 + 0.991288i $$0.457954\pi$$
$$30$$ 0 0
$$31$$ 4.68035 0.840615 0.420307 0.907382i $$-0.361922\pi$$
0.420307 + 0.907382i $$0.361922\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ −1.07838 −0.182279
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 4.34017 0.694984
$$40$$ 0 0
$$41$$ −9.41855 −1.47093 −0.735465 0.677562i $$-0.763036\pi$$
−0.735465 + 0.677562i $$0.763036\pi$$
$$42$$ 0 0
$$43$$ −7.60197 −1.15929 −0.579645 0.814869i $$-0.696809\pi$$
−0.579645 + 0.814869i $$0.696809\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −4.68035 −0.682699 −0.341349 0.939937i $$-0.610884\pi$$
−0.341349 + 0.939937i $$0.610884\pi$$
$$48$$ 0 0
$$49$$ −5.83710 −0.833872
$$50$$ 0 0
$$51$$ −7.75872 −1.08644
$$52$$ 0 0
$$53$$ 0.156755 0.0215320 0.0107660 0.999942i $$-0.496573\pi$$
0.0107660 + 0.999942i $$0.496573\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 5.26180 0.696942
$$58$$ 0 0
$$59$$ −6.15676 −0.801541 −0.400771 0.916178i $$-0.631258\pi$$
−0.400771 + 0.916178i $$0.631258\pi$$
$$60$$ 0 0
$$61$$ −4.15676 −0.532218 −0.266109 0.963943i $$-0.585738\pi$$
−0.266109 + 0.963943i $$0.585738\pi$$
$$62$$ 0 0
$$63$$ −1.07838 −0.135863
$$64$$ 0 0
$$65$$ −4.34017 −0.538332
$$66$$ 0 0
$$67$$ 8.68035 1.06047 0.530237 0.847850i $$-0.322103\pi$$
0.530237 + 0.847850i $$0.322103\pi$$
$$68$$ 0 0
$$69$$ −2.15676 −0.259643
$$70$$ 0 0
$$71$$ 4.68035 0.555455 0.277727 0.960660i $$-0.410419\pi$$
0.277727 + 0.960660i $$0.410419\pi$$
$$72$$ 0 0
$$73$$ −10.4969 −1.22857 −0.614286 0.789083i $$-0.710556\pi$$
−0.614286 + 0.789083i $$0.710556\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 1.07838 0.122893
$$78$$ 0 0
$$79$$ 8.09890 0.911197 0.455599 0.890185i $$-0.349425\pi$$
0.455599 + 0.890185i $$0.349425\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 11.0205 1.20966 0.604830 0.796355i $$-0.293241\pi$$
0.604830 + 0.796355i $$0.293241\pi$$
$$84$$ 0 0
$$85$$ 7.75872 0.841552
$$86$$ 0 0
$$87$$ −1.41855 −0.152085
$$88$$ 0 0
$$89$$ −12.8371 −1.36073 −0.680365 0.732873i $$-0.738179\pi$$
−0.680365 + 0.732873i $$0.738179\pi$$
$$90$$ 0 0
$$91$$ 4.68035 0.490634
$$92$$ 0 0
$$93$$ −4.68035 −0.485329
$$94$$ 0 0
$$95$$ −5.26180 −0.539849
$$96$$ 0 0
$$97$$ 14.6803 1.49056 0.745282 0.666750i $$-0.232315\pi$$
0.745282 + 0.666750i $$0.232315\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −15.5753 −1.54980 −0.774900 0.632083i $$-0.782200\pi$$
−0.774900 + 0.632083i $$0.782200\pi$$
$$102$$ 0 0
$$103$$ −6.83710 −0.673680 −0.336840 0.941562i $$-0.609358\pi$$
−0.336840 + 0.941562i $$0.609358\pi$$
$$104$$ 0 0
$$105$$ 1.07838 0.105239
$$106$$ 0 0
$$107$$ −6.34017 −0.612928 −0.306464 0.951882i $$-0.599146\pi$$
−0.306464 + 0.951882i $$0.599146\pi$$
$$108$$ 0 0
$$109$$ 2.31351 0.221594 0.110797 0.993843i $$-0.464660\pi$$
0.110797 + 0.993843i $$0.464660\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 2.15676 0.201118
$$116$$ 0 0
$$117$$ −4.34017 −0.401249
$$118$$ 0 0
$$119$$ −8.36683 −0.766987
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 9.41855 0.849242
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 2.24128 0.198881 0.0994406 0.995044i $$-0.468295\pi$$
0.0994406 + 0.995044i $$0.468295\pi$$
$$128$$ 0 0
$$129$$ 7.60197 0.669316
$$130$$ 0 0
$$131$$ −8.68035 −0.758405 −0.379203 0.925314i $$-0.623802\pi$$
−0.379203 + 0.925314i $$0.623802\pi$$
$$132$$ 0 0
$$133$$ 5.67420 0.492016
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −15.3607 −1.31235 −0.656176 0.754608i $$-0.727827\pi$$
−0.656176 + 0.754608i $$0.727827\pi$$
$$138$$ 0 0
$$139$$ −8.58145 −0.727869 −0.363935 0.931425i $$-0.618567\pi$$
−0.363935 + 0.931425i $$0.618567\pi$$
$$140$$ 0 0
$$141$$ 4.68035 0.394156
$$142$$ 0 0
$$143$$ 4.34017 0.362943
$$144$$ 0 0
$$145$$ 1.41855 0.117804
$$146$$ 0 0
$$147$$ 5.83710 0.481436
$$148$$ 0 0
$$149$$ −18.0989 −1.48272 −0.741360 0.671108i $$-0.765819\pi$$
−0.741360 + 0.671108i $$0.765819\pi$$
$$150$$ 0 0
$$151$$ −22.9360 −1.86651 −0.933253 0.359221i $$-0.883042\pi$$
−0.933253 + 0.359221i $$0.883042\pi$$
$$152$$ 0 0
$$153$$ 7.75872 0.627256
$$154$$ 0 0
$$155$$ 4.68035 0.375934
$$156$$ 0 0
$$157$$ −10.9939 −0.877405 −0.438703 0.898632i $$-0.644562\pi$$
−0.438703 + 0.898632i $$0.644562\pi$$
$$158$$ 0 0
$$159$$ −0.156755 −0.0124315
$$160$$ 0 0
$$161$$ −2.32580 −0.183298
$$162$$ 0 0
$$163$$ 6.52359 0.510967 0.255484 0.966813i $$-0.417765\pi$$
0.255484 + 0.966813i $$0.417765\pi$$
$$164$$ 0 0
$$165$$ 1.00000 0.0778499
$$166$$ 0 0
$$167$$ −1.97334 −0.152701 −0.0763507 0.997081i $$-0.524327\pi$$
−0.0763507 + 0.997081i $$0.524327\pi$$
$$168$$ 0 0
$$169$$ 5.83710 0.449008
$$170$$ 0 0
$$171$$ −5.26180 −0.402380
$$172$$ 0 0
$$173$$ 3.75872 0.285770 0.142885 0.989739i $$-0.454362\pi$$
0.142885 + 0.989739i $$0.454362\pi$$
$$174$$ 0 0
$$175$$ −1.07838 −0.0815177
$$176$$ 0 0
$$177$$ 6.15676 0.462770
$$178$$ 0 0
$$179$$ −15.1506 −1.13241 −0.566205 0.824264i $$-0.691589\pi$$
−0.566205 + 0.824264i $$0.691589\pi$$
$$180$$ 0 0
$$181$$ 4.83710 0.359539 0.179769 0.983709i $$-0.442465\pi$$
0.179769 + 0.983709i $$0.442465\pi$$
$$182$$ 0 0
$$183$$ 4.15676 0.307276
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ −7.75872 −0.567374
$$188$$ 0 0
$$189$$ 1.07838 0.0784404
$$190$$ 0 0
$$191$$ −2.52359 −0.182601 −0.0913003 0.995823i $$-0.529102\pi$$
−0.0913003 + 0.995823i $$0.529102\pi$$
$$192$$ 0 0
$$193$$ 0.0266620 0.00191917 0.000959586 1.00000i $$-0.499695\pi$$
0.000959586 1.00000i $$0.499695\pi$$
$$194$$ 0 0
$$195$$ 4.34017 0.310806
$$196$$ 0 0
$$197$$ 21.1194 1.50470 0.752348 0.658766i $$-0.228921\pi$$
0.752348 + 0.658766i $$0.228921\pi$$
$$198$$ 0 0
$$199$$ −10.5236 −0.745998 −0.372999 0.927832i $$-0.621670\pi$$
−0.372999 + 0.927832i $$0.621670\pi$$
$$200$$ 0 0
$$201$$ −8.68035 −0.612264
$$202$$ 0 0
$$203$$ −1.52973 −0.107366
$$204$$ 0 0
$$205$$ −9.41855 −0.657820
$$206$$ 0 0
$$207$$ 2.15676 0.149905
$$208$$ 0 0
$$209$$ 5.26180 0.363966
$$210$$ 0 0
$$211$$ −9.57531 −0.659191 −0.329596 0.944122i $$-0.606912\pi$$
−0.329596 + 0.944122i $$0.606912\pi$$
$$212$$ 0 0
$$213$$ −4.68035 −0.320692
$$214$$ 0 0
$$215$$ −7.60197 −0.518450
$$216$$ 0 0
$$217$$ −5.04718 −0.342625
$$218$$ 0 0
$$219$$ 10.4969 0.709317
$$220$$ 0 0
$$221$$ −33.6742 −2.26517
$$222$$ 0 0
$$223$$ 2.15676 0.144427 0.0722135 0.997389i $$-0.476994\pi$$
0.0722135 + 0.997389i $$0.476994\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −9.65983 −0.641145 −0.320573 0.947224i $$-0.603875\pi$$
−0.320573 + 0.947224i $$0.603875\pi$$
$$228$$ 0 0
$$229$$ −3.36069 −0.222081 −0.111040 0.993816i $$-0.535418\pi$$
−0.111040 + 0.993816i $$0.535418\pi$$
$$230$$ 0 0
$$231$$ −1.07838 −0.0709520
$$232$$ 0 0
$$233$$ −2.39803 −0.157100 −0.0785501 0.996910i $$-0.525029\pi$$
−0.0785501 + 0.996910i $$0.525029\pi$$
$$234$$ 0 0
$$235$$ −4.68035 −0.305312
$$236$$ 0 0
$$237$$ −8.09890 −0.526080
$$238$$ 0 0
$$239$$ 7.20394 0.465984 0.232992 0.972479i $$-0.425148\pi$$
0.232992 + 0.972479i $$0.425148\pi$$
$$240$$ 0 0
$$241$$ −5.20394 −0.335215 −0.167608 0.985854i $$-0.553604\pi$$
−0.167608 + 0.985854i $$0.553604\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −5.83710 −0.372919
$$246$$ 0 0
$$247$$ 22.8371 1.45309
$$248$$ 0 0
$$249$$ −11.0205 −0.698397
$$250$$ 0 0
$$251$$ −15.3197 −0.966968 −0.483484 0.875353i $$-0.660629\pi$$
−0.483484 + 0.875353i $$0.660629\pi$$
$$252$$ 0 0
$$253$$ −2.15676 −0.135594
$$254$$ 0 0
$$255$$ −7.75872 −0.485870
$$256$$ 0 0
$$257$$ 4.15676 0.259291 0.129646 0.991560i $$-0.458616\pi$$
0.129646 + 0.991560i $$0.458616\pi$$
$$258$$ 0 0
$$259$$ 2.15676 0.134014
$$260$$ 0 0
$$261$$ 1.41855 0.0878061
$$262$$ 0 0
$$263$$ 18.7070 1.15352 0.576762 0.816912i $$-0.304316\pi$$
0.576762 + 0.816912i $$0.304316\pi$$
$$264$$ 0 0
$$265$$ 0.156755 0.00962941
$$266$$ 0 0
$$267$$ 12.8371 0.785618
$$268$$ 0 0
$$269$$ 23.3607 1.42433 0.712163 0.702014i $$-0.247716\pi$$
0.712163 + 0.702014i $$0.247716\pi$$
$$270$$ 0 0
$$271$$ 5.57531 0.338676 0.169338 0.985558i $$-0.445837\pi$$
0.169338 + 0.985558i $$0.445837\pi$$
$$272$$ 0 0
$$273$$ −4.68035 −0.283267
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −26.0144 −1.56305 −0.781526 0.623872i $$-0.785558\pi$$
−0.781526 + 0.623872i $$0.785558\pi$$
$$278$$ 0 0
$$279$$ 4.68035 0.280205
$$280$$ 0 0
$$281$$ −9.41855 −0.561864 −0.280932 0.959728i $$-0.590643\pi$$
−0.280932 + 0.959728i $$0.590643\pi$$
$$282$$ 0 0
$$283$$ −14.2413 −0.846556 −0.423278 0.906000i $$-0.639121\pi$$
−0.423278 + 0.906000i $$0.639121\pi$$
$$284$$ 0 0
$$285$$ 5.26180 0.311682
$$286$$ 0 0
$$287$$ 10.1568 0.599534
$$288$$ 0 0
$$289$$ 43.1978 2.54105
$$290$$ 0 0
$$291$$ −14.6803 −0.860577
$$292$$ 0 0
$$293$$ −15.7587 −0.920634 −0.460317 0.887754i $$-0.652264\pi$$
−0.460317 + 0.887754i $$0.652264\pi$$
$$294$$ 0 0
$$295$$ −6.15676 −0.358460
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ −9.36069 −0.541343
$$300$$ 0 0
$$301$$ 8.19779 0.472513
$$302$$ 0 0
$$303$$ 15.5753 0.894778
$$304$$ 0 0
$$305$$ −4.15676 −0.238015
$$306$$ 0 0
$$307$$ 18.9216 1.07991 0.539957 0.841693i $$-0.318440\pi$$
0.539957 + 0.841693i $$0.318440\pi$$
$$308$$ 0 0
$$309$$ 6.83710 0.388949
$$310$$ 0 0
$$311$$ 20.8781 1.18389 0.591945 0.805978i $$-0.298360\pi$$
0.591945 + 0.805978i $$0.298360\pi$$
$$312$$ 0 0
$$313$$ 6.31351 0.356861 0.178430 0.983953i $$-0.442898\pi$$
0.178430 + 0.983953i $$0.442898\pi$$
$$314$$ 0 0
$$315$$ −1.07838 −0.0607597
$$316$$ 0 0
$$317$$ 31.3607 1.76139 0.880696 0.473682i $$-0.157075\pi$$
0.880696 + 0.473682i $$0.157075\pi$$
$$318$$ 0 0
$$319$$ −1.41855 −0.0794236
$$320$$ 0 0
$$321$$ 6.34017 0.353874
$$322$$ 0 0
$$323$$ −40.8248 −2.27155
$$324$$ 0 0
$$325$$ −4.34017 −0.240749
$$326$$ 0 0
$$327$$ −2.31351 −0.127937
$$328$$ 0 0
$$329$$ 5.04718 0.278260
$$330$$ 0 0
$$331$$ −19.2039 −1.05554 −0.527772 0.849386i $$-0.676972\pi$$
−0.527772 + 0.849386i $$0.676972\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 8.68035 0.474258
$$336$$ 0 0
$$337$$ 13.5031 0.735559 0.367780 0.929913i $$-0.380118\pi$$
0.367780 + 0.929913i $$0.380118\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −4.68035 −0.253455
$$342$$ 0 0
$$343$$ 13.8432 0.747465
$$344$$ 0 0
$$345$$ −2.15676 −0.116116
$$346$$ 0 0
$$347$$ −6.34017 −0.340358 −0.170179 0.985413i $$-0.554435\pi$$
−0.170179 + 0.985413i $$0.554435\pi$$
$$348$$ 0 0
$$349$$ 16.1568 0.864851 0.432426 0.901670i $$-0.357658\pi$$
0.432426 + 0.901670i $$0.357658\pi$$
$$350$$ 0 0
$$351$$ 4.34017 0.231661
$$352$$ 0 0
$$353$$ −13.2039 −0.702775 −0.351387 0.936230i $$-0.614290\pi$$
−0.351387 + 0.936230i $$0.614290\pi$$
$$354$$ 0 0
$$355$$ 4.68035 0.248407
$$356$$ 0 0
$$357$$ 8.36683 0.442820
$$358$$ 0 0
$$359$$ −3.31965 −0.175205 −0.0876023 0.996156i $$-0.527920\pi$$
−0.0876023 + 0.996156i $$0.527920\pi$$
$$360$$ 0 0
$$361$$ 8.68649 0.457184
$$362$$ 0 0
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −10.4969 −0.549434
$$366$$ 0 0
$$367$$ 36.1445 1.88673 0.943363 0.331762i $$-0.107643\pi$$
0.943363 + 0.331762i $$0.107643\pi$$
$$368$$ 0 0
$$369$$ −9.41855 −0.490310
$$370$$ 0 0
$$371$$ −0.169042 −0.00877620
$$372$$ 0 0
$$373$$ −2.81044 −0.145519 −0.0727595 0.997350i $$-0.523181\pi$$
−0.0727595 + 0.997350i $$0.523181\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −6.15676 −0.317089
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −2.24128 −0.114824
$$382$$ 0 0
$$383$$ −33.5585 −1.71476 −0.857379 0.514685i $$-0.827909\pi$$
−0.857379 + 0.514685i $$0.827909\pi$$
$$384$$ 0 0
$$385$$ 1.07838 0.0549592
$$386$$ 0 0
$$387$$ −7.60197 −0.386430
$$388$$ 0 0
$$389$$ 12.8371 0.650867 0.325433 0.945565i $$-0.394490\pi$$
0.325433 + 0.945565i $$0.394490\pi$$
$$390$$ 0 0
$$391$$ 16.7337 0.846258
$$392$$ 0 0
$$393$$ 8.68035 0.437866
$$394$$ 0 0
$$395$$ 8.09890 0.407500
$$396$$ 0 0
$$397$$ −5.31965 −0.266986 −0.133493 0.991050i $$-0.542619\pi$$
−0.133493 + 0.991050i $$0.542619\pi$$
$$398$$ 0 0
$$399$$ −5.67420 −0.284065
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ −20.3135 −1.01189
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ 26.1978 1.29540 0.647699 0.761897i $$-0.275732\pi$$
0.647699 + 0.761897i $$0.275732\pi$$
$$410$$ 0 0
$$411$$ 15.3607 0.757687
$$412$$ 0 0
$$413$$ 6.63931 0.326699
$$414$$ 0 0
$$415$$ 11.0205 0.540976
$$416$$ 0 0
$$417$$ 8.58145 0.420235
$$418$$ 0 0
$$419$$ 2.83710 0.138601 0.0693007 0.997596i $$-0.477923\pi$$
0.0693007 + 0.997596i $$0.477923\pi$$
$$420$$ 0 0
$$421$$ 11.4764 0.559326 0.279663 0.960098i $$-0.409777\pi$$
0.279663 + 0.960098i $$0.409777\pi$$
$$422$$ 0 0
$$423$$ −4.68035 −0.227566
$$424$$ 0 0
$$425$$ 7.75872 0.376353
$$426$$ 0 0
$$427$$ 4.48255 0.216926
$$428$$ 0 0
$$429$$ −4.34017 −0.209546
$$430$$ 0 0
$$431$$ −23.5708 −1.13536 −0.567682 0.823248i $$-0.692160\pi$$
−0.567682 + 0.823248i $$0.692160\pi$$
$$432$$ 0 0
$$433$$ −14.9939 −0.720559 −0.360279 0.932844i $$-0.617319\pi$$
−0.360279 + 0.932844i $$0.617319\pi$$
$$434$$ 0 0
$$435$$ −1.41855 −0.0680143
$$436$$ 0 0
$$437$$ −11.3484 −0.542868
$$438$$ 0 0
$$439$$ −4.77924 −0.228101 −0.114050 0.993475i $$-0.536383\pi$$
−0.114050 + 0.993475i $$0.536383\pi$$
$$440$$ 0 0
$$441$$ −5.83710 −0.277957
$$442$$ 0 0
$$443$$ −20.1978 −0.959626 −0.479813 0.877371i $$-0.659296\pi$$
−0.479813 + 0.877371i $$0.659296\pi$$
$$444$$ 0 0
$$445$$ −12.8371 −0.608537
$$446$$ 0 0
$$447$$ 18.0989 0.856048
$$448$$ 0 0
$$449$$ −21.5708 −1.01799 −0.508994 0.860770i $$-0.669982\pi$$
−0.508994 + 0.860770i $$0.669982\pi$$
$$450$$ 0 0
$$451$$ 9.41855 0.443502
$$452$$ 0 0
$$453$$ 22.9360 1.07763
$$454$$ 0 0
$$455$$ 4.68035 0.219418
$$456$$ 0 0
$$457$$ 28.1711 1.31779 0.658895 0.752235i $$-0.271024\pi$$
0.658895 + 0.752235i $$0.271024\pi$$
$$458$$ 0 0
$$459$$ −7.75872 −0.362146
$$460$$ 0 0
$$461$$ −1.47187 −0.0685520 −0.0342760 0.999412i $$-0.510913\pi$$
−0.0342760 + 0.999412i $$0.510913\pi$$
$$462$$ 0 0
$$463$$ 23.2039 1.07838 0.539189 0.842185i $$-0.318731\pi$$
0.539189 + 0.842185i $$0.318731\pi$$
$$464$$ 0 0
$$465$$ −4.68035 −0.217046
$$466$$ 0 0
$$467$$ −14.1568 −0.655097 −0.327548 0.944834i $$-0.606222\pi$$
−0.327548 + 0.944834i $$0.606222\pi$$
$$468$$ 0 0
$$469$$ −9.36069 −0.432237
$$470$$ 0 0
$$471$$ 10.9939 0.506570
$$472$$ 0 0
$$473$$ 7.60197 0.349539
$$474$$ 0 0
$$475$$ −5.26180 −0.241428
$$476$$ 0 0
$$477$$ 0.156755 0.00717734
$$478$$ 0 0
$$479$$ 13.8432 0.632514 0.316257 0.948674i $$-0.397574\pi$$
0.316257 + 0.948674i $$0.397574\pi$$
$$480$$ 0 0
$$481$$ 8.68035 0.395790
$$482$$ 0 0
$$483$$ 2.32580 0.105827
$$484$$ 0 0
$$485$$ 14.6803 0.666600
$$486$$ 0 0
$$487$$ 40.9939 1.85761 0.928804 0.370570i $$-0.120838\pi$$
0.928804 + 0.370570i $$0.120838\pi$$
$$488$$ 0 0
$$489$$ −6.52359 −0.295007
$$490$$ 0 0
$$491$$ −34.8371 −1.57218 −0.786088 0.618114i $$-0.787897\pi$$
−0.786088 + 0.618114i $$0.787897\pi$$
$$492$$ 0 0
$$493$$ 11.0061 0.495692
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ 0 0
$$497$$ −5.04718 −0.226397
$$498$$ 0 0
$$499$$ −15.1506 −0.678235 −0.339117 0.940744i $$-0.610128\pi$$
−0.339117 + 0.940744i $$0.610128\pi$$
$$500$$ 0 0
$$501$$ 1.97334 0.0881622
$$502$$ 0 0
$$503$$ 6.65368 0.296673 0.148337 0.988937i $$-0.452608\pi$$
0.148337 + 0.988937i $$0.452608\pi$$
$$504$$ 0 0
$$505$$ −15.5753 −0.693092
$$506$$ 0 0
$$507$$ −5.83710 −0.259235
$$508$$ 0 0
$$509$$ 41.3484 1.83274 0.916368 0.400337i $$-0.131107\pi$$
0.916368 + 0.400337i $$0.131107\pi$$
$$510$$ 0 0
$$511$$ 11.3197 0.500752
$$512$$ 0 0
$$513$$ 5.26180 0.232314
$$514$$ 0 0
$$515$$ −6.83710 −0.301279
$$516$$ 0 0
$$517$$ 4.68035 0.205841
$$518$$ 0 0
$$519$$ −3.75872 −0.164990
$$520$$ 0 0
$$521$$ 7.67420 0.336213 0.168106 0.985769i $$-0.446235\pi$$
0.168106 + 0.985769i $$0.446235\pi$$
$$522$$ 0 0
$$523$$ −23.2351 −1.01600 −0.508001 0.861357i $$-0.669615\pi$$
−0.508001 + 0.861357i $$0.669615\pi$$
$$524$$ 0 0
$$525$$ 1.07838 0.0470643
$$526$$ 0 0
$$527$$ 36.3135 1.58184
$$528$$ 0 0
$$529$$ −18.3484 −0.797757
$$530$$ 0 0
$$531$$ −6.15676 −0.267180
$$532$$ 0 0
$$533$$ 40.8781 1.77063
$$534$$ 0 0
$$535$$ −6.34017 −0.274110
$$536$$ 0 0
$$537$$ 15.1506 0.653797
$$538$$ 0 0
$$539$$ 5.83710 0.251422
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ −4.83710 −0.207580
$$544$$ 0 0
$$545$$ 2.31351 0.0990999
$$546$$ 0 0
$$547$$ 23.0661 0.986235 0.493117 0.869963i $$-0.335857\pi$$
0.493117 + 0.869963i $$0.335857\pi$$
$$548$$ 0 0
$$549$$ −4.15676 −0.177406
$$550$$ 0 0
$$551$$ −7.46412 −0.317982
$$552$$ 0 0
$$553$$ −8.73367 −0.371393
$$554$$ 0 0
$$555$$ 2.00000 0.0848953
$$556$$ 0 0
$$557$$ 10.5958 0.448960 0.224480 0.974479i $$-0.427932\pi$$
0.224480 + 0.974479i $$0.427932\pi$$
$$558$$ 0 0
$$559$$ 32.9939 1.39549
$$560$$ 0 0
$$561$$ 7.75872 0.327574
$$562$$ 0 0
$$563$$ 36.2122 1.52616 0.763080 0.646303i $$-0.223686\pi$$
0.763080 + 0.646303i $$0.223686\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ −1.07838 −0.0452876
$$568$$ 0 0
$$569$$ −27.5753 −1.15602 −0.578008 0.816031i $$-0.696170\pi$$
−0.578008 + 0.816031i $$0.696170\pi$$
$$570$$ 0 0
$$571$$ 27.9299 1.16883 0.584414 0.811456i $$-0.301324\pi$$
0.584414 + 0.811456i $$0.301324\pi$$
$$572$$ 0 0
$$573$$ 2.52359 0.105425
$$574$$ 0 0
$$575$$ 2.15676 0.0899429
$$576$$ 0 0
$$577$$ 41.4017 1.72358 0.861788 0.507268i $$-0.169345\pi$$
0.861788 + 0.507268i $$0.169345\pi$$
$$578$$ 0 0
$$579$$ −0.0266620 −0.00110803
$$580$$ 0 0
$$581$$ −11.8843 −0.493043
$$582$$ 0 0
$$583$$ −0.156755 −0.00649215
$$584$$ 0 0
$$585$$ −4.34017 −0.179444
$$586$$ 0 0
$$587$$ −8.48255 −0.350112 −0.175056 0.984558i $$-0.556011\pi$$
−0.175056 + 0.984558i $$0.556011\pi$$
$$588$$ 0 0
$$589$$ −24.6270 −1.01474
$$590$$ 0 0
$$591$$ −21.1194 −0.868737
$$592$$ 0 0
$$593$$ 7.56093 0.310490 0.155245 0.987876i $$-0.450383\pi$$
0.155245 + 0.987876i $$0.450383\pi$$
$$594$$ 0 0
$$595$$ −8.36683 −0.343007
$$596$$ 0 0
$$597$$ 10.5236 0.430702
$$598$$ 0 0
$$599$$ −5.67420 −0.231842 −0.115921 0.993258i $$-0.536982\pi$$
−0.115921 + 0.993258i $$0.536982\pi$$
$$600$$ 0 0
$$601$$ −1.31965 −0.0538298 −0.0269149 0.999638i $$-0.508568\pi$$
−0.0269149 + 0.999638i $$0.508568\pi$$
$$602$$ 0 0
$$603$$ 8.68035 0.353491
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 2.24128 0.0909706 0.0454853 0.998965i $$-0.485517\pi$$
0.0454853 + 0.998965i $$0.485517\pi$$
$$608$$ 0 0
$$609$$ 1.52973 0.0619879
$$610$$ 0 0
$$611$$ 20.3135 0.821797
$$612$$ 0 0
$$613$$ 42.8638 1.73125 0.865626 0.500692i $$-0.166921\pi$$
0.865626 + 0.500692i $$0.166921\pi$$
$$614$$ 0 0
$$615$$ 9.41855 0.379793
$$616$$ 0 0
$$617$$ 11.3607 0.457364 0.228682 0.973501i $$-0.426558\pi$$
0.228682 + 0.973501i $$0.426558\pi$$
$$618$$ 0 0
$$619$$ 45.1917 1.81641 0.908203 0.418530i $$-0.137455\pi$$
0.908203 + 0.418530i $$0.137455\pi$$
$$620$$ 0 0
$$621$$ −2.15676 −0.0865476
$$622$$ 0 0
$$623$$ 13.8432 0.554618
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −5.26180 −0.210136
$$628$$ 0 0
$$629$$ −15.5174 −0.618721
$$630$$ 0 0
$$631$$ 9.78992 0.389731 0.194865 0.980830i $$-0.437573\pi$$
0.194865 + 0.980830i $$0.437573\pi$$
$$632$$ 0 0
$$633$$ 9.57531 0.380584
$$634$$ 0 0
$$635$$ 2.24128 0.0889423
$$636$$ 0 0
$$637$$ 25.3340 1.00377
$$638$$ 0 0
$$639$$ 4.68035 0.185152
$$640$$ 0 0
$$641$$ 0.210079 0.00829764 0.00414882 0.999991i $$-0.498679\pi$$
0.00414882 + 0.999991i $$0.498679\pi$$
$$642$$ 0 0
$$643$$ −14.5236 −0.572754 −0.286377 0.958117i $$-0.592451\pi$$
−0.286377 + 0.958117i $$0.592451\pi$$
$$644$$ 0 0
$$645$$ 7.60197 0.299327
$$646$$ 0 0
$$647$$ −15.4641 −0.607957 −0.303979 0.952679i $$-0.598315\pi$$
−0.303979 + 0.952679i $$0.598315\pi$$
$$648$$ 0 0
$$649$$ 6.15676 0.241674
$$650$$ 0 0
$$651$$ 5.04718 0.197815
$$652$$ 0 0
$$653$$ −17.8310 −0.697779 −0.348890 0.937164i $$-0.613441\pi$$
−0.348890 + 0.937164i $$0.613441\pi$$
$$654$$ 0 0
$$655$$ −8.68035 −0.339169
$$656$$ 0 0
$$657$$ −10.4969 −0.409524
$$658$$ 0 0
$$659$$ 32.3135 1.25876 0.629378 0.777099i $$-0.283310\pi$$
0.629378 + 0.777099i $$0.283310\pi$$
$$660$$ 0 0
$$661$$ −5.68649 −0.221179 −0.110589 0.993866i $$-0.535274\pi$$
−0.110589 + 0.993866i $$0.535274\pi$$
$$662$$ 0 0
$$663$$ 33.6742 1.30780
$$664$$ 0 0
$$665$$ 5.67420 0.220036
$$666$$ 0 0
$$667$$ 3.05947 0.118463
$$668$$ 0 0
$$669$$ −2.15676 −0.0833850
$$670$$ 0 0
$$671$$ 4.15676 0.160470
$$672$$ 0 0
$$673$$ −21.0205 −0.810281 −0.405141 0.914254i $$-0.632777\pi$$
−0.405141 + 0.914254i $$0.632777\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 36.7526 1.41252 0.706258 0.707954i $$-0.250382\pi$$
0.706258 + 0.707954i $$0.250382\pi$$
$$678$$ 0 0
$$679$$ −15.8310 −0.607536
$$680$$ 0 0
$$681$$ 9.65983 0.370165
$$682$$ 0 0
$$683$$ 17.3074 0.662248 0.331124 0.943587i $$-0.392572\pi$$
0.331124 + 0.943587i $$0.392572\pi$$
$$684$$ 0 0
$$685$$ −15.3607 −0.586902
$$686$$ 0 0
$$687$$ 3.36069 0.128218
$$688$$ 0 0
$$689$$ −0.680346 −0.0259191
$$690$$ 0 0
$$691$$ −17.6742 −0.672358 −0.336179 0.941798i $$-0.609135\pi$$
−0.336179 + 0.941798i $$0.609135\pi$$
$$692$$ 0 0
$$693$$ 1.07838 0.0409642
$$694$$ 0 0
$$695$$ −8.58145 −0.325513
$$696$$ 0 0
$$697$$ −73.0759 −2.76795
$$698$$ 0 0
$$699$$ 2.39803 0.0907019
$$700$$ 0 0
$$701$$ −17.1050 −0.646048 −0.323024 0.946391i $$-0.604700\pi$$
−0.323024 + 0.946391i $$0.604700\pi$$
$$702$$ 0 0
$$703$$ 10.5236 0.396905
$$704$$ 0 0
$$705$$ 4.68035 0.176272
$$706$$ 0 0
$$707$$ 16.7961 0.631681
$$708$$ 0 0
$$709$$ 25.1506 0.944551 0.472276 0.881451i $$-0.343433\pi$$
0.472276 + 0.881451i $$0.343433\pi$$
$$710$$ 0 0
$$711$$ 8.09890 0.303732
$$712$$ 0 0
$$713$$ 10.0944 0.378037
$$714$$ 0 0
$$715$$ 4.34017 0.162313
$$716$$ 0 0
$$717$$ −7.20394 −0.269036
$$718$$ 0 0
$$719$$ 1.78992 0.0667528 0.0333764 0.999443i $$-0.489374\pi$$
0.0333764 + 0.999443i $$0.489374\pi$$
$$720$$ 0 0
$$721$$ 7.37298 0.274584
$$722$$ 0 0
$$723$$ 5.20394 0.193536
$$724$$ 0 0
$$725$$ 1.41855 0.0526837
$$726$$ 0 0
$$727$$ −25.9877 −0.963831 −0.481915 0.876218i $$-0.660059\pi$$
−0.481915 + 0.876218i $$0.660059\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −58.9816 −2.18151
$$732$$ 0 0
$$733$$ −41.0205 −1.51513 −0.757564 0.652761i $$-0.773610\pi$$
−0.757564 + 0.652761i $$0.773610\pi$$
$$734$$ 0 0
$$735$$ 5.83710 0.215305
$$736$$ 0 0
$$737$$ −8.68035 −0.319745
$$738$$ 0 0
$$739$$ 47.6163 1.75160 0.875798 0.482678i $$-0.160336\pi$$
0.875798 + 0.482678i $$0.160336\pi$$
$$740$$ 0 0
$$741$$ −22.8371 −0.838942
$$742$$ 0 0
$$743$$ −0.550252 −0.0201868 −0.0100934 0.999949i $$-0.503213\pi$$
−0.0100934 + 0.999949i $$0.503213\pi$$
$$744$$ 0 0
$$745$$ −18.0989 −0.663092
$$746$$ 0 0
$$747$$ 11.0205 0.403220
$$748$$ 0 0
$$749$$ 6.83710 0.249822
$$750$$ 0 0
$$751$$ −41.5585 −1.51649 −0.758245 0.651969i $$-0.773943\pi$$
−0.758245 + 0.651969i $$0.773943\pi$$
$$752$$ 0 0
$$753$$ 15.3197 0.558279
$$754$$ 0 0
$$755$$ −22.9360 −0.834726
$$756$$ 0 0
$$757$$ 1.31965 0.0479636 0.0239818 0.999712i $$-0.492366\pi$$
0.0239818 + 0.999712i $$0.492366\pi$$
$$758$$ 0 0
$$759$$ 2.15676 0.0782853
$$760$$ 0 0
$$761$$ −2.21461 −0.0802797 −0.0401399 0.999194i $$-0.512780\pi$$
−0.0401399 + 0.999194i $$0.512780\pi$$
$$762$$ 0 0
$$763$$ −2.49484 −0.0903192
$$764$$ 0 0
$$765$$ 7.75872 0.280517
$$766$$ 0 0
$$767$$ 26.7214 0.964853
$$768$$ 0 0
$$769$$ −14.3668 −0.518081 −0.259041 0.965866i $$-0.583406\pi$$
−0.259041 + 0.965866i $$0.583406\pi$$
$$770$$ 0 0
$$771$$ −4.15676 −0.149702
$$772$$ 0 0
$$773$$ 40.1568 1.44434 0.722169 0.691717i $$-0.243145\pi$$
0.722169 + 0.691717i $$0.243145\pi$$
$$774$$ 0 0
$$775$$ 4.68035 0.168123
$$776$$ 0 0
$$777$$ −2.15676 −0.0773732
$$778$$ 0 0
$$779$$ 49.5585 1.77562
$$780$$ 0 0
$$781$$ −4.68035 −0.167476
$$782$$ 0 0
$$783$$ −1.41855 −0.0506949
$$784$$ 0 0
$$785$$ −10.9939 −0.392388
$$786$$ 0 0
$$787$$ −49.5897 −1.76768 −0.883841 0.467788i $$-0.845051\pi$$
−0.883841 + 0.467788i $$0.845051\pi$$
$$788$$ 0 0
$$789$$ −18.7070 −0.665987
$$790$$ 0 0
$$791$$ 6.47027 0.230056
$$792$$ 0 0
$$793$$ 18.0410 0.640656
$$794$$ 0 0
$$795$$ −0.156755 −0.00555954
$$796$$ 0 0
$$797$$ −46.7091 −1.65452 −0.827261 0.561818i $$-0.810102\pi$$
−0.827261 + 0.561818i $$0.810102\pi$$
$$798$$ 0 0
$$799$$ −36.3135 −1.28468
$$800$$ 0 0
$$801$$ −12.8371 −0.453577
$$802$$ 0 0
$$803$$ 10.4969 0.370429
$$804$$ 0 0
$$805$$ −2.32580 −0.0819736
$$806$$ 0 0
$$807$$ −23.3607 −0.822335
$$808$$ 0 0
$$809$$ −18.5814 −0.653289 −0.326644 0.945147i $$-0.605918\pi$$
−0.326644 + 0.945147i $$0.605918\pi$$
$$810$$ 0 0
$$811$$ −27.3028 −0.958732 −0.479366 0.877615i $$-0.659133\pi$$
−0.479366 + 0.877615i $$0.659133\pi$$
$$812$$ 0 0
$$813$$ −5.57531 −0.195535
$$814$$ 0 0
$$815$$ 6.52359 0.228511
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 0 0
$$819$$ 4.68035 0.163545
$$820$$ 0 0
$$821$$ −31.2085 −1.08918 −0.544592 0.838701i $$-0.683315\pi$$
−0.544592 + 0.838701i $$0.683315\pi$$
$$822$$ 0 0
$$823$$ 50.1855 1.74936 0.874678 0.484704i $$-0.161073\pi$$
0.874678 + 0.484704i $$0.161073\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ −27.3874 −0.952352 −0.476176 0.879350i $$-0.657977\pi$$
−0.476176 + 0.879350i $$0.657977\pi$$
$$828$$ 0 0
$$829$$ −26.1978 −0.909887 −0.454943 0.890520i $$-0.650341\pi$$
−0.454943 + 0.890520i $$0.650341\pi$$
$$830$$ 0 0
$$831$$ 26.0144 0.902429
$$832$$ 0 0
$$833$$ −45.2885 −1.56915
$$834$$ 0 0
$$835$$ −1.97334 −0.0682902
$$836$$ 0 0
$$837$$ −4.68035 −0.161776
$$838$$ 0 0
$$839$$ −7.20394 −0.248708 −0.124354 0.992238i $$-0.539686\pi$$
−0.124354 + 0.992238i $$0.539686\pi$$
$$840$$ 0 0
$$841$$ −26.9877 −0.930611
$$842$$ 0 0
$$843$$ 9.41855 0.324392
$$844$$ 0 0
$$845$$ 5.83710 0.200802
$$846$$ 0 0
$$847$$ −1.07838 −0.0370535
$$848$$ 0 0
$$849$$ 14.2413 0.488759
$$850$$ 0 0
$$851$$ −4.31351 −0.147865
$$852$$ 0 0
$$853$$ 39.8043 1.36287 0.681437 0.731877i $$-0.261356\pi$$
0.681437 + 0.731877i $$0.261356\pi$$
$$854$$ 0 0
$$855$$ −5.26180 −0.179950
$$856$$ 0 0
$$857$$ −36.9504 −1.26220 −0.631100 0.775701i $$-0.717396\pi$$
−0.631100 + 0.775701i $$0.717396\pi$$
$$858$$ 0 0
$$859$$ −57.5052 −1.96205 −0.981025 0.193879i $$-0.937893\pi$$
−0.981025 + 0.193879i $$0.937893\pi$$
$$860$$ 0 0
$$861$$ −10.1568 −0.346141
$$862$$ 0 0
$$863$$ 1.89657 0.0645599 0.0322800 0.999479i $$-0.489723\pi$$
0.0322800 + 0.999479i $$0.489723\pi$$
$$864$$ 0 0
$$865$$ 3.75872 0.127800
$$866$$ 0 0
$$867$$ −43.1978 −1.46707
$$868$$ 0 0
$$869$$ −8.09890 −0.274736
$$870$$ 0 0
$$871$$ −37.6742 −1.27654
$$872$$ 0 0
$$873$$ 14.6803 0.496854
$$874$$ 0 0
$$875$$ −1.07838 −0.0364558
$$876$$ 0 0
$$877$$ 32.5380 1.09873 0.549365 0.835583i $$-0.314870\pi$$
0.549365 + 0.835583i $$0.314870\pi$$
$$878$$ 0 0
$$879$$ 15.7587 0.531529
$$880$$ 0 0
$$881$$ 18.1978 0.613099 0.306550 0.951855i $$-0.400825\pi$$
0.306550 + 0.951855i $$0.400825\pi$$
$$882$$ 0 0
$$883$$ −36.3956 −1.22481 −0.612405 0.790545i $$-0.709798\pi$$
−0.612405 + 0.790545i $$0.709798\pi$$
$$884$$ 0 0
$$885$$ 6.15676 0.206957
$$886$$ 0 0
$$887$$ −27.8699 −0.935780 −0.467890 0.883787i $$-0.654986\pi$$
−0.467890 + 0.883787i $$0.654986\pi$$
$$888$$ 0 0
$$889$$ −2.41694 −0.0810616
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 24.6270 0.824112
$$894$$ 0 0
$$895$$ −15.1506 −0.506429
$$896$$ 0 0
$$897$$ 9.36069 0.312544
$$898$$ 0 0
$$899$$ 6.63931 0.221433
$$900$$ 0 0
$$901$$ 1.21622 0.0405182
$$902$$ 0 0
$$903$$ −8.19779 −0.272805
$$904$$ 0 0
$$905$$ 4.83710 0.160791
$$906$$ 0 0
$$907$$ 27.9376 0.927653 0.463826 0.885926i $$-0.346476\pi$$
0.463826 + 0.885926i $$0.346476\pi$$
$$908$$ 0 0
$$909$$ −15.5753 −0.516600
$$910$$ 0 0
$$911$$ 11.8843 0.393744 0.196872 0.980429i $$-0.436922\pi$$
0.196872 + 0.980429i $$0.436922\pi$$
$$912$$ 0 0
$$913$$ −11.0205 −0.364726
$$914$$ 0 0
$$915$$ 4.15676 0.137418
$$916$$ 0 0
$$917$$ 9.36069 0.309117
$$918$$ 0 0
$$919$$ 45.6041 1.50434 0.752170 0.658970i $$-0.229007\pi$$
0.752170 + 0.658970i $$0.229007\pi$$
$$920$$ 0 0
$$921$$ −18.9216 −0.623489
$$922$$ 0 0
$$923$$ −20.3135 −0.668627
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ −6.83710 −0.224560
$$928$$ 0 0
$$929$$ −25.1506 −0.825165 −0.412582 0.910920i $$-0.635373\pi$$
−0.412582 + 0.910920i $$0.635373\pi$$
$$930$$ 0 0
$$931$$ 30.7136 1.00660
$$932$$ 0 0
$$933$$ −20.8781 −0.683520
$$934$$ 0 0
$$935$$ −7.75872 −0.253737
$$936$$ 0 0
$$937$$ 5.33403 0.174255 0.0871276 0.996197i $$-0.472231\pi$$
0.0871276 + 0.996197i $$0.472231\pi$$
$$938$$ 0 0
$$939$$ −6.31351 −0.206034
$$940$$ 0 0
$$941$$ 56.8203 1.85229 0.926144 0.377170i $$-0.123103\pi$$
0.926144 + 0.377170i $$0.123103\pi$$
$$942$$ 0 0
$$943$$ −20.3135 −0.661499
$$944$$ 0 0
$$945$$ 1.07838 0.0350796
$$946$$ 0 0
$$947$$ 20.9939 0.682209 0.341104 0.940025i $$-0.389199\pi$$
0.341104 + 0.940025i $$0.389199\pi$$
$$948$$ 0 0
$$949$$ 45.5585 1.47889
$$950$$ 0 0
$$951$$ −31.3607 −1.01694
$$952$$ 0 0
$$953$$ −25.2351 −0.817446 −0.408723 0.912658i $$-0.634026\pi$$
−0.408723 + 0.912658i $$0.634026\pi$$
$$954$$ 0 0
$$955$$ −2.52359 −0.0816615
$$956$$ 0 0
$$957$$ 1.41855 0.0458552
$$958$$ 0 0
$$959$$ 16.5646 0.534900
$$960$$ 0 0
$$961$$ −9.09436 −0.293367
$$962$$ 0 0
$$963$$ −6.34017 −0.204309
$$964$$ 0 0
$$965$$ 0.0266620 0.000858280 0
$$966$$ 0 0
$$967$$ −13.1317 −0.422287 −0.211144 0.977455i $$-0.567719\pi$$
−0.211144 + 0.977455i $$0.567719\pi$$
$$968$$ 0 0
$$969$$ 40.8248 1.31148
$$970$$ 0 0
$$971$$ −8.94053 −0.286915 −0.143458 0.989656i $$-0.545822\pi$$
−0.143458 + 0.989656i $$0.545822\pi$$
$$972$$ 0 0
$$973$$ 9.25404 0.296671
$$974$$ 0 0
$$975$$ 4.34017 0.138997
$$976$$ 0 0
$$977$$ 50.3956 1.61230 0.806149 0.591713i $$-0.201548\pi$$
0.806149 + 0.591713i $$0.201548\pi$$
$$978$$ 0 0
$$979$$ 12.8371 0.410276
$$980$$ 0 0
$$981$$ 2.31351 0.0738647
$$982$$ 0 0
$$983$$ 32.1978 1.02695 0.513475 0.858105i $$-0.328358\pi$$
0.513475 + 0.858105i $$0.328358\pi$$
$$984$$ 0 0
$$985$$ 21.1194 0.672921
$$986$$ 0 0
$$987$$ −5.04718 −0.160654
$$988$$ 0 0
$$989$$ −16.3956 −0.521349
$$990$$ 0 0
$$991$$ 46.7747 1.48585 0.742924 0.669376i $$-0.233438\pi$$
0.742924 + 0.669376i $$0.233438\pi$$
$$992$$ 0 0
$$993$$ 19.2039 0.609419
$$994$$ 0 0
$$995$$ −10.5236 −0.333620
$$996$$ 0 0
$$997$$ 38.2122 1.21019 0.605096 0.796153i $$-0.293135\pi$$
0.605096 + 0.796153i $$0.293135\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
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 2640.2.a.be.1.2 3
3.2 odd 2 7920.2.a.cj.1.2 3
4.3 odd 2 165.2.a.c.1.1 3
12.11 even 2 495.2.a.e.1.3 3
20.3 even 4 825.2.c.g.199.6 6
20.7 even 4 825.2.c.g.199.1 6
20.19 odd 2 825.2.a.k.1.3 3
28.27 even 2 8085.2.a.bk.1.1 3
44.43 even 2 1815.2.a.m.1.3 3
60.23 odd 4 2475.2.c.r.199.1 6
60.47 odd 4 2475.2.c.r.199.6 6
60.59 even 2 2475.2.a.bb.1.1 3
132.131 odd 2 5445.2.a.z.1.1 3
220.219 even 2 9075.2.a.cf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 4.3 odd 2
495.2.a.e.1.3 3 12.11 even 2
825.2.a.k.1.3 3 20.19 odd 2
825.2.c.g.199.1 6 20.7 even 4
825.2.c.g.199.6 6 20.3 even 4
1815.2.a.m.1.3 3 44.43 even 2
2475.2.a.bb.1.1 3 60.59 even 2
2475.2.c.r.199.1 6 60.23 odd 4
2475.2.c.r.199.6 6 60.47 odd 4
2640.2.a.be.1.2 3 1.1 even 1 trivial
5445.2.a.z.1.1 3 132.131 odd 2
7920.2.a.cj.1.2 3 3.2 odd 2
8085.2.a.bk.1.1 3 28.27 even 2
9075.2.a.cf.1.1 3 220.219 even 2