# Properties

 Label 825.2.a.k.1.3 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: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} -1.07838 q^{7} +9.04945 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.70928 q^{2} -1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} -1.07838 q^{7} +9.04945 q^{8} +1.00000 q^{9} +1.00000 q^{11} -5.34017 q^{12} +4.34017 q^{13} -2.92162 q^{14} +13.8371 q^{16} -7.75872 q^{17} +2.70928 q^{18} +5.26180 q^{19} +1.07838 q^{21} +2.70928 q^{22} +2.15676 q^{23} -9.04945 q^{24} +11.7587 q^{26} -1.00000 q^{27} -5.75872 q^{28} +1.41855 q^{29} -4.68035 q^{31} +19.3896 q^{32} -1.00000 q^{33} -21.0205 q^{34} +5.34017 q^{36} +2.00000 q^{37} +14.2557 q^{38} -4.34017 q^{39} -9.41855 q^{41} +2.92162 q^{42} -7.60197 q^{43} +5.34017 q^{44} +5.84324 q^{46} -4.68035 q^{47} -13.8371 q^{48} -5.83710 q^{49} +7.75872 q^{51} +23.1773 q^{52} -0.156755 q^{53} -2.70928 q^{54} -9.75872 q^{56} -5.26180 q^{57} +3.84324 q^{58} +6.15676 q^{59} -4.15676 q^{61} -12.6803 q^{62} -1.07838 q^{63} +24.8576 q^{64} -2.70928 q^{66} +8.68035 q^{67} -41.4329 q^{68} -2.15676 q^{69} -4.68035 q^{71} +9.04945 q^{72} +10.4969 q^{73} +5.41855 q^{74} +28.0989 q^{76} -1.07838 q^{77} -11.7587 q^{78} -8.09890 q^{79} +1.00000 q^{81} -25.5174 q^{82} +11.0205 q^{83} +5.75872 q^{84} -20.5958 q^{86} -1.41855 q^{87} +9.04945 q^{88} -12.8371 q^{89} -4.68035 q^{91} +11.5174 q^{92} +4.68035 q^{93} -12.6803 q^{94} -19.3896 q^{96} -14.6803 q^{97} -15.8143 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 + 9 * q^8 + 3 * q^9 $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{11} - 5 q^{12} + 2 q^{13} - 12 q^{14} + 13 q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + q^{22} - 9 q^{24} + 10 q^{26} - 3 q^{27} + 8 q^{28} - 10 q^{29} + 8 q^{31} + 29 q^{32} - 3 q^{33} - 30 q^{34} + 5 q^{36} + 6 q^{37} - 2 q^{39} - 14 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} + 24 q^{46} + 8 q^{47} - 13 q^{48} + 11 q^{49} - 2 q^{51} + 30 q^{52} + 6 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} + 18 q^{58} + 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} - q^{66} + 4 q^{67} - 42 q^{68} + 8 q^{71} + 9 q^{72} + 14 q^{73} + 2 q^{74} + 48 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 26 q^{82} - 8 q^{84} - 8 q^{86} + 10 q^{87} + 9 q^{88} - 10 q^{89} + 8 q^{91} - 16 q^{92} - 8 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} - 39 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 + 9 * q^8 + 3 * q^9 + 3 * q^11 - 5 * q^12 + 2 * q^13 - 12 * q^14 + 13 * q^16 + 2 * q^17 + q^18 + 8 * q^19 + q^22 - 9 * q^24 + 10 * q^26 - 3 * q^27 + 8 * q^28 - 10 * q^29 + 8 * q^31 + 29 * q^32 - 3 * q^33 - 30 * q^34 + 5 * q^36 + 6 * q^37 - 2 * q^39 - 14 * q^41 + 12 * q^42 - 4 * q^43 + 5 * q^44 + 24 * q^46 + 8 * q^47 - 13 * q^48 + 11 * q^49 - 2 * q^51 + 30 * q^52 + 6 * q^53 - q^54 - 4 * q^56 - 8 * q^57 + 18 * q^58 + 12 * q^59 - 6 * q^61 - 16 * q^62 + 13 * q^64 - q^66 + 4 * q^67 - 42 * q^68 + 8 * q^71 + 9 * q^72 + 14 * q^73 + 2 * q^74 + 48 * q^76 - 10 * q^78 + 12 * q^79 + 3 * q^81 - 26 * q^82 - 8 * q^84 - 8 * q^86 + 10 * q^87 + 9 * q^88 - 10 * q^89 + 8 * q^91 - 16 * q^92 - 8 * q^93 - 16 * q^94 - 29 * q^96 - 22 * q^97 - 39 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.70928 1.91575 0.957873 0.287190i $$-0.0927213\pi$$
0.957873 + 0.287190i $$0.0927213\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 5.34017 2.67009
$$5$$ 0 0
$$6$$ −2.70928 −1.10606
$$7$$ −1.07838 −0.407588 −0.203794 0.979014i $$-0.565327\pi$$
−0.203794 + 0.979014i $$0.565327\pi$$
$$8$$ 9.04945 3.19946
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ −5.34017 −1.54158
$$13$$ 4.34017 1.20375 0.601874 0.798591i $$-0.294421\pi$$
0.601874 + 0.798591i $$0.294421\pi$$
$$14$$ −2.92162 −0.780836
$$15$$ 0 0
$$16$$ 13.8371 3.45928
$$17$$ −7.75872 −1.88177 −0.940883 0.338730i $$-0.890003\pi$$
−0.940883 + 0.338730i $$0.890003\pi$$
$$18$$ 2.70928 0.638582
$$19$$ 5.26180 1.20714 0.603569 0.797311i $$-0.293745\pi$$
0.603569 + 0.797311i $$0.293745\pi$$
$$20$$ 0 0
$$21$$ 1.07838 0.235321
$$22$$ 2.70928 0.577619
$$23$$ 2.15676 0.449715 0.224857 0.974392i $$-0.427808\pi$$
0.224857 + 0.974392i $$0.427808\pi$$
$$24$$ −9.04945 −1.84721
$$25$$ 0 0
$$26$$ 11.7587 2.30608
$$27$$ −1.00000 −0.192450
$$28$$ −5.75872 −1.08830
$$29$$ 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 $$-0.638078\pi$$
−0.420307 + 0.907382i $$0.638078\pi$$
$$32$$ 19.3896 3.42763
$$33$$ −1.00000 −0.174078
$$34$$ −21.0205 −3.60499
$$35$$ 0 0
$$36$$ 5.34017 0.890029
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 14.2557 2.31257
$$39$$ −4.34017 −0.694984
$$40$$ 0 0
$$41$$ −9.41855 −1.47093 −0.735465 0.677562i $$-0.763036\pi$$
−0.735465 + 0.677562i $$0.763036\pi$$
$$42$$ 2.92162 0.450816
$$43$$ −7.60197 −1.15929 −0.579645 0.814869i $$-0.696809\pi$$
−0.579645 + 0.814869i $$0.696809\pi$$
$$44$$ 5.34017 0.805061
$$45$$ 0 0
$$46$$ 5.84324 0.861539
$$47$$ −4.68035 −0.682699 −0.341349 0.939937i $$-0.610884\pi$$
−0.341349 + 0.939937i $$0.610884\pi$$
$$48$$ −13.8371 −1.99721
$$49$$ −5.83710 −0.833872
$$50$$ 0 0
$$51$$ 7.75872 1.08644
$$52$$ 23.1773 3.21411
$$53$$ −0.156755 −0.0215320 −0.0107660 0.999942i $$-0.503427\pi$$
−0.0107660 + 0.999942i $$0.503427\pi$$
$$54$$ −2.70928 −0.368686
$$55$$ 0 0
$$56$$ −9.75872 −1.30406
$$57$$ −5.26180 −0.696942
$$58$$ 3.84324 0.504643
$$59$$ 6.15676 0.801541 0.400771 0.916178i $$-0.368742\pi$$
0.400771 + 0.916178i $$0.368742\pi$$
$$60$$ 0 0
$$61$$ −4.15676 −0.532218 −0.266109 0.963943i $$-0.585738\pi$$
−0.266109 + 0.963943i $$0.585738\pi$$
$$62$$ −12.6803 −1.61041
$$63$$ −1.07838 −0.135863
$$64$$ 24.8576 3.10720
$$65$$ 0 0
$$66$$ −2.70928 −0.333489
$$67$$ 8.68035 1.06047 0.530237 0.847850i $$-0.322103\pi$$
0.530237 + 0.847850i $$0.322103\pi$$
$$68$$ −41.4329 −5.02448
$$69$$ −2.15676 −0.259643
$$70$$ 0 0
$$71$$ −4.68035 −0.555455 −0.277727 0.960660i $$-0.589581\pi$$
−0.277727 + 0.960660i $$0.589581\pi$$
$$72$$ 9.04945 1.06649
$$73$$ 10.4969 1.22857 0.614286 0.789083i $$-0.289444\pi$$
0.614286 + 0.789083i $$0.289444\pi$$
$$74$$ 5.41855 0.629894
$$75$$ 0 0
$$76$$ 28.0989 3.22316
$$77$$ −1.07838 −0.122893
$$78$$ −11.7587 −1.33141
$$79$$ −8.09890 −0.911197 −0.455599 0.890185i $$-0.650575\pi$$
−0.455599 + 0.890185i $$0.650575\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −25.5174 −2.81793
$$83$$ 11.0205 1.20966 0.604830 0.796355i $$-0.293241\pi$$
0.604830 + 0.796355i $$0.293241\pi$$
$$84$$ 5.75872 0.628328
$$85$$ 0 0
$$86$$ −20.5958 −2.22090
$$87$$ −1.41855 −0.152085
$$88$$ 9.04945 0.964674
$$89$$ −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
$$92$$ 11.5174 1.20078
$$93$$ 4.68035 0.485329
$$94$$ −12.6803 −1.30788
$$95$$ 0 0
$$96$$ −19.3896 −1.97894
$$97$$ −14.6803 −1.49056 −0.745282 0.666750i $$-0.767685\pi$$
−0.745282 + 0.666750i $$0.767685\pi$$
$$98$$ −15.8143 −1.59749
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −15.5753 −1.54980 −0.774900 0.632083i $$-0.782200\pi$$
−0.774900 + 0.632083i $$0.782200\pi$$
$$102$$ 21.0205 2.08134
$$103$$ −6.83710 −0.673680 −0.336840 0.941562i $$-0.609358\pi$$
−0.336840 + 0.941562i $$0.609358\pi$$
$$104$$ 39.2762 3.85135
$$105$$ 0 0
$$106$$ −0.424694 −0.0412499
$$107$$ −6.34017 −0.612928 −0.306464 0.951882i $$-0.599146\pi$$
−0.306464 + 0.951882i $$0.599146\pi$$
$$108$$ −5.34017 −0.513858
$$109$$ 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
$$112$$ −14.9216 −1.40996
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −14.2557 −1.33516
$$115$$ 0 0
$$116$$ 7.57531 0.703350
$$117$$ 4.34017 0.401249
$$118$$ 16.6803 1.53555
$$119$$ 8.36683 0.766987
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −11.2618 −1.01960
$$123$$ 9.41855 0.849242
$$124$$ −24.9939 −2.24451
$$125$$ 0 0
$$126$$ −2.92162 −0.260279
$$127$$ 2.24128 0.198881 0.0994406 0.995044i $$-0.468295\pi$$
0.0994406 + 0.995044i $$0.468295\pi$$
$$128$$ 28.5669 2.52498
$$129$$ 7.60197 0.669316
$$130$$ 0 0
$$131$$ 8.68035 0.758405 0.379203 0.925314i $$-0.376198\pi$$
0.379203 + 0.925314i $$0.376198\pi$$
$$132$$ −5.34017 −0.464802
$$133$$ −5.67420 −0.492016
$$134$$ 23.5174 2.03160
$$135$$ 0 0
$$136$$ −70.2122 −6.02064
$$137$$ 15.3607 1.31235 0.656176 0.754608i $$-0.272173\pi$$
0.656176 + 0.754608i $$0.272173\pi$$
$$138$$ −5.84324 −0.497410
$$139$$ 8.58145 0.727869 0.363935 0.931425i $$-0.381433\pi$$
0.363935 + 0.931425i $$0.381433\pi$$
$$140$$ 0 0
$$141$$ 4.68035 0.394156
$$142$$ −12.6803 −1.06411
$$143$$ 4.34017 0.362943
$$144$$ 13.8371 1.15309
$$145$$ 0 0
$$146$$ 28.4391 2.35363
$$147$$ 5.83710 0.481436
$$148$$ 10.6803 0.877919
$$149$$ −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 $$-0.116958\pi$$
0.933253 + 0.359221i $$0.116958\pi$$
$$152$$ 47.6163 3.86220
$$153$$ −7.75872 −0.627256
$$154$$ −2.92162 −0.235431
$$155$$ 0 0
$$156$$ −23.1773 −1.85567
$$157$$ 10.9939 0.877405 0.438703 0.898632i $$-0.355438\pi$$
0.438703 + 0.898632i $$0.355438\pi$$
$$158$$ −21.9421 −1.74562
$$159$$ 0.156755 0.0124315
$$160$$ 0 0
$$161$$ −2.32580 −0.183298
$$162$$ 2.70928 0.212861
$$163$$ 6.52359 0.510967 0.255484 0.966813i $$-0.417765\pi$$
0.255484 + 0.966813i $$0.417765\pi$$
$$164$$ −50.2967 −3.92751
$$165$$ 0 0
$$166$$ 29.8576 2.31740
$$167$$ −1.97334 −0.152701 −0.0763507 0.997081i $$-0.524327\pi$$
−0.0763507 + 0.997081i $$0.524327\pi$$
$$168$$ 9.75872 0.752902
$$169$$ 5.83710 0.449008
$$170$$ 0 0
$$171$$ 5.26180 0.402380
$$172$$ −40.5958 −3.09540
$$173$$ −3.75872 −0.285770 −0.142885 0.989739i $$-0.545638\pi$$
−0.142885 + 0.989739i $$0.545638\pi$$
$$174$$ −3.84324 −0.291356
$$175$$ 0 0
$$176$$ 13.8371 1.04301
$$177$$ −6.15676 −0.462770
$$178$$ −34.7792 −2.60681
$$179$$ 15.1506 1.13241 0.566205 0.824264i $$-0.308411\pi$$
0.566205 + 0.824264i $$0.308411\pi$$
$$180$$ 0 0
$$181$$ 4.83710 0.359539 0.179769 0.983709i $$-0.442465\pi$$
0.179769 + 0.983709i $$0.442465\pi$$
$$182$$ −12.6803 −0.939930
$$183$$ 4.15676 0.307276
$$184$$ 19.5174 1.43885
$$185$$ 0 0
$$186$$ 12.6803 0.929768
$$187$$ −7.75872 −0.567374
$$188$$ −24.9939 −1.82286
$$189$$ 1.07838 0.0784404
$$190$$ 0 0
$$191$$ 2.52359 0.182601 0.0913003 0.995823i $$-0.470898\pi$$
0.0913003 + 0.995823i $$0.470898\pi$$
$$192$$ −24.8576 −1.79394
$$193$$ −0.0266620 −0.00191917 −0.000959586 1.00000i $$-0.500305\pi$$
−0.000959586 1.00000i $$0.500305\pi$$
$$194$$ −39.7731 −2.85554
$$195$$ 0 0
$$196$$ −31.1711 −2.22651
$$197$$ −21.1194 −1.50470 −0.752348 0.658766i $$-0.771079\pi$$
−0.752348 + 0.658766i $$0.771079\pi$$
$$198$$ 2.70928 0.192540
$$199$$ 10.5236 0.745998 0.372999 0.927832i $$-0.378330\pi$$
0.372999 + 0.927832i $$0.378330\pi$$
$$200$$ 0 0
$$201$$ −8.68035 −0.612264
$$202$$ −42.1978 −2.96903
$$203$$ −1.52973 −0.107366
$$204$$ 41.4329 2.90089
$$205$$ 0 0
$$206$$ −18.5236 −1.29060
$$207$$ 2.15676 0.149905
$$208$$ 60.0554 4.16409
$$209$$ 5.26180 0.363966
$$210$$ 0 0
$$211$$ 9.57531 0.659191 0.329596 0.944122i $$-0.393088\pi$$
0.329596 + 0.944122i $$0.393088\pi$$
$$212$$ −0.837101 −0.0574924
$$213$$ 4.68035 0.320692
$$214$$ −17.1773 −1.17421
$$215$$ 0 0
$$216$$ −9.04945 −0.615737
$$217$$ 5.04718 0.342625
$$218$$ 6.26794 0.424518
$$219$$ −10.4969 −0.709317
$$220$$ 0 0
$$221$$ −33.6742 −2.26517
$$222$$ −5.41855 −0.363669
$$223$$ 2.15676 0.144427 0.0722135 0.997389i $$-0.476994\pi$$
0.0722135 + 0.997389i $$0.476994\pi$$
$$224$$ −20.9093 −1.39706
$$225$$ 0 0
$$226$$ 16.2557 1.08131
$$227$$ −9.65983 −0.641145 −0.320573 0.947224i $$-0.603875\pi$$
−0.320573 + 0.947224i $$0.603875\pi$$
$$228$$ −28.0989 −1.86089
$$229$$ −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
$$232$$ 12.8371 0.842797
$$233$$ 2.39803 0.157100 0.0785501 0.996910i $$-0.474971\pi$$
0.0785501 + 0.996910i $$0.474971\pi$$
$$234$$ 11.7587 0.768692
$$235$$ 0 0
$$236$$ 32.8781 2.14018
$$237$$ 8.09890 0.526080
$$238$$ 22.6681 1.46935
$$239$$ −7.20394 −0.465984 −0.232992 0.972479i $$-0.574852\pi$$
−0.232992 + 0.972479i $$0.574852\pi$$
$$240$$ 0 0
$$241$$ −5.20394 −0.335215 −0.167608 0.985854i $$-0.553604\pi$$
−0.167608 + 0.985854i $$0.553604\pi$$
$$242$$ 2.70928 0.174159
$$243$$ −1.00000 −0.0641500
$$244$$ −22.1978 −1.42107
$$245$$ 0 0
$$246$$ 25.5174 1.62693
$$247$$ 22.8371 1.45309
$$248$$ −42.3545 −2.68952
$$249$$ −11.0205 −0.698397
$$250$$ 0 0
$$251$$ 15.3197 0.966968 0.483484 0.875353i $$-0.339371\pi$$
0.483484 + 0.875353i $$0.339371\pi$$
$$252$$ −5.75872 −0.362765
$$253$$ 2.15676 0.135594
$$254$$ 6.07223 0.381006
$$255$$ 0 0
$$256$$ 27.6803 1.73002
$$257$$ −4.15676 −0.259291 −0.129646 0.991560i $$-0.541384\pi$$
−0.129646 + 0.991560i $$0.541384\pi$$
$$258$$ 20.5958 1.28224
$$259$$ −2.15676 −0.134014
$$260$$ 0 0
$$261$$ 1.41855 0.0878061
$$262$$ 23.5174 1.45291
$$263$$ 18.7070 1.15352 0.576762 0.816912i $$-0.304316\pi$$
0.576762 + 0.816912i $$0.304316\pi$$
$$264$$ −9.04945 −0.556955
$$265$$ 0 0
$$266$$ −15.3730 −0.942578
$$267$$ 12.8371 0.785618
$$268$$ 46.3545 2.83155
$$269$$ 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 $$-0.554163\pi$$
−0.169338 + 0.985558i $$0.554163\pi$$
$$272$$ −107.358 −6.50955
$$273$$ 4.68035 0.283267
$$274$$ 41.6163 2.51414
$$275$$ 0 0
$$276$$ −11.5174 −0.693269
$$277$$ 26.0144 1.56305 0.781526 0.623872i $$-0.214442\pi$$
0.781526 + 0.623872i $$0.214442\pi$$
$$278$$ 23.2495 1.39441
$$279$$ −4.68035 −0.280205
$$280$$ 0 0
$$281$$ −9.41855 −0.561864 −0.280932 0.959728i $$-0.590643\pi$$
−0.280932 + 0.959728i $$0.590643\pi$$
$$282$$ 12.6803 0.755104
$$283$$ −14.2413 −0.846556 −0.423278 0.906000i $$-0.639121\pi$$
−0.423278 + 0.906000i $$0.639121\pi$$
$$284$$ −24.9939 −1.48311
$$285$$ 0 0
$$286$$ 11.7587 0.695308
$$287$$ 10.1568 0.599534
$$288$$ 19.3896 1.14254
$$289$$ 43.1978 2.54105
$$290$$ 0 0
$$291$$ 14.6803 0.860577
$$292$$ 56.0554 3.28039
$$293$$ 15.7587 0.920634 0.460317 0.887754i $$-0.347736\pi$$
0.460317 + 0.887754i $$0.347736\pi$$
$$294$$ 15.8143 0.922310
$$295$$ 0 0
$$296$$ 18.0989 1.05198
$$297$$ −1.00000 −0.0580259
$$298$$ −49.0349 −2.84052
$$299$$ 9.36069 0.541343
$$300$$ 0 0
$$301$$ 8.19779 0.472513
$$302$$ 62.1399 3.57575
$$303$$ 15.5753 0.894778
$$304$$ 72.8080 4.17582
$$305$$ 0 0
$$306$$ −21.0205 −1.20166
$$307$$ 18.9216 1.07991 0.539957 0.841693i $$-0.318440\pi$$
0.539957 + 0.841693i $$0.318440\pi$$
$$308$$ −5.75872 −0.328134
$$309$$ 6.83710 0.388949
$$310$$ 0 0
$$311$$ −20.8781 −1.18389 −0.591945 0.805978i $$-0.701640\pi$$
−0.591945 + 0.805978i $$0.701640\pi$$
$$312$$ −39.2762 −2.22358
$$313$$ −6.31351 −0.356861 −0.178430 0.983953i $$-0.557102\pi$$
−0.178430 + 0.983953i $$0.557102\pi$$
$$314$$ 29.7854 1.68089
$$315$$ 0 0
$$316$$ −43.2495 −2.43297
$$317$$ −31.3607 −1.76139 −0.880696 0.473682i $$-0.842925\pi$$
−0.880696 + 0.473682i $$0.842925\pi$$
$$318$$ 0.424694 0.0238156
$$319$$ 1.41855 0.0794236
$$320$$ 0 0
$$321$$ 6.34017 0.353874
$$322$$ −6.30122 −0.351154
$$323$$ −40.8248 −2.27155
$$324$$ 5.34017 0.296676
$$325$$ 0 0
$$326$$ 17.6742 0.978884
$$327$$ −2.31351 −0.127937
$$328$$ −85.2327 −4.70619
$$329$$ 5.04718 0.278260
$$330$$ 0 0
$$331$$ 19.2039 1.05554 0.527772 0.849386i $$-0.323028\pi$$
0.527772 + 0.849386i $$0.323028\pi$$
$$332$$ 58.8515 3.22989
$$333$$ 2.00000 0.109599
$$334$$ −5.34632 −0.292537
$$335$$ 0 0
$$336$$ 14.9216 0.814041
$$337$$ −13.5031 −0.735559 −0.367780 0.929913i $$-0.619882\pi$$
−0.367780 + 0.929913i $$0.619882\pi$$
$$338$$ 15.8143 0.860185
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −4.68035 −0.253455
$$342$$ 14.2557 0.770857
$$343$$ 13.8432 0.747465
$$344$$ −68.7936 −3.70910
$$345$$ 0 0
$$346$$ −10.1834 −0.547464
$$347$$ −6.34017 −0.340358 −0.170179 0.985413i $$-0.554435\pi$$
−0.170179 + 0.985413i $$0.554435\pi$$
$$348$$ −7.57531 −0.406079
$$349$$ 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
$$352$$ 19.3896 1.03347
$$353$$ 13.2039 0.702775 0.351387 0.936230i $$-0.385710\pi$$
0.351387 + 0.936230i $$0.385710\pi$$
$$354$$ −16.6803 −0.886550
$$355$$ 0 0
$$356$$ −68.5523 −3.63327
$$357$$ −8.36683 −0.442820
$$358$$ 41.0472 2.16941
$$359$$ 3.31965 0.175205 0.0876023 0.996156i $$-0.472080\pi$$
0.0876023 + 0.996156i $$0.472080\pi$$
$$360$$ 0 0
$$361$$ 8.68649 0.457184
$$362$$ 13.1050 0.688786
$$363$$ −1.00000 −0.0524864
$$364$$ −24.9939 −1.31003
$$365$$ 0 0
$$366$$ 11.2618 0.588663
$$367$$ 36.1445 1.88673 0.943363 0.331762i $$-0.107643\pi$$
0.943363 + 0.331762i $$0.107643\pi$$
$$368$$ 29.8432 1.55569
$$369$$ −9.41855 −0.490310
$$370$$ 0 0
$$371$$ 0.169042 0.00877620
$$372$$ 24.9939 1.29587
$$373$$ 2.81044 0.145519 0.0727595 0.997350i $$-0.476819\pi$$
0.0727595 + 0.997350i $$0.476819\pi$$
$$374$$ −21.0205 −1.08695
$$375$$ 0 0
$$376$$ −42.3545 −2.18427
$$377$$ 6.15676 0.317089
$$378$$ 2.92162 0.150272
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −2.24128 −0.114824
$$382$$ 6.83710 0.349817
$$383$$ −33.5585 −1.71476 −0.857379 0.514685i $$-0.827909\pi$$
−0.857379 + 0.514685i $$0.827909\pi$$
$$384$$ −28.5669 −1.45780
$$385$$ 0 0
$$386$$ −0.0722347 −0.00367665
$$387$$ −7.60197 −0.386430
$$388$$ −78.3956 −3.97993
$$389$$ 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
$$392$$ −52.8225 −2.66794
$$393$$ −8.68035 −0.437866
$$394$$ −57.2183 −2.88262
$$395$$ 0 0
$$396$$ 5.34017 0.268354
$$397$$ 5.31965 0.266986 0.133493 0.991050i $$-0.457381\pi$$
0.133493 + 0.991050i $$0.457381\pi$$
$$398$$ 28.5113 1.42914
$$399$$ 5.67420 0.284065
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ −23.5174 −1.17294
$$403$$ −20.3135 −1.01189
$$404$$ −83.1748 −4.13810
$$405$$ 0 0
$$406$$ −4.14447 −0.205687
$$407$$ 2.00000 0.0991363
$$408$$ 70.2122 3.47602
$$409$$ 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
$$412$$ −36.5113 −1.79878
$$413$$ −6.63931 −0.326699
$$414$$ 5.84324 0.287180
$$415$$ 0 0
$$416$$ 84.1543 4.12600
$$417$$ −8.58145 −0.420235
$$418$$ 14.2557 0.697267
$$419$$ −2.83710 −0.138601 −0.0693007 0.997596i $$-0.522077\pi$$
−0.0693007 + 0.997596i $$0.522077\pi$$
$$420$$ 0 0
$$421$$ 11.4764 0.559326 0.279663 0.960098i $$-0.409777\pi$$
0.279663 + 0.960098i $$0.409777\pi$$
$$422$$ 25.9421 1.26284
$$423$$ −4.68035 −0.227566
$$424$$ −1.41855 −0.0688909
$$425$$ 0 0
$$426$$ 12.6803 0.614365
$$427$$ 4.48255 0.216926
$$428$$ −33.8576 −1.63657
$$429$$ −4.34017 −0.209546
$$430$$ 0 0
$$431$$ 23.5708 1.13536 0.567682 0.823248i $$-0.307840\pi$$
0.567682 + 0.823248i $$0.307840\pi$$
$$432$$ −13.8371 −0.665738
$$433$$ 14.9939 0.720559 0.360279 0.932844i $$-0.382681\pi$$
0.360279 + 0.932844i $$0.382681\pi$$
$$434$$ 13.6742 0.656383
$$435$$ 0 0
$$436$$ 12.3545 0.591676
$$437$$ 11.3484 0.542868
$$438$$ −28.4391 −1.35887
$$439$$ 4.77924 0.228101 0.114050 0.993475i $$-0.463617\pi$$
0.114050 + 0.993475i $$0.463617\pi$$
$$440$$ 0 0
$$441$$ −5.83710 −0.277957
$$442$$ −91.2327 −4.33950
$$443$$ −20.1978 −0.959626 −0.479813 0.877371i $$-0.659296\pi$$
−0.479813 + 0.877371i $$0.659296\pi$$
$$444$$ −10.6803 −0.506867
$$445$$ 0 0
$$446$$ 5.84324 0.276686
$$447$$ 18.0989 0.856048
$$448$$ −26.8059 −1.26646
$$449$$ −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
$$452$$ 32.0410 1.50708
$$453$$ −22.9360 −1.07763
$$454$$ −26.1711 −1.22827
$$455$$ 0 0
$$456$$ −47.6163 −2.22984
$$457$$ −28.1711 −1.31779 −0.658895 0.752235i $$-0.728976\pi$$
−0.658895 + 0.752235i $$0.728976\pi$$
$$458$$ −9.10504 −0.425451
$$459$$ 7.75872 0.362146
$$460$$ 0 0
$$461$$ −1.47187 −0.0685520 −0.0342760 0.999412i $$-0.510913\pi$$
−0.0342760 + 0.999412i $$0.510913\pi$$
$$462$$ 2.92162 0.135926
$$463$$ 23.2039 1.07838 0.539189 0.842185i $$-0.318731\pi$$
0.539189 + 0.842185i $$0.318731\pi$$
$$464$$ 19.6286 0.911236
$$465$$ 0 0
$$466$$ 6.49693 0.300964
$$467$$ −14.1568 −0.655097 −0.327548 0.944834i $$-0.606222\pi$$
−0.327548 + 0.944834i $$0.606222\pi$$
$$468$$ 23.1773 1.07137
$$469$$ −9.36069 −0.432237
$$470$$ 0 0
$$471$$ −10.9939 −0.506570
$$472$$ 55.7152 2.56450
$$473$$ −7.60197 −0.349539
$$474$$ 21.9421 1.00784
$$475$$ 0 0
$$476$$ 44.6803 2.04792
$$477$$ −0.156755 −0.00717734
$$478$$ −19.5174 −0.892707
$$479$$ −13.8432 −0.632514 −0.316257 0.948674i $$-0.602426\pi$$
−0.316257 + 0.948674i $$0.602426\pi$$
$$480$$ 0 0
$$481$$ 8.68035 0.395790
$$482$$ −14.0989 −0.642187
$$483$$ 2.32580 0.105827
$$484$$ 5.34017 0.242735
$$485$$ 0 0
$$486$$ −2.70928 −0.122895
$$487$$ 40.9939 1.85761 0.928804 0.370570i $$-0.120838\pi$$
0.928804 + 0.370570i $$0.120838\pi$$
$$488$$ −37.6163 −1.70281
$$489$$ −6.52359 −0.295007
$$490$$ 0 0
$$491$$ 34.8371 1.57218 0.786088 0.618114i $$-0.212103\pi$$
0.786088 + 0.618114i $$0.212103\pi$$
$$492$$ 50.2967 2.26755
$$493$$ −11.0061 −0.495692
$$494$$ 61.8720 2.78375
$$495$$ 0 0
$$496$$ −64.7624 −2.90792
$$497$$ 5.04718 0.226397
$$498$$ −29.8576 −1.33795
$$499$$ 15.1506 0.678235 0.339117 0.940744i $$-0.389872\pi$$
0.339117 + 0.940744i $$0.389872\pi$$
$$500$$ 0 0
$$501$$ 1.97334 0.0881622
$$502$$ 41.5052 1.85247
$$503$$ 6.65368 0.296673 0.148337 0.988937i $$-0.452608\pi$$
0.148337 + 0.988937i $$0.452608\pi$$
$$504$$ −9.75872 −0.434688
$$505$$ 0 0
$$506$$ 5.84324 0.259764
$$507$$ −5.83710 −0.259235
$$508$$ 11.9688 0.531030
$$509$$ 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
$$512$$ 17.8599 0.789303
$$513$$ −5.26180 −0.232314
$$514$$ −11.2618 −0.496736
$$515$$ 0 0
$$516$$ 40.5958 1.78713
$$517$$ −4.68035 −0.205841
$$518$$ −5.84324 −0.256737
$$519$$ 3.75872 0.164990
$$520$$ 0 0
$$521$$ 7.67420 0.336213 0.168106 0.985769i $$-0.446235\pi$$
0.168106 + 0.985769i $$0.446235\pi$$
$$522$$ 3.84324 0.168214
$$523$$ −23.2351 −1.01600 −0.508001 0.861357i $$-0.669615\pi$$
−0.508001 + 0.861357i $$0.669615\pi$$
$$524$$ 46.3545 2.02501
$$525$$ 0 0
$$526$$ 50.6824 2.20986
$$527$$ 36.3135 1.58184
$$528$$ −13.8371 −0.602183
$$529$$ −18.3484 −0.797757
$$530$$ 0 0
$$531$$ 6.15676 0.267180
$$532$$ −30.3012 −1.31372
$$533$$ −40.8781 −1.77063
$$534$$ 34.7792 1.50505
$$535$$ 0 0
$$536$$ 78.5523 3.39294
$$537$$ −15.1506 −0.653797
$$538$$ 63.2905 2.72865
$$539$$ −5.83710 −0.251422
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −15.1050 −0.648817
$$543$$ −4.83710 −0.207580
$$544$$ −150.439 −6.45001
$$545$$ 0 0
$$546$$ 12.6803 0.542669
$$547$$ 23.0661 0.986235 0.493117 0.869963i $$-0.335857\pi$$
0.493117 + 0.869963i $$0.335857\pi$$
$$548$$ 82.0288 3.50409
$$549$$ −4.15676 −0.177406
$$550$$ 0 0
$$551$$ 7.46412 0.317982
$$552$$ −19.5174 −0.830718
$$553$$ 8.73367 0.371393
$$554$$ 70.4801 2.99441
$$555$$ 0 0
$$556$$ 45.8264 1.94347
$$557$$ −10.5958 −0.448960 −0.224480 0.974479i $$-0.572068\pi$$
−0.224480 + 0.974479i $$0.572068\pi$$
$$558$$ −12.6803 −0.536802
$$559$$ −32.9939 −1.39549
$$560$$ 0 0
$$561$$ 7.75872 0.327574
$$562$$ −25.5174 −1.07639
$$563$$ 36.2122 1.52616 0.763080 0.646303i $$-0.223686\pi$$
0.763080 + 0.646303i $$0.223686\pi$$
$$564$$ 24.9939 1.05243
$$565$$ 0 0
$$566$$ −38.5835 −1.62179
$$567$$ −1.07838 −0.0452876
$$568$$ −42.3545 −1.77716
$$569$$ −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 $$-0.698676\pi$$
−0.584414 + 0.811456i $$0.698676\pi$$
$$572$$ 23.1773 0.969091
$$573$$ −2.52359 −0.105425
$$574$$ 27.5174 1.14856
$$575$$ 0 0
$$576$$ 24.8576 1.03573
$$577$$ −41.4017 −1.72358 −0.861788 0.507268i $$-0.830655\pi$$
−0.861788 + 0.507268i $$0.830655\pi$$
$$578$$ 117.035 4.86800
$$579$$ 0.0266620 0.00110803
$$580$$ 0 0
$$581$$ −11.8843 −0.493043
$$582$$ 39.7731 1.64865
$$583$$ −0.156755 −0.00649215
$$584$$ 94.9914 3.93077
$$585$$ 0 0
$$586$$ 42.6947 1.76370
$$587$$ −8.48255 −0.350112 −0.175056 0.984558i $$-0.556011\pi$$
−0.175056 + 0.984558i $$0.556011\pi$$
$$588$$ 31.1711 1.28548
$$589$$ −24.6270 −1.01474
$$590$$ 0 0
$$591$$ 21.1194 0.868737
$$592$$ 27.6742 1.13740
$$593$$ −7.56093 −0.310490 −0.155245 0.987876i $$-0.549617\pi$$
−0.155245 + 0.987876i $$0.549617\pi$$
$$594$$ −2.70928 −0.111163
$$595$$ 0 0
$$596$$ −96.6512 −3.95899
$$597$$ −10.5236 −0.430702
$$598$$ 25.3607 1.03708
$$599$$ 5.67420 0.231842 0.115921 0.993258i $$-0.463018\pi$$
0.115921 + 0.993258i $$0.463018\pi$$
$$600$$ 0 0
$$601$$ −1.31965 −0.0538298 −0.0269149 0.999638i $$-0.508568\pi$$
−0.0269149 + 0.999638i $$0.508568\pi$$
$$602$$ 22.2101 0.905215
$$603$$ 8.68035 0.353491
$$604$$ 122.482 4.98373
$$605$$ 0 0
$$606$$ 42.1978 1.71417
$$607$$ 2.24128 0.0909706 0.0454853 0.998965i $$-0.485517\pi$$
0.0454853 + 0.998965i $$0.485517\pi$$
$$608$$ 102.024 4.13763
$$609$$ 1.52973 0.0619879
$$610$$ 0 0
$$611$$ −20.3135 −0.821797
$$612$$ −41.4329 −1.67483
$$613$$ −42.8638 −1.73125 −0.865626 0.500692i $$-0.833079\pi$$
−0.865626 + 0.500692i $$0.833079\pi$$
$$614$$ 51.2639 2.06884
$$615$$ 0 0
$$616$$ −9.75872 −0.393190
$$617$$ −11.3607 −0.457364 −0.228682 0.973501i $$-0.573442\pi$$
−0.228682 + 0.973501i $$0.573442\pi$$
$$618$$ 18.5236 0.745128
$$619$$ −45.1917 −1.81641 −0.908203 0.418530i $$-0.862545\pi$$
−0.908203 + 0.418530i $$0.862545\pi$$
$$620$$ 0 0
$$621$$ −2.15676 −0.0865476
$$622$$ −56.5646 −2.26803
$$623$$ 13.8432 0.554618
$$624$$ −60.0554 −2.40414
$$625$$ 0 0
$$626$$ −17.1050 −0.683655
$$627$$ −5.26180 −0.210136
$$628$$ 58.7091 2.34275
$$629$$ −15.5174 −0.618721
$$630$$ 0 0
$$631$$ −9.78992 −0.389731 −0.194865 0.980830i $$-0.562427\pi$$
−0.194865 + 0.980830i $$0.562427\pi$$
$$632$$ −73.2905 −2.91534
$$633$$ −9.57531 −0.380584
$$634$$ −84.9647 −3.37438
$$635$$ 0 0
$$636$$ 0.837101 0.0331932
$$637$$ −25.3340 −1.00377
$$638$$ 3.84324 0.152156
$$639$$ −4.68035 −0.185152
$$640$$ 0 0
$$641$$ 0.210079 0.00829764 0.00414882 0.999991i $$-0.498679\pi$$
0.00414882 + 0.999991i $$0.498679\pi$$
$$642$$ 17.1773 0.677933
$$643$$ −14.5236 −0.572754 −0.286377 0.958117i $$-0.592451\pi$$
−0.286377 + 0.958117i $$0.592451\pi$$
$$644$$ −12.4202 −0.489423
$$645$$ 0 0
$$646$$ −110.606 −4.35172
$$647$$ −15.4641 −0.607957 −0.303979 0.952679i $$-0.598315\pi$$
−0.303979 + 0.952679i $$0.598315\pi$$
$$648$$ 9.04945 0.355496
$$649$$ 6.15676 0.241674
$$650$$ 0 0
$$651$$ −5.04718 −0.197815
$$652$$ 34.8371 1.36433
$$653$$ 17.8310 0.697779 0.348890 0.937164i $$-0.386559\pi$$
0.348890 + 0.937164i $$0.386559\pi$$
$$654$$ −6.26794 −0.245096
$$655$$ 0 0
$$656$$ −130.325 −5.08835
$$657$$ 10.4969 0.409524
$$658$$ 13.6742 0.533076
$$659$$ −32.3135 −1.25876 −0.629378 0.777099i $$-0.716690\pi$$
−0.629378 + 0.777099i $$0.716690\pi$$
$$660$$ 0 0
$$661$$ −5.68649 −0.221179 −0.110589 0.993866i $$-0.535274\pi$$
−0.110589 + 0.993866i $$0.535274\pi$$
$$662$$ 52.0288 2.02215
$$663$$ 33.6742 1.30780
$$664$$ 99.7296 3.87026
$$665$$ 0 0
$$666$$ 5.41855 0.209965
$$667$$ 3.05947 0.118463
$$668$$ −10.5380 −0.407726
$$669$$ −2.15676 −0.0833850
$$670$$ 0 0
$$671$$ −4.15676 −0.160470
$$672$$ 20.9093 0.806595
$$673$$ 21.0205 0.810281 0.405141 0.914254i $$-0.367223\pi$$
0.405141 + 0.914254i $$0.367223\pi$$
$$674$$ −36.5835 −1.40915
$$675$$ 0 0
$$676$$ 31.1711 1.19889
$$677$$ −36.7526 −1.41252 −0.706258 0.707954i $$-0.749618\pi$$
−0.706258 + 0.707954i $$0.749618\pi$$
$$678$$ −16.2557 −0.624295
$$679$$ 15.8310 0.607536
$$680$$ 0 0
$$681$$ 9.65983 0.370165
$$682$$ −12.6803 −0.485556
$$683$$ 17.3074 0.662248 0.331124 0.943587i $$-0.392572\pi$$
0.331124 + 0.943587i $$0.392572\pi$$
$$684$$ 28.0989 1.07439
$$685$$ 0 0
$$686$$ 37.5052 1.43195
$$687$$ 3.36069 0.128218
$$688$$ −105.189 −4.01030
$$689$$ −0.680346 −0.0259191
$$690$$ 0 0
$$691$$ 17.6742 0.672358 0.336179 0.941798i $$-0.390865\pi$$
0.336179 + 0.941798i $$0.390865\pi$$
$$692$$ −20.0722 −0.763032
$$693$$ −1.07838 −0.0409642
$$694$$ −17.1773 −0.652040
$$695$$ 0 0
$$696$$ −12.8371 −0.486589
$$697$$ 73.0759 2.76795
$$698$$ 43.7731 1.65684
$$699$$ −2.39803 −0.0907019
$$700$$ 0 0
$$701$$ −17.1050 −0.646048 −0.323024 0.946391i $$-0.604700\pi$$
−0.323024 + 0.946391i $$0.604700\pi$$
$$702$$ −11.7587 −0.443804
$$703$$ 10.5236 0.396905
$$704$$ 24.8576 0.936857
$$705$$ 0 0
$$706$$ 35.7731 1.34634
$$707$$ 16.7961 0.631681
$$708$$ −32.8781 −1.23564
$$709$$ 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
$$712$$ −116.169 −4.35361
$$713$$ −10.0944 −0.378037
$$714$$ −22.6681 −0.848331
$$715$$ 0 0
$$716$$ 80.9069 3.02363
$$717$$ 7.20394 0.269036
$$718$$ 8.99386 0.335648
$$719$$ −1.78992 −0.0667528 −0.0333764 0.999443i $$-0.510626\pi$$
−0.0333764 + 0.999443i $$0.510626\pi$$
$$720$$ 0 0
$$721$$ 7.37298 0.274584
$$722$$ 23.5341 0.875848
$$723$$ 5.20394 0.193536
$$724$$ 25.8310 0.960000
$$725$$ 0 0
$$726$$ −2.70928 −0.100551
$$727$$ −25.9877 −0.963831 −0.481915 0.876218i $$-0.660059\pi$$
−0.481915 + 0.876218i $$0.660059\pi$$
$$728$$ −42.3545 −1.56976
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 58.9816 2.18151
$$732$$ 22.1978 0.820454
$$733$$ 41.0205 1.51513 0.757564 0.652761i $$-0.226390\pi$$
0.757564 + 0.652761i $$0.226390\pi$$
$$734$$ 97.9253 3.61449
$$735$$ 0 0
$$736$$ 41.8187 1.54146
$$737$$ 8.68035 0.319745
$$738$$ −25.5174 −0.939310
$$739$$ −47.6163 −1.75160 −0.875798 0.482678i $$-0.839664\pi$$
−0.875798 + 0.482678i $$0.839664\pi$$
$$740$$ 0 0
$$741$$ −22.8371 −0.838942
$$742$$ 0.457980 0.0168130
$$743$$ −0.550252 −0.0201868 −0.0100934 0.999949i $$-0.503213\pi$$
−0.0100934 + 0.999949i $$0.503213\pi$$
$$744$$ 42.3545 1.55279
$$745$$ 0 0
$$746$$ 7.61425 0.278778
$$747$$ 11.0205 0.403220
$$748$$ −41.4329 −1.51494
$$749$$ 6.83710 0.249822
$$750$$ 0 0
$$751$$ 41.5585 1.51649 0.758245 0.651969i $$-0.226057\pi$$
0.758245 + 0.651969i $$0.226057\pi$$
$$752$$ −64.7624 −2.36164
$$753$$ −15.3197 −0.558279
$$754$$ 16.6803 0.607462
$$755$$ 0 0
$$756$$ 5.75872 0.209443
$$757$$ −1.31965 −0.0479636 −0.0239818 0.999712i $$-0.507634\pi$$
−0.0239818 + 0.999712i $$0.507634\pi$$
$$758$$ −54.1855 −1.96811
$$759$$ −2.15676 −0.0782853
$$760$$ 0 0
$$761$$ −2.21461 −0.0802797 −0.0401399 0.999194i $$-0.512780\pi$$
−0.0401399 + 0.999194i $$0.512780\pi$$
$$762$$ −6.07223 −0.219974
$$763$$ −2.49484 −0.0903192
$$764$$ 13.4764 0.487559
$$765$$ 0 0
$$766$$ −90.9192 −3.28504
$$767$$ 26.7214 0.964853
$$768$$ −27.6803 −0.998828
$$769$$ −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
$$772$$ −0.142380 −0.00512436
$$773$$ −40.1568 −1.44434 −0.722169 0.691717i $$-0.756855\pi$$
−0.722169 + 0.691717i $$0.756855\pi$$
$$774$$ −20.5958 −0.740302
$$775$$ 0 0
$$776$$ −132.849 −4.76900
$$777$$ 2.15676 0.0773732
$$778$$ 34.7792 1.24690
$$779$$ −49.5585 −1.77562
$$780$$ 0 0
$$781$$ −4.68035 −0.167476
$$782$$ −45.3361 −1.62122
$$783$$ −1.41855 −0.0506949
$$784$$ −80.7686 −2.88459
$$785$$ 0 0
$$786$$ −23.5174 −0.838840
$$787$$ −49.5897 −1.76768 −0.883841 0.467788i $$-0.845051\pi$$
−0.883841 + 0.467788i $$0.845051\pi$$
$$788$$ −112.781 −4.01767
$$789$$ −18.7070 −0.665987
$$790$$ 0 0
$$791$$ −6.47027 −0.230056
$$792$$ 9.04945 0.321558
$$793$$ −18.0410 −0.640656
$$794$$ 14.4124 0.511477
$$795$$ 0 0
$$796$$ 56.1978 1.99188
$$797$$ 46.7091 1.65452 0.827261 0.561818i $$-0.189898\pi$$
0.827261 + 0.561818i $$0.189898\pi$$
$$798$$ 15.3730 0.544198
$$799$$ 36.3135 1.28468
$$800$$ 0 0
$$801$$ −12.8371 −0.453577
$$802$$ 5.41855 0.191336
$$803$$ 10.4969 0.370429
$$804$$ −46.3545 −1.63480
$$805$$ 0 0
$$806$$ −55.0349 −1.93852
$$807$$ −23.3607 −0.822335
$$808$$ −140.948 −4.95853
$$809$$ −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 $$-0.340867\pi$$
0.479366 + 0.877615i $$0.340867\pi$$
$$812$$ −8.16904 −0.286677
$$813$$ 5.57531 0.195535
$$814$$ 5.41855 0.189920
$$815$$ 0 0
$$816$$ 107.358 3.75829
$$817$$ −40.0000 −1.39942
$$818$$ 70.9770 2.48165
$$819$$ −4.68035 −0.163545
$$820$$ 0 0
$$821$$ −31.2085 −1.08918 −0.544592 0.838701i $$-0.683315\pi$$
−0.544592 + 0.838701i $$0.683315\pi$$
$$822$$ −41.6163 −1.45154
$$823$$ 50.1855 1.74936 0.874678 0.484704i $$-0.161073\pi$$
0.874678 + 0.484704i $$0.161073\pi$$
$$824$$ −61.8720 −2.15541
$$825$$ 0 0
$$826$$ −17.9877 −0.625873
$$827$$ −27.3874 −0.952352 −0.476176 0.879350i $$-0.657977\pi$$
−0.476176 + 0.879350i $$0.657977\pi$$
$$828$$ 11.5174 0.400259
$$829$$ −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
$$832$$ 107.886 3.74029
$$833$$ 45.2885 1.56915
$$834$$ −23.2495 −0.805065
$$835$$ 0 0
$$836$$ 28.0989 0.971821
$$837$$ 4.68035 0.161776
$$838$$ −7.68649 −0.265525
$$839$$ 7.20394 0.248708 0.124354 0.992238i $$-0.460314\pi$$
0.124354 + 0.992238i $$0.460314\pi$$
$$840$$ 0 0
$$841$$ −26.9877 −0.930611
$$842$$ 31.0928 1.07153
$$843$$ 9.41855 0.324392
$$844$$ 51.1338 1.76010
$$845$$ 0 0
$$846$$ −12.6803 −0.435959
$$847$$ −1.07838 −0.0370535
$$848$$ −2.16904 −0.0744852
$$849$$ 14.2413 0.488759
$$850$$ 0 0
$$851$$ 4.31351 0.147865
$$852$$ 24.9939 0.856275
$$853$$ −39.8043 −1.36287 −0.681437 0.731877i $$-0.738644\pi$$
−0.681437 + 0.731877i $$0.738644\pi$$
$$854$$ 12.1445 0.415575
$$855$$ 0 0
$$856$$ −57.3751 −1.96104
$$857$$ 36.9504 1.26220 0.631100 0.775701i $$-0.282604\pi$$
0.631100 + 0.775701i $$0.282604\pi$$
$$858$$ −11.7587 −0.401436
$$859$$ 57.5052 1.96205 0.981025 0.193879i $$-0.0621070\pi$$
0.981025 + 0.193879i $$0.0621070\pi$$
$$860$$ 0 0
$$861$$ −10.1568 −0.346141
$$862$$ 63.8597 2.17507
$$863$$ 1.89657 0.0645599 0.0322800 0.999479i $$-0.489723\pi$$
0.0322800 + 0.999479i $$0.489723\pi$$
$$864$$ −19.3896 −0.659648
$$865$$ 0 0
$$866$$ 40.6225 1.38041
$$867$$ −43.1978 −1.46707
$$868$$ 26.9528 0.914838
$$869$$ −8.09890 −0.274736
$$870$$ 0 0
$$871$$ 37.6742 1.27654
$$872$$ 20.9360 0.708982
$$873$$ −14.6803 −0.496854
$$874$$ 30.7460 1.04000
$$875$$ 0 0
$$876$$ −56.0554 −1.89394
$$877$$ −32.5380 −1.09873 −0.549365 0.835583i $$-0.685130\pi$$
−0.549365 + 0.835583i $$0.685130\pi$$
$$878$$ 12.9483 0.436983
$$879$$ −15.7587 −0.531529
$$880$$ 0 0
$$881$$ 18.1978 0.613099 0.306550 0.951855i $$-0.400825\pi$$
0.306550 + 0.951855i $$0.400825\pi$$
$$882$$ −15.8143 −0.532496
$$883$$ −36.3956 −1.22481 −0.612405 0.790545i $$-0.709798\pi$$
−0.612405 + 0.790545i $$0.709798\pi$$
$$884$$ −179.826 −6.04821
$$885$$ 0 0
$$886$$ −54.7214 −1.83840
$$887$$ −27.8699 −0.935780 −0.467890 0.883787i $$-0.654986\pi$$
−0.467890 + 0.883787i $$0.654986\pi$$
$$888$$ −18.0989 −0.607359
$$889$$ −2.41694 −0.0810616
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 11.5174 0.385633
$$893$$ −24.6270 −0.824112
$$894$$ 49.0349 1.63997
$$895$$ 0 0
$$896$$ −30.8059 −1.02915
$$897$$ −9.36069 −0.312544
$$898$$ −58.4412 −1.95021
$$899$$ −6.63931 −0.221433
$$900$$ 0 0
$$901$$ 1.21622 0.0405182
$$902$$ −25.5174 −0.849638
$$903$$ −8.19779 −0.272805
$$904$$ 54.2967 1.80588
$$905$$ 0 0
$$906$$ −62.1399 −2.06446
$$907$$ 27.9376 0.927653 0.463826 0.885926i $$-0.346476\pi$$
0.463826 + 0.885926i $$0.346476\pi$$
$$908$$ −51.5851 −1.71191
$$909$$ −15.5753 −0.516600
$$910$$ 0 0
$$911$$ −11.8843 −0.393744 −0.196872 0.980429i $$-0.563078\pi$$
−0.196872 + 0.980429i $$0.563078\pi$$
$$912$$ −72.8080 −2.41091
$$913$$ 11.0205 0.364726
$$914$$ −76.3234 −2.52455
$$915$$ 0 0
$$916$$ −17.9467 −0.592975
$$917$$ −9.36069 −0.309117
$$918$$ 21.0205 0.693781
$$919$$ −45.6041 −1.50434 −0.752170 0.658970i $$-0.770993\pi$$
−0.752170 + 0.658970i $$0.770993\pi$$
$$920$$ 0 0
$$921$$ −18.9216 −0.623489
$$922$$ −3.98771 −0.131328
$$923$$ −20.3135 −0.668627
$$924$$ 5.75872 0.189448
$$925$$ 0 0
$$926$$ 62.8659 2.06590
$$927$$ −6.83710 −0.224560
$$928$$ 27.5052 0.902901
$$929$$ −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
$$932$$ 12.8059 0.419471
$$933$$ 20.8781 0.683520
$$934$$ −38.3545 −1.25500
$$935$$ 0 0
$$936$$ 39.2762 1.28378
$$937$$ −5.33403 −0.174255 −0.0871276 0.996197i $$-0.527769\pi$$
−0.0871276 + 0.996197i $$0.527769\pi$$
$$938$$ −25.3607 −0.828056
$$939$$ 6.31351 0.206034
$$940$$ 0 0
$$941$$ 56.8203 1.85229 0.926144 0.377170i $$-0.123103\pi$$
0.926144 + 0.377170i $$0.123103\pi$$
$$942$$ −29.7854 −0.970460
$$943$$ −20.3135 −0.661499
$$944$$ 85.1917 2.77275
$$945$$ 0 0
$$946$$ −20.5958 −0.669628
$$947$$ 20.9939 0.682209 0.341104 0.940025i $$-0.389199\pi$$
0.341104 + 0.940025i $$0.389199\pi$$
$$948$$ 43.2495 1.40468
$$949$$ 45.5585 1.47889
$$950$$ 0 0
$$951$$ 31.3607 1.01694
$$952$$ 75.7152 2.45395
$$953$$ 25.2351 0.817446 0.408723 0.912658i $$-0.365974\pi$$
0.408723 + 0.912658i $$0.365974\pi$$
$$954$$ −0.424694 −0.0137500
$$955$$ 0 0
$$956$$ −38.4703 −1.24422
$$957$$ −1.41855 −0.0458552
$$958$$ −37.5052 −1.21174
$$959$$ −16.5646 −0.534900
$$960$$ 0 0
$$961$$ −9.09436 −0.293367
$$962$$ 23.5174 0.758233
$$963$$ −6.34017 −0.204309
$$964$$ −27.7899 −0.895053
$$965$$ 0 0
$$966$$ 6.30122 0.202739
$$967$$ −13.1317 −0.422287 −0.211144 0.977455i $$-0.567719\pi$$
−0.211144 + 0.977455i $$0.567719\pi$$
$$968$$ 9.04945 0.290860
$$969$$ 40.8248 1.31148
$$970$$ 0 0
$$971$$ 8.94053 0.286915 0.143458 0.989656i $$-0.454178\pi$$
0.143458 + 0.989656i $$0.454178\pi$$
$$972$$ −5.34017 −0.171286
$$973$$ −9.25404 −0.296671
$$974$$ 111.064 3.55871
$$975$$ 0 0
$$976$$ −57.5174 −1.84109
$$977$$ −50.3956 −1.61230 −0.806149 0.591713i $$-0.798452\pi$$
−0.806149 + 0.591713i $$0.798452\pi$$
$$978$$ −17.6742 −0.565159
$$979$$ −12.8371 −0.410276
$$980$$ 0 0
$$981$$ 2.31351 0.0738647
$$982$$ 94.3833 3.01189
$$983$$ 32.1978 1.02695 0.513475 0.858105i $$-0.328358\pi$$
0.513475 + 0.858105i $$0.328358\pi$$
$$984$$ 85.2327 2.71712
$$985$$ 0 0
$$986$$ −29.8187 −0.949620
$$987$$ −5.04718 −0.160654
$$988$$ 121.954 3.87988
$$989$$ −16.3956 −0.521349
$$990$$ 0 0
$$991$$ −46.7747 −1.48585 −0.742924 0.669376i $$-0.766562\pi$$
−0.742924 + 0.669376i $$0.766562\pi$$
$$992$$ −90.7501 −2.88132
$$993$$ −19.2039 −0.609419
$$994$$ 13.6742 0.433719
$$995$$ 0 0
$$996$$ −58.8515 −1.86478
$$997$$ −38.2122 −1.21019 −0.605096 0.796153i $$-0.706865\pi$$
−0.605096 + 0.796153i $$0.706865\pi$$
$$998$$ 41.0472 1.29933
$$999$$ −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 825.2.a.k.1.3 3
3.2 odd 2 2475.2.a.bb.1.1 3
5.2 odd 4 825.2.c.g.199.6 6
5.3 odd 4 825.2.c.g.199.1 6
5.4 even 2 165.2.a.c.1.1 3
11.10 odd 2 9075.2.a.cf.1.1 3
15.2 even 4 2475.2.c.r.199.1 6
15.8 even 4 2475.2.c.r.199.6 6
15.14 odd 2 495.2.a.e.1.3 3
20.19 odd 2 2640.2.a.be.1.2 3
35.34 odd 2 8085.2.a.bk.1.1 3
55.54 odd 2 1815.2.a.m.1.3 3
60.59 even 2 7920.2.a.cj.1.2 3
165.164 even 2 5445.2.a.z.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 5.4 even 2
495.2.a.e.1.3 3 15.14 odd 2
825.2.a.k.1.3 3 1.1 even 1 trivial
825.2.c.g.199.1 6 5.3 odd 4
825.2.c.g.199.6 6 5.2 odd 4
1815.2.a.m.1.3 3 55.54 odd 2
2475.2.a.bb.1.1 3 3.2 odd 2
2475.2.c.r.199.1 6 15.2 even 4
2475.2.c.r.199.6 6 15.8 even 4
2640.2.a.be.1.2 3 20.19 odd 2
5445.2.a.z.1.1 3 165.164 even 2
7920.2.a.cj.1.2 3 60.59 even 2
8085.2.a.bk.1.1 3 35.34 odd 2
9075.2.a.cf.1.1 3 11.10 odd 2