# Properties

 Label 1815.2.a.j.1.2 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{10} -0.618034 q^{12} +0.763932 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.85410 q^{16} +3.47214 q^{17} +1.61803 q^{18} -2.47214 q^{19} -0.618034 q^{20} +1.23607 q^{21} +8.70820 q^{23} +2.23607 q^{24} +1.00000 q^{25} +1.23607 q^{26} -1.00000 q^{27} -0.763932 q^{28} +3.23607 q^{29} +1.61803 q^{30} +6.70820 q^{31} -3.38197 q^{32} +5.61803 q^{34} +1.23607 q^{35} +0.618034 q^{36} +5.23607 q^{37} -4.00000 q^{38} -0.763932 q^{39} +2.23607 q^{40} +12.4721 q^{41} +2.00000 q^{42} -0.763932 q^{43} -1.00000 q^{45} +14.0902 q^{46} -4.70820 q^{47} +4.85410 q^{48} -5.47214 q^{49} +1.61803 q^{50} -3.47214 q^{51} +0.472136 q^{52} -7.94427 q^{53} -1.61803 q^{54} +2.76393 q^{56} +2.47214 q^{57} +5.23607 q^{58} -1.70820 q^{59} +0.618034 q^{60} +7.47214 q^{61} +10.8541 q^{62} -1.23607 q^{63} +4.23607 q^{64} -0.763932 q^{65} +6.76393 q^{67} +2.14590 q^{68} -8.70820 q^{69} +2.00000 q^{70} -5.52786 q^{71} -2.23607 q^{72} -1.52786 q^{73} +8.47214 q^{74} -1.00000 q^{75} -1.52786 q^{76} -1.23607 q^{78} -0.708204 q^{79} +4.85410 q^{80} +1.00000 q^{81} +20.1803 q^{82} -4.00000 q^{83} +0.763932 q^{84} -3.47214 q^{85} -1.23607 q^{86} -3.23607 q^{87} +9.23607 q^{89} -1.61803 q^{90} -0.944272 q^{91} +5.38197 q^{92} -6.70820 q^{93} -7.61803 q^{94} +2.47214 q^{95} +3.38197 q^{96} +17.2361 q^{97} -8.85410 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - 2q^{3} - q^{4} - 2q^{5} - q^{6} + 2q^{7} + 2q^{9} - q^{10} + q^{12} + 6q^{13} - 4q^{14} + 2q^{15} - 3q^{16} - 2q^{17} + q^{18} + 4q^{19} + q^{20} - 2q^{21} + 4q^{23} + 2q^{25} - 2q^{26} - 2q^{27} - 6q^{28} + 2q^{29} + q^{30} - 9q^{32} + 9q^{34} - 2q^{35} - q^{36} + 6q^{37} - 8q^{38} - 6q^{39} + 16q^{41} + 4q^{42} - 6q^{43} - 2q^{45} + 17q^{46} + 4q^{47} + 3q^{48} - 2q^{49} + q^{50} + 2q^{51} - 8q^{52} + 2q^{53} - q^{54} + 10q^{56} - 4q^{57} + 6q^{58} + 10q^{59} - q^{60} + 6q^{61} + 15q^{62} + 2q^{63} + 4q^{64} - 6q^{65} + 18q^{67} + 11q^{68} - 4q^{69} + 4q^{70} - 20q^{71} - 12q^{73} + 8q^{74} - 2q^{75} - 12q^{76} + 2q^{78} + 12q^{79} + 3q^{80} + 2q^{81} + 18q^{82} - 8q^{83} + 6q^{84} + 2q^{85} + 2q^{86} - 2q^{87} + 14q^{89} - q^{90} + 16q^{91} + 13q^{92} - 13q^{94} - 4q^{95} + 9q^{96} + 30q^{97} - 11q^{98} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.618034 0.309017
$$5$$ −1.00000 −0.447214
$$6$$ −1.61803 −0.660560
$$7$$ −1.23607 −0.467190 −0.233595 0.972334i $$-0.575049\pi$$
−0.233595 + 0.972334i $$0.575049\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ −1.61803 −0.511667
$$11$$ 0 0
$$12$$ −0.618034 −0.178411
$$13$$ 0.763932 0.211877 0.105938 0.994373i $$-0.466215\pi$$
0.105938 + 0.994373i $$0.466215\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ −4.85410 −1.21353
$$17$$ 3.47214 0.842117 0.421058 0.907034i $$-0.361659\pi$$
0.421058 + 0.907034i $$0.361659\pi$$
$$18$$ 1.61803 0.381374
$$19$$ −2.47214 −0.567147 −0.283573 0.958951i $$-0.591520\pi$$
−0.283573 + 0.958951i $$0.591520\pi$$
$$20$$ −0.618034 −0.138197
$$21$$ 1.23607 0.269732
$$22$$ 0 0
$$23$$ 8.70820 1.81579 0.907893 0.419202i $$-0.137690\pi$$
0.907893 + 0.419202i $$0.137690\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 1.00000 0.200000
$$26$$ 1.23607 0.242413
$$27$$ −1.00000 −0.192450
$$28$$ −0.763932 −0.144370
$$29$$ 3.23607 0.600923 0.300461 0.953794i $$-0.402859\pi$$
0.300461 + 0.953794i $$0.402859\pi$$
$$30$$ 1.61803 0.295411
$$31$$ 6.70820 1.20483 0.602414 0.798183i $$-0.294205\pi$$
0.602414 + 0.798183i $$0.294205\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 0 0
$$34$$ 5.61803 0.963485
$$35$$ 1.23607 0.208934
$$36$$ 0.618034 0.103006
$$37$$ 5.23607 0.860804 0.430402 0.902637i $$-0.358372\pi$$
0.430402 + 0.902637i $$0.358372\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −0.763932 −0.122327
$$40$$ 2.23607 0.353553
$$41$$ 12.4721 1.94782 0.973910 0.226934i $$-0.0728701\pi$$
0.973910 + 0.226934i $$0.0728701\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −0.763932 −0.116499 −0.0582493 0.998302i $$-0.518552\pi$$
−0.0582493 + 0.998302i $$0.518552\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 14.0902 2.07748
$$47$$ −4.70820 −0.686762 −0.343381 0.939196i $$-0.611572\pi$$
−0.343381 + 0.939196i $$0.611572\pi$$
$$48$$ 4.85410 0.700629
$$49$$ −5.47214 −0.781734
$$50$$ 1.61803 0.228825
$$51$$ −3.47214 −0.486196
$$52$$ 0.472136 0.0654735
$$53$$ −7.94427 −1.09123 −0.545615 0.838036i $$-0.683704\pi$$
−0.545615 + 0.838036i $$0.683704\pi$$
$$54$$ −1.61803 −0.220187
$$55$$ 0 0
$$56$$ 2.76393 0.369346
$$57$$ 2.47214 0.327442
$$58$$ 5.23607 0.687529
$$59$$ −1.70820 −0.222389 −0.111195 0.993799i $$-0.535468\pi$$
−0.111195 + 0.993799i $$0.535468\pi$$
$$60$$ 0.618034 0.0797878
$$61$$ 7.47214 0.956709 0.478354 0.878167i $$-0.341233\pi$$
0.478354 + 0.878167i $$0.341233\pi$$
$$62$$ 10.8541 1.37847
$$63$$ −1.23607 −0.155730
$$64$$ 4.23607 0.529508
$$65$$ −0.763932 −0.0947541
$$66$$ 0 0
$$67$$ 6.76393 0.826346 0.413173 0.910653i $$-0.364421\pi$$
0.413173 + 0.910653i $$0.364421\pi$$
$$68$$ 2.14590 0.260228
$$69$$ −8.70820 −1.04834
$$70$$ 2.00000 0.239046
$$71$$ −5.52786 −0.656037 −0.328018 0.944671i $$-0.606381\pi$$
−0.328018 + 0.944671i $$0.606381\pi$$
$$72$$ −2.23607 −0.263523
$$73$$ −1.52786 −0.178823 −0.0894115 0.995995i $$-0.528499\pi$$
−0.0894115 + 0.995995i $$0.528499\pi$$
$$74$$ 8.47214 0.984866
$$75$$ −1.00000 −0.115470
$$76$$ −1.52786 −0.175258
$$77$$ 0 0
$$78$$ −1.23607 −0.139957
$$79$$ −0.708204 −0.0796792 −0.0398396 0.999206i $$-0.512685\pi$$
−0.0398396 + 0.999206i $$0.512685\pi$$
$$80$$ 4.85410 0.542705
$$81$$ 1.00000 0.111111
$$82$$ 20.1803 2.22855
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0.763932 0.0833518
$$85$$ −3.47214 −0.376606
$$86$$ −1.23607 −0.133289
$$87$$ −3.23607 −0.346943
$$88$$ 0 0
$$89$$ 9.23607 0.979021 0.489511 0.871997i $$-0.337175\pi$$
0.489511 + 0.871997i $$0.337175\pi$$
$$90$$ −1.61803 −0.170556
$$91$$ −0.944272 −0.0989866
$$92$$ 5.38197 0.561109
$$93$$ −6.70820 −0.695608
$$94$$ −7.61803 −0.785740
$$95$$ 2.47214 0.253636
$$96$$ 3.38197 0.345170
$$97$$ 17.2361 1.75006 0.875029 0.484071i $$-0.160842\pi$$
0.875029 + 0.484071i $$0.160842\pi$$
$$98$$ −8.85410 −0.894399
$$99$$ 0 0
$$100$$ 0.618034 0.0618034
$$101$$ −14.4721 −1.44003 −0.720016 0.693958i $$-0.755865\pi$$
−0.720016 + 0.693958i $$0.755865\pi$$
$$102$$ −5.61803 −0.556268
$$103$$ 16.9443 1.66957 0.834784 0.550577i $$-0.185592\pi$$
0.834784 + 0.550577i $$0.185592\pi$$
$$104$$ −1.70820 −0.167503
$$105$$ −1.23607 −0.120628
$$106$$ −12.8541 −1.24850
$$107$$ 9.76393 0.943915 0.471957 0.881621i $$-0.343548\pi$$
0.471957 + 0.881621i $$0.343548\pi$$
$$108$$ −0.618034 −0.0594703
$$109$$ −2.94427 −0.282010 −0.141005 0.990009i $$-0.545033\pi$$
−0.141005 + 0.990009i $$0.545033\pi$$
$$110$$ 0 0
$$111$$ −5.23607 −0.496986
$$112$$ 6.00000 0.566947
$$113$$ 14.4164 1.35618 0.678091 0.734978i $$-0.262808\pi$$
0.678091 + 0.734978i $$0.262808\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −8.70820 −0.812044
$$116$$ 2.00000 0.185695
$$117$$ 0.763932 0.0706255
$$118$$ −2.76393 −0.254441
$$119$$ −4.29180 −0.393428
$$120$$ −2.23607 −0.204124
$$121$$ 0 0
$$122$$ 12.0902 1.09459
$$123$$ −12.4721 −1.12457
$$124$$ 4.14590 0.372313
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −9.70820 −0.861464 −0.430732 0.902480i $$-0.641745\pi$$
−0.430732 + 0.902480i $$0.641745\pi$$
$$128$$ 13.6180 1.20368
$$129$$ 0.763932 0.0672605
$$130$$ −1.23607 −0.108410
$$131$$ 8.47214 0.740214 0.370107 0.928989i $$-0.379321\pi$$
0.370107 + 0.928989i $$0.379321\pi$$
$$132$$ 0 0
$$133$$ 3.05573 0.264965
$$134$$ 10.9443 0.945441
$$135$$ 1.00000 0.0860663
$$136$$ −7.76393 −0.665752
$$137$$ 8.52786 0.728585 0.364292 0.931285i $$-0.381311\pi$$
0.364292 + 0.931285i $$0.381311\pi$$
$$138$$ −14.0902 −1.19943
$$139$$ −20.7082 −1.75645 −0.878223 0.478251i $$-0.841271\pi$$
−0.878223 + 0.478251i $$0.841271\pi$$
$$140$$ 0.763932 0.0645640
$$141$$ 4.70820 0.396502
$$142$$ −8.94427 −0.750587
$$143$$ 0 0
$$144$$ −4.85410 −0.404508
$$145$$ −3.23607 −0.268741
$$146$$ −2.47214 −0.204595
$$147$$ 5.47214 0.451334
$$148$$ 3.23607 0.266003
$$149$$ −17.7082 −1.45071 −0.725356 0.688374i $$-0.758325\pi$$
−0.725356 + 0.688374i $$0.758325\pi$$
$$150$$ −1.61803 −0.132112
$$151$$ 20.7082 1.68521 0.842605 0.538532i $$-0.181021\pi$$
0.842605 + 0.538532i $$0.181021\pi$$
$$152$$ 5.52786 0.448369
$$153$$ 3.47214 0.280706
$$154$$ 0 0
$$155$$ −6.70820 −0.538816
$$156$$ −0.472136 −0.0378011
$$157$$ 15.7082 1.25365 0.626826 0.779160i $$-0.284354\pi$$
0.626826 + 0.779160i $$0.284354\pi$$
$$158$$ −1.14590 −0.0911628
$$159$$ 7.94427 0.630022
$$160$$ 3.38197 0.267368
$$161$$ −10.7639 −0.848317
$$162$$ 1.61803 0.127125
$$163$$ −16.1803 −1.26734 −0.633671 0.773603i $$-0.718453\pi$$
−0.633671 + 0.773603i $$0.718453\pi$$
$$164$$ 7.70820 0.601910
$$165$$ 0 0
$$166$$ −6.47214 −0.502335
$$167$$ −19.1803 −1.48422 −0.742110 0.670279i $$-0.766175\pi$$
−0.742110 + 0.670279i $$0.766175\pi$$
$$168$$ −2.76393 −0.213242
$$169$$ −12.4164 −0.955108
$$170$$ −5.61803 −0.430884
$$171$$ −2.47214 −0.189049
$$172$$ −0.472136 −0.0360000
$$173$$ −14.9443 −1.13619 −0.568096 0.822962i $$-0.692320\pi$$
−0.568096 + 0.822962i $$0.692320\pi$$
$$174$$ −5.23607 −0.396945
$$175$$ −1.23607 −0.0934380
$$176$$ 0 0
$$177$$ 1.70820 0.128396
$$178$$ 14.9443 1.12012
$$179$$ 2.18034 0.162966 0.0814831 0.996675i $$-0.474034\pi$$
0.0814831 + 0.996675i $$0.474034\pi$$
$$180$$ −0.618034 −0.0460655
$$181$$ 5.41641 0.402598 0.201299 0.979530i $$-0.435484\pi$$
0.201299 + 0.979530i $$0.435484\pi$$
$$182$$ −1.52786 −0.113253
$$183$$ −7.47214 −0.552356
$$184$$ −19.4721 −1.43550
$$185$$ −5.23607 −0.384963
$$186$$ −10.8541 −0.795861
$$187$$ 0 0
$$188$$ −2.90983 −0.212221
$$189$$ 1.23607 0.0899107
$$190$$ 4.00000 0.290191
$$191$$ −21.5967 −1.56269 −0.781343 0.624102i $$-0.785465\pi$$
−0.781343 + 0.624102i $$0.785465\pi$$
$$192$$ −4.23607 −0.305712
$$193$$ −6.18034 −0.444871 −0.222435 0.974947i $$-0.571401\pi$$
−0.222435 + 0.974947i $$0.571401\pi$$
$$194$$ 27.8885 2.00228
$$195$$ 0.763932 0.0547063
$$196$$ −3.38197 −0.241569
$$197$$ 25.4164 1.81084 0.905422 0.424513i $$-0.139555\pi$$
0.905422 + 0.424513i $$0.139555\pi$$
$$198$$ 0 0
$$199$$ 11.7639 0.833923 0.416962 0.908924i $$-0.363095\pi$$
0.416962 + 0.908924i $$0.363095\pi$$
$$200$$ −2.23607 −0.158114
$$201$$ −6.76393 −0.477091
$$202$$ −23.4164 −1.64757
$$203$$ −4.00000 −0.280745
$$204$$ −2.14590 −0.150243
$$205$$ −12.4721 −0.871092
$$206$$ 27.4164 1.91019
$$207$$ 8.70820 0.605262
$$208$$ −3.70820 −0.257118
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ 6.23607 0.429309 0.214654 0.976690i $$-0.431138\pi$$
0.214654 + 0.976690i $$0.431138\pi$$
$$212$$ −4.90983 −0.337209
$$213$$ 5.52786 0.378763
$$214$$ 15.7984 1.07995
$$215$$ 0.763932 0.0520997
$$216$$ 2.23607 0.152145
$$217$$ −8.29180 −0.562884
$$218$$ −4.76393 −0.322654
$$219$$ 1.52786 0.103243
$$220$$ 0 0
$$221$$ 2.65248 0.178425
$$222$$ −8.47214 −0.568613
$$223$$ 26.9443 1.80432 0.902161 0.431400i $$-0.141980\pi$$
0.902161 + 0.431400i $$0.141980\pi$$
$$224$$ 4.18034 0.279311
$$225$$ 1.00000 0.0666667
$$226$$ 23.3262 1.55164
$$227$$ 3.29180 0.218484 0.109242 0.994015i $$-0.465158\pi$$
0.109242 + 0.994015i $$0.465158\pi$$
$$228$$ 1.52786 0.101185
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ −14.0902 −0.929078
$$231$$ 0 0
$$232$$ −7.23607 −0.475071
$$233$$ −24.8885 −1.63050 −0.815251 0.579107i $$-0.803401\pi$$
−0.815251 + 0.579107i $$0.803401\pi$$
$$234$$ 1.23607 0.0808043
$$235$$ 4.70820 0.307129
$$236$$ −1.05573 −0.0687220
$$237$$ 0.708204 0.0460028
$$238$$ −6.94427 −0.450130
$$239$$ 6.65248 0.430313 0.215156 0.976580i $$-0.430974\pi$$
0.215156 + 0.976580i $$0.430974\pi$$
$$240$$ −4.85410 −0.313331
$$241$$ −0.0557281 −0.00358976 −0.00179488 0.999998i $$-0.500571\pi$$
−0.00179488 + 0.999998i $$0.500571\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 4.61803 0.295639
$$245$$ 5.47214 0.349602
$$246$$ −20.1803 −1.28665
$$247$$ −1.88854 −0.120165
$$248$$ −15.0000 −0.952501
$$249$$ 4.00000 0.253490
$$250$$ −1.61803 −0.102333
$$251$$ −10.2918 −0.649612 −0.324806 0.945781i $$-0.605299\pi$$
−0.324806 + 0.945781i $$0.605299\pi$$
$$252$$ −0.763932 −0.0481232
$$253$$ 0 0
$$254$$ −15.7082 −0.985620
$$255$$ 3.47214 0.217434
$$256$$ 13.5623 0.847644
$$257$$ −6.88854 −0.429696 −0.214848 0.976648i $$-0.568926\pi$$
−0.214848 + 0.976648i $$0.568926\pi$$
$$258$$ 1.23607 0.0769542
$$259$$ −6.47214 −0.402159
$$260$$ −0.472136 −0.0292806
$$261$$ 3.23607 0.200308
$$262$$ 13.7082 0.846896
$$263$$ −20.1246 −1.24094 −0.620468 0.784231i $$-0.713057\pi$$
−0.620468 + 0.784231i $$0.713057\pi$$
$$264$$ 0 0
$$265$$ 7.94427 0.488013
$$266$$ 4.94427 0.303153
$$267$$ −9.23607 −0.565238
$$268$$ 4.18034 0.255355
$$269$$ 24.9443 1.52088 0.760440 0.649409i $$-0.224984\pi$$
0.760440 + 0.649409i $$0.224984\pi$$
$$270$$ 1.61803 0.0984704
$$271$$ −19.7639 −1.20057 −0.600287 0.799785i $$-0.704947\pi$$
−0.600287 + 0.799785i $$0.704947\pi$$
$$272$$ −16.8541 −1.02193
$$273$$ 0.944272 0.0571499
$$274$$ 13.7984 0.833590
$$275$$ 0 0
$$276$$ −5.38197 −0.323956
$$277$$ 12.9443 0.777746 0.388873 0.921291i $$-0.372865\pi$$
0.388873 + 0.921291i $$0.372865\pi$$
$$278$$ −33.5066 −2.00959
$$279$$ 6.70820 0.401610
$$280$$ −2.76393 −0.165177
$$281$$ 10.7639 0.642122 0.321061 0.947058i $$-0.395960\pi$$
0.321061 + 0.947058i $$0.395960\pi$$
$$282$$ 7.61803 0.453647
$$283$$ −23.8885 −1.42003 −0.710013 0.704188i $$-0.751311\pi$$
−0.710013 + 0.704188i $$0.751311\pi$$
$$284$$ −3.41641 −0.202727
$$285$$ −2.47214 −0.146437
$$286$$ 0 0
$$287$$ −15.4164 −0.910002
$$288$$ −3.38197 −0.199284
$$289$$ −4.94427 −0.290840
$$290$$ −5.23607 −0.307472
$$291$$ −17.2361 −1.01040
$$292$$ −0.944272 −0.0552593
$$293$$ 12.5279 0.731886 0.365943 0.930637i $$-0.380747\pi$$
0.365943 + 0.930637i $$0.380747\pi$$
$$294$$ 8.85410 0.516382
$$295$$ 1.70820 0.0994555
$$296$$ −11.7082 −0.680526
$$297$$ 0 0
$$298$$ −28.6525 −1.65979
$$299$$ 6.65248 0.384723
$$300$$ −0.618034 −0.0356822
$$301$$ 0.944272 0.0544269
$$302$$ 33.5066 1.92809
$$303$$ 14.4721 0.831402
$$304$$ 12.0000 0.688247
$$305$$ −7.47214 −0.427853
$$306$$ 5.61803 0.321162
$$307$$ 23.8885 1.36339 0.681696 0.731636i $$-0.261243\pi$$
0.681696 + 0.731636i $$0.261243\pi$$
$$308$$ 0 0
$$309$$ −16.9443 −0.963926
$$310$$ −10.8541 −0.616472
$$311$$ 1.23607 0.0700910 0.0350455 0.999386i $$-0.488842\pi$$
0.0350455 + 0.999386i $$0.488842\pi$$
$$312$$ 1.70820 0.0967080
$$313$$ 27.2361 1.53947 0.769737 0.638361i $$-0.220387\pi$$
0.769737 + 0.638361i $$0.220387\pi$$
$$314$$ 25.4164 1.43433
$$315$$ 1.23607 0.0696445
$$316$$ −0.437694 −0.0246222
$$317$$ 13.9443 0.783188 0.391594 0.920138i $$-0.371924\pi$$
0.391594 + 0.920138i $$0.371924\pi$$
$$318$$ 12.8541 0.720822
$$319$$ 0 0
$$320$$ −4.23607 −0.236803
$$321$$ −9.76393 −0.544970
$$322$$ −17.4164 −0.970578
$$323$$ −8.58359 −0.477604
$$324$$ 0.618034 0.0343352
$$325$$ 0.763932 0.0423753
$$326$$ −26.1803 −1.44999
$$327$$ 2.94427 0.162819
$$328$$ −27.8885 −1.53989
$$329$$ 5.81966 0.320848
$$330$$ 0 0
$$331$$ 5.18034 0.284737 0.142369 0.989814i $$-0.454528\pi$$
0.142369 + 0.989814i $$0.454528\pi$$
$$332$$ −2.47214 −0.135676
$$333$$ 5.23607 0.286935
$$334$$ −31.0344 −1.69813
$$335$$ −6.76393 −0.369553
$$336$$ −6.00000 −0.327327
$$337$$ 27.8885 1.51919 0.759593 0.650399i $$-0.225398\pi$$
0.759593 + 0.650399i $$0.225398\pi$$
$$338$$ −20.0902 −1.09276
$$339$$ −14.4164 −0.782992
$$340$$ −2.14590 −0.116378
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 15.4164 0.832408
$$344$$ 1.70820 0.0921002
$$345$$ 8.70820 0.468834
$$346$$ −24.1803 −1.29994
$$347$$ −9.76393 −0.524155 −0.262078 0.965047i $$-0.584408\pi$$
−0.262078 + 0.965047i $$0.584408\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ −0.763932 −0.0407757
$$352$$ 0 0
$$353$$ −7.00000 −0.372572 −0.186286 0.982496i $$-0.559645\pi$$
−0.186286 + 0.982496i $$0.559645\pi$$
$$354$$ 2.76393 0.146901
$$355$$ 5.52786 0.293389
$$356$$ 5.70820 0.302534
$$357$$ 4.29180 0.227146
$$358$$ 3.52786 0.186453
$$359$$ −19.4164 −1.02476 −0.512379 0.858759i $$-0.671236\pi$$
−0.512379 + 0.858759i $$0.671236\pi$$
$$360$$ 2.23607 0.117851
$$361$$ −12.8885 −0.678344
$$362$$ 8.76393 0.460622
$$363$$ 0 0
$$364$$ −0.583592 −0.0305885
$$365$$ 1.52786 0.0799721
$$366$$ −12.0902 −0.631963
$$367$$ 9.23607 0.482119 0.241059 0.970510i $$-0.422505\pi$$
0.241059 + 0.970510i $$0.422505\pi$$
$$368$$ −42.2705 −2.20350
$$369$$ 12.4721 0.649273
$$370$$ −8.47214 −0.440445
$$371$$ 9.81966 0.509811
$$372$$ −4.14590 −0.214955
$$373$$ −5.88854 −0.304897 −0.152449 0.988311i $$-0.548716\pi$$
−0.152449 + 0.988311i $$0.548716\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 10.5279 0.542933
$$377$$ 2.47214 0.127321
$$378$$ 2.00000 0.102869
$$379$$ −18.5967 −0.955251 −0.477625 0.878564i $$-0.658502\pi$$
−0.477625 + 0.878564i $$0.658502\pi$$
$$380$$ 1.52786 0.0783778
$$381$$ 9.70820 0.497366
$$382$$ −34.9443 −1.78790
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ −13.6180 −0.694942
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −0.763932 −0.0388328
$$388$$ 10.6525 0.540798
$$389$$ 27.5967 1.39921 0.699605 0.714529i $$-0.253359\pi$$
0.699605 + 0.714529i $$0.253359\pi$$
$$390$$ 1.23607 0.0625907
$$391$$ 30.2361 1.52910
$$392$$ 12.2361 0.618015
$$393$$ −8.47214 −0.427363
$$394$$ 41.1246 2.07183
$$395$$ 0.708204 0.0356336
$$396$$ 0 0
$$397$$ 9.41641 0.472596 0.236298 0.971681i $$-0.424066\pi$$
0.236298 + 0.971681i $$0.424066\pi$$
$$398$$ 19.0344 0.954110
$$399$$ −3.05573 −0.152978
$$400$$ −4.85410 −0.242705
$$401$$ 3.05573 0.152596 0.0762979 0.997085i $$-0.475690\pi$$
0.0762979 + 0.997085i $$0.475690\pi$$
$$402$$ −10.9443 −0.545851
$$403$$ 5.12461 0.255275
$$404$$ −8.94427 −0.444994
$$405$$ −1.00000 −0.0496904
$$406$$ −6.47214 −0.321207
$$407$$ 0 0
$$408$$ 7.76393 0.384372
$$409$$ −34.7771 −1.71962 −0.859808 0.510617i $$-0.829417\pi$$
−0.859808 + 0.510617i $$0.829417\pi$$
$$410$$ −20.1803 −0.996636
$$411$$ −8.52786 −0.420649
$$412$$ 10.4721 0.515925
$$413$$ 2.11146 0.103898
$$414$$ 14.0902 0.692494
$$415$$ 4.00000 0.196352
$$416$$ −2.58359 −0.126671
$$417$$ 20.7082 1.01409
$$418$$ 0 0
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ −0.763932 −0.0372761
$$421$$ −23.8328 −1.16154 −0.580770 0.814068i $$-0.697249\pi$$
−0.580770 + 0.814068i $$0.697249\pi$$
$$422$$ 10.0902 0.491182
$$423$$ −4.70820 −0.228921
$$424$$ 17.7639 0.862693
$$425$$ 3.47214 0.168423
$$426$$ 8.94427 0.433351
$$427$$ −9.23607 −0.446965
$$428$$ 6.03444 0.291686
$$429$$ 0 0
$$430$$ 1.23607 0.0596085
$$431$$ −28.1803 −1.35740 −0.678700 0.734416i $$-0.737456\pi$$
−0.678700 + 0.734416i $$0.737456\pi$$
$$432$$ 4.85410 0.233543
$$433$$ 20.7639 0.997851 0.498925 0.866645i $$-0.333728\pi$$
0.498925 + 0.866645i $$0.333728\pi$$
$$434$$ −13.4164 −0.644008
$$435$$ 3.23607 0.155158
$$436$$ −1.81966 −0.0871459
$$437$$ −21.5279 −1.02982
$$438$$ 2.47214 0.118123
$$439$$ 32.1246 1.53322 0.766612 0.642111i $$-0.221941\pi$$
0.766612 + 0.642111i $$0.221941\pi$$
$$440$$ 0 0
$$441$$ −5.47214 −0.260578
$$442$$ 4.29180 0.204140
$$443$$ −19.4164 −0.922501 −0.461251 0.887270i $$-0.652599\pi$$
−0.461251 + 0.887270i $$0.652599\pi$$
$$444$$ −3.23607 −0.153577
$$445$$ −9.23607 −0.437832
$$446$$ 43.5967 2.06437
$$447$$ 17.7082 0.837569
$$448$$ −5.23607 −0.247381
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 1.61803 0.0762749
$$451$$ 0 0
$$452$$ 8.90983 0.419083
$$453$$ −20.7082 −0.972956
$$454$$ 5.32624 0.249973
$$455$$ 0.944272 0.0442681
$$456$$ −5.52786 −0.258866
$$457$$ 10.3607 0.484652 0.242326 0.970195i $$-0.422090\pi$$
0.242326 + 0.970195i $$0.422090\pi$$
$$458$$ 11.3262 0.529240
$$459$$ −3.47214 −0.162065
$$460$$ −5.38197 −0.250935
$$461$$ 7.81966 0.364198 0.182099 0.983280i $$-0.441711\pi$$
0.182099 + 0.983280i $$0.441711\pi$$
$$462$$ 0 0
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ −15.7082 −0.729235
$$465$$ 6.70820 0.311086
$$466$$ −40.2705 −1.86550
$$467$$ −8.23607 −0.381120 −0.190560 0.981676i $$-0.561030\pi$$
−0.190560 + 0.981676i $$0.561030\pi$$
$$468$$ 0.472136 0.0218245
$$469$$ −8.36068 −0.386060
$$470$$ 7.61803 0.351394
$$471$$ −15.7082 −0.723796
$$472$$ 3.81966 0.175814
$$473$$ 0 0
$$474$$ 1.14590 0.0526328
$$475$$ −2.47214 −0.113429
$$476$$ −2.65248 −0.121576
$$477$$ −7.94427 −0.363743
$$478$$ 10.7639 0.492331
$$479$$ −13.7082 −0.626344 −0.313172 0.949696i $$-0.601392\pi$$
−0.313172 + 0.949696i $$0.601392\pi$$
$$480$$ −3.38197 −0.154365
$$481$$ 4.00000 0.182384
$$482$$ −0.0901699 −0.00410713
$$483$$ 10.7639 0.489776
$$484$$ 0 0
$$485$$ −17.2361 −0.782650
$$486$$ −1.61803 −0.0733955
$$487$$ 1.70820 0.0774061 0.0387031 0.999251i $$-0.487677\pi$$
0.0387031 + 0.999251i $$0.487677\pi$$
$$488$$ −16.7082 −0.756345
$$489$$ 16.1803 0.731700
$$490$$ 8.85410 0.399988
$$491$$ −29.5279 −1.33257 −0.666287 0.745695i $$-0.732117\pi$$
−0.666287 + 0.745695i $$0.732117\pi$$
$$492$$ −7.70820 −0.347513
$$493$$ 11.2361 0.506047
$$494$$ −3.05573 −0.137484
$$495$$ 0 0
$$496$$ −32.5623 −1.46209
$$497$$ 6.83282 0.306494
$$498$$ 6.47214 0.290023
$$499$$ 16.9443 0.758530 0.379265 0.925288i $$-0.376177\pi$$
0.379265 + 0.925288i $$0.376177\pi$$
$$500$$ −0.618034 −0.0276393
$$501$$ 19.1803 0.856914
$$502$$ −16.6525 −0.743236
$$503$$ −0.819660 −0.0365468 −0.0182734 0.999833i $$-0.505817\pi$$
−0.0182734 + 0.999833i $$0.505817\pi$$
$$504$$ 2.76393 0.123115
$$505$$ 14.4721 0.644002
$$506$$ 0 0
$$507$$ 12.4164 0.551432
$$508$$ −6.00000 −0.266207
$$509$$ −33.5967 −1.48915 −0.744575 0.667539i $$-0.767348\pi$$
−0.744575 + 0.667539i $$0.767348\pi$$
$$510$$ 5.61803 0.248771
$$511$$ 1.88854 0.0835443
$$512$$ −5.29180 −0.233867
$$513$$ 2.47214 0.109147
$$514$$ −11.1459 −0.491624
$$515$$ −16.9443 −0.746654
$$516$$ 0.472136 0.0207846
$$517$$ 0 0
$$518$$ −10.4721 −0.460119
$$519$$ 14.9443 0.655981
$$520$$ 1.70820 0.0749097
$$521$$ 40.1803 1.76033 0.880166 0.474665i $$-0.157431\pi$$
0.880166 + 0.474665i $$0.157431\pi$$
$$522$$ 5.23607 0.229176
$$523$$ 42.5410 1.86019 0.930094 0.367320i $$-0.119725\pi$$
0.930094 + 0.367320i $$0.119725\pi$$
$$524$$ 5.23607 0.228739
$$525$$ 1.23607 0.0539464
$$526$$ −32.5623 −1.41978
$$527$$ 23.2918 1.01461
$$528$$ 0 0
$$529$$ 52.8328 2.29708
$$530$$ 12.8541 0.558347
$$531$$ −1.70820 −0.0741297
$$532$$ 1.88854 0.0818788
$$533$$ 9.52786 0.412698
$$534$$ −14.9443 −0.646702
$$535$$ −9.76393 −0.422132
$$536$$ −15.1246 −0.653284
$$537$$ −2.18034 −0.0940886
$$538$$ 40.3607 1.74007
$$539$$ 0 0
$$540$$ 0.618034 0.0265959
$$541$$ 19.3050 0.829985 0.414992 0.909825i $$-0.363784\pi$$
0.414992 + 0.909825i $$0.363784\pi$$
$$542$$ −31.9787 −1.37360
$$543$$ −5.41641 −0.232440
$$544$$ −11.7426 −0.503462
$$545$$ 2.94427 0.126119
$$546$$ 1.52786 0.0653865
$$547$$ 39.5967 1.69303 0.846517 0.532361i $$-0.178695\pi$$
0.846517 + 0.532361i $$0.178695\pi$$
$$548$$ 5.27051 0.225145
$$549$$ 7.47214 0.318903
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 19.4721 0.828789
$$553$$ 0.875388 0.0372253
$$554$$ 20.9443 0.889837
$$555$$ 5.23607 0.222259
$$556$$ −12.7984 −0.542772
$$557$$ −17.4721 −0.740318 −0.370159 0.928968i $$-0.620697\pi$$
−0.370159 + 0.928968i $$0.620697\pi$$
$$558$$ 10.8541 0.459491
$$559$$ −0.583592 −0.0246833
$$560$$ −6.00000 −0.253546
$$561$$ 0 0
$$562$$ 17.4164 0.734667
$$563$$ 16.9443 0.714116 0.357058 0.934082i $$-0.383780\pi$$
0.357058 + 0.934082i $$0.383780\pi$$
$$564$$ 2.90983 0.122526
$$565$$ −14.4164 −0.606503
$$566$$ −38.6525 −1.62468
$$567$$ −1.23607 −0.0519100
$$568$$ 12.3607 0.518643
$$569$$ −6.29180 −0.263766 −0.131883 0.991265i $$-0.542102\pi$$
−0.131883 + 0.991265i $$0.542102\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −40.1246 −1.67916 −0.839581 0.543234i $$-0.817200\pi$$
−0.839581 + 0.543234i $$0.817200\pi$$
$$572$$ 0 0
$$573$$ 21.5967 0.902217
$$574$$ −24.9443 −1.04115
$$575$$ 8.70820 0.363157
$$576$$ 4.23607 0.176503
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 6.18034 0.256846
$$580$$ −2.00000 −0.0830455
$$581$$ 4.94427 0.205123
$$582$$ −27.8885 −1.15602
$$583$$ 0 0
$$584$$ 3.41641 0.141372
$$585$$ −0.763932 −0.0315847
$$586$$ 20.2705 0.837367
$$587$$ 19.6525 0.811144 0.405572 0.914063i $$-0.367072\pi$$
0.405572 + 0.914063i $$0.367072\pi$$
$$588$$ 3.38197 0.139470
$$589$$ −16.5836 −0.683315
$$590$$ 2.76393 0.113789
$$591$$ −25.4164 −1.04549
$$592$$ −25.4164 −1.04461
$$593$$ −17.0557 −0.700395 −0.350197 0.936676i $$-0.613885\pi$$
−0.350197 + 0.936676i $$0.613885\pi$$
$$594$$ 0 0
$$595$$ 4.29180 0.175946
$$596$$ −10.9443 −0.448295
$$597$$ −11.7639 −0.481466
$$598$$ 10.7639 0.440170
$$599$$ 45.0132 1.83919 0.919594 0.392870i $$-0.128518\pi$$
0.919594 + 0.392870i $$0.128518\pi$$
$$600$$ 2.23607 0.0912871
$$601$$ 19.5279 0.796558 0.398279 0.917264i $$-0.369608\pi$$
0.398279 + 0.917264i $$0.369608\pi$$
$$602$$ 1.52786 0.0622711
$$603$$ 6.76393 0.275449
$$604$$ 12.7984 0.520758
$$605$$ 0 0
$$606$$ 23.4164 0.951227
$$607$$ 19.7082 0.799931 0.399966 0.916530i $$-0.369022\pi$$
0.399966 + 0.916530i $$0.369022\pi$$
$$608$$ 8.36068 0.339070
$$609$$ 4.00000 0.162088
$$610$$ −12.0902 −0.489517
$$611$$ −3.59675 −0.145509
$$612$$ 2.14590 0.0867428
$$613$$ 38.8328 1.56844 0.784221 0.620481i $$-0.213063\pi$$
0.784221 + 0.620481i $$0.213063\pi$$
$$614$$ 38.6525 1.55989
$$615$$ 12.4721 0.502925
$$616$$ 0 0
$$617$$ −29.7771 −1.19878 −0.599390 0.800457i $$-0.704590\pi$$
−0.599390 + 0.800457i $$0.704590\pi$$
$$618$$ −27.4164 −1.10285
$$619$$ −18.4721 −0.742458 −0.371229 0.928541i $$-0.621063\pi$$
−0.371229 + 0.928541i $$0.621063\pi$$
$$620$$ −4.14590 −0.166503
$$621$$ −8.70820 −0.349448
$$622$$ 2.00000 0.0801927
$$623$$ −11.4164 −0.457389
$$624$$ 3.70820 0.148447
$$625$$ 1.00000 0.0400000
$$626$$ 44.0689 1.76135
$$627$$ 0 0
$$628$$ 9.70820 0.387400
$$629$$ 18.1803 0.724898
$$630$$ 2.00000 0.0796819
$$631$$ −13.1803 −0.524701 −0.262351 0.964973i $$-0.584498\pi$$
−0.262351 + 0.964973i $$0.584498\pi$$
$$632$$ 1.58359 0.0629919
$$633$$ −6.23607 −0.247861
$$634$$ 22.5623 0.896064
$$635$$ 9.70820 0.385258
$$636$$ 4.90983 0.194687
$$637$$ −4.18034 −0.165631
$$638$$ 0 0
$$639$$ −5.52786 −0.218679
$$640$$ −13.6180 −0.538300
$$641$$ 31.1246 1.22935 0.614674 0.788781i $$-0.289288\pi$$
0.614674 + 0.788781i $$0.289288\pi$$
$$642$$ −15.7984 −0.623512
$$643$$ 24.8328 0.979311 0.489655 0.871916i $$-0.337123\pi$$
0.489655 + 0.871916i $$0.337123\pi$$
$$644$$ −6.65248 −0.262144
$$645$$ −0.763932 −0.0300798
$$646$$ −13.8885 −0.546437
$$647$$ −11.7639 −0.462488 −0.231244 0.972896i $$-0.574280\pi$$
−0.231244 + 0.972896i $$0.574280\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ 0 0
$$650$$ 1.23607 0.0484826
$$651$$ 8.29180 0.324981
$$652$$ −10.0000 −0.391630
$$653$$ −36.8328 −1.44138 −0.720690 0.693258i $$-0.756175\pi$$
−0.720690 + 0.693258i $$0.756175\pi$$
$$654$$ 4.76393 0.186284
$$655$$ −8.47214 −0.331034
$$656$$ −60.5410 −2.36373
$$657$$ −1.52786 −0.0596077
$$658$$ 9.41641 0.367090
$$659$$ −13.5967 −0.529654 −0.264827 0.964296i $$-0.585315\pi$$
−0.264827 + 0.964296i $$0.585315\pi$$
$$660$$ 0 0
$$661$$ −25.7771 −1.00261 −0.501306 0.865270i $$-0.667147\pi$$
−0.501306 + 0.865270i $$0.667147\pi$$
$$662$$ 8.38197 0.325774
$$663$$ −2.65248 −0.103014
$$664$$ 8.94427 0.347105
$$665$$ −3.05573 −0.118496
$$666$$ 8.47214 0.328289
$$667$$ 28.1803 1.09115
$$668$$ −11.8541 −0.458649
$$669$$ −26.9443 −1.04173
$$670$$ −10.9443 −0.422814
$$671$$ 0 0
$$672$$ −4.18034 −0.161260
$$673$$ 21.5967 0.832493 0.416247 0.909252i $$-0.363345\pi$$
0.416247 + 0.909252i $$0.363345\pi$$
$$674$$ 45.1246 1.73814
$$675$$ −1.00000 −0.0384900
$$676$$ −7.67376 −0.295145
$$677$$ −21.4164 −0.823099 −0.411550 0.911387i $$-0.635012\pi$$
−0.411550 + 0.911387i $$0.635012\pi$$
$$678$$ −23.3262 −0.895839
$$679$$ −21.3050 −0.817609
$$680$$ 7.76393 0.297733
$$681$$ −3.29180 −0.126142
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ −1.52786 −0.0584193
$$685$$ −8.52786 −0.325833
$$686$$ 24.9443 0.952377
$$687$$ −7.00000 −0.267067
$$688$$ 3.70820 0.141374
$$689$$ −6.06888 −0.231206
$$690$$ 14.0902 0.536404
$$691$$ 29.5410 1.12379 0.561897 0.827207i $$-0.310072\pi$$
0.561897 + 0.827207i $$0.310072\pi$$
$$692$$ −9.23607 −0.351103
$$693$$ 0 0
$$694$$ −15.7984 −0.599698
$$695$$ 20.7082 0.785507
$$696$$ 7.23607 0.274282
$$697$$ 43.3050 1.64029
$$698$$ −24.2705 −0.918652
$$699$$ 24.8885 0.941371
$$700$$ −0.763932 −0.0288739
$$701$$ 13.8197 0.521961 0.260981 0.965344i $$-0.415954\pi$$
0.260981 + 0.965344i $$0.415954\pi$$
$$702$$ −1.23607 −0.0466524
$$703$$ −12.9443 −0.488202
$$704$$ 0 0
$$705$$ −4.70820 −0.177321
$$706$$ −11.3262 −0.426269
$$707$$ 17.8885 0.672768
$$708$$ 1.05573 0.0396767
$$709$$ −0.0557281 −0.00209291 −0.00104646 0.999999i $$-0.500333\pi$$
−0.00104646 + 0.999999i $$0.500333\pi$$
$$710$$ 8.94427 0.335673
$$711$$ −0.708204 −0.0265597
$$712$$ −20.6525 −0.773984
$$713$$ 58.4164 2.18771
$$714$$ 6.94427 0.259883
$$715$$ 0 0
$$716$$ 1.34752 0.0503593
$$717$$ −6.65248 −0.248441
$$718$$ −31.4164 −1.17245
$$719$$ −18.3607 −0.684738 −0.342369 0.939566i $$-0.611229\pi$$
−0.342369 + 0.939566i $$0.611229\pi$$
$$720$$ 4.85410 0.180902
$$721$$ −20.9443 −0.780005
$$722$$ −20.8541 −0.776109
$$723$$ 0.0557281 0.00207255
$$724$$ 3.34752 0.124410
$$725$$ 3.23607 0.120185
$$726$$ 0 0
$$727$$ −8.83282 −0.327591 −0.163796 0.986494i $$-0.552374\pi$$
−0.163796 + 0.986494i $$0.552374\pi$$
$$728$$ 2.11146 0.0782558
$$729$$ 1.00000 0.0370370
$$730$$ 2.47214 0.0914979
$$731$$ −2.65248 −0.0981054
$$732$$ −4.61803 −0.170687
$$733$$ 12.1115 0.447347 0.223673 0.974664i $$-0.428195\pi$$
0.223673 + 0.974664i $$0.428195\pi$$
$$734$$ 14.9443 0.551603
$$735$$ −5.47214 −0.201843
$$736$$ −29.4508 −1.08557
$$737$$ 0 0
$$738$$ 20.1803 0.742849
$$739$$ −29.6525 −1.09078 −0.545392 0.838181i $$-0.683619\pi$$
−0.545392 + 0.838181i $$0.683619\pi$$
$$740$$ −3.23607 −0.118960
$$741$$ 1.88854 0.0693774
$$742$$ 15.8885 0.583287
$$743$$ −2.23607 −0.0820334 −0.0410167 0.999158i $$-0.513060\pi$$
−0.0410167 + 0.999158i $$0.513060\pi$$
$$744$$ 15.0000 0.549927
$$745$$ 17.7082 0.648778
$$746$$ −9.52786 −0.348840
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ −12.0689 −0.440987
$$750$$ 1.61803 0.0590822
$$751$$ 42.0132 1.53308 0.766541 0.642195i $$-0.221976\pi$$
0.766541 + 0.642195i $$0.221976\pi$$
$$752$$ 22.8541 0.833403
$$753$$ 10.2918 0.375054
$$754$$ 4.00000 0.145671
$$755$$ −20.7082 −0.753649
$$756$$ 0.763932 0.0277839
$$757$$ 1.05573 0.0383711 0.0191855 0.999816i $$-0.493893\pi$$
0.0191855 + 0.999816i $$0.493893\pi$$
$$758$$ −30.0902 −1.09292
$$759$$ 0 0
$$760$$ −5.52786 −0.200517
$$761$$ −13.3475 −0.483847 −0.241924 0.970295i $$-0.577778\pi$$
−0.241924 + 0.970295i $$0.577778\pi$$
$$762$$ 15.7082 0.569048
$$763$$ 3.63932 0.131752
$$764$$ −13.3475 −0.482896
$$765$$ −3.47214 −0.125535
$$766$$ 25.8885 0.935391
$$767$$ −1.30495 −0.0471191
$$768$$ −13.5623 −0.489388
$$769$$ −33.4721 −1.20704 −0.603518 0.797349i $$-0.706235\pi$$
−0.603518 + 0.797349i $$0.706235\pi$$
$$770$$ 0 0
$$771$$ 6.88854 0.248085
$$772$$ −3.81966 −0.137473
$$773$$ 9.94427 0.357671 0.178835 0.983879i $$-0.442767\pi$$
0.178835 + 0.983879i $$0.442767\pi$$
$$774$$ −1.23607 −0.0444295
$$775$$ 6.70820 0.240966
$$776$$ −38.5410 −1.38354
$$777$$ 6.47214 0.232187
$$778$$ 44.6525 1.60087
$$779$$ −30.8328 −1.10470
$$780$$ 0.472136 0.0169052
$$781$$ 0 0
$$782$$ 48.9230 1.74948
$$783$$ −3.23607 −0.115648
$$784$$ 26.5623 0.948654
$$785$$ −15.7082 −0.560650
$$786$$ −13.7082 −0.488955
$$787$$ −44.9443 −1.60209 −0.801045 0.598604i $$-0.795722\pi$$
−0.801045 + 0.598604i $$0.795722\pi$$
$$788$$ 15.7082 0.559582
$$789$$ 20.1246 0.716455
$$790$$ 1.14590 0.0407692
$$791$$ −17.8197 −0.633594
$$792$$ 0 0
$$793$$ 5.70820 0.202704
$$794$$ 15.2361 0.540708
$$795$$ −7.94427 −0.281754
$$796$$ 7.27051 0.257696
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ −4.94427 −0.175025
$$799$$ −16.3475 −0.578334
$$800$$ −3.38197 −0.119571
$$801$$ 9.23607 0.326340
$$802$$ 4.94427 0.174588
$$803$$ 0 0
$$804$$ −4.18034 −0.147429
$$805$$ 10.7639 0.379379
$$806$$ 8.29180 0.292066
$$807$$ −24.9443 −0.878080
$$808$$ 32.3607 1.13844
$$809$$ −8.00000 −0.281265 −0.140633 0.990062i $$-0.544914\pi$$
−0.140633 + 0.990062i $$0.544914\pi$$
$$810$$ −1.61803 −0.0568519
$$811$$ 48.1246 1.68988 0.844942 0.534858i $$-0.179635\pi$$
0.844942 + 0.534858i $$0.179635\pi$$
$$812$$ −2.47214 −0.0867550
$$813$$ 19.7639 0.693151
$$814$$ 0 0
$$815$$ 16.1803 0.566773
$$816$$ 16.8541 0.590012
$$817$$ 1.88854 0.0660718
$$818$$ −56.2705 −1.96745
$$819$$ −0.944272 −0.0329955
$$820$$ −7.70820 −0.269182
$$821$$ 37.7771 1.31843 0.659215 0.751955i $$-0.270889\pi$$
0.659215 + 0.751955i $$0.270889\pi$$
$$822$$ −13.7984 −0.481274
$$823$$ −42.2492 −1.47272 −0.736358 0.676592i $$-0.763456\pi$$
−0.736358 + 0.676592i $$0.763456\pi$$
$$824$$ −37.8885 −1.31991
$$825$$ 0 0
$$826$$ 3.41641 0.118872
$$827$$ −8.94427 −0.311023 −0.155511 0.987834i $$-0.549703\pi$$
−0.155511 + 0.987834i $$0.549703\pi$$
$$828$$ 5.38197 0.187036
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 6.47214 0.224651
$$831$$ −12.9443 −0.449032
$$832$$ 3.23607 0.112190
$$833$$ −19.0000 −0.658311
$$834$$ 33.5066 1.16024
$$835$$ 19.1803 0.663763
$$836$$ 0 0
$$837$$ −6.70820 −0.231869
$$838$$ −22.6525 −0.782517
$$839$$ −16.3607 −0.564833 −0.282417 0.959292i $$-0.591136\pi$$
−0.282417 + 0.959292i $$0.591136\pi$$
$$840$$ 2.76393 0.0953647
$$841$$ −18.5279 −0.638892
$$842$$ −38.5623 −1.32894
$$843$$ −10.7639 −0.370730
$$844$$ 3.85410 0.132664
$$845$$ 12.4164 0.427137
$$846$$ −7.61803 −0.261913
$$847$$ 0 0
$$848$$ 38.5623 1.32424
$$849$$ 23.8885 0.819853
$$850$$ 5.61803 0.192697
$$851$$ 45.5967 1.56304
$$852$$ 3.41641 0.117044
$$853$$ 3.88854 0.133141 0.0665706 0.997782i $$-0.478794\pi$$
0.0665706 + 0.997782i $$0.478794\pi$$
$$854$$ −14.9443 −0.511382
$$855$$ 2.47214 0.0845453
$$856$$ −21.8328 −0.746230
$$857$$ 1.00000 0.0341593 0.0170797 0.999854i $$-0.494563\pi$$
0.0170797 + 0.999854i $$0.494563\pi$$
$$858$$ 0 0
$$859$$ 26.8328 0.915524 0.457762 0.889075i $$-0.348651\pi$$
0.457762 + 0.889075i $$0.348651\pi$$
$$860$$ 0.472136 0.0160997
$$861$$ 15.4164 0.525390
$$862$$ −45.5967 −1.55303
$$863$$ −2.11146 −0.0718748 −0.0359374 0.999354i $$-0.511442\pi$$
−0.0359374 + 0.999354i $$0.511442\pi$$
$$864$$ 3.38197 0.115057
$$865$$ 14.9443 0.508120
$$866$$ 33.5967 1.14166
$$867$$ 4.94427 0.167916
$$868$$ −5.12461 −0.173941
$$869$$ 0 0
$$870$$ 5.23607 0.177519
$$871$$ 5.16718 0.175083
$$872$$ 6.58359 0.222949
$$873$$ 17.2361 0.583353
$$874$$ −34.8328 −1.17824
$$875$$ 1.23607 0.0417867
$$876$$ 0.944272 0.0319040
$$877$$ −51.3050 −1.73245 −0.866223 0.499658i $$-0.833459\pi$$
−0.866223 + 0.499658i $$0.833459\pi$$
$$878$$ 51.9787 1.75420
$$879$$ −12.5279 −0.422554
$$880$$ 0 0
$$881$$ −14.8328 −0.499730 −0.249865 0.968281i $$-0.580386\pi$$
−0.249865 + 0.968281i $$0.580386\pi$$
$$882$$ −8.85410 −0.298133
$$883$$ −12.1803 −0.409901 −0.204951 0.978772i $$-0.565703\pi$$
−0.204951 + 0.978772i $$0.565703\pi$$
$$884$$ 1.63932 0.0551363
$$885$$ −1.70820 −0.0574206
$$886$$ −31.4164 −1.05545
$$887$$ −0.944272 −0.0317055 −0.0158528 0.999874i $$-0.505046\pi$$
−0.0158528 + 0.999874i $$0.505046\pi$$
$$888$$ 11.7082 0.392902
$$889$$ 12.0000 0.402467
$$890$$ −14.9443 −0.500933
$$891$$ 0 0
$$892$$ 16.6525 0.557566
$$893$$ 11.6393 0.389495
$$894$$ 28.6525 0.958282
$$895$$ −2.18034 −0.0728807
$$896$$ −16.8328 −0.562345
$$897$$ −6.65248 −0.222120
$$898$$ −16.1803 −0.539945
$$899$$ 21.7082 0.724009
$$900$$ 0.618034 0.0206011
$$901$$ −27.5836 −0.918943
$$902$$ 0 0
$$903$$ −0.944272 −0.0314234
$$904$$ −32.2361 −1.07216
$$905$$ −5.41641 −0.180047
$$906$$ −33.5066 −1.11318
$$907$$ −42.0000 −1.39459 −0.697294 0.716786i $$-0.745613\pi$$
−0.697294 + 0.716786i $$0.745613\pi$$
$$908$$ 2.03444 0.0675153
$$909$$ −14.4721 −0.480010
$$910$$ 1.52786 0.0506482
$$911$$ −45.9574 −1.52264 −0.761319 0.648378i $$-0.775448\pi$$
−0.761319 + 0.648378i $$0.775448\pi$$
$$912$$ −12.0000 −0.397360
$$913$$ 0 0
$$914$$ 16.7639 0.554502
$$915$$ 7.47214 0.247021
$$916$$ 4.32624 0.142943
$$917$$ −10.4721 −0.345820
$$918$$ −5.61803 −0.185423
$$919$$ −23.4164 −0.772436 −0.386218 0.922408i $$-0.626219\pi$$
−0.386218 + 0.922408i $$0.626219\pi$$
$$920$$ 19.4721 0.641977
$$921$$ −23.8885 −0.787154
$$922$$ 12.6525 0.416687
$$923$$ −4.22291 −0.138999
$$924$$ 0 0
$$925$$ 5.23607 0.172161
$$926$$ 9.70820 0.319031
$$927$$ 16.9443 0.556523
$$928$$ −10.9443 −0.359263
$$929$$ 33.0557 1.08452 0.542262 0.840210i $$-0.317568\pi$$
0.542262 + 0.840210i $$0.317568\pi$$
$$930$$ 10.8541 0.355920
$$931$$ 13.5279 0.443358
$$932$$ −15.3820 −0.503853
$$933$$ −1.23607 −0.0404670
$$934$$ −13.3262 −0.436048
$$935$$ 0 0
$$936$$ −1.70820 −0.0558344
$$937$$ −7.12461 −0.232751 −0.116375 0.993205i $$-0.537128\pi$$
−0.116375 + 0.993205i $$0.537128\pi$$
$$938$$ −13.5279 −0.441700
$$939$$ −27.2361 −0.888815
$$940$$ 2.90983 0.0949082
$$941$$ −40.7639 −1.32887 −0.664433 0.747348i $$-0.731327\pi$$
−0.664433 + 0.747348i $$0.731327\pi$$
$$942$$ −25.4164 −0.828111
$$943$$ 108.610 3.53683
$$944$$ 8.29180 0.269875
$$945$$ −1.23607 −0.0402093
$$946$$ 0 0
$$947$$ −13.7639 −0.447268 −0.223634 0.974673i $$-0.571792\pi$$
−0.223634 + 0.974673i $$0.571792\pi$$
$$948$$ 0.437694 0.0142156
$$949$$ −1.16718 −0.0378884
$$950$$ −4.00000 −0.129777
$$951$$ −13.9443 −0.452174
$$952$$ 9.59675 0.311032
$$953$$ −3.88854 −0.125962 −0.0629811 0.998015i $$-0.520061\pi$$
−0.0629811 + 0.998015i $$0.520061\pi$$
$$954$$ −12.8541 −0.416167
$$955$$ 21.5967 0.698854
$$956$$ 4.11146 0.132974
$$957$$ 0 0
$$958$$ −22.1803 −0.716614
$$959$$ −10.5410 −0.340387
$$960$$ 4.23607 0.136719
$$961$$ 14.0000 0.451613
$$962$$ 6.47214 0.208670
$$963$$ 9.76393 0.314638
$$964$$ −0.0344419 −0.00110930
$$965$$ 6.18034 0.198952
$$966$$ 17.4164 0.560364
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 8.58359 0.275745
$$970$$ −27.8885 −0.895447
$$971$$ 24.7639 0.794712 0.397356 0.917664i $$-0.369928\pi$$
0.397356 + 0.917664i $$0.369928\pi$$
$$972$$ −0.618034 −0.0198234
$$973$$ 25.5967 0.820594
$$974$$ 2.76393 0.0885621
$$975$$ −0.763932 −0.0244654
$$976$$ −36.2705 −1.16099
$$977$$ 1.94427 0.0622028 0.0311014 0.999516i $$-0.490099\pi$$
0.0311014 + 0.999516i $$0.490099\pi$$
$$978$$ 26.1803 0.837155
$$979$$ 0 0
$$980$$ 3.38197 0.108033
$$981$$ −2.94427 −0.0940034
$$982$$ −47.7771 −1.52463
$$983$$ −32.8197 −1.04678 −0.523392 0.852092i $$-0.675334\pi$$
−0.523392 + 0.852092i $$0.675334\pi$$
$$984$$ 27.8885 0.889054
$$985$$ −25.4164 −0.809834
$$986$$ 18.1803 0.578980
$$987$$ −5.81966 −0.185242
$$988$$ −1.16718 −0.0371331
$$989$$ −6.65248 −0.211536
$$990$$ 0 0
$$991$$ 1.76393 0.0560331 0.0280166 0.999607i $$-0.491081\pi$$
0.0280166 + 0.999607i $$0.491081\pi$$
$$992$$ −22.6869 −0.720310
$$993$$ −5.18034 −0.164393
$$994$$ 11.0557 0.350666
$$995$$ −11.7639 −0.372942
$$996$$ 2.47214 0.0783326
$$997$$ 43.2361 1.36930 0.684650 0.728872i $$-0.259955\pi$$
0.684650 + 0.728872i $$0.259955\pi$$
$$998$$ 27.4164 0.867851
$$999$$ −5.23607 −0.165662
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 1815.2.a.j.1.2 yes 2
3.2 odd 2 5445.2.a.o.1.1 2
5.4 even 2 9075.2.a.bd.1.1 2
11.10 odd 2 1815.2.a.f.1.1 2
33.32 even 2 5445.2.a.x.1.2 2
55.54 odd 2 9075.2.a.bx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.1 2 11.10 odd 2
1815.2.a.j.1.2 yes 2 1.1 even 1 trivial
5445.2.a.o.1.1 2 3.2 odd 2
5445.2.a.x.1.2 2 33.32 even 2
9075.2.a.bd.1.1 2 5.4 even 2
9075.2.a.bx.1.2 2 55.54 odd 2