# Properties

 Label 3675.2.a.ba.1.2 Level $3675$ Weight $2$ Character 3675.1 Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ 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 $$\chi$$ $$=$$ 3675.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{11} -0.618034 q^{12} +6.47214 q^{13} -4.85410 q^{16} -2.47214 q^{17} +1.61803 q^{18} -2.47214 q^{19} -3.61803 q^{22} +4.23607 q^{23} +2.23607 q^{24} +10.4721 q^{26} -1.00000 q^{27} -3.00000 q^{29} +4.00000 q^{31} -3.38197 q^{32} +2.23607 q^{33} -4.00000 q^{34} +0.618034 q^{36} -3.47214 q^{37} -4.00000 q^{38} -6.47214 q^{39} +8.94427 q^{41} +4.70820 q^{43} -1.38197 q^{44} +6.85410 q^{46} +10.4721 q^{47} +4.85410 q^{48} +2.47214 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.61803 q^{54} +2.47214 q^{57} -4.85410 q^{58} +6.47214 q^{59} +12.0000 q^{61} +6.47214 q^{62} +4.23607 q^{64} +3.61803 q^{66} -12.7082 q^{67} -1.52786 q^{68} -4.23607 q^{69} +8.23607 q^{71} -2.23607 q^{72} -14.4721 q^{73} -5.61803 q^{74} -1.52786 q^{76} -10.4721 q^{78} -2.70820 q^{79} +1.00000 q^{81} +14.4721 q^{82} +16.9443 q^{83} +7.61803 q^{86} +3.00000 q^{87} +5.00000 q^{88} +1.52786 q^{89} +2.61803 q^{92} -4.00000 q^{93} +16.9443 q^{94} +3.38197 q^{96} -4.00000 q^{97} -2.23607 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} - 2q^{3} - q^{4} - q^{6} + 2q^{9} + q^{12} + 4q^{13} - 3q^{16} + 4q^{17} + q^{18} + 4q^{19} - 5q^{22} + 4q^{23} + 12q^{26} - 2q^{27} - 6q^{29} + 8q^{31} - 9q^{32} - 8q^{34} - q^{36} + 2q^{37} - 8q^{38} - 4q^{39} - 4q^{43} - 5q^{44} + 7q^{46} + 12q^{47} + 3q^{48} - 4q^{51} + 8q^{52} - 12q^{53} - q^{54} - 4q^{57} - 3q^{58} + 4q^{59} + 24q^{61} + 4q^{62} + 4q^{64} + 5q^{66} - 12q^{67} - 12q^{68} - 4q^{69} + 12q^{71} - 20q^{73} - 9q^{74} - 12q^{76} - 12q^{78} + 8q^{79} + 2q^{81} + 20q^{82} + 16q^{83} + 13q^{86} + 6q^{87} + 10q^{88} + 12q^{89} + 3q^{92} - 8q^{93} + 16q^{94} + 9q^{96} - 8q^{97} + O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ −1.61803 −0.660560
$$7$$ 0 0
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.23607 −0.674200 −0.337100 0.941469i $$-0.609446\pi$$
−0.337100 + 0.941469i $$0.609446\pi$$
$$12$$ −0.618034 −0.178411
$$13$$ 6.47214 1.79505 0.897524 0.440966i $$-0.145364\pi$$
0.897524 + 0.440966i $$0.145364\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −2.47214 −0.599581 −0.299791 0.954005i $$-0.596917\pi$$
−0.299791 + 0.954005i $$0.596917\pi$$
$$18$$ 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 0
$$21$$ 0 0
$$22$$ −3.61803 −0.771367
$$23$$ 4.23607 0.883281 0.441641 0.897192i $$-0.354397\pi$$
0.441641 + 0.897192i $$0.354397\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ 10.4721 2.05375
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 2.23607 0.389249
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0.618034 0.103006
$$37$$ −3.47214 −0.570816 −0.285408 0.958406i $$-0.592129\pi$$
−0.285408 + 0.958406i $$0.592129\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −6.47214 −1.03637
$$40$$ 0 0
$$41$$ 8.94427 1.39686 0.698430 0.715678i $$-0.253882\pi$$
0.698430 + 0.715678i $$0.253882\pi$$
$$42$$ 0 0
$$43$$ 4.70820 0.717994 0.358997 0.933339i $$-0.383119\pi$$
0.358997 + 0.933339i $$0.383119\pi$$
$$44$$ −1.38197 −0.208339
$$45$$ 0 0
$$46$$ 6.85410 1.01058
$$47$$ 10.4721 1.52752 0.763759 0.645501i $$-0.223352\pi$$
0.763759 + 0.645501i $$0.223352\pi$$
$$48$$ 4.85410 0.700629
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.47214 0.346168
$$52$$ 4.00000 0.554700
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.61803 −0.220187
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.47214 0.327442
$$58$$ −4.85410 −0.637375
$$59$$ 6.47214 0.842600 0.421300 0.906921i $$-0.361574\pi$$
0.421300 + 0.906921i $$0.361574\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ 6.47214 0.821962
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 3.61803 0.445349
$$67$$ −12.7082 −1.55255 −0.776277 0.630392i $$-0.782894\pi$$
−0.776277 + 0.630392i $$0.782894\pi$$
$$68$$ −1.52786 −0.185281
$$69$$ −4.23607 −0.509963
$$70$$ 0 0
$$71$$ 8.23607 0.977441 0.488721 0.872440i $$-0.337464\pi$$
0.488721 + 0.872440i $$0.337464\pi$$
$$72$$ −2.23607 −0.263523
$$73$$ −14.4721 −1.69384 −0.846918 0.531724i $$-0.821544\pi$$
−0.846918 + 0.531724i $$0.821544\pi$$
$$74$$ −5.61803 −0.653083
$$75$$ 0 0
$$76$$ −1.52786 −0.175258
$$77$$ 0 0
$$78$$ −10.4721 −1.18574
$$79$$ −2.70820 −0.304697 −0.152348 0.988327i $$-0.548684\pi$$
−0.152348 + 0.988327i $$0.548684\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 14.4721 1.59818
$$83$$ 16.9443 1.85988 0.929938 0.367717i $$-0.119860\pi$$
0.929938 + 0.367717i $$0.119860\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 7.61803 0.821474
$$87$$ 3.00000 0.321634
$$88$$ 5.00000 0.533002
$$89$$ 1.52786 0.161953 0.0809766 0.996716i $$-0.474196\pi$$
0.0809766 + 0.996716i $$0.474196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 2.61803 0.272949
$$93$$ −4.00000 −0.414781
$$94$$ 16.9443 1.74767
$$95$$ 0 0
$$96$$ 3.38197 0.345170
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ 0 0
$$99$$ −2.23607 −0.224733
$$100$$ 0 0
$$101$$ 9.52786 0.948058 0.474029 0.880509i $$-0.342799\pi$$
0.474029 + 0.880509i $$0.342799\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 7.41641 0.730760 0.365380 0.930858i $$-0.380939\pi$$
0.365380 + 0.930858i $$0.380939\pi$$
$$104$$ −14.4721 −1.41911
$$105$$ 0 0
$$106$$ −9.70820 −0.942944
$$107$$ 8.94427 0.864675 0.432338 0.901712i $$-0.357689\pi$$
0.432338 + 0.901712i $$0.357689\pi$$
$$108$$ −0.618034 −0.0594703
$$109$$ 16.4164 1.57241 0.786203 0.617968i $$-0.212044\pi$$
0.786203 + 0.617968i $$0.212044\pi$$
$$110$$ 0 0
$$111$$ 3.47214 0.329561
$$112$$ 0 0
$$113$$ 10.5279 0.990378 0.495189 0.868785i $$-0.335099\pi$$
0.495189 + 0.868785i $$0.335099\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −1.85410 −0.172149
$$117$$ 6.47214 0.598349
$$118$$ 10.4721 0.964038
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −6.00000 −0.545455
$$122$$ 19.4164 1.75788
$$123$$ −8.94427 −0.806478
$$124$$ 2.47214 0.222004
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −19.1803 −1.70198 −0.850990 0.525182i $$-0.823997\pi$$
−0.850990 + 0.525182i $$0.823997\pi$$
$$128$$ 13.6180 1.20368
$$129$$ −4.70820 −0.414534
$$130$$ 0 0
$$131$$ 13.5279 1.18193 0.590967 0.806695i $$-0.298746\pi$$
0.590967 + 0.806695i $$0.298746\pi$$
$$132$$ 1.38197 0.120285
$$133$$ 0 0
$$134$$ −20.5623 −1.77631
$$135$$ 0 0
$$136$$ 5.52786 0.474010
$$137$$ 19.8885 1.69919 0.849596 0.527433i $$-0.176846\pi$$
0.849596 + 0.527433i $$0.176846\pi$$
$$138$$ −6.85410 −0.583460
$$139$$ −11.4164 −0.968327 −0.484164 0.874978i $$-0.660876\pi$$
−0.484164 + 0.874978i $$0.660876\pi$$
$$140$$ 0 0
$$141$$ −10.4721 −0.881913
$$142$$ 13.3262 1.11831
$$143$$ −14.4721 −1.21022
$$144$$ −4.85410 −0.404508
$$145$$ 0 0
$$146$$ −23.4164 −1.93796
$$147$$ 0 0
$$148$$ −2.14590 −0.176392
$$149$$ 12.8885 1.05587 0.527935 0.849285i $$-0.322966\pi$$
0.527935 + 0.849285i $$0.322966\pi$$
$$150$$ 0 0
$$151$$ 6.70820 0.545906 0.272953 0.962027i $$-0.412000\pi$$
0.272953 + 0.962027i $$0.412000\pi$$
$$152$$ 5.52786 0.448369
$$153$$ −2.47214 −0.199860
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ −0.944272 −0.0753611 −0.0376806 0.999290i $$-0.511997\pi$$
−0.0376806 + 0.999290i $$0.511997\pi$$
$$158$$ −4.38197 −0.348610
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.61803 0.127125
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 5.52786 0.431654
$$165$$ 0 0
$$166$$ 27.4164 2.12793
$$167$$ −22.4721 −1.73895 −0.869473 0.493980i $$-0.835541\pi$$
−0.869473 + 0.493980i $$0.835541\pi$$
$$168$$ 0 0
$$169$$ 28.8885 2.22220
$$170$$ 0 0
$$171$$ −2.47214 −0.189049
$$172$$ 2.90983 0.221872
$$173$$ 2.47214 0.187953 0.0939765 0.995574i $$-0.470042\pi$$
0.0939765 + 0.995574i $$0.470042\pi$$
$$174$$ 4.85410 0.367989
$$175$$ 0 0
$$176$$ 10.8541 0.818159
$$177$$ −6.47214 −0.486476
$$178$$ 2.47214 0.185294
$$179$$ −0.944272 −0.0705782 −0.0352891 0.999377i $$-0.511235\pi$$
−0.0352891 + 0.999377i $$0.511235\pi$$
$$180$$ 0 0
$$181$$ 6.47214 0.481070 0.240535 0.970640i $$-0.422677\pi$$
0.240535 + 0.970640i $$0.422677\pi$$
$$182$$ 0 0
$$183$$ −12.0000 −0.887066
$$184$$ −9.47214 −0.698295
$$185$$ 0 0
$$186$$ −6.47214 −0.474560
$$187$$ 5.52786 0.404237
$$188$$ 6.47214 0.472029
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.05573 0.221105 0.110552 0.993870i $$-0.464738\pi$$
0.110552 + 0.993870i $$0.464738\pi$$
$$192$$ −4.23607 −0.305712
$$193$$ 22.8885 1.64755 0.823777 0.566914i $$-0.191863\pi$$
0.823777 + 0.566914i $$0.191863\pi$$
$$194$$ −6.47214 −0.464672
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.0000 1.49619 0.748094 0.663593i $$-0.230969\pi$$
0.748094 + 0.663593i $$0.230969\pi$$
$$198$$ −3.61803 −0.257122
$$199$$ −21.8885 −1.55164 −0.775819 0.630956i $$-0.782663\pi$$
−0.775819 + 0.630956i $$0.782663\pi$$
$$200$$ 0 0
$$201$$ 12.7082 0.896368
$$202$$ 15.4164 1.08469
$$203$$ 0 0
$$204$$ 1.52786 0.106972
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ 4.23607 0.294427
$$208$$ −31.4164 −2.17834
$$209$$ 5.52786 0.382370
$$210$$ 0 0
$$211$$ 13.8885 0.956127 0.478063 0.878325i $$-0.341339\pi$$
0.478063 + 0.878325i $$0.341339\pi$$
$$212$$ −3.70820 −0.254680
$$213$$ −8.23607 −0.564326
$$214$$ 14.4721 0.989295
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ 0 0
$$218$$ 26.5623 1.79903
$$219$$ 14.4721 0.977936
$$220$$ 0 0
$$221$$ −16.0000 −1.07628
$$222$$ 5.61803 0.377058
$$223$$ −3.41641 −0.228780 −0.114390 0.993436i $$-0.536491\pi$$
−0.114390 + 0.993436i $$0.536491\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 17.0344 1.13311
$$227$$ −7.41641 −0.492244 −0.246122 0.969239i $$-0.579156\pi$$
−0.246122 + 0.969239i $$0.579156\pi$$
$$228$$ 1.52786 0.101185
$$229$$ 28.9443 1.91269 0.956346 0.292238i $$-0.0943999\pi$$
0.956346 + 0.292238i $$0.0943999\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.70820 0.440415
$$233$$ 4.52786 0.296630 0.148315 0.988940i $$-0.452615\pi$$
0.148315 + 0.988940i $$0.452615\pi$$
$$234$$ 10.4721 0.684585
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 2.70820 0.175917
$$238$$ 0 0
$$239$$ −20.9443 −1.35477 −0.677386 0.735628i $$-0.736887\pi$$
−0.677386 + 0.735628i $$0.736887\pi$$
$$240$$ 0 0
$$241$$ 18.4721 1.18989 0.594947 0.803765i $$-0.297173\pi$$
0.594947 + 0.803765i $$0.297173\pi$$
$$242$$ −9.70820 −0.624067
$$243$$ −1.00000 −0.0641500
$$244$$ 7.41641 0.474787
$$245$$ 0 0
$$246$$ −14.4721 −0.922710
$$247$$ −16.0000 −1.01806
$$248$$ −8.94427 −0.567962
$$249$$ −16.9443 −1.07380
$$250$$ 0 0
$$251$$ −10.4721 −0.660995 −0.330498 0.943807i $$-0.607217\pi$$
−0.330498 + 0.943807i $$0.607217\pi$$
$$252$$ 0 0
$$253$$ −9.47214 −0.595508
$$254$$ −31.0344 −1.94727
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −8.94427 −0.557928 −0.278964 0.960302i $$-0.589991\pi$$
−0.278964 + 0.960302i $$0.589991\pi$$
$$258$$ −7.61803 −0.474278
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −3.00000 −0.185695
$$262$$ 21.8885 1.35228
$$263$$ −21.1803 −1.30604 −0.653018 0.757343i $$-0.726497\pi$$
−0.653018 + 0.757343i $$0.726497\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.52786 −0.0935038
$$268$$ −7.85410 −0.479766
$$269$$ −8.94427 −0.545342 −0.272671 0.962107i $$-0.587907\pi$$
−0.272671 + 0.962107i $$0.587907\pi$$
$$270$$ 0 0
$$271$$ −3.41641 −0.207532 −0.103766 0.994602i $$-0.533089\pi$$
−0.103766 + 0.994602i $$0.533089\pi$$
$$272$$ 12.0000 0.727607
$$273$$ 0 0
$$274$$ 32.1803 1.94409
$$275$$ 0 0
$$276$$ −2.61803 −0.157587
$$277$$ −15.8885 −0.954650 −0.477325 0.878727i $$-0.658394\pi$$
−0.477325 + 0.878727i $$0.658394\pi$$
$$278$$ −18.4721 −1.10789
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −19.3607 −1.15496 −0.577481 0.816404i $$-0.695964\pi$$
−0.577481 + 0.816404i $$0.695964\pi$$
$$282$$ −16.9443 −1.00902
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 5.09017 0.302046
$$285$$ 0 0
$$286$$ −23.4164 −1.38464
$$287$$ 0 0
$$288$$ −3.38197 −0.199284
$$289$$ −10.8885 −0.640503
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ −8.94427 −0.523424
$$293$$ −16.3607 −0.955801 −0.477901 0.878414i $$-0.658602\pi$$
−0.477901 + 0.878414i $$0.658602\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.76393 0.451269
$$297$$ 2.23607 0.129750
$$298$$ 20.8541 1.20805
$$299$$ 27.4164 1.58553
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 10.8541 0.624583
$$303$$ −9.52786 −0.547361
$$304$$ 12.0000 0.688247
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ −31.4164 −1.79303 −0.896515 0.443014i $$-0.853909\pi$$
−0.896515 + 0.443014i $$0.853909\pi$$
$$308$$ 0 0
$$309$$ −7.41641 −0.421905
$$310$$ 0 0
$$311$$ 20.3607 1.15455 0.577274 0.816550i $$-0.304116\pi$$
0.577274 + 0.816550i $$0.304116\pi$$
$$312$$ 14.4721 0.819323
$$313$$ −7.41641 −0.419200 −0.209600 0.977787i $$-0.567216\pi$$
−0.209600 + 0.977787i $$0.567216\pi$$
$$314$$ −1.52786 −0.0862224
$$315$$ 0 0
$$316$$ −1.67376 −0.0941565
$$317$$ −7.94427 −0.446195 −0.223097 0.974796i $$-0.571617\pi$$
−0.223097 + 0.974796i $$0.571617\pi$$
$$318$$ 9.70820 0.544409
$$319$$ 6.70820 0.375587
$$320$$ 0 0
$$321$$ −8.94427 −0.499221
$$322$$ 0 0
$$323$$ 6.11146 0.340051
$$324$$ 0.618034 0.0343352
$$325$$ 0 0
$$326$$ −19.4164 −1.07538
$$327$$ −16.4164 −0.907829
$$328$$ −20.0000 −1.10432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −33.1803 −1.82376 −0.911878 0.410461i $$-0.865368\pi$$
−0.911878 + 0.410461i $$0.865368\pi$$
$$332$$ 10.4721 0.574733
$$333$$ −3.47214 −0.190272
$$334$$ −36.3607 −1.98957
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 46.7426 2.54246
$$339$$ −10.5279 −0.571795
$$340$$ 0 0
$$341$$ −8.94427 −0.484359
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ −10.5279 −0.567624
$$345$$ 0 0
$$346$$ 4.00000 0.215041
$$347$$ 5.29180 0.284078 0.142039 0.989861i $$-0.454634\pi$$
0.142039 + 0.989861i $$0.454634\pi$$
$$348$$ 1.85410 0.0993903
$$349$$ 23.4164 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$350$$ 0 0
$$351$$ −6.47214 −0.345457
$$352$$ 7.56231 0.403072
$$353$$ 13.8885 0.739213 0.369606 0.929188i $$-0.379493\pi$$
0.369606 + 0.929188i $$0.379493\pi$$
$$354$$ −10.4721 −0.556588
$$355$$ 0 0
$$356$$ 0.944272 0.0500463
$$357$$ 0 0
$$358$$ −1.52786 −0.0807501
$$359$$ −11.7639 −0.620877 −0.310438 0.950594i $$-0.600476\pi$$
−0.310438 + 0.950594i $$0.600476\pi$$
$$360$$ 0 0
$$361$$ −12.8885 −0.678344
$$362$$ 10.4721 0.550403
$$363$$ 6.00000 0.314918
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −19.4164 −1.01491
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ −20.5623 −1.07188
$$369$$ 8.94427 0.465620
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −2.47214 −0.128174
$$373$$ −17.4721 −0.904673 −0.452336 0.891847i $$-0.649409\pi$$
−0.452336 + 0.891847i $$0.649409\pi$$
$$374$$ 8.94427 0.462497
$$375$$ 0 0
$$376$$ −23.4164 −1.20761
$$377$$ −19.4164 −0.999996
$$378$$ 0 0
$$379$$ 8.70820 0.447310 0.223655 0.974668i $$-0.428201\pi$$
0.223655 + 0.974668i $$0.428201\pi$$
$$380$$ 0 0
$$381$$ 19.1803 0.982639
$$382$$ 4.94427 0.252971
$$383$$ −19.4164 −0.992132 −0.496066 0.868285i $$-0.665223\pi$$
−0.496066 + 0.868285i $$0.665223\pi$$
$$384$$ −13.6180 −0.694942
$$385$$ 0 0
$$386$$ 37.0344 1.88500
$$387$$ 4.70820 0.239331
$$388$$ −2.47214 −0.125504
$$389$$ 12.0557 0.611250 0.305625 0.952152i $$-0.401135\pi$$
0.305625 + 0.952152i $$0.401135\pi$$
$$390$$ 0 0
$$391$$ −10.4721 −0.529599
$$392$$ 0 0
$$393$$ −13.5279 −0.682390
$$394$$ 33.9787 1.71182
$$395$$ 0 0
$$396$$ −1.38197 −0.0694464
$$397$$ 9.88854 0.496292 0.248146 0.968723i $$-0.420179\pi$$
0.248146 + 0.968723i $$0.420179\pi$$
$$398$$ −35.4164 −1.77526
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 28.4164 1.41905 0.709524 0.704681i $$-0.248910\pi$$
0.709524 + 0.704681i $$0.248910\pi$$
$$402$$ 20.5623 1.02555
$$403$$ 25.8885 1.28960
$$404$$ 5.88854 0.292966
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.76393 0.384844
$$408$$ −5.52786 −0.273670
$$409$$ 7.41641 0.366718 0.183359 0.983046i $$-0.441303\pi$$
0.183359 + 0.983046i $$0.441303\pi$$
$$410$$ 0 0
$$411$$ −19.8885 −0.981030
$$412$$ 4.58359 0.225817
$$413$$ 0 0
$$414$$ 6.85410 0.336861
$$415$$ 0 0
$$416$$ −21.8885 −1.07317
$$417$$ 11.4164 0.559064
$$418$$ 8.94427 0.437479
$$419$$ 29.8885 1.46015 0.730075 0.683367i $$-0.239485\pi$$
0.730075 + 0.683367i $$0.239485\pi$$
$$420$$ 0 0
$$421$$ 10.4164 0.507665 0.253832 0.967248i $$-0.418309\pi$$
0.253832 + 0.967248i $$0.418309\pi$$
$$422$$ 22.4721 1.09393
$$423$$ 10.4721 0.509173
$$424$$ 13.4164 0.651558
$$425$$ 0 0
$$426$$ −13.3262 −0.645658
$$427$$ 0 0
$$428$$ 5.52786 0.267199
$$429$$ 14.4721 0.698721
$$430$$ 0 0
$$431$$ −11.0557 −0.532536 −0.266268 0.963899i $$-0.585791\pi$$
−0.266268 + 0.963899i $$0.585791\pi$$
$$432$$ 4.85410 0.233543
$$433$$ −12.9443 −0.622062 −0.311031 0.950400i $$-0.600674\pi$$
−0.311031 + 0.950400i $$0.600674\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10.1459 0.485900
$$437$$ −10.4721 −0.500950
$$438$$ 23.4164 1.11888
$$439$$ 27.4164 1.30851 0.654257 0.756272i $$-0.272982\pi$$
0.654257 + 0.756272i $$0.272982\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −25.8885 −1.23139
$$443$$ 15.0557 0.715319 0.357660 0.933852i $$-0.383575\pi$$
0.357660 + 0.933852i $$0.383575\pi$$
$$444$$ 2.14590 0.101840
$$445$$ 0 0
$$446$$ −5.52786 −0.261752
$$447$$ −12.8885 −0.609607
$$448$$ 0 0
$$449$$ −6.52786 −0.308069 −0.154034 0.988065i $$-0.549227\pi$$
−0.154034 + 0.988065i $$0.549227\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 6.50658 0.306044
$$453$$ −6.70820 −0.315179
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −5.52786 −0.258866
$$457$$ 25.9443 1.21362 0.606811 0.794846i $$-0.292449\pi$$
0.606811 + 0.794846i $$0.292449\pi$$
$$458$$ 46.8328 2.18835
$$459$$ 2.47214 0.115389
$$460$$ 0 0
$$461$$ 30.4721 1.41923 0.709614 0.704590i $$-0.248869\pi$$
0.709614 + 0.704590i $$0.248869\pi$$
$$462$$ 0 0
$$463$$ 1.88854 0.0877681 0.0438840 0.999037i $$-0.486027\pi$$
0.0438840 + 0.999037i $$0.486027\pi$$
$$464$$ 14.5623 0.676038
$$465$$ 0 0
$$466$$ 7.32624 0.339381
$$467$$ 1.88854 0.0873914 0.0436957 0.999045i $$-0.486087\pi$$
0.0436957 + 0.999045i $$0.486087\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0.944272 0.0435098
$$472$$ −14.4721 −0.666134
$$473$$ −10.5279 −0.484072
$$474$$ 4.38197 0.201270
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ −33.8885 −1.55003
$$479$$ −30.4721 −1.39231 −0.696154 0.717893i $$-0.745107\pi$$
−0.696154 + 0.717893i $$0.745107\pi$$
$$480$$ 0 0
$$481$$ −22.4721 −1.02464
$$482$$ 29.8885 1.36139
$$483$$ 0 0
$$484$$ −3.70820 −0.168555
$$485$$ 0 0
$$486$$ −1.61803 −0.0733955
$$487$$ −14.7082 −0.666492 −0.333246 0.942840i $$-0.608144\pi$$
−0.333246 + 0.942840i $$0.608144\pi$$
$$488$$ −26.8328 −1.21466
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 4.81966 0.217508 0.108754 0.994069i $$-0.465314\pi$$
0.108754 + 0.994069i $$0.465314\pi$$
$$492$$ −5.52786 −0.249215
$$493$$ 7.41641 0.334018
$$494$$ −25.8885 −1.16478
$$495$$ 0 0
$$496$$ −19.4164 −0.871822
$$497$$ 0 0
$$498$$ −27.4164 −1.22856
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ 0 0
$$501$$ 22.4721 1.00398
$$502$$ −16.9443 −0.756260
$$503$$ −12.9443 −0.577157 −0.288578 0.957456i $$-0.593183\pi$$
−0.288578 + 0.957456i $$0.593183\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −15.3262 −0.681334
$$507$$ −28.8885 −1.28299
$$508$$ −11.8541 −0.525941
$$509$$ 9.52786 0.422315 0.211158 0.977452i $$-0.432277\pi$$
0.211158 + 0.977452i $$0.432277\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −5.29180 −0.233867
$$513$$ 2.47214 0.109147
$$514$$ −14.4721 −0.638339
$$515$$ 0 0
$$516$$ −2.90983 −0.128098
$$517$$ −23.4164 −1.02985
$$518$$ 0 0
$$519$$ −2.47214 −0.108515
$$520$$ 0 0
$$521$$ −20.9443 −0.917585 −0.458793 0.888543i $$-0.651718\pi$$
−0.458793 + 0.888543i $$0.651718\pi$$
$$522$$ −4.85410 −0.212458
$$523$$ −1.88854 −0.0825803 −0.0412901 0.999147i $$-0.513147\pi$$
−0.0412901 + 0.999147i $$0.513147\pi$$
$$524$$ 8.36068 0.365238
$$525$$ 0 0
$$526$$ −34.2705 −1.49427
$$527$$ −9.88854 −0.430752
$$528$$ −10.8541 −0.472364
$$529$$ −5.05573 −0.219814
$$530$$ 0 0
$$531$$ 6.47214 0.280867
$$532$$ 0 0
$$533$$ 57.8885 2.50743
$$534$$ −2.47214 −0.106980
$$535$$ 0 0
$$536$$ 28.4164 1.22740
$$537$$ 0.944272 0.0407483
$$538$$ −14.4721 −0.623938
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −28.4164 −1.22172 −0.610858 0.791740i $$-0.709176\pi$$
−0.610858 + 0.791740i $$0.709176\pi$$
$$542$$ −5.52786 −0.237442
$$543$$ −6.47214 −0.277746
$$544$$ 8.36068 0.358461
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −14.8197 −0.633643 −0.316821 0.948485i $$-0.602616\pi$$
−0.316821 + 0.948485i $$0.602616\pi$$
$$548$$ 12.2918 0.525080
$$549$$ 12.0000 0.512148
$$550$$ 0 0
$$551$$ 7.41641 0.315950
$$552$$ 9.47214 0.403161
$$553$$ 0 0
$$554$$ −25.7082 −1.09224
$$555$$ 0 0
$$556$$ −7.05573 −0.299230
$$557$$ −20.8885 −0.885076 −0.442538 0.896750i $$-0.645922\pi$$
−0.442538 + 0.896750i $$0.645922\pi$$
$$558$$ 6.47214 0.273987
$$559$$ 30.4721 1.28883
$$560$$ 0 0
$$561$$ −5.52786 −0.233387
$$562$$ −31.3262 −1.32142
$$563$$ −4.94427 −0.208376 −0.104188 0.994558i $$-0.533224\pi$$
−0.104188 + 0.994558i $$0.533224\pi$$
$$564$$ −6.47214 −0.272526
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ −18.4164 −0.772735
$$569$$ −36.3050 −1.52198 −0.760991 0.648762i $$-0.775287\pi$$
−0.760991 + 0.648762i $$0.775287\pi$$
$$570$$ 0 0
$$571$$ 11.7639 0.492305 0.246153 0.969231i $$-0.420834\pi$$
0.246153 + 0.969231i $$0.420834\pi$$
$$572$$ −8.94427 −0.373979
$$573$$ −3.05573 −0.127655
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 4.23607 0.176503
$$577$$ 0.583592 0.0242953 0.0121476 0.999926i $$-0.496133\pi$$
0.0121476 + 0.999926i $$0.496133\pi$$
$$578$$ −17.6180 −0.732814
$$579$$ −22.8885 −0.951215
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 6.47214 0.268279
$$583$$ 13.4164 0.555651
$$584$$ 32.3607 1.33909
$$585$$ 0 0
$$586$$ −26.4721 −1.09355
$$587$$ −14.4721 −0.597329 −0.298664 0.954358i $$-0.596541\pi$$
−0.298664 + 0.954358i $$0.596541\pi$$
$$588$$ 0 0
$$589$$ −9.88854 −0.407450
$$590$$ 0 0
$$591$$ −21.0000 −0.863825
$$592$$ 16.8541 0.692699
$$593$$ 44.9443 1.84564 0.922820 0.385231i $$-0.125878\pi$$
0.922820 + 0.385231i $$0.125878\pi$$
$$594$$ 3.61803 0.148450
$$595$$ 0 0
$$596$$ 7.96556 0.326282
$$597$$ 21.8885 0.895838
$$598$$ 44.3607 1.81404
$$599$$ 12.7082 0.519243 0.259622 0.965710i $$-0.416402\pi$$
0.259622 + 0.965710i $$0.416402\pi$$
$$600$$ 0 0
$$601$$ 36.9443 1.50699 0.753494 0.657455i $$-0.228367\pi$$
0.753494 + 0.657455i $$0.228367\pi$$
$$602$$ 0 0
$$603$$ −12.7082 −0.517518
$$604$$ 4.14590 0.168694
$$605$$ 0 0
$$606$$ −15.4164 −0.626249
$$607$$ −35.4164 −1.43751 −0.718754 0.695265i $$-0.755287\pi$$
−0.718754 + 0.695265i $$0.755287\pi$$
$$608$$ 8.36068 0.339070
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 67.7771 2.74197
$$612$$ −1.52786 −0.0617602
$$613$$ −35.2492 −1.42370 −0.711851 0.702330i $$-0.752143\pi$$
−0.711851 + 0.702330i $$0.752143\pi$$
$$614$$ −50.8328 −2.05145
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 34.4164 1.38555 0.692776 0.721153i $$-0.256387\pi$$
0.692776 + 0.721153i $$0.256387\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ −17.5279 −0.704504 −0.352252 0.935905i $$-0.614584\pi$$
−0.352252 + 0.935905i $$0.614584\pi$$
$$620$$ 0 0
$$621$$ −4.23607 −0.169988
$$622$$ 32.9443 1.32094
$$623$$ 0 0
$$624$$ 31.4164 1.25766
$$625$$ 0 0
$$626$$ −12.0000 −0.479616
$$627$$ −5.52786 −0.220762
$$628$$ −0.583592 −0.0232879
$$629$$ 8.58359 0.342250
$$630$$ 0 0
$$631$$ 0.347524 0.0138347 0.00691736 0.999976i $$-0.497798\pi$$
0.00691736 + 0.999976i $$0.497798\pi$$
$$632$$ 6.05573 0.240884
$$633$$ −13.8885 −0.552020
$$634$$ −12.8541 −0.510502
$$635$$ 0 0
$$636$$ 3.70820 0.147040
$$637$$ 0 0
$$638$$ 10.8541 0.429718
$$639$$ 8.23607 0.325814
$$640$$ 0 0
$$641$$ 25.3607 1.00169 0.500843 0.865538i $$-0.333023\pi$$
0.500843 + 0.865538i $$0.333023\pi$$
$$642$$ −14.4721 −0.571170
$$643$$ 24.0000 0.946468 0.473234 0.880937i $$-0.343087\pi$$
0.473234 + 0.880937i $$0.343087\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 9.88854 0.389060
$$647$$ 1.88854 0.0742463 0.0371232 0.999311i $$-0.488181\pi$$
0.0371232 + 0.999311i $$0.488181\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ −14.4721 −0.568081
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −7.41641 −0.290449
$$653$$ −11.8885 −0.465235 −0.232617 0.972568i $$-0.574729\pi$$
−0.232617 + 0.972568i $$0.574729\pi$$
$$654$$ −26.5623 −1.03867
$$655$$ 0 0
$$656$$ −43.4164 −1.69513
$$657$$ −14.4721 −0.564612
$$658$$ 0 0
$$659$$ −15.0557 −0.586488 −0.293244 0.956038i $$-0.594735\pi$$
−0.293244 + 0.956038i $$0.594735\pi$$
$$660$$ 0 0
$$661$$ 20.3607 0.791939 0.395969 0.918264i $$-0.370409\pi$$
0.395969 + 0.918264i $$0.370409\pi$$
$$662$$ −53.6869 −2.08660
$$663$$ 16.0000 0.621389
$$664$$ −37.8885 −1.47036
$$665$$ 0 0
$$666$$ −5.61803 −0.217694
$$667$$ −12.7082 −0.492064
$$668$$ −13.8885 −0.537364
$$669$$ 3.41641 0.132086
$$670$$ 0 0
$$671$$ −26.8328 −1.03587
$$672$$ 0 0
$$673$$ −30.0000 −1.15642 −0.578208 0.815890i $$-0.696248\pi$$
−0.578208 + 0.815890i $$0.696248\pi$$
$$674$$ 3.23607 0.124649
$$675$$ 0 0
$$676$$ 17.8541 0.686696
$$677$$ −49.8885 −1.91737 −0.958686 0.284466i $$-0.908184\pi$$
−0.958686 + 0.284466i $$0.908184\pi$$
$$678$$ −17.0344 −0.654204
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 7.41641 0.284197
$$682$$ −14.4721 −0.554167
$$683$$ 27.1803 1.04003 0.520013 0.854158i $$-0.325927\pi$$
0.520013 + 0.854158i $$0.325927\pi$$
$$684$$ −1.52786 −0.0584193
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −28.9443 −1.10429
$$688$$ −22.8541 −0.871304
$$689$$ −38.8328 −1.47941
$$690$$ 0 0
$$691$$ 10.8328 0.412100 0.206050 0.978541i $$-0.433939\pi$$
0.206050 + 0.978541i $$0.433939\pi$$
$$692$$ 1.52786 0.0580807
$$693$$ 0 0
$$694$$ 8.56231 0.325021
$$695$$ 0 0
$$696$$ −6.70820 −0.254274
$$697$$ −22.1115 −0.837531
$$698$$ 37.8885 1.43410
$$699$$ −4.52786 −0.171260
$$700$$ 0 0
$$701$$ 37.7771 1.42682 0.713410 0.700746i $$-0.247150\pi$$
0.713410 + 0.700746i $$0.247150\pi$$
$$702$$ −10.4721 −0.395245
$$703$$ 8.58359 0.323736
$$704$$ −9.47214 −0.356995
$$705$$ 0 0
$$706$$ 22.4721 0.845750
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −2.70820 −0.101566
$$712$$ −3.41641 −0.128035
$$713$$ 16.9443 0.634568
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −0.583592 −0.0218099
$$717$$ 20.9443 0.782178
$$718$$ −19.0344 −0.710359
$$719$$ −0.944272 −0.0352154 −0.0176077 0.999845i $$-0.505605\pi$$
−0.0176077 + 0.999845i $$0.505605\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −20.8541 −0.776109
$$723$$ −18.4721 −0.686986
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 9.70820 0.360305
$$727$$ 17.3050 0.641805 0.320903 0.947112i $$-0.396014\pi$$
0.320903 + 0.947112i $$0.396014\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −11.6393 −0.430496
$$732$$ −7.41641 −0.274118
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ −6.47214 −0.238891
$$735$$ 0 0
$$736$$ −14.3262 −0.528072
$$737$$ 28.4164 1.04673
$$738$$ 14.4721 0.532727
$$739$$ −20.7082 −0.761764 −0.380882 0.924624i $$-0.624380\pi$$
−0.380882 + 0.924624i $$0.624380\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ 3.05573 0.112104 0.0560519 0.998428i $$-0.482149\pi$$
0.0560519 + 0.998428i $$0.482149\pi$$
$$744$$ 8.94427 0.327913
$$745$$ 0 0
$$746$$ −28.2705 −1.03506
$$747$$ 16.9443 0.619958
$$748$$ 3.41641 0.124916
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −19.7771 −0.721676 −0.360838 0.932628i $$-0.617509\pi$$
−0.360838 + 0.932628i $$0.617509\pi$$
$$752$$ −50.8328 −1.85368
$$753$$ 10.4721 0.381626
$$754$$ −31.4164 −1.14412
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 5.47214 0.198888 0.0994441 0.995043i $$-0.468294\pi$$
0.0994441 + 0.995043i $$0.468294\pi$$
$$758$$ 14.0902 0.511778
$$759$$ 9.47214 0.343817
$$760$$ 0 0
$$761$$ −52.9443 −1.91923 −0.959614 0.281319i $$-0.909228\pi$$
−0.959614 + 0.281319i $$0.909228\pi$$
$$762$$ 31.0344 1.12426
$$763$$ 0 0
$$764$$ 1.88854 0.0683251
$$765$$ 0 0
$$766$$ −31.4164 −1.13512
$$767$$ 41.8885 1.51251
$$768$$ −13.5623 −0.489388
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ 8.94427 0.322120
$$772$$ 14.1459 0.509122
$$773$$ −39.7771 −1.43068 −0.715341 0.698775i $$-0.753729\pi$$
−0.715341 + 0.698775i $$0.753729\pi$$
$$774$$ 7.61803 0.273825
$$775$$ 0 0
$$776$$ 8.94427 0.321081
$$777$$ 0 0
$$778$$ 19.5066 0.699345
$$779$$ −22.1115 −0.792225
$$780$$ 0 0
$$781$$ −18.4164 −0.658991
$$782$$ −16.9443 −0.605926
$$783$$ 3.00000 0.107211
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −21.8885 −0.780739
$$787$$ −33.5279 −1.19514 −0.597570 0.801817i $$-0.703867\pi$$
−0.597570 + 0.801817i $$0.703867\pi$$
$$788$$ 12.9787 0.462348
$$789$$ 21.1803 0.754040
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 5.00000 0.177667
$$793$$ 77.6656 2.75799
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ −13.5279 −0.479482
$$797$$ 34.4721 1.22107 0.610533 0.791991i $$-0.290955\pi$$
0.610533 + 0.791991i $$0.290955\pi$$
$$798$$ 0 0
$$799$$ −25.8885 −0.915871
$$800$$ 0 0
$$801$$ 1.52786 0.0539844
$$802$$ 45.9787 1.62356
$$803$$ 32.3607 1.14198
$$804$$ 7.85410 0.276993
$$805$$ 0 0
$$806$$ 41.8885 1.47546
$$807$$ 8.94427 0.314853
$$808$$ −21.3050 −0.749506
$$809$$ 8.52786 0.299824 0.149912 0.988699i $$-0.452101\pi$$
0.149912 + 0.988699i $$0.452101\pi$$
$$810$$ 0 0
$$811$$ 38.8328 1.36360 0.681802 0.731536i $$-0.261196\pi$$
0.681802 + 0.731536i $$0.261196\pi$$
$$812$$ 0 0
$$813$$ 3.41641 0.119819
$$814$$ 12.5623 0.440309
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ −11.6393 −0.407208
$$818$$ 12.0000 0.419570
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −45.7771 −1.59763 −0.798816 0.601576i $$-0.794540\pi$$
−0.798816 + 0.601576i $$0.794540\pi$$
$$822$$ −32.1803 −1.12242
$$823$$ 17.5410 0.611442 0.305721 0.952121i $$-0.401103\pi$$
0.305721 + 0.952121i $$0.401103\pi$$
$$824$$ −16.5836 −0.577717
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14.2361 0.495037 0.247518 0.968883i $$-0.420385\pi$$
0.247518 + 0.968883i $$0.420385\pi$$
$$828$$ 2.61803 0.0909830
$$829$$ −2.47214 −0.0858608 −0.0429304 0.999078i $$-0.513669\pi$$
−0.0429304 + 0.999078i $$0.513669\pi$$
$$830$$ 0 0
$$831$$ 15.8885 0.551167
$$832$$ 27.4164 0.950493
$$833$$ 0 0
$$834$$ 18.4721 0.639638
$$835$$ 0 0
$$836$$ 3.41641 0.118159
$$837$$ −4.00000 −0.138260
$$838$$ 48.3607 1.67059
$$839$$ −55.1935 −1.90549 −0.952746 0.303770i $$-0.901755\pi$$
−0.952746 + 0.303770i $$0.901755\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 16.8541 0.580831
$$843$$ 19.3607 0.666817
$$844$$ 8.58359 0.295459
$$845$$ 0 0
$$846$$ 16.9443 0.582556
$$847$$ 0 0
$$848$$ 29.1246 1.00014
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −14.7082 −0.504191
$$852$$ −5.09017 −0.174386
$$853$$ 6.83282 0.233951 0.116976 0.993135i $$-0.462680\pi$$
0.116976 + 0.993135i $$0.462680\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ 37.8885 1.29425 0.647124 0.762385i $$-0.275972\pi$$
0.647124 + 0.762385i $$0.275972\pi$$
$$858$$ 23.4164 0.799423
$$859$$ −25.8885 −0.883306 −0.441653 0.897186i $$-0.645608\pi$$
−0.441653 + 0.897186i $$0.645608\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −17.8885 −0.609286
$$863$$ −2.12461 −0.0723226 −0.0361613 0.999346i $$-0.511513\pi$$
−0.0361613 + 0.999346i $$0.511513\pi$$
$$864$$ 3.38197 0.115057
$$865$$ 0 0
$$866$$ −20.9443 −0.711715
$$867$$ 10.8885 0.369794
$$868$$ 0 0
$$869$$ 6.05573 0.205427
$$870$$ 0 0
$$871$$ −82.2492 −2.78691
$$872$$ −36.7082 −1.24310
$$873$$ −4.00000 −0.135379
$$874$$ −16.9443 −0.573149
$$875$$ 0 0
$$876$$ 8.94427 0.302199
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ 44.3607 1.49710
$$879$$ 16.3607 0.551832
$$880$$ 0 0
$$881$$ 40.9443 1.37945 0.689724 0.724073i $$-0.257732\pi$$
0.689724 + 0.724073i $$0.257732\pi$$
$$882$$ 0 0
$$883$$ −43.5410 −1.46527 −0.732636 0.680621i $$-0.761710\pi$$
−0.732636 + 0.680621i $$0.761710\pi$$
$$884$$ −9.88854 −0.332588
$$885$$ 0 0
$$886$$ 24.3607 0.818413
$$887$$ 26.8328 0.900958 0.450479 0.892787i $$-0.351253\pi$$
0.450479 + 0.892787i $$0.351253\pi$$
$$888$$ −7.76393 −0.260540
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.23607 −0.0749111
$$892$$ −2.11146 −0.0706968
$$893$$ −25.8885 −0.866327
$$894$$ −20.8541 −0.697466
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −27.4164 −0.915407
$$898$$ −10.5623 −0.352469
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 14.8328 0.494153
$$902$$ −32.3607 −1.07749
$$903$$ 0 0
$$904$$ −23.5410 −0.782963
$$905$$ 0 0
$$906$$ −10.8541 −0.360603
$$907$$ 45.8885 1.52370 0.761852 0.647751i $$-0.224290\pi$$
0.761852 + 0.647751i $$0.224290\pi$$
$$908$$ −4.58359 −0.152112
$$909$$ 9.52786 0.316019
$$910$$ 0 0
$$911$$ −56.0132 −1.85580 −0.927899 0.372831i $$-0.878387\pi$$
−0.927899 + 0.372831i $$0.878387\pi$$
$$912$$ −12.0000 −0.397360
$$913$$ −37.8885 −1.25393
$$914$$ 41.9787 1.38853
$$915$$ 0 0
$$916$$ 17.8885 0.591054
$$917$$ 0 0
$$918$$ 4.00000 0.132020
$$919$$ 15.1803 0.500753 0.250377 0.968149i $$-0.419446\pi$$
0.250377 + 0.968149i $$0.419446\pi$$
$$920$$ 0 0
$$921$$ 31.4164 1.03521
$$922$$ 49.3050 1.62377
$$923$$ 53.3050 1.75455
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 3.05573 0.100417
$$927$$ 7.41641 0.243587
$$928$$ 10.1459 0.333055
$$929$$ −4.58359 −0.150383 −0.0751914 0.997169i $$-0.523957\pi$$
−0.0751914 + 0.997169i $$0.523957\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2.79837 0.0916638
$$933$$ −20.3607 −0.666579
$$934$$ 3.05573 0.0999865
$$935$$ 0 0
$$936$$ −14.4721 −0.473037
$$937$$ 28.0000 0.914720 0.457360 0.889282i $$-0.348795\pi$$
0.457360 + 0.889282i $$0.348795\pi$$
$$938$$ 0 0
$$939$$ 7.41641 0.242025
$$940$$ 0 0
$$941$$ −31.7771 −1.03590 −0.517952 0.855410i $$-0.673305\pi$$
−0.517952 + 0.855410i $$0.673305\pi$$
$$942$$ 1.52786 0.0497805
$$943$$ 37.8885 1.23382
$$944$$ −31.4164 −1.02252
$$945$$ 0 0
$$946$$ −17.0344 −0.553837
$$947$$ −32.9443 −1.07054 −0.535272 0.844679i $$-0.679791\pi$$
−0.535272 + 0.844679i $$0.679791\pi$$
$$948$$ 1.67376 0.0543613
$$949$$ −93.6656 −3.04052
$$950$$ 0 0
$$951$$ 7.94427 0.257611
$$952$$ 0 0
$$953$$ −11.3607 −0.368009 −0.184004 0.982925i $$-0.558906\pi$$
−0.184004 + 0.982925i $$0.558906\pi$$
$$954$$ −9.70820 −0.314315
$$955$$ 0 0
$$956$$ −12.9443 −0.418648
$$957$$ −6.70820 −0.216845
$$958$$ −49.3050 −1.59297
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −36.3607 −1.17232
$$963$$ 8.94427 0.288225
$$964$$ 11.4164 0.367698
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43.7771 1.40778 0.703888 0.710311i $$-0.251446\pi$$
0.703888 + 0.710311i $$0.251446\pi$$
$$968$$ 13.4164 0.431220
$$969$$ −6.11146 −0.196328
$$970$$ 0 0
$$971$$ 41.3050 1.32554 0.662769 0.748823i $$-0.269381\pi$$
0.662769 + 0.748823i $$0.269381\pi$$
$$972$$ −0.618034 −0.0198234
$$973$$ 0 0
$$974$$ −23.7984 −0.762549
$$975$$ 0 0
$$976$$ −58.2492 −1.86451
$$977$$ 7.58359 0.242621 0.121310 0.992615i $$-0.461290\pi$$
0.121310 + 0.992615i $$0.461290\pi$$
$$978$$ 19.4164 0.620868
$$979$$ −3.41641 −0.109189
$$980$$ 0 0
$$981$$ 16.4164 0.524136
$$982$$ 7.79837 0.248856
$$983$$ 23.0557 0.735364 0.367682 0.929952i $$-0.380152\pi$$
0.367682 + 0.929952i $$0.380152\pi$$
$$984$$ 20.0000 0.637577
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ −9.88854 −0.314596
$$989$$ 19.9443 0.634191
$$990$$ 0 0
$$991$$ −57.5410 −1.82785 −0.913925 0.405882i $$-0.866964\pi$$
−0.913925 + 0.405882i $$0.866964\pi$$
$$992$$ −13.5279 −0.429510
$$993$$ 33.1803 1.05295
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −10.4721 −0.331822
$$997$$ 7.41641 0.234880 0.117440 0.993080i $$-0.462531\pi$$
0.117440 + 0.993080i $$0.462531\pi$$
$$998$$ −19.4164 −0.614616
$$999$$ 3.47214 0.109854
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 3675.2.a.ba.1.2 yes 2
5.4 even 2 3675.2.a.v.1.1 yes 2
7.6 odd 2 3675.2.a.bc.1.2 yes 2
35.34 odd 2 3675.2.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3675.2.a.u.1.1 2 35.34 odd 2
3675.2.a.v.1.1 yes 2 5.4 even 2
3675.2.a.ba.1.2 yes 2 1.1 even 1 trivial
3675.2.a.bc.1.2 yes 2 7.6 odd 2