# Properties

 Label 4800.2.a.bp.1.1 Level $4800$ Weight $2$ Character 4800.1 Self dual yes Analytic conductor $38.328$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -3.00000 q^{13} -6.00000 q^{17} +7.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} +1.00000 q^{27} +2.00000 q^{29} -5.00000 q^{31} -2.00000 q^{33} +10.0000 q^{37} -3.00000 q^{39} +12.0000 q^{41} +3.00000 q^{43} +10.0000 q^{47} +2.00000 q^{49} -6.00000 q^{51} +7.00000 q^{57} +6.00000 q^{59} +13.0000 q^{61} -3.00000 q^{63} +7.00000 q^{67} -6.00000 q^{69} -4.00000 q^{71} +6.00000 q^{73} +6.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +2.00000 q^{87} +16.0000 q^{89} +9.00000 q^{91} -5.00000 q^{93} +7.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −3.00000 −0.832050 −0.416025 0.909353i $$-0.636577\pi$$
−0.416025 + 0.909353i $$0.636577\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ 3.00000 0.457496 0.228748 0.973486i $$-0.426537\pi$$
0.228748 + 0.973486i $$0.426537\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 7.00000 0.927173
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 9.00000 0.943456
$$92$$ 0 0
$$93$$ −5.00000 −0.518476
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 12.0000 1.18240 0.591198 0.806527i $$-0.298655\pi$$
0.591198 + 0.806527i $$0.298655\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −9.00000 −0.862044 −0.431022 0.902342i $$-0.641847\pi$$
−0.431022 + 0.902342i $$0.641847\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ −12.0000 −1.12887 −0.564433 0.825479i $$-0.690905\pi$$
−0.564433 + 0.825479i $$0.690905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −3.00000 −0.277350
$$118$$ 0 0
$$119$$ 18.0000 1.65006
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 3.00000 0.264135
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −21.0000 −1.82093
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ 1.00000 0.0813788 0.0406894 0.999172i $$-0.487045\pi$$
0.0406894 + 0.999172i $$0.487045\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.00000 −0.718278 −0.359139 0.933284i $$-0.616930\pi$$
−0.359139 + 0.933284i $$0.616930\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ 7.00000 0.535303
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000 0.450988
$$178$$ 0 0
$$179$$ −18.0000 −1.34538 −0.672692 0.739923i $$-0.734862\pi$$
−0.672692 + 0.739923i $$0.734862\pi$$
$$180$$ 0 0
$$181$$ 19.0000 1.41226 0.706129 0.708083i $$-0.250440\pi$$
0.706129 + 0.708083i $$0.250440\pi$$
$$182$$ 0 0
$$183$$ 13.0000 0.960988
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ −3.00000 −0.212664 −0.106332 0.994331i $$-0.533911\pi$$
−0.106332 + 0.994331i $$0.533911\pi$$
$$200$$ 0 0
$$201$$ 7.00000 0.493742
$$202$$ 0 0
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −14.0000 −0.968400
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 0 0
$$213$$ −4.00000 −0.274075
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 15.0000 1.01827
$$218$$ 0 0
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 18.0000 1.21081
$$222$$ 0 0
$$223$$ −11.0000 −0.736614 −0.368307 0.929704i $$-0.620063\pi$$
−0.368307 + 0.929704i $$0.620063\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 0 0
$$233$$ 4.00000 0.262049 0.131024 0.991379i $$-0.458173\pi$$
0.131024 + 0.991379i $$0.458173\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −21.0000 −1.33620
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −4.00000 −0.249513 −0.124757 0.992187i $$-0.539815\pi$$
−0.124757 + 0.992187i $$0.539815\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000 0.979184
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 0 0
$$273$$ 9.00000 0.544705
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ 0 0
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 5.00000 0.297219 0.148610 0.988896i $$-0.452520\pi$$
0.148610 + 0.988896i $$0.452520\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −36.0000 −2.12501
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ 18.0000 1.04097
$$300$$ 0 0
$$301$$ −9.00000 −0.518751
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5.00000 0.285365 0.142683 0.989769i $$-0.454427\pi$$
0.142683 + 0.989769i $$0.454427\pi$$
$$308$$ 0 0
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 2.00000 0.113410 0.0567048 0.998391i $$-0.481941\pi$$
0.0567048 + 0.998391i $$0.481941\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 32.0000 1.79730 0.898650 0.438667i $$-0.144549\pi$$
0.898650 + 0.438667i $$0.144549\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ −16.0000 −0.893033
$$322$$ 0 0
$$323$$ −42.0000 −2.33694
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −9.00000 −0.497701
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7.00000 0.381314 0.190657 0.981657i $$-0.438938\pi$$
0.190657 + 0.981657i $$0.438938\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 0 0
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ −3.00000 −0.160128
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 18.0000 0.952661
$$358$$ 0 0
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11.0000 0.574195 0.287098 0.957901i $$-0.407310\pi$$
0.287098 + 0.957901i $$0.407310\pi$$
$$368$$ 0 0
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7.00000 0.362446 0.181223 0.983442i $$-0.441994\pi$$
0.181223 + 0.983442i $$0.441994\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ 29.0000 1.48963 0.744815 0.667271i $$-0.232538\pi$$
0.744815 + 0.667271i $$0.232538\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.00000 0.152499
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 36.0000 1.82060
$$392$$ 0 0
$$393$$ 8.00000 0.403547
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 11.0000 0.552074 0.276037 0.961147i $$-0.410979\pi$$
0.276037 + 0.961147i $$0.410979\pi$$
$$398$$ 0 0
$$399$$ −21.0000 −1.05131
$$400$$ 0 0
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 0 0
$$403$$ 15.0000 0.747203
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ 5.00000 0.247234 0.123617 0.992330i $$-0.460551\pi$$
0.123617 + 0.992330i $$0.460551\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 0 0
$$413$$ −18.0000 −0.885722
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 0 0
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −39.0000 −1.88734
$$428$$ 0 0
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ 0 0
$$433$$ −3.00000 −0.144171 −0.0720854 0.997398i $$-0.522965\pi$$
−0.0720854 + 0.997398i $$0.522965\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −42.0000 −2.00913
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 22.0000 1.04056
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 1.00000 0.0469841
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 26.0000 1.20314 0.601568 0.798821i $$-0.294543\pi$$
0.601568 + 0.798821i $$0.294543\pi$$
$$468$$ 0 0
$$469$$ −21.0000 −0.969690
$$470$$ 0 0
$$471$$ −9.00000 −0.414698
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ −30.0000 −1.36788
$$482$$ 0 0
$$483$$ 18.0000 0.819028
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11.0000 0.498458 0.249229 0.968445i $$-0.419823\pi$$
0.249229 + 0.968445i $$0.419823\pi$$
$$488$$ 0 0
$$489$$ 1.00000 0.0452216
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 0 0
$$503$$ 4.00000 0.178351 0.0891756 0.996016i $$-0.471577\pi$$
0.0891756 + 0.996016i $$0.471577\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −4.00000 −0.177646
$$508$$ 0 0
$$509$$ −22.0000 −0.975133 −0.487566 0.873086i $$-0.662115\pi$$
−0.487566 + 0.873086i $$0.662115\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ 0 0
$$513$$ 7.00000 0.309058
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −20.0000 −0.879599
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 30.0000 1.30682
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ −36.0000 −1.55933
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −18.0000 −0.776757
$$538$$ 0 0
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ 19.0000 0.815368
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 13.0000 0.554826
$$550$$ 0 0
$$551$$ 14.0000 0.596420
$$552$$ 0 0
$$553$$ 24.0000 1.02058
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 39.0000 1.63210 0.816050 0.577982i $$-0.196160\pi$$
0.816050 + 0.577982i $$0.196160\pi$$
$$572$$ 0 0
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −11.0000 −0.457936 −0.228968 0.973434i $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ 0 0
$$579$$ −19.0000 −0.789613
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 0 0
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ 0 0
$$593$$ 36.0000 1.47834 0.739171 0.673517i $$-0.235217\pi$$
0.739171 + 0.673517i $$0.235217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −3.00000 −0.122782
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 0 0
$$603$$ 7.00000 0.285062
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 40.0000 1.61034 0.805170 0.593045i $$-0.202074\pi$$
0.805170 + 0.593045i $$0.202074\pi$$
$$618$$ 0 0
$$619$$ −5.00000 −0.200967 −0.100483 0.994939i $$-0.532039\pi$$
−0.100483 + 0.994939i $$0.532039\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ −48.0000 −1.92308
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −14.0000 −0.559106
$$628$$ 0 0
$$629$$ −60.0000 −2.39236
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 0 0
$$633$$ −9.00000 −0.357718
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ 15.0000 0.587896
$$652$$ 0 0
$$653$$ 26.0000 1.01746 0.508729 0.860927i $$-0.330115\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ 18.0000 0.699062
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ −11.0000 −0.425285
$$670$$ 0 0
$$671$$ −26.0000 −1.00372
$$672$$ 0 0
$$673$$ 50.0000 1.92736 0.963679 0.267063i $$-0.0860531\pi$$
0.963679 + 0.267063i $$0.0860531\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −10.0000 −0.384331 −0.192166 0.981363i $$-0.561551\pi$$
−0.192166 + 0.981363i $$0.561551\pi$$
$$678$$ 0 0
$$679$$ −21.0000 −0.805906
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 40.0000 1.53056 0.765279 0.643699i $$-0.222601\pi$$
0.765279 + 0.643699i $$0.222601\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 19.0000 0.724895
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −72.0000 −2.72719
$$698$$ 0 0
$$699$$ 4.00000 0.151294
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 70.0000 2.64010
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 30.0000 1.12351
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ −36.0000 −1.34071
$$722$$ 0 0
$$723$$ 25.0000 0.929760
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.00000 0.185440 0.0927199 0.995692i $$-0.470444\pi$$
0.0927199 + 0.995692i $$0.470444\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −14.0000 −0.515697
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −21.0000 −0.771454
$$742$$ 0 0
$$743$$ −20.0000 −0.733729 −0.366864 0.930274i $$-0.619569\pi$$
−0.366864 + 0.930274i $$0.619569\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ −28.0000 −1.02038
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 13.0000 0.472493 0.236247 0.971693i $$-0.424083\pi$$
0.236247 + 0.971693i $$0.424083\pi$$
$$758$$ 0 0
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −52.0000 −1.88500 −0.942499 0.334208i $$-0.891531\pi$$
−0.942499 + 0.334208i $$0.891531\pi$$
$$762$$ 0 0
$$763$$ 27.0000 0.977466
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −18.0000 −0.649942
$$768$$ 0 0
$$769$$ −21.0000 −0.757279 −0.378640 0.925544i $$-0.623608\pi$$
−0.378640 + 0.925544i $$0.623608\pi$$
$$770$$ 0 0
$$771$$ −4.00000 −0.144056
$$772$$ 0 0
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −30.0000 −1.07624
$$778$$ 0 0
$$779$$ 84.0000 3.00961
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 37.0000 1.31891 0.659454 0.751745i $$-0.270788\pi$$
0.659454 + 0.751745i $$0.270788\pi$$
$$788$$ 0 0
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 36.0000 1.28001
$$792$$ 0 0
$$793$$ −39.0000 −1.38493
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −20.0000 −0.708436 −0.354218 0.935163i $$-0.615253\pi$$
−0.354218 + 0.935163i $$0.615253\pi$$
$$798$$ 0 0
$$799$$ −60.0000 −2.12265
$$800$$ 0 0
$$801$$ 16.0000 0.565332
$$802$$ 0 0
$$803$$ −12.0000 −0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ 48.0000 1.68759 0.843795 0.536666i $$-0.180316\pi$$
0.843795 + 0.536666i $$0.180316\pi$$
$$810$$ 0 0
$$811$$ −17.0000 −0.596951 −0.298475 0.954417i $$-0.596478\pi$$
−0.298475 + 0.954417i $$0.596478\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 21.0000 0.734697
$$818$$ 0 0
$$819$$ 9.00000 0.314485
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 9.00000 0.313720 0.156860 0.987621i $$-0.449863\pi$$
0.156860 + 0.987621i $$0.449863\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 1.00000 0.0346896
$$832$$ 0 0
$$833$$ −12.0000 −0.415775
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −5.00000 −0.172825
$$838$$ 0 0
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 2.00000 0.0688837
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000 0.721569
$$848$$ 0 0
$$849$$ 5.00000 0.171600
$$850$$ 0 0
$$851$$ −60.0000 −2.05677
$$852$$ 0 0
$$853$$ −25.0000 −0.855984 −0.427992 0.903783i $$-0.640779\pi$$
−0.427992 + 0.903783i $$0.640779\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.00000 0.136637 0.0683187 0.997664i $$-0.478237\pi$$
0.0683187 + 0.997664i $$0.478237\pi$$
$$858$$ 0 0
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ 0 0
$$861$$ −36.0000 −1.22688
$$862$$ 0 0
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ −21.0000 −0.711558
$$872$$ 0 0
$$873$$ 7.00000 0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31.0000 1.04680 0.523398 0.852088i $$-0.324664\pi$$
0.523398 + 0.852088i $$0.324664\pi$$
$$878$$ 0 0
$$879$$ −2.00000 −0.0674583
$$880$$ 0 0
$$881$$ 8.00000 0.269527 0.134763 0.990878i $$-0.456973\pi$$
0.134763 + 0.990878i $$0.456973\pi$$
$$882$$ 0 0
$$883$$ −53.0000 −1.78359 −0.891796 0.452438i $$-0.850554\pi$$
−0.891796 + 0.452438i $$0.850554\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 70.0000 2.34246
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 18.0000 0.601003
$$898$$ 0 0
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −9.00000 −0.299501
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −24.0000 −0.792550
$$918$$ 0 0
$$919$$ 41.0000 1.35247 0.676233 0.736688i $$-0.263611\pi$$
0.676233 + 0.736688i $$0.263611\pi$$
$$920$$ 0 0
$$921$$ 5.00000 0.164756
$$922$$ 0 0
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 14.0000 0.458831
$$932$$ 0 0
$$933$$ 2.00000 0.0654771
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −39.0000 −1.27407 −0.637037 0.770833i $$-0.719840\pi$$
−0.637037 + 0.770833i $$0.719840\pi$$
$$938$$ 0 0
$$939$$ −19.0000 −0.620042
$$940$$ 0 0
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ 0 0
$$943$$ −72.0000 −2.34464
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.00000 0.194974 0.0974869 0.995237i $$-0.468920\pi$$
0.0974869 + 0.995237i $$0.468920\pi$$
$$948$$ 0 0
$$949$$ −18.0000 −0.584305
$$950$$ 0 0
$$951$$ 32.0000 1.03767
$$952$$ 0 0
$$953$$ 8.00000 0.259145 0.129573 0.991570i $$-0.458639\pi$$
0.129573 + 0.991570i $$0.458639\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −4.00000 −0.129302
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −16.0000 −0.515593
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 0 0
$$969$$ −42.0000 −1.34923
$$970$$ 0 0
$$971$$ −30.0000 −0.962746 −0.481373 0.876516i $$-0.659862\pi$$
−0.481373 + 0.876516i $$0.659862\pi$$
$$972$$ 0 0
$$973$$ 12.0000 0.384702
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −58.0000 −1.85558 −0.927792 0.373097i $$-0.878296\pi$$
−0.927792 + 0.373097i $$0.878296\pi$$
$$978$$ 0 0
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ −9.00000 −0.287348
$$982$$ 0 0
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −30.0000 −0.954911
$$988$$ 0 0
$$989$$ −18.0000 −0.572367
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ 4.00000 0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 42.0000 1.33015 0.665077 0.746775i $$-0.268399\pi$$
0.665077 + 0.746775i $$0.268399\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 4800.2.a.bp.1.1 1
4.3 odd 2 4800.2.a.bd.1.1 1
5.2 odd 4 4800.2.f.k.3649.1 2
5.3 odd 4 4800.2.f.k.3649.2 2
5.4 even 2 4800.2.a.bc.1.1 1
8.3 odd 2 1200.2.a.q.1.1 1
8.5 even 2 600.2.a.b.1.1 1
20.3 even 4 4800.2.f.z.3649.1 2
20.7 even 4 4800.2.f.z.3649.2 2
20.19 odd 2 4800.2.a.bs.1.1 1
24.5 odd 2 1800.2.a.e.1.1 1
24.11 even 2 3600.2.a.bl.1.1 1
40.3 even 4 1200.2.f.c.49.2 2
40.13 odd 4 600.2.f.d.49.1 2
40.19 odd 2 1200.2.a.b.1.1 1
40.27 even 4 1200.2.f.c.49.1 2
40.29 even 2 600.2.a.i.1.1 yes 1
40.37 odd 4 600.2.f.d.49.2 2
120.29 odd 2 1800.2.a.t.1.1 1
120.53 even 4 1800.2.f.e.649.2 2
120.59 even 2 3600.2.a.i.1.1 1
120.77 even 4 1800.2.f.e.649.1 2
120.83 odd 4 3600.2.f.o.2449.1 2
120.107 odd 4 3600.2.f.o.2449.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.a.b.1.1 1 8.5 even 2
600.2.a.i.1.1 yes 1 40.29 even 2
600.2.f.d.49.1 2 40.13 odd 4
600.2.f.d.49.2 2 40.37 odd 4
1200.2.a.b.1.1 1 40.19 odd 2
1200.2.a.q.1.1 1 8.3 odd 2
1200.2.f.c.49.1 2 40.27 even 4
1200.2.f.c.49.2 2 40.3 even 4
1800.2.a.e.1.1 1 24.5 odd 2
1800.2.a.t.1.1 1 120.29 odd 2
1800.2.f.e.649.1 2 120.77 even 4
1800.2.f.e.649.2 2 120.53 even 4
3600.2.a.i.1.1 1 120.59 even 2
3600.2.a.bl.1.1 1 24.11 even 2
3600.2.f.o.2449.1 2 120.83 odd 4
3600.2.f.o.2449.2 2 120.107 odd 4
4800.2.a.bc.1.1 1 5.4 even 2
4800.2.a.bd.1.1 1 4.3 odd 2
4800.2.a.bp.1.1 1 1.1 even 1 trivial
4800.2.a.bs.1.1 1 20.19 odd 2
4800.2.f.k.3649.1 2 5.2 odd 4
4800.2.f.k.3649.2 2 5.3 odd 4
4800.2.f.z.3649.1 2 20.3 even 4
4800.2.f.z.3649.2 2 20.7 even 4