# Properties

 Label 4800.2.f.z.3649.1 Level $4800$ Weight $2$ Character 4800.3649 Analytic conductor $38.328$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4800.3649 Dual form 4800.2.f.z.3649.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -3.00000i q^{13} +6.00000i q^{17} +7.00000 q^{19} -3.00000 q^{21} +6.00000i q^{23} +1.00000i q^{27} -2.00000 q^{29} +5.00000 q^{31} -2.00000i q^{33} -10.0000i q^{37} -3.00000 q^{39} +12.0000 q^{41} -3.00000i q^{43} +10.0000i q^{47} -2.00000 q^{49} +6.00000 q^{51} -7.00000i q^{57} +6.00000 q^{59} +13.0000 q^{61} +3.00000i q^{63} +7.00000i q^{67} +6.00000 q^{69} +4.00000 q^{71} +6.00000i q^{73} -6.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{83} +2.00000i q^{87} -16.0000 q^{89} -9.00000 q^{91} -5.00000i q^{93} -7.00000i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{11} + 14q^{19} - 6q^{21} - 4q^{29} + 10q^{31} - 6q^{39} + 24q^{41} - 4q^{49} + 12q^{51} + 12q^{59} + 26q^{61} + 12q^{69} + 8q^{71} - 16q^{79} + 2q^{81} - 32q^{89} - 18q^{91} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 3.00000i − 0.832050i −0.909353 0.416025i $$-0.863423\pi$$
0.909353 0.416025i $$-0.136577\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 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
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ − 2.00000i − 0.348155i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 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
$$43$$ − 3.00000i − 0.457496i −0.973486 0.228748i $$-0.926537\pi$$
0.973486 0.228748i $$-0.0734631\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 7.00000i − 0.927173i
$$58$$ 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
$$63$$ 3.00000i 0.377964i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 6.00000i − 0.683763i
$$78$$ 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
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ −9.00000 −0.943456
$$92$$ 0 0
$$93$$ − 5.00000i − 0.518476i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 7.00000i − 0.710742i −0.934725 0.355371i $$-0.884354\pi$$
0.934725 0.355371i $$-0.115646\pi$$
$$98$$ 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
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 0 0
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ − 12.0000i − 1.12887i −0.825479 0.564433i $$-0.809095\pi$$
0.825479 0.564433i $$-0.190905\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.00000i 0.277350i
$$118$$ 0 0
$$119$$ 18.0000 1.65006
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ −3.00000 −0.264135
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ − 21.0000i − 1.82093i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 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
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000i 0.164957i
$$148$$ 0 0
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.00000i 0.718278i 0.933284 + 0.359139i $$0.116930\pi$$
−0.933284 + 0.359139i $$0.883070\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ 0 0
$$163$$ − 1.00000i − 0.0783260i −0.999233 0.0391630i $$-0.987531\pi$$
0.999233 0.0391630i $$-0.0124692\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ 4.00000 0.307692
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ 0 0
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 6.00000i − 0.450988i
$$178$$ 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
$$183$$ − 13.0000i − 0.960988i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ 0 0
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ − 19.0000i − 1.36765i −0.729646 0.683825i $$-0.760315\pi$$
0.729646 0.683825i $$-0.239685\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 14.0000i − 0.997459i −0.866758 0.498729i $$-0.833800\pi$$
0.866758 0.498729i $$-0.166200\pi$$
$$198$$ 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
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 6.00000i − 0.417029i
$$208$$ 0 0
$$209$$ 14.0000 0.968400
$$210$$ 0 0
$$211$$ 9.00000 0.619586 0.309793 0.950804i $$-0.399740\pi$$
0.309793 + 0.950804i $$0.399740\pi$$
$$212$$ 0 0
$$213$$ − 4.00000i − 0.274075i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 15.0000i − 1.01827i
$$218$$ 0 0
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 18.0000 1.21081
$$222$$ 0 0
$$223$$ 11.0000i 0.736614i 0.929704 + 0.368307i $$0.120063\pi$$
−0.929704 + 0.368307i $$0.879937\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 0 0
$$229$$ −19.0000 −1.25556 −0.627778 0.778393i $$-0.716035\pi$$
−0.627778 + 0.778393i $$0.716035\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 4.00000i 0.262049i 0.991379 + 0.131024i $$0.0418266\pi$$
−0.991379 + 0.131024i $$0.958173\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 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
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 21.0000i − 1.33620i
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4.00000i 0.249513i 0.992187 + 0.124757i $$0.0398150\pi$$
−0.992187 + 0.124757i $$0.960185\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 12.0000i 0.739952i 0.929041 + 0.369976i $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 16.0000i 0.979184i
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 9.00000i 0.544705i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1.00000i − 0.0600842i −0.999549 0.0300421i $$-0.990436\pi$$
0.999549 0.0300421i $$-0.00956413\pi$$
$$278$$ 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
$$283$$ − 5.00000i − 0.297219i −0.988896 0.148610i $$-0.952520\pi$$
0.988896 0.148610i $$-0.0474798\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 36.0000i − 2.12501i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −7.00000 −0.410347
$$292$$ 0 0
$$293$$ − 2.00000i − 0.116841i −0.998292 0.0584206i $$-0.981394\pi$$
0.998292 0.0584206i $$-0.0186065\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000i 0.116052i
$$298$$ 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
$$307$$ 5.00000i 0.285365i 0.989769 + 0.142683i $$0.0455728\pi$$
−0.989769 + 0.142683i $$0.954427\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ − 19.0000i − 1.07394i −0.843600 0.536972i $$-0.819568\pi$$
0.843600 0.536972i $$-0.180432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 32.0000i − 1.79730i −0.438667 0.898650i $$-0.644549\pi$$
0.438667 0.898650i $$-0.355451\pi$$
$$318$$ 0 0
$$319$$ −4.00000 −0.223957
$$320$$ 0 0
$$321$$ −16.0000 −0.893033
$$322$$ 0 0
$$323$$ 42.0000i 2.33694i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 9.00000i − 0.497701i
$$328$$ 0 0
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 7.00000i − 0.381314i −0.981657 0.190657i $$-0.938938\pi$$
0.981657 0.190657i $$-0.0610619\pi$$
$$338$$ 0 0
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 18.0000i − 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 3.00000 0.160128
$$352$$ 0 0
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 18.0000i − 0.952661i
$$358$$ 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
$$363$$ 7.00000i 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11.0000i 0.574195i 0.957901 + 0.287098i $$0.0926904\pi$$
−0.957901 + 0.287098i $$0.907310\pi$$
$$368$$ 0 0
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 7.00000i 0.362446i 0.983442 + 0.181223i $$0.0580056\pi$$
−0.983442 + 0.181223i $$0.941994\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000i 0.309016i
$$378$$ 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
$$383$$ 36.0000i 1.83951i 0.392488 + 0.919757i $$0.371614\pi$$
−0.392488 + 0.919757i $$0.628386\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.00000i 0.152499i
$$388$$ 0 0
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 0 0
$$393$$ 8.00000i 0.403547i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 11.0000i − 0.552074i −0.961147 0.276037i $$-0.910979\pi$$
0.961147 0.276037i $$-0.0890213\pi$$
$$398$$ 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
$$403$$ − 15.0000i − 0.747203i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.0000i − 0.991363i
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 0 0
$$413$$ − 18.0000i − 0.885722i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000i 0.195881i
$$418$$ 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
$$423$$ − 10.0000i − 0.486217i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 39.0000i − 1.88734i
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 34.0000 1.63772 0.818861 0.573992i $$-0.194606\pi$$
0.818861 + 0.573992i $$0.194606\pi$$
$$432$$ 0 0
$$433$$ − 3.00000i − 0.144171i −0.997398 0.0720854i $$-0.977035\pi$$
0.997398 0.0720854i $$-0.0229654\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 42.0000i 2.00913i
$$438$$ 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
$$443$$ 16.0000i 0.760183i 0.924949 + 0.380091i $$0.124107\pi$$
−0.924949 + 0.380091i $$0.875893\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 22.0000i 1.04056i
$$448$$ 0 0
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 1.00000i 0.0469841i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 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
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 26.0000i 1.20314i 0.798821 + 0.601568i $$0.205457\pi$$
−0.798821 + 0.601568i $$0.794543\pi$$
$$468$$ 0 0
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ 9.00000 0.414698
$$472$$ 0 0
$$473$$ − 6.00000i − 0.275880i
$$474$$ 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
$$483$$ − 18.0000i − 0.819028i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11.0000i 0.498458i 0.968445 + 0.249229i $$0.0801771\pi$$
−0.968445 + 0.249229i $$0.919823\pi$$
$$488$$ 0 0
$$489$$ −1.00000 −0.0452216
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 12.0000i − 0.538274i
$$498$$ 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
$$503$$ − 4.00000i − 0.178351i −0.996016 0.0891756i $$-0.971577\pi$$
0.996016 0.0891756i $$-0.0284232\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 4.00000i − 0.177646i
$$508$$ 0 0
$$509$$ 22.0000 0.975133 0.487566 0.873086i $$-0.337885\pi$$
0.487566 + 0.873086i $$0.337885\pi$$
$$510$$ 0 0
$$511$$ 18.0000 0.796273
$$512$$ 0 0
$$513$$ 7.00000i 0.309058i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 20.0000i 0.879599i
$$518$$ 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
$$523$$ − 29.0000i − 1.26808i −0.773300 0.634041i $$-0.781395\pi$$
0.773300 0.634041i $$-0.218605\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 30.0000i 1.30682i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 18.0000i 0.776757i
$$538$$ 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
$$543$$ − 19.0000i − 0.815368i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ −13.0000 −0.554826
$$550$$ 0 0
$$551$$ −14.0000 −0.596420
$$552$$ 0 0
$$553$$ 24.0000i 1.02058i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ 26.0000i 1.09577i 0.836554 + 0.547885i $$0.184567\pi$$
−0.836554 + 0.547885i $$0.815433\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 3.00000i − 0.125988i
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −39.0000 −1.63210 −0.816050 0.577982i $$-0.803840\pi$$
−0.816050 + 0.577982i $$0.803840\pi$$
$$572$$ 0 0
$$573$$ − 18.0000i − 0.751961i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 11.0000i 0.457936i 0.973434 + 0.228968i $$0.0735351\pi$$
−0.973434 + 0.228968i $$0.926465\pi$$
$$578$$ 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
$$587$$ − 16.0000i − 0.660391i −0.943913 0.330195i $$-0.892885\pi$$
0.943913 0.330195i $$-0.107115\pi$$
$$588$$ 0 0
$$589$$ 35.0000 1.44215
$$590$$ 0 0
$$591$$ −14.0000 −0.575883
$$592$$ 0 0
$$593$$ 36.0000i 1.47834i 0.673517 + 0.739171i $$0.264783\pi$$
−0.673517 + 0.739171i $$0.735217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.00000i 0.122782i
$$598$$ 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
$$603$$ − 7.00000i − 0.285062i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 0 0
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 30.0000 1.21367
$$612$$ 0 0
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 40.0000i − 1.61034i −0.593045 0.805170i $$-0.702074\pi$$
0.593045 0.805170i $$-0.297926\pi$$
$$618$$ 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
$$623$$ 48.0000i 1.92308i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 14.0000i − 0.559106i
$$628$$ 0 0
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ 25.0000 0.995234 0.497617 0.867397i $$-0.334208\pi$$
0.497617 + 0.867397i $$0.334208\pi$$
$$632$$ 0 0
$$633$$ − 9.00000i − 0.357718i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000i 0.237729i
$$638$$ 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
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ −15.0000 −0.587896
$$652$$ 0 0
$$653$$ 26.0000i 1.01746i 0.860927 + 0.508729i $$0.169885\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 6.00000i − 0.234082i
$$658$$ 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
$$663$$ − 18.0000i − 0.699062i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 12.0000i − 0.464642i
$$668$$ 0 0
$$669$$ 11.0000 0.425285
$$670$$ 0 0
$$671$$ 26.0000 1.00372
$$672$$ 0 0
$$673$$ 50.0000i 1.92736i 0.267063 + 0.963679i $$0.413947\pi$$
−0.267063 + 0.963679i $$0.586053\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10.0000i 0.384331i 0.981363 + 0.192166i $$0.0615511\pi$$
−0.981363 + 0.192166i $$0.938449\pi$$
$$678$$ 0 0
$$679$$ −21.0000 −0.805906
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ − 40.0000i − 1.53056i −0.643699 0.765279i $$-0.722601\pi$$
0.643699 0.765279i $$-0.277399\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 19.0000i 0.724895i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ 0 0
$$693$$ 6.00000i 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 72.0000i 2.72719i
$$698$$ 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
$$703$$ − 70.0000i − 2.64010i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −19.0000 −0.713560 −0.356780 0.934188i $$-0.616125\pi$$
−0.356780 + 0.934188i $$0.616125\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 30.0000i 1.12351i
$$714$$ 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
$$723$$ − 25.0000i − 0.929760i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.00000i 0.185440i 0.995692 + 0.0927199i $$0.0295561\pi$$
−0.995692 + 0.0927199i $$0.970444\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 18.0000 0.665754
$$732$$ 0 0
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 14.0000i 0.515697i
$$738$$ 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
$$743$$ 20.0000i 0.733729i 0.930274 + 0.366864i $$0.119569\pi$$
−0.930274 + 0.366864i $$0.880431\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 13.0000i − 0.472493i −0.971693 0.236247i $$-0.924083\pi$$
0.971693 0.236247i $$-0.0759173\pi$$
$$758$$ 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
$$763$$ − 27.0000i − 0.977466i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 18.0000i − 0.649942i
$$768$$ 0 0
$$769$$ 21.0000 0.757279 0.378640 0.925544i $$-0.376392\pi$$
0.378640 + 0.925544i $$0.376392\pi$$
$$770$$ 0 0
$$771$$ 4.00000 0.144056
$$772$$ 0 0
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 30.0000i 1.07624i
$$778$$ 0 0
$$779$$ 84.0000 3.00961
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 37.0000i 1.31891i 0.751745 + 0.659454i $$0.229212\pi$$
−0.751745 + 0.659454i $$0.770788\pi$$
$$788$$ 0 0
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ − 39.0000i − 1.38493i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 20.0000i 0.708436i 0.935163 + 0.354218i $$0.115253\pi$$
−0.935163 + 0.354218i $$0.884747\pi$$
$$798$$ 0 0
$$799$$ −60.0000 −2.12265
$$800$$ 0 0
$$801$$ 16.0000 0.565332
$$802$$ 0 0
$$803$$ 12.0000i 0.423471i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.00000i 0.211210i
$$808$$ 0 0
$$809$$ −48.0000 −1.68759 −0.843795 0.536666i $$-0.819684\pi$$
−0.843795 + 0.536666i $$0.819684\pi$$
$$810$$ 0 0
$$811$$ 17.0000 0.596951 0.298475 0.954417i $$-0.403522\pi$$
0.298475 + 0.954417i $$0.403522\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 21.0000i − 0.734697i
$$818$$ 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
$$823$$ − 9.00000i − 0.313720i −0.987621 0.156860i $$-0.949863\pi$$
0.987621 0.156860i $$-0.0501372\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24.0000i 0.834562i 0.908778 + 0.417281i $$0.137017\pi$$
−0.908778 + 0.417281i $$0.862983\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ −1.00000 −0.0346896
$$832$$ 0 0
$$833$$ − 12.0000i − 0.415775i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.00000i 0.172825i
$$838$$ 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
$$843$$ − 2.00000i − 0.0688837i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 21.0000i 0.721569i
$$848$$ 0 0
$$849$$ −5.00000 −0.171600
$$850$$ 0 0
$$851$$ 60.0000 2.05677
$$852$$ 0 0
$$853$$ − 25.0000i − 0.855984i −0.903783 0.427992i $$-0.859221\pi$$
0.903783 0.427992i $$-0.140779\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 4.00000i − 0.136637i −0.997664 0.0683187i $$-0.978237\pi$$
0.997664 0.0683187i $$-0.0217635\pi$$
$$858$$ 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
$$863$$ 8.00000i 0.272323i 0.990687 + 0.136162i $$0.0434766\pi$$
−0.990687 + 0.136162i $$0.956523\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 21.0000 0.711558
$$872$$ 0 0
$$873$$ 7.00000i 0.236914i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 31.0000i − 1.04680i −0.852088 0.523398i $$-0.824664\pi$$
0.852088 0.523398i $$-0.175336\pi$$
$$878$$ 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
$$883$$ 53.0000i 1.78359i 0.452438 + 0.891796i $$0.350554\pi$$
−0.452438 + 0.891796i $$0.649446\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 42.0000i − 1.41022i −0.709097 0.705111i $$-0.750897\pi$$
0.709097 0.705111i $$-0.249103\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ 70.0000i 2.34246i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 18.0000i − 0.601003i
$$898$$ 0 0
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 9.00000i 0.299501i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 0 0
$$913$$ 12.0000i 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 24.0000i 0.792550i
$$918$$ 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
$$923$$ − 12.0000i − 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.0000i 0.394132i
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −14.0000 −0.458831
$$932$$ 0 0
$$933$$ 2.00000i 0.0654771i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39.0000i 1.27407i 0.770833 + 0.637037i $$0.219840\pi$$
−0.770833 + 0.637037i $$0.780160\pi$$
$$938$$ 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
$$943$$ 72.0000i 2.34464i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 6.00000i 0.194974i 0.995237 + 0.0974869i $$0.0310804\pi$$
−0.995237 + 0.0974869i $$0.968920\pi$$
$$948$$ 0 0
$$949$$ 18.0000 0.584305
$$950$$ 0 0
$$951$$ −32.0000 −1.03767
$$952$$ 0 0
$$953$$ 8.00000i 0.259145i 0.991570 + 0.129573i $$0.0413606\pi$$
−0.991570 + 0.129573i $$0.958639\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 4.00000i 0.129302i
$$958$$ 0 0
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 16.0000i 0.515593i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 24.0000i − 0.771788i −0.922543 0.385894i $$-0.873893\pi$$
0.922543 0.385894i $$-0.126107\pi$$
$$968$$ 0 0
$$969$$ 42.0000 1.34923
$$970$$ 0 0
$$971$$ 30.0000 0.962746 0.481373 0.876516i $$-0.340138\pi$$
0.481373 + 0.876516i $$0.340138\pi$$
$$972$$ 0 0
$$973$$ 12.0000i 0.384702i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 58.0000i 1.85558i 0.373097 + 0.927792i $$0.378296\pi$$
−0.373097 + 0.927792i $$0.621704\pi$$
$$978$$ 0 0
$$979$$ −32.0000 −1.02272
$$980$$ 0 0
$$981$$ −9.00000 −0.287348
$$982$$ 0 0
$$983$$ − 4.00000i − 0.127580i −0.997963 0.0637901i $$-0.979681\pi$$
0.997963 0.0637901i $$-0.0203188\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 30.0000i − 0.954911i
$$988$$ 0 0
$$989$$ 18.0000 0.572367
$$990$$ 0 0
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ 0 0
$$993$$ 4.00000i 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 42.0000i − 1.33015i −0.746775 0.665077i $$-0.768399\pi$$
0.746775 0.665077i $$-0.231601\pi$$
$$998$$ 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.f.z.3649.1 2
4.3 odd 2 4800.2.f.k.3649.2 2
5.2 odd 4 4800.2.a.bd.1.1 1
5.3 odd 4 4800.2.a.bs.1.1 1
5.4 even 2 inner 4800.2.f.z.3649.2 2
8.3 odd 2 600.2.f.d.49.1 2
8.5 even 2 1200.2.f.c.49.2 2
20.3 even 4 4800.2.a.bc.1.1 1
20.7 even 4 4800.2.a.bp.1.1 1
20.19 odd 2 4800.2.f.k.3649.1 2
24.5 odd 2 3600.2.f.o.2449.1 2
24.11 even 2 1800.2.f.e.649.2 2
40.3 even 4 600.2.a.i.1.1 yes 1
40.13 odd 4 1200.2.a.b.1.1 1
40.19 odd 2 600.2.f.d.49.2 2
40.27 even 4 600.2.a.b.1.1 1
40.29 even 2 1200.2.f.c.49.1 2
40.37 odd 4 1200.2.a.q.1.1 1
120.29 odd 2 3600.2.f.o.2449.2 2
120.53 even 4 3600.2.a.i.1.1 1
120.59 even 2 1800.2.f.e.649.1 2
120.77 even 4 3600.2.a.bl.1.1 1
120.83 odd 4 1800.2.a.t.1.1 1
120.107 odd 4 1800.2.a.e.1.1 1

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