# Properties

 Label 4600.2.e.a.4049.1 Level $4600$ Weight $2$ Character 4600.4049 Analytic conductor $36.731$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.4049 Dual form 4600.2.e.a.4049.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +5.00000i q^{13} -6.00000i q^{17} -6.00000 q^{19} -6.00000 q^{21} -1.00000i q^{23} +9.00000i q^{27} -9.00000 q^{29} +3.00000 q^{31} -8.00000i q^{37} +15.0000 q^{39} +3.00000 q^{41} +8.00000i q^{43} +7.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} +2.00000i q^{53} +18.0000i q^{57} -4.00000 q^{59} -10.0000 q^{61} +12.0000i q^{63} +8.00000i q^{67} -3.00000 q^{69} +7.00000 q^{71} -9.00000i q^{73} +6.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} +27.0000i q^{87} -16.0000 q^{89} +10.0000 q^{91} -9.00000i q^{93} +6.00000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100})$$ 2 * q - 12 * q^9 - 12 * q^19 - 12 * q^21 - 18 * q^29 + 6 * q^31 + 30 * q^39 + 6 * q^41 + 6 * q^49 - 36 * q^51 - 8 * q^59 - 20 * q^61 - 6 * q^69 + 14 * q^71 + 12 * q^79 + 18 * q^81 - 32 * q^89 + 20 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 3.00000i − 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −1.30931
$$22$$ 0 0
$$23$$ − 1.00000i − 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 9.00000i 1.73205i
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ 15.0000 2.40192
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.00000i 1.02105i 0.859861 + 0.510527i $$0.170550\pi$$
−0.859861 + 0.510527i $$0.829450\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −18.0000 −2.52050
$$52$$ 0 0
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 18.0000i 2.38416i
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 12.0000i 1.51186i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 0 0
$$73$$ − 9.00000i − 1.05337i −0.850060 0.526685i $$-0.823435\pi$$
0.850060 0.526685i $$-0.176565\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 14.0000i 1.53670i 0.640030 + 0.768350i $$0.278922\pi$$
−0.640030 + 0.768350i $$0.721078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 27.0000i 2.89470i
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 10.0000 1.04828
$$92$$ 0 0
$$93$$ − 9.00000i − 0.933257i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.0000i 1.35343i 0.736245 + 0.676716i $$0.236597\pi$$
−0.736245 + 0.676716i $$0.763403\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ −24.0000 −2.27798
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 30.0000i − 2.77350i
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ − 9.00000i − 0.811503i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.00000i 0.443678i 0.975083 + 0.221839i $$0.0712060\pi$$
−0.975083 + 0.221839i $$0.928794\pi$$
$$128$$ 0 0
$$129$$ 24.0000 2.11308
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ 12.0000i 1.04053i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.00000i 0.341743i 0.985293 + 0.170872i $$0.0546583\pi$$
−0.985293 + 0.170872i $$0.945342\pi$$
$$138$$ 0 0
$$139$$ 23.0000 1.95083 0.975417 0.220366i $$-0.0707252\pi$$
0.975417 + 0.220366i $$0.0707252\pi$$
$$140$$ 0 0
$$141$$ 21.0000 1.76852
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 9.00000i − 0.742307i
$$148$$ 0 0
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ 7.00000 0.569652 0.284826 0.958579i $$-0.408064\pi$$
0.284826 + 0.958579i $$0.408064\pi$$
$$152$$ 0 0
$$153$$ 36.0000i 2.91043i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 12.0000i 0.957704i 0.877896 + 0.478852i $$0.158947\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 11.0000i 0.861586i 0.902451 + 0.430793i $$0.141766\pi$$
−0.902451 + 0.430793i $$0.858234\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 36.0000 2.75299
$$172$$ 0 0
$$173$$ 10.0000i 0.760286i 0.924928 + 0.380143i $$0.124125\pi$$
−0.924928 + 0.380143i $$0.875875\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ 0 0
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 30.0000i 2.21766i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 18.0000 1.30931
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 7.00000i 0.503871i 0.967744 + 0.251936i $$0.0810671\pi$$
−0.967744 + 0.251936i $$0.918933\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 27.0000i − 1.92367i −0.273629 0.961835i $$-0.588224\pi$$
0.273629 0.961835i $$-0.411776\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 24.0000 1.69283
$$202$$ 0 0
$$203$$ 18.0000i 1.26335i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000i 0.417029i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ − 21.0000i − 1.43890i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.00000i − 0.407307i
$$218$$ 0 0
$$219$$ −27.0000 −1.82449
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 10.0000i − 0.663723i −0.943328 0.331862i $$-0.892323\pi$$
0.943328 0.331862i $$-0.107677\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 15.0000i − 0.982683i −0.870967 0.491341i $$-0.836507\pi$$
0.870967 0.491341i $$-0.163493\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 18.0000i − 1.16923i
$$238$$ 0 0
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 30.0000i − 1.90885i
$$248$$ 0 0
$$249$$ 42.0000 2.66164
$$250$$ 0 0
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 13.0000i − 0.810918i −0.914113 0.405459i $$-0.867112\pi$$
0.914113 0.405459i $$-0.132888\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 54.0000 3.34252
$$262$$ 0 0
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 48.0000i 2.93755i
$$268$$ 0 0
$$269$$ −5.00000 −0.304855 −0.152428 0.988315i $$-0.548709\pi$$
−0.152428 + 0.988315i $$0.548709\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ − 30.0000i − 1.81568i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 23.0000i − 1.38194i −0.722885 0.690968i $$-0.757185\pi$$
0.722885 0.690968i $$-0.242815\pi$$
$$278$$ 0 0
$$279$$ −18.0000 −1.07763
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ 0 0
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.00000 0.289157
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 0 0
$$303$$ − 18.0000i − 1.03407i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ −42.0000 −2.38930
$$310$$ 0 0
$$311$$ −17.0000 −0.963982 −0.481991 0.876176i $$-0.660086\pi$$
−0.481991 + 0.876176i $$0.660086\pi$$
$$312$$ 0 0
$$313$$ 20.0000i 1.13047i 0.824931 + 0.565233i $$0.191214\pi$$
−0.824931 + 0.565233i $$0.808786\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 42.0000 2.34421
$$322$$ 0 0
$$323$$ 36.0000i 2.00309i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 14.0000 0.771845
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 0 0
$$333$$ 48.0000i 2.63038i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 7.00000 0.374701 0.187351 0.982293i $$-0.440010\pi$$
0.187351 + 0.982293i $$0.440010\pi$$
$$350$$ 0 0
$$351$$ −45.0000 −2.40192
$$352$$ 0 0
$$353$$ 11.0000i 0.585471i 0.956193 + 0.292735i $$0.0945655\pi$$
−0.956193 + 0.292735i $$0.905434\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 36.0000i 1.90532i
$$358$$ 0 0
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 33.0000i 1.73205i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 10.0000i − 0.521996i −0.965339 0.260998i $$-0.915948\pi$$
0.965339 0.260998i $$-0.0840516\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 45.0000i − 2.31762i
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 15.0000 0.768473
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 48.0000i − 2.43998i
$$388$$ 0 0
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 0 0
$$393$$ 27.0000i 1.36197i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 31.0000i 1.55585i 0.628360 + 0.777923i $$0.283727\pi$$
−0.628360 + 0.777923i $$0.716273\pi$$
$$398$$ 0 0
$$399$$ 36.0000 1.80225
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ 15.0000i 0.747203i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 69.0000i − 3.37894i
$$418$$ 0 0
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ − 42.0000i − 2.04211i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 20.0000i 0.967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.00000i 0.287019i
$$438$$ 0 0
$$439$$ −5.00000 −0.238637 −0.119318 0.992856i $$-0.538071\pi$$
−0.119318 + 0.992856i $$0.538071\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ 9.00000i 0.427603i 0.976877 + 0.213801i $$0.0685846\pi$$
−0.976877 + 0.213801i $$0.931415\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 42.0000i 1.98653i
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 21.0000i − 0.986666i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.0000i 1.77757i 0.458329 + 0.888783i $$0.348448\pi$$
−0.458329 + 0.888783i $$0.651552\pi$$
$$458$$ 0 0
$$459$$ 54.0000 2.52050
$$460$$ 0 0
$$461$$ 1.00000 0.0465746 0.0232873 0.999729i $$-0.492587\pi$$
0.0232873 + 0.999729i $$0.492587\pi$$
$$462$$ 0 0
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 36.0000 1.65879
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 12.0000i − 0.549442i
$$478$$ 0 0
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 0 0
$$483$$ 6.00000i 0.273009i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17.0000i 0.770344i 0.922845 + 0.385172i $$0.125858\pi$$
−0.922845 + 0.385172i $$0.874142\pi$$
$$488$$ 0 0
$$489$$ 33.0000 1.49231
$$490$$ 0 0
$$491$$ −7.00000 −0.315906 −0.157953 0.987447i $$-0.550489\pi$$
−0.157953 + 0.987447i $$0.550489\pi$$
$$492$$ 0 0
$$493$$ 54.0000i 2.43204i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 14.0000i − 0.627986i
$$498$$ 0 0
$$499$$ 1.00000 0.0447661 0.0223831 0.999749i $$-0.492875\pi$$
0.0223831 + 0.999749i $$0.492875\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ − 10.0000i − 0.445878i −0.974832 0.222939i $$-0.928435\pi$$
0.974832 0.222939i $$-0.0715651\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 36.0000i 1.59882i
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ 0 0
$$513$$ − 54.0000i − 2.38416i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 30.0000 1.31685
$$520$$ 0 0
$$521$$ 8.00000 0.350486 0.175243 0.984525i $$-0.443929\pi$$
0.175243 + 0.984525i $$0.443929\pi$$
$$522$$ 0 0
$$523$$ − 18.0000i − 0.787085i −0.919306 0.393543i $$-0.871249\pi$$
0.919306 0.393543i $$-0.128751\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 18.0000i − 0.784092i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ 15.0000i 0.649722i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 9.00000i − 0.388379i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −43.0000 −1.84871 −0.924357 0.381528i $$-0.875398\pi$$
−0.924357 + 0.381528i $$0.875398\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 19.0000i − 0.812381i −0.913788 0.406191i $$-0.866857\pi$$
0.913788 0.406191i $$-0.133143\pi$$
$$548$$ 0 0
$$549$$ 60.0000 2.56074
$$550$$ 0 0
$$551$$ 54.0000 2.30048
$$552$$ 0 0
$$553$$ − 12.0000i − 0.510292i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8.00000i 0.338971i 0.985533 + 0.169485i $$0.0542106\pi$$
−0.985533 + 0.169485i $$0.945789\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 44.0000i − 1.85438i −0.374593 0.927189i $$-0.622217\pi$$
0.374593 0.927189i $$-0.377783\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 18.0000i − 0.755929i
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ 0 0
$$573$$ 48.0000i 2.00523i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 33.0000i 1.37381i 0.726748 + 0.686904i $$0.241031\pi$$
−0.726748 + 0.686904i $$0.758969\pi$$
$$578$$ 0 0
$$579$$ 21.0000 0.872730
$$580$$ 0 0
$$581$$ 28.0000 1.16164
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 15.0000i 0.619116i 0.950881 + 0.309558i $$0.100181\pi$$
−0.950881 + 0.309558i $$0.899819\pi$$
$$588$$ 0 0
$$589$$ −18.0000 −0.741677
$$590$$ 0 0
$$591$$ −81.0000 −3.33189
$$592$$ 0 0
$$593$$ − 22.0000i − 0.903432i −0.892162 0.451716i $$-0.850812\pi$$
0.892162 0.451716i $$-0.149188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 60.0000i 2.45564i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 13.0000 0.530281 0.265141 0.964210i $$-0.414582\pi$$
0.265141 + 0.964210i $$0.414582\pi$$
$$602$$ 0 0
$$603$$ − 48.0000i − 1.95471i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ 54.0000 2.18819
$$610$$ 0 0
$$611$$ −35.0000 −1.41595
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 9.00000 0.361158
$$622$$ 0 0
$$623$$ 32.0000i 1.28205i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −48.0000 −1.91389
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ − 24.0000i − 0.953914i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 15.0000i 0.594322i
$$638$$ 0 0
$$639$$ −42.0000 −1.66149
$$640$$ 0 0
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 0 0
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 33.0000i − 1.29736i −0.761060 0.648682i $$-0.775321\pi$$
0.761060 0.648682i $$-0.224679\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −18.0000 −0.705476
$$652$$ 0 0
$$653$$ − 39.0000i − 1.52619i −0.646288 0.763094i $$-0.723679\pi$$
0.646288 0.763094i $$-0.276321\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 54.0000i 2.10674i
$$658$$ 0 0
$$659$$ −50.0000 −1.94772 −0.973862 0.227142i $$-0.927062\pi$$
−0.973862 + 0.227142i $$0.927062\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ − 90.0000i − 3.49531i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.00000i 0.348481i
$$668$$ 0 0
$$669$$ −24.0000 −0.927894
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 5.00000i 0.192736i 0.995346 + 0.0963679i $$0.0307225\pi$$
−0.995346 + 0.0963679i $$0.969277\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 22.0000i − 0.845529i −0.906240 0.422764i $$-0.861060\pi$$
0.906240 0.422764i $$-0.138940\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ −30.0000 −1.14960
$$682$$ 0 0
$$683$$ − 29.0000i − 1.10965i −0.831966 0.554827i $$-0.812784\pi$$
0.831966 0.554827i $$-0.187216\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 60.0000i − 2.28914i
$$688$$ 0 0
$$689$$ −10.0000 −0.380970
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 18.0000i − 0.681799i
$$698$$ 0 0
$$699$$ −45.0000 −1.70206
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 48.0000i 1.81035i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 12.0000i − 0.451306i
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ −36.0000 −1.35011
$$712$$ 0 0
$$713$$ − 3.00000i − 0.112351i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 39.0000i − 1.45648i
$$718$$ 0 0
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ −28.0000 −1.04277
$$722$$ 0 0
$$723$$ 12.0000i 0.446285i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 36.0000i − 1.33517i −0.744535 0.667583i $$-0.767329\pi$$
0.744535 0.667583i $$-0.232671\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −29.0000 −1.06678 −0.533391 0.845869i $$-0.679083\pi$$
−0.533391 + 0.845869i $$0.679083\pi$$
$$740$$ 0 0
$$741$$ −90.0000 −3.30623
$$742$$ 0 0
$$743$$ 6.00000i 0.220119i 0.993925 + 0.110059i $$0.0351041\pi$$
−0.993925 + 0.110059i $$0.964896\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 84.0000i − 3.07340i
$$748$$ 0 0
$$749$$ 28.0000 1.02310
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ − 30.0000i − 1.09326i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 12.0000i − 0.436147i −0.975932 0.218074i $$-0.930023\pi$$
0.975932 0.218074i $$-0.0699773\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 41.0000 1.48625 0.743124 0.669153i $$-0.233343\pi$$
0.743124 + 0.669153i $$0.233343\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 20.0000i − 0.722158i
$$768$$ 0 0
$$769$$ 28.0000 1.00971 0.504853 0.863205i $$-0.331547\pi$$
0.504853 + 0.863205i $$0.331547\pi$$
$$770$$ 0 0
$$771$$ −39.0000 −1.40455
$$772$$ 0 0
$$773$$ − 8.00000i − 0.287740i −0.989597 0.143870i $$-0.954045\pi$$
0.989597 0.143870i $$-0.0459547\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 48.0000i 1.72199i
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 81.0000i − 2.89470i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 40.0000i 1.42585i 0.701242 + 0.712923i $$0.252629\pi$$
−0.701242 + 0.712923i $$0.747371\pi$$
$$788$$ 0 0
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ − 50.0000i − 1.77555i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 0 0
$$799$$ 42.0000 1.48585
$$800$$ 0 0
$$801$$ 96.0000 3.39199
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 15.0000i 0.528025i
$$808$$ 0 0
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ −5.00000 −0.175574 −0.0877869 0.996139i $$-0.527979\pi$$
−0.0877869 + 0.996139i $$0.527979\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 48.0000i − 1.67931i
$$818$$ 0 0
$$819$$ −60.0000 −2.09657
$$820$$ 0 0
$$821$$ −54.0000 −1.88461 −0.942306 0.334751i $$-0.891348\pi$$
−0.942306 + 0.334751i $$0.891348\pi$$
$$822$$ 0 0
$$823$$ 57.0000i 1.98690i 0.114289 + 0.993448i $$0.463541\pi$$
−0.114289 + 0.993448i $$0.536459\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ −54.0000 −1.87550 −0.937749 0.347314i $$-0.887094\pi$$
−0.937749 + 0.347314i $$0.887094\pi$$
$$830$$ 0 0
$$831$$ −69.0000 −2.39358
$$832$$ 0 0
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 27.0000i 0.933257i
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 54.0000i 1.85986i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000i 0.755929i
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 23.0000i 0.785665i 0.919610 + 0.392833i $$0.128505\pi$$
−0.919610 + 0.392833i $$0.871495\pi$$
$$858$$ 0 0
$$859$$ 17.0000 0.580033 0.290016 0.957022i $$-0.406339\pi$$
0.290016 + 0.957022i $$0.406339\pi$$
$$860$$ 0 0
$$861$$ −18.0000 −0.613438
$$862$$ 0 0
$$863$$ − 49.0000i − 1.66798i −0.551780 0.833990i $$-0.686051\pi$$
0.551780 0.833990i $$-0.313949\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 57.0000i 1.93582i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −40.0000 −1.35535
$$872$$ 0 0
$$873$$ − 36.0000i − 1.21842i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ 0 0
$$879$$ −36.0000 −1.21425
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 0 0
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 27.0000i 0.906571i 0.891365 + 0.453286i $$0.149748\pi$$
−0.891365 + 0.453286i $$0.850252\pi$$
$$888$$ 0 0
$$889$$ 10.0000 0.335389
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 42.0000i − 1.40548i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 15.0000i − 0.500835i
$$898$$ 0 0
$$899$$ −27.0000 −0.900500
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ − 48.0000i − 1.59734i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 10.0000i − 0.332045i −0.986122 0.166022i $$-0.946908\pi$$
0.986122 0.166022i $$-0.0530924\pi$$
$$908$$ 0 0
$$909$$ −36.0000 −1.19404
$$910$$ 0 0
$$911$$ −50.0000 −1.65657 −0.828287 0.560304i $$-0.810684\pi$$
−0.828287 + 0.560304i $$0.810684\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.0000i 0.594412i
$$918$$ 0 0
$$919$$ −10.0000 −0.329870 −0.164935 0.986304i $$-0.552741\pi$$
−0.164935 + 0.986304i $$0.552741\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 35.0000i 1.15204i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 84.0000i 2.75892i
$$928$$ 0 0
$$929$$ −7.00000 −0.229663 −0.114831 0.993385i $$-0.536633\pi$$
−0.114831 + 0.993385i $$0.536633\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 0 0
$$933$$ 51.0000i 1.66967i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.00000i 0.130674i 0.997863 + 0.0653372i $$0.0208123\pi$$
−0.997863 + 0.0653372i $$0.979188\pi$$
$$938$$ 0 0
$$939$$ 60.0000 1.95803
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 0 0
$$943$$ − 3.00000i − 0.0976934i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 7.00000i 0.227469i 0.993511 + 0.113735i $$0.0362814\pi$$
−0.993511 + 0.113735i $$0.963719\pi$$
$$948$$ 0 0
$$949$$ 45.0000 1.46076
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ − 42.0000i − 1.36051i −0.732974 0.680257i $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 84.0000i − 2.70686i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 47.0000i − 1.51142i −0.654907 0.755709i $$-0.727292\pi$$
0.654907 0.755709i $$-0.272708\pi$$
$$968$$ 0 0
$$969$$ 108.000 3.46946
$$970$$ 0 0
$$971$$ −26.0000 −0.834380 −0.417190 0.908819i $$-0.636985\pi$$
−0.417190 + 0.908819i $$0.636985\pi$$
$$972$$ 0 0
$$973$$ − 46.0000i − 1.47469i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 14.0000i 0.446531i 0.974758 + 0.223265i $$0.0716716\pi$$
−0.974758 + 0.223265i $$0.928328\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 42.0000i − 1.33687i
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 57.0000i 1.80884i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 50.0000i − 1.58352i −0.610835 0.791758i $$-0.709166\pi$$
0.610835 0.791758i $$-0.290834\pi$$
$$998$$ 0 0
$$999$$ 72.0000 2.27798
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 4600.2.e.a.4049.1 2
5.2 odd 4 4600.2.a.a.1.1 1
5.3 odd 4 184.2.a.d.1.1 1
5.4 even 2 inner 4600.2.e.a.4049.2 2
15.8 even 4 1656.2.a.c.1.1 1
20.3 even 4 368.2.a.a.1.1 1
20.7 even 4 9200.2.a.bj.1.1 1
35.13 even 4 9016.2.a.b.1.1 1
40.3 even 4 1472.2.a.m.1.1 1
40.13 odd 4 1472.2.a.a.1.1 1
60.23 odd 4 3312.2.a.i.1.1 1
115.68 even 4 4232.2.a.j.1.1 1
460.183 odd 4 8464.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.d.1.1 1 5.3 odd 4
368.2.a.a.1.1 1 20.3 even 4
1472.2.a.a.1.1 1 40.13 odd 4
1472.2.a.m.1.1 1 40.3 even 4
1656.2.a.c.1.1 1 15.8 even 4
3312.2.a.i.1.1 1 60.23 odd 4
4232.2.a.j.1.1 1 115.68 even 4
4600.2.a.a.1.1 1 5.2 odd 4
4600.2.e.a.4049.1 2 1.1 even 1 trivial
4600.2.e.a.4049.2 2 5.4 even 2 inner
8464.2.a.b.1.1 1 460.183 odd 4
9016.2.a.b.1.1 1 35.13 even 4
9200.2.a.bj.1.1 1 20.7 even 4