# Properties

 Label 1575.2.a.q.1.1 Level $1575$ Weight $2$ Character 1575.1 Self dual yes Analytic conductor $12.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} +O(q^{10})$$ $$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.73205 q^{8} +3.46410 q^{11} -2.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} -3.46410 q^{17} -4.00000 q^{19} -6.00000 q^{22} +3.46410 q^{23} +3.46410 q^{26} -1.00000 q^{28} -4.00000 q^{31} +5.19615 q^{32} +6.00000 q^{34} -2.00000 q^{37} +6.92820 q^{38} +10.3923 q^{41} +4.00000 q^{43} +3.46410 q^{44} -6.00000 q^{46} -6.92820 q^{47} +1.00000 q^{49} -2.00000 q^{52} +6.92820 q^{53} -1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} +6.92820 q^{62} +1.00000 q^{64} +4.00000 q^{67} -3.46410 q^{68} -10.3923 q^{71} -14.0000 q^{73} +3.46410 q^{74} -4.00000 q^{76} -3.46410 q^{77} +8.00000 q^{79} -18.0000 q^{82} -6.92820 q^{86} +6.00000 q^{88} -3.46410 q^{89} +2.00000 q^{91} +3.46410 q^{92} +12.0000 q^{94} -14.0000 q^{97} -1.73205 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{7} + O(q^{10})$$ $$2 q + 2 q^{4} - 2 q^{7} - 4 q^{13} - 10 q^{16} - 8 q^{19} - 12 q^{22} - 2 q^{28} - 8 q^{31} + 12 q^{34} - 4 q^{37} + 8 q^{43} - 12 q^{46} + 2 q^{49} - 4 q^{52} - 20 q^{61} + 2 q^{64} + 8 q^{67} - 28 q^{73} - 8 q^{76} + 16 q^{79} - 36 q^{82} + 12 q^{88} + 4 q^{91} + 24 q^{94} - 28 q^{97} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.73205 0.612372
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 1.73205 0.462910
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −6.00000 −1.27920
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 3.46410 0.679366
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 6.92820 1.12390
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.3923 1.62301 0.811503 0.584349i $$-0.198650\pi$$
0.811503 + 0.584349i $$0.198650\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 3.46410 0.522233
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 6.92820 0.951662 0.475831 0.879537i $$-0.342147\pi$$
0.475831 + 0.879537i $$0.342147\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.73205 −0.231455
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 6.92820 0.879883
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −3.46410 −0.420084
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.3923 −1.23334 −0.616670 0.787222i $$-0.711519\pi$$
−0.616670 + 0.787222i $$0.711519\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 3.46410 0.402694
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −3.46410 −0.394771
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −18.0000 −1.98777
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −6.92820 −0.747087
$$87$$ 0 0
$$88$$ 6.00000 0.639602
$$89$$ −3.46410 −0.367194 −0.183597 0.983002i $$-0.558774\pi$$
−0.183597 + 0.983002i $$0.558774\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 3.46410 0.361158
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −1.73205 −0.174964
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.46410 0.344691 0.172345 0.985037i $$-0.444865\pi$$
0.172345 + 0.985037i $$0.444865\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −3.46410 −0.339683
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ −17.3205 −1.67444 −0.837218 0.546869i $$-0.815820\pi$$
−0.837218 + 0.546869i $$0.815820\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.00000 0.472456
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 3.46410 0.317554
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 17.3205 1.56813
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −13.8564 −1.21064 −0.605320 0.795982i $$-0.706955\pi$$
−0.605320 + 0.795982i $$0.706955\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ −6.92820 −0.598506
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 6.92820 0.591916 0.295958 0.955201i $$-0.404361\pi$$
0.295958 + 0.955201i $$0.404361\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 18.0000 1.51053
$$143$$ −6.92820 −0.579365
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 24.2487 2.00684
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 6.92820 0.567581 0.283790 0.958886i $$-0.408408\pi$$
0.283790 + 0.958886i $$0.408408\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −6.92820 −0.561951
$$153$$ 0 0
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −13.8564 −1.10236
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.46410 −0.273009
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 10.3923 0.811503
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 20.7846 1.60836 0.804181 0.594385i $$-0.202604\pi$$
0.804181 + 0.594385i $$0.202604\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ −17.3205 −1.31685 −0.658427 0.752645i $$-0.728778\pi$$
−0.658427 + 0.752645i $$0.728778\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −17.3205 −1.30558
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −17.3205 −1.29460 −0.647298 0.762237i $$-0.724101\pi$$
−0.647298 + 0.762237i $$0.724101\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −3.46410 −0.256776
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ −6.92820 −0.505291
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.2487 −1.75458 −0.877288 0.479965i $$-0.840649\pi$$
−0.877288 + 0.479965i $$0.840649\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 24.2487 1.74096
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −20.7846 −1.48084 −0.740421 0.672143i $$-0.765374\pi$$
−0.740421 + 0.672143i $$0.765374\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −6.92820 −0.482711
$$207$$ 0 0
$$208$$ 10.0000 0.693375
$$209$$ −13.8564 −0.958468
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.92820 0.475831
$$213$$ 0 0
$$214$$ 30.0000 2.05076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ −3.46410 −0.234619
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.92820 0.466041
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −5.19615 −0.347183
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6.92820 0.459841 0.229920 0.973209i $$-0.426153\pi$$
0.229920 + 0.973209i $$0.426153\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.92820 −0.453882 −0.226941 0.973909i $$-0.572872\pi$$
−0.226941 + 0.973909i $$0.572872\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.92820 −0.450988
$$237$$ 0 0
$$238$$ −6.00000 −0.388922
$$239$$ 10.3923 0.672222 0.336111 0.941822i $$-0.390888\pi$$
0.336111 + 0.941822i $$0.390888\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −1.73205 −0.111340
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ −6.92820 −0.439941
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.7846 1.31191 0.655956 0.754799i $$-0.272265\pi$$
0.655956 + 0.754799i $$0.272265\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 13.8564 0.869428
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ 3.46410 0.216085 0.108042 0.994146i $$-0.465542\pi$$
0.108042 + 0.994146i $$0.465542\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 24.0000 1.48272
$$263$$ 17.3205 1.06803 0.534014 0.845476i $$-0.320683\pi$$
0.534014 + 0.845476i $$0.320683\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.92820 −0.424795
$$267$$ 0 0
$$268$$ 4.00000 0.244339
$$269$$ −17.3205 −1.05605 −0.528025 0.849229i $$-0.677067\pi$$
−0.528025 + 0.849229i $$0.677067\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 17.3205 1.05021
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 27.7128 1.66210
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −20.7846 −1.23991 −0.619953 0.784639i $$-0.712848\pi$$
−0.619953 + 0.784639i $$0.712848\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −10.3923 −0.616670
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ −10.3923 −0.613438
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ 10.3923 0.607125 0.303562 0.952812i $$-0.401824\pi$$
0.303562 + 0.952812i $$0.401824\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −3.46410 −0.201347
$$297$$ 0 0
$$298$$ −12.0000 −0.695141
$$299$$ −6.92820 −0.400668
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ −13.8564 −0.797347
$$303$$ 0 0
$$304$$ 20.0000 1.14708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ −3.46410 −0.197386
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 34.6410 1.96431 0.982156 0.188069i $$-0.0602227\pi$$
0.982156 + 0.188069i $$0.0602227\pi$$
$$312$$ 0 0
$$313$$ −2.00000 −0.113047 −0.0565233 0.998401i $$-0.518002\pi$$
−0.0565233 + 0.998401i $$0.518002\pi$$
$$314$$ −17.3205 −0.977453
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −6.92820 −0.389127 −0.194563 0.980890i $$-0.562329\pi$$
−0.194563 + 0.980890i $$0.562329\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.00000 0.334367
$$323$$ 13.8564 0.770991
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 34.6410 1.91859
$$327$$ 0 0
$$328$$ 18.0000 0.993884
$$329$$ 6.92820 0.381964
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −36.0000 −1.96983
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 15.5885 0.847900
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −13.8564 −0.750366
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 6.92820 0.373544
$$345$$ 0 0
$$346$$ 30.0000 1.61281
$$347$$ 17.3205 0.929814 0.464907 0.885360i $$-0.346088\pi$$
0.464907 + 0.885360i $$0.346088\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 18.0000 0.959403
$$353$$ −3.46410 −0.184376 −0.0921878 0.995742i $$-0.529386\pi$$
−0.0921878 + 0.995742i $$0.529386\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3.46410 −0.183597
$$357$$ 0 0
$$358$$ 30.0000 1.58555
$$359$$ −24.2487 −1.27980 −0.639899 0.768459i $$-0.721024\pi$$
−0.639899 + 0.768459i $$0.721024\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −3.46410 −0.182069
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ −17.3205 −0.902894
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.92820 −0.359694
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ 20.7846 1.07475
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 42.0000 2.14891
$$383$$ 13.8564 0.708029 0.354015 0.935240i $$-0.384816\pi$$
0.354015 + 0.935240i $$0.384816\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 24.2487 1.23423
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 13.8564 0.702548 0.351274 0.936273i $$-0.385749\pi$$
0.351274 + 0.936273i $$0.385749\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ 1.73205 0.0874818
$$393$$ 0 0
$$394$$ 36.0000 1.81365
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −38.0000 −1.90717 −0.953583 0.301131i $$-0.902636\pi$$
−0.953583 + 0.301131i $$0.902636\pi$$
$$398$$ 27.7128 1.38912
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.92820 0.345978 0.172989 0.984924i $$-0.444657\pi$$
0.172989 + 0.984924i $$0.444657\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 3.46410 0.172345
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.92820 −0.343418
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 6.92820 0.340915
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.3923 −0.509525
$$417$$ 0 0
$$418$$ 24.0000 1.17388
$$419$$ 20.7846 1.01539 0.507697 0.861536i $$-0.330497\pi$$
0.507697 + 0.861536i $$0.330497\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −34.6410 −1.68630
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ −17.3205 −0.837218
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.46410 0.166860 0.0834300 0.996514i $$-0.473413\pi$$
0.0834300 + 0.996514i $$0.473413\pi$$
$$432$$ 0 0
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ −6.92820 −0.332564
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −13.8564 −0.662842
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ 3.46410 0.164584 0.0822922 0.996608i $$-0.473776\pi$$
0.0822922 + 0.996608i $$0.473776\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 13.8564 0.656120
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ 41.5692 1.96177 0.980886 0.194581i $$-0.0623348\pi$$
0.980886 + 0.194581i $$0.0623348\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 38.1051 1.78054
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 31.1769 1.45205 0.726027 0.687666i $$-0.241365\pi$$
0.726027 + 0.687666i $$0.241365\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ −6.92820 −0.320599 −0.160300 0.987068i $$-0.551246\pi$$
−0.160300 + 0.987068i $$0.551246\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 13.8564 0.637118
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3.46410 0.158777
$$477$$ 0 0
$$478$$ −18.0000 −0.823301
$$479$$ −6.92820 −0.316558 −0.158279 0.987394i $$-0.550594\pi$$
−0.158279 + 0.987394i $$0.550594\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 17.3205 0.788928
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ −17.3205 −0.784063
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10.3923 0.468998 0.234499 0.972116i $$-0.424655\pi$$
0.234499 + 0.972116i $$0.424655\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −13.8564 −0.623429
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ 10.3923 0.466159
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −36.0000 −1.60676
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −20.7846 −0.923989
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −3.46410 −0.153544 −0.0767718 0.997049i $$-0.524461\pi$$
−0.0767718 + 0.997049i $$0.524461\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ −8.66025 −0.382733
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ −3.46410 −0.152204
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.46410 0.151765 0.0758825 0.997117i $$-0.475823\pi$$
0.0758825 + 0.997117i $$0.475823\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −13.8564 −0.605320
$$525$$ 0 0
$$526$$ −30.0000 −1.30806
$$527$$ 13.8564 0.603595
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −20.7846 −0.900281
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 6.92820 0.299253
$$537$$ 0 0
$$538$$ 30.0000 1.29339
$$539$$ 3.46410 0.149209
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ −34.6410 −1.48796
$$543$$ 0 0
$$544$$ −18.0000 −0.771744
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 6.92820 0.295958
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ −17.3205 −0.735878
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ 6.92820 0.293557 0.146779 0.989169i $$-0.453109\pi$$
0.146779 + 0.989169i $$0.453109\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 36.0000 1.51857
$$563$$ −34.6410 −1.45994 −0.729972 0.683477i $$-0.760467\pi$$
−0.729972 + 0.683477i $$0.760467\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −6.92820 −0.291214
$$567$$ 0 0
$$568$$ −18.0000 −0.755263
$$569$$ 6.92820 0.290445 0.145223 0.989399i $$-0.453610\pi$$
0.145223 + 0.989399i $$0.453610\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ −6.92820 −0.289683
$$573$$ 0 0
$$574$$ 18.0000 0.751305
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 8.66025 0.360219
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 24.0000 0.993978
$$584$$ −24.2487 −1.00342
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 20.7846 0.857873 0.428936 0.903335i $$-0.358888\pi$$
0.428936 + 0.903335i $$0.358888\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000 0.410997
$$593$$ 24.2487 0.995775 0.497888 0.867242i $$-0.334109\pi$$
0.497888 + 0.867242i $$0.334109\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.92820 0.283790
$$597$$ 0 0
$$598$$ 12.0000 0.490716
$$599$$ 45.0333 1.84001 0.920006 0.391905i $$-0.128184\pi$$
0.920006 + 0.391905i $$0.128184\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 6.92820 0.282372
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −20.7846 −0.842927
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 13.8564 0.560570
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ −48.4974 −1.95720
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ 20.7846 0.836757 0.418378 0.908273i $$-0.362599\pi$$
0.418378 + 0.908273i $$0.362599\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −60.0000 −2.40578
$$623$$ 3.46410 0.138786
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 3.46410 0.138453
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 6.92820 0.276246
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 13.8564 0.551178
$$633$$ 0 0
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −48.4974 −1.91553 −0.957767 0.287547i $$-0.907160\pi$$
−0.957767 + 0.287547i $$0.907160\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ −3.46410 −0.136505
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 6.92820 0.272376 0.136188 0.990683i $$-0.456515\pi$$
0.136188 + 0.990683i $$0.456515\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ −27.7128 −1.08449 −0.542243 0.840222i $$-0.682425\pi$$
−0.542243 + 0.840222i $$0.682425\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −51.9615 −2.02876
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ −10.3923 −0.404827 −0.202413 0.979300i $$-0.564878\pi$$
−0.202413 + 0.979300i $$0.564878\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −34.6410 −1.34636
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 20.7846 0.804181
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −34.6410 −1.33730
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ 24.2487 0.934025
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 24.2487 0.931954 0.465977 0.884797i $$-0.345703\pi$$
0.465977 + 0.884797i $$0.345703\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 24.0000 0.919007
$$683$$ −24.2487 −0.927851 −0.463926 0.885874i $$-0.653559\pi$$
−0.463926 + 0.885874i $$0.653559\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.73205 0.0661300
$$687$$ 0 0
$$688$$ −20.0000 −0.762493
$$689$$ −13.8564 −0.527887
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −17.3205 −0.658427
$$693$$ 0 0
$$694$$ −30.0000 −1.13878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ −24.2487 −0.917827
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ 3.46410 0.130558
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ −3.46410 −0.130281
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ −13.8564 −0.518927
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −17.3205 −0.647298
$$717$$ 0 0
$$718$$ 42.0000 1.56743
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 5.19615 0.193381
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.00000 0.148352 0.0741759 0.997245i $$-0.476367\pi$$
0.0741759 + 0.997245i $$0.476367\pi$$
$$728$$ 3.46410 0.128388
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −13.8564 −0.512498
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ −27.7128 −1.02290
$$735$$ 0 0
$$736$$ 18.0000 0.663489
$$737$$ 13.8564 0.510407
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ 10.3923 0.381257 0.190628 0.981662i $$-0.438947\pi$$
0.190628 + 0.981662i $$0.438947\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −17.3205 −0.634149
$$747$$ 0 0
$$748$$ −12.0000 −0.438763
$$749$$ 17.3205 0.632878
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 34.6410 1.26323
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 48.4974 1.76151
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 38.1051 1.38131 0.690655 0.723185i $$-0.257322\pi$$
0.690655 + 0.723185i $$0.257322\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ −24.2487 −0.877288
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ −45.0333 −1.61974 −0.809868 0.586612i $$-0.800461\pi$$
−0.809868 + 0.586612i $$0.800461\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −24.2487 −0.870478
$$777$$ 0 0
$$778$$ −24.0000 −0.860442
$$779$$ −41.5692 −1.48937
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ 20.7846 0.743256
$$783$$ 0 0
$$784$$ −5.00000 −0.178571
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −20.7846 −0.740421
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 65.8179 2.33579
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −31.1769 −1.10434 −0.552171 0.833731i $$-0.686201\pi$$
−0.552171 + 0.833731i $$0.686201\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −12.0000 −0.423735
$$803$$ −48.4974 −1.71144
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −13.8564 −0.488071
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ −27.7128 −0.974331 −0.487165 0.873310i $$-0.661969\pi$$
−0.487165 + 0.873310i $$0.661969\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −24.2487 −0.847836
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.92820 0.241796 0.120898 0.992665i $$-0.461423\pi$$
0.120898 + 0.992665i $$0.461423\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 6.92820 0.241355
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 10.3923 0.361376 0.180688 0.983540i $$-0.442168\pi$$
0.180688 + 0.983540i $$0.442168\pi$$
$$828$$ 0 0
$$829$$ 50.0000 1.73657 0.868286 0.496064i $$-0.165222\pi$$
0.868286 + 0.496064i $$0.165222\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ −3.46410 −0.120024
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −13.8564 −0.479234
$$837$$ 0 0
$$838$$ −36.0000 −1.24360
$$839$$ 20.7846 0.717564 0.358782 0.933421i $$-0.383192\pi$$
0.358782 + 0.933421i $$0.383192\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 17.3205 0.596904
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ −34.6410 −1.18958
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −6.92820 −0.237496
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ −17.3205 −0.592696
$$855$$ 0 0
$$856$$ −30.0000 −1.02538
$$857$$ 17.3205 0.591657 0.295829 0.955241i $$-0.404404\pi$$
0.295829 + 0.955241i $$0.404404\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −6.00000 −0.204361
$$863$$ −38.1051 −1.29711 −0.648557 0.761166i $$-0.724627\pi$$
−0.648557 + 0.761166i $$0.724627\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 45.0333 1.53029
$$867$$ 0 0
$$868$$ 4.00000 0.135769
$$869$$ 27.7128 0.940093
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 3.46410 0.117309
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 27.7128 0.935262
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −51.9615 −1.75063 −0.875314 0.483555i $$-0.839345\pi$$
−0.875314 + 0.483555i $$0.839345\pi$$
$$882$$ 0 0
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 6.92820 0.233021
$$885$$ 0 0
$$886$$ −6.00000 −0.201574
$$887$$ −6.92820 −0.232626 −0.116313 0.993213i $$-0.537108\pi$$
−0.116313 + 0.993213i $$0.537108\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ 27.7128 0.927374
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 12.1244 0.405046
$$897$$ 0 0
$$898$$ −72.0000 −2.40267
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ −62.3538 −2.07616
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 6.92820 0.229920
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31.1769 1.03294 0.516469 0.856306i $$-0.327246\pi$$
0.516469 + 0.856306i $$0.327246\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −17.3205 −0.572911
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 13.8564 0.457579
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −54.0000 −1.77840
$$923$$ 20.7846 0.684134
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 55.4256 1.82140
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −45.0333 −1.47750 −0.738748 0.673982i $$-0.764582\pi$$
−0.738748 + 0.673982i $$0.764582\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ −6.92820 −0.226941
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 46.0000 1.50275 0.751377 0.659873i $$-0.229390\pi$$
0.751377 + 0.659873i $$0.229390\pi$$
$$938$$ 6.92820 0.226214
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −17.3205 −0.564632 −0.282316 0.959321i $$-0.591103\pi$$
−0.282316 + 0.959321i $$0.591103\pi$$
$$942$$ 0 0
$$943$$ 36.0000 1.17232
$$944$$ 34.6410 1.12747
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ 24.2487 0.787977 0.393989 0.919115i $$-0.371095\pi$$
0.393989 + 0.919115i $$0.371095\pi$$
$$948$$ 0 0
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.00000 0.194461
$$953$$ 41.5692 1.34656 0.673280 0.739388i $$-0.264885\pi$$
0.673280 + 0.739388i $$0.264885\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 10.3923 0.336111
$$957$$ 0 0
$$958$$ 12.0000 0.387702
$$959$$ −6.92820 −0.223723
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −6.92820 −0.223374
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 1.73205 0.0556702
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.7128 0.889346 0.444673 0.895693i $$-0.353320\pi$$
0.444673 + 0.895693i $$0.353320\pi$$
$$972$$ 0 0
$$973$$ 16.0000 0.512936
$$974$$ −69.2820 −2.21994
$$975$$ 0 0
$$976$$ 50.0000 1.60046
$$977$$ −34.6410 −1.10826 −0.554132 0.832429i $$-0.686950\pi$$
−0.554132 + 0.832429i $$0.686950\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −18.0000 −0.574403
$$983$$ −13.8564 −0.441951 −0.220975 0.975279i $$-0.570924\pi$$
−0.220975 + 0.975279i $$0.570924\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 13.8564 0.440608
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ −20.7846 −0.659912
$$993$$ 0 0
$$994$$ −18.0000 −0.570925
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 6.92820 0.219308
$$999$$ 0 0
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 1575.2.a.q.1.1 2
3.2 odd 2 inner 1575.2.a.q.1.2 2
5.2 odd 4 1575.2.d.i.1324.1 4
5.3 odd 4 1575.2.d.i.1324.4 4
5.4 even 2 63.2.a.b.1.2 yes 2
15.2 even 4 1575.2.d.i.1324.3 4
15.8 even 4 1575.2.d.i.1324.2 4
15.14 odd 2 63.2.a.b.1.1 2
20.19 odd 2 1008.2.a.n.1.1 2
35.4 even 6 441.2.e.j.226.1 4
35.9 even 6 441.2.e.j.361.1 4
35.19 odd 6 441.2.e.i.361.1 4
35.24 odd 6 441.2.e.i.226.1 4
35.34 odd 2 441.2.a.g.1.2 2
40.19 odd 2 4032.2.a.bq.1.2 2
40.29 even 2 4032.2.a.bt.1.2 2
45.4 even 6 567.2.f.j.379.1 4
45.14 odd 6 567.2.f.j.379.2 4
45.29 odd 6 567.2.f.j.190.2 4
45.34 even 6 567.2.f.j.190.1 4
55.54 odd 2 7623.2.a.bi.1.1 2
60.59 even 2 1008.2.a.n.1.2 2
105.44 odd 6 441.2.e.j.361.2 4
105.59 even 6 441.2.e.i.226.2 4
105.74 odd 6 441.2.e.j.226.2 4
105.89 even 6 441.2.e.i.361.2 4
105.104 even 2 441.2.a.g.1.1 2
120.29 odd 2 4032.2.a.bt.1.1 2
120.59 even 2 4032.2.a.bq.1.1 2
140.139 even 2 7056.2.a.cm.1.2 2
165.164 even 2 7623.2.a.bi.1.2 2
420.419 odd 2 7056.2.a.cm.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 2 15.14 odd 2
63.2.a.b.1.2 yes 2 5.4 even 2
441.2.a.g.1.1 2 105.104 even 2
441.2.a.g.1.2 2 35.34 odd 2
441.2.e.i.226.1 4 35.24 odd 6
441.2.e.i.226.2 4 105.59 even 6
441.2.e.i.361.1 4 35.19 odd 6
441.2.e.i.361.2 4 105.89 even 6
441.2.e.j.226.1 4 35.4 even 6
441.2.e.j.226.2 4 105.74 odd 6
441.2.e.j.361.1 4 35.9 even 6
441.2.e.j.361.2 4 105.44 odd 6
567.2.f.j.190.1 4 45.34 even 6
567.2.f.j.190.2 4 45.29 odd 6
567.2.f.j.379.1 4 45.4 even 6
567.2.f.j.379.2 4 45.14 odd 6
1008.2.a.n.1.1 2 20.19 odd 2
1008.2.a.n.1.2 2 60.59 even 2
1575.2.a.q.1.1 2 1.1 even 1 trivial
1575.2.a.q.1.2 2 3.2 odd 2 inner
1575.2.d.i.1324.1 4 5.2 odd 4
1575.2.d.i.1324.2 4 15.8 even 4
1575.2.d.i.1324.3 4 15.2 even 4
1575.2.d.i.1324.4 4 5.3 odd 4
4032.2.a.bq.1.1 2 120.59 even 2
4032.2.a.bq.1.2 2 40.19 odd 2
4032.2.a.bt.1.1 2 120.29 odd 2
4032.2.a.bt.1.2 2 40.29 even 2
7056.2.a.cm.1.1 2 420.419 odd 2
7056.2.a.cm.1.2 2 140.139 even 2
7623.2.a.bi.1.1 2 55.54 odd 2
7623.2.a.bi.1.2 2 165.164 even 2