# Properties

 Label 900.4.d.i.649.2 Level $900$ Weight $4$ Character 900.649 Analytic conductor $53.102$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$53.1017190052$$ Analytic rank: $$0$$ 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: $$2$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 900.649 Dual form 900.4.d.i.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{7} +O(q^{10})$$ $$q+2.00000i q^{7} +30.0000 q^{11} +4.00000i q^{13} +90.0000i q^{17} +28.0000 q^{19} -120.000i q^{23} -210.000 q^{29} -4.00000 q^{31} +200.000i q^{37} +240.000 q^{41} +136.000i q^{43} -120.000i q^{47} +339.000 q^{49} +30.0000i q^{53} +450.000 q^{59} -166.000 q^{61} +908.000i q^{67} -1020.00 q^{71} +250.000i q^{73} +60.0000i q^{77} +916.000 q^{79} +1140.00i q^{83} +420.000 q^{89} -8.00000 q^{91} +1538.00i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 60 q^{11} + 56 q^{19} - 420 q^{29} - 8 q^{31} + 480 q^{41} + 678 q^{49} + 900 q^{59} - 332 q^{61} - 2040 q^{71} + 1832 q^{79} + 840 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q + 60 * q^11 + 56 * q^19 - 420 * q^29 - 8 * q^31 + 480 * q^41 + 678 * q^49 + 900 * q^59 - 332 * q^61 - 2040 * q^71 + 1832 * q^79 + 840 * q^89 - 16 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000i 0.107990i 0.998541 + 0.0539949i $$0.0171955\pi$$
−0.998541 + 0.0539949i $$0.982805\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 30.0000 0.822304 0.411152 0.911567i $$-0.365127\pi$$
0.411152 + 0.911567i $$0.365127\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 0.0853385i 0.999089 + 0.0426692i $$0.0135862\pi$$
−0.999089 + 0.0426692i $$0.986414\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 90.0000i 1.28401i 0.766700 + 0.642006i $$0.221898\pi$$
−0.766700 + 0.642006i $$0.778102\pi$$
$$18$$ 0 0
$$19$$ 28.0000 0.338086 0.169043 0.985609i $$-0.445932\pi$$
0.169043 + 0.985609i $$0.445932\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 120.000i − 1.08790i −0.839117 0.543951i $$-0.816928\pi$$
0.839117 0.543951i $$-0.183072\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −210.000 −1.34469 −0.672345 0.740238i $$-0.734713\pi$$
−0.672345 + 0.740238i $$0.734713\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.0231749 −0.0115874 0.999933i $$-0.503688\pi$$
−0.0115874 + 0.999933i $$0.503688\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 200.000i 0.888643i 0.895867 + 0.444322i $$0.146555\pi$$
−0.895867 + 0.444322i $$0.853445\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ 0 0
$$43$$ 136.000i 0.482321i 0.970485 + 0.241161i $$0.0775280\pi$$
−0.970485 + 0.241161i $$0.922472\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 120.000i − 0.372421i −0.982510 0.186211i $$-0.940379\pi$$
0.982510 0.186211i $$-0.0596207\pi$$
$$48$$ 0 0
$$49$$ 339.000 0.988338
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 30.0000i 0.0777513i 0.999244 + 0.0388756i $$0.0123776\pi$$
−0.999244 + 0.0388756i $$0.987622\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 450.000 0.992966 0.496483 0.868046i $$-0.334624\pi$$
0.496483 + 0.868046i $$0.334624\pi$$
$$60$$ 0 0
$$61$$ −166.000 −0.348428 −0.174214 0.984708i $$-0.555738\pi$$
−0.174214 + 0.984708i $$0.555738\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 908.000i 1.65567i 0.560972 + 0.827835i $$0.310428\pi$$
−0.560972 + 0.827835i $$0.689572\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1020.00 −1.70495 −0.852477 0.522765i $$-0.824901\pi$$
−0.852477 + 0.522765i $$0.824901\pi$$
$$72$$ 0 0
$$73$$ 250.000i 0.400826i 0.979712 + 0.200413i $$0.0642284\pi$$
−0.979712 + 0.200413i $$0.935772\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 60.0000i 0.0888004i
$$78$$ 0 0
$$79$$ 916.000 1.30453 0.652266 0.757990i $$-0.273818\pi$$
0.652266 + 0.757990i $$0.273818\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1140.00i 1.50761i 0.657101 + 0.753803i $$0.271783\pi$$
−0.657101 + 0.753803i $$0.728217\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 420.000 0.500224 0.250112 0.968217i $$-0.419533\pi$$
0.250112 + 0.968217i $$0.419533\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.00921569
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1538.00i 1.60990i 0.593343 + 0.804950i $$0.297808\pi$$
−0.593343 + 0.804950i $$0.702192\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 450.000 0.443333 0.221667 0.975122i $$-0.428850\pi$$
0.221667 + 0.975122i $$0.428850\pi$$
$$102$$ 0 0
$$103$$ 1150.00i 1.10012i 0.835124 + 0.550062i $$0.185396\pi$$
−0.835124 + 0.550062i $$0.814604\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1620.00i − 1.46366i −0.681489 0.731829i $$-0.738667\pi$$
0.681489 0.731829i $$-0.261333\pi$$
$$108$$ 0 0
$$109$$ 1702.00 1.49561 0.747807 0.663916i $$-0.231107\pi$$
0.747807 + 0.663916i $$0.231107\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 1350.00i − 1.12387i −0.827181 0.561935i $$-0.810057\pi$$
0.827181 0.561935i $$-0.189943\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −180.000 −0.138660
$$120$$ 0 0
$$121$$ −431.000 −0.323817
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2450.00i 1.71183i 0.517117 + 0.855915i $$0.327005\pi$$
−0.517117 + 0.855915i $$0.672995\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 690.000 0.460195 0.230098 0.973168i $$-0.426095\pi$$
0.230098 + 0.973168i $$0.426095\pi$$
$$132$$ 0 0
$$133$$ 56.0000i 0.0365099i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2070.00i 1.29089i 0.763806 + 0.645445i $$0.223328\pi$$
−0.763806 + 0.645445i $$0.776672\pi$$
$$138$$ 0 0
$$139$$ 1924.00 1.17404 0.587020 0.809572i $$-0.300301\pi$$
0.587020 + 0.809572i $$0.300301\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 120.000i 0.0701742i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2910.00 −1.59998 −0.799988 0.600016i $$-0.795161\pi$$
−0.799988 + 0.600016i $$0.795161\pi$$
$$150$$ 0 0
$$151$$ 176.000 0.0948522 0.0474261 0.998875i $$-0.484898\pi$$
0.0474261 + 0.998875i $$0.484898\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2348.00i 1.19357i 0.802400 + 0.596786i $$0.203556\pi$$
−0.802400 + 0.596786i $$0.796444\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 240.000 0.117482
$$162$$ 0 0
$$163$$ 1996.00i 0.959134i 0.877505 + 0.479567i $$0.159206\pi$$
−0.877505 + 0.479567i $$0.840794\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3120.00i 1.44571i 0.691002 + 0.722853i $$0.257170\pi$$
−0.691002 + 0.722853i $$0.742830\pi$$
$$168$$ 0 0
$$169$$ 2181.00 0.992717
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 1770.00i − 0.777865i −0.921266 0.388932i $$-0.872844\pi$$
0.921266 0.388932i $$-0.127156\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2130.00 0.889406 0.444703 0.895678i $$-0.353309\pi$$
0.444703 + 0.895678i $$0.353309\pi$$
$$180$$ 0 0
$$181$$ −1654.00 −0.679231 −0.339616 0.940564i $$-0.610297\pi$$
−0.339616 + 0.940564i $$0.610297\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2700.00i 1.05585i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1740.00 0.659173 0.329586 0.944125i $$-0.393091\pi$$
0.329586 + 0.944125i $$0.393091\pi$$
$$192$$ 0 0
$$193$$ − 86.0000i − 0.0320747i −0.999871 0.0160373i $$-0.994895\pi$$
0.999871 0.0160373i $$-0.00510506\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2490.00i − 0.900534i −0.892894 0.450267i $$-0.851329\pi$$
0.892894 0.450267i $$-0.148671\pi$$
$$198$$ 0 0
$$199$$ 832.000 0.296376 0.148188 0.988959i $$-0.452656\pi$$
0.148188 + 0.988959i $$0.452656\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 420.000i − 0.145213i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 840.000 0.278010
$$210$$ 0 0
$$211$$ 2084.00 0.679945 0.339973 0.940435i $$-0.389582\pi$$
0.339973 + 0.940435i $$0.389582\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.00000i − 0.00250265i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −360.000 −0.109576
$$222$$ 0 0
$$223$$ 1174.00i 0.352542i 0.984342 + 0.176271i $$0.0564035\pi$$
−0.984342 + 0.176271i $$0.943597\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 3120.00i − 0.912254i −0.889915 0.456127i $$-0.849236\pi$$
0.889915 0.456127i $$-0.150764\pi$$
$$228$$ 0 0
$$229$$ 58.0000 0.0167369 0.00836845 0.999965i $$-0.497336\pi$$
0.00836845 + 0.999965i $$0.497336\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5910.00i 1.66170i 0.556494 + 0.830852i $$0.312146\pi$$
−0.556494 + 0.830852i $$0.687854\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −3300.00 −0.893135 −0.446567 0.894750i $$-0.647354\pi$$
−0.446567 + 0.894750i $$0.647354\pi$$
$$240$$ 0 0
$$241$$ −2986.00 −0.798113 −0.399056 0.916926i $$-0.630662\pi$$
−0.399056 + 0.916926i $$0.630662\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 112.000i 0.0288518i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6630.00 −1.66726 −0.833629 0.552324i $$-0.813741\pi$$
−0.833629 + 0.552324i $$0.813741\pi$$
$$252$$ 0 0
$$253$$ − 3600.00i − 0.894585i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 1530.00i − 0.371357i −0.982611 0.185679i $$-0.940552\pi$$
0.982611 0.185679i $$-0.0594483\pi$$
$$258$$ 0 0
$$259$$ −400.000 −0.0959644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 2640.00i − 0.618971i −0.950904 0.309486i $$-0.899843\pi$$
0.950904 0.309486i $$-0.100157\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 7350.00 1.66594 0.832969 0.553319i $$-0.186639\pi$$
0.832969 + 0.553319i $$0.186639\pi$$
$$270$$ 0 0
$$271$$ 3512.00 0.787228 0.393614 0.919276i $$-0.371225\pi$$
0.393614 + 0.919276i $$0.371225\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 5368.00i − 1.16437i −0.813055 0.582187i $$-0.802197\pi$$
0.813055 0.582187i $$-0.197803\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3060.00 −0.649624 −0.324812 0.945779i $$-0.605301\pi$$
−0.324812 + 0.945779i $$0.605301\pi$$
$$282$$ 0 0
$$283$$ 5044.00i 1.05949i 0.848158 + 0.529743i $$0.177712\pi$$
−0.848158 + 0.529743i $$0.822288\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 480.000i 0.0987230i
$$288$$ 0 0
$$289$$ −3187.00 −0.648687
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2010.00i 0.400769i 0.979717 + 0.200385i $$0.0642192\pi$$
−0.979717 + 0.200385i $$0.935781\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 480.000 0.0928399
$$300$$ 0 0
$$301$$ −272.000 −0.0520858
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 2752.00i − 0.511612i −0.966728 0.255806i $$-0.917659\pi$$
0.966728 0.255806i $$-0.0823409\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9540.00 1.73943 0.869717 0.493551i $$-0.164301\pi$$
0.869717 + 0.493551i $$0.164301\pi$$
$$312$$ 0 0
$$313$$ − 9254.00i − 1.67114i −0.549384 0.835570i $$-0.685137\pi$$
0.549384 0.835570i $$-0.314863\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 150.000i 0.0265768i 0.999912 + 0.0132884i $$0.00422995\pi$$
−0.999912 + 0.0132884i $$0.995770\pi$$
$$318$$ 0 0
$$319$$ −6300.00 −1.10574
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2520.00i 0.434107i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 240.000 0.0402177
$$330$$ 0 0
$$331$$ 1892.00 0.314180 0.157090 0.987584i $$-0.449789\pi$$
0.157090 + 0.987584i $$0.449789\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 7378.00i − 1.19260i −0.802763 0.596299i $$-0.796637\pi$$
0.802763 0.596299i $$-0.203363\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −120.000 −0.0190568
$$342$$ 0 0
$$343$$ 1364.00i 0.214720i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6720.00i 1.03962i 0.854282 + 0.519811i $$0.173997\pi$$
−0.854282 + 0.519811i $$0.826003\pi$$
$$348$$ 0 0
$$349$$ −5186.00 −0.795416 −0.397708 0.917512i $$-0.630194\pi$$
−0.397708 + 0.917512i $$0.630194\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 3330.00i − 0.502091i −0.967975 0.251045i $$-0.919226\pi$$
0.967975 0.251045i $$-0.0807743\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −9000.00 −1.32312 −0.661562 0.749890i $$-0.730106\pi$$
−0.661562 + 0.749890i $$0.730106\pi$$
$$360$$ 0 0
$$361$$ −6075.00 −0.885698
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8758.00i − 1.24568i −0.782350 0.622839i $$-0.785979\pi$$
0.782350 0.622839i $$-0.214021\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −60.0000 −0.00839635
$$372$$ 0 0
$$373$$ − 4724.00i − 0.655763i −0.944719 0.327881i $$-0.893665\pi$$
0.944719 0.327881i $$-0.106335\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 840.000i − 0.114754i
$$378$$ 0 0
$$379$$ −7292.00 −0.988298 −0.494149 0.869377i $$-0.664520\pi$$
−0.494149 + 0.869377i $$0.664520\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 14520.0i − 1.93717i −0.248676 0.968587i $$-0.579996\pi$$
0.248676 0.968587i $$-0.420004\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −7110.00 −0.926713 −0.463356 0.886172i $$-0.653355\pi$$
−0.463356 + 0.886172i $$0.653355\pi$$
$$390$$ 0 0
$$391$$ 10800.0 1.39688
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 11488.0i − 1.45231i −0.687532 0.726154i $$-0.741306\pi$$
0.687532 0.726154i $$-0.258694\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 780.000 0.0971355 0.0485678 0.998820i $$-0.484534\pi$$
0.0485678 + 0.998820i $$0.484534\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.00197771i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6000.00i 0.730735i
$$408$$ 0 0
$$409$$ −5402.00 −0.653085 −0.326542 0.945183i $$-0.605884\pi$$
−0.326542 + 0.945183i $$0.605884\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 900.000i 0.107230i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2190.00 0.255342 0.127671 0.991817i $$-0.459250\pi$$
0.127671 + 0.991817i $$0.459250\pi$$
$$420$$ 0 0
$$421$$ −7162.00 −0.829108 −0.414554 0.910025i $$-0.636062\pi$$
−0.414554 + 0.910025i $$0.636062\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 332.000i − 0.0376267i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9360.00 1.04607 0.523034 0.852312i $$-0.324800\pi$$
0.523034 + 0.852312i $$0.324800\pi$$
$$432$$ 0 0
$$433$$ − 12806.0i − 1.42129i −0.703552 0.710643i $$-0.748404\pi$$
0.703552 0.710643i $$-0.251596\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 3360.00i − 0.367805i
$$438$$ 0 0
$$439$$ −11288.0 −1.22721 −0.613607 0.789612i $$-0.710282\pi$$
−0.613607 + 0.789612i $$0.710282\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 8520.00i 0.913764i 0.889527 + 0.456882i $$0.151034\pi$$
−0.889527 + 0.456882i $$0.848966\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1260.00 0.132434 0.0662172 0.997805i $$-0.478907\pi$$
0.0662172 + 0.997805i $$0.478907\pi$$
$$450$$ 0 0
$$451$$ 7200.00 0.751740
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 13750.0i − 1.40744i −0.710480 0.703718i $$-0.751522\pi$$
0.710480 0.703718i $$-0.248478\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3210.00 −0.324305 −0.162152 0.986766i $$-0.551844\pi$$
−0.162152 + 0.986766i $$0.551844\pi$$
$$462$$ 0 0
$$463$$ 12850.0i 1.28983i 0.764255 + 0.644914i $$0.223107\pi$$
−0.764255 + 0.644914i $$0.776893\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 8220.00i − 0.814510i −0.913314 0.407255i $$-0.866486\pi$$
0.913314 0.407255i $$-0.133514\pi$$
$$468$$ 0 0
$$469$$ −1816.00 −0.178795
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4080.00i 0.396614i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 7020.00 0.669628 0.334814 0.942284i $$-0.391326\pi$$
0.334814 + 0.942284i $$0.391326\pi$$
$$480$$ 0 0
$$481$$ −800.000 −0.0758355
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8122.00i − 0.755735i −0.925860 0.377868i $$-0.876657\pi$$
0.925860 0.377868i $$-0.123343\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13470.0 −1.23807 −0.619035 0.785363i $$-0.712476\pi$$
−0.619035 + 0.785363i $$0.712476\pi$$
$$492$$ 0 0
$$493$$ − 18900.0i − 1.72660i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 2040.00i − 0.184118i
$$498$$ 0 0
$$499$$ −2468.00 −0.221409 −0.110704 0.993853i $$-0.535311\pi$$
−0.110704 + 0.993853i $$0.535311\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 4440.00i 0.393578i 0.980446 + 0.196789i $$0.0630514\pi$$
−0.980446 + 0.196789i $$0.936949\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 11190.0 0.974436 0.487218 0.873280i $$-0.338012\pi$$
0.487218 + 0.873280i $$0.338012\pi$$
$$510$$ 0 0
$$511$$ −500.000 −0.0432851
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 3600.00i − 0.306243i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4020.00 −0.338041 −0.169021 0.985613i $$-0.554060\pi$$
−0.169021 + 0.985613i $$0.554060\pi$$
$$522$$ 0 0
$$523$$ 9076.00i 0.758826i 0.925228 + 0.379413i $$0.123874\pi$$
−0.925228 + 0.379413i $$0.876126\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 360.000i − 0.0297568i
$$528$$ 0 0
$$529$$ −2233.00 −0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 960.000i 0.0780154i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 10170.0 0.812714
$$540$$ 0 0
$$541$$ −7486.00 −0.594914 −0.297457 0.954735i $$-0.596138\pi$$
−0.297457 + 0.954735i $$0.596138\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 7400.00i 0.578430i 0.957264 + 0.289215i $$0.0933942\pi$$
−0.957264 + 0.289215i $$0.906606\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5880.00 −0.454621
$$552$$ 0 0
$$553$$ 1832.00i 0.140876i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 11490.0i 0.874052i 0.899449 + 0.437026i $$0.143968\pi$$
−0.899449 + 0.437026i $$0.856032\pi$$
$$558$$ 0 0
$$559$$ −544.000 −0.0411606
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 19320.0i 1.44625i 0.690715 + 0.723127i $$0.257296\pi$$
−0.690715 + 0.723127i $$0.742704\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −8340.00 −0.614466 −0.307233 0.951634i $$-0.599403\pi$$
−0.307233 + 0.951634i $$0.599403\pi$$
$$570$$ 0 0
$$571$$ 21044.0 1.54232 0.771159 0.636642i $$-0.219677\pi$$
0.771159 + 0.636642i $$0.219677\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1418.00i 0.102309i 0.998691 + 0.0511543i $$0.0162900\pi$$
−0.998691 + 0.0511543i $$0.983710\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −2280.00 −0.162806
$$582$$ 0 0
$$583$$ 900.000i 0.0639351i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 22020.0i − 1.54832i −0.632991 0.774159i $$-0.718173\pi$$
0.632991 0.774159i $$-0.281827\pi$$
$$588$$ 0 0
$$589$$ −112.000 −0.00783511
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 25230.0i − 1.74717i −0.486671 0.873585i $$-0.661789\pi$$
0.486671 0.873585i $$-0.338211\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8280.00 −0.564794 −0.282397 0.959298i $$-0.591130\pi$$
−0.282397 + 0.959298i $$0.591130\pi$$
$$600$$ 0 0
$$601$$ −18874.0 −1.28101 −0.640505 0.767954i $$-0.721275\pi$$
−0.640505 + 0.767954i $$0.721275\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 10550.0i 0.705455i 0.935726 + 0.352728i $$0.114746\pi$$
−0.935726 + 0.352728i $$0.885254\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 480.000 0.0317819
$$612$$ 0 0
$$613$$ − 11000.0i − 0.724773i −0.932028 0.362386i $$-0.881962\pi$$
0.932028 0.362386i $$-0.118038\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 11310.0i − 0.737963i −0.929437 0.368982i $$-0.879706\pi$$
0.929437 0.368982i $$-0.120294\pi$$
$$618$$ 0 0
$$619$$ 17572.0 1.14100 0.570499 0.821298i $$-0.306750\pi$$
0.570499 + 0.821298i $$0.306750\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 840.000i 0.0540191i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −18000.0 −1.14103
$$630$$ 0 0
$$631$$ 1604.00 0.101195 0.0505976 0.998719i $$-0.483887\pi$$
0.0505976 + 0.998719i $$0.483887\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1356.00i 0.0843433i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 31320.0 1.92990 0.964950 0.262435i $$-0.0845254\pi$$
0.964950 + 0.262435i $$0.0845254\pi$$
$$642$$ 0 0
$$643$$ 31300.0i 1.91968i 0.280555 + 0.959838i $$0.409481\pi$$
−0.280555 + 0.959838i $$0.590519\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 10920.0i 0.663539i 0.943361 + 0.331769i $$0.107646\pi$$
−0.943361 + 0.331769i $$0.892354\pi$$
$$648$$ 0 0
$$649$$ 13500.0 0.816520
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3210.00i 0.192369i 0.995364 + 0.0961845i $$0.0306639\pi$$
−0.995364 + 0.0961845i $$0.969336\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 11910.0 0.704018 0.352009 0.935997i $$-0.385499\pi$$
0.352009 + 0.935997i $$0.385499\pi$$
$$660$$ 0 0
$$661$$ −3382.00 −0.199008 −0.0995042 0.995037i $$-0.531726\pi$$
−0.0995042 + 0.995037i $$0.531726\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 25200.0i 1.46289i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4980.00 −0.286514
$$672$$ 0 0
$$673$$ − 15950.0i − 0.913562i −0.889579 0.456781i $$-0.849002\pi$$
0.889579 0.456781i $$-0.150998\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 32190.0i 1.82742i 0.406369 + 0.913709i $$0.366795\pi$$
−0.406369 + 0.913709i $$0.633205\pi$$
$$678$$ 0 0
$$679$$ −3076.00 −0.173853
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 22140.0i − 1.24036i −0.784461 0.620178i $$-0.787060\pi$$
0.784461 0.620178i $$-0.212940\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −120.000 −0.00663518
$$690$$ 0 0
$$691$$ −6172.00 −0.339789 −0.169894 0.985462i $$-0.554343\pi$$
−0.169894 + 0.985462i $$0.554343\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 21600.0i 1.17383i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19170.0 1.03287 0.516434 0.856327i $$-0.327259\pi$$
0.516434 + 0.856327i $$0.327259\pi$$
$$702$$ 0 0
$$703$$ 5600.00i 0.300438i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 900.000i 0.0478755i
$$708$$ 0 0
$$709$$ 21898.0 1.15994 0.579969 0.814638i $$-0.303064\pi$$
0.579969 + 0.814638i $$0.303064\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 480.000i 0.0252120i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −16680.0 −0.865173 −0.432586 0.901593i $$-0.642399\pi$$
−0.432586 + 0.901593i $$0.642399\pi$$
$$720$$ 0 0
$$721$$ −2300.00 −0.118802
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 6518.00i 0.332516i 0.986082 + 0.166258i $$0.0531685\pi$$
−0.986082 + 0.166258i $$0.946832\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −12240.0 −0.619306
$$732$$ 0 0
$$733$$ 23200.0i 1.16905i 0.811377 + 0.584524i $$0.198719\pi$$
−0.811377 + 0.584524i $$0.801281\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 27240.0i 1.36146i
$$738$$ 0 0
$$739$$ 16324.0 0.812568 0.406284 0.913747i $$-0.366824\pi$$
0.406284 + 0.913747i $$0.366824\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 120.000i 0.00592513i 0.999996 + 0.00296257i $$0.000943015\pi$$
−0.999996 + 0.00296257i $$0.999057\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3240.00 0.158060
$$750$$ 0 0
$$751$$ 30548.0 1.48430 0.742152 0.670232i $$-0.233805\pi$$
0.742152 + 0.670232i $$0.233805\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 16952.0i 0.813911i 0.913448 + 0.406956i $$0.133410\pi$$
−0.913448 + 0.406956i $$0.866590\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −20220.0 −0.963173 −0.481586 0.876399i $$-0.659939\pi$$
−0.481586 + 0.876399i $$0.659939\pi$$
$$762$$ 0 0
$$763$$ 3404.00i 0.161511i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1800.00i 0.0847382i
$$768$$ 0 0
$$769$$ 20722.0 0.971722 0.485861 0.874036i $$-0.338506\pi$$
0.485861 + 0.874036i $$0.338506\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 4350.00i − 0.202404i −0.994866 0.101202i $$-0.967731\pi$$
0.994866 0.101202i $$-0.0322689\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6720.00 0.309074
$$780$$ 0 0
$$781$$ −30600.0 −1.40199
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 41972.0i 1.90107i 0.310621 + 0.950534i $$0.399463\pi$$
−0.310621 + 0.950534i $$0.600537\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2700.00 0.121367
$$792$$ 0 0
$$793$$ − 664.000i − 0.0297343i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 39510.0i − 1.75598i −0.478679 0.877990i $$-0.658884\pi$$
0.478679 0.877990i $$-0.341116\pi$$
$$798$$ 0 0
$$799$$ 10800.0 0.478193
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 7500.00i 0.329601i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −16680.0 −0.724892 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$810$$ 0 0
$$811$$ −15484.0 −0.670428 −0.335214 0.942142i $$-0.608809\pi$$
−0.335214 + 0.942142i $$0.608809\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3808.00i 0.163066i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 4170.00 0.177264 0.0886322 0.996064i $$-0.471750\pi$$
0.0886322 + 0.996064i $$0.471750\pi$$
$$822$$ 0 0
$$823$$ 30226.0i 1.28021i 0.768288 + 0.640105i $$0.221109\pi$$
−0.768288 + 0.640105i $$0.778891\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 14760.0i − 0.620623i −0.950635 0.310312i $$-0.899567\pi$$
0.950635 0.310312i $$-0.100433\pi$$
$$828$$ 0 0
$$829$$ 9934.00 0.416191 0.208095 0.978109i $$-0.433274\pi$$
0.208095 + 0.978109i $$0.433274\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 30510.0i 1.26904i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 23520.0 0.967820 0.483910 0.875118i $$-0.339216\pi$$
0.483910 + 0.875118i $$0.339216\pi$$
$$840$$ 0 0
$$841$$ 19711.0 0.808192
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 862.000i − 0.0349689i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24000.0 0.966756
$$852$$ 0 0
$$853$$ − 29816.0i − 1.19681i −0.801193 0.598406i $$-0.795801\pi$$
0.801193 0.598406i $$-0.204199\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 35430.0i − 1.41221i −0.708106 0.706106i $$-0.750450\pi$$
0.708106 0.706106i $$-0.249550\pi$$
$$858$$ 0 0
$$859$$ 36196.0 1.43771 0.718854 0.695161i $$-0.244667\pi$$
0.718854 + 0.695161i $$0.244667\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 480.000i − 0.0189332i −0.999955 0.00946662i $$-0.996987\pi$$
0.999955 0.00946662i $$-0.00301336\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 27480.0 1.07272
$$870$$ 0 0
$$871$$ −3632.00 −0.141292
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28532.0i 1.09858i 0.835631 + 0.549291i $$0.185102\pi$$
−0.835631 + 0.549291i $$0.814898\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −20340.0 −0.777834 −0.388917 0.921273i $$-0.627151\pi$$
−0.388917 + 0.921273i $$0.627151\pi$$
$$882$$ 0 0
$$883$$ 10756.0i 0.409930i 0.978769 + 0.204965i $$0.0657081\pi$$
−0.978769 + 0.204965i $$0.934292\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 600.000i 0.0227125i 0.999936 + 0.0113563i $$0.00361489\pi$$
−0.999936 + 0.0113563i $$0.996385\pi$$
$$888$$ 0 0
$$889$$ −4900.00 −0.184860
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 3360.00i − 0.125911i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 840.000 0.0311630
$$900$$ 0 0
$$901$$ −2700.00 −0.0998336
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 25400.0i 0.929871i 0.885345 + 0.464936i $$0.153923\pi$$
−0.885345 + 0.464936i $$0.846077\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −36240.0 −1.31799 −0.658993 0.752149i $$-0.729017\pi$$
−0.658993 + 0.752149i $$0.729017\pi$$
$$912$$ 0 0
$$913$$ 34200.0i 1.23971i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1380.00i 0.0496964i
$$918$$ 0 0
$$919$$ −6572.00 −0.235898 −0.117949 0.993020i $$-0.537632\pi$$
−0.117949 + 0.993020i $$0.537632\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 4080.00i − 0.145498i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2340.00 −0.0826404 −0.0413202 0.999146i $$-0.513156\pi$$
−0.0413202 + 0.999146i $$0.513156\pi$$
$$930$$ 0 0
$$931$$ 9492.00 0.334144
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2522.00i 0.0879297i 0.999033 + 0.0439649i $$0.0139990\pi$$
−0.999033 + 0.0439649i $$0.986001\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −52770.0 −1.82811 −0.914056 0.405589i $$-0.867067\pi$$
−0.914056 + 0.405589i $$0.867067\pi$$
$$942$$ 0 0
$$943$$ − 28800.0i − 0.994546i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28200.0i 0.967663i 0.875161 + 0.483832i $$0.160755\pi$$
−0.875161 + 0.483832i $$0.839245\pi$$
$$948$$ 0 0
$$949$$ −1000.00 −0.0342059
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 15570.0i 0.529236i 0.964353 + 0.264618i $$0.0852458\pi$$
−0.964353 + 0.264618i $$0.914754\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −4140.00 −0.139403
$$960$$ 0 0
$$961$$ −29775.0 −0.999463
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 8350.00i − 0.277681i −0.990315 0.138841i $$-0.955662\pi$$
0.990315 0.138841i $$-0.0443376\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 43650.0 1.44263 0.721316 0.692606i $$-0.243538\pi$$
0.721316 + 0.692606i $$0.243538\pi$$
$$972$$ 0 0
$$973$$ 3848.00i 0.126784i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18810.0i 0.615952i 0.951394 + 0.307976i $$0.0996517\pi$$
−0.951394 + 0.307976i $$0.900348\pi$$
$$978$$ 0 0
$$979$$ 12600.0 0.411336
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 25320.0i 0.821549i 0.911737 + 0.410774i $$0.134742\pi$$
−0.911737 + 0.410774i $$0.865258\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16320.0 0.524718
$$990$$ 0 0
$$991$$ −6736.00 −0.215919 −0.107960 0.994155i $$-0.534432\pi$$
−0.107960 + 0.994155i $$0.534432\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 20500.0i − 0.651195i −0.945508 0.325598i $$-0.894435\pi$$
0.945508 0.325598i $$-0.105565\pi$$
$$998$$ 0 0
$$999$$ 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 900.4.d.i.649.2 2
3.2 odd 2 900.4.d.d.649.2 2
5.2 odd 4 900.4.a.j.1.1 1
5.3 odd 4 180.4.a.e.1.1 yes 1
5.4 even 2 inner 900.4.d.i.649.1 2
15.2 even 4 900.4.a.i.1.1 1
15.8 even 4 180.4.a.b.1.1 1
15.14 odd 2 900.4.d.d.649.1 2
20.3 even 4 720.4.a.w.1.1 1
45.13 odd 12 1620.4.i.c.541.1 2
45.23 even 12 1620.4.i.i.541.1 2
45.38 even 12 1620.4.i.i.1081.1 2
45.43 odd 12 1620.4.i.c.1081.1 2
60.23 odd 4 720.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
180.4.a.b.1.1 1 15.8 even 4
180.4.a.e.1.1 yes 1 5.3 odd 4
720.4.a.h.1.1 1 60.23 odd 4
720.4.a.w.1.1 1 20.3 even 4
900.4.a.i.1.1 1 15.2 even 4
900.4.a.j.1.1 1 5.2 odd 4
900.4.d.d.649.1 2 15.14 odd 2
900.4.d.d.649.2 2 3.2 odd 2
900.4.d.i.649.1 2 5.4 even 2 inner
900.4.d.i.649.2 2 1.1 even 1 trivial
1620.4.i.c.541.1 2 45.13 odd 12
1620.4.i.c.1081.1 2 45.43 odd 12
1620.4.i.i.541.1 2 45.23 even 12
1620.4.i.i.1081.1 2 45.38 even 12