# Properties

 Label 1350.4.c.q.649.2 Level $1350$ Weight $4$ Character 1350.649 Analytic conductor $79.653$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.6525785077$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1350.649 Dual form 1350.4.c.q.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} -8.00000i q^{8} +O(q^{10})$$ $$q+2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} -8.00000i q^{8} +42.0000 q^{11} -20.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +93.0000i q^{17} -59.0000 q^{19} +84.0000i q^{22} -9.00000i q^{23} +40.0000 q^{26} +16.0000i q^{28} -120.000 q^{29} +47.0000 q^{31} +32.0000i q^{32} -186.000 q^{34} -262.000i q^{37} -118.000i q^{38} +126.000 q^{41} +178.000i q^{43} -168.000 q^{44} +18.0000 q^{46} +144.000i q^{47} +327.000 q^{49} +80.0000i q^{52} -741.000i q^{53} -32.0000 q^{56} -240.000i q^{58} +444.000 q^{59} +221.000 q^{61} +94.0000i q^{62} -64.0000 q^{64} -538.000i q^{67} -372.000i q^{68} +690.000 q^{71} +1126.00i q^{73} +524.000 q^{74} +236.000 q^{76} -168.000i q^{77} -665.000 q^{79} +252.000i q^{82} -75.0000i q^{83} -356.000 q^{86} -336.000i q^{88} +1086.00 q^{89} -80.0000 q^{91} +36.0000i q^{92} -288.000 q^{94} +1544.00i q^{97} +654.000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 84 q^{11} + 16 q^{14} + 32 q^{16} - 118 q^{19} + 80 q^{26} - 240 q^{29} + 94 q^{31} - 372 q^{34} + 252 q^{41} - 336 q^{44} + 36 q^{46} + 654 q^{49} - 64 q^{56} + 888 q^{59} + 442 q^{61} - 128 q^{64} + 1380 q^{71} + 1048 q^{74} + 472 q^{76} - 1330 q^{79} - 712 q^{86} + 2172 q^{89} - 160 q^{91} - 576 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 84 * q^11 + 16 * q^14 + 32 * q^16 - 118 * q^19 + 80 * q^26 - 240 * q^29 + 94 * q^31 - 372 * q^34 + 252 * q^41 - 336 * q^44 + 36 * q^46 + 654 * q^49 - 64 * q^56 + 888 * q^59 + 442 * q^61 - 128 * q^64 + 1380 * q^71 + 1048 * q^74 + 472 * q^76 - 1330 * q^79 - 712 * q^86 + 2172 * q^89 - 160 * q^91 - 576 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 0.707107i
$$3$$ 0 0
$$4$$ −4.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.00000i − 0.215980i −0.994152 0.107990i $$-0.965559\pi$$
0.994152 0.107990i $$-0.0344414\pi$$
$$8$$ − 8.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 42.0000 1.15123 0.575613 0.817723i $$-0.304764\pi$$
0.575613 + 0.817723i $$0.304764\pi$$
$$12$$ 0 0
$$13$$ − 20.0000i − 0.426692i −0.976977 0.213346i $$-0.931564\pi$$
0.976977 0.213346i $$-0.0684362\pi$$
$$14$$ 8.00000 0.152721
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 93.0000i 1.32681i 0.748259 + 0.663406i $$0.230890\pi$$
−0.748259 + 0.663406i $$0.769110\pi$$
$$18$$ 0 0
$$19$$ −59.0000 −0.712396 −0.356198 0.934410i $$-0.615927\pi$$
−0.356198 + 0.934410i $$0.615927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 84.0000i 0.814039i
$$23$$ − 9.00000i − 0.0815926i −0.999167 0.0407963i $$-0.987011\pi$$
0.999167 0.0407963i $$-0.0129895\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 40.0000 0.301717
$$27$$ 0 0
$$28$$ 16.0000i 0.107990i
$$29$$ −120.000 −0.768395 −0.384197 0.923251i $$-0.625522\pi$$
−0.384197 + 0.923251i $$0.625522\pi$$
$$30$$ 0 0
$$31$$ 47.0000 0.272305 0.136152 0.990688i $$-0.456526\pi$$
0.136152 + 0.990688i $$0.456526\pi$$
$$32$$ 32.0000i 0.176777i
$$33$$ 0 0
$$34$$ −186.000 −0.938198
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 262.000i − 1.16412i −0.813145 0.582061i $$-0.802246\pi$$
0.813145 0.582061i $$-0.197754\pi$$
$$38$$ − 118.000i − 0.503740i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 126.000 0.479949 0.239974 0.970779i $$-0.422861\pi$$
0.239974 + 0.970779i $$0.422861\pi$$
$$42$$ 0 0
$$43$$ 178.000i 0.631273i 0.948880 + 0.315637i $$0.102218\pi$$
−0.948880 + 0.315637i $$0.897782\pi$$
$$44$$ −168.000 −0.575613
$$45$$ 0 0
$$46$$ 18.0000 0.0576947
$$47$$ 144.000i 0.446906i 0.974715 + 0.223453i $$0.0717328\pi$$
−0.974715 + 0.223453i $$0.928267\pi$$
$$48$$ 0 0
$$49$$ 327.000 0.953353
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 80.0000i 0.213346i
$$53$$ − 741.000i − 1.92046i −0.279217 0.960228i $$-0.590075\pi$$
0.279217 0.960228i $$-0.409925\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −32.0000 −0.0763604
$$57$$ 0 0
$$58$$ − 240.000i − 0.543337i
$$59$$ 444.000 0.979727 0.489863 0.871799i $$-0.337047\pi$$
0.489863 + 0.871799i $$0.337047\pi$$
$$60$$ 0 0
$$61$$ 221.000 0.463871 0.231936 0.972731i $$-0.425494\pi$$
0.231936 + 0.972731i $$0.425494\pi$$
$$62$$ 94.0000i 0.192549i
$$63$$ 0 0
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 538.000i − 0.981002i −0.871441 0.490501i $$-0.836814\pi$$
0.871441 0.490501i $$-0.163186\pi$$
$$68$$ − 372.000i − 0.663406i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 690.000 1.15335 0.576676 0.816973i $$-0.304350\pi$$
0.576676 + 0.816973i $$0.304350\pi$$
$$72$$ 0 0
$$73$$ 1126.00i 1.80532i 0.430355 + 0.902660i $$0.358388\pi$$
−0.430355 + 0.902660i $$0.641612\pi$$
$$74$$ 524.000 0.823159
$$75$$ 0 0
$$76$$ 236.000 0.356198
$$77$$ − 168.000i − 0.248641i
$$78$$ 0 0
$$79$$ −665.000 −0.947068 −0.473534 0.880776i $$-0.657022\pi$$
−0.473534 + 0.880776i $$0.657022\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 252.000i 0.339375i
$$83$$ − 75.0000i − 0.0991846i −0.998770 0.0495923i $$-0.984208\pi$$
0.998770 0.0495923i $$-0.0157922\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −356.000 −0.446378
$$87$$ 0 0
$$88$$ − 336.000i − 0.407020i
$$89$$ 1086.00 1.29344 0.646718 0.762729i $$-0.276141\pi$$
0.646718 + 0.762729i $$0.276141\pi$$
$$90$$ 0 0
$$91$$ −80.0000 −0.0921569
$$92$$ 36.0000i 0.0407963i
$$93$$ 0 0
$$94$$ −288.000 −0.316010
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1544.00i 1.61618i 0.589059 + 0.808090i $$0.299499\pi$$
−0.589059 + 0.808090i $$0.700501\pi$$
$$98$$ 654.000i 0.674122i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −132.000 −0.130044 −0.0650222 0.997884i $$-0.520712\pi$$
−0.0650222 + 0.997884i $$0.520712\pi$$
$$102$$ 0 0
$$103$$ 892.000i 0.853314i 0.904413 + 0.426657i $$0.140309\pi$$
−0.904413 + 0.426657i $$0.859691\pi$$
$$104$$ −160.000 −0.150859
$$105$$ 0 0
$$106$$ 1482.00 1.35797
$$107$$ − 1140.00i − 1.02998i −0.857196 0.514990i $$-0.827795\pi$$
0.857196 0.514990i $$-0.172205\pi$$
$$108$$ 0 0
$$109$$ 1735.00 1.52461 0.762307 0.647216i $$-0.224067\pi$$
0.762307 + 0.647216i $$0.224067\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 64.0000i − 0.0539949i
$$113$$ 1434.00i 1.19380i 0.802316 + 0.596900i $$0.203601\pi$$
−0.802316 + 0.596900i $$0.796399\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 480.000 0.384197
$$117$$ 0 0
$$118$$ 888.000i 0.692771i
$$119$$ 372.000 0.286565
$$120$$ 0 0
$$121$$ 433.000 0.325319
$$122$$ 442.000i 0.328007i
$$123$$ 0 0
$$124$$ −188.000 −0.136152
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 686.000i 0.479312i 0.970858 + 0.239656i $$0.0770347\pi$$
−0.970858 + 0.239656i $$0.922965\pi$$
$$128$$ − 128.000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −114.000 −0.0760323 −0.0380161 0.999277i $$-0.512104\pi$$
−0.0380161 + 0.999277i $$0.512104\pi$$
$$132$$ 0 0
$$133$$ 236.000i 0.153863i
$$134$$ 1076.00 0.693673
$$135$$ 0 0
$$136$$ 744.000 0.469099
$$137$$ 159.000i 0.0991554i 0.998770 + 0.0495777i $$0.0157875\pi$$
−0.998770 + 0.0495777i $$0.984212\pi$$
$$138$$ 0 0
$$139$$ −2276.00 −1.38883 −0.694417 0.719573i $$-0.744337\pi$$
−0.694417 + 0.719573i $$0.744337\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1380.00i 0.815542i
$$143$$ − 840.000i − 0.491219i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2252.00 −1.27655
$$147$$ 0 0
$$148$$ 1048.00i 0.582061i
$$149$$ 1398.00 0.768648 0.384324 0.923198i $$-0.374434\pi$$
0.384324 + 0.923198i $$0.374434\pi$$
$$150$$ 0 0
$$151$$ 2624.00 1.41416 0.707080 0.707134i $$-0.250012\pi$$
0.707080 + 0.707134i $$0.250012\pi$$
$$152$$ 472.000i 0.251870i
$$153$$ 0 0
$$154$$ 336.000 0.175816
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 394.000i − 0.200284i −0.994973 0.100142i $$-0.968070\pi$$
0.994973 0.100142i $$-0.0319297\pi$$
$$158$$ − 1330.00i − 0.669678i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −36.0000 −0.0176223
$$162$$ 0 0
$$163$$ 3346.00i 1.60785i 0.594733 + 0.803923i $$0.297258\pi$$
−0.594733 + 0.803923i $$0.702742\pi$$
$$164$$ −504.000 −0.239974
$$165$$ 0 0
$$166$$ 150.000 0.0701341
$$167$$ − 1491.00i − 0.690881i −0.938441 0.345440i $$-0.887730\pi$$
0.938441 0.345440i $$-0.112270\pi$$
$$168$$ 0 0
$$169$$ 1797.00 0.817934
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 712.000i − 0.315637i
$$173$$ − 2403.00i − 1.05605i −0.849229 0.528025i $$-0.822933\pi$$
0.849229 0.528025i $$-0.177067\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 672.000 0.287806
$$177$$ 0 0
$$178$$ 2172.00i 0.914597i
$$179$$ 2640.00 1.10236 0.551181 0.834386i $$-0.314177\pi$$
0.551181 + 0.834386i $$0.314177\pi$$
$$180$$ 0 0
$$181$$ 1073.00 0.440638 0.220319 0.975428i $$-0.429290\pi$$
0.220319 + 0.975428i $$0.429290\pi$$
$$182$$ − 160.000i − 0.0651648i
$$183$$ 0 0
$$184$$ −72.0000 −0.0288473
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3906.00i 1.52746i
$$188$$ − 576.000i − 0.223453i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1470.00 0.556887 0.278444 0.960453i $$-0.410181\pi$$
0.278444 + 0.960453i $$0.410181\pi$$
$$192$$ 0 0
$$193$$ 4720.00i 1.76038i 0.474623 + 0.880189i $$0.342584\pi$$
−0.474623 + 0.880189i $$0.657416\pi$$
$$194$$ −3088.00 −1.14281
$$195$$ 0 0
$$196$$ −1308.00 −0.476676
$$197$$ − 765.000i − 0.276670i −0.990385 0.138335i $$-0.955825\pi$$
0.990385 0.138335i $$-0.0441751\pi$$
$$198$$ 0 0
$$199$$ −668.000 −0.237956 −0.118978 0.992897i $$-0.537962\pi$$
−0.118978 + 0.992897i $$0.537962\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 264.000i − 0.0919553i
$$203$$ 480.000i 0.165958i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1784.00 −0.603384
$$207$$ 0 0
$$208$$ − 320.000i − 0.106673i
$$209$$ −2478.00 −0.820128
$$210$$ 0 0
$$211$$ 4601.00 1.50117 0.750583 0.660777i $$-0.229773\pi$$
0.750583 + 0.660777i $$0.229773\pi$$
$$212$$ 2964.00i 0.960228i
$$213$$ 0 0
$$214$$ 2280.00 0.728307
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 188.000i − 0.0588123i
$$218$$ 3470.00i 1.07806i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1860.00 0.566141
$$222$$ 0 0
$$223$$ 2158.00i 0.648029i 0.946052 + 0.324014i $$0.105033\pi$$
−0.946052 + 0.324014i $$0.894967\pi$$
$$224$$ 128.000 0.0381802
$$225$$ 0 0
$$226$$ −2868.00 −0.844144
$$227$$ 3123.00i 0.913131i 0.889690 + 0.456566i $$0.150921\pi$$
−0.889690 + 0.456566i $$0.849079\pi$$
$$228$$ 0 0
$$229$$ −2027.00 −0.584925 −0.292463 0.956277i $$-0.594475\pi$$
−0.292463 + 0.956277i $$0.594475\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 960.000i 0.271668i
$$233$$ 438.000i 0.123152i 0.998102 + 0.0615758i $$0.0196126\pi$$
−0.998102 + 0.0615758i $$0.980387\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1776.00 −0.489863
$$237$$ 0 0
$$238$$ 744.000i 0.202632i
$$239$$ −6414.00 −1.73593 −0.867965 0.496626i $$-0.834572\pi$$
−0.867965 + 0.496626i $$0.834572\pi$$
$$240$$ 0 0
$$241$$ 3431.00 0.917055 0.458527 0.888680i $$-0.348377\pi$$
0.458527 + 0.888680i $$0.348377\pi$$
$$242$$ 866.000i 0.230035i
$$243$$ 0 0
$$244$$ −884.000 −0.231936
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1180.00i 0.303974i
$$248$$ − 376.000i − 0.0962743i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 7308.00 1.83776 0.918878 0.394541i $$-0.129096\pi$$
0.918878 + 0.394541i $$0.129096\pi$$
$$252$$ 0 0
$$253$$ − 378.000i − 0.0939314i
$$254$$ −1372.00 −0.338925
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ − 3729.00i − 0.905092i −0.891741 0.452546i $$-0.850516\pi$$
0.891741 0.452546i $$-0.149484\pi$$
$$258$$ 0 0
$$259$$ −1048.00 −0.251427
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 228.000i − 0.0537629i
$$263$$ 1956.00i 0.458601i 0.973356 + 0.229301i $$0.0736439\pi$$
−0.973356 + 0.229301i $$0.926356\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −472.000 −0.108798
$$267$$ 0 0
$$268$$ 2152.00i 0.490501i
$$269$$ −990.000 −0.224392 −0.112196 0.993686i $$-0.535788\pi$$
−0.112196 + 0.993686i $$0.535788\pi$$
$$270$$ 0 0
$$271$$ 8495.00 1.90419 0.952093 0.305808i $$-0.0989266\pi$$
0.952093 + 0.305808i $$0.0989266\pi$$
$$272$$ 1488.00i 0.331703i
$$273$$ 0 0
$$274$$ −318.000 −0.0701134
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1366.00i − 0.296300i −0.988965 0.148150i $$-0.952668\pi$$
0.988965 0.148150i $$-0.0473318\pi$$
$$278$$ − 4552.00i − 0.982053i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5520.00 1.17187 0.585935 0.810358i $$-0.300727\pi$$
0.585935 + 0.810358i $$0.300727\pi$$
$$282$$ 0 0
$$283$$ − 5438.00i − 1.14225i −0.820865 0.571123i $$-0.806508\pi$$
0.820865 0.571123i $$-0.193492\pi$$
$$284$$ −2760.00 −0.576676
$$285$$ 0 0
$$286$$ 1680.00 0.347344
$$287$$ − 504.000i − 0.103659i
$$288$$ 0 0
$$289$$ −3736.00 −0.760432
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 4504.00i − 0.902660i
$$293$$ 8253.00i 1.64555i 0.568369 + 0.822774i $$0.307575\pi$$
−0.568369 + 0.822774i $$0.692425\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2096.00 −0.411579
$$297$$ 0 0
$$298$$ 2796.00i 0.543517i
$$299$$ −180.000 −0.0348149
$$300$$ 0 0
$$301$$ 712.000 0.136342
$$302$$ 5248.00i 0.999962i
$$303$$ 0 0
$$304$$ −944.000 −0.178099
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9290.00i 1.72706i 0.504295 + 0.863531i $$0.331752\pi$$
−0.504295 + 0.863531i $$0.668248\pi$$
$$308$$ 672.000i 0.124321i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8112.00 −1.47907 −0.739533 0.673121i $$-0.764953\pi$$
−0.739533 + 0.673121i $$0.764953\pi$$
$$312$$ 0 0
$$313$$ 7900.00i 1.42663i 0.700845 + 0.713314i $$0.252807\pi$$
−0.700845 + 0.713314i $$0.747193\pi$$
$$314$$ 788.000 0.141622
$$315$$ 0 0
$$316$$ 2660.00 0.473534
$$317$$ − 4419.00i − 0.782952i −0.920188 0.391476i $$-0.871965\pi$$
0.920188 0.391476i $$-0.128035\pi$$
$$318$$ 0 0
$$319$$ −5040.00 −0.884595
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 72.0000i − 0.0124609i
$$323$$ − 5487.00i − 0.945216i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −6692.00 −1.13692
$$327$$ 0 0
$$328$$ − 1008.00i − 0.169687i
$$329$$ 576.000 0.0965225
$$330$$ 0 0
$$331$$ −8200.00 −1.36167 −0.680835 0.732437i $$-0.738383\pi$$
−0.680835 + 0.732437i $$0.738383\pi$$
$$332$$ 300.000i 0.0495923i
$$333$$ 0 0
$$334$$ 2982.00 0.488526
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 9556.00i − 1.54465i −0.635225 0.772327i $$-0.719093\pi$$
0.635225 0.772327i $$-0.280907\pi$$
$$338$$ 3594.00i 0.578366i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1974.00 0.313484
$$342$$ 0 0
$$343$$ − 2680.00i − 0.421885i
$$344$$ 1424.00 0.223189
$$345$$ 0 0
$$346$$ 4806.00 0.746740
$$347$$ − 10116.0i − 1.56500i −0.622650 0.782500i $$-0.713944\pi$$
0.622650 0.782500i $$-0.286056\pi$$
$$348$$ 0 0
$$349$$ 6751.00 1.03545 0.517726 0.855546i $$-0.326779\pi$$
0.517726 + 0.855546i $$0.326779\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1344.00i 0.203510i
$$353$$ 4062.00i 0.612460i 0.951958 + 0.306230i $$0.0990677\pi$$
−0.951958 + 0.306230i $$0.900932\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −4344.00 −0.646718
$$357$$ 0 0
$$358$$ 5280.00i 0.779488i
$$359$$ 8778.00 1.29049 0.645244 0.763977i $$-0.276756\pi$$
0.645244 + 0.763977i $$0.276756\pi$$
$$360$$ 0 0
$$361$$ −3378.00 −0.492492
$$362$$ 2146.00i 0.311578i
$$363$$ 0 0
$$364$$ 320.000 0.0460785
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 956.000i 0.135975i 0.997686 + 0.0679875i $$0.0216578\pi$$
−0.997686 + 0.0679875i $$0.978342\pi$$
$$368$$ − 144.000i − 0.0203981i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2964.00 −0.414780
$$372$$ 0 0
$$373$$ − 2300.00i − 0.319275i −0.987176 0.159637i $$-0.948968\pi$$
0.987176 0.159637i $$-0.0510325\pi$$
$$374$$ −7812.00 −1.08008
$$375$$ 0 0
$$376$$ 1152.00 0.158005
$$377$$ 2400.00i 0.327868i
$$378$$ 0 0
$$379$$ −29.0000 −0.00393042 −0.00196521 0.999998i $$-0.500626\pi$$
−0.00196521 + 0.999998i $$0.500626\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2940.00i 0.393779i
$$383$$ 8127.00i 1.08426i 0.840296 + 0.542128i $$0.182381\pi$$
−0.840296 + 0.542128i $$0.817619\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −9440.00 −1.24478
$$387$$ 0 0
$$388$$ − 6176.00i − 0.808090i
$$389$$ −7938.00 −1.03463 −0.517317 0.855794i $$-0.673069\pi$$
−0.517317 + 0.855794i $$0.673069\pi$$
$$390$$ 0 0
$$391$$ 837.000 0.108258
$$392$$ − 2616.00i − 0.337061i
$$393$$ 0 0
$$394$$ 1530.00 0.195635
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 272.000i 0.0343861i 0.999852 + 0.0171931i $$0.00547299\pi$$
−0.999852 + 0.0171931i $$0.994527\pi$$
$$398$$ − 1336.00i − 0.168260i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −4554.00 −0.567122 −0.283561 0.958954i $$-0.591516\pi$$
−0.283561 + 0.958954i $$0.591516\pi$$
$$402$$ 0 0
$$403$$ − 940.000i − 0.116190i
$$404$$ 528.000 0.0650222
$$405$$ 0 0
$$406$$ −960.000 −0.117350
$$407$$ − 11004.0i − 1.34017i
$$408$$ 0 0
$$409$$ −1001.00 −0.121018 −0.0605089 0.998168i $$-0.519272\pi$$
−0.0605089 + 0.998168i $$0.519272\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 3568.00i − 0.426657i
$$413$$ − 1776.00i − 0.211601i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 640.000 0.0754293
$$417$$ 0 0
$$418$$ − 4956.00i − 0.579918i
$$419$$ −1794.00 −0.209171 −0.104585 0.994516i $$-0.533352\pi$$
−0.104585 + 0.994516i $$0.533352\pi$$
$$420$$ 0 0
$$421$$ −16129.0 −1.86717 −0.933586 0.358354i $$-0.883338\pi$$
−0.933586 + 0.358354i $$0.883338\pi$$
$$422$$ 9202.00i 1.06148i
$$423$$ 0 0
$$424$$ −5928.00 −0.678984
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 884.000i − 0.100187i
$$428$$ 4560.00i 0.514990i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 13356.0 1.49266 0.746329 0.665577i $$-0.231814\pi$$
0.746329 + 0.665577i $$0.231814\pi$$
$$432$$ 0 0
$$433$$ 11500.0i 1.27634i 0.769896 + 0.638169i $$0.220308\pi$$
−0.769896 + 0.638169i $$0.779692\pi$$
$$434$$ 376.000 0.0415866
$$435$$ 0 0
$$436$$ −6940.00 −0.762307
$$437$$ 531.000i 0.0581263i
$$438$$ 0 0
$$439$$ 11149.0 1.21210 0.606051 0.795426i $$-0.292753\pi$$
0.606051 + 0.795426i $$0.292753\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3720.00i 0.400322i
$$443$$ 3849.00i 0.412803i 0.978467 + 0.206401i $$0.0661752\pi$$
−0.978467 + 0.206401i $$0.933825\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4316.00 −0.458225
$$447$$ 0 0
$$448$$ 256.000i 0.0269975i
$$449$$ 18048.0 1.89697 0.948483 0.316828i $$-0.102618\pi$$
0.948483 + 0.316828i $$0.102618\pi$$
$$450$$ 0 0
$$451$$ 5292.00 0.552529
$$452$$ − 5736.00i − 0.596900i
$$453$$ 0 0
$$454$$ −6246.00 −0.645681
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4264.00i − 0.436458i −0.975898 0.218229i $$-0.929972\pi$$
0.975898 0.218229i $$-0.0700280\pi$$
$$458$$ − 4054.00i − 0.413605i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10242.0 1.03475 0.517373 0.855760i $$-0.326910\pi$$
0.517373 + 0.855760i $$0.326910\pi$$
$$462$$ 0 0
$$463$$ − 3302.00i − 0.331441i −0.986173 0.165720i $$-0.947005\pi$$
0.986173 0.165720i $$-0.0529949\pi$$
$$464$$ −1920.00 −0.192099
$$465$$ 0 0
$$466$$ −876.000 −0.0870814
$$467$$ 1923.00i 0.190548i 0.995451 + 0.0952739i $$0.0303727\pi$$
−0.995451 + 0.0952739i $$0.969627\pi$$
$$468$$ 0 0
$$469$$ −2152.00 −0.211877
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 3552.00i − 0.346386i
$$473$$ 7476.00i 0.726738i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1488.00 −0.143282
$$477$$ 0 0
$$478$$ − 12828.0i − 1.22749i
$$479$$ 15246.0 1.45430 0.727148 0.686481i $$-0.240846\pi$$
0.727148 + 0.686481i $$0.240846\pi$$
$$480$$ 0 0
$$481$$ −5240.00 −0.496722
$$482$$ 6862.00i 0.648455i
$$483$$ 0 0
$$484$$ −1732.00 −0.162660
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 8206.00i − 0.763551i −0.924255 0.381776i $$-0.875313\pi$$
0.924255 0.381776i $$-0.124687\pi$$
$$488$$ − 1768.00i − 0.164003i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 16806.0 1.54469 0.772346 0.635202i $$-0.219083\pi$$
0.772346 + 0.635202i $$0.219083\pi$$
$$492$$ 0 0
$$493$$ − 11160.0i − 1.01952i
$$494$$ −2360.00 −0.214942
$$495$$ 0 0
$$496$$ 752.000 0.0680762
$$497$$ − 2760.00i − 0.249100i
$$498$$ 0 0
$$499$$ 5425.00 0.486686 0.243343 0.969940i $$-0.421756\pi$$
0.243343 + 0.969940i $$0.421756\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 14616.0i 1.29949i
$$503$$ − 19665.0i − 1.74318i −0.490236 0.871589i $$-0.663090\pi$$
0.490236 0.871589i $$-0.336910\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 756.000 0.0664196
$$507$$ 0 0
$$508$$ − 2744.00i − 0.239656i
$$509$$ −14724.0 −1.28218 −0.641090 0.767466i $$-0.721518\pi$$
−0.641090 + 0.767466i $$0.721518\pi$$
$$510$$ 0 0
$$511$$ 4504.00 0.389912
$$512$$ 512.000i 0.0441942i
$$513$$ 0 0
$$514$$ 7458.00 0.639997
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6048.00i 0.514489i
$$518$$ − 2096.00i − 0.177786i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2058.00 −0.173057 −0.0865284 0.996249i $$-0.527577\pi$$
−0.0865284 + 0.996249i $$0.527577\pi$$
$$522$$ 0 0
$$523$$ − 11912.0i − 0.995938i −0.867195 0.497969i $$-0.834079\pi$$
0.867195 0.497969i $$-0.165921\pi$$
$$524$$ 456.000 0.0380161
$$525$$ 0 0
$$526$$ −3912.00 −0.324280
$$527$$ 4371.00i 0.361297i
$$528$$ 0 0
$$529$$ 12086.0 0.993343
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 944.000i − 0.0769316i
$$533$$ − 2520.00i − 0.204790i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4304.00 −0.346837
$$537$$ 0 0
$$538$$ − 1980.00i − 0.158669i
$$539$$ 13734.0 1.09752
$$540$$ 0 0
$$541$$ −5170.00 −0.410861 −0.205430 0.978672i $$-0.565859\pi$$
−0.205430 + 0.978672i $$0.565859\pi$$
$$542$$ 16990.0i 1.34646i
$$543$$ 0 0
$$544$$ −2976.00 −0.234550
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4186.00i − 0.327204i −0.986526 0.163602i $$-0.947689\pi$$
0.986526 0.163602i $$-0.0523112\pi$$
$$548$$ − 636.000i − 0.0495777i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7080.00 0.547401
$$552$$ 0 0
$$553$$ 2660.00i 0.204547i
$$554$$ 2732.00 0.209515
$$555$$ 0 0
$$556$$ 9104.00 0.694417
$$557$$ − 13026.0i − 0.990896i −0.868637 0.495448i $$-0.835004\pi$$
0.868637 0.495448i $$-0.164996\pi$$
$$558$$ 0 0
$$559$$ 3560.00 0.269359
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 11040.0i 0.828638i
$$563$$ − 10668.0i − 0.798584i −0.916824 0.399292i $$-0.869256\pi$$
0.916824 0.399292i $$-0.130744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 10876.0 0.807690
$$567$$ 0 0
$$568$$ − 5520.00i − 0.407771i
$$569$$ −15372.0 −1.13256 −0.566281 0.824212i $$-0.691618\pi$$
−0.566281 + 0.824212i $$0.691618\pi$$
$$570$$ 0 0
$$571$$ −14989.0 −1.09855 −0.549273 0.835643i $$-0.685095\pi$$
−0.549273 + 0.835643i $$0.685095\pi$$
$$572$$ 3360.00i 0.245610i
$$573$$ 0 0
$$574$$ 1008.00 0.0732981
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 1066.00i − 0.0769119i −0.999260 0.0384559i $$-0.987756\pi$$
0.999260 0.0384559i $$-0.0122439\pi$$
$$578$$ − 7472.00i − 0.537706i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −300.000 −0.0214219
$$582$$ 0 0
$$583$$ − 31122.0i − 2.21088i
$$584$$ 9008.00 0.638277
$$585$$ 0 0
$$586$$ −16506.0 −1.16358
$$587$$ 621.000i 0.0436651i 0.999762 + 0.0218325i $$0.00695007\pi$$
−0.999762 + 0.0218325i $$0.993050\pi$$
$$588$$ 0 0
$$589$$ −2773.00 −0.193989
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 4192.00i − 0.291031i
$$593$$ − 20187.0i − 1.39794i −0.715149 0.698972i $$-0.753641\pi$$
0.715149 0.698972i $$-0.246359\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5592.00 −0.384324
$$597$$ 0 0
$$598$$ − 360.000i − 0.0246179i
$$599$$ −18228.0 −1.24337 −0.621683 0.783269i $$-0.713551\pi$$
−0.621683 + 0.783269i $$0.713551\pi$$
$$600$$ 0 0
$$601$$ −11743.0 −0.797017 −0.398508 0.917165i $$-0.630472\pi$$
−0.398508 + 0.917165i $$0.630472\pi$$
$$602$$ 1424.00i 0.0964085i
$$603$$ 0 0
$$604$$ −10496.0 −0.707080
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 24418.0i − 1.63278i −0.577503 0.816389i $$-0.695973\pi$$
0.577503 0.816389i $$-0.304027\pi$$
$$608$$ − 1888.00i − 0.125935i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2880.00 0.190691
$$612$$ 0 0
$$613$$ − 2672.00i − 0.176054i −0.996118 0.0880270i $$-0.971944\pi$$
0.996118 0.0880270i $$-0.0280562\pi$$
$$614$$ −18580.0 −1.22122
$$615$$ 0 0
$$616$$ −1344.00 −0.0879080
$$617$$ 8601.00i 0.561205i 0.959824 + 0.280602i $$0.0905342\pi$$
−0.959824 + 0.280602i $$0.909466\pi$$
$$618$$ 0 0
$$619$$ −21308.0 −1.38359 −0.691794 0.722095i $$-0.743179\pi$$
−0.691794 + 0.722095i $$0.743179\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 16224.0i − 1.04586i
$$623$$ − 4344.00i − 0.279356i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −15800.0 −1.00878
$$627$$ 0 0
$$628$$ 1576.00i 0.100142i
$$629$$ 24366.0 1.54457
$$630$$ 0 0
$$631$$ −19015.0 −1.19964 −0.599822 0.800134i $$-0.704762\pi$$
−0.599822 + 0.800134i $$0.704762\pi$$
$$632$$ 5320.00i 0.334839i
$$633$$ 0 0
$$634$$ 8838.00 0.553631
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6540.00i − 0.406788i
$$638$$ − 10080.0i − 0.625503i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4416.00 0.272108 0.136054 0.990701i $$-0.456558\pi$$
0.136054 + 0.990701i $$0.456558\pi$$
$$642$$ 0 0
$$643$$ − 7580.00i − 0.464893i −0.972609 0.232446i $$-0.925327\pi$$
0.972609 0.232446i $$-0.0746730\pi$$
$$644$$ 144.000 0.00881117
$$645$$ 0 0
$$646$$ 10974.0 0.668369
$$647$$ 14901.0i 0.905439i 0.891653 + 0.452719i $$0.149546\pi$$
−0.891653 + 0.452719i $$0.850454\pi$$
$$648$$ 0 0
$$649$$ 18648.0 1.12789
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 13384.0i − 0.803923i
$$653$$ 12915.0i 0.773971i 0.922086 + 0.386985i $$0.126484\pi$$
−0.922086 + 0.386985i $$0.873516\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2016.00 0.119987
$$657$$ 0 0
$$658$$ 1152.00i 0.0682517i
$$659$$ 28128.0 1.66269 0.831344 0.555758i $$-0.187572\pi$$
0.831344 + 0.555758i $$0.187572\pi$$
$$660$$ 0 0
$$661$$ −8362.00 −0.492049 −0.246024 0.969264i $$-0.579124\pi$$
−0.246024 + 0.969264i $$0.579124\pi$$
$$662$$ − 16400.0i − 0.962846i
$$663$$ 0 0
$$664$$ −600.000 −0.0350670
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1080.00i 0.0626953i
$$668$$ 5964.00i 0.345440i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 9282.00 0.534020
$$672$$ 0 0
$$673$$ − 29708.0i − 1.70157i −0.525511 0.850787i $$-0.676126\pi$$
0.525511 0.850787i $$-0.323874\pi$$
$$674$$ 19112.0 1.09224
$$675$$ 0 0
$$676$$ −7188.00 −0.408967
$$677$$ − 6762.00i − 0.383877i −0.981407 0.191939i $$-0.938523\pi$$
0.981407 0.191939i $$-0.0614774\pi$$
$$678$$ 0 0
$$679$$ 6176.00 0.349062
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3948.00i 0.221667i
$$683$$ − 19155.0i − 1.07313i −0.843860 0.536563i $$-0.819722\pi$$
0.843860 0.536563i $$-0.180278\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5360.00 0.298317
$$687$$ 0 0
$$688$$ 2848.00i 0.157818i
$$689$$ −14820.0 −0.819444
$$690$$ 0 0
$$691$$ −22975.0 −1.26485 −0.632424 0.774622i $$-0.717940\pi$$
−0.632424 + 0.774622i $$0.717940\pi$$
$$692$$ 9612.00i 0.528025i
$$693$$ 0 0
$$694$$ 20232.0 1.10662
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 11718.0i 0.636802i
$$698$$ 13502.0i 0.732175i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6450.00 0.347522 0.173761 0.984788i $$-0.444408\pi$$
0.173761 + 0.984788i $$0.444408\pi$$
$$702$$ 0 0
$$703$$ 15458.0i 0.829317i
$$704$$ −2688.00 −0.143903
$$705$$ 0 0
$$706$$ −8124.00 −0.433075
$$707$$ 528.000i 0.0280870i
$$708$$ 0 0
$$709$$ −34538.0 −1.82948 −0.914740 0.404042i $$-0.867605\pi$$
−0.914740 + 0.404042i $$0.867605\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 8688.00i − 0.457299i
$$713$$ − 423.000i − 0.0222181i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10560.0 −0.551181
$$717$$ 0 0
$$718$$ 17556.0i 0.912513i
$$719$$ 27114.0 1.40637 0.703186 0.711006i $$-0.251760\pi$$
0.703186 + 0.711006i $$0.251760\pi$$
$$720$$ 0 0
$$721$$ 3568.00 0.184299
$$722$$ − 6756.00i − 0.348244i
$$723$$ 0 0
$$724$$ −4292.00 −0.220319
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 236.000i 0.0120396i 0.999982 + 0.00601978i $$0.00191617\pi$$
−0.999982 + 0.00601978i $$0.998084\pi$$
$$728$$ 640.000i 0.0325824i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −16554.0 −0.837581
$$732$$ 0 0
$$733$$ − 27128.0i − 1.36698i −0.729960 0.683489i $$-0.760462\pi$$
0.729960 0.683489i $$-0.239538\pi$$
$$734$$ −1912.00 −0.0961488
$$735$$ 0 0
$$736$$ 288.000 0.0144237
$$737$$ − 22596.0i − 1.12935i
$$738$$ 0 0
$$739$$ −5249.00 −0.261282 −0.130641 0.991430i $$-0.541704\pi$$
−0.130641 + 0.991430i $$0.541704\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 5928.00i − 0.293293i
$$743$$ 13896.0i 0.686130i 0.939312 + 0.343065i $$0.111465\pi$$
−0.939312 + 0.343065i $$0.888535\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4600.00 0.225761
$$747$$ 0 0
$$748$$ − 15624.0i − 0.763730i
$$749$$ −4560.00 −0.222455
$$750$$ 0 0
$$751$$ 27665.0 1.34422 0.672111 0.740451i $$-0.265388\pi$$
0.672111 + 0.740451i $$0.265388\pi$$
$$752$$ 2304.00i 0.111726i
$$753$$ 0 0
$$754$$ −4800.00 −0.231838
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 8122.00i − 0.389959i −0.980807 0.194980i $$-0.937536\pi$$
0.980807 0.194980i $$-0.0624641\pi$$
$$758$$ − 58.0000i − 0.00277923i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10584.0 −0.504165 −0.252083 0.967706i $$-0.581115\pi$$
−0.252083 + 0.967706i $$0.581115\pi$$
$$762$$ 0 0
$$763$$ − 6940.00i − 0.329286i
$$764$$ −5880.00 −0.278444
$$765$$ 0 0
$$766$$ −16254.0 −0.766685
$$767$$ − 8880.00i − 0.418042i
$$768$$ 0 0
$$769$$ 18619.0 0.873106 0.436553 0.899679i $$-0.356199\pi$$
0.436553 + 0.899679i $$0.356199\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 18880.0i − 0.880189i
$$773$$ 22251.0i 1.03533i 0.855582 + 0.517667i $$0.173199\pi$$
−0.855582 + 0.517667i $$0.826801\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12352.0 0.571406
$$777$$ 0 0
$$778$$ − 15876.0i − 0.731597i
$$779$$ −7434.00 −0.341914
$$780$$ 0 0
$$781$$ 28980.0 1.32777
$$782$$ 1674.00i 0.0765500i
$$783$$ 0 0
$$784$$ 5232.00 0.238338
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 24854.0i 1.12573i 0.826549 + 0.562865i $$0.190301\pi$$
−0.826549 + 0.562865i $$0.809699\pi$$
$$788$$ 3060.00i 0.138335i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5736.00 0.257837
$$792$$ 0 0
$$793$$ − 4420.00i − 0.197930i
$$794$$ −544.000 −0.0243147
$$795$$ 0 0
$$796$$ 2672.00 0.118978
$$797$$ − 3681.00i − 0.163598i −0.996649 0.0817991i $$-0.973933\pi$$
0.996649 0.0817991i $$-0.0260666\pi$$
$$798$$ 0 0
$$799$$ −13392.0 −0.592960
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 9108.00i − 0.401016i
$$803$$ 47292.0i 2.07833i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 1880.00 0.0821590
$$807$$ 0 0
$$808$$ 1056.00i 0.0459777i
$$809$$ −5142.00 −0.223465 −0.111732 0.993738i $$-0.535640\pi$$
−0.111732 + 0.993738i $$0.535640\pi$$
$$810$$ 0 0
$$811$$ −18484.0 −0.800322 −0.400161 0.916445i $$-0.631046\pi$$
−0.400161 + 0.916445i $$0.631046\pi$$
$$812$$ − 1920.00i − 0.0829788i
$$813$$ 0 0
$$814$$ 22008.0 0.947641
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 10502.0i − 0.449717i
$$818$$ − 2002.00i − 0.0855725i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25014.0 −1.06333 −0.531665 0.846954i $$-0.678434\pi$$
−0.531665 + 0.846954i $$0.678434\pi$$
$$822$$ 0 0
$$823$$ 32146.0i 1.36153i 0.732502 + 0.680765i $$0.238352\pi$$
−0.732502 + 0.680765i $$0.761648\pi$$
$$824$$ 7136.00 0.301692
$$825$$ 0 0
$$826$$ 3552.00 0.149625
$$827$$ 10977.0i 0.461557i 0.973006 + 0.230779i $$0.0741273\pi$$
−0.973006 + 0.230779i $$0.925873\pi$$
$$828$$ 0 0
$$829$$ −36602.0 −1.53346 −0.766731 0.641969i $$-0.778118\pi$$
−0.766731 + 0.641969i $$0.778118\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1280.00i 0.0533366i
$$833$$ 30411.0i 1.26492i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 9912.00 0.410064
$$837$$ 0 0
$$838$$ − 3588.00i − 0.147906i
$$839$$ 11076.0 0.455764 0.227882 0.973689i $$-0.426820\pi$$
0.227882 + 0.973689i $$0.426820\pi$$
$$840$$ 0 0
$$841$$ −9989.00 −0.409570
$$842$$ − 32258.0i − 1.32029i
$$843$$ 0 0
$$844$$ −18404.0 −0.750583
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1732.00i − 0.0702624i
$$848$$ − 11856.0i − 0.480114i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2358.00 −0.0949838
$$852$$ 0 0
$$853$$ − 36848.0i − 1.47908i −0.673115 0.739538i $$-0.735044\pi$$
0.673115 0.739538i $$-0.264956\pi$$
$$854$$ 1768.00 0.0708428
$$855$$ 0 0
$$856$$ −9120.00 −0.364153
$$857$$ 26961.0i 1.07464i 0.843377 + 0.537322i $$0.180564\pi$$
−0.843377 + 0.537322i $$0.819436\pi$$
$$858$$ 0 0
$$859$$ 415.000 0.0164838 0.00824192 0.999966i $$-0.497376\pi$$
0.00824192 + 0.999966i $$0.497376\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 26712.0i 1.05547i
$$863$$ − 45501.0i − 1.79475i −0.441265 0.897377i $$-0.645470\pi$$
0.441265 0.897377i $$-0.354530\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −23000.0 −0.902508
$$867$$ 0 0
$$868$$ 752.000i 0.0294062i
$$869$$ −27930.0 −1.09029
$$870$$ 0 0
$$871$$ −10760.0 −0.418586
$$872$$ − 13880.0i − 0.539032i
$$873$$ 0 0
$$874$$ −1062.00 −0.0411015
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 35042.0i 1.34924i 0.738165 + 0.674620i $$0.235693\pi$$
−0.738165 + 0.674620i $$0.764307\pi$$
$$878$$ 22298.0i 0.857085i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −1080.00 −0.0413009 −0.0206505 0.999787i $$-0.506574\pi$$
−0.0206505 + 0.999787i $$0.506574\pi$$
$$882$$ 0 0
$$883$$ 20164.0i 0.768485i 0.923232 + 0.384243i $$0.125537\pi$$
−0.923232 + 0.384243i $$0.874463\pi$$
$$884$$ −7440.00 −0.283070
$$885$$ 0 0
$$886$$ −7698.00 −0.291895
$$887$$ − 20067.0i − 0.759621i −0.925064 0.379811i $$-0.875989\pi$$
0.925064 0.379811i $$-0.124011\pi$$
$$888$$ 0 0
$$889$$ 2744.00 0.103522
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 8632.00i − 0.324014i
$$893$$ − 8496.00i − 0.318374i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −512.000 −0.0190901
$$897$$ 0 0
$$898$$ 36096.0i 1.34136i
$$899$$ −5640.00 −0.209238
$$900$$ 0 0
$$901$$ 68913.0 2.54809
$$902$$ 10584.0i 0.390697i
$$903$$ 0 0
$$904$$ 11472.0 0.422072
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 26524.0i − 0.971020i −0.874231 0.485510i $$-0.838634\pi$$
0.874231 0.485510i $$-0.161366\pi$$
$$908$$ − 12492.0i − 0.456566i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −35568.0 −1.29355 −0.646773 0.762683i $$-0.723882\pi$$
−0.646773 + 0.762683i $$0.723882\pi$$
$$912$$ 0 0
$$913$$ − 3150.00i − 0.114184i
$$914$$ 8528.00 0.308623
$$915$$ 0 0
$$916$$ 8108.00 0.292463
$$917$$ 456.000i 0.0164214i
$$918$$ 0 0
$$919$$ 23704.0 0.850841 0.425420 0.904996i $$-0.360126\pi$$
0.425420 + 0.904996i $$0.360126\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 20484.0i 0.731675i
$$923$$ − 13800.0i − 0.492126i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 6604.00 0.234364
$$927$$ 0 0
$$928$$ − 3840.00i − 0.135834i
$$929$$ −40590.0 −1.43349 −0.716746 0.697334i $$-0.754369\pi$$
−0.716746 + 0.697334i $$0.754369\pi$$
$$930$$ 0 0
$$931$$ −19293.0 −0.679165
$$932$$ − 1752.00i − 0.0615758i
$$933$$ 0 0
$$934$$ −3846.00 −0.134738
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 12964.0i − 0.451991i −0.974128 0.225995i $$-0.927437\pi$$
0.974128 0.225995i $$-0.0725634\pi$$
$$938$$ − 4304.00i − 0.149819i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −29922.0 −1.03659 −0.518294 0.855203i $$-0.673433\pi$$
−0.518294 + 0.855203i $$0.673433\pi$$
$$942$$ 0 0
$$943$$ − 1134.00i − 0.0391603i
$$944$$ 7104.00 0.244932
$$945$$ 0 0
$$946$$ −14952.0 −0.513881
$$947$$ − 5241.00i − 0.179841i −0.995949 0.0899206i $$-0.971339\pi$$
0.995949 0.0899206i $$-0.0286613\pi$$
$$948$$ 0 0
$$949$$ 22520.0 0.770316
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 2976.00i − 0.101316i
$$953$$ 26214.0i 0.891033i 0.895274 + 0.445517i $$0.146980\pi$$
−0.895274 + 0.445517i $$0.853020\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 25656.0 0.867965
$$957$$ 0 0
$$958$$ 30492.0i 1.02834i
$$959$$ 636.000 0.0214155
$$960$$ 0 0
$$961$$ −27582.0 −0.925850
$$962$$ − 10480.0i − 0.351236i
$$963$$ 0 0
$$964$$ −13724.0 −0.458527
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 18278.0i 0.607840i 0.952698 + 0.303920i $$0.0982955\pi$$
−0.952698 + 0.303920i $$0.901705\pi$$
$$968$$ − 3464.00i − 0.115018i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24942.0 0.824333 0.412166 0.911109i $$-0.364772\pi$$
0.412166 + 0.911109i $$0.364772\pi$$
$$972$$ 0 0
$$973$$ 9104.00i 0.299960i
$$974$$ 16412.0 0.539912
$$975$$ 0 0
$$976$$ 3536.00 0.115968
$$977$$ 11226.0i 0.367607i 0.982963 + 0.183803i $$0.0588409\pi$$
−0.982963 + 0.183803i $$0.941159\pi$$
$$978$$ 0 0
$$979$$ 45612.0 1.48904
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 33612.0i 1.09226i
$$983$$ − 23073.0i − 0.748641i −0.927299 0.374321i $$-0.877876\pi$$
0.927299 0.374321i $$-0.122124\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 22320.0 0.720906
$$987$$ 0 0
$$988$$ − 4720.00i − 0.151987i
$$989$$ 1602.00 0.0515072
$$990$$ 0 0
$$991$$ 22037.0 0.706386 0.353193 0.935551i $$-0.385096\pi$$
0.353193 + 0.935551i $$0.385096\pi$$
$$992$$ 1504.00i 0.0481371i
$$993$$ 0 0
$$994$$ 5520.00 0.176141
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 19082.0i 0.606151i 0.952966 + 0.303076i $$0.0980135\pi$$
−0.952966 + 0.303076i $$0.901986\pi$$
$$998$$ 10850.0i 0.344139i
$$999$$ 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 1350.4.c.q.649.2 2
3.2 odd 2 1350.4.c.d.649.1 2
5.2 odd 4 1350.4.a.g.1.1 1
5.3 odd 4 270.4.a.i.1.1 yes 1
5.4 even 2 inner 1350.4.c.q.649.1 2
15.2 even 4 1350.4.a.u.1.1 1
15.8 even 4 270.4.a.e.1.1 1
15.14 odd 2 1350.4.c.d.649.2 2
20.3 even 4 2160.4.a.e.1.1 1
45.13 odd 12 810.4.e.h.541.1 2
45.23 even 12 810.4.e.q.541.1 2
45.38 even 12 810.4.e.q.271.1 2
45.43 odd 12 810.4.e.h.271.1 2
60.23 odd 4 2160.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.e.1.1 1 15.8 even 4
270.4.a.i.1.1 yes 1 5.3 odd 4
810.4.e.h.271.1 2 45.43 odd 12
810.4.e.h.541.1 2 45.13 odd 12
810.4.e.q.271.1 2 45.38 even 12
810.4.e.q.541.1 2 45.23 even 12
1350.4.a.g.1.1 1 5.2 odd 4
1350.4.a.u.1.1 1 15.2 even 4
1350.4.c.d.649.1 2 3.2 odd 2
1350.4.c.d.649.2 2 15.14 odd 2
1350.4.c.q.649.1 2 5.4 even 2 inner
1350.4.c.q.649.2 2 1.1 even 1 trivial
2160.4.a.e.1.1 1 20.3 even 4
2160.4.a.o.1.1 1 60.23 odd 4