# Properties

 Label 1350.4.c.p.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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.p.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -4.00000 q^{4} -23.0000i q^{7} -8.00000i q^{8} +O(q^{10})$$ $$q+2.00000i q^{2} -4.00000 q^{4} -23.0000i q^{7} -8.00000i q^{8} +30.0000 q^{11} -34.0000i q^{13} +46.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +139.000 q^{19} +60.0000i q^{22} +192.000i q^{23} +68.0000 q^{26} +92.0000i q^{28} -234.000 q^{29} -55.0000 q^{31} +32.0000i q^{32} -84.0000 q^{34} -191.000i q^{37} +278.000i q^{38} +138.000 q^{41} +53.0000i q^{43} -120.000 q^{44} -384.000 q^{46} -366.000i q^{47} -186.000 q^{49} +136.000i q^{52} -330.000i q^{53} -184.000 q^{56} -468.000i q^{58} +396.000 q^{59} +23.0000 q^{61} -110.000i q^{62} -64.0000 q^{64} -452.000i q^{67} -168.000i q^{68} +204.000 q^{71} -691.000i q^{73} +382.000 q^{74} -556.000 q^{76} -690.000i q^{77} +709.000 q^{79} +276.000i q^{82} +1098.00i q^{83} -106.000 q^{86} -240.000i q^{88} +816.000 q^{89} -782.000 q^{91} -768.000i q^{92} +732.000 q^{94} -905.000i q^{97} -372.000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 60 q^{11} + 92 q^{14} + 32 q^{16} + 278 q^{19} + 136 q^{26} - 468 q^{29} - 110 q^{31} - 168 q^{34} + 276 q^{41} - 240 q^{44} - 768 q^{46} - 372 q^{49} - 368 q^{56} + 792 q^{59} + 46 q^{61} - 128 q^{64} + 408 q^{71} + 764 q^{74} - 1112 q^{76} + 1418 q^{79} - 212 q^{86} + 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 60 * q^11 + 92 * q^14 + 32 * q^16 + 278 * q^19 + 136 * q^26 - 468 * q^29 - 110 * q^31 - 168 * q^34 + 276 * q^41 - 240 * q^44 - 768 * q^46 - 372 * q^49 - 368 * q^56 + 792 * q^59 + 46 * q^61 - 128 * q^64 + 408 * q^71 + 764 * q^74 - 1112 * q^76 + 1418 * q^79 - 212 * q^86 + 1632 * q^89 - 1564 * q^91 + 1464 * 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
$$7$$ − 23.0000i − 1.24188i −0.783857 0.620942i $$-0.786750\pi$$
0.783857 0.620942i $$-0.213250\pi$$
$$8$$ − 8.00000i − 0.353553i
$$9$$ 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
$$13$$ − 34.0000i − 0.725377i −0.931910 0.362689i $$-0.881859\pi$$
0.931910 0.362689i $$-0.118141\pi$$
$$14$$ 46.0000 0.878144
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 42.0000i 0.599206i 0.954064 + 0.299603i $$0.0968542\pi$$
−0.954064 + 0.299603i $$0.903146\pi$$
$$18$$ 0 0
$$19$$ 139.000 1.67836 0.839179 0.543856i $$-0.183036\pi$$
0.839179 + 0.543856i $$0.183036\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 60.0000i 0.581456i
$$23$$ 192.000i 1.74064i 0.492485 + 0.870321i $$0.336089\pi$$
−0.492485 + 0.870321i $$0.663911\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 68.0000 0.512919
$$27$$ 0 0
$$28$$ 92.0000i 0.620942i
$$29$$ −234.000 −1.49837 −0.749185 0.662361i $$-0.769554\pi$$
−0.749185 + 0.662361i $$0.769554\pi$$
$$30$$ 0 0
$$31$$ −55.0000 −0.318655 −0.159327 0.987226i $$-0.550933\pi$$
−0.159327 + 0.987226i $$0.550933\pi$$
$$32$$ 32.0000i 0.176777i
$$33$$ 0 0
$$34$$ −84.0000 −0.423702
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 191.000i − 0.848654i −0.905509 0.424327i $$-0.860511\pi$$
0.905509 0.424327i $$-0.139489\pi$$
$$38$$ 278.000i 1.18678i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 138.000 0.525658 0.262829 0.964842i $$-0.415344\pi$$
0.262829 + 0.964842i $$0.415344\pi$$
$$42$$ 0 0
$$43$$ 53.0000i 0.187963i 0.995574 + 0.0939817i $$0.0299595\pi$$
−0.995574 + 0.0939817i $$0.970040\pi$$
$$44$$ −120.000 −0.411152
$$45$$ 0 0
$$46$$ −384.000 −1.23082
$$47$$ − 366.000i − 1.13588i −0.823068 0.567942i $$-0.807740\pi$$
0.823068 0.567942i $$-0.192260\pi$$
$$48$$ 0 0
$$49$$ −186.000 −0.542274
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 136.000i 0.362689i
$$53$$ − 330.000i − 0.855264i −0.903953 0.427632i $$-0.859348\pi$$
0.903953 0.427632i $$-0.140652\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −184.000 −0.439072
$$57$$ 0 0
$$58$$ − 468.000i − 1.05951i
$$59$$ 396.000 0.873810 0.436905 0.899508i $$-0.356075\pi$$
0.436905 + 0.899508i $$0.356075\pi$$
$$60$$ 0 0
$$61$$ 23.0000 0.0482762 0.0241381 0.999709i $$-0.492316\pi$$
0.0241381 + 0.999709i $$0.492316\pi$$
$$62$$ − 110.000i − 0.225323i
$$63$$ 0 0
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 452.000i − 0.824188i −0.911141 0.412094i $$-0.864798\pi$$
0.911141 0.412094i $$-0.135202\pi$$
$$68$$ − 168.000i − 0.299603i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 204.000 0.340991 0.170495 0.985358i $$-0.445463\pi$$
0.170495 + 0.985358i $$0.445463\pi$$
$$72$$ 0 0
$$73$$ − 691.000i − 1.10788i −0.832556 0.553941i $$-0.813123\pi$$
0.832556 0.553941i $$-0.186877\pi$$
$$74$$ 382.000 0.600089
$$75$$ 0 0
$$76$$ −556.000 −0.839179
$$77$$ − 690.000i − 1.02121i
$$78$$ 0 0
$$79$$ 709.000 1.00973 0.504865 0.863198i $$-0.331542\pi$$
0.504865 + 0.863198i $$0.331542\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 276.000i 0.371696i
$$83$$ 1098.00i 1.45206i 0.687662 + 0.726031i $$0.258637\pi$$
−0.687662 + 0.726031i $$0.741363\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −106.000 −0.132910
$$87$$ 0 0
$$88$$ − 240.000i − 0.290728i
$$89$$ 816.000 0.971863 0.485932 0.873997i $$-0.338480\pi$$
0.485932 + 0.873997i $$0.338480\pi$$
$$90$$ 0 0
$$91$$ −782.000 −0.900834
$$92$$ − 768.000i − 0.870321i
$$93$$ 0 0
$$94$$ 732.000 0.803192
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 905.000i − 0.947308i −0.880711 0.473654i $$-0.842935\pi$$
0.880711 0.473654i $$-0.157065\pi$$
$$98$$ − 372.000i − 0.383446i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1278.00 −1.25907 −0.629533 0.776973i $$-0.716754\pi$$
−0.629533 + 0.776973i $$0.716754\pi$$
$$102$$ 0 0
$$103$$ 605.000i 0.578761i 0.957214 + 0.289381i $$0.0934493\pi$$
−0.957214 + 0.289381i $$0.906551\pi$$
$$104$$ −272.000 −0.256460
$$105$$ 0 0
$$106$$ 660.000 0.604763
$$107$$ − 1488.00i − 1.34440i −0.740371 0.672198i $$-0.765350\pi$$
0.740371 0.672198i $$-0.234650\pi$$
$$108$$ 0 0
$$109$$ −593.000 −0.521093 −0.260546 0.965461i $$-0.583903\pi$$
−0.260546 + 0.965461i $$0.583903\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 368.000i − 0.310471i
$$113$$ 324.000i 0.269729i 0.990864 + 0.134864i $$0.0430599\pi$$
−0.990864 + 0.134864i $$0.956940\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 936.000 0.749185
$$117$$ 0 0
$$118$$ 792.000i 0.617877i
$$119$$ 966.000 0.744143
$$120$$ 0 0
$$121$$ −431.000 −0.323817
$$122$$ 46.0000i 0.0341364i
$$123$$ 0 0
$$124$$ 220.000 0.159327
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 1928.00i − 1.34711i −0.739139 0.673553i $$-0.764768\pi$$
0.739139 0.673553i $$-0.235232\pi$$
$$128$$ − 128.000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2742.00 1.82878 0.914388 0.404839i $$-0.132672\pi$$
0.914388 + 0.404839i $$0.132672\pi$$
$$132$$ 0 0
$$133$$ − 3197.00i − 2.08432i
$$134$$ 904.000 0.582789
$$135$$ 0 0
$$136$$ 336.000 0.211851
$$137$$ − 1326.00i − 0.826918i −0.910523 0.413459i $$-0.864320\pi$$
0.910523 0.413459i $$-0.135680\pi$$
$$138$$ 0 0
$$139$$ −893.000 −0.544916 −0.272458 0.962168i $$-0.587837\pi$$
−0.272458 + 0.962168i $$0.587837\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 408.000i 0.241117i
$$143$$ − 1020.00i − 0.596480i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 1382.00 0.783391
$$147$$ 0 0
$$148$$ 764.000i 0.424327i
$$149$$ 2502.00 1.37565 0.687825 0.725877i $$-0.258566\pi$$
0.687825 + 0.725877i $$0.258566\pi$$
$$150$$ 0 0
$$151$$ −2767.00 −1.49123 −0.745613 0.666379i $$-0.767843\pi$$
−0.745613 + 0.666379i $$0.767843\pi$$
$$152$$ − 1112.00i − 0.593389i
$$153$$ 0 0
$$154$$ 1380.00 0.722101
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2701.00i 1.37301i 0.727123 + 0.686507i $$0.240857\pi$$
−0.727123 + 0.686507i $$0.759143\pi$$
$$158$$ 1418.00i 0.713987i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4416.00 2.16167
$$162$$ 0 0
$$163$$ 1748.00i 0.839963i 0.907533 + 0.419981i $$0.137963\pi$$
−0.907533 + 0.419981i $$0.862037\pi$$
$$164$$ −552.000 −0.262829
$$165$$ 0 0
$$166$$ −2196.00 −1.02676
$$167$$ 534.000i 0.247438i 0.992317 + 0.123719i $$0.0394822\pi$$
−0.992317 + 0.123719i $$0.960518\pi$$
$$168$$ 0 0
$$169$$ 1041.00 0.473828
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 212.000i − 0.0939817i
$$173$$ − 192.000i − 0.0843786i −0.999110 0.0421893i $$-0.986567\pi$$
0.999110 0.0421893i $$-0.0134333\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 480.000 0.205576
$$177$$ 0 0
$$178$$ 1632.00i 0.687211i
$$179$$ −1140.00 −0.476020 −0.238010 0.971263i $$-0.576495\pi$$
−0.238010 + 0.971263i $$0.576495\pi$$
$$180$$ 0 0
$$181$$ 398.000 0.163443 0.0817213 0.996655i $$-0.473958\pi$$
0.0817213 + 0.996655i $$0.473958\pi$$
$$182$$ − 1564.00i − 0.636986i
$$183$$ 0 0
$$184$$ 1536.00 0.615410
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1260.00i 0.492729i
$$188$$ 1464.00i 0.567942i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3474.00 1.31607 0.658036 0.752986i $$-0.271387\pi$$
0.658036 + 0.752986i $$0.271387\pi$$
$$192$$ 0 0
$$193$$ − 2713.00i − 1.01184i −0.862579 0.505922i $$-0.831152\pi$$
0.862579 0.505922i $$-0.168848\pi$$
$$194$$ 1810.00 0.669848
$$195$$ 0 0
$$196$$ 744.000 0.271137
$$197$$ − 4734.00i − 1.71210i −0.516894 0.856050i $$-0.672912\pi$$
0.516894 0.856050i $$-0.327088\pi$$
$$198$$ 0 0
$$199$$ −5132.00 −1.82813 −0.914065 0.405568i $$-0.867074\pi$$
−0.914065 + 0.405568i $$0.867074\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 2556.00i − 0.890295i
$$203$$ 5382.00i 1.86080i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1210.00 −0.409246
$$207$$ 0 0
$$208$$ − 544.000i − 0.181344i
$$209$$ 4170.00 1.38012
$$210$$ 0 0
$$211$$ 5240.00 1.70965 0.854826 0.518915i $$-0.173664\pi$$
0.854826 + 0.518915i $$0.173664\pi$$
$$212$$ 1320.00i 0.427632i
$$213$$ 0 0
$$214$$ 2976.00 0.950632
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1265.00i 0.395732i
$$218$$ − 1186.00i − 0.368468i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1428.00 0.434650
$$222$$ 0 0
$$223$$ − 4519.00i − 1.35702i −0.734593 0.678508i $$-0.762627\pi$$
0.734593 0.678508i $$-0.237373\pi$$
$$224$$ 736.000 0.219536
$$225$$ 0 0
$$226$$ −648.000 −0.190727
$$227$$ − 5064.00i − 1.48066i −0.672244 0.740329i $$-0.734670\pi$$
0.672244 0.740329i $$-0.265330\pi$$
$$228$$ 0 0
$$229$$ −2573.00 −0.742483 −0.371242 0.928536i $$-0.621068\pi$$
−0.371242 + 0.928536i $$0.621068\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1872.00i 0.529754i
$$233$$ 4098.00i 1.15223i 0.817370 + 0.576114i $$0.195431\pi$$
−0.817370 + 0.576114i $$0.804569\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1584.00 −0.436905
$$237$$ 0 0
$$238$$ 1932.00i 0.526189i
$$239$$ 306.000 0.0828180 0.0414090 0.999142i $$-0.486815\pi$$
0.0414090 + 0.999142i $$0.486815\pi$$
$$240$$ 0 0
$$241$$ 6482.00 1.73254 0.866270 0.499575i $$-0.166511\pi$$
0.866270 + 0.499575i $$0.166511\pi$$
$$242$$ − 862.000i − 0.228973i
$$243$$ 0 0
$$244$$ −92.0000 −0.0241381
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 4726.00i − 1.21744i
$$248$$ 440.000i 0.112661i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5850.00 −1.47111 −0.735555 0.677465i $$-0.763079\pi$$
−0.735555 + 0.677465i $$0.763079\pi$$
$$252$$ 0 0
$$253$$ 5760.00i 1.43134i
$$254$$ 3856.00 0.952547
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ − 5598.00i − 1.35873i −0.733800 0.679365i $$-0.762255\pi$$
0.733800 0.679365i $$-0.237745\pi$$
$$258$$ 0 0
$$259$$ −4393.00 −1.05393
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5484.00i 1.29314i
$$263$$ − 8286.00i − 1.94272i −0.237603 0.971362i $$-0.576362\pi$$
0.237603 0.971362i $$-0.423638\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 6394.00 1.47384
$$267$$ 0 0
$$268$$ 1808.00i 0.412094i
$$269$$ 504.000 0.114236 0.0571179 0.998367i $$-0.481809\pi$$
0.0571179 + 0.998367i $$0.481809\pi$$
$$270$$ 0 0
$$271$$ −4489.00 −1.00623 −0.503113 0.864221i $$-0.667812\pi$$
−0.503113 + 0.864221i $$0.667812\pi$$
$$272$$ 672.000i 0.149801i
$$273$$ 0 0
$$274$$ 2652.00 0.584720
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2213.00i − 0.480023i −0.970770 0.240011i $$-0.922849\pi$$
0.970770 0.240011i $$-0.0771512\pi$$
$$278$$ − 1786.00i − 0.385314i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2718.00 −0.577019 −0.288509 0.957477i $$-0.593160\pi$$
−0.288509 + 0.957477i $$0.593160\pi$$
$$282$$ 0 0
$$283$$ 5615.00i 1.17942i 0.807613 + 0.589712i $$0.200759\pi$$
−0.807613 + 0.589712i $$0.799241\pi$$
$$284$$ −816.000 −0.170495
$$285$$ 0 0
$$286$$ 2040.00 0.421775
$$287$$ − 3174.00i − 0.652806i
$$288$$ 0 0
$$289$$ 3149.00 0.640953
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2764.00i 0.553941i
$$293$$ 1488.00i 0.296689i 0.988936 + 0.148345i $$0.0473945\pi$$
−0.988936 + 0.148345i $$0.952606\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1528.00 −0.300045
$$297$$ 0 0
$$298$$ 5004.00i 0.972731i
$$299$$ 6528.00 1.26262
$$300$$ 0 0
$$301$$ 1219.00 0.233429
$$302$$ − 5534.00i − 1.05446i
$$303$$ 0 0
$$304$$ 2224.00 0.419589
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7487.00i − 1.39188i −0.718102 0.695938i $$-0.754989\pi$$
0.718102 0.695938i $$-0.245011\pi$$
$$308$$ 2760.00i 0.510603i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8118.00 1.48016 0.740080 0.672519i $$-0.234788\pi$$
0.740080 + 0.672519i $$0.234788\pi$$
$$312$$ 0 0
$$313$$ − 5002.00i − 0.903290i −0.892198 0.451645i $$-0.850837\pi$$
0.892198 0.451645i $$-0.149163\pi$$
$$314$$ −5402.00 −0.970868
$$315$$ 0 0
$$316$$ −2836.00 −0.504865
$$317$$ − 5082.00i − 0.900421i −0.892922 0.450211i $$-0.851349\pi$$
0.892922 0.450211i $$-0.148651\pi$$
$$318$$ 0 0
$$319$$ −7020.00 −1.23211
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8832.00i 1.52853i
$$323$$ 5838.00i 1.00568i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −3496.00 −0.593943
$$327$$ 0 0
$$328$$ − 1104.00i − 0.185848i
$$329$$ −8418.00 −1.41064
$$330$$ 0 0
$$331$$ 7625.00 1.26619 0.633094 0.774075i $$-0.281785\pi$$
0.633094 + 0.774075i $$0.281785\pi$$
$$332$$ − 4392.00i − 0.726031i
$$333$$ 0 0
$$334$$ −1068.00 −0.174965
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3778.00i 0.610685i 0.952243 + 0.305342i $$0.0987709\pi$$
−0.952243 + 0.305342i $$0.901229\pi$$
$$338$$ 2082.00i 0.335047i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1650.00 −0.262031
$$342$$ 0 0
$$343$$ − 3611.00i − 0.568442i
$$344$$ 424.000 0.0664551
$$345$$ 0 0
$$346$$ 384.000 0.0596646
$$347$$ 8268.00i 1.27911i 0.768747 + 0.639553i $$0.220880\pi$$
−0.768747 + 0.639553i $$0.779120\pi$$
$$348$$ 0 0
$$349$$ −1379.00 −0.211508 −0.105754 0.994392i $$-0.533726\pi$$
−0.105754 + 0.994392i $$0.533726\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 960.000i 0.145364i
$$353$$ 3072.00i 0.463190i 0.972812 + 0.231595i $$0.0743944\pi$$
−0.972812 + 0.231595i $$0.925606\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3264.00 −0.485932
$$357$$ 0 0
$$358$$ − 2280.00i − 0.336597i
$$359$$ 7446.00 1.09467 0.547333 0.836915i $$-0.315643\pi$$
0.547333 + 0.836915i $$0.315643\pi$$
$$360$$ 0 0
$$361$$ 12462.0 1.81688
$$362$$ 796.000i 0.115571i
$$363$$ 0 0
$$364$$ 3128.00 0.450417
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6496.00i 0.923947i 0.886894 + 0.461973i $$0.152858\pi$$
−0.886894 + 0.461973i $$0.847142\pi$$
$$368$$ 3072.00i 0.435161i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7590.00 −1.06214
$$372$$ 0 0
$$373$$ − 1633.00i − 0.226685i −0.993556 0.113343i $$-0.963844\pi$$
0.993556 0.113343i $$-0.0361557\pi$$
$$374$$ −2520.00 −0.348412
$$375$$ 0 0
$$376$$ −2928.00 −0.401596
$$377$$ 7956.00i 1.08688i
$$378$$ 0 0
$$379$$ −6788.00 −0.919990 −0.459995 0.887922i $$-0.652149\pi$$
−0.459995 + 0.887922i $$0.652149\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6948.00i 0.930604i
$$383$$ 6582.00i 0.878132i 0.898455 + 0.439066i $$0.144691\pi$$
−0.898455 + 0.439066i $$0.855309\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5426.00 0.715482
$$387$$ 0 0
$$388$$ 3620.00i 0.473654i
$$389$$ −2850.00 −0.371467 −0.185734 0.982600i $$-0.559466\pi$$
−0.185734 + 0.982600i $$0.559466\pi$$
$$390$$ 0 0
$$391$$ −8064.00 −1.04300
$$392$$ 1488.00i 0.191723i
$$393$$ 0 0
$$394$$ 9468.00 1.21064
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 7451.00i − 0.941952i −0.882146 0.470976i $$-0.843902\pi$$
0.882146 0.470976i $$-0.156098\pi$$
$$398$$ − 10264.0i − 1.29268i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14124.0 1.75890 0.879450 0.475991i $$-0.157911\pi$$
0.879450 + 0.475991i $$0.157911\pi$$
$$402$$ 0 0
$$403$$ 1870.00i 0.231145i
$$404$$ 5112.00 0.629533
$$405$$ 0 0
$$406$$ −10764.0 −1.31578
$$407$$ − 5730.00i − 0.697851i
$$408$$ 0 0
$$409$$ −6374.00 −0.770597 −0.385298 0.922792i $$-0.625901\pi$$
−0.385298 + 0.922792i $$0.625901\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 2420.00i − 0.289381i
$$413$$ − 9108.00i − 1.08517i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1088.00 0.128230
$$417$$ 0 0
$$418$$ 8340.00i 0.975892i
$$419$$ −3948.00 −0.460316 −0.230158 0.973153i $$-0.573924\pi$$
−0.230158 + 0.973153i $$0.573924\pi$$
$$420$$ 0 0
$$421$$ 12629.0 1.46199 0.730997 0.682380i $$-0.239055\pi$$
0.730997 + 0.682380i $$0.239055\pi$$
$$422$$ 10480.0i 1.20891i
$$423$$ 0 0
$$424$$ −2640.00 −0.302381
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 529.000i − 0.0599534i
$$428$$ 5952.00i 0.672198i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 4842.00 0.541139 0.270570 0.962700i $$-0.412788\pi$$
0.270570 + 0.962700i $$0.412788\pi$$
$$432$$ 0 0
$$433$$ 3851.00i 0.427407i 0.976899 + 0.213704i $$0.0685526\pi$$
−0.976899 + 0.213704i $$0.931447\pi$$
$$434$$ −2530.00 −0.279825
$$435$$ 0 0
$$436$$ 2372.00 0.260546
$$437$$ 26688.0i 2.92142i
$$438$$ 0 0
$$439$$ 7435.00 0.808322 0.404161 0.914688i $$-0.367564\pi$$
0.404161 + 0.914688i $$0.367564\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2856.00i 0.307344i
$$443$$ − 5760.00i − 0.617756i −0.951102 0.308878i $$-0.900047\pi$$
0.951102 0.308878i $$-0.0999535\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 9038.00 0.959555
$$447$$ 0 0
$$448$$ 1472.00i 0.155235i
$$449$$ 2190.00 0.230184 0.115092 0.993355i $$-0.463284\pi$$
0.115092 + 0.993355i $$0.463284\pi$$
$$450$$ 0 0
$$451$$ 4140.00 0.432251
$$452$$ − 1296.00i − 0.134864i
$$453$$ 0 0
$$454$$ 10128.0 1.04698
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 7202.00i − 0.737189i −0.929590 0.368594i $$-0.879839\pi$$
0.929590 0.368594i $$-0.120161\pi$$
$$458$$ − 5146.00i − 0.525015i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −13476.0 −1.36147 −0.680737 0.732528i $$-0.738341\pi$$
−0.680737 + 0.732528i $$0.738341\pi$$
$$462$$ 0 0
$$463$$ − 10843.0i − 1.08837i −0.838964 0.544187i $$-0.816838\pi$$
0.838964 0.544187i $$-0.183162\pi$$
$$464$$ −3744.00 −0.374592
$$465$$ 0 0
$$466$$ −8196.00 −0.814748
$$467$$ − 6108.00i − 0.605235i −0.953112 0.302617i $$-0.902140\pi$$
0.953112 0.302617i $$-0.0978604\pi$$
$$468$$ 0 0
$$469$$ −10396.0 −1.02355
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 3168.00i − 0.308939i
$$473$$ 1590.00i 0.154563i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3864.00 −0.372072
$$477$$ 0 0
$$478$$ 612.000i 0.0585611i
$$479$$ −9852.00 −0.939769 −0.469885 0.882728i $$-0.655704\pi$$
−0.469885 + 0.882728i $$0.655704\pi$$
$$480$$ 0 0
$$481$$ −6494.00 −0.615594
$$482$$ 12964.0i 1.22509i
$$483$$ 0 0
$$484$$ 1724.00 0.161908
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7796.00i − 0.725401i −0.931906 0.362701i $$-0.881855\pi$$
0.931906 0.362701i $$-0.118145\pi$$
$$488$$ − 184.000i − 0.0170682i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2454.00 0.225555 0.112777 0.993620i $$-0.464025\pi$$
0.112777 + 0.993620i $$0.464025\pi$$
$$492$$ 0 0
$$493$$ − 9828.00i − 0.897831i
$$494$$ 9452.00 0.860862
$$495$$ 0 0
$$496$$ −880.000 −0.0796636
$$497$$ − 4692.00i − 0.423471i
$$498$$ 0 0
$$499$$ 2953.00 0.264919 0.132459 0.991188i $$-0.457713\pi$$
0.132459 + 0.991188i $$0.457713\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 11700.0i − 1.04023i
$$503$$ − 11322.0i − 1.00362i −0.864977 0.501812i $$-0.832667\pi$$
0.864977 0.501812i $$-0.167333\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −11520.0 −1.01211
$$507$$ 0 0
$$508$$ 7712.00i 0.673553i
$$509$$ −9696.00 −0.844337 −0.422169 0.906517i $$-0.638731\pi$$
−0.422169 + 0.906517i $$0.638731\pi$$
$$510$$ 0 0
$$511$$ −15893.0 −1.37586
$$512$$ 512.000i 0.0441942i
$$513$$ 0 0
$$514$$ 11196.0 0.960767
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 10980.0i − 0.934042i
$$518$$ − 8786.00i − 0.745241i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12192.0 −1.02522 −0.512612 0.858621i $$-0.671322\pi$$
−0.512612 + 0.858621i $$0.671322\pi$$
$$522$$ 0 0
$$523$$ − 8491.00i − 0.709915i −0.934882 0.354957i $$-0.884495\pi$$
0.934882 0.354957i $$-0.115505\pi$$
$$524$$ −10968.0 −0.914388
$$525$$ 0 0
$$526$$ 16572.0 1.37371
$$527$$ − 2310.00i − 0.190940i
$$528$$ 0 0
$$529$$ −24697.0 −2.02983
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 12788.0i 1.04216i
$$533$$ − 4692.00i − 0.381300i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −3616.00 −0.291394
$$537$$ 0 0
$$538$$ 1008.00i 0.0807769i
$$539$$ −5580.00 −0.445914
$$540$$ 0 0
$$541$$ −9355.00 −0.743443 −0.371722 0.928344i $$-0.621232\pi$$
−0.371722 + 0.928344i $$0.621232\pi$$
$$542$$ − 8978.00i − 0.711509i
$$543$$ 0 0
$$544$$ −1344.00 −0.105926
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 295.000i 0.0230590i 0.999934 + 0.0115295i $$0.00367004\pi$$
−0.999934 + 0.0115295i $$0.996330\pi$$
$$548$$ 5304.00i 0.413459i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −32526.0 −2.51480
$$552$$ 0 0
$$553$$ − 16307.0i − 1.25397i
$$554$$ 4426.00 0.339427
$$555$$ 0 0
$$556$$ 3572.00 0.272458
$$557$$ 16914.0i 1.28666i 0.765589 + 0.643330i $$0.222448\pi$$
−0.765589 + 0.643330i $$0.777552\pi$$
$$558$$ 0 0
$$559$$ 1802.00 0.136344
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 5436.00i − 0.408014i
$$563$$ − 12108.0i − 0.906379i −0.891414 0.453189i $$-0.850286\pi$$
0.891414 0.453189i $$-0.149714\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −11230.0 −0.833979
$$567$$ 0 0
$$568$$ − 1632.00i − 0.120558i
$$569$$ 6960.00 0.512792 0.256396 0.966572i $$-0.417465\pi$$
0.256396 + 0.966572i $$0.417465\pi$$
$$570$$ 0 0
$$571$$ −10687.0 −0.783252 −0.391626 0.920124i $$-0.628087\pi$$
−0.391626 + 0.920124i $$0.628087\pi$$
$$572$$ 4080.00i 0.298240i
$$573$$ 0 0
$$574$$ 6348.00 0.461603
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8329.00i 0.600937i 0.953792 + 0.300469i $$0.0971431\pi$$
−0.953792 + 0.300469i $$0.902857\pi$$
$$578$$ 6298.00i 0.453222i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 25254.0 1.80329
$$582$$ 0 0
$$583$$ − 9900.00i − 0.703287i
$$584$$ −5528.00 −0.391696
$$585$$ 0 0
$$586$$ −2976.00 −0.209791
$$587$$ 25302.0i 1.77909i 0.456848 + 0.889545i $$0.348978\pi$$
−0.456848 + 0.889545i $$0.651022\pi$$
$$588$$ 0 0
$$589$$ −7645.00 −0.534816
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 3056.00i − 0.212164i
$$593$$ − 22248.0i − 1.54067i −0.637641 0.770334i $$-0.720090\pi$$
0.637641 0.770334i $$-0.279910\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10008.0 −0.687825
$$597$$ 0 0
$$598$$ 13056.0i 0.892809i
$$599$$ −24252.0 −1.65427 −0.827137 0.562001i $$-0.810032\pi$$
−0.827137 + 0.562001i $$0.810032\pi$$
$$600$$ 0 0
$$601$$ −9829.00 −0.667110 −0.333555 0.942731i $$-0.608248\pi$$
−0.333555 + 0.942731i $$0.608248\pi$$
$$602$$ 2438.00i 0.165059i
$$603$$ 0 0
$$604$$ 11068.0 0.745613
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14155.0i 0.946514i 0.880925 + 0.473257i $$0.156922\pi$$
−0.880925 + 0.473257i $$0.843078\pi$$
$$608$$ 4448.00i 0.296694i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12444.0 −0.823945
$$612$$ 0 0
$$613$$ 23051.0i 1.51879i 0.650627 + 0.759397i $$0.274506\pi$$
−0.650627 + 0.759397i $$0.725494\pi$$
$$614$$ 14974.0 0.984204
$$615$$ 0 0
$$616$$ −5520.00 −0.361051
$$617$$ 8352.00i 0.544958i 0.962162 + 0.272479i $$0.0878435\pi$$
−0.962162 + 0.272479i $$0.912157\pi$$
$$618$$ 0 0
$$619$$ 24331.0 1.57988 0.789940 0.613184i $$-0.210112\pi$$
0.789940 + 0.613184i $$0.210112\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16236.0i 1.04663i
$$623$$ − 18768.0i − 1.20694i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10004.0 0.638722
$$627$$ 0 0
$$628$$ − 10804.0i − 0.686507i
$$629$$ 8022.00 0.508518
$$630$$ 0 0
$$631$$ 2216.00 0.139806 0.0699030 0.997554i $$-0.477731\pi$$
0.0699030 + 0.997554i $$0.477731\pi$$
$$632$$ − 5672.00i − 0.356994i
$$633$$ 0 0
$$634$$ 10164.0 0.636694
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6324.00i 0.393353i
$$638$$ − 14040.0i − 0.871237i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −28500.0 −1.75613 −0.878067 0.478537i $$-0.841167\pi$$
−0.878067 + 0.478537i $$0.841167\pi$$
$$642$$ 0 0
$$643$$ − 25252.0i − 1.54874i −0.632731 0.774371i $$-0.718066\pi$$
0.632731 0.774371i $$-0.281934\pi$$
$$644$$ −17664.0 −1.08084
$$645$$ 0 0
$$646$$ −11676.0 −0.711124
$$647$$ 28920.0i 1.75728i 0.477481 + 0.878642i $$0.341550\pi$$
−0.477481 + 0.878642i $$0.658450\pi$$
$$648$$ 0 0
$$649$$ 11880.0 0.718537
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 6992.00i − 0.419981i
$$653$$ 32268.0i 1.93376i 0.255234 + 0.966879i $$0.417848\pi$$
−0.255234 + 0.966879i $$0.582152\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2208.00 0.131415
$$657$$ 0 0
$$658$$ − 16836.0i − 0.997471i
$$659$$ 10050.0 0.594070 0.297035 0.954867i $$-0.404002\pi$$
0.297035 + 0.954867i $$0.404002\pi$$
$$660$$ 0 0
$$661$$ −4561.00 −0.268385 −0.134192 0.990955i $$-0.542844\pi$$
−0.134192 + 0.990955i $$0.542844\pi$$
$$662$$ 15250.0i 0.895329i
$$663$$ 0 0
$$664$$ 8784.00 0.513381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 44928.0i − 2.60812i
$$668$$ − 2136.00i − 0.123719i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 690.000 0.0396977
$$672$$ 0 0
$$673$$ 21359.0i 1.22337i 0.791101 + 0.611686i $$0.209508\pi$$
−0.791101 + 0.611686i $$0.790492\pi$$
$$674$$ −7556.00 −0.431819
$$675$$ 0 0
$$676$$ −4164.00 −0.236914
$$677$$ − 15042.0i − 0.853931i −0.904268 0.426965i $$-0.859583\pi$$
0.904268 0.426965i $$-0.140417\pi$$
$$678$$ 0 0
$$679$$ −20815.0 −1.17645
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 3300.00i − 0.185284i
$$683$$ 27462.0i 1.53851i 0.638940 + 0.769256i $$0.279373\pi$$
−0.638940 + 0.769256i $$0.720627\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 7222.00 0.401949
$$687$$ 0 0
$$688$$ 848.000i 0.0469908i
$$689$$ −11220.0 −0.620389
$$690$$ 0 0
$$691$$ 6212.00 0.341991 0.170995 0.985272i $$-0.445302\pi$$
0.170995 + 0.985272i $$0.445302\pi$$
$$692$$ 768.000i 0.0421893i
$$693$$ 0 0
$$694$$ −16536.0 −0.904464
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 5796.00i 0.314977i
$$698$$ − 2758.00i − 0.149559i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13224.0 −0.712502 −0.356251 0.934390i $$-0.615945\pi$$
−0.356251 + 0.934390i $$0.615945\pi$$
$$702$$ 0 0
$$703$$ − 26549.0i − 1.42434i
$$704$$ −1920.00 −0.102788
$$705$$ 0 0
$$706$$ −6144.00 −0.327525
$$707$$ 29394.0i 1.56361i
$$708$$ 0 0
$$709$$ −34709.0 −1.83854 −0.919269 0.393629i $$-0.871219\pi$$
−0.919269 + 0.393629i $$0.871219\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6528.00i − 0.343606i
$$713$$ − 10560.0i − 0.554664i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4560.00 0.238010
$$717$$ 0 0
$$718$$ 14892.0i 0.774045i
$$719$$ −32070.0 −1.66343 −0.831717 0.555200i $$-0.812642\pi$$
−0.831717 + 0.555200i $$0.812642\pi$$
$$720$$ 0 0
$$721$$ 13915.0 0.718754
$$722$$ 24924.0i 1.28473i
$$723$$ 0 0
$$724$$ −1592.00 −0.0817213
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 125.000i − 0.00637688i −0.999995 0.00318844i $$-0.998985\pi$$
0.999995 0.00318844i $$-0.00101491\pi$$
$$728$$ 6256.00i 0.318493i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2226.00 −0.112629
$$732$$ 0 0
$$733$$ 32222.0i 1.62367i 0.583890 + 0.811833i $$0.301530\pi$$
−0.583890 + 0.811833i $$0.698470\pi$$
$$734$$ −12992.0 −0.653329
$$735$$ 0 0
$$736$$ −6144.00 −0.307705
$$737$$ − 13560.0i − 0.677733i
$$738$$ 0 0
$$739$$ 19240.0 0.957720 0.478860 0.877891i $$-0.341050\pi$$
0.478860 + 0.877891i $$0.341050\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 15180.0i − 0.751045i
$$743$$ 30252.0i 1.49373i 0.664978 + 0.746863i $$0.268441\pi$$
−0.664978 + 0.746863i $$0.731559\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 3266.00 0.160291
$$747$$ 0 0
$$748$$ − 5040.00i − 0.246365i
$$749$$ −34224.0 −1.66958
$$750$$ 0 0
$$751$$ −12517.0 −0.608192 −0.304096 0.952641i $$-0.598354\pi$$
−0.304096 + 0.952641i $$0.598354\pi$$
$$752$$ − 5856.00i − 0.283971i
$$753$$ 0 0
$$754$$ −15912.0 −0.768542
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2201.00i − 0.105676i −0.998603 0.0528380i $$-0.983173\pi$$
0.998603 0.0528380i $$-0.0168267\pi$$
$$758$$ − 13576.0i − 0.650531i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 8742.00 0.416422 0.208211 0.978084i $$-0.433236\pi$$
0.208211 + 0.978084i $$0.433236\pi$$
$$762$$ 0 0
$$763$$ 13639.0i 0.647136i
$$764$$ −13896.0 −0.658036
$$765$$ 0 0
$$766$$ −13164.0 −0.620933
$$767$$ − 13464.0i − 0.633842i
$$768$$ 0 0
$$769$$ 28618.0 1.34199 0.670996 0.741461i $$-0.265867\pi$$
0.670996 + 0.741461i $$0.265867\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10852.0i 0.505922i
$$773$$ 6594.00i 0.306817i 0.988163 + 0.153409i $$0.0490251\pi$$
−0.988163 + 0.153409i $$0.950975\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −7240.00 −0.334924
$$777$$ 0 0
$$778$$ − 5700.00i − 0.262667i
$$779$$ 19182.0 0.882242
$$780$$ 0 0
$$781$$ 6120.00 0.280398
$$782$$ − 16128.0i − 0.737514i
$$783$$ 0 0
$$784$$ −2976.00 −0.135569
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 15881.0i − 0.719309i −0.933085 0.359655i $$-0.882894\pi$$
0.933085 0.359655i $$-0.117106\pi$$
$$788$$ 18936.0i 0.856050i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7452.00 0.334972
$$792$$ 0 0
$$793$$ − 782.000i − 0.0350185i
$$794$$ 14902.0 0.666061
$$795$$ 0 0
$$796$$ 20528.0 0.914065
$$797$$ 26052.0i 1.15785i 0.815380 + 0.578927i $$0.196528\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$798$$ 0 0
$$799$$ 15372.0 0.680629
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 28248.0i 1.24373i
$$803$$ − 20730.0i − 0.911016i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3740.00 −0.163444
$$807$$ 0 0
$$808$$ 10224.0i 0.445147i
$$809$$ −12648.0 −0.549666 −0.274833 0.961492i $$-0.588623\pi$$
−0.274833 + 0.961492i $$0.588623\pi$$
$$810$$ 0 0
$$811$$ 27179.0 1.17680 0.588399 0.808570i $$-0.299758\pi$$
0.588399 + 0.808570i $$0.299758\pi$$
$$812$$ − 21528.0i − 0.930400i
$$813$$ 0 0
$$814$$ 11460.0 0.493456
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7367.00i 0.315470i
$$818$$ − 12748.0i − 0.544894i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11874.0 0.504757 0.252378 0.967629i $$-0.418787\pi$$
0.252378 + 0.967629i $$0.418787\pi$$
$$822$$ 0 0
$$823$$ − 18448.0i − 0.781357i −0.920527 0.390679i $$-0.872240\pi$$
0.920527 0.390679i $$-0.127760\pi$$
$$824$$ 4840.00 0.204623
$$825$$ 0 0
$$826$$ 18216.0 0.767331
$$827$$ 3234.00i 0.135982i 0.997686 + 0.0679911i $$0.0216589\pi$$
−0.997686 + 0.0679911i $$0.978341\pi$$
$$828$$ 0 0
$$829$$ 32155.0 1.34715 0.673576 0.739118i $$-0.264757\pi$$
0.673576 + 0.739118i $$0.264757\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2176.00i 0.0906721i
$$833$$ − 7812.00i − 0.324934i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −16680.0 −0.690060
$$837$$ 0 0
$$838$$ − 7896.00i − 0.325493i
$$839$$ 21996.0 0.905109 0.452554 0.891737i $$-0.350513\pi$$
0.452554 + 0.891737i $$0.350513\pi$$
$$840$$ 0 0
$$841$$ 30367.0 1.24511
$$842$$ 25258.0i 1.03379i
$$843$$ 0 0
$$844$$ −20960.0 −0.854826
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 9913.00i 0.402143i
$$848$$ − 5280.00i − 0.213816i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 36672.0 1.47720
$$852$$ 0 0
$$853$$ 3278.00i 0.131579i 0.997834 + 0.0657893i $$0.0209565\pi$$
−0.997834 + 0.0657893i $$0.979043\pi$$
$$854$$ 1058.00 0.0423935
$$855$$ 0 0
$$856$$ −11904.0 −0.475316
$$857$$ 21228.0i 0.846131i 0.906099 + 0.423066i $$0.139046\pi$$
−0.906099 + 0.423066i $$0.860954\pi$$
$$858$$ 0 0
$$859$$ 5767.00 0.229066 0.114533 0.993419i $$-0.463463\pi$$
0.114533 + 0.993419i $$0.463463\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 9684.00i 0.382643i
$$863$$ − 5322.00i − 0.209922i −0.994476 0.104961i $$-0.966528\pi$$
0.994476 0.104961i $$-0.0334718\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −7702.00 −0.302222
$$867$$ 0 0
$$868$$ − 5060.00i − 0.197866i
$$869$$ 21270.0 0.830305
$$870$$ 0 0
$$871$$ −15368.0 −0.597847
$$872$$ 4744.00i 0.184234i
$$873$$ 0 0
$$874$$ −53376.0 −2.06576
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 42097.0i 1.62088i 0.585819 + 0.810442i $$0.300773\pi$$
−0.585819 + 0.810442i $$0.699227\pi$$
$$878$$ 14870.0i 0.571570i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −17658.0 −0.675270 −0.337635 0.941277i $$-0.609627\pi$$
−0.337635 + 0.941277i $$0.609627\pi$$
$$882$$ 0 0
$$883$$ − 22297.0i − 0.849778i −0.905246 0.424889i $$-0.860313\pi$$
0.905246 0.424889i $$-0.139687\pi$$
$$884$$ −5712.00 −0.217325
$$885$$ 0 0
$$886$$ 11520.0 0.436819
$$887$$ 10542.0i 0.399059i 0.979892 + 0.199530i $$0.0639414\pi$$
−0.979892 + 0.199530i $$0.936059\pi$$
$$888$$ 0 0
$$889$$ −44344.0 −1.67295
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 18076.0i 0.678508i
$$893$$ − 50874.0i − 1.90642i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2944.00 −0.109768
$$897$$ 0 0
$$898$$ 4380.00i 0.162764i
$$899$$ 12870.0 0.477462
$$900$$ 0 0
$$901$$ 13860.0 0.512479
$$902$$ 8280.00i 0.305647i
$$903$$ 0 0
$$904$$ 2592.00 0.0953635
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 47639.0i − 1.74402i −0.489487 0.872010i $$-0.662816\pi$$
0.489487 0.872010i $$-0.337184\pi$$
$$908$$ 20256.0i 0.740329i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10326.0 −0.375539 −0.187769 0.982213i $$-0.560126\pi$$
−0.187769 + 0.982213i $$0.560126\pi$$
$$912$$ 0 0
$$913$$ 32940.0i 1.19404i
$$914$$ 14404.0 0.521271
$$915$$ 0 0
$$916$$ 10292.0 0.371242
$$917$$ − 63066.0i − 2.27113i
$$918$$ 0 0
$$919$$ −5147.00 −0.184748 −0.0923742 0.995724i $$-0.529446\pi$$
−0.0923742 + 0.995724i $$0.529446\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 26952.0i − 0.962708i
$$923$$ − 6936.00i − 0.247347i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 21686.0 0.769596
$$927$$ 0 0
$$928$$ − 7488.00i − 0.264877i
$$929$$ 47064.0 1.66213 0.831066 0.556175i $$-0.187731\pi$$
0.831066 + 0.556175i $$0.187731\pi$$
$$930$$ 0 0
$$931$$ −25854.0 −0.910130
$$932$$ − 16392.0i − 0.576114i
$$933$$ 0 0
$$934$$ 12216.0 0.427965
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 15385.0i 0.536399i 0.963363 + 0.268200i $$0.0864287\pi$$
−0.963363 + 0.268200i $$0.913571\pi$$
$$938$$ − 20792.0i − 0.723756i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −34914.0 −1.20953 −0.604763 0.796406i $$-0.706732\pi$$
−0.604763 + 0.796406i $$0.706732\pi$$
$$942$$ 0 0
$$943$$ 26496.0i 0.914982i
$$944$$ 6336.00 0.218453
$$945$$ 0 0
$$946$$ −3180.00 −0.109293
$$947$$ − 37434.0i − 1.28452i −0.766486 0.642261i $$-0.777997\pi$$
0.766486 0.642261i $$-0.222003\pi$$
$$948$$ 0 0
$$949$$ −23494.0 −0.803633
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 7728.00i − 0.263094i
$$953$$ 8778.00i 0.298371i 0.988809 + 0.149185i $$0.0476651\pi$$
−0.988809 + 0.149185i $$0.952335\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1224.00 −0.0414090
$$957$$ 0 0
$$958$$ − 19704.0i − 0.664517i
$$959$$ −30498.0 −1.02694
$$960$$ 0 0
$$961$$ −26766.0 −0.898459
$$962$$ − 12988.0i − 0.435291i
$$963$$ 0 0
$$964$$ −25928.0 −0.866270
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 54061.0i 1.79781i 0.438141 + 0.898906i $$0.355637\pi$$
−0.438141 + 0.898906i $$0.644363\pi$$
$$968$$ 3448.00i 0.114486i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3084.00 −0.101926 −0.0509631 0.998701i $$-0.516229\pi$$
−0.0509631 + 0.998701i $$0.516229\pi$$
$$972$$ 0 0
$$973$$ 20539.0i 0.676722i
$$974$$ 15592.0 0.512936
$$975$$ 0 0
$$976$$ 368.000 0.0120691
$$977$$ − 12048.0i − 0.394524i −0.980351 0.197262i $$-0.936795\pi$$
0.980351 0.197262i $$-0.0632049\pi$$
$$978$$ 0 0
$$979$$ 24480.0 0.799167
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 4908.00i 0.159491i
$$983$$ 33618.0i 1.09079i 0.838179 + 0.545396i $$0.183621\pi$$
−0.838179 + 0.545396i $$0.816379\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 19656.0 0.634863
$$987$$ 0 0
$$988$$ 18904.0i 0.608721i
$$989$$ −10176.0 −0.327177
$$990$$ 0 0
$$991$$ 25043.0 0.802742 0.401371 0.915916i $$-0.368534\pi$$
0.401371 + 0.915916i $$0.368534\pi$$
$$992$$ − 1760.00i − 0.0563307i
$$993$$ 0 0
$$994$$ 9384.00 0.299439
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 17710.0i 0.562569i 0.959624 + 0.281285i $$0.0907605\pi$$
−0.959624 + 0.281285i $$0.909240\pi$$
$$998$$ 5906.00i 0.187326i
$$999$$ 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.p.649.2 2
3.2 odd 2 1350.4.c.e.649.1 2
5.2 odd 4 1350.4.a.m.1.1 yes 1
5.3 odd 4 1350.4.a.p.1.1 yes 1
5.4 even 2 inner 1350.4.c.p.649.1 2
15.2 even 4 1350.4.a.ba.1.1 yes 1
15.8 even 4 1350.4.a.b.1.1 1
15.14 odd 2 1350.4.c.e.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1350.4.a.b.1.1 1 15.8 even 4
1350.4.a.m.1.1 yes 1 5.2 odd 4
1350.4.a.p.1.1 yes 1 5.3 odd 4
1350.4.a.ba.1.1 yes 1 15.2 even 4
1350.4.c.e.649.1 2 3.2 odd 2
1350.4.c.e.649.2 2 15.14 odd 2
1350.4.c.p.649.1 2 5.4 even 2 inner
1350.4.c.p.649.2 2 1.1 even 1 trivial