# Properties

 Label 8100.2.d.m.649.4 Level $8100$ Weight $2$ Character 8100.649 Analytic conductor $64.679$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.6788256372$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1620) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 8100.649 Dual form 8100.2.d.m.649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.73205i q^{7} +O(q^{10})$$ $$q+2.73205i q^{7} +1.73205 q^{11} +5.46410i q^{13} +4.73205i q^{17} +4.46410 q^{19} +3.46410i q^{23} +7.73205 q^{29} +5.92820 q^{31} +6.19615i q^{37} -11.1962 q^{41} +3.26795i q^{43} +1.26795i q^{47} -0.464102 q^{49} -7.26795i q^{53} +7.73205 q^{59} -4.00000 q^{61} -6.39230i q^{67} -11.1962 q^{71} -0.196152i q^{73} +4.73205i q^{77} +14.3923 q^{79} -15.1244i q^{83} -5.19615 q^{89} -14.9282 q^{91} -0.732051i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 4 q^{19} + 24 q^{29} - 4 q^{31} - 24 q^{41} + 12 q^{49} + 24 q^{59} - 16 q^{61} - 24 q^{71} + 16 q^{79} - 32 q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$$.

 $$n$$ $$4051$$ $$6401$$ $$7777$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$7$$ 2.73205i 1.03262i 0.856402 + 0.516309i $$0.172694\pi$$
−0.856402 + 0.516309i $$0.827306\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.73205 0.522233 0.261116 0.965307i $$-0.415909\pi$$
0.261116 + 0.965307i $$0.415909\pi$$
$$12$$ 0 0
$$13$$ 5.46410i 1.51547i 0.652563 + 0.757735i $$0.273694\pi$$
−0.652563 + 0.757735i $$0.726306\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.73205i 1.14769i 0.818964 + 0.573845i $$0.194549\pi$$
−0.818964 + 0.573845i $$0.805451\pi$$
$$18$$ 0 0
$$19$$ 4.46410 1.02414 0.512068 0.858945i $$-0.328880\pi$$
0.512068 + 0.858945i $$0.328880\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.46410i 0.722315i 0.932505 + 0.361158i $$0.117618\pi$$
−0.932505 + 0.361158i $$0.882382\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.73205 1.43581 0.717903 0.696143i $$-0.245102\pi$$
0.717903 + 0.696143i $$0.245102\pi$$
$$30$$ 0 0
$$31$$ 5.92820 1.06474 0.532368 0.846513i $$-0.321302\pi$$
0.532368 + 0.846513i $$0.321302\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.19615i 1.01864i 0.860577 + 0.509321i $$0.170103\pi$$
−0.860577 + 0.509321i $$0.829897\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −11.1962 −1.74855 −0.874273 0.485435i $$-0.838661\pi$$
−0.874273 + 0.485435i $$0.838661\pi$$
$$42$$ 0 0
$$43$$ 3.26795i 0.498358i 0.968458 + 0.249179i $$0.0801607\pi$$
−0.968458 + 0.249179i $$0.919839\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.26795i 0.184949i 0.995715 + 0.0924747i $$0.0294777\pi$$
−0.995715 + 0.0924747i $$0.970522\pi$$
$$48$$ 0 0
$$49$$ −0.464102 −0.0663002
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 7.26795i − 0.998330i −0.866507 0.499165i $$-0.833640\pi$$
0.866507 0.499165i $$-0.166360\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 7.73205 1.00663 0.503314 0.864104i $$-0.332114\pi$$
0.503314 + 0.864104i $$0.332114\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.39230i − 0.780944i −0.920615 0.390472i $$-0.872312\pi$$
0.920615 0.390472i $$-0.127688\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −11.1962 −1.32874 −0.664369 0.747404i $$-0.731300\pi$$
−0.664369 + 0.747404i $$0.731300\pi$$
$$72$$ 0 0
$$73$$ − 0.196152i − 0.0229579i −0.999934 0.0114790i $$-0.996346\pi$$
0.999934 0.0114790i $$-0.00365394\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.73205i 0.539267i
$$78$$ 0 0
$$79$$ 14.3923 1.61926 0.809630 0.586940i $$-0.199668\pi$$
0.809630 + 0.586940i $$0.199668\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 15.1244i − 1.66011i −0.557679 0.830057i $$-0.688308\pi$$
0.557679 0.830057i $$-0.311692\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −5.19615 −0.550791 −0.275396 0.961331i $$-0.588809\pi$$
−0.275396 + 0.961331i $$0.588809\pi$$
$$90$$ 0 0
$$91$$ −14.9282 −1.56490
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 0.732051i − 0.0743285i −0.999309 0.0371642i $$-0.988168\pi$$
0.999309 0.0371642i $$-0.0118325\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.12436 0.609396 0.304698 0.952449i $$-0.401444\pi$$
0.304698 + 0.952449i $$0.401444\pi$$
$$102$$ 0 0
$$103$$ 18.3923i 1.81225i 0.423013 + 0.906124i $$0.360973\pi$$
−0.423013 + 0.906124i $$0.639027\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 3.46410i − 0.334887i −0.985882 0.167444i $$-0.946449\pi$$
0.985882 0.167444i $$-0.0535512\pi$$
$$108$$ 0 0
$$109$$ 7.92820 0.759384 0.379692 0.925113i $$-0.376030\pi$$
0.379692 + 0.925113i $$0.376030\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 0.339746i − 0.0319606i −0.999872 0.0159803i $$-0.994913\pi$$
0.999872 0.0159803i $$-0.00508691\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −12.9282 −1.18513
$$120$$ 0 0
$$121$$ −8.00000 −0.727273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 4.19615i − 0.372348i −0.982517 0.186174i $$-0.940391\pi$$
0.982517 0.186174i $$-0.0596089\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.2679 0.897115 0.448557 0.893754i $$-0.351938\pi$$
0.448557 + 0.893754i $$0.351938\pi$$
$$132$$ 0 0
$$133$$ 12.1962i 1.05754i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 4.39230i − 0.375260i −0.982240 0.187630i $$-0.939919\pi$$
0.982240 0.187630i $$-0.0600806\pi$$
$$138$$ 0 0
$$139$$ −15.3923 −1.30556 −0.652779 0.757548i $$-0.726397\pi$$
−0.652779 + 0.757548i $$0.726397\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 9.46410i 0.791428i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −17.3923 −1.41537 −0.707683 0.706530i $$-0.750259\pi$$
−0.707683 + 0.706530i $$0.750259\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3.26795i − 0.260811i −0.991461 0.130405i $$-0.958372\pi$$
0.991461 0.130405i $$-0.0416279\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −9.46410 −0.745876
$$162$$ 0 0
$$163$$ 18.7321i 1.46721i 0.679577 + 0.733604i $$0.262163\pi$$
−0.679577 + 0.733604i $$0.737837\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 21.1244i 1.63465i 0.576176 + 0.817326i $$0.304544\pi$$
−0.576176 + 0.817326i $$0.695456\pi$$
$$168$$ 0 0
$$169$$ −16.8564 −1.29665
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 24.2487i − 1.84360i −0.387671 0.921798i $$-0.626720\pi$$
0.387671 0.921798i $$-0.373280\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.1244 0.906217 0.453108 0.891455i $$-0.350315\pi$$
0.453108 + 0.891455i $$0.350315\pi$$
$$180$$ 0 0
$$181$$ −16.4641 −1.22377 −0.611884 0.790948i $$-0.709588\pi$$
−0.611884 + 0.790948i $$0.709588\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.19615i 0.599362i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.0526 1.37859 0.689297 0.724479i $$-0.257919\pi$$
0.689297 + 0.724479i $$0.257919\pi$$
$$192$$ 0 0
$$193$$ − 10.5885i − 0.762174i −0.924539 0.381087i $$-0.875550\pi$$
0.924539 0.381087i $$-0.124450\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.8564i 0.987228i 0.869681 + 0.493614i $$0.164324\pi$$
−0.869681 + 0.493614i $$0.835676\pi$$
$$198$$ 0 0
$$199$$ −15.8564 −1.12403 −0.562015 0.827127i $$-0.689974\pi$$
−0.562015 + 0.827127i $$0.689974\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 21.1244i 1.48264i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 7.73205 0.534837
$$210$$ 0 0
$$211$$ −19.9282 −1.37191 −0.685957 0.727642i $$-0.740616\pi$$
−0.685957 + 0.727642i $$0.740616\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.1962i 1.09947i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −25.8564 −1.73929
$$222$$ 0 0
$$223$$ − 5.85641i − 0.392174i −0.980587 0.196087i $$-0.937177\pi$$
0.980587 0.196087i $$-0.0628235\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 26.1962i − 1.73870i −0.494197 0.869350i $$-0.664538\pi$$
0.494197 0.869350i $$-0.335462\pi$$
$$228$$ 0 0
$$229$$ 17.8564 1.17998 0.589992 0.807409i $$-0.299131\pi$$
0.589992 + 0.807409i $$0.299131\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.5885i 1.61084i 0.592702 + 0.805422i $$0.298061\pi$$
−0.592702 + 0.805422i $$0.701939\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −8.53590 −0.552141 −0.276071 0.961137i $$-0.589032\pi$$
−0.276071 + 0.961137i $$0.589032\pi$$
$$240$$ 0 0
$$241$$ 10.3205 0.664802 0.332401 0.943138i $$-0.392141\pi$$
0.332401 + 0.943138i $$0.392141\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 24.3923i 1.55205i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.5359 1.29621 0.648107 0.761549i $$-0.275561\pi$$
0.648107 + 0.761549i $$0.275561\pi$$
$$252$$ 0 0
$$253$$ 6.00000i 0.377217i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 15.4641i − 0.964624i −0.875999 0.482312i $$-0.839797\pi$$
0.875999 0.482312i $$-0.160203\pi$$
$$258$$ 0 0
$$259$$ −16.9282 −1.05187
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 24.2487i − 1.49524i −0.664127 0.747620i $$-0.731197\pi$$
0.664127 0.747620i $$-0.268803\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 16.2679 0.991874 0.495937 0.868358i $$-0.334825\pi$$
0.495937 + 0.868358i $$0.334825\pi$$
$$270$$ 0 0
$$271$$ 16.7846 1.01959 0.509796 0.860295i $$-0.329721\pi$$
0.509796 + 0.860295i $$0.329721\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 5.12436i − 0.307893i −0.988079 0.153946i $$-0.950802\pi$$
0.988079 0.153946i $$-0.0491983\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 29.3205 1.74911 0.874557 0.484922i $$-0.161152\pi$$
0.874557 + 0.484922i $$0.161152\pi$$
$$282$$ 0 0
$$283$$ 16.5359i 0.982957i 0.870890 + 0.491479i $$0.163543\pi$$
−0.870890 + 0.491479i $$0.836457\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 30.5885i − 1.80558i
$$288$$ 0 0
$$289$$ −5.39230 −0.317194
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 24.5885i 1.43647i 0.695799 + 0.718237i $$0.255050\pi$$
−0.695799 + 0.718237i $$0.744950\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −18.9282 −1.09465
$$300$$ 0 0
$$301$$ −8.92820 −0.514613
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.80385i 0.102951i 0.998674 + 0.0514755i $$0.0163924\pi$$
−0.998674 + 0.0514755i $$0.983608\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.5167 −0.936574 −0.468287 0.883576i $$-0.655129\pi$$
−0.468287 + 0.883576i $$0.655129\pi$$
$$312$$ 0 0
$$313$$ 14.9282i 0.843792i 0.906644 + 0.421896i $$0.138635\pi$$
−0.906644 + 0.421896i $$0.861365\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 3.12436i − 0.175481i −0.996143 0.0877406i $$-0.972035\pi$$
0.996143 0.0877406i $$-0.0279647\pi$$
$$318$$ 0 0
$$319$$ 13.3923 0.749825
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 21.1244i 1.17539i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −3.46410 −0.190982
$$330$$ 0 0
$$331$$ −29.3923 −1.61555 −0.807774 0.589493i $$-0.799328\pi$$
−0.807774 + 0.589493i $$0.799328\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 34.2487i 1.86565i 0.360335 + 0.932823i $$0.382662\pi$$
−0.360335 + 0.932823i $$0.617338\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.2679 0.556041
$$342$$ 0 0
$$343$$ 17.8564i 0.964155i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 21.8038i − 1.17049i −0.810856 0.585246i $$-0.800998\pi$$
0.810856 0.585246i $$-0.199002\pi$$
$$348$$ 0 0
$$349$$ 25.0000 1.33822 0.669110 0.743164i $$-0.266676\pi$$
0.669110 + 0.743164i $$0.266676\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 28.9808i − 1.54249i −0.636538 0.771245i $$-0.719634\pi$$
0.636538 0.771245i $$-0.280366\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −9.33975 −0.492933 −0.246466 0.969151i $$-0.579270\pi$$
−0.246466 + 0.969151i $$0.579270\pi$$
$$360$$ 0 0
$$361$$ 0.928203 0.0488528
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.3923i 1.37767i 0.724920 + 0.688834i $$0.241877\pi$$
−0.724920 + 0.688834i $$0.758123\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 19.8564 1.03089
$$372$$ 0 0
$$373$$ − 14.0526i − 0.727614i −0.931474 0.363807i $$-0.881477\pi$$
0.931474 0.363807i $$-0.118523\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 42.2487i 2.17592i
$$378$$ 0 0
$$379$$ −4.53590 −0.232993 −0.116497 0.993191i $$-0.537166\pi$$
−0.116497 + 0.993191i $$0.537166\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 18.2487i 0.932466i 0.884662 + 0.466233i $$0.154389\pi$$
−0.884662 + 0.466233i $$0.845611\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −27.4641 −1.39249 −0.696243 0.717807i $$-0.745146\pi$$
−0.696243 + 0.717807i $$0.745146\pi$$
$$390$$ 0 0
$$391$$ −16.3923 −0.828994
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 37.3205i − 1.87306i −0.350584 0.936531i $$-0.614017\pi$$
0.350584 0.936531i $$-0.385983\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.7846 −0.738308 −0.369154 0.929368i $$-0.620353\pi$$
−0.369154 + 0.929368i $$0.620353\pi$$
$$402$$ 0 0
$$403$$ 32.3923i 1.61358i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10.7321i 0.531968i
$$408$$ 0 0
$$409$$ 17.8564 0.882942 0.441471 0.897275i $$-0.354457\pi$$
0.441471 + 0.897275i $$0.354457\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 21.1244i 1.03946i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −21.4641 −1.04859 −0.524295 0.851537i $$-0.675671\pi$$
−0.524295 + 0.851537i $$0.675671\pi$$
$$420$$ 0 0
$$421$$ 13.7846 0.671821 0.335910 0.941894i $$-0.390956\pi$$
0.335910 + 0.941894i $$0.390956\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 10.9282i − 0.528853i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.5885 −1.61790 −0.808950 0.587878i $$-0.799963\pi$$
−0.808950 + 0.587878i $$0.799963\pi$$
$$432$$ 0 0
$$433$$ 11.4641i 0.550930i 0.961311 + 0.275465i $$0.0888317\pi$$
−0.961311 + 0.275465i $$0.911168\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 15.4641i 0.739748i
$$438$$ 0 0
$$439$$ −6.60770 −0.315368 −0.157684 0.987490i $$-0.550403\pi$$
−0.157684 + 0.987490i $$0.550403\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 37.2679i 1.77065i 0.464969 + 0.885327i $$0.346065\pi$$
−0.464969 + 0.885327i $$0.653935\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −24.1244 −1.13850 −0.569249 0.822165i $$-0.692766\pi$$
−0.569249 + 0.822165i $$0.692766\pi$$
$$450$$ 0 0
$$451$$ −19.3923 −0.913148
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.1962i − 1.03829i −0.854686 0.519146i $$-0.826250\pi$$
0.854686 0.519146i $$-0.173750\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.58846 0.446579 0.223289 0.974752i $$-0.428320\pi$$
0.223289 + 0.974752i $$0.428320\pi$$
$$462$$ 0 0
$$463$$ − 2.39230i − 0.111180i −0.998454 0.0555899i $$-0.982296\pi$$
0.998454 0.0555899i $$-0.0177039\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 26.1962i − 1.21221i −0.795383 0.606107i $$-0.792730\pi$$
0.795383 0.606107i $$-0.207270\pi$$
$$468$$ 0 0
$$469$$ 17.4641 0.806417
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.66025i 0.260259i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 3.33975 0.152597 0.0762984 0.997085i $$-0.475690\pi$$
0.0762984 + 0.997085i $$0.475690\pi$$
$$480$$ 0 0
$$481$$ −33.8564 −1.54372
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 30.5359i 1.38371i 0.722035 + 0.691857i $$0.243207\pi$$
−0.722035 + 0.691857i $$0.756793\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.26795 −0.192610 −0.0963049 0.995352i $$-0.530702\pi$$
−0.0963049 + 0.995352i $$0.530702\pi$$
$$492$$ 0 0
$$493$$ 36.5885i 1.64786i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 30.5885i − 1.37208i
$$498$$ 0 0
$$499$$ −27.3923 −1.22625 −0.613124 0.789987i $$-0.710087\pi$$
−0.613124 + 0.789987i $$0.710087\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 35.3205i − 1.57486i −0.616402 0.787432i $$-0.711410\pi$$
0.616402 0.787432i $$-0.288590\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 26.7846 1.18721 0.593603 0.804758i $$-0.297705\pi$$
0.593603 + 0.804758i $$0.297705\pi$$
$$510$$ 0 0
$$511$$ 0.535898 0.0237067
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2.19615i 0.0965867i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.3923 −0.455295 −0.227648 0.973744i $$-0.573103\pi$$
−0.227648 + 0.973744i $$0.573103\pi$$
$$522$$ 0 0
$$523$$ 3.60770i 0.157753i 0.996884 + 0.0788767i $$0.0251334\pi$$
−0.996884 + 0.0788767i $$0.974867\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 28.0526i 1.22199i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 61.1769i − 2.64987i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −0.803848 −0.0346242
$$540$$ 0 0
$$541$$ 2.46410 0.105940 0.0529700 0.998596i $$-0.483131\pi$$
0.0529700 + 0.998596i $$0.483131\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.78461i − 0.204575i −0.994755 0.102288i $$-0.967384\pi$$
0.994755 0.102288i $$-0.0326162\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 34.5167 1.47046
$$552$$ 0 0
$$553$$ 39.3205i 1.67208i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.39230i 0.186108i 0.995661 + 0.0930540i $$0.0296629\pi$$
−0.995661 + 0.0930540i $$0.970337\pi$$
$$558$$ 0 0
$$559$$ −17.8564 −0.755246
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.7321i 0.958042i 0.877804 + 0.479021i $$0.159008\pi$$
−0.877804 + 0.479021i $$0.840992\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 43.0526 1.80486 0.902429 0.430839i $$-0.141782\pi$$
0.902429 + 0.430839i $$0.141782\pi$$
$$570$$ 0 0
$$571$$ 0.856406 0.0358395 0.0179197 0.999839i $$-0.494296\pi$$
0.0179197 + 0.999839i $$0.494296\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11.8038i − 0.491401i −0.969346 0.245700i $$-0.920982\pi$$
0.969346 0.245700i $$-0.0790179\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 41.3205 1.71426
$$582$$ 0 0
$$583$$ − 12.5885i − 0.521361i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 3.80385i − 0.157002i −0.996914 0.0785008i $$-0.974987\pi$$
0.996914 0.0785008i $$-0.0250133\pi$$
$$588$$ 0 0
$$589$$ 26.4641 1.09043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 23.0718i − 0.947445i −0.880674 0.473723i $$-0.842910\pi$$
0.880674 0.473723i $$-0.157090\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 14.4115 0.588840 0.294420 0.955676i $$-0.404874\pi$$
0.294420 + 0.955676i $$0.404874\pi$$
$$600$$ 0 0
$$601$$ −24.3205 −0.992054 −0.496027 0.868307i $$-0.665208\pi$$
−0.496027 + 0.868307i $$0.665208\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.58846i − 0.105062i −0.998619 0.0525311i $$-0.983271\pi$$
0.998619 0.0525311i $$-0.0167289\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.92820 −0.280285
$$612$$ 0 0
$$613$$ 24.3923i 0.985196i 0.870257 + 0.492598i $$0.163953\pi$$
−0.870257 + 0.492598i $$0.836047\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 14.1962i − 0.568757i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −29.3205 −1.16909
$$630$$ 0 0
$$631$$ −0.0717968 −0.00285818 −0.00142909 0.999999i $$-0.500455\pi$$
−0.00142909 + 0.999999i $$0.500455\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.53590i − 0.100476i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.8038 −0.505722 −0.252861 0.967503i $$-0.581371\pi$$
−0.252861 + 0.967503i $$0.581371\pi$$
$$642$$ 0 0
$$643$$ 14.5885i 0.575313i 0.957734 + 0.287656i $$0.0928761\pi$$
−0.957734 + 0.287656i $$0.907124\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 0.248711i − 0.00977785i −0.999988 0.00488893i $$-0.998444\pi$$
0.999988 0.00488893i $$-0.00155620\pi$$
$$648$$ 0 0
$$649$$ 13.3923 0.525694
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 2.53590i − 0.0992374i −0.998768 0.0496187i $$-0.984199\pi$$
0.998768 0.0496187i $$-0.0158006\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 2.53590 0.0987846 0.0493923 0.998779i $$-0.484272\pi$$
0.0493923 + 0.998779i $$0.484272\pi$$
$$660$$ 0 0
$$661$$ 15.3923 0.598691 0.299346 0.954145i $$-0.403232\pi$$
0.299346 + 0.954145i $$0.403232\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 26.7846i 1.03710i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.92820 −0.267460
$$672$$ 0 0
$$673$$ − 38.3923i − 1.47991i −0.672654 0.739957i $$-0.734846\pi$$
0.672654 0.739957i $$-0.265154\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 40.6410i − 1.56196i −0.624555 0.780981i $$-0.714720\pi$$
0.624555 0.780981i $$-0.285280\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 21.4641i 0.821301i 0.911793 + 0.410651i $$0.134698\pi$$
−0.911793 + 0.410651i $$0.865302\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 39.7128 1.51294
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 52.9808i − 2.00679i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −17.8756 −0.675154 −0.337577 0.941298i $$-0.609607\pi$$
−0.337577 + 0.941298i $$0.609607\pi$$
$$702$$ 0 0
$$703$$ 27.6603i 1.04323i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.7321i 0.629274i
$$708$$ 0 0
$$709$$ −10.5359 −0.395684 −0.197842 0.980234i $$-0.563393\pi$$
−0.197842 + 0.980234i $$0.563393\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 20.5359i 0.769075i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 17.1962 0.641308 0.320654 0.947196i $$-0.396097\pi$$
0.320654 + 0.947196i $$0.396097\pi$$
$$720$$ 0 0
$$721$$ −50.2487 −1.87136
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 29.1769i 1.08211i 0.840987 + 0.541056i $$0.181975\pi$$
−0.840987 + 0.541056i $$0.818025\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −15.4641 −0.571960
$$732$$ 0 0
$$733$$ 43.5692i 1.60927i 0.593773 + 0.804633i $$0.297638\pi$$
−0.593773 + 0.804633i $$0.702362\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 11.0718i − 0.407835i
$$738$$ 0 0
$$739$$ 26.1769 0.962933 0.481467 0.876464i $$-0.340104\pi$$
0.481467 + 0.876464i $$0.340104\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 5.41154i − 0.198530i −0.995061 0.0992651i $$-0.968351\pi$$
0.995061 0.0992651i $$-0.0316492\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 9.46410 0.345811
$$750$$ 0 0
$$751$$ 40.7846 1.48825 0.744126 0.668040i $$-0.232866\pi$$
0.744126 + 0.668040i $$0.232866\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 20.3923i 0.741171i 0.928798 + 0.370585i $$0.120843\pi$$
−0.928798 + 0.370585i $$0.879157\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.1244 −1.09201 −0.546004 0.837783i $$-0.683851\pi$$
−0.546004 + 0.837783i $$0.683851\pi$$
$$762$$ 0 0
$$763$$ 21.6603i 0.784154i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 42.2487i 1.52551i
$$768$$ 0 0
$$769$$ −35.2487 −1.27110 −0.635551 0.772059i $$-0.719227\pi$$
−0.635551 + 0.772059i $$0.719227\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 17.6603i 0.635195i 0.948226 + 0.317598i $$0.102876\pi$$
−0.948226 + 0.317598i $$0.897124\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −49.9808 −1.79075
$$780$$ 0 0
$$781$$ −19.3923 −0.693911
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 36.1962i 1.29025i 0.764076 + 0.645127i $$0.223195\pi$$
−0.764076 + 0.645127i $$0.776805\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.928203 0.0330031
$$792$$ 0 0
$$793$$ − 21.8564i − 0.776144i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 23.3205i − 0.826055i −0.910719 0.413027i $$-0.864471\pi$$
0.910719 0.413027i $$-0.135529\pi$$
$$798$$ 0 0
$$799$$ −6.00000 −0.212265
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 0.339746i − 0.0119894i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 8.41154 0.295734 0.147867 0.989007i $$-0.452759\pi$$
0.147867 + 0.989007i $$0.452759\pi$$
$$810$$ 0 0
$$811$$ −25.2487 −0.886602 −0.443301 0.896373i $$-0.646193\pi$$
−0.443301 + 0.896373i $$0.646193\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 14.5885i 0.510386i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.33975 0.116558 0.0582790 0.998300i $$-0.481439\pi$$
0.0582790 + 0.998300i $$0.481439\pi$$
$$822$$ 0 0
$$823$$ − 16.9282i − 0.590080i −0.955485 0.295040i $$-0.904667\pi$$
0.955485 0.295040i $$-0.0953330\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 45.4641i 1.58094i 0.612500 + 0.790471i $$0.290164\pi$$
−0.612500 + 0.790471i $$0.709836\pi$$
$$828$$ 0 0
$$829$$ 51.7846 1.79855 0.899277 0.437380i $$-0.144093\pi$$
0.899277 + 0.437380i $$0.144093\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 2.19615i − 0.0760922i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 8.41154 0.290399 0.145199 0.989402i $$-0.453618\pi$$
0.145199 + 0.989402i $$0.453618\pi$$
$$840$$ 0 0
$$841$$ 30.7846 1.06154
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 21.8564i − 0.750995i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −21.4641 −0.735780
$$852$$ 0 0
$$853$$ − 0.196152i − 0.00671613i −0.999994 0.00335807i $$-0.998931\pi$$
0.999994 0.00335807i $$-0.00106891\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 29.0718i 0.993074i 0.868016 + 0.496537i $$0.165395\pi$$
−0.868016 + 0.496537i $$0.834605\pi$$
$$858$$ 0 0
$$859$$ −7.78461 −0.265607 −0.132804 0.991142i $$-0.542398\pi$$
−0.132804 + 0.991142i $$0.542398\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 49.5167i − 1.68557i −0.538253 0.842783i $$-0.680915\pi$$
0.538253 0.842783i $$-0.319085\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 24.9282 0.845631
$$870$$ 0 0
$$871$$ 34.9282 1.18350
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 14.2487i − 0.481145i −0.970631 0.240572i $$-0.922665\pi$$
0.970631 0.240572i $$-0.0773351\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.80385 0.229227 0.114614 0.993410i $$-0.463437\pi$$
0.114614 + 0.993410i $$0.463437\pi$$
$$882$$ 0 0
$$883$$ − 46.8372i − 1.57620i −0.615550 0.788098i $$-0.711066\pi$$
0.615550 0.788098i $$-0.288934\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 13.2679i − 0.445494i −0.974876 0.222747i $$-0.928498\pi$$
0.974876 0.222747i $$-0.0715024\pi$$
$$888$$ 0 0
$$889$$ 11.4641 0.384494
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5.66025i 0.189413i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 45.8372 1.52876
$$900$$ 0 0
$$901$$ 34.3923 1.14577
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.21539i 0.0403564i 0.999796 + 0.0201782i $$0.00642335\pi$$
−0.999796 + 0.0201782i $$0.993577\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −6.80385 −0.225422 −0.112711 0.993628i $$-0.535953\pi$$
−0.112711 + 0.993628i $$0.535953\pi$$
$$912$$ 0 0
$$913$$ − 26.1962i − 0.866966i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 28.0526i 0.926377i
$$918$$ 0 0
$$919$$ −51.3923 −1.69528 −0.847638 0.530575i $$-0.821976\pi$$
−0.847638 + 0.530575i $$0.821976\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 61.1769i − 2.01366i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −1.48334 −0.0486668 −0.0243334 0.999704i $$-0.507746\pi$$
−0.0243334 + 0.999704i $$0.507746\pi$$
$$930$$ 0 0
$$931$$ −2.07180 −0.0679004
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 19.0718i − 0.623048i −0.950238 0.311524i $$-0.899160\pi$$
0.950238 0.311524i $$-0.100840\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −8.53590 −0.278262 −0.139131 0.990274i $$-0.544431\pi$$
−0.139131 + 0.990274i $$0.544431\pi$$
$$942$$ 0 0
$$943$$ − 38.7846i − 1.26300i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 52.6410i 1.71060i 0.518130 + 0.855302i $$0.326628\pi$$
−0.518130 + 0.855302i $$0.673372\pi$$
$$948$$ 0 0
$$949$$ 1.07180 0.0347920
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 2.53590i − 0.0821458i −0.999156 0.0410729i $$-0.986922\pi$$
0.999156 0.0410729i $$-0.0130776\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 4.14359 0.133664
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 7.41154i − 0.238339i −0.992874 0.119170i $$-0.961977\pi$$
0.992874 0.119170i $$-0.0380232\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.2679 0.714612 0.357306 0.933987i $$-0.383695\pi$$
0.357306 + 0.933987i $$0.383695\pi$$
$$972$$ 0 0
$$973$$ − 42.0526i − 1.34814i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 4.39230i − 0.140522i −0.997529 0.0702611i $$-0.977617\pi$$
0.997529 0.0702611i $$-0.0223833\pi$$
$$978$$ 0 0
$$979$$ −9.00000 −0.287641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 52.7321i − 1.68189i −0.541120 0.840946i $$-0.681999\pi$$
0.541120 0.840946i $$-0.318001\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −11.3205 −0.359971
$$990$$ 0 0
$$991$$ 55.7846 1.77206 0.886028 0.463631i $$-0.153454\pi$$
0.886028 + 0.463631i $$0.153454\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 52.1962i − 1.65307i −0.562886 0.826534i $$-0.690309\pi$$
0.562886 0.826534i $$-0.309691\pi$$
$$998$$ 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 8100.2.d.m.649.4 4
3.2 odd 2 8100.2.d.l.649.4 4
5.2 odd 4 1620.2.a.g.1.1 2
5.3 odd 4 8100.2.a.t.1.2 2
5.4 even 2 inner 8100.2.d.m.649.1 4
15.2 even 4 1620.2.a.h.1.1 yes 2
15.8 even 4 8100.2.a.s.1.2 2
15.14 odd 2 8100.2.d.l.649.1 4
20.7 even 4 6480.2.a.bh.1.2 2
45.2 even 12 1620.2.i.m.1081.2 4
45.7 odd 12 1620.2.i.n.1081.2 4
45.22 odd 12 1620.2.i.n.541.2 4
45.32 even 12 1620.2.i.m.541.2 4
60.47 odd 4 6480.2.a.bp.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.a.g.1.1 2 5.2 odd 4
1620.2.a.h.1.1 yes 2 15.2 even 4
1620.2.i.m.541.2 4 45.32 even 12
1620.2.i.m.1081.2 4 45.2 even 12
1620.2.i.n.541.2 4 45.22 odd 12
1620.2.i.n.1081.2 4 45.7 odd 12
6480.2.a.bh.1.2 2 20.7 even 4
6480.2.a.bp.1.2 2 60.47 odd 4
8100.2.a.s.1.2 2 15.8 even 4
8100.2.a.t.1.2 2 5.3 odd 4
8100.2.d.l.649.1 4 15.14 odd 2
8100.2.d.l.649.4 4 3.2 odd 2
8100.2.d.m.649.1 4 5.4 even 2 inner
8100.2.d.m.649.4 4 1.1 even 1 trivial