# Properties

 Label 225.2.b.b.199.1 Level $225$ Weight $2$ Character 225.199 Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.2.b.b.199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{4} -3.00000i q^{8} +4.00000 q^{11} -2.00000i q^{13} -1.00000 q^{16} +2.00000i q^{17} -4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} -2.00000 q^{29} -5.00000i q^{32} +2.00000 q^{34} +10.0000i q^{37} +4.00000i q^{38} -10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +8.00000i q^{47} +7.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} +2.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} -7.00000 q^{64} -12.0000i q^{67} +2.00000i q^{68} +8.00000 q^{71} +10.0000i q^{73} +10.0000 q^{74} -4.00000 q^{76} +10.0000i q^{82} -12.0000i q^{83} +4.00000 q^{86} -12.0000i q^{88} -6.00000 q^{89} +8.00000 q^{94} -2.00000i q^{97} -7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} + 8q^{11} - 2q^{16} - 8q^{19} - 4q^{26} - 4q^{29} + 4q^{34} - 20q^{41} + 8q^{44} + 14q^{49} - 8q^{59} - 4q^{61} - 14q^{64} + 16q^{71} + 20q^{74} - 8q^{76} + 8q^{86} - 12q^{89} + 16q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i −0.935414 0.353553i $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 3.00000i − 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000i 0.262613i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 10.0000i 1.10432i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ − 12.0000i − 1.27920i
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 4.00000i 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 3.00000i − 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 8.00000i − 0.671345i
$$143$$ − 8.00000i − 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ 0 0
$$148$$ 10.0000i 0.821995i
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 12.0000i 0.973329i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000i 0.304997i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 6.00000i 0.449719i
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 8.00000i 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 14.0000i 0.948200i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 12.0000i − 0.713326i −0.934233 0.356663i $$-0.883914\pi$$
0.934233 0.356663i $$-0.116086\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 10.0000i 0.585206i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ − 22.0000i − 1.27443i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 28.0000i − 1.59804i −0.601302 0.799022i $$-0.705351\pi$$
0.601302 0.799022i $$-0.294649\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 0 0
$$328$$ 30.0000i 1.65647i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ − 28.0000i − 1.50312i −0.659665 0.751559i $$-0.729302\pi$$
0.659665 0.751559i $$-0.270698\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 20.0000i − 1.06600i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 20.0000i − 1.05703i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000i 0.525588i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.0000i 1.25279i 0.779506 + 0.626395i $$0.215470\pi$$
−0.779506 + 0.626395i $$0.784530\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 24.0000 1.23771
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000i 0.818631i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ − 2.00000i − 0.101535i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 21.0000i − 1.06066i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 16.0000i 0.782586i
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ 0 0
$$424$$ 30.0000 1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 4.00000i − 0.190261i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 24.0000i 1.11537i 0.830051 + 0.557687i $$0.188311\pi$$
−0.830051 + 0.557687i $$0.811689\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 12.0000i 0.552345i
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 16.0000i 0.731823i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 14.0000i 0.637683i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 32.0000i − 1.45006i −0.688718 0.725029i $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ − 4.00000i − 0.180151i
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000i 0.535586i
$$503$$ 32.0000i 1.42681i 0.700752 + 0.713405i $$0.252848\pi$$
−0.700752 + 0.713405i $$0.747152\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000i 0.354943i
$$509$$ −34.0000 −1.50702 −0.753512 0.657434i $$-0.771642\pi$$
−0.753512 + 0.657434i $$0.771642\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000i 0.486136i
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −36.0000 −1.55496
$$537$$ 0 0
$$538$$ − 14.0000i − 0.603583i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 0 0
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −6.00000 −0.254916
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −12.0000 −0.504398
$$567$$ 0 0
$$568$$ − 24.0000i − 1.00702i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000i 1.65663i
$$584$$ 30.0000 1.24141
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 10.0000i − 0.410997i
$$593$$ − 34.0000i − 1.39621i −0.715994 0.698106i $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000i 0.324710i 0.986732 + 0.162355i $$0.0519090\pi$$
−0.986732 + 0.162355i $$0.948091\pi$$
$$608$$ 20.0000i 0.811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 14.0000i − 0.554700i
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ − 36.0000i − 1.41970i −0.704352 0.709851i $$-0.748762\pi$$
0.704352 0.709851i $$-0.251238\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 32.0000i 1.25805i 0.777385 + 0.629025i $$0.216546\pi$$
−0.777385 + 0.629025i $$0.783454\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ − 46.0000i − 1.80012i −0.435767 0.900060i $$-0.643523\pi$$
0.435767 0.900060i $$-0.356477\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 12.0000i − 0.466393i
$$663$$ 0 0
$$664$$ −36.0000 −1.39707
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ − 30.0000i − 1.15642i −0.815890 0.578208i $$-0.803752\pi$$
0.815890 0.578208i $$-0.196248\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ − 4.00000i − 0.152499i
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.0000i − 0.757554i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ − 40.0000i − 1.50863i
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 18.0000i 0.674579i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 24.0000i 0.895672i
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16.0000i 0.593407i 0.954970 + 0.296704i $$0.0958873\pi$$
−0.954970 + 0.296704i $$0.904113\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 48.0000i − 1.76810i
$$738$$ 0 0
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 8.00000i 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000i 0.0719816i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ − 2.00000i − 0.0708436i −0.999372 0.0354218i $$-0.988723\pi$$
0.999372 0.0354218i $$-0.0112775\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 18.0000i 0.635602i
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 18.0000i 0.633238i
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ 26.0000i 0.909069i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −54.0000 −1.88461 −0.942306 0.334751i $$-0.891348\pi$$
−0.942306 + 0.334751i $$0.891348\pi$$
$$822$$ 0 0
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ −48.0000 −1.67216
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 0 0
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 14.0000i 0.485363i
$$833$$ 14.0000i 0.485071i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 0 0
$$838$$ − 4.00000i − 0.138178i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 26.0000i 0.896019i
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 10.0000i − 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ − 22.0000i − 0.751506i −0.926720 0.375753i $$-0.877384\pi$$
0.926720 0.375753i $$-0.122616\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 56.0000i 1.90626i 0.302558 + 0.953131i $$0.402160\pi$$
−0.302558 + 0.953131i $$0.597840\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 42.0000i 1.42230i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 30.0000i − 1.01303i −0.862232 0.506514i $$-0.830934\pi$$
0.862232 0.506514i $$-0.169066\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 46.0000 1.54978 0.774890 0.632096i $$-0.217805\pi$$
0.774890 + 0.632096i $$0.217805\pi$$
$$882$$ 0 0
$$883$$ 44.0000i 1.48072i 0.672212 + 0.740359i $$0.265344\pi$$
−0.672212 + 0.740359i $$0.734656\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000i 0.267860i
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 40.0000i 1.33185i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 0 0
$$913$$ − 48.0000i − 1.58857i
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 18.0000i − 0.592798i
$$923$$ − 16.0000i − 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 24.0000 0.788689
$$927$$ 0 0
$$928$$ 10.0000i 0.328266i
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 54.0000i 1.76410i 0.471153 + 0.882052i $$0.343838\pi$$
−0.471153 + 0.882052i $$0.656162\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 22.0000i 0.712650i 0.934362 + 0.356325i $$0.115970\pi$$
−0.934362 + 0.356325i $$0.884030\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ − 20.0000i − 0.644826i
$$963$$ 0 0
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ − 15.0000i − 0.482118i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ 0 0
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 28.0000i 0.893516i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 8.00000i 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 54.0000i − 1.71020i −0.518465 0.855099i $$-0.673497\pi$$
0.518465 0.855099i $$-0.326503\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 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 225.2.b.b.199.1 2
3.2 odd 2 75.2.b.b.49.2 2
4.3 odd 2 3600.2.f.e.2449.1 2
5.2 odd 4 45.2.a.a.1.1 1
5.3 odd 4 225.2.a.b.1.1 1
5.4 even 2 inner 225.2.b.b.199.2 2
12.11 even 2 1200.2.f.h.49.2 2
15.2 even 4 15.2.a.a.1.1 1
15.8 even 4 75.2.a.b.1.1 1
15.14 odd 2 75.2.b.b.49.1 2
20.3 even 4 3600.2.a.u.1.1 1
20.7 even 4 720.2.a.c.1.1 1
20.19 odd 2 3600.2.f.e.2449.2 2
24.5 odd 2 4800.2.f.bf.3649.2 2
24.11 even 2 4800.2.f.c.3649.1 2
35.27 even 4 2205.2.a.i.1.1 1
40.27 even 4 2880.2.a.bc.1.1 1
40.37 odd 4 2880.2.a.y.1.1 1
45.2 even 12 405.2.e.f.271.1 2
45.7 odd 12 405.2.e.c.271.1 2
45.22 odd 12 405.2.e.c.136.1 2
45.32 even 12 405.2.e.f.136.1 2
55.32 even 4 5445.2.a.c.1.1 1
60.23 odd 4 1200.2.a.e.1.1 1
60.47 odd 4 240.2.a.d.1.1 1
60.59 even 2 1200.2.f.h.49.1 2
65.12 odd 4 7605.2.a.g.1.1 1
105.2 even 12 735.2.i.e.361.1 2
105.17 odd 12 735.2.i.d.226.1 2
105.32 even 12 735.2.i.e.226.1 2
105.47 odd 12 735.2.i.d.361.1 2
105.62 odd 4 735.2.a.c.1.1 1
105.83 odd 4 3675.2.a.j.1.1 1
120.29 odd 2 4800.2.f.bf.3649.1 2
120.53 even 4 4800.2.a.t.1.1 1
120.59 even 2 4800.2.f.c.3649.2 2
120.77 even 4 960.2.a.l.1.1 1
120.83 odd 4 4800.2.a.bz.1.1 1
120.107 odd 4 960.2.a.a.1.1 1
165.32 odd 4 1815.2.a.d.1.1 1
165.98 odd 4 9075.2.a.g.1.1 1
195.77 even 4 2535.2.a.j.1.1 1
240.77 even 4 3840.2.k.m.1921.1 2
240.107 odd 4 3840.2.k.r.1921.1 2
240.197 even 4 3840.2.k.m.1921.2 2
240.227 odd 4 3840.2.k.r.1921.2 2
255.152 even 4 4335.2.a.c.1.1 1
285.227 odd 4 5415.2.a.j.1.1 1
345.137 odd 4 7935.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 15.2 even 4
45.2.a.a.1.1 1 5.2 odd 4
75.2.a.b.1.1 1 15.8 even 4
75.2.b.b.49.1 2 15.14 odd 2
75.2.b.b.49.2 2 3.2 odd 2
225.2.a.b.1.1 1 5.3 odd 4
225.2.b.b.199.1 2 1.1 even 1 trivial
225.2.b.b.199.2 2 5.4 even 2 inner
240.2.a.d.1.1 1 60.47 odd 4
405.2.e.c.136.1 2 45.22 odd 12
405.2.e.c.271.1 2 45.7 odd 12
405.2.e.f.136.1 2 45.32 even 12
405.2.e.f.271.1 2 45.2 even 12
720.2.a.c.1.1 1 20.7 even 4
735.2.a.c.1.1 1 105.62 odd 4
735.2.i.d.226.1 2 105.17 odd 12
735.2.i.d.361.1 2 105.47 odd 12
735.2.i.e.226.1 2 105.32 even 12
735.2.i.e.361.1 2 105.2 even 12
960.2.a.a.1.1 1 120.107 odd 4
960.2.a.l.1.1 1 120.77 even 4
1200.2.a.e.1.1 1 60.23 odd 4
1200.2.f.h.49.1 2 60.59 even 2
1200.2.f.h.49.2 2 12.11 even 2
1815.2.a.d.1.1 1 165.32 odd 4
2205.2.a.i.1.1 1 35.27 even 4
2535.2.a.j.1.1 1 195.77 even 4
2880.2.a.y.1.1 1 40.37 odd 4
2880.2.a.bc.1.1 1 40.27 even 4
3600.2.a.u.1.1 1 20.3 even 4
3600.2.f.e.2449.1 2 4.3 odd 2
3600.2.f.e.2449.2 2 20.19 odd 2
3675.2.a.j.1.1 1 105.83 odd 4
3840.2.k.m.1921.1 2 240.77 even 4
3840.2.k.m.1921.2 2 240.197 even 4
3840.2.k.r.1921.1 2 240.107 odd 4
3840.2.k.r.1921.2 2 240.227 odd 4
4335.2.a.c.1.1 1 255.152 even 4
4800.2.a.t.1.1 1 120.53 even 4
4800.2.a.bz.1.1 1 120.83 odd 4
4800.2.f.c.3649.1 2 24.11 even 2
4800.2.f.c.3649.2 2 120.59 even 2
4800.2.f.bf.3649.1 2 120.29 odd 2
4800.2.f.bf.3649.2 2 24.5 odd 2
5415.2.a.j.1.1 1 285.227 odd 4
5445.2.a.c.1.1 1 55.32 even 4
7605.2.a.g.1.1 1 65.12 odd 4
7935.2.a.d.1.1 1 345.137 odd 4
9075.2.a.g.1.1 1 165.98 odd 4