# Properties

 Label 1728.3.b.e.1567.2 Level $1728$ Weight $3$ Character 1728.1567 Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.2 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1567 Dual form 1728.3.b.e.1567.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.46410i q^{5} +1.00000i q^{7} +O(q^{10})$$ $$q-3.46410i q^{5} +1.00000i q^{7} -17.3205 q^{11} +1.73205i q^{13} +6.00000 q^{17} -1.73205 q^{19} -30.0000i q^{23} +13.0000 q^{25} +20.7846i q^{29} +14.0000i q^{31} +3.46410 q^{35} +19.0526i q^{37} -48.0000 q^{41} -20.7846 q^{43} +66.0000i q^{47} +48.0000 q^{49} +48.4974i q^{53} +60.0000i q^{55} +31.1769 q^{59} +43.3013i q^{61} +6.00000 q^{65} -60.6218 q^{67} -48.0000i q^{71} +49.0000 q^{73} -17.3205i q^{77} +83.0000i q^{79} -13.8564 q^{83} -20.7846i q^{85} +66.0000 q^{89} -1.73205 q^{91} +6.00000i q^{95} +107.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{17} + 52 q^{25} - 192 q^{41} + 192 q^{49} + 24 q^{65} + 196 q^{73} + 264 q^{89} + 428 q^{97}+O(q^{100})$$ 4 * q + 24 * q^17 + 52 * q^25 - 192 * q^41 + 192 * q^49 + 24 * q^65 + 196 * q^73 + 264 * q^89 + 428 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$5$$ − 3.46410i − 0.692820i −0.938083 0.346410i $$-0.887401\pi$$
0.938083 0.346410i $$-0.112599\pi$$
$$6$$ 0 0
$$7$$ 1.00000i 0.142857i 0.997446 + 0.0714286i $$0.0227558\pi$$
−0.997446 + 0.0714286i $$0.977244\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −17.3205 −1.57459 −0.787296 0.616575i $$-0.788520\pi$$
−0.787296 + 0.616575i $$0.788520\pi$$
$$12$$ 0 0
$$13$$ 1.73205i 0.133235i 0.997779 + 0.0666173i $$0.0212207\pi$$
−0.997779 + 0.0666173i $$0.978779\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 0.352941 0.176471 0.984306i $$-0.443532\pi$$
0.176471 + 0.984306i $$0.443532\pi$$
$$18$$ 0 0
$$19$$ −1.73205 −0.0911606 −0.0455803 0.998961i $$-0.514514\pi$$
−0.0455803 + 0.998961i $$0.514514\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 30.0000i − 1.30435i −0.758069 0.652174i $$-0.773857\pi$$
0.758069 0.652174i $$-0.226143\pi$$
$$24$$ 0 0
$$25$$ 13.0000 0.520000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 20.7846i 0.716711i 0.933585 + 0.358355i $$0.116662\pi$$
−0.933585 + 0.358355i $$0.883338\pi$$
$$30$$ 0 0
$$31$$ 14.0000i 0.451613i 0.974172 + 0.225806i $$0.0725017\pi$$
−0.974172 + 0.225806i $$0.927498\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.46410 0.0989743
$$36$$ 0 0
$$37$$ 19.0526i 0.514934i 0.966287 + 0.257467i $$0.0828879\pi$$
−0.966287 + 0.257467i $$0.917112\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −48.0000 −1.17073 −0.585366 0.810769i $$-0.699049\pi$$
−0.585366 + 0.810769i $$0.699049\pi$$
$$42$$ 0 0
$$43$$ −20.7846 −0.483363 −0.241682 0.970356i $$-0.577699\pi$$
−0.241682 + 0.970356i $$0.577699\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 66.0000i 1.40426i 0.712051 + 0.702128i $$0.247766\pi$$
−0.712051 + 0.702128i $$0.752234\pi$$
$$48$$ 0 0
$$49$$ 48.0000 0.979592
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 48.4974i 0.915046i 0.889198 + 0.457523i $$0.151263\pi$$
−0.889198 + 0.457523i $$0.848737\pi$$
$$54$$ 0 0
$$55$$ 60.0000i 1.09091i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 31.1769 0.528422 0.264211 0.964465i $$-0.414888\pi$$
0.264211 + 0.964465i $$0.414888\pi$$
$$60$$ 0 0
$$61$$ 43.3013i 0.709857i 0.934893 + 0.354928i $$0.115495\pi$$
−0.934893 + 0.354928i $$0.884505\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000 0.0923077
$$66$$ 0 0
$$67$$ −60.6218 −0.904803 −0.452401 0.891814i $$-0.649433\pi$$
−0.452401 + 0.891814i $$0.649433\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 48.0000i − 0.676056i −0.941136 0.338028i $$-0.890240\pi$$
0.941136 0.338028i $$-0.109760\pi$$
$$72$$ 0 0
$$73$$ 49.0000 0.671233 0.335616 0.941999i $$-0.391055\pi$$
0.335616 + 0.941999i $$0.391055\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 17.3205i − 0.224942i
$$78$$ 0 0
$$79$$ 83.0000i 1.05063i 0.850907 + 0.525316i $$0.176053\pi$$
−0.850907 + 0.525316i $$0.823947\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −13.8564 −0.166945 −0.0834723 0.996510i $$-0.526601\pi$$
−0.0834723 + 0.996510i $$0.526601\pi$$
$$84$$ 0 0
$$85$$ − 20.7846i − 0.244525i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 66.0000 0.741573 0.370787 0.928718i $$-0.379088\pi$$
0.370787 + 0.928718i $$0.379088\pi$$
$$90$$ 0 0
$$91$$ −1.73205 −0.0190335
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.00000i 0.0631579i
$$96$$ 0 0
$$97$$ 107.000 1.10309 0.551546 0.834144i $$-0.314038\pi$$
0.551546 + 0.834144i $$0.314038\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 159.349i − 1.57771i −0.614580 0.788855i $$-0.710674\pi$$
0.614580 0.788855i $$-0.289326\pi$$
$$102$$ 0 0
$$103$$ 95.0000i 0.922330i 0.887314 + 0.461165i $$0.152568\pi$$
−0.887314 + 0.461165i $$0.847432\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 17.3205 0.161874 0.0809370 0.996719i $$-0.474209\pi$$
0.0809370 + 0.996719i $$0.474209\pi$$
$$108$$ 0 0
$$109$$ 187.061i 1.71616i 0.513516 + 0.858080i $$0.328343\pi$$
−0.513516 + 0.858080i $$0.671657\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 150.000 1.32743 0.663717 0.747984i $$-0.268978\pi$$
0.663717 + 0.747984i $$0.268978\pi$$
$$114$$ 0 0
$$115$$ −103.923 −0.903679
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.00000i 0.0504202i
$$120$$ 0 0
$$121$$ 179.000 1.47934
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 131.636i − 1.05309i
$$126$$ 0 0
$$127$$ 190.000i 1.49606i 0.663663 + 0.748031i $$0.269001\pi$$
−0.663663 + 0.748031i $$0.730999\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −48.4974 −0.370209 −0.185105 0.982719i $$-0.559262\pi$$
−0.185105 + 0.982719i $$0.559262\pi$$
$$132$$ 0 0
$$133$$ − 1.73205i − 0.0130229i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −222.000 −1.62044 −0.810219 0.586127i $$-0.800652\pi$$
−0.810219 + 0.586127i $$0.800652\pi$$
$$138$$ 0 0
$$139$$ −164.545 −1.18378 −0.591888 0.806020i $$-0.701617\pi$$
−0.591888 + 0.806020i $$0.701617\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 30.0000i − 0.209790i
$$144$$ 0 0
$$145$$ 72.0000 0.496552
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 96.9948i 0.650972i 0.945547 + 0.325486i $$0.105528\pi$$
−0.945547 + 0.325486i $$0.894472\pi$$
$$150$$ 0 0
$$151$$ 109.000i 0.721854i 0.932594 + 0.360927i $$0.117540\pi$$
−0.932594 + 0.360927i $$0.882460\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 48.4974 0.312887
$$156$$ 0 0
$$157$$ 62.3538i 0.397158i 0.980085 + 0.198579i $$0.0636327\pi$$
−0.980085 + 0.198579i $$0.936367\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 30.0000 0.186335
$$162$$ 0 0
$$163$$ 84.8705 0.520678 0.260339 0.965517i $$-0.416166\pi$$
0.260339 + 0.965517i $$0.416166\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 234.000i 1.40120i 0.713555 + 0.700599i $$0.247084\pi$$
−0.713555 + 0.700599i $$0.752916\pi$$
$$168$$ 0 0
$$169$$ 166.000 0.982249
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 180.133i 1.04123i 0.853791 + 0.520616i $$0.174298\pi$$
−0.853791 + 0.520616i $$0.825702\pi$$
$$174$$ 0 0
$$175$$ 13.0000i 0.0742857i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 193.990 1.08374 0.541871 0.840462i $$-0.317716\pi$$
0.541871 + 0.840462i $$0.317716\pi$$
$$180$$ 0 0
$$181$$ 164.545i 0.909087i 0.890724 + 0.454544i $$0.150198\pi$$
−0.890724 + 0.454544i $$0.849802\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 66.0000 0.356757
$$186$$ 0 0
$$187$$ −103.923 −0.555738
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 198.000i − 1.03665i −0.855184 0.518325i $$-0.826556\pi$$
0.855184 0.518325i $$-0.173444\pi$$
$$192$$ 0 0
$$193$$ 227.000 1.17617 0.588083 0.808801i $$-0.299883\pi$$
0.588083 + 0.808801i $$0.299883\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 232.095i 1.17815i 0.808080 + 0.589073i $$0.200507\pi$$
−0.808080 + 0.589073i $$0.799493\pi$$
$$198$$ 0 0
$$199$$ 239.000i 1.20101i 0.799623 + 0.600503i $$0.205033\pi$$
−0.799623 + 0.600503i $$0.794967\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −20.7846 −0.102387
$$204$$ 0 0
$$205$$ 166.277i 0.811107i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 30.0000 0.143541
$$210$$ 0 0
$$211$$ −292.717 −1.38728 −0.693641 0.720321i $$-0.743995\pi$$
−0.693641 + 0.720321i $$0.743995\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 72.0000i 0.334884i
$$216$$ 0 0
$$217$$ −14.0000 −0.0645161
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10.3923i 0.0470240i
$$222$$ 0 0
$$223$$ 130.000i 0.582960i 0.956577 + 0.291480i $$0.0941476\pi$$
−0.956577 + 0.291480i $$0.905852\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −41.5692 −0.183124 −0.0915622 0.995799i $$-0.529186\pi$$
−0.0915622 + 0.995799i $$0.529186\pi$$
$$228$$ 0 0
$$229$$ − 332.554i − 1.45220i −0.687589 0.726100i $$-0.741331\pi$$
0.687589 0.726100i $$-0.258669\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −276.000 −1.18455 −0.592275 0.805736i $$-0.701770\pi$$
−0.592275 + 0.805736i $$0.701770\pi$$
$$234$$ 0 0
$$235$$ 228.631 0.972897
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 204.000i − 0.853556i −0.904356 0.426778i $$-0.859648\pi$$
0.904356 0.426778i $$-0.140352\pi$$
$$240$$ 0 0
$$241$$ 133.000 0.551867 0.275934 0.961177i $$-0.411013\pi$$
0.275934 + 0.961177i $$0.411013\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 166.277i − 0.678681i
$$246$$ 0 0
$$247$$ − 3.00000i − 0.0121457i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −457.261 −1.82176 −0.910879 0.412673i $$-0.864595\pi$$
−0.910879 + 0.412673i $$0.864595\pi$$
$$252$$ 0 0
$$253$$ 519.615i 2.05382i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −300.000 −1.16732 −0.583658 0.812000i $$-0.698379\pi$$
−0.583658 + 0.812000i $$0.698379\pi$$
$$258$$ 0 0
$$259$$ −19.0526 −0.0735620
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 84.0000i 0.319392i 0.987166 + 0.159696i $$0.0510513\pi$$
−0.987166 + 0.159696i $$0.948949\pi$$
$$264$$ 0 0
$$265$$ 168.000 0.633962
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 93.5307i − 0.347698i −0.984772 0.173849i $$-0.944380\pi$$
0.984772 0.173849i $$-0.0556205\pi$$
$$270$$ 0 0
$$271$$ 265.000i 0.977860i 0.872323 + 0.488930i $$0.162613\pi$$
−0.872323 + 0.488930i $$0.837387\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −225.167 −0.818788
$$276$$ 0 0
$$277$$ − 311.769i − 1.12552i −0.826620 0.562760i $$-0.809739\pi$$
0.826620 0.562760i $$-0.190261\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 540.000 1.92171 0.960854 0.277055i $$-0.0893584\pi$$
0.960854 + 0.277055i $$0.0893584\pi$$
$$282$$ 0 0
$$283$$ −353.338 −1.24855 −0.624273 0.781206i $$-0.714605\pi$$
−0.624273 + 0.781206i $$0.714605\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 48.0000i − 0.167247i
$$288$$ 0 0
$$289$$ −253.000 −0.875433
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 155.885i − 0.532029i −0.963969 0.266015i $$-0.914293\pi$$
0.963969 0.266015i $$-0.0857069\pi$$
$$294$$ 0 0
$$295$$ − 108.000i − 0.366102i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 51.9615 0.173784
$$300$$ 0 0
$$301$$ − 20.7846i − 0.0690519i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 150.000 0.491803
$$306$$ 0 0
$$307$$ 353.338 1.15094 0.575470 0.817823i $$-0.304819\pi$$
0.575470 + 0.817823i $$0.304819\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 462.000i − 1.48553i −0.669552 0.742765i $$-0.733514\pi$$
0.669552 0.742765i $$-0.266486\pi$$
$$312$$ 0 0
$$313$$ −35.0000 −0.111821 −0.0559105 0.998436i $$-0.517806\pi$$
−0.0559105 + 0.998436i $$0.517806\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 263.272i 0.830510i 0.909705 + 0.415255i $$0.136308\pi$$
−0.909705 + 0.415255i $$0.863692\pi$$
$$318$$ 0 0
$$319$$ − 360.000i − 1.12853i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −10.3923 −0.0321743
$$324$$ 0 0
$$325$$ 22.5167i 0.0692820i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −66.0000 −0.200608
$$330$$ 0 0
$$331$$ −39.8372 −0.120354 −0.0601770 0.998188i $$-0.519167\pi$$
−0.0601770 + 0.998188i $$0.519167\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 210.000i 0.626866i
$$336$$ 0 0
$$337$$ 59.0000 0.175074 0.0875371 0.996161i $$-0.472100\pi$$
0.0875371 + 0.996161i $$0.472100\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 242.487i − 0.711106i
$$342$$ 0 0
$$343$$ 97.0000i 0.282799i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 630.466 1.81691 0.908453 0.417987i $$-0.137264\pi$$
0.908453 + 0.417987i $$0.137264\pi$$
$$348$$ 0 0
$$349$$ 185.329i 0.531030i 0.964107 + 0.265515i $$0.0855420\pi$$
−0.964107 + 0.265515i $$0.914458\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.0000 0.0339943 0.0169972 0.999856i $$-0.494589\pi$$
0.0169972 + 0.999856i $$0.494589\pi$$
$$354$$ 0 0
$$355$$ −166.277 −0.468386
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 318.000i 0.885794i 0.896573 + 0.442897i $$0.146049\pi$$
−0.896573 + 0.442897i $$0.853951\pi$$
$$360$$ 0 0
$$361$$ −358.000 −0.991690
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 169.741i − 0.465044i
$$366$$ 0 0
$$367$$ − 109.000i − 0.297003i −0.988912 0.148501i $$-0.952555\pi$$
0.988912 0.148501i $$-0.0474449\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −48.4974 −0.130721
$$372$$ 0 0
$$373$$ − 729.193i − 1.95494i −0.211070 0.977471i $$-0.567695\pi$$
0.211070 0.977471i $$-0.432305\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −36.0000 −0.0954907
$$378$$ 0 0
$$379$$ 438.209 1.15622 0.578112 0.815957i $$-0.303790\pi$$
0.578112 + 0.815957i $$0.303790\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 516.000i 1.34726i 0.739069 + 0.673629i $$0.235266\pi$$
−0.739069 + 0.673629i $$0.764734\pi$$
$$384$$ 0 0
$$385$$ −60.0000 −0.155844
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 543.864i 1.39811i 0.715069 + 0.699054i $$0.246395\pi$$
−0.715069 + 0.699054i $$0.753605\pi$$
$$390$$ 0 0
$$391$$ − 180.000i − 0.460358i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 287.520 0.727900
$$396$$ 0 0
$$397$$ 685.892i 1.72769i 0.503759 + 0.863844i $$0.331950\pi$$
−0.503759 + 0.863844i $$0.668050\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −252.000 −0.628429 −0.314214 0.949352i $$-0.601741\pi$$
−0.314214 + 0.949352i $$0.601741\pi$$
$$402$$ 0 0
$$403$$ −24.2487 −0.0601705
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 330.000i − 0.810811i
$$408$$ 0 0
$$409$$ −11.0000 −0.0268949 −0.0134474 0.999910i $$-0.504281\pi$$
−0.0134474 + 0.999910i $$0.504281\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 31.1769i 0.0754889i
$$414$$ 0 0
$$415$$ 48.0000i 0.115663i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 135.100 0.322434 0.161217 0.986919i $$-0.448458\pi$$
0.161217 + 0.986919i $$0.448458\pi$$
$$420$$ 0 0
$$421$$ − 601.022i − 1.42760i −0.700347 0.713802i $$-0.746971\pi$$
0.700347 0.713802i $$-0.253029\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 78.0000 0.183529
$$426$$ 0 0
$$427$$ −43.3013 −0.101408
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 486.000i 1.12761i 0.825908 + 0.563805i $$0.190663\pi$$
−0.825908 + 0.563805i $$0.809337\pi$$
$$432$$ 0 0
$$433$$ −494.000 −1.14088 −0.570439 0.821340i $$-0.693227\pi$$
−0.570439 + 0.821340i $$0.693227\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 51.9615i 0.118905i
$$438$$ 0 0
$$439$$ − 610.000i − 1.38952i −0.719241 0.694761i $$-0.755510\pi$$
0.719241 0.694761i $$-0.244490\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 145.492 0.328425 0.164212 0.986425i $$-0.447492\pi$$
0.164212 + 0.986425i $$0.447492\pi$$
$$444$$ 0 0
$$445$$ − 228.631i − 0.513777i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −366.000 −0.815145 −0.407572 0.913173i $$-0.633625\pi$$
−0.407572 + 0.913173i $$0.633625\pi$$
$$450$$ 0 0
$$451$$ 831.384 1.84342
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 6.00000i 0.0131868i
$$456$$ 0 0
$$457$$ −398.000 −0.870897 −0.435449 0.900214i $$-0.643410\pi$$
−0.435449 + 0.900214i $$0.643410\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 710.141i − 1.54044i −0.637781 0.770218i $$-0.720147\pi$$
0.637781 0.770218i $$-0.279853\pi$$
$$462$$ 0 0
$$463$$ − 875.000i − 1.88985i −0.327289 0.944924i $$-0.606135\pi$$
0.327289 0.944924i $$-0.393865\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 349.874 0.749195 0.374598 0.927187i $$-0.377781\pi$$
0.374598 + 0.927187i $$0.377781\pi$$
$$468$$ 0 0
$$469$$ − 60.6218i − 0.129258i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 360.000 0.761099
$$474$$ 0 0
$$475$$ −22.5167 −0.0474035
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 228.000i − 0.475992i −0.971266 0.237996i $$-0.923510\pi$$
0.971266 0.237996i $$-0.0764905\pi$$
$$480$$ 0 0
$$481$$ −33.0000 −0.0686071
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 370.659i − 0.764245i
$$486$$ 0 0
$$487$$ − 251.000i − 0.515400i −0.966225 0.257700i $$-0.917035\pi$$
0.966225 0.257700i $$-0.0829647\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −703.213 −1.43220 −0.716102 0.697995i $$-0.754076\pi$$
−0.716102 + 0.697995i $$0.754076\pi$$
$$492$$ 0 0
$$493$$ 124.708i 0.252957i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 48.0000 0.0965795
$$498$$ 0 0
$$499$$ −374.123 −0.749745 −0.374873 0.927076i $$-0.622314\pi$$
−0.374873 + 0.927076i $$0.622314\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 762.000i − 1.51491i −0.652887 0.757455i $$-0.726442\pi$$
0.652887 0.757455i $$-0.273558\pi$$
$$504$$ 0 0
$$505$$ −552.000 −1.09307
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 204.382i 0.401536i 0.979639 + 0.200768i $$0.0643438\pi$$
−0.979639 + 0.200768i $$0.935656\pi$$
$$510$$ 0 0
$$511$$ 49.0000i 0.0958904i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 329.090 0.639009
$$516$$ 0 0
$$517$$ − 1143.15i − 2.21113i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 726.000 1.39347 0.696737 0.717327i $$-0.254634\pi$$
0.696737 + 0.717327i $$0.254634\pi$$
$$522$$ 0 0
$$523$$ 833.116 1.59296 0.796478 0.604667i $$-0.206694\pi$$
0.796478 + 0.604667i $$0.206694\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 84.0000i 0.159393i
$$528$$ 0 0
$$529$$ −371.000 −0.701323
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 83.1384i − 0.155982i
$$534$$ 0 0
$$535$$ − 60.0000i − 0.112150i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −831.384 −1.54246
$$540$$ 0 0
$$541$$ 646.055i 1.19419i 0.802172 + 0.597093i $$0.203678\pi$$
−0.802172 + 0.597093i $$0.796322\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 648.000 1.18899
$$546$$ 0 0
$$547$$ 313.501 0.573128 0.286564 0.958061i $$-0.407487\pi$$
0.286564 + 0.958061i $$0.407487\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 36.0000i − 0.0653358i
$$552$$ 0 0
$$553$$ −83.0000 −0.150090
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 682.428i 1.22518i 0.790399 + 0.612592i $$0.209873\pi$$
−0.790399 + 0.612592i $$0.790127\pi$$
$$558$$ 0 0
$$559$$ − 36.0000i − 0.0644007i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −651.251 −1.15675 −0.578376 0.815770i $$-0.696313\pi$$
−0.578376 + 0.815770i $$0.696313\pi$$
$$564$$ 0 0
$$565$$ − 519.615i − 0.919673i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −846.000 −1.48682 −0.743409 0.668837i $$-0.766793\pi$$
−0.743409 + 0.668837i $$0.766793\pi$$
$$570$$ 0 0
$$571$$ −375.855 −0.658240 −0.329120 0.944288i $$-0.606752\pi$$
−0.329120 + 0.944288i $$0.606752\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 390.000i − 0.678261i
$$576$$ 0 0
$$577$$ −791.000 −1.37088 −0.685442 0.728127i $$-0.740391\pi$$
−0.685442 + 0.728127i $$0.740391\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 13.8564i − 0.0238492i
$$582$$ 0 0
$$583$$ − 840.000i − 1.44082i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −273.664 −0.466208 −0.233104 0.972452i $$-0.574888\pi$$
−0.233104 + 0.972452i $$0.574888\pi$$
$$588$$ 0 0
$$589$$ − 24.2487i − 0.0411693i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −768.000 −1.29511 −0.647555 0.762019i $$-0.724208\pi$$
−0.647555 + 0.762019i $$0.724208\pi$$
$$594$$ 0 0
$$595$$ 20.7846 0.0349321
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 564.000i 0.941569i 0.882248 + 0.470785i $$0.156029\pi$$
−0.882248 + 0.470785i $$0.843971\pi$$
$$600$$ 0 0
$$601$$ −178.000 −0.296173 −0.148087 0.988974i $$-0.547311\pi$$
−0.148087 + 0.988974i $$0.547311\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 620.074i − 1.02492i
$$606$$ 0 0
$$607$$ − 1103.00i − 1.81713i −0.417740 0.908567i $$-0.637178\pi$$
0.417740 0.908567i $$-0.362822\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −114.315 −0.187096
$$612$$ 0 0
$$613$$ − 226.899i − 0.370145i −0.982725 0.185072i $$-0.940748\pi$$
0.982725 0.185072i $$-0.0592519\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 162.000 0.262561 0.131280 0.991345i $$-0.458091\pi$$
0.131280 + 0.991345i $$0.458091\pi$$
$$618$$ 0 0
$$619$$ 351.606 0.568023 0.284012 0.958821i $$-0.408335\pi$$
0.284012 + 0.958821i $$0.408335\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 66.0000i 0.105939i
$$624$$ 0 0
$$625$$ −131.000 −0.209600
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 114.315i 0.181741i
$$630$$ 0 0
$$631$$ 85.0000i 0.134707i 0.997729 + 0.0673534i $$0.0214555\pi$$
−0.997729 + 0.0673534i $$0.978545\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 658.179 1.03650
$$636$$ 0 0
$$637$$ 83.1384i 0.130516i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −720.000 −1.12324 −0.561622 0.827394i $$-0.689823\pi$$
−0.561622 + 0.827394i $$0.689823\pi$$
$$642$$ 0 0
$$643$$ 394.908 0.614164 0.307082 0.951683i $$-0.400647\pi$$
0.307082 + 0.951683i $$0.400647\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 312.000i 0.482226i 0.970497 + 0.241113i $$0.0775124\pi$$
−0.970497 + 0.241113i $$0.922488\pi$$
$$648$$ 0 0
$$649$$ −540.000 −0.832049
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 720.533i 1.10342i 0.834036 + 0.551710i $$0.186024\pi$$
−0.834036 + 0.551710i $$0.813976\pi$$
$$654$$ 0 0
$$655$$ 168.000i 0.256489i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −200.918 −0.304883 −0.152441 0.988312i $$-0.548714\pi$$
−0.152441 + 0.988312i $$0.548714\pi$$
$$660$$ 0 0
$$661$$ − 933.575i − 1.41237i −0.708028 0.706184i $$-0.750415\pi$$
0.708028 0.706184i $$-0.249585\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −6.00000 −0.00902256
$$666$$ 0 0
$$667$$ 623.538 0.934840
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 750.000i − 1.11773i
$$672$$ 0 0
$$673$$ −323.000 −0.479941 −0.239970 0.970780i $$-0.577138\pi$$
−0.239970 + 0.970780i $$0.577138\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 737.854i 1.08989i 0.838473 + 0.544944i $$0.183449\pi$$
−0.838473 + 0.544944i $$0.816551\pi$$
$$678$$ 0 0
$$679$$ 107.000i 0.157585i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1060.02 1.55200 0.775999 0.630734i $$-0.217246\pi$$
0.775999 + 0.630734i $$0.217246\pi$$
$$684$$ 0 0
$$685$$ 769.031i 1.12267i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −84.0000 −0.121916
$$690$$ 0 0
$$691$$ −561.184 −0.812134 −0.406067 0.913843i $$-0.633100\pi$$
−0.406067 + 0.913843i $$0.633100\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 570.000i 0.820144i
$$696$$ 0 0
$$697$$ −288.000 −0.413199
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 557.720i 0.795607i 0.917471 + 0.397803i $$0.130227\pi$$
−0.917471 + 0.397803i $$0.869773\pi$$
$$702$$ 0 0
$$703$$ − 33.0000i − 0.0469417i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 159.349 0.225387
$$708$$ 0 0
$$709$$ − 912.791i − 1.28743i −0.765264 0.643717i $$-0.777391\pi$$
0.765264 0.643717i $$-0.222609\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 420.000 0.589060
$$714$$ 0 0
$$715$$ −103.923 −0.145347
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 432.000i − 0.600834i −0.953808 0.300417i $$-0.902874\pi$$
0.953808 0.300417i $$-0.0971259\pi$$
$$720$$ 0 0
$$721$$ −95.0000 −0.131761
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 270.200i 0.372690i
$$726$$ 0 0
$$727$$ − 758.000i − 1.04264i −0.853361 0.521320i $$-0.825440\pi$$
0.853361 0.521320i $$-0.174560\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −124.708 −0.170599
$$732$$ 0 0
$$733$$ 270.200i 0.368622i 0.982868 + 0.184311i $$0.0590054\pi$$
−0.982868 + 0.184311i $$0.940995\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1050.00 1.42469
$$738$$ 0 0
$$739$$ 852.169 1.15314 0.576569 0.817048i $$-0.304391\pi$$
0.576569 + 0.817048i $$0.304391\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 150.000i 0.201884i 0.994892 + 0.100942i $$0.0321857\pi$$
−0.994892 + 0.100942i $$0.967814\pi$$
$$744$$ 0 0
$$745$$ 336.000 0.451007
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 17.3205i 0.0231248i
$$750$$ 0 0
$$751$$ − 457.000i − 0.608522i −0.952589 0.304261i $$-0.901591\pi$$
0.952589 0.304261i $$-0.0984095\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 377.587 0.500115
$$756$$ 0 0
$$757$$ − 621.806i − 0.821409i −0.911769 0.410704i $$-0.865283\pi$$
0.911769 0.410704i $$-0.134717\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1458.00 −1.91590 −0.957950 0.286935i $$-0.907364\pi$$
−0.957950 + 0.286935i $$0.907364\pi$$
$$762$$ 0 0
$$763$$ −187.061 −0.245166
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 54.0000i 0.0704042i
$$768$$ 0 0
$$769$$ 47.0000 0.0611183 0.0305592 0.999533i $$-0.490271\pi$$
0.0305592 + 0.999533i $$0.490271\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 706.677i − 0.914200i −0.889415 0.457100i $$-0.848888\pi$$
0.889415 0.457100i $$-0.151112\pi$$
$$774$$ 0 0
$$775$$ 182.000i 0.234839i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 83.1384 0.106725
$$780$$ 0 0
$$781$$ 831.384i 1.06451i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 216.000 0.275159
$$786$$ 0 0
$$787$$ −708.409 −0.900138 −0.450069 0.892994i $$-0.648601\pi$$
−0.450069 + 0.892994i $$0.648601\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 150.000i 0.189633i
$$792$$ 0 0
$$793$$ −75.0000 −0.0945776
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 1129.30i − 1.41693i −0.705743 0.708467i $$-0.749387\pi$$
0.705743 0.708467i $$-0.250613\pi$$
$$798$$ 0 0
$$799$$ 396.000i 0.495620i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −848.705 −1.05692
$$804$$ 0 0
$$805$$ − 103.923i − 0.129097i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 288.000 0.355995 0.177998 0.984031i $$-0.443038\pi$$
0.177998 + 0.984031i $$0.443038\pi$$
$$810$$ 0 0
$$811$$ −187.061 −0.230655 −0.115328 0.993328i $$-0.536792\pi$$
−0.115328 + 0.993328i $$0.536792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 294.000i − 0.360736i
$$816$$ 0 0
$$817$$ 36.0000 0.0440636
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 114.315i 0.139239i 0.997574 + 0.0696196i $$0.0221785\pi$$
−0.997574 + 0.0696196i $$0.977821\pi$$
$$822$$ 0 0
$$823$$ 601.000i 0.730255i 0.930957 + 0.365128i $$0.118975\pi$$
−0.930957 + 0.365128i $$0.881025\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −820.992 −0.992735 −0.496368 0.868112i $$-0.665333\pi$$
−0.496368 + 0.868112i $$0.665333\pi$$
$$828$$ 0 0
$$829$$ 642.591i 0.775140i 0.921840 + 0.387570i $$0.126685\pi$$
−0.921840 + 0.387570i $$0.873315\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 288.000 0.345738
$$834$$ 0 0
$$835$$ 810.600 0.970778
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 972.000i 1.15852i 0.815142 + 0.579261i $$0.196659\pi$$
−0.815142 + 0.579261i $$0.803341\pi$$
$$840$$ 0 0
$$841$$ 409.000 0.486326
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 575.041i − 0.680522i
$$846$$ 0 0
$$847$$ 179.000i 0.211334i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 571.577 0.671653
$$852$$ 0 0
$$853$$ − 334.286i − 0.391894i −0.980614 0.195947i $$-0.937222\pi$$
0.980614 0.195947i $$-0.0627781\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24.0000 0.0280047 0.0140023 0.999902i $$-0.495543\pi$$
0.0140023 + 0.999902i $$0.495543\pi$$
$$858$$ 0 0
$$859$$ −105.655 −0.122998 −0.0614989 0.998107i $$-0.519588\pi$$
−0.0614989 + 0.998107i $$0.519588\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 150.000i − 0.173812i −0.996217 0.0869061i $$-0.972302\pi$$
0.996217 0.0869061i $$-0.0276980\pi$$
$$864$$ 0 0
$$865$$ 624.000 0.721387
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 1437.60i − 1.65432i
$$870$$ 0 0
$$871$$ − 105.000i − 0.120551i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 131.636 0.150441
$$876$$ 0 0
$$877$$ 39.8372i 0.0454244i 0.999742 + 0.0227122i $$0.00723013\pi$$
−0.999742 + 0.0227122i $$0.992770\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1014.00 1.15096 0.575482 0.817814i $$-0.304814\pi$$
0.575482 + 0.817814i $$0.304814\pi$$
$$882$$ 0 0
$$883$$ −476.314 −0.539427 −0.269713 0.962941i $$-0.586929\pi$$
−0.269713 + 0.962941i $$0.586929\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 768.000i − 0.865840i −0.901432 0.432920i $$-0.857483\pi$$
0.901432 0.432920i $$-0.142517\pi$$
$$888$$ 0 0
$$889$$ −190.000 −0.213723
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 114.315i − 0.128013i
$$894$$ 0 0
$$895$$ − 672.000i − 0.750838i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −290.985 −0.323676
$$900$$ 0 0
$$901$$ 290.985i 0.322957i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 570.000 0.629834
$$906$$ 0 0
$$907$$ 209.578 0.231067 0.115534 0.993304i $$-0.463142\pi$$
0.115534 + 0.993304i $$0.463142\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 600.000i 0.658617i 0.944222 + 0.329308i $$0.106816\pi$$
−0.944222 + 0.329308i $$0.893184\pi$$
$$912$$ 0 0
$$913$$ 240.000 0.262870
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 48.4974i − 0.0528870i
$$918$$ 0 0
$$919$$ − 422.000i − 0.459195i −0.973286 0.229597i $$-0.926259\pi$$
0.973286 0.229597i $$-0.0737409\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 83.1384 0.0900741
$$924$$ 0 0
$$925$$ 247.683i 0.267766i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1116.00 1.20129 0.600646 0.799515i $$-0.294910\pi$$
0.600646 + 0.799515i $$0.294910\pi$$
$$930$$ 0 0
$$931$$ −83.1384 −0.0893001
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 360.000i 0.385027i
$$936$$ 0 0
$$937$$ −275.000 −0.293490 −0.146745 0.989174i $$-0.546880\pi$$
−0.146745 + 0.989174i $$0.546880\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 329.090i 0.349723i 0.984593 + 0.174862i $$0.0559478\pi$$
−0.984593 + 0.174862i $$0.944052\pi$$
$$942$$ 0 0
$$943$$ 1440.00i 1.52704i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 162.813 0.171925 0.0859624 0.996298i $$-0.472604\pi$$
0.0859624 + 0.996298i $$0.472604\pi$$
$$948$$ 0 0
$$949$$ 84.8705i 0.0894315i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1506.00 −1.58027 −0.790136 0.612931i $$-0.789990\pi$$
−0.790136 + 0.612931i $$0.789990\pi$$
$$954$$ 0 0
$$955$$ −685.892 −0.718212
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 222.000i − 0.231491i
$$960$$ 0 0
$$961$$ 765.000 0.796046
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 786.351i − 0.814872i
$$966$$ 0 0
$$967$$ 1307.00i 1.35160i 0.737084 + 0.675801i $$0.236202\pi$$
−0.737084 + 0.675801i $$0.763798\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 627.002 0.645729 0.322864 0.946445i $$-0.395354\pi$$
0.322864 + 0.946445i $$0.395354\pi$$
$$972$$ 0 0
$$973$$ − 164.545i − 0.169111i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −840.000 −0.859775 −0.429887 0.902883i $$-0.641447\pi$$
−0.429887 + 0.902883i $$0.641447\pi$$
$$978$$ 0 0
$$979$$ −1143.15 −1.16767
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 1134.00i − 1.15361i −0.816881 0.576806i $$-0.804299\pi$$
0.816881 0.576806i $$-0.195701\pi$$
$$984$$ 0 0
$$985$$ 804.000 0.816244
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 623.538i 0.630473i
$$990$$ 0 0
$$991$$ 1355.00i 1.36731i 0.729807 + 0.683653i $$0.239610\pi$$
−0.729807 + 0.683653i $$0.760390\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 827.920 0.832081
$$996$$ 0 0
$$997$$ 394.908i 0.396096i 0.980192 + 0.198048i $$0.0634602\pi$$
−0.980192 + 0.198048i $$0.936540\pi$$
$$998$$ 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 1728.3.b.e.1567.2 yes 4
3.2 odd 2 1728.3.b.b.1567.4 yes 4
4.3 odd 2 inner 1728.3.b.e.1567.1 yes 4
8.3 odd 2 inner 1728.3.b.e.1567.3 yes 4
8.5 even 2 inner 1728.3.b.e.1567.4 yes 4
12.11 even 2 1728.3.b.b.1567.3 yes 4
24.5 odd 2 1728.3.b.b.1567.2 yes 4
24.11 even 2 1728.3.b.b.1567.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.b.1567.1 4 24.11 even 2
1728.3.b.b.1567.2 yes 4 24.5 odd 2
1728.3.b.b.1567.3 yes 4 12.11 even 2
1728.3.b.b.1567.4 yes 4 3.2 odd 2
1728.3.b.e.1567.1 yes 4 4.3 odd 2 inner
1728.3.b.e.1567.2 yes 4 1.1 even 1 trivial
1728.3.b.e.1567.3 yes 4 8.3 odd 2 inner
1728.3.b.e.1567.4 yes 4 8.5 even 2 inner