# Properties

 Label 1728.3.b.f.1567.4 Level $1728$ Weight $3$ Character 1728.1567 Analytic conductor $47.085$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Learn more

## 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: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.4 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1567 Dual form 1728.3.b.f.1567.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.00000i q^{5} +5.00000i q^{7} +O(q^{10})$$ $$q+6.00000i q^{5} +5.00000i q^{7} +6.00000 q^{11} +5.19615i q^{13} -31.1769 q^{17} -25.9808 q^{19} +31.1769i q^{23} -11.0000 q^{25} -36.0000i q^{29} -26.0000i q^{31} -30.0000 q^{35} -25.9808i q^{37} -20.7846 q^{43} +31.1769i q^{47} +24.0000 q^{49} +60.0000i q^{53} +36.0000i q^{55} +18.0000 q^{59} -77.9423i q^{61} -31.1769 q^{65} -77.9423 q^{67} +25.0000 q^{73} +30.0000i q^{77} +31.0000i q^{79} +120.000 q^{83} -187.061i q^{85} +155.885 q^{89} -25.9808 q^{91} -155.885i q^{95} -85.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100})$$ 4 * q + 24 * q^11 - 44 * q^25 - 120 * q^35 + 96 * q^49 + 72 * q^59 + 100 * q^73 + 480 * q^83 - 340 * 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
$$5$$ 6.00000i 1.20000i 0.800000 + 0.600000i $$0.204833\pi$$
−0.800000 + 0.600000i $$0.795167\pi$$
$$6$$ 0 0
$$7$$ 5.00000i 0.714286i 0.934050 + 0.357143i $$0.116249\pi$$
−0.934050 + 0.357143i $$0.883751\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.00000 0.545455 0.272727 0.962091i $$-0.412074\pi$$
0.272727 + 0.962091i $$0.412074\pi$$
$$12$$ 0 0
$$13$$ 5.19615i 0.399704i 0.979826 + 0.199852i $$0.0640461\pi$$
−0.979826 + 0.199852i $$0.935954\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −31.1769 −1.83394 −0.916968 0.398961i $$-0.869371\pi$$
−0.916968 + 0.398961i $$0.869371\pi$$
$$18$$ 0 0
$$19$$ −25.9808 −1.36741 −0.683704 0.729759i $$-0.739632\pi$$
−0.683704 + 0.729759i $$0.739632\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 31.1769i 1.35552i 0.735284 + 0.677759i $$0.237049\pi$$
−0.735284 + 0.677759i $$0.762951\pi$$
$$24$$ 0 0
$$25$$ −11.0000 −0.440000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 36.0000i − 1.24138i −0.784056 0.620690i $$-0.786853\pi$$
0.784056 0.620690i $$-0.213147\pi$$
$$30$$ 0 0
$$31$$ − 26.0000i − 0.838710i −0.907822 0.419355i $$-0.862256\pi$$
0.907822 0.419355i $$-0.137744\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −30.0000 −0.857143
$$36$$ 0 0
$$37$$ − 25.9808i − 0.702183i −0.936341 0.351091i $$-0.885811\pi$$
0.936341 0.351091i $$-0.114189\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 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
$$47$$ 31.1769i 0.663339i 0.943396 + 0.331669i $$0.107612\pi$$
−0.943396 + 0.331669i $$0.892388\pi$$
$$48$$ 0 0
$$49$$ 24.0000 0.489796
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 60.0000i 1.13208i 0.824379 + 0.566038i $$0.191524\pi$$
−0.824379 + 0.566038i $$0.808476\pi$$
$$54$$ 0 0
$$55$$ 36.0000i 0.654545i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 18.0000 0.305085 0.152542 0.988297i $$-0.451254\pi$$
0.152542 + 0.988297i $$0.451254\pi$$
$$60$$ 0 0
$$61$$ − 77.9423i − 1.27774i −0.769314 0.638871i $$-0.779402\pi$$
0.769314 0.638871i $$-0.220598\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −31.1769 −0.479645
$$66$$ 0 0
$$67$$ −77.9423 −1.16332 −0.581659 0.813433i $$-0.697596\pi$$
−0.581659 + 0.813433i $$0.697596\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 25.0000 0.342466 0.171233 0.985231i $$-0.445225\pi$$
0.171233 + 0.985231i $$0.445225\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 30.0000i 0.389610i
$$78$$ 0 0
$$79$$ 31.0000i 0.392405i 0.980563 + 0.196203i $$0.0628610\pi$$
−0.980563 + 0.196203i $$0.937139\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 120.000 1.44578 0.722892 0.690961i $$-0.242813\pi$$
0.722892 + 0.690961i $$0.242813\pi$$
$$84$$ 0 0
$$85$$ − 187.061i − 2.20072i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 155.885 1.75151 0.875756 0.482754i $$-0.160363\pi$$
0.875756 + 0.482754i $$0.160363\pi$$
$$90$$ 0 0
$$91$$ −25.9808 −0.285503
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 155.885i − 1.64089i
$$96$$ 0 0
$$97$$ −85.0000 −0.876289 −0.438144 0.898905i $$-0.644364\pi$$
−0.438144 + 0.898905i $$0.644364\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 156.000i − 1.54455i −0.635286 0.772277i $$-0.719118\pi$$
0.635286 0.772277i $$-0.280882\pi$$
$$102$$ 0 0
$$103$$ − 125.000i − 1.21359i −0.794858 0.606796i $$-0.792454\pi$$
0.794858 0.606796i $$-0.207546\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −150.000 −1.40187 −0.700935 0.713226i $$-0.747234\pi$$
−0.700935 + 0.713226i $$0.747234\pi$$
$$108$$ 0 0
$$109$$ − 103.923i − 0.953422i −0.879060 0.476711i $$-0.841829\pi$$
0.879060 0.476711i $$-0.158171\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −31.1769 −0.275902 −0.137951 0.990439i $$-0.544052\pi$$
−0.137951 + 0.990439i $$0.544052\pi$$
$$114$$ 0 0
$$115$$ −187.061 −1.62662
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 155.885i − 1.30995i
$$120$$ 0 0
$$121$$ −85.0000 −0.702479
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 84.0000i 0.672000i
$$126$$ 0 0
$$127$$ 110.000i 0.866142i 0.901360 + 0.433071i $$0.142570\pi$$
−0.901360 + 0.433071i $$0.857430\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −156.000 −1.19084 −0.595420 0.803415i $$-0.703014\pi$$
−0.595420 + 0.803415i $$0.703014\pi$$
$$132$$ 0 0
$$133$$ − 129.904i − 0.976720i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 155.885 1.13784 0.568922 0.822392i $$-0.307361\pi$$
0.568922 + 0.822392i $$0.307361\pi$$
$$138$$ 0 0
$$139$$ −181.865 −1.30838 −0.654192 0.756329i $$-0.726991\pi$$
−0.654192 + 0.756329i $$0.726991\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 31.1769i 0.218020i
$$144$$ 0 0
$$145$$ 216.000 1.48966
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 24.0000i − 0.161074i −0.996752 0.0805369i $$-0.974337\pi$$
0.996752 0.0805369i $$-0.0256635\pi$$
$$150$$ 0 0
$$151$$ − 31.0000i − 0.205298i −0.994718 0.102649i $$-0.967268\pi$$
0.994718 0.102649i $$-0.0327318\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 156.000 1.00645
$$156$$ 0 0
$$157$$ 20.7846i 0.132386i 0.997807 + 0.0661930i $$0.0210853\pi$$
−0.997807 + 0.0661930i $$0.978915\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −155.885 −0.968227
$$162$$ 0 0
$$163$$ −306.573 −1.88082 −0.940408 0.340048i $$-0.889557\pi$$
−0.940408 + 0.340048i $$0.889557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 155.885i 0.933441i 0.884405 + 0.466720i $$0.154565\pi$$
−0.884405 + 0.466720i $$0.845435\pi$$
$$168$$ 0 0
$$169$$ 142.000 0.840237
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 120.000i 0.693642i 0.937931 + 0.346821i $$0.112739\pi$$
−0.937931 + 0.346821i $$0.887261\pi$$
$$174$$ 0 0
$$175$$ − 55.0000i − 0.314286i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 192.000 1.07263 0.536313 0.844019i $$-0.319817\pi$$
0.536313 + 0.844019i $$0.319817\pi$$
$$180$$ 0 0
$$181$$ − 337.750i − 1.86602i −0.359848 0.933011i $$-0.617172\pi$$
0.359848 0.933011i $$-0.382828\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 155.885 0.842619
$$186$$ 0 0
$$187$$ −187.061 −1.00033
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 155.885i 0.816150i 0.912948 + 0.408075i $$0.133800\pi$$
−0.912948 + 0.408075i $$0.866200\pi$$
$$192$$ 0 0
$$193$$ −325.000 −1.68394 −0.841969 0.539526i $$-0.818603\pi$$
−0.841969 + 0.539526i $$0.818603\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 30.0000i 0.152284i 0.997097 + 0.0761421i $$0.0242603\pi$$
−0.997097 + 0.0761421i $$0.975740\pi$$
$$198$$ 0 0
$$199$$ 259.000i 1.30151i 0.759289 + 0.650754i $$0.225547\pi$$
−0.759289 + 0.650754i $$0.774453\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 180.000 0.886700
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −155.885 −0.745859
$$210$$ 0 0
$$211$$ −25.9808 −0.123132 −0.0615658 0.998103i $$-0.519609\pi$$
−0.0615658 + 0.998103i $$0.519609\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 124.708i − 0.580036i
$$216$$ 0 0
$$217$$ 130.000 0.599078
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 162.000i − 0.733032i
$$222$$ 0 0
$$223$$ − 310.000i − 1.39013i −0.718945 0.695067i $$-0.755375\pi$$
0.718945 0.695067i $$-0.244625\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −360.000 −1.58590 −0.792952 0.609285i $$-0.791457\pi$$
−0.792952 + 0.609285i $$0.791457\pi$$
$$228$$ 0 0
$$229$$ − 207.846i − 0.907625i −0.891097 0.453812i $$-0.850064\pi$$
0.891097 0.453812i $$-0.149936\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −311.769 −1.33807 −0.669033 0.743233i $$-0.733291\pi$$
−0.669033 + 0.743233i $$0.733291\pi$$
$$234$$ 0 0
$$235$$ −187.061 −0.796006
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 311.769i 1.30447i 0.758015 + 0.652237i $$0.226169\pi$$
−0.758015 + 0.652237i $$0.773831\pi$$
$$240$$ 0 0
$$241$$ −131.000 −0.543568 −0.271784 0.962358i $$-0.587614\pi$$
−0.271784 + 0.962358i $$0.587614\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 144.000i 0.587755i
$$246$$ 0 0
$$247$$ − 135.000i − 0.546559i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −216.000 −0.860558 −0.430279 0.902696i $$-0.641585\pi$$
−0.430279 + 0.902696i $$0.641585\pi$$
$$252$$ 0 0
$$253$$ 187.061i 0.739373i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −187.061 −0.727866 −0.363933 0.931425i $$-0.618566\pi$$
−0.363933 + 0.931425i $$0.618566\pi$$
$$258$$ 0 0
$$259$$ 129.904 0.501559
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 311.769i 1.18543i 0.805411 + 0.592717i $$0.201945\pi$$
−0.805411 + 0.592717i $$0.798055\pi$$
$$264$$ 0 0
$$265$$ −360.000 −1.35849
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 414.000i − 1.53903i −0.638627 0.769517i $$-0.720497\pi$$
0.638627 0.769517i $$-0.279503\pi$$
$$270$$ 0 0
$$271$$ 317.000i 1.16974i 0.811126 + 0.584871i $$0.198855\pi$$
−0.811126 + 0.584871i $$0.801145\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −66.0000 −0.240000
$$276$$ 0 0
$$277$$ − 519.615i − 1.87587i −0.346815 0.937934i $$-0.612737\pi$$
0.346815 0.937934i $$-0.387263\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −311.769 −1.10950 −0.554749 0.832018i $$-0.687186\pi$$
−0.554749 + 0.832018i $$0.687186\pi$$
$$282$$ 0 0
$$283$$ −103.923 −0.367219 −0.183610 0.982999i $$-0.558778\pi$$
−0.183610 + 0.982999i $$0.558778\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 683.000 2.36332
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 270.000i 0.921502i 0.887530 + 0.460751i $$0.152420\pi$$
−0.887530 + 0.460751i $$0.847580\pi$$
$$294$$ 0 0
$$295$$ 108.000i 0.366102i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −162.000 −0.541806
$$300$$ 0 0
$$301$$ − 103.923i − 0.345259i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 467.654 1.53329
$$306$$ 0 0
$$307$$ 436.477 1.42175 0.710874 0.703319i $$-0.248299\pi$$
0.710874 + 0.703319i $$0.248299\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 467.654i − 1.50371i −0.659329 0.751855i $$-0.729159\pi$$
0.659329 0.751855i $$-0.270841\pi$$
$$312$$ 0 0
$$313$$ 85.0000 0.271565 0.135783 0.990739i $$-0.456645\pi$$
0.135783 + 0.990739i $$0.456645\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 120.000i 0.378549i 0.981924 + 0.189274i $$0.0606136\pi$$
−0.981924 + 0.189274i $$0.939386\pi$$
$$318$$ 0 0
$$319$$ − 216.000i − 0.677116i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 810.000 2.50774
$$324$$ 0 0
$$325$$ − 57.1577i − 0.175870i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −155.885 −0.473813
$$330$$ 0 0
$$331$$ 25.9808 0.0784917 0.0392459 0.999230i $$-0.487504\pi$$
0.0392459 + 0.999230i $$0.487504\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 467.654i − 1.39598i
$$336$$ 0 0
$$337$$ −325.000 −0.964392 −0.482196 0.876063i $$-0.660161\pi$$
−0.482196 + 0.876063i $$0.660161\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 156.000i − 0.457478i
$$342$$ 0 0
$$343$$ 365.000i 1.06414i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 300.000 0.864553 0.432277 0.901741i $$-0.357710\pi$$
0.432277 + 0.901741i $$0.357710\pi$$
$$348$$ 0 0
$$349$$ 597.558i 1.71220i 0.516811 + 0.856100i $$0.327119\pi$$
−0.516811 + 0.856100i $$0.672881\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 436.477 1.23648 0.618239 0.785990i $$-0.287846\pi$$
0.618239 + 0.785990i $$0.287846\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 467.654i 1.30266i 0.758796 + 0.651328i $$0.225788\pi$$
−0.758796 + 0.651328i $$0.774212\pi$$
$$360$$ 0 0
$$361$$ 314.000 0.869806
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 150.000i 0.410959i
$$366$$ 0 0
$$367$$ − 185.000i − 0.504087i −0.967716 0.252044i $$-0.918897\pi$$
0.967716 0.252044i $$-0.0811026\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −300.000 −0.808625
$$372$$ 0 0
$$373$$ 57.1577i 0.153238i 0.997060 + 0.0766189i $$0.0244125\pi$$
−0.997060 + 0.0766189i $$0.975588\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 187.061 0.496184
$$378$$ 0 0
$$379$$ 129.904 0.342754 0.171377 0.985206i $$-0.445178\pi$$
0.171377 + 0.985206i $$0.445178\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 187.061i − 0.488411i −0.969723 0.244206i $$-0.921473\pi$$
0.969723 0.244206i $$-0.0785272\pi$$
$$384$$ 0 0
$$385$$ −180.000 −0.467532
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 354.000i 0.910026i 0.890485 + 0.455013i $$0.150365\pi$$
−0.890485 + 0.455013i $$0.849635\pi$$
$$390$$ 0 0
$$391$$ − 972.000i − 2.48593i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −186.000 −0.470886
$$396$$ 0 0
$$397$$ 644.323i 1.62298i 0.584367 + 0.811490i $$0.301343\pi$$
−0.584367 + 0.811490i $$0.698657\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 311.769 0.777479 0.388740 0.921348i $$-0.372911\pi$$
0.388740 + 0.921348i $$0.372911\pi$$
$$402$$ 0 0
$$403$$ 135.100 0.335236
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 155.885i − 0.383009i
$$408$$ 0 0
$$409$$ −539.000 −1.31785 −0.658924 0.752209i $$-0.728988\pi$$
−0.658924 + 0.752209i $$0.728988\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 90.0000i 0.217918i
$$414$$ 0 0
$$415$$ 720.000i 1.73494i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −306.000 −0.730310 −0.365155 0.930947i $$-0.618984\pi$$
−0.365155 + 0.930947i $$0.618984\pi$$
$$420$$ 0 0
$$421$$ 25.9808i 0.0617120i 0.999524 + 0.0308560i $$0.00982333\pi$$
−0.999524 + 0.0308560i $$0.990177\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 342.946 0.806932
$$426$$ 0 0
$$427$$ 389.711 0.912673
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 155.885i − 0.361681i −0.983512 0.180841i $$-0.942118\pi$$
0.983512 0.180841i $$-0.0578818\pi$$
$$432$$ 0 0
$$433$$ 250.000 0.577367 0.288684 0.957425i $$-0.406782\pi$$
0.288684 + 0.957425i $$0.406782\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 810.000i − 1.85355i
$$438$$ 0 0
$$439$$ 22.0000i 0.0501139i 0.999686 + 0.0250569i $$0.00797671\pi$$
−0.999686 + 0.0250569i $$0.992023\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −540.000 −1.21896 −0.609481 0.792801i $$-0.708622\pi$$
−0.609481 + 0.792801i $$0.708622\pi$$
$$444$$ 0 0
$$445$$ 935.307i 2.10181i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 155.885 0.347182 0.173591 0.984818i $$-0.444463\pi$$
0.173591 + 0.984818i $$0.444463\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 155.885i − 0.342603i
$$456$$ 0 0
$$457$$ −230.000 −0.503282 −0.251641 0.967821i $$-0.580970\pi$$
−0.251641 + 0.967821i $$0.580970\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 78.0000i 0.169197i 0.996415 + 0.0845987i $$0.0269608\pi$$
−0.996415 + 0.0845987i $$0.973039\pi$$
$$462$$ 0 0
$$463$$ − 55.0000i − 0.118790i −0.998235 0.0593952i $$-0.981083\pi$$
0.998235 0.0593952i $$-0.0189172\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −150.000 −0.321199 −0.160600 0.987020i $$-0.551343\pi$$
−0.160600 + 0.987020i $$0.551343\pi$$
$$468$$ 0 0
$$469$$ − 389.711i − 0.830941i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −124.708 −0.263653
$$474$$ 0 0
$$475$$ 285.788 0.601660
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 311.769i − 0.650875i −0.945564 0.325438i $$-0.894488\pi$$
0.945564 0.325438i $$-0.105512\pi$$
$$480$$ 0 0
$$481$$ 135.000 0.280665
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 510.000i − 1.05155i
$$486$$ 0 0
$$487$$ 785.000i 1.61191i 0.591977 + 0.805955i $$0.298348\pi$$
−0.591977 + 0.805955i $$0.701652\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 618.000 1.25866 0.629328 0.777140i $$-0.283330\pi$$
0.629328 + 0.777140i $$0.283330\pi$$
$$492$$ 0 0
$$493$$ 1122.37i 2.27661i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 623.538 1.24958 0.624788 0.780795i $$-0.285185\pi$$
0.624788 + 0.780795i $$0.285185\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 654.715i − 1.30162i −0.759240 0.650810i $$-0.774429\pi$$
0.759240 0.650810i $$-0.225571\pi$$
$$504$$ 0 0
$$505$$ 936.000 1.85347
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 366.000i 0.719057i 0.933134 + 0.359528i $$0.117062\pi$$
−0.933134 + 0.359528i $$0.882938\pi$$
$$510$$ 0 0
$$511$$ 125.000i 0.244618i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 750.000 1.45631
$$516$$ 0 0
$$517$$ 187.061i 0.361821i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −779.423 −1.49601 −0.748007 0.663691i $$-0.768989\pi$$
−0.748007 + 0.663691i $$0.768989\pi$$
$$522$$ 0 0
$$523$$ 109.119 0.208641 0.104320 0.994544i $$-0.466733\pi$$
0.104320 + 0.994544i $$0.466733\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 810.600i 1.53814i
$$528$$ 0 0
$$529$$ −443.000 −0.837429
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 900.000i − 1.68224i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 144.000 0.267161
$$540$$ 0 0
$$541$$ 649.519i 1.20059i 0.799779 + 0.600295i $$0.204950\pi$$
−0.799779 + 0.600295i $$0.795050\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 623.538 1.14411
$$546$$ 0 0
$$547$$ 420.888 0.769449 0.384724 0.923032i $$-0.374297\pi$$
0.384724 + 0.923032i $$0.374297\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 935.307i 1.69747i
$$552$$ 0 0
$$553$$ −155.000 −0.280289
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 690.000i 1.23878i 0.785084 + 0.619390i $$0.212620\pi$$
−0.785084 + 0.619390i $$0.787380\pi$$
$$558$$ 0 0
$$559$$ − 108.000i − 0.193202i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −840.000 −1.49201 −0.746004 0.665942i $$-0.768030\pi$$
−0.746004 + 0.665942i $$0.768030\pi$$
$$564$$ 0 0
$$565$$ − 187.061i − 0.331082i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1091.19 −1.91774 −0.958868 0.283852i $$-0.908388\pi$$
−0.958868 + 0.283852i $$0.908388\pi$$
$$570$$ 0 0
$$571$$ 805.404 1.41051 0.705257 0.708952i $$-0.250832\pi$$
0.705257 + 0.708952i $$0.250832\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 342.946i − 0.596428i
$$576$$ 0 0
$$577$$ 385.000 0.667244 0.333622 0.942707i $$-0.391729\pi$$
0.333622 + 0.942707i $$0.391729\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 600.000i 1.03270i
$$582$$ 0 0
$$583$$ 360.000i 0.617496i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 930.000 1.58433 0.792164 0.610309i $$-0.208955\pi$$
0.792164 + 0.610309i $$0.208955\pi$$
$$588$$ 0 0
$$589$$ 675.500i 1.14686i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −498.831 −0.841198 −0.420599 0.907247i $$-0.638180\pi$$
−0.420599 + 0.907247i $$0.638180\pi$$
$$594$$ 0 0
$$595$$ 935.307 1.57195
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 311.769i 0.520483i 0.965544 + 0.260241i $$0.0838021\pi$$
−0.965544 + 0.260241i $$0.916198\pi$$
$$600$$ 0 0
$$601$$ −346.000 −0.575707 −0.287854 0.957674i $$-0.592942\pi$$
−0.287854 + 0.957674i $$0.592942\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 510.000i − 0.842975i
$$606$$ 0 0
$$607$$ − 595.000i − 0.980231i −0.871658 0.490115i $$-0.836955\pi$$
0.871658 0.490115i $$-0.163045\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −162.000 −0.265139
$$612$$ 0 0
$$613$$ 109.119i 0.178008i 0.996031 + 0.0890042i $$0.0283685\pi$$
−0.996031 + 0.0890042i $$0.971632\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −592.361 −0.960067 −0.480034 0.877250i $$-0.659376\pi$$
−0.480034 + 0.877250i $$0.659376\pi$$
$$618$$ 0 0
$$619$$ 701.481 1.13325 0.566624 0.823976i $$-0.308249\pi$$
0.566624 + 0.823976i $$0.308249\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 779.423i 1.25108i
$$624$$ 0 0
$$625$$ −779.000 −1.24640
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 810.000i 1.28776i
$$630$$ 0 0
$$631$$ 713.000i 1.12995i 0.825107 + 0.564976i $$0.191115\pi$$
−0.825107 + 0.564976i $$0.808885\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −660.000 −1.03937
$$636$$ 0 0
$$637$$ 124.708i 0.195773i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −436.477 −0.678813 −0.339407 0.940640i $$-0.610226\pi$$
−0.339407 + 0.940640i $$0.610226\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 623.538i 0.963738i 0.876243 + 0.481869i $$0.160042\pi$$
−0.876243 + 0.481869i $$0.839958\pi$$
$$648$$ 0 0
$$649$$ 108.000 0.166410
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 960.000i − 1.47014i −0.677992 0.735069i $$-0.737150\pi$$
0.677992 0.735069i $$-0.262850\pi$$
$$654$$ 0 0
$$655$$ − 936.000i − 1.42901i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −132.000 −0.200303 −0.100152 0.994972i $$-0.531933\pi$$
−0.100152 + 0.994972i $$0.531933\pi$$
$$660$$ 0 0
$$661$$ − 1013.25i − 1.53290i −0.642301 0.766452i $$-0.722020\pi$$
0.642301 0.766452i $$-0.277980\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 779.423 1.17206
$$666$$ 0 0
$$667$$ 1122.37 1.68271
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 467.654i − 0.696950i
$$672$$ 0 0
$$673$$ 445.000 0.661218 0.330609 0.943768i $$-0.392746\pi$$
0.330609 + 0.943768i $$0.392746\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 990.000i − 1.46233i −0.682199 0.731167i $$-0.738976\pi$$
0.682199 0.731167i $$-0.261024\pi$$
$$678$$ 0 0
$$679$$ − 425.000i − 0.625920i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1260.00 −1.84480 −0.922401 0.386233i $$-0.873776\pi$$
−0.922401 + 0.386233i $$0.873776\pi$$
$$684$$ 0 0
$$685$$ 935.307i 1.36541i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −311.769 −0.452495
$$690$$ 0 0
$$691$$ 103.923 0.150395 0.0751976 0.997169i $$-0.476041\pi$$
0.0751976 + 0.997169i $$0.476041\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 1091.19i − 1.57006i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 246.000i − 0.350927i −0.984486 0.175464i $$-0.943858\pi$$
0.984486 0.175464i $$-0.0561424\pi$$
$$702$$ 0 0
$$703$$ 675.000i 0.960171i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 780.000 1.10325
$$708$$ 0 0
$$709$$ 129.904i 0.183221i 0.995795 + 0.0916106i $$0.0292015\pi$$
−0.995795 + 0.0916106i $$0.970799\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 810.600 1.13689
$$714$$ 0 0
$$715$$ −187.061 −0.261624
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1247.08i 1.73446i 0.497908 + 0.867230i $$0.334102\pi$$
−0.497908 + 0.867230i $$0.665898\pi$$
$$720$$ 0 0
$$721$$ 625.000 0.866852
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 396.000i 0.546207i
$$726$$ 0 0
$$727$$ 770.000i 1.05915i 0.848264 + 0.529574i $$0.177648\pi$$
−0.848264 + 0.529574i $$0.822352\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 648.000 0.886457
$$732$$ 0 0
$$733$$ 1143.15i 1.55955i 0.626057 + 0.779777i $$0.284668\pi$$
−0.626057 + 0.779777i $$0.715332\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −467.654 −0.634537
$$738$$ 0 0
$$739$$ −727.461 −0.984386 −0.492193 0.870486i $$-0.663805\pi$$
−0.492193 + 0.870486i $$0.663805\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 592.361i 0.797256i 0.917113 + 0.398628i $$0.130514\pi$$
−0.917113 + 0.398628i $$0.869486\pi$$
$$744$$ 0 0
$$745$$ 144.000 0.193289
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 750.000i − 1.00134i
$$750$$ 0 0
$$751$$ 1099.00i 1.46338i 0.681636 + 0.731691i $$0.261269\pi$$
−0.681636 + 0.731691i $$0.738731\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 186.000 0.246358
$$756$$ 0 0
$$757$$ 420.888i 0.555995i 0.960582 + 0.277998i $$0.0896707\pi$$
−0.960582 + 0.277998i $$0.910329\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −155.885 −0.204842 −0.102421 0.994741i $$-0.532659\pi$$
−0.102421 + 0.994741i $$0.532659\pi$$
$$762$$ 0 0
$$763$$ 519.615 0.681016
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 93.5307i 0.121944i
$$768$$ 0 0
$$769$$ −1009.00 −1.31209 −0.656047 0.754720i $$-0.727773\pi$$
−0.656047 + 0.754720i $$0.727773\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 360.000i 0.465718i 0.972511 + 0.232859i $$0.0748081\pi$$
−0.972511 + 0.232859i $$0.925192\pi$$
$$774$$ 0 0
$$775$$ 286.000i 0.369032i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −124.708 −0.158863
$$786$$ 0 0
$$787$$ 514.419 0.653646 0.326823 0.945086i $$-0.394022\pi$$
0.326823 + 0.945086i $$0.394022\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 155.885i − 0.197073i
$$792$$ 0 0
$$793$$ 405.000 0.510719
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 660.000i 0.828105i 0.910253 + 0.414053i $$0.135887\pi$$
−0.910253 + 0.414053i $$0.864113\pi$$
$$798$$ 0 0
$$799$$ − 972.000i − 1.21652i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 150.000 0.186800
$$804$$ 0 0
$$805$$ − 935.307i − 1.16187i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1247.08 1.54150 0.770752 0.637135i $$-0.219881\pi$$
0.770752 + 0.637135i $$0.219881\pi$$
$$810$$ 0 0
$$811$$ −519.615 −0.640709 −0.320355 0.947298i $$-0.603802\pi$$
−0.320355 + 0.947298i $$0.603802\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1839.44i − 2.25698i
$$816$$ 0 0
$$817$$ 540.000 0.660955
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1242.00i 1.51279i 0.654116 + 0.756395i $$0.273041\pi$$
−0.654116 + 0.756395i $$0.726959\pi$$
$$822$$ 0 0
$$823$$ − 955.000i − 1.16039i −0.814478 0.580194i $$-0.802977\pi$$
0.814478 0.580194i $$-0.197023\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −90.0000 −0.108827 −0.0544135 0.998518i $$-0.517329\pi$$
−0.0544135 + 0.998518i $$0.517329\pi$$
$$828$$ 0 0
$$829$$ 181.865i 0.219379i 0.993966 + 0.109690i $$0.0349857\pi$$
−0.993966 + 0.109690i $$0.965014\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −748.246 −0.898254
$$834$$ 0 0
$$835$$ −935.307 −1.12013
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1558.85i − 1.85798i −0.370104 0.928990i $$-0.620678\pi$$
0.370104 0.928990i $$-0.379322\pi$$
$$840$$ 0 0
$$841$$ −455.000 −0.541023
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 852.000i 1.00828i
$$846$$ 0 0
$$847$$ − 425.000i − 0.501771i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 810.000 0.951821
$$852$$ 0 0
$$853$$ − 46.7654i − 0.0548246i −0.999624 0.0274123i $$-0.991273\pi$$
0.999624 0.0274123i $$-0.00872670\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 623.538 0.727583 0.363791 0.931480i $$-0.381482\pi$$
0.363791 + 0.931480i $$0.381482\pi$$
$$858$$ 0 0
$$859$$ −129.904 −0.151227 −0.0756134 0.997137i $$-0.524091\pi$$
−0.0756134 + 0.997137i $$0.524091\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 342.946i − 0.397388i −0.980062 0.198694i $$-0.936330\pi$$
0.980062 0.198694i $$-0.0636700\pi$$
$$864$$ 0 0
$$865$$ −720.000 −0.832370
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 186.000i 0.214039i
$$870$$ 0 0
$$871$$ − 405.000i − 0.464983i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −420.000 −0.480000
$$876$$ 0 0
$$877$$ 1034.03i 1.17906i 0.807747 + 0.589529i $$0.200687\pi$$
−0.807747 + 0.589529i $$0.799313\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 467.654 0.530821 0.265411 0.964135i $$-0.414492\pi$$
0.265411 + 0.964135i $$0.414492\pi$$
$$882$$ 0 0
$$883$$ −285.788 −0.323656 −0.161828 0.986819i $$-0.551739\pi$$
−0.161828 + 0.986819i $$0.551739\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 498.831i − 0.562380i −0.959652 0.281190i $$-0.909271\pi$$
0.959652 0.281190i $$-0.0907290\pi$$
$$888$$ 0 0
$$889$$ −550.000 −0.618673
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 810.000i − 0.907055i
$$894$$ 0 0
$$895$$ 1152.00i 1.28715i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −936.000 −1.04116
$$900$$ 0 0
$$901$$ − 1870.61i − 2.07615i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 2026.50 2.23923
$$906$$ 0 0
$$907$$ −1096.39 −1.20881 −0.604404 0.796678i $$-0.706589\pi$$
−0.604404 + 0.796678i $$0.706589\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 623.538i 0.684455i 0.939617 + 0.342227i $$0.111181\pi$$
−0.939617 + 0.342227i $$0.888819\pi$$
$$912$$ 0 0
$$913$$ 720.000 0.788609
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 780.000i − 0.850600i
$$918$$ 0 0
$$919$$ − 1006.00i − 1.09467i −0.836914 0.547334i $$-0.815643\pi$$
0.836914 0.547334i $$-0.184357\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 285.788i 0.308960i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −311.769 −0.335596 −0.167798 0.985821i $$-0.553666\pi$$
−0.167798 + 0.985821i $$0.553666\pi$$
$$930$$ 0 0
$$931$$ −623.538 −0.669751
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 1122.37i − 1.20039i
$$936$$ 0 0
$$937$$ 565.000 0.602988 0.301494 0.953468i $$-0.402515\pi$$
0.301494 + 0.953468i $$0.402515\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 294.000i 0.312434i 0.987723 + 0.156217i $$0.0499298\pi$$
−0.987723 + 0.156217i $$0.950070\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30.0000 0.0316790 0.0158395 0.999875i $$-0.494958\pi$$
0.0158395 + 0.999875i $$0.494958\pi$$
$$948$$ 0 0
$$949$$ 129.904i 0.136885i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −904.131 −0.948720 −0.474360 0.880331i $$-0.657321\pi$$
−0.474360 + 0.880331i $$0.657321\pi$$
$$954$$ 0 0
$$955$$ −935.307 −0.979380
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 779.423i 0.812745i
$$960$$ 0 0
$$961$$ 285.000 0.296566
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 1950.00i − 2.02073i
$$966$$ 0 0
$$967$$ − 305.000i − 0.315408i −0.987486 0.157704i $$-0.949591\pi$$
0.987486 0.157704i $$-0.0504093\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1194.00 1.22966 0.614830 0.788660i $$-0.289225\pi$$
0.614830 + 0.788660i $$0.289225\pi$$
$$972$$ 0 0
$$973$$ − 909.327i − 0.934560i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −124.708 −0.127643 −0.0638217 0.997961i $$-0.520329\pi$$
−0.0638217 + 0.997961i $$0.520329\pi$$
$$978$$ 0 0
$$979$$ 935.307 0.955370
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1278.25i 1.30036i 0.759780 + 0.650180i $$0.225306\pi$$
−0.759780 + 0.650180i $$0.774694\pi$$
$$984$$ 0 0
$$985$$ −180.000 −0.182741
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 648.000i − 0.655207i
$$990$$ 0 0
$$991$$ 511.000i 0.515641i 0.966193 + 0.257820i $$0.0830043\pi$$
−0.966193 + 0.257820i $$0.916996\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1554.00 −1.56181
$$996$$ 0 0
$$997$$ − 1060.02i − 1.06320i −0.846994 0.531602i $$-0.821590\pi$$
0.846994 0.531602i $$-0.178410\pi$$
$$998$$ 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.f.1567.4 yes 4
3.2 odd 2 1728.3.b.a.1567.2 yes 4
4.3 odd 2 1728.3.b.a.1567.4 yes 4
8.3 odd 2 inner 1728.3.b.f.1567.1 yes 4
8.5 even 2 1728.3.b.a.1567.1 4
12.11 even 2 inner 1728.3.b.f.1567.2 yes 4
24.5 odd 2 inner 1728.3.b.f.1567.3 yes 4
24.11 even 2 1728.3.b.a.1567.3 yes 4

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