# Properties

 Label 1440.2.h.a.1151.1 Level $1440$ Weight $2$ Character 1440.1151 Analytic conductor $11.498$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1151.1 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1440.1151 Dual form 1440.2.h.a.1151.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{5} -3.41421i q^{7} +O(q^{10})$$ $$q-1.00000i q^{5} -3.41421i q^{7} -2.58579 q^{11} -3.41421 q^{13} +1.17157i q^{17} -4.82843 q^{23} -1.00000 q^{25} +6.00000i q^{29} -6.48528i q^{31} -3.41421 q^{35} -9.07107 q^{37} +11.0711i q^{41} +6.82843i q^{43} +5.65685 q^{47} -4.65685 q^{49} +1.17157i q^{53} +2.58579i q^{55} -6.58579 q^{59} +12.8284 q^{61} +3.41421i q^{65} -8.00000i q^{67} -5.65685 q^{71} -10.4853 q^{73} +8.82843i q^{77} -14.4853i q^{79} -9.17157 q^{83} +1.17157 q^{85} -4.24264i q^{89} +11.6569i q^{91} +2.48528 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 16 q^{11} - 8 q^{13} - 8 q^{23} - 4 q^{25} - 8 q^{35} - 8 q^{37} + 4 q^{49} - 32 q^{59} + 40 q^{61} - 8 q^{73} - 48 q^{83} + 16 q^{85} - 24 q^{97}+O(q^{100})$$ 4 * q - 16 * q^11 - 8 * q^13 - 8 * q^23 - 4 * q^25 - 8 * q^35 - 8 * q^37 + 4 * q^49 - 32 * q^59 + 40 * q^61 - 8 * q^73 - 48 * q^83 + 16 * q^85 - 24 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$991$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$5$$ − 1.00000i − 0.447214i
$$6$$ 0 0
$$7$$ − 3.41421i − 1.29045i −0.763992 0.645226i $$-0.776763\pi$$
0.763992 0.645226i $$-0.223237\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.58579 −0.779644 −0.389822 0.920890i $$-0.627463\pi$$
−0.389822 + 0.920890i $$0.627463\pi$$
$$12$$ 0 0
$$13$$ −3.41421 −0.946932 −0.473466 0.880812i $$-0.656997\pi$$
−0.473466 + 0.880812i $$0.656997\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.17157i 0.284148i 0.989856 + 0.142074i $$0.0453771\pi$$
−0.989856 + 0.142074i $$0.954623\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.82843 −1.00680 −0.503398 0.864054i $$-0.667917\pi$$
−0.503398 + 0.864054i $$0.667917\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ − 6.48528i − 1.16479i −0.812906 0.582395i $$-0.802116\pi$$
0.812906 0.582395i $$-0.197884\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.41421 −0.577107
$$36$$ 0 0
$$37$$ −9.07107 −1.49127 −0.745637 0.666352i $$-0.767855\pi$$
−0.745637 + 0.666352i $$0.767855\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.0711i 1.72901i 0.502624 + 0.864505i $$0.332368\pi$$
−0.502624 + 0.864505i $$0.667632\pi$$
$$42$$ 0 0
$$43$$ 6.82843i 1.04133i 0.853762 + 0.520663i $$0.174315\pi$$
−0.853762 + 0.520663i $$0.825685\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.65685 0.825137 0.412568 0.910927i $$-0.364632\pi$$
0.412568 + 0.910927i $$0.364632\pi$$
$$48$$ 0 0
$$49$$ −4.65685 −0.665265
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.17157i 0.160928i 0.996758 + 0.0804640i $$0.0256402\pi$$
−0.996758 + 0.0804640i $$0.974360\pi$$
$$54$$ 0 0
$$55$$ 2.58579i 0.348667i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.58579 −0.857396 −0.428698 0.903448i $$-0.641028\pi$$
−0.428698 + 0.903448i $$0.641028\pi$$
$$60$$ 0 0
$$61$$ 12.8284 1.64251 0.821256 0.570560i $$-0.193274\pi$$
0.821256 + 0.570560i $$0.193274\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.41421i 0.423481i
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.65685 −0.671345 −0.335673 0.941979i $$-0.608964\pi$$
−0.335673 + 0.941979i $$0.608964\pi$$
$$72$$ 0 0
$$73$$ −10.4853 −1.22721 −0.613605 0.789613i $$-0.710281\pi$$
−0.613605 + 0.789613i $$0.710281\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.82843i 1.00609i
$$78$$ 0 0
$$79$$ − 14.4853i − 1.62972i −0.579657 0.814861i $$-0.696813\pi$$
0.579657 0.814861i $$-0.303187\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.17157 −1.00671 −0.503355 0.864079i $$-0.667901\pi$$
−0.503355 + 0.864079i $$0.667901\pi$$
$$84$$ 0 0
$$85$$ 1.17157 0.127075
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 4.24264i − 0.449719i −0.974391 0.224860i $$-0.927808\pi$$
0.974391 0.224860i $$-0.0721923\pi$$
$$90$$ 0 0
$$91$$ 11.6569i 1.22197i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.48528 0.252342 0.126171 0.992009i $$-0.459731\pi$$
0.126171 + 0.992009i $$0.459731\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 4.82843i 0.480446i 0.970718 + 0.240223i $$0.0772206\pi$$
−0.970718 + 0.240223i $$0.922779\pi$$
$$102$$ 0 0
$$103$$ − 17.5563i − 1.72988i −0.501877 0.864939i $$-0.667357\pi$$
0.501877 0.864939i $$-0.332643\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.8284 −1.43352 −0.716759 0.697321i $$-0.754375\pi$$
−0.716759 + 0.697321i $$0.754375\pi$$
$$108$$ 0 0
$$109$$ −7.17157 −0.686912 −0.343456 0.939169i $$-0.611598\pi$$
−0.343456 + 0.939169i $$0.611598\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 1.65685i − 0.155864i −0.996959 0.0779319i $$-0.975168\pi$$
0.996959 0.0779319i $$-0.0248317\pi$$
$$114$$ 0 0
$$115$$ 4.82843i 0.450253i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −4.31371 −0.392155
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ − 10.7279i − 0.951949i −0.879459 0.475975i $$-0.842095\pi$$
0.879459 0.475975i $$-0.157905\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.7279 −1.11204 −0.556022 0.831168i $$-0.687673\pi$$
−0.556022 + 0.831168i $$0.687673\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 19.3137i − 1.65008i −0.565073 0.825041i $$-0.691152\pi$$
0.565073 0.825041i $$-0.308848\pi$$
$$138$$ 0 0
$$139$$ 3.65685i 0.310170i 0.987901 + 0.155085i $$0.0495652\pi$$
−0.987901 + 0.155085i $$0.950435\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.82843 0.738270
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 8.82843i − 0.723253i −0.932323 0.361626i $$-0.882222\pi$$
0.932323 0.361626i $$-0.117778\pi$$
$$150$$ 0 0
$$151$$ 15.6569i 1.27414i 0.770807 + 0.637068i $$0.219853\pi$$
−0.770807 + 0.637068i $$0.780147\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.48528 −0.520910
$$156$$ 0 0
$$157$$ −9.75736 −0.778722 −0.389361 0.921085i $$-0.627304\pi$$
−0.389361 + 0.921085i $$0.627304\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.4853i 1.29922i
$$162$$ 0 0
$$163$$ 0.485281i 0.0380102i 0.999819 + 0.0190051i $$0.00604987\pi$$
−0.999819 + 0.0190051i $$0.993950\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 21.7990 1.68686 0.843428 0.537242i $$-0.180534\pi$$
0.843428 + 0.537242i $$0.180534\pi$$
$$168$$ 0 0
$$169$$ −1.34315 −0.103319
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.48528i 0.645124i 0.946548 + 0.322562i $$0.104544\pi$$
−0.946548 + 0.322562i $$0.895456\pi$$
$$174$$ 0 0
$$175$$ 3.41421i 0.258090i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 24.0416 1.79696 0.898478 0.439019i $$-0.144674\pi$$
0.898478 + 0.439019i $$0.144674\pi$$
$$180$$ 0 0
$$181$$ 1.51472 0.112588 0.0562941 0.998414i $$-0.482072\pi$$
0.0562941 + 0.998414i $$0.482072\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 9.07107i 0.666918i
$$186$$ 0 0
$$187$$ − 3.02944i − 0.221534i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.51472 0.254316 0.127158 0.991882i $$-0.459414\pi$$
0.127158 + 0.991882i $$0.459414\pi$$
$$192$$ 0 0
$$193$$ 18.9706 1.36553 0.682765 0.730638i $$-0.260777\pi$$
0.682765 + 0.730638i $$0.260777\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 21.3137i − 1.51854i −0.650776 0.759269i $$-0.725556\pi$$
0.650776 0.759269i $$-0.274444\pi$$
$$198$$ 0 0
$$199$$ − 15.6569i − 1.10988i −0.831889 0.554942i $$-0.812740\pi$$
0.831889 0.554942i $$-0.187260\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.4853 1.43778
$$204$$ 0 0
$$205$$ 11.0711 0.773237
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 26.0000i 1.78991i 0.446153 + 0.894957i $$0.352794\pi$$
−0.446153 + 0.894957i $$0.647206\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.82843 0.465695
$$216$$ 0 0
$$217$$ −22.1421 −1.50311
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 4.00000i − 0.269069i
$$222$$ 0 0
$$223$$ 2.72792i 0.182675i 0.995820 + 0.0913376i $$0.0291142\pi$$
−0.995820 + 0.0913376i $$0.970886\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5.65685 −0.375459 −0.187729 0.982221i $$-0.560113\pi$$
−0.187729 + 0.982221i $$0.560113\pi$$
$$228$$ 0 0
$$229$$ −14.9706 −0.989283 −0.494641 0.869097i $$-0.664701\pi$$
−0.494641 + 0.869097i $$0.664701\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2.14214i 0.140336i 0.997535 + 0.0701680i $$0.0223535\pi$$
−0.997535 + 0.0701680i $$0.977646\pi$$
$$234$$ 0 0
$$235$$ − 5.65685i − 0.369012i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −25.4558 −1.64660 −0.823301 0.567605i $$-0.807870\pi$$
−0.823301 + 0.567605i $$0.807870\pi$$
$$240$$ 0 0
$$241$$ 28.9706 1.86616 0.933079 0.359671i $$-0.117111\pi$$
0.933079 + 0.359671i $$0.117111\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.65685i 0.297516i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −13.8995 −0.877328 −0.438664 0.898651i $$-0.644548\pi$$
−0.438664 + 0.898651i $$0.644548\pi$$
$$252$$ 0 0
$$253$$ 12.4853 0.784943
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 30.9706i 1.92442i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 9.51472 0.586703 0.293351 0.956005i $$-0.405229\pi$$
0.293351 + 0.956005i $$0.405229\pi$$
$$264$$ 0 0
$$265$$ 1.17157 0.0719691
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 7.65685i − 0.466847i −0.972375 0.233423i $$-0.925007\pi$$
0.972375 0.233423i $$-0.0749928\pi$$
$$270$$ 0 0
$$271$$ 22.0000i 1.33640i 0.743980 + 0.668202i $$0.232936\pi$$
−0.743980 + 0.668202i $$0.767064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.58579 0.155929
$$276$$ 0 0
$$277$$ −5.27208 −0.316768 −0.158384 0.987378i $$-0.550628\pi$$
−0.158384 + 0.987378i $$0.550628\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 22.5858i − 1.34736i −0.739025 0.673678i $$-0.764714\pi$$
0.739025 0.673678i $$-0.235286\pi$$
$$282$$ 0 0
$$283$$ − 7.31371i − 0.434755i −0.976088 0.217377i $$-0.930250\pi$$
0.976088 0.217377i $$-0.0697502\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 37.7990 2.23120
$$288$$ 0 0
$$289$$ 15.6274 0.919260
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 29.3137i − 1.71253i −0.516541 0.856263i $$-0.672781\pi$$
0.516541 0.856263i $$-0.327219\pi$$
$$294$$ 0 0
$$295$$ 6.58579i 0.383439i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.4853 0.953368
$$300$$ 0 0
$$301$$ 23.3137 1.34378
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 12.8284i − 0.734554i
$$306$$ 0 0
$$307$$ − 24.0000i − 1.36975i −0.728659 0.684876i $$-0.759856\pi$$
0.728659 0.684876i $$-0.240144\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −26.1421 −1.48238 −0.741192 0.671293i $$-0.765739\pi$$
−0.741192 + 0.671293i $$0.765739\pi$$
$$312$$ 0 0
$$313$$ 7.65685 0.432791 0.216395 0.976306i $$-0.430570\pi$$
0.216395 + 0.976306i $$0.430570\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 16.6274i − 0.933889i −0.884287 0.466944i $$-0.845355\pi$$
0.884287 0.466944i $$-0.154645\pi$$
$$318$$ 0 0
$$319$$ − 15.5147i − 0.868657i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 3.41421 0.189386
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 19.3137i − 1.06480i
$$330$$ 0 0
$$331$$ 28.9706i 1.59237i 0.605056 + 0.796183i $$0.293151\pi$$
−0.605056 + 0.796183i $$0.706849\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ −26.4853 −1.44275 −0.721373 0.692547i $$-0.756488\pi$$
−0.721373 + 0.692547i $$0.756488\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.7696i 0.908122i
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 22.3431 1.19944 0.599721 0.800209i $$-0.295278\pi$$
0.599721 + 0.800209i $$0.295278\pi$$
$$348$$ 0 0
$$349$$ 31.4558 1.68379 0.841896 0.539639i $$-0.181439\pi$$
0.841896 + 0.539639i $$0.181439\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.85786i 0.0988841i 0.998777 + 0.0494421i $$0.0157443\pi$$
−0.998777 + 0.0494421i $$0.984256\pi$$
$$354$$ 0 0
$$355$$ 5.65685i 0.300235i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 18.8284 0.993726 0.496863 0.867829i $$-0.334485\pi$$
0.496863 + 0.867829i $$0.334485\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10.4853i 0.548825i
$$366$$ 0 0
$$367$$ 24.8701i 1.29821i 0.760700 + 0.649103i $$0.224856\pi$$
−0.760700 + 0.649103i $$0.775144\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ 0 0
$$373$$ −10.7279 −0.555471 −0.277735 0.960658i $$-0.589584\pi$$
−0.277735 + 0.960658i $$0.589584\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 20.4853i − 1.05505i
$$378$$ 0 0
$$379$$ − 9.31371i − 0.478413i −0.970969 0.239207i $$-0.923113\pi$$
0.970969 0.239207i $$-0.0768873\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −10.3431 −0.528510 −0.264255 0.964453i $$-0.585126\pi$$
−0.264255 + 0.964453i $$0.585126\pi$$
$$384$$ 0 0
$$385$$ 8.82843 0.449938
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 12.3431i − 0.625822i −0.949782 0.312911i $$-0.898696\pi$$
0.949782 0.312911i $$-0.101304\pi$$
$$390$$ 0 0
$$391$$ − 5.65685i − 0.286079i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −14.4853 −0.728834
$$396$$ 0 0
$$397$$ 14.7279 0.739173 0.369587 0.929196i $$-0.379499\pi$$
0.369587 + 0.929196i $$0.379499\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0.727922i 0.0363507i 0.999835 + 0.0181753i $$0.00578571\pi$$
−0.999835 + 0.0181753i $$0.994214\pi$$
$$402$$ 0 0
$$403$$ 22.1421i 1.10298i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 23.4558 1.16266
$$408$$ 0 0
$$409$$ −38.2843 −1.89304 −0.946518 0.322652i $$-0.895426\pi$$
−0.946518 + 0.322652i $$0.895426\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 22.4853i 1.10643i
$$414$$ 0 0
$$415$$ 9.17157i 0.450215i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 38.3848 1.87522 0.937610 0.347690i $$-0.113034\pi$$
0.937610 + 0.347690i $$0.113034\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 1.17157i − 0.0568296i
$$426$$ 0 0
$$427$$ − 43.7990i − 2.11958i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2.14214 −0.103183 −0.0515915 0.998668i $$-0.516429\pi$$
−0.0515915 + 0.998668i $$0.516429\pi$$
$$432$$ 0 0
$$433$$ −14.4853 −0.696118 −0.348059 0.937473i $$-0.613159\pi$$
−0.348059 + 0.937473i $$0.613159\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 18.2843i 0.872661i 0.899787 + 0.436330i $$0.143722\pi$$
−0.899787 + 0.436330i $$0.856278\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 26.6274 1.26511 0.632553 0.774517i $$-0.282007\pi$$
0.632553 + 0.774517i $$0.282007\pi$$
$$444$$ 0 0
$$445$$ −4.24264 −0.201120
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.5858i 0.877117i 0.898703 + 0.438559i $$0.144511\pi$$
−0.898703 + 0.438559i $$0.855489\pi$$
$$450$$ 0 0
$$451$$ − 28.6274i − 1.34801i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 11.6569 0.546482
$$456$$ 0 0
$$457$$ 2.48528 0.116257 0.0581283 0.998309i $$-0.481487\pi$$
0.0581283 + 0.998309i $$0.481487\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 24.8284i 1.15638i 0.815904 + 0.578188i $$0.196240\pi$$
−0.815904 + 0.578188i $$0.803760\pi$$
$$462$$ 0 0
$$463$$ 17.0711i 0.793360i 0.917957 + 0.396680i $$0.129838\pi$$
−0.917957 + 0.396680i $$0.870162\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −31.1127 −1.43972 −0.719862 0.694117i $$-0.755795\pi$$
−0.719862 + 0.694117i $$0.755795\pi$$
$$468$$ 0 0
$$469$$ −27.3137 −1.26123
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 17.6569i − 0.811863i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −27.3137 −1.24800 −0.623998 0.781426i $$-0.714493\pi$$
−0.623998 + 0.781426i $$0.714493\pi$$
$$480$$ 0 0
$$481$$ 30.9706 1.41214
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 2.48528i − 0.112851i
$$486$$ 0 0
$$487$$ 10.7279i 0.486129i 0.970010 + 0.243064i $$0.0781526\pi$$
−0.970010 + 0.243064i $$0.921847\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 31.0711 1.40222 0.701109 0.713054i $$-0.252689\pi$$
0.701109 + 0.713054i $$0.252689\pi$$
$$492$$ 0 0
$$493$$ −7.02944 −0.316590
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 19.3137i 0.866338i
$$498$$ 0 0
$$499$$ − 18.6274i − 0.833878i −0.908935 0.416939i $$-0.863103\pi$$
0.908935 0.416939i $$-0.136897\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −36.2843 −1.61784 −0.808918 0.587922i $$-0.799946\pi$$
−0.808918 + 0.587922i $$0.799946\pi$$
$$504$$ 0 0
$$505$$ 4.82843 0.214862
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 8.14214i − 0.360894i −0.983585 0.180447i $$-0.942246\pi$$
0.983585 0.180447i $$-0.0577544\pi$$
$$510$$ 0 0
$$511$$ 35.7990i 1.58365i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −17.5563 −0.773625
$$516$$ 0 0
$$517$$ −14.6274 −0.643313
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 0.727922i − 0.0318908i −0.999873 0.0159454i $$-0.994924\pi$$
0.999873 0.0159454i $$-0.00507580\pi$$
$$522$$ 0 0
$$523$$ 7.51472i 0.328596i 0.986411 + 0.164298i $$0.0525358\pi$$
−0.986411 + 0.164298i $$0.947464\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.59798 0.330973
$$528$$ 0 0
$$529$$ 0.313708 0.0136395
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 37.7990i − 1.63726i
$$534$$ 0 0
$$535$$ 14.8284i 0.641089i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0416 0.518670
$$540$$ 0 0
$$541$$ 28.8284 1.23943 0.619715 0.784827i $$-0.287248\pi$$
0.619715 + 0.784827i $$0.287248\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7.17157i 0.307196i
$$546$$ 0 0
$$547$$ − 16.4853i − 0.704860i −0.935838 0.352430i $$-0.885356\pi$$
0.935838 0.352430i $$-0.114644\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −49.4558 −2.10308
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 20.4853i − 0.867989i −0.900915 0.433995i $$-0.857104\pi$$
0.900915 0.433995i $$-0.142896\pi$$
$$558$$ 0 0
$$559$$ − 23.3137i − 0.986065i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.34315 0.267332 0.133666 0.991026i $$-0.457325\pi$$
0.133666 + 0.991026i $$0.457325\pi$$
$$564$$ 0 0
$$565$$ −1.65685 −0.0697044
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 9.21320i 0.386238i 0.981175 + 0.193119i $$0.0618603\pi$$
−0.981175 + 0.193119i $$0.938140\pi$$
$$570$$ 0 0
$$571$$ − 4.97056i − 0.208012i −0.994577 0.104006i $$-0.966834\pi$$
0.994577 0.104006i $$-0.0331660\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.82843 0.201359
$$576$$ 0 0
$$577$$ 15.4558 0.643435 0.321718 0.946836i $$-0.395740\pi$$
0.321718 + 0.946836i $$0.395740\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 31.3137i 1.29911i
$$582$$ 0 0
$$583$$ − 3.02944i − 0.125466i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −26.1421 −1.07900 −0.539501 0.841985i $$-0.681387\pi$$
−0.539501 + 0.841985i $$0.681387\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 10.3431i − 0.424742i −0.977189 0.212371i $$-0.931881\pi$$
0.977189 0.212371i $$-0.0681185\pi$$
$$594$$ 0 0
$$595$$ − 4.00000i − 0.163984i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −42.1421 −1.72188 −0.860940 0.508706i $$-0.830124\pi$$
−0.860940 + 0.508706i $$0.830124\pi$$
$$600$$ 0 0
$$601$$ 16.6863 0.680648 0.340324 0.940308i $$-0.389463\pi$$
0.340324 + 0.940308i $$0.389463\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 4.31371i 0.175377i
$$606$$ 0 0
$$607$$ − 3.21320i − 0.130420i −0.997872 0.0652100i $$-0.979228\pi$$
0.997872 0.0652100i $$-0.0207717\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −19.3137 −0.781349
$$612$$ 0 0
$$613$$ −3.21320 −0.129780 −0.0648900 0.997892i $$-0.520670\pi$$
−0.0648900 + 0.997892i $$0.520670\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.8284i 1.08007i 0.841642 + 0.540036i $$0.181589\pi$$
−0.841642 + 0.540036i $$0.818411\pi$$
$$618$$ 0 0
$$619$$ − 22.9706i − 0.923265i −0.887071 0.461632i $$-0.847264\pi$$
0.887071 0.461632i $$-0.152736\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −14.4853 −0.580341
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 10.6274i − 0.423743i
$$630$$ 0 0
$$631$$ − 7.17157i − 0.285496i −0.989759 0.142748i $$-0.954406\pi$$
0.989759 0.142748i $$-0.0455938\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −10.7279 −0.425725
$$636$$ 0 0
$$637$$ 15.8995 0.629961
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 43.5563i 1.72037i 0.509980 + 0.860186i $$0.329653\pi$$
−0.509980 + 0.860186i $$0.670347\pi$$
$$642$$ 0 0
$$643$$ − 35.1127i − 1.38471i −0.721557 0.692355i $$-0.756573\pi$$
0.721557 0.692355i $$-0.243427\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.6863 −0.498750 −0.249375 0.968407i $$-0.580225\pi$$
−0.249375 + 0.968407i $$0.580225\pi$$
$$648$$ 0 0
$$649$$ 17.0294 0.668464
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19.6569i 0.769232i 0.923077 + 0.384616i $$0.125666\pi$$
−0.923077 + 0.384616i $$0.874334\pi$$
$$654$$ 0 0
$$655$$ 12.7279i 0.497321i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.58579 −0.256546 −0.128273 0.991739i $$-0.540943\pi$$
−0.128273 + 0.991739i $$0.540943\pi$$
$$660$$ 0 0
$$661$$ −40.4264 −1.57240 −0.786202 0.617969i $$-0.787956\pi$$
−0.786202 + 0.617969i $$0.787956\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 28.9706i − 1.12174i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −33.1716 −1.28057
$$672$$ 0 0
$$673$$ 51.4558 1.98348 0.991739 0.128276i $$-0.0409443\pi$$
0.991739 + 0.128276i $$0.0409443\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 6.68629i − 0.256975i −0.991711 0.128488i $$-0.958988\pi$$
0.991711 0.128488i $$-0.0410122\pi$$
$$678$$ 0 0
$$679$$ − 8.48528i − 0.325635i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −10.8284 −0.414338 −0.207169 0.978305i $$-0.566425\pi$$
−0.207169 + 0.978305i $$0.566425\pi$$
$$684$$ 0 0
$$685$$ −19.3137 −0.737939
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 4.00000i − 0.152388i
$$690$$ 0 0
$$691$$ − 8.00000i − 0.304334i −0.988355 0.152167i $$-0.951375\pi$$
0.988355 0.152167i $$-0.0486252\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3.65685 0.138712
$$696$$ 0 0
$$697$$ −12.9706 −0.491295
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 9.51472i 0.359366i 0.983725 + 0.179683i $$0.0575072\pi$$
−0.983725 + 0.179683i $$0.942493\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.4853 0.619993
$$708$$ 0 0
$$709$$ 6.97056 0.261785 0.130892 0.991397i $$-0.458216\pi$$
0.130892 + 0.991397i $$0.458216\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 31.3137i 1.17271i
$$714$$ 0 0
$$715$$ − 8.82843i − 0.330164i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 10.8284 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$720$$ 0 0
$$721$$ −59.9411 −2.23232
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 6.00000i − 0.222834i
$$726$$ 0 0
$$727$$ 26.9289i 0.998739i 0.866389 + 0.499369i $$0.166435\pi$$
−0.866389 + 0.499369i $$0.833565\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −12.1005 −0.446942 −0.223471 0.974711i $$-0.571739\pi$$
−0.223471 + 0.974711i $$0.571739\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 20.6863i 0.761989i
$$738$$ 0 0
$$739$$ 22.3431i 0.821906i 0.911657 + 0.410953i $$0.134804\pi$$
−0.911657 + 0.410953i $$0.865196\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ −8.82843 −0.323449
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 50.6274i 1.84989i
$$750$$ 0 0
$$751$$ − 28.8284i − 1.05196i −0.850496 0.525982i $$-0.823698\pi$$
0.850496 0.525982i $$-0.176302\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 15.6569 0.569811
$$756$$ 0 0
$$757$$ −52.8701 −1.92159 −0.960797 0.277251i $$-0.910577\pi$$
−0.960797 + 0.277251i $$0.910577\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 30.8701i − 1.11904i −0.828817 0.559519i $$-0.810986\pi$$
0.828817 0.559519i $$-0.189014\pi$$
$$762$$ 0 0
$$763$$ 24.4853i 0.886427i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 22.4853 0.811896
$$768$$ 0 0
$$769$$ −2.34315 −0.0844960 −0.0422480 0.999107i $$-0.513452\pi$$
−0.0422480 + 0.999107i $$0.513452\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 41.3137i 1.48595i 0.669319 + 0.742975i $$0.266586\pi$$
−0.669319 + 0.742975i $$0.733414\pi$$
$$774$$ 0 0
$$775$$ 6.48528i 0.232958i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 14.6274 0.523410
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9.75736i 0.348255i
$$786$$ 0 0
$$787$$ − 42.1421i − 1.50220i −0.660186 0.751102i $$-0.729522\pi$$
0.660186 0.751102i $$-0.270478\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5.65685 −0.201135
$$792$$ 0 0
$$793$$ −43.7990 −1.55535
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.8284i 0.666937i 0.942761 + 0.333469i $$0.108219\pi$$
−0.942761 + 0.333469i $$0.891781\pi$$
$$798$$ 0 0
$$799$$ 6.62742i 0.234461i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 27.1127 0.956786
$$804$$ 0 0
$$805$$ 16.4853 0.581030
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 36.0416i − 1.26716i −0.773679 0.633578i $$-0.781586\pi$$
0.773679 0.633578i $$-0.218414\pi$$
$$810$$ 0 0
$$811$$ 3.37258i 0.118427i 0.998245 + 0.0592137i $$0.0188593\pi$$
−0.998245 + 0.0592137i $$0.981141\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0.485281 0.0169987
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 29.3137i − 1.02306i −0.859267 0.511528i $$-0.829080\pi$$
0.859267 0.511528i $$-0.170920\pi$$
$$822$$ 0 0
$$823$$ 6.24264i 0.217605i 0.994063 + 0.108802i $$0.0347016\pi$$
−0.994063 + 0.108802i $$0.965298\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −42.6274 −1.48230 −0.741150 0.671339i $$-0.765719\pi$$
−0.741150 + 0.671339i $$0.765719\pi$$
$$828$$ 0 0
$$829$$ −19.4558 −0.675729 −0.337865 0.941195i $$-0.609705\pi$$
−0.337865 + 0.941195i $$0.609705\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 5.45584i − 0.189034i
$$834$$ 0 0
$$835$$ − 21.7990i − 0.754385i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −42.4264 −1.46472 −0.732361 0.680916i $$-0.761582\pi$$
−0.732361 + 0.680916i $$0.761582\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.34315i 0.0462056i
$$846$$ 0 0
$$847$$ 14.7279i 0.506057i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 43.7990 1.50141
$$852$$ 0 0
$$853$$ 21.7574 0.744958 0.372479 0.928041i $$-0.378508\pi$$
0.372479 + 0.928041i $$0.378508\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 35.7990i − 1.22287i −0.791295 0.611435i $$-0.790593\pi$$
0.791295 0.611435i $$-0.209407\pi$$
$$858$$ 0 0
$$859$$ 26.0000i 0.887109i 0.896248 + 0.443554i $$0.146283\pi$$
−0.896248 + 0.443554i $$0.853717\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 6.20101 0.211085 0.105542 0.994415i $$-0.466342\pi$$
0.105542 + 0.994415i $$0.466342\pi$$
$$864$$ 0 0
$$865$$ 8.48528 0.288508
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 37.4558i 1.27060i
$$870$$ 0 0
$$871$$ 27.3137i 0.925490i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.41421 0.115421
$$876$$ 0 0
$$877$$ −1.55635 −0.0525542 −0.0262771 0.999655i $$-0.508365\pi$$
−0.0262771 + 0.999655i $$0.508365\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 35.3553i 1.19115i 0.803299 + 0.595576i $$0.203076\pi$$
−0.803299 + 0.595576i $$0.796924\pi$$
$$882$$ 0 0
$$883$$ − 42.4264i − 1.42776i −0.700267 0.713881i $$-0.746936\pi$$
0.700267 0.713881i $$-0.253064\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.8284 0.430736 0.215368 0.976533i $$-0.430905\pi$$
0.215368 + 0.976533i $$0.430905\pi$$
$$888$$ 0 0
$$889$$ −36.6274 −1.22844
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ − 24.0416i − 0.803623i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 38.9117 1.29778
$$900$$ 0 0
$$901$$ −1.37258 −0.0457274
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 1.51472i − 0.0503510i
$$906$$ 0 0
$$907$$ − 1.85786i − 0.0616894i −0.999524 0.0308447i $$-0.990180\pi$$
0.999524 0.0308447i $$-0.00981973\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18.3431 −0.607736 −0.303868 0.952714i $$-0.598278\pi$$
−0.303868 + 0.952714i $$0.598278\pi$$
$$912$$ 0 0
$$913$$ 23.7157 0.784876
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 43.4558i 1.43504i
$$918$$ 0 0
$$919$$ − 12.8284i − 0.423171i −0.977360 0.211585i $$-0.932137\pi$$
0.977360 0.211585i $$-0.0678626\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 19.3137 0.635718
$$924$$ 0 0
$$925$$ 9.07107 0.298255
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.8995i 1.50591i 0.658070 + 0.752957i $$0.271373\pi$$
−0.658070 + 0.752957i $$0.728627\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −3.02944 −0.0990732
$$936$$ 0 0
$$937$$ −14.9706 −0.489067 −0.244533 0.969641i $$-0.578635\pi$$
−0.244533 + 0.969641i $$0.578635\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 42.2843i − 1.37843i −0.724558 0.689214i $$-0.757956\pi$$
0.724558 0.689214i $$-0.242044\pi$$
$$942$$ 0 0
$$943$$ − 53.4558i − 1.74076i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −51.5980 −1.67671 −0.838355 0.545125i $$-0.816482\pi$$
−0.838355 + 0.545125i $$0.816482\pi$$
$$948$$ 0 0
$$949$$ 35.7990 1.16208
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 45.2548i 1.46595i 0.680257 + 0.732974i $$0.261868\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ − 3.51472i − 0.113734i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −65.9411 −2.12935
$$960$$ 0 0
$$961$$ −11.0589 −0.356738
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 18.9706i − 0.610684i
$$966$$ 0 0
$$967$$ 26.7279i 0.859512i 0.902945 + 0.429756i $$0.141400\pi$$
−0.902945 + 0.429756i $$0.858600\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.0711 −0.868752 −0.434376 0.900732i $$-0.643031\pi$$
−0.434376 + 0.900732i $$0.643031\pi$$
$$972$$ 0 0
$$973$$ 12.4853 0.400260
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 37.1716i − 1.18922i −0.804013 0.594612i $$-0.797306\pi$$
0.804013 0.594612i $$-0.202694\pi$$
$$978$$ 0 0
$$979$$ 10.9706i 0.350621i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 39.5980 1.26298 0.631490 0.775384i $$-0.282444\pi$$
0.631490 + 0.775384i $$0.282444\pi$$
$$984$$ 0 0
$$985$$ −21.3137 −0.679111
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 32.9706i − 1.04840i
$$990$$ 0 0
$$991$$ − 16.3431i − 0.519157i −0.965722 0.259579i $$-0.916416\pi$$
0.965722 0.259579i $$-0.0835837\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −15.6569 −0.496356
$$996$$ 0 0
$$997$$ 5.27208 0.166968 0.0834842 0.996509i $$-0.473395\pi$$
0.0834842 + 0.996509i $$0.473395\pi$$
$$998$$ 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 1440.2.h.a.1151.1 4
3.2 odd 2 1440.2.h.d.1151.3 yes 4
4.3 odd 2 1440.2.h.d.1151.2 yes 4
5.2 odd 4 7200.2.o.j.7199.4 4
5.3 odd 4 7200.2.o.b.7199.1 4
5.4 even 2 7200.2.h.c.1151.4 4
8.3 odd 2 2880.2.h.a.1151.4 4
8.5 even 2 2880.2.h.d.1151.3 4
12.11 even 2 inner 1440.2.h.a.1151.4 yes 4
15.2 even 4 7200.2.o.m.7199.4 4
15.8 even 4 7200.2.o.e.7199.1 4
15.14 odd 2 7200.2.h.i.1151.4 4
20.3 even 4 7200.2.o.m.7199.3 4
20.7 even 4 7200.2.o.e.7199.2 4
20.19 odd 2 7200.2.h.i.1151.1 4
24.5 odd 2 2880.2.h.a.1151.1 4
24.11 even 2 2880.2.h.d.1151.2 4
60.23 odd 4 7200.2.o.j.7199.3 4
60.47 odd 4 7200.2.o.b.7199.2 4
60.59 even 2 7200.2.h.c.1151.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.a.1151.1 4 1.1 even 1 trivial
1440.2.h.a.1151.4 yes 4 12.11 even 2 inner
1440.2.h.d.1151.2 yes 4 4.3 odd 2
1440.2.h.d.1151.3 yes 4 3.2 odd 2
2880.2.h.a.1151.1 4 24.5 odd 2
2880.2.h.a.1151.4 4 8.3 odd 2
2880.2.h.d.1151.2 4 24.11 even 2
2880.2.h.d.1151.3 4 8.5 even 2
7200.2.h.c.1151.1 4 60.59 even 2
7200.2.h.c.1151.4 4 5.4 even 2
7200.2.h.i.1151.1 4 20.19 odd 2
7200.2.h.i.1151.4 4 15.14 odd 2
7200.2.o.b.7199.1 4 5.3 odd 4
7200.2.o.b.7199.2 4 60.47 odd 4
7200.2.o.e.7199.1 4 15.8 even 4
7200.2.o.e.7199.2 4 20.7 even 4
7200.2.o.j.7199.3 4 60.23 odd 4
7200.2.o.j.7199.4 4 5.2 odd 4
7200.2.o.m.7199.3 4 20.3 even 4
7200.2.o.m.7199.4 4 15.2 even 4