# Properties

 Label 1216.2.c.f.609.3 Level $1216$ Weight $2$ Character 1216.609 Analytic conductor $9.710$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 609.3 Root $$-1.65831 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.609 Dual form 1216.2.c.f.609.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{3} -3.31662i q^{5} -3.31662 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{3} -3.31662i q^{5} -3.31662 q^{7} -1.00000 q^{9} +5.00000i q^{11} +6.63325 q^{15} +5.00000 q^{17} -1.00000i q^{19} -6.63325i q^{21} -6.63325 q^{23} -6.00000 q^{25} +4.00000i q^{27} +6.63325i q^{29} -10.0000 q^{33} +11.0000i q^{35} +6.63325i q^{37} -6.00000 q^{41} -1.00000i q^{43} +3.31662i q^{45} -9.94987 q^{47} +4.00000 q^{49} +10.0000i q^{51} +13.2665i q^{53} +16.5831 q^{55} +2.00000 q^{57} -6.00000i q^{59} +9.94987i q^{61} +3.31662 q^{63} +8.00000i q^{67} -13.2665i q^{69} +6.63325 q^{71} -9.00000 q^{73} -12.0000i q^{75} -16.5831i q^{77} +13.2665 q^{79} -11.0000 q^{81} -4.00000i q^{83} -16.5831i q^{85} -13.2665 q^{87} -4.00000 q^{89} -3.31662 q^{95} -12.0000 q^{97} -5.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 20q^{17} - 24q^{25} - 40q^{33} - 24q^{41} + 16q^{49} + 8q^{57} - 36q^{73} - 44q^{81} - 16q^{89} - 48q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ − 3.31662i − 1.48324i −0.670820 0.741620i $$-0.734058\pi$$
0.670820 0.741620i $$-0.265942\pi$$
$$6$$ 0 0
$$7$$ −3.31662 −1.25357 −0.626783 0.779194i $$-0.715629\pi$$
−0.626783 + 0.779194i $$0.715629\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 5.00000i 1.50756i 0.657129 + 0.753778i $$0.271771\pi$$
−0.657129 + 0.753778i $$0.728229\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 6.63325 1.71270
$$16$$ 0 0
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 0 0
$$19$$ − 1.00000i − 0.229416i
$$20$$ 0 0
$$21$$ − 6.63325i − 1.44749i
$$22$$ 0 0
$$23$$ −6.63325 −1.38313 −0.691564 0.722315i $$-0.743078\pi$$
−0.691564 + 0.722315i $$0.743078\pi$$
$$24$$ 0 0
$$25$$ −6.00000 −1.20000
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ 6.63325i 1.23176i 0.787839 + 0.615882i $$0.211200\pi$$
−0.787839 + 0.615882i $$0.788800\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −10.0000 −1.74078
$$34$$ 0 0
$$35$$ 11.0000i 1.85934i
$$36$$ 0 0
$$37$$ 6.63325i 1.09050i 0.838274 + 0.545250i $$0.183565\pi$$
−0.838274 + 0.545250i $$0.816435\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 0 0
$$45$$ 3.31662i 0.494413i
$$46$$ 0 0
$$47$$ −9.94987 −1.45134 −0.725669 0.688044i $$-0.758470\pi$$
−0.725669 + 0.688044i $$0.758470\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ 10.0000i 1.40028i
$$52$$ 0 0
$$53$$ 13.2665i 1.82229i 0.412082 + 0.911147i $$0.364802\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 16.5831 2.23607
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ − 6.00000i − 0.781133i −0.920575 0.390567i $$-0.872279\pi$$
0.920575 0.390567i $$-0.127721\pi$$
$$60$$ 0 0
$$61$$ 9.94987i 1.27395i 0.770884 + 0.636975i $$0.219815\pi$$
−0.770884 + 0.636975i $$0.780185\pi$$
$$62$$ 0 0
$$63$$ 3.31662 0.417855
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ − 13.2665i − 1.59710i
$$70$$ 0 0
$$71$$ 6.63325 0.787222 0.393611 0.919277i $$-0.371226\pi$$
0.393611 + 0.919277i $$0.371226\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ − 12.0000i − 1.38564i
$$76$$ 0 0
$$77$$ − 16.5831i − 1.88982i
$$78$$ 0 0
$$79$$ 13.2665 1.49260 0.746299 0.665611i $$-0.231829\pi$$
0.746299 + 0.665611i $$0.231829\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ − 16.5831i − 1.79869i
$$86$$ 0 0
$$87$$ −13.2665 −1.42232
$$88$$ 0 0
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.31662 −0.340279
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 0 0
$$99$$ − 5.00000i − 0.502519i
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 6.63325 0.653594 0.326797 0.945095i $$-0.394031\pi$$
0.326797 + 0.945095i $$0.394031\pi$$
$$104$$ 0 0
$$105$$ −22.0000 −2.14698
$$106$$ 0 0
$$107$$ − 6.00000i − 0.580042i −0.957020 0.290021i $$-0.906338\pi$$
0.957020 0.290021i $$-0.0936623\pi$$
$$108$$ 0 0
$$109$$ 13.2665i 1.27070i 0.772224 + 0.635350i $$0.219144\pi$$
−0.772224 + 0.635350i $$0.780856\pi$$
$$110$$ 0 0
$$111$$ −13.2665 −1.25920
$$112$$ 0 0
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ 22.0000i 2.05151i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −16.5831 −1.52017
$$120$$ 0 0
$$121$$ −14.0000 −1.27273
$$122$$ 0 0
$$123$$ − 12.0000i − 1.08200i
$$124$$ 0 0
$$125$$ 3.31662i 0.296648i
$$126$$ 0 0
$$127$$ −6.63325 −0.588606 −0.294303 0.955712i $$-0.595087\pi$$
−0.294303 + 0.955712i $$0.595087\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ − 15.0000i − 1.31056i −0.755388 0.655278i $$-0.772551\pi$$
0.755388 0.655278i $$-0.227449\pi$$
$$132$$ 0 0
$$133$$ 3.31662i 0.287588i
$$134$$ 0 0
$$135$$ 13.2665 1.14180
$$136$$ 0 0
$$137$$ 5.00000 0.427179 0.213589 0.976924i $$-0.431485\pi$$
0.213589 + 0.976924i $$0.431485\pi$$
$$138$$ 0 0
$$139$$ 11.0000i 0.933008i 0.884519 + 0.466504i $$0.154487\pi$$
−0.884519 + 0.466504i $$0.845513\pi$$
$$140$$ 0 0
$$141$$ − 19.8997i − 1.67586i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 22.0000 1.82700
$$146$$ 0 0
$$147$$ 8.00000i 0.659829i
$$148$$ 0 0
$$149$$ − 3.31662i − 0.271708i −0.990729 0.135854i $$-0.956622\pi$$
0.990729 0.135854i $$-0.0433779\pi$$
$$150$$ 0 0
$$151$$ −6.63325 −0.539806 −0.269903 0.962887i $$-0.586992\pi$$
−0.269903 + 0.962887i $$0.586992\pi$$
$$152$$ 0 0
$$153$$ −5.00000 −0.404226
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ −26.5330 −2.10420
$$160$$ 0 0
$$161$$ 22.0000 1.73384
$$162$$ 0 0
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ 0 0
$$165$$ 33.1662i 2.58199i
$$166$$ 0 0
$$167$$ 6.63325 0.513296 0.256648 0.966505i $$-0.417382\pi$$
0.256648 + 0.966505i $$0.417382\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 1.00000i 0.0764719i
$$172$$ 0 0
$$173$$ − 19.8997i − 1.51295i −0.654023 0.756475i $$-0.726920\pi$$
0.654023 0.756475i $$-0.273080\pi$$
$$174$$ 0 0
$$175$$ 19.8997 1.50428
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 18.0000i 1.34538i 0.739923 + 0.672692i $$0.234862\pi$$
−0.739923 + 0.672692i $$0.765138\pi$$
$$180$$ 0 0
$$181$$ − 19.8997i − 1.47914i −0.673081 0.739568i $$-0.735030\pi$$
0.673081 0.739568i $$-0.264970\pi$$
$$182$$ 0 0
$$183$$ −19.8997 −1.47103
$$184$$ 0 0
$$185$$ 22.0000 1.61747
$$186$$ 0 0
$$187$$ 25.0000i 1.82818i
$$188$$ 0 0
$$189$$ − 13.2665i − 0.964996i
$$190$$ 0 0
$$191$$ 16.5831 1.19991 0.599956 0.800033i $$-0.295185\pi$$
0.599956 + 0.800033i $$0.295185\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.5330i 1.89040i 0.326495 + 0.945199i $$0.394132\pi$$
−0.326495 + 0.945199i $$0.605868\pi$$
$$198$$ 0 0
$$199$$ 16.5831 1.17555 0.587773 0.809026i $$-0.300005\pi$$
0.587773 + 0.809026i $$0.300005\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 0 0
$$203$$ − 22.0000i − 1.54410i
$$204$$ 0 0
$$205$$ 19.8997i 1.38986i
$$206$$ 0 0
$$207$$ 6.63325 0.461043
$$208$$ 0 0
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ − 14.0000i − 0.963800i −0.876226 0.481900i $$-0.839947\pi$$
0.876226 0.481900i $$-0.160053\pi$$
$$212$$ 0 0
$$213$$ 13.2665i 0.909006i
$$214$$ 0 0
$$215$$ −3.31662 −0.226192
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 18.0000i − 1.21633i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −19.8997 −1.33259 −0.666293 0.745690i $$-0.732120\pi$$
−0.666293 + 0.745690i $$0.732120\pi$$
$$224$$ 0 0
$$225$$ 6.00000 0.400000
$$226$$ 0 0
$$227$$ − 8.00000i − 0.530979i −0.964114 0.265489i $$-0.914466\pi$$
0.964114 0.265489i $$-0.0855335\pi$$
$$228$$ 0 0
$$229$$ 23.2164i 1.53418i 0.641539 + 0.767091i $$0.278296\pi$$
−0.641539 + 0.767091i $$0.721704\pi$$
$$230$$ 0 0
$$231$$ 33.1662 2.18218
$$232$$ 0 0
$$233$$ 3.00000 0.196537 0.0982683 0.995160i $$-0.468670\pi$$
0.0982683 + 0.995160i $$0.468670\pi$$
$$234$$ 0 0
$$235$$ 33.0000i 2.15268i
$$236$$ 0 0
$$237$$ 26.5330i 1.72350i
$$238$$ 0 0
$$239$$ −3.31662 −0.214535 −0.107267 0.994230i $$-0.534210\pi$$
−0.107267 + 0.994230i $$0.534210\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ − 13.2665i − 0.847566i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ 5.00000i 0.315597i 0.987471 + 0.157799i $$0.0504397\pi$$
−0.987471 + 0.157799i $$0.949560\pi$$
$$252$$ 0 0
$$253$$ − 33.1662i − 2.08514i
$$254$$ 0 0
$$255$$ 33.1662 2.07695
$$256$$ 0 0
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ − 22.0000i − 1.36701i
$$260$$ 0 0
$$261$$ − 6.63325i − 0.410588i
$$262$$ 0 0
$$263$$ 3.31662 0.204512 0.102256 0.994758i $$-0.467394\pi$$
0.102256 + 0.994758i $$0.467394\pi$$
$$264$$ 0 0
$$265$$ 44.0000 2.70290
$$266$$ 0 0
$$267$$ − 8.00000i − 0.489592i
$$268$$ 0 0
$$269$$ − 13.2665i − 0.808873i −0.914566 0.404436i $$-0.867468\pi$$
0.914566 0.404436i $$-0.132532\pi$$
$$270$$ 0 0
$$271$$ −19.8997 −1.20882 −0.604412 0.796672i $$-0.706592\pi$$
−0.604412 + 0.796672i $$0.706592\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 30.0000i − 1.80907i
$$276$$ 0 0
$$277$$ 3.31662i 0.199277i 0.995024 + 0.0996383i $$0.0317686\pi$$
−0.995024 + 0.0996383i $$0.968231\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 11.0000i 0.653882i 0.945045 + 0.326941i $$0.106018\pi$$
−0.945045 + 0.326941i $$0.893982\pi$$
$$284$$ 0 0
$$285$$ − 6.63325i − 0.392920i
$$286$$ 0 0
$$287$$ 19.8997 1.17465
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ − 24.0000i − 1.40690i
$$292$$ 0 0
$$293$$ − 26.5330i − 1.55007i −0.631916 0.775037i $$-0.717731\pi$$
0.631916 0.775037i $$-0.282269\pi$$
$$294$$ 0 0
$$295$$ −19.8997 −1.15861
$$296$$ 0 0
$$297$$ −20.0000 −1.16052
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 3.31662i 0.191167i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 33.0000 1.88957
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 13.2665i 0.754705i
$$310$$ 0 0
$$311$$ 16.5831 0.940343 0.470171 0.882575i $$-0.344192\pi$$
0.470171 + 0.882575i $$0.344192\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 0 0
$$315$$ − 11.0000i − 0.619780i
$$316$$ 0 0
$$317$$ 6.63325i 0.372560i 0.982497 + 0.186280i $$0.0596432\pi$$
−0.982497 + 0.186280i $$0.940357\pi$$
$$318$$ 0 0
$$319$$ −33.1662 −1.85695
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 5.00000i − 0.278207i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −26.5330 −1.46728
$$328$$ 0 0
$$329$$ 33.0000 1.81935
$$330$$ 0 0
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ 0 0
$$333$$ − 6.63325i − 0.363500i
$$334$$ 0 0
$$335$$ 26.5330 1.44965
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ 36.0000i 1.95525i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 9.94987 0.537243
$$344$$ 0 0
$$345$$ −44.0000 −2.36888
$$346$$ 0 0
$$347$$ 27.0000i 1.44944i 0.689046 + 0.724718i $$0.258030\pi$$
−0.689046 + 0.724718i $$0.741970\pi$$
$$348$$ 0 0
$$349$$ 9.94987i 0.532605i 0.963890 + 0.266302i $$0.0858019\pi$$
−0.963890 + 0.266302i $$0.914198\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ − 22.0000i − 1.16764i
$$356$$ 0 0
$$357$$ − 33.1662i − 1.75534i
$$358$$ 0 0
$$359$$ −23.2164 −1.22531 −0.612657 0.790349i $$-0.709899\pi$$
−0.612657 + 0.790349i $$0.709899\pi$$
$$360$$ 0 0
$$361$$ −1.00000 −0.0526316
$$362$$ 0 0
$$363$$ − 28.0000i − 1.46962i
$$364$$ 0 0
$$365$$ 29.8496i 1.56240i
$$366$$ 0 0
$$367$$ −6.63325 −0.346253 −0.173126 0.984900i $$-0.555387\pi$$
−0.173126 + 0.984900i $$0.555387\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ − 44.0000i − 2.28437i
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ −6.63325 −0.342540
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 6.00000i 0.308199i 0.988055 + 0.154100i $$0.0492477\pi$$
−0.988055 + 0.154100i $$0.950752\pi$$
$$380$$ 0 0
$$381$$ − 13.2665i − 0.679663i
$$382$$ 0 0
$$383$$ −13.2665 −0.677886 −0.338943 0.940807i $$-0.610069\pi$$
−0.338943 + 0.940807i $$0.610069\pi$$
$$384$$ 0 0
$$385$$ −55.0000 −2.80306
$$386$$ 0 0
$$387$$ 1.00000i 0.0508329i
$$388$$ 0 0
$$389$$ − 16.5831i − 0.840798i −0.907339 0.420399i $$-0.861890\pi$$
0.907339 0.420399i $$-0.138110\pi$$
$$390$$ 0 0
$$391$$ −33.1662 −1.67729
$$392$$ 0 0
$$393$$ 30.0000 1.51330
$$394$$ 0 0
$$395$$ − 44.0000i − 2.21388i
$$396$$ 0 0
$$397$$ − 9.94987i − 0.499370i −0.968327 0.249685i $$-0.919673\pi$$
0.968327 0.249685i $$-0.0803271\pi$$
$$398$$ 0 0
$$399$$ −6.63325 −0.332078
$$400$$ 0 0
$$401$$ 16.0000 0.799002 0.399501 0.916733i $$-0.369183\pi$$
0.399501 + 0.916733i $$0.369183\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 36.4829i 1.81285i
$$406$$ 0 0
$$407$$ −33.1662 −1.64399
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 10.0000i 0.493264i
$$412$$ 0 0
$$413$$ 19.8997i 0.979203i
$$414$$ 0 0
$$415$$ −13.2665 −0.651227
$$416$$ 0 0
$$417$$ −22.0000 −1.07734
$$418$$ 0 0
$$419$$ − 4.00000i − 0.195413i −0.995215 0.0977064i $$-0.968849\pi$$
0.995215 0.0977064i $$-0.0311506\pi$$
$$420$$ 0 0
$$421$$ − 13.2665i − 0.646570i −0.946302 0.323285i $$-0.895213\pi$$
0.946302 0.323285i $$-0.104787\pi$$
$$422$$ 0 0
$$423$$ 9.94987 0.483779
$$424$$ 0 0
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ − 33.0000i − 1.59698i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 26.5330 1.27805 0.639025 0.769186i $$-0.279338\pi$$
0.639025 + 0.769186i $$0.279338\pi$$
$$432$$ 0 0
$$433$$ −38.0000 −1.82616 −0.913082 0.407777i $$-0.866304\pi$$
−0.913082 + 0.407777i $$0.866304\pi$$
$$434$$ 0 0
$$435$$ 44.0000i 2.10964i
$$436$$ 0 0
$$437$$ 6.63325i 0.317311i
$$438$$ 0 0
$$439$$ 19.8997 0.949763 0.474882 0.880050i $$-0.342491\pi$$
0.474882 + 0.880050i $$0.342491\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ − 13.0000i − 0.617649i −0.951119 0.308824i $$-0.900064\pi$$
0.951119 0.308824i $$-0.0999355\pi$$
$$444$$ 0 0
$$445$$ 13.2665i 0.628892i
$$446$$ 0 0
$$447$$ 6.63325 0.313742
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ − 30.0000i − 1.41264i
$$452$$ 0 0
$$453$$ − 13.2665i − 0.623315i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −29.0000 −1.35656 −0.678281 0.734802i $$-0.737275\pi$$
−0.678281 + 0.734802i $$0.737275\pi$$
$$458$$ 0 0
$$459$$ 20.0000i 0.933520i
$$460$$ 0 0
$$461$$ − 16.5831i − 0.772353i −0.922425 0.386177i $$-0.873796\pi$$
0.922425 0.386177i $$-0.126204\pi$$
$$462$$ 0 0
$$463$$ 23.2164 1.07896 0.539478 0.842000i $$-0.318622\pi$$
0.539478 + 0.842000i $$0.318622\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3.00000i 0.138823i 0.997588 + 0.0694117i $$0.0221122\pi$$
−0.997588 + 0.0694117i $$0.977888\pi$$
$$468$$ 0 0
$$469$$ − 26.5330i − 1.22518i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.00000 0.229900
$$474$$ 0 0
$$475$$ 6.00000i 0.275299i
$$476$$ 0 0
$$477$$ − 13.2665i − 0.607431i
$$478$$ 0 0
$$479$$ −6.63325 −0.303081 −0.151540 0.988451i $$-0.548423\pi$$
−0.151540 + 0.988451i $$0.548423\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 44.0000i 2.00207i
$$484$$ 0 0
$$485$$ 39.7995i 1.80720i
$$486$$ 0 0
$$487$$ 33.1662 1.50291 0.751453 0.659787i $$-0.229353\pi$$
0.751453 + 0.659787i $$0.229353\pi$$
$$488$$ 0 0
$$489$$ 40.0000 1.80886
$$490$$ 0 0
$$491$$ 20.0000i 0.902587i 0.892375 + 0.451294i $$0.149037\pi$$
−0.892375 + 0.451294i $$0.850963\pi$$
$$492$$ 0 0
$$493$$ 33.1662i 1.49373i
$$494$$ 0 0
$$495$$ −16.5831 −0.745356
$$496$$ 0 0
$$497$$ −22.0000 −0.986835
$$498$$ 0 0
$$499$$ − 3.00000i − 0.134298i −0.997743 0.0671492i $$-0.978610\pi$$
0.997743 0.0671492i $$-0.0213904\pi$$
$$500$$ 0 0
$$501$$ 13.2665i 0.592703i
$$502$$ 0 0
$$503$$ 19.8997 0.887286 0.443643 0.896204i $$-0.353686\pi$$
0.443643 + 0.896204i $$0.353686\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 26.0000i 1.15470i
$$508$$ 0 0
$$509$$ − 26.5330i − 1.17605i −0.808841 0.588027i $$-0.799905\pi$$
0.808841 0.588027i $$-0.200095\pi$$
$$510$$ 0 0
$$511$$ 29.8496 1.32047
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ − 22.0000i − 0.969436i
$$516$$ 0 0
$$517$$ − 49.7494i − 2.18797i
$$518$$ 0 0
$$519$$ 39.7995 1.74700
$$520$$ 0 0
$$521$$ 16.0000 0.700973 0.350486 0.936568i $$-0.386016\pi$$
0.350486 + 0.936568i $$0.386016\pi$$
$$522$$ 0 0
$$523$$ 6.00000i 0.262362i 0.991358 + 0.131181i $$0.0418769\pi$$
−0.991358 + 0.131181i $$0.958123\pi$$
$$524$$ 0 0
$$525$$ 39.7995i 1.73699i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 21.0000 0.913043
$$530$$ 0 0
$$531$$ 6.00000i 0.260378i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −19.8997 −0.860341
$$536$$ 0 0
$$537$$ −36.0000 −1.55351
$$538$$ 0 0
$$539$$ 20.0000i 0.861461i
$$540$$ 0 0
$$541$$ 9.94987i 0.427779i 0.976858 + 0.213889i $$0.0686132\pi$$
−0.976858 + 0.213889i $$0.931387\pi$$
$$542$$ 0 0
$$543$$ 39.7995 1.70796
$$544$$ 0 0
$$545$$ 44.0000 1.88475
$$546$$ 0 0
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ 0 0
$$549$$ − 9.94987i − 0.424650i
$$550$$ 0 0
$$551$$ 6.63325 0.282586
$$552$$ 0 0
$$553$$ −44.0000 −1.87107
$$554$$ 0 0
$$555$$ 44.0000i 1.86770i
$$556$$ 0 0
$$557$$ 3.31662i 0.140530i 0.997528 + 0.0702650i $$0.0223845\pi$$
−0.997528 + 0.0702650i $$0.977616\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −50.0000 −2.11100
$$562$$ 0 0
$$563$$ 14.0000i 0.590030i 0.955493 + 0.295015i $$0.0953246\pi$$
−0.955493 + 0.295015i $$0.904675\pi$$
$$564$$ 0 0
$$565$$ − 59.6992i − 2.51157i
$$566$$ 0 0
$$567$$ 36.4829 1.53214
$$568$$ 0 0
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ 44.0000i 1.84134i 0.390339 + 0.920671i $$0.372358\pi$$
−0.390339 + 0.920671i $$0.627642\pi$$
$$572$$ 0 0
$$573$$ 33.1662i 1.38554i
$$574$$ 0 0
$$575$$ 39.7995 1.65975
$$576$$ 0 0
$$577$$ −43.0000 −1.79011 −0.895057 0.445952i $$-0.852865\pi$$
−0.895057 + 0.445952i $$0.852865\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 13.2665i 0.550387i
$$582$$ 0 0
$$583$$ −66.3325 −2.74721
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.00000i 0.206372i 0.994662 + 0.103186i $$0.0329037\pi$$
−0.994662 + 0.103186i $$0.967096\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −53.0660 −2.18284
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 55.0000i 2.25478i
$$596$$ 0 0
$$597$$ 33.1662i 1.35740i
$$598$$ 0 0
$$599$$ 13.2665 0.542054 0.271027 0.962572i $$-0.412637\pi$$
0.271027 + 0.962572i $$0.412637\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ 46.4327i 1.88776i
$$606$$ 0 0
$$607$$ −26.5330 −1.07694 −0.538471 0.842644i $$-0.680998\pi$$
−0.538471 + 0.842644i $$0.680998\pi$$
$$608$$ 0 0
$$609$$ 44.0000 1.78297
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 3.31662i 0.133957i 0.997754 + 0.0669786i $$0.0213359\pi$$
−0.997754 + 0.0669786i $$0.978664\pi$$
$$614$$ 0 0
$$615$$ −39.7995 −1.60487
$$616$$ 0 0
$$617$$ −41.0000 −1.65060 −0.825299 0.564696i $$-0.808993\pi$$
−0.825299 + 0.564696i $$0.808993\pi$$
$$618$$ 0 0
$$619$$ 4.00000i 0.160774i 0.996764 + 0.0803868i $$0.0256155\pi$$
−0.996764 + 0.0803868i $$0.974384\pi$$
$$620$$ 0 0
$$621$$ − 26.5330i − 1.06473i
$$622$$ 0 0
$$623$$ 13.2665 0.531511
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 10.0000i 0.399362i
$$628$$ 0 0
$$629$$ 33.1662i 1.32242i
$$630$$ 0 0
$$631$$ −9.94987 −0.396098 −0.198049 0.980192i $$-0.563461\pi$$
−0.198049 + 0.980192i $$0.563461\pi$$
$$632$$ 0 0
$$633$$ 28.0000 1.11290
$$634$$ 0 0
$$635$$ 22.0000i 0.873043i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.63325 −0.262407
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ − 5.00000i − 0.197181i −0.995128 0.0985904i $$-0.968567\pi$$
0.995128 0.0985904i $$-0.0314334\pi$$
$$644$$ 0 0
$$645$$ − 6.63325i − 0.261184i
$$646$$ 0 0
$$647$$ 16.5831 0.651950 0.325975 0.945378i $$-0.394307\pi$$
0.325975 + 0.945378i $$0.394307\pi$$
$$648$$ 0 0
$$649$$ 30.0000 1.17760
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 36.4829i − 1.42769i −0.700306 0.713843i $$-0.746953\pi$$
0.700306 0.713843i $$-0.253047\pi$$
$$654$$ 0 0
$$655$$ −49.7494 −1.94387
$$656$$ 0 0
$$657$$ 9.00000 0.351123
$$658$$ 0 0
$$659$$ − 26.0000i − 1.01282i −0.862294 0.506408i $$-0.830973\pi$$
0.862294 0.506408i $$-0.169027\pi$$
$$660$$ 0 0
$$661$$ 39.7995i 1.54802i 0.633173 + 0.774011i $$0.281752\pi$$
−0.633173 + 0.774011i $$0.718248\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 11.0000 0.426562
$$666$$ 0 0
$$667$$ − 44.0000i − 1.70369i
$$668$$ 0 0
$$669$$ − 39.7995i − 1.53874i
$$670$$ 0 0
$$671$$ −49.7494 −1.92055
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 0 0
$$675$$ − 24.0000i − 0.923760i
$$676$$ 0 0
$$677$$ − 19.8997i − 0.764809i −0.923995 0.382405i $$-0.875096\pi$$
0.923995 0.382405i $$-0.124904\pi$$
$$678$$ 0 0
$$679$$ 39.7995 1.52736
$$680$$ 0 0
$$681$$ 16.0000 0.613121
$$682$$ 0 0
$$683$$ − 24.0000i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$684$$ 0 0
$$685$$ − 16.5831i − 0.633609i
$$686$$ 0 0
$$687$$ −46.4327 −1.77152
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 31.0000i 1.17930i 0.807661 + 0.589648i $$0.200733\pi$$
−0.807661 + 0.589648i $$0.799267\pi$$
$$692$$ 0 0
$$693$$ 16.5831i 0.629941i
$$694$$ 0 0
$$695$$ 36.4829 1.38387
$$696$$ 0 0
$$697$$ −30.0000 −1.13633
$$698$$ 0 0
$$699$$ 6.00000i 0.226941i
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 6.63325 0.250178
$$704$$ 0 0
$$705$$ −66.0000 −2.48570
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 13.2665i − 0.498234i −0.968473 0.249117i $$-0.919860\pi$$
0.968473 0.249117i $$-0.0801403\pi$$
$$710$$ 0 0
$$711$$ −13.2665 −0.497533
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 6.63325i − 0.247723i
$$718$$ 0 0
$$719$$ 3.31662 0.123689 0.0618446 0.998086i $$-0.480302\pi$$
0.0618446 + 0.998086i $$0.480302\pi$$
$$720$$ 0 0
$$721$$ −22.0000 −0.819323
$$722$$ 0 0
$$723$$ 52.0000i 1.93390i
$$724$$ 0 0
$$725$$ − 39.7995i − 1.47812i
$$726$$ 0 0
$$727$$ 16.5831 0.615034 0.307517 0.951543i $$-0.400502\pi$$
0.307517 + 0.951543i $$0.400502\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ − 5.00000i − 0.184932i
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 26.5330 0.978684
$$736$$ 0 0
$$737$$ −40.0000 −1.47342
$$738$$ 0 0
$$739$$ 45.0000i 1.65535i 0.561206 + 0.827676i $$0.310337\pi$$
−0.561206 + 0.827676i $$0.689663\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −53.0660 −1.94680 −0.973401 0.229107i $$-0.926420\pi$$
−0.973401 + 0.229107i $$0.926420\pi$$
$$744$$ 0 0
$$745$$ −11.0000 −0.403009
$$746$$ 0 0
$$747$$ 4.00000i 0.146352i
$$748$$ 0 0
$$749$$ 19.8997i 0.727121i
$$750$$ 0 0
$$751$$ 39.7995 1.45230 0.726152 0.687534i $$-0.241307\pi$$
0.726152 + 0.687534i $$0.241307\pi$$
$$752$$ 0 0
$$753$$ −10.0000 −0.364420
$$754$$ 0 0
$$755$$ 22.0000i 0.800662i
$$756$$ 0 0
$$757$$ − 9.94987i − 0.361634i −0.983517 0.180817i $$-0.942126\pi$$
0.983517 0.180817i $$-0.0578742\pi$$
$$758$$ 0 0
$$759$$ 66.3325 2.40772
$$760$$ 0 0
$$761$$ 1.00000 0.0362500 0.0181250 0.999836i $$-0.494230\pi$$
0.0181250 + 0.999836i $$0.494230\pi$$
$$762$$ 0 0
$$763$$ − 44.0000i − 1.59291i
$$764$$ 0 0
$$765$$ 16.5831i 0.599564i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 33.0000 1.19001 0.595005 0.803722i $$-0.297150\pi$$
0.595005 + 0.803722i $$0.297150\pi$$
$$770$$ 0 0
$$771$$ − 16.0000i − 0.576226i
$$772$$ 0 0
$$773$$ − 6.63325i − 0.238581i −0.992859 0.119291i $$-0.961938\pi$$
0.992859 0.119291i $$-0.0380620\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 44.0000 1.57849
$$778$$ 0 0
$$779$$ 6.00000i 0.214972i
$$780$$ 0 0
$$781$$ 33.1662i 1.18678i
$$782$$ 0 0
$$783$$ −26.5330 −0.948212
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ 0 0
$$789$$ 6.63325i 0.236150i
$$790$$ 0 0
$$791$$ −59.6992 −2.12266
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 88.0000i 3.12104i
$$796$$ 0 0
$$797$$ − 26.5330i − 0.939847i −0.882707 0.469924i $$-0.844281\pi$$
0.882707 0.469924i $$-0.155719\pi$$
$$798$$ 0 0
$$799$$ −49.7494 −1.76001
$$800$$ 0 0
$$801$$ 4.00000 0.141333
$$802$$ 0 0
$$803$$ − 45.0000i − 1.58802i
$$804$$ 0 0
$$805$$ − 72.9657i − 2.57170i
$$806$$ 0 0
$$807$$ 26.5330 0.934006
$$808$$ 0 0
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ 28.0000i 0.983213i 0.870817 + 0.491606i $$0.163590\pi$$
−0.870817 + 0.491606i $$0.836410\pi$$
$$812$$ 0 0
$$813$$ − 39.7995i − 1.39583i
$$814$$ 0 0
$$815$$ −66.3325 −2.32353
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9.94987i 0.347253i 0.984812 + 0.173627i $$0.0555486\pi$$
−0.984812 + 0.173627i $$0.944451\pi$$
$$822$$ 0 0
$$823$$ 29.8496 1.04049 0.520246 0.854016i $$-0.325840\pi$$
0.520246 + 0.854016i $$0.325840\pi$$
$$824$$ 0 0
$$825$$ 60.0000 2.08893
$$826$$ 0 0
$$827$$ − 32.0000i − 1.11275i −0.830932 0.556375i $$-0.812192\pi$$
0.830932 0.556375i $$-0.187808\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ −6.63325 −0.230105
$$832$$ 0 0
$$833$$ 20.0000 0.692959
$$834$$ 0 0
$$835$$ − 22.0000i − 0.761341i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −46.4327 −1.60304 −0.801518 0.597970i $$-0.795974\pi$$
−0.801518 + 0.597970i $$0.795974\pi$$
$$840$$ 0 0
$$841$$ −15.0000 −0.517241
$$842$$ 0 0
$$843$$ − 52.0000i − 1.79098i
$$844$$ 0 0
$$845$$ − 43.1161i − 1.48324i
$$846$$ 0 0
$$847$$ 46.4327 1.59545
$$848$$ 0 0
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ − 44.0000i − 1.50830i
$$852$$ 0 0
$$853$$ 53.0660i 1.81695i 0.417945 + 0.908473i $$0.362751\pi$$
−0.417945 + 0.908473i $$0.637249\pi$$
$$854$$ 0 0
$$855$$ 3.31662 0.113426
$$856$$ 0 0
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ − 41.0000i − 1.39890i −0.714681 0.699451i $$-0.753428\pi$$
0.714681 0.699451i $$-0.246572\pi$$
$$860$$ 0 0
$$861$$ 39.7995i 1.35636i
$$862$$ 0 0
$$863$$ 33.1662 1.12899 0.564496 0.825436i $$-0.309071\pi$$
0.564496 + 0.825436i $$0.309071\pi$$
$$864$$ 0 0
$$865$$ −66.0000 −2.24407
$$866$$ 0 0
$$867$$ 16.0000i 0.543388i
$$868$$ 0 0
$$869$$ 66.3325i 2.25018i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 12.0000 0.406138
$$874$$ 0 0
$$875$$ − 11.0000i − 0.371868i
$$876$$ 0 0
$$877$$ 6.63325i 0.223989i 0.993709 + 0.111994i $$0.0357239\pi$$
−0.993709 + 0.111994i $$0.964276\pi$$
$$878$$ 0 0
$$879$$ 53.0660 1.78987
$$880$$ 0 0
$$881$$ 51.0000 1.71823 0.859117 0.511780i $$-0.171014\pi$$
0.859117 + 0.511780i $$0.171014\pi$$
$$882$$ 0 0
$$883$$ − 31.0000i − 1.04323i −0.853180 0.521617i $$-0.825329\pi$$
0.853180 0.521617i $$-0.174671\pi$$
$$884$$ 0 0
$$885$$ − 39.7995i − 1.33785i
$$886$$ 0 0
$$887$$ −19.8997 −0.668168 −0.334084 0.942543i $$-0.608427\pi$$
−0.334084 + 0.942543i $$0.608427\pi$$
$$888$$ 0 0
$$889$$ 22.0000 0.737856
$$890$$ 0 0
$$891$$ − 55.0000i − 1.84257i
$$892$$ 0 0
$$893$$ 9.94987i 0.332960i
$$894$$ 0 0
$$895$$ 59.6992 1.99553
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 66.3325i 2.20986i
$$902$$ 0 0
$$903$$ −6.63325 −0.220741
$$904$$ 0 0
$$905$$ −66.0000 −2.19391
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 19.8997 0.659308 0.329654 0.944102i $$-0.393068\pi$$
0.329654 + 0.944102i $$0.393068\pi$$
$$912$$ 0 0
$$913$$ 20.0000 0.661903
$$914$$ 0 0
$$915$$ 66.0000i 2.18189i
$$916$$ 0 0
$$917$$ 49.7494i 1.64287i
$$918$$ 0 0
$$919$$ 33.1662 1.09405 0.547027 0.837115i $$-0.315760\pi$$
0.547027 + 0.837115i $$0.315760\pi$$
$$920$$ 0 0
$$921$$ −24.0000 −0.790827
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 39.7995i − 1.30860i
$$926$$ 0 0
$$927$$ −6.63325 −0.217865
$$928$$ 0 0
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ − 4.00000i − 0.131095i
$$932$$ 0 0
$$933$$ 33.1662i 1.08581i
$$934$$ 0 0
$$935$$ 82.9156 2.71163
$$936$$ 0 0
$$937$$ 1.00000 0.0326686 0.0163343 0.999867i $$-0.494800\pi$$
0.0163343 + 0.999867i $$0.494800\pi$$
$$938$$ 0 0
$$939$$ − 68.0000i − 2.21910i
$$940$$ 0 0
$$941$$ − 46.4327i − 1.51366i −0.653609 0.756832i $$-0.726746\pi$$
0.653609 0.756832i $$-0.273254\pi$$
$$942$$ 0 0
$$943$$ 39.7995 1.29605
$$944$$ 0 0
$$945$$ −44.0000 −1.43132
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −13.2665 −0.430196
$$952$$ 0 0
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 0 0
$$955$$ − 55.0000i − 1.77976i
$$956$$ 0 0
$$957$$ − 66.3325i − 2.14423i
$$958$$ 0 0
$$959$$ −16.5831 −0.535497
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 6.00000i 0.193347i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −33.1662 −1.06655 −0.533277 0.845940i $$-0.679040\pi$$
−0.533277 + 0.845940i $$0.679040\pi$$
$$968$$ 0 0
$$969$$ 10.0000 0.321246
$$970$$ 0 0
$$971$$ − 32.0000i − 1.02693i −0.858111 0.513464i $$-0.828362\pi$$
0.858111 0.513464i $$-0.171638\pi$$
$$972$$ 0 0
$$973$$ − 36.4829i − 1.16959i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16.0000 0.511885 0.255943 0.966692i $$-0.417614\pi$$
0.255943 + 0.966692i $$0.417614\pi$$
$$978$$ 0 0
$$979$$ − 20.0000i − 0.639203i
$$980$$ 0 0
$$981$$ − 13.2665i − 0.423567i
$$982$$ 0 0
$$983$$ 13.2665 0.423136 0.211568 0.977363i $$-0.432143\pi$$
0.211568 + 0.977363i $$0.432143\pi$$
$$984$$ 0 0
$$985$$ 88.0000 2.80391
$$986$$ 0 0
$$987$$ 66.0000i 2.10080i
$$988$$ 0 0
$$989$$ 6.63325i 0.210925i
$$990$$ 0 0
$$991$$ 33.1662 1.05356 0.526780 0.850001i $$-0.323399\pi$$
0.526780 + 0.850001i $$0.323399\pi$$
$$992$$ 0 0
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ − 55.0000i − 1.74362i
$$996$$ 0 0
$$997$$ 43.1161i 1.36550i 0.730651 + 0.682751i $$0.239216\pi$$
−0.730651 + 0.682751i $$0.760784\pi$$
$$998$$ 0 0
$$999$$ −26.5330 −0.839467
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 1216.2.c.f.609.3 yes 4
4.3 odd 2 inner 1216.2.c.f.609.1 4
8.3 odd 2 inner 1216.2.c.f.609.4 yes 4
8.5 even 2 inner 1216.2.c.f.609.2 yes 4
16.3 odd 4 4864.2.a.y.1.1 2
16.5 even 4 4864.2.a.y.1.2 2
16.11 odd 4 4864.2.a.r.1.2 2
16.13 even 4 4864.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.f.609.1 4 4.3 odd 2 inner
1216.2.c.f.609.2 yes 4 8.5 even 2 inner
1216.2.c.f.609.3 yes 4 1.1 even 1 trivial
1216.2.c.f.609.4 yes 4 8.3 odd 2 inner
4864.2.a.r.1.1 2 16.13 even 4
4864.2.a.r.1.2 2 16.11 odd 4
4864.2.a.y.1.1 2 16.3 odd 4
4864.2.a.y.1.2 2 16.5 even 4