# Properties

 Label 1216.2.c.c.609.1 Level $1216$ Weight $2$ Character 1216.609 Analytic conductor $9.710$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 609.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.609 Dual form 1216.2.c.c.609.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -6.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} -4.00000i q^{5} +1.00000 q^{7} -6.00000 q^{9} +5.00000i q^{13} -12.0000 q^{15} -5.00000 q^{17} -1.00000i q^{19} -3.00000i q^{21} -3.00000 q^{23} -11.0000 q^{25} +9.00000i q^{27} -7.00000i q^{29} +10.0000 q^{31} -4.00000i q^{35} -2.00000i q^{37} +15.0000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +24.0000i q^{45} +8.00000 q^{47} -6.00000 q^{49} +15.0000i q^{51} -9.00000i q^{53} -3.00000 q^{57} -1.00000i q^{59} +2.00000i q^{61} -6.00000 q^{63} +20.0000 q^{65} -7.00000i q^{67} +9.00000i q^{69} -12.0000 q^{71} +11.0000 q^{73} +33.0000i q^{75} +16.0000 q^{79} +9.00000 q^{81} -14.0000i q^{83} +20.0000i q^{85} -21.0000 q^{87} -4.00000 q^{89} +5.00000i q^{91} -30.0000i q^{93} -4.00000 q^{95} -12.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 12q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 12q^{9} - 24q^{15} - 10q^{17} - 6q^{23} - 22q^{25} + 20q^{31} + 30q^{39} - 12q^{41} + 16q^{47} - 12q^{49} - 6q^{57} - 12q^{63} + 40q^{65} - 24q^{71} + 22q^{73} + 32q^{79} + 18q^{81} - 42q^{87} - 8q^{89} - 8q^{95} - 24q^{97} + O(q^{100})$$

## 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
$$3$$ − 3.00000i − 1.73205i −0.500000 0.866025i $$-0.666667\pi$$
0.500000 0.866025i $$-0.333333\pi$$
$$4$$ 0 0
$$5$$ − 4.00000i − 1.78885i −0.447214 0.894427i $$-0.647584\pi$$
0.447214 0.894427i $$-0.352416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ −6.00000 −2.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ 0 0
$$15$$ −12.0000 −3.09839
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ − 1.00000i − 0.229416i
$$20$$ 0 0
$$21$$ − 3.00000i − 0.654654i
$$22$$ 0 0
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ −11.0000 −2.20000
$$26$$ 0 0
$$27$$ 9.00000i 1.73205i
$$28$$ 0 0
$$29$$ − 7.00000i − 1.29987i −0.759991 0.649934i $$-0.774797\pi$$
0.759991 0.649934i $$-0.225203\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 4.00000i − 0.676123i
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 15.0000 2.40192
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 24.0000i 3.57771i
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 15.0000i 2.10042i
$$52$$ 0 0
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.00000 −0.397360
$$58$$ 0 0
$$59$$ − 1.00000i − 0.130189i −0.997879 0.0650945i $$-0.979265\pi$$
0.997879 0.0650945i $$-0.0207349\pi$$
$$60$$ 0 0
$$61$$ 2.00000i 0.256074i 0.991769 + 0.128037i $$0.0408676\pi$$
−0.991769 + 0.128037i $$0.959132\pi$$
$$62$$ 0 0
$$63$$ −6.00000 −0.755929
$$64$$ 0 0
$$65$$ 20.0000 2.48069
$$66$$ 0 0
$$67$$ − 7.00000i − 0.855186i −0.903971 0.427593i $$-0.859362\pi$$
0.903971 0.427593i $$-0.140638\pi$$
$$68$$ 0 0
$$69$$ 9.00000i 1.08347i
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ 33.0000i 3.81051i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 14.0000i − 1.53670i −0.640030 0.768350i $$-0.721078\pi$$
0.640030 0.768350i $$-0.278922\pi$$
$$84$$ 0 0
$$85$$ 20.0000i 2.16930i
$$86$$ 0 0
$$87$$ −21.0000 −2.25144
$$88$$ 0 0
$$89$$ −4.00000 −0.423999 −0.212000 0.977270i $$-0.567998\pi$$
−0.212000 + 0.977270i $$0.567998\pi$$
$$90$$ 0 0
$$91$$ 5.00000i 0.524142i
$$92$$ 0 0
$$93$$ − 30.0000i − 3.11086i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ −12.0000 −1.21842 −0.609208 0.793011i $$-0.708512\pi$$
−0.609208 + 0.793011i $$0.708512\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ 0 0
$$105$$ −12.0000 −1.17108
$$106$$ 0 0
$$107$$ 9.00000i 0.870063i 0.900415 + 0.435031i $$0.143263\pi$$
−0.900415 + 0.435031i $$0.856737\pi$$
$$108$$ 0 0
$$109$$ − 9.00000i − 0.862044i −0.902342 0.431022i $$-0.858153\pi$$
0.902342 0.431022i $$-0.141847\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ 12.0000i 1.11901i
$$116$$ 0 0
$$117$$ − 30.0000i − 2.77350i
$$118$$ 0 0
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 18.0000i 1.62301i
$$124$$ 0 0
$$125$$ 24.0000i 2.14663i
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ − 1.00000i − 0.0867110i
$$134$$ 0 0
$$135$$ 36.0000 3.09839
$$136$$ 0 0
$$137$$ −15.0000 −1.28154 −0.640768 0.767734i $$-0.721384\pi$$
−0.640768 + 0.767734i $$0.721384\pi$$
$$138$$ 0 0
$$139$$ − 14.0000i − 1.18746i −0.804663 0.593732i $$-0.797654\pi$$
0.804663 0.593732i $$-0.202346\pi$$
$$140$$ 0 0
$$141$$ − 24.0000i − 2.02116i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −28.0000 −2.32527
$$146$$ 0 0
$$147$$ 18.0000i 1.48461i
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 30.0000 2.42536
$$154$$ 0 0
$$155$$ − 40.0000i − 3.21288i
$$156$$ 0 0
$$157$$ 20.0000i 1.59617i 0.602542 + 0.798087i $$0.294154\pi$$
−0.602542 + 0.798087i $$0.705846\pi$$
$$158$$ 0 0
$$159$$ −27.0000 −2.14124
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ − 10.0000i − 0.783260i −0.920123 0.391630i $$-0.871911\pi$$
0.920123 0.391630i $$-0.128089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 6.00000i 0.458831i
$$172$$ 0 0
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ 0 0
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ − 12.0000i − 0.896922i −0.893802 0.448461i $$-0.851972\pi$$
0.893802 0.448461i $$-0.148028\pi$$
$$180$$ 0 0
$$181$$ 6.00000i 0.445976i 0.974821 + 0.222988i $$0.0715812\pi$$
−0.974821 + 0.222988i $$0.928419\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 9.00000i 0.654654i
$$190$$ 0 0
$$191$$ −5.00000 −0.361787 −0.180894 0.983503i $$-0.557899\pi$$
−0.180894 + 0.983503i $$0.557899\pi$$
$$192$$ 0 0
$$193$$ 20.0000 1.43963 0.719816 0.694165i $$-0.244226\pi$$
0.719816 + 0.694165i $$0.244226\pi$$
$$194$$ 0 0
$$195$$ − 60.0000i − 4.29669i
$$196$$ 0 0
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ −21.0000 −1.48123
$$202$$ 0 0
$$203$$ − 7.00000i − 0.491304i
$$204$$ 0 0
$$205$$ 24.0000i 1.67623i
$$206$$ 0 0
$$207$$ 18.0000 1.25109
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.00000i 0.0688428i 0.999407 + 0.0344214i $$0.0109588\pi$$
−0.999407 + 0.0344214i $$0.989041\pi$$
$$212$$ 0 0
$$213$$ 36.0000i 2.46668i
$$214$$ 0 0
$$215$$ 16.0000 1.09119
$$216$$ 0 0
$$217$$ 10.0000 0.678844
$$218$$ 0 0
$$219$$ − 33.0000i − 2.22993i
$$220$$ 0 0
$$221$$ − 25.0000i − 1.68168i
$$222$$ 0 0
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 0 0
$$225$$ 66.0000 4.40000
$$226$$ 0 0
$$227$$ 17.0000i 1.12833i 0.825662 + 0.564165i $$0.190802\pi$$
−0.825662 + 0.564165i $$0.809198\pi$$
$$228$$ 0 0
$$229$$ − 22.0000i − 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ − 32.0000i − 2.08745i
$$236$$ 0 0
$$237$$ − 48.0000i − 3.11794i
$$238$$ 0 0
$$239$$ 11.0000 0.711531 0.355765 0.934575i $$-0.384220\pi$$
0.355765 + 0.934575i $$0.384220\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 24.0000i 1.53330i
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ −42.0000 −2.66164
$$250$$ 0 0
$$251$$ − 20.0000i − 1.26239i −0.775625 0.631194i $$-0.782565\pi$$
0.775625 0.631194i $$-0.217435\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 60.0000 3.75735
$$256$$ 0 0
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ − 2.00000i − 0.124274i
$$260$$ 0 0
$$261$$ 42.0000i 2.59973i
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ −36.0000 −2.21146
$$266$$ 0 0
$$267$$ 12.0000i 0.734388i
$$268$$ 0 0
$$269$$ 14.0000i 0.853595i 0.904347 + 0.426798i $$0.140358\pi$$
−0.904347 + 0.426798i $$0.859642\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 0 0
$$273$$ 15.0000 0.907841
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ 0 0
$$279$$ −60.0000 −3.59211
$$280$$ 0 0
$$281$$ −16.0000 −0.954480 −0.477240 0.878773i $$-0.658363\pi$$
−0.477240 + 0.878773i $$0.658363\pi$$
$$282$$ 0 0
$$283$$ 6.00000i 0.356663i 0.983970 + 0.178331i $$0.0570699\pi$$
−0.983970 + 0.178331i $$0.942930\pi$$
$$284$$ 0 0
$$285$$ 12.0000i 0.710819i
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 36.0000i 2.11036i
$$292$$ 0 0
$$293$$ − 7.00000i − 0.408944i −0.978872 0.204472i $$-0.934452\pi$$
0.978872 0.204472i $$-0.0655478\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 15.0000i − 0.867472i
$$300$$ 0 0
$$301$$ 4.00000i 0.230556i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 6.00000i 0.341328i
$$310$$ 0 0
$$311$$ −35.0000 −1.98467 −0.992334 0.123585i $$-0.960561\pi$$
−0.992334 + 0.123585i $$0.960561\pi$$
$$312$$ 0 0
$$313$$ −9.00000 −0.508710 −0.254355 0.967111i $$-0.581863\pi$$
−0.254355 + 0.967111i $$0.581863\pi$$
$$314$$ 0 0
$$315$$ 24.0000i 1.35225i
$$316$$ 0 0
$$317$$ 23.0000i 1.29181i 0.763418 + 0.645904i $$0.223520\pi$$
−0.763418 + 0.645904i $$0.776480\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 27.0000 1.50699
$$322$$ 0 0
$$323$$ 5.00000i 0.278207i
$$324$$ 0 0
$$325$$ − 55.0000i − 3.05085i
$$326$$ 0 0
$$327$$ −27.0000 −1.49310
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ − 11.0000i − 0.604615i −0.953211 0.302307i $$-0.902243\pi$$
0.953211 0.302307i $$-0.0977569\pi$$
$$332$$ 0 0
$$333$$ 12.0000i 0.657596i
$$334$$ 0 0
$$335$$ −28.0000 −1.52980
$$336$$ 0 0
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ 0 0
$$339$$ − 24.0000i − 1.30350i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 36.0000 1.93817
$$346$$ 0 0
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 0 0
$$349$$ − 28.0000i − 1.49881i −0.662114 0.749403i $$-0.730341\pi$$
0.662114 0.749403i $$-0.269659\pi$$
$$350$$ 0 0
$$351$$ −45.0000 −2.40192
$$352$$ 0 0
$$353$$ 29.0000 1.54351 0.771757 0.635917i $$-0.219378\pi$$
0.771757 + 0.635917i $$0.219378\pi$$
$$354$$ 0 0
$$355$$ 48.0000i 2.54758i
$$356$$ 0 0
$$357$$ 15.0000i 0.793884i
$$358$$ 0 0
$$359$$ 27.0000 1.42501 0.712503 0.701669i $$-0.247562\pi$$
0.712503 + 0.701669i $$0.247562\pi$$
$$360$$ 0 0
$$361$$ −1.00000 −0.0526316
$$362$$ 0 0
$$363$$ − 33.0000i − 1.73205i
$$364$$ 0 0
$$365$$ − 44.0000i − 2.30307i
$$366$$ 0 0
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 0 0
$$369$$ 36.0000 1.87409
$$370$$ 0 0
$$371$$ − 9.00000i − 0.467257i
$$372$$ 0 0
$$373$$ − 15.0000i − 0.776671i −0.921518 0.388335i $$-0.873050\pi$$
0.921518 0.388335i $$-0.126950\pi$$
$$374$$ 0 0
$$375$$ 72.0000 3.71806
$$376$$ 0 0
$$377$$ 35.0000 1.80259
$$378$$ 0 0
$$379$$ 11.0000i 0.565032i 0.959263 + 0.282516i $$0.0911690\pi$$
−0.959263 + 0.282516i $$0.908831\pi$$
$$380$$ 0 0
$$381$$ − 6.00000i − 0.307389i
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 24.0000i − 1.21999i
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 15.0000 0.758583
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 64.0000i − 3.22019i
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −4.00000 −0.199750 −0.0998752 0.995000i $$-0.531844\pi$$
−0.0998752 + 0.995000i $$0.531844\pi$$
$$402$$ 0 0
$$403$$ 50.0000i 2.49068i
$$404$$ 0 0
$$405$$ − 36.0000i − 1.78885i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ 45.0000i 2.21969i
$$412$$ 0 0
$$413$$ − 1.00000i − 0.0492068i
$$414$$ 0 0
$$415$$ −56.0000 −2.74893
$$416$$ 0 0
$$417$$ −42.0000 −2.05675
$$418$$ 0 0
$$419$$ − 24.0000i − 1.17248i −0.810139 0.586238i $$-0.800608\pi$$
0.810139 0.586238i $$-0.199392\pi$$
$$420$$ 0 0
$$421$$ − 21.0000i − 1.02348i −0.859141 0.511739i $$-0.829002\pi$$
0.859141 0.511739i $$-0.170998\pi$$
$$422$$ 0 0
$$423$$ −48.0000 −2.33384
$$424$$ 0 0
$$425$$ 55.0000 2.66789
$$426$$ 0 0
$$427$$ 2.00000i 0.0967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.0000 1.05970 0.529851 0.848091i $$-0.322248\pi$$
0.529851 + 0.848091i $$0.322248\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 84.0000i 4.02749i
$$436$$ 0 0
$$437$$ 3.00000i 0.143509i
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 36.0000 1.71429
$$442$$ 0 0
$$443$$ 32.0000i 1.52037i 0.649709 + 0.760183i $$0.274891\pi$$
−0.649709 + 0.760183i $$0.725109\pi$$
$$444$$ 0 0
$$445$$ 16.0000i 0.758473i
$$446$$ 0 0
$$447$$ 18.0000 0.851371
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 24.0000i 1.12762i
$$454$$ 0 0
$$455$$ 20.0000 0.937614
$$456$$ 0 0
$$457$$ 11.0000 0.514558 0.257279 0.966337i $$-0.417174\pi$$
0.257279 + 0.966337i $$0.417174\pi$$
$$458$$ 0 0
$$459$$ − 45.0000i − 2.10042i
$$460$$ 0 0
$$461$$ 20.0000i 0.931493i 0.884918 + 0.465746i $$0.154214\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ −120.000 −5.56487
$$466$$ 0 0
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ − 7.00000i − 0.323230i
$$470$$ 0 0
$$471$$ 60.0000 2.76465
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 11.0000i 0.504715i
$$476$$ 0 0
$$477$$ 54.0000i 2.47249i
$$478$$ 0 0
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ 9.00000i 0.409514i
$$484$$ 0 0
$$485$$ 48.0000i 2.17957i
$$486$$ 0 0
$$487$$ −30.0000 −1.35943 −0.679715 0.733476i $$-0.737896\pi$$
−0.679715 + 0.733476i $$0.737896\pi$$
$$488$$ 0 0
$$489$$ −30.0000 −1.35665
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 35.0000i 1.57632i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ − 28.0000i − 1.25345i −0.779240 0.626726i $$-0.784395\pi$$
0.779240 0.626726i $$-0.215605\pi$$
$$500$$ 0 0
$$501$$ 6.00000i 0.268060i
$$502$$ 0 0
$$503$$ 29.0000 1.29305 0.646523 0.762894i $$-0.276222\pi$$
0.646523 + 0.762894i $$0.276222\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 36.0000i 1.59882i
$$508$$ 0 0
$$509$$ − 22.0000i − 0.975133i −0.873086 0.487566i $$-0.837885\pi$$
0.873086 0.487566i $$-0.162115\pi$$
$$510$$ 0 0
$$511$$ 11.0000 0.486611
$$512$$ 0 0
$$513$$ 9.00000 0.397360
$$514$$ 0 0
$$515$$ 8.00000i 0.352522i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −42.0000 −1.84360
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 1.00000i 0.0437269i 0.999761 + 0.0218635i $$0.00695991\pi$$
−0.999761 + 0.0218635i $$0.993040\pi$$
$$524$$ 0 0
$$525$$ 33.0000i 1.44024i
$$526$$ 0 0
$$527$$ −50.0000 −2.17803
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 6.00000i 0.260378i
$$532$$ 0 0
$$533$$ − 30.0000i − 1.29944i
$$534$$ 0 0
$$535$$ 36.0000 1.55642
$$536$$ 0 0
$$537$$ −36.0000 −1.55351
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 22.0000i 0.945854i 0.881102 + 0.472927i $$0.156803\pi$$
−0.881102 + 0.472927i $$0.843197\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 0 0
$$545$$ −36.0000 −1.54207
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 0 0
$$549$$ − 12.0000i − 0.512148i
$$550$$ 0 0
$$551$$ −7.00000 −0.298210
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 0 0
$$555$$ 24.0000i 1.01874i
$$556$$ 0 0
$$557$$ 24.0000i 1.01691i 0.861088 + 0.508456i $$0.169784\pi$$
−0.861088 + 0.508456i $$0.830216\pi$$
$$558$$ 0 0
$$559$$ −20.0000 −0.845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 0 0
$$565$$ − 32.0000i − 1.34625i
$$566$$ 0 0
$$567$$ 9.00000 0.377964
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ − 26.0000i − 1.08807i −0.839064 0.544033i $$-0.816897\pi$$
0.839064 0.544033i $$-0.183103\pi$$
$$572$$ 0 0
$$573$$ 15.0000i 0.626634i
$$574$$ 0 0
$$575$$ 33.0000 1.37620
$$576$$ 0 0
$$577$$ −33.0000 −1.37381 −0.686904 0.726748i $$-0.741031\pi$$
−0.686904 + 0.726748i $$0.741031\pi$$
$$578$$ 0 0
$$579$$ − 60.0000i − 2.49351i
$$580$$ 0 0
$$581$$ − 14.0000i − 0.580818i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −120.000 −4.96139
$$586$$ 0 0
$$587$$ − 20.0000i − 0.825488i −0.910847 0.412744i $$-0.864570\pi$$
0.910847 0.412744i $$-0.135430\pi$$
$$588$$ 0 0
$$589$$ − 10.0000i − 0.412043i
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ −2.00000 −0.0821302 −0.0410651 0.999156i $$-0.513075\pi$$
−0.0410651 + 0.999156i $$0.513075\pi$$
$$594$$ 0 0
$$595$$ 20.0000i 0.819920i
$$596$$ 0 0
$$597$$ − 15.0000i − 0.613909i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 42.0000i 1.71037i
$$604$$ 0 0
$$605$$ − 44.0000i − 1.78885i
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ −21.0000 −0.850963
$$610$$ 0 0
$$611$$ 40.0000i 1.61823i
$$612$$ 0 0
$$613$$ − 6.00000i − 0.242338i −0.992632 0.121169i $$-0.961336\pi$$
0.992632 0.121169i $$-0.0386643\pi$$
$$614$$ 0 0
$$615$$ 72.0000 2.90332
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 34.0000i 1.36658i 0.730149 + 0.683288i $$0.239451\pi$$
−0.730149 + 0.683288i $$0.760549\pi$$
$$620$$ 0 0
$$621$$ − 27.0000i − 1.08347i
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 10.0000i 0.398726i
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 3.00000 0.119239
$$634$$ 0 0
$$635$$ − 8.00000i − 0.317470i
$$636$$ 0 0
$$637$$ − 30.0000i − 1.18864i
$$638$$ 0 0
$$639$$ 72.0000 2.84828
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ 0 0
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 0 0
$$645$$ − 48.0000i − 1.89000i
$$646$$ 0 0
$$647$$ 25.0000 0.982851 0.491426 0.870919i $$-0.336476\pi$$
0.491426 + 0.870919i $$0.336476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ − 30.0000i − 1.17579i
$$652$$ 0 0
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −66.0000 −2.57491
$$658$$ 0 0
$$659$$ − 21.0000i − 0.818044i −0.912525 0.409022i $$-0.865870\pi$$
0.912525 0.409022i $$-0.134130\pi$$
$$660$$ 0 0
$$661$$ 13.0000i 0.505641i 0.967513 + 0.252821i $$0.0813583\pi$$
−0.967513 + 0.252821i $$0.918642\pi$$
$$662$$ 0 0
$$663$$ −75.0000 −2.91276
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ 0 0
$$667$$ 21.0000i 0.813123i
$$668$$ 0 0
$$669$$ 42.0000i 1.62381i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 0 0
$$675$$ − 99.0000i − 3.81051i
$$676$$ 0 0
$$677$$ 1.00000i 0.0384331i 0.999815 + 0.0192166i $$0.00611720\pi$$
−0.999815 + 0.0192166i $$0.993883\pi$$
$$678$$ 0 0
$$679$$ −12.0000 −0.460518
$$680$$ 0 0
$$681$$ 51.0000 1.95432
$$682$$ 0 0
$$683$$ 16.0000i 0.612223i 0.951996 + 0.306111i $$0.0990280\pi$$
−0.951996 + 0.306111i $$0.900972\pi$$
$$684$$ 0 0
$$685$$ 60.0000i 2.29248i
$$686$$ 0 0
$$687$$ −66.0000 −2.51806
$$688$$ 0 0
$$689$$ 45.0000 1.71436
$$690$$ 0 0
$$691$$ 6.00000i 0.228251i 0.993466 + 0.114125i $$0.0364066\pi$$
−0.993466 + 0.114125i $$0.963593\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −56.0000 −2.12420
$$696$$ 0 0
$$697$$ 30.0000 1.13633
$$698$$ 0 0
$$699$$ − 54.0000i − 2.04247i
$$700$$ 0 0
$$701$$ 20.0000i 0.755390i 0.925930 + 0.377695i $$0.123283\pi$$
−0.925930 + 0.377695i $$0.876717\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ −96.0000 −3.61557
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 14.0000i 0.525781i 0.964826 + 0.262891i $$0.0846758\pi$$
−0.964826 + 0.262891i $$0.915324\pi$$
$$710$$ 0 0
$$711$$ −96.0000 −3.60028
$$712$$ 0 0
$$713$$ −30.0000 −1.12351
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 33.0000i − 1.23241i
$$718$$ 0 0
$$719$$ 39.0000 1.45445 0.727227 0.686397i $$-0.240809\pi$$
0.727227 + 0.686397i $$0.240809\pi$$
$$720$$ 0 0
$$721$$ −2.00000 −0.0744839
$$722$$ 0 0
$$723$$ 42.0000i 1.56200i
$$724$$ 0 0
$$725$$ 77.0000i 2.85971i
$$726$$ 0 0
$$727$$ 25.0000 0.927199 0.463599 0.886045i $$-0.346558\pi$$
0.463599 + 0.886045i $$0.346558\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ − 20.0000i − 0.739727i
$$732$$ 0 0
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ 0 0
$$735$$ 72.0000 2.65576
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 40.0000i 1.47142i 0.677295 + 0.735712i $$0.263152\pi$$
−0.677295 + 0.735712i $$0.736848\pi$$
$$740$$ 0 0
$$741$$ − 15.0000i − 0.551039i
$$742$$ 0 0
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 0 0
$$745$$ 24.0000 0.879292
$$746$$ 0 0
$$747$$ 84.0000i 3.07340i
$$748$$ 0 0
$$749$$ 9.00000i 0.328853i
$$750$$ 0 0
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ 0 0
$$753$$ −60.0000 −2.18652
$$754$$ 0 0
$$755$$ 32.0000i 1.16460i
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.00000 0.0362500 0.0181250 0.999836i $$-0.494230\pi$$
0.0181250 + 0.999836i $$0.494230\pi$$
$$762$$ 0 0
$$763$$ − 9.00000i − 0.325822i
$$764$$ 0 0
$$765$$ − 120.000i − 4.33861i
$$766$$ 0 0
$$767$$ 5.00000 0.180540
$$768$$ 0 0
$$769$$ −7.00000 −0.252426 −0.126213 0.992003i $$-0.540282\pi$$
−0.126213 + 0.992003i $$0.540282\pi$$
$$770$$ 0 0
$$771$$ − 36.0000i − 1.29651i
$$772$$ 0 0
$$773$$ 17.0000i 0.611448i 0.952120 + 0.305724i $$0.0988984\pi$$
−0.952120 + 0.305724i $$0.901102\pi$$
$$774$$ 0 0
$$775$$ −110.000 −3.95132
$$776$$ 0 0
$$777$$ −6.00000 −0.215249
$$778$$ 0 0
$$779$$ 6.00000i 0.214972i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 63.0000 2.25144
$$784$$ 0 0
$$785$$ 80.0000 2.85532
$$786$$ 0 0
$$787$$ 13.0000i 0.463400i 0.972787 + 0.231700i $$0.0744288\pi$$
−0.972787 + 0.231700i $$0.925571\pi$$
$$788$$ 0 0
$$789$$ − 72.0000i − 2.56327i
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 0 0
$$795$$ 108.000i 3.83037i
$$796$$ 0 0
$$797$$ − 27.0000i − 0.956389i −0.878254 0.478195i $$-0.841291\pi$$
0.878254 0.478195i $$-0.158709\pi$$
$$798$$ 0 0
$$799$$ −40.0000 −1.41510
$$800$$ 0 0
$$801$$ 24.0000 0.847998
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 12.0000i 0.422944i
$$806$$ 0 0
$$807$$ 42.0000 1.47847
$$808$$ 0 0
$$809$$ −29.0000 −1.01959 −0.509793 0.860297i $$-0.670278\pi$$
−0.509793 + 0.860297i $$0.670278\pi$$
$$810$$ 0 0
$$811$$ − 7.00000i − 0.245803i −0.992419 0.122902i $$-0.960780\pi$$
0.992419 0.122902i $$-0.0392200\pi$$
$$812$$ 0 0
$$813$$ − 33.0000i − 1.15736i
$$814$$ 0 0
$$815$$ −40.0000 −1.40114
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ − 30.0000i − 1.04828i
$$820$$ 0 0
$$821$$ 2.00000i 0.0698005i 0.999391 + 0.0349002i $$0.0111113\pi$$
−0.999391 + 0.0349002i $$0.988889\pi$$
$$822$$ 0 0
$$823$$ 51.0000 1.77775 0.888874 0.458151i $$-0.151488\pi$$
0.888874 + 0.458151i $$0.151488\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 43.0000i 1.49526i 0.664117 + 0.747628i $$0.268807\pi$$
−0.664117 + 0.747628i $$0.731193\pi$$
$$828$$ 0 0
$$829$$ − 5.00000i − 0.173657i −0.996223 0.0868286i $$-0.972327\pi$$
0.996223 0.0868286i $$-0.0276732\pi$$
$$830$$ 0 0
$$831$$ −18.0000 −0.624413
$$832$$ 0 0
$$833$$ 30.0000 1.03944
$$834$$ 0 0
$$835$$ 8.00000i 0.276851i
$$836$$ 0 0
$$837$$ 90.0000i 3.11086i
$$838$$ 0 0
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 48.0000i 1.65321i
$$844$$ 0 0
$$845$$ 48.0000i 1.65125i
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ 18.0000 0.617758
$$850$$ 0 0
$$851$$ 6.00000i 0.205677i
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ 0 0
$$855$$ 24.0000 0.820783
$$856$$ 0 0
$$857$$ 20.0000 0.683187 0.341593 0.939848i $$-0.389033\pi$$
0.341593 + 0.939848i $$0.389033\pi$$
$$858$$ 0 0
$$859$$ 14.0000i 0.477674i 0.971060 + 0.238837i $$0.0767661\pi$$
−0.971060 + 0.238837i $$0.923234\pi$$
$$860$$ 0 0
$$861$$ 18.0000i 0.613438i
$$862$$ 0 0
$$863$$ 10.0000 0.340404 0.170202 0.985409i $$-0.445558\pi$$
0.170202 + 0.985409i $$0.445558\pi$$
$$864$$ 0 0
$$865$$ −56.0000 −1.90406
$$866$$ 0 0
$$867$$ − 24.0000i − 0.815083i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 35.0000 1.18593
$$872$$ 0 0
$$873$$ 72.0000 2.43683
$$874$$ 0 0
$$875$$ 24.0000i 0.811348i
$$876$$ 0 0
$$877$$ − 37.0000i − 1.24940i −0.780864 0.624701i $$-0.785221\pi$$
0.780864 0.624701i $$-0.214779\pi$$
$$878$$ 0 0
$$879$$ −21.0000 −0.708312
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 0 0
$$883$$ 34.0000i 1.14419i 0.820187 + 0.572096i $$0.193869\pi$$
−0.820187 + 0.572096i $$0.806131\pi$$
$$884$$ 0 0
$$885$$ 12.0000i 0.403376i
$$886$$ 0 0
$$887$$ 26.0000 0.872995 0.436497 0.899706i $$-0.356219\pi$$
0.436497 + 0.899706i $$0.356219\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 8.00000i − 0.267710i
$$894$$ 0 0
$$895$$ −48.0000 −1.60446
$$896$$ 0 0
$$897$$ −45.0000 −1.50251
$$898$$ 0 0
$$899$$ − 70.0000i − 2.33463i
$$900$$ 0 0
$$901$$ 45.0000i 1.49917i
$$902$$ 0 0
$$903$$ 12.0000 0.399335
$$904$$ 0 0
$$905$$ 24.0000 0.797787
$$906$$ 0 0
$$907$$ 37.0000i 1.22856i 0.789086 + 0.614282i $$0.210554\pi$$
−0.789086 + 0.614282i $$0.789446\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ − 24.0000i − 0.793416i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 55.0000 1.81428 0.907141 0.420826i $$-0.138260\pi$$
0.907141 + 0.420826i $$0.138260\pi$$
$$920$$ 0 0
$$921$$ 36.0000 1.18624
$$922$$ 0 0
$$923$$ − 60.0000i − 1.97492i
$$924$$ 0 0
$$925$$ 22.0000i 0.723356i
$$926$$ 0 0
$$927$$ 12.0000 0.394132
$$928$$ 0 0
$$929$$ 29.0000 0.951459 0.475730 0.879592i $$-0.342184\pi$$
0.475730 + 0.879592i $$0.342184\pi$$
$$930$$ 0 0
$$931$$ 6.00000i 0.196642i
$$932$$ 0 0
$$933$$ 105.000i 3.43755i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 31.0000 1.01273 0.506363 0.862320i $$-0.330990\pi$$
0.506363 + 0.862320i $$0.330990\pi$$
$$938$$ 0 0
$$939$$ 27.0000i 0.881112i
$$940$$ 0 0
$$941$$ − 51.0000i − 1.66255i −0.555860 0.831276i $$-0.687611\pi$$
0.555860 0.831276i $$-0.312389\pi$$
$$942$$ 0 0
$$943$$ 18.0000 0.586161
$$944$$ 0 0
$$945$$ 36.0000 1.17108
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 0 0
$$949$$ 55.0000i 1.78538i
$$950$$ 0 0
$$951$$ 69.0000 2.23748
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ 20.0000i 0.647185i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −15.0000 −0.484375
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ − 54.0000i − 1.74013i
$$964$$ 0 0
$$965$$ − 80.0000i − 2.57529i
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ 0 0
$$969$$ 15.0000 0.481869
$$970$$ 0 0
$$971$$ − 32.0000i − 1.02693i −0.858111 0.513464i $$-0.828362\pi$$
0.858111 0.513464i $$-0.171638\pi$$
$$972$$ 0 0
$$973$$ − 14.0000i − 0.448819i
$$974$$ 0 0
$$975$$ −165.000 −5.28423
$$976$$ 0 0
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 54.0000i 1.72409i
$$982$$ 0 0
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ 0 0
$$985$$ 8.00000 0.254901
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ 0 0
$$989$$ − 12.0000i − 0.381578i
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −33.0000 −1.04722
$$994$$ 0 0
$$995$$ − 20.0000i − 0.634043i
$$996$$ 0 0
$$997$$ 2.00000i 0.0633406i 0.999498 + 0.0316703i $$0.0100827\pi$$
−0.999498 + 0.0316703i $$0.989917\pi$$
$$998$$ 0 0
$$999$$ 18.0000 0.569495
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.c.609.1 yes 2
4.3 odd 2 1216.2.c.b.609.2 yes 2
8.3 odd 2 1216.2.c.b.609.1 2
8.5 even 2 inner 1216.2.c.c.609.2 yes 2
16.3 odd 4 4864.2.a.a.1.1 1
16.5 even 4 4864.2.a.b.1.1 1
16.11 odd 4 4864.2.a.p.1.1 1
16.13 even 4 4864.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.b.609.1 2 8.3 odd 2
1216.2.c.b.609.2 yes 2 4.3 odd 2
1216.2.c.c.609.1 yes 2 1.1 even 1 trivial
1216.2.c.c.609.2 yes 2 8.5 even 2 inner
4864.2.a.a.1.1 1 16.3 odd 4
4864.2.a.b.1.1 1 16.5 even 4
4864.2.a.o.1.1 1 16.13 even 4
4864.2.a.p.1.1 1 16.11 odd 4