# Properties

 Label 1536.2.d.g.769.8 Level $1536$ Weight $2$ Character 1536.769 Analytic conductor $12.265$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.18939904.2 Defining polynomial: $$x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2$$ x^8 - 4*x^7 + 14*x^6 - 28*x^5 + 43*x^4 - 44*x^3 + 30*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 769.8 Root $$0.500000 - 0.691860i$$ of defining polynomial Character $$\chi$$ $$=$$ 1536.769 Dual form 1536.2.d.g.769.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +3.95687i q^{5} -1.63899 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +3.95687i q^{5} -1.63899 q^{7} -1.00000 q^{9} -4.82843i q^{11} -5.59587i q^{13} -3.95687 q^{15} -0.828427 q^{17} -2.82843i q^{19} -1.63899i q^{21} -7.91375 q^{23} -10.6569 q^{25} -1.00000i q^{27} +7.23486i q^{29} -1.63899 q^{31} +4.82843 q^{33} -6.48528i q^{35} -2.31788i q^{37} +5.59587 q^{39} -3.17157 q^{41} -4.48528i q^{43} -3.95687i q^{45} +7.91375 q^{47} -4.31371 q^{49} -0.828427i q^{51} -0.678892i q^{53} +19.1055 q^{55} +2.82843 q^{57} -9.65685i q^{59} +2.31788i q^{61} +1.63899 q^{63} +22.1421 q^{65} -13.6569i q^{67} -7.91375i q^{69} +3.27798 q^{71} -4.00000 q^{73} -10.6569i q^{75} +7.91375i q^{77} +1.63899 q^{79} +1.00000 q^{81} +8.82843i q^{83} -3.27798i q^{85} -7.23486 q^{87} -10.0000 q^{89} +9.17157i q^{91} -1.63899i q^{93} +11.1917 q^{95} +11.6569 q^{97} +4.82843i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^9 $$8 q - 8 q^{9} + 16 q^{17} - 40 q^{25} + 16 q^{33} - 48 q^{41} + 56 q^{49} + 64 q^{65} - 32 q^{73} + 8 q^{81} - 80 q^{89} + 48 q^{97}+O(q^{100})$$ 8 * q - 8 * q^9 + 16 * q^17 - 40 * q^25 + 16 * q^33 - 48 * q^41 + 56 * q^49 + 64 * q^65 - 32 * q^73 + 8 * q^81 - 80 * q^89 + 48 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$517$$ $$1025$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 3.95687i 1.76957i 0.466001 + 0.884784i $$0.345694\pi$$
−0.466001 + 0.884784i $$0.654306\pi$$
$$6$$ 0 0
$$7$$ −1.63899 −0.619480 −0.309740 0.950821i $$-0.600242\pi$$
−0.309740 + 0.950821i $$0.600242\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ − 4.82843i − 1.45583i −0.685670 0.727913i $$-0.740491\pi$$
0.685670 0.727913i $$-0.259509\pi$$
$$12$$ 0 0
$$13$$ − 5.59587i − 1.55201i −0.630724 0.776007i $$-0.717242\pi$$
0.630724 0.776007i $$-0.282758\pi$$
$$14$$ 0 0
$$15$$ −3.95687 −1.02166
$$16$$ 0 0
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ 0 0
$$19$$ − 2.82843i − 0.648886i −0.945905 0.324443i $$-0.894823\pi$$
0.945905 0.324443i $$-0.105177\pi$$
$$20$$ 0 0
$$21$$ − 1.63899i − 0.357657i
$$22$$ 0 0
$$23$$ −7.91375 −1.65013 −0.825065 0.565037i $$-0.808862\pi$$
−0.825065 + 0.565037i $$0.808862\pi$$
$$24$$ 0 0
$$25$$ −10.6569 −2.13137
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 7.23486i 1.34348i 0.740787 + 0.671740i $$0.234453\pi$$
−0.740787 + 0.671740i $$0.765547\pi$$
$$30$$ 0 0
$$31$$ −1.63899 −0.294371 −0.147186 0.989109i $$-0.547022\pi$$
−0.147186 + 0.989109i $$0.547022\pi$$
$$32$$ 0 0
$$33$$ 4.82843 0.840521
$$34$$ 0 0
$$35$$ − 6.48528i − 1.09621i
$$36$$ 0 0
$$37$$ − 2.31788i − 0.381058i −0.981682 0.190529i $$-0.938980\pi$$
0.981682 0.190529i $$-0.0610203\pi$$
$$38$$ 0 0
$$39$$ 5.59587 0.896056
$$40$$ 0 0
$$41$$ −3.17157 −0.495316 −0.247658 0.968847i $$-0.579661\pi$$
−0.247658 + 0.968847i $$0.579661\pi$$
$$42$$ 0 0
$$43$$ − 4.48528i − 0.683999i −0.939700 0.341999i $$-0.888896\pi$$
0.939700 0.341999i $$-0.111104\pi$$
$$44$$ 0 0
$$45$$ − 3.95687i − 0.589856i
$$46$$ 0 0
$$47$$ 7.91375 1.15434 0.577169 0.816624i $$-0.304157\pi$$
0.577169 + 0.816624i $$0.304157\pi$$
$$48$$ 0 0
$$49$$ −4.31371 −0.616244
$$50$$ 0 0
$$51$$ − 0.828427i − 0.116003i
$$52$$ 0 0
$$53$$ − 0.678892i − 0.0932530i −0.998912 0.0466265i $$-0.985153\pi$$
0.998912 0.0466265i $$-0.0148471\pi$$
$$54$$ 0 0
$$55$$ 19.1055 2.57618
$$56$$ 0 0
$$57$$ 2.82843 0.374634
$$58$$ 0 0
$$59$$ − 9.65685i − 1.25722i −0.777723 0.628608i $$-0.783625\pi$$
0.777723 0.628608i $$-0.216375\pi$$
$$60$$ 0 0
$$61$$ 2.31788i 0.296775i 0.988929 + 0.148387i $$0.0474082\pi$$
−0.988929 + 0.148387i $$0.952592\pi$$
$$62$$ 0 0
$$63$$ 1.63899 0.206493
$$64$$ 0 0
$$65$$ 22.1421 2.74639
$$66$$ 0 0
$$67$$ − 13.6569i − 1.66845i −0.551424 0.834225i $$-0.685915\pi$$
0.551424 0.834225i $$-0.314085\pi$$
$$68$$ 0 0
$$69$$ − 7.91375i − 0.952703i
$$70$$ 0 0
$$71$$ 3.27798 0.389025 0.194512 0.980900i $$-0.437688\pi$$
0.194512 + 0.980900i $$0.437688\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ − 10.6569i − 1.23055i
$$76$$ 0 0
$$77$$ 7.91375i 0.901855i
$$78$$ 0 0
$$79$$ 1.63899 0.184401 0.0922004 0.995740i $$-0.470610\pi$$
0.0922004 + 0.995740i $$0.470610\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 8.82843i 0.969046i 0.874779 + 0.484523i $$0.161007\pi$$
−0.874779 + 0.484523i $$0.838993\pi$$
$$84$$ 0 0
$$85$$ − 3.27798i − 0.355547i
$$86$$ 0 0
$$87$$ −7.23486 −0.775658
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 9.17157i 0.961442i
$$92$$ 0 0
$$93$$ − 1.63899i − 0.169955i
$$94$$ 0 0
$$95$$ 11.1917 1.14825
$$96$$ 0 0
$$97$$ 11.6569 1.18357 0.591787 0.806094i $$-0.298423\pi$$
0.591787 + 0.806094i $$0.298423\pi$$
$$98$$ 0 0
$$99$$ 4.82843i 0.485275i
$$100$$ 0 0
$$101$$ 0.678892i 0.0675523i 0.999429 + 0.0337762i $$0.0107533\pi$$
−0.999429 + 0.0337762i $$0.989247\pi$$
$$102$$ 0 0
$$103$$ −17.4665 −1.72102 −0.860512 0.509430i $$-0.829856\pi$$
−0.860512 + 0.509430i $$0.829856\pi$$
$$104$$ 0 0
$$105$$ 6.48528 0.632899
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ 16.7876i 1.60796i 0.594656 + 0.803980i $$0.297288\pi$$
−0.594656 + 0.803980i $$0.702712\pi$$
$$110$$ 0 0
$$111$$ 2.31788 0.220004
$$112$$ 0 0
$$113$$ 0.343146 0.0322804 0.0161402 0.999870i $$-0.494862\pi$$
0.0161402 + 0.999870i $$0.494862\pi$$
$$114$$ 0 0
$$115$$ − 31.3137i − 2.92002i
$$116$$ 0 0
$$117$$ 5.59587i 0.517338i
$$118$$ 0 0
$$119$$ 1.35778 0.124468
$$120$$ 0 0
$$121$$ −12.3137 −1.11943
$$122$$ 0 0
$$123$$ − 3.17157i − 0.285971i
$$124$$ 0 0
$$125$$ − 22.3835i − 2.00204i
$$126$$ 0 0
$$127$$ −14.1885 −1.25903 −0.629513 0.776990i $$-0.716746\pi$$
−0.629513 + 0.776990i $$0.716746\pi$$
$$128$$ 0 0
$$129$$ 4.48528 0.394907
$$130$$ 0 0
$$131$$ − 7.31371i − 0.639002i −0.947586 0.319501i $$-0.896485\pi$$
0.947586 0.319501i $$-0.103515\pi$$
$$132$$ 0 0
$$133$$ 4.63577i 0.401972i
$$134$$ 0 0
$$135$$ 3.95687 0.340554
$$136$$ 0 0
$$137$$ 16.1421 1.37912 0.689558 0.724231i $$-0.257805\pi$$
0.689558 + 0.724231i $$0.257805\pi$$
$$138$$ 0 0
$$139$$ 8.00000i 0.678551i 0.940687 + 0.339276i $$0.110182\pi$$
−0.940687 + 0.339276i $$0.889818\pi$$
$$140$$ 0 0
$$141$$ 7.91375i 0.666458i
$$142$$ 0 0
$$143$$ −27.0192 −2.25946
$$144$$ 0 0
$$145$$ −28.6274 −2.37738
$$146$$ 0 0
$$147$$ − 4.31371i − 0.355789i
$$148$$ 0 0
$$149$$ 3.95687i 0.324160i 0.986778 + 0.162080i $$0.0518202\pi$$
−0.986778 + 0.162080i $$0.948180\pi$$
$$150$$ 0 0
$$151$$ −14.1885 −1.15464 −0.577322 0.816516i $$-0.695902\pi$$
−0.577322 + 0.816516i $$0.695902\pi$$
$$152$$ 0 0
$$153$$ 0.828427 0.0669744
$$154$$ 0 0
$$155$$ − 6.48528i − 0.520910i
$$156$$ 0 0
$$157$$ − 18.1454i − 1.44816i −0.689717 0.724080i $$-0.742265\pi$$
0.689717 0.724080i $$-0.257735\pi$$
$$158$$ 0 0
$$159$$ 0.678892 0.0538397
$$160$$ 0 0
$$161$$ 12.9706 1.02222
$$162$$ 0 0
$$163$$ 10.1421i 0.794393i 0.917734 + 0.397197i $$0.130017\pi$$
−0.917734 + 0.397197i $$0.869983\pi$$
$$164$$ 0 0
$$165$$ 19.1055i 1.48736i
$$166$$ 0 0
$$167$$ 4.63577 0.358726 0.179363 0.983783i $$-0.442596\pi$$
0.179363 + 0.983783i $$0.442596\pi$$
$$168$$ 0 0
$$169$$ −18.3137 −1.40875
$$170$$ 0 0
$$171$$ 2.82843i 0.216295i
$$172$$ 0 0
$$173$$ − 11.8706i − 0.902507i −0.892396 0.451253i $$-0.850977\pi$$
0.892396 0.451253i $$-0.149023\pi$$
$$174$$ 0 0
$$175$$ 17.4665 1.32034
$$176$$ 0 0
$$177$$ 9.65685 0.725854
$$178$$ 0 0
$$179$$ 12.9706i 0.969465i 0.874662 + 0.484733i $$0.161083\pi$$
−0.874662 + 0.484733i $$0.838917\pi$$
$$180$$ 0 0
$$181$$ 16.7876i 1.24781i 0.781499 + 0.623906i $$0.214455\pi$$
−0.781499 + 0.623906i $$0.785545\pi$$
$$182$$ 0 0
$$183$$ −2.31788 −0.171343
$$184$$ 0 0
$$185$$ 9.17157 0.674307
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ 1.63899i 0.119219i
$$190$$ 0 0
$$191$$ −17.7477 −1.28418 −0.642089 0.766630i $$-0.721932\pi$$
−0.642089 + 0.766630i $$0.721932\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 22.1421i 1.58563i
$$196$$ 0 0
$$197$$ − 16.5064i − 1.17603i −0.808849 0.588016i $$-0.799909\pi$$
0.808849 0.588016i $$-0.200091\pi$$
$$198$$ 0 0
$$199$$ −1.63899 −0.116185 −0.0580925 0.998311i $$-0.518502\pi$$
−0.0580925 + 0.998311i $$0.518502\pi$$
$$200$$ 0 0
$$201$$ 13.6569 0.963280
$$202$$ 0 0
$$203$$ − 11.8579i − 0.832259i
$$204$$ 0 0
$$205$$ − 12.5495i − 0.876496i
$$206$$ 0 0
$$207$$ 7.91375 0.550044
$$208$$ 0 0
$$209$$ −13.6569 −0.944664
$$210$$ 0 0
$$211$$ 8.00000i 0.550743i 0.961338 + 0.275371i $$0.0888008\pi$$
−0.961338 + 0.275371i $$0.911199\pi$$
$$212$$ 0 0
$$213$$ 3.27798i 0.224604i
$$214$$ 0 0
$$215$$ 17.7477 1.21038
$$216$$ 0 0
$$217$$ 2.68629 0.182357
$$218$$ 0 0
$$219$$ − 4.00000i − 0.270295i
$$220$$ 0 0
$$221$$ 4.63577i 0.311835i
$$222$$ 0 0
$$223$$ 14.1885 0.950133 0.475066 0.879950i $$-0.342424\pi$$
0.475066 + 0.879950i $$0.342424\pi$$
$$224$$ 0 0
$$225$$ 10.6569 0.710457
$$226$$ 0 0
$$227$$ − 12.1421i − 0.805902i −0.915222 0.402951i $$-0.867985\pi$$
0.915222 0.402951i $$-0.132015\pi$$
$$228$$ 0 0
$$229$$ − 21.4234i − 1.41570i −0.706365 0.707848i $$-0.749666\pi$$
0.706365 0.707848i $$-0.250334\pi$$
$$230$$ 0 0
$$231$$ −7.91375 −0.520686
$$232$$ 0 0
$$233$$ 21.3137 1.39631 0.698154 0.715948i $$-0.254005\pi$$
0.698154 + 0.715948i $$0.254005\pi$$
$$234$$ 0 0
$$235$$ 31.3137i 2.04268i
$$236$$ 0 0
$$237$$ 1.63899i 0.106464i
$$238$$ 0 0
$$239$$ −4.63577 −0.299863 −0.149931 0.988696i $$-0.547905\pi$$
−0.149931 + 0.988696i $$0.547905\pi$$
$$240$$ 0 0
$$241$$ −16.9706 −1.09317 −0.546585 0.837404i $$-0.684072\pi$$
−0.546585 + 0.837404i $$0.684072\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ − 17.0688i − 1.09049i
$$246$$ 0 0
$$247$$ −15.8275 −1.00708
$$248$$ 0 0
$$249$$ −8.82843 −0.559479
$$250$$ 0 0
$$251$$ − 20.8284i − 1.31468i −0.753595 0.657339i $$-0.771682\pi$$
0.753595 0.657339i $$-0.228318\pi$$
$$252$$ 0 0
$$253$$ 38.2110i 2.40230i
$$254$$ 0 0
$$255$$ 3.27798 0.205275
$$256$$ 0 0
$$257$$ −19.6569 −1.22616 −0.613080 0.790020i $$-0.710070\pi$$
−0.613080 + 0.790020i $$0.710070\pi$$
$$258$$ 0 0
$$259$$ 3.79899i 0.236058i
$$260$$ 0 0
$$261$$ − 7.23486i − 0.447826i
$$262$$ 0 0
$$263$$ 15.8275 0.975965 0.487983 0.872853i $$-0.337733\pi$$
0.487983 + 0.872853i $$0.337733\pi$$
$$264$$ 0 0
$$265$$ 2.68629 0.165018
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ 0 0
$$269$$ − 7.23486i − 0.441117i −0.975374 0.220558i $$-0.929212\pi$$
0.975374 0.220558i $$-0.0707880\pi$$
$$270$$ 0 0
$$271$$ −17.4665 −1.06101 −0.530507 0.847681i $$-0.677998\pi$$
−0.530507 + 0.847681i $$0.677998\pi$$
$$272$$ 0 0
$$273$$ −9.17157 −0.555089
$$274$$ 0 0
$$275$$ 51.4558i 3.10290i
$$276$$ 0 0
$$277$$ − 14.8674i − 0.893295i −0.894710 0.446648i $$-0.852618\pi$$
0.894710 0.446648i $$-0.147382\pi$$
$$278$$ 0 0
$$279$$ 1.63899 0.0981238
$$280$$ 0 0
$$281$$ −0.343146 −0.0204704 −0.0102352 0.999948i $$-0.503258\pi$$
−0.0102352 + 0.999948i $$0.503258\pi$$
$$282$$ 0 0
$$283$$ 28.9706i 1.72212i 0.508502 + 0.861061i $$0.330199\pi$$
−0.508502 + 0.861061i $$0.669801\pi$$
$$284$$ 0 0
$$285$$ 11.1917i 0.662941i
$$286$$ 0 0
$$287$$ 5.19818 0.306839
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ 11.6569i 0.683337i
$$292$$ 0 0
$$293$$ − 0.678892i − 0.0396613i −0.999803 0.0198307i $$-0.993687\pi$$
0.999803 0.0198307i $$-0.00631271\pi$$
$$294$$ 0 0
$$295$$ 38.2110 2.22473
$$296$$ 0 0
$$297$$ −4.82843 −0.280174
$$298$$ 0 0
$$299$$ 44.2843i 2.56103i
$$300$$ 0 0
$$301$$ 7.35134i 0.423724i
$$302$$ 0 0
$$303$$ −0.678892 −0.0390013
$$304$$ 0 0
$$305$$ −9.17157 −0.525163
$$306$$ 0 0
$$307$$ − 21.6569i − 1.23602i −0.786169 0.618011i $$-0.787939\pi$$
0.786169 0.618011i $$-0.212061\pi$$
$$308$$ 0 0
$$309$$ − 17.4665i − 0.993634i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −7.31371 −0.413395 −0.206698 0.978405i $$-0.566272\pi$$
−0.206698 + 0.978405i $$0.566272\pi$$
$$314$$ 0 0
$$315$$ 6.48528i 0.365404i
$$316$$ 0 0
$$317$$ 21.7046i 1.21905i 0.792767 + 0.609525i $$0.208640\pi$$
−0.792767 + 0.609525i $$0.791360\pi$$
$$318$$ 0 0
$$319$$ 34.9330 1.95587
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 2.34315i 0.130376i
$$324$$ 0 0
$$325$$ 59.6343i 3.30792i
$$326$$ 0 0
$$327$$ −16.7876 −0.928356
$$328$$ 0 0
$$329$$ −12.9706 −0.715090
$$330$$ 0 0
$$331$$ − 10.3431i − 0.568511i −0.958749 0.284255i $$-0.908254\pi$$
0.958749 0.284255i $$-0.0917463\pi$$
$$332$$ 0 0
$$333$$ 2.31788i 0.127019i
$$334$$ 0 0
$$335$$ 54.0385 2.95244
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 0 0
$$339$$ 0.343146i 0.0186371i
$$340$$ 0 0
$$341$$ 7.91375i 0.428554i
$$342$$ 0 0
$$343$$ 18.5431 1.00123
$$344$$ 0 0
$$345$$ 31.3137 1.68587
$$346$$ 0 0
$$347$$ − 0.828427i − 0.0444723i −0.999753 0.0222361i $$-0.992921\pi$$
0.999753 0.0222361i $$-0.00707857\pi$$
$$348$$ 0 0
$$349$$ − 4.23808i − 0.226859i −0.993546 0.113430i $$-0.963816\pi$$
0.993546 0.113430i $$-0.0361836\pi$$
$$350$$ 0 0
$$351$$ −5.59587 −0.298685
$$352$$ 0 0
$$353$$ 1.31371 0.0699216 0.0349608 0.999389i $$-0.488869\pi$$
0.0349608 + 0.999389i $$0.488869\pi$$
$$354$$ 0 0
$$355$$ 12.9706i 0.688406i
$$356$$ 0 0
$$357$$ 1.35778i 0.0718616i
$$358$$ 0 0
$$359$$ −34.9330 −1.84369 −0.921846 0.387556i $$-0.873319\pi$$
−0.921846 + 0.387556i $$0.873319\pi$$
$$360$$ 0 0
$$361$$ 11.0000 0.578947
$$362$$ 0 0
$$363$$ − 12.3137i − 0.646302i
$$364$$ 0 0
$$365$$ − 15.8275i − 0.828449i
$$366$$ 0 0
$$367$$ −24.0225 −1.25396 −0.626981 0.779035i $$-0.715710\pi$$
−0.626981 + 0.779035i $$0.715710\pi$$
$$368$$ 0 0
$$369$$ 3.17157 0.165105
$$370$$ 0 0
$$371$$ 1.11270i 0.0577684i
$$372$$ 0 0
$$373$$ − 20.0656i − 1.03896i −0.854484 0.519478i $$-0.826126\pi$$
0.854484 0.519478i $$-0.173874\pi$$
$$374$$ 0 0
$$375$$ 22.3835 1.15588
$$376$$ 0 0
$$377$$ 40.4853 2.08510
$$378$$ 0 0
$$379$$ 0.485281i 0.0249272i 0.999922 + 0.0124636i $$0.00396739\pi$$
−0.999922 + 0.0124636i $$0.996033\pi$$
$$380$$ 0 0
$$381$$ − 14.1885i − 0.726899i
$$382$$ 0 0
$$383$$ −11.1917 −0.571871 −0.285935 0.958249i $$-0.592304\pi$$
−0.285935 + 0.958249i $$0.592304\pi$$
$$384$$ 0 0
$$385$$ −31.3137 −1.59589
$$386$$ 0 0
$$387$$ 4.48528i 0.228000i
$$388$$ 0 0
$$389$$ − 3.95687i − 0.200621i −0.994956 0.100311i $$-0.968016\pi$$
0.994956 0.100311i $$-0.0319837\pi$$
$$390$$ 0 0
$$391$$ 6.55596 0.331549
$$392$$ 0 0
$$393$$ 7.31371 0.368928
$$394$$ 0 0
$$395$$ 6.48528i 0.326310i
$$396$$ 0 0
$$397$$ − 18.1454i − 0.910691i −0.890315 0.455345i $$-0.849516\pi$$
0.890315 0.455345i $$-0.150484\pi$$
$$398$$ 0 0
$$399$$ −4.63577 −0.232079
$$400$$ 0 0
$$401$$ 24.1421 1.20560 0.602800 0.797892i $$-0.294052\pi$$
0.602800 + 0.797892i $$0.294052\pi$$
$$402$$ 0 0
$$403$$ 9.17157i 0.456869i
$$404$$ 0 0
$$405$$ 3.95687i 0.196619i
$$406$$ 0 0
$$407$$ −11.1917 −0.554753
$$408$$ 0 0
$$409$$ 3.65685 0.180820 0.0904099 0.995905i $$-0.471182\pi$$
0.0904099 + 0.995905i $$0.471182\pi$$
$$410$$ 0 0
$$411$$ 16.1421i 0.796233i
$$412$$ 0 0
$$413$$ 15.8275i 0.778820i
$$414$$ 0 0
$$415$$ −34.9330 −1.71479
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 15.1716i 0.741180i 0.928797 + 0.370590i $$0.120844\pi$$
−0.928797 + 0.370590i $$0.879156\pi$$
$$420$$ 0 0
$$421$$ − 21.4234i − 1.04411i −0.852912 0.522055i $$-0.825165\pi$$
0.852912 0.522055i $$-0.174835\pi$$
$$422$$ 0 0
$$423$$ −7.91375 −0.384780
$$424$$ 0 0
$$425$$ 8.82843 0.428242
$$426$$ 0 0
$$427$$ − 3.79899i − 0.183846i
$$428$$ 0 0
$$429$$ − 27.0192i − 1.30450i
$$430$$ 0 0
$$431$$ −14.4697 −0.696982 −0.348491 0.937312i $$-0.613306\pi$$
−0.348491 + 0.937312i $$0.613306\pi$$
$$432$$ 0 0
$$433$$ 24.6274 1.18352 0.591759 0.806115i $$-0.298434\pi$$
0.591759 + 0.806115i $$0.298434\pi$$
$$434$$ 0 0
$$435$$ − 28.6274i − 1.37258i
$$436$$ 0 0
$$437$$ 22.3835i 1.07075i
$$438$$ 0 0
$$439$$ −39.8499 −1.90193 −0.950967 0.309292i $$-0.899908\pi$$
−0.950967 + 0.309292i $$0.899908\pi$$
$$440$$ 0 0
$$441$$ 4.31371 0.205415
$$442$$ 0 0
$$443$$ 18.4853i 0.878262i 0.898423 + 0.439131i $$0.144714\pi$$
−0.898423 + 0.439131i $$0.855286\pi$$
$$444$$ 0 0
$$445$$ − 39.5687i − 1.87574i
$$446$$ 0 0
$$447$$ −3.95687 −0.187154
$$448$$ 0 0
$$449$$ 17.5147 0.826571 0.413285 0.910602i $$-0.364381\pi$$
0.413285 + 0.910602i $$0.364381\pi$$
$$450$$ 0 0
$$451$$ 15.3137i 0.721094i
$$452$$ 0 0
$$453$$ − 14.1885i − 0.666634i
$$454$$ 0 0
$$455$$ −36.2908 −1.70134
$$456$$ 0 0
$$457$$ 23.6569 1.10662 0.553310 0.832975i $$-0.313364\pi$$
0.553310 + 0.832975i $$0.313364\pi$$
$$458$$ 0 0
$$459$$ 0.828427i 0.0386677i
$$460$$ 0 0
$$461$$ 32.8963i 1.53213i 0.642761 + 0.766067i $$0.277789\pi$$
−0.642761 + 0.766067i $$0.722211\pi$$
$$462$$ 0 0
$$463$$ 10.9105 0.507055 0.253528 0.967328i $$-0.418409\pi$$
0.253528 + 0.967328i $$0.418409\pi$$
$$464$$ 0 0
$$465$$ 6.48528 0.300748
$$466$$ 0 0
$$467$$ 37.7990i 1.74913i 0.484910 + 0.874564i $$0.338853\pi$$
−0.484910 + 0.874564i $$0.661147\pi$$
$$468$$ 0 0
$$469$$ 22.3835i 1.03357i
$$470$$ 0 0
$$471$$ 18.1454 0.836095
$$472$$ 0 0
$$473$$ −21.6569 −0.995783
$$474$$ 0 0
$$475$$ 30.1421i 1.38302i
$$476$$ 0 0
$$477$$ 0.678892i 0.0310843i
$$478$$ 0 0
$$479$$ −32.2174 −1.47205 −0.736025 0.676954i $$-0.763300\pi$$
−0.736025 + 0.676954i $$0.763300\pi$$
$$480$$ 0 0
$$481$$ −12.9706 −0.591407
$$482$$ 0 0
$$483$$ 12.9706i 0.590181i
$$484$$ 0 0
$$485$$ 46.1247i 2.09442i
$$486$$ 0 0
$$487$$ 20.7445 0.940022 0.470011 0.882661i $$-0.344250\pi$$
0.470011 + 0.882661i $$0.344250\pi$$
$$488$$ 0 0
$$489$$ −10.1421 −0.458643
$$490$$ 0 0
$$491$$ − 1.65685i − 0.0747728i −0.999301 0.0373864i $$-0.988097\pi$$
0.999301 0.0373864i $$-0.0119032\pi$$
$$492$$ 0 0
$$493$$ − 5.99355i − 0.269936i
$$494$$ 0 0
$$495$$ −19.1055 −0.858727
$$496$$ 0 0
$$497$$ −5.37258 −0.240993
$$498$$ 0 0
$$499$$ − 30.3431i − 1.35835i −0.733978 0.679173i $$-0.762339\pi$$
0.733978 0.679173i $$-0.237661\pi$$
$$500$$ 0 0
$$501$$ 4.63577i 0.207111i
$$502$$ 0 0
$$503$$ −12.5495 −0.559555 −0.279778 0.960065i $$-0.590261\pi$$
−0.279778 + 0.960065i $$0.590261\pi$$
$$504$$ 0 0
$$505$$ −2.68629 −0.119538
$$506$$ 0 0
$$507$$ − 18.3137i − 0.813340i
$$508$$ 0 0
$$509$$ − 8.59264i − 0.380862i −0.981701 0.190431i $$-0.939011\pi$$
0.981701 0.190431i $$-0.0609886\pi$$
$$510$$ 0 0
$$511$$ 6.55596 0.290019
$$512$$ 0 0
$$513$$ −2.82843 −0.124878
$$514$$ 0 0
$$515$$ − 69.1127i − 3.04547i
$$516$$ 0 0
$$517$$ − 38.2110i − 1.68052i
$$518$$ 0 0
$$519$$ 11.8706 0.521063
$$520$$ 0 0
$$521$$ −15.1716 −0.664679 −0.332339 0.943160i $$-0.607838\pi$$
−0.332339 + 0.943160i $$0.607838\pi$$
$$522$$ 0 0
$$523$$ − 13.1716i − 0.575953i −0.957638 0.287976i $$-0.907018\pi$$
0.957638 0.287976i $$-0.0929824\pi$$
$$524$$ 0 0
$$525$$ 17.4665i 0.762300i
$$526$$ 0 0
$$527$$ 1.35778 0.0591460
$$528$$ 0 0
$$529$$ 39.6274 1.72293
$$530$$ 0 0
$$531$$ 9.65685i 0.419072i
$$532$$ 0 0
$$533$$ 17.7477i 0.768738i
$$534$$ 0 0
$$535$$ −15.8275 −0.684282
$$536$$ 0 0
$$537$$ −12.9706 −0.559721
$$538$$ 0 0
$$539$$ 20.8284i 0.897144i
$$540$$ 0 0
$$541$$ − 10.2316i − 0.439892i −0.975512 0.219946i $$-0.929412\pi$$
0.975512 0.219946i $$-0.0705882\pi$$
$$542$$ 0 0
$$543$$ −16.7876 −0.720425
$$544$$ 0 0
$$545$$ −66.4264 −2.84539
$$546$$ 0 0
$$547$$ − 7.79899i − 0.333461i −0.986003 0.166730i $$-0.946679\pi$$
0.986003 0.166730i $$-0.0533209\pi$$
$$548$$ 0 0
$$549$$ − 2.31788i − 0.0989248i
$$550$$ 0 0
$$551$$ 20.4633 0.871764
$$552$$ 0 0
$$553$$ −2.68629 −0.114233
$$554$$ 0 0
$$555$$ 9.17157i 0.389312i
$$556$$ 0 0
$$557$$ 27.6981i 1.17361i 0.809729 + 0.586804i $$0.199614\pi$$
−0.809729 + 0.586804i $$0.800386\pi$$
$$558$$ 0 0
$$559$$ −25.0990 −1.06158
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ 4.82843i 0.203494i 0.994810 + 0.101747i $$0.0324432\pi$$
−0.994810 + 0.101747i $$0.967557\pi$$
$$564$$ 0 0
$$565$$ 1.35778i 0.0571224i
$$566$$ 0 0
$$567$$ −1.63899 −0.0688312
$$568$$ 0 0
$$569$$ 27.4558 1.15101 0.575504 0.817799i $$-0.304806\pi$$
0.575504 + 0.817799i $$0.304806\pi$$
$$570$$ 0 0
$$571$$ − 0.686292i − 0.0287204i −0.999897 0.0143602i $$-0.995429\pi$$
0.999897 0.0143602i $$-0.00457115\pi$$
$$572$$ 0 0
$$573$$ − 17.7477i − 0.741421i
$$574$$ 0 0
$$575$$ 84.3357 3.51704
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ − 4.00000i − 0.166234i
$$580$$ 0 0
$$581$$ − 14.4697i − 0.600305i
$$582$$ 0 0
$$583$$ −3.27798 −0.135760
$$584$$ 0 0
$$585$$ −22.1421 −0.915465
$$586$$ 0 0
$$587$$ − 28.9706i − 1.19574i −0.801592 0.597872i $$-0.796013\pi$$
0.801592 0.597872i $$-0.203987\pi$$
$$588$$ 0 0
$$589$$ 4.63577i 0.191013i
$$590$$ 0 0
$$591$$ 16.5064 0.678982
$$592$$ 0 0
$$593$$ 30.9706 1.27181 0.635904 0.771768i $$-0.280627\pi$$
0.635904 + 0.771768i $$0.280627\pi$$
$$594$$ 0 0
$$595$$ 5.37258i 0.220254i
$$596$$ 0 0
$$597$$ − 1.63899i − 0.0670794i
$$598$$ 0 0
$$599$$ 46.1247 1.88460 0.942302 0.334763i $$-0.108656\pi$$
0.942302 + 0.334763i $$0.108656\pi$$
$$600$$ 0 0
$$601$$ 27.3137 1.11415 0.557075 0.830462i $$-0.311924\pi$$
0.557075 + 0.830462i $$0.311924\pi$$
$$602$$ 0 0
$$603$$ 13.6569i 0.556150i
$$604$$ 0 0
$$605$$ − 48.7238i − 1.98090i
$$606$$ 0 0
$$607$$ 8.19496 0.332623 0.166311 0.986073i $$-0.446814\pi$$
0.166311 + 0.986073i $$0.446814\pi$$
$$608$$ 0 0
$$609$$ 11.8579 0.480505
$$610$$ 0 0
$$611$$ − 44.2843i − 1.79155i
$$612$$ 0 0
$$613$$ − 35.8931i − 1.44971i −0.688903 0.724854i $$-0.741907\pi$$
0.688903 0.724854i $$-0.258093\pi$$
$$614$$ 0 0
$$615$$ 12.5495 0.506045
$$616$$ 0 0
$$617$$ −23.9411 −0.963833 −0.481917 0.876217i $$-0.660059\pi$$
−0.481917 + 0.876217i $$0.660059\pi$$
$$618$$ 0 0
$$619$$ 28.0000i 1.12542i 0.826656 + 0.562708i $$0.190240\pi$$
−0.826656 + 0.562708i $$0.809760\pi$$
$$620$$ 0 0
$$621$$ 7.91375i 0.317568i
$$622$$ 0 0
$$623$$ 16.3899 0.656648
$$624$$ 0 0
$$625$$ 35.2843 1.41137
$$626$$ 0 0
$$627$$ − 13.6569i − 0.545402i
$$628$$ 0 0
$$629$$ 1.92020i 0.0765633i
$$630$$ 0 0
$$631$$ 11.4729 0.456730 0.228365 0.973576i $$-0.426662\pi$$
0.228365 + 0.973576i $$0.426662\pi$$
$$632$$ 0 0
$$633$$ −8.00000 −0.317971
$$634$$ 0 0
$$635$$ − 56.1421i − 2.22793i
$$636$$ 0 0
$$637$$ 24.1389i 0.956419i
$$638$$ 0 0
$$639$$ −3.27798 −0.129675
$$640$$ 0 0
$$641$$ 27.1716 1.07321 0.536606 0.843833i $$-0.319706\pi$$
0.536606 + 0.843833i $$0.319706\pi$$
$$642$$ 0 0
$$643$$ 20.4853i 0.807861i 0.914790 + 0.403930i $$0.132356\pi$$
−0.914790 + 0.403930i $$0.867644\pi$$
$$644$$ 0 0
$$645$$ 17.7477i 0.698815i
$$646$$ 0 0
$$647$$ −5.19818 −0.204362 −0.102181 0.994766i $$-0.532582\pi$$
−0.102181 + 0.994766i $$0.532582\pi$$
$$648$$ 0 0
$$649$$ −46.6274 −1.83029
$$650$$ 0 0
$$651$$ 2.68629i 0.105284i
$$652$$ 0 0
$$653$$ 13.2284i 0.517668i 0.965922 + 0.258834i $$0.0833382\pi$$
−0.965922 + 0.258834i $$0.916662\pi$$
$$654$$ 0 0
$$655$$ 28.9394 1.13076
$$656$$ 0 0
$$657$$ 4.00000 0.156055
$$658$$ 0 0
$$659$$ 1.65685i 0.0645419i 0.999479 + 0.0322709i $$0.0102739\pi$$
−0.999479 + 0.0322709i $$0.989726\pi$$
$$660$$ 0 0
$$661$$ − 4.23808i − 0.164842i −0.996598 0.0824211i $$-0.973735\pi$$
0.996598 0.0824211i $$-0.0262653\pi$$
$$662$$ 0 0
$$663$$ −4.63577 −0.180038
$$664$$ 0 0
$$665$$ −18.3431 −0.711317
$$666$$ 0 0
$$667$$ − 57.2548i − 2.21692i
$$668$$ 0 0
$$669$$ 14.1885i 0.548559i
$$670$$ 0 0
$$671$$ 11.1917 0.432052
$$672$$ 0 0
$$673$$ 14.6863 0.566115 0.283057 0.959103i $$-0.408651\pi$$
0.283057 + 0.959103i $$0.408651\pi$$
$$674$$ 0 0
$$675$$ 10.6569i 0.410183i
$$676$$ 0 0
$$677$$ − 2.59909i − 0.0998911i −0.998752 0.0499456i $$-0.984095\pi$$
0.998752 0.0499456i $$-0.0159048\pi$$
$$678$$ 0 0
$$679$$ −19.1055 −0.733201
$$680$$ 0 0
$$681$$ 12.1421 0.465288
$$682$$ 0 0
$$683$$ 5.51472i 0.211015i 0.994419 + 0.105507i $$0.0336467\pi$$
−0.994419 + 0.105507i $$0.966353\pi$$
$$684$$ 0 0
$$685$$ 63.8724i 2.44044i
$$686$$ 0 0
$$687$$ 21.4234 0.817352
$$688$$ 0 0
$$689$$ −3.79899 −0.144730
$$690$$ 0 0
$$691$$ 38.8284i 1.47710i 0.674197 + 0.738551i $$0.264490\pi$$
−0.674197 + 0.738551i $$0.735510\pi$$
$$692$$ 0 0
$$693$$ − 7.91375i − 0.300618i
$$694$$ 0 0
$$695$$ −31.6550 −1.20074
$$696$$ 0 0
$$697$$ 2.62742 0.0995205
$$698$$ 0 0
$$699$$ 21.3137i 0.806158i
$$700$$ 0 0
$$701$$ 30.9761i 1.16995i 0.811051 + 0.584976i $$0.198896\pi$$
−0.811051 + 0.584976i $$0.801104\pi$$
$$702$$ 0 0
$$703$$ −6.55596 −0.247263
$$704$$ 0 0
$$705$$ −31.3137 −1.17934
$$706$$ 0 0
$$707$$ − 1.11270i − 0.0418473i
$$708$$ 0 0
$$709$$ 26.0591i 0.978671i 0.872096 + 0.489336i $$0.162761\pi$$
−0.872096 + 0.489336i $$0.837239\pi$$
$$710$$ 0 0
$$711$$ −1.63899 −0.0614670
$$712$$ 0 0
$$713$$ 12.9706 0.485751
$$714$$ 0 0
$$715$$ − 106.912i − 3.99827i
$$716$$ 0 0
$$717$$ − 4.63577i − 0.173126i
$$718$$ 0 0
$$719$$ −34.9330 −1.30278 −0.651390 0.758743i $$-0.725814\pi$$
−0.651390 + 0.758743i $$0.725814\pi$$
$$720$$ 0 0
$$721$$ 28.6274 1.06614
$$722$$ 0 0
$$723$$ − 16.9706i − 0.631142i
$$724$$ 0 0
$$725$$ − 77.1008i − 2.86345i
$$726$$ 0 0
$$727$$ 30.5784 1.13409 0.567045 0.823687i $$-0.308086\pi$$
0.567045 + 0.823687i $$0.308086\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 3.71573i 0.137431i
$$732$$ 0 0
$$733$$ 18.7078i 0.690988i 0.938421 + 0.345494i $$0.112289\pi$$
−0.938421 + 0.345494i $$0.887711\pi$$
$$734$$ 0 0
$$735$$ 17.0688 0.629592
$$736$$ 0 0
$$737$$ −65.9411 −2.42897
$$738$$ 0 0
$$739$$ − 4.00000i − 0.147142i −0.997290 0.0735712i $$-0.976560\pi$$
0.997290 0.0735712i $$-0.0234396\pi$$
$$740$$ 0 0
$$741$$ − 15.8275i − 0.581438i
$$742$$ 0 0
$$743$$ 33.5752 1.23175 0.615877 0.787842i $$-0.288802\pi$$
0.615877 + 0.787842i $$0.288802\pi$$
$$744$$ 0 0
$$745$$ −15.6569 −0.573623
$$746$$ 0 0
$$747$$ − 8.82843i − 0.323015i
$$748$$ 0 0
$$749$$ − 6.55596i − 0.239550i
$$750$$ 0 0
$$751$$ 1.63899 0.0598076 0.0299038 0.999553i $$-0.490480\pi$$
0.0299038 + 0.999553i $$0.490480\pi$$
$$752$$ 0 0
$$753$$ 20.8284 0.759030
$$754$$ 0 0
$$755$$ − 56.1421i − 2.04322i
$$756$$ 0 0
$$757$$ 0.960099i 0.0348954i 0.999848 + 0.0174477i $$0.00555405\pi$$
−0.999848 + 0.0174477i $$0.994446\pi$$
$$758$$ 0 0
$$759$$ −38.2110 −1.38697
$$760$$ 0 0
$$761$$ −30.7696 −1.11540 −0.557698 0.830044i $$-0.688315\pi$$
−0.557698 + 0.830044i $$0.688315\pi$$
$$762$$ 0 0
$$763$$ − 27.5147i − 0.996100i
$$764$$ 0 0
$$765$$ 3.27798i 0.118516i
$$766$$ 0 0
$$767$$ −54.0385 −1.95122
$$768$$ 0 0
$$769$$ 12.0000 0.432731 0.216366 0.976312i $$-0.430580\pi$$
0.216366 + 0.976312i $$0.430580\pi$$
$$770$$ 0 0
$$771$$ − 19.6569i − 0.707924i
$$772$$ 0 0
$$773$$ − 5.87707i − 0.211384i −0.994399 0.105692i $$-0.966294\pi$$
0.994399 0.105692i $$-0.0337057\pi$$
$$774$$ 0 0
$$775$$ 17.4665 0.627415
$$776$$ 0 0
$$777$$ −3.79899 −0.136288
$$778$$ 0 0
$$779$$ 8.97056i 0.321404i
$$780$$ 0 0
$$781$$ − 15.8275i − 0.566352i
$$782$$ 0 0
$$783$$ 7.23486 0.258553
$$784$$ 0 0
$$785$$ 71.7990 2.56262
$$786$$ 0 0
$$787$$ − 20.7696i − 0.740355i −0.928961 0.370177i $$-0.879297\pi$$
0.928961 0.370177i $$-0.120703\pi$$
$$788$$ 0 0
$$789$$ 15.8275i 0.563474i
$$790$$ 0 0
$$791$$ −0.562413 −0.0199971
$$792$$ 0 0
$$793$$ 12.9706 0.460598
$$794$$ 0 0
$$795$$ 2.68629i 0.0952729i
$$796$$ 0 0
$$797$$ − 13.2284i − 0.468574i −0.972167 0.234287i $$-0.924724\pi$$
0.972167 0.234287i $$-0.0752756\pi$$
$$798$$ 0 0
$$799$$ −6.55596 −0.231933
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 19.3137i 0.681566i
$$804$$ 0 0
$$805$$ 51.3229i 1.80889i
$$806$$ 0 0
$$807$$ 7.23486 0.254679
$$808$$ 0 0
$$809$$ −47.4558 −1.66846 −0.834229 0.551418i $$-0.814087\pi$$
−0.834229 + 0.551418i $$0.814087\pi$$
$$810$$ 0 0
$$811$$ 1.45584i 0.0511216i 0.999673 + 0.0255608i $$0.00813714\pi$$
−0.999673 + 0.0255608i $$0.991863\pi$$
$$812$$ 0 0
$$813$$ − 17.4665i − 0.612576i
$$814$$ 0 0
$$815$$ −40.1312 −1.40573
$$816$$ 0 0
$$817$$ −12.6863 −0.443837
$$818$$ 0 0
$$819$$ − 9.17157i − 0.320481i
$$820$$ 0 0
$$821$$ − 2.03668i − 0.0710805i −0.999368 0.0355403i $$-0.988685\pi$$
0.999368 0.0355403i $$-0.0113152\pi$$
$$822$$ 0 0
$$823$$ 14.1885 0.494580 0.247290 0.968941i $$-0.420460\pi$$
0.247290 + 0.968941i $$0.420460\pi$$
$$824$$ 0 0
$$825$$ −51.4558 −1.79146
$$826$$ 0 0
$$827$$ − 13.9411i − 0.484780i −0.970179 0.242390i $$-0.922069\pi$$
0.970179 0.242390i $$-0.0779314\pi$$
$$828$$ 0 0
$$829$$ − 5.59587i − 0.194352i −0.995267 0.0971762i $$-0.969019\pi$$
0.995267 0.0971762i $$-0.0309810\pi$$
$$830$$ 0 0
$$831$$ 14.8674 0.515744
$$832$$ 0 0
$$833$$ 3.57359 0.123818
$$834$$ 0 0
$$835$$ 18.3431i 0.634791i
$$836$$ 0 0
$$837$$ 1.63899i 0.0566518i
$$838$$ 0 0
$$839$$ 32.2174 1.11227 0.556134 0.831092i $$-0.312284\pi$$
0.556134 + 0.831092i $$0.312284\pi$$
$$840$$ 0 0
$$841$$ −23.3431 −0.804936
$$842$$ 0 0
$$843$$ − 0.343146i − 0.0118186i
$$844$$ 0 0
$$845$$ − 72.4650i − 2.49287i
$$846$$ 0 0
$$847$$ 20.1821 0.693464
$$848$$ 0 0
$$849$$ −28.9706 −0.994267
$$850$$ 0 0
$$851$$ 18.3431i 0.628795i
$$852$$ 0 0
$$853$$ 18.1454i 0.621286i 0.950527 + 0.310643i $$0.100544\pi$$
−0.950527 + 0.310643i $$0.899456\pi$$
$$854$$ 0 0
$$855$$ −11.1917 −0.382749
$$856$$ 0 0
$$857$$ −13.5147 −0.461654 −0.230827 0.972995i $$-0.574143\pi$$
−0.230827 + 0.972995i $$0.574143\pi$$
$$858$$ 0 0
$$859$$ 29.4558i 1.00502i 0.864571 + 0.502510i $$0.167590\pi$$
−0.864571 + 0.502510i $$0.832410\pi$$
$$860$$ 0 0
$$861$$ 5.19818i 0.177153i
$$862$$ 0 0
$$863$$ 4.63577 0.157803 0.0789017 0.996882i $$-0.474859\pi$$
0.0789017 + 0.996882i $$0.474859\pi$$
$$864$$ 0 0
$$865$$ 46.9706 1.59705
$$866$$ 0 0
$$867$$ − 16.3137i − 0.554043i
$$868$$ 0 0
$$869$$ − 7.91375i − 0.268456i
$$870$$ 0 0
$$871$$ −76.4219 −2.58946
$$872$$ 0 0
$$873$$ −11.6569 −0.394525
$$874$$ 0 0
$$875$$ 36.6863i 1.24022i
$$876$$ 0 0
$$877$$ − 58.2765i − 1.96786i −0.178558 0.983929i $$-0.557143\pi$$
0.178558 0.983929i $$-0.442857\pi$$
$$878$$ 0 0
$$879$$ 0.678892 0.0228985
$$880$$ 0 0
$$881$$ −5.02944 −0.169446 −0.0847230 0.996405i $$-0.527001\pi$$
−0.0847230 + 0.996405i $$0.527001\pi$$
$$882$$ 0 0
$$883$$ − 48.7696i − 1.64123i −0.571484 0.820613i $$-0.693632\pi$$
0.571484 0.820613i $$-0.306368\pi$$
$$884$$ 0 0
$$885$$ 38.2110i 1.28445i
$$886$$ 0 0
$$887$$ 15.8275 0.531435 0.265718 0.964051i $$-0.414391\pi$$
0.265718 + 0.964051i $$0.414391\pi$$
$$888$$ 0 0
$$889$$ 23.2548 0.779942
$$890$$ 0 0
$$891$$ − 4.82843i − 0.161758i
$$892$$ 0 0
$$893$$ − 22.3835i − 0.749034i
$$894$$ 0 0
$$895$$ −51.3229 −1.71553
$$896$$ 0 0
$$897$$ −44.2843 −1.47861
$$898$$ 0 0
$$899$$ − 11.8579i − 0.395482i
$$900$$ 0 0
$$901$$ 0.562413i 0.0187367i
$$902$$ 0 0
$$903$$ −7.35134 −0.244637
$$904$$ 0 0
$$905$$ −66.4264 −2.20809
$$906$$ 0 0
$$907$$ − 15.7990i − 0.524597i −0.964987 0.262298i $$-0.915520\pi$$
0.964987 0.262298i $$-0.0844805\pi$$
$$908$$ 0 0
$$909$$ − 0.678892i − 0.0225174i
$$910$$ 0 0
$$911$$ 1.92020 0.0636190 0.0318095 0.999494i $$-0.489873\pi$$
0.0318095 + 0.999494i $$0.489873\pi$$
$$912$$ 0 0
$$913$$ 42.6274 1.41076
$$914$$ 0 0
$$915$$ − 9.17157i − 0.303203i
$$916$$ 0 0
$$917$$ 11.9871i 0.395849i
$$918$$ 0 0
$$919$$ 4.91697 0.162196 0.0810980 0.996706i $$-0.474157\pi$$
0.0810980 + 0.996706i $$0.474157\pi$$
$$920$$ 0 0
$$921$$ 21.6569 0.713618
$$922$$ 0 0
$$923$$ − 18.3431i − 0.603772i
$$924$$ 0 0
$$925$$ 24.7013i 0.812175i
$$926$$ 0 0
$$927$$ 17.4665 0.573675
$$928$$ 0 0
$$929$$ 21.7990 0.715202 0.357601 0.933875i $$-0.383595\pi$$
0.357601 + 0.933875i $$0.383595\pi$$
$$930$$ 0 0
$$931$$ 12.2010i 0.399872i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −15.8275 −0.517615
$$936$$ 0 0
$$937$$ 40.6274 1.32724 0.663620 0.748070i $$-0.269019\pi$$
0.663620 + 0.748070i $$0.269019\pi$$
$$938$$ 0 0
$$939$$ − 7.31371i − 0.238674i
$$940$$ 0 0
$$941$$ − 52.7972i − 1.72114i −0.509334 0.860569i $$-0.670108\pi$$
0.509334 0.860569i $$-0.329892\pi$$
$$942$$ 0 0
$$943$$ 25.0990 0.817337
$$944$$ 0 0
$$945$$ −6.48528 −0.210966
$$946$$ 0 0
$$947$$ − 28.9706i − 0.941417i −0.882289 0.470708i $$-0.843998\pi$$
0.882289 0.470708i $$-0.156002\pi$$
$$948$$ 0 0
$$949$$ 22.3835i 0.726598i
$$950$$ 0 0
$$951$$ −21.7046 −0.703819
$$952$$ 0 0
$$953$$ −29.7990 −0.965284 −0.482642 0.875818i $$-0.660323\pi$$
−0.482642 + 0.875818i $$0.660323\pi$$
$$954$$ 0 0
$$955$$ − 70.2254i − 2.27244i
$$956$$ 0 0
$$957$$ 34.9330i 1.12922i
$$958$$ 0 0
$$959$$ −26.4568 −0.854335
$$960$$ 0 0
$$961$$ −28.3137 −0.913345
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ 0 0
$$965$$ − 15.8275i − 0.509505i
$$966$$ 0 0
$$967$$ −43.1279 −1.38690 −0.693450 0.720504i $$-0.743910\pi$$
−0.693450 + 0.720504i $$0.743910\pi$$
$$968$$ 0 0
$$969$$ −2.34315 −0.0752727
$$970$$ 0 0
$$971$$ 15.4558i 0.496002i 0.968760 + 0.248001i $$0.0797736\pi$$
−0.968760 + 0.248001i $$0.920226\pi$$
$$972$$ 0 0
$$973$$ − 13.1119i − 0.420349i
$$974$$ 0 0
$$975$$ −59.6343 −1.90983
$$976$$ 0 0
$$977$$ −42.7696 −1.36832 −0.684160 0.729332i $$-0.739831\pi$$
−0.684160 + 0.729332i $$0.739831\pi$$
$$978$$ 0 0
$$979$$ 48.2843i 1.54317i
$$980$$ 0 0
$$981$$ − 16.7876i − 0.535987i
$$982$$ 0 0
$$983$$ −38.2110 −1.21874 −0.609370 0.792886i $$-0.708578\pi$$
−0.609370 + 0.792886i $$0.708578\pi$$
$$984$$ 0 0
$$985$$ 65.3137 2.08107
$$986$$ 0 0
$$987$$ − 12.9706i − 0.412858i
$$988$$ 0 0
$$989$$ 35.4954i 1.12869i
$$990$$ 0 0
$$991$$ 46.4059 1.47413 0.737066 0.675821i $$-0.236211\pi$$
0.737066 + 0.675821i $$0.236211\pi$$
$$992$$ 0 0
$$993$$ 10.3431 0.328230
$$994$$ 0 0
$$995$$ − 6.48528i − 0.205597i
$$996$$ 0 0
$$997$$ 29.3371i 0.929116i 0.885543 + 0.464558i $$0.153787\pi$$
−0.885543 + 0.464558i $$0.846213\pi$$
$$998$$ 0 0
$$999$$ −2.31788 −0.0733346
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 1536.2.d.g.769.8 8
3.2 odd 2 4608.2.d.p.2305.1 8
4.3 odd 2 inner 1536.2.d.g.769.4 8
8.3 odd 2 inner 1536.2.d.g.769.5 8
8.5 even 2 inner 1536.2.d.g.769.1 8
12.11 even 2 4608.2.d.p.2305.2 8
16.3 odd 4 1536.2.a.n.1.4 yes 4
16.5 even 4 1536.2.a.n.1.1 yes 4
16.11 odd 4 1536.2.a.m.1.1 4
16.13 even 4 1536.2.a.m.1.4 yes 4
24.5 odd 2 4608.2.d.p.2305.7 8
24.11 even 2 4608.2.d.p.2305.8 8
48.5 odd 4 4608.2.a.t.1.4 4
48.11 even 4 4608.2.a.ba.1.4 4
48.29 odd 4 4608.2.a.ba.1.1 4
48.35 even 4 4608.2.a.t.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.m.1.1 4 16.11 odd 4
1536.2.a.m.1.4 yes 4 16.13 even 4
1536.2.a.n.1.1 yes 4 16.5 even 4
1536.2.a.n.1.4 yes 4 16.3 odd 4
1536.2.d.g.769.1 8 8.5 even 2 inner
1536.2.d.g.769.4 8 4.3 odd 2 inner
1536.2.d.g.769.5 8 8.3 odd 2 inner
1536.2.d.g.769.8 8 1.1 even 1 trivial
4608.2.a.t.1.1 4 48.35 even 4
4608.2.a.t.1.4 4 48.5 odd 4
4608.2.a.ba.1.1 4 48.29 odd 4
4608.2.a.ba.1.4 4 48.11 even 4
4608.2.d.p.2305.1 8 3.2 odd 2
4608.2.d.p.2305.2 8 12.11 even 2
4608.2.d.p.2305.7 8 24.5 odd 2
4608.2.d.p.2305.8 8 24.11 even 2