# Properties

 Label 1680.2.t.h.1009.1 Level $1680$ Weight $2$ Character 1680.1009 Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.t (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1009.1 Root $$0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 1680.1009 Dual form 1680.2.t.h.1009.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -2.23607 q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -4.47214i q^{13} +2.23607i q^{15} -2.47214i q^{17} +2.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -0.472136 q^{29} -8.47214 q^{31} -2.00000i q^{33} +2.23607i q^{35} -6.47214i q^{37} -4.47214 q^{39} -12.4721 q^{41} +6.47214i q^{43} +2.23607 q^{45} +2.47214i q^{47} -1.00000 q^{49} -2.47214 q^{51} -2.00000i q^{53} -4.47214 q^{55} -2.00000i q^{57} -12.4721 q^{61} +1.00000i q^{63} +10.0000i q^{65} -10.4721i q^{67} +4.00000 q^{69} -3.52786 q^{71} -16.4721i q^{73} -5.00000i q^{75} -2.00000i q^{77} -8.94427 q^{79} +1.00000 q^{81} +12.9443i q^{83} +5.52786i q^{85} +0.472136i q^{87} -9.41641 q^{89} -4.47214 q^{91} +8.47214i q^{93} -4.47214 q^{95} +12.4721i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 8q^{11} + 8q^{19} - 4q^{21} + 20q^{25} + 16q^{29} - 16q^{31} - 32q^{41} - 4q^{49} + 8q^{51} - 32q^{61} + 16q^{69} - 32q^{71} + 4q^{81} + 16q^{89} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ −2.23607 −1.00000
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 4.47214i − 1.24035i −0.784465 0.620174i $$-0.787062\pi$$
0.784465 0.620174i $$-0.212938\pi$$
$$14$$ 0 0
$$15$$ 2.23607i 0.577350i
$$16$$ 0 0
$$17$$ − 2.47214i − 0.599581i −0.954005 0.299791i $$-0.903083\pi$$
0.954005 0.299791i $$-0.0969168\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −0.472136 −0.0876734 −0.0438367 0.999039i $$-0.513958\pi$$
−0.0438367 + 0.999039i $$0.513958\pi$$
$$30$$ 0 0
$$31$$ −8.47214 −1.52164 −0.760820 0.648963i $$-0.775203\pi$$
−0.760820 + 0.648963i $$0.775203\pi$$
$$32$$ 0 0
$$33$$ − 2.00000i − 0.348155i
$$34$$ 0 0
$$35$$ 2.23607i 0.377964i
$$36$$ 0 0
$$37$$ − 6.47214i − 1.06401i −0.846740 0.532006i $$-0.821438\pi$$
0.846740 0.532006i $$-0.178562\pi$$
$$38$$ 0 0
$$39$$ −4.47214 −0.716115
$$40$$ 0 0
$$41$$ −12.4721 −1.94782 −0.973910 0.226934i $$-0.927130\pi$$
−0.973910 + 0.226934i $$0.927130\pi$$
$$42$$ 0 0
$$43$$ 6.47214i 0.986991i 0.869748 + 0.493496i $$0.164281\pi$$
−0.869748 + 0.493496i $$0.835719\pi$$
$$44$$ 0 0
$$45$$ 2.23607 0.333333
$$46$$ 0 0
$$47$$ 2.47214i 0.360598i 0.983612 + 0.180299i $$0.0577065\pi$$
−0.983612 + 0.180299i $$0.942293\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −2.47214 −0.346168
$$52$$ 0 0
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ 0 0
$$55$$ −4.47214 −0.603023
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −12.4721 −1.59689 −0.798447 0.602066i $$-0.794345\pi$$
−0.798447 + 0.602066i $$0.794345\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 0.125988i
$$64$$ 0 0
$$65$$ 10.0000i 1.24035i
$$66$$ 0 0
$$67$$ − 10.4721i − 1.27938i −0.768635 0.639688i $$-0.779064\pi$$
0.768635 0.639688i $$-0.220936\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −3.52786 −0.418680 −0.209340 0.977843i $$-0.567132\pi$$
−0.209340 + 0.977843i $$0.567132\pi$$
$$72$$ 0 0
$$73$$ − 16.4721i − 1.92792i −0.266051 0.963959i $$-0.585719\pi$$
0.266051 0.963959i $$-0.414281\pi$$
$$74$$ 0 0
$$75$$ − 5.00000i − 0.577350i
$$76$$ 0 0
$$77$$ − 2.00000i − 0.227921i
$$78$$ 0 0
$$79$$ −8.94427 −1.00631 −0.503155 0.864196i $$-0.667827\pi$$
−0.503155 + 0.864196i $$0.667827\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.9443i 1.42082i 0.703789 + 0.710409i $$0.251490\pi$$
−0.703789 + 0.710409i $$0.748510\pi$$
$$84$$ 0 0
$$85$$ 5.52786i 0.599581i
$$86$$ 0 0
$$87$$ 0.472136i 0.0506183i
$$88$$ 0 0
$$89$$ −9.41641 −0.998137 −0.499069 0.866562i $$-0.666324\pi$$
−0.499069 + 0.866562i $$0.666324\pi$$
$$90$$ 0 0
$$91$$ −4.47214 −0.468807
$$92$$ 0 0
$$93$$ 8.47214i 0.878520i
$$94$$ 0 0
$$95$$ −4.47214 −0.458831
$$96$$ 0 0
$$97$$ 12.4721i 1.26635i 0.774007 + 0.633177i $$0.218249\pi$$
−0.774007 + 0.633177i $$0.781751\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −4.47214 −0.444994 −0.222497 0.974933i $$-0.571421\pi$$
−0.222497 + 0.974933i $$0.571421\pi$$
$$102$$ 0 0
$$103$$ − 4.94427i − 0.487174i −0.969879 0.243587i $$-0.921676\pi$$
0.969879 0.243587i $$-0.0783241\pi$$
$$104$$ 0 0
$$105$$ 2.23607 0.218218
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 2.94427 0.282010 0.141005 0.990009i $$-0.454967\pi$$
0.141005 + 0.990009i $$0.454967\pi$$
$$110$$ 0 0
$$111$$ −6.47214 −0.614308
$$112$$ 0 0
$$113$$ 2.94427i 0.276974i 0.990364 + 0.138487i $$0.0442239\pi$$
−0.990364 + 0.138487i $$0.955776\pi$$
$$114$$ 0 0
$$115$$ − 8.94427i − 0.834058i
$$116$$ 0 0
$$117$$ 4.47214i 0.413449i
$$118$$ 0 0
$$119$$ −2.47214 −0.226620
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 12.4721i 1.12457i
$$124$$ 0 0
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 0 0
$$129$$ 6.47214 0.569840
$$130$$ 0 0
$$131$$ 21.8885 1.91241 0.956205 0.292696i $$-0.0945525\pi$$
0.956205 + 0.292696i $$0.0945525\pi$$
$$132$$ 0 0
$$133$$ − 2.00000i − 0.173422i
$$134$$ 0 0
$$135$$ − 2.23607i − 0.192450i
$$136$$ 0 0
$$137$$ 15.8885i 1.35745i 0.734393 + 0.678725i $$0.237467\pi$$
−0.734393 + 0.678725i $$0.762533\pi$$
$$138$$ 0 0
$$139$$ 14.9443 1.26756 0.633778 0.773515i $$-0.281503\pi$$
0.633778 + 0.773515i $$0.281503\pi$$
$$140$$ 0 0
$$141$$ 2.47214 0.208191
$$142$$ 0 0
$$143$$ − 8.94427i − 0.747958i
$$144$$ 0 0
$$145$$ 1.05573 0.0876734
$$146$$ 0 0
$$147$$ 1.00000i 0.0824786i
$$148$$ 0 0
$$149$$ 3.52786 0.289014 0.144507 0.989504i $$-0.453840\pi$$
0.144507 + 0.989504i $$0.453840\pi$$
$$150$$ 0 0
$$151$$ −17.8885 −1.45575 −0.727875 0.685710i $$-0.759492\pi$$
−0.727875 + 0.685710i $$0.759492\pi$$
$$152$$ 0 0
$$153$$ 2.47214i 0.199860i
$$154$$ 0 0
$$155$$ 18.9443 1.52164
$$156$$ 0 0
$$157$$ − 17.4164i − 1.38998i −0.719019 0.694990i $$-0.755409\pi$$
0.719019 0.694990i $$-0.244591\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ − 3.41641i − 0.267594i −0.991009 0.133797i $$-0.957283\pi$$
0.991009 0.133797i $$-0.0427170\pi$$
$$164$$ 0 0
$$165$$ 4.47214i 0.348155i
$$166$$ 0 0
$$167$$ − 1.52786i − 0.118230i −0.998251 0.0591148i $$-0.981172\pi$$
0.998251 0.0591148i $$-0.0188278\pi$$
$$168$$ 0 0
$$169$$ −7.00000 −0.538462
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 0 0
$$175$$ − 5.00000i − 0.377964i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.05573 −0.377883 −0.188941 0.981988i $$-0.560506\pi$$
−0.188941 + 0.981988i $$0.560506\pi$$
$$180$$ 0 0
$$181$$ 7.52786 0.559542 0.279771 0.960067i $$-0.409742\pi$$
0.279771 + 0.960067i $$0.409742\pi$$
$$182$$ 0 0
$$183$$ 12.4721i 0.921967i
$$184$$ 0 0
$$185$$ 14.4721i 1.06401i
$$186$$ 0 0
$$187$$ − 4.94427i − 0.361561i
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −9.41641 −0.681347 −0.340674 0.940182i $$-0.610655\pi$$
−0.340674 + 0.940182i $$0.610655\pi$$
$$192$$ 0 0
$$193$$ 4.94427i 0.355896i 0.984040 + 0.177948i $$0.0569460\pi$$
−0.984040 + 0.177948i $$0.943054\pi$$
$$194$$ 0 0
$$195$$ 10.0000 0.716115
$$196$$ 0 0
$$197$$ 15.8885i 1.13201i 0.824401 + 0.566006i $$0.191512\pi$$
−0.824401 + 0.566006i $$0.808488\pi$$
$$198$$ 0 0
$$199$$ 11.5279 0.817189 0.408594 0.912716i $$-0.366019\pi$$
0.408594 + 0.912716i $$0.366019\pi$$
$$200$$ 0 0
$$201$$ −10.4721 −0.738648
$$202$$ 0 0
$$203$$ 0.472136i 0.0331374i
$$204$$ 0 0
$$205$$ 27.8885 1.94782
$$206$$ 0 0
$$207$$ − 4.00000i − 0.278019i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 9.88854 0.680755 0.340378 0.940289i $$-0.389445\pi$$
0.340378 + 0.940289i $$0.389445\pi$$
$$212$$ 0 0
$$213$$ 3.52786i 0.241725i
$$214$$ 0 0
$$215$$ − 14.4721i − 0.986991i
$$216$$ 0 0
$$217$$ 8.47214i 0.575126i
$$218$$ 0 0
$$219$$ −16.4721 −1.11308
$$220$$ 0 0
$$221$$ −11.0557 −0.743689
$$222$$ 0 0
$$223$$ 3.05573i 0.204627i 0.994752 + 0.102313i $$0.0326244\pi$$
−0.994752 + 0.102313i $$0.967376\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ − 25.8885i − 1.71828i −0.511738 0.859142i $$-0.670998\pi$$
0.511738 0.859142i $$-0.329002\pi$$
$$228$$ 0 0
$$229$$ −18.3607 −1.21331 −0.606654 0.794966i $$-0.707489\pi$$
−0.606654 + 0.794966i $$0.707489\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 0 0
$$233$$ − 14.9443i − 0.979032i −0.871994 0.489516i $$-0.837174\pi$$
0.871994 0.489516i $$-0.162826\pi$$
$$234$$ 0 0
$$235$$ − 5.52786i − 0.360598i
$$236$$ 0 0
$$237$$ 8.94427i 0.580993i
$$238$$ 0 0
$$239$$ 24.4721 1.58297 0.791485 0.611188i $$-0.209308\pi$$
0.791485 + 0.611188i $$0.209308\pi$$
$$240$$ 0 0
$$241$$ 23.8885 1.53880 0.769398 0.638769i $$-0.220556\pi$$
0.769398 + 0.638769i $$0.220556\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 2.23607 0.142857
$$246$$ 0 0
$$247$$ − 8.94427i − 0.569110i
$$248$$ 0 0
$$249$$ 12.9443 0.820310
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 8.00000i 0.502956i
$$254$$ 0 0
$$255$$ 5.52786 0.346168
$$256$$ 0 0
$$257$$ − 3.41641i − 0.213110i −0.994307 0.106555i $$-0.966018\pi$$
0.994307 0.106555i $$-0.0339820\pi$$
$$258$$ 0 0
$$259$$ −6.47214 −0.402159
$$260$$ 0 0
$$261$$ 0.472136 0.0292245
$$262$$ 0 0
$$263$$ 4.00000i 0.246651i 0.992366 + 0.123325i $$0.0393559\pi$$
−0.992366 + 0.123325i $$0.960644\pi$$
$$264$$ 0 0
$$265$$ 4.47214i 0.274721i
$$266$$ 0 0
$$267$$ 9.41641i 0.576275i
$$268$$ 0 0
$$269$$ 14.3607 0.875586 0.437793 0.899076i $$-0.355760\pi$$
0.437793 + 0.899076i $$0.355760\pi$$
$$270$$ 0 0
$$271$$ −15.5279 −0.943251 −0.471625 0.881799i $$-0.656332\pi$$
−0.471625 + 0.881799i $$0.656332\pi$$
$$272$$ 0 0
$$273$$ 4.47214i 0.270666i
$$274$$ 0 0
$$275$$ 10.0000 0.603023
$$276$$ 0 0
$$277$$ − 3.41641i − 0.205272i −0.994719 0.102636i $$-0.967272\pi$$
0.994719 0.102636i $$-0.0327277\pi$$
$$278$$ 0 0
$$279$$ 8.47214 0.507214
$$280$$ 0 0
$$281$$ −14.9443 −0.891501 −0.445750 0.895157i $$-0.647063\pi$$
−0.445750 + 0.895157i $$0.647063\pi$$
$$282$$ 0 0
$$283$$ − 0.944272i − 0.0561311i −0.999606 0.0280656i $$-0.991065\pi$$
0.999606 0.0280656i $$-0.00893472\pi$$
$$284$$ 0 0
$$285$$ 4.47214i 0.264906i
$$286$$ 0 0
$$287$$ 12.4721i 0.736207i
$$288$$ 0 0
$$289$$ 10.8885 0.640503
$$290$$ 0 0
$$291$$ 12.4721 0.731130
$$292$$ 0 0
$$293$$ − 7.05573i − 0.412200i −0.978531 0.206100i $$-0.933923\pi$$
0.978531 0.206100i $$-0.0660772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2.00000i 0.116052i
$$298$$ 0 0
$$299$$ 17.8885 1.03452
$$300$$ 0 0
$$301$$ 6.47214 0.373048
$$302$$ 0 0
$$303$$ 4.47214i 0.256917i
$$304$$ 0 0
$$305$$ 27.8885 1.59689
$$306$$ 0 0
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ −4.94427 −0.281270
$$310$$ 0 0
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ − 1.41641i − 0.0800601i −0.999198 0.0400301i $$-0.987255\pi$$
0.999198 0.0400301i $$-0.0127454\pi$$
$$314$$ 0 0
$$315$$ − 2.23607i − 0.125988i
$$316$$ 0 0
$$317$$ − 10.9443i − 0.614692i −0.951598 0.307346i $$-0.900559\pi$$
0.951598 0.307346i $$-0.0994408\pi$$
$$318$$ 0 0
$$319$$ −0.944272 −0.0528691
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 4.94427i − 0.275107i
$$324$$ 0 0
$$325$$ − 22.3607i − 1.24035i
$$326$$ 0 0
$$327$$ − 2.94427i − 0.162819i
$$328$$ 0 0
$$329$$ 2.47214 0.136293
$$330$$ 0 0
$$331$$ 1.88854 0.103804 0.0519019 0.998652i $$-0.483472\pi$$
0.0519019 + 0.998652i $$0.483472\pi$$
$$332$$ 0 0
$$333$$ 6.47214i 0.354671i
$$334$$ 0 0
$$335$$ 23.4164i 1.27938i
$$336$$ 0 0
$$337$$ − 28.9443i − 1.57669i −0.615230 0.788347i $$-0.710937\pi$$
0.615230 0.788347i $$-0.289063\pi$$
$$338$$ 0 0
$$339$$ 2.94427 0.159911
$$340$$ 0 0
$$341$$ −16.9443 −0.917584
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ −8.94427 −0.481543
$$346$$ 0 0
$$347$$ 22.8328i 1.22573i 0.790188 + 0.612865i $$0.209983\pi$$
−0.790188 + 0.612865i $$0.790017\pi$$
$$348$$ 0 0
$$349$$ 1.41641 0.0758186 0.0379093 0.999281i $$-0.487930\pi$$
0.0379093 + 0.999281i $$0.487930\pi$$
$$350$$ 0 0
$$351$$ 4.47214 0.238705
$$352$$ 0 0
$$353$$ − 26.4721i − 1.40897i −0.709719 0.704485i $$-0.751178\pi$$
0.709719 0.704485i $$-0.248822\pi$$
$$354$$ 0 0
$$355$$ 7.88854 0.418680
$$356$$ 0 0
$$357$$ 2.47214i 0.130839i
$$358$$ 0 0
$$359$$ −13.4164 −0.708091 −0.354045 0.935228i $$-0.615194\pi$$
−0.354045 + 0.935228i $$0.615194\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 7.00000i 0.367405i
$$364$$ 0 0
$$365$$ 36.8328i 1.92792i
$$366$$ 0 0
$$367$$ − 9.88854i − 0.516178i −0.966121 0.258089i $$-0.916907\pi$$
0.966121 0.258089i $$-0.0830927\pi$$
$$368$$ 0 0
$$369$$ 12.4721 0.649273
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ − 28.3607i − 1.46846i −0.678901 0.734230i $$-0.737543\pi$$
0.678901 0.734230i $$-0.262457\pi$$
$$374$$ 0 0
$$375$$ 11.1803i 0.577350i
$$376$$ 0 0
$$377$$ 2.11146i 0.108746i
$$378$$ 0 0
$$379$$ 29.8885 1.53527 0.767636 0.640886i $$-0.221433\pi$$
0.767636 + 0.640886i $$0.221433\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 24.3607i 1.24477i 0.782710 + 0.622386i $$0.213837\pi$$
−0.782710 + 0.622386i $$0.786163\pi$$
$$384$$ 0 0
$$385$$ 4.47214i 0.227921i
$$386$$ 0 0
$$387$$ − 6.47214i − 0.328997i
$$388$$ 0 0
$$389$$ 29.4164 1.49147 0.745736 0.666242i $$-0.232098\pi$$
0.745736 + 0.666242i $$0.232098\pi$$
$$390$$ 0 0
$$391$$ 9.88854 0.500085
$$392$$ 0 0
$$393$$ − 21.8885i − 1.10413i
$$394$$ 0 0
$$395$$ 20.0000 1.00631
$$396$$ 0 0
$$397$$ − 16.4721i − 0.826713i −0.910569 0.413356i $$-0.864356\pi$$
0.910569 0.413356i $$-0.135644\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 19.8885 0.993186 0.496593 0.867983i $$-0.334584\pi$$
0.496593 + 0.867983i $$0.334584\pi$$
$$402$$ 0 0
$$403$$ 37.8885i 1.88736i
$$404$$ 0 0
$$405$$ −2.23607 −0.111111
$$406$$ 0 0
$$407$$ − 12.9443i − 0.641624i
$$408$$ 0 0
$$409$$ −27.8885 −1.37900 −0.689500 0.724286i $$-0.742170\pi$$
−0.689500 + 0.724286i $$0.742170\pi$$
$$410$$ 0 0
$$411$$ 15.8885 0.783724
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 28.9443i − 1.42082i
$$416$$ 0 0
$$417$$ − 14.9443i − 0.731824i
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 6.94427 0.338443 0.169222 0.985578i $$-0.445875\pi$$
0.169222 + 0.985578i $$0.445875\pi$$
$$422$$ 0 0
$$423$$ − 2.47214i − 0.120199i
$$424$$ 0 0
$$425$$ − 12.3607i − 0.599581i
$$426$$ 0 0
$$427$$ 12.4721i 0.603569i
$$428$$ 0 0
$$429$$ −8.94427 −0.431834
$$430$$ 0 0
$$431$$ −5.41641 −0.260899 −0.130450 0.991455i $$-0.541642\pi$$
−0.130450 + 0.991455i $$0.541642\pi$$
$$432$$ 0 0
$$433$$ 5.41641i 0.260296i 0.991495 + 0.130148i $$0.0415452\pi$$
−0.991495 + 0.130148i $$0.958455\pi$$
$$434$$ 0 0
$$435$$ − 1.05573i − 0.0506183i
$$436$$ 0 0
$$437$$ 8.00000i 0.382692i
$$438$$ 0 0
$$439$$ 40.4721 1.93163 0.965815 0.259233i $$-0.0834697\pi$$
0.965815 + 0.259233i $$0.0834697\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ − 19.0557i − 0.905365i −0.891672 0.452682i $$-0.850467\pi$$
0.891672 0.452682i $$-0.149533\pi$$
$$444$$ 0 0
$$445$$ 21.0557 0.998137
$$446$$ 0 0
$$447$$ − 3.52786i − 0.166862i
$$448$$ 0 0
$$449$$ 26.9443 1.27158 0.635789 0.771863i $$-0.280675\pi$$
0.635789 + 0.771863i $$0.280675\pi$$
$$450$$ 0 0
$$451$$ −24.9443 −1.17458
$$452$$ 0 0
$$453$$ 17.8885i 0.840477i
$$454$$ 0 0
$$455$$ 10.0000 0.468807
$$456$$ 0 0
$$457$$ 8.94427i 0.418395i 0.977873 + 0.209198i $$0.0670852\pi$$
−0.977873 + 0.209198i $$0.932915\pi$$
$$458$$ 0 0
$$459$$ 2.47214 0.115389
$$460$$ 0 0
$$461$$ −3.52786 −0.164309 −0.0821545 0.996620i $$-0.526180\pi$$
−0.0821545 + 0.996620i $$0.526180\pi$$
$$462$$ 0 0
$$463$$ 18.8328i 0.875235i 0.899161 + 0.437618i $$0.144178\pi$$
−0.899161 + 0.437618i $$0.855822\pi$$
$$464$$ 0 0
$$465$$ − 18.9443i − 0.878520i
$$466$$ 0 0
$$467$$ 36.9443i 1.70958i 0.518976 + 0.854789i $$0.326313\pi$$
−0.518976 + 0.854789i $$0.673687\pi$$
$$468$$ 0 0
$$469$$ −10.4721 −0.483558
$$470$$ 0 0
$$471$$ −17.4164 −0.802506
$$472$$ 0 0
$$473$$ 12.9443i 0.595178i
$$474$$ 0 0
$$475$$ 10.0000 0.458831
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −28.9443 −1.31975
$$482$$ 0 0
$$483$$ − 4.00000i − 0.182006i
$$484$$ 0 0
$$485$$ − 27.8885i − 1.26635i
$$486$$ 0 0
$$487$$ 21.8885i 0.991865i 0.868361 + 0.495932i $$0.165174\pi$$
−0.868361 + 0.495932i $$0.834826\pi$$
$$488$$ 0 0
$$489$$ −3.41641 −0.154495
$$490$$ 0 0
$$491$$ −18.9443 −0.854943 −0.427472 0.904029i $$-0.640596\pi$$
−0.427472 + 0.904029i $$0.640596\pi$$
$$492$$ 0 0
$$493$$ 1.16718i 0.0525673i
$$494$$ 0 0
$$495$$ 4.47214 0.201008
$$496$$ 0 0
$$497$$ 3.52786i 0.158246i
$$498$$ 0 0
$$499$$ 29.8885 1.33799 0.668997 0.743265i $$-0.266724\pi$$
0.668997 + 0.743265i $$0.266724\pi$$
$$500$$ 0 0
$$501$$ −1.52786 −0.0682599
$$502$$ 0 0
$$503$$ − 12.5836i − 0.561075i −0.959843 0.280537i $$-0.909487\pi$$
0.959843 0.280537i $$-0.0905126\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ 7.00000i 0.310881i
$$508$$ 0 0
$$509$$ 37.4164 1.65845 0.829227 0.558913i $$-0.188781\pi$$
0.829227 + 0.558913i $$0.188781\pi$$
$$510$$ 0 0
$$511$$ −16.4721 −0.728684
$$512$$ 0 0
$$513$$ 2.00000i 0.0883022i
$$514$$ 0 0
$$515$$ 11.0557i 0.487174i
$$516$$ 0 0
$$517$$ 4.94427i 0.217449i
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 17.4164 0.763027 0.381513 0.924363i $$-0.375403\pi$$
0.381513 + 0.924363i $$0.375403\pi$$
$$522$$ 0 0
$$523$$ − 24.9443i − 1.09074i −0.838196 0.545368i $$-0.816390\pi$$
0.838196 0.545368i $$-0.183610\pi$$
$$524$$ 0 0
$$525$$ −5.00000 −0.218218
$$526$$ 0 0
$$527$$ 20.9443i 0.912347i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 55.7771i 2.41597i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 5.05573i 0.218171i
$$538$$ 0 0
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −5.05573 −0.217363 −0.108681 0.994077i $$-0.534663\pi$$
−0.108681 + 0.994077i $$0.534663\pi$$
$$542$$ 0 0
$$543$$ − 7.52786i − 0.323052i
$$544$$ 0 0
$$545$$ −6.58359 −0.282010
$$546$$ 0 0
$$547$$ 25.3050i 1.08196i 0.841035 + 0.540981i $$0.181947\pi$$
−0.841035 + 0.540981i $$0.818053\pi$$
$$548$$ 0 0
$$549$$ 12.4721 0.532298
$$550$$ 0 0
$$551$$ −0.944272 −0.0402273
$$552$$ 0 0
$$553$$ 8.94427i 0.380349i
$$554$$ 0 0
$$555$$ 14.4721 0.614308
$$556$$ 0 0
$$557$$ 22.9443i 0.972180i 0.873909 + 0.486090i $$0.161577\pi$$
−0.873909 + 0.486090i $$0.838423\pi$$
$$558$$ 0 0
$$559$$ 28.9443 1.22421
$$560$$ 0 0
$$561$$ −4.94427 −0.208747
$$562$$ 0 0
$$563$$ − 18.8328i − 0.793709i −0.917882 0.396854i $$-0.870102\pi$$
0.917882 0.396854i $$-0.129898\pi$$
$$564$$ 0 0
$$565$$ − 6.58359i − 0.276974i
$$566$$ 0 0
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 45.8885 1.92038 0.960188 0.279355i $$-0.0901206\pi$$
0.960188 + 0.279355i $$0.0901206\pi$$
$$572$$ 0 0
$$573$$ 9.41641i 0.393376i
$$574$$ 0 0
$$575$$ 20.0000i 0.834058i
$$576$$ 0 0
$$577$$ 15.5279i 0.646433i 0.946325 + 0.323217i $$0.104764\pi$$
−0.946325 + 0.323217i $$0.895236\pi$$
$$578$$ 0 0
$$579$$ 4.94427 0.205477
$$580$$ 0 0
$$581$$ 12.9443 0.537019
$$582$$ 0 0
$$583$$ − 4.00000i − 0.165663i
$$584$$ 0 0
$$585$$ − 10.0000i − 0.413449i
$$586$$ 0 0
$$587$$ 32.9443i 1.35976i 0.733325 + 0.679878i $$0.237967\pi$$
−0.733325 + 0.679878i $$0.762033\pi$$
$$588$$ 0 0
$$589$$ −16.9443 −0.698177
$$590$$ 0 0
$$591$$ 15.8885 0.653567
$$592$$ 0 0
$$593$$ − 41.3050i − 1.69619i −0.529843 0.848096i $$-0.677749\pi$$
0.529843 0.848096i $$-0.322251\pi$$
$$594$$ 0 0
$$595$$ 5.52786 0.226620
$$596$$ 0 0
$$597$$ − 11.5279i − 0.471804i
$$598$$ 0 0
$$599$$ −26.3607 −1.07707 −0.538534 0.842604i $$-0.681022\pi$$
−0.538534 + 0.842604i $$0.681022\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 10.4721i 0.426458i
$$604$$ 0 0
$$605$$ 15.6525 0.636364
$$606$$ 0 0
$$607$$ − 12.9443i − 0.525392i −0.964879 0.262696i $$-0.915388\pi$$
0.964879 0.262696i $$-0.0846116\pi$$
$$608$$ 0 0
$$609$$ 0.472136 0.0191319
$$610$$ 0 0
$$611$$ 11.0557 0.447267
$$612$$ 0 0
$$613$$ − 19.4164i − 0.784221i −0.919918 0.392111i $$-0.871745\pi$$
0.919918 0.392111i $$-0.128255\pi$$
$$614$$ 0 0
$$615$$ − 27.8885i − 1.12457i
$$616$$ 0 0
$$617$$ − 12.1115i − 0.487589i −0.969827 0.243794i $$-0.921608\pi$$
0.969827 0.243794i $$-0.0783922\pi$$
$$618$$ 0 0
$$619$$ −23.8885 −0.960162 −0.480081 0.877224i $$-0.659393\pi$$
−0.480081 + 0.877224i $$0.659393\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ 9.41641i 0.377260i
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ − 4.00000i − 0.159745i
$$628$$ 0 0
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −40.9443 −1.62997 −0.814983 0.579485i $$-0.803254\pi$$
−0.814983 + 0.579485i $$0.803254\pi$$
$$632$$ 0 0
$$633$$ − 9.88854i − 0.393034i
$$634$$ 0 0
$$635$$ 8.94427i 0.354943i
$$636$$ 0 0
$$637$$ 4.47214i 0.177192i
$$638$$ 0 0
$$639$$ 3.52786 0.139560
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ − 24.9443i − 0.983706i −0.870678 0.491853i $$-0.836320\pi$$
0.870678 0.491853i $$-0.163680\pi$$
$$644$$ 0 0
$$645$$ −14.4721 −0.569840
$$646$$ 0 0
$$647$$ − 42.2492i − 1.66099i −0.557027 0.830494i $$-0.688058\pi$$
0.557027 0.830494i $$-0.311942\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 8.47214 0.332049
$$652$$ 0 0
$$653$$ − 18.9443i − 0.741347i −0.928763 0.370673i $$-0.879127\pi$$
0.928763 0.370673i $$-0.120873\pi$$
$$654$$ 0 0
$$655$$ −48.9443 −1.91241
$$656$$ 0 0
$$657$$ 16.4721i 0.642639i
$$658$$ 0 0
$$659$$ −47.8885 −1.86547 −0.932736 0.360559i $$-0.882586\pi$$
−0.932736 + 0.360559i $$0.882586\pi$$
$$660$$ 0 0
$$661$$ −32.2492 −1.25435 −0.627175 0.778879i $$-0.715789\pi$$
−0.627175 + 0.778879i $$0.715789\pi$$
$$662$$ 0 0
$$663$$ 11.0557i 0.429369i
$$664$$ 0 0
$$665$$ 4.47214i 0.173422i
$$666$$ 0 0
$$667$$ − 1.88854i − 0.0731247i
$$668$$ 0 0
$$669$$ 3.05573 0.118141
$$670$$ 0 0
$$671$$ −24.9443 −0.962963
$$672$$ 0 0
$$673$$ 13.8885i 0.535364i 0.963507 + 0.267682i $$0.0862577\pi$$
−0.963507 + 0.267682i $$0.913742\pi$$
$$674$$ 0 0
$$675$$ 5.00000i 0.192450i
$$676$$ 0 0
$$677$$ 37.8885i 1.45618i 0.685484 + 0.728088i $$0.259591\pi$$
−0.685484 + 0.728088i $$0.740409\pi$$
$$678$$ 0 0
$$679$$ 12.4721 0.478637
$$680$$ 0 0
$$681$$ −25.8885 −0.992051
$$682$$ 0 0
$$683$$ − 49.8885i − 1.90893i −0.298320 0.954466i $$-0.596426\pi$$
0.298320 0.954466i $$-0.403574\pi$$
$$684$$ 0 0
$$685$$ − 35.5279i − 1.35745i
$$686$$ 0 0
$$687$$ 18.3607i 0.700504i
$$688$$ 0 0
$$689$$ −8.94427 −0.340750
$$690$$ 0 0
$$691$$ −4.83282 −0.183849 −0.0919245 0.995766i $$-0.529302\pi$$
−0.0919245 + 0.995766i $$0.529302\pi$$
$$692$$ 0 0
$$693$$ 2.00000i 0.0759737i
$$694$$ 0 0
$$695$$ −33.4164 −1.26756
$$696$$ 0 0
$$697$$ 30.8328i 1.16788i
$$698$$ 0 0
$$699$$ −14.9443 −0.565244
$$700$$ 0 0
$$701$$ 8.47214 0.319988 0.159994 0.987118i $$-0.448852\pi$$
0.159994 + 0.987118i $$0.448852\pi$$
$$702$$ 0 0
$$703$$ − 12.9443i − 0.488202i
$$704$$ 0 0
$$705$$ −5.52786 −0.208191
$$706$$ 0 0
$$707$$ 4.47214i 0.168192i
$$708$$ 0 0
$$709$$ −0.111456 −0.00418582 −0.00209291 0.999998i $$-0.500666\pi$$
−0.00209291 + 0.999998i $$0.500666\pi$$
$$710$$ 0 0
$$711$$ 8.94427 0.335436
$$712$$ 0 0
$$713$$ − 33.8885i − 1.26914i
$$714$$ 0 0
$$715$$ 20.0000i 0.747958i
$$716$$ 0 0
$$717$$ − 24.4721i − 0.913929i
$$718$$ 0 0
$$719$$ −8.94427 −0.333565 −0.166783 0.985994i $$-0.553338\pi$$
−0.166783 + 0.985994i $$0.553338\pi$$
$$720$$ 0 0
$$721$$ −4.94427 −0.184134
$$722$$ 0 0
$$723$$ − 23.8885i − 0.888425i
$$724$$ 0 0
$$725$$ −2.36068 −0.0876734
$$726$$ 0 0
$$727$$ 36.9443i 1.37019i 0.728455 + 0.685094i $$0.240239\pi$$
−0.728455 + 0.685094i $$0.759761\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 0 0
$$733$$ 50.3607i 1.86011i 0.367415 + 0.930057i $$0.380243\pi$$
−0.367415 + 0.930057i $$0.619757\pi$$
$$734$$ 0 0
$$735$$ − 2.23607i − 0.0824786i
$$736$$ 0 0
$$737$$ − 20.9443i − 0.771492i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −8.94427 −0.328576
$$742$$ 0 0
$$743$$ − 50.8328i − 1.86488i −0.361331 0.932438i $$-0.617678\pi$$
0.361331 0.932438i $$-0.382322\pi$$
$$744$$ 0 0
$$745$$ −7.88854 −0.289014
$$746$$ 0 0
$$747$$ − 12.9443i − 0.473606i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.8328 0.979143 0.489572 0.871963i $$-0.337153\pi$$
0.489572 + 0.871963i $$0.337153\pi$$
$$752$$ 0 0
$$753$$ 24.0000i 0.874609i
$$754$$ 0 0
$$755$$ 40.0000 1.45575
$$756$$ 0 0
$$757$$ − 21.5279i − 0.782444i −0.920296 0.391222i $$-0.872053\pi$$
0.920296 0.391222i $$-0.127947\pi$$
$$758$$ 0 0
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ −15.5279 −0.562885 −0.281442 0.959578i $$-0.590813\pi$$
−0.281442 + 0.959578i $$0.590813\pi$$
$$762$$ 0 0
$$763$$ − 2.94427i − 0.106590i
$$764$$ 0 0
$$765$$ − 5.52786i − 0.199860i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −18.9443 −0.683148 −0.341574 0.939855i $$-0.610960\pi$$
−0.341574 + 0.939855i $$0.610960\pi$$
$$770$$ 0 0
$$771$$ −3.41641 −0.123039
$$772$$ 0 0
$$773$$ − 0.944272i − 0.0339631i −0.999856 0.0169815i $$-0.994594\pi$$
0.999856 0.0169815i $$-0.00540566\pi$$
$$774$$ 0 0
$$775$$ −42.3607 −1.52164
$$776$$ 0 0
$$777$$ 6.47214i 0.232187i
$$778$$ 0 0
$$779$$ −24.9443 −0.893721
$$780$$ 0 0
$$781$$ −7.05573 −0.252474
$$782$$ 0 0
$$783$$ − 0.472136i − 0.0168728i
$$784$$ 0 0
$$785$$ 38.9443i 1.38998i
$$786$$ 0 0
$$787$$ − 36.0000i − 1.28326i −0.767014 0.641631i $$-0.778258\pi$$
0.767014 0.641631i $$-0.221742\pi$$
$$788$$ 0 0
$$789$$ 4.00000 0.142404
$$790$$ 0 0
$$791$$ 2.94427 0.104686
$$792$$ 0 0
$$793$$ 55.7771i 1.98070i
$$794$$ 0 0
$$795$$ 4.47214 0.158610
$$796$$ 0 0
$$797$$ 28.0000i 0.991811i 0.868377 + 0.495905i $$0.165164\pi$$
−0.868377 + 0.495905i $$0.834836\pi$$
$$798$$ 0 0
$$799$$ 6.11146 0.216208
$$800$$ 0 0
$$801$$ 9.41641 0.332712
$$802$$ 0 0
$$803$$ − 32.9443i − 1.16258i
$$804$$ 0 0
$$805$$ −8.94427 −0.315244
$$806$$ 0 0
$$807$$ − 14.3607i − 0.505520i
$$808$$ 0 0
$$809$$ −28.8328 −1.01371 −0.506854 0.862032i $$-0.669192\pi$$
−0.506854 + 0.862032i $$0.669192\pi$$
$$810$$ 0 0
$$811$$ −30.9443 −1.08660 −0.543300 0.839539i $$-0.682825\pi$$
−0.543300 + 0.839539i $$0.682825\pi$$
$$812$$ 0 0
$$813$$ 15.5279i 0.544586i
$$814$$ 0 0
$$815$$ 7.63932i 0.267594i
$$816$$ 0 0
$$817$$ 12.9443i 0.452863i
$$818$$ 0 0
$$819$$ 4.47214 0.156269
$$820$$ 0 0
$$821$$ −43.3050 −1.51135 −0.755677 0.654945i $$-0.772692\pi$$
−0.755677 + 0.654945i $$0.772692\pi$$
$$822$$ 0 0
$$823$$ 32.9443i 1.14837i 0.818727 + 0.574183i $$0.194680\pi$$
−0.818727 + 0.574183i $$0.805320\pi$$
$$824$$ 0 0
$$825$$ − 10.0000i − 0.348155i
$$826$$ 0 0
$$827$$ − 56.7214i − 1.97239i −0.165573 0.986197i $$-0.552947\pi$$
0.165573 0.986197i $$-0.447053\pi$$
$$828$$ 0 0
$$829$$ −12.4721 −0.433175 −0.216588 0.976263i $$-0.569493\pi$$
−0.216588 + 0.976263i $$0.569493\pi$$
$$830$$ 0 0
$$831$$ −3.41641 −0.118514
$$832$$ 0 0
$$833$$ 2.47214i 0.0856544i
$$834$$ 0 0
$$835$$ 3.41641i 0.118230i
$$836$$ 0 0
$$837$$ − 8.47214i − 0.292840i
$$838$$ 0 0
$$839$$ 15.0557 0.519781 0.259891 0.965638i $$-0.416313\pi$$
0.259891 + 0.965638i $$0.416313\pi$$
$$840$$ 0 0
$$841$$ −28.7771 −0.992313
$$842$$ 0 0
$$843$$ 14.9443i 0.514708i
$$844$$ 0 0
$$845$$ 15.6525 0.538462
$$846$$ 0 0
$$847$$ 7.00000i 0.240523i
$$848$$ 0 0
$$849$$ −0.944272 −0.0324073
$$850$$ 0 0
$$851$$ 25.8885 0.887448
$$852$$ 0 0
$$853$$ 40.4721i 1.38574i 0.721063 + 0.692870i $$0.243654\pi$$
−0.721063 + 0.692870i $$0.756346\pi$$
$$854$$ 0 0
$$855$$ 4.47214 0.152944
$$856$$ 0 0
$$857$$ − 43.4164i − 1.48308i −0.670911 0.741538i $$-0.734097\pi$$
0.670911 0.741538i $$-0.265903\pi$$
$$858$$ 0 0
$$859$$ −38.9443 −1.32876 −0.664381 0.747394i $$-0.731305\pi$$
−0.664381 + 0.747394i $$0.731305\pi$$
$$860$$ 0 0
$$861$$ 12.4721 0.425049
$$862$$ 0 0
$$863$$ − 34.8328i − 1.18572i −0.805305 0.592861i $$-0.797998\pi$$
0.805305 0.592861i $$-0.202002\pi$$
$$864$$ 0 0
$$865$$ 26.8328i 0.912343i
$$866$$ 0 0
$$867$$ − 10.8885i − 0.369794i
$$868$$ 0 0
$$869$$ −17.8885 −0.606827
$$870$$ 0 0
$$871$$ −46.8328 −1.58687
$$872$$ 0 0
$$873$$ − 12.4721i − 0.422118i
$$874$$ 0 0
$$875$$ 11.1803i 0.377964i
$$876$$ 0 0
$$877$$ − 13.3050i − 0.449276i −0.974442 0.224638i $$-0.927880\pi$$
0.974442 0.224638i $$-0.0721200\pi$$
$$878$$ 0 0
$$879$$ −7.05573 −0.237984
$$880$$ 0 0
$$881$$ 36.4721 1.22878 0.614389 0.789003i $$-0.289403\pi$$
0.614389 + 0.789003i $$0.289403\pi$$
$$882$$ 0 0
$$883$$ 28.3607i 0.954413i 0.878791 + 0.477206i $$0.158351\pi$$
−0.878791 + 0.477206i $$0.841649\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 31.4164i 1.05486i 0.849599 + 0.527430i $$0.176844\pi$$
−0.849599 + 0.527430i $$0.823156\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 0 0
$$893$$ 4.94427i 0.165454i
$$894$$ 0 0
$$895$$ 11.3050 0.377883
$$896$$ 0 0
$$897$$ − 17.8885i − 0.597281i
$$898$$ 0 0
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ −4.94427 −0.164718
$$902$$ 0 0
$$903$$ − 6.47214i − 0.215379i
$$904$$ 0 0
$$905$$ −16.8328 −0.559542
$$906$$ 0 0
$$907$$ − 28.3607i − 0.941701i −0.882213 0.470850i $$-0.843947\pi$$
0.882213 0.470850i $$-0.156053\pi$$
$$908$$ 0 0
$$909$$ 4.47214 0.148331
$$910$$ 0 0
$$911$$ −1.41641 −0.0469277 −0.0234638 0.999725i $$-0.507469\pi$$
−0.0234638 + 0.999725i $$0.507469\pi$$
$$912$$ 0 0
$$913$$ 25.8885i 0.856786i
$$914$$ 0 0
$$915$$ − 27.8885i − 0.921967i
$$916$$ 0 0
$$917$$ − 21.8885i − 0.722823i
$$918$$ 0 0
$$919$$ 52.7214 1.73912 0.869559 0.493830i $$-0.164403\pi$$
0.869559 + 0.493830i $$0.164403\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 15.7771i 0.519309i
$$924$$ 0 0
$$925$$ − 32.3607i − 1.06401i
$$926$$ 0 0
$$927$$ 4.94427i 0.162391i
$$928$$ 0 0
$$929$$ 43.3050 1.42079 0.710395 0.703804i $$-0.248516\pi$$
0.710395 + 0.703804i $$0.248516\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 16.0000i 0.523816i
$$934$$ 0 0
$$935$$ 11.0557i 0.361561i
$$936$$ 0 0
$$937$$ 47.3050i 1.54539i 0.634780 + 0.772693i $$0.281091\pi$$
−0.634780 + 0.772693i $$0.718909\pi$$
$$938$$ 0 0
$$939$$ −1.41641 −0.0462227
$$940$$ 0 0
$$941$$ 44.4721 1.44975 0.724875 0.688880i $$-0.241897\pi$$
0.724875 + 0.688880i $$0.241897\pi$$
$$942$$ 0 0
$$943$$ − 49.8885i − 1.62459i
$$944$$ 0 0
$$945$$ −2.23607 −0.0727393
$$946$$ 0 0
$$947$$ 16.0000i 0.519930i 0.965618 + 0.259965i $$0.0837111\pi$$
−0.965618 + 0.259965i $$0.916289\pi$$
$$948$$ 0 0
$$949$$ −73.6656 −2.39129
$$950$$ 0 0
$$951$$ −10.9443 −0.354892
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 0 0
$$955$$ 21.0557 0.681347
$$956$$ 0 0
$$957$$ 0.944272i 0.0305240i
$$958$$ 0 0
$$959$$ 15.8885 0.513068
$$960$$ 0 0
$$961$$ 40.7771 1.31539
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 11.0557i − 0.355896i
$$966$$ 0 0
$$967$$ 9.88854i 0.317994i 0.987279 + 0.158997i $$0.0508260\pi$$
−0.987279 + 0.158997i $$0.949174\pi$$
$$968$$ 0 0
$$969$$ −4.94427 −0.158833
$$970$$ 0 0
$$971$$ −1.88854 −0.0606063 −0.0303031 0.999541i $$-0.509647\pi$$
−0.0303031 + 0.999541i $$0.509647\pi$$
$$972$$ 0 0
$$973$$ − 14.9443i − 0.479091i
$$974$$ 0 0
$$975$$ −22.3607 −0.716115
$$976$$ 0 0
$$977$$ 0.832816i 0.0266441i 0.999911 + 0.0133221i $$0.00424067\pi$$
−0.999911 + 0.0133221i $$0.995759\pi$$
$$978$$ 0 0
$$979$$ −18.8328 −0.601899
$$980$$ 0 0
$$981$$ −2.94427 −0.0940034
$$982$$ 0 0
$$983$$ − 34.4721i − 1.09949i −0.835332 0.549745i $$-0.814725\pi$$
0.835332 0.549745i $$-0.185275\pi$$
$$984$$ 0 0
$$985$$ − 35.5279i − 1.13201i
$$986$$ 0 0
$$987$$ − 2.47214i − 0.0786890i
$$988$$ 0 0
$$989$$ −25.8885 −0.823208
$$990$$ 0 0
$$991$$ 39.0557 1.24065 0.620323 0.784346i $$-0.287002\pi$$
0.620323 + 0.784346i $$0.287002\pi$$
$$992$$ 0 0
$$993$$ − 1.88854i − 0.0599311i
$$994$$ 0 0
$$995$$ −25.7771 −0.817189
$$996$$ 0 0
$$997$$ − 50.3607i − 1.59494i −0.603359 0.797469i $$-0.706172\pi$$
0.603359 0.797469i $$-0.293828\pi$$
$$998$$ 0 0
$$999$$ 6.47214 0.204769
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 1680.2.t.h.1009.1 4
3.2 odd 2 5040.2.t.u.1009.3 4
4.3 odd 2 840.2.t.c.169.3 yes 4
5.2 odd 4 8400.2.a.cz.1.1 2
5.3 odd 4 8400.2.a.db.1.2 2
5.4 even 2 inner 1680.2.t.h.1009.3 4
12.11 even 2 2520.2.t.f.1009.4 4
15.14 odd 2 5040.2.t.u.1009.4 4
20.3 even 4 4200.2.a.bj.1.2 2
20.7 even 4 4200.2.a.bk.1.1 2
20.19 odd 2 840.2.t.c.169.1 4
60.59 even 2 2520.2.t.f.1009.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.c.169.1 4 20.19 odd 2
840.2.t.c.169.3 yes 4 4.3 odd 2
1680.2.t.h.1009.1 4 1.1 even 1 trivial
1680.2.t.h.1009.3 4 5.4 even 2 inner
2520.2.t.f.1009.3 4 60.59 even 2
2520.2.t.f.1009.4 4 12.11 even 2
4200.2.a.bj.1.2 2 20.3 even 4
4200.2.a.bk.1.1 2 20.7 even 4
5040.2.t.u.1009.3 4 3.2 odd 2
5040.2.t.u.1009.4 4 15.14 odd 2
8400.2.a.cz.1.1 2 5.2 odd 4
8400.2.a.db.1.2 2 5.3 odd 4