# Properties

 Label 1225.2.b.d.99.2 Level $1225$ Weight $2$ Character 1225.99 Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1225.99 Dual form 1225.2.b.d.99.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} +2.00000 q^{4} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} +4.00000 q^{16} -3.00000i q^{17} +2.00000 q^{19} +6.00000i q^{23} +5.00000i q^{27} -3.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -5.00000 q^{39} +12.0000 q^{41} +10.0000i q^{43} -6.00000 q^{44} -9.00000i q^{47} +4.00000i q^{48} +3.00000 q^{51} +10.0000i q^{52} -12.0000i q^{53} +2.00000i q^{57} -8.00000 q^{61} +8.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -6.00000 q^{69} +2.00000i q^{73} +4.00000 q^{76} +1.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} -3.00000i q^{87} -12.0000 q^{89} +12.0000i q^{92} +4.00000i q^{93} +1.00000i q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + 4q^{9} + O(q^{10})$$ $$2q + 4q^{4} + 4q^{9} - 6q^{11} + 8q^{16} + 4q^{19} - 6q^{29} + 8q^{31} + 8q^{36} - 10q^{39} + 24q^{41} - 12q^{44} + 6q^{51} - 16q^{61} + 16q^{64} - 12q^{69} + 8q^{76} + 2q^{79} + 2q^{81} - 24q^{89} - 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 2.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 2.00000i 0.577350i
$$13$$ 5.00000i 1.38675i 0.720577 + 0.693375i $$0.243877\pi$$
−0.720577 + 0.693375i $$0.756123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000i 0.962250i
$$28$$ 0 0
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ − 3.00000i − 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 4.00000 0.666667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ 0 0
$$43$$ 10.0000i 1.52499i 0.646997 + 0.762493i $$0.276025\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ 4.00000i 0.577350i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 10.0000i 1.38675i
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 3.00000i − 0.321634i
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 12.0000i 1.25109i
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.00000i 0.101535i 0.998711 + 0.0507673i $$0.0161667\pi$$
−0.998711 + 0.0507673i $$0.983833\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 5.00000i 0.492665i 0.969185 + 0.246332i $$0.0792255\pi$$
−0.969185 + 0.246332i $$0.920775\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 10.0000i 0.962250i
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 10.0000i 0.924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 0 0
$$129$$ −10.0000 −0.880451
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ − 6.00000i − 0.522233i
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 0 0
$$143$$ − 15.0000i − 1.25436i
$$144$$ 8.00000 0.666667
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 4.00000i 0.328798i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −10.0000 −0.800641
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 2.00000i − 0.156652i −0.996928 0.0783260i $$-0.975042\pi$$
0.996928 0.0783260i $$-0.0249575\pi$$
$$164$$ 24.0000 1.87409
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.00000i 0.232147i 0.993241 + 0.116073i $$0.0370308\pi$$
−0.993241 + 0.116073i $$0.962969\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 20.0000i 1.52499i
$$173$$ − 9.00000i − 0.684257i −0.939653 0.342129i $$-0.888852\pi$$
0.939653 0.342129i $$-0.111148\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −12.0000 −0.904534
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ − 8.00000i − 0.591377i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ − 18.0000i − 1.31278i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.0000i 0.834058i
$$208$$ 20.0000i 1.38675i
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ − 24.0000i − 1.64833i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 15.0000 1.00901
$$222$$ 0 0
$$223$$ − 19.0000i − 1.27233i −0.771551 0.636167i $$-0.780519\pi$$
0.771551 0.636167i $$-0.219481\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.00000i 0.199117i 0.995032 + 0.0995585i $$0.0317430\pi$$
−0.995032 + 0.0995585i $$0.968257\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 24.0000i − 1.57229i −0.618041 0.786146i $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1.00000i 0.0649570i
$$238$$ 0 0
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 16.0000i 1.02640i
$$244$$ −16.0000 −1.02430
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 10.0000i 0.636285i
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ − 18.0000i − 1.13165i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 30.0000i − 1.87135i −0.352865 0.935674i $$-0.614792\pi$$
0.352865 0.935674i $$-0.385208\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ − 12.0000i − 0.727607i
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −12.0000 −0.722315
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 3.00000 0.178965 0.0894825 0.995988i $$-0.471479\pi$$
0.0894825 + 0.995988i $$0.471479\pi$$
$$282$$ 0 0
$$283$$ − 13.0000i − 0.772770i −0.922338 0.386385i $$-0.873724\pi$$
0.922338 0.386385i $$-0.126276\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −1.00000 −0.0586210
$$292$$ 4.00000i 0.234082i
$$293$$ − 21.0000i − 1.22683i −0.789760 0.613417i $$-0.789795\pi$$
0.789760 0.613417i $$-0.210205\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 15.0000i − 0.870388i
$$298$$ 0 0
$$299$$ −30.0000 −1.73494
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 11.0000i − 0.627803i −0.949456 0.313902i $$-0.898364\pi$$
0.949456 0.313902i $$-0.101636\pi$$
$$308$$ 0 0
$$309$$ −5.00000 −0.284440
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ − 19.0000i − 1.07394i −0.843600 0.536972i $$-0.819568\pi$$
0.843600 0.536972i $$-0.180432\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ − 6.00000i − 0.333849i
$$324$$ 2.00000 0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 7.00000i 0.387101i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 24.0000i 1.31717i
$$333$$ 4.00000i 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 18.0000i − 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −25.0000 −1.33440
$$352$$ 0 0
$$353$$ 15.0000i 0.798369i 0.916871 + 0.399185i $$0.130707\pi$$
−0.916871 + 0.399185i $$0.869293\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −24.0000 −1.27200
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 2.00000i − 0.104973i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 17.0000i − 0.887393i −0.896177 0.443696i $$-0.853667\pi$$
0.896177 0.443696i $$-0.146333\pi$$
$$368$$ 24.0000i 1.25109i
$$369$$ 24.0000 1.24939
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000i 0.414781i
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 15.0000i − 0.772539i
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 20.0000i 1.01666i
$$388$$ 2.00000i 0.101535i
$$389$$ 3.00000 0.152106 0.0760530 0.997104i $$-0.475768\pi$$
0.0760530 + 0.997104i $$0.475768\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 0 0
$$393$$ 6.00000i 0.302660i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −12.0000 −0.603023
$$397$$ 25.0000i 1.25471i 0.778732 + 0.627357i $$0.215863\pi$$
−0.778732 + 0.627357i $$0.784137\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.0000 −0.749064 −0.374532 0.927214i $$-0.622197\pi$$
−0.374532 + 0.927214i $$0.622197\pi$$
$$402$$ 0 0
$$403$$ 20.0000i 0.996271i
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 6.00000i − 0.297409i
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 10.0000i 0.492665i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 14.0000i 0.685583i
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 17.0000 0.828529 0.414265 0.910156i $$-0.364039\pi$$
0.414265 + 0.910156i $$0.364039\pi$$
$$422$$ 0 0
$$423$$ − 18.0000i − 0.875190i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 15.0000 0.724207
$$430$$ 0 0
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ 20.0000i 0.962250i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 12.0000i 0.574038i
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.0000i 0.855206i 0.903967 + 0.427603i $$0.140642\pi$$
−0.903967 + 0.427603i $$0.859358\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.00000i 0.283790i
$$448$$ 0 0
$$449$$ 9.00000 0.424736 0.212368 0.977190i $$-0.431882\pi$$
0.212368 + 0.977190i $$0.431882\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ − 12.0000i − 0.564433i
$$453$$ − 1.00000i − 0.0469841i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ −12.0000 −0.557086
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 15.0000i − 0.694117i −0.937843 0.347059i $$-0.887180\pi$$
0.937843 0.347059i $$-0.112820\pi$$
$$468$$ 20.0000i 0.924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ 0 0
$$473$$ − 30.0000i − 1.37940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 24.0000i − 1.09888i
$$478$$ 0 0
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ −10.0000 −0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −4.00000 −0.181818
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ 0 0
$$489$$ 2.00000 0.0904431
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ 24.0000i 1.08200i
$$493$$ 9.00000i 0.405340i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 16.0000 0.718421
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.0000 1.38775 0.693875 0.720095i $$-0.255902\pi$$
0.693875 + 0.720095i $$0.255902\pi$$
$$500$$ 0 0
$$501$$ −3.00000 −0.134030
$$502$$ 0 0
$$503$$ 27.0000i 1.20387i 0.798545 + 0.601935i $$0.205603\pi$$
−0.798545 + 0.601935i $$0.794397\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 12.0000i − 0.532939i
$$508$$ − 32.0000i − 1.41977i
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 10.0000i 0.441511i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −20.0000 −0.880451
$$517$$ 27.0000i 1.18746i
$$518$$ 0 0
$$519$$ 9.00000 0.395056
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 12.0000i − 0.522728i
$$528$$ − 12.0000i − 0.522233i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 60.0000i 2.59889i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 12.0000i − 0.517838i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 0 0
$$543$$ − 20.0000i − 0.858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ − 24.0000i − 1.02523i
$$549$$ −16.0000 −0.682863
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 28.0000 1.18746
$$557$$ − 24.0000i − 1.01691i −0.861088 0.508456i $$-0.830216\pi$$
0.861088 0.508456i $$-0.169784\pi$$
$$558$$ 0 0
$$559$$ −50.0000 −2.11477
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 18.0000 0.757937
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ − 30.0000i − 1.25436i
$$573$$ 9.00000i 0.375980i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 16.0000 0.666667
$$577$$ 7.00000i 0.291414i 0.989328 + 0.145707i $$0.0465456\pi$$
−0.989328 + 0.145707i $$0.953454\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 24.0000i 0.990586i 0.868726 + 0.495293i $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ − 39.0000i − 1.60154i −0.598973 0.800769i $$-0.704424\pi$$
0.598973 0.800769i $$-0.295576\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ − 16.0000i − 0.654836i
$$598$$ 0 0
$$599$$ −45.0000 −1.83865 −0.919325 0.393499i $$-0.871265\pi$$
−0.919325 + 0.393499i $$0.871265\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ − 8.00000i − 0.325785i
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 13.0000i 0.527654i 0.964570 + 0.263827i $$0.0849848\pi$$
−0.964570 + 0.263827i $$0.915015\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 45.0000 1.82051
$$612$$ − 12.0000i − 0.485071i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 0 0
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −20.0000 −0.800641
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 6.00000i − 0.239617i
$$628$$ − 28.0000i − 1.11732i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 0 0
$$633$$ − 13.0000i − 0.516704i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 24.0000 0.951662
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ 41.0000i 1.61688i 0.588577 + 0.808441i $$0.299688\pi$$
−0.588577 + 0.808441i $$0.700312\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 48.0000 1.87409
$$657$$ 4.00000i 0.156055i
$$658$$ 0 0
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ −32.0000 −1.24466 −0.622328 0.782757i $$-0.713813\pi$$
−0.622328 + 0.782757i $$0.713813\pi$$
$$662$$ 0 0
$$663$$ 15.0000i 0.582552i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 18.0000i − 0.696963i
$$668$$ 6.00000i 0.232147i
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 28.0000i 1.07932i 0.841883 + 0.539660i $$0.181447\pi$$
−0.841883 + 0.539660i $$0.818553\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −24.0000 −0.923077
$$677$$ − 45.0000i − 1.72949i −0.502211 0.864745i $$-0.667480\pi$$
0.502211 0.864745i $$-0.332520\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.00000 −0.114960
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 4.00000i − 0.152610i
$$688$$ 40.0000i 1.52499i
$$689$$ 60.0000 2.28582
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ 0 0
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ −9.00000 −0.339925 −0.169963 0.985451i $$-0.554365\pi$$
−0.169963 + 0.985451i $$0.554365\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ −24.0000 −0.904534
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ 0 0
$$713$$ 24.0000i 0.898807i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 21.0000i 0.784259i
$$718$$ 0 0
$$719$$ −30.0000 −1.11881 −0.559406 0.828894i $$-0.688971\pi$$
−0.559406 + 0.828894i $$0.688971\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 10.0000i 0.371904i
$$724$$ −40.0000 −1.48659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 8.00000i − 0.296704i −0.988935 0.148352i $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 30.0000 1.10959
$$732$$ − 16.0000i − 0.591377i
$$733$$ − 31.0000i − 1.14501i −0.819901 0.572506i $$-0.805971\pi$$
0.819901 0.572506i $$-0.194029\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 12.0000i 0.442026i
$$738$$ 0 0
$$739$$ 43.0000 1.58178 0.790890 0.611958i $$-0.209618\pi$$
0.790890 + 0.611958i $$0.209618\pi$$
$$740$$ 0 0
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.0000i 0.878114i
$$748$$ 18.0000i 0.658145i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ − 36.0000i − 1.31278i
$$753$$ − 18.0000i − 0.655956i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 16.0000i − 0.581530i −0.956795 0.290765i $$-0.906090\pi$$
0.956795 0.290765i $$-0.0939098\pi$$
$$758$$ 0 0
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 16.0000i 0.577350i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ 8.00000i 0.287926i
$$773$$ 21.0000i 0.755318i 0.925945 + 0.377659i $$0.123271\pi$$
−0.925945 + 0.377659i $$0.876729\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 15.0000i − 0.536056i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 5.00000i − 0.178231i −0.996021 0.0891154i $$-0.971596\pi$$
0.996021 0.0891154i $$-0.0284040\pi$$
$$788$$ 0 0
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 40.0000i − 1.42044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −32.0000 −1.13421
$$797$$ − 15.0000i − 0.531327i −0.964066 0.265664i $$-0.914409\pi$$
0.964066 0.265664i $$-0.0855911\pi$$
$$798$$ 0 0
$$799$$ −27.0000 −0.955191
$$800$$ 0 0
$$801$$ −24.0000 −0.847998
$$802$$ 0 0
$$803$$ − 6.00000i − 0.211735i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 6.00000i − 0.211210i
$$808$$ 0 0
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ 16.0000i 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ 20.0000i 0.699711i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −27.0000 −0.942306 −0.471153 0.882051i $$-0.656162\pi$$
−0.471153 + 0.882051i $$0.656162\pi$$
$$822$$ 0 0
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 54.0000i 1.87776i 0.344239 + 0.938882i $$0.388137\pi$$
−0.344239 + 0.938882i $$0.611863\pi$$
$$828$$ 24.0000i 0.834058i
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 40.0000i 1.38675i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −12.0000 −0.415029
$$837$$ 20.0000i 0.691301i
$$838$$ 0 0
$$839$$ −42.0000 −1.45000 −0.725001 0.688748i $$-0.758161\pi$$
−0.725001 + 0.688748i $$0.758161\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 3.00000i 0.103325i
$$844$$ −26.0000 −0.894957
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 48.0000i − 1.64833i
$$849$$ 13.0000 0.446159
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 8.00000i 0.271694i
$$868$$ 0 0
$$869$$ −3.00000 −0.101768
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ 2.00000i 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ 0 0
$$879$$ 21.0000 0.708312
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 30.0000 1.00901
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ − 38.0000i − 1.27233i
$$893$$ − 18.0000i − 0.602347i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 30.0000i − 1.00167i
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26.0000i 0.863316i 0.902037 + 0.431658i $$0.142071\pi$$
−0.902037 + 0.431658i $$0.857929\pi$$
$$908$$ 6.00000i 0.199117i
$$909$$ −12.0000 −0.398015
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ − 36.0000i − 1.19143i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ −8.00000 −0.264327
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ 11.0000 0.362462
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 10.0000i 0.328443i
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 48.0000i − 1.57229i
$$933$$ − 18.0000i − 0.589294i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 47.0000i − 1.53542i −0.640796 0.767712i $$-0.721395\pi$$
0.640796 0.767712i $$-0.278605\pi$$
$$938$$ 0 0
$$939$$ 19.0000 0.620042
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 72.0000i 2.34464i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 2.00000i 0.0649570i
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 42.0000 1.35838
$$957$$ 9.00000i 0.290929i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 20.0000 0.644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 22.0000i − 0.707472i −0.935345 0.353736i $$-0.884911\pi$$
0.935345 0.353736i $$-0.115089\pi$$
$$968$$ 0 0
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 32.0000i 1.02640i
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −32.0000 −1.02430
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 0 0
$$983$$ − 21.0000i − 0.669796i −0.942254 0.334898i $$-0.891298\pi$$
0.942254 0.334898i $$-0.108702\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 20.0000i 0.636285i
$$989$$ −60.0000 −1.90789
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −24.0000 −0.760469
$$997$$ 37.0000i 1.17180i 0.810383 + 0.585901i $$0.199259\pi$$
−0.810383 + 0.585901i $$0.800741\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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 1225.2.b.d.99.2 2
5.2 odd 4 1225.2.a.e.1.1 1
5.3 odd 4 245.2.a.c.1.1 1
5.4 even 2 inner 1225.2.b.d.99.1 2
7.6 odd 2 175.2.b.a.99.1 2
15.8 even 4 2205.2.a.e.1.1 1
20.3 even 4 3920.2.a.ba.1.1 1
21.20 even 2 1575.2.d.c.1324.2 2
28.27 even 2 2800.2.g.l.449.2 2
35.3 even 12 245.2.e.a.226.1 2
35.13 even 4 35.2.a.a.1.1 1
35.18 odd 12 245.2.e.b.226.1 2
35.23 odd 12 245.2.e.b.116.1 2
35.27 even 4 175.2.a.b.1.1 1
35.33 even 12 245.2.e.a.116.1 2
35.34 odd 2 175.2.b.a.99.2 2
105.62 odd 4 1575.2.a.f.1.1 1
105.83 odd 4 315.2.a.b.1.1 1
105.104 even 2 1575.2.d.c.1324.1 2
140.27 odd 4 2800.2.a.z.1.1 1
140.83 odd 4 560.2.a.b.1.1 1
140.139 even 2 2800.2.g.l.449.1 2
280.13 even 4 2240.2.a.k.1.1 1
280.83 odd 4 2240.2.a.u.1.1 1
385.153 odd 4 4235.2.a.c.1.1 1
420.83 even 4 5040.2.a.v.1.1 1
455.363 even 4 5915.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.a.a.1.1 1 35.13 even 4
175.2.a.b.1.1 1 35.27 even 4
175.2.b.a.99.1 2 7.6 odd 2
175.2.b.a.99.2 2 35.34 odd 2
245.2.a.c.1.1 1 5.3 odd 4
245.2.e.a.116.1 2 35.33 even 12
245.2.e.a.226.1 2 35.3 even 12
245.2.e.b.116.1 2 35.23 odd 12
245.2.e.b.226.1 2 35.18 odd 12
315.2.a.b.1.1 1 105.83 odd 4
560.2.a.b.1.1 1 140.83 odd 4
1225.2.a.e.1.1 1 5.2 odd 4
1225.2.b.d.99.1 2 5.4 even 2 inner
1225.2.b.d.99.2 2 1.1 even 1 trivial
1575.2.a.f.1.1 1 105.62 odd 4
1575.2.d.c.1324.1 2 105.104 even 2
1575.2.d.c.1324.2 2 21.20 even 2
2205.2.a.e.1.1 1 15.8 even 4
2240.2.a.k.1.1 1 280.13 even 4
2240.2.a.u.1.1 1 280.83 odd 4
2800.2.a.z.1.1 1 140.27 odd 4
2800.2.g.l.449.1 2 140.139 even 2
2800.2.g.l.449.2 2 28.27 even 2
3920.2.a.ba.1.1 1 20.3 even 4
4235.2.a.c.1.1 1 385.153 odd 4
5040.2.a.v.1.1 1 420.83 even 4
5915.2.a.f.1.1 1 455.363 even 4