# Properties

 Label 1815.2.a.g.1.2 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.73205 q^{7} +1.00000 q^{9} +2.00000 q^{12} +1.00000 q^{15} +4.00000 q^{16} -3.46410 q^{17} -5.19615 q^{19} +2.00000 q^{20} -1.73205 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.46410 q^{28} +6.92820 q^{29} +1.00000 q^{31} -1.73205 q^{35} -2.00000 q^{36} -5.00000 q^{37} +3.46410 q^{41} +10.3923 q^{43} -1.00000 q^{45} -12.0000 q^{47} -4.00000 q^{48} -4.00000 q^{49} +3.46410 q^{51} +6.00000 q^{53} +5.19615 q^{57} -2.00000 q^{60} -12.1244 q^{61} +1.73205 q^{63} -8.00000 q^{64} -5.00000 q^{67} +6.92820 q^{68} -6.00000 q^{69} -6.00000 q^{71} -1.73205 q^{73} -1.00000 q^{75} +10.3923 q^{76} -15.5885 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.92820 q^{83} +3.46410 q^{84} +3.46410 q^{85} -6.92820 q^{87} -6.00000 q^{89} -12.0000 q^{92} -1.00000 q^{93} +5.19615 q^{95} -13.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + 4q^{12} + 2q^{15} + 8q^{16} + 4q^{20} + 12q^{23} + 2q^{25} - 2q^{27} + 2q^{31} - 4q^{36} - 10q^{37} - 2q^{45} - 24q^{47} - 8q^{48} - 8q^{49} + 12q^{53} - 4q^{60} - 16q^{64} - 10q^{67} - 12q^{69} - 12q^{71} - 2q^{75} - 8q^{80} + 2q^{81} - 12q^{89} - 24q^{92} - 2q^{93} - 26q^{97} + O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −2.00000 −1.00000
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.73205 0.654654 0.327327 0.944911i $$-0.393852\pi$$
0.327327 + 0.944911i $$0.393852\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 2.00000 0.577350
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 4.00000 1.00000
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 0 0
$$19$$ −5.19615 −1.19208 −0.596040 0.802955i $$-0.703260\pi$$
−0.596040 + 0.802955i $$0.703260\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −1.73205 −0.377964
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −3.46410 −0.654654
$$29$$ 6.92820 1.28654 0.643268 0.765641i $$-0.277578\pi$$
0.643268 + 0.765641i $$0.277578\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.73205 −0.292770
$$36$$ −2.00000 −0.333333
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.46410 0.541002 0.270501 0.962720i $$-0.412811\pi$$
0.270501 + 0.962720i $$0.412811\pi$$
$$42$$ 0 0
$$43$$ 10.3923 1.58481 0.792406 0.609994i $$-0.208828\pi$$
0.792406 + 0.609994i $$0.208828\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ −4.00000 −0.571429
$$50$$ 0 0
$$51$$ 3.46410 0.485071
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.19615 0.688247
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −12.1244 −1.55236 −0.776182 0.630509i $$-0.782846\pi$$
−0.776182 + 0.630509i $$0.782846\pi$$
$$62$$ 0 0
$$63$$ 1.73205 0.218218
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 6.92820 0.840168
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −1.73205 −0.202721 −0.101361 0.994850i $$-0.532320\pi$$
−0.101361 + 0.994850i $$0.532320\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 10.3923 1.19208
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −15.5885 −1.75384 −0.876919 0.480638i $$-0.840405\pi$$
−0.876919 + 0.480638i $$0.840405\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.92820 −0.760469 −0.380235 0.924890i $$-0.624157\pi$$
−0.380235 + 0.924890i $$0.624157\pi$$
$$84$$ 3.46410 0.377964
$$85$$ 3.46410 0.375735
$$86$$ 0 0
$$87$$ −6.92820 −0.742781
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −12.0000 −1.25109
$$93$$ −1.00000 −0.103695
$$94$$ 0 0
$$95$$ 5.19615 0.533114
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.00000 −0.200000
$$101$$ −13.8564 −1.37876 −0.689382 0.724398i $$-0.742118\pi$$
−0.689382 + 0.724398i $$0.742118\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ 1.73205 0.169031
$$106$$ 0 0
$$107$$ −6.92820 −0.669775 −0.334887 0.942258i $$-0.608698\pi$$
−0.334887 + 0.942258i $$0.608698\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 12.1244 1.16130 0.580651 0.814152i $$-0.302798\pi$$
0.580651 + 0.814152i $$0.302798\pi$$
$$110$$ 0 0
$$111$$ 5.00000 0.474579
$$112$$ 6.92820 0.654654
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ −13.8564 −1.28654
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −3.46410 −0.312348
$$124$$ −2.00000 −0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 1.73205 0.153695 0.0768473 0.997043i $$-0.475515\pi$$
0.0768473 + 0.997043i $$0.475515\pi$$
$$128$$ 0 0
$$129$$ −10.3923 −0.914991
$$130$$ 0 0
$$131$$ 3.46410 0.302660 0.151330 0.988483i $$-0.451644\pi$$
0.151330 + 0.988483i $$0.451644\pi$$
$$132$$ 0 0
$$133$$ −9.00000 −0.780399
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 3.46410 0.293821 0.146911 0.989150i $$-0.453067\pi$$
0.146911 + 0.989150i $$0.453067\pi$$
$$140$$ 3.46410 0.292770
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ −6.92820 −0.575356
$$146$$ 0 0
$$147$$ 4.00000 0.329914
$$148$$ 10.0000 0.821995
$$149$$ 24.2487 1.98653 0.993266 0.115857i $$-0.0369614\pi$$
0.993266 + 0.115857i $$0.0369614\pi$$
$$150$$ 0 0
$$151$$ 17.3205 1.40952 0.704761 0.709444i $$-0.251054\pi$$
0.704761 + 0.709444i $$0.251054\pi$$
$$152$$ 0 0
$$153$$ −3.46410 −0.280056
$$154$$ 0 0
$$155$$ −1.00000 −0.0803219
$$156$$ 0 0
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 10.3923 0.819028
$$162$$ 0 0
$$163$$ −13.0000 −1.01824 −0.509119 0.860696i $$-0.670029\pi$$
−0.509119 + 0.860696i $$0.670029\pi$$
$$164$$ −6.92820 −0.541002
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.46410 0.268060 0.134030 0.990977i $$-0.457208\pi$$
0.134030 + 0.990977i $$0.457208\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ −5.19615 −0.397360
$$172$$ −20.7846 −1.58481
$$173$$ −10.3923 −0.790112 −0.395056 0.918657i $$-0.629275\pi$$
−0.395056 + 0.918657i $$0.629275\pi$$
$$174$$ 0 0
$$175$$ 1.73205 0.130931
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 12.1244 0.896258
$$184$$ 0 0
$$185$$ 5.00000 0.367607
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 24.0000 1.75038
$$189$$ −1.73205 −0.125988
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 8.00000 0.577350
$$193$$ 1.73205 0.124676 0.0623379 0.998055i $$-0.480144\pi$$
0.0623379 + 0.998055i $$0.480144\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 8.00000 0.571429
$$197$$ −6.92820 −0.493614 −0.246807 0.969065i $$-0.579381\pi$$
−0.246807 + 0.969065i $$0.579381\pi$$
$$198$$ 0 0
$$199$$ −11.0000 −0.779769 −0.389885 0.920864i $$-0.627485\pi$$
−0.389885 + 0.920864i $$0.627485\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ 0 0
$$203$$ 12.0000 0.842235
$$204$$ −6.92820 −0.485071
$$205$$ −3.46410 −0.241943
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −19.0526 −1.31163 −0.655816 0.754921i $$-0.727675\pi$$
−0.655816 + 0.754921i $$0.727675\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ −10.3923 −0.708749
$$216$$ 0 0
$$217$$ 1.73205 0.117579
$$218$$ 0 0
$$219$$ 1.73205 0.117041
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 27.7128 1.83936 0.919682 0.392664i $$-0.128446\pi$$
0.919682 + 0.392664i $$0.128446\pi$$
$$228$$ −10.3923 −0.688247
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 13.8564 0.907763 0.453882 0.891062i $$-0.350039\pi$$
0.453882 + 0.891062i $$0.350039\pi$$
$$234$$ 0 0
$$235$$ 12.0000 0.782794
$$236$$ 0 0
$$237$$ 15.5885 1.01258
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 4.00000 0.258199
$$241$$ 6.92820 0.446285 0.223142 0.974786i $$-0.428369\pi$$
0.223142 + 0.974786i $$0.428369\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 24.2487 1.55236
$$245$$ 4.00000 0.255551
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 6.92820 0.439057
$$250$$ 0 0
$$251$$ 30.0000 1.89358 0.946792 0.321847i $$-0.104304\pi$$
0.946792 + 0.321847i $$0.104304\pi$$
$$252$$ −3.46410 −0.218218
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −3.46410 −0.216930
$$256$$ 16.0000 1.00000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −8.66025 −0.538122
$$260$$ 0 0
$$261$$ 6.92820 0.428845
$$262$$ 0 0
$$263$$ −10.3923 −0.640817 −0.320408 0.947279i $$-0.603820\pi$$
−0.320408 + 0.947279i $$0.603820\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 10.0000 0.610847
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −10.3923 −0.631288 −0.315644 0.948878i $$-0.602220\pi$$
−0.315644 + 0.948878i $$0.602220\pi$$
$$272$$ −13.8564 −0.840168
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ 25.9808 1.56103 0.780516 0.625135i $$-0.214956\pi$$
0.780516 + 0.625135i $$0.214956\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 31.1769 1.85986 0.929929 0.367738i $$-0.119868\pi$$
0.929929 + 0.367738i $$0.119868\pi$$
$$282$$ 0 0
$$283$$ 15.5885 0.926638 0.463319 0.886192i $$-0.346658\pi$$
0.463319 + 0.886192i $$0.346658\pi$$
$$284$$ 12.0000 0.712069
$$285$$ −5.19615 −0.307794
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 13.0000 0.762073
$$292$$ 3.46410 0.202721
$$293$$ −31.1769 −1.82137 −0.910687 0.413096i $$-0.864447\pi$$
−0.910687 + 0.413096i $$0.864447\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 2.00000 0.115470
$$301$$ 18.0000 1.03750
$$302$$ 0 0
$$303$$ 13.8564 0.796030
$$304$$ −20.7846 −1.19208
$$305$$ 12.1244 0.694239
$$306$$ 0 0
$$307$$ −8.66025 −0.494267 −0.247133 0.968981i $$-0.579489\pi$$
−0.247133 + 0.968981i $$0.579489\pi$$
$$308$$ 0 0
$$309$$ 13.0000 0.739544
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ −1.73205 −0.0975900
$$316$$ 31.1769 1.75384
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 8.00000 0.447214
$$321$$ 6.92820 0.386695
$$322$$ 0 0
$$323$$ 18.0000 1.00155
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −12.1244 −0.670478
$$328$$ 0 0
$$329$$ −20.7846 −1.14589
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ 13.8564 0.760469
$$333$$ −5.00000 −0.273998
$$334$$ 0 0
$$335$$ 5.00000 0.273179
$$336$$ −6.92820 −0.377964
$$337$$ −29.4449 −1.60396 −0.801982 0.597348i $$-0.796221\pi$$
−0.801982 + 0.597348i $$0.796221\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −6.92820 −0.375735
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −19.0526 −1.02874
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ 0 0
$$347$$ 17.3205 0.929814 0.464907 0.885360i $$-0.346088\pi$$
0.464907 + 0.885360i $$0.346088\pi$$
$$348$$ 13.8564 0.742781
$$349$$ 5.19615 0.278144 0.139072 0.990282i $$-0.455588\pi$$
0.139072 + 0.990282i $$0.455588\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ 12.0000 0.635999
$$357$$ 6.00000 0.317554
$$358$$ 0 0
$$359$$ −24.2487 −1.27980 −0.639899 0.768459i $$-0.721024\pi$$
−0.639899 + 0.768459i $$0.721024\pi$$
$$360$$ 0 0
$$361$$ 8.00000 0.421053
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.73205 0.0906597
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 24.0000 1.25109
$$369$$ 3.46410 0.180334
$$370$$ 0 0
$$371$$ 10.3923 0.539542
$$372$$ 2.00000 0.103695
$$373$$ −19.0526 −0.986504 −0.493252 0.869886i $$-0.664192\pi$$
−0.493252 + 0.869886i $$0.664192\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ −10.3923 −0.533114
$$381$$ −1.73205 −0.0887357
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.3923 0.528271
$$388$$ 26.0000 1.31995
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −20.7846 −1.05112
$$392$$ 0 0
$$393$$ −3.46410 −0.174741
$$394$$ 0 0
$$395$$ 15.5885 0.784340
$$396$$ 0 0
$$397$$ −19.0000 −0.953583 −0.476791 0.879017i $$-0.658200\pi$$
−0.476791 + 0.879017i $$0.658200\pi$$
$$398$$ 0 0
$$399$$ 9.00000 0.450564
$$400$$ 4.00000 0.200000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 27.7128 1.37876
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.0526 0.942088 0.471044 0.882110i $$-0.343877\pi$$
0.471044 + 0.882110i $$0.343877\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 26.0000 1.28093
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 6.92820 0.340092
$$416$$ 0 0
$$417$$ −3.46410 −0.169638
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ −3.46410 −0.169031
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ −3.46410 −0.168034
$$426$$ 0 0
$$427$$ −21.0000 −1.01626
$$428$$ 13.8564 0.669775
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −38.1051 −1.83546 −0.917729 0.397206i $$-0.869980\pi$$
−0.917729 + 0.397206i $$0.869980\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −7.00000 −0.336399 −0.168199 0.985753i $$-0.553795\pi$$
−0.168199 + 0.985753i $$0.553795\pi$$
$$434$$ 0 0
$$435$$ 6.92820 0.332182
$$436$$ −24.2487 −1.16130
$$437$$ −31.1769 −1.49139
$$438$$ 0 0
$$439$$ 15.5885 0.743996 0.371998 0.928233i $$-0.378673\pi$$
0.371998 + 0.928233i $$0.378673\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 18.0000 0.855206 0.427603 0.903967i $$-0.359358\pi$$
0.427603 + 0.903967i $$0.359358\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ −24.2487 −1.14692
$$448$$ −13.8564 −0.654654
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −17.3205 −0.813788
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 34.6410 1.62044 0.810219 0.586127i $$-0.199348\pi$$
0.810219 + 0.586127i $$0.199348\pi$$
$$458$$ 0 0
$$459$$ 3.46410 0.161690
$$460$$ 12.0000 0.559503
$$461$$ −17.3205 −0.806696 −0.403348 0.915047i $$-0.632154\pi$$
−0.403348 + 0.915047i $$0.632154\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 27.7128 1.28654
$$465$$ 1.00000 0.0463739
$$466$$ 0 0
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ −8.66025 −0.399893
$$470$$ 0 0
$$471$$ 5.00000 0.230388
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −5.19615 −0.238416
$$476$$ 12.0000 0.550019
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 27.7128 1.26623 0.633115 0.774057i $$-0.281776\pi$$
0.633115 + 0.774057i $$0.281776\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −10.3923 −0.472866
$$484$$ 0 0
$$485$$ 13.0000 0.590300
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 0 0
$$489$$ 13.0000 0.587880
$$490$$ 0 0
$$491$$ 24.2487 1.09433 0.547165 0.837025i $$-0.315707\pi$$
0.547165 + 0.837025i $$0.315707\pi$$
$$492$$ 6.92820 0.312348
$$493$$ −24.0000 −1.08091
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −10.3923 −0.466159
$$498$$ 0 0
$$499$$ 1.00000 0.0447661 0.0223831 0.999749i $$-0.492875\pi$$
0.0223831 + 0.999749i $$0.492875\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ −3.46410 −0.154765
$$502$$ 0 0
$$503$$ 34.6410 1.54457 0.772283 0.635278i $$-0.219115\pi$$
0.772283 + 0.635278i $$0.219115\pi$$
$$504$$ 0 0
$$505$$ 13.8564 0.616602
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ −3.46410 −0.153695
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 5.19615 0.229416
$$514$$ 0 0
$$515$$ 13.0000 0.572848
$$516$$ 20.7846 0.914991
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 10.3923 0.456172
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 29.4449 1.28753 0.643767 0.765222i $$-0.277371\pi$$
0.643767 + 0.765222i $$0.277371\pi$$
$$524$$ −6.92820 −0.302660
$$525$$ −1.73205 −0.0755929
$$526$$ 0 0
$$527$$ −3.46410 −0.150899
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 18.0000 0.780399
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 6.92820 0.299532
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ −2.00000 −0.0860663
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 0 0
$$543$$ 7.00000 0.300399
$$544$$ 0 0
$$545$$ −12.1244 −0.519350
$$546$$ 0 0
$$547$$ −31.1769 −1.33303 −0.666514 0.745492i $$-0.732214\pi$$
−0.666514 + 0.745492i $$0.732214\pi$$
$$548$$ 24.0000 1.02523
$$549$$ −12.1244 −0.517455
$$550$$ 0 0
$$551$$ −36.0000 −1.53365
$$552$$ 0 0
$$553$$ −27.0000 −1.14816
$$554$$ 0 0
$$555$$ −5.00000 −0.212238
$$556$$ −6.92820 −0.293821
$$557$$ 3.46410 0.146779 0.0733893 0.997303i $$-0.476618\pi$$
0.0733893 + 0.997303i $$0.476618\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −6.92820 −0.292770
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 34.6410 1.45994 0.729972 0.683477i $$-0.239533\pi$$
0.729972 + 0.683477i $$0.239533\pi$$
$$564$$ −24.0000 −1.01058
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.73205 0.0727393
$$568$$ 0 0
$$569$$ −20.7846 −0.871336 −0.435668 0.900107i $$-0.643488\pi$$
−0.435668 + 0.900107i $$0.643488\pi$$
$$570$$ 0 0
$$571$$ 19.0526 0.797325 0.398662 0.917098i $$-0.369475\pi$$
0.398662 + 0.917098i $$0.369475\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ −8.00000 −0.333333
$$577$$ 7.00000 0.291414 0.145707 0.989328i $$-0.453454\pi$$
0.145707 + 0.989328i $$0.453454\pi$$
$$578$$ 0 0
$$579$$ −1.73205 −0.0719816
$$580$$ 13.8564 0.575356
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −8.00000 −0.329914
$$589$$ −5.19615 −0.214104
$$590$$ 0 0
$$591$$ 6.92820 0.284988
$$592$$ −20.0000 −0.821995
$$593$$ −10.3923 −0.426761 −0.213380 0.976969i $$-0.568447\pi$$
−0.213380 + 0.976969i $$0.568447\pi$$
$$594$$ 0 0
$$595$$ 6.00000 0.245976
$$596$$ −48.4974 −1.98653
$$597$$ 11.0000 0.450200
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ 0 0
$$601$$ 22.5167 0.918474 0.459237 0.888314i $$-0.348123\pi$$
0.459237 + 0.888314i $$0.348123\pi$$
$$602$$ 0 0
$$603$$ −5.00000 −0.203616
$$604$$ −34.6410 −1.40952
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.46410 −0.140604 −0.0703018 0.997526i $$-0.522396\pi$$
−0.0703018 + 0.997526i $$0.522396\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.92820 0.280056
$$613$$ −29.4449 −1.18927 −0.594633 0.803997i $$-0.702703\pi$$
−0.594633 + 0.803997i $$0.702703\pi$$
$$614$$ 0 0
$$615$$ 3.46410 0.139686
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 2.00000 0.0803219
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ −10.3923 −0.416359
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 17.3205 0.690614
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 0 0
$$633$$ 19.0526 0.757271
$$634$$ 0 0
$$635$$ −1.73205 −0.0687343
$$636$$ 12.0000 0.475831
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −31.0000 −1.22252 −0.611260 0.791430i $$-0.709337\pi$$
−0.611260 + 0.791430i $$0.709337\pi$$
$$644$$ −20.7846 −0.819028
$$645$$ 10.3923 0.409197
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −1.73205 −0.0678844
$$652$$ 26.0000 1.01824
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ −3.46410 −0.135354
$$656$$ 13.8564 0.541002
$$657$$ −1.73205 −0.0675737
$$658$$ 0 0
$$659$$ −10.3923 −0.404827 −0.202413 0.979300i $$-0.564878\pi$$
−0.202413 + 0.979300i $$0.564878\pi$$
$$660$$ 0 0
$$661$$ 49.0000 1.90588 0.952940 0.303160i $$-0.0980418\pi$$
0.952940 + 0.303160i $$0.0980418\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 9.00000 0.349005
$$666$$ 0 0
$$667$$ 41.5692 1.60957
$$668$$ −6.92820 −0.268060
$$669$$ 7.00000 0.270636
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −15.5885 −0.600891 −0.300445 0.953799i $$-0.597135\pi$$
−0.300445 + 0.953799i $$0.597135\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 26.0000 1.00000
$$677$$ 20.7846 0.798817 0.399409 0.916773i $$-0.369215\pi$$
0.399409 + 0.916773i $$0.369215\pi$$
$$678$$ 0 0
$$679$$ −22.5167 −0.864110
$$680$$ 0 0
$$681$$ −27.7128 −1.06196
$$682$$ 0 0
$$683$$ 42.0000 1.60709 0.803543 0.595247i $$-0.202946\pi$$
0.803543 + 0.595247i $$0.202946\pi$$
$$684$$ 10.3923 0.397360
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 14.0000 0.534133
$$688$$ 41.5692 1.58481
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 35.0000 1.33146 0.665731 0.746191i $$-0.268120\pi$$
0.665731 + 0.746191i $$0.268120\pi$$
$$692$$ 20.7846 0.790112
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.46410 −0.131401
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ −13.8564 −0.524097
$$700$$ −3.46410 −0.130931
$$701$$ 24.2487 0.915861 0.457931 0.888988i $$-0.348591\pi$$
0.457931 + 0.888988i $$0.348591\pi$$
$$702$$ 0 0
$$703$$ 25.9808 0.979883
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ −24.0000 −0.902613
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −15.5885 −0.584613
$$712$$ 0 0
$$713$$ 6.00000 0.224702
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −24.0000 −0.896922
$$717$$ 20.7846 0.776215
$$718$$ 0 0
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −22.5167 −0.838564
$$722$$ 0 0
$$723$$ −6.92820 −0.257663
$$724$$ 14.0000 0.520306
$$725$$ 6.92820 0.257307
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −36.0000 −1.33151
$$732$$ −24.2487 −0.896258
$$733$$ 27.7128 1.02360 0.511798 0.859106i $$-0.328980\pi$$
0.511798 + 0.859106i $$0.328980\pi$$
$$734$$ 0 0
$$735$$ −4.00000 −0.147542
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −15.5885 −0.573431 −0.286715 0.958016i $$-0.592563\pi$$
−0.286715 + 0.958016i $$0.592563\pi$$
$$740$$ −10.0000 −0.367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20.7846 −0.762513 −0.381257 0.924469i $$-0.624509\pi$$
−0.381257 + 0.924469i $$0.624509\pi$$
$$744$$ 0 0
$$745$$ −24.2487 −0.888404
$$746$$ 0 0
$$747$$ −6.92820 −0.253490
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −11.0000 −0.401396 −0.200698 0.979653i $$-0.564321\pi$$
−0.200698 + 0.979653i $$0.564321\pi$$
$$752$$ −48.0000 −1.75038
$$753$$ −30.0000 −1.09326
$$754$$ 0 0
$$755$$ −17.3205 −0.630358
$$756$$ 3.46410 0.125988
$$757$$ 13.0000 0.472493 0.236247 0.971693i $$-0.424083\pi$$
0.236247 + 0.971693i $$0.424083\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.46410 −0.125574 −0.0627868 0.998027i $$-0.519999\pi$$
−0.0627868 + 0.998027i $$0.519999\pi$$
$$762$$ 0 0
$$763$$ 21.0000 0.760251
$$764$$ −48.0000 −1.73658
$$765$$ 3.46410 0.125245
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −16.0000 −0.577350
$$769$$ 1.73205 0.0624593 0.0312297 0.999512i $$-0.490058\pi$$
0.0312297 + 0.999512i $$0.490058\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −3.46410 −0.124676
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ 1.00000 0.0359211
$$776$$ 0 0
$$777$$ 8.66025 0.310685
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.92820 −0.247594
$$784$$ −16.0000 −0.571429
$$785$$ 5.00000 0.178458
$$786$$ 0 0
$$787$$ −24.2487 −0.864373 −0.432187 0.901784i $$-0.642258\pi$$
−0.432187 + 0.901784i $$0.642258\pi$$
$$788$$ 13.8564 0.493614
$$789$$ 10.3923 0.369976
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 6.00000 0.212798
$$796$$ 22.0000 0.779769
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 41.5692 1.47061
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ −10.0000 −0.352673
$$805$$ −10.3923 −0.366281
$$806$$ 0 0
$$807$$ −24.0000 −0.844840
$$808$$ 0 0
$$809$$ −38.1051 −1.33970 −0.669852 0.742494i $$-0.733643\pi$$
−0.669852 + 0.742494i $$0.733643\pi$$
$$810$$ 0 0
$$811$$ 29.4449 1.03395 0.516975 0.856001i $$-0.327058\pi$$
0.516975 + 0.856001i $$0.327058\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 10.3923 0.364474
$$814$$ 0 0
$$815$$ 13.0000 0.455370
$$816$$ 13.8564 0.485071
$$817$$ −54.0000 −1.88922
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 6.92820 0.241943
$$821$$ 10.3923 0.362694 0.181347 0.983419i $$-0.441954\pi$$
0.181347 + 0.983419i $$0.441954\pi$$
$$822$$ 0 0
$$823$$ 23.0000 0.801730 0.400865 0.916137i $$-0.368710\pi$$
0.400865 + 0.916137i $$0.368710\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −41.5692 −1.44550 −0.722752 0.691108i $$-0.757123\pi$$
−0.722752 + 0.691108i $$0.757123\pi$$
$$828$$ −12.0000 −0.417029
$$829$$ −55.0000 −1.91023 −0.955114 0.296237i $$-0.904268\pi$$
−0.955114 + 0.296237i $$0.904268\pi$$
$$830$$ 0 0
$$831$$ −25.9808 −0.901263
$$832$$ 0 0
$$833$$ 13.8564 0.480096
$$834$$ 0 0
$$835$$ −3.46410 −0.119880
$$836$$ 0 0
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 19.0000 0.655172
$$842$$ 0 0
$$843$$ −31.1769 −1.07379
$$844$$ 38.1051 1.31163
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 24.0000 0.824163
$$849$$ −15.5885 −0.534994
$$850$$ 0 0
$$851$$ −30.0000 −1.02839
$$852$$ −12.0000 −0.411113
$$853$$ −43.3013 −1.48261 −0.741304 0.671170i $$-0.765792\pi$$
−0.741304 + 0.671170i $$0.765792\pi$$
$$854$$ 0 0
$$855$$ 5.19615 0.177705
$$856$$ 0 0
$$857$$ −10.3923 −0.354994 −0.177497 0.984121i $$-0.556800\pi$$
−0.177497 + 0.984121i $$0.556800\pi$$
$$858$$ 0 0
$$859$$ 23.0000 0.784750 0.392375 0.919805i $$-0.371654\pi$$
0.392375 + 0.919805i $$0.371654\pi$$
$$860$$ 20.7846 0.708749
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 10.3923 0.353349
$$866$$ 0 0
$$867$$ 5.00000 0.169809
$$868$$ −3.46410 −0.117579
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −13.0000 −0.439983
$$874$$ 0 0
$$875$$ −1.73205 −0.0585540
$$876$$ −3.46410 −0.117041
$$877$$ −29.4449 −0.994282 −0.497141 0.867670i $$-0.665617\pi$$
−0.497141 + 0.867670i $$0.665617\pi$$
$$878$$ 0 0
$$879$$ 31.1769 1.05157
$$880$$ 0 0
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 0 0
$$883$$ −11.0000 −0.370179 −0.185090 0.982722i $$-0.559258\pi$$
−0.185090 + 0.982722i $$0.559258\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −17.3205 −0.581566 −0.290783 0.956789i $$-0.593916\pi$$
−0.290783 + 0.956789i $$0.593916\pi$$
$$888$$ 0 0
$$889$$ 3.00000 0.100617
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000 0.468755
$$893$$ 62.3538 2.08659
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 6.92820 0.231069
$$900$$ −2.00000 −0.0666667
$$901$$ −20.7846 −0.692436
$$902$$ 0 0
$$903$$ −18.0000 −0.599002
$$904$$ 0 0
$$905$$ 7.00000 0.232688
$$906$$ 0 0
$$907$$ −1.00000 −0.0332045 −0.0166022 0.999862i $$-0.505285\pi$$
−0.0166022 + 0.999862i $$0.505285\pi$$
$$908$$ −55.4256 −1.83936
$$909$$ −13.8564 −0.459588
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 20.7846 0.688247
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −12.1244 −0.400819
$$916$$ 28.0000 0.925146
$$917$$ 6.00000 0.198137
$$918$$ 0 0
$$919$$ 15.5885 0.514216 0.257108 0.966383i $$-0.417230\pi$$
0.257108 + 0.966383i $$0.417230\pi$$
$$920$$ 0 0
$$921$$ 8.66025 0.285365
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −5.00000 −0.164399
$$926$$ 0 0
$$927$$ −13.0000 −0.426976
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 20.7846 0.681188
$$932$$ −27.7128 −0.907763
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 29.4449 0.961922 0.480961 0.876742i $$-0.340288\pi$$
0.480961 + 0.876742i $$0.340288\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ −24.0000 −0.782794
$$941$$ 24.2487 0.790485 0.395243 0.918577i $$-0.370660\pi$$
0.395243 + 0.918577i $$0.370660\pi$$
$$942$$ 0 0
$$943$$ 20.7846 0.676840
$$944$$ 0 0
$$945$$ 1.73205 0.0563436
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ −31.1769 −1.01258
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ −17.3205 −0.561066 −0.280533 0.959844i $$-0.590511\pi$$
−0.280533 + 0.959844i $$0.590511\pi$$
$$954$$ 0 0
$$955$$ −24.0000 −0.776622
$$956$$ 41.5692 1.34444
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −20.7846 −0.671170
$$960$$ −8.00000 −0.258199
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ −6.92820 −0.223258
$$964$$ −13.8564 −0.446285
$$965$$ −1.73205 −0.0557567
$$966$$ 0 0
$$967$$ 43.3013 1.39247 0.696237 0.717812i $$-0.254856\pi$$
0.696237 + 0.717812i $$0.254856\pi$$
$$968$$ 0 0
$$969$$ −18.0000 −0.578243
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 6.00000 0.192351
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −48.4974 −1.55236
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −8.00000 −0.255551
$$981$$ 12.1244 0.387101
$$982$$ 0 0
$$983$$ −30.0000 −0.956851 −0.478426 0.878128i $$-0.658792\pi$$
−0.478426 + 0.878128i $$0.658792\pi$$
$$984$$ 0 0
$$985$$ 6.92820 0.220751
$$986$$ 0 0
$$987$$ 20.7846 0.661581
$$988$$ 0 0
$$989$$ 62.3538 1.98274
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 19.0000 0.602947
$$994$$ 0 0
$$995$$ 11.0000 0.348723
$$996$$ −13.8564 −0.439057
$$997$$ 8.66025 0.274273 0.137136 0.990552i $$-0.456210\pi$$
0.137136 + 0.990552i $$0.456210\pi$$
$$998$$ 0 0
$$999$$ 5.00000 0.158193
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 1815.2.a.g.1.2 yes 2
3.2 odd 2 5445.2.a.q.1.2 2
5.4 even 2 9075.2.a.bl.1.1 2
11.10 odd 2 inner 1815.2.a.g.1.1 2
33.32 even 2 5445.2.a.q.1.1 2
55.54 odd 2 9075.2.a.bl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.g.1.1 2 11.10 odd 2 inner
1815.2.a.g.1.2 yes 2 1.1 even 1 trivial
5445.2.a.q.1.1 2 33.32 even 2
5445.2.a.q.1.2 2 3.2 odd 2
9075.2.a.bl.1.1 2 5.4 even 2
9075.2.a.bl.1.2 2 55.54 odd 2