# Properties

 Label 3420.2.a.i.1.1 Level $3420$ Weight $2$ Character 3420.1 Self dual yes Analytic conductor $27.309$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3420.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.3088374913$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3420.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -2.60555 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -2.60555 q^{7} +4.60555 q^{11} +4.60555 q^{13} -2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -2.60555 q^{29} +4.00000 q^{31} -2.60555 q^{35} +3.39445 q^{37} -6.60555 q^{41} +10.6056 q^{43} -6.00000 q^{47} -0.211103 q^{49} +4.60555 q^{55} -5.21110 q^{59} -7.21110 q^{61} +4.60555 q^{65} +4.00000 q^{67} +9.21110 q^{71} +6.00000 q^{73} -12.0000 q^{77} +8.00000 q^{79} +11.2111 q^{83} -2.00000 q^{85} -6.60555 q^{89} -12.0000 q^{91} -1.00000 q^{95} +16.6056 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^35 + 14 * q^37 - 6 * q^41 + 14 * q^43 - 12 * q^47 + 14 * q^49 + 2 * q^55 + 4 * q^59 + 2 * q^65 + 8 * q^67 + 4 * q^71 + 12 * q^73 - 24 * q^77 + 16 * q^79 + 8 * q^83 - 4 * q^85 - 6 * q^89 - 24 * q^91 - 2 * q^95 + 26 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.60555 −0.984806 −0.492403 0.870367i $$-0.663881\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.60555 1.38863 0.694313 0.719673i $$-0.255708\pi$$
0.694313 + 0.719673i $$0.255708\pi$$
$$12$$ 0 0
$$13$$ 4.60555 1.27735 0.638675 0.769477i $$-0.279483\pi$$
0.638675 + 0.769477i $$0.279483\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.60555 −0.483839 −0.241919 0.970296i $$-0.577777\pi$$
−0.241919 + 0.970296i $$0.577777\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.60555 −0.440419
$$36$$ 0 0
$$37$$ 3.39445 0.558044 0.279022 0.960285i $$-0.409990\pi$$
0.279022 + 0.960285i $$0.409990\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.60555 −1.03161 −0.515807 0.856705i $$-0.672508\pi$$
−0.515807 + 0.856705i $$0.672508\pi$$
$$42$$ 0 0
$$43$$ 10.6056 1.61733 0.808666 0.588268i $$-0.200190\pi$$
0.808666 + 0.588268i $$0.200190\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 0 0
$$49$$ −0.211103 −0.0301575
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 4.60555 0.621012
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −5.21110 −0.678428 −0.339214 0.940709i $$-0.610161\pi$$
−0.339214 + 0.940709i $$0.610161\pi$$
$$60$$ 0 0
$$61$$ −7.21110 −0.923287 −0.461644 0.887066i $$-0.652740\pi$$
−0.461644 + 0.887066i $$0.652740\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.60555 0.571248
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9.21110 1.09316 0.546578 0.837408i $$-0.315930\pi$$
0.546578 + 0.837408i $$0.315930\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −12.0000 −1.36753
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 11.2111 1.23058 0.615289 0.788301i $$-0.289039\pi$$
0.615289 + 0.788301i $$0.289039\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.60555 −0.700187 −0.350094 0.936715i $$-0.613850\pi$$
−0.350094 + 0.936715i $$0.613850\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 16.6056 1.68604 0.843019 0.537884i $$-0.180776\pi$$
0.843019 + 0.537884i $$0.180776\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.21110 0.717532 0.358766 0.933428i $$-0.383198\pi$$
0.358766 + 0.933428i $$0.383198\pi$$
$$102$$ 0 0
$$103$$ 18.4222 1.81519 0.907597 0.419843i $$-0.137915\pi$$
0.907597 + 0.419843i $$0.137915\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.4222 −1.78094 −0.890471 0.455040i $$-0.849625\pi$$
−0.890471 + 0.455040i $$0.849625\pi$$
$$108$$ 0 0
$$109$$ −11.2111 −1.07383 −0.536914 0.843637i $$-0.680410\pi$$
−0.536914 + 0.843637i $$0.680410\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1.21110 0.113931 0.0569655 0.998376i $$-0.481858\pi$$
0.0569655 + 0.998376i $$0.481858\pi$$
$$114$$ 0 0
$$115$$ 2.00000 0.186501
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 5.21110 0.477701
$$120$$ 0 0
$$121$$ 10.2111 0.928282
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 18.4222 1.63471 0.817353 0.576137i $$-0.195441\pi$$
0.817353 + 0.576137i $$0.195441\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.3944 1.34502 0.672510 0.740088i $$-0.265216\pi$$
0.672510 + 0.740088i $$0.265216\pi$$
$$132$$ 0 0
$$133$$ 2.60555 0.225930
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.4222 1.06130 0.530650 0.847591i $$-0.321948\pi$$
0.530650 + 0.847591i $$0.321948\pi$$
$$138$$ 0 0
$$139$$ 9.21110 0.781276 0.390638 0.920544i $$-0.372255\pi$$
0.390638 + 0.920544i $$0.372255\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 21.2111 1.77376
$$144$$ 0 0
$$145$$ −2.60555 −0.216379
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −16.4222 −1.34536 −0.672680 0.739934i $$-0.734857\pi$$
−0.672680 + 0.739934i $$0.734857\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ 15.2111 1.21398 0.606989 0.794710i $$-0.292377\pi$$
0.606989 + 0.794710i $$0.292377\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.21110 −0.410692
$$162$$ 0 0
$$163$$ −17.0278 −1.33372 −0.666858 0.745184i $$-0.732361\pi$$
−0.666858 + 0.745184i $$0.732361\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.78890 0.525341 0.262670 0.964886i $$-0.415397\pi$$
0.262670 + 0.964886i $$0.415397\pi$$
$$168$$ 0 0
$$169$$ 8.21110 0.631623
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 9.21110 0.700307 0.350154 0.936692i $$-0.386129\pi$$
0.350154 + 0.936692i $$0.386129\pi$$
$$174$$ 0 0
$$175$$ −2.60555 −0.196961
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 21.2111 1.58539 0.792696 0.609617i $$-0.208677\pi$$
0.792696 + 0.609617i $$0.208677\pi$$
$$180$$ 0 0
$$181$$ −0.788897 −0.0586383 −0.0293191 0.999570i $$-0.509334\pi$$
−0.0293191 + 0.999570i $$0.509334\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 3.39445 0.249565
$$186$$ 0 0
$$187$$ −9.21110 −0.673583
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 5.81665 0.420878 0.210439 0.977607i $$-0.432511\pi$$
0.210439 + 0.977607i $$0.432511\pi$$
$$192$$ 0 0
$$193$$ −0.605551 −0.0435885 −0.0217943 0.999762i $$-0.506938\pi$$
−0.0217943 + 0.999762i $$0.506938\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ 0 0
$$199$$ 17.2111 1.22006 0.610031 0.792377i $$-0.291157\pi$$
0.610031 + 0.792377i $$0.291157\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.78890 0.476487
$$204$$ 0 0
$$205$$ −6.60555 −0.461352
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.60555 −0.318573
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 10.6056 0.723293
$$216$$ 0 0
$$217$$ −10.4222 −0.707505
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −9.21110 −0.619606
$$222$$ 0 0
$$223$$ −10.4222 −0.697922 −0.348961 0.937137i $$-0.613466\pi$$
−0.348961 + 0.937137i $$0.613466\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 13.2111 0.876852 0.438426 0.898767i $$-0.355536\pi$$
0.438426 + 0.898767i $$0.355536\pi$$
$$228$$ 0 0
$$229$$ −3.21110 −0.212196 −0.106098 0.994356i $$-0.533836\pi$$
−0.106098 + 0.994356i $$0.533836\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 16.4222 1.07585 0.537927 0.842991i $$-0.319208\pi$$
0.537927 + 0.842991i $$0.319208\pi$$
$$234$$ 0 0
$$235$$ −6.00000 −0.391397
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −25.8167 −1.66994 −0.834970 0.550295i $$-0.814515\pi$$
−0.834970 + 0.550295i $$0.814515\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.211103 −0.0134868
$$246$$ 0 0
$$247$$ −4.60555 −0.293044
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.18335 0.390289 0.195145 0.980774i $$-0.437482\pi$$
0.195145 + 0.980774i $$0.437482\pi$$
$$252$$ 0 0
$$253$$ 9.21110 0.579097
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2.78890 −0.173967 −0.0869833 0.996210i $$-0.527723\pi$$
−0.0869833 + 0.996210i $$0.527723\pi$$
$$258$$ 0 0
$$259$$ −8.84441 −0.549565
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7.21110 −0.444656 −0.222328 0.974972i $$-0.571366\pi$$
−0.222328 + 0.974972i $$0.571366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 17.3944 1.06056 0.530279 0.847823i $$-0.322087\pi$$
0.530279 + 0.847823i $$0.322087\pi$$
$$270$$ 0 0
$$271$$ 11.6333 0.706673 0.353337 0.935496i $$-0.385047\pi$$
0.353337 + 0.935496i $$0.385047\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 4.60555 0.277725
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.2389 −1.08804 −0.544020 0.839073i $$-0.683098\pi$$
−0.544020 + 0.839073i $$0.683098\pi$$
$$282$$ 0 0
$$283$$ −18.6056 −1.10599 −0.552993 0.833186i $$-0.686514\pi$$
−0.552993 + 0.833186i $$0.686514\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 17.2111 1.01594
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −19.6333 −1.14699 −0.573495 0.819209i $$-0.694413\pi$$
−0.573495 + 0.819209i $$0.694413\pi$$
$$294$$ 0 0
$$295$$ −5.21110 −0.303402
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 9.21110 0.532692
$$300$$ 0 0
$$301$$ −27.6333 −1.59276
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.21110 −0.412907
$$306$$ 0 0
$$307$$ −11.6333 −0.663948 −0.331974 0.943289i $$-0.607715\pi$$
−0.331974 + 0.943289i $$0.607715\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.6056 −0.941614 −0.470807 0.882236i $$-0.656037\pi$$
−0.470807 + 0.882236i $$0.656037\pi$$
$$312$$ 0 0
$$313$$ 0.788897 0.0445911 0.0222956 0.999751i $$-0.492903\pi$$
0.0222956 + 0.999751i $$0.492903\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.42221 0.360707 0.180353 0.983602i $$-0.442276\pi$$
0.180353 + 0.983602i $$0.442276\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 4.60555 0.255470
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 15.6333 0.861892
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 3.02776 0.164932 0.0824662 0.996594i $$-0.473720\pi$$
0.0824662 + 0.996594i $$0.473720\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 18.4222 0.997618
$$342$$ 0 0
$$343$$ 18.7889 1.01451
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −21.6333 −1.16134 −0.580668 0.814140i $$-0.697209\pi$$
−0.580668 + 0.814140i $$0.697209\pi$$
$$348$$ 0 0
$$349$$ 34.8444 1.86518 0.932589 0.360939i $$-0.117544\pi$$
0.932589 + 0.360939i $$0.117544\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −0.422205 −0.0224717 −0.0112359 0.999937i $$-0.503577\pi$$
−0.0112359 + 0.999937i $$0.503577\pi$$
$$354$$ 0 0
$$355$$ 9.21110 0.488875
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 13.8167 0.729215 0.364608 0.931161i $$-0.381203\pi$$
0.364608 + 0.931161i $$0.381203\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 0 0
$$367$$ 25.0278 1.30644 0.653219 0.757169i $$-0.273418\pi$$
0.653219 + 0.757169i $$0.273418\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.6056 0.859803 0.429901 0.902876i $$-0.358548\pi$$
0.429901 + 0.902876i $$0.358548\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 26.4222 1.35722 0.678609 0.734500i $$-0.262583\pi$$
0.678609 + 0.734500i $$0.262583\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −36.4222 −1.84668 −0.923340 0.383984i $$-0.874552\pi$$
−0.923340 + 0.383984i $$0.874552\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −7.57779 −0.380319 −0.190159 0.981753i $$-0.560900\pi$$
−0.190159 + 0.981753i $$0.560900\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −25.3944 −1.26814 −0.634069 0.773276i $$-0.718617\pi$$
−0.634069 + 0.773276i $$0.718617\pi$$
$$402$$ 0 0
$$403$$ 18.4222 0.917675
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 15.6333 0.774914
$$408$$ 0 0
$$409$$ −12.7889 −0.632370 −0.316185 0.948698i $$-0.602402\pi$$
−0.316185 + 0.948698i $$0.602402\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 13.5778 0.668120
$$414$$ 0 0
$$415$$ 11.2111 0.550331
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7.02776 0.343328 0.171664 0.985156i $$-0.445086\pi$$
0.171664 + 0.985156i $$0.445086\pi$$
$$420$$ 0 0
$$421$$ −8.42221 −0.410473 −0.205237 0.978712i $$-0.565796\pi$$
−0.205237 + 0.978712i $$0.565796\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 18.7889 0.909258
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 9.21110 0.443683 0.221842 0.975083i $$-0.428793\pi$$
0.221842 + 0.975083i $$0.428793\pi$$
$$432$$ 0 0
$$433$$ −16.6056 −0.798012 −0.399006 0.916948i $$-0.630645\pi$$
−0.399006 + 0.916948i $$0.630645\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2.00000 −0.0956730
$$438$$ 0 0
$$439$$ 34.4222 1.64288 0.821441 0.570293i $$-0.193170\pi$$
0.821441 + 0.570293i $$0.193170\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −18.8444 −0.895325 −0.447662 0.894203i $$-0.647743\pi$$
−0.447662 + 0.894203i $$0.647743\pi$$
$$444$$ 0 0
$$445$$ −6.60555 −0.313133
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.2389 0.860745 0.430372 0.902651i $$-0.358382\pi$$
0.430372 + 0.902651i $$0.358382\pi$$
$$450$$ 0 0
$$451$$ −30.4222 −1.43253
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 32.0555 1.49949 0.749747 0.661725i $$-0.230175\pi$$
0.749747 + 0.661725i $$0.230175\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 22.8444 1.06397 0.531985 0.846754i $$-0.321446\pi$$
0.531985 + 0.846754i $$0.321446\pi$$
$$462$$ 0 0
$$463$$ 33.0278 1.53493 0.767465 0.641091i $$-0.221518\pi$$
0.767465 + 0.641091i $$0.221518\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −22.8444 −1.05711 −0.528557 0.848898i $$-0.677267\pi$$
−0.528557 + 0.848898i $$0.677267\pi$$
$$468$$ 0 0
$$469$$ −10.4222 −0.481253
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.8444 2.24587
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −28.6056 −1.30702 −0.653510 0.756917i $$-0.726704\pi$$
−0.653510 + 0.756917i $$0.726704\pi$$
$$480$$ 0 0
$$481$$ 15.6333 0.712817
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 16.6056 0.754019
$$486$$ 0 0
$$487$$ 0.366692 0.0166164 0.00830821 0.999965i $$-0.497355\pi$$
0.00830821 + 0.999965i $$0.497355\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.2389 0.552332 0.276166 0.961110i $$-0.410936\pi$$
0.276166 + 0.961110i $$0.410936\pi$$
$$492$$ 0 0
$$493$$ 5.21110 0.234696
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −24.0000 −1.07655
$$498$$ 0 0
$$499$$ −27.6333 −1.23704 −0.618518 0.785770i $$-0.712267\pi$$
−0.618518 + 0.785770i $$0.712267\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −12.7889 −0.570229 −0.285114 0.958494i $$-0.592032\pi$$
−0.285114 + 0.958494i $$0.592032\pi$$
$$504$$ 0 0
$$505$$ 7.21110 0.320890
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 35.4500 1.57129 0.785646 0.618676i $$-0.212331\pi$$
0.785646 + 0.618676i $$0.212331\pi$$
$$510$$ 0 0
$$511$$ −15.6333 −0.691577
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 18.4222 0.811779
$$516$$ 0 0
$$517$$ −27.6333 −1.21531
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4.18335 0.183276 0.0916379 0.995792i $$-0.470790\pi$$
0.0916379 + 0.995792i $$0.470790\pi$$
$$522$$ 0 0
$$523$$ −25.2111 −1.10240 −0.551202 0.834372i $$-0.685831\pi$$
−0.551202 + 0.834372i $$0.685831\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −30.4222 −1.31773
$$534$$ 0 0
$$535$$ −18.4222 −0.796461
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −0.972244 −0.0418775
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −11.2111 −0.480231
$$546$$ 0 0
$$547$$ −22.7889 −0.974383 −0.487191 0.873295i $$-0.661979\pi$$
−0.487191 + 0.873295i $$0.661979\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.60555 0.111000
$$552$$ 0 0
$$553$$ −20.8444 −0.886394
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.42221 0.187375 0.0936874 0.995602i $$-0.470135\pi$$
0.0936874 + 0.995602i $$0.470135\pi$$
$$558$$ 0 0
$$559$$ 48.8444 2.06590
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −31.6333 −1.33318 −0.666592 0.745422i $$-0.732248\pi$$
−0.666592 + 0.745422i $$0.732248\pi$$
$$564$$ 0 0
$$565$$ 1.21110 0.0509515
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −39.4500 −1.65383 −0.826914 0.562328i $$-0.809906\pi$$
−0.826914 + 0.562328i $$0.809906\pi$$
$$570$$ 0 0
$$571$$ 7.63331 0.319444 0.159722 0.987162i $$-0.448940\pi$$
0.159722 + 0.987162i $$0.448940\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.00000 0.0834058
$$576$$ 0 0
$$577$$ −23.2111 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −29.2111 −1.21188
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.4222 −0.512719 −0.256360 0.966581i $$-0.582523\pi$$
−0.256360 + 0.966581i $$0.582523\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 3.57779 0.146922 0.0734612 0.997298i $$-0.476595\pi$$
0.0734612 + 0.997298i $$0.476595\pi$$
$$594$$ 0 0
$$595$$ 5.21110 0.213634
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.8444 −1.50542 −0.752711 0.658351i $$-0.771254\pi$$
−0.752711 + 0.658351i $$0.771254\pi$$
$$600$$ 0 0
$$601$$ 4.78890 0.195343 0.0976716 0.995219i $$-0.468861\pi$$
0.0976716 + 0.995219i $$0.468861\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 10.2111 0.415140
$$606$$ 0 0
$$607$$ −23.6333 −0.959246 −0.479623 0.877475i $$-0.659227\pi$$
−0.479623 + 0.877475i $$0.659227\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −27.6333 −1.11792
$$612$$ 0 0
$$613$$ 4.78890 0.193422 0.0967109 0.995313i $$-0.469168\pi$$
0.0967109 + 0.995313i $$0.469168\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −34.0000 −1.36879 −0.684394 0.729112i $$-0.739933\pi$$
−0.684394 + 0.729112i $$0.739933\pi$$
$$618$$ 0 0
$$619$$ −4.36669 −0.175512 −0.0877561 0.996142i $$-0.527970\pi$$
−0.0877561 + 0.996142i $$0.527970\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 17.2111 0.689548
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −6.78890 −0.270691
$$630$$ 0 0
$$631$$ 36.8444 1.46675 0.733376 0.679823i $$-0.237943\pi$$
0.733376 + 0.679823i $$0.237943\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 18.4222 0.731063
$$636$$ 0 0
$$637$$ −0.972244 −0.0385217
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −35.4500 −1.40019 −0.700095 0.714050i $$-0.746859\pi$$
−0.700095 + 0.714050i $$0.746859\pi$$
$$642$$ 0 0
$$643$$ 5.39445 0.212736 0.106368 0.994327i $$-0.466078\pi$$
0.106368 + 0.994327i $$0.466078\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.36669 0.0930443 0.0465221 0.998917i $$-0.485186\pi$$
0.0465221 + 0.998917i $$0.485186\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ 15.3944 0.601511
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −31.2111 −1.21397 −0.606986 0.794713i $$-0.707621\pi$$
−0.606986 + 0.794713i $$0.707621\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2.60555 0.101039
$$666$$ 0 0
$$667$$ −5.21110 −0.201775
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −33.2111 −1.28210
$$672$$ 0 0
$$673$$ 36.6056 1.41104 0.705520 0.708690i $$-0.250713\pi$$
0.705520 + 0.708690i $$0.250713\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −10.4222 −0.400558 −0.200279 0.979739i $$-0.564185\pi$$
−0.200279 + 0.979739i $$0.564185\pi$$
$$678$$ 0 0
$$679$$ −43.2666 −1.66042
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 48.0000 1.83667 0.918334 0.395805i $$-0.129534\pi$$
0.918334 + 0.395805i $$0.129534\pi$$
$$684$$ 0 0
$$685$$ 12.4222 0.474628
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −42.0555 −1.59987 −0.799934 0.600089i $$-0.795132\pi$$
−0.799934 + 0.600089i $$0.795132\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9.21110 0.349397
$$696$$ 0 0
$$697$$ 13.2111 0.500406
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ −3.39445 −0.128024
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −18.7889 −0.706629
$$708$$ 0 0
$$709$$ −25.6333 −0.962679 −0.481340 0.876534i $$-0.659850\pi$$
−0.481340 + 0.876534i $$0.659850\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 21.2111 0.793250
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −16.2389 −0.605607 −0.302804 0.953053i $$-0.597923\pi$$
−0.302804 + 0.953053i $$0.597923\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −2.60555 −0.0967677
$$726$$ 0 0
$$727$$ −17.3944 −0.645124 −0.322562 0.946548i $$-0.604544\pi$$
−0.322562 + 0.946548i $$0.604544\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −21.2111 −0.784521
$$732$$ 0 0
$$733$$ −24.4222 −0.902055 −0.451027 0.892510i $$-0.648942\pi$$
−0.451027 + 0.892510i $$0.648942\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.4222 0.678591
$$738$$ 0 0
$$739$$ 38.4222 1.41338 0.706692 0.707521i $$-0.250187\pi$$
0.706692 + 0.707521i $$0.250187\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 20.0000 0.733729 0.366864 0.930274i $$-0.380431\pi$$
0.366864 + 0.930274i $$0.380431\pi$$
$$744$$ 0 0
$$745$$ −16.4222 −0.601663
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ 40.8444 1.49043 0.745217 0.666822i $$-0.232346\pi$$
0.745217 + 0.666822i $$0.232346\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.0000 −0.436725
$$756$$ 0 0
$$757$$ −36.0555 −1.31046 −0.655230 0.755429i $$-0.727428\pi$$
−0.655230 + 0.755429i $$0.727428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 29.2111 1.05751
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −20.4222 −0.736444 −0.368222 0.929738i $$-0.620033\pi$$
−0.368222 + 0.929738i $$0.620033\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 43.2666 1.55619 0.778096 0.628145i $$-0.216186\pi$$
0.778096 + 0.628145i $$0.216186\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6.60555 0.236668
$$780$$ 0 0
$$781$$ 42.4222 1.51799
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 15.2111 0.542908
$$786$$ 0 0
$$787$$ −19.6333 −0.699852 −0.349926 0.936777i $$-0.613793\pi$$
−0.349926 + 0.936777i $$0.613793\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −3.15559 −0.112200
$$792$$ 0 0
$$793$$ −33.2111 −1.17936
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −17.2111 −0.609649 −0.304824 0.952409i $$-0.598598\pi$$
−0.304824 + 0.952409i $$0.598598\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 27.6333 0.975158
$$804$$ 0 0
$$805$$ −5.21110 −0.183667
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 48.4222 1.70243 0.851217 0.524814i $$-0.175865\pi$$
0.851217 + 0.524814i $$0.175865\pi$$
$$810$$ 0 0
$$811$$ −40.8444 −1.43424 −0.717121 0.696949i $$-0.754540\pi$$
−0.717121 + 0.696949i $$0.754540\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −17.0278 −0.596456
$$816$$ 0 0
$$817$$ −10.6056 −0.371041
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −43.2111 −1.50808 −0.754039 0.656830i $$-0.771897\pi$$
−0.754039 + 0.656830i $$0.771897\pi$$
$$822$$ 0 0
$$823$$ −47.8167 −1.66678 −0.833392 0.552683i $$-0.813604\pi$$
−0.833392 + 0.552683i $$0.813604\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27.6333 0.960904 0.480452 0.877021i $$-0.340473\pi$$
0.480452 + 0.877021i $$0.340473\pi$$
$$828$$ 0 0
$$829$$ 15.2111 0.528303 0.264152 0.964481i $$-0.414908\pi$$
0.264152 + 0.964481i $$0.414908\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0.422205 0.0146285
$$834$$ 0 0
$$835$$ 6.78890 0.234939
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −27.6333 −0.954008 −0.477004 0.878901i $$-0.658277\pi$$
−0.477004 + 0.878901i $$0.658277\pi$$
$$840$$ 0 0
$$841$$ −22.2111 −0.765900
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 8.21110 0.282471
$$846$$ 0 0
$$847$$ −26.6056 −0.914178
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6.78890 0.232720
$$852$$ 0 0
$$853$$ 29.6333 1.01463 0.507313 0.861762i $$-0.330639\pi$$
0.507313 + 0.861762i $$0.330639\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.42221 −0.219378 −0.109689 0.993966i $$-0.534986\pi$$
−0.109689 + 0.993966i $$0.534986\pi$$
$$858$$ 0 0
$$859$$ 8.84441 0.301767 0.150884 0.988552i $$-0.451788\pi$$
0.150884 + 0.988552i $$0.451788\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −6.42221 −0.218614 −0.109307 0.994008i $$-0.534863\pi$$
−0.109307 + 0.994008i $$0.534863\pi$$
$$864$$ 0 0
$$865$$ 9.21110 0.313187
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 36.8444 1.24986
$$870$$ 0 0
$$871$$ 18.4222 0.624213
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.60555 −0.0880837
$$876$$ 0 0
$$877$$ 27.0278 0.912662 0.456331 0.889810i $$-0.349163\pi$$
0.456331 + 0.889810i $$0.349163\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −59.2111 −1.99487 −0.997436 0.0715590i $$-0.977203\pi$$
−0.997436 + 0.0715590i $$0.977203\pi$$
$$882$$ 0 0
$$883$$ 27.8167 0.936105 0.468052 0.883701i $$-0.344956\pi$$
0.468052 + 0.883701i $$0.344956\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 20.8444 0.699887 0.349943 0.936771i $$-0.386201\pi$$
0.349943 + 0.936771i $$0.386201\pi$$
$$888$$ 0 0
$$889$$ −48.0000 −1.60987
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ 21.2111 0.709009
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −10.4222 −0.347600
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −0.788897 −0.0262238
$$906$$ 0 0
$$907$$ 30.0555 0.997977 0.498988 0.866609i $$-0.333705\pi$$
0.498988 + 0.866609i $$0.333705\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10.4222 −0.345303 −0.172652 0.984983i $$-0.555233\pi$$
−0.172652 + 0.984983i $$0.555233\pi$$
$$912$$ 0 0
$$913$$ 51.6333 1.70881
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −40.1110 −1.32458
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 42.4222 1.39634
$$924$$ 0 0
$$925$$ 3.39445 0.111609
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.6333 1.49718 0.748590 0.663033i $$-0.230731\pi$$
0.748590 + 0.663033i $$0.230731\pi$$
$$930$$ 0 0
$$931$$ 0.211103 0.00691861
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −9.21110 −0.301235
$$936$$ 0 0
$$937$$ −19.2111 −0.627599 −0.313800 0.949489i $$-0.601602\pi$$
−0.313800 + 0.949489i $$0.601602\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 30.2389 0.985759 0.492879 0.870098i $$-0.335944\pi$$
0.492879 + 0.870098i $$0.335944\pi$$
$$942$$ 0 0
$$943$$ −13.2111 −0.430213
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 7.21110 0.234329 0.117165 0.993113i $$-0.462619\pi$$
0.117165 + 0.993113i $$0.462619\pi$$
$$948$$ 0 0
$$949$$ 27.6333 0.897015
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −5.21110 −0.168804 −0.0844021 0.996432i $$-0.526898\pi$$
−0.0844021 + 0.996432i $$0.526898\pi$$
$$954$$ 0 0
$$955$$ 5.81665 0.188222
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −32.3667 −1.04518
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −0.605551 −0.0194934
$$966$$ 0 0
$$967$$ −26.2389 −0.843785 −0.421892 0.906646i $$-0.638634\pi$$
−0.421892 + 0.906646i $$0.638634\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −24.0000 −0.769405
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −6.42221 −0.205465 −0.102732 0.994709i $$-0.532758\pi$$
−0.102732 + 0.994709i $$0.532758\pi$$
$$978$$ 0 0
$$979$$ −30.4222 −0.972298
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 9.21110 0.293789 0.146894 0.989152i $$-0.453072\pi$$
0.146894 + 0.989152i $$0.453072\pi$$
$$984$$ 0 0
$$985$$ −14.0000 −0.446077
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 21.2111 0.674474
$$990$$ 0 0
$$991$$ 50.4222 1.60171 0.800857 0.598856i $$-0.204378\pi$$
0.800857 + 0.598856i $$0.204378\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 17.2111 0.545629
$$996$$ 0 0
$$997$$ −61.6333 −1.95195 −0.975973 0.217891i $$-0.930082\pi$$
−0.975973 + 0.217891i $$0.930082\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 3420.2.a.i.1.1 2
3.2 odd 2 1140.2.a.e.1.1 2
12.11 even 2 4560.2.a.bl.1.2 2
15.2 even 4 5700.2.f.n.3649.3 4
15.8 even 4 5700.2.f.n.3649.2 4
15.14 odd 2 5700.2.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.e.1.1 2 3.2 odd 2
3420.2.a.i.1.1 2 1.1 even 1 trivial
4560.2.a.bl.1.2 2 12.11 even 2
5700.2.a.u.1.2 2 15.14 odd 2
5700.2.f.n.3649.2 4 15.8 even 4
5700.2.f.n.3649.3 4 15.2 even 4