# Properties

 Label 6480.2.a.bs.1.3 Level $6480$ Weight $2$ Character 6480.1 Self dual yes Analytic conductor $51.743$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6480 = 2^{4} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6480.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.7430605098$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.51414$$ of defining polynomial Character $$\chi$$ $$=$$ 6480.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +0.514137 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +0.514137 q^{7} -3.32088 q^{11} -1.32088 q^{13} +3.32088 q^{17} +1.32088 q^{19} +4.12763 q^{23} +1.00000 q^{25} +1.38650 q^{29} -8.73566 q^{31} -0.514137 q^{35} +0.292611 q^{37} +11.3492 q^{41} -10.3492 q^{43} -4.86330 q^{47} -6.73566 q^{49} +5.02827 q^{53} +3.32088 q^{55} -5.02827 q^{59} +7.34916 q^{61} +1.32088 q^{65} -9.44852 q^{67} +8.99093 q^{71} +6.05655 q^{73} -1.70739 q^{77} +8.05655 q^{79} +1.54241 q^{83} -3.32088 q^{85} +3.00000 q^{89} -0.679116 q^{91} -1.32088 q^{95} -12.2553 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - 5 q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - 5 * q^7 $$3 q - 3 q^{5} - 5 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 3 q^{23} + 3 q^{25} + 7 q^{29} - 8 q^{31} + 5 q^{35} + 6 q^{37} + 13 q^{41} - 10 q^{43} + 13 q^{47} - 2 q^{49} + 2 q^{53} + 2 q^{55} - 2 q^{59} + q^{61} - 4 q^{65} - 11 q^{67} - 10 q^{71} - 8 q^{73} - 2 q^{79} - 15 q^{83} - 2 q^{85} + 9 q^{89} - 10 q^{91} + 4 q^{95} - 18 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - 5 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 4 * q^19 + 3 * q^23 + 3 * q^25 + 7 * q^29 - 8 * q^31 + 5 * q^35 + 6 * q^37 + 13 * q^41 - 10 * q^43 + 13 * q^47 - 2 * q^49 + 2 * q^53 + 2 * q^55 - 2 * q^59 + q^61 - 4 * q^65 - 11 * q^67 - 10 * q^71 - 8 * q^73 - 2 * q^79 - 15 * q^83 - 2 * q^85 + 9 * q^89 - 10 * q^91 + 4 * q^95 - 18 * 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
$$7$$ 0.514137 0.194325 0.0971627 0.995269i $$-0.469023\pi$$
0.0971627 + 0.995269i $$0.469023\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.32088 −1.00128 −0.500642 0.865654i $$-0.666903\pi$$
−0.500642 + 0.865654i $$0.666903\pi$$
$$12$$ 0 0
$$13$$ −1.32088 −0.366347 −0.183174 0.983081i $$-0.558637\pi$$
−0.183174 + 0.983081i $$0.558637\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.32088 0.805433 0.402716 0.915325i $$-0.368066\pi$$
0.402716 + 0.915325i $$0.368066\pi$$
$$18$$ 0 0
$$19$$ 1.32088 0.303032 0.151516 0.988455i $$-0.451585\pi$$
0.151516 + 0.988455i $$0.451585\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.12763 0.860671 0.430335 0.902669i $$-0.358395\pi$$
0.430335 + 0.902669i $$0.358395\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.38650 0.257467 0.128734 0.991679i $$-0.458909\pi$$
0.128734 + 0.991679i $$0.458909\pi$$
$$30$$ 0 0
$$31$$ −8.73566 −1.56897 −0.784486 0.620147i $$-0.787073\pi$$
−0.784486 + 0.620147i $$0.787073\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.514137 −0.0869050
$$36$$ 0 0
$$37$$ 0.292611 0.0481049 0.0240524 0.999711i $$-0.492343\pi$$
0.0240524 + 0.999711i $$0.492343\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.3492 1.77244 0.886220 0.463264i $$-0.153322\pi$$
0.886220 + 0.463264i $$0.153322\pi$$
$$42$$ 0 0
$$43$$ −10.3492 −1.57823 −0.789116 0.614244i $$-0.789461\pi$$
−0.789116 + 0.614244i $$0.789461\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.86330 −0.709385 −0.354692 0.934983i $$-0.615414\pi$$
−0.354692 + 0.934983i $$0.615414\pi$$
$$48$$ 0 0
$$49$$ −6.73566 −0.962238
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 5.02827 0.690687 0.345343 0.938476i $$-0.387762\pi$$
0.345343 + 0.938476i $$0.387762\pi$$
$$54$$ 0 0
$$55$$ 3.32088 0.447788
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −5.02827 −0.654625 −0.327313 0.944916i $$-0.606143\pi$$
−0.327313 + 0.944916i $$0.606143\pi$$
$$60$$ 0 0
$$61$$ 7.34916 0.940963 0.470482 0.882410i $$-0.344080\pi$$
0.470482 + 0.882410i $$0.344080\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.32088 0.163836
$$66$$ 0 0
$$67$$ −9.44852 −1.15432 −0.577160 0.816631i $$-0.695839\pi$$
−0.577160 + 0.816631i $$0.695839\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.99093 1.06703 0.533513 0.845792i $$-0.320871\pi$$
0.533513 + 0.845792i $$0.320871\pi$$
$$72$$ 0 0
$$73$$ 6.05655 0.708865 0.354433 0.935082i $$-0.384674\pi$$
0.354433 + 0.935082i $$0.384674\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1.70739 −0.194575
$$78$$ 0 0
$$79$$ 8.05655 0.906432 0.453216 0.891401i $$-0.350277\pi$$
0.453216 + 0.891401i $$0.350277\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.54241 0.169302 0.0846508 0.996411i $$-0.473023\pi$$
0.0846508 + 0.996411i $$0.473023\pi$$
$$84$$ 0 0
$$85$$ −3.32088 −0.360200
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ −0.679116 −0.0711906
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.32088 −0.135520
$$96$$ 0 0
$$97$$ −12.2553 −1.24433 −0.622167 0.782885i $$-0.713747\pi$$
−0.622167 + 0.782885i $$0.713747\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −11.6700 −1.16121 −0.580606 0.814184i $$-0.697184\pi$$
−0.580606 + 0.814184i $$0.697184\pi$$
$$102$$ 0 0
$$103$$ −0.292611 −0.0288318 −0.0144159 0.999896i $$-0.504589\pi$$
−0.0144159 + 0.999896i $$0.504589\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.87237 0.181009 0.0905043 0.995896i $$-0.471152\pi$$
0.0905043 + 0.995896i $$0.471152\pi$$
$$108$$ 0 0
$$109$$ 5.54787 0.531390 0.265695 0.964057i $$-0.414399\pi$$
0.265695 + 0.964057i $$0.414399\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −7.80128 −0.733883 −0.366942 0.930244i $$-0.619595\pi$$
−0.366942 + 0.930244i $$0.619595\pi$$
$$114$$ 0 0
$$115$$ −4.12763 −0.384904
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1.70739 0.156516
$$120$$ 0 0
$$121$$ 0.0282739 0.00257035
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −17.8916 −1.58762 −0.793810 0.608166i $$-0.791906\pi$$
−0.793810 + 0.608166i $$0.791906\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ 0.679116 0.0588868
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.67004 −0.484424 −0.242212 0.970223i $$-0.577873\pi$$
−0.242212 + 0.970223i $$0.577873\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.38650 0.366818
$$144$$ 0 0
$$145$$ −1.38650 −0.115143
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.6610 −1.44684 −0.723422 0.690407i $$-0.757432\pi$$
−0.723422 + 0.690407i $$0.757432\pi$$
$$150$$ 0 0
$$151$$ −1.26434 −0.102890 −0.0514451 0.998676i $$-0.516383\pi$$
−0.0514451 + 0.998676i $$0.516383\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.73566 0.701665
$$156$$ 0 0
$$157$$ −15.6700 −1.25061 −0.625303 0.780382i $$-0.715025\pi$$
−0.625303 + 0.780382i $$0.715025\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2.12217 0.167250
$$162$$ 0 0
$$163$$ −15.7074 −1.23030 −0.615149 0.788411i $$-0.710904\pi$$
−0.615149 + 0.788411i $$0.710904\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.16498 0.477060 0.238530 0.971135i $$-0.423334\pi$$
0.238530 + 0.971135i $$0.423334\pi$$
$$168$$ 0 0
$$169$$ −11.2553 −0.865790
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −8.58522 −0.652722 −0.326361 0.945245i $$-0.605823\pi$$
−0.326361 + 0.945245i $$0.605823\pi$$
$$174$$ 0 0
$$175$$ 0.514137 0.0388651
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1.06562 −0.0796482 −0.0398241 0.999207i $$-0.512680\pi$$
−0.0398241 + 0.999207i $$0.512680\pi$$
$$180$$ 0 0
$$181$$ −12.6700 −0.941757 −0.470878 0.882198i $$-0.656063\pi$$
−0.470878 + 0.882198i $$0.656063\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −0.292611 −0.0215132
$$186$$ 0 0
$$187$$ −11.0283 −0.806467
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.9344 −1.22533 −0.612664 0.790343i $$-0.709902\pi$$
−0.612664 + 0.790343i $$0.709902\pi$$
$$192$$ 0 0
$$193$$ 26.7175 1.92317 0.961585 0.274509i $$-0.0885153\pi$$
0.961585 + 0.274509i $$0.0885153\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.2553 1.01565 0.507823 0.861462i $$-0.330450\pi$$
0.507823 + 0.861462i $$0.330450\pi$$
$$198$$ 0 0
$$199$$ 24.6610 1.74817 0.874085 0.485773i $$-0.161462\pi$$
0.874085 + 0.485773i $$0.161462\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0.712853 0.0500325
$$204$$ 0 0
$$205$$ −11.3492 −0.792660
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.38650 −0.303421
$$210$$ 0 0
$$211$$ 5.37743 0.370198 0.185099 0.982720i $$-0.440739\pi$$
0.185099 + 0.982720i $$0.440739\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 10.3492 0.705807
$$216$$ 0 0
$$217$$ −4.49133 −0.304891
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.38650 −0.295068
$$222$$ 0 0
$$223$$ −8.66458 −0.580223 −0.290112 0.956993i $$-0.593692\pi$$
−0.290112 + 0.956993i $$0.593692\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3.32088 0.220415 0.110207 0.993909i $$-0.464848\pi$$
0.110207 + 0.993909i $$0.464848\pi$$
$$228$$ 0 0
$$229$$ −25.3118 −1.67265 −0.836326 0.548233i $$-0.815301\pi$$
−0.836326 + 0.548233i $$0.815301\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −27.6327 −1.81028 −0.905139 0.425116i $$-0.860233\pi$$
−0.905139 + 0.425116i $$0.860233\pi$$
$$234$$ 0 0
$$235$$ 4.86330 0.317246
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4.19872 −0.271592 −0.135796 0.990737i $$-0.543359\pi$$
−0.135796 + 0.990737i $$0.543359\pi$$
$$240$$ 0 0
$$241$$ 3.60442 0.232181 0.116091 0.993239i $$-0.462964\pi$$
0.116091 + 0.993239i $$0.462964\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 6.73566 0.430326
$$246$$ 0 0
$$247$$ −1.74474 −0.111015
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.87783 −0.434125 −0.217062 0.976158i $$-0.569648\pi$$
−0.217062 + 0.976158i $$0.569648\pi$$
$$252$$ 0 0
$$253$$ −13.7074 −0.861776
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0.150442 0.00934801
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6.23606 0.384532 0.192266 0.981343i $$-0.438416\pi$$
0.192266 + 0.981343i $$0.438416\pi$$
$$264$$ 0 0
$$265$$ −5.02827 −0.308884
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −9.92345 −0.605044 −0.302522 0.953142i $$-0.597828\pi$$
−0.302522 + 0.953142i $$0.597828\pi$$
$$270$$ 0 0
$$271$$ −6.60442 −0.401190 −0.200595 0.979674i $$-0.564288\pi$$
−0.200595 + 0.979674i $$0.564288\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.32088 −0.200257
$$276$$ 0 0
$$277$$ 22.6610 1.36157 0.680783 0.732485i $$-0.261640\pi$$
0.680783 + 0.732485i $$0.261640\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.5479 0.927508 0.463754 0.885964i $$-0.346502\pi$$
0.463754 + 0.885964i $$0.346502\pi$$
$$282$$ 0 0
$$283$$ −0.645378 −0.0383637 −0.0191819 0.999816i $$-0.506106\pi$$
−0.0191819 + 0.999816i $$0.506106\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.83502 0.344430
$$288$$ 0 0
$$289$$ −5.97173 −0.351278
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1.37743 0.0804704 0.0402352 0.999190i $$-0.487189\pi$$
0.0402352 + 0.999190i $$0.487189\pi$$
$$294$$ 0 0
$$295$$ 5.02827 0.292757
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.45213 −0.315305
$$300$$ 0 0
$$301$$ −5.32088 −0.306691
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.34916 −0.420812
$$306$$ 0 0
$$307$$ 7.98546 0.455754 0.227877 0.973690i $$-0.426822\pi$$
0.227877 + 0.973690i $$0.426822\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9.63270 0.546220 0.273110 0.961983i $$-0.411948\pi$$
0.273110 + 0.961983i $$0.411948\pi$$
$$312$$ 0 0
$$313$$ −24.5369 −1.38691 −0.693455 0.720500i $$-0.743912\pi$$
−0.693455 + 0.720500i $$0.743912\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 20.3492 1.14292 0.571461 0.820629i $$-0.306377\pi$$
0.571461 + 0.820629i $$0.306377\pi$$
$$318$$ 0 0
$$319$$ −4.60442 −0.257798
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.38650 0.244072
$$324$$ 0 0
$$325$$ −1.32088 −0.0732695
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2.50040 −0.137851
$$330$$ 0 0
$$331$$ −16.4431 −0.903792 −0.451896 0.892071i $$-0.649252\pi$$
−0.451896 + 0.892071i $$0.649252\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 9.44852 0.516228
$$336$$ 0 0
$$337$$ 4.89703 0.266758 0.133379 0.991065i $$-0.457417\pi$$
0.133379 + 0.991065i $$0.457417\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 29.0101 1.57099
$$342$$ 0 0
$$343$$ −7.06201 −0.381313
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 22.2745 1.19576 0.597878 0.801587i $$-0.296011\pi$$
0.597878 + 0.801587i $$0.296011\pi$$
$$348$$ 0 0
$$349$$ −2.94345 −0.157559 −0.0787797 0.996892i $$-0.525102\pi$$
−0.0787797 + 0.996892i $$0.525102\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.8296 −1.00220 −0.501098 0.865390i $$-0.667070\pi$$
−0.501098 + 0.865390i $$0.667070\pi$$
$$354$$ 0 0
$$355$$ −8.99093 −0.477189
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −31.8770 −1.68241 −0.841203 0.540720i $$-0.818152\pi$$
−0.841203 + 0.540720i $$0.818152\pi$$
$$360$$ 0 0
$$361$$ −17.2553 −0.908172
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.05655 −0.317014
$$366$$ 0 0
$$367$$ 18.3492 0.957818 0.478909 0.877864i $$-0.341032\pi$$
0.478909 + 0.877864i $$0.341032\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2.58522 0.134218
$$372$$ 0 0
$$373$$ −2.19872 −0.113845 −0.0569226 0.998379i $$-0.518129\pi$$
−0.0569226 + 0.998379i $$0.518129\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.83141 −0.0943226
$$378$$ 0 0
$$379$$ −15.4713 −0.794709 −0.397354 0.917665i $$-0.630072\pi$$
−0.397354 + 0.917665i $$0.630072\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7.70739 0.393829 0.196915 0.980421i $$-0.436908\pi$$
0.196915 + 0.980421i $$0.436908\pi$$
$$384$$ 0 0
$$385$$ 1.70739 0.0870166
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 24.6327 1.24893 0.624464 0.781054i $$-0.285318\pi$$
0.624464 + 0.781054i $$0.285318\pi$$
$$390$$ 0 0
$$391$$ 13.7074 0.693212
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −8.05655 −0.405369
$$396$$ 0 0
$$397$$ −6.77301 −0.339928 −0.169964 0.985450i $$-0.554365\pi$$
−0.169964 + 0.985450i $$0.554365\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.4996 0.923826 0.461913 0.886925i $$-0.347163\pi$$
0.461913 + 0.886925i $$0.347163\pi$$
$$402$$ 0 0
$$403$$ 11.5388 0.574789
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.971726 −0.0481667
$$408$$ 0 0
$$409$$ −13.4148 −0.663318 −0.331659 0.943399i $$-0.607608\pi$$
−0.331659 + 0.943399i $$0.607608\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2.58522 −0.127210
$$414$$ 0 0
$$415$$ −1.54241 −0.0757140
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −33.1150 −1.61777 −0.808886 0.587966i $$-0.799929\pi$$
−0.808886 + 0.587966i $$0.799929\pi$$
$$420$$ 0 0
$$421$$ −14.6983 −0.716352 −0.358176 0.933654i $$-0.616601\pi$$
−0.358176 + 0.933654i $$0.616601\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3.32088 0.161087
$$426$$ 0 0
$$427$$ 3.77847 0.182853
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −32.7549 −1.57775 −0.788873 0.614556i $$-0.789335\pi$$
−0.788873 + 0.614556i $$0.789335\pi$$
$$432$$ 0 0
$$433$$ −11.8314 −0.568581 −0.284291 0.958738i $$-0.591758\pi$$
−0.284291 + 0.958738i $$0.591758\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5.45213 0.260811
$$438$$ 0 0
$$439$$ −8.31181 −0.396701 −0.198351 0.980131i $$-0.563558\pi$$
−0.198351 + 0.980131i $$0.563558\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −29.1751 −1.38615 −0.693076 0.720865i $$-0.743745\pi$$
−0.693076 + 0.720865i $$0.743745\pi$$
$$444$$ 0 0
$$445$$ −3.00000 −0.142214
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −18.9717 −0.895331 −0.447666 0.894201i $$-0.647744\pi$$
−0.447666 + 0.894201i $$0.647744\pi$$
$$450$$ 0 0
$$451$$ −37.6892 −1.77472
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0.679116 0.0318374
$$456$$ 0 0
$$457$$ −23.2353 −1.08690 −0.543450 0.839442i $$-0.682882\pi$$
−0.543450 + 0.839442i $$0.682882\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −4.42571 −0.206126 −0.103063 0.994675i $$-0.532864\pi$$
−0.103063 + 0.994675i $$0.532864\pi$$
$$462$$ 0 0
$$463$$ 19.5087 0.906645 0.453322 0.891347i $$-0.350239\pi$$
0.453322 + 0.891347i $$0.350239\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −24.5935 −1.13805 −0.569026 0.822320i $$-0.692679\pi$$
−0.569026 + 0.822320i $$0.692679\pi$$
$$468$$ 0 0
$$469$$ −4.85783 −0.224314
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 34.3684 1.58026
$$474$$ 0 0
$$475$$ 1.32088 0.0606063
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 32.7549 1.49661 0.748304 0.663356i $$-0.230868\pi$$
0.748304 + 0.663356i $$0.230868\pi$$
$$480$$ 0 0
$$481$$ −0.386505 −0.0176231
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.2553 0.556483
$$486$$ 0 0
$$487$$ 6.03735 0.273578 0.136789 0.990600i $$-0.456322\pi$$
0.136789 + 0.990600i $$0.456322\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 14.4431 0.651806 0.325903 0.945403i $$-0.394332\pi$$
0.325903 + 0.945403i $$0.394332\pi$$
$$492$$ 0 0
$$493$$ 4.60442 0.207373
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.62257 0.207351
$$498$$ 0 0
$$499$$ −20.9717 −0.938823 −0.469412 0.882979i $$-0.655534\pi$$
−0.469412 + 0.882979i $$0.655534\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −5.31728 −0.237086 −0.118543 0.992949i $$-0.537822\pi$$
−0.118543 + 0.992949i $$0.537822\pi$$
$$504$$ 0 0
$$505$$ 11.6700 0.519310
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −18.2270 −0.807897 −0.403949 0.914782i $$-0.632363\pi$$
−0.403949 + 0.914782i $$0.632363\pi$$
$$510$$ 0 0
$$511$$ 3.11389 0.137751
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0.292611 0.0128940
$$516$$ 0 0
$$517$$ 16.1504 0.710296
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.1232 −1.75783 −0.878915 0.476978i $$-0.841732\pi$$
−0.878915 + 0.476978i $$0.841732\pi$$
$$522$$ 0 0
$$523$$ −18.9873 −0.830257 −0.415129 0.909763i $$-0.636263\pi$$
−0.415129 + 0.909763i $$0.636263\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −29.0101 −1.26370
$$528$$ 0 0
$$529$$ −5.96265 −0.259246
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −14.9909 −0.649329
$$534$$ 0 0
$$535$$ −1.87237 −0.0809495
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 22.3684 0.963473
$$540$$ 0 0
$$541$$ 16.5279 0.710589 0.355294 0.934754i $$-0.384381\pi$$
0.355294 + 0.934754i $$0.384381\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −5.54787 −0.237645
$$546$$ 0 0
$$547$$ −17.6737 −0.755671 −0.377835 0.925873i $$-0.623331\pi$$
−0.377835 + 0.925873i $$0.623331\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.83141 0.0780208
$$552$$ 0 0
$$553$$ 4.14217 0.176143
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −17.3401 −0.734723 −0.367362 0.930078i $$-0.619739\pi$$
−0.367362 + 0.930078i $$0.619739\pi$$
$$558$$ 0 0
$$559$$ 13.6700 0.578181
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 12.9945 0.547654 0.273827 0.961779i $$-0.411710\pi$$
0.273827 + 0.961779i $$0.411710\pi$$
$$564$$ 0 0
$$565$$ 7.80128 0.328202
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 16.6802 0.699269 0.349635 0.936886i $$-0.386306\pi$$
0.349635 + 0.936886i $$0.386306\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.12763 0.172134
$$576$$ 0 0
$$577$$ −23.5953 −0.982287 −0.491144 0.871079i $$-0.663421\pi$$
−0.491144 + 0.871079i $$0.663421\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.793010 0.0328996
$$582$$ 0 0
$$583$$ −16.6983 −0.691574
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 28.1276 1.16095 0.580476 0.814277i $$-0.302867\pi$$
0.580476 + 0.814277i $$0.302867\pi$$
$$588$$ 0 0
$$589$$ −11.5388 −0.475448
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 9.17872 0.376925 0.188462 0.982080i $$-0.439650\pi$$
0.188462 + 0.982080i $$0.439650\pi$$
$$594$$ 0 0
$$595$$ −1.70739 −0.0699961
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −31.4713 −1.28588 −0.642942 0.765915i $$-0.722286\pi$$
−0.642942 + 0.765915i $$0.722286\pi$$
$$600$$ 0 0
$$601$$ −29.2654 −1.19376 −0.596880 0.802330i $$-0.703593\pi$$
−0.596880 + 0.802330i $$0.703593\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −0.0282739 −0.00114950
$$606$$ 0 0
$$607$$ 44.2034 1.79416 0.897080 0.441868i $$-0.145684\pi$$
0.897080 + 0.441868i $$0.145684\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.42385 0.259881
$$612$$ 0 0
$$613$$ −35.1715 −1.42056 −0.710282 0.703918i $$-0.751432\pi$$
−0.710282 + 0.703918i $$0.751432\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.42571 0.298948 0.149474 0.988766i $$-0.452242\pi$$
0.149474 + 0.988766i $$0.452242\pi$$
$$618$$ 0 0
$$619$$ −8.54787 −0.343568 −0.171784 0.985135i $$-0.554953\pi$$
−0.171784 + 0.985135i $$0.554953\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 1.54241 0.0617954
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0.971726 0.0387453
$$630$$ 0 0
$$631$$ 2.36836 0.0942829 0.0471415 0.998888i $$-0.484989\pi$$
0.0471415 + 0.998888i $$0.484989\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 17.8916 0.710005
$$636$$ 0 0
$$637$$ 8.89703 0.352513
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0.133096 0.00525698 0.00262849 0.999997i $$-0.499163\pi$$
0.00262849 + 0.999997i $$0.499163\pi$$
$$642$$ 0 0
$$643$$ 22.6464 0.893088 0.446544 0.894762i $$-0.352655\pi$$
0.446544 + 0.894762i $$0.352655\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 46.3912 1.82383 0.911913 0.410385i $$-0.134606\pi$$
0.911913 + 0.410385i $$0.134606\pi$$
$$648$$ 0 0
$$649$$ 16.6983 0.655466
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −36.4057 −1.42467 −0.712333 0.701842i $$-0.752361\pi$$
−0.712333 + 0.701842i $$0.752361\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 19.1414 0.745642 0.372821 0.927903i $$-0.378391\pi$$
0.372821 + 0.927903i $$0.378391\pi$$
$$660$$ 0 0
$$661$$ 39.9072 1.55221 0.776104 0.630605i $$-0.217193\pi$$
0.776104 + 0.630605i $$0.217193\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −0.679116 −0.0263350
$$666$$ 0 0
$$667$$ 5.72298 0.221595
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24.4057 −0.942172
$$672$$ 0 0
$$673$$ 23.6508 0.911673 0.455836 0.890064i $$-0.349340\pi$$
0.455836 + 0.890064i $$0.349340\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 14.8031 0.568931 0.284465 0.958686i $$-0.408184\pi$$
0.284465 + 0.958686i $$0.408184\pi$$
$$678$$ 0 0
$$679$$ −6.30088 −0.241806
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4.95252 −0.189503 −0.0947515 0.995501i $$-0.530206\pi$$
−0.0947515 + 0.995501i $$0.530206\pi$$
$$684$$ 0 0
$$685$$ 5.67004 0.216641
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −6.64177 −0.253031
$$690$$ 0 0
$$691$$ 19.2088 0.730739 0.365369 0.930863i $$-0.380943\pi$$
0.365369 + 0.930863i $$0.380943\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ 37.6892 1.42758
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 29.3492 1.10850 0.554251 0.832349i $$-0.313005\pi$$
0.554251 + 0.832349i $$0.313005\pi$$
$$702$$ 0 0
$$703$$ 0.386505 0.0145773
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 38.7266 1.45441 0.727204 0.686422i $$-0.240820\pi$$
0.727204 + 0.686422i $$0.240820\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −36.0576 −1.35037
$$714$$ 0 0
$$715$$ −4.38650 −0.164046
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −15.0848 −0.562569 −0.281284 0.959624i $$-0.590760\pi$$
−0.281284 + 0.959624i $$0.590760\pi$$
$$720$$ 0 0
$$721$$ −0.150442 −0.00560275
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.38650 0.0514935
$$726$$ 0 0
$$727$$ −12.3455 −0.457871 −0.228936 0.973442i $$-0.573525\pi$$
−0.228936 + 0.973442i $$0.573525\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −34.3684 −1.27116
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 31.3774 1.15580
$$738$$ 0 0
$$739$$ −29.7266 −1.09351 −0.546755 0.837293i $$-0.684137\pi$$
−0.546755 + 0.837293i $$0.684137\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 48.3648 1.77433 0.887165 0.461452i $$-0.152671\pi$$
0.887165 + 0.461452i $$0.152671\pi$$
$$744$$ 0 0
$$745$$ 17.6610 0.647048
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0.962653 0.0351746
$$750$$ 0 0
$$751$$ 31.8205 1.16115 0.580573 0.814208i $$-0.302829\pi$$
0.580573 + 0.814208i $$0.302829\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1.26434 0.0460139
$$756$$ 0 0
$$757$$ 4.94531 0.179740 0.0898701 0.995953i $$-0.471355\pi$$
0.0898701 + 0.995953i $$0.471355\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −35.4249 −1.28415 −0.642076 0.766641i $$-0.721927\pi$$
−0.642076 + 0.766641i $$0.721927\pi$$
$$762$$ 0 0
$$763$$ 2.85237 0.103263
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.64177 0.239820
$$768$$ 0 0
$$769$$ 49.4249 1.78231 0.891154 0.453701i $$-0.149897\pi$$
0.891154 + 0.453701i $$0.149897\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −12.6599 −0.455345 −0.227673 0.973738i $$-0.573112\pi$$
−0.227673 + 0.973738i $$0.573112\pi$$
$$774$$ 0 0
$$775$$ −8.73566 −0.313794
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 14.9909 0.537106
$$780$$ 0 0
$$781$$ −29.8578 −1.06840
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 15.6700 0.559288
$$786$$ 0 0
$$787$$ −30.9344 −1.10269 −0.551346 0.834277i $$-0.685885\pi$$
−0.551346 + 0.834277i $$0.685885\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.01093 −0.142612
$$792$$ 0 0
$$793$$ −9.70739 −0.344720
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 30.5935 1.08368 0.541839 0.840483i $$-0.317728\pi$$
0.541839 + 0.840483i $$0.317728\pi$$
$$798$$ 0 0
$$799$$ −16.1504 −0.571362
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −20.1131 −0.709776
$$804$$ 0 0
$$805$$ −2.12217 −0.0747966
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 2.89703 0.101854 0.0509271 0.998702i $$-0.483782\pi$$
0.0509271 + 0.998702i $$0.483782\pi$$
$$810$$ 0 0
$$811$$ 14.8861 0.522722 0.261361 0.965241i $$-0.415829\pi$$
0.261361 + 0.965241i $$0.415829\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 15.7074 0.550206
$$816$$ 0 0
$$817$$ −13.6700 −0.478254
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 8.95173 0.312417 0.156209 0.987724i $$-0.450073\pi$$
0.156209 + 0.987724i $$0.450073\pi$$
$$822$$ 0 0
$$823$$ −2.99454 −0.104383 −0.0521915 0.998637i $$-0.516621\pi$$
−0.0521915 + 0.998637i $$0.516621\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 31.9663 1.11158 0.555788 0.831324i $$-0.312417\pi$$
0.555788 + 0.831324i $$0.312417\pi$$
$$828$$ 0 0
$$829$$ 22.7458 0.789994 0.394997 0.918682i $$-0.370746\pi$$
0.394997 + 0.918682i $$0.370746\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −22.3684 −0.775018
$$834$$ 0 0
$$835$$ −6.16498 −0.213348
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −23.2643 −0.803174 −0.401587 0.915821i $$-0.631541\pi$$
−0.401587 + 0.915821i $$0.631541\pi$$
$$840$$ 0 0
$$841$$ −27.0776 −0.933710
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 11.2553 0.387193
$$846$$ 0 0
$$847$$ 0.0145366 0.000499485 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.20779 0.0414025
$$852$$ 0 0
$$853$$ 10.9909 0.376322 0.188161 0.982138i $$-0.439747\pi$$
0.188161 + 0.982138i $$0.439747\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 16.1504 0.551689 0.275844 0.961202i $$-0.411043\pi$$
0.275844 + 0.961202i $$0.411043\pi$$
$$858$$ 0 0
$$859$$ −28.5188 −0.973049 −0.486524 0.873667i $$-0.661736\pi$$
−0.486524 + 0.873667i $$0.661736\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 12.2890 0.418322 0.209161 0.977881i $$-0.432927\pi$$
0.209161 + 0.977881i $$0.432927\pi$$
$$864$$ 0 0
$$865$$ 8.58522 0.291906
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −26.7549 −0.907597
$$870$$ 0 0
$$871$$ 12.4804 0.422882
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −0.514137 −0.0173810
$$876$$ 0 0
$$877$$ −39.7002 −1.34058 −0.670290 0.742099i $$-0.733830\pi$$
−0.670290 + 0.742099i $$0.733830\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32.1040 −1.08161 −0.540806 0.841147i $$-0.681881\pi$$
−0.540806 + 0.841147i $$0.681881\pi$$
$$882$$ 0 0
$$883$$ −13.5051 −0.454482 −0.227241 0.973839i $$-0.572970\pi$$
−0.227241 + 0.973839i $$0.572970\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 35.1222 1.17929 0.589643 0.807664i $$-0.299268\pi$$
0.589643 + 0.807664i $$0.299268\pi$$
$$888$$ 0 0
$$889$$ −9.19872 −0.308515
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −6.42385 −0.214966
$$894$$ 0 0
$$895$$ 1.06562 0.0356198
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −12.1120 −0.403959
$$900$$ 0 0
$$901$$ 16.6983 0.556302
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 12.6700 0.421166
$$906$$ 0 0
$$907$$ −15.1186 −0.502004 −0.251002 0.967987i $$-0.580760\pi$$
−0.251002 + 0.967987i $$0.580760\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 52.5561 1.74126 0.870631 0.491936i $$-0.163711\pi$$
0.870631 + 0.491936i $$0.163711\pi$$
$$912$$ 0 0
$$913$$ −5.12217 −0.169519
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3.08482 −0.101870
$$918$$ 0 0
$$919$$ 54.5489 1.79940 0.899702 0.436505i $$-0.143784\pi$$
0.899702 + 0.436505i $$0.143784\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −11.8760 −0.390903
$$924$$ 0 0
$$925$$ 0.292611 0.00962098
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −20.3793 −0.668623 −0.334311 0.942463i $$-0.608504\pi$$
−0.334311 + 0.942463i $$0.608504\pi$$
$$930$$ 0 0
$$931$$ −8.89703 −0.291588
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 11.0283 0.360663
$$936$$ 0 0
$$937$$ 49.1979 1.60723 0.803613 0.595152i $$-0.202908\pi$$
0.803613 + 0.595152i $$0.202908\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 23.2371 0.757508 0.378754 0.925497i $$-0.376353\pi$$
0.378754 + 0.925497i $$0.376353\pi$$
$$942$$ 0 0
$$943$$ 46.8452 1.52549
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 37.1642 1.20767 0.603837 0.797108i $$-0.293638\pi$$
0.603837 + 0.797108i $$0.293638\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 23.5761 0.763706 0.381853 0.924223i $$-0.375286\pi$$
0.381853 + 0.924223i $$0.375286\pi$$
$$954$$ 0 0
$$955$$ 16.9344 0.547984
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2.91518 −0.0941360
$$960$$ 0 0
$$961$$ 45.3118 1.46167
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −26.7175 −0.860067
$$966$$ 0 0
$$967$$ −8.38290 −0.269576 −0.134788 0.990874i $$-0.543035\pi$$
−0.134788 + 0.990874i $$0.543035\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 13.2078 0.423858 0.211929 0.977285i $$-0.432025\pi$$
0.211929 + 0.977285i $$0.432025\pi$$
$$972$$ 0 0
$$973$$ −4.11310 −0.131860
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 14.3310 0.458490 0.229245 0.973369i $$-0.426374\pi$$
0.229245 + 0.973369i $$0.426374\pi$$
$$978$$ 0 0
$$979$$ −9.96265 −0.318408
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −32.3082 −1.03047 −0.515236 0.857048i $$-0.672296\pi$$
−0.515236 + 0.857048i $$0.672296\pi$$
$$984$$ 0 0
$$985$$ −14.2553 −0.454210
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −42.7175 −1.35834
$$990$$ 0 0
$$991$$ 39.6700 1.26016 0.630080 0.776530i $$-0.283022\pi$$
0.630080 + 0.776530i $$0.283022\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −24.6610 −0.781805
$$996$$ 0 0
$$997$$ 38.6874 1.22524 0.612621 0.790377i $$-0.290115\pi$$
0.612621 + 0.790377i $$0.290115\pi$$
$$998$$ 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 6480.2.a.bs.1.3 3
3.2 odd 2 6480.2.a.bv.1.3 3
4.3 odd 2 405.2.a.i.1.1 3
9.2 odd 6 720.2.q.i.481.1 6
9.4 even 3 2160.2.q.k.721.1 6
9.5 odd 6 720.2.q.i.241.1 6
9.7 even 3 2160.2.q.k.1441.1 6
12.11 even 2 405.2.a.j.1.3 3
20.3 even 4 2025.2.b.m.649.6 6
20.7 even 4 2025.2.b.m.649.1 6
20.19 odd 2 2025.2.a.o.1.3 3
36.7 odd 6 135.2.e.b.91.3 6
36.11 even 6 45.2.e.b.31.1 yes 6
36.23 even 6 45.2.e.b.16.1 6
36.31 odd 6 135.2.e.b.46.3 6
60.23 odd 4 2025.2.b.l.649.1 6
60.47 odd 4 2025.2.b.l.649.6 6
60.59 even 2 2025.2.a.n.1.1 3
180.7 even 12 675.2.k.b.199.1 12
180.23 odd 12 225.2.k.b.124.6 12
180.43 even 12 675.2.k.b.199.6 12
180.47 odd 12 225.2.k.b.49.6 12
180.59 even 6 225.2.e.b.151.3 6
180.67 even 12 675.2.k.b.424.6 12
180.79 odd 6 675.2.e.b.226.1 6
180.83 odd 12 225.2.k.b.49.1 12
180.103 even 12 675.2.k.b.424.1 12
180.119 even 6 225.2.e.b.76.3 6
180.139 odd 6 675.2.e.b.451.1 6
180.167 odd 12 225.2.k.b.124.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 36.23 even 6
45.2.e.b.31.1 yes 6 36.11 even 6
135.2.e.b.46.3 6 36.31 odd 6
135.2.e.b.91.3 6 36.7 odd 6
225.2.e.b.76.3 6 180.119 even 6
225.2.e.b.151.3 6 180.59 even 6
225.2.k.b.49.1 12 180.83 odd 12
225.2.k.b.49.6 12 180.47 odd 12
225.2.k.b.124.1 12 180.167 odd 12
225.2.k.b.124.6 12 180.23 odd 12
405.2.a.i.1.1 3 4.3 odd 2
405.2.a.j.1.3 3 12.11 even 2
675.2.e.b.226.1 6 180.79 odd 6
675.2.e.b.451.1 6 180.139 odd 6
675.2.k.b.199.1 12 180.7 even 12
675.2.k.b.199.6 12 180.43 even 12
675.2.k.b.424.1 12 180.103 even 12
675.2.k.b.424.6 12 180.67 even 12
720.2.q.i.241.1 6 9.5 odd 6
720.2.q.i.481.1 6 9.2 odd 6
2025.2.a.n.1.1 3 60.59 even 2
2025.2.a.o.1.3 3 20.19 odd 2
2025.2.b.l.649.1 6 60.23 odd 4
2025.2.b.l.649.6 6 60.47 odd 4
2025.2.b.m.649.1 6 20.7 even 4
2025.2.b.m.649.6 6 20.3 even 4
2160.2.q.k.721.1 6 9.4 even 3
2160.2.q.k.1441.1 6 9.7 even 3
6480.2.a.bs.1.3 3 1.1 even 1 trivial
6480.2.a.bv.1.3 3 3.2 odd 2