# Properties

 Label 3920.2.a.y.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{9} -2.00000 q^{11} +1.00000 q^{15} -4.00000 q^{17} -2.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +9.00000 q^{29} +4.00000 q^{31} -2.00000 q^{33} +4.00000 q^{37} -1.00000 q^{41} -9.00000 q^{43} -2.00000 q^{45} -4.00000 q^{51} -10.0000 q^{53} -2.00000 q^{55} -2.00000 q^{57} -10.0000 q^{59} -9.00000 q^{61} -5.00000 q^{67} -1.00000 q^{69} -14.0000 q^{71} -12.0000 q^{73} +1.00000 q^{75} -14.0000 q^{79} +1.00000 q^{81} +11.0000 q^{83} -4.00000 q^{85} +9.00000 q^{87} +15.0000 q^{89} +4.00000 q^{93} -2.00000 q^{95} +18.0000 q^{97} +4.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.00000 −0.156174 −0.0780869 0.996947i $$-0.524881\pi$$
−0.0780869 + 0.996947i $$0.524881\pi$$
$$42$$ 0 0
$$43$$ −9.00000 −1.37249 −0.686244 0.727372i $$-0.740742\pi$$
−0.686244 + 0.727372i $$0.740742\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ −9.00000 −1.15233 −0.576166 0.817333i $$-0.695452\pi$$
−0.576166 + 0.817333i $$0.695452\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ −14.0000 −1.66149 −0.830747 0.556650i $$-0.812086\pi$$
−0.830747 + 0.556650i $$0.812086\pi$$
$$72$$ 0 0
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 11.0000 1.20741 0.603703 0.797209i $$-0.293691\pi$$
0.603703 + 0.797209i $$0.293691\pi$$
$$84$$ 0 0
$$85$$ −4.00000 −0.433861
$$86$$ 0 0
$$87$$ 9.00000 0.964901
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −1.00000 −0.0901670
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ −9.00000 −0.792406
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 0 0
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 8.00000 0.646762
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ 17.0000 1.31550 0.657750 0.753237i $$-0.271508\pi$$
0.657750 + 0.753237i $$0.271508\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ −16.0000 −1.21646 −0.608229 0.793762i $$-0.708120\pi$$
−0.608229 + 0.793762i $$0.708120\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −10.0000 −0.751646
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 25.0000 1.85824 0.929118 0.369784i $$-0.120568\pi$$
0.929118 + 0.369784i $$0.120568\pi$$
$$182$$ 0 0
$$183$$ −9.00000 −0.665299
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −5.00000 −0.352673
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1.00000 −0.0698430
$$206$$ 0 0
$$207$$ 2.00000 0.139010
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ −14.0000 −0.959264
$$214$$ 0 0
$$215$$ −9.00000 −0.613795
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.00000 0.524097 0.262049 0.965055i $$-0.415602\pi$$
0.262049 + 0.965055i $$0.415602\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −14.0000 −0.909398
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 11.0000 0.697097
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 2.00000 0.125739
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ 1.00000 0.0616626 0.0308313 0.999525i $$-0.490185\pi$$
0.0308313 + 0.999525i $$0.490185\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 15.0000 0.917985
$$268$$ 0 0
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ −10.0000 −0.582223
$$296$$ 0 0
$$297$$ 10.0000 0.580259
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −3.00000 −0.172345
$$304$$ 0 0
$$305$$ −9.00000 −0.515339
$$306$$ 0 0
$$307$$ 21.0000 1.19853 0.599267 0.800549i $$-0.295459\pi$$
0.599267 + 0.800549i $$0.295459\pi$$
$$308$$ 0 0
$$309$$ −13.0000 −0.739544
$$310$$ 0 0
$$311$$ 26.0000 1.47432 0.737162 0.675716i $$-0.236165\pi$$
0.737162 + 0.675716i $$0.236165\pi$$
$$312$$ 0 0
$$313$$ −16.0000 −0.904373 −0.452187 0.891923i $$-0.649356\pi$$
−0.452187 + 0.891923i $$0.649356\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.0000 −0.898650 −0.449325 0.893368i $$-0.648335\pi$$
−0.449325 + 0.893368i $$0.648335\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −1.00000 −0.0553001
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ −8.00000 −0.438397
$$334$$ 0 0
$$335$$ −5.00000 −0.273179
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ 7.00000 0.375780 0.187890 0.982190i $$-0.439835\pi$$
0.187890 + 0.982190i $$0.439835\pi$$
$$348$$ 0 0
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ −14.0000 −0.743043
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ 0 0
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −28.0000 −1.44979 −0.724893 0.688862i $$-0.758111\pi$$
−0.724893 + 0.688862i $$0.758111\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 30.0000 1.54100 0.770498 0.637442i $$-0.220007\pi$$
0.770498 + 0.637442i $$0.220007\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 18.0000 0.914991
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ −8.00000 −0.403547
$$394$$ 0 0
$$395$$ −14.0000 −0.704416
$$396$$ 0 0
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −17.0000 −0.848939 −0.424470 0.905442i $$-0.639539\pi$$
−0.424470 + 0.905442i $$0.639539\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 11.0000 0.539969
$$416$$ 0 0
$$417$$ −2.00000 −0.0979404
$$418$$ 0 0
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ −27.0000 −1.31590 −0.657950 0.753062i $$-0.728576\pi$$
−0.657950 + 0.753062i $$0.728576\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −4.00000 −0.194029
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −14.0000 −0.674356 −0.337178 0.941441i $$-0.609472\pi$$
−0.337178 + 0.941441i $$0.609472\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ 0 0
$$435$$ 9.00000 0.431517
$$436$$ 0 0
$$437$$ 2.00000 0.0956730
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.0000 0.522626 0.261313 0.965254i $$-0.415845\pi$$
0.261313 + 0.965254i $$0.415845\pi$$
$$444$$ 0 0
$$445$$ 15.0000 0.711068
$$446$$ 0 0
$$447$$ −5.00000 −0.236492
$$448$$ 0 0
$$449$$ −41.0000 −1.93491 −0.967455 0.253044i $$-0.918568\pi$$
−0.967455 + 0.253044i $$0.918568\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 0 0
$$459$$ 20.0000 0.933520
$$460$$ 0 0
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ 0 0
$$463$$ −39.0000 −1.81248 −0.906242 0.422760i $$-0.861061\pi$$
−0.906242 + 0.422760i $$0.861061\pi$$
$$464$$ 0 0
$$465$$ 4.00000 0.185496
$$466$$ 0 0
$$467$$ −7.00000 −0.323921 −0.161961 0.986797i $$-0.551782\pi$$
−0.161961 + 0.986797i $$0.551782\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 18.0000 0.827641
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 0 0
$$477$$ 20.0000 0.915737
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 18.0000 0.817338
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 38.0000 1.70111 0.850557 0.525883i $$-0.176265\pi$$
0.850557 + 0.525883i $$0.176265\pi$$
$$500$$ 0 0
$$501$$ 17.0000 0.759504
$$502$$ 0 0
$$503$$ −23.0000 −1.02552 −0.512760 0.858532i $$-0.671377\pi$$
−0.512760 + 0.858532i $$0.671377\pi$$
$$504$$ 0 0
$$505$$ −3.00000 −0.133498
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 10.0000 0.441511
$$514$$ 0 0
$$515$$ −13.0000 −0.572848
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −16.0000 −0.702322
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 20.0000 0.867926
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ 0 0
$$543$$ 25.0000 1.07285
$$544$$ 0 0
$$545$$ −1.00000 −0.0428353
$$546$$ 0 0
$$547$$ 35.0000 1.49649 0.748246 0.663421i $$-0.230896\pi$$
0.748246 + 0.663421i $$0.230896\pi$$
$$548$$ 0 0
$$549$$ 18.0000 0.768221
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 4.00000 0.169791
$$556$$ 0 0
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ −45.0000 −1.89652 −0.948262 0.317489i $$-0.897160\pi$$
−0.948262 + 0.317489i $$0.897160\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 46.0000 1.92842 0.964210 0.265139i $$-0.0854179\pi$$
0.964210 + 0.265139i $$0.0854179\pi$$
$$570$$ 0 0
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ 0 0
$$573$$ −18.0000 −0.751961
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 20.0000 0.828315
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −20.0000 −0.807792 −0.403896 0.914805i $$-0.632344\pi$$
−0.403896 + 0.914805i $$0.632344\pi$$
$$614$$ 0 0
$$615$$ −1.00000 −0.0403239
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 0 0
$$621$$ 5.00000 0.200643
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.00000 0.159745
$$628$$ 0 0
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 0 0
$$633$$ −2.00000 −0.0794929
$$634$$ 0 0
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 28.0000 1.10766
$$640$$ 0 0
$$641$$ −19.0000 −0.750455 −0.375227 0.926933i $$-0.622435\pi$$
−0.375227 + 0.926933i $$0.622435\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ −9.00000 −0.354375
$$646$$ 0 0
$$647$$ 23.0000 0.904223 0.452112 0.891961i $$-0.350671\pi$$
0.452112 + 0.891961i $$0.350671\pi$$
$$648$$ 0 0
$$649$$ 20.0000 0.785069
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 0 0
$$657$$ 24.0000 0.936329
$$658$$ 0 0
$$659$$ −2.00000 −0.0779089 −0.0389545 0.999241i $$-0.512403\pi$$
−0.0389545 + 0.999241i $$0.512403\pi$$
$$660$$ 0 0
$$661$$ 3.00000 0.116686 0.0583432 0.998297i $$-0.481418\pi$$
0.0583432 + 0.998297i $$0.481418\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 18.0000 0.694882
$$672$$ 0 0
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ 24.0000 0.922395 0.461197 0.887298i $$-0.347420\pi$$
0.461197 + 0.887298i $$0.347420\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ −9.00000 −0.344375 −0.172188 0.985064i $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ 0 0
$$687$$ 10.0000 0.381524
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 50.0000 1.90209 0.951045 0.309053i $$-0.100012\pi$$
0.951045 + 0.309053i $$0.100012\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −2.00000 −0.0758643
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ 8.00000 0.302588
$$700$$ 0 0
$$701$$ 9.00000 0.339925 0.169963 0.985451i $$-0.445635\pi$$
0.169963 + 0.985451i $$0.445635\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 21.0000 0.788672 0.394336 0.918966i $$-0.370975\pi$$
0.394336 + 0.918966i $$0.370975\pi$$
$$710$$ 0 0
$$711$$ 28.0000 1.05008
$$712$$ 0 0
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 22.0000 0.818189
$$724$$ 0 0
$$725$$ 9.00000 0.334252
$$726$$ 0 0
$$727$$ −31.0000 −1.14973 −0.574863 0.818250i $$-0.694945\pi$$
−0.574863 + 0.818250i $$0.694945\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 36.0000 1.33151
$$732$$ 0 0
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.0000 0.368355
$$738$$ 0 0
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −27.0000 −0.990534 −0.495267 0.868741i $$-0.664930\pi$$
−0.495267 + 0.868741i $$0.664930\pi$$
$$744$$ 0 0
$$745$$ −5.00000 −0.183186
$$746$$ 0 0
$$747$$ −22.0000 −0.804938
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ −24.0000 −0.872295 −0.436147 0.899875i $$-0.643657\pi$$
−0.436147 + 0.899875i $$0.643657\pi$$
$$758$$ 0 0
$$759$$ 2.00000 0.0725954
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ 28.0000 1.00192
$$782$$ 0 0
$$783$$ −45.0000 −1.60817
$$784$$ 0 0
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 45.0000 1.60408 0.802038 0.597272i $$-0.203749\pi$$
0.802038 + 0.597272i $$0.203749\pi$$
$$788$$ 0 0
$$789$$ 1.00000 0.0356009
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −10.0000 −0.354663
$$796$$ 0 0
$$797$$ −8.00000 −0.283375 −0.141687 0.989911i $$-0.545253\pi$$
−0.141687 + 0.989911i $$0.545253\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 0 0
$$803$$ 24.0000 0.846942
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 21.0000 0.739235
$$808$$ 0 0
$$809$$ 13.0000 0.457056 0.228528 0.973537i $$-0.426609\pi$$
0.228528 + 0.973537i $$0.426609\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 0 0
$$813$$ 22.0000 0.771574
$$814$$ 0 0
$$815$$ 20.0000 0.700569
$$816$$ 0 0
$$817$$ 18.0000 0.629740
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ −49.0000 −1.70803 −0.854016 0.520246i $$-0.825840\pi$$
−0.854016 + 0.520246i $$0.825840\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ 17.0000 0.591148 0.295574 0.955320i $$-0.404489\pi$$
0.295574 + 0.955320i $$0.404489\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 17.0000 0.588309
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 0 0
$$839$$ 10.0000 0.345238 0.172619 0.984989i $$-0.444777\pi$$
0.172619 + 0.984989i $$0.444777\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1.00000 0.0340404 0.0170202 0.999855i $$-0.494582\pi$$
0.0170202 + 0.999855i $$0.494582\pi$$
$$864$$ 0 0
$$865$$ −16.0000 −0.544016
$$866$$ 0 0
$$867$$ −1.00000 −0.0339618
$$868$$ 0 0
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −36.0000 −1.21842
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ 0 0
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ −17.0000 −0.572745 −0.286372 0.958118i $$-0.592449\pi$$
−0.286372 + 0.958118i $$0.592449\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ −10.0000 −0.336146
$$886$$ 0 0
$$887$$ 15.0000 0.503651 0.251825 0.967773i $$-0.418969\pi$$
0.251825 + 0.967773i $$0.418969\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ 40.0000 1.33259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25.0000 0.831028
$$906$$ 0 0
$$907$$ −7.00000 −0.232431 −0.116216 0.993224i $$-0.537076\pi$$
−0.116216 + 0.993224i $$0.537076\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ −22.0000 −0.728094
$$914$$ 0 0
$$915$$ −9.00000 −0.297531
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −42.0000 −1.38545 −0.692726 0.721201i $$-0.743591\pi$$
−0.692726 + 0.721201i $$0.743591\pi$$
$$920$$ 0 0
$$921$$ 21.0000 0.691974
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 4.00000 0.131519
$$926$$ 0 0
$$927$$ 26.0000 0.853952
$$928$$ 0 0
$$929$$ 13.0000 0.426516 0.213258 0.976996i $$-0.431592\pi$$
0.213258 + 0.976996i $$0.431592\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 26.0000 0.851202
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −4.00000 −0.130674 −0.0653372 0.997863i $$-0.520812\pi$$
−0.0653372 + 0.997863i $$0.520812\pi$$
$$938$$ 0 0
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 1.00000 0.0325645
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −5.00000 −0.162478 −0.0812391 0.996695i $$-0.525888\pi$$
−0.0812391 + 0.996695i $$0.525888\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −16.0000 −0.518836
$$952$$ 0 0
$$953$$ 28.0000 0.907009 0.453504 0.891254i $$-0.350174\pi$$
0.453504 + 0.891254i $$0.350174\pi$$
$$954$$ 0 0
$$955$$ −18.0000 −0.582466
$$956$$ 0 0
$$957$$ −18.0000 −0.581857
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ 0 0
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ −55.0000 −1.76868 −0.884340 0.466843i $$-0.845391\pi$$
−0.884340 + 0.466843i $$0.845391\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −8.00000 −0.256732 −0.128366 0.991727i $$-0.540973\pi$$
−0.128366 + 0.991727i $$0.540973\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 3.00000 0.0956851 0.0478426 0.998855i $$-0.484765\pi$$
0.0478426 + 0.998855i $$0.484765\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.00000 0.286183
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 0 0
$$999$$ −20.0000 −0.632772
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 3920.2.a.y.1.1 1
4.3 odd 2 1960.2.a.e.1.1 1
7.3 odd 6 560.2.q.h.401.1 2
7.5 odd 6 560.2.q.h.81.1 2
7.6 odd 2 3920.2.a.m.1.1 1
20.19 odd 2 9800.2.a.bc.1.1 1
28.3 even 6 280.2.q.a.121.1 yes 2
28.11 odd 6 1960.2.q.k.961.1 2
28.19 even 6 280.2.q.a.81.1 2
28.23 odd 6 1960.2.q.k.361.1 2
28.27 even 2 1960.2.a.i.1.1 1
84.47 odd 6 2520.2.bi.a.361.1 2
84.59 odd 6 2520.2.bi.a.1801.1 2
140.3 odd 12 1400.2.bh.b.849.2 4
140.19 even 6 1400.2.q.e.1201.1 2
140.47 odd 12 1400.2.bh.b.249.2 4
140.59 even 6 1400.2.q.e.401.1 2
140.87 odd 12 1400.2.bh.b.849.1 4
140.103 odd 12 1400.2.bh.b.249.1 4
140.139 even 2 9800.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.a.81.1 2 28.19 even 6
280.2.q.a.121.1 yes 2 28.3 even 6
560.2.q.h.81.1 2 7.5 odd 6
560.2.q.h.401.1 2 7.3 odd 6
1400.2.q.e.401.1 2 140.59 even 6
1400.2.q.e.1201.1 2 140.19 even 6
1400.2.bh.b.249.1 4 140.103 odd 12
1400.2.bh.b.249.2 4 140.47 odd 12
1400.2.bh.b.849.1 4 140.87 odd 12
1400.2.bh.b.849.2 4 140.3 odd 12
1960.2.a.e.1.1 1 4.3 odd 2
1960.2.a.i.1.1 1 28.27 even 2
1960.2.q.k.361.1 2 28.23 odd 6
1960.2.q.k.961.1 2 28.11 odd 6
2520.2.bi.a.361.1 2 84.47 odd 6
2520.2.bi.a.1801.1 2 84.59 odd 6
3920.2.a.m.1.1 1 7.6 odd 2
3920.2.a.y.1.1 1 1.1 even 1 trivial
9800.2.a.r.1.1 1 140.139 even 2
9800.2.a.bc.1.1 1 20.19 odd 2